[    {        "title": "0909.0079v1.Directional_Dichroism_of_X_Ray_Absorption_in_a_Polar_Ferrimagnet_GaFeO_3.pdf",        "content": "arXiv:0909.0079v1  [cond-mat.str-el]  1 Sep 2009Typeset with jpsj2.cls <ver.1.2.2b > Full Paper\nDirectional Dichroism of X-Ray Absorption in a Polar Ferrim agnet GaFeO 3\nJun-ichi Igarashi1∗and Tatsuya Nagao2\n1Faculty of Science, Ibaraki University, Mito, Ibaraki 310- 8512, Japan\n2Faculty of Engineering, Gunma University, Kiryu, Gunma 376 -8515, Japan\nWe study the directional dichroic absorption spectra in the x-ray r egion in a polar ferri-\nmagnet GaFeO 3. The directional dichroism on the absorption spectra at the Fe pre -K-edge\narises from the E1-E2 interference process through the hybridization between the 4 pand\n3dstates in the noncentrosymmetric environment of Fe atoms. We pe rform a microscopic\ncalculation of the spectra on a model of FeO 6with reasonable parameter values for Coulomb\ninteraction and hybridizations. We obtain the difference in the absor ption coefficients when\nthe magnetic field is applied parallel and antiparallel to the caxis. The spectra thus ob-\ntained have similar shapes to the experimental curves as a function of photon energy in the\nFe pre-K-edge region, although they have opposite signs.\nKEYWORDS: directional dichroism, magnetoelectric effect, GaFeO 3, noncentrosymmetricity,\nE1-E2 interference, x-ray K-edge absorption\n1. Introduction\nGallium ferrate (GaFeO 3) exhibits simultaneously spontaneous electric polarizat ion and\nmagnetization at low temperatures. This compound was first s ynthesized by Remeika,1)and\nthe large magnetoelectric effect was observed by Rado.2)Recently, untwinned large single\ncrystals have been prepared,3)and the optical and x-ray absorption measurements have been\ncarried out with changing the direction of magnetization.4,5)The purpose of this paper is\nto analyze the magnetoelectric effects on the x-ray absorptio n spectra and to elucidate the\nmicroscopic origin by carrying out a microscopic calculati on of the spectra.\nThe crystal of GaFeO 3has an orthorhombic unit cell with the space group Pc21n.6)\nEach Fe atom is octahedrally surrounded by O atoms. With negl ecting slight distortions of\noctahedrons,Featoms areregardedasslightly displacedfr omthecenter of theoctahedron;the\nshift is 0.26˚A at Fe1 sites and −0.11˚A at Fe2 sites along the baxis.3)Thereby the spontaneous\nelectric polarization is generated along the baxis. Note that two kinds of FeO 6clusters exist\nforboth Fe1 and Fe2 sites, one of which is given by rotating th eother by an angle πaroundthe\nbaxis. As regards magnetic properties, the compound behaves as a ferrimagnet,3,7)with the\nlocal magnetic moments at Fe1 and Fe2 sites aligning antifer romagnetically along the caxis.\nOne reason for the ferrimagnetism may be that the Fe occupati on at Fe1 and Fe2 sites are\nslightly different from each other.3)In the present analysis, we neglect such a small deviation\n∗E-mail:jigarash@mx.ibaraki.ac.jp\n1/14J. Phys. Soc. Jpn. Full Paper\nP : polarization\n(Fe1)\n(Fe2)q : photon wave vector\nb (z) a (y)\nc (x)M loc :  local magnetization\nFig. 1. Geometry of absorption experiment.5)X-rays propagate along the aaxis with polarization\nalong the baxis or the caxis. The electric dipole moment is along the baxis. When the magnetic\nfield is applied to the positive direction of the caxis, the sublattice magnetization is directed to\nthe negative and positive directions of the caxis at Fe1 sites and at Fe2 sites, respectively. When\nthe magnetic field is reversed, the sublattice magnetization is rever sed.\nfrom a perfect antiferromagnet.\nIn theK-edge absorption experiment,5)the x-rays propagated along the positive direction\nof theaaxis, and the magnetic field was applied along the ±caxis, as illustrated in Fig. 1. The\ndifference of the absorption coefficient was measured between t he two directions of magnetic\nfield.Aswillbeshownlater[eq.(4.11)], thesedifferencespe ctrahavecharacteristic dependence\non the polarization and magnetization, and would be termed a smagnetoelectric spectra.5)\nSince the compound is a ferrimanget, reversing the directio n of applied magnetic field results\nin reversing the direction of the local magnetic moment of Fe atoms.\nIn the analysis of absorption spectra, we consider only thep rocesses on Fe atoms, since the\n1s-core state is well localized on Fe atoms. In addition to the E1-E1 andE2-E2 processes, we\nformulate the E1-E2 interference process which gives rise to the magnetoelect ric spectra. The\ncontribution of this process arises from the mixing of the 3 d61s-configuration to the 4 p3d51s-\nconfiguration, where 1 sindicates the presence of a 1 s-core hole. Such mixings exist only under\nthe noncentrosymmetric environment. In the present analys is, we describe the E1-E2 process\nby employing a cluster model of FeO 6, where all the 3 dand 4porbitals of Fe atoms and the 2p\norbitals of O atoms as well as the Coulomb and the spin-orbit i nteractions in the 3 dorbitals\nare taken into account. We obtain an effective hybridization b etween the 4 pand 3dstates in\naddition to the ligand field on the 3 dstates through the hybridization with the O 2 pstates.\nOn the basis of these frameworks, we clarify various symmetr y relations to the E1-E2 process\nand the relation between the nonreciprocal directional dic hroism and the anapole moment.\n2/14J. Phys. Soc. Jpn. Full Paper\nFurthermore, we numerically calculate the absorption spec tra as a function of photon energy\nby diagonalizing the Hamiltonian matrix in the 3 d6- and 4p3d5-configurations. The shapes of\nmagnetoelectric spectra as a function of photon energy are f ound similar to the experimental\ncurves buttheir signsare oppositetotheexperiment in thep re-K-edge region.5)Theorigin for\ntheoppositesignis not known.Note that themagnetoelectri c spectraintheoptical absorption\nhave been obtained in agreement with the experiment by using the same cluster model.8)\nThis paper is organized as follows. In §2, we introduce a cluster model of FeO 6. In§3, we\ndescribethe x-ray transition operators associated with Fe atoms. In §4, we derive the formulas\nof x-ray absorption, and present the calculated spectra in c omparison with the experiment.\nThe last section is devoted to concluding remarks.\n2. Electronic Structures\n2.1 FeO 6cluster and 4pband\nIn a FeO 6cluster, we consider the 1 s, 3dand 4pstates in the Fe atom, and the 2 pstates\nin O atoms. The Hamiltonian may be written as\nH=H3d+H2p+H3d−2p\nhyb+H4p+H4p−2p\nhyb\n+H1s+H1s−3d+H1s−4p, (2.1)\nwhere\nH3d=/summationdisplay\nmσEd\nmd†\nmσdmσ\n+1\n2/summationdisplay\nν1ν2ν3ν4g(ν1ν2;ν3ν4)d†\nν1d†\nν2dν4dν3\n+ζ3d/summationdisplay\nmm′σσ′∝angb∇acketleftmσ|L·S|m′σ′∝angb∇acket∇ightd†\nmσdm′σ′\n+Hxc·/summationdisplay\nmσσ′(S)σσ′d†\nmσdmσ′, (2.2)\nH2p=/summationdisplay\njησEpp†\njησpjησ, (2.3)\nH3d−2p\nhyb=/summationdisplay\njησmt3d−2p\nmη(j)d†\nmσpjησ+H.c., (2.4)\nH4p=/summationdisplay\nkη′σǫ4p(k)p′†\nkη′σp′\nkη′σ, (2.5)\nH4p−2p\nhyb=/summationdisplay\njηση′t4p−2p\nη′η(j)p′†\nη′σpjησ+H.c., (2.6)\nH1s=ǫ1s/summationdisplay\nσs†\nσsσ, (2.7)\nH1s−3d=U1s−3d/summationdisplay\nmσσ′d†\nmσdmσs†\nσ′sσ′, (2.8)\n3/14J. Phys. Soc. Jpn. Full Paper\nH1s−4p=U1s−4p/summationdisplay\nη′σσ′p′†\nη′σp′\nη′σs†\nσ′sσ′. (2.9)\nThe energy of 3 delectrons can be described by H3d[eq. (2.2)] where dmσrepresents an\nannihilation operatorof a3 delectron withspin σandorbital m(=x2−y2,3z2−r2,yz,zx,xy ).\nThe symbol Ed\nmrefers to the energy of 3 dstate with orbital m. The second and third terms\nof eq. (2.2) represent the intra-atomic Coulomb and spin-or bit interactions for 3 delectrons,\nrespectively. The matrix elements g(ν1ν2;ν3ν4) are expressed in terms of the Slater integrals\nF0,F2, andF4[νstands for ( m,σ)], and the spin-orbit coupling is ζ3d. We evaluate atomic\nvalues of F2,F4andζ3dwithin the Hartree-Fock (HF) approximation,9)and multiply 0 .8 to\nthese atomic values in order to take account of the slight scr eening effect. On the other hand,\nwe multiply 0.25 to the atomic value for F0, sinceF0is known to be considerably screened by\nsolid-state effects. The last term in eq. (2.2) describes the e nergy arising from the exchange\ninteraction with neighboring Fe atoms, where ( S)σσ′represents the matrix element of the spin\noperator of 3 delectrons. The exchange field Hxchas a dimension of energy, and is ∼kBTc/4\nwithTc∼250 K. Note that this term has merely a role of selecting the gr ound state by\nlifting the degeneracy and therefore the spectra depend lit tle on its absolute value. The Hxcis\ndirected to the negative direction of the caxis at Fe1 sites when the magnetic field is applied\nalong the positive direction of the caxis.\nThe energy of oxygen 2 pelectrons is given by H2pwherepjησis the annihilation operator\nof the 2pstate of energy Epwithη=x,y,zand spin σat the oxygen site j. The Coulomb\ninteraction is neglected in oxygen 2p states. The H3d−2p\nhybdenotes the hybridization energy\nbetween the 3 dand 2pstates with the coupling constant t3d−2p\nmη. The energy of the 2 plevel\nrelative to the 3 dlevels is determined from the charge-transfer energy ∆ defin ed by ∆ =\nEd−Ep+15U(3d6)−10U(3d5) withEdbeing an average of Ed\nm. The multiplet-averaged d-d\nCoulomb interaction in the 3 d6and 3d5configurations are referred to as U(3d6) andU(3d5),\nrespectively, which are defined by U=F0−(2/63)F2−(2/63)F4.\nTheH4prepresents the energy of the 4 pstates, where p′\nkη′σis the annihilation operator\nof the 4pstate with momentum k,η′=x,y,z, and spin σ. The 4pstates form an energy band\nǫ4p(k). The density of states (DOS) of the 4 pband is inferred from the K-edge absorption\nspectra5)(see Fig. 2). The H4p−2p\nhybrepresents the hybridization between the 4 pand oxygen 2 p\nstates with the coupling constant t4p−2p\nη′η. The annihilation operator of the local 4 porbitalp′\nη′σ\nmay be expressed as p′\nη′σ= (1/√N0)/summationtext\nkp′\nkη′σ(N0is the discretized number of k-points).\nThe energy of the 1 sstate is denoted as H1swheresσrepresents the annihilation operator\nof the 1sstate with spin σ. Finally, the interaction between the 1 sand 3dstates and that\nbetween the 1 sand 4pstates are denoted as H1s−3dandH1s−4p, respectively.\nTable I lists the parameter values used in this paper, which a re used in our previous paper\nanalyzing the optical spectra in GaFeO 3,8)and are consistent with the values in previous\n4/14J. Phys. Soc. Jpn. Full Paper\nTable I. Parameter values for a FeO 6cluster in the 3 d5configuration, in units of eV. The Slater-\nKoster two-center integrals are defined for the Fe atom at the ce nter of the octahedron.\nF0(3d,3d) 6.39 ( pdσ)2p,3d-1.9\nF2(3d,3d) 9.64 ( pdπ)2p,3d0.82\nF4(3d,3d) 6.03 ( ppσ)2p,4p3.5\nζ3d 0.059 ( ppπ)2p,4p-1.0\n∆ 3.3\ncalculations for Fe 3O4.10,11)\n2.2 Ligand field and effective hybridization between the 4pand3dstates\nFe atoms are assumed to be displaced from the center of the oct ahedron along the baxis.\nThe shift δis 0.26˚A at Fe1 sites and −0.11˚A at Fe2 sites. We evaluate the hybridization\nmatrices t3d−2p\nmη(j) andt4p−2p\nη′η(j) for the Fe atom at the off-center positions by modifying the\nSlater-Koster two-center integrals for the Fe atom at the ce ntral position of the octahedron\n(Table I) with the assumption that ( pdσ)2p,3d, (pdπ)2p,3d∝d−4, and (ppσ)4p,2p, (ppπ)4p,2p∝\nd−2fordbeing the Fe-O distance.12)Thereby the ligand field Hamiltonian on the 3 dstates\nis given in second-order perturbation theory,\n˜H3d−3d=/summationdisplay\nmm′σ˜t3d−3d\nmm′d†\nmσdm′σ+H.c., (2.10)\nwith\n˜t3d−3d\nmm′=/summationdisplay\njηt3d−2p\nmη(j)t3d−2p\nm′η(j)/∆, (2.11)\nwhere the sum over jis taken on neighboring O sites, and ∆ = 3 .3 eV is the charge transfer\nenergy defined in §2.1. In addition to the ligand field corresponding to the cubi c symmetry,\nwe have a field proportional to δ2. The latter causes extra splittings of the 3 dlevels. Similarly,\nwe can evaluate the effective hybridization between the 4 pand 3dstates in the form:\n˜H4p−3d=/summationdisplay\nη′mσ˜t4p−3d\nη′mp′†\nη′σdmσ+H.c. (2.12)\nHere, the effective coupling is defined by\n˜t4p−3d\nη′m=/summationdisplay\njηt4p−2p\nη′η(j)t3d−2p\nmη(j)/(E4p−E2p), (2.13)\nwhereE4pis the average of the 4 p-band energy, which is estimated as E4p−E2p≈17 eV.\nThe value of coupling coefficient ˜t4p−3d\nη′mis nearly proportional to the shift δof the Fe atom\nfrom the center of the octahedron.\n5/14J. Phys. Soc. Jpn. Full Paper\n2.3 Ground state\nWe assume that Fe ions are in the d5-configuration in the ground state, which will be\ndenoted as |Φg(d5)∝angb∇acket∇ightwith eigenenergy Eg(d5). We calculate the state by diagonalizing the\nHamiltonian H3d+˜H3d−3d, where the exchange field Hxcand the displacement of Fe atoms\nfrom the center of the octahadron are different from Fe1 and Fe2 sites. The lowest energy\nstate is characterized as6A1under the trigonal crystal field when the exchange field and th e\nspin-orbit interaction are disregarded. The inclusion of t hese interactions could induce the\norbital moment /planckover2pi1∝angb∇acketleftLx∝angb∇acket∇ight, but its absolute value is given less than 0 .004/planckover2pi1.\n3. Absorption process on Fe\nThe interaction between the electromagnetic wave and elect rons is described by\nHint=−1\nc/integraldisplay\nj(r)·A(r)d3r, (3.1)\nwherecstands for the speed of light and jrepresents the current-density operator. The\nelectromagnetic field A(r) for linear polarization is defined as\nA(r) =/summationdisplay\nq/radicalBigg\n2π/planckover2pi12c2\nV/planckover2pi1ωqecqeiq·r+H.c., (3.2)\nwherecqand/planckover2pi1ωqare the annihilation operator and the energy of photon, resp ectively. The\nunit vector of polarization is described by e. We approximate this expression into a sum of\nthe contributions from each Fe atom:\nHint=−1\nc/summationdisplay\nq,ij(q,i)·A(q,e,i) +H.c., (3.3)\nwith\nj(q,i) =/summationdisplay\nnn′/bracketleftbigg/integraldisplay\neiq·(r−ri)jnn′(r−ri)d3(r−ri)/bracketrightbigg\na†\nn(i)an′(i), (3.4)\nA(q,e,i) =/radicalBigg\n2π/planckover2pi1c2\nVωqecqeiq·ri. (3.5)\nThe local current operator may be described by13)\njnn′(r−ri) =ie/planckover2pi1\n2m/bracketleftbig\n(∇φ∗\nn)φn′−φ∗\nn∇φn′/bracketrightbig\n−e2\nmcAφ∗\nnφn′+e/planckover2pi1\nmcc∇×[φ∗\nnSφn′].\n(3.6)\nThe integration in eq. (3.4) is carried out around site i, andan(i) is the annihilation operator\nof electron with the local orbital expressed by the wave func tionφn(r−ri). The charge and\nthe mass of electron are denoted as eandm, and/planckover2pi1Sis the spin operator of electron. The\nsecond term in eq. (3.6), which describes the scattering of p hoton, will be neglected in the\nfollowing. The approximation made by taking account of the p rocess only on Fe atoms may\n6/14J. Phys. Soc. Jpn. Full Paper\nbe justified at the core-level spectra, since the core state i s well localized at Fe sites.\nThe absorption experiment5)we analyse has been carried out on the geometry that the\nphoton propagates along the a-axis with linear polarization, as illustrated in Fig. 1. Co rre-\nsponding to this situation, it is convenient to rewrite the i nteraction between the matter and\nthe photon in a form,\nHint=−e/summationdisplay\nq/radicalBigg\n2π\nV/planckover2pi1ωq/summationdisplay\niT(q,e,i)cqeiq·ri+H.c., (3.7)\nwhere the transition operator T(q,e,i) is defined as /planckover2pi1e·j(q,i)/e. The explicit expression of\nT(q,e,i) for the E1 andE2 transitions are given in the following subsections.\n3.1E1transition\nThe transition operator T(q,e,i) for the E1 transition is obtained by putting eiq·(r−rj)=\n1 in eq. (3.4). Therefore it is independent of the propagatio n direction of photon. For the\npolarization along the z-axis, the first term in eq. (3.6) is r ewritten by employing the following\nrelation\n/integraldisplay\nφ∗\nn∂\n∂zφn′d3r=−m\n/planckover2pi12(ǫn−ǫn′)/integraldisplay\nφ∗\nnzφn′d3r, (3.8)\nwhereǫnandǫn′are energy eigenvalues corresponding to the eigenstates φnandφn′, respec-\ntively. At the K-edge, we assign the 4 pstates to φnand 1sstate toφn′. Hence the transition\noperator TE1is expressed as\nTE1(q,e,i) =iBE1/summationdisplay\niσNE1\nη[p′†\nησ(i)sσ(i)−s†\nσ(i)p′\nησ(i)]. (3.9)\nwhereiruns over Fe sites. Non-vanishing values of the coefficients NE1\nη’s are given by Nη=\n1/√\n3 forthe polarization along the η(=x,y,z) axis, independentof thepropagating direction\nof photon. The coefficient BE1is defined by\nBE1= (ǫ4p−ǫ1s)/integraldisplay∞\n0r3R4p(r)R1s(r)dr, (3.10)\nwhereR1s(r) andR4p(r) are radial wave functions of the 1 sand 4pstates with energy ǫ1sand\nǫ4p, respectively, in the Fe atom. The energy difference may be app roximate as ǫ4p−ǫ1s∼\n/planckover2pi1ωq=/planckover2pi1cq.11)Within the HF approximation BE1is estimated as BE1≈1.5×10−7cm·eV in\nthe 1s23d54p0.001-configuration of an Fe atom.9)\n3.2E2transition\nThe transition operator for the E2 transition is extracted from eq. (3.4) by retaining the\nsecond term in the expansion eiq·(r−ri)≈1+iq·(r−ri)+···. Let the photon be propagating\nalong the y-axis with the polarization parallel to the z-axis. Then we could derive a relation,\n/integraldisplay\nφ∗\nny∂\n∂zφn′d3r=−m\n/planckover2pi12(ǫn−ǫn′)/integraldisplay\nφ∗\nnyz\n2φn′d3r\n7/14J. Phys. Soc. Jpn. Full Paper\n+i\n2/integraldisplay\nφ∗\nnLxφn′d3r, (3.11)\nwhere/planckover2pi1Lxis the orbital angular momentum operator. The last term shou ld be moved into\nthe terms of the M1 transition.8)At theK-edge, we assign the 3 dstates to φnand the 1 s\nstate toφn′, respectively. Hence the transition operator TE2may be expressed as\nTE2(q,e,i) =−qBE2/summationdisplay\nimσNE2\nm(q)[d†\nmσ(i)sσ(i)−s†\nσ(i)dmσ(i)]. (3.12)\nWhen thephoton is propagating along the y-axis,mis selectively yzwithNE2\nyz(q) = 1/(2√\n15)\nin the polarization along the z-axis, and mis selectively xywithNE2\nxy(q) = 1/(2√\n15) in the\npolarization along the x-axis, respectively. Note that a relation NE2\nm(−q) =−NE2\nm(q) holds.\nTheBE2are defined by\nBE2= (ǫ3d−ǫ1s)/integraldisplay∞\n0r4R3d(r)R1s(r)dr, (3.13)\nwhereR3d(r) is radial wave function of the 3 dstate with energy ǫ3din the Fe atom. An\nevaluation within the HF approximation gives BE2≈1.7×10−16cm2·eV.9)\n4. Absorption spectra\nRestricting the processes on Fe atoms, we sum up cross sectio ns at Fe sites to obtain the\nabsorption intensity I(ωq,e). Dividing it by the incident flux c/V, we have\nI(ωq,e)∝4π2e2\n/planckover2pi12cωq/summationdisplay\ni/summationdisplay\nf|∝angb∇acketleftΨf(i)|T(q,e,i)|Ψg(i)∝angb∇acket∇ight|2\n×δ(/planckover2pi1ωq+Eg−Ef), (4.1)\nwhereT(q,e,i) =TE1(q,e,i) +TE2(q,e,i), and|Ψg(i)∝angb∇acket∇ightand|Ψf(i)∝angb∇acket∇ightrepresent the ground\nand the final states with energy EgandEfat sitei, respectively. The sum over fis taken\nover all the excited state at Fe sites.\nIn the Fe pre-K-edge region, the final states are constructed by perturbation theory start-\ning from the states ( |Φm(d6),1sσ∝angb∇acket∇ight) in thed6configuration with the 1 s-core hole. Within the\nsecond-order perturbation of ˜H4p−3d, they are given by\n|Ψf(i)∝angb∇acket∇ight=|Φm(d6),1sσ∝angb∇acket∇ight\n+/summationdisplay\nnkη′|Φn(d5),kη′σ,1sσ∝angb∇acket∇ight1\nEf−E′\nnkη′\n× ∝angb∇acketleftΦn(d5),kη′σ,1sσ|˜H4p−3d|Φm(d6),1sσ∝angb∇acket∇ight, (4.2)\nwhereEfstands for the energy of the unperturbed state. It is defined a s\nEf=Em(d6)−ǫ1s−Eint(1s−d6), (4.3)\nwhereEint(1s−d6) is theinteraction energy between theelectron inthe 1 sstates andelectrons\n8/14J. Phys. Soc. Jpn. Full Paper\nin the 3d6configuration. In the second term of eq. (4.2), E′\nnkη′is defined as\nE′\nnkη′=En(d5)−ǫ1s+ǫ4p(k)−Eint(1s−4pd5), (4.4)\nwhereEint(1s−4pd5) is the interaction energy between the electron in the 1 sstates and\nelectrons in the 4 p3d5configuration. Symbols kη′σand 1sσappeared in the bras and kets\nindicate the presence of an electron in the 4 pstate (kη′σ) and the absence of a 1 s-core\nelectron with spin σ, respectively. Since the second term of eq. (4.2) is complet ely evaluated\nby|Φm(d6),1sσ∝angb∇acket∇ightandEf,thelabel fisspecifiedbythe m-theigenstates of the d6configuration\nand the core-hole spin σ. Notice that the lowest values of eqs. (4.3) and (4.4) corres pond to\nthe positions of the pre- and main-edges, respectively.\nThe sum over kmay be replaced by the integral with the help of the 4 pDOS. In our\nnumerical treatment, the position of the pre-edge energy is adjusted to the experimental\nvalue and the difference between the pre- and main-edges is cho sen as the minimum of E′\nnkη′\nto be 12 eV higher than Eg(d6)−ǫ1s−Eint(1s−d6). For simplicity, the explicit dependence on\nsiteiis omitted from the right hand side of eq. (4.2). From these wa ve-functions, we obtain\nthe expression of transition amplitudes at site iby\nM(q,e,i;f) =ME1(q,e,i;f)+ME2(q,e,i;f), (4.5)\nwith\nME1(q,e,i;f)≡ ∝angb∇acketleftΨf(i)|TE1(q,e,i)|Ψg(i)∝angb∇acket∇ight\n=/summationdisplay\nnkη′∝angb∇acketleftΦm(d6),1sσ|˜H4p−3d|Φn(d5),kη′σ,1sσ∝angb∇acket∇ight\n×1\nEf−E′\nnkη′∝angb∇acketleftΦn(d5),kη′σ,1sσ|TE1(q,e,i)|Φg(d5)∝angb∇acket∇ight,(4.6)\nME2(q,e,i;f)≡ ∝angb∇acketleftΨf(i)|TE2(q,e,i)|Ψg(i)∝angb∇acket∇ight\n=∝angb∇acketleftΦm(d6),1sσ|TE2(q,e,i)|Φg(d5)∝angb∇acket∇ight. (4.7)\nWith these amplitudes, eq. (4.1) is rewritten as\nI(ωq,q,e)∝1\n/planckover2pi1ωq/summationdisplay\ni/summationdisplay\nf|M(q,e,i;f)|2\n×Γ/π\n[/planckover2pi1ωq+Eg(d5)−Ef]2+Γ2, (4.8)\nwhere the δ-function is replaced by the Lorentzian function with the li fe-time broadening\nwidth of 1 s-core hole Γ = 0 .8 eV.\nNow we examine the symmetry relation of the amplitudes. Firs t, let the propagating\ndirection of photon be reversed with keeping other conditio ns. In eq. (3.4), iq·(r−ri) is to be\nreplacedby −iq·(r−ri).Weknowthat NE1\nηhasnodependenceon qandthat NE2\nm(−q)isequal\nto−NE2\nm(q). Since other conditions are the same, we have the new amplit udes (ME1)′=ME1\n9/14J. Phys. Soc. Jpn. Full Paper\nand (ME2)′=−ME2. Second, let the local magnetic moment at each Fe atom be reve rsed\nwith keeping the same shifts from the center of octahedron. T he reverse of the local magnetic\nmoment corresponds to taking the complex conjugate of the wa ve functions. Considering\neq. (4.6) together with eq. (3.9), we have ( ME1)′=−(ME1)∗. Similarly, considering eq. (4.7)\ntogether with eq. (3.12), we have ( ME2)′= (ME2)∗. Third, let the shifts of Fe atoms from\nthe center of octahedron be reversed with keeping the same lo cal magnetic moment, which\nmeans the reversal of the direction of the local electricdipole moment. This operation causes\nthe reversal of the sign of ˜H4p−3d. However, no change is brought about to the 3 dstates in\nthe 3d5- and 3d6-configurations, because the ligand field ˜H3d−3dvaries as δ2. As a result, we\nhave the new amplitude ( ME1)′=−ME1from eq. (3.9) while ( ME2)′=ME2.\nAs already stated, the direction of the local magnetic momen t could be reversed by revers-\ning the direction of the applied magnetic field, since the act ual material is a ferrimagnet with\nslightly deviating from a perfect antiferromagnet. Let I±(ωq,q,e) be the intensity for the ex-\nternal magnetic field along the ±caxis. Then, from the second symmetry relation mentioned\nabove, we have the average and the difference of the intensitie s as\n¯I(ωq,q,e)≡1\n2[I+(ωq,q,e)+I−(ωq,q,e)]\n∝1\n/planckover2pi1ωq/braceleftBigg/summationdisplay\ni/summationdisplay\nf/bracketleftBig\n|ME1(q,e,i;f)|2+|ME2(q,e,i;f)|2/bracketrightBig\n×Γ/π\n[/planckover2pi1ωq+Eg(d5)−Ef]2+Γ2\n+/summationdisplay\ni|BE1|22\n3/summationdisplay\nkΓ/π\n[/planckover2pi1ωq+Eg(d5)−E′\ngkη]2+Γ2/bracerightBigg\n, (4.9)\nand\n∆I(ωq,q,e)≡I+(ωq,q,e)−I−(ωq,q,e)\n∝2\n/planckover2pi1ωq/summationdisplay\ni/summationdisplay\nf/braceleftBigg\n/bracketleftbig\nME1(q,e,i;f)/bracketrightbig∗ME2(q,e,i;f)\n+/bracketleftbig\nME2(q,e,i;f)/bracketrightbig∗ME1(q,e,i;f)/bracerightBigg\n×Γ/π\n[/planckover2pi1ωq+Eg(d5)−Ef]2+Γ2, (4.10)\nrespectively. The ME1andME2represent the amplitudes when the magnetic field is applied\nparallel to the caxis. For the average intensity [eq. (4.9)], the last term de scribes the K-edge\nspectra due to the E1 transition that the 1 selectron is excited to the 4 pband. Although its\nmain contribution is restricted in the main K-edge region, its tail spreads over the pre- K-edge\nspectra due to the life-time width. We see that the difference i ntensity [eq. (4.10)] is brought\n10/14J. Phys. Soc. Jpn. Full Paper\nabout from the E1-E2 interference process. According to the symmetry relation s mentioned\nabove, it is expected to follow5)\n∆I(ωq,q,e)∝q·/summationdisplay\niPloc(i)×Mloc(i), (4.11)\nwherePloc(i) andMloc(i) are the electric and the magnetic dipole moment of Fe atom at\nsitei, respectively. Note that Ploc(i) is proportional to δi[≡(0,0,δ)]. Then, the right hand\nside of eq. (4.11) is the sum of the local toroidal moment τ(i) [≡δi×Mloc(i)].14)We have\nalready derived the same form in the optical absorption spec tra,8)which is brought about by\ntheE1-M1 interference process. The spectra change their sign if one of the vectors among q,\nPloc, orMlocreverses its direction.\nFigure 2 shows the calculated average intensity I(ωq,q,e) as a function of photon energy\n/planckover2pi1ωq, in comparison with the experiment.5)The 1s-core energy is adjusted such that the K-\nedge position corresponds to the experiment. The intensity at theK-edge (/planckover2pi1ωq>7120 eV)\nmainly comes from the E1-E1 process given by the last term of eq. (4.9). In the pre- K-\nedge region ( /planckover2pi1ωq∼7110−7115 eV), the tail of that intensity spreads due to the life-t ime\nbroadening of the core level. In addition, we have another co ntribution of E1-E1 process\nthrough the term proportional to |ME1|2, and that of the E2-E2 process through the term\nproportional to |ME2|2. The latter is found larger than the former, giving rise to a s mall\ntwo-peak structure with a weak polarization dependence (th e inset in Fig. 2). The E1-E1\nprocess through the term proportional to |ME1|2is effective only on the noncentrosymmetric\nsituation, because the 4 p3d51s-configuration has to mix with the 3 d61s-configuration. Note\nthat the E1-E1 process could also gives rise to the intensity in the pre- K-edge region through\nthe mixing of the 4 pstate with the 3 dstates at neighboring Fe sites. This process need not the\nnoncentrosymmetric situation, and has nothing to do with th e magnetoelectric spectra. It is\nknown that a substantial intensity of the resonant x-ray sca ttering (RXS) spectra is brought\nabout from this process at the pre- K-edge on LaMnO 3,15)but the present analysis could not\ninclude this process because of the cluster size.\nFigure 3 shows the magnetoelectric spectra ∆ I(ωq,q,e) as a function of photon energy\n/planckover2pi1ωq. For polarization eparallel to the baxis, the calculated spectra form a positive sharp\npeak and then change into a negative double peak with increas ing/planckover2pi1ωq. On the other hand,\nfor polarization eparallel to the caxis, the calculated spectra form a negative sharp peak and\nthen change into a positive sharp peak with increasing /planckover2pi1ωq. The corresponding experimental\ncurves look similar, but their signs are opposite to the calc ulated ones. We do not find the\norigin for the opposite sign.\n5. Concluding Remarks\nWe have studied the magnetoelectric effects on the x-ray absor ption spectra in a polar\nferrimagnet GaFeO 3. We have performed a microscopic calculation of the absorpt ion spectra\n11/14J. Phys. Soc. Jpn. Full Paper\n7100 7120 7140 7160\nhωq [eV]012345Absorption coefficients (arb. units)Theory\n04p DOS\n7100 716005\nExp.\nFig. 2. Average intensity I(ωq,q,e) as a function of photon energy /planckover2pi1ωq. Photons propagate along\nthe positive direction of the aaxis with polarization vector ealong the baxis. The solid line repre-\nsents the calculated spectra. The broken line denotes the experim ental data with the background\nintensity subtracted from the raw data given in ref. [4]. The dotted line represents the 4 pDOS;\nboth the low-energy and high-energy sides are arbitrarily cut-off.\nusingaclustermodelofFeO 6.TheclusterconsistsofanoctahedronofOatomsandanFeato m\ndisplaced from the center of octahedron. We have disregarde d additional small distortions of\nthe octahedron. We have derived an effective hybridization be tween the 4 pand 3dstates as\nwell as the ligand field on the 3 dstates by modifying the Fe-O hybridizations due to the shift s\nof Fe atoms. This leads to the mixing of the 4 p3d5-configuration to the 3 d6-configuration,\nand thereby to finite contributions of the E1-E2 interference process to the magnetoelectric\nspectra. We have derived the symmetry relations of the ampli tudesME1andME2, and have\ndiscussedthe directional dichroism of the spectra. The clu ster model used in the present paper\nis the same as the model used in the analysis of the optical abs orption spectra in GeFeO 3\nand is similar to the model used in the analysis of RXS in Fe 3O4.11)We have numerically\ncalculated the magnetoelectric spectra as a function of pho ton energy in the pre- K-edge\nregion. Although the spectral shapes are similar to the expe rimental curves, their signs are\nopposite to the experimental ones. The origin for the opposi te sign has not been clarified yet.\nWe would like to simply comment that the magnetoelectric spe ctra in the optical absorption\nare obtained in agreement with the experiment by using the sa me model.8)\nThe magnetoelectric effect on RXS has been studied experiment ally16)and theoreti-\ncally17,18)in GaFeO 3. We think the approach used in the present paper is effective al so\nto the analysis of RXS. Closely related to these studies, the magnetoelectric effects on RXS\nhave also been measured,19,20)and theoretically analyzed11)in magnetite, where A sites are\ntetrahedrally surrounded by oxygens with the local inversi on symmetry being broken.\n12/14J. Phys. Soc. Jpn. Full Paper\n7090 7110 7130 7150\nhωq [eV]−0.0004−0.000200.00020.0004∆µt [arb. units]e//b :Exp.\ne//c :Exp.\ne//b :Theory\ne//c :Theory\nFig. 3. Difference of the absorption intensities ∆ I(ωq,q,e) as a function of photon energy /planckover2pi1ωqwhen\nthe magnetic field is applied parallel and antiparallel to the caxis. The solid and broken lines\ncorrespond to ∆ I(ωq,q,e) where photons propagate along the positive direction of the aaxis\nwith polarization vector ealong the bandcaxes, respectively. Experimental data are taken from\nref. [4] and denoted as filled ( e∝ba∇dblb) and open ( e∝ba∇dblc) circles, respectively.\nAcknowledgment\nThis work was partly supported by Grant-in-Aid for Scientifi c Research from the Ministry\nof Education, Culture, Sport, Science, and Technology, Jap an.\n13/14J. Phys. Soc. Jpn. Full Paper\nReferences\n1) J. P. Remeika: J. Appl. Phys. 31(1960) S263.\n2) G. T. Rado: Phys. Rev. Lett. 13(1964) 335.\n3) T. Arima, D. Higashiyama, Y. Kaneko, J. P. He, T. Goto, S. Miyasa ka, T. Kimura, K. Oikawa, T.\nKamiyama, R. Kumai, and Y. Tokura : Phys. Rev. B 70(2004) 064426.\n4) J. H. Jung, M. Matsubara, T. Arima, J. P. He, Y. Kaneko, and Y. Tokura: Phys. Rev. Lett. 93\n(2004) 037403.\n5) M. Kubota, T. Arima, Y. Kaneko, J. P. He, X. Z. Yu, and Y. Tokur a: Phys. Rev. Lett. 92(2004)\n137401.\n6) E. A. Wood: Acta Crystallogr. 13(1960) 682.\n7) R. B. Frankel, N. A. Blum, S. Foner, A. J. Freeman, and M. Schieb er: Phys. Rev. Lett. 15(1965)\n958.\n8) J. Igarashi and T. Nagao: Phys. Rev. B 80(2009) 054418.\n9) R. Cowan: The Theory of Atomic Structure and Spectra (University of California,Berkeley,1981).\n10) J. Chen, D. J. Huang, A. Tanaka, C. F. Chang, S. C. Chung, W. B. Wu, and C. T. Chen : Phys.\nRev. B69(2004) 085107.\n11) J. Igarashi and T. Nagao : J. Phys. Soc. Jpn. 77(2008) 084706.\n12) W. A. Harrison: Elementary Electronic Structure (World Scientific,Singapore,2004).\n13) L. D. Landau and E. M. Lifshitz: Quantum Mechanics (Pergamon,Oxford,1977).\n14) Y. F. Popov, A. M. Kadomtseva, G. P. Vorob’ev, V. A. Timofeev a, D. M. Ustinin, A. K. Zvezdin,\nand M. M. Tegeranchi : Zh. Eksp. Teor. Fiz. 114(1998) 263 [Translation: Sov. Phys. JETP 87\n(1998) 146].\n15) M. Takahashi, J. Igarashi, and P. Fulde : J. Phys. Soc. Jpn. 69(2000) 1614.\n16) T. Arima, J. H. Jung, M. Matsubara, M. Kubota, J. P. He, Y. Ka neko, and Y. Tokura : J. Phys.\nSoc. Jpn. 74(2005) 1419.\n17) S. Di Matteo and Y. Joly : Phys. Rev. B 74(2006) 014403.\n18) S. W. Lovesey, K. S. Knight, and E. Balcar : J. Phys.:Condens. M atter19(2007) 376205.\n19) M. Matsubara, Y. Shimada, T. Arima, Y. Taguchi, and Y. Tokura : Phys. Rev. B 72(2005)\n220404(R).\n20) M.Matsubara,Y.Kaneko,J.P.He,H.Okamoto,andY.Tokura :Phys.Rev.B 79(2009)140411(R).\n14/14"    },    {        "title": "2105.08088v2.Fluctuation_induced_ferrimagnetism_in_sublattice_imbalanced_antiferromagnets_with_application_to_SrCu__2__BO__3____2__under_pressure.pdf",        "content": "Fluctuation-induced ferrimagnetism in sublattice-imbalanced antiferromagnets\nwith application to SrCu 2(BO 3)2under pressure\nPedro M. C^ onsoli, Max Fornoville, and Matthias Vojta\nInstitut f ur Theoretische Physik and W urzburg-Dresden Cluster of Excellence ct.qmat,\nTechnische Universit at Dresden, 01062 Dresden, Germany\n(Dated: August 16, 2021)\nWe show that a collinear Heisenberg antiferromagnet, whose sublattice symmetry is broken at the\nHamiltonian level, becomes a \ructuation-induced ferrimagnet at any \fnite temperature Tbelow the\nN\u0013 eel temperature TN. We demonstrate this using a layered variant of a square-lattice J1-J2model.\nLinear spin-wave theory is used to determine the low-temperature behavior of the uniform magne-\ntization, and non-linear corrections are argued to yield a temperature-induced qualitative change\nof the magnon spectrum. We then consider a layered Shastry-Sutherland model, describing a frus-\ntrated arrangement of orthogonal dimers. This model displays an antiferromagnetic phase for large\nintra-dimer couplings. A lattice distortion which breaks the glide symmetry between the two types\nof dimers corresponds to broken sublattice symmetry and hence gives rise to ferrimagnetism. Given\nindications that such a distortion is present in the material SrCu 2(BO 3)2under hydrostatic pressure,\nwe suggest the existence of a \ructuation-induced ferrimagnetic phase in pressurized SrCu 2(BO 3)2.\nWe predict a non-monotonic behavior of the uniform magnetization as function of temperature.\nI. INTRODUCTION\nThe \feld of quantum magnetism harbors a wealth of\nfascinating phenomena which are driven by \ructuations\n[1]. These include quantum spin liquids [2, 3] { stable\nstates of matter devoid of symmetry-breaking order {,\nseveral types of unconventional quantum phase transi-\ntions [4], as well as a variety of symmetry-breaking states\nstabilized by \ructuations. A large class of the latter are\ndescribed as \\order by disorder\", a mechanism where a\nsubset of states is selected from a classically degener-\nate manifold by either quantum or thermal \ructuations\n[5]. Order by disorder is prominent in strongly frustrated\nmagnets, one important example being the easy-plane\npyrochlore antiferromagnet where an ordered state is cho-\nsen from a one-parameter degenerate manifold [6].\nAmong the various frustrated spin systems, the\nShastry-Sutherland model [7] plays a prominent role. It\ndescribes a planar Heisenberg model of coupled pairs of\nspins 1=2 with a particular orthogonal-dimer structure.\nIts ground-state phase diagram features a dimer-singlet\nstate, a symmetry-breaking plaquette-singlet state, and\na N\u0013 eel antiferromagnet as function of increasing ratio\nof inter-dimer to intra-dimer couplings, x=J0=J[8{\n10]. Very recently, a narrow quantum spin-liquid phase\nhas been proposed in addition [11]. Local moments ar-\nranged on the Shastry-Sutherland lattice appear in a\nnumber of compounds, the most prominent one being\nthe spin-1=2 Mott insulator SrCu 2(BO 3)2[8, 12]. Re-\nmarkably, hydrostatic pressure can be used to tune x\nin SrCu 2(BO 3)2, and signatures of magnetic transitions\nhave been detected around 1 :8 GPa [13{19] and 4 :5 GPa\n[14], with various experimental aspects being under ac-\ntive debate [18, 20]. NMR experiments [13] yield evidence\nfor two distinct Cu sites in the intermediate phase, sug-\ngesting two types of inequivalent dimers. Antiferromag-\nnetic (AF) order has been detected by neutron di\u000bractionin the high-pressure phase [21].\nIn this paper we discuss the phenomenon of\n\ructuation-induced ferrimagnetism in antiferromagnets,\nand we propose that SrCu 2(BO 3)2at high pressure is in\nfact a ferrimagnet. Ferrimagnetism refers to states which\ndisplay both staggered and uniform magnetizations, and\nit commonly occurs in systems with two di\u000berent types\nof magnetic ions with unequal spin sizes [22]. Here, we\nidentify a distinct mechanism for ferrimagnetism: In a\nsystem with equal-sized spins which displays N\u0013 eel anti-\nferromagnetism in the ground state, a uniform magne-\ntization is induced at \fnite temperature solely by \ruc-\ntuation e\u000bects. More precisely, we show that thermal\n\ructuations generically produce a \fnite magnetization\nin a Heisenberg antiferromagnet once the Z2symmetry\nbetween the two sublattices is broken at the Hamiltonian\nlevel. Remarkably, quantum \ructuations do not produce\na \fnite magnetization at T= 0 due to spin conserva-\ntion, such that the uniform magnetization becomes a\nnon-monotonic function of temperature, as illustrated in\nFig. 1. We exemplify this in a layered toy model consist-\ning of two interpenetrating square-lattice ferromagnets,\nfor which we employ spin-wave theory to calculate the\ntemperature-induced magnetization. In addition, simple\nTmtot\nTN00~T 4~(TNT) \nFIG. 1: Qualitative temperature dependence of the\n\ructuation-induced uniform magnetization in sublattice-\nimbalanced Heisenberg antiferromagnets, with the critical ex-\nponent\f= 0:37 [23, 24] in d= 3 dimensions.arXiv:2105.08088v2  [cond-mat.str-el]  13 Aug 20212\nLandau theory is used to analyze the behavior near the\nN\u0013 eel temperature TN. We then consider a layered ver-\nsion of the Shastry-Sutherland model, as appropriate for\nthe material SrCu 2(BO 3)2. We predict the existence of a\nuniform magnetization in its orthorhombic high-pressure\nphase and provide a rough estimate for its amplitude.\nThe remainder of the paper is organized as follows:\nIn Sec. II we introduce the toy model and demonstrate\nthe phenomenon of \ructuation-induced ferrimagnetism.\nWe also discuss temperature-induced corrections to spin-\nwave spectrum and link them to general hydrodynam-\nics. Sec. III is devoted to the layered Shastry-Sutherland\nmodel appropriate for SrCu 2(BO 3)2where we provide\nquantitative results of relevance for its high-pressure\nphase. A discussion and outlook close the paper.\nII. FERRIMAGNETISM FROM THERMAL\nFLUCTUATIONS: TOY MODEL\nIn this section we utilize a simple toy model to discuss\nthe emergence of ferrimagnetism from thermal \ructua-\ntions in antiferromagnets with broken sublattice symme-\ntry. We also connect the results to general aspects from\nLandau theory and hydrodynamics.\nA. Model\nOur model is constructed from a bipartite square-\nlattice Heisenberg model with nearest-neighbor AF cou-\nplingJbetween spins S, displaying collinear N\u0013 eel order.\nTheZ2symmetry between the two sublattices is broken\nby adding second-neighbor couplings which are di\u000berent\nfor the two sublattices AandB; we label them J0\naandJ0\nb\nand choose them to be ferromagnetic in order to stabilize\nN\u0013 eel antiferromagnetism. Finally, we add a (small) fer-\nromagnetic inter-layer interaction J?such that magnetic\norder also appears at \fnite temperatures. The model is\ndepicted in Fig. 2, its Hamiltonian reads\nH=JX\nhijim~Si;m\u0001~Sj;m\u0000J?X\nim~Si;m\u0001~Si;m+1\n\u0000J0\naX\nhhii02Aii~Si;m\u0001~Si0;m\u0000J0\nbX\nhhjj02Bii~Sj;m\u0001~Sj0;m (1)\nwherei;jdenote in-plane lattice coordinates, mis the\nlayer index, andhijiandhhii0iidenote pairs of \frst and\nsecond neighbors, respectively. The model displays a\nglobal SU(2) spin symmetry. For J0\na6=J0\nbit features\na two-site unit cell, and we will set the lattice constant\nof the underlying square lattice to unity.\nVarious limiting cases are of interest: On one hand,\nforJ\u001dJ0\na;bwe have an antiferromagnet in which J0\na6=\nJ0\nbinduces weak sublattice symmetry breaking. On the\nother hand, J0\na;b\u001dJcorresponds to two inequivalent\nferromagnets on the two sublattices which are weakly\ncoupled byJsuch that global collinear AF order emerges.\nFIG. 2: Layered square-lattice Heisenberg antiferromagnet\nwith two inequivalent second-neighbor couplings J0\na(green)\nandJ0\nb(red). Thermal \ructuations induce a \fnite uniform\nmagnetization for 0 <T <T NonceJ0\na6=J0\nb.\nFor purposes of illustration, we will also discuss the\nmodel with unequal spin sizes on the two sublattices, i.e.,\nspinsSand\u0011Son sublattices AandB, respectively. For\n\u00116= 1, this then corresponds to the standard setting of\na ferrimagnet. Unless noted otherwise, \u0011= 1 is assumed\nbelow.\nWe note that a ferromagnetic inter-layer coupling is\ncrucial for the broken sublattice symmetry: If, instead,\nJ?were antiferromagnetic, the long-range order would\nfeature a two-layer periodicity, with J0\na(as well as J0\nb)\nacting on di\u000berent sublattices in adjacent layers, such\nthat, globally, the sublattice symmetry would be unbro-\nken.\nB. Ground-state antiferromagnetism\nIn the classical limit, S!1 , the ground state of the\nmodel (1) is obviously a two-sublattice collinear antifer-\nromagnet, which has zero total magnetization Mand\ndisplays a global U(1) symmetry. This symmetry im-\nplies that all quantum \ructuation processes occurring at\n\fniteSon top of the collinear state conserve the longi-\ntudinal component of M. As a result, the magnetization\nremains zero at any order of a 1 =Sexpansion, as will\nbe demonstrated explicitly below. In fact, this argument\nholds beyond the semiclassical limit: For in\fnitesimal J\nthe ground state consists of two saturated ferromagnets\nwhose magnetizations are opposite, resulting in M= 0.\nSince \ructuation processes on top of this state arise from\nJonly,Mremains zero at any order in perturbation the-\nory inJ=J0for allS. From this we conclude that there\nis a stable AF quantum ground state which has zero to-\ntal magnetization, protected by SU(2) symmetry, despite\nthe broken sublattice symmetry.\nWe note that these arguments do not exclude the ex-\nistence of phase transitions to other phases with non-\nzeromtotwhich might exist for \fnite S, strongly broken\nsublattice symmetry, and/or J;J0of similar magnitude.3\nHowever, for the toy model at hand such phases are not\nexpected as long as the couplings J0\na;bremain ferromag-\nnetic.\nC. Spin-wave spectrum at T= 0\nThe broken sublattice symmetry has in\ruence on\nthe spin-wave spectrum. Since the ground state is\na collinear antiferromagnet, we expect two Goldstone\nmodes which are degenerate and linearly dispersing in\nthe long-wavelength limit. This is con\frmed by an ex-\nplicit spin-wave calculation using the standard Holstein-\nPrimako\u000b technique. The model features a two-site mag-\nnetic unit cell, such that all results can be obtained an-\nalytically, for details see Appendix A. At leading order\nin 1=S, i.e., by linear spin-wave theory, we obtain the\ndispersion of the two spin-wave modes as\n!~k\u0006=S\u0010q\nP2\n~k\u0000Q2\n~k\u0006R~k\u0011\n(2)\nwith\nP~k= 2 (J0\na+\u0011J0\nb)\u0018~k+ (\u0011+ 1) [2J+J?(1\u0000coskz)];\nQ~k= 4\u00111=2J\r~k;\nR~k= 2 (J0\na\u0000\u0011J0\nb)\u0018~k+ (\u0011\u00001) [2J\u0000J?(1\u0000coskz)];\n\u0018~k= 1\u0000coskxcosky;\n\r~k= (coskx+ cosky)=2: (3)\nFor\u0011= 1 we \fnd two modes which are degenerate for\nJ0\na=J0\nb, but non-degenerate otherwise, i.e., the spectrum\nis split by broken sublattice symmetry. However, while\nthe modes split at general ~k, they have the same linear\nslope in the hydrodynamic limit of small j~kj. Denoting\nthe in-plane wavevector as ~kk= (kx;ky) andkk=j~kkj,\nthe mode energies can be expanded as\n!~k\u0006= 2Sp\n2Jq\n(J0a+J0\nb+J)k2\nk+J?k2z\n\u0006S(J0\na\u0000J0\nb)k2\nk+O\u0000\nk3\u0001\n(4)\nforJ >0. The mode dispersions are illustrated in Fig. 3\nfor parameter sets with (a) J\u001dJ0\na;band (b)J\u001cJ0\na;b.\nNote that !~k\u0006is symmetric with respect to rotations\naround the kzaxis up toO\u0000\nk2\u0001\n. Moreover, the quadratic\nterm in Eq. (4) vanishes when J0\na=J0\nb, i.e., for equivalent\nsublattices. We also see a vanishing of the quadratic term\nforkk= 0 because the interlayer coupling J?does not\ndistinguish sublattice AfromB. Hence, spin waves with\nzero in-plane momentum do not experience the sublattice\nsymmetry breaking.\nIt is instructive to discuss the limit J0\na;b\u001dJ, which, as\nmentioned earlier, describes two inequivalent and weakly\ncoupled ferromagnetic subsystems. In this setting, de-\ncreasingJshould restrict the linear portion of the spec-\ntrum to smaller and smaller values of kas the Gold-\nstone modes approach the quadratic shape expected for\n(a)\n012345\n(b)024680.000.010.02FIG. 3: Spin-wave dispersion for the toy model (1) along\na path in the Brillouin zone and parameters (a) J= 100,\nJ0\na= 10,J?= 2 and (b) J= 0:1,J0\na= 10,J?= 1 in units\nofJ0\nb, both with \u0011= 1. Blue (red) curves correspond to !~k+\n(!~k\u0000), respectively. The inset in (b) shows a zoom into the\nlow-energy part of the dispersion near ~k= (0;0;0).\ndecoupled ferromagnets. One can track this transforma-\ntion by computing the nonzero wavenumber k\u0003(\u0012), with\ntan\u0012=kk=kz, at which the magnitudes of the linear and\nquadratic terms in Eq. (4) become equal. A simple cal-\nculation yields\nk\u0003(\u0012) =2p\n2J\njJ0a\u0000J0\nbjsin\u0012\u0012\nJ0\na+J0\nb+J+J?\ntan2\u0012\u00131=2\n;(5)\nwhich con\frms our expectation: For \fxed J0\na6=J0\nband\nJ?,k\u0003indeed decreases with J.\nInspecting the Bogoliubov coe\u000ecients, explicitly listed\nin Eq. (A8), shows that the two modes have di\u000berent\nweights on the two sublattices once J0\na6=J0\nb. More specif-\nically, forJ0\na> J0\nbthe mode + (\u0000) is primarily located\non sublattice A(B).\nWe note that the low-energy behavior of the mode dis-\npersion is qualitatively di\u000berent for \u00116= 1, and we will\nget back to this in Sec. II F below. This section will also\ndiscuss corrections to the mode dispersion beyond linear\nspin-wave theory.4\nD. Uniform magnetization at low temperatures\nSpin-wave theory can be used to calculate \ructuation\ncorrections to the sublattice magnetizations via a 1 =S\nexpansion. The next-to-leading order result, obtained\nfrom linear spin-wave theory (see Appendix A for de-\ntails), reads\nmA=S\u0000m0(T) +1\nNX\n~k\u0010\nn~k\u0000\nBE\u0000n~k+\nBE\u0011\n;\nmB=\u0000\u0011S+m0(T) +1\nNX\n~k\u0010\nn~k\u0000\nBE\u0000n~k+\nBE\u0011\n: (6)\nHere,n~k\u0006\nBE= 1=(e!~k\u0006=T\u00001) is the Bose-Einstein distri-\nbution function where we have set Boltzmann's constant\nto unity,kB= 1, and\nm0(T) =X\n~kF~k\nN\u0010\n1 +n~k\u0000\nBE+n~k+\nBE\u0011\n\u00001\n2; (7)\nwithF~kbeing a temperature-independent coe\u000ecient\nspeci\fed in Eq. (A9). One can see that the quantum cor-\nrections,m0(T= 0), are equal on both sublattices, lead-\ning to vanishing total magnetization at T= 0 for\u0011= 1,\nas announced in Sec. II B above. The thermal correc-\ntions, however, are di\u000berent because the non-degenerate\nspin-wave modes experience di\u000berent thermal occupa-\ntions. As a result, we obtain a uniform magnetization\nper site,\nmtot=(1\u0000\u0011)\n2S+1\nNX\n~k\u0010\nn~k\u0000\nBE\u0000n~k+\nBE\u0011\n; (8)\nwhich is \fnite for non-zero temperature even if the spin\nsizes on the two sublattices are equal, \u0011= 1. This is a\ncentral result of this paper.\nWe expect these expressions to yield reliable results\nfor low temperatures where the occupation of the spin-\nwaves modes remains small to validate the approximation\nof noninteracting magnons. This corresponds to having\nsmall 1=Scorrections in Eq. (6) in comparison to the\nleading-order term.\nThe \ructuation-induced uniform magnetization\nemerges at order S0in the spin-wave expansion. Fig. 4\ndepicts its temperature dependence, together with the\nthermal corrections to the sublattice magnetizations,\nat a \fxed ratio J0\na=J0\nb= 10 when (a) J\u001dJ0\na;band (b)\nJ\u001cJ0\na;b. In both cases, the low-temperature corrections\ntomAandmBscale asT2, a feature which is also\nobserved in antiferromagnets without broken sublattice\nsymmetry [25]. This can be easily rationalized by power\ncounting: For linearly dispersing modes, F~kscales\nas 1=k, and hence the leading contribution to the T\ndependence of the integral in Eq. (7) has the formR\ndkkd\u00002nBE(k=T) and scales as Td\u00001.\nIn this low-temperature regime, the only accessible ex-\ncited states lie within the energy range where the magnon\n(a)\n10010110210310-1010-810-610-410-2100\n(b)\n10-210-110010110210-910-710-510-310-1101FIG. 4: Fluctuation-induced uniform magnetization,\nmtot, as function of temperature, together with the \fnite-\ntemperature corrections to the sublattice magnetizations,\n\u0001mA;B=mA;B(T)\u0000mA;B(0). The coupling constants were\nset to (a)J= 100,J0\na= 10,J?= 2 and (b) J= 0:1,J0\na= 10,\nJ?= 1 in units of J0\nb. In both plots, the horizontal grid\nline marks the value 1 =2, whereas the vertical grid line in (b)\nindicates the temperature T=JS.\nbranches are nearly degenerate, so that a di\u000berence in\nthe occupation of the spin-wave modes emerges as a sub-\nleading e\u000bect in T. Expanding the dispersions in next-\nto-leading order in kand noting that the diverging F~k\nfactor is absent from Eq. (8), we \fnd that the uniform\nmagnetization scales as T4, see Appendix A.\nIn Fig. 4(a), we see that for J\u001dJ0\na;bthis low-Tpower\nlaw continues well beyond the point where T=S matches\nthe smallest coupling in the system, J0\nb, and extends up to\ntemperatures T\u0018JS, beyond which spin-wave theory is\nno longer valid. The reason is that strong Jyields both\nlarge spin-wave velocities at small k(4)and large val-\nues ofk\u0003(\u0012) (5), such that a signi\fcant di\u000berence in the\noccupations of the spin-wave modes only appears at rela-\ntively high energies. As a result, mtotonly reaches values\nof 10\u00003at the temperature where thermal corrections to\nmA;Bbecome signi\fcant, i.e., of order 10\u00001. The oppo-\nsite limit,J\u001cJ0\na;b, leads to a markedly di\u000berent behav-\nior, shown in Fig. 4(b). The now weak coupling Jstabi-\nlizes approximately sublattice-symmetric AF order only\nat very low temperatures. Thus, once T&JS, sublattice\nBbecomes much more susceptible to \ructuations than5\n(a)10-510-410-310-210-1\n(b)\n123456789100.000.010.020.030.04\nFIG. 5: Fluctuation-induced magnetization as in Fig. 4, but\nnow as function of J0\na=J0\nbat \fxed \fnite temperature. Param-\neters were set to (a) T=S = 50,J= 100,J?= 2 and (b)\nT=S = 1,J= 0:1,J?= 1 all in units of J0\nb, and are such that\nthe results with the ratio J0\na=J0\nb= 10 correspond to data in\nFig. 4 at the speci\fed temperatures.\nsublatticeA, which explains why mB(mA) starts grow-\ning faster (slower) than T2around the vertical dashed\nline. The deviation from the low-temperature scaling is\nthen responsible for enhancing the uniform magnetiza-\ntion, which becomes as large as 5 \u000110\u00002when thermal\ncorrections to mBreach 10\u00001.\nFig. 5 illustrates how the same quantities in Fig. 4 vary\nwith the ratio J0\na=J0\nbat \fxedT=(J0\nbS) when (a) J\u001dJ0\na;b\nand (b)J\u001cJ0\na;b. The uniform magnetization vanishes\nin the limit J0\na=J0\nb!1, which corresponds to restoring\nsublattice symmetry, while mtot/(J0\na=J0\nb\u00001) for small\nimbalance. However, the increase in mtotsaturates at a\npoint where J0\na=J0\nbbecomes so large that the spin-wave\nvelocities in Eq. (4) start growing at the same rate that\nk\u0003(\u0012) decreases in Eq. (5). Put more simply, saturation\noccurs when increasing J0\na=J0\nbonly leads to further split-\nting of the spin-wave bands at energies larger than T=S.\nThe main di\u000berence between Figs. 5(a) and (b) lies in the\norder of magnitude of the \ructuation-induced e\u000bects: At\nJ0\na=J0\nb= 10,mtotis two orders of magnitude larger in\npanel (b) compared to panel (a). Once again, this di\u000ber-\nence follows from the fact that the strong AF coupling J\nbetween the sublattices in (a) balances how \ructuationsact on each of them, therefore diminishing the resulting\nuniform magnetization.\nE. Uniform magnetization at elevated\ntemperatures\nTo access elevated temperatures where spin-wave the-\nory is no longer reliable, we resort to arguments from\nLandau theory. The sublattice-imbalanced antiferromag-\nnet { equivalent to a ferrimagnet { is described by two\norder parameters, a staggered magnetization and a uni-\nform magnetization, which are linearly coupled [26]. As\na result, there is a single transition upon cooling at TN\nfrom the paramagnet to the ferrimagnet, and both order\nparameters are zero (non-zero) above (below) TN, respec-\ntively. The linear coupling also implies that the onset of\nthe uniform magnetization below TNis identical to that\nof the staggered magnetization, hence mtot/(TN\u0000T)\f\nwhere\fis the order-parameter exponent [27]. For the\nclassical phase transition at hand, the universality class\nfor the Heisenberg magnet remains O(3) independent of\nwhether the transition is into an antiferromagnetic or fer-\nrimagnetic state. Hence, \f= 0:37 ind= 3 space dimen-\nsions [23, 24]. Together with the low-temperature result\nmtot/T4, we conclude that the uniform magnetization\ndisplays a non-monotonic temperature dependence as il-\nlustrated in Fig. 1.\nAgain, it is instructive to discuss the limit J\u001cJ0\na;b.\nForJ0\na>J0\nb, the transition at TNconcerns primarily the\nonset of ferromagnetism on the A sublattice. Weak J\nproduces a small opposite magnetization on the Bsub-\nlattice, resulting in a collinear ferrimagnet. (Note that\nthe energy gained by maintaining the collinearity of the\nglobal magnetic order is extensive, whereas the entropy\nassociated with directional \ructuations of sublattice B\nis only intensive.) In the limit of large sublattice im-\nbalance,J0\na\u001dJ;J0\nb, there is hence a window of tem-\nperatures below TNwhere the uniform magnetization is\nlarge,mtot\u0019mA=2 asmB\u001cmA. In this limit, it is also\neasy to see that the sign of the uniform magnetization is\nthe same at low Tand close to TN: It is the sublattice\nwith weaker magnetism that experiences stronger ther-\nmal \ructuations, such that mtotaligns with the magne-\ntization of the more strongly ordered sublattice, i.e., the\nAsublattice if J0\na>J0\nb.\nF. Hydrodynamic modes and corrections to the\nspin-wave spectrum\nFor a broader picture, we connect our \fndings\nto general hydrodynamic considerations. A collinear\ntwo-sublattice antiferromagnet, spontaneously breaking\nSU(2) symmetry, is characterized by a non-conserved or-\nder parameter and displays two linearly dispersing Gold-\nstone modes which are degenerate in the long-wavelength\nlimit. This is in agreement with the spin-wave result (4)6\nfor\u0011= 1.\nA ferrimagnet, in contrast, has in addition a conserved\norder parameter, namely uniform magnetization mtot.\nAs a result, it features a single quadratically dispersing\nGoldstone mode [28, 29]. This can be nicely seen in the\nexplicit spin-wave expressions (2) for \u00116= 1: Here !~k\u0000\nis gapless and quadratic in kwhereas!~k+, though also\nquadratic, exhibits a gap given by 4 j\u0011\u00001jJS.\nTogether, this implies that the low-energy spectrum of\nthe model (1) must change qualitatively when going from\nT= 0 toT >0: The system turns from an antiferromag-\nnet to a ferrimagnet, such that one of the T= 0 Gold-\nstone modes must acquire a temperature-induced gap,\nand the other one must change its dispersion from linear\nto quadratic at small k. This change can be captured by\nnon-linear spin-wave theory, i.e., has the form of 1 =Scor-\nrections at \fnite T. While it is straightforward to write\ndown the quartic terms in the spin-wave Hamiltonian,\nanalyzing all terms at \fnite temperature turns out to be\nrather laborious, and therefore we refrain from doing so.\nHowever, as these corrections are suppressed as T!0,\nthey have no in\ruence on the leading low-temperature\nbehavior of the uniform magnetization, mtot/T4.\nIII. FERRIMAGNETISM IN A LAYERED\nDISTORTED SHASTRY-SUTHERLAND MODEL\nAfter having established that sublattice-imbalanced\nantiferromagnets generically display \ructuation-induced\nferrimagnetism, we now turn to an experimentally rele-\nvant example, namely the Shastry-Sutherland lattice as\nrealized in the compound SrCu 2(BO 3)2.\nA. Model and symmetries\nOur starting point is the Heisenberg model on\nthe Shastry-Sutherland lattice, consisting of orthogonal\ndimers of spins 1 =2 with intra-dimer coupling Jand inter-\ndimer coupling J0, Fig. 6(a). This model features a four-\nsite unit cell, containing two dimers, and displays, in ad-\ndition to mirror symmetries along the dimer axes, a non-\nsymmorphic glide symmetry which maps the two types\nof dimers into each other.\nThe phase diagram of the Shastry-Sutherland model\nhas been determined numerically [9, 10], Fig. 6(b): It\ncontains a paramagnetic dimer phase for x=J0=J <\n0:675, a bipartite N\u0013 eel antiferromagnet for x > 0:765,\nand a plaquette-ordered singlet paramagnet, the so-called\nempty-plaquette phase, in between [10]. In addition, a\nvery recent numerical study [11] has proposed that a\ngapless quantum spin-liquid phase is realized in a nar-\nrow range, 0 :79< x < 0:82, intervening between the\nplaquette-singlet and AF phases [30].\nThe ferrimagnetism discussed in this paper appears\nupon breaking the glide symmetry, such that two dif-\nferent types of (mutually parallel) dimers emerge. Such\n(a)\n(b)\ndimer\nsingletplaquette\nsingletNéel\nAFSL?FIG. 6: (a) Shastry-Sutherland model with intra-dimer cou-\nplingsJa;band inter-dimer coupling J0. The dashed lines\nindicate the unit cell. (b) Ground-state phase diagram of the\nS= 1=2 Shastry-Sutherland model with Ja=Jb, as reported\nin Ref. 10. An intermediate spin-liquid (SL) phase (shaded)\n[30] has been recently proposed in Ref. 11. In the present\nwork, the focus is on the antiferromagnetic phase at large\nJ0=Jshown in red.\nFIG. 7: Layered Shastry-Sutherland model (9) used to de-\nscribe SrCu 2(BO 3)2. An orthorhombic distortion is assumed\nto generate di\u000berent intra-dimer couplings Ja,Jb.\nsymmetry breaking corresponds to an orthorhombic dis-\ntortion where half of the intra-dimer bonds elongate and\nthe other half contract [31, 32]. In the AF state, each of\nthe intra-dimer couplings acts on one AF sublattice only,\nsuch that the symmetry between the two sublattices is\nbroken. In the following we will therefore consider a dis-\ntorted Shastry-Sutherland model with intra-dimer cou-7\nplingsJa;band inter-dimer coupling J0. To meaningfully\ndiscuss magnetic order at \fnite temperature, we work\nwith a layered version of the model. Guided by the struc-\nture of SrCu 2(BO 3)2, we consider a stacking of the layers\nsuch that orthogonal dimers are on top of each other, and\ninclude a (small) antiferromagnetic Heisenberg interlayer\ncouplingJ?which pairwise connects vertically stacked\ndimers [33]. The model, illustrated in Fig. 7, is described\nby the Hamiltonian\nH=JaX\nhhij2Aiim~Si;m\u0001~Sj;m+JbX\nhhij2Biim~Si;m\u0001~Sj;m\n+J0X\nhiji~Si;m\u0001~Sj;m\n+J?X\nhhijiim(~Si;m+~Sj;m)\u0001(~Si;m+1+~Sj;m+1) (9)\nwhere each term in the last sum represents four couplings\nbetween the spins of neighboring dimers in zdirection,\nand we consider spins of general size S.\nThe ground states of the single-layer version of the dis-\ntorted Shastry-Sutherland model (9) with S= 1=2 have\nbeen studied in Refs. 31, 32. While all phases of the orig-\ninal Shastry-Sutherland model appear stable against a\nsmall dimer imbalance, the main \fnding of Ref. 31 is the\nexistence of a Haldane-like phase for strongly imbalanced\ndimers and weak inter-dimer coupling J0. This phase is\ndominated by one-dimensional correlations; it is adiabat-\nically connected to a so-called full-plaquette phase and\nhas been argued [32] to be a candidate for the interme-\ndiate phase observed experimentally in SrCu 2(BO 3)2.\nB. Spin-wave theory in the antiferromagnetic phase\nAs announced, we are interested in antiferromagnets\nwith broken sublattice symmetry. Hence we focus on the\nphysics of the model (9) in the regime of larger x=J0=J,\nwhere one encounters a clear connection to the toy model\ndiscussed in Sec. II: Both systems display a collinear anti-\nferromagnetic classical ground state with two sublattices,\neach of which experiences an independent internal cou-\npling. Therefore, the two models share the same mecha-\nnism for breaking sublattice symmetry.\nAs in Sec. II, we perform a spin-wave calculation to\ndetermine its properties in a 1 =Sexpansion both at zero\nand \fnite temperature. For the standard 2D Shastry-\nSutherland model, the linear-spin-wave theory descrip-\ntion of the AF phase breaks down for x < 1, signaling\na transition to a di\u000berent phase at this level of the ap-\nproximation. Hence, we work with parameter sets corre-\nsponding to x&1.\nIn the AF phase of model (9), the symmetry-broken\nstate features four sites per unit cell, such that the Bo-\ngoliubov transformation can only be performed numer-\nically. For convenience, we employ an in-plane coordi-\nnate system corresponding to the square lattice shown in\n012345FIG. 8: Spin-wave dispersion of the distorted Shastry-\nSutherland model (9) along a path in the Brillouin zone for\nparameters Ja= 0:97,Jb= 0:9 andJ?= 0:2 in units of J0.\nFig. 6(a), such that the basis vectors of the (magnetic)\nunit cell are given by a1= (2a;0;0),a2= (0;2a;0), and\na3= (a;a;c ) whereaandcare the in-plane and out-of-\nplane lattice constants which we set to unity in the fol-\nlowing. The relevant details of the calculation are given\nin Appendix B; here we summarize the key results.\nThe spin-wave spectrum along an exemplary path in\nthe BZ is illustrated in Fig. 8. Of the four spin-wave\nmodes, two are gapped at small momenta, while the\ntwo others are linearly dispersing Goldstone modes. As\nwith the toy model, these modes are degenerate only\nforJa=Jb, i.e. when the sublattice symmetry is pre-\nserved; for Ja6=Jbthey share the same velocity, but\ndi\u000ber at quadratic order except for kk= 0. Notably,\nthe Goldstone-mode velocity is highly anisotropic, e.g., it\nis di\u000berent even for di\u000berent in-plane directions because\nJa6=Jbleaves only mirror symmetries intact.\nC. Ferrimagnetism\nThe qualitative arguments for \ructuation-induced fer-\nrimagnetism brought forward in Sec. II apply unchanged\nto the sublattice-imbalanced antiferromagnet of the\nShastry-Sutherland model. Our numerical evaluation of\n1=Scorrections to the magnetizations on the individual\nsites of the unit cell, as detailed in Appendix B, con-\n\frms this expectation. The quantum corrections are\nequal on all sites, resulting in a vanishing total magne-\ntization at T= 0. In contrast, the thermal corrections\nare di\u000berent on the A(up) andB(down) sublattices,\nwhile they are pairwise equal on the two unit-cell sites\nbelonging to the AandBsublattice, respectively. As\nbefore, the thermal corrections to the sublattice mag-8\n(a)10-1210-1010-810-610-410-2100\n(b)10-210-110010-1010-810-610-410-2100\nFIG. 9: Uniform magnetization, mtot, and thermal correc-\ntions, \u0001mA;B, to the sublattice magnetization of the distorted\nShastry-Sutherland model, with parameters (a) Ja= 0:9,\nJb= 0:8,J?= 0:1 and (b)Ja= 0:97,Jb= 0:9,J?= 0:01 in\nunits ofJ0.\nnetizations, \u0001 mA(T) and \u0001mB(T), scale proportional\ntoT2at low temperature. The uniform magnetization,\nmtot= [mA(T) +mB(T)]=2, scales as T4because two\nmode dispersions di\u000ber at quadratic order only.\nNumerical results illustrating the variation of the mag-\nnetization with temperature are shown in Fig. 9. The\nparameters in panels (a) and (b) correspond to weaker\n(stronger) \ructuation corrections, driven both by the\ndi\u000berentJ?and byJa;bbeing further away (closer) to\nthe critical value Ja;b=J0within spin-wave theory.\nConsequently, the uniform magnetization is much larger\nin (b) compared to (a) at the same temperature, even\nthough the ratio Ja=Jbis similar in both cases. While\nthe extreme limit of two weakly coupled ferromagnets,\ndiscussed for the toy model, cannot be realized in the\nShastry-Sutherland model, the magnetization neverthe-\nless can get as large as 5 \u000110\u00003at the temperature where\nthe largest \u0001 mis 10\u00001.\nFig. 10 depicts the e\u000bect of varying the sublattice im-\nbalance at a \fxed temperature while keeping J0as the\nlargest coupling in the system. Di\u000berently from the toy\nmodel, the thermal corrections are now larger on the\nstrongly coupled sublattice, since the AF couplings Ja;b\nare frustrated. Still, the uniform magnetization shows\nthe same trend as in Fig. 5, growing with increasing sub-\n1 2 3 4 5 6 7 8 9100.000.050.100.150.20FIG. 10: Same quantities as in Fig. 9, but now as a function\nofJa=Jbat \fxed temperature T=J0Sand withJa= 0:9\n(and varying Jb) andJ?= 0:1 in units of J0. The ratio\nJa=Jb= 1:125 reproduces the data in Fig. 9(a) at the speci\fed\ntemperature.\nlattice imbalance until it saturates at large Ja=Jb. For\nSrCu 2(BO 3)2, small orthorhombic distortion likely im-\nplies thatJa=Jbremains close to unity.\nD. Application to SrCu(BO 3)2under pressure\nSrCu 2(BO 3)2assumes a tetragonal structure at ambi-\nent pressure and low temperatures, where its magnetic\nproperties are in very good agreement with those of the\ntwo-dimensional Shastry-Sutherland model in the small-\nxdimer phase. The magnetic couplings have been esti-\nmated to be J\u001985 K andJ0\u001954 K [8, 33].\nHigh-pressure studies of SrCu 2(BO 3)2detected various\nsignatures of pressure-driven phase transitions. In par-\nticular, indications for a di\u000berent, but still paramagnetic,\nphase were found above 2 GPa [13], and this transition\nwas later located more precisely to be around 1 :8 GPa\n[14{19]. While it is natural to assume that this para-\nmagnetic phase represents the empty-plaquette phase of\nthe Shastry-Sutherland model, both the NMR results of\nRef. 13 and the neutron scattering results of Ref. 16\nappear to be incompatible with this idea: The empty-\nplaquette phase displays equivalent magnetic sites and\nC4symmetry while the NMR data indicate the existence\nof two inequivalent magnetic sites. To resolve this con-\ntradiction, it has been argued [32] that an orthorhombic\ndistortion, stabilizing a di\u000berent plaquette phase in the\nintermediate regime, is most compatible with the NMR\n[13] and neutron scattering [16] data.\nAt higher pressures, a structural transition to a mon-\noclinic structure occurs around 4 :5 GPa [14], and AF or-\nder with a rather high N\u0013 eel temperature of 120 K has\nbeen detected at 5 :5 GPa via neutron scattering [21]. It\nhas been suggested, but not clari\fed beyond doubt, that\nthis magnetic order in fact emerges around 4 GPa before\nthe structural transition [20]. In addition, a recent low-9\ntemperature thermodynamic study [18] found indications\nfor a previously undetected AF state below 4 K occurring\nbetween 3 and 4 :2 GPa. It has been suggested [18] that it\nis this low-temperature AF state which should be inter-\npreted as the genuine AF state of the Shastry-Sutherland\nmodel, given that the higher-pressure monoclinic sys-\ntem no longer features the orthogonal dimers character-\nistic of the Shastry-Sutherland model. The same study\nalso presented evidence for an additional phase transi-\ntion occurring above 4 :2 GP at 8 K and proposed that\nthis transition is related to the existence of yet another\nlow-temperature magnetic state, which is likely to dis-\nplay AF order as well. However, a full characterization\nof this phase is still lacking. Apparently, more work is\nneeded to discern the fascinating high-pressure physics of\nSrCu 2(BO 3)2.\nFor our purpose, we focus on the fact that pressure-\ninduced structural distortions lead to inequivalent\ndimers; this likely applies to all pressures larger than\n2 GPa [13, 21, 32]. AF order in each layer will thus be\nsublattice-imbalanced because of the broken glide sym-\nmetry. Achieving a \fnite uniform magnetization then\nrelies on the uniform magnetization in adjacent layers\nbeing parallel. Our model in Fig. 7 assumes the estab-\nlished layer stacking, with orthogonal dimers on top of\neach other [21, 33], an antiferromagnetic interlayer cou-\npling [18, 21, 33], and an orthorhombic distortion of the\ntetragonal structure as proposed in Ref. 32. Together,\nthis yields a macroscopic uniform magnetization, which\nwe can estimate to reach up to 5 \u000110\u00003\u0016Bper Cu atom\nat its temperature maximum, i.e., slightly below the N\u0013 eel\ntemperature, see Fig. 1. It would be very interesting to\ntest this prediction in future high-pressure magnetization\nmeasurements. For such an experiment, one needs to\nkeep in mind that ferrimagnets generically form magne-\ntization domains much like ferromagnets [22]; detecting\na uniform magnetization might therefore require cooling\nin an applied \feld.\nWe note that the type of dimer distortion suggested to\nexist at 5:5 GPa at elevated Tin Ref. 21, see their Figs. 5\nand 6, would lead to a vanishing total magnetization in-\nstead, because a strong up-spin dimer in one layer would\ncouple to a strong (instead of weak) down-spin dimer in\nthe next layer. The resulting \fnite magnetization per\nlayer may still be detectable as a surface e\u000bect.\nFinally, we comment on the e\u000bect of small\nDzyaloshinskii-Moriya (DM) interactions which break\nSU(2) symmetry at the Hamiltonian level and are known\nto be present in SrCu 2(BO 3)2[21, 34]. In the AF phase\nof interest here, DM interactions may lead to a small,\nbut \fnite, uniform magnetization at T= 0 and also alter\nits leading low-temperature corrections as the spin-wave\nspectrum may acquire a gap at T= 0. Nonetheless, the\nconclusion that inequivalent sublattices experience dif-\nferent thermal \ructuations remains, and for small DM\ninteractions the non-monotonic temperature dependence\nof the magnetization will be preserved.IV. CONCLUSIONS AND OUTLOOK\nIn summary, we have shown that collinear N\u0013 eel anti-\nferromagnets, whose Z2symmetry between the two sub-\nlattices is broken in the Hamiltonian, become ferrimag-\nnets at \fnite temperature. Interestingly, this is an e\u000bect\ndriven by thermal \ructuations but not by quantum \ruc-\ntuations, constituting an interesting example where both\ntypes of \ructuations produce di\u000berent physics { in con-\ntrast to many instances of order by disorder where ther-\nmal and quantum \ructuations lead to very similar state\nselection [5]. We have proposed that such a \ructuation-\ninduced ferrimagnetic phase is realized in SrCu 2(BO 3)2\nunder high pressure, where a lattice distortion breaks the\nglide symmetry of the Shastry-Sutherland lattice.\nOur work suggests a number of future directions: First,\nit is conceivable that similar thermal \ructuation ef-\nfects occur in antiferromagnets with more complicated\nground-state spin structures. Second, while we have ar-\ngued that the antiferromagnetic phase with mtot= 0\nis a stable state of matter at T= 0 despite sublat-\ntice imbalance, it is interesting to ask whether quantum\n\ructuations can generate additional, more non-trivial,\nzero-temperature phases in sublattice-imbalanced anti-\nferromagnets. Finally, considering the same phenomenol-\nogy in the presence of charge carriers will lead to a\n\ructuation-induced anomalous Hall e\u000bect.\nAcknowledgments\nWe thank L. Janssen and E. Andrade for discus-\nsions and collaborations on related work. Financial sup-\nport from the Deutsche Forschungsgemeinschaft through\nSFB 1143 (project-id 247310070) and the W urzburg-\nDresden Cluster of Excellence on Complexity and Topol-\nogy in Quantum Matter { ct.qmat (EXC 2147, project-id\n390858490) is gratefully acknowledged.\nAppendix A: Spin-wave calculations for toy model\nThe starting point for our analysis of the toy model\nproposed in Sec. II was to expand the Hamiltonian in\nEq. (1) in powers of 1 =p\nSaround a collinear N\u0013 eel state.\nSince we allow the two magnetic sublattices to have, at\nleast in principle, spins of unequal sizes, the Holstein-\nPrimako\u000b transformation reads\nSz\ni=S\u0000ay\niai;\nS+\ni=q\n2S\u0000ay\niaiai=p\n2Sai+O\u0010\n1=p\nS\u0011\n;\nS\u0000\ni=ay\niq\n2S\u0000ay\niai=p\n2Say\ni+O\u0010\n1=p\nS\u0011\n;(A1)10\nfor sitesilocated on sublattice Aand\nSz\ni=\u0000\u0011S+by\nibi\nS+\ni=by\niq\n2\u0011S\u0000by\nibi=p\n2\u0011Sby\ni+O\u0010\n1=p\n\u0011S\u0011\n;\nS\u0000\ni=q\n2\u0011S\u0000by\nibibi=p\n2\u0011Sbi+O\u0010\n1=p\n\u0011S\u0011\n;(A2)\nfor sites belonging to sublattice B. As usual, ay\niandby\ni\nare bosonic creation operators with corresponding anni-\nhilation operators aiandbi.\nAfter substituting Eqs. (A1) and (A2) into the Hamil-\ntonian, applying a Fourier transform to the bosonic op-\nerators, and neglecting terms beyond O(S), one arrives\nat a quadratic Hamiltonian of the form\nHLSW=S2Ecl+S\n2X\n~kh\n\ty\n~kM~k\t~k\u0000\u0000\nA~k+B~k\u0001i\n;(A3)\nwhere\nS2Ecl=\u0000NS2\"\n2\u0011J+J0\na+\u00112J0\nb+\u0000\n1 +\u00112\u0001\n2J?#\n(A4)\nis the classical ground-state energy for a system with N\nsites, \t~k=\u0010\na~k;b~k;ay\n\u0000~k;by\n\u0000~k\u0011T\n, and\nM~k=0\nB@A~k0 0C~k\n0B~kC~k0\n0C~kA~k0\nC~k0 0B~k1\nCA (A5)\nwith\nA~k= 4\u0000\n\u0011J+J0\na\u0018~k\u0001\n+ 2J?(1\u0000coskz);\nB~k= 4\u0000\nJ+\u0011J0\nb\u0018~k\u0001\n+ 2\u0011J?(1\u0000coskz);\nC~k= 4\u00111=2J\r~k: (A6)\nThe linear spin-wave Hamiltonian in Eq. (A3) can be\ndiagonalized by means of a Bogoliubov transformation\na~k=u~k\u000b~k\u0000v~k\fy\n\u0000~k; b~k=u~k\f~k\u0000v~k\u000by\n\u0000~k(A7)\nwith coe\u000ecients\nu~k= sgn\u0000\nC~k\u0001r\nF~k+ 1\n2; v~k=r\nF~k\u00001\n2(A8)\ngiven in terms of the function\nF~k=A~k+B~kq\u0000\nA~k+B~k\u00012\u00004C2\n~k: (A9)\nWhen expressed in terms of the Bogoliubov operators,\nEq. (A3) takes on the diagonal form\nHLSW=S2Ecl+1\n2X\n~kh\n!~k\u0000+!~k+\u0000S\u0000\nA~k+B~k\u0001i\n+X\n~k\u0010\n!~k+\u000by\n~k\u000b~k+!~k\u0000\fy\n~k\f~k\u0011\n: (A10)The resulting mode dispersions !~k\u0006are speci\fed in\nEq. (4) of the main text, with the abbreviations be-\ning related to the terms de\fned in Eq. (A6) as P~k=\n(A~k+B~k)=2,R~k= (A~k\u0000B~k)=2, andQ~k=C~k.\nWe can then use the previous relations to derive\nthe sublattice magnetizations to next-to-leading order in\n1=S. For sublattice A, we have\nmA=2\nNX\ni2AhSz\nii=S\u00002\nNX\n~kD\nay\n~ka~kE\n=S\u00002\nNX\n~k\u0010\nu2\n~kD\n\u000by\n~k\u000b~kE\n+v2\n~kD\n1 +\fy\n~k\f~kE\u0011\n=S\u00001\nNX\n~kh\u0000\nF~k+ 1\u0001\nn~k+\nBE+\u0000\nF~k\u00001\u0001\u0010\n1 +n~k\u0000\nBE\u0011i\n;\n(A11)\nwhich is equal to the \frst expression in Eq. (6). One\nrecovers the second expression straightforwardly after a\nsimilar sequence of steps for sublattice B.\nThe low-temperature behavior of the uniform magneti-\nzation can in fact be predicted analytically by expanding\nthe spin-wave dispersion up to quadratic order in kand\nretaining the leading contribution in Tto Eq. (8). Let us\nfocus on the case \u0011= 1 for concreteness. After rewriting\nEq. (4) in the form\n!~k\u0006\u0019!1(\u0012)k\u0006!2(\u0012)k2; (A12)\nwe \fnd\nn~k\u0000\nBE\u0000n~k+\nBE\u00192e\u0000\f!1ksinh\u0000\n\f!2k2\u0001\n1\u00002e\u0000\f!1kcosh (\f!2k2) +e\u00002\f!1k\n\u00192e\u0000\f!1k\f!2k2: (A13)\nThe approximation made in the last step of (A13) is\njusti\fed by the fact that we are dealing with momenta\nk.1=(\f!1), and hence \f!2k2.!2=\u0000\n\f!2\n1\u0001\n\u001c1. We then\nsubstitute Eq. (A13) into Eq. (8) to obtain\nmtot/\fZ\u0019\n0d\u0012sin\u0012!2(\u0012)Z1\n0dkk4e\u0000\f!1(\u0012)k\n/jJ0\na\u0000J0\nbj\n(\fS)4J5Z\u0019\n0d\u0012sin\u0012\u0014!2(\u0012)\njJ0a\u0000J0\nbjS\u0015\u0014JS\n!1(\u0012)\u00155\n:\n(A14)\nSince the integral in the last line above is expressed solely\nin terms of dimensionless quantities, we conclude that in\nthe low-temperature limit\nmtot/jJ0\na\u0000J0\nbj\nJ\u0012T\nJS\u00134\n; (A15)\nwhich is precisely what our numerical results indicate.\nAppendix B: Spin-wave calculations for\nShastry-Sutherland model\nThe spin-wave calculations for the Shastry-Sutherland\nmodel are based on a Holstein-Primako\u000b representation11\nof the spin operators as above. Since the unit cell consists\nof four spins which exhibit collinear N\u0013 eel order in the\nclassical limit, we assign the transformations given by\nEqs. (A1) and (A2) to two spins each, with spin size S\non all sublattices. Here, the spins on sublattice A(B)\ncorrespond to Holstein-Primako\u000b operators aiandci(bi\nanddi), respectively, see Fig. 6(a).\nAs noted in the main text, for calculation purposes\nwe choose a coordinate system where the Shastry-\nSutherland plane is located on a square lattice along the\nJ0bonds. Moreover, the planes are stacked such that\ninequivalent dimers are positioned on top of each other.\nWith lattice constants set to unity, this yields real-space\nbasis vectors a1= (2;0;0),a2= (0;2;0),a3= (1;1;1)\nand reciprocal-space basis vectors b1=\u0019(1;0;\u00001),b2=\n\u0019(0;1;\u00001),b3= 2\u0019(0;0;1).\nSubstitution of the Holstein-Primako\u000b operators into\nthe Hamiltonian given by Eq. (9) and a subsequent\nFourier transform yields the quadratic Hamiltonian\nHLSW=S2Ecl+S\n2X\n~kh\n\ty\n~kM~k\t~k\u00002B~ki\n(B1)\nwhere\nS2Ecl=NS2\u0014Ja+Jb\n4\u00002J0\u00002J?\u0015\n(B2)\nis again the classical ground state energy for a system\nconsisting of Nspins, \t~k=\u0010\na~k;c~k;by\n\u0000~k;dy\n\u0000~k\u0011T\nand\nM~k=0\nBB@A~kC~kE~kF~k\nC\u0003\n~kA~kF\u0003\n~kE\u0003\n~k\nE\u0003\n~kF~kB~kD~k\nF\u0003\n~kE~kD\u0003\n~kB~k1\nCCA(B3)\nwhere\u0003denotes the complex conjugate and\nA~k= 4J0\u0000Ja+ 4J?;\nB~k= 4J0\u0000Jb+ 4J?;\nC~k=Jaei(\u0000kx+ky);\nD~k=Jbei(kx+ky);\nE~k= 2J0cosky+J?(ei(\u0000kx+kz)+ei(\u0000kx\u0000kz));\nF~k= 2J0coskx+J?(ei(ky+kz)+ei(ky\u0000kz)): (B4)\nThe Hamiltonian in Eq. (B1) can now be diagonalized\nby means of a generalized Bogoliubov transformation [35]\n\t~k=T(~k)\b~k: (B5)with \b~k=\u0010\n\u000b~k;\r~k;\fy\n~k;\u000ey\n~k\u0011T\nbeing the normal mode\nspinor. The columns of T(~k) correspond to the eigen-\nvectors of \u0006 M~kwith\n\u0006 =\u0012\nI0\n0\u0000I\u0013\n(B6)\nwhere Idenotes the 2\u00022 unit matrix. Solving the eigen-\nvalue problem for \u0006 M~kyields two positive eigenvalues\n\u0015~k1;2and two negative eigenvalues \u0015~k3;4. The normal\nmodes of the Hamiltonian are then given by\n!~k1;2=S\u0015~k1;2;\n!~k3;4=\u0000S\u0015~k3;4: (B7)\nImportantly, the positive and negative eigenvalues are\nnotof pairwise equal magnitude. With Eqs. (B5) and\n(B7) the linear spin-wave Hamiltonian then takes the di-\nagonal form\nHLSW=S2Ecl+X\n~k\u0014!~k3+!~k4\n2\u0000SB~k\u0015\n+X\n~kh\n!~k1\u000by\n~k\u000b~k+!~k2\ry\n~k\r~k+!~k3\fy\n~k\f~k+!~k4\u000ey\n~k\u000e~ki\n:\n(B8)\nThe sublattice magnetization can then be derived from\nthe Holstein-Primako\u000b operators using the respective Bo-\ngoliubov coe\u000ecients and Eq. (B5). 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Nievaa,c\naLaboratorio de Bajas Temperaturas, Centro At´ omico Barilo che - CNEA, 8400\nBariloche, R´ ıo Negro, Argentina.\nbINFIQC-CONICET, Depto. de F´ ısico Qu´ ımica, Facultad de Ci encias Qu´ ımicas,\nUniversidad Nacional de C´ ordoba, Ciudad Universitaria, X 5000HUA C´ ordoba,\nArgentina.\ncInstituto Balseiro, CNEA and Universidad Nacional de Cuyo, 8400 Bariloche, R´ ıo\nNegro, Argentina.\nAbstract\nWe report on the magnetic and structural properties of La 2Ni(Ni1/3Sb2/3)O6\nin polycrystal, single crystal and thin film samples. We found that this\nmaterial is a ferrimagnet ( Tc≈100 K) which possesses a very distinctive and\nuncommon feature inits virgin curve of the hysteresis loops. We obs erve that\nbellow 20 K it lies outside the hysteresis cycle, and this feature was fo und to\nbe an indication of a microscopically irreversible process possibly involv ing\nthe interplay of competing antiferromagnetic interactions that hin der the\ninitial movement of domain walls. This initial magnetic state is overcome\nby applying a temperature dependent characteristic field. Above t his field,\nan isothermal magnetic demagnetization of the samples yield a groun d state\ndifferent from the initial thermally demagnetized one.\nKeywords: Ferrimagnetics, Doubleperovskites, Magnetic frustration,\nSuperexchange and super-superexchange interactions, Magne tic oxides\n1. Introduction\nThe physics and chemistry of complex oxides have been largely studie d\ndue to the rich phenomena resulting from their combined magnetic, c harge\nand orbital degrees of freedom. Among these complex oxides, tho se with\nEmail address: gnieva@cab.cnea.gov.ar (G. Nieva)\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials September 22, 2018perovskite structure ABO 3have been extensively studied. They are the\narchetype of superconducting oxides, giant magnetoresistive ox ides and the\nmaterials for the emergent field of novel device functions [1].\nThe size and oxidation state of A and B cations determine the symmet ry\nof the perovskite structure, that departs from cubic when a tiltin g of the\noctahedral arrangement of oxygen around the B cation occurs [2 ]. Also, the\npartial replacement of A or B site cations by isovalent or aliovalent A’ and B’\natoms could result in a double perovskite structure with lower symme try. In\ndouble perovskites with general formula A 2(BB’)O 6, B and B’ atoms occupy\ntwo different crystallographic sites, and an ordered or disordered occupancy\nof them depends on the oxidation state and size difference between B and B’\nions [3, 4].\nMany of the magnetic interactions found in transition metal oxide pe r-\novskites are due to superexchange and/or super-superexchan ge interactions\nmediatedthroughtheoxygenorbitals. Insomematerialstherelativ estrength\nand arrangement of these interactions determine the magnetic st ructure,\nrange of the ordering temperatures and the possibility of frustra tion [5, 6,\n7, 8, 9]. In particular, the effective spin lattice, ferro- or antiferr omagnetic\ninteractions among transition metal ions, depends on a delicate bala nce pro-\nvided by the charge in the linking orbital, the bond angle, the degree o f\norbital overlap, the distance between interacting ions and their sp in state\n[10, 11]. In the specific case of the perovskite structure, the typ ical bond\nangles and distances usually favors antiferromagnetic superexch ange interac-\ntions [11]. However, in some particular cases, when there are more t han one\nelectronic pathway in the linking orbitals, a destructive interferenc e leads to\nthe cancellation of the antiferromagnetic interaction. In these ca ses, a weak\nferromagnetic coupling becomes relevant [12].\nIn transition metal oxides, the competing effects of a ferromagne tic in-\nteraction and local frustration leads to a spin glass like behavior [13]. The\nmain signatures of this frustration are made evident in the time evolu tion\nfrom metastable magnetic states [14]. However, in bulk ferro/ferr imagnetic\nsamples, a local frustration is normally hard to visualize due to the ma gnetic\nhistory dependence of the metastable states created by the pinin g of domain\nwalls.\nAlso in some transition metal oxides the competition between a ferro -\nand antiferromagnetic ground states leads to a nonequilibrium glass y behav-\nior that arises from a kinetic arrest of a ferromagnetic to antiferr omagnetic\nphase transition [15]. These materials are identified as magnetic glass es [16]\n2consisting of ferro- (or ferri-) magnetic and antiferromagnetic c lusters frozen\nrandomly with a dynamics similar to that of structural glasses.\nThis article will present a detailed magnetic study of the ferrimagnet ic\ndouble perovskite La 2Ni(Ni1/3Sb2/3)O6[17, 18] showing evidence of a frus-\ntrated magnetic state at low temperatures. The properties of th is compound\nwere explored using polycrystalline, single crystalline and thin film samp les.\nThe main result is the observation that below 20 K there is evidence of a\nmicroscopically irreversible process involving the interplay of compet ing an-\ntiferromagnetic interactions that hinders the initial magnetic polar ization or\nthe movement of domain walls and determines the microscopic nature of the\nstrong pinning centers found in this system [19]. This material result s in a\nmodel system for studying the seldom found ferrimagnetic Ni2+oxides and\nthe low temperature glassy or magnetically disordered state prese nt in many\ncomplex transition metal oxides.\n2. Experimental details\nWe prepared polycrystalline samples of La 2Ni(Ni1/3Sb2/3)O6by conven-\ntionalsolid-statereaction. StoichiometricamountsofLa 2O3, Ni(NO 3)2.6H2O\nand Sb 2O3were ground and fired at 1400oC for 12 hours in air. The single\ncrystals were grownby thefloatingzone technique inadouble ellipsoid al mir-\nror furnace. The thin films were grown using RF magnetron sputter ing in\nan Ar/O 2atmosphere on (100) SrTiO 3substrates at 700-800oC. We checked\nthe composition of all the investigated samples by EDS. Only the thin fi lms\nshowed a 10% Ni2+deficiency not present in the stoichiometric target mate-\nrial. We carried out the structural analysis using powder X-ray diffr action\n(XRD) at room temperature. The magnetic measurements were pe rformed\nin an QD-MPMS SQUID magnetometer in the range 2 to 300 K and -5 to 5\nTesla.\n3. Results\n3.1. Structural characterization\nIn spite of an earlier report of this structure as being orthorromb ic [17],\nwith a fully disordered arrangement of Ni2+and Sb5+ions, we found the\ncrystalline symmetry to be monoclinic (space group P2 1/n) with a rock salt\narrangement of BO 6and B’O 6octahedra described by the a−b−c+system\nof three octahedral tilts in the Glazer’s notation. The (Ni2+/Sb5+)2dO6and\n3Figure 1: (color online) Left: monoclinic structure of La 2Ni(Ni1/3Sb2/3)O6double per-\novskite. Grey spheres: La3+, yellow spheres: O2−, light blue spheres: B 2dions and green\nspheres: B 2cions. Righ: two dimensional scheme of the distribution of B ions over t he\n2d and 2c sites for La 2(Ni)2d(Ni1/3Sb2/3)2cO6showing the different neighbors of the mag-\nnetic Ni2+ions (turquoise spheres): first next nearest neighbors (1 nnn, b lack), second\nnext nearest neighbors (2 nnn, dash blue line) and third next neare st neighbors (3 nnn,\nred dot line). Red spheres are Sb5+ions and yellow spheres O2−ions. For simplicity a\nsquare structure was supposed and lanthanum ions were omitted.\n(Ni2+/Sb5+)2cO6octahedra are rotated in phase along the primitive caxis\nand out-of phase along the primitive aandbaxes. We performed a Rietveld\nrefinement of the structure using the FULLPROF program [20], ob taining\nthe lattice parameters a= 5.6051(3) ˚A,b= 5.6362(3) ˚A,c= 7.9350(5) ˚A\nandβ= 89.986(4)o. The occupancy of the two crystallographic sites 2 d\nand 2cwere refined allowing the Ni2+/Sb5+distribution to vary, in order\nto model the octahedral site disorder. We found that the 2 dcation site is\nalmost fully occupied by Ni2+while the 2 csite has an occupancy close to 1/3\nof Ni2+ions and 2/3 of Sb5+. The resulting crystallographic formula can be\nwritten as La 2(Ni0.976Sb0.024)2d(Ni0.357Sb0.643)2cO6. It should be noted here\nthat this space group does not allow a further ordering of the Ni2+and Sb5+\nions at 2csite. Figure 1 shows the structure of La 2Ni(Ni1/3Sb2/3)O6and also\na schematic two dimensional square view of Ni2+/Sb5+distribution among\n2d and 2c sites. From this picture it can be seen that Ni2+ions have three\n4types of Ni2+neighbors: first next nearest neighbors (mediated by -O-, that\nis, superexchange), second next nearest neighbors (through a -O-O- bridge)\nand third next nearest neighbors (-O-Sb5+-O-, super-superexchange).\nSmall crystals with a face parallel to the (103) planes could be extra cted\nfrom the rod grown in the mirror furnace. The rocking curve on the (103)\npeak had a FWHM = 0.25oand the interplanar spacing showed a 0.05%\nreduction with respect to the bulk.\nThe thin films (100 -130 nm thick) were grown epitaxially in the c-axis\ndirection. FromXRDwemeasureda1.4% c-axisexpansion ascomparedwith\nbulk samples, considering a constant cell volume. This corresponds to a 1.5%\n(Ni2+/Sb5+)2d-(Ni2+/Sb5+)2cdistancereductioninthe ab-plane, tomatchthe\nTi-Ti distance of the substrate. Rocking curves showed typically a FWHM\n=0.4oindicating a good crystallographic quality of the La 2Ni(Ni1/3Sb2/3)O6\nfilms.\n3.2. Magnetic characterization\nWe measured the magnetization as a function of temperature ( MvsT)\nusing a ZFC-FC procedure (i.e. cooling with zero applied field or with a\nfinite applied field) for several powder samples, thin films and single cr ystals.\nThe lower curve in Figure 2(a) (open diamonds) shows a typical resu lt for\npolycrystalline samples with a Curie-Weiss behavior at high temperatu re,\nM\nH=C\nT−TCW, withCthe Curie constant and TCWthe Curie-Weiss tem-\nperature. The lower inset in Figure 2(a) shows the magnetization de rivative\ndM/dTused to determine the transition temperature to the ordered sta te,\nthe Curie temperature, TC. A Curie-Weiss fit of several polycrystalline sam-\nplesgives TCW=155(6)Kand TC=98(2)K.Theeffectivemagneticmoment,\ncalculated from the Curie constant is p= 2.3(1) µB. This value is lower but\nclose to the one expected for spin only Ni2+,p= 2.86µB.\nA typical result for the single crystals is also shown in Figure 2(a), fille d\ncircles. In this case TCW= 138(4) K and p = 2.60(4) µB. We measured\nthe single crystals with the magnetic field applied along two crystallogr aphic\ndirections, H/bardbl(103) and H/bardbl(010). In the upper inset of Figure 2(a) the\nsusceptibility extracted from magnetization measurements at 1 an d 5 kOe\n(∆M/∆H= (M(5 kOe)−M(1 kOe))/4 kOe) is shown. The inflection point\nof these curves, identified with TC, is located at 105 K which agrees very well\nwith the polycrystalline samples.\nThe thin films paramagnetic behavior ( T > T c) is very difficult to isolate\nfrom the total signal, given their small mass, typically about 40 µg. How-\n5ever, subtracting the substrate magnetization contribution, a c lear transition\nto a ferromagnetic state could be observed. Figure 2(b) shows th e FC mag-\nnetization at 1 kOe, applied parallel to the surface of the film ( H⊥(001)).\nThe low temperature signal is approximately three times smaller than the\none observed on the polycrystals and single crystals. The inset in Fig ure 2(b)\nshowsdM/dT, the peak indicates a TC= 78 K, which is about 20 % smaller\nthan for bulk samples.\nHysteresis loops ( MvsH) were measured at several temperatures for\nT < T c. Figure 3 shows typical loops at T= 2 K for a thin film, a single\ncrystal and a polycrystalline pellet. For the crystals, loops with the applied\nfield along the (103) and (010) directions are included (Figure 3(c)) . In this\ncase the initial magnetization branch, the MvsHvirgin curve measured\nafter cooling at zero field, falls always outside the loop area for temp eratures\nbellow about 20 K. This effect can be interpreted as a sign of frustra tion, is\nless visible in pellets and was not observed in films.\nFor temperatures below 20 K, we also measured the hysteresis loop s after\ncooling the sample in 1 T, observing that they coincide with those meas ured\nafter ZFC. No shift was found in the coercive fields ruling out the pre sence\nof exchange bias [21].\n4. Discussion\n4.1. The ordered state\nThe value of the Curie-Weiss constant, the shape of the MvsHcurves\nand the hysteretic behavior, all point to a ferromagnetic state of the Ni2+\nbelow 100 K. However, the saturation magnetization value, Ms, taken as\nthe asymptotic extrapolation with a Langevin function of the behav ior at\nthe largest applied field (5 T), has a lower value than the one expecte d for\nthe ferromagnetic complete polarization of the Ni2+magnetic moments, 2.67\nµB/f.u.. The experimental Msvalues range from 0.73 µB/f.u. to 1.19 µB/f.u.\nconsidering films, single crystals and polycrystalline samples. These s maller\nexperimental values are better understood if the system behave s as a ferri-\nmagnet having two Ni2+magnetic sublattices antiferromagnetically coupled,\none at the 2 dand another at the 2 csites. The near 1/3 Ni2+random occupa-\ntion of the 2 csites sublattice give as a result uncompensated Ni2+magnetic\nmoments that order at 100 K. For a perfectly stoichiometric sample and full\nNi2+occupancy of the 2 dsiteMsshould be 1.33 µB/f.u., and lower values\nare expected if Sb5+partially occupies also the 2 dsite. In particular, for\n6the refined occupancy of the octahedral sites in the powder, we c an calculate\nMs= 1.24µB/f.u., close to the measured values. A similar ferrimagnetic\norder was reported in Sr 2Fe(Fe1/3Mo2/3)O6[23] and Sr 2Fe(Fe1/3U2/3)O6[22]\nwhere the magnetic Fe3+ions and non magnetic Mo6+or U6+display a sim-\nilar structural arrangement and magnetic array as the Ni2+and Sb5+in our\ncase.\nIn order to get a microscopic understanding of the uncommon feat ures\nseen in the virgin curves we shall analyze with some detail the charac teris-\ntics of the ordered state through a study of the irreversible magn etization\nmeasurements from hysteresis loops. Figure 4 (a) shows that the coercive\nfield for the single crystals (measured along the (103) direction) ha s a similar\ntemperature behavior as that of the polycrystalline pellet at low tem pera-\ntures [19]. We have previously established that the polycrystalline ma terial\ncoercive field has an upturn at T≈20 K changing from a weak domain wall\npinning (WDWP) behavior at high temperatures to a strong domain wa ll\npinning (SDWP) one below 20 K [19]. Analyzing the coercive field and the\ntime dependence of the magnetization we have ruled out the freezin g of large\nparticles as a posible mechanism for the Hcupturn [19]. We found that in\nthe single crystals the coercivity above 30 K is negligibly small. The film co -\nercivity, measured at 2 K, is close to the bulk value, in spite of the exp ected\nHcenhancement due to barriers for the DW motion introduced by surf ace\nroughness or local strains [25]. Figure 4(b) shows the Tdependences of the\nthe ratio between remanent and saturation magnetizations ( Mr/Ms). This\nratio and Hcincrease steeply when the temperature is lowered below 20 K\nindicating increase in the energy needed to change the direction of M.\nIn the low temperature regime, T/lessorsimilar20 K, the polycrystalline pellets and\nsingle crystals ( H/bardbl103)Hcdata can be described by a model of strong\ndomain wall pinning (SDWP),\nHc=H0S/bracketleftBigg\n1−/parenleftbigg75kBT\n4bf/parenrightbigg2/3/bracketrightBigg2\n(1)\nwhereH0Sis the coercive field at zero temperature, fis the magnetic force\nneeded to depin a domain wall and bis a measure of the domain wall thick-\nness. The fitted values are shown in Table 1.\nAn estimation of bfrom the exchange stiffness, A, and anisotropy con-\nstant,K1, yield a small value of the domain wall thickness b=π(2A\nK1)1/2⋍\n10nm. The anisotropy constant was calculated at 2 K from the area betw een\n7Table 1: Fitted parameters for the SDWP model below 20 K, for polyc rystalline, PC, and\nsingle crystalline, SC, samples.\n4bf H 0S\n(10−13erg) (Oe)\nPC 3.07 780\nSC 2.54 1440\nthe anhysteretic curves MvsHand theMaxis [24] for the single crystals\nmeasured in the (103) and (010) direction. The value obtained was K1= 3\n105erg/cm3 1. Although the calculated anisotropy constant is of the order of\nthe value found in other ferrimagnets [24], the exchange stiffness c onstant is\nsmall due to the rather long average distance between uncompens ated Ni2+\nmoments in the structure.\nThe microscopic origin of the change of regime of domain wall pinning\nmechanism at 20 K remains unclear. The onset of a disordered or fru strated\nmagnetic state may result in the emergence of strong pining sites fo r DW\nmovement. The answer could be made evident analyzing the virgin cur ve in\nthe hysteresis loops. The initial branch of the MvsHcurve (cooling from\naboveTcat zero applied field) does not fall entirely within the magnetization\nloopforall thesamples, except forthe thinfilm, see Figure3. Inwha t follows\nthis uncommon behavior will be addressed.\n4.2. The virgin magnetization curve\nSimilar loopsasthoseshowninFigure3were obtainedforseveral tem per-\natures, below and above the temperature of the steep increase o f the coercive\nfield when lowering T. We have found that the virgin curve excursion outside\nthe regular loop takes place at low temperatures ( T/lessorsimilar20 K) coinciding with\nthe regime of SDWP described previously.\nIn some systems the virgin curve was observed to go outside the loo p for\na certain range of fields and temperatures [26, 27, 28, 29, 30, 31, 32]. In\nsome complex magnetic oxides this feature was related with magnetic cation\ndisorder [26, 27], which led not only to a spin-glass behavior but also to\nlocal structural distortions. These local distortions are due to a microscopic\nrearrangement of valence electrons [28], or to magnetic cations de ficiency\n1This value was obtained considering the 010 direction as the easy mag netization axis\n8changing the nature of the local crystal field [29] which in turn lead t o an\nirreversiblemovement ofdomainwalls[28,29]. Theanomalousvirgincu rvein\na ferrimagnet was also attributed to the development of an antifer romagnetic\norder at low temperature that produces a magnetic glass state [32 ].\nFigure 5 shows the Hderivative of the virgin curves dM/dH at several\ntemperatures for the thin film, single crystals and polycrystalline pe llets. In\nthe thin film case (Figure 5(a)), the derivative is approximately cons tant.\nFor all the other samples there is a maximum in the derivative that indic ates\na characteristic field, Hmax, for the magnetic moments alignment, larger than\nthe coercive field at low T(Figure 6).\nIn a spin glass scenario Hmax(T) could represent the line separating the\nglassy phase from the ordered state. Therefore a Tdependence following the\nAlmeida-Thouless [33]or Gabay-Tolouse[34] models couldbeexpect ed. This\nis not our case, indicating a more complex behavior involving the hinder ing\nof DW movement. We have also ruled out the magnetic glass [16] scena rio\nbecause we found no evidence for a long range antiferromagnetic o rder at\nlow temperatures and we did not detect differences between FC coo ling and\nFC warming magnetization measurements at 1 kOe as in a typical magn etic\nglass displaying the kinetic arrest of the phase transition [15]. The diff er-\nence between HmaxandHc(Figure 6) is larger for single crystals than for\npolycrystalline pellets suggesting an intrinsic origin for this behavior.\n4.3. The frustrated state and its removal\nTo understand themicroscopic originofthefrustrationthelocalm agnetic\ninteractions of Ni2+have to be analyzed. They depend on the nearest mag-\nnetic neighbor (1 nnn) and on the exchange path for the second an d third\nnearest neighbor (2 nnn, 3 nnn). Considering the structure (Figu re 1(left))\nwith perfect stoichiometry and full Ni2+occupancy of the 2 dsite, the inter-\naction among 1 nnn relays on the presence of a Ni2+in the closer 2 csite (1/3\nNi2+ion occupancy). If this is the case there will be an antiferromagnet ic su-\nperexchange interaction of the Ni2+-O-Ni2+kind [10]. The remaining 2/3 of\nthe 2csublattice is occupied by non magnetic (d10) Sb5+ions. Then, a Ni2+\nferrimagnetic lattice is formed at the 2 dand 2csites. The 2 dsite Ni2+sec-\nondandthirdnearest neighbor interactions aremediated by -O-O- (at∼90o)\nand-O-Sb+5-O-(at∼180o) super-superexchange paths (see Figure 1(right)),\nwhose relative strength would determine the type of magnetic orde ring in the\nregions with Sb+5in the 2csublattice. These super-superexchange interac-\ntions were found to be dominant for the antiferromagnetic struct ure of some\n9ordered double perovskites A 2BB’O6where B is a magnetic transition metal\nion and B’ is a non magnetic cation [35]. In particular, LaSrNiSbO 6was\nfound to have an antiferromagnetic structure of type I (ferrom agnetic planes\nparallel to the abandacplanes antiferromagnetically coupled along the cand\nbdirection respectively) with a transition temperature of 26 K [36]. Th ere-\nfore, for La 2Ni(Ni1/3Sb2/3)O6, at temperatures around 20 K, the Ni2+\n2d-Ni2+\n2d\ninteractions mediated through -O-O- paths and the ones mediated through\n-O-Sb5+-O-, both antiferromagnetic in nature, became important. These in-\nteractions together with Ni2+-O-Ni2+superexchange one that exists below\n100 K, create a magnetically frustrated state at lower temperatu res than 20\nK. A similar antiferromagnetic transition temperature was found in r elated\nCo based perovskites [37] and in La 2Ni(Ni1/3Nb2/3)O62. Consequently, a\nmagnetically frustrated ground state is expected when cooling the sample in\nzero field due to competing antiferromagnetic interactions among t he Ni2+\nions and Hmaxis necessary to overcome this frustration. The microscopic\norigin of the barrier which is overcome by increasing Hat lowTis still\nnot clear. It is probably related to the spin orientation of the Ni2+moments\ninteracting antiferromagneticaly with its 1 nnn trying to keep their a ntiferro-\nmagnetic arrangement in planes perpendicular to Hand the uncompensated\nNi2+moments following the field as in a normal ferromagnet. The random\norientation of the antiferromagnetic regions and the frustrated interaction\nnear Sb5+ions hinders the initial movement of the DW. When they are ori-\nented, further changes of the magnitude of Hresult only in canting of the\nmoments. However, the canting is not very important since a magne tization\nsaturation value seems to be almost achieved at 5 T with Mscorrespond-\ning to the uncompensated Ni2+moments in the ferrimagnetic structure, as\ndescribed in previous sections. This model can be tested going back to the\nzero magnetization state with a demagnetizing protocol, illustrated in the in-\nsets of Figure 7. After this demagnetizing procedure, the zero ma gnetization\nstate would involve only a random orientation of the uncompensated Ni2+\nmoments and the antiferromagnetically arranged regions would hav e the mo-\nments perpendicular to the direction of the preexisting field, provid ing a new\nground state. Indeed, the results indicate that after this demag netizing pro-\ncess, the sample initial magnetization branch lies inside the hysteres is loop.\n2La2Ni(Ni1/3Nb2/3)O6shows an antiferromagnetic ordering at 28 K and is currently\nunder study.\n10This is illustrated in Figure 7 for a single crystal at T= 2 K, where the\nvirgin magnetization, the regular loop branch and a new initial magnet iza-\ntion branch are shown. This new initial magnetization is measured aft er the\ndemagnetizing protocol shown in the inset.\nThe absence of exchange bias [21] in the magnetization loops indicate\nthat large clusters with only antiferromagnetic super exchange Ni2+-O-Ni2+\ninteractions are not likely to exist (i.e: the one third Ni2+occupancy of the\n2csite seems to be homogeneous at a microscopic level).\nFigure8showsthecomparisonbetweenthederivativesofthevirgin curves\nand the initial magnetization after a demagnetizing protocol at 2 K f or a\nsingle crystal and a polycrystalline pellet. We performed similar exper iments\nat 5 and 10 K. In all the cases the new initial magnetization lies inside th e\nloop as in a usual, non- frustrated, ferromagnet.\nWe can observe in Figure 8 that the initial susceptibility of the virgin\ncurve after isothermal demagnetization is larger than the corres ponding to\nthe thermally demagnetized sample indicating that the new ground st ate is\neasier to magnetize in the direction of the preexisting magnetic field.\nForthethinfilms, noindication ofthefrustratedstate wasfound( Figures\n3(a) and 5(a)). The superexchange and super-superexchange interactions\nare very sensitive to bond angles and lengths [11], which are likely modifi ed\nnear the surface. In our samples, the magnetic anisotropy impose d by the\ngeometry seems to overcome the disordered interactions that se ts in at low\ntemperatures.\n5. Conclusions\nIt is shown that the anomalous behavior of the virgin curve in hyster esis\nloops is a distinctive feature of a low temperature magnetic frustra ted state\nfoundinthe ferrimagnetic double perovskite oxide La 2Ni(Ni1/3Sb2/3)O6. The\nvirgin curve lies outside the loops at T/lessorsimilar20 K, at about one fifth of the ferri-\nmagnetic ordering temperature ( Tc≈100 K). This was found to be anindica-\ntion of a microscopically irreversible process possibly involving the inte rplay\nof antiferromagnetic interactions that hinder the initial movement of domain\nwalls. This feature was observed in single crystals and in polycrystallin e pel-\nletized samples but not in thin films. This initial frustrated magnetic st ate\nis overcome by applying a characteristic field. Above this field the mat erial\nbehaves macroscopically as a typical ferromagnet. The model pro posed for\nthe frustrated state is based on the competing antiferromagnet ic interaction\n11between 1 nnn and 3 nnn (ie: Ni2+-O-Ni2+and Ni2+-O-Sb5+-O-Ni2+). This\nmicroscopic scenario for the frustrated state is tested by going b ack to the\nzero magnetization state at fixed low temperature applying a demag netizing\nprotocol.\n6. Acknowledgments\nWe thank P. Pedrazzini for help with the crystal growth and E. De B iasi\nforfruitfulsuggestions. R.E.C., E.E.K., andG.N.aremembersofCONI CET,\nArgentina. D.G.F. has a scholarship from CONICET, Argentina. Work par-\ntiallysupportedbyANPCyTPICT07-00819,CONICETPIP11220090 100448\nandSeCTyP-UNCuyo 06/C381. R.E.C.thanksFONCYT(PICT200700 303),\nCONICET(PIP11220090100995)andSECYT-UNC(Res. 214/10)f orfinan-\ntial support. E.E.K. thanks ANPCyT PICT2008-1731 for finantial s upport.\nReferences\n[1] H. Takagi and H. Y. Hwang, Science 327 (2010) 1601-1602.\n[2] I. Levin, L. A. Bendersky, J. P. Cline, R. S. Roth and T. A. Vande rah,\nJ. 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Bull. 46 (2011) 62-69.\n14/s48/s51/s120/s49/s48/s53/s54/s120/s49/s48/s53/s57/s120/s49/s48/s53\n/s45/s51/s120/s49/s48/s53/s48/s51/s120/s49/s48/s53/s54/s120/s49/s48/s53\n/s57/s48 /s49/s48/s48 /s49/s49/s48/s49/s48/s49\n/s49/s48/s50\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\n/s53/s48 /s49/s48/s48/s40/s97/s41\n/s48/s48/s72/s32/s47/s32/s77/s32/s40/s79/s101/s32/s102/s46/s117/s46/s47\n/s66/s41/s54/s46/s49/s48/s45/s57\n/s51/s46/s49/s48/s45/s57\n/s50/s46/s49/s48/s45/s51/s52/s46/s49/s48/s45/s51\n/s40/s98/s41/s32/s32/s54/s46/s49/s48/s53\n/s51/s46/s49/s48/s53\n/s45/s49/s46/s53/s46/s49/s48/s53\n/s32\n/s32/s32/s84/s32/s40/s75/s41/s45 /s32/s100/s77/s47/s100/s84/s32/s40\n/s66/s47/s102/s46/s117/s46/s75/s41/s32/s72/s32/s124/s124/s32/s48/s49/s48/s72/s32/s124/s124/s32/s49/s48/s51/s77/s47 /s72/s32/s40\n/s66/s47/s102/s46/s117/s46/s79/s101/s41\n/s84/s32/s40/s75/s41/s32/s77/s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s84/s32/s40/s75/s41/s52/s46/s49/s48/s45/s51\n/s50/s46/s49/s48/s45/s51\n/s48/s45/s100/s77/s47/s100/s84/s32/s40\n/s66/s47/s102/s46/s117/s46/s75/s41\n/s84/s32/s40/s75/s41/s48\nFigure 2: (color online) (a) H/MvsTfor a single crystalline (solid symbols) and a\npolycrystalline (lower curve, open symbols) La 2Ni(Ni1/3Sb2/3)O6samples, measured at\n1000 Oe. The upper inset shows the susceptibility (defined in the tex t), ∆M/∆HvsT\nforHalong (103) and (010) directions in the single crystal. The symbols ind icate the ZFC\nand the line the FC measurement. The lower inset shows the magnetiz ation derivative\ndM/dTvsTfor a polycrystalline sample taken from a FC measurement at 1 Oe. (b )M\nvsTfor two thin films of La 2Ni(Ni1/3Sb2/3)O6, measured at 1000 Oe. The inset shows\ndM/dTvsTfor the same films FC measurement at 1000 Oe.\n15/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s56 /s45/s52 /s48 /s52 /s56/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s49 /s48 /s49 /s50/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s32\n/s32/s40/s48/s49/s48/s41/s116/s104/s105/s110/s32/s102/s105/s108/s109\n/s40/s99/s41/s32\n/s32\n/s72/s32/s40/s107/s79/s101/s41/s77/s32/s32/s32 /s40\n/s66/s47/s102/s46/s117/s46 /s41/s77/s32/s32/s32 /s40\n/s66/s47/s102/s46/s117/s46 /s41\n/s40/s102/s41/s40/s101/s41/s40/s100/s41/s40/s49/s48/s51/s41/s32\n/s112/s101/s108/s108/s101/s116/s99/s114/s121/s115/s116/s97/s108/s115/s105/s110/s103/s108/s101/s40/s98/s41\n/s32/s32\n/s32/s40/s97/s41\nFigure 3: (color online) M vs H at 2 K for a thin film (a) and (b), a single cr ystal (c)\nand (d) and a polycrystalline pellet (e) and (f) of La 2Ni(Ni1/3Sb2/3)O6. For the single\ncrystal(c), loops with the applied field along directions (103) and (01 0) are included.\n16/s48/s51/s48/s48/s54/s48/s48/s57/s48/s48/s49/s50/s48/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48 /s52 /s56/s48/s49/s48/s50/s48/s51/s48\n/s40/s98/s41/s32/s72\n/s67/s32/s40/s79/s101/s41/s40/s97/s41\n/s32/s32/s77\n/s114/s47/s77\n/s115\n/s84/s32/s40/s75/s41/s32/s72\n/s67/s49/s47/s50\n/s32/s40/s79/s101/s49/s47/s50\n/s41\n/s84/s50/s47/s51\n/s32/s40/s75/s50/s47/s51\n/s41\nFigure 4: (coloronline) (a) Coercivefield HcvsTfor La2Ni(Ni1/3Sb2/3)O6polycrystalline\npellets (yellow, open symbols), single crystals (blue, dark circles) an d thin films (open\ncircles). The lines corresponds to models described in the text (SDW P and WDWP\nmodels, full lines and dashed line respectively). Inset: H1/2\ncvs T2/3showing the linear\nbehaviorexpectedin theSDWP model. (b) Normalizedremanentmagn etization(at H=0)\nvsT, the symbols represent the same samples indicated in (a). The lines a re a guide to\nthe eye.\n17/s49/s48/s50\n/s49/s48/s51/s32/s32\n/s40/s97/s41/s50/s75\n/s32/s32\n/s49/s46/s49/s48/s45/s52\n/s40/s98/s41/s51/s48/s75\n/s49/s48/s75\n/s53/s75\n/s50/s75\n/s32/s32\n/s49/s48/s45/s52/s49/s48/s45/s51/s51/s46/s49/s48/s45/s52\n/s49/s48/s45/s51\n/s49/s48/s45/s52/s53/s75\n/s32/s72/s32/s40/s79/s101/s41/s40/s99/s41/s56/s48/s75\n/s53/s48/s75\n/s50/s48/s75\n/s49/s48/s75\n/s50/s75\n/s32/s32/s100/s77/s47/s100/s72/s32/s40\n/s66/s47/s102/s46/s117/s46/s79/s101/s41\nFigure 5: (color online) Derivative of the virgin curves in Magnetizatio n loops for (a) thin\nfilm, (b) single crystals and (c) polycrystalline pellets.\n18/s49 /s49/s48/s49/s48/s50\n/s32/s32/s72\n/s109/s97/s120/s32/s45/s32/s72\n/s99/s32/s32/s40/s79/s101/s41\n/s84/s32/s40/s75/s41/s49/s48/s51\nFigure 6: (color online) Difference of the field of the maximum slope of v irgin curves in\nmagnetization loops and the coercive field for crystals (blue, dark c ircles) and polycrys-\ntalline pellets (yellow, open squares). The lines are guide to the eye.\n19/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57\n/s48 /s50/s53/s48/s48 /s53/s48/s48/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s77 /s40\n/s66/s47/s102/s46/s117/s46 /s41\n/s72/s32/s40/s79/s101/s41/s84/s32/s61/s32/s50/s75\n/s77/s32/s40/s66/s47/s102/s46/s117/s46/s41/s72/s32/s40/s107/s79/s101/s41\n/s116/s32/s40/s115/s41\n/s32\nFigure 7: (color online) MvsHfor a single crystalline sample at 2K showing the initial\nbranchafterademagnetizingprocess(circles). The virgincurve( triangles)andthe regular\nloop ascending field branch (diamonds) are also shown. Inset: MandHvs time in a\ndemagnetizing process.\n20/s49/s48/s50\n/s49/s48/s51/s115/s105/s110/s103/s108/s101/s32/s99/s114/s121/s115/s116/s97/s108\n/s112/s101/s108/s108/s101/s116\n/s32/s32/s53/s46/s49/s48/s45/s52\n/s49/s46/s49/s48/s45/s52\n/s49/s48/s45/s51\n/s49/s48/s45/s52/s100/s77/s47/s100/s72/s32 /s40\n/s66/s47/s102/s46/s117/s46/s79/s101 /s41\n/s32\n/s32\n/s72/s32/s40/s79/s101/s41/s32\n/s32/s32\nFigure8: (coloronline)Comparisonofthederivativesofthe virgincu rvesin magnetization\nloops and the initial magnetization after a demagnetizing process, f or a single crystal(a),\na polycrystalline pellet (b). All the curves were taken at 2K. The arr ows indicate the\ncorresponding coercive field\n21"    },    {        "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. 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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": "2111.11603v1.Dynamics_of_ferrimagnetic_skyrmionium_driven_by_spin_orbit_torque.pdf",        "content": "Dynamics of ferrimagnetic skyrmionium driven by spin-orbit torque\nXue Liang,1Xichao Zhang,2Laichuan Shen,1Jing Xia,3Motohiko Ezawa,4,\u0003\nXiaoxi Liu,2and Yan Zhou1,y\n1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China\n2Department of Electrical and Computer Engineering, Shinshu University, Wakasato 4-17-1, Nagano 380-8553, Japan\n3College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China\n4Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan\n(Dated: November 24, 2021)\nMagnetic skyrmionium is a skyrmion-like spin texture with nanoscale size and high mobility. It is a topo-\nlogically trivial but dynamically stable structure, which can be used as a non-volatile information carrier for\nnext-generation spintronic storage and computing devices. Here, we study the dynamics of a skyrmionium\ndriven by the spin torque in a ferrimagnetic nanotrack. It is found that the direction of motion is jointly deter-\nmined by the internal configuration of a skyrmionium and the spin polarization vector. Besides, the deformation\nof a skyrmionium induced by the intrinsic skyrmion Hall effect depends on both the magnitude of the driving\nforce and the net angular momentum. The ferrimagnetic skyrmionium is most robust at the angular momentum\ncompensation point, whose dynamics is quite similar to the skyrmionium in antiferromagnet. The skyrmion Hall\neffect is perfectly prohibited, where it is possible to observe the position of the skyrmionium by measuring the\nmagnetization. Furthermore, the current-induced dynamics of a ferrimagnetic skyrmionium is compared with\nthat of a ferromagnetic and antiferromagnetic skyrmionium. We also make a comparison between the motion of\na ferrimagnetic skyrmionium and a skyrmion. Our results will open a new field of ferrimagnetic skyrmioniums\nfor future development of ferrimagnetic spintronics devices.\nI. INTRODUCTION\nSpintronics is an important field, which has led to many\ninformation processing applications, including memory and\nlogic computing devices [1–12]. The primary tasks in the\nfield of spintronics include the understanding of the funda-\nmental physical phenomena associated with magnetic spins\nand the exploration of various effective methods to control\nspins [13, 14]. Ferromagnetic (FM) materials are vital for\nspin-texture-based devices [1–5, 9, 12, 13]; however, the scal-\ning limitation as well as the limited operation speed may hin-\nder the use of ferromagnets in future spintronic devices [14–\n17]. Therefore, it is necessary to find some different types of\nmagnetic materials to host smaller spin textures and to realize\nfaster spin dynamics.\nAntiferromagnets are predicted theoretically to stabilize\nnanoscale spin textures showing ultra-fast spin dynamics,\nwhich have been an emerging topic in recent years [14, 16,\n18–21]. However, the direct manipulation and detection of an-\ntiferromagnetic (AFM) spin textures are still challenges in ex-\nperiments due to the fact that a perfect antiferromagnet shows\nzero net magnetic moment. Hence, recently much attention\nhas being paid to a special ordered spin system, that is, the fer-\nrimagnetic (FiM) system, where the magnetic moments from\ntwo inequivalent sublattices are coupled in an AFM configura-\ntion. This system holds desirable qualities of both ferromag-\nnets and antiferromagnets, such as the measurable net mag-\nnetization and the fast dynamics of spins, which may provide\na promising material platform for the research of spintronics\ndevices.\n\u0003Email: ezawa@ap.t.u-tokyo.ac.jp\nyEmail: zhouyan@cuhk.edu.cnOne class of ferrimagnets is the rare earth (RE)-transition\nmetal (TM) compounds [22–27], in which the RE and TM el-\nements have different Landé gfactors, corresponding to the\ngyromagnetic ratio \r. Consequently, FiM systems have two\nsignificant compensation points, that is, the magnetization\ncompensation point and the angular momentum compensa-\ntion point. In particular, at the angular momentum compen-\nsation point, the AFM dynamics can be achieved due to the\ncomplete cancellation of angular momentum, and the resul-\ntant magnetization is nonzero, which allows the manipulation\nof ferrimagnets by an external field similar to the case of ferro-\nmagnets [28, 29]. These remarkable features enable ferrimag-\nnets to solve the difficulty on the detection of antiferromag-\nnets and overcome the low-speed limitation of ferromagnets,\nthus being a candidate system for studying spin structures and\ndynamics.\nOn the other hand, future spintronic devices are expected\nto store and process digital information with an ultra-fast op-\neration speed and ultra-high storage density. The topologi-\ncally protected spin texture called the skyrmion is a promising\nbuilding block for future spintronic devices [3, 5–7, 9, 30–\n33], which can be used to encode bits in racetrack-based\ndevices and has been extensively studied in both theoretical\nand experimental approaches [3, 5, 8, 10–12]. The magnetic\nskyrmion has some unique properties, such as the nanoscale\nsize and low depinning current density, which provide a cer-\ntain possibility to build spintronic devices with high perfor-\nmance. However, skyrmions may deviate from the direction\nof the driving force induced by external stimulus, such as the\nspin-polarized current, since the topology-dependent Mag-\nnus force introduces a transverse velocity, which is known as\nthe skyrmion Hall effect [34–36]. Such a phenomenon will\nlead to a serious accumulation or annihilation of skyrmions\nat the boundary of devices. To avoid this detrimental effect,\ntremendous efforts have been devoted to the realization ofarXiv:2111.11603v1  [cond-mat.mes-hall]  23 Nov 20212\nthe in-line motion of skyrmions by using various methods,\nincluding the modification of magnetic properties [37–39],\nthe use of synthetic AFM skyrmions [40–43], and the AFM\nskyrmions [18, 21].\nnz(a) (b)\nFIG. 1. Schematic diagram of an isolated ferrimagnetic skyrmion-\nium in a two-dimensional film. (a) Illustration of a Néel-type ferri-\nmagnetic skyrmionium spin structure with the two anti-parallel mag-\nnetic sublattices. The magnetization orientation along any center-\nline through the skyrmionium core is sketched below. Red and blue\ncolors indicate the sublattices 1 and 2, respectively. (b) The simu-\nlation snapshot of a relaxed ferrimagnetic skyrmionium, where the\nz-component of Néel vector nis coded in the pixel color as shown\nin the color bar on the right.\nSimilar to a synthetic AFM bilayer skyrmion, the skyrmio-\nnium can be regarded as a combination of two skyrmions\nwith topological charges Q= +1 andQ=\u00001. Since the\nmagnitude of Magnus force is proportional to the topological\ncharge and its direction also depends on the sign of Q, these\ntwo opposite Magnus forces acting on the skyrmionium can\nbe compensated completely, eliminating the skyrmion Hall\neffect. [44, 45]. Several works have demonstrated that the\nzero net topological charge of a skyrmionium allows its mo-\ntion along the driving current without transverse drift [44–50].\nMost recently, such a spin texture has been experimentally\nobserved in FiM multilayers [51] and its topological counter-\npart, which is called the bimeronium, has also been studied\nin frustrated magnets [52]. However, the dynamics of a FiM\nskyrmionium induced by external stimulus still remain elu-\nsive.\nIn this work, we systematically investigate the spin current-\ninduced dynamics of a skyrmionium in a FiM nanotrack by\nusing both theoretical and numerical methods. It is found\nthat the FiM skyrmionium can reach a higher speed without\nshowing a severe distortion compared with the FM skyrmion-\nium. The dynamics of a FiM skyrmionium at the compensa-\ntion point of angular momentum is comparable with the case\nof an AFM skyrmionium because of the vanishing angular\nmomentum. Another prominent feature is that it is possible\nto observe the position of the skyrmionium due to a nonzero\nmagnetization at this point.II. MODEL AND METHODS\nWe consider a basic RE-TM FiM film consisting of two\nsublattices, where the unit magnetic moments are s1(r;t)and\ns2(r;t). The Néel vector and total unit magnetization are de-\nfined as n(r;t) = (s1\u0000s2)=2andm(r;t) = (s1+s2)=2, re-\nspectively. In the continuum approximation, the energy func-\ntional is given by [53]\nEtotal=Z\nf\u0015\n2m2+A\n2[(rn)2+@xn\u0001@yn]\n+Lm\u0001(@xn+@yn)\u0000K\n2n2\nz+\"DMIgdV;(1)\nwhere\u0015,AandLare the homogeneous, inhomogeneous\nand parity-breaking exchange constants, respectively [19, 53].\nKis the perpendicular magnetic anisotropy (PMA) constant.\nThe last term \"DMI = (D=2)[nz(r\u0001n)\u0000(n\u0001r)nz]\nis the interface-induced Dzyaloshinskii-Moriya interaction\n(DMI) energy density [54–56], which stabilizes the Néel-type\nskyrmionium, as illustrated in Fig. 1. By taking the functional\nderivative of the total energy density \", we obtain the effective\nfieldsfn=\u0000\u000e\"=(\u00160\u000en)andfm=\u0000\u000e\"=(\u00160\u000em)for the\nNéel vector and total unit magnetization, respectively.\nFor the spin dynamics driven by the spin-orbit torque\n(SOT), the following coupled equations of motion are con-\nstructed (see Appendix A for more details on the derivation)\n\u001a_n=fm\u0002n+\u000b(\u001an\u0002_m+\u001bn\u0002_n)\n+u2n\u0002(p\u0002n) +Tn\nnl; (2)\n\u001a_m+\u001b_n=fm\u0002m+fn\u0002n+\u000b\u001an\u0002_n\n+u1n\u0002(p\u0002n) +Tm\nnl; (3)\nwhere\u001a=M1=\r1+M2=\r2,\u001b=M1=\r1\u0000M2=\r2is the\nspin density that is used to quantify the net angular momen-\ntum, e.g.,\u001b= 0 denotes the compensation point of angular\nmomentum. \u000bis the damping coefficient, p= (px;py;0)\nis the polarization vector of the spin current, u1=\f1+\f2\nandu2=\f1\u0000\f2are the parameters of SOT with \fi=\n\u0016B\u0012SHj=(\rietz).Miand\riare the saturation magnetization\nand the gyromagnetic ratio for the sublattice i,\u0016Bis the Bohr\nmagneton,jis the driving current density, \u0012SHis the spin Hall\nangle fixed at 0:1in this work, eis the electron charge and\ntz= 1:0nm is the thickness of the FiM film. Considering\nthe fact thatjmj\u001cjnj\u00191for the colinear ferrimagnets,\nTn\nnl=u1n\u0002(p\u0002m)andTm\nnl=\u000b\u001bn\u0002_m+u2n\u0002(p\u0002m)\nare the weak nonlinear terms that will be discarded in the fol-\nlowing derivation.\nSubstituting fm=\u0000[\u0015m+L(@xn+@yn)]=\u00160into Eq. 2\nand combining with Eq. 3, we obtain the closed equation in\nterms of the Néel vector n, given as\n\u00160\u001a2\n\u0015n\u0002(n\u0002n) =\u001bn\u0002_n+\u000b\u001a_n+u1p\u0002n\u0000T0:(4)\nHere, (1 +\u000b2)\u00191,\u0015= 4A=a2andL=p\n2A=a are used\nwithabeing the lattice constant, T0= (\u00160\u001au2=\u0015)n\u0002(p\u0002\n_n) +n\u0002(f\u0003\nn\u0002n)\u0000(Lu2=\u0015)n\u0002f[p\u0002(@x+@y)n]\u0002\nng,f\u0003\nn=fA\u0003\u0001n+Knzez+D[@xnzex+@ynzey\u00003\n(@xnx+@yny)ez]g=\u00160is the reduced effective field with\nA\u0003=A=2[50].\nIn order to derive the analytical velocity from the motion\nEqs. 2 and 3, we also use the collective coordinate approach\nin FiM as proposed in the FM system [57], with which the\nmagnetic soliton is regarded as a rigid body and the time-\ndependent order parameter nneed to be rewritten by the soli-\nton position R(t),n(r;t) =n[r\u0000R(t)]. Note that the time\nderivatives of Néel vector nis_n=\u0000@in_Ri, and then nbe-\ncomes n=\u0000@inRi, discarding the quadratic term @iin_R2\ni.\nAfter taking the scalar product of Eq. 4 with @in, and integrat-\ning over the space, the motion equation of a centrosymmetric\nmagnetic soliton is given by\nMR+\u001bG\u0002_R+\u000b\u001aD_R\u0000u1Ip=0; (5)\nwhereM=\u00160\u001a2d=\u0015 is the effective mass of the magnetic\nsoliton with d=dii=R\n@in\u0001@indxdy ,G= 4\u0019Qez\nis the gyrovector with the topological charge Qdefined as\nQ= (1=4\u0019)R\nn\u0001(@xn\u0002@yn)dxdy . The third and the\nfourth terms are the dissipative force due to the nonzero\nGilbert damping and the driving force induced by SOT, re-\nspectively, where the components of tensors DandIare\ngiven byDij=\u000eijdandIij=R\n(n\u0002@in)jdxdy . The\naxisymmetric spin configurations are normally described by\nn= [sin\u0012(r)cos\b(');sin\u0012(r)sin\b(');cos\u0012(r)], where\n(r;')are the polar coordinates. Consequently, the gyrovec-\ntorG(or the topological charge Q) and the dissipative tensor\nDare determined by the angle \u0012that is the angle between\nthe Néel vector nand thez-axis, while the tensor Idepends\non both\u0012(r)and\b('), as shown in Appendix A. For the\nsteady motion, Rshould be zero, so that we obtain the ve-\nlocityv=_Rof the soliton in the following matrix\n\u0014\nvx\nvy\u0015\n=u1\n\u000b2\u001a2d2+\u001b2G2\u0014\n\u000b\u001ad \u001bG\n\u0000\u001bG \u000b\u001ad\u0015\u0014\nIxxIxy\nIyxIyy\u0015\u0014\npx\npy\u0015\n:\n(6)\nThe full derivation can be found in Appendix A.\nIn this work, we focus on the FiM skyrmionium as illus-\ntrated in Fig. 1, where the Néel vector nrotates 2\u0019from the\ncenter to the edge of the skyrmionium (i.e., \u0012(0) = 0 and\n\u0012(1) = 2\u0019), where the topological charge Q= 0. There-\nfore, for a Néel-type skyrmionium, \b =',Ixy=\u0000Iyx=\nI=\u0019R\n(r\u0012r+sin\u0012cos\u0012)drandIxx=Iyy= 0, the analytical\nsteady velocity of the FiM skyrmionium is simplified to\n\u0014\nvx\nvy\u0015\nNéel=u1I\n\u000b\u001ad\u0014\npy\n\u0000px\u0015\n: (7)\nSimilarly, for the Bloch-type skyrmionium stabilized by the\nbulk DMI\"DMI= (D=2)n\u0001(r\u0002n),\b ='+\u0019=2,Ixx=\nIyy=\u0000IandIxy=Iyx= 0, the velocity is given by\n\u0014\nvx\nvy\u0015\nBloch=\u0000u1I\n\u000b\u001ad\u0014\npx\npy\u0015\n: (8)\nTo confirm the analytical prediction on the skyrmionium\nvelocity, we numerically solve the motion Eqs. 2 and 3 by us-\ning the method shown in the Ref. 58. In addition, the positionof the FiM skyrmionium is defined as follow [50, 59, 60]\nRi=R\ni(1\u0000nz)dxdyR\n(1\u0000nz)dxdy; i =x;y: (9)\nWe can therefore get the position of the FiM skyrmionium\nevolving in time, and then obtain the velocity (vx;vy) =\n(_Rx;_Ry)numerically. According to the above analytical\nderivation, the dynamics of the spins in a ferrimagnetic sys-\ntem are independent of the specific magnetization on two\nsublattices, but are strongly related to \u001aand the total spin\ndensity\u001b. For simplicity, we assume that the magnetic mo-\nmentM1is fixed at 710 kA\u0001m\u00001, and the gyromagnetic ratio\n\r1= 2:211\u0002105m\u0001A\u00001\u0001s\u00001and\r2= 1:1\r1[23, 28, 61, 62].\nTherefore, we change the ratio \u0011=M2=M1between the mag-\nnetic moments M1andM2to modify the spin density \u001b(or\u001a).\nOther magnetic parameters are adopted from Ref. 23 and 28:\nA= 11:5pJ\u0001m\u00001,D= 1:85mJ\u0001m\u00002,K= 0:438MJ\u0001m\u00003,\n\u0015= 263:3MJ\u0001m\u00003,L= 38:91mJ\u0001m\u00002,\u000b= 0:1\u00180:3with\na default value of 0.2. The mesh size of 1\u00021\u00021nm3is used\nto discretize the FiM nanotrack with l= 500 nm andw= 150\nnm.\nIII. RESULTS AND DISCUSSION\np= +ey\np= -ey\np= -exp= +ex\np= +ey\np= -eyp= +exp= -ex\nnz ny nz ny(a) (b)\n(d) (c)\nFIG. 2. Comparison between the motion of a Néel-type skyrmionium\nand a Bloch-type skyrmionium. The top-views of the (a) Néel-type\nskyrmionium and the (b) Bloch-type skyrmionium with the in-plane\ncomponent of the order parameter nindicated by the black arrows\n(left), and the corresponding y-component of these skyrmioniums\n(right). (c-d) The motion of these two types of skyrmioniums driven\nby the spin current with different p, where the yellow dashed circle\nrepresents the initial position of the skyrmionium and green arrows\ndenote the polarization direction.\nFrom the equations ( 7) and ( 8), one can find that the veloc-\nity of a FiM skyrmionium depends on both the internal spin\nconfiguration (i.e., the Néel-type and the Bloch-type) and the\npolarization direction p. Assuming that the spin current is\ngenerated from the heavy metal by the spin Hall effect, the4\n0.91.01.11.21.3010200\n.10 .20 .3010203040( b) \nvx (Numerical) \nvy (Numerical) \nvx (Analytical) \nvy (Analytical) v (m/s)/s61544\n (a) v (m/s)/s61537\n/s61544 = 0.9\nFIG. 3. Comparison between numerical results and analytical re-\nsults. The ferrimagnetic skyrmionium velocity as a function of (a)\nthe magnetization ratio \u0011=M2=M1and (b) the damping coefficient\n\u000bfor\u0011= 0:9. Here, the driving current density jis fixed at 10\nMA\u0001cm\u00002to ensure that the skyrmionium keeps a circle shape with-\nout severe distortion. Symbols represent the results from numerical\nsimulations, and the dashed lines indicate the analytical results based\non Eq. ( 7).\npolarization vector is thus given by p=ez\u0002je, where ezis\nthe unit vector normal to the film and jeis the direction of the\nelectron flow. As shown in Fig. 2, the Néel-type FiM skyrmio-\nnium moves perpendicular to the polarization direction (or\nparallel to the driving current) without showing the skyrmion\nHall effect. However, compared with the Néel-type skyrmio-\nnium, the velocity of the Bloch-type skyrmionium does rotate\n90\u000e, parallel to the direction of polarization. These simula-\ntion results are in good agreement with our theoretical veloc-\nity equations. It should be mentioned that the y-component\nof the Néel vector nis used to distinguish these two types of\nskyrmioniums due to the fact that the out-of-plane component\nis independent of \b(')as mentioned before. Both the inter-\nnal structure of skyrmioniums and the polarization direction\njust influence the direction of the velocity, not the magnitude.\nTherefore, we focus on the dynamics of a Néel-type skyrmio-\nnium driven by the spin current in the following, where pis\nfixed at\u0000ey.\nFigure 3 illustrates the velocity of a FiM skyrmionium ver-\nsus the magnetization ratio \u0011=M2=M1and the damping\ncoefficient\u000b. Note that at the magnetization compensation\npoint, the net magnetic moment vanishes, corresponding to\n\u0011=M2=M1= 1:0, while the angular momentum compensa-\ntion point calls for \u0011=M2=M1= 1:1in this work. From the\nmotion Eqs. 2 and 3, we find that the spin dynamics strongly\ndepends on\u001aand\u001b, rather than the net magnetization. Thus,\nwe will discuss more about the angular momentum compen-\nsation point in the following sections. In Fig. 3, it is seen that\nthex-component of velocity decreases with increasing ratio\n\u0011, and is inversely proportional to the damping constant \u000b,\nwhich are in good agreement with Eq. 7. vyalso remains at 0\nas predicted analytically. It should be mentioned that the rela-\ntion between the FiM skyrmionium velocity and the ratio \u0011is\nmonotonous, which is different from that of FiM skyrmions or\nbimerons [24, 25, 27], where the velocity at \u0011= 1:1reaches\nmaximum. Such a difference mainly comes from the second\nterm in Eq. 5, which is determined by the topological charge Q\n3 04 05 06 07 06 08 01 0 01 2 01 4 0\n0 . 80 . 91 . 01 . 11 . 21 . 31 . 43 04 05 06 07 0\nj( M A⋅c m- 2)v( m / s ) /s61544 =  0 . 9\n/s61544 =  1 . 0\n/s61544 =  1 . 1\n/s61544 =  1 . 2\n/s61544 =  1 . 301 02 002 04 0\n/s61544jc( M A⋅c m- 2)\n- 1 . 0- 0 . 50 . 00 . 51 . 0( b )\n/s61555 (A2 s m-2)( a )xFIG. 4. Driving current density dependence of the ferrimagnetic\nskyrmionium velocity. (a) Ferrimagnetic skyrmionium velocity as a\nfunction of the driving current density jfor\u0011=0.9, 1.0, 1.1, 1.2 and\n1.3. The inset is zoom-in of the current-velocity relationship when\nj < 30MA\u0001cm\u00002. The cross represents the skyrmionium with se-\nvere distortion under such a driving current density. (b) The critical\ndriving current density that causes a severe distortion for a ferrimag-\nnetic skyrmionium, as a function of the magnetization ratio \u0011. The\nblue symbols indicate the spin density \u001bcorresponding to different\n\u0011.\nand the spin density \u001b. For the FiM skyrmions and bimerons\nwithQ=\u00061,vxis maximum and vy= 0 at the compen-\nsation point of the angular momentum ( \u0011= 1:1). However,\nthe topological charge Q= 0 of FiM skyrmioium leads to\nthe\u001b-independent velocity (see Eq. 7) to vary with \u0011since\nthe dissipative force is related to \u001a. Besides, the effect of the\nparity-breaking exchange constant on the motion of the FiM\nskyrmionium is also discussed [63].\nWe now study the effect of the driving current density on\nthe dynamics of a FiM skyrmionium. Figure 4(a) shows the\nvelocities of a FiM skyrmionium at five different \u0011driven by\nthe damping-like SOT. The FiM skyrmionium shows a simi-\nlar current-velocity relation at different \u0011, where the velocity\nis proportional to the driving current density j, correspond-\ning to the Eq. 7. However, a large driving force causes severe\ndeformation of the skyrmionium. Here, the level of defor-\nmation is distinguished by the aspect ratio of a skyrmionium\nwhen it moves to the same specified location. In our work,\nwe measure the aspect ratio of the skyrmionium at 400 nm of\na 500-nm-long nanotrack, and the aspect ratio larger than 1.5\nis defined as the severe distortion, otherwise it is regarded as\na slight deformation. Fig. 4(a) shows that the required cur-\nrent density jto destroy the structure of a FiM skyrmionium\nis unequal for different \u0011. For example, the skyrmionium with\n\u0011= 0:9is distorted severely at j= 39 MA\u0001cm\u00002, while the\nskyrmionium with \u0011= 1:2shows the same deformation at\nlarger current density j= 58 MA\u0001cm\u00002. Figure 4(b) illus-\ntrates the critical current density jcthat forces FiM skyrmion-\nium to be destroyed seriously versus the ratio \u0011. The skyrmio-\nnium with\u0011= 1:1is the most robust to resist the deforma-\ntion, which is attributed to the zero net angular momentum.\nAs shown in Eq. 6, the y-component of the steady velocity\nfor FiM skyrmions ( Q=\u00061) vanishes at the angular mo-\nmentum compensation point. Namely, there is no skyrmion\nHall effect on both the inner and outer skyrmions consisting\nof the FiM skyrmionium, which is the same as the case of the5\n𝜓 𝜓𝜓\nFIG. 5. Analysis of the skyrmionium deformation. (a) The components of dissipative tensor Dand tensor Ias functions of the ratio \u0011when\nthe skyrmionium moves along the nanotrack without severe distortion. Here, j= 10 MA\u0001cm\u00002. (b-d) The time evolution of DijandIijfor\n\u0011=0.9, 1.1 and 1.3 (corresponding to \u001b= +0:58;0:0and\u00000:58A2s\u0001m\u00002), where the driving current density is 40, 70 and 45 MA \u0001cm\u00002,\nrespectively. Insets show the top view of a ferrimagnetic skyrmionium marked with a deformation angle  .\nAFM skyrmionium. In addition, the relationship between the\nspin density \u001band the magnetization ratio \u0011is also shown in\nFig. 4(b), where \u001bincreases as \u0011decreases.\nConsidering that the tensors DandIare determined by\nthe magnetic structure, we extract the components of these\ntensors to describe the deformation of a FiM skyrmionium.\nAs shown in Fig. 5(a), when the driving current density jis\nrelatively small, the skyrmionium moves along the nanotrack\nkeeping a circular shape without severe distortion. Therefore,\nthe components Dxx=Dyy= 58 ,Dxy= 0,Ixx=Iyy= 0\nandIxy=\u0000Iyx=\u0000283nm are independent on the ratio \u0011\nand remain unchanged. However, for a large driving current\ndensity, the FiM skyrmioniums with different \u0011deform in dif-\nferent directions. Here, we introduce a deformation angle  \nthat is the clockwise angle between the vertical line along the\n+ydirection of the nanotrack and the major axis of the de-\nformed skyrmionium [see Fig. 5(b-d)] to parametrize the dis-\ntortion. It is found that the angle  is smaller than 90\u000ewhen\n\u0011= 0:9, while = 90\u000efor\u0011= 1:1and >90\u000efor\u0011= 1:3.\nThe varying deformation originates from the fact that the di-\nrection of the Magnus force is determined by the sign of both\nthe topological charge Qand the net spin density \u001b.\nAs shown in Fig. 5(b-d), \u0011= 0:9(\u001b= +0:58A2s\u0001m\u00002),\nthe drift speed vyof the inner skyrmion is along the \u0000ydi-\nrection, while the Magnus force acting on the outer skyrmion\npoints up. Besides, the magnitudes of vxandvyare propor-\ntional to the size of skyrmions. As a consequence,  < 90\u000e,\nand the components of tensors Das well as Iobviously vary\nwith time. On the contrary, when \u0011= 1:3(\u001b=\u00000:58\nA2s\u0001m\u00002), the drift speed vyof the inner skyrmion and the\nouter skyrmion are upward and downward respectively, which\ninduce a deformation angle  larger than 90\u000e. As discussed\nFIG. 6. Comparison between the current-induced dynamics of (a)\nFM, (b) FiM ( \u0011= 0:9) and (c) AFM skyrmioniums. Insets show the\ntop view of a skyrmionium driven by the selected current density j=\n20MA\u0001cm\u00002att= 6 ns, which is indicated by the vertical dashed\nline. Symbols represent the results from numerical simulations, and\nthe dashed lines indicate the analytical results based on Eq. ( 7).\nabove, the skyrmions have no drift speed along the y-axis\nat the compensation point of the angular momentum, so that\nthe two skyrmions consisting of the FiM skyrmionium move\nalong thex-axis at different velocities leading to  = 90\u000e. It\nis worth mentioning that, when \u0011= 1:1(\u001b= 0:0A2s\u0001m\u00002),\nthe components Dxy,IxxandIyyare still 0and do not\nvary with time. However, when \u00116=1:1, these quantities are6\nFM\nFiM25 MA· cm-2\n30 MA· cm-230 MA· cm-2\n30 MA· cm-227 m· s-1\n41 m· s-1\n61 m· s-1(a)\n(b)\n(c)\n(d)\nFIG. 7. Snapshots of (a-b) a FM skyrmion, (c) a FiM skyrmion and (d) a FiM skyrmionium during their motion driven by a spin current, where\nthe yellow dashed line represents the initial position of spin structures, and the gray lines stand for their trajectories. The time evolution of\nvelocities for these magnetic structures driven by a spin current with (e) j= 25 MA\u0001cm\u00002and (f-h)j= 30 MA\u0001cm\u00002. Here,\u0011= 0:9for a\ntypical uncompensated FiM system.\nnonzero, indicting the different deformations. In general,\nDxy<0(Ixx<0andIyy>0) corresponds to the defor-\nmation angle  <90\u000e, whileDxy>0(Ixx>0andIyy<0)\nis related to the deformation angle  >90\u000e. The final config-\nurations of these deformed FiM skyrmioniums at such current\ndensities are provided in Ref. [63].\nWe now compare the current-induced dynamics of FM,\nFiM and AFM skyrmioniums. In Fig. 6, we demonstrate\nthat the response of a skyrmionium to the driving current\nin the three magnetic systems. It is seen that a relatively\nsmall driving current density results in the distortion of a FM\nskyrmionium during its motion, while the AFM skyrmionium\nis the most difficult to be destroyed. The FiM skyrmionium\nis the intermediate one between the cases of FM and AFM\nskyrmioniums. Moreover, driven by the same current den-\nsity ofj= 20 MA\u0001cm\u00002and compared with FiM and AFM\nskyrmioniums, the FM skyrmionium is most fragile. Taking\nthe micro-structures of the three systems into account, the net\nangular momentum is completely canceled in the AFM sys-\ntem due to the two sublattices coupled antiferromagnetically\nwith the same magnetic moments and the same gyromagnetic\nratio. Different from the AFM system, there is a resultant\nangular momentum in the FiM system, where the two sub-\nlattices normally have different spin densities Mi=\ri. On the\nother hand, the net angular momentum of the FiM system is\nstill smaller than that of the FM system. Note that the Magnus\nforce depends on \u001bas shown in Eq. 5. From the point of view\nof applications, the FiM skyrmionium may be a good choice\nsince it not only suppresses the skyrmion Hall effect more ef-\nfectively than the FM skyrmionium, but also is easier to be\ndetected compared with the AFM one.\nHere, the current-induced dynamics of a FiM skyrmion-\nium is also compared with that of a FM skyrmion and a FiM\nskyrmion. During the motion driven by a spin current, the FM\nskyrmion shows an inevitable skyrmion Hall effect due to the\nMagnus force associated with the nonzero topological charge.In a confined nanotrack, as shown in Fig. 7(a), this Magnus\nforce is canceled by the repulsive force arisen from the bound-\nary so that the skyrmion moves along the edge with a steady\nspeed. However, for a large driving current density, the en-\nergy barrier of the edge is insufficient to confine a skyrmion\nin a nanotrack. Consequently, the skyrmion disappears at the\nedge [see Fig. 7(b)]. Considering that the net angular momen-\ntum of two sublattices are not completely cancelled, the FiM\nskyrmion shows the similar motion behaviors. From Figs. 7(c)\nand (d), it is seen that the steady speed of a FiM skyrmion is\nsmaller than that of a FiM skyrmionium driven by the same\ncurrent density. Moreover, the FiM skyrmionium moves along\nthe centerline of the nanotrack without drift speed as dis-\ncussed before. Figures 7(e-h) also show the time evolution\nof velocities for these spin configurations during their motion.\nFor the undamaged FM skyrmion and FiM skyrmion, the x-\ncomponent of the velocity begins with a constant, and then\nincreases as the skyrmion approaches the edge and eventually\nreaches a new steady value, while the y-component decreases\nto zero. However, the velocity of a skyrmionium remains un-\nchanged with a relatively large value. Therefore, compared\nwith skyrmions, the FiM skyrmionium as a carrier of informa-\ntion can effectively prevent the accumulation and annihilation\nof skyrmions at the edge, and potentially improve the access\nspeed of storage devices.\nIV . CONCLUSION\nIn conclusion, we have analytically and numerically inves-\ntigated the current-induced dynamics of a FiM skyrmionium\nin a nanotrack. Our results show that, at the angular momen-\ntum compensation point, the FiM skyrmionium is most robust\nto resist the deformation due to the zero intrinsic skyrmion\nHall effect, which is same as the case of an AFM skyrmio-\nnium. Nevertheless, the position of a FiM skyrmioniums is7\nobservable due to a nonzero magnetization. It is found that\nthe direction of distortion depends on the sign of the net an-\ngular momentum. Above the angular momentum compensa-\ntion point, the deformation angle is smaller than 90\u000e. How-\never, the deformation angle is larger than 90\u000ebelow the an-\ngular momentum compensation point. We have also demon-\nstrated the change in components of two tensors DandIfor a\nFiM skyrmionium during the motion to describe its deforma-\ntion. Furthermore, we have made a comparison between the\ncurrent-induced dynamics of FM, FiM and AFM skyrmion-\niums. The motion of a FiM skyrmionium is also compared\nwith that of FM and FiM skyrmions. Our results open a new\nfiled of the skyrmionium physics in the FiM system and could\nprovide guidelines for the design of future spintronic devices\nbased on ferrimagnets.\nACKNOWLEDGMENTS\nThis research was supported by Guangdong Special\nSupport Project (2019BT02X030), Shenzhen Fundamen-\ntal Research Fund (Grant No. JCYJ20210324120213037),\nShenzhen Peacock Group Plan (KQTD20180413181702403),\nPearl River Recruitment Program of Talents (2017GC010293)\nand National Natural Science Foundation of China\n(11974298, 61961136006). X.L.’s PhD study was financially\nsupported by the National Natural Science Foundation of\nChina (Grant No. 12004320). X.Z. was an International\nResearch Fellow of the Japan Society for the Promotion of\nScience (JSPS). X.Z. was supported by JSPS KAKENHI\n(Grant No. JP20F20363). J.X. acknowledges the support by\nthe National Natural Science Foundation of China (Grant No.\n12104327). M.E. acknowledges the support by the Grants-\nin-Aid for Scientific Research from JSPS KAKENHI (Grant\nNos. JP17K05490 and JP18H03676) and the support by\nCREST, JST (Grant Nos. JPMJCR16F1 and JPMJCR20T2).\nX.X.L. acknowledges the support by the Grants-in-Aid\nfor Scientific Research from JSPS KAKENHI (Grant Nos.\nJP20F20363 and JP21H01364).\nAPPENDIX A: ANALYTICAL DERIVATION OF THE\nMOTION EQUATIONS\nTo derive the motion equation of magnetization in a ferri-\nmagnetic system, we start from the Laudau-Lifshitz-Gilbert\n(LLG) equations with the spin-orbit torque for the two sublat-\ntices:\n_s1=\u0000\r1s1\u0002H1+\u000bs1\u0002_s1+\r1BD1s1\u0002(p\u0002s1);(A1)\n_s2=\u0000\r2s2\u0002H2+\u000bs2\u0002_s2+\r2BD2s2\u0002(p\u0002s2);(A2)\nwhere Hi =\u0000\u000e\"=(\u00160Mi\u000esi)andBDi =\n(\u0016B\u0012SHj)=(\rieMitz)are the effective fields associated\nwith various energies in the system and the damping-like\ntorque induced by the spin current, respectively. Multiplying\nthe Eqs. (A1) and (A2) by M1=\r1andM2=\r2, respectively,\nand then finding the addition and the subtraction of thesetwo equations, we obtain the following coupled equations of\nmotion:\n\u001a_n+\u001b_m=\u0000(m\u0002fn+n\u0002fm)\n+\u000b[\u001a(n\u0002_m+m\u0002_n) +\u001b(m\u0002_m+n\u0002_n)]\n+u1[m\u0002(p\u0002n) +n\u0002(p\u0002m)]\n+u2[n\u0002(p\u0002n) +m\u0002(p\u0002m)]; (A3)\n\u001a_m+\u001b_n=\u0000(m\u0002fm+n\u0002fn)\n+\u000b[\u001a(m\u0002_m+n\u0002_n) +\u001b(n\u0002_m+m\u0002_n)]\n+u1[n\u0002(p\u0002n) +m\u0002(p\u0002m)]\n+u2[m\u0002(p\u0002n) +n\u0002(p\u0002m)]: (A4)\nHere,\u001a=M1=\r1+M2=\r2,\u001b=M1=\r1\u0000M2=\r2,fm=\nf1+f2,fn=f1\u0000f2andu1=\f1+\f2,u2=\f1\u0000\f2with\n\fi=BDiMiandfi=HiMi. It is important to noted that\nthe rule for the linear combination of two variables is used in\nthis derivation, Ax+By= (1=2)[(A+B)(x+y) + (A\u0000\nB)(x\u0000y)]andAx\u0000By= (1=2)[(A+B)(x\u0000y) + (A\u0000\nB)(x+y)]. Considering the fact that jmj\u001cjnj\u00191for the\ncolinear ferrimagnets, and keeping leading-order terms, the\nabove equations are reduced as\n\u001a_n=fm\u0002n+\u000b(\u001an\u0002_m+\u001bn\u0002_n)\n+u2n\u0002(p\u0002n) +Tn\nnl; (A5)\n\u001a_m+\u001b_n=fm\u0002m+fn\u0002n+\u000b\u001an\u0002_n\n+u1n\u0002(p\u0002n) +Tm\nnl; (A6)\nwhere, Tn\nnl=u1n\u0002(p\u0002m)andTm\nnl=\u000b\u001bn\u0002_m+u2n\u0002\n(p\u0002m)are the weak nonlinear terms that will be discarded\nin the following derivation. The micromagnetic simulation of\nthe spin dynamics in this work is based on numerically solving\nEqs. (A5) and (A6).\nSubstituting fm=\u0000(1=\u00160)[\u0015m+L(@xn+@yn)]into\nEq. (A5), we obtain the total magnetization mthat depends\non the spatial Néel vector n.\nm=\u00160\n\u0015f\u0000\u001a(1 +\u000b2)n\u0002_n+\u000b[n\u0002fn\u0000u1n\u0002(p\u0002n)]\n\u0000u2p\u0002ng\u0000L\n\u0015(@xn+@yn); (A7)\nwhere the dissipation term \u000b[n\u0002fn\u0000u1n\u0002(p\u0002n)]\ncan be ignored. Rewriting the effective field fn=f\u0003\nn+\n(A\u0003=\u00160)(@xx+@yy+ 2@xy)n+ (L=\u0016 0)(@xm+@ym)with\nA\u0003=A=2and substituting mintofn\u0002n, we obtain\nfn\u0002n=f\u0003\nn\u0002n+A\n2\u00160(@xx+@yy+ 2@xy)n\u0002n+L\n\u00160\nh\u00160\n\u0015f\u0000\u001a( 1 +\u000b2)n\u0002(@x+@y)_n\u0000u2p\u0002(@x+@y)ng\n\u0000L\n\u0015(@xxn+@yyn+ 2@xyn)i\u0002n: (A8)\nSince\u0015= 4A=a2andL=p\n2A=a, we deduce that A=2 =\nL2=\u0015, and then the second and the last terms on the right side\nof the Eq.(A8) can be cancelled. We substitute mandfn\u0002n8\ninto Eq. (A6),\n\u00160\u001a\n\u0015f\u0000\u001a(1 +\u000b2)n\u0002n\u0000u2p\u0002_ng\u0000L\u001a\n\u0015(@x_n+@y_n)\n=\u0000\u001b_n+fm\u0002m+f\u0003\nn\u0002n+L\n\u0015f\u0000\u001a(1 +\u000b2)\nn\u0002(@x+@y)_n\u0000u2p\u0002(@x+@y)ng\u0002n\n+\u000b\u001an\u0002_n+u1n\u0002(p\u0002n): (A9)\nSupposing that (1 +\u000b2)\u00191and multiplying the Eq. (A9) by\nn, we obtain the following closed equation of the Néel vector\nn\n\u00160\u001a2\n\u0015n\u0002(n\u0002n) =\u001bn\u0002_n+\u000b\u001a_n+u1p\u0002n\n\u0000\u00160\u001au2\n\u0015n\u0002(p\u0002_n)\u0000n\u0002(f\u0003\nn\u0002n)\n+Lu2\n\u0015n\u0002f[p\u0002(@x+@y)n]\u0002ng: (A10)\nFor a centrosymmetric magnetic soliton, the order parameter\nnis described by both the position rand the azimuthal angle\n', i.e.,n(r;') = [ sin\u0012(r)cos\b(');sin\u0012(r)sin\b(');cos\u0012(r)].\nTaking the scalar product of Eq. (A10) with @in, and integrat-\ning over the space, we obtain the motion equation\nMR+\u001bG\u0002_R+\u000b\u001aD_R\u0000u1Ip=0; (A11)\nwhereM=\u00160\u001a2d=\u0015is the effective mass of the soliton, G=\n(0;0;G)is the topology-dependent gyrovector, DandIare\ntensors related to the damping term and the SOT, respectively.\nHere,\ndij=Z\n(@in\u0001@jn)dxdy =\u000eijd\n=\u000eij\u0019Z\n(\u00122\nr+sin2\u0012=r2)rdr; (A12)\nG=Z\n[n\u0001(@xn\u0002@yn)]dxdy = 2\u0019Z\nsin\u0012\u0012rdr; (A13)\nIij=Z\n(n\u0002@in)jdxdy; (A14)Z\n@in\u0001[n\u0002(p\u0002_n)]dxdy =0; (A15)\nZ\n@in\u0001[n\u0002(f\u0003\nn\u0002n)]dxdy =0; (A16)\nZ\n@in\u0001fn\u0002[p\u0002(@x+@y)n]\u0002ngdxdy = 0:(A17)\nThis term (n\u0002@in)jdenotes thej-component of the vector\n(n\u0002@in). Different from the gyrovector Gand the tensor\nD, the tensor Istrongly depends on the angle \b(')due to\nthe fact that\n(n\u0002@xn)x=\u0000sin\bcos'\u0012r+sin\u0012cos\u0012sin'cos\b\nr;(A18)\n(n\u0002@xn)y=cos\bcos'\u0012r+sin\u0012cos\u0012sin'sin\b\nr;(A19)\n(n\u0002@yn)x=\u0000sin\bsin'\u0012r\u0000sin\u0012cos\u0012cos'cos\b\nr;(A20)\n(n\u0002@yn)y=cos\bsin'\u0012r\u0000sin\u0012cos\u0012cos'sin\b\nr:(A21)\nIn general, the steady velocity is written as\n\u0014\nvx\nvy\u0015\n=u1\n\u000b2\u001a2d2+\u001b2G2\u0014\n\u000b\u001ad \u001bG\n\u0000\u001bG \u000b\u001ad\u0015\u0014\nIxxIxy\nIyxIyy\u0015\u0014\npx\npy\u0015\n:\n(A22)\nFor a Néel-type skyrmionium, \b =',Ixy=\u0000Iyx=I=\n\u0019R\n(r\u0012r+sin\u0012cos\u0012)drandIxx=Iyy= 0, above velocity\nequation becomes\n\u0014\nvx\nvy\u0015\nNéel=u1I\n\u000b\u001ad\u0014\npy\n\u0000px\u0015\n: (A23)\nSimilarly, for the Bloch-type skyrmionium, \b ='+\u0019=2,\nIxx=Iyy=\u0000IandIxy=Iyx= 0, the velocity is given by\n\u0014\nvx\nvy\u0015\nBloch=\u0000u1I\n\u000b\u001ad\u0014\npx\npy\u0015\n: (A24)\nTherefore, one can find that the velocity of a skyrmionium\ndepends on both the internal spin distribution and the polar-\nization direction of the spin current.\n[1] J. 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B 97, 220403(R) (2018).\n[63] See Supplementary Material for the details of the analytical\nderivation of the energy functional, the effect of the exchange\ninteraction on the motion of a FiM skyrmionium and the defor-\nmation process of a FiM skyrmionium at a larger driving current\nabove the critical density."    },    {        "title": "1604.01180v1.Micromagnetic_simulation_of_exchange_coupled_ferri__ferromagnetic_composite_in_bit_patterned_media.pdf",        "content": "Micromagnetic simulation of exchange coupled ferri-/ferromagnetic\ncomposite in bit patterned media\nHarald Oezelt,1,a)Alexander Kovacs,1Phillip Wohlh uter,2, 3Eugenie Kirk,2, 3Dennis Nissen,4, 5Patrick Matthes,5\nLaura Jane Heyderman,2, 3Manfred Albrecht,4, 5and Thomas Schre\r1, 6\n1)Industrial Simulation, University of Applied Sciences, 3100 St. P olten, Austria\n2)Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich,\nSwitzerland\n3)Laboratory for Micro- and Nanotechnology, Paul Scherrer Institute, 5232 Villigen PSI,\nSwitzerland\n4)Institute of Physics, Chemnitz University of Technology, 09126 Chemnitz, Germany\n5)Institute of Physics, University of Augsburg, 86159 Augsburg, Germany\n6)Centre for Integrated Sensor Systems, Danube University Krems, 2700 Wiener Neustadt,\nAustria\nFerri-/ferromagnetic exchange coupled composites are promising candidates for bit patterned media because\nof the ability to control the magnetic properties of the ferrimagnet by its composition. A micromagnetic\nmodel for the bilayer system is presented where we also incorporate the microstructural features of both\nlayers. Micromagnetic \fnite element simulations are performed to investigate the magnetization reversal\nbehaviour of such media. By adding the exchange coupled ferrimagnet to the ferromagnet, the switching\n\feld could be reduced by up to 40 % and also the switching \feld distribution is narrowed. To reach these\nsigni\fcant improvements, an interface exchange coupling strength of 2 mJ/m2is required.\nKeywords: exchange-coupled composite media; micromagnetic simulation; ferrimagnet; FEM; bit patterned\nmedia; switching \feld distribution\nI. INTRODUCTION\nIn the attempt to move forward to higher data density\nin magnetic storage devices, the concept of bit patterned\nmedia is the next logical step. In order to maintain ther-\nmal stability of the bits and keep them writeable at the\nsame time, Suess et al.1and Victora et al.2proposed\nthe idea of exchange spring media. This approach allows\nthe use of material with high magnetic anisotropy by de-\ncreasing the required switching \feld with an exchange-\ncoupled soft magnetic material. Experimental studies\ncon\frmed the feasibility of exchange coupled media in\nmultilayer structures3and later also in bit patterned me-\ndia4{6. Krone et al.7performed micromagnetic simula-\ntions of arrays consisting of exchange coupled compos-\nite stacks and also graded media, where the magnetic\nanisotropy constant was decreased quadratically across\nten layers.\nIn this paper we investigate exchange coupled bilayer\ndots where a ferrimagnetic material, such as FeTb or\nFeGd, represents the soft magnetic layer. Ferrimagnetic\nmaterials have been extensively studied8,9and used as\nmagneto-optical recording media10,11. A big advantage\nof using ferrimagnetic layers is the possibility to tailor\ntheir magnetic properties through their composition with\na)Electronic mail: harald.oezelt@fhstp.ac.at; The following ar-\nticle appeared in H. Oezelt et al., Journal of Applied Physics\n117, 17E501 and may be found at http://dx.doi.org/10.1063/\n1.4906288 . This article may be downloaded for personal use only.\nAny other use requires prior permission of the author and AIP\nPublishing. Copyright (2015) American Institute of Physics.respect to the desired working temperature12. Moreover,\nsince these layers are amorphous, the lack of crystalline\ndefects may positively in\ruence the switching \feld dis-\ntribution of the exchange coupled ferromagnetic layer.\nIn the following, we will describe the micromagnetic\nmodel used for the exchange coupled ferri-/ferromagnetic\ncomposite dots. We also consider the granular structure\nof the magnetically harder ferromagnet and material in-\nhomogeneities in the softer, amorphous ferrimagnet in\nour geometrical model. Then, micromagnetic \fnite ele-\nment simulations are used to calculate the magnetization\nreversal of dots with varying microstructure, diameter\nand interface coupling strength. This in turn enables the\ninvestigation of the magnetization con\fguration during\nreversal and the switching \feld distribution.\nII. MICROMAGNETIC MODEL\nIn this study we consider cylindrical dots composed of\na ferromagnetic layer \nFMand a ferrimagnetic layer \nFI\ncollinearly exchange-coupled at the interface \u0000. Since\na detailed explanation of a ferri-/ferromagnetic bilayer\nmodel has already been given in our previous work13, we\nprovide a brief summary here.\nWhile the \fnite element simulation of ferromagnets is\na common task, the mathematical model for ferrimag-\nnets has to be adapted. We follow Mansuripur's14ap-\nproach and assume that the ferrimagnetic sublattices are\nstrongly coupled antiparallel at all times. Therefore we\ncan substitute the magnetic moments M(a),M(b)of the\nsublattices with an e\u000bective net moment MFIby de\fn-\ning its net magnitude as MFI=M(a)\u0000M(b)and its unitarXiv:1604.01180v1  [physics.comp-ph]  5 Apr 20162\nvector as m=m(a)=\u0000m(b)(see FIG. 1). The Gilbert\nequations of both sublattices can then be summed up to\nobtain an e\u000bective Gilbert equation. Since we are only\ninterested in the static hysteresis behaviour, the damping\nconstant is set to \u000be\u000b= 1.\nFM\n¡-z\n5nm\ngranular FePt\n20nm\namorphous FeGdFIkiFM\nkiFIpiM(a)M(b)\nMFIMFM\nFIG. 1. Geometric model of the bilayer dot with a ferrimag-\nnetic phase \nFIand a ferromagnetic phase \nFMconnected at\nthe interface \u0000. The magneticaly softer, amorphous \nFIis\ndivided in patches piwith varying uniaxial anisotropic prop-\nerties kFI\niandKFI\nu;i. The granular hard magnetic phase, \nFM,\nposseses strong out-of-plane uniaxial anisotropy.\nTo take into account the exchange coupling between\nthe two layers, we extend the equation for the e\u000bec-\ntive \feld of each layer with the interface exchange \feld\nHixhg =\u00001=\u00160\u000eEixhg=\u000eMexerted by the respective\nneighbouring layer. The exchange energy across the in-\nterface \u0000 is given by Eixhg=\u0000A\u0000=aR\n\u0000uFMuFId\u0000, where\nA\u0000is the exchange sti\u000bness constant, ais the distance\nbetween spins in a simple cubic lattice and uis the unit\nvector of each spin direction. Due to the microstructural\ndi\u000berences both layers have to be meshed separately. As\nthe mesh nodes at the interface do not match, we employ\na surface integral technique and calculate Eixhgby using\na symmetric Gaussian quadrature rule for triangles15,16.\nThe geometrical model used for the simulations is de-\npicted in FIG. 1. The \nFMmodel is a 5 nm thick,\nL10chemically ordered Fe 52Pt48layer with an average\ngrain diameter of 13 nm. The layer exhibits a satura-\ntion polarization of JFM\ns= 1:257 T and an exchange\nsti\u000bness constant of AFM\nx= 10 pJ/m. Each grain has\nits own randomized anisotropic constant and uniaxial\nanisotropic direction. The average assigned anisotropic\nconstant is KFM\nu= 1:3 MJ/m3with a standard deviation\nof 0:05KFM\nuJ/m3. The uniaxial anisotropic direction is\nlimited within a cone angle of 15\u000efrom the out-of-plane\n(z-) axis. No intergrain phase is considered.\nThe \nFIphase is an amorphous, 20 nm thick Fe 74Gd26\nlayer. The ferrimagnet is characterized by a saturation\npolarization of JFI\ns= 0:268 T, an exchange sti\u000bness con-\nstant ofAFI\nx= 2 pJ/m. To incorporate material inho-\nmogeneities in the amorphous model we divide the layer\ninto patches piwith an average diameter of 13 nm as sug-\ngested by Mansuripur and Giles17. Each patch exhibitsits own randomly assigned anisotropic constant and uni-\naxial anisotropic direction. The average anisotropic con-\nstant isKFI\nu= 10 kJ/m3with a standard deviation of\n0:2KFI\nuJ/m3. The uniaxial anisotropic direction varies\nwithin a cone angle of 90\u000efrom patch to patch.\nThe micromagnetic simulations are performed by us-\ning the \fnite element micromagnetic package FEMME18.\nWe investigate the dependence of reversal curve and espe-\ncially the switching \feld Hswon the dot diameter and the\nexchange coupling at the interface. In order to calculate\nthe switching \feld distribution, 20 simulation runs for\neach dot diameter and interface coupling strength were\nperformed. Each simulation had its individual mesh for\nboth layers with randomized microstructure generated by\nthe software Neper19. Also the randomized anisotropic\nproperties of both layers were generated anew for each\nsimulation within the limits described previously. The\nmesh size for both layers was set to 2 nm.\nIII. RESULTS AND DISCUSSION\nBy applying an increasing external \feld Hextto the\nfully saturated dot in the opposite direction ( \u0000z) to the\nmagnetization, the reversal curve is computed. For each\nset of dot diameter and interface exchange strength we\ncompute the average reversal curve over the 20 random-\nized simulations. So the result can be seen as the rever-\nsal curve of an array of 20 dots of equal diameter, but\nvarying microstructure and anisotropic properties. The\naveraged reversal curves for di\u000berent dot diameters with\nan exchange coupling strength of A\u0000=a= 5 mJ/m2at the\ninterface are depicted in FIG. 2.\n-1-0.8-0.6-0.4-0.200.20.40.60.81\n-1.6-1.4-1.2 -1-0.8-0.6-0.4-0.2 0Mz[Ms]\napplied field Hext [T]5nm\n20nm\n40nm\n60nm80nm\n100nm\n120nm\nFIG. 2. Reversal curves of dot arrays for di\u000berent dot di-\nameter. The layers of the dot are strongly coupled with\nA\u0000=a= 5 mJ/m2.\nFor all diameters the soft magnetic \nFIphase switches\nat about \u00000:27 T. With increasing diameter the rever-\nsal curve of the soft magnetic phase gets \rattened. This\ncan be accredited to the shape anisotropy, since the layer3\nthickness is \fxed for all models. This can also be seen\nin FIG. 3 where, in contrast to the 5 nm dot, the 120 nm\ndot shows an inhomogeneous reversal of the magnetic\nmoments starting with an in-plane con\fguration at the\nsurface of the ferrimagnet (FIG. 3f). The reversal of the\nhard magnetic \nFMphase in FIG. 3 strongly depends\non the dot diameter. With smaller diameters the \nFM\nphase consists only of one or a few grains, which leads to\na \rattened reversal curve when averaged over the 20 sim-\nulations, i. e. a broader switching \feld distribution. The\nswitching \feld HFM\nswdrastically increases with decreasing\ndiameter. This is because with increasing diameter the\nmodel changes from the single domain to a multi domain\nregime.\nIn FIG. 3 the magnetization con\fguration during re-\nversal of the bilayer is depicted on an x-z-slice through\nthe center of a d= 5 nm and a d= 120 nm dot. The re-\ngions in bright gray are still not reversed, the dark gray\nareas are already reversed, whereas the domain walls are\nin black.\na) b)\nc) d)e)\nf)\ng)\nh)\nmz↑ mz↓ mz 0\nFIG. 3. Reversal process of a dot with d= 5 nm from a) to\nd) and of a dot with d= 120 nm from e) to h). The interface\n\u0000 is the white dashed line while the \nFIis the upper and the\n\nFMis the lower layer.\nBy applying the external \feld in down direction in the\nsmall dot, the magnetic moments of the upper region\ncoherently switch and form a domain wall due to the\nexchange coupling at the interface (FIG. 3b). With in-\ncreasing \feld the domain wall gets pushed towards the in-\nterface c), when eventually the \nFMphase nucleates as a\nsingle domain d). In the 120 nm dot the \nFIphase starts\nto rotate more inhomogeneously e) and turns in-plane at\nthe surface f). At this state the domain wall is widened\nbecause of its 90\u000econ\fguration. With increasing Hext\nthe domain wall gets narrower and is pushed through\nthe interface into the \nFMphase g). Compared to the\nsmaller dot, the reversal of the harder phase is much more\ninhomogeneous and a lateral domain wall movement can\nbe observed leading to full reversal h).\nThis translation from homogeneous to inhomogeneous\nreversal of the hard magnetic phase can be clearly rec-ognized in FIG. 4 where the HFM\nswcurves drop between\n25 nm and 30 nm dot diameter. In FIG. 4 we show the\nswitching \felds of both layers, again averaged over the 20\nrandomized simulation runs. The switching \felds, HFM\nsw\nin the upper and HFI\nswin the lower area, are de\fned as\nMFI\nz(HFI\nsw) = 0 and MFM\nz(HFM\nsw) = 0. The curves for\nthree di\u000berent interface coupling strengths are shown.\nAdditionally the standard deviation of the 20 simula-\ntion runs for each data point is shown as a grey shade:\nHsw\u0006\u001bsw.\n0.811.21.41.61.82Hswno coupling\n1 mJ/m2\n5 mJ/m2\n0.10.20.3\n20 40 60 80 100 120Hsw[T]FI FM\ndot diameter d [nm]FM FI[T]\nFIG. 4. Averaged switching \feld of the \nFMand the \nFI\nphase for di\u000berent interface coupling strengths depending on\nthe dot diameter. The standard deviation \u001bswof 20 simula-\ntions for each data point is depicted as the gray area Hsw\u0006\u001bsw.\nWithout a coupled ferrimagnet, HFM\nswis reduced by\n39 %, when moving from a 5 nm to a 120 nm dot diameter.\nFor a strongly coupled bilayer this reduction is improved\nto 50 %. If we look at a speci\fc dot diameter, intro-\nducing the coupled ferrimagnet reduces HFM\nswby 30 % to\n40 %. The higher the diameter, the higher the reduction\nof the switching \feld of the hard phase. While the switch-\ning \feld of a single ferrimagnetic layer would decrease\nwith growing diameter, an increasing interface coupling\ncan stabilize or even cause an increase of HFI\nswby about\n12 % within the investigated diameter range. FIG. 4 also\nshows that the switching \feld distribution decreases with\nincreasing diameter and interface exchange coupling for\nthe ferromagnet. The ferrimagnetic phase shows a signif-\nicant reduction of the switching \feld distribution when\nincreasing the diameter from 5 to 20 nm.4\nThis behaviour can also be seen in FIG. 5 where the\nrelative standard deviation of the switching \feld \u001bsw=Hsw\nis plotted against the exchange coupling strength. The\nsolid symbols refer to the ferromagnetic phase and the\nempty symbols to the ferrimagnetic phase for three dif-\nferent dot diameters.\n00.050.10.150.20.250.3\n012345¾sw/Hsw\n[mJ/m2]5nm\n60nm\n120nmFIFM\nA/g1/a\nFIG. 5. Relative standard deviation of the switching \feld of\nboth layers for di\u000berent dot diameters as a function of inter-\nface exchange coupling strength.\nCoupling the ferrimagnetic layer to the ferromagnet de-\ncreases the relative standard deviation \u001bFM\nsw=HFM\nswfrom\n11 % to below 7 % for 5 nm diameter. For larger diame-\nters it decreases to 4 % for 60 nm or even below 2 % for\n120 nm. The relative standard deviation for the ferrimag-\nnet is also reduced with increasing interface exchange en-\nergy, especially for the 5 nm diameter dot, where it is re-\nduced from 26 % to 7 %. The major change of the switch-\ning \feld distribution occurs below A\u0000=a= 2 mJ/m2and\nonly slightly improves above.\nIV. SUMMARY\nA micromagnetic model for exchange coupled ferri-\n/ferromagnetic bilayer dots was presented. We per-\nformed a series of simulations for the dots of diameters\nfrom 5 nm to 120 nm with varying interface exchange cou-\npling strength from 0 to 5 mJ/m2. For each parameter\nset, 20 simulations were performed with randomized mi-\ncrostructure and anisotropic properties.\nWe found that with increasing dot diameter the switch-\ning \feld of the hard phase drastically decreases and\nalso narrows the switching \feld distribution. Dots\nwith small diameters exhibit homogeneous switching be-\nhaviour, only interrupted when in the ferrimagnet a do-\nmain wall close to the exchange coupled interface is\ncreated and is slowly pushed towards it. Dots with\nlarger diameters reverse more inhomogeneously, build-ing an in-plane orientation con\fguration and show a lat-\neral domain wall movement in the hard magnetic phase.\nThe switching \feld and its distribution can also be con-\ntrolled by the exchange coupling strength at the inter-\nface. With increasing exchange coupling, the ferromag-\nnetic switching \feld is reduced by 30 % for 5 nm dots\nand 40 % for 120 nm dots. Its distribution is improved\nto\u001bFM\nsw=HFM\nsw= 7 % for 5 nm dots and \u001bFM\nsw=HFM\nsw= 2 %\nfor 120 nm dots. To reach signi\fcant improvements, an\ninterface exchange coupling strength of A\u0000=a= 2 mJ/m2\nis required.\nACKNOWLEDGMENTS\nWe gratefully acknowledge the \fnancial support pro-\nvided by the Austrian Science Fund (FWF Grant No.\nI821), the German Research Foundation (DFG Grant No.\nAL 618/17-1) and the Swiss National Science Foundation\n(SNF Grant No. 200021L 137509).\n1D. 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Kronm uller and S. Parkin (John Wiley & Sons,\nLtd, 2007) pp. 1{30.\n19R. Quey, P. Dawson, and F. Barbe, Computer Methods in Ap-\nplied Mechanics and Engineering 200, 1729 (2011)."    },    {        "title": "2304.13698v2.Direct_observation_of_Néel_type_skyrmions_and_domain_walls_in_a_ferrimagnetic_DyCo__3__thin_film.pdf",        "content": "Direct observation of Néel-type skyrmions and domain walls in a\nferrimagnetic DyCo 3thin film\nChen Luo,1, 2Kai Chen,1, 3,∗Victor Ukleev,1Sebastian Wintz,1, 4Markus\nWeigand,1, 4Radu-Marius Abrudan,1Karel Prokeš,5and Florin Radu1,†\n1Helmholtz-Zentrum-Berlin für Materialen und Energie,\nAlbert-Einstein-Straße 15, 12489 Berlin, Germany\n2Institute of Experimental Physics of Functional Spin Systems,\nTechnical University Munich, James-Franck-Straße 1,\n85748 Garching b. München, Germany\n3National Synchrotron Radiation Laboratory,\nUniversity of Science and Technology of China, Hefei, Anhui 230029, China\n4Max-Planck-Institut für Intelligente Systeme, 70569 Stuttgart, Germany\n5Helmholtz-Zentrum-Berlin für Materialen und Energie,\nHahn-Meitner Platz 1, D-14109 Berlin, Germany\n(Dated: August 11, 2023)\n1arXiv:2304.13698v2  [cond-mat.mtrl-sci]  10 Aug 2023Abstract\nIsolated magnetic skyrmions are stable, topologically protected spin textures that are at the fore-\nfront of research interests today due to their potential applications in information technology. A\ndistinct class of skyrmion hosts are rare earth - transition metal (RE-TM) ferrimagnetic materials.\nTo date, the nature and the control of basic traits of skyrmions in these materials are not fully\nunderstood. We show that for an archetypal ferrimagnetic material DyCo 3that exhibits a strong\nperpendicular anisotropy, the ferrimagnetic skyrmion size can be tuned by an external magnetic\nfield. Moreover, by taking advantage of the high spatial resolution of scanning transmission X-ray\nmicroscopy (STXM) and utilizing a large x-ray magnetic linear dichroism (XMLD) contrast that\noccurs naturally at the RE resonant edges, we resolve the nature of the magnetic domain walls of\nferrimagnetic skyrmions. We demonstrate that through this method one can easily discriminate\nbetween Bloch and Néel type domain walls for each individual skyrmion. For all isolated ferri-\nmagnetic skyrmions, we observe that the domain walls are of Néel-type. This key information is\ncorroborated with results of micromagnetic simulations and allows us to conclude on the nature of\nthe Dzyaloshinskii–Moriya interaction (DMI) which concurs to the stabilisation of skyrmions in this\nferrimagnetic system. Establishing that an intrinsic DMI occurs in RE-TM materials will also be\nbeneficial towards a deeper understanding of chiral spin texture control in ferrimagnetic materials.\n∗kaichen2021@ustc.edu.cn\n†florin.radu@helmholtz-berlin.de\n2INTRODUCTION\nMagnetic skyrmions are stable nanoscale whirls of magnetic spin textures [1–6]. Due to\ntheir topological stability, small size at the nanometer scale and controlled mobility under\nlow current densities, skyrmions hold the promise to impact significantly next-generation\ninformation storage technology [7–16]. Initially introduced in nuclear physics as soliton\nsolutions of non-linear field equations [1, 17], they hold now a distinct place in solid state\nphysics as well, following their theoretical prediction [18] and experimental observation [19].\nBroken inversion symmetry that is characteristic to certain crystal structures induces a non-\ncollinear coupling mechanism that contributes as an asymmetric term in the Hamiltonian\ndescribing the resulting magnetically chiral ground states [20]. Under certain conditions,\nthese chiral states concur in forming magnetic skyrmions as observed in archetypal cubic\nchiral crystals that exhibit a bulk Dzyaloshinskii–Moriya interaction (DMI) [19, 21–25].\nMoreover, symmetry breaking along with spin-orbit coupling present at magnetic interfaces\nlead to a weak interfacial DMI that contributes to the stabilisation of skyrmions observed\nin thin films [26–30] and multilayers [31–36].\nSkyrmion lattices that occur in single crystals fill a small pocket in the phase diagram for\ntemperatures that typically extends over few Kelvin [19, 37]. This is detrimental to applica-\ntions which require stability over a broad range around room temperature. The temperature\npocket can be eventually extended by reducing the dimensionality of the structures or by\nengineering the interfacial DMI of ferromagnetic thin films and multilayers [38]. Yet, caused\nby the skyrmion Hall effect [39, 40], the trajectories of these ferromagnetic topological units\nin devices are not straight, being deflected away by the Lorentz forces. To overcome this\nlimitation, ferrimagnetic materials are offering an advantage due to a versatile tunability of\ntheir magnetic properties.\nRare-earth-transition-metals (RE-TM) ferrimagnetic (FiM) alloys consist two anitferro-\nmagnetically coupled sublattices. At the compensation temperature (T comp), the magne-\ntizations of both sub-lattices are equal, leading to a vanishing net magnetization, just as\nfor an antiferromagnet. By the choice of the elemental composition and through tempera-\nture variation, their magnetic properties, including T comp, net magnetization and magnetic\nanisotropy, can be easily engineered, which makes FiM materials advantageous for spintron-\nics devices [41–43]. By selecting the RE element, two classes of FiM can be distinguished,\n3namely Gd-base FiM alloys that exhibit a weak perpendicular magnetic anisotropy (PMA)\ndue to the vanishing orbital moment of the RE, and Dy, Tb, Ho-based FiM alloys which\nhave a stronger PMA due to the large orbital moment of the RE. The latter category has\nthe potential to offer a higher stability of stored information, but the reports on skyrmions\nin these systems are scarce [44].\nBy contrast, for weak PMA FiM alloys, like FeGdCo, magnetic skyrmions have been\nrecently observed [45] and they can be controlled in microstructured devices [46, 47]. Since\nthen much research has been carried out to enable control and fuctionalization of ferrimag-\nnetic skyrmions: they have been observed in ferrimagnetic confined nanostructures [48];\ntopological spin memory is reported for Co/Gd multilayers exhibiting skyrmion stabil-\nity in fully compensated antiferromagnetically coupled heterostructure [49];observation of\nspin spirals and individual skyrmions in synthetic Pt/CoGd/Pt ferrimagnetic multilayers at\nroom-temperature has been achieved without the assistance of external magnetic fields [50];\nmagneto-transportmeasurementshaverevealedatopologicalcontributionresultingfromthe\noccurrence of an interfacial DMI in Ho/CoFeGd/ β−W multilayers [51]; evidence for chiral\nferrimagnetism in an ultrathin GdCo layer has been demonstrated through a combination\nof high-resolution Lorentz microscopy and XMCD [52]; Néel-type homochirality has been\nobserved over a large temperature range in Ta/Ir/Fe/GdFeCo/Pt multilayers using scanning\nelectron microscopy with polarization analysis [53]; and information on Néel versus Bloch\nDWs can be inferred by a tilt geometry with Lorentz transmission microscopy as shown for\na Mn 3Sn topological antiferromagnet [54]. However, a direct determination of the type of\nthe spin structures, namely Néel-type versus Bloch-type, was not experimentally reported,\nfor neither of the two FiM classes.\nAn unambiguous determination of the skyrmion type is crucial for understanding the sta-\nbilization mechanism of skyrmions in these materials. Indeed, one would expect stabilization\nof Néel-type skyrmions in a thin film system with an interfacial DMI induced by engineering\nof the spin-orbit coupling of the ferrimagnetic layer with neighboring heavy-metal (HM)\nlayers [8]. However, this approach usually requires stacking ultrathin magnetic and HM\nlayers into an asymmetric periodic multilayer in order to achieve a sizeable magnitude of\nDMI and, consequently, a small enough ( ∼100nm) skyrmion size [32]. On the other hand,\n\"bulk\" DMI stabilizing chiral Bloch-type skyrmions requires an intrinsic lack of inversion\nsymmetry within the material [20]. Alternatively, skyrmion bubbles having the same topo-\n4logical charge, but degenerate chirality can also be stabilized by dipolar interactions [55].\nFurthermore, dipolar interaction can compete with an interfacial DMI and change the spin\nrotation sense from Néel to Bloch type, or give rise to hybrid spin textures [56]. Therefore,\nunveiling the skyrmion type in FiM will shed light onto the microscopic spin Hamiltonian\nin this class of thin-film FiM alloys.\nIn this study, we report real space imaging of the magnetic structures in a FiM DyCo 3\nthin film by means of scanning transmission X-ray microscopy (STXM) utilizing both x-\nray magnetic circular dichroism (XMCD) and x-ray magnetic linear dichroism (XMLD)\ncontrast [57–61]. Using XMCD-STXM, we directly observe well-isolated FiM skyrmions\nand their transformation to maze-like domains as a function of the out-of-plane external\nfield. With XMLD-STXM, we demonstrate that these FiM skyrmions are Néel-type and\nthe maze-like domains also show a preference of Néel-type domain walls. We confirm our\nexperimental results to be consistent with micromagnetic simulations. Please note that the\nexperiments reported here are performed at low temperatures (26 K) which correspond to a\nnon-fully compensated magnetization state. The fully compensated magnetization for this\nferrimagnet occurs at a temperature that is well above the room temperature, therefore a\nvanishing skyrmion Hall effect is not addressed(see Supplementary Note S1.2).\nRESULTS AND DISCUSSION\nField dependence of Skyrmion size\nIn Fig. 1a we show the hysteresis loop in perpendicular geometry which was measured\nby SQUID magnetometry. The magnetic reversal of the DyCo 3film as a function of applied\nfield was investigated at 26 K using STXM. Because the maximum magnetic field for the\nSTXM experiments is limited to 260 mT which is not enough to saturate the sample at\nlow temperatures, we saturated the sample out-of-plane at room temperature prior to the\nSTXM measurements. After cooling down the sample (in a perpendicular magnetic field of\n+260 mT) to 26 K, the STXM images were obtained with the magnetic field sweeping from\n+260 mT to -260 mT. Figure 1b shows selected examples of XMCD-STXM areal images of\nthe FiM skyrmions acquired at different perpendicular magnetic fields. One can distinguish\ntheevolutionofwell-isolatedFiMskyrmionsasafunctionofthemagneticfield, rangingfrom\n5260 mT to 140 mT (within the field range that is marked by a grey rectangle in Fig. 1a). The\naverage density and radius of the skyrmions extracted from the STXM images as a function\noffieldareshowninFig.1candinFig.1d, respectively. Itincreasesfrom2.5to9.6skyrmions\nper square micrometer when the field is decreased from 260 mT to 140 mT, indicating that\nthe skyrmions can be created and annihilated by varying the field. At the same time, the\naverage skyrmion radius Rincreases from 45 nm to 65 nm, demonstrating that the skyrmion\nsize can be reduced and inflated with the external field. When the magnetic field decreases\nfurther, the skyrmions will merge into worm- and maze-like domain structures, as shown in\nthe insets of Fig. 1a.\nLateral imaging of skyrmions with XMCD contrast\nTo resolve the details of the FiM skyrmions, high resolution XMCD-STXM images were\nrecorded at the Co L 3and the Dy M 5edges at an external field of 140 mT, as shown in Fig. 2.\nThe left panels display the images of several single skyrmions, whereas in the right panels we\nshowtheirlineprofilesalongtheXandYaxes, whichrepresentthenormalizedperpendicular\nmagnetic moment of Dy ( MDy) and Co ( MCo) elements. One can easily observe that the\nmagnetic profiles of MDyandMCooverlap nicely, which clearly demonstrate the formation\nof FiM skyrmions with an antiparallel alignment of the Dy and Co moments. At this field,\nthe radius (FWHM) of the respective skyrmion is about 65 nm.\nLateral imaging of skyrmion domain walls with XMLD contrast\nTo identify the spin structure of the domain walls of the FiM skyrmions, XMLD was\nutilized by taking advantage of the strong linear dichroism of Dy at the M 5absorption edge.\nFigure 3a presents the demonstration of X-ray absorption spectroscopy (XAS) at the Dy M 5\nedgeforcircularleft(CL),circularright(CR),linearvertical(LV)andlinearhorizontal(LH)\npolarizations, respectively. The XMCD spectrum was obtained by taking the difference of\n(CR-CL), and the XMLD spectrum was obtained by taking the difference of (LV-LH). One\ncan see that the maximum XMLD signal appears enhanced at the second peak of the Dy\nM5edge whereas the maximum XMCD signal is located at the third peak. The intensity\nof the XMLD spectra at the Dy edge is sufficiently large to be exploited for the STXM\n6measurements [61]. Unlike XMCD which is sensitive to the magnetic moments collinear to\nthe x-ray propagation direction, XMLD is sensitive to the magnetic moments collinear to\nthe− →Evector of linearly polarized X-rays, which lies in the plane of the sample surface for\nour experimental geometry. To distinguish Néel-type and Bloch-type skyrmions from each\nother, we made simulations on how the two different types of skyrmions should look like in\nthe presence of circular and linearly polarized X-rays, which are shown in Fig. 3b and 3c.\nBy comparison, one can see that our experimental data match very well with the Néel-type\ncontrast, indicating that the FiM skyrmions in our DyCo 3thin film are Néel-type skyrmions.\nLateral imaging of domain walls of a maze domain state with XMLD contrast\nAfter successfully identifying Néel-type skyrmions utilizing the advantage of XMLD, we\nalso investigated the domain walls for maze-like domains using the same technique, as shown\ninFig.4(a-c). Onecaneasilyobservethatthedomainwallsatthetop/bottomsidesaremore\npronounced for LV, and that the domain walls at the left/right sides show more intensity\nfor LH, similar to the domain walls in skyrmions shown in Fig. 3d. Note that, one still\ncan see a weak contrast for some domain walls which do not follow this rule. This result\nindicates that the majority of the domain walls for maze-like domains are Néel-type with\na low mixing of Bloch-type. We also applied Fast Fourier Transform (FFT) to the STXM\nimages, which show different patterns for different polarizations (see Fig. 4(d-f). The FFT\nshows a ring pattern for circular polarization. For linear polarization, however, it shows\nan ellipse with long vertical axis for LV and an ellipse with long horizontal axis for LH,\nwhich represent the preference of Néel-type domain walls (compare also to the results of\nmicromagnetic simulations shown in the Supplementary Note S3 and Supplementary Note\nS4).\nMicromagnetic simulations\nMicromagnetic simulations were performed using the MuMax3 package [62] using mag-\nnetic parameters for a DyCo 3thin film deduced in a previous study [44] and in the present\nmagnetometry measurements that can be found in the Supplementary Information file (see\nSupplementary Note S1.1).\n7Figure 5 shows simulated magnetic structures as a function of the magnetic field applied\nperpendicular to the sample plane. The top row shows the color-coded three-dimensional\norientation of the magnetization, and the bottom row shows the in-plane component Mx. A\nmaze domain pattern shows up upon a relaxation of the random magnetization state at zero\nfield (Fig. 5a), featuring also some Néel-type skyrmions with opposite polarities. Skyrmions\nwith core magnetization parallel to the applied magnetic field collapse and disappear upon\nincreasing the field (Fig. 5b). Finally, at a higher field of µ0H= 530mT the maze domain\npattern evolves into a isolated skyrmion phase (Fig. 5c). This phase persists up to 810mT\nwhen the skyrmions collapse and the sample gets fully magnetized along the field. Bottom\npanels of Figs. 5a-d show the in-plane magnetization component within each cell of the\nsimulation. The contrast inversion from blue (left) to red (right) of each stable Néel-type\nskyrmion represents the chirality of the system given by the sign of the DMI constant, which\nis not picked up by the STXM experiment. Importantly, the simulation captures accurately\nthe impact of the domain wall type on the polarization-dependent STXM contrast. Circular\nandellipticshapesofFFTpatternsinFigs. 4d,e,fcorrespondverywelltotheonescalculated\nfrom the simulations (compare to Supplementary Figure S7) described in the Supplementary\nNote S3.\nInterestingly, the simulated skyrmion size is very tunable and can be changed by the\nexternal magnetic field by a factor of three from R= 32nm at 530 mT to R= 9nm\nat 810 mT. While the skyrmion size differs from the experimental value quite significantly,\nlowerDMIparametersdonotallowtostabilizepurelyNéel-typeskyrmionsbutratherhybrid\nones that carry Bloch caps at the surface (see Supplementary Note S5.3), being consistent\nwith the theory reported for magnetic multilayers [63]. If no DMI is assumed, the interplay\nbetween PMA and the stray field results in the formation of a bubble lattice with dominantly\nBloch-type domain walls (see Supplementary Note S5.1). Once a DMI term of sufficient\nstrength is introduced, the stability of the skyrmions is increased towards higher fields, and\nthe unique rotation sense of the domain walls gets defined. Note also that besides the DMI\nstrength, the saturation net magnetization and the magnetic anisotropy parameters play an\nimportant role for the skyrmion formation in this material (see Supplementary Figure S3\nand Supplementary Note S5.2).\nFor a discussion on the possible origin of a \"bulk\" DMI in this system see Ref. [44] and\nthe afferent Supplementary Note S6. Similar observations have recently been reported in\n8Fe3GeTe 2flakesof various thicknesswhere aninterplaybetween dipolarand DMinteractions\nresults in a complex history-dependent magnetic phase diagram of spin textures [64].\nCONCLUSIONS\nWe presented an experimental resolve of skyrmion traits in a ferrimagnetic DyCo 3thin\nfilmat26KusingSTXMimaging, utilizingbothXMCDandXMLDcontrast. WithXMCD-\nSTXM,themagneticstructuresinrealspacearerevealedasafunctionofdecreasingexternal\nfield. Well-isolated ferrimagntic skyrmions are observed between +260 mT and +140 mT,\nand the density as well as the radius of the skyrmions can be controlled by the external\nmagnetic field. When the magnetic field is further reduced, these skyrmions will merge\ninto maze-like domains, which matches very well with the results of magnetic simulations.\nUtilizing XMLD-STXM at the Dy M 5edge, we successfully identify the domain wall type\nof ferrimagnetic skyrmions to be of Néel-type. Moreover, the domain walls for the maze-like\ndomains are also investigated, revealing also a majority of Néel-type domain walls. Hence,\nwe are able to unambiguously conclude the interfacial-type symmetry [65] of DMI in DyCo 3\nthin film. Nevertheless, the origin of the strong DMI of this type remains an open question.\nThe technique of using XMLD contrast in the STXM measurements at the rare earth M\nedges provides a promising way to study complex spin textures in real-space, which is highly\nuseful for the characterization of skyrmions, chiral domain walls and various non-collinear\nmagnetic systems.\nMETHODS\nSample preparation and characterization\nThe ferrimagnetic DyCo 3film of 50 nm thickness was prepared by magnetron sputtering\nchamber (MAGSSY) at room temperature and in an argon atmosphere of 1.5×10−3mbar\nwith a base pressure of 5×10−9mbar. The stoichiometry of the DyCo 3alloy was controlled\nby varying the deposition rates of the Co and the Dy targets in a co-sputtering scheme.\nA Si 3N4membrane with a thickness of 100 nm was used as substrate for the soft X-ray\ntransmission measurements. A capping layer of 3 nm thick Ta was deposited on top of the\nsample surface to prevent surface oxidation. The magnetic properties of the sample have\n9been measured by SQUID magnetometry and by anomalous Hall effect (Tensormeter RTM1,\nHZDR Innovation, Germany), and they are described in the Supplementary Information file.\nX-ray measurements\nScanning transmission X-ray microscopy (STXM) measurements were performed at the\nMAXYMUS endstation at the Bessy II electron storage ring operated by the Helmholtz-\nZentrum Berlin für Materialien und Energie [66]. The X-ray beam was focused with a\nzone plate and an order selecting aperture on the transmissive sample in the presence of an\napplied out-of-plane magnetic field which was controlled by varying the arrangement of four\npermanent magnets. The STXM images were collected pixel by pixel using a piezoelectric\nsample stage at the Co L 3edge and the Dy M 5edge by exploiting the effects of x-ray\nmagnetic circular dichroism (XMCD) and x-ray magnetic linear dichroism (XMLD). The\nXMLD contrast represents an intensity map for LV (vertical axis in real space, parallel to\nthe sample surface) and LH (horizontal axis in real space, parallel to the sample surface)\norientations of the linear polarization axes. When the linear polarization is perpendicular to\nthe spin axis, the XAS intensity measured at the middle resonance peak of the M5 edge is\nlow (high in transmission), whereas for a parallel orientation of the linear polarization axis\nwith respect to the spin axis the intensity is high (low in transmission). (see for instance\nFigure S1, in Ref [61]). This makes the XMLD contrast easy to comprehend for the present\ntransmissiongeometryoffilmswithperpendicularmagneticanisotropy: achangeofintensity\nalong the linear direction shown on the LV, LH and L45 maps (see Supplementary Note\nS2) can be given only by Néel walls, whereas a change of intensity towards a direction\nperpendicular to the linear polarization direction can be given only by Bloch walls. Note\nthat the experimental XMLD maps shown in the manuscript are all logarithm of the raw\ndata images.\nThe XAS, XMLD and XMCD spectra for the Dy M 5edge (Fig. 3a) were performed at the\nDeimos beamline at synchrotron Soleil [67] in transmission mode using the same 50 nm thick\nDyCo 3sample grown on a Si 3N4membrane. The magnetic field of 2 T, which is much higher\nthanthesaturationfield, wasappliedalongthebeamduringtheXMCDmeasurementsusing\ncircular polarized X-rays and perpendicular to the beam during the XMLD measurements\nusing linear horizontal and linear vertical polarized X-rays.\n10Magnetic simulations\nMicromagnetic simulations were performed using MuMax3 [62]. The simulation was\nperformed on a three-dimensional grid 512×512×25voxels with the size of 2×2×2nm3.\nThe material parameters used for the simulation are given further below and can be found\nin the Supplementary Information file. A larger scale simulation with a grid of 1024×\n1024×25voxels was carried out for the simulation without DMI, in order to account for\nthe larger domain size. The computation was performed using a graphics processing unit\n(GPU) NVIDIA GeForce RTX 3080 Ti. The following material parameters were used:\nexchange stiffness Aex= 6 pJ m−1, saturation magnetization Ms= 600 kA m−1, and uniaxial\nanisotropy Ku= 130 kJ m−3. The interfacial-type DMI constant Dintwas tuned to obtain\nthe isolated field-induced Néel-type skyrmion phase without admixtures of maze domains.\nThe minimal value of DMI required for the purely Néel-type skyrmion stability was found to\nbeDint=0.0015 J m−2. It is remarkable that this value amounts about 8% of the exchange\nenergy of DyCo 3[68], in agreement with the suggestion of up to ∼20% of the isotropic\nexchange expected for DMI in disordered systems [69, 70] (see Supplementary Note S6).\nACKNOWLEDGMENTS\nWe thank the Helmholtz-Zentrum Berlin für Materialien und Energie for the allocation\nof synchrotron radiation beamtime (Proposal No. 212-10386). The authors acknowledge\nfinancial support by the German Federal Ministry for Education and Research (BMBF\nproject No. 05K19W061). F.R. acknowledges financial support by the German Research\nFoundation via Project No. SPP2137/RA 3570. We acknowledge the use of the Physical\nproperties laboratory, which is part of the CoreLab \"Quantum Materials\" operated by HZB.\nF.R. acknowledges insightful information provided by Dr. Eugen Weschke on the UE46\nundulator operation.\nCONTRIBUTIONS\nC.L., K.C. and F.R. conceived and designed the experiments. K.C. prepared the samples.\nC.L. and F.R. performed the STXM experiments with the help of S.W. and M.W., C.L. and\nK.C. analyzed the STXM data and prepared the figures. V.U. performed the micromagnetic\n11simulations. K.P. performed the magnetic characterization by SQUID. R.A., C.L., V.U.\nand F.R. performed the magneto-transport experiments. C.L., V.U. and F.R. wrote the\nmanuscript draft. All authors discussed the results and contributed to the manuscript.\n12FIG. 1. Hysteresis loop (measured by superconducting quantum interference device\n(SQUID)magnetometry)andscanningtransmissionX-raymicroscopy(STXM)images\nof ferrimagnetic (FiM) skyrmions at 26 K.\n(a)Out-of-planemagnetichysteresisloopmeasuredat26KbySQUIDmagnetometry. Theexternal\nmagnetic field µ0His expressed in units and sub-units of Tesla(T), with µ0being the vacuum\nmagnetic permeability and Hdenoting the magnetic field strength. (b) STXM images showing the\nFiM skyrmions at different perpendicular magnetic fields recorded at the Dy M 5edge and 26 K.\nThe average density (c) and radius (d) of the skyrmions as a function of external magnetic fields,\nwere extracted from the STXM images. The grey rectangle box in panel (a) represents the magnetic\nfield range where the FiM skyrmions are probed. The arrow represents the field sweeping direction.\nThe insets of panel (a) show the transition into maze-like domains. 13FIG. 2.X-ray magnetic circular dichroism imaging of isolated skyrmions.\n(a) Scanning transmission x-ray microscopy (STXM) images at the Co L 3edge and the Dy M 5edge\n(b), respectively. The color bar represents the normalized sublattice magnetization (M z) along the\nmagnetic field (and the x-ray beam) direction. (c, d) Line profiles across a skyrmion along the X\nand Y axes (see the dashed lines in panel a), representing the normalized perpendicular magnetic\nmoment for Dy and Co elements.\n14FIG.3.Resolveofthemagneticdomainwalltypeusingx-raymagneticlineardichroism.\n(a) x-ray absorption spectra (XAS) measured by circular left (CL), circular right (CR), linear\nhorizontal (LH) and linear vertical (LV) polarized X-rays, as well as x-ray magnetic linear dichroism\n(XMLD) and x-ray magnetic circular dichroism(XMCD) spectra at the Dy M 5edge at 4 K. The\nthree peaks of the Dy M 5edge are marked by dashed lines. Expected magnetic image contrast when\nusing circular, LH and LV polarized X-rays for Bloch-type (b) and Néel-type (c) skyrmions. (d)\nExperimental Scanning transmission X-ray microscopy (STXM) results. Here the STXM images\nfor LV and LH x-ray polarizations were obtained at the middle peak of the Dy M 5edge, and the\nSTXM image for circular polarized X-rays was measured at the third peak of the Dy M 5edge.\n15FIG. 4.Imaging of the magnetic domains and domain walls in a maze domain state\nand their corresponding Fast Fourier Transform.\n(a-c) Scanning transmission X-ray microscopy (STXM) images of the maze domains at -260 mT for\ncircular right (CR), linear horizontal (LH) and linear vertical (LV) polarized X-rays, respectively.\n(d-f) Fast Fourier Transform (FFT) of the top STXM images, with the dashed circles and ellipses\nas guide for the eye.\n16FIG. 5.Micromagnetic simulations demonstrating Néel-type skyrmion formation in the\npresence of a finite Dzyaloshinskii–Moriya interaction.\nTop row of (a,b,c,d) panels: Micromagnetic simulations of relaxed spin configurations for the DyCo 3\nfilm as a function of out-of-plane magnetic field. The external magnetic field µ0His expressed\nin sub-units of Tesla(T), with µ0being the vacuum magnetic permeability and Hdenoting the\nmagnetic field strength. The white and black colors represent the net normalized magnetization\nbeing parallel and antiparallel to the z-axis, respectively. The other colors represents the orientation\nof the in-plane component of the net magnetization within each cell as shown in the color wheel in\ntop panel (d). Bottom row of (a,b,c,d) panels: the magnetization component along the x-axis. The\ncolor bar shown in the bottom panel (d) represents the normalized net magnetization Mx.\n17SUPPLEMENTARY INFORMATION\nS1. MAGNETIC CHARACTERIZATION\nS1.1. SQUID magnetometry\nThe magnetic properties of the sample have been measured by SQUID magnetometry.\nThesamplehasbeencooleddownto2Kinamagneticfieldof2Teslaandmagnetichysteresis\nloops have been measured as a function of temperature, on warming. They are shown in the\nSupplementary Figure S1. The left axis displays the magnetization in kA/m as a function\nof an external magnetic field which was applied in a direction perpendicular to the sample\nsurface. At each temperature a field dependent magnetization was measured without the\nsample to correct for the eventual contributions from the sample holder itself. The absolute\nvalue of the magnetization is obtained by dividing the corrected SQUID raw response (emu)\non the volume of the layer which is area ×thickness, where the measured surface area was\nmeasured to be 6.53×10−6m2and the thickness of the film is 50×10−9m. The hysteresis\nloops exhibit a typical characteristic shape for films with perpendicular magnetic anisotropy,\nshowing the onset of the magnetization reversal at the nucleation field (the magnetic field\nwhere the global remagnetization initiates), followed by magnetic domains formation down\nto the annihilation field where the magnetization is fully reversed.\nThe nucleation field is extracted for each temperature and is shown in Figure S2. It\nexhibits a peculiar behavior: it has negative values at room temperature, changes sign at\nan intermediate temperature of about 230 K, increases in absolute values up to about 50\nK and decreases again, even to negative values, towards lower temperatures. This type of\nbehavior is expected for ferrimagnetic alloys due to the interplay between the anisotropy\nand the demagnetization energies.\nin Supplementary Figure S3 the saturation net magnetization extracted at the highest\napplied field is shown together with the magnetization at the nucleation field. The net\nmagnetization increases from room temperature towards lower temperatures in a monotonic\nway. Interestingly, the net magnetization at the nucleation field begins to deviate signifi-\ncantlyfromthesaturationmagnetizationatatemperatureofabout150K.Thistemperature\nrange (150 K to 2 K) can be considered as the phase diagram for the skyrmions formation.\nWith the magnetic parameters determined from the hysteresis loops, one can extract the\n18FIG. S1. Hysteresis loops measured by SQUID at different temperatures with field applied perpen-\ndicular to the sample’s surface.\nuniaxial magnetic anisotropy of the film as [71, 72]:\nKu=1\n2µM net(−Hn+ 4πM net) (1)\nwhere HNis the nucleation field, Mnetis the net magnetization at the nucleation field. The\nresults are shown in Supplementary Figure S4. We observe that the magnetic anisotropy\nchanges as function of temperature in an expected fashion. For ferrimagnetic films, it is\ndominated by the single ion anisotropy which further follows the character of the orbital\nmagnetic moment of the rare earth. The orbital magnetic moment was measured previously\nby soft x-ray spectroscopy [73]. It exhibits a non-monotonic increase below 200 K which\ncorrelates well with the temperature dependence of the anisotropy determined from the\nSQUID data.\n19FIG. S2. Temperature dependence of the nucleation field determined from data shown in Supple-\nmentary Figure S1\n20FIG. S3. Top panel: Saturation net magnetization extracted from the hysteresis loops at 2 Tesla\n(black down-triangles) and the magnetization at the nucleation field(blue up-triangles). Bottom\npanel: The difference between the saturation net magnetization and the magnetization at the\nnucleation field. This panel can be considered at a phase diagram for temperature range (0-150K)\nof skyrmions formation.21FIG. S4. Temperature dependence of the magnetic anisotropy calculated according to Supplemen-\ntary Equation 1.\n22S1.2. Electrical Transport\nThe characterization across the compensation magnetization was performed by magneto-\ntransport measurements as a function of temperature from 290 K to 460 K. The method\ninvolved here is spinning-current anomalous Hall magnetometry [74] which is implemented\nin a commercial device named Tensormeter (HZDR Innovation, Germany). This spinning-\ncurrent approach has the advantage that extrinsic parasitic contributions to the anomalous\nHall output signal are compensated dynamically which lifts the need of sample microstruc-\nturing.\nThe sample has been wire-bonded and connected with low resistance wires to the measur-\ning device. The transport measurements were performed under high vacuum( 3×10−8mbar)\nwith the Alice II instrument [25, 75]. The Hall resistance R xyof the sample was measured\nin a four-wire configuration using the Zero-Offset Hall preset of the Tensormeter. The tem-\nperature was calibrated by a temperature sensor mounted on the sample holder and the\nmagnetic field was applied perpendicular to the sample’s surface.\nIn Supplementary Figure S5a we show few selected hysteresis loops at temperatures be-\nlow and above the magnetization compensation. The hysteresis loops show an increased\ncoercive field close to the compensation temperature. Moreover, the sign of the hysteresis\nloops reverses as the temperature increases above the compensation. This effect reflects the\nsensitivity of the anomalous Hall resistance (R xy) whichin effectum is proportional to the\ndifference of the up and down electronic density of states at the Fermi energy. For 3d fer-\nromagnets this is proportional to the net magnetization. However, for RE-TM ferrimagnets\nalloys the 3d density of states at the Fermi energy are dominated by the TM element (in our\ncase Co), whereas the density of 4f-states of the RE ion (in our case Dy) are more localized\nbelow the Fermi energy(see Fig. 1c of [76]), therefore TM is contributing less significantly\nto the anomalous Hall resistivity.\nThe coercive field of each hysterysis loop was extracted and plotted in the Supplementary\nFigure S5b. At the divergence position, the net magnetization of the film vanishes. This\ncorresponds to magnetic compensation temperature Tcomp, which is for this sample 415 ±5\nK.\n23FIG. S5. The anomalous Hall effect (AHE) was measured at different temperatures. (a) The\nHall resistance R xyas a function of perpendicular magnetic field. The hysteresis loops change\nsign between 397.3 K and 426.1 K, which indicates the crossing of the magnetic compensation\ntemperature Tcomp. (b) Temperature dependence of the coercivity field µ0Hcexhibits a divergent\nbehavior near Tcompat≈415 K.\n24S2. STXM IMAGES FOR A 45 DEGREES ORIENTATION OF THE LINEAR PO-\nLARIZATION\nA 45◦rotation of the linear X-ray polarization was implemented to investigate the\nskyrmions for larger areas, as demonstratively shown in Supplementary Figure S6. By\ncomparing such a STXM image to those with LH and LV polarization, it can be seen that\nthe contrast of the domain walls also rotates by 45 degrees, which further confirms that the\nskyrmions are Néel-type.\nNote that in spite of a clear Néel character, the the skyrmion and domain walls shape\nexhibit distortions. Given that the atomic arrangements in amorphous alloys are not well-\ndefined, local variation of magnetic anisotropy and stiffness may impact on the homogeneity\nof the magnetic ground states. Also, local pinning at eventual defect sites may contribute\nto complex skyrmion shape distortions [77].\nFIG. S6. XMLD-STXM images of skyrmions at 140 mT and 30 K for LH and LV, and 45◦linear\nx-ray polarization.\nS3. MICROMAGNETIC SIMULATIONS: FAST FOURIER TRANSFORM OF\nTHE MAZE-DOMAIN STATE WITH NÉEL WALLS\nTheMz,Mx, and Mymagnetization components of the maze domain pattern resulting\nfrom the micromagnetic simulations are shown in Supplementary Figure S7, together with\ntheir Fast Fourier transform (FFT) patterns. The FFTs correspond well to the experimental\n25data shown in Figure 4(d-f) of the main manuscript. While the FFT of the Mzcomponent,\nwhich corresponds to the contrast mechanism to the circular light, results in a ring of\nintensity (Supplementary Figure S7a), the FFT patterns for the square of Mx,Myshow\narcs of intensity aligned in horizontal and vertical directions, respectively (Supplementary\nFigure S7b,c). On the other hand, the FFTs of the experimental data measured with linear\nhorizontal and vertical light polarizations (Figure 4e,f) appear as elongated ellipses because\nof a broader width distribution of the maze domains in the real sample, which is further\nconvoluted with the finite resolution of STXM images.\nFIG. S7. Micromagnetic simulations of the zero-field maze domain pattern for the DyCo 3film.\nMagnetization components (a) Mz, (b) Mx, and (c) Myare shown, respectively. First (from the\nleft) bottom panel shows the Fast Fourier transform (FFT) of the Mzcomponent while the other\ntwo bottom panels show the FFT of the square of the corresponding upper MxandMypatterns.\n26S4. MICROMAGNETICSIMULATIONS:FOURIERTRANSFORMOFTHEMAZE-\nDOMAIN STATE WITH BLOCH WALLS\nFFT of Mz,Mx,Mycomponents were also calculated for the micromagnetic simulations\nresults without taking a DMI into account (Supplementary Figure S8). While the ring-like\nintensity is clearly observed in the FFT of the Mzcomponent, FFTs for MxandMyare\nclearly rotated by 90◦compared to the previous case (Supplementary Figure S7). This\nclearly indicates the Bloch type character of the maze domain walls for a vanishing DMI in\ncontrast to the Néel type case when the DMI is present.\nFIG. S8. Micromagnetic simulations of the zero-field maze domain pattern for the DyCo 3film\nwithout taking DMI into account. Magnetization components (a) Mz, (b) Mx, and (c) Myare\nshown, respectively. First (from the left) bottom panel shows the Fast Fourier transform (FFT)\nof the Mzcomponent while the other two bottom panels show the FFT of the square of the\ncorresponding upper MxandMypatterns.\n27S5. MICROMAGNETIC SIMULATIONS: FEW MORE RELEVANT SHOWCASES\nS5.1. Case 1: Simulations for magnetic parameters corresponding to 26 K, without\nDMI\nThe magnetic field dependence of the maze domain pattern has been calculated also\nfor the case of a vanishing DMI, where the magnetic ground state is determined by the\ninterplay between dipolar interaction and uniaxial anisotropy only. Except for the DMI\nconstant, all the other parameters in the simulation are similar to the ones in the main text.\nSupplementary Figure S9 shows the simulated magnetization distributions at zero applied\nmagnetic field, 230mT, 340mT and 450mT. Relatively large ( ∼100nm) magnetic domains\nare observed at zero field, that evolve into worm-like stripes and, finally, into skyrmion\nbubbles as higher out-of-plane magnetic field is applied. The size of the bubbles originally\nformed at 340mT decreases to 45nm at 450mT and then the skyrmion state collapses into\na homogeneous one as the field is further increased. As a result, in the absence of DMI,\nmaze domain walls and field-induced skyrmions are clearly identified as Bloch-type, which\ncan be easily seen in the bottom panels of Supplementary Figure S9, where the intensity of\ntheMxcomponent changes from positive to negative at the top and bottom edges of the\nskyrmions. This is in contrast to the case of Néel type domain walls and skyrmions where\nthe intensity of these maps is rotated by 90 degrees (see main text).\nS5.2. Case 2: Simulations for the magnetic parameters corresponding to 200 K\nwith DMI\nMicromagnetic simulations for a saturation magnetization of Ms= 358 .5kA/m, an uni-\naxial anisotropy of Ku= 69 .8kJm−3and DMI equal to Dint= 0.0015Jm−2were carried\nout to mimic the sample behavior at 200K. The saturation net magnetization and magnetic\nanisotropy parameters were extracted from the magnetometry measurements. Here, large\nmagnetic domains (of the order of hundreds of nm) with Néel-type walls take place, evolving\ninto a very sparse skyrmion state ( ∼1per 1 µm2) at higher applied fields (Supplementary\nFigure S10). The simulation matches well with the STXM results which show similar large-\nscale textures at 200K [44] and absence of the skyrmion phase. We speculate that tuning\nthe DyCo composition in order to increase the anisotropy and saturation magnetization to-\n28FIG. S9. Micromagnetic simulations of the spin configurations for the DyCo 3film as a function\nof magnetic field without taking any DMI into account.The white and black represent the magne-\ntization parallel and anti-parallel to the z-axis. The color code represents the orientation of the\nin-plane component of the magnetization within each cell as shown in the color wheel in the panel\n(d). The bottom panels show the magnetization component along the x-axis. The color code of\nthe intensity of Mxis given in the bottom panel (d).\nwards room temperature can pave a way to stabilize technologically promising ferrimagnetic\ntopological spin textures in DyCo at 300K.\nInterestingly, a single anti-skyrmion has been spotted at an intermediate field range of\n150-180mT in the simulations (Supplementary Figure S10c), which is unexpected in the sys-\ntem with this type of DMI. Confirmation of this observation requires further computational\nand experimental studies.\nS5.3. Case 3: Simulations for the magnetic parameters corresponding to an inter-\nmediate DMI parameter at 26K\nAs we describe in the main text, the minimal value of the interfacial-type DMI con-\nstant required to stabilize Néel-type skyrmions in the micromagnetic simulations is found\natDint= 0.0015Jm−2, while lower values result in Bloch- or hybrid-type skyrmion textures.\n29FIG. S10. Micromagnetic simulations of the spin configurations for the DyCo 3film as a function\nof magnetic field with the MsandKuparameters corresponding to 200K. The black and white\ncontrast indicates the magnetization being parallel and anti-parallel to the z-axis. The color code\nrepresents the orientation of the in-plane component of the magnetization within each cell as shown\nin the color wheel in the top panel (d). The bottom panels show the magnetization component\nalong the x-axis. The color code of the intensity of Mxis given in the bottom panel (d).\nAn example of such a hybrid skyrmion is shown in Supplementary Figure S11. Here, the\nskyrmion winding changes from Bloch-type in the top layer of the film ( z= 1) to Néel-\ntype in the bottom layer ( z= 25) via an intermediate state in the middle. This behavior\ncorrespond well to predictions given by Lemesh and Beach in their analytical theory for\nmultilayers [63].\n30FIG. S11. Magnetic structure of hybrid skyrmions in DyCo 3film with an intermediate value of the\nDMI constant Dint= 0.00125Jm−2at 470mT as obtained from the micromagnetic simulations.\nThe panels show the single skyrmion structure in the top ( z= 1), middle ( z= 12) and bottom\n(z= 25) layers of the film, respectively.\n31S6. POSSIBLEORIGINOFA\"BULK\"DMIINTHEFERRIMAGNETICALLOYS\nOne main outcome of the resolve of the domain walls as Néel-type is that they require\na \"bulk\" DMI of an interfacial-type symmetry [65] to occur. In the absence of a DMI, the\ndomain walls are Bloch-type as revealed by the micromagnetic simulations. It is well known\nthat Ta or Pt buffer layers induce an interfacial DMI which has positive or negative sign,\nrespectively. However, a DMI that is localized at the interface only is too weak to stabilize\nNéel walls. For instance, in FM/Ta bilayers, the iDMI has an noticeable impact only for a\nFM layer thickness that is below 1 nm [78]. In our case, the ferrimagnetic layer is 50 nm\nthick, and therefore a DMI localized at the interface only, is not likely to stabilize skyrmions\nas observed experimentally. Instead a bulk DMI owes to be responsible for the skyrmions\nformation in this system [44].\nThe DyCo 3single crystal has a complex rhombohedral structure where Dy occupies two\nsites, whereas Co occupies three sites in the unit cell. According to Ref. [79], for one site the\nDy will align itself with the easy axis, whereas the Dy on the second site will prefer to align\nparallel to the c-axis of the crystal which is oriented perpendicular to the easy axis. As such,\na non-collinear spin arrangement is possible due to the frustrated site magnetic anisotropy\nof the rare earth ions. For the amorphous DyCo 3there is also strong experimental evidence\nfor the occurrence of a noncollinear spin state. In Ref. [80] an amourphous DyCo3 thick film\nhas been studied by Mössbauer spectroscopy. It has been observed that the Dy ion exhibit\na sperimagnetic arrangement, whereas the Co sublattice is ferromagnetically ordered. The\nabove experimental evidence for noncollinear behavior manifest intrinsically for both single\ncrystal and amourphous DyCo 3.\nCertainly, non-collinear interactions are yet not sufficient to lead to skyrmion formation.\nTwo more aspects play an important role for the formation of skyrmions in our DyCo3\nthin film, namely: an amorphous thin film naturally lacks a rotational and transnational\nsymmetry, (i. e. a lack of inversion symmetry that is similar to the B20 structures) which,\naccording to Ref. [70] may lead to a DMI that amounts up to about 10% of the isotropic\nexchange; and the dipolar interaction strength for thin ferrimagnetic films with perpendicu-\nlar magnetic anisotropy varies as a function of temperature(see section S1). 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Lett. 36, 1061\n(1976).\n38"    },    {        "title": "2208.07024v1.Two_Dimensional_Semiconducting_Metal_Organic_Frameworks_with_Auxetic_Effect__Room_Temperature_Ferrimagnetism__Chiral_Ferroelectricity__Bipolar_Spin_Polarization_and_Topological_Nodal_Lines_Points.pdf",        "content": "Two-Dimensional  Semiconduct ing Metal Organic Frameworks  with \nAuxetic Effect, Room  Temperature Ferrimagnetism , Chiral Ferroelec-\ntricity, Bipolar Spin Polarization  and Topological Nodal Lines/Point s \nXiang yang  Li,†,∆ Qing -Bo Liu,&,∆ Yongsen  Tang ,||,∆ Wei Li,‡ Ning Ding ,¶ Zhao  Liu,† Hua-Hua Fu ,& \nShuai Dong ,¶ Xingxing Li,*,†,§,| and Jinlong Yang*,†,§,| \n†Hefei National Research Center for Physical Sciences at the Microscale , University of Science and Technology of China, \nHefei, Anhui 230026, China  \n|Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China  \n§Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, China  \n‡Department of Physics, University  of Science and Technology of China, Hefei, Anhui 230026, China  \n&School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, \nWuhan 430074, China  \n||Laboratory of Solid -State Microstructures and Innovative Center of Advanced Microstructures, Nanjing University, Nanjing, \n210093, China  \n¶School of Physics, Southeast University, Nanjing 211189, China  \n \nABSTRACT:  Two-dimensional  (2D) semiconductors integrated with two or more functions are the  cornerstone  for constructi ng \nmultifunctional nanodevices, but remain  largely limited.  Here, by tuning the spin state of organic linkers and the sym-\nmetry/topology of crystal lattice,  we predict  a class of  unprecedented multifunctional  semiconductor s in 2D Cr(II)  five-membered  \nheterocyclic metal organic framework s that simultaneously possess auxetic effect , room temperature ferrimagnetism,  chiral ferroe-\nlectricity, electric ally reversible spin polarization and topological nodal lines/point s. Taking 2D Cr( TDZ)2 (TDZ =1.2.5 -thiadiazole) \nas an exemplification , the auxetic effect is produced  by the anti-tetra-chiral lattice  structure . The high temperature ferrimagnetism \noriginates from the strong  d-p direct magnetic exchange interaction between Cr cations and TDZ  doublet radical  anions . Meanwhile , \nthe clockwise -counterclockwise alignment of TDZ ’ dipoles  results in unique 2D chiral ferroelectricity with atomic -scale vortex -\nantivortex states. 2D Cr( TDZ )2 is an intrinsic bipolar magnetic semiconductor  where half-metallic conduction with switchable  spin-\npolarization direction can be induced by applying a gate voltage . Besides , the symmetry of the little group C4 of lattice structure \nendows 2D Cr(TDZ )2 with topological nodal lines and a quadratic nodal point in the Brillouin zone near the Fermi level . \nINTRODUCTION  \nTwo-dimensional (2D) multifunctional materials with \nunique atomic -scale configurations and exotic electronic prop-\nerties have aroused great interests in recent decades.1, 2 How-\never, up to now, only  limited number s of such materials  have \nbeen reported  experimentally or theoretically , such as NiI 2,3 \nReWCl 6,4 h-Ti2(O2)3,5 and AlB 6.6 In addition, m ost are concen-\ntrated in traditional inorganic compounds  with only two or \nthree  function s (Table S1) . Developing  2D multifunctiona l \nmaterials  with more functions and exotic properties remains a \npending task . \nConsidering the structur al rigidity and limited tunability of \ninorganic compounds , we turn our attention to  organometallic \nmaterials with structural varia bility and rich functionalization \npossibilities .7, 8 Organometallic frameworks  are hybrid porous \nmaterials composed of abundant metal node s and inexpensive \norganic linkers .8 By tuning  metal node s or organic linkers  or \nthe connectivity between them ,9-15 they can possess functional \nproperties with potential applications in fields of emergent \nelectromechanical, magnetoelectronic, magnetic sensing, and \ntopological quantum  technologies.  For instance,  by using the \ndicyanoquinonediimine as a rotatory unit, the Cr(dicyanoquinonediimine) 2 sheet  has been predicted to be an \nauxetic magnet .9 By changing the spin state of organic linker s \nfrom s inglet to doublet and  introduc ing a strong  d-p direct \nferrimagnetic exchange interaction , high Curie temperature \n(TC) magnetic semiconductors  Cr(pentalene) 2 (TC=560  K),10 \nCr(diketopyrrolopyrrole) 2 (TC=316  K)11 and Cr(pyrazine) 2 \n(TC=342  K)12, 13 have been  designed theoretically . Via dis-\ntorting out-of-plane K+ counterions , ferroelectric  2D magnetic \nK3M2[PcMO 8] (M = Cr -Co) sheets have been forecast ed.14 \nThrough forming a Kagome lattice on a superconduc ting sub-\nstrate, t he experimentally synthesized 2D \nCu2(dicyanoanthracene) 3 sheet has been  calculated  and found \nto possess topological Dirac  cone s coupled with substrate’s \nsuperconductivity .16  \nAmon g numerous organometallic  frameworks , transition \nmetal Cr atom as a common metal node has been widely \nused,9-14 and its related planar tetracoordinate  molecules or \ncrystals have been extensively synthesized.17-19 Particularly, \nPerlepe et al.  have prepared a layered \nLi0.7[Cr(pyrazine) 2]Cl 0.7· 0.25(THF) (THF=tetrahydrofuran) \ncrystal  with room temperature ferrimagnetism , in which  each \nCr(II) is coordinated to four pyrazine organic linkers within  the layer s forming a planar tetracoordinate d sheet .19 If the \ninversion symmetric  pyrazine rings are replaced by inversion \nsymmetry -breaking  organic linkers  to increase the tunable \ndegree of freedom of the crystal structure, the functional prop-\nerties can be further enriched.   \nIn this work , by employing the Cr(II) as nodes and  inver-\nsion symmetry -breaking  five-membered aromatic heterocycles  \n(1.2.5 -thiadiazole, 1,2,5 -oxadiazole, 1,2,5 -selenadiazole)  as \norganic linkers , a class of  unprecedented 2D semiconductor s \nwith up to five important functio ns, i.e. auxetic effect , room -\ntemperature ferrimagnetism, chiral ferroelectricity, electrical  \nfield controlled  spin polarization, and topological nodal \nlines/point s are predicted  in metal organic frameworks with a \nplanar  tetracoordinate  structure . As exemplified by  Cr(TDZ )2 \n(TDZ =1.2.5 -thiadiazole) sheet , due to  the anti-tetra-chiral  \nsquare  lattice , auxetic effect  emerges along the diagonal direc-\ntion with a negative Poisson's ratio (NPR) of about -0.12. At \nthe same time, t he strong d-p direct magnetic e xchange inter-\naction between Cr cations and TDZ  doublet radicals enables a \nroom -temperature ferrimagnetism with TC = 378 K. Moreover, \n2D chiral ferroelectricity with atomic -scale vortex -antivortex \nstates is discovered as a result of the coexistence of clockwise -\ncounterclockwise dipoles , which has previously been  shown to \nexist  only in extremely rare cases and complex heterojunc-\ntions .20, 21  The electronic band structure  indicates 2D  \nCr(TDZ )2 not only belongs to  a special class of bipolar mag-\nnetic semiconductor s (BMS s)22, 23 with the carrier s’ spin orien-\ntation readily reversible by electrical gating , but also be a top-\nological material with square nodal lines and a quadratic nodal \npoint protected by  C4 crystal symmetry in the first Brillouin \nzone near the Fermi level.  For practical applications, these \nmultifunctional materials provide an excellent platform to \nstudy the proximity effect between different properties . More-\nover, by combining different function s, some high -\nperformance spintronic devices can be designed, such as ultra -\nhigh-density data storage  device.   \nCOMPUTATIONAL METHODS  Density functional theory (DFT) calculations are performed \nby using the projector -augmented wave method and the \nPerdew -Burke -Ernzerhof (PBE) functional as implemented in \nVienna  ab initio  Simulation Package (VASP).24, 25 The ap-\nproach of Grimme (DFT -D3) with Becke -Jonson damping is \nadopted for the van der Waals (vdW) interactions.26 To treat \nthe partially fill ed 3d orbitals of transition metal atoms, the \nstrongly correlated correction is considered with PBE+U \nmethod.27 The values of effective exchange interaction param-\neter ( J) and onsite Coulomb interaction parameter ( U) are re-\nspectively set as 1.0 and 3.0 eV, which are the same as previ-\nously calculated Cr(pyz) 2 sheet .12, 13 The energy cutoff for \nplane -wave basis set is 520 eV. For the first Brillouin zone \nintegration, the Monkhorst -Pack k-point mesh is used with a \ngrid spacing less than 0.02 Å−1. The energy and force criteria \nfor convergence are set to 1 ×10-6 eV and 0.01  eV/Å, respec-\ntively. A vacuum region of about 15 Å is chosen to avoid mir-\nror interactions between periodic layers. The phonon spectrum \nis simulated by using the finite displacement method as im-\nplemented in Phonopy package interfaced with VASP.28 A \n2×2×1 supercell with a Monkhorst -Pack k-point mesh of \n2×2×1 is adop ted. The thermal stability is assessed according \nto the ab initio  molecular dynamic (AIMD) simulation at 300  \nK by using a 3 ×3×1 supercell. The climbing image nudged \nelastic band (CI -NEB) method is adopted to investigate the \nstructure of transition state an d the energy barrier between \ndifferent ferroelectric phases.29 Considering  the complicated \ntransition between different ferroelectric phases,  a series of \npossible structures on the intermediate path  are conjecture d to \nfind th e structure with the highest energy, and their phonon \nspectra are further analyze d to verify that they are  indeed sad-\ndle point s (Figure S1) . The out of plane electric polarization is \nevaluated by using the dipole correction scheme.30 To accu-\nrately calculate the electronic structure, the screened hybrid \nHSE06 functional is applied,31 which includes the accurate \nFock exchange and usually performs much better than the \nPBE and PBE+U methods.32-34 The edge states have been per-\nformed using the open -source code  WANNIERTOOLS35 \nFigure 1. (a) Geometrical structures of the three phases of Cr( TDZ )2 (TDZ  = 1.2.5 -thiadiazole). (b) Gibbs free energies per unit cell of the \nthree  phases of Cr( TDZ )2 as a function of temperature. The dotted arrow marks the temperature of the phase transition point of P-42m to \nP21/c. (c) Phonon spectrum of the most stable P4bm phase . (d) Poisson ’s ratio of the P4bm structure over a ±5% strain range along the \ndiagonal di rection.  \n based on the Wannier tight -binding model constructed using \nthe WANNIE R90 code.36 The irreps of the electr onic bands \nare computed by the program IR2TB on the electr onic Hamil-\ntonian of the tight -binding model.37  \nRESULTS AND DISCUSSION  \nDue to  the lack of inversion symmetry , each five-membered \naromatic heterocyclic structure has two different orientations \nwith respect to  the lattice plane, leading to the diversification \nof crystal structures  formed  with Cr  atom s. Taking TDZ  or-\nganic ring as an example, Figure  1(a) shows three low energy \nstructures of the Cr(TDZ )2 sheet with point groups of P4bm, \nP21/c, and P-42m symmetry, respectively.  In these structures , \nfour TDZ  organic rings form  approximately square planar \ncoordination with the Cr atoms, and each TDZ  unit is connect-\ned by two adjacent Cr atoms. The four  organic rings are ar-\nranged clockwise or counterclockwise around the Cr atoms , \nwith the ring planes presenting  an inclination angle of about \n43° with respect to the ab lattice plane. The unit cell parame-\nters are a=b=9.14  Å for the P4bm phase, a (b)=9.20  (8.66)  Å \nfor the P21/c phase, and a=b=9.24  Å for the P-42m phase. For \nthe P4bm structure, the S atoms in TDZ  rings are all on one \nside of the ab plane. The panel on the left side of Figure  1(a) \nshows one possible structure in which all S atoms are located \non the upper side of the lattice plane. Such spatial inversion \nsymmetry -breaki ng gives rise to the  electric polarization and \nferroelectricity in the  structure (to be elaborated later). In con-\ntrast, when the S atoms in two ortho or para TDZ  rings are on \nthe other side of lattice plane  [see the rings enclosed by dotted \ncircles in the middle and right panels of Figure  1(a)], the cor-\nresponding P21/c and P-42m structures are formed and tend to \nbe antiferroelectric.   \nFirst-principles calculations identify that the Gibbs free en-\nergy of P4bm crystal at 0 K is 0.26 and 0.3 0 eV per unit cell \nlower than those of P21/c and P-42m crystals, respectively. As \nthe temperature increases, the P4bm structure remains to pos-\nsess the lowest energy [Figure  1(b)], thus it is the ground  state \nconfiguration. For the other two structures , a phase  transition \nfrom P21/c to P-42m is observed at around 1040 K. In the fol-\nlowing studies, we mainly focus  on the most stable  P4bm \nphase.   \nAs shown in Figure  1(c), no obvious imaginary frequency is \nobserved from the calculated phonon spectrum, indicating that \nCr(TDZ )2 sheet  is dynamically stable. Due to the  rather large \nlattice constant, the phonon band s are dispersionless. The ex-\nistence of a la rge number of soft phonon modes illustrates the \nflexibility of Cr( TDZ )2 sheet .10 For example, the maximum Young ’s modulus of P4bm structure is only  39 GPa (Figure \nS2), which is much smaller than that of MoS 2 (170-370 \nGPa).38 Beside s the dynamic stability , the thermal stability is \nfurther examined  by performing AIMD simulation at 300 K \n(Figure S3) . It is found that d uring the simulation, the total \nenergy always fluctuates near its equilibrium value without a \nsudden drop , and t he lattice structure can maintain well with-\nout any reconstruction after 9 ps, confirming that  the structure \nis thermally stable .  \n Interestingly,  the structure of Cr(TDZ )2 sheet belongs to the \nso called anti-tetra-chiral lattice  capable of  exhibiting \nauxeticity ,39 therefore  a distinct auxetic effect  and negative \nPoisson's ratio (NPR)  along the diagon al direction  is expected  \n[see Figure  1(d)]. Here, Poisson ’s ratio is defined as 𝜕𝜀𝑡/𝜕𝜀𝑎, \nwhere  𝜀𝑡 and 𝜀𝑎 are strains in the transverse and corresponding \nlongitudinal directions, respectively. The maximum value of \nNPR can reach -0.12 in the 5% strain range. This value is \nsmaller than that of 2D Cr( dicyanoquinonediimine) 2 (-0.85),9 \nbut comparable to most reported 2D inorganic auxetic \nmaterials, such as Ag 2S (-0.12),40 Be5C2 (-0.16),41 and SnSe ( -\n0.17) .42 The auxetic  property  endows 2D Cr( TDZ )2 with \npotential applications  in nano -mechanics, defense and \naerospace aspects.    \nIn Cr(TDZ )2 sheet, each Cr atom donates formally two elec-\ntrons to neighboring TDZ  rings, forming a Cr(II) cation and \nTDZ  doublet radical anions. The total magnetic moment of \neach chemical formula is 2 μB, in which the magnetic moment \nof the Cr (II) cation is 3.4 μB and that of each TDZ  radicals  is -\n0.7 μB. Therefore, the spins of Cr and TDZ  can be approximat-\ned as 2 and 1/2, respectively. To determine  the magnetic \nground state of Cr( TDZ )2 sheet , five different magnetic states, \nincluding one ferromagnetic (FM) state, one antiferromagnetic \nstate, and three ferrimagnetic (FiM) states (Figure S4) , are \ninvestigated. The results show  that the FiM1 state is the \nground state, where the spins on Cr (II) cations  are all antipar-\nallelly aligned with the spins on TDZ  radical s. Figure  2(a) \ngives  the spin density distribution of the FiM1 state. Obvious-\nly, the spin density on the TDZ  radicals  is distributed over the \npz orbital s of all nonmetallic C, N and S atoms. Due to the \nstrong direct exchange interaction  between the π-conju gated \norbitals of TDZ  rings and  d orbitals of Cr , the FiM1 state  is \nsignificantly more stable than the FM state by as large  as 0.85 \neV per chemical formula.  \nFor practical spintronic applications, it is crucial to keep  \nthe magnetic ordering  of Cr( TDZ )2 sheet  above room  tempera-\nture. To confirm this , we perform Monte Carlo simulations \nFigure 2. (a) Spin density distribution of Cr( TDZ )2 sheet with a P4bm symmetry in the ground ferrimagnetic state. Red and blue indicate \nup and down spins, respectively. (b) The nearest neighbor and next -nearest neighbor spin exchange paths for Cr( TDZ )2 sheet . The ex-\nchange -coupling parameters Jk (k = 1~4) are also marked. J1 represents the interaction between the Cr atom and nearest neighbor TDZ . J2 \nmeans the interaction between the nearest two Cr atoms. J3 and J4 serve as the interaction between the nearest and next -nearest two TDZ . \n(c) Magnetic moment ( M) per unit cell (black) and specific heat Cv (red) as a function of temperature by using a Monte Carlo simulation \nbased on the classic Heisenberg model. The magne tic exchange parameters used here are calculated with the HSE06 functional.  \n based on the classic Heisenberg model Hamiltonian,43, 44  \n𝐻=−∑∑∑𝐽𝑘\n𝑗𝑖>𝑗𝑘×𝑆𝑖⋅𝑆𝑗+ ∑𝐷𝑖𝑆𝑖𝑧2\n𝑖               (1) \nwhere Jk are four different exchange -coupling parameters \npresented in Figure  2(b), and Si is the spin of Cr or TDZ . Di \nare magnetic anisotropy parameters with a value of 123.5  μeV \nfor Cr and 0 μeV for TDZ , as derived from the calculated \nmagnetic anisotropy energy of 494.1 μeV per formula . The \nvalues of Jk are deduced from the energy differences between \nthe above  mentioned  five different magnetic states (summa-\nrized in Table S 2). Figure  2(c) shows the temperature -\ndependent spin magnetic moment ( M) per chemical formula, \nin which the M gradually decreases from 2 μB to 0 with the \nincrease of temperature. The specific heat 𝐶𝑣=(〈𝐸2〉−\n〈𝐸〉2)/𝑇2   is obtained after the system reaching equilibrium at \na given temperature, and the peak position manifests that the \nferrimagnetic -paramagnetic transition occurs at the Curie tem-\nperature  TC of 378 K, significantly higher than room tempera-\nture.  \nMore intriguingly, due to the inversion symmetry -breaking  \nfeature of the five membered heterocycles , the negative charge \ncenter of the TDZ  ring does not coincide with the positive one, \nthus an intrinsic  electric polarization would be  present . There-\nfore, each TDZ  ring can be regarded as an electric dipole , \nwhich is inclined from the c -axis at an angle of 47° and re-\nversible in dipole direction by rotating the TDZ  rings . The \nsuperimpose d c-axis component of all TDZ  dipoles induces \nthe ferroelectric  polarization ( ±pc) of Cr( TDZ )2 sheet  along the \nc-direction. In the ab plane, four non -collinear electric dipoles \nconnecting one of the Cr atoms form an atomic -scale vortex \nstate C+, and another four electric dipoles linking the adjacent \nCr atoms form an antivortex state C-[Figure  3(a)]. Such  two \nstates assemble a 2D lattice  with chiral vortex -antivortex polar \nstates  down to  the monolayer  scale , which is distinct  from the \nnanometer -scale chiral vortex -antivortex arrays in a complex heterojunction structure with alternating lead titanate and \nstrontium tita nate layers .21  \nBased on the above analysis, the non -collinear ferroelectric \nstates of Cr( TDZ )2 sheet  can be described by two ferroelectric \norder parameters Q1 = +C++C- in the ab plane and Q2 = +pc+pc \nalong the c-direction, where Q1 and Q2 obviously have two \ndegenerate mo des ±Q1 and ±Q2, respectively. As shown in \nFigure  3(a), those four order parameters can describe four \ndegenerate ferroelectric vortex states: S1 (+Q1, +Q2), S2 (+Q1, -\nQ2), S3 (-Q1, +Q2), S4 (-Q1, -Q2). It is worth mentioning that the \nin-depth study of such ferroelectric materials with chiral vor-\ntex properties at the atomic scale is of great value for the un-\nderstanding of non -collinear ferroelectric and chiral ferroelec-\ntric physics.   \nDFT calculations indicate  that the polarization values of the \nabove four vortex states are all 1.65 μC/cm2, where the polari-\nzation directions of S1 and S2 are along the positive direction \nof c-axis, while  those of S3 and S4 are along the negative direc-\ntion of c-axis. The polarization values  are comparable to  those \nof reported  Sc2CO 2 (1.6 μC/cm2),45 ReWCl 6 (3.22 μC/cm2),4 \nand hexagonal YMnO 3 (5 μC/cm2).46 \nFigure  3(b) shows the transition path from S1 to S2. The four \nTDZ  rings in the S1 state first simultaneously rotate 47° coun-\nterclockwise to reach the transition state and then further ro-\ntate 47° counterclockwise to evolve to  the S2 state, where the \nenergy barrier is 0.37 eV per TDZ . At the transition state, all \nthe TDZ  rings are perpendicular to the ab plane, where the \npolarization reaches the maximum value of 2.04 μC/cm2 and \nthe vortex state  feature disappears. The transition from S1 to S4 \nis complicated, and the corresponding energy barrier is also \n0.37 eV per TDZ . Figure  3(c) illustrates one possible path. \nFirst the four TDZ  rings in the S1 state rotate 47° counter-\nclockwise, next the two TDZ  rings at the para position rotate \n180° clockwise, then the other two TDZ  rings rotate 180° \nclockwise, and finally the four TDZ  rings simultaneously ro-\ntate 47° clockwise to get to the S4 state. Specifically, a series \nFigure 3. (a) Four possible chiral vortex -antivortex states Si (i = 1~4) of Cr( TDZ )2 sheet with a P4bm symmetry. C+ represents a vortex \nstate comprising four clockwise dipoles around the central Cr atoms and C- means an anti -vortex state comprising four counterclockwise \ndipoles around the central Cr atoms. + pc (-pc) serves as a state of the total electric polarization that is outward (inward) perpendicular to the \nab lattice plane. (b) A possible path and energy barrier for the transition from S1 to S2. (c) A possible path and energy barrier for the transi-\ntion from S1 to S4.  of possible intermediate states exist throughout the transition, \nbut none of them is stable  after examining their phonon spec-\ntra. For example, an intermediate state without polarization \n[see the middle panel in Figure 3(c)] possesses  an energy  of \n0.27 eV highe r than the ground structure , and its phonon spec-\ntrum presents  a certain imaginary frequency; see Figure S5 . In \naddition, Figure S6  displays another possible transition path, \nwhere the energ y barrier remains to be 0.37 eV . Thus , we can \ninfer that the energy barriers from Si to Sj are all around 0.37 \neV, the value of which is higher than th at of WO 2Cl2 (0.22  \neV),47 but lower than those of K 3Fe2[PcFeO 8] (0.38  eV)14 and \nCrI 3 (0.65  eV).48 \nTo reveal the electronic properties of Cr( TDZ )2 sheet , the \nband structures and density of states are calculated by using \nthe HSE06 functional , as shown in Figure  4(a). Obviously, it \nis a direct semiconductor with a band gap of 1.60 eV. The \nvalence and conduction bands are 100% spin polarized in op-\nposite spin channels, suggesting the Cr(TDZ )2 sheet  belongs to \nan intrinsic bipolar magnetic semiconductor ( BMS ),22, 23 where \nelectrical gating can generate half -metallic conduction with \ncontrollable spin polarization directions. Moreover, t he energy \nbands near the Fermi level not only possess doubly -degenerate \nnoda l lines on the X -M path protected by a 2D irreps Γ1Γ2 in \nlittle group C4, but also exist a quadratic nodal point (QNP) \nwith a zero Chern number protected by a 2D irreps Γ5 in point \ngroup C4v at the Γ point (Note S1) . The nodal lines are distrib-\nuted at the boundary of 2D Brillouin zone, showing a square -\nshaped feature; see the inset of Figure  4(a). By investigating \nthe distributions of projected density of states, one can derive \nthat the states of square nodal lines (SNLs) and QNP are dom-\ninated by the p orbitals of N, C, and S atoms. For clarity, Fig-\nure 4(b) demonstrates the 3D band characteristics of an SNL \nand a QNP near the Fermi level .  \nTo further verify the topological properties, we calculate the \nedge states of Cr( TDZ )2 sheet along the (100) direction. As \ndisplayed in Figure  4(c), the QNP at the Γ̅ point forms a clear \nquadratic Dirac cone edge state, which proves that it is non -\ntrivial. Since the SNLs are projected into the one -dimensional \nBrillouin zone, the corresponding edge states cannot be ob-\nserved. Besides, after considering the spin -orbit coupling  \n(SOC)  effect, the QNP of Cr( TDZ )2 opens a topological gap of \n7 meV (Figure S7 ), which  is comparable to those of \nMn(C 6H5)3 (9.5 meV),49 Mn 2C6S12 (7~15 meV),50 and Cr 2Se3 \n(6.7 meV) .51 Based on the example of Cr( TDZ )2 sheet, we can easily ex-\npand int o a range of  multifunctional organometallic semicon-\nductors  by substituting the TDZ  organic linkers  with other \nfive-membered heterocycles , such as Cr( ODZ )2 (ODZ =1,2,5 -\noxadiazole) and Cr( SDZ )2 (SDZ =1,2,5 -selenadiazole) . For the \nCr(ODZ )2 sheet, the ground state is the antiferroelectric P21/c \nstate, which is 0.26 eV lower in energy  than the ferroelectric \nP4bm state.  In contrast, the ground state of Cr( SDZ )2 sheet  is \nferroelectric  with an electric polarization strength  of about \n1.01 μC/cm2 (Table S 3), slightly weaker than that of Cr( TDZ )2 \nsheet . The transition energy barrier s between different ferroe-\nlectric phases remain  around  0.37 eV.  The auxetic effect of \nthese two extended sheets is superior to that of Cr(TDZ) 2 sheet , \nwhere t he maximum a bsolute value of NPR  reach es 0.17 and \n0.13 for Cr(ODZ) 2 and Cr(SDZ) 2 sheet s, respectively  (Figure \nS8). Both  TC are above room temperature with the highest \nbeing 410 K  (Figure S9) . Meanwhile,  the Cr( ODZ)2 and \nCr(SDZ) 2 sheets are also BMS s with oppositely spin -polarized \nvalence and conduction band edges  (Figure S10) . Besides,  \ncompared with Cr( TDZ )2, the QNP of Cr( SDZ )2 opens a large r \ntopological gap of 33 meV  from the HSE+SOC band structure, \nproviding a potential platform for studying the a nomalous \nquantum Hall effect.  \nFor the experimental fabrication  of these  2D organometallic \nframeworks , one possible route is to adopt  top-down technol-\nogies . Similar to the synthesized Li 0.7[Cr(pyz) 2]Cl 0.7· 0.25(THF) \n(THF=tetrahydrofuran) crystal,19 their bulk layered crystals are \nfirstly realized  by combining redox -active coordination chem-\nistry52 and postsynthetic reduction modification,19 and then the \ncorresponding sheets are achiev ed through the  mechanical \nexfoliation.  Another possible route is  to use bottom -up meth-\nodologies . Metal atoms and organic linker molecules are de-\nposited on to a metal surface by molecular beam evaporation or \nelectron beam evaporation to induce  their self -assembly to \nform 2D organometallic frameworks.  Such preparation strate-\ngies have been widely  used to synthesize  similar coordination  \nstructures , e.g., Mn-TCNQ 4 network,53 Ni-TPyP network ,54 \nTPA -Cs, BDA -Cs, and TDA -Cs networks.55  \nIn practical application s, integrating  so many functional \nproperties in to a single  sheet  can offer  at least two advantages.  \nThe first is to provid e an ideal platform to study different \nkinds of proximity effect.56 Specifically, as shown in Figure \n5(a), the proximity effects between FiM, ferroelectricity, chi-\nrality, BMS, and topology  can be investigated  by constructing \na bilayer homojunction.  Such a homojunction can effectively \navoid additional effects caused by lattice mismatch of the het-Figure 4. (a) Spin -polarized band structures and projected density of states for Cr( TDZ )2 sheet  with the HSE06 functional. Red and black \nlines represent spin -up and spin -down bands, respectively. The inset shows high -symmetry points in the first Brillouin zone. QNP and \nSNL stand for quadratic nodal point and square nodal line, respectively. (b) Thre e-dimensional energy band structures of an SNL and a \nQNP near the Fermi level. (c) Dirac -cone edge states for QNP.  \n erojunction.  The second is to improve  the performance of re-\nlated spintronic devices  through the synergy between multiple \nfunctions.  For instance, when the five functional properties of \nFiM, ferroelectricity, chirality, BMS, and topology are simul-\ntaneously applied  to a data storage device , the storage density \ncan be increased 16 (24) folds compared to a single -function \ndevice  since each function contain s two sw itchable states . The \nswitching between different storage states  can be realized un-\nder an external electric or magnetic  field. Figure 5(b) displays \na field effect transistor  with the Cr( TDZ )2 sheet  as a channel \nmaterial to illustrate the specific modulation method  in differ-\nent functions . The orientation of spin moment (± Ms) can be \nchanged by applying a magnetic field  𝐵⃑  perpendicular to the \nsheet plane. The transition between different electrical polari-\nzation (± P) states and chiral (C±) states can be achieved by \nassign ing a certain electric field.  Both the  direction of current \nspin polarization ( I↑/↓) and the transition from trivial semicon-\nductor (SC) to topological  half-metal including boundary \nstates can be modulated by using different  gate voltage s.  \nCONCLUSIONS  \nTo summarize, on the basis of first -principles calculations, \nwe report a class of unprecedented 2D multifunctional semi-\nconductors with several unique properties including auxetic \neffect, room temperature ferrimagnetism, chiral ferroelectricity, \nelectrically controllable spin polarization and topological nod-\nal lines/points. The simultaneous realization of these functions \nrelies on the combined tuning of the spin state of organic link-\ners and the symmetry/topology of lattice struct ure in metal \norganic frameworks constructed by Cr(II) and inversion sym-metry -breaking  five-membered aromatic heterocycles (1.2.5 -\nthiadiazole, 1,2,5 -oxadiazole, 1,2,5 -selenadiazole). These ma-\nterials not only serve as promising candidates for studying \ndifferent proximity effects and design ing multifunctional \nnanodevices, but also imply the u nique abilities of metal or-\nganic frameworks in obtaining electronic/magnetic properties \nthat are difficult to achieve in inorganic materials such as chi-\nral vortex -antivortex polar states in the monolayer limit . \nASSOCIATED CONTENT   \nSupporting Information . Additional material includes phonon \nspectr a of the transition state and an intermediate state , Young’s \nmodulus, AIMD simulation, and HSE+SOC band structure for \nCr(TDZ )2; spin density of different magnetic states; another pos-\nsible pathway for t ransition from S1 to S4; Poisson’s ratio, Curie \ntemperature, HSE and HSE+SOC band structur es for Cr(ODZ) 2 \nand Cr(SDZ )2; magnetic exchange parameters  and ground  state \nproperties of different sheets ; two-band k·p Hamiltonian for SNLs \nand QNP . This material is available free of charge via the Internet \nat http://pubs.acs.org.  \nAUTHOR INFORMATION  \nCorresponding Author  \n* lixx@ustc.edu.cn  \n* jlyang@ustc.edu.cn  \n∆ These authors contributed equally to this work.  \nNotes  \nThe authors declare no competing financial interest . \nACKNOWLEDGMENT   \nThis work is supported by Anhui Initiative in Quantum Infor-\nmation Tech nologies with Grant No. AHY090400, by the Youth \nInnovation Promotion Association CAS with Grant No. 2019441, \nby the Innovation Program for Quantum Science and Technology \nwith Grant No. 2021ZD0303306, by USTC Research Funds of the \nDouble First -Class Initiat ive with Grant No. YD2060002011, by \nthe National Natural Science Foundation of China with Grant No. \n12147113, and by project funded by the China Postdoctoral Sci-\nence Foundation with Grant No. 2021M691149.  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Nanotechnol. 2021 , 16, 856 -868. \n  \n 9 Table of Contents artwork  \n \n"    },    {        "title": "0910.0422v1.Effect_of_strain_on_the_stability_and_electronic_properties_of_ferrimagnetic_Fe___2_x__Ti__x_O__3__heterostructures_from_correlated_band_theory.pdf",        "content": "arXiv:0910.0422v1  [cond-mat.mtrl-sci]  2 Oct 2009Effect of strain on the stability and electronic properties o f ferrimagnetic\nFe2−xTixO3heterostructures from correlated band theory\nHasan Sadat Nabi and Rossitza Pentcheva∗\nDepartment of Earth and Environmental Sciences,\nUniversity of Munich, Theresienstr. 41, 80333 Munich, Germ any\n(Dated: July 31, 2021)\nBased on density functional theory (DFT) calculations incl uding an on-site Hubbard Uterm we\ninvestigate the effect of substrate-induced strain on the pr operties of ferrimagnetic Fe 2O3-FeTiO 3\nsolid solutions and heterostructures. While the charge com pensation mechanism through formation\nof a mixed Fe2+, Fe3+-contact layer is unaffected, strain can be used to tune the el ectronic prop-\nerties of the system, e.g.by changing the position of impurity levels in the band gap. S training\nhematite/ilmenite films at the lateral parameters of Al 2O3(0001), commonly used as a substrate,\nis found to be energetically unfavorable as compared to films on Fe 2O3(0001) or FeTiO 3(0001)-\nsubstrates.\nPACS numbers: 73.20.-r,73.20.Hb,75.70.Cn,71.28.+d\nI. INTRODUCTION\nInthefabricationofferromagneticsemiconductors\nfor spintronics applications a lot of research focuses\non the homogeneous doping of traditional or oxide\nsemiconductors with 3 dions1,2,3,4. However, the\ncoupling between magnetic impurities and charge\ncarriers is often too weak, leading to Curie temper-\natures (TC) way below room temperature (RT). On\nthe other hand, materials like Fe 2−xTixO3exhibit\nintrinsic semiconducting and ferrimagnetic proper-\nties, althoughtheendmembers α-Fe2O3andFeTiO 3\nare antiferromagnetic insulators with TN= 948 and\n56 K, respectively. Besides applications in spintron-\nics, this materialisalsodiscussedin paleomagnetism\nas a possible cause of anomalies in the Earth’s mag-\nnetic field, as well as for electronics devices ( e.g.\nvaristors) because it is a wide band gap semiconduc-\ntor that can be either n- orp-type depending on the\ndoping concentration5. A Curie temperature above\nRT and a reduction of resistivity was observed in\nsynthetic solid solutions with Ti concentrations up\nto 70%6,7. Moreover, TCwas found to increase upon\nannealing both in these samples and in thin epitax-\nial films8. This behavior can be attributed to cation\nordering phenomena related to a miscibility gap in\nthe rather complex phase diagram of the system9.\nThe origin of ferrimagnetic behavior remained\nunclear until recently. Both materials have a\ncorundum(-related) structure (see Fig. 1) with a\nstacking of 2Fe3+/3O2−in hematite (space group\nR¯3c)and2Fe2+/3O2−/2Ti4+/3O2−inilmenite( R¯3)\nalong the [0001]-direction. Thus at an interface or in\na solid solution (SS) charge is not compensated, if\nall ions preserved their bulk valence states. DFT\ncalculations considering correlation effects within\nLDA+U10showed that the charge mismatch is ac-\ncommodated by a mixed Fe3+, Fe2+contact layer atthe interface11, providing first theoretical evidence\nfor thelamellar magnetism hypothesis12. The Fe2+-\nions at the interface give rise to uncompensated mo-\nments and also to impurity states in the band gap.\nThe incorporation of Ti in hematite13(a= 5.04\n˚A,c= 13.75˚A) introduces a substantial strain: the\nvolume of the end member ilmenite14(a= 5.18˚A,\nc= 14.27˚A) is 9.7%largerthan the oneof hematite.\nIndeed, lens-shapeddarkcontrastsaroundnanoscale\nhematite lamellae in an ilmenite host, imaged by\ntransmission electron microscopy, indicate signifi-\nFIG. 1: (Color online) Crystal structure of the 60-atom\nunit cell of Fe 2−xTixO3forx= 0.33 with a layered (a)\nand more homogeneous arrangement of the Ti-cations\nwith Ti in the same (b) and different (c) spin-sublattices.\nOxygen, Fe and Ti are shown with light grey, red and\nblack spheres respectively. Pink circles mark the Fe2+-\npositions, while the rest of the iron are Fe3+. The local\nmagnetic moments at the cation sites and the total mag-\nnetization of the system in µBare given in the right side\nand bottom of each configuration, respectively.cant strain fields12.\nEpitaxial Fe 2−xTixO3films5,8,15,16,17,18are typ-\nically grown on an Al 2O3(0001)-substrate ( a=\n4.76˚A,c= 12.99˚A) which introduces a substantial\ncompressive strain of 5.8% and 8.8% compared to\nFe2O3and FeTiO 3and only rarely, a Cr 2O3-buffer\nlayer is used19to reduce the lattice mismatch.\nEpitaxial strain can have a strong impact on the\nfilm properties, e.g.by tuning the magnetic inter-\nactions in magnetoelastic composites20, enhancing\nferroelectricity21,22or even inducing orbital recon-\nstructions23. The goal of the present study is to\nexplore the effect of strain on the properties of\nFe2−xTixO3. In particular we address its influ-\nence on ( i) the energetic stability and compensation\nmechanism as well as on ( ii) the electronic, mag-\nnetic and structural properties of the system. DFT\ncalculations are performed on SS and layered config-\nurations with x= 0.17,0.33,0.50 and 0.66, strained\nlaterally at the lattice parameters of Al 2O3, Fe2O3,\nand FeTiO 3.\nII. CALCULATIONAL DETAILS\nWe use the all-electron full-potential linear aug-\nmented plane wave (FP-LAPW) method as im-\nplemented in the WIEN2K code24and the gener-\nalized gradient approximation (GGA)25. Within\nLDA+U10U= 8.0 eV and J= 1.0 eV is applied\nto the Fe and Ti 3 dstates. These values were found\nto reproducecorrectly the ground state of FeTiO 311.\nThe systems are simulated in a hexagonal unit cell\nwith 60 atoms (Fig. 1). Besides the layered configu-\nrations(cf. Fig.1a)morehomogeneousdistributions\nare generated by substituting 50% of Fe in a bilayer\nby Ti, as shown e.g.in Fig. 1b-c. For further details\non the calculation see (Ref.11).\nIII. RESULTS AND DISCUSSION\nThe optimized c/a-ratio and volume (Fig. 2a-b)\nshow a linear increase with xIlmin accordance with\nVegard’s law, similar to what was observed exper-\nimentally in synthetic hematite-ilmenite solid solu-\ntions6. Furthermore, for a given concentration both\nc/aandVare largely independent of the distribu-\ntion of Ti-impurities. The c/a-ratio of bulk FeTiO 3\n(2.76) is slightly largerthan the one for α-Fe2O3and\nAl2O3(2.73). Due to the small tensile/compressive\nstrain when using a FeTiO 3/aFe2O3thec/a-ratio of\nFe2−xTixO3is slightly reduced (-1.1 to -2.8 %)/in-\ncreased (3.1-5.2 %), respectively. In contrast, due to\nthe high compressive strain on an Al 2O3-substrate,\nc/aincreases strongly by 14.7-16.6 % which corre-/s50/s53/s48/s50/s55/s53/s51/s48/s48/s51/s50/s53\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s120/s86/s40 /s197/s51\n/s41\n/s32/s32\n/s86\n/s65/s108\n/s50/s79\n/s51/s61/s50/s53/s52/s46/s53/s50/s86\n/s70 /s101\n/s50/s79\n/s51/s61/s51/s48/s49/s46/s57/s50/s86\n/s70 /s101/s84/s105/s79\n/s51/s61/s51/s51/s49/s46/s48/s54/s99/s47/s97/s32\n/s98/s41/s50/s46/s53/s48/s50/s46/s55/s53/s51/s46/s48/s48/s51/s46/s50/s53\n/s32/s32\n/s32/s99/s47/s97\n/s70 /s101/s84/s105/s79\n/s51/s61/s50/s46/s55/s54\n/s99/s47/s97\n/s70 /s101\n/s50/s79\n/s51/s61/s99/s47/s97\n/s65/s108\n/s50/s79\n/s51/s61/s50/s46/s55/s51/s97/s41\n/s32/s83/s83/s32/s84/s105/s52/s43\n/s32/s64/s97\n/s70/s101/s50/s79/s51\n/s32/s83/s83/s32/s84/s105/s52/s43\n/s32/s64/s97\n/s70/s101/s84/s105/s79/s51\n/s32/s83/s83/s32/s84/s105/s52/s43\n/s32/s64/s97\n/s65 /s108/s50/s79/s51\n/s32/s76/s32/s32/s32/s84/s105/s51/s43\n/s32/s64/s97\n/s70/s101/s84/s105/s79/s51\n/s32/s69/s40/s101/s86/s47/s102/s46/s117/s41/s61/s69\n/s116/s45/s40/s49/s45 /s120 /s41/s42/s69\n/s70/s101\n/s50/s79\n/s51/s45/s120 /s42/s69\n/s70/s101/s84/s105/s79\n/s51/s32\n/s32/s76/s32/s84/s105/s52/s43\n/s32/s64/s97\n/s70/s101/s50/s79/s51\n/s32/s76/s32/s84/s105/s52/s43\n/s32/s64/s97\n/s70/s101/s84/s105/s79/s51\n/s32/s76/s32/s84/s105/s52/s43\n/s32/s64/s97\n/s65/s108/s50/s79/s51\n/s32/s76/s32/s84/s105/s51/s43\n/s32/s64/s97\n/s70/s101/s50/s79/s51/s32\n/s32/s76/s32/s84/s105/s51/s43\n/s32/s64/s97\n/s65/s108/s50/s79/s51/s99/s41\nFIG. 2: (Color online) (a) c/a-ratio (b) volume and (c)\nformation energy (eV/f.u) versus ilmenite concentration\nxIlmfor Fe 2−xTixO3strained at the Al 2O3(red/dark\ngrey), Fe 2O3(grey) and FeTiO 3(black) lateral lattice\nconstants. Circles (triangles) denote compensation in-\nvolving Ti4+(Ti3+). Open/filled symbols refer to solid\nsolutions (SS)/ layered configurations (L). Horizontal\nlines mark the bulk c/aratio and volume of the end\nmembers and Al 2O3. Red (dark grey) squares indicate\nexperimental data from Takada et al.27.\nsponds to crel= 14.89−15.15˚A. Nevertheless, the\nvolume does not completely relax: The volume of\nthe system strained at the Al 2O3-lateral lattice con-\nstant is 6.8 % (10.2 %) smaller than when strained\nat aFe2O3(aFeTiO 3). The volumes of Fe 2−xTixO3\nstrained at a FeTiO 3and aFe2O3lie between the ones\nfor the end members Fe 2O3and FeTiO 3.\nX-ray diffraction data for Fe 2−xTixO3films on\n2Al2O3(0001)26,27indicate significant lateral strain\nrelaxation: already in a 10 nm thick film arelaxesto\nthe bulk value of FeTiO 3with only a small change\ninc/a(see Fig. 2a). The c/avalues and volumes ob-\ntainedbyTakada etal.27areingoodagreementwith\nthe DFT values of the systems strained at a FeTiO 3.\nNext we turn to the influence of strain on the en-\nergetic stability. The formation energy with respect\nto the end members as a function of xTiis shown\nin Fig. 2c for the three different substrate lattice\nconstants. For each Ti-concentration we have con-\nsidered several different cation arrangements, e.g.\nforx= 0.33 these include an ordered arrangement\nwith an Fe layer sandwiched between two Ti layers\n(Fig. 1a) or solid solutions with Ti ions either in the\nsame (Fig. 1b) or different spin-sublattices (Fig. 1c).\nWefindthatcompensationthroughTi4+anddispro-\nportionation in Fe2+, Fe3+is more favorable over\nmechanisms involving Ti3+. Furthermore, the for-\nmation energy increases linearly with xIlm. These\nfeatures are independent of the substrate lattice pa-\nrameters. Systems strained laterally at a FeTiO 3are\nmorestablethanthe onesona Fe2O3. Incontrast,the\nformationenergyoffilmsstrainedata Al2O3increases\nby 0.7 eV as compared to films on a Fe2O3. This im-\nplies that the strong compressive strain is energet-\nically unfavorable and gives a possible explanation\nwhy a lateralstrainrelaxationoccursin Fe 2−xTixO3\nfilms26,27. While for systems strained on hematite\nand ilmenite substrates layered arrangements (full\nsymbols) are more favorable than homogeneous dis-\ntributions (open symbols), the trend is reversed for\nx= 0.33 andx= 0.66 on an Al 2O3(0001)-substrate.\nConcerning the electronic properties of the hemo-\nilmenite system, we have plotted in Fig. 3 the den-\nsity of states of a Ti-double layer in a hematite host\n(Fig. 1a), but similar behavior is observed for all\nstudied systems. Upon Ti4+substitution, an iron-\nion from the neighboring layer turns Fe2+, as ob-\nserved also for isolated impurities by Velev et al.28.\nThe Fe2+O6and the TiO 6-octahedron are corner-\n(and not face-)sharing. The so formed Fe2+-ions\nin the contact layer have an impurity state of a1g\nsymmetry ( dz2) that is pinned at the Fermi level\nfor systems strained at a Fe2O3and aFeTiO 3. Such\na mid-gap state was recently reported from x-ray\nvalence band photoemission29and optical measure-\nments17, although it was related to the low oxygen\npressure during deposition. The main feature re-\nlated to strain is the change in band width: While\nfor tensile strain at a FeTiO 3the bands are narrowed,\nfor compressive strain at a Al2O3they are strongly\nbroadened. This results in a reduction of the band\ngap (between the impurity state defining the Fermi\nlevel and the bottom of the conduction band) from\n1.90eV for a FeTiO 3and 1.79 eVfor a Fe2O3to 1.43eV\nFIG. 3: (Color online) Densityof states ofFe 1.67Ti0.33O3\ncontaining two Ti-layers in a hematite host (shown in\nFig. 1a): a) total; b) and c) projected of the 3 dstates of\nFe2+at the interface and between the two Ti layers. The\nDOS of the system strained at the lateral lattice param-\neters of Fe 2O3, Al2O3, and FeTiO 3is shown with a grey\nshadedarea, red (darkgrey), andblackline, respectively.\nfor aAl2O3. The corresponding values for x= 66%\nshow the same trend but are smaller: 1.64 eV for\naFeTiO 3and 1.46 eV for a Fe2O3to 0.78 eV for a Al2O3.\nThe local magnetic moments and total magneti-\nzation for the three systems with x= 0.33 is dis-\nplayed in Fig. 1. Strain has only a small impact\non the magnetic moments of Fe2+(∼3.5µB) and\nFe3+(∼4.1µB) respectively which are reduced by\nless than 0.05 µBat aAl2O3. The Fe2+-layer sand-\nwiched between two Ti-layers in Fig. 1a is only\nweakly coupled to the next Fe-layer (parallel and\nantiparallel orientation of the magnetic moments is\nnearly degenerate as in the ilmenite end member).\nTherefore, at temperatures above the N´ eel temper-\nature of ilmenite, such layers will not contribute to\nthe total magnetization. In contrast, Fe2+in the\ncontact layer shows a strong antiferromagnetic cou-\npling to the neighboring Fe-layer of the hematite\nhost. These defect interface moments are respon-\nsible for the ferrimagnetic behavior of the system\n(Mtot= 8.0µB). In solid solutions, Ti substitution\nin different spin-sublattices ( e.g.in adjacent layers\nas shown in Fig. 1c), resulting in a zero net magne-\n3tization, is less favorable compared to substitution\nin the same spin-sublattice (Fig. 1b), which maxi-\nmizes the total magnetization ( Mtot=−16.0µB).\nThis trend promotes ferrimagnetic behavior in the\nsystem.\nIV. CONCLUSIONS\nDensity functional theory calculations within\nGGA+Ushow that the charge compensation in\nhematite-ilmenite heterostructures and solid solu-\ntions takes place through a mixed Fe2+, Fe3+con-\ntact layer. This mechanism is robust with respect\nto substrate-induced strain. For Fe 2O3(0001) or\nFeTiO 3(0001) substrates layered arrangements are\nmore stable than solid solutions. However, the com-\npressive strain at a Al2O3is likely to cause a stronger\ncompetition and even reverse the trend for x= 0.33\nandx= 0.66. The growth of epitaxial films onan Al2O3-substrate is connected with a high energy\ncost. Therefore, in order to release strain such films\nmay roughen or buckle in the first layers as recently\nreported by Popova et al.26. In contrast, the growth\nonlatticematchedsubstratesorevensubstratesthat\nproduce a small tensile strain like FeTiO 3is energet-\nically favored. Our DFT results indicate that strain\ncan have a strong impact on the structural and elec-\ntronic properties in the hematite-ilmenite system:\ne.g.by tuning the band width or the position of im-\npurity levels in the band gap and thus changing the\nconcentration of spin-polarized carriers.\nV. 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Rev.\nB75, 104412 (2007).\n5"    },    {        "title": "2001.08037v1.The_dynamics_of_a_domain_wall_in_ferrimagnets_driven_by_spin_transfer_torque.pdf",        "content": "The dynamics of a domain wall in ferrimagnets driven by spin-transfer torque\nDong-Hyun Kim,1Duck-Ho Kim,2Kab-Jin Kim,3Kyoung-Woong\nMoon,4Seungmo Yang,4Kyung-Jin Lee,1, 5, 6and Se Kwon Kim7\n1Department of Semiconductor Systems Engineering,\nKorea University, Seoul 02841, Republic of Korea\n2Center for Spintronics, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea\n3Department of Physics, KAIST, Daejeon 34141, Republic of Korea\n4Quantum Technology Institute, Korea Research Institute of Standards and Science, Daejeon 34113, Republic of Korea\n5Department of Materials Science and Engineering,\nKorea University, Seoul 02841, Republic of Korea\n6KU-KIST Graduate School of Converging Science and Technology,\nKorea University, Seoul 02841, Republic of Korea\n7Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n(Dated: January 23, 2020)\nThe spin-transfer-torque-driven (STT-driven) dynamics of a domain wall in an easy-axis rare-\nearth transition-metal ferrimagnet is investigated theoretically and numerically in the vicinity of\nthe angular momentum compensation point TA, where the net spin density vanishes. The partic-\nular focus is given on the unusual interaction of the antiferromagnetic dynamics of a ferrimagnetic\ndomain wall and the adiabatic component of STT, which is absent in antiferromagnets but exists\nin the ferrimagnets due to the dominant coupling of conduction electrons to transition-metal spins.\nSpeci\fcally, we \frst show that the STT-induced domain-wall velocity changes its sign across TA\ndue to the sign change of the net spin density, giving rise to a phenomenon unique to ferrimagnets\nthat can be used to characterize TAelectrically. It is also shown that the frequency of the STT-\ninduced domain-wall precession exhibits its maximum at TAand it can approach the spin-wave\ngap at su\u000eciently high currents. Lastly, we report a numerical observation that, as the current\ndensity increases, the domain-wall velocity starts to deviate from the linear-response result, calling\nfor a more comprehensive theory for the domain-wall dynamics in ferrimagnets driven by a strong\ncurrent.\nI. INTRODUCTION\nSpintronics is the \feld, in which electrons' spin is uti-\nlized in addition to charge for the advancement of infor-\nmation processing technology beyond the conventional\ncharge-based electronics, and, therefore, the interaction\nbetween charge and spin has been one of the central top-\nics in the \feld. In particular, the e\u000bect of a charge cur-\nrent on the magnetic dynamics, which is described as the\nspin-transfer torque (STT), has been intensively studied\nfor metallic ferromagnets since the \frst theoretical pre-\ndictions in 1996 [1, 2]. One of the major practical utilities\nof STT in spintronics is to drive a magnetic domain wall,\na topological defect between two uniform domains, that\ncan be used to realize racetrack memory [3]. Also, fun-\ndamental research on STT-induced domain-wall motion\nhas been allowing us to strengthen our understanding of\nspin-charge interaction in ferromagnets [4{7].\nDeparting from ferromagnets consisting of parallel\nspins, there is another class of magnets called antiferro-\nmagnets, where neighboring spins are antiparallel. An-\ntiferromagnets have been attracting great attention in\nspintronics, particularly for the last decade, owing to\ntheir much faster dynamics than ferromagnets and the\nensuing promise for ultrafast spintronic devices [8]. How-\never, the research on antiferromagnetic dynamics has\nbeen impeded by di\u000eculties in controlling and detecting\nthem due to, e.g., the absence of magnetization. For this\nreason, the previous research on STT in antiferromagnetshas been mostly theoretical [9{14].\nThere have been recent developments in understand-\ning and utilizing antiferromagnetic dynamics enabled by\nan emerging class of magnets called ferrimagnets, which\nconsist of two or more inequivalent magnetic sublattices\ncoupled antiferromagnetically [15]. These ferrimagnets,\nwhich are typi\fed by rare-earth transition-metal (RE-\nTM) ferrimagnetic alloys such as GdCo or GdFeCo, can\nexhibit antiferromagnetic dynamics owing to the anti-\nferromagnetic coupling of constituent spins, and, at the\nsame time, can be controlled and detected easily due to\nsmall, but \fnite magnetization caused by imperfect can-\ncellation of neighboring magnetic moments. This unique\ncombination of antiferromagnetic dynamics and \fnite\nmagnetization has recently allowed for achieving fast\ndomain-wall motion [16{19] and magnetization switch-\ning [20{22] in various ferrimagnets. In particular, Okuno\net al. [23] recently reported an experimental study of\nSTT in ferrimagnets through current-assisted \feld-driven\ndomain-wall motion, introducing ferrimagnets as a useful\nplatform to investigate STT in magnets with antiferro-\nmagnetic coupling [24].\nIn this work, we theoretically and numerically study\nthe domain-wall dynamics in RE-TM ferrimagnets driven\nby STT, which has not been explored yet. One of the\nunique features of ferrimagnets that are absent in ferro-\nmagnets and antiferromagnets is that their spin density\ncan be continuously tuned across zero by changing tem-\nperature or chemical composition. The temperature atarXiv:2001.08037v1  [cond-mat.mes-hall]  22 Jan 20202\n(a)(b)e\u0000\n<latexit sha1_base64=\"NcEf+Yf5aLrXREBD+N9gj13U0s4=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgxbAbBQUvAS8eI5oHJGuYnXSSIbOzy8ysEJZ8ghcPinj1i7z5N06SPWhiQUNR1U13VxALro3rfju5ldW19Y38ZmFre2d3r7h/0NBRohjWWSQi1QqoRsEl1g03AluxQhoGApvB6GbqN59QaR7JBzOO0Q/pQPI+Z9RY6R4fz7rFklt2ZyDLxMtICTLUusWvTi9iSYjSMEG1bntubPyUKsOZwEmhk2iMKRvRAbYtlTRE7aezUyfkxCo90o+ULWnITP09kdJQ63EY2M6QmqFe9Kbif147Mf0rP+UyTgxKNl/UTwQxEZn+TXpcITNibAllittbCRtSRZmx6RRsCN7iy8ukUSl75+XK3UWpep3FkYcjOIZT8OASqnALNagDgwE8wyu8OcJ5cd6dj3lrzslmDuEPnM8f5k2NhQ==</latexit>Domain-wall motionDomain-wall motione\u0000\n<latexit sha1_base64=\"NcEf+Yf5aLrXREBD+N9gj13U0s4=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgxbAbBQUvAS8eI5oHJGuYnXSSIbOzy8ysEJZ8ghcPinj1i7z5N06SPWhiQUNR1U13VxALro3rfju5ldW19Y38ZmFre2d3r7h/0NBRohjWWSQi1QqoRsEl1g03AluxQhoGApvB6GbqN59QaR7JBzOO0Q/pQPI+Z9RY6R4fz7rFklt2ZyDLxMtICTLUusWvTi9iSYjSMEG1bntubPyUKsOZwEmhk2iMKRvRAbYtlTRE7aezUyfkxCo90o+ULWnITP09kdJQ63EY2M6QmqFe9Kbif147Mf0rP+UyTgxKNl/UTwQxEZn+TXpcITNibAllittbCRtSRZmx6RRsCN7iy8ukUSl75+XK3UWpep3FkYcjOIZT8OASqnALNagDgwE8wyu8OcJ5cd6dj3lrzslmDuEPnM8f5k2NhQ==</latexit>\nFIG. 1. Schematic illustration of a domain-wall motion driven\nby the adiabatic STT in a RE-TM ferrimagnet. The red and\nthe blue arrows in the gray box represent spins of TM and RE\nelements, respectively. When a conduction electron (denoted\nbye\u0000) traverses the domain wall from the left to the right,\nits spin denoted by the red arrow follows the TM spin direc-\ntion adiabatically. After passing through the domain wall,\nthe change of electron's spin is transferred to the domain wall\nvia the adiabatic STT. (a) When TM spins are larger than\nRE spins (corresponding to T > T A), the net spin direction\nis given by the TM spin direction. The transfer of up-spin\nfrom conduction electrons to the spin texture expands the\nleft domain with net-spin up, pushing the domain wall to the\nright. (b) When RE spins are larger than TM spins (corre-\nsponding to T < T A), the net spin direction is given by the\nRE spin direction. The transfer of up-spin from conduction\nelectrons drive the domain wall to the left by expanding the\nright domain with net-spin up.\nwhich the net spin density vanishes is called the angu-\nlar momentum compensation point TA[25, 26], and it\no\u000bers one of the situations where the advantage of fer-\nrimagnets is most prominent: their dynamics is purely\nantiferromagnetic due to the vanishing spin density and\nthus is shown to be the fastest [16{18]. Therefore, in\nour study on STT-driven domain-wall dynamics, we will\nfocus on ferrimagnets in the vicinity of the angular mo-\nmentum compensation point TA.\nSTT of ferrimagnets that describes the e\u000bect of a cur-\nrent on a spatially varying spin texture is similar to STT\nof ferromagnets due to the dominant coupling of conduc-\ntion electrons' spin to one of multiple sublattices of ferri-\nmagnets, as have been invoked in Ref. [27], where the\nSTT-driven dynamics are studied for two-dimensional\nspin textures called skyrmions in ferrimagnets. It con-\nsists of the reactive and the dissipative components,\nwhich are also referred to as the adiabatic and the nonadi-\nabatic STT. The adiabatic STT, which is even under time\nreversal, describes the angular momentum transfer from\nconduction electrons to the background spin texture such\nas a domain wall when electrons' spin follows the local\nspin texture adiabatically. The nonadiabatic STT cap-\ntures the angular momentum transfer via the other pro-\ncesses deviating from the aforementioned adiabatic pro-\ncess, e.g., mistracking between conduction electrons' spin\nand the spin texture. While both terms exist in ferro-\nmagnets, antiferromagnets are lack of the adiabatic STT\nsince electrons' spin cannot follow atomically-changing\nstaggered spins adiabatically [11, 28]. STT of RE-TM\nferrimagnets possesses both terms akin to ferromagnets\nsince conduction electrons are known to interact mostlywith TM magnetic moments and thus are spin-polarized\nfollowing TM's magnetization [29, 30]. See Fig. 1 for the\nillustration. Therefore, in the vicinity of angular momen-\ntum compensation point, where the nature of dynamics is\nantiferromagnetic but STT possesses the adiabatic term,\nferrimagnets are expected to exhibit a unique phenomena\nthat can occur neither in ferromagnets nor in antiferro-\nmagnets, which we reveal theoretically and numerically\nin this work through the domain-wall dynamics.\nThis paper is organized as follows. In Sec. II, we study\nthe STT-driven domain-wall dynamics in ferrimagnets\nby varying the temperature across the angular momen-\ntum compensation point TA. Speci\fcally, by studying\nthe dependence of the velocity and the angular velocity\nof a domain wall on the net spin density and the current\nwithin the linear response, we show that the domain-wall\nvelocity changes its sign across TA(see Fig. 1 for the il-\nlustration), o\u000bering an electrical way to identify TAthat\nis known to be di\u000ecult to characterize. In addition, the\nangular velocity is shown to exhibit its maximum at TA,\nwhere most of the transferred angular momentum from\nconduction electrons is used for rotating the domain wall\nby accumulating a nonequilibrium spin density inside it.\nThe theoretical result based on the collective coordinate\napproach is supported by atomistic spin simulations. In\nSec. III, we numerically study the STT-driven dynamics\nof a domain wall exactly at TAby applying large currents\nto go beyond the linear-response regime. As the current\nincreases, the domain-wall angular velocity is shown to\nsaturate to the spin-wave gap, which is caused by the\nincrease of the domain-wall width. In addition, at large\ncurrents, we observe that the domain-wall velocity de-\nviates from what is predicted from the linear-response\ntheory, showing a limitation of the linear-response the-\nory for the domain-wall dynamics at high biases. We\nconclude the paper by providing a summary and future\noutlook in Sec. IV.\nII. STT-DRIVEN DOMAIN-WALL DYNAMICS\nWITHIN THE LINEAR RESPONSE\nIn this section, we study the dynamics of a domain\nwall in ferrimagnets driven by STT within the linear re-\nsponse, i.e., at su\u000eciently small currents. For concrete-\nness, we consider RE-TM ferrimagnets which are com-\nposed of two antiferromagnetically-coupled sublattices of\nRE spins and TM spins.\nA. Theory\nThe Landau-Lifshitz-Gilbert-like equation for a RE-\nTM ferrimagnet with STT is given by [23, 27, 31{35]\n\u000es_n\u0000\u000bsn\u0002_n\u0000\u001an\u0002n\n=\u0000n\u0002he\u000b+P(J\u0001r)n\u0000\fPn\u0002(J\u0001r)n;(1)3\nto linear order in the bias, a charge current density\nJ=J^x, where nis the unit vector along the magne-\ntization direction of RE elements, \u000esis the equilibrium\nspin density along \u0000n(i.e., along the spin direction of\nRE elements), \u000b>0 is the Gilbert damping constant, s\nis the sum of the spin densities of the two sublattices, \u001a\nis the moment of inertia representing antiferromagnetic\ndynamics of n[36],he\u000b\u0011\u0000\u000eU=\u000enis the e\u000bective \feld\nconjugate to n, andU[n] is the potential energy. The\nlast two terms on the right-hand side are the adiabatic\nand the nonadiabatic STT terms, where Pis the spin\nconversion factor given by P= (~=2e)(\u001b\"\u0000\u001b#)=(\u001b\"+\u001b#)\n(with the electron charge e>0) which characterizes the\npolarization of the spin-dependent conductivity \u001bs(s=\"\nor#with\"chosen along\u0000n), and\fis the dimensionless\nparameter characterizing the nonadiabatic torque term.\nNote that there exists the adiabatic component of STT\nsince the charge current can be spin-polarized according\nto one of two sublattices, which is in contrast with STT\nfor antiferromagnets where the adiabatic component is\nabsent [11, 28].\nWe consider a quasi-one-dimensional magnet with uni-\naxial anisotropy described by the following potential en-\nergy:\nU=Z\ndV(An02\u0000Km2\nz)=2; (2)\nwhich has been used to describe the domain-wall motion\nin magnets with perpendicular magnetic anisotropy and\nnegligible in-plane anisotropy (see, e.g., Refs. [17, 37{\n39]). Here, Ais the exchange coe\u000ecient, Kis the easy-\naxis anisotropy coe\u000ecient, and0represents the deriva-\ntive with respect to the spatial coordinate x. Here, we\nassume that the magnetic order is uniform in the yz\nplane. A domain wall is a topological soliton connecting\ntwo ground states n=\u0006^z. Its low-energy dynamics is\nknown to be well described by two collective coordinates,\npositionX(t) and angle \b( t), via the so-called Walker\nansatz [40]: n(x;t) =fsech((x\u0000X)=\u0015) cos \b;sech((x\u0000\nX)=\u0015) sin \b;\u0000tanh((x\u0000X)=\u0015)g, where\u0015represents the\ndomain-wall width. In equilibrium, the domain-wall\nwidth is given by \u00150=p\nA=K determined by the compe-\ntition between the exchange energy and the anisotropy.\nBy employing the collective-coordinate approach [5,\n41, 42], we obtain the equations of motion for Xand\n\b, which are given by\n\u000es\u0015_\b +\u000bs_X+\u001aX=\u0000\fPJ; (3)\n\u000es_X\u0000\u000bs\u0015_\b\u0000\u001a\u0015\b =\u0000PJ: (4)\nThe steady-state velocity Vand the angular velocity \nare then given by, respectively,\n_X!V=\u0000PJ(\u000es+\u000b\fs)\n\u000e2s+ (\u000bs)2; (5)\nand\n_\b!\n =PJ\n\u00150\u000bs\u0000\f\u000es\n\u000e2s+ (\u000bs)2: (6)This is our main analytical result: The domain-wall ve-\nlocityVand the angular velocity \n as a function of the\nnet spin density \u000es, which can be controlled in RE-TM\nferrimagnets by changing temperature or chemical com-\nposition.\nLet us discuss the obtained results for speci\fc cases.\nFirst, when the net spin density is su\u000eciently large, i.e.,\nwhen the temperature is su\u000eciently away from TA, the\ndomain-wall velocity can be approximated by\nV\u0019\u0000PJ=\u000e s;forj\u000esj\u001d\u000bs ; (7)\nwhile assumingj\fj\u001c1. This can be understood as the\nresult of the angular-momentum transfer /PJfrom con-\nduction electrons to the domain wall via the adiabatic\nSTT. The absorption of the transferred angular momen-\ntum translates into the expansion of one of the two do-\nmains at the velocity V[1, 2]. Note that the direction\nof the domain-wall motion changes when the sign of the\nnet spin density changes, i.e., across TA. The net spin\ndensity\u000esis de\fned with respect to \u0000n, and thus, in\nour domain-wall ansatz, the net spin densities of the left\n(x!\u00001 ) and the right ( x!+1) domains are given\nby +\u000esand\u0000\u000es, respectively, polarized in the + zdirec-\ntion. Therefore, for the given angular momentum trans-\nfer from conduction electrons, whether the left domain\nor the right domain expands is determined by the sign of\nthe net spin density. See Fig. 1 for the schematic illustra-\ntion. We would like to remark here that the analogous\nresult of the reversal of the domain-wall motion has been\nreported in the theoretical study of the spin-wave-driven\nferrimagnetic domain-wall motion [43].\nSecond, when the net spin density vanishes \u000es= 0, i.e.,\natTA, the domain-wall velocity is reduced to\nV=\u0000\fPJ=\u000bs; for\u000es= 0: (8)\nThis reproduces the known result for the STT-induced\ndomain-wall motion in antiferromagnets [11], where the\ndomain wall is driven by the nonadiabatic STT /\f.\nAlso, for\u000es= 0, the angular velocity is reduced to\n\n =PJ=\u000b\u0015 0s;for\u000es= 0: (9)\nThis can be understood as follows. Conduction electrons\ntransfer angular momentum at the rate /PJto the\ndomain wall by passing through it. When the net spin\ndensity is \fnite \u000es6= 0, the domain wall can absorb the\ntransferred angular momentum by moving, i.e., by ex-\npanding one of the two domains. However, when the net\nspin density vanishes \u000es= 0, the transferred angular mo-\nmentum cannot be absorbed by the domain-wall motion\nand thus it is accumulated inside the domain wall. This\nnonequilibrium spin density exerts the e\u000bective magnetic\n\feld [44], which in turn rotates the magnetic order inside\nthe domain wall at the angular velocity \n /PJ. The\nsteady-state amount of the nonequilibrium spin density\nand the corresponding precession frequency \n are deter-\nmined by balancing the spin dissipation rate caused by\nthe precession/\u000bs\n and the generation rate of the4\n(a)(b)•2(a) -DW velocity as a function of current density•2(b) -DW angular velocity as a function of current온도범위1~5 에서의Figure 입니다. 𝛽=0.0010.5𝛼일때의계산입니다.-Fig. 2(b) 를Eq. (6) 으로fitting 하였습니다.0246810-2-1012 1 2 3 (TA) 4 5Velocity (km/s)Current density (1011 A/m2)02468100400800120016002000 1 2 3 (TA) 4 5Angular velocity (109 rad/s)\nCurrent density (1011 A/m2)•2(a) -DW velocity as a function of current density•2(b) -DW angular velocity as a function of current온도범위1~5 에서의Figure 입니다. 𝛽=0.0010.5𝛼일때의계산입니다.-Fig. 2(b) 를Eq. (6) 으로fitting 하였습니다.0246810-2-1012 1 2 3 (TA) 4 5Velocity (km/s)Current density (1011 A/m2)02468100400800120016002000 1 2 3 (TA) 4 5Angular velocity (109 rad/s)\nCurrent density (1011 A/m2)\nFIG. 2. (a) The domain-wall velocity and (b) the domain-wall\nangular velocity as a function of the current density J\u00141012\nA/m2at various temperatures shown in Table I. The symbols\nare numerical results. The lines show the analytical results for\nthe velocity V[Eq. (5)] and the angular velocity \n [Eq. (6)]\nobtained within the linear response.\nnonequilibrium spin density /PJ. The similar phe-\nnomenon has been observed numerically and explained\ntheoretically in the dynamics of a domain wall in an anti-\nferromagnet driven by a circularly-polarized magnon cur-\nrent [37, 38].\nB. Simulation\nTo con\frm the obtained analytical results, we perform\nnumerical simulations by solving the following coupled\natomistic LLG equations for RE-TM ferrimagnets [16,\n43, 45]:\n@Ak\n@t=\u0000\rreAk\u0002Hk\ne\u000b,A+\u000breAk\u0002@Ak\n@t\n\u0000bre@Ak\n@x\u0000\frebreAk\u0002@Ak\n@x;\n@Bk\n@t=\u0000\rtmBk\u0002Hk\ne\u000b,B+\u000btmBk\u0002@Bk\n@t\n\u0000btm@Bk\n@x\u0000\ftmbtmBk\u0002@Bk\n@x;(10)\nwhere AkandBkare the normalized spins at the kth\nsites in RE and TM sublattices, respectively, Hk\ne\u000b,A =\n(1=\u0016re)\u0001@H=@ AkandHk\ne\u000b,B = (1=\u0016tm)\u0001@H=@ Bkare\nthe e\u000bective magnetic \felds, \u0016reand\u0016tmare the local\nmagnetic moments, \u000breand\u000btmare the Gilbert damp-\ning constants, \rreand\rtmare the gyromagnetic ratios,\nMreandMtmare the saturation magnetizations, bre=\n\u0000gre\u0016BPreJ=(2eMre) andbtm=\u0000gtm\u0016BPtmJ=(2eMtm)\nare the STT parameters [46], Jis the charge current den-\nsity,eis the electron charge, greandgtmare the g-factors,\nPreandPtmare the dimensionless spin polarizations, and\n\freand\ftmare the dimensionless nonadiabatic STT pa-\nrameters. Here, His the discrete Hamiltonian given by\nH=AsimP\nk(Ak\u0001Bk+Bk\u0001Ak+1)\u0000KsimP\nk[(Ak\u0001^z)2+\n(Bk\u0001^z)2].\nFor the sample geometry, we considered 3200 \u0002100\u00020:4\nnm3with cell size 0 :4\u0002100\u00020:4 nm3. Correspond-\ningly, the lattice constant in the xdirection is given\n(a)(b)•3(a) –DW velocity as a function of temperature  •3(b) –DW angular velocity as a function of temperature-4-2024-200-1000100200Velocity (m/s)ds (10-8 Js/m3)Current density (1010 A/m2) 1  2  3 4  5  6 7  8  9-4-2024060120180240300Current density (1010 A/m2) 1  2  3 4  5  6 7  8  9Angular velocity (109 rad/s)\nds (10-8 Js/m3)•3(a) –DW velocity as a function of temperature  •3(b) –DW angular velocity as a function of temperature-4-2024-200-1000100200Velocity (m/s)ds (10-8 Js/m3)Current density (1010 A/m2) 1  2  3 4  5  6 7  8  9-4-2024060120180240300Current density (1010 A/m2) 1  2  3 4  5  6 7  8  9Angular velocity (109 rad/s)\nds (10-8 Js/m3)FIG. 3. (a) The domain-wall velocity and (b) the domain-wall\nangular velocity as functions of the spin density \u000esat various\ncurrent densities. The symbols are numerical results. The\nlines show the analytical results: the velocity V[Eq. (5)] and\nthe angular velocity \n [Eq. (6)] within the linear response.\nNote the sign change of the velocity across \u000es= 0 and the\nmaximum of the angular velocity at \u000es= 0.\nbyd= 0:4 nm. The used material parameters are\nAsim= 7:5 meV,Ksim= 0:4 meV,\u000btm=\u000bre= 0:002,\nand the gyromagnetic ratios \rre= 1:76\u00021011rad/s\u0001T\nand\rtm= 1:936\u00021011rad/s\u0001T (corresponding to the\ng-factorsgtm= 2:2 andgre= 2 [26]). The used mag-\nnetic moments for \fve di\u000berent cases are shown in Ta-\nble I. For STT parameters, a current in RE-TM ferrimag-\nnets is known to interact mostly with TM magnetic mo-\nments [29, 30]. Therefore, in this work, we consider the\nsimplest case where the current interacts only with TM\nelements. Accordingly, we use the following STT param-\neters:Pre= 0;Ptm= 0:3, and\ftm= 0:001. Correspond-\ning parameters in the continuum model [Eq. (2)] are given\nbyA= 4Asim=dandK= 4Ksim=d3. In addition, we\nhave the following relations: n= (Ak\u0000Bk)=jAk\u0000Bkj,\nsre=Mre=\rre,stm=Mtm=\rtm,\u000es=sre\u0000stm,s=\nsre+stm,\u000b= (\u000bresre+\u000btmstm)=s,PJ=s(bre\u0000btm)=2,\nand\f= (\frebre+\ftmbtm)=2PJ.\nFigure 2(a) and (b) show the domain-wall velocity V\nand the angular velocity \n as a function of the current\ndensityJ\u00141\u00021012A/m2for various values of the net\nspin density \u000esshown in Table I. The analytical results\nshown as lines and the numerical results shown as sym-\nbols agree well for the current densities J.5\u00021011\nA/m2. Figure 3(a) and (b) show the velocity and the an-\ngular velocity as a function of the net spin density \u000esfor\nvarious current densities. Two main features predicted\nIndex 1 23 (TA)4 5\nMre(kA/m) 1020 1010 1000 990 980\nMtm(kA/m) 1130 1115 1100 1085 1070\n\u000es(10\u00008J\u0001s/m3)-4.13 -2.07 0 2.07 4.13\nTABLE I. The values of the magnetic moments MreandMtm\nfor transition-metal and rare-earth elements, respectively, and\nthe net spin density \u000esused in atomistic spin simulations. In-\ndex 3 represents the angular momentum compensation point\nTA.5\n(a)(b)012340.00.51.01.52.02.53.0b 0 0.0005 0.001Angular velocity (1012 rad/s)\nCurrent density (1012 A/m2)012341234567b 0 0.0005 0.001DW width (nm)Current density (1012 A/m2)At 𝑇𝐴•4(a) –DW angular velocity as a function of current density•4(b) –DW width as a function of current density012340.00.51.01.52.02.53.0b 0 0.0005 0.001Angular velocity (1012 rad/s)\nCurrent density (1012 A/m2)012341234567b 0 0.0005 0.001DW width (nm)Current density (1012 A/m2)At 𝑇𝐴•4(a) –DW angular velocity as a function of current density•4(b) –DW width as a function of current density\nFIG. 4. (a) The domain-wall angular velocity and (b) the\ndomain-wall width as functions of the current density for\nthree di\u000berent values of the nonadiabatic STT parameter\n\f= 0;0:0005;0:001 at the angular momentum compensa-\ntion point\u000es= 0. The symbols are numerical results. The\nsolid lines show the analytical results: the angular velocity \n[Eq. (12)] and the width \u0015[Eq. (13)].\nby the theory, the sign change of the domain-wall veloc-\nity across\u000es= 0 and the maximum angular velocity \nat\u000es= 0, are demonstrated in the numerical results.\nIII. STT-INDUCED DOMAIN-WALL\nDYNAMICS AT TA\nIn this section, we study the STT-induced dynamics of\na domain wall exactly at TA, where the net spin density\nvanishes and thus the nature of the magnetic dynamics\nis purely antiferromagnetic, by applying a charge-current\ndensity up to 4\u00021012A/m2to look for novel phenomena.\nA. Angular velocity of a domain wall\nLet us \frst discuss the numerical results on the\ndomain-wall angular velocity from atomistic spin simula-\ntions performed with three di\u000berent values of the nona-\ndiabatic STT parameter \f= 0;0:0005, and 0 :001. Fig-\nure 4(a) shows the angular velocity _\b as a function of the\ncurrent density. The angular velocity increases linearly\nas the current increases for the small current density as\npredicted by Eq. (6), but it deviates from the equation\nfor high current densities by showing the saturation.\nThis observed saturation of the angular velocity can\nbe understood as the e\u000bect of the change of the domain-\nwall width as follows. The width of the static domain\nwall is given by \u00150=p\nA=K determined by the competi-\ntion between the exchange energy /Aand the easy-axis\nanisotropy/K. When the domain wall is precessing uni-\nformly at the angular velocity \n in the laboratory frame,\nthe e\u000bective easy-axis anisotropy in the spin frame ro-\ntating at the domain-wall angular velocity \n is given by\nKe\u000b=K\u0000\u001a\n2as shown in Ref. [38]: the uniform spin\nrotation about the zaxis in the laboratory frame gives\nrise to the e\u000bective magnetic \feld along the zaxis in the\n(a)(b)01234-30-150Velocity (m/s)Current density (1012 A/m2)b 0 0.0005 0.001•5(a) -DW velocity as a function of current density. (low current density)•5(b) -DW velocity as a function of current density. (high current density)0.00.10.20.30.40.5-2.5-2.0-1.5-1.0-0.50.0b 0 0.0005 0.001Velocity (m/s)Current density (1012 A/m2)01234-30-150Velocity (m/s)Current density (1012 A/m2)b 0 0.0005 0.001•5(a) -DW velocity as a function of current density. (low current density)•5(b) -DW velocity as a function of current density. (high current density)0.00.10.20.30.40.5-2.5-2.0-1.5-1.0-0.50.0b 0 0.0005 0.001Velocity (m/s)Current density (1012 A/m2)FIG. 5. (a) and (b) The domain-wall velocity as a function of\nthe current density for three di\u000berent values of the nonadia-\nbatic STT parameter \f= 0;0:0005;0:001 at the angular mo-\nmentum compensation point \u000es= 0. The symbols are simula-\ntion results. The sold lines show the analytical results for the\ndomain-wall velocity Vwithin the linear response [Eq. (5)].\nrotating spin frame of reference (which is analogous to the\ncentrifugal force in a rotating frame of reference), which\nin turn decreases the easy-axis anisotropy as known for\nmagnets with antiferromagnetic coupling [36]. Therefore,\nthe width of the domain wall rotating at the angular ve-\nlocity \n is given by\n\u0015=\u00150p\n1\u0000(\n=!0)2; (11)\nwhere!0\u0011p\nK=\u001a is the spin-wave gap at TA[45]. By\nsolving Eq. (6) with \u00150replaced by \u0015[Eq. (13)] for \n, we\nobtain the domain-wall precession frequency as a func-\ntion of the current density:\n\n =PJ=\u000b\u0015 0sp\n1 + (PJ=\u000b\u0015 0s!0)2; (12)\nand the domain-wall width:\n\u0015=\u00150p\n1 + (PJ=\u000b\u0015 0s!0)2: (13)\nThe obtained expression for the angular velocity \n[Eq. (12)] is reduced to the linear-response result [Eq. (6)]\nwhen the quadratic e\u000bects in the current density Jis ne-\nglected. Note that the angular velocity converges to the\nspin-wave gap !0\u00193\u00021012rad/s as the current den-\nsity increases, but can never exceed it. The sold lines in\nFigs. 4(a) and (b) show the analytical solutions for the\nangular velocity \n [Eq. (12)] and width \u0015[Eq. (13)], re-\nspectively for several values of \f. They agree well with\nthe simulation results shown as the symbols.\nB. Velocity of a domain wall\nLet us now turn to the STT-induced translation mo-\ntion of a domain wall at large currents. Figure 5(a) shows\nthe domain-wall velocity Vas a function of the current\ndensity up to 0 :5\u00021012A/m2. For relatively small cur-\nrentsJ\u00140:2\u00021012A/m2, the simulation results shown6\nas symbols are well explained by the linear-response an-\nalytical result V=\u0000\fPJ=\u000bs [Eq. (5)] shown as solid\nlines. However, Fig. 5(b), where the current density as\nlarge as 4\u00021012A/m2is applied, shows that the domain-\nwall velocity starts to deviate signi\fcantly from Eq. (5)\nfor the current density J&1\u00021012A/m2. This de-\nviation is not due to the current-induced change of the\ndomain-wall width since V=\u0000\fPJ=\u000bs does not depend\non\u0015. There are two notable features. First, even when\nthe nonadiabatic torque is absent \f= 0, the domain\nwall exhibits a \fnite velocity at high current densities,\nwhich disagrees with the known results for antiferromag-\nnetic domain-wall motion obtained within the linear re-\nsponse [11]. Secondly, as the current density increases,\nthe domain-wall velocities corresponding to three di\u000ber-\nent values of \fappear to converge on the one univer-\nsal line, suggesting that it is not the nonadiabatic STT\n/\fPJ but the adiabatic STT /PJthat plays a main\nrole in the observed domain-wall velocity at high current\ndensities. Our numerical result demonstrates a limita-\ntion of the linear-response theory for the STT-induced\ndomain-wall motion at high currents. We leave a theoret-\nical understanding of the observed domain-wall velocity\nat higher currents as a future research topic.\nIV. DISCUSSION\nTo sum up, we have studied the STT-induced dynam-\nics of a domain wall in ferrimagnets theoretically and\nnumerically. The domain-wall velocity changes it sign\nacrossTAdue to the sign change of the net spin density,\ngiving rise to a phenomenon unique to ferrimagnets that\ncannot be found in ferromagnets and antiferromagnets.\nThe angular velocity of a domain wall is shown to exhibit\nits maximum at TA, which can be understood as the e\u000bect\nof the STT-induced accumulation of the nonequilibrium\nspin density inside the domain wall. At higher currents,\nwe have found numerically that the domain-wall velocity\ncan signi\fcantly deviate from the linear-response result,\ncalling for the development of a more general theory for\nthe dynamics of a domain wall subjected to strong cur-\nrents.\nIn this work, we have focused on the e\u000bects of STT on\nthe dynamics of a domain wall in ferrimagnets. The re-\nciprocal e\u000bects of a spin texture on a current are known\nto give rise to intriguing phenomena in ferromagnetssuch as the generation of electromotive force by domain-\nwall precession [47{50] and the topological Hall e\u000bect in\nskyrmion crystal phases [51{53]. The corresponding ef-\nfects in ferrimagnets would be worth being investigated\nin the future. In addition, our understanding of STT in\nantiferromagnetically-coupled magnetic systems can be\nadvanced further by pursuing the microscopic theory for\nthe spin-charge interaction in ferrimagnets as has been\ndone for ferromagnets within the Stoner model or the\ns-d model for itinerant ferromagnetism [34]. More gen-\nerally, we envision that the research on the spin dynam-\nics as well as the spin-charge interaction in ferrimagnets\nwill lead us to more uni\fed understanding of magnetic\nphenomena spanning various types of magnetic materi-\nals including ferromagnets and antiferromagnets as two\nspecial cases, and also it will facilitate the advancement\nof ferrimagnetic spintronics aiming at easily-controllable\nhigh-speed devices by combining the advantages of ferro-\nmagnetic and antiferromagnetic devices.\nACKNOWLEDGMENTS\nD.H.K was supported by the National Research Coun-\ncil of Science & Technology (NST) Research Fellowship\nfor Young Scientist of the National Research Council of\nScience & Technology (NST), the POSCO Science Fel-\nlowship of POSCO TJ Park Foundation, the Korea In-\nstitute of Science and Technology (KIST) institutional\nprogram (No. 2E29410 and 2E30600), and the National\nResearch Council of Science & Technology (NST) grant\n(No. CAP-16-01-KIST) funded by the Korea govern-\nment (Ministry of Science and ICT). 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B 100, 174403 (2019).\n[44] Microscopically, the nonequilibrium spin density and the\nresultant e\u000bective magnetic \feld are the manifestiations\nof relative canting of the spin densities of multiple sub-\nlattices. See Refs. [8, 36, 54] or Supplemental Material of\nRef. [27] for the detailed discussions.\n[45] S.-H. Oh, S. K. Kim, D.-K. Lee, G. Go, K.-J. Kim,\nT. Ono, Y. Tserkovnyak, and K.-J. Lee, Coherent ter-\nahertz spin-wave emission associated with ferrimagnetic\ndomain wall dynamics, Phys. Rev. B 96, 100407(R)\n(2017).[46] The STT parameters breandbtmhave units of speed.\n[47] L. Berger, Possible existence of a josephson e\u000bect in fer-\nromagnets, Phys. Rev. B 33, 1572 (1986).\n[48] G. E. Volovik, Linear momentum in ferromagnets, J.\nPhys. C: Solid State Phys. 20, L83 (1987).\n[49] S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q. Niu,\nM. Tsoi, and J. L. Erskine, Universal electromotive force\ninduced by domain wall motion, Phys. Rev. Lett. 102,\n067201 (2009).\n[50] S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao,\nZ. Zhang, M. Tsoi, Q. Niu, A. H. MacDonald, and J. L.\nErskine, Topological electromotive force from domain-\nwall dynamics in a ferromagnet, Phys. Rev. B 82, 054410\n(2010).\n[51] B. Binz and A. Vishwanath, Chirality induced\nanomalous-hall e\u000bect in helical spin crystals, Physica B\n403, 1336 (2008).\n[52] M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong,\nUnusual hall e\u000bect anomaly in mnsi under pressure, Phys.\nRev. Lett. 102, 186601 (2009).\n[53] A. Neubauer, C. P\reiderer, B. Binz, A. Rosch, R. Ritz,\nP. G. Niklowitz, and P. B oni, Topological Hall E\u000bect\nin theAPhase of MnSi, Phys. Rev. Lett. 102, 186602\n(2009).\n[54] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak,\nSuper\ruid spin transport through antiferromagnetic in-\nsulators, Phys. Rev. B 90, 094408 (2014)."    },    {        "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 <ver.1.0> 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. 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Sindzingre: arXiv:0707.4264."    },    {        "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. 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Phys.: Condens. Matter 20, 135218\n(2008)\n[*] Electronic address: naoum@phys.uni-sofia.bg"    },    {        "title": "2203.09641v1.Rare_earth_free_noncollinear_metallic_ferrimagnets_Mn4_xZxN_with_compensation_at_room_temperature.pdf",        "content": "1 \n Rare -earth -free noncollinear metallic ferrimagnet s Mn 4-xZxN wit h \ncompensa tion at room temperature  \n \nRui Zhang1, Yangkun He1,*, Daniel Fruchart2, J.M.D. Coey1, Zsolt Gercsi1 \n \n1. School of  Physic s and CRANN , Trinity College, Dublin 2, Ireland  \n2. Institut Néel, CNRS & UGA, Dept QUEST, 38042, Grenoble Cedex 9, France  \n \nE-mail: heya@tcd.ie  (Dr. Yangkun He)  \n \nAbstract:  \nCompensated ferrimagnets, like antiferromag nets, show no net magnetization but their \ntransport and magneto -optic properties resemble those of ferromagnets, thereby creating \nopportunities for applications in high -frequency spintronics and low -energy loss communications. \nHere we study the modificatio n the noncollinear ferrimagnetic spin structure of Mn 4N by a variety \nof metallic substitutions Z  (Z = Cu — Ge and Ag — Sn) to achieve compensation at room \ntemperature. The noncolli near frustrated 2.35µB moment s of Mn on 3c sites of the (111) kagome \nplanes  tilt about 20° out-of-plane in Mn 4N and are easily influenced by the substitution s on 1a sites, \nleading to different  efficiency of compensation in  Mn 4-xZxN that increases gradually from group 11 \n(Cu, Ag) to group 14 (Ge, Sn) with increasing number of valance electrons. Elements from the 5th \nperiod are more efficient to for compensation than those from the 4th period due to lattice expansion. \nThe manganese site  moment s are determined by Z, orbital hybridization, charge transfer and the  tilt \nangle, analy zed by constrained density functional theory . The Ga compound with compensation at \nroom temperature for x ≈ 0.26 is recommended for high-frequency  spintronic applications.   \n \nKeywords : Metallic perovskites, Noncollinear  magnetic structure , Kagome lattice, F errimagnet ism, \nCompensation temperature, Mn 4N,  \n \n1. Introduction  \nLow energy loss  communication s and high-frequency  spintronic  applications  could  benefit \nfrom  magnetic materials with a very low net  magnetic moment  at room temperature  [1]. Although \nantiferromagnetic  (AFM)  materials  have no net moment, apart from some exceptions with special \nsymmetry  [2,3], they usually  lack net spin polariz ation and transport properties such as anomalous \nHall effect . Ferrimagnetic (FiM) metals  exhibit transport properties just like ferromagnets, while \ntheir magnetization can also fall to zero if the net moments of the antiparallel  sublattices compensate, \nleading to the current interest in  develop ing new compensated ferrimagnet s for spintronics  [4,5]. \nFerrimagnets usually consist of two  sublattices with different  magnetic  moment s where  AFM  \nexchange  may coexist with  ferromagneti c (FM) interactions . The most common  type, collinear  \nferrimagnetism , is shown in Fig. 1a; well-studied examples are  Fe3O4, Y3Fe5O12, amorphous Gd -\nCo [6] and Mn 2Sb [6,7]. In these materials , the moment s within each sub -lattice are aligned  parallel \nwhile  the two sublattices are coupled antiferromagnetically. A less common non -collinear 2 \n ferrimagnetic (ncFIM) magnetic structure can manifest,  where the coupling is between  a non-\ncollinear antiferromagneti c sublattice,  and a ferromagneti c sublattice as  shown in Fig. 1b. The \nsignificant difference in the second type is that the ferrimagnetism is non -collinear . Examples \ninclude Ni(NO 3)2 [8], MnCr 2O4 [9] and Ho 2Fe14B [10] at low temperature.  Unlike  oxides , which are \nusually insulating, metallic ferrimagnets are often  R-T-based , where R is a heavy rare -earth and T \nis Fe or Co,  or else  Mn-based. The latter category avoids the use of rare-earth  metals  and often \nexhibit s a high Curie temperature  thus it is well suited for applications . \nThe m etallic perovskite Mn 4N crystallize s in face-centred  cubic  (fcc) structure (space group \nPm-3m) where  N atoms occupying  the body -centered  interstitial site  are coordinated by an \noctahedron six Mn atoms in the 3 c face-center positions, as shown in Fig. 2a. The Mn in the 1 a \ncorner positions is not in direct contact with the nitrogen . The magnetic order has been investigated \nboth experimentally and theoretically. Initially, it was thought to have a collinear ferrimagnetic  \nstructure  along [111]  [11,12] with a large 1a – site magnetic moment m1a = 3.8 μB and a 3c site \nmoment m3c = 0.9 µ B that was much -reduced  by p-d hybridization  with the neighboring nitrogen  \n[12]. Subsequent neutron diffraction with polarization analysis  revealed that triangles of 3 c atoms \nin (111) planes, where they from a kago me lattice  shown in Fig. 2b , add  a triangular \nantiferromagnetic component to the 3c sublattice  spin structure, where the spins are either in the \n(111) plane where the spin  axes either meet at the centre of the triangle ( Γ4g mode)  or else lie  along  \nthe sides ( Γ5g mode). Only a small anisotropy energy separates them, and the modes may coexist at \nfinite temperature  [13]. Calculations by Uhl et al. [14] confirmed a noncollinear ‘umbrella ’ structure  \nwith the net sublattice moments aligned antiparallel  along the [111] axis, producing a net moment \nclose to the value of 1.1 μB that is found experimentally  [12]. The Γ4g mode has a topological \ncharacter that accounts for the large anomalous Hall effect in many Mn 4-xZxN materials [15]. The \nassumption that Mn 4N is a collinear  ferrimagnet  is therefore  unwarranted and may  lead to \nmisleading conclusions  [16]. \nIn the Weiss mean -field model  the temperature -dependent magnetization ( M-T) curves for \nferrimagnets can be categori sed into three types depending on the moment and main molecular field  \ncoefficients  within and between sublattice s (n1a1a, n1a3c and n3c3c) [17]; curves  are shown in Figs. \n1c-1f. The M-T curve of b inary Mn 4N [18 ,19 ] is Q-type, without compensation , where the \nmagnetiz ation of the 1a sublattice  is always larger than that of the 3 c sublattice  and n1a3c is the main \nmolecular field  coefficient . An N-type M-T curve with compensation may  be achieved by  \nsubstitution Z for Mn  on 1a sites in Mn 4-xZxN (Z  = Co, Ni, Cu, Zn, Ga, Ge, As, Rh, Pd, Ag, Cd, In, \nSn, Sb, Pt, Au and Hg  with x < 1)  [20,21]. All of them  are expected to  produce compensation  and \nsome have been identified experiment ally [22,23,24,25]. Though most dopants are nonmagnetic, it \nis found  that compensation is achieved at quite different values of x for different elements  [26]. \nTherefore, it is necessary to analyze the doping efficiency in order to clarify the governing physical \nmechanisms that allow us to productively design novel Mn 4N-based compensated ferrimagnets for \nroom temperature applications.  \nIn this study, we first reveal the origin of the noncollinear ferrimagnetism of Mn 4N. We  then \ndemonstrate compensation at room temperature with various non-magnetic alloying elements and \ncompar e their efficiency. We discuss the findings in relation to valance electron count, magnetic \nmoment, tilt angle and lattice constant, based on the  exper imental data and constrained density \nfunctional theory calculations.  \n 3 \n  \nFig. 1. Ferrimagnetic prototypes . (a) Collinear ferrimagneti c (FiM)  spin structure is a combination of \nferromagnetic ( FM) and antiferromagnetic ( AFM ) spin structure s. (b) Noncollinear ferrimagneti c (ncFiM)  structure  \ncombines non -collinear antiferromagneti c (ncAFM) and FM structure s. Three  types of temperature -dependent \nmagnetization curves  (schematically)  for a ferrimagnet  with two sublattices (1a and 3c for Mn 4N) with (c) m1a > \n3m3c with  n1a3c as the main molecular field coefficient , (d) m1a < 3m3c with n1a3c as the main molecular field coefficient , \n(e) m1a > 3m3c with  n1a1a as the main molecular field coefficient and (f) m1a > 3m3c and n3c3c as the main molecular \nfield coefficient . The collinear model is used here . In a noncollinear model the chief  difference in the shape of the \ncurve is the non -zero slope at low temperatures  (see Fig. 2c) . \n \n2. Methods  \nThe high purity (> 99.99%) elements of Mn and Z = Cu, Ga, Ge, In, Sn  were arc-melted \ntogether  five times  to prepare homogeneous polycrystalline ingots. Additional Mn (2%) was added \nto compensate the loss due to  its high vapor pressure. The ingots were then ground into powder and \nreacted with N 2 (> 99.99%) at 750 – 800 ℃ at a pressure of  50 kPa for 1  day. We found that if  the \nN2 pressure is too large (100 kPa ) a Mn 2N impurity  phase  will form in some samples  with small \nvalue s of x. Nitrogen deficiency can lead nitrogen vacancies or formation of  γ- or β-Mn type  \nimpurit ies. Additional heat treatment (anneal ing at 660 ℃ in vacuum  for one day was needed  for \nMn-Cu and Mn -Ge ingots before grinding them into powder to transform γ-Mn into β-Mn, owing \nto the ductile mechanical properties  of γ-Mn which makes it difficult to grind . \nThe composition of the polycrystalline  sample  was checked by energy -dispersive X -ray \n4 \n spectroscopy. The crystal structure was  characterized by powder X -ray diffraction  (XRD)  that \nshowed  a single -phase  cubic  structure. Magnetization measurements were conducted using a \nsuperconducting quantum interference device  magnetometer (SQUID, Quantum Design).  \nAb-initio calculations based on density functional theory were carried out using norm -\nconserving pseudopotentials and pseudo -atomic localized basis function s implemented in  the \nOpenMX software package [27]. Calculations were based on a minimal 5 atom basis cell of the \ncubic structure using 13 × 13× 13 k -points to evaluate the total energies. Pre -generated fully \nrelativistic pseudo potentials and the pseudo -atomic orbitals with a typical cut -off radius of 6 atomic \nunits (a.u.) were used with s3p3d3 for the metal and s3p3d2 for the metalloid elements , respectively.  \nA energy cut -off of 300 Ry  was used  for the numerical integratio ns. The convergence criterion for \nthe e nergy minimization procedure was set to 10−8 Hartree. In the case of the non-collinear \ncalculations, we show results without spin -orbit interaction (SOI) , whose influence on the total \nenergy is negligible compared with the exchange interaction  \n \n3. Result s \n3.1 Non-collinear ferrimagnetism in Mn 4N \nThe origin of non -collinear ferrimagnetism can be deduced from the magnetic interaction s and \nthe crystal structure , identified by X -ray diffraction in Fig. 2a. The lattice parameter  a0 = 3.865 Å is \nalso the nearest -neighbour  distance between two Mn1a atoms d1a1a. The nearest distance s between \nMn3c and Mn1a or Mn3c d1a3c and d3c3c are both equal to a0/√2 = 2.733 Å. Generally, Mn atoms \nseparated by 2.5-2.8 Å have delocalized electrons and couple antiferromagnetically  while Mn atoms \nwith longer separations  (> 2.9 Å) couple ferromagnetically . Therefore, the Mn1a moment s lie parallel \nto each other, wh ereas  the small d1a3c distan ce favors antiparallel coupling between the sublattices. \nThe separation of nearest -neighbor Mn3c atoms  d3c3c is responsible for  the non -collinear triangular \nantiferromagneti sm of the 3c sublattice . Together, these interactions lead to the umbrella -like spin \nstructure, and the  overall  non-collinear ferrimagnetism.  \nMn 4N has a high Curie temperature TC (780 K) and a small saturation moment  mtot = 1.1 μB/f.u. \nalong  a [111] direction , as shown in Fig. 2c. The measured  moment mtot is the difference  of the m1a \nand three times the ferr imagnetic component of Mn3c mcFiM, which are 3.8  μB and -0.9 μB per Mn , \nrespectively  [13]. It should be noted  that the net moment  in Fig . 2c remains a constant below 50 K \nand then  drops with increasing temperature . By 160 K ( T/TC = 0.2), the moment has fallen  by 13% \nof the 4 K  value, in agreement with literature  [18]. This is quite unusual, because according to the  \ncollinear  mean -field model , the decrease  at T/TC = 0.2  should be smaller  than 1% (see Fig. 1) . The \ninability  to fit the P-type curve to a  collinear mean -field model  for Mn 4N [19] is a strong indication \nof the noncollinear nature of  the magnetic order . 5 \n  \nFig. 2. Noncollinear magnetic structure of Mn 4N. (a) Crystal  and magnetic  structure showing  the Γ4g triangular  \nferrimagnetism. Grey, blue and red atoms represent N, Mn1a and Mn3c, respectively. (b) Kagome lattice of Mn3c in a \n(111) plane  showing Γ4g-like magnetic structure . The out -of-plane magnetic component is not shown . \n(c) Temperature -dependent magnetization  M(T) for Mn 4N. The magnetization remains constant up to 50 K ( green  \narrow)  but drops significantly above . At 160 K (purple arrow ) the moment has already fallen to  87% of the base \ntemperature value . (d) Energy difference in the calculated magnetic structure as a function of tilt angle  θ between \nm3c and (111) plane. (e) Magnetic moment s with varied  tilt angle  θ. (f) Comparison of c alculated total energies as a \nfunction of lattice constant for the collinear and noncollinear ferrimagnetic structures.  \n \n6 \n The noncollinear spin structure is analysed further using  a constrained DFT approach, where \nthe direction s of the individual spins are pinned to a selected angle  but the magnitude s of the \nmoments are allowed to vary freely in the total energy minimi zation process. The direction of M n1a \nis pinned to the body diagonal [111] and the tilt angle θ of the spins the Mn3c atoms is varied (also \nsee inset of Fig. 5b ). As the  angle is rotated from the collinear ferrimagnetic configuration  (θ = 90°) \ninto the (111) plane ( θ = 0°, Γ4g spin structure ) and then towards a ferromagnetic  configuration at θ \n= -90° the energy and magnetic moments vary as shown in Fig . 2d and 2e One can visualize this \nangle change like a closed umbrella (FiM) that opens out  to close to 0 ° in regular usage and well \nbeyond it on a windy day (FM, θ = -90°). The relative total energy cha nge of the constrained angle \napproach is shown Fig. 2d. Following the total energy minimum curve from the right to left, we \nwitness a decrease in total energy Etot with the Mn3c moments canting away from the antiparallel \nspin arrangement into the (111) plane. The minimum of Etot is found at around θ = 20°. The Etot \ndifference between the collinear  ferrimagnetic  and noncollinear ferrimagnetic  ground state is \nsignificant , about 4.5  meV/atom. The calculations also confirm that the FM arrangement ( θ = -90°) \nof the spins on Mn is very unfavourable  with energy difference  ~32 meV/atom. Our results suggest \nthat the M n3c sublattice moment s make an angle close to 70° with the Mn1a moments, very far from  \nthe simplified picture of collinear ferrimagnetism often assumed . For further analysis, Fig . 2e shows \nthe calculated site -specific magnetic moments together with the total magnetic moment  per formula \nunit in our range of interest ( θ = 0 to 90°). A strong dependence of magneti zation for both magnetic \nsites as a function of  θ is revealed; the M n1a moment remains  ~4 μB down to about θ = 45° and then  \na significant reduction to ~3.2 μB occurs on closing towards the (111) plane ( θ = 0°). In contrast, the \nMn3c site moment increases from about 1.1  μB up to 2.4  μB when the angle closes from FiM towards \nthe Γ4g-like configuration. T he Etot minimum suggest magnetic values of m1a=3.65  μB, m3c=2.35  μB \nwith mtot=1.24 μB/f.u. Indeed, the collinear ferrimagnetic  spin configuration also yields a value of \nmtot that is close to the experimental one , but  we have relaxed both spin configuration for the \nequilibrium lattice parameters from DFT and find a  value  a0 = 3.75 Å  for the collinear ferrimagnetic \nstate that is smaller than  that for the non -collinear ferrimagnetic state  a0 = 3.82 Å , in Fig . 2f, which  \nis closer to the experimental value of 3.8 65 Å at 300 K. The earlier  calculation  [14] found  a greater  \ntilt angle  and a smaller  3c moment,  fixing  the lattice  parameter  and exploring  a smaller  range  \nof . The energy  difference  can be ascribed to the electronic pressure  caused by the altered magnetic \nspin configuration . This exchange striction , like that in FeRh  [28 ], explains the significantly \nexpanded lattice constant for Mn 4N (3.86 Å ) compared to its ferromagnetic cousins  such as  Fe4N \n(3.79 Å ), Co 4N (3.75 Å ) and Ni 4N (3.72 Å ) on the one hand  and o n the other hand it  is also manifest \nthroughout the rotation of the spins that alters exchange -split band energies by Coulomb repulsion. \nThis non -Heisenberg like behaviour relates to the spin split d-bands crossing the Fermi le vel that \ninfluences the band filling and calculated magnetic moments.  \n \n3.2 Doping for compensation  \nIn order to achieve compensation, namely to chang e the temperature -dependent magnetization \nfrom Q-type to N-type, the main exchange should change from n1a3c to n3c3c, meanwhile the 1 a site \nmoment  m1a should be larger than three times the axial component of the 3c site moment s 3m3cFiM. \nThis means that Mn on  the 1a site should be substituted at the appropriate  level  x in Mn 4-xZxN (Z= \nCo, Ni, Cu, Zn, Ga, Ge, As, Rh, Pd, Ag, Cd, In, Sn, Sb, Pt, Au and Hg  with x < 1). Fig. 3a shows \nthe X-ray diffraction (XRD) pattern of Mn 3.76Ga0.24N, and the  low-angle data are expanded  in Fig. 7 \n 3b. The larger intensity of the (110) superlattice  peak indicates that Mn 3.76Ga0.24N crystallize s in a \nwell-ordered structure with Ga atoms occupy ing the  1a site. Nonmagnetic Ga weakens the magnetic \nexchange  leading to a decreased TC = 610 K. The net moment of 0.17 μB/f.u. at 4 K indicat es that \neach Ga decrease s the moment  by of ~3.8 μB, matching both the moment of m1a from neutron \ndiffraction  [13] and our calculation . The compensation temperature  is then Tcomp = 408 K.  In this \ncase, the 1a sublattice dominates  the magnetization at low temperature s, while the 3c sublattice is \ndominant  above compensation.  The M-H curve s shown in Fig. 3  exhibit  very little hysteresis, \nindicating weak cubic magnetocrystalline anisotropy. Note the magnetization at 4 K is not saturated  \neven in 5 T, further supporting the non-collinear ferrimagnetic structure where  the tilt angle  θ \nchanges with magnetic field.  \nThe doping efficiency of different elements from Cu to Sn is shown  in Fig s. 3e-3i. Unlike  Ga, \nthat changes  the net moment at the rate of ~3.8 μB/atom, the rates for the other elements are \nsignificantly different.  For Sn with x = 0.26, the magnetization is reduced to 0.26 μB/f.u., much more  \nthan 0.12 μB/f.u. for Ga with the same x. The magnetization curve shows a large hysteresis at 4 K in \nFig. 3e, attributed to the  Γ5g antiferromagnetic configuration that co-exist all the way down from  the \nNéel temperature of Mn 3SnN [29]. There is  a difference in the M-T curves measured after field -\ncooling (FC) and zero -field-cooling (ZFC) shown in Fig. 3f, which is also observed in  the In-doped \nsample (x = 0.26).  The compensation temperature is around  400 K for Sn substitution for x = 0.26. \nThe magnetization is less sensitive to x for Sn  than for Ga, and this trend towards  low doping \nefficiency  is more significant for Ge than Sn, as shown in Fig. 3g. Compensation was not observed \nbelow 400 K for Ge  with x = 0.35 . On the other hand, Mn 4-xCuxN is very sensitive to the \ncomposition al changes . The M-T curve gradually changes from Q-type to N-type and finally to P-\ntype within a narrow range of x, as illustrated  in Fig. 3h. The M-T curves for Cu, Ga, Ge, In and Sn  \nwith x = 0.26 are  all compared in Fig. 3i ). The Cu-doped sample has the highest doping efficiency  \nwith a P-type M-T curve  without compensation ( Tcomp < 0 K ). Ge, Ga and Sn doped samples exhibit \nTcomp of 70 K, 291 K and ~  400 K respectively with N-type M-T curves.  Ge leads either to a much \nhigher Tcomp or else to a complete disappearance of  compensation ( Q-type M-T curve ). 8 \n  \nFig. 3. (a) XRD  pattern of Mn 3.76Ga0.24N. (b) Expanded low -angle  data with simulations showing \nthe superlattice peak  for Mn 3.76Ga0.24N. The experimental data confirm that  Ga atoms occupy  the \n1a site. ( c) Magnetization curve s for Mn 3.76Ga0.24N at different temperatures. (d) Thermomagnetic \nscans  Mn 4-xGaxN (x = 0.24 and 0.26) . (e) Magnetization curve for Mn 3.74Sn0.26N at 4 K and 300 K . \n(f) Zero-field-cooled (ZFC) and field -cooled (FC) thermomagnetic scans for Mn 4-xSnxN. (g) \nThermomagnetic scans  for Mn 4-xGexN (x = 0.26 and 0.35) . (h) Thermomagnetic scans for Mn 4-\nxCuxN (x = 0.16 to 0.34) . (i) Thermomagnetic scans  for Mn 3.74Z0.26N (Z = Cu, Ga, Ge, In and Sn).  \n \nWe plot data for different compositions of Mn 4-xZxN at 4 K in Fig. 4a) including both our own \ndata and previous reports  [26,30,31,32]. The slope changes  significantly from  Ni (-6.20 μ B/atom ), \nCu (-5.01 μB/atom ), Zn ( -4.35 μB/atom ), Ga ( -3.70 μB/atom ) to Ge (-2.52 μB/atom ) with increasing \nvalence electron for dopants in the fourth period. A similar trend is also found for dopants in the \nfifth period : Ag (-7.70 μ B/atom ), In (-4.35 μB/atom ) and Sn ( -3.08 μB/atom ). Based on this trend, the \nmagnetic  diagram s for different  types of  M-T curve are visualized in the maps of  Figs. 4c and 4d. \nWith a small concentrations  of dopan ts, the interaction between 1 a and 3 c sites dominates and there \nis no compensation below the Curie temperature leading to a Q-type M-T curve.  With suitab le x, the \nmoment if the  1a sublattice  is still larger than that of the 3c sublattice  at low temperature, while at \nhigh temperature the 3 c sublattice  wins above compensation. Therefore, an N-type M-T curve is \nfound. When heavily -doped , the 3 c sublattice  dominat es throughout  whole temperature range, and \nthere is a P-type M-T curve with no compensation. Elements from the fifth period have a greater \nability  to compensate than those from the fourth period , and the magnetization is very sensitive to \n9 \n x. Therefore, the boundary for different M-T curves are shifted to the left (lower x) and the  useful  \nN-type region is narrower . \n \nFig. 4. (a) Summary of the net moment as a function of composition  x in Mn 4-xZxN. We include our \nown data (solid points) and previous reports (open points). (b) The slope in (a) showing different \nefficienc ies. Magnetic d iagram for Q, N and P type M-T curves, depending  both on x and Z for  \nelements from  the fourth (c) and fifth (d) period s. \n \n4. Discussion  \nThe efficiency of the dopants to compensate is  analyzed from the view of magnetic moment , \nnon-collinear angle and lattice constant, from both experimental and theoretical  points of  view.   \n4.1 Lattice constant  \nCompounds with the same x but different Z from the same group have  the same  valance \nelectron number , and the main difference  in their effects  on the magnetic structure is related to the  \nlattice parameter . Fig. 5a shows a0 for Mn 4-xZxN (Z = Cu, Ga, Ge, Ag, Sn In). It is clear that Ag  [33], \nIn and Sn [26] lead to a greater increase in lattice parameter than Cu, Ga and Ge for the same x, \nbecause of the ir larger atomic radii.  The increased lattice parameter translates to a larger atomic \nseparation in the cubic crystal, leading to  reduced p-d hybridisation of Mn3c and increased Mn -Mn \nexchange that  produce  a relative increase of magnetism of the 3c sublattice . The Mn3c moment is \nlarger  and more localized , leading to improved  doping efficiency  for dopants from the 5th period . \n10 \n  \nFig. 5. (a) Lattice parameters  for Mn 4-xZxN (Z = Cu, Ga, Ge, Ag, Sn In ), compar ing our data (solid \npoints) and previous reports (open points)  [26,33]. (b) Calculated  tilt angle  θ versus x  by Eq. 6. (c) \nCalculated magnetic moment for Mn3c in Mn 3ZN ( x = 1). (d) Tilt angle  θ for different m1a and m3c \ndeduced from  DFT. \n \n4.2 Magnetic moment  \nSince the nearest -neighbor for Mn1a is always Mn3c, the magnetic coupling between Mn1a \natoms is weak. As a result,  the moment for the remaining Mn1a is not influenced  significantly , as \nalso indicated from neutron diffraction  [34]. Therefore the effect of doping on the net magnetization \ncomes  mainly from Mn3c. We buil t our DFT model to capture trends in the electronic and magnetic \nstructures and used the  simplest 5 -atom unit cell model to model  our experimental observation with \ndifferent compositions. This model  allows us to compare trends for x = 1, when  the 3 c site is fully \noccupied by Mn  and the symmetry  is cubic.  \nThe calculated Mn3c moment as a function of valence electron  count  in Mn 3ZN from Mn to Ge \nin the 4th period and from Tc to Sn in  5th period is plotted  in Fig. 5c. The magnetic behaviour shows \nthe same trend; an initial increase saturates around Ni and Pd , and drops monotonically afterward s. \nIn order to separate the electronic effects from the impact of chemical pressure on the lattice \nparameter, we first fixed a0 of all members of the series to that of Mn 4N (3.82 Å). This is shown by \nthe solid red and black  lines for the 4th and 5th period s in Fig. 5c. The peak at Ni, which has three \nelectrons more than Mn, resembles a localized moment  picture with striking  similar ities with the  \n11 \n Slater -Pauling rule.  These t hree extra  electrons are shared by  the three nearby Mn3c atoms, and hence \neach of the Mn3c atom s get one more electron  becoming like iron , which shows  the largest average \nmoment in 3 d alloys. The influence of the lattice constant on  the moment expected on M n3c without \nconstrain t is also drawn in green and  blue lines for comparison . The same trend is maintained,  with \npeak s at Z = Ni and Pd. The main difference is found on the right hand side of the curves , especially \nfor elements from 5th period , as it relates to the expanded lattice parameters with additional valence \nelectrons compared to  the Z = Mn reference. The significan t change in the amplitude of m3c is one \nof the main reason s for the different doping efficienc ies. In addition, the orientation of m3c also \ndepends on its amplitude that further impact s the efficiency of compensation as we discuss it in the \nfollowing section.  \n \n4.3 Tilt angle   \nTwo questions concerning the tilt angle  θ between m3c and the (111) plane  are: How d oes θ \nchange with x , and with the different dopings ? \nThe 3c moment has  two components , one  component m3cFi along the ferrimagnetic [111] axis \nand the other m3cAFM in triangular antiferromagnetic (111) plane . The  molecular field acting on 3 c \nsite also has two components , parallel and perpendicular to the [111] axis HFi and HAFM, \ncorresponding to the ferrimagnetism and in -plane antiferromagnetism. They  satisfy the relati onship s \nHAFM = -2 n3c3c m3c cosθ cos120°                      (1) \n     HFiM = n1a3c m1a(1 - x) - 2n3c3c m3c sinθ                   (2) \nm3cFi = m3c sinθ                                   (3) \nm3cAFM = m3c cosθ                                 (4) \ntanθ = HFiM / HAFM                                (5) \nwhere  n3c3c and n1a3c are the Weiss coefficients  for interactions between  3c-3c Mn and 1 a-3c Mn, as \nshown from the inset of Fig. 5b. In Eq s. 1 and 2 , the in-plane antiferromagnetism  considers the \ninteraction from the other two nearest neighbor Mn3c with 120 ° triangular spin structure ; the \nnegative sign of m3c is already considered. Taking Eqs. 1 and 2 into Eq. 5, we get  \nsinθ = n1a3c m1a(1 - x)/(3n3c3c m3c)                     (6) \nPreviously θ was estimated from neutron diffraction  to be  about 70° (nearly collinear) , with  \nm3cFi = 0.9 μ B, m3cAFM = 0.36  μB, m3c= 0.97  μB, m1a= 3.8  μB but with large error bar s [13]. This means \nthe doping  efficiency  should be weaker  than –(3.8+0.36) = -4.16 μB/atom, if we assum e that the  \nmagnetic structure becomes  collinear ferrimagnetic  after doping . But from both our and previous \nexperiments, dopants of Ni, Ag, Cu are much  more  effective , indicating  that m3cAFM was \nunderestimated . This is also found in  our DFT calculations, where θ = 19.5 ° for binary Mn 4N.Thus \nif we plot the relationship between x and θ in Eq. 6 (Fig. 5b ), we find  that θ decreases with x almost \nlinearly . The umbrella -like triangular spin structure of Mn3c rotates away from the [111] direction  \nand becomes in -plane  and the refore the  net moment changes at a slower rate , as confirmed by \ncomparative neutron study of Mn 3.2Ga0.8N and Mn 4N [34]. Finally, when x = 1 , Mn1a is completely \nreplaced by the nonmagnetic dopant , Mn 3ZN is a triangular  topological antiferromagnet  in the (111) \nplane  if the  crystal remains cubic . \nWhen doped with different elements from Cu to Ge, or from Ag to Sn, the decrease of m3c with \nincrease of valance electrons leads to a rise of θ according to Eq. 6. This can weaken the effect of \ndecreasing m3cFi according to Eq. 3. Similarly, considering  the increased lattice constant for dopants \nfrom 5th period, the enhanced m3c can also lead to a drop of θ, weakening the influence on m3cFi. We 12 \n further estimate θ by DFT calculation  based on differ ent m1a and m3c manganese site moments , as \nshown in Fig. 5d. We use a fixed spin moment (FSM) approach, where the amplitude of the magnetic \nmoments on both sites  is fixed . In Fig 5d , we plot the angles for min imum total energy Etot(m1a , \nm3c ). The general trend is that the larger m3c for a given m1a, the smaller θ. The larger m3c moment \ntends to stay in the (111) plane, and only the increasing moment on the 1a site could compensate \nfor this rotation. This is in qualitative agreement with Eq. 6 and the vanishing moment on m1a with \nincreasing x, from experiment.  \n  \n4.4 Best dopants for compensated ferrimagnet ism \nMn 4-xZxN thin films  are already attract ing increasing  attention for spintronics  [35,36]. Most \nstudies have been done with Ni or Co  [22,23,24,25], but they are not ideal for achieving \ncompensation. Beside the demand for compensation, additional requirements must  be considered \nwhen choosing the best dopants. Based on our analysis, Ga appears to be a suitable dopant in Mn 4-\nxZxN films for spintronics for the following reasons: First, earlier elements from 4th period like Ni \ncompensate the moment with small values  of x. As the total moment is very sensitive to the \ncomposition , it is difficult to control the composition precisely and homogeneously. Second, for \nelements that have many additional valance electrons like Ge, a large value of x is needed due to the \nlow dopi ng efficiency. As a result, the Curie temperature drops substantially, which is not beneficial \nfor room temperature applications. Third, Ga does not significantly increase the lattice constant \ncompared to elements from 5th period so that a series of thin f ilms can be grown with different \ncompositions x and a similar tetragonal distortion is expected on the same substrate. The slight \ntetragonal distortion ( c/a ~ 0.99) due to biaxial strain imposed  at the interface of the film and \nsubstrate is the origin of p erpendicular [001] anisotropy. A smaller lattice constant of the film than \ncommon substrates, such as SrTiO 3 with a001 = 3.91 Å, is the key for the in -plane tensile strain and \nperpendicular anisotropy [37,38], which can be easily realized in Ga -doped samples. Finally, the \ndoping efficiency of Ga , -3.70 μ B/atom coincides with m1a. This is a consequence of a combination \nof an increased m3c and a decreased θ rather than  simply  the nonmagnetic nature of the dopant.  \n \n5. Conclusion  \nFrom our experimental and  theoretical  study of  the rare-earth -free noncollinear ferrimagnetic  \nmetals  Mn 4-xZxN, we conclude that the  noncollinear ferrimagnetism originates from the structure  of \nthe Mn3c (111) kago me planes  with a small Mn-Mn interatomic separation  that leads to frustration \nof the  antiferromagnetic  nearest -neighbor interactions. The tilt angle of the moments from the (111) \nplanes , θ = 20° , is smaller than previous ly thought . There is a choice of substitutions to  achieve  \nmagnetic compensation  at room temperature. The efficiency of different elements in this respect  \nrises gradually with increasing valance electrons from group  11 (Cu, Ag) to group  14 (Ge, Sn). The \nMn1a moment is not sensitive to the dopants, while the Mn3c moment peaks at Ni and  Pd then  drops \nwith further  valance electron  addition . The higher efficiency of elements  from 5th period is due to \nthe lar ger lattice constant, which weakens the hybridization of Mn3c and N  and leads to an increase  \nof m3c. In addition,  the tilt angle decreases  with increasing  m3c and composition x. Based on the \nabove results,  Ga is the recommended dopant for Mn 4-xZxN for thin film spintronics with \nperpendicular anisotropy .  \n \n 13 \n Acknowledgement  \nYangkun He and Rui Zhang contribute equally to the work. We thank Yu Kang for high -temperature \nmeasurements. This work was supported by Science Foundation Ireland, under the MANIAC, SFI -\nNSF China project (17/NSFC/5294)  and ZEMS project  16/IA/4534.  \n \nReferences  \n[1] T. Jungwirth, X. Marti , P. Wadley , J. Wunderlich , Antiferromagnetic spintronics , Nat. Nanotech . \n11 (2016) 231 –241. \n[2 ] S. Nakatsuji, N. 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Suemasu,  Perpendicular magnetic anisotropy of \nMn 4N films on MgO(001) and SrTiO 3(001) substrates , J. Appl. Phys.  115 (2014) 17A935 . \n[38] T. Hirose, T. Komori, T. Gushi, A. Anzai, K. Toko, and T. Suemasu,  Strong correlation between \nuniaxial magnetic anisotropic constant and in -plane tensile strain in Mn 4N epitaxial films , AIP Adv. \n10 (2020)  025117.  "    },    {        "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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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": "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<S> 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. 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Mater 218  (2000) 121. \n[32] M. Acharyya, Phys. Rev. E 59 (1999) 218. [33] G. Korniss, P. A. Rikvold, M. A. Novotny, Phys. Rev. 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": "0711.1051v1.Low_energy_structure_of_the_intertwining_double_chain_ferrimagnets_A_3_Cu_3__PO_4___4___A_Ca_Sr_Pb_.pdf",        "content": "arXiv:0711.1051v1  [cond-mat.str-el]  7 Nov 2007Low-energy structure of the intertwining double-chain fer rimagnets\nA3Cu3(PO4)4(A=Ca,Sr,Pb)†\nShoji Yamamoto and Jun Ohara\nDepartment of Physics, Hokkaido University, Sapporo 060-0 810, Japan\n(Dated: 11 March 2007)\nMotivated by the homometallic intertwining double-chain f errimagnets A3Cu3(PO4)4(A=\nCa,Sr,Pb), we investigate the low-energy structure of their model Hamiltonian H=/summationtext\nn[J1(Sn:1+\nSn:3) +J2(Sn+1:1+Sn−1:3)]·Sn:2, whereSn:lstands for the Cu2+ion spin labeled lin thenth\ntrimer unit, with particular emphasis on the range of bond al ternation 0 < J2/J1<1. Although\nthe spin-wave theory, whether up to O(S1) or up to O(S0), claims that there exists a flat band in\nthe excitation spectrum regardless of bond alternation, a p erturbational treatment as well as the\nexact diagonalization of the Hamiltonian reveals its weak b ut nonvanishing momentum dispersion\nunlessJ2=J1orJ2= 0. Quantum Monte Carlo calculations of the static structur e factor further\nconvince us of the low-lying excitation mechanism, elucida ting similarities and differences between\nthe present system and alternating-spin linear-chain ferr imagnets.\nPACS numbers: 75.10.Jm, 75.50.Gg, 75.40.Cx\nI. INTRODUCTION\nIt is a long-standing and still challenging theme in\nmaterials science to design molecular systems ordering\nferromagnetically.1The naivest idea of ferromagnetically\ncoupling nearest-neighbor magnetic centers leads to the\nhighest spin multiplicity but critically depends on some\nstructural parameters which are hard to handle chemi-\ncally. An alternative solution to highly magnetic ground\nstates consists of aligning molecular bricks so as to ob-\ntainanonzeroresultantspininthegroundstateandthen\ncoupling the chains again in a ferromagnetic fashion. A\nvariety of quasi-one-dimensional ferrimagnets were thus\nsynthesized and not a few of them have been attracting\ntheoretical as well as experimental interest.\nBimetallic chain compounds are early examples and\namong others is MnCu(pbaOH)(H2O)3(pbaOH =\n2-hydroxy-1 ,3-propylenebis(oxamato) = C 7H6N2O7),2\nwhich retains the long-range ferromagnetic order on\nthe scale of the crystal lattice. Replacing the Mn2+\nions by Fe2+, Co2+, and Ni2+ions, Kahn and co-\nworkers further synthesized a series of isomorphous\ncompounds,3which stimulated extensive chemical ex-\nplorations of heterometallic chain magnets4,5and sys-\nFIG. 1: Cu2+trimeric chains in A3Cu3(PO4)4. The strongly\ncoupled Cu2+trimer consists of a central square planar\nCu2+(1) ion (black circle) and two pyramidal Cu2+(2) ions\n(gray circles) bridged by oxygen ions (open circles).\n†Phys. Rev. B 76, 014409 (2007)tematic theoretical investigations of alternating-spin\nchains.6,7,8,9,10,11,12,13,14In an attempt to obtain sub-\nstantially larger couplings between neighboring mag-\nnetic centers and possibly attain transitions to three-\ndimensional order at higher temperatures, Caneschi et\nal.15made a distinct attempt to bring into interac-\ntion metal ions and stable organic radicals. The repre-\nsentative materials of general formula Mn(hfac)2NIT-R\n(hfac = hexafluoroacetylacetonate = C 5H2O2F6;\nNIT-R= nitronyl nitroxide radical = C 7H12N2O2-R\nwithR= CH 3,C2H5,C3H5,C6H5) indeed exhibit anti-\nferromagneticintrachaininteractionsrangingfrom200to\n330cm−1. The metal-radical hybrid strategy, combined\nwith fabrication of novel polyradicals,16yielded various\npolymerized heterospin chain compounds.17,18\nHomometallicferrimagnetismisalsorealizable19,20but\nits mechanism is often more subtle, essentially depend-\ning on the structural features of the system. Coron-\nadoet al.21,22pioneeringly synthesized chain-structured\ncompounds of such kind, M2(EDTA)(H 2O)4·2H2O\n(M= Ni,Co; EDTA = ethylenediamminetetraacetate =\nC10N2O8), whose ferrimagnetic behavior originates from\nthe alternating gfactors and is therefore faint. Ho-\nmometallic chain compounds of more pronouncedly fer-\nrimagnetic aspect23,24,25,26were not obtained until an-\nother decade had passed, where particular topologies\nwere elaborately imposed on the intrachain exchange\ninteractions. A series of compounds, M(R-py)2(N3)2\n(M= Cu,Mn;R-py = pyridinic ligand = C 5H4N-R\nwithR= Cl,CH3,···), consists of bond-polymerized ho-\nmometallic chains, where the neighboringmetal ion spins\nare bridged by versatile azido ligands and are coupled to\neach other ferromagnetically or antiferromagnetically.\nThe homometallic intertwining double-chain com-\npoundsA3Cu3(PO4)4(A= Ca,Sr,Pb),27,28,29which are\nillustrated in Fig. 1, aretopologicalferrimagnets30in the\nstrict sense. Their hybrid analogs Ca 3−xSrxCu3(PO4)4\n(0≤x≤3)28were also fabricated in an attempt to\ntune the antiferromagnetic bridges between the Cu(1)2\nand Cu(2) sites, labeled J1andJ2, and possibly ex-\nplore how paramagnetic spins grow into bulk ferrimag-\nnets. The magnetic centers without single ion anisotropy\nand the simple crystalline structure without any organic\nligand will contribute toward revealing intrinsic features\nof one-dimensional ferrimagnetic phenomena. Thus mo-\ntivated, various experiments have been performed on\nthese copper phosphates in recent years, including high-\nfield magnetization,31specific-heat,32inelastic neutron-\nscattering,33nuclear spin-lattice relaxation-time,34and\nelectron-spin-resonance35measurements.\nIt is therefore unfortunate that theoretical investiga-\ntions of this system still stay in their early stage.30,36,37\nIndeed there exists a field-theoretical study38deserving\nspecial mention, but the authors restricted their argu-\nment to the particular case of J1=J2taking a main\ninterest in realizing organic ferromagnetism. A recent\nnumerical diagonalization study39is also a fine guide\nto this system, but the authors still devoted themselves\nto clarifying the electronic correlation effect on unsatu-\nrated ferromagnetism rather than geometrically modify-\ning this unique bipartite lattice, starting from a model\nof the Hubbard type. An introduction of bond alterna-\ntionδ≡J2/J1/ne}ationslash= 1 to this system will not only con-\ntribute toward understanding the magnetic properties of\nA3Cu3(PO4)428,30,33,34but also illuminate the character-\nisticoftheuniform point δ= 1. We arethusled toreport\nthe whole excitation mechanism of homogeneous-spin\nintertwining double-chain ferrimagnets, employing both\nanalytical and numerical tools. According to the spin-\nwave theory, there exist three modes of elementary exci-\ntation, two of which exhibit parallel dispersion relations,\nwhile the rest of which is of no dispersion, regardless\nof bond alternation. However, the exact-diagonalization\nand perturbational calculations disprove the spin-wave\nscenario that the low-lying excitation spectrum remains\nqualitatively unchanged with varying δ. Indeed there\nexist local excitations which are rigorously immobile at\nδ= 1, but they can be itinerant with δmoving away from\nunity. Except for the two particular points δ= 1 and\nδ= 0, corresponding to a plaquette chain and decoupled\ntrimers, respectively, there is no flat band in the exci-\ntation spectrum of the homogeneous-spin trimeric chain .\nWefurtherinquireintothermalexcitationsbasedonsuch\nan energy spectrum. Calculating the static structure fac-\ntor as a function of temperature for an alternating-spin\nlinear-chainferrimagnetaswellasforthe presentsystem,\nwe show what are the universal ferrimagnetic features\nand how they vary with decreasing δ.\nII. PLAQUETTE CHAINS\nThe Hamiltonian of our interest is represented as\nH ≡ H 1+H2=N/summationdisplay\nn=1/bracketleftbig\nJ1(Sn:1+Sn:3)·Sn:2\n+J2(Sn+1:1+Sn−1:3)·Sn:2/bracketrightbig\n, (1)whereSn:lsymbolizes the Cu2+ion spin (S=1\n2) labeled\nlin thenth trimer unit (see Fig. 1) and the intratrimer\n(J1) and intertrimer ( J2) exchange interactions, denoted\nbyH1andH2, respectively, are defined as 0 ≤J2≤J1.\nFirst we take a look at the particular point of J2/J1≡\nδ= 1, where the model reads a plaquette chain, bearing\nsome analogy with a linear chain of alternating spins 1\nand1\n2.\nIntroducing bosonic operators through the Holstein-\nPrimakoff transformation\nS+\nn:1=/radicalBig\n2S−a†\nn:1an:1an:1, Sz\nn:1=S−a†\nn:1an:1,\nS+\nn:2=a†\nn:2/radicalBig\n2S−a†\nn:2an:2, Sz\nn:2=a†\nn:2an:2−S,\nS+\nn:3=/radicalBig\n2S−a†\nn:3an:3an:3, Sz\nn:3=S−a†\nn:3an:3,\n(2)\ndefining their Fourier transforms as\nak:l=1√\nN/summationdisplay\nnei(−1)lk(n+l/2−1)an:l,(3)\nwith the lattice constant set equal to unity, and further\nprocessing them via the Bogoliubov transformation\nα†\nk:−=ψ−1(k)a†\nk:1+ψ−2(k)ak:2+ψ−3(k)a†\nk:3,\nα†\nk:0=ψ01(k)a†\nk:1+ψ02(k)ak:2+ψ03(k)a†\nk:3,\nα†\nk:+=ψ+1(k)ak:1+ψ+2(k)a†\nk:2+ψ+3(k)ak:3,(4)\nwe reach a spin-wave Hamiltonian,\nH=Eg+/summationdisplay\nλ=∓,0ωλ(k)α†\nk:λαk:λ, (5)\nFIG. 2: Dispersion relations of the elementary excitations\nin the spin-1\n2plaquette chain, two (circles and diamonds) of\nwhich reduce the ground-state magnetization and are thus\nof ferromagnetic character, while the rest (squares) of whi ch\nenhances the ground-state magnetization and is thus of anti -\nferromagnetic character. The exact-diagonalization resu lts at\nN= 4, 6, and 8 are presented by symbols of small, middle,\nand large sizes, respectively, whereas the up-to- O(S1) linear\n(LSW) and up-to- O(S0) interacting (ISW) spin-wave calcu-\nlations are given by dotted and solid lines, respectively.3\nFIG. 3: (Color online) Dispersion relations of the elementa ry excitations in the spin-1\n2trimeric chain with varying δ. The\nfirst-order perturbational calculations, together with th e up-to-O(S1) linear spin-wave findings, are given in the upper five,\nwhereas the second-order perturbational calculations, to gether with the up-to- O(S0) interacting spin-wave findings, are given\nin the lower five. The exact-diagonalization results at N= 4, 6, and 8 are presented in both upper and lower panels by sym bols\nof small, middle, and large sizes, respectively\nwithEg=/summationtext\ni=2,1,0E(i)\ngandωλ(k) =/summationtext\ni=1,0ω(i)\nλ(k),\nwhereE(2)\ng=−2S2(J1+J2)Nis the classical ground-\nstate energy, while E(i)\ngandω(i)\nλ(k) (i= 1,0,···) are the\nO(Si) quantum corrections to the ground-state energy\nand the dispersion relation of mode λ, respectively. Here\nwe have discarded the O(S−1) terms. There are sev-\neral ways40,41,42,43,44of treating the quartic interactions.\nWhen we diagonalize the one-body terms and then take\naccount of the two-body terms perturbationally,45the\nspin-wave energies read\nE(1)\ng\nJ1N=S(1+δ)\n2N/summationdisplay\nk/bracketleftbig\nω(k)−3/bracketrightbig\n, (6)\nE(0)\ng\nJ1N=1+δ\n2(Γ2−1)−2δ\n1+δ\n×(3Γ2+2Λ2−5ΓΛ−3Γ+3Λ), (7)\nω(1)\n∓(k)\nJ1=S(1+δ)\n2/bracketleftbig\nω(k)∓1/bracketrightbig\n,\nω(1)\n0(k)\nJ1=S(1+δ), (8)\nω(0)\n∓(k)\nJ1=1+δ\n2/bracketleftbig\nΓΓ(k)∓Γ/bracketrightbig\n−δ\n1+δ\n×/bracketleftbig\n6ΓΓ(k)−5ΓΛ(k)−5ΛΓ(k)+4ΛΛ(k)\n−3Γ(k)+3Λ(k)∓Γ±Λ/bracketrightbig\n,ω(0)\n0(k)\nJ1=−1+δ\n2(Γ−1)−2δ\n1+δ(Γ−Λ),(9)\nand their eigenvectors are given by\nψ∓1(k) =ψ∗\n∓3(k) =2(e±ik/2+δe∓ik/2)\n(1+δ)/radicalBig\n2ω(k)/bracketleftbig\n3∓ω(k)/bracketrightbig,\nψ∓2(k) =/radicalBigg\n3∓ω(k)\n2ω(k), ψ01(k) =1√\n2,\nψ02(k) = 0, ψ03(k) =−e−ik/2+δeik/2\n√\n2(eik/2+δe−ik/2),(10)\nwhere\nω(k) =/radicalBigg\n1+32δ\n(1+δ)2sin2k\n2, (11)\nΓ=1\nN/summationdisplay\nkΓ(k) =1\nN/summationdisplay\nk1\nω(k),\nΛ=1\nN/summationdisplay\nkΛ(k) =1\nN/summationdisplay\nkcosk\nω(k).(12)\nFigure 2 shows the thus-calculated spin-wave ex-\ncitation modes together with the exact eigenvalues.\nFree spin waves well describe the ferromagnetic modes4\nω−(k) andω0(k), while higher-order quantum correc-\ntions play an essential role in reproducing the antifer-\nromagnetic mode ω+(k). TheO(S0) quantum correc-\ntions significantly improve fully delocalized magnetic ex-\ncitations in general,18,41,46but the standard Holstein-\nPrimakoff magnon series expansion seems not to work\nwell for highly localized excitations. The dispersive\nbranchesω∓(k) are nothing but the elementary excita-\ntion modes of spin-alternating linear-chain Heisenberg\nferrimagnets.47They are parallel within the spin-wave\ntheory, but their difference ω+(k)−ω−(k) is momentum\ndependent in fact. On the other hand, the flat band\nω0(k) arises from further excitation degrees of freedom\nin the present system. When J1=J2, the Hamiltonian\n(1) reads\nH=J1N/summationdisplay\nn=1(Sn:2+Sn+1:2)·Tn:3;n+1:1,(13)\nwith composite spins Tn:3;n+1:1≡Sn:3+Sn+1:1, each\nlying diagonally across an elementary palquette. Since\nthe Hamiltonian (13) commutes with T2\nn:3;n+1:1≡\nTn:3;n+1:1(Tn:3;n+1:1+ 1), we have good quantum num-\nbersTn:3;n+1:1, each taking either 0 or 1. Therefore, the\nplaquette-chain Hamiltonian is block-diagonalizedby the\nset of numbers {Tn:3;n+1:1;n= 1,2,···,N}48as well as\nby the total magnetization/summationtextN\nn=1(Sz\nn:2+Tz\nn:3;n+1:1)≡\nM. The Hilbert space of/summationtextN\nn=1(Tn:3;n+1:1)2/2 =/summationtextN\nn=1(Sn:3·Sn+1:1+ 3/4)≡ N=Ncorresponds to\nthe ferrimagnetic chain of alternating spins 1 and1\n2\nand consequently we here have exactly the same disper-\nsion relations46of elementary excitations. The Hilbert\nspace of N=N−1 andM=N/2−1 consists\nofNsubspaces labeled {T1:3;2:1,T2:3;3:1,···,TN:3;1:1}=\n{0,1,···,1},{1,0,1,···,1},···,{1,···,1,0}, and they\nall givethe sameset ofeigenvalues, forming Nflat bands.\nWe find the lowest one in Fig. 2.\nThus and Thus, the spin- Splaquette chain turns out a\ncombination of the alternating-spin-(2 S,S) linear chain\nand extra excitation degrees of freedom within the com-\nFIG. 4: Probability of two spin1\n2’s constructing a spin 1 in\nthe ground state of the spin-1\n2trimeric chain of N= 64 with\nvaryingδ, wherePn:3;n+1:1≡T2\nn:3;n+1:1/2 =Sn:3·Sn+1:1+\n3/4 andPn:1;n:3≡T2\nn:1;n:3/2 =Sn:1·Sn:3+3/4 are estimated\nby a quantum Monte Carlo method.posite spins Tn:3;n+1:1. All the composite spins are sat-\nurated in the ground state, T2\nn:3;n+1:1= 2S(2S+ 1),\nand therefore, their excitations are necessarily of ferro-\nmagnetic aspect. The ferromagnetic excitations of local\ncharacter are well understandable within the spin-wave\ndescription. Equations (4) and (10) show that a†\nn:1and\na†\nn:3, creating bosonic excitations on the Cu2+(2) sites,\nindeed participate in the construction of α†\nk:0, but any of\na†\nn:2, creating bosonic excitations on the Cu2+(1) sites,\ndoes not. Without mediation of bridging spins Sn:2, any\nintraplaquette excitation is never movable. Then what\nmay happen with δmoving away from unity? The spin-\nwave theory, whether up to O(S1) or up toO(S0), pre-\ndicts that the ferromagnetic and antiferromagnetic exci-\ntation modes ω∓(k) are still parallel and the extra ferro-\nmagnetic excitation mode between them, ω0(k), remains\ndispersionless. Let us verify the true scenario.\nIII. BOND-ALTERNATING TRIMERIC\nCHAINS\nWe demonstrate in Fig. 3 several schemes of calcu-\nlating low-lying excitation modes for the spin-1\n2bond-\nalternating trimeric chain. In spite of the persistent flat\nband within the spin-wave theory, the exact diagonal-\nization reveals that it can be dispersive with varying δ.\nWhenδ/ne}ationslash= 1, the Hamiltonian (1) does not commute\nwithT2\nn:3;n+1:1. Now that there is a certain probability\nof composite spins Tn:3;n+1:1being singlet even in the\nground state, the gapped ferromagnetic excitation mode\nω0(k) is no more describable as their individual triplet-\nto-singlet flips. At δ= 0, any excitation is localized\nwithinatrimerof Sn:1,Sn:2,andSn:3, andtheexcitation\nspectrum degenerates into three flat bands, ω−(k)≡0,\nω0(k) =J1, andω+(k) = 3J1/2. Figure 3 shows that\nthe middle branch of them connects with the flat band\natδ= 1. Without J2, the Hamiltonian (1) is reduced to\nH=H1=J1N/summationdisplay\nn=1Sn:2·Tn:1;n:3, (14)\nwith intratrimer composite spins Tn:1;n:3≡Sn:1+Sn:3\nand thus commutes with T2\nn:1;n:3. The plaquette chain\n(13) and the decoupled trimers (14) both exhibit a flat\nband due to gapped ferromagnetic excitations, but their\nways of constructing local immobile excitations are dif-\nferent from each other. Triplet-to-singlet [(4 S+1)-fold-\nmultiplet-breakingingeneral]flipsofintraplaquettecom-\nposite spins Tn:3;n+1:1are the elementary excitations in\nthe former, while those of intratrimer composite spins\nTn:1;n:3are the elementary excitations in the latter. Fig-\nure 4 shows how such composite spins behave in the\nground state with varying δ. Neither Tn:3;n+1:1nor\nTn:1;n:3form complete triplets at 0 < δ <1, due to\nnonvanishing off-diagonal matrix elements /an}bracketle{tTn:3;n+1:1=\n1|H|Tn:3;n+1:1= 0/an}bracketri}htand/an}bracketle{tTn:1;n:3= 1|H|Tn:1;n:3= 0/an}bracketri}ht.5\nAt the two particular points δ= 1 andδ= 0, only\nthe Cu2+(2) ion spins Sn:1andSn:3constitute the\ngapped ferromagneticexcitationmode, but otherwisethe\nCu2+(1) ion spins Sn:2also contribute to that. Without\ninterconnecting spins Sn:2, any excitation is immobile,\nwhereas with their mediation, all the local excitations\ncan be itinerant and the resultant bands are dispersive.\nPerturbational calculations support such a scenario.\nWith increasing couplings J2between isolated trimers,\nthe energy dispersion relations grow as follows:\nEg\nJ1N=−1−δ\n9−869\n2430δ2+O(δ3), (15)\nω−(k)\nJ1=4\n9δ(1−cosk)+δ2\n2430(929−474\n×cosk−455cos2k)+O(δ3), (16)\nω0(k)\nJ1= 1−0.38970δ+δ2(0.32099−0.26736\n×cosk+0.04212cos2k)+O(δ3), (17)\nω+(k)\nJ1=3\n2+δ\n18(7−12cosk)+δ2\n810(346−100\n×cosk−109cos2k)+O(δ3). (18)\nEquations (16)-(18) are also drawn in Fig. 3. The first-\norder perturbation points out that not only ω∓(k) them-\nselves but also their difference should be dispersive, but\nit cannot reveal nonvanishing momentum dependence of\nω0(k). We cannot reproduce the dispersive middle band\nuntil we take account of the second-order perturbation.\nThe lowest ferromagnetic and antiferromagnetic excita-\ntions of decoupled trimers (14), gapless and gapped by\n3J1/2 from the ground state, respectively, are both N-\nfold degenerate and are expressed as\n|E∓(m)/an}bracketri}ht=|−(1±3)J1/4;1/2∓1/an}bracketri}htm\n⊗\nn/negationslash=m|−J1;1/2/an}bracketri}htn(m= 1,2,···N),(19)\nwhiletheirgappedferromagneticexcitationsatanenergy\ncost ofJ1areN2-fold degenerate and are expressed as\n|E0(m,m′)/an}bracketri}ht=δmm′|0;−1/2/an}bracketri}htm⊗\nn/negationslash=m|−J1;1/2/an}bracketri}htn\n+(1−δmm′)|0;1/2/an}bracketri}htm⊗|−J1;−1/2/an}bracketri}htm′\n⊗\nn/negationslash=m,m′|−J1;1/2/an}bracketri}htn(m,m′= 1,2,···N),(20)\nin terms of the eigenstates of an isolated trimer\n|Sn:1,Sn:2,Sn:3/an}bracketri}ht,\n|−J1;1/2/an}bracketri}htn=1√\n6(|↑↑↓/an}bracketri}ht−2|↑↓↑/an}bracketri}ht+|↓↑↑/an}bracketri}ht),\n|−J1;−1/2/an}bracketri}htn=1√\n6/parenleftbig\n|↓↓↑/an}bracketri}ht−2|↓↑↓/an}bracketri}ht+|↑↓↓/an}bracketri}ht/parenrightbig\n,\n|0;1/2/an}bracketri}htn=1√\n2/parenleftbig\n|↑↑↓/an}bracketri}ht−|↓↑↑/an}bracketri}ht/parenrightbig\n,\n|0;−1/2/an}bracketri}htn=1√\n2/parenleftbig\n|↓↓↑/an}bracketri}ht−|↑↓↓/an}bracketri}ht/parenrightbig\n,|J1/2;3/2/an}bracketri}htn=|↑↑↑/an}bracketri}ht,\n|J1/2;1/2/an}bracketri}htn=1√\n3/parenleftbig\n|↑↑↓/an}bracketri}ht+|↑↓↑/an}bracketri}ht+|↓↑↑/an}bracketri}ht/parenrightbig\n,\n|J1/2;−1/2/an}bracketri}htn=1√\n3/parenleftbig\n|↑↓↓/an}bracketri}ht+|↓↑↓/an}bracketri}ht+|↓↓↑/an}bracketri}ht/parenrightbig\n,\n|J1/2;−3/2/an}bracketri}htn=|↓↓↓/an}bracketri}ht. (21)\nWith perturbational interactions H2turned on, the\nN-fold degeneracy of the eigenvalue −[N−3(1∓\n1)/4]J1=/an}bracketle{tE∓(m)|H1|E∓(m)/an}bracketri}htis completely lifted,\nwhereas the N2-fold degenerate eigenvalue −(N−1)J1=\n/an}bracketle{tE0(m,m′)|H1|E0(m,m′)/an}bracketri}htonly splits into Nflat bands\nwithin the first-order corrections. The second-order cor-\nrections are necessary for reproducing the dispersion re-\nlation ofω0(k). In this context we may be reminded that\nHonecker and L¨ auchli49pioneeringly investigated anal-\nogous but frustrated Cu2+trimeric chains. The gap-\nless ferromagnetic excitation mode (16) is indeed derived\nfrom their effective Hamiltonian under strong trimeriza-\ntionδ≪1.\nIV. SUMMARY AND DISCUSSION\nWe have investigated the low-energystructure of inter-\ntwining double-chain ferrimagnets composed of homoge-\nneous spins with particular emphasis on the gapped fer-\nromagnetic excitation mode. While there exist a macro-\nscopic number of flat bands50in the excitation spectrum\natδ= 1 andδ= 0, which signify uncorrelated excita-\ntions of local spin-2 Smultiplets in any case, they become\ndispersive as soon as δmoves away from these particular\npoints. Pair excitations of corner spins Sn:3andSn+1:1\nare elementary in plaquette chains of δ= 1, while those\nofSn:1andSn:3are elementary in decoupled trimers of\nδ= 0, both of which are completely immobile without\nany mediation of joint spins Sn:2. The spin-wave theory\nsuccessfully characterizes the plaquette chain but fails\nto find arising contribution of Sn:2to gapped ferromag-\nnetic excitations with bond alternation. Such a mislead-\ning prediction has been corrected by further numerical\nand analytical investigations.\nThe spin-Splaquette chain thus shares whole the na-\nture of the alternating-spin-(2 S,S) linear chain and fur-\nther exhibits ferromagnetic excitations of its own. All\nthe findings but the flat band in Fig. 2 are indeed ex-\nactlythesameaswehaveintheferrimagneticHeisenberg\nchain of alternating spins 1 and1\n2.45Even though the\nhomogeneous-spin plaquette chain and the alternating-\nspinlinearchainareequivalentintheirgroundstates, the\nformer demonstrates its extra excitation degrees of free-\ndom and deviates fromthe latter with increasingtemper-\nature. In order to illuminate similarities and differences\nbetween them, we show in Fig. 5 quantum Monte Carlo\ncalculations of their static structure factors\nS(q) =1\nN/summationdisplay\nn,l,n′,l′eiq(xn:l−xn′:l′)Sz\nn:lSz\nn′:l′,(22)6\nFIG. 5: (Color online) Quantum Monte Carlo calculations of t he static structure factor S(q), with the distance between\nneighboring spins in the chain direction set equal to unity, as a function of temperature. The whole view and enlargement s at\nq= 0 and q=πfor the spin-1\n2plaquette chain of N= 64 (a) and the alternating-spin-(1 ,1\n2) linear chain of N= 64 (b). The\nferromagnetic [ S(0)] and antiferromagnetic [ S(π)] peaks are observed in more detail (c), where the common asy mptotic values\nin the high-temperature limit, 3 /4 and 11 /12 for the the spin-1\n2plaquette chain and the alternating-spin-(1 ,1\n2) linear chain,\nrespectively, are indicated with arrows. Thermal averages of the projection Pn:3;n+1:1≡T2\nn:3;n+1:1/2 =Sn:3·Sn+1:1+ 3/4\nin the spin-1\n2plaquette chain are also shown for reference, where the asym ptotic value in the high-temperature limit, 3 /4, is\nindicated with an arrow.\nas functions of temperature, where the chain-directional\ncoordinates xn:lare given in the unit of neighboring-spin\nspacing. The pronounced peaks at q= 0 andq=π\nreflect the ferromagnetic and antiferromagnetic double\nexcitation mechanism in common. Without any field\napplied,S(0) andS(π) are, respectively, the uniform\nFIG. 6: (Color online) Quantum Monte Carlo calculations of\nthe static structure factor S(q), with the distance between\nneighboring spins in the chain direction set equal to unity, as\na function of bond alternation and temperature for the spin-1\n2\ntrimeric chain of N= 64.and the staggered susceptibilities multiplied by temper-\nature. With decreasing temperature, they both diverge\nas 1/T.38,45,51With increasing temperature, they both\napproach the paramagnetic value/summationtext\nlSn:l(Sn:l+1)/3 but\nbehave differently at intermediate temperatures. A mini-\nmumofS(0)asafunctionoftemperatureischaracteristic\nof ferrimagnets.6,7,18,25,30,52,53,54S(0) monotonically de-\ncreases and increases with increasing temperature in fer-\nromagnets and antiferromagnets, respectively.14Though\nthe thermal as well as quantum behaviors of the spin-\nSplaquette chain and the alternating-spin-(2 S,S) lin-\near chain are very much alike, yet there grows a differ-\nence between them with pair excitations of intraplaque-\ntte spins Sn:3andSn+1:1from their highest multiplets.\nThe ferromagnetic and antiferromagnetic structures of\nS(q) less survive increasing temperature in the spin- S\nplaquette chain than in the alternating-spin-(2 S,S) lin-\near chain. The larger Sthe major difference in S(q) as\nlimT→∞[S(2S,S)(q)−S(S,S,S)(q)] = 2S2/3. Alternating-\nspin-(2S,S) ferrimagnetic chains behave like combina-\ntionsofspin- Sferromagneticandspin-(2 S)antiferromag-\nnetic chains,14while such a simple magnetic sum rule\nis not available to intertwining double-chain ferrimag-\nnets of our interest. Additional intraplaquette antifer-\nromagnetic interactions induce incommensurate peaks in\nS(q),55making corner spins Sn:3andSn+1:1frustrated.\nOnceδmoves away from unity, the homogeneous-\nspin trimeric chain never more shares any feature of the7\nalternating-spin chain. Figure 6 presents S(q) with vary-\ningδand analyzes its features at q= 0 andq=πin\nparticular. At low temperatures, S(0) andS(π) both\ndecline with decreasing δ, but they still diverge as 1 /T\nunlessδ= 0.30At high temperatures, S(π) remains de-\ncreasing, whereas S(0) turns increasing, with decreasing\nδ. Decoupled trimers are nothing more than paramag-\nnets and their structure factor is given as\nS(q) =3\n4−2\n3eJ1/kBT−e−J1/2kBT\neJ1/kBT+1+2e−J1/2kBTcosq\n+1\n6eJ1/kBT−3+2e−J1/2kBT\neJ1/kBT+1+2e−J1/2kBTcos2q,(23)\nwhich is also drawn in Fig. 6 with solid lines. Equa-\ntion (23) at q= 0 reads as the effective Curie law for a\ntrimer entity, where the Curie constant varies from 1 /4,\nattributable to the ground-state doublet, to 3 /4, simply\ncoming from free spin1\n2’s, with increasing temperature.\nArising intertrimer couplings J2immediately pronounce\na quadratic dispersion relation of the ferromagnetic exci-\ntations at small momenta and their further increasecosts\nthe antiferromagnetic excitations higher energy. That is\nwhy growing global correlations enhance and reduce the\nuniform susceptibility-temperature product at low and\nhigh temperatures, respectively.\nWeak but nonvanishing dispersion of the gapped fer-\nromagnetic excitation mode is the most remarkable find-\nings of ours and is the very characteristic of intertwining\ndouble-chain ferrimagnets. As the existent compounds\nA3Cu3(PO4)4have all been reported to exhibit ratherstrong bond alternation δ<∼0.1,30,32,33,34,35it may be\nhard to detect the dispersion relation ω0(k) there. Ni\nanalogs, if available, will present an energy structure of\nthe same type on an enlarged energy scale. Highly lo-\ncalized excitations in fully exchange-coupled bulk mag-\nnets may either arise from an accidental arrangement of\nexchange couplings or come out of a particular lattice\nstructure of geometric aspect. 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B 65, 014414 (2001)."    },    {        "title": "1212.6032v1.Magnetization_process_in_the_exactly_solved_spin_1_2_Ising_Heisenberg_model_on_decorated_Bethe_lattices.pdf",        "content": "arXiv:1212.6032v1  [cond-mat.stat-mech]  25 Dec 2012Condensed Matter Physics, 2012, Vol. 15, No 4, 43003: 1–10\nDOI: 10.5488/CMP.15.43003\nhttp://www.icmp.lviv.ua/journal\nMagnetizationprocessintheexactlysolvedspin-1/2\nIsing-HeisenbergmodelondecoratedBethelattices\nJ. Strečka1, C. Ekiz2\n1Department of Theoretical Physics and Astrophysics, Facul ty of Science, P.J. Šafárik University,\nPark Angelinum 9, 040 01 Košice, Slovak Republic\n2Department of Physics, Faculty of Science, Adnan Menderes U niversity, 090 10 Aydın, Turkey\nReceived June 22, 2012, in final form August 10, 2012\nThe spin-1/2 Ising-Heisenberg model on diamond-like decor ated Bethe lattices is exactly solved in the pres-\nence of the longitudinalmagnetic field by combiningthe deco ration-iterationmapping transformation with the\nmethodofexactrecursionrelations.Inparticular,thegro undstateandlow-temperaturemagnetizationprocess\nof the ferrimagnetic version of the considered model is inve stigated in detail. Three different magnetization\nscenarios with up to two consecutive fractional magnetizat ion plateaus were found, whereas the intermedi-\nate magnetization plateau may either correspond to the clas sical ferrimagnetic spin arrangement and/or the\nfield-inducedquantum ferrimagnetic spin ordering without any classical counterpart.\nKeywords: Ising-Heisenberg model, Bethe lattice, exact results, mag netization plateau\nPACS:05.50.+q, 75.10.-b, 75.10.Jm, 75.10.Kt 75.40.Cx, 75.60.E j\n1.Introduction\nLow-dimensionalquantumspinsystemshaveattractedmucha ttentionoverthepastfewdecades,\nsincetheyexhibitalotofstrikingquantumphenomenainclu dingfractionalmagnetizationplateaus,\nspin-Peierlsdimerization,unconventionalspin-liquidg roundstates,ormanyotherpeculiarvalence-\nbond-solidgroundstatessuchastheHaldanephase[1,2].It isworthnotingthatthemostremarkable\nexperimentalfindingsreportedforlow-dimensionalspinsy stemsweremostlysatisfactorilyinterpreted\nwiththehelpofquantumHeisenbergmodelanditsvariousext ensions.Fromthetheoreticalpointof\nview,anexacttreatmentofthequantumHeisenbergmodelrem ainsanunresolvedproblemmainlydue\ntosubstantialmathematicaldi fficulties,whicharisefromanoncommutabilityofspinoperat orsinvolved\nintherelevantHamiltonian.However,thismathematicalco mplexitycanbeavoidedbyconsideringsim-\nplerIsing-Heisenbergmodels,whichdescribehybridclass ical-quantumspinsystemsconstitutedbothby\nthe‘classical ’IsingaswellasthequantumHeisenbergspins.ThehybridIsi ng-Heisenbergmodelscanbe\nexactlytreatedbymakinguseofgeneralizedmappingtransf ormations,whichwereoriginallyintroduced\nbySyozi[3,4]andlaterongeneralizedbyFisher[5],Rojase tal.[6,7]andoneofthepresentauthors[8].\nInthiswork,thegeneralizeddecoration-iterationtransf ormationiscombinedwiththemethodofex-\nactrecursionrelationsinordertoobtainexactresultsfor thespin-1\n2Ising-Heisenbergmodelondiamond-\nlikedecoratedBethelatticesinthepresenceofthelongitu dinalmagnetic field.Itshouldbenotedthatthe\napplieddecoration-iterationtransformationestablishe sarigorousmappingequivalencebetweenthein-\nvestigatedmodelsystemandthespin-1\n2IsingmodelonacorrespondingsimpleBethelatticewiththe\neffectivenearest-neighbourinteractionandtheeffectiv emagnetic field.Owingtothisprecisemapping\ncorrespondence,exactresultsforthespin-1\n2Ising-Heisenbergmodelonthediamond-likedecoratedBeth e\nlatticescanbesubsequentlyextractedfromtherelevantex actsolutionofthespin-1\n2Isingmodelonasim-\npleBethelatticebymeansofthemethodofexactrecursionre lations[9 –11].\nTheorganizationofthispaperisasfollows.Insection2,th edetaileddescriptionoftheinvestigated\nmodelsystemispresentedtogetherwiththebasicstepsofit sexactsolution.Themostinterestingre-\n©J. Strečka, C. Ekiz, 2012 43003-1J. Strečka, C. Ekiz\nsultsarethenpresentedanddiscussedinsection3.Inparti cular,ourattentionisfocusedontheground\nstateandlow-temperaturemagnetizationprocessofthefer rimagneticversionoftheconsideredmodel.\nFinally,someconcludingremarksaredrawninsection4.\n2.Ising-HeisenbergmodelondecoratedBethelattices\nLetusintroducethespin-1\n2Ising-Heisenbergmodelonadiamond-likedecoratedBethel attice,which\nisschematicallyillustratedontheleft-hand-sideof figure1ontheparticularexampleoftheunderlying\nBethelatticewiththecoordinationnumber q=3.Inthis figure,thefullcircleslabellatticepositionsof\ntheIsingspins µ=1\n2,whiletheemptycirclesmarklatticepositionsoftheHeise nbergspins S=1\n2.One\nmayinferfrom figure1thatthemagneticstructureoftheinvestigatedmodel isformedbytheIsingspins\nplacedatlatticesitesofadeepinteriorofin finiteCayleytree(Bethelattice),whicharelinkedtogether\nthroughtheHeisenbergspinpairsplacedin-betweeneachco upleoftheIsingspins.ThetotalHamiltonian\nofthespin-1\n2Ising-Heisenbergmodelondiamond-likedecoratedBethela tticesreads\nH=− JHN q/2/summationdisplay\n(k,l)/bracketleftbig\n∆/parenleftbig\nSx\nkSx\nl+Sy\nkSy\nl/parenrightbig\n+Sz\nkSz\nl/bracketrightbig\n−JI2N q/summationdisplay\n(k,i)Sz\nkµz\ni−HAN/summationdisplay\ni=1µz\ni−HBN q/summationdisplay\nk=1Sz\nk.(1)\nHere, Sα\nk(α=x,y,z)andµz\nirepresentspatialcomponentsofthespin-1\n2operator,theparameter JHde-\nnotestheXXZinteractionbetweenthenearest-neighbourHe isenbergspins,theparameter ∆controlsa\nspatialanisotropyinthisinteractionbetweentheeasy-ax is(∆<1)andeasy-plane (∆>1)regime,and\ntheparameter JImarkstheIsinginteractionbetweenthenearest-neighbour HeisenbergandIsingspins,\nrespectively.Furthermore,twoZeeman ’sterms HAand HBdeterminethemagnetostaticenergyofthe\nIsingandHeisenbergspinsinalongitudinalmagnetic field.\nDIT\nSk1\nSk2/c109k1\n/c109k2/c109k1\n/c109k2Jeff\nq=3JI\nJH( )/c68\nFigure1. The spin-1\n2Ising-Heisenberg model on the diamond-like decorated Beth e lattice (figure on the\nleft) and its exact mapping via the decoration-iteration tr ansformation (DIT) onto the spin-1\n2Ising model\non a simple Bethe lattice (figure on the right). The full (empt y) circles denote lattice positions of the Ising\n(Heisenberg) spins, the ellipse demarcates the elementary diamond-shaped spin cluster described by the\nkth bond Hamiltonian (3).\nItisquiteevidentfrom figure1thateachpairofHeisenbergspinsissurroundedbyone coupleof\ntheIsingspinslocatedatlatticesitesofasimpleBethelat ticeandhence,themodelunderconsideration\ncanalternativelybeviewedastheBethelatticeofIsingspi nswhose( fictitious)bondsaredecoratedina\ndiamond-likefashionbytwoquantumHeisenbergspins.Invi ewoffurthermanipulations,itis,therefore,\nofpracticalimportancetorewritethetotalHamiltonian(1 )asasumofbondHamiltonians\nH=N q/2/summationdisplay\nk=1Hk, (2)\nwhereasthebondHamiltonian Hkinvolvesalltheinteractiontermsbelongingtothe kthdiamond-\n43003-2Spin-1/2 Ising-Heisenberg model in the external magnetic fi eld\nshapedclusterspeci ficallydelimitedin figure1byanellipse\nHk= − JH/bracketleftbig\n∆/parenleftbig\nSx\nk1Sx\nk2+Sy\nk1Sy\nk2/parenrightbig\n+Sz\nk1Sz\nk2/bracketrightbig\n−JI/parenleftbig\nSz\nk1+Sz\nk2/parenrightbig/parenleftbig\nµz\nk1+µz\nk2/parenrightbig\n−HB/parenleftbig\nSz\nk1+Sz\nk2/parenrightbig\n−HA\nq/parenleftbig\nµz\nk1+µz\nk2/parenrightbig\n. (3)\nOwingtothevalidityofthecommutationrelationshipbetwe endifferentbondHamiltonians [Hi,Hj]=\n0,thepartitionfunctioncanbepartiallyfactorizedintoap roductofbondpartitionfunctions\nZIHM=/summationdisplay\n{µi}N q/2/productdisplay\nk=1Trkexp(−βHk)=/summationdisplay\n{µi}N q/2/productdisplay\nk=1Zk, (4)\nwhere β=1/(kBT),kBistheBoltzmann ’sconstantand Tistheabsolutetemperature.ThesymbolTr k\ndenotesatraceoverdegreesoffreedomoftwoHeisenbergspi nsfromthe kthdiamond-shapedcluster\nandthesummation/summationtext\n{µi}runsoverallpossiblecon figurationsoftheIsingspins.Thebondpartition\nfunction Zkcanbeevaluatedinthemoststraightforwardwaybyadirectd iagonalizationofthebond\nHamiltonian(3)withintheparticularsubspaceofthe kthHeisenbergspinpairandemployingatrace\ninvarianceofthebondpartitionfunctionwithrespecttoau nitarytransformation.Afterexecutingthis\nprocedureonegainstheresultantexpression,whichimplie sapossibilityofapplyingthegeneralized\ndecoration-iterationtransformation[5 –8]\nZk/parenleftbig\nµz\nk1,µz\nk2/parenrightbig\n= 2exp/bracketleftbiggβHA\nq/parenleftbig\nµz\nk1+µz\nk2/parenrightbig/bracketrightbigg/braceleftbigg\nexp/parenleftbiggβJH\n4/parenrightbigg\ncosh/bracketleftbig\nβJI/parenleftbig\nµz\nk1+µz\nk2/parenrightbig\n+βHB/bracketrightbig\n+exp/parenleftbigg\n−βJH\n4/parenrightbigg\ncosh/parenleftbiggβJH∆\n2/parenrightbigg/bracerightbigg\n=Aexp/bracketleftbigg\nβJeffµz\nk1µz\nk2+βHeff\nq/parenleftbig\nµz\nk1+µz\nk2/parenrightbig/bracketrightbigg\n.(5)\nConsideringfouravailablecombinationsofspinstatesoft woIsingspins µz\nk1andµz\nk2,onegetsfromthe\ntransformationformula(5)threeindependentequationsth atunambiguouslydeterminethemapping\nparameters A,Jeffand Heff\nA=2/parenleftbig\nV+V−V2\n0/parenrightbig1/4,βJeff=ln/parenleftBigg\nV+V−\nV2\n0/parenrightBigg\n,βHeff=βHA+q\n2ln/parenleftbiggV+\nV−/parenrightbigg\n, (6)\nwhichareforsimplicityde finedthroughthefunctions V±and V0\nV±=exp/parenleftbiggβJH\n4/parenrightbigg\ncosh/parenleftbig\nβJI±βHB/parenrightbig\n+exp/parenleftbigg\n−βJH\n4/parenrightbigg\ncosh/parenleftbiggβJH∆\n2/parenrightbigg\n,\nV0=exp/parenleftbiggβJH\n4/parenrightbigg\ncosh/parenleftbig\nβHB/parenrightbig\n+exp/parenleftbigg\n−βJH\n4/parenrightbigg\ncosh/parenleftbiggβJH∆\n2/parenrightbigg\n. (7)\nAtthisstage,thedirectsubstitutionofthealgebraicmapp ingtransformation(5)intothefactorizedform\nofthepartitionfunction(4)leadstoarigorousmappingrel ationship\nZIHM(β,JI,JH,∆,HA,HB,q)=AN q\n2ZIM(β,Jeff,Heff,q), (8)\nwhichconnectsthepartitionfunction ZIHMofthespin-1\n2Ising-Heisenbergmodelonthediamond-like\ndecoratedBethelatticewiththepartitionfunction ZIMofthespin-1\n2Isingmodelonacorresponding\nsimple(undecorated)Bethelatticeschematicallyillustr atedontheright-hand-sideof figure1andmath-\nematicallygivenbytheHamiltonian\nHIM=− JeffN q/2/summationdisplay\n(i,j)µz\niµz\nj−HeffN/summationdisplay\ni=1µz\ni. (9)\nApparently,themappingparameters Jeffand Heffgivenbyequations(6) –(7)determinetheeffective\nnearest-neighbourinteractionandtheeffectivemagnetic fieldofthecorrespondingspin-1\n2Isingmodel\n43003-3J. Strečka, C. Ekiz\nonthesimpleBethelattice,whilethemappingparameter Aisjustasimplemultiplicativefactorinthe\nestablishedmappingrelation(8)betweenbothpartitionfu nctions.\nNow,otherphysicalquantitiesofourparticularinterestf ollowquitestraightforwardly.Forinstance,\nwiththehelpofequation(8),oneeasily findsasimilarmappingrelationbetweenthefreeenergy FIHMof\nthespin-1\n2Ising-Heisenbergmodelonthediamond-likedecoratedBeth elatticeandthefreeenergy FIM\noftheequivalentspin-1\n2IsingmodelonasimpleBethelattice\nFIHM=−kBTlnZIHM=FIM−N qk BT\n2lnA. (10)\nConsequently,thesingle-sitesublatticemagnetizationo ftheIsingspinscanbecalculatedbydifferentiat-\ningthefreeenergy(10)withrespecttotherelevantmagneti cfieldHA\nmA=−1\nN∂FIHM\n∂HA=−1\nN/parenleftbigg∂FIM\n∂βHeff/parenrightbigg∂βHeff\n∂HA=mIM(β,Jeff,Heff). (11)\nAccordingtoequation(11),thesublatticemagnetizationo ftheIsingspinsinthespin-1\n2Ising-Heisenberg\nmodelonthediamond-likedecoratedBethelatticeisequalt othemagnetizationofthecorresponding\nspin-1\n2IsingmodelonthesimpleBethelatticewiththeeffectivene arest-neighbourinteraction Jeffand\ntheeffectivemagnetic fieldHeffgivenby(6) –(7).Asimilarcalculationprocedurecanalsobeperformed\nforobtainingthesingle-sitesublatticemagnetizationof theHeisenbergspins,whichcanbeforconve-\nnienceexpressedintermsofthemagnetization mIMandthenearest-neighbourpaircorrelation εIMof\ntheequivalentspin-1\n2IsingmodelonthesimpleBethelattice\nmB= −1\nN q∂FIHM\n∂HB=1\n2∂lnA\n∂βHB−/parenleftbigg1\nN q∂FIM\n∂βJeff/parenrightbigg∂βJeff\n∂HB−/parenleftbigg1\nN q∂FIM\n∂βHeff/parenrightbigg∂βHeff\n∂HB\n=1\n8/parenleftbiggW+\nV+−W−\nV−+2W0\nV0/parenrightbigg\n+εIM\n2/parenleftbiggW+\nV+−W−\nV−−2W0\nV0/parenrightbigg\n+mIM\n2/parenleftbiggW+\nV++W−\nV−/parenrightbigg\n.(12)\nThenewlyde finedfunctions W±and W0aregivenby\nW±=exp/parenleftbiggβJH\n4/parenrightbigg\nsinh/parenleftbig\nβJI±βHB/parenrightbig\n, W0=exp/parenleftbiggβJH\n4/parenrightbigg\nsinh/parenleftbig\nβHB/parenrightbig\n. (13)\nTocompleteourexactcalculationofbothsublatticemagnet izations,itisnowsu fficienttosubsti-\ntuteintothederivedformulas(11) –(12)therelevantexactresultsforthemagnetizationandne arest-\nneighbourspin-spincorrelationofthecorrespondingspin -1\n2IsingmodelonthesimpleBethelatticewith\ntheeffectivenearest-neighbourinteraction Jeffandtheeffectivemagnetic fieldHeffgivenby(6) –(7).The\nsublatticemagnetizationandspin-spincorrelationfunct ionofthespin-1/2Isingmodelontheundeco-\nratedBethelatticecanberigorouslyfoundwithintheframe workofexactrecursionrelations[9 –13].If\nthesimpleBethelattice(see figure1, figureontheright)is ‘cut’atacentralsitewiththespin µk1,itwill\ndisintegrateinto qidenticalbranchesandthepartitionfunctionofthesystem willtaketheform\nZ=/summationdisplay\nµk1exp(βHeffµk1)/bracketleftbig\ngn(µk1)/bracketrightbigq, (14)\nwhere gn(µk1)isthepartitionfunctionofaseparatebranch\ngn(µk1)=/summationdisplay\nµk2exp(βJeffµk1µk2+βHeffµk2)/bracketleftbig\ngn−1(µk2)/bracketrightbigq−1. (15)\nByusingof(15),onecaneasilyobtainarecursionrelations hipforthevariable xn=gn(−1/2)\ngn(+1/2)\nxn=exp/parenleftBig\n−βJeff\n4+βHeff\n2/parenrightBig\n+exp/parenleftBigβJeff\n4−βHeff\n2/parenrightBig\nxq−1\nn−1\nexp/parenleftBigβJeff\n4+βHeff\n2/parenrightBig\n+exp/parenleftBig\n−βJeff\n4−βHeff\n2/parenrightBig\nxq−1\nn−1. (16)\n43003-4Spin-1/2 Ising-Heisenberg model in the external magnetic fi eld\nEventhoughtheparameter xndoesnothaveadirectphysicalsense,itplaysacrucialrole indetermining\nthecanonicalensembleaveragesofallphysicalquantities inthelimit n→ ∞.Forinstance,oneeas-\nilyobtainsthefollowingexpressionsforthemagnetizatio nandnearest-neighbourspin-spincorrelation\nfunctionofthespin-1/2IsingmodelontheBethelattice\nmIM=1\n2exp/parenleftbig\nβHeff/parenrightbig\n−xq\nexp/parenleftbig\nβHeff/parenrightbig\n+xq,\nεIM=1\n4exp/parenleftBigβJeff\n4+βHeff/parenrightBig\n−2exp/parenleftBig\n−βJeff\n4/parenrightBig\nxq−1+exp/parenleftBigβJeff\n4−βHeff/parenrightBig\nx2q−2\nexp/parenleftBigβJeff\n4+βHeff/parenrightBig\n+2exp/parenleftBig\n−βJeff\n4/parenrightBig\nxq−1+exp/parenleftBigβJeff\n4−βHeff/parenrightBig\nx2q−2.(17)\nwhichcanbebothexpressedthroughastable fixedpoint x=limn→∞xnoftherecurrencerelation(16).\n3.Resultsanddiscussion\n/s99/s50/s99/s49/s83/s80/s80\n/s67/s70/s80/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s81/s70/s80/s72/s32/s61/s32/s74/s72/s40/s45/s49/s41/s47/s50/s32/s43/s32/s124/s74/s73/s124\n/s72/s32/s61/s32/s113/s91/s74/s72/s32/s40/s45/s49/s41/s47/s50/s32/s45/s32/s124/s74/s73/s124/s93/s47/s40/s113/s45/s50/s41/s72/s32 /s61/s32 /s113/s124/s74\n/s73/s124/s32\n/s72/s32\nFigure2. The general ground-state phase diagram in\nthe∆−Hplane.Inthispart,letusproceedtoadiscussionof\nthemostinterestingresultsobtainedforthefer-\nrimagneticversionofthespin-1\n2Ising-Heisenberg\nmodelonthediamond-likedecoratedBethelat-\nticewiththeferromagneticHeisenberginterac-\ntion JH>0andtheantiferromagneticIsingin-\nteraction JI<0,which,atsu fficientlylow fields,\nwill favour the antiparallel alignment between\nthe nearest-neighbouring Ising and Heisenberg\nspins,respectively.Itisworthwhiletoremarkthat\nthecriticalbehaviouroftheconsideredmodelin\nthe absence of the external magnetic field has\nbeeninvestigatedinsomedetailinourprevious\nwork[14]andhence,theeffectofanon-zeromag-\nneticfieldwillbeatthemainfocusofourresearch\ninterest.Toreducethetotalnumberoffreepa-\nrameters,themostnotablefeaturesofthemagnetizationpr ocesswillbeillustratedforaspeci ficchoice\nH≡HA=HB,whichcoincideswithsettingLandég-factorsoftheIsinga ndHeisenbergspinsequalto\neachother.\nFirst,letuscommentonpossiblespinarrangementsemergin gatzerotemperature.Owingtothe\nvalidityofthecommutationrelationshipbetweenthediffe rentclusterHamiltonians,theground-state\nspinarrangementscaneasilybeobtainedbysearchingforth elowest-energyeigenstateofthecluster\nHamiltonian(3).Theground-statephasediagramdisplayed infigure2impliestheexistenceofthree\ndifferentgroundstates,whichcanbethoroughlycharacter izedbythefollowingeigenvectors\n|CFP〉 =N/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleµz\nk=−1\n2/angbracketrightbiggN q/2/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk1=1\n2/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk2=1\n2/angbracketrightbigg\n,\n|QFP〉 =N/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleµz\nk=sgn(H)1\n2/angbracketrightbiggN q/2/productdisplay\nk=11/rad〉callow\n2/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk1=1\n2/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk2=−1\n2/angbracketrightbigg\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk1=−1\n2/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk2=1\n2/angbracketrightbigg/parenrightbigg\n,\n|SPP〉 =N/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleµz\nk=1\n2/angbracketrightbiggN q/2/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk1=1\n2/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk2=1\n2/angbracketrightbigg\n. (18)\nAscouldbeexpected,twogroundstatescorrespondtoclassi calspinarrangementswithaperfectparal-\nlelandantiparallelalignmentsbetweenthenearest-neigh bourIsingandHeisenbergspinstobefurther\nreferredtoastheclassicalferrimagneticphase(CFP)andt hesaturatedparamagneticphase(SPP),respec-\ntively.Apartfromthoserathertrivialphases,onemayalso detectamorespectacularquantumfrustrated\nphase(QFP)withapeculiarspinfrustrationoftheIsingspi nsstemmingfromaquantumentanglement\n43003-5J. Strečka, C. Ekiz\noftheHeisenbergspinpairs.Asamatteroffact,theemergen tquantumsuperpositionoftwopossible\nantiferromagneticstatesoftheHeisenbergspinpairsisre sponsibleinQFPforacompleterandomness\noftheIsingspinsatazeromagnetic fieldasconvincinglyevidencedinourpreviousstudy[14].Du eto\nthespinfrustration,alltheIsingspinstendtoalignintot heexternal- fielddirectionforarbitrarybut\nnon-zeromagnetic fieldand,consequently,astriking quantumferrimagneticphase developsfromQFP\nwithafullpolarizationoftheIsingspinsandthenon-magne ticnatureoftheHeisenbergspinpairs.The\nexistenceofQFPaloneseemstobeaquitegeneralfeatureoft heIsing-Heisenbergmodels,whereamu-\ntualcompetitionbetweentheeasy-axisIsinginteractiona ndtheeasy-planeHeisenberginteractiontakes\nplace[15,16].Furthermore,allphasetransitionsbetween threeavailablegroundstatesareofthe first\norderandtheirexplicitformisgivenin figure2alongthedepictedphaseboundaries.\n/s48 /s49 /s50 /s51 /s52 /s53/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\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48\n/s113 /s32/s61/s32/s51\n/s40/s97 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32 /s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s48/s53\n/s48 /s49 /s50 /s51 /s52 /s53/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/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s52\n/s48 /s49 /s50 /s51 /s52 /s53/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/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s54\n/s48 /s49 /s50 /s51 /s52 /s53/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/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s50\nFigure3. (Color online) The total and sublattice magnetizations as a function of the external magnetic\nfield for the spin-1\n2Ising-Heisenberg model withthecoordinationnumber q=3,theinteractionratio\nJH/|JI|=1.0,theexchangeanisotropy ∆=1.0andfourdifferenttemperatures.\nNow,letusillustratetypicalmagnetizationscenariosasd isplayedin figures3 –5forthespin-1\n2Ising-\nHeisenbergmodelonthediamond-likedecoratedBethelatti cewiththecoordinationnumber q=3,the\nspecificvalueoftheinteractionratio JH/|JI| =1.0,threedifferentvaluesoftheexchangeanisotropy\n∆andseveraltemperatures.Itisworthwhiletoremarkthatth etotalsingle-sitemagnetization mT≡/parenleftbig\nmA+qm B/parenrightbig\n/(1+q)isalsoplottedin figures3 –5inadditiontobothsublatticemagnetizations mAand\nmBoftheIsingandHeisenbergspins,respectively.Iftheexch angeanisotropyisselectedbelowits first\ncriticalvalue ∆<∆c1≡1+2|JI|/JH,then,oneencountersarathertypicalmagnetizationcurve reflecting\nthefield-inducedtransitionfromCFPtoSPPasshownin figure3.Itisquiteclearthattheintermedi-\natemagnetizationplateauobservedatahalfofthesaturati onmagnetizationindeedcorrespondstothe\nclassicalferrimagneticspinarrangementinherenttoCFPa ndthemagnetizationplateaugraduallydi-\nminishesuponincreasingthetemperature.Themostsigni ficantchangesinthedisplayedmagnetization\ncurveevidentlyoccurifthetemperatureisselectedslight lyabovethecriticaltemperatureofCFP(note\nthat kBTc/|JI|≈0.5for∆=1).Eventhoughbothsublatticemagnetizationsalreadystar tfromzerointhis\nparticularcase,theyobviouslytendtowardstypicalmagne tizationvaluesforCFPstillbearingevidence\nofanintermediatemagnetizationplateauatmoderate fieldsandtemperatures[see figure3(c)].\n43003-6Spin-1/2 Ising-Heisenberg model in the external magnetic fi eld\n/s48 /s49 /s50 /s51 /s52 /s53/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/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/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s48/s53\n/s48 /s49 /s50 /s51 /s52 /s53/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/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s50\n/s48 /s49 /s50 /s51 /s52 /s53/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/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s51\n/s48 /s49 /s50 /s51 /s52 /s53/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/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s50\nFigure4.(Coloronline)Thetotalandsublatticemagnetizationsasa functionoftheexternalmagnetic\nfieldforthespin-1\n2Ising-Heisenbergmodelwiththecoordinationnumber q=3,theinteractionratio\nJH/|JI|=1.0,theexchangeanisotropy ∆=4.0andfourdifferenttemperatures.\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s97 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32\n/s109\n/s66/s32/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s48/s53\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s98 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32\n/s109\n/s66/s32/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s51\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s99 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32\n/s109\n/s66/s32/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s54\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s100 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32\n/s109\n/s66/s32/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s50\nFigure5.(Coloronline)Thetotalandsublatticemagnetizationsasa functionoftheexternalmagnetic\nfieldforthespin-1\n2Ising-Heisenbergmodelwiththecoordinationnumber q=3,theinteractionratio\nJH/|JI|=1.0,theexchangeanisotropy ∆=6.0andfourdifferenttemperatures.\n43003-7J. Strečka, C. Ekiz\nHowever,themostinterestingmagnetizationprocesscanbe foundiftheexchangeanisotropyisse-\nlectedfromtheinterval ∆∈(∆c1,∆c2)with∆c2≡1+2(q−1)|JI|/JH.Underthiscondition,atlowenough\ntemperatures,thetotalmagnetizationexhibitstwosucces sivefractionalmagnetizationplateausatone\nquarterandonehalfofthesaturationmagnetization[see figures4(a) –(b)],whichendupattwodifferent\nfield-inducedtransitionsfromQFPtoCFPand,respectively, fromCFPtoSPP.Thelowermagnetization\nplateauatonequarterofthesaturationmagnetizationgive saclearevidenceofQFP,becausethetotal\nmagnetizationstartsfromzeroanditbecomesnon-zeromain lyduetothe field-inducedalignmentof\nthefrustratedIsingspins.Moreover,itisquiteinteresti ngtoobservefrom figure4thattheformer field-\ninducedtransitionbetweenQFPandCFPismuchsharperatagi ventemperaturethanthelatter field-\ninducedtransitionbetweenCFPandSPP.Ofcourse,therelev antmagnetizationcurvebecomessmoother\nuponincreasingtemperatureuntilbothmagnetizationplat eauscompletelydisappearfromthemagneti-\nzationprocessaboveacertaintemperature( kBT/|JI|≈0.5for∆=4.0).\nLastbutnotleast,themagnetizationcurvewithoutthehigh erintermediatemagnetizationplateau\natahalfofthesaturationmagnetizationcanbedetectedwhe nevertheexchangeanisotropyexceedsits\nsecondcriticalvalue ∆>∆c2.Inagreementwiththeground-statephasediagramshownin figure2,the\nlow-temperaturemagnetizationcurvedisplaysadirect field-inducedtransitionfromQFPtowardsSPP\nwithoutpassingthroughanothermagnetizationplateauCFP .Forillustration,themagnetizationscenario\nofthistypeisdepictedin figure5.Itisworthnotingthatthe field-inducedpolarizationoftheHeisen-\nbergspins,whichappearsinthevicinityofthesaturation field,maycause,atmoderatetemperatures,\natransientloweringofthesublatticemagnetizationofthe Isingspinsasitcanbeclearlyseenin fig-\nures5(b) –(c).Thepartialloweringofthesublatticemagnetizationo ftheIsingspinscanbeattributed\ntoaspinreorientationoftheHeisenbergspinstowardsthee xternal- fielddirectionandatendencyof\nthenearest-neighbourIsingandHeisenbergspinstoaligna ntiparallelwithrespecttoeachotherdueto\ntheantiferromagneticinteractionin-betweenthem.Furth ermore, figure5(d)showsaninterestingcross-\ningofbothsublatticemagnetizations,whichoccursatsu fficientlyhightemperaturesonaccountofthe\nantiferromagneticcorrelationsbetweenthenearest-neig hbourIsingandHeisenbergspins.\nFigure6.(Coloronline)Acolormapofthetotalmagnetizationasafun ctionofthedimensionlesstemper-\natureandexternalmagnetic fieldforthespin-1\n2Ising-Heisenbergmodelonthediamond-likedecorated\nBethelatticewiththecoordinationnumber q=3,theinteractionratio JH/|JI|=1.0andthreedifferent\nvaluesoftheexchangeanisotropy:(a) ∆=1.0;(b)∆=4.0;(c)∆=6.0.\nLetusconcludeouranalysisofthemagnetizationprocessby fewcommentsonacolormapofthetotal\nmagnetizationdepictedin figure6asafunctionoftemperatureandexternalmagnetic field.Accordingto\nauniquecolormaplabellingusedin figure6,twofractionalvaluesofthetotalmagnetization mT=0.125\nand 0.25thatcorrespondtotheintermediatemagnetizationplateau sassociatedwiththeappearanceof\nQFPandCFParedisplayedbycyanandgreencolor,respective ly.Ascouldbeexpected,thequiteextensive\ngreenregionin figure6(a)indicatesaratherwidemagnetizationplateauata halfofthesaturation\nmagnetizationemergingforrelativelyweakexchangeaniso tropies∆<∆c1,whilethewidecyanregion\ninfigure6(c)impliestheexistenceofarelativelyrobustmagne tizationplateauatonequarterofthe\nsaturationmagnetizationforstrongenoughexchangeaniso tropies∆>∆c2.Hence,iffollowsthatthe\nmoststrikingmagnetizationpro filewithtwosuccessiveintermediatemagnetizationplateau smightbe\nindeedexpectedfortheintermediateexchangeanisotropie s∆∈(∆c1,∆c2).Infact, figure6(b)serves\n43003-8Spin-1/2 Ising-Heisenberg model in the external magnetic fi eld\ninevidenceofthepresenceofbothintermediatemagnetizat ionplateaus,whicharegraduallysmudged\nbythermal fluctuationsastemperatureincreases.Interestingly,ittu rnsoutthatthelowerfractional\nmagnetizationplateaupertinenttoQFPdiminishesmuchmor esteadilywithanincreasingtemperature\nincomparisonwiththehigherfractionalmagnetizationpla teaupertinenttoCFP,whichseemstobemuch\nmoreresistantagainstthermal fluctuations.\n4.Conclusion\nThepresentworkdealswiththespin-1\n2Ising-Heisenbergmodelondiamond-likedecoratedBethela t-\nticesinthepresenceofthelongitudinalmagnetic field.Exactsolutionfortheinvestigatedmodelhas\nbeenobtainedbycombiningthedecoration-iterationmappi ngtransformationwiththemethodofexact\nrecursionrelations.Theformertransformationmethodmak esitpossibletoestablisharigorousmapping\nrelationshipwiththeequivalentspin-1\n2IsingmodelonasimpleBethelattice,whichissubsequently ex-\nactlytreatedwithintheframeworkofthelattermethodbase donexactrecursionrelations.Exactresults\nforthepartitionfunction,Gibbsfreeenergy,totalandbot hsublatticemagnetizationswerederivedby\nmakinguseofthisrigorousapproach.\nOurparticularattentionwasfocusedonexploringthegroun dstateandlow-temperaturemagnetiza-\ntionprocessoftheferrimagneticversionofthemodelconsi dered.Themostinteresting findingstemming\nfromourpresentstudyisanexactevidenceofaratherdivers emagnetizationprocess.Asamatteroffact,\nwehavedemonstratedthreedifferentmagnetizationscenar ioswithuptotwodifferentfractionalmagne-\ntizationplateaus,whereastheintermediatemagnetizatio nplateaumayeithercorrespondtotheclassical\nferrimagneticspinarrangementand/orthequantumferrima gneticspinorderingwithoutanyclassical\ncounterpart.Theoriginofthestrikingquantumferrimagne ticphaseliesinapeculiarspinfrustrationof\ntheIsingspins,whichcomesfromthenonmagneticnatureoft heHeisenbergspinpairsgovernedbythe\nsymmetricquantumsuperpositionoftheirtwointrinsicall yantiferromagneticspinstates.\nAcknowledgements\nThisworkwassupportedbytheScienti ficGrantAgencyofMinistryofEducationofSlovakRepublic\nundertheVEGAGrantNo.1/0234/12andbyERDFEU(EuropeanUn ionEuropeanregionaldevelopment\nfund)grantunderthecontractITMS26220120005(activity3 .2.).\nReferences\n1. MillerJ.S.,DrillonM.,Magnetism:MoleculestoMateria ls,Wiley,Weinheim,2001.\n2. LacroixC.,MendelsP.,MilaF.,IntroductiontoFrustrat edMagnetism,Springer,Berlin,2011.\n3. SyoziI.,Prog.Theor.Phys.,1951, 6,341;doi:10.1143/PTP.5.341.\n4. SyoziI.,In:PhaseTransitionandCriticalPhenomena,Vo l.1,ed.byDombC.,GreenM.S.,AcademicPress,New\nYork,1972,pp.269 –329.\n5. FisherM.E.,Phys.Rev.,1959, 113,969;doi:10.1103/PhysRev.113.969.\n6. RojasO.,ValverdeJ.S.,deSouzaS.M.,PhysicaA,2009, 388,1419;doi:10.1016/j.physa.2008.12.063.\n7. RojasO.,deSouzaS.M.,J.Phys.A:Math.Theor.,2011, 44,245001;doi:10.1088/1751-8113/44/24/245001.\n8. StrečkaJ.,Phys.Lett.A,2010, 374,3718;doi:10.1016/j.physleta.2010.07.030.\n9. BaxterR.J.,ExactlySolvedModelsinStatisticalMechan ics,Academic,NewYork,1982.\n10. ThompsonC.J.,J.Stat.Phys.,1982, 27,441;doi:10.1007/BF01011085.\n11. MukamelD.,Phys.Lett.,1974, 50A,339;doi:10.1016/0375-9601(74)90050-4.\n12. OhanyanV.R.,AnanikyanL.N.,AnanikianN.S.,PhysicaA ,2007,377,501;doi:10.1016/j.physa.2006.11.034.\n13. IzmailianN.Sh.,HuChin-Kun,PhysicaA,1998, 254,198;doi:10.1016/S0378-4371(98)00193-9.\n14. StrečkaJ.,EkizC.,ActaPhys.Pol.A,2010, 118,725.\n15. StrečkaJ.,Ja ščurM.,ActaPhys.Slovaca,2006, 56,65.\n16. EkizC.,StrečkaJ.,Ja ščurM.,J.Magn.Magn.Mater.,2011, 323,493;doi:10.1016/j.jmmm.2010.10.001.\n43003-9J. Strečka, C. Ekiz\nПроцеснамагнiченостiвточнорозв’язнiйспiн-1/2моделi\nIзинга-ГайзенберганадекорованихграткахБете\nЙ.Стречка1,С.Екiз2\n1Природничийфакультет ,Унiверситет iм.П.Й.Шафарика ,Кошiце,Словацькареспубл iка\n2Природничийфакультет ,Унiверситет iм.АднанаМендереса ,Айдин090 10,Туреччина\nСпiн-1/2модель Iзинга-Гайзенберганаромбопод iбнiйдекорован iйгратц iБетерозв ’язаноточноупри -\nсутност iпоздовжньогомагн iтногополя ,поєднуючидекорац iйно-iтерацiйнеперетвореннязметодомто -\nчнихрекурсивнихсп iввiдношень .Зокрема ,детальнодосл iдженоосновнийстан iнизькотемпературне\nнамагн iченняферимагн iтноїверс iїрозглянутоїмодел i.Знайденотрир iзнiсценарiїнамагн iченостiзщо-\nнайбiльшедвомапосл iдовнимидробовимиплато ,депром iжнеплатонамагн iченостiможев iдповiдати\nкласичномуферимагн iтномусп iновомувпорядкуваннюта /абоiндукованомуполемквантовомуферима -\nгнiтномусп iновомувпорядкуванню ,якенемаєжодногокласичногоаналога .\nКлючовiслова:модель Iзинга-Гайзенберга ,граткаБете ,точнiрезультати ,платонамагн iченостi\n43003-10"    },    {        "title": "1304.4459v1.Kinetic_arrest_related_to_a_first_order_ferrimagnetic_to_antiferromagnetic_transition_in_the_Heusler_compound_Mn2PtGa.pdf",        "content": "arXiv:1304.4459v1  [cond-mat.mtrl-sci]  16 Apr 2013Nayak et al.\nKinetic arrest related to a first-order ferrimagnetic to ant iferromagnetic\ntransition in the Heusler compound Mn 2PtGa\nAjaya K. Nayak,1,a)Michael Nicklas,1Chandra Shekhar,1and Claudia Felser1,2\n1)Max Planck Institute for Chemical Physics of Solids, 01187 D resden, Germany\n2)Institut f¨ ur Anorganische und Analytische Chemie, Johann es Gutenberg Universit¨ at, 55099 Mainz,\nGermany\n(Dated: 24 July 2018)\nWe report a magnetization study of the Heusler compound Mn 2PtGa that shows the existence of a magnetic-\nglass state. Mn 2PtGa shows a first-order ferromagnetic (FM)/ferrimagnetic (FI ) to antiferromagnetic (AFM)\ntransition in contrast to the martensitic structural transition ob served in several Heusler alloys. The kinetic\narrest of this first-order FM (FI) to AFM transition leads to the ob served magnetic-glass behavior. We show\nthat the strength of the applied magnetic field, which is the primary p arameter to induce the magnetic-glass\nstate, is also responsible for the stability of the supercooled FM (FI ) phase in time.\nI. INTRODUCTION\nFirst-ordermagnetic to magnetic phase transition pos-\nsessesagreatresearchinterestduetotheexistenceofvar-\nious anomalous magnetic behaviors. Though there were\nseveral previous studies on first-order magnetic phase\ntransition (FOMT), a detailed study was reported in the\nintermetallic compounds doped CeFe 21,2Gd5Ge43and\nin some of the phase separated manganites.4,5Follow-\ning this, several studies related to FOMT have been re-\nportedin othersystemssuchasNd 7Rh36and cobaltites.7\nThe disordered-broadened first-order transition in most\nof these compounds arises mainly due to the presence of\nintrinsicdisorder,whichserveasnucleationcentersbelow\nacertaintemperature. Thisfirst-ordertransitionleadsto\nsupercooling, field-induced irreversibility and phase co-\nexistence, that can be eventually termed as a magnetic-\nglass phase.6–8This phase is different from the spin glass\none as the glassy state in these systems can be obtained\nby application of external parameters like magnetic field.\nThe Ni 2Mn1+xZ1−xbased Heusler alloys show quite\nsimilar types of magnetic properties that of systems\nshowing a magnetic-glass phase. However, these al-\nloys exhibit a structural transition from a high tem-\nperature cubic austenite phase to a low temperature\ntetragonal/orthorhombic martensitic phase with strong\nmagneto-structural coupling.9With help of field cooling\nit is possible to arrest the high temperature austenite\nphase below the martensitic transition temperature.10,11\nIt is also observed that one can get large field induced\nirreversibility in the magnetization loops when measured\nnear to the martensitic transition temperature.12,13In\ncontrast to the other systems showing magnetic-glassbe-\nhavior, the field induced structural change in the Heusler\nalloys plays a major role in inducing the irreversibili-\nties that become stronger on approaching the marten-\nsitic transition upon increasing the temperature. How-\never, the other magnetic-glass systems show irreversible\na)Electronic mail: nayak@cpfs.mpg.debehavior at the lowest temperatures that vanishes by in-\ncreasing the temperature. In order to explore more func-\ntional as well as fundamental properties in the Heusler\nfamily, we prepare a new Heusler compound Mn 2PtGa.14\nIn this report we show the existence of a magnetic-glass\nphase in Mn 2PtGa that does not show any structural\ntransition. With help of dc-magnetization measurements\nwe show that Mn 2PtGa undergoes a first-order ferro-\nmagnetic (FM)/ferrimagnetic (FI) to antiferromagnetic\n(AFM) transition below the magnetic ordering tempera-\nture, which is responsible for the observed effect.\nII. EXPERIMENTAL DETAILS\nPolycrystalline ingots of Mn 2PtGa were prepared by\narc melting stoichiometric amounts of the constituent\nelements in a high purity argon atmosphere. The as-\nprepared ingots were annealed at 1273 K in an evacu-\natedquartztubeforoneweekandsubsequentlyquenched\nin an ice-water mixture. The samples were structurally\ncharacterized by x-ray powder diffraction (XRD) using\nCu-Kαradiation. Magnetizationmeasurementswerecar-\nried out on the sample using a superconducting quan-\ntum interference device (SQUID) vibrating sample mag-\nnetometer (VSM).\nIII. EXPERIMENTAL RESULT\nThe powder XRD measurement at room temperature\nreveals that Mn 2PtGa crystallizes in a tetragonal struc-\nture with space group I-4m2. M(T) measured in zero\nfield cooled (ZFC), field cooled (FC) and field heated\n(FH) modes is shown in Fig.1. In the ZFC mode, the\nsample was initially cooled to 2K and the data were\ntaken upon increasing the temperature in applied field.\nIn the FC mode, the data were collected while cooling\nin magnetic field and subsequently in FH mode the data\nwerecollected during heating. All measurementsare per-\nformed with temperature sweep rate of 3 K/min. It is\nfound that Mn 2PtGa undergoes a paramagnetic (PM) to2\nFM (FI) transition around 230 K followed by a FM (FI)\nto AFM transition at 150 K. The small magnetic mo-\nmentobservedinvariousmagneticmeasurementssuggest\nthatMn 2PtGaordersferrimagneticallylikeotherMn 2YZ\nbased compounds.15,16This is because the Mn atoms oc-\ncupy two different crystallographic positions with oppo-\nsite spin alignment. The non-zero nature of the FC/FH\nmagnetization curves below the transition temperature\nindicates that the low temperature phase is not perfectly\nAFM in nature. The first-order nature of the low tem-\nperaturetransitionisconfirmedbythepresenceofather-\nmal hysteresis between the FC and FH curves. As the\nsample shows a tetragonalcrystalstructure at roomtem-\nperature, the low temperature transition should not be\na structural phase transition, instead a pure magnetic to\nmagnetic one.\nTo probe the existence of a magnetic-glass state in\nMn2PtGa, we haveperformed M(T) measurement in 1 T\nafter cooling the sample in different fields. For this the\nsample was cooled down to 2 K in presence of a cool-\ning fields HCFfrom room temperature. At 2 K, the field\nwaschangedtothemeasurementfield HMFandthemag-\nnetization was measured in heating mode. From Fig.2\ntwo types of magnetic behaviors are observed depending\nupon the strength of cooling fields. For HCF≤HMF,\ntheM(T) curves show an AFM to FM (FI) transition\non increasing temperature. The increase in magnetiza-\ntion with higher cooling fields at low temperatures is due\nto the increase in the super cooled FM (FI) phase in\nan AFM background. However, for HCF> HMFthe\nmagnetizationdecreaseswithincreaseintemperaturefol-\nlowed by an increase around the AFM to FM transition\ntemperature. A higher cooling field will enable a higher\namount of supercooled FM (FI) phase. Hence, when the\nfield is reduced to a lower value the system will try to\nacquire the equilibrium state for that field by a partial\nconversion of the FM (FI) to AFM phase. This results\nin an initial decrease in the magnetization at low tem-\nperature. It can be mentioned here that irrespective of\nthe strength of cooling field the AFM to FM transition\ntemperature does not change for a fixed measurement\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/s72/s61/s48/s46/s49/s32/s84/s32/s90/s70/s67\n/s32/s70/s67\n/s32/s70/s72/s77/s32/s91\n/s66/s47/s102/s46/s117/s46/s93\n/s84/s32/s91/s75/s93\nFIG. 1. (Color online)Temperature dependence of magneti-\nzation,M(T), measured at 0.1 T./s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s72\n/s67/s70\n/s32/s77/s32/s91\n/s66/s47/s102/s46/s117/s46/s93\n/s84/s32/s91/s75/s93/s32/s48/s46/s48/s49/s32/s84\n/s32/s48/s46/s49/s32/s84\n/s32/s48/s46/s51/s32/s84\n/s32/s48/s46/s53/s32/s84\n/s32/s48/s46/s55/s32/s84\n/s32/s49/s32/s84\n/s32/s49/s46/s50/s32/s84\n/s32/s49/s46/s53/s32/s84\n/s32/s50/s32/s84/s72\n/s77/s70/s61/s49/s32/s84\nFIG. 2. (Color online) FH M(T) curves measured at 1 T\nwith heating rate of 3 K/min after field cooling the sample in\nvarious fields.\nfield. The above observation confirms that there exists\na phase coexistence and a magnetic-glass state in the\npresent sample.\nFor a better knowledge of the stability of the super-\ncooled FM (FI) phase we have studied the temperature\nsweep-rate dependency of the field cooled magnetization\nas shown in Fig.3. For this a field is applied at room\ntemperature and the measurement is performed with a\ncooling rate of 5 K/min, 1 K/min and 0.2 K/min, re-\nspectively. As seen from Fig.3 the magnetization mono-\ntonically decreases with decreasing cooling rate. This\nsuggests that with higher cooling rate one can quench\nthe high temperature FM (FI) phaseresulting in a higher\namountofFM (FI) componentsin the AFM phase. With\na decrease in the cooling rate the supercooled FM (FI)\ncomponents get a sufficient amount of time to achieve\nthe equilibrium AFM state that results in a lower mag-\nnetization value. The process also gives rise to a slight\nincrease in the FM (FI) to AFM transition temperature\nwith lower cooling rate. As both field and cooling rate\nare responsible for the kinetic arrest of the supercooled\nFM (FI) phase, it is interestingto seehowthemagnetiza-\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s52/s48/s48/s46/s52/s53/s48/s46/s53/s48/s48/s46/s53/s53\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s53/s57/s48/s46/s54/s48/s48/s46/s54/s49/s48/s46/s54/s50/s48/s46/s54/s51/s48/s46/s54/s52\n/s72/s61/s48/s46/s53/s32/s84\n/s84/s32/s91/s75/s93/s77/s32/s91\n/s66/s47/s102/s46/s117/s46/s93\n/s32/s53/s32/s75/s47/s109/s105/s110\n/s32/s49/s32/s75/s47/s109/s105/s110\n/s32/s48/s46/s50/s32/s75/s47/s109/s105/s110/s72/s61/s49/s32/s84\n/s84/s32/s91/s75/s93/s77/s32/s91\n/s66/s47/s102/s46/s117/s46/s93\n/s32/s32\n/s32/s53/s32/s75/s47/s109/s105/s110\n/s32/s49/s32/s75/s47/s109/s105/s110\n/s32/s48/s46/s50/s32/s75/s47/s109/s105/s110\nFIG. 3. (Color online) FC M(T) curves measured at 0.5 T\nwith different cooling rates. Inset shows FC M(T) curves\nmeasured at 1 T with different cooling rates.3\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s72 /s32/s91/s84/s93\n/s32/s77/s32/s91\n/s66/s47/s102/s46/s117/s46/s93\n/s32/s90/s70/s67\n/s32/s70/s67/s32/s97/s116/s32/s48/s46/s53/s32/s84\n/s32/s70/s67/s32/s97/s116/s32/s49/s32/s84\n/s32/s70/s67/s32/s97/s116/s32/s53/s32/s84/s84/s61/s50/s32/s75\nFIG. 4. (Color online) Field dependenceofthemagnetizatio n,\nM(H), measured at 2 K with field sweep rate of 0.01 T/sec\nafter field cooling in 0, 0.5 T, 1 T and 5 T.\ntion responds to the cooling rate in higher cooling fields.\nThe cooling rate dependence of magnetization measured\nin a field of 1 T is shown in inset of Fig.3. It is found\nthatthemagnetizationchangesby5%asthecoolingrate\nchangesfrom 5 K/minto 0.2K/min in case ofa measure-\nment performed in 0.5 T field. Whereas, it is only about\n1.7 % when the measurement is performed in 1 T. As the\nfield stabilizes the FM (FI) phase, with higher cooling\nfield the supercooled FM (FI) phase requires more time\nto achieve the equilibrium AFM phase. Hence, the effect\nofthe coolingratebecomeslessprominent athigherfield.\nFigure4 shows ZFC and FC M(H) loops measured\nat 2 K. In case of FC measurements a field is applied\nat room temperature and subsequently the sample was\ncooled down to 2 K. It is found that the ZFC M(H) un-\ndergoes a field induced metamagnetic transition around\n4.8 T. This metamagnetic transition corresponds to the\nfirst-order AFM to FM (FI) phase transition. The pres-\nence ofa metamagnetictransitionin the lowtemperature\nZFCM(H) loop confirms the existence of a FM (FI) to\nAFM transition in Mn 2PtGa. To verify the kinetic ar-\nrest of the supercooled FM (FI) phase as observed in\ntheM(T) data in Fig.2 and Fig.3, we have measured\ntheM(H) loops in different field cooled protocols. The\nM(H) loop measured after FC in 0.5 T shows a higher\nmagnetization value than the ZFC curve. The 1 T and\n5 T FC curves follow a similar increasing trend as the\n0.5 T FC curve. The critical field for the metamagnetic\ntransition also increases to 5 T and 5.5 T for FC at 0.5 T\nand 1 T, respectively. No metamagnetic transition can\nbe observed for the M(H) loop measured after field cool-\ning in 5 T. From the above observations it is clear that\ndepending on the strength of the cooling field a certain\namount of the high temperature FM (FI) phase is su-\npercooled at low temperatures that gives rise to larger\nmagnetization value than the ZFC curve. For 5 T field\ncooling, the sample transfers to the FM (FI) phase in\nthe whole temperature range below the magnetic order-\ning temperature and hence no metamagnetic transition\nis observed.Alltheabovemeasurementsshowthatthefieldanneal-\ning acrossthe first-ordertransition forms a new magnetic\nstate at low temperatures. The observation of unequal\nmagneticbehaviorswhenthesampleiscooledwithafield\nsmaller/larger than the measuring field clearly indicates\nthe kinetic arrest of the FM (FI) phase. This is further\nsupported by the cooling rate dependence of magneti-\nzation measurements, which show that the supercooled\nFM (FI) phase is highly unstable in time. The stability\nof the supercooled phase can be increased by higher ap-\npliedfield. Theevidenceforthe FM(FI) phaseatthelow\ntemperature is also seen from the M(H) loops measured\nafter different field cooling procedure. Furthermore, the\npresent magnetic behaviors are well matched with that\nobserved in various magnetic-glass systems that show\na kinetic arrest.7?Therefore, it is confirmed that the\nMn2PtGa possesses a magnetic-glass phase.\nIV. CONCLUSIONS\nIn conclusion, we have studied the existence of a\nmagnetic-glass phase derived from the first-order FM\n(FI) to AFM transition in the Heusler compound\nMn2PtGa. With help of field annealing across the first-\nordertransitionweshowthat thekineticarrestoftheFM\n(FI) phase that gives rise to phase coexistence is respon-\nsible for the magnetic-glass phase. We also show that\nthe supercooled FM (FI) phase is more stable in time in\npresence of higher fields.\nACKNOWLEDGMENTS\nThis work was financially supported by the Deutsche\nForschungsgemeinschaft DFG (Project Nos. TP 1.2-A\nand TP 2.3-A of Research Unit FOR 1464 ASPIMATT).\n1M. A. Manekar, et al., Phys. Rev. B 64, 104416 (2001).\n2M. Manekar, et al., J. Phys.: Condens. Matter 14, 4477 (2002) .\n3E. M. Levin, K. A. Gschneidner, Jr., and V. K. Pecharsky, Phys .\nRev. B65, 214427 (2002).\n4R. Mahendiran, et al., Phys. Rev. Lett. 89, 286602 (2002).\n5V. Hardy, et al., Phys. Rev. B 68, 220402 (2003).\n6K. Sengupta and E. V. Sampathkumaran, Phys. Rev. B 73,\n020406 (2006).\n7T. Sarkar, V. Pralong, and B. Raveau, Phys. Rev. B 83, 214428\n(2011).\n8M. K. Chattopadhyay, S. B. Roy, and P. Chaddah, Phys. Rev. B\n72, 180401 (2005).\n9T. Krenke, E. Duman, M. Acet, E. F. Wassermann, X. Moya, L.\nMa˜ nosa, and A. Planes Nature mater. 4, 450 (2005).\n10W. Ito, et al., Appl. Phys. Lett. 92, 021908 (2008).\n11A. K. Nayak, K. G. Suresh, and A. K. Nigam, J. Appl. Phys.\n108, 063915 (2010).\n12A. K. Nayak, K. G. Suresh, and A. K. Nigam, Appl. Phys. Lett.\n96, 112503 (2010).\n13V. K. Sharma, M. K. Chattopadhyay, and S. B. Roy, Phys. Rev.\nB 76, 140401 (2007).\n14A. K. Nayak, et al., Phys. Rev. Lett. 110, 127204 (2013).\n15J. Winterlik, et al., Phys. Rev. B 83, 174448 (2011).\n16P. Klaer, et al., Appl. Phys. Lett. 98, 212510 (2011)."    },    {        "title": "1501.00322v4.A_hybrid_exchange_density_functional_theory_study_of_the_electronic_structure_of___mathrm_MnV__2_mathrm_O__4___Exotic_orbital_ordering_in_the_cubic_structure.pdf",        "content": "arXiv:1501.00322v4  [cond-mat.str-el]  13 Apr 2015A hybrid-exchange density-functional theory study of the e lectronic structure of\nMnV2O4: Exotic orbital ordering in the cubic structure\nWei Wu∗\nDepartment of Electronic and Electrical Engineering and Lo ndon Centre for Nanotechnology,\nUniversity College London, Gower Street, London, WC1E 6BT, United Kingdom\nThe electronic structures of the cubic and tetragonal MnV 2O4have been studied by using hybrid-\nexchangedensityfunctional theory. Thecomputedelectron ic structureof thetetragonal phase shows\nan anti-ferro orbital ordering on V sites and a ferrimagneti c ground state (the spins on V and Mn are\nanti-aligned). These results are in a good agreement with th e previous theoretical result obtained\nfrom the local-density approximation+ Umethods [S. Sarkar, et. al., Phys. Rev. Lett. 102, 216405\n(2009)]. Moreover, the electronic structure, especially t he projected density of states of the cubic\nphase has been predicted with a good agreement with the recen t soft x-ray spectroscopy experiment.\nSimilar to the tetragonal phase, the spins on V and Mn in the cu bic structure favour a ferrimagnetic\nconfiguration. Most interesting is that the computed charge densities of the spin-carrying orbitals\non V in the cubic phase show an exotic orbital ordering, i.e., a ferro-orbital ordering along [110] but\nan anti-ferro-orbital ordering along [ 110].\nPACS numbers: 71.15.Mb, 75.47.Lx, 75.50.Gg, 75.25.Dk\nI. INTRODUCTION\nMany fascinating phenomena in condensed-matter\nphysics were discovered in transition-metal oxides\n(TMOs) and their closely related compounds such as\nthe first high transition temperature superconductivity\nin YBaCuO 3[1, 2]. An important ordering phenomenon,\nOrbital ordering (OO) has a long history in the physics\nof TMOs, back to the 1930s, when the concept of charge\nordering in Fe 3O4was proposed [3]. OO occurs when the\norbital degeneracy is lifted essentially by the interaction\nwith the lattice environment [4, 5]. The best-known ex-\nample of OO is the Jahn-Teller (JT) distortion observed\nin LaMnO 3[6]. Therein the egdegenerate manifold ( dz2\nanddx2−y2) in a cubic environment is further splitted by\nthe JT distortion owing to partially filling of electrons.\nThe underlying physics in OO has been well accounted\nfor by the so-called Kugel-Khomskii model [7]. Most ex-\nperimental and theoretical works so far have been fo-\ncused on the system with partially filled egorbitals, i.e.,\negactive. This type of orbital occupancy can lead to\nthe strongest JT effect because the related d-orbitals di-\nrectly point to ligands and strongly hybridise with the\n2p-orbitals of the anion.\nRecently much attention has been paid to vanadium\nspinels AB 2O4(A = Mg, Co, Fe, Mn, and Cd, etc and\nB = V) [8–13] owing to their fascinating spin and orbital\norderings accompanied by complex structural transitions\nat low temperature. The interplay between structure, or-\nbital, and spin implies that the underlying physics could\nbeveryinteresting,andthattheremightbelargeapplica-\ntionpotentialin thesematerials. Forexample, artificially\ntuning structure by applying external magnetic field can\n∗Electronic address: wei.wu@ucl.ac.ukbe thought of as a prototype of TMO-based metamate-\nrials or even quantum metamaterials [11, 14]. Another\nunique point for vanadium spinels is that they are t2g-\nactive, in sharp contrast to most of the known TMOs\nshowing OO. In this type of compound, the V3+ions on\nthe B sites form a pyrochlorelattice. The two d-electrons\non V occupying two t2gorbitals lead to a total spin of\nS= 1, resulting in a t2g-active OO. The OO in this type\nof compounds is further complicated by their intrinsic\ngeometrical frustration, which played an important role\nin OO. In 1939, the charge ordering on the B-site py-\nrochlore lattice was proposed to cause a sharp increase\nof the electrical resistivity when cooling below ∼120 K\nin Fe3O4(the ’Verwey’ transition) [3]. Anderson was\nthe first to realise the intimate relationship between the\nproton ordering in the ice, the Verwey charge ordering,\nand the ordering of Ising spins [15]. Regardless of the\nspecific material types, geometrical frustration plays a\ncrucial role to determine the spin or charge configura-\ntions, byminimising exchangeorCoulombenergies. This\nwill in turn trigger many interesting phenomena such as\norbital-glass and orbital-ice states [16–18]. In addition,\nthe much faster response of electron charges as compared\nto electron spins will find many potential applications in\nadvanced electronics such as orbitronics [19].\nMnV2O4(See Fig.1), in which the Mn2+ion is in a d5\nhalf-filled high-spin configuration ( S=5\n2), first experi-\nences a magnetic transition to the collinear ferrimagnetic\nstate atT= 56 K, and then a structural distortion at a\nslightly lower temperature T= 53 K, along with a tran-\nsition to the non-collinear ferrimagnetic state [20, 21].\nThe observed magnetic ordering below T= 56 K is ferri-\nmagnetic, i.e., the magnetic moments on Mn2+and V3+\nare anti-aligned. In the structural transition, the sym-\nmetry is lowered from cubic to tetragonal. The OO of\nthe ground state in the tetragonal MnV 2O4structure\nhas been shown theoretically [12] as an A-type antiferro-2\norbital ordering with a propagation vector of (002).\nFIG. 1: (Color online.) the conventional cells in the cubic\n(left) and tetragonal (right) MnV 2O4crystal structures are\nshown. Mn is depicted as small yellow ball, V as large green\nball, and O as large red ball.\nPreviously,theelectronicstructureandmagneticprop-\nerties of the tetragonal MnV 2O4have been investigated\ntheoretically using LDA (local-density approximation) +\nUmethod [12], whereanorbitalorderingamongV3+was\nproposed, along with a non-collinear magnetic ordering.\nFollowing this first-principles calculation, an analytical\nmodelling [22] has further confirmed that an antiferro-\norbital ordering exists in the tetragonal MnV 2O4. On\nthe other hand, the electronic structure (especially OO)\nand magnetic properties of the cubicMnV2O4are yet\nto be studied in detail. It might be worthwhile to (i)\nemploy a computational method without any adjustable\nparameters,(ii) takeintoaccountthe electroncorrelation\nproperlyand(iii) comparethe electronicstructuresofthe\ncubic and tetragonal MnV 2O4. Hybrid-exchange func-\ntional theory(HDFT), in which the exact exchangeis hy-\nbridised with generalised gradient approximation (GGA)\nfunctionaltolocalise d-electrons,hasperformedwellfora\nwide range of inorganic and organic compounds [23, 24].\nIn this paper, HDFT with PBE0 functional [25] has been\nusedtostudytheelectronicstructureandmagneticprop-\nerties of the cubic and tetragonal MnV 2O4. The results\npresented here are not only in a good agreement with the\nprevious theoretical and recent experimental studies, but\nalso point to an exotic OO state that mixes ferro-orbital-\nand anti-ferro-orbital orderings. The rest of the discus-\nsion is organised as the following: in §II computational\ndetails are introduced, in §III the calculation results are\npresented and discussed, and in §IV some general con-\nclusions are drawn.\nII. COMPUTATIONAL DETAILS\nThe calculations of the electronic structures of the\ntetragonal and cubic MnV 2O4were carried out by us-ing DFT and hybrid-exchange functional PBE0 as im-\nplemented in the CRYSTAL 09 code [29]. The crystal\nstructures of the cubic (space group Fd3m) and tetrago-\nnal (space group I41/amd) MnV 2O4experimentally de-\ntermined in Ref.[11] have been adopted here to perform\nall the calculations. The basis sets for the atomic or-\nbitals centred on the Mn[30], V[31], and O[32] atoms,\nwhich are designed for solid-state compounds, were used.\nThe Monkhorst-Pack samplings [33] of reciprocal space\nwere carried out choosing a grid of shrinking factor to\nbe 6×6×6 (6×6×5) in order for a consistency\nwith the ratios among reciprocal lattice parameters for\nthe cubic (tetragonal) MnV 2O4. The truncation of the\nCoulomb and exchange series in direct space was con-\ntrolled by setting the Gaussian overlap tolerance crite-\nria to 10−6,10−6,10−6,10−6, and 10−12[29]. The self-\nconsistent field (SCF) procedure was converged to a tol-\nerance of 10−6a.u. per unit cell (p.u.c). To accelerate\nconvergence of the SCF process, all the calculations have\nbeen converged by using a linear mixing of Fock matrices\nby 30%.\nElectronic exchange and correlation are described us-\ning the PBE0 hybrid functional [25], free of any empir-\nical or adjustable parameter. The advantages of PBE0\ninclude a partial elimination of the self-interaction er-\nror and balancing the tendencies to delocalize and lo-\ncalize wave-functions by mixing a quarter of Fock ex-\nchange with that from a generalized gradient approxi-\nmation (GGA) exchange functional [25]. The broken-\nsymmetry method [34] is used to localize collinear op-\nposite electron spins on atoms in order to describe the\nanti-ferromagnetic state.\nIII. RESULTS AND DISCUSSIONS\nA. Projected densities of states\nThe projected densities of states (PDOS) of the cubic\nand tetragonal MnV 2O4structures for the AFM config-\nuration (the spins on Mn and V are anti-ligned) have\nbeen shown in Fig.2. The zero energy is aligned with the\nvalence band maximum (VBM).\nIn the cubic and tetragonal structures, Mn 3 dPDOS\n(Fig.2a and b) shows that the five d-orbitals including\ndx2−y2,dz2,dxz,dyz, anddxyare singly occupied, thus\ngiving a spin-5\n2. From the analysis of Mulliken spin den-\nsities and V 3 dPDOS (Fig.2c and d), the two electrons\noccupying t2gstates can be identified as the origin of the\nOOboth in the cubicandtetragonalMnV 2O4. In thecu-\nbic phase, the molecular orbitals delocalized among the\nthreet2gstates are occupied, whereas in the tetragonal\nphase, only dxzanddyzare involved. The O 2 pPDOS\n(Fig.2e) for the cubic phase are much more delocalized\nthan those for the tetragonal (Fig.2f); this might be due\nto the elongation of the lattice vector aandbin the\ntetragonal structure. The O 2 pPDOS are particularly\ndominant at ∼5 eV below the VBM for both the cubic3\nCubic Tetragonal\n-10 -5 0 5 10\nEnergy (eV)-100-50050100Density of states (states/eV/cell)dx2-y2\ndz2\ndxz\ndyz\ndxyCubic MnV2O4 AFM Mn-3d-orbital PDOS\n-10 -5 0 5 10\nEnergy (eV)-20020Density of states (states/eV/cell)dx2-y2\ndz2\ndxz\ndyz\ndxyTetragonal MnV2O4 AFM Mn-3d-orbital PDOS\n(a) (b)\n-10 -5 0 5 10\nEnergy (eV)-50050100150Density of states (states/eV/cell)dx2-y2\ndz2\ndxz\ndyz\ndxyCubic MnV2O4 AFM V-3d-orbital PDOS\n-10 -5 0 5 10\nEnergy (eV)-20020406080100Density of states (states/eV/cell)dx2-y2\ndz2\ndxz\ndyz\ndxyTetragonal MnV2O4 AFM V-3d-orbital PDOS\n(c) (d)\n-10 -5 0 5 10\nEnergy (eV)-50050Density of states (states/eV/cell)2px\n2py\n2pzCubic MnV2O4 AFM O-2p-orbital PDOS\n-10 -5 0 5 10\nEnergy (eV)-10010Density of states (states/eV/cell)2px\n2py\n2pzTetragonal MnV2O4 O-2p AFM PDOS\n(e) (f)\n-10 -5 0 5 10\nEnergy (eV)-200-1000100200300Density of states (states/eV/cell)Mn 3d\nV 3d\nO 2pCubic MnV2O4 AFM PDOS\n-10 -5 0 5 10\nEnergy (eV)-1000100200Density of states (states/eV/cell)Mn-3d\nV-3d\nO-2pTetragonal MnV2O4 AFM PDOS\n(g) (h)\nFIG. 2: (Color online.) The PDOS of the 3 d-orbitals on Mn, the 3 d-orbitals on V, and the 2 p-orbitals on O of the cubic (left)\nand tetragonal (right) MnV 2O4are shown. The PDOS onto dx2−y2is in green, dz2in red,dxzin blue,dyzin purple, and dxy\nin black, respectively. For 2 p-orbitals, the PDOS onto 2 pxis in green, 2 pyin red, and 2 pzin black, respectively. In (g) and\n(h), the total PDOS onto Mn d-orbitals is in red, V d-orbitals in green, and O 2 p-orbitals in black, respectively.4\nand tetragonal structures, which overlap with the rather\nweak Mn 3 dand V 3dPDOS. The O 2 porbitals involved\nhere will give rise to the so-called oxygen bonding states\nthat feature the hybridisation between the d-orbitals on\ntransition metals and p-orbitalson O atoms. In addition,\nit can be observed by comparing the spin-up and spin-\ndown PDOS, that the O 2 pstates have been strongly\nspin-polarized by the magnetic moments on Mn and V.\nThe PDOS onto the V sites have also been compared\nwith the previous results obtained by the LDA+ U[12],\nwhich suggests that there is a qualitative agreement be-\ntween them except that the DFT band-gap computed\nhere (∼3 eV) is larger than the previously computed\none (∼1.1 eV) [12] and the observed ( ∼1.1 eV)[36].\nThe computed O 2 pPDOS are in good agreement with\nthe intensities measured in the recent K-edge x-ray ab-\nsorption and emission spectra; the combination of the\ntheoretical work and experiments will be published in a\nforthcoming paper [37].\nThecomputedcrystal-fieldsplittingbetween t2gandeg\nstatesofMn2+andV3+inthecubicMnV 2O4canberead\nfrom Fig.2a, which is ∼1 eV. On the other hand, in the\ntetragonalMnV 2O4, forbothMn2+andV3+,dx2−y2and\ndxyare closely aligned while the other three 3 d-orbitals\noverlap well, as shown in Fig.2b. This picture is consis-\ntent with the previous calculations reported in Ref.[12].\nThe crystal-field splitting between these two groups is\n∼1 eV. The on-site Coulomb interaction Ucan be ap-\nproximatedby observingthe gap between lower Hubbard\nd-bands and uppers ones ( ∼5 eV for Mn site, and ∼7\neV for V), which is much larger than the crystal-field\nsplitting computed here. This will result in a high-spin\nstate for Mn2+and V3+ions, which is consistent with\nthe previous experiments [11].\nB. Exchange interactions\nThe magnetic structure could be much more compli-\ncated owing to the exchange interaction between the\nnearest-neighbouring Mn (V) atoms [13] and the spin-\norbit coupling (SOC), which will cause the non-collinear\nmagnetic ordering. However, this topic is beyond the\nscope of this paper that is focused on the electronic\nstructure and the collinear magnetic ground state. The\nMulliken spin densities on the Mn and V sites for\nthe cubic (tetragonal) MnV 2O4are 4.8 (4.7)µBand\n−2.0 (−1.9)µB, respectively. They are close to the ex-\npected values, i.e. 5 µBfor Mn and −2µBfor V (anti-\naligned). For the cubic (tetragonal) structure, the com-\nputed total energies for a conventional cell with the spins\non Mn and V aligned (ferromagnetic) is higher than the\nanti-aligned (AFM) by 937 (754) meV. This in turn gives\nan approximate exchange interaction of ∼38 meV (di-\nvided by the factor of 4 ×SMn×SV=20, where 4 is the\nnumber of Mn-V pairs in the unit cell for the tetragonal\nstructure). Similarly we can estimate the exchange in-\nteraction for the cubic structure, which is ∼23 meV. Onthe other hand, the exchangeinteractionbetween the Mn\nand V spins can also be estimated by using the super-\nexchange formalism, where tcan be quantified in the\nPDOS, which is ∼0.5 eV for Mn and ∼1 eV for V. By\nusing these hopping integrals, the exchange interaction\ncan be estimated astMntV\nU(U=UMn+UV\n2), which is ∼59\nmeV. This is in the same order as that calculated from\nthe total energy differences aforementioned (here an av-\nerage value is taken for on-site Coulomb interaction). A\ndetailed discussion about the exchange interaction, tak-\ning into account all the d-orbitals on both Mn and V\nwould be great, but beyond the scope of this paper.\nC. Orbital ordering\nThe OO in the cubic and tetragonal MnV 2O4can be\nillustrated by the spin densities on V (see Fig.3)– the dif-\nference between the charge densities of spin-up and spin-\ndownorbitals(essentiallythechargedensitiesofthespin-\ncarry orbitals). In the tetragonal structure, the relative\norientationrotationofneighbouringorbitalsisillustrated\nby the red arrows in Fig.3b; this is consistent with the\nprevious calculations presented in Ref.[12] in which an\nanti-ferro-orbital ordering (AFOO) has been predicted.\nThe orbital orientations for the nearest-neighbouring\n(NN) orbitals are perpendicular to each other (defined\nas AFOO). In sharp contrast, for the cubic structure the\nNN orbtibals (labeled by X and Y in Fig.3a) are orga-\nnized in an exotic way, in which the orbitals on one of\nthe four symmetry-inequivalent V atoms in the unit cell\nhas a different orbital orientation (labeled by Y), perpen-\ndicular to the others (labeled by X). The orbital orien-\ntations corresponding to Fig.3a are further illustrated in\nFig.3d, e, and f, which are the views of the spin densities\nfrom the lattice direction [100] ( /vector a), [010] ( /vectorb), and [001]\n(/vector c), respectively. This leads to an exotic OO: a ferro-\nOO (FOO) along [110] (blue arrow in Fig.3), whereas an\nAFOO along [ 110] (red arrow in Fig.3). This OO can\nprobably be attributed to (i) the crystal field environ-\nment formed by neighbouring oxygen atoms and (ii) the\nCoulomb interaction between d-orbitals. This interesting\nOOhasbeen furtherconfirmed bythe calculationswith a\ndifferent type of exchange-correlation functional, B3LYP\n[35], which relies on the empirical parameters for mixing\nexact exchange and GGA exchange functional. The spin\ndensities on V predicted by using B3LYP functional are\nshown in Fig.3c, in which FOO is along [110] and AFOO\n[110] (all the atoms are made partially transparent in\norder to illustrate OO).\nTo see the orbitals involved in the OO of the cubic\nMnV2O4more clearly, the atomic orbitalcompositionsof\nthe bands (see Fig.4) at the Γ-point that contribute most\ntothe twopeaksat −0.26eVand −0.64eVin the Vspin-\ndown PDOS (see Fig.2c) have been investigated. The\nbands VB −8 and VB −9 contribute the most to the peak\nat−0.26 eV, whereas VB −19 and VB −20 to the one at\n−0.64 eV. Although the atomic coefficients for d-orbitals5\nFIG. 3: (Colour on line.) The spin densities on V of the AFM sta te in the conventional unit cell of the cubic (a) and\ntetragonal (b) structures of MnV 2O4are shown. In the tetragonal an anti-ferro-orbital orderin g is found. In contrast, in the\ncubic structure, the rotation of the orbital orientation il lustrates an exotic OO, in which an ferro-orbital ordering i s along [110],\nwhile anti-ferro-orbital ordering along [ 110]. This exotic OO is further confirmed by the calculation f rom B3LYP functional;\nthe resulting spin densities are shown in (c). The B3LYP spin -density calculation is performed in a unit cell, but shown i n a\nconventional cell. The views from [100] ( /vector a), [010] ( /vectorb), [001] ( /vector c) lattice directions are shown in (d), (e), and (f), respecti vely.\nThe isovalue is chosen as 0 .04e/˚A3.\nare slightly different among V atoms, approximately the\nmost dominant linear combinations of d-orbitals for each\nV atoms contained in these bands read,\nFIG. 4: (Colour on line.) The spin-down band structure\n(from -0.8 to 0 eV) of the AFM configuration in the cubic\nMnV2O4is shown. Note that the two red arrows refer to\nVB-8 and VB-9, and VB-19 and VB-20, respectively.φVB−8= 0.2|dxz/angbracketright+0.1|dyz/angbracketright+0.2|dxy/angbracketright,(1)\nφVB−9=−0.1|dxz/angbracketright+0.2|dyz/angbracketright−0.2|dxy/angbracketright,(2)\nφVB−19= 0.2|dxz/angbracketright−0.1|dyz/angbracketright−0.2|dxy/angbracketright,(3)\nφVB−20= 0.2|dxz/angbracketright+0.2|dyz/angbracketright+0.1|dxy/angbracketright.(4)\nThe corresponding spherical harmonic functions have\nbeen plotted in Fig.5 to see the contributions of d-\norbitals. From the perspectives of linear combination\nof atomic orbitals (LCAO), the strong variations of the\norbital orientation illustrated in Fig.3 and Fig.5, result\nfrom the minimisation of the total energy respective to\natomic orbital coefficients. On the other hand, the pro-\ncess of energy-minisation can be seen as the orbital re-\norientation driven by Coulomb and crystal-field interac-\ntions.\nThere are two concepts that are closely related to this\nexotic orbital ordering, including orbital glass and ice\nstates. The orbital glass state is formed by the randomly\norientated orbitals, which result from Jahn-Teller distor-\ntion and orbital frustrations in a similar manner to spin\nfrustrations [16, 17]. In light that orbital glass states\nhave been observed in the spinel compounds FeCr 2S4\nand LuFe 2O4[16, 17] (which has similar chemical for-\nmula to a spinel, but a different structure), this is not\ncompletely unexpectable in the cubic MnV 2O4. There-\nfore, these calculations could suggest that at T= 53\nK, there is not only a structural transition, but also an\norbital-orderingtransitionfrom the exoticOOillustrated6\n(a) (b) (c) (d)\nFIG. 5: (Colour online.) The linear combination of harmonic spherical functions are shown. 2 Yxz+Yyz+2Yxy(corresponding\nto eq.(1)) are shown in (a), −Yxz+2Yyz−2Yxy(corresponding to eq.(2)) in (b), 2 Yxz−Yyz−2Yxy(corresponding to eq.(3)) in\n(c), and 2 Yxz+2Yyz+Yxy(corresponding to eq.(4)) in (d). Notice that the linear com bination illustrated in (c) changes the\norientation of orbitals significantly away from their origi nal ones, which is closely related to the exotic OO shown in Fi g.3a.\nin Fig.3 to anti-ferro orbital ordering [12], when lowering\ntemperature. In addition, Chern and the collaborators\nhave presented an ice model consisting of triplet orbital\nvariables (such as p-orbitals), which is however closely\nrelated to orbital-driven many-body phenomena in opti-\ncal lattice [18]. The analogue of spin-interacting Hamil-\ntonian, an orbital-exchange model describing Coulomb\ninteractions has been used therein. The exotic ordering\npresented here is closely related to the newly proposed\nconcept of the orbital ice [18], similar to the spin ice.\nIV. CONCLUSION\nThe electronic structures of the cubic and tetrago-\nnal MnV 2O4structures have been computed by using\nhybrid-exchange density functional theory PBE0 and\nB3LYP. The calculated PDOS of the tetragonal struc-\nture has been in a good agreement with previous theoret-\nical results. The most important finding here is that the\ncharge densities of spin-carrying orbitals suggest a pos-\nsible existence of an exotic orbital ordering: FOO along\n[110] and AFOO [ 110] in the cubic MnV 2O4. The theo-\nreticalpredictionpresentedherecouldbe validatedinthe\nfuture experiment. For example, the different OOs along\nthose two lattice orientations aforementioned could leadtodifferentelectrontransportproperties. Moreadvanced\ntheoretical methods such as dynamical mean-field theory\n[38] is also needed to further understand the electronic\nstructure of MnV 2O4. In the ground state the spins on\nMn and V are predicted to be anti-aligned, suggesting a\nferrimagnetic state for both structures, which is in good\nagreementwith previoustheoreticaland experimentalre-\nsults. The lower and upper Hubbard bands of Mn and\nVd-electrons have been shown clearly below and above\nthe band gap. The on-site Coulomb interaction can be\nestimated by using the gap between the lower and upper\nHubbard bands, which is ∼7 eV for V and ∼5 eV for\nMn, respectively. The transfer integral responsible for\nthe delocalization is in the order of 1 eV, which should\nbe attributed to a combination of the first-order direct\nhopping between Mn and V and the second-order effect\nvia O atoms.\nV. ACKNOWLEDGEMENT\nI would like to acknowledgethe stimulating discussions\nwith Dr. Leonardo Bernasconi, Dr. Bo Chen, and Dr.\nJude Laverock. I also thank Mr. Barry Searle for tech-\nnical supports and Rutherford Appleton Laboratory for\ncomputational resources.\n[1] J. G.Bednorz andK.A.Mller, Z.Phys.B 64, 189 (1986).\n[2] M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L.\nMeng, L. Gao, Z. J. Huang, Y. Q. Wang, and C. W. Chu,\nPhys. Rev. Lett. 58, 908 (1987).\n[3] E. J. W. Verwey, Nature 144, 327 (1939).\n[4] I. D. Khomskii and M. V. Mostovoy, J. Phys. A: Math.\nGen.36, 9197 (2003).\n[5] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.\nRev. B52, R5467(R) (1995).\n[6] J. B. Goodenough, Phys. Rev. 100, 564 (1955).\n[7] K. I. Kugel and and I. D. Khomskii, Sov. Phys. — SolidState17, 285 (1975)\n[8] E. D. Mun, Gia-Wei Chern, V. Pardo, F. Rivadulla, R.\nSinclair, H. D. Zhou, V. S. Zapf, and C. D. Batista, Phys.\nRev. Lett. 112, 017207 (2014).\n[9] Y. Huang, Z. Yang and Y. 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Phys. 78,\n865 (2006)."    },    {        "title": "1705.10062v1.Distinct_domain_wall_motion_between_creep_and_flow_regimes_near_the_angular_momentum_compensation_temperature_of_ferrimagnet.pdf",        "content": "Distinct domain -wall motion  between  creep and flow regimes  near the \nangular  momentum  compensation temperature  of ferrimagnet  \nYuushou Hirata,1† Duck -Ho Kim,1†★ Takaya Okuno,1 Woo Seung Ham,1 Sanghoon Kim,1 \nTakahiro Moriyama,1 Arata Tsukamoto,2 Kab-Jin Kim,3★ and Teruo Ono1,4★ \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 611 -0011, Japan.  \n2College of Science and Technology, Nihon University, Funabashi, Chiba 274 -8501, Japan.  \n3Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, \nRepublic of Korea.  \n4Center for Spintronics Research  Network (CSRN), Graduate School of Engineering Science, \nOsaka University, Machikaneyama 1 -3, Toyonaka, Osaka 560 -8531, Japan.  \n†These authors contributed equally to this work . \n★Correspondence to: kim.duckho.23z@st.kyoto -u.ac.jp , kabjin@kaist.ac.kr , ono@scl.kyoto -\nu.ac.jp  \n \n  We investigate a magnetic  domain -wall (DW)  motion  in two dynamic regimes, \ncreep and flow regimes,  near the angu lar momentum compensation temperature (𝑻𝐀) of \nferrimagnet . In the flow regime, t he DW speed shows sharp increase at 𝑻𝐀 due to the \nemergence of antiferromagnetic DW dynamics. In the creep regime, however, the DW \nspeed exhibits a monotonic increase with increasing the temperature . This result suggests  \nthat, in the creep regime,  the thermal activation process governs the DW dynamics  even \nnear 𝑻𝐀. Our result unambiguously shows the distinct behavio r of ferrimagn etic DW \nmotion depending on the dynamic regime, which is important for emerging ferrimag net-\nbased spintronic applications.   The rare earth (RE) –transition metal (TM) compounds , in which RE and TM moments \nare coupled antiferromagnetically, is receiving of great attention  because they have two unique \ncompensation temperatures [1–10]. One is  the magnetization compensation temperature 𝑇M \n[3, 4, 11], at which the total magnetic moment goes zero. The other is  the angular momentum  \ncompensatio n temperature 𝑇A, at which the net angular momentum vanishes  [2–4, 12 ]. The \n𝑇A is receiving of particular interest  because of the possibility to have  a fast antiferromagnetic \nspin dynamics. Recent experiment has indeed shown the fast DW dynamics at 𝑇A [12], which \nhighlights  an important role of 𝑇A in the  dynamics of ferrimagnetic  DW. To extend the \nknowledge  of ferrimagnet ic DW dynamics  near 𝑇A, here we investigate a temperature \ndependence of the DW motion in two different dynamic regimes:  the creep and flow regimes.  \nFor this study, ferrimagnetic amorphous GdFeCo films with perpendicular magnetic \nanisotropy (PMA) were prepared. 5 -nm SiN  (top) /30-nm Gd 23Fe67.4Co9.6/5-nm Cu/5 -nm SiN  \n(bottom)  films were deposited on Si wafer s using DC magnetron sputtering. The GdFeCo film \nis then patterned into microstrips having  3-𝜇m width, 100 -𝜇m length, and 3 - 𝜇m wide Hall bar \nstructures using  photolithography and Ar ion milling  [13]. The distance  between Hall bars were \nset to 30 𝜇m. 5-nm Ti/100 -nm Au electrodes are stacked onto the ends of the wire and Hall \nbars for current injection  and Hall measur ement  [see Fig. 1(a)] .  \nWe first try to  determine the magnetization compensation temperature of GdFeCo \nmicrostrip . To this end,  the anomalous Hall effect (AHE) resistance 𝑅H (≡𝑉H/𝐼) is measured \nwith respect to the magnetic field , 𝐵𝑧, for various  temperature s (160 K < T < 170  K with 1 -K \ninterval) . The inset of Fig. 1(b) shows the typical result of 𝑅H when we sweep the magnetic \nfield 𝐵𝑧. A square hysteresis loop  is observed for all temperatures examined,  indicat ing the \nperpendicular magnetic anisotropy  of the patterned  GdFeCo microstrip . The magnitude of the \nAHE resistances, which is defined as Δ𝑅H≡𝑅H(𝐵z>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. Our findings therefore suggest that the \nidentification of DW dynamic regime is important  for emerging ferrimagnet -based spintronic \napplications.  \n  References  \n1. I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. Dürr, T. A. 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Kim , B.-C. Min , C. Hwang, \nand S.-B. Choe, Sci. Rep. 5, 9166  (2015).  \n28. J. H. Franken, H. J. M. Swagten, and B. Koopmans, Nat. Nanotech. 7, 499  (2012).  \n29. K.-W. Moon , D.-H. Kim , S.-G. Je, B . S. Chun, W . Kim , Z.Q. Qiu , S.-B. Choe , and  C. \nHwang, Sci. Rep. 6, 20360  (2016).  \n  Figure Captions  \nFigure 1 . (a) Optical image of the device structure with  schematic illustration  of the \nmeasurement set -up for real-time domain wall (DW)  motion . (b) The magnitude of \nanomalous Hall resistance ( 𝚫𝑹𝐇) as a function of  temperature ( 𝑻). The red arrow \nindicates the magnetiz ation compensation temperature (𝑻𝐌). The inset shows the  𝑹𝐇 \nwith respect to the magnetic field ( 𝑩𝐳) at 𝑻= 170 K.  \nFigure 2 . (a) DW speed  𝒗 with respect to  𝑩𝐳 at 𝑻= 260 K. The blue box indicates the \ncreep regime. 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": "1911.12103v1.Spin_lattice_coupling_in_a_ferrimagnetic_spinel__The_exotic__H___T__phase_diagram_of_MnCr__2_S__4__up_to_110_T.pdf",        "content": "arXiv:1911.12103v1  [cond-mat.str-el]  27 Nov 2019Spin-lattice coupling in a ferrimagnetic spinel: The exoti cH–Tphase diagram of\nMnCr 2S4up to 110 T\nA. Miyata,1H. Suwa,2,3T. Nomura,4L. Prodan,5V. Felea,4,5,6Y. Skourski,4J. Deisenhofer,7\nH.-A. Krug von Nidda,7O. Portugall,1S. Zherlitsyn,4V. Tsurkan,5,7J. Wosnitza,4,6and A. Loidl7\n1Laboratoire National des Champs Magn´ etiques Intenses,\n(LNCMI-EMFL), CNRS-UGA-UPS-INSA, 31400 Toulouse, France\n2Department of Physics and Astronomy, The University of Tenn essee, Knoxville, TN 37996, USA\n3Department of Physics, The University of Tokyo, Tokyo 113-0 033, Japan\n4Hochfeld-Magnetlabor Dresden (HLD-EMFL) and W¨ urzburg-D resden Cluster of Excellence ct.qmat,\nHelmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Ger many\n5Institute of Applied Physics, MD 2028, Chisinau, R. Moldova\n6Institut f¨ ur Festk¨ orper- und Materialphysik, TU Dresden , 01069 Dresden, Germany\n7Experimental Physics 5, Center for Electronic Correlation s and Magnetism,\nInstitute of Physics, University of Augsburg, 86159, Augsb urg, Germany\n(Dated:)\nInantiferromagnets, theinterplayofspinfrustration and spin-latticecouplinghasbeenextensively\nstudied as the source of complex spin patterns and exotic mag netism. Here, we demonstrate that,\nalthough neglected in the past, the spin-lattice coupling i s essential to ferrimagnetic spinels as\nwell. We performed ultrahigh-field magnetization measurem ents up to 110 T on a Yafet-Kittel\nferrimagnetic spinel, MnCr 2S4, which was complemented by measurements of magnetostricti on and\nsound velocities up to 60 T. Classical Monte Carlo calculati ons were performed to identify the\ncomplex high-field spin structures. Our minimal model incor porating spin-lattice coupling accounts\nfor the experimental results and corroborates the complete phase diagram, including two new high-\nfieldphasetransitions at75and 85T. Magnetoelastic coupli nginducesstrikingeffects: Anextremely\nrobust magnetization plateau is embedded between two uncon ventional spin-asymmetric phases.\nFerrimagnetic spinels provide a new platform to study asymm etric and multiferroic phases stabilized\nby spin-lattice coupling.\nINTRODUCTION\nFerrimagnets, thanks to their finite spontaneous mag-\nnetization, have been widely used in engineering and\ntechnology, for example, as strong permanent magnets,\noptical isolators, and circulators for optical communi-\ncations. Unlike ferromagnets, by tuning the antiferro-\nmagnetic exchanges in ferrimagnets, additional function-\nalities, e.g., ferroelectricity and skyrmion controllability\n[1, 2], may emerge as a consequence of a noncollinear\nspin structure. The realization of such multifunctional-\nity opens the door for next-generation technologies.\nNoncollinear ferrimagnets have been well studied, for\ninstance, in spinels, AB2X4, where the AandBcations\nform a bipartite diamond lattice and a pyrochlore lat-\ntice, respectively, with X= O, S, or Se. The key\nessence is the competition (or frustration) of magnetic\nexchanges within or between the two AandBlattices\n(JA-A,JB-B, andJA-B). Yafet and Kittel (YK) proposed\na model for triangular-structure ground states, where\nJA-A∼JA-B[3], while Lyons, Kaplan, Dwight, and\nMenyuk (LKDM) proposed another model for conical-\nstructure ground states, where JB-B∼JA-B[4]. Both\nnoncollinear ground states have been widely observed in\nferrimagnetic spinels [5, 6].\nTo realize unconventional ferrimagnetic structures be-\nyond both the YK and the LKDM models, one can con-\nsider that spontaneous lattice deformation will modulatethese main antiferromagnetic exchanges, i.e., through a\nspin-lattice coupling mechanism [7]. Such lattice defor-\nmation has been expected in strongly frustrated antifer-\nromagnets to lift the macroscopic degeneracy (or lower\nthe free energy of the system), resulting in unconven-\ntional magneto-structural phases [8, 9]. On the other\nhand, this kind of spin-lattice coupling mechanism has\nnot been taken into account in previoustheoretical works\non ferrimagnetic spinels, because they have been consid-\nered to be less frustrated. Here, our experimental and\ntheoretical studies demonstrate that the spin-lattice cou-\npling is essential in ferrimagnetic spinels as well.\nMnCr2S4(A= Mn2+with spin S= 5/2, and B=\nCr3+withS= 3/2 in the spinel structure, see Fig. 1a)\nis a representative material for the YK model, where\nJMn-Mn∼JMn-Cr[10–14]. The interaction between the\nCrspins, JCr-Cr,isferromagneticandmuchstrongerthan\nthe other spin interactions, which has been evidenced\nby two consecutive magnetic phase transitions at TC≈\n65 K (ferromagnetic order of Cr spins) and at TYK≈5\nK (triangular-like magnetic order of Cr and Mn spins)\n[10, 14]. Recent high-field ultrasound and magnetization\nexperiments on MnCr 2S4up to 60 T revealed a complex\nphase diagram and novel magneto-structural phases [15].\nAn extremely robust magnetization plateau at 6 µB/f.u.\nbetween 25 and 50 T was observed. In analogy to the\nmodels proposed by Matsuda and Tsuneto [16] and Liu\nand Fisher [17], Tsurkan et al. proposed that the mag-2\nnetic orders of the Mn spin below and above the magne-\ntization plateau should be interpreted as spin supersolid-\nlike phases [15].\nFurthermore, it has been revealed very recently that\nthe supersolid and superfluid-like phases of MnCr 2S4are\nferroelectric, i.e., multiferroic [18]. It is, thus, impor-\ntant to clarify the magnetic structures of MnCr 2S4in\nthe sense of multiferroicity as well. The understanding\nof the phases could pave the way to exploit ferrimagnetic\nmultiferroicity, where finite magnetization moments can\nbeutilized, e.g., the electric-fieldreversalofferrimagnetic\nmoments [19].\nIn this paper, we report on the magnetization process\nof MnCr 2S4in magnetic fields up to 110 T and show\nthecomplete phase diagram of the YK-type ferrimag-\nnetic spinel. The magnetization experiments were com-\nplemented by magnetostriction and ultrasound measure-\nments up to 60 T, which evidence strongspin-lattice cou-\nplings. We propose a minimal Hamiltonian for describ-\ning the magnetic and structural properties of MnCr 2S4,\nincorporating spin-lattice coupling between the Mn and\nCr spins. Using the classical Monte Carlo method we\ncompare the theoretical and experimental data regard-\ning both the spinandlatticedegrees of freedom. Our\nanalysisevidencesthatthespin-latticecouplingstabilizes\nthe magnetization plateau and unconventional phases of\nasymmetric magnetic structures. The obtained phase di-\nagramisremarkablysymmetricwithrespecttothecenter\nof the magnetization plateau ( ∼40 T), where the exter-\nnal field perfectly cancels out the internal exchange field\nacting on the Mn spins.\nMETHODS\nMagnetization measurements were performed at the\nLNCMI-EMFL in Toulouse in pulsed magnetic fields up\nto 110 T along H∝bardbl[110] using a single-turn-coil tech-\nnique [20]. The magnetization was measured using the\ninduction method with a compensated pair of coils as\ndescribed in ref. [21].\nThe optical fibre Bragg grating (FBG) method was\nused to measure the magnetostriction [22] up to 60 T at\nthe HLD-EMFL in Dresden. The relative length change\n∆L/Lis obtained from the shift of the Bragg wavelength\nof the FBG. The magnetostriction was measured along\nH∝bardbl[110].\nUltrasound measurements were performed using the\nstandard pulse-echo technique to investigate the elastic\nproperties [23] up to 60 T at the HLD-EMFL in Dres-\nden. The longitudinal ( c11+ 2c12+ 4c44)/3 mode was\nstudied with the alignment of H∝bardblk∝bardblu∝bardbl[111], where\nkanduare propagation and polarization vector, respec-\ntively. Polyvinylidene fluoride (PVDF) film transducers\nwere used to excite and to detect ultrasonic waves at the\nfrequency of 65 MHz.Classical (Markov chain) MC simulations were per-\nformed for the system described by eq. (1). The spin\nand lattice configurations were sampled from the Boltz-\nmann distribution at finite temperatures. The number of\nsites and bonds in the simulations are 5,184 and 34,560,\nrespectively. We confirmed that the results for 5,184 and\n1,536 sites are consistent within error bars: the size ef-\nfectisnegligibleandthepresentresultsarerepresentative\nfor the thermodynamic limit. We adopted the overrelax-\nation technique both for the spin and lattice degrees of\nfreedom, which enables rejection-free updates with the\nenergy constant. MC updates with the Metropolis algo-\nrithm were combined for thermalization. We repeated\nthe whole process consisting of one Metropolis and five\noverrelaxation steps. We took the average over 221MC\nsamples after 216thermalization steps, which are much\nlonger than the autocorrelation time. The error bars\nare comparable to or smaller than the linewidths in the\npresent figures.\nRESULTS\nMagnetization measurements under ultrahigh\nmagnetic fields\nWemeasuredthemagnetization MofMnCr 2S4at4, 7,\nand14Kin pulsed fieldsup to 110T shownin Fig. 1b. In\nFig. 1c, the field derivatives of the magnetization curves\ndM/dH are presented. Focusing on the 4 K derivative\nin Fig. 1c, we easily identify five distinct steps labeled\nasH1–H5, which characterize field-induced transitions.\nThe observation of an extremely robust magnetization\nplateaubetween25( H2) and50T( H3) isconsistentwith\nthe previous study up to 60 T [15]. From the magneti-\nzation plateau at 6 µB/f.u. and the strong ferromagnetic\ninteractions between the Cr spins JCr-Cr, one can expect\nthat the observed moment of 6 µB/f.u. is attributed to\nthe full ferromagnetic moment of the Cr spins and zero\nnet moment of the antiparallel Mn spins. The spin state\nbelow 10 T ( H1) corresponds to the well-known coplanar\nYK-type structure, where the two Mn-sublattice spins\nare canted and their net magnetization moment is an-\ntiparallel to the Cr spins.\nTo connect continuously from these canted Mn spins\n(in the YK phase) to the antiferromagnetically collinear\nMn spins (in the plateau phase) without magnetization\njumps, one can assume that the canted Mn spins rotate\nin theplane with changingthe anglebetween twospinsin\nthe intermediatephasebetween H1andH2. As discussed\nin the following section, this asymmetric spin structure\nappearingin the intermediate phase is supported by clas-\nsical Monte Carlo (MC) calculations (see Fig. 2d).\nAbove the plateau phase, we found two additional\nphase transitions at approximately 75 ( H4) and 85 T\n(H5) in the ultrahigh-field experiments up to 110 T. Re-3\n—-!!\"#!\"!$%#$% \n!$%#!\"\n$%& $%' !\"\n0.3 \n0.2 \n0.1 \n0.0 \n-0.1 \n-0.2 \n-0.3 dM/dH (µB/T f.u.) \n100 80 60 40 20 0\nMagnetic field (T) H1H2\nH3H4\nH512 \n10 \n8\n6\n4\n2\n0\n-2 Magnetization ( µB/f.u.) \n100 80 60 40 20 0\nMagnetic field (T) H || [110] \n 4 K \n 7 K \n 14 K Exp. Exp. ()* (+* (,*\n—-\nFIG. 1. (a) Crystal structure of MnCr 2S4and the three main magnetic interactions, JMn-Mn,JMn-Cr, andJCr-Cr. (b)\nMagnetization and (c) its field derivative as a function of th e magnetic field along the [110] direction. The curves at 4 K ar e\nshown on the original scale, while the curves at 7 and 14 K are v ertically shifted for clarity. In (c), the arrows indicate t he phase\ntransitions, H1–H5, at which the step-like anomalies corresponding to the sudd en changes of the slope in the magnetization\nwere observed. The magnetization saturates in the fully pol arized state above 85 T.\nmarkably, the behavior observed below and above the\nplateau is symmetric: the plateau is sandwiched by two\nphases with steep slopes of the magnetization curve and\nthen by two phases with more gentle slopes. The broad-\neneddM/dH data at 14 K in Fig. 1c also show a rea-\nsonably symmetry with respect to the magnetic field of\n40 T. Based on this symmetry, one can assume that the\ntwo Mn spins rotate again through a spin-asymmetric\nphase between H3andH4and then a spin-canted phase\nappears between H4andH5. Finally, full polarization\nof 11µB/f.u. is found above H5. Our MC calculations\nsupport this assumption as shown below.\nSpin-lattice coupling model\nSpin-lattice coupling is expected to play an essential\nrole in magnetic properties of MnCr 2S4. The previous\nultrasound experiments showed strong anomalies in the\nvelocity of the longitudinal sound waves at H1,H2, and\nH3[15]. This observation indicates a significant spin-\nlattice coupling in the material.\nSpin-lattice coupling has been discussed for antiferro-\nmagnetic chromium spinel oxides, CdCr 2O4[24, 25] and\nHgCr2O4[26, 27], where only Cr is the magnetic ion.\nPencet al. [7] theoretically predicted that a collinear\nspin configuration is favored and a robust magnetization\nplateau appearsas aresult ofspin-lattice coupling, which\nintroduces biquadratic spin interactions after tracing out\nthe lattice degrees of freedom. In order to render the\nplateau robust for MnCr 2S4, where both Mn and Cr ions\nare magnetic, it is reasonableto assume collinear Mn and\nCr spins. We thus take into account a biquadratic term\nbetween Mn and Cr spins, bMn-Cr/parenleftbig\nSMn·SCr)2.Although the importance of the biquadratic terms in\nMnCr2S4has been pointed out in the past [28, 29], this\nMn-Cr biquadratic term has been neglected so far.\nWe propose the following minimal model:\nH=HMM+HCC+HMC+HZ, (1)\nwhere\nHMM=/summationdisplay\n/angbracketleftij/angbracketrightJMn-Mn/parenleftbig\nSMni·SMnj),\nHCC=/summationdisplay\n/angbracketleftij/angbracketrightJCr-Cr/parenleftbig\nSCri·SCrj),\nHMC=/summationdisplay\n/angbracketleftij/angbracketrightJMn-Cr(1−αρij)/parenleftbig\nSMni·SCrj)+K\n2ρ2\nij,\nHZ=−gµBH·(/summationdisplay\niSMni+/summationdisplay\njSCrj).\nIn this Hamiltonian, ∝angbracketleftij∝angbracketrightruns over the nearest neighbor\nwithin or between the Mn and Cr lattices, ρijdescribes\nthe lattice displacement between neighboring Mn and Cr\nions,Kandαarethespringconstantand thespin-lattice\ncoupling, g≈2 andµBare the g-factor and the Bohr\nmagneton, respectively.\nWe expect the interaction between Mn and Cr spins\nto be more sensitive to the positions of atoms than that\nbetween Mn spins. It is because the superexchange path\nof a Mn-Cr bond is involved with a single sulfur atom,\nwhile the path of a Mn-Mn bond is involved with multi-\nple sulfur atoms. The angle formed by Mn-S-Cr atoms,\nwhich is naturally parameterized by the bond phonon,\nshould affect the interaction between Mn and Cr spins\nsignificantly. It is, therefore, reasonable to include the4\n—-0.4 \n0.2 \n0.0 \n-0.2 dM/dH (µB/T f.u.) \n100 80 60 40 20 0\nMagnetic field (T) H1H2\nH3H4\nH512 \n10 \n8\n6\n4\n2\n0\n-2 Magnetization ( µB/f.u.) \n100 80 60 40 20 0\nMagnetic field (T) H || [110] \n 4 K \n 7 K \n 14 K 150 \n100 \n50 \n0Angle (deg) \n100 80 60 40 20 0\nMagnetic field (T) 4 K \n Mn1 \n Mn2 \n Cr Calc. Calc. Calc. !\"# !\"# !\"#\n!\"\n#$% #$& !\"\n#$% #$& \n#'($)*+,-+)./ !\"01234'5) 6.'*)'7234'5) \n#$% #$& !% !&\n859::)*\"+,234'5) !;\n859::)*\"+,234'5) !<\n=>$?)\"5)@201234'5) \n#$% #$& !\"!A\n!\"\n#$% #$& B/C\n—-\nFIG. 2. (a) Magnetization and (b) its field derivative as a fun ction of the magnetic field applied along the [110] direction\ncalculated from classical Monte Carlo simulations. The cur ves at 4 K are shown on the original scale, while the curves at 7\nand 14 K are vertically shifted for clarity. The arrows ( H1–H5) indicate the transitions similar to Fig. 1c. (c) Angles of t he\nCr, Mn1, and Mn2 spins with respect to the external-field dire ction. (d) Typical magnetic structures formed by Cr, Mn1, an d\nMn2 spins for each phase.\nspin-lattice coupling only in the Mn-Cr spin interaction\nas a minimal model.\nAfter tracing out ρij, we can exactly rewrite the inter-\nactions between the Mn and Cr lattices as\nHMC=/summationdisplay\n/angbracketleftij/angbracketrightJMn-Cr/parenleftbig\nSMni·SCrj)−bMn-Cr/parenleftbig\nSMni·SCrj)2,\nwhere the biquadratic term is introduced with the co-\nefficient bMn-Cr=J2\nMn-Crα2/2K, which we use as a pa-\nrameter for the spin-lattice coupling below. Despite the\nintegrability of lattice displacements, we intentionally re-\ntain them in the model (1) to compare theoretical results\nto experimental data regarding not only spin(magneti-\nzation) but also lattice(magnetostriction and sound ve-\nlocity) degrees of freedom.\nNote that the magneto-crystalline anisotropy of\nMnCr2S4is weak [30], because Cr ions have a half-filled\nt2gshell (3d3) and Mn ions have a half-filled t2gandeg\nshell (3d5) with zero orbital momentum.Classical Monte Carlo calculations: Magnetization\nand spin angles\nWe performed classical Monte Carlo simulations for\nthe model (1) and calculated the magnetization, M, and\nthe field derivatives of the magnetization, dM/dH, at 4,\n7, and 14 K as shown in Figs. 2a and 2b. Indeed, our\nsimple model already accounts for the experimental data\n(Figs. 1b and 1c). The five distinct steps, H1–H5, are\nall reproduced at 4 K. The two intermediate phases from\nH1toH2and from H3toH4, which are adjacent to\nthe magnetization plateau, have the steep slopes of the\nmagnetization curve in a similar way as the experimental\ndata.\nWe optimized the parameters in the model,\nJMn-Mn= 3.4 K, JMn-Cr= 3.1 K, JCr-Cr= -9.1 K,\nandbMn-Cr= 0.04 K, by fitting the calculated magne-\ntization to the experimental data. Some parameters\nwere simultaneously optimized to reproduce the two\ntransitions at TC≈65 K and at TYK≈5 K observed in\nprevious experiments without magnetic field [10, 14] (see\nAppendix A). Our estimates differ only by a few percent5\n!\" #$% \n#$& '$()*+\"\",-./$+()01,\"2+\"13 !%4\n5+\"\",-./$+()01,\"2+\"13 !&4!\"0.4 \n0.3 \n0.2 \n0.1 \n0.0  ∆L/L (10 -3 )\n60 40 20 0\nMagnetic field (T)  2.7 K \n-0.10 -0.05 0.00 0.05 0.10 \nq\n100 80 60 40 20 0\nMagnetic field(T) 4 K q1q2\nq\n-0.05 0.00 0.05 \n∆v/v\n60 40 20 0\nMagnetic field (T)  3.9 K \n 6.2 K \n 11 K -0.10 -0.05 0.00 0.05 \n∆v/v\n100 80 60 40 20 0\nMagnetic field (T)  4 K \n 7 K \n 14 K \n HYK \nHYK Exp.\nExp.Calc.\nCalc.\nH1H2H3\nH1H2H3\nH4H53.4 364\n304 3243+4\nFIG.3. Magnetostriction andsoundvelocityobtainedbyexp erimentandtheory. (a)Experimentallyobtainedmagnetost riction\nalongH/bardbl[110]. (b) Exchange-modification parameters q1andq2, which are proportional to the Cr-Mn1 and Cr-Mn2 bond\nlengths, respectively. The magnetostriction measurement ∆L/Lshould be proportional to q= (q1+q2)/2. In (b), q1andq2\nare identical below 20 T and above 60 T. (c, d) Relative change s of the sound velocity obtained in experiment and theory.\nThe longitudinal acoustic mode ( c11+2c12+4c44)/3 propagating along H/bardbl[111] was measured. (e) Schematic illustration of\nantiferromagnetically ordered spins with q1and ferromagnetically ordered spins with q2in the plateau phase.\nfrom the previous estimates [15], JMn-Mn= 3.1 K and\nJMn-Cr= 3.24 K [31]. In our spin-lattice model, the\neffective exchange coupling between Mn and Cr spins\nbecomes JMn-Cr(1−αρij)≈JMn-Mnaround zero mag-\nnetic field, which is consistent with the YK model [13].\nThe internal magnetic field that is created by the Cr\nspins and acting on the Mn spins is estimated to be\nabout 42 T using JMn-Cr= 3.1 K. This is consistent with\nthe observation of the symmetric behavior with respect\nto the field of 40 T.\nFromthequantitativepointofview, thetwointermedi-\nate phasesobtainedtheoreticallyarenarrowerthan those\nobserved experimentally. We incorporated additional bi-\nquadratic terms, bMn-Mn/parenleftbig\nSMn·SMn)2, but they do not\nimprove the results (not shown here). A more complex\nmodel is required for a quantitative comparison to the\nexperimental data.\nTo investigate the magnetic structures of the nontriv-\nial phases induced by magnetic field, we calculated the\nanglesofthe Cr and Mn spins from the external-fieldaxis\nat 4 K, shown in Fig. 2c. Using this data, we sketched\ntypical magnetic structures for each phase in Fig. 2d.\nThe spin-lattice coupling stabilizes not only the magne-\ntization plateau with a collinear magnetic structure but\nalso the two intermediate phases, where a triangular-like\nstructure formed by one Cr and two Mn spins rotates\ncontinuously with magnetic fields variation. The inter-\nmediate phases give rise to unconventional asymmetricmagnetic structures.\nMagnetostriction and sound velocity\nMCcalculationsforthe model(1) accountforthe mag-\nnetostriction and the sound velocity as well. The ex-\nperimentally obtained magnetostriction ∆ L/Lis shown\nin Fig. 3a. Figure 3b shows the exchange-modification\nparameters q1=αρMn1−Cr,q2=αρMn2−Cr, andq=\n(q1+q2)/2 obtained by the MC calculations.\nThe averaged displacement qis proportional to the\nchange of the sample length, which is measured in the\nmagnetostriction experiment.\nThe relative change of the sound velocity ∆ v/vob-\ntained experimentally and theoretically are shown in\nFigs. 3c and 3d, respectively. In the MC calculations,\nwe deduced the change of the sound velocity from the\neffective spring constant Keff, which differs from the\noriginal constant Kowing to the spin-lattice coupling.\nWe calculated the compressibility κin the MC simula-\ntions:κ≡βVar[q]∼β/integraltext\ndqq2e−βKeff\n2q2//integraltext\ndqe−βKeff\n2q2=\n1/Keff, where βis the inverse temperature. Although\nacousticphononmodesarenotincluded inourmodel(1),\ntheeffectivespringconstantshouldbecommontovarious\nphonon modes. The sound velocity measured in experi-\nments should be related to the effective spring constant:\nv∝√Keff. Therefore, we can estimate the velocity using6\nthe relation v∝1/√κ.\nOur theoretical model accounts for all the character-\nistic features related to the phase transitions ( H1–H3)\nobserved in the experiments up to 60 T. Moreover, the\nlow-field anomalies labelled as HYK(2 T at 6.2 K and\n9 T at 11 K in Fig. 3c), which corresponds to a phase\nboundary between the YK phase and the ferrimagnetic\nphase, where only the Cr spins order [14], are also re-\nproduced in the MC simulations as shown in Figs. 3c\nand 3d. The agreement between experiment and theory\nclearly shows the validity of our model and the relevance\nof the magnetoelastic coupling between the Mn and Cr\nions.\nThe theory reproduces the magnetostriction data very\nwell and suggests that the spin-lattice coupling produces\nstrong (q1<0) and weak ( q2>0) bonds between the\nMn and Cr spins (see sketch in Fig. 3e). In the plateau\nphase, the Cr and Mn spins are antiferromagnetically\n(ferromagnetically) ordered on the strong (weak) bonds.\nIn other words, this Z2symmetry of the Mn-Cr bond\nis spontaneously broken. Thus, a domain-wall excitation\nrelatedtothe Z2symmetryisexpectedtoappearatsome\nfinite temperatures due to the entropic effect. The broad\nminimum observedaround40Tintheultrasounddataat\nhigher temperatures (6.2 K and 11 K), which is absent in\nthe theoretical results, might be attributed to dynamics\nof the domain wall.\nPhase diagram\nCombining the present ultrahigh magnetic-field exper-\niments with previousdatain ref. [15], weobtain the com-\nplete phase diagram of MnCr 2S4in Fig. 4. The magnetic\nstructures for all phases, identified by our MC calcula-\ntions, are schematically shown in the figure. To identify\nthe higher temperature phase above TYKat zero mag-\nnetic field, we calculated the spin correlations between\nthe Mn1 and Mn2 spins using the model (1) with the\nsame parameters (see Appendix B). While the correla-\ntion between the Mn spin components parallel to the Cr\nspins develops below TC, the perpendicular components\nare almost independent down to TYK. This indicates\nthat the transverse components of the Mn spins remain\nparamagnetic in the intermediate temperature phase be-\ntweenTCandTYK, and the antiferromagnetic order of\nthe transverse components eventually takes place at TYK\nas shown in Fig. 4.\nWith increasing temperatures, both spin-asymmetric\nphases diminish on the expense of the expanding spin-\ncollinear phase. This finding indicates that thermal fluc-\ntuations favor the collinear magnetic structure rather\nthan noncollinear ones [32, 33]. In the phase diagram,\nthe higher-field phases mirror the lower-field phases; the\nphase diagram is symmetric with respect to approxi-\nmately 40 T, where the external fields cancel out theinternal exchange fields acting on the Mn spins.\n!\"#$% \n#$& #$!\"%'(100 \n80 \n60 \n40 \n20 \n0Magnetic field (T) \n20 15 10 5 0\nTemperature (K) \nFIG. 4. Magnetic-field-temperature phase diagram of\nMnCr 2S4. Thephaseboundarieswere obtainedfrom themag-\nnetization, ultrasound propagation and magnetostriction ex-\nperiments in pulsed fields. The closed circles with error bar s\nof±2 T and ±1 K were obtained in the present ultrahigh-\nfield experiments. The open squares were taken from ref.\n[15]. Typical magnetic structures revealed by the MC calcu-\nlations are illustrated in each phase. The Cr spins (orange\narrows) are almost perfectly aligned with the external field .\nThe Mn spins (two blue arrows) show a complex, strongly\nfield-dependent order.\nDISCUSSIONS\nThanks to our measurements in ultrahigh magnetic\nfields, we unraveled the complete H–Tphase diagram\nof the YK ferrimagnetic spinel compound. The experi-\nmental results are accounted for by a quite simple model\nincorporating spin-lattice coupling between the Mn and\nCr ions. Our findings indicate that the spin-lattice cou-\nplingiscrucialforestablishingthe extremelyrobustmag-\nnetization plateau. The effect of spin-lattice couplings\nhas been extensively investigated both experimentally\nand theoretically in antiferromagnetic Cr spinel oxides,\nZnCr2O4[34], CdCr 2O4[24], and HgCr 2O4[27], where\nthe pyrochlore lattice reveals strong geometrical frustra-\ntion. Here, with only weak frustration caused by the\ncompetition between JMn-MnandJMn-Crand with two\ndifferent magnetic ions, Mn and Cr, still a robust mag-\nnetization plateau was observed. Our spin-lattice model\nappears to be universally applicable to a broad range of\nferrimagnetic andantiferromagnetic spinel compounds.\nIn fact, strongspin-latticecouplingshavebeen widely ex-7\npected to exist in spinel compounds, such as in CoCr 2O4\n[35] and ZnCr 2S4[36], where unconventional magneto-\nstructural phase transitions have been observed. They\nmight be understood incorporating spin-lattice couplings\nto a simple spin model in a similar way as done in the\npresent study.\nAnother important finding is that the unconventional\ntriangular-like magnetic structures in the two intermedi-\nate phases are formed by one Cr and two Mn spins rotat-\ning continuously as magnetic fields increase. This identi-\nfication finally gives a clear solution to the long-standing\nproblemofthehigh-fieldmagneticstructuresofMnCr 2S4\n[15, 28, 29, 37]. The origin of the intermediate phases is\nthecompetitionbetweentheYK-typeinteractions,which\nfavors a noncollinear triangular structure, and the spin-\nlattice coupling, which favorsa collinear structure. Here,\nwe gain the straightforward insight that incorporation of\nspin-lattice couplings in ferrimagnetic spinel compounds\nleads to a variety of unconventional magnetic phases,\nsuchasfractionalmagnetizationplateausandspin-driven\nmultiferroic phases.\nLastly, we discuss possible improvements in our spin-\nlattice model. In the ultrasound experimental data\n(Figs. 3c and d), anomalous features were observed\naroundthe middle ofthe magnetizationplateauat higher\ntemperatures, which is absent in our theoretical con-\nsideration. This anomaly is even more pronounced in\nthe ultrasound attenuation as shown in previous experi-\nments [15]. It was speculated that this anomalousbehav-\nior might be attributed to a frustration in the Mn dia-\nmond lattice induced by next-nearest-neighbour interac-\ntions, as recently discussed for another spinel compound,\nMnSc2S4[38, 39]. The inclusion of farther spin interac-\ntions to our minimal model (1) might be appropriate.\nAnother possibility to improve the agreement may lie\ninmorecomplexinteractionsofthelatticedegreesoffree-\ndom. In our minimal model, lattice displacements are\nindependent of each other, forming a flat phonon band.\nA phonon band structure formed by connected lattice\ndegrees of freedom could more accurately describe rel-\nevant lattice displacements and reproduce the anomaly\nobserved in experiment.\nAlthough it has been believed for a long time that a\nmajority of magnetic properties of ferrimagnetic spinels\ncanbeunderstoodwithintheYKandLKDMmodels,our\nstudies have revealed that additional perturbations, such\nas the spin-lattice coupling, are crucial to explain their\nproperties under magnetic fields. Incorporating spin-\nlattice coupling, the YK and LKDM models reopen a\nnew platform to study unconventional magnetic phases,\nsuchasfractionalmagnetizationplateausandspin-driven\nmultiferroic phases.ACKNOWLEDGEMENTS\nThis research has been supported by the DFG via\nTRR 80 (Augsburg - Munich), by SFB 1143 (Dresden),\nthrough the W¨ urzburg-Dresden Cluster of Excellence on\nComplexity and Topology in Quantum Matter - ct.qmat\n(EXC 2147, project-id 39085490), and by the BMBF via\nDAAD (project-id 57457940). We acknowledge support\nby the project 16.80012.02.03F (ASM) and by HLD at\nHZDR and LNCMI at CNRS, both members of the Eu-\nropean Magnetic Field Laboratory (EMFL).\nAPPENDIX A: MAGNETIC SUSCEPTIBILITY\nAND SOUND VELOCITY\nWe calculated the inverse magnetic susceptibility χ−1\nand the relative change of the sound velocity ∆ v/vat\ntemperatures ranging from 2.5 to 400 K using the clas-\nsical Monte Carlo simulation for the model (1) in the\nmain text. The parameters were set to the same val-\nues as those in the main text: JMn-Mn= 3.4 K, JMn-Cr\n= 3.1 K, JCr-Cr= -9.1 K, and bMn-Cr=J2\nMn-Crα2/2K\n= 0.04 K. The quantum-classical crossover of the spin\nsystem needs to be correctly considered [40]: We treat\nthe spins as vectors and set the spin (vector) lengths to\n|S|cl=Sbelow 15 K and |S|cl=/radicalbig\nS(S+1) above 20 K.\nThe classical Monte Carlo simulations qualitatively re-\nproduce experimental data observed by Tsurkan et al. in\nref. [15].\nAPPENDIX B: SPIN CORRELATION\nTo investigate a magnetic phase between T YKand TC,\nwecalculatedspincorrelationsoftheMn1andMn2spins,\nS/bardbl\nMn1S/bardbl\nMn2(parallel component along the external field)\nand S⊥\nMn1S⊥\nMn2(perpendicular component against the ex-\nternal field).8\n—-\n0.9 \n0.8 \n0.7 χ-1  (mol/emu) \n15 10 5\nTemperature (K) 50 \n40 \n30 \n20 \n10 \n0χ-1  (mol/emu) \n400 300 200 100 \nTemperature (K) θCW  ~ -29 K 50 \n40 \n30 \n20 \n10 \n0χ-1  (mol/emu) \n400 300 200 100 \nTemperature (K) θCW  ~ 12 K \n1.4 \n1.3 \n1.2 \n1.1 \n1.0 \n0.9 χ-1  (mol/emu) \n15 10 5\nTemperature (K) Exp. Calc. \nExp. Calc. TYK TYK TC TC!\"# !$#\n!\"# !$#\n—-\nFIG. 5. Inverse magnetic susceptibility χ−1of MnCr 2S4ob-\ntained in the experiments [15] and the theory. An external\nmagnetic field of 1 T is applied along the [111] direction. (a)\nand (b) show the data at high temperatures of 20–400 K. (c)\nand (d) show the data at low temperatures of 2.5–15 K.\n—-\n30 \n20 \n10 \n0\n-10 \n-20 ∆v/v (10 -3 )\n15 10 5\nTemperature (K) 10 \n5\n0\n-5 ∆v/v (10 -3 )\n15 10 5\nTemperature (K) Exp. Calc. \nTYK TYK !\"# !$#\n—-\nFIG. 6. Relative changes of elastic constant ∆ c/cof MnCr 2S4\nobtained in the experiments (a) [15] and the theory (b).\nREFERENCE\n[1] T. Kimura, et al.,Nature426, 55 (2003).\n[2] S. Seki, X.Z. Yu, S. Ishiwata, and Y. Tokura, Science\n336, 198 (2012)..\n[3] Y. Yafet, and C. Kittel, Phys. Rev. 87, 290 (1952).\n[4] D.H. Lyons, T.A. Kaplan, K. Dwight, and N. Menyuk,\nPhys. Rev. 126, 540 (1962).\n[5] J. M. D. Coey, Can. J. Phys. 65, 1210 (1987).\n[6] T.A. Kaplan, and N. Menyuk, Philos. Mag. 87, 3711\n(2007).\n[7] K. Penc, N. Shannon, and H. Shiba, Phys. Rev. Lett. 93,\n197203 (2004).—-\nCr !\nMn1 ! Mn2 !\nTemperature Cr !\nMn1 ! Mn2 !Paramagnetic state !TYK  ~ 5 K ! 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Theory E. ,P05017\n(2008)."    },    {        "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 < <VDW  (km/s) \nP0Hz (mT) Solid lines : Eq. (8) vg,max \nEq. (14) \n100 200 300 400 01234567\nSolid lines : Eq. (14) < < = T A < <\nDsim  = 1 (meV) VDW  (km/s) \nP0Hz (mT) Temperature 2 4 6 8040 80 120 160 \nP0Hz (mT) VDW  (m/s) Solid lines : Eq. (5)  <  < = TA  <  < Temperature \nFIG. 2. (color online) (a) Domain wall velocity as a func-\ntion of the external field Hin the low-field regime ( µ0H <10\nmT). Symbols indicate the calculation results and solid lin es\nindicate Eq. (5). (b) Walker Breakdown field HWBas a func-\ntion of net angular momentum δsat various DMI constants\nDsim. Domain wall velocity in the high-field regime for (c)\nDsim= 0 and (d) Dsim= 1 meV. Solid straight lines in (c)\nindicate Eq. (8). A red solid line in (c) and solid curved line s\nin (d) represent relativistically modified solution of doma in\nwall velocity, Eq. (14). Horizontal dashed lines in (c) and ( d)\nrepresent vg,max[Eq. (12)].\nmentioning. First, HWBdiverges at TA(i.e.,δs= 0).\nSecond,HWBfor a finite DMI becomes much larger than\nthat for D= 0, in agreement with Eq. (6) (solid lines).\nCompared to ferromagnets, ferrimagnets exhibit larger\nHWBbecause the net magnetic moment M(=M1−M2)\nis small near TA, as explained above. Figure 2(c) shows\nVDWin the high field regime for D= 0. The numerically\nobtained values of VDWatTA(green symbols) deviate\nfrom Eq. (8) (a green solid line), which predicts a linear\nincrease in VDWwithH. This deviation is a manifes-\ntation of the approximately relativistic dynamics of fer-\nrimagnetic DW, as we will explain below in detail. For\nD= 0[Fig. 2(c)], the nonlineardependence of VDWonH\nappearsonly at TA. ForDsim= 1 meV(correspondingto\nD= 2 mJ/m2), however, the nonlinear dependence ap-\npears in all tested ranges of T[orδs; Fig. 2(d)]. It means\nthat the relativistic dynamics occurs in wide ranges of T,\nresulting from the largely enhanced HWBnearTA.\nForthecasesshowingthenonlineardependenceof VDW\nonH, we observe the spin-wave emission from DW [Fig.\n3(a)]. The spin–wave frequency is in THz ranges [Fig.\n3(b)]. To elaborate the nonlinear dependence of VDW\nonHand associated THz spin-wave emission, we derive\nthe spin-wave dispersion for ferrimagnets on top of the\nuniform ground state, n=ˆz, given as\nω±=±δs+/radicalbig\nδ2s+4ρ(Ak2+K−h)\n2ρ,(11)where the upper (lower) sign corresponds to the right-\nhanded (left-handed) circular mode and kis the\nwavenumber. We note that DMI does not contribute to\nthe spin-wave dispersion as it is effective only for the y-\ncomponent of magnetization, which is negligible for per-\npendicularly magnetized ferrimagnets. From the disper-\nsion [Eq. (11)], we obtain the maximum spin-wave group\nvelocityvg,maxas\nvg,max=A/ds, (12)\nwheres= (s1+s2)/2. We note that vg,maxis indicated\nby horizontal dashed lines in Fig. 2(c) and (d). With\nvg,max, the nonlinear dependence of VDWonHis readily\ninterpreted based on the approximately relativistic kine-\nmatics of a DW, similar to the dynamics of an antiferro-\nmagnetic DW [10]: vg,maxacts as the speed of light [13]\nand the DW width shrinks as VDWapproaches vg,maxvia\nthe Lorentz contraction. The Lorentz contraction of DW\nis described by\nλDW=λeq/radicalBig\n1−(VDW/vg,max)2,(13)\nwhereλeqis the equilibrium DW width. The in-\nertial DW mass also varies with the Lorentz factor\n1//radicalbig\n1−(VDW/vg,max)2, i.e.,M= 2ρA/λDW. Figure\n3(c) shows that the DW width decreases as VDWap-\nproaches vg,maxwhile the inertial mass increases with\nVDW. With the Lorentz contraction, we modify Eq. (5)\nrelativistically as\nVDW=vg,max/radicalBig\n1−(λDW/λeq)2,(14)\nwhich is represented by a red solid line in Fig. 2(c) and\nsolid lines in Fig. 2(d). Excellent agreement between\nnumerically obtained VDWand Eq. (14) confirms the rel-\nativistic dynamics of ferrimagnetic DWs.\nWe also show that spin-orbit torque drives a fast rel-\nativistic motion of ferrimagnetic DWs. We consider\ndamping-like torque only for simplicity. Using the col-\nlective coordinate approach, we obtain the steady-state\nsolution of ferrimagnetic DW velocity as\nVDLT=λDWπs˜BDLT\n2α, (15)\nwhere˜BDLT=/planckover2pi1θSHJ/2etzs1s2is the effective field corre-\nsponding to the damping-like torque, θSHis the effective\nspin-Hall angle of ferrimagnet/heavy metal bilayer, Jis\nthe current density, eis the electron charge, and tzis\nthe ferrimagnet-thickness. Numerical simulation includ-\ning the damping-liketorque for Dsim= 1 meV [Fig. 3(d)]\nshows that spin-orbit torque combined with DMI effect\nis highly efficient for ferrimagnetic DW motion and the\nrelativistic dynamics occurs for all tested ranges of T(or\nδs). This spin-orbit-torque-driven ferrimagnetic DW mo-\ntion also accompanies with THz spin-wave emission (not\nshown).\nC\u000b \nD\u000b \n\nE\u000b \nF\u000b 1400 1500 1600 -1 01\n1400 1450 1500 1550 1600 -0.01 0.00 0.01 0.02 0.03 n\nSite (nm)  n x n y n z\n400 450 500 1.6 1.8 2.0 2.2 \nGs = -2 \nZ+\nZ-Frequecy (THz) \nP0Hz (mT) Gs (10 -8  J s/m 3)\nGs = +2 \n0.5 1.0 1.5 2.0 2.5 0123456\n <  <   <  < VDW  (km/s) \nJ (10 11  A/m 2)Solid lines : Eq. (15) Eq. (14) \nTemperature \n0.0 0.2 0.4 0.6 0.8 2.0 2.5 3.0 \n  Eq. (13) \nVDW /v g,max ODW  (nm) \n45678(10 -26  kg) \nDW mass \nFIG. 3. (color online) (a) Configuration of domain wall\nand spin-waves for the staggered vector n. The inset shows\noverall shape of domain wall. (b) The frequency of emitted\nspin-waves as a function of Hwith finite δs.ω+andω−\nindicate the right-handed and left-handed spin-wave modes ,\nrespectively [see Eq. (11)]. (c) Domain wall width and mass\nas a function of VDW/vg,maxatTA. The orange solid line indi-\ncates Eq. (13). (d) Domain wall velocity as a function of the\ncurrent density Jat several temperatures. Symbols represent\ncalculation results and solid lines represent Eq. (15). Das hed\nlines represent relativistic modified solutions, Eq. (14).\nFinally, we discuss about the origin of spin-wave emis-\nsion from ferrimagnetic DWs. Two mechanisms have\nbeen proposed: Cherenkov-like process [35] and internal\nDW structure distortion [36]. The former occurs when\nDW velocity matches spin-wave phase velocity. In anti-\nferromagnets or ferrimagnets near TA, the phase velocity\nis alwayshigher than the groupvelocity as one finds from\nthe dispersion [Eq. (11)]. Therefore, this Cherenkov pro-\ncess is irrelevant to our case. For ferrimagnetic DWs,\ninstead, the spin-waves can be emitted by releasing the\nDW energy enhanced through the Lorentz contraction as\nin the case of antiferromagnetic DWs [10].\nIn conclusion, we have shown field-driven THz spin-\nwave emission for ferrimagnetic DWs, which is not pos-\nsible for ferromagnetic or antiferromagnetic DWs. In\nferrimagnet/heavy metal bilayers in which the interfa-\ncial DMI arises naturally, the field-driven THz spin-wave\nemission can be observed in wide ranges of T(orδs),\nthereby largely enhancing the experimental accessibility\nto our prediction. Moreover, an in-plane current can also\nexciteTHz spin-wavesinwide rangesof T(orδs) through\nthe combined effect between spin-orbit torque and DMI.\nOur finding suggests that ferrimagnetic DWs are poten-\ntially useful for high-speed and high-frequency spintron-\nics devices.\nK.-J.L. was supported by the National Research\nFoundation of Korea (NRF) (NRF-2015M3D1A1070465,\nNRF-2017R1A2B2006119)and by the DGIST R&D Pro-gram of the Ministry of Science, ICT and Future Plan-\nning (17-BT-02). S.K.K. andY.T. weresupportedby the\nArmy Research Office under Contract No. W911NF-14-\n1-0016. T.O. was supported by JSPS KAKENHI Grant\nNumbers 15H05702, 26103002. K.J.K. was supported by\nNRF-2017R1C1B2009686. G.G. was supported by NRF-\n2016R1A6A3A1193588.\n∗These two authors contributed equally to this work.\n†kjlee@korea.ac.kr\n[1] C. Sirtori, Nature 417, 132 (2002).\n[2] M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle,\nM. Ilegems, E. Gini, and H. 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Ghimire and Xiao Hu\nInternational Center for Materials Nanoarchitectonics (WPI-MANA),\nNational Institute for Materials Science,Tsukuba 305-0044, Japan\n(Dated: September 26, 2018)\nUsing the \frst-principles density functional approach, we investigate Ca 2FeOsO 6, a material of\ndouble perovskite structure synthesized recently. According to the calculations, Ca 2FeOsO 6is a\nferrimagnetic Mott-insulator in\ruenced by the cooperative e\u000bect of spin-orbit coupling (SOC) and\nCoulomb interactions of Fe-3 dand Os-5delectrons, as well as the crystal \feld. When Fe is replaced\nwith Ni, the system exhibits half metallic (HM) states desirable for spintronic applications. In\n[Ca2Fe1\u0000xNixOsO 6]2, HM ferrimagnetism is observed with \u0016tot= 2\u0016Bper unit cell for doping rate\nx= 0:5, whereas HM antiferromagnetism (HMAFM) with nearly zero spin magnetization in the\nunit cell for x= 1, respectively. It is emphasized that half metallicity is retained even with SOC\ne\u000bect due to the large exchange-splitting between spin-up and spin-down bands close to the Fermi\nlevel.\nPACS numbers: 85.75.-d, 71.30.+h, 75.70.Tj, 71.15.Mb\nI. INTRODUCTION\nDouble perovskite oxides have been widely investigated\ndue to their interesting properties useful for spintronic\napplications1,2. With the general formula A 2BB0O6, A\nis usually an alkaline earth or rare-earth element, and B\nand B0are the transition metal elements, both of which\nare composed of edge-shared octahedra. Depending on\nthe choice of B and B0cations, these compounds show\na variety of electrical and magnetic properties, namely\nmetallicity, half metallicity, insulator as well as ferro-\nmagnetism, ferrimagnetism, and antiferromagnetism1.\nThe discovery of room-temperature colossal magne-\ntoresistance and half metallicity in Sr 2FeTO 6(where\nT=Mo, Re)3{5, multiferroicity in Bi 2NiMnO 66, magneto-\ndielectricity in La 2NiMnO 67, leads to intensive study in\ndouble perovskite materials. Recent researches have been\ndevoted to understanding the electronic and magnetic\nproperties of hybrid 3 d-4d(5d) double perovskites which\nexhibit high spin polarization8,9essential for device ap-\nplications at room temperature.\nHalf metals (HMs) are a class of materials which\nare metallic in one spin channel, while insulating in\nthe opposite spin channel due to the asymmetric band\nstructure10{16. HMs allow spin polarized currents to \row\nwithout any external operation, and thus are very use-\nful for spintronics applications. The total spin-moment\nper unit cell is quantized in units of Bohr magneton ( \u0016B)\ndue to insulating state in one spin-channel. HMs have\nbeen identi\fed in several groups of materials17{30. Most\nof the experimentally available HMs are either ferromag-\nnets (FM) or ferrimagnets (FiM) which give non-zero in-\nteger moments. It was noted that spin-polarized current\ncan be hampered by stray \felds which stabilize magnetic\ndomains13,26. This drawback can be overcome by HM an-\ntiferromagnets (HMAFM), a subclass of HMs character-\nized further by zero spin magnetization per unit cell11,12.\nA number of materials with DPs structures have been\npredicted as possible candidates of HMAFMs17{22.Because of the recent success in synthesizing os-\nmium oxides, osmates have been attracting signi\f-\ncant interests yielding various unconventional phases.\nFor instance, unusual superconductivity is observed in\nA2Os2O7(A=Cs, Rb and K)31. Magnetically driven\nmetal-insulator transition is found in Cd 2Os2O732and\nNaOsO 333, and the ferroelectric-type structural transi-\ntion has been discovered in metallic LiOsO 334. Among\nthe double perovskites, magnetic insulating states in\nSr2MOsO 6(where M=Cu, Ni)35, Mott-insulating ferro-\nmagnetic state in Ba 2NaOsO 636, and half semi-metallic\nantiferromagnetism in Sr 2CrOsO 637are a few examples\nthat have been reported. A newly synthesized double\nperovskite material Ca 2FeOsO 6comes into our attention\nwhich is reported to be a FiM insulator driven by lat-\ntice distortion with high Curie temperature ( Tc)38. The\ncrystal has unique properties suitable for spintronic de-\nvice applications: Fe and Os atoms carry on opposite spin\nmagnetizations, and octahedra exhibit strong crystal dis-\ntortion which may induce strong crystal \feld that helps\nin splitting the spin-up and spin-down bands. The origin\nof novel FiM state with Tcof 320K opens new possibility\nof realizing device applications at room temperature. We\ndope Ni atom having charge state +2 to replace Fe atom\nwith charge state +3 to study its in\ruence on the parent\nmaterial Ca 2FeOsO 6.\nWe have performed \frst-principles density-functional\ncalculations on Ca 2FeOsO 6(CFOO). It is found that\nCFOO is a FiM Mott insulator with total angular mo-\nment\u0016tot= 4\u0016Bper unit cell of [Ca 2FeOsO 6]2. This ma-\nterial is interesting in the sense that the topmost valence\nstates close to Fermi level ( EF) are exclusively spin-down\nbands contributed from 5 delectrons of Os atoms, and the\nelement Fe has no in\ruence on the electronic state near\nEF. Therefore, replacing Fe by 3 delements with more\nthan \fve valence electrons one can make a \fne control\non charge and spin. Speci\fcally, we consider the replace-\nment of Fe atom by the Ni atom which shows interesting\nproperties desirable in spintronics.arXiv:1408.1771v1  [cond-mat.mes-hall]  8 Aug 20142\nFIG. 1. Double perovskite structure of [Ca 2FeOsO 6]2and\n[Ca2NiOsO 6]2. The red (blue) arrows indicate the direction\nof Fe/Ni(Os) spins along the adirection which is the easy\naxis.\nWe \fnd that the material [Ca 2Fe1\u0000xNixOsO 6]2is HM-\nFiM with\u0016tot= 2\u0016Batx= 0:5 and nearly compen-\nsated HMAFM with \u0016tot= 0:3\u0016Batx= 1. The in-\nterplay among Coulomb repulsion, SOC and the crystal\n\feld plays an important role in this material.\nThe organization of this paper is as follows: Section II\ndescribes the details on crystal structure and methods.\nSection III presents the results on the parent material\n[Ca2FeOsO 6]2, while Sec. IV discusses the results on\nthe possibility of obtaining half metallicity in the doped\nmaterials [Ca 2Fe1\u0000xNixOsO 6]2. Section V contains the\ndiscussions and the conclusions are drawn in Sec. VI.\nII. CRYSTAL STRUCTURES AND METHODS\nIn the double perovskite [Ca 2FeOsO 6]2(CFOO), the\ntransition metal Fe and Os occupy B and B0sites in an\nordered way. The crystal structure of CFOO shown in\nFig. 1 falls in the space group P21=nwith monoclinic-\ndistortion. It has structural distortions due to the tilting\nand rotation of the two corner-sharing FeO 6and OsO 6\noctahedra.\nThe electronic and magnetic structure calculations\nwere performed within density-functional theory (DFT)\nby using the full-potential linearized augmented plane\nwave plus local orbital method implemented in the\nWIEN2k code39. The atomic sphere radii RMTwere 2.17,\n1.99, 1.99, 2.0 and 1.64 Bohr for Ca, Fe, Ni, Os and O\nrespectively. A set of 1000 k-points were used in the\nfull Brillouin zone. The standard generalized-gradient\napproximation (GGA) exchange-correlation potential\nwithin the PBE-scheme40were used with Coulomb in-\nteractionUof 5eV for Fe (Ni) and 1.5eV for Os,\nrespectively41,42. Spin-orbit coupling is considered via a\nsecond variational step using the scalar-relativistic eigen-\n-20-10 0 10 20(a) CFOO-Total DOS\n-6-3 0 3 6 (b)Fe-3d\n-2-1 0 1 2 3\n(c) Os-5d\n-0.6 0 0.6\n-8 -6 -4 -2  0  2  4Density of States (states/eV)\nEnergy (eV)(d) O1-2p\nO2-2p\nO3-2pFIG. 2. Density of states for [Ca 2FeOsO 6]2in spin-up (\") and\nspin-down (#) channels: (a) total, (b) Fe-3 dstates, (c) Os-5 d\nstates and (d) three in-equivalent oxygen-2 pstates.\nfunctions as basis43. We checked di\u000berent magnetic con-\n\fgurations of Fe, Ni and Os sites and found that the FiM\nstructure shown in Fig. 1 is the ground state consistent\nwith the experiments38,44.\nIII. PARENT MATERIAL [Ca 2FeOsO 6]2\nIn CFOO, the transition element Fe nominally takes\nthe charge state +3 with 3 d5con\fguration. These \fve\nvalence electrons occupy the t2gandegorbits resulting\nto a high-spin (HS) state of Fe. On the other hand, Os\ntakes the charge state +5 with 5 d3con\fguration, where\nthree valence electrons occupy the t2gorbits giving rise\nto HS state.\nTo gain insight into the electronic properties of CFOO\nthe spin-resolved total and partial density of states\n(DOS) in spin-up and spin-down channels are shown in\nFig. 2. According to \frst-principles calculations, there is\nan energy gap of \u00180:8eV atEF, indicating clearly that\nCFOO is a Mott insulator. This result is consistent with\nthe recent experimental report of the insulating state in\nCFOO38. Fe-3d(i.e.,t3\n2gande2\ng) states in spin-up chan-\nnel are fully occupied and remain deep in the valence\nregion, whereas the states in spin-down channel lies in\nthe conduction region. Unlike Fe, Os-5 d(i.e.,t3\n2g) states\nare located in the conduction region with a broad peak\nfor spin-up channel indicating the empty states, whereas\nin spin-down channel they occupy the topmost valence3\n-20-10 0 10 20(a) CFNOO-Total DOS\n-3 0 3(b) Fe-3d\nNi-3d\n-2-1 0 1 2 3\n(c) Os1-5d\nOs2-5d\n-0.6 0 0.6\n-8 -6 -4 -2  0  2  4Density of States (states/eV)\nEnergy (eV)(d) O1-2p\nO2-2p\nO3-2p\nFIG. 3. Density of states for [Ca 2Fe0:5Ni0:5OsO 6]2in spin-up\n(\") and spin-down ( #) channels: (a) total, (b) Fe-3 d/Ni-3d\nstates, (c) Os-5 dstates, and (d) three inequivalent oxygen-2 p\nstates\nbands below EF. As observed in Fig. 2(d), oxygen bands\nbelowEFin both spin-channels hybridize with the Fe-\n3dand Os-5dstates. This is caused due to octahedral\ndistortions, where three sorts of oxygen positions with\ndi\u000berent Fe(Os)-O bond-lengths appear.\nThe magnetic property of CFOO is of particular inter-\nests. At the ground state obtained from \frst-principles\ncalculations, Fe couples antiferromagnetically with Os.\nThe calculated total moment ( \u0016tot) is 4.0\u0016Bper unit cell\n(see Table I). In an ionic picture, each Fe ion carries mo-\nment +5\u0016Bwhile Os ion carries \u00003\u0016B, giving rise to \u0016tot\n= 2\u0002(+5\u0016B) + 2\u0002(\u00003\u0016B) = 4\u0016Bin [Ca 2FeOsO 6]2,\nconsistent with the \frst-principles calculations.\nIV. DOPED MATERIALS [Ca 2Fe1\u0000XNiXOsO 6]2\nThe above properties makes CFOO a promising can-\ndidate for exploring possible HM states with \fne con-\ntrol on charge and magnetic moments. HM states can be\nachieved in CFOO by doping 3 dtransition metals having\nvalence electron larger than \fve. To be speci\fc, we con-\nsider the B-site modi\fcation by replacing one Fe atomd\nwith Ni in [Ca 2FeOsO 6]2, an element having charge state\n+2 with 3d8con\fguration. Out of eight valence electrons\n\fve occupy the t2gandegorbits of spin-up channel while\nthe remaining three occupy the t2gorbits in spin-down\nchannel giving rise to a moment of 2 \u0016Bper Ni atom.\n-20-10 0 10 20(a)CNOO-Total DOS\n-3 0 3(b) Ni-3d\n-2-1 0 1 2 3\n(c) Os-5d\n-0.6 0 0.6\n-8 -6 -4 -2  0  2  4Density of States (states/eV)\nEnergy (eV)(d) O1-2p\nO2-2p\nO3-2pFIG. 4. Density of states for [Ca 2NiOsO 6]2in spin-up (\") and\nspin-down (#) channels: (a) total, (b) Ni-3 dstates, (c) Os-5 d\nstates and (d) three in-equivalent oxygen-2 pstates.\nNi doping corresponds to addition of three extra elec-\ntrons to the system. Since the charge states of Fe(+3)\nand Ni(+2) di\u000ber from each other, the Os atom interact-\ning with Fe has a charge state +5 and 5 d3con\fguration\nwhile the other one interacting with Ni has +6 charge\nstate and 5 d2con\fguration, to maintain charge neutral\nin the system. Thus, Ni doping will in\ruence the sys-\ntem by (i) shifting the Os bands towards the conduction\nregion, and (ii) compensating magnetic moment by 2 \u0016B\nper each Fe replacement. In this way, one can modify the\nmaterial to design the desired HMFiM and HMAFM.\nWe perform \frst-principles calculations to check\nthe above idea for replacement rate x= 0:5 in\n[Ca2Fe1\u0000xNixOsO 6]2. As shown in Fig. 3, Fe-3 dstates in\nspin-up channel are fully occupied lying deep in the va-\nlence region, whereas for spin-down channel they remain\nin the conduction region. The Ni-3 dbands are also oc-\ncupied in spin-up channel while the spin-down bands are\npartially occupied with a localized peak at -1.8eV below\nEF. The remaining bands appear in the conduction re-\ngion. This happens because \fve out of eight delectrons\nfrom Ni occupy the spin-up channel and the three re-\nmaining electrons go to spin-down channel to \fll the t2g\norbits. Due to repulsive interaction from spin-down Ni-\n3delectrons, Os- t2gelectrons which were originally lying\nat the topmost valence region in the parent material (see\nFig. 2) shift towards the conduction region crossing EF\n(see Fig. 5(b)). As the results, the Os- t2gstates crossing\nEFform a continuous band and give rise to metallic state4\nFIG. 5. Band structures for doping rate (a) x= 0, (b)x= 0:5 and (c)x= 1 in [Ca 2Fe1\u0000xNixOsO 6]2for spin-up (red) and\nspin-down (black) channels.\nfor spin-down channel. Hence, with spin-up channel in-\nsulating and spin-down channel metallic, the material is\na half metal.\nLet us now focus to the most interesting case of re-\nplacement rate x= 1, where both Fe atoms are replaced\nby Ni atoms within unit-cell. From the \frst principles\ncalculations the FiM ground state is \u0018135meV less than\nthe \frst excited AFM state. Due to the presence of Ni\natom alone on the B-site, Os atom fully attains the charge\nstate +6 with 5 d2con\fguration unlike in the parent ma-\nterial [Ca 2FeOsO 6]2. For Ni of d8, threet2gand twoeg\norbits in spin-up channel and three t2gorbits in spin-\ndown channel are occupied. For Os of d2, only two t2g\norbits in spin-down channel are occupied. The system is\nthen expected to favor half metallicity with full compen-\nsation of total moment. As revealed by the DOS shown in\nFig. 4, Ni-3 dstates are fully occupied in spin-up channel\nby \fve electrons and the remaining electrons go to oc-\ncupy thet2gorbits in spin-down channel. This is evident\nwith the occupation of Ni-3 dstates in the valence region\nwith a localized peak at -1.6eV below EF. The unoccu-\npiedegband appears far in the conduction region. The\nOs-5dstates are almost empty in spin-up channel but\nhas partial occupation with a band crossing EFin spin-\ndown channel. This is caused mainly by the spin-down\nelectrons of Ni which push the Os- t2gstates up towards\nthe conduction region. Thus, two of the t2gstates of Os\natom remain in the valence region while the unoccupied\nstate moves to the conduction region crossing EF(see\nalso Fig. 5(c)). Due to crystal distortion, gap could not\nopen-up among the t2gstates of Os atom which results\nin metallic state in spin-down channel. Oxygen 2 pstates\nare found to hybridize strongly with Os-5 dstates near EF\nin both spin channels. Thus, with spin-up channel insu-TABLE I. Moments per atom of Fe[Ni] and Os, one set of\nthree in-equivalent oxygen atoms and unit cell ( \u0016tot) for re-\nplacement rate xin [Ca 2Fe1\u0000xNixOsO 6]2from \frst-principles\ncalculations. The unit of moments is the Bohr magneton \u0016B.\nThe contributions from individual atoms are within mu\u000en-\ntins while the total angular moment includes those from in-\nterstitial regime.\nx Fe [Ni] Os O \u0016tot\n0 4.13 -1.6 -0.11 4.0\n0.5 4.13 [1.68] -1.36 -0.15 2.0\n1 [1.68] -1.09 -0.14 0.3\nlating and spin-down channel metallic, the system turns\nto a HM as clearly seen in Fig. 4.\nAs summarized in Table I, two replaced Fe atoms take\naway\u0016'2\u0016B, and the charge transfer in two Os atoms\nassociated on doping reduces \u0016'2\u0016Bfurther, result-\ning in compensation of total moment to zero. The re-\nsults obtained by \frst-principles calculations are consis-\ntent with the ionic picture. These features can also be\nseen from the spin-density isosurface plot in Fig. 5(b).\nWith the zero total moment and HM property, the mate-\nrial [Ca 2NiOsO 6]2should be a HMAFM. However, SOC\ninduces an orbital moment of \u00180:17\u0016Bacross Os result-\ning in the total moment of 0.3 \u0016Bper unit cell. Hence the\nmaterial may be called a nearly compensated half metal.\nV. DISCUSSIONS\nIn order to give a clear picture on how doping changes\nthe electronic structure for x= 0, 0.5 and 1 in\n[Ca2Fe1\u0000xNixOsO 6]2, band structure plots are shown in5\nFIG. 6. Isosurface of spin magnetization density at \u00060.21\ne=\u0017A3with red (blue) for spin up (down): (a) parent ma-\nterial [Ca 2FeOsO 6]2, (b) material with full Fe replacement\n[Ca2NiOsO 6]2.\nFig. 5. It should be noted that six bands lying in the top-\nmost valence region below EFin the parent material (i.e.\nx= 0) are the t2gbands contributed by two Os atoms\nin spin-down channel (see Fig. 5(a)). It is interesting to\nnote that half replacement of Fe by Ni corresponds to\nshifting of one of the t2gband towards the conduction\nregion. As clearly observed for x= 0:5, \fve out of six\nof thet2gbands are occupied while the remaining one\nband moves to the conduction region. As similar case is\nobserved for x= 1 where four t2gbands remain below\nEFwhile the two bands go to conduction region.\nIn the present materials, SOC is crucially important\ndue to the presence of heavy elements such as Os. The or-\nbital moments obtained from \frst-principles calculations\nfor Os (0:1\u00180:2\u0016B) are in accordance with the Hund's\nthird rule45. SOC reduces the band gap by \u00180:1eV\nthrough the broadening of Os-5 dstates in the parent ma-\nterial. SOC induces the total moment of 0 :3\u0016Bper unit\ncell for [Ca 2NiOsO 6]2which is in reasonable agreement\nwith the experiment44.\nCharge-transfer e\u000bect46is prominent between Os and\noxygen via t2gandpstates in the parent material.\nTherefore, oxygens get spin-polarized in parallel with\nthe Os ions, consistent with the isosurface plot shown\nin Fig. 6(a). The isosurface of Fe-3 dstates are spherical\ndue to occupation of all the dorbitals in spin-up channel.Os on the other hand shows the t2gcharacters. The full\nreplacement of Fe with Ni shows an active eglike orbitals\n(see Fig. 6(b)). The size of the Os- t2gisosurface reduces\nin the doped material due to one electron less than in the\nparent material.\nFirst-principles calculations on magnetic anisotropy\nenergy indicates aaxis of the crystal as the easy axis (see\nFig. 1) with anisotropy energy of \u00182meV and\u00184meV\nper unit cell for Ca 2FeOsO 6and Ca 2NiOsO 6respectively.\nRobustness of half metallicity is checked for doping\nratex= 1 by considering the (i) antisite disorder and\n(ii) surface e\u000bects with a vacuum of 20 \u0017Aalong 001. We\nhave con\frmed that the HM state remains stable in both\ncon\fgurations.\nIn the present work, HMAFM and HMFiM have been\nderived from the same parent material. Thus, using them\nin an integrated system, one can construct a useful device\nfor spintronics applications without su\u000bering from the\nproblem of lattice mismatching.\nVI. CONCLUSIONS\nBased on the \frst-principles density functional ap-\nproach, we propose material tailoring on a Mott insu-\nlator [Ca 2FeOsO 6]2with double perovskite structure ex-\nploiting the cooperative e\u000bect from Coulomb interaction,\nspin-orbit coupling and the crystal \feld. It is demon-\nstrated that replacing Fe by Ni, one can achieve several\nhalf metals. Especially, [Ca 2NiOsO 6]2is found to be a\nnearly compensated half metals, which is ideal for spin-\ntronic applications. It is emphasized that the large ex-\nchange splitting between spin-up and spin-down bands at\nthe Fermi level retains the half metallicity even in pres-\nence of strong spin-orbit coupling.\nACKNOWLEDGMENTS\nThe authors thank K. Yamaura for providing the crys-\ntal information of the parent material. 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The realization will be beneficial for spintronics as a\ncandidate for spin-current generator.\nPACS numbers: 71.10.Fd, 73.22.Pr, 72.25.-b\nMassless Dirac fermions show substantially different\nnaturefromordinaryelectrons. The peculiarnatureorig-\ninates in the characteristic energy dispersion —the nodal\nstructure with linear dispersion often referred to as the\nDirac cone. While the Dirac fermions were originally\nintroduced in the relativistic quantum theory, recent dis-\ncovery of graphene [1, 2], a single layer sheet of graphite,\nhascarvedoutanewdirectionoftheirstudyincondensed\nmatter systems [3, 4]. In graphene, two Dirac cones ap-\npear in the energy dispersion of πelectrons, which are\nat theKandK′points in the Brillouin zone for the\ntwo-dimensional honeycomb lattice. The Fermi level of\nthis two dimensional conductor comes right at the nodal\npoints, and the low-energy Hamiltonian is well approx-\nimated by the Weyl equation [5]. Various remarkable\nelectronic and transport properties of graphene mainly\nowe to these Dirac cones in the band structure.\nThe extraordinary nature of Dirac fermions in\ngraphene has also attracted a great interest from appli-\ncation to electronics [3]. From the viewpoint of such po-\ntential applications, it is of great interest to control the\ncharacteristic band structure. Furthermore, it is also de-\nsired to control the electronic spin degree of freedom for\nthe application to spintronics [6]. However, there is not\nso much flexibility in graphene, as the Dirac cone is a di-\nrect consequence of the honeycomb lattice geometry and\nthe relativistic spin-orbit interaction is very weak.\nIn this Letter, we propose an alternative solution for\nmanipulating the spin degree of freedom by seeking pos-\nsible emergence of Dirac fermions from itinerant mag-\nnets. We show that itinerant electrons coupled to a\nwell-known ferrimagnet on a triangular lattice give rise\nto the Dirac nodes in their band structure, similar to\nthose of graphene. The resultant masslessDirac fermions\nare spin-polarized, and they are stable in a wide range\nof the spin-charge coupling including typical values in\nsolids. We demonstrate that, by an unbiased Monte\nCarlo (MC) simulation as well as a variational calcu-\nlation, such Dirac half-metal with ferrimagnetic order\nspontaneously emerges in a minimal Kondo-lattice type\nmodel. The results strongly suggest the possibility ofrealizing the exotic electronic state in transition-metal\nand rare-earth compounds, which generally retain much\nhigher controllable degrees of freedom than graphene.\nSuch a new family will not only add a member to the\nknown list of Dirac electrons in solids [7–10], but also\nbring a completely new aspect by the spin polarization.\nIn a half-metal, the electric current is perfectly spin-\npolarized as the low energy excitations only exist for the\nmajority spin [6]. This nature works as a spin-current\ngenerator by filtering out the minority-spin electrons.\nThus, our proposal opens a new frontier for the applica-\ntion of Dirac massless fermions, especially for spintron-\nics [11].\nLet us first discuss a naive, rather trivial approach\nto achieve a Dirac half-metal. We here consider a\nsingle-band ferromagnetic Kondo lattice model (double-\nexchange model) on a honeycomb or kagome lattice [see\nFigs. 1(a) and 1(b)]. The model consists of the nearest-\nneighbor hopping of electrons and the exchange inter-\naction between the electron spin and localized moment,\nwhose Hamiltonian is given by\nH=−t/summationdisplay\n/angbracketlefti,j/angbracketright,σ(c†\niσcjσ+H.c.)−J/summationdisplay\niσi·Si.(1)\nHere,ciσ(c†\niσ) is the annihilation(creation) operatorof an\nitinerant electron with spin σ=↑,↓atith site,σiandSi\nrepresent the itinerant and localized spin, respectively; t\nis the transfer integral and Jis the onsite Kondo cou-\npling. Hereafter we take t= 1 andJ >0.\nIn this model, when Jis sufficiently large compared\nto the bandwidth at J= 0, a ferromagnetic order is\nstabilized by the double-exchange mechanism in a wide\nrange of electron filling n=/summationtext\niσ/angbracketleftc†\niσciσ/angbracketright/2N, whereN\nis the system size [12, 13]. In the ferromagnetic phase,\nthe band is split into two by the large exchange coupling\naccording to the spin component, and each band has ex-\nactly the same form as that for the noninteracting case\nJ= 0. Hence, in principle, the Dirac half-metal arises\nfor the honeycomb and kagome lattices, as the nonin-\nteracting bands on these lattices have the Dirac nodes.\nHowever, these situations are very difficult to realize in2\n(a)\nFIG. 1. (color online). Schematic pictures of (a) a hon-\neycomb ferromagnet, (b) kagome ferromagnet, and (c) three-\nsublattice triangular ferrimagnet. The arrows at each site\nrepresent localized spins.\nsolids as neither such a strong exchange interaction nor\nthe honeycomb and kagome structures is easily realized\nin magnetic compounds.\nAs a more realistic approach, here we propose a sim-\nple, but rather nontrivial route to the half-metallic Dirac\nfermion systems. Let us consider the model in Eq. (1) on\na triangular lattice, and the situation in which a three-\nsublattice collinear ferrimagnetic order with up-up-down\nspin configuration is realized —see Fig. 1(c). By treating\nthe localized moments as classical spins with |Si|= 1,\nthe band structure is easily calculated by the exact diag-\nonalization of the Hamiltonian. The lower three bands\nof the totally six bands are shown in Fig. 2; the two red\nbands are of up spins, and the blue band is of down spin\n(the other upper three bands have the similar form with\nopposite spins).\nThe band structure has a notable feature at the energy\nε=−J; the two up-spin bands touch with each other\nat theKandK′points in the Brillouin zone to form\na Dirac-type point node with linear dispersion, and the\ndown-spinbandhas the bandtop atthe samepoints with\nan ordinary parabolic dispersion. See also the enlarged\nfigure in Fig. 2(b) and the energy dispersion along the\nsymmetric lines in Fig. 2(c).\nInthissituation, whentheelectronfillingisat n= 1/3,\nthe two lower bands are fully occupied while the remain-\ningbands(includingtheupperthree)areunoccupied; theFermi level is located at the nodes where the three bands\nmeet. Asthedown-spinbandhasanenergygap,thehalf-\nmetallic Dirac electrons are obtained by electron doping\nto the unoccupied up-spin band. Although hole doping\nhides the Dirac nature as the down-spin parabolic band\nis doped at the same time, the situation is avoided by in-\ntroducing an additional antiferromagnetic exchange cou-\npling between the neighboring sites, J′/summationtext\n/angbracketlefti,j/angbracketrightσi·Sj[14].\nA finiteJ′>0 shifts the down-spin band to the lower\nenergy and isolates the half-metallic Dirac nodes ener-\ngetically, as demonstrated in Figs. 2(d) and 2(e). Hence,\nthe simple ferrimagnetic order on the triangular lattice\nrealizes the peculiar Dirac half-metallic state near 1/3\nfilling.\nThe Dirac nodes have essentially the same structures\nas those in graphene. Under the ferrimagnetic order, the\nHamiltonian is written as\nH=/summationdisplay\nk\n−Jσz\nAτkτ∗\nk\nτ∗\nk−Jσz\nBτk\nτkτ∗\nk(J+6J′)σz\nC\n.(2)\nHere, the upper tworowscorrespondto the sites with the\nup localized moment ( A,Bsublattices) and the bottom\nrow is for the down one ( Csublattice) in the three-site\nunit cell. In Eq. (2), σzis thezcomponent of the Pauli\nmatrix for itinerant electrons, kis the wave vector, and\nτkis the Fourier transform of the hopping term given by\nτk=−t[eikx+ei/parenleftBig\n−kx\n2+√\n3\n2ky/parenrightBig\n+ei/parenleftBig\n−kx\n2−√\n3\n2ky/parenrightBig\n]. By using\nthek·pperturbation around the KandK′points in the\nBrillouin zone [5] and by expanding the result up to the\nfirst order in terms of tκx/Jandtκy/J(κis the relative\nwave vector measured from KandK′points), we end up\nwith the low-energy Hamiltonian which is factorized into\ntwo parts. One is a 2 ×2 Hamiltonian for the up-spin\nhoneycomb subnetwork of the AandBsublattices, and\nthe other is a localized state at the down-spin sites in the\nCsublattice. The former is given by\nHDirac\nk±=/parenleftbigg\n−J3\n2it(κx±iκy)\n−3\n2it(κx∓iκy)−J/parenrightbigg\n,(3)\nwhere the sign ±corresponds to the KandK′points.\nThis has an equivalent form to that of graphene.\nIt is worthy to note that the Dirac nodes are formed\nimmediately by switching on J. However, when Jis\nsmall, the low-energy physics at n= 1/3 is not char-\nacterized solely by the massless Dirac fermions because\nthere is a band overlap at the energy of the Dirac\nnodes. The band overlap comes from the second lower\nband for up spin, which has an energy minimum at\nk= (2π/3,0) points and its threefold symmetric points\nfor smallJ; the minimum energy is given by ε(2π\n3,0)=\nt/2−/radicalig\n(J+3J′−t/2)2+2t2. In order for the Dirac\nnodes to be isolated at the Fermi level, this energy\nshould be higher than that at the KandK′points,3\nKM Γ Γ01\n-1\n-2\n-3\n-4\n-5\n-6\n-7(c)\nε(a)\n2\n0\n-2\n-4\n-6\n0\n2π\n32π\n3-\n04π\n3√3-ε\nkx\nkyK\nMΓ\n-8\nK'\n4π\n3√3(b)\nε\nkykx-1.0\n-1.5\n-2.0\n-2.5\n-3.0\n00(d)\nε\nkykx-1.0\n-1.5\n-2.0\n-2.5\n-3.0\n00\n(e)\nKM Γ Γ01\n-1\n-2\n-3\n-4\n-5\n-6\n-7ε2π\n34π\n3√32π\n34π\n3√3\nFIG. 2. (color online). Band structures of the model in Eq. (1 ) under the three-sublattice ferrimagnetic order at J= 2. (a)\nThe overall band structure of the three lower-energy bands a tJ′= 0, (b) the enlarged view near the Fermi level ε=−Jat\nn= 1/3 in the first quadrant, and (c) the cut along the symmetric lin es. (d) and (e) show the results at J′= 0.05. The arrows\nindicate the spins for each band. In (a), the gray hexagon on t he basal plane shows the first Brillouin zone for the magnetic\nsupercell. The dashed line in (e) indicates the Fermi level i n the MC simulation shown in Fig. 4.\nεK=−(J+3J′). Hence, the Dirac nodes are energeti-\ncally isolated and play a decisive role when the condition\n(J+3J′)/t>1 is satisfied. This condition is important\nbecausethenecessary JandJ′aremuchsmallerthanthe\nnoninteracting bandwidth 9 t, and it is indeed satisfied in\nwide range of materials.\nSo far, we assumed the presence of three-sublattice fer-\nrimagnetic order. In the following, we show that such\norder is indeed stable in the Kondo-lattice type model\nas Eq. (1). We here simplify the model by assuming the\nlocalized moments are the Ising spins taking the values\nSi=±1.\nFirst, we investigate the ground state phase diagram\nnearn= 1/3 by a variational calculation. We com-\npare the ground state energy of the two-sublattice stripe\nphase and three-sublattice ferrimagnetic phase appeared\nin the previous study [15], in addition to the ferromag-\nnetic phase. The results at J= 2 are shown in Fig. 3 for\nJ′= 0 and 0.05. AtJ′= 0, the ground state in the plot-\nted rangeis dominated by the ferrimagneticphaseas well\nas the stripe phase. The different phases are separated\nbyphaseseparation. As showninFig. 3(b), the introduc-\ntion of small J′largely stabilizes the ferrimagnetic phase\nnearn= 1/3 as well as the stripe phase. This is because\nthe itinerant electron spins are polarized parallel to the\nlocalized spins in the ground state, leading to an energy\ngain (loss) by the antiferromagnetic J′for the two states\n(the ferromagnetic state).\nWe next examine the stability of the ferrimagnetic or-\nderatfinite temperaturesbyanunbiasedMC simulation.\nFor the simulation, a standard algorithm for fermionnJ\n012345678\n0.260.280.30 0.38 0.320.340.36\nnJ\n2-sub stripe\n0.260.280.30 0.38 0.320.340.36012345678(a)\nFIG. 3. (color online). Ground state phase diagram obtained\nby variational calculation at (a) J′= 0 and (b) J′= 0.05.\nThe schematic picture of magnetic structure in each phase\nis shown. The white region indicates the electronic phase\nseparation (PS)andthedottedvertical lines indicate n= 1/3.4\nMxy[1/√2 ]\n|Mz|[x√3]\nN =12x12\nN =12x18\nN =18x18\n0.00.20.40.60.81.01.2\n(a)Tc\nTN =12x12\nN =12x18\nN =18x18\n0.00.20.40.60.8\n0.00 0.05 0.10 0.15 0.20 0.25(c)\nψχz[x5] N =12x12\nN =12x18\nN =18x18\n020406080100120140160180(b)\nχxyTKT\nFIG. 4. (color online). MC results for (a) the pseudo mo-\nmentsMxyand|Mz|, (b) corresponding susceptibilities χxy\nandχz, and (c) azimuth parameter ψ. The data are calcu-\nlated atn= 0.34.\nsystems coupled to classical fields is used [16]. In this\nmethod, the trace overthe fermions in the partitionfunc-\ntion is calculated by the exact diagonalization, while the\ntrace over classical spin configurations is computed by\na classical MC method using the Metropolis dynamics.\nThe phase transition to ferrimagnetic phase is detected\nby using two parameter [17]. One is the pseudo-moment\ndefined by\n˜Sm=\n2√\n6−1√\n6−1√\n6\n01√\n2−1√\n21√\n31√\n31√\n3\n\nSi\nSj\nSk\n,(4)\nwheremis the index for the three-site unit cells, and\n(i,j,k) denote the three sites in the mth unit cell be-\nlonging to the sublattices ( A,B,C), respectively. We\nmeasure the summation M= (3/N)/summationtext\nm˜Smand the\nsusceptibility. The other is the azimuth parameter ψ\ndefined by ψ= (˜Mxy)3cos6φM, whereφMis the az-\nimuth angle of Min thexyplane and ˜Mxy= 3M2\nxy/8\n(M2\nxy=M2\nx+M2\ny). The ferrimagnetic ordering is sig-\nnaled byMxy→2/radicalbig\n2/3,|Mz| →1/√\n3, andψ→1 at\nlow temperature T→0, respectively [15, 18, 19].\nFigure 4 shows the MC results at J= 2 andJ′= 0.05\nin the slightly electron doped region to n= 1/3 [see alsoFig.2(e)]. Theresultsindicatetwosuccessivephasetran-\nsitions atTKT= 0.192(15) and at Tc= 0.108(9). The\ntransition temperatures are estimated by extrapolating\nthe peak of susceptibilities χxyandχzasN→ ∞. The\ntransition at TKTis considered as a Kosterlitz-Thouless\ntype with the growth of quasi-long-range order [15]. On\nthe other hand, the phase transition at Tcis a three-\nsublatticeferrimagneticordering. TheMC resultand the\nabove analysis for the ground state consistently indicate\nthat the three-sublattice ferrimagnetic order is stabilized\nin the vicinity of n= 1/3 in the wide range of parame-\nters forJandJ′, spontaneously giving rise to the Dirac\nhalf-metal.\nAs such ferrimagnetic order was indeed observed in\nseveralinsulatingmagnets[20,21], ourresultsinthemin-\nimal model will stimulatethe hunt forDirachalf-metal in\ntransition-metal and rare-earth compounds. The present\nresults will be qualitatively robust even when extend-\ning the model to more realistic situation. For instance,\ntheferrimagneticstateremainsstablewhenincluding the\ntransverse components of localized spins, at least, in the\npresence of the Ising anisotropy. Multi-band effect may\nbe avoided under a particular crystal field; for instance,\nthed-electrona1gorbital isolated by a strong trigonal\nfield is a good candidate for the realization. Interlayer\ncoupling, however, may open a gap at the Dirac nodes.\nNevertheless, a straightforward stacking of layers or suf-\nficiently isolated layers in a controlled thin film will be\npromising to preserve the massless nature.\nThe authors thank Y. Matsushita, A. Shitade, and\nY. Yamaji for helpful comments. H.I. is supported\nby Grant-in-Aid for JSPS Fellows. This research\nwas supported by KAKENHI (No.19052008, 21340090,\n22540372, and 24340076), Global COE Program “the\nPhysical Sciences Frontier”, the Strategic Programs for\nInnovative Research (SPIRE), MEXT, and the Compu-\ntational Materials Science Initiative (CMSI), Japan.\n[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nY. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A.\nFirsov, Science 306, 666 (2004).\n[2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nI. V. Katsunelson, I. V. Grigorieva, S. V. Dubonos, and\nA. A. Firsov, Nature 438, 197 (2005).\n[3] A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183\n(2006).\n[4] For a recent review, see A. H. Castro Neto, N. M. R.\nPeres, K. S. Novoselov, and A. K. Geim, Rev.Mod. Phys.\n81, 109 (2009).\n[5] G. W. Semenoff, Phys. Rev. 53, 2449 (1984).\n[6] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.\nM. Daughton, S. von Molnar, M. L. Roukes, A. Y.\nChtchelakanova, amd D. M. Treger, Science 294, 1488\n(2001).\n[7] M. H. Cohen and E. I. Blount, Phil. Mag. 5, 115 (1960).5\n[8] T. Konoike, K. Uchida, and T. Osada, J. Phys. Soc. Jpn.\n81, 043601 (2012).\n[9] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[10] S. Ishibashi, K. Terakura, and H. Hosono, J. Phys. Soc.\nJpn.77, 053709 (2008).\n[11] D. Pesin and A. H. MacDonald, Nature Mater. 11, 409\n(2012).\n[12] C. Zener, Phys. Rev. 82, 403 (1951).\n[13] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675\n(1955).\n[14] An off-site Kondo coupling may exist generally in the\nKondo lattice systems, although the magnitude is much\nsmaller than the onsite one and the sign depends on\nthe orbital nature of itinerant and localized electrons.\nThe antiferromagnetic superexchange coupling between\nneighboring localized spins, given by JAF/summationtext\n/angbracketlefti,j/angbracketrightSi·Sj,\nmay also exist, but it neither modifies the band structure\nnor harms the stability of the ferrimagnetic state.[15] H. Ishizuka and Y. Motome, Phys. Rev. Lett. 108,\n257205 (2012).\n[16] S. Yunoki,J. Hu, A.L.Malvezzi, A.Moreo, N.Furukawa,\nand E. Dagotto, Phys. Rev. Lett. 80, 845 (1998).\n[17] In principle, the ferrimagnetic ordering can be detect ed\nby the spin structure factor. In the finite-size MC cal-\nculations for the current model, however, it is useful to\nemploy the pseudo-moment and its azimuth parameter\nfor distinguishing it from a three-sublattice partial diso r-\nder and Kosterlitz-Thouless type quasi long-range order.\nSee also Ref. [15, 18, 19].\n[18] H. Takayama, K. Matsumoto, H. Kawahara, and K.\nWada, J. Phys. Soc. Jpn. 52, 2888 (1983).\n[19] S. Fujiki, K. Shutoh, S. Inawashiro, Y. Abe, and S. Kat-\nsura, J. Phys. Soc. Jpn. 55, 3326 (1986).\n[20] M. Tanaka, H. Iwasaki, K. Siratori, and I. Shindo, J.\nPhys. Soc. Jpn. 58, 1433 (1989).\n[21] J. Iida, M. Tanaka, Y. Nakagawa, S. Funahash, N.\nKimizuka, and S. Takekawa, J. Phys. Soc. Jpn. 62, 1723\n(1993)."    },    {        "title": "1006.5531v2.Magnetostatics_of_synthetic_ferrimagnet_elements.pdf",        "content": "arXiv:1006.5531v2  [cond-mat.mtrl-sci]  11 Jun 2011Magnetostatics of synthetic ferrimagnet elements\nOlivier Frucharta,∗, Bernard Diényb\naInstitut NÉEL, CNRS & Université Joseph Fourier – BP166 – F-38 042 Grenoble Cedex 9 – France\nbSPINTEC (UMR8191 CEA/CNRS/UJF/G-INP), CEA Grenoble, INAC, 3805 4 Grenoble Cedex 9,\nFrance\nAbstract\nWe calculate the magnetostatic energy of synthetic ferrima gnet (SyF) elements, con-\nsisting of two thin ferromagnetic layers coupled antiferro magnetically, e.g. through\nRKKY coupling. Uniform magnetization is assumed in each lay er. Exact formulas as\nwell as approximate yet accurate ones are provided. These ma y be used to evaluate\nvarious quantities of SyF such as shape-induced coercivity and thermal stability, like\ndemagnetizing coefficients are used in single elements.\nSynthetic antiferromagnets (SAF, resp. ferrimagnets, SyF )[1,2] consist of two thin\nferromagnetic films of moments of same (resp. different) magn itude, strongly coupled\nantiferromagnetically thanks to the RKKY interaction thro ugh an ultrathin spacer layer,\ntypically Ru 0.6−0.9nmthick[3]. Hereon we consider only the case of in-plane magne-\ntized layers. SyFs are widely used to provide spin-polarize d layers displaying an overall\nweak moment. One benefit is to minimize cross-talk of neighbo ring (e.g.memory bits) or\nstacked ( e.g.in a spin-valve) elements through stray-field coupling[ 1,2], such as in Mag-\nnetic Random Access Memory (MRAM)[ 4]. SyFs are also used to decrease the Zeeman\ncoupling with external fields, e.g.to increase coercivity in reference layers[ 5], decrease\neffects of the Oersted field in magneto-resistive or spin-tor que oscillator pillars, or more\nrecently boost the current-induced domain-wall propagati on speed in nanostripes[ 6,7].\n∗Corresponding author\nEmail address: Olivier.Fruchart@grenoble.cnrs.fr (Olivier Fruchart)\nPreprint submitted to Elsevier December 2, 2018In practice SyFs are used as elements of finite lateral size. I t has been shown[ 8] and\nit is widely used [ 9,10] that for flat and magnetically soft nanomagnets of lateral s ize\nsmaller than a few hundreds of nanometers, the macrospin app roximation (uniform mag-\nnetization) is largely correct. In this framework the coerc ive field equals the anisotropy\nfield2K/µ0Msand the energy barrier KV(Vis the volume of the dot) preventing spon-\ntaneous magnetization reversal equals the magnitude of ani sotropy of the total magnetic\nenergyE, to which all the physics therefore boils down. Elongated do ts are often used to\ninduce or contribute to an easy axis of magnetization and an e nergy barrier ∆E, based on\ndipolar energy. Dipolar energy is a quadratic form and thus i t is fully determined by its\nvalue along the two main in-plane axes. For single elements ∆Ed=KdV∆N with∆N\nthe difference between the two in-plane demagnetizing coeffic ients, and Kd= (1/2)µ0M2\ns.\nAnalytical formulas have been known for a long time to evalua te the mutual energy\nof an arbitrary set of prisms[ 11]. However while simple expressions for Nand thus ∆Ed\nhave been described, displayed and discussed for single-la yer flat elements[ 11,12], the\nanalytical expressions and the evaluation of Edin SyFs have not been discussed in detail\nso far. Instead the studies requiring estimation of the dipo lar energy in SyF, mainly\npertaining to MRAM cells[ 9,10,13], have in the best case made use of an effective so-\ncalled attenuation coefficient with respect to self-energy[ 10], which requires a numerical\nevaluation[ 13]. The meaning and scaling laws of this attenuation coefficien t have never\nbeen discussed in detail, hiding the physics at play. As ther mal stability, coercivity and\ntoggle switching fields[ 9,10,14] depend crucially on the interlayer magnetostatic cou-\npling, it is desirable to have a simple yet accurate analytic al expression for interlayer\ndipolar fields. In this manuscript we report exact analytica l expressions for the magne-\ntostatics of SyFs uniformly-magnetized in each sub-layer. From the numerical evaluation\nwe discuss the physics at play, while from the analytical for mulas we propose an approx-\nimate yet accurate scaling law for their straightforward ta ble-top evaluation.\nWe first consider SyF prisms and name F1 and F2 the two ferromag netic layers (Fig-\nure1), with magnetization aligned along z. This covers the case of both finite-size\nprisms as well as infinitely-long stripes with a rectangular cross-section. We apply for-\nmulas expressing the interaction between two parallel char ged surfaces[ 11], and adopt\nthe convenient notation of Fijkfunctions, the i,jandk-fold indefinite integrals along x,\n2Figure 1: Geometry and notations of a prismatic SyF element c omprising two ferromagnetic layers F1\nandF2.\nyandzof the Green’s function F000= 1/r[15]. The only such function needed here is\nF220=1\n2[x(v−w)Lx+y(u−w)Ly]−xyPz+1\n6r(3w−r2) (1)\nwithu=x2,v=y2,w=z2,r=√u+v+w,Lx= (1/2)ln[(r+x)/(r−x)]etc,\nPx=xarctan(yz/xr)etc, and Lx= 0andPx= 0forx= 0etc.\nThe integrated magnetostatic energy of a single prismatic e lement of thickness tis:\nEd=2Kd\nπ/summationdisplay\nδa,δt,δc∈{0,1}(−1)δa+δt+δcF220(aδa,tδt,cδc) (2)\nwhich normalized to Kdyields the demagnetizing coefficient Nz. It can be verified that\nEq. (2) coincides with the explicit formula already known[ 12]. The magnetostatic energy\nof a prismatic SyF element may be calculated using the same fo rmalism , may be written\nas:\nEd=Kd,1Nz(a,t1,c)V1+Kd,2Nz(a,t2,c)V2\n+2/radicalbig\nKd,1Kd,2Nm(a,t1,s,t2,c)/radicalbig\nV1V2 (3)\nwithNm(a,t1,s,t2,c) =1\nπa√t1t2c/summationtext\nδ1,δ2,δa,δc∈{0,1}(−1)δ1+δ2+δa+δc×F220(aδa,s+t1δ1+\nt2δ2,cδc)is a mutual magnetostatic coefficient with a negative value, a ndVi=atic(resp.\nKd,i) is the volume (resp. dipolar constant) of each single prism i. This equation of dipo-\nlar energy is a quadratic form of M1andM2, generalizing the definition of demagnetizing\ncoefficients.\n3Figure 2: Magnetostatic energy of a SyF with c= 2a= 100nm ,M1,2= 106A/m,s= 0.7nm. (a) Sum\nof the energies of two prisms without mutual interaction, an d when embedded in the SyF geometry. t1\nis kept constant at 2.5nm, whilet2is varied. (b) Energy of the general SyF. The curved lines are those\nof minimum energy for either constant t1ort2. The thick horizontal dotted line highlights the path for\nthe SyF curve shown in (a).\nFigure 2(a) shows Edupon building a SyF via the progressive thickness increase o f F2\nabove F1, considering or not the interaction between the two layers. In the latter case the\nenergy increase nearly scales with t2\n2, which is understandable because it is a self-energy\n(inF2alone). In the coupled case (a SyF) Edretains like for the uncoupled case an overall\nclose-to-parabolic convex shape as can be verified with fitti ng, however with an initial\nnegative slope. This can be understood as for low t2the extra edge charges induced by\nan infinitesimal increase δt2mainly feel the stabilizing stray field arising from F1, whil e\nfor large t2they feel more the nearby charges induced by F2 itself. Notic e that, contrary\nto what could be a first guess, the minimum of Ed(t2)occurs before the compensation of\nmoment ( t1=t2). This stems from the same argument as above, which is that ma gnetic\ncharges at an edge of F2 are closer to another than to the charg es on the nearby edge of\nF1, thus for an identical amount δt2contribute more to Ed.\nFigure 2(b) shows the full plot of Ed(t1,t2)fors= 0.7nm. The above arguments\nappear general. From this figure let us outline three take-aw ay messages. 1. For a given\nt1the minimum of Edof a SyF is found for t2/greaterorsimilart1/2. 2. At this minimum Edis reduced\nby only≈20−30%with respect to a single-layer element of thickness t1considered alone.\n3.Edroughly regains the value of the single layer at the moment co mpensation point\n(t1=t2).\n4This sheds light on results previously noticed empirically , however whose origin and\ngenerality had not been highlighted. Wiese et al. reported that the effective dipolar\nfield anisotropy of a SyF basically scales with the inverse net moment [16],i.e.like the\ninverse strength of Zeeman energy. This suggests that ∆Edis essentially independent of\nthe imbalance of moment, which goes against the widespread b elief that magnetostatic\nenergy nearly vanishes upon moment compensation. Our resul ts clarify and quantify\nthis: the dipolar energy does not differ more than 20-30 % from that of a single layer\nfort2/lessorsimilart1(Figure 2a, dotted line). Saito et al. also reported that the thermal stability\nofCo90Fe10[3]/Ru[0.95]/Co90Fe10[5]is similar to that of Co90Fe10[3]. As explained in\nthe introduction, we recall that thermal stability is deter mined by the energy barrier\nalong the hard axis direction, with respect to the easy axis d irection. In the case of\nanisotropy arising from dipolar energy and an elongated sha pe of the element, this bar-\nrier can be evaluated straightforwardly by calculating onc eEdalong the short edge of\nthe dot, and second along the longedge of the dot. Doing this we explain the findings of\nSaitoet al.., whereas a reduction of 50%would be expected on the basis of compensated\nmoments (the numbers in brackets are thicknesses in nanomet ers). Our calculations may\nalso be applicable to the cross-over of vortex versus single domain in flat disks[ 17] or\nvortex versus transverse domain walls in stripes[ 18], whose scaling law t×a= Cte may\nbe derived qualitatively by equaling the energy of a vortex ∼tand that of a single-\ndomain∼a2t(t/a)(hereV=a2tis the volume, and t/athe demagnetizing coefficient).\nInterestingly Tezuka et al. noticed that there is an optimum ferromagnetic film thick-\nness at which SyAF can obtain a single-domain structure . This minimum (related to\na minimum of demagnetization energy) is found for an imbalanced thickness in good\nquantitative agreement with Figure 2b.\nWith a view to promote the use of accurate magnetostatics for SyF while eliminating\nthe need for numerical evaluation, we derived approximate y et highly accurate expressions\nforEd. Figure 3a shows that to a very good approximation, Edis proportional to the\nwidth of the element (along x) and is independent of its length (along z). This is already\naccurate for a single-layer ( t2= 0), and is very accurate close to the compensation\nt1=t2because edges then behave as lines of dipoles, whose stray fie ld quickly decays\nwith distance ( ∼1/r2). ThusEdboils down to a single line integral along its edge:\n5Energy(10  J)-19\nDot length (nm)200 400 600 800 1000456\n200 400 600 800 1000Energy(10  J)-18\n0123456\nDot length (nm)(a) (b)\nFigure 3: (a) Energy of a single layer (full symbols) and SyF ( open symbols) as a function of dot length,\ni.e.alongz, whilea= 100nm . (b) Energy of a single layer as a function of width (along x, open\nsymbols), and length (along z, full symbols, same curve as in a), while the other in-plane d imension is\nkept constant at 100nm . The lines are linear fits. For both plots the parameters are: Mi= 106A/m,\nt1=t2= 2.5nm,s= 0.7nm.\nEd=Eλ/contintegraldisplay\n(m.n)2ds (4)\n=Eλ/contintegraldisplay|dx|/radicalbig\n1+(∂xf)2(5)\n=Eλ/integraldisplay2π\n0(rsinθ−∂θrcosθ)2\n/radicalbig\n(∂θr)2+r2dθ (6)\nEq. (4) is the general expression, expressed in the following two l ines in cartesian and\npolar coordinates (Figure 4a).Eλis the density of magnetostatic energy per unit length\nof edge, a concept once discussed in the case of single layers [19]. Equations ( 4-6) apply to\nan arbitrary shape of perimeter (not simply rectangles for p risms) by considering the in-\nplane angle ϕbetween magnetization and the normal to the edge. It can be ve rified that\nfor a SAF we have, with an accuracy better than 10%for geometrical parameters relevant\nfor practical cases, i.e.t1,2in the range of 2−10nm andsintherangeof 0.5−1nm:\nEλ≈(1/2)Kdt2(7)\nThe meaning of Eq. ( 7) is straightforward: due to the short range of interaction\n6\u0001\u0002\u0003\u0004\u0005\u0006\u0007 \b\t\n\u0002\u000b\f\r\n\u000e\u000f\f\u0010\u0011\u0012\u0006\n\u0013\u0014\u0014\u0005\u0015\u0006\t\u000e\t\u0016\u0017\u0002\r\u000b\u0014\t\u0001\u0011\u0003\nFigure 4: (a) notations for the calculation of edge energy (b ) integrated magnetostatic energy for various\nshapes. Eis the elliptical integral of the second kind. See text for th e definition of Eλ.\nbetween dipolar lines, the density of dipolar energy is non- zero only in the vicinity of\nthe edges, with a lateral range t. Thus a volume t2is concerned with a line density\nof energy of the order of Kd. Expressions for non-compensated cases (including single\nelements) may also be evaluated. This provides us with analy tical expressions for the\nmagnetostatics of SyFs for the most usual shapes (Figure 4b).\nA scaling law sometimes used as a first guess is based on the poi nt dipole approxima-\ntion. In this framework the energy gained by coupling F1andF2would roughly scale\nwithKda4/t, resulting from two point moments Msa2tinteracting like 1/t3(fors≪t,\nand assuming lateral dimensions of the order of a). The scaling arising from our exact\ncalculation is aEλ∼Kdat2(Eq. ( 7) and Figure 4a). The point dipole approximation is\nthus clearly incorrect with an extra scaling (a/t)3(see Figure 4a) which largely overesti-\nmates the dipolar coupling. This is a general argument for an y flat element, where dipolar\nfields are short-ranged[ 20] and thus the point-dipole approximation is clearly incorr ect.\nTo conclude we derived exact formulas for the magnetostatic s of prism SyF, and sim-\nple yet accurate forms for SyFs of arbitrary shapes. These si mple forms may be used\nstraightforwardly to derive scaling laws for all aspects of SyF physics pertaining with\n7dipolar energy such as thermal stability, coercivity and an isotropy field. Notice that sim-\nilar to the case of single flat elements edge roughness is liab le to reduce significantly dipo-\nlar energy[ 21,22,23], so that the theoretical predictions need to be considered as upper\nbounds to the experimental values. The non-uniformity of ma gnetization is not expected\nto have a significant impact for lateral sizes below a few hund reds of nanometers[ 8].\nWe acknowledge useful discussions with Y. Henry, IPCMS-Str asbourg.\nReferences\n[1] D. Heim, S. S. P. Parkin, US patent 5,465,185, 1995. Magne toresistive spin valve sensor with\nimproved pinned ferromagnetic layer and magnetic recordin g system using the sensor.\n[2] H. Van den Berg, US patent 5,686,838, 1997. Magnetoresis tive sensor having at least a layer system\nand a plurality of measuring contacts disposed thereon, and a method of producing the sensor.\n[3] S. S. P. Parkin, Systematic variation of the strength and oscillation period of indirect magnetic\nexchange coupling through the 3d, 4d, and 5d transition meta ls, Phys. Rev. Lett. 67 (1991)\n3598–3601.\n[4] R. C. Sousa, I. L. Prejbeanu, Non-volatile magnetic rand om access memories (mram), C. R.\nPhysique 6 (2005) 1013.\n[5] H. A. M. van den Berg, W. Clemens, G. Gieres, G. Rupp, W. Sch elter, M. Vieth, Gmr sensor\nscheme with artificial antiferromagnetic subsystem, IEEE T rans. Magn. 32 (1996) 4624.\n[6] S. Pizzini, V. Uhlir, N. Rougemaille, E. Bonet, M. Bonfim, J. Vogel, S. Laribi, V. Cros, R. Mattana,\nC. Deranlot, F. Petroff, A. Fert, E. Jimenez, J. Camarero, C. U lysse, G. Faini, C. Tieg, High domain\nwall velocity at zero magnetic field induced by low current de nsities in spin-valve nanostripes, Appl.\nPhys. Express 2 (2009) 023003.\n[7] A. Chanthbouala, R. Matsumoto, J. Grollier, V. Cros, A. A nane, A. Fert, A. V. Khvalkovskiy,\nK. A. Zvezdin, K. Nishimura, Y. Nagamine, H. Maehara, K. Tsun ekawa, A. Fukushima, S. Yuasa5,\nVertical-current-induced domain-wall motion in mgo-base d magnetic tunnel junctions with low cur-\nrent densities, Nat. Phys. (2011).\n[8] J. Z. Sun, J. C. Slonczewski, P. L. Trouilloud, D. Abraham , I. Bacchus, W. J. Gallagher, J. Hummel,\nY. Lu, G. Wright, S. S. P. Parkin, R. H. Koch, Thermal activati on-induced sweep-rate dependence\nof magnetic switching astroid, Appl. Phys. Lett. 78 (2001) 4 004–4006.\n[9] D. C. Worledge, Magnetic phase diagram of two identical c oupled nanomagnets, Appl. Phys. Lett.\n84 (2004) 2847.\n[10] H. Fujiwara, S.-Y. Wang, Critical-field curves for swit ching toggle mode magnetoresistance random\naccess memory devices, J. Appl. Phys. 97 (2005) 10P507.\n[11] P. Rhodes, G. Rowlands, Demagnetizing energies of unif ormly magnetized rectangular blocks, Proc.\nLeeds Phil. Liter. Soc. 6 (1954) 191.\n8[12] A. Aharoni, Demagnetizing factors for rectangular fer romagnetic prisms, J. Appl. Phys. 83 (1998)\n3432–3434.\n[13] J. K. Han, J. H. NamKoong, S. H. Lim, A new analytical/num erical combined method for the\ncalculation of the magnetic energy barrier in a nanostructu red synthetic antiferromagnet, J. Phys.\nD: Appl. Phys. 41 (2008) 232005.\n[14] K. Kim, K. Shin, S. Lim, Effects of induced anisotropy on t he bit stability and switching field in\nmagnetic random access memor, J. Magn. Magn. Mater. 309 (200 7) 326.\n[15] A. Hubert, R. Schäfer, Magnetic domains. The analysis o f magnetic microstructures, Springer,\nBerlin, 1999.\n[16] N. Wiese, T. Dimopoulos, M. Rührig, J. Wecker, G. Reiss, Switching of submicron-sized, antifer-\nromagnetically coupled cofeb/ru/cofeb trilayers, J. Appl . Phys. 98 (2005) 103904.\n[17] R. P. Cowburn, Property variation with shape in magneti c nanoelements, J. Phys. D: Appl. Phys.\n33 (2000) R1–R16.\n[18] R. McMichael, M. Donahue, Head to head domain wall struc tures in thin magnetic strips, IEEE\nTrans. Magn. 33 (1997) 4167.\n[19] S. W. Yuan, H. N. Bertram, J. F. Smyth, S. Shultz, Size effe cts of switching fields of thin permalloy\nparticles, IEEE Trans. Magn. 28 (1992) 3171.\n[20] O. Fruchart, J.-P. Nozières, W. Wernsdorfer, D. Givord , F. Rousseaux, D. Decanini, Enhanced\ncoercivity in sub-micrometer-sized ultrathin epitaxial d ots with in-plane magnetization, Phys. Rev.\nLett. 82 (1999) 1305–1308.\n[21] R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland , Lateral interface anisotropy in\nnanomagnets, J. Appl. Phys. 87 (2000) 7067.\n[22] W. C. Uhlig, J. Shi, Systematic study of the magnetizati on reversal in patterned co and nife\nnanolines, Appl. Phys. Lett. 84 (2004) 759.\n[23] K. W. Moon, J. C. Lee, S. B. Choe, K. H. Shin, Experimental determination technique for magnetic\nanisotropy of individual nanowires, Curr. Appl. Phys. (200 9).\n9"    },    {        "title": "0911.0078v1.Compensation_temperature_of_3d_mixed_ferro_ferrimagnetic_ternary_alloy.pdf",        "content": "arXiv:0911.0078v1  [cond-mat.mtrl-sci]  31 Oct 2009APS/123-QED\nCompensation temperature of 3d mixed ferro-ferrimagnetic ternary alloy\nEbru Kı¸ s-C ¸am1and Ekrem Aydiner2∗\n1Department of Physics, Dokuz Eylul University, 35160 `Izmir, Turkey\n2Department of Physics, Istanbul University, 34134 Istanbu l, Turkey\n(Dated: September 4, 2018)\nIn this study, we have considered the three dimensional mixe d ferro-ferrimagnetic ternary alloy\nmodel of the type AB pC1−pwhere the A and X (X=B or C) ions are alternately connected and have\ndifferent Ising spins SA=3/2, SB=1 and SC=5/2, respectively. We have investigated the dependence\nof the critical and compensation temperatures of the model o n concentration and interaction param-\neters by using MC simulation method. We have shown that the be havior of the critical temperature\nand the existence of compensation points strongly depend on interaction and concentration param-\neters. In particular, we have found that the critical temper ature of the model is independent on\nconcentration of different types of spins at a special intera ction value and the model has one or two\ncompensation temperature points in a certain range of value s of the concentration of the different\nspins.\nPACS numbers: 75.50.Gg; 75.10.Hk; 75.30.Kz; 05.10.Ln\nKeywords: Compensation temperature; ferro-ferrimagneti c ternary alloys; Monte Carlo simulation.\nI. INTRODUCTION\nMolecular-based magnetic materials have recently at-\ntracted considerable interest and study of the magnetic\nproperties [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,\n14, 15, 16, 17, 18, 19]. A special class of the these\nmaterials, the so-called Prussian blue analogs, such as\n(XII\npMnII\n1−p)1.5[CrIII(CN)6].nH2O (XII=NiII,FeII) [4,\n5] and (NiII\npMnII\nqFeII\nr)1.5[CrIII(CN)6].nH2O [6] which\nexhibitmanyunusualproperties, forinstance, occurrence\nof one [4] or even two [6] compensation points, mag-\nnetic pole inversion [5, 7], the photoinduced magnetiza-\ntion effect [8, 9], inverted magnetic hysteresis [10]. These\nternary alloys have ferromagnetic-ferrimagnetic proper-\nties since they include mixed both ferromagnetic ( J >0)\nand antiferromagnetic ( J <0) superexchange interac-\ntions between the nearest-neighbor metal ions. The the-\noretical investigations of these systems are difficult be-\ncause of their structural complexity. However, to ob-\ntain magnetic properties of the molecular-based mag-\nnetic materials, up to now, these systems havebeen stud-\nied by using effective-field theory [11], mean field theory\n[12, 13, 14] and Monte carlo simulation (MC) methods\n[15, 16, 17].\nIn this study we consider three dimensional ferro-\nferrimagnetic AB pC1−pternary alloy, consisting of three\ndifferent Ising spins A=3/2, B=1, and C=5/2, which\ncorresponds to the Prussian blue analog of the type\n(NiII\npMnII\n1−p)1.5[CrIII(CN)6].nH2O [4]. In this system,\nthe coupling Cr-Ni is ferromagnetic and Mn-Cr is anti-\nferromagnetic. Our aim, in this study, is to clarify the\neffects of the concentration and the interaction parame-\nters on the magnetic behavior of the three dimensional\nternary alloy model by using MC simulation method.\n∗Electronic address: ekrem.aydiner@deu.edu.trII. THE MODEL AND ITS SIMULATION\nThree dimensional ferro-ferrimagnetic AB pC1−pIsing\nmodel consists of two interpenetrating cubic sublattices\nas seen in Fig.1. It can be assumed that the A ions are\nlocated on the first cubic sublattice and the B and C ions\nare randomly distributed on the second cubic sublattice\nwith the concentration pand 1−p, respectively. Also, to\nconstruct a Hamiltonian for this system, the ion A can\nbe represented by spin SA, and on the other hand, ions\nB and C can be represented by Ising spins SBand SC,\nrespectively. If the interactions between nearest neigh-\nbors can be chosen such as A ions ferromagnetically in-\nteract with B, on the other hand, antiferromagnetically\ninteractwith C ions, thus, spins ofthe Prussianblue ana-\nlogofthe type (NiII\npMnII\n1−p)1.5[CrIII(CN)6].nH2Ocan be\nrepresented by this model where SA, SBand SCcorre-\nspond to Cr, Ni and Mn, respectively. In this study we\nalso consider next-nearest neighbor interactions between\nspins SA.\nThe Hamiltonian ofthe consideredsystem can be writ-\nten in the form\nH=−/summationdisplay\n<nn>SA\ni[JABSB\njεj+JACSC\nj(1−εj)]\n−JAA/summationdisplay\n<nnn>SA\niSA\nk(1)\nwhere SA=±3/2,±1/2 for A, SB=±1,0 for B and\nSC=±5/2,±3/2,±1/2 for C, on the other hand, εjis a\nrandom variable which takes the value of unity if there\nis a spin X (SBor SC) at the site j, if it not is zero.\nIn Eq.(1), the first sum is over the nearest-neighbor and\nthe second one is over the next-nearest neighbor spins.\nIn this Hamiltonian the nearest neighbor interactions are\nchosen as JAB>0 andJAC<0, and the next-nearest\nneighbor interactions are chosen as JAA>0.\nIn order to show the effects of the concentration pand\nthe interaction parameterson the compensation and crit-2\nical temperature of the three dimensional ternary alloy\nmodel, we simulate the Hamiltonian given by Eq.(1).\nTo simulate this model, we employed Metropolis Monte\nCarlo simulation algorithm [20] to the L×L×Lthree-\ndimensionallatticewith periodicboundaryconditionsfor\nL= 10, 12, 16, 20, 24. One of the cubic sublattice is fully\ndecorated with spin SA, and spins SBand SCare ran-\ndomly distributed on the other cubic sublattice with the\nconcentration por 1−p, respectively. All initial spin\nstates in the L×L×Lthree-dimensional lattice are ran-\ndomly assigned. Configurationsare generated by making\nsingle-spin-flipattempts, which wereaccepted orrejected\naccording to the Metropolis algorithm. To calculate the\naverages,data, over20 different spin configuration, is ob-\ntained by using 50000 Monte Carlo steps per site after\ndiscarding 10000 steps.\nThe sublattice average magnetizations per site are ob-\ntained by\nMA=2\nL3/angbracketleftBiggL3/2/summationdisplay\niSA\ni/angbracketrightBigg\n, (2a)\nMB=2\nL3/angbracketleftBiggNB/summationdisplay\nj=1SB\nj/angbracketrightBigg\n, (2b)\nMC=−2\nL3/angbracketleftBiggNC/summationdisplay\nj=1SC\nj/angbracketrightBigg\n(2c)\nwhereNBdenotes the number of B ions NB=pL3/2,\nwhilstNCrepresents the number of C ions NC= (1−\np)L3/2 on the same cubic lattice. Total magnetization\nper site is given by\nM=1\n2(MA+MB+MC). (3)\nIII. RESULTS AND DISCUSSION\nIn this section, we have given the simulation results of\nthe ternary alloy model AB pC1−pand we have also dis-\ncussed the dependence of the critical and compensation\ntemperature on the concentration and other interaction\nparameters in the Hamiltonian. Simulation results have\nbeen obtained for the system with lattice size L= 10,\n12, 16, 20 and 24, however, here, we have only presented\nthe results of the model with lattice size L= 20. We also\nnote that the critical temperature of the system for the\ndifferent interaction rates and concentrations have been\nobtained by using of the method of the finite-size scaling\n[20].\nIn a recent study [15] it was reported that two dimen-\nsional ternary alloy model does not show a compensation\ntemperature point when there is no next-nearest neigh-\nbor interactions term in the Hamiltonian i.e., JAA= 0.\nFIG.1: The crystallographic structureofprussian blueana log\nwith two interpenetrating cubic lattices.\n/s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48\n/s82\n/s99\n/s32/s32/s107/s84\n/s99/s47/s74\n/s65/s66\n/s82/s32/s112/s61/s48/s46/s48\n/s32/s112/s61/s48/s46/s50/s53\n/s32/s112/s61/s48/s46/s53/s48\n/s32/s112/s61/s48/s46/s55/s53\n/s32/s112/s61/s49/s46/s48\nFIG. 2: Dependenceof the critical temperature on interacti on\nratioRin the three dimensional ternary alloy AB pC1−pfor\ndifferent values of pwhenJAA= 0.0.\nHowever, our simulations show that the system has a\ncompensation point for all R(we setR=|JAC|/JAB)\nvalues in interval of 0 .1≤R≤2.642 atp= 0 when\nJAA= 0. This point will be considered below. Now,\nin order to compare with the previous results [15, 18],\nin Figs.2 and 3 we discuss the dependence of the critical\ntemperatureofthe threedimensionalternaryalloymodel\nABpC1−pon interaction rate Rand concentration pfor\nnext-nearest neighbor interactions i.e., JAA= 0.\nIn Fig.2, the critical temperature of the three dimen-\nsional ternary alloy model has been plotted as a func-\ntion ofRfor various values of pwhenJAA= 0. It can\nbe seen from Fig.2 that the critical temperature of the\nsystem has a linear dependence on the interaction ratio\nRand there is a critical behavior at a special Rvalue.\nWhenRc=R= 0.513, the critical temperature of the3\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s50/s52/s50/s54/s50/s56\n/s32/s32/s107/s84\n/s99/s32/s47/s32/s74\n/s65/s66\n/s112/s32/s82/s61/s48/s46/s49\n/s32/s82/s61/s48/s46/s50/s53\n/s32/s82/s61/s48/s46/s53\n/s32/s82/s61/s48/s46/s55/s53\n/s32/s82/s61/s49/s46/s48\n/s32/s82/s61/s49/s46/s50/s53\n/s32/s82/s61/s50/s46/s48\n/s32/s82/s61/s50/s46/s54/s52/s50\nFIG. 3: Dependence of the critical temperature on the con-\ncentration pin the three dimensional ternary alloy AB pC1−p\nfor several values of interaction ratio RwhenJAA= 0.0. The\nlines show a part of the second-order transitions separatin g\nthe ferrimagnetic and paramagnetic phases.\nsystem has a fixed value of Tc= 5.47 for all pvalues.\nAtRc, the critical temperature of the system does not\nchange with concentration p. This means that neither\nthe spin-1 ions nor spin-5/2 ions substitution to system\nchangethe criticaltemperature ofthe systemat Rc. This\ncritical behavior has been reported in theoretical and ex-\nperimental studies [15, 18, 19]. The value of the Rcfor\nternary alloy AB pC1−pwhose spins consist of SA= 3/2,\nSB= 1 and SC= 5/2 has been obtained as Rc= 0.4781\nin the study based on mean field approximation [18] and\nasRc= 0.49intheMonteCarlosimulationoftwodimen-\nsional system [15]. Furthermore, the experimental mea-\nsurements indicate that there are Prussian blue analogs\nat theR= 0.45 have a Tcwhich is almost independent\nofp[19]. Fig.2 also reveals that concentration pplays\nan important role for the ternary alloy model AB pC1−p\nsince it determine the kinds of the spins and interactions\nin the system. For example, when p= 1 and p= 0,\nthe system AB pC1−pfully reduces to the ferromagnetic\nmixed spin-3/2 and spin-1 and ferrimagnetic mixed spin-\n3/2 and spin-5/2 Ising system, respectively. As seen in\nFig.2, although Tcof the system is independent of pat\nRc, however, the total magnetization of the system may\nconsiderablychangeowingto relativelysmall variationof\nthe concentration p. Indeed, for different values of p, the\ndependence of critical temperature of the system on the\ninteraction ratio Ris very different above and below of\nRc. This behavior can be explained by the change of the\nconcentration pin the system. On the other hand, it can\nbe detected from Fig.2 that when R < R c, the critical\ntemperature of the mixed spin-3/2 and spin-5/2 system/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/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\n/s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110\n/s107/s84/s32/s112/s61/s48/s46/s48\n/s32/s112/s61/s48/s46/s49\n/s32/s112/s61/s48/s46/s49/s53\n/s32/s112/s61/s48/s46/s50\n/s32/s112/s61/s48/s46/s50/s53\n/s32/s112/s61/s48/s46/s51\nFIG. 4: Magnetization of the three dimensional ternary allo y\nABpC1−pvs temperature for different values of p(JAA= 7.5\nandR= 1.0).\nis smaller than mixed spin-3/2 and spin-1 system. On\nthe contrary, when R > R c, the critical temperature of\nthe mixed spin-3/2 and spin-5/2 system has the highest\nvalue. On the Tclines, the critical temperature of the\nmixed spin-3/2 and spin-1 Ising system is equal to that\nof the mixed spin-3/2 and spin-5/2 Ising one.\nIn Fig.3, the dependence of the critical temperature of\nthe three dimensional AB pC1−psystem on the concen-\ntrationphas been shown for several values of Rwhen\nJAA= 0. The lines represent part of the second-order\nphase transition separating the ferrimagnetic and para-\nmagnetic. Fig.3 provides the argument that the concen-\ntrationpdetermines the magnetic features of the system\nmentioned above. Indeed, Fig.3 clearly shows that the\ncritical temperature of the system is changed by the con-\ncentration pfor fixed values of R. As seen from this\nfigure that, when R < R c, the critical temperature of the\nsystem linearly increases with increasing of p, whereas,\nwhenR > R c, the critical temperature of the system lin-\nearly decreases with increasing of pfor fixed values of R.\nHowever, when the values of Rclose up Rc, the criti-\ncal temperature of the system more slowly, but linearly,\nchangewith increasing p, and at the critical Rcvalue, the\ncritical temperature of the system denoted by triangle-\nline in Fig.3 is independent of the concentration p. On\nthe otherhand, Fig.3 alsoshowsthat the interactionrate\nRplays an important role on the critical temperature of\nthe three dimensional AB pC1−psystem. Finally we state\nthat the critical temperature of the model are consistent\nwith previous result [18].\nIn this study we recognize that the three dimensional\nternary alloy model AB pC1−phas one compensation be-\nhavior for JAA= 0, however, for JAA/negationslash= 0 it has one4\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53\n/s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110\n/s107/s84/s32/s112/s61/s48/s46/s50\n/s32/s112/s61/s48/s46/s50/s53\n/s32/s112/s61/s48/s46/s51\nFIG. 5: Magnetization of the three dimensional ternary allo y\nABpC1−pvs temperature for different values of p(JAA= 7.5\nandR= 2.642).\nor multi compensation points, when other conditions are\nsatisfied. However, the appearance of the compensation\ntemperature is strongly affected by the interaction and\nconcentration parameters. Indeed we see in the present\nstudy that the model has not a compensation point for\nall values of pandR. The dependence of the compen-\nsation temperature behavior on concentration and other\ninteraction parameters has been discussed below. For\ndiscussion, although the system has been simulated in\nthe intervals of 0 .0≤p <1.0 and 0.1≤R≤2.642, the\nvalue ofJAAused in the present study is chosen based on\nprevious theoretical study [15]. The results of simulation\nforR= 1.0 andR= 2.642 are respectively represented\nin Figs.4 and 5 for chosen parameters.\nOne compensation point has been found in the in-\ntervals of 0 .0≤p <0.3 and 0.1≤R <2.642 when\nJAA= 7.5. However, it is seen that the system has not\ncompensation behavior for the same values of parame-\nters when p≥0.3. ForR= 1.0 and several values of\np, the compensation behavior of the system can be seen\nfrom Fig.4. On the other hand, as seen from Fig.5, the\nconsidered system has a multi compensation behavior at\nR= 2.642 and p= 0.3 forJAA= 7.5 while it has one\ncompensation point for 0 .2≤p <0.3. Furthermore, our\nsimulation data introduce that the system shows com-\npensation behavior at p= 0 for all values of R, when\nJAA= 0. In addition, in the case JAA= 0, the com-\npensation point has been found for R= 0.25 atp= 0.2,\n0.25; forR= 0.75 atp= 0.3; forR= 1.25 atp= 0.3;\nforR= 2.0 atp= 0.1, 0.3, 0.4; forR= 2.642 atp= 0.1,\n0.2, 0.3, 0.4. Whereas, it has been reported in previous\nstudy that there is no compensation point for JAA= 0/s50 /s52 /s54 /s56 /s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48\n/s32/s32/s107/s84\n/s99/s111/s109/s112\n/s74/s32/s32/s74/s65/s65\n/s32/s32/s74/s65/s66\n/s32/s45/s74/s65/s67\nFIG. 6: Dependence of the compensation temperature Tcomp\non the interaction parameters in Hamiltonian of ternary all oy\nABpC1−pforp= 0.25.\nin two dimensional model [15].\nThe effect of the interaction parameters on the com-\npensation behavior of the three dimensional ternary\nmodel is also discussed in Fig.6. This figure shows de-\npendence of the compensation temperature Tcompof the\nmodel on interaction parameters in Hamiltonian only for\nafixed value ofthe concentrationparameter p(p= 0.25).\nIn this figure, square-line represents the behavior of the\ncompensation point vs JAAfor fixed values of JAB= 5,\nJAC=−5 andp= 0.25, circle-line indicates the behav-\nior of the compensation point vs JABfor fixed values of\nJAA= 7.5 andJAC=−5 andp= 0.25, and on the other\nhand, the behavior of the compensation point vs JAC\nis plotted for fixed values of JAA= 7.5,JAB= 5 and\np= 0.25 with triangle-line. As seen from Fig.6 that for\nfixedp,JAB= 5 and JAC=−5, the compensation tem-\nperature decreases slowly as the strength of the JAAin-\ncreases. Similarly for fixed p,JAA= 7.5 andJAC=−5,\nthe compensation temperature decreases slowly with in-\ncreasing of JAB. However, for fixed p,JAA= 7.5 and\nJAB= 5, the compensation temperature dramatically\nincreases as |JAC|increases. These results indicate that\nthe compensation temperature has a strong dependence\non the parameter JACwhereas its dependence on JAA\nandJABis relatively weak. The characteristic behav-\nior of the dependence of the compensation temperature\non the parameters of present model consistent with the\nresults of two dimensional model [15].5\nIV. CONCLUSION\nIn thisstudy, wehaveconsideredthe threedimensional\nternarymodelAB pC1−pwhosespins consistofSA= 3/2,\nSB= 1 and SC= 5/2. We have investigated the depen-\ndence of the critical and compensation temperature be-\nhavior of the considered model on concentration and in-\nteractions by using MC simulation method. We have ob-\nserved that the behavior of the critical temperature and\nthe existence of compensation points strongly depend on\ninteraction and concentration parameters. Particularly,\nwe have found that the critical temperature of the modelisindependent onconcentrationofdifferenttypesofspins\nat a critical Rcvalue and the model has one or two com-\npensation temperature points in a certain range of values\nof the concentration of the different spins. We concluded\nthat magnetic properties of the system AB pC1−pcan be\ncontrolled by changing the relative concentration of the\ndifferent species of ions. As a result, we would like to\nstress that these theoretical results can be very useful\nfor designing molecular magnets in experimental studies\nsince the existence of compensation in the ternary alloy\nABpC1−pthat can be setup by adjusting the proportion\nof compounds.\n[1] W. M. Liu et al., Phy. Rev. B 65 (2002) 172416.\n[2] P. B. He and W. M. Liu, Phy. Rev. B 72 (2005) 064410.\n[3] M. Gmitra and J. Barnas, Phy. Rev. Lett. 96 (2006)\n207205.\n[4] S. Ohkoshi, T. Iyoda, A. Fujishima and K. Hashimoto,\nPhy.Rev. B 56 (1997) 11642.\n[5] S. Ohkoshi, S. Yorozu, O. Sato, T. Iyoda, A. Fujishima\nand K. Hashimoto, Appl. Phys. Lett. 70 (1997) 1040.\n[6] S. Ohkoshi, Y. Abe, A. Fujishima and K. Hashimoto,\nPhys. Rev. Lett. 82 (1999) 1285.\n[7] S. Ohkoshi and K. Hashimoto, J. Am. Chem. Soc. 121\n(1999) 10591.\n[8] O. Sato, T. Iyoda, A. Fujishima and K. Hashimoto, Sci-\nence 271 (1996) 49.\n[9] D. A. Pejakovic, J. L. Manson, J. S. Miller and A. J.\nEipstein, Current Appl. Phys. 1 (2001) 15.\n[10] S. Ohkoshi, T. Hozumi and K. Hashimoto, Phy. Rev. B\n64 (2001) 132404.[11] A. Bob´ ak, O. F. Abubrig and D. Horv´ ath, Physica A 312\n(2002) 187.\n[12] A. Bob´ ak and J. Dely, Physica A 341, (2004) 281.\n[13] J. Dely and A. Bob´ ak, Physica B 388 (2007) 49.\n[14] S. Ohkoshi and K. Hashimoto, Phys. Rev. B 60 (1999)\n12820.\n[15] G. M. Buend´ ıa and J. E. Villarroel, J. Magn. Magn.\nMater. 310 (2007) 495.\n[16] J. Dely, A. Bob´ ak and M. ˇZukoviˇ c, Phys. Lett. A 373\n(2009) 3197.\n[17] S. G. Carling and P. Day, Polyhedron 20 (2001) 1525.\n[18] A. Bob´ ak, F. O. Abubrig, T. Balcerzak, Phy. Rev. B 68\n(2003) 224405.\n[19] P. Zhoug, D. Xue, H. Lou and X. Chen, Nanoletters 2\n(2002) 845.\n[20] K. Binder, in: K. Binder (Ed.), Monte Carlo Methods in\nStatistical Physics, Springer, Berlin, 1979."    },    {        "title": "1607.04312v1.Room_temperature_polarization_in_the_ferrimagnetic_Ga2_xFexO3_ceramics.pdf",        "content": "Room temperature polarization in the ferrimagnetic Ga 2−xFe xO3ceramics\nB. Kundys, F. Roulland, C. Lefèvre, C. Mény, A. Thomasson, N. Viart \nInstitut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), UMR 7504 CNRS-UdS, 67034 Strasbourg, France \nAbstract\nThe effect of the Fe/Ga ratio on the magnetic and electric properties of the multiferroic Ga2−xFe xO3 compound has been studied in order to \ndetermine the composition range exhibiting magnetic and electric orders coexistence and their critical temperatures. A magnetoelectric phase \ndiagram, showing the evolution of both the Néel magnetic ordering temperature TN and the electric ordering temperature Tc, versus the iron content \nhas been established for 0.9 ≤ x ≤ 1.4. While the ferrimagnetic Néel temperature increases with the iron content, the electric ordering temperature \nshows an opposite trend. The electric polarization has been found to exist far above room temperature for the x = 1.1 composition which shows \nthe highest observed electric ordering temperature of Tc ≈ 580 K. The x = 1.3 and 1.4 compounds are ferrimagnetic–electric relaxors with both \nproperties coexisting at room temperature.\nKeywords: Dielectric permittivity; Polarization; Ferrimagnetism; Magnetoelectric phase diagram\n1. Introduction\nThe coexistence of magnetic and electric orders at the same\ntemperature and pressure regions present high research inter-\nest due to the potential to design technologically importantcross-functionalities in these materials. Such functionalities caninclude, butare not limited to, the electric field-controlled mag-\nnetization or magnetic field-controlled polarization.\n1–3There\nare however very few materials that present such propertiesat room temperature.\n4,5Until recently the only material show-\ning unambiguously both ferroelectricity and magnetoelectricityat room temperature was BiFeO\n3;6–8this explains why this\nmaterial has been the focus point of most of the researcheson multifunctional materials. Lately, convincing evidences ofthe coexistence of room temperature electric and magneticorders and room temperature magnetoelectric effect have beenobserved in a few other compounds such as solid solutionphases between lead iron based perovskites, and in particularthe PbFe\n0.5Ta0.5O3–PbZr 0.53Ti0.47O3(PFT–PZT)9,10and the\nPbFe 0.5Nb0.5O3–PbZr 0.53Ti0.47O3(PFN–PZT)11systems, the\ncomplex hexaferrite Sr 3Co2Fe24O41,12oreven /H9252-NaFeO 2.13\nThe preparation of these compounds is however rather tricky andthere is still room for improvement of their electric polarization,net magnetization and magnetoelectric effect.The Ga\n2−xFe xO3(GFO) compound, is known since\nthe 60s to present interesting pyroelectric, ferrimagneticand magnetoelectric properties near room temperature.\n14\nIt crystallizes in the orthorhombic space group Pc2 1n,\nwith a= 0.87512 ±0.00008 nm, b= 0.93993 ±0.00003 nm and\nc= 0.50806 ±0.00002 nm (Fig. 1).15This structure is based on\nan ABAC double hexagonal close-packed stacking of oxygens,and strongly differs from the usual perovskite structure observedfor most of the other multiferroic materials. The cations aredistributed among four cationic sites Ga1, Ga2, Fe1, and Fe2.\nGa1 and Fe1 are antiferromagnetically coupled to Ga2 and\nFe2. If the Fe\n3+cations only occupied the Fe1 and Fe2 sites, and\nthe Ga3+only the Ga1 and Ga2 one, the compound would be\nstrictly antiferromagnetic for x= 1.0. In fact a cationic site disor-\nder, observed by Arima et al.17by neutron diffraction, allows the\ncompound to exhibit a non-negligible net resulting magnetiza-tion (0.7 /H9262\nB/Fe for x= 1.0). The magnetic ordering temperature\nis relatively high in this family of compounds and is above\nroom temperature for x≥1.3. Although GaFeO 3was already\nreported to show magnetic field dependent polarization14,17the2 B. Kundys et al. \nFig. 1. Projection of the crystal structure of Ga 2−xFexO3along the caxis (space\ngroup Pc2 1n) produced using the VESTA16crystallographic software.\nanswer to the question whether it is electrically polar at room\ntemperature and in zero magnetic field remains unclear. A phasetransition to electrically polar state may be confirmed by a dipolereorientation induced maximum in the temperature variation ofthe dielectric permittivity. However the temperature variations ofthe dielectric properties reported so farfor pure GFO compounds\nconcern exclusively the x= 1.0 compound and the temperature\ndependence of the permittivity shows no maximum, indicatingthe absence of the transition to electrically polar state.\n18–21The\nonly evidence of a maximum in the temperature variation ofthe dielectric permittivity wasfound for a Mn-doped GaFeO\n3\nsample.22Recently, a partially reversible polarization of ca.\n0.3/H9262C/cm2washowever evidenced in polycrystalline GaFeO 3\nthrough pyroelectric measurements.23The reported ferroelec-\ntric Curie temperature (T Cf)wasbelow room temperature (ca.\n100 K). A similar value of the polarization, ca.0.2/H9262C/cm2,\nbutthis time fully reversible and at room temperature, has\nbeen observed on thin films, both for the Ga 0.6Fe1.4O3:Mg24\nand GaFeO 325compositions. It must be noted that a polar-\nization of ca.1/H9262C/cm2has also been measured on thin films\nof the isostructural compound /H9255-Fe 2O3at room temperature.26\nThe awaited polarization in GaFeO 3is however two orders of\nmagnitude bigger than the measured ones, with a value of ca.\n25/H9262C/cm2, as evaluated by Stoeffler using a simple point charge\nmodel.\nThere is therefore a need to, first, clarify the electric behavior\nof GFO compounds and, then, study the influence of the ironcontent xon the temperature of the prospective electric order. In\nthis work, we attend to the electric characterization of a series ofGa\n2−xFe xO3polycrystalline compounds (0.9 ≤x≤1.4) through\nboth the study of the variation of the dielectric constant withtemperature and pyroelectric current measurements.\n2. Materials and methods\nIn our experiment the Ga\n2−xFe xO3polycrystalline samples\n(0.9≤x≤1.4) were prepared viaan optimized ceramic pro-\ncess already published elsewhere.27High purity commercial\npowders of /H9251-Fe 2O3(Prolabo >99%) and Ga 2O3(Alfa Aesar99.999%) were first ball-milled in a Teflon jar in an optimized\ndispersive environment. The resulting slurries are subsequentlydried and the powders obtained are mixed with an organic binder\n(Rhodoviol) to be uniaxially pressed under 60 bars into 15 mmdiameter and 0.6 mm height disk shaped pellets using a hydraulic\npress. The green pellets were then sintered at the tempera-ture necessary for the formation of the desired Ga\n2−xFe xO3,\nphase exempt of any parasitic phase, which depends upon theFe content.\n27Forthe electric measurements, the temperature\nvariation wasperformed in an Instec cryostat with a possibility\nto heat up to 873 K. Dielectric measurements were performedin vacuum at different frequencies using an Agilent LCR meter.The samples were first taken to the highest available temperature(ca.600 K) and the dielectric constant wasthen measured upon\ncooling them down to 150 K. Pyroelectric current measurementswere performed with a Keithley electrometer upon cooling thesamples in a small electric field of 0.05 V/0.6 mm. The electricpolarization wasdeduced from those pyroelectric measurements\nthrough a time integration method. Magnetic measurementswere performed using a superconducting quantum interferencedevice magnetometer (SQUID MPMS XL, Quantum Design).\n3. Results and discussion\nThe temperature dependence of the dielectric permittivity\nwas measured for various compounds of the Ga\n2−xFe xO3fam-\nily (with 0.9 ≤x≤1.4). The results are shown in Fig. 2(a), and\nthe evolution of the dielectric losses with the Fe content arepresented in Fig. 2(b). A maximum in the temperature depend-\nent permittivity is clearly seen for compositions above x= 1.0,\nindicating thus the existence of an electric polarization. The posi-tion of this maximum varies from 570 K for the x= 1.1 sample to\nabout 400 K for the x= 1.3 sample. It is clear that both the electric\nordering temperature (T\nc) and dielectric losses of the GFO com-\npounds strongly depend on the Fe/Ga ratio. The maximum in thedielectric permittivity curve is within an experimentally reach-able range of temperatures for x≥1.1,butit strongly smoothens\nwith increasing x.\nThe Ga\n2−xFe xO3,x= 1.1 compound shows the most impor-\ntant dielectric anomaly. The temperature position of the observedpeak in the dielectric permittivity and dielectric loss is frequencyindependent, and allows determining a transition temperature of582 K (Fig. 3(a)). The ratio between the slopes of the 1/ε\n/primeversus\ntemperature curve above and below Tcis larger than 2 (Fig. 3(b)),\nthus indicating a first-order transition to the paraelectric stateabove 582 K.\nPyroelectric current measurements also reveal an anomaly\nnear this temperature for x= 1.1 (Fig. 4). The current integration\nwith respect to time method gives a rather large polarizationvalue of ca.33/H9262C/cm\n2, existing at room temperature. Although\npolarization loops have been measured in thin films of this mate-rial by different teams, including ours,\n24,28it has to be noted\nthat our attempts to measure ferroelectric loops in the bulk com-pound, at different temperatures between 150 and 500 K, and inelectric fields up to 60 kV/m, were unsuccessful. There is there-fore no evidence of ferroelectricity of the material in its bulkform. Although the reported value of 33 /H9262C/cm\n2has an onlyB. Kundys et al. 3\nFig. 2. (a) Temperature variation of the reduced dielectric permittivity measured at 100 kHz for various Ga 2−xFexO3compounds (0.9 ≤x≤1.4) and (b) dielectric\nloss evolution with the Fe content in Ga 2−xFexO3compounds at room temperature.\nFig. 3. (a) Frequency dispersion of the dielectric permittivity and dielectric loss (inset) as a function of temperature, (b) temperature dependence of 1/ε/primebelow and\nabove the electric ordering temperature Tc, for GFO x= 1.1.\nindicative character as it wasinduced by a specific electric field,\nit nevertheless, together with a peak in the dielectric permittiv-\nity, confirms the existence of an important electric polarizationstate in the sample, with a phase transition at 580 K. The I(T)\ncurve, measured under an applied electric field of 0.05 V/0.6 mm(Fig. 4) shows a zero current value in the temperature range of160–450 K. This therefore excludes any artifact due to a domi-nant leakage contribution. At the electric ordering temperature,the small applied electric field allows orienting the polariza-tion from random paraelectric state to polar state. It also has tobe noted that the observed polarization value perfectly matches\nFig. 4. Electric polarization (left) deduced from pyroelectric current (right)\nintegration with respect to time, for GFO (x = 1.1).with the value of 25 /H9262C/cm2calculated by Stoeffler29using first\nprinciple methods.\nFor the Ga 2−xFe xO3,x= 1.3 and 1.4 compounds, a relaxor-\nlike behavior is observed, where the temperature for which the\nmaximum permittivity is observed depends on the measurementfrequenc y(Fig. 5).\nSuch a behavior is correlated with an important increase of the\ndielectric losses with the increasing iron content (Fig. 2(b)). Thegeneral trend in the T\ncevolution can be deduced by comparing\nthe dielectric anomalies at one chosen frequency (Fig. 2(a)): itclearly decreases with increasing x.\nThe thermomagnetic curves of the samples were measured\nunder an applied field of 75 Oe, high enough to increase themagnetic response while still keeping the samples well belowmagnetic saturation. The evolution of the Néel temperature of thesamples versus their iron content is given in Fig. 6. As awaited,\n17\nthe magnetic properties are strongly dependent on the iron con-tent of the samples: the higher the iron content, the higher boththe magnetization and the magnetic transition temperature. Themost interesting results are ascribed to the two samples of high-est iron contents (x = 1.3 and 1.4) as their T\nNvalues are higher\nthan room temperature (T N= 330 K for x= 1.3 and TN= 400 K\nfor x= 1.4). The Ga 2−xFe xO3system belongs to type 1 multifer-\nroic compounds in which magnetic and electric orders occur atdifferent temperatures.4 B. Kundys et al.\nFig. 5. Temperature variation of the dielectric permittivity for (a) Ga 0.7Fe1.3O3and (b) Ga 0.6Fe1.4O3at different frequencies.\nFig. 6. M(T) dependences for the different GFO compounds (0.9 ≤x≤1.4)\nunder an applied field of 75 Oe.\nFig. 7. Magnetoelectric phase diagram of Ga 2−xFexO3compounds with x\nbetween 0.9 and 1.4.\nFig. 7 finally summarizes the magnetoelectric phase diagram\nfor Ga 2−xFe xO3compounds, showing the evolution of the Néel\nmagnetic ordering temperature TN, together with the electric\nordering temperature Tc,versus the iron content.\n4. Conclusion\nIn summary, the temperature variation (150 K < T< 600 K)\nof the dielectric properties of Ga 2−xFe xO3, 0.9≤x≤1.4, com-\npounds shows a maximum for the x> 1 compositions. The\nabsence of previous documented studies for compositions other\nthan x= 1 explains why this phenomenon had not been observed\nbefore. The composition x= 1.1 shows a polar state with anordering temperature of about 580 K and a polarization of33/H9262C/cm\n2very close to the value awaited from theoretical cal-\nculations. Compositions with x≥1.3 present a relaxor behavior,\nwith high dielectric losses. The temperature at which the maxi-mum of permittivity is observed decreases with increasing ironcontent. The dependence of the electric and magnetic propertiesof this system upon the Fe content is interestingly opposite, sinceT\nNincreases with x.Forsamples with x≥1.3, the coexistence of\nboth electric and magnetic polarizations in a wide temperaturerange including room temperature is possible. Further studiesin this area should focus on the optimal xvalues for which the\ndielectric relaxor behavior is transformed into a classical onewith low dielectric dissipation, allowing a large magnetizationand a large polarization near room temperature.\nAcknowledgments\nThis work was done with the financial support from the\ninternational ANR DFG Chemistry project GALIMEO #2011-\nINTB-1006-01. The authors are grateful to Dr. Ingrid CA ˜NERO\nINFANTE for fruitful discussions.\nReferences\n1.Binek C, Doudin B. Magnetoelectronics with magnetoelectrics. J Phys:\nCondens Matter 2005;17:L39–44.\n2.Bibes M, Barthelemy A. Multiferroics: towards a magnetoelectric memory.\nNat Mater 2008;7:425–6.\n3.Scott JF. Applications of magnetoelectrics. J Mater Chem 2012;22:4567–74.\n4.Hill NA. Why are there so few magnetic ferroelectrics? J Phys Chem B\n2000;104:6694–709.\n5.Scott JF. Room-temperature multiferroic magnetoelectrics. NPG Asia Mater\n2013:5.\n6.Wang J, Neaton JB, Zheng H, Nagarajan V,Ogale SB, Liu B, et al. Epitaxial\nBiFeO 3multiferroic thin film heterostructures. Science 2003;299:1719–22.\n7.Lebeugle D, Colson D, Forget A, Viret M, Bonville P,Marucco JF,\net al. Room-temperature coexistence of large electric polarization andmagnetic order in BiFeO\n3single crystals. Phys Rev B: Condens Matter\n2007;76:024116.\n8.Catalan G, Scott JF. Physics and applications of bismuth ferrite. Adv Mater\n2009;21:2463–85.\n9.Evans DM, Schilling A, Kumar A, Sanchez D, Ortega N, Arredondo M,\net al. Magnetic switching of ferroelectric domains at room temperature inmultiferroic PZTFT. Nat Commun 2013;4:1534.\n10.Sanchez DA, Ortega N, Kumar A, Roque-Malherbe R, Polanco R, Scott\nJF,et al. Symmetries and multiferroic properties of novel room-temperature5\nmagnetoelectrics: lead iron tantalate–lead zirconate titanate (PFT/PZT). AIP\nAdv2011;1:042169.\n11.Sanchez DA, Ortega N, Kumar A, Sreenivasulu G, Katiyar RS, Scott JF,\net al. Room-temperature single phase multiferroic magnetoelectrics: Pb(Fe,\nM) x(Zr,Ti)1−xO3M=Ta,Nb. J Appl Phys 2013;113:74105(1)–7).\n12.Kitagawa Y,Hiraoka Y,Honda T,Ishikura T,Nakamura H, Kimura\nT.Low-field magnetoelectric effect at room temperature. Nat Mater\n2010;9:797–802.\n13.Viret M, Rubi D, Colson D, Lebeugle D, Forget A, Bonville P,et al. /H9252-\nNaFeO 2, a new room-temperature multiferroic material. Mater Res Bull\n2012;47:2294–8.\n14.Rado GT. Observation and possible mechanisms of magnetoelectric effectsin a ferromagnet. Phys Rev Lett 1964;13:335–7.\n15.Abrahams SC, Reddy JM, Bernstein JL. Crystal structure of piezoelectricferromagnetic gallium iron oxide. J Chem Phys 1965;42:3957–68.\n16.Momma K, Izumi F.VESTA 3 for three-dimensional visualization of crystal,\nvolumetric and morphology data. J Appl Crystallogr 2011;44:1272–6.\n17.Arima T,Higashiyama D, Kaneko Y,He JP, Goto T,Miyasaka S, et al.\nStructural and magnetoelectric properties of Ga\n2−xFe xO3single crystals\ngrown by a floating-zone method. Phys Rev B: Condens Matter 2004;70:\n064426.\n18.Naik VB, Mahendiran R. Electrical, magnetic, magnetodielectric,and magnetoabsorption studies in multiferroic GaFeO\n3.J Appl Phys\n2009;106:123910.\n19.Shireen A, Saha R, Mandal P,Sundaresan A, Rao CNR. Multiferroic and\nmagnetodielectric properties of the Al 1−xGa xFeO 3family of oxides. J Mater\nChem 2011;21:57–9.20.Mohamed MB, Senyshyn A, Ehrenberg H, Fuess H. Structural, magnetic,\ndielectric properties of multiferroic GaFeO 3prepared by solid state reaction\nand sol–gel methods. J Alloys Compd 2010;492:L20–7.\n21.Sun ZH, Cheng BL, Dai S, Cao LZ, Zhou YL, Jin KJ, et al. Dielectric prop-erty studies of multiferroic GaFeO\n3.J Phys D: Appl Phys 2006;39:2481–4.\n22.Mohamed MB, Fuess H. Effect of Mn doping on structural and magneticproperties of GaFeO\n3.J Magn Magn Mater 2011;323:2090–4.\n23.Saha R, Shireen A, Shirodkar SN, Waghmare UV , Sundaresan A, Rao CNR.Multiferroic and magnetoelectric nature of GaFeO\n3, AlFeO 3and related\noxides. Solid State Commun 2012;152:1964–8.\n24.Thomasson A, Cherifi S, Lefevre C, Roulland F,Gautier B, Albertini D,\net al. Room temperature multiferroicity in Ga 0.6Fe1.4O3:Mg thin films. J\nAppl Phys 2013;113:214101.\n25.Sharma K, Reddy VR, Gupta A, Choudhary RJ, Phase DM, Ganesan V.\nStudy of site-disorder in epitaxial magneto-electric GaFeO 3thin films. Appl\nPhys Lett 2013;102:212401.\n26.Gich M, Fina I, Morelli A, Sanchez F,Alexe M, Gazquez J, et al. Multiferroic\niron oxide thin films at room temperature. Adv Mater 2014;26:4645.\n27.Roulland F,Lefevre C, Thomasson A, Viart N. Study of Ga (2−x)Fe xO3\nsolid solution: optimisation of the ceramic processing. J Eur Ceram Soc\n2013;33:1029–35.\n28.Mukherjee S, Roy A, Auluck S, Prasad R, Gupta R, Garg A. Room temper-ature nanoscale ferroelectricity in magnetoelectric GaFeO\n3epitaxial thin\nfilms. Phys Rev Lett 2013;111:087601.\n29.Stoeffler D. First principles study of the electric polarization and of itsswitching in the multiferroic GaFeO\n3system. J Phys: Condens Matter\n2012;24:185502."    },    {        "title": "2005.08914v3.Noncollinear_Magnetic_Modulation_of_Weyl_Nodes_in_Ferrimagnetic_Mn__3_Ga.pdf",        "content": "Noncollinear Magnetic Modulation of Weyl Nodes in Ferrimagnetic Mn 3Ga\nCheng-Yi Huang,1, 2Hugo Aramberri,2,\u0003Hsin Lin,1and Nicholas Kioussis2,y\n1Institute of Physics, Academia Sinica, Taipei 11529, Taiwan\n2Department of Physics and Astronomy, California State University, Northridge, CA 91330-8268, USA\n(Dated: July 16, 2020)\nThe tetragonal ferrimagnetic Mn 3Ga exhibits a wide range of intriguing magnetic properties.\nHere, we report the emergence of topologically nontrivial nodal lines in the absence of spin orbit\ncoupling (SOC) which are protected by both mirror and C4zrotational symmetries. In the presence\nof SOC we demonstrate that the doubly degenerate nontrivial crossing points evolve into C4z-\nprotected Weyl nodes with chiral charge of \u00062. Furthermore, we have considered the experimentally\nreported noncollinear ferrimagnetic structure, where the magnetic moment of the Mn Iatom (on the\nMn-Ga plane) is tilted by an angle \u0012with respect to the crystallographic caxis. The evolution of the\nWeyl nodes with \u0012reveals that the double Weyl nodes split into a pair of charge-1 Weyl nodes whose\nseparation can be tuned by the magnetic orientation in the noncollinear ferrimagnetic structure.\nPACS numbers: 73.20.At, 75.50.Gg\nI. INTRODUCTION\nThe discovery of topological states of matter repre-\nsents a cornerstone of condensed-matter physics that may\naccelerate the development of quantum information and\nspintronics and pave the way to realize massless particles\nsuch as Dirac and Weyl fermions. A Weyl semimetal\n(WSM) is a topological semimetallic material hosting\ndoubly-degenerate gapless nodes near the Fermi level in\nthe three-dimensional (3D) momentum space1{4. The\nnodes correspond to e\u000bective magnetic monopoles or an-\ntimonopoles which carry nonvanishing positive and neg-\native chiral charge \u0006q. Typically, qtakes values of\u00061\ncorresponding to Weyl nodes, but is also possible to have\nintegers,q=\u00062;\u00063;::: for double Weyl nodes, etc.5\nThe Weyl nodes gives rise to surface states which form\nopen Fermi arcs rather than closed loops.\nCompared to their Dirac semimetal counterparts,\nWSMs require the breakdown either of inversion symme-\ntry or time reversal symmetry (TRS) to split each four-\nfold degenerate Dirac node into a pair of Weyl nodes. A\nnumber of WSMs that break inversion symmetry have\nbeen identi\fed in the past few years1{4. Moreover the\npresence of crystalline symmetries can further protect\nmultiple Weyl nodes with large chiral charge6{8. On the\nother hand, the discovery of their broken TRS counter-\nparts, which link the two worlds of topology and spintron-\nics, remains challenging and elusive6. Many potential\nTRS-breaking WSM have been proposed. Recently, three\ngroups have provided unambiguous and direct experi-\nmental con\frmation that Co 3Sn2S29,10, which becomes\na ferromagnet below 175 K, and Co 2MnGa, a room-\ntemperature ferromagnet11, are TRS-breaking WSMs.\nThe discovery of magnetic WSMs give rise to exotic quan-\ntum states ranging from quantum anomalous Hall e\u000bect\nto axion insulators3.\nAnother remarkable and highly promising class of\nmagnetic materials is the Heusler family12,13which in-\ncludes half metals,14ferromagnets, ferrimagnets, antifer-\nromagnets, and even topological insulators15,16and Weylsemimetals. In particular the ferrimagnetic and antifer-\nromagnetic compounds with antiparallel exchange cou-\npling, have recently garnered intense interest because of\nthe faster spin dynamics (in the terahertz range) com-\npared to the gigahertz-range magnetization dynamics of\ntheir ferromagnetic counterparts.17\nThe Mn 3X (X=Ga, Ge, Sn) Heusler compounds are\nconsidered prototypes with promising applications in the\narea of spintronics13,18. These compounds can be ex-\nperimentally stabilized in either the hexagonal DO 19\nstructure ( \u000fphase) or the tetragonal DO 22structure\n(\u001cphase)19. The high-temperature hexagonal crystal\nstructure is antiferromagnetic with a high N\u0013 eel temper-\nature (\u0018470 K) and a noncollinear triangular magnetic\nstructure. Recently, several experimental and theoretical\nstudies have demonstrated20{26the emergence of large\nanomalous Hall e\u000bect (AHE) in the noncollinear AFM\nhexagonal Mn 3X family, whose origin lies on the non-\nvanishing Berry curvature in momentum space. In addi-\ntion, ab initio calculations have revealed that these chiral\nAFM materials are topological Weyl semimetals22. On\nthe other hand, the low-temperature tetragonal phase,\nwhich can be obtained by annealing the hexagonal phase,\nis ferrimagnetic at room temperature and shows a unique\ncombination of magnetic and electronic properties, in-\ncluding low magnetization,27high uniaxial anisotropy,28\nhigh spin polarization ( \u001988%),29{31low Gilbert damp-\ning constant,29high Curie temperature,32and large volt-\nage controlled magnetic anisotropy e\u000eciency33. Interest-\ningly, neutron scattering experiments have reported34a\nnoncollinear ferrimagnetic magnetic structure in Mn 3Ga,\nwhere the magnetic moment orientation of the Mn atoms\non the Mn-Ga (001) plane is tilted by about 21 °with re-\nspect to the crystallographic caxis.\nThe objective of this work is to carry out \frst-\nprinciples electronic structure calculations to investigate\nthe emergence of topological nodal lines in the absence\nor presence of SOC in tetragonal ferrimagnetic Mn 3Ga.\nFurthermore, we present results of the e\u000bect of non-\ncollinear magnetism on the evolution of the Weyl nodes.arXiv:2005.08914v3  [cond-mat.mtrl-sci]  15 Jul 20202\nII. METHODOLOGY\nThe electronic structure calculations were carried out\nby means of \frst-principles spin-polarized collinear cal-\nculations within the density functional theory (DFT)\nframework as implemented in the VASP package35.\nThe Perdew-Burke-Ernzerhof36(PBE) implementation\nof the generalized gradient approximation (GGA) for\nthe exchange-correlation functional was employed. The\nplane-wave cuto\u000b energy was set to 400 eV, which was\nenough to yield well-converged results. The Brillouin\nzone (BZ) was sampled using a \u0000-centered mesh of\n10x10x10 k-points. The structure was allowed to relax\nuntil residual atomic forces became lower than 0.01 eV/ \u0017A\nand residual stresses became smaller than 0.01 GPa. The\nelectron-electron interactions are included, where indi-\ncated, within the GGA+U approach of Dudarev et al.37.\nIn this way, the electron correlations are taken into ac-\ncount through a single e\u000becive parameter Ue\u000b=U\u0000J.\nThe values of Ue\u000bfor the Mn I, MnIIand Ga are set to 2.6\neV, 0 and 0, respectively. Previous studies have shown\nthat these values yield lattice parameters closer to their\nexperimental values38. The spin-orbit coupling (SOC) of\nthe valence electrons is in turn included self-consistently\nusing the second-variation method employing the scalar-\nrelativistic eigenfunctions of the valence states39, as im-\nplemented in VASP. Then, DFT derived wave functions\nboth without and with SOC were in turn projected to\nWannier functions using the wannier90 package40.\nIn theDO22structure (I4/mmm space group) the\ntwo (001) antiferromagnetically-coupled Mn sublattices,\nshown in Fig. 1(a), consist of Mn Iatoms at the Wycko\u000b\npositions 2b (0,0,1/2) [Mn I-Ga (001) plane] and Mn II\natoms at the 4d (0,1/2,1/4) positions [Mn II-MnII(001)\nplane]. For the noncollinear calculation, where the mag-\nnetic moment of the Mn Iis rotated by an angle \u0019\u0000\u0012with\nrespect to the [001] direction, the angular dependence of\nthe Wannier Hamiltonian in the presence of SOC is de-\ntermined from,\nH(k;\u0012) =H0(k) +U(\u0012)Hex(k)Uy(\u0012): (1)\nHere,H0(k) is the TRS preserving Hamiltonian\nwith SOC, [ TH0(k)T\u00001=H0(\u0000k)],Hex(k) is the\nTRS-breaking exchange Hamiltonian, [ THex(k)T\u00001=\n\u0000Hex(\u0000k)],T=i\u001b2Kis the TRS operator, \u001b2acts on\nthe spin degrees of freedom, Kis complex conjugation,\nU(\u0012) =e\u0000i\u0019\u0000\u0012\n2\u001b2;MnIis the spin rotation operator, and\n\u001b2;MnIis theycomponent of Pauli matrix acting on the\nspin degrees of freedom of Mn I.\nIII. RESULTS AND DISCUSSION\nA. Nodal lines in the absence of SOC\nThe calculated lattice parameters a=b= 3.78 \u0017A and\nc= 7.08 \u0017A, are in good agreement with previous calcu-TABLE I: List of ab initio and experimental lattice constants\nand magnetic moments values for the collinear case. \u00162b\nz(\u00164d\nz)\ndenotes the z-component of the magnetic moment of the Mn I\n(Mn II) atom.\nMethod a( \u0017A) c( \u0017A)\u00162b\nz(\u0016B/Mn)\u00164d\nz(\u0016B/Mn)\nGGA 3.78 7.08 -2.83 2.30\nGGA+U 3.91 7.00 -3.76 2.45\nGGA+U+SOC 3.91 7.00 -3.87 2.51\nexperiment343.92 7.08 -3.07 2.08\nE-EF(eV) \nΓ X M Γ N\nMn IGa \nMn II\nbc\na(a) (b) \n(c) \nΓ\nXM\nNkz\nkx ky\nFIG. 1: (Color online) (a) The tetragonal cell of the DO 22\nferrimagnetic structure with [001] spin polarization. Arrows\ndenote the magnetic moments of Mn I(purple) and Mn II(red)\nsublattices which are coupled antiferromagnetically. (b) First\nBrillouin zone of the primitive cell shown in panel (a). (c)\nSpin-polarized band structure without SOC along the high\nsymmetry directions of the primitive cell, where the spin-up\n(spin-down) bands are denoted by blue (red).\nlations34,41,42, which are, however, lower than the exper-\nimental values of a=b= 3.92 \u0017A andc= 7.08 \u0017A(see\nTable I). The e\u000bect of U on the topology of the band\nstructure is discussed in Sec. III. Overall, our calculated\nGGA values of the magnetic moments of -2.83 \u0016Band\n2.30\u0016Bfor the Mn Iand MnIIatoms, respectively, are in\ngood agreement with previous DFT calculations34,41,42.\nFig. 1(c) shows the spin-polarized band structure of\nthe majority- (blue) and minority-spin (red) bands of\nMn3Ga without SOC and with collinear spins along the\nsymmetry lines of the Brillouin zone (BZ) of the prim-\nitive cell, shown in Fig. 1(b). For each spin channel,\nthe energy bands can be labeled by the eigenvalues of\nthe crystalline symmetry operator of a particular high3\n(a)\n(b)\nE-EF(eV)\nkz(Å-1)(-1,-1)(1,1)\nkz(Å-1)E-EF(eV)\nFIG. 2: (Color online) (a) Spin polarized band structure along\nthekz-axis (\u0000\u0000Msymmetry direction) without SOC, where\nthe blue (black) bands denote the spin-up states calculated\nfrom GGA (GGA+U). The two nontrivial crossing points, de-\nnoted with the blue dots, are labeled with the pair of eigenval-\nues, (\u00061,\u00061), of the mirror, M[110], and four-fold rotational,\nC4z, symmetries, respectively, which protect them. Red dots\ndenote the nontrivial crossing points when U is turned on.\n(b) 3D landscape of the nodal lines where the two blue dots\ndenote the two nontrivial crossing points in (a). The color\nbar represents the energy of the nodal points relative to the\nFermi energy.\nsymmetry direction. The band structure along the M-\n\u0000-M direction, shown in Fig. 2(a), features several band\ncrossings close to the Fermi level. Thus, throughout the\nremainder of the manuscript, we only focus on the cross-\ning points, marked by blue dots in Fig. 2 (a), between\nthe majority-spin bands along the kz(\u0000\u0000M) direction.\nThese points are protected by both a mirror re\rection\nsymmetry normal to the [110] direction, M[110], and a\nfour-fold rotational symmetry, C4z, and hence can be la-\nbeled by the pair of eigenvalues, ( \u00061,\u00061), ofM[110]and\nC4z, respectively. The e\u000bective k\u0001pmodel in the basis\nfj(1;1)i;j(\u00001;\u00001)igup to order of k2in the absence ofTABLE II: The C4z-protected Weyl fermion on kzaxis.\nuc(uv) denotes the eigenvalue of C4zin conduction (valence)\nband.Cdenotes chiral charge.\nkz(\u0017A\u00001)E-EF(meV)uc=uvCDispersion\nonkx-kyplane\n0.2811 229 -1 2 k2\n-0.2811 229 -1 -2 k2\nSOC can be straightforward derived and is given by\nHNL= (m1\u0000m2k2\nz)s3+a(k2\nx\u0000k2\ny)s1; (2)\nwherekis close to the \u0000 point, m1m2>0, thesi's are\nPauli matrices and the nodal lines lie on kz=\u0006q\nm1\nm2\nandkx=\u0006ky. We have tracked the nodal lines on\ntheM[110]-invariant plane. The other nodal lines on the\nM[1\u001610]-invariant plane were determined using the C4zro-\ntational symmetry. Fig. 2(b) shows the 3D landscape of\nnodal lines in momentum space. We \fnd that the nodal\nlines are topologically nontrivial characterized by the \u0019\nBerry phase43{45. The two blue points denote the non-\ntrivial crossing points as well as the intersecting points of\nnodal lines along the kzdirection in Fig. 2(b). Notably\nthe crossing points remain gapless and robust against a\ndistortion breaking either M[110]orC4z.\nB. Weyl Nodes in the Presence of SOC\nIn the presence of SOC, the symmetry conservation\ndepends on the magnetic orientation and the crystalline\nsymmetries. More speci\fcally the [001] collinear mag-\nnetic con\fguration is invariant under (1) inversion sym-\nmetry (P), (2) fourfold rotational symmetry about the\nz-axis (C4z) and (3) mirror re\rection symmetry normal\nto thezdirection (Mz). We next discuss the e\u000bect of\nmagnetization orientation (collinear versus noncollinear)\non the topological features of the band structure.\nWeyl Nodes in Collinear Ferrimagnetism| In the\npresence of SOC, the mirror symmetry M[110]is no longer\npreserved when the magentization of the collinear fer-\nrimagentic Mn 3Ga is along the [001] direction. Con-\nsequently, in general the nodal points in Fig. 2(b) are\ngapped out except for those crossing points along kz\nwhich are protected by the C4zrotational symmetry46.\nThus, for the band structure along the C4z-invariantkz-\naxis, shown in Fig. 3(a), we can identify the states by\nthe eigenvalues of C4zand locate the nontrivial crossing\npoints associated with di\u000berent eigenvalues. The non-\ntrivial crossing points, marked by red circles in Fig. 3(a),\nare Weyl nodes protected by C4zsymmetry, whose posi-\ntion alongkz, energy relative to E F, ratio of conduction\nto valence band C4zeigenvalues, uc=uv, chiral charge, C,\nand dispersion are summarized in Table II. Interestingly,\ntheC4z-protected Weyl fermion with uc=uv= -1 carries4\n+2 -2\n+2 \n-2kz(Å -1)\nky(Å -1)E-EF(eV) \nkz(Å -1)(a) (b) \n\u0001\u0002\u0003\u0004\n\u0005 \u0001\u0006\u0002\u0004\n\u0005\nFIG. 3: (Color online) (a) Band structure along the kz-axis (\u0000\u0000Msymmetry direction) with SOC, protected by C4zsymmetry.\nThe two Weyl points, denoted with red dots, have chiral charge of \u00062. Thee\u0000i\u0019\n4andei3\u0019\n4indicate the eigenvalues of C4zfor\nthe crossing bands, respectively. (b) Two Fermi arcs on the (100) surface where green (red) color denotes the spectral weight\nof the surface (bulk) states. Solid (hollow) white circle denotes positive (negative) chiral charge, and white arrows indicate the\nFermi arcs emerging from the Weyl nodes.\nchiral charge +2 and has quadratic dispersion on the kx-\nkyplane,6in sharp contrast to the double Weyl fermion\nwith fourfold degeneracy and linear dispersion1. Its other\nparity partner has opposite chiral charge of -2. Based on\nthe above analysis, the e\u000bective k\u0001pmodel in the basis\nof theC4zeigenstates,fje\u0000i\u0019\n4i;jei3\u0019\n4ig, up to order of k2\nfor theC4z-protected Weyl nodes can be straightforward\nadapted from Ref. 6, and reads\nHWP= (m1\u0000m2k2\nz)s3+(ak2\n++bk2\n\u0000)s++(ak2\n\u0000+bk2\n+)s\u0000;\n(3)\nwherekis around the \u0000 point, k\u0006=kx\u0006iky,s\u0006=\ns1\u0006is2,jaj6=jbj,m1m2>0 and the Weyl node positions\nare atkz=\u0006q\nm1\nm2. Interestingly, if a=b,HWPreduces\ntoHNLin Eq. (2), implying that the gap opening of the\nnodal lines would close. Fig. 3(b) displays the two Fermi\narcs on the (100) surface emerging from the two charge-2\nWeyl nodes.\nEvolution of Weyl Fermions in NonCollinear\nFerrimagnetism|\nNeutron scattering experiments have reported34a non-\ncollinear ferrimagnetic magnetic structure in the DO 22\nferrimagnetic Mn 3Ga structure, where there is a signi\f-\ncant in-plane magnetic moment, \u00162b\nx= 1.19\u0016Bcarried by\nthe MnIatoms [on the Mn-Ga (001) plane] leading to a\n21\u000etilt of the Mn Imoment from the crystallographic c\naxis [see Fig. 4(a)]. This noncollinear magnetic ordering\nspontaneously breaks both the C4zandMzsymmetry op-\nerations while only preserving P. Consequently, the C4z-\nprotected double Weyl fermion on the kzaxis for the case\nof collinear ferrimagntism splits into two charge-1 Weyl\nfermions which shift away from the kzaxis.\nIn order to investigate this scenario, we have stud-\nied the evolution of the Weyl points upon rotation of\nall magnetic moments of the Mn Iatoms at the Wyck-\no\u000b positions 2b with respect to the crystallographic z\naxis by the angle \u0012,\u00162b=\u00162b(\u0000sin\u0012^x+ cos\u0012^z), while\nkz(Å-1)\nΓ+2\n-2 -1\n-1-1 +1\n+1 \n+1+1\n-1\nMnIGa\nMnII (a) (b)\nM\nMz\nx θ\nμ2bFIG. 4: (Color online) (a) Noncollinear ferrimagnetic DO 22\nstructure of Mn 3Ga,34where the Mn Iatoms [on the Mn-Ga\n(001) plane] carry a substantial in-plane magnetic moment\nleading to a tilt of their moments from the crystallographic c\naxis. (b) Evolution of Weyl nodes in the 3D BZ as a function\nof tilt angle \u0012, where the red, green and blue circles denote the\nWeyl nodes at \u0012= 180 °, 170 °and 160 °,respectively. Dashed\narrows show the motion of Weyl points with decreasing \u0012. At\n\u0012=180 °, (collinear case) the two charge-2 Weyl nodes lie on\ntheC4z-protectedkz-axis. For\u00126=180 °each charge-2 Weyl\nnode splits into two charge-1 Weyl nodes which in turn move\naway from the kz-axis. The integer above each Weyl node\ndenotes the chiral charge.\n\fxing the direction of the Mn IImagnetic moments, as\nshown in Fig. 4(a). Here, \u0012= 180 °indicates the collinear\n(001) ferrimagnetism. Using the Wannier functions we\n\fnd that at \u0012= 160 °the magnitude of the calculated\nx-component of the magnetic moment of the Mn Iatoms\nis 0.94\u0016B/Mn in good agreement with the correspond-\ning experimental values of 1.19 \u0016B. Table III summarizes\nthe comparison of the values of the magnetic moments\nof the Mn Iatom between theory and experiment. The\nmagnetic moments from the rotated Wannier approach\nagree with DFT calculations well. Fig. 4(b) shows the5\nTABLE III: Comparison of the values of the magnetic mo-\nments for the Mn Iatoms for the noncollinear case from the-\nory and experiment. \u00162b\nz(\u00162b\nx) denotes the z(x)-component\nof the magnetic moment of Mn I.\nMethod \u00162b\nz(\u0016B/Mn)\u00162b\nx(\u0016B/Mn)\u0012(°)\nrotated Wannier -2.60 -0.94 160\nGGA+SOC -2.65 -0.95 160\nexperiment34-3.07 1.19 159\nevolution of the Weyl nodes as \u0012changes from 180 °to\n170 °and \fnally to 160 °. Initially, at \u0012= 180 °, the two\ncharge-2 Weyl nodes lie on the kz-axis. As\u0012decreases\neach charge-2 Weyl node splits into two charge-1 Weyl\nnodes which move away from the kz-axis, leading to the\nemergence of four charge-1 Weyl fermions in the case\nof noncollinear ferrimagnetism. Our electronic structure\ncalculations of the Fermi arcs on the (100) surface for \u0012\n= 160 °show that the noncollinear e\u000bect is small on the\nFermi arcs in Fig. 3(b), at least for small angle.\nIV. DISCUSSION\nIn this section we discuss the e\u000bect of electron-electron\ninteractions, U, on the equilibrium lattice constants,\nmagnetic moments, and the topology of the band struc-\nture for the collinear magnetic structure. As was alluded\nearlier we employed U=2.6 eV for Mn Iatoms and U=0\nfor the remaining atoms, which were found38to give a\nvalue for the alattice constant of 3.91 \u0017A in good agree-\nment with the experimental value. However, as shown in\nTable I, this is at the expense of a worse agreement for\nboth the lattice parameter cand thez-component of the\nmagnetic moment of the Mn2b(MnI) atoms. As shown\nin Fig. 2(a) in the absence of SOC the presence of U\nshifts the position of the Weyl nodes (red dots) to higher\nenergies and slightly towards the center of BZ. Moreover,\neven in the presence of SOC, the nodes are robust and re-\nmain gapless at about the same energies. Therefore, the\ncharge-2 Weyl nodes survive in the presence of electronic\ncorrelations. Moreover, since the e\u000bect of electronic cor-\nrelations can not gap out the charge-2 Weyl nodes, the\ncharge-1 Weyl nodes splitting of the charge-2 Weyl nodesshould be robust in the noncollinear magnetic structure\nas well.\nDue to the large shift of the Weyl nodes to higher en-\nergies induced by U, it will be challenging to observe the\nFermi arcs above the Fermi level in Fig. 3(b) employing\nangle-resolved photoemission spectroscopy (ARPES). On\nthe other hand, the time-resolved ARPES (trARPES),\na fast-growing and powerful technique to observe con-\nduction electron states up to hundreds meV above the\nFermi level47,48, may be a suitable platform to observe\nthe Fermi arcs above the Fermi level on the (100) sur-\nface in future experiments. Moreover, electron doping or\nalloying that preserves the C4zsymmetry, e.g., Mn 3Ge\nin the cubic structure18, can rise the chemical potential\nwhich may allow in turn the observation of these non-\ntrivial surface states emerging from the Weyl nodes.\nV. CONCLUSION\nIn summary, our ab initio electronic structure calcula-\ntions have shown that in the absence of SOC, nontrivial\nnodal lines emerge in collinear ferrimagnetic tetragonal\nMn3Ga. The nodal lines are protected by both mirror re-\n\rection symmetry normal to the [110] direction, M[110],\nand a four-fold rotational symmetry, C4z. 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Richter4\n1Institute for Problems of Materials Science NASU, Krzhizhanovskogo 3, 03180 Kiev, Ukraine\n2Donostia International Physics Center (DIPC), ES-20018 Donostia-SanSebastian, Spain\n3Institute of Physics, AS CR, Cukrovarnicka 10, 16253 Prague, Czech Republic\n4Institut f ur Theoretische Physik, Otto-von-Guericke-Universit at Magdeburg,\nPF 4120, D - 39016 Magdeburg, Germany\n(Dated: 16.01.14)\nWe show that a superstructure of antiferromagnetically interacting Fe3+(S= 5=2) ions in double\nperovskites AFe 1=2M1=2O3exhibits a ferrimagnetic ordering below Tfe\u00195:6J1(J1=kB\u001850 K),\nwhich is close to room temperature. Small clusters of the same structure exhibit a superparamagnetic\nbehavior at T.Tfe. The possibility of formation of such clusters explains the room-temperature\n(superpara)magnetism in 3 d-metal based oxides.\nPACS numbers: 75.10.-b, 75.20.-g, 75.50.Gg, 75.50.Lk, 75.85.+t\nI. INTRODUCTION\nAn experimental quest to \fnd a room-temperature\nmultiferroic with high magnetoelectric coupling is stim-\nulated by wide prospects they open for applications in\nthe \feld of information and energy-saving technologies.\nThey may form the basis for a fabrication of novel func-\ntional devices: highly sensitive magnetic sensors, ca-\npacitance electromagnets, elements of magnetic memory\nswitched by electric \feld, nonreciprocal microwave \flters,\nand others.1,2Spintronics, an emerging branch of micro-\nand nanoelectronics which manipulates the electron spin\nrather than its charge, has need for a room-temperature\nferromagnetic semiconductor.3\nThe rich family of Fe-based double perovskites\nAFe 1=2M1=2O3=A2FeMO 6(with non-magnetic ions\nA=Pb,Ca,Sr,Ba, and M=Nb,Ta,Sb) is in the focus of\nthe studies as it includes PbFe 1=2Nb1=2O3(PFN) and\nPbFe 1=2Ta1=2O3(PFT) systems, where the multiferroic-\nity was reported more then \ffty years ago.4,5\nIn AFe 1=2M1=2O3compositions, Fe3+and M5+cation\npositions may be ordered or disordered within simple cu-\nbic B-sublattice of perovskite structure ABO 3. The de-\ngree of chemical ordering depends on the strength of elec-\ntrostatic and elastic energies and, in particular, on the\nionic radii of these cations. It is commonly accepted that\nPFN and PFT are chemically disordered compounds due\nto almost equal ionic radii of Fe3+and Nb5+or Ta5+,6\nwhile Sb-contained compounds can be chemically ordered\nup to 90% because Sb5+is much larger than Fe3+.7Mag-\nnetism of the compositions is due to Fe3+,S= 5=2 ions\nthat occupy half of octahedral sites of the perovskite lat-\ntice. The magnetic moments of the Fe3+ions interact\nvia various superexchange paths,\n^H=1\n2X\nR;rJr^SR^SR+r: (1)\nThe disorder prevents an experimental access to the val-\nues of the interactions. In a recent publication, some ofus have argued that the largest superexchange values are\nthe nearest-neighbor (NN) Fe-Fe interaction (Fe ions are\nseparated by the edge of perovskite unit cell and inter-\nact via the shortest Fe-O-Fe path) J1\u001850\u000070 K and\nthe next-nearest-neighbor interaction (Fe ions are sep-\narated by the face diagonal of the cell) J2'0:04J1.8\nThe interaction values J1,J2are similar to the values in\northoferrite RFeO 3(R=Y or a rare earth)9{13and bis-\nmuth ferrite BiFeO 314compounds. Note that both ex-\nchange couplings have antiferromagnetic sign. We thus\nhave two substantially di\u000berent magnetic energy scales:\nS(S+ 1)J1= 8:75J1, which corresponds to temperatures\nof several hundred Kelvins, and S(S+ 1)J2=kB\u001820 K.\nNote that many of Fe-based double perovskites have an\nantiferromagnetic phase transition in the latter temper-\nature range.15{20It means that the probability to \fnd\na pair of Fe ions separated by the face diagonal of the\nperovskite cell is much higher than to \fnd a nearest-\nneighbor Fe pair that is caused by partial chemical or-\ndering of cations. Two multiferroic compounds, PFN\nand PFT, exhibit a magnetic transition at TN\u0018150 K.\nThis means that the probability to \fnd a pair of NN Fe\nions is enhanced in these compounds. But it leads to the\nincrease of the temperature, at which the antiferromag-\nnetic order is established.21{23For instance, in the more\nconcentrated compound PbFe 2=3W1=3O3it increases up\nto 380 K.24\nRecent reports on room-temperature multiferroicity of\nPFT/lead zirconate titanate (PZT)25,26and PFN/PZT27\nand [Pb(Fe 2=3W1=3)O3]/PZT28solid solution systems\nare a real challenge for the solid state theory. One of the\nquestions is the nature of large room-temperature mag-\nnetic response of the systems (non-linear magnetization\ncurves and hysteresis loops) that imply the existence of\nFe spins alignment in a part of the sample with uncom-\npensated magnetic moment. On the qualitative level, it\nwas suggested that the clustering of Fe ions is responsi-\nble for the appearence of the uncompensated magnetic\nmoment.25{29We should mention that the clustering ofarXiv:1310.8079v2  [cond-mat.mtrl-sci]  15 May 20142\n 0 1 2 3 4 5 6 7 8\n 0  0.5  1  1.5  2χ-1(gµB)2\nkBT/J1S(S+1)a\n 0 1 2 3 4 5 6 7 8\n 0  0.5  1  1.5  2χ-1(gµB)2\n kBT/J1S(S+1) b\nFIG. 1. (Color online) a: Inverse subsceptibility \u001f\u00001for a periodic arrangement of PFB2 chemical order with two inequivalent\nS= 5=2 Fe3+ion positions (red solid line - [4,4] Pad\u0013 e approximant of the 8th order HTE series). The susceptibility exeeds\nthe Curie-Weiss (CW) asymptotic (green dashed line) and diverges at Tfe\u00190:640J1S(S+ 1) (shown by the vertical line)\ncorresponding to a transition into a ferrimagnetic phase. The black thin solid line shows the susceptibility of 1:1 ordered\nPFB0 con\fguration, where Fe spins interact with J2= 0:05J1. Inset: Unit cell of the PFB2 chemical ordering, only Fe\n(open circles) and M (\flled circles) cations are shown. Fe1(Fe2) positions are depicted by up(down) arrows, respectively. b:\nInverse subsceptibility \u001f\u00001(red solid line) for a small cluster of a PFB2 con\fguration (as shown in the inset) obtained by full\nexact diagonalization. The susceptibility shows a crossover between CW (dashed green) and superparamagnetic (dashed black)\nbehavior. In both parts, the blue dotted line shows the Curie law for independent spins, \u001f\u00001\np/T.\nFe ions30,31forms locally fragments of AFeO 3structure,\nwhere Fe spins form the simple cubic lattice. Thus, it can\nlead only to G-type antiferromagnetic ordering within the\nfragments, and produces a small or vanishing uncompen-\nsated magnetic moment. It can not convincingly explain\nthe observation of room-temperature hysteresis loops.\nA small canting of predominantly antiferromagnetic Fe\nspins due to the antisymmetric Dzyaloshinskii-Moriya in-\nteraction ^HDM=D\u0001[S1\u0002S2] causes weak ferromag-\nnetism in ortoferrites RFeO 3, R3+being Y or a rare\nearth ion. It was suggested that the canting may cause\nalso the uncompensated magnetic moment in AFeO 3\nstructure that is formed by the Fe ions clustering in\nthe double perovskites.25,32But the moment seems to\nbe too small to explain the e\u000bect.33,34In the ordered\nstate of RFeO 3, the canting angle \u001e\u001810 mrad results\nin the moment \u001b\u00180:05\u0016Bper Fe ion.35,36But such\na moment was never observed in the antiferromagneti-\ncally ordered state of PFN neither in magnetic18,20nor\nin neutron21{23studies. A possible reason is that the\nDzyaloshinskii-Moriya vector for a Fe-O-Fe bond may\nbe written as33,34D=d[r1\u0002r2], wheredis a scalar\nvalue, and riis a unit vector in the direction from oxy-\ngen to spin Si. Thus, its value depends on the Fe-O-Fe\nbond angle D/sin\u0012, which is substantially larger inAFe 1=2M1=2O3(170\u000e<\u0012 < 180\u000e)7,22than in the ortho-\nferrites (140 <\u0012< 157)37.\nIn this paper, we quantitatively consider another\nscenario for the room-temperature magnetism of bulk\nPFT/PZT and PFN/PZT systems25{27and superpara-\nmagnetism often observed in PFN nanoparticles or even\nceramics and thin \flms.29,38,39We explain it by the\nexistence of regions with a special chemical order (a\nsub-nano-size superstructure) that results in a ferrimag-\nnetic ordering of antiferromagnetically interacting Fe3+\nS= 5=2 spins. This explanation was implicitly as-\nsumed in Ref. 29, where the observed slightly asymmet-\nric EPR line shapes above room temperature were simu-\nlated by a model involving the presence of thermally \ruc-\ntuating superparamagneticlike nanoclusters. Note that\nour explanation does not demand the clusterization, as\nthe stoichiometry AFe 1=2M1=2O3is retained within the\n2\u00022\u00022 supercell of the superstructure. Using the\nhigh-temperature expansion (HTE),40,41we show that a\nmacroscopic number of spins orders at about the room\ntemperature, whereas small clusters (studied by exact di-\nagonalization method) exhibit a crossover between para-\nmagnetic and superparamagnetic behavior.3\na)\nb)\ncJc)\n 0 0.5 1 1.5 2\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7χ-1(gµB)2\n kBT/J1S(S+1) d)\na\nb\nc1c2\np\nFIG. 2. (Color online) ( a,b) The fragments of PFB2 con\fgu-\nration. ( c) The model simulating two interacting clusters of\nPFB2 con\fguration. The coupling strength J1(Jc) is denoted\nby black solid (red dashed) lines. Arrows indicate spin-spin\ncorrelations in the ground state of each cluster. ( d) The tem-\nperature dependence of the inverse susceptibility of all clusters\n(full exact diagonalization data for S= 1=2). The vertical\nline showsTfe. AtT.Tfethe susceptibility of the clusters\nexceeds the susceptibility of independent spins (line p) down\nto the lowest temperatures. The lines aandbcorrespond to\nclusters with 7 spins aand 13 spins brespectively. The lines\nc1andc2correspond to 14-spin cluster cwithJc= 0:25J1\nand 0:5J1respectively.\nII. METHODS\nWe use the method and the program packages pre-\nsented earlier in the Refs. 40, and 41 for the eighth-\nand tenth-order high-temperature expansion (HTE) of\nthe magnetic susceptibility \u001ffor a general Heisenberg\nmodel with up to four di\u000berent exchange parameters\nJ1;J2;J3;J4. The input for the HTE package is the de\f-\nnition \fle where all bonds within a cluster or a L\u0002L\u0002L\n(L=16,20) super-cell of a periodic Heisenberg lattice are\nenumerated with the indication of a corresponding valueof the exchange interaction. We use an originally devel-\noped C++ program, for the generation of the de\fnition\n\fles for spin structures studied in this work.\nIn order to simulate the behavior of fragments of PFB2\ncon\fguration in the Fe-based double perovskite mate-\nrial, we have performed full exact diagonalization studies\n(ED) of thermodynamic properties of clusters shown in\nFig. 1b, and in Fig. 2 using J. Schulenburg's spinpack .\nThe susceptibility \u001f(T) is calculated as the ratio of the\ninduced magnetization Mto the \feld H. We use the\n\"vanishing\" magnetic \feld H= 10\u00005J1=g\u0016Bunless oth-\nerwise noted.\nIII. RESULT AND DISCUSSION\nA. Ferrimagnetic superstructure\nThe simplest way to model the (partial) disorder in the\ndistribution of Fe and M ions between the sites of the\nB-sublattice of the perovskite structure is to consider a\nperiodic lattice with a supercell containing several per-\novskite cells and study such periodic systems with di\u000ber-\nent versions of chemical order (ion distributions). Such\nan approach was suggested in Ref. 31 for a 2 \u00022\u00022 su-\npercell, where 6 con\fgurations PFB0. . . PFB5 (see Fig. 3\nof Ref. 31, and Fig. 2 of Ref. 8) of chemical ordering are\npossible in the double perovskites. It was shown that the\ntotal energy is substantially di\u000berent for di\u000berent con\fg-\nurations. Moreover, the hierarchy of the energies depends\non the type of M-ion. In Ref. 8, it was found that the\nPFB2 con\fguration shown in the inset of Fig. 1a has an\nenergy close to the most stable con\fgurations (PFB5 for\nM=Nb,Ta and PFB0 for M=Sb), and has a ferrimag-\nnetic ground state (see Table II of Ref. 8). Below, we\nconsider the ferrimagnetism of PFB2 superstructure in\nmore detail.\nThe PFB2 chemical order has two inequivalent Fe\nsites. Within the B-sublattice of the perovskite struc-\nture, Fe1 has six Fe2 NN ions, whereas three Fe2 sites\nin the supercell has only two Fe1 NN ions (insets in Fig.\n1a,b). In other words, Fe2 sites form a superstructure\nof corner-shared octahedra, Fe1 sites being in the cen-\nter of each octahedron. The interaction value between\nthe two sublattices is J1, and within Fe2 sublattice is\nJ2\u001cJ1. Thus, the spin system satis\fes the require-\nments of the Lieb-Mattis theorem42withg2\nLM=J2=4\n(see Eq.(2) of the Ref. 42). Moreover, it is close to the\nspecial case g2\nLM= 0. According to the theorem (see\nalso the consideration of frustration J26= 0 in the Ref.\n43), the PFB2 ground state corresponds to a ferrimag-\nnetic ordering of Fe spins with a magnetic moment of\n2g\u0016BS\u001910\u0016Bper supercell, or 2 :5\u0016Bper Fe ion. This\nmoment value is much larger than the value provided by\nDzyaloshinskii-Moriya interaction for realistic values of\nlocal lattice distortions.33,34,36\nWe use the [4,4] Pad\u0013 e approximant of the HTE se-\nries to analyse the susceptibilty data.40For a magnetic4\nsuperstructure with the PFB2 spin arrangement the tem-\nperature dependence of the inverse susceptibility \u001f\u00001(T)\nforS= 5=2 is shown in the Fig. 1a. Only NN interaction\nJ16= 0 was taken into account. A reasonable estimate\nof the temperature for the transition into the ferrimag-\nnetically ordered phase Tfeis given by that point where\n\u001f\u00001(Tfe) = 0. The precision of the determination of crit-\nical temperatures by the zero of \u001f\u00001was estimated to be\nabout 10%.41The values of Tfefor di\u000berent spin values\nare given in the Table I.\nTABLE I. The ferrimagnetic transition temperature for\nPFB2 con\fguration, obtained from the [4,4] Pad\u0013 e approxi-\nmant of the 8th order HTE series.\nSpin,S k BTfe=J1S(S+ 1) kBTfe=J1\n1/2 0.61 0.46\n1 0.69 1.4\n3/2 0.64 2.4\n2 0.64 3.8\n5/2 0.64 5.6\n 0 0.5 1 1.5 2 2.5\n 0  0.2  0.4  0.6  0.8  1χ-1=H/M\n kBT/JS(S+1) S=0.5\nS=1.0\nS=1.5\nS=2.0\nS=2.5\nh=10-5J, S=2.5\nFIG. 3. (Color online) The inverse susceptibility \u001f\u00001(T) =\nH=M for the cluster shown in the Fig. 2a and the \feld\nH= 3J1=[(5S+ 1)g\u0016B] for di\u000berent spin values. For com-\nparison, an S= 5=2 curve for a vanishing \feld, i.e. \u001f\u00001\u0019\n(@M=@H )\u00001(H= 0), is given by the black solid line.\nFor Fe-based double perovskites Tfeis of the order\nof the room temperature, as J1=kB\u001850 K. From\nthe graph shown in Fig. 1a we see that in the range\nTfe< T < T\u0003\u00190:92J1S(S+ 1)=kB, the magnetic\nsusceptibility of the PFB2 phase exceeds the value for\nindependent spins, \u001f(T)> \u001f p(T) =S(S+ 1)=(3kBT),\ndespite the antiferromagnetic character of the exchange\ninteraction, which suppress the magnetic response at high\nFIG. 4. (Color online) Examples of ferrimagnetic superstruc-\ntures that may be formed by magnetic impurities (arrows)\nsubstituting for cations (blue circles) in zinc blend (left) and\nwurtzit (right) lattices.\n 0 10 20 30 40 50\n 0  2  4  6  8 10χ-1(gµB)2\nkBT/J2S(S+1)\nFIG. 5. Main panel: The inverse magnetic susceptibility (per\nspin)\u001f\u00001(red solid line - [4,4] Pad\u0013 e approximant of the 8th\norder HTE series) for the ideal 1:1 chemical order (PFB0,\nshown in the insert). It shows a minimum at T\u0018TIidicated\nby the arrow. The Curie-Weiss asymptotic is shown by the\ngreen dotted line. Black dotted line shows the bare HTE se-\nries. Inset: The super-cell for the PFB0, only M5+(closed\ncircles) and Fe3+ions (open circles) are shown. Arrows indi-\ncate the distribution of spins in the I-type ordering.\ntemperatures T\u001dJ1. For comparison, the black thin\nsolid line shows the susceptibility \u001ffcc(T) of 1:1 ordered\nPFB0 con\fguration, where Fe spins form a face centered\ncubic lattice, and interact with J2= 0:05J1. We see that\n\u001ffcc(T)<\u001f p(T) at all temperatures (see Appendix A).\nB. Superparamagnetism\nA sample of a disordered double perovskite compound\nmay contain some regions with PFB2 chemical order. In\nthe ground state, such a region possesses the total spin\nSg= (N2\u0000N1)S, whereN1,N2are the numbers of Fe1,\nand Fe2 sites in that region.42In order to simulate the be-\nhavior of fragments of PFB2 con\fguration in a Fe-based\ndouble perovskite material, we show in Figs. 1b, 2 full5\n 0 5 10 15 20 25\n 0  1  2  3  4  5  6χ-1(gµB)2\nkBT/JS(S+1)S=1.0\nS=1.5\nS=2.0\nS=2.5\nS=0.5\n 0 0.2 0.4 0.6 0.8 1\n 0.6  0.62  0.64  0.66  0.68  0.7χ-1(gµB)2\nkBT/JS(S+1)S=1.0\nS=1.5\nS=2.0\nS=2.5\nS=0.5\nFIG. 6. Temperature dependence of inverse magnetic suscep-\ntibility per spin for PFB2 chemical order for systems with\ndi\u000berent spin values S. [4,4] Pad\u0013 e approximant of the 8th\norder HTE series are shown for S > 0:5, and [4,6] Pad\u0013 e ap-\nproximant of the 10th order HTE series for S= 0:5\nexact-diagonalization data of thermodynamic properties\nof clusters shown in Figs. 1b and 2(a-c). Since we have\nfound that the dependence of the inverse susceptibility as\na function of normalized temperature kBT=J 1S(S+1) on\nthe spin value Sis weak (see Appendix A), the ED data\nfor the simplest S= 1=2 case can be considered as repre-\nsantative for higher values of S. The 7-site cluster shown\nin the Fig. 2a contains one Fe1 site interacting with six\nFe2 sites via J1exchange. This is a particular case of the\nHeisenberg star model.44,45ForT\u001dJ1S(S+ 1)=kBthe\nsusceptibility per spin tends to the Curie-Weiss asymp-\ntotic\u001fCW=\u001fp=[1+4S(S+1)J1=7kBT]. In the opposite\nlimit, the system shows a super-paramagnetic behavior,46\ni.e. it behaves as a single super-spin Sg= 5S, and the\nsusceptibility is \u001fSPM=Sg(Sg+1)\u001fp=[(N2+N1)S(S+1)]\n(see Fig. 1b). At temperatures T\u0018Tfethe system ex-\nhibits a crossover between the two regimes. The suscep-\ntibility exceeds the independent-spin value for T <T\u0003\n1\u00190:74J1S(S+ 1)=kB. Similar results for a 13-site cluster\n(Fig. 2b) are shown in Fig. 2d (see also Fig. 8 in the\nAppendix).\nIn a real sample, an interaction between the regions of\nPFB2 con\fgurations always exists. When the tempera-\nture becomes su\u000eciently low, the thermal and interac-\ntion energies become comparable, and a collective state\nof super-spins is formed. The behavior of two interact-\ning PFB2 clusters (Fig. 2c) is shown in Fig. 2d (lines\nc1andc2). For temperatures T\u001dJcthe susceptibility\nbehaves similar to the non-interacting case. In partic-\nular, it exceeds the susceptibility of independent spins\n\u001f(T)> \u001f p(T) atT.Tfeand tends to the superpara-\nmagnetic behavior down to low temperature, where it\nexhibits a maximum (minimum at \u001f\u00001(T) curve). Below\nthe maximum, a singlet ground state of two interact-\ning super-spins is formed. In reality, for large number\nof interacting clusters the disorder in the system favors\na super-spin glass formation18,23,46at temperatures gov-\nerned by the low energy scale T <S (S+ 1)J2=kB.\nA characteristic feature of large spin formation in a sys-\ntem is a non-linearity of its magnetization curve M(H),\nwhich results in the dependence of the susceptibility\n\u001f=M=H on the \feld value. The Fig. 3 shows the\n\u001f\u00001(T) for the 7-site cluster and a \fnite value of the\nmagnetic \feld. The susceptibility substantially deviates\nfrom the \"theoretical\" value \u001fth=@M=@H (H= 0) at\nlow temperatures, and it does not diverge at T!0.\nNote that we have considered here only isotropic Heisen-\nberg interactions. The magnetic anisotropy, which is al-\nways present in real compounds20would transform the\nnon-linear magnetization curves into narrow hysteresis\nloops.46\nThe model of ferrimagnetism considered here can also\nbe applied to PFN and PFT diluted by non-magnetic\nTi and Zr ions.25{28As it was mentioned above, these\n 0 2 4 6 8 10 12\n 0 0.5  1 1.5  2 2.5  3χ-1(gµB)2\n kBT/JS(S+1) S=0.5\nS=1.0\nS=1.5\nS=2.0\nS=2.5\nFIG. 7. Temperature dependence of inverse magnetic suscep-\ntibility per spin for the cluster shown in Fig. 2a and various\nspin values. Dotted line shows Curie-Weiss asymptotic.6\n 0 2 4 6 8 10 12\n 0 0.5  1 1.5  2 2.5  3χ-1(gµB)2\nkBT/JS(S+1)ED\nT>>J\nT<<J\nFIG. 8. Temperature dependence of inverse magnetic sus-\nceptibility per spin for the cluster shown in Fig. 2b, S= 0:5.\nDotted line shows the Curie-Weiss asymptotic for T\u001dJ, dot-\ndashed line shows superparamagnetic behavior with a super-\nspinSg= 9SforT\u001cJ.\nsystems show a sizable magnetic moment at room tem-\nperature in spite that the concentration of Fe ions was\ndecreased up to 10%. From a general point of view, the\nmagnetic dilution will lead to the breaking of the in\fnite\nmagnetic percolation clusters responsible for the long-\nrange antiferromagnetic order as it was pointed out in\nRef. 20. In a small magnetic cluster the probability of\ncreation of a ferrimagnetic con\fguration of spins should\nbe enhanced due to the limited number of interacting\nspins. Moreover, the ferrimagnetic ordering can be real-\nized on the edge of the (semi-)in\fnite antiferromagnetic\ncluster which size is of order of a few nanometers only be-\ncause it is controlled by local \ructuations of the 1:1 com-\nposition between magnetic Fe and non-magnetic ions.18\nObviously, the \"edge\" e\u000bect becomes substantial with Ti\nand Zr doping.\nA large room-temperature magnetic response is re-\nported in many wide-gap diluted magnetic semiconduc-\ntors, such as GaN, ZnO, and TiO 2isovalently doped by\ntransition metals,47{49\"in which no ferromagnetism was\nexpected at any temperature\".47We think that a solu-\ntion of this puzzle may be a formation of ferrimagnetic\nsuperstructure clusters similar to that we have considered\nhere. Fig. 4 shows two examples of a planar ferrimagnetic\narrangement that may be formed by magnetic impuri-\nties substituting for cations in zinc blend and wurtzit\nsemiconductors. We see that these arrangements may\nform in\fnite two-dimensional sublattices, they also may\nbe transformed into tree-dimensional superstructures if\nthey will be connected by bridging spins having antifer-\nromagnetic interactions with both planes, the planes then\nwill be ordered parallel.IV. CONCLUSIONS.\nIn summary, we have studied an example of a system\nofantiferromagnetically interacting equal spins relevant\nfor Fe-based double perovskite compounds, and having\naferrimagnetic ground state. We have estimated the\ntransition temperature Tfefor a macroscopic system,\nand have argued that it can be close to room temper-\nature. Such kind of ferrimagnetism may be the origin\nof the room-temperature magnetism of PFT/PZT and\nPFN/PZT systems. For small clusters of the same struc-\nture we have shown that their magnetic susceptibility\nexceeds the susceptibility of independent spins at tem-\nperaturesT.Tfe. This gives a possible microscopic\nexplanation for the still puzzling monotonous increase of\nthe magnetic susceptibility with decreasing temperature\nbelow N\u0013 eel temperature, which is observed practically in\nall Fe-based double perovskites. The ferrimagnetism of\nthis kind may be responsible also for numerous observa-\ntions of an unexpected large room-temperature magnetic\nresponse in 3 d-metal based oxides.\nACKNOWLEDGMENTS\nThe authors thank M. D. Kuz'min and M. Mary\u0014 sko for\nvery useful discussions. The projects GACR 13-11473S,\nand NASc of Ukraine 07-02-14 are acknowledged. The\nexact diagonalization calculations were performed using\nJ. Schulenburg's spinpack .\nAppendix A: Details of numerical calculations\nIn the ideal 1:1 chemical order, Fe3+and the non-\nmagnetic M5+ions alternate in the B position of the\nperovskite lattice ABO 3. In this con\fguration (called\nPFB0 in Ref. 31), magnetic Fe3+ions form regular face\ncentered cubic sublattice with antiferromagnetic inter-\nactionJ2between nearest spins in the sublattice. The\nHTE results for the PFB0 lattice are shown in the Fig. 5.\nFor such a lattice, a transition into so called I-type an-\ntiferromagnetic order (see insert of Fig. 5) occurs at\nTI\u0019\u0000\u0002CW;0=5:76\u00190:69S(S+ 1)J2=kB50, \u0002CW;0=\n\u00004S(S+ 1)J2=kBbeing the paramagnetic Curie-Weiss\ntemperature. Note that the whole curve \u001f\u00001(T) lies\nabove the Curie-Weiss asymptotic (CW). This is a \"nor-\nmal\" behavior when the antiferromagnetic interactions\nsuppress the magnetic response of a spin system.\nFor the PFB2 spin arrangement, the temperature de-\npendence of the susceptibility for di\u000berent spin values is\nshown in Fig. 6. For S= 1=2-plot the tenth-order HTE41\nwas used. 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Chubykalo-Fesenko2\n1Department of Physics, University of York, Heslington, York YO10 5DD United Kingdom and\n2Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n(Dated: May 31, 2022)\nUltrafast laser-induced magnetic switching in rare earth-transition metal ferrimagnetic alloys has\nrecently been reported to occur by ultrafast heating alone. Using atomistic simulations and a\nferrimagnetic Landau-Lifshitz-Bloch formalism, we demonstrate that for switching to occur it is\nnecessary that angular momentum is transferred from the longitudinal to transverse magnetization\ncomponents in the transition metal. This dynamical path leads to the transfer of the angular\nmomentum to the rare earth metal and magnetization switching with subsequent ultrafast precession\ncaused by the inter-sublattice exchange \feld on the atomic scale.\nThe behavior of magnetization dynamics triggered by\nan ultrafast laser stimulus is a topic of intense research\ninterest in both fundamental and applied magnetism [1].\nA range of studies using ultrafast laser pulses have shown\nvery di\u000berent timescales of demagnetization for di\u000berent\nmaterials; from 100 fs in Ni [2] to 100 ps in Gd [3]. Any\npotential applications utilizing such a mechanism would\nrequire, not only ultrafast demagnetization, but also con-\ntrolled magnetization switching.\nMagnetization reversal induced by an ultrafast laser\npulse has been reported in the ferrimagnet GdFeCo, to-\ngether with a rich variety of phenomena [4{8]. Several\nhypotheses have been put forward to explain the ob-\nserved magnetization switching: crossing of the angu-\nlar momentum compensation point [4], the Inverse Fara-\nday E\u000bect [5], and its combination with ultrafast heat-\ning [6]. It has been shown that the rare earth (RE)\nresponds more slowly to the laser pulse than the tran-\nsition metal (TM) [7], even though the sublattices are\nstrongly exchange coupled. Intriguingly, Radu et al. [7]\nshow experimentally and theoretically the existence of a\ntransient ferromagnetic-like state, whereby the two sub-\nlattices align against their exchange interaction, existing\nfor a few hundred femtoseconds. Recently [8], the atom-\nistic model outlined in [7, 8] predicted the phenomenon\nof magnetization reversal induced by heat alone, in the\nabsence of any external \feld; a prediction veri\fed experi-\nmentally. This remarkable result opens many interesting\npossibilities in terms of ultrafast magnetization reversal\nand potential areas of practical exploitation, however a\ncomplete theoretical understanding of this e\u000bect is cur-\nrently missing.\nIn magnets consisting of more than one magnetic\nspecies, excitation of the spins on a time scale compa-\nrable with that of the inter-sublattice exchange takes the\nsublattices out of equilibrium with each other. It is in\nthis regime where the thermally driven switching of fer-\nrimagnetic GdFeCo occurs. A recent study by Mentink\net al. [10] proposed an explanation of the process using a\nphenomenological model of the magnetization dynamics,\nwhich assumes the additive character of two relaxation\nmechanisms: one governed by the inter-sublattice ex-change and another by the relativistic contribution (cou-\npling to external degrees of freedom). The model is based\non the physically plausible argument that the switching\nis driven by angular momentum transfer in the exchange-\ndominated regime. However, the assumption of a linear\npath to reversal allows the angular momentum transfer\nto occur through longitudinal components only, since the\nperpendicular components are neglected. Additionally,\nthe dynamical equation in Ref. [10] was derived from\nthe Onsager principle, generally valid for small devia-\ntions from the equilibrium only. Thus far, a complete\nexplanation of the heat driven, ultrafast reversal process\nremains illusive.\nIn this Letter we demonstrate that the switching of\nmagnetization in a ferrimagnet after femtosecond heat-\ning is due to the transfer of angular momentum from the\nlongitudinal to the transverse magnetization components\nin the TM and consequent transfer of the angular mo-\nmentum through perpendicular components to the RE.\nWe present a general formalism, leading to a macroscopic\ndynamical equation for a ferrimagnet. This is in the form\nof a Landau-Lifshitz-Bloch (LLB) equation, in which, un-\nlike the phenomenological model of Ref. 10, the two re-\nlaxation mechanisms are not additive. Our theory gives\nthe non-equilibrium conditions necessary for this angu-\nlar momentum transfer to happen and thus to produce\nthe precessional rather than linear reversal as suggested\nin Ref. 10. These predictions are supported by calcula-\ntions using an atomistic model based on the Heisenberg\nexchange Hamiltonian with Langevin dynamics.\nIn the absence of any external stimulus, the energet-\nics of the atomistic spin model are described purely by\nexchange interactions, given by the spin Hamiltonian:\nH=\u0000X\nj<iJijSi\u0001Sj (1)\nwhereJijis the exchange integral between spins iandj\n(i;jare lattice sites), and where jruns over \frst nearest\nneighbors only, Siis the normalized magnetic moment\njSij= 1. We model the magnetization dynamics of the\nsystem using the Landau-Lifshitz-Gilbert (LLG) equa-\ntion with Langevin dynamics, as detailed in Ref. 9. ThearXiv:1207.4092v2  [cond-mat.mtrl-sci]  22 Jan 20132\n-1.00-0.500.000.501.00\n   0    1    2    3    4    5mz\nTime [ps]TM\nRE\nFIG. 1. Numerical integration of the switching behavior for\nthe non-stochastic LLB with a small angle (15 degrees) be-\ntween sublattices (solid lines). Without the angle (dashed\nlines) switching does not occur, as predicted. The time t= 0\ncorresponds to the end of the square shaped laser pulse with\nTmax= 1500 K. For the integration at temperatures above\nTCwe use the paramagnetic version of the ferrimagnetic LLB\nequation with MFA [14].\nsystem consists of N\u0002N\u0002N cells in a fcc structure lat-\ntice which we populate with a random distribution of TM\nand RE ions in the desired concentration qandx= 1\u0000q.\nTo simulate the e\u000bect of an ultrafast heat pulse we use\na step-like temperature pulse of duration 500 fs with a\nvalue ofT=Tmax. The model predicts the switching\nof GdFeCo compound under the ultrafast heat alone, as\ndemonstrated in Ref. 8. Atomistic models have proven to\nbe a powerful tool in predicting heat-induced switching,\nbut fail to provide a simple picture for the cause of its\nphysical origin.\nHowever, the macroscopic LLB equation has been\ndemonstrated to be an adequate approach, allowing a\nsimple description of ultrafast magnetization phenomena\n[11, 12], but up to now it existed only for a single species\nferromagnet [13]. Recently [14], we have derived the\nLLB equation for a two species system which describes\nthe average magnetization dynamics in each sublattice\nm\u0017=hS\u0017\nii, where\u0017stands for TM or RE sublattice in\nthis case and ifor spins in the sublattice \u0017. Importantly,\nunlike the approach used in Ref. [10], the derivation does\nnot use the Onsager principle and is thus valid far from\nequilibrium.\nIn the absence of an applied \feld and anisotropy, the\nLLB equation for the TM is written as:\n1\nj\rTjdmT\ndt=\u0000mT\u0002h\nHEX\nT+\u000b?\nT\nm2\nTmT\u0002HEX\nTi\n+\u000bk\nTHk\nTmT,\n(2)\nwith a complementary equation for the RE. The exchange\n\feld from the RE is calculated via the mean-\feld approx-\nimation (MFA) of the impurity model presented in [9] as\nHEX\nT= (J0;TR=\u0016T)mR, whereJ0;TR=xzJTR,xis the im-\npurity content, zthe number of nearest TM neighborsin the ordered lattice and JTR<0 the inter-sublattice\nexchange parameter. The TM magnetic moment is de-\nnoted\u0016T,\rTis the gyromagnetic ratio for the TM lat-\ntice,\u000bk\nT(T) and\u000b?\nT(T) are temperature-dependent TM\nlongitudinal and transverse damping parameters, linearly\nproportional to the intrinsic coupling to the bath param-\neter\u0015T[14]. The longitudinal e\u000bective \feld in Eq. (2)\nreads\nHk\nT=\u0000TT\n2\u0012\n1\u0000m2\nT\nm2e;T\u0013\n\u0000\u0000TR\n2\u0012\n1\u0000\u001c2\nR\n\u001c2e;R\u0013\n,(3)\nwhere\u001cR=j(mT\u0001mR)j=mTis the absolute value of the\nprojection of the RE magnetization onto the TM mag-\nnetization and \u001ce;Ris its equilibrium value. The rate\nparameters in Eq. (3) read\n\u0000TT=1\ne\u001fT;k\u0012\n1 +jJ0;TRj\n\u0016Te\u001fR;k\u0013\n;\u0000TR=jJ0;TRj\n\u0016T\u001ce;R\nme;T:(4)\nThey are temperature-dependent via the equilibrium\nmagnetizations and partial longitudinal susceptibilities\ne\u001fT;k= (@mT=@H)H!0,e\u001fR;k= (@mR=@H)H!0, evalu-\nated in the MFA in the presence of inter-sublattice and\nintra-sublattice exchange [14].\nIn Eq. (2) the \frst term in the r.h.s. describes the\nprecession of the TM magnetization, mT, around the ex-\nchange \feld produced by the RE sublattice. Although\nthis term conserves the magnetization modulus, mT, it\nallows transfer of angular momentum between lattices.\nThe second term in Eq. (2) describes the relaxation\nofmTtowards the antiparallel alignment between both\nsublattice magnetizations. Finally, the third term in\nEq. (2) de\fnes the longitudinal relaxation, comprised of;\nthe di\u000berence between relaxation coming from the devia-\ntions of TM magnetization from equilibrium and those of\nRE. In the ferrimagnetic LLB all three terms act on the\ntimescale given by the exchange interactions in compari-\nson to the ferromagnetic LLB case, where the longitudi-\nnal and transverse motion have very di\u000berent timescales\n[15, 17].\nFig. 1 shows the direct numerical integration of Eq. (2).\nWith initial antiparallel alignment of the RE and TM,\nmTkmR, when the temperature is raised both sublattice\nmagnetizations are reduced, followed by the linear mag-\nnetization recovery path to the expected ground state\n[see dashed lines in Fig. 1] and does notproduce switch-\ning. In this case no torque is exerted from one sub-lattice\nto another as mT\u0002mR= 0. However this torque, which\nallows transfer of angular momentum between sublat-\ntices, is always present in the full atomistic approach\nwith stochastic \felds because of the high temperatures\nreached during the reversal process. We can include in\nEq. (2) the presence of this torque by canting by a small\nangle the two sub-lattices magnetization once the heat\npulse is gone or alternatively by the integration of the\nstochastic LLB equation [16]. The solid lines in Fig. 13\n0.000.250.500.751.001.25\n 0.7  0.8  0.9  1.0Γ [ps-1]\nT/TCΓTT\nΓTR\nFIG. 2. Longitudinal relaxation rates as a function of temper-\nature in the LLB equation, evaluated for GdFeCo parameters.\nThe dashed line shows the TM-RE relaxation rate and the\nsolid line is that of the TM-TM interaction. At low tempera-\ntures \u0000 TT\u001d\u0000TR, due to the small value of the susceptibility\ne\u001fT;k, therefore the relaxation of the TM magnetization is al-\nways to its own equilibrium. However, at temperatures close\ntoTC, \u0000TT<\u0000TR, thus the TM prefers to relax towards the\nRE magnetization in this regime.\nshow the integration of Eq. (2) including this angle and\nshows reversal. This small angle generates a mutual pre-\ncessional motion which occurs due to the exchange \feld\nexerted by the opposite sub-lattice and the transverse\nrelaxation directed towards the direction of the opposite\nsub-lattice. This mutual motion leads to the switching,\nas illustrated in Fig. 1 and is presented schematically in\nFig. 3(a).\nThough the longitudinal magnetization process con-\ntributes to the timescale of reversal it does not drive the\nswitching process. Unlike the statement in Ref. [10], the\nlongitudinal relaxation itself cannot change the direction\nofmT, due to the multiplication of the longitudinal re-\nlaxation term in Eq. (2) by mT. In order to understand\nthe switching mechanism we therefore need to consider\nboth longitudinal and transverse relaxation.\nNow we demonstrate that at high temperatures the\nlongitudinal relaxation becomes unstable. This happens\nbecause close to TCthe sign of Hk\nTcan change. In\nFig. 2 we present the temperature dependence of relax-\nation rates (4) evaluated for the parameters of GdFeCo\n[9] in the MFA. One can see that close to TC: \u0000TT<\u0000TR.\nFirstly we reduce the LLB equation (2) to a dynamical\nsystem, based on information from atomistic modeling.\nWe can assume that slightly before the reversal the ini-\ntial transverse moments of the sublattices are small (but\nnot zero), and that the modulus of the TM sublattice\nis much smaller than that of the RE ( mz\nT\u001cm0\nR), ow-\ning to the faster relaxation time of the TM. In this ap-\nproximation the longitudinal \feld is positive Hk\nT>0:\nHk\nT'\u0000TRm0\nR\nme\nRfor the case before the heat pulse is re-\nFIG. 3. (a) Precession of sublattice magnetizations around\nthe exchange \feld of each other in the macroscopic (LLB) de-\nscription. After the action of an ultrafast laser pulse the large\namplitude of the TM precession causes it to cross mz= 0, and\nfor su\u000eciently large angular momentum transfer, the angle\nbetween sublattices becomes small. After cooling the dom-\ninance of the TM sublattice forces the RE to realign along\nthe opposite direction, completing the switching process. (b)\nTrajectories of the parallel and transverse magnetization com-\nponents for TM calculated via atomistic simulations of the\nHeisenberg model (1) at di\u000berent maximum pulse tempera-\nturesTmax= 1000;1200;1250;1300;1350 and 1400 K. After\nthe pulse, the temperature is removed, this moment is indi-\ncated by small circles.\nmoved (mT>me;T= 0) and because after the heat pulse\nis gone the system cools down Hk\nT'[\u0000TT\u0000\u0000TR]=2>0\nwithmT\u001cme;T(T). The LLB equation for the TM is\nreduced to the following system of equations:\ndm2\nT\ndt= 2j\rTj\u000bk\nTHk\nTm2\nT;\nd\u001a\ndt=\u00002h\n\u000b?\nT\nTp\n1\u0000\u001a=m2\nT\u0000j\rTj\u000bk\nTHk\nTi\n\u001a(5)\nwhere\u001a= (mt\nT)2= (mx\nT)2+ (my\nT)2is the TM trans-\nverse magnetization component, \n T=m0\nRj\rTjjJ0;TRj=\u0016T\nis the precessional frequency of the anti-ferromagnetic\nexchange mode.\nThe trajectory \u001a= 0 corresponds to a linear dynamical\nmode. The standard analysis of the dynamical system\n(5) shows that for Hk\nT>0 andmz\nT< \u000b?\nT\nT=(j\rTjHk\nT)\nthis trajectory becomes unstable. Before the end of the\npulse it is equivalent to mT>(\u000b?\nT=\u000bk\nT)me;Twhich is also\neasily satis\fed, taking into account that \u000b?\nT> \u000bk\nT, see\nRef. [14]. The physical interpretation is that in this\ncase very small perturbations from \u001a= 0 will not be\ndamped but will lead to the development of a perpendic-\nular magnetization component, as is indeed observed by\nthe atomistic simulations Fig. 3(b), in which we use the\natomistic model and apply heat pulses of di\u000berent tem-\nperatures to drive the system into di\u000berent states. The\natomistic simulations clearly con\frm the development of\nthe perpendicular component.\nHowever, the dynamical system (5) alone does not de-\nscribe the reversal due to the assumption of the static4\nRE magnetization. In the same approximation, the LLB\nequation for the RE reads:\ndmx(y)\nR\ndt=\u0006\nRmy(x)\nT\u0000\u000b?\nR\nm0\nR\nRmx(y)\nT\u0000j\rRj\u000bk\nRHk\nRmx(y)\nR\n(6)\nwhere the upper sign corresponds to the equation\nformx\nRand the lower sign for the my\nRone, \nR=\nzqm0\nRj\rRjjJTRj=\u0016RandHk\nRis the RE longitudinal \feld.\nEquation (6) shows that the perpendicular motion of the\nTM triggers the corresponding precessional motion of the\nRE via the angular momentum transfer (the \frst two\nterms of Eq. (2), i.e. via perpendicular components)\nwith the same frequency \n T, but di\u000berent amplitude,\nsee Fig. 3(a). During this dynamical process in some\ntime interval the RE and TM magnetization have both\nthe same sign of the z-component, forming the transient\nferromagnetic-like state seen experimentally [7]. Note\nthat the subsequent precession has a frequency which is\nproportional to the exchange \feld and thus is extremely\nfast. The motion of the TM around RE direction and\nvice versa occurs during and after the ferromagnetic-like\nstate until the system has relaxed to equilibrium.\nAn outstanding question is whether the magnetiza-\ntion precession, a central part of the process, can be\nobserved experimentally on a macroscopic sample. We\nshould recall that in non-equilibrium at high tempera-\ntures the correlation between atomic sites is weak, thus\nwe cannot expect the precession to occur with the same\nphase in the whole sample; an e\u000bect which would make\nthe precession macroscopically unobservable. To demon-\nstrate the e\u000bect we present in Fig. 4 the results of atom-\nistic switching simulations in GdFeCo for di\u000berent system\nsizes (Tmax= 2000 K). In Fig. 4 we observe that for small\nsystem sizes transverse oscillations with the frequency of\nan exchange mode are visible, consistent with the predic-\ntion of our analytical model. However, in large system\nsizes of the order of (20 nm)3it is averaged out, con-\nsistent with the excitation of localized exchange modes\nwith random phase. Note that the same e\u000bect happens\nfor very high temperatures where the observed magne-\ntization trajectory appears close to linear; although we\nstress again the importance of a small perpendicular com-\nponent to initiate the magnetization reversal, which will\noccur on a local level as demonstrated by Fig. 4.\nIn conclusion, the LLB equation for a ferrimagnet de-\nscribes the mutual relaxation of sublattices which oc-\ncurs simultaneously under internal damping and inter-\nsublattice exchange. This model allows us to present a\nsimple picture of the magnetization reversal of GdFeCo\nin response to an ultrafast heat pulse alone. The physical\norigin of this e\u000bect is revealed within the LLB equation\nas a dynamical reversal path resulting from the insta-\nbility of the linear motion. To trigger the reversal path\na small perpendicular component is necessary. In prac-\ntice this will arise from random \ructuations of the mag-\n-0.8-0.6-0.4-0.20.00.20.40.60.8\n 0  5  10  15  20  25  30mx\nTime [ps](3nm)3\n(9nm)3\n(20nm)3FIG. 4. Atomistic modeling of the system size dependence\nof the transverse magnetization components of the TM un-\nder ultrafast switching, showing cancelation of the localized\ntransverse magnetization components arising from exchange\nprecession for larger system sizes. The time t= 0 corresponds\nto the end of the laser pulse.\nnetization at elevated temperatures. The perpendicular\ncomponent grows in time resulting in ultrafast magne-\ntization precession in the inter-sublattice exchange \feld,\nalso observed in atomistic simulations for small system\nsizes. The switching is initiated by the TM which ar-\nrives at zero magnetization faster than the RE and re-\nsponds dynamically to its exchange \feld. Thus, the non-\nequivalence of the two sub-lattices is an essential part of\nthe process. Switching into the transient ferromagnetic\nstate occurs due to large-amplitude precessional motion\nof the TM in the exchange \feld from the RE and a slow\ndynamics of RE.\nThis work was supported by the European Commu-\nnity's Seventh Framework Programme (FP7/2007-2013)\nunder grant agreements NMP3-SL-2008-214469 (Ultra-\nMagnetron) N 214810 (FANTOMAS), NNP3-SL-2012-\n281043 (FEMTOSPIN) and the Spanish Ministry of Sci-\nence and Innovation under the grant FIS2010-20979-C02-\n02.\n[1] J. St ohr, and H. C. Siegmann, Magnetism: from Funda-\nmentals to Nanoscale Dynamics (Springer, Berlin, 2006).\n[2] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,\nPhys. Rev. Lett. 76, 4250 (1996).\n[3] M. Wietstruk, A. Melnikov, C. Stamm, T. Kachel, N.\nPontius, M. Sultan, C. Gahl, M. Weinelt, H. A. D urr, and\nU. Bovensiepen Phys. Rev. Lett., 106, 127401 (2011),\n[4] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, Th. Rasing, Phys. Rev. Lett. 99,\n047601 (2007).\n[5] F. Hansteen, A. Kimel, A. Kirilyuk and Th. Rasing,\nPhys. Rev. Lett. 95, 047402 (2005).\n[6] K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, D.\nHinzke, U. Nowak, R. Chantrell, A. Tsukamoto, A.\nItoh, A. Kirilyuk and Th. Rasing, Phys. Rev. Lett. 103,5\n117201 (2009).\n[7] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius,\nH. A. D urr, T. A. Ostler, J. Barker, R. F. L. Evans, R.\nW. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, Th.\nRasing and A. V. Kimel, Nature, 472, 205 (2011).\n[8] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nU. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L.\nLe Guyader, E. Mengotti, L. J. Heyderman, F. Nolting,\nA. Tsukamoto, A. Itoh, D. Afanasiev, B. A. Ivanov, A.\nM. Kalashnikova, K. Vahaplar, J. Mentink, A. Kirilyuk,\nTh. Rasing, and A. V. Kimel, Nature Commun. 3, 666\n(2012).\n[9] T. A. Ostler, R. F. L.Evans, R. W. Chantrell, U. Atxitia,\nO. Chubykalo-Fesenko, I. Radu, R. Abduran, F. Radu,\nA. Tsukamoto, A. Itoh, A. Kirilyuk, Th. Tasing and A.\nKimel, Phys. Rev. B. 84, 024407 (2011).\n[10] J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov,A. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson,\nand Th. Rasing, Phys. Rev. Lett. 108057202 (2012).\n[11] U. Atxitia, O. Chubykalo-Fesenko, J. Walowski, A. Mann\nand M. M unzenberg, Phys. Rev. B 81, 174401 (2010).\n[12] M. Sultan, U. Atxitia, A. Melnikov, O. Chubykalo-\nFesenko and U. Bovensiepen, Phys.Rev.B 85, 184407\n(2012)\n[13] D. Garanin, Phys. Rev. B 55, 3050 (1997).\n[14] U. Atxitia, P. Nieves and O. Chubykalo-Fesenko, Phys.\nRev. B 86, 104414 (2012)\n[15] O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell and\nD. Garanin, Phys. Rev. B 74, 094436 (2006).\n[16] R. F. L. Evans, D. Hinzke, U. Atxitia, U. Nowak, R. W.\nChantrell and O. Chubykalo-Fesenko, Phys. Rev. B 85,\n014433 (2012)\n[17] U. Atxitia and O. Chubykalo-Fesenko, Phys. Rev. B 84,\n144414 (2011)"    },    {        "title": "1409.8430v2.Predicting_a_Ferrimagnetic_Phase_of_Zn2FeOsO6_with_Strong_Magnetoelectric_Coupling.pdf",        "content": " \n1 \n Predicting a Ferrimagnetic  Phase of Zn 2FeOsO 6 with Strong  Magnetoelectric Coupling  \nP. S. Wang ,1 W. Ren,2 L. Bellaiche,3 and H. J. Xiang*1 \n \n1Key Laboratory of Computational Physical Sciences (Ministry of Education), State Key \nLaboratory of Surface Physics, Collaborative Innovation Center of Advanced Microstructures , \nand Department of Physics, Fudan University, Shanghai 200433, P. R. China  \n2Depa rtment of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, P. R. China  \n3Physics Department and Institute for Nanoscience and Engineering, University of Arkansas, \nFayette ville, Arkansas 72701, USA  \nAbstract  \nMultiferroic  materials, in which ferroelectric and magnetic ordering coexist, are of fundamental \ninterest for the development of novel memory devices that allow for electrical writing and non -\ndestructive magnetic readout operation. The great challenge is to create roo m temperature \nmultife rroic materials with strongly coupled ferroelectric and ferromagnetic (or ferrimagnetic) \nordering s. BiFeO 3 has been the most heavily investigated  single -phase multiferroic to date due to \nthe coexistence  of its magnetic order and ferroe lectric order at room temperature. However, there \nis no net magnetic moment in the cycloidal ( antiferromagnetic -like) magnetic state  of bulk \nBiFeO 3, which severely limits its realistic applications in electric field controlled spintronic  \ndevices. Here, we predict that double perovskite Zn 2FeOsO 6 is a new multiferroic with properties \nsuperior to BiFeO 3. First, there are strong ferroelectricity and strong ferrimagnetism at room \ntemperature in Zn 2FeOsO 6. Second, the easy -plane of the spontan eous magnetization can be \nswitched by an external electric field, evidencing the strong magnetoelectric coupling existing in  \n2 \n this system. Our results suggest that ferrimagnetic 3d -5d double perovskite may therefore be \nused to achieve voltage control of mag netism in future spintronic devices.  \nPACS:  75.85.+t,  75.50.Gg, 71.15.Mb, 71.15.Rf  \n \n \n The highly efficient control of magnetism  by an  electric field in a  solid may widen the \nbottle -neck of the state -of-the-art spin -electronics (spintronics) technology, such  as magnetic \nstorage  and magnetic random -access memory. Multiferroics  [1-7], which show simultaneous \nferroelectric and magnetic ordering s, provide an ideal platform for the electric field control of \nmagnetism because of the coupling  between their dual order parameters. For realistic \napplications, one need s to design/discover room temperature multiferroic materials with strong \ncoupled ferroelectric and ferromagnetic (or ferrimagnetic) ordering.  \n Perovskite -structure bismuth ferrite (BiFeO 3) is currently the most  studied room \ntemperature single -phase multiferroic, mostly because its large  polarization and high \nferroelectric  Curie temperature (~820 ° C) make it appealing  for applications in ferroelectric non -\nvolatile memories and high temperature  electronics. Bulk BiFeO 3 is an antiferromagnet with \nNé el  temperature T N ≈ 643 K  [8]. The Fe magnetic moments order almost in a checkerboard G -\ntype manner with a cycloidal spiral  spin structure in which the antiferromagnetic (AFM) axis \nrotates through  the crystal with an incommensurate long -wavelength period  [9]. This spiral spin \nstructure leads to a cancellation of  any macroscopic magnetization. The magnetic properties of \nBiFeO 3 thin films  were found to be  markedly different from those of the bulk: The  spiral spin \nstructure seems to be suppressed and a weak magnetization appears  [10]. Nevertheless, the \nmagnetization is too  small  for many applications  [11]. In addition, an interesting low -field  \n3 \n magnetoelectric (ME) effect at  room temperature was discovered in Z -type hexaferrite \nSr3Co2F24O41 by Kitagawa et al.  [12]. Unfortunately, the electric polarization (about 20 µC/m2) \ninduced by the spin order  is too low  [13]. \nIn searching for new multiferroic compounds, the double perovskite system was proposed \nas a promising candidate  [14, 15]. The double perovskite structure A 2BB′O6 is derived from the \nABO 3 perovskite structure. The two cations B and B ′ occupy the octahedra l B sites of perovskite  \nwith the rock salt ordering. Double perovskite Bi 2NiMnO 6 was successfully synthesized under \nhigh-pressure, which displays the multiferroic behavior with a high ferroelectric transition \ntemperature (485 K) but a low ferromagnetic transition  temperature (14 0 K) [14]. Polar LiNbO 3 \n(LN) -type Mn 2FeMO 6 (M=Nb, Ta) compounds were prepared at 1573 K under  7 GPa  [16]. \nUnfortunately, the magnetic ground state of Mn 2FeMO 6 is AFM with a rather low Né el  \ntemperature  (around 80 K). Very recently, LN -type polar magnetic Zn 2FeTaO 6 was obtained via \nhigh pressure and temperature synthesis  [17]. The AFM magnetic transition temperature (T N∼22 \nK) for Zn2FeTaO 6 is also low. In a pioneering work, Ležaić  and Spaldin proposed to design \nmultiferroics based on 3d -5d ordered  double perovskites  [18]. They found that Bi 2NiReO 6 and \nBi2MnReO 6 are insulating and exhibit a robust ferrimagnetism that persists above room  \ntemperature. Although coherent heteroepitaxy strain may stabilize the R3 ferroelectric ( FE) state, \nfree-standing bulk  of Bi2NiReO 6 and Bi 2MnReO 6 unfortunately take the non -polar P21/n \nstructure as the ground state. The magnetic properties of non -polar double perovskites \nCa2FeOsO 6 [19] and Sr 2FeOsO 6 [20] were also theoretically investigated.  Recently, Zhao et al.  \npredicted  that double perovskite superlattices R 2NiMnO 6/La 2NiMnO 6 (R is a rare -earth ion) \nexhibit an  electrical polarization and strong ferromagnetic order near room temperature  [21].  \n4 \n However, the ME coupling in these superlattices appears to be weak  and the polariz ation to be \nsmall . \nIn this work, we predict that double perovskite Zn 2FeOsO 6 takes the FE LN -type structure \nas the ground state through a global structure searching. Similar to Bi 2NiReO 6 and Bi 2MnReO 6, \nZn2FeOsO 6 exhibit a strong ferrimagnetism at room  temperature. Importantly, there is a rather \nstrong magnetic anisotropy with the easy -plane of magnetization perpendicular to the FE \npolarization  due to the presence of the significant 3d -5d Dzyaloshinskii -Moriya  (DM) interaction . \nThis suggests that the swi tching between the 71 °or 109°FE domains by the electric field will \ncause the rotation of the magnetic easy -plane. Our work therefore indicates that Zn 2FeOsO 6 may \nbe a material of choice  for realizing voltage control of magnetism at room temperature.  \nIt is well-known that there  are several lattice instabilities  including ferroelectric distortion s \nand oxygen octahedron rotation s in perovskite materials . We now examine how double -\nperovskite Zn 2FeOsO 6 distort s to lower the total energy. For this purpose, we per form a global \nsearch for the lowest energy structure based on the genetic algorithm (GA) specially designed for \nfinding the optimal structural distortion  [22]. We repeat the simulations three times. All three \nsimulations consistently show that the polar rhombohedral structure with the R3 space group \n[shown in Fig. 1( b) and ( c)] has the lowest energy for Zn 2FeOsO 6. Similar to Zn 2FeTaO 6, \nZn2FeOsO 6 with the R3 structure is based on the R3c LN -type structure. Previous experiments \nshowed that double perovskite  structure A 2BB′O6 may adopt other structures, such as the P21/n  \n[23], \n3R[24], and C2  structure s [25]. Our density functional theory ( DFT ) calculations show \nthat the R3 phase of Zn2FeOsO 6 has a lower energy than the P21/n, \n3R , and C2 structures by  \n0.22, 0.09, 0.45 eV/f.u., respectively. This strongly suggest s that double perovskite Zn 2FeOsO 6  \n5 \n adopts the R3 structure as its  ground state. This can be understood by using the tolerance factor \ndefined for the LN -type ABO 3 system. It was shown  that when the tolerance factor \n(\n2( )AO\nR\nBOrrt\nrr\n , where \nAr , \nBr and \nOr  are the ionic radii  of the A -site ion, B -site ion and O ion ) \nis smaller than 1, the polar \n3Rc  structure is more stable than the \n3Rc  ABO 3 structure due to the \nA-site instability  [26]. In the case of Zn 2FeOsO 6, the average tolerance factor ( 0.75) is smaller \nthan 1, which suggests that the \n3R  structure is more stable than the non -polar \n3R  structure.  \nNote that phonon calculation s shows that the \n3R  state of Zn2FeOsO 6 is dynamically stable  [27]. \nAdditional tests indicate that the Fe and Os ions tend to order in a rock salt manner (i.e., double \nperovskite  configuration ) to lower the Coulomb interaction  energy [27,28]. \nOur electronic structure calculation shows that the R3 phase of Zn2FeOsO 6 in the \nferrimagnetic  state is insulating  [27]. The density of states plot shows that the Fe majority 3d \nstates are almost fully occupied, while the minority states are almost empty. This suggests that \nthe Fe ion takes the high -spin Fe3+ (d5) valence state. It is also clear that the Os ion takes the \nhigh-spin Os5+ (d3) valence state, which contra sts with the case of Ba2NaOsO 6 where the Os \natom takes a 5d1 valence electron  configuration  [29]. This is also consistent with the total \nmagnetic moment of 2 µB/f.u. for the ferrimagnetic state from the collinear spin -polarized \ncalculation . Through the four -state mapping approach  which is able to deal with  spin in teractions \nbetween two different atomic types [30], we compute the symmetric exchange parameters to find \nthat the magnetic ground state of R3 Zn 2FeOsO 6 is indeed ferrimagnetic. The Fe -Os \nsuperexchange interactions mediated by the corner -sharing O ions [J 1 and J 2, see Fig. 1( c)] are \nstrongly AFM (J 1 = 31.68  meV , J2 = 29.62  meV ). Here, the spin interaction parameters are  \n6 \n effective by setting the spin values of Fe3+ and Os5+ to 1. Our results seem  to be in contradiction  \nwith the Goodenough -Kanamori rule which predicts a ferromagnetic interaction between a d5 ion \nand a d3 ion since the virtual electron transfer from a half -filled σ -bond e orbital on the d5 ion to \nan empty e orbital on the d3 ion dominates the antiferromag netic π -bonding t -electron transfer  \n[31]. This discrepancy is because the <Fe -O-Os angle  (about 140 °) in Zn 2FeOsO 6 is much \nsmaller than 180 °(which is the case for which the Goodenough -Kanamori rule applies ). The \nAFM super -superexchange interaction [J 3 and J 4, see Fig. 1( c)] between the Os ions is much \nweaker (J 3 =7.25 meV , J4 =3.24 meV ) than J 1 and J 2, but not negligible. Since the Fe 3d orbitals \nare much more localized than Os 5d orbitals, the super -superexchange interactions between the \nFe ions are negligible. The dominance of the AFM Fe -Os exchange interactions suggests that the \nmagnetic ground state of R3 Zn 2FeOsO 6 should be ferri magnetic, which is also confirmed by the \nMonte Carlo (MC) simulations to be discussed below.  \nThe magnetic anisotropy in R3 Zn 2FeOsO 6 is investigated by including spin -orbit coupling \n(SOC) in the DFT calculation. The DFT+U+SOC calculation shows that the magnetic moments \ntend to lie in the hexagonal ab -plane. Interestingly, when magnetic moments are in the ab -plane, \nthere is a spin canting between the Fe 3d moments and Os 5d moments. We perform a test \ncalculation to explicitly illustrate this point. We fix the Fe spin moments along the x -direction, \nand compute the total energies as a function of the angle ( α) between Fe and Os  moments. Fig. \n2(a) shows that the total energy has a minima at αmin≈174°  and the total energy curve is \nasymmetric with respect to the axis of α = 180°. This is solely due to the SOC effect since the \ntotal energy from the DFT+U calculations has a minima at  α = 180°.  The deviation of α min by 6°  \nfrom 180°  implies that the magnetic moment also acquires a weak component along the y -axis,  \n7 \n which can be naturally understood by the universal law [32] . As shown in Fig. 2(b), the \nferrimagnetic state with the moments  along the c -axis is higher in energy by 0.55 meV/f.u. than \nthe ferrimagnetic state with the Fe and Os moments aligned oppositely in the ab -plane ( α = 180° ). \nThe canting of the spins ( α = 174° ) further lowers the total energy by 0.97 meV /f.u.. Therefore, \nthe spin canting is extremely important for the easy -plane behavior.  \nTo understand the magnetic anisotropy in R3 Zn 2FeOsO 6, we consider the general spin \nHamiltonian  [30]: \n2\nspin ij i j ij i j i iz i j i j iH J S S D (S S ) A S       \n , which includes symmetric \nexchange interactions, antisymmetric DM interactions, and single -ion anisotropy. For the \nmagnetic structure in which the magnetic unit cell only contains one Fe ion and one Os ion, the \ntotal spin interaction energy can be written as:  \n \nz 2 z 2\nspin 1 2 Fe Os 0i Fe Os Fe Fe Os Os\ni 1,6E 3(J J ) ( ) A (S ) A (S )\n         S S D S S , \nwhere the DM interaction vectors D0i are shown in the insert of Fig. 2(a), A Fe and A Os are the \neffective single -ion anisotropy parameters. Here, the DM interaction s between two different \nkinds of atoms are considered [33-35]. Our calculations show that AFe = 0.89 meV and A Os = -\n0.34 meV, suggesting an overall weak easy -plane behavior resulting from the single -ion \nanisotropy term. The exchange interaction between the Os ions is omitted since it is a constant \nfor such magnetic structure. Our analysis shows  that the DM interaction results in the spin \ncanting, and subsequently contributes dominantly to the easy -plane anisotropy [27].  \nBy performing the parallel tempering  (PT) MC simulation using  the full spin Hamiltonian \n(containing symmetric exchange interactions, antisymmetric DM interactions, and single -ion \nanisotropy), we obtain the thermodynamic behavior of the magnetism in the R3 phase of  \n8 \n Zn2FeOsO 6 [36]. As shown in Fig. 3, there is a peak at 39 4 K in the specific heat curve, \nindicating that there is a magnetic  phase transition at 394  K. The total spin moment (M ab) in the \nab-plane is also plotted as a function of temperature. We can see that the in -plane total spin \nmoment increase s rapidly near the magnetic phase transition point. This suggests that the in -\nplane  total spin moment can be chosen as the order parameter, and the paramagnetic phase \ntransforms to the ferrimagnetic phase with moments in the  ab-plane at 39 4 K. We notice that the \nin-plane total spin moment is larger than 2 μB/f.u. at low temperature. This  is a consequence of \nthe spin canting resulting from the DM interactions. Our MC simulation suggests that R3 \nZn2FeOsO 6 is a room temperature ferrimagnet.  Note also that t est calculations show that the \nferrimagnetic transition temperatures of Zn 2FeOsO 6 with the low -energy phases \n3R  and P21/n \nare 344 K and 356 K, respectively. This indicates that the effect of the ferroelectric displacement, \noxygen octahedron tilts, and the associated magnetoelectric coupling on the ferrimagnetic \ntransition temperature is rather weak  and does not prevent Zn2FeOsO 6 from having a magnetic \ntransition temperature above 300K . \nSimilar to the LiNbO 3 case where the paraelectric structure has the \nR3c  symmetry, the \nparaelectric state corresponding to the FE R3 Zn2FeOsO 6 can be chosen to be \nR3 . Fig ure 4(a) \nshows the double well potential for Zn2FeOsO 6. We can see that the barrier between the two FE \nstate with opposite electric polarizations along the pseudo -cubic direction [111] is 0.09 eV/f.u., \nwhich is smaller than  the value (about 0.20 eV/f.u.) in the case of PbTiO 3 [37]. Taking the \nR3  \nstructure as the reference structure, the electric polarization of R3 Zn2FeOsO 6 is computed to be \n54.7 μC/cm2, which  is much higher than that (2 5.0 μC/cm2) in bulk tetragonal BaTiO 3 [38]. Thus, \nR3 Zn2FeOsO 6 should be a switchable ferroelectric with a high electric polarization.  Our tests  \n9 \n based on an effective model Hamiltonian and molecular dynamics simulations show that the \nferroelectric transition temperature of Zn2FeOsO 6 is well above room temperatu re (see Part 9 and \n10 of [27] ). \nLet us now discuss the possible ME coupling in double -perovskite Zn2FeOsO 6. Because of \nthe cubic symmetry of the perovskite structure, there are eight symmetrically equivalent <111> \ndirections. In rhombohedral  Zn2FeOsO 6, the spontaneous electric polarization is directed along \none of  the eight <111> axes of the perovskite structure. Thus, in the sample of double -perovskite \nZn2FeOsO 6, there might occur eight different FE domains [see Fig. 4( b)]. Our above calculations \nshow that the easy -plane of mag netization is always perpendicular to the direction of the electric \npolarizatio n. Although a 180°  switching  of the ferroelectric polarization should not affect the \nmagnetic state, a 71°  or 109°  switch of the FE domains by the electric field will change the  \norientation  of the easy -plane of magnetization, as shown schematically in Fig. 4( c). This could \nbe a promising  route to manipulate the orientation of the ferrimagnetism by an electric field. A \nsimilar ME coupling mechanism in BiFeO 3 thin films has been de monstrated experimentally by \nZhao et al. , who showed that the AFM  plane can be switched by an electric field  [39]. Note that \nmagnetoelectric effects can be classified in to two different types : one for which changing the \nmagnitude of the polarization  affects the magnitude of the magnetization (energy of the form\n22PM\n) and one for which changing the direction of  \nP\n changes the direction of  \nM\n (energy of \nthe form  \nPM\n ). In Zn 2FeOsO 6, the first type of ME effect is weak, while the second type of ME \neffect is strong.   \nWe now compare Zn2FeOsO 6 with the classic multiferroic BiFeO 3. First, they adopt similar \nrhombohedral  structures. Second, both compounds have high electric al polarizations. Third, both  \n10 \n compounds are room temperature multiferroics. Fourth, the ME coupling mechanism is rather \nsimilar in that the magnetic easy -plane can be manipulated by electric field. However, the \nmagnetic ground state of R3 Zn2FeOsO 6 is dramatically different from BiFeO 3. Zn2FeOsO 6 has a \nferrimagnetic ground state, while BiFeO 3 is AFM. And the magnetic anisotropy in Zn2FeOsO 6 is \nstronger than that (0.2 meV when U(Fe) = 5 eV [40]) in BiFeO 3 because of the strong SOC \neffect of the 5d Os ion. These desirable properties  make  Zn2FeOsO 6 suitable for realizing \nelectric -field control of magnetism at room temperature.  \nThe reason why we practically propose d Zn2FeOsO 6 as a possible multiferroic is two -fold. \nFirst, polar Zn 2FeTaO 6 has already been synthesized under high -pressure, as mentioned above. \nSecond, it was experimentally  showed that Ca 2FeOsO 6 crystallizes into an ordered double -\nperovskite  structure with a space gro up of P21/n under high -pressure  and high -temperature, and \nCa2FeOsO 6 presents a long -range  ferrimagnetic transition above room temperature (T c ∼320 K)  \n[24]. Therefore, we expect that Zn 2FeOsO 6 is synthesizable and would most likely exhibit both \nferroelectricity and ferrimagnetism at room temperature.  \nWork at Fudan was supported by NSFC, FANEDD, NCET -10-0351, Research Program of \nShanghai Municipality and MOE, the Special Funds for Major State Basic Research, Program \nfor Professor of Special Appoin tment (Eastern Scholar), and Fok Ying Tung Education \nFoundation.  L.B. thanks the Department of Energy, Office of Basic Energy Sciences, under \ncontract ER -46612 . \ne-mail: hxiang@fudan.edu.cn  \nReferences  \n[1] S.-W. Cheong  and M. Mostovoy,  Nature Mater.  6, 13 (2007).   \n11 \n [2] R. Ramesh  and N. Spaldin , Nature Mater.  6, 21 (2007).  \n[3] S. Picozzi  and C. Ederer,  J. Phys. : Condens. Matter  21, 303201 (2009) .  \n[4] K. F.  Wang,  J.-M. Liu, and Z. Ren, Adv. Phys.  58, 321 (2009) .  \n[5] J. van den Brink  and D. Khomskii,  J. Phys .: Condens . Matter  20, 434217 (2008).  \n[6] Y .  Tokura  and S. Seki,  Adv. 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Matter 24, 312201  (2012) ] states that the weak component of the magnetization has a \ndirection that is given by the cross -product of the oxygen octahedral  tilting vector (that is lying \nalong the c -direction in the hexagonal setting of the R3 phase) with the G -type antiferromagnetic \nvector (that is chosen here to be along the x -axis).  Note that this law also implies that choosing a \nG-type antiferromagnetic v ector being along the same direction as the ox ygen octahedral tilting \nvector (that is, the c -axis) should not produce any weak component of the magnetization \nperpendicularly to the G -type antiferromagnetic vector.  This is precisely what we further fou nd \nin the simulations when the Fe and Os moments are chosen to lie (in antiparallel fashion) along \nthe c -axis.  \n[33] A. Fert and P. M. Levy, Phys. Rev. Lett. 44, 1538 (1980).  \n[34] M. Heide, G. Bihlmayer, S.  Blügel, Physica B 404, 2678 (2009) . \n[35] Y . A. Izyumov, Sov . Phys. Usp . 27, 845 (1984) . \n[36] P. S. Wang and H. J. Xiang, Phys. Rev. X 4, 011035 (2014).  \n[37] R. E.  Cohen,  Nature (London)  358, 136 (1992); C. Ederer and N. A.  Spaldin, Phys. Rev. B  \n74, 024102 (2006).  \n[38] J. B. Neaton  and K. M.  Rabe , Appl.  Phys. Lett . 82, 1586 (2003).   \n14 \n [39] T. Zhao , A. Scholl ,  F. Zavaliche , K. Lee, M. Barry , A. Doran , M. P.  Cruz , Y . H.  Chu, C. \nEderer, N. A.  Spaldin, R. R.  Das, D. M.  Kim, S. H.  Baek , C. B.  Eom , and R. Ramesh , Nature \nMater.  5, 823 (2006).  \n[40] C. Ederer  and N . A. Spaldin , Phys. Rev. B 71, 060401 (R) ( 2005 ). \n \nFIG. 1(color online) . (a) The pseudo -cubic structure of double perovskite  Zn2FeOsO 6. (b) The \npolar rhombohedral structure of R3 Zn 2FeOsO 6.  (c ) The Fe -Os superexchange interactions \nmediated by the corner -sharing O ions (J 1 and J 2) and t he AFM super -superexchange interaction \n(J3 and J 4) between the Os ions . (d) The ferrimagnetic s tructure  with the Fe and Os spin moments \naligned in the ab -plane.   \n \n15 \n  \nFIG. 2(color online).  (a) The total energies as a function of the angle α [see the insert ] from the \ndirect DFT+U+SOC calculations  (small circles ). The total energy has a minima at αmin≈174° . \nThe total energy curve can be  described rather well by the formula \nzz\nspin 1 2 01 04E 3(J J )cos 3(D D )sin    \n (blue line) . The insert shows t he DM interaction \nvectors D0i (i =1, 6) between a Fe ion and six Os ions  and t he definition of the angl e α between \nthe Fe spin and Os spin in the ab -plane.  (b) Relative energies between different spin \nconfigurations. The ferrimagnetic state with the moments along the c -axis is higher in energy by \n0.55 meV/f.u. than the ferrimagnetic state with the Fe and Os moments aligned oppositely in the \nab-plane ( α = 180° ), and t he canting of the spins ( α = 174° ) further lowers the total energy by \n0.97 meV /f.u.. \n \n16 \n  \nFIG. 3(color online).  The specific heat and t he total in -plane spin moment (M ab) as a function of \ntemperature from the PTMC simulations. The specific heat  curve indicates that the ferrimagnetic \nCurie temperature (Tc) is 39 4 K. The in -plane total spin moment increases rapidly near T c. \n \n \n \n \n \n \n17 \n  \nFIG. 4(color online).  (a) The total energy as a function of  the electric polarization  for Zn2FeOsO 6. \nIt displays the  double well potential with an energy barrier of 0.09 eV/f.u.. (b) Eight possible \norientations of the FE polarization vector ( P) in the sample of  double -perovskite Zn2FeOsO 6. (c) \nIllustration of the ME coupling in Zn 2FeOsO 6. A 71°  or 109°  switch of the FE domains  by the \nexternal electric field will be associated with the reorientation  of the easy -plane of magnetization.  \n \n"    },    {        "title": "0902.3109v1.Majority_spin_non_quasiparticle_states_in_half_metallic_ferrimagnet_Mn__2_VAl.pdf",        "content": "arXiv:0902.3109v1  [cond-mat.mtrl-sci]  18 Feb 2009Majority-spin non-quasiparticle states in half-metallic ferrimagnet Mn 2VAl\nL. Chioncel,1,2E. Arrigoni,1M.I. Katsnelson,3and A.I. Lichtenstein4\n1Institute of Theoretical Physics, Graz University of Techn ology, A-8010 Graz, Austria\n2Faculty of Science, University of Oradea, RO-410087 Oradea , Romania\n3Institute for Molecules and Materials, Radboud University of Nijmegen, NL-6525 ED Nijmegen, The Netherlands\n4Institute of Theoretical Physics, University of Hamburg, D E-20355 Hamburg, Germany\nThe density of non-quasiparticle states in the ferrimagnet ic full-Heuslers Mn 2VAl alloy is cal-\nculated from first principles upon appropriate inclusion of correlations. In contrast to most half-\nmetallic compounds, this material displays an energy gap in the majority-spin spectrum. For this\nsituation, non-quasiparticle states are located below the Fermi level, and should be detectable by\nspin-polarizedphotoemission. Thisopensanewwaytostudy many-bodyeffectsinspintronic-related\nmaterials.\nHalf metals display a particular type of itinerant-\nelectron magnetism as well as unusual electronic prop-\nerties: they are metallic for one spin channel, and in-\nsulating or semiconducting for the opposite one [1, 2].\nElectronic structure calculations based on density func-\ntional theory offer an explanation for the half-metallicity\nbased on the interplay between the crystal structure, the\nvalence electron count, the covalent bonding, and the\nlarge exchange splitting in addition to symmetry con-\nstrains. The expected 100% spin polarization of half-\nmetals turned out to be an excellent motivation in de-\nveloping the field of spintronics both from a theoretical\nand an experimental point of view [2, 3]. In reality many\npotential half-metallic ferromagnets exhibit a dramatic\ndecrease of bulk spin polarization at temperatures well\nbelow their Curie temperature. In order to understand\nsuch a behavior from a theoretical point of view it is nec-\nessary to consider finite temperature many-body effects\n[2].\nAn important effect of dynamical electron correlations\ninhalf-metalsistheexistenceofnon-quasiparticle(NQP)\nstates [4, 5, 6]. These states contribute significantly\nin reducing the tunneling transport in heterostructures\ncontaining HMF [7, 8, 9, 10, 11], even in the presence\nof disorder. NQP states strongly influence the value\nand temperature dependence of the spin polarization in\nHMF [2, 6, 12], which is of primary interest for po-\ntential applications. These states originate from spin-\npolaron processes whereby the minority spin low-energy\nelectron excitations, which are forbidden for HMF in the\nsingle-particle picture, are possible as superpositions of\nmajority-spin electron excitations and virtual magnons\n[2, 4, 5, 6]. Recently we have applied the LDA+DMFT\n(local density approximation plus dynamical mean field\ntheory) method (for review of this approach, see Ref.13)\nto describe from first principles the non-quasiparticle\nstates in several half-metals [14, 15, 16, 17, 18]. Up to\nnow, our studies were restricted to half metals with a\ngap in the minority spin channel. In this situation NQP\nstates appear just abovethe Fermi level [6].\nOn the contrary, it was predicted that in half-metallic\nmaterials with a gap in the majority (say, “up”) spinchannel, NQP states should appear below the Fermi\nlevel [4, 5, 6]. This asymmetry can be understood in\nterms of electron-magnon scattering processes, as pre-\nsented in the followings.\nA well studied model which takes into account the in-\nteraction of charge carriers with local moments is the s-d\nexchange model. The interacting part of the Hamilto-\nnian is given by −I/summationtextSiσαβc†\niαciβ, where Iis thes-d\nexchange parameter, Sirepresents the localized spin op-\nerators,σαβare the Pauli matrices, and ciσare operators\nfor conduction electrons. The NQP picture turns out to\nbe essentially different for the two possible signs of the\ns−dexchange parameter.\nThe ground state of the system with I >0 (assum-\ning that the Fermi energy is smaller than the spin split-\nting 2IS) has maximum spin projection and, thus, the\nminority-electron band should be empty. For this case\n(I >0), NQP states in the minority spin gap develop\nas a superposition of the majority-electron states plus\nmagnon states, and of the minority-electron states, so\nthat the totalspin projection of the system is conserved.\nAs a result of this spin-polaronic effect, the minority-\nelectron density of states has a tail corresponding to\nthe virtual conduction electron spin-flip processes with\nmagnon emission. However, these virtual flips are im-\npossible below the Fermi energy EFdue to the Pauli\nprinciple (all majority-electron states are already occu-\npied and thus unavailable). Therefore, for the positive\ns-d exchange interaction the NQP states form above EF.\nContrary,fornegative Ithe minority-spinbandliesbe-\nlow the majority-spin one [2, 6]. Occupied minority-spin\nstates can be superposed with majority-electron states\nplus magnons, with conserved total spin projection, so\nthe NQP states occur below EF. At the same time, for\nI <0 the ferromagnetic ground state is non-saturated\nand thus zero-point magnon fluctuations are allowed. It\nis the fluctuations which are responsible for formation of\noccupied majority-electron NQP states there.\nFormally, the difference between I <0 and the previ-\nousI >0 cases, can be explained in terms of a particle-\nholetransformation c†\niσ→di¯σ, andciσ→d†\ni¯σ. Thismod-\nifies thes-dexchange Hamiltonian into I/summationtextSiσαβd†\niαdiβ.2\nIn other words, the Hamiltonian with I >0 for electrons\nis equivalent to that with I <0 for holes.\nThe above argument based on the s-dexchange model\ncan be generalized for arbitrary multi-band half-metallic\nelectronic structures [2, 6]. The conclusion remain un-\nchanged: for the case of minority-electron gap, NQP\nstates are situated above the Fermi energy, while for\nthe cases when the gap is present for majority-electrons,\nNQP states are formed below the Fermi energy.\nMost HMF materials have a gap in the minority spin\nchannel so that NQP states arise above the Fermi level.\nAs a consequence, these states cannot be studied by\nthe very well-developed and accurate technique of spin-\npolarized photoemission [19], which can only probe occu-\npied states. The spin-polarized Bremsstrahlung Isochro-\nmat Spectroscopy (BIS) probing unoccupied states [20]\nhas a much lower resolution. For this reason, HMF with\na gap in the majority-spin channel, and, consequently,\nNQP states in the occupied region of the spectrum, allow\nfor a detailed experimental analysis of these correlation-\ninduced states and are, therefore, potentially of great in-\nterest. Itisthepurposeofthepresentworktoperforman\nelectronic structure calculation based on a combination\nof the generalized-gradient approximation (GGA) and of\nDMFT for the half-metallic ferrimagnetic full-Heusler al-\nloy Mn 2VAl, which has a gap in the majority spin chan-\nnel. By appropriately taking into account effects due to\nelectronic correlations, we demonstrate explicitly the ex-\nistence of majority spin NQP states arising just below\nthe Fermi level, and study the temperature dependence\nof their spectral weight.\nIn full Heusler compounds with the formula X 2YZ, Mn\natomsusuallyoccupytheY-position,whilecompoundsin\nwhich Mn assumes the X-position Mn 2YZ, are very rare.\nThe prototype from the latter category is Mn 2VAl, for\nwhich a large number of theoretical and experimental in-\nvestigations have been made. Neutron diffraction experi-\nments [21] demonstrated the existence of a ferrimagnetic\nstate in which Mn has a magnetic moment of 1 .5±0.3µB\nandVmomentis −0.9µB. X-raydiffractionandmagneti-\nzationmeasurements[22] foundatotal magneticmoment\nof 1.94µBat 5K, close to the half-metallic value of 2 µB.\nThe Curie temperature of the sample was found to be\nabout 760 Kand the loss of half-metallic character was\nattributed to the small amount of disorder. Electronic-\nstructurecalculationsperformedbyIshida[23]withinthe\nlocal-density approximation (LDA), predict the ground\nstate of Mn 2VAl to be close to half-metallicity. Weht\nand Pickett [24] used the GGA for the exchange cor-\nrelation potential and showed that Mn 2VAl is a half-\nmetallic ferrimagnet with atomic moments in very good\nagreement with the experiment. Recent calculations of\nthe exchange parameters for Mn 2VAl [25] show a strong\nMn−Vexchange interaction that influence the ordering\nin the Mn sublattice. The estimated Curie temperatures\nare in good agreement with the experimental values [25].The intermixing between V and Al atoms in the Mn 2VAl\nalloyshowedthat a small degreeofdisorderdecreasesthe\nspin polarization at the Fermi level from its ideal 100%\nvalue, but the resulting alloy Mn 2V1−xAl1+xstill show\nan almost half-metallic behavior [26, 27].\nAccording to the ideal full Heusler ( L21) structure, the\nV atom occupy the (0 ,0,0) position, the Mn atoms are\nsituated at (1 /4,1/4,1/4)aand (3/4,3/4,3/4)a, and the\nAl at (1/2,1/2,1/2)a, wherea= 5.875˚Ais the lattice\nconstant of the Mn 2VAl compound. In our work, corre-\nlation effects in the valence V and Mn dorbitals are in-\ncluded via an on-site electron-electron interaction in the\nform1\n2/summationtext\ni{m,σ}Umm′m′′m′′′c†\nimσc†\nim′σ′cim′′′σ′cim′′σ. The\ninteraction is treated in the framework of dynamical\nmean field theory (DMFT) [13], with a spin-polarized T-\nmatrix Fluctuation Exchange (SPTF) type of impurity\nsolver [28]. Here, cimσ/c†\nimσdestroys/creates an elec-\ntron with spin σon orbital mon lattice site i. The\nCoulomb matrix elements Umm′m′′m′′′are expressed in\nthe usual way [29] in terms of three Kanamori parame-\ntersU,U′=U−2JandJ. Typical values for Coulomb\n(U= 2eV) and Stoner ( J= 0.93eV) parameters were\nused for Mn and V atoms. The above value of U is con-\nsiderablysmallerthanthebandwidthofMn 2VAl(7–8eV)\ntherefore the use of a perturbative SPTF-solver is justi-\nfied. In addition, the same solver was used to investi-\ngate spectroscopic properties of transition metals with\nremarkable results [30, 31, 32, 33, 34].\nSince the static contribution from correlations is al-\nready included in the local spin-density approximation\n(LSDA/GGA), so-called “double counted” terms must\nbe subtracted. To achieve this, we replace Σ σ(E) with\nΣσ(E)−Σσ(0) [35] in all equations of the DMFT pro-\ncedure [13]. Physically, this is related to the fact\nthat DMFT only adds dynamical correlations to the\nLSDA/GGA result. For this reason, it is believed that\nthis kind of double-counting subtraction “Σ(0)” is more\nappropriate for a DMFT treatment of metals than the\nalternative static “Hartree-Fock” (HF) subtraction [36].\nIn Fig. 1 we present the total density of states com-\nputed in GGA and GGA+DMFT, for T=200K. The\nGGA density of states displays a gap of about 0.4eV in\nthe majority spin channel in agreement with previous\ncalculations [24]. As expected from the s-d model calcu-\nlation, majority spin NQP states are visible just below\nthe Fermi level, with a peak around −0.25eV. In order\nto evaluate the spectral weight of these NQP states, we\nfit the low-energy density of states below the Fermi level\nin the majority channel with a Gaussian centered around\nthe peak position. The NQP spectral weight is then de-\nfined as the area below the Gaussian curve. The inset\nshows the NQP spectral weight for several temperatures\nup to 300K. It is interesting to note that within the com-\nputed temperature range 50 ≤T≤300 these values are\nalmost constant and considerably larger in comparison\nwith similar values for (NiFe)MnSb [16]. The data pre-3\n-3 -2 -1 0 1 2 3\nE-EF(eV)-12 -12-8 -8-4 -40 04 48 8DOS(states/eV/spin/unit cell)majority spinminority spinGGA\nDMFT\nGaussian fit\n0 100 200 300\nT(K)00.30.6spectral weight\nFIG. 1: (color online) Total density of states, computed\nwithin GGA (dashed/blue), and GGA+DMFT (full/red).\nThe Gaussian fit to the density of NQP states is shown as a\ndotted-dashed (black) line just below the Fermi level for th e\nmajority spin channel. The temperature dependent spectral\nweight of NQP states is displayed in the inset.\nsented in the inset can be extrapolated down to T= 0K,\nandaspectralweightof ≈0.544±0.018(states/Mn −d)is\nobtained. ThisdemonstratesthatNQPstatesarepresent\nalsoatT=0K,andobviouslytheyarenotcapturedbythe\nmean-field, GGA result. As it will be discussed below,\nNQP states predominantly consist of Mn-delectrons, so\nthe value for the integrated spectral weight (inset of Fig.\n1) only comes from Mn-dorbtials.\nThe atom resolved DOS is presented in Fig. 2. In\nGGA, the net magnetic moment per unit cell is 2 µBwith\nparallel Mn moments having values close to 1 .6µBand\noppositely oriented V moments close to −0.8µB. Be-\nlow the gap, the majority spin total DOS is mainly of\nMn character. The Mn and V moments have a strong\nt2gcharacter, and a small Al contribution to the mag-\nnetic moment is present. Most of the Vmajority spin\nstates lie above the gap, along with the Mn egstates.\nMinority-spin states below 0.5eV have roughly an equal\namounts of V(t2g) and Mn character. States around the\nFermi energy have a predominant Mn(t2g) character, in\nagreement with [24]. In contrast to the GGA results,\nthe many-body DMFT calculation (see Fig. 2) yields a\nsignificant DOS for the majority spin states just below\nthe Fermi level. These are the majority spin NQP states\ndiscussed above [12, 14, 15, 16, 17, 18]. As can be seen\nin Fig. 2 majority spin NQP states are predominantly\nofMn−3d↑character. Their spectral weight is quite-3-1.501.53\nUMn=2eV, JMn=0.9eV\nUV=2eV, JV=0.9eVGGA\nDMFT\n-5-4 -3 -2 -1 0 1 2 3 4\nE-EF(eV)-3-1.501.53DOS(states/eV)majority spin majority spinminority spin minority spinVMn\nFIG. 2: (color online) Atom resolved density of states, com-\nputed within the GGA (dashed/blue) and GGA+DMFT\n(full/red) approach, at T=200K. The majority spin NQP\nstates are visible in the Mn-3d↑DOS just below the Fermi\nlevel.\nsignificant (see inset of Fig. 1) so that accurate spin-\npolarized photoemission experiments should be able to\nidentify the existence of such states. In contrast, ma-\njority spin V(t 2g) states below the Fermi energy are not\nsignificantly changed. Above the Fermi level, the Mn(e g)\nand V(e g) states are pushed to higher energy, such that a\ngap is formed just above EF. In the minority spin chan-\nnel below EF, both V and Mn(t 2g) states are slightly\nmodified, while above EF, V(eg) states are shifted to\nhigher energies by 0 .5eV. Around the Fermi level, the\ndominant Mn−3d↓DOS is not significantly changed\nwith respect to the GGA values.\nThe applicability of the local DMFT approach to the\nproblem of the existence of NQP states has been dis-\ncussed in ref. [2] and [14]. It is essential to stress that\nthe accurate description of the magnon spectrum is not\nimportantfortheexistenceofnonquasiparticlestatesand\nfortheproperestimationoftheirspectralweight, but can\nbe important to describe an explicit shape of the density\nofstates“tail”inaveryclosevicinityoftheFermienergy.\nThe imaginary part of the atom and orbital resolved\nself-energies,forT=200KarepresentedinFig. 3. Forthe\nminority Mn- eg, V-egand V-t2g-orbitals we observe that\nthe imaginary part of the self-energy has a rather sym-\nmetric energy dependence around the Fermi level, with\na normal Fermi-liquid-type behavior −ImΣ↓\nMn/V(E)∝4\n-0.18-0.12-0.060.00\nMn(t2g) maj.\nMn(t2g) min.\n-1 0 1 2\nE-EF(eV)-0.18-0.12-0.060.00\nMn(eg) - maj.\nMn(eg) - nim.-0.18-0.12-0.060.00\nV(t2g) maj.\nV(t2g) min.\n-1 0 1 2-0.18-0.12-0.060.00\nV(eg) maj.\nV(eg) min.Σ Im Im Σ \nMn(t2g)\nIm Σ \nIm Σ Mn(eg)V(t2g)\nV(eg)Im \nIm Σ Σ \nIm \nΣ Im Σ \nFIG. 3: (color online) Imaginary part of the self-energies\nImΣσ\nMn/Vfort2g-orbitals on Mn (left upper panel) and V\n(right upper panel). The solid (red) line shows results for t he\nmajority( ↑) spins, while the dashed (blue) line for minority\nspins. The lower pannels show the corresponding ImΣσ\nMn/V\nfor theegorbitals.\n(E−EF)2. The majority spin −ImΣ↑\nMn/V(E) shows\na significant increase right below the Fermi level which\nis more pronounced for the t2g-orbitals. In addition, a\nslight kink is evidenced for an energy around -0.25eV.\nThe majority-state nonquasiparticles are visible in the\nmajority spin channel Fig. 2 at about the same energy.\nThese results shown in Fig. 3 suggests that many-body\neffects are stronger on Mn than on V sites. Therefore,\nNQP states are mainly determined by the Mn-datoms.\nThe behavior of the imaginary part of the self-energy\n(Fig. 3) and the Green function (Fig. 2) can be corre-\nlated with the analysis of the spin-resolved optical con-\nductivity. We have estimated the latter for different\ntemperatures whithin the GGA and GGA+DMFT ap-\nproaches using an approximation of constant matrix ele-\nments. Already at 50K the majority-spin optical spectra\nshows the appearance of a Drude peak signaling the clo-\nsure of the majority spin gap. With increasing temper-\natures, spectral weight is transfered towards the Drude\npeak contributing to the depolarization discussed below\nin Fig. 4.\nAs it was demonstrated previously [2, 4, 5, 6], the non-\nquasiparticlespectral weightin the density ofstates (Fig.\n2) is proportionalto the imaginarypart ofthe self-energy\n(Fig. 3), therefore it is determined by the quasiparticle\ndecay, which is the reason for the name of these states.\nFig. 4 displays the temperature dependence of the050100150200250300\nT(K)0.5 0.50.6 0.60.7 0.70.8 0.80.9 0.91 1MDMFT(T)/MGGAPDMFT(EF,T)P(EF,T)=N  (EF,T) + N  (EF,T)N  (EF,T) - N  (EF,T)\nMDMFT(T)/MGGA\nPDMFT(EF,T)\nFIG. 4: (color online) Temperature dependence of the spin\npolarization of conduction electrons (full/red) at the Fer mi\nlevelP(EF,T), and normalized magnetization M(T)/M(0)\n(dashed/black).\nmagnetization obtained directly from the GGA+DMFT\ncalculations and the spin polarization at the Fermi level,\nobtained using the relation P(EF,T) = (N↑(EF,T)−\nN↓(EF,T))/(N↑(EF,T)+N↓(EF,T)),Nσbeingtheden-\nsity of states. These results reflect a general trend\nvalid for half-metals in the presence of NQP states\n[2, 11, 14, 15, 16, 17, 18], namely that magnetization\nandpolarizationbehavedifferentlyasfunction oftemper-\nature. However, as can be seen in Fig. 4, this difference\nis not as sharp as in other HMF materials [12, 18].\nThe effect of disorder on half-metallicity was recently\ndiscussed in Mn 2V1−xAl1+xalloys, for −0.2≤x≤0.2\n[22, 26, 27]. The excess of both Al and V atoms ( x=\n0.1/−0.1 orx= 0.2/−0.2) has the effect of shrinking\nthe gap to zero, but with the Fermi level situated within\nthe gap. In addition, the Mn moment is not affected by\ndisorder and remains constant, in contrast to the V mo-\nment. Spin polarization is decreased by about 10%, with\nelectrons around the Fermi level having a dominant mi-\nnority spin character [26]. In contrast, many-body corre-\nlations have a much more dramatic effect. For non-zero\ntemperatures, all atomic magnetizations are decreased.\nFor instance, near room temperature ( T= 300K) the\nstrongest decrease occurs in V, for which the moment\ndrops almost by 47%, the Mn moment is reduced by\n32%, while the Al experiences just a small reduction by\n4%. As one can see from Fig. 4, already at 50 Kpolar-\nization drops to 75%, and is further decreased upon in-\ncreasing the temperature. As we discussed previously for5\nthe case of FeMnSb [16], many-body induced depolariza-\ntion is significantly stronger than the effect of disorder or\nof other spin-mixing mechanisms such as spin-orbit cou-\npling. This observation seems to hold also for the case of\nMn2VAl, although we can not exclude the fact that for\na larger degree of disorder, the material could possibly\ndepart from its almost half-metallic situation displayed\nfor small degree of substitution ( −0.2< x <0.2).\nInconclusion,inthispaperwehaveshownforaspecific\nmaterial that NQP states are also present in half-metals\nwith a gap in the majority spin channel, and appear\njust below the Fermi level, as predicted in model cal-\nculations [5]. In the case of Mn 2VAl, these states mainly\nconsist of Mn-3d↑electronsand have a considerablespec-\ntral weight. Although this material was reported to be a\nhalf-metal from electronic structure calculations [24], the\nexperimental evidence is not clear. Several reasons are\ninvoked such as existence of defects or the reduced sym-\nmetry at surface and interfaces. From a theoretical point\nofview, weshowthatcorrelation-inducedNQPstatessig-\nnificantly changethe majorityspin electronicstates, thus\nreducing the spin polarization at EF. The appearance of\nNQP states and its connection with tunneling magne-\ntoresistance was recently studied in Co 2MnSi-based tun-\nnel magnetic junction [12]. A great challenge would be\nto produce TMR junctions based on the ferri-magnetic\nMn2VAl. This would allow for a direct experimental in-\nvestigation of the existence of majority spin NQP states.\nPromisingcandidateHMFmaterialswithamajorityspin\ngap of similar magnitude as Mn 2VAl are the double per-\novskites Sr 2FeMO 6(M=Mo,Re) or Sr 2CrReO 6often as-\nsociated with collosal magnetoresistance behavior. In\nparticular the electronic structure of Sr 2CrReO6shows\na closure of its majority spin gap in the presence of spin-\norbit coupling [37], with states having a small spectral\nweight symmetrically distributed around the Fermi en-\nergy. 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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 film 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": "0806.1641v1.Ferrimagnetism_of_MnV_2O_4_spinel.pdf",        "content": "arXiv:0806.1641v1  [cond-mat.str-el]  10 Jun 2008Ferrimagnetism of MnV2O4spinel\nNaoum Karchev[*]\nDepartment of Physics, University of Sofia, 1164 Sofia, Bulga ria\nThe spinel MnV2O4is atwo-sublattice ferrimagnet, with site Aoccupied bythe Mn2+ion andsite\nB by the V3+ion. The magnon of the system, the transversal fluctuation of the total magnetization,\nis a complicated mixture of the sublattice AandBtransversal magnetic fluctuations. As a result,\nthe magnons’ fluctuations suppress in a different way the mang anese and vanadium magnetic orders\nand one obtains two phases. At low temperature (0 ,T∗) the magnetic orders of the MnandV\nions contribute to the magnetization of the system, while at the high temperature ( T∗,TN), the\nvanadium magnetic order is suppressed by magnon fluctuation s, and only the manganese ions have\nnon-zero spontaneous magnetization. A modified spin-wave t heory is developed to describe the two\nphases and to calculate the magnetization as a function of te mperature. The anomalous M(T) curve\nreproduces the experimentally obtained ZFC magnetization .\nPACS numbers: 75.50.Gg, 75.30.Ds, 75.60.Ej, 75.50.-y\nThis Letter is inspired from the experimental measure-\nmentsoftheZFCmagnetizationof MnV2O4[1, 2, 3]. The\nprofile of the experimental curve reproduces the anoma-\nlous magnetization curve predicted by L. Neel [4, 5] for\nferrimagnets with equal sublattice spins. This stimulates\nto model the manganese vanadate spinel in the spirit of\nthe Neel’s theory. By comparing and contrasting ZFC\nandFCmagnetizationonegainsinsightintoamagnetism\nof the manganese vanadate oxide.\nThe spinel MnV2O4is a two-sublattice ferrimagnet,\nwith site Aoccupied by the Mn2+ion, which is in the\n3d5high-spin configuration with quenched orbital an-\ngular momentum, which can be regarded as a simple\ns= 5/2 spin. The B site is occupied by the V3+ion,\nwhich takes the 3 d2high-spin configuration in the triply\ndegenerate t2gorbital,andhasorbitaldegreesoffreedom.\nThe measurements show that the set in of the magnetic\norder is at Neel temperature TN= 56K[1], and that\nthe magnetization has a maximum near T∗= 50K. Be-\nlowthistemperaturethemagnetizationsharplydecreases\nand goes to zero when temperature approaches zero.\nWe consider a system which obtains its magnetic prop-\nerties from MnandVmagnetic moments. It is shown\nthat the true magnons in this system, which are the\ntransversal fluctuations corresponding to the total mag-\nnetization, are complicated mixtures of the Mnand\nVtransversal fluctuations. The magnons interact with\nmanganese and vanadium ions in a different way, and the\nmagnons fluctuations suppress the MnandVordered\nmoments at different temperatures. As a result, the fer-\nrimagnetic phase is divided into two phases: low temper-\nature phase 0 < T < T∗, where the magnetic orders of\ntheMnandVions contribute to the magnetization of\nthe system, and high temperature phase ( T∗,TN), where\nthe vanadium magnetic order is suppressed by magnon\nfluctuations, and only the manganese ions have non-zero\nspontaneous magnetization. A modified spin-wave the-\nory is developed to describe the two phases and to cal-\nculate the magnetization as a function of temperature.\nThe anomalous M(T) curve reproduces the experimen-\ntally obtained ZFC magnetization [2, 3].Because of the strong spin-orbital interaction it is con-\nvenient to consider jjcoupling with JA=SAand\nJB=LB+SB. The sublattice Atotal angular mo-\nmentum is jA=sA= 5/2, while the sublattice Btotal\nangular momentum is jB=lB+sB, and sublattice A\nmagnetic order is antiparallel with sublattice Bone. In\nthe simplest case one can consider lB= 1 and sB= 3/2.\nThen, the g-factor for the sublattice AisgA= 2, and the\natomicvalueofthe gBisgB= 1.6. Thesaturatedmagne-\ntization is σ= 25\n2−1.65\n2. The experimental curve shows\nthat the magnetization goes to zero when the tempera-\nture approaches zero, which indicates that the real value\nofgBis not the atomic one but gB≈2. The deviation is\ndue to the anisotropy which increases the gB-factor [5].\nThe Hamiltonian of the system is\nH=−κA/summationdisplay\n≪ij≫AJA\ni·JA\nj−κB/summationdisplay\n≪ij≫BJB\ni·JB\nj\n+κ/summationdisplay\n/angbracketleftij/angbracketrightJA\ni·JB\nj (1)\nwhere the sums are over all sites of a three-dimensional\ncubic lattice: /angbracketlefti,j/angbracketrightdenotes the sum over the near-\nest neighbors, ≪i,j≫A(B)denotes the sum over the\nsites of A(B) sublattice. The first two terms describe\nthe ferromagnetic Heisenberg intra-sublattice exchange\nκA>0,κB>0, while the third term describes the inter-\nsublattice exchange which is antiferromagnetic κ >0.\nTo proceed we use the Holstein-Primakoff representa-\ntion of the total angular momentum vectors JA\nj(a+\nj,aj)\nandJB\nj(b+\nj, bj), where a+\nj, ajandb+\nj, bjare Bose fields.\nIn terms of these fields and keeping only the quadratic\nand quartic terms, the effective Hamiltonian Eq.(1)\nadopts the form , H=H2+H4where\nH2=jAκA/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\n+jBκB/summationdisplay\n≪ij≫B/parenleftbig\nb+\nibi+b+\njbj−b+\njbi−b+\nibj/parenrightbig\n(2)2\n+κ/summationdisplay\n/angbracketleftij/angbracketright/bracketleftBig\njAb+\njbj+jBa+\niai−/radicalbig\njAjB/parenleftbig\na+\nib+\nj+aibj/parenrightbig/bracketrightBig\nH4=1\n4κA/summationdisplay\n≪ij≫A/bracketleftbig\na+\nia+\nj(ai−aj)2+(a+\ni−a+\nj)2aiaj/bracketrightbig\n+1\n4κB/summationdisplay\n≪ij≫B/bracketleftbig\nb+\nib+\nj(bi−bj)2+(b+\ni−b+\nj)2bibj/bracketrightbig\n+1\n4κ/summationdisplay\n/angbracketleftij/angbracketright/bracketleftBigg/radicalBigg\njA\njB/parenleftbig\naib+\njbjbj+a+\nib+\njb+\njbj/parenrightbig\n(3)\n+/radicalBigg\njB\njA/parenleftbig\na+\niaiaibj+a+\nia+\niaib+\nj/parenrightbig\n−4a+\niaib+\njbj/bracketrightBigg\nand the terms without operators are dropped.\nThe next step is to represent the Hamiltonian in\nHartree-Fockapproximation H≈HHF=Hcl+Hqwhere\nHcl= 12NκAj2\nA(uA−1)2+12NκBj2\nB(uB−1)2\n+ 6NκjAjB(u−1)2(4)\nwhereN=NA=NBis the number of sites on a sub-\nlattice. The Hamiltonian Hqcan be obtained from the\nHamiltonian Eq.(2) replacing κAwithκAuA,κBwith\nκBuBandκwithκu, whereuA,uBanduare Hartree-\nFock parameters, to be determined self-consistently. It\nis convenient to rewrite the Hamiltonian in momentum\nspace representation\nHq=/summationdisplay\nk∈Br/bracketleftbig\nεa\nka+\nkak+εb\nkb+\nkbk−γk/parenleftbig\na+\nkb+\nk+bkak/parenrightbig/bracketrightbig\n,\n(5)\nwhere the wave vector kruns over the reduced first Bril-\nlouinzone Brofacubiclattice. Thedispersionsaregiven\nby equalities\nεa\nk= 4jAκAuAεk+ 6jBκu (6)\nεb\nk= 4jBκBuBεk+ 6jAκu\nγk= 2κu/radicalbig\njAjB(coskx+cosky+coskz)\nwithεk= 6−cos(kx+ky)−cos(kx−ky)−cos(kx+kz)−\ncos(kx−kz)−cos(ky+kz)−cos(ky−kz).\nTo diagonalize the Hamiltonian one introduces new\nBose fields αk, α+\nk, βk, β+\nkby means of the transforma-\ntion\nak=ukαk+vkβ+\nk, bk=ukβk+vkα+\nk(7)\nwith coefficients ukandvkreal functions of the wave\nvectork\nuk=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nεa\nk+εb\nk/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk+ 1\n,(8)\nvk=sign(γk)/radicalbig\nu2\nk−1. The transformed Hamiltonian\nadopts the form\nHq=/summationdisplay\nk∈Br/parenleftBig\nEα\nkα+\nkαk+Eβ\nkβ+\nkβk+E0\nk/parenrightBig\n,(9)with new dispersions Eα\nk=E+\nk,Eβ\nk=E−\nkwhere\nE±\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk±(εa\nk−εb\nk)/bracketrightbigg\n(10)\nand vacuum energy\nE0\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−εb\nk−εa\nk/bracketrightbigg\n(11)\nTo obtain the system ofequations for the Hartree-Fock\nparameters we consider the free energy of a system with\nHamiltonian HHFEqs.(4,9)\nF= 12κAj2\nA(uA−1)2+12κBj2\nB(uB−1)2\n+ 6κjAjB(u−1)2+1\nN/summationdisplay\nk∈BrE0\nk (12)\n+1\nN/summationdisplay\nk∈Br/bracketleftBig\nln/parenleftBig\n1−e−βEα\nk/parenrightBig\n+ ln/parenleftBig\n1−e−βEβ\nk/parenrightBig/bracketrightBig\n.\nThen the three equations ∂F/∂uA= 0, ∂F/∂uB= 0,\nand∂F/∂u= 0 adopt the form\nu1= 1−1\n6j11\nN/summationdisplay\nk∈Brεk/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\nu2= 1−1\n6j21\nN/summationdisplay\nk∈Brεk/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\nu= 1−1\nN/summationdisplay\nk∈Br/bracketleftbigg1\n2j1/parenleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/parenrightBig\n(13)\n+1\n2j2/parenleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/parenrightBig\n−2\n3κu/parenleftBig\n1+nα\nk+nβ\nk/parenrightBig(coskx+cosky+coskz)2\n/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk\n\nwherenα\nkandnβ\nkare the Bose functions of αandβex-\ncitations. Hartree-Fock parameters, the solution of the\nsystem of equations (13), are positive function of T/κ,\nu1(T/κ)>0, u2(T/κ)>0 andu(T/κ)>0. Utilizing\nthese functions, one can calculate the spontaneous mag-\nnetization MA=< JA\n3>andMB=< JB\n3>ofMn\nandVions respectively. In terms of the Bose functions\nof theαandβexcitations they adopt the form\nMA=jA−1\nN/summationdisplay\nk∈Br/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\n(14)\nMB=−jB+1\nN/summationdisplay\nk∈Br/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\nThe magnon excitations in the effective theory are a\ncomplicatedmixtureofthesublattices’ AandBtransver-\nsal fluctuations. As a result, the magnon fluctuations\nsuppress in a different way MnandVmagnetic orders.\nQuantitatively, this depends on the coefficients ukand3\nvkin Eq.(14). At characteristic temperature T∗Vspon-\ntaneous magnetization becomes equal to zero, while Mn\nspontaneous magnetization is still nonzero. Above T∗\nthe system of equations (13) has no solution, and one\nhas to modify the spin-wave theory.\nOnce suppressed, the magnetic moment of Vions can-\nnot be restored increasing the temperature above T*. To\nformulate this mathematically we modify the spin-wave\ntheory using the idea on description of paramagnetic\nphase of 2D ferromagnets ( T >0) by means of modi-\nfied spin-wave theory [6, 7]. We consider two-sublattice\nsystem and to enforce the magnetic moments on the two\nsublattices to be equal to zero in paramagnetic pase we\nintroduce two parameters λ1andλ2[8]. The new Hamil-\ntonian is obtained from the old one Eq.(1) adding two\nnew terms\nˆH=H−/summationdisplay\ni∈AλAJA\n3i+/summationdisplay\ni∈BλBJB\n3i(15)\nIn momentum space, the Hamiltonian adopts the form\nEq.(5) with new dispersions ˆ εa\nk=εa\nk+λAand ˆεb\nk=\nεb\nk+λB, where the old dispersions are given by equal-\nities (6). We utilize the same transformation Eq.(7) with\ncoefficients ˆ ukand ˆvkwhich depend on the new disper-\nsions in the same way as the old ones depend on the old\ndispersions Eq.(8). In terms of the αkandβkbosons, the\nHamiltonian ˆHqadopts the form Eq.(9) with dispersions\nˆEα\nkandˆEβ\nk, which can be written in the form Eq.(10)\nreplacing εa\nkandεb\nkwith ˆεa\nkand ˆεb\nk.\nWe have to do some assumptions for parameters λA\nandλBto ensure correct definition of the two-boson the-\nory. For that purpose, it is convenient to represent the\nparameters in the form λA= 6κjB(µA−1)λB=\n6κjA(µB−1). In terms of the parameters µAandµB,\nthe dispersion reads ˆ εa\nk= 4jAκAεk+ 6κjBµAˆεb\nk=\n4jBκBεk+ 6κjAµB. The conventional spin-wave the-\nory is reproduced when µA=µB= 1(λA=λb= 0).\nWe assume µAandµBto be positive ( µA>0, µB>0).\nThen, ˆεa\nk>0, ˆεb\nk>0, forallvalues ofthe wave-vector kif\nthe Hartree-Fock parameters are positive too. The Bose\ntheory is well defined if Eα\nk≥0, Eβ\nk≥0. This comes\ntrue ifµAµB≥1. In the case µAµB>1, bothαkand\nβkbosons are gapped excitations. In the particular case,\nµAµB= 1, long-range excitations (magnons) emerge in\nthe system.\nWe introduced the parameters λAandλB(µA,µB) to\nenforce the sublattice AandBspontaneous magnetiza-\ntions to be equal to zero in paramagnetic phase. We\nfind out the parameters µAandµB, as well as Hartree-\nFockparameters,asfunctionsoftemperature, solvingthe\nsystem of five equations, equations (13) and the equa-\ntionsMA=MB= 0, where the ordered moments have\nthe same representation as Eq.(14) but with coefficients\nˆuk,ˆvk, anddispersions ˆEα\nk,ˆEβ\nkintheexpressionsforthe\nBose functions. The numerical calculations show that for\nhigh enough temperature µAµB>1. When the temper-\nature decreases the product µAµBdecreases remaininglarger than one. The temperature at which the prod-\nuct becomes equal to one ( µAµB= 1) is the Neel tem-\nperature. Below TN, the spectrum contains long-range\n(magnon) excitations, thereupon µAµB= 1. It is conve-\nnient to represent the parameters in the following way:\nµA=µ, µ B= 1/µ. (16)\nIn ordered phase magnon excitations are origin of sup-\npressionofthemagnetization. Nearthezerotemperature\ntheir contribution is small and at zero temperature Mn\nandVspontaneous magnetizationreach their saturation.\nIncreasingthetemperaturemagnonfluctuationssuppress\nthe magnetic order of MnandVions in different way.\nAtT∗theVspontaneous magnetization becomes equal\nto zero. Increasing the temperature above T∗, the mag-\nnetic moment of the vanadium ions should be zero. This\nis why we impose the condition MB(T) = 0 ifT > T∗.\nFor temperatures above T∗, the parameter µand the\nHartree-Fock parameters are solution of a system of four\nequations, equations (13) and the equation MB= 0. We\nutilize the obtained function µ(T),uA(T),uB(T),u(T)to\ncalculate the spontaneous magnetization MAof theMn\nions as a function of the temperature. Above T∗,MA(T)\nis equal to the magnetization of the system.\nWe consider two-sublattice ferrimagnet with Mnions\non sublattice AandVions on sublattice B. The sub-\nlatticeAtotal angular moment is jA=sA= 5/2, and\ng-factorgA= 2. The sublattice Btotal angular momen-\ntum isjB=lB+sB= 5/2, andg-factorgB= 2 which\nis larger then atomic value, because of the anisotropy.\nThe magnetization of the system gAMA+gBMBas a\nfunction of the temperature is depicted in Fig.1 for pa-\nrameters κA/κ= 0.6 andκB/κ= 0.01\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s49/s50/s51/s52\n/s84/s42\n/s47/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78/s91\n/s66/s93\n/s84/s47\nFIG. 1: (color online)The magnetization as a function of T/κ\nfor parameters κA/κ= 0.6 andκB/κ= 0.01\nThe figure is in a very good agrement with the exper-\nimental ZFC magnetization curves [2, 3]. (see Fig.1 [9]).\nThe anomalous temperature dependence of the magne-\ntization, predicted by Neel, is reproduced, but there is4\nan important difference between Neel’s theory, interpre-\ntation of NMR results [2, 3], and the present modified\nspin-wave theory. Neel’s calculations predict a temper-\natureTNat which both the sublattice AandBmagne-\ntization become equal to zero. The modified spin wave\ntheory predicts to phases: at low temperatures (0 ,T∗)\nMnandVmagnetic orders contribute to the magnetiza-\ntion of the system, while at high temperatures ( T∗,TN)\nonlyMnions have non-zero spontaneous magnetization.\nThe vanadium electrons start to form magnetic moment\natT∗, and an evidence for this is the abrupt decrease\nof magnetization below T∗, which also indicates that the\nmagneticorderofvanadiumelectronsisanti-parallelwith\nthe order of Mnelectrons.\nTwo ferromagnetic phases where theoretically pre-\ndicted, very recently, in spin-Fermion systems, which\nobtain their magnetic properties from a system of lo-\ncalized magnetic moments being coupled to conducting\nelectrons [8]. At the characteristic temperature T∗, the\nmagnetization of itinerant electrons becomes zero, and\nhigh temperature ferromagnetic phase ( T∗< T < T C) is\na phase where only localized electrons give contribution\nto the magnetization of the system. An anomalous in-\ncreasing of magnetization below T∗is obtained in good\nagrement with experimental measurements of the ferro-\nmagnetic phase of UGe2[9].\nThe results of the present paper and the previous one\n[8] suggest that T∗transition from a magnetic phase to\nanother magnetic phase is a generic feature of the two\nmagnetic orders systems. The additional phase transi-\ntion demonstrates itself through the anomalous temper-\nature variation of the spontaneous magnetization, but\nit is important to discuss alternative experimental de-\ntections of T∗transition. This is why we consider the\nFC magnetization curves [2, 3]. For samples cooled in a\nfield (FC magnetization) the field leads to formation of\na single domain and, in addition, increases the chaotic\norder of the spontaneous magnetization of the vanadium\nelectrons, which is antiparallel with it. As a result the\naverage value of the vanadium magnetic order decreasesand does not compensate the Mnmagnetic order. The\nmagnetization curves depend on the applied field, and\ndoes not go to zero. For a larger field the (FC) curve\nincreases when temperature decreases below Neel tem-\nperature . It has a maximum at the same temperature\nT∗< TNas the ZFC magnetization, and a minimum at\nT∗\n1< T∗. Below T∗\n1the magnetization increases mono-\ntonically when temperature approaches zero.\nTheexperimentswith samplescooledinfield (FCmag-\nnetization) providea new opportunity to clarify the mag-\nnetism of the manganese vanadium oxide spinel. The\napplied field is antiparallel with vanadium magnetic mo-\nment and strongly effect it. On the other hand the\nexperiments show that there is no difference between\nZFCand FCmagnetizationcurveswhenthe temperature\nruns the interval ( T∗,TN) [2, 3]. They begin to diverge\nwhen the temperature is below T∗. This is in accor-\ndance with the theoretical prediction that the vanadium\nmagnetic moment does not contribute the magnetization\nwhenT > T∗, andT∗is the temperature at which the\nvanadium ions start to form magnetic order. Because of\nthe strong field, the vanadium bands are split and part\nof the magnetic orders are reoriented to be parallel with\nfield and magnetic order of Mnelectrons. The descrip-\ntion of this case is more complicate and requires three\nmagnetic orders to be involved. When T∗< T < T N\nonlyMnions have non zero spontaneous magnetization.\nAtT∗vanadium magnetic order antiparallel with mag-\nnetic order of Mnsets in and partially compensates it.\nBelowT∗\n1the reoriented magnetic orders give contribu-\ntion, which explains the increasing of the magnetization\nof the system when the temperature approaches zero.\nTo conclude, we note that a series of experiments with\ndifferent applied field could be decisive for the confirma-\ntion or rejection of the T∗transition. Increasing the ap-\nplied field oneexpectsincreasingof T∗\n1andwhen the field\nisstrongenough, sothat allvanadium electronsarereori-\nented, an anomalous increasing of magnetization below\nT∗would be obtained as within the ferromagnetic phase\nofUGe2[9].\n[*] Electronic address: naoum@phys.uni-sofia.bg\n[1] K. Adachi, T. Suzuki, K. Kato, K. Osaka, M. Takata, and\nT. Katsufuji, Phys. Rev. Lett., 95, 197202 (2005).\n[2] V. O. Garlea, R. Jin, D. Mandrus, B. Roessli, Q. Huang,\nM. Miller, A. J. Schultz, and S. E. Nagler, Phys. Rev.\nLett.,100, 066404 (2008).\n[3] S.-H.Baek, K.-Y Choi, A.P.Reyes, P.L. Kuhns, N.J.Curro ,\nV.Ramanchandran, N.S. Dalal, H. D. Zhou, and C.R.\nWiebe, J. Phys.:Condens.Matter, 20, 135218 (2008).[4] L. Neel, Ann. Phys., Paris, 3, 137 (1948).\n[5] W. P. Wolf, Rep. Prog. Phys., 24, 212 (1961).\n[6] M.Takahashi, Prog. Theor. Physics Supplement 87, 233\n(1986).\n[7] M.Takahashi, Phys. Rev. Lett. 58, 168 (1987).\n[8] N. Karchev, Phys. Rev. B 77, 012405 (2008).\n[9] C. Pfleiderer and A. D. Huxley, Phys. Rev. Lett., 89,\n147005 (2002)."    },    {        "title": "0906.4889v1.Magnetic_behavior_of_nanocrystalline_ErCo2.pdf",        "content": " 1      (JPCM 2009, in press) \n \nMagnetic behavior of nanocrystalline ErCo 2 \n \nSitikantha D Das, Niharika Mohapatra, Kartik K Iyer , R.D. Bapat, and E.V. \nSampathkumaran * \nTata Institute of Fundamental Research, Homi Bhabha  Road, Colaba, Mumbai 400005, \nIndia \n \nWe have investigated the magnetic behavior of the n anocrystalline form of a well-known \nLaves phase compound, ErCo 2 - the bulk form of which has been known to undergo an  \ninteresting first-order ferrimagnetic ordering near  32 K – synthesized by high-energy \nball-milling. It is found that, in these  nanocryst allites, Co exhibits ferromagnetic order at \nroom temperature as inferred from the magnetization  data.  However, the magnetic \ntransition temperature for Er sublattice remains es sentially unaffected  as though the \n(Er)4f-Co(3d) coupling is weak on Er magnetism.  Th e net magnetic moment as \nmeasured at high fields, say at 120 kOe, is signifi cantly reduced with respect to that for \nthe bulk in the ferrimagnetically ordered state and  possible reasons are outlined. We have \nalso compared the magnetocaloric behavior  for the bulk and the nano particles.  \n \nPACS  numbers: 73.63.Bd; 75.30.Kz; 75.50.Tt; 71.20. Eh \n  2 I. Introduction \nThe Laves phase compounds, RCo 2 (R= Rare-earths), attracted considerable \nattention in the literature during last few decades , as this family served as a model system \nfor itinerant electron metamagnetism (IEM) [1]. For  instance, YCo 2 and  LuCo 2, the \nexchange-enhanced Pauli-paramagnets, have been foun d to undergo IEM at about 700 \nkOe [2, 3]. With respect to those members in which R carries a localized-moment, quite \ninterestingly, the magnetic transition appears to b e first order for R= Dy, Ho, and Er, \nwhereas for many other members of this series, it i s second order. It is generally accepted \nthat the first-order nature of the transition for t he former members is due to the onset of \nIEM at the Co site induced by the molecular field a rising from the magnetic ordering of R \nsub-lattice. This finding led to intense investigat ions [4], particularly with respect to \npotential magnetic refrigeration applications [5]. \n While all the reports on these heavy rare-earth me mbers  is on bulk form (single \ncrystals and poly crystals), very little work has b een carried out on nanocrystalline form, \nas it is not usually easy to stabilize the nanopart icles of rare-earth intermetallics without \nshell protection. In this respect, there is a recen t claim on the synthesis of oxidation-\nresistant DyCo 2 nano particles by arc-discharge process [6] with a  promising low-\ntemperature refrigeration applications.  In this ar ticle, we present the results of our \nmagnetic investigation on the nanoparticles of  ErC o 2 synthesized by high-energy ball-\nmilling [7], the reason being that   this compound in particular attracted considerable \nattention in the recent literature [8-21]. This com pound has been found to order \nferrimagnetically at (T C=) 32 K due to antiparallel aligment of Er and Co m oments (with \na moment on Co being about 1 µB) [8] with extreme sensitivity of T C to small amounts of \nmetallic impurities [9, 10]. This compound is well- studied for its magnetocaloric effect \n(MCE) [11-13].  The so-called  ‘inverse IEM’ phenom enon has been noted for some \ndegree of chemical doping [20]. \n \nII.  Experimental details  \n The bulk sample was prepared by arc melting stoich iometric amounts of high-\npurity (>99.9%) constituent elements in an atmosphe re of argon. The molten ingot was \nannealed at 900 C for 60 h in an evacuated sealed q uartz tube. The ingot was then \nsubjected to high-energy ball-milling  (Fritsch pul verisette-7 premium line) for 2 ½ h \nemploying zirconia vials with balls of 5mm diameter  (balls-to-material mass ratio: 5) \nwith an operating speed of 500 rpm in an atmosphere  of toluene. The specimens were \ncharacterized by x-ray diffraction (XRD), scanning electron microscope (SEM, JEOL, \nJSM 840A), transmission electron microscope (TEM, T echnai 200 kV) and energy-\ndispersive x-ray analysis. A commercial magnetomete r (Quantum Design) was employed \nto measure magnetization. \n \nIII. Results and discussion  \n The XRD patterns are shown in figure 1 for both th e ingot and milled specimens. \nThe patterns are shifted along y-axis for the sake of clarity. Otherwise the background \nintensity and shape of the background are found to be essentially the same for both the \nspecimens, thereby indicating that the milling does  not result in amorphous formation.  \nThe sharp XRD lines in addition endorse that the mi lled samples are crystalline. We did \nnot find any noticeable change in the lattice const ants (as determined from XRD) due to  3 ball-milling. As well-known in the field of metallu rgy, the milling reduces the intensity \nof the XRD lines. The XRD lines are broadened as de monstrated in an inset of figure 1 \nand  the average particle-size, if estimated from t he width of the most intense line, turns \nout to be  about 30 nm . The nano-sized nature of the specimen was ascertain ed from \nTEM pictures (see figure 1). However, the TEM pictu re suggests that the particle size is \nsmaller than that estimated from XRD. We attribute this discrepancy to the fact that the \nstrain-contribution to broadening of XRD lines coul d not be separated due to non-linear \nWilliamson-Hall plots (not shown here).  The electron diffraction pattern inserted in  \nfigure 1   confirms that the particles belong to Er Co 2 phase. We have also carried out \nReitveld fitting of the XRD pattern for the milled specimen and the difference  between \nthe experimental and fitted pattern  is  shown  in figure 1 to endorse proper phase \nformation.   There is no evidence in the XRD for th e formation of any other phase due to \nball-milling. From the backscattered image of  SEM,  we further confirmed the absence of \nany other phase in the nanospecimen. In addition, t he stoichiometry of the nanospecimen \nwas established to correspond to ErCo 2 from energy dispersive x-ray analysis performed \nwith this SEM instrument. \n The results of magnetization measurements  as a fu nction of temperature  ( T) and \nmagnetic-field ( H) for the zero-field-cooled (ZFC from 300 K) and fi eld-cooled (FC) \nconditions of the milled specimen are shown  in fig ures 2 and 3 respectively. The data \nobtained on the parent ingot is also included for c omparison. In the ingot employed to \nprepare the fine particle, we see the features attr ibutable to the onset of magnetic ordering \nnear 36 K as inferred, say, from  the peak temperat ure in the M(T)  obtained in a field of \n100 Oe. It is straightforward to see the changes (s ee figure 2) those have occurred in the \nshapes of M(T) curves after milling. Most notably, the sharpness  of the features seen at \nTC for the bulk specimen is absent for the nano parti cles; see, for instance, the features in \nthe ZFC curve  ( H= 100 Oe). This is attributable to the broadening o f the magnetic \ntransition due to  defects introduced by ball-milli ng. In addition to above features near  \nmagnetic transition temperature, there is a bifurca tion of ZFC-FC curves extending \nbeyond 36 K in the  nano specimen, which is indicat ive of another magnetic anomaly at \nhigher temperatures. This aspect is addressed in th e next paragraph on the basis of M(H) \nanomaly.  \nFor the bulk specimen, the M(H) plots  (figure 3) are hysteretic below  20 kOe in  \nthe magnetically ordered state and the value of M  tends towards saturation beyond 30 \nkOe  with the saturation moment (say, about 7.2 µB per formula unit at 1.8 K, see also \nRef. 13) being less than that expected for trivalen t Er ions due to ferrimagnetism. As \nexpected, in the paramagnetic state, say, at 50 K, the plot of M(H) is linear.   On the other \nhand, in the nanoform, hysteretic nature of the cur ve persists even near 50 kOe, for \ninstance, at 1.8 and 30 K (see figure 3) without an y evidence for saturation; in addition, \nhigh-field magnetic moment is significantly reduced . On the basis of these M(H) \nproperties, we infer that  ferrimagnetism  is retai ned in the nano form. If one performs a \nlinear extrapolation of the high-field data to zero -field at 1.8 K, assuming that the \nmoment on Er is unchanged, we arrive at a value of   at least  2 µB on each Co ion for the \nnanoform. Thus, if one solely attributes the observ ed reduction of high-field magnetic \nmoment to Co, then there is an  enhancement in the magnetic moment on Co following \nEr sub-lattice ordering. However,  it is also possi ble that a significant surface spin \ndisorder due to lack of symmetry and reduced coordi nation  at the surface is also  4 responsible for the observed behavior, as known, fo r instance, for the nanoform of γ-\nFe 2O3 [22]. The finding we stress for the nanoform is th at, well above 36 K, the M(H) \nplots show a dramatic increase for initial applicat ions of magnetic field with a linearity at \nhigher fields, as though there  is a ferromagnetic component  superimposed over a \nparamagnetic component. The plots are hysteretic at  low fields as shown for 130 K and \n300 K in the inset of figure 3. These findings reve al that Co in the nanoparticle exhibits \nferromagnetic character at room temperature with Er  remaining paramagnetic. The value \nof the magnetic moment on Co turns out to be ~0.12 µB/Co, at least above 32 K. We \nbelieve that when Er is paramagnetic, it is possibl e that Co spins undergo ‘canted’ \nferromagnetic alignment, as magnetic moment value i s lower than that observed [7] in \nYCo 2. In support of a complex magnetic ordering of Co, the plot of M/H exhibits an \nincrease above 200 K as shown in the inset of figur e 2. It is of interest to carry out \nneutron diffraction studies to understand this issu e better. In any case, it is clear that there \nis a  magnetic ordering of Co in the nano particles  of ErCo 2 above room temperature.  It \nis to be stressed that, despite high magnetic order ing temperature of Co, the T C for the Er \nsublattice is not noticeably influenced.  \nWe find dramatic modifications of Arrott plots in t he temperature range where Er \nsub-lattice undergoes magnetic ordering (see figure  4). For the ingot, these plots have \nbeen known to be complex with negative slopes and i nflexions due to metamagnetic \ntransitions the exact nature of which appears to be  sample dependent [see, for instance, \nRefs. 13, 22]. In the nano specimen, though  M2 increases monotonically with H/M, the \nplot is still complex which is attributable to a po ssible interference from Co sub-lattice \nordering at higher temperatures.  \n We  compare  MCE properties in figure 5 for a typical change of magnetic field. \nFor this purpose, we in fact obtained M(H) curves at close intervals of temperatures and \nderived entropy change, ∆S, for a given variation of H (from zero field), on the basis of  \nwell-known Maxwell’s relation (see, for instance, R ef. 5). The results are shown in the \nfigure for selected fields. The results obtained fo r the ingot are in good agreement with \nthose known in the literature [11-13, 21]. It is di stinctly clear from figure 5 that, in the \ncase of nanospecimen,   though ∆S values at the peak are relatively reduced at T C, the \ncurve is broad extending over a wider temperature r ange. The magnetic refrigeration \ncapacity [for the definition we use, see Ref.  23],  turns out to be about half of that seen \nfor the bulk form [about 80 J/mol].  \n \nIV. Conclusion \n The compound, ErCo 2, attracted attention in the literature from the an gle of \npotential magnetic refrigeration applications as we ll as of interesting magnetic anomalies. \nWe have synthesized the nanocrystals of this materi al, to our knowledge, for the first time \nand studied its magnetic behavior. The Co sublattic e is found to be magnetically ordered  \nat room temperature in these fine particles unlike in bulk. It is notable that the transition \ntemperature for the Er sub-lattice is not influence d at all by the high-temperature \nmagnetism of Co sub-lattice, as though Er-Co magnet ic coupling is weak in the \nnanoform. There is a significant reduction in the h igh-field magnetic moment compared \nto that of bulk when Er sub-lattice orders at low t emperatures.   Though the magnetic \nrefrigeration capacity becomes half of the bulk mat erial in the temperature range of  5 interest, the entropy change is spread over a rathe r larger temperature range when \ncompared to that in bulk form.  \nThis family of binary compounds are characterized b y sharp density of 3d states \nof Co in the vicinity of Fermi level and, in the ca se of pseudo-binary compounds based \non YCo 2, even defects have been proposed to have  a profou nd effect on this feature and \nhence on magnetism [24]. It is not clear whether si milar effect is responsible for the \nmagnetism of Co in the ball-milled ErCo 2 as well. Thus, it appears that this family could \nbe a good example for probing an interplay between the defects, electronic structure  and \nstrong electron correlations.        \n      \n*E-mail: sampath@mailhost.tifr.res.in  \n1. E.P. Wohlfarth and P. Rhodes, Philos. Mag. 7, 1817 (1962). \n2. T. Goto, K. Fukamichi, T. Sakakibara, and H. Kom atsu, Solid State Commun. 72, \n945 (1989). \n3.  H. Yamada, T. Tohyama and M. Shimizu J. Magn. Magn.  Mater. 70, 44 (1987); T. \nYokoyama et al., J. Phys.: Condens. Matter. 13 , 9281 (2001). \n4.  S. Khmelevskyi and P.Mohn, J. Phys.: Condens. Matte r 12 , 9453 (2000); N.H. Duc and \nT. Goto In: K.A. Gschneidner, Jr. and L. Eyring, Ed itors, Handbook on the Physics and \nChemistry of Rare Earths  Vol. 28 , North-Holland, Amsterdam (1999), p. 177 and \nreferences therein. \n5.  See, for a review, K.A. Gschneidner Jr, V.K. Pechar sky, and A.O. Tsokol, Rep. Prog. \nPhys. 68 , 1479 (2005).  \n6.  S. Ma, W.B. Cui, D. Li, N.K. Sun, D.Y. Geng, X. Jia ng, and Z.D. Zhang, App. Phys. \nLett. 92,  173113 (2008).  \n7.  Through this procedure, in the past, we were able s ynthesize stable form of nano particles \nof  YCo 2, with ferromagnetism at room temperature. See, S. Narayana Jammalamadaka, \nE.V. Sampathkumaran, V. Satya Narayana Murthy, and G. Markandeyulu, App. Phys. \nLett. 92,  192506 (2008). \n8.  M. Moon, W.C.  Koehler, and J. Farrell, J. Appl. Ph ys. 36, 978 (1965). \n9.  X.B. Liu and Z. Altounian, J. Appl. Phys. 103,  07B304 (2008). \n10.  M. Guillot and Y. Öner, J. Appl. Phys. 103 , 07E137 (2008). \n11.  T.D. Cuong, L. Havela, V. Sechovsky, A.V. Andreev, Z. Amold, J. Kamarad, and N.H. \nDuc, J. Appl. Phys. 81, 4221 (1997). \n12.  H. Wada, S. Tomekawa, and M. Shiga, Cryogenics 39, 915  (1999). \n13.  A. Gigurre, M. Foldeaki, W. Schnelle, and E. Gmelin , J. Phys.: Condens. Matter 11,  6969 \n(1999). \n14.  J. Herrero-Albillos, D. Paudyal, F. Bartolome, L.M.  Garcia, V.K. Pecharsky, K.A. \nGschneidner, Jr., A.T. Young, N. Jaouen, and A. Rog alev, J. Appl. Phys. 103,  07E146 \n(2008). \n15.  N. Ishimatsu, S. Miyamoto, H. Maruyama, J. Chaboy, M.A. Laguna-Marco, and N. \nKawamura, Phys. Rev. B 75, 180402(R) 2007. \n16.  O. Syshchenko, V. Sechovsky, M. Divis, T. Fujita, R . Hauser, and H. Fujii, J. Appl. Phys. \n89 , 7323 (2001). \n17.  O. Syschenko, T. Fujita, V. Sechovsky, M. Divis, an d H. Fujii, Phys. Rev. B 63 , 054433 \n(2001). \n18.  R. Hauser, E. Bauer, and E. Gratz, Phys. Rev. B 57 , 2904 (1998).  6 19.  A. Podlesnyak, Th.Strässle, J. Schefer, A. Furrer, A. Mirmelstein, A. Pirogov, P. Markin, \nand N. Baranov, Phys. Rev. B 66, 012409 (2002). \n20.  R. Hauser, E. Bauer, E. Gratz, H. Mueller, M. Rotte r, H.Michor, G. Hilscher, A.S. \nMarkosyan, K. Kamishima, and T. Goto, Phys. Rev. B 61 , 1198 (2000). \n21.  Z. Jun-Ding, Shen Bao-Gen, and S. Ji-Rong, Chinese Physics 16, 1817 (2007). \n22. T.N. Shendruk, R.D. Desautels, B.W. Southern, a nd J. van Lierop, \nNanotechnology, 18, 455704 (2007); E. Tronc, D. Fiorani, M. Nogues, A.M . \nTesta, F. Lucari, F.D’Orazio, J.M. Greneche, W. Wer nsdorfer, N. Galvez, C. \nChaneac, D. Mailly, and J.P. Jolivet, J. Magn. Magn . Mater. 262, 6 (2003). \n23.  N. Mohapatra, Kartik K Iyer and E.V. Sampathkumaran , Eur. Phys. J. B 63 , 451 \n(2008). \n24.  A.T. Burkov, E. Bauer, E. Gratz, R. Resel, T. Nakam a, and K. Yagasaki, Phys. \nRev. B 78 , 035101 (2008). \n \n \nFigure 1: \n(color online) X-ray diffraction patterns of the bu lk and nano crystals of ErCo 2. The fit \nobtained by Reitveld analysis in the case of nanosp ecimen is shown by continuous line \nand the difference between fit and experimental dat a points are also shown. In the  inset \n(a), the shapes of most intense lines are compared after  normalizing to respective peak \nheights. In the  insets (b) and  (c), TEM image and electron diffraction pattern  are sho wn. \n  7 \n \nFigure 2: \n(color online) Magnetization  ( M) divided by magnetic field ( H) as a function of \ntemperature ( T) for the bulk and nanocrystals of ErCo 2 for two values of externally \napplied magnetic fields. In the case of H= 100 Oe, the curves for the zero-field-cooled \nand field-cooled conditions of the nanoparticles ar e shown. The lines through the data \npoints serve as guides to the eyes. Inset highlight s increasing M/H behavior above 200 K \nfor the nanocrystals (shown along with the data for  the bulk). \n  8 \n \nFigure 3: \n(color online) Isothermal remnant behavior at selec ted temperatures for the bulk and \nnanocrystals of ErCo 2. The lines through the data points serve as guides  to the eyes. In \nthe inset, low-field hysteresis loops at 130 and 30 0 K for the nano crystals are plotted.  \n \n \nFigure 4: \n(color online) The Arrott plots for bulk and nanocr ystals of ErCo 2.  For the sake of \nclarity, we show only the lines through the data po ints.  9 \n \nFigure 5: \n(color online) The temperature dependence of entrop y change for different variations of \nmagnetic fields for the   bulk and nanocrystals of ErCo 2. The lines through the points \nserve as guides to the eyes. "    },    {        "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. 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Brabazon, Sinteri ng behavior of cobalt ferrite ceramic, Ceram Int, \n34 (2008) 15-21. \n \n "    },    {        "title": "1810.10404v1.Long_spin_coherence_length_and_bulk_like_spin_orbit_torque_in_ferrimagnetic_multilayers.pdf",        "content": "1 \n Long spin coherence  length and b ulk-like spin-orbit torque in ferrimagnet ic \nmultilayers  \n \nJiawei Y u1, Do Bang2†, Rahul Mishra1†, Rajagopalan Ramaswamy1, Jung Hyun Oh3, Hyeon-\nJong Park4, Yunboo Jeong5, Pham Van Thach2, Dong -Kyu Lee3, Gyungchoon Go3, Seo-Won \nLee3, Yi Wang1, Shuyuan Shi1, Xuepeng Qiu6, Hiroyuki Awano2, Kyung -Jin Lee3,4,5*, and \nHyunsoo Yang1* \n1 Department of Electrical and Computer Engineering, National University of Singapore, \n117576, Singapore  \n2 Toyota Technological Institute, Tempaku, Nagoya 468 -8511, Japan  \n3 Department of Materials Science and Engineering, Korea University, Seoul  02841, Korea  \n4 KU-KIST Graduate School of Conversing Science and Technology, Korea University, Seoul \n02841, Korea  \n5 Department of Semiconductor Systems Engineering, Korea University, Seoul 02841, Korea  \n6 Shanghai Key Laboratory of Special Artificial Macro structure Materials and Technology and \nSchool of Physics Science and Engineering, Tongji University, Shanghai 200092, China  \n†These authors con tributed equally to this work.  \n*e-mail: eleyang@nus.edu.sg, kj_lee@korea.ac.kr  \nSpintronics is a multidisciplinary field whose central theme is the active manipula tion of spin \ndegrees of freedom in solid -state system s. Ferromagnetic spintronics has been a main focus \nas it offer s non-volatile memory and logic applications  through current -induced  spin-\ntransfer torque s1-4. Enabling w ider application s of such  magnetic devices requires a lower \nswitching current for a smaller cell  while keeping the thermal stability  of magnetic cells for \nnon-volatility . As the cell size reduces , however, it becomes extreme ly difficult to meet th is \nrequirement with ferromagnets because spin-transfer torque  for ferromagnets is a surface  \ntorque  due to rapid spin dephasing5,6, leading to the 1/ferromagnet -thickness  dependence of \nthe spin -torque efficiency7. Requirement  of a larger switching current for a thicker and thus 2 \n more thermally stable ferromagnetic cell is the fundamental obstacle f or high -density  non-\nvolatil e application s with ferromagnets . Theories predicted that antiferromagnets have a \nlong spin coherence length due to the staggered spin order  on an atomic scale8,9, thereby \nresolv ing the above  fundamental limitation . Despite several spin -torque experiments on \nantiferr omagnets10-12 and ferrimagnetic alloys13-16, this prediction  has remained un explored . \nHere we report  a long spin coherence  length and associated bulk -like-torque characteristic \nin an antiferromagnetically coupled ferrimagnetic multilayer . We find that a transverse spin \ncurrent can pass through > 10 nm-thick ferrimagnetic Co/Tb multilayers whereas it is \nentirely absorbed by 1  nm-thick ferromagnetic Co/Ni mult ilayer. We also find that the \nswitching efficiency of Co/Tb multilayers partially reflects a bulk -like-torque characteristic \nas it increases with the ferrimagnet -thickness  up to 8 nm  and then decreases , in clear contrast \nto 1/thickness -dependence of Co/Ni multilayers . Our results on antiferromagnetically \ncoupled system s will invigorate researches towards energy -efficient spintronic technologies . \nThe spin -transfer torque ( STT) acting on ferromagnet s (FMs) is a surface torque, based on the \naveraging effect of STT5,6. We note that the same averaging effect occurs regardless of the spin -\ncurrent source , and the spin -orbit torque  (SOT)17,18, which we use in our experiment, is also a \nsurface torque for FMs  (Extended Data Fig. 1 and Methods ). When a transverse spin current with \na spin orientation  non-collinear with the magnetization  is injected into a FM, the electron spin \nprecesses rapidly in real space because the wave vectors of the majority (↑) and minority (↓) spins \nat the Fermi surface ar e different (i.e.,  \nFFkk ). The p recession wavelength s are different for \ndifferent incident angle s of electron s (i.e., the direction of  wave vector  k), leading to rapid spin \ndephasing when summing over  all current -carrying  k-states . As a result , the k-integrated  transverse \nspin c urrent decays to zero within a distance from the FM surface , called the ferromagnetic 3 \n coherence length  (spin coherence length, more generally) , \nc F F kk .19 As \nF Fk k  \nbecomes larger for larger exchange splitting, λc is only a few angstroms i n strong  FMs (e.g. cobalt \nor iron) for which the STT is almost a surface torque.  \nTheories predicted that the spin coherence length is very long in antiferromagnets (AFMs) \nbecause of the staggered spin order on an atomic scale8,9. We use the term of “bulk -like-torque” to \ndescribe the characteristic of spin -torque for AFMs, i.e., spin -current absorption on a larger \nthickness, in contrast to the surface -torque of FMs. A semi -classical explanatio n of bulk -like-\ntorque  is that for conduction electron spins, the moments with alternating orientation on an atomic \nscale are seen as the exchange interactions with alternating signs. As a result, an ideal AFM  has \nzero net  effective exchange interaction whe n averaged ove r two sub -lattices and thus has  an \ninfinitely long  λc, yielding the bulk -like-torque characteristic.  Several experiments have \ninvestigated on STT/SOT  effect s in systems including AFMs10-12 and more recently on \nferrimagnetic alloys13-16, but not on the long spin coherence length and associated bulk -like-torque \ncharacteristic.   \nWe qualitatively illustrate the spin coherence length in FMs and FIMs  (or AFMs)  based on the \nspin precession around the local exchange field. Neglecting the spin relaxation, dynamics of non -\nequilibrium spin density s is described by \nex t Hs s  / , where \n  is the gyromagnetic ratio, \nHex is the effective exchange field  that is aligned along the local magnetic moment m. Assuming \n 0, cos,sins\n and \nz H ˆHex , this equation of motion transforms to \nH t t   / / , \nwhere the sign of  spin precession angle \n  follows the sign of H and thus the sign of m (Fig. 1a \nand b). In a FM, an electron spin propagating along the x-direction continuously precesses in the \nsame sense because of the homogeneous exchange field (Fig. 1c). On the other hand, in a FIM, an \nelectron spin precesses counter -clockwise on a lattice corresponding to a positive exchange f ield, 4 \n whereas it precesses clockwise on the next lattice corresponding to a negative exchange field. As \na result, the period (or wavelength) of spin precession in FIMs is longer than that in FMs, resulting \nin much less spin dephasing.  \nIn order t o verify th e theoretical prediction  of long spin coherence length , we perform \nexperiments with a  ferrima gnet (FIM) , i.e., Co/Tb multilayers  where both Co and Tb layers are \natomically thin and their moments are coupled antiferromagnetically. We choose a FIM, instead \nof an AFM , for following two reasons. One is that Co/Tb  multilayer s can show a longer λc than a \nFM because of the antiferromagnetic alignment of Co and Tb moments ( Extended Data Fig. 1 and \nMethods ), thereby exhibiting a feature of the bulk -like-torque characteristic. As explained above, \nthe STT efficiency of  a FM is inversely proportional to the FM -thickness whereas that of an ideal \nAFM is independent of the AFM -thickness. As λc of FIM is located between those of FM and \nAFM, it is expected that the STT efficiency of a FIM first increases and then decreases with the \nFIM-thickness. The other reason to choose FIM s is that various  measurement methods established \nfor FM s are applicable  to FIM s because of nonzero net moment13,20. However, t he choice of  FIM \nalso results in a difficulty. FIMs commonly show a thickness -dependent variation of magnetic \nproperties21, as also observed in Co/Tb multilayer s (Extended Data Fig. 4 and Methods ), which \nmakes  a quantitative analysis of spin transport difficult. Even with this difficulty, our thickness -\ndependent S OT measurement s combined with spin pumping measurement s support a long spin \ncoherence length and associated bulk -like-torque characteristic in ferrimagne tic Co/Tb multilayer s, \nas we show below . \nWe fabricate  perpendicularly magnetized ferromagnetic [Co/Ni ]N and ferrimagnetic [Co/Tb ]N \nmultilayers  (Fig.  2a, b; see Methods for details ), where the total thickness varies with changing the  \nrepetition number N. Both Co/Ni and Co/Tb multilayers have an additional Pt layer, and an in -5 \n plane current generates SOT s. We use the harmonic Hall voltage measurements to quantify the \nstrength of SOT effective fields22,23. The longitudinal and transverse measurement schematics are \nillustrated in Fig. 2c and d, respectively. Representative results for the longitudinal (blue line) and \ntransverse (red line) second h armonic voltage s (V2f) from Co/Ni ( N = 2) and Co/Tb ( N = 5) devices \nare shown in Fig. 2e and f, respectively . The anomalous Nernst effect  is corrected in  the V2f data22. \nWe observe a clear V2f in the longitudinal configuration  (H//I), which is mostly determined by the \nanti-damping SOT22,23. The opposite  V2f signs in the H//I case  for Co/Ni and Co/Tb multilayers  \nindicate  that the Pt layer is the source of spin currents, as it is pl aced on top of the Co/Tb multilayer, \nbut under the Co/Ni multilayer. In order to rule out the contribution from pure bulk Co/Tb to  SOT, \nwe conduct a control experiment without and with the spin current source, Pt ( Extended Data Fig. \n5, 6 and Methods ). We find that there is no noticeable current -induced SOT without the Pt layer, \nsuggesting that the Co/Tb bulk itself cannot directly contribute to SOTs.  \nWe extract the spin -orbit effective fields, HL and HT, by fitting  V2f,23 where HL and HT \ncorrespond to the anti -damping (longitudinal) and field -like (transverse) component s of SOT s, \nrespectively. The planar Hall effect is considered for the fitting ( Extended Data Fig. 8 and \nMethods ). Devices with different N, corresponding to different  thickness es, tFM or tFIM, have been \nmeasured. Absolute SOT effective fields normalized by the current density in the Pt layer (HL/T/J) \nare presented in Fig. 3a and b for the Co/Ni and Co/Tb systems, respectively. We find that both \nHL and HT of Co/Ni multilayers decrease as tFM increases , consistent with the surface -torque \ncharacteristic expected for FMs. However, Co/Tb multilayers  show an entirely different trend , in \nwhich both HL and HT increase  up to tFIM of 7.9  nm and then decrease for thicker samples .  \nWe estimate the effective spin Hall angle \n J teM HFIM FMS L eff / 2/  , where e is the electron \ncharge and ħ is the reduced Planck’s constant,  with considering thickness -dependent variation of 6 \n the saturation magnetization MS and current density J in the Pt layer . The Co/Tb multilayer shows \na significant MS variation with the minimum at tFIM of 6.6 nm ( inset of Fig. 3d). We find that θeff \nof Co/Ni multilayer is nearly constant with tFM (Fig. 3c). Similar to the tendency of HL/J, θeff of \nCo/Tb multilayer increases  up to tFIM = 9.9 nm and then decreases for a thicker sample  (Fig. 3d). \nBesides the distinct thickness -dependence of θeff, another interesting observation is that  the Co/Tb \nmultilayer shows a larger θeff than Co/Ni multilayer (θeff of Co/Tb multilayer = 2. 1 at tFIM of 9.9 \nnm and the average θeff of Co/Ni multilayer = 0. 2 ± 0.05). We note that a model calculation with \nconsidering a thickness -dependent variation of the sd exchange in FIMs shows qualitatively \nsimilar trends  with the experimental ones ( Extended Data Fig. 2 and Methods), even though the \nmodel is too simple to capture all the details of FIMs. Nevertheless, this qualitative agreement \nbetween model and experiment results indicat es that the distinct behavior  of θeff of Co/ Tb \nmultilayer would originate from a combined effect  of long spin coherence length and thickness -\ndependent property variation.   \nAs an independent test, we perform SOT switching experiments with applying an external field \n(Hext) in the current direction ( θ = 0°) for deterministic switching. The insets of Fig. 3e and f show \nthe representative current -induced switching data obtained from Co/Ni ( N = 2) and Co/Tb (N = 5) \nsamples , respectively . As the switching is governed by domain nucleation and propagation  in large \nsamples ( i.e., Hall bar width = 10 m), we estimate  the STT efficiency  \nJ Hp/ , where Hp is \nthe domain wall depi nning field24. Figure  3e and f show η as a function of tFM and tFIM, respectively. \nFor Co/Ni multilayers, η decreases with tFM, whereas for Co/Tb multilayers it increases  and then \ndecreases  with tFIM, following similar trends to SOT effective fields (Fig. 3a and b). We note that \nrecently reported fast dynamics at the angular momentum compensation condition in FIMs20 \nwould affect the switching data, but not the harmonic Hall data.  7 \n As different approaches for the  estimat ion of  spin torque  efficiency show  qualitatively similar \ntrends, it indicate s that S OT for Co/Tb mul tilayer s is not a surface  torque. Moreover, the observed \nthickness -dependence of spin torque  efficiency is  qualitatively  consistent with the model \ncalculation ( Extended Data Fig. 2 and Methods)  for the bulk -like-torque characteristic in FIMs ; it \nfirst increase s and then decrease s with the FIM-thickness. However, because of the thickness -\ndependent  property  variation s in Co/Tb multilayers  (inset of Fig. 3d and Methods ), this result is \nnot yet conclusive but it is still possible that another unknown mechan ism is responsible for the \ndistinct thickness -dependence observed in Co/Tb multilayers.  \nIn order t o resolve this ambiguity, we perform additional spin pumping experiment s to estimate \nthe spin coherence length λc. We measure a spin -pumping -induced inverse spin Hall voltage (VISHE) \nfor substrate/Pt(10) /[FIM or FM] /Cu(2.4)/Co(20) structure s (numbers in nanometers ; FIM  = \n[Co(0.32)/Tb(0.34)] N and FM  = [Co(0.3)/Ni(0.6)] N) as shown in Fig. 4a. In these structures, the \nCo/Ni and Co/Tb multilayers are perpendicularly magnetized , whereas the top thick Co layer has \nan in -plane magnetization. In the spin pumping setup  (Fig. 4b; see Methods for details) , the top \nCo layer generates a spin -pumping -induced spin current with an in -plane spin polarization (thus \ntransverse to Co/Ni or Co/Tb magnetization direction), which passes through the Cu layer and \nenters the Co/Ni or Co/Tb layer. If λc of the Co/Tb  multilayer is long, it is expected that a transverse \nspin current passes through the Co/Tb layer without much spin dephasing and reaches the bottom \nspin sink, Pt , and s ubsequently, VISHE is generated  by the inverse spin Hall effect of Pt . On the \nother hand, VISHE is expecte d to be negligible for a thick Co/Ni  multilayer  because a transverse \nspin current is almost absorbed at the [Co/Ni]/Cu interface . Therefore, the measurement of VISHE \nversus FIM - or FM -thickness provides an estimate of λc. 8 \n We find that the experimental results are consistent with th is expectation. In Fig. 4c, black \nsymbols are the data from a reference Pt/Cu/Co sample , which  show s the largest VISHE signal  at an \nin-plane bias field Hb. For the Co/Ni -based structure, VISHE signal becomes negligible at a Co/Ni \nthickness of 0.9 nm (blue symbols). In contrast, VISHE signal for the Co/Tb -based structure is finite \nat a much thicker Co/Tb (red symbols, Co/Tb thickness = 5.3 nm as an example). VISHE signal \ndisappears when excluding the top Co layer from the Co/Tb -based structure (green symbol), \nproving that the perpendicularly magnetized Co/Tb itself does not generate a VISHE signal. In Fig. \n4d, spin pumping results are summarized for a wide thickness range of Co/Tb multilayer. It shows \nthat VISHE signal is finite even at 13 nm -thick Co/Tb . This result evidenc es a long spin coherence  \nlength in ferrimagnetic Co/Tb multilayers .  \nThe spin pumping and spin torque are connected through the Onsager reciprocity25. Therefore, \nthe long spin coherence length observed in spin pumping experiment s suggests that the bulk-like-\ntorqu e characteristic must be present  in spin torque  experiment s at least partially . Given that the \nthickness -dependent change in the spin torque efficiency follows the trend expected for the bulk -\nlike-torque characteristic (Fig. 3), the spin pumping experiment s combined with the spin torque \nexperiment s allow us to conclude that the antiferromagnetically coupled FIMs show a long spin \ncoherence length and associated bulk -like-torque characteristic.  We note that this bulk -like-torque \ncharacteristic and equivalentl y long spin coherence length are also observed for FIM alloys \n(Extended Data Fig. 9 and Methods), which would relate to some ordering in FIM alloys26,27. The \nresults were supported by model calculation using a ferrimagnetic alloy  (Extended Data Fig. 2 and \nMethods ) in which an ordered alloy show s a longer spin coherence length than a random alloy.  \nThese salient features make antiferromagnetically coupled FIMs attractive for low -power non -\nvolatile applications. We expect the bulk-like-torque principle is also applicable to domain wall  or 9 \n skyrmion  devices28-30 operated by S OTs. In this respect, o ur findings will motivate research \nactivities to introduce FIMs as core elements in spintronics devices, which have been so far \ndominated by FMs. Therefore, our result provides an important step towards “ferrimagnetic \nspintronics”.  \n \n  10 \n References  \n1 Slonczewski, J. C. Current -driven excitation of magnetic multilayers. J. Magn. Magn. \nMater.  159, L1-L7, (1996).  \n2 Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. \nRev. 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Blue dots (Fig. 1a) and red crosses (Fig. 1b) indicate the directions \nof magnetic moments. Precession of an electron spin in the FM layer ( c) and in the FIM layer ( d). \nBlue and red curved arrows indicate \n  > 0 and  \n  <  0, respectively.  \n13 \n  \nFigure 2 | Film stacks and SOT measurements . a, b, Illustrations of Co/Ni ( a) and Co/Tb ( b) \nmultilayers. The magnetization s of Co, Ni and Tb sub -lattices are presented by the yellow, blue \nand green arrow s, respectively. c, d, The measurement schematics for longitudinal ( c) and \ntransverse ( d) SOT effective fie lds. e, f, Second harmonic voltage s (V2f) obtained from Co/Ni ( e) \nand Co/Tb ( f) multilayer devices, with the blue curves representing the longitudinal signals and \nred curves representing the transverse signals. The i nsets  correspond to first harmonic voltage s \n(Vf). \n14 \n   \nFigure 3 | SOT effective fields and switching efficiencies . a, b, Longitudinal ( HL) and transverse \n(HT) SOT effective fields as a function of Co/Ni ( a) or Co/Tb ( b) thicknesses. c, d, Effective spin \nHall angle  (eff) as a function of Co/Ni ( c) or Co/Tb ( d) thicknesses. Insets  in c and d are the \nsaturation magnetization (MS) as a function of ferromagnet - or ferrimagnet -thickness. e, f, \nSwitching efficiencies ( η) as a function of Co/Ni ( e) or Co/Tb ( f) thicknesses. Insets  in e and f are \ncurrent -induced switching  data, showing the Hall resistance ( RH) as a f unction of appli ed pulse \ncurrent.  \n0.20.40.6  HL\n HTHL/T/J (10-8 kOe cm2/A)\n1 2 34681012\n  (10-10 Oe m2/A)\ntFM (nm)\n0.00.51.0eff\n012eff\n0510 HL\n HTHL/T/J (10-8 kOe cm2/A)\n2 4 6 810 12 140100200300400500\n  (10-10 Oe m2/A)\ntFIM (nm)\n2468101214100200MS (emu/cc)\ntFIM (nm)\n1.0 1.5 2.0 2.5 3.06008001000 MS (emu/cc)\ntFM (nm)a b\nc d\ne f\n-4 -2 2 4-0.50.00.5R ()\nJ (1011 A/m2)\n-6-3 36-0.30.00.3R ()\nJ (1011 A/m2)15 \n  \n \nFigure 4 | Spin pumping measurements.  a, Spin pumping sample structure. b, Schematic of spin \npumping measurements.  S and G indicates signal and ground connection for high frequency \nmeasurements. An in -plane field ( Hb) along the waveguide direction is applied.  c, Spin pumping \nsignal s in various structures. d, Inverse spin Hall signal as a function of [Co/Tb] -thickness in \nPt/[Co/Tb]/Cu/Co structures  at various frequencies . \n  \n-2024681012140.000.020.040.060.080.10  7 GHz\n 8 GHz\n 9 GHzVISHE/R (A)\ntCoTb (nm)\nS\nGVHb\nSubstrateNi or Tb\nCo×N\nPtCu…Coa b\n-0.6 -0.4 -0.2 0.00.00.51.0 Pt/Cu/Co  Pt/[Co/Tb](5.3)/Cu/Co\n Pt/[Co/Ni](0.9)/Cu/Co\n Pt/[Co/Tb]/CuVISHE (V)\nHb (kOe)\nc d16 \n Methods  \nSample preparation  \nSubstrate /[Tb (0.3 4 nm)/Co (0.32 nm)] N/Pt (4 nm) and substrate /MgO (2 nm)/Pt (4 nm)/[Co \n(0.3 nm)/Ni (0.3 nm)] N/SiO 2 (3 nm) multilayers are fabricated on thermally oxidized silicon \nsubstrates using rf and dc magnetron sputtering system with a base pressure of ~ 10-9 Torr. N is \nthe repetition number of Tb/Co or Co/Ni bilayer pairs, which is varied from 4 to 20 for Co/Tb \nsystems and from 2 to 5 for Co/Ni systems. For Co/Tb multilayers, a 4 nm -thick Pt layer is \ndeposited on top as a spin current source which also protects the m ultilayer from being oxidized. \nFor Co/Ni multilayers, a bilayer of MgO (2 nm)/Pt (4 nm) is deposited on the bottom as a buffer \nand spin current source, and a SiO 2 (3 nm) layer is deposited as a capping layer to prevent possible \noxidation of the FM layer. S ubsequent photolithography and ion milling processes are performed \nto fabricate the films into Hall bar devices.  \nSecond harmonic and spin pumping measurements  \nFor the second harmonic measurements, an ac current Iac with a frequency of 13.7 Hz and a \nmagnitude of 5 mA is injected into the channel of the device31. An external magnetic field Hext is \napplied along (orthogonal to) the current direction with a small out -of-plane tilting of θ = 4 from \nthe film plane in the longitudinal (transverse) configuration. The first and second harmonic Hall \nvoltages are recorded simultaneously by using two lock -in amplifiers triggered at the same \nfrequency by the current source.   \nIn the spin pumping m easure ments, a microwave at 7 to 9 GHz is applied to the asymmetric \ncoplanar stripline waveguide by a signal generator. An in -plane field ( Hb) along the waveguide \ndirection is swept around the resonance field ( H0) given by the Kittel formula \n 0 0 S 42f H H M\n, where γ is the gyromagnetic ratio and MS is the saturation 17 \n magnetization. The voltage ( V) is recorded by a lock -in amplifier. V includes the asymmetric \ncomponent ( Vasym) from the anomalous Hall effect (AHE) and the anisotropic magnetoresistance, \nas well as the symmetric components ( Vsym) from the spin pumping induced inverse spin Hall \nvoltage ( VISHE). Thus the measured voltage is fitted by a sum of symmetric and asymmetric \nLorentzian function \n\n2\n0\n22 22\n00sym asymHHV V V\nH H H H \n      , from which VISHE is extracted  \nas Vsym. "    },    {        "title": "2205.03058v1.Interband_magnon_drag_in_ferrimagnetic_insulators.pdf",        "content": "Interband magnon drag in ferrimagnetic insulators\nNaoya Arakawa1,\u0003\n1The Institute of Science and Engineering, Chuo University, Bunkyo, Tokyo, 112-8551, Japan\n(Dated: May 9, 2022)\nWe propose a new drag phenomenon, an interband magnon drag, and report on interaction\ne\u000bects and multiband e\u000bects in magnon transport of ferrimagnetic insulators. We study a spin-\nSeebeck coe\u000ecient Sm, a magnon conductivity \u001bm, and a magnon thermal conductivity \u0014mof\ninteracting magnons for a minimal model of ferrimagnetic insulators using a 1 =Sexpansion of the\nHolstein-Primako\u000b method, the linear-response theory, and a method of Green's functions. We show\nthat the interband magnon drag enhances \u001bmand reduces \u0014m, whereas its total e\u000bects on Smare\nsmall. This drag results from the interband momentum transfer induced by the magnon-magnon\ninteractions. We also show that the higher-energy band magnons contribute to Sm,\u001bm, and\u0014m\neven for temperatures smaller than the energy di\u000berence between the two bands.\nI. INTRODUCTION\nMagnon transport is the key to understanding spin-\ntronics and spin-caloritronics phenomena of magnetic in-\nsulators1{3. For example, a magnon spin current is vital\nfor the spin Seebeck e\u000bect2,4{7. Magnon transport is im-\nportant also for other relevant phenomena8{13.\nThere are two key issues about magnon transport in\nferrimagnetic insulators. One is about multiband e\u000bects.\nYttrium iron garnet (YIG) is a ferrimagnetic insulator\nused in various spintronics or spin-caloritronics phenom-\nena1{3,8{12. Its magnons have been often approximated\nas those of a ferromagnet. However, a study using its\nrealistic model14showed that not only the lowest-energy\nband magnons, which could be approximated as those of\na ferromagnet, but also the second-lowest-energy band\nmagnons should be considered except for su\u000eciently low\ntemperatures. Since the experiments using YIG are per-\nformed typically at room temperature1{3,8,9,11,12, it is\nnecessary to clarify the e\u000bects of the higher-energy band\nmagnons on the magnon transport. The other is about\ninteraction e\u000bects. The magnon-magnon interactions are\nusually neglected. However, their e\u000bects may be drastic\nin a ferrimagnet because they can induce the interband\nmomentum transfer, which is expected to cause an in-\nterband magnon drag by analogy with various drag phe-\nnomena15{40. Nevertheless, it remains unclear how the\nmagnon-magnon interactions a\u000bect the magnon trans-\nport.\nIn this paper, we provide the \frst step towards re-\nsolving the above issues and propose a new drag phe-\nnomenon, the interband magnon drag. We derive three\ntransport coe\u000ecients of interacting magnons for a two-\nsublattice ferrimagnet and numerically evaluate their\ntemperature dependences. We show that the interband\nmagnon drag enhances a magnon conductivity and re-\nduces a magnon thermal conductivity, whereas its total\ne\u000bects on a spin-Seebeck coe\u000ecient are small. We also\nshow that the higher-energy band magnons contribute to\nthese transport coe\u000ecients even for temperatures lower\nthan the energy splitting of the two bands.\n\"#\nYZ[FIG. 1. Our ferrimagnetic insulator. The up or down arrows\nrepresent the spins on the AorBsublattice, respectively. The\nx,y, andzaxes are also shown.\nII. MODEL\nOur ferrimagnetic insulator is described by\nH= 2JX\nhi;jiSi\u0001Sj\u0000hN=2X\ni=1Sz\ni\u0000hN=2X\nj=1Sz\nj; (1)\nwhere the \frst term is the Heisenberg exchange interac-\ntion between nearest-neighbor spins, and the others are\nthe Zeeman energy of a weak magnetic \feld ( jhj\u001cJ).\n(The ground-state magnetization is aligned parallel to\nthe magnetic \feld.) We have disregarded the dipolar in-\nteraction and the magnetic anisotropy, which are usually\nmuch smaller than J14,41. For concreteness, we consider\na two-sublattice ferrimagnet on the body-centered cubic\nlattice (Fig. 1); i's andj's in Eq. (1) are site indices\nof theAandBsublattice, respectively. There are N=2\nsites per sublattice. Our model can be regarded as a\nminimal model of ferrimagnetic insulators because a fer-\nrimagnetic state, the spin alignments of which are given\nbySi=t(0 0SA) for alli's and Sj=t(0 0\u0000SB) for all\nj's, is stabilized for J >0 with the weak magnetic \feld.\nWe set ~= 1,kB= 1, anda= 1, where ais the lattice\nconstant.\nTo describe magnons of our ferrimagnetic insulator,\nwe rewrite Eq. (1) by using the Holstein-Primako\u000b\nmethod42. By applying the Holstein-Primako\u000b transfor-arXiv:2205.03058v1  [cond-mat.mes-hall]  6 May 20222\nmation43{45to Eq. (1) and using a 1 =Sexpansion43,44,46\nand the Fourier transformation of magnon operators, we\ncan write Eq. (1) in the form\nH=HKE+Hint: (2)\nHereHKErepresents the kinetic energy of magnons,\nHKE=X\nq\u0000\nay\nqbq\u0001\u0012\u000fAA\u000fAB(q)\n\u000fAB(q)\u000fBB\u0013 \naq\nby\nq!\n;(3)\nwhere\u000fAA= 2J0SB+h,\u000fAB(q) = 2pSASBJq,\u000fBB=\n2J0SA\u0000h, andJq= 8Jcosqx\n2cosqy\n2cosqz\n2;Hintrepre-\nsents the leading terms of magnon-magnon interactions,\nHint=\u00001\nNX\nq1;q2;q2;q4\u000eq1+q2;q3+q4(2Jq1\u0000q3ay\nq1aq3by\nq4bq2\n+r\nSA\nSBJq1aq1by\nq2bq3bq4+r\nSB\nSAJq1bq1ay\nq2aq3aq4) + (H.c.):\n(4)\nWe can also express HKEas a two-band Hamiltonian by\nusing the Bogoliubov transformation43{45:\nHKE=X\nq[\u000f\u000b(q)\u000by\nq\u000bq+\u000f\f(q)\fq\fy\nq]; (5)\nwhere\u000f\u000b(q) =h+J0(SB\u0000SA) + \u0001\u000fq,\n\u000f\f(q) =\u0000h+J0(SA\u0000SB) + \u0001\u000fq, and\n\u0001\u000fq=q\nJ2\n0(SA+SB)2\u00004SASBJ2q. ForSA> SB\nwe have\u000f\u000b(q)<\u000f\f(q). Note that the Bogoliubov trans-\nformation is given by aq= (Uq)A\u000b\u000bq+ (Uq)A\f\fy\nqand\nby\nq= (Uq)B\u000b\u000bq+ (Uq)B\f\fy\nq, where (Uq)A\u000b= (Uq)B\f=\ncosh\u0012q, (Uq)A\f= (Uq)B\u000b=\u0000sinh\u0012q, and these hyper-\nbolic functions satisfy cosh 2 \u0012q= [J0(SA+SB)]=\u0001\u000fq\nand sinh 2\u0012q= (2pSASBJq)=\u0001\u000fq. Then, by using the\nBogoliubov transformation, we can decompose Hintinto\nthe intraband and the interband components47. Because\nof these properties, our model is a minimal model to\nstudy the two key issues explained above.\nIII. DERIVATIONS OF TRANSPORT\nCOEFFICIENTS\nWe consider three transport coe\u000ecients: a spin-\nSeebeck coe\u000ecient Sm, a magnon conductivity \u001bm, and\na magnon thermal conductivity \u0014m. They are given by\nSm=L12,\u001bm=L11, and\u0014m=L22, whereL\u0016\u0011's are\nde\fned as\njS=L11ES+L12\u0010\n\u0000rT\nT\u0011\n; (6)\njQ=L21ES+L22\u0010\n\u0000rT\nT\u0011\n: (7)\nHerejSandjQare magnon spin and heat, respectively,\ncurrent densities, ESis a nonthermal external \feld, andrTis a temperature gradient. (Note that one of the\npossible choices of ESis a magnetic-\feld gradient48.)\nL21=L12holds owing to the Onsager reciprocal the-\norem. It should be noted that although \u0014mis generally\ngiven by\u0014m=L22\u0000L21L12\nL11, our de\fnition \u0014m=L22is\nsu\u000ecient to describe the thermal magnon transport at\nlow temperatures at which the magnon picture is valid\nbecause the L22gives the leading temperature depen-\ndence. Since a magnon chemical potential is zero in equi-\nlibrium, jQ=jE, where jEis a magnon energy current\ndensity. Hereafter we focus on the magnon transport\nwithESor (\u0000rT=T) applied along the xaxis.\nWe express L\u0016\u0011's in terms of the correlation functions\nusing the linear-response theory23,49{54. First,L12is\ngiven by\nL12= lim\n!!0\bR\n12(!)\u0000\bR\n12(0)\ni!; (8)\nwhere \bR\n12(!) = \b 12(i\nn!!+i\u000e) (\u000e= 0+),\n\b12(i\nn) =ZT\u00001\n0d\u001cei\nn\u001c1\nNhT\u001cJx\nS(\u001c)Jx\nEi; (9)\nand \nn= 2\u0019Tn (n > 0). HereT\u001cis the time-ordering\noperator51, andJx\nSandJx\nEare spin and energy, respec-\ntively, current operators. They are obtained from the\ncontinuity equations55{57(see Appendix A); the results\nare\nJx\nS=\u0000X\nqX\nl;l0=A;Bvx\nll0(q)xy\nqlxql0; (10)\nJx\nE=X\nqX\nl;l0=A;Bex\nll0(q)xy\nqlxql0; (11)\nwherevx\nll0(q) = (1\u0000\u000el;l0)@\u000fAB(q)\n@qx,xqA=aq,xqB=by\nq,\nex\nBB(q) =\u0000ex\nAA(q) =\u000fAB(q)@\u000fAB(q)\n@qx, andex\nAB(q) =\nex\nBA(q) =1\n2(\u000fAA\u0000\u000fBB)@\u000fAB(q)\n@qx. In deriving Eqs. (10)\nand (11), we have omitted the corrections due to Hint\nbecause they may be negligible23. Then we can obtain\nL11by replacing Jx\nEin \b 12(i\nn) byJx\nS, andL22by re-\nplacingJx\nS(\u001c) in \b 12(i\nn) byJx\nE(\u001c). Thus the derivation\nofL12is enough in obtaining L\u0016\u0017's. In addition, since\nwe can derive L12in a similar way to the derivations of\nelectron transport coe\u000ecients23,33,50,54,58, we explain its\nmain points below. (Note that the Bose-Einstein con-\ndensation of magnons is absent in our situation.)\nBy substituting Eqs. (10) and (11) into Eq. (9) and\nperforming some calculations (for the details see Ap-\npendix B), we obtain\nL12=L0\n12+L0\n12: (12)\nFirst,L0\n12, the noninteracting L12, is given by (see Ap-\npendix B)\nL0\n12=1\n\u0019NX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)I(I)\n\u0017\u00170(q); (13)3\nwherevx\n\u00170\u0017(q) =P\nl;l0=A;Bvx\nll0(q)(Uq)l\u00170(Uq)l0\u0017,\nex\n\u0017\u00170(q) =P\nl;l0=A;Bex\nll0(q)(Uq)l\u0017(Uq)l0\u00170, and\nI(I)\n\u0017\u00170(q) =Z1\n\u00001dz@n(z)\n@zImGR\n\u0017(q;z)ImGR\n\u00170(q;z):(14)\nHeren(z) = (ez=T\u00001)\u00001,GR\n\u000b(q;z) = [z\u0000\u000f\u000b(q) +i\r]\u00001,\nGR\n\f(q;z) =\u0000[z+\u000f\f(q) +i\r]\u00001, and\ris the magnon\ndamping. Next, L0\n12, the leading correction due to the\n\frst-order perturbation of Hint, is given by (see Appendix\nB)\nL0\n12=1\n\u00192N2X\nq;q0X\n\u00171\u00172;\u00173;\u00174vx\n\u00171\u00172(q)ex\n\u00173\u00174(q0)V\u00171\u00172\u00173\u00174(q;q0)\n\u0002[I(I)\n\u00171\u00172(q)I(II)\n\u00173\u00174(q0) +I(II)\n\u00171\u00172(q)I(I)\n\u00173\u00174(q0)]; (15)\nwhere\nI(II)\n\u0017\u00170(q) =Z1\n\u00001dzn(z)Im[GR\n\u0017(q;z)GR\n\u00170(q;z)];(16)\nV\u00171\u00172\u00173\u00174(q;q0) = 4Jq\u0000q0P\nl(Uq)l\u00171(Uq)\u0016l\u00172(Uq0)\u0016l\u00173(Uq0)l\u00174,\nand\u0016lisBorAforl=AorB, respectively. Then we\nobtain\nL11=L0\n11+L0\n11; L22=L0\n22+L0\n22; (17)\nwhereL0\n11,L0\n11,L0\n22, andL0\n22are obtained by replacing\nex\n\u0017\u00170(q) in Eq. (13) by\u0000vx\n\u0017\u00170(q),ex\n\u00173\u00174(q0) in Eq. (15) by\n\u0000vx\n\u00173\u00174(q0),vx\n\u00170\u0017(q) in Eq. (13) by\u0000ex\n\u00170\u0017(q), andvx\n\u00171\u00172(q)\nin Eq. (15) by\u0000ex\n\u00171\u00172(q), respectively.\nSince we suppose that the magnon lifetime \u001c= (2\r)\u00001\nis long enough to regard magnons as quasiparticles, we\nrewrite Eqs. (13) and (15) by taking the limit \u001c!1 .\nFirst, Eq. (13) reduces to\nL0\n12\u0018L0\n12\u000b+L0\n12\f; (18)\nwhere\nL0\n12\u0017\u00181\nNX\nqvx\n\u0017\u0017(q)ex\n\u0017\u0017(q)\u001c@n[\u000f\u0017(q)]\n@\u000f\u0017(q): (19)\n(The detailed derivation is described in Appendix C.)\nThis expression is consistent with that obtained in the\nBoltzmann theory with the relaxation-time approxima-\ntion59. Equation (18) shows that L0\n12\u0019L0\n12\u000bat\nsu\u000eciently low temperatures for SA> SBowing to\n@n[\u000f\u000b(q)]\n@\u000f\u000b(q)\u001d@n[\u000f\f(q)]\n@\u000f\f(q). Similarly, we obtain\nL0\n11\u0018L0\n11\u000b+L0\n11\f; L0\n22\u0018L0\n22\u000b+L0\n22\f; (20)\nwhereL0\n11\u0017andL0\n22\u0017are obtained by replacing ex\n\u0017\u0017(q) in\nEq. (19) by\u0000vx\n\u0017\u0017(q) and by replacing vx\n\u0017\u0017(q) by\u0000ex\n\u0017\u0017(q),\nrespectively. Then, as we show in Appendix C, Eq. (15)\nreduces to\nL0\n12\u0018L0\n12-intra +L0\n12-inter1 +L0\n12-inter2; (21)whereL0\n12-intra is the correction due to the intraband in-\nteractions,\nL0\n12-intra =L0\n12-intra-\u000b+L0\n12-intra-\f; (22)\nL0\n12-intra-\u0017=\u00002\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u0017\u0017(q0)\u001cV\u0017\u0017\u0017\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0); (23)\nandL0\n12-inter1 andL0\n12-inter2 are the corrections due to the\ninterband interactions,\nL0\n12-inter1 =\u00002\nN2X\nq;q0vx\n\u000b\u000b(q)ex\n\f\f(q0)\u001cV\u000b\u000b\f\f(q;q0)\n\u0002@n[\u000f\u000b(q)]\n@\u000f\u000b(q)@n[\u000f\f(q0)]\n@\u000f\f(q0)\n\u00002\nN2X\nq;q0vx\n\f\f(q)ex\n\u000b\u000b(q0)\u001cV\f\f\u000b\u000b(q;q0)\n\u0002@n[\u000f\f(q)]\n@\u000f\f(q)@n[\u000f\u000b(q0)]\n@\u000f\u000b(q0); (24)\nL0\n12-inter2 =L0\n12-inter2-\u000b+L0\n12-inter2-\f\n= (L0\nE\u000b+L0\nS\u000b) + (L0\nE\f+L0\nS\f); (25)\nL0\nE\u0017=2\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u000b\f(q0)\u001cV\u0017\u0017\u000b\f(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)n[\u000f\u000b(q0)]\u0000n[\u0000\u000f\f(q0)]\n\u000f\u000b(q0) +\u000f\f(q0);(26)\nL0\nS\u0017=2\nN2X\nq;q0vx\n\u000b\f(q)ex\n\u0017\u0017(q0)\u001cV\u000b\f\u0017\u0017(q;q0)\n\u0002n[\u000f\u000b(q)]\u0000n[\u0000\u000f\f(q)]\n\u000f\u000b(q) +\u000f\f(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0):(27)\nHere theV\u00171\u00172\u00173\u00174(q;q0)'s are given by\nV\u0017\u0017\u0017\u0017(q;q0) =V\u000b\u000b\f\f(q;q0) =V\f\f\u000b\u000b(q;q0)\n= 2Jq\u0000q0sinh 2\u0012qsinh 2\u0012q0; (28)\nV\u0017\u0017\u000b\f(q;q0) =V\u000b\f\u0017\u0017(q0;q)\n=\u00002Jq\u0000q0sinh 2\u0012qcosh 2\u0012q0: (29)\nEquation (24) shows that the interband components\nof the magnon-magnon interactions cause the energy-\ncurrent-drag correction and the spin-current-drag cor-\nrection, which are, in the case for SA> SB, the \frst\nand the second term, respectively, of Eq. (24). Further-\nmore, Eqs. (26) and (27) show that other interband com-\nponents cause the energy-current-drag corrections L0\nE\u0017's\nand the spin-current-drag corrections L0\nS\u0017's. Since these\ninterband components cause the interband momentum\ntransfer,L0\n12-inter1 andL0\n12-inter2 are the corrections due\nto the interband magnon drag. The similar corrections\nare obtained for L0\n11andL0\n22:\nL0\n11\u0018L0\n11-intra +L0\n11-inter1 +L0\n11-inter2; (30)\nL0\n22\u0018L0\n22-intra +L0\n22-inter1 +L0\n22-inter2; (31)4\nwhereL0\n11-intra andL0\n22-intra are the corrections due to\nthe intraband interactions,\nL0\n11-intra =L0\n11-intra-\u000b+L0\n11-intra-\f; (32)\nL0\n11-intra-\u0017=2\nN2X\nq;q0vx\n\u0017\u0017(q)vx\n\u0017\u0017(q0)\u001cV\u0017\u0017\u0017\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0); (33)\nL0\n22-intra =L0\n22-intra-\u000b+L0\n22-intra-\f; (34)\nL0\n22-intra-\u0017=2\nN2X\nq;q0ex\n\u0017\u0017(q)ex\n\u0017\u0017(q0)\u001cV\u0017\u0017\u0017\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0); (35)\nandL0\n11-inter1 ,L0\n11-inter2 ,L0\n22-inter1 , andL0\n22-inter2 are the\ncorrections due to the interband interactions,\nL0\n11-inter1 =4\nN2X\nq;q0vx\n\u000b\u000b(q)vx\n\f\f(q0)\u001cV\u000b\u000b\f\f(q;q0)\n\u0002@n[\u000f\u000b(q)]\n@\u000f\u000b(q)@n[\u000f\f(q0)]\n@\u000f\f(q0); (36)\nL0\n11-inter2 =L0\n11-inter2-\u000b+L0\n11-inter2-\f; (37)\nL0\n11-inter2-\u0017=\u00004\nN2X\nq;q0vx\n\u0017\u0017(q)vx\n\u000b\f(q0)\u001cV\u0017\u0017\u000b\f(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)n[\u000f\u000b(q0)]\u0000n[\u0000\u000f\f(q0)]\n\u000f\u000b(q0) +\u000f\f(q0);(38)\nL0\n22-inter1 =4\nN2X\nq;q0ex\n\u000b\u000b(q)ex\n\f\f(q0)\u001cV\u000b\u000b\f\f(q;q0)\n\u0002@n[\u000f\u000b(q)]\n@\u000f\u000b(q)@n[\u000f\f(q0)]\n@\u000f\f(q0); (39)\nL0\n22-inter2 =L0\n22-inter2-\u000b+L0\n22-inter2-\f; (40)\nL0\n22-inter2-\u0017=\u00004\nN2X\nq;q0ex\n\u0017\u0017(q)ex\n\u000b\f(q0)\u001cV\u0017\u0017\u000b\f(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)n[\u000f\u000b(q0)]\u0000n[\u0000\u000f\f(q0)]\n\u000f\u000b(q0) +\u000f\f(q0):(41)\nAs well as L0\n12-inter1 andL0\n12-inter2 ,L0\n11-inter1 ,L0\n11-inter2 ,\nL0\n22-inter1 , andL0\n22-inter2 are the interband magnon drag\ncorrections.\nIV. NUMERICAL RESULTS\nWe numerically evaluate Sm,\u001bm, and\u0014m. We setJ=\n1,h= 0:02J, and (SA;SB) = (3\n2;1).SA:SB= 3 : 2 is\nconsistent with a ratio of FeTto FeOsites in the unit cell\nof YIG41. The reason why ( SA;SB) = (3\n2;1) is considered\nis that the transition temperature derived in a mean-\feld\napproximation in this case with J= 3 meV at h= 0\n[i.e.,Tc= (16=3)JSA(SB+ 1)\u0018557 K] is close to the\nCurie temperature of YIG, TC. To perform the momen-\ntum summations numerically, we divide the \frst Brillouinzone into a Nq-point mesh and set Nq= 243(=N=2) (for\nmore details, see Appendix D). The temperature range\nis chosen to be 0 < T\u001410J(\u00180:6Tc) because a previ-\nous study60showed that the magnon theory in which the\nmagnon-magnon interactions are considered in the \frst-\norder perturbation theory can reproduce the perpendicu-\nlar spin susceptibility of MnF 2up to about 0 :6TN, where\nTNis the N\u0013 eel temperature. For simplicity, we deter-\nmine\u001cby\u001c\u00001=\r0+\r1T+\r2T2, where\r0= 10\u00002J,\n\r1= 10\u00004, and\r2= 10\u00003. (The results shown below\nremain qualitatively unchanged at h= 0:08Jand 0:16J,\nas shown in Appendix E.)\nWe begin with the temperature dependence of Sm.\nFigure 2(a) shows that in the range of 0 < T\u00142J\nL12\u0019L0\n12\u000bholds, whereas for T\u00153Jthe contribu-\ntion fromL0\n12\fis non-negligible. For example, at T= 6J\nwe haveL0\n12=L0\n12\u000b\u00180:7. This result indicates that the\nhigher-energy band magnons contribute to Smeven for\nT < [\u000f\f(q)\u0000\u000f\u000b(q)] = 7:96J. This may be surprising\nbecause their contributions are believed to be negligi-\nble at such temperatures. Then, Fig. 2(a) shows that\nthe magnitude of Smis enhanced by the intraband cor-\nrectionL0\n12-intra [=L(a)\n12\u0000L0\n12], whereas it is reduced by\nthe interband corrections L0\n12-inter2 [=L(b)\n12\u0000L(a)\n12] and\nL0\n12-inter1 [=L0\n12+L0\n12\u0000L(b)\n12] (Table I). Among these cor-\nrections,L0\n12-intra gives the largest contribution. (As we\nwill see below, this contrasts with the result of L11orL22,\nfor which the largest contribution comes from L0\n11-inter2\norL0\n22-inter2 , respectively.) The reason why the inter-\nband magnon drag corrections L0\n12-inter2 andL0\n12-inter1 are\nsmall is that the energy-current-drag contributions and\nspin-current-drag contributions [e.g., L0\nE\u000bandL0\nS\u000bin Eq.\n(25)] are opposite in sign and are nearly canceled out.\nFigure 2(a) also shows L0\n12+L0\n12\u0019L0\n12. These results\nsuggest that the total e\u000bects of the interband magnon\ndrag onSmare small.\nWe turn to \u001bmand\u0014m. Their temperature depen-\ndences are shown in Figs. 2(b) and 2(c). First, we see the\n\f-band magnons contribute to L11forT\u00154Jand toL22\nforT\u00153J. This result is similar to that of L12and indi-\ncates that the multiband e\u000bects are signi\fcant also for \u001bm\nand\u0014m. The largest e\u000bects on L22are due to the prop-\nerty thatex\n\u0017\u0017(q) includes\u000f\u0017(q) [more precisely, ex\n\u000b\u000b(q) =\nvx\n\u000b\u000b(q)\u000f\u000b(q) andex\n\f\f(q) =\u0000vx\n\f\f(q)\u000f\f(q)]. Then, Figs.\n2(b) and 2(c) show that \u001bmis enhanced by L0\n11-intra ,\nL0\n11-inter2 , andL0\n11-inter1 , and that \u0014mis enhanced by\nL0\n22-intra and reduced by L0\n22-inter2 andL0\n22-inter1 (Table I).\n[Note thatL0\n\u0016\u0011-intra =L(a)\n\u0016\u0011\u0000L0\n\u0016\u0011,L0\n\u0016\u0011-inter2 =L(b)\n\u0016\u0011\u0000L(a)\n\u0016\u0011,\nandL0\n\u0016\u0011-inter1 =L0\n\u0016\u0011+L0\n\u0016\u0011\u0000L(b)\n\u0016\u0011.] In contrast to L0\n12, the\nlargest contributions to L0\n11andL0\n22come fromL0\n11-inter2\nandL0\n22-inter2 , respectively. Since L0\n11-inter2 ,L0\n11-inter1 ,\nL0\n22-inter2 , andL0\n22-inter1 are the interband magnon drag\ncorrections, the above results suggest that the interband\nmagnon drag enhances \u001bmand reduces \u0014m. This implies\nthat the interband magnon drag could be used to enhance\nthe spin current and to reduce the energy current. Since\nthis drag results from the interband momentum transfer5\n\tC\n \tB\n \tD\n−20−15−10−5 0\n 0  2  4  6  8  10h = 0.02JSm (arb. unit)\nT/JL0\n12 α\nL0\n12 \nL(a)\n12 \nL(b)\n12 \nL0\n12 +L’12 \n 0 1 2 3 4 5\n 0  2  4  6  8  10h = 0.02J\nσm (arb. unit)\nT/JL0\n11 α\nL0\n11 \nL(a)\n11 \nL(b)\n11 \nL0\n11 +L’11 \n 0 50 100 150 200 250\n 0  2  4  6  8  10h = 0.02Jκm (arb. unit)\nT/JL0\n22 α\nL0\n22 \nL(a)\n22 \nL(b)\n22 \nL0\n22 +L’22 \nFIG. 2. The temperature dependences of (a) Sm(=L12), (b)\u001bm(=L11), and (c) \u0014m(=L22) for (SA;SB) = (3\n2;1) at\nh= 0:02J.L(a)\n\u0016\u0011andL(b)\n\u0016\u0011are de\fned as L(a)\n\u0016\u0011=L0\n\u0016\u0011+L0\n\u0016\u0011-intra andL(b)\n\u0016\u0011=L0\n\u0016\u0011+L0\n\u0016\u0011-intra +L0\n\u0016\u0011-inter2 , respectively. Note that\nL0\n\u0016\u0011=L0\n\u0016\u0011\u000b+L0\n\u0016\u0011\fandL0\n\u0016\u0011=L0\n\u0016\u0011-intra +L0\n\u0016\u0011-inter1 +L0\n\u0016\u0011-inter2 . ForSm, theL0\n12\fis non-negligible for T\u00153Jand the largest\nterm of the drag terms is L0\n12-intra , which enhances jSmj. For\u001bm, theL0\n11\fis non-negligible for T\u00154Jand the largest term of\nthe drag terms is L0\n11-inter2 , which enhances \u001bm. For\u0014m, theL0\n22\fis non-negligible for T\u00153Jand the largest term of the drag\nterms isL0\n22-inter2 , which reduces \u0014m. The e\u000bects of the other drag terms are summarized in Table I.\nTABLE I. The e\u000bects of the drag terms on L12(=Sm),L11(=\u001bm), andL22(=\u0014m).jL12jis enhanced by L0\n12-intra and reduced\nbyL0\n12-inter1 andL0\n12-inter2 .L11is enhanced by L0\n11-intra ,L11-inter1 , andL0\n11-inter2 .L22is enhanced by L0\n22-intra and reduced by\nL0\n22-inter1 andL0\n22-inter2 .\nTransport coe\u000ecient Intra term Inter1 term Inter2 term\njL12j Enhanced Reduced Reduced\nL11 Enhanced Enhanced Enhanced\nL22 Enhanced Reduced Reduced\ninduced by the magnon-magnon interactions, its e\u000bects\ncould be controlled by changing the band splitting energy\nconsiderably via external \felds. (Such control is mean-\ningful if and only if the magnon picture remains valid.)\nNote that for ferrimagnetic insulators the e\u000bects of the\nweak magnetic \feld on the band splitting energy are neg-\nligible because this energy for h= 0 is of the order of J.\n(The actual analysis about the possibility of controlling\nthe interband magnon drag is a future problem.)\nV. DISCUSSIONS\nWe discuss the validity of our theory. Since Hint\ncould be treated as perturbation except near TC, we be-\nlieve our theory is appropriate for describing the magnon\ntransport for T < T C. It may be suitable to treat the\nmagnon-magnon interactions in the Holstein-Primako\u000b\nmethod because the unphysical processes that can ap-\npear in aS= 1=2 ferromagnet61are absent in our case.\nThen the e\u000bects of the magnon-phonon interactions may\nnot change the results qualitatively. First, since the\ninteraction-induced magnon polaron occurs only at sev-\neral values of h62, its e\u000bect can be avoided. Another e\u000bect\nis to cause the temperature dependence of \u001c63,64, and it\ncould be approximately considered as the temperature-\ndependent \u001c. Although the phonon-drag contributions\nmight change Sm21, experimental results59suggest thatsuch contributions are small or negligible.\nWe make a short comment about the relation between\nour theory and the Boltzmann theory. Our theory is\nbased on a method of Green's functions, which can de-\nscribe the e\u000bects of the damping and the vertex correc-\ntions appropriately. In principle, these e\u000bects can be\ndescribed also in the Boltzmann theory if the collision\nintegral is treated appropriately65. However, in many\nanalyses using the Boltzmann theory, the collision inte-\ngral is evaluated in the relaxation-time approximation,\nin which the vertex corrections are completely omitted.\nSince our interband magnon drag comes from the ver-\ntex corrections due to the \frst-order perturbation of the\nquartic terms, the similar result might be obtained also\nin the Boltzmann theory if the interband components of\nthe collision integral are treated appropriately.\nWe remark on the implications of our results. First,\nour interband magnon drag is distinct from a magnon\ndrag in metals. For the latter, magnons drag an elec-\ntron charge current via the second-order perturbation\nof asd-type exchange interaction25. Second, the inter-\nband magnon drag is possible in various ferrimagnetic\ninsulators and other magnetic systems, such as anti-\nferromagnets47,56,66and spiral magnets57,67. Note that\nthe possible ferrimagnetic insulators include not only\nYIG, but also some spinel ferrites, such as CoFe 2O4and\nNiFe 2O468,69. Third, our theory can be extended to\nphonons and photons. Thus it may be useful for study-6\ning transport phenomena of various interacting bosons.\nFourth, our results will stimulate further studies of YIG.\nFor example, the reduction in jSmjdue to the multiband\ne\u000bect could improve the di\u000berences between the voltages\nobserved in the spin-Seebeck e\u000bect and obtained in the\nBoltzmann theory of the ferromagnet59at high temper-\natures because the voltage is proportional to Sm.\nVI. CONCLUSION\nWe have studied Sm,\u001bm, and\u0014mof interacting\nmagnons in the minimal model of ferrimagnetic insu-\nlators. We derived them by using the linear-response\ntheory and treating the magnon-magnon interactions as\nperturbation. We showed that some interband compo-\nnents of the magnon-magnon interactions give the cor-\nrections to these transport coe\u000ecients. These correc-\ntions are due to the interband magnon drag, which is\ndistinct from the magnon drag in metals. Then we nu-\nmerically calculated the temperature dependences of Sm,\n\u001bm, and\u0014mfor (SA;SB) = (3\n2;1) andh= 0:02J. We\nshowed that the total e\u000bects of the interband magnon\ndrag onSmbecome small, whereas it enhances \u001bmand\nreduces\u0014m. The latter result may suggest that the in-\nterband magnon drag could be used to enhance the spin\ncurrent and reduce the energy current. For Sm, the in-\nterband corrections become small because they lead to\nthe energy-current-drag contributions and spin-current-\ndrag contributions, which are opposite in sign and are\nnearly canceled out. We also showed that the contribu-\ntions from the higher-energy band magnons to Sm,\u001bm,\nand\u0014mare non-negligible even for temperatures lower\nthan the band splitting. This result indicates the impor-\ntance of the multiband e\u000bects.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grants\nNo. JP19K14664 and JP22K03532. The author also ac-\nknowledges support from JST CREST Grant No. JP-\nMJCR1901.\nAppendix A: Derivations of Eqs. (10) and (11)\nWe explain the details of the derivations of Jx\nSand\nJx\nE, Eqs. (10) and (11). As described in the main text,\nthey are obtained from the continuity equations. Such a\nderivation is explained, for example, in Ref. 55.\nWe begin with the derivation of Jx\nS. (Note that the fol-\nlowing derivation, which is applicable to collinear mag-\nnets, can be extended to noncollinear magnets.) We sup-\npose that the zcomponent of a spin angular momentum,Sz\nm, satis\fes\ndSz\nm\ndt+r\u0001j(S)\nm= 0; (A1)\nwhere j(S)\nmis a spin current operator at site m. Using\nthis equation, we have\nd\ndt\u0010X\nmRmSz\nm\u0011\n=\u0000X\nmRmr\u0001j(S)\nm\n=X\nmj(S)\nm=J(S)\nl: (A2)\nHerelisAorBwhen the sumP\nmtakes over sites on\ntheAor theBsublattice, respectively. In deriving the\nsecond equal in Eq. (A2) we have omitted the surface\ncontributions. Jx\nSis given by the xcomponent of JS,\nwhere\nJS=J(S)\nA+J(S)\nB: (A3)\nCombining Eq. (A2) with the Heisenberg equation of\nmotion, we obtain\nJ(S)\nl=iX\nmRm\u0002\nH;Sz\nm\u0003\n; (A4)\nwhereHis the Hamiltonian of the system considered.\nThen, since we focus on the magnon system described\nbyH=HKE+Hint, whereHKEandHintare given in\nthe main text, and treat Hintas perturbation, we replace\nHin Eq. (A4) by HKEandSz\nmin Eq. (A4) either by\nSA\u0000ay\nmamforl=Aor by\u0000SB+by\nmbmforl=B; as a\nresult, we obtain\nJ(S)\nA=iX\nhi;jiX\nmRm\u0002\nh0\nij;SA\u0000ay\nmam\u0003\n; (A5)\nJ(S)\nB=iX\nhi;jiX\nmRm\u0002\nh0\nij;\u0000SB+by\nmbm\u0003\n; (A6)\nwhereHKE=P\nhi;jih0\nijandh0\nij= (2JSB+\u000ei;jh)ay\niai+\n(2JSA\u0000\u000ei;jh)by\njbj+2JpSASB(ay\niby\nj+aibj). Note that the\nreplacement of HbyHKEmay be suitable because the\ncorrections due to Hintare next-leading terms; and that\nthe replacement of Sz\nmbySA\u0000ay\nmamor by\u0000SB+by\nmbm\ncorresponds to the Holstein-Primako\u000b transformation of\nthe ferrimagnet. After some algebra, we can write Eqs.\n(A5) and (A6) as follows:\nJ(S)\nA=\u0000i2Jp\nSASBX\nhi;jiRi(aibj\u0000ay\niby\nj); (A7)\nJ(S)\nB=i2Jp\nSASBX\nhi;jiRj(aibj\u0000ay\niby\nj): (A8)\nCombining these equations with Eq. (A3), we have\nJS=\u0000i2Jp\nSASBX\nhi;ji(Ri\u0000Rj)(aibj\u0000ay\niby\nj):(A9)7\nThen, by using the Fourier coe\u000ecients of the magnon\noperators,\nai=r\n2\nNX\nqaqeiq\u0001Ri; by\nj=r\n2\nNX\nqby\nqeiq\u0001Rj;(A10)\nwe can rewrite Eq. (A9) as follows:\nJS=\u00002Jp\nSASBX\nq@Jq\n@q(aqbq+ay\nqby\nq)\n=\u0000X\nq@\u000fAB(q)\n@q(xy\nqBxqA+xy\nqAxqB); (A11)\nwhereJq=JPz\nj=1eiq\u0001(Ri\u0000Rj)= 8Jcosqx\n2cosqy\n2cosqz\n2,\n\u000fAB(q) = 2JpSASBJq,xqA=aq, andxqB=by\nq. Note\nthatzis the number of nearest-neighbor sites ( z= 8).\nThexcomponent of Eq. (A11) gives Eq. (10).\nIn a similar way we can obtain the expression of Jx\nE.\n(The following derivation is similar to that for an anti-\nferromagnet56.) First, we suppose that the Hamiltonian\nat sitem,hm, satis\fes\ndhm\ndt+r\u0001j(E)\nm= 0; (A12)\nwhere j(E)\nmis an energy current operator at site m. Be-\ncause of this relation, the energy current operator JEcan\nbe determined from\nJE=J(E)\nA+J(E)\nB; (A13)\nwhere J(E)\nlis given by\nJ(E)\nl=iX\nm;nRn\u0002\nhm;hn\u0003\n; (A14)\nthe sumP\nmtake over sites on the Aor theBsublattice,\nand the sumP\nntake over sites on sublattice l. Then, to\ncalculate the commutator in Eq. (A14), we consider the\ncontributions only from HKEand neglect the corrections\ndue toHint, as in the derivation of J(S)\nl. As a result, hm\nform2Ais given by\nh0\nmA= (2SBzJ+h)ay\nmam+p\nSASBX\njJmj(ambj+ay\nmby\nj);\n(A15)\nand that for m2Bis given by\nh0\nmB= (2SAzJ\u0000h)by\nmbm+p\nSASBX\niJim(aibm+ay\niby\nm):\n(A16)\nHerem2AorBmeans that mis on theAorBsublat-\ntice, respectively, and Jij=Jji=Jfor nearest-neighbor\nsitesiandj. Note thatPN=2\ni=1h0\niA+PN=2\nj=1h0\njB=HKE. In\nour de\fnition, the energy current operator includes theconribution from the Zeeman energy [see Eq. (A14){\n(A16)]. Combining Eqs. (A15) and (A16) with Eqs.\n(A13) and (A14), we have\nJE=iX\nm;nRn\u0002\nh0\nmA;h0\nnA\u0003\n+iX\nm;nRn\u0002\nh0\nmB;h0\nnB\u0003\n+iX\nm;nRn\u0002\nh0\nmA;h0\nnB\u0003\n+iX\nm;nRn\u0002\nh0\nmB;h0\nnA\u0003\n:(A17)\nThen we can calculate the commutators in Eq. (A17) by\nusing the commutation relations of the magnon opera-\ntors and the identities [ AB;C ] =A[B;C] + [A;C]Band\n[A;BC ] = [A;B]C+B[A;C]; the results are\n\u0002\nh0\nmA;h0\nnA\u0003\n=SASBX\njJmjJnj(ay\nnam\u0000ay\nman);(A18)\n\u0002\nh0\nmB;h0\nnB\u0003\n=SASBX\niJimJin(bmby\nn\u0000bnby\nm);(A19)\n\u0002\nh0\nmA;h0\nnB\u0003\n=SASBX\njJmnJmj(bjby\nn\u0000bnby\nj)\n+ [2Jz(SA\u0000SB)\u00002h]p\nSASBJmn\n\u0002(ambn\u0000ay\nmby\nn)\n+SASBX\niJmnJni(ay\niam\u0000ay\nmai);(A20)\n\u0002\nh0\nmB;h0\nnA\u0003\n=SASBX\njJmnJnj(bmby\nj\u0000bjby\nm)\n+ [\u00002Jz(SA\u0000SB) + 2h]p\nSASBJnm\n\u0002(anbm\u0000ay\nnby\nm)\n+SASBX\niJnmJmi(ay\nnai\u0000ay\nian):(A21)\nBy substituting these equations into Eq. (A17) and per-\nforming some calculations, we obtain\nJE= 2iX\nm;n;j(Rn\u0000Rm)SASBJnjJjmay\nnam\n\u00002iX\nm;n;i(Rn\u0000Rm)SASBJniJimbnby\nm\n+iX\nm;n(Rn\u0000Rm)[2Jz(SA\u0000SB)\u00002h]p\nSASB\n\u0002Jmn(ambn\u0000ay\nmby\nn): (A22)\nAs in the derivation of JS, we can rewrite Eq. (A22)\nby using the Fourier coe\u000ecients of the magnon operators\n[Eq. (A10)]; as a result, we have\nJE=\u0000X\nq2p\nSASBJq2p\nSASB@Jq\n@q(ay\nqaq\u0000bqby\nq)\n\u0000[J0(SA\u0000SB)\u0000h]2p\nSASB@Jq\n@q(aqbq+ay\nqby\nq):\n(A23)8\nSince\u000fAA= 2J0SB+h,\u000fBB= 2J0SA\u0000h, and\u000fAB(q) =\n2pSASBJq, we can write Eq. (A23) as follows:\nJE=\u0000X\nq\u000fAB(q)@\u000fAB(q)\n@q(ay\nqaq\u0000bqby\nq)\n+X\nq1\n2(\u000fAA\u0000\u000fBB)@\u000fAB(q)\n@q(aqbq+ay\nqby\nq)\n=X\nqX\nl;l0=A;Bell0(q)xy\nqlxql0; (A24)\nwhere eAA(q) =\u0000eBB(q) =\u0000\u000fAB(q)@\u000fAB(q)\n@qand\neAB(q) =eBA(q) =1\n2(\u000fAA\u0000\u000fBB)@\u000fAB(q)\n@q. Equation\n(A24) for the xcomponent is Eq. (11).\nAppendix B: Derivations of Eqs. (13) and (15)\nWe derive Eqs. (13) and (15). As described in the\nmain text, their derivations can be done in a simi-\nlar way to the derivations of electron transport coe\u000e-\ncients23,50,54,58: the transport coe\u000ecients can be derived\nby using a method of Green's functions51. We \frst de-\nriveL0\n12, the noninteracting L12, and then derive L0\n12,\nthe leading correction to L12due to the \frst-order per-\nturbation of Hint.\nFirst, we derive L0\n12, Eq. (13). Substituting Eqs. (10)\nand (11) into Eq. (9), we have\n\b12(i\nn) =\u00001\nNX\nq;q0X\nl1;l2;l3;l4=A;Bvx\nl1l2(q)ex\nl3l4(q0)\n\u0002ZT\u00001\n0d\u001cei\nn\u001chT\u001cxy\nql1(\u001c)xql2(\u001c)xy\nq0l3xq0l4i\n=\u00001\nNX\nq;q0X\nl1;l2;l3;l4=A;Bvx\nl1l2(q)ex\nl3l4(q0)\n\u0002G(II)\nl1l2l3l4(q;q0;i\nn); (B1)\nwhere \nn= 2\u0019Tn withn > 0. (Note that the nand\nmused in this section are di\u000berent from those used in\nAppendix A.) Equation (B1) provides a starting point\nto deriveL0\n12andL0\n12. To derive L0\n12, we calculate\nG(II)\nl1l2l3l4(q;q0;i\nn) in the absence of Hintby using Wick's\ntheorem51; the result is\nG(II)\nl1l2l3l4(q;q0;i\nn) =\u000eq;q0TX\nmGl2l3(q;i\nn+i\nm)\n\u0002Gl4l1(q;i\nm); (B2)\nwhereGll0(q;i\nm) is the magnon Green's function in thesublattice basis with \n m= 2\u0019Tm and an integer m,\nGll0(q;i\nm) =\u0000ZT\u00001\n0d\u001cei\nm\u001chT\u001cxql(\u001c)xy\nql0i:(B3)\nThen the magnon operators in the sublattice basis, xql\nandxy\nql, are connected with those in the band basis, xq\u0017\nandxy\nq\u0017, through the Bogoliubov transformation,\nxql=X\n\u0017=\u000b;\f(Uq)l\u0017xq\u0017; (B4)\nwherexq\u000b=\u000bq,xq\f=\fy\nq, (Uq)A\u000b= (Uq)B\f= cosh\u0012q,\nand (Uq)A\f= (Uq)B\u000b=\u0000sinh\u0012q; as described in the\nmain text, these hyperbolic functions satisfy cosh 2 \u0012q=\nJ0(SA+SB)\n\u0001\u000fqand sinh 2\u0012q=2pSASBJq\n\u0001\u000fq. ThusGll0(q;i\nm)\nis related to the magnon Green's function in the band\nbasis,G\u0017(q;i\nm):\nGll0(q;i\nm) =X\n\u0017=\u000b;\f(Uq)l\u0017(Uq)l0\u0017G\u0017(q;i\nm);(B5)\nwhere\nG\u000b(q;i\nm) =1\ni\nm\u0000\u000f\u000b(q); G\f(q;i\nm) =\u00001\ni\nm+\u000f\f(q):\n(B6)\nCombining Eq. (B5) with Eqs. (B2) and (B1), we have\n\b12(i\nn) =\u00001\nNX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)\n\u0002TX\nmG\u0017(q;i\nn+m)G\u00170(q;i\nm)\n=\u00001\nNX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)G(II)\n\u0017\u00170(q;i\nn);\n(B7)\nwhere\nvx\n\u00170\u0017(q) =X\nl1;l2=A;Bvx\nl1l2(q)(Uq)l1\u00170(Uq)l2\u0017; (B8)\nex\n\u0017\u00170(q) =X\nl3;l4=A;Bex\nl3l4(q)(Uq)l3\u0017(Uq)l4\u00170: (B9)\nThen we can rewrite G(II)\n\u0017\u00170(q;i\nn) in Eq. (B7) as follows:\nG(II)\n\u0017\u00170(q;i\nn) =Z\nCdz\n2\u0019in(z)G\u0017(q;i\nn+z)G\u00170(q;z)\n+T[G\u0017(q;i\nn)G\u00170(q;0) +G\u0017(q;0)G\u00170(q;\u0000i\nn)];\n(B10)\nwheren(z) is the Bose distribution function, n(z) =\n(ez=T\u00001)\u00001, and C is one of the contours shown in Fig.\n3. Using Eqs. (B10) and (B6), we obtain9\n\tB\n \tC\n \tD\n \tE\nFIG. 3. The contours used for the integrations in (a) G(II)\n\u000b\u000b(q;i\nn), (b)G(II)\n\u000b\f(q;i\nn), (c)G(II)\n\f\u000b(q;i\nn), and (d)G(II)\n\f\f(q;i\nn).\nThe horizontal dashed lines correspond to Im z=\u0000\nn.\nG(II)\n\u0017\u00170(q;i\nn) =Z1\n\u00001dz\n2\u0019in(z)n\nGR\n\u0017(q;z+i\nn)[GR\n\u00170(q;z)\u0000GA\n\u00170(q;z)] + [GR\n\u0017(q;z)\u0000GA\n\u0017(q;z)]GA\n\u00170(q;z\u0000i\nn)o\n;(B11)\nwhereGR\n\u0017(q;z) is the retarded magnon Green's function,\nGR\n\u000b(q;z) =1\nz\u0000\u000f\u000b(q) +i\r; GR\n\f(q;z) =\u00001\nz+\u000f\f(q) +i\r; (B12)\nGA\n\u0017(q;z) is the advanced one, and \ris the magnon damping. By combining Eq. (B11) with Eq. (B7) and performing\nthe analytic continuation i\nn!!+i\u000ewith\u000e= 0+, we have\n\bR\n12(!) = \b 12(i\nn!!+i\u000e) =\u00001\nNX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)Z1\n\u00001dz\n2\u0019in(z)\n\u0002n\nGR\n\u0017(q;z+!)[GR\n\u00170(q;z)\u0000GA\n\u00170(q;z)] + [GR\n\u0017(q;z)\u0000GA\n\u0017(q;z)]GA\n\u00170(q;z\u0000!)o\n:\n(B13)\nBy usingG(z+!) =G(z) +!@G(z)\n@z+O(!2) and performing the partial integration, we obtain\nL0\n12= lim\n!!0\bR\n12(!)\u0000\bR\n12(0)\ni!\n=\u00001\n4NX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)Z1\n\u00001dz\n\u0019@n(z)\n@zh\nGR\n\u0017(q;z)GR\n\u00170(q;z)\u00002GR\n\u0017(q;z)GA\n\u00170(q;z) +GA\n\u0017(q;z)GA\n\u00170(q;z)i\n=1\nNX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)Z1\n\u00001dz\n\u0019@n(z)\n@zImGR\n\u0017(q;z)ImGR\n\u00170(q;z): (B14)\nIn deriving this equation we have used the symmetry relations vx\n\u00170\u0017(q) =vx\n\u0017\u00170(q) andex\n\u0017\u00170(q) =ex\n\u00170\u0017(q). Equation\n(B14) is Eq. (13).\nNext, we derive L0\n12, Eq. (15). By using Eq. (B1), we can write the correction due to the \frst-order perturbation\nofHintas follows:\n\u0001\b12(i\nn) = +1\nNX\nq;q0X\nl1;l2;l3;l4=A;Bvx\nl1l2(q)ex\nl3l4(q0)ZT\u00001\n0d\u001cei\nn\u001cZT\u00001\n0d\u001c1hT\u001cxy\nql1(\u001c)xql2(\u001c)xy\nq0l3xq0l4Hint(\u001c1)i:(B15)\n[Note that Hinthas been de\fned in Eq. (4).] By using Wick's theorem51, we can calculate\nhT\u001cxy\nql1(\u001c)xql2(\u001c)xy\nq0l3xq0l4Hint(\u001c1)i; the result is\nhT\u001cxy\nql1(\u001c)xql2(\u001c)xy\nq0l3xq0l4Hint(\u001c1)i=\u00001\nNX\nl5;l6;l7;l8=A;BVl5l6l7l8(q;q0)Gl5l1(q;\u001c1\u0000\u001c)Gl2l6(q;\u001c\u0000\u001c1)\n\u0002Gl7l3(q0;\u001c1)Gl4l8(q0;\u0000\u001c1); (B16)10\nwhereGll0(q;\u001c) =TP\nme\u0000i\nm\u001cGll0(q;i\nm),\nVl5l6l7l8(q;q0) =8\n>>>>>>>>>>><\n>>>>>>>>>>>:4J0 (l5=l6=l;l7=l8=\u0016l);\n4Jq\u0000q0 (l5=l8=l;l6=l7=\u0016l);\n2Jq0q\nSA\nSB(l5=l6=B;l7=l;l8=\u0016l);\n2Jqq\nSA\nSB(l5=l;l6=\u0016l;l7=l8=B);\n2Jq0q\nSB\nSA(l5=l6=A;l7=l;l8=\u0016l);\n2Jqq\nSB\nSA(l5=l;l6=\u0016l;l7=l8=A);(B17)\nand\u0016l=BorAforl=AorB, respectively. Then, by substituting Eq. (B16) into Eq. (B15) and carrying out the\nintegrations, we obtain\n\u0001\b12(i\nn) =\u00001\nN2X\nq;q0X\nl1;l2;\u0001\u0001\u0001;l8=A;Bvx\nl1l2(q)ex\nl3l4(q0)Vl5l6l7l8(q;q0)T2X\nm;m0Gl5l1(q;i\nm)Gl2l6(q;i\nn+m)\n\u0002Gl7l3(q0;i\nn+m0)Gl4l8(q0;i\nm0): (B18)\nFurthermore, we can rewrite this equation by using the Bogoliubov transformation [i.e., Eq. (B4)]; the result is\n\u0001\b12(i\nn) =\u00001\nN2X\nq;q0X\n\u00171;\u00172;\u00173;\u00174=\u000b;\fvx\n\u00171\u00172(q)ex\n\u00173\u00174(q0)V\u00171\u00172\u00173\u00174(q;q0)\u0001G(II)\n\u00171\u00172\u00173\u00174(q;q0;i\nn); (B19)\nwhere\nV\u00171\u00172\u00173\u00174(q;q0) =X\nl5;l6;l7;l8=A;BVl5l6l7l8(q;q0)(Uq)l5\u00171(Uq)l6\u00172(Uq0)l7\u00173(Uq0)l8\u00174; (B20)\n\u0001G(II)\n\u00171\u00172\u00173\u00174(q;q0;i\nn) =T2X\nm;m0G\u00171(q;i\nm)G\u00172(q;i\nn+m)G\u00173(q0;i\nn+m0)G\u00174(q0;i\nm0): (B21)\nSincevx\n\u00171\u00172(q) andex\n\u00173\u00174(q0) are odd functions in term of qxandq0\nx, respectively, and G\u0017(q;i\nm)'s are even functions,\nthe \fnite terms of V\u00171\u00172\u00173\u00174(q;q0) in Eq. (B19), i.e., the terms which are \fnite even after carrying outP\nq;q0, come\nonly fromVABBA (q;q0) =VBAAB (q;q0) = 4Jq\u0000q0[Eq. (B17)]; because of this property, we can replace Eq. (B20) by\nV\u00171\u00172\u00173\u00174(q;q0) =X\nl=A;B4Jq\u0000q0(Uq)l\u00171(Uq)\u0016l\u00172(Uq0)\u0016l\u00173(Uq0)l\u00174: (B22)\nThen, as in G(II)\n\u0017\u00170(q;i\nn) [Eq. (B10)], we can replace the sums in Eq. (B21) by the corresponding integrals:\n\u0001G(II)\n\u00171\u00172\u00173\u00174(q;q0;i\nn) =hZ\nCdz\n2\u0019in(z)G\u00171(q;z)G\u00172(q;z+i\nn) +AihZ\nC0dz0\n2\u0019in(z0)G\u00173(q0;z0+i\nn)G\u00174(q0;z0) +A0i\n=G(II)\n\u00172\u00171(q;i\nn)G(II)\n\u00173\u00174(q0;i\nn); (B23)\nwhereA=T[G\u00171(q;0)G\u00172(q;i\nn)+G\u00171(q;\u0000i\nn)G\u00172(q;0)],A0=T[G\u00173(q0;i\nn)G\u00174(q0;0)+G\u00173(q0;0)G\u00174(q0;\u0000i\nn)],\nandCorC0is one of the contours shown in Fig. 3. By substituting Eq. (B11) into Eq. (B23) and performing the\nanalytic continuation i\nn!!+i\u000e(\u000e= 0+), we have\n\u0001\bR\n12(!) = \u0001\b 12(i\nn!!+i\u000e)\n=\u00001\nN2X\nq;q0X\n\u00171;\u00172;\u00173;\u00174=\u000b;\fvx\n\u00171\u00172(q)ex\n\u00173\u00174(q0)V\u00171\u00172\u00173\u00174(q;q0)\n\u0002Z1\n\u00001dz\n2\u0019in(z)n\n[GR\n\u00171(q;z)\u0000GA\n\u00171(q;z)]GR\n\u00172(q;z+!) +GA\n\u00171(q;z\u0000!)[GR\n\u00172(q;z)\u0000GA\n\u00172(q;z)]o\n\u0002Z1\n\u00001dz0\n2\u0019in(z0)n\nGR\n\u00173(q0;z0+!)[GR\n\u00174(q0;z0)\u0000GA\n\u00174(q0;z0)] + [GR\n\u00173(q0;z0)\u0000GA\n\u00173(q0;z0)]GA\n\u00174(q0;z0\u0000!)o\n:(B24)11\nThen, by performing the calculations similar to the derivation of Eq. (B14), we obtain\nL0\n12= lim\n!!0\u0001\bR\n12(!)\u0000\u0001\bR\n12(0)\ni!\n=1\n4\u00192iN2X\nq;q0X\n\u00171;\u00172;\u00173;\u00174=\u000b;\fvx\n\u00171\u00172(q)ex\n\u00173\u00174(q0)V\u00171\u00172\u00173\u00174(q;q0)h\nF(I)\n\u00171\u00172(q)F(II)\n\u00173\u00174(q0) +F(II)\n\u00171\u00172(q)F(I)\n\u00173\u00174(q0)i\n;(B25)\nwhere\nF(I)\n\u0017\u00170(q) =\u00001\n2Z1\n\u00001dz@n(z)\n@zh\nGR\n\u0017(q;z)GR\n\u00170(q;z) +GA\n\u0017(q;z)GA\n\u00170(q;z)\u00002GA\n\u0017(q;z)GR\n\u00170(q;z)i\n= 2Z1\n\u00001dz@n(z)\n@zImGR\n\u0017(q;z)ImGR\n\u00170(q;z); (B26)\nF(II)\n\u0017\u00170(q0) =Z1\n\u00001dz0n(z0)h\nGR\n\u0017(q0;z0)GR\n\u00170(q0;z0)\u0000GA\n\u0017(q0;z0)GA\n\u00170(q0;z0)i\n= 2iZ1\n\u00001dz0n(z0)h\nReGR\n\u0017(q0;z0)ImGR\n\u00170(q0;z0) + ImGR\n\u0017(q0;z0)ReGR\n\u00170(q0;z0)i\n: (B27)\nA combination of Eqs. (B26), (B27), and (B25) gives Eq. (15).\nAppendix C: Derivations of Eqs. (18), (19),\n(21){(27)\nWe explain the details of the derivations of Eqs. (18),\n(19), (21){(27). These equations are obtained by deriving\nthe expressions of L0\n12andL0\n12in the limit \u001c!1 , where\n\u001c= (2\r)\u00001is the magnon lifetime.\nFirst, we derive Eqs. (18) and (19). Using Eq. (B12),\nwe have\nImGR\n\u000b(q;z) =\u0000\r\n[z\u0000\u000f\u000b(q)]2+\r2; (C1)\nImGR\n\f(q;z) =\r\n[z+\u000f\f(q)]2+\r2: (C2)\nSince\u001c! 1 corresponds to \r!0, we can express\nI(I)\n\u0017\u00170(q) [i.e., Eq. (14)] in this limit as follows:\nI(I)\n\u000b\u000b(q)\u0018@n[\u000f\u000b(q)]\n@\u000f\u000b(q)Z1\n\u00001dz\r2\nf[z\u0000\u000f\u000b(q)]2+\r2g2\n=\u0019\n2\r@n[\u000f\u000b(q)]\n@\u000f\u000b(q); (C3)\nI(I)\n\f\f(q)\u0018\u0019\n2\r@n[\u000f\f(q)]\n@\u000f\f(q); (C4)\nI(I)\n\u000b\f(q) =I(I)\n\f\u000b(q)\u00180: (C5)Combining these equations with Eq. (13), we have\nL0\n12\u0018L0\n12\u000b+L0\n12\f; (C6)\nL0\n12\u0017=1\nNX\nqvx\n\u0017\u0017(q)ex\n\u0017\u0017(q)@n[\u000f\u0017(q)]\n@\u000f\u0017(q)\u001c: (C7)\nThese are Eqs. (18) and (19).\nNext, we derive Eqs. (21){(27). Since L0\n12is given by\nEq. (15), the remaining task is to derive the expression of\nI(II)\n\u0017\u00170(q) in the limit \u001c!1 . By performing the similar\ncalculations to the derivations of Eqs. (C3){(C5), we\nobtain12\nZ1\n\u00001dzn(z)ReGR\n\u000b(q;z)ImGR\n\u000b(q;z) =\u0000\rZ1\n\u00001dzn(z)z\u0000\u000f\u000b(q)\nf[z\u0000\u000f\u000b(q)]2+\r2g2\n=\u0000\rZ1\n\u00001dzn(z)@\n@zn\n\u00001\n21\n[z\u0000\u000f\u000b(q)]2+\r2o\n\u0018\u0000\u0019\n2@n[\u000f\u000b(q)]\n@\u000f\u000b(q); (C8)\nZ1\n\u00001dzn(z)ReGR\n\u000b(q;z)ImGR\n\f(q;z) =\rZ1\n\u00001dzn(z)z\u0000\u000f\u000b(q)\nf[z\u0000\u000f\u000b(q)]2+\r2gf[z+\u000f\f(q)]2+\r2g\u0018\u0000\u0019n[\u0000\u000f\f(q)]\n\u000f\u000b(q) +\u000f\f(q);\n(C9)\nZ1\n\u00001dzn(z)ReGR\n\f(q;z)ImGR\n\u000b(q;z) =\rZ1\n\u00001dzn(z)z+\u000f\f(q)\nf[z+\u000f\f(q)]2+\r2gf[z\u0000\u000f\u000b(q)]2+\r2g\u0018\u0019n[\u000f\u000b(q)]\n\u000f\u000b(q) +\u000f\f(q);(C10)\nZ1\n\u00001dzn(z)ReGR\n\f(q;z)ImGR\n\f(q;z) =\u0000\rZ1\n\u00001dzn(z)z+\u000f\f(q)\nf[z+\u000f\f(q)]2+\r2g2\n=\u0000\rZ1\n\u00001dzn(z)@\n@zn\n\u00001\n21\n[z+\u000f\f(q)]2+\r2o\n\u0018\u0000\u0019\n2@n[\u000f\f(q)]\n@\u000f\f(q): (C11)\nBy combining these equations with Eq. (16), we can\nexpressI(II)\n\u0017\u00170(q) in the limit \u001c!1 as follows:\nI(II)\n\u000b\u000b(q)\u0018\u0000\u0019@n[\u000f\u000b(q)]\n@\u000f\u000b(q); (C12)\nI(II)\n\f\f(q)\u0018\u0000\u0019@n[\u000f\f(q)]\n@\u000f\f(q); (C13)\nI(II)\n\u000b\f(q) =I(II)\n\f\u000b(q)\u0018\u0019n[\u000f\u000b(q)]\u0000n[\u0000\u000f\f(q)]\n\u000f\u000b(q) +\u000f\f(q):(C14)\nSubstituting these equations and Eqs. (C3){(C5) into\nEq. (15), we obtain\nL0\n12\u0018L0\n12-intra +L0\n12-inter1 +L0\n12-inter2; (C15)\nwhere\nL0\n12-intra =X\n\u0017=\u000b;\fL0\n12-intra-\u0017; (C16)\nL0\n12-intra-\u0017=\u00002\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u0017\u0017(q0)\u001cV\u0017\u0017\u0017\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0); (C17)\nL0\n12-inter1 =X\n\u0017=\u000b;\fn\n\u00002\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u0016\u0017\u0016\u0017(q0)\u001cV\u0017\u0017\u0016\u0017\u0016\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0016\u0017(q0)]\n@\u000f\u0016\u0017(q0)o\n; (C18)and\nL0\n12-inter2 =X\n\u0017=\u000b;\f(L0\nE\u0017+L0\nS\u0017); (C19)\nL0\nE\u0017=2\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u000b\f(q0)V\u0017\u0017\u000b\f(q;q0)\u001c\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)n[\u000f\u000b(q0)]\u0000n[\u0000\u000f\f(q0)]\n\u000f\u000b(q0) +\u000f\f(q0);(C20)\nL0\nS\u0017=2\nN2X\nq;q0vx\n\u000b\f(q)ex\n\u0017\u0017(q0)V\u000b\f\u0017\u0017(q;q0)\u001c\n\u0002n[\u000f\u000b(q)]\u0000n[\u0000\u000f\f(q)]\n\u000f\u000b(q) +\u000f\f(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0):(C21)\nIn Eq. (C18), \u0016 \u0017=\for\u000bfor\u0017=\u000bor\f, respectively.\nEquations (C15){(C21) are Eqs. (21){(27).\nAppendix D: Remark on the numerical calculation\nTo calculate L0\n\u0016\u0011andL0\n\u0016\u0011numerically, we perform the\nmomentum summations using a Nq-point mesh of the\n\frst Brillouin zone. Since the sublattice of our ferrimag-\nnetic insulator is described by a set of primitive vectors,\na1=t(1 0 0), a2=t(0 1 0), and a3=t(0 0 1), the prim-\nitive vectors for the reciprocal lattice are b1=t(2\u00190 0),\nb2=t(0 2\u00190), and b3=t(0 0 2\u0019). Thus, in the periodic\nboundary condition, momentum qis written in the form\nq=mx\nNxb1+my\nNyb2+mz\nNzb3; (D1)\nwhere 0\u0014mx< Nx, 0\u0014my< Ny, and 0\u0014mz<\nNzwithNxNyNz=Nq=N=2. As a result, the \frst\nBrillouin zone is divided into the ( NxNyNz)-point mesh.\nIn the numerical calculation we set Nx=Ny=Nz= 24\n(i.e.,Nq= 243).13\nAppendix E: Numerical results at h= 0:08Jand 0:16J\nWe present the additional results of the numerical cal-\nculations, the temperature dependences of Sm,\u001bm, and\n\u0014math= 0:08Jand 0:16J. They are shown in Figs.4(a){4(f). Comparing these \fgures with Fig. 2, we see\nthe results obtained at h= 0:08Jand 0:16Jare similar\nto those obtained at h= 0:02J. Namely, the properties\nobtained at h= 0:02Jremain qualitatively unchanged\nfor other values of h.\n\u0003arakawa@phys.chuo-u.ac.jp\n1Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. Saitoh, Nature (London)\n464, 262 (2010).\n2K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J.\nIeda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E.\nW. Bauer, S. Maekawa, and E. Saitoh, Nat. 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B 74, 174431 (2006)."    },    {        "title": "0910.2377v1.Monte_Carlo_Study_of_Mixed_Spin_S__1_2_1__Ising_Ferrimagnets.pdf",        "content": "arXiv:0910.2377v1  [cond-mat.mtrl-sci]  13 Oct 2009Monte Carlo Study of Mixed-Spin S=(1/2,1) Ising\nFerrimagnets\nW Selke1,2and J Oitmaa2\n1Institut f¨ ur Theoretische Physik, RWTH Aachen, 52056 Aachen, Germany\n2School of Physics, The University of New South Wales, Sydney, NSW 2052,\nAustralia\nAbstract. WeinvestigateIsingferrimagnetsonsquareandsimple–cubiclattice swith\nexchange couplings between spins of values S=1/2 and S=1 on neighb ouring sites and\nan additional single–site anisotropy term on the S=1 sites. Based ma inly on a careful\nand comprehensive Monte Carlo study, we conclude that there is no tricritical point in\nthe two–dimensional case, in contradiction to mean-field prediction s and recent series\nresults. However, evidence for a tricritical point is found in the thr ee–dimensionalcase.\nIn addition, a line of compensation points is found for the simple–cubic , but not for\nthe square lattice.\nPACS numbers: 75.10.-b, 75.10.Hk, 75.40.Mg, 75.50.Gg\nSubmitted to: Institute of Physics Publishing\nJ. Phys.: Condens. Matter\n1. Introduction\nMixed-spin Ising models have been studied for some time as simple mode ls of\nferrimagnets, and there has been renewed interest recently in co nnection with\n’compensation points’. These are temperatures, below the critical temperature, at\nwhich the sublattice magnetizations cancel exactly, giving zero tot al moment. As the\ntemperature is tuned through such a point the total magnetizatio n changes sign, which\nmay be used in technological applications. In this context, Ising mod els may be exactly\nsolvable in special cases [1, 2, 3, 4] or they may be studied by a variet y of powerful\napproaches, including Monte Carlo [5, 6, 7, 8, 9, 10] or other [11, 1 2, 13, 14, 15] methods.\nIn the present work we revisit oneof the simplest such models, a mixe d–spin Ising model\nwith spins S=1/2 and 1 occupying the sites of a bipartite square or sim ple cubic lattice\nwith the Hamiltonian\nH=−J/summationdisplay\n/angbracketlefti,j/angbracketrightσiSj+D/summationdisplay\nj∈BSj2(1)\nwith couplings Jbetween spins σi=±1 on the sites of sublattice ’A’, and\nneighbouring spins Sj= 1,0,−1 on sites forming the sublattice ’B’. D denotes theMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 2\nstrength of a single–ion term acting only on the S=1 spins of sublattic e B. Following\nprevious convention, we choose σi=±1 rather than ±1/2, which has to be taken into\naccountwhencalculatingsublatticemagnetizationsandwhendefinin gthecompensation\npoint. Theconventionsimplyamountstoarescalingoftheexchange coupling. Notethat\nthe nearest neighbour coupling Jmay be either antiferromagnetic, J <0, as assumed\noften for ferrimagnets, or ferromagnetic, J >0. Both cases are completely equivalent\nby a simple spin reversal on either sublattice. We shall use in this artic le ferromagnetic\ncouplings. As a consequence, in our case the magnetizations of bot h sublattices are\nidentical at the compensation point, while in the antiferromagnetic c ase, at the same\ncompensation point, the sublattice magnetizations have equal mag nitude but different\nsign leading to the above mentioned vanishing of the total magnetiza tion.\nThe model on the square lattice has been studied by several autho rs. Kaneyoshi\nand Chen [13], via a mean-field treatment, found a line of compensatio n points in a\nnarrow region 4 > D/J≥2 ln6 (= 3.583..) and a tricritical point at Dt/J= 3.72,\ni.e. a first-order transition for D > D t. Buendia and Novotny [8], using transfer matrix\nmethods, supplemented by Monte Carlo simulations, found no eviden ce of either a\ncompensation point or a tricritical point, although a compensation p oint was observed\nin an extended model with additional ferromagnetic interactions be tweenσspins. More\nrecently, Oitmaa and Enting [16] studied the same model using a comb ination of high-\nand low-temperature series. No compensation point was found, bu t evidence for a\nfirst-order transition, and hence a tricritical point was observed from an apparent\ncrossing of the high- and low-temperature branches of the free e nergy with different\nslopes, for D/J≥3.2. Thus the phase diagram of this simple model remained\nuncertain, motivating partly the present extensive Monte Carlo st udy, improving\nprevious simulations substantially. In fact, our study provides clea r evidence that the\nmodel in two dimensions has no compensation point or tricritical point . Moreover, the\nmodel is found to exhibit very interesting thermal behaviour, both for the specific heat\nand the magnetization, especially in the low–temperature region nea rD= 4, which\nhas not been discussed in detail before. This behaviour is the likely ex planation for the\napparent ’first-order’ behaviour observed in Ref. 16.\nFor the simple cubic lattice, to our knowledge, no detailed analyses ha ve been done\nso far. Of course, mean–field theory may be easily applied, leading ag ain to a tricritical\npoint and a line of compensation points.\nThe outline of the article is as follows. In Section 2 we present and disc uss our\nresults for the square lattice. In Section 3 we consider the simple–c ubic lattice. Here, in\ncontrasttothetwo–dimensional case, wefindaclearoccurrence ofalineofcompensation\npoints. Furthermore, we obtain clear evidence of transitions of fir st-order, and thence of\na tricritical point, which we locate approximately. In the final sectio n, a brief summary\nwill be given.Monte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 3\n2. The model on the square lattice\nLet us first consider the ferrimagnet, eq. (1), in the case of a squ are lattice. We have\nperformed mainly standard Monte Carlo simulations, using the Metro polis algorithm\nwith single–spin flips, providing, indeed, the required accuracy, so t hat there was no\nneed to apply other techniques like cluster–updates or the Wang–L andau approach [17].\nWe studied lattices with LxLsites, employing full periodic boundary conditions. L\nranged from 4 to 80, to study finite–size effects. Typically, runs of 107Monte Carlo\nsteps per spin have been done, with averages and error bars obta ined from evaluating a\nnumber of such runs, at least three, using different random numbe rs. These rather long\nruns lead to very good statistics, improving appreciably results of p revious simulations\n[7, 8]. The estimated errors, unless shown otherwise, are smaller th an the symbols\ndepicted in the figures.\n33.13.2 3.3 3.4 3.53.63.7 3.8 3.9 4\nD/J00.20.40.60.811.21.41.6kBT/J\nFigure 1. Phase diagram of the mixed–spin model on a square lattice.\nWe recorded the energy per site, E, the specific heat, C, both from the energy\nfluctuationsandfromdifferentiating Ewithrespecttothetemperature, andtheabsolute\nvalues of the sublattice magnetizations of the two sublattices\n|mA|=<|/summationdisplay\nAσi|> /(2(L2/2)) (2)\nand\n|mB|=<|/summationdisplay\nBSj|> /(L2/2) (3)\nas well as the absolute value of the total magnetization,\n|m|=<|/summationdisplay\nAσi+/summationdisplay\nBSj|> /L2(4)\nwhere the brackets <>denote the thermal average. Note the factor of 1/2 in\nthe definition of |mA|, taking into account the correct length of the S=1/2 spins, soMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 4\nthat|mA(T= 0)|= 1/2, while |mB(T= 0)|= 1 for the ferromagnetic ground state.\nIn addition, the corresponding susceptibilities, χA,χB, andχ, have been computed\nfrom the fluctuations of the magnetizations. We also analysed histo grams for the\ntotal magnetization, p(m), i.e. the probability to encounter a configuration with the\nmagnetization m, as well as the fourth–order cumulant of the order parameter, t he\nBinder cumulant [18], defined by\nU= 1−< m4> /(3< m2>2) (5)\nwith< m2>and< m4>being the second and fourth moment of the total\nmagnetization. Finally, we monitored typical equilibrium Monte Carlo co nfigurations,\nillustrating the microscopic behaviour of the system.\nTo test the accuracy of the simulations, we computed numerically ex act results\nfor various quantities by enumerating all possible configurations fo r small lattices with\nL= 4.\n0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8 2 2.2\nkBT/J00.20.40.60.811.2 C\n0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8 2 2.2 2.4 2.6\nkBT/J00.050.10.150.20.250.3C\nFigure 2. (a) Left: Specific heat at D/J= 3.0, showing numerically exact, for\nL= 4(solid line), and Monte Carlo data for sizes L= 10 (circles), 20 (squares),\n40 (diamonds) and 60 (triangles). (b) Right: Specific heat at D/J=3.8, showing\nnumerically exact, for L= 4 (solid line), and Monte Carlo data for sizes L= 20\n(circles), 40 (squares) and 80 (diamonds).\nIn agreement with previous work, the model is observed to display a ferromagnetic\nground state and low–temperature phase for D/J <4. The energy to flip a B spin\nfrom its ferromagnetic orientation, ’+’ or ’ −’, surrounded by four A spins of the same\norientation, to the state 0 is obviously ∆ E= 4J−Dwhich vanishes at D= 4J.\nHence the ground state at D/J= 4 will comprise configurations with ’0’ states on B\nsites and arbitrarily oriented spins on the neighbouring A sites, as we ll as ferromagnetic\nplaquettes (of either sign) on B sites and neighbouring A sites. Due t o the resulting\nhigh degeneracy, one may call ( D/J= 4,T= 0) the ’degeneracy point’. For D >4J\nat zero temperature, all B spins will be in the state 0, with the A spins being randomlyMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 5\noriented. This leads to a lower, but still macroscopic degeneracy. A tD/J≥4, there is\nno ordered phase even at zero temperature.\nMost of our Monte Carlo work deals with the interesting range 3 ≤D/J <4,\nwhich had been discussed controversially before, augmented by so me simulations at\nlower values of D/J. The resulting phase diagram is depicted in Fig. 1, based on\nmonitoring the size–dependence of the position of the (critical) max ima in the specific\nheat and susceptibility, and the intersection points of the Binder cu mulant, see below.\nOur findings are in accordance with a continuous transition in the Isin g universality\nclass for all values of D/Jwe studied, D/J≤3.98. There is no compensation point.\nIn the following, we shall discuss main properties of the physical qua ntities\nmentioned above.\nThe specific heat C, for negative or relatively small positive D/J, is observed to\nresemble qualitatively that of the nearest–neighbour Ising model o n a square lattice.\nThere is a unique maximum in C(T), for finite L, turning into a logarithmic singularity\nin the thermodynamic limit. Indeed, in the limit D/J→ −∞, one recovers the simple\nIsing model. Increasing D/J, as displayed in Fig. 2a for D/J= 3.0, an additional\nshoulder or maximum evolves at a lower temperature, Tl, being largely independent of\nlattice size and being non–critical. Its origin becomes clear by furthe r increasing D/J,\nas shown in Fig. 2bfor D/J= 3.8. In fact, one finds kBTl/J≈0.42(4−D/J), reflecting\nthe thermally activated flipping of B spins from the ferromagnetic st ate ’1’ (or –’1’) to\nthe state zero, requiring, as stated above, an energy proportio nal to 4 −D/J. It is\ninteresting to note that the height of the pronounced non–critica l peak, signalling the\npartial disordering of the B sublattice, depends only very weakly on D/J. In the range\n3.5≤D/J <4, one has C(Tl)≈0.22.\nAs illustrated in Fig. 3b for D/J= 3.8, the critical peak, located at Tm, may\nseparate from the upper maximum, at Tu, when increasing the strength of the single–ion\nterm. Thus, the specific heat may display a three–peak structure , with two non–critical\nmaxima and a critical peak in between. The origin of the maximum at Tuis due to\nthe fact that at the critical point, the σspins on the A sublattice form rather large\nclusters of different orientations, leading to the vanishing of the or der parameter. That\nbehaviour may be seen by monitoring typical equilibrium configuration s. These clusters\nshrink quickly near Tu, due to thermally activated flipping of σspins, determined by the\ncoupling constant J. Indeed, Tuis essentially independent of D. As seen in Fig. 3b, the\nmaximum in CatTudepends rather weakly on the size of the lattice, L, demonstrating\nits non–critical character.\nThe height of the critical maximum at Tmis expected, for Ising universality,\nto increase logarithmically with Lfor sufficiently large values of L. Our results are\nconsistent with this expectation. However, on approach to the de generacy point, the\nbackground contribution to the specific heat becomes more and mo re relevant. Then\nlarger and larger lattices, with L > L 0, are needed to see the anticipated logarithmic\nbehaviour. For example, at D/J= 3.6, one gets L0≈40, andL0≈60 atD/J= 3.95.\nIn fact, in that range, the Ising–like character of the transition m ay be inferred moreMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 6\nclearly from other quantities, as discussed below.\n0 0.1 0.2 0.3 0.4 0.5 0.6\nkBT/J00.10.20.30.40.50.60.70.80.91|mA,B|\nFigure 3. Sublattice magnetizations |mA|and|mB|atD/J= 3.8 (circles) and 3.95\n(diamonds), with lattices of size L= 40 (dashed lines) and 60 (solid lines).\nThe partial disordering of the B sublattice, near Tl, leads to a rapid decrease\nof the magnetization |mB|, as illustrated in Fig. 3. Actually, the anomaly in |mB|\nbecomes more and more dramatic on approach to the degeneracy p oint. In contrast,\nthe magnetization of the A sublattice, |mA|, is hardly affected by the disordering of the\nB sublattice. Indeed, this behaviour may open the possibilty of a com pensation point,\nat which the two sublattice magnetizations, |mA|and|mB|, would coincide. However, as\ndepicted in Fig. 3, we find no evidence for such a compensation point in two dimensions\nfor all cases we studied, with D/Jgoing up to 3.95.\nThe susceptibility χis found to show, in all cases we studied, only one maximum,\nclose to the critical temperature. The background term is much we aker than for the\nspecific heat, allowing an analysis of critical properties for smaller lat tices. In fact, as\nillustrated in Fig. 4, the size dependence of the height of the maximum inχ,χmax(L),\nis observed to be nicely compatible with the asymptotic form χmax∝L7/4, expected for\nthe Ising universality class, for all cases studied and sufficiently larg e lattices. Note that\nthe susceptibility shows a rather mild anomaly near Tl, where the specific heat shows a\npronounced maximum, close to the degeneracy point. At that anom aly,χ(T) exhibits\na maximal slope, as may be easily identified using exact enumeration fo r small lattices.\nThe shrinking of the A clusters, as indicated by the broad maximum in CatTu, leads\nto no obviously unusual features in the susceptibility.\nAs usual, one may estimate the bulk transition temperature, Tc, from the size\ndependent position of the corresponding peaks in χandC. We obtain consistent\nestimates, shown in Fig. 1, with the location of the maxima varying, fo r largeL,\nproportionally to 1 /L, as expected for Ising–like transitions. Of course, one gets distin ct\nproportionality factors for the two quantities.Monte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 7\n2 2.5 3 3.5 4 4.5 5\nln (L)02468ln (χmax(L))\nFigure 4. Log–log plot of the susceptibility χmaxversus system size Lfor the square\nlattice at D/J= 3.6 (circles), 3.8 (squares) and 3.95 (diamonds). For comparison, the\ndashed line shows χmax∝L7/4.\nThe transition temperature may be also conveniently estimated fro m the Binder\ncumulant, U. Indeed, the estimates follow from the location of the intersection\ntemperatures of the cumulants for different lattice sizes [18]. Finite size corrections\noften turn out to be rather small. Actually, this is also true for the p resent model, as\nshown in Fig. 5 for D/J= 3.95. We find very good agreement with the estimates\nofTcbased on the susceptibilty and the specific heat. Note that the valu e ofUat the\nintersection temperature is, already for fairly small systems sizes , close to the accurately\nknown [19] critical Binder cumulant U∗=U(Tc,L=∞) for isotropic Ising models,\nU∗= 0.6069.... One may emphasize that anisotropic interactions and correlations may\nlead to non–trivial dependences of U∗on such interactions [20, 21]. However, here we\nare dealing with an isotropic system, and excellent agreement with th e known critical\nvalue is observed, demonstrating that the the transition belongs t o the Ising universality\nclass.\nAdditional insight into the phase transition is provided by the histogr ams for the\ntotal magnetization, p(m). An example is displayed in Fig. 6. As expected for a\ncontinuous transition, p(m) shows, in the ferromagnetic low–temperature phase, two\nsymmetric peaks, at ±m0, moving closer and closer to each other on approach to Tcand\nwhen increasing the lattice size. Above Tc,p(m) tends to acquire a Gaussian shape [18].\nWe emphasize that Fig.6 refers to the case D/J= 3.98, i.e. very close to the degeneracy\npoint. There is no indication of a transition of first order, which might be signalled by a\ncentral peak, in addition to the two peaks at ±m0, as would be the case for coexistence\nof the disordered and ordered phases. Accordingly, we may safely conclude, based on\nthe analysis of several quantities, that we have clear evidence for continuous transitions\nof Ising type along the boundary of the ferromagnetic phase, at le ast for the regionMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 8\n0.11 0.115 0.12 0.125 0.13 0.135 0.14\nkBT/J0.50.520.540.560.580.60.620.640.66 U\nFigure 5. Binder cumulant U(L,T) at D/J=3.95 for L=20 (circles), 40 (squares),\n60 (diamonds), and 80 (triangles). The horizontal line indicates the critical Binder\ncumulant of an isotropic Ising model in the thermodynamic limit [19].\nD/J≤3.98.\n-400 -300 -200 -100 0100200 300 400m00.0010.0020.0030.0040.0050.0060.0070.0080.009p(m)\nFigure 6. Histogram of the total magnetization, p(m), for the square lattice\nwithL= 20 at D/J= 3.98 and temperatures below and above the transition,\nkBT/J= 0.04,0.05,0.06, and 0.07 (peak positions moving towards the center), with\nkBTc/J≈0.051.\n3. The model on the simple–cubic lattice\nLet us now turn to the analysis of the mixed–spin model, eq. (1), on a simple cubic\nlattice. In complete analogy to the two–dimensional case, we did sta ndard Monte CarloMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 9\nsimulations, applying the Metropolis algorithm. We studied lattices with L3sites, with\nLranging from 4 to 32. Full periodic boundary conditions were employe d. Typically,\nruns of 2 ×106to 5×106Monte Carlo steps per spin were performed, averaging over a\nfew, at least three, such runs to estimate thermal averages and error bars.\n3.63.8 4 4.2 4.4 4.64.85 5.2 5.4 5.6 5.8 6\nD/J00.511.522.533.54kBT/J\nFigure 7. Phasediagramofthe mixed–spinmodelonasimple–cubiclattice. The s olid\nline denotes the boundary of the ferromagnetic phase, while the da shed line denotes\ncompensation points.\nAs for the square lattice, the energy E, the specific heat C, magnetizations\n|mA|,|mB|, and|m|, as well as corresponding susceptibilities, the Binder cumulant U,\nand histograms for the total magnetization, p(m), were recorded. Typical Monte Carlo\nequilibrium configurations were generated to illustrate the microsco pic behaviour.\nFor the cubic lattice, one has a ferromagnetic ground state at D/J <6. The\ndegeneracy point occurs now at D/J= 6, with ground states comprising local\nferromagnetic plaquettes of neighbouring A and B spins as well as B s pins in the state\n0 with surrounding A spins being randomly oriented. For D/J >6, a high, but reduced\ndegeneracy prevails, with all B spins being zero, and the A spins point ing randomly ’up’\nor ’down’.\nForD/Jsmall or negative, a continuous transition of Ising type is expected to\noccur, as we confirm in simulations with moderate efforts. Most of ou r work has been\ndone for 3 .5≤D/J <6, to identify possible deviations from that kind of transition.\nIndeed, significant deviations from Ising universality have been obs erved for D/J≥5.9,\nwhile for smaller values of D/Jthe simulational data are consistent with an Ising–like\ntransition. In addition, we identified and located a line of compensatio n points in the\nrange 5.5< D/J < 6. The main features of the phase diagram are summarized in Fig.\n7. The phase transition line is based on analyzing various quantities an d taking into\naccount finite–size effects, as for the square lattice. Details of ou r Monte Carlo findings\nwill be discussed in the following.Monte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 10\nThe specific heat C(T) shows for small and negative values of D/Ja single\nmaximum, giving rise to critical behaviour in the thermodynamic limit. In case of an\nIsing–like transition, itsheight isexpected [22] togrowlike Cmax∝Lα/νwith thecritical\nexponents of the Ising universality class, α≈0.11 andν≈0.63 [23]. Our simulational\nfindings confirm this scenario. As in the case of the square lattice, u pon increasing D/J,\none encounters, eventually, three maxima in C(T), see Fig. 8. In complete analogy to\nthe two–dimensional case, the peak at the lower temperature, Tl, is rather sharp and\ndepends only very weakly on lattice size. It signals the partial disord ering of the B\nsublattice, with B spins being flipped thermally from the ferromagnet ic (’+’ or ’ −’)\nstate to 0. The maximum occurs at kBTl/J≈0.6(6−D/J). The upper, rather broad\nmaximum, at Tu, is non–critical as well, stemming from dissolving the, at criticality\nstill quite large spin clusters on the A sublattice. Tuis only very weakly affected by\nthe strength of D, being determined by the ferromagnetic coupling J. In between the\ntwo non–critical maxima in C(T), a critical peak shows up. It signals the transition, at\nwhich both sublattice magnetizations vanish, with quite pronounced local spin order on\nthe A sublattice.\nThe type of the transition may be inferred from the size dependenc e of the critical\npeak,Cmax(L). Indeed, for single–ion terms up to D/J= 5.8, we find agreement with\nan Ising–type transition, α/ν≈0.17. On further approach to the degeneracy point,\naccurate Monte Carlo data with a fine temperature resolution are r equired, due to the\nrather large nonanalytic background term in Cand the sharpness of the peak. In\nfact, other quantities may provide more easily and clearly reliable clue s on the type of\ntransition for that part of the transition line of the ferromagnetic phase.\n0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2\nkBT/J00.050.10.150.20.250.3C\nFigure 8. Specific heat versus temperature for the model on the simple–cub ic lattice\natD/J= 5.9for systems with L= 4(circles), 10 (squares), 16(diamonds), 20(triangles\nup) and 32 (triangles left).\nBefore discussing further the type of the phase transition close t o the degeneracyMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 11\npoint, we shall deal with the compensation points. Indeed, we ident ified such points\nin the range 5 .5< D/J < 6. The resulting line is depicted in Fig. 7. Two concrete\nexamples are shown in Fig. 9, for D/J= 5.7 and 5.9. As may be inferred from that\nfigure, the sublattice magnetization at the compensation point dec reases monotonically\nwith decreasing single–ion term. Therefore, when the compensatio n occurs at low\nmagnetizations, the accurate location of the compensation point is difficult, because\nof strong finite–size effects in the critical region. On the other han d, with increasing\nD/J, the compensation point moves towards lower temperatures, and finite size effects\nplayusuallynosignificant role. Inanyevent, incontrasttothetwo– dimensional case, we\nfindalineofcompensationpointsforthesimple–cubic lattice. Obvious ly, thedecreasein\nthe magnetization of the B sublattice, |mB|, occurs in three dimensions more drastically\nthan for the square lattice, while |mA|changes there rather mildly in both cases.\n0 0.2 0.4 0.60.81 1.2 1.4 1.6 1.8\nkBT/J00.10.20.30.40.50.60.70.80.91|mA,B|\nFigure 9. Sublattice magnetizations |mA|and|mB|for the simple–cubic lattice with\nL= 20 atD/J= 5.7 (circles) and 5.9 (squares).\nLet us now turn back to the discussion on the type of phase transit ion. For\nD/J≤5.8, the data on the susceptibilty χconfirm the Ising–like character of the\ntransition. In particular, the size dependence of the height of the maximum in χ,\nχmax(L), isfoundtobeconsistent withIsingcriticality, χmax∝Lγ/ν, whereγ≈1.24and\nν≈0.63, thusγ/ν≈1.97. Indeed, from our simulational data we obtain characteristic\nexponents close to 2. However, at D/J= 5.9, we observe, for systems sizes ranging from\nL= 8 toL= 32, a substantially lower (effective) exponent, of about 1.7. Beca use\nthe peak in χgets extremely sharp, very accurate simulational data with a very fine\ntemperature mesh are needed to arrive at safe conclusions. A mor e convenient way to\nmonitor the possible change in the type of the transition will be discus sed below.\nInterestingly, our analysis of the Binder cumulant Useems to indicate substantial\ndeviations from an Ising–like transition at about D/J≈5.9 as well. For smaller values\nofD/Jthe intersection values of the cumulant curves for different syste m sizes, alreadyMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 12\n1.9 1.92 1.94 1.96 1.98 2 2.02 2.04\nkBT/J0.320.360.40.440.480.520.560.6U\n1.2 1.24 1.28 1.32 1.36 1.4\nkBT/J0.320.360.40.440.480.520.560.6U\nFigure 10. (a) Left: Binder cumulant Uversus temperature at D/J= 5.5 for lattices\nwithL= 8 (circles), 12 (squares), 16 (diamonds) and 20 (triangles). (b) Right:U\nversus temperature at D/J= 5.9 for lattices with L= 10 (circles), 16 (squares),\n20 (diamonds), and 32 (triangles). The horizontal lines indicate the critical Binder\ncumulant of an isotropic three–dimensional Ising model in the therm odynamic limit\n[24].\nfor fairly small systems, seem to agree with the expected asympto tic value of the critical\nBinder cumulant for isotropic Ising systems [24], U∗≈0.465. An example is depicted in\nFig. 10a, for D/J= 5.5, with the intersection points, for the simulated finite lattices,\napproachingtheasymptoticvaluefrombelow, whenincreasingthes ystemsize. Atlarger\nsingle–ion anisotropy, D/J≥5.9, the intersection points of the curves are appreciably\nlower than U∗, as shown in Fig. 10b for D/J= 5.9. However, it is not completely clear,\nwhether the tendency reflects stronger finite–size effects or a c hange in the type of the\nphase transition.\nTo get more evidence for a possible change of the nature of the tra nsition, the\nhistograms for the magnetization, p(m), turned out to be most instructive. Already\nfor small lattices, L= 4, one sees, close to the transition, a qualitative change of the\nhistograms. We did simulations close to the transitions in the range 5 .85≥D/J≥5.98,\nusing an increment of 0.01. We observe a dramatic change in the form of the histograms\naroundD/J≈5.91. Below that value, there is no central peak and thus no indication\nof phase coexistence when crossing the transition, in contrast to the situation closer to\nthe degeneracy point, where a central peak, in addition to the sym metric peaks at ±m0,\nindicates coexistence of the ordered and disordered phases and, accordingly, a transition\nof first order. That distinction persists for larger system sizes. E xamples are displayed\nin Figs. 11 a, for D/J= 5.85, and 11 b, for D/J= 5.975. Based on these observations,\nwe may tentatively locate the tricritical point at D/J= 5.91±0.03. Note that such\na change in the form of the histograms does not occur in two dimensio ns, as has been\ndiscussed above, see also Fig. 6.\nInsummary, thepresent analysis onthemixed–spin model onasimple –cubic latticeMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 13\n-500-400 -300 -200 -1000100200 300 400 500m00.0010.0020.003p(m)\n-500-400 -300 -200 -1000100200 300 400 500m00.0010.0020.0030.0040.005p(m)\nFigure 11. (a) Left: Histogram of the total magnetization, p(m), forL=8 and\nD/J= 5.85, at temperatures crossing the transition, kBT/J= 1.32, 1.36, 1.40, 1.44,\nand 1.48, where the maxima move towards the center with increasing temperature.\n(b) Right: p(m) forL= 8 and D/J= 5.975, at temperatures crossing the transition,\nkBT/J= 0.47, 0.51, 0.55, 0.59, and 0.63, where the central peak grows in he ight with\nincreasing temperature.\nshows clearly a line of compensation points, and allows to locate appro ximately the\ntricritical point.\n4. Summary\nWe have studied a mixed–spin Ising model with ferromagnetic coupling s,J, between\nspins 1/2 and 1 on neighbouring sites of square and simple–cubic lattic es, the two types\nof spins forming a bipartite lattice. An additional quadratic single–ion term,D, acts\nupon the S=1 spins. We mainly used standard Monte Carlo simulations t o compute\nvarious thermodynamic properties as well as the Binder cumulants a nd histograms of\nthe total magnetization.\nThe model on the square lattice has been shown to display a continuo us phase\ntransition of Ising–type, presumably up to the degeneracy point a tD/J=4. No\ncompensation point has been found. Close to the degeneracy point , the model displays\nan intriguing three–peak structure in the specific heat as a functio n of temperature.\nThe sharp, but non–critical anomaly at low temperatures arises fr om flipping S=1 spins\ninto the state 0, while the broad non–critical maximum at high temper atures stems\nfrom thermal activation of spins in fairly large clusters of S=1/2 spin s persisting above\nthe phase transition. At temperatures in between, the critical pe ak shows up. Both\nanomalies may cause difficulties in low– and high–temperature expansio ns, which have\npredicted, incorrectly, the existence of a tricritical point. The su ggestion on the absence\nof a compensation point has been confirmed, albeit the magnetizatio n on the S=1\nsublattice decreases rapidly near the anomaly of the specific heat a t low temperatures.\nInthecaseofthesimple–cubic lattice, thespecificheatdisplaysasim ilarthree–peakMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 14\nstructure, with two non–critical maxima and the critical peak in bet ween. Sufficiently\nfar away from the degeneracy point, the ferromagnetic phase dis orders via a continuous,\nIsing–like transition. In the vicinity of the degeneracy point, D/J= 6, this transition\nseems to beof first order. The evidence forthat kind of transition ismainly based onthe\ntype of the histograms of the magnetization, showing phase coexis tence. We tentatively\nlocate the tricritical point at D/J= 5.91±0.03. In addition, we determined a line of\ncompensation points, arising from the degeneracy point. Thus, in t hree dimensions, the\nmean–field theory appears to give at least qualitatively correct pre dictions. However,\nin two dimensions the mean–field theory is found to be incorrect, eve n qualitatively.\n5. Acknowledgement\nW.S. thanks the School of Physics at the University of New South Wa les for the very\nkind hospitality during his stay there, which was supported by the Go rdon Godfrey\nFund.\nReferences\n[1] Goncalves L L 1985 Phys. Scripta 32248.\n[2] Lipowski A and Horiguchi T 1995 J. Phys. A: Math. Gen. 28L261.\n[3] Jascur M 1998 Physica A 252217.\n[4] Dakhama A 1998 Physica A 252225.\n[5] Oitmaa J and Zheng W-H 2003 Physica A 328185.\n[6] Jascur M and Strecka J 2005 Condens. Mat. Phys. 8869.\n[7] Zhang G-M and Yang C-Z 1993 Phys. Rev. B489452.\n[8] Buendia G M and Novotny M 1997 J. Phys.C: Cond Mat. C95951.\n[9] Nakamura Y 2000 J. Phys.C: Cond. Mat. 124067.\n[10] Godoy M and Figueiredo W 2004 PhysicaA339392.\n[11] Godoy M, Leite V S, and Figueiredo W 2004 Phys. Rev. B69054428.\n[12] Oitmaa J 2005 Phys. Rev. B72224404.\n[13] Kaneyoshi T and Chen J C 1991 J. Mag. Mag. Mat. 98201.\n[14] Boechat B, Filgueiras R, Marins L, Cordeiro C, and Branco NS 200 0Mod.Phys. Lett. B 14749.\n[15] Sarmento E F, Cressoni J C, and dos Santos R J V 2000 Int. J. Mod. Phys. B 14521.\n[16] Oitmaa J and Enting I G 2006 J. Phys. C: Cond. Mat 1810931.\n[17] Landau DP and Binder K 2005 A Guide to Monte Carlo Simulations in Statistical Physics\n(Cambridge: University Press)\n[18] Binder K 1981 Z. Physik B43119.\n[19] Kamieniarz G and Bl¨ ote HWJ 1993 J. Phys. A: Math. Gen 26201.\n[20] Chen XS and Dohm V 2004 Phys. Rev. E70056136; Dohm V 2008 Phys. Rev. E77061128.\n[21] Selke W and Shchur LN 2005 J. Phys. A: Math. Gen. 38L739; Selke W 2006 Eur. Phys. J. B\n51223.\n[22] Fisher ME and Barber MN 1972 Phys. Rev. Lett 281516.\n[23] Pelissetto A and Vicari E 2002 Phys. Rep. 368549.\n[24] Hasenbusch M, Pinn K, and Vinti S 1999 Phys. Rev. B5911471."    },    {        "title": "2102.00716v1.Real_time_Hall_effect_detection_of_current_induced_magnetization_dynamics_in_ferrimagnets.pdf",        "content": "1 \n Real -time Hall-effect  detection of current -induced magnetization dynamics  \nin ferrimagnets  \nG. Sala1*, V. Krizakova1, E. 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−𝑡0\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 . We chose a piecewise  linear function because of its \nsimplicity and its robustness with respect to the fitting routine as opposed to, e.g., the cumulative \nfunction of the Gaussian distribution, which is more prone to errors for small values of 𝑡0. \n \nDATA AVAILABILITY  \nThe datas ets generated and/or analysed during the current study are available from the corresponding \nauthors on reasonable request.  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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": "1203.4569v1.Is_the_Yb2Ti2O7_pyrochlore_a_quantum_spin_ice_.pdf",        "content": "arXiv:1203.4569v1  [cond-mat.str-el]  20 Mar 2012Is the Yb 2Ti2O7pyrochlore a quantum spin ice?\nR. Applegate,1N. R. Hayre,1R. R. P. Singh,1T. Lin,2A. G. R. Day,2,3and M. J. P. Gingras1,2,4\n1Physics Department, University of California at Davis, Dav is, CA 95616\n2Department of Physics and Astronomy, University of Waterlo o, Waterloo, Ontario, N2L 3G1, Canada\n3D´ epartement de Physique, Universit´ e de Sherbrooke, Sher brooke, Qu´ ebec, J1L 2R1, Canada\n4Canadian Institute for Advanced Research, 180 Dundas St. W. , Toronto, Ontario, M5G 1Z8, Canada\n(Dated: September 2, 2018)\nWe use numerical linked cluster (NLC) expansions to compute the specific heat, C(T), and en-\ntropy,S(T), of a quantum spin ice model of Yb 2Ti2O7using anisotropic exchange interactions\nrecently determined from inelastic neutron scattering mea surements and find good agreement with\nexperimental calorimetric data. In the perturbative weak q uantum regime, this model has a ferri-\nmagnetic ordered ground state, with two peaks in C(T): a Schottky anomaly signalling the para-\nmagnetic to spin ice crossover followed at lower temperatur e by a sharp peak accompanying a first\norder phase transition to the ferrimagnetic state. We sugge st that the two C(T) features observed\nin Yb 2Ti2O7are associated with the same physics. Spin excitations in th is regime consist of weakly\nconfined spinon-antispinon pairs. We suggest that conventi onal ground state with exotic quantum\ndynamics will prove a prevalent characteristic of many real quantum spin ice materials.\nPACS numbers: 74.70.-b,75.10.Jm,75.40.Gb,75.30.Ds\nThe experimental search for quantum spin liquids\n(QSLs), magnetic systems disordered by large quan-\ntum fluctuations, has remained unabated for over twenty\nyears [1]. One direction that is rapidly gathering mo-\nmentum is the search for QSLs among materials that are\nclose relatives to spin ice systems [2], but with additional\nquantum fluctuations, or quantum spin ice [3, 4].\nSpinicesarefoundamonginsulatingpyrochloreoxides,\nsuch as R 2M2O7(R=Ho, Dy; M=Ti, Sn) [5]. In these\ncompounds,themagneticRrareearthionssitonalattice\nof corner-sharing tetrahedra, experiencing a large single-\nion anisotropy forcing the magnetic moment to point\nstrictly “in” or “out” of the two tetrahedra it joins (see.\nFig. 1a). Consequently, the directionofamomentcanbe\ndescribed by a classicalIsing spin [2]. In these materials,\nthe combination of nearest-neighbor exchange and long-\nrangemagnetostaticdipolar interactionslead to an expo-\nnentially large number of low-energy states characterized\nby two spins pointing in and two spins pointing out on\neach tetrahedron (see Fig. 1a). This energetic constraint\nis equivalent to the Bernal-Fowler ice rule which gives\nwater ice a residual entropy SP∼kB(1\n2)ln(3/2) per pro-\nton, estimatedby Pauling[6] andin goodagreementwith\nexperiments on water ice [7]. Since they share the same\n“ice-rule”, the (Ho,Dy) 2(Ti,Sn) 2O7pyrochlores also pos-\nsess a residual low-temperature Pauling entropy SP[8],\nhencethe namespinice. The spinicestateisnotthermo-\ndynamically distinct from the paramagnetic phase. Yet,\nbecause of the ice-rules, it is a strongly correlated state\nof matter – a classical spin liquid of sorts [1, 2].\nFor infinite Ising anisotropy, quantum effects are ab-\nsent [2]. However, these can be restored when consid-\nering the realistic situation of finiteanisotropy. In two\nclosely related papers, Hermele et al.[9] and Castro-Neto\net al.[10] considered effective spins one-half on a py-\nFIG. 1: (a) Two neighboring tetrahedra with spins in their\ntwo-in/two-out ground state, (b) spinon/antispinon pair, (c)\nspinon/antispinon pair separated by a (green) string of mis -\naligned spins in the pyrochlore lattice.\nrochlorelattice wherethe highly degenerateclassicalspin\nice state is promoted via quantum fluctuations to a QSL\nwith fascinating properties. This QSL is described by\na compact lattice quantum electrodynamics (QED) -like\ntheory. In this QSL state inherited from the parent clas-\nsical spin ice, the ice-rules amount to a divergence-free\ncoarse-grained fictitious electric field whose sources are\ndeconfined spinons while the sources of the canonically\nconjugate field are deconfined monopoles [11], along with\na gauge boson (“artificial photon”).\nRecent numerical studies have found evidence that\nQED-like phenomena may be at play in some minimal\nquantum spin ice (QSI) lattice models [13] – but does\nthe QSI picture apply to real materials? Also, should\na QSI state be solely defined by whether or not a QSL\nstate is realized? While a QSI picture has been sug-\ngested relevant to the QSL behavior in Tb 2Ti2O7[3]2\nand Pr 2M2O7[4], intense experimental [14–21] and the-\noretical [16–18, 21–26] interest has recently turned to\nYb2Ti2O7(YbTO), which has been argued to be on the\nverge of realizing a QSL originating from QSI physics. In\nfact, the combination of (i) an unexplained transition at\nTc∼0.24 K [14, 27], (ii) the controversial evidence for\nlong-rangeorderbelow Tc[28,29]and(iii)thehighsensi-\ntivity of the low-temperature ( T <300 mK) behavior to\nsample preparation conditions [19, 20] are all tantalizing\nevidence that YbTO has a fragile and perhaps uncon-\nventional ground state. Thus, explaining YbTO is a key\nmilestone in the study of QSI in a materials context.\nIn this paper, we first use the numerical linked clus-\nter (NLC) method [30, 31] to calculate the heat capac-\nity,C(T), and entropy, S(T), of a microscopic model\nfor YbTO with exchange parameters, {Je}, taken from\nRef. [18]. This calculation, which converges down to\nabout1K,agreeswellwith experiments. It demonstrates\nthat YbTO is indeed a spin-half, anisotropic exchange\nmodel, with {Je}determined from magnon energies in\nthe strong-field polarized paramagnet regime [18]. Our\nwork suggests that a two-peaked C(T) structure is natu-\nralinYbTOandshouldbepresentinthebest(“quality”)\nsamples [19, 20]. Below the higher temperature C(T)\nhump near 2 K, the system has a residual S(T) com-\nparable to SP, but without a clean S(T)≈SPplateau\ndevelopingupon cooling. We proposethat the lowertem-\nperaturesharppeakin C(T) isassociatedwith astrongly\nfirst order transition to a ferrimagnetic state. Such a\nbehavior is indeed found in our study when the quan-\ntum (non-Ising) exchanges are small. Finally, we argue\nthat despite a conventional ground state, the spin excita-\ntions consist of spinon/antispinon pairs connected with\n(Dirac-like [12]) strings of reversed spins, whose confine-\nment length lsdiverges in the limit of small quantum\nexchanges. We propose that these excitations should ul-\ntimately form the basis for describing what we expect to\nbe highly unconventional inelastic neutron spectra [26].\nModel & Method – The anisotropic exchange QSI\nmodel is defined by the nearest-neighbor Hamiltonian\n[18, 25] on the pyrochlore lattice\nHQSI=/summationdisplay\n<i,j>{JzzSz\niSz\nj−λJ±(S+\niS−\nj+S−\niS+\nj)\n+λJ±±[γijS+\niS+\nj+γ∗\nijS−\niS−\nj]\n+λJz±[(Sz\ni(ζijS+\nj+ζ∗\ni,jS−\nj)+i↔j]}.(1)\nγijis a 4×4 complex unimodular matrix, and ζ=−γ∗\n[18]. The ˆ zquantizationaxisisalongthelocal[111]direc-\ntion, and ±refers to the two orthogonal local directions.\nWe take λ= 1, except when stated otherwise.\nRecently Ross et al.[18] used inelastic neutron scat-\ntering data in high fields to deduce the {Je}exchangepa-\nrameters for YbTO: Jzz= 0.166±0.04,J±= 0.05±0.01,\nJ±±= 0.05±0.01, andJz±=−0.14±0.01, all in meV.\nThese parameters have also been determined through ananalysis of the zero-field energy-integrated paramagnetic\nneutron scattering [17, 21], but the values of the {Je}pa-\nrameters disagree significantly – an issue that we address\nin the supplementary material [32].\nNLC expansions provide a controlled way of calculat-\ning macroscopic properties of a thermodynamic system\n[30,31]. Bysummingup contributionsfromclustersupto\nsome size, one can obtain properties in the thermody-\nnamic limit, which include all terms in high temperature\nexpansions upto some order. Furthermore, since the con-\ntributions ofthe clusters areentirelyincluded forall tem-\nperatures, all shortdistance physicsis fully incorporated,\nandthuscanconvergedowntolowertemperaturesthana\n(high-temperature, T) series expansion [30] in 1 /T. NLC\nis particularly suited to the study of spin ice systems. It\nwas recently shown that for classicalspin ice models, just\nfirstorderNLC basedonasingletetrahedron, gives C(T)\nandS(T) for allTwithin a few percent accuracy [33].\nHere, we calculate the thermodynamic properties of\nthe exchange QSI model of Eq. (6) using tetrahedra-\nbased NLC upto 4thorder [32]. Euler extrapolations [34]\nare used to eliminate some alternating pieces in the ex-\npansion, which further improves the convergence of the\ncalculations to lower T. In zero field, there is only one\ncluster in each of the first three orders, and three clusters\nin the fourth order [32]. The different g-tensor elements\non different sites (expressed in a global frame) [24] mean\nthat many more clusters are needed for calculating field-\ndependent C(T), magnetization and susceptibility, and\nthese will be presented elsewhere.\nFigure 2 shows C(T) calculated with different NLC\norders. By 4thorder, there is good convergence to tem-\nperatures below the C(T) peak at ∼2 K. Applying Eu-\nler transformations [34] improves the convergence down\nto slightly below 1 K. The experimental data from Refs.\n[27], shown for comparison, agree well with the NLC re-\nsults. Here, we used the mean values of the {Je}from\nRef. [18] and did not adjust any parameters. Given the\nvariabilityin the experimental C(T) data from onegroup\ntoanother[19–21,32], itdoesnotseemusefulatthistime\nto search for {Je}parameters giving a better fit. This\nagreement shows that the {Je}parameters are not sub-\nstantially renormalized compared to the high (5 Tesla)\nfield values [18]. Using the {Je}of Refs. [17, 21] gives\nsubstantially different C(T) results [32].\nFigure 3 shows S(T) calculated by NLC, together\nwith the entropy obtained by integrating C(T)/Tdata\nof Ref. [27]. We found the data from Ref. [27] ideally\nsuited to perform this comparison [32]. The entropy con-\nverges to lower temperature slightly better than C(T)\nwhere, with Euler transformations, S(T) converges down\nto about 0.7 K, matching well with the experimental en-\ntropy values over the overlapping temperature range.\nPerturbative considerations – In order to better un-\nderstand the properties of this system, we turn to the\nperturbative regime λ≪1 in Eq. 1 [18, 25]. To second3\nFIG. 2: Specific heat, C(T), per mole of Yb for the model\nparameters in Ref. [18], in units of the Boltzmann constant\nkB, calculated via NLC (up to 4thorder NLC together with\nEuler extrapolations) are compared with experimental data\nfor Yb 2Ti2O7. The black circles are data from Ref. [27].\norder in λ, onlyJz±, by far the largest quantum term\nfor YbTO, leads to a degeneracy-lifting classical poten-\ntial for different spin-ice configurations. It amounts to\na fluctuation-induced ferromagnetic exchange constant\nJ3≡ −3λ2J2\nz±/Jzz[25] between shortest distance spins\non the same tetrahedral sublattice that share a neigh-\nbor [35]. It leads to the selection of a q= 0 long-range\nordered ground state in which all tetrahedra are in the\nsame configuration and the spins develop a small ferro-\nmagnetic moment alongone ofthe ∝an}bracketle{t100∝an}bracketri}htcubic directions.\nThisq= 0ferrimagnet(FM)lackstheCoulombicphysics\noriginally present in the Jzz-only spin ice model [36].\nTo calculate C(T) andS(T) in the perturbativeregime\nat lowT, we turn to classical loop Monte Carlo simula-\ntions [37] of the J3−Jzzmodel [32]. These reveal a\nvery sharp lower temperature peak signalling a first or-\nder phase transition to a q= 0 state (see Fig. S5 [32]).\nExcited states in the perturbative regime: spinons and\nstrings– A surpriseof the perturbativetreatment is that,\nwhile the ground state is classical, the spin-flip excita-\ntions remain non-trivial and of quantum nature. This is\nbecause, oncea spin is flipped in a spin-icestate, creating\na spinon/antispinon pair [11], the pair can hop through\nJz±acting through first order degenerate perturbation\ntheory. Thus, the dispersion in the excited state man-\nifold isλJz±,much larger than the dispersion within\nthe low-energy manifold of spin ice states, which is only\nλ2J2\nz±/Jzz.\nA sketch of a spinon/antispinon pair is shown in Fig.\n1b and 1c. Note that only spins inside the tetrahedron\n“already” containing spinons are flippable in first orderFIG. 3: Entropy, S(T), per mole of Yb, in units kBfollowing\nthe methods described in the caption of Fig. 2. The black\ncircles are obtained by integrating the data from Ref. [27]\nexcluding the nuclear (hyperfine) contribution. The Paulin g\nentropySP∼kB\n2ln3\n2is shown as a horizontal line. The inset\nshowsS(T) in the perturbative regime with J3/Jzz=−0.001.\nA clear plateau at S(T)≈SPis seen, followed at lower Tby\na precipitous drop of S(T) (i.e. latent heat) accompanying\nthe transition to long range FM order [32].\ndegenerate perturbation theory. Hence, the connecting\nstring of misaligned spins can only fluctuate by higher\norder processes involving closed loops with alternating\nin-out spins [26]. Thus the renormalized string tension\nper unit length remains finite and of order J3. One can\nestimate the typical string length as the length, ls, at\nwhich the cost of the string becomes comparable to the\ndelocalization energy of the spinon/antispinon pair. The\nstring energy per unit length goes as ∼J3∼λ2, whereas\nthe delocalization energy (spinon bandwidth) goes as λ.\nThis leads to lsscaling as 1 /λ, which diverges as λ→0.\nA detailed theory of neutron scattering in this ferri-\nmagnetic phase is not attempted here, but we anticipate\nit to follow the proposal of Ref. [26]. At temperatures\nabove the transition to the q= 0 long-range ordered\nstate, the system explores the classical two-in/two-out\nspin ice states and should display singularities (pinch\npoints, PPs) in neutron scattering [36] rounded off by\nthe finite density of thermally excited spinon/antispinon\ndefects [11, 36]. While the system has thermally smeared\nPPs above the ferrimagnetic transition and no static PPs\nwell below the transition, it may display some remnant\nof PPs in the spin dynamics at higher energies. These\ninteresting issues deserve further attention.\nBeyond the λ≪1regime– Why is the transition\ntemperature of YbTO so low? As discussed by Ross et\nal.[18], the low Tpeak inC(T) is at a temperature lower\nthan mean-field theory by an order of magnitude. Com-4\nFIG. 4: Monopole defect density, ρ(T), calculated using NLC,\nshown down to a temperature where 3rd and 4th order Euler\nTransforms agree. Here, quantum exchanges are scaled with\nrespect to YbTO parameters by different values of λ.\nparingC(T) for the quantum model with different λwith\nthe corresponding classical model with the perturbative\nJ3/Jzzvalue provides a hint of the reason why [32]. It\nshows that, in the classical model, the long-range order\nkeeps steadily moving up with increased J3, even beyond\nthe short-range order C(T) peak. In contrast, the quan-\ntum systems, with different λcontinue to displaya short-\nrange order C(T) peak and presumably long-range order\nonly occurs at a much lower T. Perturbative considera-\ntions here have an analogy with strong coupling studies\nof Mott physics in the Hubbard model, where the N´ eel\ntemperature first increases with tast2/Ubut then be-\ngins decreasing when the system moves away from the\nperturbative small t/Uregime. We propose that a sim-\nilar non-monotonic Tcarises in this QSI model due to\nenhanced quantum fluctuations.\nAnother argument for a reduced Tccomes from\nconsidering the temperature dependence of the defect\n(spinon/antispinon) monopole density, ρ(T), as calcu-\nlated by NLC (see Fig. 4 and Figs. S3 and S4 [32]). To\nillustrate the point, we show the behavior for several dif-\nferentλvalues. Convergence increases to lower T, with\ndecreasing λ, as expected. One finds that as Tdrops\nbelow the hump in C(T),ρ(T) displays a plateau-like\nregion, whose value increases steadily with increasing λ.\nThis indicates that the states withinthe spin-ice man-\nifold develop large spinon/antispinon spectral weight,\nthus strongly renormalizing all low energy scales and,\npresumably, leading to reduced Tc.\nDiscussion: What constitutes an exchange QSI? –\nWe suggest that a double-peaked C(T) with an en-\ntropy between the peaks comparable to SPis the hall-\nmark of an exchange quantum spin ice (QSI). How-\never, one is unlikely to find an exact plateau at S(T)≈SPoutside the perturbative (small λ) regime. Such a\ndouble-peaked structure and quasi-separation of the en-\nergy/temperature scales associated with short and long-\nrange physics has also been suggested for other systems\nwhere quantum spin liquid physics may apply [38].\nAccording to the gauge mean-field theory of Ref. [25],\nat low temperature below which short-range spin ice cor-\nrelations develop, a system may exhibit either a conven-\ntional ferrimagnetic (FM) order, a Coulombic ferromag-\nnet (CFM) or a full-blown quantum spin-liquid (QSL),\ndepending on its quantum exchange parameters. The\nlargest quantum exchange terms in YbTO is Jz±, which\nfavors the FM state, which we believe is the origin of\nthe 0.24 K transition in the best samples [28]. It re-\nmains to be seen if there are real materials for which J±,\nwhich favors the QSL [9, 10, 25], is the dominant quan-\ntum term. Nevertheless, even when the ground state is\nFM, the excitations remain highly exotic, consisting of\nspinon-antispinon pairs separated by long strings. This\nnon-trivialfeature is derivedfrom the underlying spin-ice\nphysics. Finally, as one notes that Jz±is strictly zero for\nnon-Kramers ions (e.g. Pr, Tb) and that virtual crystal\nfield excitations [3] in Tb-based pyrochlores are a fun-\ndamentally different pathway from anisotropic superex-\nchange[4]togenerateanisotropic {Je}couplingsbetween\neffective spins one-half [3, 4], the prospect to ultimately\nfind a QSI-based QSL among rare-earth pyrochlores [5]\nis perhaps promising.\nThis work is supported in part by NSF grant number\nDMR-1004231, the NSERC of Canada and the Canada\nResearchChairprogram(M.G., Tier1). Weacknowledge\nvery useful discussions with B. Javanparast, K. Ross and\nJ. Thompson. We thank P. Dalmas de R´ eotier for pro-\nviding specific heat data of Ref. [19].\nSupplementary Material\nThis supplement provides the reader with further ma-\nterial to assist with some of the technical materials of the\nmain part paper\nNumerical Linked Cluster Method\nFor the proposed QSI Hamiltonian [18], the numeri-\ncal linked cluster (NLC) method [30, 31] gives reliable\nquantitative properties of the system in the thermody-\nnamic limit down to some temperature by developing an\nexpansion in connected tetrahedra that embed in the py-\nrochlore lattice. For each cluster, we perform an exact\ndiagonalization (ED) and calculate physical quantities\nfrom the resulting spectrum and states. Once a prop-\nerty is calculated, the properties of all subclusters are\nsubtracted to get the weight of the cluster cdenoted as5\nW(c). In the thermodynamic limit, an extensive prop-\nerty,Pis expressed as\nP/N=/summationdisplay\ncL(c)×W(c), (2)\nwhereLcis the count of the cluster, per lattice site.\nWe consider all clusters up to four tetrahedra, the\nlargest diagonalization being a 13-site system. All states\nare required to calculate the partition function and ther-\nmodynamic quantities presented below. The particular\nclusters to fourth order in our expansion are shown in\nFigure S1.\nComputational Requirements\nNLC using the tetrahedral basis requires exact diago-\nnalization of increasingly large tetrahedral clusters. Us-\ning modern hardware and freely-available linear algebra\nroutines, diagonalizations for clusters of one tetrahedron\n(foursites)andtwotetrahedra(sevensites)couldbedone\nin less than a second, while the three-tetrahedron (10-\nsite) cluster still required less than 10 seconds. Comput-\ning only the spectrum for a single four-tetrahedron (13-\nsite)clusterrequiredabout1200secondsandmorethan1\nGBofmemory, whilegeneratingthefull setofeigenstates\nrequired approximately 8 GB of memory. Note that the\nHamiltonian of an N-site cluster is a 2N×2Ncomplex\nHermitian matrix. Exact diagonalizations of larger sys-\ntems are, in practice, limited by memory requirements.\nThe next order calculation will have 3 more sites and the\nmemory requirement will grow by a factor of 64.\nEuler Summation\nNLC generates a sequence of property estimates {Pn}\nwith increasing order n, wherePn=/summationtextn\ni=1SiandSiis\nsomephysicalquantitycalculatedatthe ith order. When\nsuch a sequence is found to alternate, its convergencecan\nbe improved by Euler Transformation [34]. In general,\ngiven alternating terms Si= (−1)iui, the Euler Trans-\nform method amounts to estimates,\nu0−u1+u2−...−un−1+/summationdisplay\ns=0(−1)s\n2s+1[∆sun],(3)\nwhere ∆ is the forward difference operator\n∆0un=un,\n∆1un=un+1−un,\n∆2un=un+2−2un+1+un,\n∆3un=un+3−3un+2+3un+1−un,.... (4)\nUsually, a small number of terms are computed directly,\nand the Euler transformation is applied to rest of theFIG. 5: S1: Clusters used for the zero-field NLC expansion\nin the tetrahedral basis, up to fourth order. Each graph is\naccompanied by its lattice constant L.\nseries. In our case, where direct terms are available\nto fourth order, we begin the Euler transform after the\nsecond order, so that the third and fourth order Euler-\ntransformed property estimates are\nP3,E=S0+S1+S2+1\n2S3,\nP4,E=P3,E+S3+S4\n4. (5)\nVarious Hamiltonians and perturbative limit\nWe use the notation of Ross et al.[18] and define the\nquantum spin ice Hamiltonian as\nHQSI=/summationdisplay\n<i,j>{JzzSz\niSz\nj−J±(S+\niS−\nj+S−\niS+\nj)\n+J±±[γijS+\niS+\nj+γ∗\nijS−\niS−\nj]\n+Jz±[(Sz\ni(ζijS+\nj+ζ∗\ni,jS−\nj)+i↔j]}.(6)\nThe parametersforYb 2Ti2O7determined byfitting from\nhigh-field inelastic neutron (magnon) spectra in Ref. [18]\nare, measured in meV, Jzz= 0.166±0.04,J±= 0.05±\n0.01,J±±= 0.05±0.01, andJz±=−0.14±0.01. Two\nother sets of parameter estimates for Yb 2Ti2O7were de-\ntermined by fitting the diffused (energy-integrated) neu-\ntron scattering using the random phase approximation\n(RPA) [17, 21]. The values obtained by Thompson et\nal.[17] are: Jzz= 0.023,J±= 0.038,J±±= 0.007,\nandJz±=−0.040, while those obtained by Chang et6\n1 10\nT(K)00.10.20.30.4C(T)(kB)Ross-E4\nRoss-E3\nThompson-E4\nThompson-E3\nChang-E4\nChang-E3\nBlote\nYaouanc-1\nYaouanc-2\nFIG. 6: S2: Molar heat capacity for YbTO reported by Bl¨ ote\net al.[27] and by Yaouanc et al.[19] compared with cal-\nculated values using exchange parameters from Ross et al.\n(Ross-E3,E4) [20], Thompson et al.(Thompson-E3,E4) [17]\nand Chang et al.(Chang-E3,E4) [21]. Third (E3) and Fourth\n(E4) order Euler Transforms of the NLC results using the pa-\nrameters are shown.\nal.[21] are Jzz= 0.059,J±= 0.023,J±±= 0.006,\nandJz±=−0.029. In all cases, the values of the {Je}\nexchange parameters are given in meV. The calculated\nheat capacity for all these parameters, together with the\nexperimental data on Yb 2Ti2O7from difference groups\n[19, 20], are shown in Fig. S2. It is clear that the latter\ntwo parametrizations by Thompson et al.and Chang et\nal.do not give a good description of the heat capacity of\nthe material. It is not clear at this time why RPA calcu-\nlations find such {Je}parameters compared to high-field\nparamagnon spectra [20]. This problem warrants further\nattention.\nInordertoexploretowhatextentquantummechanical\neffects are at play in HQSI, we introduce a Hamiltonian\nwith rescaled quantum terms as\nHλ=H0+λH1, (7)\nwhereH0is the classical spin-ice Hamiltonian consisting\nofJzzterms only, while all other terms are included in\nH1. The value λ= 1 corresponds to the parameters of\nRosset al.[18] In the perturbative regime ( λ≪1), this\nmodel maps on to a J1−J3model with J1=Jzzand\nJ3=−3λ2J2\nz±/Jzz.\nSpecific heat and entropy of the system with different\nvalues of λin 4th order Euler Transform, down to a tem-\nperaturewhere3rd and4th orderEulerTransformsagree\nwith each other are shown in Fig. S3 and Fig. S4. Heat\ncapacity of the perturbative classical J1−J3model, cal-\nculated by classical loop Monte Carlo simulations [37] is0.1 1 10\nT(K)00.10.2C(kB)λ=1λ=0.5λ=0\nλ=0.2\nFIG. 7: S3: Heat capacity where quantum terms are scaled\nbyλ.\n0.1 1 10\nT(K)00.20.40.6S(kB)λ=0\nλ=0.2\nλ=0.5λ=1\nSP\nFIG. 8: S4: Entropy of the system, when quantum terms are\nscaled by λ. The orange line is the Pauling entropy Sp.\nshown in Fig. S5. Note that while the models with differ-\nentλalwayshave a short-rangeorderpeak, in the J1−J3\nmodel, long-range order temperature increases well past\nthe short-range order peak with increasing J3/J1.\nComparison of the experimental entropy vs NLC\nresults\nThe entropy difference, S(T2)−S(T1) between two\ntemperatures T1andT2can be obtained by integrating7\nC(T)/Tbetween those two temperatures:\nS(T2)−S(T1) =/integraldisplayT2\nT1C(T)\nTdT\nThe number of experimental specific heat, C(T), re-\nsults on Yb 2Ti2O7has rapidly accumulatedoverthe past\nyear or so [19–21]. Most of these data are somewhat\nproblematic in wanting to assess whether those thermo-\ndynamic data hide spin ice phenomenology, associated\nwith a rapid diminution of spinon/antispinon excitation\nand the concurrent C(T) hump at a temperature ∼2 K\nas we now discuss.\nAll of the published C(T) data [19–21, 27] do not go\nto sufficiently high temperature to extract reliably the\nlimiting C(T)∝1/T2high temperature behaviour that\nwould allow one to determine the residual magnetic en-\ntropy by integrating C(T)/Tupon decreasing Tstarting\nfrom the infinite kBln(2) value. One must therefore in-\ntegrateC(T)/Tfrom low temperature, and assume an\nentropy value, Slowat some reference (low) temperature,\nTlow. The apparent large amount of residual entropy be-\nlow∼0.2 K in the single crystal samples of Refs. [19–\n21] make difficult ascribing a reasonable value to Slow.\nThis problem is further compounded by the rising low-\ntemperature nuclear contribution to the total specific\nheat below about 0.1 K. The very sharp 1st order transi-\ntion seen in powder powder sample of Ref. [20], without\na precise measurement of the associated latent heat also\nmake difficult using those data for comparison of experi-\nmental entropywith the S(T)calculated byNLC. On the\notherhand,thedataofBl¨ ote et al.[27]seemthemostade-\nquate for comparison with NLC: there is a sharp specific\nheat peak at Tc∼0.24 K with sufficient temperature\nresolution that allows integration of C(T)/Tover the\npeak without concern about an associated latent heat.\nTheC(T) data are dropping rapidly below Tc, suggest-\ning the opening of an excitation gap, ultimately reaching\na low-valuethat is limited by the “high temperature tail”\n(T∼0.1 K) of the nuclear contribution. Using the data\nfrom Ref. [27], we thus assume that the magnetic part of\nthe specific heat is zero at T= 0.1 K, and integrate up-\nward(increasing temperature) C(T)/Tup to the highest\ntemperature point available from those data ( ∼3.5 K).\nThis results in the data (filled black circles in Fig. 3 in\nthe body of the paper).\nIt would be highly desirable to repeat this procedure\nfrom the C(T) data of Refs. [19–21] which show a sharp\npeak, but including (magnetic specific heat) data for T\nup to 20 K where the limiting high-temperature regime\nC(T)≈A\nT2+B\nT3can be fitted and compared with NLC,\nalong with measurements of the magnetic entropy, S(T).\nlimit available dataThe data from Working from the\nreasonable presumption high temperatureMonte Carlo Simulation of the Jzz−J3Model\nIn the perturbative regime of the QSI, we consider the\neffective Hamiltonian\nH=/summationdisplay\n<i,j>Jzzσiσj+/summationdisplay\n<i,j>′J3σiσj (8)\nwhereσ=±1 are the Ising variables. ∝an}bracketle{t...∝an}bracketri}htdenotes\nthe sum over the nearest neighbors, ∝an}bracketle{t...∝an}bracketri}ht′denotes the\nsum over the third nearest neighbors which share a near-\nest neighbour. Distance-wise there exists another type\nof third nearest neighbors which do not share a nearest\nneighbor. For any given spin, there are six third near-\nest neighbors for both types. Antiferromagnetic Jzz>0\ndrives the spin ice formation in the classical spin ice sys-\ntem, and a small fluctuation-induced ferromagnetic ex-\nchangeJ3≡ −3J2\nz±/Jzz<0 favors the q= 0 ordering\nwithin the spin ice manifold, i.e., all tetrahedra on the\nsame primitive FCC lattice have the same one of the six\nspin ice states.\nMonte Carlo simulations are performed using the\nMetropolis algorithm. Single spin flip updates are used\nalong with the non-local loop algorithm [37], which re-\nstores the ergodicity of the system once it is frozen into\nthe spin ice states. Systems of 128 spins are simulated\nin a cubic box with periodic boundary conditions. Up\nto about 78,000 Monte Carlo steps per spin are used in\nequilibratingthe systemat a giventemperature, with the\nsame number of steps in data sampling. To investigate\nthe calorimetric quantities, fluctuations of the energy are\nrecorded to give the heat capacity:\nC=< E2>−< E >2\nkBT2(9)\nCalculation of Monopole density\nThe defect (spinon/antispinon) monopole number\nM(T), for a cluster, is evaluated as\nM(T) = tr(ˆme−βˆH)/Z\n=1\nZ/summationdisplay\nαe−βEα∝an}bracketle{tα|ˆm|α∝an}bracketri}ht\n=1\nZ/summationdisplay\nα,ke−βEα|∝an}bracketle{tα|k∝an}bracketri}ht|2mk(10)\nwheremkis the monopole count in the local Szbasis\nstate|k∝an}bracketri}ht. This count is a sum over all the tetrahedra in\na cluster, mk=/summationtext\nimki, where\nmki=\n\n2 all in/out,\n1 three in/out and one out/in,\n0 two in and two out.(11)8\n0.01 0.1 1\nT/J100.511.52C(kB)J3=-0.001\nJ3=-0.01\nJ3=-0.02\nJ=-0.05\nJ3=-0.10\nFIG. 9: S5: Heat capacity of the classical Jzz−J3model,\nwith different Jzz/J1ratios.\nThe monopole density ρ(T) is defined as number of\nmonopoles present per site, giving\nρ(T) =M(T)/N. (12)\n[1] L. Balents, Nature 464, 199 (2010).\n[2] M. J. P. Gingras, in Introduction to Frustrated Mag-\nnetism, (Springer, 2011) arXiv:0903.2772 .\n[3] H.R.Molavian et al., Phys.Rev.Lett. 98, 157204 (2007).\n[4] S. Onoda and Y. Tanaka, Phys. Rev. Lett. 105, 047201\n(2010).\n[5] J. S. Gardner et al., Rev. Mod. Phys. 82, 53 (2010).\n[6] L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935).\n[7] W. F. Giauque and J. W. Stout, J. Am. Chem. Soc. 58,\n1144 (1936).\n[8] A. P. Ramirez et al., Nature 399, 333 (1999); A. L. Cor-\nneliusandJ.S.Gardner, Phys.Rev.B 64, 060406 (2001).\n[9] M. Hermele et al., Phys. Rev. B 69, 064404 (2004).\n[10] A. H. Castro Neto et al., Phys. Rev. B 74, 024302 (2006).\n[11] To relate our presentation more directly to the compact\nlattice QED context set in Refs. [9, 10], in which con-finement in three-dimensions is traditionally referred to\nthe strong (electric charge) coupling, we refrain from us-\ning the language of “monopoles” employed in Ref. [12] to\nlabel local defects in the ice rule of the parent classical\nspin ice. We use instead the more traditional wording of\nspinon/antispinon to label finite energy excitations out\nof the 2in/2out spin ice manifold.\n[12] C. Castelnovo et al., Nature 451, 42 (2008).\n[13] A. Banerjee et al., Phys. Rev. Lett. 100, 047208 (2008);\nN. Shannon et al., ibid108, 067204 (2012).\n[14] J. A. Hodges et al., Phys. Rev. Lett. 88, 077204 (2002).\n[15] K. A. Ross et al., Phys. Rev. Lett. 103, 227202 (2009).\n[16] H. B. Cao et al., J. Phys. Condens. Matter 21, 492202\n(2009).\n[17] J. D. Thompson et al., Phys. Rev. Lett. 106, 187202\n(2011).\n[18] K. A. Ross et al., Phys. Rev. X 1, 021002 (2011).\n[19] A. Yaouanc et al., Phys. Rev. B 84, 172408 (2011).\n[20] K. A. Ross et al., Phys. Rev. B 84, 174442 (2011).\n[21] L.-J. Chang et al., arXiv:1111.5406\n[22] B. Z. Malkin et al., J. Phys. Cond. Matter 22, 276003\n(2010).\n[23] S. Onoda, J. Phys.: Conf. Series., 320, 012065 (2011).\n[24] J. D. Thompson et al., J. Phys. Condens. Matter 23,\n164219 (2011).\n[25] L. Savary and L. Balents, Phys. Rev. Lett. 108, 037202\n(2012)\n[26] Y. Wan and O. Tchernyshyov, arXiv:1201.5314\n[27] H. W. J. Bl¨ ote et al., Physica 43, 549 (1969).\n[28] Y. Yasui et al., J. Phys. Soc. Jpn. 72, 3014 (2003).\n[29] J. S. Gardner et al., Phys. Rev. B 70, 180404(R) (2004).\n[30] J. Oitmaa, C. Hamer and W. Zheng, Series Expansion\nMethods for strongly interacting lattice models (Cam-\nbridge University Press, 2006).\n[31] M. Rigol et al., Phys. Rev. Lett. 97, 187202 (2006); Phys.\nRev. E 75, 061118 (2007); Phys. Rev. E 75, 061119\n(2007).\n[32] See Supplementary Material.\n[33] R. R. P. Singh and J. Oitmaa, arXiv:1112.4439.\n[34] See for example, Numerical Recipes , by W. H. Press et\nal, Cambridge University Press (1989), Page 133.\n[35] These are geometrically 3rd neighbors on the pyrochlor e\nlattice but not all 3rd neighbors belong to this category.\n[36] C. L. Henley, Annu. Rev. Cond. Matt. Phys. 1, 179\n(2010).\n[37] R. G. Melko and M. J. P. Gingras, J. Phys.: Condens.\nMatter16, R1277 (2004).\n[38] V. Elser, Phys. Rev. Lett. 62, 2405 (1989). N. Elstner\nand A. P. Young, Phys. Rev. B 50, 6871 (1994)."    },    {        "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. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat Nanotechnol 11, 231 (2016). \n[3] P. Wadley  et al. , Science 351, 587 (2016). \n[4] V. Baltz, A. Manchon, M. Tsoi, T. Moriya ma, T. Ono, and Y. Tserkovnyak, Rev Mod Phys 90, \n015005 (2018). [5] J. Zelezny, P. Wadley, K. Olejnik, A. Hoffmann, and H. Ohno, Nature Physics 14, 220 (2018). \n[6] M. Binder  et al. , Phys Rev B 74, 134404 (2006). \n[7] C. D. Stanciu, A. V. Kimel, F. Hansteen, A. Tsuk amoto, A. Itoh, A. Kirilyuk, and T. Rasing, Phys Rev \nB 73, 220402 (2006). \n[8] J. Finley and L. Q. Liu, Phys Rev Appl 6, 054001 (2016). \n[9] R. Mishra, J. Yu, X. Qiu, M. Motapothul a, T. Venkatesan, and H. Yang, Phys Rev Lett 118, 167201 \n(2017). \n[10] N. Roschewsky, T. Matsumura, S. Cheema, F. He llman, T. Kato, S. Iwata, and S. Salahuddin, Appl \nPhys Lett 109, 112403 (2016). \n[11] K. Ueda, M. Mann, C.-F. Pai, A.-J. Tan, and G. S. D. Beach, Appl Phys Lett 109, 232403 (2016). \n[12] J. Han, A. Richardella, S. A. Siddiqui, J. 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Beach, Nat Mater 12, 611 (2013). \n[24] K. S. Ryu, L. Thomas, S. H. Yang, and S. Parkin, Nat Nanotechnol 8, 527 (2013). \n[25] A. Thiaville, S. Rohart, É. Jué, V. Cros, and A. Fert, Europhys. Lett. \n100, 57002, 57002 (2012). \n[26] K. S. Ryu, S. H. Yang, L. Thomas, and S. S. Parkin, Nat Commun 5, 3910, 3910 (2014). \n[27] B. I. Min and Y. R. Jang, J Phys-Condens Mat 3, 5131 (1991). \n[28] Y. Ogata, H. Chudo, M. Ono, K. Harii, M. Matsuo, S. Maekawa, and E. Saitoh, Appl Phys Lett 110, \n072409 (2017). \n[29] J. M. D. Coey, Rare-Earth Iron Permanent Magnets (Clarendon Press, 1996). \n[30] Y. Mimura, N. Imamura, an d Y. Kushiro, J Appl Phys 47, 3371 (1976). \n[31] See Supplementary Material. [32] J. Torrejon, J. Kim, J. Sinha, S. Mitani, M.  Hayashi, M. Yamanouchi, and H. Ohno, Nat Commun 5, \n4655 (2014). \n[33] S. H. Yang, K. S. Ryu, and S. Parkin, Nat Nanotechnol 10, 221 (2015). \n[34] S. Woo  et al. , Nat Commun 9, 959 (2018). \n "    },    {        "title": "1903.10271v1.Octahedral_tilting_and_emergence_of_ferrimagnetism_in_cobalt_ruthenium_based_double_perovskites.pdf",        "content": "arXiv:1903.10271v1  [cond-mat.mtrl-sci]  25 Mar 2019Octahedral tilting and emergence of ferrimagnetism in coba lt-ruthenium based double\nperovskites\nManjil Das, Prabir Dutta, Saurav Giri and Subham Majumdar∗∗\nSchool of Physical Sciences, Indian Association for the Cul tivation of Science,\n2A & B Raja S. C. Mullick Road, Jadavpur, Kolkata 700 032, INDI A\nRare earth based cobalt-ruthenium double perovskites A 2CoRuO 6(A = La, Pr, Nd and Sm)\nwere synthesized and investigated for their structural and magnetic properties. All the compounds\ncrystallize in the monoclinic P21/nstructure with the indication of antisite disorder between Co\nand Ru sites. While, La compound is already reported to have a n antiferromagnetic state below\n27 K, the Pr, Nd and Sm systems are found to be ferrimagnetic be lowTc= 46, 55 and 78 K\nrespectively. Field dependent magnetization data indicat e prominent hysteresis loop below Tcin\nthe samples containing magnetic rare-earth ions, however m agnetization does not saturate even\nat the highest applied fields. Our structural analysis indic ates strong distortion in the Co-O-Ru\nbond angle, as La3+is replaced by smaller rare-earth ions such as Pr3+, Nd3+and Sm3+. The\nobserved ferrimagnetism is possibly associated with the en hanced antiferromagnetic superexchange\ninteraction in the Co-O-Ru pathway due to bond bending. The P r, Nd and Sm samples also show\nsmall magnetocaloric effect with Nd sample showing highest v alue of magnitude ∼3 Jkg−1K−1at\n50 kOe. The change in entropy below 20 K is found to be positive in the Sm sample as compared\nto the negative value in the Nd counterpart.\nI. INTRODUCTION\nSince the discovery of low field room temperature\nmagneto-resistance in Sr 2FeMoO 61, the double per-\novskites (A 2BB′O6) have been intensely studied2. These\nquaternary compounds can be synthesized with a vary-\ning combinations of cations at the A (alkaline earth or\nrare earth metals) and B/B′(3d, 4dor 5dtransition met-\nals) sites, which provides a scope to access diverse ma-\nterial properties within the similar crystallographic en-\nvironment. Elements with partially filled dlevel at the\nB/B′can give rise to wealth of magnetic ground states\nincluding ferromagnetism, antiferromagnetism, ferrimag-\nnetism as well as glassy magnetic phase. Double per-\novskites are also associated with intriguing electronic\nproperties3such as half-metallic behaviour4, tunneling\nmagneto-resistance5, metal-insulator transition6and so\non.\nIn case of insulating A 2BB′O6double perovskites, the\nmagnetic interaction between B and B′ions is primarily\nB-O-B′superexchange type and Goodenough-Kanamori\nrule can predict the sign of the interaction7. Ideal dou-\nble perovskites have cubic symmetry, but the presence of\ncations with small ionic radii at the A site can distort the\nstructure. Such distortion lowers the lattice symmetry to\ntetragonal or monoclinic, where the BO 6/B′O6octahe-\ndra get tilted through the bending of B-O-B′bond angle.\nIt has been found that for two fixed B and B′, the mag-\nnetic ground state is very much sensitive to this bond\nangle. Doping at the A site can change the bond an-\ngle and henceforth the nature of the ordered magnetic\nstate. For example, substitution of Ca at the Sr site\nof Sr2CoOsO 6drives the system from an antiferromag-\nnetic (AFM) insulator to spin-glass (SG) and eventually\nto a ferrimagnetic (FI) state on full replacement of Sr\nby Ca8,9. There is also report of drastic change in fer-\nrimagnetic coercivity in (Ca,Ba) 2FeReO 6under hydro-static pressure due to the buckling of Fe-O-Re bond10.\nIt has been argued that the magnetic ground state in\nthese insulating systems is an outcome of the compe-\ntition between the interactions along the paths B-O-B′\nand B-O-B′-O-B (and similarly, B′-O-B-O-B′). When\nthe octahedral tilting is minimal, the B-O-B′-O-B type\ninteraction dominates, giving rise to strong AFM corre-\nlations within B sublattice11. However, with increasing\ndistortion, B-O-B′becomes stronger with the simultane-\nous weakening of B-O-B′-O-B coupling, and a simple FI\nstate emerges (provided the magnetic moments at B and\nB′ions are unequal) due to the strong AFM coupling be-\ntween B and B′ions. The intermediate spin-glass state\npossibly arises from these competing interactions.\nRecently, La 2CoRuO 6compound has been shown to\nhave an AFM ground state, which crystallizes in the dis-\ntortedmonoclinicstructure12. Interestingly, the isostruc-\ntural Y 2CoRuO 6shows ferrimagnetism, and turns into a\nspin-glass on La doping at the Y site13. It is therefore\nworthwhile to study the magnetic states of A 2CoRuO 6\nwith different rare-earth atoms at the A site. It is well\nknownthattheionicradiusofrare-earth(inthe3+state)\ndiminishes with increasing atomic number, which can\ntune the lattice distortion. The main goal of the present\nwork is to study the effect of lattice distortion and as-\nsociated change in the magnetic properties with varying\nrare-earth ion at the A site. It is also worthwhile to note\nhow the rare-earth moment is affecting the ground state\nmagnetic properties. In a recent report, the magnetic\nproperties of A 2CoMnO 6compounds were found to get\nstrongly affected by the variation of rare-earth ion at the\nA-site14.2\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/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/s115\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121 /s40/s97/s46/s117/s46/s41\n/s50 /s40/s100/s101/s103/s114/s101/s101/s41/s98\n/s99\n/s97/s91/s97/s93 /s91/s98/s93\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/s115\n/s32/s32\n/s50 /s40/s100/s101/s103/s114/s101/s101/s41\nFIG. 1. (a) and (b) show powder x-ray diffraction data of LCRO a nd SCRO respectively collected at room temperature.\nThe inset of (a) shows the crystal structure of the sample. Gr een, pink, blue and spheres indicate La, Ru, Co and O atoms\nrespectively. The bottom panel indicates the octahedral ti lt of four compositions.\nII. EXPERIMENTAL DETAILS\nSingle phase polycrystalline A 2CoRuO 6samples (for\nA = La, Pr, Nd and Sm) were synthesized by solid state\ntechnique. Stiochiometric amounts of A 2O3(except Pr-\nsample, where Pr 6O11was used), Co 3O4and RuO 2were\nwell mixed in a agate morter pestle and calcined at 1073\nK for12h and then sinteredat 1473K for24h in the pel-\nlet form with one intermediate grinding. The powder X-\nray diffraction (PXRD) data were obtained in RIGAKU\nX-ray diffractometer with Cu-K αradiation in the range\n15◦to 80◦. Magnetization ( M) measurements were car-\nried out using a SQUID magnetometer (Quantum De-\nsign, MPMS-3) up to 70 kOe. High field magnetic mea-\nsurements (up to 150 kOe) was performed on a vibrat-\ning sample magnetometer from Cryogenic Ltd., UK. The\ntemperature dependence of the electrical resistivity ( ρ)\nwas measured by DC four-probe method in the temper-\nature range between 50 K and 300 K.III. SAMPLE CHARACTERIZATION\nAll four compositions, La 2CoRuO 6(LCRO),\nPr2CoRuO 6(LCRO) Nd 2CoRuO 6(NCRO) and\nSm2CoRuO 6(SCRO), crystallize in monoclinic rock salt\nstructure (space group P21/n)15. For double perovskite\nA2BB′O6, one can define a Goldschmidt tolerance factor\nt=rA+rO √\n2(rB+rO), whererA, andrOare the ionic radii of\nA and O respectively, while rBstands for the average\nradius of B and B′. It has been found that if t <1,\nthere can be distortion from the ideal cubic structure2.\nFor the present case, r3+\nLa= 103.2 pm, r3+\nPr= 99 pm r3+\nNd=\n98.3 pm, r3+\nSm= 95.8 pm, r2+\nCo= 65(74.5) pm for low-spin\n(high-spin), r4+\nRu= 62 pm and r2−\nO= 140 pm, which give\ntin the range of 0.81-0.83 for four samples. Evidently\nfor these compositions, tis significantly lower than unity,\nand this explains the observed monoclinic symmetry\nrather than ideal cubic one. This lower symmetry is\nassociated with the tilting of the (B,B′)O6octahedra.\nIn order to determine the crystallographic parameters\nof our samples, we have performed Reitveld refinement\non the room temperature PXRD data using MAUD\nsoftware package16. The data converges well with3\n/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48 /s51/s50/s48/s48/s46/s48/s50/s48/s46/s48/s52\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s49/s50/s48/s46/s48/s49/s56/s48/s46/s48/s50/s52\n/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48 /s51/s50/s48/s48/s56/s49/s54\n/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48 /s51/s50/s48/s48/s49/s54/s51/s50\n/s50/s52 /s52/s56 /s55/s50/s48/s50/s48/s52/s48\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s50/s52/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48 /s51/s50/s48/s48/s56/s49/s54\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s50/s53/s48/s46/s48/s53/s48/s40/s101/s109/s117/s47/s109/s111/s108/s41/s40/s101/s109/s117/s47/s109/s111/s108/s41/s32/s49/s48/s48/s32/s79/s101\n/s32/s90/s70/s67\n/s32/s70/s67/s32/s90/s70/s67\n/s32/s70/s67\n/s32\n/s84 /s32/s40/s75/s41/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s91/s97/s93/s76/s67/s82/s79\n/s84/s32 /s40/s75/s41/s32/s49/s48/s48/s32/s79/s101\n/s32/s32\n/s84/s32 /s40/s75/s41\n/s78/s67/s82/s79\n/s32/s49/s48/s48/s32/s79/s101/s84/s32 /s40/s75/s41/s40/s101/s109/s117/s47/s109/s111/s108/s41/s32\n/s32\n/s32/s32\n/s91/s99/s93/s84\n/s80 /s84\n/s80/s83/s67/s82/s79\n/s32/s49/s48/s48/s32/s79/s101/s32/s90/s70/s67\n/s32/s70/s67/s32\n/s32\n/s32/s32\n/s91/s100/s93\n/s32/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s84/s32 /s40/s75/s41/s32/s49/s48/s48/s32/s79/s101\n/s32/s53/s48/s48/s32/s79/s101\n/s32/s49/s48/s48/s48/s32/s79/s101\n/s32/s32/s32\n/s32/s32/s49/s32/s107/s79/s101/s84\n/s80\n/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s32/s49/s48/s48/s32/s79/s101/s32/s90/s70/s67\n/s32/s70/s67/s80/s67/s82/s79\n/s91/s98/s93\n/s32/s32\n/s32/s49/s48/s48/s32/s79/s101\n/s32/s32\nFIG. 2. (a)-(d) represent temperature variation of magneti c susceptibility for LCRO, PCRO, NCRO and SCRO respectively\nfor an applied field of 100 Oe both in ZFC and FC protocols. The i nsets of (a) and (b) show the modified Curie-Weiss fit to\nthe susceptibility data of respective samples. The inset of (c) shows the susceptibility of NCRO measured under 1 kOe of fi eld.\nThe inset in (d) shows an enlarged view of the ZFC susceptibil ity of SCRO recorded at different applied fields.\nmonoclinic space group P2 1/n for all the samples along\nwith antisite disorder between Co and Ru sites [Figs. 1\n(a) and (b)]. Our calculations show that there are 1-5%\nantisite defects, i.e., the fractional occupancy of B site\nconsists of 99-95% Co and 1-5% Ru. The antisite defect\nis found to be large in Nd and Sm compounds and it\nis low in case of La and Pr counterparts. The refined\ncrystallographic parameters are depicted in Table 1,\nand they match well with the previous reported data17.\nWe have also added the structural data of Y 2CoRuO 6\n(YCRO) from reference13for comparison. The crystal\nstructure of LCRO, as obtained from the refinement of\nour PXRD data, is shown in the inset of fig. 1 (a). It is\nclearly evident that the (Co,Ru)O 6octahedra are tilted.\nThe average tilting angle is defined as /angbracketleftΨ/angbracketright=1\n2[π−/angbracketleftΦ/angbracketright],\nwhere/angbracketleftΦ/angbracketrightis the average inter-octahedral Co-O-Ru\nangle. Clearly, the tilt angle increases systematically aswe movefrom La to Sm, which is the effect of the gradual\nbending of Co-O-Ru bond (see Table 1). It is interesting\nto note that all the crystallographic parameters vary\nsystematically with the ionic radius for La, Pr, Nd and\nSm samples.\nIV. RESULTS\nIV.1. Magnetic studies\nFigs. 2 (a)-(d) depict the temperature ( T) dependence\nof susceptibility ( χ=M/H) measured under different\nvalues of Hvalues for all the four samples, where both\nzero-field-cooled (ZFC) and field-cooled (FC) measure-\nments were performed. LCRO [fig.2 (a)] shows well de-\nfined peak at the Ne´ el temperature TN= 27 K, indicat-4\nParameters LCRO PCRO NCRO SCRO YCRO\nr3+\nA(˚A) 1.032 0.990 0.983 0.958 0.900\na(˚A) 5.575(2) 5.497(1) 5.440(5) 5.390(2) 5.266\nb(˚A) 5.638(7) 5.689(7) 5.717(7) 5.722(8) 5.711\nc(˚A) 7.886(2) 7.802(4) 7.739(2) 7.685(7) 7.558\nβ(◦) 90.01(1) 89.88(7) 89.99(8) 89.93(2) 90.03\n/angbracketleft∠Co-O-Ru /angbracketright(◦)152.4 150.9 143.4 139.5 141.7\n/angbracketleftΨ/angbracketright(◦) 13.8 14.6 18.3 20.3 19.7\nMag. state AFM FI FI FI FI\nTran. temp. TN= 27 K Tc= 46 K Tc= 55 K Tc= 78 K Tc= 82 K\nHcoer –9 kOe (2 K) 8 kOe (2 K) 22 kOe (2 K) 22.5 kOe (5 K)\nTABLE I. Crystallographic lattice parameters ( a,b,c, andβ), average Co-O-Ru bond angle, average octahedral tilt angl e (Ψ),\nmagnetic state, magnetic transition temperatures and coer civity are depicted for A 2CoRuO 6(A = La, Pr, Nd, Sm and Y). The\nparameters for Y compound are obtained from reference13. The ionic radii of rare-earth ions (in the 3+ state with coor dination\nnumber VI) at the A site are also shown18.\ning an AFM ground state and it matches well with the\nprevious report7,12,19. TheTvariation of susceptibility\n(χ=M/H) of LCRO in the paramagnetic (PM) state\ncannot be fitted with a simple Curie-Weiss law. How-\never, the χ(T) data above 100 K can be fitted well with\na modified Curie-Weiss law, χCW(T) =C/(T−θ)+χ0,\nwherean additional Tindependent term ( χ0) is included.\nHereCis the Curie constant and θis the Curie-Weiss\ntemperature. The effective PM moment, µeff, obtained\nfrom Curie-Weiss fitting, is found to be 6.63 µB/f.u. The\nvalue ofθis−140 K, signifying strong AFM correlations.\nThe FC and ZFC data show weak divergence below TN.\nWe also observed an upward rise in the χ(T) data below\n6 K. The observed value of µeffis higher than expected\nfor a Co2+-high-spin and Ru4+-low-spin states7, which\nwas attributed to extended 4 d-orbitals of Ru19.\nFor PCRO, NCRO and SCRO, the χ(T) data are dras-\ntically different from that of LCRO [figs. 2 (b), (c) and\n(d) respectively], and it isquite eventful. χshowsasharp\nrise below Tc= 46, 55 and 78 K for these magnetic rare-\nearth containing samples respectively. The FC and ZFC\nsusceptibilities show strong irreversibility below Tc. The\ndivergence exists even in measurement at H= 1 kOe [see\ninset of fig. 2 (c)], however the extend of divergence re-\nduces. The point of bifurcation also moves to lower T\nwith increasing H. The ZFC data show a well defined\npeak at a temperature TP, which lies below Tc. Notably,\nwe observe a change in the value of TPwith increasing\nH. TheHdependence of TPis particularly significant\nfor SCRO, where we observe a shift of Tpby 16 K when\nHis changed from 100 Oe to 1 kOe [see inset of fig. 2\n(c)]. Similar shift in the observed peak in ferrimagnetic\nNd2CoMnO 6was also reported previously, which was at-\ntributed to the presenceofferromagnetic(FM) and AFM\nclusters20. TheFCsusceptibility, ontheotherhand, rises\nmonotonically for all the samples with decreasing T.\nThe susceptibility data of PCRO and NCRO can be\nwell fitted with χCW(T) above 100 K, which gives the\nvalues of µeffto be 6.59 and 6.47 µBrespectively. Thevalue of θis -25 (-22) K for Pr(Nd) sample. These val-\nues ofµeffare slightly lower than the expected value of\n6.97 (7.01) µBwith Pr3+(Nd3+), Co2+(high-spin) and\nRu4+(low-spin) states. This mismatch can be caused\nby the presence of antisite disorder. Such disorder is ex-\npected to cause a decrease in the magnetic moment, and\nempirically Mactual=Mobs/(1−2D)21whereDis the\ndegree of disorder between Co and Ru atoms. For exam-\nple, in case of NCRO with D= 0.05 and Mobs= 6.47\nµB,Mactualis found to be 7.18 µB, which is very close\nto the theoretically predicted value.\nFor SCRO, we failed to achieve good fit using χCW(T)\nin the temperature range 100-315 K. The separation be-\ntween ground ( J= 5/2) and first excited ( J= 7/2) mul-\ntiplets in Sm3+is small, and their mixing can be respon-\nsible for the observed non-Curie-Weiss behaviour22.\nThe isothermal MversusHdata are shown in figs. 3\n(a)to(d). ForLCRO,alinear M−Hcurveisobtainedat\n2 K [see fig. 3 (a)], indicating AFM state. On the other\nhand, Pr, Nd and Sm compounds show significant hys-\nteresis with large coercive field ( Hcoer). Non-zero Hcoer\nis observed for these FI systems just below Tc, and it in-\ncreases with decreasing T. The values of Hcoerare found\nto be 9, 8 and 22 kOe at 2 K for PCRO, NCRO and\nSCRO respectively. However, Mdoes not saturate even\nat 70 kOe of field for none the samples. We have also\nrecorded high field magnetization for SCRO, as shown in\nthe inset of fig. 3 (d). The M−Hcurve at 5 K does\nnot fully saturate even at 150 kOe of applied field. Mat-\ntains a value close to 2 µBat the highest H. The value of\nHcoerfor SCRO at 5 K is found to be 10.6 kOe, which is\nsmaller than the value of Hcoerreported for Y 2CoRuO 6\n(∼22.5 kOe) at the same T.\nConsideringnon-zerocoercivityandthepresenceofan-\ntisite disorder,wehaverecorded M−Hhysteresisloopat\n2 K after the sample being field-cooled from room tem-\nperature. In case of inhomogeneous magnetic systems, a\nshift in the hysteresis loop along the field axis may be\nobserved due to the interfacial coupling of two magnetic5\n/s45/s55/s48 /s45/s51/s53 /s48 /s51/s53 /s55/s48/s45/s51/s48/s51/s45/s55/s48 /s45/s51/s53 /s48 /s51/s53 /s55/s48/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52\n/s45/s55/s48 /s45/s51/s53 /s48 /s51/s53 /s55/s48/s45/s49/s46/s54/s48/s46/s48/s49/s46/s54\n/s45/s49/s53/s48 /s45/s55/s53 /s48 /s55/s53 /s49/s53/s48/s45/s50/s48/s50/s45/s55/s48 /s45/s51/s53 /s48 /s51/s53 /s55/s48/s45/s51/s48/s51/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46 /s41\n/s52/s53/s32/s75/s50/s48/s32/s75/s50/s32/s75\n/s32\n/s32/s32/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46 /s41\n/s91/s99/s93/s78/s67/s82/s79/s91/s97/s93/s32/s50/s32/s75\n/s32\n/s32/s32\n/s76/s67/s82/s79\n/s32/s83/s67/s82/s79/s50/s32/s75/s32\n/s91/s100/s93\n/s72 /s40/s107/s79/s101/s41\n/s32/s32\n/s53/s32/s75\n/s32/s32 /s32/s32/s80/s67/s82/s79/s52/s48/s32/s75/s49/s48/s32/s75/s50/s32/s75\n/s91/s98/s93/s32 /s32/s32\nFIG. 3. (a) to (d) show isothermal magnetization data up to fie ld 70 kOe at different temperatures for LCRO, PCRO, NCRO\nand SCRO respectively. The inset of (d) shows the M−Hcurve for SCRO at 5 K for maximum field of 150 kOe.\nphases, and it is referred as exchange bias effect23. Many\ndouble perovskites show exchange bias effect due to the\npresence of antisite disorder24. However, we failed to ob-\nserve such exchange bias in NCRO and SCRO samples,\nwhich possibly rule out the existence of large magnetic\ninhomogeneity in the system.\nIn orderto investigatethe effect ofexternal field on the\nmagnetic state, we have measured magneto-caloric effect\n(MCE) ofthe samples in terms ofentropy-change(∆ SM)\nbyH. In the recent past, MCE has emerged out to be\nan important technique for green refrigeration25. In the\npresent work, we have obtained MCE from our isother-\nmal magnetization data recorded at different constant\ntemperatures. From the theory of thermodynamics,\n∆SM(0→H0) =/integraldisplayH0\n0/parenleftbigg∂M\n∂T/parenrightbigg\nHdH,\nwhere ∆SM(0→H0) denotes the entropy change for thechange in Hfrom 0 to H026. Figs. 4 (a) to (c) show\n∆SM(T,H0) versus Tplot at different values of H0for\nPCRO, NCRO and SCRO samples respectively.\nThe magnitude of MCE is found to be low for all three\nsamples. For the Pr and Nd samples, ∆ SM(T) is mostly\nnegative with its magnitude peaking around 12 and 8\nK (peak magnitude: 1.7 and 2.9 Jkg−1K−1atH0=\n50 kOe) respectively. A broad feature is also observed\nin ∆SM(T,H0) data around 35 K. On the other hand,\nSCRO shows a contrasting behaviour as far as the MCE\nis concerned. ∆ SM(T) for SCRO is positive below 20 K,\nand increases with decreasing temperature (at least for\nH0>10 kOe). ∆ SMattains a value of 3.1 Jkg−1K−1for\nH0= 50 kOe at 2 K.\nAs already discussed, double perovskite systems can\nshow glassy magnetic state27. An intermediate SG state\nis observed when the AFM state is transformed into an\nFI state by suitable A-site doping. In order to investi-6\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s51/s45/s50/s45/s49/s48/s49\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s45/s49/s48/s49/s50/s51\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s50/s45/s49/s48/s49\n/s91/s98/s93\n/s32/s32\n/s84 /s32/s40/s75/s41/s32/s50/s48/s32/s107/s79/s101\n/s32/s51/s48/s32/s107/s79/s101\n/s32/s52/s48/s32/s107/s79/s101\n/s32/s53/s48/s32/s107/s79/s101\n/s91/s97/s93/s78/s67/s82/s79\n/s91/s99/s93/s32\n/s32/s83/s67/s82/s79\n/s32/s32\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/s32/s53/s48/s32/s107/s79/s101/s32 /s32/s83\n/s77/s32/s40/s74/s107/s103/s45/s49\n/s75/s45/s49\n/s41\n/s32/s32/s50/s48/s32/s107/s79/s101\n/s32/s51/s48/s32/s107/s79/s101\n/s32/s52/s48/s32/s107/s79/s101\n/s32/s53/s48/s32/s107/s79/s101/s80/s67/s82/s79\n/s32/s32\nFIG. 4. (a) to (c) respectively show the Tvariation of ∆ SMof PCRO, NCRO and SCRO at different magnetic fields.\ngate the possibility of a glassy state, particularly those\nwhich lie acrossthe AFM-FI boundary, field-cooled-field-\nstopmemorymeasurementwereperformedon Prand Nd\nsamples28,29. In this protocol, the samples were cooled\nin 100 Oe of field down to 2 K with intermediate stops at\nseveral temperatures below Tc. Subsequently, the sam-\nples were heated back in 100 Oe and dc magnetization\nwas measured. We do not observe any anomaly at the\nstopping temperatures during heating, which rules out\nthe possibility of any glassy (spin glass or cluster glass)\nor super paramagnetic state in PCRO and NCRO sam-\nples.\nIV.2. Electrical transport\nLikemanyotherA 2BB′O6compounds, LCRO,PCRO,\nNCRO and SCRO show semiconducting behaviour as ev-\nident from the transport data depicted in figs. 5 (a) to\n(d) respectively. The values of ρat room temperature\n(∼300 K) are found to be 8.86, 8.81, 4.16 and 6.68 Ω-\ncm for La, Pr, Nd and Sm compounds. Our analysis on\ntheρ(T) data indicates that all three compositions show\nMott Variable Range (VRH) hopping conduction, where\nρ(T)∼exp/bracketleftBig/parenleftbigT0\nT/parenrightbig1\n4/bracketrightBig\n. This is also quite common among\ndisordered double perovskite such as Sr 2MnRuO 630or\nSr2CoSbO 631. The VRH type conduction is quite clearly\nvisible from the log ρversusT−1/4plots in the respec-\ntive insets of figs. 5 (a), (b) and (c). While for La and\nNd compounds, the VRH nature is present almost over\nthe full range of temperature (it deviates only below 65\nK), Pr and Sm compounds show VRH conduction only\nin the range 160 to 60 K. The values of the parameter T0\nassociated with the VRH conduction are found to be 6.9\n×107, 5.1×107, 8.8×107and 6.8×107K for La, Pr, Nd\nand Sm compounds respectively.V. DISCUSSIONS\nIt turns out that the magnetic properties of samples\ncontaining magnetic rare-earth are drastically different\nfrom that of LCRO. Naively, one can relate the FI state\nwith the magnetic moment of A site. However, FI state\nis also observed in Y 2CoRuO 6, where A site contains\nnonmagnetic Y3+ions. In order to address the issue, let\nus first discuss various salient observations made on the\nstudied samples.\n1. The Co-O-Ru bond distortion (and consequently,\nthe octahedral tilt) is found to increase as we move\nfrom LCRO to heavier rare-earth containing com-\npounds. This is due to the reduction of ionic radius\nat the A-site (lanthanide contraction), as we pro-\nceed from La to Sm. It has been alreadymentioned\nthat the smaller radius of A-element leads to larger\nB-O-B′bond distortion2.\n2. LCRO with relatively smaller bond distortion\nshows AFM ground state. On the other hand,\nPCRO, NCRO and SCRO show large increase in\nMbelow the magnetic transition at Tc. Isothermal\nmagnetization curves show large hysteresis with\nthe presenceofsignificant remanentmagnetization.\nIsostructural Y 2CoRuO 6shows similar magnetic\nbehaviour and the Co-O-Ru superexchange inter-\naction is found to be AFM in nature leading to a\nFI ground state (Co and Ru sublattices have differ-\nent moment values)13. In analogy with the Y com-\npound, we can conclude that the ground states of\nPCRO, NCRO and SCRO are ferrimagnetic. Sim-\nilar to Y compound, FI state in PCRO and subse-\nquent compounds emerges possibly due to the Co-\nO-Ru bond distortion, rather than A-site magnetic\nmoment formation.\n3. All four studied compounds are found to be semi-\nconductingwithreasonablyhighresistivity( ∼MΩ-\ncm)aroundthemagneticanomalies. Therefore,the\nmagnetic interaction is likely to be mediated by the7\n/s55/s53 /s49/s53/s48 /s50/s50/s53 /s51/s48/s48/s48/s46/s48/s48/s48/s46/s53/s50/s49/s46/s48/s52\n/s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s52/s56/s49/s50/s55/s53 /s49/s53/s48 /s50/s50/s53 /s51/s48/s48/s48/s46/s48/s48/s48/s46/s51/s51/s48/s46/s54/s54\n/s48/s46/s50/s52 /s48/s46/s51/s48 /s48/s46/s51/s54/s52/s56/s49/s50\n/s55/s53 /s49/s53/s48 /s50/s50/s53 /s51/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s52/s56/s49/s50/s55/s53 /s49/s53/s48 /s50/s50/s53 /s51/s48/s48/s48/s46/s48/s48/s48/s46/s50/s56/s48/s46/s53/s54\n/s48/s46/s50/s52 /s48/s46/s51/s48 /s48/s46/s51/s54/s52/s56/s49/s50\n/s83/s67/s82/s79/s78/s67/s82/s79/s91/s99/s93/s32\n/s32/s32/s32/s108/s110/s32\n/s84 /s32/s45/s49/s47/s52\n/s32/s40/s75/s32/s45/s49/s47/s52\n/s41 /s32/s76/s67/s82/s79/s40 /s45/s99/s109/s41\n/s40 /s41/s40 /s45/s99/s109/s41\n/s84 /s32/s45/s49/s47/s52\n/s32/s40/s75/s32/s45/s49/s47/s52\n/s41 /s32/s108/s110/s32\n/s32/s32\n/s91/s97/s93\n/s32\n/s32\n/s32/s32\n/s91/s100/s93\n/s108/s110/s32\n/s32/s32\n/s32/s32\n/s84 /s32/s45/s49/s47/s52\n/s32/s40/s75/s32/s45/s49/s47/s52\n/s41 /s32\n/s32/s32/s80/s67/s82/s79/s91/s98/s93\n/s84 /s32/s45/s49/s47/s52\n/s32/s40/s75/s32/s45/s49/s47/s52\n/s41 /s32/s108/s110/s32\n/s32/s32\n/s32/s32\nFIG. 5. (a) to (d) respectively show resistivity data as a fun ction of temperature for LCRO, PCRO, NCRO and SCRO. The\ninsets show the VRH-type fittings to the data.\nsuperexchange, rather than the double-exchange\nmechanism.\n4. The coercivity associated with M−Hhysteresis\nloop is found to be much higher in case of SCRO,\nhowever it is lower than the value reported for\nY2CoRuO 6. It appears that the coercivity is not\ndirectly connected to the fact whether A site con-\ntains a magnetic (such as Pr, Nd or Sm) or non-\nmagnetic (here Y) ion.\n5. ∆SMversusTplots for PCRO, NCRO and SCRO\nshow further anomaly at around 8-12 K. It is diffi-\ncult to guess the origin of such anomaly. However,\nconsideringthe presence of rare-earthat the A-site,\nit may signify the low- Tordering of the magnetic\nrare-earth ions.\nAs already mentioned, a transition from AFM to FI\nvia a glassy magnetic state has been observed in sev-\neral double perovskites with the bending of the B-O-\nB′bond8–10,13, where the lattice distortion was cre-\nated by systematic doping at the A site or by ap-\nplying hydrostatic pressure. In the present case, wefound similar effect when one rare-earth ion is replaced\nby another one with smaller ionic radius. In analogy\nwith the idea mooted in case of Sr 2−xCaxFeOsO 6and\nSr2−xCaxCoOsO 68,9, the AFM state in LCRO is due to\nthe strong AFM correlation along the long bonds Co-O-\nRu-O-Co and Ru-O-Co-O-Ru when the Co-O-Ru bond\ndistortion is low. Replacement of La by Pr initiates\nstrong bending in Co-O-Ru bond (see Table 1), which\npossibly strengthen the Co-O-Ru superexchangeover the\nmagnetic interaction on longer Co-O-Ru-O-Co and Ru-\nO-Co-O-Ru pathways leading to FI state. The bending\nfurther enhances in Sm compound, and a significantly\nlarge coercive field is observed.\nThe above argument is also supported by the drastic\nchange in the values of paramagnetic Curie temperature\nθinPrandNdcompounds(-25and-22Krespectively)as\ncompared to the antiferromagnetically ordered La coun-\nterpart. This may be an indication of the weakening\nof the exchange interaction along longer Co-O-Ru-O-Co\nand Ru-O-Co-O-Ru pathways. However, θstill remains\nnegative due to the presence of Co-O-Ru AFM interac-\ntion.\nIn case of SrCaCoOsO 6and La 2−xYxCoRuO 6(x≈8\n0.25 to 1.5) SG states are observed8,13, when Ca(Y) is\ndoped at the Sr(La) site, which has been assigned due to\nthe frustration between long (B-O-B′-O-B) and short(B-\nO-B′) exchangepaths. However, wedo notsee anyglassy\nmagnetic state in neither of the Pr and Nd compounds.\nFor La 1.25Y0.75CoRuO 6with tilt angle 15.5◦, a promi-\nnent frequency dispersion is observed in the ac suscep-\ntibility data. On the other hand PCRO with lower tilt\nangle (14.6◦) shows ordered FI state. Possibly, the emer-\ngence of glassy state in doped samples is also connected\nwith the doping induced disorder. It is to be noted that\nLa1.25Y0.75CoRuO 6sample has huge antisite disorder of\n20%, as compared to 1% antisite disorder in PCRO.\nIt is now pertinent to address the role of 4 fmoment\nfrom the rare-earth present at the A site. In case of\nisostructural Er 2CoMnO 6, rare-earth moment orders at\na relatively lower Tthan the Co-Mn ordering tempera-\nture32. From our magnetization data, it is hard to iden-\ntify the ordering of rare-earth moment. The ∆ SMversus\nTdata depicted in fig.4 show peak like anomalies be-\ntween 8 and 12 K in PCRO, NCRO and SCRO , and\nthey can be probable ordering points of rare-earth mo-\nments. In order to shed more light on this issue, we have\ncompared the moments of A 2CoRuO 6(A = Pr, Nd, Sm\nand Y), where the magnetic data of Y 2CoRuO 6is ob-\ntained from reference13. The moments at 5 K ( M5K), on\napplying 50 kOe of field, is found to be 2.3, 2.9, 1.4 and\n0.8µB/f.u. on the virgin line of the M−Hcurve for Pr,\nNd, Sm and Y compounds. Y does not carry any mo-ment, while the total angular momentum J= 4, 9/2 and\n5/2 for Pr3+, Nd3+and Sm3+states respectively. The\nmagnetic moments of these three ions are 3.58, 3.62 and\n0.84µBrespectively. Clearly, the variation of M5Kcor-\nresponds well with the variation of rare-earth moment.\nThis signifies that the A site rare-earth moment remains\nin ordered state at 5 K. It is important to perform a\nneutron diffraction study to ascertain the true magnetic\nstructure of these compounds.\nIn summary we have studied the structural, magnetic\nas well as transport properties of the double perovskites\nLa2CoRuO 6, Pr2CoRuO 6, Nd2CoRuO 6, Sm2CoRuO 6.\nWe observe a systematic change in the magnetic ground\nstate as La is replaced by Pr, Nd and Sm. This matches\nwell with the case of Fe-Os and Co-Os based double per-\novskites,wherelatticedistortiontunesthestrengthofthe\nmagnetic interactions in different exchange pathways.\nVI. ACKNOWLEDGMENT\nThe work is supported by the financial grant from\nDST-SERB project (EMR/2017/001058). MD would\nlike to thank CSIR, India for her research fellowship,\nwhile PD thanks DST-SERB for his NPDF fellowship\n(PDF/2017/001061).\nREFERENCES\n∗ ∗sspsm2@iacs.res.in\n1Kobayashi K -I, Kimura T, Sawada H, Terakura K and\nTokura Y 1989 Nature395677\n2Vasala S and Karppinen 2015 Prog. Solid State Chem. 43\n1\n3Nag A, Jana S, Middey S and Ray S 2017 Ind. J. Phys. 91\n883\n4Serrate D, Teresa J M De and Ibarra M R 2007 J. Phys.:\nCondens. Matter 19023201\n5SarmaDD,RaySugata, TanakaK,KobayashiM,Fujimori\nA, Sanyal P, Krishnamurthy H R and Dasgupta C 2007\nPhys. Rev. Lett. 98157205\n6Kato H, Okuda T , Okimoto Y, Tomioka Y, Oikawa K,\nKamiyama T and Tokura Y 2002 Phys. Rev. B 65144404\n7Dass R I, Yan J -Q and Goodenough J B 2004 Phy. Rev.\nB69094416\n8Morrow R, Yan Jiaqiang, McGuire Michael A, Freeland\nJohn W, Haskel Daniel and Woodward Patrick M 2015\nPhys. Rev. 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Chem. 15776\n32Blasco J, Subas G, Garca J, Stankiewicz J, Rodr´ ıguez Ve-\nlamaz´ an J A, Ritter C and Garc´ ıa-Mu˜ noz J L 2017 Solid\nState Phenom. 25795"    },    {        "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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Hirata, H. \nYoshikawa, A. Tsukamoto, K. -J. Kim, Y. Shiota, K. -J. Lee, and T. Ono, Temperature \ndependence of magnetic resonance in ferrimagneti c GdFeCo alloys, Appl. Phys. Express 12, \n093001 (2019) . \n37. S. Mizukami, Y. Sasaki, D. -K. Lee, H. Yoshikawa, A. Tsukamoto, K. -J. Lee, and T. Ono, \nLaser -induced antiferromagnetic -like resonance in amorphous ferrimagnets, \narXiv:1808.05707v1 . \n38. M. I. Dyakonov and V. I. Perel,  Possibility of orienting electron spins with current,  Sov. \nPhys. JETP Lett. 13, 467 (1971) . \n39. J. E. Hirsh,  Spin Hall Effect,  Phys. Rev. Lett. 83, 1834 (1999) . 16 \n 40. J. W. Lee, J. Y. Park, J. M. Yuk, and B. -G. Park, Spin -orbit torque in a perpendicularly           \nmagnetized ferrimagnetic  Tb-Co single layer, Phys. Rev. Applied  13, 044030 (2020).  \n41. V. E. Demidov and S. O. Demokritov,  Magnonic waveguides studied by m icrofocus \nBrillouin light scattering,  IEEE Trans. Mag. 51, 0800215 (2015) . \n42. A. Mitrofanov and S. Urazhdin, Energy and momentum conservation in spin transfer, \narXiv:2004.01957.  \n43. F. Schlickeiser, U. 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": "1705.06398v1.Magnetic_vortex_nucleation_annihilation_in_artificial_ferrimagnet_microdisks.pdf",        "content": " \n1 \n Magnetic vortex nuclea tion/annihilation in artificial -ferrimagnet micro disks \n \nPavel N. Lapa ,1,2 Junjia  Ding,1 Charudatta Phatak,1 John  E. Pearson,1 J. S. Jiang,1 Axel \nHoffmann1 and Valentine  Novosad1 \n1Material Science Division, Argonne National Laboratory, Argonne, IL 60439, USA  \n2Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843 -4242, USA  \nThe topological nature of magnet ic-vortex state gives rise to peculiar magnetization reversal observed in magnetic microdisks. \nInterestingly, magnetostatic and exchange energies which drive  this reversal can be effe ctively controlled in artificial \nferrimagnet heterostructures composed of rare -earth and transition met als. [Py( t)/Gd( t)]25 (t=1 or  2 nm) superlattices \ndemonstrate a pronounced change of the magnetization and exchange stiffness in a 10–300 K temperature range as well as very \nsmall magnetic anisotropy . Due to these properties , the magnetization of  cylindrical micro disks composed of these artificial \nferrimagnets  can be transformed from the vortex to uniform ly-magnetized  states in a permanent magnetic field by changing \nthe temperature . We explore d the behavior of magnetization  in 1.5 -µm [Py( t)/Gd( t)]25 (t=1 or 2 nm) disk s at different \ntemperatures and magnetic fields  and observed  that due to the energy barrier separating vortex and uniform ly-magnetized  \nstates , the vortex  nucleation and annihilation occur  at different temperatures. T his causes  the temperature dependences  of the \nPy/Gd disks  magnetization to demonstrate unique hysteretic behavior  in a narrow temperature range . It was discovered that for \nthe [Py(2 nm)/Gd(2 nm)] 25 microdisks the vortex can be metastable at a certain temperature rang e. \n \nI. INTRODUCTION  \nIn order to minimize magnetic stray field  and magnetostatic energy associated  with it, the magnetization of  a macroscopic  \nferromagnetic object  typically develops  an inhomogeneous domain structure . For  nano -sized  objects, the  cost of  exchange \nenergy due to  the domain wall s makes multi -domain state energ etically unfavorable. H ence, for these objects, a uniformly -\nmagnetized single -domain s tate is stable , even in a zero magnetic field . For cylindrical micro disks, which have sizes in between \nmacro - and nano -size regimes , another mechanism of a magnetic -flux elimination  can be realized. Namely, in a zero magnetic \nfield, magnetization curls  azimuthally  around the disk geometrical center , forming a so-called magnetic vortex.1 To avoid \ndiscontinuity  of magnetization  in the geometrical center of the vortex , the magnetization smoothly points perpendicular to  the \ndisk plane within a narrow  area containing the vortex center. The area with a nonzero out-of-plane component of magnetization \nis usually called a vortex core.2 An increasing magnetic  field caus es the vortex  to shift away  from the geometrical center of the \ndisk in the  direction perpendicular to the magneti c field.  When the magnetic field exceeds a critical value, which is defin ed by \nmicroscopic  parameters of the material (exchange stiffness and magnetization)  and geometrical parameters of the disks \n(diameter and height ), the vortex annihilates.  Vice versa, a decreasing magnetic field drives the vortex nucleation and it \nsubsequently moves  towards the geometrical center of the disk.  \n2 \n It is argued that  the drastically reduced dipole -dipole interaction  between the disks and the topological nature  of the vortex \nstate can be utilized  for storing binary information .3, 4 For this, t he static5 and dynamic6-8 behavior of magnetic vortices  have  \nbeen studied extensively. More fundamental aspects , like a magnetic structure of a vortex core or microscopic  mechanisms of \nvortex  nucleation/ annihilation , have also attracted  a lot of attention and have been investigate d using a variety of tools .9-15 Thus, \nfabrication of multilayered microdisks in which different layers are either coupled to each other16, 17 or exchange biased  with \nan antiferromagnet18 has proved to be an efficient way to control magnetization reversal.  \n One of the main requirement for an observation of a vortex state in a magnetic micro disk is that the magnetic material s \nhave  very low magnetic anisotropy.19 A typical  choice of materials which satisfy this requirement includes  magnetically soft  \nPermalloy (Py =Ni 0.81Fe0.19),20 polycrystalline Ni21 and Fe .15 For micro disks compos ed of these materials, a conventional \napproach to switch the magnetization from the vortex to uniformly -magnetized states is to change  the amplitude of the \nexternally -applied magnetic field. Since the  Curie temperature s for these materials  are high, their  magnetic parameters \n(magnetization and exchange stiffness ) are almost constant  below room temperature. This  means t hat for the microdisk s \ncomposed of these materials,  the nucleation and annih ilation fields demonstrate an insignificant change while  temperature is \nvaried within a  10–300 K range.14 \nDue to an antiferromagnetic coupling between rare -earth Gd and transition metal s, the heterostructures composed of these \nmaterials are widely used for artificial ferrimagnet application s.22-27 As we observed previously,26, 27 if an artificial ferrimagnet  \nis composed of thin layers of Py and Gd, the heterostructure has very low anisotropy which is comparable with that of Py. \nHence, it is possible that the magnetic vortex may  be a stable magnetization configuration for the micro disks composed of \nthese artificial ferrimagnets . Importantly, the Curie temperature  of Gd (293  K for bulk ) is much lower than that for Py which \nmeans that the Gd/Py superlattice s have significant  change s of magnetization between 10 and 300 K. This suggests  that the \nvortex nucle ation and annihilation fields for  the Py/Gd superlattice micro disks may also var y significantly  within  this \ntemperature range . Our motivation  for this work  was to detect and study vortex nucleation /annihilation  transitions that occur \nin a permanent magnetic field  due to  the change  of the microscopic  parameters of the material  for different temperatures . \nII. EXPERIMENTAL DETAILS  \nWe prepared two groups of  disk array s for the study. T he disks from the first and second group s have \n[Py(1  nm)/Gd(1  nm)]25 and [Py(2  nm)/Gd(2  nm)] 25 structures along the height , respectively. The nominal  diameter of each disk \nis 1.5  µm. In addition, two different approaches  were used fo r fabrication  of each group: etching and lift-off. For  fabrication  of \nthe lift -off technique,  Si/SiO 2 wafers were spin -coated with photoresist, and an array of 1.5  µm holes were patterned by means \nof optical lithography. Then, after magnetron sputtering of the multilayers , the lift -off process yielded array s of 1.5  µm disks.  \n3 \n For fabrication of the etched disks , the artificial ferrimagnet films were sputtered on top of the Si/SiO 2 wafer . After that,  the \narrays of holes  were patterned  on top of the films. T his pattern was transferred into arrays of photoresist disks using an image \nreversal technique (baking the wafer at 100ºC in NH 3 environment for 30 minutes followed by flood exposure and \ndevelopment).  The final step  of this process  was Ar-ion milling. The deposition rates  used for the sputtering of Gd and Py  are \n1.4 Å/sec  and 0.7  Å/sec, respectively. For seeding and a capping , 5-nm thick  Ta layer s were deposited before and after \nsputtering each multilayer . In addition to the disk array s, two control films with the same  structure s were sputtered for the \nstudy. The wafer s containing fabricated disk arrays were cut into 6  mm×6  mm pieces , and their magnetic behavior was studied \nusing a superconducting quantum interference device (SQUID) magnetometer.  We also fabricated the  [Py(1  nm)/Gd(1  nm)] 25 \ndisk array on top of a Si3N4 membrane window  for a Lorentz transmission  electron microscopy ( TEM ) study  using the lift -off \ntechnique . TEM images at different temperatures were taken in Fresnel imaging mode .28 \nIII. MATERIAL  PROPERTIES AND SIMULATION S DETAILS  \nMagnetic and structural properties of the Py/Gd  multilayers have been studied by us26, 27 and other groups25, 29, 30 previously. \nImportantly for this work,  a polycrystalline Gd film has comparative ly high anisotropy,31 and only the Py/Gd multilayers with \nGd-layer  thickness  of 2 nm or less are magnetically soft. At the same time, it was  demonstrated that the influence  of material \nintermixing becomes significant  for multilayers w ith thin Py and Gd  layers .26, 27 The temperature dependences  of magnetization \nfor the [Py(t)/Gd( t)]25 (t=1 or 2 nm) control films  are shown in Fig.  1(a). It is seen that only the magnetization for the film with \nt=2 nm demonstrates ferrimagnet -like behavior. Its magnetization is small at room temperature . Then when the Gd layers  \nmagnetization s begin  to rise  the total magnetization decrease s; at around 172  K, the magnetic moments  of the Py and Gd layers \ncompensate each other. At low temperatures, the total magnetization in the multilayer is Gd -aligned. In contrast, th e \n[Py(1  nm)/Gd(1  nm)] 25 film becomes ferromagnetic only at around 275 K, which  is below  the Curie temperature s of bulk Gd \nand Py, and then  its magnetization rises monotonically with decreasing temperature.  Due to the strong intermixing of Py and \nGd, the [Py(1  nm)/Gd(1  nm)] 25 superlattice  can be considered as a homogeneous  film composed of a PyGd alloy . We adopt ed \nthis model for micromagnetic simulation s of magnetization behavior for the [Py(1  nm)/Gd(1  nm)] 25 disks. The analysis26, 27 \nshowed that the magnetic and at omic structure s of the [Py(2  nm)/Gd(2  nm)] 25 superlattice  is more complicated . According to \nour estimates ,26, 27 only the 0.5-nm thick core parts of  the Py and Gd lay ers are  not subjected to intermixing, while the rest of \nthe film is  the PyGd alloy.  Taking into account this magnetization profile over the [Py(2  nm)/Gd(2  nm)] 25 film thickness would \nlead to a very complicated  micromagnetic model which would require  a very small mesh , and hence, long calculation time. To \navoid this , similar ly to the microm agnetic model used for the [Py(1  nm)/Gd(1  nm)] 25 disks, we assume d that the  \n4 \n [Py(2  nm)/Gd(2  nm)] 25 disks are composed of a homogeneous material. Microscopically, the effective magnetization  of the \nPy/Gd  disks , Meff, is expected  to be equal to the magnetization s of the corresponding control films at a given temperature.   \n \nFIG. 1. a) Magnetization and b) effective exchange stiffness, Aeff, as a function of temperature for the [Py(1  nm)/Gd(1  nm)] 25 (black dots) \nand [Py(2  nm)/Gd(2  nm)] 25 (red dots) control films. Magnetization was measured in the 100 -Oe magnetic field applied in -plane. The lines \nrepresents dependences used for the micromagnetic simulations.  \n \nOnce  it is assumed  that the disks are composed of a homogene ous material, the effective exchange stiffness of this material  \nat different temperatures must  be estimated . For this, we utilize  a technique used by us previously.26, 27 Shortly, the  films with \nstructures Py(50  nm)/ [Py(t)/Gd( t)]25 (t=1 or  2 nm) were sputtered and their magnetization curve s were measured  at different \ntemperatures. The total magnetic moment of the [Py(t)/Gd( t)]25 stack is  coupled antife rromagnetically  with the magnetic \nmoment  of the 50 -nm thick Py layer . An applied magnetic field results in  an alignment of  these magnetic moments along the \nfield, and since  the effective exchange stiffness of the stack s is much smalle r than  that of Py, the process is realized through an \nexchange -spring -like magnetization twist  in the Py/Gd stacks.  Modeling the observed non -linear dependences  of the \nmagnetization on magn etic field, and assuming that the exchange stiffness ( Aeff) is constant across the Py/Gd stacks, allowed  \nus to estimate  Aeff in the [Py(t)/Gd( t)]25 (t=1 or  2 nm) multilayers for  different temperatures  [Fig.  1(b)] .  \nSummarizing, for a micromagnetic simulation  at a given temperature, T, it was assumed that a disk is composed of an \nisotropic homogeneous material with magnetization Meff(T) and exchange stiffness Aeff(T). The simulation s were  conducted \nusing a GPU -accelerated micromagnetic simulation program , mumax3.32 The size of the mesh  cell is 2.9  nm×2.9  nm in plane \nof a disk and 12.5  nm over  its thickness.  \nIV. EXPERIMENTAL RESULTS  \nFor both superlattice s composition s, Meff is at a maximum at low temperatures. Magnetization curves for the etched \n[Py(t)/Gd( t)]25 (t=1 or  2 nm) disks measure d at 10  K are shown in Fig.  2(a). Both curves correspond  to vortex -like behavior5 \n \n5 \n in low magnetic fields, i.e. the magnetization  changes linearly with  the field and the coercivity is negligibly small (the coercive \nfields are  0.5 and 2.5  Oe for t=1 and 2  nm, respectively ). The micromagnetically -simulated curves shown in  Fig. 2(b), \nquantitatively and qualitatively resemble the experimental ones. According to the simulations, the vortex  configuration is stable \nin the magnetic field s of up to 340  Oe for  the [Py(1  nm)/Gd(1  nm)] 25 disks, and up  to 580 Oe for the  [Py(2  nm)/Gd(2  nm)] 25 \ndisks. Due to an energy barrier separating  the vortex and uniformly -magnetized state s, the simulated and experimental loops \nhave  hysteretic  regions  in higher  magnetic  fields [Fig.  2(a, b)]. Because the disks diameter is 1.5  µm and Aeff is low, the vortex \nnucleation and annihilation are very different fr om those  observed in submicron Py disks.33 First, i ncreasing magnetic field \ncauses the magnetic vortex  to deform  in addition to shifting . In magnetic fields below the annihilation  fields, the vortices are \nmoon -shaped [schematics in Fig.  2(b)]. Second, upon magnetic field decrease,  the magnetization does not transform  from  a \nuniform state to a vortex state  directly, but instead , two vortices appear which move  toward each other and merge into a single \nvortex in 210 and 360 -Oe magnetic fields for t=1 and 2  nm, respectively [schematics in Fig.  2(b)] . Only the vortex annihilatio n \nlooks like a sharp transition . \n \nFIG. 2. a) Experimental b) micromagnetically -simulated hysteresis loops (10  K) for the etched [Py(1  nm)/Gd(1  nm)] 25 (black line) and \n[Py(2 nm)/Gd(2  nm)] 25 (red line) disks; b) contains the schematics of magnetization configurations realized in the disks at different \nmagnetic fields.  \n \nTo determine which magnetizatio n configurations are realized at  different temperatures , magnetization  of the etched  arrays \nwas measured in a 100 -Oe magnetic field while the temperature was slowly (1  K/min) swept from 300 to 10  K and back to \n300 K [Fig.  3(a)].  At the initial stage of the cooling down, the curves demonstrate the behavior almost identical to  that \n \n6 \n demonstrated by  the control films. T hen, at 112  K for t=1 nm and 66  K for t=2 nm, a phase  transition occurs  and the \nmagnetization begin s to decrease when  the temperature  is increased . At low temperatures, the magnetization is almost constant.  \nWhen the temperature is ramped up , the magnetization  decrease s but not monotonically. For the [Py(1  nm)/Gd(1  nm)] 25 disks, \nthe magnetization falls slowly until 143 K, and then the curve merge s with the one obtained while cooling down  the sample . A \nvery similar decrease  of the magnetizati on is demonstrated by the [Py(2  nm)/Gd(2  nm)] 25 disks in the initial stage  of the \nwarm -up, but right before the merging with the cool-down curve , the magnetization  kinks up. Interestingly , the magne tic phase \ntransition s yield  the temperature dependences  of the magnetization  to demonstrate  hysteretic behavior within the  40-160 K and \n40-110 K ranges, for t=1 and 2  nm, respectively.  \n \nFIG. 3. a) Experimental b) micromagnetically -simulated temperature dependences of magnetization in the 100 -Oe magnetic field for the \netched [Py(1  nm)/Gd(1  nm)] 25 (black line) and [Py(2  nm)/Gd(2  nm)] 25 (red line) disks; b) contains the schematics of magnetization \nconfigurations realized in the disks at  different temperatures.  \n \nThe tem perature dependences  of the disks magnetization in the 100-Oe magnetic field  were si mulated micromagnetically \n[Fig.  3(b)].  Importantly,  it is a zero-temperature  simulation s without  thermal field fluctuations  and any dependence on \ntemperature is only introduced by temperature -dependent parameters,  Meff(T) and Aeff(T). This simplification is reasonable \nbecause any fluctuation effect s, such as a thermal excitation over an energy barrier ,34 are expected to be insignificant for 1.5  µm \ndisks. The s imulated curves demonstrate hysteretic behavior  similar to the one  observed in the experiment  [Fig.  3(a, b)]. The \nsimulation s reveal that the phase transition s observed in the experiment are  due to the vortex nucleation and annihilation.  At \n \n7 \n the high temperatures, the disks magnetization s are  low and the disks are in the uniformly -magnetized state. W hen the \ntemperature  is decreased  the disks magnetization increase s, and at a critical temperature , it becomes energet ically favorable to \nnucleate vortices,  thus minimizing  the magnetic  flux. Again, similar ly to the nucleation processes observed at 10  K, the disks \nmagnetization does not switch  from uniform ly-magnetized  to single -vortex states  directly ; instead, the magnetization  transfer s \nthrough a series of intermediate states in which the doub le vortices configurations are  stable.  The transition s between each \nintermediate state cause s an abrupt decrease  of magnetization. Since the arrays consist of disks with slightly differe nt diameters, \nthe abrupt transitions observed in the simulation s are smeared out for the experimental curves. According to the simulations , \nincreasing the temperature causes the vortex core  shifting  from the center of the disk and its deformation, eventually the disks \nswitch  to the quasi -uniformly -magnetized state  [schematics in Fig.  3(b)] .  \nIt was of particular  interest to investigate the stability of the vortex state  at the temperatures which are within the thermal \nhysteresis regions , e.g. at 115 K for the [Py(1  nm)/Gd(1  nm)] 25 disk array and 75 K for the [Py(2  nm)/Gd(2  nm)] 25 disk array.  \nFor this, the disks were coole d down to 10  K, and the magnetic field was set to 0 Oe. T his procedure allowed to ensure  that the \nvortex  is the  initial magnetic state for the disks . Then , the temperature was increased to 115  K and 75  K for the \n[Py(1  nm)/Gd(1  nm)] 25 and for [Py(2  nm)/Gd(2  nm)] 25 disk array s, respectively, and  the arrays  magnetization  was measured \nwhile  the magnetic field was ramped  up from 0  Oe to 500  Oe and then cycle d between 500 and -500 Oe [Fig.  4(a)]. The \nexperiment s were  simulated microma gnetically [Fig.  4(b)].  For the [Py(1  nm)/Gd(1  nm)] 25 disk array , the magnetization  shows  \nthe behavi or identical to that observed for the array at  10 K, i.e. the magnetization rises l inearly in low magnetic field s and the \nhysteresis is  present only i n the high  fields. T he initial part of the magnetiza tion curve measured between 0 and  500 Oe coincides \nwith its  main part (500 Oe → -500 Oe → 500  Oe) indicat ing that the v ortex state is stable for these  disks in low magnetic field . \nThe simulated curve is  in qualitative agreement with the experimental one. In contrast, the main  part of the magnetization curve \nfor the [Py(2  nm)/Gd(2  nm)] 25 disk array has a double -waist ed shape, while its initial  curve demonstrates typical vortex -like \nbehavior.  Moreover, the initial  magnetization curve passes outside the area under the main magnetization curve. This unusual \nbehavior indicate s that the vortex state  is stable  at the initial stage of the measurements , but after the saturation of the disk \nmagnetization, the magnetization  does not switch back to the vortex state while the main hysteresis loop is measured. It was \nalso observed that, if the disk is demagnetized after measuring the main hysteresis loop, and after that, the magnetization i s \nmeasured while the magnetic field is  increased, the resulting curve passes within the main hysteresis loop and does not c oincide \nwith the i nitial magnetization curve.  Thus,  the (0,0) point at the M -H plane can correspond to two different magnetization \nstates, one of which is the vortex  state. This means  that the vortex is a metastable  state for the [Py(2  nm)/Gd(2  nm)] 25 disk at \n75 K. By a metastable state , it is implied that the state cannot be accessed by isothermal changing of an external magnetic field.   \n8 \n Interesting, the magnetic field depe ndence of magnetization for  the [Py(2  nm)/Gd(2  nm)] 25 disk array was not completely \nreproduced in the s imulation [Fig.  4(b) red line]. It is possible that  the fine magn etic structure within the [Py(2  nm)/Gd(2  nm)] 25 \nsuperlattice  as well as its very low magnetic anisotropy  must be taken into account  for an adequate modeling of the \nmagnetization reversal for  the corresponding disks. \n \nFIG. 4. a) Experimental b) micromagnetically -simulated hysteresis loops for the etched [Py(1  nm)/ Gd(1  nm)] 25 disks at 115  K (black lines) \nand the etched [Py(2  nm)/Gd(2  nm)] 25 disks at 75  K (red lines); the initial parts of the magnetization curves (from 0 to 500  Oe) are shown \nwith thin lines while the main parts of hysteresis curves (500  Oe → -500 Oe → 500  Oe) with thick lines; b) contains the schematics of \nmagnetization configurations realized in the disks at different magnetic fields. The inset in the upper -left corner of (b) shows the hysteretic \npart of the loop for the [Py(2  nm)/Gd(2  nm)] 25 disks.  \n \nAn important factor that strongly affects magnetic properties of the Py/Gd superlattice disks is the intermixing of Py and \nGd. Due to a shadow effect, t he intermixing is much more pronounced for the lift -off disks than that for the etched  disks. Since \nthe intermixing is almost c omplete for the [Py(1  nm)/Gd(1  nm)] 25 superlattice, t he lift -off and etched  [Py(1  nm)/Gd(1  nm)] 25 \ndisks demonstrate almost identical temperature and field dependences . At the same time, due to the intermixing, the \ncompen sation tempe rature of the lift -off [Py(2  nm)/Gd(2  nm)] 25 disks is about 50  K higher than that for the etched  disks and \nthe corresponding  control films. Although  the same physical phenomena are observed in the  lift-off [Py(2  nm)/Gd(2  nm)] 25 \ndisks as that  observed in the etched  ones ( vortex nucleation and annihilation ), some quantitative parameters, like  the \nnucleation/annih ilation fields and temperatures, are different for the  [Py(2  nm)/Gd(2  nm)] 25 disks prepared using di fferent \nfabrication techniques.  \n \n9 \n  \nFIG. 5. Under -focused Lorentz transition electron microscopy images of a lift-off [Py(1  nm)/Gd(1  nm)] 25 disk on a Si 3N4 membrane \nwindow acquired at 153  K (a) and 98  K (b). A white dot at the center of the disk at 98  K indicates the presence of magnetic vor tex; c) \nmagnetization of the etched [Py(1  nm)/Gd(1  nm)] 25 disk array as a function of temperature measured in the 10 -Oe magnetic field.  \n \nTo get a direct confirm ation that vortices nucleate  in the Py/Gd superl attice disks at low temperatures, we acquired a  series \nof under -focused Lorentz TEM images of a lift -off [Py(1  nm)/Gd(1  nm)] 25 disk on a Si 3N4 membrane at different temperatures. \nFig. 5(a) and (b) shows  the Lorentz T EM obtained at 152 and 98  K, respectively. A white dot at the center of the disk in \nFig. 5(b), which appears when the temperature drops below 129  K, indicates the presence of a  magnetic vortex . Fig. 5(c) shows  \na temperature dependence of  the magnetization measured in very low magnetic field (10 Oe), accordi ng to  which the \nmagnetization of  the [Py(1  nm)/Gd(1  nm)] 25 disks switches to the vortex state below 180  K in the low magnetic field.  \nImportantly, during the TEM imaging the temperature was s wept continuously. T he real temperature of the disk on the Si 3N4 \nmembrane  can be very different from the readings of  the microscope’s stage thermo couple . This yields slightly different  vortex \nnucleation temperatures detected using  Lorentz TEM and  using  magnetometry . \nV. CONCLUSION  \nWe observed that the 1.5 -µm disks composed of the [Py( t)/Gd( t)]25 (t=1 or  2 nm) artificial ferrimagnets experience phase \ntransitions.  For these disks, t he resulting temperature dependences  of magnetization measured in the 100 -Oe magnetic field are  \nhysteretic in narrow temperature regions . Micromagnetic  simulation s revealed that these phase transitions are due to nucleation \nand annihilation of magnetic vortices which happen because the magnetic properties of the artificial ferrimagnet change  \nsignificantly within  the 10 –300 K temperature range. It was shown that  at 75 K,  the vortex  in the [Py(2  nm)/Gd(2  nm)] 25 disk \nis in a meta stable state. I t is impossible to nucleate the vortices  in the [Py(2 nm)/Gd(2  nm)] 25 disks by changing the external \n \n10 \n magnetic  field isothermally , at the same time,  if the disk  with the magnetic vortex is brought to 75  K, the vortex does not \nannihilate until the disk magnetization is saturated.  \nREFERENCES  \n1. K. Y. Guslienko, J. Nanosci. Nanotechnol. 8, 2745 -2760 (2008).  \n2. T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto and T. Ono, Science 289 (5481), 930 -932 (2000).  \n3. K. Bussmann, G. A. Prinz, S. -F. Cheng and D. Wang, Applied Physics Letters 75 (16), 2476 -2478 (1999).  \n4. R. 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Waeyenberge, AIP Advance s 4 (10), \n107133 (2014).  \n33. G. Gubbiotti, G. Carlotti, F. Nizzoli, R. Zivieri, T. Okuno and T. Shinjo, IEEE Transactions on Magnetics 38 (5), 2532 -\n2534 (2002).  \n34. K. Liu, J. Nogués, C. Leighton, H. Masuda, K. Nishio, I. V. Roshchin and I. K. Schuller, Ap plied Physics Letters 81 \n(23), 4434 -4436 (2002).  "    },    {        "title": "2011.02001v1.An_attempt_to_simulate_laser_induced_all_optical_spin_switching_in_a_crystalline_ferrimagnet.pdf",        "content": "arXiv:2011.02001v1  [cond-mat.mtrl-sci]  3 Nov 2020An attempt to simulate laser-induced all-optical spin swit ching in a crystalline\nferrimagnet\nG. P. Zhang∗, Robert Meadows, and Antonio Tamayo\nDepartment of Physics, Indiana State University, Terre Hau te, IN 47809, USA†\nY. H. Bai\nOffice of Information Technology, Indiana State University, Terre Haute, Indiana 47809, USA\nThomas F. George\nDepartments of Chemistry & Biochemistry and Physics & Astro nomy\nUniversity of Missouri-St. Louis, St. Louis, MO 63121, USA\n(Dated: November 5, 2020)\nInterest in all-optical spin switching (AOS) is growing rap idly. The recent discovery of AOS in\nMn2RuGa provides a much needed clean case of crystalline ferrim agnets for theoretical simulations.\nHere, we attempt to simulate it using the state-of-the-art fi rst-principles method combined with the\nHeisenberg exchange model. We first compute the spin moments at two inequivalent manganese\nsites and then feed them into our model Hamiltonian. We emplo y an ultrafast laser pulse to switch\nthe spins. We find that there is a similar optimal laser field am plitude to switch spins. However, we\nfind that the exchange interaction has a significant effect on t he system switchability. Weakening\nthe exchange interaction could make the system unswitchabl e. This provides a crucial insight into\nthe switching mechanism in ferrimagnets.\nPACS numbers: 75.78.Jp, 75.40.Gb, 78.20.Ls, 75.70.-i\nCentral to the magnetic storage device is the writ-\ning/reading speed of magnetic bits in a storage medium.\nTraditionally, these operations are mostly driven by an\nexternal magnetic field. A full-optical driven spin manip-\nulation could break the speed barrier of several hundred\npicoseconds set by the Zeeman interaction and magnetic\ndipole-dipole interaction. In 1996, Beaurepaire et al.[1]\nshowed that when they shone a 60-fs pulse on the ferro-\nmagnetic nickel thin film, they found a sharp decrease in\nthe Kerr signal within 1 ps. This finding received imme-\ndiate attention worldwide, and a new research field, fem-\ntomagnetism, wasborn[2,3]. Researchintensified, andis\nfar beyond the scope of the original research interest. In\n2007, Stanciu et al.[4] showed that a left-circularly po-\nlarized laser pulse can switch an up-spin to down, while a\nright-circularly polarized laser pulse can switch a down-\nspinup. Thisremarkablepropertyrepresentsaninterest-\ningnew magnetic phenomenononan ultrafasttime scale,\nalthough their compound, GdFeCo, is not new. GdFeCo\nhas been used in traditional magneto-optical recording\n(see the references cited in [5, 6]). However, being to\nable to switch spins on a picosecond time scale optically\nis new, and has raised the possibility for a real applica-\ntion. However,itisunclearhowthelaserpulsecanswitch\nspins directly. Ostler et al.[7] further showed that if the\nlaser intensity is increased above a certain level, regard-\nless of laser helicity, each pulse can flip spins from one\ndirection to another deterministically. They argued that\n∗Author to whom correspondence should be addressed.\n†Electronic address: guo-ping.zhang@outlook.com.there is a threshold intensity that one has to exceed to\nchange from all-optical helicity dependent spin switch-\ning to all-optical helicity independent switching, but the\nactual picture is more complicated [8, 9].\nFor a long time, GdFeCo was the only material that\nshows AOS. Soon, many more materials were found [10–\n14]. However, these materials are mostly amorphous,\nwhich introduces an uncertainty in theoretical simula-\ntions and represents a formidable task. In 2017, Vomir\net al.[15] reported the first observation of AOS in a\nPt/Co/Pt ferromagnetic stack, but the switching is not\ncomplete. Very recently, Banerjee et al.[16] showed\nsingle-pulse all-optical toggle switching of magnetization\nin Mn2RuGa. Mn 2RuGa is a ferrimagnetic Heusler com-\npound, with a cubic structure. Two Mn atoms are not\nequivalent, and have different spin moments. They are\nantiferromagnetically coupled. As shown before [17], fer-\nrimagnets have a big advantage over ferromagnets and\nantiferromagnets. This offers an ideal theoretical model.\nIn this paper, we investigate all-optical switching in\nMn2RuGa. Different from prior studies, we compute the\nspin moments at two Mn sites using the first-principles\ndensity functional theory. These spin moments are fed\ninto the Heisenberg exchange model with both spin-orbit\ncoupling and a harmonic potential [18, 19]. We find that\nnot any arbitrary laser field amplitude can switch spins.\nThere is a narrowwindow of opportunity where the spins\nat two Mn sites can be switched into their respective op-\nposite directions. Because of the strong spin moments\nat two Mn sites, its switching is very stable. Quite dif-\nferent from other systems, we find that if we reduce the\nexchange interaction, the spins precess strongly at both\nMn sites, very much like a regular antiferromagnet, in-2\nstead of a ferrimagnet. These strong spin oscillation oc-\ncur at both Mn 1and Mn 2sites. Their oscillation period\nis inverselyproportionalto the laser field amplitude, sim-\nilar to the Rabi frequency in a two-level system. For a\nweak exchange interaction case, regardless of the magni-\ntude ofthe laserfield amplitude, the spin switchingis not\nobserved. This points out an entirely different scenario\nfrom ferromagnetic cases [19] and demagnetization [20].\nThe rich picture that is found here revealsa crucial effect\nof the effect of exchange interaction on AOS, and should\nmotivate further experimental and theoretical investiga-\ntions in the future.\nMn2RuGa is a Heusler ferrimagnet, with a stoichio-\nmetric composition of X2YZand space group F¯43m.\nTwo Mn atoms, Mn 1and Mn 2, are situated at (4 a)\nand (4c), which are magnetically inequivalent [21, 22].\nTheir spins are antiferromagnetically coupled. To prop-\nerlyinvestigatemagneticpropertiesofMn 2RuGa, weem-\nploythe density functional theoryusing the full-potentialaugmented plane wave method as implemented in the\nWien2k code [23, 24]. Our first-principles calculation\nshows that Mn 1has a spin moment of 3.17232 µBand\nMn2has -2.30765 µB. This agreeswith priorcalculations\n[21, 22]. So each cell has a net spin moment of 1.02394\nµB, close to unity, which is consistent with the nature of\na stoichiometric half-metal [25]. The spin moments on\nRu and Ga are very small, and will be ignored below.\nIt is not often recognized that the large spin mo-\nments on Mn atoms are advantageous since the spin-\norbit torque is proportional to the spin moment [18, 19].\nSinceitisnotpossibletosimulateall-opticalspinreversal\nat the first-principles level, in the following we will feed\nthese two spin moments into our Heisenberg-exchange\ncoupled harmonic model [17, 19, 26, 27], and limit our-\nselves to a small system with 101 lattice sites along the x\naxis and yaxis, respectively, with two monolayers along\nthezaxis. Our Hamiltonian is\nH=/summationdisplay\ni/bracketleftbiggp2\ni\n2m+V(ri)+λLi·Si−eE(r,t)·ri/bracketrightbigg\n−/summationdisplay\nijJexSi·Sj, (1)\nwhere terms from the left to right are respectively the\nkinetic energy operator of the electron, the potential en-\nergyoperator,the spin-orbitcoupling, the interactionbe-\ntween the laser and system, and the exchange interaction\nbetween spins. Our exchange parameter Jexis still time-\nindependent, although prior studies have shown that the\nexchange interaction itself could be affected by the elec-\ntricfield[28,29] .λisthe spin-orbitcouplingconstant, Li\nandSiare the orbital and spin angular momenta at site\ni, respectively, and pandrare the momentum and posi-\ntion operators of the electron, respectively. We choose a\nspherical harmonic potential V(ri) =1\n2mΩ2r2\niwith sys-\ntem frequency Ω. E(r,t) is the laser field. This model is\nthe only magnetic field-free model currently available to\nsimulate spin reversal, while the commonly used model\nemploys an effective magnetic field [7], which should be\navoided. It represents a small step towards a complete\nmodel.\nIn order to compute the spin change, we solve the\nHeisenberg equation of motion [20] for each spin oper-\nator at every site under laser excitation [17]. Figure 1(a)\nshows the spin zcomponent at two Mn sites as a func-\ntion of time. We employ a laser pulse of 60 fs, with a\nfield amplitude of 0.017 V /˚A. We see that the spin at\nthe Mn 1site starts from the positive zaxis (see the cir-\ncles). Upon laser excitation, it switches over the negative\nzaxis, while the spin at the Mn 2site switches up from\nits−zdirection. This is consistent with the experimen-\ntal observation [16]. The strong spin moment stabilizes\nthe entire switching process. However, not any arbitrarylaser field amplitude can lead to faithful switching. Fig-\nure 1(b) shows how the final spin changes with the laser\nfield amplitude. The dependence is highly nonlinear. If\nwe use a weak laser pulse, there is little change in spins\nat both sites. But if the laser field is too strong, the\nspins overturn toward the xyplane, so there is no spin\nreversal either. We find that the optimal field amplitude\nis 0.017 V /˚A, whose result is shown in Fig. 1(a). In this\nregard, Mn 2RuGa is pretty much similar to other ferri-\nmagnets where there is an optimal amplitude [17, 19].\nMicroscopically, the real situation is more complicated.\nTo this end, there is no generic understanding of spin\nswitching in both ferromagnets [14, 15] and ferrimagnets\n[4]. It has been often argued that the angular momen-\ntum exchange between two spin sublattices in ferrimag-\nnetic GdFeCo [30] is the key to AOS. Theoretically, this\nmomentum exchange picture is interesting, but such mo-\nmentum exchange between sublattices, if it exists, occurs\nall the time through the exchange interaction, with or\nwithout the laser. In other words, it must be something\nextra due to the laser that switches the spin. In GdFeCo,\narigoroustesting isdifficult because itis amorphous,and\nit is difficult to tell whether a model system really repre-\nsents a true GdFeCo sample. This brings ambiguity to a\ntheoretical simulation. Mn 2RuGa removes this ambigu-\nity completely.\nAs a first test, we investigate the effect of the exchange\ninteraction on AOS. We reduce the exchange interaction\nfrom 0.1 eV to 0.001 eV. From prior studies, we know\nsuch a reduction does not constitute a major issue for3\n0 0.01 0.02 0.03 0.04\nA0(V/Å)−2−1012Sz(h− )−400 −200 0 200 400 600Time (fs)\n−2−1012Sz(h− ) Mn1\nMn2(a)\n(b)\nFIG. 1: (a) The zcomponent ofthe spins at the Mn 1and Mn 2\nsites as a function of time at the optimal laser field amplitud e.\nHere, thelaser amplitudeis0.017 V /˚A, andthepulseduration\nis 60 fs. The empty circles denote the spin at the Mn 1site,\nwhile the boxes refer to the spin at the Mn 2site. We see there\nis a clear spin reversal upon laser excitation. (b) Dependen ce\nof theSzas a function of the laser field amplitude A0. The\nempty circles refer to the spin at Mn 1site, and the boxes refer\nto the spin at Mn 2site. A weak laser field does not reverse\nthe spins, but a too strong laser field can not either. There is\na narrow window that one can switch spins.\n0 0.010.020.030.04\nA0(V/Å)4006008001000Period (fs)\n0 0.010.020.030.04\nA0(V/Å)−1012Sz(h− )−500 0 500 1000 1500 2000Time (fs)\n−2−1012S(h− )Sx\nSy\nSz(a)\n(b) (c)\nFIG. 2: (a) Spin precession at Mn 1site under a reduced\nexchange interaction. Here we choose J= 0.001 eV. The rest\nof parameters are the same as those in Fig. 1. The solid,\ndotted, and dashed lines denote the x,y, andzcomponents\nof the spin, respectively. (b) The spin oscillation period d e-\ncreases with laser field amplitude A0. (c) At any of the laser\nfield amplitudes, spin reversal is not found. Only a strong\noscillation is noticed. The solid line is the time-average o f the\nspin, and the dotted and dashed lines refer to the maximum\nand minimum spin values, respectively. The effect of the ex-\nchange interaction on spin reversal is much more pronounced\nin Mn 2RuGa than in other materials.demagnetization in a system with a small spin moment\n[20]. Figure 2(a) shows the spin change at Mn 1(the sit-\nuation is similar at Mn 2), where the solid, dotted, and\ndashed lines denote the x,y, andzcomponents, respec-\ntively. Both the laser duration and amplitude are exactly\nthe same as those in Fig. 1. We see that there is a strong\noscillation in all these three components. We note in\npassing that these three components must obey the op-\nerator permutation [8], where they can not be considered\na linear reversal [31]. In principle, we need to cut off the\nsimulation around 1 ps, after which we need to introduce\ndamping, but to demonstrate the high accuracy of our\ncalculation, we do not use the damping. These strong\noscillations resemble a pure antiferromagnetic case. The\nspins at two neighboring sites are out of phase and re-\nmain antiferromagnetically coupled, even upon laser ex-\ncitation. The laser pulse essentially initiates the spin dy-\nnamics, and the exchange interaction takes over, without\nswitching the spins. For this reason, the angular momen-\ntum exchange picture for AOS can not explain this even\nin the same ferrimagnet. The period of the oscillation\nis not determined by the exchange interaction and spin\nmoment alone. Figure 2(b) shows that as we increase the\nlaser field amplitude, the period becomes shorter. The\nsmall fluctuation at the largest amplitudes is due to the\nperiodsamplingbecausetheoscillationisnotstrictlyhar-\nmonic. This laser-field dependence of the oscillation pe-\nriod is very similar to the Rabi period dependence. For\nall the field amplitudes that we investigate, we do not\nsee a case where the spins are reversed. Figure 2(c) illus-\ntrates the average (solid line), maximum (dotted line),\nand minimum (dashed line) of the final spin. We see the\naverage spin never becomes negative (the initial spin is\nalong the + zaxis). The maximum and minimum values\nshow the limits of spin. Our results point out an impor-\ntant fact: In a ferrimagnet, the effect of the exchange\ninteraction is far more complicated than thought.\nNow, we have two cases: One shows AOS, and the\nother does not. We can directly check whether the prior\ncriteria proposed by Mentink et al.[30] apply to them.\nTheir argument is based on a two-spin system, so for the\npure exchange interaction, the spins at two sublattices\nmust obey the scalar form of spins, ∂S1/∂t=−∂S2/∂t,\nwith the extra term from demagnetization. In our sys-\ntem, each spin is coupled with more than four neighbor-\ningspins, sowetaketwoneighboringspinsasanexample.\nFor the above nonswitchable case (Fig. 2), we find that\nS1x+S2xandS1y+S2yare not constant, so they do not\nobey∂S1/∂t=−∂S2/∂t. OurS1x+S2xdecreasesfrom 0\nto about −1¯hwith oscillations, while S1y+S2yincreases\nfrom 0 to about +1¯ hat the same rate. For our switch-\nable case (Fig. 1), ∂S1/∂t=−∂S2/∂tis not fulfilled\neither. Instead, we find that our result obeys the vector\nformS1(t)/|S1(0)|=−S2(t)/|S2(0)|. This shows that\nthe simple argument based on a two-spin model is not\napplicable to our realistic case. We plan to investigate\nthis issue further in a much larger system.\nIn conclusion, we have carried out a joint first-4\nprinciples density functional theory and model simula-\ntion of all-optical spin reversal in Mn 2RuGa. We are\nable to find a case that the spins can be switched with-\nout employing a magnetic field. The system also shows\nan optimal laser electric field amplitude, with the same\nprofile like those in other systems. The spins at Mn 1site\nare switched from the + zaxis to−zaxis, while those at\nMn2site are switched from the −zaxis to + zaxis. This\nisfullyconsistentwiththeexperimentalfindings[16]. We\nfind that the exchange interaction has a significant effect\non the switching. When we reduce the exchange to 0.001\neV, we find the system becomes unswitchable. It behaves\nlike a regular antiferromagnet. Our finding is expected\nto motivate further theoretical and experimental investi-\ngations in the future.\nAcknowledgments\nThis work was solely supported by the U.S. De-\npartment of Energy under Contract No. DE-FG02-06ER46304. Part of the work was done on Indiana\nState University’s Quantum Cluster and High Perfor-\nmance computers. This research used resources of the\nNational Energy Research Scientific Computing Center,\nwhich is supported by the Office of Science of the U.S.\nDepartment of Energy under Contract No. DE-AC02-\n05CH11231. Our calculations also used resources of the\nArgonne Leadership Computing Facility at Argonne Na-\ntional Laboratory, which is supported by the Office of\nScience of the U.S. Department of Energy under Con-\ntract No. DE-AC02-06CH11357.\nAvailability of data. 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Lett. 103, 117201 (2009)."    },    {        "title": "0706.0749v1.Peculiar_Ferrimagnetism_Associated_with_Charge_Order_in_Layered_Perovskite_GdBaMn2O5.pdf",        "content": "arXiv:0706.0749v1  [cond-mat.mtrl-sci]  6 Jun 2007Peculiar Ferrimagnetism Associated with Charge Order in La yered Perovskite\nGdBaMn 2O5.0\nA. A. Taskin and Yoichi Ando\nCentral Research Institute of Electric Power Industry, Kom ae, Tokyo 201-8511, Japan\nThe magnetic properties of GdBaMn 2O5.0, which exhibits charge ordering, are studied from 2 to\n400 K using single crystals. In a small magnetic field applied along the easy axis, the magnetization\nMshows a temperature-induced reversal which is sometimes fo und in ferrimagnets. In a large\nmagnetic field, on the other hand, a sharp change in the slope o fM(T) coming from an unusual\nturnabout of the magnetization of the Mn sublattices is obse rved. Those observations are essentially\nexplained by a molecular field theory which highlights the ro le of delicate magnetic interactions\nbetween Gd3+ions and the antiferromagnetically coupled Mn2+/Mn3+sublattices.\nPACS numbers: 75.30.Cr, 75.47.Lx, 75.50.Gg, 75.30.Et\nFerrimagnetism is a complex but intriguing type of\nmagnetic ordering. It occurs when antiferromagnetically\naligned spins have different local moments, resulting in\ntheir incomplete cancellation. The characteristics of this\ntype of orderingstems from the fact that it combines fea-\ntures of both ferromagnetic (FM) and antiferromagnetic\n(AF) systems. Moreover, if more than two spin sublat-\ntices are involved, a new level of complexity, and hence\nnew physics, can emerge [1]. So far, only a few families of\nferrimagnetics with three magnetic sublattices — spinel\nferrites and rare-earthgarnets being the most famous ex-\namples [2] — have been discovered. These materials are\nproved to be not only fundamentally interesting, but also\ntechnologicallyimportant[3], inspiringthesearchfornew\npromising magnetic compounds.\nRecently, half-doped A-site ordered manganite per-\novskitesRBaMn 2O5+x(whereRis a rare-earth element)\nhave been synthesized in an effort to clarify the role of\nthe random potential effect in the colossal magnetoresis-\ntance (CMR) phenomena[4, 5, 6, 7, 8]. It has been found\nthat in addition to the interesting electronic properties,\nthese compounds possessanother potentially useful qual-\nity, a variability of the oxygen content [4, 5, 6, 9]. By\nvarying the oxygen concentration from x=0 tox=1, one\ncan readily change the valence state of Mn ions from\n2+ through 3+ to 4+, generating a variety of possi-\nble magnetic states. In particular, Mn ions adopt two\nvalence states, Mn2+and Mn3+, and develop a charge\norder in the reduced composition RBaMn 2O5.0(x=0)\n[4, 5]. Since most rare-earth ions ( R3+) are also mag-\nnetic,RBaMn 2O5.0can be a three-sublattice magnetic\nsystem with potentially ferrimagnetic type of ordering;\namong the rare-earths, Gd with the largest spin and\nzero orbital angular momentum would make a bench-\nmark compound of this family. It is worth noting that\nthe layered crystal structure of these compounds natu-\nrallyimpliesananisotropyinitsmagneticpropertiesand,\ntherefore, single crystals would have a great advantage\nfor studying their magnetic behavior; however, previous\nstudieson RBaMn 2O5.0[4, 5, 6] onlyused polycrystallinesamples.\nIn this Letter, we present the first study of the mag-\nnetic properties of GdBaMn 2O5.0single crystals, which\nshow unusual behavior: At TN=144 K, a long-range or-\nder of Mn2+and Mn3+magnetic moments is established.\nUpon decreasing temperature in a small magnetic field\nalong the caxis, a magnetization reversal occurs at the\ncompensation point Tcomp, below which the net magnetic\nmoment is opposite to the magnetic-field direction. On\nthe other hand, in a large magnetic field the magnetiza-\ntionMis positive at all temperature, and shows a sharp\nkink in the M(T) curve at Tcomp. We show that the\nobserved magnetic behavior is essentially understood if\nGd spins, which remain paramagnetic through TNand\nTcomp, are weakly coupled ferromagnetically (antiferro-\nmagnetically) with Mn3+(Mn2+) neighbors and gradu-\nally align their spins parallel (antiparallel) to the Mn3+\n(Mn2+) sublattice with decreasing Tin low fields. What\nis special here is that in high magnetic fields, due to the\nweak Gd-Mn coupling, the Gd spins are aligned along\nthe external magnetic field, which eliminates the magne-\ntization reversal; intriguingly, in this situation an abrupt\nturnabout of the magnetization of the Mn sublattices oc-\ncurs atTcomp, which, to our knowledge, has never been\nobserved in any ferrimagnets. Hence, the peculiar ferri-\nmagnetism in GdBaMn 2O5.0is quite distinct from other\ntransition-metal oxides showing magnetization reversals\n[1, 2, 10, 11].\nThehigh-qualitysinglecrystalsofGdBaMn 2O5+xused\nfor this study were grown by the floating-zone technique.\nThe as-grown crystals were annealed in flowing argon-\nhydrogen mixture at 600◦C for several days to obtain\nthe stoichiometry of x= 0, which is confirmed by the\nthermogravimetric analysis. Parallelepiped samples were\ncut and polished with all faces adjusted to the crystal-\nlographic planes to within 1◦. Magnetization measure-\nments were carried out using a SQUID magnetometer at\nfields up to 70 kOe applied parallel or perpendicular to\nthecaxis.\nFigure 1 shows the temperature dependences of the2\n0 50 100 150 -0.4 -0.2 0.0 0.2 0.4 0.6 \nH = 100 Oe  H || the c axis, cooling \n H || the c axis, heating \n H    the c axis \nTcomp TNM ( µB / f.u.) \nT (K) \nFIG. 1: (Color online) Temperature dependences of Munder\ndifferent conditions. Upon cooling in H= 100 Oe applied\nalong the caxis (circles), M(T) becomes negative at Tcomp\nand eventually returns to positive at lower temperature; up on\nheating, nearly a “mirror image” is observed after applying a\nhigh magnetic field ( H= 3 kOe at T= 35 K for the curve\nshown by triangles). The squares show M(T) measured in\nH= 100 Oe applied along the abplane for both cooling and\nheating.\nmagnetization of GdBaMn 2O5.0in a magnetic field of\n100 Oe applied parallel or perpendicular to the spin easy\naxis. Upon cooling in a magnetic field along the caxis,\nM(T) (shown by circles) rapidlyincreasesbelow the N´ eel\ntemperature, TN=144 K, indicating the onset of a mag-\nnetic ordering. About 10 K below TN,M(T) reaches a\nmaximum, then starts to decrease and eventually goes to\nzero atTcomp=95 K. This temperature, which is called\nthe compensation point, marks the state where mag-\nnetic contributions of all sublattices cancel each other.\nBelowTcomp,M(T) becomes negative, indicating that\nthe net magnetic moment is opposite to the external\nmagnetic-field direction. This phenomenon is called the\ntemperature-induced magnetization reversal [1, 2, 10, 11].\nUpon further decrease in temperature and concomi-\ntant development of sublattice magnetizations, the net\nmagnetic moment grows too large to remain in this\nmetastable state. Eventually a magnetic field (even as\nsmall as 100 Oe) can overcome the coercive force, lead-\ning to a rotation of the net magnetization. As a conse-\nquence, another sign change of M(T) occurs at around\n20–30 K. The effect of this rotation of the net magnetic\nmoment can be illustrated even more clearly in the fol-\nlowing experiment: First, the direction of the sublattice\nmagnetizations in the sample cooled down to an inter-\nmediate temperature in 100 Oe are switched by applying\na large magnetic field (for example, H= 3 kOe at T=\n35 K as shown in Fig. 1). Then, the magnetic field is\nreduced down to 100 Oe again and M(T) is measured\nupon heating. As can be seen in Fig. 1, M(T) shows\nalmost a “mirror image” of the magnetization recordedupon cooling, indicating that the 180◦-spin-rotated state\nis kept intact up to TNonce it is formed.\nA comparison of M(T) measured along different crys-\ntallographicaxesrevealsanotherimportantfeatureofthe\nmagnetic behavior in GdBaMn 2O5: The magnetization\nis strongly anisotropic with an easy direction along the\ncaxis (see Fig 1), simplifying significantly a possible ar-\nrangement of Mn3+, Mn2+, Gd3+spins.\nIn order to obtain further insight into the observed\nphenomena, the field dependence of the magnetization\nhas been measured at 2 K where the magnetizations of\nall sublattices are expected to be fully developed. As\nshown in Fig. 2(a), a magnetic field of about 1 kOe ap-\nplied along the caxis is enough to reach the saturation.\nA much higher magnetic field (about 50 kOe) must be\napplied perpendicular to the easy axis to reach the same\nsaturation level. The equality of the saturation mag-\nnetization Msat≈5.85µB/f.u. achieved along different\ncrystallographicaxes suggeststhat the obtained value re-\nflects the net saturated moment, which should be simply\ncomposed of the sublattice magnetizations.\nObviously, there is a unique combination of spin-\nonly moments of the three spin sublattices — Mn3+\n(SMn3+=2), Mn2+(SMn2+=5/2),andGd3+(SGd3+=7/2)\n— that can be consistent with the experimentally ob-\nserved value of the saturated magnetization:\ngµB(SMn3+−SMn2++SGd3+) = 6µB,(1)\nwhereg=2is the Land´ e g-factorand µBis the Bohrmag-\nneton. This equation suggests an AF coupling between\nMn2+andMn3+andaFM (AF) interactionofGd3+ions\n-60 -40 -20 0 20 40 60 -6 -4 -2 0246-60 -40 -20 0 20 40 60 \nT = 2 K                          \n (a)   M ( µB / f.u.)  H || the c axis \n H    the c axis \n Gd 3+  (PM) \nH (kOe) (b) \n   \nFIG. 2: (Color online) (a) M(H) curves of GdBaMn 2O5.0\nwithHalong the caxis (circles) and the abplane (squares).\nThe calculated paramagnetic response from the Gd3+sublat-\ntice is shown by the dashed line. (b) A sketch of the crys-\ntal and magnetic structure of GdBaMn 2O5.0, showing three\nmagnetic sublattices composed of Mn2+, Mn3+, and Gd3+.\nNonmagnetic Ba and O are omitted for the sake of clarity.3\nwith the Mn3+(Mn2+) sublattice. Note that the param-\nagnetic (PM) contribution of Gd3+alone [shown by the\ndashed line in Fig 2(a)] would give a larger saturated\nmagnetization than observed. The spin-only magnetic\nmoments of Mn3+, Mn2+, and Gd3+ions are expected\nto be a good approximation in GdBaMn 2O5.0, because\nthe high-spin(HS) Gd3+(4f7) and Mn2+(3d5) havezero\norbital angular momentum. For HS-Mn3+(3d4), a state\nof theegsymmetry is unoccupied, also implying a lack\nof the orbital contribution to the total magnetic moment\nof Mn3+.\nThus, both M(T) andM(H) data point to a ferrimag-\nnetic type of ordering among the three magnetic sub-\nlattices of GdBaMn 2O5.0. Neutron powder diffraction\nstudies [4, 5] as well as density-functional calculations\nof electronic structure [12, 13] have reported a rock-salt\narrangementofMn2+andMn3+withaG-typeAForder-\ning in YBaMn 2O5.0and LaBaMn 2O5.0, compounds with\nrare-earth ions from opposite ends of the lanthanide se-\nries. It would be natural to assume that the same types\nof charge and magnetic ordering among Mn ions are real-\nized in GdBaMn 2O5.0. Magnetic Gd3+ions with a half-\nfilled 4fshell are expected to interact ferromagnetically\nwith Mn3+and antiferromagnetically with Mn2+sublat-\ntices according to the Goodenough-Kanamori rules [14],\nalthough this interaction should be rather weak.\nThe magnetic structure most likely realized in\nGdBaMn 2O5.0is shown in Fig. 2(b). Below TN, Mn2+\n(S=5/2) and Mn3+(S=2) ions develop a long-range fer-\nrimagneticorderwith thespin directionsalongthe caxis.\nBecause of the weak magnetic interaction between Gd3+\nand Mn2+/Mn3+, the alignment of Gd spins ( S=7/2)\ngrows rather slowly with the development of the magne-\ntization of the Mn sublattices below TN, but eventually\nalmost all Gd spins are aligned along the Mn3+spins at\nlow enough Tand add to the large Msat. As can be seen\nin Fig 2(a), the magnetization vectorsof the orderedsub-\nlattices can be 90◦rotated by applying a magnetic field\nof∼50 kOe perpendicular to the easy axis.\nAnother piece of information about the magnetic be-\nhavior of GdBaMn 2O5.0comes from the susceptibility\nmeasurements in the paramagneticphase above TN. Fig-\nure 3 shows the inverse molar susceptibility, measured in\nH=100Oeparallel(circles)andperpendicular(squares)\nto thecaxis. Both curves coincide at high temperature,\ndemonstrating the isotropic nature of the paramagnetic\nphase. In the molecular field theory [1], the tempera-\nture dependence of the inverse susceptibility of a three-\nsublattice ferrimagnetic material above TNis given by\n1\nχ=1\nχ0+T\nC−σ(T)\nT−TN, (2)\nwithCthe effective Curie constant, TNthe N´ eel temper-\nature,and σ(T) = (σ0T+m)/(T+θ);χ0,σ0,m,andθare\nall constants. The slope of the high-temperature inverse\nsusceptibility shown by the straight dash-dotted line in0 100 200 300 400 -5 0510 15 20 25 30 35 40 \nTcomp \n  H || the c axis \n  H    the c axis TN\n 1 / χm  (mol / emu) \nT (K) \nFIG. 3: (Color online) Inverse molar susceptibility 1 /χm,\nmeasured in H= 100 Oe parallel (circles) and perpendicular\n(squares) to the caxis, is in agreement with a ferrimagnetic\ntype of ordering. The dashed line is a calculation within the\nmolecular field theory (see text). The dash-dotted line is a\nguide to the eye.\nFig. 3 is solely given by the effective Curie constant C,\nwhich is the sum of the Curie constants of the three mag-\nnetic sublattices. The obtained value of15.1emuK/mole\nis only 1% less than what is expected for Mn3+(S=2),\nMn2+(S=5/2), and Gd3+(S=7/2) [15], giving further\nsupport to the proposed magnetic structure.\nWhen one assumes that only Mn sublattices develop a\nlong-range magnetic order at TN, the exchange coupling\nconstant Jcanbeestimated, tofirstapproximation,from\nthe following equation [16]:\nTN=q(2+α+β)CMn3+CMn2+\nCMn3++CMn2+, (3)\nwhereCMn3+andCMn2+are the Curie constants for\nMn3+and Mn2+. Here, q= (zJ)/(Nag2µ2\nB) is the\nmolecular field constant related to the Mn3+-Mn2+ex-\nchange interaction, where z= 6 is the number of nearest\nMn2+neighbors of Mn3+(and vice-versa), Nais Avo-\ngadro number, and the coefficients αandβgive the\nstrengths of the nearest-neighbor exchange interactions\nin each sublattice of Mn3+and Mn2+, respectively. Note\nthat all constants in Eq. (2) are determined entirely by\ntheseexchangeinteractionparametersandtheCuriecon-\nstants. The fitting curve shown by the dashed line in\nFig. 3 is calculated with J/kB= 30.5 K, α=−0.63, and\nβ=−0.69, and it obviously reproduces the data very\nwell, supporting the assumption that Gd spins remain\nessentially paramagnetic across TN.\nWith the obtained exchange interaction parameters,\nthe molecular field theory [1, 16] can be applied to the\nanalysis of the temperature dependence of Min the or-\ndered state below TN. The temperature-induced mag-\nnetization reversal observed at low magnetic fields can\nbe reproduced [as shown by the solid line in Fig. 4(a)]4\n50 100 150 -0.4 -0.3 -0.2 -0.1 0.0 0.1 TN Tcomp (a) \n M ( µB / f.u.) \nT (K) 0 100 200 300 400 0123456 \nTcomp \nTN(b) \n M ( µB / f.u.) \nT (K) 0 50 100 150 -4 -2 0246 \nMn 2+ \nMn 3+ Gd 3+ \n M ( µB / f.u.) \nT (K) \n50 100 150 -4 -2 024 \nMn 2+ \nMn 3+ Gd 3+ \n M ( µB / f.u.) \nT (K) \nFIG. 4: (Color online) Temperature dependence of the easy-\naxis magnetization in (a) H= 10 Oe and (b) H= 70 kOe.\nSolid lines are calculations within the molecular field theo ry\nwith the same parameters as for Fig. 3 (see text). Insets\nshow the contributions of individual sublattices to the ove rall\nmagnetization.\nonly if there is a weak FM (AF) coupling between Gd3+\nions and the Mn3+(Mn2+) sublattice. Furthermore, the\nvalue of the compensation temperature is very sensitive\nto the strength of this interaction. For Tcomp= 95 K,\nthe molecular field constant related to the Gd3+-Mn3+\nFM exchange is found to be about 0.01 q, i.e., two orders\nof magnitude smaller than the AF exchange interaction\nbetween Mn3+and Mn2+. This value corresponds to an\neffective magnetic field of approximately 35 kOe, while\nfor the ferrimagnetic spin order in the Mn3+/Mn2+sub-\nlattices it is ∼5000 kOe.\nThe large difference in the exchange interactions\nbetween Mn3+-Mn2+and Gd-Mn is crucial for the\nunderstanding of the observed magnetic behavior in\nGdBaMn 2O5.0. Just below TN, Mn3+spins, being an-\ntiferromagnetically coupled to the Mn2+sublattice, are\nopposite to the applied magnetic field, and at low fields\nGd spins tend to align in the same direction as the Mn3+\nsublattice [see inset of Fig. 4(a)], though their alignment\nis weak; below Tcomp, the sum of the magnetic moments\nof Gd3+and Mn3+overgrows the magnetic moment of\nMn2+, giving rise to a negative magnetization. Intrigu-\ningly,thesituationchangescompletelyinahighmagnetic\nfield which is strong enough to compete with the Gd-Mn\nexchange interaction: In this case, Gd spins tend to align\nalongtheexternalmagnetic-fielddirection,causingapos-\nitive magnetization at all temperature as shown in Fig.\n4(b), and the magnetization reversal is eliminated. It is\nnoteworthythat the Mn spins also behave differently and\nshowsanovelturnabout; namely, justbelow TNtheirori-\nentation is the same asin the lowmagnetic field case, but\nwith decreasing temperature this state becomes energet-\nically unfavorable as the influence of Gd spins grows. As\na result, near the compensation point, an abrupt turn-\nabout of the magnetization of the Mn sublattices takesplace [see inset of Fig. 4(b)], which is manifested in the\nsharp change of the slope of M(T). Note that in the\npresent case the strong Ising anisotropy probably plays a\nkey role in the abrupt turnabout.\nTo conclude, the present study shows that un-\nusual magnetic behavior of a new three-sublattice fer-\nrimagnetic manganite GdBaMn 2O5.0— a temperature-\ninduced magnetization reversal at low magnetic fields\nand a novel turnabout of Mn sublattice magnetizations\nat high magnetic fields — are governed by an elabo-\nrate interplay between its magnetic sublattices that are\nformed by the A-site ordering as well as the charge or-\ndering of Mn into Mn2+and Mn3+. In the present case,\nthe weak coupling of Gd spins with magnetically or-\ndered Mn2+/Mn3+sublattices is the source of novel fer-\nrimagnetism not found elsewhere before. This result not\nonly enriches our knowledge about ferrimagnetics, but\nalso points to the potential of charge-order-susceptible\ntransition-metal oxides for discovering physically inter-\nesting and technologically useful properties.\nWe thank I. Tsukada for helpful discussions.\n[1] L. N´ eel, Ann. Phys. (Paris) 3, 137 (1948); Science 174,\n985 (1971).\n[2] K. P. Belov, Phys. Usp. 39, 623 (1996); ibid.42, 711\n(1999);ibid.43, 407 (2000).\n[3] M. Pardavi-Horvath, JMMM 215-216 , 171 (2000).\n[4] F. Millange, E. Suard, V. Caignaert, and B. Raveau\nMater. Res. Bull. 34, 1 (1999).\n[5] F. Millange, V. Caignaert, B. Domeng` es, B. Raveau, and\nE. Suard, Chem. Mater. 10, 1974 (1998).\n[6] S.V.Trukhanov,I.O.Troyanchuk,M.Hervieu, H.Szym-\nczak, and K. B¨ arner, Phys. Rev. B 66, 184424 (2002).\n[7] D. Akahoshi, M. Uchida, Y. Tomioka, T. Arima, Y. Mat-\nsui, and Y. Tokura, Phys. Rev. Lett. 90, 177203 (2003).\n[8] Y. Kawasaki, T. Minami, Y. Kishimoto, T. Ohno, K.\nZenmyo, H. Kubo, T. Nakajima, and Y. Ueda, Phys.\nRev. Lett. 96, 037202 (2006).\n[9] A. A. Taskin, A. N. Lavrov, and Y. Ando, Appl. Phys.\nLett.86, 091910 (2005).\n[10] Y. Ren, T. T. M. Palstra, D. I. Khomskii, E. Pellegrin,\nA. A. Nugroho, A. A. Menovsky, and G. A. Sawatzky,\nNature396, 441 (1998).\n[11] J. Hemberger, S. Lobina, H.-A. Krug von Nidda, N. Tris-\ntan, V. Yu. Ivanov, A. A. Mukhin, A. M. Balbashov, and\nA. Loidl Phys. Rev. B 70, 024414 (2004).\n[12] R. Vidya, P. Ravindran, A. Kjekshus, and H. Fjellv˚ ag,\nPhys. Rev. B 65, 144422 (2002).\n[13] R. Vidya, P. Ravindran, P. Vajeeston, A. Kjekshus, and\nH. Fjellv˚ ag, Phys. Rev. B 69, 092405 (2004).\n[14] H. Weihe and H. U. G¨ udel, Inorg. Chem 36, 3632 (1997).\n[15] The Curie constant Ciof an individual sublattice is\nCi=Na(gµB)2Si(Si+ 1)/3kB, where Siis the spin of\nthe magnetic ion i.\n[16] See, for example, R. Kubo, H. Ichimura, T. Usui, N.\nHashitsume, Statistical Mechanics: An Advanced Course\nwith Problems and Solutions (North-Holland, Amster-5\ndam, 1965)."    },    {        "title": "2212.02807v1.Dynamics_of_hybrid_magnetic_skyrmion_driven_by_spin_orbit_torque_in_ferrimagnets.pdf",        "content": " \n \nDynamics  of hybrid magnetic skyrmion driven by spin-orbit torque  in ferrimagnets  \nY . Liu1, T. T. Liu1, Z. P. Hou1, D. Y . Chen1, Z. Fan1, M. Zeng1, X. B. Lu1, X. S. Gao1,  \nM. H. Qin1,*, and J. –M. Liu1,2 \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, Nanjing University, Nanjing 210093, China  \n \n[Abstract]  Magnetic s kyrmion s are magnetic texture s with topolog ical protection , which  are \nexpected to be information carriers in future spintronic devices. In this work,  we propose a \nscheme to implement hybrid magnetic skyrmion s (HMS)  in ferrimagne ts, and we study \ntheoretically and numerically the dynamics of  the HMS  driven  by spin-orbit torque . It is \nrevealed  that the  skyrmion  Hall effect  depends on  the skyrmion  helicity and the net angular \nmomentum ( δs), allowing the effective modulation of  the HMS  motion  through  tuning  \nDzyaloshinskii -Moriya interaction  and δs. Thus , the H all effect can be suppressed through \nselecting suitable materials to better control the HMS motion . Moreover,  Magnus force for \nfinite δs suppresses the transverse motion and enhances the longitudinal propagation, resulting \nin the HMS dynamics  in ferrimagnets  faster than that in antiferromagnets.  \n \nKeywords: ferrimagne ts, hybrid  magnetic  skyrmion, spin-orbit torque , spintronics  \n  \n \n*Email: qinmh@scnu.edu.cn  I. Introduction   \nMagnetic skyrmion s are localized  spin texture s with a topological number  [1–5], which  \nhave been observed in a series of chiral magnets and heavy metal/ferromagnetic films where \nDzyaloshinskii -Moriya interaction (DMI)  due to the broken inversion symmetry plays an \nimportant role in stabilizing the skyr mion s. Specifically, the bulk DMI in chiral magnets \nstabilize s Bloch skyrmion s (typical spin configuration is shown in Fig. 1(a))  [6], and the \ninterfacial  DMI in films favors the formation of  Néel  skyrmion s (Fig. 1(b) ) [7]. Importantly, \nskyrmions are relatively steady  with small s ize owing to the topological protection  [8–12], and \nthey can be driven by electric current with a rather low density, making them exciting candidates \nfor high -density and low -power -consuming information storage  devices . \nSpintronic  devices based on skyrmion require precise modulation of the skyrmion s, while \nthe skyrmion Hall effect remains a key challenge  [13–16]. In detail , when skyrmion is driven \nby spin -polarized current, it suffers a transverse Magnus force and deviates from the current \ndirection. As a result, the skyrmion can be driven out of the track  under a high current , resulting \nin a loss of information.  To overcome this deficiency , a number of so lutions including the use \nof antiferromagnets  [7], boundary effect  [17], and  other similar spin structures  [18] have been \nproposed.   \nMost recently, hybrid magnetic skyrmion  (HMS)  in ferromagnets with a structure \ninterpolating between Néel and Bloch skyrmions  stabilized by the hybrid DMI  (a mixture of \ninterfacial and bulk DMIs) or by coupling with vortex structure s has been  reported , which is  \nrevealed  to has an enhanced mobility and a reduced skyrmion Hall effect  [19–22]. Interestingly, \nthe HMS  dynamics driven by spin -orbit torque  (SOT)  significantly depends on the skyrmion \nhelicity, providing another parameter in modulating the Hall effect. For example, one can tune  \nthe ratio of the interfacial and bulk DMIs through various methods such as applying  \nvoltage  [23], electric field  [24] and strain  [25,26] , and in turn modulates the skyrmion helicity \nand dynamics. As a matter of fact, the skyrmion Hall effect induced by SOT  in ferromagnets \ncan be eliminated through delicate tuning the skyrmion helicity , while the strong stray field and \nrelatively slow spin dynamics are still disadvantages  in applications .   In antiferromagnets,  on the other hand, the Magnus forces of two -sublattices induced by \nspin-orbit /spin -transfer torque on Néel  or B loch skyrmion are well canceled out, and the \nskyrmion moves along the cu rrent direction  [27,28] . Thus, besides zero stray field and fast spin \ndynamics, antiferromagnets also provide an important platform  for achieving skyrmion long -\nrange transmission . However, effectively detection and modulation of antiferromagnetic \nskyrmion s are still challenging in practice due to zero net magnetization  of the system . \nMoreover,  in antiferromagnets, a strong  Hall motion  of HMS is also revealed when it is driven \nby SOT, which is different from the case of spin -transfer torque driven HMS motion with zero \nHall angle  [29]. This phenomenon attributes to the fact  that the effective SOT  depending on the \nhelicity of t he HMS generally deviate s from the current direction , resulting in the HMS  Hall \neffect .  \nAlternatively , ferrimagnets are suggested to combine the benefits of ultra -high working \nfrequencies with ease of detecting and modulating  [30–32], noting that ferrimagnet has a \nnonzero magnetic moment and ultrafast spin dynamics comparable to antiferromagnets around \nthe angular momentum compensation temperature ( TA). Importantly , the net angular \nmomentum δs can be adjusted  through tuning temperature and ion doping, which has a \nsignificant effect on skyrmion dynamics  [33–38]. For example, δs can effectively modulate the \nMagnus force induced by SOT and control the skyrmion Hall motion in ferrimag nets [18,32,39] .  \nNaturally , it is expected that the joint action of δs and skyrmion helicity gives rise to \ninteresting skyrmion dynamics  in ferrimagnets , which urgently deserves to be uncovered to \nprovide a clear scenario for skyrmion motion manipulation . On the one hand, this study \nuncovers new dynamic behavior, contributing to the development of spintronics. For example, \nour earlier work has  revealed that δs determines the momentum onto the skyrmion in \nferrimagnets imposed by the injected ma gnons , and consequently affects the Hall angle  [12]. In \nsome extent , δs probably has an important effect on the  SOT-driven HMS dynamics in \nferrimagnets . On the other hand,  the Hall effect of the HMS in ferrimagnets  can be suppressed \nwhen  the Magnus force depending on δs suppresses  the transverse  motion induced  by the SOT.  Thus, the HMS motion with zero Hall effect could  be available for certain  δs, which is \ninstructive for experiment and device design to achieve the straight skyrmion motion .  \n In this work, we study the dynamics  of HMS  driven by SOT in ferrimagnets  using Thiele \ntheory and numerical simulations based on solving the two coupled Landau -Lifshitz -Gilbert \n(LLG)  equations . The dependence of the skyrmion Hall angle on δs and the helicity is \ninvestigated, which exhibits the gradual transition in Hall angle o n δs and the helicity. Thus, \nHall motion of the HMS can be suppressed through choosing suitable δs, allowing one to select \nsuitable materials to better control the skyrmion motion.  Furthermore, the mobility of the HMS \nin ferrimagnets could be enhanced attr ibuting to Magnus force .  \n \n \nII. Theory for SOT -driven  HMS dynamic s in ferrimagnets  \nIn this section, we  study theoretically the HMS dynamics in ferrimagnets  driven by SOT \nthrough deriv ing the equations of motion for a HMS based on Thiele theory . We consider a spin \nmodel  composed  of two  unequal sublattices  which are antiferromagnetically coupled , as shown \nin Fig. 1( d). The unit magnetization vectors of the two sublattices are m1 and m2, respectively.  \nIn the continuum approximation  [40], one introduce s the Néel vector  n = (m1 − m2)/2 and the \nmagnetization vector  m = (m1 + m2)/2 to deal with the dynamic equations of ferrimagnets . \nConsidering the SOT term, t he dynamics of vectors m and n can be described by the following \nequation , \n( )()() 2s m n p ss + =−  +  +  +  m n m f n f n n n m n\n, (1a) \n()()()ms ss  =−  +  +  n n f n n n m\n, (1b) \nwhere s = (s1 + s2) with the angular momentum densit ies of the two sublattices  s1 and s2, δs = \n(s1 − s2), sα = (s1α1 + s2α2) with the damping constants  α1 and α2, and α = (s1α1 + s2α2)/s is the \ndamping coefficient . β = ħjθSH / et is the coefficient of the SOT term, where ħ is the reduced \nPlanck constant, e is the electron charge , and t is the  thickness  of the magnetic  layer . mp is the \npolarization  vector  of the  spin polarized  current , θSH is the spin Hall angle , and j is the current \ndensity.  fn = − δU / δn and fm = −δU / δm denote the effective fields of n and m, respectively , with the free energy density  U. Considering  the fact  that |m| << |n| in slowly evolving system, \nthe dynamic  equation  can be represented by  n after neglecting the weak term s: \n()() 20s n p s    − −  +  +   =n n n n f n n n m n\n, (2) \nwhere ρ = s2/a is the inertia coefficient . \nOwing to the topological protection , the HMS  behaves as a soliton or  a rigid body that \nkeeps  its shape unchanged during the propagation.  For convenience, the HMS  motion  is \nintroduced into the dynamic  equation in the form of collective coordinates R(t), n(r, t) = n(r − \nR(t)). Projecting the dynamic  equation onto the  HMS  translational mode, the following Thiele \nequation  of the HMS dynamics  is obtained  [41]: \n() 20sp M s IR −  −  +  =R G R R m\n, (3) \nwhere M = −ρ∫(∂in ∙ ∂in)dxdy is the effective mass of the HMS,  and G = (0,  0, 4πQ) is the \ngyromagnetic  coupling vector with the topological charge  Q. Here, Q, the viscous coefficient \n, and tensor  I read \n()() 1/ 4 d dxy Q x y =   n n n\n, (4a) \n()\n()d d 0\n0 d dii\njjxy\nxy =  \nnn\nnn\n, (4b) \n()() sin cos dr I r r r r  =  +\n, (4c) \nwhere θ is the polar angle of the Néel vector in the spherical coordinate, and r is the radius in \nthe polar coordinate, as shown i n Fig. 1 (c). The tensor  R(Θ) reads  \n()sin cos\ncos sinR−  =−  − \n, (5) \nwhere the skyrmion helicity Θ = arctan( Db/Di) is determined by the bulk DMI magnitude (Db) \nand the interfacial DMI magnitude (Di), which could  be altered between 0 and 2π. It is worth  \nnoting that the tensor R(Θ) and SOT are directly coupled , allowing one to modulate the SOT \nacting on the HMS through tuning the HMS  helicity.  First, w e focus on  the dynamics of the HMS at TA with δs = 0. In this case, the Magnus  \nforce  denoted by t he second term in  Eq. (3) is elimin ated, and the acceleration\nR\n can be safely \nignored considering  the s teady  motion of the HMS.  When  the current is polarized along the x-\ndirection mp = ex, one obtains the velocity  components of the HMS,  \n2 sinxv I s =− \n, (6a) \n2 cosyv I s =− \n. (6b) \nIt is clearly shown that t he motion of the HMS  depends on  its helicity . Particularly , the \ndependence of  vx/vy on the  DMI coefficient s reads : \nn / / tabi xyv DD v= =\n, (7) \nthe same as that obtained  through solving the LLG equation in the earlier report  [29]. \nSubsequently , we derive the velocit y of the HMS  in uncompensated  ferrimagnets  for a \nfinite δs, which is  given by:  \n( )( )2 2 2 2 28 cos 2 sin 16x s a sv I Q s Q s     =− +  +\n, (8a) \n( )( )2 2 2 2 28 sin 2 cos 16y s a sv I Q s Q s     =− − +  +\n. (8b) \nThus , the dependence of the velocity on both δs and Θ is clearly demonstrated, allowing one to \nmodulate the dynamics through tuning these parameters. Taking  Θ = π/4 as an example, the \nMagnus force ~Qδs for negative δs enhances the longitudinal motion and suppresses the \ntransverse motion, resulting in the decrease of the Hall angle ~vx/vy. Subsequently, the HMS \nspeed  v and Hall angle θH are derived , respectively,   \n2 2 2 2 22\n16sIv\nQs\n=\n+\n, (9a) \n8 sin 2 cosarctan( )8 cos 2 sinsa\nH\nsaQs\nQs− + =+ \n. (9b) \nIt is demonstrated that v linearly depends on the current intensity and hardly be affected by the \nHMS helicity. Importantly , the helicity  and δs effectively modulates the skyrmion Hall angle , \ndemonstrating the important role of the DMI magnitude s and δs in controlling  the HMS dynamics in ferrimagnets . For tanΘ = sα/4πQδs, zero Hall angle is achieved, and the skyrmion \nmoves straightly along the current direction  with a speed of 2βI/(16πQ2δs2 + sα22)1/2. \nInterestingly , sα decreases as δs increases, which may enhanc e the speed  of the HMS . Thus, it \nis suggested theoretically that the Hall motion of the HMS in ferrimagnets can be completely \nsuppressed in accompany of  the speed enhancement by tunin g δs, which definitely favors future \napplications.  \nTo check these predictions, a comparison betw een the theoretical analysis with the \nnumerical simulations is indispensable.  In Sec. III, we introduce numerical simulation s of the \nHeisenberg spin model , and then give the calculated results and discussion . \n \n \nIII. Numerical simulations and discussion  \nIn this section, we first introduce the simulation method based on the standard  Heisenberg  \nmodel through solving the LLG equation, and then give the simulated and calculated results. A \nbrief discussion on potential  application of the HMS is disc ussed at last.  \n \nA. Model and simulation method  \nThe micromagnetic simulations are performed based on the  classical Heisenberg model , \nand t he model Hamiltonian is given by  \n()() ()()2\n,,ˆ ˆAB\nABAB i j A i j B i j\n<i,j> i,j i,j\nAB\ni ij i j b ij i j i\ni j i j iH J J J\nD z +D K z   \n   =  +  + \n+      +   \n  s s s s s s\nu s s u s s s\n, (10) \nwhere si is the normalized spin at lattice site i, JAB is the antiferromagnetic  interlayer interaction  \nbetween sublattice A and sublattice B, JA and JB are the ferromagnetic intra -sublattice coupling  \nfor sublattice A and sublattice B between the nearest neighboring spins , respectively . DA \niand D\nB \nbare the interfac ial DMI and b ulk DMI coefficients  for two sublattices , respectively, uij is the \nunit vector connecting  two spins in  sublattices , and K is the anisotropy constant . The coupling parameters are the same as those in the theoretical analysis. Actually, the \nstabilization of ferrimagnetic  skyrmions have been experimentally reported in GdCo films  [42]. \nHerein, we consider a GdCo/Co heterostructure to produce HMS, where the coupling between \nthe two magnetic layers can be modulated by tuning the thickness of the spacer  [43]. The \ninterfacial DMI in Co laye r is induced by coupling with a heavy metal layer  which generate s \nstrong  spin-orbit coupling s [44], and  SOT can be easily applied by injecting in -plane current \ninto the heavy metal layer  [8,15] . Withou t loss of generality, we set the intra -sublattice \nexchange stiffness AGd-Gd = 5 pJ/m, ACo-Co = 5 pJ/m , the perpendicular magnetic anisotropy \nconstant K = 0.16 MJ/m3, the bilinear surface exchange coefficient  σ = −1 mJ/m2, and t he DMI \ncoefficients Di and Db vary within a reasonable range.  \nThen , the dynamics of the HMS  is investigated by solving the LLG equation,  \n() ,Mi i i\ni i eff i i i i p i\ni tt=−  +  +  sss H s s m s\n, (11) \nwhere Heff,i = Mi−1H/si is the effective field with the magnetic moment  Mi at site i, and the \ngyromagnetic ratio γi = giμB/ħ with the g-factors g1 = 2.2 and  g2 = 2. We set  the spin Hall angle  \nθSH = 0.2 , and the damping constants α1 = α2 = 0.4. \nHere, the micromagnetic simulations are performed using the Object -Oriented \nMicroMagnetic Framework (OOMMF) with extended DMI module. We start from a discrete \nlattice with the size of 100 nm × 100 nm × 9 nm and cell size of 1 nm × 1 nm × 3 nm, and set \nthe time step to 10−13 s. The used magnetic moments Mi for nine different cases are shown in \nTable 1, correspond ing to nine different δs.  \n \nB. Results and discussion  \nIn Fig. 2(a), w e present the simulated and theoretical calculated vx/vy as function s of tan Θ \nat TA. The simulations coincide well with the theory, confirming the validity  of the theory \nanalysis . A linear dependence of vx/vy on tanΘ is observed for  mp = ex, the same as that in \nantiferromagnets  [29]. Similarly, when the polarization  vector  of the injected spin current is \ntuned from ex to ey, the HMS moves along the direction perpendicular to the motion  for mp = ex, and vy/vx linearly increases with tanΘ. Therefore, one can modulate the HMS motion through \ntuning t he helicity and spin polarization.   \nSubsequently, the effect of the helicity on the speed of the HMS v is investigated, and the \ncorresponding results are shown in Fig. 2 (b) where present the simulated v as a function of \ncurrent density j for various Di/Db with a fixed \n22\nib D D D=+ . v linearl y increases with the \nincreasing j due to the enhancement  of SOT , consistent with the earlier report. For a fixed j, the \nHMS moves at a highest speed for Di = Db, and speed down when Di and Db deviate from each \nother.  This phenomenon can be understood from the following aspects.  It is well noted that \nskyrmion size mainly depends on DMI and anisotropy . For two uncoupled magne tic layers,  \nDMIs with a same magnitude stabilize the isolated skyrmion s with a same size , while the sizes \nof the skyrmions are different for DMIs with different magnitudes . Thus, for Di = Db, the \nskyrmions in two sublattices  are well coupled  due to the sam e skyrmion size. However, when \nDi deviates from Db, the interlayer antiferromagnetic coupling competes with the DMI stronger \nthan that for Di = Db, resulting in the increase of the internal energy and the reduction of the \nskyrmion mobility.  \nIn some extent, this phenomenon is analogous to the barrel effect, i.e., the size and speed  \nof the HMS are mainly determined on the smaller one of Di and Db. As a matter of fact, this \nqualitative explanation is confirmed in the calculated I/shown  in the insert of Fig. 2(b) , noting \nthat I/ depends on the HMS structure and determines the dynamics . I/reaches its maximum \nfor Di = Db, and decreases with the deviation between Di and Db. Moreover, for a same deviation \nmagnitude, Di/Db = 1:4 or 4:1 as an example, the HMS have a same speed, as shown in our \nsimulations.  \nSubsequently, the effect of δs on HMS dynamics in uncompensated ferrimagnets is \nnumerically investigated, and the corresponding  results are shown in Fig. 3. Generally, during \nthe propagation of the HMS, Magnus force proportional to δs is generated via the  gyrotropic \ncoupling , while the inertia coefficient ρ is decreased  with the increase of δs. As a result, the \nspeed of the HMS slightly increases as δs increases, as shown in Fig. 3(a) where presents the simulated and calculated v as a function of δs for various j with mp = ex. Furthermore,  the \nsimulations coincide well with the theory analysis for low current density  (j ~ 6 × 1011 A/m2), \nwhile deviate s from the theory for large j. It is noted that a deformation of the HMS may occur \nduring the propagation with a high speed, attributing to the slight discrepancy between \nsimulations and theory.  \nImportantly, a significant effect of δs on the HSM Hall motion is revealed, as shown in Fig . \n3(b) where gives the simulated velocity components vx and vy as functions of δs for various j. \nHere,  without loss of generality, we consider the case of Db = Di, corresponding to  the helicity \nof –3π/4. For δs = 0, vx equals to vy, and the HMS moves along the [1, 1] direction. vy increases \nwith the increase of δs, while vx less depends on δs. Thus, the Hall angle of the HMS could be \nmodulated through tuning δs. On the one hand, the Magnus force along the [-1, 1]/[1, -1] \ndirection is generated during the HMS propagation for positive/negative δs, which is enhanced \nas |δs| increases. Thus, the longitudinal/transverse  motion vx/vy is suppressed/enhanced by \nMagnus force with the increase of δs. On the other hand, the parameter sα also decreases, and \nthe contribution of the dissipative term to vx/vy is enhanc ed. Moreover , the dissipative term \ncompetes with the Magnus force  in determining  vx, as demonstrated in Eq. 8, resulting in a \nrather stable  vx for various δs. However , both the two terms contribute to vy, and vy extensively \nincreases with the increasing δs. As a result, the HMS Hall motion can be suppressed and even \neliminated by choosing suitable δs.  \nSimilar effect of δs on the Hall motion of HMS with other helicity has been revealed, and \nthe results are summarized in Fig. 4 where gives the simulated Hall a ngle in the ( Di/Db, δs) \nparameter plane. It is clearly shown that both the HMS helicity and δs can be used in modulating \nthe Hall angle, and zero Hall angle can be achieved for these parameter values  shown with the \ndashed line . For exampl e, the Bloch skyrmion with the helicity ~ −π/2 stabilized by Db exhibit s \na Hall motion, as shown in Fig. 5(a) where gives the trajectory of the skyrmion for δs = -3.1×10-\n7 J∙s/m3. The Hall motion is suppressed for introducing addition Di = Db /30 which stabilize the \nHMS with the helicity of -91.9° , and the HMS moves straightly along the x-direction, as shown  \nin Fig. 5(b) . Figs. 5(c) and 5(d) gives the trajectories of the Bloch skyrmion and HMS for δs = 3.1×10-7 J∙s/m3, respectively, which also dem onstrate the suppression of Hall motion by \nintroducing suitable Di. Interestingly , the HMS moves faster  than the case of negative δs, \nconsistent with the theory analysis in Eq. ( 9a). In details, the speed is  enhanced up to ~ 20% due \nto the enlargement of the Magnus  force , noting that Magnus force suppresses the transverse \nmotion and enhances the longitudinal propagation.  Thus, ultrafast dynamics of HMS in \nferrimagnets is available in the absence of the Hall motion, confirming ag ain the advantages of \nferrimagnets in future spintronic applications.  \n \nC. Possible applications  \nSo far, this study unveils the important role of the skyrmion helicity and δs in modulating \nthe SOT-driven dynamics  of the HMS in ferrimagnets , which is helpful in guiding future \nexperiments and device design.   \nGenerally , most of the parameters chosen  in this study are comparable to those in \nGdCo /Co heterostructure [42–44]. The current density is in the order of 1011 A/m2 which is the \ntypical magnitude in experiments  [8,45,46] , and the DMI  magnitudes  are set within a \nreasonable range . Importantly, the ratio Di/Db has been proven to be a core control parameter \nin modulating the skyrmion Hall effect , allowing one to better control the HMS  dynamics \nthrough tuning the DMI coefficients  in ferrimagnets.  In particular, the DMI can be easily \nadjusted through applying voltage, electric field, or strain.  Of course, the theoretically revealed \nHMS  dynamics in ferrimagnet s deserves to be checked in future  experiments.  \nA precise  manipulation of skyrmion is import ant for implementing skyrmion -based logic \ndevices such as logic arrays. For the application of skyrmionic logic circuit, we show here a \nskyrmion diversion  operation in a Y -junction , as depicted in Fig . 6. Here, a gate voltage is \napplied to tune  the DMI  [24], and the e lectrodes are set at the node s to control the  interfac ial \nDMI in the local area.  When the input signal is “0” with low potential , no additional DMI is \ngenerated at the  node  and the skyrmion will move along  its original  track , as shown in fig. 6(a). \nOne tunes  the input voltage to “1” with high potential, a significant skyrmion Hall motion is \ninduced  and the skyrmion move s to another track , as depicted  in fig. 6(b). Moreover, e ach voltage node can be considered as a logical unit , and t he skyrmion -based programmable arrays \ncan be implemented by connecting nodes in a series.  \n \n \nV . Conclusion  \nIn conclusion , we have studied theoretically and numerically the dynamics of HMS  in \nferrimagnets driven by SOT . The dependence of the skyrmion Hall angle on the net angle \nmomentum δs and the helicity is unveiled by numerical simulations,  which demonstrates the \ngradual transition in Hall angle on δs and the helicity . Particularly, Hall motion of the HMS can \nbe eliminated through tuning  δs, allowing one to select suitable materials to better control the \nskyrmion motion . Moreover, the HMS for finite δs may exhibit faster dynamics than that of \nantiferromagnets, attributing to the Magnus force which suppresses the transverse motion and \nenhances the longitudinal propagation.  Thus, this work demonstrates the advantages of the \nferrimagnetic hybrid skyrmions for spintronic application, which is very meaningful for \nexperiments and device design.  Acknowledgment  \nWe sincerely appreciate the insightful discussions with Zhejunyu Jin and Xue Liang.  The work \nis supported by the Natural  Science Foundation of China (Grants No. U22A20117, No. \n51971096 , No. 92163210,  and No. 51721001), the Guangdong Basic and Applied Basic \nResearch Foundation (Grant No. 2022A1515011727) , and Funding by Science and Technology \nProjects in Guangzhou (Grant No.  202201000008).  References:  \n[1] N. Nagaosa and Y. Tokura, Topological Properties and Dynamics of Magnetic Skyrmions , \nNat. Nanotechnol. 8, 899 (2013).  \n[2] A. Fert, V. Cros, and J. Sampaio, Skyrmions on the Track , Nat. Nanotechnol. 8, 152 (2013).  \n[3] X. Zhang, Y. Zhou, and M. Ezawa, High -Topological -Number Magnetic Skyrmions and \nTopologically Protected Dissipative Structure , Phys. Rev. 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Schematic spin configuration  of (a) Bloch -skyrmion, (b) Néel -skyrmion and (c) hybrid \nskyrmion , and  (d) sketch of a system with hybrid skyrmion  where  red and blue arrows represent \nthe magnetization direction of CoGd and Co/Pt alloy , respectively . \n \n \n \n \n  \n \nFIG. 2. (a) The calculated (lines) and simulated ( symbols ) vx/vy as function s of the skyrmion \nhelicity tanΘ for various mp, and  (b) the simulated HMS speed as a fu nction of the current \ndensity  j for various  DMI ratios . The insert shows the calculated I/ for j = 5 × 1011 A/m2 for \nvarious DMI ratios.  \n \nFIG. 3. (a) The calculated (lines) and simulated (symbols)  v, and (b) the simulated vx and vy of \nthe HMS as a function of  δs for various j.  \n \nFIG. 4. The simulated Hall angle of the HMS for mp = ex in the (Di/Db, δs) parameter plane. T he \nblack dashed line corresponds to  the absence of the skyrmion Hall effect . \n \nFIG. 5. Trajectories of the skyrmion  for (a) δs = −3.1 ×10-7 J∙s/m3 and Θ = −90º, and (b) δs = \n−3.1 ×10-7 J∙s/m3 and Θ = −91.9º, and  (c) δs = 3.1 ×10-7 J∙s/m3 and Θ = −90º, and (d) δs = 3.1 \n×10-7 J∙s/m3 and Θ = −86.8º. The magnetic texture s in (a) and (c) are Bloch -type skyrmions, \nwhile those in (b) and (d) are HMS stabilized by  the additional interfacial DMI.  \n \nFIG. 6 . The propagation of HMS  driven by SOT in the Y -junctions  with the input of (a) low \npotential “0”, and (b) high potential “1”. Here, the interfacial DMI is tuned by the applied \nvoltage.  \n"    },    {        "title": "1712.05622v1.Effect_of_the_Canting_of_Local_Anisotropy_Axes_on_Ground_State_Properties_of_a_Ferrimagnetic_Chain_with_Regularly_Alternating_Ising_and_Heisenberg_Spins.pdf",        "content": "arXiv:1712.05622v1  [cond-mat.str-el]  15 Dec 2017Vol.XXX (201X) CSMAG‘16 No.X\nEffect of the Canting of Local Anisotropy Axes on Ground-Stat e Properties of a\nFerrimagnetic Chain with Regularly Alternating Ising and H eisenberg Spins\nJ. Torrico,1,∗M.L. Lyra,1O. Rojas,2S.M. de Souza,2and J. Streˇ cka3\n1Instituto de F´ ısica, Universidade Federal de Alagoas, 570 72-970 Maceio, AL, Brazil\n2Departamento de F´ ısica, Universidade Federal de Lavras, 3 7200-000, Lavras-MG\n3Institute of Physics, Faculty of Science, P. J. ˇSaf´ arik University, Park Angelinum 9, 040 01 Koˇ sice, Slov akia\nThe effect of the canting of local anisotropy axes on the groun d-state phase diagram and mag-\nnetization of a ferrimagnetic chain with regularly alterna ting Ising and Heisenberg spins is exactly\nexamined in an arbitrarily oriented magnetic field. It is sho wn that individual contributions of Ising\nand Heisenberg spins to the total magnetization basically d epend on the spatial orientation of the\nmagnetic field and the canting angle between two different loc al anisotropy axes of the Ising spins.\nPACS numbers: 75.10.Pq ; 75.10.Kt ; 75.30.Kz ; 75.40.Cx ; 75. 60.Ej\nIntroduction\nIn spite of a certain over-simplification, a few\nexactly solved Ising-Heisenberg models capture es-\nsential magnetic features of some real polymeric\ncoordination compounds as for instance Cu(3-\nClpy)2(N3)2[1], [(CuL) 2Dy][Mo(CN) 8] [2, 3] and\n[Fe(H2O)(L)][Nb(CN) 8][Fe(L)] [4]. The rigorous solu-\ntions for the Ising-Heisenberg models thus afford an\nexcellent playground for experimental testing of a lot\nof intriguing magnetic properties such as quantized\nmagnetization plateaus, anomalous thermodynamics,\nenhanced magnetocaloric effect, etc. [1–4]\nRecently, it has been verified that the bimetallic coor-\ndination polymer Dy(NO 3)(DMSO) 2Cu(opba)(DMSO) 2\n(to be further abbreviated as DyCu) can be satisfactorily\ndescribed by the spin-1/2 Ising-Heisenberg chain with\nregularly alternating Ising and Heisenberg spins, which\ncapture the magnetic behavior of Dy3+and Cu2+mag-\nnetic ions, respectively [5]. However, a closer inspection\nof available structural data reveals two crystallographi-\ncally inequivalent orientantions of coordination polyhe-\ndra of Dy3+magnetic ions, which regularly alternate\nalong the DyCu chain [6]. Motivated by this fact, we\nwill investigate in the present work the effect of the cant-\ning between two different local anisotropy axes on the\nground-state properties of the spin-1/2 Ising-Heisenberg\nchain with regularly alternating Ising and Heisenberg\nspins in an arbitrarily oriented magnetic field.\nModel and its Hamiltonian\nLet us introduce the spin-1/2 Ising-Heisenberg chain\nschematically illustrated in Fig. 1, in which the Ising\n∗corresponding author; e-mail: jordanatorrico@gmail.comspins with two different local anisotropy axes z1andz2\nregularly alternate with the Heisenberg spins. The local\nanisotropy axis z1(z2) of the Ising spins σ= 1/2 on odd\n(even) lattice positions is canted by the angle α(-α) from\nthe global frame z-axis. Hence, it follows that the angle\n2αdetermines the overall canting between two coplanar\nlocal anisotropy axes z1andz2. The Heisenberg spins\nS= 1/2 are coupled to their nearest-neighbor Ising spins\nthrough the antiferromagnetic coupling J <0 projected\nintothe respectiveanisotropyaxis. Furthermore, we take\ninto account the effect of the external magnetic field B,\nwhose spatial orientation is given by the angle θdeter-\nmining its tilting from the global frame z-axis. Under\nthese circumstances, the spin-1/2 Ising-Heisenberg chain\ncan be defined through the following Hamiltonian\nH=−JN/2/summationdisplay\ni=1(Sz1\n2i−1σz1\n2i−1+Sz2\n2i−1σz2\n2i+Sz2\n2iσz2\n2i+Sz1\n2iσz1\n2i+1)\n−hz1N/2/summationdisplay\ni=1σz1\n2i−1−hz2N/2/summationdisplay\ni=1σz2\n2i\n−hzN/summationdisplay\ni=1Sz\ni−hxN/summationdisplay\ni=1Sx\ni, (1)\nwherehz1=gz1\n1µBBcos(α−θ) andhz2=gz2\n1µBBcos(α+\nθ) determine projections of the external magnetic field B\ntowards the anisotropy axes of the Ising spins on odd\nand even lattice positions, respectively, gz1\n1andgz2\n1are\nthe respective Land´ e g-factors of the Ising spins and µB\nis the Bohr magneton. Similarly, hz=gz\n2µBBcosθand\nhx=gx\n2µBBsinθdetermine two orthogonal projections\nof the external magnetic field for the Heisenberg spins,\nwhereasgz\n2andgx\n2are the respective spatial components\nof the Land´ e g-factors of the Heisenberg spins.\nThetotalHamiltonianofthespin-1/2Ising-Heisenberg\nchaincanberewrittenasthesumofthecellHamiltonians\nH=N/2/summationdisplay\ni=1(H2i−1+H2i), (2)Ferrimagnetic Chain of Alternating Ising and Heisenberg Sp ins 2\nz z z z z\nxyz α α αz1 z1 z1 z2z2/vectorBθ\nσ2i−1σ2iS2i S2i−1\nSi= 1 /2 Heisenberg spins σi= 1 /2 Ising spins α′=−αα′α′\nFIG. 1: (Color online) Schematic representation of a spin\nchain with regularly alternating Ising and Heisenberg spin s.\nThe angle α(-α) determines the canting of the local\nanisotropy axis z1(z2) from the global frame z-axis for odd\n(even) Ising spins so that 2 αis the canting angle between two\ncoplanar anisotropy axes. The angle θdetermines the tilting\nof the magnetic field from the global frame z-axis.\neach of which involves all the interaction and field terms\nof exactly one Heisenberg spin\nH2i−1=−hz1\n2σz1\n2i−1−hz2\n2σz2\n2i−hz\n2i−1Sz\n2i−1−hx\n2i−1Sx\n2i−1,\nH2i=−hz2\n2σz2\n2i−hz1\n2σz1\n2i+1−hz\n2iSz\n2i−hx\n2iSx\n2i.(3)\nIn above, we have introduced the following notation for\nthe effective longitudinal and transverse fields acting on\nthe Heisenberg spins\nhz\n2i−1=Jcosα/parenleftbig\nσz1\n2i−1+σz2\n2i/parenrightbig\n+gz\n2µBBcosθ,\nhz\n2i=Jcosα/parenleftbig\nσz2\n2i+σz1\n2i+1/parenrightbig\n+gz\n2µBBcosθ,\nhx\n2i−1=Jsinα/parenleftbig\nσz1\n2i−1−σz2\n2i/parenrightbig\n+gx\n2µBBsinθ,\nhx\n2i=−Jsinα/parenleftbig\nσz2\n2i−σz1\n2i+1/parenrightbig\n+gx\n2µBBsinθ.(4)\nIt is noteworthy that the cell Hamiltonians (3) commute\nand hence, they can be diagonalized independently of\neach other by performing a local spin-rotation transfor-\nmation following the approach worked out previously [5].\nIn this way, one obtains the full spectrum of the eigenval-\nues, which can be subsequently utilized for the construc-\ntion of the ground-state phase diagram and magnetiza-\ntionprocess. Thefulldetailsofthiscalculationprocedure\nwill be published elsewheretogetherwith a morecompre-\nhensive analysis of the thermodynamic properties.\nResults and discussion\nLet us illustrate a few typical ground-state phase di-\nagrams and zero-temperature magnetization curves for\nthe most interesting particular case with the antiferro-\nmagnetic coupling J <0, equal Land´ e g-factors of the\nIsing spins gz1\n1=gz2\n1= 20 and equal components of the\nLand´ e g-factor of the Heisenberg spins gx\n2=gz\n2= 2,\nwhich nearly coincide with usual values of gyromagnetic\nratioforDy3+andCu2+magneticions, respectively. Un-\nder these circumstances, one finds four different ground\nstates: two ground states CIF 1and CIF 2with the cantedferromagnetic alignment of the Ising spins\n|CIF1/angbracketright=N/2/productdisplay\ni=1| ր/angbracketright2i−1|ψ/angbracketright2i−1| տ/angbracketright2i|ψ/angbracketright2i,(5)\n|CIF2/angbracketright=N/2/productdisplay\ni=1| ւ/angbracketright2i−1|ψ/angbracketright2i−1| ց/angbracketright2i|ψ/angbracketright2i,(6)\nand two ground states CIA 1and CIA 2with the canted\nantiferromagnetic alignment of the Ising spins\n|CIA1/angbracketright=N/2/productdisplay\ni=1| ր/angbracketright2i−1|ψ/angbracketright2i−1| ց/angbracketright2i|ψ/angbracketright2i,(7)\n|CIA2/angbracketright=N/2/productdisplay\ni=1| ւ/angbracketright2i−1|ψ/angbracketright2i−1| տ/angbracketright2i|ψ/angbracketright2i.(8)\nIt is noteworthythat the state vector | ր/angbracketright2i−1(| ւ/angbracketright2i−1)\ncorrespondstothespinstate σz1\n2i−1= 1/2(σz1\n2i−1=−1/2)\noftheodd-siteIsingspins, thestatevector | տ/angbracketright2i(| ց/angbracketright2i)\ncorresponds to the spin state σz2\n2i= 1/2 (σz2\n2i=−1/2)\nof the even-site Ising spins, while each Heisenberg spin\nunderlies a quantum superposition of both spin states\n|ψ/angbracketrighti=1/radicalbig\na2\ni+1(| ↓/angbracketrighti−ai| ↑/angbracketrighti), (9)\nwhich depends on the orientation of its two nearest-\nneighbor Ising spins via ai=hx\ni/[hz\ni−/radicalbig\n(hz\ni)2+(hx\ni)2].\nFIG. 2: (Color online) The ground-state phase diagram in\npolar coordinates for two different canting angles between t he\nlocal anisotropy axes: (a) 2 α=π/6; (b) 2α=π/4. The\nrelative size of the magnetic field µBB/|J|is represented by\nthe radius of the polar coordinates and the angle θdetermines\nits inclination with respect to the global frame z-axis.\nThe overall ground-state phase diagram in polar co-\nordinates is illustrated in Fig. 2 for two different canting\nangles 2αbetween both local anisotropyaxes. The phase\ndiagram has an obvious symmetry with respect to θ= 0\nandπ/2 axes, the former symmetry axis θ= 0 merely\ninterchanges CIA 1↔CIA2, while the latter symmetry\naxisθ=π/2 is responsible for CIF 1↔CIF2interchange.\nWithout loss of generality, our further discussion will be\ntherefore restricted just to the first quadrant θ∈[0,π/2].\nThe coexistence line between CIF 1and CIA 1phases3 Ferrimagnetic Chain of Alternating Ising and Heisenberg Sp ins\nis macroscopically degenerate with the residual entropy\nper Ising-Heisenberg pair S=kBln(2)/2, whereas CIF 1\nand CIF 2phases coexist together at θ=π/2 up to a\ntriple point (diamond symbol) with the residual entropy\nS=kBln[(√\n5 + 3)/2]/2. In addition, the Heisenberg\nspinsarecompletelyfreetoflip atmacroscopicallydegen-\nerate points (blue circles) with the residual entropy S=\nkBln(2) given by the coordinates B=|J|cos(α)/(2µB),\nθ= 0 andB=|J|sin(α)/(2µB),θ=π/2 forα>∼π/9.\nThese highly degenerate points correspond to a novel-\ntype spin frustration ’half ice, half fire’ [7], which origi-\nnates from the difference between Land´ e g-factors being\nresponsible for a fully frozen (ordered) character of the\nIsing spins and fully floppy (disordered) character of the\nHeisenberg spins.\nFinally, the individual contributions of the Ising and\nHeisenberg spins to the total magnetization are depicted\nin Fig. 3 as a function of the magnetic-field strength for\nseveral spatial orientations θof the applied field. As one\ncan see, any deviation of the magnetic field from its lon-\ngitudinal direction θ= 0 destroys the sharp stepwise de-\npendence in the longitudinalprojectionofthe magnetiza-\ntionmz\n2of the Heisenberg spins. In fact, the longitudinal\ncomponent mz\n2is gradually smeared out upon increasing\nofthe tiltingangle θ, whileits transversepart mx\n2risesup\nto its global maximum successively followed by a gradual\ndecline [cf. Fig. 3(a)-(b)]. The most notabledependences\nofthemagnetizationoftheHeisenbergspinscanbefound\nfor greater tilting angles θof the magnetic field, which\ncause an abrupt jump in both components mx\n2andmz\n2\nof the Heisenberg spins due to a sudden reorientation of\nIsing spins at the phase transition from the canted fer-\nromagnetic phase CIF 1to the canted antiferromagnetic\nphase CIA 1(see the curves for θ= 2π/5). The sharp\nstepwise dependence of the magnetization of the Heisen-\nberg spins is afterwards recovered for the special case of\nthe transverse field θ=π/2, for which it appears in the\ntransverse projection mx\n2while its longitudinal part mz\n2\nequals zero. The coexistence of the canted ferromagnetic\nphases CIF 1and CIF 2is manifested through a quasi-\nlinear dependence of mx\n2at low magnetic fields, where\nother contributions mz\n2=mz1\n1=mz2\n1= 0 vanish.\nConclusion\nIn the present work we have examined in detail the\nground-state phase diagram and zero-temperature mag-\nnetization process of the spin-1/2 Ising-Heisenberg chain\nwith two different local anisotropy axes in an arbitrar-\nily oriented magnetic field. It has been shown that the\nphase diagram involvesin total two canted ferromagnetic\nand two canted antiferromagnetic ground states. An-\nother interesting finding concerns with the existence of a\nfewmacroscopicallydegeneratepoints, atwhichaperfect\norder of the Ising spins accompanies a complete disorder/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48\n/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s109/s120 /s50/s32/s32/s91/s103/s120 /s50\n/s66/s93\n/s32/s32\n/s32/s32 /s32\n/s32 /s32\n/s32\n/s40/s98/s41\n/s32/s32\n/s32/s109/s122 /s50/s32/s32/s91/s103/s122 /s50\n/s66/s93\n/s40/s99/s41\n/s32/s32/s109/s122\n/s49\n/s49/s32/s32/s32/s32/s91/s103/s122\n/s49\n/s49 /s66/s93\n/s66/s47/s124/s74/s124/s40/s97/s41\n/s40/s100/s41\n/s32/s32/s109/s122\n/s50\n/s49/s32/s32/s32/s32/s91/s103/s122\n/s50\n/s49 /s66/s93\n/s66/s47/s124/s74/s124\nFIG. 3: (Color online) Zero-temperature magnetizations ve r-\nsus the relative strength of the magnetic field for the cantin g\nangle 2α=π/4 and several spatial orientations θof the ap-\nplied field: (a)-(b) the transverse mx\n2and longitudinal mz\n2\nprojections of the Heisenberg spins; (c)-(d) the local proj ec-\ntionsmz1\n1andmz2\n1of the Ising spins towards their easy axes.\nof the Heisenberg spins within the so-called ’half ice, half\nfire’ frustrated ground state [7]. It has been also convinc-\ningly evidenced that the canting angle between two local\nanisotropy axes of the Ising spins and the spatial orien-\ntation of the applied magnetic field basically influences\nthe overall shape of the magnetization curves.\nAcknowledgments\nThis work was partially supported by FAPEAL\n(Alagoas State Research agency), CNPq, CAPES,\nFAPEMIG, VEGA 1/0043/16 and APVV-14-0073.\n[1] J. Streˇ cka, M. Jaˇ sˇ cur, M. Hagiwara, K. Minami, Y.\nNarumi, K. Kindo, Phys. Rev. B 72, 024459 (2005).\nDOI:10.1103/PhysRevB.72.024459.\n[2] W. Van den Heuvel, L.F. Chibotaru, Phys. Rev. B 82,\n174436 (2010). DOI:10.1103/PhysRevB.82.174436.\n[3] S. Bellucci, V. Ohanyan, O. Rojas, EPL105, 47012\n(2014). DOI: 10.1209/0295-5075/105/47012\n[4] S. Sahoo, J.P. Sutter, S. Ramasesha, J. Stat. Phys. 147,\n181 (2012). DOI: 10.1007/s10955-012-0460-7.\n[5] J. Streˇ cka, M. Hagiwara, Y. Han, T. Kida, Z. Honda,\nM. Ikeda, Condens. Matter Phys. 15, 43002 (2012). DOI:\n10.5488/CMP.15.43002.\n[6] G. Calvez, K. Bernot, O. Guillou et al., Inorg. Chim. Acta\n361, 3997 (2008). DOI: 10.1016/j.ica.2008.03.040.\n[7] W. Yin, Ch. Roth, A. Tsvelik, arxiv: 1510.00030v2."    },    {        "title": "1503.07893v1.Enhancing_Magnetic_Ordering_in_Cr_doped_Bi2Se3_using_High_TC_Ferrimagnetic_Insulator.pdf",        "content": " \n   Enhancing Magnetic Ordering in Cr-doped Bi2Se3 using High-TC Ferrimagnetic Insulator Wenqing Liu,1, 2 Liang He,1, 3 Yongbing Xu,1, 2 Koichi Murata,3 Mehmet C. Onbasli,4 Murong Lang,3 Nick J. Maltby,2 Shunpu Li,2 Xuefeng Wang,1 Caroline A. Ross,4 Peter Bencok,5 Gerrit van der Laan, 5 Rong Zhang,1 Kang. L. Wang3 1 York-Nanjing Joint Center for Spintronics and Nano Engineering (YNJC), School of Electronics Science and Engineering, Nanjing University, Nanjing 210093, China 2 Spintronics and Nanodevice Laboratory, Department of Electronics, University of York, York YO10 5DD, UK  3 Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA  4 Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA  5 Diamond Light Source, Didcot OX11 0DE, UK  ABSTRACT: We report a study of enhancing the magnetic ordering in a model magnetically doped topological insulator (TI), Bi2-xCrxSe3, via the proximity effect using a high-TC ferrimagnetic insulator Y3Fe5O12. The FMI provides the TI with a source of exchange interaction yet without removing the nontrivial surface state. By performing the elemental specific X-ray magnetic circular dichroism (XMCD) measurements, we have unequivocally observed an enhanced TC of 50 K in this magnetically doped TI/FMI heterostructure. We have also found a larger (6.6 nm at 30 K) but faster decreasing (by 80% from 30 K to 50 K) penetration depth compared to that of diluted ferromagnetic semiconductors (DMSs), which could indicate a novel mechanism for the interaction between FMIs and the nontrivial TIs surface. KEY WORDS: Magnetic topological insulator, XMCD, proximity effect, ferrimagnetic insulator YIG, spintronics          DOI: 10.1021/nl504480g Nano Lett., 2015, 15 (1), pp 764–769\nLiu et al. Nano Lett. 15, 764 (2015) \n 1 Three-dimensional TIs are insulating bulk materials that carry a conducting surface state, arising from the intrinsically strong spin-orbit interaction in the bulk band structure protected by time-reversal symmetry (TRS). While such unique systems offer nontrival surface states that can be utilized to perform dissipationless spin transport, it is equally important to break the TRS of TIs to realize novel physical phenomena. The newly discovered quantum anomalous Hall (QAH) effect,1, 2 the hedgehog-like spin textures,3 magnetoresistance switch effect,4 carrier-independent long-range ferromagnetic order,5 the predicted giant magneto-optical Kerr effect,6 and magnetic monopole effect7 are some of the fascinating examples.  Two categories of route for breaking TRS or introducing ferromagnetic order in TIs have been developed. One route is to dope the TI host with specific elements, by which ferromagnetism has been observed in Cr- and Mn-doped single crystals of Sb2Te3,8, 9, 10 Fe-, and Mn-doped single crystals of Bi2Te3, 11, 12 and Mn- and Cr-doped thin films of Bi2Se3.13, 14 The other routine is to engineer layered heterostructures, where the surface states of TIs experience the exchange interaction from an adjacent ferro- or ferri- magnetic material. This route subsequently can be divided into two ways in terms of ferro- or ferri- magnetic metal (FM) and ferro- or ferri- magnetic insulator (FMI) induction. Pioneering theoretical work15, 16, 17 suggests that suitable FMIs have the potential to achieve a strong and uniform exchange coupling in contact with TIs without significant spin-dependent random scattering of helical carriers on magnetic atoms. Progresses are made experimentally in FMI/TI heterostructures including GdN/Bi2Se3 by Kandala et al.,18 EuS/Bi2Se3 by Yang et al.,19 and Wei et al.,20 respectively, although the effect observed is limited to low temperature (< 22 K) due to the low TC of EuS. The interface magnetism of (anti-) FM/TI heterostructures, such as Fe/Bi2Se3, 21, 22, 23 Co/Bi2Se3,22 and Cr/Bi2Se3,24 has also been investigated. Remarkably, Vobornik et al.25 demonstrated that long-range ferromagnetism at ambient temperature can be induced in Bi2-xMnxTe3 by a deposited Fe overlayer. However, in the presence of a metallic layer, the nontrivial surface states of the TI can be significantly altered due to their hybridization with the bulk states of the (anti-) FM in contact. Besides, the metallic layer naturally short circuits the TI layer and therefore fundamentally restrict the device design.  Although magnetically doped TIs have demonstrated a pronounced capability with the magnetic proximity effect, very limited successful experimental demonstrations, especially by means of direct measurements like XMCD19, 22, 25 have been reported. As shown in Figure 1 among all the building blocks within the research framework of FM or FMI/(magnetically doped) TI heterostructures, investigations of FMI/magnetically doped TI remains absent. Here, we present a work in realizing the proximity effect in an epitaxial Bi2-Liu et al. Nano Lett. 15, 764 (2015) \n 2 xCrxSe3/Y3Fe5O12 heterostructure to fill up the void. Garnet-type Y3Fe5O12 (YIG) is a well-known FMI with TC (~550 K) well above RT and a desirable large spin diffusion length. It contains two Fe ions occupying octahedral sites and three Fe ions occupying tetrahedral sites with opposite spin, resulting in ferrimagnetic ordering. The proximity effect has been demonstrated in PdPt/YIG,26 Pt/YIG,27 and Nb/YIG,28 where interesting spin-transport properties were observed. While YIG-based heterostructures can consist of various materials, the best chance to realize strong exchange coupling may exist in the candidates with two-dimensional quantum surface states, such as TIs, as we have demonstrated in this report with a model TI/FMI heterostcrture Bi2-xCrxSe3/Y3Fe5O12.  The 10 nm Bi1.89Cr0.11Se3 thin films used in this study were grown in ultra-high vacuum using a Perkin-Elmer molecular-beam epitaxy (MBE) system on 50 nm YIG (111) film, which was pre-deposited on gallium gadolinium garnet (GGG) (111) substrate using pulsed-laser deposition (PLD).29, 30 The X-ray diffraction (XRD) and magneto-optical Kerr effect (MOKE) magnetometry characterization of the YIG/GGG substrate have been published elsewhere.31 High-purity Bi (99.9999%) and Cr (99.99%) were evaporated from conventional effusion cells at 470°C, while Se (99.99%) was formed from a cracker cell from SVTA at 240 °C, and the YIG/GGG (111) substrate was kept at 200 °C during growth. Interdiffusion of materials at the interface is not expected due to the high stability of YIG and the relatively low growth temperature of Bi1.89Cr0.11Se3. 2 nm Al was then in-situ evaporated immediately after the growth of Bi1.89Cr0.11Se3 to protect it from oxidation and environmental doping during transport to the synchrotron facility. Further details of the real-time reflection high-energy electron diffraction (RHEED) and the scanning transmission electron microscopy (STEM) characterization of the sample can be found in the supplementary materials. The magnetic response of the epitaxial Bi1.89Cr0.11Se3/YIG (111) thin film samples was first examined by the magneto-transport measurements by patterning into standard Hall bar devices, using conventional optical photolithography and a subsequent CHF3 dry etching for 20 s. As shown in Figure 2A, six Hall channel contacts (10 nm Ti and 100 nm Au) were defined by e-beam evaporation. Standard four-terminal electrodes were fabricated to eliminate the contact resistance. A constant AC current of 0.05 ~ 0.1 µA with a frequency of 1300 Hz is fed through two outer contacts, and the voltage drop across the inner pads is measured to determine the resistance. By subtracting the ordinary Hall component, we plotted the anomalous Hall resistance (RAHE = Rxy - R0⋅H) 32   as a function of field applied perpendicularly to the film in Figure 2B. Non-zero RAHE was observable up to 50 K and vanished above 90 K. Figure 2C presents the temperature dependent RAHE of the Bi1.89Cr0.11Se3 thin films on YIG, to which a Bi1.89Cr0.11Se3 epitaxial thin film of the same thickness grown on highly resistive Si Liu et al. Nano Lett. 15, 764 (2015) \n 3 (111) substrate was attached for comparison purpose. It can be seen that both these Cr-doped Bi2Se3 thin films exhibit Curie-like behavior, however, their magnetic ordering disappears at different temperatures, namely 30 K for Bi1.89Cr0.11Se3/Si and up to 50 K for Bi1.89Cr0.11Se3/YIG. The ferromagnetic ordering of Bi1.89Cr0.11Se3/YIG was also observed from the field dependent longitudinal resistance (Rxx). In the low field region, weak anti-localization (WAL) with a clear cusp was observed at low temperatures, which is a characteristic feature associated with the gapped topological surface states below the critical temperature.9, 14, 33 A typical Rxx obtained at 3 K is presented in the inset of Figure 2D. The valleys of the WAL cusp exhibit a shift under the opposite field scanning directions, with the Rmin occurring approximately at the coercive field (Hc). We repeated the hysteretic longitudinal magnetic resistance measurement at elevated temperatures up to 90 K and found that Hc remains observable till beyond 50 K, as plotted in Figure 2D, which is consistent with the TC estimated from RAHE.  The element-specific technique of X-ray absorption spectroscopy (XAS) and XMCD at the Cr L2,3 absorption edges were performed on beamline I10 at Diamond Light Source, UK, to probe the local electronic character of the magnetic ground state of the Bi1.89Cr0.11Se3/YIG. Circularly polarized X-rays with ~100% degree of polarization were used in normal incidence with respect to the sample plane and parallel to the applied magnetic field, i.e., in Faraday geometry, as schematically shown in Figure 3A.  XAS measurements were carried out at 6 - 300 K using total-electron yield (TEY) detection. XMCD was obtained by taking the difference of the XAS spectra, σ- - σ +, obtained by flipping the X-ray helicity at fixed magnetic field of 10 kOe, under which the sample is fully magnetized with little paramagnetic contribution.  A typical XAS and XMCD of the bilayer sample obtained by total-electron yield (TEY) at 6 K, normalized to the incident beam intensity, is presented in Figure 3C. The XAS spectra of Cr for both left- and right- circularly polarized X-rays show a white line at each spin-orbit split core level without prominent multiplet structure, except for a shoulder structure for the L2 peak. The XAS spectral line shape resembles that of the ferromagnetic spinel-type Cr chalcogenides, i.e., CdCr2Se4, reported by Kimura et al.,34 suggesting that the sample contains predominately Cr3+ cations. Features of the obtained XMCD spectra are also in good agreement with those obtained for CrFe2O4 spinel ferrite powders, which can be well reproduced by multiplet calculations using a charge-transfer model with trivalent Cr cations on octahedral sites.35  Consistent with the reported transport measurements,14 the observed XAS and XMCD line shape also suggests that a majority of the Cr ions is incorporated within the crystal lattice by substituting onto the Bi sites (with a formal valance of 3+) in the TI Liu et al. Nano Lett. 15, 764 (2015) \n 4 matrix. The XAS and XMCD measurements were repeated at elevated temperatures and the dichroism at the Cr L3 edge (575.3eV) was observable up to 50 K, as shown in Figure 3C, despite the decreasing intensity with increasing temperature. For clarity, the partial enlarged XAS of Cr L3 edge at 30, 50, and 100 K, respectively, are presented in Figure 3B.  One of the most powerful aspects of the XMCD technique is that the average magnetic moment of the each element under interrogation can be quantitatively related to the integrated intensity of the XAS and XMCD spectra by applying the sum rules.36, 37 Here, the orbital (ml) and spin (ms) moments of Cr were calculated according to equations ml=−43nh(σ−−L2,3∫σ+)dE(σ−+L2,3∫σ+)dEms=−nh6(σ−−L3∫σ+)dE−4(σ−−L2,3∫σ+)dE(σ−+L2,3∫σ+)dE×SC−<Tz>      (1) where E, nh, SC, and  <Tz>, respectively, represents the photon energy, the number of d holes, the spin correction (SC) factor and the magnetic dipole term. In order to exclude the non-magnetic contribution of the XAS spectra an arctangent-based step function is used to fit the threshold.38 The spectral overlap or j-j mixing36 was taken into account because of the relatively small spin-orbit coupling in the Cr 2p level. The value of SC, i.e. 2.0±0.2 for Cr, was estimated by calculating the L2,3 multiplet structure for a given ground state, applying the sum rule on the calculated XMCD spectrum, and comparing the result with the spin moment calculated directly for this ground state.39 Furthermore, ms needs to be corrected for the magnetic dipole term <Tz>, however, its contribution is small for a Cr 32gtconfiguration, giving an error < 5%. Figure 4A-B presents the calculated ms, ml, and total magnetic moment (ms+l) of Cr in Bi1.89Cr0.11Se3/YIG bilayer at 6-300 K. Consistent with the magneto-transport results, the derived ms+l also exhibits a Curie-like behavior, pointing to a ferromagnetic phase of Bi1.89Cr0.11Se3 at low temperatures. We obtained a remarkable ms = 1.38 ± 0.10 µB/Cr and a small negative ml = -0.03 ± 0.02 µB/Cr at 6 K. Noting that ms retains a sizable value of 0.58 ± 0.10 µB/Cr at 30 K, we claim a pronounced increase of the TC in the Bi1.89Cr0.11Se3 from 30 K, since otherwise ms should have nearly vanished at, or below, this point. With increasing temperature, ms reduces to 0.10 ± 0.10 µB/Cr at 50 K, suggesting that the Bi1.89Cr0.11Se3 is close to its TC here. Note that although the Fe dichroism in YIG remains sizably large up to RT (see supplementary materials), the Cr dichroism is no longer distinguishable from the Liu et al. Nano Lett. 15, 764 (2015) \n 5 noise at and above 100 K. As listed in Table 1, the reported TC of various kinds of magnetic TIs remained so far below ∼30 K. Our demonstration of the ferromagnetic phase up to 50 K is significant in enhancing magnetic ordering of the magnetically doped TIs by the proximity effect with high-TC FMI, where the surface states of the TIs can be preserved with the insulating YIG. The derived ml and ms have opposite signs, corresponding to antiparallel alignment of the spin and orbital moment in Cr. This agrees with the Hund’s rule for trivalent Cr, whose 3d shell is less-than-half full.34 The octahedral crystal-field interaction quenches ml, since the three d electrons occupy the threefold degenerate majority-spin t2g orbitals, leading to a nearly vanishing ml as observed here. For similar reasons as for ml, the magnetic-dipole term is small. Our observation of the total Cr magnetic moment is close to the value reported by Haazen et al.,13 who performed superconducting quantum interference device (SQUID) measurements on a series of epitaxial Bi2Se3 with different Cr doping concentration (maximum TC = 20 K). In their work, the magnetic moment per Cr decreases significantly for x > 5.2%, which coincides with a loss of Bi2-xCrxSe3 crystallinity. Such dependence is further evidence that the magnetization originates from the crystalline Bi2-xCrxSe3 phase. Cr clustering would not give rise to non-zero XMCD, since Cr is antiferromagnetic, as are the CrxSey compounds, but therefore could have led to a reduced average Cr magnetic moment in the Bi2-xCrxSe3. However, since transport measurements are less sensitive to isolated ferromagnetic particles, our observed ferromagnetic behavior is still ascribed to be from the entire Bi1.89Cr0.11Se3 thin film instead of magnetic clusters, if any. Both the electrical transport and XMCD results point to the fact that between 30 and 50 K the magnetization of Cr can be attributed to the magnetic exchange coupling with YIG. We now address the ability of the YIG underlayer to induce magnetic ordering in Bi1.89Cr0.11Se3 utilizing a model that was developed in the study of the proximity effect in dilute magnetic semiconductors (DMSs), 40, 41 as schematically sketched in Figure 4C. It is generally accepted that the XMCD intensity measured by TEY is attenuated by an exponentially decaying electron-escape probability, exp(-x/le), 42 we obtained  CrXMCD=δ(x)ρ(x)e−x/λe0∞∫dxρ(x)e−x/λe0∞∫dx      (2) where λe is the mean electron escape length. Provided (i) a sharp interface (see supplementary materials), (ii) a uniform distribution of Cr in Bi2Se3, and (iii) a steplike dichroism profile versus thickness d, we have δ(x) = δexp for dYIG  < x < dmin and δ(x) = 0 elsewhere. Here dmin Liu et al. Nano Lett. 15, 764 (2015) \n 6 represents the lower limit, or the thickness of Bi1.89Cr0.11Se3 contributing to the ferromagnetic signal at 30 - 50 K. Integration gives   dmin=dBi2−xCrxSe3+λeln[δexpδsat+(1−δexpδsat)e−dBi2−xCrxSe3/λe]     (3) Quoting the value ms = 1.38 µB/Cr at 6 K, whose magnetic moment is considered to be intrinsic of the Bi1.89Cr0.11Se3 without the effect of YIG, we obtain δexp/δsat = 43% and dmin = 6.6 nm at 30 K. This value quickly reduces to dmin = 1.8 nm at 50 K, where δexp/δsat = 7%. For the calculation of dmin, we adopted λe = 5 nm for the bulk mean electron escape length of Bi2Se3 and d = 10 nm for the thin film thickness. Compared with the FM/DMS bilayer systems investigated by Maccherozzi et al.40 using the same model, we have observed a larger penetration depth of the magnetic proximity effect at the interface of TI/FMI. The penetration depth decreases sharply with increasing temperature (i.e., > 80% from 30 K to 50 K). Typically, the proximity-induced magnetization in DMSs reduces only by ∼10% within a comparable temperature range.40 Such contrast may imply a unique type of interaction of the nontrivial surface state of TI with FMI. In other words, the penetration depth of the magnetic proximity effect in DMSs may have been limited by the contact barrier, while the conducting surface states of TIs may lift this limitation. It is generally believed that the origin of the proximity effect in a nonmagnetic/ferro- or ferri- magnetic (NM/FM) heterostructure arises from (spin-polarized) charge carriers propagating from the FM into the NM metal and vice versa, such that a finite spin polarization builds up close to the interface. A substantial reduction of such a spin polarization accumulation can be expected in a DMS, where charge carriers can hardly penetrate. In contrast, our results suggest that such charge carrier propagation can be less suppressed in the TI/FMI system due to the presence of the conducting surface states, though whose ability is more sensitive to the temperature variation.  To summarize, we have observed strongly dichroic XAS spectra of Cr in Bi1.89Cr0.11Se3/YIG up to 50 K, corresponding to an enhanced TC in this magnetically doped TI/FMI exchange system. The unique elemental selectivity of the XMCD technique has enabled a direct determination of the proximity-induced magnetization of Bi1.89Cr0.11Se3 at its interface with YIG. We have found a larger but faster dropping penetration depth in such a magnetic TI/FMI heterostructure compared to that in DMS/FM, which may be due to a novel mechanism of the interaction of FMIs and nontrivial TIs surface. The result is furthermore the first demonstration of XMCD in Cr-doped Bi2Se3 epitaxial thin films, presenting an unambiguous picture of the electronic and magnetic state of the magnetic dopants in the TIs.  Our study takes an important step towards realizing TI-based spintronics. Future work to Liu et al. Nano Lett. 15, 764 (2015) \n 7 explore the coupling mechanism of the TI/FMI interface and its dependence on the band filling will help to find experimental approaches to further increase the Tc in the magnetically doped TI/FMI hybrid material systems, which has strong implications for both fundamental physics and emerging spintronics technology.   ASSOCIATED CONTENT Supplementary materials accompany this paper.   AUTHOR INFORMATION  Corresponding Authors  *E-mail: yongbing.xu@york.ac.uk. *E-mail: rzhang@nju.edu.cn. *E-mail: wang@seas.ucla.edu.  Author Contributions Wenqing Liu and Liang He contributed equally to this work. Notes  The authors declare no competing financial interest.  ACKNOWLEDGEMENT This work is supported by the State Key Programme for Basic Research of China (Grants No. 2014CB921101), NSFC (Grants No. 61274102), UK STFC, DARPA Meso program under contract No.N66001-12-1-4034 and N66001-11-1-4105. We thank Y. Wang of Zhenjiang University for the help with the TEM characterization. C. A. Ross and M. C. Onbasli acknowledge support of FAME, a STARnet Center of DARPA and MARCO, and by the NSF. Diamond Light Source is acknowledged for beamtime on I10.  Liu et al. Nano Lett. 15, 764 (2015) \n 8 TABLES  System TC Ref. Bi2-xCrxSe3/YIG  50 K  [*] Bi2-xCrxSe3  20 K [13] Sb2−xCrxTe3  20 K [8] Crx(BiySb1−y)2Te3  11 K [9] Bi2-xMnxTe3  12 K [11] Bi2-xFexTe3 12 K [12]  Table 1. The TC of the magnetically doped TIs from the literatures. Typical values of TC from reports, which remains generally below ∼30 K till today. Our work [*] has demonstrated an enhanced TC up to ∼50 K via the proximity effect using a YIG under layer.        Liu et al. 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Nano Lett. 15, 764 (2015) \n 12                                                                                                                                                                40 Maccherozzi, F.; Sperl, M.; Panaccione, G.; Minár, J.; Polesya, S.; Ebert, H.; Wurstbauer, U.; Hochstrasser, M.; Rossi, G.; Woltersdorf, G.; Wegscheider, W.; Back, C. Phys. Rev. Lett. 2008, 101, 267201.  41 Nie, S. H.; Chin, Y. Y.; Liu, W. Q.; Tung, J. C.; Lu, J.; Lin, H. J.; Guo, G. Y.; Meng, K. K.; Chen, L.; Zhu, L. J.; Pan, D.; Chen, C. T.; Xu, Y. B.; Yan, W. S.; Zhao, J. H. Phys. Rev. Lett. 2013, 111, 027203. 42 Frazer, B. H.; Gilbert, B.; Sonderegger, B. R. 2003, 537, 161–167.  Liu et al. Nano Lett. 15, 764 (2015) \n 13 FIGURES     \n Figure 1. Schematic diagram of the research framework of magnetic TI hybrid systems. Squares representing FM (top right) and FMI (bottom left) integrate with those representing TI (top right) and magnetically doped TI (bottom right), encompassing four categories of subjects of magnetic TI by engineering layered heterostructures, namely, investigations of (A) TI/FM including Fe/Bi2Se3,21 Co/Bi2Se3,22 and Cr/Bi2Se3;24 (B) TI/FMI including MnSe/Bi2Se3,15, 16 GdN/Bi2Se3,18 EuS/Bi2Se3;19, 20 (C) doped TI/ FM Fe/Bi2-xMnxTe3;25 and (D) doped TI/FMI, the remaining unexplored area, on which this letter reports on.          \nLiu et al. Nano Lett. 15, 764 (2015) \n 14 \n Figure 2. Magneto-transport measurement. (A) Schematic diagram of the experimental set up for the transport measurements. (C) AHE of the Bi1.89Cr0.11Se3/YIG thin film versus magnetic field at 20-90 K. (B) Comparison of the AHE versus temperature of the Bi1.89Cr0.11Se3 thin films grown on YIG (111) and Si (111), respectively. (D) The Hc of Bi1.89Cr0.11Se3/YIG versus temperature. Inset: the shift of the valleys of WAL cusp obtained at 3 K, associated with the Hc. The arrows represent the scanning direction of the magnetic field.    \nLiu et al. Nano Lett. 15, 764 (2015) \n 15 \n Figure 3. XAS and XMCD measurement. (A) Schematic diagram of the experimental set up of the XMCD experiment. (B) The partial enlarged XAS of Cr L3 edge at 30, 50, and 100 K, respectively. (C) Typical pair of XAS and XMCD spectra of the Bi1.89Cr0.11Se3/YIG bilayer sample obtained at 6 K and their integrals (the XAS and XMCD spectra are offset for clarity) and that at elevated temperatures, where dichroism at the Cr L3 edge (spectra at different temperatures are offset vertically for clarity). The dash lines indicate the integration of the spectra.     \nLiu et al. Nano Lett. 15, 764 (2015) \n 16 \n Figure 4. The enhanced magnetic ordering of Bi1.89Cr0.11Se3 via the proximity effect. (A)-(B) The ms and ml of Cr at 6-300 K derived from the sum rules. The dashed line is a guide to the eye. (b) A schematic diagram of the model used to estimate the proximity length, showing the Cr distribution ρ(x) and the ferromagnetically ordered Cr distribution δ(x) at given temperature, as described in the text.   \n"    },    {        "title": "0707.2133v2.Martensitic_transition__ferrimagnetism_and_Fermi_surface_nesting_in_Mn_2NiGa.pdf",        "content": "arXiv:0707.2133v2  [cond-mat.other]  15 Oct 2007Martensitic Transition, Ferrimagnetism and Fermi Surface\nNesting in Mn 2NiGa\nS. R. BARMAN1, S. BANIK1, A. K. SHUKLA1,\nC. KAMAL2, and APARNA CHAKRABARTI2\n1UGC-DAE Consortium for Scientific Research,\nIndore, 452017, Madhya Pradesh, India and\n2Raja Ramanna Centre for Advanced Technology,\nIndore, 452013, Madhya Pradesh, India\nAbstract\nPACS. 81.30.Kf - Martensitic transformations\nPACS. 71.20.Be - Electron density of states and band structu re of transition metals and alloys\nPACS. 71.18.+y - Fermi surface: calculations\nPACS. 71.15.Nc - Total energy calculations\nThe electronic structure of Mn 2NiGa has been studied using density functional theory and ph o-\ntoemission spectroscopy. The lower temperature tetragona l martensitic phase with c/a=1.25 is\nmore stable compared to the higher temperature austenitic p hase. Mn 2NiGa is ferrimagnetic in\nboth phases. The calculated valence band spectrum, the opti mized lattice constants and the mag-\nnetic moments are in good agreement with experiment. The maj ority-spin Fermi surface (FS)\nexpands in the martensitic phase, while the minority-spin F S shrinks. FS nesting indicates occur-\nrence of phonon softening and modulation in the martensitic phase.\nPACS numbers:\n1Introduction: Recentadvent ofmultiferroicshapememoryalloys(SMA)likeNi-Co-M n-In,\nNi-Mn-Ga that exhibit both ferroelastic and ferromagnetic proper ties has ushered a flurry\nof activity in this field1,2,3,4,5,6,7,8,9. In particular, Ni-Mn-Ga has generated immense interest\nbecause of very large strain (10%) in a moderate magnetic field ( ≈1 Tesla)3,4. Moreover,\nin Ni-Mn-Ga the actuation is much faster ( ≈2kHz) than conventional SMA5. However,\nNi2MnGa are brittle and so search for materials with better mechanical properties exhibit-\ningsimilarmagneticfieldinducedstrainisbeingactivelypursued10,11. Mn2NiGaisarecently\ndiscovered ferromagnetic SMA in the Ni-Mn-Ga family. It has Curie an d martensitic start\ntemperatures of 588 and 270 K, respectively11. Ferromagnetism in Mn 2NiGa is surprising\nbecause direct Mn-Mn interaction normally leads to antiferromagne tic alignment12,13. More-\nover, the origin of the martensitic transition involving a relatively larg e tetragonal distortion\n(c/a=1.21) has not been studied theoretically till date. Recently, a dens ity functional the-\nory (DFT) study on Mn 2NiGa shows a large enhancement of the density of states (DOS)\nnear the Fermi level ( EF) and quenching of Mn and Ni magnetic moments in the marten-\nsitic phase14. However, such large change in the magnetic moments or DOS has no t been\nobserved in any other SMA either from experiment9,15,16or theory8,17,18.\nThegeometryoftheFermisurface(FS)isresponsible foravariet yofphenomena likespin\norchargedensity waves, Kohnanomalies, Friedel oscillationsinmeta ls. IftheFShasparallel\nplanes, strong electronic response can occur at the wave vector that translates one parallel\nplane of the FS to the other. This wave vector is called the nesting ve ctor (n.v.). FS nesting\nhas been reported to cause softening of the transverse-acous tic (TA 2) phonon mode along\n[110] direction resulting in modulated pre-martensitic phase of SMA’s like Ni 2MnGa and\nNi-Ti19. Recently, an inelastic neutron scattering study on Ni 2MnGa showed the presence\nof charge density wave in the martensitic phase resulting from FS ne sting7. Thus, it is\nworthwhile to study the FS of Mn 2NiGa, particularly because the relatively large tetragonal\ndistortion is likely to modify the FS substantially.\nIn this work, a DFT study of the electronic structure of Mn 2NiGa using full poten-\ntial linearized augmented plane wave method (FPLAPW) is presented . The valence band\n(VB) spectrum, calculated from the theoretical DOS, is in agreeme nt with the ultra-violet\nphotoemission spectroscopy (UPS). We find that the total energ y (Etot) is lower in the\nmartensitic phase with a tetragonal distortion of c/a=1.25. We show that Mn 2NiGa is an\nitinerant ferrimagnet in both the martensitic and austenitic phases . The equilibrium lattice\n2constants and the magnetic moments are in agreement with x-ray d iffraction and magneti-\nzation data, respectively. The FS in the martensitic phase is drastic ally different from the\naustenitic phase. A highly nested hole-type majority-spin cuboidal FS sheet around the Γ\npoint appears in the martensitic phase that is absent in the austenit ic phase.\nMethodology: FirstprinciplesDFTcalculationswereperformedusingtheWIEN97co de20.\nGeneralized gradient approximation (GGA) for the exchange corre lation that accounts for\nthe density gradients was used21. An energy cut-off for the plane wave expansion of 16 Ry\nis used (RMTKmax= 9). The cut-off for charge density is Gmax= 14. The maximum l(lmax)\nfor the radial expansion is 10, and for the non-spherical part: lmax,ns=6. The muffin-tin\nradii are Ni: 2.1364, Mn: 2.2799, and Ga: 2.1364 a.u. The number of kpoints for self-\nconsistent field cycles in the irreducible Brilloiun zone is 256 and 484 in th e austenitic and\nmartensitic phase, respectively. The convergence criterion for Etotis 0.1 mRy, which implies\nthat accuracy of Etotis±0.34 meV/atom. The charge convergence is set to 0.001. FS has\nbeen calculated using XcrySDen22. Mn2NiGa ingot was prepared by arc furnace melting and\nannealing at 1100K9. It was characterized by x-ray diffraction (XRD), energy dispers ive\nanalysis of x-rays and differential scanning calorimetry16. Atomically clean specimen surface\nwas prepared by in situscraping using a diamond file and the chamber base pressure was\n6×10−11mbar. UPS was performed with a HeI (h ν=21.2 eV) photon source using electron\nenergy analyzer from Specs GmbH, Germany. The overall resolutio n was 120meV.\nMn2NiGa has a cubic L21structure in the austenitic phase that consists of four inter-\npenetrating f.c.c. lattices at (0,0,0), (0.25,0.25,0.25), (0.5,0.5, 0.5), and (0.75,0.75,0.75)\n(Fig. 1a)11,16. The structure of Mn 2NiGacan be better explained incomparison to Ni 2MnGa\nthat also has L21structure. In Ni 2MnGa, the Ni atoms are at (0.25,0.25,0.25) and\n(0.75,0.75,0.75), while Mn and Ga are at (0.5,0.5,0.5) and (0,0,0), r espectively and there\nis no direct Mn-Mn interaction, with Mn having eight Ni atoms as neare st neighbours. In\ncontrast, Mn 2NiGa has one Mn atom at (0.5,0.5,0.5) (referred to as MnII), while th e other\nMn atom (MnI) occupies the Ni atom position (0.75,0.75,0.75) of Ni 2MnGa. Thus, MnI\nand MnII occupy inequivalent sites in the unit cell, and there is a direct Mn-Mn interaction\nsince MnI and MnII are nearest neighbours. In the martensitic pha se, the XRD pattern for\nMn2NiGa has been indexed by a tetragonal unit cell with c/a=1.21 (Fig. 1b)11,16.\nTotal energy and magnetic moment calculation: To determine whether minimiza-\ntion ofEtotcauses the structural transition, we have calculated Etotfor both phases as a\n3function of the lattice parameters in the lowest energy magnetic st ate (discussion about the\nmagnetic state is given later). In the austenitic phase, Etotas a function of cell volume\n(V) exhibits a parabolic behaviour and the minimum (shown by arrow) det ermines the op-\ntimized lattice constant ( a=11.059 a.u.=5.85 ˚A) (Fig. 2a). The agreement is within 1% of\nthe experimental value of 5.9072 ˚A11. For the martensitic phase, in the first step, Etot(V)\nis calculated to obtain optimized V=1330 a.u.3at fixedc/a=1.21 (XRD value). Next,\nEtot(c/a) is calculated at V=1330 a.u.3. This gives the optimized c/ato be 1.25. In the\nfinal step, Etot(V) is calculated again with c/a=1.25 (Fig. 2a). Least square fitting of the\ndata8,23gives the Etotminimum at 1335.2 a.u.3(shown by arrow). From Fig. 2a, the Etot\nminimum in the martensitic phase is 6.8 meV/atom lower than the austen itic phase. This\ndemonstrates that the martensitic phase is stabilized through a siz able tetragonal distortion\n(c/a=1.25). The optimized lattice constants ( a=5.409 and c=6.762, ˚A) are within 2.1%\nand0.85%oftheexperimental latticeconstants a=5.5272 ˚A andc=6.7044 ˚A,respectively11.\nThus, the agreement of the lattice constants for both the phase s is satisfying, considering\nthat even forfree-electron-like non-magneticmetals therecould beabout 2%discrepancy be-\ntween experiment andGGAbased DFTtheory24. The decrease of Vby 1.2%is inagreement\nwith the experimental volume decrease of 0.64% in the martensitic ph ase11.\nThelowestenergymagneticstateisobtainedbyperforming Etotminimizationovervarious\npossible starting MnI and MnII magnetic moment combinations, as dis cussed in details in\nRef.25. For both austenitic and martensitic phase, the anti-parallel star ting spin (equal or\nunequal)configurationsofMnIandMnIIconvergetoaferrimagne ticstatethathasminimum\nEtot. We have used starting Mn magnetic moments for structure optimiz ation runs to be\n3µBfor both Mn atoms in anti-parallel orientation. However, when the s tarting MnI and\nMnII moments are parallel (equal or unequal), Etotconverges to different magnetic moments\nrelated to local minima at higher energies. For example, in the austen itic phase there are\nthree local minima25. Also in the martensitic phase, multiple local minima are obtained\nwith parallel starting moments of MnI and MnII. In particular, a loca l minimum that is 108\nmeV/atom higher in Etot, gives MnI and MnII moments to be 0.24 and 2.38 µB25. Thus,\none Mn moment is small, as has been reported in Ref.14. Our calculation based on the\nmagnetic moments reported in Ref.14converges at 193 meV/atom higher energy than the\nEtotminimum25. This gives an idea why the results from Ref.14are in disagreement with\nexperimental data, as discussed later.\n4The spin magnetic moment distribution in the martensitic phase clearly shows that it is\nferrimagnetic with MnI magnetic moment anti-parallel and smaller tha n MnII (Fig. 2b). Ni\nmoment is small and is parallel to MnII moment. For the martensitic (a ustenitic) phase,\nthe local spin magnetic moments are -2.21 (-2.43), 2.91 (3.2), 0.27 (0 .32), 0.01 (0.01) µB\nper formula unit ( µB/f.u.) for MnI, MnII, Ni, and Ga, respectively. The moment related\nto the interstitial charge is small (-0.04 µB). The total moment for the martensitic phase\n(1.01µB/f.u.) is 11% less than the austenitc phase (1.14 µB/f.u). The lowering of the mag-\nnetic moment in the martensitic phase has been reported by Liu et al.from magnetization\nstudies: 1.21 µB/f.u. (28.28 emu/g) and 1.29 µB/f.u. (30.3 emu/g) in the martensitic and\naustenitic phase, respectively11. Thus, the magnetic moment values and the trend that\nmagnetization is lower in the martensitic phase are in agreement with o ur calculations.\nDensity of states and photoemission spectroscopy: The stabilization of the tetrago-\nnally distorted martensitic phase in Ni 2MnGa has been related to band Jahn-Teller effect,\nwhere a DOS peak at EFin the cubic phase splits into two peaks below and above EFin\nthe tetragonal phase, resulting in a lowering of the total energy17. Splitting and shift of the\nDOS peaks just below EFhave also been observed in Ni 2.25Mn0.75Ga9. For Mn 2NiGa, the\ndifferences inthetotalDOSnear EFareinteresting: apeakat-0.1eVintheausteniticphase\nshifts to lower energy (-0.35 eV) and diminishes in intensity in the mart ensitic phase (both\npeaks indicated by arrows). The peak above EFat 0.35 eV (tick) does not shift but is en-\nhanced in intensity in the martensitic phase indicating a transfer of D OS from the occupied\ntotheunoccupiedstates. FromthepartialDOS(PDOS),itiscleart hatthepeaksat-0.1and\n-0.35eVarise primarily due to Ni 3 dand MnI3 dhybridization. The shift of the-0.1eV peak\nto lower energy in the martensitic phase results from enhanced Ni 3 d- MnI 3dhybridization\ncaused by decrease in Ni-MnIdistance from2.925 ˚A(austenitic) to 2.701 ˚A(martensitic) and\nis a possible reason for the stabilization of the martensitic phase. Th e DOS at EFis sub-\nstantially reduced in the martensitic phase (1.29 states/eVf.u.) com pared to the austenitic\nphase (3.39). Thus, decrease in electronic specific heat in the mart ensitic phase could be\nexpected.\nThe antiferromagnetic alignment of MnI and MnII spin moments can b e understood from\nthe 3dspin resolved PDOS (Fig. 3b). MnI 3 dminority-spin states appear below EFbetween\n-1to -3.5eV, whereas MnII 3 dmajority-spin states appear below EFwith two well separated\nhigh PDOS region around -1.5 and -2.7eV. MnI 3 dmajority-spin states appear primarily\n5aboveEFcentered around 0.7eV; while MnII 3 dminority-spin states appear above EF\nwith the main peak at 1.1eV and a smaller peak at 0.35eV. Thus, while the minority-spin\nstates are mostly excluded from the MnII 3 dshell, the majority-spin states are excluded\nfrom the MnI 3 dshell resulting in large but oppositely aligned moments. MnI and MnII a re\nnearest neighbors (n.n.) with n.n. distance of 2.549 (2.533) ˚A in the martensitic (austenitic)\nphase. The exchange pair interaction as a function of Mn-Mn separ ation was calculated by a\nHeisenberg-like model and an antiferromagnetic coupling at short in teratomic distances was\nfound that becomes ferromagnetic at larger distances12. Thus, direct Mn-Mn interaction\nat short interatomic distance is responsible for their opposite alignm ent12,13. The energy\nseparation between the centroid of the occupied and the unoccup ied spin states of opposite\npolarization gives an exchange splitting of 2.7eV (3.1eV) for MnI (MnI I) in the martensitic\nphase. In the austenitic phase, the exchange splittings are 2.8 and 3.6 eV for MnI and MnII,\nrespectively. Thus, the Stoner parameter (ratio of exchange sp litting and magnetic moment)\nis roughly about 1 eV/ µBin both phases, which is characteristic of itinerant magnetism.\nIt was shown for Mn excess Ni 2Mn1+xGa1−xthat the magnetic moments of Mn atom in\nGa site is equal but anti-parallel to the Mn atom at Mn site26. This would tend to suggest\nthat in Mn 2NiGa, the Mn moments would cancel and a small total moment might re sult\nfrom Ni. However, this does not happen and the difference of MnI an d MnII moments is key\nto the larger total moment ( ≈1µB). This originates fromthe stronger hybridization between\nthe majority-spin Ni and MnII 3 dstates in comparison to hybridization between Ni and MnI\n3dminority-spin states. Note that Ni and MnII are n.n. separated by 2.549 (2.533) ˚A in\nthe martensitic (austenitic) phase and stronger hybridization pulls down almost all the MnII\n3dmajority-spin states below EFresulting in strong spin polarization and larger moment.\nOn the contrary, hybridization between Ni and MnI 3 dminority-spin states is relatively\nweaker, distance being larger: 2.701 (2.925) ˚A in the martensitic (austenitic) phase, and\nthere are sizable MnI 3 dminority-spin states above EFincluding the 0.35 eV peak, resulting\nin smaller moment on MnI.\nPhotoemission spectroscopy is a direct probe of the DOS in the VB re gion. In Fig. 4, the\nmain peak of the UPS VB spectrum appears at -1.4 eV and the Fermi c ut-off is at 0 eV.\nIn order to calculate the VB spectrum, we note because of the ord er of magnitude larger\nphotoemission cross-sections of Ni 3 dand Mn 3 d(4.0 and 5.3 mega barns at h ν=21.2 eV,\nrespectively)27, these PDOS determine the shape of VB28. So, we have added the Ni and\n6Mn 3dPDOS in proportion to their cross-sections, multiplied by the Fermi f unction and\nbroadened by the instrumental Gaussian resolution and the life-tim e width related energy\ndependent Lorenzian to obtain the calculated VB (Fig. 4). This is a st andard procedure of\ncomparing the photoemission spectrum from a polycrystalline sample with the calculated\nDOS28,29. The position of the main peak at -1.4 eV and the ratio between the ma in peak\nand the intensity at EFare in good agreement with UPS VB spectrum. It is clear from\nFig. 4 that the main peak is dominated by Mn 3 d-Ni 3dhybridized states that have almost\nequal contribution. States near EFare dominated by Mn 3 dstates, and the MnI 3 din\nparticular.\nThe martensitic phase DOS from Ref.14, obtained by adding up the majority and\nminority-spin DOS from Fig. 5 of Ref.14, is in clear disagreement with our DOS (Fig. 3a).\nThis prompted us to calculate the VB spectrum from the PDOS of Ref .14following the same\nprocedure as discussed above and compare it with the experimenta l UPS VB. As shown in\nFig. 4, the calculated VB based on Ref.14is in obvious disagreement with UPS VB: no\nclear peak is observed in the former; a weak broad feature is prese nt at -2 eV and the in-\ntensity near EFis highest. This shows that the martensitic phase DOS reported in Re f.14\nis inconsistent with experiment. Moreover, the large change of loca l moments (austenitic\nMnI=-2.2, MnII=3.15, Ni 0.27 µBto martensitic MnII=Ni ≈0, MnI=1.4 µB) obtained in\nRef.14is physically unexpected25, since the MnI-MnII distance change by only 0.6% in the\nmartensitic phase. Thus, it is no wonder why the total moment repo rted in Ref.14is higher\nin the martensitic phase compared to the austenitic phase, in contr adiction to their own\nmagnetization data11,14.\nElectronic bands and Fermi surface: Austenitic phase majority spin states: We now\nturn to the discussion of the electronic bands and Fermi surface o f Mn2NiGa. The majority-\nspin bands in the austenitic phase show that band 29 forms electron pockets (Fig. 5b). The\ncorresponding FS, shown in Fig. 5d, is distorted prolate ellipsoidal in s hape and occurs\naround the Xpoint of the Brillouin zone (BZ) with the long axis along the Γ Xdirection.\nThe BZ is shown in Fig. 5a. The projection of the FS along Γ Xis a square (inset, Fig. 5d),\nwhich indicates that the FS nests onto itself with n.v. 0.44(1,0,0) and 0 .44(0,1,0), in units\nof 2π/a(=1 a.u.). The nested portion of the FS is a rhombus (shown by black lin es in\nFig. 5d) of area 0.052 a.u.2with an opening angle of about 15◦.\nMartensitic phase majority spin states: In the martensitic phase, the majority-spin FS\n7TABLE I: Nesting vectors for the Fermi surface of Mn 2NiGa, in units of 2 π/a(=1 a.u.).\nAustenitic phase Martensitic phase\nBandno. Majority spin Minority spin Majority spin Minority\nspin\n29 0.44(1,0,0),\n0.44(0,1,0)– 0.34(1,0,0),\n0.34(0,1,0)–\n28 – 0.31{1,0,0}0.75(1,1,0),\n0.75(1,-1,0),\n1.13(0,0,1)–\n27 – 0.4{1,0,0}– –\nexhibits interesting modification (Fig. 5e). The majority-spin band 2 9 related electron type\nFS is now connected as continuous pipes along (1,0,0) direction, but w ith varying cross-\nsection with flat parallel parts that nest onto each other (green/ pink sheet in Fig. 5e). The\nn.v. are 0.34(1,0,0) and 0.34 (0,1,0), and compared to the austenitic p hase the direction\nis same but the magnitude of the n.v.’s is reduced. Interestingly, a se cond majority-spin\nband (28) crosses EFthat results in a hole-type cuboid FS around the Γ point that has no\ncounterpart in the austenitic phase (blue sheet, Fig. 5e). Two mut ually perpendicular n.v.’s\n0.75(1,1,0) and 0.75(1,-1,0) are identified, along with a larger n.v. of 1.1 3(0,0,1). The n.v.’s\nalong the {1,0,0}, identified above, are not expected to contribute to phonon soft ening\nbecause these hardly contribute to the electron-phonon coupling matrix element19. On the\nother hand, the 0.75(1,1,0) and 0.75(1,-1,0) n.v.’s might be responsible for the softening\nof the TA 2[110] phonon resulting in a modulated martensitic phase. The differen t nesting\nvectors are shown in Table I.\nFrom Fig. 5d and e, the majority spin FS is clearly enlarged in the marte nsitic phase\ncompared to the austenitic phase. In the contrary, for the minor ity-spin states (Fig. 5f-i),\nthe FS clearly shrinks in the martensitic phase.\nAustenitic phase minority spin states: In the austenitic phase, minority spin band 27 is\nhole-type dispersing above (below) EFat 0.2ΓL(0.5LW) and generates distorted cubic FS,\nwhere one pair of diagonally opposite corners taper out (Fig. 5f). F S nesting is observed\nbetween the cube faces with n.v. 0.4 {1,0,0}, as shown by the yellow arrows. The second\n8sheet of the FS (band 28) is electron-like, consisting of multiply conn ected pipes of square\ncross-section (inset, Fig. 5h). The parallel surfaces of the pipes nest onto each other with a\nn.v. of 0.31 {1,0,0}a.u. and a nesting area of 0.16a.u.2\nMartensitic phase minority spin states: In the martensitic phase, the minority spin hole\ntype FS (band 27) has a flower-like shape with a perforation in the mid dle (Fig. 5g). The\nelectron type FS sheet shrink to disconnected pipes of varying diam eter (Fig. 5i). These\nminority-spin FS sheets (Fig. 5g,i) in the martensitic phase do not exh ibit nesting.\nConclusion: We observe FS nesting in the martensitic phase along [1,1,0] direction in the\nmajority-spin FS that might lead to the instability of the TA 2phonon mode in Mn 2NiGa.\nThe austenitic phase FS is drastically modified in the martensitic phase . The majority spin\nFS expands in the martensitic phase, while the minority-spin FS shrink s. We show that\nMn2NiGa is an itinerant ferrimagnet in both austenitic and martensitic pha se, and that the\nMnII or Ni moments do not become zero in the martensitic phase, re futing a recent work by\nLiuet al.14. The unequal spin magnetic moments in the two inequivalent Mn atoms (MnI\nand MnII) arise from the difference in the hybridization of the MnI 3 d-Ni 3dand MnII 3 d-\nNi 3dstates, which inturn isrelated to theinteratomic distances. We fur thermoreshow that\nin Mn 2NiGa a large tetragonal distortion ( c/a=1.25) decreases the total energy, stabilizing\nthe lower temperature martensitic phase. Mn 2NiGa would be an ideal system to study\ndifferent models of magnetization in metals since it has a simple L21structure and three\nsublatticemagnetizationwithparallel(betweenMnIIandNi)andant i-parallel(betweenMnI\nand MnII) magnetic moment alignment. Possibility of incommensurate magnetic phase or\ncharge density wave instabilities could be expected at low temperatu res due to presence\nof FS nesting and ferrimagnetism. Low temperature x-ray diffract ion might be able to\ndetect possible occurrence of a charge density wave state. Neut ron scattering, angle resolved\nphotoemission or Compton scattering experiments can verify the t heoretically predicted FS.\nIn fact, FS nesting, ferrimagnetism and large magnetoelastic coup ling makes Mn 2NiGa a\nhighly interesting material that has remained largely unexplored so f ar.\nWe thank K. KUNC and A. DE SARKAR for fruitful discussions. P. CHA DDAH, V.\nC. SAHNI, K. HORN, A. GUPTA and S. M. OAK are thanked for suppor t. Ramanna\nFellowship Research Grant and D.S.T.-Max Planck Partner Group Proj ect are thanked for\n9funding.\n1KAINUMA R. et al., Nature, 439(2006) 957.\n2TAKEUCHI I. et al., Nature Materials, 2(2003) 180.\n3SOZINOV A., LIKHACHEV A. A., LANSKA N., AND ULLAKKO K., Appl. Phys. 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Rev. B, 75(2007) 134403.\n16BANIK S., AHUJA B. L., CHAKRABARTI A., PANDEY D. AND BARMAN S. R. (to be\npublished).\n17FUJII S., ISHIDA S., ASANO S., J. Phy. Soc. Japan, 58(1989) 3657.\n18GODLEVSKY V. V. AND RABE K. M., Phys. Rev. B 63(2001) 134407; AYUELA A., ENKO-\nVAARA J., AND NIEMINEN R. M., J. Phys. : Condens.Matter 14(2002) 5325; BIHLMAYER\nG., EIBLER R., AND NECKEL A., J. Phys. Condens. Matter 5(1993) 5083.\n19BUNGARO C., RABE K. M., AND DAL CORSO A., Phys. Rev. B, 68(2003) 134104; ZHOU\nG. L. AND HARMON B. N., Phys. Rev. B, 48(1993) 2031; ZAYAK A. T., ENTEL P.,\n10ENKOVAARA J., AYUELA A., AND NIEMINEN R. M., Phys. Rev. B, 68 (2003)\n132402; LEE Y., RHEE J. Y., AND HARMON B. N., Phys. Rev. B, 66 (2002)\n054424.\n20BLAHA P., SCHWARTZ K., AND LUITZ J., WIEN97, (1999, ISBN 3-95 01031-0-4.\n21PERDEW J. P., BURKE K., AND ERNZERHOF M., Phys. Rev. Lett., 77(1996) 3865.\n22A. KOKALJ, J. Mol. Graphics Modelling, 17(1999) 176; Comp. Mater. Sci. 28(2003) 155.\n23CHAKRABARTI, A., Phys. Rev. B 62(2000) 1806.\n24PERDEW J. P. et al., Phys. Rev. B, 46(1992) 6671.\n25BARMAN S. R. AND CHAKRABARTI A., Phys. Rev. B (Comment, submitted ).\n26ENKOVAARA J., HECZKO O., AYUELA A., AND NIEMINEN R. M., Phys. Rev. B, 67\n(2003) 212405.\n27YEH J. J. AND LINDAU I., At. Data Nucl. Data Tables, 32(1985) 1.\n28CHAKRABARTIA. , BISWASC., BANIKS., DHAKAR.S., SHUKLAA.K. , ANDBARMAN\nS. R., Phys. Rev. B, 72(2005) 073103.\n29SARMA D. D. et al., Phys. Rev. Lett., 75(1995) 1126; FUJIMORI A. AND MINAMI A.,\nPhys. Rev. B, 30(1984) 957; BARMAN S. R. AND SARMA D. D., Phys. Rev. B, 51(1995)\n4007; BARMAN, S. R. et al., Phys. Rev. B 53(1996) 3746; BROWN D. et al., Phys. Rev. B\n57(1998) 1563.\n11Figure Captions\nFig. 1The structure of Mn 2NiGa in the (a) austenitic and (b) martensitic phase; the blue,\ngreen, red, and brown spheres represent Ni, MnI, MnII and Ga, r espectively.\nFig. 2 (a) The calculated total energies ( Etot) of Mn 2NiGa as a function of cell volume\nof the austenitic and martensitic phase. (b) Three dimensional plot of the spin magnetic\nmoment distribution (in unit of e ˚A−3) in the (110) plane in the martensitic phase, a contour\nplot is shown in the bottom.\nFig. 3(a) Comparison of total density of states (DOS) and Ni 3 dand Mn 3 dpartial DOS\nof Mn 2NiGa between the martensitic and austenitic phases (b) minority- an d majority-spin\ncomponents of the DOS in the martensitic phase.\nFig. 4UPS valence band (VB) spectrum of Mn 2NiGa in the martensitic phase compared\nwith theoretical VB spectrum calculated from the DOS in Fig. 3a. The contributions from\nthe Mn 3 dand the Ni 3 dpartial DOS are also shown. The spectra have been shifted along\nthe vertical axis for clarity of presentation.\nFig. 5 (a) The f.c.c. Brillouin zone showing the high symmetry directions. (b) Majority\nand(c)minority-spinenergybandsofMn 2NiGaintheausteniticphase. Majority-spinFermi\nsurface (FS) of the (d) austenitic phase compared to the (e) mar tensitic phase FS related to\nbands 28 and 29. Minority-spin austenitic phase FS related to (f) ba nd 27 and (h) band 28.\nInsets show the FS in a different orientation. Martensitic phase mino rity-spin FS related to\n(g) band 27 and (i) band 28. All the FS are shown in the repeated zon e scheme and yellow\narrows represent the nesting vectors. Black arrows relate the F S of the two phases.\n12This figure \"Fig1_mngth.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2This figure \"Fig2m_mngth.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2This figure \"Fig3_mngth.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2This figure \"Fig4m_mngth.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2This figure \"Fig5m_mng.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2"    },    {        "title": "0810.0449v1.Structural_phase_stability_and_Magnetism_in_Co2FeO4_spinel_oxide.pdf",        "content": "Structural phase stab ility and Magnetism in Co 2FeO 4 spinel oxide \n \nI. Panneer Muthuselvam and R.N. Bhowmik*\n \nDepartment of Physics, Pondicherry University, R. Venkataraman Nagar, \nKalapet, Pondiche rry-605014, India \n \n*E-mail address for corresponding au thor: rnbhowmik.phy@pondiuni.edu.in \n \nAbstract \nWe report a correlation between structural phase stability and magnetic properties of \nCo2FeO 4 spinel oxide. We employed mechanical alloying and subsequent annealing to \nobtain the desired samples. The particle size  of the samples changes from 25 nm to 45 \nnm. The structural phase separation of samples, except sample annealed at 9000C, into \nCo rich and Fe rich spinel phase has been examined from XRD spectrum, SEM picture, \nalong with EDAX spectrum, and magnetic m easurements. The present study indicated \nthe ferrimagnetic character of Co 2FeO 4, irrespective of structur al phase stability. The \nobservation of mixed ferrimagnetic phases, asso ciated with two Curie temperatures at \nTC1 and T C2 (>T C1), respectively, provides the additional support of the splitting of single \ncubic spinel phase in Co 2FeO 4 spinel oxide.  \n \nKey words: A. Spinel Ferrite; B. Mechanical alloyed nanomaterial; C. Structural phase \nstability; D. Ferrimagnetism. \n \nPACS:  61.66.Fn, 61.46.-w, 75.75.+a, 75.50.Kj \n \n \n \n \n \n I. INTRODUCTION \n Spinel ferrites are rep resented by  the form ula unit AB 2O4.  Most of the spinel \nferrites for m cubic spinel structure with oxyge n anions in fcc position s and cations in the \ntetrahedral and octah edral coordin ated in terstitial lattice  sites, f orming the A and  B \nsublattices [1]. Depending upon the nature (m agnetic or non-m agnetic) and distribution \nof cations among A and B sublattices, spinel ferrites can exhibit pr operties of different \ntype m agnets, like: ferrim agnet, antiferro magnet and param agnet. The inter-sublattice \ninteractions (J AB: A-O-B) is m uch stronger than th e intra-sublattice interactions (J AA: A-\nO-A and J BB: B-O-B) in spinel f errites with co llinea r ferrimagnetic str ucture. Exte nsive  \nworks on certain spin el ferrites  have been carr ied out for the last few decades, becau se of \ntheir theoretical understanding and potential ap plications in science and technology. For \nexam ple, Fe 3O4 (magnetite) derived com pounds (Fe 3-xMxO4: M is  magnetic or  non-\nmagnetic elem ents-Co, Mn, Zn, etc.) [2-4] have drawn a lo t of research interes ts for their \nexhibition of m any unusual phys ical properties [5-7], in cluding high m agnetic m oment, \nmagneto-res istance, half  metallic behaviour. In particula r, the cob alt subs tituted \nmagnetite (F e3-xCoxO4) could be m ore attractive due to ty pical anisotrop y character of Co \nions. However, m ost of the reports [4, 8, 9] are  limited to the Fe rich regions (x ≤1) of \nFe3-xCoxO4 series. A fe w reports [10] are only av ailable for the Co rich regions (x ≥ 2), \nalthough these com pounds have potential app lications in chem ical sensors [11, 12], \ncatalytic activity [13] and phot o-conductive m aterials [14-16]. Co 3O4 is an  \nantiferromagnet (T N∼30 K) but its derivative compound Co 3-xMxO4 [M =  Al, Mn, Fe] are \ngenerating a lot of re search interest in recen t tim es [17, 18]. The study of Co rich \ncompounds  could be of interesting in view of the diversity in magnetic properties \n(ferrim agnet, antiferromagnet and a ntiferrom agnetic spin glass) and magneto-transport \nphenom ena (colossal m agnetoresistance) in Co xMn 3-xO4 (0 <x <3) spinel oxides [19, 20].  \nIn the present work, we focus on the spinel compound Co 2FeO 4. The experim ental \nwork on this com pound is very lim ited. The probable reason could be  the spinel phase \nstability over a sm all temperature range about 900oC [21, 22].  A few works m ay be \nmentioned here, which indicated that  the distribution of cations in Co 2FeO 4 is very much \nsensitive on  the prep aration techniques. Murray et al.  [23] suggested that the probable \ndistribution of cations in Co 2FeO 4 could be (Co2+\n0.7Fe3+\n0.3)A[Co2+\n0.3Fe3+\n0.7Co3+ ]BO4. \nBased on the Mössbauer study [24], Magnetic measure ment [25] a nd Neutron diffraction \n 2study [22], the estim ated dist ribution of cations in Co 2FeO 4 are \n(Co2+\n0.55Fe3+\n0.45)A[Co2+\n0.45Fe3+\n0.55Co3+]BO4, (Co2+\n0.6Fe3+\n0.4)A[Co2+\n0.4Fe3+\n0.6Co3+]BO4, and \n(Co2+\n0.82Fe3+\n0.18)A[Co2+\n0.18 Fe3+\n0.82Co3+]BO4, respective ly. The presence of  multi valen ce \ncations in B sublattice m ay attribute to  a good m agnetic sem iconducting property in \nCo2FeO 4, although the experim ental re sults [16-18, 26] indicated an increase of band gap \nenergy (0.14 eV to 1.1 e V) with the increase of Co content in Fe 3-xCoxO4. It is also noted \nthat Co 2FeO 4 is prepared in m ost of the reports using different chem ical m ethods, e.g., \ncoprecipitation [21, 27] and com bustion reaction [28].  The fact is that ph ysical properties  \nmay depend upon the preparation technique as well as particle m orphology (size, shape) \nof the sam ple [29]. \nConsidering the theoretical interest [ 10] and technological applications of \nCo2FeO 4, detailed inform ation of t he correlati on between crystal structure and physical \nproperties are very m uch essential. However, lim ited inform ation on this aspect of \nCo2FeO 4 can be obtained due to lack of suffi cient ex perim ental work. Here,  we \nemphasize on the understanding of structural pha se stabilization and m agnetic properties \nof mechanical alloyed C o2FeO 4 nanoparticles. \n \nII. EXPERI MENTAL  \nA. Sample preparation  \nStoichiom etric am ounts of high purity (>99.5) Fe 2O3 and Co 3O4 were m ixed to \nobtain the required com pound. The m ixture wa s ground using m ortar and pestle for 2 \nhours. The m ixed powder was m echanical al loyed using Fritsch Planetary Micro Mill \n“Pulverse tte 7”. The m illing was carr ied ou t in a 50 m l silicon nitr ide bow l in \natmospheric  conditions.  The m ass ratio of  the ball (10 mm Silicon Nitrid e and 5 mm \nTungsten Carbide) and  material w as maintain ed to 4:1.  The m illing was contin ued at \nrotational speed 300 rpm up to 100 hours with  interm ediate stopping for proper m ixing \nand m onitoring the phas e evolution of th e alloy ed com pound. After 100 hours m illing, \nthe alloyed powder was m ade into pellets of se veral batches. Each pellet was annealed at \ndifferent temperatur es in the range 700oC to 1000oC. The annealed samples hav e been  \ndenoted as SX,  where X =70, 80, 86, 90, 95 and 100 for annealing tem perature at 700oC \n(3 hours), 800oC (6 hours), 860oC (12 hours), 900oC (12 hours), 950oC (12 hours) and  \n1000oC (3 hours), respectively.  \n 3B. Sample Characterization \nThe crystallographic ph ase of the s amples has been exam ined from  the X-ray \ndiffraction (XRD) spectrum  using X-Pert PANa lytical dif fractom eter. The spec trum of \neach sam ple was reco rded at 300  K using Cu K α radia tion in th e 2θ range 10 to 90 \ndegrees with step size 0.01 degrees. The s canning electron m icroscope (SEM) (m odel: \nHITACHI S-3400N, Japan) was em ployed to  study the surface m orphology of the \nsamples. The elem ental analys is of the sam ples was carried out us ing E nergy Dispersive \nanalysis of X-ray (EDAX) spect rometer (Therm o electron cor poration Instrum ent, USA).   \nMagnetic p roperties of  the sam ples were in vestig ated f rom the dc m agnetiza tion \nmeasure ments using Vibrating Sam ple Ma gnetom eter (Model: 7404 LakeShore, U SA), \nwith high temperature o ven attachment.  The temperature depende nce of m agnetization \nwas carried out at 1 kOe m agnetic field by increasing the temperature from  300 K to 900 \nK (ZFC m ode) and  reversing  back  the tem perature to 30 0 K in the p resence of s ame \napplied field 1 kOe (FC m ode). It should be noted that the ZFC m ode denoted here is \nlittle bit different from  the c onventional zero filed cooling (ZFC) m easurem ent, where the \nsample is first cooled without applying m agnetic field from  the tem perature greater than \nTC to the tem perature lower than T C and m agnetization m easure ment starts in the \npresence of m agnetic field by increasing th e tem perature. The field dependence of \nmagnetization of the sa mples was m easured at 300 K in the field range ± 15 kOe. It m ay \nbe m entioned that th e VSM was calib rated using a standard Ni ferrom agnet before \nstarting the m easurem ent of sam ple. \n \nIII. RESULTS AND DISCUSSION \nThe X-ray diffraction  pattern of  Co 2FeO 4 samples are shown in Fig. 1. The XRD \nspectrum  of 100 hours m illed sam ple (d ata not shown) is  largely  dom inated by sp inel \nphase, along with a sm all fraction of unreacted α-Fe 2O3 phase. There is n o trace of peak  \nlines corresponds to α-Fe 2O3 phase after ann ealing the a lloyed sample at different \ntemperatures. It is found (Fig .1) that XR D pea ks of the annealed sam ples, except 900oC \nand 950oC, are splitted into two components. We are confir med, from  the com parative \npeak positions of XRD spectra of our sa mples and spectrum  of standard spinel \ncompounds (Co 3O4, CoFe 2O4) using the sam e X-Ray diffractom eter, that additional peak \nlines are not contributed due to Cu K α2 radiations. The splitted peaks are, in fact, \n 4matching to the spe ctra of two cubi c spinel phase s with a  shift in 2 θ values. The splitting \nof spectra l lines is  noted  in Fig. 1 (left hand side: 2 θ range 10 to 900) and clearly shown \nfor (311) XRD peak line alone in the right ha nd side of Fig. 1. The peak positions at \nhigher and lower 2 θ values are denoted at 2 θ2 and 2 θ1, respectiv ely.  As the ann ealing  \ntemperature increases from  700oC to 1000oC, the peaks  at 2θ1 and 2 θ2 are com ing closer \nto each oth er and there is no splitting at 900oC (S90) sam ple, suggesting single phased \ncompound. Although there is no clear sp litting in (311) XRD line for 950oC (S95) sam ple \n(i.e., the sample is appeared to be in single phase ), but there is a tendency of m inor \nsplitting in other XRD lines of 950oC sam ple in com parison with 900oC sam ple.  This \nindicates that a m inor s econdary phase m ay coe xist at 950oC. The peaks correspond to \n2θ1 and 2 θ2 are again well separated at 1000oC, indicatin g the reapp earance of phase \nseparation in the m aterial. The results of our mechanical alloyed and subsequent ann ealed  \nsamples also confirm  the earlier reports [21, 22] that Co 2FeO 4 is stab ilized in to a s ingle \ncubic spinel phase at about 900oC (S90), although our preparation technique (m echanical \nalloy ing) is com pletely dif ferent f rom the re ported works. It m ay be m entioned that we \ndid not find any significant change in the XR D spectrum when slow cooled sample is  \ncompared with th e direct air quench ed sam ples. Sm ith et al. [24] has reported the sam e \neffect from  X-ray diffraction and M össbauer m easurem ents. \nThe lattic e param eter was calcu lated by m atching the XRD peaks at 2 θ1 and \n2θ2 separately with cubic spinel structure, considering the coexis tence  of two spinel \nphases in the spectrum .  The la ttice param eter corresponds to 2 θ1 and 2 θ2 are denoted as  \na1 and a2, respectively. The data are shown in Fi g. 2a. The lattice param eter a1 decreases \nas the annealing tem perature increases from  700 t o 9000C, unlike the incre ase of  a2. Both \n(a1 and a2) are coin ciding for 900oC (S90) sam ple. In the absence of clear splitting for the \n950oC (S95) sam ple, we have f itted the spec trum assum ing the sing le phase. The la ttice \nparam eter of 950oC sa mple is  little bit high er in com parison with 90 0oC sam ple. The \nlattice pa rameters a1 and a2, again, separated for the 1000oC sam ple with a1 higher than \na2.  The calculated lattice param eter a (~8.24 Å) for the  900oC sample is in good \nagreem ent with the rep orted data both from  theoretical calculation ( a =8.27 Å) [10] and \nexperim ental work ( a =8.24 Å) [24]. The lattice param eter for single phased Co 3O4 and \nCoFe 2O4 is found ~ 8.082 Å and 8.40 Å, respectively. On the other hand, a1 and a2 are ~  \n 58.37 Å and 8.13 Å for S70 sam ple and ~ 8.31 Å and 8.16 Å for S100 sample. In betw een, \nthe sys tem stabilize s to single  phased cubic spinel structure of Co 2FeO 4 with a ~8.24 Å. \nViewing the different values of a1 and a2, we suggest that a1 (corresponds to peaks at 2 θ1) \nis contributed from  Fe-rich cubic spinel structure and a2 (corresponds to peaks at 2 θ2) is \ncontributed from  Co-rich cubic spinel stru cture. W e have attem pted to es timate the  \nfraction of coexisting tw o phases from the relative peak heights at 2 θ1 and 2θ2  positions, \nassum ing that the add ition of  two peak heights will contrib ute to th e total peak h eight of \nsingle phase cubic spinel structure. Fig.2b shows the % height of splitted p eaks at 2 θ1 and \n2θ2  with respect to (311), (004) and (333) p eaks of single phased cu bic spinel structure \n(S90 sam ple). W e observed that Fe-rich ph ase increas es fro m ∼60% (S70) to 100% (S90 \nand S95) by decreasing the Co-rich phase from ∼40% to 0%. For S100 sa mple, the Co-\nrich phase increas es by decreasing the Fe-rich p hase. The change of relative peak heights  \nof the two phases with annealing temperature is  found to be sam e within the error lim it of \npeak heigh t determ inatio n for all th e three p eaks ( Fig. 2b).  \nThe partic le size of  the sam ples was de termined using Debye-Scherrer for mula: \n<d> = 0.89 λ/βcosθ  (λ the wavelen gth of  the X-ray, 2 θ corresponds to position of pea k \nheight, <d> is particle size, β is the f ull width  at half  maximum  of peak height) on f our \n((311), (004), (333), (044)) XRD peaks at 2θ1. The par ticle size (in Table I) showed the \nusual increase ( ∼25 nm  to 45 nm ) with annealing tem peratures, as an effect of therm al \ninduced grain growth kinetics. The SEM pictur es of selected sam ples are shown in Fig 3 \n(a-d). The general observation from the Fig. 3 (a-d) is tha t therm al annealing incr eases \nthe pa rticle  size of  the samples. The particles are in agg lomerated state in the as alloyed \nsample (MA100), whereas a better hom ogeneity  in the particle size distribution is \nobserved in  the anne aled sam ples. The elem ental com position  of the sam ples is \ndeterm ined from the EDAX spectrum over a selected zone.  The spectru m of each sample \nwas recorded for 10 points and the selected spec trum  is shown in Fig. 3e-h.  The expected \natom ic ratio of Co and Fe in Co 2FeO 4 must be 2:1. The spectrum  of M A100 sam ple (Fig. \n3e) suggested an inhomogeneous elem ental distribution. This m eans som e points are \nhaving m ore Fe atom  (Co: Fe=2:1.35) and ot her points are having less Fe atom s (Co: Fe= \n2:0.85) compared to the expected value (2:1). The Fe rich region in MA100 is probably \ndue to the p resence of a fraction of unreacted Fe 2O3 in the sa mple, as see n from the XRD \n 6spectrum . The chem ical inhom ogene ity is agai n noted in the EDAX spectrum  of anne aled \nsamples, except S90 sample (Fig. 3g). For S8 0 sample (Fig. 3f), there are som e points \nwhich are Co-rich (Co: Fe~2.078:1) and some  points are Co-deficient (Co:Fe~1.78:1). \nThe S100 sample (Fig. 3h) also showed sim ilar atom ic distribution with Co rich (Co: Fe  \n~3.053:1) zone and Co-deficient (Co:Fe ~1.64:1)  zone.  On the other hand, the Co and Fe  \natom s are almost hom ogeneously distributed over the selected zone of S90 sa mple and \nthe atom ic ratio (Co:Fe~1.95:1) is close to  the expected value (2:1). The atom ic \npercentage of the non-m agnetic impurity atom s (Si ~ 0.7 % and W  ~ 0.4 %) from  the \nmilling bowl and balls is very insignif icant in  the alloyed  as well as in the ann ealed \nsamples.  \nThe tem perature (T) dependence of m agnetization, m easured under ZFC and FC \nmodes at 1 kOe, is shown in Fig. 4a-e. The m agnetic  irrev ersibility between  FC \nmagnetization (MFC) and ZFC m agnetization (MZF C) is seen f or all the sam ples. The \nnature of  irreversib ility is distin ct for individual sam ple. The  mixed ferrim agnetic phase \nof the sa mples, except S90, is reflected from  the m agnetization plots. The param agnetic \nto ferrim agnetic ordering tem perature (T C1) of one phas e is determ ined from  the \ninflection p oint of  MZFC curves. It m ay be noted th at the irreversibility between  MFC \nand MZFC occured at tem perature that is much higher than T C1. We define the \nirrev ersibility point as the param agnetic to f errimagnetic tr ansition tem peratu re (T C2) of \nthe second ferrim agnetic phase (T C2 > T C1). The  coexis tence of two m agnetic  phases  is \nalso indicated in  the MFC(T) cu rve. The MFC curves continued  to inc rease with \ndecreasing the tem perature below irrevers ibilit y point, but a change of slope is m arked \nnear to T C1 of the sam ple. The signature  of changing slope was also  verified from t he first \norder de rivative of  MFC(T) (da ta are not shown).  The m agnetic irreve rsibility o ccurs  at T \n≤ 450 K for S90 sa mple and there is no change of slope in M FC(T) curve of the sa mple. \nThis indicates a rem arkable ch ange in th e ferrim agnetic behaviour of S90 sam ple in \ncomparison with other sam ples. The  TC1 of S90 sam ple, determ ined f rom the inf lection  \npoint of MZFC(T), is ∼ 453 K. This is consistent w ith the reported value ~ 450 K of \nsingle phase Co 2FeO 4 [22]. The typical MFC and MZF C behaviour of the samples, \nassociated with m ixed m agnetic phases, are also understood from the tem perature \ndependence of nor malized therm oremanent m agnetization (NTRM = ∆M(T)/ ∆M(300 K), \nwhere ∆M = MFC-MZFC) data (Fig. 4f). W e, now, look at the m agnetic behaviour of the \n 7samples below T C1. The zero  field  cooled m agnetiza tion exhibited blocking behaviour \nbelow the tem perature T m (∼ 420 K for S80, ∼ 360 K for S86 and ∼ 400 K for S100 \nsamples, respectively). The continuous in crease of MZFC down to 300 K for S90 and \nS95 sam ples without blocking of m agnetizat ion suggests that th e possible blocking \nbehaviour for these two nanoparticle sam ples might occur below room  temperature.   \nWe have seen som e interesting m agnetic f eatures of  the sam ples from the f ield \n(H) depend ence of  magnetiz ation  (M) curve a t 300 K (Fig.  5a). The  magnetiz ation  of the \nsamples, af ter rapid  increase within  5 kOe, is  tending to saturate at  higher m agnetic field. \nThe feature suggests a typical long ranged ferrim agnetic character of the sa mples \nirrespective of phase stability. The spontaneous m agnetization (M S) at 300 K is calculated \nfrom  the extrapolation of high field m agnetiza tion da ta to  H = 0 value. The M S value \n(Table I) sh ows decreas ing trend with incr easing the annealing te mperature from  8000C \n(∼23.5 em u/g) to 9000C (∼16 em u/g). After attaining the m inimum value for S90 sa mple, \nthe M S value again is increas ing with anneal ing tem perature. The variation of M S with  \nannealing temperature s hows a close relati on to  the variatio n of lattice param eter a1 (due \nto Fe rich spinel pha se). We, fur ther, try to understand  such correlation from the \nfollowing argum ents. The calculat ed m agnetic mom ent 16 emu/g ( ∼0.68 µB per formul a \nunit of Co 2FeO 4) for S90 sam ple at 300 K is comparable to the reported value (0.70 µB) \nfor single phase com pound [22]. We also note d that m agnetic m oment of S80 and S100 \nsamples ∼1.0 µB per formula unit of Co 2FeO 4 is well be low of  the m agnetic  moment ∼4.2 \nµB per for mula unit of CoFe 2O4. On the other hand, higher Curie tem perature (T C2) of the  \nbi-phase samples, e.g., ~752 K for S80 a nd ~772 K for S100, are close to the Curie \ntemperature (T C ~785 K) of single phase CoFe 2O4 [30]. This m eans the f errimagnetic  \nphase with Curie tem perature T C2 is associated  with the p hase that may not be d ue to \ntypical CoFe 2O4, but definitely due to a Fe-rich spin el ph ase coexis ting with Co-rich  (low \nmagnetic mom ent) phase. Now, we exam ine the M-H loop of the sam ples in Fig. 5 (b-f). \nAlthough measurem ent is  done within H = ± 15 kOe, the data are shown within H = ± 7 \nkOe for clarity of the Figure. It is intere sting to note that M-H loop for S90 and S95 \nsamples is d istingu ished  with m ore symmetric n ature in com parison  with  other  annea led \nsamples. Such interestin g magnetic featur e clearly reflects the undergo ing com petitive \nspin order of two ferrimagnetic domains, aris ing from  the coexistence o f two differen t \ntype cubic spinel phases in the m aterial. The calculated values of coercivity (H C) and \n 8remanent magnetization (M R) from M-H loop (also known as  hysteresis loop) of the \nsamples are shown in Table I.  The coerciv ity (H C) and rem anent m agnetization (M R) are \nshowing sim ilar kind of variation with  annea ling tem perature s imilar to M S, except H C \nand M R attains m inimum value for S 95 and S90 sam ples, respectively. T he small increase \nof M R in S86 sam ple com pared to S80 sa mple is m ost probably related to the more \nasymm etric shape of the M-H loop in S86 sa mple. The M R/MS ratio ( ∼0.42, 0.55, 0.46, \n0.40 and 0.44 for S80, S86, S90, S95 and S100 sam ples, respec tively) suggests that about \n40 to 50% of the spontaneous m agnetization is  retained in the m aterial as soon as the \nmaximum applied field (+15 kOe) is reduced to zero.   The variation  of dM/dH with \napplied m agnetic field (Fig. 6) showed   p eaks, which are alm ost symmetric about the H \n= 0 axis. The peaks are separated by m agnetic field 2H m, as shown for sa mple S80. It ma y \nbe noted  (Table I) that the peak position at H m, i.e., the inflection point in M-H curve,  is \nvery close to the coercivity (H C) of the sam ples, excep t S86 sa mple that has shown more \nasymm etric M-H loop. The peak height of dM/dH at H m (∼ 0.0165, 0.021, 0.026, 0.060 \nand 0.027 in em u/g Oe unit for S80, S86, S90, S95 and S 100 sam ples, respectively) is \nincre asing to attain the  maximum value f or S95 sa mple in com parison with other \nsamples. We noted that the in itial su sceptibility ( dM/dH) H→0) of the sam ple is nearly half  \nof the dM/dH peak height at H m, i.e, at the inflection point of the M-H curves. The peaks \nare com paratively narrowed for S90 and S95 sam ples. This, further, indicates that \nferrim agnetic dom ains a re atta ining a better homoge neous structure in the tem perature \nrange 900-9500C and can be realized from  the single p hase character of the XRD \nspectrum .  \n \nIV. SUMMARY AND CONCL USIONS \nThe present work successfully  applied the novel technique of m echanical alloy ing \nto synthesize Co 2FeO 4 spinel oxide. The structural pha se evolution of the synthesized \ncompound with annealing tem perature consistent with earlier reports, based on chem ical \nrouted sam ples. The experim ental results sugge st that crystal structure of the sam ples, \nexcept 9000C annealed sam ple, is separa ted into th e structure of two cubic spinel phase, \nconsisting of Co-rich and Fe-rich ph ase. The blocking of MZFC below T m is relate d to \nparticle  size  of the sam ples in nano meter range. However, the m aterial does not exhibit \nthe conventional variation (eithe r increase or decrease) of T m with the  increase of  particle  \n 9size, rather the m agnetic blocking of nanopartic les is affected by the phase stability of  \ncubic spinel structure. The detailed low temperature (below 300 K) study for  \nunderstanding the m agnetic blocking behaviour is not within the scope of the pr esent \nwork. The high tem perature study indicated that  the sam ples are f errimagnet, irre spective \nof structu ral phase stability. The co -existence of two f errimagnetic ph ases in the  samples, \nexcept 9000C, further supported the structural pha se separation and confirm ed a strong \ncorrelation between cub ic spinel  structure and m agnetis m in Co 2FeO 4 spinel oxide. 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Curie tem peratures (T C1(K) and T C2(K)) and blocking tem perature (T m(K)),  \nSpontaneous m agnetization (M S), Coerciv ity (H C), rem anent m agnetization (M R) and \ncritical field (H m) of Co 2FeO 4 samples. \nSample  Particle \nSize (nm ) TC1 (K) TC2 (K) Tm (K)    M S \n(emu/g) HC (Oe)  MR \n(emu/g) Hm (Oe)\nS80 \nS86 \nS90 \nS95 \n S100 25 \n29 \n31 \n36 \n42 645 \n555 \n453 \n449 \n581 752 \n750 \n-- \n560 \n772 423 \n358 \n-- \n-- \n396 23.5 \n19 \n16 \n21.6 \n22 696 \n540 \n280 \n125 \n430 9.85 \n10.53 \n7.28 \n8.57 \n9.75 700 \n415 \n300 \n130 \n460 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 12 \n \n \n \n \n 13 \n \n 14 \n \n \n \n \n \n \n \n 15 \n \n \n \n \n 16 \n \n \n \n 17 \n 18"    },    {        "title": "1412.6944v1.Frustration_effects_and_role_of_selective_exchange_coupling_for_magnetic_ordering_in_the_Cairo_pentagonal_lattice.pdf",        "content": "Frustration e\u000bects and role of selective exchange coupling for magnetic ordering\nin the Cairo pentagonal lattice\nA. Chainani1,\u0003and K. Sheshadri2,y\n1RIKEN SPring-8 Centre, Sayo-cho, Sayo-gun, Hyogo 679-5148, Japan\n2226, Bagalur, Bangalore North Taluk, Karnataka - 562149, India\n(Dated: June 1, 2022)\nThe Cairo pentagonal lattice, consisting of an irregular pentagonal tiling of magnetic ions on\ntwo inequivalent sites (3- and 4-co-ordinated ones), represents a fascinating example for studying\ngeometric frustration e\u000bects in two-dimensions. In this work, we investigate the spin S= 1=2 Cairo\npentagonal lattice with respect to selective exchange coupling (which e\u000bectively corresponds to a\nvirtual doping of x= 0;1=6;1=3), in a nearest-neighbour antiferromagnetic Ising model. We also\ndevelop a simple method to quantify geometric frustration in terms of a frustration index \u001e(\f;T),\nwhere\f=J=~J, the ratio of the two exchange couplings required by the symmetry of the Cairo lattice.\nAtT= 0, the undoped Cairo pentagonal lattice shows antiferromagnetic ordering for \f\u0014\fcrit= 2,\nbut undergoes a \frst-order transition to a ferrimagnetic phase for \f > \f crit. The results show that\n\u001e(\f;T= 0) tracks the transition in the form of a cusp maximum at \fcrit. While both phases show\nfrustration, the obtained magnetic structures reveal that the frustration originates in di\u000berent bonds\nfor the two phases. The frustration and ferrimagnetic order get quenched by selective exchange\ncoupling, and lead to robust antiferromagnetic ordering for x= 1/6 and 1/3. From mean-\feld\ncalculations, we determine the temperature-dependent sub-lattice magnetizations for x= 0;1=6\nand 1=3. The calculated results are discussed in relation to known experimental results for trivalent\nBi2Fe4O9and mixed valent BiFe 2O4:63. The study identi\fes the role of frustration e\u000bects, the ratio\n\fand selective exchange coupling for stabilizing ferrimagnetic versus anti-ferromagnetic order in\nthe Cairo pentagonal lattice.\nPACS numbers: 81, 75.10.-b, 61.14.-x\nI. INTRODUCTION\nGeometric frustration on a lattice is the inability to\nconsistently satisfy all pair-wise spin interactions be-\ntween lattice sites as de\fned by a Hamiltonian. The most\ncommon example of geometric frustration is the clas-\nsical nearest-neighbour antiferromagnetic (NNAF) Ising\nmodel on a triangular lattice in two dimensions.1This\nseemingly simple case does not show magnetic order\ndown to the lowest temperature, but instead exhibits a\n\fnite entropy at T= 0. Interestingly, the more general\nor complex case, with the Ising-spins replaced by spin\n1/2 Heisenberg-spins has been shown to exhibit an or-\ndered ground state, which corresponds to the quantum\nanalogue of the classical Neel ground state.2This may be\ncompared with another case of frustration-induced zero-\npoint entropy and absence of ordering, namely, the quan-\ntum spin-liquid known for the S= 1=2 kagome lattice\nHeisenberg antiferromagnet.3These contradictory results\ni.e. presence or absence of ordering, typify the uncer-\ntain role of frustration in two dimensional systems with\ntriangular plaquettes. Extensive studies on a variety of\nlattice spin models in two and three dimensions have in-\ndeed revealed and established a plethora of exotic phases\nand properties due to frustration.4{6. Well-known exam-\nples include resonating valence bond superconductivity7,\n'order by disorder'8, spin-ice9, spin liquid with charge\nfractionalization10, magnetic monopoles11, and so on.\nAlthough the triangular plaquette is the smallest unit\nwith intrinsic frustration, any larger plaquette with anodd number of edges (or vertices) would also naturally\ngive rise to frustration. In particular, a pentagonal pla-\nquette also leads to frustration. While a regular pen-\ntagonal plaquette cannot form a Bravais lattice, there\nare 14 known pentagonal tesselations based on irregular\npentagons. Early studies12{14discussed frustration ef-\nfects for the two-dimensional pentagonal lattice obtained\nfrom the hexagonal lattice by cutting each hexagon into\nhalves with parallel lines(Fig. 1 of ref. 12). In the follow-\ning, we call this pentagonal lattice the P1 lattice. Using\na transfer matrix approach, Waldor et al. showed that\nthe NNAF Ising model for the P1 lattice had a \fnite\nground state entropy due to frustration.12For the NNAF\nHeisenberg model on the P1 lattice, Bhaumik and Bose\nidenti\fed the possibility of a collinear Neel type ground\nstate order.13Moessner and Sondhi investigated the P1\nlattice in terms of a NNAF Ising model with a trans-\nverse \feld, and they identi\fed a novel sawtooth state\nconsisting of zigzag antiferromagnetic stripes coexisting\nalternately with frustrated ferromagnetic stripes.14Uru-\nmov, on the other hand, investigated15a di\u000berent tesel-\nlation, the so-called two-dimensional Cairo pentagonal\nlattice (See Fig. 1). Urumov presented an exact solu-\ntion of the nearest-neighbour ferromagnetic Ising model\nfor the two dimensional Cairo pentagonal lattice by map-\nping it onto a Union Jack lattice with nearest and second\nnearest-neighbour non-crossing interactions.15\nHowever, more recently, since the discovery of an ap-\nproximate experimental realization of the Cairo pen-\ntagonal lattice in Bi 2Fe4O9, which exhibits magneto-arXiv:1412.6944v1  [cond-mat.mtrl-sci]  22 Dec 20142\nelectric coupling16and frustration induced non-collinear\nmagnetism,17there has been a signi\fcant resurgence of\ninterest in the Cairo pentagonal lattice.18{24Ralko dis-\ncussed the phase diagram of the XXZ spinS= 1=2\nsystem under an applied magnetic \feld and showed that\nfrustration leads to unconventional phases such as a ferri-\nmagnetic super\ruid.18Rojas et al. employed a direct dec-\noration transformation approach in order to investigate\nantiferromagnetic as well as ferromagnetic coupling for\nan exact solution of the Cairo pentaonal lattice.19They\ncould thereby show that the phase diagram includes a so\ncalled disordered/frustrated state, in addition to ferro-\nmagnetic and ferrimagnetic phases. In an extensive study\nof the NNAF Heisenberg model on the Cairo pentagonal\nlattice, Rousochatzakis et al. revealed the role of an or-\nder by disorder mechanism and a possible spin-nematic\nphase with d-wave symmetry.20The authors addressed\nthe evolution of the phase diagram from the classical to\nthe quantum limit as a function of \f=J=~J, whereJ\nand ~J(J43andJ33, respectively, in ref. 20 ) are the two\ntypes of exchange couplings which originate in the two\ntypes of lattice sites. Jis the exchange coupling between\na 4-co-ordinated and a 3-co-ordinated site, while ~Jis the\nexchange coupling between two 3-co-ordinated sites, re-\nspectively, of the Cairo pentagonal lattice(see Fig. 1(a)\nand its caption).\nOn the experimental front, Retuerto et al. addressed21\nthe role of oxygen non-soichiometry in BiFe 2O5\u0000\u000eand\ncon\frmed that the mixed valence of iron, with a nominal\nvalency of Fe3:2+in BiFe 2O4:63, leads to an antiferro-\nmagnetic ground state with TN= 250K. In the related\nsystem Bi 4Fe5O13F, Abakunov et al. showed22that the\npresence of frustrated exchange couplings lead to a se-\nquence of magnetic transitions at T1= 62 K,T2= 71 K\nandTN= 178 K. Using neutron di\u000braction and thermo-\ndynamic measurements, the authors could show the for-\nmation of a non-collinear antiferromagnet below T1= 62\nK, while the structure between T1andTNwas partially\ndisordered. Pchelkina and Streltsov carried out ab-initio\nband structure calculations23for Bi 2Fe4O9and showed\nthat a complete description required going beyond the\ntwo-dimensional Cairo pentagonal lattice. However, con-\nsidering only the two largest in-plane exchange coupling\nparameters, the obtained ground state was found to be\nconsistent with experiment. And very recently, Nakano\net al. addressed24the magnetization process in the two\ndimensional spin S= 1=2 Heisenberg model on the Cairo\npentagonal lattice. Using a numerical diagonalization\nmethod, they discussed the role of the ratio of the two\ntypes of exchange couplings in driving a quantum phase\ntransition, and very interestingly, found a 1/3 magneti-\nzation plateau usually associated with triangular plaque-\nttes.\nThe results described above clearly show that frustra-\ntion e\u000bects play an important role in determining the\nmagnetic ordering in the Cairo pentagonal lattice. How-\never, the quanti\fcation of frustration and the role of spe-\nci\fc or selective exchange couplings associated with the\nFIG. 1. (Color online) (a) The unit cell of the two-dimensional\nCairo pentagonal lattice(thick blue lines) used in this paper.\nThe undoped ( x= 0) case corresponds to ~J1=~J2=~J, and\nJ1=J0\n1=J2=J0\n2=J, and the magnetic structure depends\non the ratio \f=J/~J. The two magnetic structures obtained\nforx= 0, with \fnite but di\u000berent values of frustration index\n\u001e(T= 0), are shown : (b) for \f= 1, it is antiferromagnetic,\nand (c) for \f= 3, it is ferrimagnetic. For x= 1/6, the ob-\ntained 'star' antiferromagnetic structure is the same as shown\nin (b), while (d) shows the 'boat' antiferromagnetic structure\nobtained for x= 1/3.\ntwo types of sites in the lattice has not been addressed\nyet. Further, while experimental results for the hole-\ndoped system BiFe 2O5\u0000\u000ehave been reported, it is im-\nportant to theoretically investigate the role of doping for\nmagnetic ordering in the Cairo pentagonal lattice. In\nthe present study, we address these issues for the spin\nS= 1=2 Cairo pentagonal lattice in a NNAF Ising model.\nWe \frst develop a method to quantify geometric frustra-\ntion in terms of a frustration index \u001e(\f;T). We \fnd\nthat, for the undoped case (x = 0), \u001e(\f;T= 0) exhibits\na cusp maximum as a function of \f=J=~J. Further,\nin the presence of \fnite frustration, we identify antiferro-\nmagnetic ordering at low \f, which undergoes a \frst-order\ntransition to a ferrimagnetic phase for \f >\fcrit= 2. The\nfrustration and ferrimagnetic order get suppressed by se-\nlective exchange coupling, and lead to antiferromagnetic\nordering for x = 1/6 and 1/3. Mean-\feld calculations\nare carried out to determine the temperature-dependent\nmagnetization for x= 0, 1/6 and 1/3. The results are\ndiscussed in relation to known experimental results for\ntrivalent Bi 2Fe4O9and mixed valent BiFe 2O4:63. The\nstudy identi\fes the role of frustration e\u000bects, the ratio\n\fand selective exchange coupling in relation to ferri-\nmagnetic versus anti-ferromagnetic ordering in the Cairo\npentagonal lattice.3\nII. CALCULATIONAL DETAILS\nThe unit cell is as shown in Fig. 1 (thick blue lines)\nand consists of six Ising spins: four 3-co-ordinated spins\ns1;s2;s3;s4, a 4-co-ordinated spin \u001b0at the center, and\nanother 4-co-ordinated spin \u001b1at the bottom-left corner.\nEach of these can take values of S=\u00061=2. For the most\ngeneral case, the couplings (that are symmetric in the\nindices) are denoted as follows:\nJs1;s3=~J1; Js2;s4=~J2;\nJs1;\u001b0=Js3;\u001b0=J0\n1;\nJs2;\u001b0=Js4;\u001b0=J0\n2;\nJs1;\u001b=Js3;\u001b=J1;\nJs2;\u001b=Js4;\u001b=J2: (1)\nWhile the above detailed notations for the couplings\nare necessary for describing selective exchange couplings,\nwhich e\u000bectively represent virtual doping content ( x), the\nundoped case is obtained by setting ~J1=~J2=~J, and\nJ1=J0\n1=J2=J0\n2=J. As detailed in the following,\nappropriate limits of these parameters allow us to calcu-\nlate physical quantities for x= 1/6 and 1/3. The Ising\nHamiltonian is\nH=X\niHi; (2)\nwhere\nHi=X\n\u000bK(i)\n\u000bs(i)\n\u000b (3)\ncorresponds to the ithunit cell of the two-dimensional\nCairo lattice. Here the symbols K(i)\n\u000bare de\fned by\nK(i)\n1=J1\u001b(i)+J0\n1\u001b0(i)+~J1s(i\u0000y)\n3;\nK(i)\n2=J2\u001b(i+x)+J0\n2\u001b0(i)+~J2s(i+x)\n4;\nK(i)\n3=J1\u001b(i+x+y)+J0\n1\u001b0(i)+~J1s(i+y)\n1;\nK(i)\n4=J2\u001b(i+y)+J0\n2\u001b0(i)+~J2s(i\u0000x)\n2; (4)\nin which the superscripts i\u0006x; i\u0006y; i +x+yare\nindices of unit cells neighboring i. We perform a mean-\n\feld decoupling of the Hamiltonian Eq.(2) according to\nSiSj'hSiiSj+hSjiSi\u0000hSiihSji (5)\nfor a product of any two spins SiandSj, wherehSi=\nTr(Se\u0000HMF=kBT)=Tr(e\u0000HMF=kBT) for anyS. This ap-\nproximation linearizes the unit cell Hamiltonian Eq.(3)\nin the spin variables:\nHMF=c0+4X\n\u000b=1c\u000bs\u000b+c5\u001b0+c6\u001b: (6)\nFIG. 2. (Color online) Plots of (a) frustration index \u001e(\f;0)\n(Eq. (14)), (b) zero-temperature free energy f(\f;0) (Eq.\n(10)), and (c) Emin(\f) (Eq. (13)) as a function of \f=J=~Jfor\nthe undoped case, x= 0 ;\u001e(\f;0) exhibits a non-monotonic\nbehaviour and a cusp at \f= 2.\nWe have suppressed the unit cell index i. The coe\u000ecients\nare\nc0=\u00002~J1m1m3cos(ky)\u0000J1m3m6cos(kx+ky)\n\u00002~J2m2m4cos(kx)\u0000J1m1m6\n\u0000J2m2m6cos(kx)\u0000J2m4m6cos(ky)\n\u0000(J0\n1m1+J0\n1m3+J0\n2m2+J0\n2m4)m5;\nc1= 2~J1m3cos(ky) +J1m6+J0\n1m5;\nc2= (2 ~J2m4+J2m6) cos(kx) +J0\n2m5;\nc3= 2~J1m1cos(ky) +J1m6cos(kx+ky) +J0\n1m5;\nc4= 2~J2m2cos(kx) +J2m6cos(ky) +J0\n2m5;\nc5=J0\n1m1+J0\n1m3+J0\n2m2+J0\n2m4;\nc6=J1m1+J2m2cos(kx) +J1m3cos(kx+ky)\n+J2m4cos(ky): (7)\nFor the sub-lattice magnetizations, we have introduced\nthe notation m\u000b=hs\u000bi(for\u000b= 1;2;3;4),m5=\nh\u001b0i; m 6=h\u001bi. Since the single unit-cell Hamiltonian\n(Eq.(3)) involves couplings with spins in the neighboring\nunit cellsi\u0006x\u0006y, we have introduced a spin density wave\n(SDW) vector kwhose components kx= 2\u0019=\u0015 1; ky=\n2\u0019=\u0015 2appear in the expressions for the coe\u000ecients. The\nsix magnetizations m\u000b(\u000b= 1;\u0001\u0001\u0001;6), and the two wave\nlengths\u00151;\u00152together form an eight-component mean-\n\feld order parameter vector \u0016:\n\u0016= (m1; m 2; m 3; m 4; m 5; m 6; \u00151; \u00152):(8)\nThe thermodynamics of the model is determined by the\nunit cell free energy f(T):\ne\u0000f(T)=kBT=Tr(e\u0000HMF=kBT); (9)4\nFIG. 3. (Color online) Components of the order parameter\n\u0016at zero temperature plotted as a function of \fatx= 0.\n(a) Plots of the sub-lattice magnetizations m1tom6. The\njumps seen for m2,m4andm5indicate a change in the mag-\nnetic order from an antiferromagnetic phase for \f\u00142 to a\nferrimagnetic phase for \f > 2. (b) The SDW lengths \u00151;\u00152\nplotted as a function of \findicate a doubling of the magnetic\nlattice along x and y axes for the antiferromagnetic phase\nfor\f\u00142(panel c) compared to the ferrimagnetic phase for\n\f >2(panel d).\nwhere the trace is performed over the six spins of the unit\ncell, each of which takes the values \u00061=2. Therefore\nf(T) =c0\u00006kBTln(2)\u0000kBT6X\n\u000b=1ln\u0014\ncosh\u0012c\u000b\n2kBT\u0013\u0015\n:\n(10)\nThe magnetizations are determined using the self-\nconsistency equations\n2m\u000b+ tanh\u0012c\u000b\n2kBT\u0013\n= 0; \u000b = 1;\u0001\u0001\u0001;6; (11)\nFIG. 4. Sub-lattice magnetizations mi;i= 1\u00006, plotted as a\nfunction of temperature for the undoped case ( x= 0), for (a)\n\f= 1 (antiferromagnetic phase) and (b) \f= 3 (ferrimagnetic\nphase).\nwhile the SDW lengths are determined by free energy\nminimization @f=@\u0015k= 0; k= 1;2:\n@c0\n@\u0015k+6X\n\u000b=1m\u000b@c\u000b\n@\u0015k= 0; k = 1;2: (12)\nWe solve the eight equations in (11) and (12) numerically\nto obtain the order parameter vector \u0016(fJijg;T).\nWe de\fne a frustration index \u001e(\f;T) in the follow-\ning manner. Consider a reference state for which each\nbond between neighboring spins were to be fully satis-\n\fed. Then, Eq.(3) would give a zero temperature energy\nof\nEmin=\u00001\n2(J1+J0\n1+J2+J0\n2+~J1+~J2): (13)\nThe zero temperature free energy f(\f;T = 0) deviates\nfromEmin(\f) because of frustration, so if we de\fne\n\u001e(\f;T) = 1\u0000f(\f;T)\nEmin(\f); (14)5\nFIG. 5. Sub-lattice magnetizations mi;i= 1\u00006, plotted as\na function of temperature for the antiferromagnetic phases of\nx= 1=6) for (a)\f= 1 and (b) \f= 3.\nthen\u001e(\f;T = 0) is a convenient measure of frustration.\nIt provides a quantitative measure of frustration as a\nfunction of \fand selective exchange coupling.\nIII. RESULTS AND DISCUSSIONS\nWe \frst compute the order parameter vector \u0016by nu-\nmerically solving the eight equations in (11) and (12)\nfor the undoped case with ~J1=~J2=~J, andJ1=\nJ0\n1=J2=J0\n2=J. In Fig. 2, we plot the frustration\nmeasure\u001e(\f;T = 0) (Eq. (14)), the zero-temperature\nfree energy f(\f;T = 0) (Eq. (10)), and Emin(\f) (Eq.\n(13)) as a function of \f=J=~J. The frustration measure\n\u001e(\f;T = 0) exhibits a clear cusp maximum at \f= 2.\nThe cusp originates in the zero-temperature free energy\nf(\f;T= 0) which exhibits a derivative discontinuity at \f\n= 2. This indicates a \frst-order transition as a function\nof\f. As shown in Fig. 3, the order parameter plots as\na function of \findicate an antiferromagnetic phase for\n\f\u00142, which transforms into a ferrimagnetic phase for\n\f >2.\nFIG. 6. Sub-lattice magnetizations mi;i= 1\u00006, plotted as\na function of temperature for the antiferromagnetic phases of\nx= 1=3) for (a)\f= 1 and (b) \f= 3.\nWe see discontinuities in the sub-lattice magnetizations\nm2,m4andm5at\f= 2: while m2andm4\rip from\n1=2 to\u00001=2,m5responds by changing from 0 to 1 =2 to\nminimize the free energy (Fig. 3(a)). The SDW lengths\n\u00151;\u00152also have discontinuities (Fig. 3(b)): \u00151;\u00152= 2\nfor\f\u00142, and\u00151;\u00152= 1 for\f >2. Thus, the magnetic\nunit cell of the antiferromagnetic phase is doubled along\nthe x and y axes, compared to the ferrimagnetic unit cell\nas shown in Fig. 3(c) and (d), respectively. Interestingly,\nwhile the system remains frustrated in both these phases,\nthe frustration index \u001e(T= 0)!0 as\f!0, but goes to\na \fnite value at large \fasymptotically. Thus, in the limit\n\f= 0, we have isolated dimers on the 3-co-ordinated sites\nwhile for\f=1, we retain the ferrimagnetism.\nThe ferrimagnetic phase obtained for \f >2 with a to-\ntal magnetization M(T= 0) = 1=3, is identical to earlier\nstudies from (i) exact calculations for the Ising15,19, (ii)\nhard-core bosons18and (iii) the Heisenberg20models. It\nis noted that, for the antiferromagnetic phase, the sub-\nlattice magnetization m5=h\u001b0i= 0 for\f\u00142, (see Fig.\n3(a)) due to the fact that out of its 4 nearest-neighbour\nsites, two sites have up-spins and two have down-spins.6\nThis leads to an e\u000bective cancellation of the magnetiza-\ntion contribution from the \u001b0site. In earlier work, Rojas\net al.19reported a frustrated phase of Ising spins from ex-\nact calculations, corresponding to the antiferromagnetic\nphase obtained from our mean-\feld calculations, while\nRousochatzakis et al.20obtained a so called orthogonal\nphase for the Heisenberg case.\nWe now turn to investigating the role of selective ex-\nchange coupling at T= 0. We carry out this exercise\nmainly to identify the role of speci\fc couplings of the 3-\nand 4-co-ordinated sites in the Cairo lattice, and to e\u000bec-\ntively simulate virtual dopings of x = 1/6 and 1/3. From\nthe way we have de\fned our couplings (see Eq. (1)), it\ncan be observed that if we set J0\n1=J0\n2= 0, the central\nspin\u001b0gets disconnected from the lattice. Consequently,\nthe central spin \u001b0does not participate in the magnetic\nordering and the system e\u000bectively represents a virtual\nhole doping content of x= 1=6. In the same way, it can\nbe observed that when J2=~J2=J0\n2= 0, the spins s2\nands4get disconnected, corresponding to a virtual hole\ndoping ofx= 1=3. Our results indicate that for both\nthese cases, the system is antiferromagnetic for all \f, i.e.\n\u00151;\u00152= 2. Forx= 1=6, the sub-lattice magnetizations\nare exactly the same as those for x= 0;\f\u00142 (Fig. 3),\nand we therefore refrain from showing these plots in a\n\fgure. For x= 1=3, the actual magnetic structure gets\nmodi\fed as discussed below. Another important di\u000ber-\nence from the x= 0 case is that \u001e(\f;0) = 0 for both\nvalues ofx= 1=6 and 1=3 and is independent of \f, indi-\ncating the absence of frustration.\nThe absence of a \f-driven transition for the cases\nx= 1=6;1=3 is easy to understand. In the x= 1=6\ncase, the central spin \u001b0does not participate in the mag-\nnetic ordering. The remaining spins can be looked upon\nas forming a \\star\" con\fguration with 12 bonds on its\nperiphery, all of which can be satis\fed in the antifero-\nmagnetic phase, thus removing frustration fully. An in-\ncrease of\fonly strengthens the antiferromagnetic order,\nincreasingTN(see Fig. 7 and associated description). In\nthex= 1=3 case, the spins \u001b2;\u001b4do not participate in the\nmagnetic ordering. The remaining spins can be looked\nupon as forming a \\boat\" con\fguration with 10 bonds\non its periphery, all of which can be satis\fed in the antif-\neromagnetic phase, again fully removing frustration(Fig.\n1(d)). An increase of \fin this case also, merely strength-\nens the antiferromagnetic order by increasing TN. While\nboth thex= 1=6 andx= 1=3 cases exhibit antifer-\nromagnetic order, the actual spin ordering is found to\nbe di\u000berent: the former corresponds to a repetition of\n\\star\"(Fig. 1(b)) unit cells, and the latter, a repetition\nof \\boat\"(Fig. 1(d)) unit cells.\nIn Figs. 4(a) and 4(b), we plot the sub-lattice magne-\ntizationsmi;i= 1;\u0001\u0001\u0001;6 for the undoped case ( x= 0),\nas a function of temperature in units of ~Jfor\f= 1\nand\f= 3, i.e. in the antiferromagnetic and ferrimag-\nnetic phases, respectively. We \fnd that the transition\ntemperature TCandTN(the ferrimagnetic and antifer-\nromagnetic transition temperatures, respectively) for de-\nFIG. 7. (a) Plots of the transition temperature Tc;TNas a\nfunction of \ffor x = 0, 1/6 and 1/3. A jump in the ordering\ntemperature is seen only for x = 0 at \f= 2. In (b) we present\na magni\fed view of the region near Tc.\nstruction of magnetic order primarily depends on \f. This\nis further borne out by the temperature-dependence plots\nforx= 1=6 presented in Figs. 5(a) and 5(b) for \f= 1 and\n\f= 3, respectively. The temperature-dependence plots\nforx= 1=3, presented in Figs. 6(a) and 6(b) for \f= 1\nand\f= 3, respectively, also re\rect this. In particular,\nTNincreases on increasing \fforx= 1=6 and 1=3. In\nfact, the ordering temperatures increase smoothly with\n\fas can be seen in Fig. 7(a), where we plot TCand\nTNas a function of \ffor the three cases x= 0;1=6;1=3.\nThe discontinuity at the \frst-order transition for x= 0\nat\f= 2 is quite clear and remarkable. In Fig. 7(b), we\npresent a magni\fed view of the plot around \f= 2.\nWe can also see in Fig. 7(a) that the dependence of\nTNonxis negligible for all \f\u00142. But for \f > 2,\nwhile thex= 1=6;1=3 plots follow the same course as\n\f\u00142, thex= 0 curve follows a completely di\u000berent\ncourse because of the discontinuity. The larger TNof7\nthe antiferromagnetic phase compared to the TCof the\nferrimagnetic phase, just above \f= 2, suggests a higher\nstability of the antiferromagnetic phase. However, for\n\fvalues greater than \u00182:5, the ferrimagnetic TCfor\nx= 0 becomes larger than the antiferromagnetic TNfor\nx= 1=6 and 1=3. The results also show that TNgoes to\na \fnite value as \f!0, but increases linearly for high \f.\nWe now describe the physical picture of the vari-\nous phases obtained by varying the model parameters.\nFirstly, the crucial role that frustration plays in the \f-\ndriven \frst-order transition is seen from the results of the\nx= 0 case. In the x= 0 antiferromagnet region obtained\nfor\f\u00142, the frustration is con\fned to the core of the\nunit cell, i.e. two out of the four J0-bonds (either the\nJ0\n1or theJ0\n2bonds) connecting to the central spin \u001b0are\nfrustrated; in this situation, the frustration measure \u001e(0)\nis an increasing function of \f(see Fig. 2, \f\u0014\fcrit= 2 ).\nWith further increase in \fbeyond\fcrit= 2, it becomes\nenergetically favorable to \\eject\" the frustration from the\ncore to the ~Jbonds lying on the periphery of the unit\ncell i.e. the the four ~J-bonds connecting 3-co-ordinated\nsites become frustrated. Thus, the \f-driven transition is\nattended by a change in the location of the frustrated\nbonds in the unit cell. Secondly, it is surprising to \fnd\nthat the obtained TNvalues do not depend on xbut only\non\f. However, experimental results have also indicated\nthatTNdoes not depend signi\fcantly on doping content.\nFor example, studies on Bi 2Fe4O9(\u0011BiFe 2O4:5; nominal\nvalency of Fe3:0+), the material recently recognized as\na realization of the Cairo pentagonal lattice, reported a\nTN= 238 K for single crystals17,25, while for polycrys-\ntals,TNwas reported to be \u0018260 K.16,26,27For mixed\nvalent polycrystalline BiFe 2O4:63with a nominal valency\nof Fe3:2+, which corresponds to a hole doping of \u001820%,\nRetuerto et al. reported a value of TN= 250 K.21Fur-\nther, forx= 1=6 and 1=3, since the magnetic structures\nhave no frustration, the zero temperature free energy\nf(T= 0) =Emin. However, the two magnetic structures\nhave di\u000berent values of Emin. From Eq.(3), we obtain\nEmin=\u0000(J+~J) forx= 1=6, andEmin=\u0000(J+~J=2 for\nx = 1/3. Thus, even with di\u000berent values of the ground\nstate energies for x= 1=6 and 1=3, our results suggest\nthatTNdepends only on \f=J=~J.\nFinally, the present results also show that the ab-\nsolute values of the sub-lattice magnetizations at high\ntemperatures(\u00180:25\u00000:5TN/TCtoTN/TC)) depend on\nthe co-ordination of the sites and the value of \f. For ex-\nample, as can be seen in Fig. 4(a) for x= 0 and\f= 1,\nthe absolute values of sub-lattice magnetizations of the\n3-co-ordinated sites mi;i= 1;\u0001\u0001\u0001;4 are exactly the same,\nbut for\u00180:5TNtoTN, they are slightly lower than m6,\nwhich is a 4-co-ordinated site. Similarly, for the ferrimag-\nnetic phase with \f= 3, Fig. 4(b) shows that absolutevalues ofmi;i= 1;\u0001\u0001\u0001;4(3-co-ordinated sites) are exactly\nthe same, but for \u00180:25TCtoTC, they are lower than\nm5andm6which are 4-co-ordinated sites. This was also\npointed out by Urumov for the ferrimagnetic phase us-\ning an exact calculation, although the changes were very\nsmall.15Forx= 1=6, the changes are the same as for\nx= 0 and\f= 1, but the di\u000berence between 3 and 4-co-\nordinated sites get enhanced on increasing \f= 3. Very\ninterestingly, we see the opposite behaviour for x= 1=3\ncompared to x= 1=6. For\f= 1(Fig. 6(a)), due to selec-\ntive exchange coupling, the structurally 4-co-ordinated\nsites become magnetically 2-co-ordinated sites(see Fig.\n1(d)), and consequently, the absolute values of m5and\nm6get suppressed compared to m1andm3(structurally\nand magnetically 3-cordinated sites) for temperatures be-\ntween\u00180:25TNtoTN. In contrast, for \f= 3, the dif-\nference between absolute values of m5,m6compared to\nm1,m3become smaller as the magnetization pro\fles get\ndominated by the larger value of Jcompared to ~J.\nIV. CONCLUSIONS\nIn conclusion, we have investigated the spin S= 1=2\nCairo pentagonal lattice with respect to selective ex-\nchange coupling, in a nearest-neighbour antiferromag-\nnetic Ising model. We have developed a simple method\nto quantify geometric frustration in terms of a frustration\nindex\u001e(\f;T), where\f=J=~J, the ratio of the two ex-\nchange couplings required by the symmetry of the Cairo\nlattice. At T= 0, the undoped Cairo pentagonal lattice\nshows a \frst order phase transition with antiferromag-\nnetic order for \f\u0014\fcrit= 2, which transforms to a\nferrimagnet for \f > \fcrit.\u001e(\f;T = 0) exhibits a cusp\nmaximum at \fcrit. The obtained magnetic structures\nreveal that the frustration originates in di\u000berent bonds\nfor the two phases. The frustration and ferrimagnetic\norder get suppressed by selective exchange coupling, and\nthe system shows antiferromagnetic ordering for a virtual\ndoping ofx= 1/6 and 1/3. From mean-\feld calculations,\nwe obtained the temperature-dependent sub-lattice mag-\nnetizations for x= 0;1=6 and 1=3. The calculated results\nwere discussed in relation to known experimental results\nfor trivalent Bi 2Fe4O9and mixed valent BiFe 2O4:63. The\nresults show the fundamental role of frustration and se-\nlective exchange coupling in determining the kind of spin\nordering and how they transform in the Cairo pentagonal\nlattice.\nV. ACKNOWLEDGEMENT\nWe sincerely thank Professor Viktor Urumov, Institute\nof Physics, Macedonia, for sending us Reference 15.\n\u0003chainani@spring8.or.jpykshesh@gmail.com8\n1G. H. Wannier, Phys. Rev. 79, 357(1950).\n2B. Bernu, C. Lhuillier, and L. Pierre, Phys. Rev. Lett. 69,\n2590(1992).\n3S. Depenbrock, I.P. McCulloch, and U. Schollwock, Phys.\nRev. Lett. 109, 067201 (2012)\n4L. Balents, Nature 464, 199(2010).\n5Introduction to Frustrated Magnetism : Materials, Exper-\niments, Theory edited by C. Lacroix, P. 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Solid State 6,\n963(1964)."    },    {        "title": "1808.06690v1.Tunable_Magnonic_Thermal_Hall_Effect_in_Skyrmion_Crystal_Phases_of_Ferrimagnets.pdf",        "content": "Tunable Magnonic Thermal Hall E\u000bect in Skyrmion Crystal Phases of Ferrimagnets\nSe Kwon Kim,1, 2Kouki Nakata,3Daniel Loss,4, 5and Yaroslav Tserkovnyak1\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n3Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan\n4Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: August 22, 2018)\nWe theoretically study the thermal Hall e\u000bect by magnons in skyrmion crystal phases of ferri-\nmagnets in the vicinity of the angular momentum compensation point (CP). To this end, we start\nby deriving the equation of motion for magnons in the background of an arbitrary equilibrium spin\ntexture, which gives rise to the \fctitious electromagnetic \feld for magnons. As the net spin density\nvaries, the resultant equation of motion interpolates between the relativistic Klein-Gordon equation\nat CP and the nonrelativistic Schr odinger-like equation away from it. In skyrmion crystal phases,\nthe right- and the left-circularly polarized magnons with respect to the order parameter are shown\nto form the Landau levels separately within the uniform skyrmion-density approximation. For an\nexperimental proposal, we predict that the magnonic thermal Hall conductivity changes its sign\nwhen the ferrimagnet is tuned across CP, providing a way to control heat \rux in spin-caloritronic\ndevices on the one hand and a feasible way to detect CP of ferrimagnets on the other hand.\nIntroduction. |Magnetic skyrmions are swirling spin\ntextures, which are characterized by the topological\nskyrmion number de\fned in terms of the real-space spin\ncon\fguration [1]. Their topological characteristic in\ru-\nences not only the dynamics of themselves, e.g., by en-\ngendering the Magnus force, but also the dynamics of\nelectrons moving through them [2]. In particular, when\nferromagnetic skyrmions form a crystal lattice, electrons\nwhose spin follows the local spin texture adiabatically ex-\nperience the Lorentz force due to the \fctitious magnetic\n\feld proportional to the skyrmion density and thereby\nexhibit the so-called topological Hall e\u000bect [3]. Recently,\nthere has been a growing interest in skyrmion crystals in\nantiferromagnets and their electronic transport proper-\nties [4, 5] because of their fundamental di\u000berences from\nferromagnetic counterparts as well as technological ap-\nplications for THz-speed magnetic devices [6].\nMagnons, which are quanta of spin waves [7], can\ntransport information and exhibit topological phenom-\nena similarly to electrons. Their potential ability to\nrealize devices based on insulating magnets, which are\nfree from drawbacks of conventional electronics such as\nsigni\fcant energy loss due to Ohmic heating, has led\nto the emergence of magnon-based spintronics [8]. In\nskyrmion crystal phases of ferromagnets, magnons have\nbeen shown to experience the \fctitious magnetic \feld\nby keeping their spin antiparallel to the local spin tex-\nture [9]. As a result, magnons form the approximate\nLandau levels with the \fnite Berry curvature [10], caus-\ning the thermal Hall e\u000bect [11{13]. However, the magnon\nbands and their transport properties in antiferromagnetic\nskyrmion crystals have not been studied.\nIn this Letter, we \fll this gap by investigating a\ntheoretically more general problem: The dynamics of\nmagnons in the presence of skyrmion crystals in ferrimag-\nferrimagnet in skyrmion crystal phasexyrT\njQs=0s<0s>0FIG. 1. Schematic illustration of the magnonic heat \rux\njQthrough a ferrimagnet in its skyrmion crystal phase sub-\njected to a temperature gradient rT. The colored small ar-\nrows depict a single skyrmion texture of the order parameter\nn. Magnons can exhibit the thermal Hall e\u000bect since the\nskyrmion crystal gives rise to the \fctitious magnetic \feld,\nwhich magnons of left and right circular polarization (with\nrespect to the order parameter) experience as if they carry\nthe positive and the negative charge, respectively. The in-\nduced transverse heat \rux changes its direction as the net\nspin density salong nvaries across 0.\nnets exhibiting the angular momentum compensation\npoint (CP) [14], at which the net spin density vanishes,\nbut the magnetization can be \fnite. One class of such\nferrimagnets is rare-earth transition-metal alloys such as\nGdFeCo or CoGd whose net spin density can be tuned\nby varying either temperature [15] or chemical composi-\ntion [16]. To this end, we start by deriving the equation\nof motion for magnons in the presence of an arbitrary\nspin texture, which includes the \fctitious electromag-\nnetic \feld. The obtained equation of motion is reduced\nto the nonrelativistic Schr odinger-like equation for ferro-\nmagnetic magnons away from CP and to the relativistic\nKlein-Gordon equation for antiferromagnetic magnons at\nCP, interpolating between the dynamics of ferromagnets\nand that of antiferromagnets as previously shown for thearXiv:1808.06690v1  [cond-mat.mes-hall]  20 Aug 20182\ndynamics of domain walls and skyrmions [17, 18]. In the\npresence of a skyrmion crystal, two species of magnons\nwith right and left circular polarization (with respect to\nthe order parameter) will be shown to experience the \fc-\ntitious magnetic \felds of opposite directions and form\nthe Landau levels separately, realizing two-dimensional\nmagnonic topological insulators [19]. As an experimen-\ntal proposal, we will show that the thermal Hall conduc-\ntivity changes its sign as the ferrimagnet is tuned across\nCP. See Fig. 1 for a schematic illustration. One promis-\ning platform is o\u000bered by GdFeCo \flms, where isolated\nskyrmions have been observed [20] and the antiferromag-\nnetic domain-wall dynamics has been demonstrated at\nCP [17]. The proposal provides not only a feasible way\nto control the direction of the thermal \rux, which can\nbe useful in spin caloritronics [21], but also a thermal\ntransport measurement for determining CP, which can\ncomplement the other methods based on magnetic reso-\nnances [15, 22] or domain-wall speed measurements [17].\nGeneral formalism. |Our model system is a collinear\nferrimagnet, whose potential energy density is given by\nU[n] =A(rn)2=2 +Dn\u0001(r\u0002n)\u0000h\u0001n;(1)\nwhere n(r;t) is the three-dimensional unit vector repre-\nsenting the direction of the magnetic order. Here, the\n\frst term is the exchange energy; the second term is\nthe Dzyaloshinskii-Moriya interaction (DMI), which ex-\nists when the inversion symmetry is broken [23]; the last\nterm represents the Zeeman coupling between the exter-\nnal \feld hand the magnetization along the direction of\nthe order parameter. Here, we are neglecting the other\nterms such as the dipolar interaction by following the\nprevious literature on chiral magnets [24]. With a suit-\nable choice of the coe\u000ecient values, the ground state\nis a skyrmion crystal [24], which will be the phase of\nour main interest for later discussions. The equilibrium\norder-parameter con\fguration will be denoted by n0.\nThe dynamics of the order parameter nof the ferrimag-\nnet can be described by the following Landau-Lifshitz-\nlike equation [17, 18, 25]:\ns_n+\u001an\u0002n=n\u0002\u0000\nAr2n\u00002Dr\u0002n+h\u0001\n;(2)\nwheresis the equilibrium spin density and \u001a\nparametrizes the inertia associated with the dynamics\nof the order parameter. The left-hand side is the time\nderivative of the net spin density, s=sn+\u001an\u0002_n, the\nformer and the latter of which are the longitudinal and\nthe transverse component of the spin density with re-\nspect to the order parameter, respectively [26]. Conven-\ntional ferromagnets and antiferromagnets have only the\n\frst and the second term, respectively, on the left-hand\nside. The parameter of focus in this work is the spin den-\nsitys, which can be varied across zero. We are interested\nin the change of the magnon bands as a function of it.\nTo obtain the equation of motion for a magnon, which\nis a quantum of small-amplitude \ructuations from theequilibrium state, we use the local coordinate system\nn0, where the equilibrium state is in the positive zdi-\nrection, n0\n0\u0011^z[11, 27]. The transformation can be\nimplemented by a three dimensional rotation matrix R\nsatisfying n0=Rn0\n0. We will use one explicit realiza-\ntion of it in this work: R= exp(\u001e0Lz) exp(\u00120Ly) for\nn0= (sin\u00120cos\u001e0;sin\u00120sin\u001e0;cos\u00120) whereLyandLz\nare the generators of the rotations about the yand thez\naxis, respectively [28].\nThe equation of motion for magnons can be obtained\nfrom Eq. (2) to linear order in the \ructuation \u000en0\u0011n0\u0000\n^z=n0\nx^x+n0\ny^y. Since the equation is second order in\ntime derivative, there are two types of solutions. It is\nconvenient to represent the two monochromatic solutions\nwith the complex \felds:  +=n0\nx\u0000in0\ny/exp(\u0000i\u000ft=~) for\nright-circularly polarized magnons and  \u0000=n0\nx+in0\ny/\nexp(\u0000i\u000ft=~) for left-circularly polarized magnons where\n\u000fis the magnon energy. We will refer the former and the\nlatter to the positive-chirality ( q= 1) and the negative-\nchirality (q=\u00001) solutions, respectively. The equation\nof motion for a magnon of chirality qis given by\n\u0000qs\u0012\ni@t\u0000q\u001e\n~\u0013\n q+\u001a\u0012\ni@t\u0000q\u001e\n~\u00132\n q=A\u0012r\ni\u0000qa\n~\u00132\n q;\n(3)\nwhich is our \frst main result. See Supplemental Ma-\nterial for its derivation [29]. Here, \u001eis the texture-\ninduced scalar potential given by \u001e=~(R\u00001@tR)12=\n\u0000~cos\u00120@t\u001e0, whereM12represents a corresponding\nmatrix element of M. The vector potential consists of\ntwo contributions [11]: a=at+ad, where the \frst term\nis from the exchange energy, at\ni=\u0000~(R\u00001@iR)12=\n~cos\u00120@i\u001e0, and the second term is from the DMI,\nad=\u0000(~D=A)n0. The texture-induced \fctitious elec-\ntric and magnetic \felds are given by\net\ni=\u0000@i\u001e\u0000@tat\ni=~n0\u0001(@tn0\u0002@in0); (4)\nbt\ni=\u000fijk@jat\nk= (~=2)\u000fijkn0\u0001(@kn0\u0002@jn0);(5)\nin the Einstein summation convention. The obtained\n\felds are identical to those for electrons [2, 30] and\nmagnons [11, 27] in ferromagnets. The DMI-induced\nvector potential adgives rise to another contribution\nto the \fctitious \felds, ed=\u0000@tad= (~D=A)@tn0and\nbd=\u0000(~D=A)r\u0002n0[11]. Note that the chirality\nq=\u00061 of a magnon serves as its charge with respect\nto the electromagnetic \felds, which can be understood\nas follows. Since the positive- and negative-chirality\nmagnons carry spin whose directions are locked paral-\nlel and antiparallel with respect to the background spin\ntexture n0[31], they pick up the Berry phase with the op-\nposite signs and experience the opposite \fctitious electro-\nmagnetic \felds [32]. During the derivation, we neglected\nthe second and higher order terms in \u001eandaand the\nterm proportional to the external \feld, by focusing on\nhigh-energy magnons whose wavelength is much smaller3\nthan the spatial extension of the texture and whose ki-\nnetic energy dominates the Zeeman energy, which we will\nrefer to as the exchange approximation [33].\nAt CP, the equilibrium spin density vanishes, s= 0,\nand the nature of the dynamics becomes antiferromag-\nnetic. The equation of motion is then reduced to the fol-\nlowing Klein-Gordon equation [34] which describes the\ndynamics of a relativistic particle with charge qin the\npresence of an electromagnetic \feld:\n(i~@t\u0000q\u001e)2 q=c2\u0012~\nir\u0000qa\u00132\n q; (6)\nwherec\u0011p\nA=\u001a is the characteristic speed that is the\nmagnon speed in the absence of electromagnetic \felds.\nThis equation describing the dynamics of magnons in\nantiferromagnets moving through a general spin texture\nhas not been derived before except for a special case of a\none-dimensional domain wall [31, 34].\nWhen su\u000eciently distant from CP, a ferrimagnet has\nenough spin density sto neglect the inertial term /\u001a\nin Eq. (2) for the low-energy dynamics. The equation of\nmotion (3) for ferrimagnetic magnons is then reduced to\nthat for ferromagnetic magnons [11, 27]:\n\u0000sgn(qs)i~@t q=\"\n1\n2m\u0012~\nir\u0000qa\u00132\n\u0000sgn(s)\u001e#\n q;\n(7)\nwith the e\u000bective mass m=~jsj=2A, which resembles the\nSchr odinger equation for a nonrelativistic charged parti-\ncle subjected to an electromagnetic \feld.\nIt is instructive to discuss the solutions to Eq. (3) in\nthe absence of the scalar and the vector potentials \u001e= 0\nanda=0, which are given by the plane-wave solutions,\n q/exp(ik\u0001r\u0000i\u000ft=~) [35]. The energy-momentum\nrelation is given by\n\u000fq(k) =p\n(mc2)2+~2c2k2+ sgn(qs)mc2: (8)\nThe solution to Eq. (6) for an antiferromagnetic magnon\nis given by the high-kinetic energy limit, i.e., ~jkj\u001dmc:\n\u000f\u0006\u0019~cjkj. The two solutions are degenerate at the\nlevel of approximation taken in Eq. (6) where the time\nreversal symmetry is respected by having vanishing spin\ndensitys= 0. The lower-energy solution to Eq. (7) for a\nferromagnetic magnon is given by the low-kinetic energy\nlimit, i.e., ~jkj\u001cmc:\u000fq\u0019~2k2=2mwith the chirality\nq=\u0000sgn(s). Note that spin of the low-energy magnons\nis locked antiparallel to the direction of the background\nspin density sn0. Here, the momentum scale governing\nthe separation between a nonrelativistic and a relativistic\nregime is given by mc=~jsj=2pA\u001a.\nMagnon in a skyrmion crystal. |Now, let us apply the\nabove formalism to one speci\fc example: Magnons in a\nskyrmion crystal of a quasi-two-dimensional ferrimagnet.\nWe will assume that the skyrmion crystal is static, for\nn=0n=1n=2n=3n=0n=1n=2n=3\nsl/2⇢c✏l/~c\n012\u00001\u000021p3p5p7q=\u0000q=+q=+q=-FIG. 2. The plot of the Landau levels [Eq. (11)] of magnon\nbands in ferrimagnetic skyrmion crystals in terms of the\nrescaled energy \u000fl=~cand the rescaled spin density sl=2\u001ac.\nThe solid gold and the dashed blue lines represent the right-\ncircularly polarized ( q= +) and the left-circularly polarized\n(q=\u0000) magnon bands, respectively.\nwhich the \fctitious electric \feld vanishes. A skyrmion is\ncharacterized by its integer skyrmion number [36]:\nQ=1\n4\u0019Z\ndxdyn0\u0001(@xn0\u0002@yn0)\u0011Z\ndxdy\u001asky;(9)\ncounting how many times the order parameter n0wraps\nthe unit sphere. Under suitable conditions, skyrmions\nwith the de\fnite skyrmion number can crystalize in the\ntriangular lattice as observed in several ferromagnetic\nmaterials [2], giving rise to the \fnite skyrmion number\ndensity per unit area, which we denote by \u001asky. The\nassociated \fctitious magnetic \feld [Eq. (5)] is given by\nbt=\u00004\u0019~\u001asky^z: (10)\nThe spatial pro\fle of the magnetic \feld depends on the\ndetailed values of material parameters, making it cum-\nbersome to take into account analytically. Therefore,\nbelow, we will account for its e\u000bects by spatially aver-\naging it: b=\u00004\u0019~h\u001askyi^z. The corresponding mag-\nnetic length is given by l=p\n~=jbzj= 1=p\n4\u0019jh\u001askyij,\nwhich is proportional to the distance between neighbor-\ning skyrmions. The DMI-induced contribution bdvan-\nishes after spatial averaging. In addition, we will assume\nthe negative skyrmion density \u001asky<0 and thus bz>0\nwithout loss of generality for subsequent discussions.\nTo solve Eq. (3), we adopt the known results for the\nLandau levels of a nonrelativistic charged particle sub-\njected to a uniform magnetic \feld [29, 37]. Plugging the\nmonochromatic function,  q(r;t) = exp(\u0000i\u000ft=~) nn0(r)\ninto Eq. (3), where  nn0(r) is the known eigenfunction of\nthe right-hand side of Eq. (3) for the nth Landau level ( n0\nis the index for states within each Landau level), yields\nthe following solutions:\n\u000fq\nn=p\n(mc2)2+~c2bz(2n+ 1) + sgn( qs)mc2:(11)4\n012\u00001\u0000201\u00001sl/2⇢c+rT-jQs<0s>0+rT-jQxy(a)(b)(c)-+2\u00002¯yx(⇥102)\n-2 -1 0 1 2-200-1000100200\n-200-1000100200\nFIG. 3. (a) The plot of the rescaled thermal Hall conduc-\ntivity \u0016\u0014yx\u0011(2\u0019~t=k2\nBT)\u0014yx, which parametrizies the ratio\nof the induced transverse heat \rux jy\nQto the applied lon-\ngitudinal temperature gradient @xTfor the ferrimagnet \flm\nof thickness t. (b) and (c) the schematic illustrations of the\npositive-chirality (+) and the negative-chirality ( \u0000) magnon\nmotions subjected to a temperature gradient rT= (@xT)^x\nand the resultant transverse energy \rux jQ=jy\nQ^y.\nThese magnon bands in a ferrimagnetic skyrmion crystal\nwithin the approximation of the uniform skyrmion den-\nsity is our second main result. The number of states\nin one Landau level is given by the total number of\nthe \fctitious magnetic \rux quanta through the plane,\nwhich is twice the total number of skyrmions in the sys-\ntem. The massless relativistic limit is given by \u000f\u0006\nn\u0019\ncp\n~bz(2n+ 1), which agrees with the known result for\nthe Klein-Gordon equation [38]. The lower band in the\nmassive limit, mc2\u001dcp\n~bz(2n+ 1), is reduced to the\nsolution for nonrelativistic particles: \u000fn\u0019(~bz=2m)(2n+\n1). The solution can be cast into the dimensionless form\nin terms of the rescaled energy \u0018\u0011\u000fl=~cand the rescaled\nspin density \u0010\u0011sl=2\u001ac:\u0018\u0006\nn=p\n\u00102+ 2n+ 1\u0006\u0010. See\nFig. 2 for the plot. The solid gold and the dashed blue\nline represent the right-circularly polarized ( q= +) and\nthe left-circularly polarized ( q=\u0000) magnon bands, re-\nspectively. The xaxis of Fig. 2 can be swept through the\norigin by varying the temperature across CP, the physical\nimplication of which is discussed below.\nTunable thermal Hall e\u000bect. |Each \rat Landau level of\nmagnons [11, 12] has the Chern number \u00170=\u0000q, which\nis the integral of the uniform Berry curvature de\fned in\nterms of the magnonic wavefunction over the Brillouin\nzone [29]. Magnons with the \fnite Berry curvature can\ngive rise to the thermal Hall e\u000bect [39, 40], a phenomenon\nof the generation of the transverse energy \rux jy\nQupon\nthe application of the longitudinal temperature gradient\n@xT:jy\nQ=\u0000\u0014yx@xT, which is quanti\fed by the thermal\nHall conductivity \u0014yx[41]. Within the linear response\ntheory, the thermal Hall conductivity for our case is given\nby\u0014yx= (k2\nBT=2\u0019~t)P\nn[c2(\u001aB(\u000f\u0000\nn))\u0000c2(\u001aB(\u000f+\nn))],wheretis the thickness of the ferrimagnet, c2(x) =\n(1+x)(log(1+x\u00001))2\u0000(logx)2\u00002Li2(\u0000x), Li 2(z) is the\npolylogarithm function, and \u001aB(\u000f) = [exp(\u000f=kBT)\u00001]\u00001\nis the Bose-Einstein distribution [39]. Figure 3(a) shows\nthe plot of the rescaled thermal Hall conductivity \u0016 \u0014yx\u0011\n(2\u0019~t=k2\nBT)\u0014yxas a function of the rescaled spin den-\nsity\u0010=sl=2\u001ac. The plot is drawn with the following\nvalue for the ratio of the characteristic relativistic energy\nscale to the temperature: ~cl\u00001=kBT\u00180:03, which is ob-\ntained from the magnon speed c\u0018104m/s calculated for\nGdFeCo [18], the magnetic length l= 1=p\n4\u0019jh\u001askyij\u0018\n10 nm for the inter-skyrmion distance \u001850 nm of tri-\nangular lattice of skyrmions observed in chiral ferromag-\nnets [42], and the temperature T= 70 K. The induced\ntransverse heat \rux changes its direction as the net spin\ndensity varies across zero, at which magnons of two chi-\nralities are degenerate and thus the thermal Hall e\u000bect is\nabsent similarly to antiferromagnets [19]. When the spin\ndensity is negative, for example, there are more positive-\nchirality magnons than the others, causing the negative\nthermal Hall conductivity as shown in Fig. 3(b). Here,\nwe remark that, although the numerical results shown in\nFig. 3(a) are obtained within the approximation of the\nuniform \fctitious magnetic \feld, the sign change of \u0014yx\nats= 0 is the generic property of Eq. (3) under the time\nreversal and thus does not rely on the approximation.\nNext, let us estimate the change of the thermal Hall\nconductivity \u0001 \u0014yxas the net spin density svaries by\n\u0001s\u00185\u000210\u00008J\u0001s/m3, which can be achieved by changing\nthe temperature by \u0001 T= 10 K around CP of GdFeCo\n\flms according to the numerical results in Ref. [17]. Here,\nwe assume that all the parameters except the spin den-\nsity are constant. By using the inertia \u001a\u0018~2=Jd3ob-\ntained with the Heisenberg exchange energy J\u00185 meV\nand the lattice constant d\u00180:5 nm, the above \u0001 syields\n\u0001\u0014yx\u00180:05 W/K\u0001m for 50-nm thick \flms, which is com-\nparable to the large thermal Hall conductivities observed\nin frustrated magnets [43].\nDiscussion. |By investigating the magnon bands in\na skyrmion crystal of a ferrimagnet in the vicinity of\nCP, we have shown that, under certain approxima-\ntions, the positive-chirality and the negative-chirality\nmagnons form Landau levels separately by experiencing\nthe skyrmion-induced \fctitious magnetic \feld, leading\nto the identi\fcation of ferrimagnets in their skyrmion-\ncrystal phases as magnonic topological insulators. We\nhave predicted that the sign of the resultant thermal Hall\nconductivity changes its sign across CP. 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Lett. 117, 217202 (2016).7\nSupplemental Material: Tunable Magnonic Thermal Hall E\u000bect in Skyrmion Crystal Phases of\nFerrimagnets\nSe Kwon Kim,1;2Kouki Nakata,3Daniel Loss,4;5and Yaroslav Tserkovnyak1\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n3Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan\n4Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: August 20, 2018)\nThis Supplemental Material contains the derivation of the equations of motion for magnons and the\nsummary of the known results for the Landau levels of a charged particle.\nTHE DERIVATION OF THE EMERGENT ELECTROMAGNETIC FIELDS.\nWe use the three-dimensional rotation matrix Rthat transforms the zaxis to the equilibrium con\fguration\nn0:n0=Rn0\n0with n0\n0\u0011^z. One explicit form for Ris given byR= exp(\u001e0Lz) exp(\u00120Ly) for n0=\n(sin\u00120cos\u001e0;sin\u00120sin\u001e0;cos\u00120) whereLx;Ly, andLzare the generators of the three-dimensional rotations about\nthex;y; andzaxis given by\nLx=0\n@0 0 0\n0 0\u00001\n0 1 01\nA;Ly=0\n@0 0 1\n0 0 0\n\u00001 0 01\nA;Lz=0\n@0\u00001 0\n1 0 0\n0 0 01\nA: (S1)\nNote that they can be de\fned in terms of the Levi-Civita symbol: ( La)bc=\u0000\u000fabc. The derivatives of n=Rn0can\nthen be expressed in terms of the covariant derivatives of n0as follows:\nR\u00001@\u0016n= (@\u0016+R\u00001@\u0016R)n0\u0011(@\u0016+At\n\u0016)n0; (S2)\nwhere\u0016= 0 denotes the temporal derivative and \u0016= 1;2, and 3 denote the spatial derivatives with respect to the\ncoordinates x;y, andz, respectively. The e\u000bects of the background spin texture on top of which a magnon lives are\ncaptured by the matrices At\n\u0016, which is skew-symmetric due to the orthonormality of the rotation matrix R.\nExplicitly, we have the following transformation of each derivative term in the equations of motion by keeping the\nterms that include the derivative of the order parameter n0at least once:\n@\u0016n=R(@\u0016+At\n\u0016)n0; (S3)\n@2\n\u0016n=R(@\u0016+At\n\u0016)2n0: (S4)\nThe transformation of the DMI is as follows:\nr\u0002n=RRT(r\u0002n) (S5)\n=R\u0002\nRT(r\u0002Rn0)\u0003\n: (S6)\nThe factor inside the square bracket is given by\n\u0002\nRT(r\u0002Rn0)\u0003\na=RT\nac\u000fcde@dRefn0\nf (S7)\n=\u0002\nAd\nb@bn0\u0003\na; (S8)\nwhere\nAd\nb=RTLbR: (S9)\nIn terms of the n0, the Landau-Lifshitz-Gilbert Eq. (3) is given by\ns(@t+At\n0)n0+\u001an0\u0002(@t+At\n0)2n0=n0\u0002A\u0000\n@i+At\ni\u0000(D=A)Ad\ni\u00012n0; (S10)8\nwhen we keep the terms that include the temporal or spatial derivative of n0at least once by focusing on high-energy\nmagnons. For small deviations from the equilibrium, n0\u0019^z+n0\nx^x+n0\ny^ywithjn0\nxj;jn0\nyj\u001c1, we obtain the following\nequation of motion for n+\u0011n0\nx\u0000in0\ny:\n\u0000is(@t+i\u001e=~)n+\u0000\u001a(@t+i\u001e=~)2n+=\u0000A\u0002\n@i\u0000i(at\ni+ad\ni)=~\u00032n+; (S11)\nwhere the scalar potential \u001e, the vector potential atfrom the spin texture, the vector potential adfrom the DMI are\ngiven by\n\u001e\u0011~(At\n0)12; at\ni\u0011\u0000~(At\ni)12; ad\ni\u0011(~D=A)(Ad\ni)12: (S12)\nOne set of explicit expressions of them is given by \u001e=\u0000~cos\u00120@t\u001e0,at=~cos\u00120r\u001e0, and ad=\u0000(~D=A)n0.\nTHE DYNAMICS OF A NONRELATIVISTIC CHARGED PARTICLE IN THE PRESENCE OF A\nUNIFORM MAGNETIC FIELD\nIn the presence of a strong perpendicular magnetic \feld, the bands of a charged particle in two-dimensional systems\nform the Landau levels. Here, we summarize the known results for the Landau levels of a nonrelativistic charged parti-\ncle in the presence of a uniform magnetic \feld [37], which has been adopted for a ferromagnetic magnon previously [11].\nThe pertinent Schr odinger equation is given by\ni~@t =\"\n1\n2m\u0012~\nir\u0000qa\u00132#\n ; (S13)\nwhereq=\u00061 parametrizes the charge of the particle. The external magnetic \feld bz= (r\u0002a)\u0001^zperpendicular\nto the plane is characterized by the magnetic length, which is given by l=p\n~=jbzj. The corresponding cyclotron\nfrequency is given by !c=~=ml2.\nThe solution to Eq. (S13) is then given by  (r;t) = nn0(r) exp(\u0000i\u000fnt=~), with the energy of the nth Landau level\n(n= 0;1;2;\u0001\u0001\u0001)\n\u000fn=~!c(n+ 1=2) = ( ~jbzj=2m)(2n+ 1); (S14)\nand the spatial part  nn0(r) is the eigenfunction of the right-hand side of Eq. (S13) with the eigenvalue \u000fn. See\nRef. [37] for the explicit construction of the spatial part  nn0. The number of each Landau level's states, which are\nindexed by the integer n0in nn0(r), is given by the total number of magnetic \rux quanta through the plane. The\nsemiclassical equations of motion for the wavepacket localized at the position rand the momentum kare given by\n_r=1\n~@\u000fn(k)\n@k\u0000_k\u0002\nn(k) (S15)\n~_k=\u0000rU(r); (S16)\nwhere \nis the Berry curvature de\fned in terms of the magnetic Bloch wavefunctions [39, 40]. For \rat Landau levels,\nthe Berry curvature is uniform, and the Chern number, de\fned as the integral of the z-component of the Berry\ncurvature, is determined by the product of the charge and the direction of the magnetic \feld: \u0017n=R\nBZdkxdky\nn\u0001\n^z=2\u0019=\u0000sgn(qbz)."    },    {        "title": "2307.13522v1.Lattice_structure_dependence_of_laser_induced_ultrafast_magnetization_switching_in_ferrimagnets.pdf",        "content": "Lattice structure dependence of laser-induced ultrafast magnetization\nswitching in ferrimagnets\nJ. A. V´ elez,1, 2R. M. Otxoa,3, 1and U. Atxitia4,a)\n1)Donostia International Physics Center, 20018 San Sebasti´ an, Spain\n2)Polymers and Advanced Materials Department: Physics, Chemistry, and Technology,\nUniversity of the Basque Country, UPV/EHU, 20018 San Sebasti´ an, Spain\n3)Hitachi Cambridge Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom\n4)Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n(Dated: July 26, 2023)\nThe experimental discovery of single-pulse ultrafast magnetization switching in ferrimagnetic alloys, such as\nGdFeCo and MnRuGa, opened the door to a promising route toward faster and more energy efficient data stor-\nage. A recent semi-phenomenological theory has proposed that a fast, laser-induced demagnetization below\na threshold value puts the system into a dynamical regime where angular momentum transfer between sub-\nlattices dominates. Notably, this threshold scales inversely proportional to the number of exchange-coupled\nnearest neighbours considered in the model, which in the simplest case is directly linked to the underly-\ning lattice structure. In this work, we study the role of the lattice structure on the laser-induced ultrafast\nmagnetization switching in ferrimagnets by complementing the phenomenological theory with atomistic spin\ndynamics computer simulations. We consider a spin model of the ferrimagnetic GdFeCo alloy with increasing\nnumber of exchange-coupled neighbours. Within this model, we demonstrate that the laser-induced magneti-\nzation dynamics and switching depends on the lattice structure. Further, we determine that the critical laser\nenergy for switching reduces for decreasing number of exchange-coupled neighbours.\nFast, reliable, and inexpensive data manipulation and\nstorage is the cornerstone for innovation and progress\nof our information-technology-based society. Ultrafast\nmagnetism holds promise for fast and low energy data\nmanipulation solutions1–5. The field was initiated by\nthe discovery of femtosecond laser pulse induced sub-\npicosecond demagnetization in Ni6. To this break-\nthrough followed the demonstration of field-free magne-\ntization switching in GdFeCo alloys using a train of cir-\ncularly polarized pulses7. Later on, a combined theo-\nretical/experimental study showed the possibility of sin-\ngle pulse switching using linearly polarized light8,9. This\nfinding uncovered the purely thermal origin of the switch-\ning process, which in turn was used to demonstrate that\nultrafast heating by picosecond electric pulses is suffi-\ncient to achieve switching in GdFeCo10. Single-pulse\nswitching can also be accomplished in CoTb alloys11,\nGd-based ferrimagnetic multilayers12,13, magnetic tunnel\njunctions of Tb/Co14, and the rare-earth-free Heusler al-\nloy Mn 2RuxGa15,16. For future information technologies\nultrafast magnetization manipulation promises high po-\ntential, as such, further understanding of the microscopic\norigin of single-pulse magnetization switching is key for\nultrafast spintronics applications17.\nSingle pulse magnetization switching in ferrimagnets\nhas been described using computer simulations based on\natomistic spin dynamics (ASD)18–21and phenomenolog-\nical models22–25. A recent work has merged these models\ninto an unified macroscopic theory that describes magne-\ntization dynamics and switching of two-sublattice ferri-\nmagnets upon femtosecond laser excitation26,27. Within\na)Electronic mail: u.atxitia@csic.esthis theory, the switching process becomes possible due\nto an enhancement of the exchange relaxation – angu-\nlar momentum exchange between sublattices – when the\nmagnetization of the sublattices is reduced below a cer-\ntain threshold that depends on material parameters. Af-\nter femtosecond laser photo-excitation, the electron sys-\ntem enters a high temperature regime at which angular\nmomentum dissipation into electron or phonon degrees of\nfreedom dominates. This so-called relativistic relaxation\nis related to the spin-orbit coupling connecting spin and\norbital degrees of freedom. Depending on the laser power,\nthe sublattice magnetization can reduce down the thresh-\nold that leads to magnetic switching. Once the laser pulse\nis gone, the electron system starts to cool down, which\nleads to a local recovery of magnetic order due to the\nexchange coupling between spins. In two sublattice mag-\nnets, the so-called exchange relaxation, through local ex-\nchange of angular momentum between sublattices, drives\nmagnetic switching. Previous computational works us-\ning ASD methods have investigated how switching de-\npends on a variety of parameters, such as element-specific\ndamping20,21, rare-earth concentration11, duration of the\nlaser pulse20,28or the role of the initial temperature29.\nThe impact of the number of exchange-coupled neigh-\nbours on the switching behaviours in GdFeCo has re-\nmained however unexplored.\nIn this work, we provide insights about how the\nsingle-pulse magnetic switching of the ferrimagnetic al-\nloy GdFeCo depends on the lattice structure, in terms\nof number of exchange-coupled spins. We use both a\nsemi-phenomenological theory as well as atomistic spin\ndynamics simulations to demonstrate that by reducing\nthe number of exchange-coupled spins, the laser energy\nnecessary to switch the magnetic state of GdFeCo reducesarXiv:2307.13522v1  [cond-mat.mtrl-sci]  25 Jul 20232\nsignificantly.\nThe magnetization dynamics of a ferrimagnet com-\nposed of two magnetic sublattices, such as GdFeCo, can\nbe described in terms of the sublattice angular momen-\ntumSa=µa⟨sa⟩/γ, where ma=⟨sa⟩andµaits atomic\nmagnetic moment27,30\ndSa\ndt=αaµaHa+αex(µaHa−µbHb) (1)\nwhere a= Fe and b= Gd. Here, αastands for the\nmacroscopic relativistic damping parameter26\nαa= 2λaL(ξa)\nξa. (2)\nHere, L(ξ) stands for the Langevin function and λais\nthe element-specific intrinsic damping parameter. The\nso-called thermal field is defined as ξa=βµaHMFA\na (β=\n1/kBT) within the mean-field approximation (MFA),\nwhere\nµaHMFA\na =zaJaama+zabJabmb. (3)\nThe non-equilibrium fields are defined as\nHa=(ma−m0,a)\nµaβL′(ξa), (4)\nwhere, L′(ξ) =dL/dξ andm0,a=L(ξa). We note that\nin the underlying model behind the MFA the Gd spins\nare randomly located in a regular Fe spin lattice with\na concentration q, and zcorresponds to the number of\nexchange-coupled neighbours. Thus, za=zqrepresents\nthe average number of nearest neighbours (n.n.) of spins\nof type a, and similarly zab=z(1−q) represents the\naverage number of n.n. of type b. Since single-pulse\nswitching has been observed mostly for Gd concentra-\ntion of around 25%, in our model we restrict to that\nconcentration of Gd spins. Within this model, one can\ndefine J0,a=zqJaaandJ0,ab=z(1−q)Jab, that is it,\nµaHMFA\na =J0,ama+J0,abmb. Within the MFA the equi-\nlibrium magnetization me=L(ξe) only depends on the\nvalues of J0,aandJ0,abbut it is insensitive to the lattice\nstructure. In comparison to the MFA, ASD simulations\ncan capture the small differences coming from the lattice\nstructure and as a consequence the equilibrium magne-\ntization slightly depends on the lattice structure31. For\nthe MFA calculations we assume a fixed values for J0,a\nandJ0,abfor all values of z. These assumptions give the\nsame temperature-dependent equilibrium magnetization\nfor all cases.\nThe number of exchange-coupled neighbours zshows\nup explicitly in the so-called exchange relaxation param-\neter\nαex=1\n2z\u0012αa\nma+αb\nmb\u0013\n. (5)\nThe number of neighbours zwill therefore become rele-\nvant for the description of the magnetization dynamics.\n00.20.40.60.81\n0 0.2 0.4 0.6 0.8 1αex=αma\nmbz= 2\nz= 3\nz= 4\nz= 6\nz= 8\nz= 20\n00.20.40.60.81\n0 0.2 0.4 0.6 0.8 1αex=αaαex=αb\nαex=αz= 6ma\nmbFigure 1. (a) Lines separate the regions where αex< α(right\nside) and the region where αex> α (left side) for a range\nof nearest neighbours number z(λa=λb). (b) Similar for\nthe case z= 6, for αa=αbcoincident to (a), and two lines\ncorresponding to αa/αb= 5, one for the case αex=αaand\nanother for αex=αb.\nFor example, in the exact MFA limit, where z→ ∞\nwhile J0,aremains constant, αex→0. The vanishing\nof the exchange relaxation parameter for large number\nof exchange-coupled neighbours directly affects the abil-\nity of the system to switch. By contrast, the exchange\nrelaxation parameter increases as the number of neigh-\nbours reduce, which could make the switching process\nmore energy efficient.\nAs a first approximation, we can assume that the mag-\nnetization relaxation pathway, either of relativistic or ex-\nchange nature, is determined by the value of their corre-\nsponding relaxation parameters. Therefore, the crossover\nbetween relativistic- to exchange-dominated regimes can\nbe estimated by finding the conditions at which αex> αa.\nFor simplicity we consider first the case λa=λbin Eq.\n(2), for which αa≈αb=α32. Under this assumption,\none can simplify the condition α=αexto\nz=1\nma+1\nmb(6)\nFigure 1 (a) shows the lines separating the regions αex<\nαandαex> α. As the number of neighbours zreduce,\nthe region ( ma, mb) for which αex> α aincreases. For\ninstance, for the limit case of z= 2 (spin chain), αex=α\nalready for relatively large values of maandmb. For a\nlarger number of neighbours z, corresponding to simple\ncubic ( z= 6) or face-centered cubic ( z= 12) lattices, αex\nis relatively smaller than αfor a large region ( ma, mb).\nExperimental observations combined with ASD simu-\nlations suggest that in rare earth transition metal alloys,\nthe damping values are element specific20,21. Specifically,\nit was found that using λFe= 0.06 and λGd= 0.01 in\nthe ASD simulations, one can qualitatively reproduce the\nultrafast magnetization dynamics and switching of the\nGdFeCo alloys in a range of Gd concentrations20. Sim-\nilarly, in Gd 22−xTbxCo78alloys, it was found element-\nspecific damping values ( λCo=λTb= 0.05 and λGd=\n0.005−0.05) could describe switching as a function of Tb\ncontent21. For the sake of simplicity, we illustrate the im-\npact of element-specific damping on the relation between\nrelaxation parameters for a particular case, z= 6. Figure3\n30060090012001500\n−1−0.8−0.6−0.4−0.200.20.40.60.81\n0 1 2 3 4 5 6 7T(K)electronmz(Fe)\ntime (ps)z= 4\nz= 6\nz= 8\nz= 20\nFigure 2. (Top) Electron temperature dynamics driven by a\nfemtosecond laser pulse. (Bottom) Iron sublattice magnetiza-\ntion dynamics and switching for systems with different num-\nber of exchange-coupled nearest neighbours z. The sublattice\nmagnetization dynamics is calculated using Eq. (1) with the\nelectron temperature profile in the top figure as an input.\n1 (b) shows αex=αin solid lines, which corresponds to\nFig. 1(a), and αa/αb= 5, which agree to experimental\nobservations. For element-specific damping values, each\nelement (sublattice) will enter the exchange dominated\nregime under different conditions, namely, sublattice a\nwhen αex=αaand sublattice bwhen αex=αb. Our\nmodel predicts that under those circumstances the mag-\nnetization relaxation of the Gd sublattice is dominated\nby the exchange relaxation during the whole demagneti-\nzation process, i.e., by transfer of angular momentum to\nthe Fe sublattice, whereas Fe sublattice angular momen-\ntum relaxation remains mostly relativistic – transfer of\nangular momentum to other degrees of freedom.\nIt is worth noting that the magnetization relaxation\ndynamics described by Eq. (1) not only scales with the\nvalue of the relaxation parameters but also with the non-\nequilibrium effective field Ha[Eq. (4)]. In particular,\nthe relaxation of the sublattice magnetization Eq. (1)\ncan be split into two contributions, the relativistic re-\nlaxation rate: Γr\na=αaµaHa, and exchange relaxation\nrate Γex=αex(µaHa−µbHb). After the application of a\nlaser pulse the temperature quickly increases beyond the\ncritical temperature and the dynamics is dominated by\nthe thermal fields, which translates into Γr\na∼λakBTma\nwhereas Γex\na∼λakBT/z. Thus, a prerequisite for switch-\ning is that the laser pulse produces a high temperature\nprofile (see Fig. 2 (Top)) to quickly reduce the magne-\ntization ma, so that the exchange relaxation takes over,\nΓex>Γr\na. One would expect a more efficient switch-\ning process for lower values of z, and a highly non-\nefficient for high values of z. One can rationalize this\nby analysing the equations of motion for some limiting\nsituations. For the same value of the intrinsic damp-\ning parameter λFe=λGd, by assuming that one of thesublattice demagnetizes faster (Fe in GdFe alloys) than\nthe other, soon after the application of a fs laser pulse\none finds that ma≪mb, from the condition Γex>Γr\na,\none gets ma<p\nmb/2z, and from Γex>Γr\nbone gets\nma≤1/2z27. For example, for a lattice with fcc+bcc\nstructure, z= 20, ma<p\n1/20 = 0 .15, while for a\ntwo-dimensional magnet with a square lattice structure,\nz= 4, ma<p\n1/8 = 0 .35, or a spin chain, z= 2,\nma<p\n1/4 = 0 .5.\nThis finding is illustrated in Fig. 2 (Bottom), where\nthe magnetization dynamics of each sublattice is cal-\nculated using Eq. (1) coupled to the so-called two-\ntemperature model (TTM), which describes the dynam-\nics of the electron and phonon temperature33,34. For all\nzvalues, we use the same parameters for the TTM, and\nthus the electron and phonon temperatures dynamics are\nthe same (Fig. 2 (Top)). In the TTM, we use the fol-\nlowing values for the electron specific heat ce=γeTe\n(γe= 700 J/Km3), phonon specific heat cph= 3×106\nJ/Km3and electron-phonon coupling ge−ph= 6×1017\nJ/sKm327. The heat-bath to which the spins are coupled\nis represented by the electron system. Figure 2(Bottom)\nshows the magnetization dynamics of the Fe sublattice for\ndifferent number of neighbours z. We set the exchange\nparameters in such a way the Curie temperature of all\nsystems is the same. For the same energy input from the\nlaser pulse, the system with less exchange-coupled spins\nz= 4, switches faster than for example z= 6. For larger\nnumber of z, switching does not occur since the magne-\ntization threshold for switching is not achieved. In the\nlimit z→ ∞ , the exchange relaxation parameter is null,\nand consequently switching becomes impossible.\nWithin the MFA used so far, the temperature-\ndependence of the equilibrium magnetization is indepen-\ndent of the number of neighbours. Differently, a model\nbased on atomistic spins correctly accounts for the dif-\nferences in the spin correlations for the different lat-\ntice structures. In the one hand, for the same values\nof the exchange parameters, due to the higher number\nof exchange-coupled spins, the critical temperature, Tc,\nwill also be higher. For example, in a simple ferromag-\nnet with zn.n. and exchange coupling J, one finds\nthat in the MFA, kBTc=zJ/3, while in ASD simula-\ntions εkBTc=zJ/3, where ε <1 and accounts for the\nspin fluctuations neglected in the MFA31. For ferrimag-\nnets, the same arguments apply, and more n.n. leads\nto a higher Tcand the shape of the temperature depen-\ndent equilibrium magnetization is sensitive to the lat-\ntice dimension and structure and to the form of spin\ninteractions35. For this reason, we rely on atomistic spin\ndynamics simulations to conduct more accurate compar-\nison between different lattice structures.\nWe consider a classical Heisenberg spin\nHamiltonian36,37:\nH=−X\ni̸=j⟨ij⟩Jijsi·sj−X\nidz\ni(sz\ni)2. (7)\nHere, |si|= 1 and it represent the normalized classical4\nTable I. Exchange parameters used in the atomistic spin dy-\nnamics model.\nz J Fe−Fe[meV] JGd−Gd[meV] JFe−Gd[meV]\n6 21.18 8.30 -7.38\n8 15.50 6.06 -5.39\n20 5.22 1.97 -1.80\nspin vector at site i. The two sublattices have differ-\nent and antiparallel atomic magnetic moments µaand\nµb. The exchange coupling parameters have the same\nphysical meaning as in the MFA [see Eq. (3)]. The\nsecond term in Eq. (7) describes the on-site uniaxial\nanisotropy, dz\ni= 0.5 meV. We consider that the species\nof each atomic spin are randomly located in the lattice\nstructure. The dynamics of each atomic spin follows the\nstochastic Landau-Lifshitz-Gilbert (LLG) equation36\n∂si\n∂t=−|γ|\n(1 +λ2\ni)µi[(si×Hi)−λi(si×(si×Hi))].\n(8)\nλiis the local intrinsic atomic damping and it coincides\nwith those in Eq. (2). The effective field Hi=ζi−\n∂H\n∂si, where thermal fluctuations are represented by the\nstochastic field ζi. Specifically, we use for the element-\nspecific atomic magnetic moments, µFe= 1.92µBand\nµGd= 7.63µB(µBis the Bohr magneton) and intrinsic\ndamping parameters, λFe=λGd= 0.01. In Table I the\nvalues of the exchange coupling parameters for each lat-\ntice structure are given. We have re-scaled the exchange\ncoupling parameters to obtain the same value for Tcfor\nall three structures. This is convenient since it allows to\ndirectly compare the energy efficiency of the switching\nprocess for all three structures as kBTc∼J0sets the ex-\nchange energy scaling on magnetic systems. Figure 3(a)\nshows the temperature-dependent total equilibrium mag-\nnetization Mtot=MGd−MFe, where Ma=qµama(qa\nconcentration of element a) for the three cases studied\nhere. In the three cases the so-called magnetization com-\npensation temperature TM, the temperature at which\nMtot= 0, slightly depends on the lattice structure, in\ncomparison to the MFA for which all three cases be-\nhave the same. We use ASD computer simulations to\ninvestigate the minimum laser energy Pswnecessary to\nswitch the magnetic state in GdFeCo. Since the crit-\nical role of TMin the switching behaviour of GdFeCo\nhas been discussed in the literature before, we determine\nPswas a function of the initial temperature. We find\nthree scenarios depending on the final magnetization of\nthe sublattices, (i) switching is achieved when the mag-\nnetization of the sublattices have changed sign and are\nlarger than 0.1 in length 30 picoseconds after the laser\npulse is gone, (ii) no-switching, the sign of the sublattice\nmagnetization remains the same and larger than 0.1, (iii)\nthermal demagnetization, otherwise, namely, the mag-\nnetization remains lower than 0.1 for large time scales.\nThe colored area in Fig. 3(b) represents the laser energy\nthat induces switching. For a particular system with z\n−0.4−0.200.20.4\n2.533.544.5501 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0Mtot[µB]z=6z=8z=2 0Psw[×108J/m3]\ninitial temperature [K]z=6z=8z=2 0\nThermal  demagnetizationno-switchingswitching(a)(b)Figure 3. (a) Temperature-dependent equilibrium magneti-\nzation, Mtot=MGd−MFefor three different number of\nexchange-coupled nearest neighbours, z= 6,8 and 20. (b)\nColored area corresponds to the values of the laser energy\nPswfor switching. For temperature above around 400 K, only\nthermal demagnetization is observed. Below 400 K, for laser\nenergy less than the minimum, no-switching happens.\nexchange-coupled neighbours, the minimum energy nec-\nessary to switch decreases almost linearly as the initial\ntemperature increases. For z= 8, Psw(z= 8) is larger\nthan Psw(z= 6) but proportional to it. By looking at\nEq. (5), one sees that the exchange relaxation rate scales\nasz, that is, αex(z1)/αex(z2). We find that the ration\nPsw(z1)/Psw(z2)∼p\nz1/z2describes well this propor-\ntionality. For initial temperature above around 400 K,\nthe system is close to the critical temperature Tc, and\nthe final temperature after the laser pulse goes beyond\nTc. Therefore, the magnetic system ends up into a ther-\nmal demagnetization state. For even larger number of\nneighbours, z= 20, the laser energy would be too large\nso that the system will end up also into a thermal de-\nmagnetization state.\nTo summarize, we have demonstrated that for fer-\nrimagnetic GdFeCo alloys, the single-pulse magnetic\nswitching is sensitive to the lattice dimension and struc-\nture and to the form of spin interactions. We have used\na semi-phenomenological theory for a detailed discussion\nof the dependence of the relaxation terms – exchange and\nrelativistic – on the number of exchange-coupled spins.\nWe have validated these insights by computational sim-\nulations of the switching process using an atomistic spin\nmodel for GdFeCo. For example, for the limiting case of\nan infinity number of exchange-coupled spin, we predict\nthat switching is not possible in ferrimagnets. At the\nsame time, by reducing the number of exchange-coupled\nspins, the laser energy necessary to switch the magnetic\nstate of GdFeCo reduces significantly.\nThe authors have no conflicts to disclose.\nU. A. gratefully acknowledges support by grant\nPID2021-122980OB-C55 and the grant RYC-2020-\n030605-I funded by MCIN/AEI/10.13039/501100011033\nand by ”ERDF A way of making Europe” and ”ESF In-5\nvesting in your future”.\nThe data that support the findings of this study are\navailable from the corresponding authors upon reason-\nable request.\nREFERENCES\n1A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82,\n2731 (2010).\n2K. Carva, P. Bal´ aˇ z, and I. Radu (Elsevier, 2017) pp. 291–463.\n3A. El-Ghazaly, J. Gorchon, R. B. Wilson, A. Pattabi, and\nJ. Bokor, Journal of Magnetism and Magnetic Materials 502,\n166478 (2020).\n4J. Barker and U. Atxitia, Journal of the Physical Society of Japan\n90, 081001 (2021), https://doi.org/10.7566/JPSJ.90.081001.\n5C. Davies, J. Mentink, A. 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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\nηℓ∥∂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": "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\n2𝐽c𝜂\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. 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Magn. 48, 3223 (2012).  \n[26] F. Cutugno, L. Sanchez -Tejerina, R. Tomasello, M. Carpentieri, G. Finocchio, Appl. Phys. Lett. 118, \n052403 (2021).  \n[27] K. Cai, Z. Zhu, J. M. Lee, R. Mishra, L. Ren, S. D. Pollard, P. He, G. Liang, K. L. Teo, H. Yang, Nat. \nElectron. 3, 37 (2020).  \n[28] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).  \n[29] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical recipes in C++ : The Art of \nScientific Computing , 2nd edition (Cambridge University Press, Cambridge, 2002).  \n[30] Z. Zhu, K. Cai, J. Deng, V. P. K. Miriyala, H. Yang, X. Fong, G.  Liang, Phys. Rev. Appl. 13, 034040 \n(2020).  \n[31] C. Graves, A. Reid, T. Wang, B. Wu, S. De Jong, K. Vahaplar, I. Radu, D. Bernstein, M. Messerschmidt, \nL. Müller, Nat. Mater. 12, 293 (2013).  \n \n \n \n \n \n \n  \n \nFig. 1.  Illustration of  (a) the 2D atomistic model  which consists of 100 atoms  and (b) the d evice \nstructure.  Phase diagram of magnetization dynamics at ( c) x = 0.1 and ( d) x = 0.15. 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": "1610.09200v1.Spin_Orbit_Torque_Efficiency_in_Compensated_Ferrimagnetic_Cobalt_Terbium_Alloys.pdf",        "content": "1 \n Spin-Orbit Torque Efficiency in Compensated Ferrimagnetic Cobalt -Terbium Alloys  \nJoseph Finley1 and Luqiao Liu1 \n1Department of Electrical Engineering and Computer Science,  \nMassachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA  \n \n Despite the potential advantag es of information storage in antiferromagnetically coupled  \nmaterials, it remains unclear whether one can control the magnetic moment orientation efficiently because \nof the cancelled magnetic moment. Here,  we report spin -orbit torque induced magnetization switching of \nferrimagnetic Co 1-xTbx films with perpendicular magnetic anisotropy. Current induced switching is \ndemonstrated  in all of the studied film compositions, including those near the magnetization  \ncompensation point . The spin-orbit torque induced effective field is further quantified  in the domain wall \nmotion regime . A divergent behavior that scales with the inverse of magnetic moment is confirmed close \nto the compensation point, which is consistent with angular momentum conservation . Moreover , we also \nquantify the Dzyaloshinskii -Moriya interaction  energy in  the Ta/ Co1-xTbx system  and we find that  the \nenergy density increases as a function of the Tb concentration . The demonstrated spin -orbit torque \nswitching, in combinatio n with the fast magnetic dynamics and minimal net magnetization of \nferrimagnetic alloys, promises spintronic devices that are faster and with higher density than traditional \nferromagnetic systems.  \n  \n  2 \n I. Introduction  \n There has been great interest recently in using antiferromagnet ically coupled  materials as opposed \nto ferromagnet ic materials (FM) to store information . Compared with FM, antiferromagnet ically coupled  \nsystems exhibit  fast dynami cs, as well as immunities  against perturbation s from external magnetic field s, \npotentially ena bling  spintronic devices with higher speed and density  [1,2] . Rare earth (RE) – transition  \nmetal (TM) ferrimagnetic alloys are one potential candidate material for realizing such devic es. Inside \nRE-TM alloys, the moments o f TM elements (such as Fe, Co, Ni) and the RE elements (e.g., Gd, Tb , Ho, \netc) can be aligned with anti -paralle l orientations due to the exchange interaction between the f and d \nelectrons  [3]. By varying the relative concentrations of the two species, one can reach compensation \npoints where the net magnet ic moment or angular momentum goes  to zero  [4–7]. Moreover, because of \nthe different origins of magnetism in the two species, transport related properties are dominated by TM in \nthese alloys, providing a way to read out the magnetic state even in a compensated system. In this work, \nwe show that by uti lizing the current induced spin -orbit torque, one can switch magnetic moments in \nTa/Co 1-xTbx bilayer films. Particularly, we found that effective fields generated from the spin -orbit torque  \nscaled with the inverse of magnetization and reached maximum when the composition approaches  the \nmagnetic  compensation point. The large effective spin -orbit torque and the previously demonstrated fast \ndynamics  [8,9]  in these ferrimagnetic systems provide a prom ising platform for high speed spintronic \napplications.  \n \nII. Characterization  of Magnetic Properties  \n Spin-orbit torque  (SOT) originates from spin -orbit interaction  (SOI)  induced spin generation in \nthe bulk (i.e., the spin Hall effect)  [10,11] or the surface (i.e., the Rashba -Edelstein effect)  [12,13]  of solid \nmaterials. So far , SOTs have proven to be an efficient method of controlling the ferromagnetic state of \nnanoscale  devices  [14–17]. Recently, it was demonstrated that one can utilize SOT to  switch TM-\ndominant Co FeTb ferrimagnet s with perpendicular magnetic anisotropy (PMA)  [18]. It is therefore \ninteresting to ask what the relationship is between the chemical composition of RE -TM alloys and the 3 \n SOT effi ciency, and whether or not one can switch compensated ferrimagnets using SOT.  To answer \nthese questions, we grew a series of Ta(5)/Co 1-xTbx(t)/Ru(2 ) (thickness in nm) films using  magnetron \nsputtering. The Co1-xTbx alloys were deposited by co -sputtering Co and Tb sources with different \nsputtering powers. The concentration  of Tb x was calculated from the deposition rates  and varied between \n0.1 and 0.3 , while the layer thickness t ranges from 1.7 nm to 2.6 nm.  The magnetic properties of the \ndeposited films were examined using vibra ting sample magnetometry (VSM), and PMA was observed for \nall samples [Fig. 1 (a)].  Furthermore, the magnetic moment goes through zero and the coercive fields \nreach their maximum  around x ≈ 0.22, which is consistent with the room temperature magnetic moment \ncompensation point xcM  reported earlier  [6,7] . The dependence of the magnetic moment on the Tb \nconcentration is summarized in Fig. 1(c), which agrees well with t he trend  line calculated by assuming an \nanti-parallel alignment between the Co and Tb moment  (dashed lines).  The samples were then patterned \ninto Hall b ars with dimensions  4 x 44 μm2 [Fig. 1(d)]. The anomalous Hall resistance ( RAH) vs magnetic \nfield H curves measured from these samples are plotted in Fig. 1(b) and summarized in Fig. 1(e). The \npolarity of the hysteresis loops changes sign across xcM, consistent with previous studies where RAH is \nshown to be dominated by the momentum of the Co sublattice  [19,20] .  \n \nIII. Current -Induced Switching  \nFig. 2 (a) illustrates the current -induced magnetic switching for the series of samples. During \nthese measurements, in -plane magnetic fields of ±2000 Oe were applied in the current flowing direction \n[y axis in Fig. 1 (d )]. Previous studies showed that an in -plane field is necessary to ensure dete rministic \nmagnetic switching of  PMA films , as it can break the symmetry between two equivalent final states  [21–\n26]  .  As shown in Fig. 2(a ), the current -induced switching shows opposite p olarities under the positive \nand negative applied fields, consistent with the model  of SOT induced switching  [26]. Furthermore, under \nthe same in -plane field, the switching polarity changes sign as the samples go from being Co -dominant to \nTb-dominant . This phenomen on can be explained by considering a macro spin model as shown in Fig. \n2(b). In SOT switching,  the Slonczewski torque  [27] is proportional to 𝒎̂×(𝝈̂×𝒎̂), where 𝒎̂ is the unit 4 \n vector along the magnetic moment direction and 𝝈̂ is the orientation of electron spins generated from SOI  \n[along the 𝒙̂ direction in Fig. 1(d )]. Because the torque is an even  function of the local magnetic moment \n𝒎̂, effects from both sublattices in the RE -TM alloy add constructively  [1,28] . At equilibrium positions, \n𝒎̂ and 𝝈̂ are perpendicular to each other , and it is usually conve nient to use an effective field  [22] 𝐻𝑆𝑇∝\n𝝈̂×𝒎̂ to analyze  the SOT effect on magnetic switching. The equilibrium position of 𝒎̂ can then be \ndetermined by balancing the anisotropy field 𝐻𝑎𝑛, the applied in -plane field 𝐻𝑦, and 𝐻𝑆𝑇. As shown in \nFig. 2(b ), when 𝐻𝑦 and 𝝈̂ are given by the illustrated directions, the final orientation of the Co  sublattice  \nmagnetic moment will be close to  the +𝒛̂ direction f or the Co dominant sample and  close to  the −𝒛̂ \ndirection for the Tb dominant sample, giving rise to opposite Hall voltages.  Based upon this analysis, the \ncurrent induced switching should exhibit the same polarity change across the compensation point as the \nmagnetic field induced switching. We note that all of the samples follow this rule except for  the \nCo0.77Tb0.23 sample, where the field switching data had determined it to be Tb -dominant  but the current \ninduced switching corresponds to a Co -dominant  sample. A careful study on this sample reveals that this  \nchange  simply arises  from Joule heating induced temperature change. In RE-TM alloys , the magnetic \nmoments of the RE atom s have stronger temperature  dependence compared with TM atoms . \nConsequently , when the temperature increases, a higher RE concentration is ne cessary  to achieve the \nsame magnetic moment compensation point  [3–9,29] . By measuring the RAH  vs H curves of the \nCo0.77Tb0.23 sample under different applied currents , we found that the polarity of the field induced \nswitching did change si gn when the current density is higher than 2 × 107 A∙cm-2, suggest ing that the \nsample underwent a  (reversible) transition from Tb -dominant to Co -dominant . \n \nIV. Quantitative Determination of Spin-Orbit -Torque Efficiency  \n The critical current of SOT induced switching in a multi -domain sample is influenced by defect -\nrelated factors such as domain nucleation and domain wall  (DW)  pinning  [24]. Therefore, the SOT \nefficiency cannot  be simply extracted using the switching current values determined in Fig. 2 ( a). To 5 \n quantify the SOT in our samples, we measured the SOT induced effective  field in the DW motion regime \nby comparing it with the applied perpendicular field, using  the approach developed by C. F. Pai et \nal. [25]. It has been shown that in PMA films with N éel DWs , the Slonczewski term acts on the DW as an \neffective perpendicular magnetic field and induces DW motion [Fig. 3(g)]  [21–25].Therefore, by \nmeasuring the current induced shift in the RAH vs Hz curves, one can determine the magnitude of the SOT.  \nFig. 3 (a) and (b) show  typical  field induced switching curves  for a Co-dominant  sample Co 0.82Tb0.18 and a \nTb-dominant sample Co0.75Tb0.25. Under the  applied current of ±3 mA  and in -plane field of 2000 Oe, the  \ncenters of hysteresis loops  are offset from zero , with opposite values for opposite current directions. The \ncurrent dependence of the offset fiel ds are summarized in Fig. 3 (c ) and (d) for 𝐻𝑦=0 and ±2000 Oe, \nwhere a linear relationship between the offset field and the applied current is obtained. In these plots, the \nratio between the offset field 𝐻𝑧𝑒𝑓𝑓 and current density 𝐽𝑒 curve represents the efficiency of the SOT at the \nTa/RE -TM interface, defined as 𝜒 ≡𝐻𝑧𝑒𝑓𝑓\n𝐽𝑒.  𝜒 as a function of  applied  𝐻𝑦 for Co 0.82Tb0.18 and Co 0.75Tb0.25 \nsamples are plotted in Fig. 3 (e ) and (f), respectively . 𝜒 grows linearly in magnitud e for small values of \n𝐻𝑦, until reaching the saturation efficiency 𝜒𝑠𝑎𝑡 at a large  in-plane field 𝐻𝑦𝑠𝑎𝑡. The evolution of 𝜒 as a \nfunction of 𝐻𝑦 comes from the chirality change of the DWs in the sample. It is known that because of the \nDzyalosh inskii -Moriya interaction  (DMI) mechanism  at the heavy metal/magnetic metal interface  [30] or \ninside the bulk of RE -TM alloy  [31], stable Néel DW with spontaneous chiralities are formed.  Under zero \n𝐻𝑦, the DWs do not favor either sw itching polarities, leading to a zero offset field. As 𝐻𝑦 increases,  the \nDMI induced effective field 𝐻𝐷𝑀𝐼 is partially canceled, and DWs start to move in directions that facilitate \nmagnetic switching.  𝐻𝑦𝑠𝑎𝑡 therefore represents the minimum  field that is required to completely overcom e \n𝐻𝐷𝑀𝐼 and 𝜒𝑠𝑎𝑡 represents the maximum efficiency of the SOT .  \n Fig. 4 (a) illustrates the dependence of 𝜒𝑠𝑎𝑡 on x in Co1-xTbx samples. It can be seen that 𝜒𝑠𝑎𝑡 \ndiverges near xCM, with the larges t value occurring for the sample with smallest magnetization.  This result \nis consistent with the spin torque theory, where the ratio between the SOT effective field and applied 6 \n charge current is 𝜒𝑠𝑎𝑡=(𝜋/2)(𝜉ℏ/2𝑒𝜇0𝑀𝑠𝑡)  [25,32]. Here 𝜉=𝐽𝑆𝐽𝑒⁄ represents the effective spin Hall \nangle,  ℏ\n2𝑒𝐽𝑠 is the spin current density,  ℏ is Planck’s constant,  𝜇0 is the vacuum permeability , and 𝑀𝑠 is \nthe saturation magnetization. Note that this model of spin torque is based upon the conservation of total \nangular momentum. Previously it has been suggested to utilize ferrimagnetic materials with minimized \n𝑀𝑠 to increase the efficiency of spin torqu e induced switching  [33]. However, it was not verified if an \nefficient spin absorption could be achieved at the surface of a ferrimagnet material with antiparallel \naligned  sublattices. Moreover, because of the mixture between the spin angular momentum and orbital \nangular momentum in RE -TM alloys, there have been debates over the conservation of total angular \nmomentum in these systems  [34]. Our experiment al results provide clear evidence on the strong \nefficiency of spin orbit torques in antiferromagnetically  coupled materials. Within the experimenta l \naccuracy, we found that the effective field from the SOT does follow the simple trend given by 1/𝑀𝑆𝑡 \n[dashed lines in Fig. 4(a)], reflecting total angular momentum conservation. 𝜉 in our samples is \ndetermined  to be ~0.0 3, smaller than previously repo rted values from Ta/magnetic layer devices, possibly  \ndue to the relatively smaller spin-mixing conductance at the Ta/CoTb interface  [35].  \nIn addition to  the magnetic moment compensation point xcM, inside RE -TM systems there  also \nexists an  angular momentum  compensation point  xcJ due to the different g factors associated with spin and \norbit al angular moment um. For our Co 1-xTbx system, using the g factors of Co (~2.2) and Tb (~1.5) \natoms  [36,37]  , along with the relation 𝐽𝐶𝑜(𝑇𝑏)= 𝑀𝐶𝑜(𝑇𝑏)/𝛾𝐶𝑜(𝑇𝑏), where 𝛾𝐶𝑜(𝑇𝑏)= −𝑔𝐶𝑜(𝑇𝑏)𝜇𝐵/ℏ (𝜇𝐵 \nbeing the Bohr magneton, 𝛾𝐶𝑜(𝑇𝑏) the gyromagnetic ratio, and 𝐽𝐶𝑜(𝑇𝑏) the total angular mo mentum per \nunit volume), we determine xcJ  to be ~17%, which is within the range of the studied samples and lower \nthan xcM. Previously  it was demonstrated that ultrafast field-driven magnetic dynamics could be excited \naround xcJ [8,9] . According to  Landau -Lifshiz -Gilbert equation of a ferr imagnetic system  [27,29,38] , the \nspin torque term leads to  𝑑𝒎̂\n𝑑𝑡~−𝛾𝑒𝑓𝑓ℏ𝐽𝑠\n2𝑒𝜇0𝑀𝑆𝑡(𝒎̂×𝝈̂×𝒎̂) , where 𝛾𝑒𝑓𝑓=(𝑀𝐶𝑜−𝑀𝑇𝑏)/(𝐽𝐶𝑜−𝐽𝑇𝑏) is \nthe effective gyromagnetic ratio  and 𝑀𝑆=𝑀𝐶𝑜−𝑀𝑇𝑏 [8,9] . When 𝐽𝐶𝑜−𝐽𝑇𝑏 approaches zero, the time 7 \n evolution of 𝒎̂ diverg es if 𝐽𝑠 remains finite at xcJ. As observed in Fig. 4(a), 𝜒𝑠𝑎𝑡 remains roughly \nunchanged across xcJ, suggesting that similar to the field -driven experiment, SOT could also be used as an \nefficient drive force for achieving fast dynamics at this concentration. Finally, we find the switching \npolarity keeps the same sign  across xcJ, differ ing from the current induced switching of CoGd spin valves \nstudied in Ref.  [29], where a sw itching polarity reversal was observed between xcJ and xcM. This \ndifference is due to the presence of different switching mechanisms: in SOT induced switching of PMA \nfilms, the two  competing  torques are the field torque 𝛾𝑒𝑓𝑓𝑴×𝑯𝑒𝑓𝑓 and the spin tor que. Because the two \nterms have the same pre -factor 𝛾𝑒𝑓𝑓 and are only functions of 𝑴, 𝑯𝑒𝑓𝑓, and 𝝈̂,  under the same applied 𝝈̂ \nand 𝑯𝑦, the orientation of 𝒎̂ will remain the same (Fig. 2b ), regardless of the sign of  𝛾𝑒𝑓𝑓. In contrast, \nthe anti -damping switching of spin valves changes polarity for regions with 𝛾𝑒𝑓𝑓<0, as explained in \nRef. [29].  \n \nV. Measurement of the Dzyaloshinskii -Moriya I nteraction  Energy  \n The in -plane field needed for saturating the SOT, 𝐻𝑦𝑠𝑎𝑡, is plotted against the Tb concentration in \nFig. 4(b). First of all, we notice that 𝐻𝑦𝑠𝑎𝑡 is largest  near xcM. This  result  is consistent with fact that  the \neffective  DMI field  [32] 𝐻𝐷𝑀𝐼 =𝐷/𝑀𝑠𝑡𝜇0∆, where 𝐷 is the DMI energy density and ∆ is the DW width, \nwould become divergent when 𝑀𝑠 approaches zero . Secondly, 𝐻𝑦𝑠𝑎𝑡 is generally larger for the Tb-\ndominant  sample s than the Co -dominant ones. For example, the sample with the highest Co \nconcentration,  Co0.87Tb0.13, shows  𝐻𝑦𝑠𝑎𝑡~100 Oe, which is close to the  reported saturation field of Ta/FM \nstacks  [25]. However, in the Tb dominant sample Co0.71Tb0.29, which has similar magnetic moment, 𝐻𝑦𝑠𝑎𝑡 \nis found to be ~1500 Oe. In the inset of Fig. 4(b) we plot 𝐻𝑦𝑠𝑎𝑡𝑀𝑠𝑡, which increases roughly linearly as a \nfunction of x. By calculating the DW width ∆ =√𝐴𝐾𝑢⁄ using the determined anisotropy energy 𝐾𝑢 =  6.4 \n× 104 J∙m-2 from Tb - and Co -dominant samples  and the reported exchange stiffness  [39] A ~ 1.4 × 10-11 8 \n J/m, we get D in the range of 0.05~0.66 pJ/m. The increasing DMI energy with increasing Tb \nconcentration can  be explained by  the strong  spin-orbit coupling and large deviation from the free \nelectron g factor  [30] in the Tb atoms. The generation of magnetic textures such as chiral DWs and \nmagnetic skyrmion s  [40] relies on the competition between the DMI energy and other magnetostatic \nenergies. Therefore, the tunable DMI through chemical composition provides a useful a knob for \ncontrolling magnetic phases.  \n \nVI. Conclusion  \n To summarize, we demonstrated SOT induced switching in Co1-xTbx thin films with a wide range \nof chemical compositions. 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Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura,“Real -\nspace observation of a two -dimens ional skyrmion crystal,” Nature 465, 901 (2010).  \n \n  11 \n  \n \nFig. 1 (a) Out of plane magnetization curves  of Co 1-xTbx films . (b) RAH as a function of perpendicular \nmagnetic field. (c ) Magnetic  momen ts of Co1-xTbx alloys as a function of Tb concentration . (d) Schematic \nof the device geometry for  RAH measurement. (e) RAH as a function of Tb concentration .  \n \n \n \n \n \n \n \n \n \n \n12 \n  \n \nFig. 2 (a) Current induced SOT switching of Co 1-xTbx for in -plane fields of ± 2000 Oe. The current \ndensity inside Ta is calculated based on the conductivity of Ta thin films.  (b) Schematic of effective fields \nin a ferrimagnetic system. Fields acting on the moment  consist of the in -plane field  Hy, the anisotropy \nfield Han, and the SOT field  HST. \n \n  \n13 \n  \n \nFig. 3 (a), (c),(e) Measurements on Co-dominant  sample Co 0.82Tb0.18. (b),(d), (f) Measurements on Tb-\ndominant  sample Co 0.75Tb0.25. (a),(b ) RAH  vs. applied perpendicular field under  a DC current of ±3 mA. \n(c),(d ) SOT effective field as a function of applied current density  under in -plane fields of ±2000, and 0 \nOe. (e ),(f) SOT efficiency vs.  Hy. Efficiency saturates at the field  Hsaty. (g) SOT induced DW motion in \nthe ferrimagnetic system for Tb-dominant  and Co-dominant  films , showing that the effective \nperpendicular field has the same sign in both cases.  \n14 \n  \n \nFig. 4 (a ) Saturation efficiency and (b) in-plane saturation field for differe nt Co 1-xTbx films. Both the \nsaturation efficiency  and in-plane saturation field  are largest near the magnetic moment compensation \npoint.  The dashed line in (a) shows the trend calculated from χsat ~1/ Mst. Inset of (b) plots the product of \nthe in -plane sat uration field and magnetic moment M st, indicating an increase in DMI energy density D \nwith increasing Tb concentration.  \n \n \n \n"    },    {        "title": "1101.5943v1.Complex_room_temperature_ferrimagnetism_induced_by_zigzag_oxygen_vacancy_stripes_in_Sr3YCo4O10_72.pdf",        "content": "arXiv:1101.5943v1  [cond-mat.mtrl-sci]  31 Jan 2011Complex room temperature ferrimagnetism induced by zigzag oxygen-vacancy stripes\nin Sr3YCo4O10.72\nD.D. Khalyavin,1L.C. Chapon,1E. Suard,2J.E. Parker,3S.P. Thompson,3A.A. Yaremchenko,4and V.V. Kharton4\n1ISIS facility, Rutherford Appleton Laboratory-STFC, Chil ton, Didcot, Oxfordshire, OX11 0QX, UK.\n2Institut Laue-Langevin, 6 Jules Horowitz, BP156, 38042 Gre noble Cedex 9, France.\n3Diamond Light Source, Harwell Science and Innovation Campu s, Didcot, Oxfordshire OX11 0DE, UK.\n4Department of Ceramics and Glass Engineering,\nCICECO, University of Aveiro, 3810-193 Aveiro, Portugal.\n(Dated: December 4, 2018)\nThe high temperature ferromagnetism in Sr 3YCo4O10+δperovskite, whose origin has been the\nsubject of a considerable debate, has been studied by neutro n powder diffraction and synchrotron X-\nray diffraction measurements. Oxygen vacancy ordering crea tes a complex pattern of zigzag stripes\nin the oxygen-deficient CoO 4+δlayers, where the Co ions are found in three distinct coordin ations.\nThe symmetry of this unprecedented structural modulation, in conjunction with the existence of\ndifferent Co spin states, provide a straightforward explana tion for the appearance of ferrimagnetism.\nA model for the magnetic structure compatible with these str uctural features is proposed, based on\nthe refinement of powder neutron data. The macroscopic momen t as a function of temperature that\ncan be calculated from the values of the ordered spins extrac ted from refinements, is in excellent\nagreement with bulk magnetization. Unlike previous models , a collinear G-type magnetic structure\nwith uncompensated moments due to distinct spin-states of C o imposed by different coordination\nis found.\nPACS numbers: 75.25.-j, 75.50.Gg, 61.05.F-\nThe rich physical properties of cobalt oxides compared\nto other 3dtransition metal oxides originate in the vari-\nouselectronicstatesofcobaltions. Probablythefirstand\nmost famous example is LaCoO 3perovskite that under-\ngoes a diamagnetic to paramagnetic transition on warm-\ning. The phenomenon, interpreted by Goodenough1as\na thermally activated crossover of Co3+from a low-spin\nto a high-spin state, is due to a subtle balance between\ninteratomic exchange energy and crystal field splitting.\nNowadays many complex cobalt-oxides with fascinating\nelectrical and magnetic properties are known, displaying\nsuperconductivity, near room-temperature giant magne-\ntoresistance, high ionic/electronic conductivity and large\nthermoelectric power, making them attractive and tech-\nnologically relevant.2Practically in all cases the Co elec-\ntronic configuration, primarily determined by the crys-\ntalline electric field created by first-neighbour oxygen\nions, plays a central role in the underlying physics. The\nlocalCoenvironmentisthereforeagreatlevertotunethe\nelectric properties and consequently the magnetic prop-\nerties (spin-state) of such systems.\nRecently, much attention has been devoted to the\noxygen-deficient perovskites Sr 3RCo4O10+δ(R=rare\nearth or Y)3–6in particular systems with oxygen con-\ntent 0.5< δ <1 which display unconventional ferro-\nmagnetism with the highest critical temperature (T m∼\n360K) among the known cobalt-perovskites. The basic\ncrystal structure of Sr 3RCo4O10of tetragonal I4/mmm\nsymmetry, involvesboth cation ordering (Sr/ R) and oxy-\ngen vacancy ordering with a 2 ap×2ap×4apsuperstruc-\nture with respect to the pseudo-cubic perovskite unit-cell\n(Fig. 1). Thelatterorderingproducesanalternatestack-\ning ofoxygen-rich octahedral (CoO 6) layers and oxygen-deficient tetrahedral (CoO 4) layers along the c-axis. For\ncompositions with 0 .5< δ <1, an additional superstruc-\nture has been identified (2√\n2ap×2√\n2ap×4ap) and at-\ntributed to the ordering of the extra oxygen ions in the\nCoO4+δlayers.3–7It has been shown6,8,9that the for-\nmation of such superstructure is key to the appearance\nof ferromagnetism; however a clear understanding of its\nmicroscopic origin is still missing in spite of several neu-\ntron diffraction studies.10–15The main difficulty lies in\nthe simultaneous determination of the oxygen ordering\n(CoO 4+δ)(CoO 6)R\nSr \nO(8 j)O(8 i)\nFIG. 1: (Color online) Schematic representation of the\ntetragonal I4/mmm (2ap×2ap×4ap) crystal structure\nof Sr3RCo4O10+δperovskites as alternation of the oxygen-\noccupied (CoO 6) and oxygen-deficient (CoO 4+δ) layers\nstacked along the caxis (left). Expanded view of the CoO 4+δ\nlayer with partially occupied 8 joxygen position (right).2\nsuperstructure and the magnetic structure from diffrac-\ntion data, required to built a comprehensive picture.\nIn the present letter, we report a model for the mag-\nnetic structure and oxygen vacancy superstructure of\nSr3YCo4O10.72obtained by neutron diffraction and sym-\nmetry considerations. The magnetic configuration com-\npatible with the superstructure formed by oxygen va-\ncancy ordering, explains the origin of the high temper-\nature ferromagnetism. We show that the oxygen vacan-\ncies create unconventional zigzag stripes in the CoO 4+δ\nlayers with three distinct Co environments. Although\nall nearest-neighbour exchange interactions are strongly\nantiferromagnetic, the symmetry and presence of three\ninequivalent magnetic sites in the oxygen-deficient lay-\ners result in a net spontaneous moment. The magni-\ntudeofthisferromagneticcomponentcalculatedfromthe\nmagnetic configuration as a function of temperature is in\nremarkable quantitative agreement with the magnetiza-\ntion.\nA powder sample ofSr 3YCo4O10+δwas synthesized by\nsolid-state reaction as described previously.3The oxygen\ncontent determined by reduction in a 10% H 2- 90% N 2\nmixture usinga SetaramSetsys16/18thermogravimetric\nequipment was found to be 10.72(3). Neutron diffraction\ndata were collected at the ILL (Grenoble, France) us-\ning the high-resolution D2B (10K <T<575K) and high\nintensity D1B (10K <T<330K) two-axis powder diffrac-\ntometers. Synchrotron X-ray diffraction data were col-\nlected at the Diamond light source (RAL, UK) on the\nhigh resolution powder beamline I1116(10K<T<600K).\nThe program FullProf17was employed for Rietveld re-\nfinements and group-theoretical calculations were done\nwith the aid of the ISOTROPY software.18\nAnalysis of the synchrotron X-ray diffraction data re-\nvealed three structural phase transitions at T S1∼550K,\nTS2∼330K and T S3∼280K. Above T S1, the sym-\nmetry is tetragonal I4/mmmwith a 2 ap×2ap×4ap\nsupercell as originally proposed by Istomin et al.4for\nSr0.7Y0.3CoO2.63and by Withers, James and Goossens3\nforR1/3Sr2/3CoO3−γ(R=Y, Ho and Dy). The model\nimplies fully occupied 8 iand partially occupied 8 joxy-\ngen sites in the oxygen deficient CoO 4+δlayers (Fig.\n1). Below T S1, a set of new reflections associated with\ntheX(k=a*/2+b*/2) reciprocal point appears, as al-\nready reported for the analogue5Sr3.12Er0.88Co4O10.5.\nIn agreement with previous studies,3–7, we found that\nthe primary distortion involves oxygen vacancy order-\ning in the CoO 4+δlayers on the partially occupied 8 j\nWyckoff site. Group-theoretical arguments dictate that\nthe symmetry of the superstructure should be compati-\nble with one of the isotropy subgroups given in Table I.\nExtinctionconditions andstructuralrefinementsarecon-\nsistent with a unique subgroup, Cmmaassociated with\ntheX−\n4irreduciblerepresentation(irrep)ofthe I4/mmm\nspace group. This symmetry coincides with that deduced\nby Withers, James and Goossens in their original work3\nbased on electron diffraction and used recently by James\net al.7to refine the room temperature diffraction data\n8 4 4 pyr tet oct \nm g g F S S S \n\u0001 \u0000\u0003 \u0002 = − + \u0004 \u0005\u0006 \u0007∑ ∑ ∑Sr \nY\nCo tet Co pyr Co oct \nFIG. 2: (Color online) Schematic representation of the\nCoO4+δlayer with oxygen vacancy superstructure associated\nwith the X−\n4irrep. Magnetic moments localized on Co sites\nwith six-fold (4 g), five-fold (8 m) and four-fold (4 g) coordina-\ntion are shown as arrows of different length. For clarity they\nare shown in the ( bc) plane but in the actual refinement the\nmoments were perpendicular to this plane.\nof Sr0.8R0.2CoO2.5+δ/4. The corresponding superstruc-\nture is shown in Figure 2. Note that the c-axis of the\ntetragonal structure becomes a-axis in the Cmmaset-\nting (see Table I). For clarity all atoms are displayed\nin their highly symmetric positions. The oxygen va-\ncancies create zigzag stripes in the ( bc)-plane providing\nthree different Co coordinations in the CoO 4+δlayers,\nnamely octahedral (position 4 g), square pyramidal (8 m)\nand tetrahedral (4 g). The presence of these distinctly\ncoordinated Co ions is the crucial ingredient to success-\nfully model the low temperature magnetic scattering and\nadequately interpret the magnetic properties of this ma-\nterial. There is no evidence from the diffraction data\nthat the vacancy ordering pattern changes with temper-\nature as the critical temperatures of subsequent transi-\ntions T S2and T S3are too low for a superstructure re-\nconstruction which requiresatomic diffusion on distances\ncomparable to ap. Instead, these transitions are likely to\nbe displacive. At T S2, one identifiesa doublingofthe cell\nTABLE I: Isotropy subgroups associated with order-disorde r\nmodes on the 8 jWyckoff position, along a single-arm direc-\ntion of the irreducible representations of the I4/mmmspace\ngroup, conjugated with the Xpoint of symmetry.\nSubgroup Irrep Lattice vectors Origin\nCmmm X+\n1(0,0,1),(1,1,0),(-1,1,0) (0,0,0)\nCmca X+\n2(0,0,1),(1,1,0),(-1,1,0) (0,0,0)\nCmcm X−\n3(0,0,1),(1,1,0),(-1,1,0) (0,1/2,0)\nCmma X−\n4(0,0,1),(1,1,0),(-1,1,0) (0,1/2,0)3\n0.0 1.0 2.0 3.0 4.0 5.0 \nTS3 \nTS2  Octahedral \n Pyramidal \n Tetrahedral \n  Sublattice moment ( µB ) \nTm\nCoO 4 0.0 TS3 TS2 \nTmMagnetization ( µB per Co) \n0 100 200 300 400 0.00 0.03 0.06 0.09 0.12 \n \nTemperature ( K ) \nCoO 6\nCoO 5CoO 4\nFIG. 3: (Color online) Magnetic order parameters for octa-\nhedral, pyramidal and tetrahedral sublattices as a functio n\nof temperature (top). SQUID magnetization measured at\nH=0.5T at warming after cooling in this field (solid line).\nOpen circles represent net magnetic moment obtained from\nthe neutron diffraction data as a difference between spin val-\nues in pyramidal and summarized octahedral and tetrahedral\nsublatttices (bottom). Inset shows apart of the magnetic un it\ncell.\nalong the caxis with respect to the Cmmaspace group\n(2√\n2ap×4ap×4√\n2apsupercell). The appropriate struc-\ntural models with possible monoclinic symmetry involve\nmore than sixty independent atoms in the unit-cell and\ncannot be properly refined from the powder data.\nIn agreement with magnetization measurements (Fig.\n3 bottom), the magnetic transition in Sr 3YCo4O10.72\ntakes place at T m∼360K. The transition is charac-\nterized by the onset of a spontaneous moment whose\nmagnitude reaches a maximum at 290K and then gradu-\nally decreases becoming temperature independent below\n200K. The magnetic phase transition takes place at a no-\ntably higher temperature (T m∼360K) than T S2. This\nobservation contradicts previous reports5of a simultane-\nous structural and magnetic transition which suggested\nthat ferromagnetism was induced by the change in crys-\ntal structure. In the neutron data, additional intensity\non top of several nuclear reflections is observed in the\nmagnetically ordered phase, indicating a k=0 propaga-\ntion vector. The largest contribution appears on the 220\nreflection (d-spacing ∼4.4˚A) in agreement with previous\nneutron diffraction studies.10–15 12.00 12.75 13.50 14.25 \n2Θ ( degree ) (001)  10 K \n 200 K \n 330 K \n  Intensity ( arb. units ) \n2Θ ( degree ) NPD \nλ=2.52 Å\n4.00 4.25 4.50 4.75 λ=0.826 ÅXRPD  10 K \n 200 K \n 330 K  \n(001) \n10 20 30 40 50 60 70 Intensity ( arb. units ) \n2Θ ( degree ) 10 20 30 40 Rnuc = 4.01% \nRmag = 5.92% λ=2.52 Å (220)+(202) \n(221) (021) (020)+(002) (201)  \n (001) \nFIG. 4: (Color online) A part of the neutron (left) and X-ray\n(right) diffraction patterns in a vicinity of the (001) reflec -\ntion collected at different temperatures (top). An example\nof the D1B neutron diffraction pattern refined in the model\ndescribed in the text (bottom). The cross symbols and (red)\nsolid line represent the experimental and calculated inten si-\nties, respectively, and the line below is the difference betw een\nthem. Tick marks indicate the positions of Bragg peaks for\nthenuclearscattering( Cmmaspacegroup, topline)andmag-\nnetic scattering ( k=0 propagation vector, bottom line).\nThe common interpretation of the magnetic neutron\ndiffraction pattern involves the so called G-type mag-\nnetic ordering, where neighbour spins aligned along the\ncaxis of the I4/mmmspace group have opposite di-\nrections. In early reports, the magnetic structure was\nrefined assuming equal values for all the spins in a unit\ncell.10–12Later, it was established that the Co ions have\ndifferent spin valuesin the CoO 6and CoO 4+δlayers.13–15\nIn all these models, ferromagnetism was inferred from\nspin-canting. This explanation is incompatible with our\ndiffraction data: specifically, the appearance of mag-\nnetic intensity on top of the nuclear (001) reflection (Fig.\n4 top) below T m, whose magnetic origin is confirmed\nby the absence of a corresponding intensity in the X-\nray data, directly indicates instead the presence of non-\nequivalent magnetic moments in the CoO 4+δlayers. In\nother words, the magnetic arrangement corresponds to a\nsimple collinear G-type antiferromagnet (no canting) but\nwith a non-compensated moment due to the inequivalent\nspin states imposed by the different Co environments.\nAccordingly, the refinements of the neutron diffraction4\ndata were performed using a model with three indepen-\ndent parameters for the Co spin values in octahedral,\npyramidal and tetrahedral coordination. In spite of the\nsimplification, since there are actually six independent\nCo positions even in the Cmmastructure, the model\nworks very well and describes correctly all features of\nthe magnetic scattering pattern (Fig. 4 bottom). The\nnuclear scattering was satisfactorily modeled in the or-\nthorhombic Cmmaspace group. Refined occupancies for\nthe oxygen positions in CoO 4+δlayers yield an average\noxygen content 10.81(3) to be sightly higher than from\nthe TGA data. At T=10K, the magnetic moment mag-\nnitudes for the four-, five- and six-fold coordinated Co\nare 4.0(1), 2.7(1) and 1.16(5) µB, respectively. A lower\nspin-state is found for the more coordinated ions. This\nis in agreement with a recent study showing a certain\ninstability for octahedrally coordinated Co in these ma-\nterials, for which the application of a moderate pressure\n(∼2GPa) induces a switching to a non-magnetic low-spin\nstate.19\nThe variation of the moments as a function of temper-\nature has been extracted from all Rietveld refinements\n(Fig. 3 bottom). It is straightforward to deduce the\nnet macroscopic moment and its temperature variation\nfrom the latter values, which agree remarkably well with\nthe magnetization curve (Fig. 3 bottom). In this ferri-\nmagnetic structure, the net moment directed along the a\naxis (caxis in the parent I4/mmmspace group) is sim-\nply the differencebetween the moments onthe pyramidal\nsublattice (8 m) and the sum of the moments on the octa-\nhedral (4 g) and tetrahedral (4 g) sublattices (Fig. 2 and\n3 inset). The octahedral layers CoO 6do not produce\nany net moment in the proposed model. The magnetiza-\ntion peak observed around 290K is naturally explained\nby the different critical behaviours of the distinct sub-\nlattices (Fig. 3) in the CoO 4layer, which seem directly\ncorrelated to the number of nearest neighbour magnetic\ninteractions. It appears that the sublattice magnetiza-tion has a lower critical exponent (faster saturation) for\nthe octahedral site. The critical behaviours are also af-\nfected by the structural phase transition at T S3, but in\nthe absence of a detailed model for the crystal structure\nbelow this temperature, it is impossible to relate this ef-\nfect tothe displacivemodes. Finally, it shouldbe pointed\nout that in previous models, the presence of a ferromag-\nnetic component due to a canted arrangement required\nthe admixture of at least two irreps, whereas our model\ninvolves the single Γ+\n3irrep of the Cmmaspace group in\nagreement with symmetry requirements for second-order\ntransitions.\nIn conclusion, the complex superstructure in the\nSr3YCo4O10.72perovskite is related to the ordering of\noxygen vacancies in a pattern forming zigzag stripes in\nthe oxygen-deficient CoO 4.72layers. The superstructure\nis built from a corner-sharing network of Co ions in\nfour-, five- and six-fold coordination, each with distinct\nspin-states. The magnetic ordering occurring at T m∼\n360K involves a simple antiferromagnetic arrangement\nof neighbour spins but the symmetry of the superstruc-\nture and the existence of distinct spin states result in\na net ferrimagnetic moment. The nature of the oxygen\nordering superstructure is key to understand the ferro-\nmagnetism, previouslyascribedto a cantedarrangement.\nRemarkably, the complex temperature dependence of the\nmagnetizationcanbe quantitativelyexplained by the dif-\nferent moment values extracted from neutron data. Its\npeculiar shape can be directly related to the different\ncriticalities of the magnetic order parameters for the Co\nsublattices. The proposed model can be easily extrapo-\nlated for other compositions of the series exhibiting the\noxygen-vacancy superstructure but with different R-ions\nand oxygen content. In all cases, the presence of the\nsuperstructure results in an effectively different Co coor-\ndinationin theCoO 4+δlayersandthereforeinaresultant\nuncompensated moment.\n1J. B. Goodenough, J. Phys. Chem. Solids 6, 287 (1958).\n2N. B. Ivanova, S. G. Ovchinnikov, M. M. Korshunov,\nI. M. Eremin, and N. V. Kazak, Phys. Usp. 52, 789 (2009).\n3R. L. Withers, M. James, and D. J. Goossens, J. Solid\nState Chem. 174198 (2003).\n4S. Y. Istomin, J. Grins, G. Svensson, O. A. Drozhzhin,\nV. L. Kozhevnikov, E. V. Antipov, and J. P. Attfield,\nChem. Mat. 21, 4012 (2003).\n5S. Ishiwata, W. Kobayashi, I. Terasaki, K. Kato, and\nM. Takata, Phys. Rev. B 75, 220406 (2007).\n6S. Fukushima, T. Sato, D. Akahoshi, and H. Kuwahara, J.\nPhys. Soc. Jpn. 78064706 (2009).\n7M. James, M. Avdeev, P. Barnes, L.Morales, K.Wallwork,\nand R. Withers, J. Solid State Chem. 180, 2233 (2007).\n8S. Fukushima, T. Sato, D. Akahoshi, and H. Kuwahara, J.\nAppl. Phys. 10307F705 (2008).\n9S. Balamurugan, K. Yamaura, A. Asthana, A. Ubaldini,\nY. Matsui, and E. Takayama-Muromachi, J. Appl. Phys.10307B903 (2008).\n10S.Y.Istomin, O.A.Drozhzhin, G.Svensson, andE.V.An-\ntipov, Solid State Sci. 6, 539 (2004).\n11D. J. Goossens, K. F. Wilson, M. James, A. J. Studer, and\nX. L. Wang, Phys. Rev. B 69, 134411 (2004).\n12A. Baszczuk, S. Kolesnik, B. Dabrowski, O. Chmaissem,\nand J. Mais, Phys. Rev. B 76, 134407 (2007).\n13D. V. Sheptyakov, V. Yu. Pomjakushin, O. A. Drozhzhin,\nS. Ya. Istomin, E. V. Antipov, I. A. Bobrikov, and\nA. M. Balagurov, Phys. Rev. B 80, 024409 (2009).\n14I. O. Troyanchuk, D. V. Karpinsky, A. P. Sazonov,\nV. Sikolenko, V. Efimov, and A. Senyshyn, J. Mater. Sci.\n44, 5900 (2009).\n15I. O. Troyanchuk, D. V. Karpinsky, V. M. Dobryanskii,\nA. N. Chobot, G. M. Chobot, and A. P. Sazonov, J. Exp.\nTheor. Phys. 108, 428 (2009).\n16S. P. Thompson, J. E. Parker, J. Potter, T. P. Hill, A. Birt,\nT. M. Cobb, F. Yuan, and C. C. Tang, Rev. Sci. Instrum.5\n80, 075107 (2009).\n17J. Rodriguez Carvajal, Physica B 192, 55 (1993).\n18H. T Stokes, D. M Hatch, B. J Campbell, ISOTROPY,\nstokes.byu.edu/isotropy.html (2007).19N. O. Golosova, D. P. Kozlenko, L. S. Dubrovinsky,\nO. A. Drozhzhin, S. Ya. Istomin, and B. N. Savenko, Phys.\nRev. B79, 104431 (2009)."    },    {        "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": "2001.02602v2.Non_equilibrium_spin_dynamics_in_the_temperature_and_magnetic_field_dependence_of_magnetization_curves_of_ferrimagnetic_Co___1_75__Fe___1_25__O__4__and_its_composite_with_BaTiO__3_.pdf",        "content": "1\n \n \nN\non\n-\nequilibrium spin dynamics in \nthe \ntemperature and magnetic field dependen\nce of\n \nmagnetization curves of \nferrimagnetic Co\n1.75\nFe\n1.25\nO\n4\n \nand \nits composite with \nBaTiO\n3\n \n \nR.N. Bhowmik\n*\n1\n, and R.Ranganathan\n2\n      \n \n1\nDepartment of Physics, Pondicherry University, R\n. 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. 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Bhaumik, \nRSC Adv.\n \n6\n, 457\n01 (2016).\n \n \u0000 \u0001 \u0002 \u0003 \u0004 \u0005 \u0006 \u0007\b \t \n \u000b \u0001 \u0006 \f \r \u000b \u0007 \b \u0006 \f \u000e \b \u000f \u0010 \u0007\b \u0011 \u0011 \u0010 \u0012 \u0001 \u0013 \u0010 \u000b \u000e \u0014\u0006 \u000b \u0014 \u000e \b \r \f \u000e \u0015\u0000 \u0016 \u0017 \u0018 \n \u0019 \n \u0013\u001a \u0015\u0000 \u001b \u000f \u001c \u001d \u001b \u0006 \f \t \u0012\f \u0010 \u0001 \u000b \b \u0018 \u001e\u0019 \u001f \u0001 \u000b \u0007 \u0010 \u0012 \n \u0006 \b ! \" # ! $ # % & % ' ( ) * + ! & , ( # & $ ( \" ! * $ # % & - ./ 0\n1 2 3 4 5 6 7 8 4 9 4 :; 6 < 8 = ; = > = ? 6 @ = 8 ; 4 9 < 4 8 5 ; 7 8 4 A > B 4 1 5 : C @ D 4 ? 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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": "2303.04649v2.Spin_valve_nature_and_giant_coercivity_of_a_ferrimagnetic_spin_semimetal_Mn__2_IrGa.pdf",        "content": "Spin-valve nature and giant coercivity of a ferrimagnetic spin semimetal Mn 2IrGa\nAkhilesh Kumar Patel,1, 2,∗Y. Venkateswara,1, 3,∗S. Shanmukharao Samatham,4\nArchana Lakhani,5Jayita Nayak,3K. G. Suresh,1,†and Aftab Alam1,‡\n1Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India\n2Research Centre for Magnetic and Spintronic Materials,\nNational Institute for Materials Science, Tsukuba, Ibaraki 305 0047, Japan\n3Spectroscopic Investigations of Novel Systems Laboratory, Department of Physics,\nIndian Institute of Technology Kanpur, Kanpur 208016, India\n4Department of Physics, Chaitanya Bharathi Institute of Technology, Gandipet, Hyderabad 500 075, India\n5UGC-DAE Consortium for Scientific Research, University Campus, Indore 452001 Madhya Pradesh, India\n(Dated: June 2, 2023)\nSpin semimetals are amongst the most recently discovered new class of spintronic materials, which\nexhibit a band gap in one spin channel and semimetallic feature in the other, thus facilitating tunable\nspin transport. Here, we report Mn 2IrGa to be a candidate material for spin semimetal along with\ngiant coercivity and spin-valve characteristics using a combined experimental and theoretical study.\nThe alloy crystallizes in an inverse Heusler structure (without any martensitic transition) with a\npara- to ferri-magnetic transition at TC∼243 K. It shows a giant coercive field of about 8.5 kOe (at 2\nK). The negative temperature coefficient, relatively low magnitude and weak temperture dependance\nof electrical resistivity suggest the semimetallic character of the alloy. This is further supported\nby our specific heat measurement. Magnetoresistance (MR) confirms an irreversible nature (with\nits magnitude ∼1%) along with a change of sign across the magnetic transition indicating the\npotentiality of Mn 2IrGa in magnetic switching applications. In addition, asymmetric nature of\nMR in the positive and negative field cycles is indicative of spin-valve characteristics. Our ab-initio\ncalculations confirm the inverse Heusler structure with ferrimagnetic ordering to be the lowest energy\nstate, with a saturation magnetization of 2 µB.<100>is found to be the easy magnetic axis with\nconsiderable magneto-crystalline anisotropy energy. A large positive Berry flux at/around Γ point\ngives rise to an appreciable anomalous Hall conductivity ( ∼-180 S/cm).\nIntroduction : In the last two decades, the search for\npromising energy efficient multifunctional materials has\ngained tremendous momentum. Of these, new classes of\nmagnetic materials such as spin gapless semiconductors,\nbipolar magnetic semiconductors, fully compensated fer-\nrimagnetic and spin semimetals play a crucial role in\ntaking the spintronic research to a different height.[1, 2]\nHeusler alloys (HA) are one of the potential classes of ma-\nterials, which host all of the above properties in different\nmaterials. They have also gained lot of attention due to\ntheir high magnetic ordering temperature (either TCor\nTN), flexibility of doping/alloying enabling a wide plat-\nform for tunable band structure engineering. However,\nmany of these alloys suffer from anti-site disorder, which\noften hinders their potential for application. HAs are\nmostly composed of two or more 3 dtransition elements.\nOne of the ways to minimize the anti-site disorder is to\nreplace one of the 3 dtransition elements by a 4 dor 5d\nelement. Some of the 4 dbased HAs reported for spin-\ntronics applications are CoRhMnGe,[3] CoRuMnSi,[4]\nFe2RhSi,[5] VNbRuAl[6] etc. There are only a few 5 d\nbased HAs that are experimentally reported to crystal-\nlize in cubic structure e.g. Fe 2IrSi[7, 8], LiGa 2Ir[9] and\nRu2TaAl,[10] though there are several theoretical reports\n∗These authors contributed equally to this work.\n†suresh@phy.iitb.ac.in\n‡aftab@phy.iitb.ac.insuch as Ir 2ScGa[11], Ir 2YSi (Y=Sc,Ti,V, Mn, Fe, Co,\nand Ni)[12], Ir 2LuSb[13], Ir 2ScAl[14], TiZrIrZ (Z=Al,\nGa or In)[15], CoCrIrSi[16], ZnCdIrMn[17], Zn 2IrMn[18],\nCoIrMnAl[19], Ru 2TaGa[20], Mn 2YZ (Y=Ta, W, Os, Ir,\nPt, Au; Z=Ga,In).[21–23] Tetragonal phase of some of\nthese alloys, e.g. Mn 2YZ, acquire high magnetic order-\ning temperature and show large exchange bias, spin-orbit\ncoupling (SOC), anti-skyrmionic nature, giant magne-\ntocaloric and large magnetoresistance (MR).[24–26]\nMn2IrGa is an interesting system which is reported\nto stabilize in both cubic ( F-43m , #219) and tetragonal\n(I-4m2 , #119) structures.[21] In the cubic phase, the cal-\nculated equilibrium lattice constant is 5.97 ˚A and the net\nmagnetization ( m) is 2 µB/f.u. While, tetragonal phase\nshows low mand large magneto-crystalline anisotropy,\nthus promising for spin-transfer magnetization-switching\napplications. First principles calculations by Hellal et\nal.[22] reported ferromagnetic transition temperatures\n(TC) of 368 K and 244 K for tetragonal and cubic phases\nrespectively. As such, although there exist some theoret-\nical studies on Mn 2IrGa, the experimental investigation\non the structural, magnetic and electrical transport of\nthis alloy is lacking. Specially, exploring its potential for\nspintronics related applications is highly desired.\nIn this letter, we report the structural, magnetic\nand electrical transport properties of polycrystalline\nMn2IrGa using a combined experimental and theoret-\nical study. Mn 2IrGa crystallizes in an inverse cubic\nHeusler structure with no signature of martensitic tran-arXiv:2303.04649v2  [cond-mat.mtrl-sci]  1 Jun 20232\n204 06 08 01 001 20\n Bragg Position Ycalc \nYobs-Ycalc( 622)(531)(620)Instensity (arb. units)2\nθ (deg.) Yobs(111)(\n220)(200)(\n311)(\n222)(\n400)(\n420)(331)(\n422)(\n333)(\n440)(\n531)(\n442)\nFIG. 1. Room temperature XRD pattern of Mn 2IrGa along\nwith Rietveld refinement in Config. 1 whose structure is\nshown in the inset.\nsition throughout the temperature. Magnetic measure-\nments reveal ferrimagnetic nature with giant coercive\nfield (∼8.5 kOe) below TC, possibly due to strong SOC\ninduced by Ir. Zero-field-cooled (ZFC) and field-cooled\n(FC) T-dependent magnetization(M) shows large bifur-\ncation. Real part of AC-susceptibility ( χ′\nAC) exhibits a\nsharp peak at/around 246 K and a broad peak at 25 K.\nInterestingly, the position of sharp peak is independent\nof frequency, indicating the long-range magnetic order-\ning, while the broad peak suggests the short range mag-\nnetic ordering at low T. Transport measurements reveal\na semimetallic nature of Mn 2IrGa. MR data indicates\ngiant hysteresis with asymmetric nature, indicating spin\nvalve behavior. Specific heat data yields a moderate den-\nsity of states (DOS) at the Fermi level along with a kink\nat∼243 K indicating a possible transition. Ab-initio sim-\nulation predicts a ferrimagnetic spin semimetal behavior\nwith a net moment of 2.0 µB.<100>is found to be the\neasy magnetization axis with large magnetocrystalline\nanisotropic energy. Simulated Berry curvature shows few\nhot spots in the Brillouin zone yielding a reasonably high\nanomalous Hall conductivity (-180 S/cm).\nMethods : Polycrystalline Mn 2IrGa was prepared by\narc-melting method with constituent elements of 4N pu-\nrity. Further experimental details are given in Sec. A of\nthe supplementary material (SM)[27] (see also Refs. [28–\n32] therein) First-principles calculations are performed\nusing full potential linearised augmented plane- wave\n(FLAPW) method as implemented in FLEUR code.[33–\n36] Other computational details are provided in Sec. B of\nSM.[27] Mn 2IrGa belongs to a full Heusler alloy (X 2YZ)\nwhere Z is a main group element.[5, 37] There exists two\nnon-degenerate crystal configurations for this alloy, in-\nverse Heusler structure (Config. I) and normal Heusler\nstructure (Config. II). In the former, the X atoms (here\nMn) occupy two distinct Wyckoff sites namely the tetra-\nhedral (4c) and the octahedral (4b) sites, while in the\n01002003004000100200300400500600700χ-1 (kOe f.u./µB)T\n (K)(f) \nZFC1\n00 Oeθ\nCW ~ 260 Kµ\neff ~ 1.98 µB/f.u.\n01002003004000.00.40.81.21.6M (µΒ/f.u.)T\n (K) ZFC \nFCC \nFCW50 kOe(e)\n01002003004000.00.20.40.60.81.01.2M (µΒ/f.u.)T\n (K) ZFC \nFCC \nFCW10 kOe( d)\n01002003004000.00.20.40.60.81.01.2M (µΒ/f.u.)T\n (K) ZFC \nFCC \nFCW5 kOe( c)\n01002003004000.00.20.40.60.81.0M (µΒ/f.u.)T\n (K) ZFC \nFCC \nFCW1 kOe( b)\n01002003004000.00.10.20.30.40.50.6M (µΒ/f.u.)T\n (K) ZFC \nFCC \nFCW100 Oe( a)FIG. 2. For Mn 2IrGa, Mvs.Tat various applied fields (a)\n100 Oe, (b) 1 kOe, (c) 5 kOe, (d) 10 kOe and (e) 50 kOe. (f)\nCurie-Weiss fitting at 100 Oe.\nlater, X atoms occupy the tetrahedral and Y occupy the\noctahedral sites.[5]\nResults and Discussion : Figure 1(a) shows the\nroom temperature X-ray diffraction (XRD) pattern of\nMn2IrGa along with its Rietveld refined data. It crys-\ntallizes in the fcc structure (space group F ¯43m) with a\nlattice parameter of 6.03 ˚A. The best fit is achieved for\nthe inverse Heusler structure with Ga at 4a, Mn at 4b\n(say Mn1) and 4c (say Mn2), and Ir at 4d site, as shown\nin the inset of Fig. 1. Note that, it is only the minor\n(222) peak which has not fitted well due to some texture\nalong this direction. To further confirm the structural de-\ntails, a high resolution transmission electron microscopy\n(HR-TEM) measurement is carried out (see Sec. C of\nSM[27]). The estimated d-spacing agrees fairly well with\nthose obtained using XRD.\nFigures 2(a)-(e) show the T-dependence of magnetiza-\ntion ( M) at five different magnetic fields (H). Clearly,\nthe field cooled cooling (FCC) and field cooled warm-\ning (FCW) curves coincide with each other, indicating\nthe absence of first-order magnetic phase transition. In-\nterestingly, a bifurcation between ZFC and FCC/FCW\nis noticed. A dimensionless quantity δM= (MFCW−\nMZFC)/MZFC, quantifying the bifurcation is found to de-\ncrease with H and becomes zero at 50 kOe. Such bifurca-\ntion may originate from ferrimagnetic and/or spin glass\nbehavior. Our theoretical simulations confirm the pres-\nence of ferrimagnetism and strong magneto-crystalline\nanisotropy (MCA) in the system. The bifurcation of\nZFC and FCW can be explained with the help of MCA.\nTheoretically, <100>crystal direction is found to be\nthe easy axis while <110>and<111>are intermedi-\nate and hard axis respectively. In the polycrystalline\nsample, <100>axes directions are oriented randomly.\nHence, when the sample is cooled in the ZFC mode,\nthe moments freeze randomly along <100>directions.\nThis phenomenon resembles a spin glass nature. The\nfield applied for measuring the magnetization is not suf-3\n8\n6\n4\n2\n02468\nH (10 kOe)1.5\n1.0\n0.5\n0.00.51.01.5M (B/f.u.)\n(a)2K\n150K\n250K\n0 100 200 300\nT (K)185190195200205210 (cm)\n(b)0 kOe\n20 kOe\n80 kOe\n0 100 200 300\nT (K)0.3\n0.1\n0.10.30.50.7/0 (%)\n(c)20 kOe\n60 kOe\n80 kOe\n8\n6\n4\n2\n02468\nH (10 kOe)0.5\n0.00.51.0MR (%)\n(d)5K\n150K\n250K\n8\n6\n4\n2\n02468\nH (10 kOe)0.00.20.40.6MRSym (%)\n(e)5K\n150K\n250K\n8\n6\n4\n2\n02468\nH (10 kOe)0.5\n0.00.5MRAsym (%)\n(f)5K\n150K\n250K\nFIG. 3. For Mn 2IrGa, (a) Mvs.Hat 2, 150 and 250 K (b)\nT-dependence of ρ, (c)T-dependence of magnetoresistance at\ndifferent fields ( H) (d) Isothermal magnetoresistance (MR)\nvs. field at 5, 150 and 250 K. (e,f) field dependance of sym-\nmetric and asymmetric parts of MR.\nficient to reorient the moments along the field direction\ndue to the anisotropy and hence leads to low magneti-\nzation compared to that in FCC mode. As the sam-\nple is cooled in the FCC mode, the moments freeze\nonly along one of the preferred <100>directions which\nleads to higher FCC magnetization. Figure 2(f) shows\nthe Curie-Weiss fit for susceptibility using the expression\nχ−1(T) = (T−θCW)/(χ0(T−θCW)+C). Here χ0,Cand\nθCWare the T-independent part of χ, Curie constant and\nCurie-Weiss temperature respectively. The effective mag-\nnetic moment can be estimated using µeff=p\n3CkB/NA\nwhere NAandkBare the Avogadro’s number and Boltz-\nmann constants respectively. Curie-Weiss fitting yields\nθCW∼260 K and µeff∼1.97µB/f.u., which are in good\nagreement with other reports.[21] The real and imagi-\nnary part of susceptibility ( χ) indicating the magnetic\nand spin-glass transition is presented in Sec. D of SM.[27]\nFigure 3(a) shows the M-H curves measured at few\nrepresentative T. A finite hysteresis along with a non-\nsaturating behavior of Mindicate the ferrimagnetic na-\nture of the alloy. It shows a large coercive field (8.5 kOe)\nat 2 K which decreases with T. At 300 K, Meases lin-\nearly with H(not shown here) confirming the onset of\nparamagnetic behavior. Figure 3(b) shows the electrical\nresistivity ( ρ) vs. Tmeasured at few applied fields. It\nshows the negative temperature coefficient of resistivity,\nindicating the possibility of semimetal behavior. ρ(T) un-\ndergoes slope change at two different temperatures ∼100\nand∼240 K. This categorises three different tempera-\nture regions. The crossover temperature ( ∼240 K) is\nin accordance with the inferences made from suscepti-\nbility and specific heat (C( T)) data (see Sec. D and G\nof SM[27] and footnote[38]). The H-dependence of re-\nsistivity at different Tis shown in Fig. S4 of SM[27].\nρ(H) shows quite contrasting behavior in different H-\nrange with varying T-values (see footnote[39]).\nNext, we estimated the T-dependance of magnetoresis-tance using MR(T)= [ ρ(T, H)−ρ(T, H = 0)] /ρ(T, H =\n0), as shown in Fig. 3(c). MR( T) clearly shows two turn-\ning points in accordance with the two slope changes in\nρ(T). It has a positive magnitude below TCwhile changes\nsign above TC. In region I ( <100 K), MR(T) increases\nwith increasing Tand take a downhill in region II (100 <\nT <240 K). The field ( H) dependance of MR, as shown\nin Fig. 3(d) at different T, shows giant hysteresis simi-\nlar to M-Hcurves, with a clear slope change at higher\nH. The symmetric and asymmetric parts of MR can\nbe estimated as, MRtype(H) = [MR( H)±MR(−H)]/2,\nwhere type=Sym (Asym) takes positive (negative) sign\non the right hand side. Note that due to the hysteri-\nsis, MR acquires symmetric and asymmetric components\nfor both raising and lowering fields. Figures 3(e) and\n3(f) display the symmetric and asymmetric component\nof MR respectively. A more detailed MR data and its\nsymmetric and asymmetric components at numerous T\nare shown in Fig. S4 of SM[27]. The symmetric part\nof MR is negative for T ≥250 K. It is governed by the\ns-d scattering interaction with an almost linear variation\nwith H. Below 250 K, although MRSymremains +ve,\nit exhibits a crossover behavior with a sudden change in\nslope at 100 K, reflecting a similar change in ρvs. T\nbehavior. Such variation of MR due to Lorentz contribu-\ntion could arise if the product of cyclotron frequency and\nrelaxation time is large. The MRAsymhas a clear hys-\nteresis in regions I and II (i.e. 0-100 K and 100-240 K),\nwhose magnitude decreases with increasing T, similar to\nthat of M−Hcurves (Fig. 3(a)) while its hysteresis goes\naway for T >240. In fact, the MRAsymcurves resemble\nvery much with that of M−Hdata indicating the elec-\ntronic states that are responsible for magnetization, and\ncontributing to its electrical transport.\nImportantly, an asymmetric behavior of ρon positive\nand negative H-axis is clearly evident at all tempera-\ntures below 300 K (see Fig. S4 of SM). For all T < T C,\nMn2IrGa shows a two step asymmetric MR with hys-\nteresis similar to Mvs.Hcurves. This type of behav-\nior generally arises in thin film heterostructures, where\ntwo ferromagnetic layers are separated by a nonmagnetic\nconducting layer [40], giving rise to the well-known spin-\nvalve characteristics. Similar type of asymmetry is also\nreported in bulk Mn 2NiGa alloy.[31] Such behavior could\nbe due to the minor anti-site disorder between a Ga and\nMn atoms, which can be responsible for the formation\nof ferromagnetic (FM) nanoclusters (with parallel Mn\nspins) in a matrix of Mn 2IrGa bulk lattice having an-\ntiparallel Mn spin moments. The direction of Mn mo-\nments in the soft FM cluster reverses its direction with\nthe application of field. This causes a rotation or tilt in\nthe antiparallel Mn moments at the cluster-lattice inter-\nface, resulting in the observed asymmetry in MR.[31]\nAb-initio total energy calculations are performed for\ntwo different crystal configurations I, II (see Methods sec-\ntion) considering different magnetic arrangements (ferro-\n, antiferro- and ferri-magnetic). In case of Config. I, a\nunique ferrimagnetic (FiM) ordering got stabilized while4\nTABLE I. For Mn 2IrGa, theoretically optimized lattice pa-\nrameters ( aeq), total and atom projected moments and\nthe relative energies (∆ E) of different magnetic ordering\n(ferrimagnetic(FiM), ferromagnetic(FM) and antiferromag-\nnetic(AFM)) in the two structural configurations (I and II).\nConfig. 1 with FiM ordering is set as the reference energy.\nConfig. aeq(˚A)Moment ( µB)\n∆E(meV /atom)\n4b 4c 4d Total\nMn1 Mn2 Ir\nI(FiM) 5.96 3.00 -1.41 0.22 2.00 0.0\nIr Mn1 Mn2\nII(FM) 6.12 0.69 3.30 3.30 7.74 170.7\nII (AFM) 6.09 0.00 -3.08 3.08 0.00 186.3\nfor Config. II, two different magnetic ordering (FM and\nAFM) were realized with energy difference of 16 meV.\nFiM is the lowest energy magnetic ordering with the net\nmoment of 2 µB/f.u. Table I shows the optimized lattice\nparameters, total and atom projected moments and the\nrelative energies for different magnetic states in the two\nconfigurations. The theoretically optimized lattice pa-\nrameter for FiM is 5 .97˚Awhich matches fairly well with\nthe experimental value. The local moments at the octa-\nhedral Mn (Mn1), tetrahedral Mn (Mn2) and Ir are 3.00,\n-1.41 and 0.22 µBrespectively for the FiM state. Figure\n4 (lower panel) shows the spin polarized band structure\nand density of states for the lowest energy FiM state (the\nsame for AFM and FM ordering is shown in Sec. H of\nSM[27]). The spin down band shows a narrow band gap\nwhile the spin up band has an indirect overlap between\nvalence and conduction bands. For spin up channel, the\nvalence band involves three hole mediated metallic states\nwhile the conduction band contain two flat bands. The\nflat bands generally gives rise to hole and electron pock-\nets near EF. With increase in temperature, these pockets\nhelp to create/hold more charge carriers thereby domi-\nnating phonon contribution to electrical resistivity. Even\nthough the valence band contains hole mediated metal-\nlic states, its contribution is restricted by several factors\nsuch as (i) high effective mass, (ii) trapping or scatter-\ning due to disorder in the lattice etc. The three bands\ncrossing the EFforming the hole pockets around the Γ\npoint are labelled as ‘Band 1’, ‘Band 2’ and ‘Band 3’\nwhose Fermi surfaces are show in Fig. 4 (top). ‘Band 4’\ngives rise to the electron pocket arising out of one of the\nconduction band crossing the EF, whose Fermi surface is\nalso shown in Fig. 4.\nWe further performed the calculations including the\neffect of spin orbit coupling with different magnetiza-\ntion vector orientations <100>,<010>,<001>,<110>,\n<101>,<011>and<111>.<100>is found to be the\neasy axis, while <110>and<111>are the intermedi-\nate and hard axes respectively. For cubic symmetry, the\nWLΓXU,KΓW\nk  (Å−1) →2\n1\n012E−EF (eV) →\nWLΓXU,KΓW\nk  (Å−1) →2\n1\n012\n864202468\nDOS (states/(eVf.u.))2\n1\n012Mn2IrGa\naeq\nBand 1\n Band 2\n Band 3\n Band 4\nBands 1−4FIG. 4. (Bottom) Spin resolved band structure and density\nof states of Mn 2IrGa in its lowest energy Config. I (with FiM\nordering). (Top) Fermi surfaces due to spin up bands #1, 2,\n3 and 4.\nmagnetic anisotropic energy can be expressed as[41]\nEMAE(θ, ϕ) =k0+k1(α2β2+β2γ2+γ2α2) +k2α2β2γ2\nwhere α,βandγare the direction cosines of magneti-\nzation axis with respect to the crystallographic axes. k1\nandk2are estimated using the eqns, k1= 4(E<110>−\nE<100>),k2= 9(E<111>+3E<100>−4E<110>) with the\nvalues 1 .99×105J/m3, 5.20×105J/m3respectively.\nUsing these values of k1andk2,EMAE is simulated as a\nfunction of θandϕ, as shown in Sec. H of SM.[27] This\nclearly confirms the anisotropic nature of the intrinsic\nmagneto-crystalline energy. Such large anisotropy can be\nresponsible for narrow domain walls, which in turn causes\n0.6\n0.4\n0.2\n0.00.20.40.6E−EF [eV]\n12 3 4(a)\n1.0\n 0.8\n 0.6\n 0.4\n 0.2\n 0.0 0.2 0.4 0.6 0.8 1.0sz\nKΓ W LΓ XU10000\n5000\n0−Ωz(k)  [bohr2]\n(b)(c)−Ωz(b1,b2,0)\nΓΓL\nXLL\nΓΓL-105-104-103-102-101 0 101 102 103\n(d)−Ωz(b1,b2,0.5)\nLLX\nLX X\nLLX-102-101 0 101 102\nBand 1\n Band 2\n Band 3\n Band 4\nBands 1−4\nFIG. 5. For Mn 2IrGa, (a) wannierized band structure includ-\ning SOC, with z−component of spin moment shown by the\ncolor profile. 1, 2, 3 and 4 indicate the band # crossing EF\nwhose Fermi surfaces are shown at the bottom. (b) Berry\ncurvature along the high symmetry k−paths. (c,d) 2D pro-\njected Berry curvature contours and Fermi lines (shown by\nblack lines) on b3= 0 and b3= 0.5(2π\na) planes.5\nhysteresis in the MR and magnetization isotherms.\nNext, we simulated the band structure including SOC\nconsidering the easy axis ( <100>). Figure 5(a) shows\nthe Wannierized orbital projected band structure along\nwith the z−component of spin moment ( sz). Spin down\ncharacter is clearly visible along Γ to Xwith relatively\nreduced band gap, while mixed spin character for bands\n1 and 3 appear along several directions. Only band 2\nretains its pure spin up character maintaining its Fermi\nsurface unchanged (see ‘Band 2’ Fermi surface in bottom\nfigure). A slight pinched Fermi surface is noticed for\nbands 1, 3 and 4 due to the SOC mediated band splitting.\nFigure 5(b) shows the intrinsic Berry curvature along the\nhigh symmetry k-paths, with strong spikes at Γ-point\nwhere the band crossing near EFoccur. Figure 5(c,d)\nshow the 2D projected Berry curvature contours on the\nb3= 0 and b3=π/aeqplanes. The black solid lines\nhighlight the Fermi lines. Along with the large negative\nvalue, −Ωzalso has large positive values around the Γ\npoint (encircled like a toroid in Fig. S6 of SM[27]). As\nsuch, the Berry flux at/around Γ resembles the magnetic\nflux in a bar magnet. The calculated intrinsic anomalous\nHall conductivity is found to be ∼ −180 S/cm.\nSummary : We report Mn 2IrGa to be a potential\ncandidate for the recently discovered ferrimagnetic spinsemimetal (SSM). In SSM, one spin channel shows\nsemimetallic behavior while the other has a narrow band\ngap. Apart from spin semimetallic feature, Mn 2IrGa also\nshows giant coercivity and spin-valve like characteristics.\nIt crystallizes in an inverse Heusler structure with fer-\nrimagnetic (FiM) ordering and a net magnetization of\n2µB/f.u., with no signature of martensitic transition.\nNegative temperature coefficient of resistivity with a very\nweak variation with temperature indicate the semimetal-\nlic behavior of the alloy. Asymmetric nature of magneto-\nresistance with hysteresis and a change in sign across\nthe transition temperature indicate its potential to be\nused in magnetic switching applications. Ab-initio cal-\nculations confirm the SSM behavior with a unique FiM\nordering. <100>is simulated to be the easy mag-\nnetic axis with considerable anisotropy energy, possibly\ncausing narrow domain walls responsible for appreciable\nhysteresis in magnetization and MR. Our simulation con-\nfirms a large intrinsic Berry curvature mediating a rea-\nsonably high anomalous Hall conductivity ( ∼-180 S/cm).\nAcknowledgments : SSS acknowledges SERB for\nthe financial support through Core Research Grant\n(CRG/2022/0007993). AA acknowledges DST-SERB\n(Grant No. CRG/2019/002050) for funding to support\nthis research\n[1] T. Graf, C. Felser, and S. S. Parkin, Progress in Solid\nState Chemistry 39, 1 (2011).\n[2] S. Ouardi, G. H. Fecher, C. Felser, and J. K¨ ubler, Phys.\nRev. Lett. 110, 100401 (2013).\n[3] D. Rani, Enamullah, K. G. Suresh, A. K. Yadav, S. N.\nJha, D. Bhattacharyya, M. R. Varma, and A. Alam,\nPhys. Rev. B 96, 184404 (2017).\n[4] Y. Venkateswara, D. Rani, K. Suresh, and A. 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At\n50 K, it increases linearly up to 11.5 kOe beyond which it\nbecomes H-independent. However, for temperatures 100,\n150, 200 and 250 K, ρincreases rapidly (almost linear) up\nto a certain field and then increases again (but with shal-\nlow positive slope) up to 80 kOe (see Sec. F of SM[27]).\nAt 300 K (in the paramagnetic regime), a linear increase\nofρwith His attributed to the Lorentz force. ().\n[40] B. Park, J. Wunderlich, and X. Marti et al., Nat. Mater\n10, 347 (2011).\n[41] B. D. Cullity and C. D. Graham, Introduction to mag-\nnetic materials (John Wiley & Sons, 2011).\n[42] R. Y. Umetsu, X. Xu, W. Ito, and R. Kainuma, Metals\n7(2017), 10.3390/met7100414.\n[43] C. Kittel, Introduction to Solid State Physics , seventh ed.\n(Wiley India Pvt. Limited, 2007)."    },    {        "title": "1601.05681v1.Spin_pumping_in_strongly_coupled_magnon_photon_systems.pdf",        "content": "Spin pumping in strongly coupled magnon-photon systems\nH. Maier-Flaig1;2, M. Harder3, R. Gross1;2;4, H. Huebl1;2;4, S. T. B. Goennenwein1;2;4\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2 and\n4Nanosystems Initiative Munich, Schellingstra\u0019e 4, D-80799 M unchen, Germany\n(Dated: October 17, 2018)\nWe experimentally investigate magnon-polaritons, arising in ferrimagnetic resonance experiments\nin a microwave cavity with a tuneable quality factor. To his end, we simultaneously measure the\nelectrically detected spin pumping signal and microwave re\rection (the ferrimagnetic resonance sig-\nnal) of a yttrium iron garnet (YIG) / platinum (Pt) bilayer in the microwave cavity. The coupling\nstrength of the fundamental magnetic resonance mode and the cavity is determined from the mi-\ncrowave re\rection data. All features of the magnetic resonance spectra predicted by \frst principle\ncalculations and an input-output formalism agree with our experimental observations. By changing\nthe decay rate of the cavity at constant magnon-photon coupling rate, we experimentally tune in\nand out of the strong coupling regime and successfully model the corresponding change of the spin\npumping signal. Furthermore, we observe the coupling and spin pumping of several spin wave modes\nand provide a quantitative analysis of their coupling rates to the cavity.\nI. INTRODUCTION\nMotivated by the vision of hybrid quantum information\nsystems combining the fast manipulation rates of super-\nconducting qubits and the long coherence times of spin\nensembles, strong spin-photon coupling is a major goal\nof quantum information memory applications. Coher-\nent information exchange between microwave cavity pho-\ntons and a spin ensemble was initially demonstrated for\nparamagnetic systems1{3, but only recently has this con-\ncept been transferred to magnetically ordered systems,\nwhere coupling rates of hundreds of megahertz can be\nachieved.4{7Utilizing the \rexibility of exchange coupled\nmagnetically ordered systems, more complex architec-\ntures involving multiple magnetic elements have already\nbeen developed8,9. Additionally, magnetically ordered\nsystems allow to study classical strong coupling physics\neven at room temperatures.5{10\nMoreover, a key advantage of magnetically ordered sys-\ntems over their paramagnetic counterparts { which has\nyet to be fully explored { is the ability to probe mag-\nnetic excitations electrically through spin pumping and\nthe inverse spin Hall e\u000bect. Spin pumping, in general, re-\nlies on ferromagnet-normal metal (FM/NM) heterostruc-\ntures and has been demonstrated for a wide variety of ma-\nterial combinations11. Under resonant absorption of mi-\ncrowaves, the precessing magnetisation in the ferromag-\nnet sources a spin current into the normal metal, where it\nis converted into a charge current via the inverse spin Hall\ne\u000bect. This spin Hall charge current is then detected.\nIn ferromagnetic insulator (FMI)-based FMI/NM het-\nerostructures, charge current signals from the recti\fca-\ntion of the microwave electric \feld are very small12, lead-\ning to a dominant spin pumping/spin Hall signal. This\nhas led to much research on FMI/NM heterostructures, of\nwhich the Yttrium Iron Garnet (YIG)/Platinum(Pt) bi-\nlayers we use are a prime example. Spin pumping is a well\nunderstood e\u000bect for weak photon-magnon coupling11,13,i.e. for situations where the decay rates of the cavity\nand the magnetic system are larger than the photon-\nmagnon coupling strength. However, the large spin den-\nsity of YIG and the resulting large e\u000bective coupling\nstrength allows one to reach the strong coupling regime\nalso in typical spin pumping experiments. The exper-\nimental observation14and theoretical treatment15,16of\nspin pumping in a strongly coupled magnon-photon sys-\ntem has only recently been performed. These results sug-\ngest that combining spin pumping and strong magnon-\nphoton coupling may enable the transmission and elec-\ntrical read out of quantum states in ferromagnets using\na hybrid architecture. Experiments directly linking spin\npumping in the weak and strong coupling regime are,\nhowever, still missing. Such experiments are one impor-\ntant step towards understanding the functional principle\nand key requirements for such a hybrid architecture.\nIn this paper, we present a systematic study of the\nmagnon-photon coupling in magnetic resonance exper-\niments in a YIG/Pt bilayer mounted in a commer-\ncially available EPR cavity. We measure both the mi-\ncrowave re\rection spectra and the electrically detected\nspin pumping signal in the system. The tuneable cavity\nquality allows us to systematically move in and out of the\nstrong coupling regime. Measurements with high mag-\nnetic \feld and frequency resolution allow us to clearly\nobserve the coupling of spin wave modes with the hy-\nbridized cavity{fundamental FMR mode. We explore a\ndi\u000berent approach as recently used by Zhang et al.7: In\nour setup, instead of tuning the cavity frequency we tune\nits decay rate while the e\u000bective magnon-photon coupling\nrate and the magnon decay rate stay constant. We thus\nachieve a transition from the strongly coupled regime\nwhere the decay rates of spin and cavity system are both\nconsiderably smaller than the e\u000bective coupling rate, to\nthe weakly coupled regime where the cavity decay rate\nis much higher than the magnon-photon coupling rate.\nThis regime is also called the regime of magnetically in-arXiv:1601.05681v1  [cond-mat.mtrl-sci]  21 Jan 20162\nduced transparency (MIT)7.\nThis paper is organized as follows: In Sec. II we re-\nview the general theory of the coupled magnon-photon\nsystem and the main features of spin pumping in the case\nof strong coupling. In Sec. III we describe the experi-\nmental details of recording the microwave re\rection of\nthe system as a function of frequency and applied mag-\nnetic \feld while simultaneously recording the DC spin\npumping voltage across the Pt. Finally in Sec. IV we\npresent our observation of strong coupling between the\ncavity mode and both the fundamental magnetic reso-\nnance and standing spin wave modes. We also demon-\nstrate the transition from strong to weak coupling by\ntuning the cavity line width and discuss the di\u000berence\nin the experimental spin pumping signature in both the\nstrong and weak regimes.\nII. THEORY\nA. Photon-Magnon Dispersion\nConventionally, ferromagnetic resonance (FMR) is\nmodeled in terms of the Landau-Lifshitz-Gilbert (LLG)\nequation which describes the dynamics of a magnetic mo-\nment in the presence of a magnetic \feld. In a static\nmagnetic \feld H0, the magnetic moment will precess\nwith the Larmor frequency !s. In detail, !sdepends\non the static \feld strength and on its orientation due to\nanisotropy17. This precessional motion can be resonantly\nexcited by a time varying microwave magnetic \feld H1\nwith a frequency close to !s. To observe spin pumping\nin FM/NM heterostuctures, the \feld H0should be ap-\nplied perpendicular to the surface normal (i.e. in the\ninterface plane)11,13,18,19. In this case, the FMR disper-\nsion (in the absence of crystalline magnetic anisotropy)\nis!s=\r\u00160p\nH0(H0+Ms)20. Here,Msis the mate-\nrial speci\fc saturation magnetization, \ris the material\nspeci\fc gyromagnetic ratio and \u00160is the vacuum perme-\nability. In the limit H0\u001dMsthe resonance frequency\nis thus linear in magnetic \feld. Contrary to the spin\nresonance frequency !s, the resonance frequency !cof a\nmacroscopic cavity is determined by geometrical and di-\nelectric parameters only and therefore does not depend\non the magnetic \feld. However, since the magnonic and\nthe photonic mode interact in resonance, we expect mod-\ni\fcations to the pure FMR and pure cavity dispersions.\nTo be speci\fc, we will observe an anticrossing of the\nFMR and the cavity dispersion for a su\u000eciently strong\nmagnon-photon coupling.\nTo describe the coupling between the cavity mode\nand the spin excitation the quantum mechanical Tavis-\nCummings model21,22and classical \frst principles15ap-\nproaches using the input-output formalism5have success-\nfully been used. For the dipolar interaction assumed\nin the models, the single spin-single photon coupling\nstrengthg0is proportional to the vacuum microwave\nmagnetic \feld H0\n1and the dipole moment mof the spin.In the scope of the Tavis-Cummings model, it has been\nshown that the collective coupling strength ge\u000bto an en-\nsemble of spins is proportional to the square root of the\nnumber of polarized spins for the coupling to the vac-\ncum microwave magnetic \feld. In a classical theory, Cao\net al.15derived that thisp\nNbehaviour prevails also for\nthe magnon-photon coupling in magnetically ordered sys-\ntems. Here, the total magnetization and thus the \flling\nfactor of the ferromagnetic material in the cavity, can be\nused as a measure for the total number of spins.\nThe characteristic \fngerprint of strong coupling is the\nformation of an observable anti-crossing of the cavity and\nthe spin dispersion relation close to resonance. Note,\nthat the presence of stong coupling and the accompanied\nvisible anti-crossing of the dispersion relations requires\nthat the e\u000bective coupling ge\u000bexceeds the loss rates of\nthe spins (\rs) and the cavity ( \u0014i+\u0014e). Experimentally,\nwe tune the spin resonance frequency !sacross the cavity\nresonance frequency !cvia an externally applied static\nmagnetic \feld. The coupled system can most simply be\nmodelled in the vicinity of the resonance frequency using\ntwo coupled harmonic oscillators, where the resonance\nfrequency is5:\n!\u0006=!c+\u0001\n2\u00061\n2q\n\u00012+ 4g2\ne\u000b(1)\nHere, \u0001 =\r(\u00160H0\u0000\u00160Hres) is the spin-cavity detuning\nwithHresstatisfying the spin resonance condition for a\ngiven cavtiy frequency !c.\nIn ferromagnetic \flms, additional magnetic modes, so-\ncalled perpendicular standing spin waves modes, appear\ndue to magnetic boundary conditions. For the condition\nwhere the magnetization is pinned at least at one sur-\nface of the \flm (and in the absence of any anisotropies\nor magnetic gradients) the magnon spectrum can easily\nbe calculated20. The di\u000berence of the resonance \feld of\nthenth-mode from the fundamental mode Hn\nres\u0000H1\nresis\nproportional to n2. Cao et al.15also calculated the ex-\npected coupling strength for di\u000berent modes and found\nthat the coupling decreases with increasing mode number\nasge\u000b/1=n. This can be understood when considering\nthe microwave mode pro\fles and the fact that the spa-\ntial mode pro\fle of the microwave \feld H0\n1in a cavity\nis typically homogeneous and in phase throughout the\nthickness of the (thin \flm) sample. Therefore only every\nsecond mode can be excited and the e\u000bective magnetiza-\ntion to which the microwave can couple to is reduced to\nM\nn.\nB. Spin pumping and strong coupling\nSpin pumping in ferromagnet/normal metal bilayers\nin the weak coupling regime is well understood11,13,19:\nAn additional mechanism which damps the magnetiza-\ntion precession becomes available by spin pumping, as\nthe precessing magnetisation is driving a spin current\ninto the adjacent normal metal.13In electrically detected3\nVNA 9-10 GHZ\nLNA\n1 ADC\nA B\nelectro magnetfeed\nlineDC lines\nYIG sample\nsample holderDC meas. lines\ncavity\ngeffκi κe γsp\nγiH0\ncryostatA-B\nFIG. 1. Block diagram of the experimental setup and sample\nmounting. (Inset) Schematic of the coupling scheme illus-\ntrating cavity decay due to intrinsic losses viz. losses to the\nfeedline\u0014c=\u0014i+\u0014e, spin system decay consisting of intrinsic\ndamping and spin pumping damping \ri=\rs+\rspas well the\ncollective coupling rate ge\u000b\nspin pumping, this spin current is then converted into a\ncharge current via the inverse spin Hall e\u000bect (ISHE). For\nelectrical open circuit conditions, one thus obtains a volt-\nage which scales as11,19VSP/g\"#\u0015SDtanhtN\n2\u0015SDsin2\u0012.\nIt, thus, contains information on the spin mixing con-\nductanceg\"#, the spin di\u000busion length \u0015SD, the mag-\nnetization precession cone angle \u0012and depends on the\nthickness of the normal metal layer tN. The maximal\nprecession cone angle \u0012and thus the maximal expected\nspin pumping voltage depends on the microwave power\nbut also on the coupling strength between cavity and spin\nsystem. For strong coupling, the cone angle is expected\nto be reduced as compared to the weak coupling case due\nto the hybridized nature of the excitation at its maximal\nintensity.\nThe other contributions in the equation for VSPare\nmaterial constants: The spin mixing conductance g\"#de-\nscribes the the transparency of the ferromagnet/normal\nmetal interface und limits the spin pumping e\u000eciency\ngenerally; the spin di\u000busion length \u0015SDin conjunction\nwith the normal metal thickness tNaccounts for a spin\naccumulation in the normal metal and reduces the spin\npumping e\u000eciency if tN/\u0015SD.\nIII. EXPERIMENTAL DETAILS\nA. Sample preparation\nIn our experiments we used YIG/Pt heterostruc-\ntures grown by liquid phase epitaxy on (111)-oriented\nGadolinum Gallium Garnet (GGG) substrates. The YIG\flm thickness was 2 :8µm. In order to produce a high\nquality interface between YIG and Pt, and thus a large\nspin mixing conductance g\"#, we followed the work of\nJung\reisch et al.23and \frst treated the YIG surface\nby piranha etching for 5 minutes in ambient conditions.\nThereafter, the sample was annealed at 500\u000eC for 40 min-\nutes in an oxygen atmosphere of 25 µbar. Under high\nvacuum, it was then transferred into an Electron Beam\nEvaporation (EVAP) chamber where 5 nm Pt was de-\nposited. The exact Pt thickness was determined using\nX-Ray re\rectometry. However, we note that for our anal-\nysis the Pt layer thickness is of minor importance as it\nwas consistently larger than the spin di\u000busion length \u0015SD\nof Pt such that the Pt layer simply serves as a perfect spin\nsink.\nIn order to achieve collective strong coupling between\nmagnons and cavity photons, the number of magnetic\nmoments must be su\u000eciently high. Therefore, we diced\nthe sample into several pieces of di\u000berent lateral dimen-\nsions. Magnetic resonance experiments in the strong cou-\npling regime showed that thep\nNscaling of the coupling\nstrength discussed in Section II is indeed obeyed upon\ncomparing samples with di\u000berent volume and thus dif-\nferent total magnetic moment. In the following, we will\nfocus on a sample with lateral dimensions 2 mm \u00023 mm\nwhich, with the e\u000bective spin density \u001aS= 2:1\u00021022\u0016B\ncm3\nof iron atoms in YIG24, contains on the order of 4 \u00021017\nspins. Finally, the sample was mounted on a PCB sample\ncarrier and wire bonded as depicted in the inset of Fig. 1.\nThe carrier itself was mounted on a sample rod which al-\nlowed the sample to be accurately positioned in the elec-\ntrical \feld node of a Bruker Flexline MD5 dielectric ring\ncavity in an Oxford Instruments CF935 gas \row cryo-\nstat. Shielded DC cabling allowed for the measurement\nof the ISHE voltage. The detailed design blueprints of\nthe sample rod and chip carrier can be retrieved online25.\nB. Experimental setup\nThe Bruker cavity exhibits a TE 011mode with an elec-\ntric \feld node at the sample position. Its quality factor\nQ=!=\u0001!FWHM\nc (\u0001!FWHM\nc being the full width half\nmaximum line width of the cavity) is dominated by the\ndissipative losses in the dielectric and its \fnite electrical\nresistance ( \u0014i) as well as radiation back into the cavity\nfeed line (\u0014c). By changing the cavity's coupling ratio to\nthe feed line, unloaded coupled quality factors Qcfrom 0\nto 8000 can be achieved. This allows tuning in and out\nof the strong coupling regime easily. Using the gas \row\ncryostat, di\u000berent temperatures can be stabilized. All the\nfollowing experiments have, however, been performed at\nroom temperature.\nTo measure ferromagnetic resonance (FMR) the cavity\nwas connected to the port of an Agilent N5242A vector\nnetwork analyzer (VNA). The driving power of 15 dBm\nexcites at maximum on the order of NPh= 1:3\u00021014pho-\ntons in the cavity which is considerably smaller than the4\nnumber of spins in the sample (4 \u00021017). In this case,\nthe theory presented in Sec. II is well justi\fed26. The\nfrequency dependent cavity re\rection S11was measured\nwhile sweeping the external \feld \u00160Hthat is created by\na water cooled electromagnet. The IF bandwidth was\nchosen to be 100 Hz which leads to a frequency sweep\ntime of approximately 2 s for each magnetic \feld step. A\ncalibration of the microwave leads up to the resonator's\nSMA connector was performed. The calibration did not\ninclude the feed line inside the resonator mount, which\ngave rise to a background signal in the re\rection param-\neter. However, by utilizing the full complex S-parameter\nfor the background subtraction with the inverse map-\nping technique outlined by Petersan and Anlage27and\na subsequent Lorentzian \ft to the magnitude, a reliable\nmeasurement of Qis still possible, even for a completely\nuncalibrated setup. We note that even though standing\nwaves in the mirowave feed line will not appear in the\ncalibrated re\rection measurement they will still change\nthe total power in the cavity and therefore may compli-\ncate the electrically detected DC spin pumping signal.\nUncalibrated measurements did not show sharp feed line\nresonances in the frequency range studied here but only\nsmooth oscillations with an amplitude of less than 1 dB\nand there was no correlation in the DC signal resolved.\nIn order to \ft the data and as it improves clarity, we only\ndiscuss calibrated measurements in the following.\nThe DC voltage from the sample was measured along\nthe cavity axis and thus perpendicular to the external\nmagnetic \feld and the sample normal. It was ampli-\n\fed with a di\u000berential voltage ampli\fer model 560 from\nStanford Research Systems. The ampli\fer was operated\nin its low noise (4 nV/p\nHz) mode and set to a gain of\n2\u0002104. The analog high-pass \flter of the ampli\fer was\ndisabled, however, a low-pass \flter with a 6dB roll-o\u000b at\n1 kHz was employed. Limiting the bandwidth of the am-\npli\fer by \fltering is required in order to achieve a good\nsignal-to-noise ratio. Care has, however, to be taken as\nthe lineshape may be quickly distorted by inappropriate\nsettings and thus the signature of spin pumping might be\nmasked. High-pass \fltering can easily lead to a dispersive\nlike contribution to the signal, whereas low-pass \fltering\nwill give rise to asymmetric line shapes depending on the\nratio of IF bandwidth and low-pass frequency. We made\nsure that no such distortions contribute to the presented\nmeasurements. The ampli\fed voltage signal was \fnally\nrecorded using the auxiliary input of the VNA simulta-\nneously with the cavity re\rection S11.\nIV. RESULTS AND DISCUSSION\nWe \frst focus on the case of the so-called critical cou-\npling of the feed line to the cavity in which most FMR\nexperiments are conducted. In this case, the internal loss\nrate of the cavity equals the loss rate to the feed line and\nthe quality factor is Qc=Qinternal=2. Note that inserting\na sample and holder into the cavity will reduce the cavity\n275 267 259\nStatic magnetic0H(mT)\n259 267 275\n 180 0180\nV9.509.559.609.659.709.759.80\n0.20.40.6 240 0 240\n9.559.609.659.709.759.80|S   |11 VDC(µV)\n57911=nFrequency (GHz)(a)\n(b) 2geff/2π = 64 MHzFIG. 2. (a)Re\rection parameter S11recorded while sweeping\nthe magnetic \feld. Strong coupling of the collective spin ex-\ncitations is indicated by a clear anticrossing, spin wave modes\nto the low \feld side of the main resonance are visible. Black\nnumbers indicated the spin wave mode number. (b)Simul-\ntaneously recorded DC voltage. Fundamental and spin wave\nmodes are visible where the latter couple less strongly and\nthus pump spin current more e\u000eciently. Insets: Detail of\nn=5 spin wave mode including the dispersion relation of the\nstrong coupling between the fundamental FMR mode and the\ncavity as solid red line and the anti-crossing of this hybrid and\nthe spin wave mode (#5) as white lines.\nQby an amount which depends on the sample and holder\ndetails such as conductivity and dielectric losses. Based\non our measured loaded Qc= 706, the cavity decay rate\nis calculated to be \u0014c=2\u0019=!r\n2\u0019=2Qc= 6:8 MHz.\nStrong coupling of the magnon and cavity system man-\nifests itself in a characteristic anti-crossing of the (mag-\nnetic \feld independent) cavity resonance frequency and\nthe magnon dispersion that is (approximately) linear in\nmagnetic \feld. This anti-crossing corresponding to two\ndistinct peaks in a line cut at the resonance \feld, are im-\nmediately visible in the re\rection spectrum in Fig. 2. The\nminimal splitting gives the collective coupling strength\nge\u000b=2\u0019= 31:8 MHz of the fundamental mode to the cav-\nity. Taking into account the number of spins in the\nsample, the single spin coupling rate is on the order of\ng0=2\u0019= 0:1 Hz which is in agreement with experiments\non paramagnetic systems28.\nIn our setup, even the coupling of higher order spin\nwave modes to the cavity can be resolved. We number\nthe spin waves as noted in Fig. 2 taking into account\nthat with an uniform driving \feld only odd modes can\nbe excited. Analysis of the resonance position of the spin\nwave modes reveals that Hn\nres\u0000H1\nresin our sample is\nproportional to nrather than n2. This indicates a non-\nsquare like potential well. Similarly, complicated mode\nsplittings have been reported in literature29. The low-\nest order spin wave mode that can be easily observed in\nour setup is shown in the inset of Fig. 2 (a) in detail.5\nIt exhibits the largest e\u000bective coupling (3 MHz) of all\nspin wave modes. The red and white lines in Fig. 2 (a)\ncorrespond to the harmonic-oscillator model (Eqn. 1) for\nthe fundamental mode and the lowest order spin wave\nmode, respectively. As the spin wave couples to an al-\nready hybridized sytem, we superimposed the dispersion\n!c=!r(B) of the hybridized system of fundamental\nmode and unperturbed cavity as the \"cavity\" mode in\nthe modelling of the spin wave mode couplings.\nIn order to quantify the coupling strength of the higher\norder modes which only interact weakly with the hy-\nbridized cavity{fundamental FMR mode, we follow the\napproach of Herskind et al.30. For each \feld, we \ft\na Lorentzian to the magnitude of the cavity absorp-\ntion. From this \ft we extract the resonance frequency\n!cand the half width half maximum of the absorption\n\u0001!which, in the weakly coupled spin waves reads as30\n\u0001!= \u0001!c+ge\u000b\rs=\u0000\n\r2\ns+ \u00012\u0001\n:\nThe coupling of the spin waves to the already hy-\nbridized cavity resonance decreases with the order of the\nmode. This can be understood by taking into account\nthat the e\u000bective magnetization to which the homoge-\nneous microwave \feld can couple decreases with increas-\ning mode number. The extracted values, gn=7;9;11;13=\n[3:65;2:49;1:64;1:16] MHz, match accurately with the ex-\npected1\nndependence of the coupling strength15.\nWe attribute the pronounced feature that is seen to\nthe right of the anti-crossing to an unidenti\fed spin\nwave mode. A similar feature was found in other\nexperiments14and has been interpreted in the same man-\nner. In our data, we can clearly distinguish between the\nfundamental mode and this additional mode { simply\nby remembering that the relative intensity and coupling\nstrength is expected to be higher for the fundamental\nmode. Possible origins for the additional mode are an in-\nhomogeneous sample or a gradient in the magnetic prop-\nerties across the \flm thickness31. This would be consis-\ntent with the unusual spin wave mode splitting. Lastly,\nwe note that the recorded signal in the re\rection parame-\nter is completely symmetric upon magnetic \feld reversal.\nThe simultaneously recorded DC voltage is shown in\nFig. 2 (b). Contrary to the re\rection parameter, the volt-\nage signal reverses sign on reversing \u00160H0. The lineshape\nthat we record for all modes is completely symmetric as\nfar as they can be clearly distinguished from each other.\nWe thus conclude that we observe a signal purely caused\nby spin pumping and not by any rectifying e\u000bect. In a\nFMI/NM bilayer ( \u001aYIG\u001510 G\n m)32recti\fcation can\nonly arize from a change of the spin Hall magnetoresis-\ntance (SMR) in the normal metal. According to model\ncalculations12this e\u000bect is negligible for the system we\ninvestigate because of the small magnitude of the SMR\ne\u000bect (<0:1%). This notion is further corroborated by\nthe fact that the change in lineshape expected for recti\f-\ncation type signals is not visible in our data. Apart from\nthe spin wave modes which are clearly resolved in the\nDC voltage signal, we can also clearly see the electricallydetected spin pumping voltage originating from the hy-\nbridized system of cavity and fundamental FMR mode\n(the main anti-crossing). The hybridized cavity eigen-\nmodes can, however, pump spin current into the normal\nmetal only very ine\u000eciently and thus the DC voltage we\nobserve is very low.\nThe upper panels of Fig. 3 show the change in cav-\nity re\rection as we gradually increase the coupling of the\ncavity to the feed line and thus increase the cavity decay\nrate. Starting from the critically coupled case (inter-\nnal cavity losses are equal to losses into the feedline) in\nthe left panel to a highly overcoupled cavity (losses into\nthe cavity feed line dominate the cavity's decay rate) in\nthe right panel, we clearly see an increase in the cavity\nlinewidth up to the point were the unperturbed cavity\nis no longer recognizable. Correspondingly, the cavity\ndecay rate increases from left to right and, in turn, the\nmicrowave magnetic \feld strength H1in the cavity de-\ncreases. For the already weakly coupled spin wave modes\nthe spin pumping voltage decreases with decreasing mi-\ncrowave magnetic \feld strength H1resp. available mi-\ncrowave power(indicated by the higher S11parameter)\nin the cavity. The DC spin pumping voltage amplitude\ncorresponding to the fundamental mode (lower panels of\nFig. 3) does, however, not decrease for lower Q-factors\nbut stays approximately constant. Considering that the\nabsorbed power of the cavity-spin system stays approxi-\nmately constant when changing the cavity decay rate as\ncan easily be seen in the line cuts in the upper panels of\nFig. 3 this behaviour can also be understood.\nThe best measure of the true magnon spectrum and\nline widths of the spin system can be extracted from\nthe highly overcoupled case (right panels of Fig. 3 and\nFig. 4). There, the magnon-photon coupling is negligi-\nble compared to the cavity loss rate and therefore, the\nmagnon-cavity mode hybridization does not distort the\nline shape. A mode that strongly couples with the cav-\nity, on the contrary, can vanish completely in the \fxed-\nfrequency spectrum. We \fnally note that we observe\nthe described anti-crossing due to the magnon-photon\ncoupling and thus the distortion of the lines in a \fxed-\nfrequency experiment (with the cavity tuned to high Q,\nas usually done in cavity-based FMR experiments) al-\nready for sample volumes as small as V= 2\u000210\u00003mm3\nin the case of YIG ( MS= 140 kA m\u00001). These sample\nvolumes are easily achieved for LPE grown samples and\nwe note that in most cavity FMR experiments33the ef-\nfects of the coupling need to be taken into account in\norder to yield accurate results especially when automatic\nfrequency control is employed.\nV. CONCLUSIONS\nIn summary, we presented systematic measurements\nof spin pumping in di\u000berent regimes of the magnon-\nphoton coupling strength. For the fundamental mode of\na YIG/Pt bilayer, strong coupling with an e\u000bective cou-6\n0.0 0.4 0.8\n260 265 270 275 280\n 260 265 270 275 280\nStatic magnetic field µ0H0  (mT)\n260 265 270 275 280\n −250 0250\nVDC (µV)99.559.609.659.709.759.Frequency (GHz)0.16 0.32 0.48 0.64|S11|\n−240 −800 80 240\n9.559.609.659.709.759.80\nVDC (µV)\n|S11|\nFIG. 3. Increasing the coupling of the cavity to the feed line (from left to right) increases the cavity loss rate \u0014cand thus line\nwidth \u0001!=2\u0019. This enables experimental control of the transition between strong and weak coupling. The line cuts at positive\n\feld (intense colors) and negative \feld (pale colors) again con\frm the symmetry, V(\u0000H0) =\u0000V(H0) andS11(\u0000H0) =S11(H0)\nand also show the merging of the two dispersion curves during the strong/weak transition.\n−4\n−8\n−12|S11| (dB)weak coupling\nstrong coupling\n255 260 265 270 275\nMagnetic Field (mT)050100150VDC(µV)(a)\n(b)\nFIG. 4. Line cuts of (a) re\rection parameter and (b) DC volt-\nage at the resonator frequency !c(H0= 0). In the strongly\ncoupled magnon-photon case (green lines), the fundamental\nmode vanishes as opposed to the weakly coupled case where\nthe magnon spectrum is accurately reproduced\npling strength of ge\u000b=2\u0019= 31:8 MHz has been achieved\nat room temperature in a standard EPR cavity. The\ncharacteristics of the coupled magnon-photon system \ft\nwell to the established theory and are consistent with\nrecent results. Simultaneously, we recorded the electri-\ncally detected spin pumping signal of the fundamental\nmode. We were able to tune the system from the strongto the weak coupling regime by changing the cavity's de-\ncay rate. The evolution of the spin pumping signal of the\nfundamental mode has been analyzed qualitatively and\nfollows the predictions of Lotze16: In the strongly cou-\npled magnon-photon system the spin pumping e\u000eciency\nis reduced as the precession cone angle is smaller than in\nthe weakly coupled case. Additionally, we were able to\nobserve coupling and electrically detected spin pumping\nof several spin wave modes with distinctly di\u000berent cou-\npling strengths and observe for the \frst time their 1 =n\ndependence predicted by Cao et al.15. Furthermore, we\ndirectly demonstrated the implications of strong coupling\non \fxed-frequency FMR experiments. We conclude that\nsmall sample volumes or an highly overcoupled cavity are\nmandatory for a qualitatively and quantitatively correct\nevaluation of the magnon spectrum.\nACKNOWLEDGEMENTS\nWe thank Christoph Zollitsch and Johannes Lotze\nfor many valuable discussions and Michaela Lammel for\nassistance in the sample preparation. M. Harder ac-\nknowledges support from the NSERC MSFSS program.\nWe gratefully acknowledge funding via the priority pro-\ngramme Spin Caloric Transport (spinCAT) of Deutsche\nForschungsgemeinschaft (Project GO 944/4), SFB 631\nC3 and the priority programm SPP 1601 (HU 1896/2-1).\n1D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo,\nL. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B.Buckley, D. D. Awschalom, and R. J. Schoelkopf, Physical7\nReview Letters 105, 140501 (2010).\n2Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques,\nD. Zheng, A. Dr\u0013 eau, J.-F. Roch, A. Au\u000beves, F. Jelezko,\nJ. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve,\nPhysical Review Letters 105, 140502 (2010).\n3C. W. Zollitsch, K. Mueller, D. P. Franke, S. T. B. Goen-\nnenwein, M. S. Brandt, R. Gross, and H. Huebl, Applied\nPhysics Letters 107, 142105 (2015).\n4O. O. Soykal and M. E. Flatt\u0013 e, Phys. Rev. Lett. 104,\n077202 (2010).\n5H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. 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Kittel, Introduction to Solid State Physics (John Wiley\n& Sons, New York, 1995).\n21M. Tavis and F. W. Cummings, Physical Review 170, 379\n(1968).\n22J. M. Fink, R. Bianchetti, M. Baur, M. G oppl, L. Ste\u000ben,\nS. Filipp, P. J. Leek, A. Blais, and A. Wallra\u000b, Physical\nReview Letters 103, 083601 (2009).\n23M. B. Jung\reisch, V. Lauer, R. Neb, a. V. Chumak, and\nB. Hillebrands, Applied Physics Letters 103, 2011 (2013).\n24M. A. Gilleo and S. Geller, Physical Review 110, 73 (1958).\n25The designs of the sample holder are pub-\nlished under the CERN Open Hardware License\nhttp://hannes.maier-flaig.de/flexline-sample-rod .\n26I. Chiorescu, N. Groll, S. Bertaina, T. Mori, and\nS. Miyashita, Physical Review B 82, 024413 (2010).\n27P. J. Petersan and S. M. Anlage, Journal of Applied\nPhysics 84, 3392 (1998).\n28E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, Applied\nPhysics Letters 98, 251108 (2011).\n29S. T. B. Goennenwein, T. Graf, T. Wassner, M. S. Brandt,\nM. Stutzmann, J. B. 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Saitoh, Nature 464,\n262 (2010)."    },    {        "title": "1804.08719v1.Unidirectional_Loop_Metamaterials__ULM__as_Magnetless_Artificial_Ferrimagnetic_Materials__Principles_and_Applications.pdf",        "content": "1\nUnidirectional Loop Metamaterials (ULM)\nas Magnetless Artificial Ferrimagnetic Materials:\nPrinciples and Applications\nToshiro Kodera, Senior Member, IEEE, and Christophe Caloz, Fellow, IEEE\nAbstract —This paper presents an overview of Unidirectional\nLoop Metamaterial (ULM) structures and applications. Mimick-\ning electron spin precession in ferrites using loops with unidi-\nrectional loads (typically transistors), the ULM exhibits all the\nfundamental properties of ferrite materials, and represents the\nonly existing magnetless ferrimagnetic medium . We present here\nan extended explanation of ULM physics and unified description\nof its component and system applications.\nIndex Terms —Unidirectional Loop Metamaterials (ULM),\nnonreciprocity, ferrimagnetic materials and ferrites, gyrotropy,\nFaraday rotation, metamaterials and metasurfaces, transistors,\nisolators, circulators, leaky-wave antennas.\nI. I NTRODUCTION\nOver the past decades, nonreciprocal components (isola-\ntors, circulators, nonreciprocal phase shifters, etc.) have been\nhave been almost exclusively implemented in ferrite tech-\nnology [1]–[8]. This has been the case in both microwaves\nand optics, despite distinct underlying physics, namely the\npurely magnetic effect (electron spin precession) in the former\ncase [3], [9], [10] and the magneto-optic effect (electron\ncyclotron orbiting) [11]–[13] in the latter case. However,\nferrite components suffer from the well-known issues high-\ncost, high-weight and incompatibility with integrated circuit\ntechnology, and magnetless nonreciprocity has therefore long\nbeen consider a holy grail in this area [14], [15].\nThere have been several attempts to develop magnetless\nnonreciprocal components, specifically 1) active circuits [16]–\n[21], and space-time [15] 2) modulated structures [22]–[27]\nand 3) switched structures [28] (both based on 1950ies para-\nmetric (e.g. [29], [30]) or commutated (e.g. [31]) microwave\nsystems). All have their specific features, as indicated in Tab. I.\nTABLE I\nCOMPARISON (TYPICAL AND RELATIVE TERMS )BETWEEN DIFFERENT\nMAGNETLESS NONRECIPROCITY TECHNOLOGIES PLUS FERRITE .\nmaterialPconsum. bias cost noise\nferrite yes zero magnet high N/A\nact. circ. no low DC low low\nswitched no med. RF high high\nmodulated no med. RF med. med.\nULM YES low DC low med\nWe introduced in 2011 [32] in a Unidirectional Loop\nMetamaterial (ULM) mimicking ferrites at microwaves and\nrepresenting the only artificial ferrite material, or metamaterial,\nexisting to date. This paper presents an overview of the ULM\nand its applications reported to date.\nT. Kodera is with the Department of Electrical Engineering, Meisei Univer-\nsity, Tokyo Japan (e-mail: toshiro.kodera@meisei-u.ac.jp). C. Caloz is with the\nDepartment of Electrical and Engineering, ´Ecole Polytechnique de Montr ´eal,\nMontr ´eal, QC, H2T 1J3 Canada.II. O PERATION PRINCIPLE\nA Unidirectional Loop Metamaterial (ULM) may be seen\nas a physicomimetic1artificial implementation of a ferrite in\nthe microwave regime . Its operation principle is thus based\nonmicroscopic unidirectionality , from which the macroscopic\ndescription is inferred upon averaging.\nA. Microscopic Description\nMicrowave magnetism in a ferrite is based on the precession\nof the magnetic dipole moments arising from unpaired electron\nspins about the axis of an externally applied static magnetic\nbias field, B0, as illustrated in Fig. 1(a), where B0k^ z. This is\na quantum-mechanical phenomenon, that is described by the\nLandau-Lifshitz-Gilbert equation [3], [10], [33]\ndm\ndt=\u0000\rm\u0002B0+\u000b\nMsm\u0002dm\ndt; (1)\nwhere mdenotes the magnetic dipole moment, \rthe gyromag-\nnetic ratio,Msthe saturation magnetization, and \u000bthe Gilbert\ndamping term. Equation (1) states that the time-variation rate\nofmdue to a transverse2RF magnetic field signal, HRF\nt\n(k^t;^t?^ z), is equal to the sum of the torque exerted by B0\nonm(directed along +^\u001e,^\u001e: azimuth angle), and a damping\nterm (directed along\u0000^\u0012,^\u0012: elevation angle) that reduces\nthe precession angle,  , to zero along a circular-spherical\ntrajectory (conserved jmj) when the RF signal is suppressed\n(relaxation).\nClassically, magnetic dipole moments can be associated\nwith current loop sources , according to Amp `ere law. Decom-\nposing a ferrite magnetic moment, m, into its longitudinal\ncomponent, mz, and transverse component, m\u001a, as shown in\nFig. 1(a), one may thus invoke the effective current loops Imz\neff\nandim\u001a\neffassource models for the corresponding moments.\nAmong these currents, only im\u001a\neffmatters in terms of mag-\nnetism, since Imz\neff, as the source associated with HRF\nz, does\nnot induce any precession (Footnote 2). im\u001a\neffis thus the current\none has to mimic to devise an “artificial ferrite.” This current,\nas seen Fig. 1, has the form of a loop tangentially rotating on\nan imaginary cylinder of axis z.\n1The adjective “physicomimetic” is meant here, from etymology, as “mim-\nicking physics.”\n2The longitudinal ( z) component does not contribute to precession, and\nhence to magnetism. Indeed, since B0k^ z, thez-component of mproduced by\nHRF\nzwould lead to mRF\nz\u0002(B0+\u00160HRF^ z) = [mRF\nz(B0+\u00160HRF)](^ z\u0002^ z) =\n0, the only torque being produced by the transverse component ( HRF\nt,^t2xy-\nplane), mRF\nt\u0002(B0+\u00160HRF^t) = (mRF\ntB0)(^t\u0002^ z)6= 0. In the rest of the\ntext, we shall drop the superscript “RF,” without risk of ambiguity since mt,\nis exclusively produced by the RF signal.arXiv:1804.08719v1  [physics.app-ph]  16 Apr 20182\nFig. 1. “Physicomimetic” construction of the Unidirectional Loop Metama-\nterial (ULM) “meta-molecule” or particle. (a) Magnetic dipole precession,\narising from electron spinning in a ferrite material about the axis (here z) of an\nexternally applied static magnetic bias field, B0, with effective unidirectional\ncurrent loops Imz\neffandim\u001a\neff, and transverse radial rotating magnetic dipole\nmoment m\u001aassociated with im\u001a\neff. (b) ULM particle [32], typically (but not\nexclusively [34]) consisting of a pair of broadside-coupled transistor-loaded\nrings supporting antisymmetric current and unidirectional current wave (shown\nhere with exaggeratedly small wavelength for the sake of visibility), with\nresulting radial rotating magnetic dipole moment emulating that in (a).\nGiven its complexity, the current im\u001a\neffmay a priori seem\nimpossible to emulate. However, what fundamentally matters\nfor magnetism is not this current itself, but the moment m\u001a,\nfrom which magnetization will arise at the macroscopic level\n(Sec. II-B). This moment may be fortunately also produced\nby a pair of antisymmetric \u001e-oriented currents, rotating on the\nsame cylinder, which can be produced by a pair of conducting\nrings operating in their odd mode [35], as shown in Fig. 1(b).\nIf this ring-pair structure is loaded by a transistor [32], as\ndepicted in the figure, or includes another unidirectionality\nmechanism such as the injection of an azimuthal modula-\ntion [34], m\u001awillunidirectionally rotate about zwhen excited\nby an RF signal, and hence mimic the magnetic behavior of the\nelectron in Fig. 1(b). The structure in Fig. 1 constitutes thus\ntheunit-cell particle of the ULM at the microscopic level .\nOne may argue there there is a fundamental difference\nbetween the physical system in Fig. 1(a) and its presumed\nartificial emulation in Fig. 1(b): the ferrite material also\nsupports the longitudinal moment mzwhereas the ULM does\nnot include anything alike. However, as we have just seen\nabove, particularly in Footnote 2, mzdoes not contribute to\nthe magnetic response. It is therefore inessential and does thus\nnot need to be emulated. So, the particle in Fig. 1(b), with its\nmoment m\u001ais all that is needed for artificial magnetism !\nDoes this mean that the ULM particle includes no counter-\npart to the static alignment of the dipoles due to B0(and\nproducing mz) in the ferrite medium? In fact, there isa\ncounterpart, although mz= 0 in the ULM. The fundamental\noutcome of the static alignment of the dipoles along zin\nthe ferrite is the alignment of the relevant magnetic dipoles\nm\u001ain the plane perpendicular to ^ z(or within the xy-plane)\nacross the medium, for otherwise the m\u001a’s of the different\ndomains [3], [10] would macroscopically cancel out. Such an\norientation of the m\u001a’s perpendicularly to ^ zis essential to\nemulate magnetism. How is this provided in the ULM? Simply\nbyfixing the rings in a mechanical support , such as a substrate,\nas will be seen later. So, the counterpart of the ferrite static\nalignment of dipoles is simply mechanical orientation in the\nULM.B. Macroscopic Description\nSince it mimics the relevant magnetic operation of a ferrite\nat the microscopic level, the unit-cell particle in Fig. 1(b) must\nlead to the same response as bulk ferrite at the macroscopic\nlevel when repeated according to a subwavelength 3D lattice\nstructure so as to form a metamaterial as shown in Fig. 2(a).\nThe ULM in Fig. 2(a), just as a ferrite3, forms a 3D\narray of magnetic dipole moments, mi, whose average over a\nsubwavelength volume V,\nM=1\nVX\ni=1mi= \n1\nVX\ni=1m\u001a;i!\n^\u001a=M\u001a^\u001a (2)\ncorresponds to the density of magnetic dipole moments, or\nmagnetization , as the fundamental macroscopic quantity de-\nscribing the metamaterial4.\n(a) (b)\nFig. 2. ULM structures obtained by periodically repeating the unit cell with\nthe particle in Fig. 1. (a) Metamaterial (3D), described by the Polder volume\npermeability (3a). (b) Metasurface (2D metamaterial), described by a surface\npermeability [36].\nFrom this point, one may follow the same procedure as in\nferrites [8], [37] to obtain the Polder ULM permeability tensor\n\u0016=2\n4\u0016 j\u0014 0\n\u0000j\u0014 \u0016 0\n0 0\u00160;3\n5; (3a)\nwith\u0016=\u00160\u0012\n1 +!0!m\n!2\n0\u0000!2\u0013\nand\u0014=\u00160!!m\n!2\n0\u0000!2;(3b)\nwhere!0and!mare the ULM resonance frequency (or\nLarmor frequency) and effective saturation magnetization fre-\nquency , respectively5, that will be derived in the next section.\nAs in ferrites, the effect of loss can be accounted for by\nthe substitution !0 !0+j\u000b!, where\u000ba damping factor\nin (1) [8].\nSo, a ULM may really be seen as a an artificial ferrite\nmaterial producing magnet-less artificial magnetism . However,\nitsnonreciprocity is achieved from breaking time-reversal\n(TR) symmetry by a TR-odd current bias , originating in the\ntransistor (DC) biasing, instead of a TR-odd external magnetic\nfield [14].\nULMs have been implemented only in a 2D format so far.\nThe corresponding structure is shown in Fig. 2(b), and may\n3The difference is essentially quantitative : while in the ferrite p=\u0015< 10\u00006\n(p: molecular lattice constant), in the ULM p=\u0015\u00191=10\u00001=5(p:\nmetamaterial lattice constant or period), but homogeneization works in both\ncases.\n4Whereas in a ferrite, we have M=Ms+MRF= (Ms+MRF\nz)^ z+MRF\nt\u0019\nMs^ z+MRF\n\u001a^\u001a, whereMsis the saturation magnetization of material, in the\nULMMs= 0. We shall subsequently drop the superscript “RF” also in M\u001a.\n5In a ferrite,!0=\rB0and!m=\r\u00160Ms.3\nbe referred to as a Unilateral Loop Metasurface (ULMS) .\nSection IV will present ULMS Faraday rotation and Sec. V-A\nwill discuss related applications.\nIII. ULM P ARTICLE AND DESIGN\nULMs may be implemented in different manners. Figure 3\nshows a ULM particle implemented in the form of a microstrip\ntransistor-loaded single ring placed on PEC plane. Assuming\na distance much smaller than the wavelength between the\nring and the PEC plane, the structure is equivalent, by the\nimage principle, to the antisymmetric double-ring structure in\nFig. 1(b) [35].\nFig. 3. ULM particle microstrip implementation in the form of a transistor-\nloaded single ring on a PEC plane, supporting the odd effective current\ndistribution and unidirectional current wave as in Fig. 1(b). The transistor\nbiasing circuit is not shown here.\nThe transistor-loaded ULM particle in Fig. 3, or Fig. 1(b),\nis essentially a ring resonator , whose total electrical size is\ngiven by [38]\n\fmsa(2\u0019\u0000\u000bTR) +'TR= 2\u0019; \f ms=k0p\u000fe=!\ncp\u000fe;(4)\nwhere\fmsis the microstrip line wavenumber ( \u000fe: effective\nrelative permittivity), ais the average radius of the ring, \u000bTR\nis the geometrical angle subtending the transistor chip, and\n'TRis the phase shift across it. Solving Eq. (4) for !provides\nthe resonance frequency of the resonator, and hence the ULM\nresonance frequency ,\n!0=\f\f\f\f(2\u0019\u0000'TR)c\nap\u000fe(2\u0019\u0000\u000bTR)\f\f\f\f; (5)\nin (3). The parameter !min the same relations follows from\nthe mechanical orientation of the moments, as explained in\nSec. II-B: although we do not have here a saturation magne-\ntizationMsleading to the frequency parameter !m=\r\u00160Mm\nin the ferrite, we have have an equivalent phenomenological\nparameter!massociated with the orientation of the rings,\nwhich may be found by extraction, as will be seen in Sec. IV.\nNote that ULMs may be designed for multi-band operation\nand enhanced-bandwidth operation. The former, in contrast\nto ferrites that are restricted to a single ferromagnetic reso-\nnance!0=\rB06, can in principle accommodate multiple\nresonances by simply incorporating rings of different sizes,\nas illustrated in Fig. 4. The latter, in contrast to ferrite whose\nbandwidth is inversely proportional to loss due to causality, can\nbe achieved by leveraging overlapping coupled resonances [2].\n6This restriction can be somewhat overcome in a structured ferromagnetic\nstructure, such as a ferromagnetic nanowire membrane supporting a remanent\nbistable population of up and down magnetic dipole moments with corre-\nsponding resonances !\"\n0=\r\u00160H\"\n0and!#\n0=\r\u00160H#\n0[39], [40].\n(a) (b)Fig. 4. Unique magnetic properties of the ULM attainable by using multiple\nrings of resonance frequencies !0n, here two rings with resonance frequencies\n!01and!02. (a) Multiband operation using independent rings with separate\nresonance frequencies. (b) Enhanced bandwidth using coupled-resonant rings\nwith overlapping resonance frequencies.\nThe single-ring-PEC ULM implementation of Fig. 3 is\nideal for microstrip components [38] and reflective metasur-\nfaces [32]. However, for 3D ULM [Fig. 2(a)] structures and\ntransmissive metasurfaces (Sec. IV), a transparent version of\nthat ULM is required. This could theoretically be realized with\na pair of rings, as in Fig. 1(b), but may be more conveniently\nimplemented in the form of circular slots in a Coplanar Wave-\nGuide (CPW) type technology, as reported in [41].\nIV. F ARADAY ROTATION\nFaraday rotation is one of the most fundamental and useful\nproperties of magnetic materials. Given their artificial ferrite\nnature (Sec. III), ULMs can readily support this effect. The\nFaraday angle is given by [3], [8]\n\u0012F(z) =\u0000\u0012\f+\u0000\f\u0000\n2z\u0013\n;with\f\u0006=!p\n\u000f(\u0016\u0006\u0014);(6)\nwhere\u0016and\u0014are the Polder tensor components in (3b),\nwith the resonance ( !0) given by (5) and the saturation\nmagnetization frequency ( !m) discussed in Sec. III. Inter-\nesting, the ULM allows the option to reverse the direction\nof Faraday rotation by simple voltage control (instead of\nmagnet mechanical flipping in a conventional ferrite) using\nan antiparallel transistor pair load, as demonstrated in [42].\nFigure 5 shows a reflective Faraday ULM metasurface\n(ULMS) structure, based on the particle in Fig. 3, and re-\nsponse, initially reported in [32]. The results confirm that the\nULM works exactly as a ferrite, whose equivalent parameters\nare given in the caption.\nFig. 5. Reflective Faraday rotating ULMS based on the particle in Fig. 3 [32].\n(a) Perspective representation of the metasurface with rotated plane of polar-\nization. (b) Theoretical [Eq. (6)] and experimental polarization rotation angle\nversus frequency. Here, \u000fr= 2:6,!0=2\u0019= 7:42GHz (Bequiv.\n0= 0:265 T),\n!m=2\u0019= 28 MHz (\u00160Mequiv.\ns = 1 mT) and\u000b= 1:9\u000210\u00003Np\n(\u0001H=\u000b!0=\r\u0016 0= 0:4mT.)\nULM Faraday rotation has also been reported in transmis-\nsion, using the circular-slot ULM structure mentioned at the4\nend of Sec. III. Using slots, and hence equivalent magnetic\ncurrents, instead of rings supporting electric currents, that\nstructure really operates as an artificial magneto-optic material,\nwith a permittivity tensor replacing the magnetic tensor in (3a).\nA similar Faraday rotation effect may also be achieved using\narrays of twisted dipoles loaded by transistors [43].\nV. A PPLICATIONS\nA. Metasurface Isolators\nThe transmissive ULMS in [41] can be straightforwardly\napplied to build a Faraday isolator [3], [4], [44], [45], as shown\nin Fig. 6. As the wave propagates from the left to the right, its\npolarization is rotated 45\u000eby the left ULMS in the rotation\ndirection imposed by the transistors (here, clock-wise). It thus\nreaches the polarizer with its electric field perpendicular to\nthe conducting strips and therefore unimpededly crosses it. It\nis finally rotated back to its initial (vertical) direction by the\nright ULMS, whose rotation direction is opposite to the left\none (here, counter-clockwise). In the opposite direction, the\nright ULMS rotates the wave polarization in such a manner\nthat its electric field is parallel to the conducting strips of the\npolarizer, so that the wave is completely reflected. It is then\nrotated again by the right polarizer and gets back to the right\ninput orthogonal to the original wave7.\nFig. 6. Isolator using two transmissive Faraday rotation ULMSs [41] and a\n45\u000epolarizer. The bottom-left inset shows the transmissive “magneto-optic”\nslot-ULM demonstrated in [41], that may be used for this application.\nFaraday rotation is not the only approach to realize spatial\nisolation , as in Fig. 6. Such isolation may be simply achieved,\nwithout any gyrotropy but still magnetlessly, with a metasur-\nface consisting of back-to-back antenna arrays interconnected\nby transistors [47]; this nonreciprocal metasurface may exhibit\nan ultra wideband response and provide transmission gain.\nB. Nonreciprocal Antenna Systems\nThe ULM structure may be used in various nonreciprocal\nradiating (antenna, reflector and metasurface) applications.\nFigure 7 shows a nonreciprocal antenna system and its re-\nsponse [48]. The structure [Fig. 7(a)] is a ULM magnet-\nless version of the nonreciprocal ferrite-loaded Composite\nRight/Left-Handed (CRLH) open-waveguide leaky-wave an-\ntenna introduced in [49], [50], with the ferrite material re-\nplaced by a 1D ULM structure. This structure may be used\nas a nonreciprocal full-space scanning antenna [Figs. 7(b)\nand (c)], whose unidirectionality provides protection against\ninterfering signals, or as an antenna diplexer system , where\nnonreciprocity effectively plays the role of a circulator with\nhighly isolated uplink ( 3!1) and downlink ( 2!3) paths.\n7Lossy polarizers can be added, if necessary (The final wave could be\nreflected back), for dissipative (rather than reflective) isolation [46].\nFig. 7. ULM CRLH leaky-wave isolated-antenna or antenna-duplexer sys-\ntem [48]. (a) Prototype with port definitions. (b) Measured radiation patterns.\n(c) Measured scattering parameters.\nC. Isolators and Circulators\nULM technology also enables various kinds of nonre-\nciprocal components. Figure 8(a) shows a ULM microstrip\nisolator [38]. The ULM structure below the microstrip line is\ncomposed of two rows of transistor-loaded rings with opposite\nbiasing, and hence opposite allowed wave rotation directions.\nAs the wave from the microstrip line reaches a ring pair,\nits mode is coupled into a stripline mode with strip pair\nconstituted by the longitudinal sections of the overlapping\nrings, and usual antisymmetric currents. In the propagation\ndirection where these currents are co-directional, with allowed\nrotation direction of the ULM, the stripline mode is allowed\nto propagate, whereas in the opposite propagation direction, it\nis inhibited and dissipates in matching resistors on the rings.\nFigure 8(b) shows a ULM microstrip circulator [38], which is\nbased on mode-split counter-rotating modes as all circulators.\nFig. 8. ULM components [38]. (a) Isolator [51]. (b) Circulator [38].\nVI. C ONCLUSION\nWe have presented an overview of ULM structures and\napplications. The ULM physics has been described in great\ndetails, revealing that the ULM really represents an artificial\nferrite medium. It is in fact the only existing medium of\nthe kind. It has been pointed out that the ULM may offer\nunique extra benefits compared to ferrites, such a multiband\noperation, ultra broadband and electronic Faraday rotation\ndirection switching.5\nREFERENCES\n[1] A. G. Fox, S. E. Miller, and M. T. Weiss, “Behavior and applications\nof ferrites in the microwave region,” The Bell System Technical Journal ,\nvol. 34, no. 1, pp. 5–103, Jan 1955.\n[2] G. L. Matthaei, E. M. T. Jones, and S. B. Cohn, “A nonreciprocal,\nTEM-mode structure for wide-band gyrator and isolator applications,”\nIRE Transactions on Microwave Theory and Techniques , vol. 7, no. 4,\npp. 453–460, October 1959.\n[3] B. Lax and K. J. Button, Microwave ferrites and ferrimagnetics .\nMcGraw-Hill, 1962.\n[4] L. J. Aplet and J. W. 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Caloz, “Uniform ferrite-loaded open waveguide\nstructure with CRLH response and its application to a novel backfire-to-\nendfire leaky-wave antenna,” IEEE Transactions on Microwave Theory\nand Techniques , vol. 57, no. 4, pp. 784–795, April 2009.\n[50] ——, “Low-profile leaky-wave electric monopole loop antenna using\nthe\f= 0 regime of a ferrite-loaded open waveguide,” IEEE Trans.\nAntennas Propag. , vol. 58, no. 10, pp. 3165–3174, Oct. 2010.\n[51] T. Kodera, D. L. Sounas, and C. Caloz, “Isolator utilizing artificial\nmagnetic gyrotropy,” in 2012 IEEE/MTT-S International Microwave\nSymposium Digest , June 2012, pp. 1–3."    },    {        "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\nhi<jisi\u0001sj (1)\nHere, sis the Mn spin operator with magnitude s= 1.\nFig. 2(a) shows the dispersion for a single Mn kagome\nlayer given byHintrawithin linear spin wave theory. The\noverall bandwidth is 6 sjJjwith a Dirac band crossing at\nthe K-point with energy 3 sjJj. As described below, our\ndata analysis does not bene\ft from the introduction of\nlonger-ranged intralayer interactions, although we cannot\nexclude them.\nTo describe the interlayer interactions, Fig. 1(c) shows\nthat nearly equidistant FM Mn kagome layers are stacked\nalong thec-axis. Tb layers are inserted after every two\nMn layers and with opposite magnetization, forming a\nMn(\")-Mn(\")-Tb(#)-Mn(\")-Mn(\")-Tb(#) pattern. Sev-\neral unique interlayer magnetic couplings between Mn\nlayers and between Mn and Tb layers are possible, giv-\ning\nHinter=X\nkX\ni<jJMM\nksi\u0001sj+k+JMTX\nhi<jisi\u0001Sj:(2)\nHere,JMT>0 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. 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College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China  \nEmail:  *ljzhu@semi.ac.cn  \nAbstract:  Despite the blooming interest , the transition -metal rar e-earth ferrimagnets have not been comprehensive ly \nunders tood in terms of their  coercivity and transport properties. Here, w e report a systematic study of the magnetic and \ntransport properties of ferrimagnetic FeTb alloy by varying the layer thickness and temperature. The FeTb is tuned from  the \nTb- dominated regime to the Fe-dominate d regime via the layer thickness , without varying the compos ition. The coercivity  \nclosely follows the 1/cos θH scaling (where θH is the polar angle of the external magnetic field ) and increases quasi -\nexponentially upon cooling  (exceeding 90 kOe  at low temperatures) , revealing that the nature of the coercivity is  the \nthermally -assisted domain wall depinning field. The resistivity exhibits a quasi -linear upturn upon cooling  possibly due to \nthermal vibrations of the structure factor  of the amorphous alloy . The existing scaling laws of the anomalous Hall effect in \nthe literature break down for the amorphous FeTb that are either Fe- or Tb -dominated. These findings should advance the \nunderstanding of the transition -metal -rare-earth ferrimagnets  and the associated  ferrimagnetic  phenomen a in spintronics . \n \nI. Introduction  \n \nFerrimagnetic materials  (FIMs) , which have two \nantiferromagnetically coupled  sublattices, are of \nconsiderable interest in the field of spintronics  [1-5]. FIMs \nare potentially advantageous for dense magnetic recording \napplications  [6] because of their tunable magnetism , less \nsensitivity to stray magnetic fields than ferromagnets  (FMs) , \nand easier and fast detection than antiferromagnets  (AFs) . \nMoreover , FIMs are an exotic  platform to study the interplay \nof spin -orbit physics and  AF coupling as a function of the \ndegree of magnetic compensation. Several  striking spin -\norbit -coupling (SOC)  phenomena  have been demonstrated  to \narise from the magnetization compensation within the bulk \nof the FIMs,  such as large magnetic domain wall velocities  \nnear compensation  [3-5], strong compensation -dependen ce \nand sign reversal of bulk spin-orbit torques (SOTs) [7], strong \nvariation of the “interfacial” SOTs with t he relative spin \nrelaxation rates  within the bulk of FIMs  [8,9] , lack of \ncurrent -driven magnetization switching at full magnetization \ncompensation  [10-12]. However, the spin -mixing \nconductance of the interfaces of metallic FIMs is insensitive \nto temperature and magnetic compensation or the areal \ndensity of the magnetic moment of the interface  [8], in \ncontrast to the insulat ing FIM case  [13-14].  \nAn i n-depth understanding of the ferrimagnetic  \nphenomen a in magnetic heterostructures requires  insights \ninto the mechanisms of the coercivity  (Hc), the electron \nmomentum scattering , and the anomalous Hall effect (AHE) \nof the  FIMs . Note that t he coercivity  of a perpendicular \nmagnetization represents  the switching barrier to overcome \nby the driving magnetic field or SOT [15-17], while electron \nmomentum scattering  affects  the generation  [18-20] and \nrelaxation  of spin current via SOC [8,21,22] . The AHE  \ntypically functions as the indicator of t he magnetization \norientation in a variety of experiments (e.g., harmonic Hall \nvoltages  [7-9] and magnetization switching  [14-\n15,19, 23,24]). However, the magneti zation of transition -metal rare -earth FIMs arises from two competing sublattices \nthat have been suggested to contribute to the AHE in distinct \nmanners  (i.e., the 3d states of the transition metal governed \nthe transport properties , while  the 4 f states of Tb  were  less \ninvolved  [23,25,26]). The scaling behavior of the AHE in \nsuch transition -metal rare -earth FIMs thus become s \nstimulating open question s.  \nIn this work , we systematically examine t he magnetic \nand transport properties of FeTb, a  representative FIM, as a \nfunction of layer thickness and temperature.  We show that  \nthe coercivity can be rather high and increase quasi -\nexponentially upon cooling  due to the nature of the \nthermally -assisted domain wall depinning field . The large \nlongitudinal resistivity (ρxx) increases  quasi -linearly upon \ncooling . The anomalous Hall resistivity ( ρAH) cannot be \ndescribed by the existing scaling laws [27-29].  \n \nII. Samples and magnetization  \n \nFor this work, we deposit four FeTb films  with different \nthicknesses  (t = 8 nm, 16 nm, 32 nm, and 48 nm) on Si/SiO 2 \nsubstrates by co -sputtering at room temperature. Each \nsample is capped by a MgO ( 2 nm)/Ta ( 2 nm) bilayer that is \nfully oxidized upon exposur e to the atmosphere.  The \ncomposition of all the FeTb layers  is Fe0.55Tb0.45 in volume \npercentage as calibrated using the deposition rate s of the Fe \nand the Tb. This composition correspon ds to a Tb/Fe atomic \nratio of ≈ 0.3 (as calculated using the atomic volumes of bulk \nFe and Tb crystals) , which is consistent with the energy \ndispersive spectr oscopy (EDS) result s (0.32± 0.01)  in Figs. \n1(a) and 1(b). Such  FeTb films  are amorphous and \nhomogeneous as indicated by scanning trans mission electron \nmicroscopy and electron energy loss spectra  results  of the \nsamples prepared by the same sputtering tool using similar \nparameters  within the same few days  [7].  2 \n  \nFig. 1. (a) Energy dispersive  spectra collected using a Bruker \nEDS  tool and (b) the Tb/Fe atomic ratio calculated using the \nTb L α and Fe K α EDS spectra for the FeTb films with \ndifferent thicknesses, revealing that the composition is the \nsame for the FeTb films studied in this work.  \n \nThe saturation magnetization  (Ms) for each FeTb  film, \nthe sum contribution of the Fe and the Tb sublattices ( Fig. \n2(a)), is measured using  a superconducting quantum \ninterference device (SQUID). Figure 2(b) shows the \ntemperature profile of the saturation magnetization of the \nFeTb films with dif ferent thicknesses.  Ms for the 8 nm FeT b \nvaries non -monotonically with temperature and peaks at  250 \nK.  For the 16 nm FeTb, Ms increases monotonically from \nbeing negligibly small  at temperatures below 50 K to ≈60 \nemu/cm3 as the temperature approaches 300 K. In contrast, \nfor both the  32 nm and 48 nm FeTb, Ms decreases first slowly \nand then more rapidly as the temperature increases. The \ndiverse temperature profile s of the saturation magnetization \nof the FeTb films impl y a strong tu ning of the magnetization \ncompensation configuration by thickness.  As plotted in Fig. \n2(b), the magnetization exhibits full compensation at the \n“compensation thickness” of 16 nm at 5 K and of ≈20 nm at \n300 K. Thus, the magnetization  compensation  of the FeTb \nalloy is not only a function of substrate [7], composition  \n[7,30,31 ], and strain [ 32] but also of temperature  and \nthickness.  \nThe films  are then patterned into 5×60 μm2 Hall bars for \nelectrical measurements using a physical properties \nmeasurement system (PPMS -9T). From the  AHE \nmeasurements (see Figs. 2(c) and 2(d)), the FeTb  films have \nfairly square hysteresis loops  for the t ransverse resistivity  \n(ρxy), implying good magnetic uniformity of these films. The \npolarity of the hysteresis loop is opposite for the 8 nm and \n16 nm FeTb samples compared to that for the 32 nm and 48 nm ones, suggesting that the films are Tb -dominated at 8 nm \nand 16 nm but Fe -domin ated at 32 nm and 48 nm. Such \nstriking thickness dependences of the magnetization and the \ntransverse resistivity are interesting observations  and worth \nfuture investigation . While the unambiguous identification \nof the exact mechanism is beyond the scope of  this paper, we \nspeculate that the striking thickness dependence of the \nmagnetic and transport properties might be related to some \nhidden short -range ordering within the amorphous films . \n \nIII. Giant, strongly temperature -dependent  coercivity  \n \nFigure 3(a) shows the out -of-plane coercivity and the \neffective perpendicular magnetic anisotropy field ( Hk) at 300 \nK. Here, the out -of-plane coercivity is estimated  from the \nswitching of the transverse resistivity by a perpendicular \nmagnetic field ( Hz, Fig. 2(d)), while Hk is estimated from the \nparabolic scaling of  ρxy with the in-plane magnetic field  (Hxy) \ndue to tilting of the magnetization ( Fig. 3(b)), i.e.,  \n ρxy = ρAH cos(arcsin( Hxy/Hk)) ≈ ρAH [1-1/2(Hxy/Hk)2].      (1) \nBoth Hc and Hk vary as a function of the layer thickness and \ntend to increase upon approaching the “compensation \nthickness”. As shown in Fig. 3(c) and 3(d), Hc follows a \n1/cos θH scaling (θH is the polar angle of the driving magnetic \nfield Hxz) and is typically much smaller than the \nperpendicular magnetic anisotropy field at θH = 0o. This is i n \ncontrast  to the case of a perpendicular macrospin , for which \nthe coercivity varies as  \nHc = H k (cos2/3θH + sin2/3θH)-3/2,                        (2) \nand is equal to Hk at θH = 0o and 90 o. As plotted in Fig. 3(e), \nthe out-of-plane coercivity of each FeTb  film increases upon \ncooling in a quasi -exponential manner . These observations \nconsistently reveal that the coercivity of the FeTb represents \nthe thermally -assisted  depinning field  of the magnetic \ndomain walls rather than the bulk magnetic anisotropy field.  \nThis is  general ly the case for magnetic systems in which the \nreversed domain nucleation and domain wall propagation  \n(with the energy barrier of the domain wall pinning field ) \nrequire less energy than coherent rotation  (with the energy \nbarrier of Hk) [17].  \nWe also  note that the out -of-plane field required to \nswitch these films, i.e., the coercivity , exceeds 90 kOe at \ntemperatures below 175 K for the 16 nm FeTb and at \ntemperature s below 25 K for the other three samples.  The \nrapid  increase of the coercivity upon cooling is distinct from \nthe enhancement of coercivity at the magnetization \ncompensation points [ 1-5,7, 33] because it occurs in the \nwhole temperature region, includ ing the temperatures at \nwhich the magnetization is very hi gh (Fig. 2(b) ). The giant \ncoercivity and square hysteresis loops ( Fig. 3(f)) may make \nthese FeTb interesting hard magnets for some specific \nspintronic applications  [34]. \n3 \n  \nFig. 2. (a) Schematic depict of the ferrimagnetism and the anomalous Hall effect in the FeTb. (b) Dependence of the saturation \nmagnetization  on temperature , (c) Dependence of  the saturation magnetization on the thickness at 5 K and 300 K, a nd (d) \nTransverse resistivity at 300 K for the FeTb with different layer thicknesses.   \n \nFig. 3. (a) Perpendicular coercivity and perpendicular magnetic anisotropy field and (b) Parabolic scaling of the transverse \nresistivity with in -plane magnetic field for FeTb with different thicknesses (300 K). (c) Dependence on the polar angle ( θH) \nof the room -temper ature coercivity. (d) Transverse resistivity hysteresis loops for the 8 nm FeTb  measured at θH = 0o and 85o \nand 300 K.  (e) Dependence on the temperature of the perpendicular coercivity  (θH = 90o) of the FeTb films. (f) Transverse \nresistivity hysteresis loop at 25 K for the 32 nm FeTb, displaying a giant coercivity of 65 kOe . \n4 \n IV. Resistivity upturn  \nResistivity or electron momentum scattering is also a \nkey property of a spintronic material. For ins tance, e lectron \nmomentum scattering affects  spin-dependent scattering , the \ngeneration and relaxation of spin current  via SOC . To provide \ninsight into the electron momentum scattering mechanism, \nwe measure the resistivity of the FeTb  samples as a function \nof temperature  (Fig. 4(a)). ρxx varies between 210  µΩ cm and \n250 µΩ  cm. In analog y to the magnetic properties and the \nanomalous Hall resistivity (see below) , ρxx shows also an \ninteresting non -monotonic thickness dependence  at each \nfixed temperature  due to some exotic mechan ism yet to know . \nHere, i nterfacial scattering is unlikely to  play any significant  \nrole in the determination of ρxx of these thick , resistive FeTb \nbecause they should have a very short mean -free path.  \nMore interestingly,  ρxx shows a quasi -linear upturn upon \ncooling for each sample . Similar resistivity upturn has also \nbeen observed in 200 nm thick FeTb films  [35]. In general, a \nresistivity upturn  can arise from weak localization, hopping \nconductance, orbital one -channel Kondo effect, orbital two -\nchannel Kondo effect, electron -electron scattering, magnetic \nBrillouin zone scattering , or scattering of electrons by \nthermal vibration of structure factor . However,  none of these \nmechanisms appear  to explain the resistivity upturn of these \nFeTb films. First, w eak localization , which diminishes under  \nan external  magnetic field  or a strong internal exchange field,  \nis not expected in the ferrimagnetic FeTb that have giant  \nperpendicular magnetic anisotropy  (Fig. 3(a)). Hopping \nconductance is known to occur in Mott -Anderson insulators \nwith extremely high resistivity  (e.g., 106-109 µΩ cm for \nquasicrystal AlPdRe [36], amorphous GeTe, and GeSb 2Te4 \nannealed at 150 oC [37], 1011-1017 µΩ cm  for the Pt -SiO 2 \ngranular film with Pt concentration of 0.11  [38,39]) but not \nin metals like FeTb with several orders of magnitude lower \nresistivity . The absence of hopping conductance is reaffirmed \nby the lack of a T-1/4 scaling  (so-called Mott’s law  [40]) in the \nresistivity ( Fig. 4(b)). The orbital one -channel Kondo effect \n[41], if important, should increase the resistivity as a function \nof ln T, which is not the case for the FeTb ( Fig. 4(c)). The \nresistivity upturn due to e lectron -electron interaction would \nfollow a T1/2 scaling  at low temperatures  [42], which is not \nconsistent w ith the evident deviation from the T1/2 scaling at \ntemperatures below 150 K ( Fig. 4(d)).  \nThe orbital two -channel Kondo effect [41,43-45] is also \nless likely to explain the resis tivity upturn  in the  amorphous  \nFeTb films. This is because it would imply a Kondo \ntemperature of >300 K (below which the T1/2 scaling emerges)  \nand a deviation temperature of 150 K (below which the \nresistivity deviates from the T1/2 scaling), both of which are \nsurprisingly high. Note that the  Kondo temperature is only 23 \nK for the L10-MnAl [42] and 14.5 K for L10-MnGa films [45], \na few K for glasslike ThAsSe  [43] and Cu point contacts [41], \nwhile the deviation temperature is typically below 1 K  for all \nthe previously studied two -channel Kondo systems  [41-43]. \nAnother possible mechanism for a resistivity upturn is \nthe magnetic Brillouin zone scattering (the periodic potentials \ndue to antiferromagnetic alignment of the magnetic \nsublattices can produce an additional magnetic Brillouin zone, of smaller volume in k-space than the ordinary lattice \npotential, whose planes further incise and contort the Fermi \nsurface  [47]). While this possibility cannot be quantitatively \ntested due to a lack of knowledge about the exact functional \ndependence  on temperature, magnetic Brillouin zone \nscattering should be  weak in the amorphous FeTb which has \nno long -range periodicity in the crystalline and magnetic \nlattices. The resistivity upturn is also absent in epitaxial \nferrimagnets of Mn 1.5Ga [28] and Mn 2Ga [48] in which the \ntwo Mn sublattices are also AF coupled.  \nAfter we have excluded any important role of weak \nlocalization, hopping conductance, orbital one -channel \nKondo effect, orbital two -channel Kondo effect, electron -\nelectron scattering, and magnetic Brillouin zone scattering , \nscattering of electrons by thermal vibration s of the structure \nfactor  [49,50] is left as the most likely mechanism for the \nincrease of resistivity with decreasing temperature over a \nwide range of temperature in our amorphous FeTb films. \nThermal vibration s of the structure factor  have been reported \nto explain the resistivity upturn in many liquid transition \nmetals and metallic glass alloys  [35,49,50]. Note that such \nthermal vibration s of the structure factor  in disordered alloys \nare distinct from the phonon scattering that increases t he \nresistivity with increasing temperature in ordered crystalline \nmaterials [ 27-29,44 ]. Future theoretical calculations of the \nstructure factor  as a function of temperature would be \ninformative for a more quantitatively understanding of the \nresistivity of the FeTb samples, which is, however, beyond \nthe scope of this article.   \n \nFig. 4. Resistivities ( ρxx) of the FeTb films plotted as a \nfunction of (a) temperature T, (b) T (in log plot), (c) T-1/4, and \n(d) T1/2. In (b) -(d) the resistivity data for the 8 nm FeTb is \nmultiplied by 1.05 for clarity. In (b) the l og plot of ρFeTb as a \nfunction of T-1/4 indicates a lack of Mott’s law for hopping \nconduction  [38], the latter predicts ln ρxx to vary linearly with \nT-1/4. In (d) the straight lines represent the best linear fits to \nthe data in the high-temperature  regime.  \n5 \n  \nFig. 5. Scaling of the anomalous Hall effect. (a) Dependence on the temperature of ρAH of the FeTb films with different \nthicknesses. (b) ρAH vs ρxx2 and (c) ρAH vs ρxx for the FeTb films and the control Mn 1.5Ga sample. The solid straight lines in \n(b) represent f its of the data to Eq. (3)  and the solid curves in (c) represent the fits of the data to Eq. (4) . \n \nV. Scaling of the s trong Anomalous Hall effect  \n \nWe now discuss the scaling of the anomalous Hall \nresistivity ( ρAH) with the longitudinal resistivity ( ρxx). The \nscaling analysis is interesting as it can disentangle the \nintrinsic and extrinsic contributions of the anomalous Hall \nresistivity of magnetic materials in which the electron \nscattering is dominated by impurity and phonon  scattering. In \nthat case,  ρAH of a given sample is simply a linear function of \nρxx2, i.e.,  \n ρAH = αρxx0 + β0 ρxx02 +bρxx2,                                       (3) \nwhere α, β0, ρxx0, and b are constant  for a given sample and a \n= αρxx0 + β0 ρxx02 goes to zero when the residual resistivity  ρxx0 \n(due to static impurity scattering at low temperatures) is zero. \nEquation (3) describes the AHE scaling of epitaxial FIM \nMn 1.5Ga [28] and some other 3 d ferromagnets. Hou et al. [29] \nalso proposed a multivariable scaling relation for the AHE in \nmagnetic materials in very high -conductivity  regime by \nassuming two major competing scattering sources: i.e. \n  ρAH = αρxx0 + β0 ρxx02 + γ0 ρxx0 ρxxT + β1ρxxT2,            (4)                                             \nwhere ρxx0 is also the residual resistivity  and ρxxT = ρxx-ρxx0 is \nresistivity due to dynamic phonon scattering at high \ntemperatures , α, β0, γ0, and  β1 are fitting parameters. Equation \n(4) describes well the AHE scaling in epitaxial ferromagnetic \nFe [27] grown by molecular -beam epitaxy. For convenience, \nwe rewrite Equation (4)  as  \nρAH = αρxx0 + (γ 0-2 β1) ρxx0ρxx +( β0+ β1-γ0) ρxx02 + β1ρxx 2, (5)                                                  \nWe note that Eq. (3) -Eq. ( 5) predict that ρAH is a monotonic \nfunction of ρxx and scales smoothly to zero at zero ρxx (ρxx0 = \nρxxT=0). In Fig. 5(a), we plot the values of ρAH for the FIM FeTb \nas a function of temperature. While the anomalous Hall \nresistivities of the Fe -dominated and Tb -dominated films are \nof opposite signs, the magnitude increases monotonically, by \n50%, for each sample. A similar  increase of ρAH with \ntemperature has also been reported in  ferromagnetic MnAl \nwith orbital two -channel Kondo effect  [51] and is distinct \nfrom that of FMs (e.g., Fe  [27], Co [52], Ni [53], and FePt  \n[54]) and FIMs (MnGa  [28]) in which the electron scattering \nis dominated by impurity scattering and p honon scattering.  \nMore surprisingly, ρAH of the FeTb is not a linear function of \nρxx2 and even  does not have  an obvious monotonic scaling \ntowards zero as ρxx decreases.   This observation suggest s the \nbreakdown of the conventional AHE scaling  for the dirty \nmetal of amorphous FeTb ferrimagnets . This breakdown is \nnot a general case for FIMs as the Mn 1.5Ga with AF -coupled \nMn sublattices does follow Eq. (3)  (see Fig. 4(b) ). \nWe show i n Fig. 5(c) that the anomalous Hall resistivity  \nphenomenally foll ows the law  \n ρAH = α +βρxx + γρxx2,                                               ( 6) \nwhere α, β, and  γ are non-zero constant s. Equation  (6) \npredicts a peak and decay in ρAH at very high resistivities \nwhich might be consistent with the expectation that ρAH \nshould reduce to wards  zero in the limit of infinite ρxx \n(insulators). Note that we have tested that the data in Fig. 5(c) \ncannot be fit by a monotonically varying exponential, \nlogarithm, hyperbola, and othe r functions. The underlying \nphysics  and the precise application regime  of the new scaling , \nEq. ( 6), require theoretical  and experimental investigations in \nthe future and is beyond the scope of this work.   \nFinally, we mention that the anomalous Hall resistivity \nof the FeTb is giant compared to that of the 3 d magnets Fe \n[27], Co [52], Ni [53], Co40Fe40B20 [55], Mn 1.5Ga [28], MnAl \n6 \n [51], and Mn 3Ge [56] (Fig. 6(a)) due to the large anomalous \nHall angle ( ρAH/ρxx, see Fig. 6(b)) and the high resistivity. As \nplotted in Fig. 6(c), the anomalous Hall conductivity of the \nFeTb is also stronger than that of MnAl, MnGa, Mn 3Ge, and \nCoFeB  with significantly higher longitudinal conductivities. \nSuch giant anomalous Hall effect is highly preferred for \nsensor applications .  \n \nFig. 6. Dependence on the longitudinal resistivity  of (a) the \nanomalous Hall resistivity  and (b)  the anomalous Hall angle \nof representative magnetic films . (c) The anomalous Hall \nconductivity of the same materials plotted as a function of the \nlongitudinal conductivity.  \n \nConclusion  \nWe have presented a systematic study of the magnetic and \ntransport prop erties of the ferrimagnetic FeTb alloy by \nvarying the layer thickness and temperature. The FeTb is \ntuned from the Tb -dominated regime to the Fe -dominated \nregime simply via the increase of the layer thickness , without \nvarying the composition . For each of th e studied FeTb \nsamples, the coercivity  closely follows the 1/cos θH scaling \n(where θH is the polar angle of the external magnetic field ) \nand increases quasi -exponentially upon cooling and exceeds \n90 kOe  below a certain low temperature, revealing that the \nnature of the coercivity is thermally -assisted domain wall \ndepinning field. The resistivity increases quasi -linearly with \ntemperature  upon cooling  likely due to thermal fluctuations of the structure fact or of the amorphous FeTb . The anomalous \nHall resistivities of both Fe - or Tb -dominated FeTb layers \nthat are in the dirty limit cannot be described by any of the \nexisting AHE scaling laws proposed in the literature. These \nexotic findings should advance the understanding of the \nmagnetic and transport behaviors of the transition -metal -rare-\nearth ferrimagnets.  \n  \nThe authors thank Changmin Xiong for help with PPMS \nmeasurements. 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Piresa†\naDepartment of Physics, Federal University of Minas Gerais, Belo Horizonte, MG, Brazil.\nAbstract\nThe copper fluoride Cu2F5is a proposed stable compound that can be seen as a layered mag netic lattice of\nS= 1 andS= 1/2 sites, corresponding to copper ions. Intending to cast lig ht on the transport properties of\nferrimagnetic magnons, we use the linear spin wave approach to study the magnon band structure of the 2D\nlattice in a ferrimagnetic off-plane order, as well as the tra nsverse transport of magnons in the crystal bulk.\nThat transverse (Hall-like) transport can be induced by a ma gnetic field or temperature gradient, and within\nthe linear response theory is generated by the Berry curvatu re of the eigenstates. As in most of the cases\nfor magnons, the Berry curvature here is related to Dzyalosh inskii-Moriya interactions between next-near-\nneighbors. The band structure of the system is non-degenera te and the transport coefficients are non-null. We\nalso determine the condition for two transport coefficients t o change sign in response to temperature.\n1 Introduction\nWhen a crystal has magnetic order, perturbations in this\norder propagate through the crystal in the form of spin\nwaves. From a quasiparticle point of view, these spin waves\nare called magnons , which are chargeless bosons with def-\ninite spin, momentum and energy. In the past years there\nhas been a great effort to study the creation and manipula-\ntionof magnons [1–3], as well as their interaction with othe r\nparticles or quasiparticles [4–8] including the formation of\nhybrid modes [9–16]. These efforts rely on the exciting\npossibility of using the spin degree of freedom of magnons\nto transport information in a field known as magnonics\n(magnon spintronics) [17–19]. Magnonics has a significant\nadvantage over electron-based spintronics: as magnons are\nnaturally chargeless, they do not present Joule heating and\ncan propagate large distances with low dissipation.\nWithin this context it became essential to study the\ntransport properties of magnons in different lattices and\nmagnetic orders. It is theoretically established that a mag -\nnetic field or temperature gradient can generate longitudi-\nnal and transverse (Hall-like) transport of magnons in the\nbulk of a crystal [20–24]. These transport effects have been\nintensely studied in ferromagnets (FM) [21–41] and antifer -\nromagnets (AFM) [42–68]. The theoretical study of these\nthermomagnetic properties is well developed, andnowadays\nwe can describe magnonic analogs for the quantum Hall\neffectandquantum spin Hall effect of electrons [24,48],\namong other exotic transport effects. Experimental evi-\ndence have demonstrated the existence of transverse trans-\nport in the bulk of 3D and 2D lattices [69–73].\nIt has also been established that magnonic systems canshow protected edge or surface states, which are robust\nagainst non-magnetic impurities. That comes from the\nwell-known bulk-edge correspondence for topological sys-\ntems, which indicates the existence of topological magnon\ninsulators (TMI) [19,27,33,35,74]. That term has been\nused as an analogy toelectronic topological insulators. Ju st\nlike electronic topological insulators, the TMI are charac -\nterized by topological indices like the Chern number or Z2\ninvariant [24,48,75,76]. Despite that, the bosonic nature\nof magnons excludes the existence of a Fermi level, so the\nsystem is not an insulator strictu sensu . Topological and\nHall-like effects of magnons are related to the spin-orbit\ncoupling of the crystal lattice, and both rely on the Berry\ncurvature of the system, which acts like a fictitious mag-\nnetic field and imparts helical movement to the magnons.\nAlthough widely studied in FM and AFM systems, the\nHall transport of magnons has not been well investigated in\nferrimagnetic (FiM) lattices [77,78]. With that in mind, in\nthis work we investigate the magnon Hall transport in the\n2D layers of the proposed copper fluoride complex Cu2F5,\nwhich stability was predicted recently with first-principl e\nmethods [79–81]. In this crystal, copper ions have two dif-\nferent spin states ( S= 1 ands= 1/2), forming a ferri-\nmagnetic system that we investigate using the spin wave\napproach.\nThis paperisorganized asfollows. Wepresentthecrystal\nstructure and the Hamiltonian which models the 2D mag-\nnetic lattice in Section 2. In Section 3 we discuss the Berry\ncurvature of magnon bands and its implications to the\ntransverse transport of magnons. In Section 4 we present\nthe results of the systems’s band structure and transverse\n1transport coefficients, and in Section 5 we make our final\nremarks.\n2 Model\nThe proposed Cu2F5crystal is composed of CuF6distorted\noctahedra and CuF4plaquettes as shown in Figure 1 [79,\n80]. The inequivalent Cuions form a magnetic crystal. We\ncallCu1 theS= 1 ions in the center of the octahedra,\nandCu2 thes= 1/2 ions in the center of the plaquette.\nThe most energetically favorable configuration is a G-type\nferrimagnet, and DFT+U calculations show that the 3D\ncrystal can beseen as a layeredstructure withthe interlaye r\nexchange parameter five times smaller than the intralayer\nones [80]. That inspires us to study the 2D layers from a\nspin wave point of view. We chose a ferrimagnetic off-plane\nspin configuration, with the S= 1 spins pointing in the + ˆ z\ndirection and the s= 1/2 in the −ˆ zdirection. The stacked\nlayers form a ferrimagnetic C-type configuration, and the\nmagnetic lattice has two inequivalent sites. That is not the\nmost stable configuration, but it is much simpler than the\naforementioned G-type FiM, which has four inequivalent\nsites.\nFigure 1: The crystal structure of the proposed Cu2F5lat-\ntice.Cuions inside the blue octahdera ( Cu1) have spin\nS= 1.Cuions inside the magenta plaquettes ( Cu2) have\nspins= 1/2. Reproduced with permission from Ref. [80].\nThe exchange model which stabilizes the spin order of\nthe layer comprises a FM exchange bond between Cu1\nsites and an AFM exchange bond between Cu1−Cu2\nsites. We add a Dzyaloshinskii-Moriya interaction (DMI)\nbetween next-near-neighbors (NNN) sites and a single-ion\nanisotropy (SIA) in the z-direction. The DMI is responsi-\nble for the transverse transport, and the SIA stabilizes the\noff-plane configuration. The Hamiltonian of the model is:H=−J1/summationdisplay\n/angbracketleftij/angbracketrightSi·Sj+J2/summationdisplay\n/angbracketleftij/angbracketrightSi·sj\n+D/summationdisplay\n/angbracketleft/angbracketleftij/angbracketright/angbracketrightνij/parenleftbig\nSx\nisy\nj−Sy\nisx\nj/parenrightbig\n−A/summationdisplay\ni/bracketleftBig\n(Sz\ni)2+(sz\ni)2/bracketrightBig\n(1)\nc (x)b (y)J2J2J1\nFigure 2: 2D model studied in this paper, corresponding to\na layer in the b-c plane of the crystal structure in Figure 1.\nExchange interactions are represented as J1andJ2. There\nis a Dzyaloshinskii-Moriya interaction between NNN with\nνij= +1(−1) along (against) the arrow. Blue sites have\nspinS= 1, and magenta sites, s= 1/2. The gray region\ncorresponds to the unit cell.\nThe upper (lower) case Si(si) operator denotes the spin\noperators for S= 1 (s= 1/2) sites. The first term ( J1>0)\nrepresents the FM exchange between S= 1 (Cu1) sites,\nand the second term ( J2>0) represents the AFM exchange\nbetweenS= 1 ands= 1/2 (Cu1−Cu2) sites. Both in-\nteractions happen between near-neighbors (NN). The third\nterm is the DMI between NNN sites, where νij=±1 fol-\nlowing the arrow convention in Figure 2. The last term is\nthe SIA. We note that the SIA between 1/2 spins is ineffec-\ntive [68], so only the first term inside the square brackets\nneeds to be considered.\nWe use the linearized Holstein-Primakoff representation\nfor up/down spins:\nS+\ni=√\n2Sai, S−\ni=√\n2Sa†\ni, Sz\ni=S−a†\niai\ns+\ni=√\n2sb†\nj, s−\ni=√\n2sbj, sz\ni=−s+b†\njbj(2)\nso the Hamiltonian is written in terms of the magnon\ncreation and annihilation operators in configuration space .\nThis representation can be applied to collinear AFM and\nFiM with off-plane spins. Performing a Fourier transform\n2XX'M\nkx ky\u0001(k)\u0001(k)(a) (b)\nFigure 3: (a) Band structure of the system when J1=J2= 1.0,A= 0.1, andD= 0.2. The blue (red) line corresponds\nto theω↓(↑)(k) band. (b) 3D band structure.\nenables us to write the momentum-space 4x4 Hamiltonian\nin the block matrix form:\nHk=ψ†\nk/parenleftbiggHI\nk0\n0HII\nk/parenrightbigg\nψk (3)\nwithHII\nk=/bracketleftbig\nHI\nk/bracketrightbig∗. The basis is ψ†\nk=/parenleftBig\na†\nk,b−k,a−k,b†\nk/parenrightBig\n.\nThe first block of the Hamiltonian matrix is:\nHI\nk=/parenleftbiggJ1S(1−γk)+J2s+A˜S√\nsS(J2ηk−2iDmk)√\nsS(J2ηk+2iDmk) J2S/parenrightbigg\n≡/parenleftbiggr(k)+∆(k)f∗(k)\nf(k)r(k)−∆(k)/parenrightbigg\n(4)\nwith˜S≡(2S−1)/2 = 1/2 and structure fac-\ntorsγk=cos(kx/2),ηk=cos(ky/2), andmk=\n−sin(kx/2)sin(ky/2).\nThe Hilbert space was doubled in this procedure, so it\ncarries not only the magnon (pseudo-) particle states but\nalso non-physical hole states. To diagonalize Hamiltonian\n(1) we use a Bogoliubov transformation, where the two\nphysical solutions have eigenvalues:\nω↑↓(k) =/radicalBig\nr(k)2−|f(k)|2∓∆(k) (5)\nThat is the band structure of the system. The up/down\nmagnons carry magnetic dipole momentum σgσµB, with\nσ=±1. For AFM systems, the g-factor gσis the same for\nboth magnon species, so it is usually absorbed into another\nconstant. For ferrimagnets, however, usually g↑/negationslash=g↓due\nto the inequivalence of the spins in the two sublattices, and\nit is essential to carry the g-factors in the expressions [78 ].\nWe must remember that magnons are bosons, so\nthere is no Fermi level. The thermal population nλ(k)\nis given by the Bose-Einstein distribution: nλ(k) =/parenleftBig\ne/planckover2pi1ωλ(k)/kBT−1/parenrightBig−1\n. Both bands are always populated.\nEven if a gap exists between the bands, the system is not a\ntrueinsulator (thisterm canbeapplied onlyas an analogy).3 Berry curvature and transverse\ntransport\nThe Berry curvature of the λ-bandΩλ(k) is a property\nof the energy band responsible (among other things) for\ntransversal transport and topological effects. It can be ob-\ntained from the eigenstates uλ(k) of the Hamiltonian [82]:\nΩλ(k) =i/angb∇acketleft∇kuλ(k)|×|∇ kuλ(k)/angb∇acket∇ight (6)\nThe Berry curvature acts like a fictitious magnetic field\non the magnons and is related to the geometrical Berry\nphaseaccumulatedbythegroundstateeigenfunctionswhen\nevolving in the Brillouin zone. In the case of a two-band\nAFM/FiM Hamiltonian, both bands have the same Berry\ncurvature, whose off-plane component can be obtained an-\nalytically from\nΩ↑↓(k) =−1\n2sinhθk/parenleftbigg∂φk\n∂kx∂θk\n∂ky−∂φk\n∂ky∂θk\n∂kx/parenrightbigg\n(7)\nwhereθkandφkare parameters which can be written in\nterms ofr(k), ∆(k) andf(k) [41].\nWhen an external in-plane field gradient is applied to\nsome systems, it is possible to observe magnon transport\nin a direction transverse to the field gradient. This phe-\nnomenon can be generically called the Hall-like transport\nof magnons . When the external perturbation is a magnetic\nfield gradient we observe a transverse spin current given\nby [20,23]:\njS,B\ny=σxy(−∂xB) (8)\nwhereσxyis the spin Hall conductivity, and this phe-\nnomenon is called the spin Hall effect of magnons.\nWhen we apply a temperature gradient, we observe both\nspin and thermal transverse currents, given respectively b y\n[21–23]:\njS,T\ny=αxy(−∂xT) (9)\njQ,T\ny=κxy(−∂xT) (10)\n3These are, respectively, the spin Nernst effect and the\nthermal Hall effect ofmagnons. The coefficient αxyis called\nthe spin Nernst coefficient, and κxyis the thermal Hall con-\nductivity. Within the linear response theory, the transpor t\ncoefficients for each band are given by integrals (in the con-\ntinuum limit) that involve the Berry curvature [48,78]:\n[σxy]↑↓=−(g↑↓µB)2\n/planckover2pi1VBZ/integraldisplay\nBZd2k n↑↓(k)Ω↑↓(k) (11)\n[αxy]↑↓=−(g↑↓µB)kB\n/planckover2pi1VBZ/integraldisplay\nBZd2k c1[n↑↓(k)]Ω↑↓(k) (12)\n[κxy]↑↓=−k2\nBT\n/planckover2pi1VBZ/integraldisplay\nBZd2k c2[n↑↓(k)]Ω↑↓(k) (13)\nHere,n(k) is the thermal population of the band given\nby the Bose-Einstein distribution, and the functions c1and\nc2are defined as\nc1(x) = (1+x)ln(1+x)−xln(x) (14)\nc2(x) = (1+x)/bracketleftbigg\nln/parenleftbigg1+x\nx/parenrightbigg/bracketrightbigg2\n−(lnx)2−2Li2(−x).\n(15)\nwhereLi2(x) is Spence’s dilogarithm function.\nIt is important to remark that in Refs. [48] and [78] the\nauthors write the transport coefficients without the explici t\nintegrals and in terms of the Chern number of the band. In\nthose references it is assumed that the bands are almost flat\nand the functions of n(k) are factored out of the integrals\nin Eqs. (11)-(13), so the coefficients become proportional\nto the Chern number C=/integraltext\nd2kΩ/2π. We do not make\nthe flat band approximation here, so the functions of n(k)\nare not factored out.\n-0.04-0.0200.02\nkxky\n0 2\u0001 -2\u0001-\u0001\u0001\n0\nFigure 4: Berry curvature of both bands in the Brillouin\nzone. Same parameters as figures above.\n\u0001xy\u0002\u0003\u0004\u0005\u0006\u0007\u0002ℏ\u0006\u0004\b\t\n\u000b\n ℏT/J1 D = 3.0\n D = 2.0\n D = 1.0\n\fxy (10-3 ℏ-1kB\bB)\n\u0002ℏT/J1 D = 3.0\n D = 2.0\n D = 1.0\n\rxy (10-3 kB2\n ℏ-1)\n ℏT/J1 D = 3.0\n D = 2.0\nD = 1.0(a)\n(b)\n(c)\nFigure 5: (a) Spin Hall conductivity σxy, (b) thermal Hall\nconductivity and (c) spin Nernst coefficient as functions\nofT/J1. The parameters are J1=J2= 1.0,A= 0.1,\ng↓= 1.0,g↑= 1.2 and three values of D.\nThetransverse currentsofbothmagnons combinetogen-\nerate the total conductivities of the system:\nσxy= [σxy]↓+[σxy]↑ (16)\nαxy= [αxy]↓−[αxy]↑ (17)\nκxy= [κxy]↓+[κxy]↑ (18)\nThe difference in sign between αxyandκxycan be un-\nderstood as follows. If a perturbation drives both magnons\nin the same direction, the thermal current is additive while\n4the spin current is subtractive, and vice-versa if the pertu r-\nbation drives the magnons in opposite directions (which\nis the case of a thermal gradient in the system studied\nhere). Furthermore, in degenerate systems both currents\nhave the same magnitude, and we can observe a pure spin\ncurrent without thermal current ( pure spin Nernst effect\nof magnons ) [45,46,48,68]. These systems are Z2topo-\nlogical magnon insulators, protected by an effective time-\nreversal symmetry. They are analogs of the quantum spin\nHall states of electrons [83].\n4 Results\nThe band structure of the system can be seen in Figure 3.\nFor thetypical values of A<J ithebands cross in twosepa-\nrate paths in the Brillouin zone. The Berry curvature of the\nsystem is represented in Figure 4. The symmetrical shape\nof the Berry curvature show us that its integration in the\nBrillouin zone is zero, so the system is not a Chern insula-\ntor (C= 0). We can see that the points of higher (positive)\nBerry curvature are located in the Γ −Xpath, in regions\nwhereω↓dominates. Onthe other hand, thepoints of lower\n(negative) Berry curvature lay in the Γ −X′path, which\ncorresponds to points where ω↑dominates. The result is\nthat, from Eqs. (11)-(13), the transport coefficients are\npositive for the ↓-band and negative for the ↑-band. We see\nthat the total spin Hall conductivity and thermal Hall con-\nductivity (Eqs. (16) and (18)) become subtractive, while\nthe spin Nernst coefficient (Eq. (17)) becomes additive.\nThat is in accordance with the literature, which states that\na magnetic field gradient drives both species of magnons\nin the same transverse direction so the spin currents com-\npete with each other. Also, a temperature gradient drives\nthe magnons in opposite directions, so the spin currents\nare reinforced, and the thermal current is subtractive [48] .\nFor some collinear antiferromagnetic systems ( Z2topologi-\ncal magnon insulators), an effective time-reversal symmetr y\nresults in degenerate bands and we can have transverse spin\ncurrent without heat current, realizing a magnonic version\nof the quantum Hall effect [45,46,48,68]. In a ferrimag-\nnet, such symmetry does not exist and the degeneracy is\nbroken [77,78,84]. We have non-null and tunable heat and\nspin currents.\nThe transport coefficients versus temperature are plotted\nin Figure 5. We see the well-known asymptotical behavior\nofαxy(T) andκxy(T). The coefficients increase with D,\nand vanish when D= 0 (as well as the Berry curvature),\nmeaning that the Dzyaloshinkii-Moriya interaction is re-\nsponsible for the transverse transport. All three coefficien ts\nare in general non-null, as expected from a non-degenerate\nsystem.\nForA > J iit is possible to observe a sign change in\ntheσxyandκxy(subtractive) coefficients, while it does not\noccur forαxy(Fig. 6). That can be explained as follows.\nWhenA>J i, the↑-band is always lower than the ↓-band\n(the bands are gapped, Figure 7). We know that in the\n\u0001xy\u0002\u0003\u0004\u0005\u0006\u0007\u0002ℏ\u0006\u0004\b\t\n\u000b\n ℏT/J1\n ℏT/J1\fxy (10-9 kB2\n ℏ-1)\n ℏT/J1\rxy (10-9 ℏ-1kB\bB)(a)    \n(b)    \n(c)    \nFigure 6: (a) Spin Hall conductivity σxy, (b) thermal Hall\nconductivity and (c) spin Nernst coefficient as functions\nofT/J1. The parameters are J1=J2= 1.0,D= 0.3,\nA= 1.1,g↓= 1.0, andg↑= 1.2. Note that the first two\ncoefficients changes for low temperatures, which is a result\nof the competing conductivities of each individual band.\nlow-temperature limit, the lower band dominates, so the\n↑-band (which has negative transport coefficients) is more\npopulated. That results in negative total transport coeffi-\ncients. As the temperature rises, the population in the top\n5band surpasses the lower band’s, and the total transport\ncoefficients become positive. That behavior does not occur\nforA< J i, as↓-band is the lower one, and its population\ndominates in all temperatures. The change of sign of the\ntransport coefficients, which happens also in other magnon\nsystems, opens the exciting possibility of controlling the\ndirection of magnon flow with the temperature.\n\u0001(k)\u0002XX'M\nFigure 7: Band structure for J1=J2= 1.0,D= 0.3, and\nA= 1.1. ForA>J 1=J2, a gap appears.\n5 Conclusion\nWe have studied a 2D magnetic lattice which describes a\nlayer of the Cu2F5crystal [79–81] using the spin wave ap-\nproach. The magnetic order is ferrimagnetic, with spin-up\nsites (S= 1) pointing in the off-plane direction opposite to\nspin-down ( s= 1/2). The magnon bands are not gener-\nate, as expected for ferrimagnets and in contrast to AFM\ncollinear order, where an effective time-reversal symmetry\nmakes the bands degenerate. A non-null Berry curvature\nis generated by the Dzyaloshinskii-Moriya interaction be-\ntween next-near neighbors. That enables the system to\nshow transverse transport effects. 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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": "1506.02532v2.Geometric__electronic_and_magnetic_structure_of_Fe___x__O___y_______clusters.pdf",        "content": "arXiv:1506.02532v2  [physics.atm-clus]  20 Nov 2015Geometric, electronic, and magnetic structure of FexO+\nyclusters\nR. Logemann, G.A. de Wijs, M.I. Katsnelson, and A. Kirilyuk\nRadboud University, Institute for Molecules and Materials , NL-6525 AJ Nijmegen, The Netherlands\n(Dated: August 5, 2021)\nCorrelation between geometry, electronic structure and ma gnetism of solids is both intriguing and\nelusive. This is particularly strongly manifested in small clusters, where a vast number of unusual\nstructures appear. Here, we employ density functional theo ry in combination with a genetic search\nalgorithm, GGA+ Uand a hybrid functional to determine the structure of gas pha se FexO+/0\nyclus-\nters. For Fe xO+\nycation clusters we also calculate the corresponding vibrat ion spectra and compare\nthem with experiments. We successfully identify Fe 3O+\n4, Fe4O+\n5, Fe4O+\n6, Fe5O+\n7and propose struc-\ntures for Fe 6O+\n8. Within the triangular geometric structure of Fe 3O+\n4a non-collinear, ferrimagnetic\nand ferromagnetic state are comparable in energy. Fe 4O+\n5and Fe 4O+\n6are ferrimagnetic with a resid-\nual magnetic moment of 1 µBdue to ionization. Fe 5O+\n7is ferrimagnetic due to the odd number of\nFe atoms. We compare the electronic structure with bulk magn etite and find Fe 4O+\n5, Fe4O+\n6, Fe6O+\n8\nto be mixed valence clusters. In contrast, in Fe 3O+\n4and Fe 5O+\n7, all Fe are found to be trivalent.\nPACS numbers: 36.40.Cg, 36.40.Mr, 61.46.Bc, 73.22.-f\nIn nano technology there is an ever increasing demand\nfor increasing the density of electronic and magnetic de-\nvices. This continuous downscaling trend drives the in-\nteresttoelectronicandmagneticstructuresatthe atomic\nscale. In essence, two things are required: first, novel\nmaterials and building blocks with exotic physical prop-\nerties. Second, a fundamental knowledge of the physical\nmechanism of magnetism at the sub-nanometer scale.\nAtomicclusters,havinghighlynon-monotonousbehav-\nior as a function of size, are a promising model system to\nstudy the fundamentals of magnetism at the nanoscale\nand below. Such clusters consist of only tens of atoms.\nQuantum mechanics starts to play an essential role at\nthis small scale, adding extra degrees of freedom. Since\nthese clusters are studied in high vacuum, they are com-\npletely isolated from their environment.\nTo use these clusters as a model system, as a starting\npoint, a detailed understanding of the relation between\ntheir geometry and electronic structure is required.\nEvenin thebulk, ironoxidehasawidevarietyofchem-\nical compositions and phases with many interesting phe-\nnomena, such as the Verwey transition in magnetite.1,2\nExperimentsperformedonsmallgasphaseFe xOyclus-\nters beyond the two-atom case are scarce. The structure\nof one and two Fe atoms with oxygen has been studied\nin an argon matrix using infrared spectra.3,4The corre-\nsponding vibration frequencies have been identified using\ndensity functional theory (DFT).\nIron-oxide nanoparticles have been investigated for\ntheirpotentialuseascatalystinchemicalreactions.5Fur-\nthermore, since the iron-oxygen interaction has a funda-\nmental role in many chemical and biological processes,\nthere have been quite some studies, both experimen-\ntal and theoretical, of the chemical properties of Fe xOy\nclusters.6–12\nThe possible coexistence of two structural isomers for\nstoichiometric iron-oxide clusters in the size range n≥5\nwas experimentally measured using isomer separation by\nion mobility mass spectroscopy for FenOnand FenOn+1(n= 2-9).13Furthermore, the formation of Fe xOyclus-\nters has been studied in the size range ( x= 1-52).14\nThe number of theoretical studies is, however, man-\nifold. The magic cluster Fe 13O8was extensively stud-\nied and identified as a cluster with C1but close to D4h\npoint group symmetry.15–19However, also the geometry\nand electronic structure of other cluster sizes have been\nstudied theoretically.15,20–25The prediction of geometric\nstructures requires a systematic search of the potential\nenergy surface to find the global minimum.\nThe majority of theoretical studies were performed\nusing DFT.4,6,9,10,13–17,20–24,26The number of works in\nwhichFe mOnclusterswerestudied with methods beyond\nDFT is very limited and restricted to very small cluster\nsizes. For FeO+its reactivity towards H2was studied\non a wave-function-based CASPT2D level.12For Fe2O2\nthemolecularandelectronicstructurewerecalculatedus-\ning both DFT and wave-function-based CCSD(T) meth-\nods and a7B2uground state was found.25Furthermore,\nRef. 25 reports that B3LYP functional and CCSD(T)\ncalculations give the same energy ordering of different\nstates, although the energy differences are overestimated\nby the B3LYP approach.\nRecently, the structural evolution of (Fe 2O3)n\nnanoparticles was systematically investigated from the\nFe2O3cluster towards nano particles with n= 1328.9,26\nIn the size range of n= 1-10, an interatomic potential\nwas developed and combined with a genetic algorithm in\nsearchofthe lowest-energyisomer. The isomerslowest in\nenergy were further optimized using DFT and the hybrid\nfunctional B3LYP. This way, a systematic prediction of\nthe cluster structure was done for neutral (Fe 2O3)nclus-\nters.\nBecause of its high computational burden, in DFT the\ngeometric structure is often only relaxed into its nearest\nlocal minimum on the potential energy surface (PES).\nThere is no guarantee that this local minimum corre-\nsponds to the global minimum. Almost all previous2\nworksonlyconsidereitherrandomstructuresormanually\nconstructed geometries. However, for increasing cluster\nsize these methods become less successful in finding the\nlowest-energy isomer. Genetic algorithms, in which sta-\nble geometries are used to create new structures, proved\nto be efficient in finding the global energy minimum.27\nThis method has been successfully used for transition-\nmetal oxide clusters.28,29\nIdentification of the geometric cluster structure is a\ndelicate and computationally demanding task. There-\nfore,comparisonwithanexperimentalmethodtoconfirm\nthe theoreticalfindingsis essential. In this work, wecom-\nbine previously reported experimental vibration spec-\ntra30with first-principles calculations and a genetic al-\ngorithm to determine the geometric structure of cationic\nFexO+\nyclusters. Of the nine cluster sizes reported in\nRef. 30, only the geometric structure of Fe 4O+\n6was iden-\ntified. In this work, we will also identify the geomet-\nric, electronic, and magnetic structure of Fe 3O+\n4, Fe4O+\n5,\nFe5O+\n7and propose structures for Fe 6O+\n8.\nI. COMPUTATIONAL DETAILS\nWe employ a genetic algorithm (GA) as is described in\nRef. 27 in combination with DFT to optimize the cluster\nstructures. For this we use the Vienna ab-initio simu-\nlation package ( vasp)31using the projector augmented\nwave (PAW) method.32,33Since the geometry optimiza-\ntion is the most computationally expensive part of the\ngenetic algorithm, we use the PBE+ Umethod34with\nlimited accuracy for the genetic algorithm. For all ob-\ntained isomers low in energy, we reoptimized the geo-\nmetric structure using the hybrid B3LYP functional with\nhigher accuracy and consider different magnetic config-\nurations. We then calculate the vibration spectra and\ncompare them with experimental results.\nWithin the DFT framework, functionals based on the\nlocal density approximation (LDA) or general gradient\napproximation (GGA) fail to describe strongly interact-\ning systems such as transition-metal oxides.35,36Due to\nthe overestimation of the electron self-interaction, they\npredict metallic behavior instead of the (correct) wide-\nband-gap insulator. In an attempt to correct for this\nself-interaction, one can, for example, employ a hybrid\nfunctional, where a typical amount of 20% of Hartree-\nFockenergyisincorporatedintotheexchange-correlation\nfunctional. Especially for the B3LYP functional it has\nbeen shown that this results in good agreement be-\ntween the geometric structure and vibrational spectra\nfor clusters.28,30,37However, hybrid functionals are quite\ncomputationally expensive compared to LDA and GGA\nfunctionals. Therefore, in the genetic algorithm we em-\nploy the GGA+ Umethod to take into account that FeO\nclusters are strongly interacting systems. We use the\nrotational invariant implementation introduced by Du-\ndarev and a plane wave cutoff energy of 300 eV for these\ncalculations.38The differences between GGA and GGA+ Ufor iron-\noxide cluster calculations have been analyzed in Ref. 15.\nThisstudystressestheimportancetogobeyondGGAfor\ntransition-metal oxide clusters calculations. Aside from\nthe well-known difference for the electronic and magnetic\nstructure, it even finds a different lowest energy isomer\nthan GGA for Fe 32O33. In our genetic algorithm cal-\nculations we use an Ueff=U−Jof 3 eV for the Fe\natoms, based on a comparison between B3LYP calcula-\ntions and PBE+ Ucalculations for the smallest cluster,\nFe3O4(see Sec. IIB). For this comparison we also cal-\nculated the mean absolute difference (∆) between the\noccupied Kohn-Sham energies ( Ei) using B3LYP and\nPBE+U:\n∆ =n/summationdisplay\ni=1|EPBE+U\ni−EB3LYP\ni|\nn, (1)\nwherenis the number of occupied Kohn-Sham levels.\nNote that, the binding distances are only weakly depen-\ndent on the used Ueffand our value of 3 eV is close to val-\nues used in other works (e.g., 5 eV15, 3.6 eV20, 3.6 eV39).\nWe used the genetic algorithm as described in detail\nin Ref. 27. New geometries are formed by the Deaven-\nHo cut and splice crossover operation. To determine the\nfitness we used an exponential function. A generation\ntypically consists of 20 clusters. It has been shown that\nthe geometry of Fe xOyclusters only weakly depends on\nthe magnetic degree of freedom.26Therefore, we restrict\nourselves to the ferromagnetic case in our genetic algo-\nrithm.\nFor all obtained isomers low in energy, we reopti-\nmized the geometric structure using the hybrid B3LYP\nfunctional40,53and consider all possible collinear ori-\nentations of the Fe magnetic moments by constraining\nthe difference in majority and minority electrons. All\nforces were minimized below 10−3eV/˚A. Standard rec-\nommended PAWs with an energy cutoff of 400.0 eV are\nused. The clusters are placed in a periodic box of a\nsize between 11 and 17 ˚A, which we checked to be suf-\nficiently large to eliminate inter cluster interactions for\neach cluster size. For the cluster calculations, a single\nk-point (Γ) is used. Since we also consider cationic clus-\nters, a positive uniform background charge is added and\nwe correct the leading errors in the potential.41,42All\nsimulations were performed without any symmetry con-\nstraints. The reported symmetry groups are determined\nafterwardswithin 0.03 ˚A. Forthe density ofstates(DOS)\ncalculations we used a Gaussian smearing of 0.1 eV for\nvisual clarity.\nTo obtain the vibration spectra, the Hessian matrix\nof an optimized geometry is calculated by considering\nfinite ionic displacements of 0.015 ˚A for all Cartesian co-\nordinates of each atom. The vibration frequencies are\nobtained by diagonalization of the Hessian matrix. The\nabsorption intensity Aiis calculated using43,44\nAi= 974.86gi/parenleftbigg∂µ\n∂Qi/parenrightbigg\n, (2)3\nwheregiis the degeneracy of the vibration mode, Qi\nthe mass weighted vibrational mode, µthe electric dipole\nmoment, and974.86anempiricalfactor. Amethodbased\non four displacements for each ion was also tested but\nyielded the same frequencies and absorption intensities.\nZero-point vibrational energies (ZPVE) were calculated\nfor the isomers lowest in energy of which the vibration\nspectra are also shown.\nFor a quantitative comparison between experimen-\ntal and calculated vibrational spectra, we calculate the\nPendry’s reliability factor.45The Pendry’s reliability fac-\ntor is a well-established method in low-energy electron\ndiffraction (LEED) to quantify the agreement in contin-\nuous spectra and has also been applied to vibrational\nspectroscopy.46\nThe experimental used infrared multiphoton dissoci-\nation method (IR-MPD) does not only depend on the\nabsorption cross section of a vibrational mode, but also\non the dissociation cross section. Therefore, we use the\nPendry’s reliability factor to quantify the comparison of\nvibration spectrasince it is mainly sensitive to peak posi-\ntions opposed to a comparison of squared intensity. This\npeak sensitivity is achieved by comparing the renormal-\nized logarithmic derivative of the intensity I(ω):\nY(ω) =L−1(ω)\nL−2(ω)+W2, (3)\nwhereL(ω) =I′(ω)/I(ω) andWis the typicalFWHM of\nthe peaks in the spectra. The Pendry’s reliability factor\nis defined as:\nRP=/integraldisplay/bracketleftbig\nYth(ω)−Yexpt(ω)/bracketrightbig2\nY2\nth(ω)+Y2\nexpt(ω)dω, (4)\nwhere we integrate over the experimental range of fre-\nquencies. RPvalues range from 0 to 2, where 0 means\nperfect agreement, 1 uncorrelated spectra, and 2 per-\nfect anticorrelation. In practice, RPvalues of 0.3 are\nconsidered acceptable agreement within LEED. Y(ω) is\nstrongly dependent on experimental noise and values\nclose to zero, hence, we calculate Yexpt(ω) by fitting the\nexperimental spectrum with multiple Lorentzian peaks\nand extract the corresponding W. The theoretical fre-\nquencies are also convoluted with Lorentzian peaks with\nthe same W.RPis always minimized as function of a\nrigid shift of all theoretical frequencies.\nFor the calculations on magnetite we used the vasp\ncode. We used a Monkhorst grid of 6 ×6×2 and an\nenergy cutoff of 400 eV. We used the rotationally in-\nvariantLSDA+ UimplementationbyLichtenstein et al.47\nwith effective on-site Coulomb and exchange parameters:\nU= 4.5 eV48andJ= 0.89 eV for the Fe ions.\nWe used the monoclinic structure as described in\nRefs. 39,49, and calculated the electron density with 56\natoms in the unit cell. In Ref. 39, the charge and mag-\nnetic moment were calculated by integrating the density\nand spin density in a sphere with a radius of 1 ˚A forFe. This radius appears to be chosen such that compara-\nble valueswith neutronand x-raydiffraction experiments\nwere obtained.\nNote, there is no unambiguous way to define these\nradii in systems consisting of two or more atom types.\nTherefore, we checked the correspondence of our results\nto the earlier reported ones and also performed calcula-\ntionswith alargerradiusof1.3 ˚AforFeand0.82 ˚AforO.\nThis is a reasonable choice for Fe mO+\nnclusters since the\noverlapbetween different spheres is minimal, but most of\nthe intra cluster space is covered.\nII. RESULTS AND DISCUSSION\nA. Magnetite\nEven in the bulk, iron oxide is well known for its wide\nvarietyofphasesandtransitions. Magnetite(Fe 3O4), the\nmost stable phase of FemOn, is for example well known\nfor its Verweytransition.1,2Above the transitiontemper-\natureTV, the structure is a cubic inverse spinel. Upon\ncooling below TV, the conductivity decreases by two or-\nders of magnitude due to charge ordering. Furthermore,\nthe structure changes to monoclinic.\nMagnetite has the formal chemical formula\n(Fe3+\nA[Fe2+,Fe3+]BO4) where tetrahedral Asites\nare occupied by Fe3+andBsites contain both divalent\n(Fe2+) and trivalent (Fe3+) iron atoms. Since magnetite\nis a mixed valence system, it is an excellent reference\nsystem for our cluster calculations to determine their\nvalence state and corresponding magnetic moment.\nTABLE I: Spin moments within atomic spheres of 1.3 ˚A for\nthe Fe ions in monoclinic Fe 3O4. For reference the values\nwithin a sphere of 1.0 ˚A are also shown. A and B labels are\nconsistent with Ref. 39.\nSite Spin moment ( µB) Spin moment ( µB)\nRadius sphere 1 .3˚A 1 .0˚A\nFe3+(A) −4.02 −3.78\nFe2+(B1) 3 .69 3 .45\nFe3+(B2) 4 .15 3 .93\nFe3+(B3) 4 .06 3 .84\nFe2+(B4) 3 .64 3 .40\nIn Table I, the spin moments are shown for the dif-\nferent iron ions. The magnetic moments on the Aand\nBsites are antiparallel creating a ferrimagnetic struc-\nture. Within the atomic spheres of 1.3 ˚A the Fe2+and\nFe3+ions have a distinct magnetic moment of 4.0 µB\nand 3.7µBrespectively. Note the difference of 0.3 µBis\nmuch smaller than the 1 µBatomic value and does not\ndepend on the size of the atomic sphere used in the range\nbetween 1.0 and 1.3 ˚A.4\nB. GGA+U\nTo determine the optimal Ueffin comparison to the\nB3LYP functional for the genetic algorithm, we per-\nformed PBE+ Ucalculations on the neutral Fe 3O4clus-\nter. The results for the electronic DOS are shown in\nFig. 1 and compared with the hybrid B3LYP functional.\n/gl507=B==U8nt=/gl72/gl57\n[/gl721[/gl72S=38st/gl3386=\n[/gl721/gl50fS=f8yi=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=/gl56/gl72/gl73/gl73=B=n/gl72/gl57\n/gl507=B==U8nU=/gl72/gl57\n[/gl721[/gl72S=38sf/gl3386=\n[/gl721/gl50fS=f8ys=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=/gl56/gl72/gl73/gl73=B=D/gl72/gl57\n/gl507=B==U8ni=/gl72/gl57\n[/gl721[/gl72S=38sU/gl3386=\n[/gl721/gl50fS=f8ys=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=/gl56/gl72/gl73/gl73=B=3/gl72/gl57H/gl72/gl81/gl86/gl76/gl87/gl92=/gl82/gl73=/gl54/gl87/gl68/gl87/gl72/gl86/gl507=B==U8sn=/gl72/gl57 /gl56/gl72/gl73/gl73=B=f/gl72/gl57\n[/gl721[/gl72S=38sU/gl3386=\n[/gl721/gl50fS=f8yn=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=\n/gl507=B==U8it=/gl72/gl57 /gl56/gl72/gl73/gl73=B=U/gl72/gl57\n[/gl721[/gl72S=38sU/gl3386=\n[/gl721/gl50fS=f8yn=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=\n[/gl721[/gl72S=38sf/gl3386=\n[/gl721/gl50fS=f8yn=/gl3386=\n[/gl721/gl503S=f8oo=/gl3386=dD/gl47/gl60/gl51\n[/gl72=/gl86\n[/gl72=/gl83\n[/gl72=/gl71\n/gl50=/gl86\n/gl50=/gl83M1M−/gl50/gl48/gl50=/gl62/gl72/gl57/gl64=/gl237fU /gl237t8s /gl237s /gl23738s U 38s s\nFIG. 1: (Color online) The density of states for the hybrid\nB3LYP functional and PBE+ Ufor different values of Ueff.\nThe average inter atomic distances are shown on the right,\nwhere Fe-O 1and Fe-O 2refer to the Fe-O distances between\nbridging O atoms (side) and the capping O atom (center),\nrespectively. The mean absolute difference ∆ [Eq. 1] between\nthe PBE+ Uand B3LYP energy levels is also shown and is\nminimal for Ueff= 3 eV, indicating the best match in DOS.\nThe valence states within -4 and 0 eV are formed by\nhybridized orbitals between the dorbitals of iron and\ntheporbitals of oxygen. For increasing U, the majority\nspindorbitals of Fe decrease in energy, whereas HOMO-\nLUMO gap increases. Note that the HOMO-LUMO gap\nof 1.5 eV for Ueff= 4 eV still is 0.9 eV smaller than the\n2.4 eV gap for B3LYP. Furthermore, for Ueff= 2 and\n3 eV the Fe dDOS features are very similar to those of\nthe B3LYP result. To quantify this we also calculated\nthe mean absolute difference ∆ [Eq. 1] for the occupied\nlevels; the results are shown in Fig. 1. ∆ is minimal\nforUeff= 3 eV, indicating the best DOS correspondence\nto B3LYP. We also show the corresponding bonding dis-\ntances within the cluster, where Fe-O 1and Fe-O 2refer\nto the Fe-O distances between bridging O atoms (side)\nand the capping O atom (center), respectively. Note the\ninteratomicdistancesonlychangeverylittlewithincreas-\ningUeff. ForUeff= 3eV,the bindingdistancesarewithin0.01˚A; furthermore, for Ueff= 3 eV and B3LYP the oc-\ncupieddorbitalsofFeareatcomparableenergieswithre-\nspect to the HOMO level. We therefore used Ueff= 3 eV\nfor our genetic algorithm calculations.\nC. Fe 3O0\n4\nAlthough the possible number of isomers increases\nrapidly with cluster size, for small systems such as Fe 3O4\nthe number of possibilities is still small. In Fe 3O4, the Fe\natomscaneitherformatriangleorachain. Forthe trian-\ngular configuration, two isomers are low in energy. The\nfirst isomer consists of a ring like structure where the O\natomsoccupybridgingstatesandoneOatomcapstheFe\ntriangle as is shown in Fig. 2 (a). In the second isomer,\nthe additional O atom is not located above the center\nbut forms an extra bridge between the two ferromagnetic\n(FM) ordered Fe atoms as is shown in Fig. 2 (b).\n3e/gl72/gl57 3e/gl72/gl57 [Sae/gl80/gl72/gl57\n/gl71\n/gl69\n/gl68/gl40/gl81/gl72/gl85/gl74/gl92e/gl62/gl72/gl57/gl64\n335tSS5tpp5ti\n/gl54/gl83/gl76/gl81e/gl80/gl68/gl74/gl81/gl72/gl87/gl76/gl93/gl68/gl87/gl76/gl82/gl81e/gl62/gl541 /gl37/gl643 p a z µ S3 Sp Sa/gl41/gl72/gl22/gl50/gl23/gl19\nFIG. 2: (Color online) The energy as function of spin mag-\nnetization for different neutral Fe 3O4isomers. The geometric\nfigures on the right show the corresponding geometric struc-\nture. O atoms are shown in red, Fe spin up and Fe spin\ndown are indicated with orange (red) and green (blue) colors\n(arrows), respectively. For the lowest magnetic states the rel-\native energy differences are also shown in black. Isomers (a)\n(black line) and (b)(red line) are equally low in energy with\na ferrimagnetic and ferromagnetic ground state, respectiv ely\n(0 eV). The M= 6µBstate of isomer (a)is 14 meV higher\nin energy.\nFigure 2 shows the energy as a function of spin mag-\nnetic moment for the neutral Fe 3O4cluster with four\ndifferent isomers. For all spin magnetizations, the ge-\nometric structure is optimized and shown on the right\nwith its magnetic structure lowest in energy. In Fig. 2\nandthe restofthiswork, Fespin upandFe spindownare\nindicated with orange (red) and green (blue) colors (ar-\nrows), respectively. O atoms are shown in red. For the\nneutral cluster, the two triangular isomers are equally\nlow in energy with two different magnetic configurations.\nThe difference is smaller than 1 meV and therefore be-5\nyond the accuracyofDFT. In isomer (a), as indicated by\nthe black line in Fig. 2, the magnetic ground state corre-\nsponds to ferromagneticalignment between the magnetic\nmoments on the Fe atoms and a total magnetic moment\nof 14µB. The Fe-Fe distances are 2.51 ˚A, the Fe-O dis-\ntances for the bridging O atoms and capping O atom are\n1.84 and 1.99 ˚A, respectively. Aside from the FM ground\nstate, also the ferrimagnetic state with a spin magneti-\nzation of 4 µBis low in energy and only 14 meV higher\nthan the ferromagnetic state. Note we also considered\na noncollinear magnetic state with M= 0µB, but this\nmagnetic configuration did not turn out to be energeti-\ncally stable.\nIsomer(b)is equally low in energy and shown in red\nin Fig. 2. The magnetic ground state corresponds to a\nferrimagnetic alignment where the two ferromagnetically\naligned Fe atoms have Fe-O-Fe angles of approximately\n90◦.\nWe also considered zero point vibrational energies for\nthe three lowest-energy levels. When we include these\ninto our consideration, the ferromagneticstate, indicated\nby the black line, is lowest in energy, and the M= 4µB\nandM= 6µBstatesare17and19meVhigherin energy,\nrespectively.\nD. Fe 3O+\n4\nFor the cation Fe 3O+\n4cluster we also considered ring\nand chain configurations with different oxygen locations.\nFor all four isomers we calculated all possible different\ncollinear magnetic states. Since an antiferromagnetic\n(AFM) triangle is the most simple example of geometri-\ncally frustrated magnetism, we also considered the non-\ncollinear state with M= 0µBwhere all magnetic mo-\nments have 120◦angles with respect to each other. The\nresults are shown in Fig. 3. For the charged Fe 3O+\n4clus-\nter, the isomer with a Fe triangle where the fourth O\natom caps the triangle is, like in the neutral cluster, low-\nestin energy,asisshowninFig.4. Threemagneticstates\nare low in energy: 0, 5 and 15 µB, with the M= 5µB\nstate being lowest in energy, and the non-collinear 0 µB\nand ferromagnetic 15 µBare 20 meV and 58 meV higher\nin energy respectively.\nThe ferrimagnetic state which is lowest in energy, has\na reduced symmetry ( Cv) with respect to the ferromag-\nnetic state ( C3v) and the antiferromagnetic state. This\ncould indicate a Jahn-Teller distortion, but could also\nbe the result of the inability of DFT to correctly model\nthe antiferromagnetic ground state.50,51However, to dis-\ntinguish between these two cases, methods beyond DFT\nsuch as CASPT2 and CCSD(T) are required and there-\nfore beyond the scope of this work. Note that different\nmagnetic states only lead to minor differences in the vi-\nbrational frequencies.\nInterestingly, the typical classical displacement during\na zero-point vibration in these clusters is of the order\nof 0.03˚A. This is of the same order as the typical dif-ytµ8/gl80/gl72/gl57 yp18/gl80/gl72/gl57 18/gl72/gl57\n/gl40/gl81/gl72/gl85/gl74/gl928/gl62/gl72/gl57/gl64\n11.pt1.t1.otSS.ptS.tS.otp\n/gl54/gl83/gl76/gl818/gl80/gl68/gl74/gl81/gl72/gl87/gl76/gl93/gl68/gl87/gl76/gl82/gl818/gl62/gl541 /gl37/gl641 S i t o B SS Si St/gl71\n/gl69\n/gl68/gl41/gl72/gl22/gl50/gl23/gl14\nFIG. 3: (Color online) Energy of the Fe 3O+\n4isomers as func-\ntion of spin magnetization. Figures on the right indicate th e\ncorresponding structure. The isomer lowest in energy (a)is\na Fe triangle with three bridge O atoms and one O atom cap-\npingthetriangle. Forthisisomer, theferrimagnetic 5 µBstate\nis lowest in energy. The antiferromagnetic 0 µBand ferromag-\nnetic 15 µBstate are 20 and 58 meV higher in energy, respec-\ntively. Note the antiferromagnetic 0 µBstate corresponds to a\nnon-collinear orientation with 120◦angles between the spins.\nFIG. 4: (Color online) The neutral (left) and cation (right)\nFe3O4lowest-energy isomers. FespinupandFespindown are\nindicated with orange (red) and green (blue) colors (arrows ),\nrespectively. O atoms are shown in red. The interatomic\ndistances are shown in black. The neutral and cation cluster\nhaveC3vandCvpoint group symmetry, respectively.\nference in inter atomic distances between different mag-\nneticstates. Therefore,this couldleadtointerestingphe-\nnomena in which, for example, there is a strong coupling\nthrough exchange between vibrations and magnetism.\nThe second triangular isomer of Fe 3O+\n4is 154 meV\nhigherinenergyandalsoconsistsofaringstructure. The\nmagnetic state lowest in energy has a magnetic moment\nof 5µB. The Fe-Fe bonding distances are 2.5 and 3.0 ˚A\nbetween the AFM and FM bonds within the structure.\nThe Fe-O distances vary between 1.7 and 1.9 ˚A. The\nisomer has a C2vpoint group symmetry.\nThe third and fourth isomers consist of a linear chain\nof Fe atoms with two O bridging atoms between each Fe\npair. The two planes can be parallel or perpendicular,6\n/gl41/gl72/gl22/gl50/gl23/gl14\n/gl53/gl51 [ I1pbViio /gl80/gl72/gl57 /gl70\n/gl53/gl51 [ I1rnVasb /gl80/gl72/gl57 /gl69\n/gl53/gl51 [ I1sI/gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81 /gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl809a/gl64oII pII iII aIII abII\nFIG. 5: (Color online) The experimental vibration spectra\nof Fe3O+\n4and the calculated isomers lowest in energy. The\nreported energy differences include ZPVE. The Pendry’s reli -\nability factor [Eq. 4] is also shown for each isomer.\nwhere the latter is lower in energy. Both isomers have a\nmagnetic moment of 5 µB.\nIn Fig. 5, both the experimental and calculated vibra-\ntion spectra for the different isomers are shown. The\nexperimental spectrum consists of three peaks at 540,\n610 and 670 cm−1. The best match is given by isomer\n(a)with calculated vibrations at 505, 630 and 660 cm−1\nand a corresponding lowest- RPfactor of 0 .30, indicat-\ning a reasonable match with the experimental spectrum.\nSince isomer (a)is also the lowest in energy, it is identi-\nfied as the experimentally observed structure.\nE. Fe 4O0/+\n5\nFe4O5also consists of a ring structure in which the\nO atoms occupy the bridging sites and one O atom is\nlocated above the center, as is shown in Fig. 6. The clus-\nter has antiferromagnetic order. However, not all Fe-Fe\nbonds are antiferromagnetic, but also two ferromagnet-\nically aligned bonds are present. Therefore, the cluster\nhas noC2vpoint group symmetry but C2, since Fe-Fe\nand Fe-O distances vary between 2.72-2.74 ˚A and 1.79-\n2.33˚A respectively. The magnetic state with four AFM\nFe-Fe bonds is 308 meV higher in energy.\nFor Fe4O+\n5the isomer lowest in energy consists of the\nsame ring structure but is more symmetry broken, since\nthe O atom above the ring is off-center as is shown in\nFig. 6. Therefore the two Fe-Fe distances are 2.69 and\n3.07˚A, the Fe-O distances vary between 1.76 and 2.01 ˚A.\nThe isomer has Cspoint group symmetry. Two Fe 2O2\nsquares are present within the cluster. Isomer (a)has\na magnetic moment of 1 µBdue to ionization. Interest-\ningly, the ionized cluster has a different magnetic ground\nstate with four AFM Fe-Fe bonds opposed to the neutral\ncluster.\nFIG. 6: (color online) The neutral (left) and cation (right)\nFe4O5lowest energy isomers. The neutral cluster has C2sym-\nmetry, whereas the cation cluster has Cssymmetry.\nIn Fig. 7 (b), we also show the vibration spectrum of\nthe ferromagnetic state of this cluster. The Fe-Fe dis-\ntancesareincreasedto2.74and3.11 ˚A, respectively. The\nferromagneticstructureis514meVhigherin energy. The\nvibration spectrum is similar but slightly shifted to the\nblue due to the increased bonding distances.\n/gl53/gl51 - bIisV pvp /gl80/gl72/gl57 /gl71\n/gl53/gl51 - bIttV ptv /gl80/gl72/gl57 /gl70\n/gl53/gl51 - bItbV tsp /gl80/gl72/gl57 /gl69\n/gl53/gl51 - bIpo/gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81 /gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl802s/gl64obb pbb ibb Wbb sbbb sobb/gl41/gl72/gl23/gl50/gl24+\nFIG.7: (Color online)Theexperimentalandcalculated vibr a-\ntion spectra of Fe 4O+\n5. The isomer shown in (a)is both the\nlowest in energy and RP[Eq. 4] and can therefore be iden-\ntified as the experimentally observed geometrical structur e.\nThe reported energy differences include ZPVE.\nThe second isomer, 459meV higher in energy, is shown\nin Fig. 7(c). This cage-like structure has Cvpoint group\nsymmetry and a magnetic moment of 9 µB. Figure 7 (d)\nshows the third isomer which is 494 meV higher in en-\nergy compared to Fig. 7 (a). The isomer has almost no\nsymmetry ( C1), and consists of a ring where one Fe-Fe\nbond has two bridging O atoms. The Fe-Fe binding dis-\ntances vary between 2.62 and 3.13 ˚A. The isomer has a\nmagnetic moment of 1 µB.\nIn the experimental vibration spectrum of Fe 4O+\n5\nshown in Fig. 7, five vibration frequencies can be ob-7\nserved: 450, 615, 760, 810, and 1070 cm−1. The vibra-\ntion at 1070 cm−1can be identified as a shifted vibration\nin the O 2messenger attached to the cluster-messenger\ncomplex and is therefore omitted in the RPcalculation.3\nThe best fit is given by isomer Fig. 7 (a)withRP= 0.42,\nwhich is also the isomer lowest in energy. The calculated\nfrequencies: 479, 630, 637, 772 and 796 cm−1match all\nwithin 30 cm−1to the experimental spectrum. Also, the\nrelative intensities between different vibrations are very\nsimilar. Although the ferromagnetic order increases the\nbinding distances within the cluster, the changes in the\nvibration spectrum of Fig. 7 (b)are small and therefore\nthestructurecorrespondingtoFigs.7 (a)and7(b)canbe\nidentified as the experimentally observed structure and\nthe IR-MPD method is not able to resolve the magnetic\nstate in this case.\nF. Fe 4O0/+\n6\nIn Ref.30, the Fe4O+\n6cluster was already identified as\nthe structure shown in Fig. 8 (b). The reported magnetic\nstructure was ferrimagnetic with a magnetic moment of\n9µB.\n/gl53/gl51 - bIrvesWn /gl80/gl72/gl57 /gl69\n/gl53/gl51 - bIpW/gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81/gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl804s/gl64obb pbb ibb Wbb sbbb sobb/gl41/gl72/gl23/gl50/gl25/gl14\nFIG. 8: (color online) The experimental and calculated vibr a-\ntion spectra of Fe 4O+\n6for both theprevious and newmagnetic\nground state. The vibration frequencies are very similar bu t\ndiffer in absorption intensity. The M= 1µBstate in (a)is\n187 meV lower in energy.\nIn our calculations a magnetic state lower in energy\nwas found for the same geometric structure for both\nFe4O6and Fe 4O+\n6. In this state Fe 4O6and Fe 4O+\n6have a\nmagnetic moment of 0 and 1 µBrespectively as is shown\nin Fig. 9. These structures are 194 and 187 meV lower\nin energy for Fe4O6and Fe4O+\n6in comparison to the\npreviously reported state.30The antiferromagnetic mag-\nnetic ground state of Fe 4O6was also previously reported\nin Ref. 26. For Fe 4O6we also calculated a noncollinear\nstate where all magnetic moments point towardsthe cen-\nter of mass, such state with M= 0µBis 30 meV higher\nin energy compared to the collinear M= 0µBstate.\nFor the neutral cluster, minima in energy are obtained\nforM= 0, 10, 20 µBcorresponding to flips of atomic/gl41/gl72t/gl50z1\n/gl41/gl72t/gl50z[/gl40/gl81/gl72/gl85/gl74/gl924/gl62/gl72/gl57/gl64\n11.i22.iMM.i\n/gl48/gl68/gl74/gl81/gl72/gl87/gl76/gl93/gl68/gl87/gl76/gl82/gl814/gl62/gl541 /gl37/gl641 M t z µ 21 2M 2t 2z 2µ M1/gl41/gl72t/gl50z15[\nFIG. 9: (color online) Energy as function of magnetization o f\nthe neutral Fe 4O6and cationic Fe 4O+\n6clusters. The magnetic\nground state corresponds to a total spin magnetic moment of\nM= 0 and M= 1µBfor Fe 4O6, and Fe 4O+\n6respectively.\nmagnetic moments of 5 µBfor each Fe atom. Note this\nalso matches with an ionic picture in which the Fe atoms\nin Fe4O6have a Fe3+valence state resulting in an atomic\nmagnetic moment of 5 µB. The corresponding structure\nis shown in Fig. 10. In Ref. 30 is mentioned that the\nsymmetry in the M= 10µBstate is reduced from Tdfor\nthe ferromagnetic state to C3v. In this antiferromagnetic\nground state, the neutral cluster has D2dsymmetry. In\nFe4O+\n6the symmetry is reduced even further to Csas is\nshown in Fig. 10.\nFIG. 10: (Color online) The neutral (left) and cation (right )\nFe4O6lowest energy isomers. The neutral cluster has D2d\nsymmetry, whereas the cation cluster has Cssymmetry.\nFigure 8 shows both calculated and experimental spec-\ntra for Fe4O+\n6. The vibration spectra for the two calcu-\nlated magnetic states in Figs. (a)and 8(b)show very\nsimilar behavior. The RPvalues of isomer Fig. 8 (a)\n(0.48) and Fig. 8 (b)(0.39) are both large and indicate a\nbetter match for isomer Fig. 8 (b). Although the spectra\nfor Figs. 8 (a)and 8(b)are very similar, the ferrimag-\nnetic structure has an extra vibration at 720 cm−1with\nsmall IR absorption. Furthermore, around 550 cm−1,\nvibrations differ slightly in frequency. Since the men-\ntioned differences cannot be experimentally resolved, the\nIR-MPD method is unable to resolve between different\nmagnetic states and another type of experiments such\nas Stern-Gerlach deflection is required to determine the\nmagnetic moment.8\nG. Fe 5O0/+\n7\nThe neutral Fe 5O7cluster has a “basket” geometry\nas is shown in Fig. 11. The magnetic ground state is\nferrimagnetic with a total moment of 4 µBdue to the\nodd number of Fe atoms. The cluster has C2vsymmetry.\nFIG. 11: (Color online) The neutral (left) and cation (right )\nFe5O7lowest-energy isomers. The neutral cluster has C2v\nsymmetry, whereas the cation cluster has no symmetry.\nThe cationic structure of Fe 5O+\n7is very different and\nshown in Fig. 11. Like Fe 4O+\n6, it consists of a cage-like\nstructure. The Fe-Fe distances range from 2.7 to 3.1 ˚A.\nExcept for the triple bound O atom, all O atoms form\nbridges between two Fe atoms. The ground state has a\nmagnetic moment of 5 µB. The second isomer is similar\n/gl53/gl51 o vWc[/gl70 3nWvc /gl72/gl57\n/gl53/gl51 o vWrb/gl69 3rv- /gl80/gl72/gl57\n/gl53/gl51 o vWc[/gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl80mn/gl64uvv rvv cvv ]vv nvvv nuvv /gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81/gl41/gl72/gl24/gl50/gl26/gl14\nFIG. 12: (Color online) The experimental and calculated vi-\nbration spectra of Fe 5O+\n7. The reported energy differences\ninclude ZPVE.\nto the neutral ”basket” structure and is 394 meV higher\nin energy as is shown in Fig. 12 (b). The structure has\nCssymmetry and a magnetic moment of 5 µB. However,\nthe atomic spin moments have a different arrangement\nfor the neutral and cationic state.\nThe third isomer is shown in Fig. 12 (c)and is 1.04 eV\nhigher in energy. It contains two triple bonded O atoms\nand is ferrimagnetic with M= 5µB.\nThe experimental vibration spectrum shown in Fig. 12has eight distinct vibrations at 375, 490, 520, 570, 615,\n710, 780, and 830 cm−1which are best resembled by the\nisomer lowest in energy shown in Fig. 12 (a), although\nthegapbetween615and710cm−1seemstobe underesti-\nmated. Note that this also explains the high- RPfactorof\n0.65 for isomer Fig. 12 (a). Similar to Fe 4O+\n5and Fe 4O+\n6\nthe absorption intensities of vibrations in the range of\n300-500 cm−1are systematically underestimated. The\nindividualvibrationsofisomerFig. 12 (a)areallin agree-\nment within 35 cm−1. Although isomer Fig. 12 (b)has a\nlowerRP= 0.43, the energy difference of 407 meV with\nisomer Fig. 12 (a)is large and isomer Fig. 12 (b)has a\nvibration at 450 cm−1which is not present in the exper-\nimental spectrum and lacks the experimental 375 cm−1\nvibration. Therefore, isomer Fig. 12 (a)can be identified\nas the most probable ground state.\nH. Fe 6O+\n8\nThe isomer lowest in energy found for Fe6O+\n8is shown\ninFig.13andhas Cssymmetrywherethereflectionplane\nis located through Fe atoms 1, 3, and 6. The magnetic\nmoment of this isomer is 1 µB.\nFIG. 13: (Color online) The cation Fe 6O+\n8isomer lowest in\nenergy. The cluster has Cssymmetry.\nThesecondisomerlowinenergyisshowninFig.14 (b).\nIn this isomer no symmetry is present. Compared to the\nlowest found isomer in Fig. 14 (a)it is 413 meV higher in\nenergy and also has a magnetic moment of 1 µB.\nFigure 14 (c)shows the third isomer, which is a dis-\ntorted octahedral of Fe atoms in which the O atoms cap\nthe Fe triangles. The structure is slightly distorted due\nto the AFM order between spins, which lead to slightly\naltered Fe-Fe distances. This isomer is 483 meV higher\nin energy than isomer Fig. 14 (a).\nFigure 14 also shows the corresponding vibration spec-\ntra of the mentioned isomers and the experimental spec-\ntrum. The experimental spectrum has vibrations at 392,\n420, 500, 730 and 763 cm−1. Note that none of the\nprovided isomers match the experimental vibration spec-\ntrum completely. This is also shown by the large- RP\nvalues of 0.56-0.61 for all calculated isomers. The isomer\nlowest in energy Fig. 14 (a)is the best match since it also\nhasvibrationsat 420and500cm−1, but the vibrationsat9\n/gl53/gl51 o a1r]/gl70 eb-u /gl80/gl72/gl57\n/gl53/gl51 o a1[v/gl69 ebvu /gl80/gl72/gl57\n/gl53/gl51 o a1r[/gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl806v/gl64naa baa [aa -aa vaaa vnaa /gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81/gl41/gl72/gl25/gl50/gl27/gl14\nFIG. 14: (color online) The experimental and calculated vi-\nbration spectra of Fe 6O+\n8. The isomer shown in (a)is the low-\nest in energy. The reported energy differences include ZPVE.\n804 and 825 are considerably shifted with respect to 730\nand 763 cm−1. Furthermore, the vibrations at 640, 671,\nand 713 cm−1are not present in the experimental spec-\ntrum. The vibration spectra shown in Figs. 14 (b)and\n14(c)fit even worse. Therefore, we can not successfully\nidentify the Fe 6O+\n8structure.\nNote that our genetic algorithm implementation only\nuses geometry optimization at the DFT level. At cluster\nsizes of Fe 6O+\n8and larger, preselection using empirical\npotentials instead of immediate geometry optimization\nusing DFT might be more efficient in generating possible\nisomers.\nI. Electronic structure\nIn the bulk, iron-oxide materials have many different\ncrystal structures such as hematite, wustite, and mag-\nnetite with all corresponding different electronic struc-\ntures. While in hematite only trivalent Fe3+is present,\nthe mixed valence state (Fe3+\nA[Fe2+,Fe3+]BO4) in mag-\nnetite leads to interesting physical phenomena such as\nferrimagnetic ordering between the sublattices Aand\nBand the Verwey transition in which orbital ordering\nleads to a first-order phase transition in the electrical\nconductivity.1,2\nIn clusters, stoichiometries corresponding to both\nhematite (Fe4O6) and magnetite (Fe3O4, Fe6O8) and\nother combinations (Fe 4O5, Fe5O7) occur. We therefore\nexpect divalent and trivalent Fe cations to be present in\nthe reported clusters. There is no unique method to de-\ntermine the valence state in materials consisting of mul-\ntiple types of elements. We therefore compare both the\nlocal magnetic moments and the local density of states\n(LDOS) for our cluster calculations with bulk magnetite\nresults shown in Section IIA. Since the Fe2+and Fe3+\nfeatures in the LDOS are very similar for different clustersizes, we show the LDOS of Fe 4O+\n5which contains both\nFe2+and Fe3+in Fig. 15. The LDOS for other cluster\nsizes can be found in the Appendix.\n/gl41/gl720/gl502 /gl47/gl39/gl50/gl54 /gl76/gl81/gl87s4/gl39/gl50/gl54 /gl55/gl82/gl87s4/gl39/gl50/gl54/gl41/gl724/gl86\n/gl41/gl724/gl83\n/gl41/gl724/gl71\n/gl504/gl86\n/gl504/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\n2udu2wdw21d12\n/gl41/gl72u\n/gl41/gl72w\n/gl41/gl721\n/gl41/gl720\n/gl40o/gl40/gl43/gl50/gl48/gl504/gl62/gl72/gl57/gl64/gl237- /gl2373 /gl2370 /gl237w d w 0 3\nFIG. 15: (Color online) The total, integrated and local den-\nsity of states of the Fe atoms for the Fe 4O+\n5cluster. The\ntrivalent Fe(1), Fe(2) and Fe(3) all show 3 dlevels at -6 eV\nand small hybridization with O. The divalent Fe(4), however ,\nshows strong hybridization and a single level at E HOMO.\nTable II shows the local spin moments of the clus-\nters: Fe 3O+\n4, Fe4O+\n5, Fe4O+\n6, Fe5O+\n7and Fe 6O+\n8. For\nFe3O+\n4all three Fe atoms have a similar spin moment\nwithin 0.04 µB. A comparison with magnetite suggests\nall Fe atomsare trivalent. This agreeswith an ionic bond\nmodel. Furthermore, this is confirmed by the integrated\nand local density of states shown in Appendix A. The 3 d\npeaksaround-6eVcorrespondto15electrons,indicating\nthe hybridization between Fe and O is small. Note that,\nthe central oxygen atoms O(4) and O(7) are partially\nspin polarized.\nFor Fe4O+\n5, the spin moment of Fe(4) is 0.5 µBlower\nthan the other Fe atoms, indicating three trivalent and a\nsingle divalent atom. The difference is also in agreement\nwith the magnetite results. The Fe(4) also breaks the C2\nsymmetry as is shown in Fig. 6. The local (LDOS) and\nintegrated density of states are shown in Fig. 15. Note\nthat all Fe3+have 3dpeaks around −6 eV and small\nhybridization with O is present, similar to the Fe 3O+\n4\ncluster. The LDOS of the divalent Fe(4) atom however\nshows strong hybridization with O and a single minority\nlevel at E HOMO.\nWhereasFe 4O6onlycontains trivalent Fe,26for Fe4O+\n6\nthis is no longer the case due to ionization. As can be\nseen from Table II, three trivalent Fe atoms are present,\ntogether with a single Fe4+atom. The spin moment is10\nreduced with respect to Fe3+, consistent with a higher\noxidation state than Fe3+.\nIn Fe5O+\n7, only trivalent Fe atoms are present, con-\nsistent with an ionic model and the ionized state of the\ncluster. Fe 6O+\n8, on the other hand, is again a mixed\nvalence cluster where the magnetic moment of Fe(4) is\n0.4µBlower than the other Fe atoms, indicating Fe(4)\nis divalent. This is also consistent with the LDOS shown\nin Appendix A.\nFigure 16 shows the density of states for the different\ncationicclustersandmagnetite. Thecalculatedbandgap\nof 0.2 eV in magnetite is considerably smaller than for\nthereportedclusters: around3eVforFe 3O+\n4andslightly\nsmaller for Fe 4O+\n5and Fe 4O+\n6. Furthermore, whereas\nmagnetite has a t2gorbital of Fe2+just below the Fermi\nenergy,39in the reported clusters Fe4O+\n5and Fe6O+\n8have\nasimilarlevelduetoadivalentFeatom. Notethatthe3 d\norbitals of Fe3+in the clusters are located around 5.5 eV\nbelow the HOMO level, which is 2 eV higher in energy\ncompared to magnetite.\n/gl39/gl72/gl81/gl86/gl76/gl87/gl92n/gl82/gl73n/gl54/gl87/gl68/gl87/gl72/gl86/gl41/gl72g/gl50-6\n/gl41/gl72M/gl50E6/gl41/gl72n/gl86\n/gl41/gl72n/gl83\n/gl41/gl72n/gl71\n/gl50n/gl86\n/gl50n/gl83n\n/gl41/gl723/gl50g6\n/gl41/gl723/gl50M6\n/gl41/gl724/gl5036\n/gl48/gl68/gl74/gl81/gl72/gl87/gl76/gl87/gl72\n/gl408/gl40/gl43/gl50/gl48/gl50n/gl62/gl72/gl57/gl64/gl237- /gl237g /gl2373 /gl237d 7 d 3 g\nFIG. 16: (Color online) The density of states for Fe xO+\nyclus-\nters. For these calculations a smearing of 0.15 eV was used\nfor convenience of the reader. The HOMO level is located at\n0 eV and the small occupation above the HOMO level is due\nto smearing.III. CONCLUSION\nIn this work, we have studied the geometric, elec-\ntronic and magnetic structure of Fe xO+\nyclusters using\ndensity functional theory. For Fe 3O4we compared bind-\ning distances and electronic structure between the hybrid\nB3LYP functional, and different Ueffin the PBE+ Ufor-\nmalism. We found the best match for Ueff= 3 eV. Using\nthe PBE+ Uformalism and a genetic algorithm, many\npossible isomers were considered. For isomers low in en-\nergy, all different magnetic configurations were further\ngeometrically optimized. Finally, for the cationic clus-\nters we calculated the vibration spectra and compared\nthem with experiments to identify the geometric struc-\nture of Fe 3O+\n4, Fe4O+\n5, Fe4O+\n6, Fe5O+\n7and Fe 6O+\n8. All\ncationic clusters with an even number of Fe atoms have\na small magnetic moment of 1 µBdue to ionization. Fur-\nthermore, comparison with bulk magnetite reveals that\nFe4O+\n5, Fe4O+\n6and Fe 6O+\n8are mixed valence clusters.\nIn contrast, in Fe3O+\n4and Fe5O+\n7all Fe are found to be\ntrivalent.\nIV. 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Figures 17, 18, 19, and 20 show the\ntotal, integrated and local density of states of Fe3O+\n4,\nFe4O+\n6, Fe5O+\n7, and Fe 6O+\n8, respectively. Of these clus-\nters, Fe3O+\n4and Fe5O+\n7are pure trivalent and the LDOS\ncontains 3 dpeaks at -6 eV and small hybridization be-\ntween Fe and O. Fe 4O+\n6contains a single tetravalent Fe\natom, with a similar LDOS compared to Fe3+. The ion-\nized electron is not removed from the 3d levels at -6 eV,\nbut from the hybridized levels with oxygen, as can be\nseen from the integrated density of states. Fe 4O+\n5and\nFe6O+\n8contain a single divalent Fe atom, which has a\ndistinct LDOS, in which there are no peaks around -6 eV12\nbut strong spin polarized hybridization with oxygen and\nasingleoccupiedminoritylevelattheHOMOlevel. Even\nin bulk magnetite, as is shown in Fig. 21, the same fea-\ntures between divalent and trivalent Fe atoms exist.\n/gl41/gl72u/gl50w /gl47/gl39/gl50/gl54 /gl76/gl81/gl87a3/gl39/gl50/gl54 /gl55/gl82/gl87/gl68/gl793/gl39/gl50/gl54/gl41/gl723/gl86\n/gl41/gl723/gl83\n/gl41/gl723/gl71\n/gl503/gl86\n/gl503/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\n5psp5dsd5\n/gl41/gl72p\n/gl41/gl72d\n/gl41/gl72u\n/gl40o/gl40/gl43/gl50/gl48/gl503/gl62/gl72/gl57/gl64/gl237ps /gl2372 /gl2371 /gl237w /gl237d s d w 1\nFIG.17: (Color online)Thetotal, integratedandlocal dens ity\nof states of the Fe 3O+\n4cluster.\n/gl41/gl720/gl502 /gl47/gl39/gl50/gl54 /gl76/gl81/gl87s4/gl39/gl50/gl54 /gl55/gl82/gl87s4/gl39/gl50/gl54/gl41/gl724/gl86\n/gl41/gl724/gl83\n/gl41/gl724/gl71\n/gl504/gl86\n/gl504/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\nudu5wdw51d15\n/gl41/gl72u\n/gl41/gl72w\n/gl41/gl721\n/gl41/gl720\n/gl40o/gl40/gl43/gl50/gl48/gl504/gl62/gl72/gl57/gl64/gl237E /gl2372 /gl2370 /gl237w d w 0 2\nFIG. 18: (Color online) The total, integrated, and local den -\nsity of states of the Fe 4O+\n6cluster. Fe(1) is tetravalent as is\nshown in Table I.\n/gl47/gl39/gl50/gl54/gl41/gl722/gl504 /gl76/gl81/gl87sO/gl39/gl50/gl54 /gl55/gl82/gl87sO/gl39/gl50/gl54/gl41/gl72O/gl86\n/gl41/gl72O/gl83\n/gl41/gl72O/gl71\n/gl50O/gl86\n/gl50O/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\nudwd1d0d2d\n/gl41/gl72u\n/gl41/gl72w\n/gl41/gl721\n/gl41/gl720\n/gl41/gl722\n/gl40o/gl40/gl43/gl50/gl48/gl50O/gl62/gl72/gl57/gl64/gl237E /gl2373 /gl2370 /gl237w d w 0 3\nFIG. 19: (Color online) The total, integrated, and local den -\nsity of states of the Fe 5O+\n7cluster.13\nTABLE II: The spin moment for Fe xO+\nyclusters. The atom numbers correspond to the atom numbers sh own in Figures 4,6,10,\n11, and 13. The spin moment is calculated using atomic sphere s of 1.3 and 0.82 ˚A for Fe and O, respectively.\nCluster Spin moment [ µB]\n1 2 3 4 5 6 7 8\nFe3O+\n4 Fe −3.84 3 .88 3 .88\nO 0 .56 0 .00 0 .00 0 .22\nFe4O+\n5 Fe 3 .89 −3.84 3 .89 −3.40\nO −0.05 0 .13 0 .20 −0.05 0 .20\nFe4O+\n6 Fe −3.22 3 .85 3 .85 −3.79\nO 0 .01 0 .54 0 .01 −0.25 0 .00 0 .00\nFe5O+\n7 Fe 3 .85 3 .87 3 .89 −3.83 −3.80\nO 0 .01 0 .10 0 .03 0 .51 −0.09 0 .05 0 .12\nFe6O+\n8 Fe 3 .80 −3.84 3 .85 −3.47 −3.84 3 .88\nO 0 .01 0 .51 0 .01 0 .01 −0.10 0 .17 −0.10 0 .0114\n/gl41/gl723/gl505 /gl47/gl39/gl50/gl54 /gl76/gl81/gl87s6/gl39/gl50/gl54 /gl55/gl82/gl87s6/gl39/gl50/gl54/gl41/gl726/gl86\n/gl41/gl726/gl83\n/gl41/gl726/gl71\n/gl506/gl86\n/gl506/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\nudwd1d0d2d\n/gl41/gl72u\n/gl41/gl72w\n/gl41/gl721\n/gl41/gl720\n/gl41/gl722\n/gl41/gl723\n/gl40o/gl40/gl43/gl50/gl48/gl506/gl62/gl72/gl57/gl64/gl2375 /gl2373 /gl2370 /gl237w d w 0 3\nFIG. 20: (Color online) The total, integrated, and local den -\nsity of states of the Fe 6O+\n8cluster. All Fe atoms are trivalent\nexcept for Fe(4), which is divalent./gl48/gl68/gl74/gl81/gl72/gl87/gl76/gl87/gl72/gl47/gl39/gl50/gl54 /gl55/gl82/gl873g/gl39/gl50/gl54/gl41/gl722.S/gl36T\n/gl41/gl72).S/gl37AT\n/gl41/gl722.S/gl37)T\n/gl41/gl722.S/gl372T\n/gl41/gl72).S/gl37BT\n/gl40F/gl40/gl41g/gl62/gl72/gl57/gl64/gl237- /gl2374 /gl237B /gl237) ( ) B 4\nFIG. 21: (Color online) The total and local density of states\nof the different Fe atoms in magnetite. The numbering is\nconsistent with Table I. Fe2+and Fe3+have a similar LDOS\nto clusters although the symmetry is very different."    },    {        "title": "1506.01738v1.Low_moment_ferrimagnetic_phase_of_the_Heusler_compound_Cr2CoAl.pdf",        "content": "Low-moment ferrimagnetic phase of the Heusler compound Cr 2CoAl\nMichelle E. Jamer,1Luke G. Marshall,2George E. Sterbinsky,3Laura H. Lewis,2and Don Heiman1\n1Department of Physics, Northeastern University, Boston, MA 02115 USA\n2Department of Chemical Engineering, Northeastern University, Boston, MA 02115 USA\n3Photon Sciences Directorate, Brookhaven National Laboratory, Upton, NY, 11973 USA\nSynthesizing half-metallic fully-compensated ferrimagnets that form in the inverse Heusler phase\ncould lead to superior spintronic devices. These materials would have high spin polarization at\nroom temperature with very little fringing magnetic fields. Previous theoretical studies indicated\nthat Cr 2CoAl should form in a stable inverse Heusler lattice due to its low activation energy. Here,\nstoichiometric Cr 2CoAl samples were arc-melted and annealed at varying temperatures, followed by\nstudies of their structural and magnetic properties. High-resolution synchrotron X-ray diffraction\nrevealed a chemically ordered Heusler phase in addition to CoAl and Cr phases. Soft X-ray magnetic\ncircular dichroism revealed that the Cr and Co magnetic moments are antiferromagnetically oriented\nleading to the observed low magnetic moment in Cr 2CoAl.\nIntroduction\nTheoretical calculations have predicated the exis-\ntence of several inverse Heusler compounds that ex-\nhibit zero-moment magnetization while retaining half-\nmetallicity.[1–5] These compounds are a subset of SGS\nmaterials, where the density of states (DOS, shown\nin Fig. 1(a)) acts as both a half-metal and a\ngapless semiconductor.[6–8] Such compounds are zero-\nmomentferrimagnetsastheirspinsarecompensatednon-\nsymmetrically, which does not prohibit spin-polarization.\nThese compounds are known as half-metallic fully-\ncompensated ferrimagnets (HMFF)[9] and would be at-\ntractive for spintronic devices, since their magnetic tran-\nsition temperatures are often higher than room tempera-\nture (400-1000 K). In contrast traditional Néel antiferro-\nmagnets cannot be spin-polarized due to their symmet-\nric anti-aligned moments resulting in a symmetric DOS,\nwhich was demonstrated in the DOS of the gapless semi-\nconductor V 3Al.[10, 11] Unfortunately, recent research\nhas shown that some HMFF compounds tend to be un-\nstable and decompose into more stable states, in addi-\ntion to possessing properties that are adversely affected\nby structural disorder.[12–14]\nThis current work focuses on the synthesis and char-\nacterization of Cr 2CoAl, which has been predicted to be\na HMFF when it adopts the inverse Heusler structure.\nTheoretical calculations have shown that this compound\nshould have a net magnetization of 0.01 \u0016B/f.u. and neg-\native energy of formation (-0.27 eV/f.u.), indicating that\nitislikelytoforminthebulk.[15]Thenonsymmetricspin\nstructure of the transition metal atoms (with moments\nCr1(1.36\u0016B); Cr 2(-1.49\u0016B); Co(0.30\u0016B))[15] indicates\nthatthiscompoundcanhavefullspin-polarization. How-\never, calculations have shown that these Cr-based in-\nverse Heusler compounds tend to decompose into other\ncompounds.[15]Experimental details\nStoichiometric polycrystalline Cr 2CoAl ingots (1 g)\nwere synthesized using high-purity elements ( \u001599.995\n%) via arc melting in an Ar environment. The ingots\nwere annealed at 1000oC for 48 hours to homogenize\ntheir composition. The samples were cooled to various\ntemperaturesbetween600-1000oCfor72hourstostudy\nthe effects of annealing on the compound’s properties.\nUsing EDS the synthesized ingots’ composition varied by\n< 1 at %across the samples. Crystallographic proper-\nties were probed with high-resolution synchrotron x-ray\npowder diffraction (XRD) with average wavelength \u0015=\n0.459 Å using beamline 11-BM at the Advanced Pho-\nton Source at Argonne National Laboratory. The degree\nof phase segregation was quantified using GSAS for the\nRietveld refinement.[16–18] Magnetic properties were de-\ntermined using a SQUID magnetometer for temperatures\nbetween2-400K.Magnetotransportmeasurementswere\nmeasured in the temperature range 2 - 400 K using the\nvan der Pauw method in a refrigerated cryostat placed in\nthe room-temperature bore of a cryogen-free 14 T mag-\nnet. XMCD measurements were taken using the total\nelectron yield mode at 300 K using 70 %circular polar-\nization of the beam. The measurements were taken at\nbeamline U4B at the Brookhaven National Synchrotron\nLight Source. The dichroism patterns were taken using\nbothpositiveandnegativefields, andcomparedwellwith\ndatausingbothpositiveandnegativecircularlypolarized\nlight. XMCDmeasuresthetotalvectoratom-specificmo-\nment <m> at the atom’s absorption edges.\nStructural properties\nThe inverse Heusler structure has a space group F ¯43m,\nas seen in Fig. 1(b) left. The Cr atoms occupy the (0,\n0, 0) and (1\n4,1\n4,1\n4) Wyckoff positions. The Co and Al\natoms occupy the (1\n2,1\n2,1\n2) and (3\n4,3\n4,3\n4) Wyckoff posi-\ntions, respectively.[19]. The position of these atoms willarXiv:1506.01738v1  [cond-mat.mtrl-sci]  4 Jun 20152\nE\"n(E)\"\nEF\"\nInverse\"Heusler\"(F43m)\"Heusler\"La4ce\"(Fm3m)\"XA\"La4ce\"\"L21\"La4ce\"\"(b)\"\nCr\"Co\"Al\"(a)\"\nFIG. 1: Schematic illustration of the density of states for the half-metallic semiconductor. The spin-up (orange) electrons act as\na gapless semiconductor whereas the offset in the spin-down (green) carriers allow for half-metallicity. (b) The lattice of both the\ninverse Heusler (also known as the XA structure) (left) and the L2 1Heusler lattice (right). The atom arrangements along the <111>\ndiagonal are Cr-Cr-Co-Al for the XA structure and Cr-Co-Cr-Al for the L2 1structure.\nlead to a 4-atom Cr-Cr-Co-Al basis along the <111>\ndiagonal. This structure is similar to that of the L2 1\nHeusler phase Fm ¯3m seen in Fig. 1(b) right, which has a\n4-atom Cr-Co-Cr-Al basis.[20] Co 2CrAl in the L2 1struc-\nture was chosen for study due to its large spin polariza-\ntion and high Curie temperature.[21] Calculations indi-\ncate that increased Cr concentration in L2 1-structured\nCo2\u0000xCrxAl drives the magnetization to zero while re-\ntaining its half-metallic properties.[22] Recent work has\nshown that CoFeCrAl forms in a semi-ordered Heusler\nlattice with a large atomic magnetic moment.[23] It is\nchallenging to distinguish between the XRD pattern of\ntheXAandtheL2 1structureasonlyafewofthetheoret-\nical Bragg peak heights are different. In addition, when\natomic mixing occurs, the increased symmetry causes the\nreduction of the number peaks.[24]\nFigure 2(a) shows the XRD spectrum of a sample an-\nnealed at 1000oC measured using synchrotron radiation\n(\u0015= 0.459 Å) indicating that Cr 2CoAl coexists with\ncubic Cr and CoAl phases (seen in Fig. 2(b)). Previ-\nous calculations predicted that Cr and CoAl would form\nupon the decomposition of Cr 2CoAl.[15] The results of\nRietveld refinement (Table 11.1) indicate that the sam-\npleconsistsmainlyofdisorderedCr 2CoAl(48to61vol %)\nfollowed by cubic Cr (0.1 to 14 vol %). The only defini-\ntive observation of chemically-ordered Cr 2CoAl was in\nthe sample annealed at 1000oC, which is identified by\nthe <111> peak at 7.9 degrees shown in the inset of Fig.\n2(a). The <111> reflection is a necessary condition of\nthe chemically-ordered Heusler phase.[25]\nThe XRD analysis leads to a lattice constant a= 5.794\nÅ for the ordered phase of Cr 2CoAl, about 0.3 %ex-\npanded from the disordered phase. This experimental\nlattice constant matches well with the expected a= 5.79\nÅ for the theoretically-optimized magnetic moment and\nactivation energy for the XA structure.[15] The Cr lat-\ntice parameter varied between a= 2.876 - 2.881 Å, de-\npending on the annealing temperature. The CoAl latticeTC\nFIG. 2: (a) The Rietveld refined XRD patterns of the\nCr2CoAl sample annealed at 1000oC using \u0015= 0.459 Å.\nThe XRD pattern was refined via GSAS and indicated that\nthere was Cr and CoAl mixed with Cr 2CoAl. The fitting pa-\nrameters are listed in the figure, indicating a good fit. (Inset)\nThe (111) lattice peak is from the Cr 2CoAl ordered phase and\nwould not appear if the atoms were highly mixed. (b) The\ncubic Cr and CoAl bcc lattices.\nconstant was found to be a= 2.868 \u00060.001 Å and did\nnot vary with annealing temperature. The anisotropic\nstrain S400 and S220 was found for the Cr 2CoAl lattice,\nwhich broadens the disordered Cr 2CoAl peaks.3\nTABLE I: Results of Rietveld refinement of the XRD spectra of Cr 2CoAl for various annealing temperatures. The table displays the\npercentage of each compositional phase and lattice constants. The Cr 2CoAl ordered Heusler structure is only achieved for 1000oC\nannealing and has a lattice constant expanded by 0.3 %. The anisotropic strains (S400 and S220) are also shown.\nMagnetic and transport properties\nSQUID magnetometry was used to measure the mag-\nnetic properties of the Cr 2CoAl samples as a function\nof applied magnetic field (H) and temperature. Two\nmagnetic components are identified: ( i) a paramagnetic\n(PM) component that varies at the lowest temperatures;\nand ( ii) a ferrimagnetic (FiM) component that varies at\nhigher temperatures. It is assumed that these two com-\nponents are from the two main phase segregates CoAl\n(PM) and Cr 2CoAl (FiM). Fig. 3(a) shows the mag-\nnetization M(H) for the chemically-ordered sample an-\nnealed at 1000oC where it is seen that the PM com-\nponent increases strongly with decreasing temperature\nin the range 50 to 10 K. Alternatively, the FiM com-\nponent (after subtracting the PM component, Fig. 3(a\ninset) saturates quickly and the saturation moment de-\ncreases moderately over a large temperature range, 250\nto 400 K. Fig. 3(b) plots the temperature dependence of\nthe two components. The PM component attributed to\nCoAl rises quickly for T < 50 K. Fitting this component\nto the Curie-Weiss formula, M =C\nT\u0000\u0012, whereC= 6.7\nemu/gK and \u0012= 11.5 K. The positive value for \u0012can\nalso been found by extrapolating \u001f(1/M), confirming\nthe presence of FM interactions. The FiM component\nattributed to Cr 2CoAl (inset,Fig. 3(b)) was determined\nby subtracting the PM component that was nearly inde-\npendent of temperature at high temperatures. The data\nwas fit to a mean-field model for T < T C, where M =\nMo(1-T\nTC)1\n2, shown by the curve, and the Curie temper-\nature was found to be T C\u0018750 K. The large Curie\ntemperature and the low magnetic moment of the ferri-\nmagnetic data indicates that some ordered Cr 2CoAl was\nsuccessfully formed in this sample.\nCsCl-type CoAl can have varying magnetic properties\ndepending on local defects (vacancies and antisites) and\nsmall changes in concentration.[24, 26, 27] It has been\nfound that Co xAl1\u0000xshows paramagnetic behavior with\na negative Curie-Weiss temperature at compositions in\nthe range x = 50.3-50.6 %, and a positive Curie-Weiss\ntemperature for x = 50.9-51.7 %. The Curie-Weiss con-\nstant for our PM data is \u0012= 12.9 K, indicating that the\nCoAl phase might be slightly Co-rich and the Co atomsferromagnetically coupled in CoAl. Fig. 3(c) plots the\nfield-cooled (FC) and zero-field-cooled (ZFC) magnetic\ndata taken at low-field (10 mT). The difference in the FC\nand ZFC magnetizations indicates that the CoAl phase\nhasaspinglasscomponentatallannealingtemperatures.\nThe ZFC curve shows a clear maxima at T* = 55 K, co-\nincident with a spin-glass transition in Co xAl1\u0000x, and\nreported to occur for a slightly Co-rich environment (x\n= 50.9-51.7 %).[26] The low-field ZFC magnetization is\ncompared to the electrical resistivity ( \u001a(T)) in Fig. 3(c)\nwhich also shows a cusp at 55 K similar to the spin-\nglass cusp at T* = 55 K in the ZFC moment. The\nspin glass transition temperature measured here is con-\nsistent with previous measurements of slightly Co-rich\nCoAl.[27, 28] Previous resistivity measurements on Co-\nrich CoxAl1\u0000xdetermined that the composition affects\nboth the resistivity value and temperature minima cor-\nresponding to the Kondo effect.[29, 30] The overall trend\nin\u001a(T) in Fig. 3(c) shows a metallic-like increase for\nincreasing temperature, with a room temperature resis-\ntivity of\u001a= 73\u0016\ncm. This value matches with previ-\nous data on Co xAl1\u0000x(x = 50.9).[30] The resistivity of\npure CoxAl1\u0000xfollows Matthiessen’s rule [31] where the\nresistivity is the sum of temperature-dependent and in-\ndependent parts, but in our case the presence of the Cr\nand Cr 2CoAl phases can also affect the impurity scat-\ntering and electron-electron scattering terms.[29] The ef-\nfects from the Cr phase are minimal since the metallc Cr\nis a known spin-density-wave antiferromagnet with T N\n= 310 K, which contributes minimally to magnetometry\nmeasurements.[32, 33] There is no evidence of magnetic\neffects from the Cr segregates due to their low concen-\ntration and we find no evidence of a Néel transition.\nXMCD indicating antiferromagnetically coupled Cr\nand Co\nXAS results of the L 3and L 2edges, where the top\nleft and right shows the spectra for Cr and Co shown\nin Figure 4. The XAS data were first fit to determine\nthe valence states of the Cr and Co d-orbitals[34, 35]\nand indicate non-integer values that due to the mixed-4\nFIG. 3: Magnetic properties and resistivity of the Cr 2CoAl sample annealed at 1000oC, which are similar to samples annealed\nbetween 600 and 900oC. (a) Magnetization versus field for the low temperature and high temperature (inset) components. At\nlow temperatures the dominant PM trend increases below 50 K. In the inset, at high temperatures the FiM component, with PM\nsubtracted, changes moderately (scale in units of \u0016B/f.u per formula unit). (b) M(T) measured at 1 T, where points are data and\ncurves are model fits. The PM component (grey curve) was fit to Curie-Weiss model M = C/(T- \u0012), where \u0012= 11.5 K. The inset\nshows the FiM component fit to the mean-field model M = M o(1-T/T C)1/2, where T C\u0018750 K. (c) Comparison of M(T) measured\nat 10 mT with the resistivity. The M(T) data shows the spin glass behavior typical of the CoAl FM phase at low temperatures where\nT*= 55 K. The upper purple curve is the resistivity, which also shows the spin glass transition with a room temperature resistivity of\n\u001a= 73 \u0016\ncm.\nFIG. 4: (left upper) The average Cr (left) and Co (right) L-edge XAS patterns for various annealing temperatures. (left lower)\nThe XMCD signal at 1.5 Tesla for the Cr and Co L-edges. (right) The extracted total moment of the Cr and Co atoms. The\ntotal moment (labeled \"Co*\") is the sum of the orbital and spin moments.[25] The magnetic moment of Co was multiplied by\nthe phase fraction of the Cr 2CoAl found through Rietveld refinement, which is depicted by the solid FM-Co line.\nphase composition. The Cr d-orbitals were found to be\na mixture of 3 d3and 3 d4valence states. The effective\nnumber of electrons in the d-orbitals ranges between 3.2\n- 3.6 for the Cr atoms and 5.3 - 6 for the Co atoms.\nX-raymagneticcirculardichroism(XMCD)spectrafor\nCr and Co atoms are shown by the lower curves in Fig-\nure 4.[36] The integrated Cr L 3/L2integrating branch-\ning ratios were found to be 1.66 for the ingots, con-\nsistent with previous values resulting from strain andsmall crystallite size determinations.[37–39] The total ex-\ntracted atomic magnetic moments are plotted in Fig.\n4(b), which is the sum of the orbital and spin magnetic\nmoments.[14,27]TheresultantXMCDmomentistheav-\neragemomentsofthetwoCratomsintheinverseHeusler\nlattice. The Cr magnetic moment is small and nega-\ntive, and within the range of the expected total moment\n(mTCr= -0.13\u0016B).[15] (This Cr moment is not antici-\npated to be affected by the small percentage of antifer-5\nromagnetic Cr, and would not contribute to the XMCD\nsignal.[40]) The total Co moment is shown by the square\npoints and (dashed) solid line in Fig. 4(right), labeled\n\"Co*\". However, the measured Co moments are affected\nby the presence of the paramagnetic CoAl phase in these\nsamples.[41] In order to extract the Co moment asso-\nciated with the Cr 2CoAl phase, the moment associated\nwith the CoAl was subtracted from the total signal by\nconsidering the respective phase fractions and assuming\nthat the Co moments have equal contributions to the to-\ntal Co magnetic signal. This extracted Cr 2CoAl moment\nis represented by the circles and the solid (blue) curve\nlabeled \"Co.\" It is seen that the Cr and Co magnetic mo-\nments are antiferromagnetically coupled, with equal but\nopposite moment, producing a low moment in Cr 2CoAl,\nand agrees with the low total moment observed in the\nmagnetometry results.\nSummary and outlook\nCr2CoAl bulk samples were prepared and annealed at\nvarious temperatures from 600 to 1000oC. Rietveld re-\nfinement performed on high-resolution synchrotron XRD\nindicated that Cr 2CoAl was formed at all annealing tem-\nperatures. Magnetic measurements indicated the exis-\ntence of both ferrimagnetic and paramagnetic compo-\nnents contributed by the two main phases in the samples.\nXMCD measurements allowed extraction of the Cr and\nCo magnetic moments and confirmed antiferromagneti-\ncally coupling. This study indicates that while Cr 2CoAl\ncan be synthesized in bulk form, there is a clear need for\nimproved synthesis methods to improve phase purity. In\nthe future, non-equilibrium synthesis of thin films could\nbe expected to provide a better route to single-phase ma-\nterials that would be required for electronic devices.\nAcknowledgements\nWe thank T. Hussey for assistance with magnetome-\ntry and P. Wei for assistance with preparing samples for\ntransport measurements. We thank D. Arena at NSLS\nbeamline U4B for his guidance. The work was supported\nbytheNationalScienceFoundationgrantsDMR-0907007\nand ECCS-1402738. Use of the National Synchrotron\nLight Source, Brookhaven National Laboratory, was sup-\nported by the U.S. Department of Energy, Office of Sci-\nence, Office of Basic Energy Sciences, under Contract\nNo. DE-AC02-98CH10886. Use of the Advanced Photon\nSource at Argonne National Laboratory was supported\nby the U. S. Department of Energy, Office of Science,\nOffice of Basic Energy Sciences, under Contract No. DE-\nAC02-06CH11357. We thank the team at 11-BM for the\nXRD measurements. M.E.J. thanks T. Jamer and K.\nJamer for their transportation assistance. M.E.J. is sup-ported by the International Centre for Diffraction Data’s\nLudo Frevel Scholarship.\nReferences\n[1] M. Meinert. Modified Becke-Johnson potential investiga-\ntion of half-metallic Heusler compounds. Phys. Rev. B ,\n87:045103, 2013.\n[2] S. Skaftouros, K. Özdoğan, E. Sasiğlu, and I. Galanakis.\nGeneralized Slater-Pauling rule for the inverse Heusler\ncompounds. Phys. Rev. B , 87:022420, 2013.\n[3] I. Galanakis and E. Sasiğlu. High T chalf-metallic fully-\ncompensated ferrimagnetic Heusler compounds. Appl.\nPhys. Lett. , 99:052509, 2011.\n[4] X.L. Wang. Proposal for a new class of materials: Spin\ngapless semiconductors. Phys. Rev. Lett. , 100:156404,\n2008.\n[5] G.Y. Gao and K.L. Yao. 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Nordlund.\nThickness dependence of the L 2;3branching ratio of Cr\nthin films. J. All. Comp. , 508:233–237, 2010.\n[40] J. Stöhr and H.C. Seigmann. Magnetism: From Fun-\ndamentals to Nanoscale Dynamics . Springer, 1 edition,\n2006.\n[41] E. Beaurepaire. Magnetism: A synchrotron radiation ap-\nproach. Springer, 2006."    },    {        "title": "1911.05270v3.Variety_of_order_by_disorder_phases_in_the_asymmetric__J_1_J_2__zigzag_ladder__From_the_delta_chain_to_the__J_1_J_2__chain.pdf",        "content": "Variety of order-by-disorder phases in the asymmetric J1\u0000J2zigzag ladder: From the delta chain\nto the J1\u0000J2chain\nTomoki Yamaguchi,1Stefan-Ludwig Drechsler,2Yukinori Ohta,1and Satoshi Nishimoto2, 3\n1Department of physics, Chiba University, Japan\n2Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany\n3Department of Physics, Technical University Dresden, 01069 Dresden, Germany\n(Dated: March 13, 2020)\nWe study an asymmetric J1-J2zigzag ladder consisting of two different spin-1\n2antiferromagnetic (AFM; J2,\n\rJ2>0) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM; J1<0) inter-leg interaction. On the\nbasis of density-matrix renormalization group based calculations the ground-state phase diagram is obtained as\nfunctions of \randJ2=jJ1j. It contains four kinds of frustration-induced ordered phases except a trivial FM\nphase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather\nconventional order by disorder. Still, it is interesting to note that the VBS states possess an Affleck-Kennedy-\nLieb-Tasaki-type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which\nis distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic\norders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is\nbroken. In other words, the system lowers its energy via the FM inter-leg interaction by polarizing both of the\nAFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state\nappears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger-liquid\nbehavior in the AFM Heisenberg legs.\nI. INTRODUCTION\nLow-dimensional frustrated quantum magnets, in which a\nmacroscopic number of quasi-degenerate states compete with\neach other, provide an ideal playground for the emergence of\nexotic phenomena1. For instance, the interplay of frustration\nand fluctuations could lead to unexpected condensed matter\norders at low temperatures by spontaneously breaking some\nsort of symmetry order by disorder2. Long-range ordered\n(LRO) magnetic state with breaking a spatial symmetry as\nwell as valence bond solid (VBS) state with a formation of\ndisentangled-unit–like local spin-singlet pair are typical ex-\namples of order by disorder. Moreover, when quantum fluctu-\nations between the quasi-degenerate states prevent a selection\nof particular order, one ends up with spin liquids. Modern the-\nories have brought us new insight by identifying spin liquids\nas topological phases of matter3,4. In recent years the realiza-\ntion of topological phases in frustrated spin systems has been\none of the central topics in condensed matter physics5–7.\nIn one-dimensional (1D) and spin-1\n2case, quantum fluc-\ntuations are maximized so that we may place more expec-\ntations on the discovery of novel ground states by the co-\noperative effects with magnetic frustration. A most simply-\nstructured 1D frustrated system is the so-called spin-1\n2J1-\nJ2chain consisting of nearest-neighbor J1and next-nearest-\nneighborJ2couplings. When both J1andJ2are antiferro-\nmagnetic (AFM), the ground state is a VBS, the nature of\nwhich can be grasped by the Majumdar-Ghosh (MG) model8,\natJ2=J1>\u00180:249,10. The idea of MG model was generalized\nto the Affleck-Kennedy-Lieb-Tasaki (AKLT) model11exhibit-\ning spin- 1VBS ground state with a symmetry protected topo-\nlogical order12.\nMeanwhile, the J1-J2chain with ferromagnetic (FM)\nJ1and AFMJ2, which is known as a standard magnetic\nmodel for quasi-1D edge-shared cuprates such as Li 2CuO 213,\nLiCuSbO 414, LiCuVO 415, Li2ZrCuO 216, Rb 2Cu2Mo3O1217,and PbCuSO 4(OH) 218, encloses wider array of states of mat-\nter. Theoretically, this model has been extensively studied:\nOther than a trivial FM state in the dominant J1region, the\nground state is a topological VBS accompanied by sponta-\nneous multiple dimerization orders19,20(More details are de-\nscribed in Sec. III A of this article). It is also intriguing that\na vector chirality and multimagnon bound states are induced\nin the presence of magnetic field21–24. Especially, the detec-\ntion of nematic or higher multipolar phases is one of the most\nexciting experimental current issues14,25–29. Sensitive features\nto even tiny interchain couplings are another characteristic of\nthis system30–32.\nAnother typical example of frustrated 1D system is\nthe delta chain (or sawtooth chain). The lattice structure\nis a series of triangles, as shown in Fig. 1(b), which is\nsimilar to that of the J1-J2chain but certain parts of\nJ2bonds are missing. There have been several candi-\ndates for delta-chain and related materials: YCuO 0:2533,34,\n[Cu(bpy)(H2O)][Cu( bpy)(mal)(H2O)](ClO 4)235, Zn L2S4\n(L= Er, Tm, and Yb)36, Cu(AsO 4)(OH)\u00013H2O37,\nMn2GeO 438, Rb 2Fe2O(AsO 4)239, CuFe 2Ge240, Fe10Gd1041,\nCu2Cl(OH) 342, Fe2O(SeO 3)243, and V 6O1344. In these mate-\nrials, a wide array of complex phases has been experimentally\nobserved. Delta-chain systems may offer an outlook towards\npromising prospects on novel magnetic phenomena.\nThe magnetic properties of delta-chain systems are totally\ndifferent in different signs of J1andJ2. For the case of\nJ1<0andJ2>0(typically, referred as FM-AFM delta\nchain), only two corresponding materials have been recog-\nnized. One of them is malonatobridged copper complexes\n[Cu(bpy)(H2O)][Cu( bpy)(mal)(H2O)](ClO 4)235. The mag-\nnetic Cu2+ions with effective spin-1\n2form into a delta-chain\nnetwork. The base and the other exchange couplings in a tri-\nangle made of malonate were estimated, respectively, as AFM\n(J2= 6:0K) and FM ( J1=\u00006:6K) from the analysis of mag-\nnetic susceptibility \u001f(T); and, as AFM ( J2= 10:9K) and FMarXiv:1911.05270v3  [cond-mat.str-el]  12 Mar 20202\n(J1=\u000012:0K) from the fitting of magnetization curve (typ-\nically, referred as FM-AFM delta chain). In either case, the\nratio of AFM and FM couplings is close to 1. This means that\nthe material would be in the region of strong frustration. The-\noretically, the ground state was predicted to be a ferrimagnetic\nstate but the detailed spin structures are less understood45.\nIn fact, only qualitative behavior of measured magnetization\ncurve could be explained by assuming a ferrimagnetic ground\nstate46,47. A deeper understanding of the ferrimagnetic state\nis necessary to resolve the remaining discrepancy between ex-\nperiment and theory.\nThe second candidate of the FM-AFM delta-chain ma-\nterials is a mixed 3 d=4fcyclic coordination cluster system\nFe10Gd1041. In these days, the delta-chain physics is increas-\ningly attracting attention due to the synthesis of Fe 10Gd10.\nThis cluster consists of 10 + 10 alternating Gd and Fe ions.\nThe exchange couplings were estimated as FM ( J1=\u00001:0K)\nbetween Fe and Gd ions, AFM ( J2= 0:65K) between Fe\nions, and nearly zero between Gd ions; the magnetic ions\nform an FM-AFM delta chain short ring. The parameter ratio\nJ2=jJ1j= 0:65seems to be very close the FM quantum criti-\ncal pointJ2=jJ1j= 0:748Although the spin values of Fe and\nGd ions are higher than S= 1=2(S= 5=2andS= 7=2, re-\nspectively), quantum fluctuations would play important roles\nto determine the ground state because of the quantum critical-\nity49. This means that the magnetic properties can be drasti-\ncally changed upon even a small variation of external influ-\nences such as magnetic field, pressure, chemical means, and\ngating current. So, this delta chain material is drawing atten-\ntion also from the perspective of controlling magnetic states\nin molecular spintronics50.\nFor comparison, a few examples of delta-chain materials\nonly with AFM interactions ( J1;J2>0) have been also\nreported. With the help of MG-like projection method, the\nmagnetic properties of the AFM-AFM delta-chain are better\nunderstood than those of the FM-AFM one33,51–53. A pecu-\nliarly interesting feature is the dispersionless kink-antikink\ndomain wall excitations to the dimerized VBS ground state.\nA kink is highly localized only in the range of one triangle.\nThe first candidate of AFM-AFM delta-chain materials was\nthe delafossite YCuO 2:533. However, a first-principle calcula-\ntion revealed that the ratio of J2=J1in YCuO 2:5is out of the\nrange of the dimerized VBS ground state and additional in-\ntrachain FM interaction is significantly large34. Very recently,\nthe other candidate materials Cu 2Cl(OH) 3(S= 1=2)42and\nFe2O(SeO 3)2(S= 5=2) have been reported. They indeed\nexhibit characteristic features of AFM-AFM delta chain: a\nmagnetization plateau at half-saturation46,54in Cu 2Cl(OH) 3\nand an almost flat-band one-magnon excitation spectrum in\nFe2O(SeO 3)2.\nAs mentioned above, the research of frustrated 1D systems\nwithJ1-J2or delta-chain structures has become more and\nmore active. Interestingly, each of the J1-J2chain and the\ndelta chain is expressed as a limiting case of an asymmetric\nJ1-J2zigzag ladder, defined as two different AFM Heisen-\nberg chains coupled by zigzag-shaped interchain FM interac-\ntion [see Fig. 1(a)]. When one of the Heisenberg chains van-\nishes, it is the delta chain; and, when the Heisenberg chainsare equivalent, it is the J1-J2chain. However, it is known that\ntheir ground states are completely different. Then, one may\nsimply question how the two limiting cases are connected.\nOf particular interest is that the effect of exchange coupling\ntowards the J1-J2chain can be a likely perturbation in real\ndelta-chain compounds, e.g., the effect of tiny coupling be-\ntween Gd ions in Fe 10Gd10.\nIn this paper, we therefore study an asymmetric FM-AFM\nJ1-J2zigzag ladder using the density-matrix renormalization\ngroup (DMRG) technique. We first clarify the detailed spin\nstructure and low-energy excitations of ferrimagnetic state in\nthe delta chain limit. We suggest that the ferrimagnetic state is\na rare type of order by disorder, where the energy is lowered\nby FM fluctuation between two polarized AFM Heisenberg\nchains with spontaneous breaking of the global spin-rotation\nsymmetry. Then, we examine how the ferrimagnetic state is\ncollapsed and connected to the well-known incommensurate\nspiral state in the J1-J2chain. We also find there exist two\nkinds of VBS phases in the spiral region. Finally, we obtain\nthe ground-state phase diagram of asymmetric J1-J2zigzag\nladder with interpolating between the delta chain and J1-J2\nchain.\nThe paper is organized as follows: In Sec. II our spin\nmodel is explained and the applied numerical methods are de-\nscribed. In Sec. III we briefly mention to-date knowledge on\nthe ground state for two limiting cases of our spin model. In\nSec. IV we present our numerical results and discuss how the\ntwo limiting cases are connected. Finally we end the paper\nwith a summary in Sec. V .\n(a)\n(b)A\nB\nA\nB\n(c)\nA\nB\nFIG. 1. (a) Lattice structure of the asymmetric J1-J2zigzag ladder.\nThe indices ‘ A’ and ‘ B’ denote apical and basal chains, respectively.\nThe lattice spacing ais set as a distance between neighboring sites\nalong the chains. The AFM interaction in the apical chain is con-\ntrolled by\r. (b) Lattice structure of the so-called delta chain (or saw-\ntooth chain) which is realized in the limit of \r= 0. (c) Schematic\nrepresentation of the ferrimagnetic state with global spin-rotation-\nsymmetry breaking.3\nII. MODEL AND METHOD\nA. Model\nThe asymmetric J1-J2zigzag ladder is defined as two\nHeisenberg chains coupled by zigzag-shaped interchain inter-\naction. The lattice structure is sketched in Fig. 1(a). We call\na leg chain with larger interaction “basal chain” and the other\nwith smaller interaction “apical chain”. The Hamiltonian is\nwritten as\nH=J1X\niSA;i\u0001(SB;i+SB;i+1)\n+J2X\ni(SB;i\u0001SB;i+1+\rSA;i\u0001SA;i+1); (1)\nwhere SB;iis spin-1\n2operator at site ion the basal chain and\nSA;iis that on the apical chain. We focus on the case of\nFM interchain coupling ( J1<0) and AFM intrachain cou-\npling (J2>0). The intrachain interaction of apical chain is\ncontrolled by \r(0\u0014\r\u00141). The system (1) corresponds\nto the so-called delta chain (or sawtooth chain) at \r= 0\n[Fig. 1(b)] and the so-called J1-J2chain at\r= 1. The exist-\ning knowledge on the ground-state properties of these chains\nis briefly summarized in the next section. In our numerical\ncalculations the chain lengths of basal and apical chains are\ndenoted asLBandLA, respectively. The total number of\nsites isL=LB+LA. In this paper, we call the system\nfor0< \r < 1“asymmetric J1-J2zigzag ladder”, which has\nbeen little or not studied. In the case of J1>0andJ2>0,\nthere are a few studies55,56.\nB. DMRG methods\nIn order to examine the ground state and low-energy\nexcitations of asymmetric J1-J2zigzag ladder, we em-\nploy the DMRG techniques; namely, conventional DMRG\n(hereafter referred to simply as DMRG), dynamical DMRG\n(DDMRG), and matrix-product-state-based infinite DMRG\n(iDMRG) methods. They are used in a complementary fash-\nion to further confirm our numerical results.\nThe DMRG method is a very powerful numerical method\nfor various (quasi-)1D quantum systems57. However, some\ndifficulties are often involved in the DMRG analysis for\nstrongly frustrated systems like Eq. (1). First, the system\nsize dependence of physical quantities is usually not straight-\nforward. Therefore, relatively many data points are required\nto perform a reasonable finite-size scaling analysis. We thus\nstudy systems with length up to L= 161 (LB= 81;LA=\n80) under open boundary conditions (OBC) and systems with\nlength up to L= 64 (LB= 32;LA= 32 ) under periodic\nboundary conditions (PBC). Either OBC or PBC is chosen\ndepending on the calculated quantity. Second, a lot of nearly-\ndegenerate states are present around the ground state. To\nobtain results accurate enough, a relatively large number of\ndensity-matrix eigenstates mmust be kept in the renormal-\nization procedure. In this paper, we keep up to m= 8000density-matrix eigenstates, which is much larger than that kept\nin usual DMRG calculations for 1D systems, and extrapolate\nthe calculated quantities to the limit m!1 if necessary. In\nthis way, we can obtain quite accurate ground states within the\nerror of \u0001E=L = 10\u00008jJ1j.\nFor the calculation of dynamical quantities, we use the\nDDMRG method which has been developed for calculating\ndynamical correlation functions at zero temperature in quan-\ntum lattice models58. Since the DDMRG algorithm performs\nbest for OBC, we study a open cluster with length up to\nL= 129 (LB= 65;LA= 64 ). The DDMRG approach is\nbased on a variational principle so that we have to prepare\na ‘good trial function’ of the ground state with the density-\nmatrix eigenstates. Therefore, we keep m= 1200 to ob-\ntain the ground state in the first ten DMRG sweeps and keep\nm= 600 to calculate the excitation spectrum. In this way, the\nmaximum truncation error, i.e., the discarded weight, is about\n1\u000210\u00005, while the maximum error in the ground-state and\nlow-lying excited states energies is about 10\u00004jJ1j.\nThe iDMRG method is very useful because it enables us to\nobtain the physical quantities directly in the thermodynamic\nlimit59,60, if the matrix product state is not too complicated\nand the simulation can be performed accurately enough. In\nour iDMRG calculations, typical truncation errors are 10\u00008\nusing bond dimensions \u001fup to 6000 . In this way, the effective\ncorrelation length near criticality is less or at most equal to\n500, so that most of interesting parameter region of the system\n(1) can be reasonably examined by our iDMRG simulations.\nIII. PREVIOUS STUDIES FOR LIMITING CASES\nSo far, our system for two limiting cases, namely, J1-J2\nchain (\r= 1) and delta chain ( \r= 0), has been extensively\nstudied. In this section, we briefly summarize to-date knowl-\nedge on the ground state for the limiting cases.\nA.J1-J2chain (\r= 1)\nAt\r= 1, we are dealing with the J1-J2chain, which may\nbe also recognized as symmetricJ1-J2zigzag ladder. In the\nlimit ofJ2=jJ1j= 0, the system is a simple FM Heisenberg\nchain with FM ordered ground state. Increasing J2=jJ1j, the\nFM state persists up to J2=jJ1j= 1=4; then, a first-order\nphase transition from the FM to an incommensurate (“spiral”)\nstate occurs61. The total spin in the incommensurate phase\nis zero (Stot= 0). Since quantum fluctuations disappear at\nthe FM critical point, the critical value J2=jJ1j= 1=4can\nbe recognized similarly both in the quantum as well as in the\nclassical model62,63.\nAtJ2=jJ1j>1=4, the incommensurate correlations are\nshort ranged in the quantum model64,65. Instead, the sys-\ntem exhibits a spontaneous nearest-neighbor FM dimeriza-\ntion with breaking of translation symmetry, as a consequence\nof the quantum fluctuations typical of magnetic frustration,\ni.e, order by disorder. By regarding the ferromagnetically4\ndimerized spin-1/2 pair as a spin- 1site, the system is effec-\ntively mapped onto a spin- 1Heisenberg chain and an Affleck-\nKennedy-Lieb-Tasaki (AKLT)-like hidden topological order\nprotected by global Z2\u0002Z2symmetry is naively expected as\na Haldane state11,66(also see Sec. IV C 6). In fact, the hidden\norder has been numerically confirmed19,20.\nFurthermore, the existence of (exponentially small) singlet-\ntriplet gap at J2=jJ1j>\u00183:3was predicted by the field-theory\nanalysis67and its verification had been a longstanding issue.\nOnly recently, the gap was numerically estimated: with in-\ncreasingJ2=jJ1jit starts to open at J2=jJ1j= 1=4, reaches its\nmaximum\u00180:007jJ1jaroundJ2=jJ1j= 0:65, and exponen-\ntially decreases20. The ground state is a kind of VBS state with\nspin-singlet formations between third-neighbor sites. There-\nfore, the magnitude of gap basically scales to the strength of\nthird-neighbor valence bond.\nB. Delta chain ( \r= 0)\nAt\r= 0, the system is the delta chain consisting of a lin-\near chain of corner-sharing triangles [Fig. 1(b)]. The ground\nstate properties are less understood than those of the J1-J2\nchain. One main reason is that numerical investigation of the\ndelta chain is particularly difficult due to the strong magnetic\nfrustration and a number of nearly-degenerate states near the\nground state. Especially at the FM critical point is macroscop-\nically degenerate and consists of multi-magnon configurations\nformed by independent localized magnons and the special lo-\ncalized multi-magnon complexes48.\nNonetheless, a numerical study could identify the ground\nstate atJ2=jJ1j<1=2to be FM; that at J2=jJ1j>1=2\nto be ferrimagnetic45. The total spins of the ferromagnetic\nand ferrimagnetic phases are L=2andL=4, respectively. To\nunderstand the origin and the properties of this ferrimagnetic\nstate, the delta chain in the large limit of easy-axis exchange\nanisotropy was studied68. In this limit the system can be re-\nduced to a 1D XXZ basal chain under a static magnetic field\ndepending on the magnetic structure of apical chain. The\nground state was identified as ferrimagnetic with fully po-\nlarized apical spins and weakly polarized basal spins. It is\nexpected that some essential features may be inherent in the\nisotropic SU(2) limit. In fact, the spin structure agrees quali-\ntatively to the ferrimagnetic state determined in this paper [see\nFig. 1(c)].\nIV . RESULTS\nA. classical limit\nAs mentioned above, the FM critical point is known to be\nJ2=jJ1j= 1=4for theJ1-J2chain (\r= 1) andJ2=jJ1j= 1=2\nfor the delta chain ( \r= 0). To be examined first is how\nthe critical point changes between the limiting cases, i.e.,\n0< \r < 1. Since the quantum fluctuations vanish at the FM\ncritical point, the critical value can be exactly estimated by\nthe classical spin wave theory (SWT). The Fourier transform\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\n(a)\n(b)\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\n0.1\n0.4 0.6 0.800.2\n0.6 0.8FIG. 2. (a) Classical ground-state phase diagram of the asymmetric\nJ1-J2zigzag ladder. The phases are characterized by propagation\nvectorq=qminminimizingJq[Eq. (3)]: FM ( qmin= 0); commen-\nsurate (qmin=\u0019); incommensurate ( 0< qmin< \u0019). (b) Ground-\nstate phase diagram of the asymmetric J1-J2zigzag chain [Eq. (1)]\ndetermined by DMRG calculations. Inset: enlarged view around the\nPF phase.\nof our Hamiltonian (1) reads\nH=1\n2X\nqJq~Sq\u0001~S\u0000q (2)\nwith\nJq=(1 +\r)J2cosq\n\u0006q\n(1\u0000\r)2J2\n2(1\u0000cosq)2+ 4J2\n1cos2(q=2):(3)\nIf Eq. (3) has a minimum at q= 0, the system is in an FM\nground state. The FM critical point is thus derived as\nJ2;c\njJ1j=1\n2(1 +\r): (4)5\nAs shown in Fig. 2(a), the FM region is simply shrunk with\nincreasing\rbecause the AFM interaction is increased in the\napical chain. We have confirmed this FM critical boundary\nnumerically by calculating the total spin Stotof the whole\nsystem, which is defined as\nh~S2i=Stot(Stot+ 1) =X\ni;jh~Si\u0001~Sji: (5)\nIt can be also verified by finding the absence of LRO FM state\nin the spin-spin correlation functions. These results are shown\nin Appendix A.\nBy evaluating q(\u0011qmin) value to minimize Eq. (3), a\nclassical ground-state phase diagram is obtained as Fig. 2(a).\nThere are three kinds of LRO phases: FM phase with qmin=\n0, incommensurate phase 0< q min< \u0019 , and commensu-\nrate phase with qmin=\u0019. Since the ferrimagnetic state in\nFig. 1(c) is of commensurate with q=\u0019and the propagation\nnumber of the J1-J2chain is incommensurate, the SWT re-\nsults are consistent with those of the quantum system (1) in\nthe two limiting cases \r= 0 and\r= 1. Therefore, even in\nthe quantum system an incommensurate-commensurate phase\ntransition is naively expected at finite \rwithJ2=jJ1jfixed.\nB.\r= 0: delta chain\nAlthough the ground state of the delta chain is most prob-\nably ferrimagnetic at J2=jJ1j>1=2, the detailed magnetic\nstructure and properties have not been fully settled. To gain\nfurther insight into them, we here calculate the total spin, spin-\nspin correlation functions, and stabilization energy of ferri-\nmagnetic state for the delta chain. We need to pick through\nthe system-size dependence of the quantities to deal with non-\ntrivial finite-size effects of the delta chain.\n1. total spin\nInvestigating the total spin Stotis a simple way to explore\nthe possibility of ferrimagnetic state. As shown below, the\nsystem-size dependence of Stotis significantly different be-\ntween applying OBC and PBC. This obviously implies the\ndifficulty of performing numerical calculations for this sys-\ntem. Nevertheless, the fact that they should coincide in the\nthermodynamic limit L!1 makes the finite-scaling analy-\nsis even more reliable. We here use the DMRG method.\nLet us first see the case of OBC. In Fig. 3(a) the total spin\nper siteStot=Lis plotted as a function of inverse system size\n1=Lfor several values of J2=jJ1j. WhenJ2=jJ1jis order\nof1, the effect of strong frustration is explicitly embedded\nin the finite-size scaling; namely, due to the Friedel oscilla-\ntion,Stot=Lawkwardly oscillates with 1=L. However, as ex-\npected, such an oscillation no longer appears in a case of large\nJ2=jJ1j= 100 . WhenJ2=jJ1j\u001d1, a straightforward scaling\nis allowed since the frustration is much weaker. Eventually,\nfor all theJ2=jJ1jvalues,Stot=Lseems to be extrapolated to\n1=4in the thermodynamic limit. Although one may think that\n(a)\n(b)0 0.01 0.02 0.03 0.04 0.050.10.20.3\n0 0.02 0.04 0.06 0.0800.10.20.30.4FIG. 3. Finite-size scaling analysis of the total spin per site for the\ndelta chain ( \r= 0), where (a) OBC and (b) PBC are applied. The\ndotted lines are guide for eyes.\nthe PBC should be applied if the Friedel oscillation is prob-\nlematic, the situation is not so simple as explained below.\nWe then turn to the case of PBC. In Fig. 3(b) Stot=Lis plot-\nted as a function of 1=Lfor several values of J2=jJ1j. Unlike\nin the case of OBC, the oscillation is not seen in Stot=Lvs.\n1=L. Instead, there exists a ‘critical’ system size to achieve\nfiniteStot=Lin the ground state. This is caused by a kind\nof typical finite-size effects: Under the PBC, the basal chain\nforms a plaquette singlet and the basal spins are more or less\nscreened. This screening leads to the reduction of FM in-\nteraction between the basal and apical chains although the\nFM fluctuations between the two chains are essential to sta-\nbilize the ferrimagnetic state (see Sec. IV B 2). Besides, only\na small screening may be sufficient to collapse the ferrimag-\nnetic state because the stabilization gap of ferrimagnetic state\nis very small (see Sec. 5). Consequently, the ferrimagnetic\nstate can be readily prevented under the PBC. For reference,\nthe system-size dependence of energies for spin-singlet and\nferrimagnetic states is shown in Appendix B. Since the triplet\nexcitation gap of plaquette singlet roughly scales to J2=jJ1j\nwith a fixed system size, the critical system size is larger for\nlargerJ2=jJ1jas seen in Fig. 3(b). Once the system size goes\nbeyond the critical one, Stot=Lapproaches smoothly to 1=4\natL!1 . The fitting function is Stot=L= 1=4 + 1=Lfor\nanyJ2=jJ1j.6\n0 50 100\n|i−j|−0.10.00.10.2/angbracketleftBig\nSz\nα,iSz\nβ,j/angbracketrightBig\n(a)\nA-A\nA-BB-B\n0 50 100\n|i−j|−0.10.00.10.2/angbracketleftBig\nSz\nα,iSz\nβ,j/angbracketrightBig\n(b)\n0 50 100\n|i−j|−0.10.00.10.2/angbracketleftBig\nSz\nα,iSz\nβ,j/angbracketrightBig\n(c)\n0.6 1.0 1.5 2.0\nJ2/|J1|0.00.10.20.30.40.5\n/angbracketleftBig\nSz\nA,i/angbracketrightBig\n/angbracketleftBig\nSz\nB,i/angbracketrightBig(d)\nFIG. 4. Spin-spin correlation function hSz\n\u000b;iSz\n\f;jifor the delta chain\n(\r= 0) as a function of distance ji\u0000jjat (a)J2= 0:6, (b)1, and\n(c)2. The totalSzsector is set to be Sz\ntot=L=4. The legends\ndenote as A-A: (\u000b;\f) = (A;A), B-B: (\u000b;\f) = (B;B), and A-B:\n(\u000b;\f) = (A;B). (d) Averaged values of hSz\n\u000b;iion the apical and\nbasal sites as a function of J2=jJ1j.\nThus, we have confirmed that the total spin of the delta\nchain atJ2=jJ1j>1=2isStot=L=4, indicating the ferri-\nmagnetic state, and the ferrimagnetic phase persists up to the\nlargeJ2=jJ1jlimit.\n2. spin-spin correlation\nThen, in order to determine the magnetic structure of the\nferrimagnetic state, we examine spin-spin correlation func-\ntionshS\u000b;i\u0001S\f;jibetween apical sites: (\u000b;\f) = (A;A),\nbetween basal sites: (\u000b;\f) = (B;B), and between apical and\nbasal sites: (\u000b;\f) = (A;B)=(B;A). For convenience, we\nfix the z-component of total spin at the total spin retained in\nthe ferrimagnetic ground state, i.e., Sz\ntot=Stot=L=4. It\nlifts the ground-state degeneracy due to the SU(2) symmetry\nbreaking and the (spontaneous) magnetization direction is re-\nstricted to the z-axis direction. Hence, we need only to see\nhSz\n\u000b;iSz\n\f;jiinstead ofhS\u000b;i\u0001S\f;ji.\nIn Fig. 4(a)-(c) iDMRG results of the correlation function\nhSz\n\u000b;iSz\n\f;jiare plotted as a function of distance ji\u0000jjfor\nJ2=jJ1j= 0:6,1, and 2. We clearly see long-ranged cor-\nrelations indicating a magnetic order for all the J2=jJ1jval-\nues. If no magnetic order exists, all the correlation func-\ntions converge to (Sz\ntot=L)2= 1=16at long distance limit\nji\u0000jj ! 1 . The intrachain correlations hSz\nA;iSz\nA;jiand\nhSz\nB;iSz\nB;jiare both positive at the long distance, and the for-\nmer correlation is much larger than the latter one. This means\nthat the apical spins are nearly-fully polarized and the basalspins are only weakly polarized. In addition, a N ´eel-like stag-\ngered oscillation is clearly seen leastwise at the short distance\ninhSz\nB;iSz\nB;ji. These correlations immediately correspond\nto a ferrimagnetic state, which is schematically sketched in\nFig. 1(c). This picture seems to be valid in the whole region\nofJ2=jJ1j>0:5.\nNow, another question may arise: Is the N ´eel-like staggered\nspin alignment on the basal chain LRO? The answer is NO.\nAlthoughhSz\nB;iSz\nB;jiindeed exhibits an AFM oscillation at\nshort distancesji\u0000jj, it vanishes at the long distance limit.\nIn fact, the oscillating part of hSz\nB;iSz\nB;jiexhibits a power-\nlaw decay indicating a critical behavior of the Tomonaga-\nLuttinger liquid (TLL) (see Appendix C). This also excludes\nthe possibility of VBS formation in the basal chain because\nan exponential decay of the spin-spin correlation should be\nfound in a VBS state. It means that, very surprisingly, the\npresent ferrimagnetic order is an order by disorder but without\nany geometrical symmetry breaking. Instead, the magnetic\nfrustration is relaxed by a spontaneous breaking of the global\nspin-rotation symmetry. In other words, the system can gain\nenergy from the FM interaction between the apical and basal\nchains by polarizing both of the chains. This is a very rare\ntype of order by disorder.\nThis picture of order by disorder may be further convinced\nby looking at the relation between the polarization level of\nbasal spins and the stabilization of ferrimagnetic order. Thus,\nto see theJ2=jJ1j-dependence of polarization level in more\ndetail, we examine the expectation values of local spin mo-\nmentSz\nA;iandSz\nB;i. In Fig. 4(d) the averaged values of\nhSz\nA;iiandhSz\nB;iiare plotted as a function of J2=jJ1j. We\nfind that with increasing J2=jJ1j,hSz\nA;iiincreases and sat-\nurates at 1=2; while,hSz\nB;iidecreases and goes down to 0\nin the large J2=jJ1jlimit. The decrease of hSz\nB;iimay be\nnaively expected by the fact that the basal chain approaches\na 1D SU(2) Heisenberg model at large J2=jJ1j; while, the\nincrease ofhSz\nB;iiis a simple consequence of the condition\nhSz\nA;ii+hSz\nB;ii= 1=2. If the order-by-disorder picture asso-\nciated by interchain FM interaction is true, the ferrimagnetic\nstate at larger J2=jJ1jshould be fragile because of smaller po-\nlarization of basal spins. Actually, the stabilization energy of\nferrimagnetic order decreases with increasing J2=jJ1jas de-\nscribed in the following subsection.\n3. stabilization gap\nTo figure out the stability of the ferrimagnetic state, we\ncalculate a stabilization gap defined by energy difference be-\ntween the ferrimagnetic state and a lowest state in the Stot= 0\nsector:\n\u0001(L) =E0(0;L)\u0000E0(Sg:s:;L);\u0001 = lim\nL!1\u0001(L);(6)\nwhereE0(S;L)is the lowest-state energy of L-site periodic\nsystem in the Stot=Ssector, andSg:s:is a total spin of the\nground state. The energy of ferrimagnetic state can be sim-\nply obtained as a ground-state energy, though, as mentioned7\n0.5 1 1.5 21 210-310-20 0.05 0.1 0.1500.010.020.030.04\n00.0050.010.015(a)\n(b)\nFIG. 5. Energy difference between the lowest Stot= 0 state and\nferrimagnetic ground state for the delta chain, as a stabilization gap\nof the ferrimagnetic state. (a) Finite-size scaling and (b) the extrapo-\nlated values \u0001=jJ1jas a function of J2=jJ1j. Inset: Semi-log plot of\n\u0001=jJ1jas a function of J2=jJ1j. The dotted line is a fit for the large\nJ2=jJ1jregion: \u0001=jJ1j= 0:037 exp(\u00001:3J2=jJ1j).\nabove, the total spin Sg:s:=Lin the ferrimagnetic state can de-\nviate from 1=4for a finite cluster. Whereas, it is nontrivial\nto estimate the energy of a lowest state in the Stot= 0 sec-\ntor because it is an excited state and there are many quasi-\ndegenerate states near the ferrimagnetic ground state. A\nproper way needs to be provided to extract the Stot= 0state.\nWe therefore consider the following augmented Hamiltonian\nH0=H+\u0015~S2; (7)\nwhereHis our original Hamiltonian (1) and the additional\nterm corresponds to the total spin operator ( \u0015>0). By setting\n\u0015to be large enough, all the states with Stot>0are lifted.\nEventually, we can find a state with Stot= 0 as the lowest\nstate.\nIn Fig. 5(a) the energy difference \u0001(L)is plotted as a func-\ntion of 1=Lfor severalJ2=jJ1jvalues. Since the magnetic\nfrustration causes a sine-like oscillation in \u0001(L)vs.1=L20,\nthe finite-size scaling analysis is not very straightforward.\nStill, the plotted values show the sine-like oscillation with\nroughly more than one period, an acceptable scaling with lin-\near function \u0001(L)=jJ1j= \u0001=jJ1j+A=L(Ais fitting parame-\nter) may be possible. Actually, even if we assume a more gen-\neral fitting function \u0001(L)=jJ1j= \u0001=jJ1j+A=L\u0011, the extrap-\n012\n0\n00.10.20.3\n0\n00.10.20.30.4\n0\n00.10.20.3\n0(a) (b)\n(c) (d)0 1 Intensity (arb. units)FIG. 6. Dynamical spin structure factors for (a) apical and (b) basal\nchains atJ2=jJ1j= 0:6. (c)(d) The same spectra at J2=jJ1j=\n1. Finite broadening \u0011is introduced: \u0011= 0:03jJ1jin (a) and (c),\n\u0011= 0:02jJ1jin (b), and\u0011= 0:05jJ1jin (d). The dotted lines\nare approximate analytical expressions of the main dispersions (see\ntext).\nolated values of \u0001=jJ1jare almost unchanged because \u0011\u00191\nis always achieved. In Fig. 5(b) the extrapolated values of \u0001\nare plotted as a function of J2=jJ1j. At the FM critical point\nJ2=jJ1j= 1=2, the lowest energies for all Stotsectors are de-\ngenerate and it leads to \u0001 = 0 . As soon as the system goes\ninto the ferrimagnetic phase, the stabilization gap \u0001steeply\nincreases, reaches a maximum around J2=jJ1j= 1, and de-\ncreases with further increasing J2=jJ1j. The magnetic frustra-\ntion would be strongest at the maximum position J2=jJ1j\u00181\nbecause each triangle is fully frustrated. This is another in-\ndication of the fact that the ferrimagnetic state is originated\nfrom order by disorder. As shown in the inset of Fig. 5(b), the\nstabilization gap seems to decay exponentially with J2=jJ1jin\nthe largeJ2=jJ1jregion. It means that the ferrimagnetic state\nis rapidly destabilized in the large J2=jJ1jregime although it\npersists up to J2=jJ1j=1in a precise sense. This is con-\nsistent with the rapid decrease of hSz\nB;iiwithJ2=jJ1j. The\nsystem can gain only little energy from the interchain FM in-\nteraction in case where the basal spins are not really polarized.\nIn short, the quantum fluctuations between the apical and the\nbasal chains play an essential role to stabilize the ferrimag-\nnetic state.\n4. dynamical spin structure factor\nIn order to provide further insight into the ferrimagnetic\nstructure, we investigate the low-energy excitations of the8\ndelta chain. We calculate dynamical spin structure factors\nfor both the apical and basal chains with using the DDMRG\nmethod. The dynamical spin structure factor is defined as\nS\u000b(q;!) =1\n\u0019Imh 0jSz\n\u000b;q1\n^H+!\u0000E0\u0000i\u0011Sz\n\u000b;qj 0i\n=X\n\u0017jh \u0017jSz\n\u000b;qj 0ij2\u000e(!\u0000E\u0017+E0); (8)\nwhere\u000bdenotes either apical (A) or basal (B) chain, j \u0017iand\nE\u0017are the\u0017-th eingenstate and the eigenenergy of the system,\nrespectively ( \u0017= 0 corresponds to the ground state). Under\nOBC, we define the momentum-dependent spin operators as\nSz\n\u000b;q=r\n2\nL\u000b+ 1X\nleiqriSz\n\u000b;i; (9)\nwith (quasi-)momentum q=\u0019Zx=(L\u000b+ 1) for integers 1\u0014\nZx\u0014L. We use open clusters with LA= 31;LB= 32\nforSB(q;!)and withLA= 64;LB= 65 forSA(q;!). In\nFig. 6, DDMRG results of the dynamical spin structure factors\nare shown for J2=jJ1j= 0:6and1.\nLet us see the spectrum for the apical chain. Since the\nspins are fully polarized on the apical chain, the main dis-\npersion ofSA(q;!)is basically described by that of the 1D\nFM Heisenberg chain. For J2=jJ1j= 0:6, a precise fitting\nleads to!q=jJ1j=J0(cos(q)\u00001) +J00(cos(2q)\u00001)with\nnearest-neighbor FM coupling J0=\u00000:11and next-nearest-\nneighbor AFM coupling J00= 0:016. Interestingly, we find\nthat the dominant FM coupling is effectively induced on the\napical chain in spite of no direct interaction between apical\nsites. A similar fitting for J2=jJ1j= 1 givesJ0=\u00000:075\nJ00= 0:018. The reduction of jJ0jwith increasing J2=jJ1j\nis naturally expected because the apical spins are completely\nfree in the large J2=jJ1jlimit. This reduction also reflects the\nweakening of ferrimagnetic state.\nWe then turn to the spectrum for the basal chain. Although\nthe basal chain is weakly polarized, we expect that the funda-\nmental excitations could be at least qualitatively described by\nthose of the 1D SU(2) Heisenberg chain because the dom-\ninant short-range correlation is AFM. For J2=jJ1j= 0:6,\nthe magnon dispersion (lower bound of the continuum) of\nSB(q;!)is certainly sine-like function and the well-known\nshaped two-spinon continuum is seen. However, surpris-\ningly, such the weak spin polarization ( SB;tot=LB= 0:11)\ndrastically suppresses the dispersion width down to 0:15jJ1j\nfrom that of the 1D SU(2) Heisenberg chain \u0019jJ1j=269. It\nis interesting that the dispersion width is rapidly recovered\nto0:75jJ1jwhen the spin polarization is slightly reduced to\nSB;tot=LB= 0:085atJ2=jJ1j= 1. Another effect of the\nweak polarization on the dispersion is a shift of node. Due to\nthe dominant AFM fluctuation on the basal chain in the whole\nregion of ferrimagnetic phase, a largest peak always appears\nat(q;!) = (\u0019;0). If there is no spin polarization, the other\nnode should be at q= 0 but it is actually shifted to higher q\nvalue as seen in Fig. 6(b)(d). This behavior is similar to the\ncase in the presence of magnetic field. Namely, the node po-\nsition can be expressed as q= 2\u0019hSB;toti=LBwherehSB;toti\nis the total spin of the basal chain with length LB.\n0 0.01 0.02 0.0300.10.20.3\n0 0.1 0.200.10.20.3(a)\n(b)FIG. 7. (a) System-size dependence of total spin per site Stot=Lfor\nseveral\rvalues withJ2=jJ1j= 0:6fixed. The dashed line indicates\nthe value ofStot=Lfor the full ferrimagnetic state. (b) The L!1\nextrapolated values of Stot=Las a function of \r.\nC. Finite\r: asymmetric J1-J2zigzag ladder\nAs described above, we have confirmed that the ferrimag-\nnetic state is indeed stabilized at J2=jJ1j>1=2in the delta\nchain (\r= 0). Then, let us see what happens when the apical\nsites are connected by AFM interaction, the strength of which\ncan be controlled by \r. Since the system is in a singlet ground\nstate, i.e.,Stot= 0, in theJ1-J2chain (\r= 1), the collapse of\nferrimagnetic state is naively expected at some \r(<1). Inci-\ndentally, the ferrimagnetic state is trivially enhanced if an FM\ninteraction is introduced between apical sites.\n1. total spin\nSimply, we examine the \r-dependence of the total spin\nto identify when and how the ferrimagnetic state is destabi-\nlized. In Fig. 7(a) the total spin per site Stot=Lis plotted as\na function of 1=Lfor several\rvalues with J2=jJ1j= 0:6\nfixed. Due to the strong frustration, the value of Stot=Los-\ncillates with 1=L; however, we may perform a reasonable\nfinite-size scaling analysis with finer data points using large\nenough clusters. We here use open clusters with length up to\nL=LA+LB= 100+101 = 201 . TheL!1 extrapolated9\nvalue ofStot=Lis plotted as a function of \rin Fig. 7(b).\nAt\r= 0, the ground state is in the ferrimagnetic state\nwith nearly-fully polarized apical spins and the total spin per\nsite isStot=L=4. Hereafter, we call this state “ fullferri-\nmagnetic (FF) state” to discriminate it from another ferrimag-\nnetic state with Stot< L= 4(denoted as “ partial ferrimag-\nnetic (PF) state”) which appears below. When the AFM inter-\naction between apical spins is switched on, one may naively\nexpect a collapse or weakening of the full spin polarization in\nthe apical chain; nevertheless, interestingly, the FF condition\nStot=L=4survives up to \r\u00190:08. This can be interpreted\nas follows: Roughly speaking, since the system can gain more\nenergy from interchain FM interaction than AFM interaction\nbetween apical spins, the FM LRO in the apical chain is still\nmaintained at \r<\u00180:08.\nWith increasing \rfrom 0:08, the competition between in-\nterchain FM interaction and apical intrachain AFM interaction\nderives a new state. As seen in Fig. 7(a), at \r= 0:09the value\nofStot=Lseems to converge at a finite but smaller value than\n1=4as1=Ldecreases. This clearly suggests that some sort\nof collapse of the FF state happens around \r= 0:09. With\nfurther increasing \r, the value of Stot=Lappears to be con-\ntinuously reduced and reaches zero around \r= 0:14[see\nFig. 7(b)]. Surprisingly, we find that there exists a finite \r-\nrange exhibiting 0< S tot=L < 1=4. Since the basal chain\nbasically keeps its weakly polarized or nearly singlet state,\nit would be a good guess that spin polarization on the api-\ncal sites is gradually collapsed with increasing \rin this PF\nphase ( 0:08<\u0018\r<\u00180:14). In other words, the ferrimagnetic\norder by disorder in association with the global spin-rotation-\nsymmetry breaking disappears around \r= 0:14. Actually, as\nstated below, the system has the other order by disorder, i.e.,\ndimerization order, at \r>\u00180:14. The region of the PF phase\nis shown as a shaded area in the ground-state phase diagram\n[Fig. 2(b)].\nWe make some remarks on the existence of PF phase. Such\na ‘halfway’ magnetization 0< S tot=L < 1=4in a ferrimag-\nnetic state is generally prohibited by the Marshall-Lieb-Mattis\n(MLM) theorem70,71. There is an exception to this, however,\nwhen the ferrimagnetic order and a quasi-long-range order of\nTLL compete72. This corresponds to the competition between\nsmall FM polarization and dominant AFM fluctuations in the\nbasal chain of our system. As confirmed in Appendix C, the\nbasal chain in the FF state indeed exhibits a TLL behavior.\n2. spin-spin correlation\nWe then consider the evolution of spin-spin correlation\nfunction with \rin the FF phase. In Fig. 8(a)-(c) iDMRG\nresults of the spin-spin correlation function hSz\n\u000b;iSz\n\f;jifor\n\r= 0,0:04, and 0:08with fixedJ2=jJ1j= 0:6are plotted\nas a function of distance ji\u0000jj. We keepSz\ntot= 4=Las done\nin Fig. 4.\nAs far as the FF state is maintained up to \r\u00190:08, the cor-\nrelation functions seem to be almost independent of \r. Ac-\ncordingly, the expectation values of Sz\nA;iandSz\nB;iare un-\nchanged up to \r\u00190:08, as shown in Fig. 8(d). Perhaps, one\nFIG. 8. Spin-spin correlation function hSz\niSz\njiof the asymmetric\nJ1-J2zigzag ladder as a function of ji\u0000jjwith fixedSz\ntot=L= 1=4\nandJ2=jJ1j= 0:6for (a)\r= 0, (b)0:04, and (c) 0:08. (d) Averaged\nvalues ofhSz\niion the apical and basal sites as a function of \rfor\nJ2=jJ1j= 0:6. The circles and crosses denote iDMRG and DMRG\nresults, respectively.\nmay naively expect the reduction of hSz\nA;iiwith increasing\n\r, i.e., with increasing AFM coupling between apical spins.\nHowever, this is not actually the case. This can be understood\nas follows: as estimated by the fitting of low-energy excitation\nspectrum, an FM coupling with the magnitude \u00180:11jJ1jis\neffectively induced between neighboring apical sites at \r= 0.\nThus, the apical chain may be effectively mapped onto an FM\nchain withJe\u000b=\u00000:11jJ1j. Therefore, it would be rather\nnatural that the (nearly) full polarization is free of the influ-\nence of additional AFM coupling \rJ2until it reaches around\n\u00180:11jJ1j.\nNo\r-dependence of the spin structure up to \r\u00190:08then\nindicates that the total energy of FF state is simply lifted by\nthe AFM interaction \rJ2between nearly-fully polarized api-\ncal spins; it switches into the energy level with a metastable\nPF state at\r\u00190:09. Hence, the FF to PF phase transition\nis of the first order. It can be also confirmed by a steep (or\nalmost discontinuous) change of StotandhSz\n\u000b;iiat\r\u00190:09.\nOn the other hand, both of hSz\nA;iiandhSz\nB;iismoothly ap-\nproach to zero around \r\u00190:14. Thus, the transition from PF\nto the spiral singlet ( Stot= 0) region is of the second order or\ncontinuous.\n3. stabilization gap\nTo quantify the stability of ferrimagnetic state at finite \r,\nwe calculate the stabilization energy \u0001[Eq.(6)] as done in\nthe case of delta chain. In Fig. 9(a) the finite-size scal-10\n0 0.05 0.1 0.1500.010.020.03\n0 0.02 0.04 0.06 0.08 0.100.0020.0040.0060.008(a)\n(b)\nFIG. 9. (a) Finite-size scaling of energy difference between lowest\nStot= 0 state and ferrimagnetic ground state for the asymmetric\nJ1-J2zigzag ladder with fixed J2=jJ1j= 0:6. The solid and dotted\nlines show the fitting results with \u0001(L)=jJ1j= \u0001=jJ1j+ A=Land\n\u0001(L)=jJ1j= \u0001=jJ1j+ A=L\u0011, respectively. (b) Extrapolated val-\nues of \u0001=jJ1jas a function of \r. The width of error bar means the\ndifference of \u0001=jJ1jobtained by the two fitting functions.\ning analysis of \u0001(L)is performed for several \rvalues with\nJ2=jJ1j= 0:6fixed. Although a sine-like oscillation is\npresent as in the case of delta chain, an acceptable scaling may\nbe possible in the FF phase ( \r<\u00180:09). We here employ two\nkinds of fitting functions: \u0001(L)=jJ1j= \u0001=jJ1j+ A=Land\n\u0001(L)=jJ1j= \u0001=jJ1j+ A=L\u0011. Since the former (latter) func-\ntion seems to underestimate (overestimate) the extrapolated\nvalue of \u0001=jJ1j, their averaged value is plotted as a function\nof\rin Fig. 9(b). The width of error bar means the differ-\nnce between two values of \u0001=jJ1jobtained by the two fitting\nfunctions. The stabilization gap is approximately linearly re-\nduced by\rup to the critical point \r\u00190:08. This also sup-\nports the above speculation that the total energy of FF state is\nsimply lifted by \rJ2.\nIn the PF phase ( \r>\u00180:09), however, the scaling analysis\nof\u0001(L)is virtually impossible. As an example, \u0001(L)=jJ1j\nvs.1=Lfor\r= 0:1is shown in Fig. 9(a). This difficulty is\ncaused by the following several factors: (i) As shown in the\nnext subsection, an incommensurate oscillation is involved in\nthe PF state. (ii) The total spin per site Stot=Lis strongly de-\npendent on system size since the states in different Stotsectors\nare extremely quasi-degenerate around the ground state. (iii)\n0 0.1 0.2 0.30basal cha/g3444 n\nap/g3444 cal cha/g3444 n\n0 0.2 0.4 0.6 0.8 10\n0.06 0.1 0.14-12.7-12.6(a)\n(b)FIG. 10. (a) Static spin structure factor for J2=jJ1j= 0:6as a\nfunction of\r. The lattice spacing ais set as shown in Fig. 1(a). (b)\nEnlarged figure of (a) for 0\u0014\r\u00140:3. Inset: Ground-state energy\nas a function of \r.\nThe available system size is strictly limited because the sec-\nond term of Eq.(7) includes long-range interactions and a pe-\nriodic cluster must be used. Nevertheless, the stabilization gap\nshould be positive due to the nonzero total spin of the ground\nstate [Fig. 7(b)]. We can, at least, confirm that the PF order is\nvery fragile with the stabilization gap \u0001<4:3\u000210\u00004jJ1jat\n\r= 0:08. This small stabilization gap also tells us that there\nare a macroscopic number of quasi-degenerate states belong-\ning to different Stotsectors, since the total spin is continu-\nously varied from L=4to0with\rin the PF phase.\n4. Static spin structure factor\nIt is important to see how the intrachain spin modulation\nchanges with \r. A most significant quantity to know it would\nbe the propagation number which can be extracted as a max-\nimum position of static spin structure factor. Therefore, we\ncalculate the static spin structure factor for each of the apical\nand basal chains. It is defined as\nS\u000b(q) =1\nL2\u000bL\u000bX\ni:j=1eiq(ri\u0000rj)hS\u000b;i\u0001S\u000b;ji (10)\nfor the apical ( \u000b= A ) or basal (\u000b= B) chains. The lattice\nspacingais set as shown in Fig. 1(a). In Fig. 10 the propaga-11\ntion numbers qmaxforJ2=jJ1j= 0:6using a 64-site periodic\ncluster are plotted as a function of \r.\nAt\r= 0, the system is in the FF state whose spin structure\nis commensurate and simple as shown in Fig. 1(c). The apical\nspins are nearly-fully polarized and the propagation number\nisqmax= 0, whereas for basal chain the dominant correlation\nis AFM (qmax=\u0019=a) although it is slightly polarized. It is\nobvious that this spin structure persists in the whole region of\nthe FF phase at 0<\u0018\r<\u00180:08.\nIn the singlet ( Stot= 0) phase at\r>\u00180:14, the propagation\nnumbers of apical and basal chains are both incommensurate,\ni.e.,0< q max< \u0019=a . At\r= 1, they coincide and it is\nestimated as qmax= 0:958. This value is in good agreement\nwith our previous estimation qmax= 0:95873in the thermo-\ndynamic limit. With decreasing \r, the propagation number is\nreduced because short-range FM correlation is relatively en-\nhanced. Interestingly, they are equal or very close down to the\ncritical point at \r\u00190:14and obviously split for smaller \r. It\nwould be a good guess that the short-ranged spiral structure of\ntheJ1-J2chain (\r= 1) is approximately maintained down to\n\r\u00190:14. In a broad sense, this incommensurate region can\nbe referred to as a spiral singlet phase.\nIn Fig. 10(b) an enlarged figure around the PF phase\n(0:08<\u0018\r<\u00180:14) is shown. As shown in the inset of\nFig. 10(b) the phase boundaries are recognized by level cross-\ning of the ground state energies. It clearly indicates the exis-\ntence of an intermediate phase between the FF and spiral sin-\nglet phases, though the region ( 0:08>\u0018\r>\u00180:11) is a bit nar-\nrower than that in the thermodynamic limit ( 0:08>\u0018\r>\u00180:14)\ndue to finite size effects. The intermediate phase is the PF\nphase as described above.\nIn the PF phase, the dominant correlation of the basal chain\nseems to be incommensurate and the propagation number of\nthe apical chain keeps qmax= 0. It is a natural consequence of\nthe global spin-rotation-symmetry breaking because the total\nspin can be no longer finite if both of the propagation num-\nbers are nonzero. This incommensurate propagation is a con-\nsequence of the halfway magnetization so that the prohibition\nby the MLM theorem is also avoided by the TLL characteris-\ntic of basal chain. Similar incommensurate features have been\nreported for PF state in frustrated systems74,75.\n5. Dimerization order\nSo, let us see more about the magnetic properties of the\nspiral singlet ( Stot= 0) phase at larger \r. It is known that\nthe system has LRO with spontaneous dimerization in the J1-\nJ2chain (\r= 1)19,20. Therefore, as a starting point it would\nbe reasonable to examine the evolution of dimerization order\nparameters with decreasing \rfrom 1. The dimerization order\nparameter between sites distant \u000ealong theJ1zigzag chain is\ndefined as\nOdimer(\u000e) = lim\nL!1jhSA;i\u0001SB;i\u0000(\u000e\u00001)=2i\n\u0000hSA;i\u0001SB;i+(\u000e+1)=2ij;(11)\nδ\nδ = \nδ = \nδ = (a)δ = 1\nδ = 2\nδ = 3\n(b)FIG. 11. (a) Schematic pictures of possible dimerization order.\nThe states for \u000e= 2 and\u000e= 3 are characterized as VBS. A solid\n(dotted) ellipse denotes a spin-singlet (spin-triplet) dimer. (b) Aver-\naged dimerization order parameters Odimer(\u000e)as a function of \rat\nJ2=jJ1j= 1. Inset: each contribution to Odimer(2)from the apical\nand basal chains.\nfor odd\u000e, and\nOdimer(\u000e) = lim\nL!1jhS\u000b;i\u0001S\u000b;i+\u000e=2i\u0000hS\u000b;i\u0001S\u000b;i+\u000e=2ij;\n(12)\nfor even\u000e. IfOdimer(\u000e)is finite for\u000e, it signifies a long-range\ndimerization order associated with mirror-symmetry breaking\nfor odd\u000eor translation-symmetry breaking for even \u000e. We\nhere study the case of \u000e= 1,2, and 3. Schematic pictures of\nthe possible dimerization orders are shown in Fig. 11(a). A\nfiniteOdimer(\u000e)for\u000e= 2and3indicates a valence bond for-\nmation, i.e., spin-singlet formation, between two sites on the\ndimerized bond. In Fig. 11(b) the iDMRG results of dimeriza-\ntion order parameter Odimer(\u000e)for\u000e= 1,2, and 3are plot-\nted as a function of \ratJ2=jJ1j= 1. For confirmation, we\nalso estimateOdimer(\u000e)in the thermodynamic limit for some\n\rvalues using DMRG under OBC. We can find their excel-\nlent agreement with the iDMRG results. Note that the FM\ninteractionJ1at both edges in the open clusters is set to be\nzero. It enables us to perform the finite-size scaling analysis\nmore easily because the competing two translation-symmetry\nbreaking states are explicitly separated20. For confirmation,\nwe have checked that the ground state in the thermodynamic\nlimit does not depend on the choice of boundary conditions.\nAt\r= 1, two dimerization orders with \u000e= 1 and\u000e= 3\ncoexist20. With decreasing \r, interestingly,Odimer(3)is sig-\nnificantly enhanced and Odimer(1)is slightly increased down12\nto\r\u00190:195. With fixed J2=jJ1j= 1, we may deduce that\nthe magnetic frustration is largest in the limit of \r= 0where\nthe system is a series of isotropic triangles with uniform mag-\nnitude of interactions. The valence bond pair may be strength-\nened to screen spins more strongly for the relaxation of larger\nmagnetic frustration at smaller \r. SinceOdimer(2) = 0 down\nto\r\u00190:195, this state is dominantly characterized as a VBS\nstate with\u000e= 3 dimerization order (we call it “ D3-VBS”\nstate).\nEven more surprisingly, Odimer(2)exhibits a steep increase\n(almost jump) at \r\u00190:195. The other order parameters\nOdimer(1)andOdimer(3)are also not differentiable with \r\nat this point. This clearly indicates another first-order tran-\nsition at\r\u00190:195. We note that both of the apical and\nbasal chains are spontaneously dimerized along the chain di-\nrection in the region of Odimer(2). The order parameter for\neach chain is plotted in the inset of Fig. 11(b) (The main fig-\nure shows the averaged value). Since the value of Odimer(2)\nis much larger than the other dimerization order parameters at\n0:035<\u0018\r<\u00180:195, the state is recognized as a VBS one with\n\u000e= 2 dimerization order (we call it “ D2-VBS” state). Since\nthis\u000e= 2dimerization order is associated with a translation-\nsymmetry breaking, the magnetic structure consists of a su-\npercell with four sites. More detailed analysis is given in Ap-\npendix D.\nThe dimerization order can be also detected by studying\nthe topological properties of the system. Then, let us see\nthe entanglement spectrum (ES)76which can be obtained by\na canonical representation of the an infinite matrix-product-\nstate in the iDMRG calculations12. Using Schmidt decompo-\nsition, the ground-state wave function can be expressed as\nj i=X\nie\u0000\u0018i=2j\u001eA\nii\nj\u001eB\nii; (13)\nwhere the statesj\u001eS\niicorrespond to an orthonormal basis\nfor the subsystem S(either A or B). We study the ES for\nseveral kinds of splitting pattern between subsystems A and\nB. The splitting patterns are sketched in Fig. 12(a). In our\niDMRG calculations, the ES f\u0018\u000bgis simply obtained as \u0018\u000b=\n\u00002 ln\u0015\u000b, wheref\u00152\n\u000bgare the singular values of the reduced\ndensity matrices after the bipartite splitting. The low-lying\nfour ES levels are plotted as function of \rin Fig. 12(b)-(e).\nWhen the one dimerized singlet pair straddles the subsys-\ntems A and B [Fig. 12(b) and (d)], the lowest entanglement\nlevel is doubly degenerate as a reflection of the edge state.\nOn the other hand, when this is not the case [Fig. 12(c) and\n(e)], the lowest entanglement level is non-degenerate. Our re-\nsult for theD3-VBS state is consistent with previous research\non the symmetric case19. These facts would strongly support\nthe formation of long-range dimerization order. We can also\nfind a discontinuous change of @(\u00002 ln\u0015\u000b)=@\r at\r\u00180:195,\nwhich seems to correspond to the transition point from the\nD2-VBS toD3-VBS state. We note that the difference be-\ntween (b) and (d) as well as (c) and (e) in Fig. 12 comes from\nthe asymmetric nature of our system. More details about the\nasymmetric nature are discussed in Appendix D.\nIn the spiral singlet phase, the system is in either D2-VBS\norD3-VBS state. The phase boundary between them is shown\n(a)\n(e)(b) (c)\n(d)(b) (c) (d) (e)D2-VBS\nD3-VBS\nγ γ21210121\nα=λ λFIG. 12. (a) Schematic pictures of considered splitting of the sys-\ntem into two subsystems in the D2-VBS andD3-VBS state. A solid\nellipse denotes a spin-singlet pair. The number of singlet pair cross-\ning with each cut is shown in the green square. (b)-(e) Entangle-\nment spectrum for the corresponding splitting as a function of \rat\nJ2=jJ1j= 1.\nin Fig. 2(b). In general, the spin gap, namely, energy differ-\nence between spin-singlet ground state and spin-triplet first\nexcited state, is expected to be finite when the system is in a\nVBS state. In theD3-VBS phase, the spin gap simply scales to\nan energy to break a valence bond for \u000e= 3. In theD2-VBS\nphase, each of the apical and basal chains has a different va-\nlence bond. Nevertheless, it is easy to imagine that the valence\nbond in the apical chain is more fragile because of smaller\nAFM interaction, although Odimer(2)for the apical chain is\nlarger than that for the basal chain. Thus, the spin gap in the\nD2-VBS phase scales to an energy to break a valence bond in\nthe apical chain. This means that a larger energy than the spin\ngap is needed to break a valence bond in the basal chain. It\nwould provide a 1/2-plateau in the magnetization process with\nmagnetic field.13\n0.2 0.4 0.6 0.8 1.0\nγ0.000.010.020.030.040.050.060.070.08Ostring\nFIG. 13. String order parameter as a function of \ratJ2=jJ1j= 1\nusing iDMRG (circles) and DMRG (crosses) methods. The DMRG\nresults are extrapolated values to the thermodynamic limit.\n6. String order\nWe have confirmed the existence of nearest-neighbor ( \u000e=\n1) FM dimerization order in the whole spiral singlet region. A\nspin-triplet pair may be effectively formed in the each ferro-\nmagnetically dimerized bond: By relating three states j\"\"i ,\nj\"#i +j#\"i )=p\n2, andj##i toSz= 1,0, and\u00001states,\nrespectively, the resultant spin on the dimerized bond can be\nreduced to a spin- 1degree of freedom. Consequently, the sys-\ntem could be mapped onto a S= 1 Heisenberg chain ac-\ncompanied by the emergent effective spin- 1degrees of free-\ndom with the dimerized two spin-1\n2’s19. Furthermore, the\npresence of third-neighbor AFM dimerization order ensures\na valence bond formation between the neighboring effective\nS= 1 sites20. It leads to finite spin gap as a Haldane gap\nin symmetry-protected VBS state11,12. Although the spin gap\nis a good indicator to measure the stability of VBS state, it\nwould be too small to correctly estimate with DMRG method\nin most of the parameter region of system (1). Alternatively,\nthe stability of VBS state associated with the Haldane picture\ncan be evaluated by examining the string order parameter77:\nOz\nstring =\u0000lim\njk\u0000jj!1h(Sz\nA;k+Sz\nB;k)\nexp[i\u0019j\u00001X\nl=k+1(Sz\nA;l+Sz\nB;l)](Sz\nA;j+Sz\nB;j)i:\n(14)\nFor our system (1), Eq. (14) can be simplified as\nOz\nstring =\u0000lim\njk\u0000jj!1(\u00004)j\u0000k\u00002h(Sz\nA;k+Sz\nB;k)\nj\u00001Y\nl=k+1Sz\nA;lSz\nB;l(Sz\nA;j+Sz\nB;j)i:i (15)\nThe finite value of Oz\nstring suggests the formation of a VBS\nstate having a hidden topological long-range string order.Since two-fold degeneracy of the ground state due to the FM\ndimerization is lifted in our numerical calculations, jOz\nstringj\ncan have two different values depending on how to select k\nandj. We then take their average.\nIn Fig. 13 iDMRG and DMRG results for the string order\nparameter are plotted as a function of \ratJ2=jJ1j= 1. We\ncan see a good agreement between the iDMRG and DMRG\nvalues. With decreasing \rfrom 1,Oz\nstring is significantly\nincreased and has a pointed top at \r\u00190:195, which is\nthe first-order transition point between D3-VBS andD2-VBS\nphases. With further decreasing \r, it decreases and vanishes\nat\r= 0:035, which is the second-order transition point be-\ntweenD2-VBS and ferrimagnetic phases. We notice that the\noverall trend ofOz\nstring is similar to that of the \u000e= 3 dimer-\nization order parameter Odimer(3). It means that the stability\nof string order is dominated by the strength of valence bond\nwith\u000e= 3. In other words, a Haldane state is produced as\na structure where all the neighboring effective spin- 1sites are\nbridged by the \u000e= 3 valence bonds. Even though it is a\nHaldane state, the maximum value Oz\nstring\u00180:065is much\nsmaller thanOz\nstring =4\n9'0:4444 for the perfect VBS state\nfor the AKLT model11andOz\nstring'0:3743 for theS= 1\nHeisenberg chain78. This small value of Oz\nstring is interpreted\nas a sign of fragility of the D3-VBS state, which is however\ncomparable with the maximum value for the J1-J2chain at\nJ2=jJ1j\u00190:6(Oz\nstring\u00180:06)19,20.\nV . SUMMARY\nWe studied the asymmetric S=1\n2J1-J2zigzag ladder,\ndefined as two different AFM Heisenberg chains coupled by\nzigzag-shaped interchain FM interaction, using the DMRG-\nbased techniques. The AFM chain with larger (smaller) inter-\naction is referred to as apical (basal) chain.\nFirst, a classical phase diagram was obtained by the spin-\nwave theory. It contains three phases: FM, commensu-\nrate, and incommensurate phases. It offers the possibility of\ncommensurate-incommensurate phase transition by tuning the\nratio of AFM interaction of the apical and basal chains in the\nquantum case.\nNext, we revisited the ferrimagnetism in the so-called delta\nchain as the vanishing limit of AFM interaction in the apical\nchain. The ferrimagnetic state is characterized by total spin\nStot=L=4. By carefully examining the long-range spin-spin\ncorrelation functions and low-energy excitations, we pointed\nout that the origin of ferrimagnetic state is order by disor-\nder without geometrical symmetry breaking but with a global\nspin-rotation-symmetry breaking. Accordingly, the system\ncan gain energy from the FM interaction between the polar-\nized apical and basal spins. So to speak, FM fluctuations play\nan essential role to lower the ground state energy against the\nmagnetic frustration. This is a rare type of order by disor-\nder. And yet the basal chain is essentially a critical AFM\nHeisenberg chain as a TLL and its polarization is rather ill-\nconditioned as a state of chain itself. In this regards, one could\ninterpret this to mean that the ferrimagnetic order competes14\nwith a quasi-long-range AFM order of TLL.\nThen, we examined how the ferrimagnetic state is affected\nby AFM interaction of the apical chain, which is controlled by\n\r. We found that the ferrimagnetic state with Stot=L=4is\nmaintained up to a finite value of \r; and with further increas-\ning\rthe system goes into spiral singlet ( Stot= 0) phase at a\ncertain amount of \r. Of particular interest is the appearance of\nanother ferrimagnetic phase characterized by 0<Stot<L= 4\nbetween the Stot=L=4ferrimagnetic and Stot= 0 phases.\nSuch a ‘halfway’ magnetization, which is generally prohibited\nby the MLM theorem, is now allowed under the competition\nbetween ferrimagnetic and quasi-long-range order of TLL.\nFinally, the detailed magnetic properties of spiral singlet\nphase was investigated. With evaluating various dimerization\norder parameters, we confirmed that the Stot= 0 region is\ncovered by two kinds of VBS phases depending on the pa-\nrameter values. One is the D2-VBS phase where the valence\nbond is formed along the apical and basal chains; the other is\ntheD3-VBS phase where the valence bond is formed between\nthe apical and basal spins. Interestingly, a hidden topologi-\ncal long-range string order exists in the whole region of VBS\nphases and its strength scales only to the stability of D3-VBS\nstate.\nWe summarized the above numerical results as a ground-\nstate phase diagram, which contains a variety of frustration-\ninduced ordered phases. Especially, it is quite curious that a\nsimple frustrated system, like the asymmetric zigzag ladder,\nexhibits very different kinds of order by disorder – associated\nwith geometrical symmetry and global spin-rotation symme-\ntry.\nACKNOWLEDGEMENTS\nWe thank S. Ejima for fruitful discussions and U. Nitzsche\nfor technical assistance. This work was supported by Grants-\nin-Aid for Scientific Research from JSPS (Projects Nos.\nJP17K05530 and JP19J10805) and by SFB 1143 of the\nDeutsche Forschungsgemeinschaft (project-id 247310070).\nThe iDMRG simulations were performed using the ITensor\nlibrary79.\nAppendix A: Numerical confirmation of ferromagnetic critical\npoint\nIn the main text, we have obtained the \r-dependence of FM\ncritical point by classical SWT. It is given as\nJ2;c\njJ1j=1\n2(1 +\r)(A1)\nfor\r >0. This derivation can be easily extended to negative\n\r. Then, when \r <0we obtain the FM critical point as\nJ2;c\njJ1j=1\n2; (A2)\nwhich is independent of \r.\n-1 0 10.20.30.40.5\nDMRG\nsp/g3444 n-wave theory(a)\n(b)FIG. 14. (a) iDMRG results for spin-spin correlation function\nhSi\u0001Sjias a function of distance ji\u0000jjwith\r= 0:2fixed. (b)\nComparison of the \r-dependence of FM critical points J2;c=jJ1jes-\ntimated by DMRG and spin-wave theory.\nSince the quantum fluctuations vanish at the FM critical\npoint, Eqs. (A1) and (A2) should be exact. We here confirm\nthem numerically. A simplest way is to check the presence\nor absence of long-range FM order. In Fig. 14(a) we show\niDMRG result for the spin-spin correlation function hSi\u0001Sji\nat\r= 0:2as a function of distance ji\u0000jj. For convenience,\nthe site indices iandjare taken along the J1zigzag chain.\nAtJ2=jJ1j= 0:4we can clearly see an FM long-range or-\nder withhSi\u0001Sji=1\n4for any pair of iandjat the large\ndistance. However, the FM long-range order seems to be al-\nready destroyed at only a slightly larger J2=jJ1j= 0:42. In\nfact, this iDMRG result is consistent with the SWT estimation\nJ2;c=jJ1j= 0:417for\r= 0:2. Additionally, since the sys-\ntem is in theD3-VBS state at J2=jJ1j= 0:42, an exponential\ndecay ofhSi\u0001Sjiis thus expected.\nFurthermore, we also numerically estimate the FM critical\npoint for\r <0. In Fig. 14(b) we compare the \r-dependence\nofJ2;c=jJ1jvalues estimated by DMRG and SWT. We can\nfind a good agreement between them. The reason why the FM\ncritical point does not depend on \ris as follows: For any neg-\native\rthe apical chain is fully polarized and this FM order\naffects the basal chain as an external field via the interchain\ncouplingJ1. Therefore, the FM phase transition boils down15\nto the question of a competition between FM interchain inter-\nactionJ1and AFM interaction J2in the basal chain, namely,\nindependent of \r. Even intuitively, Eq. (A2) can be obtained\nby comparing the numbers of J1andJ2bonds.\nAppendix B: Artificial enhancement of spin-singlet state with\nshort chains\n0 0.05 0.1-0.1-0.09-0.08-0.07\nFIG. 15. Energies per site for lowest-lying spin-singlet and ferri-\nmagnetic states at J1=\u00001andJ2= 0:8as a function of inverse\nsystem size. The PBC is used.\nIn Fig. 3(b) of the main text, the total spin per site Stot=Lis\nplotted as a function of 1=Lunder PBC. With decreasing 1=L,\nwe see a drastic change of the total spin from Stot=L= 0 to\nStot=L= 1=4 + 1=Lat some value of 1=L. This is the conse-\nquence of a typical finite-size effect: If the system size is suf-\nficiently small, the basal chain forms a short AFM ring under\nPBC and a plaquette singlet is highly stabilized. On the other\nhand, the ferrimagnetic state may be readily prevented due\nto the screening of basal spins. This finite-size effect could be\neliminated by increasing the system size because the plaquette\nsinglet is rapidly destabilized for larger systems. To confirm\nit, we compare the energies for spin-singlet and ferrimagnetic\nstates in Fig. 15. When the system size is small, the energy\nfor spin-singlet state is much lower than that for ferrimagnetic\nstate. They approach with increasing system size and cross at\nsome value of L(\u0011Lc). Roughly speaking, one could inter-\npret this to mean that the energy gain from ferrimagnetic for-\nmation becomes larger than that from the plaquette singlet for-\nmation of basal chain around the critical system size L=Lc.\nTherefore, we take Stot=L= 1=4 + 1=Lfor the finite-size\nscaling and obtain Stot=L= 1=4in the thermodynamic limit\nL!1 .\nAppendix C: Tomonaga-Luttinger-liquid behavior of basal\nchain in full ferrimagnetic phase\nAs mentioned in the main text, the basal chain in the fer-\nrimagnetic states should be a TLL because it only allows the\nsystem to have an incommensurate ‘halfway’ magnetization\n100101102\n|i−j|10−810−710−610−510−410−310−210−1|/angbracketleftbig\nSz\nB,jSz\nB,i/angbracketrightbig\n−/angbracketleftbig\nSz\nB,j/angbracketrightbig/angbracketleftbig\nSz\nB,i/angbracketrightbig\n|\nγ= 0.0\nγ= 0.04\nγ= 0.08\n100 20010−510−310−1\nγ= 0.2FIG. 16. Spin-spin correlation function as a function of distance ji\u0000\njjwithJ2=jJ1j= 0:6.\nother than the commensurate one. We can check it by look-\ning at the spin-spin correlation in the basal chain. We plot\niDMRG results of the spin-spin correlations jhSz\nB;iSz\nB;ji\u0000\nhSz\nB;iihSz\nB;jijin the FF state are plotted as a function of dis-\ntanceji\u0000jjin Fig. 16. The z-component of total spin is fixed\natSz\ntot=L=4. We can see a power-law decay with distance,\ni.e.,jhSz\nB;iSz\nB;ji\u0000hSz\nB;iihSz\nB;jij\u00181=ji\u0000jj, as a signature of\nTLL. For comfirmation, we also plot the spin-spin correlation\nfor theD3-VBS state in the inset of Fig. 16. It clearly exhibits\nan exponential decay indicating a gapped VBS state.\nAppendix D: Dimerization order with translation symmetry\nbreaking\nIn the main text, we have estimated the dimerization order\nparameters defined as the difference between spin-spin corre-\nlations for the ‘neighboring’ bonds. To obtain further insight\ninto the dimerization structure, we here look at the bare spin-\nspin correlations. In Fig. 17(b)-(d) we plot the spin-spin cor-\nrelations for various pairs of spins as a function of \r. Each\nsymbol corresponds to a bond marked by the same symbol in\nFig. 17(a).\nAt\r>\u00180:195, the valence bond structure is relatively sim-\nple. The correlation h(Si\u0001Sj)Iihas two different values and\nthe difference between them provides the dimerization order\nparameter. This is the same for h(Si\u0001Sj)IIIi. No splitting of\nthe correlationh(Si\u0001Sj)IIiindicates the absence of dimeriza-\ntion order parameter for \u000e= 2.\nAt\r<\u00180:195, the situation is a bit more complicated.\nThe translation symmetry is broken due to the presence of\ndimerization order for \u000e= 2 and all the three dimerization\norders coexist. As a result, the two values of h(Si\u0001Sj)Iiand\nh(Si\u0001Sj)IIIiare further split into four values. 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B 48, 3844 (1993).\n79https://itensor.org/."    },    {        "title": "1710.07779v1.Correlation_between_Compensation_Temperatures_of_Magnetization_and_Angular_Momentum_in_GdFeCo_Ferrimagnets.pdf",        "content": "  \n1 Correlation between Compensation Temperatures of Magnetization and \nAngular Momentum in GdFeCo Ferrimagnets \nYuushou Hirata1, Duck-Ho Kim1†, Takaya Okuno1, Tomoe Nishimura1, Dae-Yun Kim2, \nYasuhiro Futakawa3, Hiroki Yoshikawa3, Arata Tsukamoto3, Kab-Jin Kim4, Sug-Bong Choe2, \nand Teruo Ono1,5† \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 6 11-0011, Japan. \n2Department of Physics and Institute of Applied Physics, Seoul N ational University, Seoul \n08826, Republic of Korea. \n3College of Science and Technology, Nihon University, Funabashi,  Chiba 274-8501, Japan. \n4Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea. \n5Center for Spintronics Research Network, Graduate School of Eng ineering Science, Osaka \nUniversity, Machikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan.  \n†Correspondence to: kim.duckho.23z@st.kyoto-u.ac.jp , ono@scl.kyoto-u.ac.jp \nDetermining the angular momentum  compensation temperature of fe rrimagnets \nis an important step towards ferrimagnetic spintronics, but is not generally easy to \nachieve it experimentally. We propose a way to estimate the ang ular momentum \ncompensation temperature of ferrimagnets. We find a linear rela t i o n  b e t w e e n  t h e  \ncompensation temperatures of the magnetization and angular mome ntum in GdFeCo \nferrimagnetic materials, which is proved by theoretically as we ll as experimentally. The \nlinearity comes from the power-l aw criticality and is governed by the Curie \ntemperature and the Landé g factors of the elements composing t he ferrimagnets.   \n2 Therefore, measuring the magnetization compensation temperature  and the Curie \ntemperature, which are easily ass essable experimentally, enable s to estimate the angular \nmomentum compensation temperature of ferrimagnets. Our study pr ovides efficient \navenues into an exciting world of ferrimagnetic spintronics.  \nAntiferromagnets came into the spotlight in the last decade [1– 6] as a promising \nmaterial for spintronics devices because they exhibit fast magn etic dynamics and low \nsusceptibility to magnetic fields. These advantages originate f rom the antiferromagnetic \nordering in which the magnetic moments are compensated on an at o m i c  s c a l e .  T h i s  a l s o  \nimplies that it is difficult to efficiently manipulate antiferr omagnets using external magnetic \nfields, hindering the study of antiferromagnetic spin dynamics.  However, recently, magnetic \nfield-controlled antiferromagnetic spin dynamics has been achie ved using ferrimagnets [7]. \nH e n c e ,  f e r r i m a g n e t s  h a v e  b e c o m e  a  p r o m i s i n g  m a t e r i a l  i n  t h e  e m e rging field of \nantiferromagnetic spintronics. \nFerrimagnetic materials comprise rare earth (RE) and transition  m e t a l  ( T M )  \ncompounds, wherein the spins of two inequivalent sublattices ar e coupled \nantiferromagnetically [8-10]. Because of the different Landé g factors of RE and TM \nelements, ferrimagnets exhibit compensation temperatures of mag netization and angular \nmomentum [11], at which the magnetizations (angular momenta) of  the RE and TM \nsublattices have the same magnit ude but opposite directions. Co nsequently, the net \nmagnetization (angular momentum) is compensated. The compensati on temperatures have \nbeen studied experimentally and theoretically [7, 12–23]. In pa rticular, Kim et al . recently \nobserved fast field-driven domain wall (DW) motion in the vicin ity of the compensation \ntemperature of the angular momen tum [7]. This observation revea ls that ferrimagnets exhibit \nthe antiferromagnetic dynamics because of the zero net angular momentum  a t  the   \n3 compensation temperature of the angular momentum, even though t hey have magnetic m\noments. Although remarkable efforts have been made theoreticall y and experimentally [7, \n12–23] in understanding the role of angular momentum compensati on in DW dynamics, it is \ndifficult to determine the angular momentum compensation temper ature because of the \nmethodological complexities, impeding the rapid development of this exciting research field \nas well as the fundamental understanding of the compensation te mperatures. \nI n  t h i s  l e t t e r ,  w e  r e p o r t  t h e  c o r r e l a t i o n  b e t w e e n  t h e  a n g u l a r  m omentum and \nmagnetization compensation temper atures in ferrimagnetic GdFeCo  a l l o y s .  I t  i s  \nexperimentally demonstrated that  the angular momentum compensat ion temperature is \ndirectly related to the magnetization compensation temperature,  regardless of the sample \nstructures. The results show that there exists a strong correla tion between the two types of \ncompensation temperatures. We theoretically verified the correl ation on the basis of a simple \nmodeling technique. Moreover, the proposed approach is a novel method of determining the \nangular momentum compensation temperature. \nFor this study, we prepared six types of amorphous ferrimagnet GdFeCo films. Table \nI lists the detailed sample struc tures. The films were grown by  co-sputtering, and the \ncompositions were estimated fr om the relative deposition rates of Gd and FeCo. The samples \nexhibit perpendicular magnetic anisotropy (PMA) with circular d omain expansion. As shown \nin Fig. 1( a), an e-beam lithography technique was applied to the structura l devices with a \nHall bar geometry in order to de tect the anomalous Hall resista nce. \nFirst, we characterized the magne tic properties of the GdFeCo s amples. Figure 1( b) \nshows the hysteresis loop of the GdFeCo microstrip. The anomalo us Hall resistance ܴୌ i s  \nmeasured as a function of the perpendicular magnetic field ܪ௭ at room temperature. The \nclear, square, hysteresis loop shows that the GdFeCo samples ha ve strong PMA. The orange-  \n4 colored arrow represents the magnetization switching field, whi c h  i s  r e f e r r e d  t o  a s  t h e  \ncoercive field ܪୡ. \nTo determine the magnetization compensation temperature ܶ, we measured ܴୌ b y  \nsweeping ܪ௭ at each temperature [7, 18, 20, 23]. The magneto-transport pro perties are \ndominated by the FeCo moment because the energy of the 4f shell  of Gd is located far below \nthe Fermi energy level [23]. A sign change in ܴୌ indicates a change in the direction of the \nFeCo moment. Using ܴୌ as a function of ܪ௭, we define the Hall resistance difference as \n∆ܴୌ≡ܴୌ൫ܪ,ୱୟ୲൯െܴୌ൫െܪ,ୱୟ୲൯ ( s e e  F i g .  1 ( b)). Figure 2( a) shows the Hall resistance \ndifference ∆ܴୌ as a function of the temperature ܶ f o r  S a m p l e  I I ,  w h e r e  ܪ,ୱୟ୲ and \nെܪ,ୱୟ୲ are the saturation fields with the condition ܪୡ൏ܪ,ୱୟ୲ (see inset of Fig. 1( b)). ∆ܴୌ \nto zero determines ܶൌ\t160 K, indicated by a blue dot. \nTo determine ܶ, we measured the field-driven DW speed as a function of the \ntemperature, as proposed elsewhere [7]. We first applied a suff iciently strong magnetic field \nwith a magnitude of –200 mT ( െܪ,ୱୟ୲) to saturate the magnetization along the –z \ndirection, and subsequently, a constant ܪ௭ for driving the DW. ܪ௭ i s  s e l e c t e d  l o w e r  t h a n  \nܪୡ, to eliminate the nucleation of  the domain. Next, we applied a  DC current ܫ௫ a l o n g  t h e  \nwire to detect the Hall signal (see red arrow in Fig. 1( a)), where ܫ௫ is sufficiently small to \nprevent spin torques and Joule heating effect [24–26]. We then injected a current pulse ܫ௬ \n(12 V, 100 ns) through the writing line (see blue arrow in Fig.  1(a)) to nucleate the reversed \ndomain, thereby creating two DWs in the wire. The created DW mo ves along the wire \nbecause of the presence of ܪ௭, and then, passes through the Hall bar; the DW arrival time \ncan be detected by monitoring the change in the Hall voltage us ing an oscilloscope. The DW \nspeed can be calculated from the arrival time and the distance traveled between the writing \nline and the Hall bar (400 µm).   \n5 Figure 2( b) shows the DW speed ݒ as a function of the temperature ܶ a t  ߤܪ௭ൌ\n\t80 mT for Sample II. This figure clearly shows that ݒ exhibits a peak at a certain \ntemperature (indicated by purple arrow). This tendency of ݒ with respect to ܶ is consistent \nwith the results given elsewhere [7]. Accordingly, ܶ can be determined, as shown in Fig. \n2(b). Here, the difference between ܶ and ܶ is defined as ∆\tܶሺ≡ܶെܶሻ, indicated by \nblack double arrows. \nFor a quantitative comparison, the values of ܶ are directly plotted with respect to \nܶ for all the samples, as shown in Fig. 3. It is interesting to note that all the values ሺܶ,ܶሻ \nlie on a single curve with linearity. The red line represents t he best linear fit with a slope of \n0.87 and a y-axis intercept of 101.1 K. From this result, we ex perimentally found that there \nexists a strong correlation between ܶ and ܶ for all the GdFeCo films. \nTo understand the correlation of ሺܶ,ܶሻ, we employ a theory based on a power-law \ncriticality, given that the variation in the magnetization as a  function of the temperature can \nbe reasonably well approximated [27, 28, 29]. This function des cribes the temperature \ndependence of the magnetization, which can be expressed as ܯሺܶሻ~ሺܶେെܶሻఉ, where ܯ i s  \nthe saturation magnetization, ܶେ is the Curie temperature, and ߚ is the critical exponent. \nAccordingly, the temperature dependencies of the magnetization f o r  G d  a n d  F e C o  c a n  b e  \nwritten as ܯୋୢሺܶሻൌܯୋୢሺ0ሻሺ1െܶ/ܶ େሻఉృౚ and ܯୣେ୭ሺܶሻൌܯୣେ୭ሺ0ሻሺ1െܶ/ܶ େሻఉూి, \nrespectively, where ߚୋୢ ( o r  ߚୣେ୭) is the critical exponents of Gd (or FeCo) and ܯୋୢሺ0ሻ \n(or ܯୣେ୭ሺ0ሻ) is the saturation magnetization of Gd (or FeCo) at zero tempe rature, where \nߚୋୢߚୣେ୭ and ܯୋୢሺ0ሻܯୣେ୭ሺ0ሻ [27, 28]. The total saturation magnetization ܯ୲୭୲ୟ୪ \ncan be determined using the relation, ܯ୲୭୲ୟ୪ሺܶሻൌܯୣେ୭ሺ0ሻሺ1െܶ/ܶ େሻఉూిെܯୋୢሺ0ሻሺ1െ\nܶ/ܶେሻఉృౚ. As ܯ୲୭୲ୟ୪ൌ0 \tat ܶൌܶ, the following equation can be written. \n\t\t\t\t\t\t\tܶେെܶൌܶେሾܯୋୢሺ0ሻ/ܯୣେ୭ሺ0ሻሿଵ/ሺఉూిିఉృౚሻ.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \t\t\t\t\t\t\t\tሺ1ሻ    \n6 Similarly, the total angular momentum ܣ୲୭୲ୟ୪ሺܶሻ can be given as ܣ୲୭୲ୟ୪ሺܶሻൌሺܯୣେ୭ሺ0ሻ/\nߛୣେ୭ሻሺ1െܶ/ܶ େሻఉూిെሺܯୋୢሺ0ሻ/ߛୋୢሻሺ1െܶ/ܶ େሻఉృౚ, where ߛୋୢ ( o r  ߛୣେ୭ ) is the \ngyromagnetic ratio of Gd (or FeCo). The gyromagnetic ratio of G d (or FeCo) can be defined \nas ߛୋୢൌ݃ୋୢఓా\n ( o r  ߛୣେ୭ൌ݃ୣେ୭ఓా\n\t), where ݃ୋୢ ( o r  ݃ୣେ୭) is the Landé g factor of Gd \n(or FeCo), \tߤ is the Bohr magneton and  is the reduced Plank’s constant. As ܣ୲୭୲ୟ୪ൌ0 \tat \nܶൌܶ, the following equation is obtained. \n\t\t\t\t\t\t\tܶେെܶൌܶେሾሺܯୋୢሺ0ሻ݃ୣେ୭ሻ/ሺܯୣେ୭ሺ0ሻ݃ୋୢሻሿଵ\nሺఉూిିఉృౚሻ.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tሺ2ሻ  \nBy subtracting Eq. (1) from Eq. (2), we can write the relations hip between ܶ a n d  ܶ a s  \nfollows. \n\t\t\t\t\t\t\tܶൌܶܶେቈ1െሺ݃ୣେ୭/݃ୋୢሻଵ\nሺఉూిିఉృౚሻ൫ܯୋୢሺ0ሻ/ܯୣେ୭ሺ0ሻ൯ଵ\nሺఉూిିఉృౚሻ.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tሺ3ሻ  \nBecause of the spin-orbit coupling of FeCo and zero orbital ang ular momentum of Gd, it is \nknown that ݃ୣେ୭ (~ 2.2) is slightly greater than ݃ୋୢ (~ 2) [30–32]. Consequently, we can \nexpect that ܶܶ due to the condition of ߚୋୢߚୣେ୭, ܯୋୢሺ0ሻܯୣେ୭ሺ0ሻ, and \n݃ୣେ୭݃ୋୢ [27, 28, 30–32]. From Eq. (3), the linearity of ሺܶ,ܶሻ can be easily \nunderstood. It is noteworthy that ܶ depends on ܶେ in Eq. (3), and therefore, ܶେ a f f e c t s  \n∆ܶ .This results in a slight deviation from linearity. A standard  scaling treatment is employed \nto examine the universal behaviors. By scaling Eq. (3) and divi ding it by ܶେ, we can obtain \nthe relation ܶ/ܶେൌܶ/ܶେߟ ,where \n\t\t\t\t\t\t\tߟ ≡ ቈ1െ ሺ݃ୣେ୭/݃ୋୢሻଵ\nሺఉూిିఉృౚሻ൫ܯୋୢሺ0ሻ/ܯୣେ୭ሺ0ሻ൯ଵ\nሺఉూిିఉృౚሻ.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tሺ4ሻ  \nAs ߟ is decided by the given material parameters, Eq. (4) shows tha t ܶ/ܶେ i s  d i r e c t l y  \nproportional to ܶ/ܶେ.   \n7 To confirm the above theoretical prediction, we measured ܶେ for each sample, as \nlisted in Table II by performing the temperature dependence of the ܴୌ [33]. Figure 4( a) \nshows the variation in ܶ/ܶେ with respect to ܶ/ܶେ. This relationship is clearly linear. The \nslope and y-axis intercept of the best linear fit are 0.99 ± 0. 06 and 0.19 ± 0.02, respectively, \nimplying that ߟ i s  a p p r o x i m a t e l y  c o n s t a n t  f o r  t h e  s a m p l e s  w i t h  d i f f e r e n t  ܶେ and ܶ. \nFigure 4( b) shows the results of ߟ for each sample. This result confirms that ߟ is invariant \n(= 0.19) irrespective of the sam ples. Therefore, the results pr ove the validity of the general \nassumption used in the theory. \nFor a better insight, we study ߟ for each parameter: ܯୋୢሺ0ሻ, ܯୣେ୭ሺ0ሻ, ߚୋୢ, and \nߚୣେ୭. Previous studies show that ܯୋୢሺ0ሻ/ܯୣେ୭ሺ0ሻ ranges from 1.1 to 1.2, and ߚୋୢ and \nߚୣେ୭ a r e  0 . 4 5 ൏ߚୣେ୭൏0.5 and 0.65 ൏ߚୋୢ൏0.7, respectively [27, 28]. Using these values, \nwe can numerically calculate ߟ . The blue dashed lines, shown in Fig. 4( b), indicate the \ncalculation results of the upper and lower limits of ߟ on the basis of the reported parameters \n[27, 28]. For the case of typical ranges, the figure shows that  the experimental results are in \ngood agreement with the numerical calculation. Therefore, ߟ is approximately constant \nwithin the experimental accuracy. \nFrom the relation ܶ/ܶେൌܶ/ܶେߟ , we can estimate the angular momentum \ncompensation temperature. We denoted the estimated angular mome ntum compensation \ntemperature as ܶ∗, which can be calculated using the relation ܶܶߟେ. Here, we used ߟൌ\n\t0.19, from Fig. 4( b). To confirm the accordance with the measured angular momentum  \ncompensation temperature ܶ, ܶ∗ was calculated for all the samples and is plotted with \nrespect to ܶ, as shown in Fig. 4( c). The solid red line ( ܶ∗ൌܶ) indicates a good conformity \nof the estimated angular momentum compensation temperature. Thi s observation proves that \nthe experimental inaccuracy in determining ܶ∗ i s  w i t h i n  a  f e w  K e l v i n ,  w h i c h  c o u l d  b e    \n8 acceptable for estimating ܶ. In Fig. 4( c), the inaccuracy remains lower than approximately 5 \nK. \nFinally, it is worthwhile to discuss the general application of  the estimation method \n(of the ܶ∗) to other ferrimagnetic materials, e.g., TbFeCo. Because the L andé g factors of RE \nand TM elements, ݃ୖ and ݃, are different for other elements, ߟ can be changed \ndepending on the type of ferrimagnetic material. If ߟ can be determined for a given RE–TM \nferrimagnet, ܶ∗ can be easily estimated for any ferrimagnetic material by dete rmining ܶୡ \nand ܶ. \nIn conclusion, we investigate the correlation between ܶ and ܶ i n  G d F e C o  \nferrimagnets. The results show a strong correlation between ܶ and ܶ, which can be \ndemonstrated experimentally and theoretically. Moreover, simple  yet efficient method was \nemployed for estimating ܶ by measuring ܶେ and ܶ. Therefore, this observation will help \nin easily determining the angular momentum compensation tempera ture. Accordingly, the \nproposed scheme can be potentially applied to for ferrimagnet-b ased spintronics devices. \n    \n9 References \n1. A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A 369, 3098 (2011). \n2. S. H. Yang, K.-S. Ryu, and S. Parkin, Nat. Nanotech. 10, 221 (2015). \n3. T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. N anotech. 11, 231 (2016). \n4. O. Gomonay, T. Jungwirth, and J. Sinova, Phys. Rev. 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Mater. \n10, 853 (2011). \n \n    \n12 Figure Captions \nF i g u r e  1  ( a )  Schematic of the GdFeCo microwire device, and (b) Anomalous Hall effect \nresistance ܴୌ as a function of the perpendicular magnetic field ߤܪ௭ at room temperature \n(300 K). The orange arrow indicates the coercive field ߤܪୡ and the black up–down arrow \nindicates the Hall resistance difference ∆ܴୌ≡ܴୌ൫ܪ,ୱୟ୲൯െܴୌ൫െܪ,ୱୟ୲൯, where ܪ,ୱୟ୲ \nand െܪ,ୱୟ୲ are the saturation fi elds with the condition ܪୡ൏ܪ,ୱୟ୲. \nFigure 2 (a)  ∆ܴୌ as a function of the temperature ܶ for Sample II. The blue dot indicates \nthe magnetization compensation temperature ܶ, and (b) DW  speed ݒ as a function of ܶ \nfor Sample II at ߤܪ௭ൌ\t80 mT. The blue dot indicates the magnetization compensation \ntemperature ܶ, and the purple arrow indicates the angular momentum compensat ion \ntemperature ܶ. \nFigure 3  \tܶ with respect to ܶ. The red line is the best linear fit. \nFigure 4  (a) ܶ/ܶୡ as a function of ܶ/ܶୡ (the red line is the best linear fit), (b) ߛ w i t h  \nrespect to the sample number (the red line indicates ߛൌ \t0.19, and the blue dashed lines \nindicate the calculation of the upper and lower limits of ߛ based on the reported parameters), \nand (c) ܶ∗ with respect to ܶ (the red line is the best linear fit). \n   \n13 Acknowledgements  \nThis work was supported by JSPS KAKENHI (Grant Numbers 15H05702 , 26870300, \n26870304, 26103002, 26103004, 25220604, and 2604316). Collabora tive Research Program \no f  t he  I n s tit ut e f o r  C he m ica l R e sea r ch ,  K yo to  U n iv e r s i t y ,  an d  R  & D  p r o j e ct  fo r  IC T  Ke y \nTechnology of MEXT from the Japan Society for the Promotion of Science (JSPS). This \nwork was partly supported by The Cooperative Research Project P rogram of the Research \nInstitute of Electrical Communi cation, Tohoku University. D.H.K . was supported as an \nOverseas Researcher under Postdoctoral Fellowship of JSPS (Gran t Number P16314).  D.Y.K. \nand S.B.C. were supported by a National Research Foundations of  Korea (NRF) grant funded \nb y  t h e  M i n i s t r y  o f  S c i e n c e ,  I C T  a n d  F u t u r e  P l a n n i n g  o f  K o r e a  ( M SIP) \n(2015R1A2A1A05001698 and 2015M3D1A1070465). K.J.K. was supporte d by the National \nResearch Foundation of Korea (NRF) grant funded by the Korea Go vernment (MSIP) (No. \n2017R1C1B2009686, NRF-2016R1A5A1008184) and by the DGIST R&D Pr ogram of the \nMinistry of Science, ICT and Future Planning (17-BT-02). \nContributions of Authors \nD.H.K. conceptualized the work. D.H.K. and T.O. supervised the study. Y .F., H.Y., and A.T. \nprepared the films and Y.H. and T.O. made the devices. Y .H., D. H.K., T.O., T.N., and D.Y.K. \nconducted the experiments. D.H.K. and Y .H. performed the analys is, and D.H.K. modeled the \ndata. D.H.K., T.O., S.B.C., and K.J.K. wrote the manuscript. Al l authors discussed the results \nand commented on the manuscript. \n    \n14 Table I. Summary of the sample structures \n Sample Structures \n۷\t܍ܔܘܕ܉܁  5-nm SiN/20-nm Gd 23Fe67.4Co9.6/100-nm SiN/Si substrate \n۷۷\t܍ܔܘܕ܉܁  5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/100-nm SiN/Si substrate  \nSample III  5-nm SiN/1-nm Gd/ 5-nm Gd 23Fe67.4Co9.6/1-nm Gd/100-nm SiN/Si substrate \nSample IV  5-nm SiN/30-nm Gd 23Fe67.4Co9.6/5-nm Cu/5-nm SiN/Si substrate \nSample V  5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/5-nm SiN/Si substrate \nSample VI  5-nm SiN/20-nm Gd 23Fe67.3Co9.7/5-nm Pt/100-nm SiN/Si substrate \n \nTable II. Curie temperature ܶେ for each sample \n                 ( u n i t s :  K )  \n Sample I Sample II Sample III Sample IV Sample V Sample VI \n 475 ± 10 450 ± 10 355 ± 10 412 ± 10 412 ± 10 441 ± 10 \n -0.4 -0.2 0.0 0.2 0.4-3.0-1.50.01.53.0RH []\n0H [T]RH0HC\nFigure 1\nV\nIyIx y\nxz\nba\nHzSiN\nGdFeCo\nBufferBuffer0 100 200 300-6-3036\nRH []\nT [K]\nFigure 20 100 200 30003006009001200\nv [m/s]\nT [K]Tba0 100 200 3000150300450\n TA [K]\nTM [K]\nFigure 30.0 0.2 0.4 0.6 0.80.00.20.40.60.81.0\n1234560.000.250.50b\nTA/TC\nTM/TCa\n  \n\n# of Sample\nFigure 4150 200 250 300 350150200250300350\n T *\nA [K]\nTA [K]c"    },    {        "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<jJinterj, the low-\n\feld collinear antiferromagnetic phase (blue AFM region)\ndirectly transforms into the \feld-polarized state (green\nFM region) at the saturation \feld g\u00160HS=S=Jintra +\nJinter\u00002KT. Thus, for H < H Sthe magnetization is\nzero and it jumps to the saturation value at HS.\nWith increasing KT, two intermediate states appear\nbetween the AFM and FM regions, where either one or\nboth of the tetrahedral S2andS4spins are canted to-\nwards their easy plane. In the asymmetric canted state\n(yellow, ASC) the equivalency of these two spins is bro-\nken, since the exchange \feld and the external magnetic\n\feld are parallel for one spin, while they are antiparallel\nfor the other. Correspondingly, only the spin pointing\nopposite to the \feld in the collinear AFM state can gain\nZeeman energy by canting towards the easy plane, which\nleads further gain in anisotropy energy. The other tetra-\nhedral spin stays co-aligned with the Ising spins. Conse-\nquently, the ASC phase exhibits a spontaneous transverse\nmagnetization. Interestingly, for the canting-angle range\nof 0< # < \u0019= 2 the magnetization component trans-\nverse to the \feld exceeds the component along the \feld,\nMtrans=Mlong= sin#=(1\u0000cos#), thus, a signi\fcant mag-\nnetic torque is expected. On the contrary, in the spin-\n\rop phase (red SF region) the transverse magnetization\nvanishes due to the opposite (symmetric) canting of the\ntwo tetrahedral spins. Please note that in our simplis-\ntic model the ASC phase is the realization of the desired\nhybrid state with alternating AFM and ferrimagnetic lay-\ners.\nFor even stronger easy-plane anisotropy, 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 forHc2<H <H sin the perpendicular geometry and\nwithout any sign of metamagnetic transitions. This se-\nquence of phase transitions, with two intermediate phases\nbetween the zero-\feld AFM state and the \feld-polarized\nFM state, is in line with the prediction of our spin model,\nas clear from Fig. 2.\nIn order to show that the phase between Hc1and\nHc2has the ASC spin texture with \fnite transverse mo-\nment, we also performed high-\feld torque measurements.\nFig. 3(c) displays the transverse magnetization data in\narbitrary units, since we could not properly calibrate our\ncantilever torque magnetometry setup due to technical\nlimitations in pulsed \felds. Nevertheless, these experi-\nmental data con\frm that only this phase exhibits \fnite\ntransverse magnetization, in agreement with our simplis-\ntic model. In fact, as demonstrated in Figs. 3(a)-(c), a\nfully quantitative description of the magnetization datacan be achieved by re\fning the parameters of this spin\nmodel. (The only feature not captured by our mean-\n\feld model is the \fnite longitudinal susceptibility of the\ncollinear AFM state below Hc1, which likely has an in-\nherently quantum origin and arises from spin-stretching\nmodes of the S=3/2 spins42.) As the main step of this\nre\fnement, the Ising spins at the octahedral sites have\nbeen replaced by spins with strong but \fnite easy-axis\nanisotropy, KO= 26:8 K. The full parameter set best\nreproducing the magnetization data is given in Table I.\nAs described before, the four sublattices in our model\ncorrespond to two neighboring Co sites within a single\nhoneycomb layer (one with tetrahedral and one with oc-\ntahedral coordination) and the two adjacent Co sites in\nthe next layer, as sketched in Figs. 1 and 3(a). There-\nfore, the material-speci\fc calculations on Co 2Mo3O8re-\nveal that the phase between Hc1andHc2is composed\nof alternating layers of distinct spin patterns. Namely,\nevery second layer hosts collinear AFM order, similar to\nthe zero-\feld ground state, while spins in the interme-5\ndiate layers are canted away from the caxis. Moreover,\nthe spin canting at the tetrahedral sites is larger than\nthe canting of spins to the opposite direction at the oc-\ntahedral sites, due to the weak easy-plane and strong\neasy-axis anisotropy of the respective sites. The alternat-\ning stacking of collinear AFM and canted ferrimagnetic\nlayers is reproduced by the simulation also for larger sys-\ntem sizes along the caxis, excluding \fnite-size artifacts.\nAboveHc2, spins in all layers become canted, thereby\ncompleting the transition to the conventional spin-\rop\n(SF) phase. Due to their opposing anisotropies, the oc-\ntahedral and tetrahedral spins exhibit di\u000berent canting,\nbut the spontaneous magnetic moment perpendicular to\nthecaxis is cancelled. The spin arrangement in the dif-\nferent phases is visualized in Fig. 3(a).\nNext, we investigate the nature of the metamagnetic\ntransitions induced by \felds along the caxis by calculat-\ning the energy of the four-spin magnetic unit cell for a\nlarge set of spin con\fgurations. To reduce the number of\nindependent (spin) variables, we constrained this analy-\nsis to spin con\fgurations having the same total magne-\ntization as observed in the experiments. Speci\fcally, the\nspins in one layer ( S1andS2) were \fxed, while the orien-\ntation of the spins in the other layer ( S3andS4) was var-\nied to minimize the energy. The heatmaps in Fig. 4 show\nthe energy landscape obtained at selected magnetic \felds\non the plane of the azimuthal angles, #Oand#T, spanned\nrespectively by S1andS2and thecaxis. Here the diag-\nonal, connecting the bottom left and top right corners,\ncorresponds to the collinear AFM order of S1andS2. In\nH <H c1magnetic \felds, as presented in Fig. 4(b), there\nare two equivalent energy minima, corresponding to anti-\nphase domains of the collinear AFM state. In the ASC\nphase [Figs. 4(a) and (c)], the two minimuma again rep-\nresent anti-phase domains, where adjacent layers show\nasymmetric canting and host collinear AFM spin texture\nin reversed order. The continuous evolution of the energy\nlandscape accross Hc1[compare Figs. 4(b) and (c)] indi-\ncates a second-order transition, while the abrupt change\natHc2[see Figs. 4(c) and (d)] implies a \frst-order tran-\nsition.\nBesides the sequence of spin patterns, we also studied\nstructural changes induced via the spin-lattice coupling\nthroughout the metamagnetic transitions. Figs. 3(e)\nand (f) show that the magnetostriction, \u0001L/L, and the\nmagnetically-induced polarization, \u0001 P, both measured\nalong thecaxis, follow the same trend. In the collinear\nAFM both are constant. Above Hc1the unit cell ex-\npands along the caxis and \u0001L/L reaches a value of\n5\u000210\u00004at saturation, while \u0001 Plevels o\u000b at approx-\nimately -2350 \u0016C/m2, close to the magnitude reported\nfor the giant-magnetoelectric compound Fe 2Mo3O830,35.\nThese changes indicate a considerable magnetoelastic\ncoupling. Based on the \feld-dependent spin con\fgu-\nrations obtained from our model, the magnetostriction\nas well as the polarization change were reproduced by\nutilizing their proportionality to the spin-spin correla-\ntions43,44. We found intra-layer hSOSTi-like correlations\nFIG. 4.jStability of magnetic phases in Co 2Mo 3O8.\nHeatmap plots of the energy, obtained for spin con\fgurations\nwith magnetization constrained to the experimental value, on\nthe#O{#Tplane. #Oand#Tcorrespond to the azimuthal\nangles of the octahedrally ( S1) and tetrahedrally coordinated\n(S2) spins in the same layer, respetively. (For more details,\nsee the main text.) The angular ranges are common for each\nmap. Energies are obtained from the mean-\feld spin model\nfor magnetic \feld applied along the caxis. The energy min-\nima correspond to the collinear antiferromagnetic phase at\n25 T (b), the asymmetric canted phase at 27 T (c) and 29 T\n(a), and symmetric spin-\rop phase at 31 T (d). (a) The di-\nagonal dashed line in panel (a) corresponds to the collinear\nantiferromagnetic allignement of S1andS2. At the two global\nminimuma, the orientations of S1andS2are illustrated by\nblack arrows, while spins on the next layer are depicted by\ngray arrows. In these cartoons the vertical axis corresponds\nto the cdirection.\nto be dominant, implying a strong distance dependence\nof the intra-layer exchange, Jintra. Indeed, DFT struc-\nture relaxations for di\u000berent magnetic states suggest the\nshortening of the lattice constant within the layers and\nthe simultaneous increase in the inter-layer spacing upon\ngoing from AFM to FM order. This leads to a 1.8 % in-\ncrease inJintra, whereasJinter remains unchanged within\nthe accuracy of our calculations. (For more details, see\nthe supplement.)\nTo gain further insight into the magnetoelastic prop-\nerties, we measured the \feld dependence of the sound\nvelocity, \u0001v=v, with propagation vector and polarization\nalong and perpendicular to the caxis, respectively. The6\ndata are shown in Fig. 3(f). In analogy with the mag-\nnetostriction, the sound velocity stays constant up to\nHc1. However, as the most prominent feature, it shows\na large dip in the ASC state. Magnon-phonon coupling,\ne.g. via the exchange-striction mechanism, often results\nin a renormalization of the acoustic properties at mag-\nnetic transitions45. Since acoustic magnon modes play\na dominant role in such processes, the strong reduction\nof the sound velocity in the ASC state likely indicates\nthe emergence of soft spin-wave excitations. Indeed, due\nto its transverse ferromagnetic moment, the ASC phase\nbreaks the rotational symmetry of the spin-spin interac-\ntions about the caxis and consequently turns one of the\noptical magnon branches to an acoustic one. The emer-\ngence of such a zero-energy Goldstone mode in the center\nof the Brillouin zone, also con\frmed by our spin model,\nis likely responsible for the dip of the \feld-dependent\nsound velocity in the ASC state. In the ordinary spin-\n\rop phase above Hc2, the rotational symmetry is still\nbroken by the in-plane moments, hence the sound veloc-\nity further varies with an overall change of \u0001 v=v=-1.5 %\nup to the saturation \feld. In the \feld-polarized FM state\nthe sound velocity shows a sudden upturn, as expected\ndue to the lack of a Goldstone mode. The above scenario\nis also supported by the adiabatic temperature change in\npulsed magnetic \felds, the so-called magnetocaloric ef-\nfect (MCE), shown in Fig. 3(g). Besides sharp peaks at\nthe critical \felds, the most striking feature in the MCE\ncurve is the drop in the ASC state. Since MCE originates\nfrom the \feld-derivative of the speci\fc heat and/or the\nmagnetization, this dip again implies the emergence of\nsoft magnons.\nFinally, we investigated the thermal stability range of\nthe ASC phase by repeating the \feld-dependent mea-\nsurements at various temperatures between 1.5-50 K. As\nclear from Fig. 5, the ASC phase persist up to approxi-\nmately 33 K, i.e. it keeps separating the collinear AFM\nphase and the SF phase up to the paramagnetic state.\nIn conclusion, we observed a sequence of metamag-\nnetic transitions in Co 2Mo3O8with magnetically weakly\ncoupled honeycomb layers. We found that, due to a\ndelicate interplay of strong intra-layer and weak inter-\nlayer couplings with competing easy-axis and easy-plane\nanisotropies, the collinear AFM order is only partially\ndestabilized at the lowest critical \feld: the collinear AFM\nstate is preserved in every second honeycomb layer, while\na ferrimagnetic spin con\fguration emerges in the inter-\nmediate layers. Based on a mean-\feld spin model, sup-\nplemented by ab initio calculations and quantum Monte\nCarlo simulations, we achieve a consistent re\fnement of\nthe full set of magnetic interactions relevant in this proto-\ntype honeycomb antiferromagnet, and quantitatively de-\nscribe the magnetic and magnetoelectric/elastic proper-\nties throughout the four magnetic phases. We also show\nthat the hybrid phase with AFM and ferrimagnetic lay-\ners can extend to zero \feld if the easy-plane anisotropy\nis su\u000eciently strong. The concept to realize distinct spin\nstates in atomic-scale proximity, achieved in this honey-\nFIG. 5.jMagnetic phase diagram of Co 2Mo 3O8\nfor \felds along the c-axis, comprising antiferromagnetic\n(AFM, blue), asymmetric canted (ASC, mustard), spin \rop\n(SF, wine), spin-polarized ferromagnetic (FM, green), and\nparamagnetic (PM, white) phases. The critical \felds Hc1\nandHc2as well as the saturation \feld Hsare determined\nfrom magnetization (M), magnetostriction (\u0001 L=L) and mag-\nnetocaloric e\u000bect (MCE) measurements in pulsed \felds, from\npolarization (\u0001 P) and ultrasound velocity (\u0001 v=v) measure-\nments in static and pulsed \felds. For ultrasound experiments\nboth the velocity and the attenuation are plotted with full and\nopen red symbols, respectively, for longitudinal and transverse\nacoustic waves.\ncomb system with quasi-2 dimensional magnetic interac-\ntions, can be naturally extended to van der Waals mag-\nnets, to quasi-1 dimensional systems and crystals with\nmolecular-magnet-like building blocks.\nI. METHODS\nPolycrystalline Co 2Mo3O8was prepared by solid-state\nreactions of binary oxides CoO (99.999 %) and MoO 2\n(99 %) in evacuated quartz ampoules by repeated sinter-\ning at 1000\u000eC. Single crystals were grown by a chemical\ntransport reaction method between 950 and 900\u000eC us-\ning anhydrous TeCl 4as source of the transport agent.\nCrystals up to a size of 7 mm were obtained after four\nweeks of transport. Phase purity of the samples was con-\n\frmed by x-ray powder di\u000braction on crushed single crys-\ntals. Magnetization, magnetoelectric, magnetostriction,\nmagnetocaloric and sound velocity measurements were\nperformed in static and pulsed \felds up to 65 T at low\ntemperatures on single crystals of Co 2Mo3O8.\nMagnetic interactions and magnetic anisotropy were\nobtained by relativistic DFT band-structure calculations\nperformed using the FPLO46and VASP47codes utiliz-7\ning the Perde-Burke-Ernzerhof \ravor of the exchange-\ncorrelation potential48. Correlation e\u000bects in the Co\n3dshell were taken into account on the mean-\feld level\nvia DFT+Uwith the on-site Coulomb repulsion param-\neterUd=5 eV, Hund's exchange Jd=1 eV, and double-\ncounting correction in the atomic limit49. Magnetic pa-\nrameters of the spin Hamiltonian\nH=X\n<i;j>JijSiSj\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": "1705.02871v1.Non_local_magnon_transport_in_the_compensated_ferrimagnet_GdIG.pdf",        "content": "Non-local magnon transport in the compensated ferrimagnet GdIG\nKathrin Ganzhorn,1, 2,\u0003Tobias Wimmer,1, 2Joseph Barker,3Gerrit E. W.\nBauer,3, 4, 5Zhiyong Qiu,4Eiji Saitoh,3, 4, 6, 7Nynke Vlietstra,1, 2Stephan Gepr ags,1\nRudolf Gross,1, 2, 8Hans Huebl,1, 2, 8and Sebastian T.B. Goennenwein1, 2, 8, 9, 10\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n4WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n5Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands\n6CREST, Japan Science and Technology Agency, Tokyo 102-0076, Japan\n7Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n8Nanosystems Initiative Munich, 80799 Munich, Germany\n9Institut f ur Festk orper- und Materialphysik, Technische Universit at Dresden, 01062 Dresden, Germany\n10Center for Transport and Devices of Emergent Materials,\nTechnische Universit at Dresden, 01062 Dresden, Germany\nWe study the di\u000busive transport of magnons through the compensated ferrimagnetic insulator\nGd3Fe5O12(GdIG). The magnons are injected and detected electrically in a non-local measurement\ncon\fguration via two parallel Pt strips deposited on top of the ferrimagnet. GdIG exhibits a\nrich magnon spectrum, with several thermally populated magnon bands at room temperature. We\nobserve a strong temperature and \feld dependence of the non-local voltage in the detector strip. Just\nbelow the magnetization compensation temperature we \fnd that the increasing magnetic \feld causes\nan unexpected enhancement of the non-local signal. A comparison with GdIG spin wave spectra\nobtained from atomistic modeling indicates that the thermal magnon population is important for\nunderstanding the non-local voltage signal.\nSeveral experimental and theoretical investigations of\ndi\u000busive magnon transport through the ferrimagnetic in-\nsulator yttrium iron garnet (Y 3Fe5O12, YIG) have re-\ncently been conducted, using two electrically isolated Pt\nstrips for electrical injection and detection [1{7]. Upon\nrunning a dc charge current through a Pt injector strip,\nan electron spin accumulation is generated at the YIG jPt\ninterface via the spin Hall e\u000bect (SHE) [8, 9]. This spin\naccumulation induces a non-equilibrium magnon accu-\nmulation in the magnet beneath the injector which dif-\nfuses away from the injector and is detected in a sec-\nond Pt strip via the inverse spin Hall e\u000bect (ISHE). This\nso-called non-local magnon mediated magnetoresistance\n(MMR) has been studied as a function of magnetic \feld\n[10] and temperature in YIG/Pt bilayers [2, 11], giving\ninsight into the length scales associated with magnon\ntransport. However, the microscopic mechanisms respon-\nsible for the MMR are still under investigation. One key\nquestion is which magnons or magnon bands provide the\nmost signi\fcant contributions to di\u000busive spin transport\nand the non-local voltage signal induced in the detector\nstrip. Furthermore, the MMR has only been measured in\ncollinear magnetic systems such as YIG or NiFe 2O4[12]\nso far. The in\ruence of non-collinear magnetic con\fgu-\nrations, such as a canted ferrimagnet, has not yet been\ninvestigated.\nHere, we study the non-local magnon transport in\nthe compensated ferrimagnet gadolinium iron garnet\n(Gd 3Fe5O12, GdIG), where the non-magnetic Y3+in\nYIG is substituted with magnetic Gd3+. GdIG there-\nfore consists of three magnetic sublattices: two antifer-romagnetically coupled Fe3+sublattices (FeA and FeD),\nand a Gd sublattice which is weakly antiferromagneti-\ncally coupled to the FeD moments [13, 14]. GdIG ex-\nhibits a magnetization compensation temperature Tcomp,\nwhere the remanent magnetization exactly vanishes due\nto the di\u000berent temperature dependencies of the antifer-\nromagnetically coupled net Fe and Gd sublattice magne-\ntizations [13]. Far away from the compensation temper-\nature, GdIG is a collinear ferrimagnet and the sublattice\nmagnetizations are all aligned (anti)parallel to the exter-\nnal magnetic \feld. Close to Tcomp, a canted phase can\nbe induced with magnetic \feld magnitudes accessible in\nexperiments, where the magnetic moments on the dif-\nferent sublattices are no longer collinear with the \feld,\nor one another [13{16]. The corresponding magnetic\nphase diagram can be found in Ref. [14]. Recently it was\nshown that the spin Hall magnetoresistance e\u000bect [17{20]\n(SMR) can electrically probe the interface magnetization\ntexture, since the SMR is sensitive to the orientation of\nthe individual sublattice magnetic moments relative to\nthe polarization of the spin accumulation in the adjacent\nPt [14, 21]. Non-local magnon transport or spin di\u000bu-\nsion has not yet been studied in a non-collinear magnetic\nstructure and will be presented in this Article. Aside\nfrom the magnetic compensation point, GdIG also di\u000bers\nfrom YIG in that the spin wave spectrum is richer in the\nlow THz regime. Several additional modes are present\ndue to the magnetic Gd sublattice. Local spin Seebeck\ne\u000bect (SSE) measurements in GdIG showed strong evi-\ndence that two spin wave modes with opposite polariza-\ntion dominate the thermal and dynamical behaviour ofarXiv:1705.02871v1  [cond-mat.mes-hall]  8 May 20172\nthe spin system, resulting in a sign change of the SSE\nvoltage [22, 23]. GdIG is therefore a good material to\ninvestigate the in\ruence of di\u000berent magnon modes and\ntheir polarization on spin di\u000busion. Our experimental\nresults suggest that the complex magnon spectrum of\nGdIG indeed qualitatively impacts the \feld and temper-\nature dependence of the MMR.\nThe investigated sample is a 2 :6µm thick GdIG \flm\ngrown on top of a (111)-oriented Gd 3Ga5O12substrate\nvia liquid phase epitaxy (LPE). After cleaning in Pi-\nranha solution and annealing at 500\u000eC in an oxygen at-\nmosphere of 75 µbar for 40 min, a 10 nm thick Pt \flm\nwas deposited onto the GdIG via electron beam evapo-\nration (see Refs. 2 and 24 for details). Two Pt strips\nwith edge-to-edge separation d= 200 nm and strip width\nw= 500 nm (Fig. 1 (a)) were de\fned by electron beam\nlithography followed by Ar ion etching. The sample was\nmounted in the variable temperature insert of a super-\nconducting magnet cryostat allowing dc transport mea-\nsurements at temperatures between 5 K < T < 300 K\nand magnetic \felds up to 15 T. For the electric trans-\nport measurements a charge current Ic= 100 µA (cor-\nresponding to a current density of 2 \u00021010A m\u00002) was\napplied to the left strip using a Keithley 2400 sourceme-\nter. The local ( Vloc) and non-local voltage drop ( Vnl) in\nthe injector and detector strip were simultaneously mea-\nsured with two Keithley 2182 nanovoltmeters (see Fig. 1\n(a)). To eliminate thermal signals due to Joule heat-\ning in the local strip (e.g. spin Seebeck e\u000bect [25]) and\nto increase the signal-to-noise ratio, a current switching\nmethod was used, as described in Ref. [2]. To obtain\nthe SMR and MMR amplitude, angle dependent mag-\nnetoresistance (ADMR) measurements were carried out\nby rotating the external magnetic \feld of constant mag-\nnitude in the thin \flm plane, while recording Vlocand\nVnl.\nThe local response Vlocmeasured at 300 K as a func-\ntion of the angle \u000bbetween IcandHis shown in\nFig. 1 (b) for \u00160H= 1 T (grey) and 7 T (green). From\nthese ADMR measurements, the SMR amplitude de\fned\nas \u0001Vloc=Vloc;min= (Vloc(0\u000e)\u0000Vloc(90\u000e))=Vloc(90\u000e) was\nextracted and plotted as a function of temperature for\nmagnetic \feld strengths \u00160H= 1;3;5 and 7 T in Fig. 1\n(d). Cooling to lower temperatures, the SMR ampli-\ntude decreases by a factor 2 compared to room tem-\nperature, consistent with measurements in YIG/Pt [26]\nand (In, Y) doped GdIG/Pt bilayers [14]. In a narrow\ntemperature range around the compensation tempera-\nture (Tcomp = 268 K determined via SQUID magnetom-\netry), the SMR decreases to zero. A similar behaviour\nhas been observed in (In, Y) doped GdIG [14] and was\nattributed to the formation of the canting phase in GdIG\nclose to the compensation temperature, where the sublat-\ntice magnetizations are no longer collinear. In Ref. [14] a\nsign change of the SMR was observed and interpreted as a\nperpendicular alignment of the sublattice moments with\n090180270360-1.50E-007-1.00E-007-5.00E-0080.00E+0005.00E-0080901802703600.66440.6646\n0501001502002503000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-007\n0 100 200 300\nTemperature (K) /uni0394Vloc /Vloc, min (x 10-4)\n0246\n1T\n3T\n5T\n7T200\n100150\n050/uni0394Vnl (nV )\n0 100 200 3000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-007\n0 100 200 3000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-007\n0 100 200 3000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-007\n0 100 200 3000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-0079T\n11T\n13T\n15T(a)\n(c)\n50 150 250TcrossTcomp0° 90° 180° 270° 360°664.4664.6Vloc (mV)\n0\n-50\n-100Vnl (nV)\n(d)\n(e)α\nVloc+\n-Ic+\n- Vnl+\n-(b)\nH\n α\n7T7T1T\n1T\nα (deg)026\n260 280\nTemperature (K) /uni0394Vloc /Vloc, min\n(x 10-4)4FIG. 1. (a) Schematic representation of the GdIG(green)/Pt\n(grey) nanostructures consisting of two parallel Pt strips of\nwidthwand edge-to-edge separation d. (b) Local voltage\nVlocas a function of the angle \u000bbetween IcandHat 300 K.\nDue to the weak \feld dependence of the local signal only\nexperimental data taken at \u00160H= 1 T and 7 T are shown.\n(c) Non-local voltage Vnlas a function of \u000bat 300 K for\n\u00160H= 1;3;5 and 7 T. (d) Temperature dependent SMR am-\nplitude (Vloc(0\u000e)\u0000Vloc(90\u000e))=Vloc(90\u000e) extracted from the\nlocal voltage amplitude for di\u000berent magnetic \felds up to\n7 T. The inset is a close-up of the SMR amplitude around\nthe compensation temperature Tcomp = 268 K (red vertical\nline). (e) Temperature dependence of the MMR amplitude\n\u0001Vnl=Vnl(0\u000e)\u0000Vnl(90\u000e) obtained from ADMR measure-\nments for magnetic \felds up to 15 T.\nrespect to the external \feld. The absence of such a sign\nchange in the present data may be attributed to magnetic\ndomain formation for temperatures close to Tcomp, such\nthat on average the SMR modulation vanishes. How-\never the broadening of the dip in the SMR vs. temper-3\nature curves with increasing magnetic \feld (see inset of\nFig. 1 (d)) is in good agreement with the observations in\nRef. [14] and is expected for the magnetic canting phase\nin GdIG [15]. As reported in Ref. [14], in the collinear\nphase the SMR in Fig. 1 (d) hardly depends on the ap-\nplied magnetic \feld. This is consistent with the estab-\nlished SMR model [18, 27], where only the orientation of\nthe sublattice moments is relevant [14] [28].\nThe angular dependence of the non-local response Vnl\nat 300 K is shown in Fig. 1 (c) for di\u000berent magnetic \feld\nstrengths. The non-local voltage signature has the same\ndependence on the external magnetic \feld orientation\nas in previous MMR measurements in YIG/Pt nanos-\ntructures for the same wiring scheme [2]. The MMR ef-\nfect amplitude extracted as \u0001 Vnl=Vnl(0\u000e)\u0000Vnl(90\u000e) =\n\u0000Vnl(90\u000e)>0 is plotted as a function of temperature in\nFig. 1 (e) for \felds up to 7 T. In contrast to the local\nSMR, where no substantial \feld dependence is observed\nin the collinear phase, the non-local MMR signal displays\na more complex \feld and temperature dependence. We\nhave therefore taken additional non-local data in \felds\nup to 15 T for temperatures between 150 K and 300 K,\nas shown in Fig. 1 (e). We \frst focus on the data taken\nat 1 T (grey open squares in Fig. 1 (e)). At low tem-\nperatures \u0001 Vnlvanishes, similar to recent observations\nin YIG/Pt [2] where a T3=2dependence of \u0001 Vnlwas ob-\nserved. This temperature dependence in YIG is in agree-\nment with theoretical expectations based on the magnon\ndensity of states and distribution function [3, 4, 29],\ni.e. general properties of the magnonic system. At low\ntemperatures, we therefore expect GdIG to behave very\nsimilar to YIG.\nWith increasing temperature, the non-local signal mea-\nsured at 1 T increases up to 150 nV just below the com-\npensation temperature. The MMR amplitude then drops\nbelow the experimental resolution of about 5 nV [2] at\nTcomp and recovers a \fnite value above Tcomp, similar\nto what is observed in the local SMR (see Fig. 1 (d)).\nWe attribute the vanishing MMR signal at Tcomp to the\nchange of the magnetic structure of GdIG into the canted\nphase. A vanishing non-local signal in the canted phase\nsuggests two possible scenarios, either (i) the magnon\ninjection/detection becomes ine\u000ecient due to the non-\ncollinear alignment of the magnetic sublattice moments\nand/or (ii) the damping close to the compensation point\nis enhanced [30] leading to shorter magnon lifetimes and\nthus shorter di\u000busion lengths. Additional magnon scat-\ntering e\u000bects may arise due to magnetic domain forma-\ntion in the canting phase as discussed above, which fur-\nther suppress magnon transport.\nWe now turn to the magnetic \feld dependence of the\nMMR e\u000bect. The relative \feld dependence \u000e(\u00160H) of the\nMMR amplitude \u0001 Vnlis de\fned as\n\u000e(\u00160H) =\u0001Vnl(\u00160H)\u0000\u0001Vnl(1 T)\n\u0001Vnl(1 T)(1)\n050100150200250300\n1.00.50.0-0.5\nRelativesuppression\n7T\n15T\n0 100 200 300\nTemperature (K)50 150 2500\n-10050\n-50Tcomp\nTcross7T\n15TRelative suppression \nδ(µ0H) (%)FIG. 2. Relative magnetic \feld dependence \u000e(7 T) (green tri-\nangles) and \u000e(15 T) (purple hexagons) of the MMR e\u000bect de-\n\fned by Eq. (1). A negative (positive) value corresponds to\na suppression (enhancement) with increasing \feld. The grey\nshaded region indicates the temperature range around Tcomp\nin which the MMR drops to values close to or even below the\nnoise level, impeding a reliable quanti\fcation of \u000e.\nand plotted as a function of temperature in Fig. 2 for\n\u00160H= 7 T (green triangles). A negative value corre-\nsponds to a suppression of the non-local voltage, while\na positive value represents an increase of the MMR with\nincreasing magnetic \feld. The \feld dependence is calcu-\nlated relative to the measurements at 1 T, which is the\nlowest magnetic \feld studied here. We distinguish three\ntemperature regimes: (i) for 0 < T < T cross\u0019210 K,\nthe MMR amplitude is suppressed with applied mag-\nnetic \feld by about \u000e(7 T) = 25%, (ii) in the range\nTcross< T < T comp an enhancement with \feld is ob-\nserved, while (iii) for Tcomp<T the MMR e\u000bect is sup-\npressed by up to \u000e(7 T) = 50%. The MMR suppression\nin the canted phase (grey shaded area in Fig. 2) appears\nto be very large. However, since the MMR amplitude is\nclose to the noise level, the relative \feld e\u000bect is prone\nto large errors. Additional measurements of \u000e(\u00160H) up\nto\u00160H= 15 T between T= 150 K and 300 K (purple\nhexagons in Fig. 2) qualitatively con\frm the data taken\nat 7 T, displaying a more pronounced suppression and\nenhancement of the MMR. In particular, the non-local\nsignal can be almost completely suppressed above Tcomp\nat 15 T.\nA magnetic \feld induced reduction of the MMR at\nroom temperature has also been observed in YIG/Pt\nand was attributed to a \feld-dependent magnon di\u000busion\nlength and a reduced magnon injection e\u000eciency [10].\nFurther temperature dependent MMR measurements on\nYIG/Pt (not shown here) con\frm a \feld suppression of\nup to\u000e(7 T)\u0019\u000025% in the entire investigated temper-\nature range from 50 K to 300 K. While the low temper-\nature \feld suppression of the MMR in GdIG/Pt is simi-\nlar to that of YIG/Pt, the \feld dependence above Tcomp4\nis much stronger. Furthermore, the enhancement of the\nMMR with magnetic \feld in the temperature range 200 K\nto 250 K as shown in Fig. 1 and 2 is also not observed in\nYIG/Pt.\nIn the following we discuss our experimental observa-\ntions in terms of the magnon injection e\u000eciency which\ndepends on the available magnon states in the ferrimag-\nnet [29] and is thus related to the thermal magnon popu-\nlation. In YIG, only the lowest energy ferromagnetic-like\nmode is thermally populated within the studied tempera-\nture range [31, 32] (only near room temperature the high\nenergy exchange mode starts to become populated and\nis expected to a\u000bect the SSE in YIG [32]). An external\nmagnetic \feld shifts the ferromagnetic mode to higher\nfrequencies as the Zeeman gap opens, thereby freezing\nout the lowest energy magnons. In this picture, the mag-\nnetic \feld suppression observed in SSE experiments in\nYIG/Pt was interpreted in terms of the importance of low\nenergy magnons at elevated temperatures [33, 34]. The\nsuppression of the detected MMR signal in YIG when\nincreasing the external magnetic \feld at a \fxed injector-\ndetector separation [10] is also consistent with a freeze\nout of magnons from the ferromagnetic mode.\nTo better understand the observed complex \feld and\ntemperature dependence of the MMR in GdIG we there-\nfore study the characteristic changes in the GdIG spin\nwave spectrum as a function of temperature and mag-\nnetic \feld using atomic spin dynamics simulations. The\ncalculations are based on a classical Heisenberg model\nwhere the Landau-Lifshitz-Gilbert equation is used to\nsolve the spin dynamics and a Langevin thermostat in-\ntroduces temperature. Details of the model including the\nparameters used for GdIG can be found in Refs. [14, 22].\nThe spin wave spectrum is calculated from the space-\ntime Fourier transform of the spin \ructuations. The spin\nwave modes in ferrimagnets have a polarization which\ndescribes the sense of rotation with respect to an ap-\nplied \feld [32], i.e. counterclockwise/clockwise, which is\nencoded by a red/blue coloring in the \fgures. We label\nthe modes \u000b- the lowest frequency dispersive mode at\n75 K and\f- the parabolic optical mode with opposite po-\nlarization (see Fig. 3 (a)). The modes move in frequency\nspace with temperature and \feld and change polarization\nacross the compensation point at Tsim\ncomp = 310 K, but we\nretain the same \u000b,\fdesignations throughout. Note that\nTsim\ncomp = 310 K is higher than the value Tcomp = 268 K\nexperimentally observed in our sample. The \rat, broad-\nened bands around 1 THz do not contribute signi\fcantly\nto transport because of their small group velocity and\nlarge linewidth. This was con\frmed by the temperature\ndependence of the spin Seebeck e\u000bect in GdIG/Pt bilay-\ners [22]. As detailed in Ref. [22], the SSE can be under-\nstood considering that the transport of thermal magnons\nin GdIG is dominated by the \u000band\fmagnon modes.\nAt low temperatures only the \u000b-mode is thermally pop-\nulated (red mode in Fig. 3 (a)), while with increasing\nµ0H=7T µ0H=1T\n(a)\n(e)(d) (c)\n(f)0 001 k -k 001/uni02C9\n0 001 k -k 001/uni02C9\n0 001 k -k 001/uni02C95\n4\n3\n2\n10f (THz)\n5\n43\n2\n10f (THz)\n5\n43\n2\n10f (THz)(b)\n T=75K < Tcross , Tcomp \nTcross < T = 300K < Tcomp \nTcross , Tcomp < T = 350KkBT\nα-mode\n0 001 k -k 001/uni02C9\n0 001 k -k 001/uni02C9\n0 001 k -k 001/uni02C9sim\nsimsim\nβ-modeFIG. 3. Magnon spectrum of GdIG obtained from atomistic\nsimulations, calculated (a)-(b) at T= 75 K, (c)-(d) just below\nTsim\ncomp = 310 K and (e)-(f) above the compensation point. The\nred and blue color indicates counter-clockwise and clockwise\nprecession directions relative to the external magnetic \feld\ndirection, i.e. positive and negative polarization, respectively.\nThe spectra are calculated for \u00160H= 1 T and 7 T revealing\nthe e\u000bect of the Zeeman shift of the red mode. The black\ndashed line corresponds to the thermal energy kBT.\ntemperature the \f-mode shifts below kBT(blue mode in\nFig. 3 (c)) and dominates the SSE, leading to a low tem-\nperature SSE sign change [22]. At Tcomp the orientation\nof the magnetic sublattices is inverted and consequently\nthe two magnon modes exchange roles (and polarization),\nas shown in Fig. 3 (c) and (e), leading to an abrupt sec-\nond sign change of the SSE signal at the compensation\npoint [22]. These sign changes can be understood consid-\nering that magnons with opposite polarization carry op-\nposite angular momentum. The FMR-like magnons with\npositive polarization (red modes) reduce the net mag-\nnetization. The gapped magnons at higher frequencies\narise from the precession of magnetic sublattice moments\nin the exchange \feld provided by the other sublattices,\nsimilar to the excitations in an antiferromagnet [35, 36].\nMagnons with negative polarization (blue modes) corre-\nspond to the precession of sublattice moments aligned\nantiparallel to the external \feld, such that their excita-\ntion increases the net magnetization. The MMR arises5\nfrom transfer of angular momentum from the electron to\nthe magnon system (and vice versa) [29], implying that\nmagnons with opposite polarization (carrying opposite\nangular momentum) may also lead to an opposite sign of\nthe detected MMR signal. Since we observe no such sign\nchange in the MMR in Fig. 1 (e), we conclude that either\nboth modes contribute with the same sign or that the red\nmode dominates the MMR for the Pt strip separation of\n200 nm and the temperatures studied here.\n00.050.10.150.20.250.30.350.4\n220 240 260 280 300 320 340 360 380\n\n\n α\nfk=0(THz)\nTemperature (K)Tsim\ncomp\nβ\nβ α7T\n1T H\nMFeAMFeD\nMGdMnetMnet\nFIG. 4. Temperature and \feld dependence of the k= 0\nfrequency for the \u000band\fmodes. The red and blue color\nmarks positive and negative polarization, respectively. The\norientation of the sublattice and net magnetization direction\nwith respect to the external \feld His depicted by arrows\nfor temperatures above and below the compensation point\nTsim\ncomp = 310 K.\nThe contributions from di\u000berent modes can be distin-\nguished by studying the \feld and temperature depen-\ndence of the MMR. The frequency values at k= 0 for the\n\u000band\fmodes from Fig. 3 are compiled in Fig. 4. The\norientation of the sublattice magnetizations with respect\nto the external \feld Habove and below the compensa-\ntion point is indicated by arrows. Similar to the FMR-\nlike mode in YIG, the fundamental (red) mode in Fig. 3\nshifts up in frequency proportional to the applied \feld as\nthe Zeeman gap opens, but has a very weak temperature\ndependence (red symbols in Fig. 4). The blue `exchange'\nmode has a much stronger temperature dependence (blue\nsymbols in Fig. 4) caused by relative changes in sublat-\ntice magnetization altering the e\u000bective exchange \felds\nwhich determine this mode [22].\nAt low temperatures (Fig. 3 (a) and (b)), the \f-mode\nis not populated and does not contribute to the spin\ntransport. The applied magnetic \feld freezes out the \u000b-\nmagnons, thereby reducing the MMR signal (see Fig. 1\n(e) and 2). With increasing temperature, the exchange\ngap of the\f-mode decreases (see Fig. 3 (c) and 4) due to\nthe thermally induced disorder in the Gd system, leading\nto a shift of the exchange mode to lower frequencies [22].\nBelowTcomp, this trend is partly reverted by a magnetic\feld that increases the Gd order and thereby increases\nthe\f-mode frequencies again (see \feld dependence of\nthe blue lines in Fig. 4). The spin current carried by\nmagnons with opposite polarization reduces the detected\nMMR amplitude. A magnetic \feld induced depopula-\ntion of the blue \fmagnons below Tcomp should there-\nfore enhance the MMR signal. Although the frequency\nshift of the beta-mode predicted by simulations is small\ncompared to the Zeeman shift of the alpha-mode, such a\ntrend is indeed observed in the experimental MMR data\ndisplayed in Fig. 1 between Tcross andTcomp.\nAbove the compensation temperature, the Gd mo-\nments are oriented antiparallel to the external \feld and\ndepolarize when the latter is increased [14], which implies\na frequency reduction by magnetic \feld. In other words,\nthe red (blue) mode shifts to higher (lower) frequencies\nand is depopulated (populated) with magnetic \feld. The\nobserved strong suppression of the MMR above Tcomp in\nFig. 1 and 2 is consistent with this picture.\nThe study of the magnon spectrum therefore helps to\nunderstand the \feld and temperature dependence of the\nMMR in GdIG. While for a quantitative analysis addi-\ntional factors such as the magnon transport properties\nneed to be carefully considered, our results suggest that\nthe magnon population plays an important role for the\nMMR e\u000bect.\nIn summary, we measured the non-local magnon me-\ndiated magnetoresistance e\u000bect (MMR) in the collinear\nas well as in the canted phase of the compensated fer-\nrimagnet GdIG. The data taken close to the compensa-\ntion temperature suggest that the MMR is suppressed in\nthe canted phase, either due to ine\u000ecient magnon injec-\ntion or to magnetic domain formation. The MMR signal\nis furthermore suppressed by magnetic \feld at low tem-\nperatures and above Tcomp, but is enhanced just below\ncompensation. We relate these changes to the magnon\nspectrum of GdIG by comparison with atomistic simula-\ntions. 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Department of Materials Science and Engineering, Kyoto University, Kyoto 606 -8501, Japan   \n2. Department of Physics, Faculty of Science, Assiut University, 71516 Assiut, Egypt  \n3. Lab. of Smart Materials for Energy Futures, Faculty of Science, Assiut Univers ity, Assiut 71516, Egypt  \n* Electronic email: makassem@aun.edu.eg  \nAbstract  \n Spinel chromites ACr2O4 with inherent magnetic geometrical frustration usually exhibit a \nnoncollinear ferrimagnetic ground state when A2+ are magnetic ions, with possibl y crystallite -size \ndependent intriguing magnetic features. Here, we report single -domain magnetic properties of \nACr2O4 (A = Ni, Co, and Mn ) nanocrystals , with an average crystallite size of 18, 15 and 10 nm , \nexhibiting an optical energy gap of ~ 2.87, 3.05 and 2.9 eV, respectively. The temperature \ndependence of magnetization indicates  the main bulk magnetic transitions with a commonly \ncoexisting spin -glass -like state and finite -size effects on the noncolinear ferrimagnet ic transitions . \nAn anomaly observed at Ts = 15, 24 and 10 K is attributed to the  bulk magnetic transition to a \ncanted antiferromagnetic state  in NiCr 2O4 and incommensurate spiral order s in CoCr 2O4 and \nMnCr 2O4 NCs, respectively . A further bulk magnetic transition to a commensurate spiral order is \nobserved for CoCr 2O4 NCs at a ‘lock -in’ temperature Tl = 5 K much lower than  that reported using \nbulk samples , while it is completely suppressed in the MnCr 2O4 NCs. Finite-size effects and \nsingle -domain  magnetic behavior s indicated by anomalous temperature -dependences of the \ncoercive field and the hysteresis -loop squareness, mainly driven by a magnetocrystalline  \nanisotropy , are discussed in comparison to results  reported using bulk counterparts .   \n \nKey words:  Spinel c hromite, Magnetic nanoparticles, Microwave combustion,  Single domain, \nCoercive field . \n \n   2 1. Introduction  \nInsulating magnets of noncollinear spin order are platforms of related phenomena, such as \nmultiferroicity, with possible functionalities in spintronic devices for high -density data storage \n[1]–[9]. Among the noncollinear magnetic insulators, ferrimagneti c (FIM) spinel chromites \nACr2O4, are of great interests due to their exotic electronic and magnetic properties such as the \nlarge magnetooptical and magnetodielectric couplings with a spontaneous dielectric polarization \n[6], [7], [10], [11] . The ACr2O4 compounds crystalize  in the norma l spinel structure with the \ndivalent A2+ cation s rest in the tetrahedrally coordinated A site  while the Cr3+ ions fully occupy \nthe octahedrally coordinated B site forming a 3D pyrochlore sublattice, as schematically \nrepresented in  Figs. 1(a), (b) for the common  spinel 𝐹𝑑3̅𝑚 cubic structure . The spinel structure is \ngenerally known with its essential antiferromagnetic (AFM) exchange interactions  JA-A, JB-B and \nJA-B those cannot simultaneously be negative [12]. In the normal spinel chromites, the AFM JCr-Cr \nlonely occurs with highly effective geometrical spin frustration when the A2+ cations  are \nnonmagnetic. In case the A site is occupied by magnetic ions, the negative JA-Cr becomes dominant \nand both JA-A and JCr-Cr become positive resulting in a FIM order, however, the effects of the \ninherent geometrical spin frustration remain in FIM spinel chromites. The geometrical magnetic \nfrustration is assumed as the key force for the stabilization of noncollinear spin order in these FIM \ninsulating spinels  [13]–[15]. \nThe FIM spinel chromites ACr2O4 (A = Ni, Cu, Fe, Mn, and Co) were extensively studied by \nmagnetization, dielectric polarization, and microscopic measurements [6], [7], [10], [14], [16] . FIM \nchromites with Jahn -teller active A2+ ions,  such as  Ni2+, Cu2+and Fe2+, exhibit symmetry lowering \nfrom the cubic 𝐹𝑑3̅𝑚 to a tetragonal I4 1/amd structure by a Jahn−Teller distortion at different \ntemperatures,  TJT. For example, it occurs below room temperature at  TJT = 140 K in case of \nFeCr 2O4 (lattice parameters ratio, c/a < 1)[17] while above room temperature at TJT = 310 K for \nNiCr 2O4 (c/a > 1)  [18] and at TJT = 850 K for CuCr 2O4 (c/a < 1)[18]. At TJT, the crystal -field \nsplitting  of the 3d levels are modified from those in the spinel cubic phase, as shown for Ni2+ in \nFig. 1(c ). On the other hand,  these FIM spinel chromites exhibit noncollinear FIM ground states, \ncanted AFM states in Ni Cr2O4 and CuCr 2O4 and conical FIM states in Fe, Co, and Mn spinel \nchromites . A FIM transition occurs at a Curie temperature TC ~ 74, 152, 97, 94 and 48 K for A = \nNi, Cu , Fe, Co, and Mn, respectively [5], [9], [19] . Except for CuCr 2O4 that is a noncollinear FIM  \nbelow TC [19], these FIM chromites exhibit a further magnetic transition from a collinear to a non -\ncollinear FIM state  below a temperature  Ts (< TC )[5], [14], [16] . In case of A = Mn, Fe and Co, a \nspiral component emerges below  Ts ~ 19.5, 35, and 24 K, respectively, that changes from \nincommensurate to commensurate spiral spin structure in Co and Mn chromites at a ‘lock -in’ \ntemperature, Tl ~ 13 and 17.5 K, respectively [5], [7], [14], [16] . While  in NiCr 2O4, a collinear \ntransverse antiferromagnetic (AFM) component develops, resulting in a canted AFM state, below  3 Ts ~ 31 K [16]. The successive structural and magnetic transitions between multiple states as \ndescribed above  are associated with exotic magnetic, optical and magnetoelectrical properties of \nthese compounds . For instance, the magnetic transitions at Ts are accompanied by an  emergent \npolarization  state, i.e., multiferroic effect, below Ts in the FIM spinel chromites  [1], [3], [6], [20] .  \nIn a sensitivity to the nearest -neighbor arrangement and the antiferromagnetic exchanges between \nA–Cr and Cr–Cr  sites, changing the A cation type significantly  changes the physical  properties of \nthe spinel chromites.  \n \nFigure 1 . (a) C ubic c rystal structure of the spinel chromites ACr2O4 schematically represented \nby a unit cell of the cationic ions A2+and Cr3+ with the Cr3+ ions forming a pyrochlore sublattice. \n(b) Crystallographic  polyhedrons of cation s in the A and B sites , and (c) crystal -field splitting \nschemes of the 3d -levels and their occupation for Ni2+, Co2+, and Mn2+ in the tetrahedral A site \nin the possible tetragonal ( c/a > 1) and cubic  (c/a = 1) spinel phase s [21]. \nThe magnetic properties of spinel chromites has been mainly studied using poly - and single -\ncrystalline  samples, however, the study of these materials  as magnetic nanoparticles , hitherto rare \nrelative to ferrites, can elucidate their property variations by finite size effects. Recently we have \nexplored t he magnetic properties of nanoparticles of the AFM spinel chromites MCr2O4 (M = Zn, \nMg and Cd) in comparison to the FIM CoCr 2O4 nanoparticles [22]–[24]. In this manuscript, we \nstudy the optical and magnetic properties of single -phase and high -quality nanocrystals (NCs) of \nACr2O4 (A = Ni, Co, and Mn) with average particle (and crystallite) sizes of 15, 13 and 10 nm, \nrespectively, synthesized by a facile microwave combustion method. The spectroscopic \ncharacteristics, corresp onding to the ionic bonds (vibration) and electronic bands (transition), are \nstudied by FT -IR and UV -Vis absorbance spectroscopies. Finite size effects  on the bulk magnetic \ntransitions and single -domain magnetic properties including high and largely temper ature -\ndependent coercivity are explored. The observed single -domain nanomagnetism of these spinel \n 4 noncollinear FIM chromites are discussed with an interrelation between  their structural and optical \ncharacteristics, and in a comparison view to the results reported using bulk  poly- and single \ncrystalline samples . \n2. Experimental procedures  \n2.1.  Synthesis  \nStarting with salts of a divalent A2+ cation, Ni(NO 3)2.6H 2O, Mn(NO 3)2.6H 2O, or \nCo(NO 3)2.6H 2O, and trivalent Cr3+ cation,  Cr(NO 3)3.9H 2O, nanoparticles of ACr2O4 (A = Ni,  Co \nor Mn) have been synthesized by a microwave combustion [24]. Glycine NH 2CH 2COOH was \nemployed in this synthesis as a combustion fuel that burns at a sufficient temperature initiating the \nreaction and completing the product oxidation process. As well as cations source, metal nitrates \nact as oxidant, while the fuel is a source of carbon and hydrogen and act as a partial reducing agent \nusually yielding gaseous products such as CO 2 and H 2O. However, th e fuel ratio is crucial for the \nsynthesis and is selected based on a required complete redox reaction [25]. The ratio of glycine to \ncomplete the reaction is calculated with respect to the gaseous product from the combustion [26], \n[27] and this reaction synthesis of ACr2O4 can be described as:  \n 𝐴(NO3)2.6H2O+2 Cr(NO3)3.9H2O+ δ C2H5O2N +(9δ−40 \n4)O2        ∆       →     𝐴Cr2O4\n+(5δ+48\n2)H2O +(δ+5\n2)N2 +2δ CO2 , (1) \nwhere A is a divalent cation and  is the amount required  of the fuel glycine. The divalent and \ntrivalent metal nitrates  were mixed with glycine, NH 2CH 2COOH, in a molar ratio of 1: 2: , \nrespectively. Based on a possible reaction described by equation (1), oxygen appears as a reactant \n(from air) or byproduct gas when the glycine ratio is different from  = 4.44. In our synthesis, \nglycine molar ratio of about  = 2.4 was added in the synt hesis of NiCr 2O4 and CoCr 2O4 while it \nwas almost doubled while the synthesis  of MnCr 2O4 in a total mass of reactants of about 6 g for \neach pot synthesis. The mixed raw materials were dissolved in double distilled water by means of \nmagnetic stirrer at room temperature till we obtained homogenous clear solution. The mixture \nsolution was then introduced to a microwave oven operating for ~15 mins at a power of 800 watt. \nAfter complete evaporation of water, combustion occurs resulting in a dry fluffy solid product that \nwas ground and prepared for characterization and measurements without further heat treatments.  \n2.2. Characterization and measurements  \nThe structure at room temperature has been investigated by powder x-ray diffraction ( XRD) \nusing x -ray diffractometer (X’Pert Pro, PANalytical) with Cu K α1 radiation (  = 1.5405 Å) \nmonochromated by a Ge (111) -Johansson -type monochromator. The microstructure  and the  5 particles shape and size have been investigated by a JEOL JEM -2100F electron  microscope. An \nattached unit  of energy dispersive x -ray spectroscopy (EDX -TEM)  has been employed to \ninvestigate the elemental compositions in many particles of each sampl e. FTIR spectroscopy has \nbeen performed on the samples via a NICOLET FTIR 6700 spectrometer for a pellet of each \nsample mixed with KBr powder. The measurements were performed in the wavenumber range \n400-800 cm-1. The absorption spectra of the samples were collected by UV-Vis spectrophotometer \n(Jasco v -750) in the wavelength range 200 -800 nm for particles suspended in Dimeth yl sulfoxide \n(DMSO) solvent with equivalent concentrations. For this purpose, equal amounts of the suspended \nnanoparticles were treated by ultrasonic waves directly before measurement for absorbance in a \nhomogeneous suspension of the particles.  \nMagnetizat ion has been measured by using a superconducting quantum interference device \n(SQUID) MPMS5 (Quantum Design) in the temperature range of 2 – 300 K with magnetic fields \nup to 5 T for each sample free particles charged in a gelatin capsule. Temperature depend encies  of \nzero-field-cooled (ZFC ) and field -cooled (FC) magnetization were measured at 0.01 and 1 T after \nwaiting for thermal equilibrium at each temperature. The temperature dependencies of the \nspontaneous magnetization, coercivity, and the observed magne tic squareness ratio are in -\ncorrelation discussed for these chromite nanocrystals with finite size effects in comparison to \nreported results using samples in bulk forms . \n3. Experimental Results  \n3.1. Structure and particle size  \nThe powder XRD patterns of the synthesized nanoparticles were refined by means of RIETAN -\nFP software [28]. The XRD patterns observed at room temperature and corresponding Rietveld \nrefinement results are shown in Fig. 2(a -c). The refinement reveals the forma tion of a spinel -\nstructured single phase. The structural analysis of the XRD patterns indicated the formation of a \ncubic spinel structure for the three compounds nanoparticles, with peaks indexed in Fig.2(a) . \nAlthough the bulk spinel chromite NiCr 2O4 crystallize in a tetragonal structure at room \ntemperature [29], we found NiCr 2O4 fine NCs crystalize in a cubic structure at  room temperature \nwhich agrees with previously reported results [30]. The refinement has been performed with \nassuming the cubic s pace group 𝐹𝑑3̅𝑚 (setting #2) with sites 8a (1/8, 1/8, 1/8) and 16d (½, ½, ½) \nfor atoms at the tetrahedrally coordinated ( A) and octahedral ly coordinated  (B) sites, respectively, \nwhile oxygen atoms occupy the 32e ( x, y, z) site. Assuming stochiometric compositions, the lattice   6  \nFigure 2 . Room -temperature powder XRD patterns and Rietveld refinement for (a) NiCr 2O4, (b) \nCoCr 2O4 and (c) MnCr 2O4 nanoparticles with labels for corresponding weighted -profile and expected \nrefinement reliability indicators ( Rwp and R e), goodness -of-fit factor S = Rwp/Re and lattice constant, a. \nOpen circles represent the experimental data, while solid lines for calculated patterns, dashed lines for \nthe difference between them, and vertical bars for corresponding Bragg diffraction angle s. Inset of (c) \nshows a selected area electron diffraction (SAED) of MnCr 2O4 NCs. (d) the variation of both \nexperimentally measured and theoretically calculated lattice constant, a, with the ionic radius of A2+ ions \nin the tetrahedral site, R(A2+). (e) Williamson -Hall plots, and their linear fits, as estimated from the XRD \ndiffraction patterns . \nconstant ( a), oxygen position parameter ( u = x = y = z), cations distribution among the A and B \nsites indicated by ions occupancy ( g) and peaks shape parameters (FWHM, position, intensity, \netc.) were refined.  The tendency of occupying different sites by the cations A2+ and Cr3+ is \ndependent on the calculated stabilization energy; simply the highest magnitude of crystal field \nstabilization energy (CFSE) when the cations freely occupy possible sites [31]. We found that the \nrefinement assuming free occupancy of both cations A2+ and Cr3+ in the A and B sites  results in \n10 nm-12500\n2000\n1500\n1000\n500\n0NiCr2O4Rwp=  3.299,  Re=  2.558,   S = 1.2893\na = 8.3145(21) Å  \n \n (a)\n2500\n2000\n1500\n1000\n500\n0\n100 90 80 70 60 50 40 30 20 10\n2q  / deg.MnCr2O4(c)Rwp=  2.960, \nRe=  2.830,\nS = 1.0458\na = 8.4428(55) Å  3500\n3000\n2500\n2000\n1500\n1000\n500\n0IntensityCoCr2O4(b)Rwp=  2.441,  Re=  2.397,   S = 1.0183\na = 8.3345(29) Å  (111)\n(220)\n(311)\n(400)\n(422)\n(522)\n(440)(222)\n(531)\n(620)\n(533)\n(444)\n(731)\n(800)(642)8.45\n8.40\n8.35\n8.30\n8.25lattice constant a (Å)\n0.70 0.65 0.60 0.55 0.50\nR(A2+) @ tetrahedral site (Å) ACr2O4\nNi Mn Co(d)\n Exp eriment\n Calculated\n25\n20\n15\n10\n5\n0cos  (x10-3 )\n0.5 0.4 0.3 0.2 0.1 0.0\nsin ACr2O4\nNiMn\nCo(e) 7 almost g = 1 for both A2+ in the A site, 8a (1/8, 1/8, 1/8),  and Cr3+ in the B site, 16d (½, ½, ½), in \na normal spinel configuration  for the three compounds, that agrees with the well -known high CFSE \nof Cr3+ in the octahedral site [32], [33] . The main fitting parameters and refinement results are \npresente d in Fig. 2(a -d) and Table 1. [34], [35]  Another confirmation of the lack of cations \ninversion in the ACr2O4 (A = Ni, Co, and Mn)  nanoparticle spinel structure can be achieved by \ncomparing the experimental value of lattice constant a with that calculated for a normal spinel \nACr2O4 as functions of the cation radius of Ni2+, Co2+ and Mn2+ at the tetrahedral site. The \ntheoretical latti ce constant 𝑎cal. is calculated using the equation [36]: \n𝑎cal.=8[(𝑟A+𝑟O)+√3(𝑟Cr+𝑟O)]\n3√3, (2) \nwhere  𝑟Ais the radius of a cation in the tetrahedral A site, 𝑟Cr is the radius of the Cr3+ in the \noctahedral B site, and 𝑟O is the radius of the oxygen anion, assumed as reported by Shannon [35]. \nBoth experimentally observed and theoretically calculated lattice constants are presented in Fig. \n2(d) and Table 1. The experimental lattice constant almost matches the theoretical values \ncalculated by assuming the normal spinel structure and both show linear dependence on the ionic \nradius of  A2+ in a tetrahedrally coordinated site, as shown in Fig. 2(d) .  \nThe powder XR D patterns of the different compounds show variations in the peaks ’ \nbroadening ranging from broad peaks for MnCr 2O4 to relatively  sharp peaks for NiCr 2O4, \nreflecting the larger crystallite size of NiCr 2O4 compared to the other two samples. The crystallite \nsize can be estimated from the powder XRD patterns, with correction for the contribution of lattice \nstrain to the peak broadening, by employing the Williamson -Hall (W -H) method of a formula [34]: \n𝛽ℎ𝑘𝑙𝑐𝑜𝑠𝜃=𝑘𝜆\n𝐷ℎ𝑘𝑙+4𝜀𝑠𝑖𝑛𝜃 (3) \nwhere  𝛽hkl is the full width at half maximum of the ( hkl) peak, 𝜃 is its Bragg’s angle, λ is the x -\nray wavelength, k is a constant (taken as 0.9), 𝜀 is the strain in the lattice unit cell and 𝐷ℎ𝑘𝑙 is the \ncorresponding crystallite size. The average crystallite size 𝐷XRD can be calculated from a linear \nfitting of the W -H plot between 𝛽hklcos𝜃 and sin𝜃 for different peaks in the XRD pattern. The \nW-H plot, shown in Fig. 2 (e), elucidates linear fits for the three compounds data according to \nequation (3) and the estimated 𝐷XRD values are presented in Table 1. \nHigh resolution TEM images of the synthesized nanoparticles are presented in  Fig. 3 . The \nsynthesized  particles were confirmed to be within the nanoscale with different particle size \ndistributions. The obtained largest particles of the three compounds are sized be low 30 nm. As \nshown in Figs. 3 , nanoparticles  with particle size distribution ranges of 10 –30, 5 – 27, and 3 – 18 \nnm and average particle size 𝐷TEM of ~1 6, 13 and 10 nm are observed for NiCr 2O4, CoCr 2O4, and  8 MnCr 2O4 NCs, respectively. The average particle size matches the crystallite size estimated from \nXRD analysis indicating successful growth of nanocrystals (NCs). As an example, high \nmagnification images show the fringes of NiCr 2O4 NCs indicating a high degree of crystallinity. \nThe MnCr 2O4 NCs are grown in lowest particle size in a relatively narrow size distribution, Fig. \n3(c), which indicates that using a large amount of the fuel while the microwave -induced -\ncombustion synthesis may reduce the particle size of the synthesized chromite na noparticles.  \n  \nFigure 3. (HR)TEM micrographs for (a) NiCr 2O4 at different magnifications showing the \ninterplanar fringes corresponding indexed planes, and for (b) CoCr 2O4 and (c) MnCr 2O4 \nnanoparticles, all with particle size histograms.   \nTable 1. Experimental structural parameters: lattice  constant ( a), oxygen position parameter ( u) \ncrystallite size (𝑫𝐗𝐑𝐃) and lattice strain ( 𝜺), estimated based on XRD Rietveld refinement calculated \nlattice constant ( 𝒂𝐜𝐚𝐥.) of ACr2O4 (A = Ni, Co, and Mn)  NCs, and TEM -determined particle size (𝑫𝐓𝐄𝐌). \nSample  𝑎exp. (Å) 𝑢  <DXRD> (nm)  𝜀×10−3 𝑎cal. (Å) <DTEM> \n(nm)  \nNiCr 2O4 8.314(2)  0.2582(4)  18.1( 1.3) 2.99(3)  8.291  16(2) \nCoCr 2O4 8.334(3)  0.2624(6)  15.5( 0.28)  0.41(9)  8.338  13(1) \nMnCr 2O4 8.434(65)  0.2627(5)  8.2(0.05)  3.64(6)  8.461  10 (1) \n0 5 10 15 20 25 30 35 40 45 50 55 60\n  Counts (Arb. units)\nParticle size (nm)\n0 5 10 15 20 25 30 35 40 45 50 55 60\n  Counts (Arb. units)\nParticle size (nm)0 5 10 15 20 25 30 35 40 45 50 55 60\n  Counts (Arb. units)\nParticle size (nm)\n(c)\nMnCr2O4\n50 nm(b)\nCoCr2O4\n(a)\nNiCr2O4 9 3.2. Spectroscopic Characteristics: FT -IR and UV -Vis spectroscopies  \n Further identification evidence of the spinel phase can be provided by an investigation of \nthe IR absorbance spectra corresponding to  its cation -anion bonds vibrational modes. Mainly, there \nare two characteristic stretching vibrations in the spinel structure attributed to its cation -anion \nbonds in the  available tetrahedrally and octahedrally coordinated sites. The main two bands occur \nabove 400 cm-1 are the stretching vibration frequency 𝜐1 corresponding the A-O bond at the \ntetrahedral site and the frequency   𝜐2 of the Cr-O bond at the octahedral site [37]. Other spinel \ncharacteristic bands can be detected in the far -IR region below 400 cm-1 [38], however, the \ndetection of υ1 and υ2 can be considered as strong evidence for the formation of the spinel phase.  \nFurthermore, the shape of the developed vibrational bands may indicate some information about \nthe cation distributions between the A and B sites.   \n Figure 4 shows the FT -IR spectra of ACr2O4 (A = Ni, Co, and Mn)  NCs exhibiting the main \ncharacteristic bands around 600 cm-1 and 500 cm-1. The assignments of the observed vibrational \nbands are presented in Table 2. The relative position of the two characteris tic bands slightly shifts \nto lower frequencies with varying the divalent cation from Co2+, Ni2+ to Mn2+, respectively. \nCharacteristic bands of bulk CoCr 2O4 were observed at 630 and 525  cm-1, for the tetrahedral and \noctahedral cation -anion bonds, respectively, those bands slightly shift to higher wavenumbers for \nNiCr 2O4[39]. We found a decrease of 𝜐1 and 𝜐2 from  630 cm-1 to 611 cm-1 and from 525 cm-1 to \n491 cm-1, in the case of  CoCr 2O4. This shift in bands position can be attributed to the smaller \nparticle or grain size as previously reported [40]. Similar band shift for powdered samples of ionic \ncrystals was explained by a depolarization field related to the small crystallites [41].  \nThe vibrational  characteristic bands in Fig 4  do not show any splitting, however, a small \nshoulder is observed around 675 cm-1 and it is more prominent for the NiCr 2O4 and CoCr 2O4 NCs, \nthat agree with previous observations [30], [42] . The formation of mixe d cations in the octahedral \nsite are excluded in chromites , as Cr3+ cation have high CFSE in the octahedral site compared to \nother cations in different spinel compounds . A possible explanation is the scattering from \nrandomly oriented crystallites with large distribution in shape and size in NiCr 2O4 and CoCr 2O4 as \nseen in the TEM results [43]. \nThe force  constant K for the tetrahedral site ( KTet) and octahedral site ( KOct) can be determined \nfrom the relation [44]: \n𝐾=4π2𝑐2𝜐2𝜇. (4) \nWhere c is the speed of light, 𝜐 is the vibration band frequency, and 𝜇 is the reduced mass of the \ncorresponding cation and anion with masses mC and mO, respectively. which can be simply  10 calculated as 𝜇=mOmC\nmO+mC [45]. In this normal spinel structure, the reduced mass at the octahedral \nsite 𝜇Octg is common and equals to 2.032 10-23 g, while 𝜇Tet values at the tetrahedral site are \n2.087 10-23 g, 2.057 10-23, 2.089 10-23 g for A= Ni, Mn, and Co, respectively.  The obtained \nforce constant values are listed in Table 2. We note that the vibrational energies and corresponding \nvalues of the force constant do not show systematic trends with the cationic radii of the divalent \ncations and their effect on the bon d length. However, the difference ( 𝜐1− 𝜐2) gets along with the \nradii of the cations used which is a further indication of the formation of almost complete normal \nspinel.  \n The optical absorption characteristics of the synthesized nanoparticles were inves tigated in \nthe UV -Vis region of a wavelength range 300-800 nm. The absorbance ( A) spectra are shown in \nthe inset of Fig. 4 (b). The absorption coefficient ( 𝛼) was estimated using the Beer -Lambert law \n[46]: \n 𝐴=𝛼 𝑐 𝑙, (5) \nand by considering the theoretical density 𝜌 in gm/cm3 of the material, the absorption coefficient \n(𝛼) in cm-1 can be obtained as:  \n𝛼=𝐴 𝜌\n𝑐 𝑙.  (6) \nwhere c is the concentration of the sample in the solvent and l is the width of the suspension. The \noptical energy gap can be determined from the absorption coefficient 𝛼 and the incident photon \nenergy ℎ𝜐 using the Tauc’s equation [47]: \n   𝛼ℎ𝜐=𝐵(ℎ𝜐−𝐸𝑔)𝑛.  (7) \nWhere B is a probability constant, and n is an exponent that determines the transition type. The \noptical performance of the samples demonstrates a typical allowed direct transition behavior with \nn=1/2.  \n Tauc’s plots, (𝛼ℎ𝜐)2 vs ℎ𝜐, are shown in the mai n panel of Fig. 4(b)  for the three \ncompounds NCs. The values of 𝐸g were estimated by linear extrapolation of data at high photon \nenergy region, where the optical absorption is stable, to (𝛼ℎ𝜐)2 = 0. The estimated values of 𝐸g, \npresented in Table 2, are consistent with results of optical conductivity investigations for bulk  \nchromites  with a direct bandgap of CoCr 2O4 nanoparticles reported  as ~3.1 eV [5], [9], [48] . It is \nreported that a spinel chromite with a large optical energy gap undergoes first -order phase \ntransition into spiral spin ordering at a low temperature [5], which is in consistence with the spiral \nspin transition observed in CoCr 2O4 rather than in MnCr 2O4 fine NCs as described below in section \n3.3.  11  \nFigure 4 . (a) FTIR spectra of NiCr 2O4, CoCr 2O4 and  MnCr 2O4 NCs presented in the wavenumber range \nbetween 700 nm and 400 cm-1. (b) Tauc’s plots established from UV -Vis absorbance data of NiCr 2O4, \nCoCr 2O4 and  MnCr 2O4 NCs, assuming allowed direct electronic transitions. Dashed lines represent \nextrapolations to corre sponding optical energy gap values for each compound NCs. Inset shows the \ndimensionless absorbance spectra with small vertical offsets for clarity.  \nAdditional absorption  bands are observed at low photon energies that vary in position and \nintensity for the different three chromites, as seen in the inset of Fig.4(b).  CoCr 2O4 and NiCr 2O4 \nexhibit absorption bands in the wavelength region 600 – 700 nm while MnCr 2O4 nanoparticl es do \nnot develop anomaly. The absorption band of the CoCr 2O4 is an asymmetric  band with maxima \naround ~665 nm, more examination of this band draws the attention to a damped maxima around \n~620 nm which overlapped and gave the asymmetric shape of the band. These are characteristic \nbands for the spin allowed transitions of Co2+ in the spinel structure [42], [49]  and can be assigned \n 12 using Tanabe -Sugano diagrams of spinel crystal field [50]. The bands forming the big hump in the \nCoCr 2O4 spectrum give evidenc e of the d7 configuration of Co2+, and they can be assigned to 4A2(F) \n→ 4T1(P) transitions of the Co2+ ions in tetrahedral coordination. The 4A2g → 4T2g and 4A2g → 2T1g \ntransitions of Cr3+ in octahedral coordination occur at approximate positions which give more \nprobability of overlapping between the transitions in the spectrum, that may justify the broadness \nof the bands in the spectrum. In the case of NiCr 2O4 weaker band is observed in th e absorbance \nspectrum also broadened with maxima around ~685 nm, which can be attributed mainly to the 3d -\n3d transition between Ni2+ tetrahedrally coordinated ions.  The broadening of the band developed \nfor NiCr 2O4 spectrum may be attributed to the splittin g in the 3d level of Ni2+ cation as highlighted \nin Fig. 1(c ), when compared to bulk the room temperature phase is tetragonal but for our samples \nthe main phase is cubic which affects the symmetry and hence the optical transitions between \nbands in the UV-Vis spectral range. The splitting in the 3d energy levels in case of NiCr 2O4 may \njustify the intensity of the optical band when compared to CoCr 2O4 nanoparticles.  In the case of \nMnCr 2O4 sample there were no noticeable bands. Such behavior was predicted  as the 3d -3d \ntransitions are spin forbidden for the tetrahedrally coordinated Mn2+[9]. \nTable 2. Spectroscopic characteristics of  ACr2O4 (A = Ni, Co, and Mn) NCs . \nSample  FTIR Assignments  Force constant  Optical Energy \ngap \n𝜐1 \n(cm–1) 𝜐2 \n(cm–1) 𝐾Oct \n(×105dyne/cm)  𝐾Tet \n(×105dyne/cm)  𝜐1−𝜐2 \n(cm–1) 𝐸g (eV) \nCoCr 2O4 611 487 1.729  2.753  120 3.055(13)  \nMnCr 2O4 604 491 1.701  2.649  117 2.903 (4)  \nNiCr 2O4 602 481 1.659  2.670  121 2.872 (6)  \n3.3. Magnetic properties  \n To investigate the magnetic properties of ACr2O4 NCs, we have measured their \nmagnetization as a function of temperature at low and high magnetic fields and its hysteresis loops \nat different temperatures after ZFC and FC processes. Figure 5(a-c) show the temperature \ndependence of the magnetic susceptibility , M(T)/H, measured at H = 100 Oe after ZFC and FC \nprocesses. The magnetic transitions are determined  from the temperature dependence of the FC \nmagnetization derivative, d M(T)/d T, correspondingly shown in the insets of Figure 5 (a-c).  While \nthe temperature decreases, the onset of FIM ordering is observed at a Curie temperature TC = 69, \n92 and 40 K , as observed at low fields of 100 Oe , for NiCr 2O4, CoCr 2O4, and MnCr 2O4 NCs, \nrespectively . The observed TC values agree  with those reported using bulk polycrystalline \nsamples [6]. A bifurcation of the ZFC curve from the FC one is commonly observed below a  13 freezing temperature Tf of ~68, 88 and 20 K observed at 100 Oe , that decreases to 60, 75 and 7 K \nat 10 kOe , for NiCr 2O4, CoCr 2O4, and MnCr 2O4 NCs, respectively. It is noticed tha t the spin \nfreezing is almost released by applying a field of 1 T for the Mn chromite NCs, unlike the other \ntwo chromite NCs, as shown in Figs. 5(d )–(f). This may indicate a spin -glass -like (SGL) state \nrather than a surface disorder in Ni and Co chromite nanoparticles [14], [51], [52] . Based on \nmagnetization and ac susceptibility analysis results, we have revealed the SGL in CoCr 2O4 \nNCs[23]. The obser ved SGL state may result in large coercivity in these ch romite NCs.  With \nfurther decrease of temperature, a subsequent anomaly which can be attributed to a bulk  \nnoncollinear magnetic transition is observed at a temperature Ts ~15 K for NiCr 2O4, 24 K for \nCoCr 2O4 and is barely observed around 10 K for MnCr 2O4 NCs.  \nThe transition of NiCr 2O4 at Ts is attributed to a bulk canted antiferromagnetic ordering by an \nemergent transverse AFM component [16]. We observed that Ts around 15 K for NiCr 2O4 NCS of \naverage size ~1 5 nm is significantly lower than that reported for particles of size larger  than ~30 \nnm as well as bulk samples with Ts of about 30 K [6], [14], [53] . Additional slight anomalies \nobserved at temperatures up to 𝑇s∗ ~30 K, seen in M(T) measured at 100 Oe as in the inset of  Fig. \n5(a), can be ascribed to a canted -AFM transition in particles with larger size , as elucidated by a \nrelatively wide particle -size distribution (8 - 30 nm) shown in Fig. 3(a) . On the other hand, the \nanomaly at Ts ~24 K for CoCr 2O4 NCs is attributed to a transition from the FIM state to an \nincommensurate FIM spiral order  [14], [54] . With further decrease of the temperature, another \nanomaly is observed for CoCr 2O4 NCs at Tl ∼ 5 K which is corresponding to a ‘lock -in’ transition \nto a commensurate FIM spiral order at which the spiral order becomes fully developed [14], [54] . \nAlthough the incommensurate  spiral transition temperature  Ts of CoCr 2O4 NCs is almost identical \nto that of its bulk forms, the ‘lock -in’ temperature Tl becomes much lower than corresponding bulk \nvalues, probably, by structural and finite -size effects. Similar results  were observed for CoCr 2O4 \nNCs with a critical size limit of ~5 nm for the spin -spiral order and disappearance of the ‘lock -in’ \ntransition fo r sizes up to ~14 nm [55]. For the other spiral FIM chromite MnCr 2O4 NCs,  with \naverage crystallite size of ~ 10 nm, the anomaly at Ts is barely seen around 10 K only at low fields \nof 100 Oe, while the ‘lock -in’ transition completely disappears. These results indicate that our FIM \nspine l chromite NCs are very close to the crystallite -size limit corresponding  a suppressed  \nnoncollinear FIM order . Figs.  5(d)–(f) show the temperature dependence of magnetization data \nmeasured at a higher ma gnetic field of 10 kOe . We observe that Ts is completely suppressed in the  \nMn chromite  NCs while it is  robust against high magnetic fields  in Co and Ni chromites as the \ncase of bulk samples [15], [56], [57] .   14  \nFigure 5. Temperature dependence of the magnetic susceptibility ( M /H) measured at 0.1 and 10 \nkOe after ZFC (open) and FC (solid) processes for (a), (d) NiCr 2O4; (b), (e) CoCr 2O4 and  (c), (f) \nMnCr 2O4 NCs, respectively. Corresponding insets of (a ) – (c) show the magnetic transition \ntemperatures appear in the derivative of the FC magnetization, dM/dT(T), at different fields , while \ninsets in (d ) – (f) are the T-dependence of the inverse susceptibility (H /M) measured at 10 kOe  \nafter a FC process . Solid lines in (d ) – (f) are the difference between the FC and ZFC \nmagnetizations measured at 10 kOe, while in (d) – (f) insets are the best fits to Eq. (8) , in the text.   \nThe insets of Figs.  5(d)–(f) show the inverse susceptibility, 𝜒−1= 𝐻𝑀⁄, as a function of \ntemperature for the three compounds NCs, respectively. In the Néel molecular -field theory, the \nmagnetization of FIM materials of nonequivalent sublattices behaves above TC with a hyperbolic \ncharacteristic of the inverse susceptibility, 𝜒−1= 𝐻𝑀⁄, as [58], [59] : \n𝜒−1=𝑇−𝜃𝑊\n𝐶−  \n𝑇−𝜃′ . (8) \nThe first term in Eq. (8)  is the hyperbolic high -temperature linear part described in a Curie -Weiss \n(CW) form, with W is the Weiss temperature and C is the Curie constant The second term is the  \n 15 Table (3):  Particle size and magnetic parameters and transitions of ACr2O4 (A= Ni, Co, Mn) \nnanocrystals compared to previously reported data of bulk poly - and single -crystalline samples . \nChromite  NiCr 2O4 CoCr 2O4 MnCr 2O4 \nForm  NCs \n[this \nwork]  Bulk a \n[refs.]  NCs \n[this \nwork]  Bulk a \n[refs.]  NCs \n[this \nwork] Bulk a \n[refs.]  \nDTEM (nm)  16 (±2) >100  13 (±1) >100  10 (±1) >100  \nMagnetic \nground \nstate [5] Canted  Conical  Conical  \nTC (K)  68(2) b \n 68 – 75 \n[60],[16],[6] 92(2) b \n 92– 97 [14],[6] 40(2) b \n 40 – 51 [7],[6], \n[61], [14] \nTs (K)b  15 31 [6], [16]  24 24 [14], 27 [6] 10 18 [6], 19.5  [14] \nTl (K)b   5 13, 15  [6]  14 [14], 15 [6], \n17.5 [7] \nTf(K)b 68 68 [60] 88 88 [60] 20 20 \nCore \nfraction \n(%)d 19 (±2) 100 36 (±2)  100 26 (±2)  100 \nDead layer \nt (nm)  3.37 \n(±0.44)  1.88 \n(±0.23)  1.81 \n(±0.26)  \nMs (µB/f.u.)c 0.05 – \n0.06 0.2 – 0.3[6], \n0.3e[16] 0.05 – \n0.1 0.15 – 0.25 [6], \n[15], 0.33e[14] 0.26 –\n0.31 1 – 1.2 [5], [6] , \n1.2e[14], [62]  \neff (µB/f.u.)  \nExperiment   \n5.88(6)   \n10.08 [60]  \n6.10(2)   \n9.52 [60]  \n7.43(7)   \n8.06 [61], 8.25 [7] \nTheory  6.164  6.708  8.062  \nW ( ) - 397(7) - 536 [60] - 375(3)  -600[15], -638[60] - 310 \n(6) - 400 [61], - \n411[7] \n|𝐖|/TC 5.84(22)  7.86 [60] 4.08(5)  6[15], 6.78 [60] 7.76(29)  8.7 [61], 10[7] \na. previously reported results for bulk poly - and single -crystalline  samples.  \nb. as indicated by the derivative dM(T)/dT and ZFC M(T) data measured at H = 100 Oe.  \nc. reported approximate values observed at low temperatures (2 - 5 K) using bulk samples.  \nd. results of fitting data of M(H) to Eq. 10  \ne. results based on neutron diffraction  measurements  [16]. \nhyperbolic low -T asymptote with   and θ′ are constants. The observed behavior of the magnetic \nsusceptibility of FIM ACr2O4 NCs above  TC fits well to Eq. (8) as shown by solid lines in insets of \nFigs. 5(d -f) with C =  4.33(4), 4.65(1) and 6.90(7) emu K /mol , W = -397(7), -375(2) and -310(6) \nK,  = 1000(41), 604(8) and 823(27) mol K/emu, and θ′ = 54.5(6), 87.3(1) and 25.6(7) K. As the \nmagnetic Cr3+ cations are arranged in a pyrochlore sublattice with an inherent tendency of  16 antiferromagnetic Cr -Cr interactions, ACr2O4 are geometrically spin -frustrated even if the A site \nis occupied by magnetic ions [14]. The frustration degree is indicated by a ratio |W|/TC > 1. As \nshown in Table 3 , a frustration factor , f =  |W| /TC  5.85, 4.1 and 7.76 are observed for Ni, Co, \nand  Mn spinel chromite NCs, respectively . The effective paramagnetic moment eff is \nexperimentally estimated as, 𝜇eff=√3 𝑘B𝐶/𝑁A, where kB and NA are the Boltzmann’s constant \nand Avogadro’s number, respectively. The expected values of eff can be theoretically calculated \nas, \n𝜇eff=√(𝜇eff)𝐴2+2(𝜇eff)𝐵2= 𝑔√𝑆𝐴(𝑆𝐴+1)+2𝑆𝐵(𝑆𝐵+1), (9) \nwhere g is the Landé g -factor , assumed as 2, and SA and SB are the spin quantum number of ions \nlocated  in the A and B sites, assumed as 1, 3/2, 5/2 and 3/2 for Ni2+, Co2+, Mn2+ and Cr3+, \nrespectively . The experimentally determined magnetic parameters of ACr2O4 (A = Ni, Co, and  \nMn) NCs are presented  with their average particle size , and in a comparison to correponding results \nreported  using bulk poly - and single crystalline samples , in Table 3.   \n The isothermal magnetic  hysteresis loops of ACr2O4 NCs were measured at different \ntemperatures down to 2 K  after a ZFC process and  are shown for selected temperatures in Figs. \n6(a) − (c) for A = Ni, Co, and Mn, respectively. In agreement with bulk results  [15], [52], [57] , the \nmagnetization of the three compounds NCs does not saturate with magnetic fields up to 50 and 70 \nkOe, with the magnetization of Ni and Co chromites are much lower than th at of Mn chromite as  \nthe case of bulk counterparts. However, the magnetization values are significantly lower than that \nof bulk sample  as shown in Figs. 6(d) – (f). This decrease in M(H) is attributed to an expect ed \nsignificant paramagnetic -like contribution from  the surface  uncompensated spin s in a core -shell \nFIM particles .  \n To approximately estimate  the contribution fraction of the bulk core magnetization \nMBulk(H) to the total net magnetization  of the core -shell nano particle  MNano(H), we assume that the \nfield dependence of MNano(H) is given by,  \n𝑀Nano(𝐻)=𝑓 𝑀Bulk(𝐻)+(1−𝑓)𝜒 𝐻, (10) \nwhere f measures the approximate fraction of the core magnetization and  is a susceptibility of \nthe surface magnetically disordered  (dead) layer . Fitting experimental data M(H) to Eq. (10)  at \ndifferent T, where the magnetic state is most stable, i.e. Ts < T < TC, employing the bulk data \nreported by Mufti et al, Ali and Singh, and Bhowmik et al [6], [62], [63] , indicates  core contribution \nfraction with values of  about 0.20, 0.35 and 0.25 for Ni, Co and Mn chromite nanoparticles, \nrespectively.  As shown in Figs. 6(d) – (f), the estimated f is almost constant with the temperature \nchange .  17  \nFigure 6.  The ZFC isothermal magnetic hysteresis loops , (a) – (c), and initial magnetization curves  \nM(H), (d) – (f), for NiCr 2O4, CoCr 2O4, MnCr 2O4 NCs, respectively,  shown at selected temperatures. \nThe inset  figures  of (a) – (c) are magnifications for corresponding main panels at low fields  with \nsolid lines for guidance . Bulk  M(H) results [6], [62], [63]  are shown for comparison in (d) – (f) with \ndashed lines for guidance while solid lines in (d) – (f) are for best fits to Eq. (10)  in the text.  \n One further finite -size effective property  is the magnetically dead layer at the particles \nsurface  discussed in a propose d core-shell model , with a particle -size independent thickness that \ncan be subsequently approximated for these chromite NCs [64]–[66]. As the magnetization is \nproportional to the corresponding volume , we can simply assume that  𝑓=𝑀Bulk(0)\n𝑀Nano(0)=(𝐷core\n𝐷)3, \nwhere Dcore and D are sizes of the core and a total particle , respectively .  Hence, the dead layer \nthickness , t = (D – Dcore)/2 can generally be estimated as:  \n 18 𝑡=𝐷\n2(1−𝑓1/3) . (11) \nWithin the accuracy limits of the determination of particle/crystallite size D as well as volume \nfraction f, average  values  of t are approximately estimated as 3.4(4), 1.9(2) and 1.8 (2) nm for the \nNi, Co, and Mn chromite nanoparticles of average particle size 15, 13 and 10 nm, respectively. \nValues of t (~ 1 – 3 nm) close to values approximated here in reasonable estimates have been \nreported for manganite and ferrite  nanoparticles [66]–[68]. Referring to the  lattice constant  values , \nTable  1, a shell layer of ~4a in thickness for the Ni  and ~2a for the Co and Mn chromite NCs is \nmagnetically dead at the surface of these nanoparticles . For example, by a ssuming the spherical \nshape of these nanoparticles , roughly about 2500  among 3000  unit cells per a particle in the \nNiCr 2O4, 1300 among 2000 in the CoCr 2O4, and 750 among 1000 in the MnCr 2O4 NCs have no net \nmagnetic moment, which reflect s the highly reduced spontaneous magnetization in t hese \nnanoparticles.  \n Characteristics  of the magnetic hysteresis loop  such as the squareness shape and ratio and \nthe coercive field, and their evolution with temperature in an interrelation to the freezing features \ncan provide information about the driving  interactions  in these spinel FIM nanoparticles. T he \nstrength of a spin-glass -like state observed below a freezing temperature, Tf, can be indicated by \nthe difference MFC-MZFC at low fields. The squareness ratio is estimated as SQR =  Mr / Ms, where  \nMr is the observed remanent magnetization and Ms is the high-field ‘saturation’ magnetization . Ms \nis approached here by extrapolating linear fits of the high-field magnetization M(H) data to 1/H2 = \n0 in M(H) vs 1/ H2 plots , as shown in the inset of Fig. 6(d)  for the data at 2 K .  Figure 7 show s the \ntemperatures dependence of the MFC-MZFC (at 100 Oe) , SQR  and Hc(T) of ACr2O4 (A = Ni, Co, and \nMn) NCs  accompanied with reproduced results previously reported using bulk single and \npolycrystalline samples  [6], [14], [37], [52], [56], [57], [60], [62], [69] . Below Tf, the difference MFC-\nMZFC qualitatively reproduces those of bulk polycrystalline samples showing anomalies \ncorresponding to each compound magnetic transitions, as well as for MFC-MZFC at 10 kOe shown \nin Figs. 5(d ) and (f). Small variation of the MFC-MZFC of Mn NCs from that of the bulk behavior \nexhibiting anomaly features around the bulk Ts, Figs. 7(g), can be attributed to the suppression of \nits spiral spin order by finite -size effects .  19  \nFigure 7. The temperature dependences of the FC magnetization splitting from ZFC magnetization, MFC-\nMZFC, measured at 100 Oe, the estimated squareness ratio (SQR =  Mr / Ms), and observed coercivity, Hc, \nfor (a - c) NiCr 2O4, (d – f) CoCr2O4 and (g– i) MnCr 2O4 nanocrystals (closed symbols), presented in \ncomparison to those of bulk samples  (open symbols) , reproduced from Refs.  [6], [52], [57], [60], [62], \n[69]  for polycrystalline  samples and from Ref.  [14] for single crystals . Solid lines are for a purpose of \neye guidance.  \n Unlike the bulk results,  unusual nonmonotonic behavior s are observed in  the SQR( T) and \nHc(T) of Ni and Co chromite NCs  below Tf. Broad peak with large coercivity values  is around the \nonset of the transitions to noncollinear , canted AFM and spiral FIM,  in Ni and Co chromite NCs, \nrespectively . The SQR increases up to 0.15 before it decreases with decreasing the temperature  \nbelow  Ts for Ni and Co chromite NCs, while it continuo usly increases to ~ 0.3 at the lowest \ntemperature for the Mn chromite NCs.  Similar  behavior is observed for Hc(T) with maximum \nvalues of about 10 and 6 kOe around Ts for Ni and Co chromite NCs,  respectively, and continuous \nincrease up to ~ 2 kOe at 2 K,  for the Mn chromite NCs .  \n4. Discussion  \nIn this section we discuss the interrelations and effects of the geometric frustration, the \nelectronic structure, and the finite size on the magnetic properties of the non -collinear FIM Ni, Co, \nand Mn chromite nanocrystals in a view to results reported for  bulk samples. The spinel chromites \n 20 crystalize in a normal spinel structure with high stabilization energy when Cr3+ fully occupy the \noctahedral B sites. As indicated by  of results XRD analysis in section 3.1 , the normal spinel \nstructure is not affected by decreasing the particle size of these materials down to 10 nm, in \nagreement with previously reported results [53], [62], [70] . In this structure, a pyrochlore sublattice \npurely formed by Cr3+ is associated with a geometrical magnetic frustration usually yielding to a \nnontrivial magnetic ground state in a FIM spinel chromites. We found that the synthesized NCs \nexhibit a wide optical energy gap for CoCr 2O4 NCs (3.1 eV) and gradually decreases for MnCr 2O4 \n(2.9 eV) and NiCr 2O4 NCs (2.87 eV). In a previously reported study of the magnetic properties of \nspinel chromites in relation to their electronic structures, it was found that  the spinel  chromite with \na relatively large optical energy gap can under go a first -order phase transition into a spiral spin \nground state [5]. In this sense, the spiral spin -order transition is significantly observed for CoCr 2O4 \nNCs, with a relatively wide r energy gap, rather than for the MnCr 2O4 NCs. On the other hand,  \nNiCr 2O4 NCs exhibit their bulk canted AFM ground state as described above in section 3.3 .  \nHowever, the bulk magnetic ground state of a spinel chromite was basically attributed to the degree \nof geometrical spin frustration in the Cr3+ pyrochlore sublattice that is determined by the A2+ ion \nand its magnetic moment [13]. \n In a theory proposed by Lyons , Kaplan, Dwight, and Menyuk (LKDM) [13], the magnetic \nground state of spinels with spiral or conical structures can be well described b y a parameter u is \ngiven by 𝑢=4 𝐽BB𝑆B\n3 𝐽AB𝑆A. In this theory, JBB and JAB are the only nearest -neighbor antiferromagnetic \ninteractions among spins SB and SA at the A and B sites , assuming a weak A -A magnetic  \ninteraction. As the spin order is modified by the inherent geometrical magnetic frustration o f the \nspinel B site, the parameter u takes possible values ranges from 0 to infinity corresponding to spin \nconfigurations between the Neél type FIM state and a state of highly frustrated spins , \nrespectively [6]. In a spinel with u < 8/9, the ground state is a long -range -ordered (LR O) Neél -type \nFIM state. With larger u, a LR O spiral state evolves and further it becomes locally unstable \nyielding to a SRO  spiral state in the region where u >1.298. The values of u corresponding ground \nstates of CoCr 2O4 and  MnCr 2O4 are 2 and 1.5, respectively, while that of the canted NiCr 2O4 is \nexpected to be relatively larger. Bulk CoCr 2O4 and  MnCr 2O4 exhibit a conical mag netic structure \nthat is stabilized by a SRO  incommensurate spiral component develops below Ts and becomes \ncommensurate to the lattice at a ‘lock -in’ transition temperature Tl. Based on neutron scattering \nmeasurements using single -crystal samples of CoCr 2O4 and  MnCr 2O4 [14], [54] ,  the spiral \ncomponent emerges below Ts = 25 and 18 K, with coherence length reaches 3.5 and 10 nm, in \npropagation vectors Q = (0.62, 0.62, 0) and (0.59, 0.59, 0), respectively. On the other hand, in \nNiCr 2O4, a transverse antiferromagnetic component appears below Ts with a propagation vector Q \n= (0, 0, 1) and coplanar (but non -collinear) spin configuration has been revealed by neutron  21 scattering [16]. The variation of NiCr 2O4 from CoCr 2O4 and  MnCr 2O4 in the magnetic ground state \nmay be attributed to the large tetragonal distortion occurs by a Jahn−Teller transition, observed at \n310 K in bulk samples [16]. Among the divalent cations A2+ in the three compounds studied here, \nonly Ni2+, with the spin co nfiguration shown in Fig. 1(a) , exhibits an orbital contribution to its \nmagnetic moment with a net ionic moment of 3  µB. Considering the magnitude of magnetic \nmoments of Ni2+, Co2+ and Cr3+ at its theoretical value of 3 µB and that of Mn2+ at 5 µB, spin \nstructure models represented by Tomiyasu et al based on neutron scattering results [14], [16] , \nalmost reproduce the experimental bulk magnetic moments observed  for Ni, Co and Mn spinel  \nchromites.  \n The nanoparticles form of these FIM spinel chromites interestingly enables to study their \nmagnetocrystalline anisotropy in a single domain and its evolution with temperature rather than in \nbulk-form samples. Very small magnetic particles must be single -domain and below a certain \ncritical size they do not benefit  energetically from the wall formation.  Assuming almost spherical \nparticles , we can approximately estimate the single -domain maximum  size as 2Rsd = 18√𝐴 𝐾𝑢\n𝜇0𝑀𝑠2 [71], \nwhere 𝐴 =3 𝐾𝐵𝑇𝐶\n𝑧 𝑎 and 𝐾𝑢 = 𝐻c 𝑀𝑠\n0.96, are the exchange stiffness  and the anisotropy constant, \nrespectively, assuming  the Stoner –Wohlfarth model for randomly oriented identical particles [72], \n[73]. Ms is the saturation magnetization,  KB is the Boltzmann constant, and z is the effective \nnumber of the nearest neighbors , with all  quantities are in SI unit s. Applying these criteria  to \nMnCr 2O4 NCs as the most stable FIM spinel chromite in this study, indicates a critical size of its \nsingle domain in the order of 0.1 μm. This approximate estimation indicates  that our spinel \nchromite  nano particles are single -domain  magnetic particles that free  from domain walls. \n The anomalous magnetic behavior s observed in the temperature dependence of the \ncoercive field, Hc(T), and the hysteresis -loop squareness , Fig. 7 , are indicative for the \nmagnetocrystalline anisotropy  of thes e FIM spinel chromite NCs . As the nanoparticles \nqualitatively reproduce the freezing behavior of the bulk samples, we can exclude the effect of the \nsurface spin disorder on the magnetic anisotropy. For NiCr 2O4 and Co Cr2O4 single -domain  NCs, \nwith decreasing  the temperature below TC a net moment of a collinear FIM order increases in a \npreferred direction simply resulting in an increased anisotropy and hence the coercive field \nrequired for a coherent magnetization reversal, Figs. 7(b) and (e) . At and below Ts the emergence \nof a noncollinear magnetic order, canted AFM and spiral FIM in NiCr 2O4 and Co Cr2O4, \nrespectively, reduces the magnetic anisotropy,  and hence the coercive field, that gradually \ndecreases with a possible increase in the cantin g angle with decreasing the temperature. The \nimplied softness of the magnetic structure below  Ts influences the saturation and remanent \nmagnetizations resulting in similar temperature dependencies  of the hysteresis -loop squareness   22 shape, Figs. 7(b) and (e) . In addition, the unusual behavior of the squareness ratio and the coercive \nfield at low temperatures could be related to some exchange bias of the surface moments with the \ncore moments. On the other hand, the MnCr 2O4 fine NCs with most stable spin structure exhibits  \na monotonic magnetic behavior almost similar to its corresponding bulk results with addition \neffects of the surface disorder that suppresses the spiral transition, Figs. 7(g -i). \n Finally , the nanocrystalline FIM s pinel chromite of current study exhibits crystallite size \neffects on the complex non -collinear spin order.  The approximate fraction of the core \nmagnetization  with estimated values  of about 20, 36 and 25 % , for Ni, Co, and Mn chromites, \nrespectively,  can ex plain the observed suppression of the spiral order in these chromite \nnanoparticles, with different extends in the different chromites.  For example, a clear crystallite \nsize dependent transition ( Ts) to a canted AFM state is observed in NiCr 2O4 and the coexistence of \nparticles with different sizes may result  in additional anisotropic exchanges. For CoCr 2O4 NCs, \nalmost bulk transition to the incommensurate spiral spin order was observed, however, its lock -in \nmagnetic transition ( Tl) exhibits a crystallite  size effect. On the other hand, the magnetic transitions  \nto incommensurate and commensurate spiral order s are suppressed in MnCr 2O4 NCs with \nrelatively smaller particle sizes . Instead, a coexisting spin -glass -like state is observed with a FIM \nstate below TC. The presented results spotlight on the interrelation between the crystal and \nelectronic structures in finite sizes and the unusual magnetic behavior of FIM spinel chromites. A \nstudy of the particle -size dependence of these magnetic features and related  phenomena such as \nthe exchange bias effect in these materials are stimulated by the currently presented results.  \n5. Conclusion  \nSingle -phase FIM ACr2O4 (A= Ni, Co, and Mn) NCs were synthesized by a one-pot microwave -\ncombustion  method . Room -temperature cubic Fd3̅m normal spinel -structured phase was indicated  \nfor the three compounds NCs by the XRD and FT-IR results . Compared to results using bulk \nsamples, the UV -Vis absorption demonstrated additional  3d-3d transitions in the Co and Ni \nchromite NCs spectra, unlike the Mn chromite NCs, with  an estimated  wide  optical band gap of \n~3 eV. Finite size effects on the low -temperature noncollinear FIM transitions  and intriguing \nsingle -domain anisotropic features  have been revealed in this study . NiCr 2O4 NCs of average \nparticle size 15 nm exhibit a canted AFM transition around Ts = 15 K which is much lower than \nthat in single crystals , while  CoCr 2O4 NCs of 13 nm size show almost the bulk transition to an \nincommensurate spiral spin order  at Ts = 24 K, however, its lock -in magnetic transition appears at \nTl = 5 K is shifted below the bulk transition (15 K) . On the other han d, the incommensurate spiral \nFIM transition  appears at 10 K , shifted below the bulk transition ( 20 K), while  its lock-in is \ncompletely  suppressed in MnCr 2O4 NCs of average particles size 10 nm . The observed anomalous \nbehaviors of the coercive field Hc(T) and hysteresis -loop squareness ratio SQR( T), with broad \npeaks around Ts of NiCr 2O4 and CoCr 2O4 NCs, indicate a single -domain magnetic behavior and \nelucidate the magnetotcrystalline anisotropy  of these materials .  23 Acknowledgment  \nThe authors would like to thank N. Sasaki and K. Kazumi from the Department of Materials \nScience and Engineering, Kyoto University, for their help with the TEM analyses. M.A.K \nsincerely thanks  the Japan Society for the Promotion of Science (JSPS ) for Postdoctoral \nFellowship for Research in Japan  (Kyoto University) . 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Lond A , vol. 240, pp. 599 –642, 1948.  \n  "    },    {        "title": "1207.6771v1.Ferrimagnetism_of_the_Heisenberg_Models_on_the_Quasi_One_Dimensional_Kagome_Strip_Lattices.pdf",        "content": "arXiv:1207.6771v1  [cond-mat.str-el]  29 Jul 2012Ferrimagnetism of the Heisenberg Models on the Quasi-One-D imensional Kagome\nStrip Lattices\nTokuro Shimokawa1and Hiroki Nakano1\n1Graduate School of Material Science, University of Hyogo, K amigori, Hyogo 678-1297, Japan\n(Dated: May 27, 2018)\nWe study the ground-state properties of the S= 1/2 Heisenberg models on the quasi-one-\ndimensional kagome strip lattices by the exact diagonaliza tion and density matrix renormalization\ngroup methods. The models with two different strip widths sha re the same lattice structure in their\ninner part with the spatially anisotropic two-dimensional kagome lattice. When there is no mag-\nnetic frustration, the well-known Lieb-Mattis ferrimagne tic state is realized in both models. When\nthe strength of magnetic frustration is increased, on the ot her hand, the Lieb-Mattis-type ferrimag-\nnetism is collapsed. We find that there exists a non-Lieb-Mat tis ferrimagnetic state between the\nLieb-Mattis ferrimagnetic state and the nonmagnetic groun d state. The local magnetization clearly\nshows an incommensurate modulation with long-distance per iodicity in the non-Lieb-Mattis ferri-\nmagnetic state. The intermediate non-Lieb-Mattis ferrima gnetic state occurs irrespective of strip\nwidth, which suggests that the intermediate phase of the two -dimensional kagome lattice is also the\nnon-Lieb-Mattis-type ferrimagnetism.\nPACS numbers: 75.10.Jm, 75.30.Kz, 75.40.Mg\nI. INTRODUCTION\nFerrimagnetism is a fundamental phenomenon in the\nfield of magnetism. The simplest case of the conven-\ntional ferrimagnetism is the ground state of the mixed\nspin chain. For example, there is an ( s,S)=(1/2, 1)\nmixed spin chain with a nearest-neighbor antiferromag-\nnetic (AF) isotropic interaction. In this system, the\nso-called Lieb-Mattis (LM)-type ferrimagnetism1–6is re-\nalized in the ground state because two different spins\nare arranged alternately in a line owing to the AF in-\nteraction. Since this type of ferrimagnetism has been\nstudied extensively, the magnetic properties and the oc-\ncurrence mechanism of the LM ferrimagnetism are well\nknown. In particular, the ferrimagnetism in the quan-\ntum Heisenberg spin model on the bipartite lattice with-\nout frustration is well understood within the Marshall-\nLieb-Mattis (MLM) theorem1,2. In the LM ferrimagnetic\nground state, the spontaneous magnetization occurs and\nthe magnitude is fixed to a simple fraction of the satu-\nrated magnetization.\nIn recent years, a new type of ferrimagnetism, which is\nclearlydifferent fromLM ferrimagnetism, has been found\nin the ground state of several one-dimensional frustrated\nHeisenberg spin systems7–13. The spontaneous magneti-\nzation in this new type of ferrimagnetism changes grad-\nually with respect to the strength of frustration. The\nincommensurate modulation with long-distance period-\nicity in local magnetizations is also a characteristic be-\nhavior of the new type of ferrimagnetism. Hereafter, we\ncall the new type of ferrimagnetism the non-Lieb-Mattis\n(NLM) type. The mechanism of the occurrence of the\nNLM ferrimagnetism has not yet been clarified.\nOn the other hand, some candidates of the NLM ferri-\nmagnetism among the two-dimensional (2D) systems, for\nexample, the mixed-spin J1-J2Heisenberg model on the\nsquarelattice14andtheS= 1/2HeisenbergmodelontheUnion Jack lattice15, were reported. These 2D frustrated\nsystems have the intermediate ground state, namely, the\n“canted state”, in which the spontaneous magnetization\nis changed when the inner interaction of the system is\nvaried. It has not been, however, determined whether\nthe incommensurate modulation with long-distance pe-\nriodicity exists in the local magnetization of the canted\nstate owing to the difficulty of treating these 2D frus-\ntrated systems numerically and theoretically. Therefore,\nthe relationships between the canted states of these 2D\nfrustrated systems and the NLM ferrimagnetic state are\nstill unclear.\nUnder these circumstances, recently, another candi-\ndate of the NLM ferrimagnetism among the 2D systems\nwas reported in ref. VI in which the S= 1/2 Heisen-\nberg model on the spatially anisotropic kagome lattice\ndepicted in Fig. 1(a) was studied. A region of the\nintermediate-magnetization states is observed between\nthe LM ferrimagnetism that is rigorously proved by the\nMLM theorem1,2and the nonmagnetic state of the spa-\ntially isotropic kagome-lattice antiferromagnet in the ab-\nsence of magnetic field17–27. The local magnetization in\nthe intermediate state of the kagome lattice was investi-\ngated by the exact diagonalization method, and it was\nreported that the local magnetization greatly depends\non the position of the sites, although it is difficult to\ndetermine clearly whether the incommensurate modula-\ntion with long-distance periodicity is present. This result\nleads to the high expectation that the intermediate state\nof the spatially anisotropic kagome lattice is the NLM\nferrimagnetic state. Additional research is desirable to\nconclude that the intermediate state of this 2D system is\nthe NLM ferrimagnetism.\nIn this paper, we study the ground-state properties\nof theS= 1/2 Heisenberg models on the quasi-one-\ndimensional (Q1D) kagome strip lattices depicted in\nFigs. 1(b) and 1(c) instead of the 2D lattice depicted2\n+'\n+\u0013+\u0012\n+\u0012+\u0013\n\"` \"\n#\n$\n%\n\"` #` \n$` \"\n# +\u0012\n+\u0013+'\n$\tB\n \tC\n \tD\nFIG. 1: Structures of the lattices: the spatially anisotrop ic kagome lattice (a) and the quasi-one-dimensional kagome strip\nlattices (b) and (c) with different widths. An S= 1/2 spin is located at each site denoted by a black circle. Antif erromagnetic\nbondsJ1(bold straight line) and J2(dashed line), and ferromagnetic bond JF(dotted line). The sublattices in a unit cell of\nlattice (b) are represented by A, A′, B, B′, C, C′, and D. The sublattices in a unit cell of lattice (c) are repre sented by A, A′,\nB, and C.\nin Fig. 1(a). Note that the inner parts of the lattices in\nFigs. 1(b) and 1(c) are common to a part of the 2D lat-\ntice in Fig. 1(a). We also note that the lattice shapes of\nstripsinthepresentstudyaredifferentfromsomekagome\nstrips (chains) studied in refs. VI-VI, where the nontriv-\nial properties of kagome antiferromagnets were reported.\nAccordingtothestudyinref. VI,itwasalreadyknown\nthat the NLM ferrimagnetism is realized in the ground\nstate of the kagome strip lattice in Fig. 1(c). In the\npresent study, we show that both the lattice in Fig. 1(c)\nand the lattice in Fig. 1(b) reveal the NLM ferrimag-\nnetism in the ground state. Note also that the lattice\nshape at the edge under the open boundary condition\ndepicted in Fig. 1(c) is different from that in ref. VI [see\nFig. 1(b) in ref. VI]. Thus, one can recognize that the\nresults of the strip lattice with small width are irrespec-\ntive of boundary conditions. We also present clearly the\nexistence of the incommensurate modulation with long-\ndistance periodicity in the local magnetizations of both\nmodelsinFigs.1(b)and1(c). Ournumericalcalculations\nsuggest that the intermediate state of the 2D lattice in\nFig. 1(a) is the NLM ferrimagnetism.\nThis paper is organized as follows. In §2, we first\npresent our numerical calculation methods. In §3, we\nshow the ground-state properties of the lattice depicted\nin Fig. 1(c) in finite-size clusters. In §4, we show the\nground-state properties of the lattice depicted in Fig.\n1(b). Sections 5 and 6 are devoted to discussion and\nsummary, respectively.\nII. NUMERICAL METHODS\nWe employ two reliable numerical methods: the ex-\nact diagonalization (ED) method and the density matrixrenormalization group (DMRG) method32,33.\nThe ED method is used to obtain precise physical\nquantities of finite-size clusters. This method does not\nsuffer from the limitation of the cluster shape. It is ap-\nplicable even to systems with frustration, in contrast to\nthe quantum Monte Carlo (QMC) method coming across\nthe so-callednegative-signproblemforsystemswith frus-\ntration. The disadvantage of the ED method is the lim-\nitation that available system sizes are very small. Thus,\nwe should pay careful attention to finite-size effects in\nquantities obtained by this method.\nOntheotherhand,theDMRGmethodisverypowerful\nwhen a system is (quasi-)one-dimensionalunder the open\nboundary condition. The method can treat much larger\nsystemsthan theEDmethod. Notethatthe applicability\nof the DMRG method is irrespective of whether or not\nthe systems include frustrations. In the present research,\nwe use the “finite-system” DMRG method.\nIII. KAGOME STRIP LATTICE WITH SMALL\nWIDTH\nIn this section, we study the magnetic properties in\nthe ground state of the S= 1/2 Heisenberg model on the\nkagome strip lattice depicted in Fig. 1(c). The Hamilto-\nnian of this model is given by\nH=J1/summationdisplay\ni[Si,B·Si,C+Si,C·Si,A′+Si,C·Si+1,A+Si,C·Si+1,B]3\n+J2/summationdisplay\ni[Si,A·Si,B+Si,B·Si,A′]+JF/summationdisplay\ni[Si,A·Si+1,A+Si,A′·Si+1,A′], (1)\n\tB\n\tC\n0 10 20\nStotz–39.5–39–38.5–38E nergy kagome strip in Fig. 1(c) \nN=96, MS=900, SW=15\nJF/J 1=–1\nJ2/J 1=0.50J2/J 1=0.58\n0 0.5 1 1.5 2\nJ2/J 100.20.40.6M/M sN=24, periodic\nN=48, open\nN=96, open\nJF/J 1=–1kagome strip in Fig. 1(c) \nFIG. 2: (Color)(a) Dependence of the lowest energy on Sz\ntot.\nThe results of J2/J1= 0.5 and 0.58 for the system size of\nN= 96 are presented. Arrows indicate the values of the\nspontaneous magnetization Min eachJ2/J1. The position\nof an arrow is given by the highest Sz\ntotvalue among those\nthat give the common lowest energy. (b) J2/J1dependence\nofM/Msobtained from ED calculations for N= 24 (black\nsquare) under the periodic boundary condition and DMRG\ncalculation for N= 48 (red triangle) and 96 (blue inverted\ntriangle) under the open boundary condition. Note that for\nN= 96 in (a) and (b), we use MS= 900 and SW= 15 and,\nthat for N= 48 in (b), we use MS= 400 and SW= 15.\nwhereSi,ξistheS= 1/2spinoperatoratthe ξ-sublattice\nsite in the i-th unit cell. The positions of the four sub-\nlattices in a unit cell are denoted by A, A′, B, and C in\nFig. 1(c). Energies are measured in units of J1; we fixed\nJ1= 1 hereafter. In what follows, we examine the region\nof 0< J2/J1<∞in the case of JF=−1. Note that\nthe number of total spin sites is denoted by N; thus, the\nnumber of unit cells is N/4.\nWe examine the J2/J1dependence of the ratio M/Ms,\nwhereMandMsare the spontaneous and saturation\nmagnetizations, respectively. Let us explain the method\nusedtodetermine Masafunction of NandJ2/J1. First,\nwe calculate the lowest energy E(J2/J1,Sz\ntot,N), where\tB\n\tC\n0 10 20\ni–0.4 –0.2 00.20.4<S i, ξz>\nN=96 J F/J 1=–1 J 2/J 1=0.3 M=24\nAA’ BC\n0 10 20\ni–0.4 –0.2 00.20.4<S i, ξz>\nN=96 J F/J 1=–1 J 2/J 1=0.57 M=20 \nAA’ BC\nFIG. 3: (Color) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublattice\nξ; A (black square), A′(red triangle), B (blue square), and C\n(purple inverted triangle). Panels (a) and (b) show results for\nJ2/J1= 0.3 and 0.57, respectively. These results are obtained\nfrom our DMRG calculations for N= 96 (i= 1,2,···,24).\nSz\ntotvalue is the z-component of the total spin. For ex-\nample, the energies for various Sz\ntotin the two cases of\nJ2/J1are presented in Fig. 2(a); the results are obtained\nby our DMRG calculations of the system of N= 96 with\nthe maximum number of retained states ( MS) of 600\nand the number of sweeps ( SW) of 15. The spontaneous\nmagnetization M(J2/J1,N) is determined as the highest\nSz\ntotamong those at the lowest common energy [see ar-\nrows in Fig. 2(a)]. Our results of the J2/J1dependence\nofM/Msare depicted in Fig. 2(b). We find the existence\nof the intermediate magnetic phase of 0 < M/M s<1/2\nbetween the LM ferrimagnetic phase of M/Ms= 1/2\nand the nonmagnetic phase. In order to determine the\nspin state of this intermediate phase, we calculate the\nlocal magnetization /angbracketleftSz\ni,ξ/angbracketright, where/angbracketleftA/angbracketrightdenotes the expec-\ntation value of the physical quantity AandSz\ni,ξis the\nz-component of Si,ξ. Figure 3 depicts our results for a\nsystem size N= 96 on the lattice depicted in Fig. 1(c)\nunder the open boundary condition; Figs. 3(a) and 3(b)\ncorrespondtothe caseofthe LMferrimagneticphaseand4\nthat of the intermediate phase, respectively.\nThe results clearly indicate the existence of the incom-\nmensurate modulation with long-distance periodicity in\nthe intermediate phase. We also confirm that the period-\nicities of the local magnetizations in the NLM ferrimag-\nnetism in the present model depend on the J2/J1value\nbut not on the length of the system, as reported in the\ncase of ref. VI.\nItisworthemphasizingthe pointthatthe intermediate\nphase commonly exists irrespective of the shape at the\nedge of the strip lattice when one compares the result\nof the present lattice depicted in Fig. 1(c) and that in\nref. VI. Therefore, wecan concludethat the intermediate\nphase exists and that it is NLM ferrimagnetic.\nAlthough there exists an intermediate NLM ferrimag-\nnetic phase in the case of the kagome strip lattice de-\npicted in Fig. 1(c), we should note that there is a large\ndiscrepancy in dimensionality between the kagome striplattice depicted in Fig. 1(c) and the 2D kagome lattice\ndepicted in Fig. 1(a). In the next section, we treat the\nkagome strip lattice depicted in Fig. 1(b) whose width is\nlarger than that of the kagome strip lattice depicted in\nFig. 1(c).\nIV. KAGOME STRIP LATTICE WITH LARGE\nWIDTH\nA. Hamiltonian\nIn this section, we study the ground-state properties\nof theS= 1/2 Heisenberg model on the kagome strip\nlattice depicted in Fig. 1(b). The Hamiltonian of this\nmodel is given by\nH=J1/summationdisplay\ni[Si,B·Si,C+Si,C·Si,D+Si,C·Si+1,A+Si,C·Si+1,B\n+Si,C′·Si,B′+Si,C′·Si,A′+Si,C′·Si+1,D+Si,C′·Si+1,B′]\n+J2/summationdisplay\ni[Si,A·Si,B+Si,B·Si,D+Si,D·Si,B′+Si,B′·Si,A′]\n+JF/summationdisplay\ni[Si,A·Si+1,A+Si,A′·Si+1,A′]\n−h/summationdisplay\ni[Sz\ni,A+Sz\ni,A′+Sz\ni,B+Sz\ni,B′+Sz\ni,C+Sz\ni,C′+Sz\ni,D]. (2)\nHere, the positions ofseven sublattices are denoted by A,\nA′, B, B′, C, C′, and D in Fig. 1(b). Note that the last\nterm of eq. 2 is the Zeeman term. The number of spin\nsitesisdenotedby N. Thenumberofunitcellsis N/7; we\nconsider N/14 as an integer. We mainly use the DMRG\nmethod for investigating the magnetic properties in the\nground state of this Q1D system under the open bound-\nary condition. We also investigate the properties under\nthe periodic boundary condition by the ED method, al-\nthough the size treated by this method is only in the case\nofN= 28. Hereafter, we consider J1= 1 as an energy\nscale and we investigate the region of 0 < J2/J1<∞in\nthe case of JF=−1.\nB. Phase diagram\nFirst, let us examine the J2/J1dependence of the ratio\nM/Msin the absence of the external magnetic field h.\nThe procedure for determining Mis the same as that\nmentioned in §3 [see also Fig. 4(a)]. We present our\nresults of the spontaneous magnetization in Fig. 4(b).\nWe successfully observe the intermediate-magnetizationregion irrespective of the boundary conditions. A careful\nobservation of Fig. 4(b) enables us to observe the eight\nregions at least in the finite-size system. As a matter\nof convenience, hereafter, we call these regions R 1, R2,\n···, R7, and R 8. In the case of N= 112 under the open\nboundarycondition, forexample, Fig. 5(a)illustratesthe\nregions R 1to R8: R1is the region of M/Ms= 3/7, R2is\nthe region of 11 /28≤M/Ms<3/7, R3is the region of\n1/8< M/M s<11/28, R4is the region of M/Ms= 1/8,\nR5is the region of 0 < M/M s<1/8, R6is the region of\nM/Ms= 0, R 7is the region of 0 < M/M s<1/7, and R 8\nis the region of M/Ms= 1/7. Here, the dashed lines in\nFig. 5(a) indicate the boundaries of these regions.\nIt should be noted that the values of M/Msin the\nR4region and that at the lower edge of the R 2region\nchange with increasing N, as shown in Fig. 4(b); the for-\nmer value is M/Ms= (N−14)/7Nand the latter value\nisM/Ms= (3N−28)/7N. These changes due to the in-\ncrease in system size come from the finite-size effect. We\nfind that the value of M/Msin the R 4region under the\nopen boundary condition increases and approaches the\nvalue of M/Ms= 1/7 whenNincreases. Furthermore,\nthe magnetization value in the R 4region is M/Ms= 1/75\n\tB\n \n\tC\n 0 10 20\nStotz–20020 E nergy kagome strip in Fig. 1(b) \nN=56, MS=600, SW=15 \nJF/J 1=–1\nJ2/J 1=0.20J2/J 1=0.57\n0 0.5 1 1.5 2\nJ2/J 100.20.40.6M/M skagome strip in Fig. 1(b) \nN=28,  periodic \nN=56,  open \nN=112, open\nJF/J 1=–1\nFIG. 4: (Color)(a) Lowest energy in each subspace divided by\nSz\ntot. The results of the DMRG calculations obtained when\nthe system size is N= 56 for J2/J1= 0.20 and 0.57 are\npresented. Arrows indicate the spontaneous magnetization M\nfor a given J2/J1. (b)J2/J1dependence of M/Msobtained\nfrom the ED calculations for N=28 (blue triangle) under the\nperiodic boundary condition and the DMRG calculations for\nN=56 (red square) and 112 (black pentagon) under the open\nboundary condition. Note that for N= 56 in (a) and (b),\nwe useMS= 600 and SW=15, and that for N= 112 in\n(b), we use MS≥900 and SW=15. Here, MSdenotes the\nmaximum number of retained states and SWthe number of\nsweeps used in DMRG calculations.\nin the case of N= 28 under the periodic boundary condi-\ntion. Therefore, it is expected that the value of M/Msin\nthe R4regionis 1 /7in the thermodynamic limit. We also\nconfirm that the value of M/Msat the lower edge of the\nR2region gradually increases and approaches the value\nofM/Ms= 3/7 with increasing N. In addition, we can-\nnot confirm the R 2region in the case of N= 28 under\nthe periodic boundary condition. These circumstances\nindicate a possibility that the R 2region merges with the\nR1region of M/Ms= 3/7 in the thermodynamic limit.\nNext, to determine whether each region survives in\nthe thermodynamic limit, we study the size depen-\ndences of the boundaries between the regions under the\nopen boundary condition. Figure 5(b) shows the re-\nsults of N=42, 56, 84, and 112 from DMRG calcu-\nlations. Note here that we define R 2as the region of(3N−28)/7N < M/M s<3/7 and R 4as the region of\nM/Ms= (N−14)/7Nin the finite-size system. One\ncan find immediately from Fig. 5(b) that all regions, ex-\ncept the R 7region, survive in the limit N→ ∞. To\ndetermine whether the R 7region survives in the thermo-\ndynamic limit, we investigate the size dependence of the\nwidth of the R 7region in Fig. 6. This plot shows us that\nthe width of the R 7region decreases with increasing N.\nIt is difficult to determine whether the R 7region survives\nin the thermodynamic limit. The convex downward be-\nhavior is observed for large sizes so that the region might\nsurvive; however, the observed behavior may be one of\ntheseriousfinite-sizeeffects. Theissueofestablishingthe\npresence or absence of the R 7region should be clarified\nin future studies.\n\tB\n\tC\n JF/J 1=–1\n0 0.01 0.02 0.03\n1/N00.511.52J2/J 1\nR1–R 2R2–R 3R3–R 4R4–R 5R5–R 6R6–R 7R7–R 80 0.5 1 1.5 2\nJ2/J 100.20.40.6M/M skagome strip in Fig. 1(b) \nN= 112, open  \nR1 R2\nR3\nR4R5\nR6R7R8\nFIG. 5: (a) Definitions of the R 1-R8regions in the case of\nN= 112 underthe open boundary condition: R 1is the region\nofM/Ms= 3/7, R2is the region of 11 /28≤M/Ms<3/7,\nR3is the region of 1 /8< M/M s<11/28, R4is the region of\nM/Ms= 1/8, R5is the region of 0 < M/M s<1/8, R6is the\nregion of M/Ms= 0, R 7is the region of 0 < M/M s<1/7,\nand R 8is the region of M/Ms= 1/7. Dashed lines indi-\ncate the boundaries of these regions. (b) Size dependences\nof boundaries under the open boundary condition. The re-\nsults presented are those of N=42, 56, 84, and 112 from\nDMRG calculations. The curve line named R l-Rl+1indicates\nthe boundary line between the R land R l+1regions, where l\nis integer.6\n0 0.01 0.02 0.03 0.04\n1/N00.10.20.30.4Width of region R7 region\nFIG. 6: Size dependence of the width of the R 7region. It is\ndifficult to determine whether the R 7region survives in the\nthermodynamic limit.\nC. Magnetic properties in each region\nIn this subsection, we investigate the local magnetiza-\ntion/angbracketleftSz\ni,ξ/angbracketrightto study the magnetic structures in various\nregions except the R 2and R 6regions. Note that we cal-\nculate/angbracketleftSz\ni,ξ/angbracketrightwithin the subspace of the highest Sz\ntotcor-\nresponding to the spontaneous magnetization Mwhen\nJ2/J1is given. Considering the fact that the present\nlattice depicted in Fig. 1(b) has seven sublattices in the\nsystem, we will use different colors or symbols for each\nsublattice ξfor presenting our results of /angbracketleftSz\ni,ξ/angbracketright, as de-\npicted in the inset of Fig. 7(a); we use a black square for\nξ=A, a red triangle for ξ= A′, a blue cross for ξ= B, a\ngreen pentagon for ξ= B′, a purple inverted triangle for\nξ=C, an aqua diamond for ξ= C′, and a black circle for\nξ=D.\nFirst, we examine the R 1and R 8regions. We present\nour DMRG results of /angbracketleftSz\ni,ξ/angbracketrightof the system of N= 112 in\nFigs. 7(a) and 7(b) for J2/J1= 0.2 and 1.9, respectively.\nIn Fig. 7(a), we observethe uniform behavior of upward-\ndirection spins at the sublattice sites A, A′, B, B′, and\nD, and downward-direction spins at the sublattice sites\nC and C′. In Fig. 7(b), we also observe the uniform\nbehavior of upward-direction spins at the sublattice sites\nB, B′, C, and C′, and downward-direction spins at the\nsublatticesites A, A′, andD. Therefore, weconcludethat\nthe LM ferrimagnetic states are realized in the regions of\nR1and R 8.\nOur understanding of the origins of these LM ferri-\nmagnetic phases is based on the Marshall-Lieb-Mattis\n(MLM) theorem. In the case of J2/J1= 0, no frustra-\ntion occurs; thus, the spin state depicted in Fig. 8(a) is\nrealized. This state shows the LM ferrimagnetism with\nM/Ms= 3/7. The R 1region is directly connected to the\nLM ferrimagnetic state of J2/J1= 0. Therefore, the R 1\nregion of M/Ms= 3/7 is regarded as the LM ferrimag-\nnetic phase. In the limit J2→ ∞, on the other hand, the\npresent model becomes equal to a model of an S= 1/2\ndiamond chain depicted in Fig. 8(b). The value of mag-\tB\n\tC\n%\"\n# $\n#` $` \n\"` \n5 10 15\ni–0.2 00.20.4<S i, ξz>N=112 J F/J 1=–1 J 2/J 1=0.2 M=24\n5 10 15\ni–0.2 00.20.4<S i, ξz>N=112 J F/J 1=–1 J 2/J 1=1.9 M=8 \nFIG. 7: (Color) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublattice\nξ. Panels (a) and (b) show results for J2/J1=0.2 and 1.9,\nrespectively. These results are obtained from our DMRG cal-\nculations for N= 112 (i=1, 2,···, 16). The inset in the\npanel (a) presents the correspondence relationship betwee n\neach colored symbol and each sublattice ξused in Figs. 7, 9,\nand 10.\nnetization takes M/Ms= 1/7 in the ground state of this\ndiamond chain according to the MLM theorem. There-\nfore, the R 8region of M/Ms= 1/7 is regarded as the LM\nferrimagnetic phase.\nNext, we investigate the R 3, R5, and R 7regions. Our\nresultsobtainedfromthe DMRG calculationsof N= 168\nare depicted in Figs. 9(a)-9(c) for J2/J1= 0.57, 1.14,\nand 1.69, respectively. We clearly observe incommensu-\nratemodulationswithlong-distanceperiodicitiesinevery\ncase in Fig. 9. In addition, we confirm from Fig. 4(b)\nthat the ratio M/Mschanges gradually with the varia-\ntion inJ2/J1in the R 3, R5, and R 7regions. Since the\nwidths of the R 3and R 5regions survive in the thermo-\ndynamic limit as was clarified in the previous subsection,\nwe conclude that the R 3and R 5regions are NLM ferri-\nmagnetic phases. Although it is unclear whether the R 7\nregion survives in the thermodynamic limit, this region\nis an NLM ferrimagnetic phase if it survives.\nFinally, in this subsection, we examine the R 4region.\nOur result of /angbracketleftSz\ni,ξ/angbracketrightforJ2/J1= 1 in the system of\nN= 168 is depicted in Fig. 10. We do not detect the in-\ncommensuratemodulation in this R 4region. In addition,\nwe confirm from Fig. 4(b) that the ratio M/Msin the\nR4region does not change with the variation in J2/J1in\ncontrast to the cases in the R 3and R 5regions. These7\n%\n\"` #` \n$` \"\n#\n$\n$\n$` \tB\n\tC\nFIG. 8: (a) Kagome strip lattice depicted in Fig. 1(b) in the\nlimit of J2/J1= 0. White arrowheads denote the classical\nspin configuration in the LM ferrimagnetic state of M/Ms=\n3/7. (b) Kagome strip lattice depicted in Fig. 1(b) in the\nlimit of J2→ ∞. White circles represent the effective S=\n1/2 spins formed by five S= 1/2 spins at the sublattices\nξ=A, A′, B, B′, and D. The black bold and black dotted lines\nshow the antiferromagnetic and ferromagnetic interaction s,\nrespectively.\ncircumstances suggest that the R 4region is the LM ferri-\nmagnetic phase. However, one cannot speculate the spin\nstate from the result of /angbracketleftSz\ni,ξ/angbracketrightbecause it is difficult to\nclarify the spin state on the basis of the results of deter-\nmining whether the spins are up or down, as was success-\nfully observed in the R 8region. We further investigate\nthe magnetization curve in this region to know whether\nthe R4region is the LM ferrimagnetic phase. The mag-\nnetization curves of J2/J1= 1 forN= 56 and N= 112\ncalculated by the DMRG method are presented in Fig.\n11(a)34. Figure 11(b) is obtained by zooming the region\nofhnearh= 0. One can observe the existence of the\nmagnetization plateaus at the height of the spontaneous\nmagnetization for both system sizes. We also confirm\nthat the difference in the width of the plateau between\nN= 112 and 56 is very small. These features indicate\nthat the spin gap exists in the R 4region in the thermo-\ndynamic limit. If the R 4region is the NLM ferrimagnetic\nphase, no spin gap is present in this region. (It was re-\nported in ref. VI that the NLM ferrimagnetism is gapless\nas a response to a uniform magnetic field.) Therefore,\nwe conclude that the R 4region is the LM ferrimagnetic\nphase.\nV. DISCUSSION\nWe discuss the relationships between the interval 0 <\nJ2/J1≤1 of the kagome strip lattice depicted in Fig.\n1(b) and those of the spatially anisotropic kagome lat-\tB\n\tC\n\tD\n10 20 \ni–0.2 00.20.4<S i, ξz>\nN=168 J F/J 1=–1 J 2/J 1=0.57 M=28 \n10 20 \ni–0.2 00.20.4<S i, ξz>\nN=168 J F/J 1=–1 J 2/J 1=1.14 M=7\n10 20 \ni–0.2 00.20.4<S i, ξz>N=168 J F/J 1=–1 J 2/J 1=1.69 M=2\nFIG. 9: (Color) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublat-\nticeξ. The correspondence relationship between each colored\nsymbol and each sublattice ξis described in the inset in Fig.\n7(a). Panels (a), (b), and (c) show results for J2/J1=0.57,\n1.14, and 1.69, respectively. These results are obtained fr om\nour DMRG calculations for N= 168 (i=1, 2,···, 24).\ntice studied in ref. VI, while we consider the case of\nanother new lattice, with a larger but finite width along\nthe direction perpendicular to the bonds of interaction\nJFin Fig. 1(b), namely, the strip width.\nThe ratio M/Msof the R 1and R2regionsis commonly\nM/Ms= 3/7 in the thermodynamic limit of the lattice\ndepicted in Fig. 1(b). The difference of M/Ms= 3/7\nfromM/Ms= 1/3 in the case of the LM ferrimagnetic\nphase of the spatially anisotropic kagome lattice in ref.\nVIis attributedtothe finitenessofthestripwidth. Thus,\nthe ratio M/Msapproaches M/Ms= 1/3 when the strip8\n10 20 \ni–0.2 00.20.4<S i, ξz>\nN=168 J F/J 1=–1 J 2/J 1=1 M=11\nFIG. 10: (Color) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublattice\nξ. This result is obtained from our DMRG calculations for\nJ2/J1= 1 in the system of N= 168 (i=1, 2,···, 24). The\ncorrespondence relationship between each colored symbol a nd\neach sublattice ξis described in the inset in Fig. 7(a).\nwidth increases; the R 1and R 2regions of the present\nmodel depicted in Fig. 1(b) correspond to the LM ferri-\nmagnetic phase of the spatially anisotropic kagome lat-\ntice studied in ref. VI.\nThe ratio M/Ms= 1/7 atJ2/J1= 1 in the present\nmodel depicted in Fig. 1(b) may be related to the fact\nthat the model includes seven sublattices, namely, the\nstrip width is finite. At least at J2/J1= 1, on the other\nhand, it is widely believed that the spontaneous mag-\nnetization disappears in the limit of infinite width17–27.\nAlthough the relationship should be clarified in the ex-\namination of the case of the lattice with an even larger\nstrip width, there are two possibilities of the behavior\nnear the case of J2/J1= 1. One is the case when the\nratioM/Msdecreases with increasing strip width and fi-\nnally vanishes, while the LM ferrimagnetic phase, such\nas R4, survives in systems with larger strip widths. In\nthis case, the R 3region corresponds to the NLM phase\nofthe two-dimensionalmodel onthe spatiallyanisotropic\nkagome lattice. In the other case, the LM ferrimagnetic\nphase, such as R 4, becomes narrower with increasing\nstrip width, while the ratio M/Msdoes not vanish; fi-\nnally, the R 3and R 5regions merge with each other. In\nthis case, the value of J2/J1at the boundary between\nthe R5and R 6regions decreases across J2/J1= 1. In\nany cases, it is important to note that from our finding\nof an intermediate phase in all the three cases in Fig. 1\nbetween the ferrimagnetic phase and the nonmagnetic\nphase, the phase is considered to exist irrespective of the\nstrip width.\nFinally, one should note that the intermediate phase\nbetween the Lieb-Mattis ferrimagnetic and nonmagnetic\nstates is observed in other cases. However, this phase\nis not always ferrimagnetic. One of the cases observed is\nthecaseofthethree-legladdersystemformingastriplat-\ntice obtained by cutting off from the spatially anisotropic\ntriangular lattice in ref. VI, in which the properties of\ntheintermediatephasewereunclear. Sakai’sunpublished\tB\n\tC\n0 1 2 3\nh00.51M/M skagome strip in Fig. 1(b)   \nN=56  J F/J 1=–1 J 2/J 1=1 \nN=112 J F/J 1=–1 J 2/J 1=1 \n0 0.2 0.4\nh00.10.20.3M/M sN=56 \nN=112\nFIG. 11: Magnetization curve in the R 4region. Panel (a) is\nobtained from the DMRG calculations of J2/J1= 1 forN=\n56 (triangle) and N= 112 (square). Panel (b) is obtained by\nzooming the region near h= 0 in panel (a).\nstudy suggests that the intermediate phase is nematic38.\nCareful examinations are required to investigate such an\nintermediate phase if it is found.\nVI. SUMMARY\nWe have studied the ground-state properties of the\nS= 1/2 Heisenberg models on the kagome strip lattices\ndepicted in Figs. 1(b) and 1(c) by the ED and DMRG\nmethods. As a common phenomenon in the ground state\nofboth cases,wehaveconfirmedthe existenceofthe non-\nLieb-Mattis ferrimagnetism between the Lieb-Mattis fer-\nrimagnetic phase and the nonmagnetic phase. We have\nclearly found incommensurate modulations with long-\ndistance periodicity in the non-Lieb-Mattis ferrimagnetic\nstate. The occurrence of the non-Lieb-Mattis ferrimag-\nnetism irrespective of strip width strongly suggests that\nthe intermediate state found in the case of the spatially\nanisotropic kagome lattice in two dimensions is the non-\nLieb-Mattis ferrimagnetism.9\nAcknowledgments\nWe thank Prof. Toru Sakai for letting us know his\nunpublished results. This work was partly supported\nby Grants-in-Aid (Nos. 20340096, 23340109, 23540388,\nand 24540348) from the Ministry of Education, Culture,\nSports, Science and Technology of Japan. This work\nwas partly supported by a Grant-in-Aid (No. 22014012)\nfor Scientific Research and Priority Areas “Novel Statesof Matter Induced by Frustration” from the Ministry of\nEducation, Culture, Sports, Science and Technology of\nJapan. Diagonalization calculations in the present work\nwere carried out using TITPACK Version 2 coded by\nH. Nishimori. DMRG calculations were carried out us-\ning the ALPS DMRG application39. 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B78(2008) 014418.\n12T.ShimokawaandH.Nakano: J. Phys.Soc.Jpn. 80(2011)\n043703.\n13T.ShimokawaandH.Nakano: J. Phys.Soc.Jpn. 80(2011)\n125003.\n14N. B. Ivanov, J. Richter, and D. J. J. Farnell: Phys. Rev.\nB66(2002) 014421.\n15R. F. Bishop, P. H. Y. Li, D. J. J. Farnell, and C. E.\nCampbell: Phys. Rev. B 82(2010) 024416.\n16H. Nakano, T. Shimokawa, and T. Sakai: J. Phys. Soc.\nJpn.80(2011) 033709.\n17P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, and\nP. Sindzingre: Phys. Rev. B 56(1997) 2521.\n18Ch. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuil-\nlier, P. Sindzingre, P. Lecheminant, and L. Pierre:\nEur. Phys. J. B 2(1998) 501.\n19K. Hida: J. Phys. Soc. Jpn. 70(2001) 3673.\n20D. C. Cabra, M. D. Grynberg, P. C. W. Holdsworth, and\nP. Pujol: Phys. Rev. B 65(2002) 094418.\n21J. Schulenberg, A. Honecker, J. Schnack, J. Richter, and\nH.-J. Schmidt: Phys. Rev. Lett. 88(2002) 0167207.\n22A.Honecker, J.Schulenberg, andJ.Richter: J.Phys.:Con-\ndens. Matter 16(2004) S749.\n23O. Cepas, C. M. Fong, P. W. Leung, and C. Lhuillier:\nPhys. Rev. B 78(2008) 140405(R).24P. Sindzingre and C. Lhuillier: Europhys. Lett. 88(2009)\n27009.\n25H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 79(2010)\n053707.\n26T. Sakai and H. Nakano: Phys. Rev. B 83(2011)\n100405(R).\n27H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 80(2011)\n053704.\n28P. Azaria, C. Hooley, P. Lecheminant, C. Lhuillier, and A.\nM. Tsvelik: Phys. Rev. Lett. 81(1998) 1694.\n29Ch. Waldtmann, H. Kreutzmann, U. Schollwock, K.\nMaisinger, andH.-U. Everts: Phys. Rev.B 62(2000) 9472.\n30J. Schulenburg, A. Honecker, J. Schnack, J. Richter, and\nH.-J. Schmidt: Phys. Rev. Lett. 88(2002) 167207.\n31T. Shimokawa and H. Nakano: J. Phys.: Conf. Ser. 320\n(2011) 012007.\n32S. R. White: Phys. Rev. Lett. 69(1992) 2863.\n33S. R. White: Phys. Rev. B 48(1993) 10345.\n34One can find the macroscopic jump in this magnetization\ncurvenear the saturation fieldas well as in the case ofsome\nfrustrated systems30,35. It is known that the occurrence\nmechanism of the magnetization jump can be understood\nfrom the viewpoint of the localized-magnon picture (see\nrefs. VI, VI, and VI).\n35J. Schnack, H.-J. Schmidt, A. Honecker, J. Schulenburg,\nand J. Richter: J. Phys.: Conf. Ser. 51(2006) 43.\n36M. E. Zhitomirsky and H. Tsunetsugu: Phys. Rev. B 70\n(2004) 100403.\n37S. Abe, T. Sakai, K. Okamoto, and K. Tutsui: J. Phys.:\nConf. Ser. 320(2011) 012015.\n38T. Sakai: private communications.\n39A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal, A.\nFeiguin, L. Gamper, E. Gull, S. Gurtler, A. Honecker, R.\nIgarashi, M. Korner, A. Kozhevnikov, A. Lauchli, S. R.\nManmana, M. Matsumoto, I. P. McCulloch, F. Michel, R.\nM. Noack, G. Pawlowski, L. Pollet, T. Pruschke, U. Scholl-\nwock, S. Todo, S. Trebst, M. Troyer, P. Werner, and S.\nWessel: J. Magn. Magn. Mater. 310(2007) 1187 (see also\nhttp://alps.comp-phys.org)."    },    {        "title": "1705.09049v2.Fast_Vortex_Oscillations_in_a_Ferrimagnetic_Disk_near_the_Angular_Momentum_Compensation_Point.pdf",        "content": "Fast Vortex Oscillations in a Ferrimagnetic Disk near the Angular Momentum\nCompensation Point\nSe Kwon Kim and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n(Dated: October 15, 2018)\nWe theoretically study the oscillatory dynamics of a vortex core in a ferrimagnetic disk near its\nangular momentum compensation point, where the spin density vanishes but the magnetization is\n\fnite. Due to the \fnite magnetostatic energy, a ferrimagnetic disk of suitable geometry can support\na vortex as a ground state similar to a ferromagnetic disk. In the vicinity of the angular momentum\ncompensation point, the dynamics of the vortex resemble those of an antiferromagnetic vortex, which\nis described by equations of motion analogous to Newton's second law for the motion of particles.\nOwing to the antiferromagnetic nature of the dynamics, the vortex oscillation frequency can be an\norder of magnitude larger than the frequency of a ferromagnetic vortex, amounting to tens of GHz\nin common transition-metal based alloys. We show that the frequency can be controlled either by\napplying an external \feld or by changing the temperature. In particular, the latter property allows\nus to detect the angular momentum compensation temperature, at which the lowest eigenfrequency\nattains its maximum, by performing FMR measurements on the vortex disk. Our work proposes a\nferrimagnetic vortex disk as a tunable source of fast magnetic oscillations and a useful platform to\nstudy the properties of ferrimagnets.\nFerromagnets are used as platforms to produce GHz\nmagnetic oscillations, e.g., spin-torque and spin-Hall os-\ncillators [1], which can be used as microwave signal gen-\nerators. Among them, the core oscillation of a vortex in a\nmicrodisk [2, 3], which is a curling magnetization texture\nstabilized by magnetostatic energy, stands out because of\nits high coherence [4, 5]. This advantage allows it to be\nused in areas requiring highly precise oscillations such as\nbiomedicine [6]. Aside from the dynamics of an isolated\nvortex, coupled oscillations of vortices in an array of disks\nhave been studied as magnonic vortex crystals [7, 8]. The\noscillation can be driven in several ways such as with an\nexternal \feld [9, 10] or with an electric current [4, 5]. Its\ncharacteristic frequency is given by\nfFM\u0018\r\u00160Ms; (1)\nup to the geometric factor, where \ris the gyromagnetic\nratio,\u00160is the permeability constant, and Msis the sat-\nuration magnetization. The observed frequency ranges\nfrom several hundred MHz up to 2 GHz [3, 5, 10, 11]. See\nFig. 1 for some schematic illustrations of vortex disks.\nRecently, antiferromagnets have been gaining atten-\ntion as possible hosts of magnetic oscillations that\nare orders-of-magnitude faster than their ferromagnetic\ncounterparts [12, 13]. For example, a spin Hall oscillator\nbased on an antiferromagnet/heavy-metal heterostruc-\nture has been proposed as a possible THz signal gener-\nator [14]. However, the possibility of fast oscillations of\nan antiferromagnetic vortex has not been investigated be-\ncause, di\u000bering from ferromagnetic cases, antiferromag-\nnetic disks do not support a vortex as an equilibrium\nstate due to the absence of magnetostatic energy.\nHere, we show that a ferrimagnetic disk can host a sta-\nble vortex whose oscillations can be orders-of-magnitude\nfaster than those of ferromagnetic vortices. A ferrimag-\nnetic disk can harbor a vortex as a ground state due to\nthe \fnite magnetostatic energy [15], which con\fnes the\n(a)(b)p=1\nLRp=\u00001FIG. 1. Schematic illustrations of the magnetization of ferri-\nmagnetic disks harboring (a) a vortex with polarization p= 1\nand (b) a vortex with polarization p=\u00001. The thickness and\nthe radius of disks are denoted by LandR, respectively.\nvortex core to the center of the disk. We show that at\nthe angular momentum compensation point (CP), where\nthe spin density vanishes, the nature of the vortex-core\ndynamics is antiferromagnetic and in this way the core\nbehaves like a classical particle in a potential well [16].\nThe characteristic frequency of an oscillation of the vor-\ntex core at the CP, which will be derived below, is given\nby\nfAFM\u0018r\nJ\n~\u0003\rt\u00160Ms; (2)\nup to a dimensionless geometric factor. Here, Jis the\nantiferromagnetic exchange energy between neighboring\nmagnetic moments and \rtis the transverse gyromagnetic\nratio with respect to the direction of the magnetization.\nWe will show that this frequency is in the range of tens of\nGHz, e.g.,fAFM\u001930 GHz for the material parameters\nof CoTb [17]. As a fast-oscillating source, a ferrimagnetic\nvortex in a microdisk has two advantages over other pro-\nposals using antiferromagnets. First, it can be easily op-\nerated by an external magnetic \feld due to the \fnite mag-\nnetization. Secondly, since the spin density can be con-arXiv:1705.09049v2  [cond-mat.mes-hall]  30 Jun 20172\ntrolled by changing the temperature [18], the oscillation\nfrequency can be thermally tuned. In particular, the lat-\nter advantage allows us to detect the angular momentum\ncompensation temperature, where the lowest frequency\nwill exhibit its peak, by performing FMR measurements.\nThe high coherence of the vortex oscillation [4, 5] will\nyield a narrow FMR linewidth. A recent realization of\na vortex in an arti\fcial-ferrimagnet microdisk composed\nof Py/Gd superlattices [19] supports the experimental\nfeasibility of our proposal.\nLet us begin by deriving the equations of motion for a\nvortex in a ferrimagnetic disk. We will describe the low-\nenergy dynamics of a ferrimagnet using coarse-grained\nvariables over the constituent sublattices, following the\napproach taken by Andreev and Marchenko [20] and sub-\nsequently used in Refs. [21{23]. This approach is based\non approximate exchange symmetry and does not use any\nmodel representations of the state of the magnet, e.g.,\nthe number of sublattices as pointed in Ref. [20]. The\ncollinear order of a ferrimagnet much below the Curie\ntemperature can be described by the unit vector nde-\nnoting its direction. Along this direction, a ferrimag-\nnet has the net magnetization, M=Msn, and the net\nspin density, s=sn, which are related by the (longi-\ntudinal) gyrotropic coe\u000ecient \rasM=\rs. Here,Ms\nis the net saturation magnetization density. There is a\nclass of ferrimagnets such as rare-earth transition-metal\nalloys, which exhibit two distinct special values of tem-\nperature or chemical composition: the angular momen-\ntum CP, where the spin density vanishes, s= 0, and\nthe magnetization CP, where the magnetization vanishes,\nMs= 0 [15]. These two points are di\u000berent due to the\ndistinct gyromagnetic ratios of constituent sublattices.\nIn particular, at the angular momentum CP, these ferri-\nmagnets have the \fnite magnetization and the zero spin\ndensity [18], allowing the unconventional coexistence of\nmagnetostatic energy and antiferromagnetic dynamics.\nThe former stabilizes a vortex in the microdisk and the\nlatter yields fast oscillations of the vortex, as will be ex-\nplained below.\nThe dynamics of a generic collinear ferrimagnet can be\ndescribed by the following Lagrangian density [21, 22]:\nL=\u0000sa[n]\u0001_n+\u001a_n2=2\u0000U[n]; (3)\nwhere nis the unit vector in the direction of the local\nmagnetization, ais the vector potential for the magnetic\nmonopole, rn\u0002a=n, and\u001ais the e\u000bective inertia\ndensity. Here, the \frst term represents the Berry phase\nterm associated with the spin density; the second term\nrepresents the inertia of the dynamics of nwhich can\narise due to the relative canting of sublattice spins; the\nthird term is the free-energy density. This Lagrangian\nfor ferrimagnets can be interpreted as a hybrid of those\nfor ferromagnets and antiferromagnets. We model the\nfree-energy density in the absence of an external \feld by\nU=A(rn)2+\u00160H2\nms\n2: (4)Here, the \frst term is the exchange energy parametrized\nby the sti\u000bness coe\u000ecient A > 0; the second term is\nthe magnetostatic energy, where Hmsis the dipolar \feld\ninduced by the magnetization satisfying the equations\nr\u0001(Hms+M) = 0 and r\u0002Hms= 0. At the angular\nmomentum CP, s= 0, the kinetic part of the Lagrangian\nis antiferromagnetic, but the magnetostatic energy is \f-\nnite owing to the \fnite magnetization, Ms6= 0. Recog-\nnizing this peculiar coexistence of antiferromagnetic dy-\nnamics and magnetostatic energy motivated this work.\nThe energy dissipation associated with the dynamics can\nbe accounted for by considering the Rayleigh dissipation\nfunction,R=s\u000b_n2=2, which is half of the energy (den-\nsity) dissipation rate. The parameter s\u000bis reduced to\nthe product of the Gilbert damping constant [24] and\nthe spin density in the ferromagnetic limit, \u001a!0.\nThere is a length scale associated with the energy den-\nsityU, which is given by \u0015\u0011p\nA=\u0016 0M2s. It is referred to\nas the exchange length at which the magnetostatic energy\nis comparable to the exchange energy. When a dimension\nof the sample is larger than the exchange length, the mag-\nnetostatic energy dominates the total energy and gener-\nates nonuniform ground states. Let us consider a circular\nferrimagnetic disk of radius Rand thickness Las shown\nin Fig. 1. When the thickness is order of the exchange\nlength,L\u0018\u0015, and the aspect ratio of the thickness to\nthe radius is small enough, g\u0011L=R.0:5, the disk sup-\nports a vortex as a ground state [25]. For a quantitative\nestimate of the geometry of the physical system, let us\ntake the example of Co 1\u0000xTbx, which exhibits its angu-\nlar momentum CP at x\u00190:17. According to the exper-\nimental results in Ref. [17], the parameters are given by\nA\u00191:4\u000210\u000011J/m andMs\u0019105A/m at the angular\nmomentum CP, which yields the exchange length \u0015\u001930\nnm. Therefore, for example, a disk made of CoTb with\nthicknessL= 100 nm and radius R= 1000 nm would\nhave a vortex as a ground state.\nThe observed vortex states of magnetic disks\ncan be described by the following ansatz in\nthe spherical-coordinate representation, n =\n(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012):\u0012= (1\u0000p)\u0019=2 +\n2 arctan(p\nx2+y2=Rc) forp\nx2+y2< Rcand\n\u0012=\u0019=2 otherwise, and \u001e='+c\u0019=2, whereRc\nis the vortex-core radius that is on the order of the\nexchange length \u0015[26]. Here, p=\u00061 is the direction\nof the magnetization at the core, n=p^z, which is\nreferred to as the polarization; c=\u00061 corresponds to\nthe counter-clockwise (+) or clockwise ( \u0000) rotation of\nthe magnetization in the plane, which is referred to\nas the chirality. The low-energy dynamics of a vortex\ncan be described by the dynamics of its core position,\nR(t) = (X;Y ), with the aforementioned ansatz. The\nequations of motion for the position can be derived\nfrom the above Lagrangian and the Rayleigh dissipation\nfunction within the collective-coordinate approach,\nassuming a rigid magnetization pro\fle [22, 27]:\nMR+G_R\u0002^z+D_R=\u0000KR; (5)3\n0G/Gco12\u00001\u00002!/!AFM2!+!\u0000\n-2 -1 0 1 2121\nFIG. 2. The eigenfrequencies !\u0006=!AFM as functions of the gy-\nrotropic coe\u000ecient G=G co. The solid and dashed lines show\nthe eigenfrequencies for the counter-clockwise ( !+) and clock-\nwise (!\u0000) rotations of the core. See the main text for discus-\nsions.\nwhere\nM= 2\u0019\u001aLln(R=Rc) (6a)\nG= 2\u0019psL; (6b)\nD= 2\u0019s\u000bLln(R=Rc); (6c)\nK=\u00160LM2\ns[F1(L=R)\u0000(\u0015=R)2]: (6d)\nHere,Mrepresents the mass of the vortex, which orig-\ninates from the inertial term in the Lagrangian; G\nparametrize the gyrotropic force on the vortex, which\nis rooted in the spin Berry phase; Dparametrizes the\nviscous force on the vortex. The right-hand side is the\nrestoring force on the vortex, which has been obtained\nin Ref. [9] for ferromagnetic vortices. When the vortex\ncore is away from the center of the disk, magnetostatic\ncharges are created on the boundary. The corresponding\nmagnetostatic energy engenders a con\fning potential for\nthe vortex core, which is parametrized by K. The fac-\ntor [F1(L=R)\u0000(\u0015=R)2] is a dimensionless number that\nis on the order of 0.1 for L=R\u00180:1. The de\fnition of\nthe function F1(x), which is an increasing function of x,\ncan be found in Ref. [28]. Note that the gyrotropic co-\ne\u000ecient is proportional to the spin density, G/s, and\nthus it vanishes at the angular momentum CP.\nLet us study the excitation mode for the core dynam-\nics described by the above equation. To this end, it is\nconvenient to express the equations of motion in terms\nof the complex variable, \t \u0011X+iY:\nM\t\u0000iG_\t +D_\t =\u0000K\t: (7)\nWhen the damping constant is small, \u000b\u001c1, the main ef-\nfect of the viscous force is to broaden the linewidth of the\nexcitation spectrum, not to change the spectrum itself,\nand so we will neglect it henceforth by setting D= 0.\nThe oscillation frequencies of the monochromatic solu-\ntion, \t(t)/exp(i!t), are given by\n!\u0006=G\n2M\u0006s\u0012G\n2M\u00132\n+K\nM: (8)\nAt the angular momentum CP, where the gyrotropic\nforce vanishes G= 0, the eigenfrequencies are given by\nTTCP!!AFMH=0H=HcoH=\u0000HcoFIG. 3. Schematic illustrations of the lowest eigenfrequency\n!of the vortex-core oscillation as a function of temperature\nT, in the vicinity of the angular momentum CP ( TCP) sub-\njected to an external \feld, H=H^z. The vortex with po-\nlarizationp= 1 is considered. For CoTb disks, the maxi-\nmum eigenfrequency and the crossover \feld are estimated as\n!AFM\u00192\u0019\u000230 GHz and Hco\u00192 T, respectively. See the\nmain text for discussions.\n!\u0006=\u0006!AFM with!AFM\u0011p\nK=M , which is reduced\nto Eq. (2) if we recast it in terms of the microscopic pa-\nrameters using \u001a\u0018~2=Ja3withathe lattice constant.\nThe absence of any gyrotropic coupling between Xand\nYis one characteristic of the antiferromagnetic dynamics\nof two-dimensional solitons [29] such as vortices [16] and\nskyrmions [30]. Note that two circularly polarized modes\nare degenerate at the CP, where the spin con\fgurations\nrespect time-reversal symmetry. Far away from the an-\ngular momentum CP, where the gyrotropic force domi-\nnates the dynamic part in the equations of motion, the\nlowest eigenfrequencies are given by !=\u0000psgn(\r)!FM\nwith!FM\u0011K=jGj, which corresponds to the ferromag-\nnetic case [9]. The crossover between antiferromagnetic\nand ferromagnetic dynamics occurs when the two fre-\nquencies are comparable, !FM\u0018!AFM, corresponding\ntojGj\u0018Gco\u0011p\nMK. The above equation (8) for gen-\neral cases can be written as a function of the gyrotropic\ncoe\u000ecientG:!\u0006=!AFM =G=2Gco\u0006p\n(G=2Gco)2+ 1,\nwhich is shown in Fig. 2.\nLet us provide a numerical estimate for the antiferro-\nmagnetic oscillation at the angular momentum CP. The\nalloy Co 1\u0000xTbxat its CP, x\u001917, has the exchange-\nsti\u000bness coe\u000ecient, A\u00191:4\u000210\u000011J/m, and the lat-\ntice constant, a\u00190:4 nm, which yield the microscopic\nexchange constant J=Aa\u001935 meV [15]. By us-\ning the inertia, \u001a=~2=2Jza3[23, 31], where z= 6\nis the coordination number for three-dimensional bipar-\ntite lattices, and the additional parameters, L=R = 0:1\nandRc\u0019\u0015, we can estimate the eigenfrequency at\nthe CP:fAFM\u0011!AFM=2\u0019\u001930 GHz, which is one\norder-of-magnitude larger than the observed frequencies\nin ferromagnetic disks of several hundred MHz up to 2\nGHz [3, 5, 10, 11]. For example, for a cobalt disk of the\nsame shape, the ferromagnetic resonance frequency is cal-\nculated as fFM\u0011!FM=2\u0019\u0019700 MHz when using the\nsaturation magnetization Ms\u00191:2\u0002106A/m measured\nfor 30-nm-thick \flms [32].\nThe dependence of the eigenfrequency on the gy-\nrotropic coe\u000ecient can be used to infer the angular mo-4\nmentum CP. For example, when we measure the ferro-\nmagnetic resonance (FMR) frequency of the vortex os-\ncillation by varying the temperature across the angular\nmomentum CP denoted by TCP, the lowest resonance fre-\nquency should attain its maximum at TCP. In addition,\nsince the rotational directions of the core oscillation be-\nlow and above TCPare opposite, TCPcan be measured\nby detecting the change of the oscillation direction, which\ncan be probed by time-resolved scanning transmission X-\nray microscopy [8]. See Fig. 3 for schematic illustrations\nof the lowest oscillation frequency as a function of a tem-\nperature. These methods using a vortex oscillation in\na ferrimagnetic disk to determine TCPcan be an alter-\nnative to a recent proposal based on domain-wall speed\nmeasurements [33].\nWe now study the e\u000bects of an external \feld applied\nperpendicular to the disk. In Refs. [16], it has been shown\nthat the application of an external \feld can induce a gy-\nrotropic force on antiferromagnetic solitons. The same\nmechanism works for ferrimagnets. In the presence of an\nexternal \feld, H, the inertial term in the Lagrangian (3)\nis changed to \u001a(_n\u0000\rtn\u0002H)2=2 [20]. The \feld-induced\ndynamical term, \u0000\u001a\rt_n\u0001(n\u0002H), is linear in the time\nderivative of n, and thus can be interpreted as the e\u000bec-\ntive geometric phase associated with the magnetic dy-\nnamics. When the \feld is perpendicular to the disk,\nH=H^z, it gives rise to an extra force term in the\nequations of motion (5) for a vortex within the collec-\ntive coordinate approach [16]:\nMR+G_R\u0002^z+GH_R\u0002^z+D_R=\u0000KR;(9)\nwhere\nGH\u00112\u0019p\u001a\rtLH (10)\nis the contribution to the gyrotropic coe\u000ecient induced\nby an external \feld. Its origin can be understood by writ-\ning\u001a\rtHin the \feld-induced kinetic term as \u001fH=\rtby\nusing the relation, \u001a=\u001f=\r2\nt[23], where \u001fis the trans-\nverse magnetic susceptibility. This induced spin density\ncontributes to the gyrotropic coe\u000ecient G[Eq. (6b)]. The\ncrossover \feld corresponding to the crossover gyrotropic\ncoe\u000ecient is given by Hco=Gco=2\u0019\u001a\rtL, which is esti-\nmated to be Hco\u00192T for the aforementioned CoTb disk.\nThe \feld-induced gyrotropic force can be inferred by per-\nforming FMR measurements. See Fig. 3 for schematic\nillustrations [34].\nTo conclude, we have studied the oscillation dynamics\nof a vortex core in a ferrimagnetic disk. We have shownthat the oscillation frequency can be orders-of-magnitude\nlarger than that of a ferromagnetic vortex in the vicinity\nof the angular momentum CP, where the nature of the\ndynamics is antiferromagnetic due to the vanishing spin\ndensity. This is similar to the recently observed enhance-\nment of the ferrimagnetic domain-wall speed around the\nangular momentum CP [33]. The vortex oscillation fre-\nquency can be tuned by changing the temperature or by\napplying an external \feld. Our results exemplify the pos-\nsibility of using ferrimagnets to realize desired functional-\nities that have been di\u000ecult to achieve with conventional\nferro- and antiferromagnets. One possible research topic\nrelated to this is the dynamics of a vortex domain wall\nin a long ferrimagnetic strip, which can be faster than its\nferromagnetic counterpart in the vicinity of the angular\nmomentum CP and thus may realize a better racetrack\nmemory [35].\nWe made several approximations in this work to de-\nscribe the dynamics of a ferrimagnetic vortex. First, we\nobtained the mass of a vortex by assuming that its mag-\nnetization pro\fle is rigid. There can be contributions\nto the mass from deformations of the pro\fle as shown\nfor ferromagnetic magnetic bubbles [36]. Secondly, we\nneglected the magnetic crystalline anisotropy by assum-\ning that the magnetostatic energy dominates it. We re-\nmark that rare-earth transition-metal ferromagnetic al-\nloys are known to have uniaxial crystalline anisotropy.\nThe anisotropy type can be easy axis or easy plane, de-\npending on, e.g., the deposition parameters controlling\nthe structure of the \flm [37]. Thirdly, the derived equa-\ntions of motion for the dynamics of a vortex are valid to\nlinear order in the vortex-core displacement. There can\narise nonlinear e\u000bects beyond our description when the\namplitude of oscillations is su\u000eciently large. 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In ReFe 2O4, all Re and half of the Fe\nions occupy the octahedral sites while the remaining Fe ions occupy the tetrahedral sites. All Re\nions are predicted to be at a +4 oxidation state with a low spin con\fguration ( S= 3=2), while all\nFe ions are predicted to be at a +2 oxidation state with a high spin state con\fguration ( S= 2).\nMagnetically, ReFe 2O4adopts an unconventional ferrimagnetic state in which the magnetic moment\nof Re opposes the magnetic moments of both tetrahedral and octahedral Fe ions. The spin-orbit\ncoupling is found to cause a slight spin canting of \u00181:5\u000e. The predicted magnetic ground state\nis unlike the magnetic alignment usually observed in ferrites, where the tetrahedral cations oppose\nthe spin of the octahedral cations. Given that the density of states analysis predicts a half-metallic\ncharacter driven by the presence of Re t2gstates at the Fermi level, this compound shows promise\ntowards potential spintronics applications.\nKeywords: Rhenium ferrite, unconventional magnetism, ferrimagnetism, spin canting, density functional\ntheory, spin-orbit coupling\nINTRODUCTION\nSearching for and investigating exotic magnetic phases\ndeepens our fundamental knowledge of complex func-\ntional materials and opens new horizons for novel ap-\nplications [1]. Ferrites are among the most studied mag-\nnetic materials, with broad applications in spintronics\n[2], magnetic data storage [3], magnetically recoverable\ncatalysts [4-7], and microwave guides [8]. The utility\nof ferrites stems from their high magnetic saturation,\nhigh Curie temperatures and controllable coercivity. Fer-\nrites, most commonly synthesised by reactions of Fe 2O3\nwith a smaller proportion of other metal oxides, encom-\npass a wide range of chemical compositions, stoichiome-\ntries, and crystal structures. However, they mostly crys-\ntallise into cubic spinel structures, distorted spinels with\nlower symmetry, [9] or hexagonal structures [10]. Fer-\nrite cations are situated between octahedral and tetra-\nhedral voids, created through the oxygens' closed packed\narrangement [11]. Alongside Fe, cation sites may be oc-\ncupied by Sr, Ba, Pb, or transition metals (TMs). Gen-\nerally, ferrites are ferrimagnetic where the spin of the\ntetrahedral cations opposes but does not cancel the spin\nof the octahedral cations [12]. The second cation's type\nand its abundance determine the magnetic hardness and\nsaturation of the resultant ferrite, enabling the design of\nmaterials with desirable magnetic phase transition tem-\nperatures and magnetic coercivity [13].\nTypically, in complex materials containing multiple\nmagnetic cations, the magnetic ground state is stabilisedthrough competing ferromagnetic and antiferromagnetic\nsuperexchange interactions, resulting in frustrated sys-\ntems with multiple magnetic phase transitions. In fer-\nrites with only fourth row 3d TM ions, these competi-\ntions simply stabilise the ferrimagnetic state where the\ncations on the octahedral site align antiparallel to the\ncations on the tetrahedral sites [12]. However, in other\nclasses of complex oxides, such as double perovskites,\nmagnetic interactions between 3d and heavier 4d and\n5d elements has demonstrated substantially more com-\nplex magnetic behaviour [14,15], often deviating from the\nrules of thumb established by Goodenough and Kanamori\n[16,17], where exotic magnetic behaviours are often the\nresult of the strong spin-orbit coupling, structural distor-\ntions and higher bond covalency between heavier TM ions\nand oxygen. A detailed review of Goodenough-Kanamori\nrules can be found in the literature [18,19].\nWith all these exciting developments in perovskite\nmagnetism, one wonders if there are any similar coun-\nterparts in ferrites. The idea of harnessing heavier 4d\nand 5d TM ion ferrites to control anisotropy and magne-\ntostriction in ferrite was proposed by Hansen and Krish-\nnan in 1977 [20]. However, since then, this idea has not\nattracted the attention it deserves. In the present work,\nwe demonstrate the stability of a rhenium-based ferrite\nReFe 2O4and discusses its unconventional ferrimagnetic\nground state through comprehensive density functional\ncalculations.arXiv:2202.08115v1  [cond-mat.mtrl-sci]  16 Feb 20222\nCOMPUTATIONAL SETTINGS\nSpin-polarised collinear and noncollinear density func-\ntional calculations were performed with VASP code\n[21,22], using the projector augmented wave method\n(PAW) [23] and the Perdew{Burke{Ernzerhof (PBE)\nexchange-correlation functional [24,25]. To improve the\nelectronic band description, adequate intra-atomic inter-\naction terms ( Ue\u000b), based on the Liechtenstein et al. ap-\nproach [26], were added to the Fe 3d electrons. The Uand\nJparameters were 3.5 eV and 0.5 eV, respectively, result-\ning in an e\u000bective U(Ue\u000b) of 3 eV. Comparable values\nwere reported to improve the band description accuracy\nof ferrites [27,28]. More speci\fcally a Ue\u000bvalue of 3 eV\nis necessary to adequately describe the Fe 3d electrons in\noxides [29]. Furthermore, our comprehensive test demon-\nstrated the adequacy of these values (Fig. S1). Accu-\nrate electronic localisation through the GGA+ Uformal-\nism is essential for obtaining reliable structures as the\natomic forces are sensitive to magnetic moments borne\non cations [30]. The energy cut-o\u000b was set at 650 eV.\nThe precision key for the rest of the parameters was set\nACCURATE . The noncollinear calculations were initi-\nated with the WAVECAR \fles calculated with the spin-\npolarised collinear method to facilitate convergence.\nTo simulate the ReFe 2O4structure, as shown in Fig.\n1, two Fe ions in the primitive magnetite cell, with the\nchemical formula Fe 6O8[31], were substituted with Re\nions. As shown in Fig. 1a{j, we considered all possible\nrhenium placement scenarios, and for each scenario, we\nexamined various spin alignments, searching for the most\nstable structure. Substitution at the octahedral sites con-\nsistently resulted in lower total energy, so we further in-\nvestigated all plausible spin alignments in ReFe 2O4with\nonly octahedral Re. A dense 7 \u00027\u00027 k-point mesh, gen-\nerated with the Monkhorst-Pack scheme of \u00180:015\u0017A\u00001\nspacing, consisting of 172 irreducible sampling points in\nthe Brillouin zone, was used for geometry optimisation.\nFor geometry optimisation, the internal coordinates and\nthe lattice parameters were relaxed to energy and force\nthresholds smaller than 10\u00006eV and 0.02 eV \u0017A\u00001, re-\nspectively. No symmetry restriction was applied in ge-\nometry optimisation to allow relaxation to lower symme-\ntry, should it be more stable. It is well-known that even\nthe simplest of the spinel ferrites, Fe 3O4, although cu-\nbic (Fd\u00163m) at room temperature, transforms to a lower\nsymmetry monoclinic structure ( P21=c) below\u0018125 K,\nthrough a Verwey phase transition [32].\nRESULTS AND DISCUSSION\nTwo cation types based on coordination exist in the\nspinel ferrite structure: one that is tetrahedrally coordi-\nnated and another that is octahedrally coordinated withoxygen ions. The \frst type represents one-third, while\nthe second represents two-thirds of the total cations in\nthe crystal. We examined three possible rhenium place-\nments to identify the most stable position of the Re ions\nin ReFe 2O4. First, both Re ions were placed at the tetra-\nhedral sites while all Fe ions were left at the octahedral\nsite, which is the typical cationic distribution in spinels.\nSecondly, one Re ion was placed at the tetrahedral site,\nand the other Re ion was placed at the octahedral site,\na con\fguration usually referred to as an intermediate\nspinel. Lastly, both Fe ions were placed at tetrahedral\nsites, while the octahedral sites were equally occupied\nwith Re and Fe, which is referred to as an inverse spinel.\nWe investigated all these cationic distributions by cal-\nculating the total energy for two possible ferrimagnetic\nand ferromagnetic spin alignments. In the ferromagnetic\nstructure, all cations' spin was set parallel, while in the\nferrimagnetic structure, the tetrahedral cations' spin was\nset antiparallel to the spin of the octahedral cations. As\nshown in Fig. 1a and b, when Re ions are located at\ntetrahedral sites, the total energy is relatively high re-\ngardless of the direction of the Re spin. However, the\nferrimagnetic state is still more stable than the ferro-\nmagnetic one. For intermediate rhenium occupancy, as\nin Fig. 1c, d, the total energies for both spin alignments\nwere slightly higher than the previous case of complete\ntetrahedral Re occupation.\nWhen Re ions are located at octahedral sites (Fig. 1e{\nj), the compound's total energy generally decreased sig-\nni\fcantly, indicating greater stability. For instance, the\naforementioned ferrimagnetic spin alignment of Fig. 1e\nwas more stable than the counterparts with tetrahedral\nRe (Fig. 1a) and the mixed Re (Fig. 1c) con\fgurations\nby approximately 2 eV per unitcell. The stability of the\noctahedral Re warranted further investigations of other\npossible spin alignments of ReFe 2O4with octahedrally\ncoordinated rhenium. Accordingly, we calculated all dif-\nferent possible ferrimagnetic spin alignments for this fer-\nrite compound with octahedral Re [see con\fgurations (g)\nand (h)]. In particular, con\fguration (h) was the most\nstable among all. In this ferrimagnetic con\fguration,\nthe Re ions' spin direction is antiparallel to the spins of\nboth octahedral Fe and tetrahedral Fe ions. In this case,\nthe spins of the octahedral Fe and tetrahedral Fe ions\nwere parallel. Con\fguration (e), with similar spin align-\nment to a conventional ferrimagnetic inverse spinel such\nas magnetite, was higher in total energy relative to con-\n\fguration (h) by 0.4446 eV/u.c. (u.c. is unitcell). Con-\n\fguration (g), representing the other possible realisation\nof the ferrimagnetic alignment, also had higher total en-\nergy than con\fguration (h) by 2.1606 eV/u.c. Likewise,\nthe ferromagnetic state in con\fguration (f) had higher\ntotal energy of 1.1251 eV/u.c.\nTo unambiguously con\frm the stability of con\fgura-\ntion (h), we further calculated the total energy of con-\n\fguration (i), which is quite similar to con\fguration (h)3\nexcept that the spins of the two Re ions were set an-\ntiparallel to examine the strength of the magnetic cou-\npling among Re ions. We also examined con\fguration\n(j), which is antiferromagnetic; that is, every cation has\nan antiparallel spin to the one adjacent. Both con\fgu-\nrations (i) and (j) had total energies higher than that of\ncon\fguration (h) by more than 1 eV/u.c. The stability\nof con\fguration (h) relative to con\fgurations (i) and (j)\ndemonstrates that pairs of Re Oct, Fe Octand Fe Tetions\nhave a strong tendency towards parallel spin alignment\namong themselves.\nFor the most stable placement of Re which can be\nexpressed as Fe Tet(ReFe) OctO4, con\fguration (h), which\nrepresents the magnetic ground state, is remarkably sta-\nble as \ripping to even the second most stable spin con-\n\fguration (e) has an energy cost of 0.4446 eV/u.c. This\nlevel of stability is likely to correspond to an ambient\nCurie temperature in ReFe 2O4. This prediction is based\non a comparison with CoFe 2O4's stability margin. A\nspin-\rip from ferrimagnetic to ferromagnetic order in\nCoFe 2O4costs 0.536 eV/u.c. [33], resulting in a Curie\ntemperature ( TC) of 793 K in bulk cobalt ferrite [34].\nFurthermore, the mean-\feld approximation can estimate\nthe Curie temperature based on the magnetic exchange\nbetween Fe and Re sublattice systems as follows [35]:\n3\n2kBTC=X\ni6=jJij; (1)\nin whichkBis the Boltzmann constant and Jijis the\npair exchange coupling parameter between sites iand\nj. Assuming the nearest neighbour interaction between\nRe and Fe to be the most signi\fcant, Equation 1 yields\nTC= 734:9 K, which is close to the comparison made\nearlier.\nGiven that Re is a relatively heavier sixth-row tran-\nsition element, we anticipate that the role of spin-orbit\ncoupling (SOC) is potentially signi\fcant in ReFe 2O4[36].\nTo examine the signi\fcance of SOC, we re-optimised the\nmost stable con\fguration (h) with spin-orbit coupling\ntaken into consideration. The structural relaxation was\nminor as none of the internal coordinates and the lattice\nparameters changed by more than \u00181%. However, the\ntotal energy was lowered to \u0000107:7975 eV/u.c., indicat-\ning that SOC accounts for \u00180:67% of the total energy.\nSimilarly, SOC is not anticipated to change the magnetic\nalignment in ReFe 2O4as other competing con\fgurations\nare also estimated to have their total energy lowered by\napproximately the same amount when SOC is considered\n(Table S1). SOC brings about magnetic noncollinearity\nby coupling the spin to the orbital degrees of freedom, of\nwhich the latter depends on the lattice environment. The\nnoncollinear magnetic alignment of ReFe 2O4is shown in\nFig. 2a. More precisely, the spins of the Re ions formed a\ntight angle of 1 :329\u000ebetween each other, i.e., nearly par-\nallel; and the net spin of the Re ions formed an angle of177:593\u000ewith the net spin of the Fe ions, i.e., nearly an-\ntiparallel. It can be seen that the degree of noncollinear-\nity is relatively small as the spin directions are to a great\nextent similar to that of the collinear alignment of con-\n\fguration (h). The net magnetisation of the whole com-\npound was calculated to be 6.090 \u0016B/f.u. (f.u. stands for\nformula unit), which is approximately 1.5 times larger\nthan that of the saturation magnetic moment of mag-\nnetite [37]. A more qualitative description of the e\u000bect\nof theUe\u000bchoice on the total magnetisation and the local\nmagnetisation of the Re and Fe ions in this con\fguration\nis given in Table S2.\nThe density of states (DOS) in ReFe 2O4, calculated\nconsidering SOC, are presented in Fig. 3. Here, DOS is\nprojected onto the axes of an orthogonal frame of which\nthezaxis is parallel to the cpdirection of the ReFe 2O4\nprimitive cell. First, we notice that the DOS magnitude\nalong thexandyaxes is\u00183% of the DOS magnitude\nalong thezaxis, indicating a slight deviation from linear-\nity and corroborating the magnetic moment orientations\nobtained in Fig. 2a. Along the zaxis, (Fig. 3c), we can\nsee that both Fe Tetand Fe Octare in high-spin and +2\noxidisation states as one spin channel, comprising of \fve\nelectrons per Fe, for both Fe Tetand Fe Oct, is fully occu-\npied while the other spin channel is only partly occupied.\nTherefore, the electronic con\fguration for Fe2+\nTetise2\"t3\n2\n\"e1#, while for Fe2+\nOct, the electronic con\fguration is t3\n2g\n\"e2\ng\"t1\n2g#. Moreover, Re is in low-spin state with +4\noxidation. According to its partial DOS, Re has a fully\noccupiedt2g\"states, which are immediately followed by\nits emptyt2g#states. The Fermi level crosses the tail\noft2g\"states, giving rise to half-metallic conduction, as\nno states are available at the Fermi level with opposing\nnet spin direction. Moreover, because of the larger crys-\ntal \feld acting on Re's 5d states, Re's empty egstates\nare located at\u00184 eV above the Fermi level (Fig. S2).\nThe ReFe 2O4half-metallicity can be utilised for magneto-\nresistive response [38,39] or near-perfect spin-polarised\ncurrent injection [40,41].\nAs shown in Fig. 3, the Fe 3d states are spread through\nthe conduction band, hybridising extensively with O,\nwhile Re 5d states are mainly concentrated within 2\neV below the Fermi level. Nonetheless, in the region of\n\u00002< E\u0000EFermi<0, the spin-down Re and Fe states\nand O 2p states hybridise together, facilitating the mag-\nnetic superexchange interactions that stabilise the mag-\nnetic ground state. Furthermore, the net magnetic mo-\nments borne on all cations, as shown in Fig. 2, are smaller\nthan ideal ions. For high spin Fe2+(3d6), either in tetra-\nhedral or octahedral coordination, the magnetic moment\nshould have been 4 \u0016B. For octahedral Re4+(5d3), the\nmagnetic moment should have been 3 \u0016B. However, since\nFe{O or Re{O bonds are not purely ionic but possess a\ndegree of covalency, the magnetisation of transition metal\nions is expected to be lower than the purely ionic values\n[42]. The reduction in magnetisation is more profound4\n(b)\nEt = –103.7775 eV (FM)\n(f)\nEt = –105.9444 eV (FM)\n(j)\nEt = –104.2201 eV (AFM)ap\nbpcp\nEt = –105.9039 eV\n(i)\nFe\nRe\nO\n(e)\nEt = –106.6249 eV (FiM-a) Et = –107.0695 eV (FiM-c)\n(h)\nOctTet\n(g)\nEt = –104.9089 eV (FiM-b)\nOctTetOctTet(d)\nEt = –103.5364 eV (FM)\nOctTet(c)\nEt = –103.7854 eV (FiM) Et = –104.6162 eV (FiM)\n(a)\nTet\nOct\nAll starting at\nap = bp = cp = 5.961 Å\nαp = βp = γp =  60º\n(             )Fd¯3m\nFig. 1. The spin con\fgurations used for determining the magnetic ground state of ReFe 2O4. In con\fgurations a\nandb, Re ions are at the tetrahedral sites. In candd, one Re ion is located at the tetrahedral site while the other\nis located at the octahedral site. In e,f,g,h,i, and j, both Re ions are located at the octahedral sites. The density\nfunctional total energy ( Et) of each con\fguration is also shown. FM, FiM, and AFM refer to ferromagnetic,\nferrimagnetic and antiferromagnetic, respectively.\n∠ReOct–O–ReOct: 86.626º, 86.634º∠FeOct–O–FeOct: 93.52º, 97.13º∠FeOct–O–ReOct: 91.29º, 91.32º, 98.81º, 98.84º∠ReOct–O–FeTet: 123.55º, 124.98º, 128.61º, 134.50º∠FeOct–O–FeTet: 111.94º, 112.10º, 114.02º, 114.05º, 127.39º, 127.42º\nccacbc(b)ReOct4+\nFeTet2+\nap = 6.243, bp = 6.198, cp = 6.243 Åαp = 60.24º, βp = 54.43º, γp = 60.24º\nac = 5.710 , bc = 6.198, cc = 6.243 Åαc = 119.76º, βc = 117.21º, γc = 90.00ºFeOct2+\nReOct4+FeOct2+FeTet2+\nap\nbpcp3.549 μB3.549 μB\n3.614 μB3.614 μB1.280 μB\n1.092 μB(a)\nFeTet2+\nFeOct2+\nFe\nRe\nO\nFig. 2. aThe optimised primitive cell of ReFe 2O4with SOC taken into account, along with the magnetic moments\nborne on all cations. bThe conventional representation of the optimised ReFe 2O4, which has a triclinic symmetry.\nInb, all possible bond angles between cations are also presented. The corresponding yellow marks show a\nrepresentative of each given angle. The lattice parameters shown are either indexed with porc, indicating the\nprimitive and conventional cell dimensions, respectively.5\nin heavier transition metal ions are their bonds are more\ncovalent [33]. A more quantitative description of the elec-\ntronic localisation function is provided in Fig. S3.\nAs shown in Fig. 2a, the primitive cell of ReFe 2O4\nis substantially transformed by geometry optimisation.\nThe lattice parameters of the optimised primitive cell do\nnot conform to the high symmetry of the initial structure\nas its lattice parameters asymmetrically changed and its\nvolume expanded. The initial structure volume, which\nwas based on magnetite, was 155.225 \u0017A3, while the opti-\nmised structure had a volume of 163.025 \u0017A3. The ionic\nradius of Fe can explain the expansion of the volume\nupon Re substitution at the tetrahedral site. The radius\nof high-spin Fe2+\nTetin ReFe 2O4is 0.63 \u0017A. In magnetite,\nthe tetrahedral site is occupied by Fe3+with a smaller\nradius of 0.49 \u0017A. The Re4+radius in octahedral coordi-\nnation is 0.63 \u0017A which is quite close to the replaced Fe3+\nradius (0.65 \u0017A) and is not expected to be a substantial\ndrive in the structural transformation.\nWe examined the optimised ReFe 2O4primitive cell's\nsymmetry to investigate the magnetic exchange among\nall cations. A triclinic symmetry was detected through\nthe FINDSYM symmetry detection algorithm [43] with a\ntight tolerance of 0.00001 \u0017A for lattice parameters (CIF\nprovided in the supplementary information). The con-\nventional cell with triclinic symmetry is shown in Fig. 2b.\nDetecting all possible magnetic exchanges that stabilise\nthe predicted ground state magnetism|con\fguration\n(h) re-optimised with SOC considered{is easier in the\nsymmetry-imposed structure. The TM{O{TM bond an-\ngles for this structure are all listed in Fig. 2. The TM{\nO{TM bond angles between cations on tetrahedral and\noctahedral sites are obtuse and thus dominate the mag-\nnetic exchange interactions, as the superexchange inter-\naction magnitude is proportional to cos2(6TM{O{TM).\nThe higher total energy of con\fguration (g) indicates the\nsuperexchange between Re4+\nOctand Fe2+\nTetis antiferromag-\nnetic, while the higher total energy of con\fguration (e)\nindicates that the superexchange between Fe2+\nOctand Fe2+\nTet\nis ferromagnetic. The TM{O{TM bonds among tetrahe-\ndral sites are all nearly right angles, indicating a minimal\norbital overlap favouring weaker ferromagnetic superex-\nchange. The strength of this ferromagnetic exchange can\nbe estimated from the total energy of con\fgurations (g)\nand (h) of Fig. 1, showing that setting adjacent cations\nto antiferromagnetic coupling raises the total energy.\nFinally, we examine the stability of ReFe 2O4in com-\npeting metallic and oxide phases. The compound's for-\nmation enthalpy (\u0001 Hmetallic ) was calculated relative to\nthe metallic Re (hexagonal paramagnetic) and Fe (body-\ncentred ferromagnetic) phases, and gaseous O 2was cal-\nculated as\n\u0001Hmetallic =Et(ReFe 2O4)\u0000Et(Re)\u00002Et(Fe)\u00002Et(O2):\n(2)\nHere,Etis the density functional total energy. \u0001 Hmetallic\n-0.40.00.4 \nO 2p \nFeTet 3d \nFeOct 3d \nReOct 4dE\n − EFermi (eV)(a) ReFe2O4 (x)-\n0.30.00.3(\nc) ReFe2O4 (z)(b) ReFe2O4 (y)-\n8- 6- 4- 20 2 -303DOS (eV−1)D OS (eV−1)t\n2gt\n2gt2g & egt\n2ge  & t2et2g & ege & t2DOS (eV−1)Fig. 3. Partial density of states of ReFe 2O4at its most\nstable magnetic state [con\fguration ( h) of Fig. 1],\ncalculated with spin-orbital coupling considered. The\ndensity of states is projected along an orthogonal frame\nhaving the x,y, andzaxes. Thezaxis of this frame\ncoincides along the lattice parameter cpof the primitive\nlattice parameter, shown in the lower row of Fig. 1.\nwas found to be\u00005:1924 eV/f.u. The formation enthalpy\nrelative to the competing oxide phase ( \u0001 Hoxide) was\ncalculated as\n\u0001Hoxide =Et(ReFe 2O4)\u0000Et(ReO 2){2Et(FeO):(3)\nHere, ReO 2was the most stable Re4+oxide in orthorhom-\nbic structure (materials project identi\fer mp-7228 [44]),\nand FeO was the most stable Fe2+oxide in monoclinic\nstructure (materials project identi\fer mp-1279742 [44]).\n\u0001Hoxide was found to be\u00001:1903 eV/f.u. Given the neg-\native \u0001Hvalues, we can conclude that ReFe 2O4is stable\nagainst decomposition to oxides of its constituent ele-\nments and Re4+and Fe2+. For the future synthesis of\nReFe 2O4, one can draw inspiration from the recently de-\nveloped green fabrication methods for ferrites [45].\nCONCLUSIONS\nUsing density functional calculations, considering spin-\norbit coupling, we predict that a Re-based ferrite\nReFe 2O4is stable in a distorted spinel structure with\nreduced triclinic symmetry ( P\u00161), and adopts an uncon-\nventional magnetic ordering in where Re spin opposes\nthe spin of both Fe Tetand Fe Oct, while Fe Tetand Fe Oct6\nhave parallel spin alignment among themselves. The net\nmagnetic moment of this compound is evaluated at 6.090\n\u0016B=f:u:which is about 1.5 times greater than that of\nmagnetite. The magnetic ground state is remarkably sta-\nble as \ripping any spin incurs an energetic cost of at least\n0.2223 eV/f.u., which is likely to correspond to an am-\nbient Curie temperature. The compound is predicted to\nbe half-metallic, which implies that this compound may\nbe useful towards applications in spintronics where the\nspin polarisation of conduction electrons is desired.\nCONFLICTS OF INTEREST\nThe authors declare that there is no con\rict of interest.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge the funding of this\nproject by computing time provided by the Paderborn\nCenter for Parallel Computing (PC2).\nVERSION OF RECORD\nM. Hussein N. Assadi, Marco\nFronzi and Dorian A. H. Hanaor\nUnusual ferrimagnetic ground state in rhenium ferrite\nEur. Phys. J. Plus (2022) 137, 21. https:\n//doi.org/10.1140/epjp/s13360-021-02277-z\nREFERENCES\n[1] A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Pre-\njbeanu, B. Dieny, P. Pirro, and B. Hillebrands, J. Magn.\nMagn. Mater. 509, 166711 (2020). https://doi.org/\n10.1016/j.jmmm.2020.166711\n[2] R. K. Kotnala and J. Shah, in Handbook of Mag-\nnetic Materials, edited by K. H. J. 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Li, Nanoscale\nHoriz. 4 (2), 434 (2019). http://dx.doi.org/10.1039/\nC8NH00278A"    },    {        "title": "2112.07233v1.Ground_State__Magnetization_Process_and_Bipartite_Quantum_Entanglement_of_a_Spin_1_2_Ising_Heisenberg_Model_on_Planar_Lattices_of_Interconnected_Trigonal_Bipyramids.pdf",        "content": "arXiv:2112.07233v1  [cond-mat.stat-mech]  14 Dec 2021Ground State, Magnetization Process and Bipartite Quantum Entanglement\nof a Spin-1/2 Ising–Heisenberg Model on Planar Lattices of Interconnec ted\nTrigonal Bipyramids\nLucia G´ alisov´ aa,∗, Michał Kaczorb,c\naInstitute of Manufacturing Management, Faculty of Manufac turing Technologies with the Seat in Preˇ sov, Technical Uni versity of Koˇ sice,\nBayerova 1, 080 01 Preˇ sov, Slovakia\nbThe Doctoral School of University of Rzesz´ ow, University o f Rzesz´ ow, Rejtana 16C, 35-935 Rzesz´ ow, Poland\ncInsitute of Physics, College of Natural Sciences, Universi ty of Rzesz´ ow, Rejtana 16A, 35-935 Rzesz´ ow, Poland\nAbstract\nThe ground state, magnetization scenario and the local bipa rtite quantum entanglement of a mixed spin-1 /2 Ising–\nHeisenberg model in a magnetic field on planar lattices forme d by identical corner-sharing bipyramidal plaquettes is\nexamined by combining the exact analytical concept of gener alized decoration-iteration mapping transformations wit h\nMonte Carlo simulations utilizing the Metropolis algorith m. The ground-state phase diagram of the model involves\nsix different phases, namely, the standard ferrimagnetic phase, fu lly saturated phase, two unique quantum ferrimag-\nnetic phases, and two macroscopically degenerate quantum f errimagnetic phases with two chiral degrees of freedom\nof the Heisenberg triangular clusters. The diversity of gro und-state spin arrangement is manifested themselves in\nseven different magnetization scenarios with one, two or three fracti onal plateaus whose values are determined by the\nnumber of corner-sharing plaquettes. The low-temperature values of the concurrence demonstrate that the bipartite\nquantum entanglement of the Heisenberg spins in quantum fer rimagnetic phases is field independent, but twice as\nstrong if the Heisenberg spin arrangement is unique as it is t wo-fold degenerate.\nKeywords: Ising–Heisenberg model, chiral degrees of freedom, magnet ization process, bipartite quantum\nentanglement, rigorous results\n1. Introduction\nQuantum entanglement has been attracting a lot of attention in the last few years mainly due to its crucial role\nin the development of quantum computers, superdense coding , quantum communication, quantum teleportation, as\nwell as quantum information theory [1–3]. The application p otential of this unique phenomenon also exceeds into the\nquantum biology [4, 5] and quantum metrology [6, 7].\nIn quantum theory, quantum entanglement provides a novel pl atform for exploring long-range quantum correla-\ntions, quantum phase transitions as well as exotic properti es of many-body systems [8–11]. The low-dimensional\nHeisenberg spin models, involving quantum fluctuations bet ween spins, play a significant role in this regard because\nthey have been proven to be ideal candidates for a rigorous in vestigation of the entangled states under the influ-\nence of the external stimuli such as magnetic field (homogene ous or inhomogeneous) and /or temperature [12–24].\nMoreover, many analytical and numerical calculations have been performed to examine the tuning of the quantum\nand thermal bipartite entanglement by varying the exchange anisotropy parameter [19–28], the uniaxial single-ion\nanisotropy [16, 17], the Dzyaloshinskii–Moriya interacti on (spin-orbit coupling) [18–20, 26, 27], the next-nearest -\nneighbour interaction [13, 14, 29], as well as by introducin g impurities into the system [28, 30].\nHowever, the rigorous investigation of the bipartite entan glement in the pure Heisenberg models represents a\ncomplex task, which is considerably limited due to a non-com mutability of spin operators in the Hamiltonian. This\n∗Corresponding author\nEmail address: lucia.galisova@tuke.sk (Lucia G´ alisov´ a )\nPreprint submitted to Entropy December 15, 2021computational problem makes the rigorous study of the pheno menon in general inaccessible across whole parame-\nter space of the systems. On the other hand, replacing some of the Heisenberg spins with three spin components\nby the Ising ones with only one ( z-) component at the nodal lattice sites is the alternative wa y to exactly exam-\nine the entanglement in various simpler mixed-spin Ising–H eisenberg models by using the standard transfer-matrix\nmethod [31] and/or the concept of generalized mapping transformations [32– 35]. Taking into account the fact that the\nfinite Heisenberg clusters formed by three-component Heise nberg spins are indirectly coupled with each other through\nthe intermediate one-component Ising spin(s), one finds tha t the eigenstates of two adjacent Heisenberg clusters are\nseparable. Thus, any quantity measuring the local bipartit e entanglement in the considered mixed-spin model can be\nrigorously calculated for each quantum Heisenberg cluster separately.\nTo date, the bipartite entanglement has been rigorously exa mined in several one- (1D) and two-dimensional (2D)\nmixed-spin Ising–Heisenberg models formed by the identica l Heisenberg dimers or triangular clusters which interact\nwith each other via the intermediate nodal Ising spin(s) [36 –45]. The investigations brought a deeper insight into the\nthermal and magnetic-field-driven changes of the phenomeno n [36–43], the impact of the model’s parameters on the\nphenomenon [36, 39–45], as well as the evolution of the pheno menon near and above second-order (continuous) phase\ntransitions [44, 45] without any artefacts arising from app roximations. Despite their simplicity and the general opin ion\nthat the simpler mixed-spin Ising–Heisenberg systems invo lving isolated local quantum correlations are artificial\nmodels, some of the results were in a very good correspondenc e with ones obtained for more complex Heisenberg\ncounterparts [40] and also with experiments [38, 46, 47].\nIn the present paper, we will rigorously solve a spin-1 /2 Ising–Heisenberg model in a longitudinal magnetic field\non 2D lattices formed by identical corner-sharing trigonal bipyramidal plaquettes. Our recent studies [45, 48] of the\nmodel without magnetic field on the particular lattice with f our inter-connected bipyramidal units have shown that this\nquantum mixed-spin model represents a suitable playground for a rigorous study of various unconventional physical\nphenomena such as the macroscopic degeneracy of the spontan eous long-range order caused by chiral spin degrees\nof freedom, the spin frustration, and the bipartite entangl ement. The aforementioned findings motivated us to extend\nthe investigation of the model also to the e ffect of the longitudinal magnetic field. The goals of the prese nt paper are\nto shed a light on the nature of ground states invoked by the ap plied field, to identify the actual fractional plateaus in\nthe zero-temperature magnetization process, to find out a ge neral formula describing how the values of these plateaus\ndepend on the current number of interconnected bipyramidal plaquettes and, finally, to quantify the bipartite quantum\nentanglement between the Heisenberg spins in individual gr ound states.\nIn addition to the academic interest, our investigation of t he spin-1/2 Ising-Heisenberg model on 2D lattices\nformed by interconnected trigonal bipyramids is motivated by the existence of a class of geometrically frustrated\nstructures, namely cobaltates YBaCo 4O7(Y denotes a rare-earth ion) [49] and anion-radical salts (M DABCO+)(C•−\n60)\n(MDABCO+represents N-methyldiazabicyclooctanium cation, C•−\n60is a radical anion) [50], in which one can clearly\nidentify corner-sharing trigonal bipyramidal clusters. A lthough the mentioned compounds do not represent a precise\nexperimental realization of the magnetic structure propos ed in the present paper, we hope that a targeted design of the\nmagnetic material with a magnetic structure of interconnec ted trigonal bipyramids is feasible. The targeted chemical\nsynthesis involving highly anisotropic spin carriers such Dy3+or Co2+magnetic ions and anion-radical salts could\npossibly afford desiring such a quantum mixed-spin system. The findings p resented in this paper could serve as a\nmotivation for chemists to achieve this goal.\nThe outline of the paper is as follows: in Section 2, a magneti c structure of the investigated model is described and\nthe most important steps of its rigorous treatment combinin g the analytical and numerical approaches are clarified. In\nSection 3, we present the most interesting numerical result s for the ground state and the magnetization process of the\nmodel. The section also includes an analysis of the bipartit e quantum entanglement in the individual ground states.\nFinally, the summary of the most important findings are prese nted in Section 4.\n2. Model and Its Rigorous Treatment\nWe consider a mixed spin-1 /2 Ising–Heisenberg model in a longitudinal magnetic field on 2D lattices consisting\nof identical corner-sharing trigonal bipyramidal plaquet tes, as is schematically depicted in Figure 1 for one particu lar\nlattice with four such plaquettes. In this figure, the common vertices of plaquettes (white circles) are occupied by\nthe Ising spinsσ=1/2 that interact with other spins solely through their z-components. The rest ones (red circles),\nforming internal equilateral triangles oriented perpendi cularly to the plaquette axes, are occupied by the Heisenber g\n2spins S=1/2 that are coupled to each other via x-,y-, and z-components. Assuming qbipyramidal plaquettes share\na common vertex, the total Hamiltonian of the mixed spin-1 /2 Ising–Heisenberg model can be written as a sum of\nplaquette (five-spin cluster) Hamiltonians ˆH=/summationtextNq/2\nj=1ˆHj, where Nlabels the total number of the nodal lattice sites\noccupied by the Ising spins (we consider the thermodynamic l imit N→∞ ). Each plaquette Hamiltonian ˆHjcontains\nall exchange interactions realized within the jth Ising–Heisenberg trigonal bipyramid and Zeeman terms th at describe\nthe influence of the applied external magnetic field on magnet ic moments of the individual spins:\nˆHj=−JH3/summationdisplay\nk=1/bracketleftig\n∆(ˆSx\nj,kˆSx\nj,k+1+ˆSy\nj,kˆSy\nj,k+1)+ˆSz\nj,kˆSz\nj,k+1/bracketrightig\n−JI3/summationdisplay\nk=1ˆSz\nj,k( ˆσz\nj+ˆσz\nj+1)\n−HH3/summationdisplay\nk=1ˆSz\nj,k−HI\nq( ˆσz\nj+ˆσz\nj+1).(1)\nIn the above, ˆSα\nj,k(α=x,y,z) and ˆσz\njare spatial components of the spin-1 /2 operator of the Heisenberg spin from the\njth triangle and z-component of the Pauli operator with the eigenvalues ±1/2 at the jth nodal lattice site, respectively,\nwhich satisfy the periodic boundary conditions ˆSα\nj,4≡ˆSα\nj,1and ˆσz\nNq/2+1≡ˆσz\n1. The parameter JHmarks the XXZ\nHeisenberg interaction within the Heisenberg triangles, ∆is the exchange anisotropy parameter in this interaction,\nandJIlabels the Ising-type interaction between the nearest-nei ghbouring Ising and Heisenberg spins. The last two\nterms HHandHIin the second line of Equation (1) are Zeeman terms, which acc ount for the magnetostatic energy of\nthe Heisenberg and Ising spins in an applied longitudinal ma gnetic field, respectively.\nAs we have shown in our recent work on the zero-field case of the model [45], it is convenient for further calcula-\ntions to introduce the composite spin operators:\nˆtj=3/summationdisplay\nk=1ˆSj,k,ˆtα\nj=3/summationdisplay\nk=1ˆSα\nj,k(α=x,y,z), (2)\nwhich determine the total spin of the Heisenberg triangular clusters and its spatial components, respectively. From\nthe definition of the latter operators, one can easily obtain the spin identity ( ˆtα\nj)2=3/4+2/summationtext3\nk=1ˆSα\nj,kˆSα\nj,k+1. This, in\ncombination with the identity for the square of the total com posite spin operator ( ˆtj)2=ˆtj·ˆtj=(ˆtx\nj)2+(ˆty\nj)2+(ˆtz\nj)2,\nallows one to find the following two relations for the Heisenb erg spin operators from the same triangular cluster:\n3/summationdisplay\nk=1/parenleftigˆSx\nj,kˆSx\nj,k+1+ˆSy\nj,kˆSy\nj,k+1/parenrightig\n=1\n2/bracketleftig\n(ˆtj)2−(ˆtz\nj)2/bracketrightig\n−3\n4,3/summationdisplay\nk=1ˆSz\nj,kˆSz\nj,k+1=1\n2(ˆtz\nj)2−3\n8. (3)\nBearing in mind the above relations and the definition of the z-component of the composite spin operator ˆtz\njlisted\nFigure 1: A schematic representation of the j-th trigonal bipyramidal plaquette and the mixed spin-1 /2 Ising–Heisenberg model on the particular\n2D lattice with four ( q=4) corner-sharing bipyramidal plaquettes. White circles l abel lattice sites occupied by the Ising spin σ=1/2 and red\ncircles denote lattice sites occupied by the Heisenberg spi nS=1/2. Black dashed lines illustrate the Ising-type interactio nJIbetween the Ising\nand Heisenberg spins and red solid lines indicate XXZ Heisen berg exchange interaction JH(∆) between the Heisenberg spins in the plaquette.\n3in Equation (2), the plaquette Hamiltonian (1) can be expres sed in the alternative form:\nˆHj=3JH\n8(2∆+1)−JH∆\n2(ˆtj)2+JH\n2(∆−1)(ˆtz\nj)2−JIˆtz\nj( ˆσz\nj+ˆσz\nj+1)\n−HHˆtz\nj−HI\nq( ˆσz\nj+ˆσz\nj+1).(4)\nIt is easy to prove that the operators ( ˆtj)2,ˆtz\njappearing in Equation (4) satisfy the commutation relation s/bracketleftbigˆHj,(ˆtj)2/bracketrightbig=0\nand/bracketleftbigˆHj,ˆtz\nj/bracketrightbig=0, which implies that they are both conserved quantities wit h well defined quantum spin numbers\ntj(tj+1) and tz\nj={−tj,−tj+1,..., tj}fortj={3/2,1/2}, respectively. In this regard, Equation (4) represents a\nfully diagonal form of the plaquette Hamiltonian (1), which implies that the corresponding energy eigenvalues can be\nexpressed in terms of the respective quantum spin numbers:\nEtj,tz\nj=3JH\n8(2∆+1)−JH∆\n2tj(tj+1)+JH\n2(∆−1)(tz\nj)2−JItz\nj(σz\nj+σz\nj+1)\n−HHtz\nj−HI\nq(σz\nj+σz\nj+1).(5)\nAt this calculation stage, the partition function of the con sidered spin-1/2 Ising–Heisenberg model can be partially\nfactorized due to commuting character of di fferent plaquette Hamiltonians and written in terms of the eig envalues (5)\nof the plaquette Hamiltonian:\nZ=Tr exp/parenleftig\n−βˆH/parenrightig\n=/summationdisplay\n{σn}Nq/2/productdisplay\nj=1Trjexp/parenleftig\n−βˆHj/parenrightig\n=/summationdisplay\n{σn}Nq/2/productdisplay\nj=1/summationdisplay\ntj,tz\njgtjexp/parenleftbig−βEtj,tz\nj/parenrightbig. (6)\nHere,β=1/(kBT) (kBis the Boltzmann’s constant and Tis the absolute temperature of the system), and the summatio n\nsymbol/summationtext\n{σn}denotes a summation over all possible spin configurations of the Ising spins, the product symbol/producttextNq/2\nj=1\nruns over all trigonal bipyramids, and the double summation symbol/summationtext\ntj,tz\njruns over all possible values of the quantum\nnumbers tj,tz\njof the composite spins. Finally, gtjis the degeneracy factor, which takes the value 1 for the quan tum\nnumber tj=3/2 and 2 for the quantum number tj=1/2. After performing double summations over tjandtz\nj,\none gains the effective Boltzmann’s weight w(σz\nj,σz\nj+1), which depends only on the Ising spin states σz\nj,σz\nj+1, and,\nthus, it can be replaced by a simpler but equivalent expressi on using the generalized decoration-iteration mapping\ntransformation [32–35]:\nw(σz\nj,σz\nj+1)=/summationdisplay\ntj,tz\njgtjexp/parenleftbig−βEtj,tz\nj/parenrightbig=2 exp/bracketleftiggβHI\nq(σj+σj+1)−βJH\n4/bracketrightigg\n×/braceleftigg/bracketleftigg\nexp(βJH∆)+2 exp/parenleftigg\n−βJH∆\n2/parenrightigg/bracketrightigg\ncosh/bracketleftbiggβJI\n2(σz\nj+σz\nj+1)+βHH\n2/bracketrightbigg\n+exp(βJH)cosh/bracketleftigg3βJI\n2(σz\nj+σz\nj+1)+3βHH\n2/bracketrightigg/bracerightigg\n=Aexp/bracketleftigg\nβJeffσz\njσz\nj+1+βHeff\nq(σz\nj+σz\nj+1)/bracketrightigg\n.(7)\nThe novel effective parameters A,Jeff, and Heffemerging on the right-hand side of Equation (7) are determin ed by\n’self-consistency’ of the algebraic approach used:\nA=4/radicalig\nw+w−w2\n0,Jeff=kBTlnw+w−\nw2\n0,Heff=kBTq\n2ln/parenleftiggw+\nw−/parenrightigg\n. (8)\nHere, w±=w(±1/2,±1/2) and w0=w(±1/2,∓1/2). After substituting Equation (7) into Equation (6), one o btains the\nrigorous equivalence between the partition function Zof the spin-1/2 Ising–Heisenberg model given by the Hamilto-\nnian (1) and the partition function ZIMof the effective spin-1/2 Ising model on the corresponding q-coordinated 2D\n4lattice given by the Hamiltonian HIM=−Jeff/summationtextNq/2\n/an}bracketle{tj,n/an}bracketri}htσz\njσz\nn−Heff/summationtextN\nj=1σz\nj:\nZ(T,JH,JI,∆,HH,HI)=ANq/2ZIM(T,Jeff,Heff). (9)\nThe mapping relation (9) represents the crucial result of th e rigorous solution of the considered 2D spin-1 /2 Ising–\nHeisenberg model in an external magnetic field because of all important physical quantities clarifying a ground-state\narrangement, magnetization process, and quantum bipartit e entanglement between the Heisenberg spins, namely,\nthe local magnetization mI=/an}bracketle{tˆσz\nj/an}bracketri}htandmH=/an}bracketle{t/summationtext3\nk=1ˆSz\nj,k/an}bracketri}htper nodal Ising spin and Heisenberg triangular cluster,\nrespectively, the total magnetization mper bipyramidal plaquette, as well as the pair correlation f unctions Cxx(yy)\nHH=\n/an}bracketle{tˆSx\nj,kˆSx\nj,k+1/an}bracketri}ht=/an}bracketle{tˆSy\nj,kˆSy\nj,k+1/an}bracketri}ht,Czz\nHH=/an}bracketle{tˆSz\nj,kˆSz\nj,k+1/an}bracketri}htandCzz\nIH=/an}bracketle{tˆσz\njˆSz\nj,k/an}bracketri}ht=/an}bracketle{tˆσz\nj+1ˆSz\nj,k/an}bracketri}ht, can be directly derived from the\nformula for the Gibbs free energy G=−kBTlnZby means of the differential calculus:\nmI=−1\n2N∂G\n∂HI,mH=−2\nNq∂G\n∂HH,m=qmH+2mI\nq, (10a)\nCzz\nHH=2\n3NqJ H/parenleftigg\n∆∂G\n∂∆−∂G\n∂JH/parenrightigg\n,Cxx(yy)\nHH=−1\n3NqJ H∂G\n∂∆,Czz\nIH=−1\n3Nq∂G\n∂JI. (10b)\nThe final analytical expressions of all the physical quantit ies listed in\nEquations (10a) and (10b) depend on the on-site magnetizati onmIM=/an}bracketle{tσz\nj/an}bracketri}htIMand the pair correlation function\nCzz\nIM=/an}bracketle{tσz\njσz\nj+1/an}bracketri}htIMof the effective 2D q-coordinated spin-1 /2 Ising lattice with the temperature-dependent nearest-\nneighbour interaction Jeffin the temperature-dependent magnetic field Heff. Because an exact solution for the 2D\nspin-1/2 Ising model in an external magnetic field still belongs to un resolved issues of condensed matter physics, one\nhas to resort to some numerical algorithm applicable to the 2 D Ising lattices to gain the accurate results for mIMand\nCzz\nIM. In the present paper, we will employ the classical Monte Car lo (MC) simulations implementing the standard\nMetropolis algorithm [51, 52] for the e ffective spin-1/2 Ising lattice of a su fficiently large linear size L.\n3. Discussion of the Numerical Results\nIn this section, we will proceed to a discussion of the most in teresting numerical results for the 2D spin-1 /2 Ising–\nHeisenberg model in an external magnetic field with the antif erromagnetic Ising-type interaction JI<0 between\nthe Ising and Heisenberg spins. For simplicity, we will assu me that the local magnetic fields acting in the Ising and\nHeisenberg spins are identical HI=HH=H. The absolute value of the interaction JIwill be used as an energy unit\nfor defining a relative strength of the Heisenberg interacti onJH/|JI|and the magnetic field H/|JI|.\n3.1. Ground-State Phase Diagrams\nFirst, we take a look at possible magnetic ground-state arra ngement of the model, which can be determined\nby a systematic inspection of the eigenvalues (5) of the plaq uette Hamiltonian (1) for all possible combinations of\nquantum spin numbers tj,tz\njentering therein. The typical ground-state phase diagrams are depicted in Figure 2 in the\nJH/|JI|−H/|JI|parameter plane for two representative values of the exchan ge anisotropy∆=0.5 and 2 by assuming\nfour different numbers qof corner-sharing trigonal bipyramidal plaquettes formin g 2D lattices. As one can see from\nFigure 2a, the ground-state phase diagram of the model with t he easy-axis exchange anisotropy ∆=0.5 contains four\ndifferent ground states. Specifically, two ground states are mac roscopically degenerate quantum ferrimagnetic phases\n|−1/2; 1/2/an}bracketri}htR,Land|1/2; 1/2/an}bracketri}htR,L, which differ from each other only by the orientation of Ising spins with respect to\nthe applied magnetic field as indicated by the corresponding eigenvectors and eigenenergies per plaquette:\n|±1/2; 1/2/an}bracketri}htR,L=Nq/2/productdisplay\nj=1|±/an}bracketri}htσz\nj⊗|1/2,RorL/an}bracketri}ht△j, (11a)\nE|±1/2;1/2/an}bracketri}htR,L=JH\n4+JH∆\n2∓JI\n2−(q±2)H\n2q. (11b)\n5The state vector|1/2,RorL/an}bracketri}ht△jin Equation (11a) describes a quantum superposition of thre e different up-up-down\nspin states of the j-th Heisenberg triangular cluster with two opposite ( Right- and Left-hand side) chiral degrees of\nfreedom:\n|1/2,R/an}bracketri}ht△j=1√\n3/parenleftig\n|↑↑↓/an}bracketri}ht+e2πi\n3|↑↓↑/an}bracketri}ht+e4πi\n3|↓↑↑/an}bracketri}ht/parenrightig\n△j,\n|1/2,L/an}bracketri}ht△j=1√\n3/parenleftig\n|↑↑↓/an}bracketri}ht+e4πi\n3|↑↓↑/an}bracketri}ht+e2πi\n3|↓↑↑/an}bracketri}ht/parenrightig\n△j,(12)\nThe two-fold degeneracy of each Heisenberg triangle result s in the field-independent macroscopic degeneracy 2Nq/2\nof the phases|−1/2; 1/2/an}bracketri}htR,Land|1/2; 1/2/an}bracketri}htR,L, which is obviously highly sensitive to the current number qof inter-\nconnected trigonal bipyramidal plaquettes (Heisenberg tr iangular clusters). The direct relation between the number\nof plaquettes sharing a common vertex and the macroscopic de generacy of the phases |−1/2; 1/2/an}bracketri}htR,L,|1/2; 1/2/an}bracketri}htR,Lis\nalso reflected in a current value of the residual entropy per n odal Ising spin observed in both the phases. Specifically,\nit proportionally grows with q[53]:\nSres\nNkB=lim\nN→∞1\nNln 2Nq/2≈0.347q. (13)\nThe other two ground states are the classical ferrimagnetic phase|−1/2; 3/2/an}bracketri}htand the fully saturated phase |1/2; 3/2/an}bracketri}ht.\nThese two phases again di ffer from each other only by the orientation of the Ising spins w ith respect to the applied\nmagnetic field, while the Heisenberg spins are fully polariz ed into the magnetic field direction without any quantum\ncorrelations between their x- and y-components in both phases:\n|±1/2; 3/2/an}bracketri}ht=Nq/2/productdisplay\nj=1|±/an}bracketri}htσz\nj⊗|↑↑↑/an}bracketri}ht△j, (14a)\nE|±1/2;3/2/an}bracketri}ht=−3JH\n4∓3JI\n2−(3q±2)H\n2q. (14b)\nThe uniqueness of the classical spin arrangements in the pha ses|−1/2; 3/2/an}bracketri}htand|1/2; 3/2/an}bracketri}htgiven by Equation (14a) is\nreflected in the zero entropy per Ising spin S/(NkB)=0 in parameter regions corresponding to these phases.\nIt is obvious from Figure 2a that the classical ferrimagneti c phase|−1/2; 3/2/an}bracketri}htcan be detected in the whole pa-\nrameter region with the ferromagnetic Heisenberg coupling JH/|JI|>0 and partially also in the region with the\nantiferromagnetic Heisenberg interaction JH/|JI|<0. By contrast, the macroscopically degenerate quantum pha ses\n|−1/2; 1/2/an}bracketri}htL,Rand|1/2; 1/2/an}bracketri}htL,Rare stable solely for the antiferromagnetic Heisenberg cou plings JH/|JI|<0. Finally,\nthe saturated phase |1/2; 3/2/an}bracketri}htrepresents the actual ground state at high enough magnetic fi elds regardless of whether\nthe ferro- or antiferromagnetic Heisenberg interaction JH/|JI|is considered.\nOn the other hand, the ground-state phase diagram correspon ding to the model with the easy-plane anisotropy\n∆= 2 is a little more complex (see Figure 2b). It contains two mor e ground states in addition to the previous four,\nnamely, the unique quantum ferrimagnetic phases |−1/2; 1/2/an}bracketri}htand|1/2; 1/2/an}bracketri}htwith the Heisenberg triangular clusters in\na symmetric quantum superposition of three possible up-up- down spin states but an opposite orientation of the Ising\nspins:\n|±1/2; 1/2/an}bracketri}ht=Nq/2/productdisplay\nj=1|±/an}bracketri}htσz\nj⊗1√\n3(|↑↑↓/an}bracketri}ht+|↑↓↑/an}bracketri}ht+|↓↑↑/an}bracketri}ht )△j, (15a)\nE|±1/2;1/2/an}bracketri}ht=JH\n4−JH∆∓JI\n2−(q±2)H\n2q. (15b)\nAs shown in Figure 2b, both the quantum ferrimagnetic phases |−1/2; 1/2/an}bracketri}htand|1/2; 1/2/an}bracketri}htemerge in the ground-state\nphase diagram exclusively in the parameter region of the fer romagnetic Heisenberg interaction JH/|JI|>0. Naturally,\nthe zero entropy per Ising spin S/(NkB)=0 solely can be detected in their stability regions due to uni que quantum\nspin arrangement given by Equation (15a).\n6/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s48/s50/s52/s54/s56/s49/s48/s32/s61/s32/s48/s46/s53/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124/s49/s47/s50/s59/s49/s47/s50\n/s62\n/s82/s44/s76/s72 /s32 /s47/s32/s124 /s74\n/s73/s124\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s124/s45/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124/s45/s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76\n/s40/s97/s41/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s48/s50/s52/s54/s56/s49/s48/s32/s61/s32/s50/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124/s49/s47/s50/s59/s49/s47/s50\n/s62\n/s82/s44/s76/s72 /s32 /s47/s32/s124 /s74\n/s73/s124\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s124/s49/s47/s50/s59/s49/s47/s50/s62\n/s124/s45/s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s124/s45/s49/s47/s50/s59/s49/s47/s50 /s62/s124/s45/s49/s47/s50/s59/s51/s47/s50 /s62/s32/s32 /s113 /s32/s61/s32/s51\n/s32/s32 /s113 /s32/s61/s32/s52\n/s32/s32 /s113 /s32/s61/s32/s53\n/s32/s32 /s113 /s32/s61/s32/s54\n/s40/s98/s41\nFigure 2: The ground-state phase diagram of the spin-1 /2 Ising–Heisenberg model on 2D lattices with three ( q=3), four ( q=4), five ( q=5),\nand six ( q=6) corner-sharing trigonal bipyramidal plaquettes in the JH/|JI|−H/|JI|parameter plane for two representative values of the exchan ge\nanisotropy parameter: ( a)∆=0.5 and ( b)∆=2.\nIn addition to their location in the zero-temperature JH/|JI|−H/|JI|parameter plane, it is also possible to un-\nderstand from Figure 2 how the individual phases develop dep ending on the number qof corner-sharing bipyramidal\nplaquettes. Namely, the quantum ferrimagnetic phases |−1/2; 1/2/an}bracketri}htR,L,|−1/2; 1/2/an}bracketri}htand the classical one |−1/2; 3/2/an}bracketri}ht\nare gradually extended to stronger magnetic fields with incr easing number qof the corner-haring plaquettes. More-\nover, the classical phase |−1/2; 3/2/an}bracketri}htsimultaneously spreads to the regions of stronger antiferr omagnetic (ferromag-\nnetic) Heisenberg interactions JH/|JI|<0 (JH/|JI|>0). The remaining three phases |1/2; 1/2/an}bracketri}htR,L,|1/2; 1/2/an}bracketri}htand\n|1/2; 3/2/an}bracketri}htfaithfully follow the evolution of the adjacent ones |−1/2; 1/2/an}bracketri}htR,L,|−1/2; 1/2/an}bracketri}ht,|−1/2; 3/2/an}bracketri}ht: the quantum\nphases|1/2; 1/2/an}bracketri}htR,L,|1/2; 1/2/an}bracketri}htare gradually shifted to stronger antiferromagnetic and fe rromagnetic Heisenberg in-\nteractions JH/|JI|<0 and JH/|JI|>0, respectively, and the saturated one |1/2; 3/2/an}bracketri}htis shifted to stronger magnetic\nfields.\n3.2. Magnetization Process\nThe rich ground-state phase diagrams depicted in Figure 2 su ggest various zero-temperature magnetization scenar-\nios of the studied spin-1 /2 Ising–Heisenberg model either with one, two or three di fferent plateaus at fractional values\nof the saturation magnetization msat=(3q+2)/(2q). In accordance with the definition of the total magnetizati onm\nper plaquette listed in Equation (10a), the values of these p lateaus are given by a current number qof the trigonal\nbipyramids sharing a common vertex:\nm\nmsat=q−2\n3q+2,q+2\n3q+2,3q−2\n3q+2. (16)\nThe first (lowest) magnetization plateau at m/msat=(q−2)/(3q+2) can be identified at low magnetic fields in\nthe stability regions of the macroscopically degenerate qu antum ferrimagnetic phase |−1/2; 1/2/an}bracketri}htR,Land the unique\nquantum ferrimagnetic phase |−1/2; 1/2/an}bracketri}ht. The second one at m/msat=(q+2)/(3q+2) is a result of the spin\narrangements present in the quantum phases |1/2; 1/2/an}bracketri}htR,Land|1/2; 1/2/an}bracketri}ht, and therefore it can be found at moderate\nmagnetic fields. The highest fractional plateau at m/msat=(3q−2)/(3q+2) relates to the classical ferrimagnetic\nphase|−1/2; 3/2/an}bracketri}ht.\nTo illustrate the above statements, two three-dimensional (3D) plots of the isothermal magnetization curves for the\nparticular version of the lattice with four ( q=4) corner-sharing bipyramidal plaquettes are depicted in F igure 3 at the\nsufficiently low temperature kBT/|JI|=1.5×10−3. The plots are the outcomes of MC simulations for 100 ×100 nodal\nlattice sites (Ising spins), which corresponds to 19,800 co rner-sharing trigonal bipyramidal plaquettes. The adequa te\nnumerical accuracy was achieved by 12 ×104MC steps per node. For easy reference, the interaction ratio JH/|JI|is\nfixed to the same ranges and the anisotropy parameter ∆to the same values as were used in Figure 2. It can be under-\nstood from a comparison of these plots with corresponding gr ound-state phase diagrams in Figure 2 that the displayed\nlow-temperature magnetization curves faithfully reflect u p to seven different types of zero-temperature magnetization\nscenarios with the real 1 /7-, 3/7-, and/or 5/7-plateaus satisfying the general formulas listed in Equat ion (16):\n7i.|−1/2; 1/2/an}bracketri}htR,L−|1/2; 1/2/an}bracketri}htR,L−|1/2; 3/2/an}bracketri}ht,\nii.|−1/2; 1/2/an}bracketri}htR,L−|1/2; 1/2/an}bracketri}htR,L−|−1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\niii.|−1/2; 1/2/an}bracketri}htR,L−|−1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\niv.|−1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\nv.|−1/2; 1/2/an}bracketri}ht−|− 1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\nvi.|−1/2; 1/2/an}bracketri}ht−|1/2; 1/2/an}bracketri}ht−|−1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\nvii.|−1/2; 1/2/an}bracketri}ht−|1/2; 1/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht\n(see the magnetization curves of di fferent colors). Steep continuous rises between di fferent fractional plateaus as well\nas between fractional plateaus and the saturation magnetiz ation indicate a presence of the discontinuous magnetizati on\njumps that exist at the critical fields corresponding to the fi rst-order phase transitions only at zero temperature. In\nagreement with the ground-state analysis performed in Sect ion 3.1, the first three magnetization processes i.–iii., wh ich\ncontain the macroscopically degenerate quantum phases |±1/2; 1/2/an}bracketri}htR,L, can be observed for the easy-axis exchange\nanisotropy∆= 0.5 and also the easy-plane exchange anisotropy ∆= 2, but only in the parameter region of the\nantiferromagnetic Heisenberg interactions JH/|JI|<0. On the other hand, the magnetization scenario iv., reflect ing\nthe single field-induced transition from the classical ferr imagnetic phase|−1/2; 3/2/an}bracketri}htto the saturated one |1/2; 3/2/an}bracketri}ht,\nappears for both the antiferromagnetic ( JH/|JI|<0) and ferromagnetic ( JH/|JI|>0) Heisenberg couplings. The\nlast three magnetization processes v.–vii., which involve unique quantum ferrimagnetic phases |±1/2; 1/2/an}bracketri}ht, emerge\nin the parameter region of the ferromagnetic Heisenberg cou plings JH/|JI|>0 under the condition ∆>1. It should\nbe noted for completeness that the steep staircase dependen ces of all magnetization curves plotted in Figure 3 are\ngenerally gradually smoothing upon increasing of temperat ure due to a thermal activation of excited states, until they\ncompletely disappear.\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124/s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124/s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124 /s49/s47/s50/s59/s51/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124 /s49/s47/s50/s59/s51/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62/s124 /s49/s47/s50/s59/s51/s47/s50 /s62\n/s105/s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s105/s32/s124 /s49/s47/s50/s59/s51/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s105/s32/s124/s49/s47/s50/s59/s49/s47/s50 /s62 /s32 /s32/s124 /s49/s47/s50/s59/s51/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s105/s32/s124/s49/s47/s50/s59/s49/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\nFigure 3: 3D plots of the total magnetization mof the spin-1/2 Ising–Heisenberg model on the regular lattice with four co rner-sharing bipyramidal\nplaquettes reduced to its saturation value msatas a function of the magnetic field H/|JI|and the interaction ratio JH/|JI|for the exchange anisotropy\n(a)∆= 0.5 and ( b)∆= 2 at the temperature kBT/|JI|=1.5×10−3obtained by MC simulations for the lattice with 100 ×100 nodal Ising spins\n(19,800 bipyramidal plaquettes) by using 12 ×104MC steps per node. The curves of distinct colors refer to di fferent magnetization scenarios listed\nin the legend.\n83.3. Quantum Bipartite Entanglement\nThe discussion in the last subsection will be devoted to a bip artite quantum entanglement of the Heisenberg spins\nin the individual ground states. It is obvious from the plaqu ette Hamiltonian (1) that the spins may be quantum-\nmechanically entangled only within the Heisenberg triangu lar clusters in individual plaquettes. Those from di fferent\nplaquettes can never be entangled due to the Ising spin at the ir common vertices.\nIn general, a degree of the bipartite quantum entanglement b etween the Heisenberg spins at k-th and ( k+1)-th\nvertex of the j-th plaquette can be quantified by the quantity referred to as concurrence [54]. For the studied 2D\nspin-1/2 Ising–Heisenberg model, the concurrence can be simply cal culated from the local magnetization mHof the\nHeisenberg triangular cluster and the corresponding pair c orrelation functions Cxx(yy)\nHH,Czz\nHHdefined by Equations (10a)\nand (10b) through the following formula [55, 56]:\nCk,k+1=max0,4|Cxx(yy)\nHH|−2/radicaligg/parenleftigg1\n4+Czz\nHH/parenrightigg2\n−/parenleftbiggmH\n3/parenrightbigg2. (17)\nOf course, the identical XXZ exchange coupling JH(∆) in a given Heisenberg triangle results in the same degree of\nthe bipartite entanglement of the spin pairs. This is reflect ed in the same values of the corresponding concurrences:\nC1,2=C2,3=C3,1=C. (18)\nThe global picture on a degree of the bipartite quantum entan glement between the Heisenberg spin pairs in the\nindividual ground-state phases is illustrated in Figure 4, which shows the low-temperature ( kBT/|J|=1.5×10−3)\ndensity map of the concurrence Cof the spin-1/2 Ising–Heisenberg model on the regular 2D lattice with four corner-\nsharing bipyramidal plaquettes in the JH/|JI|−H/|JI|plane for the fixed value of the exchange anisotropy paramete r\n∆= 2. The plotted data have again been obtained by MC simulation s performed for the lattice of 19,800 corner-\nsharing bipyramidal plaquettes, whereas 2 ×107MC steps per nodal Ising spin were used to achieve the accurac y\nbetter than 10−3. It is clear from a direct comparison of Figure 4 with the corr esponding ground-state phase diagram\ndepicted in Figure 2b that the Heisenberg spins forming tria ngular clusters are quantum-mechanically entangled only\nif the macroscopically degenerate phases |±1/2; 1/2/an}bracketri}htR,Land the unique phases |±1/2; 1/2/an}bracketri}htare ground states. Due to\nthe macroscopic degeneracy caused by two possible chiral de grees of freedom of each triangular cluster, the bipartite\nentanglement of the Heisenberg spins in former two phases is half weaker than that in latter ones. This is also proven\nby the corresponding zero-temperature asymptotic values o f the concurrenceC|±1/2;1/2/an}bracketri}htR,L=1/3 andC|±1/2;1/2/an}bracketri}ht=2/3.\nOn the other hand, the remaining white region with the zero co ncurrenceC|±1/2;3/2/an}bracketri}ht=0 confirms completely non-\nentangled arrangements of the Heisenberg spins in the class ical ferrimagnetic phase |−1/2; 3/2/an}bracketri}htand the fully saturated\nphase|1/2; 3/2/an}bracketri}ht.\nTo get a deeper insight onto a role of pair correlations betwe en the Heisenberg spins in their bipartite quantum\nentanglement, the concurrence Cas function of the magnetic field H/|JI|and the corresponding dependencies of the\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s48/s50/s52/s54/s56/s49/s48\n/s32/s61/s32/s50\n/s67\n/s124/s32/s32/s49/s47/s50/s59/s49/s47/s50\n/s62\n/s82\n/s44\n/s76/s61/s32/s49/s47/s51/s72 /s32/s47/s32/s124 /s74\n/s73/s124\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s67/s124/s32/s32/s49/s47/s50/s59/s49/s47/s50/s62/s32/s61/s32/s50/s47/s51/s67\n/s124 /s32/s32/s49/s47/s50/s59/s51/s47/s50 /s62/s32/s61/s32/s48\n/s43/s43\n/s43\nFigure 4: The low-temperature ( kBT/|J|=1.5×10−3) density map of the concurrence Cof the spin-1/2 Ising–Heisenberg model on the regular\nlattice with four corner-sharing trigonal bipyramidal pla quettes in the JH/|JI|−H/|JI|plane for the exchange anisotropy parameter ∆=2 constructed\nfrom MC simulations performed for the lattice of 19,800 bipy ramidal plaquettes by using 2 ×107MC steps per nodal Ising spin.\n9/s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s32/s61/s32/s45/s52\n/s109\n/s72\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s67\n/s40/s97/s41/s109\n/s73/s32/s44/s32/s32 /s109\n/s72/s32/s44/s32/s32 /s67/s32/s120/s120 /s40/s121/s121 /s41\n/s72/s72/s44/s32/s32 /s67/s32/s122/s122\n/s72/s72\n/s48/s46/s48/s48/s48/s48/s46/s51/s51/s51/s48/s46/s54/s54/s55\n/s109\n/s73/s67/s32/s120/s120 /s40/s121/s121 /s41\n/s72/s72/s67/s32/s122/s122\n/s72/s72/s67\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s32/s61/s32/s54\n/s109\n/s72\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s67\n/s40/s98/s41/s109\n/s73/s32/s44/s32/s32 /s109\n/s72/s32/s44/s32/s32 /s67/s32/s120/s120 /s40/s121/s121 /s41\n/s72/s72/s44/s32/s32 /s67/s32/s122/s122\n/s72/s72/s109\n/s73/s67/s32/s120/s120 /s40/s121/s121 /s41\n/s72/s72\n/s67/s32/s122/s122\n/s72/s72/s67/s48/s46/s51/s51/s51/s48/s46/s54/s54/s55\n/s48/s46/s48/s48/s48\nFigure 5: The low-temperature ( kBT/|JI|=1.5×10−3) dependencies of the concurrence C, the sub-lattice magnetization mI,mH, and the pair\ncorrelation functions Cxx(yy)\nHH,Czz\nHHon the magnetic field H/|JI|of the spin-1/2 Ising–Heisenberg model on the regular lattice with four co rner-\nsharing bipyramidal plaquettes for the exchange anisotrop y∆= 2 and two particular interaction ratios ( a)JH/|JI|=−4 and ( b)JH/|JI|=6. The\ncurves are results of the MC simulations for the lattice of 10 0×100 nodal Ising spins by using 2 ×107MC steps per node.\npair correlation functions Cxx(yy)\nHH,Czz\nHHare plotted in Figure 5 for the anisotropy parameter ∆= 2 and two selected\ninteraction ratios JH/|JI|=−4 and 6 at the temperature kBT/|J|=1.5×10−3. The variations are completed by low-\ntemperature dependences of the local magnetization mIandmHto facilitate identification of the current ground-state\nspin arrangement. We note for completeness that all the curv es are results of the MC simulations for the lattice of\n100×100 nodal Ising spins by using 2 ×107MC steps per node to achieve accuracy better than 10−3.\nFigure 5a captures the sequence of field-induced phase trans itions|−1/2; 1/2/an}bracketri}htR,L−|1/2; 1/2/an}bracketri}htR,L−|1/2; 3/2/an}bracketri}ht. Ev-\nidently, the nonzero concurrence C=1/3, which can be found at the magnetic fields H/|JI|<9 due to stability of\nthe macroscopically degenerate quantum phases |±1/2; 1/2/an}bracketri}htR,L, is a result of the negative pair correlation functions\nCxx(yy)\nHH=Czz\nHH=−1/12 and the reduced local magnetization mH=1/2. Identical values of the transverse and lon-\ngitudinal correlation functions and their minus sign clear ly indicate that the macroscopically degenerate unsaturat ed\nbipartite entanglement of the Heisenberg spins from the sam e XXZ triangle comes from antiferromagnetic xx(yy)\ncorrelations of these spins, which are of the same strength a nd character as those in z-axis direction.\nA different situation can be found in Figure 5b, which illustrates the sequence of field-induced phase transitions\n|−1/2; 1/2/an}bracketri}ht−|1/2; 1/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht. Here, the low-temperature concurrence C=2/3, which can be observed at the\nmagnetic fields H/|JI|<7 due to the presence of the unique quantum phases |±1/2; 1/2/an}bracketri}ht, comes from the positive\nvalue of the transverse pair correlation function(s) Cxx(yy)\nHH=1/6, the negative longitudinal correlation function Czz\nHH=\n−1/12, and the reduced local magnetization mH=1/2. It is easy to understand from the values of Cxx(yy)\nHHandCzz\nHHthat\nthe origin of the local quantum bipartite entanglement of th e Heisenberg spins peculiar to the unique quantum phases\n|±1/2; 1/2/an}bracketri}htlies in ferromagnetic xx(yy) correlations and these are twice as strong as the antiferro magnetic ones along\nz-axis.\n4. Conclusions\nIn the present work, we have comprehensively studied the gro und-state properties, possible magnetization sce-\nnarios, and the local bipartite quantum entanglement of the Heisenberg spins in the individual quantum ground states\nof the mixed spin-1 /2 Ising–Heisenberg model in a longitudinal magnetic field on 2D lattices formed by identical\ncorner-sharing bipyramidal plaquettes. The numerical res ults have been obtained by combining the exact analytical\napproach called the decoration-iteration mapping transfo rmation [32–35] with numerical Monte Carlo simulations\nutilizing the Metropolis algorithm [51, 52].\nIt has been demonstrated that the ground-state phase diagra m of the investigated quantum mixed-spin model\nqualitatively does not depend on its lattice topology (the n umber qof corner-sharing plaquettes). In general, it involves\nin total six different ground states, namely the standard ferrimagnetic pha se, fully saturated phase, two unique quantum\nferrimagnetic phases and two macroscopically degenerate q uantum ferrimagnetic phases with two chiral degrees of\nfreedom of the Heisenberg spins forming triangular cluster s in plaquettes. It is also proven that the diversity of the\n10ground-state phase diagram gives rise to seven di fferent magnetization scenarios with one, two or up to three fr actional\nplateaus. The magnitudes of these plateaus are determined b y the current number qof the corner-sharing plaquettes.\nOther interesting findings are concerned with the bipartite quantum entanglement, which has been quantified by\nthe concept of the concurrence. We have verified that the Heis enberg spins of the same XXZ triangular cluster of a\ngiven plaquette can be entangled either due to stability of t he unique quantum ferrimagnetic phases, where they are in a\nsymmetric quantum superposition of three possible up-up-d own states, or due to macroscopically degenerate quantum\nferrimagnetic phases characterized by two chiral degrees o f freedom of each Heisenberg triangle. The strength of the\nentanglement in all the phases does not depend on the applied magnetic field. Moreover, the corresponding values of\nconcurrence clearly indicate that the entanglement of the H eisenberg spins is twice as strong when their arrangement\nis unique that when it is two-fold degenerate. Thus, it can be concluded that the macroscopic degeneracy of the\nHeisenberg triangles proportionally reduces the bipartit e quantum entanglement of their spins.\nFollowing our recent paper [45], dealing with the spin-1 /2 Ising–Heisenberg model on the planar lattice formed\nby trigonal bipyramids without a magnetic field, there is str ong indication that the bipartite entanglement between\nthe Heisenberg spins observed in the unique ferrimagnetic a nd macroscopically degenerated ferrimagnetic ground-\nstate phases in the present paper will also persist at finite t emperatures. 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B 2013 ,27, 1345018.\n12"    },    {        "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": "1610.02550v1.Perpendicularly_magnetized_CoFeB_multilayers_with_tunable_interlayer_exchange_for_synthetic_ferrimagnets.pdf",        "content": "arXiv:1610.02550v1  [cond-mat.mtrl-sci]  8 Oct 2016Perpendicularly magnetized CoFeB multilayers with tunabl e interlayer\nexchange for synthetic ferrimagnets\nP. Pirro,1,a)A. Hamadeh,1M. Lavanant-Jambert,1T. Meyer,2B. Tao,1E. Rosario,1Y. Lu,1M. Hehn,1S.\nMangin,1and S. Petit Watelot1\n1)Institut Jean Lamour, Université de Lorraine, UMR 7198 CNRS , 54506 Vandoeuvre-lès-Nancy,\nFrance\n2)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern,\n67663 Kaiserslautern, Germany\n(Dated: 22 September 2018)\nA study of the multilayer system MgO/CoFeB(1.1nm)/Ta( t)/CoFeB(0.8nm)/MgO is presented, where the two\nCoFeB layers are separated by a Ta interlayer of varying thic knesst. The magnetization properties deduced\nfrom complementary techniques such as superconducting qua ntum interference magnetometry, ferromagnetic\nresonance frequency measurements and Brillouin light scat tering spectroscopy can be tuned by changing the\nTa thickness between t=0.25 nm, 0.5 nm and 0.75 nm. For t=0.5 nm, a ferromagnetic coupling is observed,\nwhereas for t=0.75 nm, the antiferromagnetic coupling need ed to construct a synthetic ferrimagnet is realized.\nIn the later case, the shape of magnetic domain walls between two ferrimagnetic alignments or between a\nferro- and a ferrimagnetic alignment is very different. This behavior can be interpreted as a result of the\nchange in dipolar as well as interlayer exchange energy and d omain wall pinning, which is an important\nconclusion for the realization of data storage devices base d on synthetic ferri- and antiferromagnets.\nI. INTRODUCTION\nThe demand to develop faster and more reliable mag-\nnetic memories with large data storage capabilities has\nlead to the proposition of the racetrack memory concept1,\nwhere data is stored in form of domain walls in magnetic\nnanowires. Since these domain walls can be moved via\nspin torque effects to the magneto-resistive readout ele-\nments, this concept allows for a three-dimensional data\nstorage device. However, dipolar effects like, e.g., the\nWalker breakdown limit the velocity and storage den-\nsity of magnetic domain walls in single layer wires. To\novercome this limitation, the use of synthetic ferrimag-\nnets (SFI) with perpendicularly-to-plane magnetized lay-\ners has proven to be promising2,3. Thus, the development\nand careful characterization of suitable SFI systems with\nlow magnetic damping to reach high domain wall speeds\nand high device efficiencies is of paramount interest.\nIn this context, we present a study of the mul-\ntilayer system MgO(2.5nm)/Co 40Fe40B20(1.1nm\n)/Ta(t)/Co 40Fe40B20(0.8nm)/MgO(2.5nm) (all nominal\nlayer thicknesses), where the two CoFeB layers are\nseparated by a Ta interlayer4,5of varying thickness\noft=0.25, 0.5 and 0.75 nm. This system is directly\ngrown on a GaAs substrate and can thus easily be\nintegrated into semiconductor based devices, e.g. spin\nLED structures6,8. In the investigated layer stack, the\nhybridization of the 3d orbitals of the transition metals\nwith the O2p orbitals of MgO provides the perpendicular\nmagnetic anisotropy (PMA). To establish PMA, thermal\nannealing is needed to crystalline the CoFeB starting\na)currently at: Fachbereich Physik and Landesforschungszen trum\nOPTIMAS, Technische Universität Kaiserslautern, 67663 Ka iser-\nslautern, Germanyfrom the CoFeB/MgO interface. During this process,\nthe diffusing boron is absorbed by the Ta interlayer. In\naddition, Ta provides an RKKY-like coupling5,9which\nis needed to create a SFI.\nUsing a superconducting quantum interference de-\nvice SQUID, magneto-optical Kerr effect microscopy\n(MOKE), inductive ferromagnetic resonance frequency\n(FMR) measurements and Brillouin light scattering spec-\ntroscopy (BLS), the magnetic properties of the systems\nhave been investigated. The experimental results are\nconsistently reproduced by a simple macrospin model.\nThe interlayer exchange coupling as well as the mag-\nnetic properties of the two individual CoFeB layers vary\nstrongly with the Ta thickness t. Fort=0.75 nm, a SFI\nwith an antiferromagnetic (AFM) exchange coupling is\nobserved and the nucleation of domain walls in between\nthe two ferrimagnetic configurations as well as in between\nthe ferri- and the ferromagnetic (FM) configuration are\nrealized.\nII. SAMPLE PREPARATION\nThe investigated multilayer stacks are deposited on\nGaAs substrates which are first desorbed in a MBE\nchamber at 300◦C by monitoring reflection high energy\nelectron diffraction patterns to remove the As capping\nlayer, which is used to passivate the surface of the sub-\nstrate. Then, the samples are transferred to the sputter-\ning chamber without breaking the vacuum to deposit the\nmultilayer at room temperature. The magnetization of\nthe samples prior to an annealing is in-plane, as usual for\nTa/CoFeB/MgO systems. To establish PMA, the sam-\nples were annealed at 250◦C for 3 mins in a rapid ther-\nmal annealing system (for details, see6. Prior studies of a\nsimilar system have show that a magnetically dead layer2\ncan be formed at the Ta/CoFeB interface7. According\nto6, the effective magnetic CoFeB layer thickness dis sig-\nnificantly reduced compared to the nominally deposited\nthickness if t≥0.5nm. The estimated values for dbased\non the findings in6are given in Table I.\nIII. SQUID MEASUREMENTS\nFigure 1 shows the magnetization along the field axis\nas a function of an applied out-of-plane (Fig.1 (a)-(c))\nand in-plane field (Fig.1 (d)-(e)) as obtained from SQUID\nmeasurements at 300 K. The sample with t=0.25 nm of\nTa shows an in-plane easy axis with a small coercive field\nof approximately 1 mT and a hysteresis similar to a single\nlayer, thus the two layers of CoFeB seem to be strongly\nFM coupled, either by strong RKKY-like exchange or by\na direct coupling due to a discontinuous Ta layer. The\nPMA of this system is not sufficient to overcome the de-\nmagnetization field and to consequently orient the mag-\nnetization out of plane. For t=0.5 nm, a single hysteresis\nloop with small coercivity for both field directions is ob-\nserved, indicating that either one of the layers has an\nin-plane easy axis and the other one an out-of-plane easy\naxis, or a multidomain state is occurring in remanence\n(2 mT coercivity field for both directions, not visible in\nFig. 1 for the in-plane field due to the scaling). Since\nthe hysteresis loops are both centered around zero field,\nwe conclude that the coupling between the layers is FM.\nFort=0.75 nm, a clear three-step hysteresis is observed\nfor the out-of-plane field (see also Fig. 3(a)), whereas for\na field in plane a continuous transition without any de-\ntectable hysteresis is revealed. Thus, in this case, both\nlayers have an out-of-plane easy axis and are AFM cou-\npled, which leads to the two additional hysteresis loops\ncentered around BRKKY=±20 mT.\nTo achieve a more quantitative characterization, the\nvalues of the saturation magnetization Msfrom the\nSQUID measurements have been extracted. To cal-\nculateMsfrom the measured magnetic moment per\narea, the effective magnetic thicknesses dgiven in Ta-\nble I have been used. To For t=0.25 nm, a saturation\nmagnetization averaged over both layers of Ms=1019\nkA/m is measured, whereas for t=0.5 nm, Ms= 1063\nkA/m is obtained. In the case of the AFM coupling\nfort=0.75 nm, assuming that the smaller magnetic mo-\nment can be attributed to the thinner layer, a satura-\ntion magnetization of M1\ns=1006 kA/m for the lower layer\nandM2\ns=973 kA/m for the upper layer (thus averaged\nMs=993 kA/m) can be deduced.\nIV. MACROSPIN MODELING\nTo access further material parameters, we model the\ntotal energy Etotof the system including Zeeman en-\nergyEZee, demagnetization energy EDM, uniaxial magne-\ntocrystalline anisotropy energy EAniwith the anisotropy-400 -200 0 200 400-1000-50005001000\n-100 -50 0 50 100 -40 -20 0 20 40(b) (c) (a)\nTa 0.75 nm Ta 0.25 nm Ta 0.5 nm\n-20 -10 0 10 20-1000-50005001000\n-600 -300 0 300 600 -600 -300 0 300 600(e) (d) (f)Mop(kA/m)\nBop(mT) Bop(mT) Bop(mT)\nMip(kA/m)\nBip(mT) Bip(mT) Bip(mT)Ta 0.75 nm Ta 0.25 nm Ta 0.5 nm\nFIG. 1. (color online) Magnetization as a function\nof the applied field along the out-of-plane plane direc-\ntion (a-c) and along the in-plane direction (d-f) for\nCoFeB(1.1 nm)/Ta( t)/CoFeB(0.8 nm) with t= 0.25nm (a,d),\nt= 0.5nm (b,e) and t= 0.75nm (c,f). (c) shows that for\nt=0.75 nm, the magnetization of both layers is out of plane\nwith a three step hysteresis loop indicating an AFM coupling\nwith a direct transition between the two ferrimagnetic stat es.\neasy axis ualong the out-of-plane direction and the in-\nterlayer exchange energy ERKKY between the two layers:\nEtot=i=1,2/summationdisplay\nlayer[EZee(Mi)+EDM(Mi)+EAni(Mi)]\n+ERKKY(M1,M2) (1)\nEZee(Mi) =−Mi·B·Vi (2)\nEDM(Mi) =−µ0\n2(Mi·u)2·Vi (3)\nEAni(Mi) =−(m·u)2·Ks\ni·S (4)\nERKKY(Mi) =−(m1·m2)·J·S (5)\nwith the normalized magnetization m=M\nMs, the out-of-\nplane unit vector u, the interlayer exchange coupling J\nand the volume Vi=S·di, which depends on the surface\nSand effective thickness diof the individual CoFeB layer.\nIn the case of AFM coupling ( t=0.75 nm), the switching\nfrom the ferromagnetic to the ferrimagnetic states takes\nplace at the field BRKKY, so we can conclude that at\nthis field EZee=ERKKY, which leads to the estimate of\nthe exchange constant J=−M2·BRKKY·d2≈ −0.011\nmJ/m2, which is on the same order of magnitude than J\nof Ref.5,9. Furthermore, we reconstructed the hard axis\nSQUID loops by numerically minimizing Etot(M1,M2)\nas a function of the applied field and subsequently ex-\ntracting the average magnetization along the field (blue3\ntd1d2M1M2Ks\n1Ks\n2J\n(nm) (nm) (kA/m) (mJ/m2)\n0.25 1.1 0.8101910190.54 0.54≥0.4\n0.50.975 0.675 106310630.690.565 0.045\n0.75 0.84 0.54 1006 9730.580.385 -0.011\nTABLE I. Overview of the obtained material parameters. The\nvalues for MandKsas well as the value for the interfacial\nexchange coupling Jfort=0.75 nm have been obtained from\nthe SQUID measurements, whereas Jin the case of t=0.25nm\nand 0.5 nm are estimates based on the FMR frequency mea-\nsurements. The effective magnetic thickness ddepends on\nthe Ta interlayer thickness t. If a distinction between the two\nlayers has not been possible, an average value is given.\ncurves in Fig. 1(a),(e) and (f)). From this reconstruc-\ntions, the magnetocrystalline surface anisotropy con-\nstantsKs\nihave been obtained which are listed in Table I.\nV. FMR EXPERIMENTS AND MODELING\nTo check the validity of the obtained material constants\nand to estimate the exchange coupling for the thicknesses\ntwith FM coupling, measurements of the frequencies of\nthe small angle precession eigenmodes have been per-\nformed (see Fig. 2). The external magnetic field has\nbeen applied along the in-plane axis and the eigenmode\nfrequencies have been obtained by standard FMR mea-\nsurements using a vector network analyzer (VNA). In\naddition, for the case of AFM coupling t=0.75 nm, the\nthermal spin-wave spectrum with a wave vector k→0\nequivalent to the ferromagnetic resonances has been ac-\nquired by Brillouin light scattering spectroscopy11,12.\n0 200 400 60005101520\n0 200 400 60005101520\n0 200 400 60005101520\nFMR\nModel\nBip(mT)FMR-VNA\nModel\nModel 1°FMR-VNA\nBLS\nModel\nModel 1°\n(c) (b) (a)\nBip(mT) Bip(mT)f(GHz)\nFIG. 2. (color online) Measurements of the ferromagnetic re s-\nonance frequencies (red dots) together with the eigenfrequ en-\ncies predicted by the macrospin model (blue and black lines)\nusing the material parameters presented in Table I obtained\nfrom SQUID. Black curves represent the macrospin model for\na perfect in-plane field whereas blue curves assume a small\nout-of-plane tilt of 1◦.\nTo compare the experimental results with our\nmacrospin model, we transform all energy expressions in\na new base given by the equilibrium magnetizations me\n1andme\n2. In this base, we calculate the effective fields\nBeff\ni=−1\nVi·Mi∇me\niEtot\ni(me\n1,me\n2) (6)\nand subsequently linearize the two coupled Landau-\nLifshitz equations:\ndme\n1\ndt=−γme\n1×Beff\n1,dme\n2\ndt=−γme\n2×Beff\n2 (7)\nwithme\nz→1 (external field axis along z) and solve the\nsystem numerically as an eigenvalue problem. The fre-\nquencies of two modes are obtained, which correspond\nto the in-phase and out-of-phase oscillations of the two\nlayers when assuming two identical layers.\nFort=0.25 nm (Fig. 2(a)), only one mode within the\nexperimental accessible range from 1 to 20 GHz has been\nfound, again indicating a strong coupling between the\nlayers. The blue curve shows the frequency evolution\npredicted by the macrospin model when using the pa-\nrameters obtained from the SQUID measurement with\nan exchange coupling J= 0.4mJ/m2. This is the min-\nimal value of Jneeded to shift the second mode above\n20 GHz and thus constitutes the lower boundary for the\ncoupling. A good agreement between the model and the\nexperiment is observed and the small deviation is proba-\nbly due to a residual misalignment of the applied fields.\nFort=0.5 nm (Fig. 2(b)), two modes with a frequency\nspacing of about 5 GHz have been observed. Using an\nexchange coupling of J=0.045 mJ/m2and the parame-\nters obtained from the SQUID, the macrospin model has\nbeen evaluated for a perfect in-plane alignment of the\nexternal field (dotted black curve) and for a small out-\nof-plane tilt of the external field of 1◦(continuous blue\nline). The tilt of the field influences especially the local\nfrequency minimum close to 200 mT. Since the trend for\nthe tilted field reproduces the experimentally obtained\nvalues more accurately, a slight misalignment from the\nin-plane axis during the FMR experiment can be con-\ncluded. No clear susceptibility peaks have been observed\nfor fields below the in-plane saturation field of 200 mT\nindicating an inhomogeneous magnetization state which\ncannot be described by the macrospin model.\nFort=0.75 nm (Fig. 2(c)), again two modes have been\nobserved. Here, with the macrospin model, we only use\nvalues obtained from the SQUID measurements. The\nmodel nicely reproduces the experimental trend if we al-\nlow for a 1◦misalignment of the external applied field\n(continuous blue curve). A perfect alignment of the field\nalong the in-plane direction would again lead to a pro-\nnounced frequency minimum, which has not been ob-\nserved (dotted black curve). Due to the smaller inter-\nlayer coupling J, the frequency gap which is occurring at\nabout 250 mT with an opening of about 800 MHz is much\nsmaller than in the t=0.5 nm case. Due to the compa-\nrably large frequency linewidth of more than 1 GHz, the\ngap could not be observed in the FMR and BLS exper-\niments. Since the frequency linewidth shows no system-\natic frequency dependence, the underlying broadening4\nmechanism is probably arising from an inhomogeneous\ndistribution of the material parameters.\nThe good general agreement between the experimen-\ntally obtained frequencies and the calculated eigenfre-\nquencies based on the material parameters of the SQUID\nmeasurements demonstrates that a comprehensive and\nself-consistent characterization of the system has been\nachieved.\nVI. MOKE MICROSCOPY\nIn the following, we will address the possible domain\nconfigurations in the case of the AFM coupling of the\nlayers (t=0.75 nm), since this configuration is especially\npromising for future high density storage elements2. The\nnucleation of DWs was studied experimentally in the\npresence of an out-of-plane magnetic field. Figure 3\npresents magneto-optical Kerr microscopy (MOKE) hys-\nteresis loops and images showing the nucleation of a do-\nmain in the different configurations of the two magneti-\nzations in the thinner upper and thicker lower magnetic\nlayer. In Fig. 3(a), the magnetic configurations are de-\npicted for the different levels of the MOKE signal by a\npair of arrows. Three jumps are observed in the hys-\nteresis loop, similar to the observations using SQUID.\nStarting from positive fields, the first step (marked by\n(b) in Fig. 3(a)) corresponds to the switching of the up-\nper thinner layer to minimize the interlayer exchange en-\nergy, which becomes more important than the Zeeman\nenergy at this field. Thus, a transition from the paral-\nlel (P) to the antiparallel (AP) state occurs. The sec-\nond step (marked by (c)) corresponds to the switching of\nboth layer’s magnetization, so a transition between the\ntwo AP states is taking place. Here, the Zeeman energy\nis reduced by aligning the residual magnetic moment of\nthe AP state along the external field, while the interlayer\nexchange energy is unaffected. With further decreasing\nfield, the Zeeman energy reaches again the value of the\ninterlayer exchange and a switching to the P state occurs\n(around -25 mT in Fig. 3(a)). In contrast to the SQUID\nmeasurement, a minor hysteresis loop around zero field\nwith a clear separation between the two AP states is ob-\ntained using MOKE. We attribute this behavior to the\ndifferent time scales of the measurements: MOKE is sig-\nnificantly faster, so the observed coercivity increases as\nthe system has less time to undergo a thermal activated\nswitching.\nFigures 3(b)-(e) present MOKE images after the nu-\ncleation of domains at different external fields around a\ndefect in the thin film. The nucleation takes place at\nthe transitions indicated in Fig. 3(a), whereas the fields\nwhere the domains are stabilized and imaged are indi-\ncated below the pictures. The MOKE contrast is deter-\nmined by the net magnetization direction of the domains\naveraged over the two layers. The orientation of the mag-\nnetization is again indicated by arrows.\nTwo magnetic domains are depicted in Figure 3(c),\n100 µm(d)\n (e)\n(c)\n24 mT(b)(a)\n(b)\n(c)\n(e)\n(d)\n-2.9 mT\n-24 mT 1.2 mTMOKE signal (a.u.)\nBop(mT)-50 -40 -30 -20 -10 0 10 20 30 40 50-0.04-0.020.000.020.04\nFIG. 3. (color online) (a) Hysteresis loop ob-\ntained from MOKE microscopy for AFM coupling in\nCoFeB(1.1nm)/Ta(0.75nm)/CoFeB(0.8nm). The arrows\nindicate the alignment of the two layers for the four differen t\nlevels of the MOKE signal. (b)-(e) MOKE microscopy images\nof different domain configurations which have been initially\nnucleated at the transition fields marked in (a) and imaged\nat the indicated field values.\neach of which includes MOKE contributions from the\nupper and lower magnetic layer. In the bright domain,\nthe magnetization is oriented down in the lower layer\nand up in the upper layer. In the darker surroundings,\nthe magnetic orientations of the two layers are both in-\nverted. Consequently, a domain wall is formed between\nthe two AP states which can be considered as a domain\nwall in a synthetic ferrimagnet. In Fig. 3(e), the role of\nthe domain around the defect and the surroundings is\njust inverted due to the different sign of the nucleation\nfield. For higher fields, the domain configurations shown\nin Fig.3(b) and (d) are stabilized in the center of the\nouter hysteresis loops and show both a domain with AP\nalignment of the two layers surrounded by an area with\nP alignment. Thus, depending on the external field, two\ndifferent kinds of domain walls can be nucleated in this\nsystem.\nFinally, it is clear that the shape of the domain de-\npends on the configuration between the upper and lower\nmagnetic layer. When the transition occurs between the5\nP and AP orientation (Fig. 3(b) and (d)), the domain\nwall shows a zigzag shape. For the transitions between\ntwo AP orientations of the layers (Fig. 3(c) and (e)),\nthe domain grows as a circle. The difference of these\nshapes might be attributed to a combination of different\neffects which all can lead to distinct domain wall dy-\nnamics for the two configurations: first, in the P to AP\ntransition, a domain wall is present only in the upper\nlayer. Thus, it could experience different pinning com-\npared to the domain wall between the two AP states,\nwhich is also present in the lower layer. In addition, the\ndipolar stray fields of a domain with P configuration are\nmuch stronger than those generated in the AP configu-\nration, which leads to different contributions of dipolar\nand interlayer exchange energy in the two cases.\nTo conclude, we have demonstrated that\nCoFeB/Ta/CoFeB multilayers grown on GaAs/MgO\npresent strong perpendicular magnetocrystalline\nanisotropy. The magnetization properties could be\ntuned by changing the Ta thickness. A 0.5 nm Ta\nthickness leads to a ferromagnetic coupling whereas a\n0.75 nm Ta thickness leads to a synthetic ferrimagnetic\nbilayer with an antiferromagnetic interlayer coupling.\nIn the later case, the shape of magnetic domain walls\nbetween two ferrimagnetic alignments and between a\nferromagnetic and a ferrimagnetic alignment is very\ndifferent, possibly due to the change in dipolar and\ninterlayer exchange energy as well as domain wall pin-\nning. Our results show that with the proper interlayer\nthickness, the CoFeB/Ta/CoFeB system is a promising\ncandidate for the realization of data storage devices\nbased on synthetic ferrimagnets.\nACKNOWLEDGMENTS\nThis work was supported by the ANR-NSF Project,\nANR-13-IS04-0008-01, COMAG by the ANR-LabcomProject LSTNM and by the Université de la Grande\nRegion (UniGR funded P. Pirro Post-Doc). Y. Lu also\nacknowledges the support by the joint French National\nResearch Agency (ANR)-National Natural Science Foun-\ndation of China (NSFC) SISTER project (Grants No.\nANR-11-IS10-0001 and No. NNSFC 61161130527) and\nENSEMBLE project (Grants No. ANR-14-0028-01 and\nNo. NNSFC 61411136001). Experiments were performed\nusing equipment from the TUBE - Daum funded by\nFEDER (EU), ANR, the Region Lorraine, and Grand\nNancy.\n1S. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008).\n2S.-H. Yang, K.-S. Ryu, and S. Parkin, Nature Nanotech 10, 221\n(2015).\n3M. Kuteifan, M.V. Lubarda, S. Fu, R. Chang, M.A. Escobar, S.\nMangin, E. Fullerton, and V. Lomakin, arXiv 1406, 3253 (2014 ).\n4C.-W. Cheng, T.-I. Cheng, C.H. Shiue, C.-L. Weng, Y.-C. Tsai ,\nand G. Chern, IEEE T Magn 49, 4433 (2013).\n5C.-W. Cheng, C.H. Shiue, T.-I. Cheng, and G. Chern, J. Appl.\nPhys.112, 033917 (2012).\n6B. S. Tao, P. Barate, J. Frougier,P. Renucci, B. Xu, A. Djeffal , H.\nJaffre‘s, J.-M. George, X. Marie, S. Petit-Watelot, S. Mangi n, X.\nF. Han, Z. G. Wang, and Y. Lu, Appl. Phys. Lett. 108, 152404\n(2016).\n7S. Y. Jang, C.-Y. You, S. H. Lim, and S. R. Lee, J. Appl. Phys.\n109, 013901 (2011).\n8S.H. Liang, T.T. Zhang, P. Barate, J. Frougier, M. Vidal, P.\nRenucci, B. Xu, H. Jaffres, J.M. George, X. Devaux, M. Hehn, X.\nMarie, S. Mangin, H.X. Yang, A. Hallal, M. Chshiev, T. Amand,\nH.F. Liu, D.P. Liu, X.F. Han, Z.G. Wang, and Y. Lu, Phys Rev\nB90, 085310 (2014).\n9S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 (1991).\n10R.R. Gareev, V. Zbarsky, J. Landers, I. Soldatov, R. Schäfer ,\nM. Münzenberg, H. Wende, and P. Grünberg, Appl. Phys. Lett.\n106, 132408 (2015).\n11T. Sebastian, Y. Kawada, B. Obry, T. Brächer, P. Pirro, D.A.\nBozhko, A.A. Serga, H. Naganuma, M. Oogane, Y. Ando, and\nB. Hillebrands, J. Phys. D Appl. Phys. 48, 164015 (2015).\n12J.V. Harzer, B. Hillebrands, R.L. Stamps, G. Güntherodt, C. D.\nEngland, and C.M. Falco, J. Appl. Phys. 69, 2448 (1991)."    },    {        "title": "1905.03521v1.Bidirectional_spin_wave_driven_domain_wall_motion_in_antiferromagnetically_coupled_ferrimagnets.pdf",        "content": "1 \n Bidirectional spin-wave -driven domain wall motion in \nantiferromagnetically coupled ferrimagnets  \nSe-Hyeok Oh1, Se Kwon Kim2, Jiang  Xiao3,4,5, and Kyung- Jin Lee1,6,7 * \n1Department of Nano -Semiconductor and Engineering, Korea University, Seoul 02841, Korea  \n2Department of Physics and Astronomy, University of Missouri, Columbia , Missouri 65211 , USA \n3Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shang hai \n200433, China  \n4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China  \n5Institute for Nanoelectronics Devices and Quantum Computing, Fudan University, Shanghai 200433, \nChina  \n6Department of Materials Science and Engineering, K orea University, Seoul 02841, Korea  \n7KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, \nKorea  \n \n* corresponding email: kj_lee@korea.ac.kr   \n   2 \n Abstract  \nWe investigate  ferrimagnetic domain wall dynamics induced by circular ly polarized \nspin wave s theoretically and numerically . We find that the direction of domain wall \nmotion depends on both the circular polarization of spin  wave s and the sign of net spin \ndensity of ferrimagnet. Below the angular momentum compensation point, left - (right-) \ncircularly  polarized spin wave s push a domain wall toward s (away from ) the spin -wave \nsource. Above the angular momentum compensation point, on th e other hand, the \ndirection of domain wall motion is reversed . This bidirectional motion  originate s from  \nthe fact that the  sign of spin -wave -induced magnon ic torque depends on the circular \npolarization and the subsequent response of the domain wall to the magnonic torque is \ngoverned by the net spin density . Our finding provides a way to utilize a spin wave as a \nversatile driving force for bidirectional domain wall motion . \n \nI. Introduction \nOne class of ferrimagnet s of emerging interest is a  rare- earth (RE) -transition metal (TM) \ncompound where the RE and TM moments are coupled antiferromagnetically. Owing to  \ndifferent Land é-g factors between the RE and TM elements,  RE-TM ferr imagnet s exhibit two \nunique compensation temperatures : the magnetic moment compensation  temperature 𝑇𝑇𝑀𝑀 at \nwhich  net magnetic moment vanishes , and the  angular momentum compensation temperature \n𝑇𝑇𝐴𝐴 at which  net angular momentum vanishes  [1-3].  \nResearch on ferr imagnetic materials ha s focused on the understanding of their \nfundamental magnetism [4]  and optical switching of magnetization [5 -11]. Recently, RE -TM \nferrimagnet s attract renewed interest as they offer a material platform to investigate the \nantiferromagnetic spintronics [ 12-15]. Compared to ferromagnets that have served as  core \nmaterial s for spintronics research, antiferromagnets exhibit several distinct features such as \nthe immunity to external field perturbations and fast dynamics due to antiferromagnetic \nexchange interaction.  However, the external -field immunity of true antiferromagnets results \nin the experimental difficulty in both creating and controlling  antiferromagnetic textures. On \nthe other hand, RE -TM ferrimagnets have finite magnetic moment at the angular momentum 3 \n compensation point  at which the antiferromagnetic dynamics is realized. As a result, \npreviously established creation and detection schemes for ferromagnets is  directly appli cable  \nto RE -TM ferrimagnets. This simple but strong benefit of RE -TM ferrimagnets have recently \ninitiated extensive studies  on ferrimagnets, which include magnetization switching [ 16-19], \ndomain wall motion [ 20-24], skyrmion (or bubble domain) motion [ 25-28], low damping [ 29], \nand efficient spin -transfer and spin- orbit torque s due to antiferromagnetic alignment of \natomic spins  [30,31 ]. \nAmong the previous studies listed above, the low damping of RE -TM ferrimagnets [2 9] \nis of particular interest from the view point of magnonic applications based on ferrimagnets \nbecause it enables  a long -distance propagation of spin waves (SWs) . For ferromagnets [ 32-39] \nand antiferromagnets [ 40-43], it was reported that a SW can move a DW by transferring its  \nangular momentum  or linear momentum . Though the SW property in ferrimagnet s was \nestablished  [44-47], the effect of SWs  on ferrimagnetic domain wall  (DW) motion remains  \nunexplored. In comparison to ferromagnets and antiferromagnets, antiferromagnetically \ncoupled ferrimagnets exhibit  a distinguishing feature of  SW eigenmode s. In ferromagnet s, a \nspin wave (SW) with only one type of polarization is permitted , which drives a DW toward s \nthe SW source through the angular momentum trans fer [3 4-37]. In antiferromagnet s, however, \nboth the left - and right -circular ly polarized SWs are allowed and energetically degenerate,  \nwhich can transfer the linear momentum to a DW through the SW  reflection [40,41,43] , \nresulting in the  DW motion away from the SW source . In antiferromagnetically coupled \nferrimagnet s, on the other hand, the degeneracy of the  two circular ly polarized SWs can be  \nlifted  depending on the net spin density of ferrimagnet . Given that SW -induced DW motion \nin ferrimagnets has been unexplored, interesting and important question s remain unanswered : \nhow a SW m oves a ferrimagnetic DW  and what the role of circular polarization of SW is . \nIn this paper, we study the dynamics of a  ferrimagnetic DW induced by a SW in the \nvicinity of the ang ular momentum compensation temperature 𝑇𝑇𝐴𝐴. We inve stigate DW \ndynamics induced by left - and right -circular ly polarized SWs [see Fig. 1(a) for an illustration \nof the two eigenmodes ]. We begin with theoretical analysis based on the Lagrangian density \nand SW dispersion. We then conduct numerical  simulation based on the atomistic Landau -\nLifshitz -Gilbert (LLG) equation to confirm the analytical results . Our model system is shown  4 \n in Fig. 1(b).  \n \nII. Model  \nOur model system is a simple bipartite ferrimagnet which consists of two sublattices \nlabeled by A and B. We introduce the staggered vector 𝒏𝒏=(𝑨𝑨k−𝑩𝑩k)/2, and 𝒎𝒎=𝑨𝑨k+\n𝑩𝑩k, where  𝑨𝑨k and 𝑩𝑩k are the unit vectors of spin moment at a site k that belongs to the \nsublattice s A and B, respectively.  The Lagrangian density for the ferrimagnet is given by  \n[26,47- 49] \nℒ=[−s𝒏𝒏̇∙(𝒏𝒏×𝒎𝒎)−𝛿𝛿𝑠𝑠𝐚𝐚(𝒏𝒏)∙𝒏𝒏̇]−𝒰𝒰,                    (1) \nwhere 𝑠𝑠=(𝑠𝑠𝐴𝐴+𝑠𝑠𝐵𝐵)/2, 𝛿𝛿𝑠𝑠=𝑠𝑠𝐴𝐴−𝑠𝑠𝐵𝐵, 𝑠𝑠𝑖𝑖=𝑀𝑀𝑖𝑖/𝛾𝛾𝑖𝑖 is the angular momentum  density , 𝑀𝑀𝑖𝑖 \nis the magnetic moment, 𝛾𝛾𝑖𝑖 is the gyromagnetic ratio for sublattice 𝑖𝑖, 𝐚𝐚(𝒏𝒏) is the vector \npotential for the magnetic monopole . The total energy 𝒰𝒰 includes the exchange energy and \nanisotropy energy as  \n𝒰𝒰=𝑎𝑎\n2|𝒎𝒎|2+𝐴𝐴\n2(∇𝒏𝒏)2−𝐾𝐾\n2(𝒛𝒛�∙𝒏𝒏)2+𝜅𝜅\n2(𝒙𝒙�∙𝒏𝒏)2,                (2) \nwhere 𝑎𝑎 is the homogeneous exchange, 𝐴𝐴 is the inhomogeneous exchange, 𝐾𝐾 is the easy-\naxis anisotropy constant , and 𝜅𝜅 is the hard-axis anisotropy constant. The Rayleigh function \naccounting  for the dissipation is  given by  ℛ=𝛼𝛼𝑠𝑠𝒏𝒏̇2 where 𝛼𝛼 is the Gilbert damping \nconstant. From the  Lagrangian  density and the Rayleigh dissipation , we obtain the equation \nof motion in terms of staggered vector 𝒏𝒏 by integrating out  the net magnetization variable  \n𝒎𝒎 [26]: \n𝜌𝜌𝒏𝒏×𝒏𝒏̈+2𝛼𝛼𝑠𝑠𝒏𝒏×𝒏𝒏̇+𝛿𝛿𝑠𝑠𝒏𝒏̇=𝒏𝒏×𝒇𝒇𝒏𝒏,                      (3) \nwhere 𝜌𝜌=𝑠𝑠2/𝑎𝑎 paramet rizes the inertia  and 𝒇𝒇𝒏𝒏=−𝛿𝛿𝒰𝒰𝛿𝛿𝒏𝒏⁄ is the effective field.  After \nlinearizing the equations for small- amplitude fluctuations from the uniform state, w e consider \nthe SW ansatz as 𝒏𝒏 (𝑥𝑥,𝑡𝑡)=Re��𝑛𝑛𝑥𝑥exp[𝑖𝑖(𝜔𝜔𝑡𝑡−𝑘𝑘𝑥𝑥)],𝑛𝑛𝑦𝑦exp[𝑖𝑖(𝜔𝜔𝑡𝑡−𝑘𝑘𝑥𝑥)],1��, where \n𝑛𝑛𝑥𝑥,𝑛𝑛𝑦𝑦 are the amplitude s of SW (|𝑛𝑛𝑥𝑥|,�𝑛𝑛𝑦𝑦�≪1), 𝜔𝜔 is the SW frequency, and 𝑘𝑘 is the \nwavevector.  By solving the linearized equations with this ansatz, we obtain the d ispersion 5 \n relation as  \n𝜔𝜔±=±𝛿𝛿𝑠𝑠+�𝛿𝛿𝑠𝑠2+4𝜌𝜌(𝐴𝐴𝑘𝑘2+𝐾𝐾+𝜅𝜅/2)\n2𝜌𝜌.                   (4). \nHere the u pper (lower) sign corresponds to the left- (right -) circular ly polarized SW. The \nresonance frequencies for left - and right -circular ly polarized SWs are different except at the \nangular mom entum compensation point  𝑇𝑇𝐴𝐴 where the net spin density 𝛿𝛿𝑠𝑠 is zero . Figure 2 \nshows the agreement between the analytic SW dispersion relations  [Eq. (4) ; lines ] and \nnumerical results  that will be discussed below  (symbols) . Below  or above  𝑇𝑇𝐴𝐴, the energy of \nright -circular ly polarized SW differs from that of left- circular ly polarized SW [see Fig. 2(a) \nand Fig. 2(c) ]. At 𝑇𝑇𝐴𝐴 [Fig. 2(b) ], two circular ly polarized SW s are degenerate, which is \nanalogous with antiferromagnetic SW s.  \nWe n ext look into the dynamics of ferrimagnetic DW  induced by SW s. We consider \n𝒏𝒏 as 𝒏𝒏=𝒏𝒏𝟎𝟎+𝜹𝜹𝒏𝒏 with DW texture 𝒏𝒏𝟎𝟎 and small fluctuation 𝜹𝜹𝒏𝒏 (|𝜹𝜹𝒏𝒏|≪|𝒏𝒏𝟎𝟎|) with the \nconstraint 𝒏𝒏𝟎𝟎∙𝜹𝜹𝒏𝒏=0 to keep the unit length of 𝒏𝒏 to linear order in 𝜹𝜹𝒏𝒏. We introduc e two \ncollective coordinates  [50], the DW position 𝑋𝑋(𝑡𝑡) and center angle 𝜙𝜙(𝑡𝑡), and define  a DW \n𝒏𝒏𝟎𝟎 by Walker ansatz  [51], 𝒏𝒏𝟎𝟎(𝑥𝑥,𝑡𝑡)=(sin𝜃𝜃sin𝜙𝜙,sin𝜃𝜃cos𝜙𝜙,𝑐𝑐𝑐𝑐𝑠𝑠𝜃𝜃 ) where 𝜃𝜃=\n2tan−1[exp{(𝑥𝑥−𝑋𝑋)/𝜆𝜆}] and 𝜆𝜆 is the DW width . We consider  the magnonic torque  𝝉𝝉𝒎𝒎 \nthat is given by  [34,37,39,52]  \n𝝉𝝉𝒎𝒎=−𝐴𝐴[(𝑱𝑱𝒎𝒎∙𝜵𝜵)𝒏𝒏𝟎𝟎−(𝜕𝜕𝑥𝑥𝜌𝜌𝑚𝑚)𝒏𝒏𝟎𝟎×𝜕𝜕𝑥𝑥𝒏𝒏𝟎𝟎],                 (5) \nwhere magnon -flux density  𝐽𝐽𝑚𝑚𝑥𝑥=𝒏𝒏𝟎𝟎∙〈𝜹𝜹𝒏𝒏×𝜕𝜕𝑥𝑥𝜹𝜹𝒏𝒏〉, and magnon number density 𝜌𝜌𝑚𝑚=\n〈𝜹𝜹𝒏𝒏〉2/2. The first term in Eq. (5) represents the adiabatic magnonic torque  rooted in the \nmagnon current , and the second term  represents the non-adiabatic magnonic torque  caused by \nthe gradient of the magnon density . Inserting Eq. (5) into the staggered LLG equation Eq. (3), \nwe derive  two coupled equations of motion as  \n𝑀𝑀𝑋𝑋̈−𝐺𝐺𝜙𝜙̇+𝑀𝑀𝑋𝑋̇𝜏𝜏⁄=𝐹𝐹𝑚𝑚,                              (6) \n𝐼𝐼𝜙𝜙̈+𝐺𝐺𝑋𝑋̇+𝐼𝐼𝜙𝜙̇𝜏𝜏⁄=−𝜅𝜅𝜆𝜆sin2𝜙𝜙+𝑇𝑇𝑚𝑚,                    (7) \nwhere 𝑀𝑀=2𝜌𝜌𝜌𝜌𝜆𝜆⁄, 𝐼𝐼=2𝜌𝜌𝜆𝜆𝜌𝜌, 𝐺𝐺=2𝛿𝛿𝑠𝑠𝜌𝜌, and 𝜏𝜏=𝜌𝜌/𝛼𝛼𝑠𝑠 are the mass, the moment of 6 \n inertia, the gyrotropic coefficient, and the relaxation time , respectively, and 𝜌𝜌 is the cross \nsectional area of the DW . Here,  𝐹𝐹𝑚𝑚=(2𝐴𝐴𝜆𝜆⁄)∫𝑑𝑑𝑑𝑑[(𝜕𝜕𝑥𝑥𝜌𝜌𝑚𝑚)𝒏𝒏𝟎𝟎×𝜕𝜕𝑖𝑖𝒏𝒏𝟎𝟎] and 𝑇𝑇𝑚𝑚=\n−2𝐴𝐴∫𝑑𝑑𝑑𝑑[(𝑱𝑱𝒎𝒎∙𝛁𝛁)𝒏𝒏𝟎𝟎] correspond to the  magnon- induced force and torque , respectively. \nWe note that the sign of 𝑇𝑇𝑚𝑚 is different for left - and right -circular ly polarized SWs whereas \nthe sign of 𝐹𝐹𝑚𝑚 is independent of the circular  polarization of SW. It is because  the sign of 𝐽𝐽𝑚𝑚𝑥𝑥 \nis different for left - and right -circularly polarized SWs whereas the sign of 𝜕𝜕𝑥𝑥𝜌𝜌𝑚𝑚 is \nindependent of the circular  polarization of SW. From Eqs. (6) and (7), we finally obtain the \nsteady -state velocity of DW below the Walker breakdown [ 53] as \n𝑣𝑣𝐷𝐷𝐷𝐷=𝑠𝑠\n2𝜌𝜌(𝛼𝛼2𝑠𝑠2+𝛿𝛿𝑠𝑠2)�𝛼𝛼𝜆𝜆𝐹𝐹𝑚𝑚+𝛿𝛿𝑠𝑠\n𝑠𝑠𝑇𝑇𝑚𝑚�,                     (8) \nwhich is the central  result of this  work.  The first and second term s originate from non -\nadiabatic and adiabatic contributions , respectively. In Eq. (8), t he ratio 𝛿𝛿𝑠𝑠/𝑠𝑠 is an estimate  of \nthe degree how the dynamics of the system is close to that of ferromagnets. The condition of \n𝛿𝛿𝑠𝑠2𝑠𝑠⁄→±1 represent s the ferromagnetic limit, whereas that of 𝛿𝛿𝑠𝑠2𝑠𝑠⁄→0 represent s the \nantiferromagnet ic limit. In the ferromagnetic limit (𝛿𝛿𝑠𝑠2𝑠𝑠⁄→±1), the second term in Eq. (8) \nbecomes  dominant  for the DW motion. On the other hand, in the antiferromagnetic limit \n(𝛿𝛿𝑠𝑠2𝑠𝑠⁄→0), the second term vanishe s and the only first term  is responsible for DW  motion . \nTo verify Eq. (8), we perform micromagnetic simulation s with the atomistic LLG \nequation. We start with the initial condition that the DW  is located at the center of one -\ndimensional nanowire as shown in Fig. 1(b). SW  is excited  by an external AC field  𝐁𝐁𝐀𝐀𝐀𝐀 on \nthe left side of DW . The atomistic LLG equation including the external AC field is given by  \n𝜕𝜕𝑺𝑺𝒊𝒊\n𝜕𝜕𝑡𝑡=−𝛾𝛾𝑖𝑖𝑺𝑺𝒊𝒊×�𝐁𝐁𝐞𝐞𝐞𝐞𝐞𝐞,𝒊𝒊+𝐁𝐁𝐀𝐀𝐀𝐀�+𝛼𝛼𝑖𝑖𝑺𝑺𝒊𝒊×𝜕𝜕𝑺𝑺𝒊𝒊\n𝜕𝜕𝑡𝑡,                 (9) \nwhere 𝑺𝑺𝒊𝒊 is the normalized spin moment vector, 𝛾𝛾𝑖𝑖=𝑔𝑔𝑖𝑖𝜇𝜇𝐵𝐵ℏ⁄ is the gyromagnetic ratio,  \n𝜇𝜇𝐵𝐵 is the Bohr magneton, and 𝛼𝛼𝑖𝑖 is the damping constant at a lattice site 𝑖𝑖. The o dd (even) \nsite 𝑖𝑖 corresponds to the TM (RE) element. 𝐁𝐁𝐞𝐞𝐞𝐞𝐞𝐞,𝒊𝒊=−1\n𝜇𝜇𝑖𝑖𝜕𝜕ℋ\n𝜕𝜕𝑺𝑺𝒊𝒊 is the effective field  at each \nsite, where 𝜇𝜇𝑖𝑖 is the magnetic mome nt per atom , one dimensional discrete Hamiltonian \nℋ=𝐴𝐴𝑠𝑠𝑖𝑖𝑚𝑚∑𝑺𝑺𝒊𝒊∙𝑺𝑺𝒊𝒊+𝟏𝟏 𝑖𝑖 −𝐾𝐾𝑠𝑠𝑖𝑖𝑚𝑚∑(𝑺𝑺𝒊𝒊∙𝒛𝒛�)2\n𝑖𝑖 +𝜅𝜅𝑠𝑠𝑖𝑖𝑚𝑚∑(𝑺𝑺𝒊𝒊∙𝒙𝒙�)2\n𝑖𝑖 , 𝐴𝐴𝑠𝑠𝑖𝑖𝑚𝑚 and 𝐾𝐾𝑠𝑠𝑖𝑖𝑚𝑚 (𝜅𝜅𝑠𝑠𝑖𝑖𝑚𝑚) are the 7 \n exchange constant and easy  (hard) axis anisotropy for simulations , respectively . To excite the \nSW, an external AC field 𝐁𝐁𝐀𝐀𝐀𝐀=B0[cos𝜔𝜔𝑡𝑡𝒙𝒙�±sin𝜔𝜔𝑡𝑡𝒚𝒚�] is applied on two cells at \n252  nm away  from the DW. We use the following simulation parameters: 𝐴𝐴𝑠𝑠𝑖𝑖𝑚𝑚=\n1.64 meV , 𝐾𝐾𝑠𝑠𝑖𝑖𝑚𝑚=6.47 𝜇𝜇eV, 𝜅𝜅𝑠𝑠𝑖𝑖𝑚𝑚=0.02𝐾𝐾𝑠𝑠𝑖𝑖𝑚𝑚, B0=100  mT, the lattice constant is \n0.42 nm, and the Land é 𝑔𝑔–factor 𝑔𝑔𝑅𝑅𝑅𝑅=2 for rare earth  and 𝑔𝑔𝑇𝑇𝑀𝑀=2.2 for transition  \nmetal  [54]. We consider the damping constant is uniform for all site s, i.e., 𝛼𝛼RE=𝛼𝛼TM=5×\n10−4 for simplicity.  We use m agnetic moment s 𝑀𝑀RE and 𝑀𝑀TM as listed in TABLE 1.   \n \nIII. Results and Discussion  \nFigure 3(a) -(c) show the simulation results of  DW velocity as a function of the SW \nfrequency.  Figure 3(a) represents the results for the case below  the angular momentum \ncompensation point 𝑇𝑇𝐴𝐴. As the SW gap is different for left - and right -circular ly polarized  \nSWs [Fig. 2(a)], the threshold SW frequenc y for the DW  motion  is also different for left - and \nright -circular ly polarized SWs. An interesting observation for the DW motion is that the \nmoving  direction of DW  depends on the circular  polarization of SW . Left- (Right -) circular ly \npolarized SW moves the  DW toward s (away  from ) the SW source. This bi -directional DW \nmotion is understood by the fact that l eft- and right-circular ly polarized SWs  carry the \nangular momentum with opposite  signs. When the SW passes through the DW, the angular \nmomentum of SW is transferred to the DW so that the DW moving  direction depends on the \ncircular polarization of SW . This is directly related to the fact that the sign of 𝑇𝑇𝑚𝑚 is different \nfor left - and right -circular ly polarized SWs , whereas the sign of 𝐹𝐹𝑚𝑚 is independent of the \ncircular  polarization of SW. Given that  𝑇𝑇𝑚𝑚 and 𝐹𝐹𝑚𝑚 in Eq. (8) respectively correspond to \ncontributions from the adiabatic and non- adiabatic magnon ic torques , the bi -directional DW \nmotion depending on the circular polarization of SW evidences  that the adiabatic magnon ic \ntorque is dominant over the non- adiabatic one.  \nSolid and dashed lines in Fig. 3(a) are calculated from Eq. (8), with 𝐹𝐹𝑚𝑚 and 𝑇𝑇𝑚𝑚 \nobtained from numerical calculation s. We find that the numerically obtained bi -directional \nbehavior (symbols) is reasonably described by Eq. (8) in high- frequency ranges.  In low-\nfrequency  ranges , a discrepancy between Eq. (8) and numerical results appears possibly 8 \n because of nonlinear effects , which are  not captured by our current analytical models . The \nresults for the case above the angular momentum compensation point 𝑇𝑇𝐴𝐴 [Fig. 3(c) ] can be \nunderstood in a similar way . Contrary to the case below 𝑇𝑇𝐴𝐴, overall spin moment s in the \nsystem are reversed so that left - (right -) circular ly polarized SW makes DW move away from \n(toward s) the source.  \nTo further elucidate SW -induced ferrimagnetic DW motion  below and above  𝑇𝑇𝐴𝐴, we \ninvestigate the spin current 𝐽𝐽𝑆𝑆, which is defined as 𝑱𝑱𝒔𝒔=−𝐴𝐴〈𝒏𝒏×𝜕𝜕𝑥𝑥𝒏𝒏〉. Figure 4 shows the \nschematic of SW transmission through a DW (top panel) and the z  component of the spin \ncurrent 𝐽𝐽𝑆𝑆𝑧𝑧 along the propagation direction (i.e., the 𝑥𝑥 axis, bottom panel). For the left - \ncircular ly polarized SW (solid line) , the spin current in the left domain part decreases \ngradually due to the damping. After the SW passes through the DW, the spin current abruptly \nflips its sign due to overall reversal of spin moments. The spin -current  change is transferred \nto the DW, resulting in the DW motion. For the right -circular ly polarized SW (dashed line), \noverall sign of the spin current is reversed. It is the reason that  the direction of DW \npropagation is the opposite for left - and right -circularly polarized SWs.  The sign of the spin-\ncurrent change is the same below and above 𝑇𝑇𝐴𝐴, but 𝛿𝛿 𝑠𝑠 changes its sign  [see Eq. (8)]  \nbecause the spin directions in the domain part changes accordingly , which results in the sign \ndifference of the DW velocity below and above 𝑇𝑇𝐴𝐴.  \nFor the case at the angular momentum compensation point 𝑇𝑇𝐴𝐴 (i.e., 𝛿𝛿𝑠𝑠=0), both \nleft- and right-circular ly polarized SWs  drive the DW to the same direction ( i.e., toward s the \nSW source)  as shown in Fig. 3(b) . We note that this DW moving direction at 𝑇𝑇𝐴𝐴 is the \nopposite to the direction of the DW motion induced by circularly polarized spin waves  in true \nantiferromagnets [ 40,41]. In true antiferromagnets where the shape anisotropy is absent, \ncircular ly polarized SWs make the DW precess, which result s in the SW reflection. The \nreflected SWs transfer  linear momentum to DW, and push the DW away from the SW source. \nFor the case at 𝑇𝑇𝐴𝐴 of ferrimagnets, however, the net magnetic moment is finite so that the \nshape anisotropy does not vanish. As a result, the DW experiences the hard- axis anisotropy, \nwhich prevents the DW precession. Therefore, the ferr imagnetic DW still serves as a \nreflect ionless potential called the Pöschl -Teller potential [ 55] and its motion is governed by \nthe force from the  magnonic torque [ 𝛼𝛼𝜆𝜆𝐹𝐹𝑚𝑚 term in Eq. (8)]. As the sign of 𝐹𝐹𝑚𝑚 is 9 \n independent of the SW circular polarization, both left - and right -circular ly polarized SWs pull \nthe DW along the same direction , i.e., towards the SW source. This force -induced motion of \nthe ferrimagnetic DW toward the spin -wave source at 𝑇𝑇𝐴𝐴 is similar  to the motion of the \nantiferromagnetic DW toward the spin -wave source for lin early -polarized spin waves \nreported in Ref. [40] . \nDependence of the DW velocity on the SW circular polarization is summarized in \nFig. 3(d), which  shows the DW velocity as a function of the net spin density 𝛿𝛿𝑠𝑠 at a fixed \nSW frequency  (𝜔𝜔 = 0.7 THz) . The sign of DW velocity depends not only on the circular \npolarization, but also on the sign of the net spin density. We note that the DW velocity is not \nzero at 𝑇𝑇𝐴𝐴 because the adiabatic and non- adiabatic contributions are not compensated at 𝑇𝑇𝐴𝐴.  \n \nIV. Summary  \nWe have investigate d the SW circular -polarity dependence of  ferrimagnetic DW \ndynamics theoretically and numerically.  We find that the DW moves along  the opposite \ndirection depending on the circular polarization of SW. This bi -directional DW motion is \ncaused by the fact that  the signs of the spin current  and the angular momentum transferred  to \nDW are opposite for left - and right -circular ly polarized SWs . The o verall tendency of DW \nmoving  direction is reversed when the sign of the net spin density  𝛿𝛿𝑠𝑠 is reversed. At 𝑇𝑇𝐴𝐴 \nwhere the angular momentum vanishe s, the dissipative  non-adiabatic magnonic torque is the \nmain driving force so that DW moves along the same direction (towards the SW source) \nregardless of the SW circular polarization .  \nOur finding of bi-directional  ferrimagnetic DW driven by SWs can be generalized to \nother ferrimagnetic topological excitations  such as magnetic skyrmions and vortices. This bi-\ndirectionality  of ferrimagnetic DW motion depending either on the SW circular polarization \nor on the sign of the net spin density will be useful for magnonic spintronics [ 56] because \nsuch bi -directional motion, which makes the device functionality versatile, can be realized \nwithout moving the location of a  SW source.  \n 10 \n Acknowledgement  \nThis work was supported by the National Research Foundation of Korea (NRF) \n(Grants No. 2015M3D1A1070465 and No. 2017R1A2B2006119), the KIST Institutional \nProgram (Project No. 2V05750). S.K.K. was supported by the startup fund at the University \nof Missouri. J.X. was supported by the National Natural Science Foundation of China (Grants No. 11722430).  \n  11 \n Table 1. Used magnetic moments  MTM and MRE for transition metal and rare earth elements , \nrespectively , in simulation. Index 5 coincides  with the angular momentum compensation \npoint TA. \nIndex  1 2 3 4 5 6 7 8 9 \n𝑀𝑀𝑇𝑇𝑀𝑀(kA/m) 460 455 450 445 440 435 430 425 420 \n𝑀𝑀𝑅𝑅𝑅𝑅(kA/m) 440 430 420 410 400 390 380 370 360 \n \n  12 \n  \nFigure 1. (a) Illustration  of left- and right -circular ly spin wave s in an antiferromagnetically \ncoupled ferrimagnet. (b) Schematic graphic of one -dimensional ferrimagnet ic nanowire  with \na domain wall  (DW). Domain wall is positioned at the center of nanowire . Spin wave is \nexcited by an external AC field ( 𝐁𝐁𝐀𝐀𝐀𝐀) on the left side  (252 nm  apart from DW ). \n  \n13 \n  \nFigure 2. Spin- wave dispersion relations  (a) below  TA, (b) at  TA, and (c) above  TA. Symbols \nrepresent numerical simulation  results and lines represent Eq. (4). Solid  (Open ) triangular \nsymbols  correspond to left - (right -) circularly  polarized spin wave . \n  \n14 \n  \nFigure 3. Calculated domain wall velocity results  (a) below  TA, (b) at  TA, and (c) above TA \nwith various spin- wave frequencies . Symbols are the simulation results and lines are Eq. (8) \n(arbitrary unit). (d) Domain wall velocity as a function of the net spin density δs at a fixed \nfrequency ω = 0.7 THz . Negative  δs corresponds to the case below TA. \n  \n15 \n  \nFigure 4. Schematic of  transm itted SW (top)  and the z component of the spin current  Js along \nthe wire length. A DW is positioned at the atomic site i  = 2000, and SW source is at  i = \n1700. Assumed parameters are those with the index 3 (i.e., below TA) listed in the TABLE  1 \nand the SW  frequency is 0.6  THz. \n  \n16 \n Reference  \n1. R. Wangness, Phys. Rev. 91, 1085 (1953).  \n2. M. Binder et al., Phys. Rev. B 74, 134404 (2006).  \n3. C. Stanciu, A. Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, \nPhys. Rev. B 73, 220402 (2006).  \n4. J. M. D. 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B 99, \n054423 (2018).  \n53. N. L. Schryer and L. R. Walker, J. Appl. Phys. 45 , 5406 (1974).  \n54. J. Jensen and A. R. Mackintosh, Rare Earth Magnetism (Clarendon, Oxford, UK, 19 \n 1991). \n55. G. Pöschl and E. Teller, Zeitschrift für Physik 83, 143 (1933).  \n56. A. V . Chumak, V . I. Vasyuchka, A. A.  Serga, and B. Hillebrands, Nat. Phys. 11, 453 \n(2015).  \n "    },    {        "title": "2209.11562v1.Magnetostatics_of_Room_Temperature_Compensated_Co_Gd_Co_Gd_based_Synthetic_Ferrimagnets.pdf",        "content": "arXiv:2209.11562v1  [cond-mat.mes-hall]  23 Sep 2022Magnetostatics of Room Temperature Compensated\nCo/Gd/Co/Gd-based Synthetic Ferrimagnets\nThomas J. Kools,1Marnix C. van Gurp,1Bert Koopmans,1and Reinoud Lavrijsen1\nDepartment of Applied Physics, Eindhoven University of Tec hnology\nP . O. Box 513, 5600 MB Eindhoven, The Netherlands\n(*Electronic mail: t.j.kools@tue.nl)\n(Dated: 26 September 2022)\nFlexibility for interface engineering, and access to all-o ptical switching of the magnetization, make synthetic ferr i-\nmagnets an interesting candidate for advanced opto-spintr onic devices. Moreover, due to their layered structure and\ndisordered interfaces they also bear promise for the emergi ng field of graded magnetic materials. The fastest and most\nefficient spin-orbit torque driven manipulation of the magn etic order in this material system generally takes place at\ncompensation. Here, we present a systematic experimental a nd modeling study of the conditions for magnetization\ncompensation and perpendicular magnetic anisotropy in the synthetic ferrimagnetic Co/Gd/Co/Gd system. A model\nbased on partial intermixing at the Co/Gd interfaces of this system has been developed which explains the experiments\nwell, and provides a new tool to understand its magnetic char acteristics. More specifically, this work provides new\ninsight in the decay of the Co proximity-induced magnetizat ion in the Gd, and the role the capping layer plays in the\nGd magnetization.\nThe ever expanding rate of data generation and consump-\ntion propels research into new material systems to use for\nprocessing and storage of information. Therefore, one ma-\njor challenge of contemporary research in spintronics is to de-\nvelop material systems of which the magnetization can be ma-\nnipulated both time- and energy-efficiently. 3d-4f ferrima g-\nnetic material systems, like GdFeCo and CoTb alloys, and\nmultilayers based on a combination of these metals, are attr ac-\ntive due to their antiferromagnetically coupled sublattic es1–7.\nThese materials aim to combine favorable properties of thei r\nferromagnetic and antiferromagnetic counterparts, and be ar\npromise for the emerging field of graded magnetism8. They\nhave garnered a great amount of attention from the sci-\nentific community due to their access to single-pulse all-\noptical switching (AOS) of the magnetization5,9–11, efficient\nspin-orbit torque (SOT)-driven manipulation of the mag-\nnetic order12–16and exchange torque driven current induced-\ndomain wall motion (CIDWM) with velocities over 1000\nm/s2,3,7. Hence, these developments push the search for ma-\nterial platforms for domain wall-based memory in advanced\nsolid state devices like racetrack memory17–19. Interestingly,\nthe combination of AOS and efficient CIDWM in this mate-\nrial system is also very promising to bridge the gap between\nphotonics and spintronics20–23.\nCo/Gd-based synthetic ferrimagnetic bilayers, where the\n3d and 4f-material are grown as discrete layers, have a few\ndistinct advantages over 3d-4f alloys. The layered structu re\nof these synthetic ferrimagnets allows for easier adaption to\nwafer scale production. Also, contrary to alloys, a much\nwider composition range between the 3d and 4f-metal ex-\nhibits AOS24,25. Combined with the increased access to in-\nterfacial engineering, this leads to more flexibility and tu n-\nability of its magnetic properties. Moreover, the Pt/Co/Gd\ntrilayer displays strong interfacial spintronic effects, such as\nperpendicular magnetic anisotropy (PMA), the spin-Hall ef -\nfect, and the interfacial Dzyaloshinskii–Moriya interact ion, all\nimportant aspects for applications based on efficient domai nwall motion19,26–28. Despite these favorable properties, the\nengineering relevance of the Co/Gd bilayer system has been\nlimited due to the absence of both magnetization and angu-\nlar momentum compensation, where the two magnetic sub-\nlattices cancel each other, at room temperature. For it is we ll\nknown that CIDWM2,3,21and in general SOT-driven ferrimag-\nnetic spin dynamics4,12–14are most effective close to the an-\ngular momentum or magnetization compensation point.\nIn this work, we therefore investigate the conditions for\ncompensation in Co/Gd/Co/Gd, which we from now on\ndub the quadlayer system. Compared to the Co/Gd bi-\nlayer, we double the magnetic volume of the Co while\ntripling the number of Co/Gd interfaces where magnetiza-\ntion is induced in the Gd through direct exchange with\nthe Co11,19. This is expected to enhance the contribution\nof the Gd to the net magnetic moment, while still main-\ntaining PMA. The samples nominally consist of stacks of\nTaN(4 nm)/Pt(4)/Co(0.6)/Gd( tGd1)/Co( tCo2)/Gd( tGd2)/TaN(4)\nas schematically drawn in Fig.2c, which were grown on\nSi/SiO 2substrates through magnetron sputtering in a cham-\nber with a typical base pressure of 5 ×10−9mBar. The first\nsample is fabricated using wedge sputtering in order to con-\nfirm that compensation is achieved. Specifically, in the first\nsample the middle Gd thickness tGd1is varied between 0 and\n1.5 nm over a few mm (see inset Fig. 1a), whereas tCo2and\ntGd2are constant and set to 0.7 and 1.5 nm, respectively.\nThe magnetic properties of this wedge were investigated\nby the polar magneto-optic Kerr effect (pMOKE), where we\nare only sensitive to out-of-plane (OOP) components of the\nCo magnetization, as Gd does not contribute appreciably to\nthe pMOKE signal at our used wavelength of 658 nm29. We\nscan the sample locally using a focused laser spot. At mag-\nnetic compensation (e.g. from a Co-dominated to a Gd dom-\ninated region or vice-versa) two effects are expected: a di-\nvergence of the coercivity and a sign change in the pMOKE\nsignal. The former can be observed in Fig. 1a, where the\ncoercivity extracted from hysteresis loops measured acros s2\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s77/s79/s75/s69/s32/s115/s105/s103/s110/s97/s108\n/s32/s67/s111/s101/s114/s99/s105/s118/s105/s116/s121 \n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s77/s79/s75/s69/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s46/s117/s46/s41\n/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48\n/s32/s67/s111/s101/s114/s99/s105/s118/s105/s116/s121/s32/s40/s109/s84/s41Co Gd Co Gd (a)\n/s45/s51/s48/s48 /s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s49/s48/s49/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s77/s79/s75/s69/s32/s115/s105/s103/s110/s97/s108\n/s109\n/s48/s72 /s32/s40/s109/s84/s41/s32/s116\n/s71/s100/s61/s48/s46/s56/s51/s32/s110/s109\n/s32/s116\n/s71/s100/s61/s48/s46/s57/s52/s32/s110/s109(b)Right axisLeft axis\nFIG. 1. Polar MOKE characterization of a Co/Gd/Co/Gd sample\nwhere the middle Gd layer is wedged between 0 and 1.5 nm. a):\nRemanent polar MOKE signal normalized by its value at tGd1= 0 nm\n(black), and coercive field (red) as a function of Gd layer thi ckness.\nb): Two sample hysteresis loops measured by polar MOKE at a Co\n(black) and Gd-dominated magnetic composition (red) on the wedge\nin a).\nthe wedge is plotted in red. The divergence follows from\nthe inefficiency of the Zeeman interaction in a compensated\nsystem. This divergence coincides with a change in sign of\nthe remanent pMOKE signal (Kerr rotation, normalized to its\nvalue at tGd1= 0 nm) which is plotted in black in Fig. 1a.\nTo understand this sign change, we must consider that in the\nGd-dominated regime the Zeeman energy dictates that the Gd\nmagnetization aligns with the magnetic field. The measured\nCo-magnetization will consequently align antiparallel to the\nfield, leading to the change in sign of the pMOKE signal. The\nchange in the hysteresis is illustrated in Fig. 1b, where the\nblack and red loops are measured in the Co ( tGd1= 0.83 nm)\nand Gd-dominated magnetic regime ( tGd1= 0.94 nm), respec-\ntively. The 100% remanence observed indicates the PMA in\nthis sample.\nIn order to obtain information on the tunability of the com-\npensation point and PMA, as a low net magnetization would\nimply a large effective anisotropy, we use orthogonal doubl e\nwedge samples. In Fig. 2a we illustrate the Co/Gd/Co/Gd\ndouble wedge sample structure. After deposition of the first/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s67/s111/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41Gd \nCo \nGd \nCo (a) (b)\nmCo >m Gd \nmGd >m Co \nIn plane\nM\nz\n0Co 1\nGd 1 Co 2 \nGd 2 \nEd\nz\n0Co 1 Co 2 \nGd 2 \nTaN Pt TaN \na0\na0\na0\nGd 1 \na0 a0a0Co 1Gd 1Co 2Gd 2(c) (d)\n(e)λ0MCo0 \nMGd0\nFIG. 2. a): Schematic of the double wedge samples under inves -\ntigation in this report. b): Polar MOKE scan of a double wedge\nCo(0.6 nm)/Gd(x-axis)/Co(y-axis)/Gd sample(1.5). The co mpensa-\ntion boundary is indicated by the green line. c): Schematic i llus-\ntration of the model used to describe the compensation and SR T-\nboundary for the magnetostatic phase diagrams. Magnetic la yers are\nmodelled with the inclusion of an intermixing region with a w idth\na0. d) and e) respectively, illustrate the magnetization and s hape\nanisotropy energy as modelled throughout the four magnetic layers.\nGd wedge, the sample is rotated by 90 degrees and the Co\nwedge is deposited. After the sample is saturated with an\nOOP magnetic field of 1 T we scan the sample surface and de-\ntermine the remanence from the pMOKE signal at each point\nwhen no magnetic field is applied. Using this method allows\nus to scan the full parameter space of nominal layer thick-\nnesses of the middle two layers in a single sample. A typical\nresulting diagram of the remanent pMOKE signal is shown in\nFig. 2b for a sample where tGd1=0−3 nm and tCo2=0−2\nnm, keeping the top Gd thickness at tGd2= 1.5 nm at which\nwe anticipate, based on earlier work, the proximity-induce d\nmagnetization in the Gd to be saturated11,19. In the diagram,\nwe can distinguish between three basic states. The red and\ndark blue regions indicate stack compositions where the mag -\nnetization points OOP, with the Co or Gd magnetization being\ndominant, respectively. The light blue region indicates st ack\ncompositions where the magnetization points in-plane (IP) ;\nthis is above the spin reorientation transition (SRT), wher e the\ninterfacial PMA is not sufficient to keep the full stack OOP.\nThese three regions define two major transitions of interest :\nthe compensation boundary (red to dark blue) and the SRT\nboundary (red/dark blue to light blue).\nTo obtain a quantitative understanding of the shapes of\nthese boundaries, a model has been developed to simulta-\nneously describe the compensation boundary and the SRT\nboundary. Furthermore, the model will be used to get insight\nin the basic properties of the proximity induced magnetizat ion3\nin the Gd and level of intermixing, a quantity that has not bee n\ninvestigated using these double wedged samples. We set out\nto model the net magnetization, which is zero at the compen-\nsation boundary, as well as the magnetostatic free energy of\nthe anisotropy, which is zero at the SRT boundary. One of\nthe main assumptions of the model relies on the experimental\nobservation that the interface between Co and Gd thin films\nare intermixed30–32. The assumed Co and Gd concentration\nas a function of thickness is illustrated in Fig. 2c. The lay-\ners are modelled by means of the four magnetic layers in our\nCo/Gd/Co/Gd structure, with each layer assumed to be sepa-\nrated by an intermixing region with a constant and identical\nwidth of a0.\nIn order to find the magnetization compensation point, we\nthen describe the net magnetization of this multilayer stru c-\nture, which vanishes at compensation. We use typical assump -\ntions for the magnetization profile of the Co/Gd bilayer to de -\nscribe the magnetization in our Co/Gd/Co/Gd system, which\nare illustrated in Fig. 2d11,19. The Co magnetization is crudely\nassumed constant throughout the nominal thickness of the Co\nlayer, with a value MCo0, giving:\nMCo1=MCo0, (1)\nand\nMCo2=MCo0Feq,Co, (2)\nwhere, in order to implement the intermixing regions into th e\nchange of the magnetization with layer thickness, we empir-\nically define the continuous function Feq,Co(see Sup. A for\ndetails). It describes the transition to an equilibrium mag netic\nstate with middle Co layer thickness caused both by the effec t\nof intermixing on the magnetization, as well as percolative be-\nhavior. The latter of which describes the minimum thickness\nneeded to stabilize a coherent ferromagnetic state.\nIn contrast to the Co magnetization, the magnetization in\nthe Gd layers is mainly induced at the interface with the Co\nlayer11,19. Therefore, the magnetization as a function of the\ndistance to the Co/Gd interface z∗will be described by an ex-\nponentially decaying profile, which is typical to describe m ag-\nnetization induced at an interface between a ferromagnet an d\na non-magnetic metal33–35, given by:\nMGd1=MGd0(exp/parenleftbigg\n−z∗\nλ0/parenrightbigg\n+exp/parenleftbiggz∗−tGd1\nλ0/parenrightbigg\n)Feq,CoFeq,Gd,\n(3)\nand\nMGd2=MGd0exp(−z∗/λ0)Feq,CoFeq,Gd, (4)\nwhere MGd0is the magnitude of the magnetization at the in-\nterface, λ0is the characteristic decay length of the magneti-\nzation, and Feq,Gdis a similar empirical function to Feq,Co, de-\nscribing the development of the Gd magnetization with mid-\ndle Gd layer thickness (see Sup. A). Note that the effective\nexponential decay constant λ0is influenced by many parame-\nters, like surface roughness, the actual degree of intermix ing,local ratios between Co and Gd atoms and the actual decay\nof the magnetization induced in the Gd, and should hence be\ninterpreted as an effective parameter describing the colle ctive\nbehavior of all these effects. The resulting total magnetic mo-\nment per unit area mtotcan then be extracted by integrating the\nmagnetizations over the respective layer thicknesses:\nmtot=4\n∑\ni=1/integraldisplayti\n0Midz∗. (5)\nNext, the SRT-condition needs to be implemented. There\nare two main contributions to the effective anisotropy: the in-\nterfacial anisotropy energy and demagnetization energy. T he\nmagnetocrystalline anisotropy energy per unit area KSdue to\nthe Co/Pt interface is assumed constant. The free energy den -\nsity per unit area due to the shape anisotropy Edis schemati-\ncally plotted in Fig. 2e. It is calculated by treating the sys tem\nas a continuous magnetic system and integrating the typical\nexpression for the volume demagnetization energy density o f\na thin film with OOP magnetization MS:E∗\nd=1\n2µ0M2\nS, where\nµ0is the magnetic permeability of vacuum. In order to ac-\ncount for the smaller demagnetizing field in the intermixing\nregions where the magnetization is inherently lower, we sub -\ntract the demagnetization energy of the intermixing region ,\nwith a characteristic width of a0, from the total demagnetiza-\ntion energy leading to the following expression for the tota l\narea-normalized demagnetization energy Ed:\nEd=4\n∑\ni=1/integraldisplayti\n01\n2µ0M2\nidz∗\n−/parenleftbigg/integraldisplaya0/2\n01\n2µ0M2\nCo1dz∗+Feq,mix/integraldisplaya0\n01\n2µ0M2\nGd1dz∗\n+Feq,mix/integraldisplaya0\n01\n2µ0M2\nCo2dz∗\n+/integraldisplaya0/2\n01\n2µ0M2\nGd2dz∗/parenrightbigg\n, (6)\nwhere Feq,mixis an identical empirical function to Feq,Gdand\nFeq,Coused to describe the onset and saturation of the inter-\nmixing regions in the Gd layer upon changing the layer thick-\nness (see Sup. A). The resulting total free energy density pe r\nunit area is Etotcan then be calculated by adding EdandKS\ntogether.\nWe will use this model for the magnetization and anisotropy\nenergy to test our physical understanding and make an esti-\nmate of the (effective) physical parameters underpinning t hese\nsystems by fitting it to the measured phase diagrams, again\nconsidering the SRT-boundary and compensation boundary to\nbe at Etot= 0 and mtot= 0, respectively. To test the quantita-\ntive applicability of our model to these Co/Gd/Co/Gd system s,\nthree double-wedged samples are considered: Co(0.6)/Gd(0 -\n3)/Co(0-2)/Gd( tGd2) with tGd2= 0.7, 1.0 and 1.5 nm, respec-\ntively, where the thicknesses chosen are expected to probe\ndifferent degrees of decay of the induced magnetization in t he\nGd. The phase diagrams measured on the three samples are\nshown in Fig. 3 a, b and c for tGd2= 0.7, 1.0 and 1.5 nm,4\n/s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s108\n/s48/s32/s40/s110/s109/s41\n/s116\n/s71/s100/s50/s32/s40/s110/s109/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s67/s111/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s67/s111/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s67/s111/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41(a) (b) (c)\ntCo1= 0.6 nm \ntGd2 = 0.7 nm tCo1 = 0.6 nm\ntGd2 = 1.0 nmtCo1 = 0.6 nm\ntGd2 = 1.5 nm\n(d) (e) (f)\n/s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s97\n/s48/s32/s40/s110/s109/s41\n/s116\n/s71/s100/s50/s32/s40/s110/s109/s41/s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s77\n/s71/s100/s48/s32/s40/s77/s65/s47/s109/s41\n/s116\n/s71/s100/s50/s32/s40/s110/s109/s41\nFIG. 3. Magnetostatic phase diagrams with model fits of the ma gnetization (orange) and demagnetization energy (green) f ortGd2=a): 0.7 nm,\nb): 1.0 nm, and c): 1.5 nm. The main magnetic parameters extra cted from the fitting procedure for the three phase diagrams: d): Magnetization\nof Gd at the Co/Gd interface MGd0, e): Magnetization decay length λ0, and f): intermixing region width a0.\nrespectively, where we can immediately observe that with in -\ncreasing top Gd thickness the area of the Gd-dominated regio n\nincreases.\nBefore fitting the model to explain the features of these\nphase diagrams, we experimentally characterize the interf a-\ncial anisotropy strength due to the Co/Pt interface KSto be\n1.22 mJ/m2(see Appendix B), and set the Co saturation mag-\nnetization MCo0equal to the bulk magnetization of Co at 1.4\nMA/m. The other parameters in the model are left uncon-\nstrained. In Fig. 3a,b,c we give the resulting fits for mtot=0\nand Ed=0 in orange and green, respectively. We find a\ngood correspondence between the model and the experiment.\nSpecifically, The curvature of the magnetization profile, an d\nthe corresponding magnetostatic energy balance character ized\nby the peaked shape, are both generally well described. Par-\nticularly for the samples with tGd2= 0.7 and 1 nm the cor-\nrespondence is good across the whole phase diagrams. For\ntGd2= 1.5 nm in Fig. 3c, the model correspondence on the\nSRT-boundary becomes worse for tGd1>1.5 nm. It is not yet\nbeen unequivocally established what causes this differenc e be-\ntween experiment and theory. We speculate that it might be\ndue to finite size effects affecting the Curie temperature in the\nGd and hence the total amount of magnetization induced be-\nyond what is currently implemented in the model. Based on\nthe model we can attribute the peak in effective anisotropy\nin the phase diagrams to the first ∼1 nm of Gd contribut-ing mainly to the intermixing regions, leading to the initia l\nincrease, after which pure Gd is found, which decreases the\neffective anisotropy, leading to the decline from ∼1 nm on-\nwards. Moreover, the change in curvature of the compensatio n\nboundary for tCo<0.5 nm between the three samples can now\nbe attributed to the percolation limit approach of the Co lay er\nleading to either one or three interfaces which induce a net\nmagnetization in the Gd layers.\nTo also illustrate the quantitative value of the model, we\nwill now discuss the magnetic parameters extracted from the se\nfits. In particular, the parameters fixing the hitherto unkno wn\nGd magnetization profile λ0,MGd0, and a0are of interest here.\nThese parameters are plotted for the three fitting procedure s in\nFig. 3d, e and f, respectively. All other parameters found in\nthe fitting procedure are listed in appendix C. The extracted\nGd interfacial magnetization of about 1.3 MA/m is compa-\nrable with values found in earlier work11,19,26. Next, the λ0\nvalues in the order of 1 nm suggest that the magnetization pro -\nfile extends well beyond the first monolayer affected by direc t\nexchange with the Co layer. This observation is further cor-\nroborated when considering the difference between identic al\nphase diagrams capped with Ta and TaN (see Appendix D).\nThere we observe that the Gd- dominated OOP regime (dark\nblue in earlier diagrams) in the phase diagram extends all th e\nway to Co-thicknesses of 0 nm in the TaN-capped sample,\nwhereas in Ta-capped samples of otherwise identical compo-5\nsition a minimum middle Co thickness is always required to\nreach the Gd-dominated regime, indicating an overall reduc -\ntion in the total magnetization of the Gd. We postulate that\nthis difference is caused by magnetization quenching in the\nGd due to intermixing between the capping layer and the Gd,\na process that will likely be more severe for atomic Ta than fo r\ncovalently bound TaN36. Finally, we will discuss the resulting\nvalues for a0(Fig. 3f). Regarding the growth of Gd on Co\nand vice versa, earlier work demonstrated that the interfac es\nbetween multilayers of Co and Gd are disordered30,31,37, and\nthat the exact growth and intermixing dynamics also depend\non the order of growth of the two layers30,38. This indicates\nthat the found intermixing region width a0of around 0.8 nm\nis in line with earlier work. A reasonable comparison can be\nmade to the [Pt/Co/Gd]-multilayers investigated by Nishim ura\net al., where using transmission electron microscopy inves ti-\ngations similar typical intermixing region widths were fou nd\nas we find from fitting our model here, i.e., in the 0.5-1 nm\nrange32.\nIn conclusion, we have experimentally demonstrated mag-\nnetic compensation in the synthetic ferrimagnetic quadlay er\nCo/Gd/Co/Gd system. It is found that compensation can be\neffectively tuned by layer thickness. We also demonstrated\nthe utility of orthogonally wedged samples to characterize the\nnominal thickness parameter space in order to investigate t he\nmagnetostatics of these systems and consequently find stack\ncompositions with favorable magnetic properties. Finally , a\ncrude model for the net magnetization and PMA was devel-\noped which described the experiments well, providing an ef-\nfective framework to discuss the magnetostatics in these co m-\npensated multilayered ferrimagnetic systems with PMA. We\nnote that this is probably an oversimplified model to describ e\nthe real intermixing profiles, e.g. the constant Co magneti-\nzation with thickness. It however provides a good qualitati ve\nframework to build more detailed models which will require\nrefinement of the assumed magnetization profiles using high-\nresolution depth sensitive magnetometry. This work improv es\nthe understanding of basic magnetostatic properties and gi ves\ninsight in the more fundamental aspects of the design and\nphysics of these promising and flexible multilayer systems.\nACKNOWLEDGMENTS\nThis work was part of the research program Foundation for\nFundamental Research on Matter (FOM) and Gravitation pro-\ngram “Research Center for Integrated Nanophotonics,” whic h\nare financed by the Dutch Research Council (NWO). This\nwork was suported by the Hedrik Casimir Institute.\nAUTHOR DELCARATIONS\nConflict of Interest\nThe authors have no conflicts to discloseAuthor Contriubtions\nThomas J. Kools: Conceptualization (equal); Investigation\n(lead); Methodology (lead); Writing – original draft (lead );\nWriting – review and editing (lead). Bert Koopmans: Con-\nceptualization (equal); Funding acquisition (equal); Sup er-\nvision (equal); Writing – review and editing (supporting).\nReinoud Lavrijsen: Conceptualization (equal); Funding ac-\nquisition (equal); Supervision (equal); Writing – review a nd\nediting (supporting).\nDATA AVAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.6\n/s45/s49/s53/s48/s48 /s45/s49/s48/s48/s48 /s45/s53/s48/s48 /s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s77/s65/s47/s109/s41\n/s109\n/s48/s72 /s32/s40/s109/s84/s41\nFIG. 4. In-plane SQUID characterization of the magnetic mom ent of\na Ta(4 nm)/Pt(4)/Co(1)/TaN(4) as a function of in-plane fiel d. The\nred dot indicates the extracted anisotropy field.\nAppendix A: Percolation functions\nIn order to implement the intermixing regions into the\nchange of the magnetization with layer thickness, we empir-\nically define continuous functions Feq,GdandFeq,Codescrib-\ning the transition to an equilibrium state with respective l ayer\nthickness. These describe both percolative behavior; the m in-\nimum thickness needed to stabilize a coherent ferromagneti c\nstate, and the effect of intermixing on particularly the Gd m ag-\nnetization:\nFeq,Gd(tGd1)=erf((tGd1−t0,Gd)/LGd1)+1\n2, (A1a)\nFeq,Co(tCo2)=erf((tCo2−t0,Co)/LCo2)+1\n2, (A1b)\nwhere t0,Gd1andt0,Co2, and LGd1andLCo2, are parameters\ndefining the critical thickness and characteristic width of the\npercolation, respectively.\nFeq,mixis an identical empirical function to those presented\nin Eq. A1 used to describe the onset of the intermixing region s\nin the Gd layer upon changing the layer thickness:\nFeq,mix(tGd1)=erf((tGd1−t0,mix)/Lmix)+1\n2. (A2)\nAppendix B: Characterization Ks\nIn order to estimate the anisotropy constant KS, we per-\nformed IP VSM-SQUID measurements (see Fig. 4) on a\nTa(4)/Pt(4)/Co(1)/TaN(4) sample. The resulting hard-axi s re-\nsponse to the IP field is typical of a sample with PMA like\nthis multilayer. We find an anisotropy field of 900 mT. To es-\ntimate the corresponding KS, we use a simple Stoner-Wolfarththeory, considering three contributions to the magnetosta tic\nfree energy: Zeeman energy Ez, interfacial anisotropy from\nthe Co/Pt interface EK, and the shape anisotropy Es. The re-\nsulting total energy Etotis then given by the sum of these three\ncontributions:\nEtot=1\n2µ0M2\nScos2(θ)+sin(θ)/parenleftbigg\n−µ0HM S+KSsin(θ)\nt/parenrightbigg\n,\n(B1)\nwhere µ0is the permeability of vacuum, MSis the saturation\nmagnetization, θis the angle between the magnetization and\nthe thin film sample normal, His the applied magnetic field,\nandtis the thickness of the magnetic layer. By minimiz-\ningEtotwith respect to θand setting θ=90◦we find the\nanisotropy field Ha:\nHa=1\n2µ0t/parenleftbig\nHM S+M2\nS/parenrightbig\n. (B2)\nForMS=1.4 MA/m (SQUID), t=1 nm and Ha=900 mT\n(SQUID), we find KS=1.22 mJ/m2.\nAppendix C: Fitting parameters\nThe parameters for the best fit of the model described in\nthe main text to the double wedge Co/Gd/Co/Gd samples as\nshown in Figs. 3a, 3b and 3c are described in this section.\nTables I, II and III show these fitting parameters for top Gd\nthickness tGd2=0.7 nm, tGd2=1.0 nm, tGd2=1.5 nm, re-\nspectively.\nTABLE I. Summary of the fitting parameters found to best descr ibe\nthe compensation boundary and boundary from OOP to IP magne-\ntization in the double wedge with top Gd thickness tGd2=0.7 nm\n(see Fig. 3a). For MS,CoandKSwe choose 1.4 MA/m and 1.22\nmJ/m2as explained in the main text and appendix B. Errors repre-\nsent 95% confidence intervals extracted from the fitting proc edure.\nParameter Value Error Unit\nMGd1 1.2 0.2 MA/m\nt0,Co 0.21 0.01 nm\nt0,Gd1 1.2 0.2 nm\nt0,mix 0.47 0.01 nm\nLCo 0.43 0.03 nm\nLGd1 0.83 0.02 nm\nLmix 1.3 0.1 nm\nλ0 0.97 0.16 nm\na0 0.66 0.01 nm\nAppendix D: Comparison phase diagram capping layers\nIn Fig.5 a and b the magnetostatic phase diagrams of a\nCo(0.6)/Gd(x)/Co(0.7)/Gd(1.5) stack with a 4 nm thick cap-\nping layer of TaN and Ta are plotted, respectively. The\nmost important difference to note here is that the region\nwhere the magnetization is OOP and the Gd-contribution is7\nTABLE II. Summary of the fitting parameters found to best de-\nscribe the compensation boundary and boundary from OOP to IP\nmagnetization in the double wedge with top Gd thickness tGd2=\n1.0 nm (see Fig. 3b). For MS,CoandKSwe choose 1.4 MA/m\nand 1.22 mJ/m2as explained in the main text and appendix B. Er-\nrors represent 95% confidence intervals extracted from the fi tting\nprocedure.\nParameter Value Error Unit\nMGd1 1.29 0.17 MA/m\nt0,Co 0.13 0.01 nm\nt0,Gd1 0.72 0.15 nm\nt0,mix 0.39 0.01 nm\nLCo 0.63 0.02 nm\nLGd1 0.56 0.06 nm\nLmix 0.27 0.01 nm\nλ0 1.3 0.2 nm\na0 0.83 0.02 nm\nTABLE III. Summary of the fitting parameters found to best de-\nscribe the compensation boundary and boundary from OOP to IP\nmagnetization in the double wedge with top Gd thickness tGd2=\n1.5 nm (see Fig. 3c). For MS,CoandKSwe choose 1.4 MA/m\nand 1.22 mJ/m2as explained in the main text and appendix B. Er-\nrors represent 95% confidence intervals extracted from the fi tting\nprocedure.\nParameter Value Unit\nMGd1 1.57 0.16 MA/m\nt0,Co 0.10 0.02 nm\nt0,Gd1 0.66 0.21 nm\nt0,mix 0.43 0.01 nm\nLCo 1.14 0.03 nm\nLGd1 3.1 1.1 nm\nLmix 0.28 0.01 nm\nλ0 1.50 0.35 nm\na0 0.90 0.03 nm\ndominant (dark blue) extends all the way to zero Co thick-\nness for the TaN cap. In contrast, the Ta-capped sample\nmagnetization only becomes dominated by the Gd magne-\ntization for a minimum Co thickness of about 0.4 nm. In\nFig. 5 a and b we plot magnetostatic phase diagrams for a\nTa(4)/Pt(4)/Co(0.6)/Gd(x)/Co(0.7)/Gd(1.5) with a TaN an d Ta\ncapping layer, respectively.\nBIBLIOGRAPHY\n1S. K. Kim, G. S. D. Beach, K.-J. Lee, T. Ono, T. Rasing, and H. Ya ng, “Fer-\nrimagnetic spintronics,” Nature Materials , vol. 21, pp. 24–34, Jan 2022.\n2K.-J. Kim, S. K. Kim, Y . Hirata, S.-H. Oh, T. Tono, D.-H. Kim, T . Okuno,\nW. S. Ham, S. Kim, G. Go, Y . Tserkovnyak, A. Tsukamoto, T. Mori yama,\nK.-J. Lee, and T. 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Parkin, “Exchange coupling torqu e in ferri-\nmagnetic Co/Gd bilayer maximized near angular momentum com pensation\ntemperature,” Nature Communications , vol. 9, p. 4984, Nov 2018.\n27K.-S. Ryu, L. Thomas, S.-H. Yang, and S. Parkin, “Chiral spin torque at\nmagnetic domain walls,” Nature Nanotechnology , vol. 8, pp. 527–533, Jul\n2013.\n28S.-H. Yang, K.-S. Ryu, and S. Parkin, “Domain-wall velociti es of up to 750\nm s−1driven by exchange-coupling torque in synthetic antiferro magnets,”\nNature Nanotechnology , vol. 10, pp. 221–226, Mar 2015.\n29J. L. Erskine and E. A. Stern, “Magneto-optic kerr effects in gadolinium,”\nPhys. Rev. B , vol. 8, pp. 1239–1255, Aug 1973.\n30J. P. Andrés, J. L. Sacedón, J. Colino, and J. M. Riveiro, “Int erdiffusion up\nto the eutectic composition and vitrification in Gd/Co multi layers,” Journal\nof Applied Physics , vol. 87, no. 5, pp. 2483–2489, 2000.\n31J. A. Alonso, R. H. de Tendler, D. A. Barbiric, and J. M. 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Brennan,\n“Structural characterization of multilayers using x-ray d iffraction,” MRS\nProceedings , vol. 239, p. 475, 1991."    },    {        "title": "0808.2001v1.High_output_CPP_GMR_sensor_with_synthetic_ferrimagnet_free_layer_and_enhanced_spin_torque_critical_currents.pdf",        "content": " \n \nHigh-output CPP-GMR sensor w ith synthe tic-ferrimagnet free layer and \nenhanced spin-torque critical currents \n \n \nM.J. Carey, N. Sm ith, S. Maat and J.R. Childress. \n \nSan Jose Research Center, Hitachi Global Stor age Technologies, 3403 Yerba Buena Rd, San Jose, \nCA 95135 \n \nAbstract: \n \nIt is shown that the m aximum stable output of a CPP-GMR sensor is increas ed significantly by using \na synthetic f errimagnet f ree layer, provided the el ectron cu rrent flows from  free l ayer t o reference  \nlayer.  This  free laye r allows a lar ger m agneto resistance ratio for a gi ven free layer m agnetic \nmom ent, and in addition results in  a greater th an three-fold increa se in the cr itical curr ent ab ove \nwhich spin- torque ins tability of  the f ree laye r occurs.  In read heads with net free layer m oments \nequivalent to only 4.5nm of Ni 80Fe20, this effect is show n to result in sustainable sense current  \ndensities above 2x108 A/cm2. \n Spin-valves (SV) have been used in m agnetic  recording since 1998, whe n the current-in-plane \n(CIP) gian t magnetoresistance (G MR) SV sensor was introduced 1 .  More recently the cu rrent-\nperpendicular to the plane (CPP) t unneling m agnetoresistance (TMR) sensor2 has becom e standard. \nHowever, in the CPP geom etry the sensor resist ance increas es with the sm aller sizes required for \nincreasing areal densities.  T oday’s sensor dimensions are al ready below 60nm  for >300 Gb/in2 \nrecording which has prom pted a major effort to reduce the large resistance-area (RA) product of  \nTMR sensors to  below 1 Ω-µm2.  Even so, t he high im pedence of TMR sensors at foreseeably \nsmaller dim ensions, and the accom panying ex cessive noise and degraded high-frequency \nperformance, m otivates a return to a  metallic -GMR SV sensor in the CPP geom etry (RA < 0.1 Ω-\nµm2).  However at the se low RA values, s ubstantial sense current densities (>1x108 A/cm2) are  \nrequired to generate sufficient output voltages, a nd the interaction of th e conduction electron spin \nwith th e sen se and  reference layer m agnetizations can resu lt in sp in-torque (ST) -induced in stability3 \nwhich can render a CPP -GMR SV sensor nonfunctional.   Therefore m ethods of retarding the onset \nof such ins tability is a key goal for the application of CPP-GMR spin -valves to recording sensors.   \nIn th is Le tter, we show that th e critical sen se current for ST- induced ins tability of  the free lay er can \nbe dram atically increased by the use of a  synthe tic-ferrim agnet free layer and the correct choice of \ncurrent direction.  This approach is shown to enable CPP-GMR read heads with sense current \ndensities up to 2x108 A/cm2.  \n Several m ethods have already been dem onstrat ed to reduce spin-torque effects in CPP-GMR \nsensors.  One m ethod is to use a \"dual-SV\" struct ure with a symm etric arrangem ent of two reference \nlayers,  whic h cance ls the net spin -torque on the central free layer4,5.  However the thicker sensor \nstacks of  a dual-SV lim its the linea r resolution of  the sensor, problem atic for reaching high densities.  \nAnother m ethod is to increase the m agnetic dam ping of the free layer, eith er by rare-earth dopants6 \nor cap layers7, or by the spin-pum ping effect using cap layers such as Pt8. Spin-torque effects can \nalso be reduced by increasing free  layer sa turation magnetiza tion (M s) and/or total m agnetic m oment, \nbut these v alues are typically con strained by the de sired m agnetic and/or transport properties of the \nsensor within existing recording syst em and other design considerations.  \nA known alternative free layer structure, but wh ose behavior under spin-t orque excitations is \nfirst stud ied here, is th e synthetic-f errimagnet (SF) f ree layer  or antipa rallel (AP) f ree layer9.  In the \nSF-FL structure (Fig.1 inset) the free layer is a multilay er of the type FL1/APC/FL2, where APC is \nan antiparallel coupling layer such as Ru and FL 1 and FL2 are two separate m agnetic layers. T his \n configuration is sim ilar to the widely-used synt hetic–antiferrom agnet (SAF) pinned layer structure \nformed by pinned layer/APC/reference layer, ex cept that the SF-FL stru cture has a nonzero net \nmagnetic mom ent mFL=m1-m2, whe re m1 and m2 are the m agnetic mom ents/area of FL1 and FL2, \nrespec tively. This al lows, for a given m FL, to use a relatively thick FL 1 thickness, which can increase \nthe CPP m agnetores istance of  the s pin-va lve10.  In addition, it is of intere st to evaluate the influence \nof the additional FL2 layer on the overall ST -induced excitations of the free layer. \nSpin-valv e film s for t he present work we re deposited by m agnetron sputtering onto Si  \nsubstrates as discussed  elsewhere4.  For m agneto- transpo rt measurem ents, m ultilayer f ilms were  \npatterned into pillars with nom inal diam eters between 300 and 50nm  using e-beam  lithography and  \nAr+ ion m illing.   The full structure for the devi ces described here was underlayer/5 Ta/1.5 Cu/ 7 \nIrMn/3 CoFe/0.55 Ru/1 CoFe/0.4 Cu/1 CoFe/0.4 Cu/1 CoFe/5 Cu/FL or SF-FL/1 R u/cap layer, with \nall thickn esses in nm .  The CoFe composition is  Co 50Fe50 (atom ic %).  T he SF-FL structure was 0.6  \nCoFe/4+t NiFe/0.2 CoFe/0.55 Ru/0.2 CoFe/t NiFe, resulting in a constant net m FL of 0.36m emu/cm2, \nequivalent to 4.5nm  of Ni 80Fe20 (Ms=800 em u/cc).   A control struct ure with a single free m agnetic \nlayer (FL) of 0.6 CoFe/3.8 Ni Fe/0.2 CoFe ( also with a m FL=0.36m emu/c m2,) was used for  \ncomparison.  Magnetoresistance m easurem ents were perform ed quasistatically in +1.5kOe applied \nfields using a constant voltage, where a positive vo ltage is defined to produce an electron current as \nshown in Fig.1. ∆RA values were derived from  the resulting R-H curves, where Rmin is the res istance \nof the device when the reference lay er and FL1 m agntizatio ns are  parallel (P -state), ∆R is the  change \nin resistance upon switching FL1 to be antiparalle l with the reference layer (AP-state), and A is the \ndevice area.    Rmin vs. 1/ A plots were used to correct the resi stance valu es for lead resistances and \nlithographic windage effects.  The RAmin value of all the structures is about 40 m Ω-µm2.  \nFig.1 shows the value of the m agnetoresistive signal ∆RA (m easured at -10m V) as a function  \nof the FL2 NiFe thickn ess t.  ∆RA is notably higher for the AP-free lay er structu res as com pared to \nthe control structure,  increasing from 0.68 +0.04 to 0.93+0.03 m Ω−µ m2 for devices w ith a FL2 NiFe \nthickness t > 1.5nm .   This corresponds to ∆R/Rmin values increasing from  1.7 to 2.7%.  For CPP-\nGMR devices, it m ight be expected  that the FL2 layer will contr ibute a  negative  value to the to tal \n∆RA due to its an tiparallel m agnetiz ation o rientat ion co mpared to FL1.  However th is is not \nnotice able h ere, con sistent with the s hort sp in-diffusion length ~4nm  previously estimated for NiF e11 \nas the NiFe thickness of FL1 is >4.5nm for  all SF-FL sam ples (FL1 also includes two CoFe \nnanolayers).   \n  \n \nFig.1:  ∆RA vs. NiFe thickness in F L1 (bottom  axis) and FL2 (top axis ).  Inse t: Sp in-\nvalve structure with synthetic-ferrim agnet free layer.  For transpo rt measurem ents, \npositiv e elec tron cur rent is def ined as  electron tr aveling f rom reference la yer to f ree layer. \n \nSpin-torqu e stability was exam ined using lock-in m easure ment of edI dV R / =′  vs.  (tickle-\ncurren t = 40 µA) in \"hexagonal\" device with cross sectional area sim ilar to that of a 75nm  diameter  \ncircle. Two exam ples of these m easur ements, along with several corresponding eI\nHR-∆  loops (with \nH collinear to the long hexagon axis ), are shown in Fig. 2. At sufficiently sm all H and/or , these \ndevices are bi-stable due to the uniaxial shape anisotropy and both eI\nHR-δ  (at low ) and eIeIR-′δ  \nloops (at ) show the expected hysteresis. The latt er, in partic ular, ind icate that the overall ST-\nstability of the FL, when employing positive (negati ve) sense current, is  then limite d by its pos itive \n(negative)  critical cu rrent  in th e AP-state ( P-state). 0=H\ncrit\neI\n  \nFig.2: transfer curves at discrete  as indicated (left), and  transfer \ncurves  at (right), where  HR−δeIeIR−′δ\nOe750,0±=HedIdVR / ≡′ . The value of minR  or minR′ is \nsubtracted out, and the  curves are further alig ned at HR−δ R R ∆→ δmax  to remove \ntherm al shif t. nm5.02FL= t NiFe for (a) and (c ), nm5.22FL= t  NiFe for (b) and (d) The  \narrows in (c) and (d) ind icate the ons et of spin-torque instability for   Oe750±=H\n \nHowever,  can depend on the device coercivity  (which itself is dependent  \non device geom etry that is irregular at 50 nm  dimensions), and is susceptible to therm al)0 (crit=H Ie )0 (→ecIH\n12  and self-\nfield effects . A m ore reliab le measurem ent of critical cu rrents can be achieved  using external \nfields , such as shown in Fig. 2c,d withcrit\neI\n)0 (→ >>e cIH H Oe750±≅H , where at most  only one \nstate (P or AP) is both m agnetostatically  and S T-stable. Here, The values of , denoting the start \n(from  ) of continuous instabilit y, are determ ined by observing the onset of significant \ndeviation from  the otherwise parabolic shape of crit\neI\n0=eI\n)(eIR′δ  due to Joule heating. These deviations, \ndiscernable by inspection for the (nonhysteretic) )(eIR′δ shown in Fig. 2, are sym ptomatic of  \ncontinuous, precessional-lik e motion of the FL magneti zation . It is generally accom panied by a rapid \nincrease in low-frequency 1/ f-like noise which  serves  as an alte rnative (usually more sensitive) \ndetection technique as w as shown earlier13 and employed recently14. \n Fig. 3 shows a summ ary of all m easured  vs FL2 thickness, for devices initially stable in \neither P o r AP states.  The enhancem ent in negative  (in P-state) with in creasing FL 2 thickness is  \ndramatic , and dram atically larger  than that ob served f or positive  (in AP-state). Th ese resu lts \nare highly consistent (both qual itatively and qu antitatively ) with  further m easurements on 75-nm \ndiam eter circular dev ices of very sim ilar cross s ection al area14. In addition, the enhancem ent with FL2 \nthickness of negative values of  for the unid irectiona lly P-s table dev ices with crit\neI\ncrit\neI\ncrit\neI\ncrit\neI Oe750−=H  is \nalso s een to be substantially  larger than that observed when unia xially bi-stable at . A physical \nexplanation, along with a sim ple macrospin m odel which reproduces all of  these observations, is \ndescribed elsewhere0=H\n14. \n \nFig.3:  Spin-torque critical current  as a function of FL2 thickness t for spin-v alve with  \nsynthetic-ferrim agnet free la yer structure and constant net m agnetic mom ent. Two or \nthree dev ices are tes ted for each valu e of t, with t=0 corresponding to a single-FL control \nstructure. The m ean resistance of these devices is about 10 Ω. \n \nAs a practical m atter, one is free to choose the po larity of the sense curr ent, for which negative \n is clearly preferable for the single m agnetic la yer FL design. To test device perform ance under \nrecording test conditions, fil ms similar to those desc ribed above were fabricated into stabilized read-eI\n only heads using a com bination of ebeam  and opt ical lithography down to ~ 30n m track-width sizes.  \nFor this experim ent the FL2 NiFe thickness t was chosen to be 2.5 nm, so that the corresponding \nFL1 NiFe t hickness was 6.5 nm .  This resulted in device-level ∆R/R min values of about 2.5%.  \nFabrication details and test conditions were sim ilar to those previously described4. Under these \nconditions, the behavior of the sensor as a function of voltage bias  was tested to evaluate the sensor \nbehavior with respect to ST-induced excitations.  Fig. 4 shows the output am plitude vs. bias for  \nvarious heads with ~30-40nm  phys ical track-widths and an RA value of about 40 m Ω-µm2.  For \npositiv e bias, transf er curves  beco me rapidly di storted with voltages  as low as  15-20 m V, as \nevidenced b y the decreased am plitude with in creas ing bias.  For negative bias  however, the \namplitude continues to increase with bias up to values in excess of 100 m V.  This  corresponds to \ncurrent densities of about Jmax ~ 2.5x108 A/cm2, which is m ore than twic e the m aximum  values we  \nhave previously reported for CPP dual-spin valves, which, in turn, are higher than those observed in \nsingle-spin valves with non s ynthetic-ferrim agnet free layers4.  These high values of Jmax illustrate  the \neffectiveness  of the SF-FL structure in suppressi ng the effect of spin-torque on the free layer for  \nCPP-GMR s ensors.   \n \nFig.4:  Read -only record ing head ou tput am plitude vs. bias v oltage applied to the sensor.  \nThese head s include a sy nthetic- FL structure similar to that described here with FL2 NiFe \n thickness ~ 2.5nm .  Only with negative bias can  substantial current de nsities be achieved \nbefore the onset of excessive ST noise. \n \nIn conclusion, we have shown that the use of a synthetic ferrim agnet free layer structure for \nCPP-GMR sensors, tog ether with a  sensor curr ent where electrons flow  from  the free layer to the \nreference la yer, resu lts in a substan tial increase in the critical current density Jmax to greater than  \n2x108 A/cm2.   Along with increased spin-valve signal ∆RA, increasin g Jmax in he ads results in an \nincrease in sensor output voltage ∆V = ∆RA x Jmax,  a key requirem ent for im proving the \napplic ability  of CPP-GMR to high -density m agnetic  recording sensors.  Consequently, synthetic–\nferrim agnetic free layers in CPP-GMR sensors consti tute a prom ising route for the next generation \nmagnetic recording sensors at 500 G b/in2 and beyond. \n \nThe authors thank Ching Tsang and Jim  Moore fo r valuable assis tance with reco rding head  \ncharacterization. \n  \n REFERENCES \n                                                 \n1 C. Tsang, R.E. Fontana, T. Lin, D.E. Heim , V.S. Speriosu, B.A. Gurney, M.L. Mason, \nIEEE Trans Mag 30, 3801 (1994). \n2 S. Mao, Y. Yonghua, F. Liu, X. Chen,  B. Xu,  P. Lu, M. Patwari, H. Xi, C. Chang, B. \nMiller, D. Menard, B. Pant, J. Loven, K.  Duxstad, S. Li, Z. Zhang, A. Johnston, R. \nLamberton, M. Gubbins, T. McLaughlin, J. Gadboi s, J. Ding1, B. Cross, S. Xue, and P. \nRyan.  IEEE Trans. Mag. 42, 97 (2006).  \n3 S.I  Kiselev, J.C Sankey, I.N. Krivorotov, N.C. Em ley, R.J. Schoelkopf,R.A. Buhr man, \nD.C. Ralph, Nature 425, 380 (2003). \n4 J.R. Childress, M.J. Carey, M.-C. Cyrille, K. Carey, N. Sm ith, J.A. 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Phys. 99, 08Q703 (2006). \n14 N. Sm ith, S. Maat, M.J. Carey, J. R. Childress, arXiv:cond-m at/0808.2015.  \n \n "    },    {        "title": "1810.00763v1.Trimers_of_MnO6_octahedra_and_ferrimagnetism_of_Ba4NbMn3O12.pdf",        "content": "1 \n Trimer s of MnO 6 octahedra and ferri magnetism of  Ba4NbMn 3O12 \nLoi T. Nguyen , Tai Kong and R.J. Cava  \nDepartment of Chemistry, Princeton University, Princeton, New Jersey 08544, USA  \n \n \nAbstract  \nBa4NbMn 3O12 is reported, synthesized by a solid state method  in air . The crystal structure , \ndetermined by performing refinements on room temperature powder X -ray diffraction data  by the \nRietveld method,  consists of  Mn 3O12 trimers  in the configuration of  three face -sharing MnO 6 \noctahedra , with the trimers arranged in triangular planes . An effective moment of 4.82 μB/f.u is \nobserved and  competing antiferromagnetic and ferromagnetic  interaction s between Mn  ions are \ninferred from  the Weiss temperature of -4 K and the ferrimagnetic ordering transition of \napproximately 4 2 K. Ba4NbMn 3O12 is a semiconductor with  a transport  activation energy of 0.37 \neV. \n \n  2 \n Introduction  \nMangan ese-based  perovskites have been intensively researched due to  the metal -insulator (MI) \ntransition s and colossal magnetoresistance ( CMR )1,2,3 ,4 in this family . The interesting properties \nhave been attributed in part to  the competition between  a ferromagnetic metallic state and an \nantiferromagnetic insulating state , and the presence of charge -ordering . Manganates are also well \nknown due to other  structural, electronic , and magnetic characteristics5,6. In the hexagonal \nperovskite  RMnO 3 phases,  for example, the electric polarization is reported to be the result of a \nstructural transition7 and the improper  nature of ferroelectricity  to a network of coupled structural \nand magnetic vortices that induce domain wall magnetoelectricity and magnetization8.  \nOf particular interest for  CMR are materials  with lower structural dimension  based on the \nRuddlesden -Popper series9, but quasi -1D materials are also of  interest10. Instead  of sharing corner s \nin a 3D network as  in corner -sharing  perovskites, the MnO 6 octahedra  in the quasi -1D materials \ncan share faces with  each other  to form Mn 2O9 dimers or Mn 3O12 trimers. These Mn mini-clusters \nare isolated from other clusters in the structure. This type of MnO 6 configuration is relatively \nuncommon  because  the Mn-Mn distance s of about 2.5  Å within the cluster s are shorter than the \n3.5 Å typically found in corner -sharing geometries11. Oxygen deficient BaMnO 3-x and \n(Ba,Sr)MnO 3-x phases are good examples of materials where clusters of face -sharing MnO 6 \noctahedra are found12. \nIn th e current  work, we report the synthes is and initial characterization of the  mixed valent \nMn3+/Mn4+ compound  Ba4NbMn 3O12. The crystal structure  of this previously unreported phase , \nwhich is much like that of Ba 4REMn 3O12 (RE = Ce, Pr )13, consists of face -sharing MnO 6 octahedra  \nform ing Mn 3O12 trimers  with their long axes aligned along the hexagonal c axis of the trigonal \nsymmetry crystallographic cell.  From  magnetic susceptibility measurement s, the effective 3 \n magnetic moment is found  to be  4.82 μB/f.u., consistent with a simple spin configuration within \nthe trimers. Resistivity measurement s show that Ba 4NbMn 3O12 is semi conducting with a transport \ngap of 0.37 eV . The current phase is chemically and structurally distinct from recently reported \nBa8MnNb 6O24, which, among other differences, does not appear to contain any face shared \noctahedra14.  \nMethod s \nPolycrystalline samples of Ba 4NbMn 3O12 were synthesized by solid -state reaction using BaCO 3, \nMnO2, and Nb2O5 (Alfa Aesar, 99.9 %, 99.999 %, and 99.5 %, respectively ) as starting materials . \nReagents  were mixed thoroughly in the appropriate ratio, placed in an alumina crucible, and heated \nin air at 900 ℃ for 24 hours. The resulting powder was re -ground, pressed into a pellet and heated \nin air at 1100 ℃ for 24 hours, and then at 1300 ℃  for 12 hours. The phase purity and crystal \nstructure  were determined through powder X -ray diffraction (PXRD) using a Bruker D8 Advance \nEco with Cu Kα radiation and a LynxEye -XE detector. The structural refinements were performed  \nwith GSAS15. The crystal structure drawing was  created by using the program VESTA16. \nThe magnetic susceptibility of Ba 4NbMn 3O12 powder was measured by a Quantum Design \nPhysical Property Measurement System (PPMS) DynaCool equipped with a VSM option. The \nmagnetic susceptibility of Ba 4NbMn 3O12, defined as M/H, where M is the sample magnetization \nand H is the applied field, was measured at the field of H = 1 kOe from 1.8 K to 300 K; some \nadditional measurements were performed in an applied field of 100 Oe. (Relatively low fields are \nemployed so as to not overly disrupt the magnetic s ystem through the measurements.) The \nresistivity of Ba 4NbMn 3O12 was measured by the DC four -contact method in the temperature range \n200 K to 3 50 K with the PPMS. The sample was pressed, sintered, and cut into pieces with the \napproximate size 1.0 × 2. 0 × 1.0 mm3. Four Pt contact wires were connected to the sample using 4 \n silver paint. The specific heat was measured  from 1.8 K to 200 K by a PPMS DynaCool equipped \nwith a heat capacity  option . \nResults and discussion  \nThe powder x-ray diffraction pattern and structural refinement of Ba 4NbMn 3O12 are shown in \nFigure 1 . The structure of Ba 4NbRu 3O1217 was used as the starting model.  Ba4NbMn 3O12 \ncrystallizes in a rhombohedral structure with the space group R-3m (No. 166). Refinements in \nwhich the Mn to Nb ratio was allowed to vary from 3:1 or Nb/Mn site mixing was permitted were \nnot satisfactory. The lattice parameters and structural parameters for Ba4NbMn 3O12 are \nsummarized in Table 1 . The structure consists of three MnO 6 octahedra connected by face -sharing \nto form Mn 3O12 trimers , as shown in Figure 2 . The crystal structure of Ba 4NbMn 3O12 is similar to \nthose of two known rare-earth containing compounds Ba 4REMn 3O12 (RE = Ce, Pr)13 in the same \nfamily. The corner -sharing NbO 6 and Mn 3O12 trimers in Ba 4NbMn 3O12 alternate  along c to \ngenerate the 12 -layer hexagonal perovskite structure . Individual Mn 3O12 trimers are co rner-sharing \nwith the non -magnetic NbO 6 octahedra , and not to other trimers, such that the magnetic coupling \nbetween trimers is of the Mn -O-O-Mn super -super exchange type . Within  each trimer, the distance \nbetween Mn atoms is 2.4694(1) Å, quite short due to the face-sharing11, favoring  strong magnetic \ninteractions between them. The Mn -O distances are 1.903(4)  Å and 1.882(4)  Å for the outer MnO 6 \noctahedra , reflecting a small distortion,  and 1.895(4)  Å for the inner MnO 6 octahedron , which has \nan ideal shape . These Mn-Mn and Mn-O bond lengths are in agreement with those found for the  \n10-layer hexagonal  perovskite Ba5Sb0.64Mn 4.36O15-δ18, for example .  \nThe temperature -dependent magnetic susceptibility of Ba 4NbMn 3O12 and its reciprocal are plotted \nin Figure  3. The magnetic data for Ba 4NbMn 3O12 from 100 K to 275 K are well fit to the Curie -5 \n Weiss law 𝜒=𝐶\n𝑇−𝛳𝐶𝑊+𝜒𝑜, where 𝜒𝑜 is the temperature -independent part of the susceptibility, C \nis the Curie constant, and ϴCW is the Curie -Weiss temperature.  The least squares fitting yields  𝜒𝑜= \n2.79x10-3 emu Oe-1 mol-f.u.-1, 𝜇𝑒𝑓𝑓= 4.82 μB/f.u, and 𝛳𝐶𝑊= -4 K. The magnetic hysteresis loops \nfor Ba 4NbMn 3O12 from -2 to 2 T at 2 K, 25 K and 150 K   are shown in Figure S 1 (See SI) . A \ncoercive force of 0.2 T and the remnant magnetization of 0.04 μ B/f.u. are observed at 2 K and 0.04 \nT and 0.02 μ B/f.u. at 25 K. The overall behavior is like that of a ferrimagnet. At 150 K, the \nmagnetization is linearly proportional to the field and there is no hysteresis loop.  \nBecaus e Ba2+ and Nb5+ (which has no electrons in d orbitals) are non -magnetic, the magnetic \nproperties of Ba 4NbMn 3O12 are determined by the intertrimer and intratrimer interactions of the \nMn 3O12 units. Formally, one  Mn3+ and two Mn4+ are found in each Mn 3O12 trimer . Thus, i n the \nparamagnetic state,  where the moments fluctuate, the total magnetization  per formula unit  is \nexpected to  be the sum of  the magnetization s of two Mn4+ and one Mn3+, which, should all the \nmoments be free to respond to an applied field  for temperatures below 300 K , yield a  total effective \nmoment of 6.16  (low spin Mn3+) or 7.35 (high spin Mn3+) μB/f.u. This magnetic state is not \ncompatible with the observed data. Figure  4 shows , in contrast,  a schematic of two simple  \nhypothetical magnetic  states within each trimer , which, to be consistent with our magnetic data, in \nthese simple scenarios , either  two Mn4+ bearing spin -3/2 point in the same direction  and opposite \nto the low spin  (S=1) Mn3+ between them  or two Mn4+ bearing spin -3/2 point in opposite direction s \nand the Mn3+ (high spin) with the spin -2 is between them . With this proposed magnetic coupling \nwithin the trimer , each trimer has spin -2 and a calculated effective moment of 4.90 μB/f.u. This is  \nin excellent agreement  with the observed effective moment of 4. 82 μB/f.u in the susceptibi lity \nmeasurement s, indicating  that either of these trimer -based spin models is likely to successfully \ndescribe the magnetic state of Ba 4NbMn 3O12 above the three -dimensional magnetic ordering 6 \n temperature  near 42 K . Compared to  Bi3Mn 3O1119, for example, which has mixed -valent Mn, an  \neffective magnetic moment of 6.27 μB/f.u and a Curie -Weiss temperature of 222 K, both the \neffective moment and Curie -Weiss temperature  of Ba4NbMn 3O12 are smaller.  The strong coupling \nof the Mn spins within the trimers to create a magnetic Mn “molecule” made of three Mn’s in the \ncurrent material is consistent with what has previously been found for (Ba,Sr)MnO 3 phases12,20.  \nThe fitted χ0 in this scenario is then a representation of the remnant susceptibility of the magnetic \nsystem after the local -only magnetic ordering has set in within the individual trimers at \ntemperatures above 300 K.   \nFigure  5 shows the field -cooled (FC) and zero -field-cooled (ZFC) DC susceptibility in an applied \nfield of 100 Oe for Ba 4NbMn 3O12. The increases of the  FC and ZFC susceptibility below  42 K \nreconfirm the magnetic transition temperature observed in the magnetic susce ptibility data shown  \nin Figure 3, and the χT vs. T data shown in the inset defines the temperature of the magnetic \ntransition at the same temperature through the dramatic change in the Curie constant . \nThe resistivity of Ba 4NbMn 3O12 is plotted as a function of reciprocal temperature in Figure 6. \nResistivity data from 300 to 350 K were fit to the standard model 𝜌=𝜌𝑜𝑒𝐸𝑎\n𝑘𝑏𝑇, and the transport \nactivat ion energy E a was calculated to be 0.37 eV. The inset shows the increase in resistivity when \ncooling. With the activation energy of 0.37 eV, Ba 4NbMn 3O12 is a semiconductor, similar to other \ntrimer -based compounds (Ba 4NbRu 3O12 and Ba 4LnRu 3O12 and Ba 4LnIr 3O12)17,21. \nFigu re 7A shows the specific heat divided by temperature of Ba 4NbMn 3O12 measured from 1.8 K \nto 200 K in its main panel, with the raw C p data shown in the inset . At 200 K, Cp has not yet \nreached the saturation value of  3NR  (N is the number of atoms) , but this is often encountered in \nmaterials where d ifferent atomic mass es and strong bonds between atoms  lead to very high 7 \n vibrational frequenc ies 22. Three  characteristic features are seen. The λ-anomaly corresponds to  \nthe magnetic  transition around 42 K and is shown in Figure 7B . In this case we employed a \npolynomial function fit to the data above and below the anomaly to estimate the  phonon  \nbackground ; at this level of analysis we do not attach any significance to the function employed to \nestimate the phonon part . In this way the magnetic entropy change in the higher temperature range \nis estimated  to be 0.45 J mol-1 K-1. This  relatively weak  anomaly implies  that there is a significant \nremnant  of magnetic entropy in Ba 4NbMn 3O12 below the 42 K transition.  The specific heat in the \nlower temperature range  is shown in more detail in Figure 7C, which shows another entropy loss \nat 5-6 K. In this case the phonon part of the specific heat is estimated by a Debye -Einstein model, \nagain to which we attribute no physical significance  at this level of analysis . The magnetic entropy  \nat this lower temperature , which d oes not clearly yield a feature  in the magnetic susceptibility, is \nfound to be 2.67  J mol-1 K-1. For these two temperature regions only, o nly one -fourth  of the \nmagnetic entropy expected for an S=2 Heisenberg system is recovered.  There is a more subtle \nfeature in C p/T at around 30 K that may hold a significant amount of entropy,  but we cannot \nreasonably  analyze it at this time. Aside from the most general conclusions  described here , analysis \nof the specific heat cannot be considered as well -established  at present , as we have not been able \nto make a non -magnetic analog to use to best model the phonon contribution to the specific hea t \nof this material . A straightforward analysis (See the SI , Figure S 2 and S 3) shows that the observed \nmagnetic transition near 42 K cannot possibly be due to the  presence of an Mn 3O4 impurity.  \n  8 \n Conclusion  \nBa4NbMn 3O12, a previously unreported material synthesized by a solid state method, crystallizes \nin a 12-layer hexagonal perovskite unit cell with the R-3m space group. Differences in the Mn -O \nbond lengths in the octahedra, although subtle, may reflect the presence of charge ordering , \nalthough we do not believe that the distinction is great enough to warrant  assigning specific charge \nstates to specific octahedra at the present time . The material has a high, temperature -dependent \nresistivity, indicating that it is semiconducting , with a transport activation energy of 0 .37 eV \nmeasured on a sintered polycrystalline pellet . From magnetic susceptibility measurements, the \nCurie -Weiss tempe rature is calculated to be -4 K, indicat ing the presence of competing \nferromagnetic and antiferromagnetic interaction s between Mn trimers . The effective moment of \n4.82 μB/f.u agrees with the value of 4.90 μB/f.u that can be deduced from a simple hypothetical \nmagnetic model  in which the spins within the trimer are essentially already ordered by 300 K, \nalthough the trimer -trimer ordering does not occur until much lower temperatures . The ordering \nof the trimer spins with respect to those in other trimers is what most likely gi ves rise to the \nmagnetic transition  observed  around  42 K.  Heat capacity measurement s show  a weak anomaly at \naround 42 K, supporting  the magnetic  ordering  transition  seen in magnetic susceptibility.  Only \nsmall amounts of magnetic entropy  are observed  through the  specific heat, consistent, in principle, \nwith the proposal that most of the magnetic entropy is lost at higher temperature than is studied \nhere. A non -magnetic analog for this system would be of interest  to account for the phonon \ncontribution i n the specific heat  so that a more detailed interpretation of the entropy will be \npossible . Magnetic neutron diffraction will also be of future interest to establish the nature of the \nmagnetism in this material.  As for Mn -based conventional perovskites, he xagonal Mn -based \nperovskites appear to d isplay rich magnetic phenomena.  9 \n Conflicts of interest  \nThere are no conflicts of interest to declare.  \nAcknowledgement  \nThis work was supported in its entirety by the Department of Energy Division of Basic Energy \nSciences, through the Institute for Quantum Matter, grant DE -FG02-08ER46544.  The authors \nthank  Daniel  Khomskii for valuable discussion s about manganite magnetism . 10 \n Reference s \n(1) E. O. Wollan and W. C. Koehler, Physical Review 100, 545 (1955).  \n(2) R. V. Helmolt, J. Wecker, B. Holzapfel, L. Schultz, and K. Samwer, Physical Review Letters \n71, 2331 (1993).  \n(3) S. Jin, T. H. Tiefel, M. Mccormack, R. A. Fastnacht, R. Ramesh, and L. H. Chen, Science \n264, 413 (1994).  \n(4) I. Álvarez -Serrano, M. López, C. Pico, an d M. Veiga, Solid State Sciences 8, 37 (2006).  \n(5) A. Sundaresan, A. Maignan, and B. Raveau, Physical Review B 55, 5596 (1997).  \n(6) R. Laiho, K. G. Lisunov, E. Lähderanta, P. Petrenko, J. Salminen, V. N. Stamov, and V. S. \nZakhvalinskii, Journal of Physics:  Condensed Matter 12, 5751 (2000).  \n(7) B. B. V. Aken, T. T. Palstra, A. Filippetti, and N. A. Spaldin, Nature Materials 3, 164 (2004).  \n(8) H. Das, A. L. Wysocki, Y. Geng, W. Wu, and C. J. Fennie, Nature Communications 5,  2998  \n(2014).  \n(9) Y. Morimoto, A. Asamitsu, H. Kawahara, Y. Tokura, Nature 380 141 (1996).  \n(10) J. Vente, K.V. Kamenev, D.A. Sokolov, Phys. Rev. B64 214403  (2001) . \n(11) A. Furrer, T. Strässle, J. P. Embs, F. Juranyi, V. Pomjakushin, M. Schneider, and K. W. \nKrämer, Physical Review Letters 107, 115502  (2011).  \n(12) J.J. Adkin and M.A. Hayward, Chem. Mat. 19 755 (2007).  \n(13) A. F. Fuentes, K. Boulahya, and U. Amador, Journal of Solid State Chemistry 177, 714 \n(2004 ). \n(14) F. Tao, C. Liang, X. Wang, X. Li, F. Porcher, M. Allix, F. Lu, H. Gong, L. Lu and X Kuang \nInorganic Chemistry 57 5732 (2018).  \n(15) B. H. Toby, J. Appl. Crystallogr. 34, 210 (2001).  \n(16) K. Momma and F.  Izumi, J. Appl. Crystallogr. 44, 1272 (2011).  \n(17) L. T. Nguyen, T. Halloran, W. Xie, T. Kong, C. L. Broholm, and R. J. Cava, Physical \nReview Materials 2, 054414  (2018).  \n(18) C. Yin, G. Li, W. A. Kockelmann, F. Liao, J. P. Attfield, and J. Lin, Chemistry of Materials \n22, 3269 (2010).  \n(19) A. A. Belik and E. Takayama -Muromachi, Journal of the American Chemical Society 131, \n9504 (2009).  \n(20) K. Kuroda, N. Ishizawa, N. Mizutani, and M. Kato, Journal of Solid State Chemistry 38, \n297 (1981).  11 \n (21) Y. Shimoda, Y.  Doi, M. Wakeshima, and Y. Hinatsu, Journal of Solid State Chemistry 183, \n1962 (2010).  \n(22) M. Akaogi and E. Ito, Geophysical Research Letters 20, 105 (1993).  12 \n Table 1. Structural parameters for Ba4NbMn 3O12 at 300 K . Space group R-3m (No. 166).  \nAtom  Wyckoff.  Occ.  x y z Uiso \nBa1 6c 1 0 0 0.1284 7(4) 0.0344(3) \nBa2 6c 1 0 0 0.2859 2(4) 0.0304(3) \nNb 3a 1 0 0 0 0.0083(3) \nMn1 3b 1 0 0 ½ 0.0195(3) \nMn2 6c 1 0 0 0.4121 8(4) 0.0146(3) \nO1 18h 1 0.4776 (6) 0.522 4(6) 0.12239 (4) 0.0212(3) \nO2 18h 1 0.4908 (6) 0.509 2(6) 0.29327 (4) 0.0127(3) \na = 5.7182 5(3) Å, c = 28.1158 (3) Å \nRwp = 10.30%, R p = 8.02%, R F2 = 10.80 % \n \n \n  13 \n  \n \nFigure 1  (Color online) : Rietveld P owder X -ray diffraction  refinement for Ba 4NbMn 3O12 in \nspace group R-3m. The observed X -ray pattern is shown in black, calculated in red, \ndifference (I obs-Icalc) in blue, the background in green, and tick marks denote allowed peak \npositions in pink (Ba 4NbMn 3O12) and in cyan (BaNb 0.5Mn 0.5O3).  \n \n14 \n  \nFigure  2 (Color online) : The crystal  structure of Ba 4NbMn 3O12. The NbO 6 octahedra (dark \ngreen)  share corners with  the Mn 3O12 (purple) trimers made from face -shared MnO6 \noctahedra, to form what is frequently referred to as a “12 -layer” hexagonal perovskite \nstructure. Barium is green, and oxygen is red.  \n15 \n  \nFigure  3 (Color online) : The temperature dependence of the magnetic susceptibility and the \ninverse of the difference between the magnetic susceptibility and the temperature \nindependent magnetic susceptibility ( 𝝌𝒐  = 2.79 ×10-3 emu Oe-1 mol- f.u.-1) for Ba 4NbMn 3O12. \nThe applied field was 1 kOe. The red solid line is the susceptibility fit to the Curie -Weiss law \nfor T from 100 K-275 K.  \n  \n16 \n  \nFigure 4 (Color online): Two s chematic s of possible simple  hypothetical  arrangement s for \nMn3+ (either S=1 for low spin (left model) or S=2  for high spin  (right model) ) and Mn4+ \n(always S=3/2) in Ba 4NbMn 3O12. Both hypothetical arrangements  yield  spin-2 for each \nMn 3O12 trimer, μ eff = 4.90 μ B/f.u., which compares very  favorably to the observed μ eff = 4.82 \nμB/f.u. \n  \n17 \n  \nFigure 5  (Color online) : Field  Cooled ( FC) and Zero Field Cooled ( ZFC ) DC magnetic \nsusceptibility in an applied field of 100 Oe for Ba 4NbMn 3O12 from 2 -300 K. The inset shows \nthe first derivative of magnetic susceptibility  multiplied by temperature as function of \ntemperature  (a measure of the Curie constant) . The magnetic transition temperature is \nfound to be 4 2.4 K by extrapolating the derivative curves.  \n   \n \n \n18 \n  \nFigure  6 (Color online) : The  resistivity of a sintered polycrystalline pellet of Ba 4NbMn 3O12 \nas a function of temperature ( Inset) and inverse temperature (Main Panel). The data was fit \nto the model 𝝆=𝝆𝒐𝒆𝑬𝒂\n𝒌𝒃𝑻 (red line) with E a = 0.37 eV. \n \n  \n19 \n  \nFigure 7  (Color online) : (A) Molar  heat capacity divided by  temperature  of Ba 4NbMn 3O12 \nmeasured from 1.8 K to 200 K. Two transitions resulting in entropy losses are marked. A \nthird one near 30 K is unmarked. Inset: the raw C p vs. T data. (B) The lower temperature \ntransition showing a detai l of the experimental data and an estimate of the phonon \nbackground. Inset: the C p/T values in excess of the estimated phonon heat capacity. (C) The \nhigher temperature transition showing a detail of the experimental data and an estimate of \nthe phonon backg round. Inset the C p/T values in excess of the estimated phonon heat \ncapacity. (D) The entropy sums through the higher and lower temperature transitions, the \nformer being 0.45 J mol-1 K-1 and the latter being 2.67 J mol-1 K-1. \n \n"    },    {        "title": "1909.09085v1.Magnetization_dynamics_of_the_compensated_ferrimagnet__Mn__2_Ru__x_Ga_.pdf",        "content": "Magnetisation dynamics of the compensated ferrimagnet Mn 2RuxGa\nG. Bon\fglio\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands\nK. Rode, K. Siewerska, J. Besbas, G. Y. P. Atcheson, P. Stamenov, and J.M.D. Coey\nCRANN, AMBER and School of Physics, Trinity College Dublin, Ireland\nA.V. Kimel, Th. Rasing, and A. Kirilyuk\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands and\nFELIX Laboratory, Radboud University, Toernooiveld 7c, 6525 ED Nijmegen, The Netherlands\nHere we study both static and time-resolved dynamic magnetic properties of the compensated\nferrimagnet Mn 2RuxGa from room temperature down to 10 K, thus crossing the magnetic compen-\nsation temperature TM. The behaviour is analysed with a model of a simple collinear ferrimagnet\nwith uniaxial anisotropy and site-speci\fc gyromagnetic ratios. We \fnd a maximum zero-applied-\n\feld resonance frequency of \u0018160 GHz and a low intrinsic Gilbert damping \u000b\u00180:02, making it a\nvery attractive candidate for various spintronic applications.\nI. INTRODUCTION\nAntiferromagnets (AFM) and compensated ferrimag-\nnets (FiM) have attracted a lot of attention over the last\ndecade due to their potential use in spin electronics1,2.\nDue to their lack of a net magnetic moment, they are\ninsensitive to external \felds and create no demagnetis-\ning \felds of their own. In addition, their spin dynamics\nreach much higher frequencies than those of their ferro-\nmagnetic (FM) counterparts due to the contribution of\nthe exchange energy in the magnetic free energy3.\nDespite these clear advantages, AFMs are scarcely\nused apart from uni-directional exchange biasing rela-\ntively in spin electronic applications. This is because\nthe lack of net moment also implies that there is no\ndirect way to manipulate their magnetic state. Fur-\nthermore, detecting their magnetic state is also compli-\ncated and is usually possible only by neutron di\u000braction\nmeasurements4, or through interaction with an adjacent\nFM layer5.\nCompensated, metallic FiMs provide an interesting al-\nternative as they combine the high-speed advantages of\nAFMs with those of FMs, namely, the ease to manipu-\nlate their magnetic state. Furthermore, it has been shown\nthat such materials are good candidates for the emerging\n\feld of All-Optical Switching (AOS) in which the mag-\nnetic state is solely controlled by a fast laser pulse6{8.\nA compensated, half-metallic ferrimagnet was \frst en-\nvisaged by van Leuken and de Groot9. In their model\ntwo magnetic ions in crystallographically di\u000berent po-\nsitions couple antiferromagnetically and perfectly com-\npensate each-other, but only one of the two contributes\nto the states at the Fermi energy responsible for elec-\ntronic transport. The \frst experimental realisation of\nthis, Mn 2RuxGa (MRG), was provided by Kurt et al.10.\nMRG crystallises in the XAHeusler structure, space\ngroupF\u001643m, with Mn on the 4 aand 4csites11.\nSubstrate-induced bi-axial strain imposes a slight tetrag-\nonal distortion, which leads to perpendicular magneticanisotropy. Due to the di\u000berent local environment of\nthe two sublattices, the temperature dependence of their\nmagnetic moments di\u000ber, and perfect compensation is\ntherefore obtained at a speci\fc temperature TMthat\ndepends on the Ru concentration xand the degree of\nbiaxial strain. It was previously shown that MRG ex-\nhibits properties usually associated with FMs: a large\nanomalous Hall angle12, that depends only on the mag-\nnetisation of the 4 cmagnetic sublattice13; tunnel magne-\ntoresistance (TMR) of 40 %, a signature of its high spin\npolarisation14, was observed in magnetic tunnel junc-\ntions (MTJs) based on MRG15; and a clear magneto-\noptical Kerr e\u000bect and domain structure, even in the ab-\nsence of a net moment16,17. Strong exchange bias of a\nCoFeB layer by exchange coupling with MRG through\na Hf spacer layer18, as well as single-layer spin-orbit\ntorque19,20showed that MRG combined the qualities of\nFMs and AFMs in spin electronic devices.\nThe spin dynamics in materials where two distinct\nsublattices are subject to di\u000bering internal \felds (ex-\nchange, anisotropy, . . . ) is much richer than that of a\nsimple FM, as previously demonstrated by the obers-\nvation of single-pulse all-optical switching in amorphous\nGdFeCo21,22and very recently in MRG8. Given that the\nmagnetisation of MRG is small, escpecially close to the\ncompensation point, and the related frequency is high,\nnormal ferromagnetic resonance (FMR) spectroscopy is\nunsuited to study their properties. Therefore, we used\nthe all-optical pump-probe technique to characterize the\nresonance frequencies at di\u000berent temperatures in vicin-\nity of the magnetic compensation point. This, together\nwith the simulation of FMR, make it possible to deter-\nmine the e\u000bective g-factors, the anisotropy constants and\ntheir evolution across the compensation point. We found,\nin particular, that our ferrimagnetic half-metallic Heusler\nalloy has resonance frequency up to 160 GHz at zero-\feld\nand a relatively low Gilbert damping.arXiv:1909.09085v1  [cond-mat.mtrl-sci]  19 Sep 20192\nFIG. 1. Net moment measured by magnetometry and coercive\n\feld measured by static Faraday e\u000bect. The upturn of the net\nmoment below T\u001850 K is due to paramagnetic impurities\nin the MgO substrate. TMis indicated by the vertical dotted\nline. As expected the maximum available applied \feld \u00160H=\n7 T is insu\u000ecient to switch the magnetisation close to TM.\nII. EXPERIMENTAL DETAILS\nThin \flm samples of MRG were grown in a `Sham-\nrock' sputter deposition cluster with a base pressure of\n2\u000210\u00008Torr on MgO (001) substrates. Further infor-\nmation on sample deposition can be found elsewhere23.\nThe substrates were kept at 250\u000eC, and a protective\n\u00183 nm layer of aluminium oxide was added at room tem-\nperature. Here we focus on a 53 nm thick sample with\nx= 0:55, leading to TM\u001980 K as determined by SQUID\nmagnetometry using a Quantum Design 5 T MPMS sys-\ntem (see FIG. 1). We are able to study the magneto-\noptical properties both above and below TM.\nThe magnetisation dynamics was investigated using an\nall-optical two-colour pump-probe scheme in a Faraday\ngeometry inside a \u00160Hmax= 7 T superconducting coil-\ncryostat assembly. Both pump and probe were produced\nby a Ti:sapphire femtosecond pulsed laser with a cen-\ntral wavelength of 800 nm, a pulse width of 40 fs and\na repetition rate of 1 kHz. After splitting the beam in\ntwo, the high-intensity one was doubled in frequency by\na BBO crystal (giving \u0015= 400 nm) and then used as\nthe pump while the lower intensity 800 nm beam acted\nas the probe pulse. The time delay between the two was\nadjusted by a mechanical delay stage. The pump was\nthen modulated by a synchronised mechanical chopper\nat 500 Hz to improve the signal to noise ratio by lock-in\ndetection. Both pump and probe beams were linearly\npolarized, and with spot sizes on the sample of 150 µm\nand 70 µm, respectively. The pump pulse hit the sample\nat an incidence angle of \u001910\u000e. After interaction with\nthe sample, we split the probe beam in two orthogonally\npolarized parts using a Wollaston prism and detect the\nchanges in transmission and rotation by calculating the\nFIG. 2. Comparison of hysteresis loops obtained by Faraday,\nAHE, and magnetometry recorded at room temperature. The\ntwo former were recorded with the applied \feld perpendicular\nto the sample surface, while for the latter we show results for\nboth \feld applied parallel and perpendicular to the sample.\nsum and the di\u000berence in intensity of the two signals.\nThe external \feld was applied at 75\u000eto the easy axis of\nmagnetization thus tilting the magnetisation away from\nthe axis. Upon interaction with the pump beam the mag-\nnetisation is momentarily drastically changed24and we\nmonitor its return to the initial con\fguration via remag-\nnetisation and then precession through the time depen-\ndent Faraday e\u000bect on the probe pulse.\nThe static magneto-optical properties were examined\nin the same cryostat/magnet assembly.\nIII. RESULTS & DISCUSSION\nA. Static magnetic properties\nWe \frst focus on the static magnetic properties as\nobserved by the Faraday e\u000bect, and compare them to\nwhat is inferred from magnetometry and the anomalous\nHall e\u000bect. In FIG. 2 we present magnetic hysteresis\nloops as recorded using the three techniques. Due to the\nhalf metallic nature of the sample, the magnetotrans-\nport properties depend only on the 4 csublattice. As the\nmain contribution to the MRG dielectric tensor in the\nvisible and near infrared arises from the Drude tail16,\nboth AHE and Faraday e\u000bect probe essentially the same\nproperties (mainly the spin polarised conduction band of\nMRG), hence we observe overlapping loops for the two\ntechniques. Magnetometry, on the other hand, measures\nthe net moment, or to be precise the small di\u000berence\nbetween two large sublattice moments. The 4 asublat-\ntice, which is insigni\fcant for AHE and Faraday here\ncontributes on equal footing. FIG. 2 shows a clear di\u000ber-\nence in shape between the magnetometry loop and the3\nFIG. 3. Time resolved Faraday e\u000bect recorded at T= 290 K\nin applied \felds ranging from 1 T to 7 T. After the initial\ndemagnetisation seen as a sharp increase in the signal at t\u0018\n0 ps, magnetisation is recovered and followed by precession\naround the e\u000bective \feld until fully damped. The lines are\n\fts to the data. The inset shows the experimental geometry\nfurther detailed in the main text.\nAHE or Faraday loops. We highlight here that the ap-\nparent `soft' contribution that shows switching close to\nzero applied \feld, is not a secondary magnetic phase, but\na signature of the small di\u000berences in the \feld-behaviour\nof the two sublattices. We also note that this behaviour\nis a result of the non-collinear magnetic order of MRG.\nA complete analysis of the dynamic properties therefore\nrequires knowledge of the anisotropy constants on both\nsublattices as well as the (at least) three intra and in-\nter sublattice exchange constants. Such an analysis is\nbeyond the scope of this article, and we limit our anal-\nysis to the simplest model of a single, e\u000bective uniaxial\nanisotropy constant Kuin the exchange approximation\nof the ferrimagnet.\nB. Dynamic properties\nWe now turn to the time-resolved Faraday e\u000bect and\nspin dynamics. Time-resolved Faraday e\u000bect data were\nrecorded at \fve di\u000berent temperatures 10 K, 50 K, 100 K,\n200 K and 290 K, with applied \felds ranging from 1 T to\n7 T.\nFIG. 3 shows the \feld-dependence of the Faraday ef-\nfect as a function of the delay between the pump and\nthe probe pulses, recorded at T= 290 K. Negative de-\nlay indicates the probe is hitting the sample before the\npump. After the initial demagnetisation, the magneti-\nsation recovers and starts precessing around the e\u000bec-\ntive \feld which is determined by the anisotropy and the\napplied \feld. The solid lines in FIG. 3 are \fts to the\ndata to extract the period and the damping of the pre-cession in each case. The \ftting model was an expo-\nnentially damped sinusoid with a phase o\u000bset. We note\nthat the apparent evolution of the amplitude and phase\nwith changing applied magnetic \feld is due to the quasi-\nresonance of the spectrum of the precessional motion\nwith the low-frequency components of the convolution\nbetween the envelope of the probe pulse and the phys-\nical relaxation of the system. The latter include both\nelectron-electron and electron-lattice e\u000bects. A rudimen-\ntary model based on a classical oscillator successfully re-\nproduces the main features of the amplitude and phase\nobserved.\nIn two-sublattice FiMs, the gyromagnetic ratios of the\ntwo sublattices are not necessarily the same. This is par-\nticularly obvious in rare-earth/transition metal alloys,\nand is also the case for MRG despite the two sublat-\ntices being chemically similar; they are both Mn. Due\nto the di\u000berent local environment however, the degree\nof charge transfer for the two di\u000bers. This leads to two\ncharacteristic temperatures, a \frst TMwhere the mag-\nnetic moments compensate, and a second TAwhere the\nangular momenta compensate. It can be shown that for\nthe ferromagnetic mode, the e\u000bective gyromagnetic ratio\n\re\u000bcan then be written25\n\re\u000b=M4c(T)\u0000M4a(T)\nM4c(T)=\r4c\u0000M4a(T)=\r4a(1)\nsubscripti= 4a;4cdenotes sublattice i,Mi(T)\nthe temperature-dependent magnetisation, and \rithe\nsublattice-speci\fc gyromagnetic ratio. \re\u000bis related to\nthe e\u000bective g-factor\nge\u000b=\re\u000bh\n\u0016B(2)\nwherehis the Planck constant and \u0016Bthe Bohr magne-\nton.\nThe frequency of the precession is determined by the\ne\u000bective \feld, which can be inferred from the derivative\nof the magnetic free energy density with respect to M.\nFor an external \feld applied at a given \fxed angle with\nrespect to the easy axis this leads to the Smit-Beljers\nformula26\n!FMR =\re\u000bvuut1\nM2ssin2\u001e\"\n\u000e2E\n\u000e\u00122\u000e2E\n\u000e\u001e2\u0000\u0012\u000e2E\n\u000e\u0012\u000e\u001e\u00132#\n(3)\nwhere\u0012and\u001eare the polar and azimuthal angles of the\nmagnetisation vector, and Ethe magnetic free energy\ndensity\nE=\u0000\u00160H\u0001M+Kusin2\u0012+\u00160M2\nscos2\u0012=2 (4)\nwhere the terms correspond to the Zeeman, anisotropy\nand demagnetising energies, respectively, and Msis the\nnet saturation magnetisation. It should be mentioned\nthat the magnetic anisotropy constant Kuis related to\nM, which is being considered constant in magnitude, via\nKu=\f\u00160M2\ns=2,\fa dimensionless parameter.4\nFIG. 4. Observed precession frequency as a function of the\napplied \feld for various temperatures. The solid lines are \fts\nto the data as described in the main text.\nBased on Eqs. (1) through (4) we \ft our entire data set\nwith\re\u000bandKuas the only free parameters. The exper-\nimental data and the associated \fts are shown as points\nand solid lines in FIG. 4. At all temperatures our simple\nmodel with one e\u000bective gyromagnetic ratio \re\u000band a\nsingle uniaxial anisotropy parameter Kureproduces the\nexperimental data reasonably well. The model systemat-\nically underestimates the resonance frequency for inter-\nmediate \felds, with the point of maximum disagreement\nincreasing with decreasing temperature. We speculate\nthis is due to the use of a simple uniaxial anisotropy in\nthe free energy (see Eq. 4), while the real situation is\nmore likely to be better represented as a sperimagnet. In\nparticular, the non-collinear nature of MRG that leads\nto a deviation from 180\u000eof the angle between the two\nsublattice magnetisations, depending on the applied \feld\nand temperature.\nFrom the \fts in FIG. 4 we infer the values of ge\u000band\nthe anisotropy \feld \u00160Ha=2Ku=Ms. The result is shown\nin FIG. 5. The anisotropy \feld is monotonically increas-\ning with decreasing temperature as the magnetisation\nof the 4csublattice increases in the same temperature\nrange. We highlight here the advantage of determining\nthis \feld through time-resolved magneto-optics as op-\nposed to static magnetometry and optics. Indeed the\nanisotropy \feld as seen by static methods is sensitive to\nthe combination of anisotropy and the netmagnetic mo-\nment, as illustrated in FIG. 1, where the coercive \feld\ndiverges as T!TM. In statics one would expect a di-\nvergence of the anisotropy \feld at the same temperature.\nThe time-resolved methods however distinguish between\nthe net and the sublattice moments, hence better re\rect-\ning the evolution of the intrinsic material properties of\nthe ferrimagnet.\nThe temperature dependence of the anisotropy con-\nstants was a matter for discussion for many years27,28.\nFIG. 5. E\u000bective g-factor,ge\u000b, and the anisotropy \feld\nas determined by time-resolved Faraday e\u000bect. ge\u000b, orange\nsquares, increases from near the free electron value of 2 to 4\njust belowTM, while the anisotropy \feld, blue triangles, in-\ncreases near-linearly with decreasing temperature. A M3\ft,\nred dashes line, of the anisotropy behaviour shows the almost-\nmetallic origin of it, indicating the dominant character of the\n4c sublattice.\nWritten in spherical harmonics the 3 danisotropy can\nbe expressed as, k2Y0\n2(\u0012) +k4Y0\n4(\u0012)29wherek2/\nM(T)3andk4/M(T)10. The experimental measured\nanisotropy is then, K2(T) =ak2(T)+bk4(T), withaand\nbthe contributions of the respective spherical harmonics.\nFIG. 5 shows that a reasonable \ft of our data is ob-\ntained with M(T)3which means, \frst, that the contri-\nbution of the 4thorder harmonic can be neglected, and\nsecond, that the contribution of the TMand 2ndsublat-\ntice is negligible, indicating the dominant character of\nthe 4c sublattice.\nIn addition, we should note here that the high fre-\nquency exchange mode was never observed on our exper-\niments. While far from TMits frequency might be too\nhigh to be observable, in the vicinity of TM, in contrast,\nits frequency is expected to be in the detection range.\nMoreover, given the di\u000berent electronic structure of the\ntwo sublattices, it is expected that the laser pulse should\nselectively excite the sublattice 4c, and therefore lead to\nthe e\u000bective excitation of the exchange mode. We argue\nthat it is the non-collinearity of the sublattices (see sec-\ntion III A) that smears out the coherent precession at\nhigh frequencies.\nThe e\u000bective gyromagnetic ratio, ge\u000b, shows a non-\nmonotonic behaviour. It increases with decreasing Tto-\nwardsTM, reaching a maximum at about 50 K before\ndecreasing again at T= 10 K. We alluded above to\nthe di\u000berence between the magnetic and the angular mo-\nmenta compensation temperatures. We expect that ge\u000b\nreaches a maximum when T=TA30, here between the\nmeasurement at T= 50 K and the magnetic compensa-\ntion temperature TM\u001980 K.5\nFIG. 6. Intrinsic and anisotropic broadening in MRG across\ntheTM. The inset shows the evaluation process of the two\ndamping parameters. A linear \ft is used to evaluate intercept\n(anisotropic broadening) and slope (intrinsic damping) of the\nfrequencies versus the inverse of the decay time. The data\npoint are obtained from the \ft of time-resolved Faraday e\u000bect\nmeasurements (an example is shown in Fig.4).\nFrom XMCD data11, we could estimate spin and or-\nbital moment components of the magnetic moments of\nthe two sublattices, what allowed us to derive the ef-\nfective g-factors for the sublattices as g4a= 2:05 and\ng4c= 2:00. In this case we expect the angular momentum\ncompensation temperature TAto be below TM, opposite\nto what is observed for GdFeCo21. Given this small dif-\nference however, TAandTMare expected to be rather\nclose to each other, consistent with the limited increase\nofge\u000bacross the compensation points.\nWe turn \fnally to the damping of the precessional mo-\ntion of Maround the e\u000bective \feld \u00160He\u000b. Damping is\nusually described via the dimensionless parameter \u000bin\nthe Landau-Lifshiz-Gilbert equation, and it is a measure\nof the dissipation of magnetic energy in the system. In\nthis model, \u000bis a scalar constant and the observed broad-\nening in the time domain is therefore a linear function of\nthe frequency of precession31{33. We infer \u000b0, the total\ndamping, from our \fts of the time-resolved Faraday e\u000bect\nas\u000b0= (\u001cd)\u00001, where\u001cdis the decay time of the \fts. We\nthen, for each temperature, plot \u000b0as a function of the\nobserved frequency and regress the data using a straight\nline \ft. The intrinsic \u000bis the slope of this line, while the\nintercept represents the anisotropic broadening.\nFIG. 6 shows the intrinsic damping \u000band the\nanisotropic broadening as a function of temperature.\nAnisotropic broadening is usually attributed to a vari-\nation of the anisotropy \feld in the region probed by the\nprobe pulse34. For MRG this is due to slight lateral vari-\nations in the Ru content xin the thin \flm sample. Such a\nvariation leads to a variation in e\u000bective TMandTAand\ncan therefore have a large in\ruence on the broadening asa function of temperature. Despite this, the anisotropic\nbroadening is reasonably low in the entire temperature\nrange above TM, and a more likely explanation for its\nrapid increase below TMis that the applied magnetic\n\feld is insu\u000ecient to completely remagnetize the sam-\nple between two pump pulses. As observed in Fig.5, the\nanisotropy \feld reaches almost 4 T at low temperature,\ncomparable to our maximum applied \feld of 7 T. The\nintrinsic damping \u000bis less than 0.02 far from TM, but\nincreases sharply at T= 100 K. We tentatively attribute\nthis to an increasing portion of the available power be-\ning transferred into the high-energy exchange mode, al-\nthough we underline that we have not seen any direct\nevidence of such a mode in any of the experimental data.\nIV. CONCLUSION\nWe have shown that the time-resolved Faraday e\u000bect\nis a powerful tool to determine the spin dynamic proper-\nties in compensated, metallic ferrimagnets. The high spin\npolarisation of MRG enables meaningful Faraday data to\nbe recorded even near TMwhere the net magnetisation\nis vanishingly small, and the dependence of the dynamics\non the sublattice as opposed to the net magnetic prop-\nerties provides a more physical understanding of the ma-\nterial. Furthermore, we \fnd that the ferromagnetic-like\nmode of MRG reaches resonance frequencies as high as\n160 GHz in zero applied \feld, together with a small in-\ntrinsic damping. This value is remarkable if compared\nto well-known materials such as GdFeCo which, at zero\n\feld, resonates at tens of GHz21or [Co/Pt] nmultilay-\ners at 80 GHz35but with higher damping. We should\nhowever stress that, in the presence of strong anisotropy\n\felds, higher frequencies can be reached. Example of that\ncan be found for ferromagnetic Fe/Pt with \u0019280 GHz\n(Ha= 10T)36, and for Heusler-like ferrimagnet (Mn 3Ge\nand Mn 3Ga) with\u0019500 GHz (Ha= 20T)37,38. Never-\ntheless, the examples cited above show a considerably\nhigher intrinsic damping compared to MRG. In addi-\ntion, it was recently shown that MRG exhibits unusu-\nally strong intrinsic spin-orbit torque20. 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Yildirim,\net al. , Applied Physics Letters 109, 032403 (2016)."    },    {        "title": "1806.07719v2.Magneto_optic_Kerr_effect_in_a_spin_polarized_zero_moment_ferrimagnet.pdf",        "content": "Magneto-optic Kerr e\u000bect in a spin-polarized zero-moment ferrimagnet\nKarsten Fleischer,1, 2Naganivetha Thiyagarajah,3Yong-Chang Lau,3Davide Betto,3Kiril\nBorisov,3Christopher C. Smith,1Igor V. Shvets,1J.M.D. Coey,3and Karsten Rode3\n1School of Physics, Trinity College Dublin\n2Dublin City University\n3CRANN and School of Physics, Trinity College Dublin\n(Dated: November 10, 2021)\nThe magneto-optical Kerr e\u000bect (MOKE) is often assumed to be proportional to the magnetisation\nof a magnetically ordered metallic sample; in metallic ferrimagnets with chemically distinct sublat-\ntices, such as rare-earth transition-metal alloys, it depends on the di\u000berence between the sublattice\ncontributions. Here we show that in a highly spin polarized, fully compensated ferrimagnet, where\nthe sublattices are chemically similar, MOKE is observed even when the net moment is negligible.\nWe analyse the spectral ellipsometry and MOKE of Mn 2RuxGa, and show that this behaviour is\ndue to a highly spin-polarized conduction band dominated by one of the two manganese sublattices\nwhich creates helicity-dependent re\rectivity determined by a broad Drude tail. Our \fndings open\nnew prospects for studying spin dynamics in the infra-red.\nI. INTRODUCTION\nSpin valves, where angular momentum can be trans-\nferred from a reference layer to a free layer, are expected\nto form the basis of future memory devices and oscil-\nlators operating in the far infra-red frequency domain.1\nThe key materials properties necessary to achieve these\nfrequencies are a high Fermi-level spin polarization P;\nlow net magnetization Msand a high e\u000bective magnetic\nanisotropy \feld \u00160Hk. A compensated ferrimagnetic half\nmetal where negligible magnetization can be obtained,\ncoupled with full ( P= 100 %) spin polarization, would\nbe an ideal choice.2\nSuch a material was suggested by van Leuken and\nde Groot3in 1995, but the \frst experimental evidence\nof one was only provided by the work of Kurt et al.4\nin 2014. We have recently shown that the material,\nMn2RuxGa (MRG), shows negligible Msover a wide\ntemperature and thermal treatment range;5,6high spin\npolarization4as re\rected in the anomalous Hall e\u000bect;7\nproduces sizeable magnetoresistive e\u000bects in spin elec-\ntronic structures8such as magnetic tunnel junctions;9\nand proposed an ab initio model reconciling the ex-\nperimental observations with density functional theory\n(DFT) band structure calculations.10\nTime-resolved magneto-optical Kerr/Faraday e\u000bect is\nparticularly suited to determine the Larmor frequency\nof thin-\flm samples, especially when the resonance fre-\nquency exceeds the experimental range of cavity- or\nnetwork-analyser-based techniques. The unique proper-\nties of MRG indicate that the material may be a candi-\ndate for low-power all-optical switching as the magnetic\ncompensation temperature Tcomp can be tuned to lie just\nabove room temperature,11but these measurements on\nnew materials are complicated by the lack of an accepted,\nwavelength-dependent optical model.\nFerrimagnets with two chemically distinct sublattices,\nnotably the rare-earth transition metal alloys have been\ninvestigated by spectroscopic MOKE.12,13The spectralresponse of the 4 fand 3delements allow the temperature\ndependence of the Kerr e\u000bect of each sublattice to be\ndetermined. The new aspect we present here is that in\na ferrimagnet with chemically identical, but structurally\ninequivalent, sublattices it is possible to track the spin\npolarization associated with only one of the sublattices.\nWe \frst show that even at magnetic compensation,\nMRG exhibits a clear Kerr rotation ( \u0012K). Spectroscopic\nellipsometry, in combination with the spectroscopic polar\nMOKE14allow us to determine the diagonal (~ \"xx) and\no\u000b-diagonal (~ \"xy) dielectric tensor components as a func-\ntion of Ru concentration x. Polar MOKE is often used\nsimply to measure hysteresis loops using monochromatic\nlight, and the spectroscopic variant provides more spe-\nci\fc information on the sites responsible for the magnetic\nresponse. Our measurements show a clear correlation\nbetween the infrared Kerr rotation and the electrically-\nmeasured Hall angle, providing contactless, optical mea-\nsurements of this quantity. The experimentally deter-\nmined, optical model developed here will facilitate mea-\nsurements in the time- and frequency-domains, that can\nadvance the development of spin electronic devices oper-\nating in the far infra-red frequency range.\nII. SAMPLE DETAILS\nThe thin-\flm samples studied here were grown on\nMgO (001) substrates by dc magnetron sputtering at\n250\u000eC substrate temperature in a 'Shamrock' system\nwith a base pressure of 2 \u000210\u00008Torr. The \flms were\nco-sputtered from Ru and Mn 2Ga targets. The Ru con-\ncentration was controlled by varying the Mn 2Ga target\nplasma power while keeping that of the Ru \fxed. The\nsamples were then capped with a protective 2 nm layer\nof amorphous aluminium oxide. The crystal structure,\nlattice parameters and strain of the \flms were charac-\nterised by 2 \u0012\u0000\u0012X-ray di\u000braction and reciprocal space\nmapping using a Bruker D8 Discover instrument, with aarXiv:1806.07719v2  [cond-mat.mtrl-sci]  3 Sep 20182\nFIG. 1. Schematic of the MRG crystal structure. There are\nthree \flled fcc sublattices (red: Mn-site (4 a), grey: Ga-site\n(4b), green: Mn-site (4 c)). Ruthenium occupies the remaining\nfcc sublattice (blue: Ru-site (4 d)). The two Mn sites are\nnon equivalent in the structure as the 4 a-site has only Ga as\nnearest neighbour, while the 4 c-site Ru, or vacancies (dashed\ncircles) depending on the Ru content x.\nmonochromatic copper K \u000b1source. We found that the\nin-plane lattice constant ais 0:5956 nm =p\n2aMgO for\nall samples, indicating that MRG grows epitaxially on\nMgO with its basal plane rotated by 45\u000ewith respect\nto the substrate. The out-of-plane parameter cvaries\nbetween 0:598 nm and 0 :618 nm, depending on Ru con-\ncentration and \flm thickness ( \u001826 to 81 nm). This con-\n\frms a slight, substrate-induced, tetragonal distortion of\nthe cubic structure. An illustration of the crystal is given\nin FIG. 1.\nIn order to determine the Ru concentration, we de-\nposited four MRG samples with varying Mn 2Ga tar-\nget power and a pure Ru \flm. The density and thick-\nness were then measured using X-ray re\rectivity (XRR).\nBased on the measured density and lattice parameters\nof these \fve control samples, we established a rela-\ntion between X-ray density and Ru concentration x,\nagainst which all the samples were calibrated. The trans-\nport properties of the thin \flms were measured on non-\npatterned \flms in a Quantum Design physical properties\nmeasurement system (PPMS) at temperatures ranging\nfrom 10 to 400 K in an applied magnetic \feld, \u00160H, of\nup to 12 T.\nIII. EXPERIMENTAL DETAILS\nBoth ellipsometry and magneto-optical Kerr spec-\ntroscopy were used to determine the optical and\nmagneto-optical properties of the \flms. Spectroscopic\nMOKE was recorded at near-normal incidence with a in-\nhouse built re\rectance anisotropy spectrometer, based\non the design of Aspnes.15Light from a Xe lamp passes\nthrough a Rochon Mg 2F polariser and is re\rected from\nthe sample. The beam then passes through a photo-\nelastic modulator (PEM), an analysing polariser and a\nmonochromator before \fnally reaching a diode detector\nsystem. All measurements were performed from 0 :35 to\n5 eV, using a Bentham TMc300 triple-grating monochro-mator and three individual photo-diodes (InAs, InGaAs,\nSi). The procedure for extracting the MOKE rotation\nhas been discussed elsewhere.16{18All samples where\nmeasured at room temperature in remanence (no exter-\nnal \feld). Samples were magnetised before these mea-\nsurements in \felds of up to 12 T at room temperature.\nSelected sample have also been magnetised at 200 K.\nTo correct for any set-up or non-magnetic anisotropy,\nwe recorded spectra for two orthogonal in-plane direc-\ntions as well as for samples magnetized in opposite di-\nrections. The spectra obtained by reversal of ~M, were\nstrictly equal to those derived from a rotation of the sam-\nple of 90\u000earound the optical axis.\nDue to larger errors in the Kerr-ellipticity measure-\nment in a PEM setup without in-situ magnetisation re-\nversal, only the Kerr rotation ( \u0012K) will be discussed.\nA Sopra GESP 5 spectroscopic ellipsometer was used\nover an energy range from 1 :5 to 5 eV with incidence\nangles of 70\u000e, 75\u000eand 80\u000e. Both, the thin \flm dielectric\nfunction (~\"xx, ~\"xy), as was derived from the raw measure-\nments by minimising the deviation between measured el-\nlipsometric data (tan \t ;cos \u0001) over the whole spectral\nrange, as well as all three angles. Full transfer matrix\ncalculations of the air/AlO x/MRG/MgO layer stack have\nto be evaluated, as in contrast to simple bulk samples no\nanalytical expression for the relationship between mea-\nsured tan \t ;cos \u0001, and \u0012Kexists. The spectral shape of\n~\"was simulated by a standard Drude-like component and\nthree harmonic oscillators (Drude-Lorentz oscillator). To\nreduce uncertainties in ~ \"xx, only the MRG layer was mod-\nelled. The sample thickness and interface roughness, as\nmeasured by XRR was used as input parameters together\nwith the known bulk dielectric functions of the MgO sub-\nstrate and the AlO xcapping layer.\nIn thin \flm structures, the \u0012Kspectra depend not\nonly on the o\u000b-diagonal dielectric tensor components\n~\"xy=iQmz~\"xxof the magnetic material but also on the\noverall re\rectance of the stack. Knowing the diagonal\ncomponents ~ \"xxfrom the ellipsometric model, the mea-\nsured\u0012Kspectral signature can then also be modelled,\nand hence ~\"xydetermined using the same methodology\nwithout skewed results caused by variations in ~ \"xxand\nsample thickness/roughness and modulation of the over-\nall re\rectivity.19{21\nFor all optical models, the MRG and AlO xthicknesses\ndetermined by XRR were treated as \fxed parameters.\nAll interfaces have been treated as sharp, while the di-\nelectric function of the AlO xcoating has been modelled\nby a Bruggeman e\u000bective medium of crystalline Al 2O3\nand air with a \fll factor set to the density ratio of the\nXRR \ftted capping layer and crystalline Al 2O3. We treat\nthe MRG to be perfectly cubic, as the deviation from cu-\nbic symmetry due to epitaxial strain of \u00181 %22is well\nbelow the sensitivity of our instrument.3\n−50510152025\n180 200 220 240 260 280 300 320 340 360051015\n200 250 300 350\nM(kA m−1)\nT(K)µ0Hc(T)\nT(K)\nFIG. 2. SQUID magnetometry of sample with Tcomp close\nto RT. The net magnetisation of the sample was measured at\nremnance on warming after saturation at low temperature. In\nthe inset we show an estimate of the coercive \feld, \u00160Hc, as\na function of temperature. Around room-temperature, even\n15 T is insu\u000ecient to saturate the magnetisation.\nIV. MOKE: DIRECT MEASUREMENTS\nDue to the di\u000berent crystallographic sites occupied by\nthe two magnetic sublattices in ferrimagnets, the temper-\nature dependence of the magnetisation of the two sub-\nlattices di\u000ber. Generally, perfect compensation and zero\nnet moment occurs only at a speci\fc temperature ( Tcomp)\nwhere the magnitudes of the two magnetic sublattices are\nequal but opposite in sign. For MRG this temperature\ndepends on both Ru content xand the biaxial substrate-\ninduced strain.\nUpon crossing Tcomp in zero applied \feld the net mag-\nnetisation changes sign whereas the spin-dependent DOS\nremains unchanged because the individual sublattices\nhave not changed direction. We therefore expect that the\nsign of any property that depends on the spin-resolved\nDOS such as MOKE, tunnel magneto-resistance (TMR)\nand anomalous Hall e\u000bect, will exhibit opposite signs de-\npending on whether the sample was magnetised below or\naboveTcomp.7,9\nIn FIG. 2 we show the temperature dependence of the\nnet magnetic moment of an MRG sample with Tcomp\u0018\n295 K. The inset gives an estimate of the coercive \feld as\na function of temperature. Close to Tcomp, the anisotropy\n\feld\u00160Hkdiverges, and as the coercive \feld is directly\nrelated to the anisotropy \feld, the sample cannot be mag-\nnetically saturated in this temperature range.\nWe now compare MOKE spectra of two samples with\nTcomp of 295 K and 250 K, respectively. The data are\nshown in FIG. 3. When magnetised at RT, i.e.above\nTcomp, both samples exhibit clear MOKE negative in\nsign. The sample with Tcomp closest to RT appears to\nhave a reduced magnitude. We then magnetise both\nsamples well below their respective Tcomp,Tsat= 200 K,\n−1.5−1−0.500.511.5\n0 1 2 3 4 5\nθK(mrad)\nhν(eV)Tcomp∼250 K\nTcomp∼295 KFIG. 3. Room-temperature Kerr rotation as a function of\nphoton energy for samples with Tcomp\u0018295 K and 250 K, re-\nspectively. The samples were \frst magnetised at RT in a \feld\nof\u00160H= 2 T (above Tcomp, open symbols). Subsequently,\nthe samples were remagnetised, this time well below Tcomp\n(Tsat= 200 K, \flled symbols). Although all measurements\nare done at RT, the sign of the MOKE signal is reversed\ndepending on whether magnetisation is carried out above or\nbelowTcomp. WhenTcomp\u0018295 K, full saturation is only\nachieved when magnetised well below Tcomp.\nand subsequently warm to RT in zero applied \feld. This\nprocedure ensures that while the net magnetic moment\nchanges sign when crossing Tcomp, the spin-dependent\ndensity of states (DOS) remains unchanged. We then\nrecord MOKE spectra at RT. As can be seen from FIG. 3,\nboth samples exhibit a change of sign as expected, and\ncrucially, the sample with Tcomp\u0018295 K and thus\u0018zero\nnet moment now displays as strong or stronger MOKE\nsignal as the sample measured further away from Tcomp.\nThis set of measurements proves that MOKE in MRG\ndepends on the spin-dependent DOS, and not the net\nmagnetic moment; and furthermore even when the net\nmagnetic moment is zero, the MOKE signal is clearly\nobserved.\nWe note here that disproportionally between the rem-\nnant magnetization and the polar MOKE measured dur-\ning a hysteresis loop have been observed earlier.23,24This\nwas attributed coupling, during switching , of the in-plane\nmagnetization and the polar MOKE through higher or-\nder terms. It was only observed in highly anisotropic\nexchange-biased materials and only during switching.\nThe samples measured here do not exhibit uniaxial in-\nplane anisotropy, and are all measured at remanence.\nIn order to pinpoint the origin of MOKE in MRG, we\nrecorded angle and energy dependent ellipsometry, which\nin combination with the measured Kerr rotation allow us\nto determine all matrix elements of the dielectric tensor.4\n354045505560\n0 1 2 3 4 5(a)\n0 1 2 3 4 5 6−4−202468\n(b)\n−2−1012\n0 1 2 3 4 5 6(c)\nσ1(1014s−1)\n/epsilon11x= 0.44\nx= 0.51x= 0.61\nx= 0.76x= 1.19θK(mrad)\nhν(eV)\nFIG. 4. (a) and (b): Imaginary and real components of\nthe diagonal tensor elements ~ \"xxof the dielectric function\nof MRG as a function of photon energy and Ru concentration\nx. The imaginary part is shown in terms of optical conduc-\ntivity (\u001b1=\"2!=4\u0019)25, to highlight the small di\u000berences be-\ntween samples.The behaviour is dominated by the Drude-like\nmetallic response, with smaller deviations caused by inter-\nband transitions. (c) MOKE spectra for the same series of\nsamples used to infer the o\u000b-diagonal ~ \"xytensor elements.\nV. OPTICAL MODEL\nWe have shown that MOKE is not determined by the\nnet magnetic moment, but depends on the spin depen-\ndent DOS in MRG. MOKE measurements alone are in-\nsu\u000ecient to determine ~ \"xxand ~\"xy. We therefore \frst\nuse spectroscopic ellipsometry to determine ~ \"xxand sub-\nsequently develop the optical model including the o\u000b-\ndiagonal tensor elements ~ \"xyfor MRG.\nIn FIG. 4 we show the dielectric function ~ \"xxas ex-\ntracted from least square \fts of the measured tan \t and\ncos \u0001 for a set of samples with varying Ru concentration\nx. Our optical model, used for both ~ \"xxand ~\"xyis based\non the following considerations:\nFor metallic materials such as MRG the dielectric re-\nsponse is usually dominated by a strong Drude-like tail\ncaused by free electron absorption, also known as intra-\nband transitions.26This contribution is the reason why\nthe imaginary part of ~ \"xxof MRG does not tend towards\nzero (\u001b\u001845\u00021014s\u00001ath\u0017= 1 eV) in the infra-red\nregion. Furthermore, there are absorption features in\nthe visible (VIS - peaks \u00182:3 eV) which we attribute\nto interband transitions on both Mn sites, ultra-violet\n(UV\u00183:2 eV) due to Ga, and a deep ultra-violet con-\ntribution (DUV >5:5 eV) necessary to account for the\nline shape but that cannot be assigned to a speci\fc el-\nement. The assignment of the VIS structure to Mn in-terband transitions is motivated by several experimental\nand theoretical \fndings, including similar spectral fea-\ntures in bulk Mn,25a strong peak in the unoccupied den-\nsity of states in ab-initio calculations for bulk Mn,27and\n\fnally the similarity in the shift of the energetic posi-\ntion of the VIS peak with increasing Ru concentration,\nfollowing the same trend as that seen in site-selective\nXAS/XMCD measurements on Mn 2RuxGa.5\nTo minimize the number of free parameters, we there-\nfore describe the diagonal part of the dielectric function\nvia one Drude-like oscillator at != 0 eV, and three broad\nharmonic oscillators (\u0000 >6 eV) in the VIS, UV and DUV\nranges, respectively.\nThe Kerr spectra show strong features in the IR and\nthe VIS regions, associated with the Drude-tail and the\nMn, respectively, but no separate features in the UV\nand DUV. As the magnetism in MRG is dominated by\nMn in two antiferromagnetically coupled sub-lattices, ~ \"xy\ncan be described in more detail. We associate the sin-\ngle, broad, VIS oscillator observed in ellipsometry at\nh\u0017\u00182:3 eV with two individual components in the Kerr\nspectra (NIR and VIS in FIG. 5, bottom panel), which\nwe assign to the spin-dependent transitions on the Mn-4 c\nand Mn-4asite. In agreement with our assignment above\nof the UV and DUV transitions to non-magnetic compo-\nnents, no separate UV and DUV oscillators are required\nto describe ~ \"xy.\nThis assignment is motivated by our earlier X-ray ab-\nsorption measurements5where we \fnd that the \fnal\nstates of the 2 p!3dtransitions of Mn in the two sites\nare only separated by approximately 1 eV, and no circu-\nlar magnetic dichroism on the Ga L2;3edges nor on the\nRuM4;5edges, as well as the dielectric functions of Mn25\nand Ga28.\nIn FIG. 5 we show an example of a \ft to the model\ndescribed above. Having established the diagonal tensor\nelements ~\"xxwe then model the o\u000b-diagonal terms which\nwe plot as the expected Kerr rotation \u0012K(the real part\nof the complex Kerr rotation). The agreement between\nthe experimental data and the optical model is excellent,\nindicating that our model is well justi\fed as well as high-\nlighting the predominant role of Mn in MRG magnetisa-\ntion. The (extrapolated) steep increase in MOKE in the\nIR and towards DC ( h\u0017!0 eV) is clearly an e\u000bect of\nthe highly spin-polarised conduction band, as re\rected by\nthe Drude contribution to both diagonal and o\u000b-diagonal\ntensor elements.\nOur modelling allows us to study in \fner detail the\nbehaviour of the spectral components of the ellipsometry\nand MOKE as a function of Ru concentration x(FIG. 4).\nWhen the Ru concentration xchanges from 0 :44 to\n1:19, the oscillator associated with Ga (UV) changes po-\nsition only slightly from 3 :4 to 3:2 eV whereas the oscilla-\ntor associated with Mn (VIS), exhibits a stronger trend\nand moves from 3 :0 to 2:2 eV with increasing x. With\nincreasing Ru content, the electronic occupation of Mn\nis increasing linearly, wheres that of Ga remains con-\nstant. The (small) changes associated with Ga are thus5\n00.511.522.5\n0 1 2 3 4 5 6\nNIR\nVIS0102030405060\nθK(mrad)\nhν(eV)Exp\nFitσ1(1014s−1) Drude\nVIS\nUV\nDUV\nExp\nFIG. 5. Complete optical model and experimental data for an\nMRG sample with x= 0:61. (Top) Experimentally observed\noptical conductance (line) and the \ftted contributions from\nthe four harmonic oscillators (areas) described in the text.\n(Bottom) Experimental Kerr rotation (points) and \ft to the\ndata (line) based on the optical model described in the main\ntext.\ndue to increasing tetragonality of the crystal structure\nwhile those associated with Mn correlate directly with\nthe band \flling. As described above, the MOKE in MRG\nis dominated by the contributions of Mn, both through\nthe spin polarisation of the conduction band as well as\nsingle-ion transitions. The two oscillators used to de-\nscribe these transitions in MOKE can therefore be asso-\nciated with Mn in the 4 cposition (\u00181:7 eV - NIR) and in\nthe 4aposition (\u00182:5 eV - VIS). Due to the half-metallic\nnature of the sample, when Ru content is increased, the\nelectronic occupation of Mn in the 4 cposition increases\nleading to a blue shift (from 1 :4 to 1:8 eV) of the oscilla-\ntor associated with this position. Simultaneously, adding\nRu decreases the spin-gap10thereby inducing a red shift\n(from 2:7 to 2:4 eV) of the oscillator associated with Mn\nin the 4aposition. More details on the optical model pa-\nrameter as function of Ru concentration xare found in\nthe supplemental information.\nIn FIG. 6 we show the o\u000b-diagonal component of the\ndielectric tensor as a function of photon energy, extracted\nfrom the \ft of the experimental ellipsometry and MOKE\ndata, in terms of \u001bxy. Unlike the raw MOKE spectra, dif-\nferences between di\u000berent compositions are now clearly\nvisible, and highlight, independently of our discussion fo-\ncussed on MRG, the need to employ robust optical mod-\nelling when reporting MOKE data on thin \flm samples\nof even slightly varying thickness.\nWe \fnally turn to a comparison of the DC limit of the\nmagneto-optical properties with the anomalous Hall an-\ngle measured by standard electronic magnetotransport.\nAs outlined above, our optical model describes the be-\n−1.5−1−0.500.511.52\n0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\nσxy(1014s−1)\nhν(eV)x= 0.44\nx= 0.61\nx= 0.76FIG. 6.\u001bxyextracted from the optical model for three dif-\nferent Ru concentrations x. The real and imaginary parts are\nplotted in solid and broken lines, respectively. In the light\nand dark grey areas, the curves are extrapolated from the\nexperimentally available range in ellipsometry and MOKE,\nrespectively. Please not that Im( \u001bxy) is only determined up\nto a potential integrational constant\n00.511.522.5\n0 1 2 3 4 5 6 7\nRe(σxy)/Re(σxx) (%)\nρxy/ρxx(%)\nFIG. 7. Ratio of the real part of the o\u000b-diagonal and on-\ndiagonal terms of the dielectric tensor extrapolated to 0 :4 eV\nplotted as a function of the electrically measured anomalous\nHall angle. The straight line is a \ft to the data, and provides\na guide to the eye. As expected, the DC limit of the magneto-\noptical characterisation agrees very well with the values ob-\ntained through electronic transport measurements.\nhaviour of MRG fully in the spectral region measured,\nand we therefore expect that extrapolation outside this\nregion should accurately predict the behaviour at di\u000ber-\nent photon energies. The lower limit is of course DC\nwhere optical measurements are not possible, but where\nwe can record magnetotransport, in particular anoma-\nlous Hall e\u000bect.7The ratio\u001axy=\u001axxis the Hall angle, the\nDC limit of the ratio Re( \u001bxy)/Re(\u001bxx). In FIG. 7 we plot\nthe ratio determined from the extrapolation of our opti-\ncal model as a function of the Hall angle determined by6\nmagnetotransport measurements as described in II. The\nratio was evaluated at h\u0017= 0:4 eV as we only need to\nextrapolate \u001bxxand not\u001bxyat this energy. As expected\nwe observe a strong correlation between the two mea-\nsurements, most certainly because the two e\u000bects share a\ncommon origin; the high spin polarisation of the conduc-\ntion band. The deviations from a perfectly proportional\nbehaviour is due to small di\u000berences in the scattering\nmatrices that have a big in\ruence in transport measure-\nments whereas they do not in\ruence the values obtained\nby optical means. We also emphasize that whereas elec-\ntronic transport probes the entire thickness of a metallic\n\flm, MOKE and ellipsometry only probe the skin-depth,\nwhich varies with photon energy, found here to be 5 nm\nat 2 eV and 40 nm at 0 :4 eV.\nVI. CONCLUSION\nWe have shown that MOKE in zero-moment half met-\nals persists, even when the net magnetic moment crosses\nzero, and that its origin is dominated by the highly spin-\npolarised conduction band associated in MRG with Mn in\nthe 4csite.5The sign of the MOKE depends on whether\nthe sample was magnetised above or below its compensa-\ntion temperature, as the process of magnetisation sets the\ndirection of the axis of quantization for the spin-polarised\ndensity of states.\nBy combining spectroscopic ellipsometry and MOKE\nit was possible to construct an optical model that identi-\n\fes the optically active components of MRG, and we use\nthis to infer the full parameterized dielectric tensor as a\nfunction of photon energy in the range 0 to 5 eV. The\nmodel is deliberately constructed to minimize the num-\nber of free parameters, yet it captures with reasonableaccuracy both the nonmagnetic and the magnetic contri-\nbutions for MRG in the spectral range measured. Based\non the model, we compare the results obtained by optical\nmeans with the anomalous Hall angle recorded by elec-\ntronic transport measurements. Although the DC limit\nis far beyond our experimentally available range, the ex-\ntrapolated values of \u001bxyagree remarkably well with those\nobtained by transport measurements.\nDi\u000berences between magnetic properties observed by\nanomalous Hall e\u000bect and MOKE are often used nowa-\ndays as a \fngerprint of topological electronic transport.29\nAn extension of this study with a more in-depth compar-\nison between the two in high and zero applied magnetic\n\felds, may allow an unambiguous determination of the\nscattering coe\u000ecients that lead to topologically driven\nspin structures in for example non-collinear ferrimagnets,\nas a complete optical model is necessary to disentangle\ne\u000bects that depend solely on di\u000berences in sample thick-\nness and re\rectivity from those that are purely magnetic\nin nature.\nFinally, this result interestingly clears a path to study\nthe behaviour of magnetic domains in antiferromagnets\nby MOKE microscopy, while the shown spectral depen-\ndence highlights photon energies most suitable for pump-\nprobe based time dependent measurements.30\nACKNOWLEDGMENTS\nThis project has received funding from the Euro-\npean Union's Horizon 2020 research and innovation pro-\ngramme under grant agreement No 737038. 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Gensch, Applied Physics Letters 109,\n032403 (2016), https://doi.org/10.1063/1.4958855."    },    {        "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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(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": "2206.01586v1.Engineering_the_spin_orbit_torque_efficiency_and_magnetic_properties_of_Tb_Co_ferrimagnetic_multilayers_by_stacking_order.pdf",        "content": "Engineering the spin-orbit torque efficiency and magnetic properties of Tb/Co\nferrimagnetic multilayers by stacking order\nMickey Martini,1, 2Can Onur Avci,1Silvia Tacchi,3Charles-Henri Lambert,1and Pietro Gambardella1\n1Department of Materials, ETH Zürich, Hönggerbergring 64 CH-8093 Zürich, Switzerland\n2Department of Physics, Politecnico di Milano, P.za Leonardo da Vinci 32, I-20133 Milano, Italy\n3Istituto Officina dei Materiali del CNR (CNR-IOM), Sede Secondaria di Perugia,\nc/o Dipartimento di Fisica e Geologia, Università di Perugia, I-06123 Perugia, Italy\n(Received June 6, 2022)\nWe measured the spin-orbit torques (SOTs), current-induced switching, and domain wall (DW)\nmotion in synthetic ferrimagnets consisting of Co/Tb layers with differing stacking order grown\non a Pt underlayer. We find that the SOTs, magnetic anisotropy, compensation temperature and\nSOT-induced switching are highly sensitive to the stacking order of Co and Tb and to the element in\ncontact with Pt. Our study further shows that Tb is an efficient SOT generator when in contact with\nCo, such that its position in the stack can be adjusted to generate torques additive to those generated\nby Pt. With optimal stacking and layer thickness, the dampinglike SOT efficiency reaches up to 0.3,\nwhich is more than twice that expected from the Pt/Co bilayer. Moreover, the magnetization can be\neasily switched by the injection of pulses with current density of about 0:5\u00002\u0002107A=cm2despite the\nextremely high perpendicular magnetic anisotropy barrier (up to 7.8 T). Efficient switching is due to\nthe combination of large SOTs and low saturation magnetization owing to the ferrimagnetic character\nof the multilayers. We observed current-driven DW motion in the absence of any external field,\nwhich is indicative of homochiral Néel-type DWs stabilized by the interfacial Dzyaloshinkii-Moriya\ninteraction. These results show that the stacking order in transition metal/rare-earth synthetic\nferrimagnets plays a major role in determining the magnetotransport properties relevant for spintronic\napplications.\nI. INTRODUCTION\nMagnetization switching and domain wall (DW) mo-\ntion driven by spin-orbit torques (SOTs) have attracted\nextensive research interest due to their potential applica-\ntions in nonvolatile memory and logic technologies [ 1–14].\nTo improve the scalability and thermal stability of de-\nvices in these applications, materials with perpendicular\nmagnetic anisotropy (PMA) are often preferred. Ultra-\nthin films of 3dtransition metals and alloys (e.g., Co or\nCoFeB) have been widely investigated for such purposes,\nwhere the interfacial spin-orbit coupling gives rise to the\nPMA [15–18]. Recently, however, increasing interest has\nbeen devoted to ferrimagnetic alloys of rare earth metals\nand transition metals possessing bulk PMA [ 19–37]. In\nthese compounds, the rare-earth-metal and transition-\nmetal magnetic sublattices are coupled antiparallel to\neach other, resulting in a reduced total magnetic moment\nrelative to the individual sublattices [ 38,39]. The low\nmagnetization allows for efficient SOT-induced switching,\nwhereas the large PMA increases the thermal stability\nof the devices. Moreover, the possibility of obtaining\nangular momentum compensation allows for antiferro-\nmagnetlike magnetization dynamics that, e.g., leads to\nultrafastswitchingandDWmotion[ 31–33]. Todate, stud-\nies of SOTs and related phenomena have mainly focused\non the alloy form of the rare-earth-metal–transition-metal\nsystems [ 19–33], whereas multilayer structures have been\nless explored [ 34–37]. In such structures, the stacking\norder and the thickness of each layer offer additional de-\ngrees of freedom to tune the SOTs and understand the\nrole of each element in interface-driven phenomena [ 13].Theoretical [ 40] and experimental [ 34,37,41–44] stud-\nies suggest that the rare-earth elements with partially\nfilled forbitals are candidates for generating large spin\nHall effect, and hence can be used as a source of SOTs.\nThe strong spin-orbit interaction in the rare earth metals\ncombined with the inversion symmetry breaking at their\ninterfaces with transition metals can also produce sizeable\nRashba-Edelstein effect and Dzyaloshinkii-Moriya inter-\naction (DMI). The latter stabilizes homochiral Néel-type\ndomain walls with efficient and ultrafast current-induced\ndynamics [9–11, 45–47].\nMotivated by the above considerations, we investigate\nthe influence of the stacking order and layer thickness\non current-induced SOTs, switching, and DW motion\nin Pt/Co/Tb-based multilayers. Harmonic Hall and\nmagneto-optic Kerr effect measurements reveal strong\nSOTs in the perpendicularly magnetized Tb/Co multi-\nlayers, which lead to efficient current-induced switching\nand DW motion, comparable to those of TbCo alloys in\ncontact with Pt. Moreover, the SOTs and other mag-\nnetic properties, such as the saturation magnetization,\nmagnetic anisotropy, compensation temperature, and the\nanomalous Hall effect (AHE) are highly sensitive to the\nstacking order as well as to the element in contact with the\nPt underlayer. We find that Tb can be an effective source\nof SOTs with additive contributions to the ones due to Pt\nwhen placed on top of Pt/Co. These results show that the\nstacking order in rare-earth-metal/transition-metal multi-\nlayers offers an additional degree of freedom to tune the\nefficiency of SOT-controlled switching and DW motion.arXiv:2206.01586v1  [cond-mat.mtrl-sci]  3 Jun 20222\nII. SAMPLE PREPARATION AND\nCHARACTERIZATION\nWe grow Pt(4)/Tb( tTb)/[Co(0.5)/Tb(0.65)/Co(0.5) ]\nand Pt(4)/ [Co(0.5)/Tb(0.65)/Co(0.5) ]/Tb(tTb) multilay-\ners with PMA as well as Pt(5)/Tb( tTb)/Co(5) and\nPt(5)/Co(5)/Tb( tTb) trilayers with in-plane anisotropy\non thermally oxidized silicon substrates by dc magnetron\nsputtering (numbers indicate the thickness in nm). For all\nlayers, aTicappingof 5 nmandaTaseedlayerof 3 nmare\nused. The base pressure and Ar partial pressure of the de-\nposition chamber are 6\u000210\u00008mbarand4\u000210\u00003mbar,\nrespectively. The first two series, schematically illustrated\ninFig.1(a), areusedfortheSOTs, current-inducedswitch-\ning, and DW motion studies, whereas in the last two series\nonly the SOTs are quantified. In the samples with PMA\nwe fix the layers between square brackets and change the\nTb thickness ( tTb) and position relative to them; similarly,\nin the trilayers with in-plane anisotropy we changed the\nTb thickness and position relative to Co. Hall bar devices\nare fabricated by patterning the substrates using optical\nlithography before deposition and subsequent lift-off.\nIn Fig. 1(b) we show representative magnetization\ncurves measured by a superconducting quantum inter-\nference device at room temperature with an out-of-plane\nfield sweep for the two different positions of the Tb(1)\nlayer with respect to the fixed [Co/Tb/Co] trilayer. Both\nsets of data show clear hysteresis and nearly 100 %rema-\nnence at zero field, indicative of strong PMA. However,\ntheir saturation magnetization ( Ms) differs significantly\neven though the amount of Tb and Co is the same in\nboth multilayers. Measurements of all the six samples\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\nFIG. 1. (a) Schematic illustration of the Co/Tb/Co multilay-\ners with different stacking order. (b) Magnetization curves of\nPt/Tb(1)/[Co/Tb/Co] and Pt/[Co/Tb/Co]/Tb(1). (c) Satu-\nration magnetization versus Tb thickness.with PMA [Figure 1(c)] show that Msis larger when Co\nis in direct contact with Pt except for the thickest Tb\nsamples. In both multilayer systems, Msis significantly\nsmaller relative to that expected for Co (>106A/m), as\nexpected due to the antiferromagnetic coupling of Co and\nTb. The large difference in Msand its dependence on Tb\nthickness indicate that the net magnetization is highly\nsensitive to the stacking order.\nFigure 2(a) shows the device layout with the electrical\nconnections and the coordinate system. Figure 2(b) re-\nports the sheet resistance Rsmeasured using a four-point\ngeometry in the structures with PMA. Rsis about 20 %\nlarger in samples with Pt/Tb interface compared with\nthose with Pt/Co interface. The larger Rsand its in-\ncrease with tTbin Pt/Tb(tTb)/[Co/Tb/Co] might be due\nto increased interfacial electron scattering and/or inter-\nmixing between Tb and Pt in this system. In contrast, in\nPt/[Co/Tb/Co]/Tb( tTb),Rsdoesnotsignificantlychange\nwith the Tb thickness, suggesting that the electric con-\nductivity is predominantly dictated by the Pt and Co\nlayers. The resistivity \u001aof all six multilayers with PMA,\nestimated by considering only the conduction through\nPt, Co, and Tb and neglecting the contribution from the\nbuffer and capping layers is reported in Table II along\nwith the other properties measured in this study.\nIn alloys of rare earth metals and transition metals and\nin multilayers, the magnetotransport properties are domi-\nnatedbythe3 d4sorbitalsofthetransition-metalelements\nnear the Fermi level; thus the AHE is mainly driven by\nthe transition-metal sublattice [ 20,34,48]. In Fig. 2(c) we\nreport the Hall resistance RHas a function of out-of-plane\nmagnetic field. As evident from the data, the AHE is\npositive and large in the Pt/[Co/Tb/Co]/Tb series as op-\nposed to the Pt/Tb/[Co/Tb/Co] series, where the AHE is\nnegative and relatively small. This means that i) at room\ntemperature the total magnetization is dominated by the\nCo (Tb) sublattice for the layers with Pt/Co (Pt/Tb) in-\nterface and that the magnetic compensation temperature\nTMis below (above) room temperature; ii) the Pt/Co\ninterface has a significant contribution to the AHE, in\nagreement with literature data [ 49]. Our data clearly show\nthat the stacking order plays a crucial role in determining\nthe magnetic compensation point as well as the AHE,\nin addition to MsandRsdiscussed earlier. We further\nstudy the coercive field ( Hc) as a function of temperature\nto characterize TMof all six samples. As the temperature\ncrossesTM, the anomalous Hall resistance RHchanges\nsign; further, Hcdiverges because a stronger external\nfield is required to reverse the vanishing net magnetiza-\ntion [50], as shown for Pt/Tb(1; 1.5 nm)/[Co/Tb/Co] in\nFig. 2(d).TMdetermined from the change of sign of RH\nis shown in Fig. 2(e) for all the six samples. The stack-\ning order gives rise to a substantial difference of more\nthan 100 K in TMbetween Pt/Tb( tTb)/[Co/Tb/Co] and\nPt/[Co/Tb/Co]/Tb( tTb), which is likely related to the\nlargerMsof Pt/[Co/Tb/Co]/Tb( tTb).\nFinally, we characterize the effective magnetic ansitropy\nfieldHanby measuring RHin the samples with PMA as3\na function of an external magnetic field applied nearly in\nplane at polar and azimuthal angles \u0012H=88°and'H=\n90°[see Fig. 2(a) for the definition of the angular coordi-\nnates]. We use a macrospin approximation to calculate\nHan=H\u0000sin\u0012H\nsin\u0012\u0000cos\u0012H\ncos\u0012\u0001\n, where\u0012=arccos\f\fRH\u000e\nRAHE\f\f\n[51], as reported in Fig. 2(f). We find \u00160Han= 7:8\u00060:5T\nin Pt/Tb(1)/[Co/Tb/Co], which rapidly decreases withincreasing Tb thickness due to the shift of TMto higher\ntemperatures. In the Pt/[Co/Tb/Co]/Tb(1) series, on\nthe other hand, Hanis relatively constant, which is consis-\ntent with the behavior of TMin these samples. We note\nthat extremely large anisotropy fields in excess of 3 T\nare measured in all of our samples, with the exception of\nPt/Tb(2)/[Co/Tb/Co].\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\nFIG. 2. (a) Schematics of a Hall bar device and coordinate system. The Hall bar channel is 100µmlong and 5µmwide. (b)\nSheet resistance versus thickness of Tb layer in Pt/Tb( tTb)/[Co/Tb/Co] and Pt/[Co/Tb/Co]/Tb( tTb). (c) Hall resistance\nas a function of out-of-plane magnetic field in samples with different Tb thickness. (d) Coercive field versus temperature in\nPt/Tb(1,1.5)/[Co/Tb/Co]. The regions shaded in blue and magenta represent temperature ranges in which the magnetization is\nstrongly compensated. (e) Magnetization compensation temperature as a function of Tb thickness. The black arrow indicates\nthatTM>370 K, which is the largest temperature reached in our setup; the green solid line indicates room temperature. (f)\nAnisotropy field at room temperature versus Tb thickness.\nIII. RESULTS AND DISCUSSION\nA. SOTs in multilayers with PMA\nThe SOT-induced effective fields are quantified by per-\nforming harmonic Hall effect measurements in the PMA\nsamples [ 52,53]. An alternating current with frequency\n!=2\u0019=10 Hzand amplitude j= 0:3\u00001\u0002107A=cm2is ap-\nplied along the xaxis, and the first- and second-harmonic\nHall resistances ( R1!\nHandR2!\nH) were measured during an\nin-plane field sweep along the xand yaxes (Hx;Hy). The\ncurrent density is estimated by dividing the current by\nthe total cross section of the Pt, Co, and Tb layers. We\nneglect the conductivity of the highly resistive Ta buffer\nand Ti capping layers. In multilayers with inversion sym-\nmetry along the zaxis and current injection along x, the\ndampinglike and fieldlike SOT effective fields are given\nbyHDL=HDL(y\u0002^M)andHFL=HFLy, respectively,\nwhere Mis the unit vector of the magnetization and\nthe current-induced spin accumulation is polarized along\ny[13]. In order to define the sign of the field ampli-\ntudesHDL;FL, we adopt the convention that HDL(HFL)\nis positive (negative) when the field points in the samedirection as in Pt/Co (opposite direction to the Oersted\nfield), where Pt is deposited before Co [ 52]. Due to the\nlarge PMA in our samples, we opt for the small-angle ap-\nproximation method where the magnetization is slightly\ntilted away from equilibrium by the applied field. In this\nregime, the effective fields can be estimated using the\nfollowing relations [54, 55]:\n\u0001Hx(y)= \n@R2!\nH\n@Hx(y)\u001e@2R1!\nH\n@H2\nx(y)!\n; (1)\nHDL(FL) =\u00002\u0001Hx(y)\u0006\u0017\u0001Hy(x)\n1\u00004\u00172; (2)\nwhere\u0017is the ratio of the planar Hall resistance RPHEto\nthe anomalous Hall resistance RAHEand\u0006corresponds\nto the measurements taken when the magnetization is\npointing along the \u0006zdirection. To characterize RAHE\nandRPHE, we use the first-harmonic signal given by R!\nH=\nRAHEcos\u0012+RPHEsin2\u0012sin 2'. Since the magnetization\ncannot be fully saturated in plane by the available field,\nwe quantify RPHEby measuring R!\nHwith an external field\napplied at (\u0012H;'H)\u0011(88°;45°). We then separate the\nantisymmetric and symmetric signal contributions (with4\nrespect to the field reversal), which originate from the first\nand second terms in the above equation, respectively. The\nantisymmetric signal is then plotted as a function of sin2\u0012,\nwhere\u0012is estimated using the macrospin approximation\nas\u0012=arccos\f\fR1!\nH\u000e\nRAHE\f\f. Finally the slope of the linear\nfit represents RPHE. The details of this approach can be\nfound in Ref. [52].\nFigure 3 shows representative R1!\nHandR2!\nHmea-\nsurements taken from the Pt/[Co/Tb/Co]/Tb(1.5) and\nPt/Tb(1)/[Co/Tb/Co] samples. R1!\nHhas a parabolic field\ndependence due to the coherent rotation of the magneti-\nzation. Note that we only plot the data corresponding\nto theHxsweep since the Hysweep shows an identical\nbehavior. The sign of R!\nHis opposite in these two samples\nfor a given magnetization orientation due to the opposite\nAHE [see Fig.2(c)]. R2!\nHvaries linearly with both Hx\nandHyas expected from the action of SOTs. We note\nthat the slopes of R2!\nHare opposite in Co-like and Tb-like\nsamples. Considering also the opposite sign of R1!\nHin\nthese two samples we conclude that the SOT effective\nfields have the same sign in both systems. Thus, the SOTs\nact on the net magnetization, in agreement with previous\nworks [21,23]. The SOT effective fields normalized by the\ncurrent density, \u001fDL(FL) =\u00160HDL(FL)\u000e\nj, are summarized\nin Figs. 4(a)-(4b) as a function of Tb thickness. The\nvalues of\u001fDL(FL)are obtained by performing a linear\nfit ofHDL(FL)versusjfor both magnetization orienta-\ntions (not shown). The fieldlike component includes alsothe Oersted field contribution due to the current flowing\nthrough Pt, which we estimate as \u00160HOe\u000e\nj=0:3 mT\n10\u00007A\u00001cm2by Ampere’s law [ 51]. The Oersted field\npoints in the opposite direction to the fieldlike component\nof the SOT field, and can be neglected due to its small am-\nplitude compared with HFL. We observe that \u001fDLis very\nstrong in Pt/Tb(1)/[Co/Tb/Co], reaching up to +72 mT\n10\u00007A\u00001cm2but rapidly decreasing as the Tb thickness\nis increased. Also \u001fFLreaches remarkable values, about\n+25 mT 10\u00007A\u00001cm2in both Pt/Tb(1)/[Co/Tb/Co] and\nPt/[Co/Tb/Co]/Tb(1). To compare the SOTs in dif-\nferent layers on an equal footing we calculate the SOT\nefficiencies \u0018j\nDL(FL)=2e\n~MstM\u001fDL(FL)[56], whereeis\nthe electron charge, ~is the reduced Planck’s constant,\nandtMis the magnetic layer thickness. The SOT ef-\nficiencies are plotted in Figs. 4(c,-d) as a function of\nTb thickness. We observe that \u0018j\nDLand\u0018j\nFLvary be-\ntween +0.1 and +0.3 and between +0.03 and +0.2, respec-\ntively. Whereas \u0018j\nDLand\u0018j\nFLin Pt/[Co/Tb/Co]/Tb(1)\nand Pt/Tb(1)/[Co/Tb/Co] are comparable to those mea-\nsured in Pt/Co bilayers [ 13,52], we observe that both\nefficiencies increase strongly as a function of Tb thickness\nin Pt/[Co/Tb/Co]/Tb( t), reaching values that are about\ntwo times larger compared with Pt/Co. Moreover, \u0018j\nFLis\nsignificantly larger in Pt/[Co/Tb/Co]/Tb( tTb) compared\nwith that in Pt/Tb( tTb)/[Co/Tb/Co], and has a different\nthickness dependence in the two series. Similar trends are\nalso observed if we consider the SOT efficiency per unit\nelectric field \u0018E\nDL(FL)=\u0018j\nDL(FL)\u000e\n\u001a(see Table II).\n(a)updown\n-400 0 400Hx (mT)RH-430-420-410410420430\n-400 0 400Hx (mT)-2000200RH-400 0 400Hy (mT)-40040RH+Mz-Mz\n(b)updown\n-800 0 800Hx (mT)RH-124-123123124\n-800 0 800Hx (mT)-15015RH-800 0 800Hy (mT)-505RH\nFIG. 3. First and second harmonic of the Hall resistance ( R1!\nHandR2!\nH, respectively) versus in-plane longitudinal and transverse\nmagnetic fields ( HxandHy) measured using j=0:8\u0002107A=m2in (a) Pt/[Co/Tb/Co]/Tb(1.5) and (b) Pt/Tb(1)/[Co/Tb/Co].5\n0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)\nFIG. 4. (a) Dampinglike and (b) fieldlike SOT effective field per unit current density \u001fDL;FLversus Tb thickness in\nPt/[Co/Tb/Co]/Tb( tTb) and (b) Pt/Tb( tTb)/[Co/Tb/Co]. (c) Dampinglike and (d) fieldlike SOT efficiency \u0018DL;FLver-\nsus Tb thickness. The error bars take into account the errors of the linear and parabolic fits of R1!\nHandR2!\nHrespectively,\naveraged according to Eq. 2 for each current density, propagated in the linear fits of HDL(FL) (j)for the +Mzand\u0000Mz\nconfigurations.\nWe note that \u0018j\nDLand\u0018j\nFLare obtained by normalization\nto the average current density jflowing in the multilayers,\nsuch that even larger efficiencies would be obtained by\nnormalization to the current density flowing in Pt, as\noften done in the literature. The dependence of \u0018j\nDL(FL)\non the stacking order and Tb thickness can have several\norigins. First, the structural details of the multilayers\nand of the interface with Pt generally change with the\nstacking order, which may ultimately lead to changes\nof the SOTs. This point is supported by the resistivity\nmeasurements, which suggest that Tb deposited on Pt\ninduces intermixing between the two species rather than\nfavoring sharp interfaces. Second, the Tb layer can be a\nsource of dampinglike and fieldlike SOTs in itself, with\na contribution opposite to that of Pt. Such a scenario,\nwhich agrees with a previous study of Pt/[Co/Ni]/Co/Tb\nmultilayers [ 37], is consistent with the enhancement of\n\u0018j\nDL(FL)when Pt and Tb are placed on opposite sides of\nthe [Co/Tb/Co] trilayer. Third, placing the Tb layer on\ntop of the stack might provide an efficient sink for the spin\ncurrent generated by Pt and the Pt/Co interface [ 57]. The\nmonotonic increase of the dampinglike and fieldlike SOTs\nin this series is consistent with both the second and third\nexplanation taking into account the finite spin diffusionlength of Tb. Overall, even if the exact mechanism giving\nrise to the enhancement of \u0018j\nDLand\u0018j\nFLcannot be singled\nout with certainty, our data indicate that Tb and the\nstacking order of Co/Tb multilayers play a crucial role in\ndetermining the strength of the SOTs.\nB. SOTs in Pt/Tb/Co and Pt/Co/Tb trilayers\nwith in-plane magnetic anisotropy\nWe quantify the dampinglike SOT efficiency in samples\nwith in-plane anisotropy following the method outlined in\nRef. [58]. Injection of a relatively small alternating cur-\nrent (!=2\u0019=10 Hz,j= 0:3\u00001\u0002107A=cm2) gives rise to\nperiodic oscillations of the magnetization about its equi-\nlibrium position. The second-harmonic Hall resistance\nrelated to the current-induced fields and the thermoelec-\ntric signal due to Joule heating can be written as:\nR2!\nH=\u0012\nRAHEHDL\nHext+I0\u000brT\u0013\ncos'+\n2RPHE(2 cos3'\u0000cos')HFL+HOe\nHext+Hdem(3)6\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\nFIG. 5. (a) In-plane angular dependence of the second-harmonic Hall resistance of Pt/Co/Tb(3) for j=0:8\u0002107A=cm2. (b)\nDampinglike and thermal contribution to the second-harmonic Hall resistance. Inset: linear fit of the cosine component of\nthe second-harmonic Hall resistance versus the inverse of the external field and the demagnetizing field acting against the the\ncurrent-induced field. (c) Dampinglike SOT efficiency versus Tb thickness. The error bars take into account the uncertainty of\nthe linear fit of HDLversusj.\nwhereI0is the amplitude of the injected current, \u000bthe\nanomalous Nernst coefficient, rTthe perpendicular ther-\nmal gradient and Hdemthe demagnetizing field. Figure\n5(a) shows a representative angle scan measurement of\nR2!\nHin Pt/Co/Tb(3) for different external in-plane fields\nalong with the fitting curves (dashed lines) given by Eq. 3.\nIn Fig. 5(b) we plot the cos'component of R2!\nH, which in-\ncludes the combined contributions of dampinglike torque\nand the Nernst (and/or Seebeck) effect. The amplitude\nof the cos'component of R2!\nHdecreases on increasing\nthe external field. The latter tends to rigidly align the\nmagnetization along its direction, suppressing the peri-\nodic oscillations. This results in a reduction of the SOT\nand therefore of R2!\nHand its cos'component. This field\ndependence can be exploited to further separate the damp-\ninglike torque and the thermoelectric effects in the cos'\ncomponent of R2!\nH. The anomalous Nernst effect depends\nonly on the magnetization direction and not on the exter-\nnal field amplitude (provided that the magnetization is\nsaturated), whereas the SOT decreases at higher field, as\nalready discussed. Thus, the dampinglike effective field\nHDLcan be quantified by performing a linear fit of the\ncos'component of R2!\nHversus the inverse of the external\nfield plus the demagnetizing field Hdem[58], as shown\nin the inset of Fig. 5(b). This experiment is repeated\nfor all the samples for several current densities to esti-\nmate\u001fDLas the slope of HDLversusj, as done for the\nmultilayers with PMA in Sect. IIIA. The values of \u001fDL\nare reported in Table I. Compared with the multilayers\nwith PMA, the dampinglike effective field in the samples\nwith in-plane anisotropy is 2 orders of magnitude lower.\nThis is due to the large magnetization on which the SOT\nacts, that is dominated by the bulk magnetic moments\nof the Co layer (the ferromagnetic layer is now thick).\nThe dampinglike SOT efficiency, calculated also here as\n\u0018j\nDL=2e\n~MstM\u001fDL, is plotted in Fig. 5(c) as a function of\ntTb. Although \u0018j\nDLis significantly lower in these samplescompared with the samples with PMA, it scales in the\nsame way with tTb, increasing (decreasing) in the trilayer\nwith Pt/Co (Pt/Tb) interface. The comparison between\nPt/Tb/Co and Pt/Co/Tb structures provides further ev-\nidence that the SOTs are very sensitive to tTband the\nstacking order of Co/Tb.\nC. Current-induced magnetization switching\nThe large SOTs evidenced by the harmonic Hall effect\nmeasurements in Sect. IIIA can be exploited to realize\ncurrent-induced magnetization switching with enhanced\nefficiency relative, e.g., to Pt/Co bilayers. In order to\nstudy SOT-induced switching in our multilayers with\nPMA, we injected 10- ms-long current pulses of variable\namplitude through the Hall bar devices in the presence of\nan in-plane bias field Hxapplied parallel to the current\nline. This field is required to break the SOT symmetry\nand uniquely define the switching polarity [ 1,13]. Figure\n6(a) shows representative AHE hysteresis loops as a func-\ntion of current measured for Pt/Tb(2)/[Co/Tb/Co] and\nPt/[Co/Tb/Co]/Tb(2). In both samples, the net mag-\nTABLE I. Saturation magnetization, demagnitizing field,\ndampinglike effective field per unit current density and SOT\nefficiency in trilayers with in-plane anisotropy. Errors are a\nfraction of the least significant digit.\nMs\u00160Hdem \u001fDL \u0018j\nDL\n(106A m\u00001)(T) ( mT 10\u00007A\u00001cm2)\nPt/Tb(1)/Co 1.13 1.80 0.19 0.040\nPt/Tb(3)/Co 0.81 1.59 0.12 0.023\nPt/Tb(5)/Co 0.48 1.48 0.10 0.014\nPt/Co/Tb(1) 1.22 0.75 0.27 0.061\nPt/Co/Tb(3) 0.81 0.72 0.37 0.073\nPt/Co/Tb(5) 0.68 0.56 0.44 0.0917\nnetization switches from +Mzto\u0000Mzwhen the current\nis parallel to the in-plane field and from \u0000Mzto+Mz\nwhen it is antiparallel to it. The loops of Fig. 6(a) have\nopposite polarity because the AHE has opposite sign in\nthese two samples at room temperature [see Fig. 2(c)].\nWe repeate the measurements summarized in Fig. 6(a)\nfor all of the multilayers with PMA and several in-plane\nmagnetic fields to characterize the threshold current den-\nsityjthrequired to switch the magnetization. The data\nare shown in Fig. 6(b) as a function of \u00160Hx. In the\nmacrospin approximation, the threshold switching cur-\nrent is given by [29, 59]\njth=2e\n~\u00160MstM\n\u0018j\nDL\u0012Hk\n2\u0000Hxp\n2\u0013\n; (4)\nwheretMis the thickness of the magnetic layer. Although\nEq. 4 does not apply to switching by domain nucleation\nand propagation, which is the preferred mode of magneti-\nzationreversalinmesoscopicsamples[ 47], onestillexpects\njthto scale with HkandHxgiven that these two fields\n-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)\nFIG. 6. (a) Change of the anomalous Hall resistance\ndue to SOT-induced switching in Pt/Tb(2)/[Co/Tb/Co]\nwith constant in-plane field \u00160Hx=182 mT and in\nPt/[Co/Tb/Co]/Tb(2) with \u00160Hx=333 mT. The Hall resis-\ntance is plotted versus the amplitude Ipof 10- ms-long current\npulses. (b) Threshold current density for switching as function\nof the in-plane field \u00160Hx.determine the domain nucleation barrier. Figure 6(b)\nshows that, despite the very large Hkof our multilayers\n(see Sect. II), we are able to switch all of the samples with\ncurrent densities below 2:5\u0002107A=cm2and in-plane field\nHx\u001cHk. This is due to the combined effect of strong\ndampinglike SOTs and low M sowing to the ferrimagnetic\ncharacter of our multilayers. The strongest dependence\nofjthonHxis found in the sample with weakest mag-\nnetic anisotropy, namely Pt/Tb(2)/[Co/Tb/Co], as ex-\npected. The samples with strongest anisotropy, where\n\u00160Hk\u00156T, present a much weaker dependence of jth\nonHx, suggesting that the in-plane field is not the main\ncause of reduction of the domain nucleation barrier in\nsuch a case. On the other hand, jthscales with \u0018DLand\nMsin samples with comparable anisotropy, such as in the\nPt/[Co/Tb/Co]/Tb( tTb) series.\nTo investigate the role of Joule heating in the switch-\ning experiments, we measure the Hall effect by injecting\ndifferent current densities up to j'2\u0002107A=cm2. We\nfind that the coercivity decreases by about 30% upon\nincreasingjfrom 0:5to2\u0002107A=cm2(not shown). Like-\nwise, we also estimate the anisotropy field \u00160Hkin all the\nmultilayers and record a drop smaller than 25% from the\ncase ofj <1\u0002107A=cm2to the case of j= 2\u0002107A=cm2.\nTherefore we conclude that the Joule heating does not\nplay a major role in determining jthforj <1\u0002107A=cm2,\nwhereas it partially assists the switching at higher current\nby decreasing the magnetic anisotropy energy barrier and\nfavoring DW depinning. Indeed, jthis determined by the\ninterplay of several factors beyond Hk,Hx,\u0018DL, andMs,\nwhich include Joule heating as well as \u0018FL, the DMI and\nDW pinning properties of the multilayers. Overall, these\ndata show that SOTs can switch ferrimagnetic multilayers\nwith very strong PMA at a current density of the order of\n107A=cm2, similarly to what is observed in CoTb alloys\n[28].\nD. Current-induced domain wall motion\nSOTs can efficiently move DWs in suitable devices\nand heterostructures [ 13]. Compensated ferrimagnets\nare particularly interesting for the SOT-induced DW mo-\ntion because the DWs can be propelled at very high\nspeeds with moderate currents owing to their partic-\nular dynamical properties near the magnetic and an-\ngular momentum compensation [ 11,27,31,32,60–62].\nWe investigate the current-induced DW motion in the\nPt/[Co/Tb/Co]/Tb( tTb) samples. We use a wide-field\nKerr microscope configured in the polar geometry to\ndetect the out-of-plane magnetization component in\nPt/[Co/Tb/Co]/Tb( tTb) racetracks and study the DW\nresponse to external fields and currents. The experimental\nprocedure can be briefly described as follows. First, the\nmagnetization is saturated along +zor\u0000zby applying\nan out-of-plane magnetic field larger than the coercivity.\nNext, a domain with the opposite magnetization is nu-\ncleated by the Oersted field produced by a short current8\npulseInucinjected into an electrically insulated wire cross-\ning the racetrack [Fig. 7(a)]. Finally, the DW is displaced\nby injecting current pulses along the DW track. We per-\nform two types of experiments: i) determination of the\ncritical current for depinning a single DW in the presence\nor absence of an out-of-plane field ( Hz);ii) determination\nof the DW velocity in the absence of any applied field.\nFigure 7(b) shows the critical current density jdepin\nrequired to depin a DW as function of Hzfor both the\nup-down and down-up DWs in Pt/[Co/Tb/Co]/Tb(1).\nThe linear variation of jdepin(Hz)shows that the current\ninduces a dampinglike SOT that acts as an effective mag-\nnetic field along\u0006z, assisting or hindering Hzto depin\nthe DWs [ 9,11,63,64]. In the one-dimensional DW\nmodel, neglecting the spin-transfer and fieldlike torques,\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jcr (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8j (107 A/cm2)-200200Hzeff (mT)(c)0Hd UP-DN DW0Hd DN-UP DW/20HDL  +Mz/20HDL  -Mz\n(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW/20HDL  +Mz/20HDL  -Mz(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW-/20HDL  +Mz-/20HDL  -Mz(a)\nDomainWalljInucl10 μm\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jcr (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8j (107 A/cm2)-200200Hzeff (mT)(c)0Hd UP-DN DW0Hd DN-UP DW/20HDL  +Mz/20HDL  -Mz\n(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW/20HDL  +Mz/20HDL  -Mz(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW-/20HDL  +Mz-/20HDL  -Mz(a)\nDomainWalljInucl10 μm\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jcr (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8j (107 A/cm2)-200200Hzeff (mT)(c)0Hd UP-DN DW0Hd DN-UP DW/20HDL  +Mz/20HDL  -Mz\n(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW/20HDL  +Mz/20HDL  -Mz(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW-/20HDL  +Mz-/20HDL  -Mz(a)\nDomainWalljInucl10 μm\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jcr (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8j (107 A/cm2)-200200Hzeff (mT)(c)0Hd UP-DN DW0Hd DN-UP DW/20HDL  +Mz/20HDL  -Mz\n(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW/20HDL  +Mz/20HDL  -Mz(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW-/20HDL  +Mz-/20HDL  -Mz(a)\nDomainWalljInucl10 μm\nFIG. 7. (a) Differential Kerr microscopy image showing\nthe nucleation of a magnetic domain (white region) in a\nPt/[Co/Tb/Co]/Tb(1) racetrack delimited by the red dotted\nline. (b) Critical current density required to depin up-down\nand down-up DWs as a function of applied out-of-plane mag-\nnetic field in Pt/[Co/Tb/Co]/Tb(1). (c) Effective out-of-plane\nmagnetic field corresponding to the depinning field \u0001Hdepin\n(squares) and dampinglike SOT effective field - \u0019=2\u0002HDL(cir-\ncles) versus current density for both DWs and magnetic states\nin Pt/[Co/Tb/Co]/Tb(1).\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 nsFIG. 8. (a) Series of Kerr differential images showing current-\ndriven DW motion in Pt/[Co/Tb/Co]/Tb(2) as hundreds of\n20-ns-long current pulses with jp=3:6\u0002107A=cm2amplitude\nare injected along the track. (b) DW velocity as a function\nof current density for different pulse widths \u001cp=15-100 ns\nin Pt/[Co/Tb/Co]/Tb(2). The errors in the DW velocity are\ndue to the arrival time uncertainty, which is larger when a\nsmall number of pulses is used.\nthe effective field induced by the current is given by\nHe\u000b\nz= \u0001Hdepin, where \u0001Hdepinis the change in the\nDW depinning field under dc current injection. Alterna-\ntively, we estimate He\u000b\nz=\u0000\u0019\n2HDLcos\t, whereHDLis\nthe dampinglike SOT effective field obtained by the har-\nmonic Hall effect measurements in Sect. IIIA and \tis the\nangle between the DW magnetization and the xaxis [65].\nFigure 7(c) compares He\u000b\nzobtained using \u0001Hdepinand\n-\u0019\n2HDLfor different current densities. The two measure-\nments are in excellent agreement, which implies that the\nDWs are of purely homochiral Néel type (i.e., \t = 0 deg).\nMoreover, the DW motion along the current flow indicates\nthat they have left-handed chirality. This is consistent\nwith the interfacial DMI occurring at the Pt/Co inter-\nface with possible additional contributions from the inner\ninterfaces between Co and Tb and the top Tb/Ti inter-\nface [66]. Qualitatively the same results are obtained\nfor Pt/[Co/Tb/Co]/Tb(1.5) and Pt/[Co/Tb/Co]/Tb(2),\nshowing that the current-induced DW depinning measure-\nments can be effectively used to identify the DW type\nand estimate the DL-SOT effective field in this system.\nNext, we characterize the current-induced DW velocities\nin Pt/[Co/Tb/Co]/Tb(1, 1.5, 2). To do so, we capture a\nsequence of Kerr images while injecting a train of current\npulses through the track to move a pre-nucleated DW, as\nshown in Fig. 8(a). To estimate the DW velocity ( vDW),\nwe divide the total displacement by the total pulse length.9\nFigure 8(b) shows the results from Pt/[Co/Tb/Co]/Tb(2),\nwherevDWis plotted as function of current density jp\nfor different pulse lengths \u001cp. The data exhibit several\nimportant features. First, the current required to initi-\nate the DW motion, i.e., the onset of the linear regime,\nis strongly dependent on the pulse width and is lower\nfor longer pulses. This can be understood by the effect\nof thermal activation on the DW depinning, which is\nstronger for longer pulses. Second, in the linear regime\nthe slopevDW=jis approximately the same for all pulse\nwidths, suggesting that Joule heating plays a minor role\ninvDWunlike in the depinning process. In all of the\ncases, the depinning current in the Pt/[Co/Tb/Co]/Tb\nmultilayers is comparable to that found in Pt/Co 1-xTbx\nferrimagnetic alloys [ 27]. Third, the extrapolation of the\nlinear fits yields velocities in the range of 10-15 m/s for\njp= 108A=cm2. Thesevaluesaremoderateincomparison\nwith those of other ferrimagnetic [ 27,31,67] and synthetic\nantiferromagnetic [ 68] systems despite the strong SOTs.However, we refrain from drawing definitive conclusions\non the DW velocity behavior of our films based on the\navailable data because we cannot conclude whether the\nupturn of the vDWis due to the onset of a wide creep\nregime or the flow regime. In the former case, it is not\npossible to extrapolate the velocities at higher current\ndensity.\nFinally, we attempt to quantify the effective DMI field\nHDMIin all the samples studied above. To do so, we\nmeasurejdepinas function of applied in-plane field up\ntoHx=40 mT, the maximum in-plane field available in\nthe Kerr microscope. If Hxis applied opposite to HDMI,\nwe expect that the DW magnetization rotates first along\ntheyaxis, i.e., the DW becomes Bloch-like for jHxj'\njHDMIj, and then Néel-like with right-handed chirality\nforjHxj>jHDMIj[11]. In the applicable in-plane field\nrange, however, we do not observe any change in the sign\nofjdepin, indicating that the chirality is not affected by\nHxand thatjHDMIj>40 mT. These measurements show\nthatHDMIis relatively strong in Pt/[Co/Tb/Co]/Tb.\nTABLE II. Summary of the magnetic properties and SOT efficiency of Pt/Tb( tTb)/[Co/Tb/Co] and Pt/[Co/Tb/Co]/Tb( tTb)\nmultilayers with PMA. All values are measured in ambient conditions. Errors are a fraction of the least significant digit, except\nforTM, whose error is of the order of 1 K.\n\u001a M sTMRAHERPHE\u00160Hk\u001fDL\u001fFL\u0018j\nDL\u0018j\nFL\u0018E\nDL\u0018E\nFL\n(µ\n cm) (105(K) (m\n) (m\n) (T) (mT 10\u00007(mT 10\u00007(105(105\nA m\u00001) A\u00001cm2) A\u00001cm2) \n\u00001m\u00001)\n\u00001m\u00001)\nPt/Tb(1)/[Co/Tb/Co] 72.5 0.24 310 129 18 7.8 +72 +23 +0.14 +0.04 +1.92 +0.61\nPt/Tb(1.5)/[Co/Tb/Co] 80.3 0.22 313 154 22 5.2 +55 +21 +0.12 +0.04 +1.44 +0.56\nPt/Tb(2)/[Co/Tb/Co] 91.9 0.50 >370 120 16 1.1 +17 +7 +0.10 +0.04 +1.02 +0.41\nPt/[Co/Tb/Co]/Tb(1) 63.5 0.74 193 375 89 3.2 +22 +20 +0.13 +0.12 +2.09 +1.94\nPt/[Co/Tb/Co]/Tb(1.5) 67.0 0.85 187 425 101 3.5 +22 +20 +0.18 +0.17 +2.63 +2.53\nPt/[Co/Tb/Co]/Tb(2) 74.4 0.53 212 341 62 2.7 +50 +35 +0.29 +0.21 +3.93 +2.79\nIV. CONCLUSIONS\nOur systematic study of synthetic ferrimagnets based\non Co/Tb layers shows that the magnetic properties, SOT\nefficiency, and switching depend strongly on the stack-\ning sequence of Co and Tb in the heterostructures as\nwell as on the thickness of Tb. In Pt/[Co/Tb/Co]/Tb,\nthe dampinglike SOT efficiency reaches up to 0.3, which\nwe attribute to the additive SOT contributions from the\nbottom Pt and the top Tb layer. Our conclusion is also\nsupported by harmonic Hall measurements performed in\nthe Tb/Co multilayers with in-plane anisotropy. The\nhigh value of the SOT efficiency in Pt/[Co/Tb/Co]/Tb is\nabout two and three times larger than that in Pt/Co and\nPt/Tb/[Co/Tb/Co], respectively. The PMA in these mul-\ntilayers is very strong with effective magnetic anisotropy\nfields estimated between 1 and 8 T. Despite the large\nPMA, the critical current for magnetization switching\nis relatively low, of the order of 0:5\u00002:5\u0002107A=m2, de-pending on the assisting in-plane field. Current-induced\nDW motion measurements reveal that the DWs are of\nNéel type with left-handed chirality, stabilized by inter-\nfacial DMI similar to Pt/Co bilayers. Our experiments\nshow that synthetic ferrimagnetic heterostructures have\ncomparable or improved PMA, SOT, and DMI relative\nto ferromagnetic systems and ferrimagnetic alloys. The\nstacking order offers an additional degree of freedom to\ntune and maximize these parameters in spintronic devices.\nV. ACKNOWLEDGMENTS\nThis work was supported by the Swiss National Sci-\nence Foundation through grants no. 200020_200465 and\nPZ00P2_179944. The European Metrology Programme\nfor Innovation and Research (EMPIR), under the Grant\nAgreement 17FUN08 TOPS, is also acknowledged.\n[1]I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. 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Nanotech. 10,\n221 (2015)."    },    {        "title": "1402.4634v1.Theory_of_Ferrimagnetism_in_the_Hubbard_Model_on_Bipartite_Lattices_with_Spectrum_Symmetry.pdf",        "content": "arXiv:1402.4634v1  [cond-mat.str-el]  19 Feb 2014Theory of Ferrimagnetism in the Hubbard Model on Bipartite L attices with Spectrum\nSymmetry\nYang Xue,1Jing He,2Xing-Hai Zhang,1and Su-Peng Kou1,∗\n1Department of Physics, Beijing Normal University, Beijing , 100875, P. R. China\n2Department of Physics, Hebei Normal University, HeBei, 050 024, P. R. China\nInthis paperwe developedtheoryoftheferrimagnetism inth eHubbardmodel on bipartitelattices\nwith spectrum symmetry. We then study the defect-induced fe rrimagnetic orders in three models\nand explored the universal features.\nThe magnetic orders are important magnetic proper-\nties for two dimensional (2D) Hubbard model, to which\nresearchers also have payed much attention. At half fill-\ning case, the ground state is known to be the long-range\n(LR)antiferromagnetic(AF) order. Nagaokamadeasur-\nprising discovery about the induced ferromagnetic (FM)\nby a single hole for infinite coupling limit U→ ∞[1].\nAt heavily doping region a possible FM order may exist.\nThe starting point on the issue of ferrimagnetism of the\nHubbard model is a theorem by Lieb[2]. In this theo-\nrem, it is pointed out that the total spin Sof the ground\nstate for the Hubbard model on bipartite lattice with\nparticle-hole (PH) symmetry and real hopping parame-\nters is givenby S=|NA−NB|/2 (we call it Lieb spin mo-\nment) whereNAandNBare the numbers of lattice sites\non A and B sublattice, respectively. Then, in Ref.[3], it\nis pointed out that the ground state of the 2D sublattice-\nunbalanced Hubbard model on bipartite lattice with the\nsame conditions (PH symmetry and real hopping param-\neters) is really the ferrimagnetic (FR) order that pos-\nsesses a finite total magnetic moment and the LR AF\norder and obeys m(0)≤m(Q) (we call it Shen-Qiu-Tian\n(SQT) inequality[4]) where m(0) =/summationdisplay\nij/angbracketleftbig\nˆs+\niˆs−\nj/angbracketrightbig\n/Nand\nm(Q) =/summationdisplay\nij(−1)i+j/angbracketleftbig\nˆs+\niˆs−\nj/angbracketrightbig\n/N(ˆsi=1\n2ˆc†\niσˆci, Nis the\ntotal number of the lattice sites). However, the FR order\nwas seldom studied [3, 5, 7, 8] and the detailed proper-\nties of the FR order in 2D Hubbard model have not been\nexplored.\nThe 2D Hubbard Model : Our starting point is the fol-\nlowing Hamiltonian\nH=−/summationdisplay\ni,jtij/parenleftBig\nˆc†\niˆcj+h.c./parenrightBig\n+U/summationdisplay\niˆni↑ˆni↓−µ/summationdisplay\ni,σˆc†\niσˆciσ,\n(1)\nwhere ˆci= (ˆci,↑,ˆci,↓)Twith ˆci,σrepresenting the\nfermion annihilation operator at site i.σ=↑,↓denote\nspin index. For this bipartite lattice, we have two sub-\nlattices, A and B. tijis the hopping amplitude. Uis the\nstrength of the repulsive interaction. µis the chemical\npotential which is set to µ= 0 for the half filling case.\n∗Corresponding author; Electronic address: spkou@bnu.edu .cnThe spectrum symmetry : Firstly, we defined the spec-\ntrum symmetry . For the Hamiltonian in Eq.(1) with the\nspectrum symmetry (S symmetry), the denisty of state\n(DOS)ρ(E) is alwayssymmetricvia Easρ(E) =ρ(−E).\nAsaresult, eachenergylevelwithpositiveenergy Emust\nbe paired with an energy level with negative energy −E.\nTo make it clearer, we define an operator of the S sym-\nmetry as S=P · DwherePis the PH transformation\noperator P=R · Kintroduced in Ref.[9, 10] and Ris\nan operator that leads to ˆ ci↔(−1)iˆci,Kis the com-\nplex conjugate operator, Dis a discrete transformation\noperator that commutes with lattice translation opera-\ntorsTx,yas, [D,Tx] = [D,Ty] = 0. For a Hamiltonian\nwith the S symmetry, we have H=−S†HS. Thus, each\nenergy level |ψ/angbracketrightwith positive energy Eis paired with an\nenergy level S|ψ/angbracketrightwith negative energy −E. So, the PH\nsymmetry is a special case of the S symmetry and for the\ncase ofD= 1,the S symmetry is reduced into the PH\nsymmetry.\nVacancy-induced zero-modes : We then consider the\ncase of free Hamiltonian ( U= 0) with lattice-defect -\nthe vacancy by adding a potential on given lattice site\nR,HR=−/summationtext\ni,jtij/parenleftBig\nˆc†\niˆcj+h.c./parenrightBig\n+VRˆc†\nRˆcR. In the uni-\ntary limit, the lattice-defect becomes a missing lattice\nsite that is just a vacancy, on which we have an infinite\non-site potential, i.e., VR→ ∞. It is pointed out that the\nvacancydoesn’tbreaktheSsymmetry( HR=−S†HRS).\nDue to the S symmetry, there exists a zero energy state\n(the so-called zero-mode) |ψ0/angbracketrightwhen we add a vacancy\non A sublattice. For the zero-mode (ZM) |ψ0/angbracketright,we have\n|ψ0/angbracketright=S|ψ0/angbracketright. We denote the wave-function of the\nZM byψ0(ri−R) whereRis the position of the va-\ncancy. When there exists n-vacancy on A sublattice,\nn=|NA−NB|zero energy modes will necessarily ap-\npear. In addition, for the case with nearest neighbor\nhoppingtij=t/angbracketleftij/angbracketright(t/angbracketleftij/angbracketrightis the real or complex near-\nest neighbor (NN) hopping parameter), these vacancy-\ninduced (VI) zero-modes (ZMs) localize only on the B\nsublattice and are orthotropic each other.\nFor some models with S symmetry, the DOS may be fi-\nnite atthe Fermi level. Exceptforthe VI ZMs, theremay\nexist additional zero energy states |ψ/angbracketright /negationslash=S|ψ/angbracketright. However,\nthe additional zero energy states are not protected by\nsymmetry and are fragile against perturbations. On the\ncontrary, since the VI ZMs are protected by S symme-\ntry (|ψ0/angbracketright=S |ψ0/angbracketright), they are fixed precisely at the Fermi2\nlevel, the interaction term is highly relevant. In particu-\nlar, arbitrary small (repulsive) interaction will drive the\nspin moments of the VI ZMs into an FM ordered state.\nConsequently, the ground state of the original Hamilto-\nnian turns into a LR FR order.\nFerrimagnetism : Let us show the universal features of\nthe FR order in the Hubbard model in the small Ulimit.\nFor a system with Lx×Lyvacancy-lattice ( Lxalong\nxdirection and Lyalongydirection), in a unit cell\n(UC) there are Lx×Ly/a2−1 lattice sites. In gen-\neral, we have Lx×Ly/a2−1 magnetic order parame-\ntersMi=/angbracketleftˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketright/2 to denote the local mag-\nnetizations on the lattice sites in a UC. To simply\ncharacterize the FR order, we introduce two order pa-\nrameters, the total FM moment in a unit cell M=/summationtext\ni∈unit-cell/angbracketleftˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketrightand the total AF moment in\na unit cell N=/summationtext\ni∈unit-cell(−1)i/angbracketleftˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketright, re-\nspectively. One can see that m(0) =/summationdisplay\nij/angbracketleftbig\nˆs+\niˆs−\nj/angbracketrightbig\n/N=\n/summationdisplay\ni/angbracketleftBig\nˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketrightBig\n/N=M/(Lx×Ly/a2−1) and\nm(Q) =N/(Lx×Ly/a2−1). We then define the spin\noperatorofthe VI ZM as ˆSR(i) =1\n2ˆψ†\n0(ri−R)σˆψ0(ri−R)\nwhereˆψ0= (ˆc0,↑,ˆc0,↓)Tψ0and ˆc0,σis the particle anni-\nhilation operator of the zero-mode with spin σ. When/angbracketleftBig\nˆSz\nR/angbracketrightBig\n/negationslash= 0,the FR order is a direct physics consequence\nof the FM order of the spin moments of the different\nZMs. The total FM moment in a UC is given by M=\n2/summationtext\ni∈unit-cellMi= 2/summationdisplay\nR/angbracketleftBig\nˆSz\nR/angbracketrightBig\n,and the total AF moment\ninaUCis N= 2/summationtext\ni∈unit-cell(−1)iMi= 2/summationdisplay\nR/angbracketleftBig\n(−1)iˆSz\nR/angbracketrightBig\n.\nFrom above discussion, we already know that there ex-\nists a ZM for each vacancy. In the small Ulimit,U→0,\nthe low energy physics is dominated by these ZMs. Be-\ncause the ZMs induced by two different vacancies at R\nandR′are orthotropic each other, when considering the\non-site interaction there exists the Hund rule’s coupling\nas−Jeff(R,R′)ˆSR·ˆSR′whereJeff(R,R′)>0 is the effec-\ntive FM spin coupling constant. Thus, the low energy ef-\nfective Hamiltonian becomes Heff=−/summationdisplay\nR,R′Jeff(R,R′)ˆSR·\nˆSR′,of which the ground state is a long range FM order\ndenoted by/angbracketleftBig\nˆSz\nR/angbracketrightBig\n=1\n2/angbracketleftBig\nˆψ†\n0,R↑ˆψ0,R↑−ˆψ†\n0,R↓ˆψ0,R↓/angbracketrightBig\n=1\n2or\nM= 1. As a result, the total spin moment of the ground\nstate must be the total number of spin moments of the\nVI ZMsS=1\n2/summationdisplay\nMthat is just the Lieb spin moment\nS=|NA−NB|/2. The local magnetizations are given by\nMi→/summationdisplay\nR/angbracketleftBig\nˆSz\nR(i)/angbracketrightBig\n.\nFor the case with only NN hopping tij=t/angbracketleftij/angbracketrightin small\nUlimit, it is obvious that Mi∈A= 0.Thus, we have\nN=M= 1.M=Nmeans the ground state is an FM-\nAF-balanced FR order. On the contrary, in the strong/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s52/s54/s56/s49/s48\n/s85/s47/s116/s69/s47/s116\nFIG. 1: (Color online) (a) Part of particle density distribu tion\nof VI ZM on a 60 a×60asquare lattice (a vacancy at the white\nspot); (b) The illustration of the uniform FM-AF-balanced\nferrimagnetic order for the Hubbard model on square lattice\nandU= 0.01t; (c) The two order parameters; (d) The density\nof state for case of Hubbard model on square lattice and U=\nt.\ncoupling limit, U→ ∞,the low energy effective Hamil-\ntonian turns into the (un-frustrated) Heisenberg model\nwith vacancy-lattice as Heff=J/summationdisplay\n/angbracketleftij/angbracketrightˆ si·ˆ sj(J→4t2\nU). The\nground state is characterized by the AF ordered stag-\ngered magnetization, M= 2(−1)iMi→1/2. So, we\nhaveN= 2(Lx×Ly/a2−1)M→Lx×Ly/a2−1 and\nM= 1. The LR FR order is really a defect-diluted AF\norder[5, 17].\nFor case with both NN hopping tij=t/angbracketleftij/angbracketrightand next\nnearest neighbor (NNN) hopping, tij=t/angbracketleft/angbracketleftij/angbracketright/angbracketright/negationslash= 0, the\nsituation becomes complex. The VI ZM still exists due\nto S symmetry. However, the wave-functions of the ZM\nmay distribute on both sublattices. As a result, in small\nUlimit, we have M= 1 and the total FM moment is also\nLieb spin moment |NA−NB|/2. In the large Ulimit, the\nlowenergyeffectiveHamiltonianturnsintothefrustrated\nHeisenberg model, of which the ground state may be not\nan AF order.\nExample 1 - the Hubbard model on square lattice : For\nthe Hubbard model on square lattice, the hopping pa-\nrameters are t/angbracketleftij/angbracketright=t. For this model, the operator of\nthe S symmetry is S=P. The S symmetry protected\nVI ZM is an extended state, of which the wave-function\nΨ0can be naturally be ψ0,i∈A= 0, ψ0,i∈B=1\nNB. See\npart of the particle density distribution of this ZM on a\n60a×60asquare lattice in Fig.1(a).\nTo check the validity of above discussion, we use the\nmean field approach to study the Hubbard model on\nsquare lattice with a vacancy-lattice. The lattice con-\nstant of the vacancy-lattice is set to be d= 6a.We\nchoose 6a×6asites to be a UC (6 aalongxdirection\nand 6aalongydirection). To search the ground state3\nwith the lowest energy, we need to solve 6 ×6−1 = 35\norder parameters that denote the local magnetizations\nMi=/angbracketleftˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketright/2 on 35 lattice sites in a UC.\nFrom the mean field calculation, we find that for the\nweakcouplinglimit, U→0,thegroundstateisa uniform\nFerrimagnetic order, Mi∈A= 0, Mi∈B=1\n6×6−1=1\n35.\nThe FR ordered state is illustrated in Fig.1(b). From\nFig.1(c) one can see that the total FM moment in a\nUC is indeed a constant, M ≡1, which is consistent\nto the prediction of Lieb spin moment |NA−NB|/2. On\nthe other hand, the total AF moment in a UC is also\nN= 1. The FM-AF-balance character ( M=Nor\nm(0) =m(Q)) comes from the fact that the VI ZMs\nonlydistributeononesublattice. With theincreasingthe\ninteraction strength, the average magnetizations on the\nsitesof Bsublatticebecomefinite valueswithanopposite\npolarization to those on the sites of A sublattice Mi∈A:\nMi∈A\n|Mi∈A|=−Mi∈B\n|Mi∈B|/negationslash= 0. Because the amplitudes of Mion\nall lattice sites increase, we have an AF-dominated FR\norder with M<N(orm(0)< m(Q)). Now, the SQT\ninequalitym(0)≤m(Q) is satisfied. In the large Ulimit,\nwe have a saturated value, N →35 butM ≡1. Fig.1(d)\nshows the DOS of the FR order, of which there exists an\nenergy gap. Near the gap, the DOS is enhanced due to\nthe VI ZMs.\nExample 2 - the staggered-flux Hubbard model on\nsquare lattice : Recently, people had realized the photon-\nassisted tunneling on optical lattice and then generated\na large effective (staggered) magnetic flux on optical\nlattice[11–13]. When two-component fermions with re-\npulsive interaction are put into such optical lattice, one\ncangetaneffectivestaggered-fluxHubbardmodel. These\nprogresses may provide new research platform to learn\ntheferrimagnetism. Itiseasytochangethepotentialbar-\nrier by varying the laser intensities to tune the Hamilto-\nnian parametersincluding the hopping strength ( t-term),\nthe staggered flux ( φ) and the particle interaction ( U-\nterm). For this reason, we take the Hubbard model on\nsquare lattice with staggered-flux (SF) as the second ex-\nample, where ti,i±δy=ei±φtfori∈A,ti,i±δx=tfor\ni∈A andti,i±δy=ti,i±δx=tfori∈B. See the illustra-\ntion in Fig.2(a). In particular, we only consider the NN\nhoppings. For this model, the operator of the S symme-\ntry isST=P ·TwhereTis the time-reversal transfor-\nmation operatorwhich commutes with lattice translation\noperators [ T,Tx] = [T,Ty] = 0.\nAfter diagonalization of the Hamiltonian in\nmomentum space, the energy spectra are ob-\ntained as E±=−t[cos(ky−φ)+cosky]±/radicalBig\nt2[cos(ky−φ)−cosky]2+4t2cos2kx.For the case\nofφ= 0, the SF disappears and we get a uni-\nform Hubbard model on square lattice. In the BZ\nkx∈(−π,π),ky∈(−π,π), the energy S turns into\nE=−2t(coskx+ cosky).Now the Fermi surface at\nhalf filling has perfect nesting condition. Away from\nthis case,φ/negationslash= 0, the BZ is reduced into a half one.\nThe system becomes a semi-metal and also has the\nFIG. 2: (Color online) (a) The illustration of the staggered -\nflux lattice; (b) The dispersion of the π/2-flux case: there are\na hole pocket and an electron pocket in a reduced BZ. The\nred plane denotes the position of chemical potential.\nperfect nesting condition. For the + band, there exists\na hole pocket; For the −band, there exists an electron\npocket. See the illustration in Fig.2(b). The density\nof state (DOS) ρ(E) near Fermi surface is reduced\nwith increasing φ. For the case of π-flux, the energy S\nturns intoE±=±2t/radicalbig\ncos2ky+cos2kxand the electron\npocket and hole pocket shrinks into two Dirac nodes at\nK1= (π/2,π/2) andK2= (π/2,−π/2).\nBecause the SF Hubbard model on bipartite lattices\nat half-filling is unstable against antiferromagnetic (AF)\ninstability, the ground state becomes an insulator with\nAF order for the case of finite U. Such AF order is de-\nscribed by the following mean field ansatz /angbracketleftˆc†\ni,σˆci,σ/angbracketright=\n1\n2/parenleftbig\n1+(−1)iσM/parenrightbig\nwhereMis the staggered magnetiza-\ntion. For the cases of spin up and spin down, we have\nσ= +1 and σ=−1, respectively. Then in the mean\nfield theory, by minimizing the ground state energy in\nthe reduced Brillouin zone, we could solve the staggered\nmagnetization. Due to the perfect nesting condition, the\narbitrary small interaction term leads to an AF spin-\ndensity-wave (SDW) order and then the BZ of φ/negationslash= 0,\nπcase is reduced into a quarter one. For the π-flux\ncase, the DOS near Fermi surface is zero and the crit-\nical point between the semi-metal and AF insulator is\naboutUc= 3.11t[14, 15].\nFor the case of φ/negationslash= 0,the VI ZM is a quasi-localized\nstate. See the particle density distribution of this ZM\nfor the case of φ=π/2 in Fig.3(a). The VI ZMs are\nanisotropicdue to the rotation-symmetrybreaking of the\noriginal Hamiltonian. In the continuum limit, for the\ncase ofφ=π,the wave function of VI ZM distributes\non B sublattice that has a simple form of ψ0(x,y)≃\neiK1.r\nx+iy+eiK2.r\nx−iy[16]. The amplitude of this state decays\nwith the distance to the vacancy as 1 /r. It is needed to\npoint out that these VI ZMs are all protected by the S4\n/s85/s47/s116\n/s69/s47/s116\nFIG. 3: (Color online) (a) Part of particle density distribu tion\nof VI ZM on a 60 a×60asquare lattice for φ=π/2 andU= 0;\n(b) The illustration of the cluster AF-balanced ferrimagne tic\norder for φ=π/2 andU= 0.001t; (c) The two order param-\neters for φ=π/2; (d) The density of state for φ=π/2 and\nU=t.\nsymmetry.\nWe use the mean field approach to study the Hubbard\nmodel onSF squarelattice with a 6 a×6avacancy-lattice.\nWe focus on the case of φ=π/2. Now the ground state\nis aclusterFerrimagnetic order for weak coupling case.\nThe word ”cluster” means that the local magnetic order\nparameters Miis larger near the vacancy but smaller\nfar from it. See the illustration Fig.3(b). This ”cluster”\nbehaviorobviouslyis aphysicalconsequenceofthe quasi-\nlocalized VI ZMs. From Fig.3(c) one can also see that\nN →1,M ≡1 (U→0) andN →35,M ≡1 (U→\n∞). From the DOS of the system (Fig.3(d)), one can see\nthat there exists energy gap of the mid-gap states (zero-\nmodes) that dominate the low energy physics. All these\nfeatures indicate an AF-dominated FR order from the\nFM order of spin moments of the quasi-localized ZMs.\nFrom the mean field calculation, for the case of φ=π,\nwe find that the quantum phase transition between the\nmetallic(orsemi-metallic)statesandthemagneticorders\nshiftsfromU=UctoU= 0. Arbitraryinteractiondrives\nthe systeminto a longrangeFR order, which is obviously\ninduced by the vacancy-lattice. At U∼Uc, there is no\ntrue phase transition, instead, a crossover occurs. For\nU < U c,the ground state can be regarded as an FR\norder from the FM ordered spin moments of ZMs. Now,\natinyenergygapopenswhichisduetothe spinpolarized\neffect of the ZMs. On the other hand, for U > U c,the\nground state can be regarded as the defect-diluted AF\norder, and a big energy gap opens which is just AF-Mott\ngap for the double occupied particles on one site.\nExample 3 - the spinful Haldane model on square lat-\ntice: An interesting issue is the FR order for the case\nfrom a topological insulator with NNN hoppings. Now\nwe consider the spinful Haldane model on square lat-\n/s85/s47/s116/s69/s47/s116/s85/s47/s116\nFIG. 4: (Color online) (a) Part of particle density distribu tion\nof VI ZM for the interacting spinful Haldane model on 60 a×\n60asquare lattice at φ=π, t′= 0.1,andU= 0; (b) The\nillustration of the cluster FM-balanced ferrimagnetic ord er\nfor the interacting spinful Haldane model on square lattice at\nφ=π, t′= 0.1 andU= 0.001t; (c) The two order parameters\nof the case of φ=π, t′= 0.1; (d) The density of state of the\ninteracting spinful Haldane model on square lattice at φ=π,\nt′= 0.1 andU=t.\ntice, of which the Hamiltonian is given by HT=H(φ=\nπ) +H(t′) whereH(φ=π) is the Hamiltonian of\nthe SF Hubbard model on square lattice and t′is the\nNNN hopping. The Hamiltonian has PH symmetry from\nHT=−P†HTP. In addition, the free Hamiltonian is a\ntopological Chern insulator.\nWe numerically calculated the free Hamiltonian with\na vacancy, and found a (S or PH symmetry protected)\nVI ZM on a 60 a×60alattice[10]. Fig.4(a) shows the\nparticle density of the ZMs, localized around the defect\ncenter within a length-scale of ∼(∆E)−1, where ∆Eis\nthe energy gap of the vacancy-free case. In particular,\nthe wave-function of the ZM distribute not only on B\nsublattice but also on A sublattice.\nWe then use the mean field approach to study the FR\norder. From the results given in Fig.4(b), we find that\nthegroundstateisalsoaclusterFR.Inthe small Ulimit,\nthe average magnetizations on the sites of A sublattice\nbecomefiniteasMi∈A\n|Mi∈A|=Mi∈B\n|Mi∈B|/negationslash= 0. Now, the magnetic\norder in A sublattice has the same polarized direction to\nthat of B sublattice but the value of it is much smaller\nto that of B sublattice. The total spin Sof the ground\nstate still obeys the prediction of Lieb spin moment as\n|NA−NB|/2. However, we have M>Nthat mean\nm(0)> m(Q). The ground state is an FM-dominated\nFR orderand the SQT inequality is violent. We conclude\nthat the violence of SQT inequality for this case is due\nto the NNN hoppings.\nWhen we increase the interaction strength, m(0) =\nm(Q) atU0. When we further increase the interaction\nstrength, the ground state turns into an AF-dominated5\nFR order with m(0)< m(Q). For larger interaction\nstrength, a topological quantum phase transition occurs.\nThe energy gap closes and opens again. The system then\nhas no nontrivial topological properties and becomes a\ndefect-diluted AF order.\nConclusion : In this paper, we developed a universal\nformula of the ferrimagnetism in the Hubbard model be-\nyond Lieb’s theorem by taking into account for the S\nsymmetry with larger universality than traditional PH\nsymmetry. Then, by taking three models as exam-\nples, we study the defect-induced FR orders that emerge\nfrom three typical fermionic systems - metal, semi-metal,\n(Chern) insulator. We found that there may exist vari-\nous FR orders (uniform FM-AF-balanced FR order, uni-\nform AF-dominated FR order, cluster FM-AF-balanced\nFR order, cluster AF-dominated FR order, cluster FM-\ndominated FR order...). The total spin of all these FR\norders is equal to the Lieb spin moment ( |NA−NB|/2).From the common feature of these FR orders, we con-\njecture that it is the S symmetry protected VI ZMs\n(|ψ0/angbracketright=S|ψ0/angbracketright) that dominate the low energy physics of\nthe system in small Ulimit. However, we found that the\nSQT inequality ( m(0)≤m(Q)) is valid for the model\nwith only the NN hoppings and can be violent by the\nNNN hoppings. In addition, we may point out that the\nformula can be straightforwardly applied to other Hub-\nbard models on bipartite lattices with S symmetry.\n* * *\nThis work is supported by National Basic Research\nProgram of China (973 Program) under the grant No.\n2011CB921803, 2012CB921704 and NSFC Grant No.\n11174035.\n[1] Y. Nagaoka, Phys. Rev. 147, 392-405 (1966).\n[2] E. H. Lieb, Phys. Rev. Lett. 62, 1201-1204 (1989), [Er-\nrata62, 1927 (1989)].\n[3] S. Q. Shen, Z. M. Qiu and G. S. Tian, Phys. Rev. Lett.\n72, 1280 (1994).\n[4] After a (partial) particle-hole transformation, the re pul-\nsive Hubbard model can be transformed into an attrac-\ntive Hubbard model, of which the ground state always\nhas singlet superconducting (SC) pairing. This singlet\nSC just corresponds to the ground state with a total spin\n|NA−NB|/2[2, 18, 19]. Then, the SQT inequality corre-\nsponds to non-negative off-diagonal long-range SC pair-\ning of the attractive Hubbard model in earlier paper[18].\n[5] G. S. Tian, J. Phys. A: Math. Gen. 272305 (1994).\n[6] G. S. Tian, Phys. Rev. B 50,6246 (1994).\n[7] G. S. Tian and T. H. Lin, Phys. Rev. B 53, 8196 (1996).\n[8] S. Q. Shen, Int. J. Mod. Phys. B 12, 709 (1998).[9] C. N. Yang and S. C. Zhang, Mod. Phys. Lett. B 4,759\n(1990).\n[10] J. He, et al, Phys. Rev. B 87, 075126 (2013).\n[11] M. Aidelsburger, et al, Phys. Rev. Lett. 107,255301\n(2011).\n[12] M. Aidelsburger, et al, Phys. Rev. Lett. 111, 185301\n(2013).\n[13] H. Miyake, et al, Phys. Rev. Lett. 111, 185302 (2013).\n[14] T. C. Hsu, Phys. Rev. B. 41, 11379 (1990).\n[15] G. Y. Sun and S. P. Kou, Europhys. Lett. 87,67002\n(2009).\n[16] V. M. Pereira, et.al, Phys. Rev. Lett. 96,036801 (2006).\n[17] E. H. Lieb, D. C. Mattis, J. Math. Phys. 3,749 (1962).\n[18] G. S. Tian, Phys. Rev. B 45, 3145 (1992).\n[19] S. Q. Shen and Z. M. Qiu, Phys. Rev. Lett. 71, 4238\n(1993)."    },    {        "title": "2205.14342v1.Helicity_independent_all_optical_switching_of_magnetization_in_ferrimagnetic_alloys.pdf",        "content": "Helicity-independent all-optical switching of magnetization in ferrimagnetic\nalloys\nC. S. Daviesa,\u0003, J. H. Mentinkb, A. V. Kimelb, Th. Rasingb, A. Kirilyuka\naFELIX Laboratory, Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands\nbRadboud University, Institute for Molecules and Materials, 135 Heyendaalseweg, 6525 AJ Nijmegen, The Netherlands\nAbstract\nWe review and discuss the process of single-shot helicity-independent all-optical switching of magnetization\nby which a single suitably-ultrafast excitation, under the right conditions, toggles magnetization from one\nstable state to another. For almost a decade, this phenomenon was only consistently observed in speci\fc rare-\nearth-transition-metal ferrimagnetic alloys of GdFeCo, but breakthrough experiments in recent years have\nrevealed that the same behavior can be achieved in a wide range of multi-sublattice magnets including TbCo\nalloys doped with minute amounts of Gd, Gd/Co and Tb/Co synthetic ferrimagnets, and the rare-earth-free\nHeusler alloy Mn 2RuxGa. Aiming to resolve the conditions that allow switching, a series of experiments have\nshown that the process in the ferrimagnetic alloys GdFeCo and Mn 2RuxGa is highly sensitive to the pulse\nduration, starting temperature and the alloy composition. We argue here that the switching displayed by\nthese two very di\u000berent ferrimagnetic alloys can be generally understood within a single phenomenological\nframework describing the \row of angular momentum between the constituent sublattices and from the\nsublattices to the environment. The conditions that facilitate switching stem from the properties of these\nchannels of angular momentum \row in combination with the size of the angular momentum reservoirs. We\nconclude with providing an outlook in this vibrant research \feld, with emphasis on the outstanding open\nquestions pertaining to the underlying physics along with noting the advances in exploiting this switching\nprocess in technological applications.\n1. Introduction\nThe possibility to store information in the binary orientation of non-volatile magnetic bits has fueled,\nalongside the development of computing machinery, the emergence of a multi-billion-dollar industry aimed\nat developing ever smaller and faster data-storage technologies. The most popular conventional method\nof switching magnetization in a hard-disk-drive involves the application of a local magnetic \feld by a\nrecording head that precessionally drives magnetization across the barrier posed by magnetocrystalline\nanisotropy. However, the magnetic trilemma { whereby the constant downscaling of magnetic bits leads to\nstronger coercive \felds that must be localized ever tighter { has essentially obstructed any further progress\nin improving the speci\fcations of conventional magnetic recording technologies [1, 2], particularly in terms\nof writing speeds, storage densities and energy-e\u000eciency. This has motivated intense exploration of new\nmethods facilitating magnetic switching, with heat- and microwave-assisted magnetic recording being the\n\ragship approaches currently adopted in state-of-the-art hard-disk-drives [3, 4, 5, 6].\nEven until the 1990s, it was commonly assumed that the laws of thermodynamics adequately described\nthe interaction between light and spins. This belief was upended by the seminal discovery of Bigot and\nBeaurepaire et al. in 1996 that found that femtosecond laser pulses can destroy magnetization in ferro-\nmagnetic nickel on sub-picosecond timescales [7]. This result gave birth to the research \feld of ultrafast\n\u0003Corresponding author\nEmail address: carl.davies@ru.nl (C. S. Davies )arXiv:2205.14342v1  [cond-mat.mtrl-sci]  28 May 2022magnetism, devoted to learning how to manipulate magnetic materials on non-equilibrium timescales. Just\nover a decade later, Stanciu et al. [8] observed that circularly-polarized femtosecond laser pulses are capable\nnot only of destroying magnetization in gadolinium-iron-cobalt alloys (hereafter referred to as GdFeCo), but\ncould actually reverse it without the help of any external magnetic \felds. In 2012, it was even established\nthat the circular polarization could be disregarded completely, leading to the process of helicity-independent\nall-optical switching [9, 10]. These seemingly paradoxical results { disproving the long-held notion that mag-\nnetic \felds must be at least somewhat involved to reverse magnetization - has since triggered intense research\ne\u000borts [11] aimed at engineering ever-faster switching capabilities under non-equilibrium conditions in a wide\narray of materials, ultimately targeting practical application within ultrafast memory technologies [12, 13].\nIn this review, we brie\ry summarize recent measurements and our current understanding of helicity-\nindependent all-optical switching of magnetization. First, we provide a historical introduction of the two\npivotal experiments revealing both the feasibility and peculiar dynamics of the switching in GdFeCo alloys.\nSecond, we summarize (in our view) the most important conditions that have been experimentally identi-\n\fed as governing the switching process in GdFeCo and the Heusler alloy Mn 2RuxGa. Third, we present\na phenomenological framework that allows us to broadly understand why the switching occurs in these\ntwo materials, and why it is governed by certain experimentally-observed restrictions. Fourth and \fnally,\nwe provide an outlook aiming to draw attention to outstanding questions that remain in describing the\nunderlying mechanism and manifestation of switching in other systems.\n2. All-optical switching of magnetization: An experimental overview\n2.1. Di\u000berent types of all-optical switching of magnetization\nThe umbrella term \\all-optical switching (AOS) of magnetization\" can refer to several distinct processes\ndi\u000berentiated by the type of stimulus required and the switching that is displayed. In speci\fc highly-\nabsorbing media with uniaxial magnetocrystalline anisotropy, AOS can be achieved using either a single\nlaser pulse of arbitrary polarization, whereby the magnetization of a ferrimagnet is toggled between two\nstates [9, 10], or using multiple circularly-polarized laser pulses, which steers the one-way direction of\nswitching as prescribed by the optical helicity [14, 15, 16, 17]. Alternatively, non-absorbing dielectrics also\ndisplay resonance-driven AOS, importantly in the entire absence of heating. By pumping speci\fc electronic\ntransitions at resonance, one can precessionally switch magnetization with the direction of reversal controlled\nby the orientation of the linearly-polarized pulse's electric \feld [18, 19, 20]. Alternatively, the selective\nexcitation of a longitudinal optical phonon at resonance enables magnetization to be permanently switched\nvia transient strain of the lattice [21]. In this review, we restrict ourselves to discussing the process of\nsingle-shot helicity-independent all-optical switching found in metallic ferrimagnets.\n2.2. First measurements of helicity-independent all-optical magnetic switching\nHelicity-independent all-optical switching (HI-AOS) of magnetization refers to how a single optical pulse\ntoggles the direction of magnetic polarity in a system with uniaxial anisotropy. The possibility of achieving\nsuch behavior was \frst suggested by Radu et al. [22] with a study of the dynamics of ultrafast demagne-\ntization of a Gd 25Fe65:6Co9:4\flm. Using time-resolved x-ray magnetic circular dichroism (XMCD) mea-\nsurements, Radu et al. distinguished the magnetization of the antiferromagnetically-coupled Gd and Fe\nsublattices. The breakthrough result reproduced in Fig. 1 reveals that the 60 fs-long pump pulse demag-\nnetizes the Fe sublattice four times faster than the Gd sublattice. Moreover, the magnetization of the Fe\nsublattice actually crosses zero within 300 fs and subsequently reforms parallel to the still-demagnetizing\nGd sublattice, giving rise to a transient \\ferromagnetic-like\" state. This state is absolutely forbidden to\nexist in equilibrium due to the strong antiferromagnetic coupling between the two magnetic sublattices.\nFerromagnetic-like alignment persists for a further 1 :2 ps, after which the magnetization of the Gd sublat-\ntice also crosses zero, with the sublattices then having antiparallel magnetic polarity in compliance with\ntheir antiferromagnetic coupling.\nFollowing the observation of the non-equilibrium ferromagnetic-like state, the process of HI-AOS was\ndiscovered [9]. In these experiments, Ostler et al. focused a single optical pulse, with wavelength 800 nm\n2and duration 100 fs, on to the surface of a Gd 24Fe66:5Co9:5\flm at room temperature. The perpendicular\nmagnetic anisotropy of GdFeCo alloys allows magnetization to point either parallel or antiparallel to the\nsample normal, allowing magneto-optical microscopy to conveniently detect magnetic domains with binary\ncontrast. The images shown in Fig. 2, taken after exposing the GdFeCo \flm to consecutive optical pulses,\nclearly and irrefutably show that each pulse switches the magnetization deterministically. The lack of\ndependence on the optical helicity con\frms that the switching, dynamically evolving in the manner shown\nin Fig. 1, is driven by the ultrashort thermal load delivered by the pulse.\nPump-probe delay (ps)(a) (b)\nPump-probe delay (ps)0 3 6 9 12 -3 0 1 2 3 -10Normalized XMCD (%)-50100\n50\n-100Gd\nFeGd\nFe\nFigure 1: (a)-(b) Transient dynamics of the Fe (open circles) and Gd (\flled circles) magnetic moments measured within the\n\frst 3 ps and 12 ps respectively. Error bars of the experimental data represent the statistical standard error. The measurements\nwere performed at a sample temperature of 83 K for an incident laser \ruence of 4 :4 mJ=cm2. The solid lines are \fts according\nto a double exponential \ft function. To facilitate comparison, the dashed line in both panels depicts the magnetization of the\nFe sublattice inverted in sign. Adapted with permission from Ref. [22].\nInitial state Shot 1 Shot 2 Shot 3 Shot 4 Shot 5\n(a)\n(b)\nMM\n25 μm\nFigure 2: Typical magneto-optical images taken after exposing a Gd 24Fe66:5Co9:5\flm to consecutive 100 fs-long optical pulses\nat room temperature. Each pulse has a \ruence of 2 :30 mJ=cm2. The images shown in rows (a)-(b) have di\u000berent initial\npolarities of magnetization Mas indicated. Adapted with permission from Ref. [9].\nSince these two landmark experiments, HI-AOS has been unambiguously identi\fed in ferrimagnetic alloys\nof GdCo [23], (Gd,Tb)Co, [24, 25] and the Heusler alloys Mn 2RuxGa (hereafter referred to as MRG) [26, 27].\nThe discovery of such switching in ferrimagnetic alloys also led to predictions that synthetic ferrimagnets\ncould exhibit similar behavior [28, 29], which has been identi\fed thus far in multilayered nanostructures\nof Gd/FeCo [30], Gd/Co [31, 32] and Tb/Co [33, 34]. Moreover, it is possible to supplement HI-AOS in\nGdFeCo with directionality of reversal using circularly-polarized pulses. This originates from preferential\noptical absorption via MCD, which gives rise to a slight mismatch in energy absorption between left- and\nright-handed circularly-polarized light. These pulses thus become capable of switching speci\fc magnetic\ndomains [35]. We emphasize that this is completely distinct from the other form of switching found in\nalternative ferri- and ferro-magnets, in which many circularly-polarized optical pulses, of left- or right-\nhanded helicity, steer magnetic switching from an initially-demagnetized state [14, 15, 36, 16, 17].\n2.3. Experimental conditions for HI-AOS in GdFeCo and MRG\nIn the following section, we present (in our view) the most important experimental conditions that\nprovide for HI-AOS in GdFeCo and MRG. Both of these materials feature relatively strong inter-sublattice\n3and dissimilar intra-sublattice exchange couplings. The magnetic moments of Gd and Fe di\u000ber by a factor\nof about four [22]. In contrast, the two sublattices of MRG correspond to Mn atoms at Wycko\u000b positions\n4a and 4c, and so the sublattice-speci\fc magnetic moments are rather similar with a ratio \u00191:1.2 [37, 38].\nA critically important parameter for HI-AOS is the length \u001cof the excitation pulse. Initial experiments\nstudying HI-AOS predominantly used ultrashort (100 fs-long) optical pulses with wavelengths in the vicinity\nof 800 nm, re\recting the output of ampli\fed Ti:Sa lasers. The possibility of using longer optical pulses\nwas \frst suggested by Steil et al. as far back as 2011, in experiments where helicity-dependent all-optical\nswitching in a Gd 26Fe64:7Co9:3sample was achieved using 13 ps-long pulses [39]. At the time, this result was\nmired in an overarching controversy pertaining to the switching's origin, but this measurement represents\nthe \frst sign indicating that AOS does not necessarily demand ultrashort optical pulses. A year later,\nVahaplar et al. identi\fed a pulse-duration threshold \u001ccfor a Gd 24Fe66:5Co9:5sample [40], whereby pulses\nshorter (longer) than \u001cc= 1:7 ps were able (unable) to activate helicity-dependent switching. In contrast,\nthe switching could be achieved in a \flm of composition Gd 26Fe64:7Co9:3using 2:1 ps-long pulses. In 2016,\nexperiments by Gorchon et al. sparked substantial interest in the use of long optical pulses [41], particularly\nwith the pivotal discovery that \u001cccan vary from 1 :5 ps for Gd 24Fe66Co10to as long as 15 ps for Gd 26Fe64Co10.\nThis dependence on alloy composition was further explored by Davies et al. , with experiments \fnding that\n\u001ccincreases monotonically with the percentage of Gd in Gd x(FeCo) 100\u0000xalloys [42], as shown in Fig. 3.\nThis behavior has also been reproduced in both phenomenological modelling and atomistic simulations of\nHI-AOS [42, 43].\n22 23 24 25 26 270246\nxτc (ps)\nτ < τc\nτ > 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\nSbwithjSbj<Sa, the ferrimagnet is driven in to the blue shaded manifold. This is a special region where\nthe angular momentum of each sublattice will move in to the top-right quandrant, corresponding to the\nferromagnetic-like state detected experimentally in Fig. 1. From this quadrant, HI-AOS follows.\nGoing further, we can express the requirement for sublattice reversal in the blue manifold to occur when\ndSb\ndt\f\f\f\f\nSb=0>0,2@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": "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. 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Zvyagin and A. Anders, JETP 40, 154 (1975).\n[30] S.-W. Cheong, A. S. Cooper, L. W. Rupp, B. Batlogg,\nJ. D. Thompson, and Z. Fisk, Phys. Rev. B 44, 9739\n(1991).\n[31] J. F. Niven, M. B. Johnson, A. Bourque, P. J. Murray,\nD. D. James, H. A. Dabkowska, B. D. Gaulin, and M. A.\nWhite, Proceedings of the Royal Society A: Mathemat-\nical, Physical and Engineering Sciences 470, 20140387\n(2014).\n[32] J. H. Van Vleck, Rev. Mod. Phys. 17, 27 (1945).\n[33] J. Sakurai, W. J. L. Buyers, R. A. Cowley, and\nG. Dolling, Phys. Rev. 167, 510 (1968).\n[34] M. Janoschek, B. Roessli, L. Keller, S. N. Gvasaliya,\nK. Conder, and E. Pomjakushina, Journal of Physics:\nCondensed Matter 17, L425 (2005).\n[35] N. E. Brese and M. 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": "2007.06805v2.Three_dimensional_Ising_Ferrimagnetism_of_Cr_Fe_Cr_trimers_in_FeCr2Te4.pdf",        "content": "arXiv:2007.06805v2  [cond-mat.str-el]  18 Aug 2020Three-dimensional Ising Ferrimagnetism of Cr-Fe-Cr trime rs in FeCr 2Te4\nYu Liu,1R. J. Koch,1Zhixiang Hu,1,2Niraj Aryal,1Eli Stavitski,3Xiao\nTong,4Klaus Attenkofer,3E. S. Bozin,1Weiguo Yin,1and C. Petrovic1,2\n1Condensed Matter Physics and Materials Science Department ,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n2Department of Physics and Astronomy, Stony Brook Universit y, Stony Brook, New York 11790, USA\n3National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA\n4Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA\n(Dated: August 19, 2020)\nWe carried out a comprehensive study of magnetic critical be havior in single crystals of ternary\nchalcogenide FeCr 2Te4that undergoes a ferrimagnetic transition below Tc∼123 K. Detailed critical\nbehavior analysis and scaled magnetic entropy change indic ate a second-order ferrimagentic transi-\ntion. Critical exponents β= 0.30(1) with Tc= 122.4(5) K,γ= 1.22(1) with Tc= 122.8(1) K, and\nδ= 4.24(2) at Tc∼123 K suggest that the spins approach three-dimensional Isi ng (β= 0.325, γ\n= 1.24, and δ= 4.82) model coupled with the attractive long-range intera ctions between spins that\ndecay as J(r)≈r−4.88. Our results suggest that the ferrimagnetism in FeCr 2Te4is due to itinerant\nferromagnetism among the antiferromagnetically coupled C r-Fe-Cr trimers.\nINTRODUCTION\nTernaryACr 2X4(A =transitionmetal, X=S, Se, and\nTe) exhibit a variety of magnetic and electronic proper-\nties. The family includes metallic CuCr 2X4and semi-\nconducting Hg(Cd)Cr 2Se4ferromagnets [1–3], semicon-\nducting Fe(Mn)Cr 2S4ferrimagnets [4–6], and insulating\nZnCr2S4antiferromagnet [7]. The FeCr 2X4compounds\nshow competing spin-orbit and exchange interactions [8].\nFeCr2S4is ferrimagnetic (FIM) insulator below Tc= 165\nK, shows a crossover transition from insulator to metal\nnearTcand colossal magnetoresistance behavior [9–11].\nFeCr2Se4is an insulating antiferromagnet (AFM) with\nTN= 218 K and ferrimagnetic with a small magnetic\nmoment of 0.007 µBbelow 75 K [12–14]. It should be\nnoted that FeCr 2Se4crystallizes in the Cr 3S4-type mon-\noclinic structure described within C2/mspace group, in\ncontrast to the cubic spinel-type of FeCr 2S4. However,\nFeCr2S4and FeCr 2Se4have similar electronic structure\nwith nearly trivalent Cr3+and divalent Fe2+states [15].\nThe magnetic moments of Cr ions are antiparallel to\nthoseofFeionsinFeCr 2X4, andthereisstronghybridiza-\ntion between Fe 3 d-states and X p-states [15].\nFeCr2Te4has not been studied much presumably due\nto the difficulty in sample preparation [16–18]. Demeaux\net al. first grew the single crystals of FeCr 2Te4[16]. The\ncrystal structure was reported as a defective NiAs-type\nwithinP63/mmcspace group, where Fe and Cr occupy\nthe same site with alloying ratio of 1 : 2 and a net oc-\ncupancy of 0.75 [16]. In contrast, Valiev et al. reported\nthat FeCr 2Te4crystalizes in a CoMo 2S4-type structure\nwithinI2/mspace group, in which Fe and Cr are octa-\nhedrally coordinated by six Te [17]. Recently, a series of\npolycrystals FeCr 2Se4−xTexwere synthesized [18]. Sub-\nstitution with Te gradually suppresses the AFM order of\nFeCr2Se4, and leads to a short range ferromagnetic(FM)\ncluster metallic state in polycrystal FeCr 2Te4[18]. In or-der to study the intrinsic physical property, high quality\nsingle crystal is required.\nIn this work, we successfully fabricated single crystals\nof FeCr 2Te4and performed a comprehensive study of the\nstructural and magnetic properties. Our analysis of crit-\nicality around Tcindicates that FeCr 2Te4displays the\n3D-Ising behavior, with the magnetic exchange distance\ndecaying as J(r)≈r−4.88. Our first principles calcula-\ntions suggest that the ferrimagnetism in FeCr 2Te4stems\nfrom the itinerant ferromagnetism among the antiferro-\nmagnetically coupled Cr-Fe-Cr trimers. Since transition-\nmetal chalcogenides represent model systems for explor-\ning local structure-related relationship between the bro-\nken symmetry and d-orbital magnetism [19], detailed lo-\ncal stucture investigation of this system would be highly\ndesirable and would bring important new insights.\nEXPERIMENTAL DETAILS\nSingle crystals of FeCr 2Te4were fabricated by melting\nstoichiometric mixture of Fe (99.99%, Alfa Aesar) pow-\nder, Cr (99.95%, Alfa Aesar) powder, and Te (99.9999%,\nAlfa Aesar) pieces. The starting materials were vacuum-\nsealed in a quartz tube, heated to 1200◦C over 12\nh, slowly cooled to 900◦C at a slow rate of 1◦C/h,\nand then quenched into iced water. The single crystal\nx-ray diffraction (XRD) data were taken with Cu Kα\n(λ= 0.15418 nm) radiation of a Rigaku Miniflex pow-\nder diffractometer. In order to obtain more comprehen-\nsive crystallographic information, the powder XRD mea-\nsurements were performed at the PDF beamline (28-ID-\n1) at National Synchrotron Light Source II (NSLS II)\nat Brookhaven National Laboratory (BNL) using Perkin\nElmer image plate detector. The setup utilized x-ray\nbeam with a wavelength of 0.1666 ˚A, and the sample to\ndetectordistance of1.25m, ascalibrated usinga Ni stan-2\ndard. Sample was cooled with an Oxford Cryosystems\n700 cryostream using liquid nitrogen. Raw data were\nintegrated and converted to intensity vs. scattering an-\ngle using the software pyFAI [20]. The average structure\nwas assessed from raw powder diffraction data using the\nGeneralStructureAnalysisSystemII(GSAS-II) software\npackage[21]. Theelementalanalysiswasperformedusing\nenergy-dispersive x-ray spectroscopy (EDS) in a JEOL\nLSM-6500 scanning electron microscope (SEM). The x-\nray absorption spectroscopy (XAS) measurements were\nperformed at 8-ID beamline of the NSLS II (BNL) in a\nfluorescencemode. Thex-rayabsorptionnearedgestruc-\nture (XANES) and extended x-ray absorption fine struc-\nture (EXAFS) spectra were processed using the Athena\nsoftware package. The AUTOBK code was used to nor-\nmalize the absorption coefficient, and separate the EX-\nAFS signal, χ(k), from the atom-absorption background.\nThe extracted EXAFS signal, χ(k), wasweighed by k2to\nemphasize the high-energy oscillation and then Fourier-\ntransformed in krange from 2 to 10 ˚A−1to analyze\nthe data in Rspace. The x-ray photoelectron spec-\ntroscopy (XPS) experiment was carried out in an ultra-\nhigh vacuum system with base pressures <5×10−9Torr,\nequipped with a hemispherical electron energy analyzer\n(SPECS, PHOIBOS 100) and a twin anode x-ray source\n(SPECS, XR50). Al Kα(1486 eV) radiation was used at\n13 kV and 30 mA. The angle between the analyzer and\nthe x-ray source was 45◦and photoelectrons were col-\nlectedalongthesamplesurfacenormal. TheXPSspectra\nwereanalyzedanddeconvolutedusingtheCasaXPSsoft-\nware. The dc/ac magnetic susceptibility were measured\nin Quantum Design MPMS-XL5 system. The applied\nfield (Ha) was corrected as H=Ha−NM, whereMis\nthe measured magnetization and Nis the demagnetiza-\ntion factor. The corrected Hwas used for the analysis of\nmagnetic entropy change and critical behavior.\nRESULTS AND DISCUSSIONS\nIn the single-crystal XRD pattern (inset to Fig. 1),\nonly (00 l) peaks were observed. Synchrotron powder\nXRD pattern of pulverized crystal of FeCr 2Te4can be\nwell fitted by using a monoclinic structure with the I2/m\nspace group (Fig. 1), confirming main phase of FeCr 2Te4\nwith less than 4% FeTe impurity. The determined lattice\nparameters at 300 K are a= 6.822(2)˚A,b= 3.938(1)˚A,\nc= 11.983(5)˚A, andβ= 90.00(5)◦, close to the reported\nvalues [17, 18]. No structural transition was observed on\ncooling, showing a compression along caxis with c=\n11.867(5) ˚A and a slight expansion in the abplane with\na= 6.825(1)˚A andb= 3.943(1)˚A down to 105 K (Table\nI).\nThe ratio of elements in the single crystal as deter-\nmined by EDS is Fe : Cr : Te = 0.99(2) : 1.90(2) : 4.0(1)\n[Fig. 2(a)], and it is referred to as FeCr 2Te4through-\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48\n/s50 /s113 /s32/s40/s100/s101/s103/s46/s41/s40/s48/s48/s50/s41\n/s40/s48/s48/s52/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s32/s117/s46/s41\nFIG. 1. (Color online) Synchrotron powder x-ray diffraction\n(XRD) data and structural model refinements. The data are\nshown by (+) and ( ◦), structural model fit at 300 K and\n105 K is shown by red and blue solid line, respectively. The\ndifference curves are given by green solid lines, offset for cl ar-\nity. The vertical tick marks represent Bragg reflections of t he\nI2/mspace group andupto4% of FeTe impurity. Insetshows\nsingle crystal XRDpattern of FeCr 2Te4at room temperature.\nout this paper. The information on the valence states of\nFe, Cr and Te atoms can be obtained from the element\ncore-level XPS spectra [Fig. 2(b)]: Fe2+(2p3/2∼710\neV, 2p1/2∼723 eV), Cr3+(2p3/2∼575 eV, 2 p1/2∼586\neV), Te2−(3d5/2∼572eV, 3 d3/2∼583 eV)]. Figure 2(c)\nshows the normalized Cr and Fe K-edge XANES spectra,\nin which a similar prepeak feature is observed, indicating\nsimilar local atomic environment for Fe and Cr atoms.\nThe prepeak feature for Fe K-edge is somewhat weaker\nthan that of Cr K-edge, suggesting a weaker lattice dis-\ntortion in FeTe 6when compared with CrTe 6. The edge\nfeatures are close to the standard compounds with Cr3+\nand Fe2+oxidation states [22, 23], in line with the XPS\nresult.\nThe local environment of Fe and Cr atoms is revealed\nin the EXAFS spectra of FeCr 2Te4measured at room\ntemperature [Figs. 2(d) and 2(e)]. In a single-scattering\napproximation, the EXAFS can be described by [24]:\nχ(k) =/summationdisplay\niNiS2\n0\nkR2\nifi(k,Ri)e−2Ri\nλe−2k2σ2\nisin[2kRi+δi(k)],\nwhereNiis the number of neighbouring atoms at a dis-\ntanceRifromthephotoabsorbingatom. S2\n0isthepassive\nelectrons reduction factor, fi(k,Ri) is the backscattering\namplitude, λis the photoelectron mean free path, δiis\nthe phase shift, and σ2\niis the correlated Debye-Waller\nfactor measuring the mean square relative displacement\nof the photoabsorber-backscatter pairs. The corrected3\n/s53/s46/s57/s56 /s54/s46/s48/s48 /s55/s46/s49/s48 /s55/s46/s49/s50 /s55/s46/s49/s52/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s53/s54/s48 /s53/s56/s48 /s55/s48/s48 /s55/s50/s48\n/s50 /s51 /s52/s48/s49/s50\n/s50 /s51 /s52/s48/s46/s48/s48/s46/s54/s49/s46/s50\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48/s45/s49/s48/s49\n/s50 /s52 /s54 /s56 /s49/s48/s45/s49/s48/s49\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48\n/s40/s97/s41\n/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32 /s109 /s40/s69/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s107/s101/s86/s41/s67/s114/s32/s75/s45/s101/s100/s103/s101/s40/s99/s41\n/s70/s101/s40/s98/s41\n/s40/s100/s41 /s40/s101/s41/s51/s48\n/s50/s48\n/s49/s48\n/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s66/s105/s110/s100/s105/s110/s103/s32/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s70/s101/s32/s50/s112\n/s51/s47/s50\n/s70/s101/s32/s50/s112\n/s49/s47/s50/s84/s101/s32/s51/s100\n/s53/s47/s50\n/s84/s101/s32/s51/s100\n/s51/s47/s50/s67/s114/s32/s50/s112\n/s51/s47/s50\n/s67/s114/s32/s50/s112\n/s49/s47/s50\n/s231/s99 /s40/s82/s41 /s231 /s32/s40/s97/s46/s117/s46/s41\n/s82/s40/s197/s41/s40/s103/s41\n/s67/s114/s32/s45/s32/s84/s101/s50/s46/s55/s48/s32/s197/s231/s99 /s40/s82/s41 /s231 /s32/s40/s97/s46/s117/s46/s41/s107/s101/s86/s49/s48 /s49/s50 /s49/s52/s49/s54 /s49/s56/s50/s48/s56/s54/s52 /s50/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s32/s117/s46/s41/s53/s48/s54/s48\n/s52/s48\n/s70/s101 /s67/s114/s84/s101/s70/s101\n/s67/s114/s84/s101\n/s82/s40/s197/s41/s40/s102/s41\n/s70/s101/s32/s45/s32/s84/s101/s50/s46/s53/s56/s32/s197\n/s107/s50\n/s99 /s40/s107/s41/s32/s40/s197/s45/s50\n/s41\n/s107/s32/s40/s197/s45/s49\n/s41\n/s107/s50\n/s99 /s40/s107/s41/s32/s40/s197/s45/s50\n/s41\n/s107/s32/s40/s197/s45/s49\n/s41\nFIG. 2. (Color online) (a) Crystal structure and results of\nEDS analysis on FeCr 2Te4. Room-temperature Fe-2p, Cr-2p,\nand Te-3d core-level XPS (b) and normalized Cr and Fe K-\nedge XANES spectra (c). Fe and Cr K-edge EXAFS oscilla-\ntions (d,e) and Fourier transform magnitudes (f,g) of EXAFS\ndata measured at room temperature. The experimental data\nare shown as blue symbols alongside the model fit plotted as\nred line. Corresponding first coordination shell of Fe and Cr\nare shown in the insets.\nmain peak in the Fourier transform magnitudes of Fe K-\nedge EXAFS around R∼2.58˚A is clearly smaller than\nthat of Cr K-edge EXAFS at R∼2.70˚A [Figs. 2(f) and\n2(g)]. The different local Fe-Te and Cr-Te bond lengths\nsuggest that the Fe and Cr atoms might occupy different\ncrystallographic sites, ruling out the possibility of NiAs-\ntype structure with the same Fe/Cr sites. Then we focus\non the first nearest neighbors of Fe and Cr atoms rang-\ning from 1.5 to 3.5 ˚A. The main peak corresponds to two\nFe-Te bond lengths of 2.69 ˚A and 2.75 ˚A, and four dif-\nferent Cr-Te bond lengths with 2.58 ˚A, 2.67˚A, and 2.80\n˚A, 2.86 ˚A, respectively, extracted from the model fits\nwith fixed coordination number CN. The peaks above\n3.25˚A are due to longer Fe-Cr, Fe-Te, and Cr-Te bond\ndistances, and the multiple scattering involving different\nnear neighbours of the Fe/Cr atoms.\nFigure3(a)showsthe temperaturedependenceofmag-TABLE I. Average and local structural parameters extracted\nfrom the powder XRD and the EXAFS spectra of FeCr 2Te4.\nCN is coordination number based on crystallographic value,\nR is interatomic distance, and σ2is Debye Waller factor.\n300 K 105 K\na(˚A) 6.822(2) 6.825(1)\nb(˚A) 3.938(1) 3.943(1)\nc(˚A) 11.983(5) 11.867(5)\nβ(◦) 90.00(5) 90.01(15)\natom site x y z\nFe 2 a 0 0 0\nCr 4 i 0.001(3) 0 0.2541(8)\nTe1 4 i 0.328(4) 0 0.3726(4)\nTe2 4 i 0.339(4) 0 0.8785(4)\nbond CN R ( ˚A) ∆R ( ˚A) σ2(˚A2)\nCr-Te1 2 2.58 0.01 0.001\nCr-Te2 1 2.67 0.01 0.001\nCr-Te3 1 2.80 0.11 0.003\nCr-Te4 2 2.86 0.11 0.003\nFe-Te1 4 2.69 0.36 0.02\nFe-Te2 2 2.75 0.24 0.02\nnetization measured in out-of-plane field µ0H= 0.1 T,\nin which χincreases with decreasing temperature and in-\ncreases abruptly near Tcdue to the paramagnetic (PM)-\nFIM transition. The in-plane χ(T) [inset in Fig. 3(a)]\nis much smaller than that in out-of-plane field, indicat-\ning the presence of large magnetic anisotropy with easy\ncaxis. The average susceptibility χave= (2/3)χab+\n(1/3)χcfrom 150 to 300 K can be fitted by the Curie-\nWeiss law χave=χ0+C/(T−θ) [Fig. 3(c)], which yields\nχ0= 0.87(1) emu/mol-Oe, C= 7.91(6) emu-K/mol-Oe\n[µeff= 7.95(9)µBper formulaunit], and θ= 126.6(2)K.\nThe positive value of θindicates dominance of ferromag-\nnetic or ferrimagnetic exchangeinteractions in FeCr 2Te4.\nFor monoclinic FeCr 2Se4with a similar layered struc-\nture, the individual spins of Fe and Cr ions have AFM\ncoupling along the caxis with the distance of 2.956 ˚A,\nwhile FM coupling along the baxis with the distance of\n3.617˚A [25]. At low temperature, the FM interaction\ndominating over the AFM interaction results in a FIM\nground state. For FeCr 2Te4, the enhanced hybridization\nbetween d-orbital of Fe and Cr with p-orbital of Te plays\nan important role in magnetic coupling. Based on our\nfirst-principle calculation (see below), the FIM structure\nwhere Fe and Cr atoms have opposite spin orientations\nare the most stable state in FeCr 2Te4, when compared\nwith the FM and AFM structures, which needs further\nverificationby neutron scatteringexperiment. The bifur-\ncation between ZFC and FC curves [Fig. 3(a)] might be\ndue to strong magnetic anisotropy and/or multidomain\nstructure, which has also been observed in other long-4\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/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\n/s45/s53/s46/s48 /s45/s50/s46/s53 /s48/s46/s48 /s50/s46/s53 /s53/s46/s48/s45/s52/s45/s50/s48/s50/s52/s32/s90/s70/s67\n/s32/s70/s67/s32/s40/s101/s109/s117/s47/s109/s111/s108/s45/s79/s101/s41\n/s84/s32/s40/s75/s41/s48/s72/s47/s47/s99/s32/s97/s116/s32/s48/s46/s49/s32/s84/s40/s97/s41\n/s48/s72/s47/s47/s97/s98/s32/s97/s116/s32/s48/s46/s49/s84/s32/s90/s70/s67\n/s32/s70/s67/s32/s40/s101/s109/s117/s47/s109/s111/s108/s45/s79/s101/s41\n/s84/s32/s40/s75/s41\n/s97/s118/s101/s32/s40/s109/s111/s108/s45/s79/s101/s47/s101/s109/s117/s41\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s32/s61/s32/s48/s46/s49/s32/s84/s97/s118/s101/s32/s61/s32 /s32/s67/s47/s40/s84/s45 /s41/s109/s39/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s84/s32/s40/s75/s41/s84\n/s99/s32/s61/s32/s49/s50/s52/s32/s75\n/s72\n/s97/s99/s61/s32/s51/s46/s56/s32/s79/s101/s32\n/s102/s32/s61/s32/s52/s57/s57/s32/s72/s122/s40/s100/s41 /s40/s99/s41\n/s32\n/s48/s72/s47/s47/s97/s98\n/s32\n/s48/s72/s47/s47/s99\n/s32/s32/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s48/s72/s32/s40/s84/s41/s84/s32/s61/s32/s50/s32/s75\nFIG. 3. (Color online) Temperature-dependent dc magnetic\nsusceptibility χ(T) in zero-field cooling (ZFC) and field cool-\ning (FC) modes taken at µ0H= 0.1 T for µ0H/bardblc(a) and\nµ0H/bardblab(inset), respectively. (b) 1 /χ(T) taken at µ0H=\n0.1 T along with Curie-Weiss fit from 150 to 300 K. (c) Field\ndependence of magnetization for FeCr 2Te4measured at T= 2\nK. (d) Ac susceptibility real part m′(T) measured with oscil-\nlating ac field of 3.8 Oe and frequency of 499 Hz. The chosen\nexperimental parameters (field and frequency) allow for wel l\ndefined moment and adequate frequency resolution.\nrange FM single crystals, such as U 2RhSi3[26]. The\nmagnetization loops of FeCr 2Te4for both field directions\natT=2Kconfirmsalargemagneticanisotropyandeasy\ncaxis [Fig. 3(c)]. The sudden jumps around µ0H≈ ±1\nTalongthe caxiscanbeascribedtothemagneticdomain\ncreeping behavior, i.e., the magnetic domain walls jump\nfrom one pinning site to another. Then we estimated the\nRhodes-Wohlfarth ratio (RWR) for FeCr 2Te4, which is\ndefined as Pc/PswithPcobtained from the effective mo-\nmentPc(Pc+2) =P2\neffandPsis the saturation moment\nobtained in the ordered state [27–29]. RWR is 1 for a lo-\ncalized system and is larger in an itinerant system. Here\nwe obtain RWR ≈1.69 for FeCr 2Te4, indicating a weak\nitinerant character. To obtain the accurate Curie tem-\nperature Tc, out-of-plane ac susceptibility was measured\nat oscillating ac field of 3.8 Oe and frequency of 499 Hz.\nThe single sharp peak in the real part m′(T) [Fig. 3(d)]\ngives the Tc= 124 K.\nIn the following we discuss the nature of the PM-FIM\ntransition for FeCr 2Te4. The magnetization isotherms\nalong easy caxis were measured at various temperatures\nin the vicinity of Tc[Fig. 4(a)]. We first considered the/s48 /s49 /s50 /s51 /s52 /s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s48/s50/s52/s54/s56\n/s49/s49/s48 /s49/s50/s48 /s49/s51/s48 /s49/s52/s48 /s49/s53/s48 /s49/s54/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s48/s49/s50/s51/s52/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s109\n/s48/s72/s32/s40/s84/s41/s40/s97/s41\n/s84/s32/s61/s32/s49/s48/s48/s32/s75\n/s84/s32/s61/s32/s49/s53/s48/s32/s75\n/s68 /s84/s32/s61/s32/s50/s32/s75/s109\n/s48/s72/s32/s47/s47/s32/s99\n/s77/s50\n/s32/s40/s49/s48/s32/s101/s109/s117/s47/s103/s41/s50\n/s109\n/s48/s72/s47/s77/s32/s40/s84/s45/s103/s47/s101/s109/s117/s41/s40/s98/s41\n/s49/s53/s48/s32/s75/s49/s48/s48/s32/s75/s45/s68 /s83\n/s77/s32/s40/s74/s47/s107/s103/s45/s75/s41\n/s84/s32/s40/s75/s41/s109\n/s48/s72/s32/s40/s84/s41/s40/s100/s41\n/s53/s46/s48\n/s52/s46/s48\n/s51/s46/s53\n/s49/s46/s50/s49/s46/s54/s50/s46/s48/s50/s46/s53/s51/s46/s48/s52/s46/s53\n/s48/s46/s52/s32/s84/s48/s46/s56/s77/s49/s47/s98\n/s32/s40/s49/s48/s52\n/s32/s40/s101/s109/s117/s47/s103/s41/s49/s47 /s98\n/s41\n/s40/s109\n/s48/s72/s47/s77/s41/s49/s47 /s103\n/s32/s40/s84/s45/s103/s47/s101/s109/s117/s41/s49/s47 /s103/s40/s99/s41\n/s49/s48/s48/s32/s75\n/s49/s53/s48/s32/s75/s98 /s32/s61/s32/s48/s46/s51/s50\n/s103 /s32/s61/s32/s49/s46/s50/s49\nFIG. 4. (Color online) (a) Typical initial isothermal magne -\ntization curves from 100 K to 150 K with a temperature step\nof 2 K for FeCr 2Te4single crystal. (b) Arrott plot ( β= 0.5,\nγ= 1) and (c) the modified Arrott plot with the optimum ex-\nponents β= 0.32 andγ= 1.21. (d) Temperature-dependent\nmagnetic entropy change −∆SM(T) at various fields change.\nwell-known Arrott plot [30]. From the Landau theory,\nthe Arrott plot of M2vsH/Mshould appear as paral-\nlel straight lines above and below Tc, and the line passes\nthrough the origin at Tc. It is clear that the mean field\ncritical exponent does not work for FeCr 2Te4, as illus-\ntrated by the set of curved lines shown in Fig. 4(b).\nFor a second-order phase transition, the spontaneous\nmagnetization MsbelowTc, the inverse initial suscepti-\nbilityχ−1\n0aboveTc, and the field-dependent magnetiza-\ntionM(H) atTcare [31–33]:\nMs(T) =M0(−ε)β,ε <0,T < T c, (1)\nχ−1\n0(T) = (h0/m0)εγ,ε >0,T > T c,(2)\nM=DH1/δ,T=Tc, (3)\nwhereε= (T−Tc)/Tcis the reduced temperature, and\nM0,h0/m0andDare the critical amplitudes. In a\nmore general case, the modified Arrott plot ( H/M)1/γ=\naε+bM1/βwith self-consistent method was considered\n[34, 35]. Figure 4(c) presents the final modified Arrott\nplot ofM1/βvs (H/M)1/γwithβ= 0.32 andγ= 1.21,\nshowing a set of quasi-parallel lines at high field region.\nThen we extracted χ−1\n0(T) andMs(T) as the intercepts5\n/s50 /s51 /s52 /s53/s49/s54/s50/s52\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s45/s56 /s45/s52 /s48 /s52 /s56/s48/s49/s50/s51/s52/s49/s48/s56 /s49/s49/s52 /s49/s50/s48 /s49/s50/s54 /s49/s51/s50 /s49/s51/s56/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s84/s32/s40/s75/s41/s77\n/s115/s32/s40/s101/s109/s117/s47/s103/s41/s84\n/s99/s61/s32/s49/s50/s50/s46/s57/s40/s51/s41/s32/s75\n/s32/s61/s32/s48/s46/s51/s51/s40/s50/s41\n/s84\n/s99/s61/s32/s49/s50/s50/s46/s55/s40/s49/s41/s32/s75\n/s32/s61/s32/s49/s46/s50/s48/s40/s49/s41/s40/s97/s41\n/s48/s50/s52/s54/s56/s49/s48\n/s40/s49/s48/s45/s50\n/s32/s84/s45/s103/s47/s101/s109/s117/s41\n/s49/s48/s56 /s49/s49/s52 /s49/s50/s48 /s49/s50/s54 /s49/s51/s50 /s49/s51/s56/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48\n/s84/s32/s40/s75/s41/s77\n/s115/s40/s100/s77\n/s115/s47/s100/s84/s41/s45/s49\n/s32/s40/s75/s41/s84\n/s99/s61/s32/s49/s50/s50/s46/s52/s40/s53/s41/s32/s75\n/s32/s61/s32/s48/s46/s51/s48/s40/s49/s41\n/s84\n/s99/s61/s32/s49/s50/s50/s46/s56/s40/s49/s41/s32/s75\n/s32/s61/s32/s49/s46/s50/s50/s40/s49/s41/s40/s98/s41\n/s48/s51/s54/s57/s49/s50\n/s40 /s100/s84/s41/s45/s49\n/s32\n/s40/s75/s41/s32/s61/s32/s52/s46/s56/s51/s40/s54/s41/s77/s32/s40 /s101/s109/s117/s32/s103/s45/s49\n/s41\n/s48/s72/s32/s40/s84/s41/s84\n/s99/s32/s61/s32/s49/s50/s51/s32/s75\n/s77 /s40/s101/s109/s117/s32/s103/s45/s49\n/s41\n/s48/s72 /s40 /s41/s84/s32/s60/s32/s84\n/s99\n/s84/s32/s62/s32/s84\n/s99/s40/s99/s41/s77/s40\n/s48/s72/s41/s45/s49/s47\n/s49/s48/s45/s51\n/s40\n/s48/s72/s41/s45/s84\n/s99/s32/s61/s32/s49/s50/s51/s32/s75/s40/s100/s41\nFIG. 5. (Color online) (a) Temperature dependence of the\nspontaneous magnetization Ms(left) and the inverse initial\nsusceptibility χ−1\n0(right)withsolid fittingcurves. Insetshows\nlogMvs log(µ0H) collected at Tc= 123 K with linear fitting\ncurve. (b) Kouvel-Fisher plots of Ms(dMs/dT)−1(left axis)\nandχ−1\n0(dχ−1\n0/dT)−1(right axis) with solid fitting curves.\n(c) Scaled magnetization mvs scaled field hbelow and above\nTcfor FeCr 2Te4. (d) The rescaling of the M(µ0H) curves by\nM(µ0H)−1/δvsε(µ0H)−1/(βδ).\non theH/Maxis and the positive M2axis, respectively.\nThe magnetic entropy change can be estimated using the\nMaxwell’s relation [36]:\n∆SM(T,H) =/integraldisplayH\n0/bracketleftbigg∂M(T,H)\n∂T/bracketrightbigg\nHdH. (4)\nFigure 4(d) presents the calculated −∆SMas a function\noftemperature. The −∆SMshowsabroadpeakcentered\nnearTcand the peak value monotonically increases with\nincreasing field. The maximum value of −∆SMreaches\n1.92 J kg−1K−1with a field change of 5 T. There is a\nslight shift of −∆SMpeak towards higher temperature\nwith increasing field, which also excludes the mean field\nmodel [37].\nFigure 5(a) presents the extracted Ms(T) andχ−1\n0(T)\nas a function of temperature. According to Eqs. (1) and\n(2), thecriticalexponents β= 0.33(2)with Tc= 122.9(3)\nK, andγ= 1.20(1) with Tc= 122.7(1) K, are obtained.\nThe exponent βdescribes the rapid increase of the or-\nder parameter below Tc. The exponent γdescribes how\nmagnetic susceptibility diverges at Tc. Here the obtained\nresult describes critical behavior of the net spontaneous\nmagnetization that arises in ferrimagnet. In the Kouvel-\nFisher (KF) relation [38]:\nMs(T)[dMs(T)/dT]−1= (T−Tc)/β, (5)χ−1\n0(T)[dχ−1\n0(T)/dT]−1= (T−Tc)/γ. (6)\nLinear fittings to the plots of Ms(T)[dMs(T)/dT]−1and\nχ−1\n0(T)[dχ−1\n0(T)/dT]−1in Fig. 5(b) yield β= 0.30(1)\nwithTc= 122.4(5) K, and γ= 1.22(1) with Tc=\n122.8(1) K. The third exponent δcan be calculated from\nthe Widom scaling relation δ= 1+γ/β[39]. From βand\nγobtainedwith the modified Arrottplotand the Kouvel-\nFisher plot, δ= 4.6(2) and 5.1(1) are obtained, respec-\ntively, which are close to the direct fit of δ= 4.83(6) tak-\ning into account that M=DH1/δatTc= 123K [inset in\nFig. 5(a)]. The obtained critical exponents of FeCr 2Te4\nare very close to the theoretically predicted values of 3D\nIsing model ( β= 0.325, γ= 1.24, and δ= 4.82) (Table\nII).\nScaling analysis can be used to estimate the reliability\nof the obtained critical exponents and Tc. The magnetic\nequation of state in the critical region is expressed as\nM(H,ε) =εβf±(H/εβ+γ), (7)\nwheref+forT > T candf−forT < T c, respectively,\nare the regular functions. Eq. (7) can be further written\nin terms of scaled magnetization m≡ε−βM(H,ε) and\nscaled field h≡ε−(β+γ)Hasm=f±(h). This suggests\nthat for true scaling relations and the right choice of β,\nγ, andδ, scaled mandhwill fall on universal curves\naboveTcand below Tc, respectively. As shown in Fig.\n5(c), allthe datacollapseon twoseparatebranchesbelow\nand above Tc, respectively. The scaling equation of state\ntakes another form,\nH\nMδ=k/parenleftBigε\nH1/β/parenrightBig\n, (8)\nwherek(x) is the scaling function. From the above equa-\ntion, all the data should also fall into a single curve.\nThis is indeed seen [Fig. 5(d)]; the M(µ0H)−1/δvs\nε(µ0H)−1/(βδ)experimental data collapse into a single\ncurve and the Tclocates at the zero point of the horizon-\ntal axis. The well-rescaled curves confirm the reliability\nof the obtained critical exponents and Tc.\nFurthermore, it is important to discuss the nature as\nwell as the range of magnetic interaction in FeCr 2Te4. In\na homogeneous magnet the universality class of the mag-\nnetic phase transition depends on the exchange distance\nJ(r). In renormalization group theory analysis the inter-\naction decays with distance rasJ(r)≈r−(3+σ), where\nσis a positive constant [41]. The susceptibility exponent\nγis:\nγ= 1+4\nd/parenleftbiggn+2\nn+8/parenrightbigg\n∆σ+8(n+2)(n−4)\nd2(n+8)2\n×/bracketleftBigg\n1+2G(d\n2)(7n+20)\n(n−4)(n+8)/bracketrightBigg\n∆σ2,(9)\nwhere ∆σ= (σ−d\n2) andG(d\n2) = 3−1\n4(d\n2)2,nis the spin\ndimensionality [42]. When σ >2, the Heisenberg model6\nTABLE II. Comparison of critical exponents of FeCr 2Te4with different theoretical models.\nReference Technique Tc− Tc+ β γ δ\nFeCr2Te4 This work Modified Arrott plot 122.9(3) 122.7(1) 0.33(2) 1.2 0(1) 4.6(2)\nThis work Kouvel-Fisher plot 122.4(5) 122.8(1) 0.30(1) 1.2 2(1) 5.1(1)\nThis work Critical isotherm 4.83(6)\n3D Heisenberg 28 Theory 0.365 1.386 4.8\n3D XY 28 Theory 0.345 1.316 4.81\n3D Ising 28 Theory 0.325 1.24 4.82\nTricritical mean field 36 Theory 0.25 1.0 5.0\nTABLE III. The first-principles total energy (in meV) per for -\nmula unit of different magnetic patterns as shown in Fig. 6(a)\nusing the 300 K and 105 K experimental structures. In the\nE-AF-2 pattern, a Cr atom is antiferromagnetically and fer-\nromagnetically aligned with the closest and farthermost of its\nsix nearest Cr neighbors, respectively; the opposite holds in\nthe E-AF-1 pattern. FM means the ferromagnetic configura-\ntion.\nStructure Ferri FM C-AF E-AF-1 E-AF-2\n300 K 8 90 197 31 82\n105 K 0 56 171 72 32\nis valid forthe 3Disotropicmagnet, where J(r) decreases\nfaster than r−5. Whenσ≤3/2, the mean-field model is\nsatisfied, expectingthat J(r) decreasesslowerthan r−4.5.\nFor the 3D-Ising model with d= 3 and n= 1,σ= 1.88\nis obtained, leading to spin interactions J(r) decaying as\nJ(r)≈r−4.88. This calculation suggests that the spin\ninteraction in FeCr 2Te4is close to the 3D Ising localized-\ntype coupled with a long-range ( σ= 1.88) interaction,\nin line with its weak itinerant character. Meanwhile, the\ncorrelation length ( ξ) correlates with the critical expo-\nnentν(ν=γ/σ), where ξ=ξ0[(T−Tc)/Tc]−ν. It gives\nthatν= 0.64(1) and α= 0.08 (α= 2−νd).\nTo get further insight into the magnetism, we per-\nformed first-principles calculations using density func-\ntion theory. We applied the WIEN2K implementation\n[43] of the full potential linearized augmented plane-wave\nmethod in generalized-gradient approximation using the\nPBEsolfunctional [44]. The basis size wasdetermined by\nRmtKmax= 7 and the Brillouin zone was sampled with\n115 irreducible kpoints to achieve energy convergence of\n1 meV. As shown in Table III, we found that for the ex-\nperimental structure refined at 105 K, the ferrimagnetic\nstate where the Cr and Fe atoms have opposite spin ori-\nentationsis56meVperformulaunitlowerintotalenergy\nthan the ferromagnetic phase and 32 meV lowered than\nthe most stable antiferromagnetic structure (i.e., E-AF-\n2) as observed in FeCr 2Se4[45]. A similar trend of the\nresults holds for the the experimental structure refined\nat 300 K (Table III). A weak easy c-axis anisotropy was\nobtained by inclusion of spin-orbit coupling in the calcu-\nFIG. 6. (Color online) (a) The antiferromagnetic structure s\nused in the first-principles total energy calculations. Ato m-\nresolved density of states (b) in the nonmagnetic state and\n(c) in the ferrimagnetic state where the Cr and Fe atoms\nhave opposite spin orientations, which are shown in the rati o\nof two Cr ions to one Fe ion.\nlations, namely the total energy per formula unit is lower\nby less than 1 meV for the magnetizationalong the caxis\nthan along the aorbaxis. Thus, other sources of mag-\nnetic anisotropy such as dipole-dipole interaction are im-\nportant. The calculated atom-resolved density of states\n(DOS) is shown in Figs. 6(b) and 6(c). For the nonmag-\nnetic case, the Cr-derived DOS has a sharp peak at the\nFermi level [Figs. 6(b)], suggesting a strong Stoner insta-\nbility that yields an itinerant ferromagnetism in the Cr\nlayers. This yields the splitting of about 2.8 eV between\nthe spin-majority and spin-minority bands of Cr charac-7\nter with the dramatic reduction of the Cr-derived DOS\nat the Fermi level [Fig. 6(c)], indicative of the localized\nspin picturefor the Cratoms. Whereas, the nonmagnetic\nFe-derived DOS peaks at about 0 .6 eV below the Fermi\nlevel [Fig. 6(b)] and experiences little reduction at the\nFermi level upon entering the magnetic phase [Fig. 6(c)].\nWe infer that the magnetism of the Fe ions is established\nvia antiferromagnetic superexchange with the neighbor-\ning two Cr ions. The Cr magnetic moment within the\natomic Muffin tins is about 2.88 µB, which is close to\nthe nominal Cr3+S= 3/2 state. With the octahedral\ncoordination, the splitting of the five 3 dorbitals between\nthe high-lying egand low-lying t2gorbitals is substantial.\nTheS= 3/2 state of the Cr3+ion (i.e., 3 d3ort3\n2ge0\ngelec-\ntronconfiguration)meansthattheCr t2gorbitalsarehalf\nfilled, rendering a vanishing orbital angular moment and\na negligible spin-orbit coupling effect. The Fe magnetic\nmoment is about 2.80 µB, significantly deviated from the\nnominal high-spin Fe2+S= 2 state, which indicates its\ndual characters with both localized spins and itinerant\nelectrons. This reveals an interesting interplay of Cr and\nFeelectronicstates,whichallowsthespin-majoritybands\nof the system at the Fermi level to be of Fe character\nrather than Cr character [Fig. 6(c)]. We thus picture\nthe FIM in FeCr 2Te4as itinerant ferromagnetism among\nthe antiferromagnetically coupled Cr-Fe-Cr trimers. The\ntrimers centered at the Fe sites form a body-centered\northorhombic lattice of magnetic dipoles with effective\nmoment of 2 µBS= 4µB[Fig. 3(c)]. In addition to the\neffective Heisenberg exchange interaction, the ith trimer\nis coupled with its ten neighboring trimers [Fig. 6(a)]\nvia dipole-dipole interaction ∝ −(Si·rij)(Sj·rij)/|rij|5,\nwhich tends to align the magnetic moments along the\nbond direction rij=ri−rjwherejdenotes one of the\nneighboring trimers and riis the spatial vector of the ith\ntrimer. Since the body-centered orthorhombic structure\nof the trimers is substantially elongated along the caxis,\neasyc-axis magnetic anisotropy has the overall minimum\ndeviation from the neighboring bond directions. We thus\npredict that the 3D Ising-like ferrimagnetism is sensitive\nto changes in the lattice structure, especially the tilting\nof the Cr-Fe-Cr trimers, which will be verified by future\npressure experiments and computer simulations.\nCONCLUSIONS\nIn summary, we systematically investigated structural\nand magnetic properties of stoichiometric FeCr 2Te4that\ncrystallizes in the I2/mspace group. The second-order\nPM-FIM transition is observed at Tc∼123 K. The\ncritical exponents β,γ, andδestimated from various\ntechniques match reasonably well and follow the scaling\nequation. The analysis of critical behavior suggests that\nFeCr2Te4isa3D-Isingsystemdisplayingalong-rangeex-\nchange interaction with the exchange distance decayingasJ(r)≈r−4.88. Combined experimental and theoret-\nical analysis attributes the ferrimagnetism in FeCr 2Te4\nto itinerant ferromagnetism among the antiferromagneti-\ncallycoupledCr-Fe-Crtrimers. 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Singh1,y\n1Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India\n2School of Physics and Materials Science, Thapar Institute of Engineering and Technology, Patiala 147004, India\n3Laboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,\n6 bd du Marechal Juin, 14050 Caen Cedex 4, France\nGeometrically frustrated structures combined with competing exchange interactions that have dif-\nferent magnitudes are known ingredients for achieving exotic properties. Herein, we studied detailed\nstructural, magnetic, thermal (specific heat), magneto-dielectric, and magnetic exchange bias prop-\nerties of a mixed 3d - 4d spinel oxide with composition CoFeRhO 4. Detailed magnetization, heat\ncapacity, and neutron powder diffraction studies (NPD) highlight long-range ferrimagnetic order-\ning with an onset at 355 K. The magnetic structure is established using a ferrimagnetic model\n(collinear-type) that has a propagation vector k = 0, 0, 0. The magneto-dielectric effect appears\nbelow the magnetic ordering temperature, and the exchange bias (EB) effect is observed in field\ncooled (FC) conditions below 355 K. The magneto-dielectric coupling in CoFeRhO 4originates due\nto the frustration in the structure, collinear ferrimagnetic ordering, and uncompensated magnetic\nmoments. The unidirectional anisotropy resulting from the uncompensated magnetic moments\ncauses the room-temperature exchange bias effect. Remarkably, the appearance of technologically\nimportant properties (ferromagnetism, magnetodielectric effect, and EB) at room temperature in\nCoFeRhO 4indicates its potential use in sensors or spintronics.\nI. INTRODUCTION\nSpinel oxides are an intriguing class of materials, not\nonly for their potentiality for a wide range of applications\nbut also because of a variety of new and exciting physics\n(such as frustrated magnetism, multiferroic properties,\norbital glass system, spintronics applications, and spin-\norbital liquids) that continues to arise from the strong\ninteractions among spin, orbital, and structural degrees\nof freedom [1–14]. They have a unique structure with\na general formula of AB 2O4, where A and B are metal\nions (Fig.1). This structure comprises an array of metal\ncations in octahedral and tetrahedral coordination, sur-\nroundedbyoxygens, creatingtwosetsofmagneticsublat-\ntices [15]. B cations generally form a pyrochlore-like lat-\ntice by residing in the octahedral sites and originate frus-\ntrated magnetic interactions [16–19]. On the other hand,\nthe metal A ions occupy the tetrahedral sites (eightfold)\nand construct a diamond lattice [21–25]. This bipar-\ntite lattice can be interpreted as two face-centered inter-\npenetrated cubic (fcc) sublattices. These sublattices are\nshifted diagonally by one-quarter. Variation of magnetic\nand nonmagnetic cations on the tetrahedral A sites and\ntheoctahedralBsitescanoriginatecomplexmagneticin-\nteractions by affecting the magnitudes of superexchange\ninteractions (J AA, JBB, and JAB). Therefore, exotic\nproperties (and states) such as a spiral spin liquid phase\n[26], unique, glassy magnetic behavior [22, 27], and spin-\norbital liquids [28, 29] emerge in spinel materials. The\n\u0003soumarik@thapar.edu\nyrpsingh@iiserb.ac.inpyrochlore lattice is a fertile playground for theoretical\nandexperimental researchtoexplore newphysics. Atthe\nsame time, the bipartite diamond lattice (in spinel sys-\ntems) is a fruitful platform for realizing exotic quantum\nbehavior. In particular, the bipartite nature of the dia-\nmond lattice can be useful in designing the recently pro-\nposed 3D topological paramagnetism for the frustrated S\n= 1 diamond lattice [22, 23].\nIn addition to geometrical frustration and competing\nexchange interactions having different magnitudes, spin-\norbit coupling (SOC) is a known ingredient that favors\nmore exotic spin order and dynamics. In this connection,\n4d and 5d elements containing oxides attracted remark-\nable research interest in recent times. Being spatially\nmore extended (4d/5d orbitals), the on-site Coulomb re-\npulsion energy (U) is smaller in 4d and 5d elements than\ntheir 3d analogues. However, in the spinel oxide family,\nonly one 5d-containing material has been reported so far.\nThe iridium-containing spinel oxide with composition\nCu[Ir 1:5Cu0:5]O4highlights a highly frustrated magnetic\nstate [30]. Among 4d-rhodium-based diamond lattice\nspinel oxides, in diamond-lattice Heisenberg antiferro-\nmagnet CoRh 2O4, a combined experimental and theoret-\nical work showed that the S = 3/2 spins are unfrustrated\nand display static and dynamic properties [21]. Tetrago-\nnallydistortedCuRh 2O4showsanincommensuratemag-\nnetic order for the S = 1/2 spins and the presence of siz-\nable quantum effects [21]. A spin-orbit entangled param-\nagnetic state is suggested in NiRh 2O4[22, 31, 32]. Stud-\nies on magnetically diluted Cu 1\u0000xZnxRh2O4highlight\nspin transition triggered by an enhancement of preced-\ning spin fluctuations [33]. At the same time, it shows\nthe suppression of orbital order on the octahedral sitesarXiv:2304.13983v1  [cond-mat.mtrl-sci]  27 Apr 20232\nthrough the percolative manner. In general, geometri-\ncally frustrated structures with 4d/5d elements (SOC)\nare ideal for realizing exotic phenomena. Therefore, ex-\nploringnewfrustratedstructures(geometricalfrustration\nand competing magnetic interactions) coupled with SOC\nis essential.\nIn this article, we present the detailed structural (us-\ning X-ray powder diffraction, neutron powder diffrac-\ntion, and electron microscopy), magnetic (magnetiza-\ntion), thermal (specific heat), and magneto-dielectric\nstudies on a mixed 3d - 4d spinel oxide with composition\nCoFeRhO 4. In this material, non-magnetic Rh3+cations\noccupy octahedral B sites; however, they can be a source\nof SOC in the structure. An insulating and ferrimagnetic\nground state is observed near room temperature (T C=\n355 K). Detailed magnetic and magneto-dielectric mea-\nsurements highlight the room-temperature exchange bias\neffect and the magnetodielectric effect in this material.\nII. EXPERIMENTAL DETAILS\nSynthesis. We have used the standard solid-state re-\naction route to synthesize the polycrystalline CoFeRhO 4\nmaterials. Stoichiometric amounts of Co 3O4(99.9%)\nFe2O3(99.999%) and Rh (99.9%) metal powders were\nused for the synthesis. The stoichiometric amounts of\nthe starting materials were mixed with a mortar pestle.\nThe materials were heated several times and the final\nsintering was performed at 1473 K for 36 h.\nX-ray Powder Diffraction and Neutron Diffrac-\ntion.X-ray diffraction (XRD) using the powder form of\nthe material was collected at ambient temperature (RT)\nusing a PANalytical diffractometer (Cu-K \u000b,\u0015= 1.54056\nÅ). Neutron diffraction (NPD) data for the powdered\nCoFeRhO 4samplewerecollectedatvarioustemperatures\n(12 K, 100 K and 250 K) in JEEP II, Kjeller Reactor,\nNorway (\u0015= 1.5538 Å). We have performed the Ri-\netveld refinements of the diffraction patterns (XRD and\nNPD) using the fullProf suite software.\nElectron Microscopy. We have performed electron\nmicroscopy measurements (high-angle annular dark-field\nscanning transmission electron microscopy (HAADF-\nSTEM) and electron diffraction) to explore the crystal\nstructure of CoFeRhO 4. HAADF-STEM and electron\ndiffraction were carried out using a JEOL ARM-200F\ncold FEG probe image aberration corrected 200 kV mi-\ncroscope. The instrument is also equipped with a solid-\nangle CENTURIO EDX detector.\nMagnetism and Specific Heat. Direct current\nmagnetization (temperature and magnetic field depen-\ndent) measurements were done in a Quantum Design su-\nperconducting quantum interference device (MPMS 3).\nMagnetic measurements were conducted in field-cooled\n(FC) and zero-field-cooled (ZFC) modes. Specific heat\nwas measured (2 K - 400 K) in a physical property mea-\nsurement system (PPMS, Quantum Design) without a\nb\naFIG. 1. crystal structure for the CoFeRhO 4. Green spheres\nrepresenttheFe/Rhcations(OctahedralBsitesintheAB 2O4\nstructure), BluespheresrepresenttheCocations(Tetrahedral\nA sites), and red spheres represent oxygen.\nmagnetic field.\nDielectric Measurements. The temperature and\nmagnetic field-dependent dielectric measurements were\nperformed employing an LCR meter (Agilent 4284A) and\na sample insert for a PPMS (Quantum Design).\nIII. RESULTS AND DISCUSSION\nPreliminary Rietveld refinement uses the room-\ntemperature XRD pattern of CoFeRhO 4. It shows\nthat the sample crystallizes in a cubic structure with a\nspace group Fd3m, which is isostructural with CoFe 2O4\n[34, 35]. However, a small amount of Rh is detected\n(main peak at 2 \u0012'41 degrees, Fig. S1 in the sup-\nporting information (SI)) [20] in the RT XRD pattern\nfor CoFeRhO 4. Further, to explore the detailed nuclear\nand magnetic structure, we have collected the NPD pat-\nterns at 12 K, 100 K, and 250 K. Therefore, we will use\nthe results of the NPD Rietveld refinements to describe\nthe crystal structure of CoFeRhO 4. Similar to the XRD\npattern, a small amount of Rh is also detected in the\nNPD patterns. Fig.2 illustrates the plot of the NPD\nRietveld refinement for the pattern collected at 12 K.\nThe plots of the NPD Rietveld refinements for the pat-\nterns collected at different temperatures are provided in\nthe supporting information (Figure S2 in SI) [20]. The\nstructural parameters obtained from the 12 K - NPD re-\nfinement for CoFeRhO 4are summarized in Table 1. The\ncrystal structure for CoFeRhO 4is highlighted in Fig.1.\nRh atoms and most of the Fe cations occupy the octa-\nhedrally coordinated B site (16d (0.5, 0.5, 0.5)) of the\nstructure. Co-cations occupy the eightfold tetrahedral A\nsites (8a (0.125, 0.125, 0.125)). Oxygen atoms occupy\nthe 32e (x, x, x) Wycoff positions. The refinement of3\n3000\n2000\n1000\n0Intensity (a. u.)\n100 80 60 40 20\n2θ (º) Obs\n Calc\n Diff.\n Bragg Position900\n450\n0\n191817\n[111]300\n200\n100\n313029 12 K\n 100 K\n 250 K\n[220]\nFIG. 2. Rietveld refinement plot of the Neutron powder\ndiffraction (NPD) pattern collected at 12 K for CoFeRhO 4.\nThe lower ticks highlight the magnetic peak positions (k = 0,\n0, 0). Insets in the figure show the enhancement of the [111]\nand [220] NPD peaks with a lowering of the temperature.\nthe oxygen occupancy indicates that there is no devia-\ntion from the full occupancy. Rh shows a small deviation\nfromfulloccupancy (Occupancy=0.98(2)). Fig.3shows\nthe STEM- HAADF image along the [011] direction for\nCoFeRhO 4. The observed STEM images agree with the\nexpected crystal structure of the spinel material. The\nlattice parameter obtained from STEM images (8.4 Å)\nmatched the values found from the NPD (Table 1) and\nRT-XRD refinements.\nThestructural(Rietveld)refinementofthe250KNPD\nhighlights the existence of magnetic peaks at that tem-\nperature. These magnetic ordering-related peaks show\nan increase in intensity with decreasing temperature (in-\nset in Fig.2). The position of all magnetic peaks for\nCoFeRhO 4coincides with the allowed nuclear reflections.\nTherefore, we describe the magnetic structure using the\npropagation vector k = (0,0,0). For the long-range mag-\nnetic order, the best fit to the NPD magnetic reflections\nis achieved using a collinear ferrimagnetic ordering model\n(Fig. 4). A similar magnetic ordering scheme was pro-\nposed for the CoFe 2O4material [34]. The absence of a\n(200) magnetic peak discards the non-collinear arrange-\nment of the spins. In general, any long-range spin cant-\ning would result in the appearance of the (200) magnetic\nBragg peak. We do not observe the (200) peak at 21.27\n\u000e((400) peak is observed at 2\u0012= 43.43\u000e).\nMagnetic moments in different sublattices (M (T d)\n- tetrahedral A sites, M (Oct) - octahedral B sites)\nobtained from refinements are listed in Table 1. The\nresultant magnetic moment per formula unit obtained\nfrom the NPD refinements is 0.6 (1) \u0016Bat T = 12\nK. However, a discrepancy is observed between the\nmagnetic moments’ experimental and theoretical values\nin the tetrahedral and octahedral sites. Theoretically\ncalculated magnetic moments in different sublattices\nFIG. 3. HAADF image and crystal structure along the [011]\nzone axis for CoFeRhO 4, collected at room temperature. Fe,\nRh, and Co cations are highlighted in the figure. The corre-\nsponding electron diffraction pattern is shown in the inset.\nc\nb\na\nFIG. 4. The observed magnetic structure (k = 0, 0, 0)\nfor CoFeRhO 4. A collinear ferrimagnetic ordering scheme is\nobserved in the NPD refinements.\nTABLE I. Structural parameters for Ba 2ScRuO 6at RT ex-\ntracted from Rietveld refinement of powder XRD diffraction\ndata\nSpace group Fd3m,a = b = c = 8.4207 (1) Å\nRP= 6.98, R WP= 9.01,\u001f2= 1.03,R Bragg= 2.79\nRMag= 5.29, M (T d) = 3.4 (1) \u0016B, M (Oct.) = 4.1 (1) \u0016B\nAtom Wyck. Pos. x y z Occu. B iso\nCo(T d) 8a 0.125 0.125 0.125 0.88 (2) 0.1 (1)\nFe(T d) 8a 0.125 0.125 0.125 0.12(2) 0.1(1)\nCo(Oct.) 16d 0.5 0.5 0.5 0.065(5)0.11(3)\nFe(Oct.) 16d 0.5 0.5 0.5 0.44(1) 0.11(3)\nRh(Oct.) 16d 0.5 0.5 0.5 0.49(1) 0.11(3)\nO 32e 0.249(1)0.249(1)0.249(1) 1 0.10(3)\nare MA=0:88\u00023:5 + 0:12\u00025= 3.66\u0016Band MB=\n0:87\u00025 + 0:13\u00023:5)= 4.81\u0016B, calculated with M Fe3+\n= 5\u0016Band MCo2+= 3.5\u0016B. Reduced magnetic4\n(a)\n(b)\nFIG. 5. (a) Temperature-dependent magnetization (M -\nT) and (b) magnetic field variation of the magnetization (M\n- H) for CoFeRhO 4. Ferrimagnetic behavior is observed for\nCoFeRhO 4.\nmoments could result from local disorder (and/or local\ncanting) of spins. The complex cationic distribution\nand the existence of nonmagnetic Rh in the structure\ncan originate competing magnetic interactions and can\ncreate nonuniform spin canting. A similar reduced\nmagnetic moment due to the local spin canting is\nalso observed for the CoFe 2O4and Ti-doped CoFe 2O4\nmaterials [34, 36]. However, a collinear ferrimagnetic\nlong-range magnetic ordering scheme is suggested for\nboth of these materials.\nThe FC and ZFC magnetic moment (M-T) tempera-\nture variation measured at 0.1 T for CoFeRhO 4is pre-\nsented in 5. The M-T shows a sharp upturn at 355 K,\nhighlighting the onset of the ferrimagnetic transition at\nthat temperature. Upon further lowering the temper-\nature bifurcation between FC and ZFC magnetization\n(MFCand MZFC) is visible at 250 K. Below 250 K,\nMFCshows an increasing trend with decreasing temper-ature. Several competing magnetic interactions exist in\nCoFeRhO 4. Thecomplexcrystalstructure(diamondand\npyrochlore lattice) and the distribution of magnetic and\nnonmagnetic cations in two different sublattices can cre-\nate competing magnetic interactions and frustration. For\nexample,A-sitemagneticcationsaresurroundedby12B-\nsite magnetic and non-magnetic first neighbours, which\ncanbeFe3+orCo2+orRh3+; anyB-cationissurrounded\nby 6 A and 6 B-sited cations. Therefore, the bifurcation\nand the rise in M FCat a lower temperature could be due\nto the frustration in the spinel structure. However, the\nferrimagnetictransition inCoFeRhO 4ismuchlowerthan\nin CoFe 2O4. In CoFe 2O4, the collinear ferrimagnetic\ntransition is observed at 800 K with a strong intersub-\nlattice AFM superexchange interaction (J AB= - 12.39\nkB) [34]. Introducing non-magnetic Rh into the struc-\nture could dilute the magnetic interactions by decreasing\nthe strength of inter- and intra-sublattice superexchange\ninteractions and, therefore, can reduce the magnetic or-\ndering temperature. Figure 5(b) shows the magnetiza-\ntion as a function of the magnetic field (M - H) at var-\nious temperatures measured in the ZFC mode. Above\nthe magnetic transition temperature (at 400 K), a linear\nparamagnetic type M-H behavior is observed. However,\nat 350 K, the M-H loop illustrates non-linearity at low\nmagnetic fields, indicating theappearance of themagnet-\nicallyorderedstate. Aswefurtherlowerthetemperature,\nsaturated ferrimagnetic-type M - H loops are observed.\nThe saturation magnetization and coercive field (H C) en-\nhance with decreasing the temperature.\nFigure 6 highlights the temperature variation of to-\ntal specific heat for CoFeRhO 4. Magnetic ordering usu-\nally involves an entropy change, resulting in specific heat\nanomalies. The C Pvs T plot shows an anomaly at 355\nK, indicating a true phase transition. This also confirms\nthe magnetic ordering temperature for CoFeRhO 4. The\nlow-temperature part of the specific heat (2-20 K) can be\nwell represented by eqn. 1:\nCP=\rT+\fT3; (1)\nwhere\r= Sommerfeld coefficient and \f= lattice con-\ntributions to the specific heat. The obtained \r= 0.015\nJmol\u00001K\u00002. However, the total specific heat at higher\ntemperatures contains both the phonon and magnetic\nparts (CP=Cph+Cm). A combined Einstein-Debye\nmodel can generally estimate the total phonon contri-\nbution in the specific heat. Therefore, to estimate the\nmagnetic contributions ( Cm) in the specific heat, we fit\nthe data with the Einstein-Debye model [37]:\nCph= 9Rx\u00003\nDZxD\n0x4ex\n(ex\u00001)2dx+R2X\ni=1aix2\nEiex\nEi\n(ex\nEi\u00001)2\n(2)\nWhere xD=\u0012D/T and x Ei=\u0012Ei/T and, Debye tem-\nperature = \u0012Dand\u0012Ei= Einstein temperature. R =5\n150\n100\n50\n0C (J/mole-K)\n400 300 200 100 0\nT (K) Cexp\n fit (eqn. 1)\n fit (eqn. 2)\n6\n4\n2\n0C (J/mole-K)\n20100\nT (K)\nFIG. 6. Temperature variation of the specific heat (C) for\nCoFeRhO 4. An anomaly due to the magnetic ordering is ob-\nserved at 355 K. The green line highlights the fitted curve\nusing Eqn.2. Low-temperature C is also fitted using eqn.1\nand shown in the inset of the figure.\n8.31 J/mol K, and a i= degree of freedom for each Ein-\nstein mode. The fitted curve is shown in figure 6. The\nbest fit is obtained with \u0012D= 632 K,\u0012E1= 177 K and\n\u0012E2= 740 K. Then the contribution related to the mag-\nnetic ordering (C m) in the specific heat is calculated by\nsubtracting C phfrom the measured total specific heat\ndata. Figure 7 highlights the C m- T plot, showing a\nsharp maximum at T = 355 K. The entropy change due\nto the magnetic ordering can be calculated by using the\nfollowing formula:\nSm(T) =ZT\n0Cm(T)\nTdT (3)\nTheSm(T)vs T plot is shown in figure 7. Magnetic\nentropy increases with increasing temperature and shows\na saturation value of '4.4 J mole\u00001K\u00001(above 355 K),\nand this is less than the theoretically estimated magnetic\nentropy for CoFeRhO 4(Sm(T)= Rln (2S+ 1)= 26.41 J\nmole\u00001K\u00001, Co2+adopts the e g4t2g3electronic config-\nuration with S = 3/2, Fe3+adopts the e g2t2g3electronic\nconfiguration with S = 5/2). In general, the frustra-\ntion of the magnetic cations near and above the magnetic\ntransition temperature can reduce the entropy contribu-\ntiontothemagneticordering. InCoFeRhO 4, geometrical\nfrustration in the structure (diamond and pyrochlore lat-\ntice) and the distribution of magnetic and non-magnetic\ncations in two different sublattices could cause frustra-\ntionofmagneticcations. Thereducedmagneticmoments\nin the neutron powder diffraction experiments point to\nspins’ local disorder (and/or local canting). Therefore,\nthe reduced entropy related to magnetic ordering can be\nattributed to the frustration of the magnetic cations [38].\nOxides geometrical frustration and ferrimagnetic\ntransition often result in exciting magnetoelec-\ntric/magnetodielectric coupling. Geometric frustration\n12\n10\n8\n6\n4\n2\n0Cm (J/mole-K)\n350 300 250 200\nT (K)5\n4\n3\n2\n1\n0Sm (J/mole-K)\nCoefficient values ± one standard deviation\nA=0.02189 ± 0.00177\nB=0.00012499 ± 1.41e-006FIG. 7. Temperature variation of the magnetic component\nof the specific heat (C) (left axis) and calculated magnetic\nentropy (right axis) for CoFeRhO 4\n(local spin canting) in a magnetic system is proven to be\na key ingredient for magneto(di)electric coupling. For\nexample, the triangular Ising lattice Ca 3Co2O6shows\nmagneto dielectric coupling below the ferrimagnetic\nordering temperature (24 K) [39]. Haldane spin-chain\nsystem Dy 2BaNiO 5shows a magneto-(di)electric effect\nbelow the long-range ordering temperature (58 K)\n[40]. In spinels, the magnetodielectric effect is observed\nfrom the beginning of the ferrimagnetic ordering in\nCoCr 2O4(TC= 96 K) [41, 42] and NiCr 2O4[43].\nMnCr 2O4shows magneto dielectric effect below the\nferrimagnetic ordering temperature (43 K) [44]. How-\never, the magneto-(di)electric effect in ferrimagnetic\nmaterials near room temperature is rare. The near-\nroom-temperature magnetodielectric effect is important\ndue to its possible applications in spintronics devices,\nmagnetically accessible ferroelectric random-access\nmemories, and communication technology. Figure 8\nshows the temperature variation of the real part of the\ndielectric constant ( \u000fr) for CoFeRhO 4. The dielectric\nloss (tan\u000e) is highlighted in the inset of Figure 8. The ex-\ntremely low values of dielectric loss (tan \u000e) highlight the\ninsulating behaviour of CoFeRhO 4. It also excludes the\npossibility of extrinsic Maxwell-Wagner-like behaviour\nappearing because of the leakage currents. However,\nthe frequency dispersion of \u000frand tan\u000estarted above\n320 K, indicating the growing Maxwell-Wagner-type\nrelaxation or other sources of conductivity around 320\nK. Nonetheless, the dielectric constant is intrinsic below\n320 K. In addition to a small hump near the magnetic\nordering temperature (335 K), two anomalies at 220 K\nand 50 K are observed in the \u000fr- T plots. In particular,\nthe magnetisation measurements also observed similar\nanomalies at 220 K and 50 K.\nAs observed from our nuclear and magnetic structure\nrefinements, heat capacity, and magnetization studies,\nCoFeRhO 4hosts an exciting combination of complex6\n575\n570\n565\n560\n555  εr\n400 300 200 100\n  T (K) 5 kHz\n 10 kHz\n 20 kHz\n 50 kHz\n 80 kHz\n0.12\n0.08\n0.04\n0.00  tanδ\n300 150\n  T (K)\nFIG. 8. Temperature variation of the real part of the dielec-\ntric constant ( \u000fr) at several fixed frequencies for CoFeRhO 4.\nThe inset in the figure shows the dependence of the dielectric\nloss as a temperature function.\ncrystal structure (diamond and pyrochlore lattice) and\ncompeting magnetic interactions. The reduced magnetic\nmoments observed in the NPD refinements highlight the\npossibility of local spin canting. Therefore, the geomet-\nrical frustration, competing magnetic interactions, and\nthe local spin canting can originate magnetic frustra-\ntion in the structure. The magnetic frustration could be\nwhy these anomalies are observed below the long-range\nmagnetic ordering. To understand the effect of the mag-\nnetic field on the dielectric constant, we have performed\nmagnetic field dependence measurements of the dielectric\nconstant. Figure 9 shows the temperature variation of \u000fr\nmeasured in the presence of different magnetic fields. In-\nterestingly, the dielectric constant increases in the pres-\nence of a magnetic field below the magnetic ordering\ntemperature, and a positive magnetodielectrictance ef-\nfect is observed. The starting of the magneto(di)electric\neffect below the long-range ferrimagnetic ordering tem-\nperature indicates that the observed magneto(di)electric\neffect is related to the magnetic ordering of CoFeRhO 4.\nThe change in the dielectric data with the magnetic field\nis most pronounced near 220 K. The magneto-dielectric\nconstant ( \u0001\u000fr= [\u000fr(H)\u0000\u000fr(H= 0)]=\u000fr(H= 0)) and its\ndependence on the magnetic field are shown in the inset\nof figure 9. As discussed, the magnetic frustration in the\nstructure (geometric frustration, competing magnetic in-\nteractions, and local spin canting) could be the origin of\nthe increase in \u000frwith the magnetic field at that temper-\nature.\nIn recent times, complex magnetic materials (for\ninstance, ferrimagnetic systems Mn 3\u0000xPtxGa [45],\nBa2Fe1:12Os0:88O6[46] and SrFe 0:15Co0:85O2:62) [47], in\nparticular, those having a magnetic transition temper-\nature near room temperature have attracted significant\ninterest in exploring the exchange bias effect (EB). The\nEB effect shifts the isothermal magnetization loop, be-\ncomingasymmetricandshiftingalongthefieldaxis. This\n575\n570\n565\n560  εr\n40035030025020015010050\n  T (K)     H \n 0 T\n 5 T\n 14 T\n0.6\n0.4\n0.2\n0.0  Δεr  (%)\n1050-5-10\n  H (T) 300 K\n 220 K\n 100 KFIG. 9. Temperature dependent dielectric constant (real\npart,\u000fr) at different magnetic fields (H = 0 T, 5 T, and 14\nT) for CoFeRhO 4. The inset highlights the magnetic field\nvariation of the dielectric constant ( \u0001\u000fr= [\u000fr(H)\u0000\u000fr(H=\n0)]=\u000fr(H= 0)) at different temperatures.\neffecthasimportanttechnologicalapplications, suchasin\nthe development of magnetic sensors, magnetic recording\nread heads [48], random access memories [49], and other\nspintronic devices [50, 51]. Figure 10 highlights the mag-\nnetic field dependence of magnetization at 100 K, mea-\nsured in FC mode (cooling field, HFC= 3 T). It shows a\nclear shift towards the left field (negative magnetic field).\nThe EB field (H EB) can be calculated from the shift of\nthe hysteresis loop using the following equation\nHEB=\u0000(HC(L)+HC(R))=2 (4)\n(HC(L)andHC(R)are the intercepts with the positive\n(right) and negative (left) field axis. The calculated ex-\nchange bias field H EB= 65 Oe at 100 K. The exchange\nbias field is almost constant with varying cooling fields\n(HFC, ranging from 1 T to 7 T). To investigate the tem-\nperature evolution of the exchange bias effect and to ex-\nplore whether the exchange bias property in CoFeRhO 4\nis correlated with magnetic ordering, we have measured\nthe temperature variation of the EB field. The material\nwas cooled to the measuring temperatures in a magnetic\nfield of 3 T. Then the M-H loops were measured between\n\u00060.5 T. Fig. 11 highlights the temperature evolution of\nthe EB field for CoFeRhO 4. Interestingly, the EB effect\nemerges just below the long-range ferrimagnetic order-\ning (350 K); however, with decreasing temperature, H EB\nshows almost constant behavior down to 10 K. In ferri-\nmagnetic systems, the presence of uncompensated mag-\nnetic moments in a compensated AFM host is proven to\nbe very effective in designing the exchange bias effect (for\ninstance, Mn 3\u0000xPtxGa [45], Ba 2Fe1:12Os0:88O6) [46].\nIn this spinel structure, there is an uncompensated\nmagnetic moment due to the distribution of two differ-\nent magnetic cations (Co2+and Fe3+) in two different\nsublattices. In CoFeRhO 4, as observed from the neutron\ndiffraction refinements (table 1), below the magnetic or-7\n-30-20-100102030 M (emu/g)\n-0.4 -0.2 0.0 0.2 0.4\nH (T)-404M (emu/g)\n-400-2000200400\n H (Oe) 100 K\nHFC = 3 T\nFIG. 10. M-Hloop of CoFeRhO 4at 100 K measured in\nfield cooled (FC) mode ( HFC= 3 T). The inset highlights\nthe enlarged central part of the FC M-Hloop. A shift along\nthe left field axis is highlighted here.\nderingtemperature, themagneticmomentofBsublattice\n(M (Oct)) dominates over the M (T d). Under FC condi-\ntions, themagneticfieldorientsthenetmagneticmoment\nof the sublattices along the field direction. Now, as we\ngradually reverse the direction of the magnetic field, the\nmagnetic moments of the B sublattice (M (Oct)) do not\neasily reverse its direction as they are antiferromagneti-\ncally coupled with the A-sublattice (M (T d)). This pin-\nning effect of irreversible uncompensated spins (Octahe-\ndral B sublattice) develops the unidirectional anisotropy,\nand therefore, exchange bias is observed in CoFeRhO 4.\nIV. CONCLUSION\nTo summarize, we synthesized and examined the\ndetailed structural, magnetic, thermal (specific heat),\nmagneto-dielectric, and magnetic exchange bias proper-\nties of CoFeRhO 4, a mixed 3d-4d spinel oxide. We used\nvarious diffraction techniques, such as RT-XRD, NPD,\nRT-electron diffraction, and STEM, to explore the ma-\nterial’s structural details. The material crystallizes in a\ncubic structure with a space group of Fd3m. Our mea-\nsurements show a long-range ferrimagnetic ordering with\nan onset at 355 K, which is explained by a collinear fer-\nrimagnetic ordering model with k = [0, 0, 0]. However,\nwe observe reduced magnetic moments in the refinement,\npossibly due to the spins, local disorder, local canting, or\nfrustration. Magnetic frustration can also reduce mag-\nnetic entropy, as reflected in the specific heat measure-\nment. The magnetic entropy increases with temperature\nandreachesasaturationvalueof4.4Jmole\u00001K\u00001(above\n355 K), but it is less than the theoretically estimated\nvalue. Furthermore, our dielectric and FC magnetiza-\ntion measurements show the appearance of two techno-\nlogically significant phenomena near room temperature:\nthe magnetodielectric effect and the exchange bias ef-\n65\n52\n39\n26\n13\n0HEB(Oe)\n400 300 200 100 0\nT (K)HFC = 3 TFIG. 11. Temperature variation of the exchange bias field\n(HEB) for CoFeRhO 4. The exchange bias effect started to\nappear below the long-range ferrimagnetic ordering (350 K).\nfect. These effects appear below the magnetic ordering\ntemperature (355 K) and can originate from magnetic\nfrustration, collinear ferrimagnetic ordering, and uncom-\npensated magnetic moments. The uncompensated mag-\nnetic moments create unidirectional anisotropy, result-\ning in an exchange bias effect. Importantly, the pres-\nence of room temperature ferrimagnetism, EB, and the\nmagneto(di)electric effect demonstrates the potential of\nCoFeRhO 4in developing materials for sensors or spin-\ntronics applications operating at room temperature.\nV. ACKNOWLEDGMENTS\nR. P. S. acknowledges the Science and Engineering\nResearch Board (SERB), Government of India, for the\nCRG/2019/001028 Core Research Grant. S. M. ac-\nknowledges the SERB, Government of India, for the\nSRG/2021/001993 Start-up Research Grant.\n[1] V. W. J. Verhoeven, F. M. Mulder, and I. M. 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Alloys Compounds 448, 59 (2008)."    },    {        "title": "1109.2705v2.Interplay_between_non_equilibrium_and_equilibrium_spin_torque_using_synthetic_ferrimagnets.pdf",        "content": "arXiv:1109.2705v2  [cond-mat.mes-hall]  10 Apr 2012Interplay between non equilibrium and equilibrium spin tor que\nusing synthetic ferrimagnets\nChristian Klein\nSPSMS, UMR-E 9001 CEA / UJF-Grenoble 1, INAC, Grenoble, F-38 054, France,\nPhysikalisches Institut, Universit¨ at Karlsruhe (TH), Wo lfgang-Gaede Strasse 1, 76131 Karlsruhe, Germany\nCyril Petitjean and Xavier Waintal\nSPSMS, UMR-E 9001 CEA / UJF-Grenoble 1, INAC, Grenoble, F-38 054, France\n(Dated: November 27, 2018)\nWe discuss the current induced magnetization dynamics of sp in valves F 0|N|SyF where the free\nlayer is a synthetic ferrimagnet SyF made of two ferromagnet ic layers F 1and F 2coupled by RKKY\nexchange coupling. In the interesting situation where the m agnetic moment of the outer layer\nF2dominates the magnetization of the ferrimagnet, we find that the sign of the effective spin\ntorque exerted on the free middle layer F 1is controlled by the strength of the RKKY coupling:\nfor weak coupling one recovers the usual situation where spi n torque tends to, say, anti-align the\nmagnetization of F 1with respect to the pinned layer F 0. However for large coupling the situation\nis reversed and the spin torque tends to align F 1with respect to F 0. Careful numerical simulations\nin the intermediate coupling regime reveal that the competi tion between these two incompatible\nlimits leads generically to spin torque oscillator (STO) be havior. The STO is found in the absence\nof magnetic field, with very significant amplitude of oscilla tions and frequencies up to 50 GHz or\nhigher.\nPACS numbers: 72.25.Ba, 75.47.-m, 75.70.Cn, 85.75.-d\nSince the first successful experimental manipulation of\nmagnetic configurations by spin-polarized currents [1, 2],\nthe interest in spintronics devices entirely controlled by\nelectrical currents has rapidly increased [3–10, 12, 13].\nThe key concept behind theses experiments is the no-\ntion of spin transfer torque (STT) predicted by Slon-\nczewski[14] andBerger[15]: acurrentgetsspinpolarized\nbyafirst(pinned)magneticlayerandthentransferssome\nof its polarization to a second free layer, hence exerting\na torque on its magnetization. As a result the magneti-\nzation can switch to a different static configuration [2, 3]\nor even to a dynamical stationary regime where the mag-\nnetization of the free layers shows a sustained preces-\nsion [4–7]. This dynamical regime, known as the spin\ntorque oscillation (STO), allows to convert a dc voltage\ninto a microwaves signal at several GHz and has a real\npotential for applications (nanoscale microwave source\nor detector, high frequencies possibly above CMOS tech-\nnology, narrow band). There are still technological is-\nsues with STO based devices, however, such as very lim-\nited output power and the need of rather strong external\nfields. Indeed, the main mechanism for producing STO\nis based on a detailed tuning of the angular dependance\nof the spin torque [5] and requires the use of an exter-\nnal magnetic field. Recently, different mechanisms have\nemerged to enable STO behavior in the absence of exter-\nnal field: the first approach [9] relies on the precession\nof a non uniform magnetic structure, a magnetic vortex,\nthat can be excited at sub GHZ frequencies; a second\napproach [8, 16] uses a strongly asymmetric spin valve to\nincrease the anharmonicity of the angular dependance of\nthe spin torque, leading to the so-called ”wavy” behaviorof the STT [16–19]. Other approaches use an orthogonal\npolarizer [10] or a perpendicular free layer [11, 12].\nIn this letter we propose an entire new mechanism\nwhich naturally induces strong STO behavior, even in\nthe absence of magnetic field, using a synthetic ferrimag-\nnet (SyF) as the free layer of the spin valve. The SyF is\nmade of two ferromagnetic layers coupled through anti-\nferromagneticRKKYexchangeinteraction[20–22]. Here,\nwe predict that by simply tuning the effective strength\nof the RKKY exchange coupling (through the different\nthicknesses of the layers for instance), the signof the\neffective STT can be changed. At intermediate effective\ncoupling, the STT generically induces strong STO be-\nhavior irrespectively of the presence of applied magnetic\nfield.\nIn the following we first provide a simple physical pic-\nture that captures the essential features of our setup.\nOur main prediction uses very general conservation ar-\nguments and should therefore be quite robust and inde-\npendent of the details of the model. In a second step,\nwe provide more quantitative arguments to describe the\nstrong coupling regime. Last, we present a detailed nu-\nmerical study of the phase diagram of our model. We\nfind strong STO behavior at rather large frequencies in a\nwideportionofthephasediagramhintingthatSyFbased\nSTOs could be good candidates for application purposes.\nPhysical mechanism. We consider a nanopillar spin\nvalveF 0|N0|SyF made of apinned ferromagneticlayerF 0\nand a free synthetic ferrimagnetic layer SyF = F 1|N1|F2\n(where N 0and N 1are normal spacers), see Fig. 1a for a\ncartoon of the full stack. Upon injecting a current den-\nsityjthrough the nanopillar, a spin torque τττis exerted2\nFigure 1: Panel (a) is a cartoon of the magnetic multilayer\nstack, in which Miis the the magnetization of the layer F i.\nThe angle between M1andM0(M2) is respectively θ(α).\nPanels (b,c,d) show the stationary magnetization mz\n1along\nM0as a function of the current density jin A.cm−2, for a\nstack with d1= 2.nm,d2= 4d1and three different coupling\nstrength J; (b) weak coupling limit ( J= 0), (c) Intermediate\ncoupling limit ( J=−10−3J.m−2) (d) Strong coupling limit\n(J=−6.10−3J.m−2) Upper (Lower) inset : Sketch of the\ntorqueτττ1actionon M1(MSyF)fortheweak(strong)coupling\nregime.\non the magnetic moments of the layers [23]. This torque\nhas been extensively discussed in the literature and can\nbe strong enough to destabilize the initial magnetic con-\nfiguration. On the other hand, the magnetization M1\nandM2of F1and F 2are coupled by an oscillatory ex-\nchange (RKKY) interaction Jwhich is nothing but the\nspin torque present in the pillar at equilibrium [24, 25].\nThe coupling Jis reminiscent of Friedel oscillations and\ntypically behaves as J∼(cos2kFdN)/dNwherekFis\nthe Fermi momentum and dNthe width of the spacer\nN1.Jcan be tuned from negative (antiferromagnetic\ncoupling) to positive (ferromagnetic coupling), hence the\nname ”oscillatory”. Here we focus on negative values of\nJwhichstabilizean anti-alignedconfigurationof M1and\nM2. The goal of this letter is to discuss new phenomena\nthat arise from the competition between the equilibrium\nand the non equilibrium torques.\nIn order to gain a qualitative understanding of the un-\nderlying physics, let us discuss two extreme limits. We\nstart from a configuration where M1is aligned with M0whileM2is anti-aligned, and inject electrons from the\nright. Upon crossing the pinned layer F 0, the electrons\nget a polarization anti-parallel to M0. Denoting Jithe\nspincurrent(perunit surface)flowingin the normallayer\nNi, we have J0=/planckover2pi1pjm0/(2e) withm0=M0/|M0|and\npthe polarization of the current (0 ≤p≤1). Let us fo-\ncus on the weak coupling limit where J→0. When there\nis a finite angle θbetween M0andM1, the spin current\non the left J0and right J1of F1become different and\na torque τττ1=J0−J1is exerted on the magnetization\nof F1: conservation of angular momentum implies that\nwhatever spin lost by the conducting electrons is gained\nby magnetic degrees of freedom [14]. This conservation\nof angular momentum reads,\n∂\n∂t/bracketleftbiggd1M1\nγ1/bracketrightbigg\n=τττ1+... (1)\nwhered1is the width of F 1,γ1its gyromagnetic ratio\nand the ...indicate other contribution to the dynam-\nics (magnetic anisotropy, damping) to be discussed later\n(upper inset of Fig. 1). Ignoring (momentarily [26]) the\nrole of F 2, we arrive at ∂θ/∂t=γ1/planckover2pi1pjsinθ/(2ed1|M1|)\nand recover the usual phenomenology of spin torque: the\ntorque tends to stabilize the configuration where M1is\nanti-parallel to M0(or destabilize it if one reverses the\nsign of the current).\nLet us now discuss how this picture is modified when\none considers the opposite strong coupling J→ −∞\nlimit. Upon switching on J, the exchange energy (per\nunit surface of the pillar) EJ=−J(m1·m2) induces\na field like term in the RHS of Eq.(1) of the form\n+Jm2×m1. One needs also to consider the correspond-\ning equation for the dynamics of M2(see below Eq.(3)\nfor the full model) and one obtains a potentially rich cou-\npled dynamics for M1andM2. The situation simplifies\nin the limit where the exchange energy J→ −∞domi-\nnates: the relative configuration of M1andM2becomes\nfrozen in the anti-aligned position. Conservation of an-\ngular momentum now leads to a spin torque on the total\nmagnetic moment:\n∂\n∂t/bracketleftbiggd1M1\nγ1+d2M2\nγ2/bracketrightbigg\n=τττ1+... (2)\nwhere we have used the fact that τττ2=J1−J2vanishes\nwhenM1andM2are collinear. Note that the exchange\nfield, which conserves the total angular momentum, re-\ndistributes the torque τττ1onM1andM2so that they\nremain anti-aligned.\nEq.(2) allows two very different regimes: if\nd1|M1|/γ1> d2|M2|/γ2then the effective magnetiza-\ntion direction MSyFof the SyF layer points along M1\nand the torque is very similar to the weak coupling\nregime. This case has attracted some attention recently\nboth experimentally by Smith et al. [27] and theoreti-\ncally by Balaz et al. [28]. Here, we focus on the other3\nlimitd1|M1|/γ1< d2|M2|/γ2whereMSyFpoints in the\nopposite direction as M1(lower inset of Fig. 1). In\nthis limit, the torque tends to stabilize the configura-\ntion where MSyFis anti-aligned with M0hence where\nM0andM1are aligned: this is the opposite situation\nfrom the weak coupling J→0 limit.\nTo summarize, when going from J→0 toJ→ −∞,\nwithoutchangingthesignofthecurrent,thetorquetends\nto favor two different stationary situations. In the rest\nof this letter, we study this crossover in more details. In\nparticular, we find that the intermediate coupling regime\nreveals interesting, STO like, dynamical behavior.\nModel.We now turn to the modelisation of our sys-\ntem. The full stack we are considering has the form (see\nFig. 1a): AF |F0|N0|F1|N1|F2|N2, where AF is an antifer-\nromagnetic layer (typically IrMn) that pins the magne-\ntization of F 0. We set our reference spin axis ezparallel\ntoM0. Magnetization dynamics is described by two cou-\npled Landau Lifshiz Gilbert (LLG) equations that read\n(in SI units),\n∂m1\n∂t=γ1B1×m1+α1m1×∂m1\n∂t+γ1\n|M1|d1τττ1(3a)\n∂m2\n∂t=γ2B2×m2+α2m2×∂m2\n∂t+γ2\n|M2|d2τττ2,(3b)\nwheremi=Mi/|Mi|is the unit vector describing the\nmagnetization orientation of the layer Fiandαiis the\ndamping factor of the corresponding magnetic material.\nThe effective field Biseen by the magnetization in the\nlayer (i= (1,2)) is\nBi=Ki\nu\n|Mi|[mi·ez]ez+J\n|Mi|dim¯i,(4)\nwhere¯i= 3−icorresponds to the index of the other\nlayer. The effective field includes an uniaxial anisotropy\nKi\nufield and the RKKY exchange field. In the physical\nregimes studied below, RRKY coupling is the dominat-\ning energy so that our results are rather insensitive to\nthe presence of less important terms (for instance dipo-\nlar coupling between layers that we do not take into ac-\ncount). The spin torque τττiis calculated in the framework\nof CRMT (Continuous Random Matrix Theory) [16, 29].\nCRMT is a semiclassicalapproachthat generalizesValet-\nFert theory to non collinear situations. For discrete sys-\ntems, It is formally equivalent to the generalized circuit\ntheory [30] and is well adapted to the treatment of metal-\nlic magnetic multilayers.\nStrong coupling limit. When the RKKY coupling is\nlarge, one can derive an effective LLG equation for the\ndynamics of the SyF. It reads,\n∂m1\n∂t=γeffBeff×m1+αeffm1×∂m1\n∂t+γeff\n|M1|d1τττ1,(5)\nwhere the various effective parameters get renormalized\nas follows: γeff=γ1/(1−δ),αeff= (α1+δα2)/(1−δ),Keff\nu=K1\nu+K2\nud2/d1,Beff= (Keff\nu/|M1|)[m1·ez]ezand\nthe renormalization parameter δhas been defined as,\nδ=γ1|M2|d2\nγ2|M1|d1(6)\n(Eq.(5) is obtained by calculating ∂/∂t[m1+δm2] which\ncancels the RKKY contribution, and then setting m2=\n−m1andτττ2=0). From this equation we can clearly\nsee the inversion of the sign of the effective torque γeffτττ1\ndiscussed previously when δ >1.\nFigure 2: Upper: phase diagram of the system as a func-\ntion of the current density jand the thickness d1for a fixed\nδ= 4 and J=−5.10−3J.m−2. The various symbols indi-\ncate the observed stationary states in the corresponding re -\ngion (defined by different colors): ↑(↓) stands for mz\n1= 1\n(mz\n1=−1), while STOcorresponds to the presence a pre-\ncessional state. In the 2 STOregion, two different STO\nstates can be observed depending on the hysteresis history.\nLower: Resistance Ras function of time tfor a pillar surface\nof 1000 nm2andd1= 3nm. Upper (lower) plots correspond\nto initial conditions in the up (down) configuration. Left pa n-\nels:j= 2.8 107A.cm−2. Right panels: j= 5.8 107A.cm−2.\nThe amplitude of the oscillation with the up initial conditi on\nis small and invisible on the scale of the graphics\nNumerical simulations. Let us now go back to the full\nmodel Eq.(3) and study it numerically. We focus on a\nstack defined as IrMn 5|Py5|Cu10|Cod1|Ru0.3|Cod2|Cu10\nwhere the index indicate the widths of the layers in\nnm. We have used KCo\nu= 8.104J.m−3,MCo=\n1.42 106A.m−1and the transport parameters (spin re-\nsolved bulk resistivities, interface resistivities, spin flip\nlengths) have been taken from the database established\nin MSU and CNRS/Thales laboratory[31]. The coupling\nconstant Jcan be tuned by changing the Ruthickness\nwith typical values JRu≈ −5.10−3J.m−2[21]. The\ntorqueand resistanceshavebeen calculatedusing CRMT\nand the corresponding parametrization (as a function of\nthe angles θandα, see Fig.1a) incorporated into our nu-\nmerical LLG integrator.\nA first set of curve is presented in Fig. 1 b,c and d\nwhere the stationary magnetization mz\n1along the zaxis4\nis plotted against the current density jfor three values\nof the coupling constant: small (b), intermediate (c) and\nlarge coupling (d). For small coupling (b) , we recover\nthe usual behavior of current induced magnetic reversal\nin spin valves: at zero current the system has two sta-\nble configurations where m1is aligned ( mz\n1= 1) or anti\naligned ( mz\n1=−1) withm0. Upon injecting a strong\npositive current, one ”pushes” toward the anti-aligned\nconfiguration which becomes stable above a critical value\nof the current density j(0.5 107A.cm−2in this example).\nAt large coupling (d), this hysteretic curve is reversed, in\nagreement with the arguments developed above. A very\ninteresting regime is found at intermediate couplings (c)\nwhere one observes a rather large window where the sta-\ntionary value of mz\n1corresponds to a finite angle between\nm0andm1, i.e. to a dynamical state where a sustained\nprecession of m1is present (with frequencies around 15\nGHz in this example).\nThe coupling Jcan be varied by tuning the width dN\nof the normal (Ru) spacer. However, this is not very\neasy to control in practice, as the oscillatory character\nof the RKKY interaction make this variation non mono-\ntonic. Alternatively, one can explore the phase diagram\nof our stack by varying the thickness d1for a fixed ra-\ntioδ=d2/d1, hence varying the relative importance of\nbulk versus surface terms in the LLG equation. Fig. 2,\nshows the resulting phase diagram in the ( j,d1) plane for\nδ= 4. The different symbols indicate the possible sta-\ntionary states in the various regionsof the phase diagram\n(defined by the various colors). The presence of two dif-\nferent symbols (most regions)indicate that depending on\nthe hysteresis history, one or the other state is observed.\nIn particular, in the 2 STO region, two different STO\nstatescanbeobserveddepending, forinstance, ontheini-\ntial condition mz\n1(t= 0) =±1. These two different STO\nstates merge upon entering the 1 STO region. We have\ninvestigated other stacks with different material parame-\nters (not shown) and found very similar phase diagrams,\nincluding the presence of high frequency STO phases, as\nlong as one remains in the δ >1 regime. We observe two\nkind of STO behaviors, as evident from the Resistance\nversus time R(t) traces plotted in Fig. 2. When the am-\nplitude of the oscillating signal is small (lower left of Fig.\n2), we observe some beating behaviors with one well de-\nfined frequency and a second much smaller frequency less\nwell defined (corresponding to the precession of the two\nlayers respectively). When these two frequencies become\ncloser, the time dependent signal gets bigger and R(t)\nbecomes much more sinusoidal indicating phase locking\nbetween the precessional dynamics of the two magneti-\nzationsm1andm2.\nIn the last Fig.3, we focus on the region of the phase\ndiagram (lower right corner of Fig.2) where the ampli-\ntude of the STO signal is the highest. The upper curves\nshow the amplitude of the time dependent signal while\nthe lower curves show the corresponding frequency. Wefind that this large angle STO phase, which is stable in\nthe absence of magnetic field, leads to very high frequen-\ncies up to several tens of GHz while sustaining high pre-\ncession angles (of the order of π/6 ). The observed high\nfrequencies is tightly linked with the high value of the\nRRKY coupling which can be up to two orders of magni-\ntude larger than typical anisotropy or Zeeman energies.\nEveniftherelativeanglebetweenthetwomagnetizations\nof the synthetic ferrimagnet remain close to π, this gives\nrise to frequencies of several tens of GHz.\nToconclude, wehaveshownthatthe interplaybetween\nequilibriumandnonequilibriumtorqueinsyntheticferri-\nmagnets based spin valves can lead to interesting physics\nand potentially a new route to high frequency STOs.\nFigure 3: Upper plot: amplitude (in /permil) of the time depen-\ndent signal of the resistance as a function of d1for fixed\nj= 6.107A.cm−2. Lower plot: idem for the main oscillating\nfrequency of the resistance. Insets: corresponding colorp lots\nin the (j,d1) plane. δ= 4 and J=−5.10−3J.m−2.\nWe thank T. Rasing, W. Wulfhekel for very useful dis-\ncussions. This work was supported by , CEA NanoSim\nprogram, CEA Eurotalent and EC Contract Macalo.\n[1] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang,\nM. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80,\n4281 (1998).\n[2] E. Myers, D. Ralph, J. Katine, R. Louie, and\nR. Buhrman, Science 285, 867 (1999).\n[3] J. Katine, F. Albert, R. Buhrman, E. Myers, and\nD. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[4] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang,\nV. Tsoi, and P. Wyder, Nature 406, 46 (2000).\n[5] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).5\n[6] W. Rippard, M. Pufall, S. Kaka, S. Russek, and T. Silva,\nPhys. Rev. Lett 92, 027201 (2004).\n[7] I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kise-\nlev, D. C. Ralph, and R. A. Buhrman, Science 307, 228\n(2005).\n[8] O.Boulle, V.Cros, J.Grollier, L.G.Pereira, C.Deranlo t,\nF. Petroff, G. Faini, J. Barna, and A. 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Waintal, Phys. Rev. Lett. 103, 066602 (2009).\n[17] J. Manschot, A. Brataas, and G. Bauer, Phys. Rev. B\n69, 092407 (2004).\n[18] J. Barna´ s, A. Fert, M. Gmitra, I. Weymann, and\nV. Dugaev, Phys. Rev. B 72, 024426 (2005).\n[19] M. Gmitra and J. Barnas, Phys. Rev. Lett. 96, 207205\n(2006).\n[20] S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev.\nLett.64, 2304 (1990).\n[21] S. Parkin, Phys. Rev. Lett. 67, 3598 (1991).\n[22] P. Bruno and C. Chappert, Phys. Rev. B 46, 261 (1992).\n[23]j >0correspondstoinjectinganelectrons from theright.\n[24] J. Slonczewski, Phys. Rev. B 39, 6995 (1989).\n[25] X. Waintal and P. W. Brouwer, Phys. Rev. B 65\n054407(2002).\n[26] In the case considered here, the effect of τττ2will only\nslightly increase the predicted effect.\n[27] N. Smith, S. Maat, M. J. Carey, and J. R. Childress,\nPhys. Rev. Lett. 101, 247205 (2008).\n[28] P. Bal´ aˇ z and J. Barna´ s, Phys. Rev. B 83, 104422 (2011).\n[29] S. Borlenghi, V. S. Rychkov, C. Petitjean, and X. Wain-\ntal, Phys. Rev. B 84, 035412 (2011).\n[30] G. E. W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando,\nand A. Brataas, Phys. Rev. B 67, 094421 (2003).\n[31] H. Jaffres private communication."    },    {        "title": "1005.0480v1.Competition_between_Ferrimagnetism_and_Magnetic_Frustration_in_Zinc_Substituted_YBaFe4O7.pdf",        "content": " 1 Competition between Ferrimagnetism and Magnetic Frustration in Zinc \nSubstituted YBaFe 4O7 \n \nTapati Sarkar*, V. Pralong, V. Caignaert and B. Raveau  \n \n \nLaboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,  \n6 bd Maréchal Juin, 14050 CAEN, France  \n \nAbstract  \n \nThe substituti on of zinc for iron in YBaFe 4O7 has allowed the oxide series YBaFe 4-\nxZnxO7, with 0.40 \n  x \n 1.50, belonging to the “114” structural family to be synthesized. These \noxides crystallize in the hexagonal symmetry ( P63mc), as opposed to the cubic symmetry ( F-\n43m) of YBaFe 4O7. Importantly, the d.c. magnetization shows that the zinc substitution \ninduces ferrimagnetism, in contrast to the spin glass behaviour of YBaFe 4O7. Moreover, a.c. \nsusceptibility measurements demonstrate  that concomitantly these oxides exhibit  a spin glass \nor a cluster glass behaviour, which increases at the expense of ferrimagnetism, as the zinc \ncontent is increased. This competition between ferrimagnetism and magnetic frustration is \ninterpreted in terms of lifting of the geometric frustration , inducing the magnetic ordering , and \nof cationic disordering , which favours the glassy state.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n* Corresponding author: Dr. Tapati Sarkar  \ne-mail: tapati.sarkar @ensicaen.fr  \nFax: +33 2 31 95 16 00   \nTel:  +33 2 31 45 26 32  \n \n  2 Introduction  \n \nTransiti on metal oxides, involving a mixed valence of the transition element , present a \ngreat potential for the generation of strongly correlated electron systems. An extraordinary \nnumber of studies have been carried out in the last two decades on systems such as \nsuperconducting cupr ates, colossal magnetoresistive  manganites, and magnetic cobaltites, \nwhose oxygen lattice exhibits a “square” symmetry derived from the perovskite structure. In \ncontrast, the number of oxides with a triangular symmetry of the oxygen fra mework is much \nmore limited, if one excludes the large family of spinels which exhibit exceptional \nferrimagnetic properties and unique magnetic transitions  [1 – 3] as for Fe 3O4, but also magnetic \nfrustration [4 – 5] due to the peculiar triangular geometry of their framework as for  spinel \nferrite thin films.  \nThe recent discovery of the new series of cobaltites (Ln,Ca) 1BaCo 4O7 [6 – 8] and \nferrites (Ln,Ca) 1BaFe 4O7 [9 – 11], termed “114 ” oxides , have opened up a new field for the \ninvestigation of strongly corre lated electron systems. These oxides are closely related to spinels \nand barium hexaferrites, by their close packing of “O 4” and “BaO 3” layers. Importantly, they \ndiffer from all the other mixed valent transition metal oxides by the fact that cobalt or iron \nexists only in the tetrahedral coordination. Moreover, the symmetry of the structure may vary \naccording to the temperature as well as  the nature of the transition element, cobalt or iron, from \nhexagonal to cubic or orthorhombic. This change of symmetry see ms to dramatically  influence  \nthe magnetic properties, ranging from spin glass behaviour for cubic LnBaFe 4O7 oxides [10 – \n11] to ferrimagnetism for orthorhombic CaBaCo 4O7 [8] and hexagonal CaBaFe 4O7 [9] oxides. \nVery complex magnetic transitions are observed , as exemplified for YBaCo 4O7, for which a \nspin glass behaviour was first reported around 66 K  [7], whereas long range magnetic order \nwas observed below T N = 110 K [12], and a magnetic transition with short range correlations \nwas revealed above T N [13]. M oreover, such magnetic transitions are often induced by \nstructural transitions (T S), as shown by the lifting of the magn etic frustration at T S in \nLnBaCo 4O7 oxides [6, 7, 14].  \nThe remarkable feature of the “114” ferrites is the existence of the cubic symme try of \nLnBaFe 4O7 oxides [10 – 11] with Ln = Dy, Ho, Er, Yb, Lu, which has never been observed in \nthe cobaltite series, whereas the hexagonal symmetry is observed for both types of oxides, \nLnBaCo 4O7 [6 – 7] oxides whatever Ln, as well as  CaBaFe 4O7 [9] and L nBaFe 4O7 [11] with \nLn = Tb, Gd. Understanding the relative stability of these two structural types of ferrites is of \ngreat importance since it appears that it governs their magnetic properties, leading to spin glass  3 behaviour and ferrimagnetism for cubic a nd hexagonal LnBaFe 4O7 oxides respectively [11]. In \nfact, the two structures are closely related, i.e. built up of an ordered 1:1 stacking of identical \ntetrahedral layers, i.e. kagomé (K) and triangular (T) layers. The cubic structure ( Fig. 1(a) ) can \nthen be deduced from the hexagonal one ( Fig. 1 (b) ) by translation of one T layer out of two by \n2a\n. Bearing in mind that the structural transition from cubic to hexagonal symmetry is very \nsensitive to geometric factors, as shown from the ef fect of the size of Ln3+ cations [11], we \nhave explored the effects of cationic substitutions upon the structural and magnetic properties \nof the cubic spin glass YBaFe 4O7. In this stud y, we show that the substitution of Zn2+ for Fe2+ \nin this phase allows t he hexagonal symmetry to be stabilized for the series YBaFe 4-xZnxO7 with \n0.40 \n  x \n 1.50. Moreover, it is observed that this substitution by a diamagnetic cation induces \na ferrimagnetic ground state, in agreement with the change of symmetry. However, since  Zn2+ \nis a diamagnetic cation, it does not participate in the magnetic interactions , but rather dilutes \nthe magnetic sublattice and introduces cationic disordering, which in turn is responsible for the \nappearance of magnetic frustration. The combined measu rements of a.c. and d.c. magnetic \nsusceptibility throw light on the competition between ferrimagnetic interactions and magnetic \nfrustration in this series, suggesting a possible phase separation.  \n \nExperimental  \n \nPhase -pure samples of YBaFe 4-xZnxO7  [x = 0. 40 – 1.50] were prepared by solid state \nreaction technique. The precursors used were Y 2O3, BaFe 2O4, ZnO, Fe 2O3 and metallic Fe \npowder. First, the precursor BaFe 2O4 was prepared from a stoichiometric mixture of BaCO 3 \nand Fe 2O3 annealed at 1200°C for 12 hrs in air. In a second step, a stoichiometric mixture of \nY2O3, BaFe 2O4, ZnO, Fe 2O3 and metallic Fe powder was intimately ground and pressed in the \nform of rectangular bars. The bars were then kept in an alumina finger, sealed in silica tubes \nunder vacuum and annealed at 1100°C for 12 hrs. Finally, the samples were cooled to room \ntemperature at a rate of 200°C / hr.  \nThe X -ray diffraction patterns were registered with a Panalytical X’Pert Pro \ndiffractometer under a continuous scanning mode in the 2\n  range 10° - 120° and step size \n 2\n \n= 0.017°. The d.c. magnetization measurements were performed using a superconducting \nquantum interference device (SQUID) magnetometer with variable temperature cryostat \n(Quantum Design, San Diego, USA). The a.c. susceptibility, \n ac(T) was measured with a \nPPMS from Quantum Design with the frequency ranging from 10 Hz to 10 kHz (H dc = 0 Oe  4 and H ac = 10 Oe). All the magnetic properties were registered on dense ceramic bars of \ndimensions ~ 4 \n  2 \n 2 mm3. \n \nResults and discussion  \n \nFor the above synthesis conditions, a monophasic system was obtained for YBaFe 4-\nxZnxO7 with 0.40 \n  x \n 1.50. The EDS analysis of the samples ( Table 1 ) allowed the cationic \ncompositions to be confirmed.  \nStructural characterization  \nThe X -ray diffraction (XRD) patte rns show that all the samples with 0.40 \n  x \n 1.50 \nexhibit the hexagonal symmetry, as exemplified for the two extreme members of the series, x = \n0.40 and x = 1.50 ( Fig. 2 ), whose fits have been achieved by  Rietveld analysis using the \nFULLPROF refinement p rogram [15]. The extracted cell parameters ( Table 1 ) show that “ a” \nincreases slightly as x increases, whereas “ c” decreases so that the cell volume does not vary \nsignificantly, in agreement with the size of Zn2+ (\n2Znr\n  0.60 Å) being  very similar to that of \nFe2+ (\n2Fer\n  0.63 Å). Thus, the stabilization of the hexagonal phase with respect to the cubic \none is probably not due to the smaller size of the Zn2+ cation . The rather different distortion of \nthe ZnO 4 tetrahedra comp ared to the FeIIO4 tetrahedra may be at the origin of this structural \ntransition allowing a different tuning of the kagomé and triangular layers in the hexagonal \nphase compared to the cubic one.  \nA prefer ential occupancy of one of the  two different kinds of  sites by Zn2+ in the \nhexagonal structure is possible, but cannot be detected here from XRD data  due to the very \nclose values of the scattering factors of Zn and Fe. In any case, the c / a ratio of the hexagonal \ncell of the series YBaFe 4-xZnxO7 (Table 1 ) clearly shows that the introduction of zinc in the \nstructure tends to destroy the close packing of the “BaO 3” and “O4” layers, leading to a c / a \nvalue significantly larger than the ideal value for a perfect close packed structure (1.6333),  and \nin the same way ruling out a possible cubic close packed symmetry. Note, however, that the \ndecrease of c / a ratio as the zinc content increases is  not yet understood.  \nIt is also quite remarkable that attempts to synthesize phase pure samples for 0 < x < \n0.40 were uns uccessful , instead leading to a mixture of the cubic and hexagonal oxides. This \nobservation is important, since it demonstrates that in this region of cross -over from cubic to \nhexagonal  symmetry , no cubic solid solution can be obtained , confirming that Zn2+ cannot \naccommodate the unique tetrahedral site of the cubic structure of YBaFe 4O7, most likely due to \nthe different distortion of ZnO 4 tetrahedra compared to FeO 4 tetrahedra.   5  \nD. C. magnetization studies  \nThe temperature dependence of d.c. magnetization w as registered according to the \nstandard zero field cooled (ZFC) and field cooled (FC) procedures. A magnetic field of 0.3 T \nwas applied during the measurements. The measurements were done in a temperature range of \n5 K to 400 K. The M ZFC(T) and M FC(T) curve s of all the samples are shown in Fig. 3 . The \nhighest value of the magnetization was observed for the sample with the lowest zinc content, x \n= 0.40. For this sample, the magnetization increases sharply as the sample is cooled below 150 \nK, as is seen from t he FC and ZFC curves. The FC curve shows saturation at about 0.78 \n B/f.u. \nat 5 K, with the typical shape of ferro (or ferri) magnetic behaviour, whereas the ZFC curve \nexhibits a maximum at ~ 80 K  reflecting the existence of strong irreversibility. This beh aviour \nis in contrast to the spin glass behaviour of YBaFe 4O7 [10]. Rather, i t is comparable to \nCaBaFe 4O7 which was found to be ferrimagnetic with a hexagonal symmetry. Nevertheless, \nthe T C value of YBaFe 3.6Zn0.4O7 is signi ficantly smaller than the value  obtained for CaBaFe 4O7 \n(270 K). Also, the Curie -Weiss temperature (\n CW) for YBaFe 3.6Zn0.4O7, obtained by \nextrapolation of the linear \n-1(T) region, is 63.7 K. This is substantially lower than the value \nthat was obtained for CaBaFe 4O7 (268.3 K).  This sugges ts that the dilution introduced by the \ndiamagnetic cation Zn2+ weakens the ferrimagnetic interaction in this system, and the final \nground state of the oxide may not be a simple ferrimagnet, but a more complex system \ninvolving magnetic frustration.  \nWith an increase in the substitution level (x), the value of the magnetization attained by \nthe sample at the lowest measured temperature of 5 K decreases monotonically. (This decrease \nis shown in the inset of Fig. 3 .) This feature is again a reflection of the fact  that the Zn2+ ions \ndo not carry any magnetic moment, and thus, serve to dilute the ferrimagnetic coupling. This \ndilution of the ferrimagnetic interaction is also seen in the fact that the temperature below \nwhich the magnetization starts rising, as well as  the sharpness of the transition keeps \ndecreasing with an increase in the zinc content (x)  (Fig. 3 ). This has been shown quantitatively \nin Fig. 4 , where we have plotted the first derivative of magnetization (dM/dT) versus \ntemperature curves. The inflection  points of the curves (T inf) can be roughly considered to be \nthe temperatures at which the paramagnetic to ferrimagnetic transition takes place. (Later on \nwe will show that the ground state obtained at low temperature in these oxides is not a simple \nferrim agnet. Instead, it is rather complex , as will become clear from the a.c. magnetic  6 susceptibility measurements, which we detail in the next section ). Inset (a) of Fig. 4  shows the  \nmonotonic decrease of T inf with increase in the substitution level (x).  \nIn in set (b) of Fig. 4 , we show the variation of \n T1/2 with the doping concentration (x). \nT1/2 is the full width at half maximum (F.W.H.M.) of the dM/dT vs T  curves. It gives a \nmeasure of the width of the transition (a lower value of \n T1/2 indicates a sharper transition). As \nis seen in the figure, the width of the transition shows a systematic increase with an increase in \nthe substitution level (x), thereby indicating the dilution of the ferrimagnetic interactions.  \nIn view of the observations on the irreversibi lity of the FC and ZFC M(T) curves seen \nat low temperature, detailed investigations were carried out on the d.c. magnetization M(H) \ncurves of the various samples at 5 K. The isothermal magnetization curves recorded at 5 K \n(Fig. 5 ) show that the x = 0.40 sa mple exhibits the largest hysteresis loop i.e. the largest values \nof the coercive field (HC ~ 1 T) and of the remanent magnetization (M r ~ 0.8 \n B/f.u.), \nindicating that YBaFe 3.6Zn0.4O7 is, like CaBaFe 4O7, a hard ferrimagnet. Importantly, the shape \nof the M (H) loop of this phase is much smoother than for CaBaFe 4O7, and rather similar to that \nobserved for CaBaFe 4-xLixO7 oxides, that were shown to be glassy ferrimagnets [16]. These \nsamples show similar magnetic hysteresis loops, characterized by  a lack of magn etic \nsaturation. Moreover, one observes that the coercive field H C (inset (a) of Fig. 5 ) and the \nremanent magnetization M r (inset (b) of Fig. 5 ) exhibit a monotonic decrease as the \nsubstitution level increases.  \nThe lack of magnetic saturation in these samp les can be quantified by taking the ratio of \nthe value of the magnetization attained at a field of 5 T and M r [M (H = 5 T) / M r]. This is \nshown in Fig. 6  (a). As can be seen from the figure, the quantity M (H = 5 T) / M r increases by \nnearly 5 times as the doping concentration is increased from x = 0.40 to x = 1.50. This again \nindicates the strong weakening of the ferrimagnetic interaction with increasing zinc \nconcentration.  The spontaneous magnetization (M at H = 0, as read out from the virgin curve) \nshows a monotonic decrease with an increase in the substitution level (x) ( Fig. 6 (b) ). As x \nincreases from 0.40 to 1.50, the spontaneous magnetization decreases by 4 orders.  \nThus, these results suggest that in these  hexagonal oxides, ferrimagnetism competes \nwith magnetic frustration due to the triangular geometry of the kagomé layers. Such a \nstatement is also supported by the recent study of the isostructural “114” hexagonal cobaltite \nYBaCo 4O7 [13] which shows that 1 D magnetic ordering competes with 2 D magneti c \nfrustration.  \n  7 A. C. magnetic susceptibility studies  \nThe measurements of the a.c. magnetic susceptibility \n ac(T,f) were performed in zero \nmagnetic field (H dc = 0) at different frequencies ranging from 10 Hz to 10000  Hz, using a \nPPMS facility. The amplitu de of the a.c. magnetic field was ~ 10 Oe. The evolution of the real \npart \n(T) and the imaginary part \n (T) of the susceptibility versus temperature at different \nfrequencies for the two end members of the series, x = 0.40 ( Fig. 7 ) and x = 1.50 ( Fig. 8 ) sheds \nlight on the complex nature of the magnetic ground state of these oxides. The \n (T) curves \n(Fig. 7 (a)  and Fig. 8 (a) ) exhibit a maximum at T max(a.c.) which is slightly higher than the peak \nvalues  obtained from d.c. magnetic measurements  (Tmax(d.c.) ) i.e. T max(a.c.)  = 10 5 K at 10 Hz \n(Fig. 7  (a) and Table 2 ) against T max(d.c.)  = 83 K (Fig. 3) for the x = 0.40 sample, and T max(a.c.) = \n45 K  (Fig. 8 (a)  and Table 2 ) against T max(d.c.)  = 36  K for the x = 1.50 sample.  Both samples \nshow frequency dependent pe aks in the a.c. susceptibility measurements i.e. as the frequency is \nincreased, T max(a.c.) shifts to higher values associated with a decrease in the magnitude of \n ac. \nThis behaviour strongly suggests a spin glass [17] or superparamagnetic behaviour, which  will \nbe discussed below. The peak observed in the real part (in -phase component) of the magnetic \nsusceptibility (\n ac(T)) at a frequency of 10 Hz decreases from 10 5 K to 45 K as the doping \nconcentration is increased from x = 0.4 0 to x = 1.5 0. \nThe plot of the imaginary part of the a.c. susceptibility ( Fig. 7 (b)  and Fig. 8 (b) ) show \nthat the maximum of \n ac(T) appears at a lower temperature than the maximum of \n ac(T), and \nis also frequency dependent. We note here that for the x = 0.40 sample, the magnitud e of the \npeak in \n ac is largest for the lowest frequency measured (11 Hz). On the other hand, for t he x \n= 1.5 sample , the magnitude of the peak in \n ac is largest for the highest frequency measured \n(10 kHz). This indicates that the inverse mean relaxati on time of the spin system for the x = \n0.40 sample is lower than that for the x = 1.50 sample. This is in accordance with the values of \nthe spin relaxation time \n 0 obtained later from the fitting of the a.c. susceptibility data  (\n0(x = 0.40)  \n> \n 0(x = 1.50) )  (see Table 2).  \nTo check the existence of a spin glass behaviour or of cluster glass behaviour, we have \nanalyzed the frequency dependence of the peak in \n ac(T) using the method previously  \ndeveloped by Bréard  et.al. [18], starting from the dynamic scalin g theory [17] which predicts a \npower law of the form \nz\nSGSG f\nTTT\n0 , where, \n 0 is the shortest relaxation time available \nto the system, TSG is the underlying spin -glass transition temperature determined by the \ninteractions in the system, z is the dynamic critical exponent and \n  is the critical exponent of  8 the correlation length. The actual fittings were done using the equivalent form of the power \nlaw: \nSGSG f\nTT Tzln ln ln0\n . The remarkable linearity obtained for the plots of \nln\nversus \nSGSG f\nTTTln  using the fit parameters , shown in Fig. 9 , indicate  that these samples \nwell obey the behaviour expected for a spin glass like system. Similar fittings were performed \nfor all the other samples of the series, and the extracted  parameters of this study have been \nlisted in Table 2 .  \nThe parameter \nfTT\npff\nlog\n  is known to  vary from 0.004 to 0.02 for canonical  spin \nglass systems. The value of p that we obtain  for the two end members of the series under \ninvestigation are p = 0.015 (for the x = 0.40 sample) and p = 0.020 (for the x = 1.50 sample). \n[Note: For superparamagnetic materials, the value of p is higher (~ 0.1), and so we discard the \npossibility of these oxides being superparamagnetic.]  In spite of the fact that the  value of p is \nquite similar for the two end members of the series, the fact that the shape of the a.c. \nsusceptibility versus temperature is quite different in the two cases gets reflected in the values \nof the parameters \n 0 and z\n. In fact, for the lower z inc concentrations (x = 0.40 to x = 0.90), the \nspin relaxation time \n 0 lies in the range of values typical of canonical spin glasses (10-10 – 10-12 \nsec) [19 – 21], and the z\n values lie between 6 and 10. However, as the zinc concentration is \nincreased beyo nd x = 1, the \n 0 values are much smaller, and the z\n values obtained are above \n10.  Thus, the power law model provides a satisfactory description of our a.c. susceptibility \ndata. \n \nConclusion  \nThe present study of the zinc substituted “114” oxides YBaFe 4-xZnxO7 shows that the \nsubstitution of zinc for iron in the YBaFe 4O7 oxide induces a ferrimagnetic behaviour, in spite \nof the diamagnetic character of Zn2+, but that these magnetic interactions compete with a spin \nglass or a cluster glass behaviour as the zinc  content is further increased.  In fact, the shape of \nthe M -T curve obtained for this series of oxides (sharp decrease in the ZFC  curve at lower \ntemperatures with  saturation of FC curve) is very similar to that reported for various alloys \nshowing mictomagne tism[22,23,24] , i.e. coexistence of frustrated spin clusters and \nferrimagnetic spin region s. This competition between ferrimagnetism and magnetic frustration  9 can be understood by considering the structure, or more exactly the iron lattice of these oxides. \nThe cubic structure of YBaFe 4O7 (Fig. 1 (a) ) forms a tetrahedral [Fe 4]\n lattice of Fe 4 tetrahedra \nshari ng apices ( Fig. 10  (a)), very similar to the [Ln 4]\n lattice observed in pyrochlores. As \npreviously shown for these compounds, this tetrahedral framework exhibits a complete 3 D \ngeom etric frustration and consequen tly, one observes, like for the pyrochlores [2 5], a spin glass \nbehaviour [10].  As soon as a part of Zn2+ is substituted for Fe2+, i.e., for YBaFe 3.6Zn0.4O7 (x = \n0.40), the structure becomes hexagon al (Fig. 1 (b) ), forming a different [Fe 8]\n lattice ( Fig. 10  \n(b)), built up of rows of corner – sharing “Fe 5” trigonal bipyramids running along “ c”, and \ninterconnected through “Fe 3” triangles in the “ a b” plane. As a consequence, the zinc \nsubstitution indu ces a lifting of the geometric frustration, favouring the appearance of magnetic \nordering, as previously observed for hexagonal CaBaFe 4O7 which is known to be ferrimagnetic \n[9]. In fact, such a hexagonal framework could be regarded as composed of ferri (or  ferro) \nmagnetic rows running along “ c” formed by the “Fe 5” bipyramids, whereas in the “ a b” plane, \nthe triangular geometry of the iron lattice is maintained so that a bidimensional magnetic \nfrustration is still possible, and may compete with the 1 D magne tic ordering along “ c”. \nHowever, t he presence of zinc on the iron sites tends to weaken the magnetic interactions due \nto its diamagnetic character by dilution effect , and the cationic disordering on the Fe / Zn sites \nfavours the spin glass or cluster glass  behaviour. As a consequence the ferrimagnetic behaviour \ndecreases at the benefit of the spin glass or cluster glass behaviour as the zinc content increases \nbeyond x = 0.40.  \nA neutron diffraction study of these glassy ferrimagnets will be necessary to det ermine \nthe exact distribution of zinc on the two different sites of the structure and to understand the \ncomplex nature of the magnetic states that arise from the competition between ferrimagnetism \nand magnetic frustration. The possibility of phase separati on should also be considered.  \n \nAcknowledgements  \n \nWe gratefully acknowledge Dr. Vincent Hardy, CRIS MAT, ENSICAEN, France for \nhelpful discussions. We also acknowledge the CNRS and the Conseil Regional of Basse \nNormandie for financial support in the frame of Emergence Program . \n \n \n \n \n  10 References  : \n      [1]    Friedrich Walz, J. Phys.: Condens. Matter ., 2002 , 14 , R285  \n      [2]     E. J. W. Verwey and P. W. Haayman, Physica , 1941 , 8, 979  \n      [3]    J.B. Goodenough, Magnetism and the Chemical Bond , Interscience  Monographs on  \n            Chemistry  vol. 1 , Huntington, New -York (1976)  \n      [4]     Yuji Muraoka, Hitoshi Tabata and Tomoji Kawai, J. Appl. Phys ., 2000 , 88, 7223  \n      [5]     G. E. Delgado, V. Sagredo and F. Bolzoni, Cryst. Res. Technol . 2008 , 43, 141  \n      [6]     Martin Valldor and Magnus Andersson, Solid State Sciences , 2002 , 4, 923  \n      [7]     Martin Valldor, J. Phys.: Condens. Matter ., 2004 , 16, 9209  \n      [8]     V. Caignaert, V. Pralong, A. Maignan and B. Raveau, Solid State Communications ,  \n             2009 , 149, 453  \n      [9]     B. Raveau, V. Caignaert, V. Pralong, D. Pelloquin and A. Maignan, Chem. Mater .,  \n             2008 , 20, 6295  \n      [10]  V. Caignaert, A. M. Abakumov, D. Pelloquin, V. Pralong, A. Maignan, G. Van  \n             Tendel oo and B. Raveau, Chem. Mater ., 2009 , 21, 1116  \n      [11]  V. Pralong, V. Caignaert, A. Maignan and B. Raveau, J. Mater. Chem ., 2009 , 19,  \n              8335  \n      [12]   L. C. Chapon, P. G. Radaelli, H. Zheng and J. F. Mitchell, Phys. Rev. B , 2006 , 74,  \n              172401  \n      [13]   P. Manuel, L. C. Chapon, P. G. Radaelli, H. Zheng and J. F. Mitchell, Phys. Rev.  \n              Lett., 2009 , 103, 037202  \n      [14]    A. Maignan, V. Caignaert, D. Pelloquin, S. Hébert, V. Pralong, J. Hejtmanek and D.  \n              Khomskii, Phys. Rev. B , 2006 , 74, 165110  \n      [15]  J. Rodriguez -Carvajal, An Introduction to the Program FULLPROF 2000; Laboratoire  \n              Léon Brillouin, CEA -CNRS: Saclay, France (2001)  \n      [16]    K. Vijayanandhini, Ch. Simon, V. Pralong, V. Caignaert and B. Raveau, Phys. Rev.  \n             B, 2009 , 79, 224407  \n      [17]     P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys ., 1977 , 49, 435  \n      [18]     Y. Bréard, V. Hardy , B. Raveau, A. Maignan, H -J. Lin, L -Y. Jang, H. H. Hsieh and   \n              C.  T. Chen, J. Phys.: Condens. Matter, 2007 , 19, 216212  \n      [19]     J. Souletie and J. L. Tholence, Phys. Rev. B , 1985 , 32, 516  \n      [20]     K. Gunnarsson, P. Svedlindh, P. Nordblad, L. Lundgren, H. Aruga and A. Ito, Phys.  \n             Rev. Lett ., 1988 , 61, 754  \n      [21]    K. H. Fischer and J. A. Hertz, Spin Glasses , Cambridge University Press,   11                Cambridge, 1993  \n      [22]     Yuntao Wang, Huaiyu Shao, Xingguo Li, Lefu Zhang and Seiki Takahashi, J. \n             Magn. M agn. Mater. , 2004 , 284, 13 \n      [23]      N Saito, H Hiroyoshi, K Fukamichi and Y Nakagawa, J. Phys. F: Met. Phys.  1986 ,  \n            16, 911  \n      [24]      Hidetoshi Hiroyoshi, Kou Noguchi, Kazuaki Fukamichi and Yasuaki Nakagawa, J. \n           Phys. Soc . Japan , 1985 , 54, 3554  \n      [25]        John E. Greedan, Journal of Alloys and Compounds , 2006 , 408 – 412, 444  \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  12 Table caption s \n \nTable 1 : Refined crystal structure parameters as obtained from the Rietveld refinement of X -\nray powde r diffraction data.  \nTable 2 : Parameters extracted from fitting of a.c. magnetic susceptibility data.   \n \nFigure captions  \n \nFigure 1:  Relative positions of the T layers in (a) cubic YBaFe 4O7 and (b) hexagonal \nCaBaFe 4O7 (adapted from Ref. 10). For details, see text. \nFigure 2:  X-ray diffraction patterns along with the fits for (a) YBaFe 3.6Zn0.4O7 and (b) \nYBaFe 2.5Zn1.5O7. \nFigure 3:  MZFC(T) and M FC(T) curves of YBaFe 4-xZnxO7 measured at H = 0.3 T. The empty \nsymbols are for MZFC(T) and the solid symbols are for MFC(T). The inset shows the variation of \nMFC at T = 5 K with the substitution level (x).  \nFigure 4 : dM/dT vs T for YBaFe 4-xZnxO7. Inset (a) shows the variation of T inf with x, inset (b) \nshows the variation of \n T1/2 with x (see text for details).  \nFigure 5:  M (H)  for YBaFe 4-xZnxO7 at T = 5 K . Inset (a) shows the variation of HC with x, \ninset (b) shows the variation of M r with x.  \nFigure 6:  (a) M (H = 5 T) / M r vs x, (b) M (H = 0 T) vs x for YBaFe 4-xZnxO7. \nFigure 7 : (a) Real (in -phase) and (b) Imaginary (out -of-phase) component of a.c. \nsusceptibilities  for YBaFe 3.6Zn0.4O7 as a function of temperature measured using a frequency \nrange 10 Hz – 10 kHz.  \nFigure 8 : (a) Real (in -phase) and (b) Imaginary (out -of-phase) component of a.c. \nsusceptibilities  for YBaFe 2.5Zn1.5O7 as a function of temperature measured using a frequency \nrange 10 Hz – 10 kHz.  \nFigure 9 : Plot of ln \n  vs \nSGSG f\nTTTln  for (a) YBaFe 3.6Zn0.4O7 and (b) YBaFe 2.5Zn1.5O7 (for \ndetails, see text).  \nFigure 10 : Schematic representation of (a) cubic YBaFe 4O7 and (b) hexagonal CaBaFe 4O7 \n(adapted from Ref. 11). For details, see text.  \n \n \n  13 Table 1  \nDoping \nconcentration \n(x)  \nCrystal \nsystem \n(Space \ngroup)  \n  \nUnit cell parameters  \n c / a Reliability \nfactor (R F) x as obtained \nfrom EDS \nanalysis  a (Å) c (Å) \n0.40 Hexa gonal \n(P63mc)  \n6.317(2)  \n 10.380(2)  1.6432  3.11 0.36(0.02)  \n0.50 Hexagonal \n(P63mc)  \n6.319(2)  \n 10.379(2)  1.6425  8.31 0.53(0.03)  \n0.60 Hexagonal \n(P63mc)  \n6.321(1)  \n 10.377(1)  1.6417  5.30 0.65(0.04)  \n0.80 Hexagonal \n(P63mc)  \n6.321 (2) \n 10.371 (2) 1.6407  9.28 0.80(0.04)  \n0.90 Hexagonal \n(P63mc)  \n6.323 (1) \n 10.367 (1) 1.6396  7.97 0.90(0.04)  \n1.25 Hexagonal \n(P63mc)  \n6.325 (3) \n 10.355 (3) 1.6372  8.31 1.27(0.03)  \n1.50 Hexagonal \n(P63mc)  \n6.328 (1) \n 10.348 (1) 1.6353  8.24 1.56(0.03)  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  14 Table 2   \nDoping \nconcentration \n(x) Tmax  from \n(T) at f = \n10 Hz (K)  p Spin relaxation time \n 0 \n(sec)   \nSpin glass \ntransition \ntemperature \nTSG (K) \n z\n \n \n0.40 \n 104.5 0.015  (1.95 \n  0.04) \n  10-10 100.44 (7)  6.2 \n1) \n \n0.50 \n 95.4 0.014  (8.14 \n  0.04) \n  10-11 92.30 (3)  6.6 \n2) \n \n0.60 \n 91.8 0.013  (3.09 \n  0.08) \n  10-11 88.04 (2)  6.9 \n2) \n \n0.80 \n 79.8 0.011  (1.11 \n  0.04) \n  10-11 76.05 (2)  7. 3 \n 1) \n \n0.90 \n 75.8 0.011  (2.93 \n  0.01) \n  10-12 72.96 (6)  7.4 \n1) \n \n1.25 \n 56.0 0.015 (8.75 \n  0.09) \n  10-13 54.26 (2)  10.8 \n2) \n \n1.50 \n 44.8 0.020  (3.75  \n 0.04) \n 10-15 40.63 (6)  13.6 \n4) \n \n \n \n \n \n \n \n \n \n \n \n \n \n  15 \n \n \nFigure 1  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  16 \n \nFigure 2  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  17 \n \nFigure 3  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  18 \n \nFigure 4  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  19 \n \nFigure 5  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  20 \n \nFigure 6  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  21 \n \nFigure 7  \n \n \n \n \n \n \n \n  22 \n \nFigure 8 \n \n \n \n \n \n  23 \n \nFigure 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  24 \n \n \nFigure 10  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n "    },    {        "title": "2205.13802v3.Magnonic_Casimir_Effect_in_Ferrimagnets.pdf",        "content": "Magnonic Casimir Effect in Ferrimagnets\nKouki Nakata1,\u0003and Kei Suzuki1,y\n1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan\n(Dated: March 2, 2023)\nQuantum fluctuations are the key concepts of quantum mechanics. Quantum fluctuations of\nquantum fields induce a zero-point energy shift under spatial boundary conditions. This quantum\nphenomenon, called the Casimir effect, has been attracting much attention beyond the hierarchy\nof energy scales, ranging from elementary particle physics to condensed matter physics together\nwith photonics. However, the application of the Casimir effect to spintronics has not yet been in-\nvestigated enough, particularly to ferrimagnetic thin films, although yttrium iron garnet (YIG) is\none of the best platforms for spintronics. Here we fill this gap. Using the lattice field theory, we\ninvestigate the Casimir effect induced by quantum fields for magnons in insulating magnets and find\nthat the magnonic Casimir effect can arise not only in antiferromagnets but also in ferrimagnets\nincluding YIG thin films. Our result suggests that YIG, the key ingredient of magnon-based spin-\ntronics, can serve also as a promising platform for manipulating and utilizing Casimir effects, called\nCasimir engineering. Microfabrication technology can control the thickness of thin films and realize\nthe manipulation of the magnonic Casimir effect. Thus, we pave the way for magnonic Casimir\nengineering.\nIntroduction. —Toward efficient transmission of infor-\nmation that goes beyond what is offered by conven-\ntionalelectronics, thelasttwodecadeshaveseenasignifi-\ncant development of magnon-based spintronics [1], called\nmagnonics. The main aim of this research field is to use\nthe quantized spin waves, magnons, as a carrier of in-\nformation in units of the reduced Planck constant ~. A\npromising strategy for this holy grail is to explore insu-\nlating magnets. Thanks to the complete absence of any\nconducting metallic elements, insulating magnets are free\nfrom drawbacks of conventional electronics, such as sub-\nstantial energy loss due to Joule heating. This is the\nadvantage of insulating magnets. Thus, exploring quan-\ntum functionalities of magnons in insulating magnets is\na central task in the field of magnonics.\nQuantum fluctuations of photon fields induce a zero-\npoint energy shift, called the Casimir energy, under spa-\ntial boundary conditions. This Casimir effect is a funda-\nmentalphenomenonofquantumphysics, andtheoriginal\nplatform for the Casimir effect was the photon field [2–\n4], which is described by quantum electrodynamics. The\nconcept can be extended to various fields such as scalar,\nvector, tensor, and spinor fields. Nowadays, the Casimir\neffect has been attracting much attention beyond the hi-\nerarchyofenergyscales, rangingfromelementaryparticle\nphysics to condensed matter physics [5, 6]. As an exam-\nple, see Refs. [7–14] for Casimir effects in magnets [15].\nHowever, the application of the Casimir effect to spin-\ntronics has not yet been studied enough, particularly to\nferrimagnetic thin films (see Fig. 1), although yttrium\niron garnet (YIG) [16] has been playing a central role in\nspintronics.\nHere we fill this gap. In terms of the lattice field the-\nory, we investigate the Casimir effect induced by quan-\n\u0003nakata.koki@jaea.go.jp; equal contribution.\nyk.suzuki.2010@th.phys.titech.ac.jp; equal contribution.\nFIG. 1. Schematic of the ferrimagnetic thin film for the\nmagnonic Casimir effect. Two kinds of magnons (circles) arise\nfrom the alternating structure of up and down spins (arrowed\nlines). Wavy lines represent quantum-mechanical behaviors\nof magnons in the discrete energy.\ntum fields for magnons in insulating magnets and refer to\nit as the magnonic Casimir effect. We study the behav-\nior of the magnonic Casimir effect with a particular fo-\ncus on the thickness dependence of thin films, which can\nbeexperimentallycontrolledbymicrofabricationtechnol-\nogy [17, 18]. Then, we show that the magnonic Casimir\neffect can arise not only in antiferromagnets (AFMs) but\nalso in ferrimagnets with realistic model parameters for\nYIG thin films. Our study indicates that YIG, an ideal\nplatform for magnonics, can serve also as a key ingredi-\nent of Casimir engineering [19] which aims at exploring\nquantum-mechanical functionalities of nanoscale devices.\nAntiferromagnets. —We consider insulating AFMs de-\nscribed by the quantum Heisenberg Hamiltonian which\nhas U (1)symmetry about the quantization axis andarXiv:2205.13802v3  [quant-ph]  1 Mar 20232\nstudy the behavior of the magnonic Casimir effect with\na focus on the thickness dependence. The AFM is a\ntwo-sublattice system, and the ground state has the Néel\nmagnetic order [20]. From the spin-wave theory with the\nBogoliubov transformation, elementary excitations are\ntwo kinds of magnons designated by the index \u001b=\u0006\nhaving the spin angular momentum \u001b~. Owing to the\nU(1)symmetry, the Hamiltonian can be recast into the\ndiagonal form with the magnon energy dispersion for the\nwave number k= (kx;ky;kz)as\u000f\u001b;kwhere the total spin\nangular momentum of magnons is conserved. Two kinds\nof magnons ( \u001b=\u0006) are in degenerate states. Hence, we\nstudy the low-energy magnon dynamics of the insulat-\ning AFM by using the quantum field theory of complex\nscalar fields, i.e., the complex Klein-Gordon field the-\nory [21, 22]. Then, we can see that there exists a zero-\npoint energy [23]. This is the origin of the Casimir effect.\nNote that throughout this study, we focus on clean insu-\nlating magnets and work under the assumption that the\ntotal spin along the quantization direction is conserved\nand thus remains a good quantum number.\nThrough the lattice regularization, the Casimir energy\nECasis defined as the difference between the zero-point\nenergyEint\n0for the continuous energy \u000f\u001b;kand the one\nEsum\n0for the discrete energy \u000f\u001b;k;nwithn2Z. In two-\nsublattice systems, such as AFMs and ferrimagnets (see\nFig. 1), the wave numbers on the lattice are replaced by\n(kja)2!2[1\u0000cos(kja)]in thejdirection for j=x;y;z,\nwhereais the length of a magnetic unit cell. Here, by\ntaking the Brillouin zone (BZ) into account, we set the\nboundary condition for the zaxis in wave number space\nk= (kx;ky;kz)askz!\u0019n=Lz, i.e.,kza!\u0019n=Nz,\nwhereLz:=aNzis the thickness of magnets, Nj2N\nis the number of magnetic unit cells in the jdirection\nforj=x;y;z, andn= 1;2;:::;2NforN2N. The\nnumberofunitcellsonthe xyplaneis 4NxNy,andthatof\nmagneticunitcellsis NxNy. Thus, themagnonicCasimir\nenergy per the number of magnetic unit cells NxNyon\nthe surface for Nz=Nis described as [24–28]\nECas:=Esum\n0(N)\u0000Eint\n0(N); (1a)\nEsum\n0(N) =X\n\u001b=\u0006Z\nBZd2(k?a)\n(2\u0019)2\"\n1\n2\u00101\n22NX\nn=1j\u000f\u001b;k;nj\u0011#\n;(1b)\nEint\n0(N) =X\n\u001b=\u0006Z\nBZd2(k?a)\n(2\u0019)2\"\n1\n2NZ\nBZd(kza)\n2\u0019j\u000f\u001b;kj#\n;\n(1c)\nwherek?:=q\nk2x+k2y,d2(k?a) =d(kxa)d(kya), the\nintegral is over the first BZ, and the factor 1=2arises\nfrom the zero-point energy for the scalar field. Since\nthe constant terms which are independent of the wave\nnumber do not contribute to the Casimir energy, we drop\nthemthroughoutthisstudy. Toseethedependenceofthe\nCasimir energy ECason the thickness of magnets Lz:=\naNz, it is convenient to introduce the rescaled Casimir\n−1−0.8−0.6−0.4−0.2 0\n 0  2  4  6  8 10  12  14  16  18  20Antiferromagnetic thin film\nNzCasimir energy ECas [meV]\nNz: Thickness of magnet in units of aδ=2.0×10−3\nδ=1.0×10−3\nδ=0 −1−0.8−0.6−0.4−0.2 0\n 0  5  10  15  20CCas[3]=Nz3ECas [meV]FIG. 2. Plots of the magnonic Casimir energy ECasand its\ncoefficient (inset) C[3]\nCas=ECas\u0002N3\nzin the AFM [see Eq. (3a)]\nas a function of Nzfor the thickness of magnets Lz=aNz.\nenergyC[b]\nCasin terms of Nb\nzforb2Ras\nC[b]\nCas:=ECas\u0002Nb\nz: (2)\nThen, we refer to C[b]\nCasas the magnonic Casimir coeffi-\ncient in the sense that ECas=C[b]\nCasN\u0000b\nz.\nHere, we consider magnons in AFMs on a cubic lattice\nwith the energy dispersion \u000f\u001b;k=\u000fAFM\n\u001b;k[29]:\n\u000fAFM\n\u001b;k=~!0r\n\u000e+\u0010ka\n2\u00112\n; (3a)\n~!0:=2p\n3JS; (3b)\n\u000e:=3h\u0010K\n6J\u00112\n+ 2\u0010K\n6J\u0011i\n; (3c)\nwherek:=jkj,J >0parametrizes the exchange interac-\ntion between the nearest-neighbor spins of the spin quan-\ntum number S, andK\u00150is the easy-axis anisotropy,\nwhich provides the magnon energy gap and ensures the\nNéel magnetic order. Two kinds of magnons ( \u001b=\u0006)\nare in degenerate states. In the absence of the spin\nanisotropy, K= 0, the energy gap vanishes \u000e= 0, and\nthegaplessmagnonmodebehaveslikearelativisticparti-\nclewiththelinearenergydispersion. Fromtheresultsob-\ntainedinRefs.[30–32],weroughlyestimatethemodelpa-\nrameter values for Cr 2O3, as an example, as follows [29]:\nJ= 15meV,S= 3=2,K= 0:03meV, anda= 0:496 07\nnm. These parameter values provide ~!0\u001877:94meV\nand\u000e\u00182\u000210\u00003.\nFigure 2 shows that the magnonic Casimir effect arises\nin the thin film of the AFM. The magnonic Casimir en-\nergyECasof the magnitude O(10\u00002)meV is induced\nforNz\u00152. Even in the presence of the magnon en-\nergy gap\u000e6= 0, the absolute value amounts to O(10\u00002)\nmeV and decreases as the magnon energy gap increases.\nThus, the magnonic Casimir energy takes a maximum3\nabsolute value in the gapless mode \u000e= 0, where the\nmagnon behaves like a relativistic particle with the lin-\near energy dispersion. We remark that in the case of\nthe gapped magnon modes, the absolute value of the\nmagnonicCasimircoefficient C[3]\nCas=ECas\u0002N3\nzdecreases\nas the thickness of the thin film increases. This behav-\nior is similar to the Casimir effect known for massive\ndegrees of freedom [33, 34]. In the case of the gapless\nmode, the magnonic Casimir coefficient C[3]\nCasapproaches\nasymptotically to a constant value, and the magnonic\nCasimir energy exhibits the behavior of ECas/1=N3\nzas\nthe thickness increases. The asymptotic value of C[3]\nCas\nfor the gapless magnon mode \u000e= 0given in the nu-\nmerical result (see Fig. 2) is estimated approximately as\n(\u0000\u00192=720)\u0002(~!0=2)\u0018\u00000:5341meV from an analyt-\nical calculation. The factor of \u0000\u00192=720is well known\nas the analytic solution for the conventional Casimir\neffect of a massless complex scalar field in continuous\nspace [34]. Thus, although the magnonic Casimir effect\nis realized on the lattice, it is qualitatively and quanti-\ntatively analogous to the continuous counterpart, except\nfora-dependent lattice effects.\nFerrimagnets. —We develop the study of AFMs into\nferrimagnets where the ground state has an alternating\nstructure of up and down spins on a cubic lattice (see\nFig. 1). In contrast to AFMs, the spin quantum num-\nber on the two-sublattice is different from each other in\nferrimagnets. Hence, the degeneracy for two kinds of\nmagnons (\u001b=\u0006) is intrinsically lifted. In ferrimagnetic\nthinfilms, dipolarinteractionsduetothenonzeromagne-\ntization play a key role. Still, at low temperatures where\nthe magnon-magnon interaction of the fourth order in\nmagnon operators is negligibly small [35], the number of\nmagnons and the total spin angular momentum are con-\nserved, and the Hamiltonian for the ferrimagnetic thin\nfilm can be diagonalized with the magnon energy disper-\nsion\u000f\u001b;k=\u000fferri\n\u001b;k.\nHere, we consider magnons in the thin film of clean\ninsulating ferrimagnets on a cubic lattice subjected to\nin-plane magnetic fields at such low temperatures. Still,\ndue to the competition between dipolar and exchange\ninteractions, the minimum energy point shifts from the\nzero mode of magnons, k= 0, to a finite wave number\nmode which is characterized by the thickness of the thin\nfilmLz=aNz(see Fig. 1).\nThe magnon energy dispersion along the in-plane di-\nrection is provided in Refs. [36, 37], whereas the disper-\nsion along the zaxis in the thin film has not yet been\nestablished [38]. Hence, taking into account the compe-\ntition between dipolar and exchange interactions in the\nthin film, we phenomenologically assume the behavior\nthat the power of kz,l2R, approaches asymptotically\ntol= 2in the bulk limit, whereas it slightly differs from\nl= 2as long as we consider the thin film (see Fig. 1).\nUsing this assumption and Refs. [36, 37], the magnon\n−120−100−80−60−40−20 0 20\n 0  2  4  6  8 10  12  14  16  18  20Ferrimagnetic thin film\nNzCasimir energy ECas [µeV]\nNz: Thickness of magnet in units of al=2.1\nl=2.0l=1.99\nl=1.9\n−100−50 0 50 100\n 0  5  10  15  20CCas[l]=NzlECas [µeV]FIG. 3. Plots of the magnonic Casimir energy ECasand its\ncoefficient (inset) C[l]\nCas=ECas\u0002Nl\nzforl= 2:1,l= 2:0,l=\n1:99, and l= 1:9in the ferrimagnetic thin film [see Eq. (4a)]\nas a function of Nzfor the thickness of magnets Lz=aNz\nwith the model parameter values for YIG of Dz=D.\nenergy dispersions are\n\u000fferri\n\u001b;k=r\n\u001bH0+ \u0001\u001b+D\na2(k?a)2+Dz\na2(jkzja)l(4a)\n\u0002r\n\u001bH0+ \u0001\u001b+D\na2(k?a)2+Dz\na2(jkzja)l+\u001b~!MFk;\nFk:=Pk?(1\u0000Pk?)\u001b~!M\n\u001bH0+ \u0001\u001b+D\na2(k?a)2+Dz\na2(jkzja)l\u0010kx\nk?\u00112\n+1\u0000Pk?\u0010ky\nk?\u00112\n; (4b)\nPk?:=1\u00001\u0000e\u0000k?Lz\nk?Lz; (4c)\nwhere the external magnetic field is applied along the y\naxis (see Fig. 1) and \u001bH0represents the resulting Zee-\nman energy, \u0001\u001b\u00150is the (intrinsic) magnon energy\ngap in ferrimagnets, D(z)>0is the spin stiffness con-\nstant,k?:=q\nk2x+k2y,~!M:= 4\u0019\rMswith the sat-\nurated magnetization density Msand the gyromagnetic\nratio\r, and the termFkis responsible for the shift of\nthe minimum energy point from the zero mode to a finite\nwave number mode due to the competition between dipo-\nlar and exchange interactions in the ferrimagnetic thin\nfilm: The first term of Fk[see Eq. (4b)] reproduces the\nDamon-Eshbach magnetostatic surface mode [39], and\nthe last term reproduces the backward volume magneto-\nstatic mode [39].\nFrom the results obtained in Refs. [40–42], the model\nparameter values for YIG thin films are estimated as fol-\nlows:D=a2\u00183:376 45meV [35] with a= 1:2376nm [43],\nH0\u00188:103 73\u0016eV [35], and ~!M\u001820:3369\u0016eV [35].\nThen, we estimate the magnon energy gap \u0001\u001bas [44]\n\u0001\u001b=\u0000\u0000\u0001\u001b=+\u001839:848 81meV with \u0001\u001b=+\u00182:131 91\nmeV and \u0001\u001b=\u0000\u001841:980 72meV, which satisfy the con-4\ndition \u0001\u001b\u001d~!M. In this condition, we study the low-\nenergy magnon dynamics of the ferrimagnetic thin film\nby using the quantum field theory of real scalar fields,\ni.e., the real Klein-Gordon field theory [21, 22]. Then,\nwe can see that there exists a zero-point energy [35].\nThe magnonic Casimir energy through the lattice reg-\nularization is given as Eq. (1a). We remark that the\nzero-point energy arises from quantum fluctuations and\ndoes exist even at zero temperature. The zero-point en-\nergy defined at zero temperature does not depend on the\nBose-distribution function [Eqs. (1b) and (1c)]. Hence,\nnot only the low-energy mode ( \u001b= +) but also the high-\nenergy mode ( \u001b=\u0000) contribute [45] to the magnonic\nCasimir energy.\nUnderthephenomenologicalassumptionthatthevalue\nofl[see Eq. (4a)] approaches asymptotically to l= 2in\nthe bulk limit, in this work focusing on the thin film,\nwe study the behavior of the magnonic Casimir effect\nby changing the value slightly from l= 2. As exam-\nples, we consider the cases of l= 2:1,l= 1:99, and\nl= 1:9. Figure 3 shows that the magnonic Casimir ef-\nfect arises in the ferrimagnetic thin film. There is the\nmagnonic Casimir energy ECasof the magnitude O(1),\nO(10), andO(10)\u0016eV forl= 1:99,l= 1:9, andl= 2:1,\nrespectively, in Nz\u00152. As the thickness increases, the\nmagnonic Casimir coefficient C[l]\nCasapproaches asymptot-\nically to a constant value, and the magnonic Casimir en-\nergy exhibits the behavior of ECas/1=Nl\nz. We also\nfind from Fig. 3 that the sign of the magnonic Casimir\ncoefficient and energy for l= 2:1is positive C[2:1]\nCas =\nECas\u0002N2:1\nz>0inNz\u00152, whereas that for l= 1:9is\nnegativeC[1:9]\nCas=ECas\u0002N1:9\nz<0. This means that the\nCasimir force works in the opposite direction.\nNote that even if l= 2, the magnonic Casimir effect\narises in the ferrimagnetic thin film. We have numer-\nically confirmed that although the value is small, there\ndoes exist the magnonic Casimir energy of the magnitude\njECasj\u0014O(0:1)neV forl= 2. This strong suppression of\nthe Casimir energy is a general property of the Casimir\neffect for quadratic dispersions on the lattice [28], and\nthe survival values originate from the dipolar interaction\nin the ferrimagnetic thin film.\nWe remark that as long as we consider thin films,\nthe value of Dzcan differ from D. Even in that case,\nthe magnonic Casimir effect arises. When the value\nofDzchanges from Dto0:8Das an example, the\nmagnonic Casimir energy ECasincreases approximately\nby0:8times. For more details about its dependence on\nthe parameters Dzandl, see the Supplemental Material.\nProposal for experimental observation. —Themagnonic\nCasimir energy of the ferrimagnetic thin film depends\nstronglyonexternalmagneticfieldsthroughZeemancou-\npling as in Eq. (4a) and contributes to magnetization of\nmagnets, whereas the photon and phonon Casimir ef-\nfects [46] do not usually. On the other hand, in the\npresenceofmagnetostriction[47–50], thephononCasimir\neffect is influenced by magnetostriction, and its correc-\ntion for the phonon Casimir energy depends on magneticfields and contributes to magnetization. However, such a\ncontribution to magnetization from the phonon Casimir\neffect should be negligibly small by the factor of 10\u00006\ncompared with that from the magnonic Casimir effect\nof ferrimagnets because the magnetostriction constant\n(i.e., the correction for the lattice constant) is known to\nbe10\u00006for YIG [47, 48]. Hence, even in the presence\nof magnetostriction, the magnonic Casimir effect can be\ndistinguished from the others. Thus, we expect that our\ntheoreticalprediction, themagnonicCasimireffectinfer-\nrimagnets, can be experimentally observed through mea-\nsurement of magnetization and its film thickness depen-\ndence. For more details, see the Supplemental Material.\nForobservation, afewcommentsareinorder. First, we\nremark on edge/surface magnon modes. The magnonic\nCasimir effect in our setup (see the thin film of Fig. 1)\nis induced by magnon fields with wave numbers kzdis-\ncretized by small Nz, and its necessary condition is a\nkz-dependent dispersion relation under the discretization\nofkz. Throughout this study, we consider thin films of\nNz\u001cNx;Ny. Even if additional edge/surface magnon\nmodesexistaswellastheDamon-Eshbachmagnetostatic\nsurface mode and the backward volume magnetostatic\nmode [see Eq. (4b)], they are confined only on the x-\nyplane. Then, their wave number in the zdirection\nis always zero, i.e., kz= 0, and its energy dispersion\nrelation is independent of kz. Since a kz-independent\ndispersion relation cannot induce the Casimir effect, the\nedge/surface modes cannot contribute to the magnonic\nCasimir effect. In this sense, our magnonic Casimir effect\nis not affected by the existence of edge/surface magnon\nmodes.\nNote that details of the edge condition, such as the\npresence or absence of disorder, may change the bound-\nary condition for the wave function of magnons, but the\nexistence of the magnonic Casimir effect remains un-\nchanged. Even if there is a change in the spectrum near\nthe edge, the magnonic Casimir effect is little influenced\nas long as one does not assume an ultrathin film such\nasNz= 1;2;3. In this sense, we expect that the fol-\nlowing size of thin films is appropriate for observation of\nour prediction: Nz\u001810, i.e., the film thickness of YIG\nisLz=aNz\u001812:376nm. Note that microfabrication\ntechnology [17, 18] can control the thickness of thin films\nand realize the manipulation of the magnonic Casimir\neffect.\nNext, we remark on the magnon band structure.\nSince our magnonic Casimir effect is induced by the kz-\ndependent dispersion, its Casimir energy of ferrimagnets\nis mainly characterized by the Dz-term in Eq. (4a), i.e.,\nDz(jkzja)l. Hence, we have investigated its dependence\non bothlandDz(see Fig. 3 and the Supplemental Mate-\nrial). Even if the magnon band structure is affected due\nto some reasons, the magnonic Casimir effect of ferrimag-\nnets is little influenced by other details of the magnon\nband structure except for landDz.\nLastly, we remark on thermal effects. At nonzero tem-\nperature, thermal contributions to the Helmholtz free en-5\nergy arise. However, at low temperatures compared to\n\u000f\u001b;k[51], the thermal contribution is exponentially sup-\npressed due to the Boltzmann factor and becomes negli-\ngibly small. Hence, at such low temperatures, the contri-\nbution of the magnonic Casimir energy given as Eq. (1a)\nis dominant.\nConclusion. —We have shown that the magnonic\nCasimir effect can arise not only in antiferromagnets but\nalso in ferrimagnets with realistic model parameters for\nYIG. Since the lifetime of magnons in YIG thin films is\nthelongestamongknownmaterials, andmagnonsexhibit\nlong-distance transport over centimeter distances [52],\nYIG is the key ingredient of magnonics [1, 16], which\nhas already realized the magnon transistor [53]. Ourresult suggests that YIG can serve also as a promis-\ning platform for Casimir engineering [19]: Because the\nmagnonic Casimir effect contributes to the internal pres-\nsure of thin films, it will provide the new principles of\nnanoscale devices such as highly sensitive pressure sen-\nsors and magnon transistors. 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Volovik, arXiv:1003.4889 .7\nSupplemental Material\nIn this Supplemental Material, first, we provide some\ndetails about the dependence of the magnonic Casimir\neffect on the parameters landDzin ferrimagnetic thin\nfilms. Next, we remark on its film thickness dependence.\nThen, we provide another point of view for its robust-\nness against disorder effects. Lastly, we comment on the\ndistinction between the Casimir effect and the thermal\nCasimir effect.\nI. THE PARAMETER l- AND Dz-DEPENDENCE\nIn the main text, we have studied the magnonic\nCasimirenergy ECasandthecoefficient C[l]\nCas=ECas\u0002Nl\nz\nforl= 2:1,l= 2:0,l= 1:99, andl= 1:9in the ferrimag-\nnetic thin film by using the model parameter values for\nYIG with fixed Dz=D. Here, we provide more details\nabout its dependence on the parameters landDz.\nFirst, we consider the cases of l= 1:5andl= 1:0with\nsettingDz=D. Figure S1 shows that the magnonic\nCasimir effect still arises in the ferrimagnetic thin film.\nThere is the magnonic Casimir energy ECasof the mag-\nnitudeO(10\u00002)meV,O(10\u00001)meV, andO(10\u00001)meV\nforl= 1:9,l= 1:5, andl= 1:0, respectively, in Nz\u00152.\nAs the value of ldecreases from l= 2and approaches\ntol= 1, the magnitude of the magnonic Casimir energy\nincreases. Note that it amounts to O(10\u00001)meV even in\nNz=O(10)forl= 1:0. As the thickness increases, the\nmagnonic Casimir coefficient C[l]\nCasapproaches asymptot-\nically to a constant value and the magnonic Casimir en-\nergy exhibits the behavior of ECas/1=Nl\nz.\nThen, we consider the cases of Dz=D= 0:3,Dz=D=\n0:5, andDz=D= 0:8by fixingl= 1:99. Figure S2 shows\nthat the magnonic Casimir effect still arises in the ferri-\nmagnetic thin film. When the value of Dzchanges from\nDto0:8Das an example, the magnonic Casimir energy\nECasincreases approximately by 0:8times. Thus, the\nvalue ofECasis approximately proportional to Dz.\nII. REMARKS ON THE THICKNESS\nDEPENDENCE OF MAGNETIZATION\nIn the main text, we have remarked that our predic-\ntion, the magnonic Casimir effect in ferrimagnets, can\nbe observed through measurement of magnetization and\nits film thickness dependence. Here, we add an expla-\nnation about it. At zero temperature, the Helmholtz\nfree energy of magnon fields in thin films (i.e., the\nsum over discrete kz) isEsum\n0(Nz)NxNy, and that un-\nder the bulk approximation (i.e., the integral with re-\nspect to continuous kz) isEint\n0(Nz)NxNy[see Eqs. (1a)-\n(1c)]. The difference between them is characterized by\nthe magnonic Casimir energy ECasasEsum\n0(Nz)NxNy=\nEint\n0(Nz)NxNy+ECasNxNy, where the magnon energy\n−1−0.8−0.6−0.4−0.2 0\n 0  2  4  6  8 10  12  14  16  18  20Ferrimagnet\nNzCasimir energy ECas [meV]\nNz: Thickness of magnet in units of al=2.0\nl=1.9l=1.5\nl=1.0\n−1−0.8−0.6−0.4−0.2 0\n 0  5  10  15  20CCas[l]=NzlECas  [meV]FIG. S1. Plots of the magnonic Casimir energy ECasand\nthe coefficient (inset) C[l]\nCas=ECas\u0002Nl\nzforl= 2:0,l= 1:9,\nl= 1:5, and l= 1:0intheferrimagneticthinfilmasafunction\nofNzfor the thickness of magnets Lz=aNzunder the model\nparameter values for YIG with fixed Dz=D.\n−14−12−10−8−6−4−2 0\n 0  2  4  6  8 10  12  14  16  18  20Ferrimagnet\n(l=1.99)\nNzCasimir energy ECas [µeV]\nNz: Thickness of magnet in units of aDz=0.3D\nDz=0.5DDz=0.8D\nDz=D\n−14−12−10−8−6−4−2 0\n 0  5  10  15  20CCas[l]=NzlECas [µeV]\nFIG. S2. Plots of the magnonic Casimir energy ECasand the\ncoefficient (inset) C[l]\nCas=ECas\u0002Nl\nzforDz=D= 0:3,Dz=D=\n0:5,Dz=D= 0:8, and Dz=D= 1:0in the ferrimagnetic thin\nfilmasafunctionof Nzforthethicknessofmagnets Lz=aNz\nunder the model parameter values for YIG with l= 1:99.\ndispersion of ferrimagnets (i.e., magnets including dipo-\nlar interactions) is Eq. (4a). Note that the magnetic-\nfield derivative (i.e., H0-derivative) of the Helmholtz free\nenergy is magnetization. Then, magnetization of thin\nfilms consists of two parts: The bulk component and the\nmagnonic Casimir energy. Since Eint\n0(Nz)/Nz, whereas\nECas/1=(Nz)l, magnetization of thin films exhibits a\ndifferentNz-dependence from the bulk component, and\nits difference is characterized by the magnonic Casimir\nenergy. In other words, magnetization of thin films ex-\nhibits a different film thickness dependence from the bulk\ncomponent due to the magnonic Casimir effect. Hence,\nour prediction, the magnonic Casimir effect in ferrimag-8\nnetic thin films (i.e., magnetic thin films including dipo-\nlar interactions), can be observed through measurement\nof magnetization and its film thickness dependence.\nNote that if dipolar interactions are relevant also in\nantiferromagnets, its low-energy magnon dynamics is es-\nsentially described by Eq. (4a) given for ferrimagnets.\nThe only difference is that the spin quantum number for\neachsublatticeisidenticalinantiferromagnets, wherethe\n(intrinsic) magnon energy gap for each mode \u001b=\u0006can\nbe identical \u0001\u001b=+= \u0001\u001b=\u0000[see Eq. (4a)]. In this sense,\nitsCasimireffectexhibitsqualitativelythesamebehavior\nas Fig. 3.\nIII. REMARKS ON DISORDER EFFECTS\nIn the main text, we have remarked that details of the\nedge condition, such as the presence or absence of dis-\norder, may change the boundary condition for the wave\nfunction of magnons, but the existence of the magnonic\nCasimir effect remains unchanged. Here, we add a com-\nment on disorder effects. Since the magnonic Casimir\nenergy does not depend on the Bose-distribution func-\ntion [see Eqs. (1a)-(1c)], not only the low-energy magnon\nmode (\u001b= +) but also its high-energy mode ( \u001b=\u0000) in\nferrimagnets contributes to the magnonic Casimir effect.\nTherefore, it can be expected that as long as disorder\neffects on the bulk are weak enough that the high-energy\nmode is little influenced, the existence of the magnonic\nCasimir effect in ferrimagnets remains unchanged.\nIV. THERMAL CASIMIR EFFECT\nIn the main text, we have explained that thermal con-\ntributions to the Helmholtz free energy arise at nonzero\ntemperature. Here, we add a remark on it. Although its\nthermal contribution is called the “thermal Casimir en-\nergy”, there is a crucial distinction between the Casimir\neffect and the thermal Casimir effect: The zero-point en-\nergy, which is the key concept of quantum mechanics and\nplays a crucial role in the Casimir effect, is absent in the\nthermal Casimir effect. It should be noted that the zero-\npoint energy is one of the most striking phenomenon of\nquantum mechanics in the sense that there are no clas-\nsical analogs. The Casimir effect arises from the zero-\npoint energy due to quantum fluctuations and is not af-\nfected by temperatures, whereas the thermal Casimir ef-\nfect arises from thermal fluctuations and is exponentially\nsuppressed at low temperatures. The thermal Casimir\neffect vanishes at zero temperature, whereas the Casimir\neffect does exist even at zero temperature. Thus, there is\na significant distinction between the Casimir effect and\nthe thermal Casimir effect."    },    {        "title": "1903.00221v2.Entangling_two_magnon_modes_via_magnetostrictive_interaction.pdf",        "content": "Entangling two magnon modes via magnetostrictive interaction\nJie Li1and Shi-Yao Zhu1\n1Zhejiang Province Key Laboratory of Quantum Technology and Device,\nDepartment of Physics and State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou, Zhejiang, China\n(Dated: July 24, 2019)\nWe present a scheme to entangle two magnon modes in a cavity magnomechanical system. The two magnon\nmodes are embodied by collective motions of a large number of spins in two macroscopic ferrimagnets, and\ncouple to a single microwave cavity mode via magnetic dipole interaction. We show that by activating the\nnonlinear magnetostrictive interaction in one ferrimagnet, realized by driving the magnon mode with a strong\nred-detuned microwave field, the two magnon modes can be prepared in an entangled state. The entanglement\nis achieved by exploiting the nonlinear magnon-phonon coupling and the linear magnon-cavity coupling, and is\nin the steady state and robust against temperature. The entangled magnon modes in two massive ferrimagnets\nrepresent genuinely macroscopic quantum states, and may find applications in the study of macroscopic quantum\nmechanics and quantum information processing based on magnonics.\nI. INTRODUCTION\nFerrimagnetic systems, for example yttrium iron garnet\n(YIG), provide a unique platform for the study of the strong\ninteractions between light and matter. Owing to their high\nspin density (several orders of magnitude larger than those of\nprevious spin ensembles) and low dissipation rate, in recent\nyears the strong [1–6] and ultrastrong [7, 8] coupling between\nthe Kittel mode [9] in the YIG sphere and the microwave cav-\nity photons have been realized leading to cavity-magnon po-\nlaritons. This strong coupling o \u000bers a possibility to enable\ncoherent information transfer between drastically di \u000berent in-\nformation carriers, and thus may find potential applications in\nquantum information processing, especially when the system\nbecomes hybrid [10], such as by coupling magnons to a su-\nperconducting qubit [11, 12], to phonons [13, 14], or to both\nmicrowave and optical photons [15]. Furthermore, various\ninteresting phenomena have been explored in the system of\ncavity-magnon polaritons, such as the observation of magnon\ngradient memory [16], the exceptional point [17, 18], manip-\nulation of distant spin currents [19], and bistability [20], to\nname but a few.\nCavity-magnon systems of YIG spheres provide also a\npromising and completely new platform for the study of\nmacroscopic quantum states [14, 21]. A magnon mode can\nget squeezed by driving the cavity with a squeezed vacuum\nmicrowave field, and the squeezing can further be transferred\nto the mechanical mode if the magnetostrictive interaction is\nactivated by driving the magnon mode with a red-detuned mi-\ncrowave field [21]. Such a hybrid cavity magnomechanical\nsystem can also be prepared in a genuinely tripartite entangled\nstate by suitably driving the YIG sphere and essentially utiliz-\ning the nonlinear magnetostrictive interaction [14]. Both the\nentangled and squeezed states of the magnon and mechanical\nmodes are macroscopic quantum states due to a large size of\nthe YIG sphere and an extremely large number of spins con-\ntained (more than 1016for a 250-\u0016m-diam YIG sphere that\nhas been employed in Refs. [14, 21]). Alternatively, nonclas-\nsical magnon states can be prepared by coupling magnons to\na superconducting qubit which provides necessary nonlinear-\nity [11, 12]. We note that the observation of quantum e \u000bectsin massive systems has achieved significant progress in the\nfield of cavity optomechanics [22], where quantum squeez-\ning of mechanical motion [23], nonclassical correlations be-\ntween single photons and phonons [24, 25], and quantum\nentanglement between mechanics and a cavity field [26], as\nwell as between two massive mechanical oscillators [27, 28]\nhave been observed. As an analogue, cavity magnomechanics\nholds a potential for realizing quantum states in more massive\nobjects. Apart from quantum e \u000bect, magnomechanically in-\nduced transparency [13] and slow light e \u000bect [29] have been\nexplored in such a system.\nHere we present a scheme to prepare two magnon modes\nin two massive YIG spheres in an entangled state. To date,\nthe proposals of preparing entangled magnon modes in cavity-\nmagnon systems are still missing [30]. The two magnon\nmodes couple to a single microwave cavity mode via linear\nbeamsplitter interactions, and it is known that such interac-\ntions will not yield any entanglement between the two magnon\nmodes. Nevertheless, by activating the magnetostrictive (ra-\ndiation pressure-like) interaction in one YIG sphere, we show\nthat such an interaction can be utilized to generate entangle-\nment between two magnon modes if one of them is suitably\ndriven by a microwave field. The entanglement arises from\nthe magnon-phonon coupling and partially transfers to cavity-\nmagnon and cavity-phonon subsystems. Further, by using\nthe e\u000bective state-swap interaction between the cavity and the\nother magnon mode, the two magnon modes therefore get en-\ntangled. The entanglement is in the stationary state and robust\nagainst environmental temperature. Di \u000berently from our pre-\nvious work [14], where three modes of di \u000berent natures get\nentangled, here we aim to entangle two magnon modes in two\nYIG spheres.\nThe remainder of this paper is organized as follows. In sec-\ntion II, we introduce a cavity magnomechanical system of two\nYIG spheres, and provide its Hamiltonian and the correspond-\ning quantum Langevin equations (QLEs). In section III, we\nstudy the system dynamics by adopting the linearization treat-\nment, show how to obtain the steady-state solutions of the sys-\ntem, and present the main results of the entanglement between\ntwo magnon modes. In section IV, we analyse the Kerr non-\nlinear e \u000bect due to the strong drive of one magnon mode, and\nprovide possible strategies to measure the entanglement. Fi-arXiv:1903.00221v2  [quant-ph]  23 Jul 20192\n(c)(a)\n(b)\nFIG. 1: (a) Sketch of the system. Two YIG spheres are, respectively,\nplaced inside a microwave cavity near the maximum magnetic fields\nof the cavity mode, and simultaneously in uniform bias magnetic\nfields, which excite the magnon modes in the spheres and couple\nthem to the cavity mode. The directions of the bias magnetic fields\nare adjusted such that only in one sphere, say the right sphere, the\nmagnetostrictive (magnon-phonon) interaction is activated, and this\ncoupling can be enhanced by directly driving the magnon mode with\na microwave source (not shown). The bias magnetic field ( zdirec-\ntion), the drive magnetic field ( ydirection) and the magnetic field ( x\ndirection) of the cavity mode are mutually perpendicular at the site\nof the right sphere. (b) Interactions among the subsystems. The cav-\nity mode alinearly couples to the two magnon modes m1andm2\nwith coupling constants g1andg2, respectively. Besides, the magnon\nmode m1couples to the mechanical mode bvia the nonlinear magne-\ntostrictive interaction with an e \u000bective coupling rate G. The magne-\ntostrictive interaction generates magnomechanical entanglement [14]\nwhich can be used to entangle two magnon modes. (c) Frequencies\nand linewidths of the system. The magnon mode m1with frequency\n!1is driven by a strong microwave field at frequency !0, and the\nmechanical motion of frequency !bscatters photons onto the two\nsidebands at !0\u0006!b. If the magnon mode m1is resonant with the\nblue (anti-Stokes) sideband, and both the cavity with frequency !a\nand the magnon mode m2with frequency !2are resonant with the\nred (Stokes) sideband, the two magnon modes are prepared in an en-\ntangled state.\nnally, we reserve section V for conclusions.\nII. THE MODEL\nWe consider a hybrid cavity magnomechanical system\nwhich consists of a microwave cavity mode, two magnon\nmodes, and a mechanical mode, as shown in Fig. 1 (a). The\ntwo magnon modes are embodied by collective motions of a\nlarge number of spins in two macroscopic YIG spheres, and\nsimultaneously couple to a single microwave cavity. Such a\nsystem of two YIG spheres (without involving the mechani-\ncal mode) has been used to study magnon dark modes [16]\nand high-order exceptional points [18]. The magnetic dipoleinteraction mediates the coupling between magnons and cav-\nity photons [see Fig. 1 (b)], and this coupling can be very\nstrong [1–8]. The mechanical mode is represented by the\nvibrations of the YIG sphere caused by the magnetostrictive\nforce, which leads to the deformation of the geometry struc-\nture of the sphere and establishes the magnon-phonon cou-\npling. In general, the magnetostrictive interaction is of dif-\nferent types depending on the resonance frequencies of the\nmagnon and phonon modes [32], but the dispersive magnon-\nphonon interaction becomes dominant when the mechanical\nfrequency is much smaller than the magnon frequency [13],\nwhich is the case to be considered in the present work. This\nmagnon-phonon coupling is currently small [13], but it can\nbe e\u000eciently enhanced by driving the sphere with a strong\nmicrowave field [14, 20]. The magnomechanical coupling\nstrength is sensitive to the direction of the bias magnetic\nfield [13], and we adjust the directions of the two bias mag-\nnetic fields [see Fig. 1 (a)] such that only in one sphere the\nmagnetostrictive interaction is e \u000bectively activated [33]. The\nHamiltonian of the system reads\nH=~=!aaya+X\nj=1;2!jmy\njmj+!b\n2(q2+p2)+G0my\n1m1q\n+X\nj=1;2gj(a my\nj+aymj)+i\n(my\n1e\u0000i!0t\u0000m1ei!0t);\n(1)\nwhere aanday(mjandmy\nj), [O;Oy]=1,O=a(mj), are, re-\nspectively, the annihilation and creation operators of the cav-\nity mode (magnon modes), qandp, [q;p]=i, are the dimen-\nsionless position and momentum quadratures of the mechan-\nical mode, and !a,!j, and!bare the resonance frequen-\ncies of the cavity, magnon, and mechanical modes, respec-\ntively. The magnon frequencies are determined by the bias\nmagnetic fields Hjvia!j=\rHj, where\r=2\u0019=28 GHz /T is\nthe gyromagnetic ratio. The coupling rate gjdenotes the lin-\near coupling between the cavity and the jth magnon mode,\nandG0represents the single-magnon magnomechanical cou-\npling rate. The Rabi frequency \n =p\n5\n4\rp\nNB 0[14] denotes\nthe coupling strength of the drive magnetic field (with ampli-\ntude B0and frequency !0) with the first magnon mode, where\nthe total number of spins N=\u001aVwith\u001a=4:22\u00021027m\u00003\nthe spin density of the YIG and Vthe volume of the sphere.\nNote that the boson (oscillator) operators mjandmy\njdescribe\nthe collective motion of the spins via the Holstein-Primako \u000b\ntransformation [34] under the assumption of the low-lying ex-\ncitationshmy\njmji\u001c2Ns(without losing generality we assume\nthe two spheres are of the same size), where s=5\n2is the spin\nnumber of the ground state Fe3+ion in YIG.\nFrom the system Hamiltonian (1), one would see the\nmechanism of entanglement generation between the two\nmagnon modes: the radiation pressure-like interaction Hm1b=\n~G0my\n1m1qallows one to create entanglement between the\nmagnon mode m1and the mechanical mode by suitably driv-\ning the magnon mode, similarly to creating optomechanical\nentanglement [35]. Such an interaction can also lead to cavity-\nmagnon ( m1) and cavity-phonon entanglement via the lin-3\near coupling Ham1=~g1(a my\n1+aym1), which is the main re-\nsult of Ref. [14]. By introducing the second magnon mode\nm2interacting with the cavity via the state-swap interaction\nHam2=~g2(a my\n2+aym2), the two magnon modes therefore\nget entangled.\nIn the frame rotating at the magnon drive frequency !0, the\nQLEs describing the system are given by\n˙a=\u0000(i\u0001a+\u0014a)a\u0000iX\nj=1;2gjmj+p\n2\u0014aain;\n˙m1=\u0000(i\u00011+\u00141)m1\u0000iG0m1q\u0000ig1a+ \n +p\n2\u00141min\n1;\n˙m2=\u0000(i\u00012+\u00142)m2\u0000ig2a+p\n2\u00142min\n2;\n˙q=!bp;\n˙p=\u0000!bq\u0000\rbp\u0000G0my\n1m1+\u0018;(2)\nwhere \u0001a=!a\u0000!0,\u0001j=!j\u0000!0(j=1;2),\u0014a,\u0014j, and\n\rbare the dissipation rates of the cavity, magnon, mechan-\nical modes, respectively, and ain,min\njare input noise oper-\nators a \u000becting the cavity and magnon modes, respectively,\nwhich are zero mean and characterized by the following corre-\nlation functions [36]: hain(t)ainy(t0)i=\u0002Na(!a)+1\u0003\u000e(t\u0000t0),\nhainy(t)ain(t0)i=Na(!a)\u000e(t\u0000t0), andhmin\nj(t)miny\nj(t0)i=\n\u0002Nj(!j)+1\u0003\u000e(t\u0000t0),hminy\nj(t)min\nj(t0)i=Nj(!j)\u000e(t\u0000t0). The\nLangevin force operator \u0018, which accounts for the Brow-\nnian motion of the mechanical mode, is autocorrelated as\nh\u0018(t)\u0018(t0)+\u0018(t0)\u0018(t)i=2'\rb\u00022Nb(!b)+1\u0003\u000e(t\u0000t0), where\nwe have made the Markov approximation, which is a good\napproximation for a mechanical oscillator of a large quality\nfactor Qb=!b=\rb\u001d1 [37], and Nk(!k)=h\nexp\u0010~!k\nkBT\u0011\n\u00001i\u00001,\nk=a;j;b, are the equilibrium mean thermal photon, magnon,\nand phonon number, respectively, with kBthe Boltzmann con-\nstant and Tthe environmental temperature.\nIII. STEADY-STATE SOLUTIONS AND THE RESULTS OF\nMAGNON ENTANGLEMENT\nSince the magnon mode m1is strongly driven by an external\nmicrowave field, and owing to the beamsplitter interactions\nbetween the cavity and the two magnon modes, both the cavity\nand magnon modes are of large amplitudes, jhaij;jhmjij\u001d 1.\nThis allows us to linearize the dynamics of the system around\nthe steady-state values by writing any operator as O=hOi+\n\u000eO, (O=a;mj;q;p), and neglecting small second-order fluc-\ntuation terms. Since we are particularly interested in the quan-\ntum correlation properties of the two magnon modes, we fo-\ncus on the dynamic of the quantum fluctuations of the system.\nThe linearized QLEs describing the fluctuations of the sys-\ntem quadratures ( \u000eX;\u000eY;\u000ex1;\u000ey1;\u000ex2;\u000ey2;\u000eq;\u000ep), with\u000eX=\n(\u000ea+\u000eay)=p\n2,\u000eY=i(\u000eay\u0000\u000ea)=p\n2,\u000exj=(\u000emj+\u000emy\nj)=p\n2,\nand\u000eyj=i(\u000emy\nj\u0000\u000emj)=p\n2, can be written in the form of\n˙u(t)=Au(t)+n(t); (3)\nwhere u(t)=\u0002\u000eX(t);\u000eY(t);\u000ex1(t);\u000ey1(t);\u000ex2(t);\u000ey2(t);\u000eq(t);\u000ep(t)\u0003T,\nn(t)=\u0002p2\u0014aXin(t);p2\u0014aYin(t);p2\u00141xin\n1(t);p2\u00141yin\n1(t);p2\u00142xin\n2(t);p2\u00142yin\n2(t);0;\u0018(t)\u0003Tis the vector of input noises, and the drift\nmatrix Ais given by\nA=0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@\u0000\u0014a\u0001a 0 g10 g2 0 0\n\u0000\u0001a\u0000\u0014a\u0000g10\u0000g20 0 0\n0 g1\u0000\u00141˜\u000110 0\u0000G 0\n\u0000g10\u0000˜\u00011\u0000\u001410 0 0 0\n0 g2 0 0\u0000\u00142\u000120 0\n\u0000g20 0 0\u0000\u00012\u0000\u001420 0\n0 0 0 0 0 0 0 !b\n0 0 0 G 0 0\u0000!b\u0000\rb1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA;(4)\nwhere ˜\u00011= \u0001 1+G0hqi, withhqi=\u0000G0\n!bjhm1ij2, is the e \u000bec-\ntive detuning for the first magnon mode, which includes the\nfrequency shift due to the magnon-phonon interaction, and\nG=ip\n2G0hm1iis the e \u000bective magnomechanical coupling\nrate, wherehm1i'\u0000\u0000g1hai+i\n\u0001=˜\u00011, which implies that the\ncoupling can be significantly enhanced by a strong driving\nfield. This is a result of the nonlinear term G0my\n1m1qin the\nHamiltonian Eq. (1). The average haiis given by\nhai'ig1\u00012\n\u0001a˜\u00011\u00012\u0000g2\n1\u00012\u0000g2\n2˜\u00011; (5)\nand therefore we can also obtain hm2iviahm2i'\u0000 g2hai=\u00012.\nThe magnomechanical coupling Gis a key parameter and we\nthus provide its specific expression, i.e.,\nG'p\n2G0\n(g2\n2\u0000\u00012\u0001a)\ng2\n2˜\u00011+ \u0001 2(g2\n1\u0000˜\u00011\u0001a): (6)\nThe above expressions of hai,hmji, and Gare achieved un-\nder the condition that j\u0001aj;j˜\u00011j;j\u00012j\u001d\u0014a;\u00141;\u00142, and in this\ninstance,haiandhmjiare pure imaginary numbers. The drift\nmatrix Aof Eq. (4) is given under this condition. In fact, we\nwill show later that j\u0001aj;j˜\u00011j;j\u00012j'!b\u001d\u0014a;\u00141;\u00142[see Fig. 1\n(c)] are optimal for generating the entanglement between the\ntwo magnon modes.\nOwing to the linearized dynamics and the fact that all noises\nare Gaussian, the dynamical map of the system preserves the\nGaussian nature of any input state. The steady state of the\nquantum fluctuations of the system is thus a continuous vari-\nable four-mode Gaussian state, which is completely character-\nized by an 8\u00028 covariance matrix (CM) V, which is defined\nasVi j=1\n2hui(t)uj(t0)+uj(t0)ui(t)i(i;j=1;2;:::;8). The sta-\ntionary CMVcan be straightforwardly obtained by solving\nthe Lyapunov equation [35, 38]\nAV+VAT=\u0000D; (7)\nwhere D=diag\u0002\u0014a(2Na+1);\u0014a(2Na+1);\u00141(2N1+1);\u00141(2N1+\n1);\u00142(2N2+1);\u00142(2N2+1);0;\rb(2Nb+1)\u0003is the di \u000busion ma-\ntrix, which is defined by hni(t)nj(t0)+nj(t0)ni(t)i=2=Di j\u000e(t\u0000\nt0). We adopt the logarithmic negativity [39] to quantify the\nmagnon entanglement, which is a full entanglement monotone\nunder local operations and classical communication [40] and\nsets an upper bound for the distillable entanglement [39]. The\nlogarithmic negativity is defined as [41]\nEN\u0011max[0;\u0000ln 2˜\u0017\u0000]; (8)4\nwhere ˜\u0017\u0000=min eigji\n2˜V4j(with the symplectic matrix \n2=\n\b2\nj=1i\u001byand the y-Pauli matrix \u001by) is the minimum symplectic\neigenvalue of the CM ˜V4=P1j2V4P1j2, whereV4is the 4\u00024\nCM of the two magnon modes, obtained by removing in V\nthe rows and columns of the cavity and mechanical modes,\nandP1j2=diag(1;\u00001;1;1) is the matrix that realizes partial\ntransposition at the level of CMs [42].\nFigure 2 shows the entanglement of the two magnon modes\nversus some key parameters of the system, which is in the\nsteady state guaranteed by the negative eigenvalues (real\nparts) of the drift matrix A. We have employed experimentally\nfeasible parameters [13]: !a=2\u0019=10 GHz,!b=2\u0019=10 MHz,\n\rb=2\u0019=102Hz,\u0014a=2\u0019=1 MHz,\u00141(2)=\u0014a,g1=2\u0019=3:2\nMHz, G=2\u0019=4:8 MHz, and at low temperature T=10 mK.\nWe take an optimal detuning ˜\u00011'!b[14] for the first\nmagnon mode, which is responsible for significantly cool-\ning the mechanical mode as quantum entanglement survives\nwith only small thermal phonon occupancy. Combining with\nastrong magnomechanical coupling G[43], magnon-phonon\nentanglement ( Em1b) is created and then partially transferred\nto the cavity-magnon ( m1) subsystem, i.e., the cavity and\nmagnon mode m1get entangled, Eam1>0 [14]. This e \u000bect is\nprominent when the cavity detuning \u0001a'\u0000!b. By coupling\nthe second magnon mode ( m2) to the cavity, and using their\nstate-swap interaction, the two magnon modes are expected to\nbe entangled. This is confirmed by Fig. 2 (a) and it manifests\nthat the optimal situation is that the magnon mode is resonant\nwith the cavity, \u00012'\u0001a'\u0000!b[see Fig. 1 (c)]. This is also\nthe case for generating squeezed states of magnons by driv-\ning the cavity with a squeezed microwave field [21]. Figure 2\n(b) and (c) denote the cavity-magnon ( m1) entanglement Eam1\nwith g2=0 and the magnon-magnon entanglement Em1m2with\ng2,0, respectively. The similar patterns of Fig. 2 (b) and (c)\n(d)(a)\n(c)(b)\nFIG. 2: Density plot of the entanglement Em1m2between two magnon\nmodes vs (a) \u0001aand\u00012, (c)\u0001aand˜\u00011, and (d) the ratios of g2=g1and\nG=g1(g1is fixed). (b) The entanglement Eam1between cavity and\nmagnon mode m1vs\u0001aand˜\u00011with g2=0. We take ˜\u00011=0:85!bin\n(a) and (d), g2=2\u0019=2:6 MHz in (a) and (c), \u0001a=\u00000:9!bin (d), and\n\u00012= \u0001 ain (c) and (d). See text for details of the other parameters.\nFIG. 3: Magnon entanglement Em1m2vs (a) \u0001aat 10 mK, and (b)\ntemperature for the two cases of \u0014a=2\u0019=\u00141(2)=2\u0019=1 MHz (solid\nlines) and\u0014a=2\u0019=5\u00141(2)=2\u0019=3 MHz (dashed lines). (c) Critical\ntemperature (below which Em1m2>0) vs\u00141(2)=2\u0019, with always \u0014a=\n5\u00141(2). We take an optimal detuning \u0001a=\u00000:9!bin (b) and (c). The\nother parameters are as in Fig. 2 (a) with \u00012= \u0001 a.\nshow more clearly the magnon entanglement Em1m2is trans-\nferred from the cavity-magnon ( m1) entanglement Eam1due to\nthe state-swap interaction between the cavity and the magnon\nmode m2. We take g2=2\u0019=2:6 MHz in Fig. 2 (a) and (c), and\ntherefore we have g2\n1;g2\n2\u001cj˜\u00011\u0001aj;j\u00012\u0001aj'!2\nb, which leads\nto a rather simple expression of the coupling G'p\n2G0\n!b[see\nEq. (6)]. G=2\u0019=4:8 MHz implies the drive magnetic field\nB0'3:9\u000210\u00005T for G0=2\u0019'0:3 Hz, corresponding to the\ndrive power P'8:9 mW [44]. A larger coupling Gwould\nyield a larger entanglement. However, we take a moderate\nvalue to keep the system stable and avoid unwanted nonlinear\ne\u000bect (we analyse this in the next section). There are opti-\nmal couplings of g2andGfor fixed g1, as shown in Fig. 2\n(d). Since all bipartite entanglements of the subsystems orig-\ninate from the magnon-phonon coupling, there is an interplay\namong the three couplings, g1,g2, and G, of the four modes\nof the system [see Fig. 1 (b)], which results in a maximum\nmagnon entanglement. The magnon entanglement is robust\nagainst environmental temperature and survives up to about\n200 mK, as shown in Fig. 3 (solid lines).\nIV . ANALYSIS OF NONLINEAR EFFECT AND\nSTRATEGIES FOR ENTANGLEMENT DETECTION\nIt should be noted that the results of section III are\nvalid only when the magnon excitation numbers hmy\njmji \u001c\n2Ns=5N. For what we used a 250- \u0016m-diam YIG sphere,\nthe number of spins N'3:5\u00021016, and the coupling\nstrength G=2\u0019=4:8 MHz corresponds to jhm1ij'1:1\u0002\n107, and Rabi frequency \n'7:1\u00021014Hz, such that\nhmy\n1m1i'1:3\u00021014\u001c5N=1:7\u00021017, which is well sat-\nisfied.hmy\n2m2i\u001c 2Nsis also well fulfilled due to the fact\nthatjhm2ij\u001cjh m1ijfor the parameters used in Figs. 2 and 3.5\nThe intense magnon drive may bring about unwanted nonlin-\near e\u000bects due to the Kerr nonlinear term Kmymmymin the\nHamiltonian [20, 45]. The Kerr coe \u000ecientKis inversely pro-\nportional to the volume of the sphere. For a 1-mm-diam YIG\nsphere used in Refs. [20, 45], K=2\u0019\u00190:1 nHz, which im-\nplies thatK=2\u0019\u00196:4 nHz for the spheres used in this paper.\nTo keep the Kerr e \u000bect negligible,Kjhm1ij3\u001c\nmust hold.\nThe parameters in Fig. 2 lead to Kjhm1ij3'5:8\u00021013Hz\n\u001c\n'7:1\u00021014Hz, which means that the nonlinear e \u000bects\nare negligible and our model is valid and a good approxima-\ntion.\nFinally, we discuss how to detect and measure the entangle-\nment. The generated magnon entanglement can be quantified\nby measuring the CM of the two magnon modes, following\nthe strategies used in Refs. [26, 35]. In contrast with Ref. [14],\nwhere the detection of a phonon state cannot be avoided, here\nwe only need to measure the states of two magnon modes.\nThe state of each magnon mode can be read out by coupling\nthe magnons to an additional cavity, and by sending a weak\nmicrowave probe field and homodyning the cavity output of\nthe probe field. This requires that the magnon dissipation\nrates\u00141;2should be much smaller than the cavity decay rate\n\u0014a, such that when the magnon drive is switched o \u000band all\ncavity photons decay the magnon states remain almost un-\nchanged, and then two probe fields are sent. The dashed lines\nin Fig. 3 (a) and (b) show the magnon entanglement for the\ncase of\u0014a=5\u00141(2), where the entanglement is still there and\nsurvives up to about 150 mK. In Fig. 3 (a) and (b) we take\n\u00141(2)=2\u0019=0:6 MHz, which is the lowest value demonstrated\nin the experiment [13]. It would be useful to study the entan-\nglement for larger magnon dissipation rates. In Fig. 3 (c) we\nplot the critical temperature versus \u00141(2)=2\u0019starting from 0.6\nMHz to 3 MHz. The critical temperature means that above\nwhich the entanglement becomes zero due to the degradation\nof the thermal noise. We see that the entanglement is quite\nrobust against the magnon dissipation rates and survives up to\nabout 80 mK for \u00141(2)=2\u0019=3 MHz.\nV . CONCLUSIONS\nWe have presented a protocol to entangle two magnon\nmodes in a cavity magnomechanical system, where the twomagnon modes in two YIG spheres couple to a microwave\ncavity mode, and one of them also couples to a vibrational\nmode of the sphere via magnetostrictive force. We have\nshown that with experimentally reachable parameters the two\nmagnon modes can be prepared in a steady-state entangled\nstate. The entanglement is achieved by exploiting the nonlin-\near magnetostrictive interaction and the linear cavity-magnon\ncoupling. The magnon entangled states in massive YIG\nspheres represent genuinely macroscopic quantum states, and\nare thus useful for the study of quantum-to-classical transi-\ntions and tests of decoherence theories [46]. In either the con-\ntinuous spontaneous localization theory [47, 48], or the gravi-\ntationally induced collapse model [49], the strength of the pos-\ntulated collapse noise is proportional to the size of the object\n(either mass or the number of particles it contains). The col-\nlapse e \u000bect becomes prominent when the size of the object is\nin macroscopic scale, which leads to spatial decoherence and\nlocalization of the superposition states, while it reproduces the\nstandard results of quantum mechanics in microscopic scale.\nTherefore, preparing macroscopic quantum states is a key\nstep to test those decoherence theories in macroscopic scale.\nFurthermore, the magnon entangled states can be applied to\nthe quantum information processing based on magnonic sys-\ntems [10], and can also be used for creating entangled states\nof microwave fields, e.g., by coupling each magnon mode to a\nmicrowave cavity and utilizing their beamsplitter (state-swap)\ninteraction.\nNote added : After the completion of this work, independent\nproposals have been put forward, using di \u000berent mechanisms,\nfor entangling two magnon modes, either in ferrimagnetic\nYIG spheres [50–52] or in an antiferromagnetic system [53].\nVI. ACKNOWLEDGMENTS\nWe thank G. S. Agarwal for continuous discussion and J. Q.\nYou for carefully reading the article and providing construc-\ntive feedback. This work has been supported by the National\nKey Research and Development Program of China (Grants\nNo. 2017YFA0304200 and No. 2017YFA0304202).\n[1] H. Huebl et al., Phys. Rev. Lett. 111, 127003 (2013).\n[2] Y . Tabuchi et al., Phys. Rev. Lett. 113, 083603 (2014).\n[3] X. Zhang et al., Phys. Rev. Lett. 113, 156401 (2014).\n[4] M. Goryachev et al., Phys. Rev. Appl. 2, 054002 (2014).\n[5] L. Bai et al., Phys. Rev. Lett. 114, 227201 (2015).\n[6] D. Zhang et al. , npj Quantum Information 1, 15014 (2015).\n[7] J. Bourhill et al., Phys. Rev. B 93, 144420 (2016).\n[8] N. Kostylev, M. Goryachev, and M. E. Tobar, Appl. Phys. Lett.\n108, 062402 (2016).\n[9] C. Kittel, Phys. Rev. 73, 155 (1948).\n[10] See D. Lachance-Quirion et al ., Appl. Phys. Express 12,\n070101 (2019), for a recent review on hybrid quantum systemsbased on magnonics.\n[11] Y . 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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|<mc)  is \n     \n             2\n2\n0 0 021 3sin cos  cotzxmH\nH m . \nThus, at the point | m|=mc  the system exhibits the transition characterized by zeroing of the \none of the susceptibility tensor element, so that it is similar to the transition from the \nferromagnetic to the paramagnetic state at the Curie point. The magnetic susceptibility tensor \nelement  zx z xdM dH  may be treated as the order parameter of the system.    \nFigure S3 shows how these cases appear in the experimentally studied \n(Bi0.6Gd2.4)(Fe4.28Ga0.57Ge0.15)O12 film with the parameters, discussed in Section I. The transition \nbetween the collinear and non-collinear phases determined as 2 m H K H   is shown by the \nblack line. The value of m ≈ Ms (see Section I) since in the non-collinear phase the angle between \nMFe and MR is less than 0.03 rad (see Fig. S4), and in the collinear state it equals 0. Therefore, \n|MFe-MR| ≈ |MFe|-|MR|=m. \n \n(a) \n \n (b)\n \n(c) \n \n \nFigure S3. 𝜃(𝑇,𝐻) diagrams showing (I)-(V) cases. (a) 𝜃(𝑇,𝐻) 3D plot showing the degeneracy \nand non-collinearity in the vicinity of the magnetization compensation point. (b), (c) false-color \nplots showing 𝜃(𝑇,𝐻) dependencies for 𝜃≥ 0 and 𝜃≤ 0 solution branches. m=mc condition is \nshown by the black line.  \n \nThe cases (II), (III) and (IV) are accompanied by the physical phenomenon of static non-\ncollinearity, which has a dual meaning. On the one hand, the antiferromagnetic vector L is non-\ncollinear to the external magnetic field vector H, as described above, and, on the other hand, the \nmagnetization vectors of the MR and MFe of the sublattices are also non-collinear to each other. It \nis convenient to discuss in terms of the bevel angle Δ between MFe and - MR: R Fe    . Using \nEq. (2), Eq. (7), and Eq. (10) Δ is expressed as: \n    022 sinHM  (11) \nThe angle Δ reaches its maximum Δ=2 MH/δ at the magnetization compensation point m=0 \n(Fig. S4). With the increase of m, Δ decreases reaching Δ=0 at | m|=mc. For |m|≥mc where 𝜃= 0  \nsublattice magnetizations are collinear to each other, and the case can be called the «collinear \nphase». At the same time, in both the collinear and non-collinear states MR and MFe lie in the XZ \nplane and 𝛽= 0. \n \n \n \nFigure S4. The Δ angle (in radians) vs temperature, according to Eq. (11).    \n \nWe have considered the configuration where the external magnetic field H is applied strictly \nin-plane so that 𝜃> 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": "2204.11595v1.Ultrafast_racetrack_based_on_compensated_Co_Gd_based_synthetic_ferrimagnet_with_all_optical_switching.pdf",        "content": "Ultrafast racetrack based on compensated Co/Gd-based\nsynthetic ferrimagnet with all-optical switching\nPingzhi Li1*, Thomas J. Kools1, Bert Koopmans1and Reinoud Lavrijsen1\n1*Department of Applied Physics, Eindhoven University of Technology, Eindhoven, 5612\nAZ, Netherlands.\n*Corresponding author(s). E-mail(s): p.li1@tue.nl;\nKeywords: synthetic ferrimagnet, angular momentum compensation, current-induced domain wall motion,\nall-optical switching\nSpin-orbitronics [1, 2] and single pulse\nall-optical switching (AOS) [3] of magne-\ntization are two major successes of the\nrapidly advancing \feld of nanomagnetism\nin recent years, with high potential for\nenabling novel, fast and energy-e\u000ecient\nmemory and logic platforms. Fast current-\ninduced domain wall motion (CIDWM) [4]\nand single shot AOS [5] have been individu-\nally demonstrated in di\u000berent ferrimagnetic\nalloys. However, the stringent requirement\nfor their composition control [5, 6] makes\nthese alloys challenging materials for wafer\nscale production [3]. Here, we simultane-\nously demonstrate fast CIDWM and energy\ne\u000ecient AOS in a synthetic ferrimagnetic\nsystem based on multilayered [Co/Gd] 2.\nWe \frstly show that AOS is present in\nits full composition range. We \fnd that\ncurrent-driven domain wall velocities over\n2000 m/s at room temperature, achieved by\ncompensating the total angular momentum\nthrough layer thickness tuning. Further-\nmore, analytical modeling of the CIDWM\nreveals that Joule heating needs to be\ntreated transiently to properly describe the\nCIDWM for our sub-ns current pulses. Our\nstudies establish [Co/Gd]-based syntheticferrimagnets to be a unique materials plat-\nform for domain wall devices with access to\nultrafast single pulse AOS.\nResearch in spintronics over the last decades\nhas demonstrated a possibility for further evo-\nlution of solid-state electronic application plat-\nforms beyond CMOS. The racetrack memory [7],\nan envisioned novel data storage device concept\nbased on using magnetic domain walls (DWs) [8],\nutilizes frontier spintronic mechanisms to func-\ntion as an ultra-dense on-chip memory with an\noperation speed comparable to low-level cache\n[9]. The e\u000eciency of racetrack memory relies\non the fast CIDWM in a material with per-\npendicular magnetic anisotropy. The maximum\nvelocity was signi\fcantly enhanced over the years\nby combining spin-orbit torques (SOTs) and the\nDzyaloshinskii-Moriya interaction (DMI) [10{12]\nwith synthetic antiferromagnets (SAFs), which\nresulted in reported DW velocities close to 750\nm/s [13].\nDespite the large improvement, the energy e\u000e-\nciency is still limited due to the weak strength\nof the antiferromagnetic (AF) coupling. There-\nfore, the materials platform of rare earth (RE)-\ntransition metal (TM) compounds garnered con-\nsiderable attention, promising faster CIDWM due\nto the much stronger direct AF coupling than the\n1arXiv:2204.11595v1  [cond-mat.mes-hall]  25 Apr 20222 Article Title\nindirect exchange coupling [14] utilized in SAFs.\nFurthermore, the SOTs used to drive the DW\npromise to be highly e\u000ecient in the RE-TM sys-\ntems as a result of the long spin coherence length\n[15]. Cosequently, high velocity CIDWM has been\nreported in Co-Gd-based ferrimagnetic alloy sys-\ntems [4, 16] when the angular momentum in the\nmagnetic material is compensated, being at least\na factor of three faster than that of the previously\nreported SAFs.\nBesides the e\u000ecient CIDWM, single pulse all-\noptical switching (AOS) of the magnetization\n[5] in the RE-TM systems has obtained signi\f-\ncant attention thanks to its sub-picosecond [17]\nenergy e\u000ecient [3, 18, 19] magnetization switch-\ning enabled by the ultrafast angular momentum\ntransfer upon laser excitation [20]. This can be\nuseful as a new generation of ultrafast magnetic\nmemory, as well as a data bu\u000ber between electron-\nics and integrated photonics [3, 21, 22]. Recently,\na synthetic ferrimagnetic system based on a Pt/-\nCo/Gd [19, 23] layered structure has shown high\nrobustness [24] for such a hybrid integration.\nThese kinds of synthetic ferrimagnets have\nsome distinct advantages over RE-TM alloys. For\ninstance, AOS is not limited by the exact compo-\nsition [25]. They also withstand thermal annealing\n[26] and o\u000ber easier magnetic composition control\nat wafer scale than the alloy system, as well as bet-\nter access to interface engineering. Therefore, it\nhas been proposed that such a materials platform\nhas high potential to realize a hybrid integration\nof DW memory in photonic platforms to further\nenhance their storage density [21, 27].\nSo far, the CIDWM of Co/Gd bilayers [27,\n28] has been investigated. However, the highest\nreported velocity, achieved at cryogenic conditions\n[28], was several times lower than that reported\nin alloys [4], in part due to large net angu-\nlar momentum, low compensation temperature as\nwell as DW pinning e\u000bects. In this report, we\ntherefore propose a materials platform based on\nthe [Co/Gd] 2synthetic ferrimagnet capable of\naccommodating both e\u000ecient CIDWM at room\ntemperature (RT) and single-pulse AOS, with\ncompatibility for wafer scale production as well as\nrobustness for engineering.\nThe magnetostatic and AOS properties of\nthe [Co/Gd] 2materials platform under investi-\ngation in this work are shown in Fig. 1. ATa/Pt/[Co/Gd] 2/TaN multilayer is deposited on\na Si/SiO 2substrate, where the \frst Gd layer is\nwedged (see Fig 1.a and Methods) to tune the\nnet magnetic moment. Due to the proximity e\u000bect\nby the Co, a net magnetization is induced in the\nGd which drops o\u000b steeply away from the Co\ninterface [29]. Compared to the bilayer Co/Gd\n[23, 28], we double the magnetic volume of the Co\nin [Co/Gd] 2, while tripling the number of inter-\nfaces at which magnetization is induced in the Gd.\nThis allows us to more easily compensate both\nthe angular momentum and magnetic moment of\nthe system at RT by varying the individual layer\nthicknesses.\nWe measured hysteresis loops by polar MOKE\nat RT across the thickness range of Gd between\n0-1.6 nm. In Fig. 1.b we plot the MOKE sig-\nnal intensity, de\fned as the di\u000berence in signal\nbetween positive and negative saturation, as well\nas the coercivity obtained from these hysteresis\nloops. The loops corresponding to the colored dots\nin Fig. 1.b are presented in Fig. 1.c.\nThe 100% remanence indicates a well-de\fned\nperpendicular magnetic anisotropy as well as tight\nexchange coupling between the layers. Impor-\ntantly, at the Gd thickness of 0.97 nm, the MOKE\nsignal switches sign (see black curve in Fig. 1.b).\nThis can be observed as well in Fig. 1.c as an\ninverted hysteresis loop. At this thickness the\nmagnetization is compensated, i.e. the Co and\nGd magnetization cancel each other. This is fur-\nther evidenced by the divergence in coercivity,\nand con\frmed by superconducting quantum inter-\nference device measurements (see Sup. A). Thus,\nupon increasing the thickness of Gd, the mag-\nnetic balance is shifted from being Co-dominated\nto Gd-dominated.\nWith respect to the magnetostatics, we note\nthat the magnetization and angular momentum\ncompensation thickness in these [Co/Gd]-based\nferrimagnets are similar but not identical due to\nthe di\u000berent Land\u0013 e g-factors of Co ( \u00182.2) and Gd\n(\u00182.0). Therefore, when discussing compensation\nwithout further speci\fcation in the remainder of\nthis text, we refer to angular momentum compen-\nsation as this is the relevant condition for e\u000ecient\nCIDWM [4, 28, 30].\nTo investigate whether our 4-layer Co/Gd sys-\ntem is all-optically switchable, and how it depends\non compensation conditions, we performed AOSArticle Title 3\na) b) c)\nCo(0.6)Gd(0-1.6)Co(0.6)Gd(1.5)TaN(4)\nPt(4)\nTa(4)d)\n1234567\nCo-dominated\nGd-dominated\nFig. 1 a): Schematic illustration of the layer stack used in our study (thickness in nm). b): Result of static MOKE study,\nwhere the MOKE signal (black) is de\fned by the di\u000berence in signal intensity between saturation at positive and negative\napplied \feld. Coercivity (red) was measured at a \fxed scanning speed of 10 mT/s (red). c): Hysteresis loops measured at the\nsample thicknesses marked by the coloured dots in b). d): Threshold \ruence for all-optical switching across the wedge shown\nin a). Inset shows toggle switched domains at two sides of the compensation thickness for varying amount of subsequent\npulses.\nexperiments by illuminating the wedge with sin-\ngle femtosecond laser pulses with varying pulse\nenergy, and examined the sample under a polar\nKerr microscope. We \fnd that single pulse AOS is\npresent at every thickness on the [Co/Gd] 2wedge.\nAn example of the resulting toggle switched\ndomains around the compensation boundary is\nshown in the inset of Fig. 1.d, which can be\nobserved from the contrast inversion (see Sup. B\nfor more images). This distinct feature that the\nAOS occurs both in the deep Co-dominated and\nGd-dominated composition is unlike that of alloys\n[18] and Co/Tb multilayers [31], in which AOS\nis only present within \u00061.5 % of the composition\nwindow. This shows the \rexibility of our system\nin terms of AOS engineering.\nWe further plot the threshold \ruence as a func-\ntion of Gd thickness, calculated from the switched\narea and the corresponding pulse energy [23],\nin Fig. 1.d. We \fnd that the threshold \ruence\ndepends signi\fcantly on the material composition,\nand is at a minimum around the compensation\npoint. Initially, the threshold \ruence decreases\nmonotonously with Gd thickness up to 1 nm, with\na reduction of the threshold \ruence of more than\n50% reaching 0.9 mJ/cm2, which corresponds to\n25 fJ for a 502nm2domain. Such a switching\nenergy is an order of magnitude lower than that\nof typical SOT and spin-transfer torque switching\nof a ferromagnet [32].\nAlthough a full dissection of the exact mech-\nanism behind the thickness-dependence of the\nthreshold \ruence is beyond the scope of this work,\nwe note that previous studies have shown thatreducing the Curie temperature of layered sys-\ntems can lead to a reduction of the threshold\n\ruence [6, 19, 23, 25]. In our case, as the Gd layer\nincreases, the Curie temperature of the total stack\nis expected to drop, because the mutual exchange\nstabilization of the two Co layers is weakened,\nleading to a reduction of threshold \ruence.\nWith compensation and AOS at RT con\frmed,\nin the remainder of this work we demonstrate\nand explain the DW velocities of over 2000 m/s,\nas achieved in [Co/Gd] 2. In the case of such an\nAF coupled system with DMI, two contributions\nmainly drive the coherent motion of the DW: the\nDMI-torque and the exchange coupling torque.\nThese two torques give rise to spin dynamics\nupon excitation by the spin Hall-e\u000bect-induced\nspin accumulation, and both lead to DW motion\nalong the current direction (for positive DMI and\nspin Hall angle [11, 33]).\nIt is well understood that the exchange cou-\npling torque only drives the DW motion e\u000eciently\nnear the compensation point [13, 28]. Adding\nadditional AF coupled layers to a ferromagnet\ncan facilitate this, however, can also increase the\nangular momentum of the DW to be translated.\nNonetheless, following the discussion in Sup. C, we\nstill expect a signi\fcant increase of the DW veloc-\nity in our study, as the enhancement of exchange\ncoupling torque due to compensation signi\fcantly\noverwhelms the e\u000bect of added magnetic inertia.\nFor this reason, we experimentally determined\nthe DW velocity from the CIDWM in [Co/Gd] 2\nas a function of lower Gd thickness. This study\nis performed in the same stack showcased in Fig.\n1.a. The wedged thin \flm was patterned into4 Article Title\na)\nb)c)\nd) e) f)\n0.4 ns\nFig. 2 a): Example image from DW motion experiments performed by di\u000berential polar Kerr microscopy, where the setup\nis illustrated schematically as a pulse generator and a 6 GHz bandwidth oscilloscope. The DW position after applying\nconsecutive pulses is shown. b): The measured DW velocity as a function of current density for three Gd layer thicknesses.\nc): The DW velocity as a function of Gd layer thickness for various current densities, where 10 % error bars are not shown.\nHere we marked the quanti\fed magnetization compensation thickness tMobtained from polar MOKE measurements as\nwell as the corresponding angular momentum compensation thickness tAbased on the estimation from magnetometry\nmeasurement. d): Compensation thickness as obtained from polar MOKE measurements of the [Co/Gd] 2at various ambient\ntemperatures. e): The temporal pro\fle of the current pulse used in the experiment (normalized by the peak current density),\nand its subsequent temperature rise (as simulated with COMSOL) at 3.6 TA/m2. f): Result of the theoretical 1-D modeling:\nthe DW velocity as a function of Gd thickness, where the magnetization pro\fle as well as its temperature dependence is in\nline with experimenes, while the current pulse and temperature pro\fle follow the pro\fle given in d). The velocity maximum\nis marked with a grey dot. The vertical dashed line indicates the angular momentum compensation thickness.\nmicro-structures as shown in Fig. 2.a (for details\nsee Methods and Sup. Fig. E4.b). The magnetic\ndomains were imaged by polar Kerr microscopy\n(see Fig. 2.a). Short current pulses of around 1 ns\nduration with varying amplitudes are injected into\nthe structures to determine the DW velocity (see\nSup. D). We observe that the DW moves coher-\nently along the current direction (see Fig. 2.a).\nThis con\frms a left-handed N\u0013 eel-type DW in our\nsystem as a result of a positive DMI, and a posi-\ntive spin Hall angle [11, 33], as is characteristic of\nthe Pt/Co interface [11].In Fig. 2.b we compare the resulting DW veloc-\nity as a function of peak current density for a\nCo-dominated (0.69 nm), compensated (0.93 nm)\nand Gd-dominated stack (1.28 nm), considering\nthe magnetic composition at RT. We demonstrate\nthat DW velocities of over 2000 m/s can be\nachieved at the highest current densities, at least\nthree times larger than that observed in SAFs [13].\nFurthermore, we notice from the crossing point\nbetween the red and black curve that the Gd\nthickness at which the highest velocity is observed\nbecomes larger with increasing current density. We\nattribute this to the e\u000bect of Joule heating inducedArticle Title 5\nby the current pulses. Earlier studies show that the\nGd magnetization is quenched more severely dur-\ning Joule heating than that of Co [28, 29, 34, 35].\nThe consequence is that at higher temperature rel-\natively more Gd than at RT is needed to achieve\ncompensation. This fact is con\frmed for our stacks\nby the upward shift of the compensation thickness\nwith increasing sample temperature using polar\nMOKE (see Fig. 2.d). We \fnd that this hypothesis\nof a Joule-heating-induced shift of the compensa-\ntion thickness explains the CIDWM experiments\nwell. To understand this, it useful to consider two\nregimes in Fig. 2.b in more detail.\nForJ < 2:3 TA/m2, before saturation e\u000bects\nstart to occur and for limited heating, we observe\na steady rise of the velocity, consistent with earlier\nwork on CIDWM in materials with AF coupling.\nAs expected, the compensated sample is driven\nmost e\u000eciently. We visualize this further by plot-\nting the DW velocity as a function of Gd thickness\nacross the Co and Gd-dominated regimes, as is\nshown in Fig. 2.c. Here, we observe that the DW\nvelocity in this range of current densities spikes\nat the RT compensation thickness, further prov-\ning the origin of the velocity increase is due to the\nenhanced exchange coupling torque.\nIn contrast, for J >2:3 TA/m2, the DW veloc-\nity saturates both for the Co-dominated and the\ncompensated sample. This is typical behaviour of\nthe CIDWM of a ferromagnet [11, 13, 28, 33, 36],\nwhich is limited by the DMI-torque. Contrar-\nily, the Gd-dominated sample shows a consistent\nvelocity rise typically associated with a compen-\nsated system [13]. This is consistent with the\nhypothesis that the increased Joule heating shifts\nthe compensation point to larger Gd thicknesses\nduring the pulse, and is supported by the velocity\npeak shift observed in Fig. 2.c.\nIn order to quantify the e\u000bect of the change\nin compensation thickness due to Joule heating\non the CIDWM, we \frst numerically investigate\nthe temperature transient for the current pulses\napplied in the experiment (see Sup. E). In Fig.\n2.e we plot a typical current density pro\fle and\nthe corresponding modelled temperature change\nin the device. We observe a delay of \u00180:4 ns\nbetween the peak current density, and the peak\ntemperature. This is a consequence of the factthat the pulse duration approaches the character-\nistic time scale of the heat transport between the\nsample and the substrate (See Sup.E).\nImportantly, around the peak current den-\nsity where the driving torque exerted on the DW\nis largest, the sample temperature is changing\nrapidly from 0 to 50 % of maximum temperature\nwithin 0.4 ns. Therefore, the net angular momen-\ntum changes rapidly during the application of\nthe pulse. This makes assigning an e\u000bective tem-\nperature during DW motion like previous studies\n[4, 28, 35, 37] inaccurate.\nTherefore, we incorporated the transient cur-\nrent density pro\fle, and the corresponding tem-\nperature rise into an analytical 1-dimensional\nmodel of the CIDWM (see Sup. F), where we\nintroduce a thickness and temperature dependent\nmagnetization pro\fle following our SQUID mea-\nsurements (see Sup. A). The resulting modelled\nDW velocity as a function of thickness and current\ndensity is plotted in Fig. 2.f.\nWe \fnd that the velocity peak shift with\nincreased current density as shown in Fig. 2.b is\nwell described by our model. Furthermore, we also\n\fnd that the rapidly changing angular momentum\nduring the current pulse gives rise to a broad-\nening of the velocity peak compared to the case\nwith constant temperature previously described in\n[4, 28](see Sup. G).\nThese results suggest that in general transient\ntreatment of the temperature pro\fle is neces-\nsary to properly address the DW dynamics for\nultrashort current pulses. Furthermore, to bene\ft\nfrom large exchange coupling torque at elevated\ntemperature, which is a likely scenario in high-\nspeed electronics, pushing the composition into\nthe Gd-dominated regime is required.\nIn summary, we have presented a materi-\nals platform of a synthetic ferrimagnet based\non [Co/Gd] 2which can be e\u000bectively tuned\nnear the compensation point via layer thickness,\nand exhibits e\u000ecient single pulse AOS. We also\nexperimentally and numerically showed that fast\nCIDWM with velocities over 2000 m/s can be\nachieved with the help of angular momentum\ncompensation. Finally, we addressed the transient\nJoule heating e\u000bect during CIDWM, which gives\nimportant insight into our experimental obser-\nvations and potential technological implementa-\ntion. Our results therefore lay the foundation\nfor an e\u000bective materials platform that exhibits6 Article Title\nboth ultrafast CIDWM and AOS, paving the way\nfor synthetic ferrimagnets to become the new\nparadigm of ferrimagnetic spintronics and pro-\nviding a jumping board for further integration\nbetween photonics and spintronics.\nMethods. Magnetic \flms were deposited on Si\nsubstrates with a 100 nm thermally oxidized SiO 2\nlayer by D.C. magnetron sputtering in a system\nwith a base pressure of \u00185\u000210\u00009mBar. A\nthickness wedge is created with the help of a\nmoving wedge shutter during sputtering. From\nthese thin \flms, nanostrips were fabricated using\nelectron-beam lithography and lift-o\u000b. The gold\ncontacts were made by wire bonding. The out-\nof-plane component of the magnetization (M z)\nof the nanostrips was measured by polar Kerr\nmicroscopy. An in-plane \feld of 200 mT along\nthe current direction, combined with a 3 TA/m2\ncurrent pulse was used to nucleate domains.\nBoth MOKE and Kerr Microscopy characteriza-\ntion were performed using a 700 nm continuous\nwave laser in the polar con\fguration. Here, we\nchose the wavelength of the laser light (around 700\nnm) such that the detected signal is only sensitive\nto the magneto-optical response from the Co layer\nand Kerr rotation is maximized.\nSupplementary information. All supplemen-\ntary information except for digital \fles are pre-\nsented in the Appendix.\nContribution. P.L. and T.J.K. contributed\nequally to this work in terms of design and con-\nduct of the project. B.K. and R.L. supervised the\nproject. All authors contributed to the writing of\nthe manuscript.\nAcknowledgments. This project has received\nfunding from the European Union's Horizon\n2020 research and innovation programme under\nthe Marie Sklodowska-Curie grant agreement\nNo.860060. This work is also part of the Gravita-\ntion programme `Research Centre for Integrated\nNanophotonics', which is \fnanced by the Nether-\nlands Organisation for Scienti\fc Research (NWO).\nCompeting Interests. Authors declare no\ncon\ricts of interest.Appendix A SQUID\ncharacterization\nIn order to characterize the magne-\ntostatic properties of the sample, a\nseries of 6 samples of composition\nTaN(4)/Pt(4)/Co(0.6)/Gd( tGd)/Co(0.7)/Gd(1.5)\n/TaN(4), with tGd= [0:2;0:4;::;1:2;1:4] nm were\ngrown. Using VSM-SQUID, out-of-plane hystere-\nsis loops were measured, which are shown in Fig.\nA1.a. The decreasing moment when going from\na middle Gd thickness of 0.4 to 1.4 nm, and the\ndiverging coercivity at the minimum of the mea-\nsured moment are indicative of a compensated\nsystem. This can also be seen in the plot of the\nmagnetic moment normalized by the sample area\nin Fig. A1.b. The temperature dependence of the\nmagnetization was also investigated, and a typical\nresult is shown in Fig. A1.c. In order to estimate\nthe Curie temperature of the stack and obtain an\nexpression for the magnetization Mtot, we follow\nearlier work on 3d-4f ferrimagnetic alloys in \ft-\nting a critical exponent to the data of the form\n[30, 38] given by:\nMtot=MCo(T)\u0000MGd(T) =\nMCo0\u0012\n1\u0000T\nTC\u0013\u0018Co\n\u0000MGd0\u0012\n1\u0000T\nTC\u0013\u0018Gd\n;\n(A1)\nwhereMGd0andMCo0and\u0018Gd,\u0018Coare the\nmagnetization at zero temperature and the critical\nexponent of Co and Gd, respectively. We assume\nMCo0= 1:4 MA/m, and \fnd that the critical\nexponents\u0018Co= 0.50,\u0018Gd= 0.72, and TC= 450 K\ndescribe the temperature dependence of the mag-\nnetic moment well (Fig. A1.c). We note that the\nmagnetization in the Gd is an ill de\fned quantity,\nsince the Gd transitions from partially magnetized\ndue to the proximity e\u000bect, to truly ferromag-\nnetic at low temperature. We therefore estimate\nthe value of MGd0by considering the known thick-\nness dependence of the magnetic moment in the\nstackmtot. This quantity is calculated as the sum\nof that of the individual layers as follows:\nmtot=mCo1+mGd1+mCo2+mGd2\n=MCo(T) (tCo1+tCo2)\u0000Article Title 7\na) b) c)\nFig. A1 VSM-SQUID characterization of the out-of-plane moment of the Co(0.6)/Gd(x)/Co(0.7)/Gd(1.5) material system.\na): Hysteresis loops across the compensation point at room temperature. b): Room temperature magnetic moment per unit\narea as a function of middle Gd thickness. c): Typical temperature dependence of the magnetic moment in a Co-dominated\nquadlayer sample. The red line indicates the best \ft of equation A1.\nMGd(T) (2tGd1+tGd2);(A2)\nwhere:\nmCo1=MCo(T)tCo1\nmCo2=MCo(T)tCo2\nmGd1= 2MGd(T)tGd1\nmCo1=MGd(T)tGd2: (A3)\nHere,tCo1,tCo2,tGd1,tGd2are the thicknesses of\nthe bottom (coming from the substrate) and top\nCo and Gd layers, respectively. We \fnd the red-\ncurve in Fig. A1 b, which represents equation (A2)\nfor the stack dimensions used in the experiment,\nwhich follows the SQUID data reasonably well by\nchoosingMGd0= 0.66 MA/m. It should be noted\nhere that this is about a factor of 2 lower than\nvalues typically found in CoFeGd alloys [30]. We\nargue that this is a consequence of the inhomoge-\nneous proximity induced magnetization in the Gd,\nwhich decays when moving away from the Co/Gd\ninterface, and is therefore concentrated at the\ninterface. In order to describe the magnetization\nfully, an intermixing pro\fle and proximity-induced\nmagnetization pro\fle needs to be assumed and\nsubstantiated, which is beyond the scope of this\nwork. We nonetheless choose this simple approach\nof homogeneous magnetization in the Gd, as the\nintra-layer exchange present in the Gd is expected\nto be strong enough to have the total angular\nmomentum in the Gd layer respond coherently to\nthe excitation of the DW, making the total angu-\nlar momentum present in the DW the relevant\nparameter and not the exact way it is distributed.Appendix B All optical\nswitching in\n[Co/Gd] 2\nIn addition to the meta-data of all-optical switch-\ning as shown in the main text (Fig. 1.d, repeated\nin Fig. B2.b), in Fig. B2.a we present a typical\nexcerpt of the polar Kerr microscope images used\nto obtain the threshold \ruence as a function of\nGd-thickness.\nAppendix C Velocity\nenhancement\nexpectation\nTo argue why a signi\fcant velocity enhancement is\nexpected regardless of the added magnetic volume\nfrom the multiple magnetic layers, it is instructive\nto discuss the ratio between the expected DMI-\ntorque-driven DW velocity in an uncompensated\nsamplevucto the exchange coupling torque-driven\nDW velocity in a compensated sample vc. Here we\ndiscuss this using an analytical approach based on\na 1-D model of DW motion [13, 28, 39] (see also\nsection F), the ratio is given by:\nvc\nvuc=~J\u0012SH\u0001\n2e\u000bDAuc\nAc: (C4)\nHere, we consider the fact that vuc, is limited\nby the strength of the DMI, following [4, 36]:\nvuc=\u0019D\n2P\nnjAnj;=\u0019D\n2Auc; (C5)8 Article Title\na)\nb)\n200 μm\n1.50\n1.30 \n1.05 \n0.96 \n0.80 \n0.50 \n0.30 \n0.10 Gd thickness (nm)\n180 160 140 120 100 80 60 40\nPulse enery (nJ)\nFig. B2 (a) Di\u000berential Kerr images (background taken\nat a Gd thickness of 0.1 nm) taken at di\u000berent locations\non the Gd wedge sample, which was illuminated by single\nlaser pulses with varying pulse energy. The inverted back-\nground contrast around 1 nm is due to the transition from\nCo-dominated to Gd-dominated. b): Calculated threshold\n\ruence as a function of Gd layer thickness. The threshold\n\ruence below and above compensation is calculated with\nthe average of 4 neighboring rows.\nwhereDis the surface DMI energy density, and A n\nandAucare the total angular momentum of the\nnthlayer and the full stack, respectively. On the\nother hand, vcis for strong enough exchange cou-\npling limited by the amount of angular momentum\ntransferred to the DW via the spin Hall-e\u000bect, and\ncan for a multilayered system be expressed as:\nvc=~J\u0012SHE\u0001\n4e\u000bP\nnjAnj=~J\u0012SHE\u0001\n4e\u000bA c:; (C6)where ~is the reduced Planck constant, Jis the\ncurrent density, \u0012SHE is the spin Hall angle for\nanti-damping-lik torque, eis the electron charge, \u000b\nis the Gilbert damping parameter, Acis the total\nangular momentum of the full stack, and \u0001 is the\nDW width, which shows no dependence on the\nDMI.\nTo estimate the expected velocity enhance-\nment, we consider equation (C4) for the simple\ncase where Auc=Ac= 1=2. This corresponds to\nthe situation where one antiferromagnetically cou-\npled layer is added to a magnetic system, which\nexactly compensates the angular momentum in\nthe original system. Within this simple model,\nand using equation (C4), the condition for DW\nvelocity enhancement then becomes:\nvc\nvuc=~J\u0012SH\u0001\n4e\u000bD>1: (C7)\nWe \fnd that for the parameters used in the\nsimulations (see Sup. F), a net gain in DW veloc-\nity can already be obtained at a current density\nof 0.28 TA/m2. Therefore, the exchange cou-\npling torque in our system can be expected to\nresult in a much higher velocity compared with\nthe uncompensated case where the DMI-torque\ndominates regardless of the inevitable increase in\nmagnetic volume for the synthetic ferrimagnets\nunder discussion.\nAppendix D Domain wall\nvelocity charac-\nterization\nIn this section, we discuss the experimental details\nof the characterization of the current-induced DW\nvelocity.\nWe \frst discuss the principle of the measure-\nment. To magnetically saturate the sample, we\n\frst reverse the magnetization in the wire with\nrespect to the rest of the device by a combined\naction of an intense current pulse and an in-\nplane \feld along the current direction. Then, N\ncurrent pulses with amplitude J, and pulse dura-\ntion \u0001t, displace the DW by a distance \u0001 L,\nwhich is recorded by polar Kerr microscopy. The\nresulting DW velocity, vDW, is then calculated as\nvDW=\u0001L\nN\u0001t. We applied multiple pulses of the\nsame amplitude to allow the DW to travel enough\ndistance from the start to the end of the wire toArticle Title 9\nminimize the errors of the length measurement\nand statistical errors caused by variation between\npulses as a result of wire edge roughness and other\npotential fabrication imperfections. For the case of\nhigh current density or high velocity, this process\nis repeated several times for better averaging due\nto lowNbeing needed to translate the DW along\nthe full wire length.\nThe method mentioned above requires the\nideal case of a current pulse with a square pro-\n\fle. Next, we discuss the pulse pro\fle used in our\nexperiment, based on which we de\fne our e\u000bec-\ntive pulse width and amplitude. The amplitude\nand duration of the pulse needs to be calibrated\ncarefully in order to clearly de\fne the DW veloc-\nity. In our measurement set-up, we terminate\nthe device with a 6 GHz bandwidth oscilloscope\n(characteristic impedance 50 \n), allowing us to\ncharacterize the electrical pulse pro\fle directly\nduring the measurements. The resulting voltage\nversus time pro\fle obtained from the oscilloscope,\nis then converted to current density values versus\ntime based on the DC resistance and the geom-\netry of the devices. Here, we have to note that\nin our calculations, only the thickness of Pt was\nconsidered in our calculation (assuming no cur-\nrent \rowing in both the ferromagnetic layers and\nTa). Such an assumption is made because Pt is\nknown to be more conductive than the other met-\nals used in the stack [28]. Therefore, the energy\ne\u000eciency of CIDWM in our study is a conservative\nestimate, since the current is likely also \rowing\nin the other metallic layers. To avoid dramatic\nheat e\u000bects induced by long current pulses, which\nmight screen much of the physics, we used ultra-\nshort pulses down to 1 ns in our study. The pulse\namplitude can reach up to 4 \u00021012A/m2without\nfully demagnetizing the sample. A typical mea-\nsurement of the current pulse pro\fle is shown in\nFig. D3.a. In our measurement, we de\fne the pulse\namplitude to be the peak value (1.8 \u00021012A/m2in\nthis case). The pulse exhibits a spike shape, which\nrequires us to de\fne an e\u000bective pulse width. We\nwe de\fne the e\u000bective pulse width in our mea-\nsurement by taking the full width half maximum\n(0.8 ns) of the active area (see gray region in\nFig. D3.a) of the pulse obtained by subtracting\nthe pulse shape by a strong, experimentally deter-\nmined, pinning threshold (0.5 TA/m2). Thus, the\ntime needed for the current pulse to pull up andthe decay of the pulse below the pinning thresh-\nold is not considered, since in this region the DW\nis not displaced or to a limited extent. To de\fne\nthe pinning threshold, we applied over 500 current\npulses when the DW is at various positions of the\nwire. The pinning threshold is then de\fned as the\ncurrent density where no detectable movement of\nthe DW is found. A further reason for using short\npulses is the fact that the temperature rise during\npulse time can be easily suppressed by the heat\ndissipation and capacity of the substrate since\nthe heat front is still in the propagation regime.\nThis also leads to a transient delay of temperature\nrise (see the next sections for its in\ruence on the\nCIDWM).\nTo justify the e\u000bective pulse parameters,\nwe compare the measurement result with that\nobtained from a longer pulse (L) and a middle\nlong pulse (M) shown in Fig. D3.b, of which the\nshape is closer to a square pulse. As seen from\nFig. D3.b, the L and M pulses both have an over-\nshoot and a slanted decay at the end of the pulse.\nWe compensate for this e\u000bect by subtracting the\nDW displacement induced by a shorter pulse from\nthat of a longer pulse. Since the di\u000berence between\nthese pulse shapes resembles a square pulse shape\nwith known pulse amplitude and width, the DW\nvelocity can be more accurately calculated. How-\never, in order to neglect the contribution from\nthe part of the pulse that was not subtracted, it\nrequires the thermal e\u000bect to be low (since the ini-\ntial spike contributes heat as well). Thus, it is only\napplicable for low current density.\nNow, we introduce a reference pulse (R)\n(shown in red), which resemble the spike of L and\nM, such that its di\u000berence with L and M will\nresults in a square pulse with a duration longer\nthan 1 ns as shown in Fig. D3.b. We calculated the\nDW velocity for the samples used in our experi-\nment (shown in Fig. 2) based on the pulse time\ndi\u000berence between L and M (L-M), L and R (L-R),\nas well as M and R (M-R), which are plotted in\nFig. D3.c (current amplitude used as shown in Fig.\nD3.b). We compare the obtained results with that\ncalculated based on the short pulse (S) used in\nthe experiment shown in the main paper (see blue\ncurve in Fig. D3. b), which di\u000bers in amplitude\nfrom the pulse pro\fle in Fig. D3. a.\nWe observe that the DW velocities using these\napproaches are in the same order of magnitude,\nfollowing the same general dependence on Gd10 Article Title\na) b) c)\nFig. D3 a): The pulse pro\fle of a typical pulse used in the DW velocity measurements. The strong pinning region which\nwas excluded during the pulse width estimation was marked as brown, the active region from where the e\u000bective pulse width\nis obtained is mark as gray. The pulse width and amplitude obtained for this characteristic pulse pro\fle are marked in the\nplot. b): The pulse pro\fle of a short pulse (S) used for the experimental results shown in the main text, a middle long pulse\n(M), a long pulse (L) and a reference pulse (R). The estimated time di\u000berence between those pulses are given and will be\nused to calculate the DW velocity. c): The DW velocity obtained from a short pulse, and other di\u000berential pulse method.\nthickness. The DW velocity calculated using S is\nfound to be systematically lower. Here we choose\nto be conservative instead of trying to compensate\nfor this discrepancy, as \fnite thermal activation\nis expected to increase the DW velocity by reduc-\ning the DW friction [11, 28, 40], as is the case for\nlong pulses. So far, little study has been spent on\nsuch an issue in the \row regime of the CIDWM,\nalthough the absolute magnitude of enhancement\nwas once shown to be 20 % for a SAF [41] for 40 K\nof temperature increase. So we make a compromise\nby taking a large but safe margin avoiding over-\nestimation of the DW velocity as well as possible\nover-claim (for large velocity).\nAppendix E COMSOL\nsimulations of\nJoule heating\nIn this section, we present our study on the tem-\nperature rise during the application of the current\npulses. This is a crucial ingredient for the anal-\nysis of the experimentally observed DW motion,\nas well as design considerations for future appli-\ncations. Here, we based our study on COMSOL\nsimulations, which employs multiphysics modeling\nincluding thermal transport and electrical con-\nduction. In our model, we de\fned the geometry\nbased on the real physical dimensions of the device\nused in this study (see Fig. E4.b) and the mate-\nrial parameters either from the COMSOL material\nlibrary or from our measurements (see Table E1),\nwhich takes into account temperature-dependente\u000bects. We model the metallic stack as a dou-\nble layer consisting of a 8 nm non-conductive\nlayer, and a 6 nm conductive layer, of which\nthe conductivity is obtained from RT DC mea-\nsurements, while the density, heat capacity and\nthermal conductivity of the two layers are kept the\nsame.\nTable E1 Parameters used in COMSOL\nsimulation for Joule heating induced by the\ncurrent pulse.\nParameter @ RT Value Unit\nThickness of the Si substrate 0.5 mm\nThickness of SiO 2 100 nm\nDiameter of wire bond 50 \u0016m\nHeat Capacity of Si Substrate 678 J/(kg \u0001K)\nDensity of Si 2320 kg/m3\nThermal Conductivity of Si 134 W/(m \u0001K)\nHeat Capacity of SiO 2Substrate 730 J/(kg \u0001K)\nDensity of SiO 2 2200 kg/m3\nThermal Conductivity of SiO 2 1.4 W/(m \u0001K)\nThickness of conductive metal \flm 6 nm\nThickness of non-conductive metal \flm 8 nm\nElectrical Conductivity of Metal 0.893 106\u0001S/m\nHeat Capacity of the Metal 133 J/(kg \u0001K)\nDensity of the metal 21450 kg/m3\nThermal Conductivity of the Metal 71.6 W/(m \u0001K)\nRelative permittivity 10000 1\nMinimum Mesh Dimension 0.5 nmArticle Title 11\nElectrical stimuli are applied through a circu-\nlar region (yellow disks in Fig. E4.b), of which\nthe size is close to the physical size of the electri-\ncal contacting wire bonds in our experiments. The\nwire is signi\fcantly heated compared with the rest\nof the metal structure owing to its large current\ndensity. Air convection is present only at the top\nsurface to emulate the experimental conditions.\nFollowing these notions, we computed the temper-\nature rise relative to RT as a result of a current\npulse similar to the one used in the experiment,\nas indicated in Fig. E4.a. In order to describe\nthis pulse pro\fle, we \ft an empirical function J(t)\nconsisting of the sum of three Gaussians:\nJ(t) =3X\ni=1Ai\nwip\n\u0019=2exp \n\u00002(t\u0000tc,i)2\nw2\ni!\n:(E8)\nThe best-\ft parameters are summarized in\ntable F3. The same pro\fle is used to de\fne\nthe voltage pulse that is applied to the system\nin COMSOL. The resulting temperature rise for\nvarious peak current densities is plotted in Fig.\nE4.c. One important observation here, is that the\nheating of the magnetic layer is delayed by approx-\nimately 0.4 ns with respect to the current pulse.\nConsequently, when the SOT is maximized at the\npeak of the current pulse, the angular momen-\ntum is still rapidly changing. Since the exchange\ncoupling torque e\u000eciency is linked to the angu-\nlar momentum balance between the Co and Gd,\nthis transient heating leads to a transient DW\nmobility.\nFor Joule heating, the expected temperature\nrise depends quadratically on the current den-\nsity. In order to describe the temperature rise for\nany value of the peak current density J0, we \ft\nan asymmetric sigmoidial function to the tem-\nperature rise at various current densities given\nby:\nT(t) =y0+b01\u0000exp\u0010\nt\u0000tc\nl2\u0011\n1 + exp\u0010\n\u0000t\u0000tc\nl1\u0011: (E9)\nThe resulting \fts for various current densities are\nshown in Fig. E4.d. During this \ft all parameters\nexcept the amplitude of the peak b0were taken\nconstant. The found amplitude, which describesthe heating is then plotted as a function of cur-\nrent density in Fig. E4.e. We \ft a simple quadratic\nfunction to this quantity:\nb0(J0) =k0J2\n0; (E10)\nwhich is shown as the red line. The correspondence\nbetween the \ft and the temperature rise modelled\nby COMSOL suggests that Joule heating plays the\ndominant role in our system.\nIn summary, based on the best \ft to COMSOL\nsimulations and the quadratic dependence of the\ntemperature change on current density, we made\nan estimation of the temporal pro\fle of the tem-\nperature rise in our sample. This asymmetric sig-\nmoidal pro\fle with the quadratic prefactor given\nin equation E9 and E10, will be used to dynam-\nically describe the change in temperature in the\nnumerical 1-dimensional simulations of CIDWM\nin our [Co/Gd] 2samples.\nTable E2 Summary of the \ftting parameters\nin equations E8 and E9, found to best describe\nthe experimental current pro\fle and the\nmodeled temperature rise due to Joule\nheating, respectively.\nParameter Value Unit\ntc,1 -2.85 ns\nw1 0.42 ns\nA1 0.54 TA m\u00002ns\ntc,2 -3.29 ns\nw2 1.23 ns\nA2 0.29 TA m\u00002ns\ntc,3 -2.29 ns\nw3 0.59 ns\nA3 0.31 TA m\u00002ns\ny0 293.15 K\ntc 2.15 ns\nl1 0.11 ns\nl2 1.94 ns\nk0 20.7 K m4TA\u0000212 Article Title\na) b)\nc) e)\nSiO2 (100 nm)\nSi (500 μm)200 μm 600 μm 100 μm \n50 μm 50 μm 2 μm \n10 nm \nd)\nFig. E4 Joule heating analysis and current pulse de\fnition for the model. a): Fit of three Gaussians (eq. E8) to a typical\nexperimental current pulse pro\fle. b): Schematic of the device geometry used in the experiment, as well as in COMSOL. c):\nTransient temperature modeled using COMSOL for the experimental current pulse pro\fle at various peak current densities.\nd): Fits of equation E9 to the temperature pro\fle at various peak current densities J0. e): Parabola \ft to the prefactor b0\nin eq. E9 obtained from the analysis from d).\nAppendix F 1-dimensional\nmodel of\ncurrent induced\ndomain wall\nmotion\nIn this section the analytical 1D-model for\ncurrent-induced DW motion in the [Co/Gd] 2mag-\nnetic system is described. This model is an exten-\nsion to earlier work by Bl asing et al. [28], who\nintroduced the model to describe the DW dynam-\nics of a Co/Gd bilayer. We will \frst summarize\nthe main points of this model, noting that a full\ndiscussion, can be found elsewhere [41].\nThe model discussed below describes the\ndynamics of an up-down DW in a nanowire device\n[length (x-direction)\u001dwidth (y-direction)\u001d\nheight (z-direction)]. The orientation of the spins\nin the DW will be discussed in terms of spherical\ncoordinates \u0012and\u001e, which describe the polar and\nazimuthal angles of spins in the system, respec-\ntively. We de\fne \u0012= 0 and\u0012=\u0019to be the\npositive and negative out-of-plane ( z) direction\nrespectively, and \u001e= 0 and\u001e=\u0019to be pointingto the positive and negative x-direction, respec-\ntively. Within the DW, the spins rotate gradually\nfrom\u0012= 0 to\u0012=\u0019. In equilbrium, the DW angle,\nwhich we de\fne as the azimuthal angle of the spin\nin the middle of the DW will point along \u001e=\u0019\n(-x) due to the counterclockwise DMI.\nIn order to then derive the equations of\nmotion of the spins in the DW in the presence\nof non-conservative forces, like Gilbert damping\nand current-induced torques like the spin transfer\ntorque (STT) and spin Hall e\u000bect (SHE) torque,\nthe Rayleigh-Lagrange equation is solved:\n@L\n@pi\u0000d\ndt@L\n@_pi\u0000@F\n@_pi: (F11)\nHerepirefers to the free variables in the system,\nLandFdescribe the Lagrangian and the dissi-\npation function, respectively. Within our model\nof [Co/Gd] 2there are \fve free variables: The\nazimuthal angles \u001efor the separate layers, and the\nDW position q, from which the DW velocity _ qis\nderived:\n_q=\u0000\u0001\nsin\u0012_\u0012: (F12)\nHere it is assumed that the DW positions are\ntightly coupled, such that qis always the same forArticle Title 13\nthe di\u000berent layers. The quantities LandFare\nde\fned as:\nL=Z1\n\u00001!DW\u0000Tdx; (F13)\nand\nF=Z1\n\u00001P\u000b+PSTT+PSHEdx; (F14)\nrespectively, where P\u000b,PSTTandPSHE, are the\ndissipation functions for Gilbert Damping, STT\nand SHE respectively, Tis the \"kinetic energy\"\nof the DW which we take identical to Blaesing et\nal. [41], and \fnally !DWis the energy density of\nthe DW. In order to de\fne !DW,P\u000b,PSTTand\nPSHE, it is useful to introduce some compact nota-\ntion for parameters corresponding to a particular\nmagnetic layer. We will refer to parameters corre-\nsponding to speci\fc magnetic layers through the\nsubscripts 1, 2, 3 and 4, where the numbering\nrefers to the layer in the Pt/[Co/Gd] 2structure\nstarting with layer 1 from the bottom upwards\n(i.e. layer 1 is the Co layer interfaced with the Pt\netc.). Using this notation !DWis given by:\n!DW=D1\n\u0001cos\u001e1sin\u0012\n+4X\ni=1Kitisin2\u0012+4X\ni=1Aex,iti\n\u00012sin2\u0012\n+3X\ni=1Jex\u0000\ncos2\u0012\u0000cos (\u001ei\u0000\u001ei+1) sin2\u0012\u0001\n:(F15)\nHere,Dis the DMI constant, \u0001 is the DW width,\nKiis the anistropy energy density, tiis again the\nthickness,Aexis the exchange sti\u000bness, and Jex\nis the exchange coupling strength. The DMI is\nassumed to only interact with the bottom Co layer\n(i= 1) as it originates from the Co/Pt inter-\nface, and no DMI at the Co/Gd interface has yet\nbeen reported. The \fnal term, describing the anti-\nferromagnetic exchange coupling has been chosen\nsuch that negative Jexpromotes antiferromagnetic\ncoupling.\nNext, the dissipation functions are de\fned as:P\u000b+PSTT+PSHE=4X\ni=1\u0000\u000bi;mi\u0001\n\ri\u0010\n_\u001ei2+ _q2=\u00012\u0011\n+4X\ni=1\u001diuimi\n\u0001\risin2\u0012_\u001ei\u0000\fiuimi\n\u00012\risin2\u0012_q\n\u0000~\u0012SHJ\n2esin\u0012\u0012cos\u001e1_q\n\u0001+ sin\u001e1cos\u0012_\u001e1\u0013\n;\n(F16)\nwhere\u000bis the Gilbert damping coe\u000ecient, \ris the\ngyromagnetic ratio, \fis the parameter describ-\ning the non-adiabatic component to the STT, ~is\nthe reduced Planck constant, \u0012SHis the spin-Hall\nangle,Jis the current density, eis the electron\ncharge, and \u001diis a variable that is equal to 1 and\n-1 in a Co layer ( i= 1;3) and Gd layer ( i= 2;4)\nrespectively. Finally, uiis given by:\nui=~Pi\ridi\n2emiJ; (F17)\nwherePiis the degree of polarization of the\ncharge current. It should be noted that the spin-\nHall current is in this work modelled to only\ninteract with the Co layer interfaced with the Pt.\nRecent work has indicated that the spin coherence\nlength in ferrimagnetic multilayers can be larger\nthan the thickness of the individual layers in this\nstudy [15]. Hence, it is possible that part of the\nspin current injected along the z-direction can also\ninteract with the other magnetic layers when it is\n(partially) transmitted through the \frst Co layer.\nIn order to then solve equation (F11) for the\n\fve degrees of freedom in our system, we evalu-\nate the integrals in equations (F13) and (F14) by\nassuming a typical Bloch pro\fle of the polar angle\n\u0012as a function of x:\n\u0012(x) = 2 tan\u00001\u0012\nexp\u0014x\u0000q(t)\n\u0001\u0015\u0013\n: (F18)\nThe resulting equations of motion in absence of an\napplied \feld for q,\u001e1,\u001e2,\u001e3,\u001e4are then \fnally\ngiven by:\n_q=\u0001\u0010P4\ni=1mi\u000bi\n\ri\u0001\u0011\u0012\n\u00004X\ni=1miui\fi\n\ri\u0001\n\u0000\u0019~\u0012SHJ\n4ecos\u001e1\u00004X\ni=1\u001dimi_\u001ei\n\ri\u0013\n(F19)14 Article Title\n_\u001e1=u1\n\u000b1\u0001+D1\u0019\r1sin\u001e1\n2m1\u000b1\u0001\n\u0000Jex\r1sin\u001e1\u0000\u001e2\nm1\u000b1+_q\n\u000b1\u0001(F20)\n_\u001e2=\u0000u2\n\u000b2\u0001+Jex\r2sin\u001e2\u0000\u001e3\n\u000b2m2\n\u0000Jex\r2sin\u001e1\u0000\u001e2\n\u000b2m2+_q\n\u000b2\u0001(F21)\n_\u001e3=u3\n\u000b3\u0001\u0000Jex\r3sin\u001e3\u0000\u001e2\nm3\u000b3\n\u0000Jex\r3sin\u001e4\u0000\u001e3\n\u000b3m3+_q\n\u000b3\u0001(F22)\n_\u001e4=u4\n\u000b4\u0001\u0000Jex\r4sin\u001e4\u0000\u001e3\n\u000b4m4+_q\n\u000b4\u0001:(F23)\nWe extended the model described earlier by\nBlaesing in three ways. First, we implement\nthe thickness and temperature-dependence of the\nmagnetization based on the SQUID measurements\ndiscussed in section A, in order to account for the\ndi\u000berent thicknesses of the samples investigated in\nthe experiment. Second, the temperature of the\nsample, contrary to earlier work, is also taken as\na dynamic parameter since we \fnd that for the\nultrashort current pulses investigated in the main\ntext, the sample is still heating up rapidly at the\npeak of the current pulse, when the driving torque\nis largest. Finally, we trivially extended the mod-\neled magnetic system from two to four coupled\nmacrospins.\nIn order to arrive at the results presented in the\nmain text, we numerically solve these \fve coupled\ndi\u000berential equations. For this we use the tempo-\nral current density pro\fle Jas given in equation\nE8 and table E2, taking into account the exper-\nimentally observed threshold current density of\n0.5 TA/m2(see Sup. D). The magnetization and\nits temperature dependence are implemented as\ndescribed in section A. All other physical parame-\nters used in the model are summarized in table F3.\nIn order to \fnally characterize the velocity fromthe model, we divide the DW displacement by the\nsame e\u000bective pulse width of 0.8 ns used to cal-\nculate the experimental domain wall velocity (see\nSup. D).\nWe \fnally note that for the realistic mate-\nrial parameters obtained from literature, the 1-D\nmodel generally predicts lower DW velocities than\nthe experiments. However, an increase of the DW\nwidth from 8 to 12 nm (both reasonable for\nthese kind of ferrimagnetic systems [47, 48]) could\nalready address this discrepancy in velocity with-\nout impacting the qualitative dependence of the\nDW velocity on the Gd thickness and current\ndensity.\nAppendix G Velocity peak\nbroadening\ndescribed by\ndynamic\ntemperature\nTo further substantiate the claim that dynamic\nsimulation of the temperature is important to\nproperly interpret the CIDWM experiments in our\n[Co/Gd] 2multilayers, we make a more elaborate\ncomparison between simulation with a dynamic\ntemperature pro\fle, and a constant temperature.\nFor this we use the same simulations as pre-\nsented in other parts of this work, and the same\ntemperature pro\fle for the dynamic temperature\nsimulations. For the constant temperature simu-\nlations we take the temperature to be 85% of the\nmaximum temperature of the dynamic tempera-\nture pro\fle, as this gives rise to a similar shift of\nthe thickness at which the peak velocity occurs.\nA typical comparison of the modelled DW veloc-\nity between static and dynamic temperature can\nbe seen in Fig. G5 for peak current densities J0of\n1.6, 2.6 and 3.6 TA/m2. It can be seen that for low\ncurrent densities the resulting velocity pro\fles are\nalmost identical, whereas for high current densi-\nties the velocity is more sharply peaked around the\ncompensation thickness. The main e\u000bect at high\ncurrent density of the dynamic temperature pro-\n\fle is a simultaneous lowering of the peak velocity\nand a broadening of the velocity peak, both of\nwhich can be understood from the magnetic com-\nposition changing rapidly throughout the peak ofArticle Title 15\nTable F3 Parameters used in 1D-modeling of CIDWM in [Co/Gd] 2.\nParameter Unit Co 1 Co2 Gd1 Gd2\ng - 2.2(a)2.2(a)2.0(b)2.0(b)\n\r 1011rad/s/T 1.93(c)1.93(c)1.76(c)1.76(c)\n\u000b - 0.1(d)0.1(d)0.1(e)0.1(e)\n\u0001 nm 8(f)8(f)8(f)8(f)\nJex mJ/m2-0.9(g)-0.9(g)-0.9(g)-0.9(g)\n\f - -0.5(h)-0.5(h)-0.5(h)-0.5(h)\nP - 0.1(i)0.1(i)0.1(i)0.1(i)\n\u0012SH - 0.08(j)0(k)0(k)0(k)\nD pJ/m 0.25(l)0(m)0(m)0(m)\naFrom [42, 43].\nbFrom [44].\ncCalculated using the layer's respective Land\u0013 e g-factor as \r=g\u0016B\n~.\ndFrom [11].\neAssumed similar to that of Co [45].\nfSimilar to [4, 28]. Assumed constant throughout the layers due to strong\nexchange.\ngFrom [46].\nhEstimate obtained from [16].\niEstimate obtained from [16].\njEstimate obtained from [47, 48].\nkSpin current interaction with the 4f moments in Gd is very small [49, 50].\nlChosen such that the simulations \ft the _ qvs.Jexperimental data.\nmNo DMI has been reported at the Co/Gd interface.\na) b) c)\nFig. G5 Comparison of using dynamic and static temperature in the 1D model of CIDWM. The constant temperature is\nchosen to be 85% of the maximum temperature from the COMSOL simulations at a given peak current density, as it leads\nto a comparable shift of the velocity peak. a): 1D CIDWM simulations comparing the velocity for the two temperature\npro\fles at 1.6, 2.6 and 3.6 TA/m2. b): Normalized DW velocity for the 1D CIDWM simulation with the constant and\ndynamic temperature pro\fles, compared with the normalized velocities found from the experiments at peak current density\nJ0= 2:6 TA/m2. c): Same as b), but for J0= 3:6 TA/m2.\nthe current density. As the degree of compensa-\ntion changes, the resulting translation of the DW\nbecomes more of an average over many di\u000berent\nmagnetic compositions which have varying degrees\nof DW mobility.In the experiments a similar phenomenon can\nbe observed, and is qualitatively shown in Fig.\nG5.b and G5.c. Again, for the lower current den-\nsity (J0= 2:6 TA/m2, \fg. G5.b), the heating\ne\u000bect within the temporal width of the current\npulse is limited, and the velocity with thickness is16 Article Title\nwell described by both the constant and dynamic\ntemperature. A drastic change is observed how-\never for large current densities ( J0= 3:6 TA/m2,\n\fg. G5.c), where we \fnd that the dynamic tem-\nperature pro\fle describes the trend of the exper-\niment well. Contrarily the constant temperature\napproximation does not work anymore beyond\npotentially pinpointing the shift in the velocity\npeak.\nReferences\n[1] S. K. Kim, G. S. D. Beach, K.-J. Lee, T. Ono,\nT. Rasing, and H. Yang, \\Ferrimagnetic spin-\ntronics,\" Nature Materials , vol. 21, no. 1,\npp. 24{34, 2022.\n[2] B. Dieny, I. L. Prejbeanu, K. Garello,\nP. Gambardella, P. Freitas, R. Lehndor\u000b,\nW. Raberg, U. Ebels, S. O. Demokritov,\nJ. Akerman, A. Deac, P. Pirro, C. Adel-\nmann, A. Anane, A. V. Chumak, A. Hirohata,\nS. Mangin, S. O. Valenzuela, M. C. 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Si\nSchool of Materials and Engineering,\nLanzhou University, Lanzhou 730000, China\nThomas F. George\nDepartments of Chemistry & Biochemistry and Physics & Astro nomy\nUniversity of Missouri-St. Louis, St. Louis, MO 63121, USA\n(Dated: July 22, 2022)\n1(July 22, 2022)Abstract\nAll-optical spin switching (AOS) represents a new frontier in magnetic storage technology –\nspin manipulation without a magnetic field, – but its underly ing working principle is not well\nunderstood. Many AOS ferrimagnets such as GdFeCo are amorph ous and renders the high-level\nfirst-principles study unfeasible. The crystalline half-m etallic Heusler Mn 2RuGa presents an op-\nportunity. Here we carry out hitherto the comprehensive den sity functional investigation into the\nmaterial properties of Mn 2RuGa, and introduce two concepts - the spin anchor site and th e optical\nactive site - as two pillars for AOS in ferrimagnets. In Mn 2RuGa, Mn(4 a) serves as the spin anchor\nsite, whose band structure is below the Fermi level and has a s trong spin moment, while Mn(4 c)\nis the optical active site whose band crosses the Fermi level . Our magneto-optical Kerr spectrum\nand band structure calculation jointly reveal that the deli cate competition between the Ru-4 dand\nGa-4pstates is responsible for the creation of these two sites. Th ese two sites found here not only\npresent a unified picture for both Mn 2RuGa and GdFeCo, but also open the door for the future\napplications. Specifically, we propose a Mn 2RuxGa-based magnetic tunnel junction where a single\nlaser pulse can control magnetoresistance.\nPACS numbers:\nKeywords:\n2(July 22, 2022)I. INTRODUCTION\nLaser-induced ultrafast demagnetization1changes the landscape of spin manipulation,\nwhere the laser field plays a central role in magnetism. All-optical spin switching (AOS)2is\na prime example, where a single laser pulse can turn spins from one dire ction to another,\nfree of an external magnetic field. As more and more materials are d iscovered3–5, a critical\nquestion on the horizon is what properties are essential to AOS. Ea rlier studies have focused\non magnetic orderings such as ferrimagnetic versus ferromagnet ic6, sample composition7,\ncompensation temperature8, magnetic domains9and others10, but most AOS materials are\namorphous and difficult to simulate within state-of-the-art density functional theory. This\ngreatly hampers the current effort to decipher the mystery of AO S at a microscopical level\nthat goes beyond the existing phenomenological understanding11.\nHeusler compounds represent a new opportunity12,13. Their properties can be systemat-\nically tailored, only subject to the structure stability. Different fro m rare-earth-transition\nmetals2,14, one has an empirical Slater-Pauling rule to predict spin moments15–22. Although\nthis rule is simple23, the actual synthesis of a desired material is a monumental task o f\ndecades in making24, because many materials are unstable experimentally. In 2002, Hor i\net al.25successfully synthesized various (Mn 1−xRux)3Ga alloys with x= 0.33−0.67 and\ndetermined the spin moment of 1.15 µBper formula. In 2014, Kurt et al.26demonstrated\nthat Ru can significantly reduce the spin moment in ferrimagnet Mn 2RuxGa. Because\none can tune composition x, Mn2RuxGa is likely to be a half-metal and fully-compensated\nferrimagnet27,28, with no stray field, ideal for spintronics13,29,30. Research has intensified\nimmediately21,31–35. Lenneet al.36found that the spin-orbit torque reaches 10−11Tm2/A in\nthe low-current limit. Banerjee et al.37reported that a single 200-fs/800-nm laser pulse can\ntoggle the spin from one direction to another in Mn 2RuGa within 2 ps or less. Just as found\nin GdFeCo2,11, for every consecutive pulse, the spin direction is switched. This dis covery37\ndemonstrates the extraordinary tunability of Heusler compounds , which now changes the\ntrajectory of AOS research38–40, with Mn 2RuGa as a crystalline model system where the\nfirst-principles investigation is now possible.\nIn this paper, we carry out the comprehensive first-principles den sity-functional study\nto pin down the material properties essential to all-optical-spin swit ching in ferrimagnet\nMn2RuGa. We introduce two concepts - the spin anchor site (SAS) and t he optical active\n3(July 22, 2022)site (OAS) as two essential pillars of AOS in ferrimagnets. SAS has a s trong spin moment,\nand in Mn 2RuGa it is the Mn(4 a) site. Our band structure reveals that the Mn(4 a)’s band\nis 0.5 eV below the Fermi level. By contrast, OAS has a smaller spin mome nt, easier to be\nswitched optically41, and in Mn 2RuGa it is the Mn(4 c) site. Its band is around the Fermi\nlevel and accessible to optical excitation42. The creation of SAS and OAS is the making\nof Ru and Ga. The Ru-4 delectrons set up the initial spin configuration with a strong\nspin moment concentrated on the distant Mn(4 c), but Ga tips this balance and reverses\nthe relative spin magnitude between Mn(4 a) and Mn(4 c). Although Ru and Ga are weakly\nmagnetic, their energy bands appear in the same energy window as t wo Mn atoms, which is\nmanifested in the magneto-optical Kerr spectrum. Guided by two e ssential sites, we can now\nunify Mn 2RuGa with GdFeCo, despite of their apparent structural differenc es, and extract\nthree essential properties for AOS. (i) A ferrimagnet must have a spin anchor site, i.e. Gd\nin GdFeCo and Mn(4 a) in Mn 2RuGa. (ii) It must have an optical active site, i.e. Fe in\nGdFeCo and Mn 2(4c) in Mn 2RuGa. (iii) Its spin anchor site and optical active site must\nbe antiferromagnetically coupled to minimize the potential energy ba rrier43. We propose a\nlaser-activated magnetic tunnel junction based on the same mate rial Mn 2RuxGa, but with\ndifferent compositions xwhich form optical activation, spin filtering and reference layers.\nThisdevice, ifsuccessful, representsanidealintegrationoffully- compensatedhalf-metallicity\nin spintronics into all-optical spin switching in femtomagnetism44,45.\nThe rest of the paper is arranged as follows. In Sec. II, we presen t our theoretical\nformalism. Section III is devoted to the results and discussion, whic h includes the crystal\nstructure, electronic band structure, ultrafast demagnetizat ion, and Kerr rotation angle.\nFinally, we conclude this paper in Sec. IV.\nII. THEORETICAL FORMALISM AND CALCULATION\nElement Mn lies in the middle of 3 dtransition metals, with a half-filled 3 dshell and\nzero orbital moment, just as Gd in the middle of 4 frare-earth metals. Mn is the only\n3dtransition metal element in inverse Heusler compounds, which is similar to a rare-earth\nelement46. Mn2RuGa crystallizes in an inverse XAHeusler structure21,22,31,35,46(see Fig.\n1(a)), where two manganese atoms, Mn 1and Mn 2, are situated at two distinct Wyckoff\npositions 4a(0,0,0) and 4c(1\n4,1\n4,1\n4), and are antiferromagnetically coupled. Ru and Ga sit at\n4(July 22, 2022)4d(3\n4,3\n4,3\n4) and4b(1\n2,1\n2,1\n2), respectively. The inverse Heusler XAstructure hastwo Mn atoms\nseparated by a vector (1\n4,1\n4,1\n4), while the L21structure by (1\n2,1\n2,1\n2)26,32,33. The experimental\nlattice constants in this nearly cubic material are a=b=c= 5.97˚A26. Viewing along the\ndiagonal direction, four atoms form chains Mn 1-Mn2-Ga-Ru-Mn 1···. Therefore, Mn 2RuGa\nloses both inversion and time reversal symmetries due to the antife rromagnetic coupling\nbetween two Mn atoms.\nWe employ the state-of-the art density functional theory and th e full-potential linearlized\naugmented plane wave (FLAPW), as implemented in the Wien2k code47. We first self-\nconsistently solve the Kohn-Sham equation\n/bracketleftBigg\n−¯h2∇2\n2me+VNe+VH+Vxc/bracketrightBigg\nψnk(r) =Enkψnk(r), (1)\nwhereψnk(r) is the wavefunction of band nat the crystal momentum kandEnkis its band\nenergy. The terms on the left are the kinetic energy operator, th e attraction between the\nnuclei and electrons, the Hartree term, and the exchange-corr elation48, respectively. The\nspin-orbit coupling is included using a second-variational method in th e same self-consistent\niteration.\nIII. RESULTS AND DISCUSSIONS\nA. Crystal structure\nOrderedHeusler alloyshavethreedistinctivekindsofstructures49: (1)normalfull-Heusler\nX2YZalloys with group symmetry L21, (2) half Heusler XYZcompounds with group\nsymmetryC1b, and (3) inverse Heusler X2YZalloys with group symmetry XA. (1) has the\nspace group No. 225. (2) and (3) have the same space group No. 2 16.L21has (8c) site,\nwhich is split into two different sites in XA46. However, over the years, various Wyckoff\npositions are adopted in the literature. For the XAstructure, Wollmann et al.46used a\ndifferent set of Wyckoff positions for Mn at 4 d(1\n4,1\n4,1\n4),Yat 4c(3\n4,3\n4,3\n4), Mn at 4b(1\n2,1\n2,1\n2),\nandZat 4a(0,0,0). So in their paper, their (4 a),(4b),(4c),(4d) positions have different\nmeanings from those in31. In order to convert Wollmann’s notation to the latter notation,\none has to shift the entire cell by 4 d(1\n4,1\n4,1\n4). For theL21structure, Wollmann et al.46also\nadopted different positions, which were again used in their review pap er13(see Table I).\n5(July 22, 2022)In Mn 2RuGa, several versions have also been used. It adopts an XAstructure. Kurt et\nal.26correctly assigned the space group symmetry L21to the full Heusler compound, but\ninappropriately assigned the same group to Mn 2RuGa, and so did Zic et al.32. Both Zic\net al.32and Fleischer et al.33had the correct notations for all the atoms, but their figure\nswitched the positions for Ru and Mn 2, where Ru(4 d) appears at position 4 c(1\n4,1\n4,1\n4) and\nMn2(4c) at 4d(3\n4,3\n4,3\n4). Since they never used the figure to characterize their experime ntal\ndata, this changedoes not affect their results. Wealso noticethat Bettoet al.50assignedC1b\ngroup symmetry to Mn 2RuGa, where two Mn atoms are at 4 a(0,0,0) and 4c(3/4,3/4,3/4)\nwhile Ru at 4 d(1/4,1/4,1/4) and Ga at 4 b(1/2,1/2,1/2). One can see from Table I that\nC1bhas no 4csite.\nGalanakis et al.31exchanged the positions for Ru and Ga, so Ru is at site 4 band Ga is\nat site 4d. Although in general such an exchange is allowed, they do not match the existing\nexperimental results42. For instance, only Mn 1(4a) has Ru as its neighbor. If we exchange\nthe positions for Ru and Ga, then Mn 2(4c) would have Ru as its neighbor.\nWe summarize those used Wyckoff positions in the same table, so the r eader can see\nthe difference. We adopt the common convention, as listed in the last line in Table I. This\nconvention matches the experimental results better33. In particular, the magneto-optics\nsignal agrees with the experimental one.\nB. Band structure\nWe choose a big kmesh of 44 ×44×44, with 11166 irreducible kpoints in the Brillouin\nzone. The product of the Muffin-tin radius RMTand the planewave cutoff is 7, where\nRMT(Mn1,Mn2,Ru) = 2.42 bohr, and RMT(Ga) = 2.28 bohr. We find that Mn 1has spin\nmoment of M4a= 3.17µB. We call Mn 1(4a) the spin anchor site, SAS, as it pins the\nmagnetic configuration, so the magnetic structure can be stabilize d and is immune to optical\nexcitation. Mn 2(4c) atoms form another spin sublattice with a smaller spin moment of\n−M4c=−2.31µB. The entire cell has the spin moment of 1.027 µB, in agreement with prior\nstudies21,31. Figure 1(a) shows the spatial valence spin density integrated fro m 2 eV below\nthe Fermi level for each atom, where the red(blue) color refers t o the majority(minority)\nspin. One can see the spin density is mainly localized on these two Mn ato ms, where Mn 1\nhas a larger spin in the spin up channel and Mn 2has the spin density in the spin down\n6(July 22, 2022)channel, so they are antiferromagnetically coupled.\nOur first finding is that the above spin configuration hinges on the de licate balance\nbetween Ru and Ga. Figure 1(b) shows their respective atomic ener gies. Ru’s 4 d75s1states\nare close to Mn’s unoccupied 5 d0states. Without Ga, when Mn and Ru form a solid, the\nspin moment on Ru increases by five times to 0.39 µBand is antiferromagnetically coupled\nto the Mn 1(4a)’s spin, but its spin is now ferromagnetically coupled to the distant Mn 2(4c),\nwhichisoppositetothenativeMn 2RuGa(seeTableII).Adding Gatipsthebalance, because\nGa’s 4p1is higher than Mn’s unoccupied 5 d0orbitals (see Fig. 1(b)), so Ga can transfer\nelectrons to Mn atoms more easily than Ru. We integrate the atom-r esolved density of\nstates around each sphere, and find that the number of the Ru 4 delectrons is 5.87, reduced\nby 1.13 with respect to its atomic 4 d7, while the 4 pelectron of Ga is 0.81, reduced by 0.2\nfrom 4p1. The total number of electrons within the Mn 1(4a) and Mn 2(4c) spheres are almost\nexactly the same, 6.04, but the number of 3 delectrons in each spin channel is very different.\nTable II shows that Mn 1(4a) has 4.09 3 delectrons in the majority channel and 1.01 in the\nminority channel, in contrast to Mn 2(4c) where 1.44 and 3.72 electrons are present. The\ntotal number of 3 delectron is still close to 5. Table II summarizes these results. In gen eral,\nthe orbital moment on Mn 1is small, around 0.025 µB, and that on Mn 2is slightly larger,\nreaching -0.046 µB, which is beneficial to the spin-orbit torque38,43, important for AOS37.\nFigure 2(a) shows the band structure, superimposed with the Mn 1-3dorbital character\nfrom its spin majority channel. The orbital characters are highlight ed by the circles, whose\nradius is proportional to the weight of the Mn 1-3dcharacter, and the lines are the actual\nband dispersion. Bands with a clear dominance of a single orbital are h ighlighted, and in\nthe figure,dz2anddx2−y2are denoted by z2andx2−y2for simplicity; and this is the same\nfor other orbitals. The entire set of detailed orbital characteriza tion is presented in51. We\nsee that the Mn 1’s occupied majority band centers around -0.6 eV below the Fermi lev elEF\n(horizontal dashed line), with a smaller contribution close to the Fer mi level. This feature\nis reflected in the 3 d-partial density of states (pDOS) in Fig. 2(b), where a small peak\nat the Fermi level is found, consistent with two prior studies21,31, indicative of structural\ninstability46. Figure 2(c) shows that the Mn 1spin minority band has a single dxz/dyzband,\nwhich crosses the Fermi level from the L to Γ, and then to X point, b ut this single band\ncrossing does not constitute a major contribution to the density o f states (DOS). Figure 2(d)\nshows the partial 3 ddensity of states at the Fermi level is very tiny but not zero. The o ther\n7(July 22, 2022)occupied minority dband is at -1.5 eV below the Fermi level. Because Mn 1(4a)’sdband is\naway from EFand has a small density of states around the Fermi level, optical ex citation\nat Mn 1is weak42.\nMn2(4c) is quite different from Mn 1(4a). Figure 2(f) shows that its majority bands cross\nthe Fermi level at multiple points, have mixed dcharacters, and are highly dispersive. Its\n3d-pDOS (Fig. 2(e)) has a larger peak at the Fermi level than Mn 1, quantitatively 1.80-1.81\nstates/eV for the former and 0.65-0.69 states/eV for the latter . This explains why Mn 2(4c)\nis more optically active than Mn 1(4a)42, where we call Mn 2(4c) the optical active site. In\nthe minority channel, Mn 2has a strong admixture of orbital characters (see Fig. 2(h)), an d\nits overall density of states at the Fermi level is also small (see Fig. 2(g)). We note that the\nminority band structure is very similar to that of Mn 3Ga16, and they both have a flat dz2\nband along the Γ-X direction.\nBefore we move on to ultrafast demagnetization, we must emphasiz e that the band struc-\nture is not solely contributed by these two Mn atoms. Both Ru and Ga significantly affect\nthe magnetic properties of Mn atoms. Thin lines in Figs. 2(e) and 2(g) are the Ru’s 4 d\npDOS for the spin majority and minority channels, respectively. One can see that the Ru- d\nmajority density of states follows the Mn 1(4a)’s pDOS (compare Figs. 2(b) and (e)), but its\nminority state follows the Mn 2(4c)’s pDOS (compare the thin and thick lines in Fig. 2(g)).\nThe split role from the same atom is remarkable.\nC. Ultrafast demagnetization\nThe circles in Fig. 3(a) are the experimental ultrafast demagnetiza tion42, and consist of\ntwo regions. Region I is from 0 to 0.26 ps, highlighted by the red arrow in Fig. 3(a), and\nregion II starts from 0.26 ps to 5 ps. This time separation of 0.26 ps is consistent with a\nprior study52. In region I, a sharp decrease in spin moment is observed, but in reg ion II\nthere is a peak. We can fit these two regions with the same equation,\n∆M(t)\nM=A/parenleftBiggM4ae−α4a(t−T)−M4ce−α4c(t−T)\nM4a−M4c/parenrightBigg\n−B, (2)\nwheretis the time. Ais necessary, since without it the laser field amplitude cannot enter t he\nequation.Bdetermines the net amount of demagnetization. Tsets the characteristic time\nfordemagnetizationorremagnetization. Sinceourspinmomentsar efixedbyourcalculation,\n8(July 22, 2022)we only have four fitting parameters for each region, where α4a(4c)is the demagnetization\nrate for site 4 a(4c). Table II shows that in region I, αis site-dependent, α4a= 4.5/ps and\nα4c= 2.8/ps, demonstrating that the larger the spin moment is, the larger αbecomes,\nα=cM,or,τM=1\ncM, (3)\nwherecis a constant. This equation is consistent with the empirical formula p roposed by\nKoopmans and coworkers53. Fromα, we find the demagnetization times τM(4a) = 222 fs,\nandτM(4c) = 357 fs. These intrinsic demagnetization times, called the H¨ ubner times10, are\nwell within the times for other transition and rare-earth metals: 58 .9 fs (Fe), 176 fs (Ni),\n363 fs [Gd(5 d)], 690 fs [Gd(4 f)]. An extreme point will appear if ∂/parenleftBig∆M(t)\nM/parenrightBig\n/∂t= 0, and the\nsecond-order time-derivative determines whether the extreme is a maximum or minimum,\n∂2/parenleftBig∆M(t)\nM/parenrightBig\n∂t2=Aα4cM4ce−α4c(t−T)(α4a−α4c). (4)\nIfα4a>α4c, weonlyhaveaminimumwhich explains thespinchangeinregionI. Inreg ionII,\nbothα4aandα4carereduced, but α4aisreducedmuch more, so α4a<α4c, whichcorresponds\nto a peak in region II. Table II shows that region II has α4a= 0.6/ps andα4a= 1.5/ps. The\ndemagnetization on the 4 asite slows down significantly.\nD. Kerr rotation angle\nUnderlying ultrafast demagnetization and subsequent all-optical s pin switching is the\nmagneto-optical property of Mn 2RuGa, which is characterized by the conductivity54in units\nof (Ωm)−1,\nσαβ(ω) =i¯he2\nm2\neV/summationdisplay\nk;m,nfnk−fmk\nEmk−Enk/angb∇acketleftnk|pα|mk/angb∇acket∇ight/angb∇acketleftmk|pβ|nk/angb∇acket∇ight\n(¯hω+iη)+(Enk−Emk), (5)\nwheremeis the electron mass, Vis the unit cell volume, fnkis the Fermi distribution\nfunction,Emkisthebandenergy ofstate |mk/angb∇acket∇ight,/angb∇acketleftnk|pα|mk/angb∇acket∇ightisthemomentum matrixelement\nbetween states |mk/angb∇acket∇ightand|nk/angb∇acket∇ight, andηis the damping parameter. The summation is over the\ncrystal momentum kand all the band states |mk/angb∇acket∇ightand|nk/angb∇acket∇ight, andωis the incident photon\nfrequency. Here αandβrefer to the directions, such as the xandydirections, not to be\nconfused with the above demagnetization rate. The anomalous Hall conductivity is just the\noff-diagonal term. In the limit of η,ω→0, the term behind the summation over kis the\n9(July 22, 2022)Berry curvature\nΩk,n\nα,β=/summationdisplay\nm/negationslash=n¯h2(fmk−fnk)/angb∇acketleftnk|vα|mk/angb∇acket∇ight/angb∇acketleftmk|vβ|nk/angb∇acket∇ight\n(Emk−Enk)2. (6)\nThe general expression given in Eq. 5 is better suited for metals with partial occupation\nthan the treatment with a separate sum over occupied and unoccu pied states55,56, though\nthe latter is faster. The intraband transition with n=mis included by replacing ( fnk−\nfmk)/(Emk−Enk)byitsderivative −∂fnk/∂Enk, whichis −β/2\ncoshβ(Enk−EF)+1, withoutresorting\nto more complicated numerics57. Hereβ= 1/(kBT), wherekBis the Boltzmann constant,\nTis the temperature, and EFis the Fermi energy. The Kerr effect is characterized by the\nKerr rotation θand ellipticity ǫ, in the small angle limit and with magnetization along the\nzaxis,\nθ+iǫ=−σxy\nσxx/radicalBig\n1+4πiσxx/ω, (7)\nwhereσxxmust be converted to 1/s. The SI version of/radicalBig\n1+4πiσxx/ωis/radicalBig\n1+iσxx/(ωǫ0).\nExperimentally, Fleischer et al.33measured the magneto-optical Kerr effect for a series of\nMn2RuxGa samples with compositions x= 0.61,0.62,0.69,0.83 and with thickness from 26\nto 81 nm. Figure 3(b) reproduces two sets of data from their supp lementary materials. One\ncan see that both the thickness and composition affect the Kerr ro tation angle. The thicker\nsamplehasalargerangle(comparedottedandlong-dashedlineswith x= 0.61,0.62),andthe\nangle peaks between 1.6-1.9 eV. Our theoretical Kerr angles with th ree different dampings\nare three solid lines with η= 0.8, 0.6 and 0.4 eV from the bottom to top, respectively. One\nnotices that the overall shape is similar to the experimental data, a nd the main peak is also\naround 2 eV, slightly higher than the experimental one, but a more d irect comparison is not\npossible since there are no experimental data at x= 1. The best agreement in terms of the\nKerr angle is obtained with η= 0.8 eV. The convergence of our spectrum is tested against\nthe mesh of 92 ×92×92, and there is no visual difference between this much bigger mesh\nand the one used in Fig. 3(b).\nWe can pinpoint the origin of the main peak by removing some atoms. We useη= 0.4\neV since it gives us more structures. First, we remove Mn 2(4c), without changing the lattice\nstructure and the rest of atoms, so we have Mn(4 a)RuGa. The solid line in Fig. 3(c) shows\nthat the Kerr rotation angle for the new Mn(4 a)RuGa is very different from the one in\nFig. 3(b), highlighting the fact that Mn 2(4c), not Mn 1(4a), contributes significantly to the\noverall signal. To verify this, we remove Mn 1(4a) but keep Mn 2(4c). The red long-dashed\n10 (July 22, 2022)line shows clearly thatthe overall shapeiswell reproduced, but the Kerr angleislarger. This\nconcludes that Mn 2(4c) is optically active and plays a decisive role in the magneto-optical\nresponse as OAS, consistent with the experiment42, but the role of Ru and Ga should not\nbe underestimated. Magnetically, they are silent and do not contrib ute to the spin moment\nsignificantly, but when we remove Ru, the spectrum changes comple tely (see the dotted\nline in Fig. 3(c). The same thing happens to the removal of Ga atom. T his reveals the\nsignificant contributions of Ru and Ga to the optical response of Mn 2RuGa. The discovery\nof two sites (spin anchor site and optical active site) found here ha s some resemblance to\nthe laser-induced intersite spin transfer58. In their system, Dewhurst et al.found that the\nspins of two Mn atoms are aligned and coupled ferromagnetically, not antiferromagnetically\ncoupled as found here. Additional calculations are necessary since their materials are not\nMn2RuGa. Mentink et al.59proposed a two-sublattice spin model where AOS is realized\nthrough the angular momentum exchange between sublattices. Bu t their did not reveal the\ndifferent roles played by two spin sublattices. Our mechanism makes a clear distinction\nbetween two sublattices, and thus ensures that two sublattices d o not compete optically and\nmagnetically.\nIV. CONCLUSION\nThrough Mn 2RuGa, our state-of-the-art first-principles density functional calculation es-\ntablishes two concepts: the spin anchor site and the optical active site as the key to AOS\nin ferrimagnets. The formation of SAS and OAS in Mn 2RuGa is accomplished by weakly\nmagnetic Ru and Ga atoms. In GdFeCo2Gd is SAS while Fe is OAS. Switching starts with\nOAS52; because the ferrimagnetic coupling between SAS and OAS is frustr ated and has a\nlower potential barrier to overcome if the spin moment is smaller60, SAS is dragged into\nthe opposite direction by OAS through the spin torque JSi×Sj38,41, to realize all-optical\nspin switching. Because the Heusler compounds have excellent tuna bility13,17,19,22,29,33, the\nfuture research can investigate the effect of the spin moments at SAS and OAS on the spin\nswitchability8. We envision an integrated device based on Mn 2RuGa as illustrated in Fig.\n1(c). All three parts of the device are made of the same materials b ut with different concen-\ntrationx. In the middle, xis close to 0.6, so we have a full-compensated half-metal, while\nat two ends, xis close to 1, whose spin is designed to be optically switched. This forms an\n11 (July 22, 2022)ideal magnetic tunnel junction that a light pulse can activate. A fut ure experimental test\nis necessary. We should add that experimentally, concentrations o f both Mn and Ru can\nbe already tuned in several different experimental groups. Chatt erjeeet al.35were able to\nadopt two concentrations x= 0.2,0.5 in Mn 2−xRu1+x, while Siewierska et al.34were able to\nindependently change xandyin Mn yRuxGa films. In fact, Banerjee et al.37already used\n13 samples with x= 0.5 up to 1.0 in Mn 2RuxGa.\nAcknowledgments\nThe authors appreciate the numerous communications with Dr. K. R ode (Dublin) and\nDr. K. Fleischer (Dublin). Dr. Fleischer provided the original experim ental data in text\nform, which is very convenient to plot, with a small correction to the thickness of their sam-\nples. G.P.Z. and Y.H.B. were supported by the U.S. Department of Ene rgy under Contract\nNo. DE-FG02-06ER46304. Part of the work was done on Indiana St ate University’s high\nperformance Quantum and Obsidian clusters. 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Pearson, The Crystal Chemistry and Physics of Metals and Alloys , Wiley-Interscience,\nNew York, London, Sydney, Toronto, (1958).\n63Theo Hahn, International Tables for Crystallography (Dordrecht: Kluwer Academic Publishers,\n2002).\n17 (July 22, 2022)TABLE I: Summary of the space group symmetries for X 2YZ used in the literature. Two X atoms\nare denoted as X 1and X2, and “share” means that they share the same positions. For Mn 2RuGa,\nX1is Mn1, X2is Mn2, while Y is Ru and Z is Ga. The full-Heusler compound has L2 1symmetry,\nthe half-Heusler one has C1bsymmetry, and the inverse Heusler compound has XAsymmetry.\nGroup symmetry Prototype X1 X2 Y Z Ref.\nL21(No. 225, Fm¯3m)Cu2MnAl8c(1\n4,1\n4,1\n4) share 4b(1\n2,1\n2,1\n2) 4a(0,0,0)46\nL21(No. 225, Fm¯3m) (0,0,0) (1\n2,1\n2,1\n2)(1\n4,1\n4,1\n4) (3\n4,3\n4,3\n4)61\nL21(No. 225, Fm¯3m)Cu2MnAl8c(1\n4,1\n4,1\n4) (3\n4,3\n4,3\n4)4a(0,0,0) 4b(1\n2,1\n2,1\n2)62\nC1b(No. 216, F¯43m)MgAgAs 4a(0,0,0) vacant 4b(1\n2,1\n2,1\n2) 4c(1\n4,1\n4,1\n4)62\nXA(No. 216, F¯43m)Li2AgSb4d(1\n4,1\n4,1\n4) 4b(1\n2,1\n2,1\n2)4c(3\n4,3\n4,3\n4) 4a(0,0,0)46\nXA(No. 216, F¯43m) 4a(0,0,0) 4c(3\n4,3\n4,3\n4)4b(1\n2,1\n2,1\n2) 4d(1\n4,1\n4,1\n4)31\nNo. 216 4a(0,0,0) 4c(1\n4,1\n4,1\n4)4d(3\n4,3\n4,3\n4) 4b(1\n2,1\n2,1\n2)63\nTABLE II: Spin and orbital moments of Mn 1, Mn2, Ru and Ga. The electron populations in their\nmajority and minority 3 dstates are listed as n3d,↑andn3d,↓. The demagnetization rate in regions\nI and II is denoted as αIandαII, respectively. In two artificial structures, Mn 2Ru and Mn 2Ga,\nonly the spin moments are given.\nElement Ms(µB)Mo(µB)n3d↑n3d↓αI(1/ps)αII(1/ps)Ms(µB)Ms(µB)\n(Mn2RuGa) (Mn 2RuGa) (Mn2Ru) (Mn 2Ga)\nMn1(4a) 3.17 0.025 4.09 1.01 4.5 0.6 2.80 3.35\nMn2(4c) -2.31 -0.046 1.44 3.72 2.8 1.5 -3.64 -3.27\nRu(4d) 0.076 -0.035 -0.39 NA\nGa(4b) 0.032 -0.000 NA -0.04\n18 (July 22, 2022)FIG. 1: (a) Structure of Mn 2RuGa and the spatial spin densities on the Mn 1(4a) (red, positive)\nand Mn 2(4c) (blue, negative). The Ru and Ga atoms have a small spin densi ty. (b) Atomic energy\nlevels of Ru, Mn and Ga. The energy splitting is due to the spin -orbit coupling in atoms. (c) Our\nproposed device has a junction structure and consists of thr ee layers of the same Mn 2RuxGa, but\nwith different composition x. The layer on the left is an optically active layer, the middl e is the\nspin filter, and the right layer is a spin reference layer. The magnetoresistance is controlled by\nlight.\n19 (July 22, 2022)W LΓXWK−2−1012Energy (eV)Mn1(4a)−2−1012Energy (eV)Mn1(4a)\nW LΓXWK−2−1012Mn2(4c)−2−1012Mn2(4c) 0 2 4 0 2 4\n0 2 4 0 2 4pDOS (states/eV)\n(a) (b)\n(c) (d)(e) (f)\n(g) (h)x2−y2\nz2\nz2\nxz,yzxz,yzxz,yz\nxyz2\nx2−y2\nz2\nxz,yzEF\nEFRuMn2\nRu\nMn2\nFIG. 2: (a) and (c) Orbital-resolved band structure with the 3dstate characters for the Mn 1(4a)\nspin-majority and spin-minority channels, respectively. (b) and (d) Partial density of states for the\nMn1spin-majority and spin-minority channels, respectively. Bands with clear orbital characters\nare denoted by their orbitals. The Fermi level is set at 0 eV (h orizontal dashed line). (f) and\n(h) Band structure with the 3 dstate characters for the Mn 2(4c) spin-majority and spin-minority\nchannels, respectively. (e) and (g) Partial density of stat es for the Mn 2’s 3d(thick lines) and Ru’s\n4d(thin lines) spin-majority and spin-minority, respective ly.\n20 (July 22, 2022)0 1 2 3 4 5\nEnergy (eV)−2−10123Kerr rotation (mrad) x=0.61, d=55 nm\nx=0.62, d=81 nm\n0 1 2 3 4 5\nEnergy (eV)−4−20246\nKerr rotation (mrad)Mn(4a)RuGa\nMn(4c)RuGa\nMn2Ga0 1 2 3 4 5 6Time (ps)\n−20−15−10−50∆M/M (%)Exp\nfitting1\nfitting2\n(a)\n(b) (c)II I\nη=0.4 eV\nη=0.6 eV\nη=0.8 eVη=0.4 eVlaser pulse\nFIG. 3: (a) Experimental demagnetization fitted by Eq. 2, wit h two sets of fitting parameters given\nin Table II, provides a crucial insight that demagnetizatio n rates at two Mn spin-sublattices change\nbetween region I (between 0 to 0.26 ps) and region II (between 0.26 ps to 5 ps). The experimental\ndata are extracted from Ref.42. The thick red curve is the laser pulse of duration 40 fs. (b) T he\nexperimental (dotted and dashed lines from Ref.33) and our theoretical Kerr rotation angles. The\nthree solid lines are our theoretical results with three diffe rent dampings η= 0.4,0.6,0.8 eV. (c)\nElement-resolved Kerr rotation angles when Mn 2(4c) (solid line), or Mn 1(4a) (dashed line), or Ru\n(dotted line) is removed separately. η= 0.4 eV is used.\n21 (July 22, 2022)"    },    {        "title": "1403.1513v2.Spontaneously_magnetized_Tomonaga_Luttinger_liquid_in_frustrated_quantum_antiferromagnets.pdf",        "content": "Spontaneously magnetized Tomonaga-Luttinger liquid in frustrated quantum\nantiferromagnets\nShunsuke C. Furuya and Thierry Giamarchi\nDPMC-MaNEP, University of Geneva, 24 Quai Ernest-Ansermet CH-1211 Geneva, Switzerland\n(Dated: November 5, 2018)\nWe develop a theory of spontaneously magnetized Tomonaga-Luttinger liquid (SMTLL) in ge-\nometrically frustrated quasi-one-dimensional quantum magnets by taking an S= 1=2 ferrimagnet\non a union-jack lattice as an example. We show that a strong frustration leads to a spontaneous\nmagnetization because of the ferrimagnetic nature of lattice. Due to the ferrimagnetic order, the\nlocal magnetization has an incommensurate oscillation with the position. We show that the spon-\ntaneously magnetized TLL is smoothly connected to the existence of a Nambu-Goldstone boson in\nthe canted ferrimagnetic phase of a two-dimensional frustrated antiferromagnet.\nPACS numbers: 75.10.Jm, 75.30.Kz, 75.50.Gg\nI. INTRODUCTION\nSpontaneous symmetry breaking is one of the most\nfundamental concepts in physics. It provides the mech-\nanism to generate mass of elementary particles and al-\nlows macroscopic alignment of magnetic moments in fer-\nromagnets. Spontaneous breaking of global continuous\nsymmetry is accompanied by a massless excitation, the\nNambu-Goldstone boson [1{3]. Since Nambu-Goldstone\nboson governs low-energy physics at long distance and\nthe low-energy physics is a\u000bected by the geometry of sys-\ntem, the dimensionality of the system has strong in\ru-\nences on Nambu-Goldstone boson.\nSuch e\u000bects are most prominent in one dimension (1D)\nbecause of the large suppression of ordering at \fnite tem-\nperatures [4, 5] and even at zero temperature [6] due to\nquantum e\u000bects. As a result the breaking of a continuous\nsymmetry in 1D is deemed impossible. For systems such\nas a 1D super\ruid, indeed no long range order exists,\nand the proper description is the one of a Tomonaga-\nLuttinger liquid (TLL) [7]. Despite the absence of the\ntrue long-range order [8, 9], Goldstone modes exist, and\nhave a dynamical origin [10]. However in some rare cases,\nsuch as a ferromagnet, the ground state can, even in\n1D spontaneously break a continuous symmetry. This\nprompts immediately for the question of why and for\nwhich systems such phenomena can occur.\nIn order to shed light on the possibility of sponta-\nneous symmetry breaking in 1D, dynamical aspects of\nthe system need to be carefully considered. In addition,\nin view of the recent experimental progresses in realiz-\ning 1D quantum liquids in various situations [11{14], it\nis worthwhile to search for novel manifestations of spon-\ntaneous symmetry breaking in 1D.\nFor quantum magnetism in 1D, one expects a system\nwith quasi-long range antiferromagnetic order to have a\nrelativistic dispersion, which is the case of the TLL, while\na ferromagnet would have a quadratic one. This behav-\nior of the Nambu-Goldstone boson has been formulated\nin a quite general context [15{17]. Finding in 1D a sys-\ntem that would spontaneously break the continuous ro-tational symmetry of the spins, while at the same time\nretaining some TLL behavior would thus be interesting\nand an example of the more general character of Nambu-\nGoldstone boson.\nA very good possibility to realize such a spontaneously\nmagnetized TLL (SMTLL) is o\u000bered by ferrimagnetic\nsystems. Several numerical studies [18{24] have fol-\nlowed this route and found incommensurate ferrimag-\nnetic phases which can be candidates for the SMTLL.\nHowever besides the numerical results, there is still no\nmicroscopic theory that explains the nature of the in-\ncommensurate phase and could relate it to a SMTLL.\nIn this paper, we present such a theory of the SMTLL,\nshowing that one can have simultaneously a spontaneous\nbreaking of the spin-rotation symmetry leading to a \f-\nnite magnetization and a TLL behavior. We demonstrate\nthat this SMTLL phase is realized in the ground state of\nanS= 1=2 geometrically frustrated quantum antifer-\nromagnet on a 1D array of the union jack lattice [see\nFig. 1].\nJ1J1Sj,1Sj,2Sj,3J2↵J2<latexit sha1_base64=\"4avvin4QIgWF6bhwNhUv+6bnFB8=\">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</latexit><latexit sha1_base64=\"4avvin4QIgWF6bhwNhUv+6bnFB8=\">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</latexit>\nFigure 1. The three-leg union-jack ladder (1).arXiv:1403.1513v2  [cond-mat.str-el]  27 May 20142\nII. INSTABILITY OF THE\nTOMONAGA-LUTTINGER LIQUID\nWe consider the union jack (UJ) spin Hamiltonian:\nH=J1X\nj3X\na=1Sj;a\u0001Sj+1;a+J1X\njSj;2\u0001(Sj;1+Sj;3)\n+J2X\njS2j;2\u0001(S2j\u00001;3+S2j+1;1)\n+\u000bJ2X\njS2j;2\u0001(S2j\u00001;1+S2j+1;3); (1)\nwhereJ1;2>0 and 0<\u000b\u001c1. The parameter \u000bdenotes\nthe imbalance of the diagonal interactions. Throughout\nthe Paper, we \fx \u000band change the ratio J2=J1from\n0 to +1. Note that this model has a priori full spin-\nrotational symmetry.\nA. Classical ground state\nWe \frst consider the classical ground state minimiz-\ning the energy of a unit cell. The classical analysis on\nthe UJ ladder (1) is similar to the 2D UJ antiferromag-\nnet [25{27]. For 0 \u0014J2=J1<1=2, the classical ground\nstate is the N\u0013 eel state. For 1 =2< J 2=J1, spins on the\n\flled sites in Fig. 1 (a) become canted with a polar angle\n#= cos\u00001(J1=2J2) and the classical ground state in the\ncanted phase has an incommensurate magnetization,\nhSz\nj;ai=~S\n2\u0012\n1\u0000J1\n2J2\u0013\n:\nHereafter we use ~= 1 for simplicity. At the classi-\ncal level, a spontaneous magnetization occurs. However,\nsince quantum \ructuation usually destroys long-range or-\nder in 1D systems (1), we have to take them into account\nto conclude on the existence of a spontaneous magneti-\nzation in 1D.\nB. The Tomonaga-Luttinger liquid\nTo do so we derive the low-energy e\u000bective \feld theory\nof the UJ ladder (1). When the diagonal interaction is\nFigure 2. The shaded area depicts the unit cell of the O(3)\nnonlinear\u001bmodel.small enough, J2=J1\u001c1, the low-energy e\u000bective \feld\ntheory is written as a function of two slowly varying \felds\nn=1\n2S3X\na=1(S2j+2\u0000a;a\u0000S2j+3\u0000a;a); (2)\nl=1\n2a03X\na=1(S2j+2\u0000a;a+S2j+3\u0000a;a): (3)\nHerea0is the lattice spacing. We take a diagonal unit\ncell (Fig. 2) to de\fne nandlalong theJ2bond [28]. The\nnandl\felds denote respectively staggered and uniform\nmagnetization densities and satisfy the constraints\nf(n;l)\u0011n2\u00001\u00001\nS\u0000l2\nS2= 0 (4)\nandn\u0001l= 0. The constraint (4) is usually replaceable to\nn2= 1. However, the l2term will play an essential role\nfor our purpose.\nThe Hamiltonian (1) in the low-energy limit is given\nby\nH=Zdx\n2\u00141P\na;bM\u00001\na;bl2+ 2S2X\napa(@xn)2\n+2SP\na;bpaM\u00001\na;bP\na;bM\u00001\na;b(l\u0001@xn+@xn\u0001l)\u0015\n(5)\n=Z\ndx\u0014gv\n2\u0012\nl\u0000\u0002\n4\u0019@xn\u00132\n+v\n2g(@xn)2\u0015\n;(6)\nwheregis a coupling constant, vis the velocity, \u0002 is a\ntopological angle given by\ng=1\nS\u0014\n2X\na;b;cpaM\u00001\nb;c\u0000\u0012\u0002\n4\u0019S\u00132\u0015\u00001=2\n; (7)\nv=Sa0\u00142P\napaP\nb;cM\u00001\nb;c\u0000\u0012\u0002\n4\u0019S1P\na;bM\u00001\na;b\u00132\u00151=2\n;(8)\n\u0002 = 6\u0019S; (9)\nandpa= 3J1=2 + (J1=2)\u000ea;2. Whilegandvdepend on\na 3\u00023 matrix of microscopic parameters,\nM=0\n@5J1\u0000~J2J1\u0000~J2 0\nJ1\u0000~J26J1\u00002~J2J1\u0000~J2\n0J1\u0000~J25J1\u0000~J21\nA (10)\nwith ~J2= (1 +\u000b)J2=2, the topological angle (9) is deter-\nmined only by the number of legs. The derivation of the\ne\u000bective \feld theory (6) is explained in the case of the\nthree-leg spin ladder in Refs. 28 and 29. We obtain the\ne\u000bective \feld theory (6) by replacing the rung coupling of\nthe three-leg ladder to J1\u0000~J2in the matrix (10). We can\nsee from Eq. (5) that information on the structure of the\nUJ lattice (1) is encoded in the matrix (10). Integrating\nlout, we obtain [30]\nH=v\n2gZ\ndx\u00141\nv2(@\u001cn)2+ (@xn)2\u0015\n+i\u0002Q; (11)3\nwhere n2= 1 and\nQ=1\n4\u0019Z1\n\u00001dxn\u0001@\u001cn\u0002@xn\ngives an integer after integrating with an imaginary time\n\u001c, that is,R1\n\u00001d\u001cQ2Z. The Hamiltonian (11) is the\nO(3) nonlinear \u001bmodel (NLSM). The \u0002 term controls\nthe low-energy limit of the NLSM (11) [31]. When \u0002 \u0011\u0019\n(mod 2\u0019), namely when Sis a half integer, the O(3)\nNLSM (11) is identical to the TLL as a conformal \feld\ntheory with a central charge c= 1 [32],\nH=v\n2\u0019Z\ndx\u0014\nK(@x\u0012)2+1\nK(@x\u001e)2\u0015\n: (12)\nWe use here the notation for the TLL of Ref. 7. The\nnature of the \u001eand\u0012\felds will be discussed in detail\nlater. Since \u0002 is independent of J2, for small enough J2,\nthe diagonal interaction is irrelevant and the UJ ladder\n(1) has the TLL ground state (12), and thus in particular\nzero spontaneous magnetization.\nC. Instability at k= 0\nHowever the diagonal interaction has a serious impact\non the ground state, and lead to an instability of the TLL.\nThe diagonal interaction J2partly compensates J1in the\nmatrix (10) and it reduces the velocity down to v= 0\nwhere the linearization of the dispersion relation !=vk\nbecomes invalid. Let us denote the instability point as\nJc1\n2. The instability point is determined from the zeros\nof the matrix (10). The matrix Mis positive de\fnite for\n~J2= 0. As we increase ~J2, the positive de\fniteness \frst\nbreaks down at ~J2= 7J1=3, namely\nJc1\n2=14J1\n3(1 +\u000b): (13)\nWhen 0<Jc1\n2\u0000J2\u001cJ1,gandvcan be expanded with\nrespect to ( Jc1\n2\u0000J2)=J1. In fact, sinceP\na;bM\u00001\na;b\u0018\n[(J2\u0000Jc1\n2)=J1]\u00001, we obtain g/S\u00001[(Jc1\n2\u0000J2)=J1]1=2\nand\nv/J1S\u0012Jc1\n2\u0000J2\nJ1\u00131=2\n: (14)\nWe can easily see that the velocity (14) approaches zero\nwhenJ2%Jc1\n2. Note that the TLL parameter Kmust\nbe 1=2 to ensure that the Hamiltonian (12) preserves the\nSU(2) rotational symmetry [7]. Thus, the susceptibility\n\u001f=K=\u0019v of the TLL (12) diverges as J2%Jc1\n2.\nNear the instability point J2=Jc1\n2, a careful treat-\nment of the interaction is required. The interaction of\ntheO(3) NLSM (11) is non-perturbatively included in\nthe constraint (4). Namely, the partition function Zof\ntheO(3) NLSM (11) is given as a path integral,\nZ=Z\nDnDl\u000e\u0000\nf(n;l)\u0001\n\u000e(n\u0001l) exp\u0012\n\u0000Z\nd\u001cH\u0013\n:(15)The constraint (4) generates a strong repulsion \u0015(l2)2.\nIndeed, one can add a strongly repulsive interaction\nUff(n;l)g2withU%+1to the Hamiltonian (6) in-\nstead of imposing the constraint (4). Then, the partition\nfunction (15) is approximated as\nZ=Z\nDnDl\u000e(n\u0001l)\n\u0002exp\u0012\n\u0000UZ\nd\u001cdx\b\nf(n\u0001l)\t2\u0013\nexp\u0012\n\u0000Z\nd\u001cH\u0013\n=Z\nDnDl\u000e(n\u0001l) exp\u0012\n\u0000Z\nd\u001c\u0016H\u0013\n; (16)\nwhere we obtain the Hamiltonian \u0016Hwith the e\u000bective\nrepulsion,\n\u0016H=Z\ndx\u0014gv\n2L2+\u0015(L2)2+v\n2g(@xn)2+U(n2\u00001)2\u0015\n;\n(17)\nwhere\u0015=U=S4>0 and\nL\u0011l\u0000\u0002\n4\u0019@xn: (18)\nWe introduced the quartic interaction ( L2)2instead of\n(l2)2because their di\u000berence (e.g. f(@xn)2g2) is negligi-\nble as we see below.\nWhenJ2>Jc1\n2, the following inequalities are valid:\ngv=1P\na;bM\u00001\na;b<0; (19)\nv\ng= 2S2X\napa\u0000\u0012\u0002\n4\u0019\u001321P\na;bM\u00001\na;b>0: (20)\nThus, the interaction that the L\feld feels takes a form\nof the wine bottle:\ngv\n2L2+\u0015(L2)2=\u0015\u0012\nL2\u00001\n\u0015jgvj\u00132\n+ const: (21)\nThe potential (21) leads to a nonvanishing expectation\nvalue of L. Note that the instability occurs only in\nthe uniform part of the O(3) NLSM (17) because of\nthe inequalities (19) and (20). The diagonal coupling\nonly changes the sign of the coupling constant of the\nL2[Eq. (19)] keeping that of ( @xn)2positive (Eq. (20)).\nNamely, the instability only occurs at the wave number\nk= 0 and the nonzero expectation value of the L\feld\n(18) is attributed to the magnetization density l. There-\nfore the ground state has a spontaneous magnetization\nper site, M\u0011hli=3, with\njMj=1\n3\u00121\n\u0015jgvj\u00131=2\n/\u0012J2\u0000Jc1\n2\nJ1\u00131=2\n: (22)\nThe transition around Jc1\n2is described by a Ginzburg-\nLandau like theory of second order transitions. Con-\ntrarily to the ( L2)2term, the higher-order interaction4\nf(@xn)2g2is negligible because the lower-order term\n(@xn)2is stable. Since \u0015 > 0 the transition cannot\nbe \frst order. We emphasize that our derivation of the\nspontaneous magnetization (22) fully respects the SU(2)\nrotational symmetry and Mcan point in an arbitrary\ndirection.\nD. Nambu-Goldstone bosons\nLet us explain the nature of Nambu-Goldstone boson\ngenerated from this phase transition. First we focus on\nthek= 0 part. We rewrite the Hamiltonian in terms of\n\ructuation\nm\u0011L\u00003M: (23)\nWe can assume M=M(0 0 1)Twithout loss of gen-\nerality. The longitudinal component mzhas a mass\n\u0001 = 12\u0015M2because of the interaction (21), that is,\n\u0015(L2\u00009M2)2'12\u0015M2(mz)2. The transverse com-\nponent m?\u0011(mx;my;0) is the Nambu-Goldstone bo-\nson generated from the spontaneous magnetization and\nit possesses a nonrelativistic dispersion relation [33, 34].\nDispersion relations of the longitudinal and the trans-\nverse modes are respectively\nEk(k) = \u0001 +k2\n2mk; E?(k) =k2\n2m?: (24)\nWe can \fnd another massless excitation near k=\u0019.\nThen the Hamiltonian (17) turns into\n\u0016H=Z\ndx\u0014\n4\u0015M2m2+v\n2g(@xn)2\u0015\n+H0; (25)\nThe repulsion U(n2\u00001)2is transformed into the con-\nstraint n2= 1 again. The last term of Eq. (25) denotes\nthe anisotropy that the spontaneous magnetization in-\nduces,\nH0=VZ\ndx\b\n2(mz)2\u0000(mx)2\u0000(my)2\t\n; (26)\nwithV= 4\u0015M2, which seemingly equals to the coupling\nconstant of m2. However, after including renormaliza-\ntion due to irrelevant operators, the coupling constant\nVof the anisotropic interaction (26) actually deviates\nfrom that of the isotropic part m2. Here we \frst omit\nthe anisotropy (26) and include it later perturbatively\nbecause it does not modify qualitative features of the ef-\nfective Hamiltonian of the Nambu-Goldstone boson. Let\nus rewrite Hamiltonian (25) in terms of nandl,\n\u0016H=Z\ndx\u0014\u0016gu\n2\u0012\nl\u0000\u0002\n4\u0019@xn\u00132\n+u\n2\u0016g(@xn)2\u00003\u0016guM\u0001l\u0015\n:\n(27)\nHere we introduced the coupling constant \u0016 gand the ve-\nlocityuin a parallel manner as Eq. (6). They are givenby\n\u0016g=2M\nS\u0014\n2\u0015X\napa\u0000\u0012\u0002\n4\u0019S\u00132\u0015P\na;bM\u00001\na;b\u0015\u00001=2\n(28)\nu= 2MSa 0\u0014\n2\u0015X\napa\u0000\u0012\u0002\n4\u0019S\u00132\u0015P\na;bM\u00001\na;b\u00151=2\n:(29)\nNote that both \u0016 ganduare positive and proportional to\nthe magnitude of the spontaneous magnetization jMj.\nThe last term of the Hamiltonian (27) can be seen as\nthe Zeeman energy \u0000he\u000b\u0001Sj;l. For further understanding\nof the e\u000bective Hamiltonian (27) of the Nambu-Goldstone\nboson atk=\u0019, we integrate lout:\n\u0016H=u\n2\u0016gZ\ndx\u00141\nu2(@\u001cn+ihe\u000b\u0002n)2+ (@xn)2\u0015\n+ 9\u0016guZ\ndx(M\u0001n)2+i\u0002Q; (30)\nwhere he\u000bis written as\nhe\u000b= 3\u0016guM: (31)\nIf we include the perturbation H0at lowest order, it gives\na correction\nH0'VZ\ndx(M\u0001n)2[3(nz)2\u00001] (32)\nto the Hamiltonian (30). If we use the value V= 4\u0015M,\nH0replaces the term 9\u0016 gu(M\u0001n)2= 9\u0016guM2(nz)2of\nEq. (30) with 27\u0016 guM2(nz)4, which has no impact on\nqualitative aspects of the e\u000bective \feld theory (30).\nTheO(3) NLSM (30) leads to three important conse-\nquences. First, the spontaneous magnetization Mleaves\n\u0002\u0011\u0019(mod 2\u0019) intact. Second, Mgenerates the easy-\nplane anisotropy. Finally, the O(3) NLSM (30) is semi-\nclassical. While the NLSM (11) in the TLL phase has\na coupling g/1=S, the NLSM (30) has \u0016 g/M=S [see\nEq. (28)]. Thus, the NLSM (30) behaves similarly to a\nspin-Se\u000bHeisenberg antiferromagnetic chain with a large\nhalf-integer spin Se\u000b\u0018S=M .\nThe e\u000bective \feld theory at k=\u0019is thus, the TLL\nunder an e\u000bective magnetic \feld he\u000b=he\u000b(0 0 1)T; that\nis,\n\u0016H=u\n2\u0019Z\ndx\u0014\n\u0016K(@x\u0012)2+1\n\u0016K(@x\u001e)2\u0015\n\u0000he\u000b\n\u0019Z\ndx@x\u001e:\n(33)\nwhich is the e\u000bective model for the SMTLL. The TLL\nparameter \u0016Kis determined from the relation [7]\nM=he\u000b\u0016K\n\u0019u: (34)\nThe TLL parameter\n\u0016K=\u0019\n\u0016g/\u0012J2\u0000Jc1\n2\nJ1\u0013\u00001=2\n(35)5\n⇡<latexit sha1_base64=\"uI2OpSLK9oTdka9QXrzyACVgPV8=\">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</latexit><latexit 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Dynamical structure factors (a) Sk(k;!) and (b)\nS?(k;!) in the low-energy region. Outside of the shaded\narea, the dynamical structure factor has zero intensity. The\nred lines represent the linear dispersion of the SMTLL and the\nblue dashed curves represent the quadratic dispersions (24) of\nthe nonrelativistic Goldstone mode [ E?(k)] and the massive\nmode [Ek(k)].\ndiverges at the instability point Jc1\n2. The point Jc1\n2brings\nabout a divergence of the susceptibility,\n\u001f/\u0012jJc1\n2\u0000J2j\nJ1\u0013\u0000\r\n; (36)\nwith the critical exponent \ris given by\r= 1=2 forJ2%\nJc1\n2and\r= 1 forJ2&Jc1\n2.\nE. Dynamical structure factors\nWe now use this theory to compute the dynami-\ncal structure factors in the SMTLL phase. We fo-\ncus on longitudinal and transverse dynamical structure\nfactors,Sk(k;!) =R1\n\u00001dtdxei(!t\u0000kx)hSz\nj(t)Sz\n0(0)iand\nS?(k;!) =R1\n\u00001dtdxei(!t\u0000kx)hS+\nj(t)S\u0000\n0(0)i.\nThe longitudinal and transverse dynamical structure\nfactors near k=\u0019are the same as those of the S=\n1=2 Heisenberg antiferromagnetic chain under magnetic\n\feld [7, 35],\nSk(k=\u0019(1\u00002M) +\u000ek;! )\n=\u00192Ck\nu\u00002(\u0016K)\u0012H(!\u0000uj\u000ekj)\u00124u2\n!2\u0000u2(\u000ek)2\u00131\u0000\u0016K\n;(37)\nS?(k=\u0019+\u000ek;! )\n=\u00192C?\nu\u00002(1\n4\u0016K)\u0012H(!\u0000uj\u000ekj)\u00124u2\n!2\u0000u2(\u000ek)2\u00131\u00001=4\u0016K\n:\n(38)\n\u0012H(z) is the Heaviside's step function and CkandC?are\nnonuniversal constants. Equations (37) and (38) hold\nwhenj\u000ekj \u001c 1. The dynamical structure factor near\nk= 0 is given by\nS\u0017(k=\u000ek;! ) =\u0012H\u0000\n!\u0000E\u0017(\u000ek)\u0001C0\n\u0017\n!\u0000E\u0017(\u000ek);(\u0017=k;?);\n(39)whereC0\nkandC0\n?are constants. Figure 3 shows the longi-\ntudinal and transverse dynamical structure factors in the\nlow-energy region. The dynamical structure factor near\nk= 0 (39) clearly shows di\u000berence between the SMTLL\nand either a TLL under magnetic \feld (e.g. the S= 1=2\nantiferromagnetic chain [7]) or a \feld-induced TLL (e.g.\ntheS= 1=2 two-leg spin ladder [36]), for which the sym-\nmetry has been externally broken.\nIII. COMMENSURATE PHASE\nA. Commensurability condition\nLet us now examine the behavior upon increasing J2\nfurther. The spontaneous magnetization saturates at a\ncertain point Jc2\n2(> Jc1\n2). In the case of the UJ lad-\nder, we can \fnd a saturation condition in the spirit of\nthe Oshikawa-Yamanaka-A\u000feck theory [37]. To do so,\nwe need to clarify the physical meanings of the \u001eand\u0012\n\felds of the SMTLL (33). The de\fnitions (2) and (3) of\nnandlindicate that nandl, equivalently \u001eand\u0012, rep-\nresent a \\center-of-mass\" mode. When one consider the\nUJ ladder (1) as a system of three spin chains weakly cou-\npled by the rung and the diagonal interactions [38], each\nspin chain is equivalent to a TLL written in a compact-\ni\fed boson \u001eaand its dual \u0012a(a= 1;2;3). The bosons\n\u001eand\u0012represent the center-of-mass mode because they\nare given by \u001e=\u001e1+\u001e2+\u001e3and\u0012=\u00121+\u00122+\u00123. The\nother \\relative-motion\" modes, \u001e1\u0000\u001e2and\u001e1+\u001e2\u00002\u001e3,\nare massive and negligible in the low-energy e\u000bective \feld\ntheories (12) and (33). The two-site translational symme-\ntryj!j+2 of the UJ ladder (1) requires the invariance\nof the e\u000bective \feld theory under the translation\n\u001e!\u001e+ 6(S\u0000M)\u0019: (40)\nGiven an incommensurate magnetization Msatisfying\n6(S\u0000M)62Z; (41)\nthe incommensurability condition (41) prohibits relevant\ninteractions of \u001e, for instance cos(2 \u001e), to appear in the\ne\u000bective \feld theory (33). Relevant interactions of \u0012are\nnot allowed from another reason, that is, the U(1) sym-\nmetry of the ground state [37].\nEquation (41) shows that the \u001e\feld can be massive\nwhen the incommensurability condition (41) is violated.\nIncreasingMfrom zero, the condition (41) \frst breaks\ndown when\nM\nMs=1\n3: (42)\nHereMs=Sis the saturated value of M. Thus, a com-\nmensurate phase as the 1 =3-plateau (42) should exist.\nThe commensurate phase has only one massless Nambu-\nGoldstone boson near k= 0 because the SMTLL acquires\na mass from a relevant interaction cos(2 \u001e). The condition\n(42) gives the saturation condition of the UJ ladder.6\nB. Trimer-spin chain\nIn order to complete the above derivation of the spon-\ntaneous magnetization, we show that for large J2it can\nbe shown to occur directly from the lattice model (1).\nThe commensurate phase is identi\fed as a ferromag-\nnetic phase of trimers formed on diagonal J2bonds\n(Fig. 4 (a)). Let us consider the case J2=J1\u001d1. When\nJ1= 0, the UJ ladder (1) is composed of an S= 1=2\ndiamond chain [39, 40] and isolated spins. Three spins\nS2j+1;1;S2j;2andS2j\u00001;3form a trimer [a solid rectangle\nin Fig. 1 (b)] on the strongest J2bond. To describe the\nground state and the lowest-energy excitation, we may\nreplace the three spins with an S= 1=2 pseudo spin [40],\nS2j+1;1=S2j\u00001;3=2\n3Tj;S2j;2=\u00001\n3Tj: (43)\nThe eigenstates, j *ijwithTz\nj= 1=2 andj +ijwith\nTz\nj=\u00001=2 are written as\nj*ij=1p\n6(j#ij;1j\"ij;2j\"ij;3\u00002j\"ij;1j#ij;2j\"ij;3\n+j\"ij;1j\"ij;2j#ij;3); (44)\nj+ij=1p\n6(j\"ij;1j#ij;2j#ij;3\u00002j#ij;1j\"ij;2j#ij;3\n+j#ij;1j#ij;2j\"ij;3): (45)\nHerej\"ij;aandj#ij;aare the eigenstate of Sz\n2j+2\u0000a;a\n(a= 1;2;3). Figure 4 (a) depicts that the mapping from\nspins to a trimer metamorphoses an antiferromagnetic in-\nteraction\u000bJ2S2j;2\u0001(S2j\u00001;1+S2j+1;3) to a ferromagnetic\ntrimer-trimer interaction,\n\u000bJ2S2j;2\u0001(S2j\u00001;1+S2j+1;3) =\u0000JFTj\u0001Tj+1 (46)\nwithJF=4\u000bJ2\n9+O(\u000b2J2) [40]. 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sha1_base64=\"s2YsxZdkOO08wQB/rHpZxxNbq10=\">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</latexit>\nFigure 4. (a) A con\fguration of trimers (solid rectangles) at\nJ2=J1\u001d1. Trimers are surrounded by spins (circles). In the\nlow-energy limit, one can regard the trimer as the S= 1=2\npseudo spin. The trimer-trimer interaction is ferromagnetic\nand the trimer-spin interaction is antiferromagnetic. (b) The\ne\u000bective model that describes the commensurate phase. The\nsolid and blank circles represent trimers ( S= 1=2 pseudo\nspins) and S= 1=2 spins, respectively. The thick line repre-\nsents the ferromagnetic coupling (46) of trimers and the thin\nlines are the antiferromagnetic coupling of trimers and spins.AtJ1= 0, the residual spins [depicted as blank circles\nin Fig. 4 (a) and (b)] are isolated from the trimers. The\nnonzeroJ1switches on trimer-spin interactions. The low-\nenergy e\u000bective Hamiltonian for J2=J1\u001d1 is given by\nHferri=\u0000JFX\nj2ZTj\u0001Tj+1+X\ni;j2ZJij~Si\u0001Tj; (47)\nwhere spins not participating in forming trimers are rela-\nbeled as ~Sj. Fig. 4 (b) shows interactions of the e\u000bective\nmodel (47). If we are concerned only with the ground\nstate magnetization of the trimer-spin chain (47), we do\nnot even need the details of coupling constants. We use\nonly three facts, JF>0,Jij\u00150 andP\nl2ZJil>0\nfori;j2Z. These conditions enable us to apply the\nMarshall-Lieb-Mattis theorem [41, 42] to the model (47).\nThis theorem imposes that the ground state of the trimer-\nspin chain (47) must have a \fxed magnetization irrespec-\ntive of parameters JFandJij. The ground state magne-\ntization of the commensurate phase is exactly Eq. (42).\nSince the ferromagnetic order of the trimer is exactly\nthe ferrimagnetic order of spins, the commensurate phase\nis exactly the ferrimagnetic phase. Figure 5 shows the\nground state magnetization of the UJ ladder, which re-\nproduces the numerically derived one [24] in the \u000b= 1\ncase. The Marshall-Lieb-Mattis theorem allows us even\nto take a limit \u000b!1. However, compared to the nu-\nmerical result [24], we overestimated Jc1\n2because of the\nimbalance\u000b\u001c1.\nIV. RELATION TO THE GENERAL THEOREM\nNow that we have a description of the SMTLL and its\nNambu-Goldstone boson, we can compare our result with\nthe general theorem [15{17] claiming that the number of\nbroken generators of the symmetry group determines the\nnumber of Nambu-Goldstone boson. The canted phase\ngenerally has a nonrelativistic Nambu-Goldstone boson\nand a relativistic Nambu-Goldstone boson [16, 43], which\nis true in the 2D UJ antiferromagnet [26]. In the 1D case\n(1), theU(1) symmetry is recovered for the ground state\nas a result of quantum \ructuations. In the TLL phase,\neven the full SU(2) symmetry is recovered. Therefore, we\nconclude that the general theorem [16, 17] is applicable\nto 1D systems at the classical level.\nTLLSMTLLferriMMs<latexit sha1_base64=\"XrFD0GozP0Cmr/u0MwrXMlLoJUU=\">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</latexit><latexit 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sha1_base64=\"6DPbbG+r7mz1ssQMhdn/MHyGz7U=\">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</latexit>\nFigure 5. A schematic magnetization curve of the UJ lad-\nder (1).Jc1\n2andJc2\n2represent quantum critical points. The\nSMTLL phase exists in the region Jc1\n2<J 2<Jc2\n2.7\nThe SMTLL phase results from the competition of the\nquasi-long-range N\u0013 eel order of the TLL and the ferri-\nmagnetic order. The geometrical frustration is necessary\nfor the SMTLL to exist because, in the absence of ge-\nometrical frustration, the Marshall-Lieb-Mattis theorem\nprohibits the existence of an incommensurate magneti-\nzation (41). However, frustration alone is not su\u000ecient.\nA frustrated diamond chain [19] has no incommensurate\nphase. This is because the diamond chain cannot have\nthe TLL and the ferrimagnetic structures simultaneously.\nBy contrast, the UJ ladder is a superposition of the three-\nleg ladder leading to the TLL and the diamond chain\nleading to the ferrimagnetic order. In this respect an in-\nvestigation of itinerant ferrimagnet [44] in 1D would be\nvery interesting.\nIn conclusion, in this Paper, we showed the existenceof a spontaneously magnetized phase, with TLL prop-\nerties, the SMTLL. We gave an e\u000bective theory for this\nphase and computed the magnetization and the dynam-\nical structure factors. We derived the nonrelativistic\nNambu-Goldstone boson near k= 0 and the SMTLL\nneark=\u0019.\nACKNOWLEDGMENTS\nWe thank T. Shimokawa for giving us the motivation\nfor this study. We are also grateful to M. Oshikawa and\nH. Watanabe for in-depth discussions. This work is sup-\nported by the Swiss National Foundation under MaNEP\nand division II.\n[1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345\n(1961).\n[2] J. Goldstone, Nuovo Cimento 19, 154 (1961).\n[3] J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev.\n127, 965 (1962).\n[4] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133\n(1966).\n[5] P. C. Hohenberg, Phys. Rev. 158, 383 (1967).\n[6] T. Momoi, J. Stat. Phys. 85, 193 (1996).\n[7] T. Giamarchi, Quantum Physics in One Dimension (Ox-\nford University Press, Oxford, 2004).\n[8] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and\nM. Rigol, Rev. Mod. Phys. 83, 1405 (2011).\n[9] A. Del Maestro, M. Boninsegni, and I. A\u000feck, Phys.\nRev. Lett. 106, 105303 (2011).\n[10] T. Eggel, M. A. Cazalilla, and M. Oshikawa, Phys. Rev.\nLett. 107, 275302 (2011).\n[11] T. St oferle, H. Moritz, C. Schori, M. K ohl, and\nT. Esslinger, Phys. Rev. Lett. 92, 130403 (2004).\n[12] J. Taniguchi, Y. Aoki, and M. Suzuki, Phys. Rev. B 82,\n104509 (2010).\n[13] M. Klanj\u0014 sek, H. Maya\u000bre, C. Berthier, M. Horvati\u0013 c,\nB. Chiari, O. Piovesana, P. Bouillot, C. Kollath,\nE. Orignac, R. Citro, and T. Giamarchi, Phys. Rev.\nLett. 101, 137207 (2008).\n[14] M. Jeong, H. Maya\u000bre, C. Berthier, D. Schmidiger,\nA. Zheludev, and M. Horvati\u0013 c, Phys. Rev. Lett. 111,\n106404 (2013).\n[15] H. Nielsen and S. Chadha, Nucl. Phys. 105, B445 (1976).\n[16] H. Watanabe and H. Murayama, Phys. Rev. Lett. 108,\n251602 (2012).\n[17] Y. Hidaka, Phys. Rev. Lett. 110, 091601 (2013).\n[18] N. B. Ivanov, J. Richter, and U. Schollw ock, Phys. Rev.\nB58, 14456 (1998).\n[19] S.-i. Yoshikawa and S. Miyashita, J. Phys. Soc. Jpn. 74,\n71 (2005).\n[20] K. Hida, J. Phys.: Condens. Matter 19, 145225 (2007).\n[21] K. Hida and K. Takano, Phys. Rev. B 78, 064407 (2008).\n[22] R. R. Montenegro-Filho and M. D. Coutinho-Filho, Phys.\nRev. B 78, 014418 (2008).\n[23] K. Hida, K. Takano, and H. Suzuki, J. Phys. Soc. Jpn.\n79, 114703 (2010).[24] T. Shimokawa and H. Nakano, J.Korean Phys. Soc. 63,\n591 (2013).\n[25] A. Collins, J. McEvoy, D. Robinson, C. J. Hamer, and\nZ. Weihong, Phys. Rev. B 73, 024407 (2006).\n[26] W. Zheng, J. Oitmaa, and C. J. Hamer, Phys. Rev. B\n75, 184418 (2007).\n[27] R. F. Bishop, P. H. Y. Li, D. J. J. Farnell, and C. E.\nCampbell, Phys. Rev. B 82, 024416 (2010).\n[28] G. Sierra, arXiv:cond-mat/9610057 (1996).\n[29] G. Sierra, J. Phys. A: Math. Gen. 29, 3299 (1996).\n[30] I. A\u000feck, J. Phys.: Condens. Matter 1, 3047 (1989).\n[31] F. Haldane, Phys. Lett. 93A, 464 (1983).\n[32] I. A\u000feck and F. D. M. Haldane, Phys. Rev. B 36, 5291\n(1987).\n[33] T. Brauner, Symmetry 2, 609 (2010).\n[34] T. Brauner, Phys. Rev. D 75, 105014 (2007).\n[35] R. Chitra and T. Giamarchi, Phys. Rev. B 55, 5816\n(1997).\n[36] P. Bouillot, C. Kollath, A. M. L auchli, M. Zvonarev,\nB. Thielemann, C. R uegg, E. Orignac, R. Citro,\nM. Klanj\u0014 sek, C. Berthier, M. Horvati\u0013 c, and T. Gia-\nmarchi, Phys. Rev. B 83, 054407 (2011).\n[37] M. Oshikawa, M. Yamanaka, and I. A\u000feck, Phys. Rev.\nLett. 78, 1984 (1997).\n[38] Strictly speaking, our model (1) is out of scope of such a\nweak coupling approach. However, it is known that the\nweak coupling approach still gives qualitatively reason-\nable results [45] in the TLL phase. As we will see later,\nthis approach is qualitatively valid even in the magne-\ntized phases.\n[39] K. Takano, K. Kubo, and H. Sakamoto, J. Phys.: Con-\ndens. Matter 8, 6405 (1996).\n[40] T. Tonegawa, K. Okamoto, T. Hikihara, Y. Takahashi,\nand M. Kaburagi, J. Phys. Soc. Jpn. 69, Suppl. A, 332\n(2000), arXiv:cond-mat/9912482v1.\n[41] W. Marshall, Proc. R. Soc. A 232, 48 (1955).\n[42] E. Lieb and D. Mattis, J. Math. Phys. 3, 749 (1962).\n[43] J. M. Rom\u0013 an and J. Soto, Phys. Rev. B 62, 3300 (2000).\n[44] H. Ishizuka and Y. Motome, Phys. Rev. Lett. 109,\n237207 (2012).\n[45] H. J. Schulz, Phys. Rev. B 34, 6372 (1986)."    },    {        "title": "2312.13911v1.Altermagnetic_ferroelectric_LiFe2F6_and_spin_triplet_excitonic_insulator_phase.pdf",        "content": "Altermagnetic ferroelectric LiFe 2F6and spin-triplet excitonic insulator phase\nPeng-Jie Guo1,∗Yuhao Gu2,†Ze-Feng Gao1, and Zhong-Yi Lu1‡\n1. Department of Physics, Renmin University of China, 100872, Beijing, China and\n2. Department of Physics, University of Science and Technology Beijing, 100083, Beijing, China\n(Dated: December 22, 2023)\nAltermagnetism is a new magnetic phase with k-dependent spin polarization and may exist in\nan insulatinging state with a high N´ eel temperature. This provides a new opportunity to obtain\nboth spin and electric polarization in one material. 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. Due to the abundance of novel physical properties,\nLiFe2F6will certainly attract a wide range of theoretical\nand experimental interest.5\nACKNOWLEDGMENTS\nWe thank Z.-X. Liu, S. Qu and S. Dong for valu-\nable discussions. This work was financially supportedby the National Natural Science Foundation of China\n(No.11934020 and No.12204533). Computational re-\nsources have been provided by the Physical Laboratory\nof High Performance Computing at Renmin University\nof China.\n[1] W. Eerenstein, N. D. Mathur, and J. F. Scott, Multi-\nferroic and magnetoelectric materials, Nature 442, 759\n(2006).\n[2] S.-W. Cheong and M. Mostovoy, Multiferroics: A mag-\nnetic twist for ferroelectricity, Nat. Mater. 6, 13 (2007).\n[3] S. Dong, J.-M. Liu, S.-W. Cheong, and Z. Ren, Multifer-\nroic materials and magnetoelectric physics: symmetry,\nentanglement, excitation, and topology, Adv. Phys. 64,\n519 (2015).\n[4] M. Fiebig, T. Lottermoser, T. Lottermoser, and\nM. 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B 6, 1968 (1972)."    },    {        "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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Recent observation of helicity -independent all-optical  \nmagnetization switching (HI -AOS) in an exchange -coupled ferromagnet ferrimagnet  \n(FM-FEM) heterostructures expanded the range and applicability of such ultrafast heat -\ndriven magnetization switching. Here we report the element -resolved HI -AOS \ndynamics of such an exchange -coupled system, using a modified microscopic three -\ntemperature model. We have studied the effect of i) the Curie temperature of the FM, \nii) FEM composition, ii i) the long -range Ruderman –Kittel –Kasuya –Yosida (RKKY) \nexchange -coupling strength, and iv) the absorbed optical energy on the element -\nspecific time -resolved magnetization dynamics. The phase -space of magnetization \nillustrates how the RKKY coupling strength  and the absorbed optical energy influence \nthe switching time. Our analysis demonstrates that the threshold switching energy \ndepends on the composition of the FEM and the switching time depends on the Curie \ntemperature of the FM as well as RKKY coupling st rength. This simulation anticipates \nnew insights into developing faster and more energy -efficient spintronics devices.  \n \n1. Introduction  \nUltrafast helicity -independent all -optical toggle switching (HI -AOS) has been an \nessential topic of research in the spintronics community for its potential applications in \ndigital data storage [1]. It is being explored in ferrimagnet (FEM) alloys and multilayers \n[2, 3, 4, 5, 6, 7, 8, 9, 10, 11] over the last decade. The magnetization switches in FEMs \ndue to the direct e xchange coupling between the transition metal and rare -earth \nelements. However, the switching is limited within a small range of optical energy and \nthe rare -earth element (mostly Gd) concentration and the utilization of synthetic FEMs \nlift these restrictio ns. [4] Theoretically, time -resolved and element -specific \nmagnetization  dynamics in such FEMs have been modeled by various means namely; \ni) atomistic Landau -Lifshitz -Gilbert equation [7, 12], ii) phenomenological mean -field \ntheory [6, 13] , and iii) a modif ied version of the microscopic three -temperature model 2 (M3TM) [14, 15, 4, 5]. Beens et al. [4, 5] have used M3TM to incorporate the direct \nexchange coupling between the transition metal and rare earth and described the time -\nresolved magnetization dynamics of FEM alloy as a function of rare earth concentration \nand absorbed optical energy.  \nOn the other hand, magnetization switching in ferromagnets (FM) is generally \nachieved using current -induced directional spin -transfer and spin -orbit torques \n(different from  HI-AOS) [16, 17, 18, 19]. The Landau -Lifshitz -Gilbert equation in \ncombination with the terms for spin currents was used to analyze such a switching \nmechanism. Only recently, it was demonstrated that a conventional FM can also exhibit \nHI-AOS either due to i) the non -local ultrafast spin current  [20, 21] or due to ii) the \nlong-range Ruderman –Kittel –Kasuya –Yosida (RKKY) exchange coupling [22, 23, \n24]. HI -AOS in such structures is promising as the exchange -coupled FM -FEM \nstructure can work as a free (storage) layer in magnetic memory devices [11, 25] for \nefficient electrical reading due to the enhanced tunneling magneto -resistance. RKKY \nexchange -mediated ultrafast switching in FMs has a rich physics that has hitherto \nlargely remained unexplored via theoretical modeling. Gorchon et al. [22] \nexperimentally demonstrated ultrafast switching of an exchange -coupled FM layer \n(Co/Pt) in ∼7 ps. Recently, Chatterjee et al. [24] have obtained a significantly faster \nswitching in ∼3.5 ps using an optimized structure and  introduced a modified M3TM \nsimulation to analyze the switching phenomenon. In these structures, RKKY -type \nexchange coupling [26, 27, 28] has been discussed as the primary source of angular \nmomentum transfer channel between the FM and the FEM for obtaining ultr afast \nmagnetization switching [22, 24].  \nIn this paper, we discuss the theoretical modeling of the modified M3TM in greater \ndetail and simulated the experimentally observed ultrafast magnetization switching in \n∼7 ps by Gorchon et al. [22]. We have analyzed the effect of i) the Curie temperature \n(Tc), ii) the composition of the FEM, iii) RKKY coupling strength, and iv) absorbed \noptical energy on the ultrafast magnetization dynamics of the exchange -coupled \nheterostructure. The main addition in this model compa red to the M3TM model, \n(introduced by Beens et al. [4] to study HI -AOS in FEM alloys and multilayers) is the \nincorporation of a long -distance RKKY exchange coupling term, which connects the \nmagnetization dynamics of the FM and FEM layer and enables us to e xamine the \nexchange coupled heterostructure. The FM is magnetically exchange -coupled with the \nFEM while physically separated by a non -magnetic spacer layer (Pt) of high spin -\nscattering strength. The spacer layer significantly blocks any possible transmissi on of \nspin-current which can affect the magnetization dynamics [22, 24]. We observe that \nRKKY exchange strength, which is two orders of magnitude less than the direct \nexchange between the transition metal and rare earth elements of the FEM, is sufficient \nfor switching the FM. Experimentally the Curie temperature of a Co/Pt ML system can \neasily be tuned by changing the thickness and the repetition of the individual layers. \nWe detect a notable slowing of the Co/Pt  switching timescale upon reducing the Tc of \nthe FM. The alloy composition of the FEM mainly affects the switching energy \nthreshold and the dependence of the switching timescale on the RKKY exchange \ncoupling.  3  \n \nFigure 1: Schematic diagram of the simulated sample structure and (b) the different exchange interactions \nacting on the sub -lattices of the sample.  \n2. Sample Structure and Simulation Method  \nWe have simulated the exchange coupled FM -FEM heterostructure \n(Sub/FMFEM/capping layer) experimentally studied by Gorchon et al. [22 ] and  \nreplaced their FEM alloy  (Fe65Gd 28Co7) with Fe72Gd 28 for the ease of calculation and \nthe Tc of the FM (Co/Pt) is taken to be 470 K as estimated in their experiment. We have \nextended the simulation to explore the effect of different FEM compositions namely; \nFe72Gd 28 and Co72Gd 28, and henceforth, these two FEMs are denoted as FeGd and \nCoGd. We have simulated the effect of different Tc (470 K, 530 K, and 580 K) of the \nCo/Pt on the magnetization dynamics. The configuration of the sample is \nSubstrate/Ta(5 nm)/ Fe72Gd 28 (or Co72Gd 28)(20 nm)/Pt(1.5 nm)/Co(0.6nm)/Pt(0.7 \nnm)/Co(0.6 nm)/Pt(3 nm), where the thicknesses of each layer are given in nanometers \nand schematically shown in Fig. 1a. The thickness of the Pt spacer layer, which \nseparates the FM and FEM sub -lattices, determines the stren gth and sign of RKKY \nexchange coupling. Experimentally it was varied between 1 to 4 nm to obtain both \nferromagnetic and anti -ferromagnetic coupling. The coupling strength varies non -\nlinearly with the thickness of the Pt layer according to the characteristi c oscillatory \nnature of the RKKY exchange interaction [26, 28] and the effect is simulated by \nvarying the RKKY coupling within a predefined range.  \n \nThe different types of exchange interactions acting on the heterostructure are \nschematically represented in Fig. 1b. The electron and phonon sub -systems of the FM \nand the two sub -lattices of the FEM have been described by a two-temperature  model \nexpressed by the equations 1 and 2. These two sub -systems (electrons and phonons) of \nthe FM and FEM layer get excited by the ultrafast optical pulse depending on the  \nrelative optical absorption of the respective layers. First, we solve the two -temperature \nSi  Substrate  \nTa (5 nm)  \nFe \n72 \nGd \n28 \n/Co \n72 \nGd \n28 \n nm \n) \n (20 \nPt (1.5 nm)  \nCo (0.6 nm)  \nPt (0.7 nm)  \nCo (0.6 nm)  \nPt (3.0 nm)  \na \n) \n( \n ) \n( \nb \nCo/Pt  \nFe  \n( \nCo \n) \nGd \nElliot \n- \nYafet \nScattering  \nElliot \n- \nYafet \nScattering  \nElliot \n- \nYafet \nScattering  \nUltrafast Laser  \nRKKY Exchange  \nJ \nCo \n/Pt \n- \nFe(Co)  4 model to m easure the temporal evolution of the electron ( Te) and phonon temperatures \n(Tp) for the FM and FEM using the following equations,  \n \n𝐶𝑒,𝑖𝛿𝑇𝑒,𝑖\n𝛿𝑡= −𝑔𝑒𝑝,𝑖(𝑇𝑒,𝑖−𝑇𝑝,𝑖)+ 𝑃𝑖; 𝑃𝑖=𝐴𝑖\n𝜎√2𝜋𝑒−(𝑡\n𝜎𝜋)2\n  \n(1) \n𝐶𝑝,𝑖𝛿𝑇𝑝,𝑖\n𝛿𝑡= 𝑔𝑒𝑝,𝑖(𝑇𝑒,𝑖−𝑇𝑝,𝑖)+ 𝐶𝑝,𝑖(𝑇𝑎𝑚𝑏 −𝑇𝑝,𝑖)\n𝜏𝐷,𝑖; (𝑖∈𝐹𝑀,𝐹𝐸𝑀 )  \n(2) \n   \nwhere Ce and Cp are electron and phonon specific heat, τD is the heat dissipation \nconstant which represents the thermal diffusion to the substrate, Tamb is the ambient \ntemperature (295 K), gep is the electron -phonon interaction strength and Ai is the \nabsorbed optical energy in each layer. The laser pulse is simulated as a Gaussian pulse \nwith 100 fs full -width -half-maximum which acts as the energy source to  the electronic \nsub-system for each layer. The laser pulse gets absorbed differently in each of the layers \ndepending on the thickness and their optical constants, which has been evaluated by \nsolving the transfer matrix [29] (Fig. S1 in the supplemental inf ormation) of the entire \nheterostructure. The complex refractive indices applied in the calculation are given in \nTable S1 in the supplemental information. It is assumed that the electron sub -system \ninstantly reaches thermal equilibrium after optical excitat ion due to a large Coulomb \ninteraction [30], therefore, it can be represented by specific electron temperatures, \nwhich are assumed to be homogeneous inside each layer. The phonon temperatures \nremain in equilibrium due to phonon -phonon interaction. All the sub-systems are \ninitially at room temperature (295 K). The material parameters for the two -temperature \nmodel are listed in Table S2 in the supplemental information. Heat diffusion to the \nsubstrate is added to the phonon subsystem as an energy dissipation t erm with a 50 ps \ntime constant ( τD) for both layers. There are two magnetic elements such as Fe(Co) and \nGd comprising the FeGd (CoGd) alloy, whereas, for the Co/Pt, there is only one \nmagnetic element. The electrons are modeled as a spinless free electron g as and the \nphonons are described by the Debye model according to the basic postulates of the \nM3TM [31]. For simplicity, it is assumed that all the spin subsystems have the same \nspin half quantum number ( si = 1/2) [4, 5] with their respective atomic magneti c \nmoments. We have used the following analytical expressions of exchange splitting for \ncalculating the exchange coupling using Fermi’s golden rule and the details of the \nmagnetization dynamics are given in section S2 in the supplemental information.  \n∆𝐹𝑒(𝐶𝑜)= 𝑥𝐹𝑒(𝐶𝑜)𝛾𝐹𝑒(𝐶𝑜)𝑚𝐹𝑒(𝐶𝑜)+(1−𝑥𝐹𝑒(𝐶𝑜))𝛾𝐹𝑒(𝐶𝑜)−𝐺𝑑𝑚𝐺𝑑 \n∆𝐺𝑑= 𝑥𝐹𝑒(𝐶𝑜)𝛾𝐺𝑑−𝐹𝑒(𝐶𝑜)𝑚𝐹𝑒(𝐶𝑜)+(1−𝑥𝐹𝑒(𝐶𝑜))𝐺𝑑𝑚𝐺𝑑  \n∆𝐶𝑜𝑃𝑡= 𝛾𝐶𝑜𝑃𝑡𝑚𝐶𝑜𝑃𝑡 +𝑥𝐹𝑒(𝐶𝑜)𝛾𝐶𝑜𝑃𝑡 −𝐹𝑒(𝐶𝑜)𝑚𝐹𝑒(𝐶𝑜)+(1−𝑥𝐹𝑒(𝐶𝑜))𝛾𝐶𝑜𝑃𝑡\n−𝐺𝑑𝑚𝐺𝑑   \n \n(3) \nwhere γij = zJijµj;(i,j ∈ Fe(Co), Gd, Co/Pt ) can be calculated from the exchange \ninteraction strength Jij and atomic magnetic moment µj. The percentage of Fe (Co) in 5 the FEM alloy is given by xFe (xCo); where xFe(Co) = 0.72. We apply the Weiss mean field \ntheory to determine the exchange strength which has the form  𝐽𝑖=3𝑘𝐵𝑇𝑐,𝑖\n2𝑧𝑆𝑖𝑆𝑖+1,𝑖=\n𝐹𝑒(𝐶𝑜),𝐺𝑑,𝐶𝑜𝑃𝑡  [6]. Here z  = 12 is the number of next nearest neighbors and Tc,i is \nthe Curie temperature of individual elements  (Tc,Co = 1388  K, T c,Fe = 1043  K, T c,Gd = 292  \nK, T c,Co/Pt = 470 K, 530 K, and 580 K). The short -range exchange interaction between \nFe and Gd sub -lattices is antiferromagnetic and it has the form: JGd−Fe = − 0.348 × JFe \nwhereas , for CoGd, the coupling between Co and Gd is; JGd−Co = − 0.388 × JCo [3, 24]. \nRKKY exchange coupling strength ha s been varied up to ∼5 % of the FM exchange \ni.e., JRKKY  = ± 0.05 × JCoPt for both the heterostructures to include the effect of varying \nPt thicknesses which leads to the oscillating RKKY interaction. There are three \nchannels for angular momentum transfer i n the FEM, i) Elliott -Yafet type spin -flip \nscattering in Fe (Co), Gd and Co/Pt [31, 30, 32], ii) a short -range exchange coupling \nbetween Fe (Co) and Gd and iii) a long -range RKKY type exchange coupling between \nCoPt -Fe(Co) and CoPt -Gd layer [22]. As describ ed in equation 3, we have used the \nRKKY exchange between CoPt -Fe(Co) and CoPt -Gd. However, the dominant RKKY \nexchange contribution comes from the interaction of Gd and Co/Pt , and by including \nFe(Co) in the scheme, the dynamics are only minimally affected ( Fig. S3 of \nsupplemental information). This is because Gd has a much larger atomic magnetic \nmoment  (µGd = 7.55µB, µFe = 2.20µB, µCo = 1.72µB, and µCo/Pt = 1.30µB) moment \ncompared to Fe(Co) and thereby dominates the angular momentum transfer process.  \n \nThe electronic parameters of the materials in the simulation are obtained from the \nliterature [33, 4, 34] and are given in the Table S3 of supplemental information. The \ntime evolution of the magnetization dynamics has been calculated up to 40 ps after the \noptical excitation with 10 fs time resolution. The FM exchange varies linearly with Tc \nand takes the following values;  JRKKY,T c,CoPt =470 K = 1.079×1 0-21 J , JRKKY,T c,CoPt =530 K = \n1.217×1 0-21 J and JRKKY,T c,CoPt =580 K  = 1.332×10−21 J. Assuming a typical atomic spacing of \n0.4 nm, the RKKY exchange coupling is varied between ±  5% of the direct exchange, \nwhich results in JRKKY,T c,CoPt =470 K = ± 0.337 mJ.m-2, JRKKY,T c,CoPt =530 K = ± 0.394mJ.m-2, and \nJRKKY,T c,CoPt =580 K  = ± 0.416 mJ.m-2 for Tc = 470 K, 530 K, and 580 K respectively. S. S. P. \nParkin [27] studied the exchange coupling strength of various 3d, 4d, and 5d transition \nmetals at room temperature and found that the exchange coupling varies from 0.1 to 5 \nmJ.m−2. Thus, the value of the RKKY exchange coupling strength in our simulation is \nwell within the known range. Although the specific thickness of the Pt layer determines \nthe sign and strength of the RKKY exchange coupling, a small thickness difference of \na couple of nanometers doe s not change the relative absorbed energy between the FM \nand FEM layer by more than 4%. Therefore, for simplicity, we assume that the relative \nabsorbed energy between the FM and FEM layer remains the same while changing the \nPt spacer layer thickness (i.e. with the variation of the RKKY exchange coupling). The \ntypical thickness variation of the Pt spacer layer (or any other spacer layer) would result \nin the modification of the RKKY coupling. Therefore, we have simulated the \nmagnetization dynamics within an a cceptable range of this coupling strength  which is \ngenerally obtained using standard RKKY coupling layers [23, 24].  6 3. Results and Discussions  \n3.1. Effect of the Curie Temperature of the Co/Pt Layer  \nThe phase diagrams of the magnetization as a function of absorbed optical energy \nand RKKY exchange coupling strength are shown in Fig. 2a -c for different values of \nTc (470 K, 530 K, and 580 K) of the Co/Pt layer. The color scale represents the \nmagnetizatio n reversal time of Co/Pt. The final magnetization state has been calculated \nby measuring the magnetization of each element (Fe, Gd, and Co/Pt) at 20 ps after the \nultrafast laser excitation. The initial magnetization of Fe and Co/Pt is positive and it is \nnegative for Gd before the arrival of the pump laser pulse (0 ps). Ideally, switching can \nbe defined when the magnetization of a particular element crosses zero, however, it has \nbeen noticed that the magnetization tends to stay close to zero for some time be fore \nswitching. Consequently, to avoid any ambiguity in the switching time, we have set a \nsmall non -zero magnetization value as the measure of switching [7]. The set conditions \nare the following in order to determine the different cases which may occur as a final \nmagnetization state: i) relaxed if mFe(t = 20 ps) > 0.01, mGd(t = 20 ps) < − 0.01 and \nmCoPt(t = 20 ps) > 0.01, ii) switched if mFe(t = 20 ps) < − 0.01, mGd(t = 20 ps) > 0.01 \nand mCoPt(t = 20 ps) < − 0.01, and iii) completely demagnetized if 0 .01 ≥ mFe(t = 20 ps) \n≥ −0.01, 0 .01 ≤ mGd(t = 20 ps) ≤− 0.01, and 0 .01 ≥ mCoPt(t = 20 ps) ≥ −0.01. \n \nThe phonon temperature increases beyond the Curie temperature at very high \noptical energy, which makes the system unstable i.e. it doesn’t remember its initial \nstate, and the final state randomly switches between up and down, especially at longer \ntimescale s (> 40 ps). Such behavior has also been identified in the simulated HI -AOS \nproperties of [Tb/Co] multilayers [35]. Hence, we have limited the calculation of the \nfinal switched state to 20 ps after the laser excitation. The FEM undergoes partial \ndemagnetiz ation and the FM shows either partial or full demagnetization before \nremagnetization at smaller absorbed optical energy. Consequently, the final state is the \nsame as the initial state for all the elements suggesting the system is in a relaxed state \nirrespe ctive of the RKKY coupling as represented by the pale -yellow region in Fig. 2a -\nc. In this energy range, an example of the magnetization dynamics of Fe, Gd, and Co/Pt \nis plotted in Fig. S4 of the supplemental information, where we find a partial \ndemagnetiza tion of Fe and Gd, and simultaneously Co/Pt gets fully demagnetized even \nthough it absorbs only ∼23% of the total optical energy.  However, we can’t switch \nCo/Pt due to the unavailability of the angular momentum transfer channel with the FEM \n(as FEM doesn’t  switch at smaller optical energy). With increasing optical energy, a \nswitched region appears as depicted by the color map in combination with bright blue \nand magenta colors, where the magnetization reversal of both the FM and FEM sub -\nlattices is obtained at the same absorbed energy.  The green region suggests the \ncomplete switching of the FEM layer, however, the Co/Pt layer doesn’t switch. \nDetailed discussions about these two regions are provided later. On the other hand, with \nintense optical energy (dark b rown region of the phase space in Fig. 2a -c), only a \npermanent demagnetization of all the sub -lattices is observed as the Tp crosses the Tc \nfor all the magnetic components.  Experimentally, in this region, the sample gets \ndamaged and its magnetic properties can be permanently lost if we expose the sample \nto such high optical energy for a longer time.  7  \nFigure 2: The phase -space of magnetization for (a) F1FeGd,Tc,CoPt =470 K, (b) F1FeGd,T c,CoPt =530 K, and (c) \nF1FeGd,T c,CoPt =580 K. The pale -yellow region shows a relaxed state, the green region depicts the switching of \nonly FEM, the bright blue and magenta region suggests the switching of both the FM and FEM, and the dark \nbrown region depicts complete demagnetizatio n of the sample. The color bar of the blue and magenta region \nshows the switching time of Co/Pt. The time -resolved element -specific magnetization dynamics of the \ncorresponding elements of (d -f) the FEM and (g -i) the FM sub -lattices in the FM -FEM heterostru cture at the \nabsorbed optical energy of 33 .2 × 108 J.m−3 for a specific RKKY exchange coupling ( JRKKY = + 0.01×JCoPt). \nThe dynamics are measured at the corresponding black dots in Fig. 2a -c. The yellow and cyan -filled region \nin Fig. 2d -i shows the evolutio n of electron and phonon temperatures of the FM and FEM layers. The red, \ncyan, and green dotted lines are the Tcs of the individual elements (Fe, Gd, and Co/Pt). The laser pulse is \nschematically displayed by the black -filled Gaussian pulse at time zero.  \nThe time -resolved element -specific magnetization dynamics of the magnetic \nelements of the FM and FEM layers measured at the black dot on the phase space is \nplotted in Fig. 2d -i for absorbed optical energy of 33 .2 × 108 J.m−3 at a specific RKKY \nexchange coupling strength of JRKKY = + 0.01 × JCoPt. Fe, Gd, and Co/Pt sub -lattices \ndemagnetize at an ultrafast timescale upon ultrafast optical excitation where the \nF1 \nFeGd, Tc(CoPt) = 470 K  \n F1 \nFeGd, Tc(CoPt) = 530 K  \n F1 \nFeGd, Tc(CoPt) = 580 K  \n( \na \n) \n ( \nb \n) \n ( \nc \n) \n( \nd \n) \n ) \n( \ng \n F1 \nFeGd, Tc(CoPt) = 470 K  \n( \ne \n) \n ( \nh \n) \n F1 \nFeGd, Tc(CoPt) = 530 K  \n( \nf \n) \n ( \ni \n) \n F1 \nFeGd, Tc(CoPt) = 580 K  8 demagnetization rate (at a constant fluence) is proportional to  𝑇𝑐2\n𝜇 [31]. The transition \nmetal (Fe) of the FEM layer, demagnetizes much faster than that of the rare earth \nelement (Gd), and the FEM shows a transient ferromagnetic -like state between ∼0.7-\n1.9 ps when the magnetization of both Fe and Gd points along the same  direction. As \nthe intrinsic coupling between Fe -Gd is antiferromagnetic, Gd switches later due to \nangular momentum exchange with Fe. The Co/Pt layer exhibits a significantly faster \ndemagnetization in ∼500 fs due to enhanced Elliott -Yafet scattering via sp in-orbit \ntorque as published earlier [33].  \n \nIn the bright blue and magenta colored region of Fig. 2a -c, the electron temperature \nof Co/Pt rises faster and reaches significantly larger than its Tc (can be seen from the \npeak of the yellow -shaded region much larger than Tc in Fig. 2g -i). As a result, Co/Pt \ngets completely demagnetized and then its magnetization stays close to zero for a \nlonger timescale as the electron -lattice temperature equilibrates and slowly cools down. \nThe switched FEM kicks in to deliver  enough angular momentum to ultimately switch \nits magnetization. In this region, the Co/Pt and the FEM switch at the same absorbed \nenergy. The magnetization dynamics of Co/Pt show a two -step process and the \nswitching occurs at a much longer timescale as de tected in Fig. 2g -h measured at the \nrespective black dots in Fig. 2a -b. The observation of such a two -step switching \ncharacteristic of the Co/Pt layer (exchange coupled with a FEM) has been explained as \na signature of the RKKY exchange coupling mediated sw itching of the hot/softened \nCo/Pt [22, 24].  \n \nOne of the main differences among the three phase -space figures in Fig. 2ac is the \nexistence of the green region which increases with increasing Tc of the Co/Pt layer, \nsuggesting that the Co/Pt and the FEM layer  doesn’t switch at the same absorbed \nenergy. FEM switches at a smaller energy (when Co/Pt remains demagnetized) and \nthen the Co/Pt requires a larger absorbed optical energy to switch. It is relatively easy \nto understand that Co/Pt cannot be switched in a d ecoupled structure, where there is no \nRKKY exchange coupling with the FEM layer. Hence, we notice the green region in \nFig. 2a -c, where the RKKY exchange is zero (i.e. at the middle of the x -axis). However, \nfor larger Tc (530 K, and 580 K) the green region extends towards a finite RKKY \ncoupling strength  (− 0.009 × JCoPt ≤ JRKKY ≤ 0.009 × JCoPt for 530 K, and −  0.045 × JCoPt \n≤ JRKKY ≤ 0.045 × JCoPt for 580 K) as shown in Fig. 2b -c. Now, for a smaller Tc, the \nelectron/phonon temperature requires a long time to cool below Tc, and the Co/Pt layer \nstays completely demagnetized for a longer time. The inter -sublattice (between FM and \nFEM) angular momentum exchange depends on the RKKY exchange strength and the \navailable magnetization in the FEM layer, which is dominated by Gd due to its much \nlarger atomic magnetic moment. At a longer timescale, Gd recovers a larger \nmagnetization value as shown in Fig 2d -f, hence the effective angular momentum \nexchange is larger,  and Co/Pt can switch at a smaller RKKY exchange strength \nhowever the switching speed gets slower. Now, with increasing Tc, the electron/phonon \ntemperature gets below Tc much faster as plotted in Fig. 2h, and we get Co/Pt switching \nat a faster timescale. T he effective exchange coupling strength increases with \nincreasing Tc. The value of JRKKY is displayed at the top x -axis of the phase space \ndiagrams in Fig. 2a -c. Next, for an even larger Tc, the electron/phonon temperature 9 cools down much more quickly to b elow Tc, as shown in Fig. 2i, when the available \nmagnetization of Gd for the RKKY exchange is tiny. Then Co/Pt can’t switch and starts \nto remagnetize along its initial direction. Hence, Co/Pt either needs a larger RKKY \nexchange coupling or a larger optical  energy to switch. In case of larger RKKY \nexchange strength, the Co/Pt switches faster as discussed later. For larger absorbed \noptical energy, the electron/phonon temperature rise is higher and it takes a longer time \nto cool down below Tc. Within that time , Gd gets sufficiently magnetized in the \nopposite direction to provide the necessary angular momentum transfer even at a \nsmaller RKKY exchange coupling strength. Hence, in general, Co/Pt switches at a \nslower speed with increasing energy as observed in Fig.  S5 in the supplemental \ninformation. However, this contradicts the experimental observation by Gorchon et al. \n[22], where the switching speed of Co/Pt got faster with increasing optical energy. The \nswitching dynamics also depend on the RKKY exchange coupli ng strength and we do \nnotice a tiny energy range at smaller RKKY interactions, where the switching speed \nincreases with increasing energy as plotted in Fig. S5 of the supplemental information. \nAnother difference in the phase space among these three structu res, is the larger \nswitched region (bright blue and magenta color region), with increasing Tc. The Co/Pt \nlayer can sustain larger absorbed optical energy before getting damaged, as the Tc \nincreases, hence we detect switching up to larger optical energy.  \n \nThe variation of the Co/Pt switching time with the RKKY exchange strength is \nplotted in Fig. 3 for different Tc. The switching timescale slows down exponentially \nupon reducing the exchange coupling. As Co/Pt gets demagnetized much more quickly \n(plotted in F ig. 2d -i) than the FEM elements, it waits for the FEM layer (mainly Gd) to \nremagnetize sufficiently and provide the necessary angular momentum transfer to \nswitch. As discussed earlier, at fixed absorbed optical energy, larger exchange coupling \nstrength mea ns the possibil ity of a larger amount of angular momentum transfer even \nif the Gd recovers a small magnetization in the opposite direction (in early timescale), \nand eventually, it translates to a faster switching. The Co/Pt layer switches ∼10 ps for  \nJRKKY = + 0.01 × JCoPt, for F1FeGd,T c,CoPt =470 K, and by increasing the exchange amplitude \nto JRKKY = + 0.05 × JCoPt, the switching gets faster at ∼6 ps as shown in Fig. 3a.  \n \n \nFigure 3: The variation of the switching time of Co/Pt with the RKKY exchange coupling strength for (a) Tc \n= 470 K, (b) Tc = 530 K, and (b) Tc = 580 K. All the simulations are performed at absorbed optical energy of \n33.2 × 108 J.m−3. \nF1 \nFeGd, Tc(CoPt) = 530 K  \n( \na \n) \nF1 \nFeGd, Tc(CoPt) = 580 K  \n F1 \nFeGd, Tc(CoPt) = 470 K  \n( \nc \n) \n ( \nb \n) 10 On the other hand, as we have discussed earlier, the FM with higher Tc can’t be \nswitched at a smaller exchange as observed in Fig. 3b -c. A significant difference in the \nswitching timescale (at a fixed exchange strength a nd absorbed optical energy) of the \nCo/Pt is detected for the different values of Tc. The switching speed of the Co/Pt layer \ngets faster with increasing Tc. At the largest RKKY exchange strength and for the fixed \nabsorbed optical energy, with increasing Tc, the electron/phonon temperature reduces \nbelow Tc much faster, and if sufficient angular momentum transfer channel is available, \nCo/Pt switches faster. Similarly, it needs a longer time, for its electron temperature to \ncool sufficiently below a smaller Tc when the transferred angular momentum from the \nswitched FEM sub -lattice ultimately switches its magnetization. Therefore, it is evident \nthat the dueling time of Co/Pt in the demagnetized state (zero magnetization state) is \nsmall for larger Tc, which ultima tely fastens the switching time. Previously we \nmeasured and simulated a differently stacked heterostructure and found the switching \ntimescale of the Co/Pt to be ∼3.5 ps [24]. We have examined the effect of Tc in that \nheterostructure as well and found simil ar dependence of the switching timescale of \nCo/Pt, which is plotted in Fig. S6 of supplemental information. Hence, we can also \nconclude that the effect of the Tc on the switching time of the Co/Pt layer is universal \nand only minimally dependent on the samp le geometry.  \n \n3.2. Effect of the Composition of the Ferrimagnet Layer  \nNext, the effect of the composition of the FEM on the magnetization switching \ndynamics has been explored by changing the FEM to CoGd (from FeGd). The phase \nspace of magnetization is displayed in Fig. 4a considering CoGd  as the FEM while \nkeeping the Tc of the Co/Pt fixed at 580 K. We don’t see the extended green region (for \na finite RKKY coupling) in Fig. 4a suggesting both FM and FEM switches \nsimultaneously even for a larger Tc (significantly different from Fig. 2c).  \n \nFigure 4: The phase -space of switching for F1CoGd,Tc,CoPt =580 K. The pale -yellow region indicates a relaxed state, \nthe green region shows the switching of only FEM alloy, the bright blue and magenta region exhibits the \nswitching of both the FM and FEM and the dark brown region shows complete demagnetization of the \nsample. The color bar of the blue and magenta region describes the switching time of Co/Pt. (b) Switching \ntime of Co/Pt as a function of RKKY exchange coupling at the absorbed optical energy o f 50.4 × 108 J.m−3. \n( \na \n) \n ( \nb \n) \n50.4 \n  \n× \n10 \n8 \nJ.m \n- \n3 \nRelaxed  \nSwitching of Co  \n72 \nGd \n28 \nSwitching of both CoPt and Co  \n72 \nGd \n28 \nComplete Demagnetization  \nF1 \nCoGd, Tc(CoPt) = 580 K  11 The Tc of Co in the CoGd sub -lattice is much larger than that of Fe in the FeGd sub -\nlattice, hence larger absorbed optical energy is required to switch the CoGd FEM itself. \nOn the other hand, the Tc of the Co/Pt layer is the same as bef ore, and ∼23% of the \noptical energy gets absorbed by it. Consequently, a larger optical energy is also \ndelivered in the Co/Pt layer, and the electron/phonon temperature needs a long time to \ncool below its Tc. Within this timescale, Gd in the CoGd sub -lattice attains adequate \nmagnetization value (discussed later), which ultimately leads to the switching of the \nFM and FEM layer at a finite RKKY exchange at the same absorbed optical energy. \nThe dependence of the switching time as a function of the RKKY couplin g is shown in \nFig. 4b and we distinguish a similar (to Fig. 3) fastening of the Co/Pt switching time \nwith increasing RKKY coupling strength, however, the much larger optical energy of \n50.4 × 108 J.m−3 is required for switching due to the high Tc of Co in t he FEM layer.  \n \nThe time -resolved magnetization dynamics of the FM -FEM heterostructure with \ndifferent FEM sub -lattices are plotted in Fig. 5a -b at their respective threshold \nswitching energies of 50 .4 × 108 J.m−3 and 33 .2 × 108 J.m−3. The red, cyan, and green \nlines of the figures respectively denote the magnetization dynamics of Fe (Co), Gd, and \nCo/Pt. The RKKY exchange strength is  JRKKY = + 0.05 × JCoPt for both systems. The \ndemagnetization rate of Co is slightly faster than Fe (and Gd) due to its higher  Tc [31].  \n \n \nFigure 5: The time -resolved magnetization dynamics simulated for (a) F1CoGd,Tc,CoPt =580 K and (b) \nF1FeGd,Tc,CoPt =580 K measured at a particular exchange coupling strength  (JRKKY = +0.05 × JCoPt) and for respective \nabsorbed threshold optical energy of 50 .4 × 108 J.m−3 and 33 .2 × 108J.m−3. The yellow and cyan -filled region \ndenotes the evolution of electron and phonon temperatures of the respective layers. The red, cyan, and green \ndotted lines are the Curie temperatures of the individual elements. The optical p ulse is schematically shown \nby the black -filled Gaussian pulse at time zero.  \n( \na \n) \n( \nb \n) \n50.4 \n  \n× \n10 \n8 \nJ.m \n- \n3 \n33.2 \n  \n× \n10 \n8 \nJ.m \n- \n3 \nF1 \nCoGd, Tc(CoPt) = 580 K  \nF1 \nF \ne \nGd, Tc(CoPt) = 580 K  12 On top of that, the anti -ferromagnetic intra -sublattice exchange (between rare earth \nand transition metal) is larger for CoGd as; JCo−Gd = −0 .388× JCo = −1 .236×10−21 J than \nin FeGd as; JFe−Gd = −0 .348 × JFe = −0 .821 × 10−21 J. Hence, the angular momentum \ntransfer between Co -Gd is faster than Fe -Gd which results in a shorter transient \nferromagnetic -like state of ∼0.6 ps in CoGd than ∼1.2 ps for the FeGd. Co switches at \n∼0.7 ps and Gd switches at ∼1.2 ps for the CoGd alloy (wher eas Fe switches ∼0.7 ps \nand Gd switches ∼1.9 ps for FeGd alloy). This also helps Gd to attain a significant \nmagnetization in the opposite direction within a shorter time. Now, Co/Pt switches in \n∼4.5 ps in Fig. 5a, which is slower compared to Fig. 5b ( ∼3.8 ps) and can be attributed \nto the increased threshold energy required for F1CoGd,T c,CoPt =580 K and faster recovery time \nof Gd. The maximum electron temperature rise in Co/Pt is larger at the higher th reshold \noptical energy, and it takes longer to cool below its Tc, when the RKKY coupling can \nkick in to switch its magnetization. Therefore, the Co/Pt moments stay longer in the \ndemagnetized state before they are flipped in the opposite direction, resultin g in slightly \nlower and two -step switching dynamics of Co/Pt. Hence, we conclude that the effect \nof the composition is important for the occurrence of reversal of FM at lower RKKY \nexchange, and the effect of Tc of plays an important role in controlling the  dynamics \ndetermining the switching time of the different types of exchange coupled \nheterostructures.  \n4. Conclusion  \nWe have simulated the time -resolved and element -specific magnetization dynamics \nin an exchange -coupled FM -FEM system using a modified M3TM. Ult rafast switching \nof the FM (Co/Pt) isn’t possible if it is decoupled from the FEM sub -lattice, as it needs \nadditional angular momentum transfer from the optically switched FEM sub -lattice. \nFor a lower Tc, both the FM and FEM sub -lattice switches at the sam e threshold energy, \nhowever, with increasing Tc, either larger optical energy or larger RKKY coupling \nstrength is required to switch the FM layer. Our analysis depicts the need for i) the \nelectron/lattice temperature of the Co/Pt layer to go below Tc after optical excitation \nand ii) the RKKY coupling strength and available magnetization from the rare -earth \n(Gd) element of the switched FEM alloy for the efficient angular momentum transfer \nto obtain ultrafast magnetization switching of the FM.  \n \nWe have reprod uced the experimentally observed ultrafast switching of the Co/Pt \nlayer in at ∼7 ps by Gorchon et al. [22]. We found that the switching can be made much \nfaster ( ∼3.8 ps) by increasing the Tc of the FM to 580 K. We have also explored the \neffect of FEM compo sition on the magnetization dynamics. While CoGd switches at a \nlarger threshold energy due to its high Tc, it displays a narrower transient ferromagnetic -\nlike state. However, for FeGd, the transient ferromagnetic -like state lasts longer due to \nits weak int ra-sublattice exchange interaction. On the other hand, the switching \ndynamics of the FM layer only get minimally affected due to the composition of the \nFEM layer , and that too can be attributed to the difference in the threshold optical \nenergy.  \n 13 We can con clude that the Curie temperature of the FM and RKKY exchange \ncoupling strength are the two most important factors in tailoring the switching timescale \nof the FM in such exchange -coupled structures, while the composition of the FEM alloy \ndetermines the requ ired strength of the RKKY interaction for magnetization switching. \nWe believe our results will help in refining the current understanding of the HI -AOS \nphenomenon in exchange coupled structures and lead the spintronics community \ntowards achieving a faster and more energy -efficient  magnetization switching.  \nAcknowledgements  \nThis work was primarily supported by the Director, Office of Science, Office of \nBasic Energy Sciences, Materials Sciences and Engineering Division, of the U.S. \nDepartment of Energy under Contract No. DE -AC02 -05-CH11231 within the \nNonequilibrium Magnetic Materials Program (MSMAG) (theoretical analysis). This \nwork was also supported by ASCENT, one of the six centers in JUMP, a Semiconductor \nResearch Corporation (SRC) program also spons ored by DARPA (instrumentation and \ndata acquisition). We also acknowledge support from the National Science Foundation \nCenter for Energy Efficient Electronics Science and the Berkeley Emerging \nTechnology Research (BETR) Center (instrumentation and data acq uisition).  \nReferences  \n[1] D. Polley, A. Pattabi, J. Chatterjee, S. Mondal, K. Jhuria, H. Singh, J. Gorchon, J. \nBokor, Progress towards picosecond on -chip magnetic memory, Appl. Phys.Lett. \n120 (14) (2022) 140501 –140501.  \n[2] I. Radu, K. Vahaplar, C. Stamm, T. Kachel , N. Pontius, H. A. Du¨rr, T. A. Ostler, \nJ. Barker, R. F. Evans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. \nRasing, A. V. Kimel, Transient ferromagnetic -like state mediating ultrafast \nreversal of antiferromagnetically coupled spins, Nature 472  (7342) (2011) 205 –\n209. \n[3] T. A. Ostler, J. 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"    },    {        "title": "1310.8586v2.Ground_State_Phase_Diagram_of_S_2_Heisenberg_Chains_with_Alternating_Single_Site_Anisotropy.pdf",        "content": "arXiv:1310.8586v2  [cond-mat.str-el]  17 Feb 2014Journal of the Physical Society of Japan FULL PAPERS\nGround-State Phase Diagram of S= 2 Heisenberg Chains with\nAlternating Single-Site Anisotropy\nKazuo Hida∗\nDivision of Material Science, Graduate School of Science an d Engineering,\nSaitama University, Saitama, Saitama 338-8570, Japan\n(Received November 1, 2013)\nThe ground-state phase diagram of S= 2 antiferromagnetic Heisenberg chains with coexisting un iform\nand alternating single-site anisotropies is investigated by the numerical exact diagonalization and density\nmatrix renormalization group methods. We find the Haldane, l arge-D, N´ eel, period-doubled N´ eel, gapless\nspin fluid, quantized and partial ferrimagnetic phases. The Haldane phase is limited to the close neighbor-\nhood of the isotropic point. Within numerical accuracy, the transition from the gapless spin-fluid phase\nto the period-doubled N´ eel phase is a direct transition. Ne vertheless, the presence of a narrow spin-gap\nphase between these two phases is suggested on the basis of th e low-energy effective theory. The ferri-\nmagnetic ground state is present in a wide parameter range. T his suggests the realization of magnetized\nsingle-chain magnets with a uniform spin magnitude by contr olling the environment of each magnetic ion\nwithout introducing ferromagnetic interactions.\n1. Introduction\nAmong various exotic ground states in quantum mag-\nnetism, the Haldane state in integer spin antiferromag-\nnetic Heisenberg chains1has been most extensively stud-\nied both experimentally and theoretically. In the case\nofS= 1, this state is characterized by a hidden anti-\nferromagnetic string order accompanied by the Z2×Z2\nsymmetry breakdown in spite of the presence of the en-\nergy gap and the exponential decay of the spin-spin cor-\nrelation function. The easy-plane single-site anisotropy\nD(>0) destroys the Haldane phase, leading to the large-\nD(LD) phase with a finite energy gap and an exponen-\ntially decayingspin-spin correlationfunction without the\nstringorder.The easy-axissingle-siteanisotropy( D <0)\ndrives the Haldane state into the N´ eel state.2,3On the\nother hand, in the case of S= 2, the Haldane and LD\nphases belong to the same topological phase, as pointed\nout by Pollmann et al.4and Tonegawa et al.5\nIn this context, it is interesting to investigate how the\nground states of the quantum spin chains are modified if\nthe easy-axis and easy-plane single-site anisotropies co-\nexist in a single chain. In a previous work,6the present\nauthor and Chen investigated the S= 1 chain with co-\nexisting uniform ( D0) and alternating ( ±δD) single-site\nanisotropies and found the period-doubled N´ eel (PDN)\nphase with a |↑0↓0∝angbracketrightstructure for large values of δD. In\nthis model, the Haldane phase is stable as long as one of\nthe single-site anisotropies D0±δDis not much larger\nthan the exchange coupling J.\nThe present author also investigated this problem in\nthe case of S= 2 with only the alternating single-\nsite anisotropy ±δD.7We found not only the nonmag-\nnetic and period-doubled N´ eel phase, but also the ferri-\nmagnetic phases with quantized and unquantized spon-\n∗E-mail address: hida@mail.saitama-u.ac.jptaneous magnetization for intermediate values of δD.\nThese quantized values of magnetization also satisfy the\nOshikawa-Yamanaka-Affleck condition,8well-known for\nthe magnetization plateau in the magnetic field. How-\never,thismodelisratherspecial,becausethemagnitudes\nof the easy-axis and easy-plane anisotropies are assumed\nto be equal. In the present work, we investigate the case\nofS= 2 in a more general situation with coexisting uni-\nform and alternating single-site anisotropies. A similar\nalternation in Dis experimentally realized in a mixed\nspin molecular magnet in which the magnitude of spins\nalso alternatesbetween S= 1 and 2.9This type of model\nhas also been investigated theoretically.10In the present\nwork, we fix S= 2 for all sites to single out the effect of\nalternation in the single-site anisotropy.\nThis paper is organized as follows. In the next section,\nthe model Hamiltonian is presented. In sect. 3, the nu-\nmerical results for the ground-state phase diagram are\npresented and the property of each phase is discussed.\nDetails of the numerical analysis are explained in sect. 4.\nThe last section is devoted to a summary and discussion.\n2. Model Hamiltonian\nWe investigate the ground state of the S= 2 antiferro-\nmagnetic Heisenberg chains with the alternating single-\nsite anisotropy described by the Hamiltonian\nH=N/summationdisplay\nl=1JSlSl+1+(D0+δD)N/2/summationdisplay\nl=1Sz2\n2l−1\n+ (D0−δD)N/2/summationdisplay\nl=1Sz2\n2l, (1)\nwhereSiis theS= 2 spin operator on the i-th site.\nWe consider the antiferromagnetic case J >0. We also\ntakeδD >0 without the loss of generality. We investi-\n1J. Phys. Soc. Jpn. FULL PAPERS\ngate this model using the numerical exact diagonaliza-\ntion (NED) and density matrix renormalization group\n(DMRG) methods.\n0 2 4 6−2024\nδD/JD0/J\nSFLD\nFerri↑0↓0\nQFPFIPFIIHaldane\n↑↓↑↓PDN\nN′eel(a)\n0 0.5 1−0.100.1\nδD/JD0/J\nSF\nFerriPFI\n↑↓↑↓Haldane\nN′eel(b)\nFig. 1. Ground-state phase diagram. Open symbols are deter-\nmined by extrapolation from the NED data for 4 ≤N≤12.\nFilled symbols are determined from DMRG data with N= 60.\n(a) Overall phase diagram. (b) Enlarged phase diagram aroun d\n(D0,δD)∼(0,0).\n3. Ground-State Phase Diagram\nOur results are summarized in the phase diagram of\nFig. 1. As in the case of the uniform D, the Haldane\nphase is fragile against anisotropy. It occupies only a\nsmall region around the isotropic point ( D0,δD) = 0,\nas shown in Fig. 1(b). Instead of the Haldane phase, a\nwide gapless spin-fluid (SF) phase and a conventional\nN´ eel-ordered phase are realized for D0>0 andD0<0,\nrespectively.\nFor a large positive D0, all spins are confined in the\nstateSz\ni= 0. As a result, a phase transition to the LD\nphase occurs with an increase in D0. As pointed out by\nPollmann et al.4and Tonegawa et al.,5the Haldane and\nLDphasesofthe S= 2chainbelongtothesametopolog-\nical phase. Within the present model, however, we find\nno direct continuous path to connect these two phases.\nThe natures of the Haldane, LD, and N´ eel phases are\nessentially the same as those in the uniform case. In thefollowing subsections, we explain the natures of the SF,\nPDN, and ferrimagnetic phases separately in detail.\n3.1 SF phase\nEven for a largepositive D0, the easy-planeanisotropy\ndecreases on the 2 l-th site with an increase in δD, and\nthe gapless SF phase is recovered. To obtain insight into\nthe nature of the SF phase in this regime, we examine\nthe limit D0+δD≫J,|D0−δD|. In this limit, only\nthe spins S2lsurvive.The effective Hamiltonian forthese\nspins is obtained by the second-order perturbation in J\nas\nHeff=N/2/summationdisplay\nl=1/bracketleftBig\nJxy\neff/parenleftBig\nSx\n2lSx\n2(l+1)+Sy\n2lSy\n2(l+1)/parenrightBig\n+(D0−δD)Sz2\n2l/bracketrightBig\n, (2)\nwhereJxy\neff=−6J2/(D0+δD).Hence,theeffectivemodel\nis equivalent to the S= 2 XY chain with a single-\nion anisotropy up to the second order in J. The ferro-\nmagnetic XY coupling implies that the ferromagnetic\nquasi-long-range order in the x- andy-components of\nthe spins S2lis dominant in the SF phase. Hence, the\ncorrelation functions should behave as/angbracketleftBig\nSx\n2lSx\n2(l+j)/angbracketrightBig\n=/angbracketleftBig\nSy\n2lSy\n2(l+j)/angbracketrightBig\n∼j−η(D0,δD). On the other hand, in the\nSF phase with δD= 0, the dominant correlation is an-\ntiferromagnetic in the x- andy-components of all spins\nas/angbracketleftBig\nSx\nlSx\nl+j/angbracketrightBig\n=/angbracketleftBig\nSy\nlSy\nl+j/angbracketrightBig\n∼(−1)jj−η(D0,0). In the pres-\nence of a finite δD, however, the unit cell is doubled. In\nthis case, the correlation should be calculated between\nequivalent sites in different unit cells. To compare the\ncorrelations for the general values of D0andδDwith\nthose in the limiting case of D0+δD≫J,|D0−δD|, we\nconcentrate on the correlations between the spins S2l.\nFor a small δD, they should behave as/angbracketleftBig\nSx\n2lSx\n2(l+j)/angbracketrightBig\n=/angbracketleftBig\nSy\n2lSy\n2(l+j)/angbracketrightBig\n∼j−η(D0,δD), taking into account the con-\ntinuity to the case of δD= 0. Thus, the dominant corre-\nlation is commonin both limiting cases.This observation\nsuggests that the SF phase is a single phase irrespective\nofD0andδD. We have also confirmed that the corre-\nlations/angbracketleftBig\nSx\n2lSx\n2(l+j)/angbracketrightBig\nand/angbracketleftBig\nSy\n2lSy\n2(l+j)/angbracketrightBig\nare always ferro-\nmagnetic within the SF phase by NED calculation for\nN= 12.\n3.2 PDN phase\nWith further increase in δD, the anisotropy on the\n2l-th site changes into the easy-axis type and a phase\ntransition to the PDN phase that has a spin configu-\nration↑0↓0 occurs. In the limit of δD≫D0, J,\nthe origin of the effective antiferromagnetic Ising inter-\naction between Sz\n2landSz\n2l+2can be understood in the\nsame way as in the case of S= 1.6Let us denote the\nspin state of three successive sites of the Hamiltonian\n(1) by/vextendsingle/vextendsingleSz\n2lSz\n2l+1Sz\n2l+2/angbracketrightbig\n. In the absence of the exchange\ntermJ,thegroundstatesare |±2 0±2∝angbracketright,whicharefour-\n2J. Phys. Soc. Jpn. FULL PAPERS\nfold degenerate. Owing to the spin flip term in (1), the\n|2 0 2∝angbracketrightstate is mixed with the |2 1 1∝angbracketrightand|1 1 2∝angbracketrightstates,\nwhereas the |2 0−2∝angbracketrightstate is mixed with the |2−1−1∝angbracketright\nand|1 1−2∝angbracketrightstates.Hence,the energyofthe |2 0 2∝angbracketrightstate\nis raised, whereas that of the |2 0−2∝angbracketrightstate is lowered by\nthe Ising component of the nearest-neighbour coupling.\nThismeansthattheeffectiveinteractionbetween Sz\n2land\nSz\n2l+2is antiferromagnetic.\nIf the SF-PDN transition is a single transition, it cor-\nresponds to the Brezinskii-Kosterlitz-Thouless (BKT)\ntransition with a Z2symmetry breakdown. However, it\nis possible that the BKT transition and the Ising-type\ntransition with a Z2symmetry breakdown take place\nseparately with a narrow intermediate spin-gap phase in\nbetween. This issue will be examined in detail in sect.\n4.4.\n3.3 Ferrimagnetic phases\nFor−∞< D0< D0c≃2.26J, ferrimagnetic phases\nalso appear in a wide parameter range, as in the case\nofD0= 0.7In the middle of the ferrimagnetic phase,\nthere exists a quantized ferrimagnetic (QF) phase, where\nthe total magnetization Mis quantized to M=Ms/4,\nexcept in the regime D0/lessorsimilarD0c. Here,Ms= 2Nis the\nsaturation magnetization. This value of Msatisfies the\nquantization condition by Oshikawa et al.8given by\np(S−m) =q, (3)\nwherepis the size of the unit cell, qis an integer, and\nmis the magnetization per site ( m=M/N=MS/M s).\nAlthough this condition is originally proposed for the\nmagnetization plateau in the magnetic field, it also holds\nfor the spontaneous magnetization in the ferrimagnetic\nphase.7The present QF state corresponds to the case of\np= 2,S= 2,m= 1/2, andq= 3 in (3).\nOn both sides of the quantized ferrimagnetic phase,\nwe find two kinds of partial ferrimagnetic (PF) phases,\nnamely, the PFI and PFII phases. In the PFI phase, the\nspontaneous magnetization varies continuously from 0\nat the PFI-SF phase boundary to Ms/4 at the PFI-QF\nphaseboundary.Inthe PFIIphase,itvariescontinuously\nfromMs/4 at the QF-PFII phase boundary to a critical\nvalueMcrat the PFII-PDNphase boundary. The critical\nvalueMcrvaries from point to point on the PFII-PDN\nboundary. The properties of the ferrimagnetic phases are\nqualitatively the same as those in the case of D0= 0,\nwhich have been discussed in detail in Ref. 7.\n4. Numerical Determination of Phase Bound-\naries\n4.1 Phase boundary between the phases with different\nspontaneous magnetization\nThe ground-state spontaneous magnetization is calcu-\nlated by NED with a periodic boundary condition and\nby DMRG with an open boundary condition for various\nvalues of D0andδD. For the DMRG calculation, ap-\npropriate end spins are added to reduce the boundary\neffects.The phase boundaries between the ground states with\ndifferent values of spontaneous magnetization are deter-\nmined by the level crossing among them. The NED cal-\nculation is carried out for N= 4,8, and 12. For small\nD0andδD, however, the phase boundary between the\nSF and PFI phases, and that between the N´ eel and PFI\nphases strongly depends on the system size. Hence, we\nemploy the DMRG calculation for N= 60 with an open\nboundary condition. For large values of D0, the phase\nboundary thus obtained almost coincides with that ob-\ntained by NED with a periodic boundary condition for\nN= 12.\n4.2 N´ eel-Haldane phase boundary\nIn the N´ eel phase, the Z2symmetry is spontaneously\nbroken. This implies that the ground state of a large\nbut finite chain is an antisymmetric superposition of two\nordered states with M= 0 that are interchanged with\neach other by spin inversion. The lowest excited state\nis their symmetric superposition that degenerates with\nthe ground state in the thermodynamic limit. Therefore,\nthe lowest excitation gap with M= 0 exponentially de-\ncreases with the system size. In the Haldane phase, all\nexcitations are gapped. Hence, the phase boundary is\ndetermined by phenomenological renormalization group\nanalysis for the excitation energy ∆ E(N,M= 0) with\nthe total magnetization M= 0. The finite size critical\npoint is determined by the condition\nN1∆E(N1,M= 0) =N2∆E(N2,M= 0),(4)\nfor (N1,N2) = (6,8),(8,10), and (10,12). The extrapo-\nlation to the thermodynamic limit is carried out assum-\ning the principal size dependence 2 /(N1+N2) based on\nthe Ising exponent ν= 1 for the correlation length, as\nshown in Fig. 2. For N= 8 and 12, the number of unit\ncells is even, whereas it is odd for N= 6 and 10. There-\nfore, we are concerned about an even-odd oscillation in\nthe extrapolation procedure. Fortunately, for the present\ntransition, no distinct oscillation is observed.\n4.3 LD-SF and Haldane-SF phase boundaries\nThese transitions are conventional BKT transitions.\nTherefore,weemploythelevelspectroscopymethodwith\na twisted boundary condition proposed by Nomura and\nKitazawa.11The finite-size critical point is determined\nby\nETW(N,M= 0) =E(N,M= 2), (5)\nwhereE(N,M) andETW(N,M) are the ground-state\nenergies in the sector with the magnetization Munder\nperiodic and twisted boundary conditions, respectively.\nFor the LD-SF transition, the extrapolation to the\nthermodynamic limit is carried out using the data for\nN= 8,10,and 12. In this case, the critical point is insen-\nsitive to the system size. For the Haldane-SF transition,\nthe extrapolation to the thermodynamic limit is carried\noutasshowninFig.3usingthedatafor N= 6,8,10,and\n12, assuming that the finite size correction is O(N−2).11\n3J. Phys. Soc. Jpn. FULL PAPERS\n0 0.1 0.2−0.03−0.02−0.010\n2/(N1+N2)D0/J δD/J\n0\n0.1\n0.2\n0.3\n0.4\nFig. 2. Extrapolation procedure for the N´ eel-Haldane critical\npoints using ( N1,N2) = (6,8),(8,10), and (10,12).\n0 0.02 0.0400.1\n1/N2δDc/J\n0.05\n0.1\n0.15\n0.2\n0.25\n0.3\n0.35\n0.4D0/J\n0\nFig. 3. Extrapolation procedure for the Haldane-SF critical\npoints using N= 6,8,10, and 12.\nIn this case, the even-odd oscillation is also not harmful\nto the extrapolation.\n4.4 SF-PDN boundary\nIf this transition is a single transition, it corresponds\nto the BKT transition with a Z2symmetry breakdown.\nThe level spectroscopy method for this type of transi-\ntion was proposed by Okamoto and Nomura.12–14In the\nPDN phase, the Z2symmetry is spontaneously broken.\nHence, the lowest excited state has M= 0, as in the case\nof the conventional N´ eel phase discussed in sect. 4.2. In\nthe gapless SF phase, the lowest excitation is the spin\nwave with M= 1. As a result, the excitation energy\nwithM= 0 and that with M= 1 degenerate at the\nphase boundary. This also implies that the low-energy\nexcitation spectrum recovers its SU(2) symmetry at the\nphase boundary in spite of the apparent absence of the0 0.1345\n3\n2.8\n2.6\n2.5\n2.4δD/J\n1/N2D0/J\nFig. 4. Extrapolation procedure for the critical points of SF-\nPDN direct transition using N= 4,8, and 12 (open symbols). The\nfilled symbols are the estimates obtained assuming the SF-sp in-gap\nBKT transition.\nSU(2) symmetry in the Hamiltonian (1). The extrapola-\ntion procedures for the critical value of δDfor different\nvalues of D0are shown in Fig. 4 by open symbols. The\nphase boundary thus obtained is plotted in Fig. 1.\nHowever, it is more natural to assume that the BKT\ntransition and the Ising-type transition with a Z2sym-\nmetry breakdown take place separately with a narrow\nspin-gap phase between them. To obtain more insight\ninto this point, we estimate the central charge cin the\nSF phase from the behavior of the entanglement entropy\nSE(χ) of the optimized matrix product state with a di-\nmension χ.15It is given by\nSE(χ) =c\n6lnξχ, (6)\nwhereξχis the correlation length of the matrix product\nstate that behaves as\nξχ∝χκ, (7)\nforχ→ ∞. The exponent κis related to the central\ncharge as\nκ=6\nc1/radicalBig\n12\nc+1. (8)\nHence, the χdependence of SE(χ) is expected to behave\nas\nSE(χ)≃1/radicalBig\n12\nc+1lnχ+C0+C1\nlnχ,(9)\nforχ→ ∞, whereC0andC1are constants. We have\nestimated SE(χ) by the infinite-size DMRG method. The\nχdependence of SE(χ) is consistent with (9), assuming\nc= 1 within the SF phase, as shown in Fig. 5.\nTherefore, we may assume that the low-energy effec-\ntive Hamiltonian in the SF phase is a Gaussian model\n4J. Phys. Soc. Jpn. FULL PAPERS\n5.5 61.822.22.4SE\nlnχ4.0\n4.1δD/J\n3.0\n3.23.4\n3.6\n3.8D0=2.5J\nFig. 5. χdependence of entanglement entropy SEforD0= 2.5J\nand various values of δDwithin the SF phase.\ngiven by\nH0=1\n2π/integraldisplay\ndx/bracketleftBig\nvsK(πΠ)2+vs\nK/parenleftbigg∂φ\n∂x/parenrightbigg2/bracketrightBig\n,(0≤φ <2π√\n2),\n(10)\nwhereφ(x) is a bosonic field and Π( x) is the momentum\ndensity conjugate to φ(x). The parameters vsandKare\nthe spin wave velocity and Luttinger liquid parameter,\nrespectively.Consideringthe periodicity in φ, the generic\nperturbation of the form\nH1=y1vs\n2πa2/integraldisplay\ndxcos√\n2φ+y2vs\n2πa2/integraldisplay\ndxcos2√\n2φ(11)\nis allowed. Here, ais the short distance cutoff. The\nhigher-order terms cos n√\n2φ(n≥3) are not considered\nhere, because they are less relevant than H1. Here,y1\nforms a spin gap and y2forms a PDN order. Because y1\nis always more relevant than y2, a transition from the\ngapless SF phase to a spin-gap phase should take place\nbefore establishing the PDN order, unless y1identically\nvanishes. Hence, within this scenario, it is natural to ex-\npect a spin-gap phase between the SF phase and the\nPDN phase.\nThe same scenario is also justified by considering the\nlimiting case of D0+δD≫J,|D0−δD|. In this limit,\nthe effective Hamiltonian for the spins S2lis given by\n(2) within the second-order perturbation in J. The an-\ntiferromagnetic Ising coupling arises in the third order\ninJ. Although other third-order terms also arise, con-\nsidering that the PDN phase is stabilized on the ordered\nside of the phase boundary, the spins Sz\n2landSz\n2l+2tend\nto align antiferromagnetically. In addition, the sign of\nJxy\neffcan be reversed by the π-rotation of the spins S4l\naround the z-axis. Therefore, the ground state is ex-\npected to be similar to that of the S= 2 XXZ chain\nwith a single-ion anisotropy and a small antiferromag-\nnetic Ising coupling. In this model, assuming continu-\nity to the isotropic case, a narrow Haldane phase is ex-\npected between the SF phase and the N´ eel phase,5,16,17\nalthough this has not been confirmed numerically. The\nHaldane phase of the effective Hamiltonian (2), which\ncorresponds to the spin-gap phase between the SF and\nPDN phases in the original Hamiltonian (1), is topolog-ically equivalent to the Haldane phase of (1) near the\nisotropic point ( D0,δD) = (0,0), since both do not have\nhalf-integer edge spins under the open boundary condi-\ntion. From the continuity, this topological equivalence\nshould hold within this phase beyond the limit of appli-\ncability of the effective Hamiltonian (2). Thus, we may\nconclude that the spin-gap phase between the SF and\nPDN phases is topologically equivalent to the Haldane\nphase near the isotropic point, if the former phase ex-\nists.\nMotivated by this consideration, we have also esti-\nmated the phase boundary, assuming the conventional\nBKT transition between the SF phase and the spin gap\nphase using the level spectroscopy with a twisted bound-\nary condition.11The finite-size critical point and their\nextrapolation procedure are also shown in Fig. 4 by filled\nsymbols. From the result, we find that the phase bound-\nary estimated in this way almost coincides with that de-\ntermined by the level crossing of the M= 0 and M= 1\nexcitations.Hence, within the presentlyavailablecompu-\ntational resource, we could not reach a definite conclu-\nsion regarding the presence of a spin-gap phase between\nthe SF phase and the N´ eel phase, although its presence\nis naturally expected from the low-energy effective theo-\nries.\n5. Summary and Discussion\nThe ground-state phases of the S= 2 Heisenberg\nchains with coexisting uniform and alternating single-\nsite anisotropies are investigated by the NED and\nDMRG methods. The nonmagnetic phase consists of the\nLD, Haldane, gapless SF, PDN, and conventional N´ eel\nphases. In addition, partial and quantized ferrimagnetic\nphases are observed in a wide parameter range. In con-\ntrast to the case of S= 1, for which the Haldane phase is\nquite robust,6the Haldane phase in the present model is\nlimited to the close neighborhood of the isotropic point,\nas in the uniform case.5Instead ofthe Haldane phase, we\nfind a wide gapless SF phase for positive values of D0.\nThe presence of a narrow spin-gap phase between the\nSF and PDN phases is suggested on the basis of the\nbosonization argument and mapping onto an effective\nS= 2 uniform XXZ chain. We could not, however, con-\nfirm the presence of this spin-gap phase within the avail-\nable numerical data. As for the possibility of experimen-\ntal observation, this intermediate spin-gap phase is so\nnarrow that it would be difficult to detect it, even if\na corresponding material is synthesized. Therefore, the\nSF-PDN transition would be observed as a direct tran-\nsition practically. Our argument also applies to the SF-\nPDN boundary in the spin-alternating chain with com-\npetingsingle-ionanisotropiesstudiedinRef.10,although\nit would not be numerically detectable.\nAs another scenario for the intermediate phase be-\ntween the conventional gapless SF phase and the PDN\nphase, a multipolar gapless SF phase with a quasi-long-\nrange order in ( S+\ni)4, which is analogous to the XY4\nphase in the S= 2 XXZ chain,5,16may be considered.\n5J. Phys. Soc. Jpn. FULL PAPERS\nIn this phase, the lowest excitation should have M= 4.\nHowever,we do not find this type of behavioraround the\nSF-PDN phase boundary.\nIn the partial ferrimagnetic phase, the spontaneous\nmagnetization varies continuously with D0andδD,\nwhereas it is locked to a fractional value of the saturated\nmagnetization that satisfies the Oshikawa-Yamanaka-\nAffleck condition in the quantized ferrimagnetic phase.\nThe presence of both ferrimagnetic phases in a wide\nparameter range suggests the realization of magnetized\nsingle-chain magnets with a uniform spin magnitude by\ncontrollingtheenvironmentofeachmagneticionwithout\nintroducing ferromagnetic interactions.\nThe computation in this work has been performed us-\ning the facilities of the Supercomputer Center, Institute\nfor Solid State Physics, University of Tokyo and Yukawa\nInstitute Computer Facility in Kyoto University. This\nwork is supported by Grants-in-Aid for Scientific Re-\nsearch(C) (Nos.25400389and21540379)fromtheJapan\nSociety for the Promotion of Science. The numerical di-\nagonalization program is based on the TITPACK ver.2\ncoded by H. Nishimori.1) F. D. M. Haldane, Phys. Lett. A 93, 464 (1983); Phys. Rev.\nLett.50, 1153 (1983).\n2) M. den Nijs and K. Rommelse, Phys. Rev. B 40, 4709 (1989).\n3) H. Tasaki, Phys. Rev. Lett. 66, 798 (1991).\n4) F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Phys.\nRev. B85, 075125 (2012).\n5) T. Tonegawa, K. Okamoto, H. Nakano, T. Sakai, K. Nomura,\nand M. Kaburagi, J. Phys. Soc. Jpn. 80, 043001 (2011).\n6) K. Hida and W. Chen, J. Phys. Soc. Jpn. 74, 2090 (2005).\n7) K. Hida, J. Phys. Soc. Jpn. 76, 024714 (2007).\n8) M. Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev. Lett.\n78, 1984 (1997).\n9) A.Saitoh,H.Miyasaka,M.Yamashita,and R.Cl´ erac,J.Ma ter.\nChem.17, 2002 (2007).\n10) T. Tonegawa, K. Okamoto, T. Sakai, and M. Kaburagi, J.\nPhys.: Conf. Ser. 145, 012066 (2009).\n11) K.Nomura and A.Kitazawa, J.Phys.A: Math.Gen. 31, 7341\n(1998).\n12) K. Okamoto and K. Nomura, Phys. Lett. A 169, 433 (1992).\n13) K.NomuraandK.Okamoto, J.Phys.Soc.Jpn. 62,1123 (1993).\n14) K. Nomura and K. Okamoto, J. Phys. A: Math. Gen 27, 5773\n(1994).\n15) F. Pollmann, S. Mukerjee, A. M. Turner, and J. E. Moore,\nPhys. Rev. Lett. 102, 255701 (2009).\n16) H. Aschauer and U. Schollw¨ ock, Phys. Rev. B 58, 359 (1998).\n17) U.Schollw¨ ock and T.Jolicœur,Europhys.Lett. 30, 493 (1995).\n6"    },    {        "title": "1702.03609v4.Lieb_and_hole_doped_ferrimagnetism__spiral__resonating_valence_bond_states__and_phase_separation_in_large_U__AB__2___Hubbard_chains.pdf",        "content": "Lieb and hole-doped ferrimagnetism, spiral, resonating valence-bond states, and phase separation\nin large-U AB 2Hubbard chains\nV . M. Martinez Alvarez1and M. D. Coutinho-Filho1\n1Departamento de F ´ısica, Laborat ´orio de F ´ısica Te ´orica e Computacional,\nUniversidade Federal de Pernambuco, Recife 50670-901, Pernambuco, Brazil\nThe ground state (GS) properties of the quasi-one-dimensional AB2Hubbard model are investigated taking\nthe effects of charge and spin quantum fluctuations on equal footing. In the strong-coupling regime, we derive\na low-energy Lagrangian suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-\ndoped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice\nmagnetizations, and Lieb total spin per unit cell of the effective quantum Heisenberg model, in very good\nagreement with previous results. In the challenging hole doping regime away from half filling, we derive the\ncorresponding t-JHamiltonian. Under the assumption that charge and spin quantum correlations are decoupled,\nthe evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially\nmodulated spin structures and the emergence of phase separation in the presence of resonating-valence-bond\nstates. We also calculate the doping-dependent GS energy and total spin per unit cell, in which case it is shown\nthat the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in\nthe doped regime are in very good agreement with density matrix renormalization group studies, where our\nassumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in\nanti-phase with the modulation of the magnetic structure.\nI. INTRODUCTION\nMuch attention has been given to quantum phase transi-\ntions [1, 2], which are phenomena characterized by the change\nof the nature of the ground state (GS) driven by a non-thermal\nparameter: pressure, magnetic field, doping, Coulomb repul-\nsion, or competitive interactions. In this context, the study\nof quasi-one-dimensional (quasi-1D) compounds with ferri-\nmagnetic properties [3, 4] has attracted considerable theoreti-\ncal and experimental interest because of their unique physical\nproperties and very rich phase diagrams. In particular, the GS\nof quasi-1D quantum ferrimagnets with AB2orABB0unit\ncell topologies (diamond or trimer chains) described by the\nHeisenberg or Hubbard models [5] exhibit unsaturated spon-\ntaneous magnetization, ferromagnetic and antiferromagnetic\nspin-wave modes, effect of quantum fluctuations, and field-\ndependent magnetization plateaus, among several other fea-\ntures of interest.\nOf special interest is the topological origin of GS mag-\nnetic long-range order associated with the unit cell structure\nof the lattice [5–12]. These studies have been motivated\nand supported by exact solutions and rigorous results [13–\n18]; in particular, at half filling, the total spin per unit cell\nobeys Lieb-Mattis [13] (Heisenberg model) or Lieb’s theo-\nrem [15] (Hubbard model). On the other hand, it has been\nverified that the ferrimagnetic GS of spin- 1=2Heisenberg and\nHubbard=t-J AB 2chains, under the effect of frustration [19–\n23] or doping [7, 11, 24, 25], are strongly affected by quan-\ntum fluctuations that might cause its destruction and the occur-\nrence of new exotic phases: spiral incommensurate (IC) spin\nstructures, Nagaoka ( U!1 ) and resonating-valence-bond\n(RVB) states, phase separation (PS), and Luttinger-liquid be-\nhavior. These features can enhance the phenomenology in\ncomparison with a linear chain, which is dominated by the\nnontrivial Luttinger-liquid behavior that exhibits fractional\nexcitations [26, 27], emergent fractionalized particles [28],and fractional-exclusion statistic properties [29] in the spin-\nincoherent regime [30]. In addition, investigations of transport\nproperties in AB2chains, and related structures, have also un-\nveiled very interesting features [31].\nOn the experimental side, studies [32–34] of the magnetic\nproperties of homometallic phosphate compounds of the fam-\nilyA3Cu3(PO 4)4(A = Ca ,Sr,Pb) suggest that in these\nmaterials the line of trimers formed by spin- 1=2 Cu+2ions\nantiferromagnetically coupled do exhibit ferrimagnetism of\ntopological origin. Further, compounds Ca3M3(PO 4)4(M =\nNi;Co) with a wave-like layer structure built by zigzag M-\nchains exhibit antiferromagnetic ordering ( M = Ni ) or param-\nagnetic behavior ( M = Co ) [35]. On the other hand, bimetal-\nlic compounds, such as CuMn (S 2C2O2)2\u00017:5H2O[36], can\nbe modeled [36–38] by alternate spin- 1=2- spin- 5=2chains\nand support interesting field-induced quantum critical points\nand Luttinger-liquid phase [37]. In addition, frustrated dia-\nmond (AB2topology) chains can properly model the com-\npound azurite, Cu3(CO 3)2(OH) 2, in which case the occur-\nrence of the 1=3magnetization plateau is verified at high\nfields [39] in agreement with topological arguments [40] akin\nto those invoked in the quantum Hall effect. The spin-1/2\ntrimer chain compound Cu3(P2O6OH) 2, with antiferromag-\nnetic interactions only, also display the 1/3 magnetization\nplateau [41]. Interestingly, it has been established that in azu-\nrite the magnetization plateau is a dimer-monomer state [42],\ni.e., the chain is formed by pairs of S= 1=2monomers and\nS= 0dimers, with a small local polarization of the diamond\nspins [43], in agreement with density functional theory [44].\nThese dimer-monomer states have been found previously in\nthe context of modeling frustrated AB2chains [45–47], and\nconfirmed through a modeling using quantum rotors [48]. In\ncontrast to azurite, whose dimers appear perpendicular to the\nchain direction, in the spin-1/2 inequilateral diamond-chain\ncompounds [49] A3Cu3AlO 2(SO 4)4(A = K;Rb;Cs), the\nmagnetic exchange interactions force the dimers to lie along\nthe sides of the diamond cells and the monomers form a 1DarXiv:1702.03609v4  [cond-mat.str-el]  13 Dec 20182\nHeisenberg chain. In fact, the low-energy excitations of these\nnew compounds have been probed and a Tomonaga-Luttinger\nspin liquid behavior identified [50]. It is worth mentioning\nthat strongly frustrated AB2chains can exhibit ladder-chain\ndecoupling [20], in which case the ladder is formed via the\ncoupling between dimer spins in neighboring AB2unit cells.\nOn the other hand, besides the above-mentioned quasi-\n1D compounds and related magnetic properties, consider-\nable efforts have been devoted to the study of supercon-\nductivity and intriguing magnetic/charge ordered phases in\ndoped materials [51, 52], in particular the formation of spin-\ngapped states in compounds such as the family of doped\n(La;Sr;Ca)14Cu24O41. This compound is formed by one-\ndimensional CuO 2diamond chains, (Sr;Ca)layers, and two-\nlegCu2O3ladders [53]. These results certainly stimulate\nexperimental and theoretical investigations of quasi-1D com-\npounds in the hole-doped regime, which is the main focus of\nour work, as described in the following.\nIn this work, we shall employ an analytical approach suit-\nable to describe the strongly coupled Hubbard model on doped\nAB2chains, which were the object of recent numerical studies\nthrough density matrix renormalization group (DMRG) tech-\nniques [25]. Our functional integral approach, combined with\na perturbative expansion in the strong-coupling regime, was\noriginally proposed to study the doped Hubbard chain [54],\nand later adapted to describe various doped-induced phase\ntransitions in the U=1AB2Hubbard chain [55]. In ad-\ndition, this approach was used to describe the doped strongly\ncoupled Hubbard model on the honeycomb lattice [56], whose\nresults are very rewarding, particularly those for the GS en-\nergy and magnetization in the doped regime, which com-\npare very well with Grassmann tensor product numerical stud-\nies [57].\nThe paper is organized as follows: in Sec. II we review the\nfunctional integral representation of the Hubbard Hamiltonian\nin terms of Grassmann fields (charge degrees of freedom) and\nspinSU(2)gauge fields (spin degrees of freedom). In Sec. III\nwe diagonalize the Hamiltonian associated with the charge de-\ngree of freedom and obtain a perturbative low-energy theory\nsuitable to describe the ferrimagnetic phase at half filling and\nthe phases in the hole-doped regimes. In Sec. IV, we show\nthat the resultant Hamiltonian at half filling and large-U maps\nonto the spin- 1=2quantum Heisenberg model. In this regime,\na perturbative series expansions in powers of 1=Sof the spin-\nwave modes is presented, which allows us to calculate the\nGS energy, sublattice magnetizations, and Lieb GS total spin\nper unit cell in very good agreement with previous estimates.\nIn Sec. V, we derive the low-energy effective t-JHamilto-\nnian, which accounts for both charge and spin quantum fluc-\ntuations. We also present the evolution of the second-order\nspin-wave modes, GS energy and total spin per unit cell un-\nder hole doping, thus identifying the occurrence of spatially\nmodulated spin structures, with non-zero and zero GS total\nspin, and phase separation involving the later spin structure\nand RVB states at hole concentration 1=3. Remarkably, these\npredictions are in very good agreement with the DMRG data\nreported in Ref. [25]. Lastly, in Sec. VI, we present a sum-\nmary and concluding remarks concerning the reported results.II. FUNCTIONAL-INTEGRAL REPRESENTATION\nThe Hamiltonian of the one-band Hubbard model on chains\nwithAB2unit cell topology is given by [7, 8, 10]:\nH=\u0000X\nhi\u000b;j\fi\u001bft\u000b\f\nij^cy\ni\u000b\u001b^cj\f\u001b+H.c.g+UX\ni\u000b^ni\u000b\"^ni\u000b#;(1)\nwherei= 1;:::;Nc(=N=3)is the specific position of the\nunit cell, whose length is set to unity, Nc(N) is the num-\nber of cells (sites), \u000b;\f=A; B 1; B2denote the type of site\nwithin the unit cell, ^cy\ni\u000b\u001b(^ci\u000b\u001b) is the creation (annihilation)\noperator of electrons with spin \u001b(=\";#) at site\u000bof celli,\nand^ni\u000b\u001b= ^cy\ni\u000b\u001b^ci\u000b\u001bis the occupancy number operator. The\nfirst term in Eq. (1) describes electron hopping, with energy\nt\u000b\f\nij\u0011t, allowed only between nearest neighbors A-B1and\nA-B2linked sites of sublattices AandB(bipartite lattice),\nand the second one is the on-site Coulombian repulsive inter-\nactionU > 0, which contributes only in the case of double\noccupancy of the site i\u000b.\nAt this point, it is instructive to digress on some fundamen-\ntal aspects of the formalism used in our work [54–56]. With\nregard to the large-U doped Hubbard chain [54], U=1AB2\nHubbard chain [55] and the Hubbard model on the honeycomb\nlattice [56], it has been shown that the particle density product\nin Eq. (1) can be treated through the use of a decomposition\nprocedure, which consists in expressing ^ni\u000b\"^ni\u000b#in terms of\ncharge and spin operators:\n^ni\u000b\"^ni\u000b#=1\n2^\u001ai\u000b\u00002(^Si\u000b\u0001ni\u000b)2; (2)\nwhere\n^Si\u000b= 1=2X\n\u001b\u001b0^cy\ni\u000b\u001b0\u001b\u001b0\u001b^ci\u000b\u001b; (3)\nand\n^\u001ai\u000b= ^ni\u000b\"+ ^ni\u000b#; (4)\nare the spin-1/2 and charge-density operators, respectively,\n\u001b\u001b0\u001bdenotes the Pauli matrix elements (~\u00111), andni\u000bis\nan arbitrary unit vector. In fact, Eq. (2) follows from the iden-\ntity:1\n2^\u001ai\u000b\u0000^ni\u000b\"^ni\u000b#= 2( ^Sx;y;z\ni\u000b)2= 2( ^Si\u000b\u0001ni\u000b)2. The\nconvenience of using the decomposition defined in Eq. (2),\nwith explicit spin-rotational invariance for the large-U Hub-\nbard model, was discussed at length in Refs. [54–56].\nWe start by using the Trotter-Suzuki formula [58, 59],\nwhich allows us to write the partition function,\nZ=Tr[exp(\u0000\fH)], at a temperature kBT\u00111=\f, as\nZ=Trf^TQM\nr=1exp[\u0000\u000e\u001cH(\u001cr)]g, where ^Tdenotes the\ntime-ordering operator, the total imaginary time interval\nis formally sliced into Mdiscrete intervals of equal size\n\u000e\u001c=\u001cr\u0000\u001cr\u00001,r= 1;2;:::;M; with\u001c0= 0;and\n\u001cM=\f=M\u000e\u001c , under the limits M!1 and\u000e\u001c!0. We\nshall now introduce, between each discrete time interval, an\novercomplete basis of fermionic coherent states [58, 59], 1 =\u0001Q\ni\u000b\u001bdcy\ni\u000b\u001bdci\u000b\u001bexp\u0010\n\u0000P\ni\u000b\u001bcy\ni\u000b\u001bci\u000b\u001b\u0011\njfci\u000b\u001bgihfci\u000b\u001bgj,3\nwherefcy\ni\u000b\u001b;ci\u000b\u001bgdenotes a set of Grassmann fields satisfy-\ning anti-periodic boundary conditions: cy\ni\u000b\u001b(0) =\u0000cy\ni\u000b\u001b(\f)\nandci\u000b\u001b(0) =\u0000ci\u000b\u001b(\f); while the set of unit vectors\ndefines the vector field fni\u000bg, satisfying periodic ones:\nni\u000b(0) = ni\u000b(\f), under a weight functional (see below).\nThereby, following standard procedure [58, 59], the partition\nfunction reads:\nZ=\u0002Y\ni\u000b\u001bDcy\ni\u000b\u001bDci\u000b\u001bY\ni\u000bD2ni\u000bW(fni\u000bg)e\u0000\u0001\f\n0L(\u001c)d\u001c;\n(5)\nwhere the pertinent measures are defined by\nDcy\ni\u000b\u001bDci\u000b\u001b\u0011 lim\nM!1;\u000e\u001c!0M\u00001Y\nr=1dcy\ni\u000b\u001b(\u001cr)dci\u000b\u001b(\u001cr);(6)\nD2ni\u000b\u0011 lim\nM!1;\u000e\u001c!0M\u00001Y\nr=1d2ni\u000b(\u001cr); (7)\nthe weight functional, W(fni\u000bg), satisfies a normalization\ncondition at each discrete imaginary time \u001cr:\n\u0002Y\ni\u000bd2ni\u000bW(fni\u000b(\u001cr)g) = 1; (8)\nand the Lagrangian density L(\u001c)is written in the form:\nL(\u001c) =X\ni\u000b\u001bcy\ni\u000b\u001b@\u001cci\u000b\u001b\u0000X\nij\u000b\f\u001b(t\u000b\f\nijcy\ni\u000b\u001bcj\f\u001b+H.c.)\n+UX\ni\u000b[\u001ai\u000b\n2\u00002(Si\u000b\u0001ni\u000b)2]: (9)\nIn order to fix W(fni\u000bg)one should notice that, in the op-\nerator formalism: ^\u001a2\ni\u000b= ^\u001ai\u000b+ 2^ni\u000b\"^ni\u000b#. Therefore, using\nEq. (2), the following identity holds [54]:\n2(^Si\u000b\u0001ni\u000b)2=^\u001ai\u000b(2\u0000^\u001ai\u000b)\n2; (10)\nwhich means that the square of the spin component opera-\ntor along the ni\u000bdirection has zero eigenvalues if the site\nis vacant or doubly occupied, and a nonzero value only for\nsingly occupied sites, i.e., (^Si\u000b\u0001ni\u000b)2= 1=4. Now, taking\nadvantage of the choice of ni\u000b, the local spin-polarization and\nspin-quantization axes are both chosen along the ni\u000bdirec-\ntion. Therefore, for singly occupied sites, we find Si\u000b\u0001ni\u000b=\npi\u000b=2, withpi\u000b=\u00061, corresponding to the two possible\nspin-1/2 states. Further, by incorporating vacancy and double\noccupancy possibilities, corresponding to the four possible lo-\ncal states of the Hubbard model, one can write [54]\npi\u000b^Si\u000b\u0001ni\u000b=^\u001ai\u000b(2\u0000^\u001ai\u000b)\n2; (11)\nwithp2\ni\u000b= (\u00061)2. We stress that, due to fermion operator\nproperties, the square of Eq. (11) reproduces Eq. (10), and\na comparison between them implies, at arbitrary doping andU value, the formal equivalence between 2(^Si\u000b\u0001ni\u000b)2and\npi\u000b(^Si\u000b\u0001ni\u000b). In this context, we remark that the origi-\nnal Coulomb repulsion term of the Hubbard Hamiltonian in\nEq. (1) is formally and energetically (eigenvalues) equivalent\nto both that in Eq. (9) or in its linear version through the fol-\nlowing replacement: 2(^Si\u000b\u0001ni\u000b)2!pi\u000b(^Si\u000b\u0001ni\u000b). In-\ndeed, using the constraint in Eq. (11) we find, UP\ni\u000b[\u001ai\u000b\n2\u0000\npi\u000b(Si\u000b\u0001ni\u000b)] =UP\ni\u000b[\u001ai\u000b\n2\u00001\n2\u001ai\u000b(2\u0000\u001ai\u000b)], which is zero\nfor\u001ai\u000b= 0;1; whereas, as expected, for double occupied\nsites,\u001ai\u000b= 2, the local energy is U. Therefore, Eq. (11) in its\nGrassmann version, can be enforced by a proper choice of the\nnormalized weight functional:\nW(fni\u000bg) = lim\nM!1;\u000e\u001c!0MY\nr=1W(fni\u000b(\u001cr)g)\n=Cexpn\n\u0000\u0002\f\n0d\u001c\rX\ni\u000b[pi\u000bSi\u000b\u0001ni\u000b\u0000\u001ai\u000b\n2(2\u0000\u001ai\u000b)]2o\n;\n(12)\nwhere\r!1 in the continuum limit ( M!1 ,\u000e\u001c!0),\nwith delta-function peaks at the four local states of the Hub-\nbard model, andCis a normalization factor such that Eq. (8)\nholds. In fact, the product of W(fni\u000b(\u001cr)g)in Eq. (12) gener-\nates a sum in rin the exponential of the suitable chosen Gaus-\nsian function, i.e., W(fni\u000bg)is such that in the continuum\nlimit,M!1;\u000e\u001c!0, Eq. (12) obtains with a diverging \r,\nas pointed out in Ref. [54]. In this way, using Eq. (12) for the\nweight functional in Eq. (5) for the partition function Z, and\nintegrating overfni\u000bg, the Lagrangian density L(\u001c)in Eq.\n(9) can thus be written in the following linearized form [54]:\nL(\u001c) =X\ni\u000b\u001bcy\ni\u000b\u001b@\u001cci\u000b\u001b\u0000X\nij\u000b\f\u001b(t\u000b\f\nijcy\ni\u000b\u001bcj\f\u001b+H.c.)\n+UX\ni\u000b[\u001ai\u000b\n2\u0000pi\u000b(Si\u000b\u0001ni\u000b)]; (13)\nwhere the constraint in Eq. (11) was explicitly used.\nNow, since we are interested in studying the GS properties\nof theAB2Hubbard chains, we choose the staggered factor\npi\u000b= +1 (\u00001)at sites\u000b=B1; B2(A), consistent with the\nlong-range ferrimagnetic GS predicted by Lieb’s theorem at\nhalf filling and for any Uvalue [7, 8, 15], in which case we\nassume broken rotational symmetry along the z-axis. In this\ncontext, by considering the symmetry exhibited by the ferri-\nmagnetic order, let us define the SU(2)=U(1)unitary rotation\nmatrix [60]\nUi\u000b=2\n4cos\u0010\u0012i\u000b2\u0011\n\u0000sin\u0010\u0012i\u000b2\u0011\ne\u0000i\u001ei\u000b\nsin\u0010\u0012i\u000b2\u0011\nei\u001ei\u000b cos\u0010\u0012i\u000b2\u00113\n5;(14)\nwhere\u0012i\u000bis the polar angle between the z-axis and the unit lo-\ncal vector ni\u000band\u001ei\u000b2[0;2\u0019)is an arbitrary azimuth angle\ndue to theU(1)gauge freedom of choice for Ui\u000b. Moreover,\na new set of Grassmann fields, fay\ni\u000b\u001b;ai\u000b\u001bgcan be obtained,\naccording to the transformation:\nci\u000b\u001b=X\n\u001b0(Ui\u000b)\u001b\u001b0ai\u000b\u001b0; (15)4\nthat locally rotates each unit vector ni\u000bto thez-direction. On\nthe other hand, if we express the product \u001b\u0001ni\u000bin matrix\nform:\n\u001b\u0001ni\u000b=\u0014\ncos (\u0012i\u000b) sin (\u0012i\u000b)e\u0000i\u001ei\u000b\nsin (\u0012i\u000b)ei\u001ei\u000b\u0000cos (\u0012i\u000b)\u0015\n; (16)\nwe obtain, after using Eq. (14),\nUy\ni\u000b(\u001b\u0001ni\u000b)Ui\u000b=\u001bz; (17)\nwhich explicitly manifest the broken rotational symmetry\nalong thez-axis. In this way, by substituting Eqs. (14) and\n(15) into Eq. (3), and using the above result, we find\nSi\u000b\u0001ni\u000b=1\n2X\n\u001b\u001b0ay\ni\u000b\u001b[Uy\ni\u000b(\u001b\u0001ni\u000b)Ui\u000b]\u001b\u001b0ai\u000b\u001b0\n=1\n2X\n\u001b\u001b0ay\ni\u000b\u001b(\u001bz)\u001b\u001b0ai\u000b\u001b0\u0011Sz\ni\u000b; (18)\nthereby, the constraint in Eq. (11) can be written in the form\nSi\u000b\u0001ni\u000b=pi\u000b\u001ai\u000b(2\u0000\u001ai\u000b)\n2=1\n2(ay\ni\u000b\"ai\u000b\"\u0000ay\ni\u000b#ai\u000b#);\n(19)\nwherepi\u000b= +1 (\u00001)at sites\u000b=B1; B2(A). The choice\nofpi\u000babove implies Lieb’s ferrimagnetic ordering with the set\nf\u0012iA=\u0012iB1=\u0012iB2= 0g, for alli, at half filling. However,\nin the hole doped regime away from half filling, the \u0012i\u000b’s can\nbe nonzero (e.g., \u0012i\u000b=\u0019for a spin flip, leading to a change\nin the sign of Sz\ni\u000b); further,Sz\ni\u000bcan be zero either by the pres-\nence of holes or doubly occupied sites ( ay\ni\u000b\"ai\u000b\"=ay\ni\u000b#ai\u000b#).\nLastly, using Eqs. (15) and (19) into the Lagrangian, Eq. (13),\nwe find, after suitable rearrangement of terms,\nL(\u001c) =L0(\u001c) +Ln(\u001c); (20)\nwhere both Lagrangians are quadratic in the Grassmann fields:\nL0(\u001c) =X\ni\u000b\u001bay\ni\u000b\u001b@\u001cai\u000b\u001b\u0000X\ni\u000bj\f\u001b(t\u000b\f\nijay\ni\u000b\u001baj\f\u001b+H.c.)\n+U\n2X\ni\u000b\u001b(1\u0000pi\u000b\u001b)ay\ni\u000b\u001bai\u000b\u001b; (21)\nand\nLn(\u001c) =X\ni\u000b\u001b\u001b0ay\ni\u000b\u001b0(Uy\ni\u000b@\u001cUi\u000b)\u001b0\u001bai\u000b\u001b\n\u0000X\ni\u000bj\f\u001b\u001b0t\u000b\f\nij[ay\ni\u000b\u001b0(Uy\ni\u000bUj\f\u00001)\u001b0\u001baj\f\u001b+H.c.];(22)\nwith the first term in both Eqs. (21) and (22) being originated\nfrom the first term in Eq. (13), the second ones come from\nthe hopping term in Eq. (13), after a rearrangement of terms,\nwhile the last one in Eq. (21) (proportional to U) is obtained\nby using Eq. (19) in the last term of Eq. (13). It is worth\nmentioning that only charge degrees of freedom (Grassmann\nfields) appear inL0(\u001c), and spin degrees of freedom under theconstraint in Eq. (19) [ SU(2)gauge fieldsfUy\ni\u000b;Ui\u000bg, which\ncarry all the information on the vector field fni\u000bg] are now\nrestricted toLn(\u001c), which includes both spin and charge de-\ngrees of freedom.\nIn the large-U regime, double occupancy is energetically\nunfavorable and the factor 2\u0000\u001ai\u000bis no longer needed in\nEq. (19), i.e., Si\u000b\u0001ni\u000b=pi\u000b\u001ai\u000b\n2, with\u001ai\u000b= 0 or1. In\nthis case, a proper perturbative analysis will allow us to study\nhole doping effects in Sec. V in a macroscopic fashion, so we\ndefine\n\u000e= 1\u00001\nNX\ni\u000bh\u001ai\u000bi; (23)\nwhich measures the thermodynamic average of hole doping\naway from half filling. In this context (strong-coupling limit),\nwe take advantage of results derived from L0(\u001c)(charge ef-\nfects in Sec. III), and at half filling (Sec. IV), in which case\ncharge degrees of freedom are frozen.\nIII. CHARGE DEGREES OF FREEDOM AND THE\nSTRONG-COUPLING LIMIT\nIn this section, we shall first diagonalize the Hamiltonian\nassociated with the Lagrangian L0(\u001c)through the use of a\nspecial symmetry property of the AB2chains and a canonical\ntransformation in reciprocal space. Then, by introducing a\nperturbative expansion in the strong-coupling regime, a low-\nenergy effective Lagrangian for the AB2Hubbard chains at\nhalf filling and in the doped regime will be obtained.\nA. Charge degrees of freedom\nWe begin our discussion by considering the Lagrangian L0\nin Eq. (21), and its corresponding Hamiltonian H0, free of\ntheSU(2)gauge fields. By performing the Legendre trans-\nformation:H0=\u0000P\ni\u000b\u001b@L0\n@(@\u001cai\u000b\u001b)@\u001cai\u000b\u001b+L0, where\n@L0\n@(@\u001cai\u000b\u001b)=ay\ni\u000b\u001b, the resultingH0is given by\nH0=\u0000X\nhi\u000b;j\fi\u001b(t\u000b\f\nijay\ni\u000b\u001baj\f\u001b+H.c.)\n+U\n2X\ni\u000b\u001b(1\u0000pi\u000b\u001b)ay\ni\u000b\u001bai\u000b\u001b: (24)\nFurther, sinceH0(L0) is quadratic in the Grassmann fields,\nthe solution for the energy of the system is given by H0in its\ndiagonalized form [59].\nTheAB2unit cell topology exhibits a symmetry [9, 11, 24,\n25, 55] under the exchange of the labels of the Bsites in a\ngiven unit cell. Thus, we can construct a new set of Grass-\nmann fields possessing this symmetry, i.e., either symmet-\nric or antisymmetric with respect to the exchange operation\nB1$B2:\n(di\u001b;ei\u001b) =1p\n2(aiB1\u001b\u0006aiB2\u001b); bi\u001b=aiA\u001b: (25)5\nIn addition, as a signature of the quasi-1D structure of the\nAB2chains, we notice that the B1andB2sites are located at\na distance 1=2(in units of length) ahead of the Asite. There-\nfore, after Fourier transforming the above Grassmann fields,\ni.e.,fdi;\u001b;ei;\u001b;bi;\u001bg=1pNcP\nkeikxifdk;\u001b;ek;\u001b;bk;\u001bg, it is\nconvenient to introduce a phase factor eik\n2through the follow-\ning transformation [55]: (Ak\u001b;Bk\u001b) =1p\n2(dk\u001b\u0006eik\n2bk\u001b), so\nthatH0in Eq. (24) thus becomes\nH0=X\nk\u001b\"k[Ay\nk\u001bAk\u001b\u0000By\nk\u001bBk\u001b] +U\n2X\nk\u001b(1\u0000\u001b)ey\nk\u001bek\u001b\n+U\n2X\nk\u001b[Ay\nk\u001bAk\u001b+By\nk\u001bBk\u001b\u0000\u001b(Ay\nk\u001bBk\u001b+By\nk\u001bAk\u001b)];\n(26)\nwhere\n\"k=\u00002p\n2tcos(k=2); (27)\nwithk= 2\u0019j(3\nN)\u0000\u0019, andj= 1;:::;N= 3. We can now ex-\nactly diagonalizeH0through the following Bogoliubov trans-\nformation:\nAk\u001b=uk\u000bk\u001b\u0000\u001bvk\fk\u001b; Bk\u001b=\u001bvk\u000bk\u001b+uk\fk\u001b;(28)\nwithukandvksatisfying the canonical constraint: (uk)2+\n(vk)2= 1 , to maintain the anticommutation relations of\nthe Grassmann fields. Due to the ferrimagnetic order of the\nGS, the above transformation is subject to a 4\u0019periodicity\nof the Bogoliubov functions fuk;vkgand Grassmann fields\nf\u000bk\u001b;\fk\u001bg. The diagonalized H0thus reads:\nH0=\u0000X\nk\u001b(Ek\u0000U\n2)\u000by\nk\u001b\u000bk\u001b+X\nk\u001b(Ek+U\n2)\fy\nk\u001b\fk\u001b\n+U\n2X\nk\u001b(1\u0000\u001b)ey\nk\u001bek\u001b; (29)\nwhere\n(uk;vk) =1p\n2\u0012\n1\u0006j\"kj\nEk\u00131=2\n; (30)\nand\nEk=q\n\"2\nk+U2=4: (31)\nAs one can see from Eq. (29), the non-interacting tight bind-\ning (U= 0) spectrum ofH0present three electronic bands:\na nondispersive flat band (related to the Grassmann fields\nfey\nk\u001b;ek\u001bg, macroscopically degenerate), and two dispersive\nones. InAB2chains, flat bands are closely associated with\nferrimagnetism (unsaturated ferromagnetism) [5, 7, 8] at half\nfilling, in agreement with Lieb’s theorem [15, 16], or fully\npolarized ferromagnetism [17] associated with the flat lowest\nband. We also stress that even at this level of approximation\nand in the weak coupling regime ( U= 2t), it was shown [7]\nthat hole doping [parametrized by \u000edefined in Eq. (23)] candestroy the ferrimagnetic order and/or induce phase separa-\ntion inAB2chains. As depicted in Fig. 1(a), the U= 0 spin\ndegeneracy of the flat bands is removed by the Coulombian\nrepulsive interaction, in which case a gap Uopens between\ntheek\u001bmodes:ek\"= 0, where spins at sites B1andB2are\nup, andek#=U, where these spins are down. On the other\nhand, the two dispersive bands are spin degenerated, and also\ndisplay a Hubbard gap Useparating the low (\u000bk\u001b)-energy and\nhigh(\fk\u001b)-energy modes [55].\nB. Strong-coupling limit\nIn this subsection, we shall introduce a perturbative expan-\nsion in the strong-coupling regime ( U\u001dt) in order to obtain\na low-energy effective Lagrangian for the AB2Hubbard chain\nat half filling and in the doped regime. First, we resume the re-\nsults of the previous section by writing the Grassmann fields\ndi\u001bandbi\u001bin terms of the Grassmann (Bogoliubov) fields\n\u000bk\u001band\fk\u001b:\n(di\u001b;bi\u001b) =1p2NcX\nk(eikxi;eik(xi\u00001\n2))\n\u0002[(uk\u0006\u001bvk)\u000bk\u001b\u0006(uk\u0007\u001bvk)\fk\u001b];(32)\nwhere the phase factor e\u0000ik\n2signalizes the quasi-1D AB2\nstructure, and the antisymmetric Grassmann field ei;\u001bremains\nas defined in Eq. (25). In the strong-coupling limit, however,\nit will prove useful to define a set of auxiliary spinless Grass-\nmann fields [54, 55] in direct space associated with di\u001band\nbi\u001b:\n(\u000bi;\fi) =r\n1\nNcX\nk;\u001b\u0012(\u0006\u001b)eikxi(\u000bk\u001b;\fk\u001b); (33)\nand a similar equation for (\u000b1\n2\ni;\f1\n2\ni)$(\u000bk\u001b;\fk\u001b)is obtained\nby the replacements: \u0012(\u0006\u001b)!\u0012(\u0007\u001b)andxi!xi\u00001=2,\nwhere\u0012(\u001b)is the Heaviside function, while for the antisym-\nmetric component, one has\nei;\u001b=r\n1\nNcX\nkeikxiek;\u001b: (34)\nNow, by expanding (uk;vk)in Eq. (30) in powers of t=U:\n(uk;vk)\u00191p\n2\u0014\n1\u0006j\"kj\nU+O\u0012t2\nU2\u0013\u0015\n; (35)\nsubstituting these results into the Eq. (32), and using the in-\nverse transformation of Eq. (33), we can derive a perturbative\nexpansion in powers of t=U for the Grassmann fields di\u001band\nbi\u001bin terms of the spinless Grassmann fields as follows:\ndi\u001b=\u0012(\u001b)\u000bi+\u0012(\u0000\u001b)\fi+p\n2t\nU\u0012(\u0000\u001b)(\u000b1\n2\ni+\u000b1\n2\ni+1)\n+t\nU\u0012(\u001b)[p\n2(\f1\n2\ni+\f1\n2\ni+1)\u0000t\nU(2\u000bi+\u000bi+1+\u000bi\u00001)]\n+O(t2=U2); (36)6\nbi\u001b=\u0012(\u0000\u001b)\u000b1\n2\ni\u0000\u0012(\u001b)\f1\n2\ni+p\n2t\nU\u0012(\u001b)(\u000bi+\u000bi\u00001)\n\u0000t\nU\u0012(\u0000\u001b)[p\n2(\fi+\fi\u00001) +t\nU(2\u000b1\n2\ni+\u000b1\n2\ni+1+\u000b1\n2\ni\u00001)]\n+O(t2=U2): (37)\nIn the above derivation, we have used that \u0012(\u001b)\u0012(\u001b0) =\n\u0012(\u001b)\u000e\u001b;\u001b0. Notice that, sincet\nU\u001c1, in Eqs. (36) and (37)\nwe can identify the fields \u000b1\n2\ni\u0019aiA#and\u000bi\u0019(aiB1\"+\naiB2\")=p\n2, a result fully consistent with the low-energy spin\nconfiguration of the ferrimagnetic state discussed previously.\nAnalogously, for the high-energy bands, the opposite spin\nconfiguration is observed, with spin up (down) present at sites\nA(B1,B2).\nIntroducing Eqs. (36) and (37) into Eq. (24), with the aid\nof Eq. (25), we obtain a perturbative expression for H0(low-\nenergy sector) in terms of the spinless Grassmann fields up to\norderJ= 4t2=U:\nH0=\u0000JX\ni[\u000by\ni\u000bi+\u000b(1\n2)y\ni\u000b1\n2\ni\u0000\fy\ni\fi\u0000\f(1\n2)y\ni\f1\n2\ni]\n\u0000J\n2X\ni[\u000by\ni\u000bi+1+\u000b(1\n2)y\ni\u000b1\n2\ni+1\u0000\fy\ni\fi+1\u0000\f(1\n2)y\ni\f1\n2\ni+1+H.c:]\n+UX\ni[\fy\ni\fi+\f(1\n2)y\ni\f1\n2\ni+ey\ni#ei#]:(38)\nBy applying Fourier transform to the above expression and\nrearranging the terms, we obtain\nH0=\u0000X\nk2Jcos2(k=2)(\u000by\nk\u000bk+\u000b(1\n2)y\nk\u000b1\n2\nk)\n+X\nk[2Jcos2(k=2) +U](\fy\nk\fk+\f(1\n2)y\nk\f1\n2\nk)\n+U\n2X\nk\u001b(1\u0000\u001b)ey\nk\u001bek\u001b: (39)\nIn Fig. 1 we plot the electronic spectrum of the Hamilto-\nnianH0, both in the weak and strong-coupling regime: (a)\nEq. (29) for U= 2tand (b) Eqs. (29) and (39) for U= 12t\n(J= 4t2=U= 1=3), respectively, with t\u00111. We can notice\nthe presence of the shrinking phenomenon [7] as Uincreases\nfrom 2tto12t(strong-coupling regime) and that, for U= 12t,\nEq. (39) is a very good approximation to Eq. (29). Notice-\nably, thet\u001cUexpansion of the fields allow us to identify\n\u000b1\n2\nk\u0019akA#,\u000bk\u0019(akB1\"+akB2\")=p\n2(triplet state) and\nek\"\u0019(akB1\"\u0000akB2\")=p\n2(singlet state), as the low-energy\nspin configuration of the ferrimagnetic state with single occu-\npancy, where spins at sites A(B1;B2) are down (up), in agree-\nment with Lieb’s theorem [7, 8, 15].\nIn order to describe the most relevant low-energy pro-\ncesses that take place in this regime, one has to additionally\nproject out the high-energy bands from H0, that is, terms con-\ntaining only fields related to the high-energy bands are ex-\ncluded. Therefore, after the Legendre transformation, H0=\n\u0000P\ni;\u0011i@L0\n@(@\u001c\u0011i)@\u001c\u0011i+L0, where\u0011i=\u000bi;\u000b1\n2\ni, andei\"(fields\n-1 0 1\nk/π-2024 E(a)\nβkσ\nek↓\nek↑\nαkσU= 2t\n-1 0 1\nk/π-2061214(b)\nU= 12tβkβ1\n2\nk\nek↓\nek↑\nαkα1\n2\nkFigure 1. (Color online) Electronic spectrum of the Hamiltonian H0:\n(a) Eq. (29) for U= 2tand (b) Eqs. (29) and (39) for U= 12t(J=\n4t2=U= 1=3), witht\u00111. Notice the band shrinking phenomenon\nasUincreases from 2tto12t(strong-coupling regime). The t\u001c\nUexpansion of the fields identifies \u000b1\n2\nk\u0019akA#,\u000bk\u0019(akB1\"+\nakB2\")=p\n2andek\"\u0019(akB1\"\u0000akB2\")=p\n2, where spins at sites\nA(B1;B2) are down (up), in agreement with Lieb’s theorem [15].\nrelated to the low-energy bands), with@L0\n@(@\u001c\u0011i)=\u0011y\ni, the La-\ngrangian associated with H0(up to order J) is given by\nL0=X\ni;\u0011i\u0011y\ni@\u001c\u0011i\u0000JX\ni(\u000by\ni\u000bi+\u000b(1\n2)y\ni\u000b1\n2\ni)\n\u0000J\n2X\ni(\u000by\ni\u000bi+1+\u000b(1\n2)y\ni\u000b1\n2\ni+1+H.c:):(40)\nWe shall now focus on the U\u001dtperturbative expansion of\nLn, Eq. (22), which amounts to consider the most significant\nlow-energy processes, after the use of Eqs. (36) and (37) for\ndi\u001bandbi\u001bin terms of the spinless Grassmann fields. How-\never, terms allowing interband transitions between low- and\nhigh-energy bands do exist in Ln. In this context, we apply a\nsuitable second-order Rayleigh-Schr ¨odinger perturbation the-\nory [54, 55], consistent with the strong-coupling expansion,\nso that the modes associated with the high-energy bands are\neliminated. Lastly, by adding L0to the perturbative expansion\nofLn, which leads to the cancellation of the exchange terms\nin Eq. (40), the effective low-energy Lagrangian density of the\nAB2Hubbard model in the strong-coupling limit (up to order\nJ) reads:\nLeff(\u001c) =L(I)+L(II)+L(III)+L(IV); (41)\nwhere\nL(I)=X\ni\u000by\ni@\u001c\u000bi+X\ni\u000b(1\n2)y\ni@\u001c\u000b(1\n2)\ni+X\niey\ni\"@\u001cei\";(42a)7\nL(II)=X\ni\u001bn\n\u0012(\u0000\u001b)(U(b)y\ni@\u001cU(b)\ni)\u001b;\u001b\u000b(1\n2)y\ni\u000b(1\n2)\ni\n+\u0012(\u001b)1\n2[(U(d)y\ni@\u001cU(d)\ni)\u001b;\u001b+ (U(e)y\ni@\u001cU(e)\ni)\u001b;\u001b]\n\u0002(\u000by\ni\u000bi+ey\ni\"ei\") +\u0014\n\u0012(\u001b)1\n2[(U(d)y\ni@\u001cU(e)\ni)\u001b;\u001b\n+(U(e)y\ni@\u001cU(d)\ni)\u001b;\u001b]\u000by\niei\"+H.c.io\n; (42b)\nL(III)=\u0000tX\ni\u001bn\n\u0012(\u0000\u001b)(U(b)y\niU(d)\ni)\u001b;\u0000\u001b\u000b(1\n2)y\ni\u000bi\n+\u0012(\u001b)(U(d)y\niU(b)\ni+1)\u001b;\u0000\u001b\u000by\ni\u000b(1\n2)\ni+1\n+\u0012(\u0000\u001b)(U(b)y\niU(e)\ni)\u001b;\u0000\u001b\u000b(1\n2)y\niei\"\n+\u0012(\u001b)\u0010\nU(e)y\niU(b)\ni+1\u0011\n\u001b;\u0000\u001bey\ni\"\u000b(1\n2)\ni+1+H.c.\u001b\n;(42c)\nL(IV)=\u0000J\n4X\ni;i0=i;i+1;\u001b\u0012(\u001b)j(U(d)y\niU(b)\ni0)\u001b;\u001bj2\u000by\ni\u000bi\n\u0000J\n4X\ni;i0=i;i+1;\u001b\u0012(\u001b)j(U(e)y\niU(b)\ni0)\u001b;\u001bj2ey\ni\"ei\"\n\u0000J\n4X\ni;i0=i;i\u00001;\u001b\u0012(\u0000\u001b)[j(U(b)y\niU(d)\ni0)\u001b;\u001bj2\n+ (U(b)y\niU(e)\ni0)\u001b;\u001bj2]\u000b(1\n2)y\ni\u000b(1\n2)\ni; (42d)\nwhere\nU(b)\ni=UiA; U(d;e)\ni =1p\n2(UiB1\u0006UiB2); (43)\nin which case we took advantage of the symmetry of the AB2\nchain under the exchange operation B1$B2, in correspon-\ndence with Eq. (25). From the above equations, we see that the\nkinetic term is represented by L(I)and is related to the charge\ndegrees of freedom only, whereas L(II)describes the dynam-\nics of the spin degrees of freedom coupled to the charge fields.\nOn the other hand, L(III)exhibit first-neighbor hopping con-\ntributions between charge degrees of freedom in the presence\nofSU(2)gauge fields, while L(IV)is the spin exchange term\nin the presence of the charge Grassmann fields.\nIV . HALF-FILLING REGIME\nLet us now discuss some basic aspects of the localized mag-\nnetic properties related to the spin degrees of freedom. At half\nfilling, i.e.,\u000e= 0, we haveh\u000by\ni\u000bii= 1,h\u000b(1=2)y\ni\u000b(1=2)\nii= 1,\nhey\ni\"ei\"i= 1, andh\u000by\niei\"i= 0(no band hybridization) as the\nelectrons tend to fill up the lower-energy bands, whereas the\nhigher-energy ones remain empty. As a consequence, a fer-\nrimagnetic configuration of localized spins emerges, i.e., the\ncharge degrees of freedom are completely frozen, such that\nh\u000by\ni@\u001c\u000bii=h\u000b(1=2)y\ni@\u001c\u000b(1=2)\nii=hey\ni\"@\u001cei\"i= 0, with for-\nbidden hopping. Therefore, only terms from LIIandLIVinEqs. (42b) and (42d), respectively, give nonzero contributions\nand the resulting effective strong-coupling Lagrangian at half\nfilling, defined in Eq. (41), reads:\nLJ\neff=X\ni\u000b\u001b\u0012(pi\u000b\u001b)(Uy\ni\u000b@\u001cUi\u000b)\u001b;\u001b\n\u0000J\n4X\nhi\u000b;j\fi\u001b\u0012(pi\u000b\u001b)\f\f\f(Uy\ni\u000bUj\f)\u001b;\u001b\f\f\f2\n;(44)\nwhere the staggered factor pi\u000bwas defined in Eq. (11), and use\nwas made of the matrix transformations defined in Eq. (43) in\norder to sum up the squares of the SU(2)gauge field prod-\nucts in the exchange contribution from LIVin Eq. (42d).\nNow, using the following Legendre transform: HJ\neff =\n\u0000P\ni\u000b\u001b@LJ\neff\n@(@\u001cUi\u000b)\u001b;\u001b(@\u001cUi\u000b)\u001b;\u001b+LJ\neff;where@LJ\neff\n@(@\u001cUi\u000b)\u001b;\u001b=\n\u0012(pi\u000b\u001b)(Uy\ni\u000b)\u001b;\u001b;we get the respective quantum Heisenberg\nHamiltonian written in terms of the SU(2)gauge fields at half\nfilling as\nHJ\neff=\u0000J\n4X\nhi\u000b;j\fi\u001b\u0012(pi\u000b\u001b)\f\f\f(Uy\ni\u000bUj\f)\u001b;\u001b\f\f\f2\n: (45)\nFurther, using the definition of the SU(2)=U(1)unitary\nrotation matrix Eq. (14), it is possible to write [54–56]\f\f\f(Uy\ni\u000bUj\f)\u001b;\u001b\f\f\f2\n=1\n2(1 + ni\u000b\u0001nj\f);where ni\u000b=\nsin(\u0012i\u000b) [cos(\u001ei\u000b)^x+ sin(\u001ei\u000b)^y] + cos(\u0012i\u000b)^zis the unit vec-\ntor pointing along the local spin direction. Lastly, by using the\nconstraint as given in Eq. (19), we can identify the spin field\nfSi\u000bgat the single occupied sites:\nSi\u000b=pi\u000bni\u000b=2; (46)\nwherepi\u000b= +1 (\u00001)at sites\u000b=B1; B2(A), in order to\nobtain\nHJ\neff=JX\nih\n(SB1\ni+SB2\ni)\u0001(SA\ni+SA\ni+1)i\n\u0000JNc:(47)\nThe above expression is indeed that of the quantum antifer-\nromagnetic Heisenberg spin-1/2 model on the AB2chain in\nzero-field, which takes into account the effects of zero-point\nquantum spin fluctuations. In fact, to achieve this goal, we an-\nalyze the Hamiltonian, Eq. (47), by means of the spin-wave\ntheory, which has proved very successful in describing the\nproperties of the GS and low-lying excited states of spin mod-\nels. The predicted results provide a check of the consistency\nof our approach and will be fully used in our description of\nthe doped regime.\nWe shall first introduce boson creation and annihilation op-\nerators via the Holstein-Primakoff [58] transformation:\nSA;z\ni=\u0000S+ay\niai;\nSA;+\ni= (SA;\u0000\ni)y=p\n2Say\nifA(S);(48)\nfor a down-spin on the Asite, and\nSBl;z\ni=S\u0000by\nlibli;\nSBl;+\ni = (SBl;\u0000\ni)y=p\n2SfB(S)bli;(49)8\nfor an up-spin on the Blsite, withl= 1;2, and\nfr(S) =\u0010\n1\u0000nr\n2S\u00111=2\n= 1\u00001\n2nr\n2S+:::; (50)\nwhereSis the spin magnitude, and nr=ay\niaiorby\nlibli. The\noperatorsay\niandai(orby\nli,bli) satisfy the boson commutation\nrules. Under the above transformation, the spin Hamiltonian,\nEq. (47) is mapped onto the boson Hamiltonian:\nHJ\neff=E0\u0000JNc+H1+H2+O(S\u00001); (51)\nwhere\nE0=\u00004S2JNc; (52)\nis the classical GS energy and H1andH2are the quadratic\nand quartic (interacting) terms of the boson Hamiltonian, suit-\nable to describe the quantum AB2Heisenberg model via a\nperturbative series expansion in powers of 1=S. By Fourier\ntransforming the boson operators, we find\nH1= 2JSX\nk(2ay\nkak+X\nlby\nlkblk)\n+X\nk;l=1;22JS\rk(ay\nkby\nlk+akblk); (53)\nwhere we have defined the lattice structure factor as\n\rk=1\nzX\n\u001aeik\u001a= cos\u0012k\n2\u0013\n; (54)\nwithzdenoting the coordination number ( z= 4for theAB2\nchain), while \u001a=\u00061=2connects the nearest neighbors A-B1\nandA-B2linked sites of sublattices AandB, and\nH2=\u00003J\n2NX\n1234;l=1;2\u000e12;34n\n4\r1\u00004ay\n1a4by\nl3bl2\n+(\r1ay\n1by\nl4by\nl3bl2+\r1+2\u00003ay\n1ay\n2a3by\nl4+H.c.)o\n:(55)\nFor simplicity, we use the convention 1fork1,2fork2, and so\non. Also, the \u000e12;34=\u000e(k1+k2\u0000k3\u0000k4)is the Kronecker\n\u000efunction, and expresses the conservation of momentum to\nwithin a reciprocal-lattice vector G.\nWe shall consider H1first, which is the term leading to\nlinear spin-wave theory (LSWT). In fact, H1is diagonalized\nusing the following Bogoliubov transformation:\nak=uk\fk\u0000vk\u000by\nk;\nblk=1p\n2[uk\u000bk\u0000vk\fy\nk+ (\u00001)l\u0018k];withl= 1;2;(56)\n(uk;vk) =(3 +p\n9\u00008\r2\nk;2p\n2\rk)q\n(3 +p\n9\u00008\r2\nk)2\u00008\r2\nk; (57)\nwhereukandvksatisfy the constraint u2\nk\u0000v2\nk= 1. Thus,\nH1=E1+X\nk(\u000f0(\u000b)\nk\u000by\nk\u000bk+\u000f0(\f)\nk\fy\nk\fk+\u000f0(\u0018)\nk\u0018y\nk\u0018k);(58)E1=JSX\nk(q\n9\u00008\r2\nk\u00003); (59)\n\u000f0(\u000b;\f)\nk=JS(q\n9\u00008\r2\nk\u00071); \u000f0(\u0018)\nk= 2JS; (60)\nwhereE1is theO(S1)quantum correction to the GS energy,\nand\u000f0(\u000b;\f)\nk,\u000f0(\u0018)\nkare the three spin-wave branches provided\nby LSWT, both in agreement with previous results [19, 61].\nIn fact, it is well known that systems with a ferrimagnetic GS\nnaturally have ferromagnetic and antiferromagnetic spin-wave\nmodes as their elementary magnetic excitations (magnons).\nFor theAB2chain, there are three spin-wave branches: an\nantiferromagnetic mode ( \u000f0(\f)\nk) and two ferromagnetic ones\n(\u000f0(\u000b)\nkand\u000f0(\u0018)\nk). The mode \u000f0(\u000b)\nkis gapless at k= 0, i.e., the\nGoldstone mode, with a quadratic (ferromagnetic) dispersion\nrelation\u000f0(\u000b)\nk\u0018k2. The other two modes are gapped. No-\ntice that the gapped ferromagnetic mode \u000f0(\u0018)\nkis flat, and is\nclosely associated with ferrimagnetic properties at half fill-\ning [7, 17]. Since the dispersive modes preserve the local\ntriplet bond, they are identical to those found in the spin- 1=2\n- spin-1 chains [62–65]. These chains also exhibit interesting\nfield-induced Luttinger liquid behavior [66].\nNow, our aim is to obtain the leading corrections to LSWT,\ni.e., second-order spin-wave theory to the GS energy, sublat-\ntice magnetizations and Lieb GS total spin per unit cell. In do-\ning so, we develop a perturbative scheme for the description\nof this quartic term. First, we decompose the two-body terms\nby means of the Wick theorem, via normal-ordering protocol\nfor boson operators. Conservation of momentum to within a\nreciprocal-lattice vector, implies: k1=k+q,k2=p\u0000q,\nk3=kandk4=p. Then, we need to look at the possible\npairings of the 4 operators, as for example, in the first term of\nEq. (55):\nay\nk+qapby\nl;kbl;p\u0000q;ay\nk+qapby\nl;kbl;p\u0000q;ay\nk+qapby\nl;kbl;p\u0000q:\nUnder this procedure, and by substituting the Bogoliubov\ntransformation, Eqs. (56)-(57), into Eq. (55), we find\nH2=E2+X\nk(\u000e\u000f(\u000b)\nk\u000by\nk\u000bk+\u000e\u000f(\f)\nk\fy\nk\fk+\u000e\u000f(\u0018)\nk\u0018y\nk\u0018k);(61)\nwhere\nE2=Nc=\u00002J(q2\n1+q2\n2\u00003p\n2q1q2); (62)\nand the corresponding corrections for the spin-wave disper-\nsion relations read:\n\u000e\u000f(\u000b)\nk=J[u2\nk(p\n2q2\u00002q1) + 2v2\nk(p\n2q2\u0000q1)]\n+ 4J\rkukvk\u00143\n2p\n2q1\u0000q2\u0015\n+O(S\u00001);(63)\n\u000e\u000f(\f)\nkis obtained from \u000e\u000f(\u000b)\nkthrough the exchange of uk$\nvk, and\n\u000e\u000f(\u0018)\nk=J(p\n2q2\u00002q1) +O(S\u00001): (64)9\nIn Eqs. (62)-(64) above, the quantities q1andq2are defined\nby (thermodynamic limit)\nq1=1\n2\u0019\u0002\u0019\n\u0000\u0019dk(v2\nk); q 2=1\n2\u0019\u0002\u0019\n\u0000\u0019dk(\rkukvk):(65)\nWe remark that in deriving Eqs. (62)-(64), we have neglected\nterms containing anomalous products, such as, \u000by\nk\fy\nkand ver-\ntex corrections.\nLastly, the above results of our perturbative 1=Sseries ex-\npansion lead to the effective Hamiltonian:\nHJ\neff=EJ\nGS\u0000JNc+X\nk(\u000f\u000b\nk\u000by\nk\u000bk+\u000f(\f)\nk\fy\nk\fk+\u000f(\u0018)\nk\u0018y\nk\u0018k);\n(66)\nwhere\nEJ\nGS=E0+E1+E2; (67)\nwhich can be read from Eqs. (52), (59), and (62), respectively,\nis the second-order result up to O(1=S)for the GS energy,\nand\n\u000f(s)\nk=\u000f0(s)\nk+\u000e\u000f(s)\nk;withs=\u000b; \f; \u0018; (68)\nare the corresponding second-order spin-wave modes, where\nthe linear and the second-order correction terms are given by\nEq. (60) and Eqs. (63)-(65), respectively.\nA. Second-order spin-wave analysis\nOur perturbative 1=Sseries expansion approach is able to\nimprove the LSWT result for the gap \u0001 =Jof the antiferro-\nmagnetic mode, which should be compared with the second-\norder result derived from \u000f(\f)\nk, Eqs. (60), (63) and (68), at\nk= 0:\u0001 = (1 +p\n2q2)J'1:676J, in full agreement\nwith similar spin-wave calculations for AB2[19] and spin-\n1=2-spin- 1[64, 65] chains, and in agreement with numeri-\ncal estimates using exact diagonalization, \u0001 = 1:759J, for\nbothAB2[5] and spin- 1=2-spin- 1[63] chains. On the other\nhand, the LSWT predicts a gap \u0001flat=Jfor the flat fer-\nromagnetic mode ( \u000f(\u0018)\nk) inAB2chain, whereas our second-\norder spin-wave theory finds, using Eqs. (60), (64) and (68):\n\u0001flat= (1\u00002q1+p\n2q2)J'1:066J, in full agreement\nwith a similar spin-wave procedure [19]. Surprisingly, the es-\ntimated value from Exact Diagonalization (ED) [5]: \u0001flat=\n1:0004J, lies between these two theoretical values. In fact,\nanalytical approaches are still unable to reproduce the ob-\nserved level crossing found in numerical calculations [5, 19]\nfor the two ferromagnetic modes. This is probably due to the\nfact that the different symmetries exhibited by the localized\nexcitation (flat mode) and the ferromagnetic dispersive mode\nare not explicitly manifested in the analytical approaches, so\nthe levels avoid the crossing.B. Ground state energy\nIn the thermodynamic limit, the second-order result for the\nGS energy of the AB2chain per unit cell reads:\nEJ\nGS\nNc=\u00004JS2+JS\n2\u0019\u0002\u0019\n\u0000\u0019dk\u0012q\n9\u00008\r2\nk\u00003\u0013\n\u00002J(q2\n1+q2\n2\u00003p\n2q1q2): (69)\nWe remark that, at half filling, we shall not consider the con-\nstant term\u0000JNcin Eq. (51), with the purpose of compari-\nson with preceding results. Performing the integration over\nthe first BZ and taking S= 1=2, we obtain that the GS en-\nergy per site at zero-field is given by \u00000:4869J. This result\nagrees very well with values obtained using exact diagonal-\nization [45] (\u00000:485J) and DMRG [67] ( \u00000:4847J) tech-\nniques. For the spin- 1=2- spin-1 chain, the value obtained\nusing DMRG [62] is \u00000:72704J. To compare it with our find-\ning, we need to multiply this value by 2=3(ratio between the\nnumber of sites of the two chains), yielding \u00000:48469J.\nC. Sublattice magnetizations and Lieb GS total spin per unit\ncell\nIn order to derive results beyond LSWT, we intro-\nduce staggered magnetic fields coupled to spins SA;z\ni\nandSBl;z\ni, withl= 1;2, through the Zeeman terms:\n\u0000hAP\niSA;z\niand\u0000hBlP\niSBl;z\ni, which are added to HJ\neff\nin Eq. (47). Thus, hSA;ziandhSBl;zicorresponding\nto sublattices AandBlare obtained from hSA;zi=\n\u0000(1=Nc)P\ni=1;2[@Ei(hA)=@hA]jhA=0, and an analogous\nequation forhSBl;ziusing Eqs. (59) and (62):\n(hSA;zi;hSBl;zi) =\u0007S\u0006\u00121\n2;1\n4\u00131\n\u0019\u0002\u0019\n\u0000\u0019dkv2\nk\n\u0007\u00121\n2;1\n4\u0013q1\n\u0019S\u0002\u0019\n\u0000\u0019dk\r2\nk\n(9\u00008\r2\nk)3=2+O(1\nS2):(70)\nCarrying out the above integration, we obtain hSA;zi=\n\u00000:316343 andhSBl;zi= 0:408172 . These results are in\ngood agreement with those obtained using DMRG [11] and\nED [5] techniques: hSA;zi=\u00000:2925 andhSBl;zi=\n0:3962 , respectively, and with values for hSA;ziand\n2hSBl;zifor the spin- 1=2- spin-1 chain [62–65]. Although\nat zero temperature, the sublattice magnetizations are strongly\nreduced by quantum fluctuations, as compared with their clas-\nsical values, the unit cell magnetization remains SL\u00111=2,\nwhereSLis the Lieb GS total spin per unit cell, in full agree-\nment with Lieb’s theorem [5, 15] for bipartite lattices:\nSL=1\n2kNA\u0000NBk; (71)\nwithNA(NB)denoting the total number of spins in sublattice\nA(B)per unit cell.10\nV .t-JHAMILTONIAN: DOPING-INDUCED PHASES,\nGROUND STATE ENERGY AND TOTAL SPIN\nIn this section, we shall derive the corresponding t-J\nHamiltonian suitable to describe the strongly correlated AB2\nHubbard chain in the doped regime, in which case both charge\n(Grassmann fields) and spin [ SU(2)gauge fields] quantum\nfluctuations are considered on an equal footing. Indeed, the\nt-JHamiltonian can be derived by means of the following\nLegendre transformation to Eq. (41):\nHt-J\neff=\u0000X\ni;\u0016=b;d;e@Leff\n@(@\u001cU(\u0016)\ni)\u001b;\u001b(@\u001cU(\u0016)\ni)\u001b;\u001b\n\u0000X\ni;\u0017i@Leff\n@(@\u001c\u0017i)@\u001c\u0017i+Leff; (72)\nwhere@Leff\n@(@\u001c\u0017i)=\u0017y\niwith\u0017i=\u000bi;\u000b1\n2\ni;ei\";@Leff\n@(@\u001cU(b)\ni)\u001b;\u001b=\n\u0012(\u0000\u001b)(U(b)y\ni)\u001b;\u001b\u000b(1\n2)y\ni\u000b(1\n2)\ni, and@Leff\n@(@\u001cU(d;e)\ni)\u001b;\u001b=\n\u0012(\u001b)1\n2[(U(d;e)y\ni )\u001b;\u001b(\u000by\ni\u000bi+ey\ni\"ei\") + (U(e;d)y\ni )\u001b;\u001b(\u000by\niei\"+\ney\ni\"\u000bi)], from which we can write the effective t-JHamilto-\nnian as\nHt-J\neff=Ht+HJ; (73)\nwhere\nHt=\u0000tX\ni\u001bf\u0012(\u0000\u001b)(U(b)y\niU(d)\ni)\u001b;\u0000\u001b\u000b(1=2)y\ni\u000bi\n+\u0012(\u001b)(U(d)y\niU(b)\ni+1)\u001b;\u0000\u001b\u000by\ni\u000b(1=2)\ni+1\n+\u0012(\u0000\u001b)(U(b)y\niU(e)\ni)\u001b;\u0000\u001b\u000b(1=2)y\niei\"\n+\u0012(\u001b)(U(e)y\niU(b)\ni+1)\u001b;\u0000\u001bey\ni\"\u000b(1=2)\ni+1+H.c.g; (74)\nand\nHJ=\u0000J\n4X\ni;i0=i;i+1;\u001b\u0012(\u001b)j(U(d)y\niU(b)\ni0)\u001b;\u001bj2\u000by\ni\u000bi\n\u0000J\n4X\ni;i0=i;i+1;\u001b\u0012(\u001b)j(U(e)y\niU(b)\ni0)\u001b;\u001bj2ey\ni\"ei\"\n\u0000J\n4X\ni;i0=i;i\u00001;\u001b\u0012(\u0000\u001b)[j(U(b)y\niU(d)\ni0)\u001b;\u001bj2\n+j(U(b)y\niU(e)\ni0)\u001b;\u001bj2]\u000b(1\n2)y\ni\u000b(1\n2)\ni: (75)\nNotice that Eqs. (74) and (75) are identical to Eqs. (42c) and\n(42d), since Eqs. (42a) and (42b) were eliminated through the\nLegendre transformation.\nSome digression on Ht-J\neffis in order. One of the key prop-\nerties of quasi-1D interacting quantum systems is the phe-\nnomenon of spin-charge separation, leading to the formation\nof spin and charge-density waves, which move independently\nand with different velocities. It has been demonstrated [24]\nthat for\u000e > 2=3the low-energy physics of the doped AB2\nHubbard chain in the U=1coupling limit is described interms of the Luttinger-liquid model, with the spin and charge\ndegrees of freedom decoupled. Most importantly, it has been\nshown that for the AB2t-JHubbard chains [25] charge and\nspin quantum fluctuations are practically decoupled, as sug-\ngested by the emergence of charge-density waves in anti-\nphase with the modulation of the ferrimagnetic order. One\ncan make use of this feature to formally split each term of the\nt-JHamiltonian, Eq. (73)-(75) , into a product of two inde-\npendent terms acting on different Hilbert spaces, i.e., we can\nenforce spin-charge separation and calculate the charge and\nspin correlation functions in a decoupled fashion.\nTherefore, from the above discussion, we shall consider\nthat the charge correlation functions are well described by an\neffective spinless tight-binding model [24, 55, 68], since the\nhole (charge) density waves develop along the x-axis and in\nanti-phase with the modulation of the ferrimagnetic structure,\nas numerically observed in Fig. 2(b) of Ref. [25]. So, using\nEqs. (33), with a=2!a(effective lattice spacing of the lin-\near chain: distance between AandBsites, see Fig. 2(a) of\nRef. [25]), we find\nh\u000b(1=2)y\ni\u000bii=1\nNcX\nkk0e\u0000ik(xi\u00001)eik0xih\t0j\u000by\nk\u000bk0j\t0i\n=1\n\u0019\u0002kF(\u000e)\n\u0000kF(\u000e)eikdk=2\n\u0019sin[kF(\u000e)]; (76)\nwithj\t0ibeing the hole-doped ferrimagnetic GS,\nwherekF(\u000e) =\u0019Nh\nN\u0011\u0019\u000e is the Fermi wave vec-\ntor of the spinless tight-binding holes. In the\nsame fashion: h\u000by\ni\u000b(1=2)\ni+1i=2\n\u0019sin[kF(\u000e)] and\nh\u000b(1=2)y\niei\"i=hey\ni\"\u000b(1=2)\ni+1i= 0; whileh\u000by\ni\u000bii =\nh\u000b(1=2)y\ni\u000b(1=2)\nii=hey\ni\"ei\"i= (1\u00001\n2\u0019\u0001kF(\u000e)\n\u0000kF(\u000e)dk) =\n(1\u0000\u000e). Here, we remark that the itinerant holes away from\nhalf filling are associated with the lower-energy dispersive \u000bk\nand\u000b(1=2)\nkbands [see Fig. 1(b) in Sec. (II)], thus contributing\nto the kinetic Hamiltonian in Eq. (74). On the other hand,\nthe local correlations related to the lower-energy bands\n\u000bk,\u000b(1=2)\nk, andek\", contribute equally to the exchange\nHamiltonian in Eq. (75). Thereby, using the above tight-\nbinding results for the charge correlation functions, Ht-J\neff\nin Eqs. (73)-(75) gives rise to the \u000e-dependent Hamiltonian,\nHt-J\neff(\u000e) =Ht\neff(\u000e) +HJ\neff(\u000e), written below:\nHt-J\neff(\u000e) =\u0000t2\n\u0019sin[kF(\u000e)]X\ni[(U(b)y\niU(d)\ni)#\"\n+ (U(d)y\niU(b)\ni+1)\"#+H.c.]\n\u0000J(1\u0000\u000e)\n4X\nhi\u000b;j\fi\u001b\u0012(pi\u000b\u001b)j(Uy\ni\u000bUj\f)\u001b;\u001bj2;(77)\nwhere the sum over \u001bwas evaluated in Eq. (74) and the square\nof theSU(2)gauge field products in the exchange contribu-\ntion have been summed up in Eq. (75), so that this contribu-\ntion is just (1\u0000\u000e)timesHJ\neffat half filling, Eq. (45), or\nalternatively, in terms of spin fields, Eq. (47), or spin-waves,\nEqs. (66)-(68). On the other hand, the SU(2)gauge fields ma-11\ntrix elements:\u0010\nU(b)y\niU(d)\ni\u0011\n#\"and\u0010\nU(d)y\niU(b)\ni+1\u0011\n\"#, that ap- pear in the kinetic contribution of Eq. (77), can be written in\nterms of the spin fields [55, 56] as\n(U(b)y\niU(d)\ni)#\"+H.c.=X\nl1p\n2\u0012q\n1\u00002SBl;z\ni\u00002SA;z\ni+ 4SA;z\niSBl;z\ni+q\n1 + 2SA;z\ni+ 2SBl;z\ni+ 4SA;z\niSBl;z\ni\u0013\n;(78)\nand\u0010\nU(d)y\niU(b)\ni+1\u0011\n\"#is obtained from Eq. (78) through the replacement SA;z\ni!SA;z\ni+1, in which case we took advantage of the\nU(1)gauge freedom and Eq. (46). Notice that these square-root matrix elements depend on z-spin components only.\nAt this stage, it will prove useful, in the calculation of the GS total spin in the doped regime, to consider Ht-J\neff(\u000e;h)which\ndescribes the system in the presence of a homogeneous magnetic field h=h^ z= (\u0000hA+hB1+hB2)^ z, where the staggered fields\npoint along the local corresponding magnetizations in the ferrimagnetic phase have the same magnitude h. The magnetic field\ncouple with the spin fields through the Zeeman term (see Sec. IV C) andwith the charge degrees of freedom through the magnetic\norbital coupling in the Landau gauge: A=hx^ y. Since our aim is to study doping effect on the magnetization, we shall assume\nvanishingly small magnetic field in the context of linear response theory and perturbative expansion in the strong-coupling\nregime. Additionally, the magnetic orbital coupling can be considered through the so-called Peierls substitution [27, 69]: t!\ntei\u0001j\f\ni\u000bA\u0001dl, wherei\u000bandj\fare first-neighbor sites, and the flux quantum \u001e0=hc=e\u00111. If one consider that the carrier is\nat the siteiA, we have four hopping possibilities: iA!iB1;2andiA!(i+ 1)B1;2, so the total phase \u001eacquired by the\ncarrier in this prescription satisfies Stokes’ theorem: \u001e=\u0017\nunit cellA\u0001dl=s\nSh\u0001dS=ha2(a\u00111). We also remark that,\nin order to obtain real values for the zero-field staggered magnetizations, we have considered, for convenience, an imaginary\ngauge transformation [56, 70]: A!iA. Therefore, by placing Eq. (78) and the similar matrix element into the kinetic term in\nEq. (77), making the above Peierls substitution, and using the Holstein-Primakoff and Bogoliubov transformations introduced in\nEqs. (48)-(50) and Eqs. (56)-(57), respectively, up to order O(S\u00001), we arrive at the following diagonalized kinetic Hamiltonian\nHt\neff(\u000e;h):\nHt\neff(\u000e;h) =\u00004p\n2\n\u0019te\u0000(\u0000hA+hB1+hB2)sin[kF(\u000e)]X\nk[4S\u00003v2\nk\u0000(u2\nk+ 2v2\nk)\u000by\nk\u000bk\u0000(2u2\nk+v2\nk)\fy\nk\fk\u0000\u0018y\nk\u0018k];(79)\nwhere the doped-induced contributions for the spin dispersion relations are evidenced in the last three terms. On the other hand,\nby adding the Zeeman terms (see Sec. IV C) to the exchange contribution HJ\neff(\u000e), given in Eq. (77), we obtain HJ\neff(\u000e;h).\nLastly, by adding the kinetic and the exchange contributions, we arrive at the effective t-JHamiltonian in the presence of a\nmagnetic field:\nHt-J\neff(\u000e;h) =\u00004p\n2\n\u0019te\u0000(\u0000hA+hB1+hB2)sin[kF(\u000e)]X\nk(4S\u00003v2\nk) +J(1\u0000\u000e)(EJ\nGS\u0000JNc)\n+X\nk[\u000f(\u000b)\nk(\u000e)\u000by\nk\u000bk+\u000f(\f)\nk(\u000e)\fy\nk\fk+\u000f(\u0018)\nk(\u000e)\u0018y\nk\u0018k]\u0000hAX\niSA;z\ni\u0000hB1X\niSB1;z\ni\u0000hB2X\niSB2;z\ni;(80)\nwhereEJ\nGSis given by Eq. (67) and (69), and the corresponding spin-wave modes [see Eqs. (79), (60), (63)-(65), and (68)] of\nthe dopedAB2t-Jchain read:\n\u000f(\u000b)\nk(\u000e) =4p\n2\n\u0019tsin(\u0019\u000e)[u2\nk+ 2v2\nk] + (1\u0000\u000e)(\u000f0(\u000b)\nk+\u000e\u000f(\u000b)\nk); (81)\n\u000f(\f)\nk(\u000e)is obtained from \u000f(\u000b)\nk(\u000e)through the exchange uk$vkand the replacement \u000b!\f, while\n\u000f(\u0018)\nk(\u000e) =4p\n2\n\u0019tsin(\u0019\u000e) + (1\u0000\u000e)(\u000f0(\u0018)\nk+\u000e\u000f(\u0018)\nk): (82)\nWe find it instructive to comment on the analytical structure of the above equations. Firstly, we mention the presence of the\nBogoliubov parameters [see Eqs. (57)] in a symmetric form in the kinetic terms of Eq. (81) and its analogous for \u000f(\f)\nk(\u000e); besides,\nalthough the flat mode is strongly affected by the presence of holes, it remains dispersionless. In addition, using Eqs. (80) and\n(65), the total GS energy (no spin-wave excitations) per unit cell in the thermodynamic limit is readily obtained:\nEt-J\nGS(\u000e;h)=Nc=\u00004p\n2\n\u0019te\u0000(\u0000hA+hB1+hB2)sin(\u0019\u000e)(4S\u00003q1)\n+ (1\u0000\u000e)(EJ\nGS=Nc\u0000J)\u0000hSA;zihA\u0000X\nl=1;2hSBl;zihBl; (83)12\nwherehSA;ziandhSBl;ziare the calculated sublattice magnetizations, at half filling and zero-field, given by Eqs. (70).\nIn subsections V A, V B, and V C, we will show that the\nunderlying competing physical mechanisms: the magnetic\norbital response and the Zeeman contribution embedded in\nEqs. (80)-(83) will dramatically affect the behavior of the sys-\ntem under hole doping and, in particular, will lead to spiral\nIC spin structures, the breakdown of the spiral ferrimagnetic\nGS at a critical value of the hole doping, a region of phase\nseparation, and RVB states at \u000e\u00191=3.\nA. Doped regime: Spin-wave modes\nBefore we go one step further to discuss relevant macro-\nscopic quantities, i.e., the GS energy and total spin in the\ndoped regime, we shall first undertake a detailed study, at a\nmicroscopic level, of the hole-doping effect on the calculated\nspin-wave branches given by Eqs. (81)-(82).\nFig. 2 depicts the second-order spin-wave dispersion rela-\ntions atJ=t= 0:3and for the indicated values of \u000e. Without\nloss of generality, we set t= 1in our numerical computations.\nAt half filling, the antiferromagnetic mode \u000f(\f)\nk, together with\nthe two ferromagnetic modes: the dispersive \u000f(\u000b)\nkand the flat\none\u000f(\u0018)\nk, are shown in Fig. 2(a), which are defined in Eq. (68),\nand can be plotted using Eqs. (60) and (63)-(65).\nAs the hole doping increases slightly, the abrupt decrease\nof the peaks at k= 0 andk=\u0019of the numerical DRMG\nstructure factor (see Fig. 3 of Ref. [25]), associated with the\nferrimagnetic order, manifests itself here through the opening\nof a gap in the ferromagnetic Goldstone mode \u000f(\u000b)\nk, as seen in\nFig. 2(b), thus indicating that the system loses its long-range\norder. Note that the antiferromagnetic mode \u000f(\f)\nkis also sim-\nilarly shifted. On the other hand, although the dispersion re-\nlation is modified for small values of the wave vector k, the\nminimum value of \u000f(\u000b)\nkstill remains at k= 0 up to the onset\nof the formation of spiral IC spin structures at \u000ec(IC) = 0:043\n(a value that should be compared with the numerical DMRG\nestimate of\u000e\u00190:055\u00060:012), characterized by the flatten-\ning of the dispersive spin-wave branches around zero. Upon\nfurther increase of \u000e, two minima form (around k= 0) and\nmove away from each other as one enhances the hole doping.\nThis behavior is the signature of the occurrence of spiral IC\nspin structures (see Fig. 3 of Ref. [25]).\nFig. 2(c) shows the onset of phase separation (PS) at\n\u000e(PS) = 0:165forJ= 0:3, which is characterized by the\noverlap of the two ferromagnetic modes at k= 0. The sig-\nnature of this regime is the spatial coexistence of two phases:\nspiral IC spin structures at \u000e(PS) = 0:165and RVB states\nat\u000e\u00191=3, in very good agreement with the numerical es-\ntimate of\u000eIC\u0000PS\u00190:16[25]. At\u000e\u00191=3, the flat mode\nhas the lowest energy, as illustrated in Fig. 2(d). This be-\nhavior indicates that the RVB state is the stable phase at\n\u000e\u00191=3andJ= 0:3[25], and also in agreement with the\nnumerical DMRG studies [11, 24] and analytical prediction at\nU=1[55].\nOnset of IC\nOnset of PS\n(b) (a)\n(c) (d)Figure 2. (Color online) Evolution of the zero-field second-order\nspin-wave dispersion relations of the AB2t-Jchain as a function of\nhole doping ( \u000e): dispersive ferromagnetic \u000f(\u000b)\nkand antiferromagnetic\n\u000f(\f)\nkmodes and the flat ferromagnetic one \u000f(\u0018)\nk, at (a) half filling; (b)\nthe onset of the spiral IC spin structures at \u000ec(IC) = 0:043, in which\ncase the flattening of the gap of \u000f(\u000b)\nkaroundk= 0 is observed; (c)\nthe onset of PS at \u000e(PS) = 0:165, characterized by the overlap of\nthe two ferromagnetic modes at k= 0and by the spatial coexistence\nof two phases: spiral IC spin structures, with modulation fixed at\n\u000e(PS) , and RVB states at \u000e\u00191=3. (d) At\u000e= 1=3the flat mode\npresents the lowest energy, thus indicating that the short-range RVB\nstate is the stable phase.\nIn order to better understand the rich variety of doping-\ninduced phases in the system, in Fig. 3 we plot the evolution of\nthe wave vector kmincorresponding to the local minimum of\n\u000f(\u000b)\nk(\u000e), upon increasing the hole doping \u000efrom 0to1=3. The\nwave vector kminremains zero until it hits the onset doping\nvalue\u000ec(IC) = 0:043, beyond which a square-root growth be-\nhavior takes place [71]: [\u000e\u0000\u000ec(IC)]1=2(blue line), for \u000eclose\nto\u000ec(IC). The square-root growth behavior is the signature\nof the occurrence of a second-order quantum phase transition\nfrom the doped ferrimagnetic phase to the IC spiral ferrimag-\nnetic state with a non-zero value of the total GS spin, SGS.\nThis result is supported by the behavior of \u0001k\u0011kmax\u0000\u0019\nat which the local maximum of the numeric DMRG structure\nfactorS(k)neark=\u0019is observed, as shown in the inset of\nFig. 3 (taken from the inset of Fig. 3(b) of Ref. [25]). For\nfurther increase of hole doping our result deviates from the\nsquare-root growth behavior and some very interesting fea-\ntures are to be noticed. The value of \u000ec= 0:08indicates\nthe breakdown of the total SGSin the IC phase, as will be\nconfirmed by the explicit calculation of SGS, a macroscopic\nquantity, in Section V C. Thus, for 0:08<\u000e < 0:165the sys-\ntem displays an IC phase with zero SGS, in agreement with\nthe DMRG data (see Fig. 1(c) of Ref. [25]). At \u000e(PS) = 0:16513\n0 0.043 0.08 0.165 1/300.10.20.3kmin/π\nDoped FerriIC\nSGS/negationslash= 0IC\nSGS= 0PS\nδc(IC)δcδδ(PS)∼\nRVB\nAnalytic\n[δ−δc(IC)]1/2\n0 0.05 0.1δ00.10.20.30.4\n∆k/π\nFigure 3. (Color online) Evolution of kmin(value ofkat the local\nminimum of \u000f(\u000b)\nk(\u000e)neark= 0) as a function of \u000e: doped ferri-\nmagnetism for 0< \u000e < \u000ec(IC)\u00190:043; spiral IC spin structures\nwith non-zero (zero) SGSfor\u000ec(IC)< \u000e < \u000ec\u00190:08(\u000ec< \u000e <\n\u000e(PS)\u00190:165), with a second-order quantum phase transition at\n\u000ec(IC) characterized by a square-root behavior [\u000e\u0000\u000ec(IC)]1=2(blue\nline), and a first-order transition at \u000e(PS) involving the IC spin struc-\nture, with modulation fixed at \u000e(PS) , and short-range RVB states at\nhole concentration 1/3. The inset shows DMRG data from Ref. [25]\nfor\u0001k\u0011kmax\u0000\u0019as a function of \u000e, wherekmaxis the value of\nkat the local maximum of the structure factor S(k)neark=\u0019, in\nqualitative agreement with the second-order transition at \u000ec(IC) .\nthe system exhibits a first-order transition accompanied by the\nspatial phase separation regime: the IC phase with zero SGS\nand modulation fixed by \u000e(PS) in coexistence with the short-\nrange RVB states at \u000e\u00191=3, also consistent with the DMRG\ndata plotted in Fig. 4 of Ref. [25].\nLastly, we emphasize that, despite the occurrence of several\ndoping-induced phases in the DMRG studies [25]: Lieb fer-\nrimagnetism, spiral IC spin structures, RVB states with finite\nspin gap, phase separation, and Luttinger-liquid behavior, it\nis surprising and very interesting that the second-order spin-\nwave modes remain stable up to \u000e\u00191=3, with predictions\nin very good agreement with the DMRG studies [25]. In this\ncontext, it is worth mentioning the long time studied case of\nrare earth metals [72], where an external magnetic field can in-\nduce non-trivial phase transitions involving spiral spin struc-\ntures, well described by spin-wave theory.\nB. Doped regime: Ground state energy\nPerforming the integration over the first BZ in Eqs. (65) and\n(69) and setting S= 1=2in Eq. (83), we find that the AB2t-J\nground state energy per unit cell as a function of hole doping\nin zero-field reads:\nEt-J\nGS(\u000e)=JNc=\u00001:9543t\nJsin (\u0019\u000e)\u00002:4608 (1\u0000\u000e):(84)\nWe shall now examine the case of small hole doping away\nfrom half filling, i.e., with hole concentration ranging from\n\u000e= 0 up to\u000e= 0:2for two values of J:0:1and0:3. In\nFig. 4, we show the evolution of the GS energy per unit cell\n0 0.05 0.1 0.15 0.2\nδ0\n-5\n-10\n-15Et-J\nGS(δ)/JNc\nAnalyticJ= 0.3\nNumericJ= 0.3\nAnalyticJ= 0.1\nNumericJ= 0.1Figure 4. (Color online) Analytical prediction for the GS energy per\nunit cell of the AB2t-Jchain as a function of doping, and compar-\nison with numerical data from DMRG technique for J=t= 0:1and\nJ=t= 0:3[25]. At half filling ( \u000e= 0), both results meet at the\nexpected prediction [25]: \u0019\u00002:4678 . Note that we have added the\nterm\u0000JNcwith the intention of comparison with numerical calcu-\nlation.\nof theAB2t-Jmodel as a function of hole doping for both\nmentioned values of J, and the comparison was made with the\nnumerical DMRG data [25]. From the two results at J= 0:3,\nthe only quantitative difference induced by the increase of the\nhole concentration is a crossing feature around \u000e\u00190:1, where\nour analytical result slightly change its behavior by lowering\nthe energy with respect to the numerical data [25]. In fact, be-\ncause our model assumes a ferrimagnetic state as the starting\npoint, this change of behavior suggests that we have entered in\na region of strong magnetic instabilities, and possibly indicat-\ning a smooth transition to an incommensurate phase with zero\nGS total spin beyond \u000e\u00190:1, as confirmed by the numerical\ndata in Ref. [25] and illustrated in Fig. 3. On the other hand,\natJ= 0:1, although our results reproduce the numerical data\nwith an acceptable agreement, we observe a discrepancy that\nincreases with \u000e. The cause of such discrepancy will be dis-\ncussed in the next subsection.\nWith the purpose of determining the interplay between the\ncontribution of magnetic exchange and the itinerant kinetic\nenergy to the zero-field GS energy Eq. (83), we take J= 0:3\nand show its evolution with doping in Fig. 5. We can see\nin the insets, Fig. 5(a) and Fig. 5(b), the competitive behav-\nior of the two energetic contributions, i.e., the contribution of\nthe exchange energy increases linearly with \u000e, while a practi-\ncally linear decrease of the hopping term is observed as one\nenhances the hole doping. This competition indicates that a\nphase transition to a paramagnetic phase should occur at some\ncritical concentration value.\nC. Doped regime: Ground state total spin\nThe existence of a transition from an IC spiral ferrimag-\nnetic phase to an IC paramagnetic one is a most interest-14\n0 0.05 0.1 0.15 0.2\nδ0\n-2\n-4\n-6Et-J\nGS(δ)/JNc\nJ= 0.3\nExchange\nHopping0 0.05 0.1 0.15 0.2\nδ-2\n-2.2\n-2.4EJ(δ)/JNc\n0 0.05 0.1 0.15 0.2\nδ0\n-1\n-2\n-3Et(δ)/JNc\nFigure 5. (Color online) Ground-state energy per unit cell for the\nAB2t-Jchain as a function of \u000eforJ= 0:3. In the insets, we\nillustrate the two energetic contribution due to (a) exchange and (b)\nhopping terms.\ning feature observed numerically in doped AB2t-JHubbard\nchains [25]. In order to firmly corroborate the mentioned\ntransition, we have calculated the GS total spin per unit cell\nas a function of hole doping, SGS(\u000e) =P\n\u000bhS\u000b;zi(\u000e),\nwith\u000b=A;B 1;B2, by means of the zero-field derivative\nof Eq. (83):\nhS\u000b;zi(\u000e) =\u0000(1=Nc)[@Et-J\nGS(\u000e;h)=@h\u000b]jh\u000b=0:(85)\nWe thus find\nhS\u000b;zi(\u000e) =hS\u000b;zi\u00064p\n2\n\u0019tsin(\u0019\u000e)(4S\u00003q1);(86)\nwhere +(\u0000) corresponds to sublattice \u000b=A(\u000b=B1;2),\nandhSA;ziandhSBl;ziare given by Eqs. (70). Therefore,\nby performing the integration over the first BZ of the three\ncontributions in Eq. (86), we finally obtain:\nSGS(\u000e)\nSL= 1\u00003:9086 sin(\u0019\u000e); (87)\nwhereSL=P\n\u000bhS\u000b;zi= 1=2is Lieb’s reference value for\nthe GS total spin per unit cell at half filling and zero-field (see\nSection IV C).\nIn Fig. 6 we plot the evolution of SGS, normalized by SL,\nas a function of \u000e, and compare it with the numerical data\nfrom DMRG and Lanczos techniques [25], for J= 0:3(red\nsquares) and J= 0:1(blue circles). In the latter (former) case,\nthe system undergoes a transition from the modulated itin-\nerant ferrimagnetic phase to an incommensurate phase with\nzero (nonzero) SGS. Notice that, in both cases, the transition\nis characterized by a decrease of SGSfromSLto0or to a\nresidual value, regardless of the value that SGStakes after the\ntransition. Indeed, at J= 0:1and\u000e > 0:1, the formation\nof magnetic polarons (onset of the Nagaoka phenomena that\nsets in asU!1 ) with charge-density waves in phase with\nthe modulation of the ferrimagnetic structure, as indicated by\nFigure 6. (Color online) Ground-state total spin SGSper unit cell\n(solid magenta line), normalized by its value in the undoped regime:\nSL=1\n2, as a function of hole doping \u000efor the indicated values of\nJ. In the figure, \u000ec\u00190:08indicates the critical value of doping\nat which the magnetic order is suppressed and a second-order phase\ntransition takes place.\nthe DMRG data [25], leads to an incommensurate phase with\nnonzeroSGS.\nMost importantly, we can observe in Fig. 6 that the value of\nSGSdecreases practically linearly with \u000euntil the magnetic\norder is completely suppressed at \u000ec\u00190:08. This behavior is\nsupported by numerical results [25], particularly in the regime\nwhere the Nagaoka phenomenon is not manifested, that is, at\nJ= 0:3, as indicated in Fig. 3. In this regime, spin and charge\nquantum fluctuations destabilize the ferrimagnetic structure\nand trigger a transition to an incommensurate paramagnetic\nphase at\u000ec, withSGS\u0018(\u000e\u0000\u000ec)!0.\nVI. CONCLUSIONS\nIn summary, we have presented a detailed analytical study\nof the large-U Hubbard model on the quasi-one-dimensional\nAB2chain. We used a functional integral approach combined\nwith a perturbative expansion in the strong-coupling regime\nthat allowed us to properly analyze the referred system at and\naway from half filling.\nAt half filling, our model was mapped onto the quantum\nHeisenberg model, and analyzed through a spin-wave pertur-\nbative series expansion in powers of 1=S. We have demon-\nstrated that the GS energy, spin-wave modes, and sublat-\ntice magnetizations are in very good agreement with previ-\nous results. In the challenging hole doping regime away\nfrom half filling, the corresponding t-J(= 4t2=U)Hamilto-\nnian was derived. Further, under the assumption that charge\nand spin quantum correlations are decoupled, the evolution\nof the second-order spin-wave modes in the doped regime\nhas unveiled the occurrence of spatially modulated spin struc-\ntures and the emergence of phase separation (first-order tran-\nsition) in the presence of resonating-valence-bond states. The\ndoping-dependent GS energy and total spin per unit cell are15\nalso calculated, in which case the collapse of the spiral mag-\nnetic order at a critical hole concentration was observed.\nRemarkably, our above-mentioned analytical results in the\ndoped regime are in very good agreement with density matrix\nrenormalization group studies, where our assumption of spin-\ncharge decoupling is numerically supported by the formation\nof charge-density waves in anti-phase with the modulation of\nthe ferrimagnetic structure.\nFinally, we stress that our reported results evidenced that\nthe present approach, also used in a study on the compatibil-\nity between numerical and analytical outcomes of the large-U\nHubbard model on the honeycomb lattice, was proved suitablefor theAB2chain (a quasi-1D system), where the impact of\ncharge and spin quantum fluctuations are expected to manifest\nin a stronger way. 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Elliott\n(Plenum Press, 1972), Chapter 1, p. 187."    },    {        "title": "1007.1856v1.Second_harmonic_generation_in_a_polar_ferrimagnet_GaFeO3.pdf",        "content": "arXiv:1007.1856v1  [cond-mat.str-el]  12 Jul 2010Second harmonic generation in a polar ferrimagnet GaFeO 3\nJun-ichi Igarashi1and Tatsuya Nagao2\n1Faculty of Science, Ibaraki University, Mito, Ibaraki 310- 8512, Japan\n2Faculty of Engineering, Gunma University, Kiryu, Gunma 376 -8515, Japan\nAbstract\nWe have studied second harmonic generation (SHG) in a polar f errimagnet GaFeO 3, employing a\nFeO6cluster model in which the Fe atom is slightly shifted from th e center of the octahedron. The\nelectric-dipole transition could take place between the 3dstates through the effective hybridization\nof the4pstates with the 3dstates, due to the breaking of the space-inversion symmetry . In the\nthird-order perturbation with Hint=−1\ncj·A, we calculate the probability per unit time, Iηaa, for\nthe process that two photons are absorbed with polarization parallel to the aaxis and one photon\nis emitted with polarization parallel to the η(=a,b,c) axis. The calculated SHG intensities consist\nof several peaks as a function of two-photon energy in agreem ent with the experiments. It is found\nthat the corresponding amplitude Saaaat each Fe site changes its sign while Sbaaremains the same\nwith the reversal of the direction of the local magnetic mome nt. This implies that Iaaawould\ndisappear while Ibaawould survive in the paramagnetic phase in accordance with t he experiment.\nPACS numbers: 78.20.Ls, 78.20.Bh, 78.40.-q\n1I. INTRODUCTION\nWhen the space-inversion symmetry and the time-reversal sy mmetry are simultaneously\nbroken, novel magneto-optical effects are expected to come o ut in the optical spectroscopy.\nSuch effects are known as the Kerr effect, the Faraday effect, th e reciprocal dichroism, the\nmagneto-chiral dichroism, and so on.1–5\nSince GaFeO 3exhibits simultaneously spontaneous electric polarizati on and magnetiza-\ntion at low temperatures, this compound is quite suitable to investigate such magneto-electric\neffects. Remeika was the first who synthesized the compound fo r decades ago.6Rado ob-\nserved the large magneto-electric effect.7Recently, untwinned large single crystals have been\nprepared.8The optical absorption measurement has been carried out wit h changing the di-\nrection of magnetization,9and the magneto-electric effects on the absorption coefficien t have\nbeen observed. That is, the absorption spectra depend on the direction of the magnetization\nin the region of photon energy 1.0−2.5eV. In our previous paper,10we have analyzed the\nspectra through the microscopic calculation. The calculat ed spectral shape has agreed well\nwith the experimental curve.9\nIn the field of the nonlinear optics, it is known that the break ing of the spatial in-\nversion symmetry in polar or chiral materials gives rise to s econd harmonic generation\n(SHG).11–13When the time-reversal symmetry is simultaneously broken i n polar magnetic\nmaterials, the SHG intensities with some specific polarizat ion are activated by the presence\nof magnetization.14–18In this context, the SHG spectra have been observed and analy zed in\nplenty of systems such as Cr 2O3and multiferroic materials,14,19–24and magnetic field induced\nSHG has been investigated in several semiconductors.25–27Recently, the SHG experiments\nhave been carried out in GaFeO 3.28–31Ogawa et al. have measured the magnetization-\ninduced SHG spectra of GaFeO 3in the region of the two-photon energy 2.5−4.5eV.28The\npurpose of this paper is to elucidate the origin of the SHG spe ctra through the microscopic\ncalculation.\nIt is known that the crystal of GaFeO 3belongs to the space group Pc21nand has an\northorhombic unit cell.32Each Fe atom is octahedrally surrounded by O atoms, and is\nslightly displaced from the center of the octahedron along t hebaxis. There are “Fe1\" and\n“Fe2\" sites for Fe atoms, where the displacement is 0.26Å at Fe1 sites and −0.11Å at Fe2\nsites.8Thereby the spontaneous electric polarization is generate d along the baxis. The local\n2magnetic moments at Fe1 and Fe2 sites are known to align antif erromagnetically along the\n±caxis. Note that the actual compound deviates slightly from a perfect antiferromagnet\nand behaves as a ferrimagnet.33The origin for this deviation is not fully understood, but\nis inferred that the Fe occupation at Fe1 and Fe2 sites are sli ghtly different from each\nother.8We assume the system as a perfect antiferromagnet in the foll owing analysis. As\nshown later, we could obtain the SHG intensities in the antif erromagnetic phase, since the\nsystem is breaking the space-inversion symmetry. This situ ation is different from those of\nEuTe and EuSe,34where no SHG intensities exist in the antiferromagnetic pha se because Eu\natoms reside on centrosymmetric sites. We expect that the sl ight deviation from the perfect\nantiferromagnet would cause merely minor quantitative cha nge in the SHG intensities.\nWe introduce a FeO 6cluster model in which the Fe atom is slightly displaced from the\ncenter of the octahedron.10We neglect the slight distortion of octahedron. We take acco unt\nof the Coulomb interaction between the 3dstates, the spin-orbit interaction on the 3dstates,\nand the hybridization of the oxygen 2pstates with the Fe 3dand4pstates. The same cluster\nmodel has been successfully applied to investigating the ma gneto-electric spectra in the\noptical absorption10and the directional dichroic spectra in the K-edge x-ray absorption35in\nGaFeO 3. We evaluate the matrix elements of the electric dipole ( E1) transition between the\n3dand the4pstates using the atomic Hamiltonian from the conventional form Hint=−1\ncj·A\nwherejis the current operator and Ais the vector potential. As shown in Sec. III, the matrix\nelements thus evaluated are found larger than those evaluat ed from another conventional\nformHint=−P·EwherePis the electric dipole operator and Eis the electric field. The\nsituation may become different without using the atomic Hami ltonian in the band structure\ncalculation. The E1transition between the 3dstates could eventually take place through the\neffective 4p-3dhybridization due to the breaking of the space-inversion sy mmetry. Thereby\nthe effective E1transition matrix elements become larger than those of the m agnetic dipole\n(M1) transition in the 3d5configuration. In the present cluster model analysis, we reg ard\nHint=−1\ncj·Amore fundamental.\nBearing the above situation in mind, we avoid to use the form Hint=−P·Ein the\ncluster model analysis. Then, using perturbation theory to third-order with Hint=−1\ncj·A,\nwe formulate the SHG intensity from the probability per unit time of the process that the\nincident two photons with the frequency ωare absorbed and one photon with the frequency\n2ωis emitted. Although this approach seems different from the c onventional analysis using\n3the nonlinear susceptibility, the expression for the proba bility amplitude is quite close to\nthat obtained from the nonlinear susceptibility. The differ ence is that the E1transition\nmatrix elements in the formula are not evaluated from Hint=−P·E. On the basis of this\nformula on the cluster model, we calculate the SHG intensiti es as a function of two-photon\nenergy on various polarization conditions in the so-called phase-matching condition. The M1\ntransition is found to give a minor contribution to the SHG in tensity. The calculated SHG\nspectra show multi-peak structure in the 1.0-4.5eV range in agreement with the available\nexperimental data. Another finding is that, when the polariz ations of both the incident and\nemitted photons are all parallel to the aaxis, the amplitude (complex number) at each Fe site\nchange its sign with the reversal of the direction of the loca l magnetization. This indicates\nthat the SHG intensity, which will be denoted as Iaaa, would disappear by passing through\nfrom the antiferromagnetic phase to the paramagnetic phase . Therefore, this spectrum may\nbe called as magneto-electric one because it is activated by the magnetization. On the other\nhand, we confirm that, when the polarization of incident phot ons is parallel to the aaxis\nwhile that of the emitted photon is parallel to the baxis, the corresponding amplitude at\neach Fe site keeps the same value with the reversal of the dire ction of the local magnetic\nmoment. This implies that the SHG intensity, which will be de noted as Ibaa, would change\nlittle by passing through from the antiferromagnetic phase to the paramagnetic phase. These\ncharacteristics of the polarization dependence as well as t he spectral shape agree with the\nSHG spectra observed in the reflection measurement,28,31although our results is not for the\nreflection spectra.\nThis paper is organized as follows. In Sec. II, we introduce a cluster model FeO 6. In Sec.\nIII, we describe the optical transition operators associat ed with Fe atoms. In Sec. IV, we\nderive the formula of the SHG intensity, and present the calc ulated spectra in comparison\nwith the experiment. The last section is devoted to concludi ng remarks.\nII. HAMILTONIAN FOR A FeO 6CLUSTER\nConsidering a FeO 6cluster, here we briefly summarize the corresponding model H amil-\ntonian. The details are found in our previous papers.10,35It may be expressed as\nH=H3d+H2p+H4p+H3d−2p\nhyb+H4p−2p\nhyb. (2.1)\n4TheH3drepresents the energy of Fe 3delectrons, which includes the intra-atomic Coulomb\ninteraction expressed in terms of the Slater integrals, the spin-orbit interaction, and the\nenergy arising from the exchange interaction via the exchan ge field from neighboring Fe\natoms. The energy of the Fe 4pstates and that of the oxygen 2pstates are represented by\nH4pandH2p, respectively.\nTheH3d−2p\nhybandH4p−2p\nhybrepresent the hybridization energies of the O 2pstates with the\nFe3dand4pstates, respectively:\nH3d−2p\nhyb=/summationdisplay\njησmt3d−2p\nmη(j)d†\nmσpjησ+H.c., (2.2)\nH4p−2p\nhyb=/summationdisplay\njηση′t4p−2p\nη′η(j)p′†\nη′σpjησ+H.c., (2.3)\nwhered†\nmσandp′†\nησstand for creation operators of electron with the 3dorbital (m=x2−\ny2,3z2−r2,yz,zx , andxy) having spin σand of the local 4porbital (η=x,y, andz) having\nspinσ, respectively. The pjησis the annihilation operator of electron with the oxygen\n2porbital at neighboring site j. The sum over jis taken on neighboring O sites. The\nhybridization parameters t3d−2p\nmη(j)andt4p−2p\nη′η(j)are expressed in terms of the Slater-Koster\ntwo-center integrals.\nThe Fe atom is slightly displaced from the center of the octah edron; the shift δis0.26Å at\nFe1 sites and −0.11Å at Fe2 sites along the baxis. Therefore, the hybridization parameters\nare modified. We evaluate the modified Slater-Koster two-cen ter integrals for the Fe atom\nby assuming that (pdσ)2p,3d,(pdπ)2p,3d∝d−4, and(ppσ)4p,2p,(ppπ)4p,2p∝d−2fordbeing\nthe Fe-O distance.36With these modified values, we obtain the ligand field Hamilto nian on\nthe3dstates, which is deviated from the cubic symmetry. In the sec ond-order perturbation,\nit may be given by\n˜H3d−3d=/summationdisplay\nmm′σ˜t3d−3d\nmm′d†\nmσdm′σ, (2.4)\nwith\n˜t3d−3d\nmm′=/summationdisplay\njηt3d−2p\nmη(j)t3d−2p\nm′η(j)/∆, (2.5)\nwhere∆denotes the charge transfer energy. In addition to the ligan d field Hamiltonian, we\nhave the effective hybridization between the 4pand3dstates, due to the breaking of the\nspace-inversion symmetry. In the second-order perturbati on, the hybridization energy may\n5be given by\n˜H4p−3d=/summationdisplay\nη′mσ˜t4p−3d\nη′mp′†\nη′σdmσ+H.c., (2.6)\nwith\n˜t4p−3d\nη′m=/summationdisplay\njηt4p−2p\nη′η(j)t3d−2p\nmη(j)\nE4p−E2p, (2.7)\nwhereE4pandE2pare the average of the 4p-band energy and the energy of the O 2pelectron,\nrespectively. The denominator in Eq. (2.7) is approximatel y estimated as E4p−E2p≈17\neV.\nIn the following numerical calculation, we use the same para meter values as in our pre-\nvious papers,10,35except for ∆ = 4.0eV, which is slightly larger than the previous value 3.3\neV.\nIII. INTERACTION BETWEEN ELECTROMAGNETIC WAVE AND ELECTRON\nWe concentrate our attention on Fe atoms. Then, the interact ion between electrons and\nthe electromagnetic wave with polarization vector eand wave vector qis approximated as\nHint=−1\nc/summationdisplay\nij(q,i)·A(q,i)+H.c., (3.1)\nwith\nj(q,i) =/summationdisplay\nnn′/bracketleftbigg/integraldisplay\neiq·(r−ri)jnn′(r−ri)d3(r−ri)/bracketrightbigg\na†\nn(i)an′(i), (3.2)\nA(q,i) =/radicalBigg\n2π/planckover2pi1c2\nVωqecqeiq·ri, (3.3)\nwhereriis the position vector of the Fe atom at site i, andan(i)is the annihilation operator\nof electron with the local wave function φn(r−ri). Thejnn′(r−ri)in Eq. (3.2) is given by\njnn′(r−ri) =ie/planckover2pi1\n2m[(∇φ∗\nn)φn′−φ∗\nn∇φn′]−e2\nmcAφ∗\nnφn′\n+e/planckover2pi1\nmcc∇×[φ∗\nnSφn′], (3.4)\nwheree,mand/planckover2pi1Smean the charge, the mass, and the spin operator of electron, respectively.\nThe interaction Hamiltonian can be rewritten as\nHint=−e/radicalBigg\n2π\nV/planckover2pi1ωq/summationdisplay\niT(q,e,i)cqeiq·ri+H.c., (3.5)\n6where the transition operator T(q,e,i)is defined as /planckover2pi1e·j(q,i)/e.\nFirst, we consider the E1transition. Putting eiq·(r−rj)= 1in Eq. (3.2), we evaluate the\nfirst term in Eq. (3.4) by using the relation\n/integraldisplay\nφ∗\nn∂\n∂zφn′d3r=−m\n/planckover2pi12(ǫn−ǫn′)/integraldisplay\nφ∗\nnzφn′d3r, (3.6)\nwhereǫnandǫn′describe the energy eigenvalues with eigenfunctions φnandφn′, respectively.\nIn the present study, φnandφn′are assigned to the 4pand3dstates. Then, the transition\noperator TE1is summarized as\nTE1(q,e,i) =iBE1/summationdisplay\niηmσNE1\nηm[p′†\nησ(i)dmσ(i)−d†\nmσ(i)p′\nησ(i)], (3.7)\nwith\nBE1= (ǫ4p−ǫ3d)/integraldisplay∞\n0r3R4p(r)R3d(r)dr, (3.8)\nwhereR3d(r),R4p(r)are radial wave-functions of 3d,4pstates with energy ǫ3d,ǫ4pin the\nFe atom. Within the HF approximation in the 1s23d54p0.001-configuration of an Fe atom,37\nwe estimate it as BE1≈7.7×10−8cm·eV. Coefficient NE1\nηm’s are given by NE1\nx,x2−y2= 1/√\n5,\nNE1\nx,3z2−r2=−1/√\n15,NE1\ny,xy= 1/√\n5,NE1\nz,zx= 1/√\n5for polarization parallel to the xaxis,\nNE1\nx,xy= 1/√\n5,NE1\ny,x2−y2=−1/√\n5,NE1\ny,3z2−r2= 1/√\n15,NE1\nz,yz= 1/√\n5for polarization parallel\nto theyaxis, and NE1\nx,zx= 1/√\n5,NE1\ny,yz= 1/√\n5,NE1\nz,3z2−r2= 2/√\n15for polarization parallel\nto thezaxis, respectively.\nThe matrix elements of the E1transition are sometimes evaluated from the dipole inter-\naction,\n˜Hint=−/summationdisplay\niPi·E(q,i), (3.9)\nwherePiis the dipole operator for electrons of the Fe atom at site i, andE(q,i)is the electric\nfield. In Eq. (3.9), the matrix element of the dipole operator between the 4pand3dstates\nmay be estimated as Eq. (3.8) divided by ǫ4p−ǫ3d, whileE(q,i) =−1\nc∂A(q,i)\n∂t=iω\ncA(q,i)for\nthe oscillating field with the frequency ω. Therefore, the use of Eq. (3.9) underestimates the\nE1-transition matrix elements by a factor /planckover2pi1ω/(ǫ4p−ǫ3d)withǫ4p−ǫ3d>10eV and/planckover2pi1ω∼\nseveral eV’s. We think Eq. (3.1) more fundamental from the mi croscopic standpoint.\nNow we evaluate the matrix elements of the E1-transition between the 3dstates. The\nmatrix elements between the 3dstates take finite values when the 3dstates mix with the 4p\nstates. As a first step of the evaluation, we calculate the ene rgy eigenstates |Φn(d5)∝angb∇acket∇ightwith\n7eigenenergy En(d5)in the3d5-configuration, and |Φn(d4)∝angb∇acket∇ightwith eigenenergy En(d4)in the\n3d4-configuration, by diagonalizing the Hamiltonian H3d+˜H3d−3d. If the displacement is\nneglected, Fe atoms are under the cubic symmetry. The displa cement gives rise to an addi-\ntional trigonal field, which makes the energy levels split fu rther. The spin-orbit interaction\nand the exchange field further modify these states. Note that the matrix elements of the E1\ntransition do not exist between these states.\nNext, we treat the effective hybridization ˜H4p−3dwithin the first order perturbation. The\nmodified wave function |Ψn(i)∝angb∇acket∇ightmay be written as\n|Ψn(i)∝angb∇acket∇ight=|Φn(d5)∝angb∇acket∇ight\n+/summationdisplay\nmkησ|Φm(d4),kησ∝angb∇acket∇ight∝angb∇acketleftΦm(d4),kησ|˜H4p−3d|Φn(d5)∝angb∇acket∇ight\nEn(d5)−[Em(d4)+ǫ4p(k)], (3.10)\nwhere|Φm(d4),kησ∝angb∇acket∇ightrepresents the state of four electrons in the 3d states and on e electron\nin the4pstates specified by η(=x,y, andz), spinσ, and momentum k. The sum over\nkmay be replaced by the integral with the 4pDOS, which is explicitly given in Ref. 10.\n>From Eq. (3.10), the matrix elements of the E1transition between the states in the 3d5-\nconfiguration are given by\n[TE1(q,e,i)]n′,n≡ ∝angb∇acketleftΨn′(i)|TE1(q,e,i)|Ψn(i)∝angb∇acket∇ight\n=/summationdisplay\nmkησ∝angb∇acketleftΦn′(d5)|TE1(q,e,i)|Φm(d4),kησ∝angb∇acket∇ight∝angb∇acketleftΦm(d4),kησ|˜H4p−3d|Φn(d5)∝angb∇acket∇ight\nEn(d5)−Em(d4)−ǫ4p(k)\n+/summationdisplay\nmkησ∝angb∇acketleftΦn′(d5)|˜H4p−3d|Φm(d4),kησ∝angb∇acket∇ight∝angb∇acketleftΦm(d4),kησ|TE1(q,e,i)|Φn(d5)∝angb∇acket∇ight\nEn′(d5)−Em(d4)−ǫ4p(k).\n(3.11)\nFinally, we close this section by evaluating the matrix elem ents of the M1transition. In\ndoing so, we approximate the third term in Eq. (3.4) as\n/integraldisplay\neiq·(r−ri)∇×(φ∗\nnSφn′)d3r=−iq×/integraldisplay\nφ∗\nnSφn′eiq·(r−ri)d3r\n≈ −iq×/integraldisplay\nφ∗\nnSφn′d3r. (3.12)\nIn addition to this term, we have, from the first term of Eq. (3. 4), the similar term to\nEq. (3.12), in which Sis replaced by L/2(Lis the orbital angular momentum). See Ref. 10\nfor the derivation. Hence, the corresponding transition op erator may be expressed as\nTM1(q,e,i) =i|q|BM1/summationdisplay\nimm′σσ′NM1\nmσ,m′σ′d†\nmσ(i)dm′σ′(i), (3.13)\n8whereBM1=/planckover2pi12/2m= 3.8×10−16cm2·eV. When the photon propagates along the\nzaxis,NM1\nmσ,m′σ′=∝angb∇acketleftmσ| −(Ly+ 2Sy)|m′σ′∝angb∇acket∇ightfor polarization parallel to the xaxis, and\nNM1\nmσ,m′σ′=∝angb∇acketleftmσ|Lx+2Sx|m′σ′∝angb∇acket∇ightfor polarization parallel to the yaxis. The matrix elements\nbetween the 3dstates in the d5-configuration are given by\n[TM1(q,e,i)]n′,n≡ ∝angb∇acketleftΨn′(i)|TM1(q,e,i)|Ψn(i)∝angb∇acket∇ight\n≈ ∝angb∇acketleftΦn′(d5)|TM1(q,e,i)|Φn(d5)∝angb∇acket∇ight. (3.14)\nThese values are found much smaller than the matrix elements of theE1transition given\nby Eq. (3.11).\nIV. SECOND HARMONIC GENERATION\nWe consider the process that two photons with the frequency ω, wave vector q, and\npolarization eare absorbed and one photon with ˜ω,˜q, and˜eis emitted with ˜ω= 2ω, as\nillustrated in Fig. 1. In the third-order perturbation with Hint=−1\ncj·A, the probability\nper unit time for the process may be expressed as\nI(e,ω,q;˜e,˜ω,˜q)∝/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\niS(e,ω,q;˜e,˜ω,˜q;i)ei(2q−˜q)·ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n˜ω2δ(˜ω−2ω), (4.1)\nwhere the amplitude Sis given by\nS(e,ω,q;˜e,˜ω,˜q;i)\n∝1\nω√\n˜ω/summationdisplay\nn′,n/braceleftbigg[T∗(˜q,˜e,i)]g,n′[T(q,e,i)]n′,n[T(q,e,i)]n,g\n(ǫn′−ǫg−2/planckover2pi1ω−iΓ)(ǫn−ǫg−/planckover2pi1ω−iΓ)\n+[T(q,e,i)]g,n′[T∗(˜q,˜e,i)]n′,n[T(q,e,i)]n,g\n(ǫn′−ǫg+/planckover2pi1˜ω−/planckover2pi1ω−iΓ)(ǫn−ǫg−/planckover2pi1ω−iΓ)\n+[T(q,e,i)]g,n′[T(q,e,i)]n′,n[T∗(˜q,˜e,i)]n,g\n(ǫn′−ǫg+/planckover2pi1˜ω−/planckover2pi1ω−iΓ)(ǫn−ǫg+/planckover2pi1˜ω−iΓ)/bracerightbigg\n. (4.2)\nTheΓrepresents the life-time broadening width by other random p erturbation on the ma-\nterial system. The ǫgis the energy of the ground state in the 3d5-configuration, and ǫnis\nthe abbreviation of En(d5). Note that Eq. (4.2) resembles the conventional expression of\nnonlinear susceptibility, which is based on the interactio n˜Hint=−/summationtext\niPi·E(q,i). The sum\noveriin Eq. (4.1) is made on all Fe sites, and the so-called phase-m atching condition ˜q= 2q\nhas to be satisfied in order to get finite intensities. Note als o that, if the wave interaction\n9lengthℓis finite, the momentum conservation would be relaxed within ∆˜q∼1\nℓ. In reflection,\nthe surface layer with 1/qthickness could contribute significantly to generating refl ection\nwave without phase matching-condition.12Since the matrix elements of the E1transition\nis much larger than those of the M1transition as discussed in Sec. III, all the transition\nmatrix-elements in Eq. (4.2) could be replaced by those of th eE1transition.\n e,ω,q\nSample\ne,ω,qe,ω,q\nFIG. 1: Schematic description of second harmonic generatio n. Two photons with e,ω,qare\nabsorbed, and one photon with ˜e,˜ω,˜qis emitted.\nWe consider the situation that the polarization of incident photons is parallel to the\naaxis. When the polarization of the emitted photon is paralle l to the caxis, the SHG\nintensity, which will be denoted as Icaa(2ω), is found to vanish. On the other hand, when\nthe polarization of the emitted photon is parallel to the aaxis, non-zero SHG intensity\nIaaa(2ω)is obtained as shown in Fig. 2. The broken line represents the result with replacing\nT∗(˜q,˜e,i)byTM1∗(˜q,˜e,i)in Eq. (4.2), demonstrating that the contribution of the M1\ntransition is much smaller than that of the E1transition as estimated in Sec. III. The\nspectral shape is composed of multi-peak structure: a small peak around 2ω= 1.75eV,\na small peak around 2.5eV, and a two-peak structure around /planckover2pi1ω= 3−4eV. The two-\npeak structure reasonably captures a whole aspect of the exp erimental SHG intensity shown\nin the inset of Fig. 2, although the experimental spectrum is obtained on the \" SinSout\"\nconfiguration in the reflection geometry.28\nFor the polarization of the emitted photon parallel to the baxis, we obtain the spectra\nIbaa(2ω)as shown in Fig. 3. In comparison with the spectral shape of Iaaa(2ω), the peak\naround2ω= 1.75eV becomes larger, but the other peaks remain similar to thos e ofIaaa(2ω).\nAs shown in the inset of Fig. 3, a large peak is observed around 2ω= 1.5eV in the SHG\nexperiment,29which is reproduced by the present theory. Our calculation a lso implies that\nanother peak in the 3.3−4.0eV range is anticipated if corresponding experiment will be\navailable. Note that the SHG intensities have been observed around2.5−4eV on the\n10\"SinPout\" configuration in the reflection experiment.28,31The obtained intensity is about\n1/4ofIaaa(2ω)in this energy region, corresponding well to the ratio of int ensity on the\n\"SinPout\" configuration to that on the \" SinSout\" configuration in the reflection experiment.\n1 2 3 4\ntwo photon energy [eV]048Iaaa(2ω) (arb. units)\nM1\nE1+M12 3 4\ntwo photon energy [eV]03060SHG Intensity (a. u.)10 K\n220 KExp.\nFIG. 2: The second harmonic generation (SHG) intensity Iaaa(2ω)as a function of two-photon\nenergy2ωwitheand˜eparallel to the aaxis. Inset: experimental SHG intensity in the reflection\ngeometry; the s-polarized light is radiated with incident angle 5degree on the surface of acplane,\nand thes-polarized reflected SHG light is observed (Ref. 28).\nNow we examine what happens to the SHG intensity when the syst em undergoes a phase\ntransition into the paramagnetic phase. The probability am plitudeSaaa(2ω;i)is found to\nchange its sign, when the direction of the local magnetic mom ent is reversed. This indicates\nthat/summationtext\niSaaa(2ω;i)would be canceled out when the local magnetic moment is rando mly\noriented, and that Iaaa(2ω)would disappear in the paramagnetic phase, in agreement wit h\nthe experiment. For this reason, Iaaa(2ω)may be called as the magnetization-induced SHG\nintensity, which corresponds well to the spectra around 2/planckover2pi1ω∼3−4eV for the \" SinSout\"\nconfiguration in the reflection experiments.28,31On the other hand, the amplitude Sbaa(2ω;i)\nis found to remain the same when the direction of the local mag netic moment is reversed.\n111 2 3 4\ntwo photon energy [eV]048Ibaa(2ω) (arb. units)1 2\ntwo photon energy [eV]048SHG Intensity (a. u.)100K\n300KExp.\nFIG. 3: The SHG intensity Ibaa(2ω)as a function of two-photon energy 2ωwitheand˜eparallel\nto theaandbaxes, respectively. Inset: experimental SHG intensity (Re f. 29).\nThis leads to that/summationtext\niSbaa(2ω;i)would notbe canceled out when the local magnetic moment\nis randomly oriented, and that Ibaa(2ω)remains finite in the paramagnetic phase, which\ncorresponds well to the spectra around 2/planckover2pi1ω∼3−4eV for the “ SinPout\" configuration in the\nreflection experiments.28,31\nFinally we comment on the energy levels. In the present calcu lation scheme, if we disre-\ngard the displacement of Fe atoms and the spin-orbit interac tion, we have the ground state\ncharacterized as6A1, and excited states characterized as4T1,2T2and4T2with excitation\nenergies 2.09,2.16and2.8eV, respectively. These states, however, have no E1transition\nmatrix elements from the ground state. Only after taking acc ount of the displacement of Fe\natoms, they have finite E1transition matrix elements, but the energy levels may be shi fted\nand split. To demonstrate this point, we calculate the absor ption coefficient Iabs(ω)from\nthe formula\nIabs(ω)∝1\nω/summationdisplay\ni,e/summationdisplay\nn|[TE1(q,e,i)]n,g|2δ(ǫn−ǫg−/planckover2pi1ω). (4.3)\n12Assuming the photon propagates along the caxis, the polarization is summed over eparallel\nto theaandbaxes. Figure 4 shows the calculated spectrum (solid line). I t is clearly seen\nthat the peaks are considerably separate from the positions of4T1,2T2and4T2shown by\nthe vertical solid lines. Therefore, it does not seem meanin gful to assign directly these peaks\nto the fictitious levels which have no E1transition matrix elements. These peak positions\ndepend on parameters such as Slater integrals and the hybrid ization between Fe and O\natoms. The broken line represents the spectrum calculated f rom a different parameter set\nthatF2andF4are reduced by multiplying a factor 0.81instead of 0.9and∆is set to be 4.4\neV instead of 4.0eV. Vertical broken lines indicate the corresponding energ ies of fictitious\nlevels. In the experiment,28a peak is found around 1.6eV, and a shoulder structure around\n2.0eV. The calculated second peak may correspond to the shoulde r in the experiment.\nFurther adjustment of parameter sets may improve the calcul ated spectra, but we would\nnot seek optimal parameter sets in this paper.\n1 2 3\none photon energy [eV]0123Absorption (arb. units)4T14T22T2\nFIG. 4: Absorption coefficient as a function of one photon ener gy. The solid line represents the\ncalculated spectrum. Vertical solid lines indicate the ene rgies of the fictitious levels with disre-\ngarding the displacement of Fe atoms. The broken line repres ents the calculated spectrum with a\ndifferent parameter set (see the text). The broken vertical l ines indicate the energies of fictitious\nlevels corresponding to the latter parameter set.\n13As regards the fiction levels with higher energies, we have4A1and4Eboth with energy\n3.68eV,2A2with3.70eV,2T1with3.75eV, and so on. As the same as the low energy\nlevels mentioned above, these levels have no E1transition matrix elements from the ground\nstate, and would be shifted and split due to the displacement of Fe atoms and the spin-orbit\ninteraction. Therefore, it would be difficult to assign direc tly these levels to the peaks on\nthe SHG spectra.\nV. CONCLUDING REMARKS\nWe have analyzed the SHG spectra in a polar ferrimagnet GaFeO 3, using the FeO 6cluster\nmodel where the Fe atom is displaced from the center of the oct ahedron. We have fully taken\naccount of the Coulomb interaction between the 3dstates, the spin-orbit interaction on the\n3dstates, and the hybridization of the oxygen 2pstates with the 3dand4pstates. The\nE1matrix elements between the 3dand4pstates are evaluated on the Fe atom with the\ninteraction Hint=−1\ncj·A. They are larger than those evaluated with the conventional\nform˜Hint=−P·E. TheE1matrix elements between the 3dstates could become finite\nthrough the effective hybridization between the 4pand3dstates owing to the breaking of the\nspace-inversion symmetry. Note that the same cluster model has been successfully applied\nto analyzing not only the optical absorption10but also the K-edge x-ray absorption.35\nIn the third-order perturbation with Hint=−1\ncj·A, we have derived the formula of the\nprobability per unit time for the process that two photons ar e absorbed and one photon is\nemitted. On the basis of this formula, we have calculated the spectra as a function of the\ntwo-photon energy in the phase-matching condition. The cal culated SHG intensities exhibit\nmulti-peak structure, which is in accordance with the exper iments.28,29,31The intensities\nalso signify that another peak structures will be found outs ide of the published experimental\nsurveys28,29,31; one centered around 1.5−2.0eV inIaaa(2ω;i)and the other centered around\n3.0−4.0eV inIbaa(2ω,i). We have found that the amplitude Saaa(2ω;i)changes its sign\nwhileSbaa(2ω;i)retains the same value, when the direction of the local magne tic moment is\nreversed. This indicates that Iaaa(2ω)would vanish but Ibaa(2ω)would remain finite in the\nparamagnetic phase. Hence our results have reproduced the e xperimental observations and\nIaaa(2ω)could be called as the magnetization-induced SHG intensity , which is completely\ngoverned by the E1 process since the M1 contribution is negligible.\n14The quantum-mechanical treatment in the present paper has n ot directly been applied\nto the reflection and the refraction problem, since the phase coherence is not appropriately\ntreated. More elaborate treatments using the coherent stat e might be required.11Researches\nalong this line are relegated to the future study. On the othe r hand, the semi-classical\ntreatment is known to cope well with the reflection and the ref raction through the nonlinear\nsusceptibility.11,12We have not directly used the conventional semi-classical m ethod, since\nthe form ˜Hint=−P·Eunderestimates considerably the E1transition. 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Cowan, The Theory of Atomic Structure and Spectra (University of California, Berkeley,\n1981).\n17"    },    {        "title": "1710.02700v2.Dissecting_spin_phonon_equilibration_in_ferrimagnetic_insulators_by_ultrafast_lattice_excitation.pdf",        "content": "Dissecting spin-phonon equilibration in ferrimagnetic \ninsulators by ultrafast lattice excitation  \n \nSebastian F. Maehrlein1,2,#, Ilie Radu3,4,5*, Pablo Maldonado6, Alexander Paarmann1, Michael \nGensch7, Alexandra M. Kalashnikova8, Roman V. Pisarev8, Martin Wo lf1, Peter M. Oppeneer6, \nJoseph Barker9, Tobias Kampfrath1,2* \n \n1. Fritz Haber Institute of the Max Planck Society, Faradayweg 4 -6, 14195 Berlin, Germany  \n2. Department of Physics, Freie Universität Be rlin, Arnimallee 14, 14195 Berlin, Germany  \n3. Max Born  Institute  for Nonlinear  Optics  and Short  Pulse  Spectroscopy,   \nMax-Born -Straße  2A, 12489 Berlin, Germany  \n4. Helmholtz -Zentrum Berlin für Materialien und  Energie, Albert -Einstein -Straße 15, 12489 Berlin, Germany  \n5. Institute for Optics and Atomic  Physics,  Technical  University  Berlin,   \nHardenbergstraße  36, 10623  Berlin,  Germany  \n6. Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden  \n7. Helmholtz -Zentrum  Dresden -Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany  \n8. Ioffe Institute, 26 Polytechnicheskaya , 194021 St. Petersburg, Russia  \n9. Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan  \n# Present address: Department of Chemistry, Columbia University,  \n3000 Broadway,  New York, NY 10027, USA  \n*  Corresponding author s. Email: radu@mbi -berlin.de, tobias.kampfrath@fu -berlin.de  \n \n \nABSTRACT  \nTo gain  contro l over magnetic order  on ultrafast time scales,  a fundamental understanding of the \nway electron spins interact with the surrounding crystal lattice  is required . However, \nmeasurement and analysis even of basic collective processes such as spin -phonon equili bration \nhave remained challenging. Here, we directly probe the flow of energy and angular momentum in \nthe model insulating ferrimagnet yttrium iron garnet . Following ultrafast resonant lattice \nexcitation , we observe that magnetic order reduces on distinct time scales of 1  ps and 100  ns. \nTemperature -dependent measurements, a spin-coupling  analysis and simulations show that the \ntwo dynamics directly reflect two stages  of spin -lattice equilibration.  On the 1 -ps scale, spins and \nphonons reach quasi -equilibrium in terms of energy through phonon -induced modulation of the \nexchange interaction. This mechanism leads to identical demagnetization of the ferrimagnet’s two \nspin-sublattices and a novel ferrimagnetic state of increased temperature yet unchanged total \nmagne tization . Finally, on the much slower, 100 -ns scale, the excess of spin angular momentum \nis released to the crystal lattice, resulting  in full equilibrium. Our findings  are relevant for all \ninsulating ferrimagnets and indicate that spin manipulation by pho nons, including the spin \nSeebeck effect, can be extended to antiferromagnets and into the terahertz frequency range.  \n  INTRODUCTION  \nIn solids, vibrations of the crystal lattice have a significant  impact on the orbital dynamics of the \nelectrons ( Fig. 1A). They strongly modify properties such as electrical conductivity and may even \ncause insulator -to-metal transitions1. Likewise, the interplay between phonons and electron spins \n(Fig. 1A) is relevant for equally drastic phenomena including colossal magnetores istance, \nfemtosecond  magnetization control2,3,4,5,6 and the spin Seebeck effect7,8. Recently, ultrafast optical \ntechniques have provided new insights into the ultrafast coherent coupling of individual phonon \nand spin modes9,10,11,12,13,14.  \nDespite this progres s, the microscopic origins of spin -phonon interaction s remain an intriguing \nproblem. Even the equilibration  of crystal lattice and electron spins , arguably the conceptually \nsimplest collective process, is far from being understood. This notion is highlight ed in the model \nsystem yttrium iron garnet Y 3Fe5O12 (YIG), which is one of the best -studied magnetically ordered \nsolids15,16 and ubiquitous in the field of magnonics8. Only estimates of the time constant of spin-\nphonon equilibration exist, and they extend over  as many as  6 orders of magnitude from ~1  µs \n(Ref.  17) to ~250  ps (Refs.  18 and 19) and down to ~1  ps (Ref.  20). Here, we introduce an \napproach based on resonant phonon excitation that allows us to directly probe  the interaction s \nbetween the crystal lattice and electron spins over multiple time scales.  \nExperiment.  A schematic of our terahertz (THz) -pump magnetooptic -probe experiment is  shown \nin Fig. 1A and detailed in the Materials and Methods  section  and Fig.  S1A. An incident, intense, \nultrashort THz pump pulse21 (photon energy of ~0.1 eV, duration of ~25 0 fs, see Fig.  S1B) \nselectively excites the crystal lattice by resonantly driving infrared -active transverse -optical (TO) \nphonons. The impact on the sample’s magne tic order is monitored from femtoseconds to \nmilliseconds  by measuring the magnetooptic Faraday rotation  of a time -delayed probe pulse (see \nFigs. 1A and S1A). In addition , we measure isotropic  changes in the  optical  transmittance of the \nsample on ultrafast time scales . \nAs samples, we ch oose model systems for spin -wave dynamics in magnetic insulators7,8,16,22,23: \npure YIG and  bismuth/gallium -substituted YIG (BiGa:YIG) . In these ferrimagnets, magnetic Fe3+ \nions at a - and d -sites in the unit cell ( Fig. 1B) comprise two inequivalent, ferromagnetic \nsublattices with magnetization Ma and Md, respectively,  which couple antiferromagne tically. The \n2:3 ratio of a - to d-sites results in a nonzero net magnetization Ma+Md below the Curie \ntemperature  TC. The Faraday rotation  θ of the probe pulse is determined by24 \nθ=a aMa + adMd (1) \nwhere aa and ad are the sublattice magnetooptic coefficient s. Compared to YIG, the Faraday effect \nof BiGa:YIG is enhanced by about one order of magnitude  and of opposite s ign24 (see Materials \nand Methods  and Fig. S1C). \nIn both materials, we expect that the analysis of the pump -induced dynamics is simplified due to \nthe following featur es. First, the sublattice magnetizations Ma and Md almost entirely arise from \nthe spin rather the orbital angular momentum of the electrons . At room temperature, the g-factor \nof YIG differs by only 0.13% from that of the free electron25. Second, the sizeable electronic band \ngap (2.85  eV for YIG) impli es that the electron ic orbital degrees of freedom  remain in their \nground state, thereby significantly reducing the complexity of the pump -induced processes.  \nFigure 2 A displays the absorptance spectrum of a 15  µm thick BiGa:YIG film from 10 to 45  THz \nwhere absorption is known to be due solely to infrared -active phonons26. Our ab initio  \ncalculations show that the pump pulse centered around 21  THz (red spectrum in Fig. 2A) \npredominantly excites long -wavelength TO( ) lattice normal modes characterized by an \nasymmetric Fe -O stretch vibration (see Figs. 1B, S4 and Materials and Methods ). RESULTS  AND DISCUSSION  \nUltrafast spin dynamics. Figure 2 A shows the relative pump -induced change Δθ/θ0=θ(t)/θ0−1 in \nthe Faraday rota tion as a function of the delay t since excitation . Here, θ0=θ(−2 ps) refers to the \nequilibrium case. When pumping off the TO( ) phonon resonances , at around 38  THz (blue pump \nspectrum in Fig. 2 A), a relatively small signal is found (blue curve in Fig.  2B). In marked \ncontrast, we witness a response more than one order of magnitude stronger for resonant phonon \nexcitation around 20  THz (red pump spectrum in Fig. 2A) : an ultrafast single -exponential drop of \nthe Faraday signal with a time constant as short as fast=1.6 ps (red curve in Fig.  2B). On much \nlonger time scales, an additional exponential reduction of θ with a time constant of slow=90 ns is \nfound (Fig.  2C). Recovery back to the initial state occurs over about 1  ms (Fig. S2D). \nSince the pump -probe sign al grows linearly with the pump fluence (inset of Fig.  2C), excitation is \ndominated by one -photon absorption, whereas strong -field effects such as field or impact \nionization are negligible27. This notion is further supported by Fig. 2 A which demonstrates th at \nthe sample absorptance and transient Faraday rotation are found  to depend on the pump frequency \nin a very similar way.  \nWe emphasize that almost identical dynamics are observed when a pure YIG sample instead of \nthe BiGa:YIG film is used (see Fig. S2B-D). Additional control experiments confirm that, as soon \nas the pump pulse has left the sample, the transient Faraday signal Δθ(t) reliably reflects the \ndynamics of the sublattice magnetizations Ma and Md with time -independent coefficients aa and ad \nin Eq.  (1) (see Materials and Methods  and Fig. S3). \nFigure 2B also shows the relative optical transmittance change of the BiGa:YIG sample  for the \ncase of resonant pumping . The signal starts with a peak -like feature whose wi dth coincides with \nthat of the THz pump pulse. Already for delays t>1 ps, the signal is almost constant, changing by \nless than 10% in the subsequent evolution . Such  relaxation to a quasi -steady state is substantially  \nfaster than seen for  the transient Fara day rotation which still doubles its value at t=1 ps in the \nfollowing 5  ps (Fig.  2B). Furthermore, the transmittance signal is  found to be independent of the \nsample magnetization (Fig.  S3C). These features suggest that the transmittance change \npredominantl y monitors the redistribution dynamics of the pump -deposited energy in the crystal \nlattice . \nAlthough the relative signal changes are small and still in the perturbative regime , our results in \nFig. 2 show a proof of concept that resonant phonon excitation provides an  ultrafast  manipulation \nof magnetic order . It only involves the crystal lattice and electron spins, yet no electron ic orbital \ndegrees of freedom (Fig. 1 A). The picosecond spin dynamics observed here (Fig.  2B) are \nunexpected because they are  five orders of magnitude faster than the spin coherence lifetime \n(>0.1  µs) of YIG, which is known to be one of the longest amongst magnetically ordered \nmaterials8,15. \nTemperature dependence.  To characterize the transient states established on the fast \n(fast=1.6 ps) and slow ( slow=90 ns) time scale, respectively, we increase the sample \ntemperature  T0 from 300  K to 420  K. Simultaneously, we  measure the equilibrium Faraday  \nrotation θ0 as well as its pump -induced change Δθ at ultrashort ( t=10 ps, Fig.  2B) and long delays \n(t=1 µs, Fig.  2C) after phonon excitation.  \nThe resulting θ0 versus T0 has the typical shape15 of a ferrimagne t’s static magnetization curve \n(Fig. 3 A). Its slope  ∂θ0/∂T0 steepens with rising T0 until the transition into the paramagnetic phase  \noccurs at the Curie temperature TC=398 K  of BiGa:YIG . In contrast, the pump -induced Faraday \nsignal at t=1 µs (Fig. 3B) incr eases with T0 and reaches a maximum right below TC, reminiscent \nof the derivative of the static curve ( Fig. 3A). Indeed, we find that Δθ(1 µs) versus T0 closely \nfollows (∂θ0/∂T0)ΔT (Fig. 3B). Here, ΔT=0.39 K  is the increase in equilibrium temperature as calculated from the energy density deposited by the 1  µJ pump pulse (see Materials and \nMethods ). The good agreement of both curves shows that ~1  µs after pumping, the BiGa:YIG \nfilm is in full thermodynamic equilibrium characterized by temperature T0+ΔT. \nRema rkably, Fig.  3B reveals that the changes in the Faraday signal at 10 ps are systematically  yet \nnonuniformly  smaller than at 1 µs, in agreement with Figs.  2B,C. While for temperatures \nT0<380  K, Δθ(10 ps) amounts to  roughly 80% of Δ θ(1 µs), this ratio decrea ses strongly when T0 \nis increased . For example, slightly above 380  K (where both curves reach their maxima) and \nslightly below 400  K, Δθ(10 ps) has reduced to only 50% and 20% of Δ θ(1 µs), respectively. \nConsequently , Δθ(10 ps) vs T0 in Fig.  3B cannot be ma de to agree with Δ θ(1 µs) vs T0 by a \nsimple global rescaling of the Δ θ(10 ps) values.  In particular, the shape of Δ θ(1 µs) vs T0 agrees \nmuch better with a suitably scaled derivative ∂θ0/∂T0 of the static Faraday rotation (black curve in \nFig. 3B) than Δθ(10 ps) vs T0. Therefore,  at time t=10 ps after pump excitation , the spin system is \nin a state that is significantly different from the equilibrium state found at t=1 µs. \nAnalysis of spin couplings.  To understand the microscopic mechanism driving the ultrafas t \nchange in magnetic order  following resonant phonon excitation  (Fig.  2B), we note that solids \nexhibit only three fundamental spin couplings. They can be understood as effective magnetic \nfields exerting torques on spins . In the following , we discuss all of  them in terms of their \ncapability to modify the sublattice magnetizations Ma and/or Md on ultrafast time scales.  \nFirst, spin -orbit (SO) coupling is often considered to dominate the ultrafast demagnetization of \nlaser -excited ferromagnetic metals  in the ab sence of transport28. In our experiment, however, we \nfind identical ultrafast dynamics for pure YIG and BiGa:YIG  (see Fig. S2B), even though the Bi \nions are known to increase SO  coupling . This notion is manifested in the magnetooptic effects \nwhich are enha nced by more than one order of magnitude24. Therefore, SO  coupling plays a \nnegligible role in the ultrafast spin dynamics seen in Fig.  2B. \nThe same conclusion can be drawn for the second type of coupling, spin -spin magnetic -dipole  \n(SSMD)  interaction, which has a comparable strength to SO  coupling in YIG15. We note that  SO \nand SSMD  coupling  also account  for magnetic anisotropy , which can undergo strong photo -\ninduced c hanges in the garnet YIG:Co  (Ref.  29). Ultrafast laser -induced  modification of the \nanisotropy field  of a YIG:Co  film was even shown to drive precessional magnetization \nswitching29. The observed time constant of this process (~20  ps) is, however,  still more than one \norder of magnitude longer than the ultrafast dynamics (fast=1.6 ps) observed here . In addition, we \ndo not observe any sign of coherent precession which is typical of anisotropy changes in the \nperturbative regime. Consequently, neither SO nor SSMD coupling can explain the spin dynamics \nobserved on the fast scale here.   \nWe finally consider the only remaining fundamental spin coupling, isotropic exchange, which is \nthe strongest spin interaction in most magnets2. In YIG, it is responsible for the ferrimagnetic \norder a nd the high magnon frequencies extending to more than 20 THz at room temperature15,30. \nTherefore, exchange interaction  may well account for the picosecond spin dynamics observed \nhere (Fig.  2B). Importantly, sinc e it conserves the total spin angular momentum, the \nmagnetization changes of a - and d -sublattice cancel each other. Despite ΔMa+ΔMd=0, the \nresulting change in Faraday rotation is nonzero  because  the magnetooptic coefficients aa and ad of \nYIG differ signifi cantly24 (see Eq. (1)). In summary, our analysis of the fundamental spin \ncouplings shows that only isotropic exchange interaction is capable of explaining the picosecon d \nspin dynamics seen in our experiment.  \nModel.  We suggest the following scenario of exchange -mediated spin -phonon coupling: the \nresonantly excited TO( ) phonons decay into other modes significantly faster than the time \nconstant fast=1.6 ps of the measured ultrafast spin dyna mics26, thereby heating up the crystal lattice. The assumption of ultrafast phonon r elaxation is consistent with our transient \ntransmittance data (Fig.  2B) which indicate that the phonons reach an approximately stationary \nstate within less than 1  ps. As the heat capacity of the crystal lattice of YIG is two orders of \nmagnitude higher than  that of the spin system31, the lattice acts akin to a bath whose temperature \nis suddenly increased from T0 to T0+ΔT.  \nAs a consequence of the ultrafast lattice heating , the O2- ions, which are the lightest, will undergo \nadditional random deflection Δu(t) (Fig.  4A). This perturbation modulates the superexchange32 of \nadjacent a -Fe3+ and d -Fe3+ spins and the associated coupling constant  Jad by16,33,34 \nΔJad(t)=(∂ Jad/∂u) Δu(t). (2) \nWe now put this model to test  by conduct ing calculations  of both dynamic and equilibrium states  \nand by their comparison to  our experimental observations  on the ultrafast ( fast, Fig. 2B) and the \nslower time scale  (slow, Fig. 2C).  \nDynamics  on the fast scale . To determine the ultrafast rate of change of Ma und Md, atomistic \nspin-dynamics simulations based on ~106 coupled spi n equations of motion  are performed30 (see \nMaterials and Methods ). We include time -dependent exchange parameters through Eq. (2) \nwhere  Δu(t) is assumed to be random wit h a variance given by the pump -induced temperature \nincrease  ΔT of the ionic lattice.  \nSimulation results are shown in Fig.  4B. At times t<0, the magnetizations of both sublattices \nfluctuate around their constant mean s Ma0 and Md0. However, when fluctuations  ΔJad(t) of the \nexchange parameter are switched on  at t=0, Ma decreases linearly with time, until ΔJad(t) is \nswitched off. We find the opposite behavior for ΔMd. The sum curve ΔMa+ΔMd has a more than \none order of magnitude smaller amplitude . It arises from  the much weaker random field s of the \nbath due to  spin couplings other than isotropic exchange  (see Materials and Methods ). \nFigure  4B demonstrates that thermal modulation of Jad induces demagnetization of the two s pin-\nsublattices by the same amount and, th us, transfers energy into the spin system.  \nThe slope of the simulated ΔMa(t)/Md0 (Fig.  4B) can directly be compared to the initial slope of \nthe pump -probe signal Δθ(t)/θ0 (Fig.  2B), which amounts to approximately  −1% ps-1. \nAgreement between experiment and theory is obtained by choosing  ∂Jad/∂u~10 Jad Å-1 (see \nMaterials and  Methods ). This value and the theory of superexchange32 imply that the overlap \nintegral of the ground -state wavefunctions of adjacent O2- and Fe3+ ions changes by 100% wh en \ntheir distance changes by ~0. 4 Å. As the distance between the two ions is one order of magnitude \nsmaller than the lattice constant  a of YIG15, we estimate ∂u/∂a~0.1 and obtain \n∂Jad/∂a~(∂Jad/∂u)(∂u/∂a)~1 Jad Å-1. This result is in excellent agreeme nt with recent ab initio  \ncalculations34 of the exchange -coupling constants as a function of  a, which yielded \n∂Jad/∂a~0.7 Jad Å-1. Therefore, phonon -modulated exchange coupling successfully and \nconsistently expl ains the rate (∂θ/∂t)/θ0 at which magnetic order of YIG is observed to reduce \nimmediately  after resonant phonon excitation .  \nAccording to our model, ∂Ma/∂t, ∂Md/∂t and, thus, ∂ θ/∂t (Eq. (1)) are proportional to t he \ndifference between lattice and spin temperature  (Eq. (10)), in agreement with the linear pump -\nfluence dependence of the transient Faraday rotation θ seen in the experiment  (inset of Fig.  2C). \nConsequently, the measured exponential decay of θ to a constant value (Fig.  2B) indicate s that \nthe electron spins approach  the lattice temperature  T0+ΔT with the time constant  fast=1.6 ps.  \nNote that the subsequent slower dynamics evolve with a time constant of slow=90 ns. We \nconclude that for all pump -probe delays t≪slow, the transient chan ges in Ma and Md arise solely \nfrom inter-sublattice exchange coupling . As this interaction conserves the total spin angular \nmomentum, the dynamics are under the constraint Ma+Md=0 throughout th at time window . Dynamics on  the slow scale.  As indicated by our temperature -dependent measurements  (Fig.  3), \nthe garnet film approaches full equilibrium at temperature T0+ΔT with the time constant slow. The \ntotal magnetization Ma+Md of this s tate is lower than for t≪slow where  the change  Ma+Md is \nstill zero . Thus, transfer of angular momentum from the spin system to the lattice has certainly \noccurred  for times  t~slow and above .  \nThe only microscopic spin couplings capable of changing the t otal spin and, thus, magnetization  \nMa+Md are SO and SSMD coupling. If SO coupling were dominant, one would expect a \ndifference in the spin dynamics in pure YIG and in BiGa:YIG in which SO coupling is enhanced \nby partial Bi substitution . However, the signal -to-noise ratio of the Faraday signal seen on the \nslow scale of pure YIG (Fig.  S2B) does currently not allow us to draw a conclusion about which \nof the two couplings is dominant . \nConstrained spin state.  As discussed above, t he spin system is under the constraint of constant \nmagnetization Ma+Md=Ma0+Md0 for times t≪slow whereas it is in full, unconstrained equilibrium \nfor t≫slow (Fig.  2C). Assuming that for t≫fast the spins and crystal lattice are  characterized by a \nsingle temperature T0+ΔT, both the constrained and unconstrained spin state can be fully \ndescribed by equilibrium statistical physics  (see Materials and Methods ).  \nUsing a mean -field approximation, we obtain the change  (∂Ml/∂T0)ΔT in the sublattice \nmagnetizations vs T0 for heating with and without the constraint of conserved Ma+Md (Fig.  5A). \nInterestingly, the  constrained  ∂Ma/∂T0 of minority s pin-sublattice  a (green line in Fig.  5A) follows \nclosely its unconstrained counterpart. In contrast, the constrained ∂ Md/∂T0 of majority s pin-\nsublattice  d (blue line in Fig.  5A) is systematically smaller t han the unconstrained ∂ Md/∂T0 \nbecause sublattice  d can only demagnetize as much as sublattice  a (ΔMd=−ΔMa). In particular, t he \ndifference between constrained and unconstrained ∂ Md/∂T0 increases considerably when the \ntemperature  T0 approache s the Curie poin t (Fig.  5A). We emphasize  that this behavior is in \nexcellent qual itative  agreem ent with the measured Faraday signals Δ θ(10 ps) and Δθ(1 µs) vs T0 \n(Fig.  3B). \nIt is interesting to note that in the equilibrium formalism used here, the constant total \nmagnetiza tion is reinforced by a Lagrange multiplier which  can be interpreted as a virtual \nhomogeneous magnetic field (see Eq. (13)). One could also term this field “spin pressure”, in \nanalogy to the pressure that is requir ed to keep a gas of particles in a bottle of constant volume \nwhile the gas temperature is increased. In our experiment, the “spin pressure” builds up on the \ntime scale given by fast and is released by angular -momentum transfer to the crystal lattice on the \ntime scale slow. \nPicture of spin -phonon equilibration. We summarize that our dynamic and equilibrium \ncalculations are fully consistent with the time scale s, fluence dependenc e, temperature \ndependence and magnitude of the transient Faraday rotation found in the experiment. This \nagreement leads us to the following picture of the flow of energy and angular momentum of \nphonon -pumped YIG (Fig.  5B): [1] the pump pulse excites zone -center TO() phonons which \n[2] decay within the pump duration, thereby increasing the crystal -lattice temperature. The \nadditional thermal modulation of a -d-exchange induces  [3a] transfer of angular momentum \nbetween a - and d -spin-sublattices and implies [3b] energy t ransfer from the phonon to the spin \nsystem with the time constant fast=1.6 ps (Fig.  2B). The excess magnetization  of this constrained \nstate decays  by [4] transfer of angular momentum and energy between crystal lattice  and spins \nwith the time constant slow=90 ns, resulting in f ull, unconstrained  equilibrium (Fig.  2C). The \nangular -momentum transfer is mediated by  SO and/or SSMD  interactions22 which do not \nconserve Ma+Md. \n CONCLUSION  \nWe employ  a THz -pump magnetoo ptic-probe experiment to investigate  spin-lattice equilibration \nfrom  the femtosecond  to the microsecond time scale.  Combined with a spin-coupling analysis and \natomistic spin -dynamics simulations, o ur results reveal that the speed of spin -phonon relaxation in \nferrimagnetic insulators critically depends on whether one refers to energy or angular momentum .  \nOn a picosecond scale, spins and phonons reach a quasi -equilibrium through phonon -modulated \nexchange interaction. It induces redistribution  of energy among  phonons and spins , whereas \nangular momentum is only transferred between spins. On a much longer time scale in the \nnanosecond range, the additional transfer of angular  momentum between spins and crystal lattice \nresults in full equilibration. The quasi -equilibrium persists on the intermediate time scale and can \nbe understood as a transient spin state with elevated temperature yet unchanged net \nmagnetization. It could be considered as a realization of magnon populations with nonzero \nchemical potential  which w ere recently introduced in the theory of magnon transport by the spin \nSeeb eck effect19. \nTransfer of angular momentum between spin -sublattices  is also a key process in magnetic \nswitching of ferrimagnetic metalli c alloys  by optical femtosecond laser pulses35,36,37,38. This \nprocess  is considered to follow the preferential ultrafast demagnetization of one of the two  spin-\nsublattices . Note that the flow of angular momentum can proceed through various mechanisms \nsuch as d irect exchange torque36 or nonlocal processes38 including the  transport of spin -polarized \nelectrons.  Our results reveal a new mechanism for manipulating the exchang e interaction on \nultrafast timescales and highlight the important role phonons can play in the direct exchange \ndynamics between spin-sublattices.  \nIn terms of applications, our results suggest  that resonant phonon excitation is a new pathway to \nthe prepar ation of spin states with increased temperature yet unchanged total magnetization. This \nroute may be  particul arly interesting for  the manipulation of antiferromagnetic order2, where spin \nangular momentum is inher ently conserved. Finally, the ultrafast spin -lattice coupling of YIG \nimplies that the magnon temperature follows the phonon  temperature with a delay on the order of \nonly 1  ps, thereby shifting the cutoff frequency of the bulk spin Seebeck effect7 to the THz range.  \n \n  MATERIALS AND METHODS  \nSamples . Our iron garnet films are grown by liquid -phase epitaxy on substrates of gadolinium \ngallium garnet Gd 3Ga5O12 (GGG). To prevent pump absorption by the substrate, several iron-\ngarnet films are transferred on diamond windows after mechanically removing the GGG. Test \nmeasurements confirm that GGG pump absorption is irrelevant to the ultrafast dynamics.  \nWe study two types of garnet samples: pure yttrium iron garnet Y 3Fe5O12 (YIG) and \nbismuth/gallium -substituted iron garnet Bi xY3xGayFe5−yO60 (x=1.53, y=1.33, BiGa:YIG) . They \nhave,  respectively, a Curie temperature of 545  K and 398  K, and in -plane and out -of-plane \nmagnetic anisotropy. Film thicknesses cover a range from 7 to 20  µm. Chemical composition of \nthe garnet films is determ ined by X -ray fluorescence measurements. Substitution of Y by Bi in \nthe BiGa:YIG sample enhances the magnetooptic Faraday effect by one order of magnitude24. \nTypical hysteresis loops obtained by measuring the s tatic Faraday rotation of the probe beam are \nshown in Fig. S1C. \nTHz spectra of the sample absorptance (which equals one minus the reflected and transmitted \npower, both normalized to the incident power) are obtained by combined reflection and \ntransmission m easurements with a broadband THz time -domain spectrometer. Figure  2A shows \nthat maximum absorptance occurs at 21  THz, which is 2  THz above the highest -frequency TO ()-\nphonon resonance26. According to Ref.  26, absorption in this frequency range solely arises from \nTO phonons, while crystal -field and charge -transfer -gap transitions are located at much higher \nfrequencies24. \nExperimental design.  A THz pump pulse is used to resonantly excite long -wavelength TO \nphonons of the iron-garnet film  under study . The instantaneous magnetic state of the sample is \ndetermined by a subsequent optical probe pulse measuring the magnetooptic Fa raday effect. In \nthis way, spin dynamics are monitored over a large range of pump -probe delays ranging from \nfemtoseconds to milliseconds. Details of our setup are shown in Fig. S1. \nTo generate intense THz pump pulses, we employ an amplified Ti:sapphire las er system \ndelivering pulses (energy of 15  mJ, center wavelength of 800  nm, duration of  40 fs, repetition rate \nof 1 kHz) to drive two optical parametric amplifiers that, in turn, generate intense infrared pulses \n(pulse energy of 1.5 mJ and 1.2  mJ, center wa velength of 1280  nm and 1400  nm, duration of 50 fs \nand 50  fs, respectively). By difference -frequency mixing of the two infrared pulses in a nonlinear -\noptical  GaSe crystal21 (thickness of 1  mm), we obtain intens e THz pulses (tunable from 15 to \n60 THz, energy of typically 3 to 20  µJ, pulse duration around 250  fs) with stable carrier -envelope \nphase. For sample excitation, the THz pulse is focused onto the sample surface to a spot with \ndiameter 180  µm. Its transient  electric field \nEt  is measured by electrooptic sampling (see \nbelow). The pulse spectrum is obtained by Fourier transformation of \nEt  and shown in Fig. 2A. \nLow-noise probe pulses (8 fs, 800  nm, 0. 5 nJ) are derived from the pulse train of the Ti:sapphire \nseed laser (repetition rate  of \nrep80MHz f ). They traverse the sample collinearly with the pump \nafter a delay  \nt. To measu re the probe’s transient polarization rotation \n Δt , we employ a \npolarimeter consisting of a Wollaston -prism polarizer followed by two fast photodiodes. \nMeasurement of the probe ellipticity  \n Δt  is accomplished by putting a quarter -wave plate \nbefore the Wollaston prism. The resulting photodiode current is a train of temporally isolated \nelectrical pulses (duration of 5  ns each, pulse -to-pulse distance of \nrep1/ 12.5nsf ), which is \nsampled using a computer -controlled fast analog -digital converter  card39. In this way, we are able \nto measure signals at delays of \nrep τ/t j f  (with integer \nj ) between −100 ns and ~1  ms within a single pump shot and with a delay spacing of \nrep1/ 12.5nsf . The ultrashort offset \nτ  is set \nfrom −2 ps to 200  ps by a variable mechanical delay stage.  \nPump -probe signal traces  \nt  are taken for the sample magnetization saturated in opposite \ndirections using an external magnetic field \nextB  of 15 mT, respectively. To measure samples \nwith in -plane or out -of-plane anisotropy, the magnetic field is oriented 45° with respect to the \nincident pump and probe beams. To vary the sample temperature, the sample is mounted on a \nresistive heater. The sample temperature is measured with two thermocouples and an imaging \ninfrared thermometer, all of them yielding consistent temperature values. To measure the THz \nelectric field, we substitute the sample by a GaSe crystal and measure the transient ellipticity \n Δt E t\n induced by the electrooptic effect21. \nOptionally, our setup also permits measurement of  the transient  isotropic  transmittanc e of the \nsample . For this purpose, we remove all polarization -sensitive components such as the Wollaston \nprism and wave  plate  and detect the probe -pulse energy with one of the two fast photodiodes of \nthe polarimeter.  Based on the small attenuation length26 (~1 µm) of the resonant pump pulse into \nthe iron -garnet sample, we estimate that the temporal blurring of the pump -probe signals due to \npump -probe velocity mismatch is smaller than 10  fs and, therefore, neglig ible. \nSignal analysis.  Up to first order in magnetization, the Faraday rotation  \n (and likewise the \nellipticity \n ) of the probe’s outgoing polarization is a weighted average24 of the magnetization of \nthe two s pin-sublattices \na,dl , that is,  \n.lla M d b \n (3) \nHere, \nla  are the local magnetooptic coupling coefficients  (Verdet constants) of YIG,  \npr llMeM\n is the magnetization of sublattice \nl  projected on the propagation -direction unit \nvector \npre  of the probe, \nd  is the sa mple thickness, and \nb  is an offset arising from any \nmagnetization -independent optical anisotropy. The angular brackets \n  denote averaging over the \nprobed sample volume.  \nIn equilibr ium, the \nla  are constant throughout the probed magnetic volume, \nl l l la M a M    . \nThe impact of the pump pulse makes all quantities in Eq.  (3) time-dependent, resulting in \nrelatively small changes \n0 t  of the signal \n0 2 ps  measured 2  ps before arrival of the \npump pulse. According to Eq.  (3), the nonmagnetic offset \nb  cancels by calculating the \nasymmetric  part \n   00ΔΔΔ Δ Δ .2l l l l t a M M a  \n (4) \nThe first term on the right -hand side scales with a weighted average of the sublattice \nmagnetizations along \npre , which is the quantity we are intereste d in. The second term, however, \nmakes a contribution independent of the \nΔlM  and arises from a possible pump -induced \nmodulation \nΔla  of the Verdet constants. Figure  S3 shows that the second term is negligible and  \nthat \nΔ  reflects the true magnetization dynamics,  \n0 ΔΔllaM , once the pump pulse has \nleft the sample.  Excitation profile and heating . Since the absorption length of the THz pump pulse26 (~1 µm) is \nsmaller than the sample thickness (~10  µm), pump excitation is inhomogeneous along the \nz -axis \nnormal to the film plane. Because the pump -induced signal \n Δt  depends linearly on the \nabsorbed pump fluence \nabsF  (inset of  Fig. 2C), the pump -induced change in the local Faraday \nrotation \n Δ,zt  is proportional to the pump energy density \nabswz  absorbed in the vicinity of \nthe plane at  \nz. In other words, one has   \nabs abs\n0 d d\nF z w z  and \nabs Δ,z t C t w z  where \nCt\n captures the temporal dynamics . As a consequence, the total pump -induced signal  \n abs\n0Δ Δ ,  d  Δ , .d\nt z t d z z t C t F    \n (5) \nis independent of the shape of the absorption profile \nabswz . Therefore, without loss of \ngenerality, we can consider the absorption profile to be homogeneous with \nabs abs / w F d . Our \nargumentation is valid as long as transport of heat from the YIG film into the substrat e is \nnegligible. Figure  S2D shows this assumption is fulfil led for pump -probe delays up to at least \n1 µs. \nAssuming the sample has reached (local) thermal equilibrium at \n1µst , we estimate the average \ntemperature increa se \n Δ 1 µsT  of the BiGa:YIG film by dividing \nabsF  by the pumped sample \nvolume [ d(pump diameter)2\n/4 ], by the mass density and by the specific heat capa city31 of \nYIG.  We obtain a value of \n Δ 1 µs 0.39 KT  for an incident pump energy of  1 µJ, which can be \ninterpreted as the temperature increase of a homogeneously excited YIG film.  \nWe checked effects due to accumulative heating of the sample by reducing the repetition rate of \nthe pump pulses from 1000  Hz to 500  Hz by means of a mechanical chopper. The magnitude  of \npump -induced signal that persists after 1  ms is found to be smal ler than 10% of the pump -induced \nsignal at \n~ 1µst . It corresponds to a static homogeneous temperature increase of  the probing \nvolume of only ~39  mK, which has a negligible influence on our results, including the \ntemperature dependence s een in Fig. 3B.  \nYIG ab initio  calculations.  Our ab initio  phonon calculations are performed using the finite -\ndisplacement method implemented in the open -source package phonopy40. The electronic -\nstructure calculation and equilibrium geometry search of YIG ar e performed within the Density -\nFunctional Theory (DFT) framework, using the Vienna Ab-initio  Simulation Package41 (VASP) \nand adopting the generalized gradient approximation42 of the exchange -correlation functional. \nOur calculations and resulting phonon dispe rsion relations are detailed in the Supplementary \nMaterial .  \nFrom the element -resolved vibrational density of states  (Fig.  S4), we find that a continuum of \nmodes from ~7 to 20  THz contributes to the displacement of the O2- ion with approximately \nfrequency -independent weight. This result implies that the vibration of the O2- ion has a mean \nfrequency of \nOΩ / 2π 14 THz  and a correlation time of \nO10 fs . \nDynamic spin model.  Our consideration of all possible  spin-coupling mechanisms ( SO, SSMD, \nexchange, see main text) leads us to the view that the ultrafast regime of the spin dynamics of \nYIG following phonon excitation arises from isotropic  exchange interaction. More precisely, the \nthermal vibrations of the ionic lattice modulate the a -d-exchange interaction  (Fig.  4A), thereby transferring spin angular momentum from the a - to d-spin-sublattice and quenching each \nsublattice magnetization at the same rate.  \nTo test this hypothesis, we calculate both the rate of spin -angular -momentum transfer between \nsublattices (based on numerical simulations) and the temperature dependence of the sublattice \nmagnetizations (based on an analytical treatment ). Starting point is the Heisenberg spin \nHamiltonian of YIG15, \nB extˆ ˆ ˆ ˆ ,1\n2jj j j j\nll l l l\nll jj ljH J g \n\n    S S B S\n (6) \nwhere \nˆj\nl\nS  is the spin -angular -momentum operator of the total electron spin of an Fe3+ ion (spin \nquantum number \n5 / 2S ) located at site \nj  in spin-sublatti ce \nl. Orbital contributions to the \nmagnetic moments are negligibly small25. In YIG, there are two different crystallographic sites \na,dl\n for the Fe3+ ions which are, respectively, surrounded by six O2- ions (octahedral sites) and \nfour O2- ions (tetrahedral sites). The exchange constants have been determined to be as follows15: \nif for given \nl , \nl, the \nj  and \nj  represent next neighbors, then \njj\nll llJJ\n . Otherwise, \n0jj\nllJ\n . \nThus, the exchange part of the Hamiltonian is fully determined by the three constants \naaJ , \nddJ  \nand \nad daJJ . \nFollowing Eq.  (2), the phonon -induced modulation of the exchange interaction is modeled as \nvariation  \n ad adΔΔu J t J u t\n (7) \nof the exchang e coupling constant between \nal - and \ndl -spin-sublattices. Here, \n/u u   , \nand \nΔu  is the pump -heat-induced deflection of the O2- ion mediating the superexchange of \nadjacent a - and d -type Fe3+ ions, as depicted in Fig. 4A. This assumption is justified becau se \nadJ  \npredominantly involves the a -Fe3+ and d -Fe3+ ions and the O2- ion in between. Since the O2- ion is \nthe by far lightest ion in the YIG unit cell, its motion is expected to modulate the a -d \nsuperexchange most st rongly.  \nAtomistic spin -dynamics simulations.  To calculate the rate of spin -angular -momentum transfer \nbetween sublattices, we conduct numerical simulations based on Eqs.  (6) and (7). For the \nimplementation, we assume an ensemble of classical spins and replace each spin operator \nˆ\niS  by \n1iSS s\n. Here, \nis is a vector of length one , and the compound index \nlj  is summarized in \none index \ni . Likewise, the Hamilton operator \nˆH  turns into the Hamilton function \nH . For the \nexchange constants \naaJ , \nddJ  and \nadJ , we use the values 0.33 meV , 1.15 meV and −3.43 meV, \nrespectively15. \nBy applying Langevin theory, a stochastic Landau -Lifshitz -Gilbert -type equation is derived for \neach spin30,43,44, \n eff , eff , 2.\n1i\nt i i i i i i i\nii\n      s s B s s B\n (8) \nwhere \ni  is the gyromagnetic constant of a spin with magnetic moment \niis . Damping due to \ncoupling to non -spin degrees of freedom (the so -called bath ) is quantified by  \n52 10i . The \neffective field \neff ,iB  acting on  \nis has three contributions: a deterministic part due to the exchan ge fields \n00\nexch, /ii H  Bs  of the adjacent spins for temporally constant \n0\nijJ  and two stochastic \ncomponents \nitξ  and \nexch,Δit B  that arise from coupling of the spins to the bath. The first field \nitξ\n transfers spin angular momentum and energy between the spin system and the bath having \ntemperature \n0T . According to the fluctuation -dissipation theorem, in equilibrium, the fluctuations \nitξ\n are balanced by a friction force, the second, Gilbert -type term in Eq.  (8), thereby driving \nthe spin system into an equilibrium state with temperature \n0T  (Refs.  30, 43, 44). The time \nconstant of this relaxation process is set by the damping parameters \ni  and found to be irrelevant \non the picosecond scale considered here.  \nThe second stochastic field \nexch,ΔΔi ij j\njJ Bs  arises from the phonon -induced modulation of the \nexchange i nteraction (Eq. (7)). It is distinctly di fferent from \nitξ  as it causes energy transfer into \nthe spin system, but leaves the total spin angular momentum unchanged. Since the noise field at \ntemperature \n0T  is already accounted for by \nitξ , the variance of \n ΔijJt  scales with the \ntemperature increase \nΔT  of the crystal lattice induced by the pump pulse. More precisely, owing \nto the equipartition theorem, the pump -induced deflection \nΔu  of the O2- ion obeys \n2 2 2 2\nO O O O BΩ Δ Ω Δ Δm u m u k T   \n, where \nOΩ / 2π 14 THz  is the mean O2--ion vibration \nfrequency (see above).  \nTo determine the initial rate of phonon -driven angular -momentum transfer betwe en the s pin-\nsublattices, phonons and spins are assumed to be thermalized at temperatures \n0ΔTT  and \n0T , \nrespectively  (see main text) . Following our ab initio  result s (see above), \n Δut  is assumed to \nhave a correlation function with a width of \nO10 fs , which is a ccordingly  modeled as  \n B\nO 2\nOOΔΔ Δ .ΩkTu t u t t tm  \n (9) \nThe equation of motion  (Eq. (8)) is integrated numerically for \n3564 2.6 10N    unit cells at \nequilibrium temperature \n0300 KT . At ea ch time step  \nt, we calculate the two sublattice \nmagnetizations \n1/j\nll\njS S N Ms  with \na,dl . In our numerical implementation, the \npump-induced thermal noise of the exchange constants (Eqs. (7) and (9)) can be switched on or \noff over arbitrary time intervals and with variable strength \n2\nadΔuJT . \nResults of our atomistic simulations are shown in Figs. 4B and S5. We find the demagnetization \nrate of each sublattice scales according to  \n2 d0\nad/\nΔtl\nu fMMJT\n (10) \nwith constant \n16 2 2 1 12 10 m J K psf   . On the other hand, our experiment shows that the \nFaraday signal \nΔ  and, thus,  \n0/t l lMM  scale with  the pump fluence (see inset of Fig.  2C) \nand, thus, the initial \nΔT  as well. In other words, we have  00/ Δ/\nΔΔt l l tMMgTT  (11) \nwhere \n3 1 11.9 10 K psg    has been determined from the initial slope of \n00 Δ / Δ /llMM  \n(Fig.  2B) seen in our experimental results. Comparison of Eqs.  (10) and (11) yields the estimate \nad ad / 10uJ g f J  \nÅ -1. \nAccording to the theory of superexchange32, \nadJ  is proportional  to the fourth power of the overlap \nintegral of the Fe3+ and the O2- ion. Therefore, upon changing the distance of the two ions, the \noverlap integral undergoes relative changes on the order of 2.5  Å -1. \nMean -field sublattice magnetization.  Our previous res ults strongly indicate that spins and \nphonons reach quasi -equilibrium for times t≫fast following phonon excitation. As exchange \ninteraction leaves the total spin angular momentum of  the electrons unchanged, this thermal state \nis constrained by the boundar y condition of conserved total spin angular momentum. Such \nconstraints are ubiquitous in thermostatics (such as for a gas contained in a closed bottle) and can \nbe treated by equilibrium statistical physics45.  \nOur goal is to determine the magnetization of t he d- and a-spin-sublattice of YIG in thermal \nequilibrium for two cases: the unconstrained (UC) situation without boundary condition and the \nconstrained (C) situation with the boundary condition of constant total spin angular momentum,  \nconstant. ˆj\nl\nljS\n (12) \nIn both cases, we need to calculate the statistical operator \nˆD  of the system, from which the \nexpectation value \n Tˆˆ ˆr A DA   of an ob servable  \nˆA (such as a sublattice magnetization) can be \nderived. In thermal equilibrium and for both the UC and C case, the statistical operator is given \nby45 \nˆ ˆˆ1expj\nl\nljDHZ   pS\n (13) \nwhere \nB1/kT  with \nBk  being the Boltzmann constant. The partition function \nZ  is determined \nby the normalization condition \nTr ˆ1D . In the UC case, we simply set \n0p  in Eq.  (13), thereby \nresulting in the standard canonical statist ical operator45. In the C case, the constraint of conserved \nspin angular momentum (Eq. (12)) has to be fulfilled by proper choice of the Lagrange \nmultiplier  \np in Eq. (13). Interestingly, \np  can be interpreted as a virtual homogeneous magnetic \nfield that is adjusted such to reinforce the boundary condition of Eq. (12). It is analogous to the \nchemical potential that keeps the number of particles in a giv en macroscopic system constant.  \nThe general Heisenberg spin Hamiltonian \nˆH  of Eq.  (6) is too complex for an analytical \ncalculation of  \nˆD. Consequently, we apply the mean -field appro ximation32, \nˆ ˆ ˆ ˆ ˆ ˆ ˆ  ˆ j j j j j j j j\nl l l l l l l l   \n             S S S S S S S S\n, omit the number \nˆˆjj\nll\nSS , and allow for alignment \nexclu sively along the \nz -axis unit vector \nze , \n,ˆ ˆ ˆj j j\nl l z z l z SS S e e , as set by the external \nmagnetic field \next ext zBBe . In addition, we assume the spins order homogeneously on each \nsublattice  \nl, that is,  \n0 ˆˆj\nllSS   (sublattice approximation32). By evaluating the \njj\nllJ\n  of YIG (see \nabove), we finally obtain the mean -field (MF) Hamiltonian  0\nMF B ext1\n2ˆ ˆ ˆ ˆjj\nll ll l l l\nll j ljH J N S S g B S   \n      (14) \nwhere the \nllJ  are defined as above, and  denotes the number of the next neig hbors a site in \nsublattice  \nl possesses in sublattice  \n'l. For YIG, one has \naa8 N , \ndd4 N , \nad6 N  and \nda4 N  \n(Ref.  15). We can now write the statistical operator of Eq.  (13) in a very compact form as  \nMF1exp ˆ ˆj\nll\nljD F SZ\n (15) \nwhere the effective field  \n0\nB ext1\n2ˆ\nl ll ll l\nlF J N S g B p   \n   \n (16) \ncaptures the mean field (first and second term) and the scalar virtual field \nz ppe  imposed by \nthe constraint of Eq.  (12). By taking advantage of the fact that all spin operators in Eq.  (15) \ncommutate with each other, we obtain the expectation value  \n00 1Tr exp . ˆˆ\nl l l\nlS F SF  \n (17) \nfor any Fe3+ spin of sublattice  \nl. Evaluation of this equation leads to an implicit relationship for \n0ˆ\nlS\n, \n0, ˆ\nl S lS S SF  \n (18) \nwhere \n5 / 2S  and \nS\n is the  Brillouin function (Ref.  32). The magnetization  \nlM of sublattice  \nl \nis proportional to  \n0ˆ\nllS  where \nl  is the number of \nl -Fe3+ ions per unit cell with \nd12  and \na8\n.  \nFor the UC case (\n0p ), Eq.  (18) is a system of two coupled equations and solved numerically, \nthereby yielding the unconstrained \nU0\nCˆ\nlS  for each temperature \n0T . For the C case, the boundary \ncondition of Eq.  (12), rewritten as  \nC0\na C0\nd d a const ˆ ant, ˆSS \n (19) \nadds a third equation to Eq.  (18) which determines  \np. Note that in our experiment, we start from \nan UC equilibrium given by the \nU0\nCˆ\nlS  at temperature \n0T . Subsequently, the pump pulse \nincreases \n0T  by a small amount \nΔT  while keeping the total spin angular momentum constant. \nSince we are only interested in small relative changes of the \n0ˆ\nlS , we linearize Eqs.  (18) and (19) \nin terms of \nΔT  and solve analytically for the small changes \n0\nC Δˆ\nlS .  \nTo be consistent with our atomistic spin -dynamics model (see above), we choose the ratio of the \ncoupling parameters \nadJ , \naaJ  and \nddJ  to be identical to the values given in Ref.  15, but rescaled \nby a factor of 0.36 to fit the experimentally observed critical temperature of \nC398K T . For \ncomparison with our experiment, we finally determine the Farada y rotation resulting from the \nllNcalculated sublattice magnetizations \nlM  (see Eq.  (1)). The results of this procedure are shown and \ndiscussed in Figs. 5A and S6. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nACKNOWLEGMENT  \n \nGeneral : We thank G.E.W.  Bauer and S.M.  Rezende for stimulating discussions .  \nFunding :  \nT.K. acknowledges funding through the grants European Researc h Council H2020 CoG  \n“TERAMAG” (grant no. 681917) and H2020 FET -OPEN project “ASPIN ” (grant no. 766566)  \nand DFG SFB/TRR 227 “Spin Dynamics” (project A05) . I.R. acknowledges  funding through \nGerman Federal Ministry of Education and Research (BMBF) grant 05K16BCA  “Femto -THz-X.” \nP.M. and P.M.O. acknowledge financi al support from the Swedish Research Council  (VR), the K. \nand A. Wallenberg Foundation (grant no. 2015.0060), and the Swedish National  Infrastructure for \nComputing. M.G. thanks for funding through BMBF grant no. 05K10KEB. R.V.P.  acknowledges \nfinancial support from the Russian Scientific  Foundation (grant no. 16 -12-10456).  M.W. \nacknowledges funding by the Max Planck Society.  \nAuthor contributions : I.R. conceived the  original  experiment idea. S.F.M ., I.R. and T.K. \ndesigned the experiment.  S.F.M. built the setup , performed measurement s and analyzed data . I.R., \nA.M.K. and R.V.P. prepared the samples, which were characterized by S.F.M., I.R., A.M.K., A.P. \nand M.G. The th eoretical model was developed by T.K. and S.F.M., with contributions from J.B., \nP.M., A.P. and  P.M.O. Atomistic spin -dynamics simulations were conducted by J.B.  Equilibrium \nmagnetization curves were calculated by S.F.M. and T.K.  Ab initio  calculations of electronic and \nvibrational properties  were performed by P.M. and P.M.O. The manuscript was written by T.K., \nJ.B. and S.F.M. , with contributions from A.P. , P.M.O., P.M. , M.W.  and I.R . All authors \ncontributed to the discussion s of the results and commented on  the manuscript .  \n \n FIGURES  \n \n \n \nFig. 1. Ultrafast probing of spin -phonon interaction s. (A) Experimental principle. A THz \npump pulse resonantly and exclusively excites optical phonons of a ferrimagnet. The impact \non the sample magnetization is monitored by the Faraday rotation  θ of a subsequent \nfemtosecond probe pulse. By using an electric insulator, the electron ic orbital degrees of \nfreedom remain unexcited (see red crosses). ( B) Part of the unit cell of ferrimagnetic YIG. \nMagnetic Fe3+ ions at tetrahedral d -sites and octahedra l a-sites comprise, respectively, the \nmajority and minority spin-sublattice of the ferrimagnet. The pump pulse resonantly excites a \nTO(Γ) optical phonon associated with a Fe -O stretch vibration at the tetrahedral d -site. \n \n \n \n \nFig. 2. Ultrafast phonon -induced dynamics of magnetic order.  (A) THz absorptance of the \nBiGa:YIG film (black solid line) . Pump intensity spectra  are either resonant (red) or non-resonant \n(blue) with the TO( Γ) phonon absorption band. Open circles show the pump -induced Faraday \nsignal 10 ps after sample excitation as a function of the pump pulse center frequency. ( B) Pump -\ninduced change Δθ in Faraday rotation for resonant and off -resonant pumping on ultrafast and \n(C) microsecond time scales normalized to the equilibrium Faraday signal θ0=θ(−2 ps). The \nincident fluence is 10  mJ cm-2. Panel  (B) also shows the  isotropic transient change in the sample \ntransmittance for resonant pumping  (thin black  solid line) . Dashed lines in  panels  (B) and (C) are \nsingle -exponential fits with time constants  of fast=1.6 ps and slow=90 ns, respectively. The inset \nof panel  (C) displays the ultrafast Faraday signal Δθ(10 ps) as a function  of the incident pump -\npulse fluence. Data are taken at a temperature of 296  K. \n \n \n \n \n \n \n \nFig. 3. Tw o regimes of spin -lattic e equilibration. (A) Equilibrium Faraday rotation \nθ0=θ(−2 ps)  vs ambient temperature  T0 along with a fit to an analytical function (thin solid \nline). ( B) Pump -induced change Δθ in the Faraday rotation at t=10 ps (red symbols) and 1  µs \n(blue) after pump -pulse arrival. The black curve is the change  (∂θ0/∂T0)ΔT in the Faraday  \nrotation expected from the increase ΔT of the sample temperature due to heating by the pump \npulse. θ0(T0) is taken from panel (A) (thin solid line), and ΔT=0.39 K  is calculated from the \nabsorbed pump energy and the heat capacity of the excited volume.  \n \n \n  \n1\n0.8\n0.6\n0.4\n0.2\n00(T0)0(296 K)\n420 400 380 360 340 320\nTemperature T0 (K)Equilibrium\nFaraday signal 0(T0)TcA\nB\n12\n10\n8\n6\n4\n2\n0Faraday rotation  0(296 K)   (10-3)\n420 400 380 360 340 320\nTemperature T0 (K)    at 1 µs\n    at 10 ps\n \n  (0T0) T,\n     T  0.39 K \n \n \nFig. 4. Atomistic spin -dynamics simulations . (A) Schematic of our model of ultrafast spin -\nphonon coupling. Thermal motion of the O2- ion modulates the superexchange constant  Jad of \nthe adjacent a -Fe3+ and d -Fe3+ spins, thereby enabling transfer of spin angular momentum \nbetween a - and d -spin-sublattices. (B) Evolution of a - and d -spin-sublattice magnetizations \nas obtained by atomistic spin -dynamics simulations. From 0 to 0.5  ps, ther mal modulation of \nthe exchange constant  Jad is switched on (orange square ). The variance of th e Jad fluctuation \nis proportional to the difference ΔT between crystal -lattice and spin temperature. To obtain \nagreement of the slope of ΔMa(t) with that found in  the experiment directly after pump \nexcitation ( −0.1% ps-1, see Fig.  2B) where Δ T=0.39 K (see Fig.  3B), an approximately three  \ntimes smaller  ∂Jad/∂u of ~10 Jad Å-1 than used here has to be chosen.  \n \n  \n \n \nFig. 5.  Constrained state and  spin-phonon  equilibra tion in YIG. (A) Calculated change in \nsublattice magnetization  Ma and Md per increase of temperature  T0, without and with the \nconstraint of constant spin angular momentum.  The constrained and unconstrained ∂Md/∂T0 \ncurves exhibit good qualitative agreement with the measured Faraday signals Δθ(10 ps) and \nΔθ(1 µs) vs T0 shown in Fig.  3B. (B) Schematic of spin -phonon equilibration in YIG. \n[1] Pump -excited TO(Γ ) phonons [2]  increase the population of other lattice modes. The \nincreased thermal modulation   of the a-d-exchange by ΔJad(t) leads to [3a]  transfer of angular \nmomentum between a - and d -spin-sublattices , accompanied by [3b]  energy transfer from the \nphonon to the spin system on the time scale  fast=1.6 ps. The resulting state is constrained by \nΔMa+ΔMd=0 and decays by [4]  transfer of angular momentum and energy bet ween crystal \nlattice  and electron spins on the slow=90 ns scale. 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Renormalized spin-wave theory, which accounts\nfor the magnon-magnon interaction, and its extension are de veloped to describe the two ferrimag-\nnetic phases (0 ,T∗) and (T∗,TN) in the system, and to calculate the magnetization as a funct ion of\ntemperature.\nThe influence of the parameters in the theory on the character istic temperatures TNandT∗is\nstudied. It is shown that, increasing the inter-sublattice exchange interaction, the ratio TN/T∗>1\ndecreases approaching one, and above some critical value of the exchange constant there is only\none phase TN=T∗, and the magnetization-temperature curve has the typical C urie-Weiss profile.\nWhen the intra-exchange constant of sublattice with strong er intra-exchange interaction increases\ntheNe`eltemperature increases while T∗remains unchanged. Finally, when the magnetic order of\nthe sublattice with smaller magnetic order decreases, T∗decreases. The theoretical predictions are\nutilize to interpret the experimentally measured magnetiz ation-temperature curves.\nPACS numbers: 75.50.Bb, 75.30.Ds, 75.60.Ej, 75.50.-y\nI. INTRODUCTION\nThe present paper is a sequel to the paper [1]. A two-\nsublattice ferrimagnet, with spin- s1operators S1iat the\nsublattice Asite and spin- s2operators S2iat the sub-\nlatticeBsite. The true magnons of a two-spin system\nare transversal fluctuations of the total magnetization\nwhich includes both the magnetization of the sublattice\nAandBspins. The magnon excitation is a complicate\nmixture of the transversal fluctuations of the sublattice\nAandBspins. As a result the magnons’ fluctuations\nsuppress, in different way, the magnetic orders on the\ndifferent sublattices and one obtains two phases. At low\ntemperature (0 ,T∗) the magnetic orders of the Aand\nBspins contribute to the magnetization of the system,\nwhile at the high temperature ( T∗,TN) the magnetiza-\ntion of the spins with a weaker intra-sublattice exchange\nis suppressed by magnon fluctuations, and only the spins\nwiththestrongerintra-sublatticeexchangehavenon-zero\nspontaneous magnetization.\nRenormalizedspin-wavetheory, which accountsforthe\nmagnon-magnon interaction, and its extension are de-\nveloped to describe the two ferrimagnetic phases in the\nsystem and to calculate the magnetization as a function\nof temperature. It is impossible to require the theoret-\nically calculated N´eeltemperature and magnetization-\ntemperaturecurvestobe in exactaccordancewith exper-\nimental results. The models are idealized, and they do\nnotconsidermanyimportanteffects: phononmodes, sev-\neral types of disorder, Coulomb interaction, etc. Because\nof this it is important to formulate theoretical criteria for\nadequacy of the method of calculation. In my opinion\nthe calculations should be in accordance with Mermin-\nWagner theorem [2]. It claims that in two dimensions\nthere is not spontaneous magnetization at non-zero tem-\nperature. Hence, thecriticaltemperatureshouldbeequal\nto zero. It is well known that the Monte Carlo methodof calculation does not satisfy this criteria, and ”weak\nz-coupling” 3D system is used to mimic a 2D layer. It is\ndifficult within Dynamical Mean-Field Theory (DMFT)\nto make a difference between two dimensional and three\ndimensional systems. DMFT is a good approximation\nwhen the dimensionality goes to infinity. The present\nmethods of calculation, being approximate, capture the\nbasic physical features and satisfy the Mermin-Wagner\ntheorem.\nThere is an important difference between N´eeltheory\n[3] and the results in the present paper. N´eel’s calcula-\ntions predict a temperature TNat which both the sublat-\nticeAandBmagnetizations become equal to zero and\nT∗is a temperature at which the magnetic moment has\na maximum.\nThe influence of the parameters in the theory on the\ncharacteristic temperatures TNandT∗is studied. It\nis shown that, increasing the inter-sublattice exchange\ninteraction, the ratio TN/T∗>1 decreases approach-\ning one, and above some critical value of the exchange\nconstant there is only one phase TN=T∗, and the\nmagnetization-temperature curve has the typical Curie-\nWeiss profile. When the intra-exchange constant of the\nsublattice with stronger intra-exchange interaction in-\ncreases the Ne`eltemperature increases while T∗remains\nunchanged. Finally, when the magnetic order of the\nsublattice with smaller magnetic order decreases, T∗de-\ncreases.\nTo compare the theoretical results and the experimen-\ntalmagnetization-temperaturecurvesonehas, firstofall,\nto interpretadequatelythe measurements. The magnetic\nmoments in some materials are close to ”spin only” value\n2µBSand the sublattice spins s1ands2can be obtained\nfrom the experimental curves. As an example I consider\nthesulpho-spinel MnCr 2S4−xSex[4]. Onthe otherhand\nthere are ferrimagnets with strong spin-orbital interac-\ntion. Itisconvenient,inthatcase,toconsider jjcoupling2\nwithJA=LA+SAandJB=LB+SB. As an example\nI consider the vanadium spinel MnV2O4[5, 6, 7, 8].\nThe paper is organized as follows. In Sec. II the\nmodel is presented and a renormalized spin-wave theory\nisworkedouttocalculatethemagnetization-temperature\ncurves for different parameters of the model. The influ-\nence of the theory parameters on the N´eelandT∗tem-\nperatures is studied in Sec. III. I consider three cases:\ni) when the inter-sublattice exchange constant increases\nandallthe otherparametersarefixed, ii) oneoftheintra-\nsublattice parameters is changed and iii) when one of the\nspins decreases. Applications and analyzes of experimen-\ntal magnetization-temperature curves are given in Sec.\nIV. A summary in Sec. V concludes the paper.\nII. SPIN-WAVE THEORY\nA. Renormalized spin-wave (RSW) theory\nThe Hamiltonian of the system is\nH=−J1/summationdisplay\n≪ij≫AS1i·S1j−J2/summationdisplay\n≪ij≫BS2i·S2j\n+J/summationdisplay\n/angbracketleftij/angbracketrightS1i·S2j (1)\nwhere the sums are over all sites of a three-dimensional\ncubic lattice: ∝angbracketlefti,j∝angbracketrightdenotes the sum over the nearest\nneighbors, ≪i,j≫Adenotes the sum over the sites of\nthe A sublattice, ≪i,j≫Bdenotes the sum over the\nsites of the B sublattice. The first two terms describe\nthe ferromagnetic Heisenberg intra-sublattice exchange\nJ1>0,J2>0, while the third term describes the inter-\nsublattice exchangewhichisantiferromagnetic J >0. To\nstudy a theory with the Hamiltonian Eq.(1) it is conve-\nnient to introduce Holstein-Primakoff representation for\nthe spin operators\nS+\n1j=S1\n1j+iS2\n1j=/radicalBig\n2s1−a+\njajaj\nS−\n1j=S1\n1j−iS2\n1j=a+\nj/radicalBig\n2s1−a+\njaj(2)\nS3\n1j=s1−a+\njaj\nwhen the sites jare from sublattice Aand\nS+\n2j=S1\n2j+iS2\n2j=−b+\nj/radicalBig\n2s2−b+\njbj\nS−\n2j=S1\n2j−iS2\n2j=−/radicalBig\n2s2−b+\njbjbj(3)\nS3\n2j=−s2+b+\njbj\nwhen the sites jare from sublattice B. The operators\na+\nj, ajandb+\nj, bjsatisfytheBosecommutationrelations.\nIn terms of the Bose operators and keeping only the\nquadratic and quartic terms, the effective Hamiltonian\nEq.(1) adopts the form\nH=H2+H4 (4)where\nH2=s1J1/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\n+s2J2/summationdisplay\n≪ij≫B/parenleftbig\nb+\nibi+b+\njbj−b+\njbi−b+\nibj/parenrightbig\n(5)\n+J/summationdisplay\n/angbracketleftij/angbracketright/bracketleftbig\ns1b+\njbj+s2a+\niai−√s1s2/parenleftbig\na+\nib+\nj+aibj/parenrightbig/bracketrightbig\nH4=1\n4J1/summationdisplay\n≪ij≫A/bracketleftbig\na+\nia+\nj(ai−aj)2+(a+\ni−a+\nj)2aiaj/bracketrightbig\n+1\n4J2/summationdisplay\n≪ij≫B/bracketleftbig\nb+\nib+\nj(bi−bj)2+(b+\ni−b+\nj)2bibj/bracketrightbig\n+1\n4J/summationdisplay\n/angbracketleftij/angbracketright/bracketleftbigg/radicalbiggs1\ns2/parenleftbig\naib+\njbjbj+a+\nib+\njb+\njbj/parenrightbig\n(6)\n+/radicalbiggs2\ns1/parenleftbig\na+\niaiaibj+a+\nia+\niaib+\nj/parenrightbig\n−4a+\niaib+\njbj/bracketrightbigg\nand the terms without operators are dropped.\nThe next step is to represent the Hamiltonian in the\nHartree-Fock approximation\nH≈HHF=Hcl+Hq (7)\nwhere\nHcl= 12NJ1s2\n1(u1−1)2+12NJ2s2\n2(u2−1)2\n+ 6NJs1s2(u−1)2, (8)\nH2=s1J1u1/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\n+s2J2u2/summationdisplay\n≪ij≫B/parenleftbig\nb+\nibi+b+\njbj−b+\njbi−b+\nibj/parenrightbig\n(9)\n+Ju/summationdisplay\n/angbracketleftij/angbracketright/bracketleftbig\ns1b+\njbj+s2a+\niai−√s1s2/parenleftbig\na+\nib+\nj+aibj/parenrightbig/bracketrightbig\nandN=NA=NBis the number of sites on a sublattice.\nEquation (9) shows that the Hartree-Fock parameters\nu1, u2andurenormalize the intra-exchange constants\nJ1, J2and the inter-exchange constant J, respectively.\nIt is convenient to rewrite the Hamiltonian in momen-\ntum space representation\nHq=/summationdisplay\nk∈Br/bracketleftbig\nεa\nka+\nkak+εb\nkb+\nkbk−γk/parenleftbig\na+\nkb+\nk+bkak/parenrightbig/bracketrightbig\n,\n(10)\nwhere the wave vector kruns over the reduced first Bril-\nlouinzone Brofacubiclattice. Thedispersionsaregiven\nby equalities\nεa\nk= 4s1J1u1εk+ 6s2Ju\nεb\nk= 4s2J2u2εk+ 6s1J u (11)\nγk= 2J u√s1s2(coskx+cosky+coskz)3\nwith\nεk= 6−cos(kx+ky)−cos(kx−ky)−cos(kx+kz)\n−cos(kx−kz)−cos(ky+kz)−cos(ky−kz).(12)\nTo diagonalize the Hamiltonian one introduces new Bose\nfieldsαk, α+\nk, βk, β+\nkby means of the transformation\nak=ukαk+vkβ+\nka+\nk=ukα+\nk+vkβk\n(13)\nbk=ukβk+vkα+\nkb+\nk=ukβ+\nk+vkαk\nwhere the coefficients of the transformation ukandvk\nare real function of the wave vector k\nuk=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nεa\nk+εb\nk/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk+ 1\n\n(14)\nvk=sign(γk)/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nεa\nk+εb\nk/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−1\n\nThe transformed Hamiltonian adopts the form\nHq=/summationdisplay\nk∈Br/parenleftBig\nEα\nkα+\nkαk+Eβ\nkβ+\nkβk+E0\nk/parenrightBig\n,(15)\nwith new dispersions\nEα\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−εb\nk+εa\nk/bracketrightbigg\n(16)\nEβ\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk+εb\nk−εa\nk/bracketrightbigg\nand vacuum energy\nE0\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−εb\nk−εa\nk/bracketrightbigg\n(17)\nFor positivevalues ofthe Hartree-Fockparametersand\nall values of k∈Br, the dispersions are nonnegative\nEα\nk≥0, Eβ\nk≥0. For definiteness I choose s1> s2.\nWith these parameters, the αkboson is the long-range\n(magnon) excitation in the two-spin system with Eα\nk∝\nρk2, near the zero wavevector, while the βkboson is a\ngapped excitation.\nTo obtain the system ofequations for the Hartree-Fock\nparameters we consider the free energy of a system with\nHamiltonian HHFequations (8) and (15)\nF= 12J1s2\n1(u1−1)2+12J2s2\n2(u2−1)2\n+ 6Js1s2(u−1)2+1\nN/summationdisplay\nk∈BrE0\nk (18)\n+1\nβN/summationdisplay\nk∈Br/bracketleftBig\nln/parenleftBig\n1−e−βEα\nk/parenrightBig\n+ ln/parenleftBig\n1−e−βEβ\nk/parenrightBig/bracketrightBig\n.whereβ= 1/Tis the inverse temperature. Then the\nthree equations\n∂F/∂u1= 0, ∂F/∂u2= 0, ∂F/∂u= 0 (19)\nadopt the form (see the appendix)\nu1= 1−1\n6s11\nN/summationdisplay\nk∈Brεk/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\nu2= 1−1\n6s21\nN/summationdisplay\nk∈Brεk/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\nu= 1−1\nN/summationdisplay\nk∈Br/bracketleftbigg1\n2s1/parenleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/parenrightBig\n(20)\n+1\n2s2/parenleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/parenrightBig\n−2\n3Ju/parenleftBig\n1+nα\nk+nβ\nk/parenrightBig(coskx+cosky+coskz)2\n/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk\n\nwherenα\nkandnβ\nkare the Bose functions of αandβ\nexcitations. The Hartree-Fock parameters, the solution\nof the system of equations (20), are positive functions\nofT/J,u1(T/J)>0, u2(T/J)>0 andu(T/J)>0.\nUtilizing these functions, one can calculate the sponta-\nneous magnetization of the system, which is a sum of\nthe spontaneous magnetization on the two sublattices\nM=MA+MB, where\nMA=< S3\n1j> j is from sublattice A\n(21)\nMB=< S3\n2j> j is from sublattice B\nIn terms of the Bose functions of the αandβexcitations\nthey adopt the form\nMA=s1−1\nN/summationdisplay\nk∈Br/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\n(22)\nMB=−s2+1\nN/summationdisplay\nk∈Br/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\nThemagnonexcitation- αkin the effectivetheoryequa-\ntion(15)-isacomplicatedmixtureofthetransversalfluc-\ntuations of the AandBspins. As a result the magnons’\nfluctuationssuppressinadifferentwaythemagnetization\non sublattices AandB. Quantitatively this depends on\nthe coefficients ukandvkin equations (22). At char-\nacteristic temperature T∗spontaneous magnetization on\nsublattice Bbecomes equal to zero, while spontaneous\nmagnetization on sublattice Bis still nonzero. Above\nT∗the system of equations (20) has no solution and one\nhas to modify the spin-wave theory. The magnetization\ndepends on the dimensionless temperature T/Jand di-\nmensionless parameters s1, s2, J1/JandJ2/J. For pa-\nrameters s1= 1.5, s2= 1, J1/J= 0.94 andJ2/J= 0.01\nthe functions MA(T/J) andMB(T/J) are depicted in\nfigure 1. The upper (blue) line is the sublattice Amag-\nnetization, the bottom (red) line is the sublattice Bmag-\nnetization.4\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52\n/s45/s49/s44/s48/s45/s48/s44/s53/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53\n/s32/s77/s65\n/s77/s66/s84/s47/s74/s115\n/s49/s61/s49/s46/s53/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s115\n/s50/s61/s49\n/s74\n/s49/s47/s74/s61/s48/s46/s57/s52/s32/s32/s32/s32/s32/s32/s74\n/s50/s47/s74/s61/s48/s46/s48/s49/s83/s80/s79/s78/s84/s65/s78/s69/s79/s85/s83/s32/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78/s84/s42\n/s47/s74\nFIG. 1: (color online)The spontaneous magnetization MA-\nupper (blue) line and MB-bottom (red) line as a function of\nT/Jfor parameters s1= 1.5, s2= 1, J1/J= 0.94 and\nJ2/J= 0.01.T∗is the temperature at which sublattice B\nmagnetization becomes equal to zero\nB. Modified RSW theory\nOnce suppressed, the sublattice Bmagnetization can-\nnot be restored increasing the temperature above T*. To\nformulate this mathematically we modify the spin-wave\ntheory using the idea of a description of the paramag-\nnetic phase of 2D ferromagnets ( T >0) by means of\nmodified spin-wave theory [10, 11] and its generalization\n[1]. We consider a two-sublattice system and, to enforce\nthe magnetization on the two sublattices to be equal to\nzero in paramagnetic phase, we introduce two parame-\ntersλAandλB[1]. The new Hamiltonian is obtained\nfrom the old one equation (1) by adding two new terms:\nˆH=H−/summationdisplay\ni∈Aλ1S3\n1i+/summationdisplay\ni∈Bλ2S3\n2i(23)\nIn momentum space the new Hamiltonian adopts the\nform\nˆH=/summationdisplay\nk∈Br/bracketleftbig\nˆεa\nka+\nkak+ ˆεb\nkb+\nkbk−γk(bkak+b+\nka+\nk)/bracketrightbig\n(24)\nwhere the new dispersions are\nˆεa\nk=εa\nk+λ1,ˆεb\nk=εb\nk+λ2.(25)\nUtilizing the same transformation equations (13) with\nparameters\nˆuk=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nˆεa\nk+ ˆεb\nk/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk+ 1\n\n(26)\nˆvk=sign(γk)/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nˆεa\nk+ ˆεb\nk/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk−1\none obtains the Hamiltonian in diagonal forma\nˆH=/summationdisplay\nk∈Br/parenleftBig\nˆEα\nkα+\nkαk+ˆEβ\nkβ+\nkβk+ˆE0\nk/parenrightBig\n,(27)\nwhere\nˆEα\nk=1\n2/bracketleftbigg/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk−ˆεb\nk+ ˆεa\nk/bracketrightbigg\nˆEβ\nk=1\n2/bracketleftbigg/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk+ ˆεb\nk−ˆεa\nk/bracketrightbigg\n(28)\nˆE0\nk=1\n2/bracketleftbigg/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk−ˆεb\nk−ˆεa\nk/bracketrightbigg\nIt is convenient to represent the parameters λ1andλ2\nin the form\nλ1= 6Jus2(µ1−1), λ2= 6Jus1(µ2−1).(29)\nIn terms ofthe new parameters µ1andµ2the dispersions\nˆεa\nkand ˆεb\nkadopt the form\nˆεa\nk= 4s1J1u1εk+ 6s2J uµ1\n(30)\nˆεb\nk= 4s2J2u2εk+ 6s1J uµ2\nThey are positive (ˆ εa\nk>0, ˆεb\nk>0) for all values of the\nwavevector k, if the parameters µ1andµ2are positive\n(µ1>0, µ2>0). The dispersions Eq.(28) are well de-\nfined if square-roots in equations (28) are well defined.\nThis is true if\nµ1µ2≥1. (31)\nTheβkexcitation is gapped ( Eβ\nk>0) for all values of\nparameters µ1andµ2which satisfy equation (31). The\nαexcitation is gapped if µ1µ2>1, but in the particular\ncase\nµ1µ2= 1 (32)\nˆEα\n0= 0, and near the zero wavevector\nˆEα\nk≈ˆρk2(33)\nwith spin-stiffness constant\nˆρ=8(s2\n2J2u2µ1+s2\n1J1u1µ2) + 2s1s2Ju\n(s1µ2−s2µ1)(34)\nIn the particular case equation (32) αkboson is the long-\nrange excitation (magnon) in the system.\nWeintroducedtheparameters λ1andλ2(µ1,µ2)toen-\nforce the sublattice AandBspontaneousmagnetizations\nto be equal to zero in the paramagnetic phase. We find\nouttheparameters µ1andµ2, aswellastheHartree-Fock\nparameters, as functions of temperature, solving the sys-\ntem of five equations, equations (20) and the equations\nMA=MB= 0, where the spontaneous magnetizations5\nhave the same representation as equations (22) but with\ncoefficients ˆ uk,ˆvk, and dispersions ˆEα\nk,ˆEβ\nkin the ex-\npressions for the Bose functions. The numerical calcula-\ntions show that for high enough temperature µ1µ2>1.\nWhen the temperature decreases the product µ1µ2de-\ncreases, remaining larger than one. The temperature at\nwhich the product becomes equal to one ( µ1µ2= 1) is\ntheN´eeltemperature. Below TN, the spectrum contains\nlong-range(magnon) excitations, thereupon µ1µ2= 1. It\nis convenientto representthe parametersin the following\nway:\nµ1=µ, µ 2= 1/µ. (35)\nIn the ordered phase magnon excitations are the ori-\ngin of the suppression of the magnetization. Near the\nzero temperature their contribution is small and at zero\ntemperature sublattice AandBspontaneous magneti-\nzation reach their saturation. On increasing the tem-\nperature magnon fluctuations suppress the sublattice A\nmagnetization and sublattice Bmagnetization in differ-\nent ways. At T∗the sublattice Bspontaneous magne-\ntization becomes equal to zero. Increasing the tempera-\nture above T∗, the sublattice Bmagnetization should be\nzero. This is why we impose the condition MB(T) = 0\nifT > T∗. For temperatures above T∗, the parameter µ\nand the Hartree-Fock parameters are solution of a sys-\ntem of four equations, equations (20) and the equation\nMB= 0. The Hartree-Fockparameters, as a functions of\ntemperature T/J, are depicted in figure 2 for parameters\ns1= 1.5, s2= 1, J1/J= 0.94 andJ2/J= 0.01. The\nvertical dotted (green) line corresponds to T∗/J.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51/s48/s44/s52/s48/s44/s53/s48/s44/s54/s48/s44/s55/s48/s44/s56/s48/s44/s57/s49/s44/s48/s49/s44/s49\n/s117\n/s49\n/s117\n/s117\n/s50\n/s84/s47/s74/s72/s65/s82/s84/s82/s69/s69/s45/s70/s48/s67/s75/s32/s80/s65/s82/s65/s77/s69/s84/s69/s82/s83\nFIG. 2: (color online)Hartree-Fock parameters u1,u2andu\nas a function of T/Jfors1= 1.5, s2= 1, J1/J= 0.94 and\nJ2/J= 0.01. The vertical dotted (green) line corresponds to\nT∗/J\nThe function µ(T/J) is depicted in figure 3 for the\nsame parameters.\nWe utilize the obtained function µ(T),u1(T),u2(T),\nu(T) to calculate the spontaneous magnetization as a/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48\n/s84/s42\n/s47/s74/s84\n/s78/s47/s74\nFIG. 3: (color online) µ(T/J) for parameters s1= 1.5, s2=\n1, J1/J= 0.94andJ2/J= 0.01. Theverticaldotted(green)\nline corresponds to T∗/J, while (red) dashed lines to TN/J\nandµ(TN/J).\nfunction of the temperature. Above T∗, the magneti-\nzation of the system is equal to the sublattice Amagne-\ntization. For the same parameters as above the functions\nMA(T/J) andMB(T/J) are depicted in figure 4a. The\nupper (blue) line is the sublattice Amagnetization, the\nbottom (red) line is the sublattice Bmagnetization. The\ntotal magnetization M=MA+MBis depicted in fig-\nure 4b.\nIII.TNANDT∗DEPENDENCE ON MODEL’S\nPARAMETERS\nThe existence of two ferromagnetic phases (0 ,T∗) and\n(T∗,TN) is a generic feature of two spin systems. The\ncharacteristic temperatures TNandT∗strongly depend\non the parameters of the model. Intuitively, it is clear\nthat, if the inter-exchange is much stronger than intra-\nexchanges, the ferromagnetic order sets in simultane-\nously on both sublattices. This is not true, if inter-\nexchange is not so strong. To demonstrate this I study\na system with sublattice Aspins1= 1.5, and sublat-\nticeBspins2= 1. For parameters J1/J= 0.5 and\nJ2/J= 0.005 the magnetization-temperature curve is\ndepicted in FIG.5 curve ”c”. The ratio of the charac-\nteristic temperatures equals TN/T∗= 1.722. Increasing\nthe inter-exchange coupling, J1/J= 0.3,J2/J= 0.003\n(curve ”b”), the ratio decreases TN/T∗= 1.229, and\nabove some critical value of the inter-exchange constant\nJ1/J= 0.05,J2/J= 0.0005N´eel’stemperature becomes\nequal to T∗. There is only one ferromagnetic phase, and\nmagnetization-temperature curve ”a” is a typical Curie-\nWeiss curve. Despite this the system does not describe\nferromagnet, because the spin wave excitations are su-\nperposition of the sublattice AandBspin excitations.6\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53\n/s45/s49/s44/s48/s45/s48/s44/s56/s45/s48/s44/s54/s45/s48/s44/s52/s45/s48/s44/s50/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\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s49/s44/s52/s49/s44/s54/s32/s77/s65\n/s77/s66/s84/s47/s74\n/s115\n/s49/s61/s49/s46/s53/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s115\n/s50/s61/s49\n/s74\n/s49/s47/s74/s61/s48/s46/s57/s52/s32/s32/s32/s32/s32/s32/s74\n/s50/s47/s74/s61/s48/s46/s48/s49/s83/s80/s79/s78/s84/s65/s78/s69/s79/s85/s83/s32/s77 /s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78/s97\n/s84/s47/s74 /s84/s42\n/s47/s74/s98\n/s77/s65\n/s43/s77/s66\nFIG. 4: (color online)a) The sublattice Aspontaneous mag-\nnetization MA-upper (blue) line and sublattice Bsponta-\nneous magnetization MB-bottom (red) line as a function of\nT/Jfor parameters s1= 1.5, s2= 1, J1/J= 0.94 and\nJ2/J= 0.01.\nb) The total spontaneous magnetization MA+MB.T∗/J-\nvertical dotted (green) line\nNext, I consider a system with sublattice Aspins1=\n1.5, and sublattice Bspins2= 1. The ratio of sublat-\nticeBexchange constant J2and inter-exchange constant\nJis fixedj2=J2/J= 0.01, while the ratio j1=J1/J\nvaries. When the sublattice Aexchange constant J1in-\ncreasesj1=J1/J= 0.64,0.84,0.94, the magnetization-\ntemperature curve at temperatures below T∗does not\nchange. There is no visible difference between T∗tem-\nperatures for the three values of the parameter J1/J.\nThe difference appears when the temperature is above\nT∗. Increasing sublattice Aexchange constat increases\ntheN´eeltemperature. The three curves are depicted in\nfigure 6.\nFinally, I consider three systems with equal exchange\nconstants J1/J= 0.4,J2/J= 0.004 and sublattice A\nspins1= 4, but with three different sublattice Bspins\n(figure 7). The calculations show that decreasing the\nsublattice Bspindecreases T∗temperature, increasesthe\nmaximum of magnetization at T∗and zero temperature/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48\n/s97\n/s98/s99\n/s84/s47/s74\n/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78\n/s91\n/s93\nFIG. 5: (color online) The magnetization 2 MA+ 2MBas\na function of T/Jfors1= 1.5 ands2= 1, curve a:J1/J=\n0.05,J2/J= 0.0005, curve b:J1/J= 0.3,J2/J= 0.003,\ncurvec:J1/J= 0.5,J2/J= 0.005.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48/s106\n/s49/s61/s48/s46/s57/s52/s32/s32/s84\n/s78/s47/s84/s42/s61/s51/s46/s48/s48/s52\n/s106\n/s49/s61/s48/s46/s56/s52/s32/s32/s84\n/s78/s47/s84/s42/s61/s50/s46/s54/s57/s50\n/s106\n/s49/s61/s48/s46/s54/s52/s32/s32/s84\n/s78/s47/s84/s42/s61/s50/s46/s48/s56/s57\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s106\n/s50/s61/s48/s46/s48/s49\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s115\n/s49/s61/s49/s46/s53/s32/s32/s115\n/s50/s61/s49\n/s84/s47/s74/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78 /s91 /s93\nFIG. 6: (color online) The magnetization 2 MA+ 2MBas a\nfunction of T/Jfors1= 1.5,s2= 1,j2=J2/J= 0.01 and\nthree values of the parameter j1=J1/J;j1= 0.94 (black)\nsquares, j1= 0.84 (red) circles, j1= 0.64 (blue) triangles\nmagnetization.7\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56\n/s84/s47/s74\n/s84\n/s99/s42\n/s47/s74/s84\n/s98/s42\n/s47/s74/s84\n/s97/s42\n/s47/s74/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78 /s91\n/s66/s93/s97\n/s98\n/s99\nFIG. 7: (color online) The magnetization 2 MA+ 2MBas\nfunction of T/JforJ1/J= 0.4,J2/J= 0.004,s1= 4, and\ns2= 0.5-curvea(green), s2= 1.5-curveb(red),s2= 2.5-\ncurvec(black)\nIV. THEORY AND EXPERIMENT\nA. Sulpho-spinel MnCr 2S4−xSex\nThe sulpho-spinel MnCr 2S4−xSexhas been investi-\ngated by measurements of the magnetization at 15 .3kOe\nas a function of temperature (figure 94 in [4]). The max-\nimum in the magnetization versus temperature curve,\nwhich is typical of MnCr 2S4(x= 0), increases when\nxincrease, and disappears at x= 0.5. TheN´eeltem-\nperature decreases from 74 Katx= 0 to 56 Katx= 2.\nThe authors’ conclusion is that the observed change of\nthe magnetic properties is attributed to a decrease of the\nstrength of the negative Mn2+−Cr3+superexchange\ninteraction with increasing Seconcentration.\nWe obtained, see figure 5, that the maximum of the\nmagnetization is at T∗. Above T∗the magnetization of\nthe system is equal to the magnetization of sublattice A\nspins. If we extrapolate this curvebelow T∗down to zero\ntemperature we will obtain a value close to 2 s1µB, where\ns1is the spin of the sublattice Aspin operators. The\nexperimentalfigures[4]showthatextrapolationsgiveone\nand the same result for all values of x. One can accept\nthe fact that the Seconcentration do not influence over\nthe value of sublattice Aspin and s1= 1.5.\nBelowT∗the magnetization is a sum of sublattice\nAandBmagnetization. Hence, the magnetization at\nzero temperature is equal to 2( s1−s2)µB. Therefore,\none can determine the sublattice Bspins2. The re-\nsults of the theoretical calculations of magnetization, in\nBohr magnetons, are depicted in figure 8 for parameters\ns1= 1.5, J1/J= 0.47, J2/J= 0.001 and s2= 1-curve\na(black); s2= 0.7-curveb(red), and s2= 0.4-curvec(blue). The temperature and magnetization axis are\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/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/s50/s44/s50/s50/s44/s52/s50/s44/s54/s50/s44/s56/s51/s44/s48/s51/s44/s50/s51/s44/s52/s51/s44/s54\n/s97/s98/s99\n/s84/s47/s74/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78 /s91\n/s66/s93\nFIG. 8: (color online) The magnetization 2 MA+ 2MBas\nfunction of T/JforJ1/J= 0.47,J2/J= 0.001,s1= 1.5,\nands2= 1-curve a(black), s2= 0.7-curveb(red),s2= 0.4-\ncurvec(blue)\nchosen in accordance with experimental figure. Compar-\ning figure 94 in [4] and figure 7 in the present paper, one\nconcludesthatthe effectivesublattice Bspins2decreases\nwith increasing Se concentration, and this is the origin\nof the anomalous temperature variation of magnetiza-\ntion. The figure 8 shows that the present calculations\ncapture the essential features of the system; increasing\ntheSeconcentration (decreasing s2) leads to a decrease\nofN´eeltemperature, T∗temperature decreases too, and\nthe maximum of the magnetization increases. Compar-\ning the figure 8 in the present paper and figure 5 in [1]\none realizes the importance of the present method of cal-\nculation for adequate reproducingthe characteristictem-\nperatures TN,T∗, and the shape of the magnetization-\ntemperature curves.\nB. Vanadium spinel MnV2O4\nThe spinel MnV2O4is a two-sublattice ferrimagnet,\nwithsite Aoccupiedbythe Mn2+ion, whichisinthe3 d5\nhigh-spin configuration with quenched orbital angular\nmomentum, which can be regarded as a simple s= 5/2\nspin. The B site is occupied by the V3+ion, which takes\nthe 3d2high-spin configuration in the triply degenerate\nt2gorbital and has orbital degrees of freedom. The mea-\nsurements show that the setting in of the magnetic order\nis atN´eeltemperature TN= 56.5K[5] and that the\nmagnetization has a maximum near T∗= 53.5K. Be-\nlowthistemperaturethemagnetizationsharplydecreases\nand goes to zero when temperature approaches zero.8\nWe consider a system which obtains its magnetic prop-\nertiesfrom MnandVmagneticmoments. Becauseofthe\nstrongspin-orbitalinteractionit is convenienttoconsider\njjcoupling with JA=SAandJB=LB+SB. The sub-\nlatticeAtotal angular momentum is jA=sA= 5/2,\nwhile the sublattice Btotal angular momentum is jB=\nlB+sB, withlB= 3, and sB= 1 [5]. Then the g-factor\nfor the sublattice AisgA= 2, and the atomic value of\nthegBisgB=5\n4. The sublattice Amagnetic order is\nantiparallel to the sublattice Bone and the saturated\nmagnetization is σ= 25\n2−5\n44 = 0, in agreement with\nthe experimental finding that the magnetization goes to\nzero when the temperature approaches zero. The Hamil-\ntonian of the system is\nH=−κA/summationdisplay\n≪ij≫AJA\ni·JA\nj−κB/summationdisplay\n≪ij≫BJB\ni·JB\nj\n+κ/summationdisplay\n/angbracketleftij/angbracketrightJA\ni·JB\nj (36)\nThe first two terms describe the ferromagnetic Heisen-\nberg intra-sublattice exchange κA>0,κB>0, while\nthe third term describes the inter-sublattice exchange\nwhich is antiferromagnetic κ >0. To proceed we use\nthe Holstein-Primakoff representation of the total angu-\nlar momentum vectors JA\nj(a+\nj,aj) andJB\nj(b+\nj, bj), where\na+\nj, ajandb+\nj, bjare Bose fields, and repeat the calcula-\ntions from sections II and III. The magnetization of the\nsystemgAMA+gBMBas a function of the tempera-\nture is depicted in figure 9 for parameters κA/κ= 0.45\nandκB/κ= 0.001. The parameters are chosen so that\nthe calculations to reproduce the experimental value of\nthe ratio TN/T∗.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53 /s54/s48 /s54/s53 /s55/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48/s51/s44/s53\n/s84/s47\n/s84/s42/s47/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78 /s91\n/s66 /s93/s65 /s47 /s61/s48/s46/s52/s53\n/s66/s47 /s61/s48/s46/s48/s48/s49\n/s84\n/s78/s47/s84/s42/s61/s49/s46/s48/s56/s53\nFIG. 9: (color online)The magnetization gAMA+gBMBas\na function of T/κfor parameters κA/κ= 0.45 andκB/κ=\n0.001.\nThe profile of the magnetization-temperature curve is\nin a very good agrement with the experimental zero-\nfield cooling (ZFC) magnetization curves [6, 7]. Theanomaloustemperaturedependence ofthe magnetization\nis reproduced, but there is an important difference be-\ntween the interpretation of the experimental results in\n[5, 6, 7, 8, 9], and the present theoretical results. In\nthe experimental papers TNis the temperature at which\nboth the MnandVmagnetization become equal to zero.\nThe present theory predicts two phases: at low temper-\natures (0 ,T∗) sublattice Mnmagnetization and sublat-\nticeVmagnetization contribute to the magnetization of\nthe system, while at high temperatures ( T∗,TN) only\nMnions have non-zero spontaneous magnetization. The\nvanadium sublattice magnetization set in at T∗, and ev-\nidence for this is the abrupt decrease of magnetization\nbelowT∗, which also indicates that the magnetic order\nof vanadium electrons is anti-parallel to the order of Mn\nelectrons.\nFor samples cooled in a field (FC magnetization) the\nfield leads to formation of a single domain and, in addi-\ntion, increases the chaotic order of the spontaneous mag-\nnetization of the vanadium sublattice, which is antipar-\nallel to it. As a result the average value of the vana-\ndium magnetic order decreases and does not compensate\ntheMnmagnetic order. The magnetization curves de-\npend on the applied field, and do not go to zero. For\na larger field the (FC) curve increases when tempera-\nture decreases below N´eeltemperature . It has a max-\nimum at the same temperature T∗< TNas the ZFC\nmagnetization, and a minimum at T∗\n1< T∗. Below T∗\n1\nthe magnetization increases monotonically when temper-\nature approaches zero.\nTheexperimentswith samplescooledinfield (FCmag-\nnetization) providea new opportunity to clarify the mag-\nnetism of the manganese vanadium oxide spinel. The\napplied field is antiparallel with vanadium magnetic mo-\nment and strongly effect it. On the other hand, the ex-\nperiments show that there is no difference between ZFC\nand FC magnetization curveswhen the temperature runs\nover the interval ( T∗,TN) [6, 7]. They begin to diverge\nwhen the temperature is below T∗. This is in accor-\ndance with the theoretical prediction that the vanadium\nmagnetic moment does not contribute the magnetization\nwhenT > T∗andT∗is the temperature at which the\nvanadium ions start to contribute the magnetization of\nthe system. Because of the strong field, the two vana-\ndium bands are split and the magnetic moment of one of\nthet2gelectrons is reoriented to be parallel with the field\nand magnetic orderof the Mnelectrons. The description\nof this case is more complicate and requires three mag-\nnetic orders to be involved. When T∗< T < T Nonly\nMnions have non zero spontaneous magnetization. At\nT∗vanadium magnetic orderantiparallelto the magnetic\norder ofMnsets in and partially compensates it. Below\nT∗\n1the reoriented electron gives contribution, which ex-\nplains the increasing of the magnetization of the system\nwhen the temperature approaches zero. A series of ex-\nperiments with different applied field could be decisive\nfor the confirmation or rejection of the T∗transition. In-\ncreasing the applied field one expects increasing of T∗\n19\nand when the field is strong enough, so that all vana-\ndium electrons are reoriented, an anomalous increasing\nof magnetization below T∗would be obtained as within\nthe ferromagnetic phase of UGe2[12].\nV. SUMMARY\nIn summary, I have worked out a renormalized spin-\nwave theory and its extension to describe the two phases\n(0,T∗) and (T∗,TN) of a two sublattice ferrimagnet.\nComparing the figure 4 in the present paper and figure 4\nin [1] and figure 8 in the present paper and figure 5 in [1]\nonebecomesawareoftherelevanceofthepresentcalcula-\ntions for the accurate reproduction of the basic features\nof the system near the characteristic temperatures TN\nandT∗.\nThe present theory of ferrimagnetism permits to con-\nsider more complicate systems such as CeCrSb 3com-\npound [13] or the spinel Fe3O4which are two sublattice\nferrimagnets but with three spins.\nVI. ACKNOWLEDGMENTS\nThis work was partly supported by a Grant-in-Aid\nDO02-264/18.12.08 from NSF-Bulgaria.\nAPPENDIX A\nTo make more transparent the derivation of the equa-\ntions for the Hartree-Fock parameters Eq.(20) I consider\nthe first term (the sublattice Aterm) in the Hamiltonian\nof the magnon-magnon interaction Eq.(6). To write this\nterm in the Hartree-Fock approximation one represents\nthe product of two Bose operators in the form\na+\niaj=a+\niaj−< a+\niaj>+< a+\niaj>(A1)\nand neglects all terms ( a+\niaj−< a+\niaj>)2in the four\nmagnon interaction Hamiltonian. The result is\n1\n2a+\niaja+\niai≈ −< a+\niaj>< a+\niai>\n+< a+\niaj> a+\niai+a+\niaj< a+\niai>\n1\n2a+\njaia+\njaj≈ −< a+\njai>< a+\njaj>\n+< a+\njai> a+\njaj+a+\njai< a+\njaj>\n1\n2a+\njaja+\niaj≈ −< a+\njaj>< a+\niaj> (A2)\n+< a+\njaj> a+\niaj+a+\njaj< a+\nraj>a+\niaia+\njaj≈ −< a+\niai>< a+\njaj>\n+< a+\niai> a+\njaj+a+\niai< a+\njaj>\n−< a+\niaj>< a+\njai>\n+< a+\niaj> a+\njai+a+\njai< a+\niaj>\nThe Hartree-Fockapproximationofthe sublattice Apart\nof the Hamiltonian of magnon-magnon interaction reads\n1\n4J1/summationdisplay\n≪ij≫A/bracketleftbig\na+\nia+\nj(ai−aj)2+(a+\ni−a+\nj)2aiaj/bracketrightbig\n≈12NJ1s2\n1(u1−1)2(A3)\n+J1s1(u1−1)/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\nwhere the Hartree-Fock parameter u1is defined by the\nequation\nu1= 1−1\n6s11\nN/summationdisplay\nk∈Brek< a+\nkak>(A4)\nCombining the sublattice Apart of the Hamiltonian\nEq.(5) (the first term) and Eq.(A3) one obtaines the\nHartree-Fock approximation for the sublattice Apart of\nthe Hamiltonian\nHA≈12NJ1s2\n1(u1−1)2(A5)\n+J1s1u1/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\nIn the same way one obtains the Hartree-Fock approx-\nimation of the sublattice Band inter sublattices parts\nof the Hamiltonian. The result is the HHFHamiltonian\nEqs.(7,8,9).\nTo calculate the thermal average < a+\nkak>, in the\nEq.(A4), one utilizes the Hamiltonian HHF. Therefor,\nthe matrix element depends on the Hartree-Fock param-\neters, andequation(A4)isoneoftheselfconsistentequa-\ntions for these parameters.\nThe matrix element can be represented in terms of\nαk(α+\nk) andβk(β+\nk) Eq.(13)\n< a+\nkak>=u2\nknα\nk+v2\nknβ\nk+v2\nk(A6)\nwherenα\nk=< α+\nkαk>, nβ\nk=< β+\nkβk>are the Bose\nfunctions of αandβexcitations. Substituting the ther-\nmal average in Eq.(A4) with Eq.(A6), one obtains that\nequation (A4) is exactly the first equation of the system\nEq.(20) which in turn is obtained from the first of the\nequations (19).\n[1] N.Karchev, J.Phys.:Condens.Matter, 20, 325219 (2008). [2] N. D. Mermin and H. Wagner, Phys. Rev. Let t.17, 113310\n(1966).\n[3] L.N´eel, Ann. Phys., Paris, 3, 137 (1948).\n[4] P. P. Van Stapele, in Handbook of Magnetic Materials ,\nVolume3, 603, EditedbyE.P Wohlfarth, (North-Holland\nPublishing Company, 1982).\n[5] K. Adachi, T. Suzuki, K. Kato, K. Osaka, M. Takata,\nand T. Katsufuji, Phys. Rev. Lett., 95, 197202 (2005).\n[6] V. O. Garlea, R. Jin, D. Mandrus, B. Roessli, Q. Huang,\nM. Miller, A. J. Schultz, and S. E. Nagler, Phys. Rev.\nLett.,100, 066404 (2008).\n[7] S.-H.Baek, K.-Y Choi, A.P.Reyes, P.L. Kuhns,\nN.J.Curro, V.Ramanchandran, N.S. Dalal, H. D.\nZhou, and C.R. Wiebe, J. Phys.:Condens.Matter, 20,135218 (2008).\n[8] H. D. Zhou, J. Lu, and C. R. Wiebe, Phys. Rev., B 76,\n174403 (2007).\n[9] Vincent Hardy, Yohan Br´ eard, and Christine Martin,\nPhys. Rev. B 78, 024406 (2008).\n[10] M.Takahashi, Prog. Theor. Physics Supplement 87, 233\n(1986).\n[11] M.Takahashi, Phys. Rev. Lett. 58, 168 (1987).\n[12] C. Pfleiderer and A. D. Huxley, Phys. Rev. Lett., 89,\n147005 (2002).\n[13] D. D. Jackson, S. K. McCall, A. B. Karki, and D. P.\nYoung, Phys. Rev. B 76, 064408 (2007)."    },    {        "title": "2202.02834v1.Enhancing_Perpendicular_Magnetic_Anisotropy_in_Garnet_Ferrimagnet_by_Interfacing_with_Few_Layer_WTe2.pdf",        "content": "   \n \n1 \n Enhancing Perpendicular Magnetic Anisotropy in Garnet Ferrimagnet \nby Interfacing with Few -Layer WTe 2 \nGuanzhong Wu1*, Dongying Wang1, Nishchhal 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,  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. The change in cantilever oscillation \nis detected optically by laser fiber interferometry. Different LMs have different mode radi i. For this \nmicroscopy study, we focus on 𝑛=1 LM since  it gives the highest spatial resolution.  The 𝑛=1 LM \nresonance peak is fit to a Lorentzian line shape, from which the peak position and peak height are \nextracted, which correspond to the resonance field 𝐻r,1 and signal amplitude ∆𝐴.   \n \n     \n \n19 \n References:  \n  \n(1) Dieny, B.; Chshiev, M. Perpendicular magnetic anisotropy at transition metal/oxide interfaces and \napplications. Rev. Mod. Phys. 2017,  89, (2), 025008.  \n(2) Lee, K. S.; Lee, S. W.; Min, B. C.; Lee, K. J. 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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 = (𝐾𝑥𝑥0 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": "1107.0983v2.Theory_of_Half_metallic_Ferrimagnetism_in_Double_Perovskites.pdf",        "content": "arXiv:1107.0983v2  [cond-mat.str-el]  7 Jul 2011Theory of Half-metallic Ferrimagnetism in Double Perovski tes\nOnur Erten,1O. Nganba Meetei,1Anamitra Mukherjee,1Mohit Randeria,1Nandini Trivedi,1and PatrickWoodward2\n1Department of Physics, The Ohio State University\n2Department of Chemistry, The Ohio State University\n(Dated: August 21, 2018)\nWe present a comprehensive theory of the temperature- and di sorder-dependence of half-metallic\nferrimagnetism in the double perovskite Sr 2FeMoO 6(SFMO) with Tcabove room temperature. We\nshow that the magnetization M(T) and conduction electron polarization P(T) are both propor-\ntional to the magnetization MS(T) of localized Fe spins. We derive and validate an effective sp in\nHamiltonian, amenable to large-scale three-dimensional s imulations. We show how M(T) andTc\nare affected by disorder, ubiquitous in these materials. We s uggest a way to enhance Tcin SFMO\nwithout sacrificing polarization.\nDouble perovskites (DPs) A 2BB′O6are an important\nfamily of complex oxides, derived from the simple ABO 3\nperovskitestructurebyathree-dimensional(3D)checker-\nboard ordering of B and B′ions. One of the best stud-\nied examples is Sr 2FeMoO 6(SFMO), a half-metallic fer-\nrimagnet with Tc≃420K, well above room tempera-\nture [1–3]. Clearly SFMO, and other DPs, can have\nenormous technological impact if their stoichiometry and\nordering can be controlled.\nFrom a theoretical point of view, we argue that DPs\naresimplesystems for understanding metallic ferromag-\nnetism, despite their apparent complexity. First, in con-\ntrast to iron, there is a clear separation of the local-\nized (B) and itinerant degrees of freedom (coming from\nB′) in the DPs. Second, in contrast to the mangan-\nites, DPs have neither Jahn-Teller distortions nor com-\npeting superexchange, given the large distance between\nB-sites. Third, in contrast to dilute magnetic semicon-\nductors, disorderis not an essentialaspect ofthe theoret-\nical problem. Important early theoretical work on half-\nmetallic DPs includes T=0 electronic structure calcula-\ntions [3], and model Hamiltonians analyzed using various\nmean-field theories [4–6] and two-dimensional (2D) sim-\nulations [7].\nIn this Letter, we present a comprehensive theory\nthat gives insight into the temperature- and disorder-\ndependence of the magnetic properties of metallic DPs.\nWe make detailed comparisons with and predictions for\nSFMO. Our main results are:\n(1) We show that both the total magnetization M(T)\nand the conduction electron polarization P(T) atEfare\nproportional to the magnetization MS(T) of localized Fe\nspins. This result is significant because, while M(T) is\neasy to measure, it is P(T) that is of crucial importance\nfor spintronic applications.\n(2) Our main theoretical advance is the derivation and\nvalidation of an effective classical spin Hamiltonian Heff\n[see eq. (2)] for DPs, which differs from both the Heisen-\nberg and Anderson-Hasegawa models [8]. We show that\nHeffdescribes the full T-dependence of the magnetiza-\ntionMS(T), and hence that of M(T) andP(T).\n(3) We present the results of simulations of Heffon large3D lattices, including disorder effects, thus going beyond\nall previous theoretical calculations on SFMO.\n(4) We compute M(T) andTc, using microscopic band-\nstructure parameters as input, and see how these are af-\nfected by deviations from stoichiometry and by anti-site\n(AS) disorder, ubiquitous in real materials. Ours is the\nfirst theory to show that Tcis insensitive to AS disorder,\nin excellent agreement with experiments, even though\nM(0) is suppressed.\n(5) We conclude with a novel proposal to enhance Tcof\nSFMO without sacrificing polarization, using a combina-\ntion of disorder and doping.\nModel Hamiltonian: For large Hund’s coupling JH,\nthe Fe3+(3d5) site has a S=5/2 “core spin” or local\nmoment. The Mo5+(4d1) contributes a t 2gelectron\nwhich hybridizes via O with the Fe t 2gstates. Sym-\nmetry implies that dαβelectrons delocalize only in the\n(α,β)-plane [10]. Thus the motion of electrons in the 3D\nsystem decouples into three 2D planes. The “double ex-\nchange” Hamiltonian [5, 7] describing itinerant electrons\ninteracting with core spins is:\nH=−t/summationdisplay\n/angbracketlefti,j/angbracketright,σ(ǫiσd†\ni↓cjσ+h.c.)\n−t′/summationdisplay\n/angbracketleftj,j′/angbracketright,σc†\njσcj′σ+∆/summationdisplay\nid†\ni↓di↓(1)\nHerediσ(ciσ) are fermion operators on the Fe (Mo)\nsites with spin σ. At the Fe sites i, we choose lo-\ncal axes of quantization along Siand Pauli exclusion\nprohibits an ↑electron. For all the Mo sites j, we\nchoose the same (global) axis of quantization. The ori-\nentation ( θi,ϕi) of the classical spins Sithen affects\nthe Mo-Fe hopping via ǫi↑=−sin(θi/2)exp(iϕi/2) and\nǫi↓= cos(θi/2)exp(iϕi/2).\nThe parameters in Hare the hopping amplitudes t,\nbetween nearest neighbor (Fe-Mo) sites, and t′, between\ntwo Mo sites, and the charge transfer energy ∆ between\nFe t2g↓and Mo t 2gstates; see Fig. 1(a). The results\nare independent [9] of the sign of t. For now, we choose\nt= 1 as our unit of energy. Symmetry dictates t′>0.\nWe choose t′/t= 0.1 and ∆/t= 2.5, using realistic band2\nFIG. 1: (a) Schematic showing energy levels at transition\nmetal sites in two unit cells (formula units) of SFMO. The Fe\nsites have localized S= 5/2 core spins, treated as classical\nvectors with orientation ( θ,ϕ). The parameters t,t′and ∆\nof the Hamiltonian (1), governing the dynamics of the itiner -\nant electrons in t 2gorbitals, are also shown. (b) Calculated\nelectronic structure E(k) and the spin-resolved DOS N↑(E)\nandN↓(E) in the ferromagnetic ground state with all core\nspins up. Note the half-metallic ground state in SFMO with\nconduction electrons polarized opposite to core spins.\nparameters for SFMO as input [3]. We will show below\nthattsets the scale for the magnetic Tc, andt= 0.27\neV, consistent with ref. [3], leads to the experimental\nTc= 420 K of SFMO.\nWe treat the quantum mechanics of “fast” itinerant\nelectrons using exact diagonalization (ED) in the back-\nground of “slow” S=5/2 spins, for which we use a T/negationslash= 0\nclassical Monte Carlo (MC) simulation. This separation\nof time scales is justified below. Using standard ED-\nMC techniques, supplemented by T=0 variational calcu-\nlations, we obtain the results shown in Figs. 1 and 2.\nThese are limited to 2D systems for computational rea-\nsons.\nWe show in Fig. 1(b) the band structure and the spin-\nresolved density of states (DOS) Nσ(E) for the ferri-\nmagnetic ground state for the band-filling correspond-\ning to SFMO with n= 0.33 electrons per unit cell per\nplane. Here the conduction electrons, with magnetiza-\ntionMel(T= 0) = 1 µB(per unit cell), are polarized\nopposite to the Fe spins, with core spin magnetization\nMS(0) = 5µB. Thehalf-metallic ground state, with\nN↑(0) = 0 and N↓(0)/negationslash= 0, has a net magnetization\nM(0) =MS(0)−Mel(0) = 4µB.\nWe see from Fig. 2(a) that MS(T),Mel(T), and hence\nthetotalmagnetization M(T) =MS(T)−Mel(T), all\nhave essentially the same T-dependence. Another quan-tity of great interest is the conduction electron polariza-\ntionP(T)atEfwhich determines the tunneling mag-\nnetoresistance. This is defined by P(T) = [N↓(0)−\nN↑(0)]/[N↓(0)+N↑(0)], with E= 0 in the DOS is mea-\nsured from Ef. From Fig. 2(b), we find that P(T) also\nfollows the T-dependent MS(T).\nEffective Spin Hamiltonian: The results of\nFig. 2(a) and (b) imply that ifwe had a reliable theory\nfor the core spin magnetization MS(T), we could under-\nstandall the magnetic properties of SFMO, including the\npolarization P(T). With this motivation, we derive an\neffective Hamiltonian Hefffor the core spins by gener-\nalizing the two-site Anderson-Hasegawa [8] analysis for\nmanganites to double perovskites.\nTo derive Heff, we find the exact solution of (1)for\ntwo unit cells . The Hilbert space, for a given t 2gorbital,\nhas three states per unit cell: Fe t 2g↓and Mo t 2g,↑,↓.\nThe smallest accessible filling is n= 0.5 (per plane), i.e.,\none electron in two unit cells [11]. We analytically find\nthe lowest eigenvalue as a function of the angle ( θi−\nθj) between spins. Working in two different geometries,\nwe find [12] the nearest-neighbor ( J1) and next-nearest-\nneighbor ( J2) interaction energies; see inset in Fig. 2(c).\nExpressing these in terms of Si·Sj, where each Siis a\nunit vector, we obtain the effective Hamiltonian\nHeff=−J1/summationdisplay\n/angbracketlefti,j/angbracketrightF1(Si·Sj)−J2/summationdisplay\n/angbracketleft/angbracketlefti,j/angbracketright/angbracketrightF2(Si·Sj) (2)\nwhere the functions F1(x) = 8/radicalbig\n2+√2+2xand\nF2(x) = (5+√\n5)/radicalbig\n6+2√3+2x. Our two-unitcell anal-\nysis gives explicit expressions [12] for J1andJ2, both of\nwhich are ferromagnetic with their scale set by the ki-\nnetic energy tof delocalization. We emphasize that the\ndouble square-root form of Heffis quite different from\nthe (single square-root) Anderson-Hasegawa model.\nNext we need to understand how we can use Heffgoing\nbeyond the simple two-unit cell derivation. Specifically:\n(i) How can we relate J1,J2tot,t′,∆ and the filling\nn? (ii) To what extent does Heffcapture the essential\nphysics of the full Hamiltonian H?\nThe dependence of J1andJ2on microscopic parame-\nters can be obtained by matching the spin-wave (SW)\nspectra of HeffandH. This comparison is shown in\nFig. 2(c) along certain symmetry directions. We find SW\ndispersion for Husing ED to compute the energy of elec-\ntrons moving in a “frozen” spin-wave background. The\nSW analysis for Heffis straightforward, since for small\ndeviations from the FM ground state, Heffreduces to a\nnearest and next nearest neighbor FM Heisenberg model\nHHeis. The lowenergyscaleof0 .1tforspindynamics(see\nFig. 2(c)) justifies a posteriori our assumption of “slow”\nspins and “fast” electrons, whose bandwidth is of order t\n(see Fig. 1(b)). The same separationof energy scalesalso\njustifies the use of T-independent exchange couplings J1\nandJ2for allT < T c, as we discuss next.3\nFIG. 2: (a) Magnetization of the core spins MS(T) and of\nthe conduction electrons Mel(T) calculated using ED-MC\nmethod. (b) Conduction electron polarization (at Ef)P(T)\nand normalized MS(T). We see that all magnetic properties\nto be proportional to MS(T). (c) Spin wave dispersion of full\nHamiltonian H, obtained by exact diagonalization, compared\nwith that of the effective Hamiltonian Heff. Inset: Fe-Mo lat-\ntice showing nearest-neighbor ( J1) and next-nearest-neighbor\n(J2) interactions of Heff. (d) The normalized magnetization\nM(T) for three Hamiltonians: H,Heffand the Heisenberg\nmodel. All results are obtained on 8 ×8 systems with error\nbars no larger than the symbol size.\nTovalidatetheeffectiveclassicalmodel Heffweshowin\nFig. 2(d) that it reproduces the magnetization M(T) of\nthe full Hamiltonian Hover the entire range of tempera-\ntures. We note that the corresponding Heisenberg model\nHHeisgives quite different results, except in the T→0\nlimit with small spin deviations. In other words, the T-\ndependent magnetization of DP’s cannot be modeled by\na Heisenberg Hamiltonian. But the (highly non-linear)\nHeffthat we have derived gives an excellent description\nof the ED-MC result for the full H.\nThe classical Heffcan be easily simulated on large 3D\nlattices, unlike the full H, and the results are shown in\nFig. 3. We note the linear drop in M(T) at lowT, due to\nclassicalspin-waves,followedbyarapidsuppressionof M\natthephasetransition[13]. Weestimate Tcintheinfinite\nvolume limit using finite size scaling; see Fig. 3(b). For\nt′/t= 0.1 and ∆/t= 2.5, we find Tc= 0.14t. Comparing\nthis toTc= 420K for pure SFMO, we obtain t= 0.27\neV, consistent with ref. [3].\nDisorder: Heffpermits us to model various kinds of\ndisorder and deviations from stoichiometry [12]. (i) Ex-\ncess Fe is modeled with an extra spin (at a Mo site)\nthat interacts with its neighboring spins with a large an-1 50\nεL1/ν1M(ε)Lβ/νL=16\nL=12\nL=8\n0150300450 600\nT(K)01234µB0 0.1 0.2T/t\nT>TCT<TC\n(a)TC\n(b)M(T)\nFIG. 3: (a) Magnetization M(T) from 3D simulations of Heff\non 163systems. The infinite system Tcis obtained from the\nscaling analysis of panel (b). (b) Finite size scaling of dat a for\nL3systems. Here ε=|T−Tc|/Tcandβ= 0.36 andν= 0.70\nare the well-known 3D O(3) critical exponents.\n0 0.1 0.2\nT/t00.51M(T)δ = 0\n0.05\n0.10\n0.15\n0.20\n0.25\n0 0.1 0.2\nδ00.51\nTC (Theory)\nM(0) (Theory)\nTC (Expt)\nM(0) (Expt)(a) (b)\nFIG. 4: (a) M(T),normalized by theM(0) for the disorder-\nfree system, for different degrees of anti-site (AS) disorde rδ\n(see text). (b) Comparison of theoretical results for Tcand\nT=0 magnetization M(0) with experiments [17]. M(0) drops\nlike 2δ, whileTcis insensitive to AS disorder.\ntiferromagnetic (AF) superexchange S(S+ 1)JAF≃34\nmev [14]. (ii) Excess Mo is modeled by removing an Fe\nspin fromthe lattice. Both(i) and (ii) alsorequirereeval-\nuation of J1andJ2due to change in carrier density n.\n(iii) Here we focus on anti-site (AS) disorder, the most\ncommon form of disorder in DPs, with Fe and Mo inter-\nchanged and no change in n.\nWe see from Fig. 4 that AS disorder systematically re-\nducesM(0) without affecting Tc, in excellent agreement\nwith experiments [16, 17]. We quantify AS disorderusing\nδthe fraction of Fe atoms that are on the Mo sublattice.\nThe observed magnetization M(0)[1−2δ] arises from the\nloss oftwomoments for each AS defect. One from the\nmoment lost at the Mo (on the Fe sublattice) and the\nother from the Fe (on the Mo sublattice) antiferromag-\nnetically coupled to its neighbors via JAF.\nTherearetwooppositeeffectsofASdisorderon Tcthat\nappear to balance each other. The strong Fe-Fe superex-\nchangeJAFpins the spins surrounding the Fe-defect, and\nmakes the magnetic order more robust against thermal\nfluctuations. On the other hand, the Mo-defect leads to4\n0 0.1 0.2y200300400500\nTC(K)\nCompensated\nUncompensated\nExpt\n0 0.1 0.2\nT/t012345µBCompensated\nUncompensated\nMS(T)\nFIG. 5: (a) Core spin magnetization MS(T) for\nLaxSr2−xFe1+yMo1−yO6for uncompensated( x,y) = (0,0.25)\nand compensated ( x,y) = (0.75,0.25) Fe-rich systems. (b)\nThe uncompensated Tc(y) is compared with experiments [20].\nOur prediction for compensated ( x= 3y) system shows large\nenhancement in TC.\nbrokenJ1,J2bonds which weaken the magnetism. The\nnet effect is a Tcinsensitive to δfor moderate levels of\nAS disorder, which is exactly what experiments observe.\nWe note that while the lossof M(0) with AS disorderhas\nbeen explained earlier [16, 18, 19], ours is the first theory\nto correctlyaccountfor Tc; previoustheorieseither found\na drop [16] in Tcor an increase [5].\nRaising Tc:We conclude with a proposal to raise\nthe ferromagnetic Tcwithout sacrificing the conduction\nelectron polarization P; see Fig. 5. In brief, it involves\nadding excess Fe and compensating for the loss of mobile\ncarriers by La substitution on the Sr site.\nTo see how this works, let us first consider excess Fe\nwithout any compensation: Sr 2Fe1+yMo1−yO6. In this\ncase,MS(0) decreases like ydue to the AF alignment of\nexcess spins. Tcalso decreases with ybecause the loss\nof carriers n= (1−3y)/3 dominates over the enhanced\npinningofmomentsby JAFinthevicinityofdefects . For\ntheuncompensatedcase, thecalculated Tc(y)inFig.5(b)\nis in reasonable agreement with experiments [20].\nWe can compensate for the carriers by A-site substi-\ntution: La xSr2−xFe1+yMo1−yO6with electron density\nn= (1+x−3y)/3 per plane. Choosing x= 3ycounters\nthe doping-dependent drop in Tcthat dominated above\nand we find that Tccan be significantly enhanced over\nthat of pure SFMO due to just the local pinning of mo-\nments at excess Fe sites. This increase in Tcgoes hand-\nin-hand with an unchanged MS(0); see Fig. 5(a).\nWe also note that the polarization P(0) remains 100%.\nWhen the Fe spins cant at finite T, electrons depolar-\nize by mixing of up and down states, however, no such\nprocesses are permitted at T=0, even in a disordered Fe-\nrich system. We have checked using the full Hamiltonian\n(1) that, even though the strict proportionality between\nMS(T) andP(T) is not observed in the presence of dis-\norder, the high temperature Pis still enhanced over the\nclean system due to the large increase in Tc.\nAn alternativewayto enhance Tcis to add mobile elec-\ntrons using La doping, which, however, has been shownto lead to considerable increase in AS disorder [15]. In\nprinciple, compensated doping as proposed here should\nintroduce less AS disorder [21].\nIn conclusion, while we have focused here on SFMO,\nour theory provides a general framework for understand-\ning half metallic ferrimagnetism in DPs. Interesting\ndirections for future work include A-site substitution,\nCoulomb correlations on B′, which may become increas-\ningly important for larger carrier concentrations, and\nspin-orbit coupling on B′for 5d elements.\nAcknowledgments: Our research was supported by\nthe Center for Emergent Materials, an NSF MRSEC\n(Award Number DMR-0820414). We thank R. Mishra,\nO. Restrepo, and W. Windl for discussions and acknowl-\nedge use of the computational facilities of the Ohio Su-\npercomputer Center.\n[1] For a recent review, see: D. Serrate, J. M. De Teresa and\nM. R. Ibarra, J. Phys. Cond. Mat. 19, 023201 (2007).\n[2] K. I. Kobayashi et al., Nature 395, 677 (1998).\n[3] D. D. Sarma et al., Phys. Rev. Lett. 85, 2549 (2000).\n[4] A. Chattopadhyay and A. J. Millis, Phys. Rev. B 64,\n024424 (2001).\n[5] J. L. Alonso et al., Phys. Rev. B 67, 214423 (2003).\n[6] L. Brey et al., Phys. Rev. B 74, 094429 (2006).\n[7] P. Sanyal and P. Majumdar Phys. Rev. B 80, 054411\n(2009).\n[8] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675\n(1955).\n[9] Flipping the sign of tonly interchanges the bonding and\nantibonding bands shown in Fig. 1(b).\n[10] A. B. Harris et al., Phys. Rev. B 69, 035107 (2004).\n[11] For the two unit cell geometry, n= 0.5 is the closest we\ncan get to the SFMO filling n= 0.33. We also ignore t′\nin this geometry.\n[12] A detailed derivation will be published elsewhere.\n[13] Note: (i) the qualitative difference between the sharp\ntransition found in 3D (Fig. 3(a)) with the smooth be-\nhavior in 2D (Fig. 2(d)), (ii) the factor of three difference\ninTscales between 3D (FCC) and 2D (square lattice)\narising from the increased number of neighbors.\n[14] We use S(S+ 1)JSE=kBTN/2 for the AF insulator\nLaFeO 3withTN= 750K.\n[15] J. Navarro et al., Phys. Rev. B 64, 092411 (2001).\n[16] A. Ogale et al., Appl. Phys. Lett. 75, 537 (1999).\n[17] J. Navarro et al., Phys. Rev. B 67, 174416 (2003).\n[18] B. Aguilar, O. Navarro and M. Avignon, Europhys. Lett.\n8867003 (2009).\n[19] R. Mishra et al., Chem. Mater. 22, 6092 (2011).\n[20] D. Topwal et al., Phys. Rev. B 73, 094419 (2006).\n[21] La doping of x= 1 gives rise to a 15% increase of Tc[15],\nbut also a large change in Mo valence (+5 to +4), which\nadversely impacts cation ordering, leading to large AS\ndisorder. In contrast, in our proposal a 25% increase in\nTcis obtained for y= 0.25 andx= 0.75, with an average\nMo valence of only +4.66."    },    {        "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         J<J*Non-stationary mode\n            J>J*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": "2308.12410v1.Magnetic_analogue_of_liquid_gas_phase_transition_of_water__case_study_of_a_spin_1_2_Ising_Heisenberg_model_on_a_diamond_decorated_square_lattice.pdf",        "content": "1\nMAGNETIC ANALOGUE OF LIQUID-GAS PHASE TRANSITION OF WATER:\nCASE STUDY OF A SPIN-1/2 ISING-HEISENBERG MODEL ON A DIAMOND-\nDECORATED SQUARE LATTICE\nJ. Streˇ cka, jozef.strecka@upjs.sk, K. Karl’ov´ a, katarina.karlova@upjs.sk, Institute of Physics, Faculty of Science, P. J.\nˇSaf´ arik University, Park Angelinum 9, 04001 Koˇ sice, Slovakia, T. Verkholyak, werch@icmp.lviv.ua, Institute for Condensed\nMatter Physics, NASU, Svientsitskii Street 1, 790 11, L’viv, Ukraine, N. Caci, caci@physik.rwth-aachen.de, S. Wessel,\nwessel@physik.rwth-aachen.de, Institute for Theoretical Solid State Physics, JARA FIT, and JARA CSD, RWTH Aachen\nUniversity, 52056 Aachen, Germany and A. Honecker, andreas.honecker@cyu.fr, Laboratoire de Physique Th´ eorique et\nMod´ elisation, CNRS UMR 8089, CY Cergy Paris Universit´ e, Cergy-Pontoise, France\nINTRODUCTION\nPhase transitions of diverse physical systems have\ncaptured significant attention due to the occurrence of\nabrupt discontinuities or divergences in several physi-\ncal quantities, which consequently preclude the proper\ndefinition at the relevant phase transitions [1]. One\nof the most extensively studied physical substances is\nwater, whose phase diagram encompasses solid, liq-\nuid, and gaseous phases separated from each other\nby lines of discontinuous phase transitions. Among\nthese phase-transition lines, the most intriguing is the\nline of liquid-gas phase transitions, which starts at a\ntriple coexistence point of all three phases and ends\nat the critical point where a discontinuous phase tran-\nsition changes to a continuous one. A similar phase-\ntransition line has been recently found in the pressure-\ntemperature phase diagram of the quantum magnetic\nmaterial SrCu 2(BO 3)2, which provides an experimen-\ntal realization of the spin-1/2 Heisenberg model on the\nShastry-Sutherland lattice [2].\nThe line of discontinuous phase transitions termi-\nnating at an Ising critical point is not unique to the\nShastry-Sutherland model, but it may be encountered\nin other frustrated two-dimensional quantum spin sys-\ntems such as the spin-1/2 Heisenberg model on a\nfully frustrated bilayer [3–5], trilayer [6] and diamond-\ndecorated square lattice [7]. To capture thermal phase\ntransitions of the spin-1/2 Heisenberg model on frus-\ntrated two-dimensional lattices one has to resort to\nstate-of-the-art numerical calculations as for instance\nquantum Monte Carlo simulations in a dimer or trimer\nbasis in order avoid the notorious sign problem [8]. In-\nterestingly, it has been recently verified that the sim-\npler spin-1/2 Ising-Heisenberg model on the diamond-\ndecorated square lattice already captures the essential\nfeatures of thermal phase transitions of the spin-1/2\nHeisenberg model on the diamond-decorated square\nlattice [7,9].\nThe spin-1/2 Ising-Heisenberg diamond-decorated\nsquare lattice in a magnetic field can be rigorously\nmapped to an effective zero-field spin-1/2 Ising square\nlattice in a particular parameter subspace correspond-\ning to the thermal phase transitions [9]. From this per-\nspective, the spin-1/2 Ising-Heisenberg model on the\ndiamond-decorated square lattice represents a valu-\nable example in the realm of exactly solved lattice-\nstatistical models [10], which allows us to rigorously\nexamine the thermal phase transitions in the pres-\nence of an external magnetic field quite similarly as\nS 1,i,jJ2\nJ1S 2,i,jS 3,i,jS 4,i,jS 5,i,jFig. 1. Schematic illustration of the spin-1/2 Ising-\nHeisenberg model on a diamond-decorated square lattice.\nLarge (light blue) circles determine nodal lattice sites oc-\ncupied by the Ising spins S1,i,j, while small (violet) circles\ndetermine decorating lattice sites occupied by the Heisen-\nberg spins Sk,i,j(k= 2−5).\nfor Fisher’s superexchange antiferromagnet [11] and its\ndifferent variants [12–16].\nIn the present paper we provide a more compre-\nhensive understanding of the thermal phase transitions\nof the spin-1/2 Ising-Heisenberg diamond-decorated\nsquare lattice. Except temperature and magnetic-field\nvariations of the local and total magnetization, we will\nbring insight into the respective changes of the mag-\nnetic susceptibility and specific heat at the discontin-\nuous as well as continuous thermal phase transitions.\nThe paper is organized as follows. In the following\nsection we will briefly recall the definition of the stud-\nied quantum spin model and review the mapping to\nthe effective spin-1/2 Ising model on a square lattice.\nThe subsequent section is devoted to a detailed de-\nscription of basic magnetic and thermodynamic quan-\ntities (magnetization, susceptibility, and specific heat)\nin close vicinity of the thermal phase transitions. Fi-\nnally, the most important findings are summarized in\nthe concluding part.\nISING-HEISENBERG DIAMOND-DECORA-\nTED SQUARE LATTICE\nAt first, let us recall the definition of the spin-\n1/2 Ising-Heisenberg model on the diamond-decorated\nsquare lattice, which is schematically illustrated in2\nFig. 1 and mathematically given by the Hamiltonian\nˆH=J1LX\ni=1LX\nj=1h\u0010\nˆSz\n1,i,j+ˆSz\n1,i+1,j\u0011\u0010\nˆSz\n2,i,j+ˆSz\n3,i,j\u0011\n+\u0010\nˆSz\n1,i,j+ˆSz\n1,i,j+1\u0011\u0010\nˆSz\n4,i,j+ˆSz\n5,i,j\u0011i\n+J2LX\ni=1LX\nj=1\u0010\nˆS2,i,j·ˆS3,i,j+ˆS4,i,j·ˆS5,i,j\u0011\n−h5X\nk=1LX\ni=1LX\nj=1ˆSz\nk,i,j. (1)\nThe first term accounts for an anisotropic (Ising-type)\nexchange interaction J1between nearest-neighbor Ising\nand Heisenberg spins schematically shown in Fig. 1\nas large and small circles, respectively, while the sec-\nond term stands for the isotropic exchange interac-\ntion between nearest-neighbor Heisenberg spins. The\nlast term takes into account the Zeeman energy of the\nHeisenberg and Ising spins in an external magnetic field\nhandLdenotes the linear size of the considered two-\ndimensional lattice.\nIt has been verified in our previous paper [9] that\nthe partition function of the spin-1/2 Ising-Heisenberg\nmodel on the diamond-decorated square lattice given\nby the Hamiltonian (1) can be related via the general-\nized decoration-iteration transformation [17–19] to the\npartition function of the effective spin-1/2 Ising model\non a square lattice:\nZ(β, J1, J2, h) =A2NZeff(β, Jeff, heff), (2)\nwhich is defined through the effective Hamiltonian in-\nvolving temperature-dependent nearest-neighbor inter-\nactions Jeffand magnetic field heff:\nHeff=−JeffLX\ni=1LX\nj=1(Sz\n1,i,jSz\n1,i+1,j+Sz\n1,i,jSz\n1,i,j+1)\n−heffLX\ni=1LX\nj=1Sz\n1,i,j. (3)\nAn explicit form of the mapping parameters A,Jeff,\nandheffis given by Eqs. (7)–(10) in Ref. [9]. It follows\nfrom the exact mapping between the partition func-\ntions (2) that the spin-1/2 Ising-Heisenberg model on\nthe diamond-decorated square lattice becomes exactly\nsoluble in the particular parameter subspace with zero\neffective field heff= 0 due to the famous exact solution\nby Onsager [20], while in the parameter space with\nnonzero effective field heff̸= 0 one may obtain precise\nnumerical results by exploiting classical Monte Carlo\nsimulations.\nExcept for a trivial fully saturated paramagnetic\nphase, the spin-1/2 Ising-Heisenberg model on the\ndiamond-decorated square lattice displays two further\nground states, which can be classified as the classical\nferrimagnetic (FRI) phase:\n|FRI⟩=LY\ni,j=1|↓1,i,j⟩⊗|↑ 2,i,j↑3,i,j⟩⊗|↑ 4,i,j↑5,i,j⟩(4)and the quantum monomer-dimer (MD) phase:\n|MD⟩=LY\ni,j=1|↑1,i,j⟩ ⊗1√\n2(|↑2,i,j↓3,i,j⟩ − |↓ 2,i,j↑3,i,j⟩)\n⊗1√\n2(|↑4,i,j↓5,i,j⟩ − |↓ 4,i,j↑5,i,j⟩). (5)\nAt zero temperature the FRI and MD ground states\ncoexist along the phase boundary given by:\nhMD−FRI= 2(J2−J1), J 2∈[J1,3J1]. (6)\nThe ground-state phase boundary (6) represents a\nlower boundary for the wall of discontinuous (first-\norder) phase transitions between the FRI and MD\nphases, which are bounded from above by a line of con-\ntinuous (second-order) phase transitions from the uni-\nversality class of the two-dimensional Ising model [20]\nas dictated by the constraint βcJeff= 2 ln(1 +√\n2) and\nheff= 0.\nTHERMAL PHASE TRANSITIONS BE-\nTWEEN FERRIMAGNETIC AND MONO-\nMER-DIMER PHASES\nIn this section our particular attention will be fo-\ncused on typical signatures of discontinuous and con-\ntinuous thermal phase transitions between the FRI\nand MD phases. To this end, we will limit our fur-\nther analysis to the spin-1/2 Ising-Heisenberg model\non the diamond-decorated square lattice with the spe-\ncific value of the interaction ratio J2/J1= 1.5, which\ncaptures all typical features of both types of thermal\nphase transitions emergent in the close vicinity of the\ncoexistence point between the FRI and MD ground\nstates ( hMD−FRI/J1= 1 for J2/J1= 1.5).\nFigure 2 reports the finite-temperature phase dia-\ngram of the spin-1/2 Ising-Heisenberg model on the\ndiamond-like decorated lattice in the magnetic field\nversus temperature plane for this ratio. It can be seen\nfrom this figure that the line of discontinuous ther-\nmal phase transitions between the FRI and MD phases\n(blue broken line) actually starts from the relevant\ncoexistence point h/J1= 1 before it bends towards\nlower magnetic fields as temperature rises. This line\nof discontinuous thermal phase transitions finally ter-\nminates at the Ising critical point at h/J1= 0.9279\nandkBT/J 1= 0.2403 for J2/J1= 1.5, which can be\nascribed to a continuous thermal phase transition be-\ntween the FRI and MD phases (red circle).\nNext, let us illustrate a few typical features of\nthe magnetization and magnetic susceptibility when\nthe magnetic field drives the spin-1/2 Ising-Heisenberg\nmodel on the diamond-like decorated lattice across the\nthermal phase transition. For this purpose, a few typ-\nical magnetic-field dependencies of the local and total\nmagnetization are plotted in Fig. 3 for three selected\ntemperatures. The calculated values of the local mag-\nnetization of the Ising spins mI(red lines) and the local\nmagnetization of the Heisenberg spins mH(blue lines)3\n0.850.900.951.001.051.100.00.10.20.3M\nD                  FRI[0.9279; 0.2403]h\n / J1J2 / J1 = 1.5 \nkB T / J1\nFig. 2. Finite-temperature phase diagram in the magnetic\nfield versus temperature plane for the fixed value of the\ninteraction ratio J2/J1= 1.5. The blue broken curve dis-\nplays the line of discontinuous thermal phase transitions\nbetween the quantum monomer-dimer (MD) phase and the\nclassical ferrimagnetic (FRI) phase, which terminates at\nthe Ising critical point (red circle) with the coordinates\nh/J 1= 0.9279 and kBT/J 1= 0.2403.\nare indeed consistent with the presence of the MD and\nFRI phase for h/J1≲1 and h/J1≳1, respectively.\nAt the lowest temperature kBT/J 1= 0.2 one detects a\ndiscontinuous thermal phase transition from the MD\nphase towards the FRI phase accompanied with an\nabrupt magnetization jump [see Fig. 3(a)]. The dis-\ncontinuous magnetization jump gradually shrinks upon\nincreasing temperature until it completely vanishes in\nvicinity of the critical temperature kBT/J 1≈0.2403,\nwhere the magnetization discontinuity is replaced by\nan inflection point with a vertical (infinite) tangent [see\nFig. 3(b)]. The inflection point finally acquires a finite\nslope at the even higher temperature kBT/J 1= 0.3,\nwhere one finds a rather simple crossover from the\nMD phase towards the FRI phase. A similar crossover\ncan be observed around the saturation field h/J1= 4,\nwhere the local and total magnetization undergo a sub-\nstantial magnetic-field-driven change from the values\ntypical for the FRI phase towards their fully saturated\nvalues.\nAll aforementioned features of the isothermal mag-\nnetization curves of the spin-1/2 Ising-Heisenberg\ndiamond-decorated square lattice can be unambigu-\nously corroborated by the magnetic-field-induced\nchanges of the isothermal magnetic susceptibility pre-\nsented in Fig. 4. The finite cusp of the magnetic\nsusceptibility encountered at the lowest temperature\nkBT/J 1= 0.2 verifies the existence of a discontin-\nuous thermal phase transition between the MD and\nFRI phases [Fig. 4(a)]. On the other hand, there are\nstrong indications that in proximity of the critical tem-\nperature kBT/J 1≈0.2403 of the continuous thermal\nphase transition the magnetic susceptibility displays a\npronounced power-law divergence [Fig. 4(b)]. Last but\nnot least, the magnetic susceptibility exhibits a rather\n01 2 3 4 5 -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nTmHm\nI MD  FRIk\nB T / J1 = 0.2J\n2 / J1 = 1.5 \nmI , mH , mT(\na)                            h / J1\n01 2 3 4 5 -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nTmHm\nI MD  FRIk\nB T / J1 = 0.2403J\n2 / J1 = 1.5 \nmI , mH , mT(\nb)                            h / J1\n01 2 3 4 5 -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nTmHm\nI k\nB T / J1 = 0.3J\n2 / J1 = 1.5 \nmI , mH , mT(\nc)                            h / J1Fig. 3. A few typical magnetic-field dependencies of the\nlocal and total magnetization for the fixed value of the in-\nteraction parameter J2/J1= 1.5 and three selected values\nof temperature: (a) kBT/J 1= 0.2; (b) kBT/J 1= 0.2403;\n(c)kBT/J 1= 0.3.\nsharp but rounded finite maximum at higher tempera-\ntures such as kBT/J 1= 0.3 [see the inset in Fig. 4(c)].\nIt is also worth mentioning that the magnetic suscepti-\nbility displays for arbitrary temperature another round\nmaximum located around h/J1≈4, which confirms a\nsimple crossover from the FRI phase towards the fully\nsaturated paramagnetic phase rather than a true phase\ntransition.\nIn the following we will examine in detail typical4\n01 2 3 4 5 0.00.20.40.60.81.0M\nD    FRIkB T / J1 = 0.2 J\n2 / J1 = 1.5 \n/s61539T(\na)                            h / J1\n01 2 3 4 5 0.00.20.40.60.81.0M\nD    FRIkB T / J1 = 0.2403 J\n2 / J1 = 1.5 \n/s61539T(\nb)                            h / J1\n0123450.00.20.40.60.81.0M\nD    k\nB T / J1 = 0.3 J\n2 / J1 = 1.5 \n/s61539T(\nc)                            h / J10.70.80.91.01.10.00.20.40.60.81.0 \n   \nFig. 4. A few typical magnetic-field dependencies of the\nisothermal magnetic susceptibility for the fixed value of the\ninteraction parameter J2/J1= 1.5 and three selected values\nof temperature: (a) kBT/J 1= 0.2; (b) kBT/J 1= 0.2403;\n(c)kBT/J 1= 0.3.\nfeatures of the magnetization and magnetic specific\nheat induced by temperature , when driving the spin-\n1/2 Ising-Heisenberg model on the diamond-decorated\nsquare lattice across the thermal phase transitions. To\nstart with, Fig. 5 depicts typical temperature varia-\ntions of the local and total magnetization for three\n0.100.150.200.250.300.350.40-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nTm\nH mI M\nD                 FRIh / J1 = 0.95J2 / J1 = 1.5 \nmI , mH , mT(\na)                            kB T / J1\n0.10 .20 .30 .40 .5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nHm\nTm\nIMD    FRIh / J1 = 0.9279J2 / J1 = 1.5 \nmI , mH , mT(\nb)                             kB T / J1\n0.10 .20 .30 .40 .5-0.3-0.2-0.10.00.10.20.30.40.5m\nTmHm\nI MDh / J1 = 0.9J2 / J1 = 1.5 \nmI , mH , mT(\nc)                           kB T / J1Fig. 5. A few typical temperature variations of the local and\ntotal magnetization for the fixed value of the interaction\nratio J2/J1= 1.5 and three different magnetic fields: (a)\nh/J 1= 0.95; (b) h/J 1= 0.9279; (c) h/J 1= 0.9.\nselected values of the magnetic field. The discontinu-\nous nature of the local and total magnetization, ob-\nservable at the selected value of the magnetic field\nh/J1= 0.95, corroborates the presence of a discon-\ntinuous thermal phase transition between the FRI and\nMD phases [Fig. 5(a)]. If the magnetic field is fixed suf-\nficiently close to the critical value h/J1≈0.9279, the\nlocal and total magnetization show an inflection point\nwith a vertical tangent serving in evidence of the con-\ntinuous thermal phase transition between the FRI and5\n0.10 .20 .30 .40.00.51.01.52.0h\n / J1 = 0.95M\nD            FRIJ2 / J1 = 1.5 \nC / N kB   (\na)                            kB T / J1\n0.10 .20 .30 .40246810h\n / J1 = 0.9279M\nD               FRIJ2 / J1 = 1.5 \nC / N kB   (\nb)                            kB T / J1\n0.10 .20 .30 .40246810h\n / J1 = 0.9M\nDJ2 / J1 = 1.5 \nC / N kB   (\nc)                          kB T / J1\nFig. 6. A few typical temperature variations of the mag-\nnetic specific heat for the fixed value of the interaction\nratio J2/J1= 1.5 and three different magnetic fields: (a)\nh/J 1= 0.95; (b) h/J 1= 0.9279; (c) h/J 1= 0.9.\nMD phases [Fig. 5(b)]. At even lower magnetic field\nh/J1= 0.9 the local and total magnetization display a\nsmooth continuous thermally-assisted change due to a\ncrossover from the MD phase towards the FRI phase.\nLet us conclude our discussion by a detailed anal-\nysis of the temperature dependencies of the magnetic\nspecific heat of the spin-1/2 Ising-Heisenberg model\non the diamond-decorated square lattice presented inFig. 6. The finite cusp of the magnetic specific heat seen\nin Fig. 6(a) around the temperature kBT/J 1≈0.216\nevidences the discontinuous thermal phase transition\nfor the relevant choice of the magnetic field h/J1=\n0.95. Contrary to this, the temperature dependence\nof the magnetic specific heat is quite reminiscent of\na steep logarithmic divergence if the magnetic field\nh/J1≈0.9279 is tuned sufficiently close to the contin-\nuous thermal phase transition [Fig. 6(b)]. Finally, the\nmagnetic specific heat may be completely free of any\nthermal phase transition as exemplified by the tem-\nperature dependence with a sizable but finite round\nmaximum shown in Fig. 6(c) for the magnetic field\nh/J1= 0.9.\nCONCLUSION\nIn the present article we have examined in detail\ndiscontinuous and continuous thermal phase transi-\ntions of the spin-1/2 Ising-Heisenberg model on the\ndiamond-decorated square lattice between the classical\nFRI phase and the quantum MD phase in the presence\nof an external magnetic field. It has been demonstrated\nthat the spin-1/2 Ising-Heisenberg diamond-decorated\nsquare lattice shows for a fixed value of the interac-\ntion ratio a line of discontinuous thermal phase transi-\ntions, which terminates at an Ising-type critical point\ninherent to a continuous thermal phase transition. It\ncould be thus concluded that the phase boundary be-\ntween the FRI and MD phases of the spin-1/2 Ising-\nHeisenberg model on the diamond-decorated square\nlattice is reminiscent of the liquid-gas phase boundary\nof water. In addition, the results for discontinuous and\ncontinuous thermal phase transitions of the spin-1/2\nIsing-Heisenberg diamond-decorated square lattice are\nexact as they can be directly descended from a rigorous\nmapping correspondence to an exactly solved spin-1/2\nIsing model on a square lattice with a temperature-\ndependent nearest-neighbor interaction Jeff̸= 0 and\nzero effective field heff= 0 [20]. Out of the param-\neter region corresponding to thermal phase transi-\ntions the spin-1/2 Ising-Heisenberg diamond-decorated\nsquare lattice can be rigorously mapped onto the spin-\n1/2 Ising model on a square lattice with a nonzero\nnearest-neighbor interaction Jeff̸= 0 and effective field\nheff̸= 0, which can be subsequently treated by classical\nMonte Carlo simulations.\nBy making use of extensive Monte Carlo simu-\nlations we have evidenced that the spin-1/2 Ising-\nHeisenberg model on the diamond-decorated square\nlattice displays a discontinuous magnetization jump\nwhen temperature or magnetic field drives the investi-\ngated spin system across a discontinuous thermal phase\ntransition. The magnetic susceptibility and specific\nheat consequently exhibit at the discontinuous thermal\nphase transitions finite cusps related to a discontinuous\nchange of both quantities. On the contrary, the mag-\nnetization of the spin-1/2 Ising-Heisenberg diamond-\ndecorated square lattice varies continuously when tem-6\nperature or magnetic-field changes drive the investi-\ngated spin system across a continuous thermal phase\ntransition. Under this condition, the magnetic suscep-\ntibility and specific heat display a strong divergence at\nthe continuous thermal phase transition.\nThe present accurate results are reminiscent of\nthe experimental findings, which recently reported un-\nconventional thermal phase transitions of the two-\ndimensional quantum magnet SrCu 2(BO 3)2[2]. In the\ncase of the diamond-decorated square lattice, the FRI-\nMD phase transition in the spin-1/2 Ising-Heisenberg\nmodel is not only qualitatively but even quantitatively\nclose to the Lieb-Mattis to MD phase transition in the\nfull spin-1/2 Heisenberg model [7,9]. The exact solution\nof the spin-1/2 Ising-Heisenberg model thus permits\nus to unveil, for example, a reentrance in the finite-\ntemperature transition line [9] that would be beyond\nthe accuracy of the numerical methods used to solve\nthe spin-1/2 Heisenberg model [7].\nIt is our hope that the present essentially exact re-\nsults will stimulate further exploration in this exciting\nresearch field.\nACKNOWLEDGEMENT\nThis work was funded by the Slovak Research\nand Development Agency and the French Ministry for\nEurope and Foreign Affairs, the French Ministry for\nHigher Education and Research under the ˇStef´ anik\nprogramme for Slovak-France bilateral projects under\ncontract Nos. SK-FR-19-0013/45125RC and SK-FR-\n22-0011/49880PG. J.S. and K.K. acknowledge the par-\ntial financial support by a grant of the Slovak Re-\nsearch and Development Agency under the contract\nNo. APVV-20-0150. We acknowledge support from\nthe Deutsche Forschungsgemeinschaft (DFG) through\nGrant No. WE/3649/4-2 of the FOR 1807 and RTG\n1995.\nREFERENCES\n1. C. Domb, M. S. Green: Phase Transitions and Criti-\ncal Phenomena: Volume 1 (Academic Press, London\n- 1972); ibid. Volume 2 (Academic Press, London -\n1972); ibid. Volume 3 (Academic Press, London -\n1976); ibid. Volume 4 (Academic Press, London -\n1977); ibid. Volume 5 (Academic Press, London -\n1983).\n2. J. Larrea Jim´ enez, S. P. G. Crone, E. Fogh,\nM. E. Zayed, R. Lortz, E. Pomjakushina, K. Con-\nder, A. M. L¨ auchli, L. Weber, S. Wessel, A. Ho-\nnecker, B. Normand, Ch. R¨ uegg, P. Corboz,\nH. M. Ronnow, F. Mila, Nature 592, 370 (2021).\n3. J. Stapmanns, P. Corboz, F. Mila, A. Honecker,\nB. Normand, S. Wessel, Phys. Rev. Lett. 121,\n127201 (2018).\n4. L. Weber, A. Y. D. Fache, F. Mila, S. Wessel, Phys.\nRev. B 106, 235128 (2022).\n5. Y. Fan, N. Xi, C. Liu, B. Normand, R. Yu, arXiv:\n2306.16288.6. L. Weber, A. Honecker, B. Normand, P. Corboz,\nF. Mila, S. Wessel, SciPost Phys. 12, 054 (2022).\n7. N. Caci, K. Karl’ov´ a, T. Verkholyak, J. Streˇ cka,\nS. Wessel, A. Honecker, Phys. Rev. B 107, 115143\n(2023).\n8. J. Gubernatis, N. Kawashima, P. Werner: Quan-\ntum Monte Carlo Methods: Algorithms for Lattice\nModels (Cambridge University Press, Cambridge -\n2016).\n9. J. Streˇ cka, K. Karl’ov´ a, T. Verkholyak, N. Caci,\nS. Wessel, A. Honecker, Phys. Rev. B 107, 134402\n(2023).\n10. R. J. Baxter: Exactly Solved Models in Statistical\nMechanics (Academic, London, 1982).\n11. M. E. Fisher, Proc. R. Soc. London A 254, 66\n(1960); ibid.256, 502 (1960).\n12. M. Hattori, H. Nakano, Prog. Theor. Phys. 40, 958\n(1968).\n13. H. Mashiyama, S. Nara, Phys. Rev. B 7, 3119\n(1973).\n14. W. T. Lu, F. Y. Wu, Phys. Rev. E 71, 046120\n(2005).\n15. L. ˇCanov´ a, M. Jaˇ sˇ cur, Condens. Matter Phys. 9, 47\n(2006).\n16. L. G´ alisov´ a, J. Phys.: Condens. Matter 28, 476005\n(2016).\n17. M. E. Fisher, Phys. Rev. 113, 969 (1959).\n18. O. Rojas, J. S. Valverde, S. M. de Souza, Physica\nA,388, 1419 (2009).\n19. J. Streˇ cka, Phys. Lett. A 374, 3718 (2010).\n20. L. Onsager, Phys. Rev. 65, 117 (1944)."    },    {        "title": "1311.3067v1.Exchange_bias_up_to_room_temperature_in_the_antiferromagnetic_bulk_hexagonal_Mn3Ge.pdf",        "content": "Exchange bias up to room temperature in the antiferromagnetic bulk \nhexagonal Mn 3Ge \nJ. F. Qian,1,a A. K. Nayak,1 G. Kreiner,1 W. Schnelle,1 and C. Felser1,b \nMax Planck Institute for Chemical Physics of Solids, N öthnitzer Str. 40, 01187 Dresden, Germany  \n(Dated: 12 November 2013)  \nThis work reports an exchange bias (EB) e ffect up to room temperature in the binary \nintermetallic bulk compound Mn 3.04Ge0.96. The sample annealed at 700 K crystallizes in a \ntetragonal structure with ferromagnetic ordering, whereas , the sample annealed at 1073 K \ncrystallizes in a hexagonal structure with antiferromagnetic ordering. The hexagonal \nMn 3.04Ge0.96 sample exhibits an EB of around 70 mT at 2 K that continues with a non -zero \nvalue up to room temperature. The exchange anisotr opy is proposed to be originating from the \nexchange interaction between the triangular antiferromagnetic host and the embedded \nferrimagnetic like clusters. The ferrimagnetic clusters develop when excess Mn atoms occupy \nempty Ge sites in the original triang ular antiferromagnet structure  of Mn 3Ge. \nPACS numbers: 75.60.Ej, 75.30.Gw, 72.15.Jf, 75.50.Cc  \n   Binary Heusler compounds Mn3Z, containing man- ganese with a main group \nelement Z, exhibit various structural and magnetic properties.1-4 In this family , Mn 3Ga \nhas attracted a lot of attention due to its possible application in spin transfer torque \n(STT) device and hard magnetic behavior.5,6 In addition, doped with a second \ntransition element the structure and magnetic properties can be tuned to fi t for various \napplications. For example, the Co substitution in place of Mn in Mn 3-xCoxGa transfers \nthe tetragonal distorted Heusler structure known as a hard magnet to a cubic soft \nmagnet with half -metallic behavior.7 Similarly, Mn 2PtGa has been found to remain in \nthe tetragonal phase and exhibits a large zero fi eld cooled exchange bias (EB).8 The \ntetragonal compounds are interesting as they show perpendic ular magnetic anisotropy \nin thin fi lms which opens a way  for application in STT magnetic random access \nmemories (MRAM).9-12 Moreover, a hexagonal triangular antiferromagnetic fi lm of \nhexagonal- Mn3Ga has  been used as a pinned layer in magnetic tunnel junctions  \n(MTJ).13 Besides the high tunneling magneto -resistance (TMR), a large EB fi eld also \nhas been observed in this system.13 They attributed this EB to the frustrated triangular \nantiferromagnetic structure easily perturbed by exchange coupling to the adjacent  \nferromagnetic layer.  \n   In the present work, we focus on another binary compound Mn3Ge, which was fi rst \nsynthesized as a hexagonal structure in 1949 by Zwicker et al.14 An antiferromagnetic \n(AFM) ordering with feeble parasitic ferro magnetic (FM) behavior was observed in \n                                                           \na Jin-Feng.Qian@cpfs.mpg.de  \nb felser@cpfs.mpg.de  Mn 3.4Ge.4 A neutron diff raction study in Mn 3.1Ge0.9 have revealed  a triangular AFM \nstructure with magnetic moment lying in the hexagonal basal plane.15 A Néel \ntemperature  TN = 395 ±10 K has been reported for this compound.15 The small FM \ncomponent was attributed to the free rotation of the spin triangle in the basal-plane \ncaused by  a sma ll hexagonal distortion.16 The hexagonal structure transforms to a face-\ncentered tetragonal phase by a long annealing at 450 °C  for more than ten days.4 A \nlarge perpendicular magnetic anisotropy and TMR have been reported in the tetragonal  \nMn 3Ge thin fi lm.17 However, there is not enough study  on the magnetic properties of \nthe hexagonal Mn 3Ge. In this letter we present a detailed structural and magnetic  study \non the bulk Mn 3Ge. We show that the hexagonal-Mn 3Ge sample exhibits exchange \nbias up to room temperature.  \n   Polycrystalline ingots of Mn 3Ge were carefully synthesized by repeated inductive \nmelting the target amounts of high purity elements in a high purity argon atmosphere  \nunder 10-2 mbar pressure in a glove box. During melting, the metal pieces were \ncontained in a cleaned alumina  crucible. In consideration of the high vapor pressure of  \nMn, and avoiding Mn loss, the induction current was controlled under an appropriate \nvalue. The weight loss after the whole process was less than 1%. Afterward, the \nsamples were sealed in evacuated quartz tubes to anneal at 1073 K for 4 days. In order \nto get higher  degree of atomic ordering a pa rt of the sample was further annealed at \n823 K for 10 days and quenched into a mixture of ice and water. For comparison, \nanother part of  the sample was annealed at 700 K for 14 days to obtain the low \ntemperature tetragonal -Mn3Ge phase. The sample quality and composition were \nstudied by utilizing a scanning electron microscope equipped with an energy \ndispersive x-ray spectroscopy (EDXS) detection system and inductively coupled \nplasma optical emission spectrometry (ICP-OES). The crystal structure was \ninvestigated by using powder x-ray diffraction (XRD), with  Cu-Kα (λ = 1.540598 Å ) \nradiation. The magnetic properties were studied using a superconducting quantum \ninterference device ( SQUID, Quantum Design MPMS) magnetometer between 1.8 K \nand 400 K.  \n   From the EDXS  and ICP- OES analysis it is found  that the hexagonal Mn3Ge \nconsists of a composition of  76.58:23.42, whereas, the tetragonal Mn3Ge sample has a \ncomposition of 76.62:23.38. In both cases the compositions are very close to the initial \ncomposition of 75:25. The phase purity of both the samples is confi rmed  by theoretical \nfitting to the experimental XRD data as  shown in Fig. 1. In case of 1073 K annealed \nsample (Fig.  1a) the refi nement has been performed with a hexagonal space group \nP63/mmc, which gives the lattice parameters a = 5:34 Å and c = 4:32 Å. The structure \nmatches  to the typical D0 19 hexagonal unit cell as expected. Similarly, for the 700 K \nannealed sample (Fig. 1b) the refi nement has been performed with a tetragonal space \ngroup I4/mmm, which results in lattice parameters a = 3:80 Å and c = 7:24 Å. From \nthe fitting it is clear that both samples don’t show reflexes of  secondary phases or impurities. The hexagonal Mn3Ge has three pairs of Mn atoms and one pair of Ge \natoms for unit cell (inset of Fig. 1a).  The basic magnetic structure of Mn atoms is \ninverse triangular, which means the chirality of the two spins triangles in a unit cell are \nopposite.  \n   The temperature variation of the magnetization, M(T) , for Mn 3Ge measured in zero \nfield cooled (ZFC) and fi eld cooled (FC) modes are shown in Fig. 2. In the ZFC  mode, \nthe sample was initially cooled from 400 K down to 2 K in zero fi eld and data were \ntaken up on increasing temperature in ap plied field, whereas, in the FC mode, data \nwere collected while cooling in fi eld. The ZFC and FC M(T)  measured at various \nfields for hexagonal Mn3Ge are shown  in Fig. 2a. The sample undergoes a magnetic \norder disorder transition around 370 K. This transition was previously recognized as \nthe Curie temperature T C connected with the residual ferromagnetic (FM) component \nof the AFM structure,18 which is very close to the Né el temperature (T N = 395 ± 10 K) \nof its antiferromagnetic structure.15 The residual FM component arises from the \nparent AFM phase as a result of small distortion of the hexagonal structure giving rise \nto a tiny uncompensated  FM component.15 The ZFC M(T)  curve measured in 0.1 T \nfollows a lower magnetization path than the FC curve, which makes a large \nirreversibility between the two curves. This irreversibility continues for the M(T) \nmeasured up to 1 T. This indicates the presence of a magnetic inhomogeneous state in \nthe whole temperature regime though the sample stays homogeneous structurally. The  \ndeviation between ZFC and FC diminishes with increasing the measuring fi eld but \ndoes not disappear completely  up to 7 T (Fig. 2b). To compare the magnetic state of  \nthe tetragonal Mn 3Ge, we have measured the high temperature M(T)  in a fi eld of 1 T \nas shown in Fig. 2c. The sample shows a TC of around 800 K, which is interesting for \napplication prospective.  \n   The magnetic hysteresis loops measured at 2 K after zero field cooling the samples \nfrom 400 K for both the hexagonal and the tetragonal phases are shown in Fig. 3. As \nexpected the hexagonal sample shows a very small magnetic moment of around \n0.09µ B/f.u. at 7 T as it arises from the little uncompensated moment of the AFM \nstructure. In addition, the linear and unsaturated magnetic behavior at higher fi elds \nclearly indicates the AFM nature of the sample. Due to the strong magneto-crystalline  \nanisotropy of the hexagonal structure, the sample shows a large coercive fi eld, HC, of \naround 0.5 T even though a small FM component present in the structure. The neutron \ndiffraction study in the hexagonal-Mn 3Ge suggests MS =0.02 µB/Mn.15 A little higher \nmagnetic moment in the present case indicates that an additional FM component \npresent in the sample. The tetragonal sample shows a much larger magnetization of \n0.7µ B/f.u. at 7 T. This indicates a normal ferrimagnetic ordering as observed in other \nMn rich compounds.19,20  In tetragonal phase the sample shows a HC of  only 0.27 T.     The presence of magnetic irreversibility in the hexagonal Mn 3Ge (Fig. 2) indicates \nthat there might be some extra FM component in the nearly AFM background. \nTherefore we have measured fi eld cooled hysteresis loops to map out any possible \ninteraction between the AFM host and FM clusters. The fi eld cooled hysteresis loops \nmeasured at different temperatures are shown in Fig. 4a. In the fi eld cooled mode a \nfield of 5 T was ap plied at 400 K before cooling the sample to measurement  \ntemperature. All measurements were performed in ± 7 T range. It is found that all loops \nare shifted in negative field direction indicating the presence of exchange bias (EB) in \nthe sample. The temperature dependence of the EB fi eld (H EB) and the coercive fi eld \n(HC) calculated from the FC hysteresis loops are shown in Fig. 4b. H EB and HC are \ncalculated using HEB = ǀH1+H 2ǀ/2 and H C = ǀH1-H2ǀ/2, where H 1 and H 2 are the lower \nand upper cut -off fields. A maximum HEB =70 mT has been calculated at 2 K. The EB \nmonotonically decreases with temperature that holds a small non-zero value at room \ntemperature. Similarly, H C of 0.47 T and 0.25 T have been calculated at 2 K and 300 K, \nrespectively. The cooling fi eld, HCF, dependence of EB and coercive fi eld are plotted \nin the inset of Fig. 4b. Both H EB and HC increase with cooling field up to 5 T that show \nsaturation behavior for higher fi elds. The tetragonal Mn3Ge sample does not show any \ndetectable exchange bias behavior. It has to be mentioned here that though EB  has \nbeen observed in various bulk materials including several Heusler alloys, the effect in \nmost of the cases has been limited to the low temperatures.21-27 The present finding of \nEB up to room temperature certainly is a subject of interest for application point of \nviews.  \n   Now that exchange bias has been established in the  hexagonal Mn 3Ge sample, it is \nclear that there must exist two different magnetically anisotropic states in the sample, \none being the AF M host, the other must be with a higher magnetic state. As mentioned \nearlier the EDXS and ICP -OES  analysis give an average composition of 3.04:0.96 Mn \nto Ge ratio. This implies that for some unit cells the extra Mn atoms occupy the Ge \nsites. As shown in Fig. 1a the magnetic moments of three Mn atoms lying in the basal \nplane constitute a triangular antiferromagnetic structure. In case of Mn replacing Ge \natom in the same plane, it will feel a strong magnetic frustration from the three basal \nplane Mn  atoms a nd the other three in the interlayer. In this process the overall \nmagnetic moment will increase to give a ferrimangetic like situation to decrease the \nfrustration. Since the distances between the Mn atoms are not equal, a ferromagnetic-\nlike structure is feasible. The present scenario will result in some ferrimagnetic \nclusters in the host AFM background. Hence, the exchange interaction between the \nAFM host and the ferrimagnetic clusters is proposed to rise to the observed exchange  \nanisotropy.  \n   In conclusion, we present the structural and magnetic properties of bulk Mn 3Ge \n(with excess Mn atomes). With different annealing conditions one can obtain a \nhexagonal and a tetragonal structure that display antiferromagnetic and ferrimagnetic  ordering, respectively. We demonstrate the presence of exchange bias (EB) up to room \ntemperature in the hexagonal sample. The EB in the present case arises from the \nexchange interaction between the triangular AFM host and the ferrimagnetic clusters. \nThe excess Mn atoms occupying the empty Ge sites convert the original triangular \nantiferromagnet to a ferrimagnet without changing the  crystal structure. The existence \nof EB up to room temperature in the bulk hexagonal Mn 3Ge is undoubtedly  promising \nfor potential application as most of the bulk materials show EB only  below room \ntemperature. As the EB possesses a great importance in memory devices the present \nwork will encourage further studies in thin films for application purpose.  \n \n    This work was financially  supported by the Deutsche Forschungsgemeinschaft DFG \n(Project No.TP 2.3-A of  Research Unit FOR 1464 ASPIMATT) and by the ERC  \nAdvanced Grant (291472) \"Idea Heusler\".  \n  1E. Kren, J. Paitz, G. Zimmer and E. 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Rev. \nB 83, 174448 (2011).  \n20P. Klaer, C. A. Jenkins, V. Alijani, J. Winterlik, B. Balke, C. Felser, and H. J. Elmers, Appl. Phys. \nLett. 98, 212510 (2011).  \n21M. Khan, I. Dubenk o, S. Stadler, and N. Ali, Appl. Phys. Lett. 91, 072510 (2007).  \n22A. K. Nayak, K. G. Suresh, and A. K. Nigam, J. Phys. D: Appl. Phys. 42, 115004 (2009).  23H. C. Xuan, Q. Q. Cao, C. L. Zhang, S. C. Ma, S. Y. Chen, D. H. Wang, and Y. W. Du, Appl. Phys. \nLett. 96, 202502 (2010).  \n24X. D. Tang, W. H. Wang, W. Zhu, E. K. Liu, G. H. Wu, F. B. Meng, H. Y. Liu, and H. Z. Luo, Appl. \nPhys. Lett. 97 (2010).  \n25A. K. Nayak, R. Sahoo, K. G. Suresh, A. K. Nigam, X. Chen, and R. V. Ramanujan, Appl. Phys. Lett. \n98, 232502 (20 11). \n26A. K. Nayak, C. Shekhar, J. Winterlik, A. Gupta, and C. Felser, Appl. Phys. Lett. 100, 152404 \n(2012).  \n27R. Machavarapu and G. Jakob, Appl. Phys. Lett. 102, 232406 (2013).  \n  Figure Captions  \n \nFIG. 1. (Color online) Room temperature powder x-ray diffraction (XRD) of Mn3Ge \n(a) annealed at 1073 K and (b) annealed at 700 K. The black lines show the \nexperimental data and the red lines correspond to the fitting. The inset of (a) shows the crystal structure of the hexagonal phase, whereas, the inset of (b) shows the crystal \nstructure of tetragonal phase. Mn atoms (blue spheres), Ge atoms (green spheres).  \n \nFIG. 2. (Color online) Temperature dependence of magnetization M(T) for the \nhexagonal and tetragonal Mn3Ge. (a) Zero field cooled (ZFC) and field cooled (FC) \nM(T)  of hexagonal Mn3Ge measured in 0.1 T, 0.5 T and 1 T. (b) ZFC and FC M(T)  of \nhexagonal Mn3Ge measured in 7 T. (c) High temperature M(T) for tetragonal Mn3Ge.  \n \nFIG. 3. (Color online) Magnetic isotherms M(H)  measured at 2 K for (a) hexagonal \nMn3Ge and (b) tetragonal Mn3Ge.  \n \nFIG. 4. (Color online) (a) M(H) loops for hexagonal-Mn\n3Ge measured at different \ntemperatures after field cooling the sample in 5 T. (b) Temperature dependence of \nexchange bias field HEB and coercive field H C calculated from the field cooled M(H ) \nloops. Inset of (b) shows the cooling field, H CF, dependence of H EB and HC measured \nat 2 K.  \n   \n"    },    {        "title": "1106.3966v2.Magnetic_Order_of_the_Hexagonal_Rare_Earth_Manganite_Dy_0_5_Y_0_5_MnO3.pdf",        "content": "Magnetic order of the hexagonal rare-earth manganite Dy 0:5Y0:5MnO 3\nJoel S. Helton1;\u0003, Deepak K. Singh1;2, Harikrishnan S. Nair3, and Suja Elizabeth4\n1NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA\n2Department of Materials Science and Engineering,\nUniversity of Maryland, College Park, MD 20742, USA\n3J ulich Centre for Neutron Science, Forschungszentrum J ulich,\nOutstation at FRM-II, D-85747 Garching, Germany and\n4Department of Physics, Indian Institute of Science, Bangalore 560012, India\n(Dated: November 4, 2021)\nHexagonal Dy 0:5Y0:5MnO 3, a multiferroic rare-earth manganite with geometrically frustrated\nantiferromagnetism, has been investigated with single-crystal neutron di\u000braction measurements.\nBelow 3.4 K magnetic order is observed on both the Mn (antiferromagnetic) and Dy (ferrimagnetic)\nsublattices that is identical to that of undiluted hexagonal DyMnO 3at low temperature. The Mn\nmoments undergo a spin reorientation transition between 3.4 K and 10 K, with antiferromagnetic\norder of the Mn sublattice persisting up to 70 K; the antiferromagnetic order in this phase is distinct\nfrom that observed in undiluted (h)DyMnO 3, yielding a qualitatively new phase diagram not seen in\nother hexagonal rare-earth manganites. A magnetic \feld applied parallel to the crystallographic c\naxis will drive a transition from the antiferromagnetic phase into the low-temperature ferrimagnetic\nphase with little hysteresis.\nPACS numbers: 75.25.-j, 75.85.+t, 75.50.Ee, 75.47.Lx\nI. INTRODUCTION\nThe crystalline structure of rare-earth manganites\n(RMnO 3withR= Y, Sc, or a lanthanide) is deter-\nmined by the ionic radius of the R3+cation. Materi-\nals with a large R3+ionic radius ( R= La through Tb)\ncrystallize in an orthorhombic perovskite structure, while\nmaterials with a smaller R3+ionic radius ( R= Y, Sc,\nor Ho through Lu)1{6crystallize in a hexagonal struc-\nture. DyMnO 3typically crystallizes in the orthorhombic\nstructure,7but with proper growth conditions hexagonal\n(h)DyMnO 3can be stabilized.8,9The hexagonal rare-\nearth manganites, (h) RMnO 3, are paraelectric at very\nhigh temperatures but display a structural transition\n(TC\u00191000 K) to a ferroelectric phase with the noncen-\ntrosymmetric P63cmspace group.10The (h)RMnO 3ma-\nterials feature slightly distorted triangular lattice planes\nof Mn ions; the antiferromagnetic nearest-neighbor ex-\nchange interaction leads to geometrically frustrated mag-\nnetism. Below a N\u0013 eel temperature of \u0019100 K these ma-\nterials are magnetically ordered, with easy-plane antifer-\nromagnetic order of the Mn sublattice coexisting with\nthe ferroelectric order. Many hexagonal rare-earth man-\nganites display one or more spin reorientation transitions\nof the Mn moments at lower temperatures11,12or under\nan applied \feld;13,14theRions also form distorted tri-\nangular lattice planes and, for magnetic ions, will order\nalong thecaxis. Several of these materials have attracted\ninterest because of signi\fcant magnetoelectric15,16or\nmagnetoelastic17e\u000bects. Despite the structural similari-\nties between the various members of the (h) RMnO 3fam-\nily, the magnetically ordered structures often di\u000ber, with\nstructures that transform according to each of the four\none-dimensional irreducible representations of the point\ngroup observed in at least one material. The dependenceof the magnetic ordering of the Mn sublattice on the\nrare-earth element has been attributed to the ionic ra-\ndius of theR3+cation18or a biquadratic 3 d-4fmagnetic\ncoupling.19\nHexagonal (h)DyMnO 3, with the largest rare-earth\nionic radius of this series, features an interesting mag-\nnetic phase diagram.9,19,20In the low-temperature phase\n(below\u00198 K) both the Mn and Dy sublattices are mag-\nnetically ordered according to the \u0000 2irreducible rep-\nresentation. At a temperature of 8 K the Mn mo-\nments undergo a spin reorientation transition and are\nordered in the \u0000 4representation up to 68 K.19X-ray\nresonant magnetic scattering measurements also \fnd a\nweak Dy moment in this temperature range, ordered ac-\ncording to the \u0000 3representation.20This incompatible\norder, with di\u000berent irreducible representations present\non the two sublattices, calls into question long stand-\ning assumptions about the rigidity of the 3 d-4finter-\naction in (h) RMnO 3materials.19YMnO 3features mag-\nnetism only on the Mn sublattice and is known to order\nin the \u0000 3irreducible representation below TN\u001975 K.21\nYMnO 3has also been reported to feature large magne-\ntoelastic e\u000bects17and coupling between electric and mag-\nnetic domains.22Compounds where magnetic rare-earth\nions are partially replaced with nonmagnetic Y3+ions,\nsuch as Ho 1\u0000xYxMnO 323,24and Er 1\u0000xYxMnO 3,25,26al-\nlow for further examination of the role that the rare-\nearth ions play in determining the magnetic structure and\nopen up the possibility of novel phase diagrams not ob-\nserved in undoped compounds. We report single-crystal\nneutron di\u000braction studies of hexagonal Dy 0:5Y0:5MnO 3\n(DYMO) and \fnd evidence for a spin reorientation tran-\nsition not seen in other hexagonal rare-earth manganites\nat zero \feld.arXiv:1106.3966v2  [cond-mat.str-el]  24 Aug 20112\nII. EXPERIMENT\nLarge, high-quality single-crystal samples of\nDy0:5Y0:5MnO 3were prepared as previously reported.27\nDYMO crystallizes in the hexagonal P63cm space\ngroup (#185) with lattice parameters (at 300 K) of\na=b= 6.161(1) \u0017A andc= 11.446(2) \u0017A. As in other\nhexagonal rare-earth manganites, Mn3+ions (S= 2)\noccupy the 6 cpositions at ( x, 0, 0) with x\u00191\n3. A\nmaterial with x=1\n3would feature perfect triangular\nlattice planes; in DYMO x= 0.3379(4) yielding a\nslightly distorted triangular lattice. The rare-earth R3+\nions occupy two crystallographically distinct sites, at\nthe 2aand 4bWycko\u000b positions, and also form distorted\ntriangular lattice planes. In DYMO the rare-earth sites\nare occupied with equal probability by nonmagnetic Y3+\nions and6H15=2Dy3+ions (gJ= 10\u0016B), as determined\nby x-ray powder di\u000braction on crushed single crystals.27\nPreviously reported speci\fc heat measurements found\npeaks at 3 K and 68 K; analogously with other hexag-\nonal rare-earth manganites such as DyMnO 3it was\nsuggested that these peaks correspond to the onset of\nantiferromagnetic order of the Mn lattice, TMn\nN\u001968 K,\nand ferrimagnetic order of Dy moments on the rare-earth\nlattice,TDy\nN\u00193 K.\nNeutron di\u000braction experiments were carried out using\nthe BT9 thermal triple-axis spectrometer at the NIST\nCenter for Neutron Research. The neutron initial and\n\fnal energies were selected using the (0 0 2) re\rection\nof the pyrolytic graphite (PG) monochromator and ana-\nlyzer. We used 400-470-400-open collimation as well as PG\n\flters to reduce contamination of the beam with higher-\norder neutron wavelengths. The detailed temperature\ndependence of four magnetic re\rections (shown in Fig. 1)\nwas measured using a large ( \u00191 g) single crystal mounted\nin the (H0L) scattering plane at a \fxed neutron energy\nof 30.5 meV ( \u0015= 1.64 \u0017A). Re\fnement of the ordered\nmoments utilized di\u000braction measurements taken at tem-\nperatures of 1.6 K, 25 K, and 120 K at a \fxed neutron\nenergy of 30.5 meV. In order to minimize absorption of\nneutrons by the sample, these measurements were taken\nwith small single crystals: an 8-mg sample mounted in\nthe (H0L) scattering plane and a 6-mg sample mounted\nin the (H K 0) scattering plane. Some magnetic re-\n\rections also featured weak nuclear contributions which\nwere removed by subtracting the 120 K intensity; the\nDebye-Waller factor has been ignored, which is justi\f-\nable, given that the intensities of nuclear Bragg peaks\nwith no magnetic intensity [such as (0 0 L) whereLis\neven] remained constant within the statistical uncertain-\nties between 1.6 K and 120 K. The scattering intensity\nin absolute units, and from this the value of the ordered\nmoments, was determined by normalizing the intensities\nof nuclear Bragg peaks measured at 120 K to the calcu-\nlated nuclear intensities. The intensities of the (1 0 0)\nand (3 \u00161 0) re\rections in a magnetic \feld were measured\non the 6-mg sample mounted in the ( H K 0) scattering\nplane; the sample was placed inside a helium \row dewarwith a minimum temperature of 4 K inserted into a 7-T\nvertical \feld superconducting magnet. These data were\ntaken at a \fxed neutron energy of 14.7 meV ( \u0015= 2.36 \u0017A).\nAll neutron di\u000braction data are reported in terms of the\nintegrated intensity, integrated over a rocking curve ( \u0012\nscan) through the peak position.\nThe allowed magnetic structures in DYMO correspond\nto the six irreducible representations of the P63cmspace\ngroup with propagation vector ~k= 0; as in other hexago-\nnal rare-earth manganites, only the four one-dimensional\nirreducible representations (designated as \u0000 1through \u0000 4)\nare required to describe the observed structures.28,29For\neach of these representations the Mn sublattice displays\nantiferromagnetic order within the abplane, with the\nthree spins around any triangle oriented 120\u000eapart. The\nR3+sublattice is ordered along the crystallographic c\naxis. The rare-earth 4 bsites are antiferromagnetically or-\ndered for the \u0000 1, \u00003, and \u0000 4representations; the 2 arare-\nearth sites are antiferromagnetically ordered in the \u0000 3\nrepresentation and paramagnetic for the \u0000 1and \u0000 4rep-\nresentations. The 2 aand 4bsites are each ferromagneti-\ncally ordered in the \u0000 2representation, with the coupling\nbetween the two sites yielding either ferromagnetism or\nferrimagnetism. In phases where only the Mn sublattice\nis magnetically ordered a hexagonal rare-earth mangan-\nite withx=1\n3(featuring undistorted triangular lattice\nplanes) would have perfectly identical neutron scattering\nstructure factors in either the \u0000 1or \u0000 3representations, as\nwell as in the \u0000 2or \u0000 4representations. Unambiguous dif-\nferentiation between these homometric magnetic struc-\ntures requires complementary measurement techniques\nsuch as optical second harmonic spectroscopy.1,21,30The\ndistortion of the triangular planes leads to some vari-\nation in the neutron scattering structure factors; how-\never, these results, from single-crystal di\u000braction mea-\nsurements on a strongly absorbing sample, will not be\nable to distinguish between the \u0000 1and \u0000 3structures.\nIII. RESULTS\nA. Zero-\feld magnetic order\nFigure 1 displays the temperature dependence of the\nintegrated intensities of the (1 0 0), (1 0 1), (2 0 0), and\n(2 0 1) Bragg re\rections measured while warming from\n2.1 K to 130 K. The background and any high temper-\nature nuclear contributions to the intensities have been\nsubtracted. At 2.1 K all four re\rections display mag-\nnetic intensity, with the (1 0 1) re\rection the strongest.\nBetween 2.1 K and 3.4 K the measured intensities of\nthe (H0 1) re\rections remain relatively constant while\nthose of the ( H0 0) re\rections steadily decrease with\nincreasing temperature, with an intensity at 3.4 K that\nis about 68% of the base temperature intensity. On in-\ncreasing the temperature beyond 3.4 K, the intensities of\nthe (H0 0) peaks steadily increase while the intensities\nof the (H0 1) peaks decrease so that the (1 0 0) re\rec-3\n02 04 06 08 0100120140051015200\n2468101214051015 \n  \n(1 0 0) \n(1 0 1) \n(2 0 0) \n(2 0 1)Integrated Intensity (arb. units)T\nemperature (K) \n Int. Intensity (arb. units)T\nemperature (K)\nFIG. 1: (Color online) Integrated magnetic intensities of the\n(1 0 0), (1 0 1), (2 0 0), and (2 0 1) Bragg re\rections as a func-\ntion of temperature. All data were measured while warming.\nThe inset shows a closer view of the low temperature portion\nof the graph. Transitions are observed at TMn\nN= 70 K and\nTDy\nN= 3.4 K; these temperatures are designated with dashed\nvertical lines. The red dotted line re\rects a \ft of the (1 0 0)\nintensity to an order parameter form with \f= 0.25 \u00060.05.\nError bars throughout this article are statistical in nature and\nrepresent one standard deviation.\nTABLE I: Calculated magnetic scattering intensities for the\nre\rections displayed in Fig. 1. The most intense re\rection\nin each column has been normalized to 100. In the low-\ntemperature phase below 3.4 K the data are consistent with\n\u00002order for both the Mn and Dy sublattice. In the anti-\nferromagnetic phase between 10 K and 70 K the data are\nconsistent with Mn sublattice order in either the \u0000 1or \u0000 3\nirreducible representation.\nMn order Dy order\n\u00001 \u00002 \u00003 \u00004 \u00002\n(1 0 0) 100 0 100 0 100\n(1 0 1) 17 100 15 100 0\n(2 0 0) 32 0 27 0 38\n(2 0 1) 1 26 1 33 0\ntion becomes the strongest measured; this rapid change\nin intensity persists only to around 10 K. This will be\nshown to correspond to a spin reorientation transition of\nthe Mn moments and perhaps explains the 10 K anomaly\nreported27in the derivative of 1/ \u001f. Above 10 K the in-\ntensities of all peaks slowly decrease until reaching zero\nat 70 K. The intensity can be \ft to an order parameter\nform:I(T)/(jT\u0000TNj=TN)2\f. The dotted red line in\nFig. 1 is a \ft of the (1 0 0) intensity to this form with\n\f= 0.25\u00060.05.\nTable I displays the calculated magnetic intensities for\nthe re\rections shown in Fig. 1 with the largest re\rec-tion in each column normalized to 100. The reported\nintensities re\rect the calculated magnetic cross section\ncorrected for the resolution function of a triple-axis neu-\ntron spectrometer. Calculated intensities are shown for\nMn moments ordered in each of the four one-dimensional\nirreducible representations, as well as for Dy moments\nordered in the \u0000 2representation. Previous magnetiza-\ntion measurements on DYMO27with~Hjj~ crevealed\na low-temperature state with a spontaneous magnetiza-\ntion of\u00190.5\u0016Bper formula unit (measured at 2 K). Of\nthe four irreducible representations present in hexagonal\nrare-earth manganites only the \u0000 2structure is consistent\nwith either ferrimagnetism or ferromagnetism; undoped\nhexagonal DyMnO 3is ferrimagnetic at low temperatures\nand is known to order in the \u0000 2representation.20When\nordering in the \u0000 2structure the magnetic cross sections\nof the (H0 1) re\rections depend only on the ordered\nmoment of the Mn ions while those of the ( H0 0) re-\n\rections depend only on the ordered moment of the Dy\nions. The Mn ordered moment is therefore found to be\nalmost temperature independent below 3.4 K, while the\nDy ordered moment decreases continuously with increas-\ning temperature between 2.1 K and 3.4 K.\nAssuming a ferrimagnetic structure with ordered mo-\nments on the 2 aand 4bsites that are antiparallel but\nequal in magnitude, a re\fnement of magnetic re\rec-\ntions measured at 1.6 K reveals ordered moments of\n3.7\u00060.4\u0016Bfor the Mn ions and 3.1 \u00060.3\u0016Bfor the Dy\nions (1.6\u00060.2\u0016Bfor each rare-earth site). This struc-\nture is displayed in Fig. 2(a). (The crystal structures\nshown in Figure 2 were produced using VESTA .31) When\nmagnetic domains are fully aligned this ordered moment\nwill lead to a net magnetization of 3.1 \u00060.3\u0016Bper unit\ncell (or 0.52\u00060.05\u0016Bper formula unit) along the crys-\ntallographic caxis, which is consistent with the reported\nbulk spontaneous magnetization.27Allowing for di\u000berent\nordered moments on the Dy 4 band 2asites did not ap-\npreciably improve the \ft of the 1.6 K data, nor could\nallowing for di\u000berent moments on these sites produce a\ncomparably good \ft while yielding a net moment consis-\ntent with magnetization measurements. A ferromagnetic\nground state can likewise be excluded. A ferromagnetic\nstate with equal ordered moments on the 2 aand 4bsites\nwould give no intensity for the ( H0 0) re\rections; when\nordering in the \u0000 2structure the intensities of these re-\n\rections depend on the di\u000berence in moment at the rare-\nearth sites: I(~QH00)/ j~M4b\u0000~M2aj2, where the ordered\nmoments on the 2 aand 4bsites are~M2aand~M4b. A fer-\nromagnetic state with di\u000berent ordered moments on the\n2aand 4bsites is consistent with the neutron di\u000braction\ndata only while yielding a net moment inconsistent with\nmagnetization measurements. The Mn ordered moment\nat 1.6 K is close to the full moment value; however, the\nDy ordered moment is considerably reduced from the full\n10\u0016Bvalue of the6H15=2Dy3+ions. The low tempera-\nture magnetization in undoped DyMnO 3would suggest\na similarly reduced ordered moment for the Dy ions in\nthe ferrimagnetic phase.84\nT< 3.4 K\n2 (P63c′m′)\n(a)10 K < T< 70 K\n1(P63cm)\n3  (P6′3cm′)\n(b)\nFIG. 2: (Color online) (a) Magnetic structure in the low-\ntemperature ferrimagnetic phase, present below TDy\nN= 3.4 K.\nThe structure belongs to the P63c0m0magnetic space group\n(\u00002irreducible representation). Mn ions at z= 0 are shown in\npurple, Mn ions at z= 1/2 are shown in orange, and the rare-\nearth ions are shown in blue. The re\fned ordered moments\nare 3.7 \u00060.4\u0016Bon the Mn ions and 3.1 \u00060.3\u0016Bon the\nDy ions. (b) The two possible magnetic structures for the\nantiferromagnetic phase: P63cmmagnetic space group (\u0000 1\nirreducible representation) on the left and P60\n3cm0magnetic\nspace group (\u0000 3) on the right. The Mn ordered moment is\n3.5\u00060.4\u0016B. Any Dy ordered moment in this temperature\nrange is too small to be measured by neutron di\u000braction. The\nstructural unit cell is outlined in gray\nThe magnetic structure in the antiferromagnetic phase\n(10 K<T < 70 K) is consistent with an ordered moment\non only the Mn sublattice, with order according to either\nthe \u0000 1or \u0000 3irreducible representation. These magnetic\nstructures are shown in Fig. 2(b). It should be noted\nthat neither of these possibilities are consistent with the\nstructure of the Mn moments in (h)DyMnO 3, which or-\nder in the \u0000 4representation above the low-temperature\nferrimagnetic phase.19A re\fnement of magnetic re\rec-\ntions measured at 25 K gives an ordered moment of\n3.5\u00060.4\u0016Bfor the Mn sublattice. The ordered moment\nat 25 K is comparable in size to that determined at 1.6 K,\nsuggesting a spin reorientation transition where Mn spins\nrotate between 3.4 K and 10 K with little change in\nmagnitude. Element speci\fc x-ray resonant magnetic\nscattering measurements on hexagonal DyMnO 320and\nHoMnO 332have reported a weak rare-earth moment at\ncomparable temperatures, induced by a splitting of the\nground-state crystal \feld doublet. While DYMO might\nsimilarly feature a weak Dy ordered moment in this tem-\nperature range, neutron scattering measurements will be\nsensitive to only the much larger ordered moment of the\nMn sublattice and the re\fnement is not improved by al-\nlowing for an ordered moment on the Dy sublattice at\n25 K.\nThe magnetic structure of YMnO 3is known to be the\n\u00003representation.21If the antiferromagnetic phase of\nDYMO were likewise \u0000 3, the spin reorientation transi-\ntion into the low-temperature \u0000 2phase would di\u000ber fromthe spin reorientation transitions previously observed in\nhexagonal rare-earth manganites. A magnetic struc-\nture in the \u0000 1irreducible representation can be trans-\nformed into the \u0000 2representation through a 90\u000ecoun-\nterclockwise rotation of all Mn moments. However, a\ntransformation of a structure in the \u0000 3representation\nas shown in Fig. 2(b) into the \u0000 2representation would\nrequire a 180\u000erotation of only the Mn moments in the\nz=1\n2plane. In other hexagonal rare-earth mangan-\nites, it has been suggested30that spin reorientation tran-\nsitions between the four irreducible representations oc-\ncur via in-phase or antiphase rotations where the mo-\nments in adjacent Mn layers rotate with an equal (in-\nphase) or opposite (antiphase) direction. For example,\nthe Mn moments in HoMnO 3display an in-phase re-\norientation transition11(\u00004to \u0000 3) atT\u001940 K and\nan antiphase transition (\u0000 3to \u0000 1) atT\u00198 K, while\nthe Mn moments in (h)DyMnO 3display an antiphase\nrotation19(\u00004to \u0000 2) atT\u00198 K. Further, while materials\nsuch as ScMnO 3,2ErxY1\u0000xMnO 3,25and YMnO 3under\npressure18have been reported to feature broad tempera-\nture ranges where the magnetic structure is not one of the\nfour irreducible representations, the observed structures\ncan still be described as an in-phase or antiphase rota-\ntion of all Mn moments away from one of the principal\nstructures.\nB. Field dependence of the magnetic order\nBelowTDy\nN= 3.4 K DYMO is ferrimagnetic with\na spontaneous net magnetization. Above TDy\nN,M(H)\ncurves display symmetric magnetization steps at a criti-\ncal \feld value that increases with temperature; the size\nof these steps decreases with increasing temperature, be-\ncoming immeasurably small around 40 K.27This behav-\nior is remarkably similar to that previously reported in\n(h)DyMnO 3,8but the moment increase associated with\nthe magnetization steps in DYMO is about one half of the\nincrease in (h)DyMnO 3. Figure 3 displays the intensity\nof the (1 0 0) magnetic Bragg re\rection as a function of\n\feld with~Hjj~ cat temperatures between 4 K and 50 K,\nmeasured while increasing the \feld after zero \feld cool-\ning. For each of these temperatures, we \fnd a sudden\ndrop in the (1 0 0) intensity at a critical \feld value that\nincreases with temperature. As was shown in Fig. 1, the\nintensity of the (1 0 0) re\rection will be much higher in\nthe antiferromagnetic phase than in the low-temperature\nphase, such that this behavior is easily associated with a\n\feld-induced transition into the ferrimagnetic phase. The\ndata de\fnitively show a spin reorientation of the Mn mo-\nments along with the \feld-induced ferrimagnetic order-\ning of the Dy moments, as the intensity of this re\rection\nwould not drop if the Mn moments remained ordered in\nthe original structure. The intensity of this re\rection in\nthe \feld-induced phase is weaker than in zero \feld at low\ntemperature [the dotted horizontal line represents the ex-\npected intensity of the (1 0 0) re\rection at 2 K and zero5\n0.00 .51 .01 .52 .02 .50369121518 \n  \nT = 4 K \n6 K \n11 K \n15 K \n20 K \n50 KIntegrated Intensity (arb. units)µ\n0H (T)\nFIG. 3: (Color online) Integrated intensity of the (1 0 0) peak\nas a function of \feld. The data were taken while increasing\n\feld. Lines are a guide to the eye. The dotted horizonal line\nrepresents the expected intensity at 2 K in zero \feld.\n\feld]. This likely arises from a considerably smaller Dy\nordered moment, consistent with the smaller magnitude\nof the magnetization steps at higher temperatures in the\nM(H) curves. Interestingly, neutron di\u000braction is sensi-\ntive enough that this transition is still clearly measurable\nin the 50 K data, while it could not be measured above\n40 K in the magnetization data.27\nThe \feld dependence of the intensities at the (1 0 0)\nand (3 \u00161 0) re\rections at T= 15 K is displayed in Fig. 4.\nIn Fig. 4(a), both re\rections show the expected tran-\nsition at\u00160Hc\u00191 T; data taken in rising and falling\n\feld show only a slight hysteresis, consistent with the\nsmall hysteresis of the magnetization steps in the bulk\nmagnetization data. In addition to DYMO, undoped\n(h)DyMnO 3displays little hysteresis while magnetiza-\ntion steps in ErMnO 333display considerable hysteresis.\nThe (3 \u00161 0) re\rection displays scattering in the low-\ntemperature \u0000 2phase and very little scattering in the\nantiferromagnetic phase, such that this change in inten-\nsity is likewise consistent with a \feld-induced transition\nbetween the two. A more detailed view of the (1 0 0)\nintensity in the high-\feld region is shown in Fig. 4(b).\nInterestingly, the intensity of this peak rises slowly with\nincreasing \feld up to about 4 T but then decreases with\nincreasing \feld beyond 5 T. This likely re\rects the evolu-\ntion of the system from ferrimagnetic to ferromagnetic as\nthe \feld is increased, as the fully polarized state (ferro-\nmagnetic with an equal moment for the 4 aand 2bsites)\nwould display no scattering at (1 0 0).\nIV. SUMMARY\nBelowTDy\nN= 3.4 K Dy 0:5Y0:5MnO 3is magnetically or-\ndered in the \u0000 2irreducible representation with antiferro-\nmagnetic Mn order and ferrimagnetic Dy order. Between\n3.4 K and 10 K we observe a spin reorientation transition\nof the Mn moments. Antiferromagnetic order of the Mn\nsites persists up to TMn\nN= 70 K. The magnetic structure\n01234560.00.61.21.805101520(\n3 -1 0) Integrated Intensity (arb. units)(\nb)µ\n0H (T) \n (a) (1 0 0) Integrated Intensity (arb. units) \n            (1 0 0) \n            (3 -1 0)    FieldD\necreasing    FieldI\nncreasing0\n.00.51.01.52.02.5FIG. 4: (Color online) (a) Integrated intensities of the (1 0 0)\n(left scale) and (3 \u00161 0) (right scale) peaks measured at\nT= 15 K. Solid symbols represent measurements taken with\nincreasing \feld, and open symbols represent those taken with\na decreasing \feld. (b) Detail of the high \feld region for the\n(1 0 0) intensity.\nof this phase is either \u0000 1or \u0000 3, inconsistent with the \u0000 4\norder observed in undiluted (h)DyMnO 3.19In the anti-\nferromagnetic phase, a magnetic \feld applied parallel to\nthecaxis will drive a \feld-induced transition into the\nlow-temperature ferrimagnetic phase with little hystere-\nsis. Further knowledge of the di\u000berent magnetic struc-\ntures and reorientation transitions displayed by the Mn\nmoments in various (h) RMnO 3materials should lead to\nimproved understanding of the unusual 3 d-4fmagnetic\ncoupling present in these materials.19While the mag-\nnetic structure of the antiferromagnetic phase cannot be\nde\fnitively determined from this experiment, it is clear\nthat Dy 0:5Y0:5MnO 3displays a zero-\feld spin reorien-\ntation transition not previously observed in any other\nhexagonal rare-earth manganite. If the magnetic struc-\nture is \u0000 3, then the \u0000 3to \u0000 2spin reorientation transition\nwould involve only half of the Mn ions in a manner not\nobserved in other (h) RMnO 3materials. The H-Tphase\ndiagram of HoMnO 3features a boundary between \u0000 1and\n\u00002phases13,29at\u00160H\u00192 T, yet a zero \feld \u0000 1to \u0000 2\nspin reorientation transition has not been reported. 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Phys. 2015, 00000 (26 pages)\nDOI: 10.1093 /ptep/0000000000\nEffective field theory and the scattering process\nfor magnons in the ferromagnet,\nantiferromagnet, and ferrimagnet\nShinya Gongyo1, Yuta Kikuchi2, Tetsuo Hyodo3, and Teiji Kunihiro4\n1Yukawa Institute for Theoretical Physics, Kyoto Universit y, Kyoto 606-8502, Japan\n∗E-mail: shinya.gongyo@yukawa.kyoto-u.ac.jp\n1Theoretical Research Division, Nishina Center, RIKEN, Sai tama 351-0198, Japan\n1CNRS, Laboratoire de Math´ ematiques et Physique Th´ eoriqu e, Universit´ e de Tours,\n37200 France\n2Department of Physics, Kyoto University, Kyoto 606-8502, J apan\n2Department of Physics and Astronomy, Stony Brook Universit y, Stony Brook, New\nYork 11794-3800, USA\n3Yukawa Institute for Theoretical Physics, Kyoto Universit y, Kyoto 606-8502, Japan\n4Department of Physics, Kyoto University, Kyoto 606-8502, J apan\n................................................... ............................\nWe discuss that a low-energy effective Lagrangian relying on SO(3) →SO(2) is appli-\ncable for a ferrimagnet as well as a ferromagnet and an antiferrom agnet. We derive the\nHamiltonian from the Lagrangian using the Poisson and Dirac bracket s, and deter-\nmine the particle states. The analysis of the particle states shows t hat there exist\nnot only massless modes with the dispersion relations ω∝ |k|,|k|2, i.e., the so-called\ntype-I and type-II Nambu-Goldstone modes, respectively, but a lso gapped modes with\nω∝m2+|k|2. We clarify how the coefficients of the terms with one time derivative\nand those with two time derivatives in the effective Lagrangian deter mine the order\nparameters specifying whether the system is in a ferromagnetic, a ntiferromagnetic or\nferrimagnetic state; we stress that the gapped mode resulting fo rm the spontaneous\nsymmetry breaking appears only in the ferrimagnetic system and no t in the ferromag-\nnetic and antiferromagnetic systems. We also establish the power c ounting scheme and\ncalculate the scattering amplitudes and thereby the scattering len gths between the two\nNambu-Goldstone bosons. We show that the scattering length of t he gapped mode is\nfinite and proportional to the gap. This characteristic property o f the gapped NG mode\ncan be used to discriminate it from gapped excitations which originate in other mech-\nanisms. Finally, we study the effects of the explicit symmetry breakin g that are given\nby an external magnetic field and a single-ion anisotropy, and show t hat the external\nmagnetic fields do not have any effects on the scattering amplitudes in all the spin sys-\ntems as was known for the ferromagnet system. In contrast, th e anisotropy does affect\nthe scattering amplitudes, the phase shift, and the scattering len gth except for spin\n1/2 systems. This result supports the possibility of the Efimov effec t in spin systems\ndiscussed in previous studies.\n................................................... ...........................................\nc/circlecopyrtThe Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.\nThis is an Open Access article distributed under the terms of the Creative Commons Attribution License\n(http://creativecommons.org/licenses/by-nc/3.0), whi ch permits unrestricted use,\ndistribution, and reproduction in any medium, provided the original work is properly cited.Subject Index B31, B36, I46\n1. Introduction\nThe method of effective field theory (EFT) is a powerful and succ essful tool to understand\nthe low-energy behaviors in various systems extending from particle physics to condensed\nmatter physics [1–7]. The EFT is based on the spontaneous sym metry breaking pattern in\nthe underlying theory, and is constructed in terms of the cor responding Nambu-Goldstone\n(NG) fields, which leads to the systematic and model-indepen dent analysis of the low-energy\nbehavior. The method has been originally employed in the fiel d of particle physics. More\nspecifically, chiral perturbation theory, which is an EFT re lying on chiral symmetry breaking\nand the NG modes, i.e. pions, has succeeded in describing the low-energy behaviors in quan-\ntum chromodynamics (QCD) such as the pion scattering proces ses and hadron structures\n[2, 3, 7]. Moreover, the EFT has been used for the analysis of t he hadron properties at finite-\ndensity QCD, in various phases including the chiral-symmet ry broken or partially restored\nphases, the kaon-condensed phase, the color superconducti vity phase and so on. Recently,\nan EFT has been developed to incorporate spatial symmetry br eaking [8, 9] for describing\nthe inhomogeneous chiral phases, which may appear in high-d ensity quark matter.\nThe EFT approach has also been widely applied to condensed ma tter physics [4–6, 10–\n14] : For instance, the translational and rotational symmet ries are spontaneously broken\nin the ground state of the solid states, and the spin rotation al symmetry is also broken\nin spin systems with some order. Then according to the symmet ry breaking patterns, the\nEFTs have been constructed in terms of spin waves, i.e. magno ns, as the NG modes in spin\nsystems [4, 12] and the sound wave, i.e. phonon, in solid-sta te systems [11]. In particular,\nusing the EFT for spin systems enables us to analyze the low-e nergy properties such as the\nnature of the low-energy modes i.e., magnons and their scatt ering processes in ferromagnets\nand antiferromagnets [13]: It was shown that the difference in the order parameters between\nferromagnet and antiferromagnet leads to different characte ristics of the low-energy (NG)\nmodesandamplitudesofthemagnon-magnonscatteringeveni fthepatternsofthesymmetry\nbreakings are the same.\nIn this paper, we develop a unified EFT for the collinear phase s of spin systems which\ninclude ferromagnets, antiferromagnets and ferrimagnets where the spin directions at lattice\nsites are all aligned or anti-aligned. On the basis of this EF T, we elucidate the characteris-\ntics of the low-energy magnon-magnon scattering, which sho uld be of basic importance in\nexamining the possible Efimov effect of magnons as the NG modes i n all the spin-ordered\nphases. We remark that Efimov effects of NG bosons have been disc ussed for magnons in\nthe ferromagnet [15, 16] and pions in QCD [17].\nThe EFT approach has also been utilized to count the number of NG modes as well\nas to understand the dispersion relations. In systems witho ut Lorentz invariance including\nfinite density QCD, spin systems, and solid systems the numbe r of NG modes can be less\nthan that of the broken generators, while in Lorentz-invari ant systems, the number of NG\nmodes coincides with that of the broken generators [18–23]. The NG modes in systems\nwithout Lorentz invariance have been classified into type-I and type-II, whose energies are\n2/26proportional to odd and even powers of the momentum, respect ively, while in Lorentz-\ninvariant systems, the NG modes always have a linear dispers ion relation. In fact, in the\nferromagnet, the number of NG modes is less than that of the br oken generators, and the\nNGmodeshowsthequadraticdispersionrelation.Ontheothe rhand,intheantiferromagnet,\nthe number of NG modes coincides with that of the broken gener ators, and the NG modes\nshow the linear dispersion relation.\nQuite recently, it has been clarified on the basis of the EFT th at gapped modes may\nalso appear in addition to the type-II NG modes as a result of t he spontaneous symmetry\nbreaking [24–28]. Therefore the NG modes can be classified in to at least three types, i.e.,\ntype-I, type-II and gapped modes. The gapped NG modes are acc ompanied by the type-\nII modes, so that a reduction of the number of NG modes does not occur [26]. A general\nrelation between the appearance of gapped NG modes and some s pecific nature of the order\nparameters has been discussed[27, 28]. Inthis paper, wedis cuss that thegappedNG modeis\nidentified with the magnon in ferrimagnets from the perspect ive of the order parameters and\ndispersion relations. We make clear how the EFT works by cons tructing the power counting\nscheme and that the gapped mode is reliable and affects the magn on-magnon scattering in\nferrimagnets. We find nonzero scattering lengths only for pr ocesses involving the gapped NG\nmode. The strength of the scattering length is given by the ga p of the mode multiplied by\na universal factor.\n2. Effective field theory for SO(3) →SO(2) and the scattering amplitude\n2.1. Construction of the effective Lagrangian for SO(3) →SO(2)\nWe consider the spin systems described by the Hamiltonian wh ich is invariant under the\nglobal SO(3) transformation of spins Sa\nm(a= 1,2,3):\nH=Hinv(Sa\nm), (1)\nSa\nm→gabSb\nm, gab∈SO(3). (2)\nWe assume that the ground state is in the collinear phases whe re all of the spins defined\non each lattice site mare aligned or anti-aligned. When we set the 3-axis in the dir ection\nof the spin alignment and anti-alignment, the order paramet er for the alignment is given\nby the magnetization,/angbracketleftbig/summationtext\nmS3\nm/angbracketrightbig\nwhile that for the anti-alignment is given by the staggered\nmagnetization,/angbracketleftbig/summationtext\nm(−1)mS3\nm/angbracketrightbig\n. When the order parameter is nonzero, the global SO(3)\nsymmetry is spontaneously broken down to SO(2) correspondi ng to the rotation around the\n3-axis.\nAs a concrete example, let us consider the Heisenberg model i n a ferromagnet system,\nH=−J/summationdisplay\n/angbracketleftm,n/angbracketrightSa\nmSa\nn, (3)\nwhere∝an}bracketle{tm,n∝an}bracketri}htdenotesthesummationoverthenearestneighbors.TheHamil tonianisinvariant\nunder the global SO(3) transformation (2). The Hamiltonian also has spatial translational\nandrotational symmetryinthecontinuumlimit.However, in thegroundstate, allofthespins\nare aligned in one direction, and the spin rotational symmet ry SO(3) breaks down to SO(2)\nspontaneously. We emphasize that the following effective fiel d theory (EFT) is applicable\nto any underlying theory with the symmetry breaking pattern being SO(3) →SO(2). The\n3/26EFT describes the low-energy behavior model-independentl y, just using the information on\na symmetry breaking pattern and the order parameter.\nThe construction of EFT relying on a general symmetry breaki ng pattern has been dis-\ncussed intermsof theMaurer-Cartan oneform[29, 30]. Herew e brieflyreview this procedure\nto give a basis of the main results of this paper. Motivated by the symmetry breaking pat-\ntern of the collinear phases, SO(3) →SO(2), we focus on the low-energy EFT using the\nMaurer-Cartan one form [31, 32]:\nαµ(π) =−iU−1(π)∂µU(π), (4)\nwith∂µ=/parenleftig\n∂\n∂t,∂\n∂xi/parenrightig\n(µ= 0,...,3,i= 1,...,3) and the representative of the coset space\nSO(3)/SO(2) being U(π). The representative U(π) is characterized by two NG fields\nπα(α= 1,2), corresponding to the number of broken generators, dim(S O(3)/SO(2)) = 2.\nUnder the SO(3) global transformation g∈SO(3),U(π) is transformed into\nU(π)→gU(π)h−1(π,g), (5)\nwithh(π,g)∈SO(2). Although in this paper, we focus on the system in (1+3) dimensions,\nit is easy to generalize the analysis to other dimensions.\nThe transformation behavior of αµ(π) is given by\nαµ(π)→h(π,g)αµ(π)h−1(π,g)\n+i−1h(π,g)∂µh−1(π,g). (6)\nUsing the derivative expansion appropriate to discuss the l ow-energy behavior, we construct\nthe effective Lagrangian as is done in chiral perturbation the ory [2, 3]. Note that the system\nhas spatial translational and rotational symmetry but not L orentz symmetry, as in the\nnonrelativistic system.1According to a previous study [4], the effective Lagrangian ma y\npossess invariant terms up to the total derivative, which ca n lead to condensates of the\ncharge densities; as also discussed in Sec.2.4. To construc t the Lagrangian explicitly, we\nmake the following decomposition:\nαµ/bardbl(π)≡1\n2Tr(T3αµ(π))T3, (7)\nαµ⊥(π)≡1\n2Tr(Tααµ(π))Tα, (8)\nwithTα(α= 1,2) andT3broken and unbroken generators, respectively,\n[Ta]bc=−iǫabc(a= 1,2,3). (9)\nThe decomposition leads to the transformation rule of αµ/bardbl(π) andαµ⊥(π),\nαµ/bardbl(π)→h(π,g)αµ/bardbl(π)h−1(π,g)\n+i−1h(π,g)∂µh−1(π,g), (10)\nαµ⊥(π)→h(π,g)αµ⊥(π)h−1(π,g). (11)\n1In Refs. [10, 12, 14], the effect of the microscopic structure of th e underlying theory that breaks\ntranslational and rotational symmetry enters in the higher order terms of the derivative expansion.\nBecause here we concentrate on the leading order calculation, the EFT can be constructed under the\ninvariance of spatial translational and rotational symmetry.\n4/26Using the decomposition, the invariant effective Lagrangian reads up to O/parenleftbig\n∂2\n0/parenrightbig\nandO/parenleftbig\n∂2\ni/parenrightbig\norders,\nL=−Σ\n2Tr/bracketleftbig\nT3α0/bardbl/bracketrightbig\n+F2\nt\n4Tr[α0⊥]2\n−F2\n4Tr[αi⊥]2+O(∂3\n0,∂4\ni)\n=−Σ\n2Tr/bracketleftbig\nT3α0/bracketrightbig\n+F2\nt\n8Tr[Tαα0]Tr[Tαα0]\n−F2\n8Tr[Tααi]Tr[Tααi]+O(∂3\n0,∂4\ni)\n=iΣ\n2Tr/bracketleftbig\nT3U−1∂0U/bracketrightbig\n−F2\nt\n8Tr/bracketleftbig\nTαU−1∂0U/bracketrightbig\nTr/bracketleftbig\nTαU−1∂0U/bracketrightbig\n+F2\n8Tr/bracketleftbig\nTαU−1∂iU/bracketrightbig\nTr/bracketleftbig\nTαU−1∂iU/bracketrightbig\n+O(∂3\n0,∂4\ni), (12)\nwiththreeparameters,Σ ,Ft,andF.Thefirsttermisinvariantonlyuptothetotalderivative.\nWe take /planckover2pi1= 1 and c∝ne}ationslash= 1, and the dimensions of the three parameters are given as fo llows:\n[Σ] =L−3,[F2\nt] =T·L−3,[F2] =T−1·L−1. The relation between the coefficients and the\norder parameter is discussed in Sec.2.4.\nBy using the coordinates of the coset space, U=eiπαTα/Fwith the NG fields πα, and\nexpanding the exponential, the Lagrangian reduces to\nL ≡ L(1,0)+L(2,0)+L(0,2)+···, (13)\nL(1,0)=−Σ\n2F2ǫαβπα∂0πβ+Σ\n24F4ǫαβπα∂0πβπγπγ+···, (14)\nL(2,0)=1\n2v2∂0πα∂0πα\n−1\n6v2F2/bracketleftig\n∂0πα∂0παπβπβ−πα∂0παπβ∂0πβ/bracketrightig\n+···, (15)\nL(0,2)=−1\n2∂iπα∂iπα\n+1\n6F2/bracketleftig\n∂iπα∂iπαπβπβ−πα∂iπαπβ∂iπβ/bracketrightig\n+···, (16)\nwithv≡F/Ft. Note that the terms with odd-power of πfields do not appear in the\nLagrangian because of the invariance under the “parity” tra nsformation,\nhpU(π)h−1\np=U(−π), hp=\n−1\n−1\n1\n, (17)\nwithhp∈SO(2). The “parity” invariance is guaranteed by the fact tha t the coset space is\na symmetric space, as a consequence of [ G −H,G −H]⊂ H.\n2.2. Dispersion relation and power counting\nAs we shall show in Sec.2.4, the nonzero values of 1 /vand Σ are the consequence of the\nexistence of the corresponding order parameters. Let us con sider the following three cases\n5/26separately: (a) 1 /v= 0,Σ∝ne}ationslash= 0, (b) 1 /v∝ne}ationslash= 0,Σ = 0, and (c) 1 /v∝ne}ationslash= 0,Σ∝ne}ationslash= 0. In the case of (a),\nthe dispersion relation of the NG mode derived from the equat ion of motion is given by\nω=F2\nΣk2, (18)\nand the number of degrees of freedom (DOF) of the mode is one. I n the case of (b), the\ndispersion relation is given by ω=v|k|, which is similar to that of pions in QCD, and the\nnumber of DOF is two. In the case of (c), the dispersion relati ons are given by\nωII≡v2Σ\n2F2\n−1+/radicaligg\n1+/parenleftbigg2F2\nvΣ/parenrightbigg2\nk2\n≃F2\nΣk2,\nωM≡v2Σ\n2F2\n1+/radicaligg\n1+/parenleftbigg2F2\nvΣ/parenrightbigg2\nk2\n≃Σ\nF2\nt+F2\nΣk2, (19)\nand the number of DOF is two [19, 20, 24, 26, 30]. In fact, these dispersion relations coincide\nwith the behaviors of the long wavelength modes obtained by t he Heisenberg models in\na ferromagnet, an antiferromagnet, and a ferrimagnet, resp ectively, where the Holstein-\nPrimakoff transformation is utilized [33–35]. Thus we stres s that the EFT describes not only\nthe low-energy behavior of the ferromagnet and antiferroma gnet [4, 13], but also that of the\nferrimagnet.\nNote that in the case of (c), the dispersion relation of one of the NG modes tends to\ncoincide with that of the NG mode in the case of (a), i.e., the m agnon in a ferromagnet,\nwhereas the other tends to be a gapped mode with the gap energy νM= Σ/F2\ntin the\nlow energy regime. The same dispersion relation has been dis cussed in the context of the\nkaon-condensed phase for finite density QCD [19, 20]. The gap ped mode at finite density\ncomes from the spontaneous breaking of a symmetry generated by the charge coupled to the\nchemical potential in an underlying Hamiltonian [25]. The g apped mechanism also occurs if\na symmetry is both spontaneously and explicitly broken and i f the broken charge couples to\nthe external field [36]. In contrast, the gapped mode in the fe rrimagnet appears without the\nchemical potential or the external field. The gapped mode bec omes the type-I NG mode in\nthe limit of Σ →0, which also indicates that the gapped mode is related to the spontaneous\nsymmetry breaking.\nIt should be instructive to note that the free part of the effect ive Lagrangian for the\nferromagnet is analogous to the classical Lagrangian of a fa st rotating rigid pendulum under\nthe gravitational force, and the dispersion relation of the type-II NG mode, Eq.(18), is\nidentifiedwiththefrequencyoftheprecessionmotionbyreg ardingthegravitational constant\ng∼k2.2Whentheangular velocity of thependulumdecreases, nutati on is visibleinaddition\nto the precession motion [37]. In terms of the Lagrangian, th e nutation motion can be\ndescribed by including the higher order time derivatives of the rigid-body coordinate, as in\nthe case of the ferrimagnet. The dispersion relation of the g apped mode is then identified\nwith the frequency of the nutation, which remains finite in th eg→0 limit.\nOncethedispersionrelation isdetermined,wecanestablis hthepowercountingscheme[12].\nThe underlying theory contains the natural scales of the ene rgy and momentum. We specify\n2Y.Hidaka, informal lecture at Kyoto University.\n6/26the energy scale νby the first gapped excitation whose origin is not related to t he spon-\ntaneous symmetry breaking. The momentum scale Λ is determin ed by the inverse of the\nlattice spacing in the underlying theory ∼1/a. Generically, the EFT description is based on\nthe double expansion of ∂0/νand∂i/Λ. The temporal and spatial components are related\nto each other through the dispersion relation. We define the s mall dimensionless parameter\npas\n\n\n∂0\nν∼(∂i\nΛ)2∼p21/v= 0,Σ∝ne}ationslash= 0 case (a)\n∂0\nν∼∂i\nΛ∼p 1/v∝ne}ationslash= 0,Σ = 0 case (b)\n∂0\nν∼νM\nν∼(∂i\nΛ)2∼p21/v∝ne}ationslash= 0,Σ∝ne}ationslash= 0 case (c), (20)\nand sort out the effective Lagrangian in powers of p≪1. The low energy phenomena can\nbe calculated with the terms with small powers of p. We note that the gapped NG mode in\nEq. (19) is meaningful only when νM≪ν.\n2.3. Construction of the Hamiltonian and particle states\nTo calculate the scattering amplitudes, we construct the as ymptotic states. In this section,\nwe derive the Hamiltonian from the free part of the Lagrangia n, and define the particle\nstates. The Lagrangian at O(p2) in Eq. (13) is given by\nL0=−Σ\n2F2ǫαβπα∂0πβ+1\n2v2∂0πα∂0πα−1\n2∂iπα∂iπα. (21)\nHow to construct the Hamiltonian depends on whether 1 /vis zero or not, because if it is\nzero, the EFT must be formulated as that for a constrained sys tem [26]. Thus we consider\ntwo cases: (i) 1 /v= 0 (ferromagnet case) and (ii) 1 /v∝ne}ationslash= 0 (antiferromagnet and ferrimagnet\ncases). According to the Dirac-Bergmann method [38, 39], th e construction of the (total)\nHamiltonian in the EFT has been generally discussed [26]. In this paper, we focus on the\nconstruction of the Hamiltonian in the spin systems.\n(i) The canonical momenta are given by\nPα≡∂L\n∂˙πα=Σ\n2F2ǫαβπβ. (22)\nThe canonical momenta are not expressed with ˙ πα, and thus primary constraints φαappear,\nφα≡Pα−Σ\n2F2ǫαβπβ. (23)\nThe constraints φαare second class,\nCαβ(x,y)≡/braceleftig\nφα(x),φβ(y)/bracerightig\n=−Σ\nF2ǫαβδ3(x−y), (24)\nwith{...}being the Poisson bracket. The total Hamiltonian HT\n0is given by\nHT\n0≡˙παPα−L0+λαφα\n=1\n2∂iπα∂iπα+λαφα\n≡ H0+λαφα, (25)\n7/26whereλαare Lagrange multipliers, and H0is the canonical Hamiltonian. In the second line,\n˙παφαhas been absorbed into the last term λαφα. The Lagrange multipliers λαare given by\nλα=−F2\nΣǫαβ∂2\niπβ, (26)\nusing the consistent condition of the constraints for the ti me evolution,\n/braceleftbigg\nφα,/integraldisplay\nHT\n0d3y/bracerightbigg\n=/braceleftbigg\nφα,/integraldisplay/parenleftbigg1\n2∂iπα∂iπα+λαφα/parenrightbigg\nd3y/bracerightbigg\n≈∂2\niπα−λβΣ\nF2ǫαβ≈0, (27)\nwhere “≈” is defined as an equality under the constraint conditions. T hus the total\nHamiltonian reduces to\nHT\n0=1\n2∂iπα∂iπα−F2\nΣǫαβ∂2\niπβ/parenleftbigg\nPα−Σ\n2F2ǫαγπγ/parenrightbigg\n=−F2\nΣǫαβPα∂2\niπβ. (28)\nThe total Hamiltonian leads to the same equation of motion fo r the NG fields obtained from\nthe Lagrangian L0.\nThe quantization of the constraint system is performed by re placing the Dirac bracket\nwith the commutation relation,\n{F(x),G(y)}D→1\ni[F(x),G(y)], (29)\nfor arbitrary functions F(x) andG(y). Instead of thetotal Hamiltonian HT\n0and thePoisson\nbracket, we use the canonical Hamiltonian H0and the Dirac bracket defined by\n{F(x),G(y)}D≡ {F(x),G(y)}\n−/integraldisplay\nd3zd3w/braceleftig\nF(x),φβ(w)/bracerightig\n×/parenleftbig\nC−1/parenrightbigβγ(w,z){φγ(z),G(y)}. (30)\nThe commutation relations between the NG fields and the momen ta are given as follows:\n[πα(x),πβ(y)] =iǫαβF2\nΣδ3(x−y), (31)\n[πα(x),Pβ(y)] =i\n2δαβδ3(x−y), (32)\n[Pα(x),Pβ(y)] =iΣ\n4F2ǫαβδ3(x−y). (33)\nThe Heisenberg equation corresponds to the equation of moti on obtained by the Lagrangian.\nThe NG fields and the momenta are expressed with the creation a nd annihilation operators,\nπα(x,t) =F√\nΣ/integraldisplayd3k\n(2π)3/braceleftig\nǫαa(k)e−ikx+ǫ∗αa†(k)eikx/bracerightig\n, (34)\nPα(x,t) =i√\nΣ\n2F/integraldisplayd3k\n(2π)3/braceleftig\nǫαa(k)e−ikx−ǫ∗αa†(k)eikx/bracerightig\n, (35)\n8/26where (ǫ1,ǫ2) = (1,−i)/√\n2, andkx=ωx0−k·xwithωdefined in Eq. (18). The creation\nand annihilation operators satisfy the following commutat ion relations:\n/bracketleftig\na(k),a†(q)/bracketrightig\n=(2π)3δ3(k−q),\n/bracketleftig\na(k),a(q)/bracketrightig\n=/bracketleftig\na†(k),a†(q)/bracketrightig\n= 0. (36)\nNote that although the number of NG fields is two, the number of of creation operators\n(annihilation operators) is one, corresponding to the numb er of physical degrees of freedom\nin the constrained system with two NG fields and two constrain ts, (2×2−2)/2 = 1.\nUsing these operators, we define the vacuum and n-particle states:\na(k)|0∝an}bracketri}ht= 0, (37)\n|k1,k2,···,kn∝an}bracketri}ht ≡a†(k1)a†(k2)···a†(kn)|0∝an}bracketri}ht. (38)\n(ii) In the case of 1 /v∝ne}ationslash= 0, the corresponding momenta are given by\nPα≡∂L\n∂˙πα=Σ\n2F2ǫαβπβ+1\nv2˙πα. (39)\nThe ˙παare replaced by Pα, and no constraints appear. The Hamiltonian H0is obtained by\nH0≡˙παPα−L0\n=v2\n2/parenleftbigg\nPα−Σ\n2F2ǫαβπβ/parenrightbigg2\n+1\n2∂iπα∂iπα. (40)\nWe remark that a general analysis on the relation among terms with two time derivatives,\nconstraints, and dispersion relations has been given in [26 ].\nThen we replace the Poisson brackets between the NG fields and the corresponding\nmomenta with commutation relations:\n[πα,πβ] = 0,\n[πα,Pβ] =iδαβδ3(x−y),\n[Pα,Pβ] = 0. (41)\nThe Heisenberg equations are now calculated to be\ni˙πα(x,t) =/bracketleftbigg\nπα(x,t),/integraldisplay\nH0(y,t)d3y/bracketrightbigg\n=iv2/parenleftbigg\nPα(x,t)−Σ\n2F2ǫαβπβ(x,t)/parenrightbigg\n, (42)\ni˙Pα(x,t) =/bracketleftbigg\nPα(x,t),/integraldisplay\nH0(y,t)d3y/bracketrightbigg\n=iv2Σ\n2F2ǫβα(x,t)/bracketleftigg\nPβ(x,t)\n−Σ\n2F2ǫβδπδ(x,t)/bracketrightigg\n+i∂2\niπα(x,t), (43)\n9/26The NG fields and the momenta are expanded in terms of the creat ion and annihilation\noperators:\nπα(x,t) =F√\nΣ/integraldisplayd3k\n(2π)3/radicalbig\n¯ω(k)\n×/braceleftig\nǫαaII(k)e−ikIIx+ǫ∗αa†\nII(k)eikIIx\n+ǫ∗αaM(k)e−ikMx+ǫαa†\nM(k)eikMx/bracerightig\n, (44)\nPα(x,t) =i√\nΣ\n2F/integraldisplayd3k/radicalbig\n¯ω(k)\n(2π)3\n×/braceleftig\n−ǫαaII(k)e−ikIIx+ǫ∗αa†\nII(k)eikIIx\n−ǫ∗αaM(k)e−ikMx+ǫαa†\nM(k)eikMx/bracerightig\n, (45)\n¯ω(k)≡/parenleftigg\n1+/parenleftbigg2F2\nvΣ/parenrightbigg2\nk2/parenrightigg1/2\n, (46)\nwherekIIx=ωIIx0−k·xandkMx=ωMx0−k·xwithωIIandωMbeing defined in\nEq. (19), and ( a†\nII, aII) and (a†\nM, aM) are the corresponding creation and annihilation\noperators, satisfying the commutation relations,\n/bracketleftig\naII(k),a†\nII(q)/bracketrightig\n=(2π)3δ3(k−q),\n/bracketleftig\naII(k),aII(q)/bracketrightig\n=/bracketleftig\na†\nM(k),a†\nII(q)/bracketrightig\n= 0,\n/bracketleftig\naM(k),a†\nM(q)/bracketrightig\n=(2π)3δ3(k−q),\n/bracketleftig\naM(k),aM(q)/bracketrightig\n=/bracketleftig\na†\nM(k),a†\nM(q)/bracketrightig\n= 0,\n/bracketleftig\naII(k),a†\nM(q)/bracketrightig\n=/bracketleftig\naM(k),a†\nII(q)/bracketrightig\n= 0. (47)\nNote that two particle states may appear in contrast to the ca se of 1/v= 0. In this system,\nthe vacuum is defined by\naII(k)|0∝an}bracketri}ht= 0,\naM(k)|0∝an}bracketri}ht= 0, (48)\nand the particle states are given by\n/vextendsingle/vextendsingleπII\nk1,···,πII\nkn,πM\nk1,···,πM\nkm/angbracketrightbig\n≡a†\nII(k1)···a†\nII(kn)a†\nM(k1)···a†\nM(km)|0∝an}bracketri}ht. (49)\nIn the limit of vanishing Σ, Eqs.(44)-(46) are reduced to\nπα(x,t) =/integraldisplayd3k\n(2π)3/radicalbiggv\n2|k|\n×/braceleftig\nǫαaII(k)e−ikIx+ǫ∗αa†\nII(k)eikIx\n+ǫ∗αaM(k)e−ikIx+ǫαa†\nM(k)eikIx/bracerightig\n, (50)\n10/26Pα(x,t) =i√\n2/integraldisplayd3k\n(2π)3/radicalbigg\n|k|\nv\n×/braceleftig\n−ǫαaII(k)e−ikIx+ǫ∗αa†\nII(k)eikIx\n−ǫ∗αaM(k)e−ikIx+ǫαa†\nM(k)eikIx/bracerightig\n, (51)\nwherekIx=ωIx0−k·xwithωI=v|k|.a†\nIIanda†\nMcreate a particle with the same dis-\npersion relation ωI=v|k|, known as the type-I NG mode in the antiferromagnet. From now\non, we change the notation of a†\nIIanda†\nM(aIIandaM) to that of a†\n1anda†\n2(a1anda2) for\nthe magnons in the antiferromagnet.\n2.4. Charges, Order parameters, and Coefficients\nTo grasp the physical meaning of the coefficients in the effectiv e Lagrangian, we discuss the\nrelation among the charges, order parameters, and coefficien ts of the effective Lagrangian for\nthespinsystems.Theorderparameterisdefinedastheexpect ation valueofthecommutation\nrelation between the broken Noether charge Qαand a local operator O(x),\nlim\nV→∞1\nV/integraldisplay\nVd3x∝an}bracketle{t0|[iQα,O(x)]|0∝an}bracketri}ht, (52)\nwhere the commutation relation with the broken charge shoul d be understood as\n[iQα,O(x)]≡/integraldisplay\nd3x′/bracketleftbig\nijα\n0(x′),O(x)/bracketrightbig\n. (53)\nThe order parameter of the ferromagnet is the magnetization , which is given by the space\nintegration of the expectation value of the commutation rel ation between the Noether charge\nand the charge density,/angbracketleftbig\n0/vextendsingle/vextendsingle/bracketleftbig\niQα,jβ/bracketrightbig/vextendsingle/vextendsingle0/angbracketrightbig\n, which leads to the quadratic dispersion relations of\ntheNGmodes and areduction of the numberof NGmodes [4, 22, 23 ]. Furthermore, thecoef-\nficient Σ in the effective Lagrangian of the ferromagnet is iden tified with the magnetization\n[4, 23],\nΣ = lim\nV→∞1\nVN/summationdisplay\nm∝an}bracketle{t0|S3\nm|0∝an}bracketri}ht, (54)\nwithVbeing the total volume and Nthe total number of lattice sites.\nIn this subsection, we first discuss that the relation is sati sfied also in the case of the ferri-\nmagnet, the effective Lagrangian for which contains terms wit h one and two time derivatives\noftheNGfieldsinEq.(13). Thechargedensities jα\n0arederivedfromtheeffective Lagrangian,\njα\n0=Σ\nFǫαβπβ+F\nv2∂0πα+O(π2). (55)\nWe have used the transformation rule of the NG fields obtained from Eq. (5),\nδαπβ= [iQα,πβ] =Fδαβ+O(πα). (56)\nNote that the charge densities include the influence of the in variance only up to the total\nderivative in Eq. (12). Using the charge densities, the expe ctation value of the commutation\n11/26relation between the charges and the charge densities reduc es to\nlim\nV→∞1\nV/integraldisplay\nVd3x/angbracketleftig\n0/vextendsingle/vextendsingle/vextendsingle/bracketleftig\niQα,jβ\n0/bracketrightig/vextendsingle/vextendsingle/vextendsingle0/angbracketrightig\n= lim\nV→∞1\nV/integraldisplay\nVd3x/angbracketleftbigg\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\niQα,Σ\nFǫβγπγ+F\nv2∂0πβ/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle0/angbracketrightbigg\n≃lim\nV→∞1\nV/integraldisplay\nVd3x/angbracketleftbigg\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingleΣ\nFǫβγδαπγ+F\nv2∂0/parenleftig\nδαπβ/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingle0/angbracketrightbigg\n≃ −ǫαβΣ. (57)\nThis does not change by the redefinition of the NG fields, πα→πα/Z, with a constant\nZ. By matching the charge density in an underlying theory, the expectation value of the\ncommutation relation is also rewritten as\nlim\nV→∞1\nV/integraldisplay\nVd3x/angbracketleftig\n0/vextendsingle/vextendsingle/vextendsingle/bracketleftig\niQα,jβ\n0/bracketrightig/vextendsingle/vextendsingle/vextendsingle0/angbracketrightig\n=−ǫαβlim\nV→∞1\nV/integraldisplay\nVd3x/angbracketleftbig\n0/vextendsingle/vextendsinglej3\n0/vextendsingle/vextendsingle0/angbracketrightbig\n=−ǫαβlim\nV→∞1\nVN/summationdisplay\nm/angbracketleftbig\n0/vextendsingle/vextendsingleS3\nm/vextendsingle/vextendsingle0/angbracketrightbig\n. (58)\nThus,thecoefficient Σisidentifiedwiththemagnetization al sointhecaseoftheferrimagnet;\nseeEq. (54). Note that in thecase of theantiferromagnet, th emagnetization vanishes,Σ = 0,\nand the effective Lagrangian does not involve terms with one ti me derivative[4, 13].\nWe may consider the commutation relation between the Noethe r charge and the NG fields:\nlim\nV→∞1\nV/integraldisplay\nVd3x/angbracketleftig\n0/vextendsingle/vextendsingle/vextendsingle/bracketleftig\niQα,πβ/bracketrightig/vextendsingle/vextendsingle/vextendsingle0/angbracketrightig\n= lim\nV→∞1\nV/integraldisplay\nVd3x/angbracketleftig\n0/vextendsingle/vextendsingle/vextendsingleδαπβ/vextendsingle/vextendsingle/vextendsingle0/angbracketrightig\n=δαβF. (59)\nThe relation changes by redefinition of the NG fields. If we red efine the NG fields as πα→\n(F/Ft)πα, the relation changes and is given in terms of the coefficient o f the terms with two\ntime derivatives of the NG fields in the effective Lagrangian as\nlim\nV→∞1\nV/integraldisplay\nVd3x/angbracketleftig\n0/vextendsingle/vextendsingle/vextendsingle/bracketleftig\niQα,πβ/bracketrightig/vextendsingle/vextendsingle/vextendsingle0/angbracketrightig\n=δαβFt. (60)\nBecause 1 /v=Ft/F, the nonzero expectation value leads to another order param eter in\nthe case of the EFT containing terms with one and two time deri vatives of the NG fields,\nwhich leads to the appearance of staggered magnetization in the ferrimagnet. The relation\nEq. (59) is also satisfied in the case of the EFT with one time de rivative but without two\ntime derivatives, where, however, we cannot distinguish th e NG field παfrom the charge\ndensityjα\n0. In fact, an explicit calculation using Eq. (34) shows that/integraltext\nd3xπαis conserved;\nd\ndt/integraldisplay\nd3xπα\n=/integraldisplay\nd3xF√\nΣ/integraldisplayd3k\n(2π)3\n12/26×/braceleftig\nǫα(−iω)a(k)e−ikx+ǫ∗α(iω)a†(k)eikx/bracerightig\n=F√\nΣ/integraldisplay\nd3kδ(k)\n×/braceleftig\nǫα(−iω)a(k)e−iωt+ǫ∗α(iω)a†(k)eiωt/bracerightig\n= 0, (61)\nwhich does not vanish in the case of the EFT with both one and tw o time derivatives of\nthe NG fields, i.e.,d\ndt/integraltext\nd3xπα∝ne}ationslash= 0. Thus in the case of a ferromagnet, we have only one\norder parameter given by the magnetization. In other words, the effective Lagrangian for\nthe ferromagnet does not have terms with two time derivative s even if we take into account\nthe contribution from higher derivatives. Similar relatio ns have been obtained for a general\nsystem on the basis of the Langevin equation [27].\n2.5. Scattering amplitudes and the scattering lengths\nIn this subsection, we calculate the scattering amplitudes between two NG modes at tree\nlevel, and their scattering lengths. These are derived from theπ4terms in Eq. (13). Cor-\nresponding to the case of (a) ferromagnet (1 /v= 0,Σ∝ne}ationslash= 0), (b) antiferromagnet (1 /v∝ne}ationslash=\n0,Σ = 0), and (c) ferrimagnet (1 /v∝ne}ationslash= 0,Σ∝ne}ationslash= 0), we discuss the amplitudes and the scat-\ntering lengths, respectively. As for the cases of ferromagn et and antiferromagnet, a similar\ncalculation was already performed using different coordinat es system describing the EFT\n[13]: In fact, the form of the resulting Lagrangian is differen t from ours given in Eqs. (13)-\n(16), and our calculation ensures that the scattering ampli tudes do not depend on the choice\nof the coordinates. Furthermore, our calculation includes the first study on the case of the\nferrimagnet.\n(a) In the case of the ferromagnet where 1 /v= 0,Σ∝ne}ationslash= 0, only one magnon mode with the\ndispersion relation ω=F2\nΣk2appears. We denote the particle with momentum kbyπ(k). A\nscattering process for π(k1)+π(k2)→π(k3)+π(k4) occurs, and the scattering amplitude\nis given by\nM[π(k1)+π(k2)→π(k3)+π(k4)]\n=−1\n6Σ(ω1+ω2+ω3+ω4)\n+F2\n3Σ2/bracketleftbig\nk1·k3+k2·k3+k1·k4+k2·k4\n+2(k1·k2+k3·k4)/bracketrightbig\n=2F2\nΣ2(k1·k2), (62)\nwhere we have used momentum and energy conservation, k1+k2=k3+k4, andω1+ω2=\nω3+ω4withωi=F2\nΣk2\ni. This result coincides with the previous study [13], and is i n\naccordance with Dyson’s microscopic analysis [40].\nThe definition of the scattering length and phase shift throu gh the scattering amplitude in\nthe nonrelativistic notation are given in Appendix A. In the case of the ferromagnet system\nwithout external fields, the scattering length vanishes, th ough it may not be well-defined\n13/26in the process, because of the appearance of scattering proc esses with massless particles.\nThis is a consequence of the derivative couplings of EFT, in a ccordance with the Goldstone\ntheorem.\n(b) In the case of the antiferromagnet where 1 /v∝ne}ationslash= 0,Σ = 0, we have two magnons\nwith the same dispersion relation ωI=v|k|as mentioned in 2.3. Let πα(k)(α= 1,2)\ndenote the particle states created by a†\nα. The unbroken SO(2) symmetry leads to the\nconservation of the number of particles, and only the follow ing three scattering pro-\ncesses are allowed: π1(k1)+π1(k2)→π1(k3)+π1(k4),π1(k1)+π2(k2)→π1(k3)+π2(k4),\nandπ2(k1)+π2(k2)→π2(k3)+π2(k4). The scattering amplitudes are given by\nM/bracketleftbig\nπ1(k1)+π1(k2)→π1(k3)+π1(k4)/bracketrightbig\n=M/bracketleftbig\nπ2(k1)+π2(k2)→π2(k3)+π2(k4)/bracketrightbig\n=−v2\n2F2/radicalbig\n|k1||k2||k3||k4|(|k1||k2|−k1·k2) (63)\nM/bracketleftbig\nπ1(k1)+π2(k2)→π1(k3)+π2(k4)/bracketrightbig\n=v2\n2F2/radicalbig\n|k1||k2||k3||k4|(|k2||k3|−k2·k3), (64)\nwhere we have used momentum and energy conservation.3Also in the case of the\nantiferromagnet, the scattering length vanishes.\n(c) In the case of the ferrimagnet where 1 /v∝ne}ationslash= 0,Σ∝ne}ationslash= 0, we also have two magnons with\nthe dispersion relations Eq.(19), denoted by πII(k) andπM(k). The following three scat-\ntering processes are allowed; πII(k1)+πII(k2)→πII(k3)+πII(k4),πII(k1)+πM(k2)→\nπII(k3)+πM(k4),andπM(k1)+πM(k2)→πM(k3)+πM(k4). Other processes such as\nπII(k1)+πII(k2)→πM(k3)+πM(k4) are prohibited because of the residual symmetry\nSO(2), as is the case for the antiferromagnet.\nThe scattering amplitudes are as follows:\nM/bracketleftbig\nπII(k1)+πII(k2)→πII(k3)+πII(k4)/bracketrightbig\n=1√¯ω1¯ω2¯ω3¯ω4/bracketleftig\n−1\n3Σ/parenleftbig\nωII\n1+ωII\n2/parenrightbig\n+F2\n3Σ2/parenleftbig\n4k1·k2+2k3·k4+k2\n1+k2\n2/parenrightbig/bracketrightig\n, (65)\nM/bracketleftbig\nπII(k1)+πM(k2)→πII(k3)+πM(k4)/bracketrightbig\n=1√¯ω1¯ω2¯ω3¯ω4/bracketleftig1\n6Σ/parenleftbig\nωM\n2+ωM\n4−ωII\n1−ωII\n3/parenrightbig\n−F2\n3Σ2/parenleftbig\n4k2·k3+2k1·k4−k2\n2−k2\n3/parenrightbig/bracketrightig\n, (66)\nM/bracketleftbig\nπM(k1)+πM(k2)→πM(k3)+πM(k4)/bracketrightbig\n=1√¯ω1¯ω2¯ω3¯ω4/bracketleftig1\n3Σ/parenleftbig\nωM\n1+ωM\n2/parenrightbig\n3ThescatteringamplitudesinRef.[13]areobtainedbychangingthede finition oftheparticlestates.\nThe creation operator in Ref.[13] corresponds to ˜ a†\n1=/parenleftig\na†\n1+a†\n2/parenrightig\n/√\n2 and ˜a†\n2=i/parenleftig\na†\n1−a†\n2/parenrightig\n/√\n2.\n14/26+F2\n3Σ2/parenleftbig\n4k1·k2+2k3·k4+k2\n1+k2\n2/parenrightbig/bracketrightig\n, (67)\nwith ¯ωi≡/bracketleftbigg\n1+/parenleftig\n2F2\nvΣ/parenrightig2\nk2\ni/bracketrightbigg1/2\n,ωII\ni≡v2Σ\n2F2(−1+ ¯ωi), andωM\ni≡v2Σ\n2F2(1+ ¯ωi) (i= 1,...,4) .\nThe scattering lengths for πII(k1)+πM(k2)→πII(k3)+πM(k4) and πM(k1)+\nπM(k2)→πM(k3)+πM(k4) are finite in contrast to the ferromagnet case:\naII+M→II+M=Σ\n12F2F2\nt=νM\n12F2, (68)\naM+M→M+M=Σ\n6F2F2\nt=νM\n6F2, (69)\nwherethemassgap is definedas νM= Σ/F2\nt. Because Σis identified withthemagnetization,\nF2can bedetermined experimentally by measuringthe k2dependenceof thedispersionrela-\ntion of the gapped NG mode through Eq. (19). In this way, the ma gnitude of the scattering\nlength is used to determine the coefficients in the dispersion relation. Moreover, the positive\nsign of the scattering lengths aII+M→II+M,aM+M→M+M>0 meaning an attractive inter-\naction and the ratio aII+M→II+M/aM+M→M+M= 1/2 are independent of the low energy\nconstants. These properties are consequences of the low ene rgy theorem for the gapped NG\nmode.\n3. Analysis in the presence of external fields: magnetic field s and anisotropic\neffects\nIn chiral perturbation theory, terms involving explicit sy mmetry breaking can be introduced\nby assuming appropriate transformation properties of the s ymmetry-breaking terms so as to\nmake the effective Lagrangian invariant under chiral symmetr y. In much the same way, we\ncan involve the explicit breaking in the EFT for magnons. Fir st we introduce the explicit\nsymmetry-breaking terms in the underlying theory, in terms of an external magnetic field\nand single-ion anisotropy:\nH=Hinv(Sa\nm)−µHa/summationdisplay\nmSa\nm−Dab/summationdisplay\nmSa\nmSb\nm, (70)\nwhereHinvis the invariant term under a rotation of the spins, and the te rms including H\nandDabrepresent the effect of the magnetic field and single-ion aniso tropy, respectively.\nWhenH=Dab= 0, as discussed in Sec. 2, the underlying theory is invarian t under SO(3),\nand the symmetry is broken down to SO(2) in the ground state. W hen the magnetic field is\napplied, the spins couple to the magnetic field, and the expli cit symmetry breaking occurs.\nWe assume that the direction of the magnetic field is in the 3-a xis,Ha=δa3H. Also the\nanisotropic effect leads to an explicit breaking, and thesqua red spinson thesame site, Sa\nmSb\nm\ncouple to the matrix Dab. Now we consider the diagonal case Dab=Daδabfor simplicity.\nLet us first discuss the effect of the magnetic field on the EFT;se e also [4, 12, 13] where the\ncases are discussed. In Eq. (70), the magnetic field couples t o the SO(3) charges,/summationtext\nmSa\nm∼/integraltext\nd3xja\n0(x), which leads to the identification of the magnetic fields wit h the time component\nof SO(3) gauge fields A0(x) =Aa\n0(x)Ta. In fact, if we consider the continuum limit, the\nterm is reduced to/integraltext\nd3xAa\n0(x)ja\n0(x), which is invariant for the partial gauge transformation\n15/26for SO(3),\nA0(x)→g(t)A0(x)g−1(t)+1\nig(t)∂0g−1(t), (71)\nwithg(t) =eiθ3(t)T3∈SO(2). Thus, the magnetic fields are handled in the EFT by the i den-\ntification Aa\n0(x) =δa3Hafter we construct the SO(3) gauge-invariant effective Lagra ngian\nonly up to a total derivative [4]. In practice, this is easily performed by replacing the time\nderivatives with the following form in Eq. (12),\n∂0→∂0−iµHT3. (72)\nNext, we discuss the effect of the single-ion anisotropy. In th e Heisenberg model, the\ntransformation of the coefficient Dabis considered as\nDab→/parenleftbig\ngDg−1/parenrightbigab, (73)\nwith the global transformation g∈SO(3). In the EFT, there are the following invariant\nterms at leading order O(Dab):\nα1[U−1DU]33≃α1D33+···, (74)\nα2[U−1DU]αα≃α2Dαα+···, (75)\nα3Tr/bracketleftbig\nT3U−1DU/bracketrightbig\n≃α3iǫαβDαβ+···, (76)\nwhere we have shown the invariance of the first term by using h3a=/bracketleftbig\neiθT3/bracketrightbig3a=δ3a. Note\nthat the last term, Tr/bracketleftbig\nT3U−1DU/bracketrightbig\n, is always zero in the present case with Dab=Daδab. The\ncoefficients α1andα2are determined by matching the condensates in the underlyin g theory\nwithout external fields:\nlim\nV→∞1\nVN/summationdisplay\nm/angbracketleftbig\nS3\nmS3\nm/angbracketrightbig\n= lim\nV→∞1\nVδZeff(H,D)\nδD33/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH=0,D=0, (77)\nlim\nV→∞1\nVN/summationdisplay\nm/angbracketleftig\nSα\nmSβ\nm/angbracketrightig\n= lim\nV→∞1\nVδZeff(H,D)\nδDαβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH=0,D=0, (78)\nwithZeff(H,D) being the partition function of the effective field theory wit h external fields,\nNthe total number of lattice sites, and Vthe total volume.\nWe give several examples of the underlying theories as a summ ary. The simplest case is a\nspin-1/2 quantum Heisenberg model (3), in which the condens ates are given by\n/summationdisplay\nm/angbracketleftbig\nS3\nmS3\nm/angbracketrightbig\n=N\n4, (79)\n/summationdisplay\nm/angbracketleftig\nSα\nmSβ\nm/angbracketrightig\n=δαβN\n4+iǫαβ\n2/summationdisplay\nm/angbracketleftbig\nS3\nm/angbracketrightbig\n, (80)\nwhich leads to\nα1=α2≡α. (81)\nThe term involving the anisotropic effect is reduced to\nLD=αTrU−1DU=αTrD. (82)\nThus in the case of the spin-1/2 Heisenberg model, the anisot ropic effects do not appear in\nthelow-energy regime.Notethattheresultisreadilyobtai nedintheconventional Heisenberg\n16/26model because/summationtext\naDaSaSa=/summationtext\naDa/4. What we have shown is that the result holds in a\nmodel-independent way for any spin-1/2 underlying theorie s having the same expectation\nvalues of the spin operators.\nIn a spin-S Heisenberg ferromagnet, the condensates are giv en by\n/summationdisplay\nm/angbracketleftbig\nS3\nmS3\nm/angbracketrightbig\n=S2N, (83)\n/summationdisplay\nm/angbracketleftig\nSα\nmSβ\nm/angbracketrightig\n=δαβ\n2SN+iǫαβ\n2SN, (84)\nand the coefficients α1andα2are found to be\nα1=N\nVS2, α2=N\n2VS. (85)\nNote that the inequality α1> α2except for the case of a spin-1/2 ferromagnet, and the\ncoefficients are related to the magnetization as Σ =N\nVS.\nIn a spin-( SA,SB) Heisenberg ferrimagnet, the condensates are given by\n/summationdisplay\nm/angbracketleftbig\nS3\nmS3\nm/angbracketrightbig\n=N\n2(S2\nA+S2\nB), (86)\n/summationdisplay\nm/angbracketleftig\nSα\nmSβ\nm/angbracketrightig\n=δαβ\n4N(SA+SB)+iǫαβ\n4N(SA−SB), (87)\nand the coefficients α1andα2are found to be\nα1=N\n2V(S2\nA+S2\nB). (88)\nα2=N\n4V(SA+SB). (89)\nAs in the ferromagnetic case, the inequality α1> α2holds except for the case with\nSA=SB= 1/2, i.e., a spin-1/2 antiferromagnet. Using the explicit exp ressions of the mag-\nnetization Σ = ( SA−SB)N/2Vand the staggered magnetization Σ h= (SA+SB)N/2V,the\ncoefficients are rewritten as α1= (Σ2+Σ2\nh)V/Nandα2= Σh/2.\nIn terms of the NG fields πα, the effective Lagrangian involving the magnetic field and\nanisotropic effect is expressed as\nL=L2π+L4π+···\nL2π=−Σ\n2F2ǫαβπαD0πβ+1\n2v2D0παD0πα−1\n2∂iπα∂iπα\n+D2−D3\nF2(α1−α2)π1π1+D1−D3\nF2(α1−α2)π2π2,\nL4π=Σ\n24F4ǫαβπαD0πβπγπγ\n−1\n6v2F2/bracketleftig\nD0παD0παπβπβ−παD0παπβD0πβ/bracketrightig\n+1\n6F2/bracketleftig\n∂iπα∂iπαπβπβ−πα∂iπαπβ∂iπβ/bracketrightig\n+D3−D1\n3F4(α1−α2)παπαπ2π2\n+D3−D2\n3F4(α1−α2)παπαπ1π1, (90)\n17/26with the covariant derivative for the time component D0πα=/parenleftbig\n∂0δαβ−ǫαβµH/parenrightbig\nπβ. We\nneglect the terms only with external fields, HandD. Because we assume that the spins\nare (anti-)aligned in the 3-axis direction, the anisotropy effect should be maximum in the\n3-direction, D3≫D1,D2. In this section, we show the magnon-magnon scattering in th e\npresence of the only 3-axis anisotropic effect D1=D2= 0,D3∝ne}ationslash= 0. As in Sec. 2, we classify\nthe cases by the value of 1 /v.\nWe now consider the power counting scheme with the external fi elds. For a reasonable\napplication of EFT, a definite power counting scheme should b e established for all the\nfields including the external fields, for which the strength o f the external fields should be\nconstrained. Because the external magnetic field is introdu ced in the form of Eq. (72), its\nstrength should be µH∼∂0≪νwhereνis the first gapped excitation other than the NG\nmodes. When Σ ∝ne}ationslash= 0, because the anisotropy effect provides the gap ( α1−α2)D3/Σ, this\nquantity should also be of the order of ∂0. We thus consider that the energy scale of the\nexternal field Eis counted as the same order as that of the NG boson field as\nE=µH,α1−α2\nΣD3 (91)\n/braceleftigg∂0\nν∼E\nν∼(∂i\nΛ)2∼p21/v= 0,Σ∝ne}ationslash= 0\n∂0\nν∼E\nν∼νM\nν∼(∂i\nΛ)2∼p21/v∝ne}ationslash= 0,Σ∝ne}ationslash= 0. (92)\nAnalyzing the dispersion relation for Σ = 0 in the same manner , we obtain the counting rule\nas\n(∂0\nν)2∼(µH\nν)2∼(∂i\nΛ)2∼(α1−α2)D3\nF2∼p21/v∝ne}ationslash= 0,Σ = 0. (93)\n(a) In the case of 1 /v= 0 and Σ ∝ne}ationslash= 0, the asymptotic fields involving the external fields\nare given by\nπα(x,t) =F√\nΣ/integraldisplayd3k\n(2π)3/braceleftig\nǫαa(k)e−ikx+ǫ∗αa†(k)eikx/bracerightig\n,\nω≡µH+α1−α2\nΣD3+F2\nΣk2. (94)\nWe note that the NG mode acquires the mass gap which is proport ional to the external\nfields,HandD3. The amplitude for the scattering process, π(k1)+π(k2)→π(k3)+π(k4),\nis given by\nM[π(k1)+π(k2)→π(k3)+π(k4)]\n=2F2\nΣ2(k1·k2)+14\n3(α1−α2)D3\nΣ2(95)\nNote that the scattering amplitude does not depend on the mag netic field H, but on the\nanisotropic effect D3. The scattering length, which is defined in Appendix A reads:\na=7\n6(α1−α2)D3\nΣF2, (96)\nwhich is finite and proportional to the anisotropy D3, as long as α1∝ne}ationslash=α2. In QCD, pion\nscattering length is proportional to the pion mass squared [ 41], implying that the nonzero\nscattering length is caused by the explicit symmetry breaki ng due to the current quark\nmasses in QCD. In the magnon scattering, we consider two type s of the explicit symmetry\n18/26breaking, theexternal magnetic field Hand the anisotropy D3. While both effects contribute\nto the mass gap in Eq. (95), the scattering length depends onl y on the anisotropy D3. This\nis because of the difference in the breaking patterns of the SO( 3) symmetry. In the spin\nsystem, it is the anisotropy D3that corresponds to the explicit symmetry breaking by the\nquark mass term in QCD.\nIf we write the mass gap by the anisotropic effect as νD= (α1−α2)D3/Σ, we obtain\na=7νD\n6F2. (97)\nThis is an analogous relation to Eqs. (68) and (69). Note, how ever, that the coefficient 7 /6\nis an order of magnitude larger than those of the gapped NG mod es, 1/12 and 1 /24. If the\nanisotropic effect induces a comparable amount of the gap νDwith the gapped modes, the\nscattering length should be much larger.\n(b) In the case of 1 /v∝ne}ationslash= 0 and Σ = 0, when the external fields are applied, the asymptot ic\nNG fields are expanded as follows:\nπα(x,t) =/integraldisplayd3k\n(2π)3/radicalbiggv\n2¯k\n×/braceleftig\nǫαa1(k)e−ik1x+ǫ∗αa†\n1(k)eik1x\n+ǫ∗αa2(k)e−ik2x+ǫαa†\n2(k)eik2x/bracerightig\n, (98)\n¯k≡/radicalbigg\nk2+D3\nF2(α1−α2), (99)\nwherekαx=ωαx0−k·xwith\nω1=µH+v¯k,\nω2=−µH+v¯k. (100)\nNote that ¯kdoes not depend on the magnetic field. The πα(k) denote the parti-\ncle states with the dispersion relation ωαcreated by a†\nα. The scattering amplitudes\nforπ1(k1)+π1(k2)→π1(k3)+π1(k4),π2(k1)+π2(k2)→π2(k3)+π2(k4),andπ1(k1)+\nπ2(k2)→π1(k3)+π2(k4) are given by\nM/bracketleftbig\nπ1(k1)+π1(k2)→π1(k3)+π1(k4)/bracketrightbig\n=M/bracketleftbig\nπ2(k1)+π2(k2)→π2(k3)+π2(k4)/bracketrightbig\n=−v2\n6F2/radicalbig¯k1¯k2¯k3¯k4/bracketleftig\n3/parenleftbig¯k1¯k2−k1·k2/parenrightbig\n−7(α1−α2)D3\nF2/bracketrightig\n(101)\nM/bracketleftbig\nπ1(k1)+π2(k2)→π1(k3)+π2(k4)/bracketrightbig\n=v2\n6F2/radicalbig¯k1¯k2¯k3¯k4/bracketleftig\n3/parenleftbig¯k1¯k3−k1·k3/parenrightbig\n+7(α1−α2)D3\nF2/bracketrightig\n. (102)\nNote that the scattering amplitudes do not depend on the magn etic field, as in the case of\nthe ferromagnet. A different result in a previous study [13] is due to the dependence of the\n19/26magnetic field for ¯kin Eq.(99). Ourresult for ¯kis consistent with the quantization conditions\n(31) - (33). The scattering lengths are given by\na1+1→1+1=a2+2→2+2=νD\n3F2(103)\na1+2→1+2=5νD\n6F2(104)\nwith the mass gap νD=v/radicalbig\n(α1−α2)D3/F2.\n(c) In the case of 1 /v∝ne}ationslash= 0 and Σ ∝ne}ationslash= 0, the asymptotic fields involving the external fields are\ngiven by\nπα(x,t) =F√\nΣ/integraldisplayd3k\n(2π)3/radicalbig\n¯ω(k)\n×/braceleftig\nǫαaII(k)e−ikIIx+ǫ∗αa†\nII(k)eikIIx\n+ǫ∗αaM(k)e−ikMx+ǫαa†\nM(k)eikMx/bracerightig\n, (105)\n¯ω(k)≡/bracketleftigg\n1+/parenleftbigg2F2\nvΣ/parenrightbigg2/parenleftbigg\nk2+D3\nF2(α1−α2)/parenrightbigg/bracketrightigg1/2\n,\nωII≡µH+v2Σ\n2F2(−1+ ¯ω)\n≃µH+α1−α2\nΣD3+F2\nΣk2,\nωM≡ −µH+v2Σ\n2F2(1+ ¯ω)\n≃Σ\nF2\nt−µH+α1−α2\nΣD3+F2\nΣk2. (106)\nTheamplitudesforthescatteringprocesses, πII(k1)+πII(k2)→πII(k3)+πII(k4),πII(k1)+\nπM(k2)→πII(k3)+πM(k4),andπM(k1)+πM(k2)→πM(k3)+πM(k4), are given by\nM/bracketleftbig\nπII(k1)+πII(k2)→πII(k3)+πII(k4)/bracketrightbig\n=1√¯ω1¯ω2¯ω3¯ω4/bracketleftigg\n−1\n3Σ/parenleftbig\nωII\n1+ωII\n2/parenrightbig\n+F2\n3Σ2/parenleftbig\n4k1·k2+2k3·k4+k2\n1+k2\n2/parenrightbig\n+2\n3ΣµH+16\n3(α1−α2)D3\nΣ2/bracketrightigg\n≃1√¯ω1¯ω2¯ω3¯ω4/bracketleftigg\n2F2\nΣ2(k1·k2)\n+14\n3(α1−α2)D3\nΣ2+O(k4\ni,H2,D2)/bracketrightigg\n, (107)\nM/bracketleftbig\nπII(k1)+πM(k2)→πII(k3)+πM(k4)/bracketrightbig\n20/26=1√¯ω1¯ω2¯ω3¯ω4/bracketleftigg\n1\n6Σ/parenleftbig\nωM\n2+ωM\n4−ωII\n1−ωII\n3/parenrightbig\n−F2\n3Σ2/parenleftbig\n4k2·k3+2k1·k4−k2\n2−k2\n3/parenrightbig\n+2\n3ΣµH+16\n3(α1−α2)D3\nΣ2/bracketrightigg\n≃1√¯ω1¯ω2¯ω3¯ω4/bracketleftigg\n1\n3F2\nt\n+F2\n6Σ2/parenleftbig\n−k2\n1+3k2\n2+k2\n3+k2\n4−4k1·k4−8k2·k4/parenrightbig\n+16\n3(α1−α2)D3\nΣ2/bracketrightigg\n, (108)\nM/bracketleftbig\nπM(k1)+πM(k2)→πM(k3)+πM(k4)/bracketrightbig\n=1√¯ω1¯ω2¯ω3¯ω4/bracketleftigg\n1\n3Σ/parenleftbig\nωM\n1+ωM\n2/parenrightbig\n+F2\n3Σ2/parenleftbig\n4k1·k2+2k3·k4+k2\n1+k2\n2/parenrightbig\n+2\n3ΣµH+16\n3(α1−α2)D3\nΣ2/bracketrightigg\n,\n≃1√¯ω1¯ω2¯ω3¯ω4/bracketleftigg\n2\n3F2\nt+2F2\n3Σ2/parenleftig\n(k1+k2)2+k1·k2/parenrightig\n+6(α1−α2)D3\nΣ2/bracketrightigg\n, (109)\nwith ¯ωi≡¯ω(ki). Note that the scattering amplitudes do not depend on the ma gnetic field\nbutontheanisotropiceffect. Usingthedefinitiongiven inApp endixA,thescattering lengths\nare given by\naII+II→II+II\n=Σ\n4F2¯ω(0)/bracketleftbigg\n−v2\n3F2(¯ω(0)−1)+16\n3(α1−α2)D3\nΣ2/bracketrightbigg\n≃7\n6F2(α1−α2)D3\nΣ, (110)\naII+M→II+M\n=Σ\n4F2¯ω(0)/bracketleftbigg1\n3F2\nt+16\n3(α1−α2)D3\nΣ2/bracketrightbigg\n≃Σ\n4F2/bracketleftbigg1\n3F2\nt+16\n3(α1−α2)D3\nΣ2/bracketrightbigg\n, (111)\naM+M→M+M\n21/26=Σ\n4F2¯ω(0)/bracketleftbiggv2\n3F2(¯ω(0)+1)+16\n3(α1−α2)D3\nΣ2/bracketrightbigg\n≃Σ\n4F2/bracketleftbigg2\n3F2\nt+6(α1−α2)D3\nΣ2/bracketrightbigg\n. (112)\nAgain, all the scattering lengths are finite and only depend o n the anisotropy.\nTheexistenceofthenonzeromagnonscatteringlengthhasan importantimplicationinfew-\nbody systems. The nonzero scattering length is induced by th e four-magnon contact vertex\nwithout derivatives in Eq. (90), because the scattering len gth is defined at zero momentum.\nThe above results are obtained by the perturbative calculat ion in the counting scheme (92)\nwhere the strength of the external field is counted as O(p). As a consequence, the scattering\nlength obtained should not be very large.\nOn the other hand, we can modify the power counting scheme in t he presence of the\ncontact interaction. Considering the small momentum limit with a fixed strength of the\nexternal field, we may regard the external field as O(1). This is analogous to the chiral EFT\nfor the nuclear force, where the leading order term contains theO(1) contact interaction. In\nthis case, systematic perturbation theory to describe the t wo-nucleon system with a large\nscatteringlengthcanbeformulated,bythenonperturbativ eresummationofthe O(1)contact\nterm [42, 43]. In general, nonrelativistic EFT with a contac t interaction has two fixed points,\ncorresponding to the vanishing scattering length (non-int eracting limit) and the infinitely\nlarge scattering length (unitary limit) [44, 45]. While the original counting scheme (92)\ncorresponds to the expansion around the non-interacting li mit, the new counting scheme\ngives the expansion around the unitary limit [42, 43]. We thu s conclude that the EFT in this\nwork can describe the magnon systems with a large scattering length, with an appropriately\nmodified power counting scheme.\nWhen the two-body scattering length is infinitely large, the three-body system is known to\nexhibit the Efimov effect [44]. In fact, the Efimov effect for magno ns is shown to be induced\nby the anisotropic effect [15, 16]. This is consistent with the presence of the contact term in\nthe EFT with the anisotropic effect.\nMoreover, wenotethatthescatteringlengthofthegappedmo deintheferrimagnetremains\nfinite without the anisotropic effect as in Eq. (69). In this cas e, although the contact term\ndoes not explicitly appear in the Lagrangian, the time deriv ative term generates an effective\ncontact interaction which is proportional to the gap energy .4We expect that the Efimov\neffect for magnons in a ferrimagnet can be realized even in the a bsence of the anisotropic\neffect.\n4. Summary and concluding remarks\nIn this paper, we have discussed the low-energy effective field theory for spin systems includ-\ningaferrimagnet, andthescattering processesofthemagno nsas theNambu-Goldstone(NG)\nmodes. On the basis of the Lagrangians, we have obtained the d ispersion relations of the\nmagnonsastheNGmodes,whichcoincidewiththemicroscopic results.Inparticularthecase\nof the ferrimagnet is worth mentioning, where the NG modes ha ve two-types of dispersion\n4Strictly speaking, this kind of effective contact term also contribut es to the scattering length with\nthe anisotropic effect.\n22/26relations, ω∝k2,k2+m2, i.e., a massless mode with a quadratic momentum dependence\nand a gapped mode, both of which come from the spontaneous sym metry breaking. We have\nshown that the Lagrangian including only terms with one time derivative describes the sys-\ntem with just one order parameter given by the commutation re lation between the broken\ncharge and the charge density (magnetization), while the La grangian including both terms\nwith one and two time derivatives describes the system with t wo order parameters (magne-\ntization and staggered magnetization). This result also su pports the good correspondence\nbetween the EFTs and the microscopic theories of the spin sys tems. Furthermore, we have\nestablished the power counting scheme for the Lagrangian wh ich confirmed the validity of\nthe gapped mode, and clarified the systematic expansion.\nTo determine the particle (magnon) states and discuss the sc attering process, we have\nderived the Hamiltonian from the Lagrangian. In the ferroma gnet case, where no term\nwith two time derivatives exists in the Lagrangian, the Hami ltonian has been constructed\nusing the Dirac-Bergmann method for constraint systems, wh ile in the antiferromagnet and\nthe ferrimagnet cases the Hamiltonian has been constructed by the conventional method.\nFurthermore, the quantization of the system has been done by equating the commutation\nrelation with the Dirac brackets in the ferromagnet and the P oisson bracket in the antifer-\nromagnet and the ferrimagnet cases. Thus we have found that o ne magnon state as the NG\nmode appears in the ferromagnet, while two magnon states do s o in the antiferromagnet and\nthe ferrimagnet.\nThen we have calculated the scattering amplitudes for magno n-magnon scattering pro-\ncesses in the nonrelativistic notation. For the magnon-mag non scattering in the ferromagnet\nand antiferromagnet, we have reproduced the results given i n a previous study [13]. In addi-\ntion, we have obtained the amplitudes for the three types of t he magnon-magnon scattering\nprocesses in the ferrimagnet. Remarkably enough, the scatt ering lengths of the processes\ninvolving the gappedstate are finite even without explicit s ymmetry breaking. Thestrengths\nof the scattering lengths are related to the gap of the mode, b ecause both originate in the\nspontaneous symmetry breaking. The relation of the scatter ing length and the gap will be\nuseful to identify the gapped NG mode in real materials. The fi nite scattering length also\nimplies that the Efimov effects come into play in the ferrimagne t even without the external\nfields, in contrast to the ferromagnet discussed in Refs.[15 , 16].\nFinally, we have discussed the effects of the magnetic fields an d the anisotropy as explicit\nsymmetrybreakingeffects. Theeffective Lagrangian and thephy sical states of magnons have\nbeen constructed, and the scattering amplitudes have been c alculated. All the amplitudes\nfor the magnon-magnon scattering processes are affected by th e anisotropy but not by the\nmagnetic field. The irrelevance of the magnetic field may be at tributed to the identifica-\ntion of the magnetic field with the SO(3) gauge-field. In a rece nt cold atom experiment,\nthe dispersion relation of the magnon in the ferromagnetic p hase of spinor Bose-Einstein\ncondensate was successfully measured [46]. In addition, re alization of the ferrimagnetic state\nis also possible, for instance, using Kagome lattices [47, 4 8]. These developments, together\nwith the Efimov effect, will be useful to experimentally realiz e the findings in this paper.\n23/26Acknowledgments\nThe authors are grateful to Yoshimasa Hidaka for suggesting that we examine the relation\nbetween the order parameter and the coefficients from the pers pective of the EFT as dis-\ncussed in Sec.2.4. The authors thank Yoshimasa Hidaka, Tomo ya Hayata, Satoshi Fujimoto,\nand Sinya Aoki for useful discussions and comments. S. G. is s upported by JSPS KAK-\nENHIGrantsNo.25287046, SPIRE(Strategic ProgramforInno vativeREsearch)andagrant\nfrom LaR´ egion Centre(France). Y. K. issupportedby aGrant -in-Aid forScientific Research\nfrom the JSPS Fellows (No.15J01626). T. H. is supported by JS PS KAKENHI Grants No.\n24740152. T.K. is supportedbyJSPSKAKENHIGrants No. 24340 054. T. H. and T.K. are\nsupported by the Yukawa International Program for Quark-Ha dron Sciences (YIPQS).\nA. Scattering length, phase shift, and scattering amplitud e in the nonrelativistic\nnotation\nIn the nonrelativistic notation, the normalization of the s tates is defined by\n∝an}bracketle{tf|i∝an}bracketri}ht=δfi. (A1)\nTheSmatrix element Sfiand the scattering amplitude Mfifor the scattering process\ni→fare related as follows:\nSfi=δfi−i(2π)4δ4(Pf−Pi)Mfi, (A2)\nwherePiandPfdenote the total momentum of the initial and final states, res pectively. The\nunitary condition, SS†= 1, is reduced to\n−2ImMfi=/summationdisplay\nn(2π)4MfnM†\nniδ4(Pn−Pi). (A3)\nFor two-particle scattering processes below inelastic thr esholds in the center of mass frame,\nthe summation is reduced to the form given in terms of energie sEk, Eias,\n/summationdisplay\nnδ4(Pn−Pi)\n→/integraldisplayd3k1\n(2π)3/integraldisplayd3k2\n(2π)3δ3(k1+k2)δ(Ek−Ei), (A4)\nand the scattering amplitude is found to be written as Mfi=M(qf,qi) with the tree-\ndimensional initial and final momenta, qi,qf. The unitary condition of the scattering\namplitude is rewritten as\n−2ImM(qf,qi)\n=/integraldisplayd3k\n(2π)2M(qf,k)M†(k,qi)δ(E(|k|)−Ei). (A5)\nUsing the spherical harmonic functions Ylmand the Legendre polynomials Pl, the partial\nwave amplitude Ml\nfiis given by\nM(qf,qi) =/summationdisplay\nl(2l+1)Pl(cosθ)Ml(|qf|,|qi|)\n= 4π/summationdisplay\nl,mYlm(Ωf)Y†\nlm(Ωi)Ml(|qf|,|qi|), (A6)\n24/26with scattering angle θand solid angles Ω f,Ωifor the final and initial state. By performing\nthe partial wave expansion and the angular integral, the uni tary condition is reduced to\n−2/summationdisplay\nl(2l+1)Pl(cosθ)ImMl(qf,qi)\n=/integraldisplay∞\n0|k|2d|k|/summationdisplay\nl,mYlm(Ωf)Y†\nlm(Ωi)\n×Ml(|qf|,|k|)M†\nl(|k|,|qi|)δ(E(|k|)−Ei)\n=/summationdisplay\nl2l+1\n4πPl(cosθ)/integraldisplay∞\n0|k|2d|k|\n×Ml(|qf|,|k|)M†\nl(|k|,|qi|)δ(E(|k|)−Ei). (A7)\nThus the unitary condition is given in terms of the partial wa ve amplitude,\n−2ImMl(q,q)\n=1\n4π/integraldisplay∞\n0|k|2d|k|Ml(q,|k|)M†\nl(|k|,q)δ(E(|k|)−Ei)\n=1\n4π/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂E(q)\n∂q/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\nq2Ml(q,q)M†\nl(q,q), (A8)\nwhere we used q≡ |qf|=|qi|. Using the unitary condition, the phase shift δlis defined\nthrough the partial wave amplitude Ml:\nMl(q) =8π\nq2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂E(q)\n∂q/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinδleiδl. 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Using the\nself-consistent mean-field theory, we find that the spin-orb ital coupling will enhance the localization\nof the spin moments near a single vacancy. We further study th e magnetic structures along the\nzigzag edges formed by a chain of vacancies. We find that the sp in-orbital coupling tends to sup-\npress the counter-polarized ferrimagnetic order on the upp er and lower edges, because of the open\nof the spin-orbital gap. As a result, in the case of the balanc e number of sublattices, it will suppress\ncompletely this kind of ferrimagnetic order. But, for the im balance case, a ferrimagnetic order along\nboth edges exists because additional zero modes will not be a ffected by the spin-orbital coupling.\nI. INTRODUCTION\nGraphene and related nanostructured materials have\nattractedmuchinterestin solidstatephysicsrecentlydue\nto their bidimensional character and a host of peculiar\nproperties1. Among them, the investigation of the mag-\nnetic properties in graphene is one of the fascinating top-\nics, as no dandfelements are necessaryin the induction\nofmagnetism in comparisonwith the usual magnetic ma-\nterials. Theoretical predictions and experimental investi-\ngations have revealed that a nonmagnetic defect such as\nanimpurity oravacancycan induce the non-triviallocal-\nized magnetism2–6. Similarly, a random arrangement of\na large number of vacancies which are generated by the\nhigh-dose exposure of graphene to strong electron irradi-\nation7can also induce magnetism theoretically8. These\nstudies not only have the fundamental importance, but\nalso open a door for the possibility of application in new\ntechnologies for designing nanoscale magnetic and spin\nelectronic devices.\nOn the other hand, the topological insulating elec-\ntronic phases driven by the spin-orbital (SO) interaction\nhave also attracted much interest recently. The Kane-\nMele model for the topological band insulator is defined\non the honeycomb lattice9,10which is the same lattice\nstructure as graphene. Possible realization of an appre-\nciable SO coupling in the honeycomb lattice includes the\ncold fermionic atoms trapped in an extraordinary optical\nlattice11, the transition-metal oxide Na 2IrO312and the\nternaty compounds such as LiAuSe and KHgSb13. Topo-\nlogical band insulator has a nontrivial topological order\nandexhibitsabulkenergygapwithgapless,helicalstates\nat the edge14–16. These edge states are protected by the\ntime reversal symmetry and are robust with respect to\nthe time-reversal symmetric perturbations, such as non-\nmagnetic impurities. It is shown that a vacancy, acting\nas a minimal circular inner edge, will induce novel time-\nreversal invariant bound states in the band gap of the\ntopological insulator17–19. Theoretically, it is also shown\nthat the SO coupling suppresses the edge magnetism in-\nduced in the zigzag ribbon of the honeycomb lattice inthe presence of electron-electron interactions20. Thus, it\nis expected that the SO coupling would also affect the\nlocal magnetism in the bulk induced by vacancies.\nIn this paper, we study theoretically the effects of the\nSO coupling on the local magnetism induced by a sin-\ngle and a multi-site vacancy on the honeycomb lattice,\nbased on the Kane-Mele-Hubbard model where both the\nSO coupling and the Hubbard interaction between elec-\ntrons are taken into consideration. This model has been\nextensively studied to explore the effect of the strong\ncorrelation on the topological insulators21–27. Making\nuse of the self-consistent mean field approximation, we\ncalculate the local spin moments and their distribution\naround the vacancies. For a single vacancy, we find that\nthe main effect of the SO coupling is to localize the spin\nmoments to be near the vacancy, so that it will enhance\nthe local spin moments. For a large stripe vacancy by\ntaking out a chain of sites from the lattice, we find that\nthe SO coupling tends to suppress the counter-polarized\nferrimagnetic order induced along the zigzag edges, be-\ncause of the open of the SO gap. As a result, in the case\nof the balance number of sublattices (with even number\nof vacancies), the SO coupling will suppress completely\nthe counter-polarized ferrimagnetic order along the up-\nper and lower edges. While, in the case of the imbalance\nnumber of sublattices (with odd number of vacancies), a\nferrimagneticorder along both edges exists because addi-\ntionalzeromodeswill notbe affectedbythe SOcoupling.\nWewillintroducethemodelandthemethodoftheself-\nconsistent mean-field approximation in Sec.II. In Sec.III\nand IV, we present the results for a single vacancy and a\nmulti-site vacancy, respectively. Finally, a briefsummary\nwill be given in Sec.V.\nII. MODELS AND COMPUTATIONAL\nMETHODS\nWe start from the Kane-Mele model9, in which the\nintrinsic SO coupling with a coupling constant λis in-2\ncluded.\nH0=−t/summationdisplay\n/angbracketleftij/angbracketright,σc†\niσcjσ+iλ/summationdisplay\n/angbracketleft/angbracketleftij/angbracketright/angbracketrightσσ′vijσz\nσσ′c†\niσcjσ′,(1)\nwherec†\niσ(cjσ) is the creation(annihilation) operator of\nthe electron with spin σon the lattice site i,/angbracketleftij/angbracketrightrep-\nresents the pairs of the nearest neighbor sites (the hop-\nping ist) and/angbracketleft/angbracketleftij/angbracketright/angbracketrightthose of the next-nearest neighbors.\nvij= +1(−1) if the electron makes a left(right) turn to\nget to the second bond. The size of our system is consid-\nered to be finite with periodic boundary condition. So,\nthe position of each lattice site can be described specif-\nically by i= Γ(m,n), representing that the lattice site\niis in the mth column and the nth row, and Γ = A,B\nthe sublattice labels. The number of the unit cells is\ndenoted by Nc=L2, therefore the total number of the\nlattice sites is Nl= 2L2. To consider the correlation be-\ntween electrons, we will include the Hubbard term in the\nHamiltonian, which is given by HI,\nHI=U/summationdisplay\niˆni↑ˆni↓, (2)\nwhere ˆniσ=c†\niσciσ. When vacancies are introduced, the\nhoppings between the vacancy and the nearest neigh-\nbors and the on-site interaction on that vacancy are sub-\ntracted from the overall Hamiltonian. Hence the corre-\nsponding number of the lattice sites is Nl= 2L2−Nv,\nwhereNvis the number of vacancies. The total number\nof electrons Neis fixed to be at the half-filling ( Ne=Nl).\nThe Hubbard interaction term is treated with the self-\nconsistent mean field approximation, so that we will ob-\ntain an effective single-particle Hamiltonian where the\nelectrons interact with a spin-dependent potential,\nHI≃U/summationdisplay\ni,σ/angbracketleftˆni−σ/angbracketrightˆniσ−U/summationdisplay\ni/angbracketleftˆni↑/angbracketright/angbracketleftˆni↓/angbracketright.(3)\nAnd the overall mean field Hamiltonian Hmfis then\ngiven by,\nHmf=U/summationdisplay\niσ/angbracketleftˆni−σ/angbracketrightˆniσ+H0. (4)\nAfter diagonalizing the Hamiltonian Hmf, we can de-\ntermine the occupation number /angbracketleftˆni−σ/angbracketrightat each site with\ndifferent spins using the eigenvectors of Hmf, and this\nprocess is carried out iteratively until a required accu-\nracy is reached. Then the magnetic moment of each site\nmi=/angbracketleftˆni↑−ˆni↓/angbracketrightcan be calculated. We note that a\ncollinear magnetic texture is assumed in our system, as\nused before for the investigations of Kane-Mele-Hubbard\nmodel20,21. We have checked the results with the non-\ncollinear magnetic texture and found that the collinear\nmagnetic texture is favored.\nIII. MAGNETISM WITH ONE VACANCY\nThe calculation is carried out on the lattice with Nc=\n14×14 unit cells in which a single vacancy is introduced/s49 /s50 /s51 /s52 /s53 /s54/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54\n/s32 /s61/s48/s46/s48/s48\n/s32 /s61/s48/s46/s48/s53\n/s32 /s61/s48/s46/s49/s48/s32/s109\n/s105\n/s32\n/s32/s114/s47/s97/s40/s99/s41/s40/s98/s41\n/s32/s48/s46/s48/s48\n/s48/s46/s48/s52\n/s48/s46/s48/s55\n/s48/s46/s49/s49\n/s48/s46/s49/s52\n/s48/s46/s49/s56/s32/s40/s97/s41\nFIG. 1: (color online). (a) and (b): Distribution of the spin\nmoments mion lattice sites around asingle vacancy at A(7,7)\nwithU= 1.0t, in which (a) corresponds to the SO coupling\nconstant λ= 0.0 and (b) λ= 0.1t. The area and color of the\nhollow circles represent the magnitude of the spin moments.\n(c)mion theBsublattice as a function of the distance r\naway from the vacancy. The unit ais the distance between\nthe nearest sites.\non the site A(7,7). Figure 1 displays the distribution of\nthe magnetic moment when the Hubbard interaction is\ntaken to be U= 1.0t, in which the size and the color\nof the circle on each lattice site denote the magnitude\nof the local spin moment. From Fig.1(a) where the SO\ncouplingisturnedoff, onecanseethatlocalizedmagnetic\nmoments are induced around the vacancy in the presence\nof a finite Hubbard interaction U. This is in agreement\nwith the prediction of the Lieb theorem28regarding the\ntotal spin Sof the exact ground state of the Hubbard\nmodel on bipartite lattices. It states that the total spin\nSis given by the sublattice imbalance 2 S=|NA−NB|,\nwithNAandNBthe number of atoms belonging to each\nsublattice. With the introducing of a single vacancy on\ntheAsublattice, an imbalance NB−NA= 1 appears\nand a magnetic structure near the vacancy with the total\nspinS= 1/2 will form. Similar results have also been\nobtained in recent studies in graphene2–4.\nIn the presence of the SO coupling, the magnitude of\nthe magnetic moments around the vacancy increases, as\nshown in Fig. 1(b) for λ= 0.1t. At the meantime, if\nwe check the distribution of the magnetic moments, as\nshown in Fig.1(c) where the magnitude of the magnetic\nmoments on sublattice Bas a function of the distance r\nawayfromthe vacancyis presented, one will find that the\nmagnetic moments are more localized with the increase\nof the SO coupling. These features demonstrate that the\nSOcoupling will enhancethe magneticmoments nearthe\nvacancy notably.\nIn order to show the emergence of the magnetism in-\nduced by the vacancy in more detail, we calculate the3\nspin resolved local density of state(LDOS) as defined by,\nDσ(ǫ) = Σn,i|un\ni,σ|2δ(ǫ−ǫn), (5)\nwhereiruns over the lattice sites surrounding the va-\ncancy up to the third-nearest neighbors, as those linked\nby the green line in Fig.1(a) and (b). un\ni,σis the single-\nparticle amplitude on the ith site with spin σand the\ncorresponding eigenvalue is ǫn. The Delta function in\nEq.(2)isreplacedbytheLorentzianfunction forplotting.\nThe results for the LDOS are presented in Fig. 2(a)-(h)\nfor different Hubbard interaction Uand SO interaction\nλ. The red and blue lines represent the LDOS for the\nspin up and spin down components respectively, and the\ndash lines show the LDOS away from the vacancy for\na comparison. In the case of U=λ= 0.0 as shown\nin Fig. 2(a), the LDOS shows a V-shape linear behavior\nnear the Fermi level for those lattice sites far away from\nthe vacancy (denoted by the dashed line) which is the\nconsequence of the linear dispersion relation of the elec-\ntrons, the so-called Dirac fermions. For those around the\nvacancy, a peak at the Fermi level emerges as shown by\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s40/s101/s41/s76/s68/s79/s83\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s40/s103/s41/s76/s68/s79/s83\n/s69/s47/s124/s116/s124/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s40/s104/s41\n/s69/s47/s124/s116/s124/s40/s102/s41/s40/s100/s41\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s40/s99/s41/s76/s68/s79/s83/s40/s98/s41\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s76/s68/s79/s83/s40/s97/s41\nFIG. 2: (color online). LDOS for λ= 0.0 [left column, includ-\ning (a),(c),(e),(g)] and for λ= 0.1t[right column, including\n(b),(d),(f),(h)], where the Hubbard interaction U= 0.0 for\n(a) and (b), U= 1.6tfor (c) and (d), U= 2.6tfor (e) and\n(f), and U= 3.6tfor (g) and (h), respectively. LDOS for\ndifferent spins is resolved, those with the spin up are denote d\nby the blue lines and the spin down the red lines. The grey\ndash lines represent the LDOS on the lattice site away from\nthe vacancy./s49 /s50 /s51 /s52/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s40/s98/s41\n/s77\n/s32 /s108/s111/s99\n/s85/s32/s47/s32/s116/s32 /s61/s48/s46/s48/s48\n/s32 /s61/s48/s46/s48/s52\n/s32 /s61/s48/s46/s49/s48\n/s32 /s61/s48/s46/s49/s54\n/s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s56 /s48/s46/s49/s50 /s48/s46/s49/s54/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48\n/s32/s32\n/s32/s85/s61/s49/s46/s48\n/s32/s85/s61/s50/s46/s50\n/s32/s85/s61/s50/s46/s52\n/s32/s85/s61/s50/s46/s56/s77\n/s32 /s108/s111/s99\n/s32/s47/s32/s116/s40/s97/s41\nFIG. 3: (color online). Local moments Mloc(see text) are\nshownasafunctionoftheSOcoupling λfordifferentHubbard\ninteraction U(a) and of Ufor different λ(b).\nthe solid line, which corresponds to the localized states\ninduced by the vacancy29. After turning on the SO cou-\npling, such as that for λ= 0.1t[see Fig.2(b)], we can see\nthat an energy gap opens for those lattice sites far away\nfrom the vacancy9,10, so that now a U-shape LDOS near\nthe Fermi level occurs. In this way, the mid-gap peak is\nenhanced noticeably because the decay rate of the local-\nized states into the continuum is reduced largely due to\nthe open of the energy gap. This will lead to the increase\nin the spectral weight of the localized states around the\nvacancy. However, for both cases, one will find that the\nLDOS for the spin up and spin down components degen-\nerates, so that the system will not show magnetism as a\nwhole without the Hubbard interaction.\nThe effect of a finite Hubbard interaction Uis to split\nthe spin degenerate LDOS, so that two peaks occur cor-\nresponding to different spins, as shown in Fig. 2(c)-(f).\nConsequently, the localized spin up and down moments\nwill not cancel out in this case, and a net magnetism\naround the vacancy is induced.\nThe magnetism may be quantified by the local mo-\nmentMloc=/summationtext\nimi, where the sum runs over the lat-\ntice sites surroundingthe vacancy up to the third-nearest\nneighbors as used above in the calculation for the LDOS.\nThe results are presented in Fig. 3(a) and (b) for dif-\nferentUandλ, respectively. The local moment Mloc\nshows a monotonic increase with the SO coupling λ, so\nit reinforces our observation that the local magnetism\nis enhanced by the SO coupling as shown in Fig.1. On\nthe other hand, Mlocshows a nonmonotonic dependence\non the Hubbard interaction U, namely it increases with\nUfirstly and then decreases with a further increase of\nUafter a critical value Uc. As discussed above, the lo-\ncal magnetism is determined by the spin-split localized\nstates induced by the vacancy, and it is the Hubbard in-\nteraction Uto split the spin-degenerate states. Because\nthe open of the gap due to the SO coupling will decrease\nthe decay rate of the localized states into the continuum,\nso it will enhance the spectral weight of the localized\nstates[see also Fig. 2], consequently the localized mag-\nnetism. The splitting between the two localized states\nwith different spins is proportional to U, so the two split4\nlocalized states will situate in the SO gap for a small\nU[Fig. 2(c)-(f)]. However, when U > U cthe splitting\nwill be larger than the SO gap, and it pushes the local-\nizedstatestomergeintothecontinuum[Fig.2(g)and(h)],\nso the local magnetism will decrease.\nIV. THE CASE OF MULTI-SITE VACANCY\nThe multi-site vacancy can be formed by removing the\nsites continuously. Here, we consider a large stripe va-\ncancy by taking out a chain of sites from the lattice as\nilluminated in Fig. 4. In this way, the stripe vacancy\nconsists of one upper and one lower zigzag edges. As\nclarified by the Lieb theorem28, the sublattice imbalance\nbetween the number of atoms belonging to different sub-\nlattices will have significant effect on the magnetism. For\nthe stripe vacancy considered here, the imbalance is ex-\npressed by the parity of the number of vacancies, where\nthe number is even ( NA=NB) in Fig. 4(a) and (b), and\nodd (NA/negationslash=NB) in Fig. 4(c) and (d), thus the total spin\nof the system is S= 0 and 1 /2 respectively.\nInthecaseofevennumberofvacancies,aferrimagnetic\nspin order emerges on both the upper and lower zigzag\nedges around the stripe vacancies when there is no SO\ncoupling, as shown in Fig. 4(a). The ferrimagnetic ar-\nrangement and the magnitude of the spin moments on\nthese two edges are symmetric, but they are counter-\npolarized, so they cancel out exactly and the whole sys-\ntem will not show magnetism. This is consistent with\nthe Lieb theorem28. The ferrimagnetic order on a suffi-\nciently long zigzag edge around the stripe vacancies here\nissimilartothespin orderformedattheouteredgeofthe\nzigzagribbon30–33and the graphenenanoisland34. In the\ncase of odd number of vacancies, a similar ferrimagnetic\nspin order is also induced with a slightly large magnitude\n[Fig. 4(c)]. Interestingly, this ferrimagnetic order occurs\nonly on the upper zigzag edge, not on the lower edge.\nThis phenomenon is ascribed to the presence of an extra\nspin when a sublattice imbalance NA/negationslash=NBexists, as\n/s40/s97/s41\n/s32\n/s32\n/s32/s32 /s32/s32\n/s40/s99/s41/s40/s98/s41\n/s32\n/s32/s45/s48/s46/s49/s56/s45/s48/s46/s49/s49/s45/s48/s46/s48/s52/s48/s46/s48/s52/s48/s46/s49/s49/s48/s46/s49/s56\n/s40/s100/s41\n/s32\n/s32/s45/s48/s46/s50/s54/s45/s48/s46/s49/s54/s45/s48/s46/s48/s53/s48/s46/s48/s53/s48/s46/s49/s54/s48/s46/s50/s54\n/s66/s65/s66/s65\nFIG. 4: (color online). Distribution of the spin moments mi\non the lattice sites surrounding the vacancies for U= 1.0tand\nL= 14. A cluster of vacancies is formed with the number of\nmissing sites for (a), (b) Nv= 8 and (c), (d) Nv= 7. SO\ncoupling is set to be λ= 0.0 for (a), (c) and λ= 0.1tfor\n(b), (d). The area and color of hollow circles represent the\nmagnitude of the moments./s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s40/s98/s41\n/s32/s78 /s118/s61/s51\n/s32/s78 /s118/s61/s53\n/s32/s78 /s118/s61/s55\n/s32/s78 /s118/s61/s57\n/s32/s78 /s118/s61/s50\n/s32/s78 /s118/s61/s52\n/s32/s78 /s118/s61/s54\n/s32/s78 /s118/s61/s56\n/s32/s78 /s118/s61/s49/s48/s77\n/s101\n/s47/s32/s116/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48\n/s32/s78 /s118/s61/s51\n/s32/s78 /s118/s61/s53\n/s32/s78 /s118/s61/s55\n/s32/s78 /s118/s61/s57\n/s32/s32/s77\n/s108/s111/s99\n/s47/s32/s116/s40/s97/s41\nFIG. 5: (color online). (a)Local moments Mlocare plotted as\na function of λwhileU= 1.0tandL= 14. The function in\ndifferent size of vacancy is distinguished by different color and\nshape of points. The cases of even Nvare not plotted as local\nmoments are always zero obeying Lieb theorem28. (b) The\nfunction of edge moments Meversusλare given in different\nNv.\ndescribed by the Lieb theorem28.\nAfter turning on the SO coupling, such as for λ= 0.1t,\nthe ferrimagnetic spin order on both the upper and lower\nzigzag edges around the stripe vacancies disappears com-\npletely in the case of even number of vacancies[Fig. 4(b)].\nHowever, the effect of the SO coupling on local mag-\nnetism is quite different for the case of an odd number\nof vacancies. Here, a ferrimagnetic spin order similar\nto that on the upper edge emerges on the lower edge,\nthough the magnitude of the individual spin moment is\nreduced[Fig. 4(d)]. To show variation of the total mag-\nnetism, we plot the quantity Mlocas a function of the\nSO coupling λin Fig. 5(a), here Mlocis the sum of the\nspin moments on the sites which are on the zigzag edges\naround the vacancies. Since Mlocis always zero in the\ncase of even Nv, it is not plotted here. With an odd\nNv, the local moment Mlocincreases with the increase\nofλ, which shows a similar behavior as that in the case\nof a single vacancy. This indicates that the total local\nmagnetism shown in Fig. 4(d) is in fact enhanced with\nthe introduction of the SO coupling and approaches the\nsaturation value 1 finally. From Fig. 5(a), one can also\nfind that Mlocincreases with the increase of the number\nof vacancies Nv. This suggests that the SO coupling will\nlocalize the induced spin moments to those lattice sites\nwhich are neighboring the vacancies.\nTo quantify the variation of the spin moments with\nλon the upper zigzag edge, we also present Meas a\nfunction of λin Fig. 5(b), here Meis the sum of the\nspin moments only on the sites on the upper zigzag edge.\nLet us consider firstly the case of even number of Nv,\nfor a small number of even vacancies, Meis always zero.\nUp toNv≥8, a finite Meoccurs and it increases with\nNvby the formation of the zigzag edges. However, Me\ndrops rapidly to zero after turning on the SO coupling.\nThese results quantify the physical picture derived from\nFig. 4(a) and (b). Now let us turn to the case of odd\nnumber of Nv. Without the SO coupling, Mealso shows5\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s40/s98/s41\n/s32/s101/s110/s101/s114/s103/s121/s47/s116\n/s32/s109/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s40/s97/s41/s32/s32/s101/s110/s101/s114/s103/s121/s47/s116\n/s32/s109\nFIG. 6: (color online). The single-particle energy levels l a-\nbeled with m(see text) near the Fermi level of the non-\ninteracting systems for (a) Nv= 8 and (b) Nv= 7. SO cou-\npling is set to be λ= 0.0 for the red circles and λ= 0.1tfor\nthe blue squares.\nan increase with Nv. With the introduction of the SO\ncoupling, Meshows a decrease with λand saturates to\nnear one half of Mloc.\nIn fact, we can make an analogy between the stripe\nvacancy and the graphene ribbon with zigzag edges. A\nremarkable feature of the graphene ribbon with zigzag\nedges is that it has a flat band localized on the zigzag\nedge1. An important effect of this flat band is that a\ncounter-polarized ferromagnetic order along the upper\nand lower edges will be induced when the Hubbard in-\nteraction between electrons is included30–33. In view of\nthis, we plot the single-particle spectra for the systems\nwith the stripe vacancy without the Hubbard interaction\nin Fig.6(a) and (b) for Nν= 8 and Nν= 7, respectively.\nEach energy level is labeled with m=n−Ne−1/2 in\norder to indicate that the energy level with m <0 is\noccupied by electron. For the systems without SO cou-\npling, we find that there are four near-degeneracy local-\nizedstates[redcirclesindicatedbyarrowsinFig.6(a)and\n(b)] which is near the Fermi level for both Nν= 8 and\nNν= 7. These states will have the same effect as the flat\nband in the zigzag ribbon when a suitable Hubbard U\nis turned on. So, a counter-polarized ferrimagnetic order\nas shown in Fig.4(a) will emerge. However, we note that\nthere are two additional zero modes for Nν= 7 relative\ntoNν= 8, due to the imbalance between the sublattices(NA> NB). Thesezeromodeswillinduceextraspinmo-\nments on both edges, which counteract the antiparallel\nmoments on the lower edge. Thus, in the case of Nν= 7,\nonly the ferrimagnetic order on the upper edge appears.\nAfter turning on the SO coupling, those localized states\n[blue squares indicated by arrows in Fig.6(a) and (b)] are\npushed away from the Fermi level due to the open of the\nSO gap. Thus, as shown in Fig.4(b), a small Hubbard U\nis not enough to induce the counter-polarized ferrimag-\nnetic order on the upper and lower edges. However, the\nzero modes originating from the imbalance of sublattices\nare not affected by the SO coupling [Fig.6(b)]. So, the\nadditional ferrimagnetic order on both edges induced by\nthese zero modes will remain for Nν= 7.\nV. CONCLUSION\nIn a summary, we have studied the local magnetism\ninduced by vacancies on the honeycomb lattice based on\nthe Kane-Mele-Hubbard model. It is shown that the SO\ncoupling tends to localize and consequently enhances the\nlocal magnetic moments near a single vacancy. Further-\nmore, along the zigzag edges formed by a chain of va-\ncancies, the SO coupling will suppress completely the\ncounter-polarized ferrimagnetic order along the edges.\nTherefore, the system will not show any local magnetism\nin the case ofeven number ofvacancies. For an odd num-\nber of vacancies, a ferrimagnetic order along both edges\nexists and the total magnetic moments along both edges\nwill increase.\nAcknowledgments\nThis work was supported by the National Natural\nScience Foundation of China (Grant Nos. 91021001,\n11190023 and 11204125) and the Ministry of Science\nand Technology of China (973 Project grant numbers\n2011CB922101 and 2011CB605902).\n1A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. 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Lett.\n99, 177204 (2007)."    },    {        "title": "2003.07750v1.Electrical_generation_and_detection_of_terahertz_signal_based_on_spin_wave_emission_from_ferrimagnets.pdf",        "content": "1 \n Electrical g eneration and detection of terahertz signal \nbased on spin-wave emission  from ferrimagnets   \nZhifeng Zhu1,2,*, Kaiming Cai1, Jiefang Deng1, Venkata  Pavan Kumar Miriyala1, Hyunsoo \nYang1, Xuanyao Fong1, Gengchiau Liang1,† \n1Department of Electrical and Computer Engineering , National University of Singapore , \nSingapore  117576   \n2School of Information Science and Technology, ShanghaiTech University, Shanghai, China  \n201210  \n \nTerahertz (THz) signals , mainly  generated by  photonic or electronic approaches,  are being  \nsought for various applications,  whereas  the development of magnetic source might be a \nnecessary step to harness the magnetic natu re of electromagnetic radiation . We show that the \nrelativistic effect  on the  current -driven  domain -wall motion induces THz  spin-wave  emission \nin ferrimagnets . The required current density increases dramatically in materials with strong \nexchange interaction  and rapidly exceeds  1012 A m–2, leading  to the device breakdown and thus \nthe lack of experimental evidence . By translating  the collective magnetization oscillation s into \nvoltage signals , we propose a three -terminal device for  the electrical  detection of THz spin \nwave . Through material engineering,  wide frequency range  from 264 GHz to 1.1 THz and \nuniform continuous signal s with improved output power  can be obtained . As a reverse effect, \nthe spin wave generated in this system is able to move ferrimagnetic domain wall. Our work \nprovides guidelines for the experimental verification of THz spin wave, and could  stimulate \nthe design of THz spintronic oscillator s for wideband  applications  as well as the all-magnon \nspintronic devices . \n  2 \n I. Introduction  \nThe genera tion of terahertz signals (THz) , approximately from 300 GHz to 3 THz, is of \ngreat interest due to the extensive applications in biology, security, material characterization, \nand hi gh speed wireless communication  [1-4]. Since the THz  spectrum lies between two well -\nestablished domains  (that is microwaves and light),  it can be generated indirectly using solid -\nstate electronic devices combine d with the frequency multiplier  [5-7] or the photonic \napproaches using the quantum cascade laser  [8,9] , requiring additional peripheral circuits for \nthe frequency conversion s. On the other hand, the direct generation using Josephson junction s \n[10,11]  and resonant -tunneling diode  [12-14] are constrained by the cryogenic cooling and \nlarge device size  (i.e., due to the capacitor structure) , respectively.  \nIn contrast to the electronic and photonic means , the strong exchange interaction (> 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.  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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": "1902.00449v1.Quantum_thermodynamics_of_complex_ferrimagnets.pdf",        "content": "Quantum thermodynamics of complex ferrimagnets\nJoseph Barker1and Gerrit E.W. Bauer2, 3\n1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n2Institute for Materials Research & AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\n3Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands\nHigh-quality magnets such as yttrium iron garnet (YIG) are electrically insulating and very complex. By\nimplementing a quantum thermostat into atomistic spin dynamics we compute YIG’s key thermodynamic prop-\nerties, viz. the magnon power spectrum and specific heat, for a large temperature range. The results differ\n(sometimes spectacularly) from simple models and classical statistics, but agree with available experimental\ndata.\nIntroduction The spin dynamics of electrically insulating\nmagnets often has high quality because the dissipation chan-\nnel by conduction electron scattering is absent. With few ex-\nceptions, they are complex ferrimagnets. Yttrium iron garnet\n(YIG) with 80 atoms in the unit cell rules with a record low\nGilbert damping of long wavelength spin wave excitations or\nmagnons, even at room temperature [1, 2]. The implied ex-\nceptionally low disorder and weak coupling with phonons re-\nmains a mystery, however. Recently, magnon heat and spin\ntransport were measured in YIG thin films in a non-local\nspin injection and detection configuration with Pt contacts by\nmeans of the spin Hall effect [3] and modelled by spin dif-\nfusion [4]. Key parameters of this model are linked to the\nthermodynamics of the magnetic order, such as the magnon\nheat capacity, which is difficult to measure because it is or-\nders of magnitude smaller than the phonon heat capacity—at\n10 K the magnon and phonon heat capacities are Cm\u00190:009\nJ kg\u00001K\u00001andCp\u00190:270J kg\u00001K\u00001[5]. They can be\nseparated by magnetic freeze-out of the magnon contribution\nat temperatures up to a few Kelvin [5, 6]. The magnon heat\ncapacity at higher temperatures has been estimated by extrap-\nolating models that agree with experimental low-temperature\nresults [4, 7]. YIG is often treated as a single-mode ferromag-\nnet with quadratic !/Dk2(or isotropic cosine function)\ndispersion, thereby ignoring higher frequency acoustic and\noptical modes and temperature dependence of the exchange\nstiffness D. Furthermore, magnon-magnon interactions are\nalso commonly neglected or treated in a mean field approx-\nimation. Statistical approaches also have issues, such as the\nuse of classical (Johnson-Nyquist) thermal noise at low tem-\nperatures [8].\nIn this Letter we introduce a numerical method that avoids\nall of these shortcomings. It allows us to carry out material-\ndependent thermodynamic calculations that are quantitatively\naccurate with a small number of parameters that can be de-\ntermined independently. The crucial ingredient is a thermo-\nstat for Planck quantum (rather than Rayleigh–Jeans classical)\nstatistics in an atomistic spin dynamics framework [9].\nWith the inclusion of quantum thermal statistics we find\nquantitative agreement for YIG with available experiments at\nlow temperatures. The computed spin wave dispersion as a\nfunction of temperature agree well with results from neutron\nscattering. This low temperature quantitative benchmarkingimbues trust in the technique for calculating thermodynamic\nfunctions and allows access to quantities such as the magnon\nheat capacity at room temperature that turns out to be an order\nof magnitude larger than previous estimates.\nMethod We address the thermodynamics by computing\nthe atomistic spin dynamics in the long (ergodic) time limit\nto generate canonical ensembles of spins. The magnetic mo-\nments (‘spins’) in this model are treated as classical unit vec-\ntorsS, an excellent approximation for the half-filled 3d-shell\nof the iron cations in YIG with S= 5=2and magnetic mo-\nment\u0016s=g\u0016BS, whereg\u00192is the electron g-factor and\n\u0016Bthe Bohr magneton.\nThe Heisenberg Hamiltonian H=\u00001\n2P\nijJijSi\u0001Sjcon-\ntains the (super)-exchange parameters Jijbetween spins on\nsitesiandj, which are determined by fits to inelastic neu-\ntron scattering data [10]. Recently, the magnon dispersions\nwere measured again with higher resolution [11], allowing\nan improved parameterization of the six nearest-neighbors ex-\nchange constants, which we adopt in the following. We add a\nZeeman term H=\u0000P\ni\u0016s;iHext\u0001SiwithHext=Hz=\n0:1 T, to fix the quantization axis. On each lattice site ‘ i’\nthe spin dynamics obey the Landau-Lifshitz equation of mo-\ntion [12]:\n@Si\n@t=\u0000j\rj(Si\u0002Hi+\u0011Si\u0002(Si\u0002Hi)); (1)\nwhere\r=g\u0016B=~is the gyromagnetic ratio and \u0011is a damp-\ning constant. Each spin feels an effective magnetic field Hi=\n\u0018i\u0000(1=\u0016s;i)@H=@Si;where \u0018iare stochastic processes\ncontrolled by the thermostat at temperature T.h\u0018i\u000bi= 0\nand the correlation function in frequency space is governed\nby the fluctuation-dissipation theorem (FDT) h\u0018i\u000b\u0018j\fi!=\n2\u0011\u000eij\u000e\u000b\f'(!;T)=\u0016s;i;where the Kronecker \u000e0s reflect the as-\nsumption that the fluctuations between lattice sites i;jand\nCartesian coordinates \u000b;\f are uncorrelated. '(!;T)de-\nscribes the temperature dependence of the noise power and is\nchosen such that the steady-state distribution functions obey\nequilibrium thermal statistics. By not approximating the spin\nHamiltonian by a truncated Holstein-Primakoff expansion,\nour approach includes magnon-magnon interactions to all or-\nders [13].\nAtomistic spin dynamics methods generally assume the\nclassical limit of the FDT with frequency independent (white)\nnoise'(!;T) =kBT, i.e. allmagnons are stimulated.arXiv:1902.00449v1  [cond-mat.mtrl-sci]  1 Feb 20192\nThe energy equipartition of the coupled system results in the\nRayleigh-Jeans magnon distribution. However, this is only\nvalid when the thermal energy is much larger than that of the\nmagnon mode kunder consideration, i.e. when kBT\u001d~!k,\nwhile the energies of the YIG magnon spectrum–and that of\nmost room temperature magnets–extend up to ~!k=kB\u0019\n1000 K [1]. A classical thermostat therefore generates too\nmany high energy magnons, which, for example, overesti-\nmates the broadening by magnon scattering and leads to other\npredictions that can be very wrong.\nIn the proper quantum FDT [14]\n'(!;T) =X\nk~!k\nexp ( ~!k=kBT)\u00001; (2)\nwhich means that equipartition is replaced by Planck statis-\ntics of the magnons at temperature T. Quantum statistics\nin classical spin systems can partially be mimicked through\na post-process rescaling of the temperature [15] or by using\ntemperature-dependent frequencies that rely on analytic ex-\npressions for the low temperature spectrum [16]. These ap-\nproaches cannot be used to evaluate all thermodynamic prop-\nerties and are not suitable to treat complex magnets such as\nYIG. We therefore adopt here the ‘quantum thermostat’ as in-\ntroduced earlier in molecular dynamics [17, 18], i.e., a corre-\nlated noise source that obeys the quantum FDT. This is a “col-\nored” noise, but very different from the one used to describe\nclassical memory effects in the heat bath [19, 20].\nWe implement the quantum statistics by generating corre-\nlated fluctuating fields \u0018i(t)numerically in time that obey\nthe FDT in the frequency domain. Savin et al. [18] employ\na set of stochastic differential equations that produce the re-\nquired distribution function. We adjust this method for the\nspin dynamics problem, but can refer the reader to Ref. [18]\nfor the technical details. The solution provides a dimension-\nless stochastic process \bi\u000b(t)with the spectrum of Eq. (2).\nThe dimensionful noise in the spin dynamics reads\n\u0018i\u000b(t) =kBTr2\u0011\u0016s;i\n\r~\bi\u000b(t): (3)\nWhen we agitate the model of classical spins with these\nstochastic fields, the excitations of the ground state (magnons)\nobey quantum statistics, quite analogous to quantized phonons\nin a classical ball-spring lattice. This approach may loosely\nbe called a ‘semi-quantum’ method which should work very\nwell for the large Fe3+spin in YIG with S= 5=2, but requires\nmore scrutiny for spin S= 1=2.\nWe integrate equation (1) using the Heun method with time\nstep\u0001t= 0:1fs. The stochastic differential equations of the\nthermostat are integrated using the fourth-order Runge-Kutta\nmethod with the same time step.\nMagnon spectrum We compare now the magnon spec-\ntrum computed with the quantum thermostat with our previ-\nous work with classical statistics (and older exchange con-\nstants from Ref. [1]) [9]. Results for low ( 5K) and room\n(300 K) temperature are shown in Fig. 1a. The classical\nN H\n N Γ H020 40 60 80 100 E(meV) T=5K T=300K \nΓ\nMagnonGap(meV) \nT(K) semi-quantum \nexperiment\n0262830323436\n100 200 300a) \nb) FIG. 1. a) YIG magnon spectrum at T= 5 K andT= 300 K\ncalculated using the quantum thermostat and the exchange parame-\nters of Ref. 11. The color intensity is adjusted on a log scale such\nthat all modes are visible (even for extremely low occupation) and is\ndifferent for both figures. The red/blue color shows the +/- polariza-\ntion of the magnons. b) Magnon gap between optical and acoustic\nmodes at \u0000. Experimental data are adopted from neutron scattering\nexperiments [10].\nthermostat overestimated the number of high-energy magnons\nand therefore the broadening of the optical modes at higher\ntemperatures. With quantum statistics, the high-energy opti-\ncal modes are well resolved at room temperature and should\nbe observable by inelastic neutron scattering with large fre-\nquency transfer. The agreement between the calculated and\nmeasured [10] temperature dependence of the exchange gap\nbetween optical and acoustic modes at the \u0000point, shown in\nFig. 1b, is improved, especially in the low temperature regime.\nMagnetization The magnetization at low temperatures\nmz= 1\u00001\nSP\nk\u0017hnk\u0017iT;wherehnk\u0017iTis the distribution of\nmagnons with wave vector kand band index \u0017in the first Bril-\nlouin zone, cannot be calculated correctly with classical statis-\ntics [21] (at higher temperatures the expression does not hold\nsince magnon-magnon interactions are important). This is ob-\nvious already for the single parabolic band, non-interacting\nmagnon gas model for which\n1\u0000mz(T) =vws1\nS\u0000\u00003\n2\u0001\n\u0010\u00003\n2\u0001\n2\u00192\u0012kBT\nD\u00133=2\n(4)\nwhere!k=Dk2;spin-wave stiffness D= 2SJa2;lattice\nconstanta;vwsvolume of the Wigner-Seitz cell, while \u0000(x)\nand\u0010(x)are the gamma and Riemann zeta functions. The\nT3=2dependence is known as Bloch’s law [22].3\nIn the ferrimagnet YIG the total magnetization is made\nup by two oppositely aligned sublattices with slightly dif-\nferent temperature dependent magnetizations. At low tem-\nperatures they are rigidly locked to an antiparallel configu-\nration by the strong nearest neighbor exchange. At ener-\ngies~!k=kB/30 K YIG’s magnon dispersion is known\nto be quadratic and its magnetization obeys Bloch’s T3=2\nlaw [23]. The expected deviations at higher temperatures\ncan be assessed by our method. We calculate the mag-\nnetization at temperature Tas an averageh\u0001\u0001\u0001iTover the\nspin configurations at many times over a 1 ns trajectory\nm(T) =hN\u00001PN\ni\u0016s;iSiiT=hN\u00001PN\ni\u0016s;iSiiT=0, where\nN= 655;360is the total number of spins in the simulation.\nFig. 2 exposes the obvious problem of classical statistics\nto compute magnetizations at low temperatures: The magne-\ntization decreases much more rapidly with temperature than\nBloch’s law (and as observed in experiments). The results\nwith the quantum thermostat, on the other hand, adhere to\nBloch’s law for T <30K (see inset) but also agree well with\nexperiments that signal a breakdown of T3=2scaling, at least\nuntil\u0018300K.\nThe Curie temperatures for the classical ( TC= 420 K) and\nquantum thermostated systems ( TC= 680 K) are quite differ-\nent, while the observed TC= 550 K lies between the theoret-\nical values. In contrast to classical results that obey equipar-\ntition, the Curie temperature of quantum approaches depend\nonSand we find this also in our semi-quantum approach. For\na simple ferromagnetic BCC lattice our computed Curie tem-\nperatures (not shown) agree well with those obtained by semi-\nanalytic approaches [24] for a large range of S. The overesti-\nmation ofTCcompared to the experiment might be caused by\nexchange parameters that are slightly too large since the neu-\ntron scattering data are fitted only up to 90meV which does\nnot cover the magnon modes with highest energy. Also, the\nchoice of S= 5=2(\u0016s= 5\u0016B) in extracting the exchange\nparameters does not fully agree with with measured values of\n\u0016s;a= 4:11\u0016Band\u0016s;d= 5:37\u0016Bfor the octahedral and\ntetrahedral sites [25]. Hence, a more accurate set of parame-\nters, fitted to neutron scattering data for large energy transfers\nor calculated from first principles, should solve this discrep-\nancy.\nHeat Capacity The magnon heat capacity per unit vol-\numeCm=V\u00001(@Um=@T)Vis the change in the inter-\nnal magnetic energy Umwith temperature at constant vol-\numeV. It can be calculated from the magnon spectrum\nasCm=V\u00001(@=@T )P\nk\u0017~!k\u0017hnk\u0017i, wherehnk\u0017iis the\nPlanck distribution. In the low temperature limit magnons oc-\ncupy only states close to k= 0, where the magnon dispersion\nof ferromagnets is parabolic. For a single parabolic magnon\nband [13]\nCm(T) =1\nV5\n8\u0000\u00005\n2\u0001\n\u0010\u00005\n2\u0001\n\u00192kB\u0012kBT\nD\u00133=2\n; (5)\nwhere \u0000(x)and\u0010(x)are the Gamma and Riemann zeta func-\ntions.\n0 200 400 600\nT (K)0.00.20.40.60.81.0m\nexperiment\nsemi-quantum\nclassical\nBloch's law0 20 400.99500.99751.0000\nFIG. 2. Temperature dependent magnetization of YIG calculated\nusing classical and semi-quantum spin dynamics. The experimen-\ntal points are from [26] and Bloch’s law from Eq. (4) with D=\n85:2\u000210\u000041Jm2[11], which in YIG is temperature independent\nuntil close to the Curie temperature. The inset is a close up of the\nsemi-quantum method (blue circles) in the low temperature regime\nwhere Bloch’s law (dashed red line) is valid.\nThe proportionality Cm/T3=2should hold for YIG up to\nenergies of ~!k=kB/30 K . Rezende and López Ortiz [7]\ncalculated the heat capacity for acoustic magnons with finite\nband-width, but neglected optical magnons that contribute to\nthe heat capacity at elevated temperatures. They found that\nCmsaturates at 150 K, i.e. when the magnon occupation\nreaches the upper band edge.\nHere we calculate the heat capacity including all\nmagnon modes and their interactions. We calculate Cm\nfrom the energy fluctuations in the canonical ensemble\nhUmiT= (1=Zm)P\nk\u0017~!k\u0017exp(\u0000~!k\u0017=kBT);where\nZm=P\nk\u0017exp(\u0000~!k\u0017=kBT)is the partition function. Then\nCm=\u0000\nhU2\nmiT\u0000hUmi2\nT\u0001\n=(VkBT2), where in a simulation\nh\u0001\u0001\u0001iTis an average over a large time interval at a constant\ntemperature and Vis the volume of the system.\nFigure 3 shows the low temperature region where the\nmagnon dispersion is, to a good approximation, parabolic\nandCm/T3=2. Calculations using quantum statistics\ngive an excellent agreement with Bloch’s law. The exper-\nimental data in Figure 3 have been collected in the range\nT= 2\u00009K [5], high enough that dipolar field effects can\nbe disregarded. The measurements were made by freezing\nthe magnons in a 7 Tesla field. Even this large field how-\never does not completely remove the magnon contribution to\nthe heat capacity, especially at the higher end of the temper-\nature range [7]. To make a proper comparison we repeat the\nexperimental procedure in our simulation by computing the\ndifference \u0001Cm=Cm(H= 0T)\u0000Cm(H= 7T) . Our cal-\nculations agree well with the observations as well as the single\nmagnon-band model.\nFigure 4 illustrates a pronounced difference between the\nclassical and semi-quantum models: classical statistics over-4\n0 2 4 6 8 10\nT (K)0.0000.0050.0100.0150.020Cm, ∆Cm (J kg−1 K−1)\nCm(H=0) Bloch's law\nCm(H=0) semi-quantum\n∆Cm Rezende\n∆Cm semi-quantum\n∆Cm experiment\nFIG. 3. Low-temperature magnon heat capacity of YIG calculated\nwith quantum statistics (red circles) compared to Bloch’s law (red\nsolid line). \u0001Cm=Cm(H= 0T) \u0000Cm(H= 7T) calculated with\nquantum statistics (red open circles) is compared with experimental\ndata from Boona and Heremans Ref. 5 (orange open squares) as well\nas a single magnon band model. [7] (blue dashed line).\nestimate the heat capacity by 5 orders of magnitude at low\ntemperatures, and do not depend on temperature in contrast\nto the quantum statistical result which approaches zero like\nT3=2. In spite of this spectacular (and rather obvious) failure,\nclassical statistics have traditionally been used (and still are)\nin both Monte-Carlo and atomistic spin dynamics.\nAtT > 30K non-parabolicities begin and Cm/Tpwith\npowerp > 3=2. At room temperatures Fig. 4 reveals dif-\nferences between the approaches of two orders of magnitude.\nThe finite-width magnon band model [7] (dashed line on fig-\nure 4) saturates prematurely with increasing Tbecause op-\ntical and higher acoustic modes become significantly occu-\npied when approaching room temperature [9]. The parabolic\nband model without high-momentum cut-off (Bloch’s law)\nalso strongly underestimates Cmbecause YIG’s magnon den-\nsity of states is strongly enhanced by the flat bands observed\nin Fig. 1. The semi-quantum calculation is an order of\nmagnitude larger than both of these heavily approximated\napproaches, benefiting from the complete description of the\nmagnon spectrum as well as magnon-magnon interactions,\nwhile the classical statistics strongly overestimates the heat\ncapacity up to the Curie temperature.\nConclusions By enforcing Planck statistics for the\nmagnons in the complex ferrimagnet YIG, we obtain excel-\nlent agreement with available inelastic neutron scattering and\nmagnon heat capacity experiments. Our results prove that fun-\ndamental thermodynamic equilibrium properties can be pre-\ndicted with confidence when experimental data are not avail-\nable, but only when quantum statistics and the full spin wave\nspectrum is taken into account. The method is not limited to\nYIG or ordered magnets, but can be directly applied to other\ncomplex materials with local magnetic moments such as spin\nglasses or paramagnets. Our results are a necessary first step\nto compute non-equilibrium properties such as magnon con-\n0 200 400 600 800\nT (K)10-410-310-210-1100101102Cm (J kg−1 K−1)\nRezende\nBloch's law\nclassical\nsemi-quantumFIG. 4. YIG magnon heat capacity calculated over a larger tem-\nperature range with the semi-quantum model (red circles), classical\nmodel (green squares), compared with Bloch’s law (solid red line)\nand the single-band model [7] (dashed blue line).\nductivities and spin Seebeck coefficients, which are essential\nparameters for future applications of magnonic devices.\nACKNOWLEDGEMENTS\nThis work was supported by JSPS KAKENHI Grant No.\n26103006, the Graduate Program in Spintronics (GP-Spin),\nTohoku University and DAAD project ‘MaHoJeRo’. The\nauthors thank Jiang Xiao, Yaroslav Tserkovnyak, Rembert\nDuine and Jerome Jackson for valuable discussion.\n[1] Vladimir Cherepanov, Igor Kolokolov, and Victor L’vov, “The\nsaga of YIG: spectra, thermodynamics, interaction and relax-\nation of magnons in a complex magnet,” Phys. Rep. 229, 81\n(1993).\n[2] Mingzhong Wu and Axel Hoffmann, eds., Recent Advances in\nMagnetic Insulators - From Spintronics to Microwave Applica-\ntions (Academic Press, 2013).\n[3] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J.\nVan Wees, “Long-distance transport of magnon spin informa-\ntion in a magnetic insulator at room temperature,” Nat. Phys.\n11, 1022–1026 (2015), arXiv:1505.06325.\n[4] L. J. Cornelissen, K. J.H. Peters, G. E.W. Bauer, R. A. Duine,\nand B. J. Van Wees, “Magnon spin transport driven by the\nmagnon chemical potential in a magnetic insulator,” Phys. Rev.\nB94, 014412 (2016), arXiv:1604.03706.\n[5] Stephen R. Boona and Joseph P. Heremans, “Magnon thermal\nmean free path in yttrium iron garnet,” Phys. Rev. B 90, 064421\n(2014).\n[6] R. Douglass, “Heat Transport by Spin Waves in Yttrium Iron\nGarnet,” Phys. Rev. 129, 1132 (1963).\n[7] S. M. Rezende and J. C. López Ortiz, “Thermal properties of\nmagnons in yttrium iron garnet at elevated magnetic fields,”\nPhys. Rev. B 91, 104416 (2015), arXiv:1504.00895.5\n[8] J. Oitmaa and Thomas Falk, “Ferrimagnetism in the rare-earth\niron garnets: A Monte Carlo study,” J. Phys. Condens. Matter\n21, 124212 (2009).\n[9] Joseph Barker and Gerrit E. W. Bauer, “Thermal spin dynamics\nof yttrium iron garnet,” Phys. Rev. Lett. 117, 217201 (2016),\narXiv:1607.03263.\n[10] J S Plant, “Spinwave dispersion curves for yttrium iron garnet,”\nJ. Phys. C Solid State Phys. 10, 4805–4814 (1977).\n[11] Andrew J. Princep, Russell A. Ewings, Simon Ward, Sandor\nTóth, Carsten Dubs, Dharmalingam Prabhakaran, and An-\ndrew T. Boothroyd, “The full magnon spectrum of yttrium iron\ngarnet,” npj Quantum Mater. 2, 63 (2017).\n[12] We prefer the Landau-Lifshitz rather than the Gilbert damp-\ning, because latter affects the frequencies (here governed exclu-\nsively by the exchange parameters) by a factor 1=(1 +\u00112).\n[13] Charles Kittel, Quantum Theory of Solids (Wiley, New York,\n1963).\n[14] L D Landau and E M Lifshitz, Statistical Physics , 3rd ed. (El-\nsevier, 1980).\n[15] R. F L Evans, U. Atxitia, and R. W. Chantrell, “Quantitative\nsimulation of temperature-dependent magnetization dynamics\nand equilibrium properties of elemental ferromagnets,” Phys.\nRev. B 91, 144425 (2015), arXiv:1409.7397.\n[16] C H Woo, Haohua Wen, A A Semenov, S L Dudarev, and Pui-\nwai Ma, “Quantum heat bath for spin-lattice dynamics,” Phys.\nRev. B 91, 104306 (2015).\n[17] Hichem Dammak, Yann Chalopin, Marine Laroche, Marc Hay-\noun, and Jean Jacques Greffet, “Quantum Thermal Bath for\nMolecular Dynamics Simulation,” Phys. Rev. Lett. 103, 190601(2009).\n[18] Alexander V . Savin, Yuriy A. Kosevich, and Andres\nCantarero, “Semiquantum molecular dynamics simu-\nlation of thermal properties and heat transport in low-\ndimensional nanostructures,” Phys. Rev. B 86, 064305 (2012),\narXiv:arXiv:1112.5919v2.\n[19] U. Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U. Nowak,\nand A. Rebei, “Ultrafast spin dynamics: The effect of colored\nnoise,” Phys. Rev. Lett. 102, 057203 (2009), arXiv:0809.4595.\n[20] Andreas Rückriegel and Peter Kopietz, “Rayleigh-Jeans Con-\ndensation of Pumped Magnons in Thin-Film Ferromagnets,”\nPhys. Rev. Lett. 115, 157203 (2015), arXiv:1507.01717.\n[21] M.D. Kuz’min, “Shape of Temperature Dependence of Sponta-\nneous Magnetization of Ferromagnets : Quantitative Analysis,”\nPhys. Rev. Lett. 94, 107204 (2005).\n[22] F. Bloch, “Zur Theorie des Ferromagnetismus,” Zeitschrift für\nPhys. 61, 206–219 (1930).\n[23] C. M. Srivastava and R. Aiyar, “Spin wave stiffness constants\nin some ferrimagnetics,” J. Phys. C Solid State Phys. 20, 1119–\n1128 (1987).\n[24] J. Oitmaa and Weihong Zheng, “Curie and Néel temperatures\nof quantum magnets,” J. Phys. Condens. Matter 16, 8653–8660\n(2004), arXiv:0409041 [arXiv:cond-mat].\n[25] D Rodic, M Mitric, R Tellgren, H Rundlof, and A Kremenovic,\n“True magnetic structure of the ferrimagnetic garnet Y 3Fe5O12\nand magnetic moments of iron ions,” J. Magn. Magn. Mater.\n191, 137–145 (1999).\n[26] Elmer E. Anderson, “Molecular field model and the magnetiza-\ntion of YIG,” Phys. Rev. 134, A1581 (1964)."    },    {        "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": "1002.0236v1.Crystal_and_electronic_structure_of_the_room_temperature_organometallic_ferrimagnet_V_TCNE___2___Analysis_of_numerical_DoS_and_magnetic_properties_as_related_to_orbital_and_spin_Hamiltonian_models.pdf",        "content": "arXiv:1002.0236v1  [cond-mat.str-el]  1 Feb 2010Crystal and electronic structure of the room\ntemperature organometallic ferrimagnet\nV(TCNE) 2. Analysis of numerical DoS and\nmagnetic properties as related to orbital and\nspin-Hamiltonian models.\nAndrei L. Tchougr´ eeff1,2\nand\nRichard Dronskowski2\n1Poncelet Laboratory, Independent University of Moscow,\nMoscow Center for Continous Mathematical Education,\nBolshoy Vlasyevskiy Pereulok 11, 119002, Moscow, Russia\n2Institut f¨ ur Anorganische Chemie, RWTH Aachen,\nLandoltweg 1, 52056 Aachen, Germany\nDedicated to Prof. I.A. Misurkin on occasion of his 75-th\nbirthday\nAbstract\nWe present a detailed analysis of the results of our numerica l\nstudy of the crystal and electronic structure of the room tem perature\norganometallic ferrimagnet of general composition V(TCNE )xwith\nx≈2. The results of the LSDA+ Ustudy show that the experimen-\ntally determined structure complies with the magnetic meas urements\nand thus can serve as a prototype structure for the entire fam ily of\nthe M(TCNE) 2organometallic magnets. The results of the numerical\nstudy and of the magnetic experiments are interpreted using model\nHamiltonians proposed here. This allowed us to obtain estim ates of\n1NC\nNCCN\nCN❜❜C\nC✧✧✧✧\n❜❜ ✧✧❜❜❜❜✧✧✧✧\n❜❜NN\n❜❜CN✧✧CN❜❜NC\n✧✧NC ✧✧❜❜✧✧❜❜NC❜❜C✧✧CN\nC✧✧NC❜❜CN✧✧❜❜✧✧❜❜\n✖✕✗✔✧✧CN\n❜❜CN✧✧NC❜❜NC\n1 2 3 4\nthe critical temperature in three- and two-dimensional reg imes and to\ngive an explanation of the differences in behavior of probably isostruc-\ntural V(TCNE) 2and Fe(TCNE) 2species.\n1 Introduction\nThe most spectacular room temperature organometallic magnet of composi-\ntion V(TCNE) x·ysolvent (where TCNE – 1– stands for tetracyanoethylene\n– a well known organic electron acceptor; x≈2 andydepends on the type of\nthe solvent) attracts a lot of attention since the time it had been sy nthesized\nyet in the beginning of the 1990’s [1]. It is an amorphous moisture sens itive\nprecipitate with the outstanding critical temperature of the tran sition to the\nmagnetically ordered state estimated to be of ca.400 K.1It is higher than\nthe decomposition temperature ( ca.350 K ) which singles it out among\nits numerous analogs with a variety of involved organic acceptors (R ef. [2] –\ntetracyanopyrazine – 2; Ref. [3] – 7,7,8,8-tetracyano-p-quinodimethane – 3;\nRef. [4] – tetracyanobenzene – 4) and of the metals (Ref. [5] – iron), syn-\nthesized in the following years, since none of them manifested as fas cinating\nmagnetic properties as the very first V(TCNE) 2compound (see Table 1 [6]).\nGenerally one has to say that not only the critical temperature, bu t also\nother properties of the compounds of the considered class are se nsitive to\n1The magnetic momenta are spontanelusly predominantly aligned in one direction be-\nlow the critical temperature.\n2the details of the preparation procedure and/or the solvent emplo yed. For\nexample, the V-TCNE compound is known in two forms. The original of\nRef. [1] coming from the reaction of V(C 6H6)2with TCNE in the CH 2Cl2\nsolution or obtained from V(CO) 6with use of the CVD technique exhibits\nthe saturation magnetization at the zero temperature which corr esponds to\napproximatelyoneunpairedelectronperformulaunitandinfactisso mewhat\nlower than this value. However in Ref. [7] (see also a review [6]) a V(TCN E)2\ncompound was reported containing tetrahydrofurane as a solven t with the\nsaturation magnetization almost twice as strong compared to that of the\noriginal compound. For that reason hereinafter we shall refer to these two\nforms of the V(TCNE) 2compound as the highly magnetic (HM) and the low\nmagnetic(LM) ones.\nFor more than one decade the amorphousity of the compound of int er-\nest did not allow anyone to make whatever definitive conclusion conce rning\nits structure. The critical breakthrough became possible with the recent\nwork [8] where the authors were able to establish the structure of the Fe2+\nanalog (presented in Fig. 1) of the V(TCNE) 2compound using Rietveld re-\nfinement of the synchrotron powder diffraction data and to revea l its most\nremarkable features: the presence of the dimer form of the TCNE˙−radical-\nanion: [TCNE]2−\n2= C4(CN)2−\n8playing an important role in shaping the loose\nthree-dimensional structure and assuring as well the three-dime nsional char-\nacter of magnetic interactions in the system.\nIn our previous paper Ref. [10] we were able to demonstrate that t he\nexperimental structure represented in Fig. 1 can be easily related with one\nrepresented in Fig. 2a proposed yet in Ref. [9] in order to conform w ith the\nmagnetic data on the V-TCNE compound available that time. That latt er\nrepresented a simple cubic lattice with the vertices occupied by the v ana-\ndium ions. Two ab-faces of each cube remain empty whereas four others are\nfilled by the TCNE units forming channels extended in the c-direction. In\nthe structure presented in Fig. 2a the C=C bonds of the TCNE units are\nlocated in the same plane (as well perpendicular to the c-axis of the lattice)\nso that they are orthogonal to each other. It is not the unique po ssibility.\nAn alternative structure differing from that of Fig. 2a by rotating t he TCNE\nunits placed in the bc-faces by 90◦in their respective planes is also possible.\nThe result of such a rotation is presented in Fig. 2b. In this structu re the\ncentral C=C bonds of the TCNE units are as previously orthogonal, but\nnow they lay in orthogonal planes so that the axes of these bonds d o not\n3intersect rather cross each other. This structure has been calle d the ”prin-\ncipal” structure in Ref. [10]. The principal structure Fig. 2b can be e asily\nput in the relation with the experimental one. If in the quadrupled un it\ncell 2a,2b,cof the principal structure four TCNE units extended in the b-\ndirection are allowed to pairwisely rotate towards each other so tha t single\nC–Cbonds canformbetween respective ethylenic carbonatomsth usyielding\nthe [TCNE]2−\n2=C4(CN)2−\n8dimers and the V-TCNE sheets originally laying\nin theac-planes areaccordingly ruffled one finally arrives to the experimenta l\nstructure of Fe(TCNE) 2(Fig. 1). Intermediate structures along this hypo-\nthetical reaction path are presented in Fig. 3. For this sequence o f structures\nwe performed in Ref. [10] the LSDA + Ucalculations with use of the VASP\nprogram suite Ref. [11] of the respective electronic structures a nd energies.\nSpecifically, the PAW potential hasbeen used and the values of the Uparam-\neter for the V, N, and C atoms were respectively taken as 5.344, 4.8 40, and\n4.428 eV. The calculations have been performed as follows: first the principal\nstructure with the equalized lattice parameters have been quadru pled. This\nstructure has been connected with the experimental structure by a straight\nline in the configuration space. Further details of the numerical tre atment\ncan be found in Ref. [10]. Its results are reproduced in Fig. 4.\nThe overall densities of states in two spin channels as given in Fig. 4 ar e\nobviously difficult to understand. One can only see some spin polarizat ion\nof the upper filled bands as well as an evolution of a noticeable density of\nstates near the Fermi level present in bothspin channels in the initia l state to\nthe final state where the expected spin-polarized structure can be recognized\nwhich doesnotshow anyDoSineither ofthespinchannels at theFerm i level.\nIn this paper we present a detailed analysis of the results of our num erical\nstudies Ref. [10] of the models of V-TCNE room temperature organ ometallic\nferrimagnet and expressed them in terms of the effective spin-Ham iltonian\nfor a selection of interacting atomic/molecular states. The propos ed model is\nthen applied to analysis of a wider collection of experimental data ava ilable\nfor this fascinating object and its Fe analog.\n2 Detailed analysis of DoS\nIn order to analyse the details of the evolution of the DoS features along the\npath going from quadrupled principal structure to the experiment al one we\n4performed a detailed study of projections of the DoS. In the initial structure\nthe three bands in the −9÷−11 eV rangeare predominantly C-bands. Other\nbands are the well hybridized C-N-bands except the ”spin-left” ba nd right\nbelow the Fermi level, which is predominantly contributed by the vana dium\nstates. As one can see from Fig. 6 these are the d-states of vanadium atoms.\nSome vanadium stemming DoS in the range −8÷ −10 eV are the s- and\np-states of vanadium involved in the bonding with the nitrogen atoms. This\ncan be seen on the corresponding N-projection of DoS in Fig. 7 from which\nonecan deduce that the s-andp-states ofvanadium hybridize predominantly\nwith the N-states coming from the V-TCNE layers (see below). In or der to\ndeepen our understanding of the DoS presented in Fig. 5 we notice t hat\neven in that complex system the atoms involved are relatively easily cla ssi-\nfied into types. One can distinguish transition metal ions whose d-density is\nexpected to contribute significantly to the spin polarized bands, th e nitrogen\natomswhere onecanexpect significant DoSchanges due toexpect ed break of\nthe pairs of excessive V-N bonds while going from the quadrupled prin cipal\nstructure to the experimental one. Analogously along the same ro ute one\ncan expect remarkable variations in the projection of the DoS to th e carbon\natoms forming the C-C bonds in the [TCNE]2−\n2units and to the carbon and\nnitrogen atoms in the ruffled V-TCNE layers. Other features of the DoS,\nhowever, are not expected to significantly vary along the path. Co nsidering\nthe evolution of the above-mentioned projections of the DoS along the path\nFig. 3 as represented in Figs. 4 - 6 one can see the expected recons truc-\ntion of the projected DoS’s. For example, almost nothing happens t o the\nd-bands along this path. They remain rigid triply degenerate ones and do\nnot change their position relative to the Fermi level despite the fac t that the\ncoordination number of vanadium ions changes from eight in the quad rupled\nprincipal structure to six in the experimental one. This agrees with the nu-\nmerical result concerning the distribution of spin polarization in the d irect\nspace: for all structures depicted in Fig. 3 the magnetic moments r esiding\nin thed-shells of vanadium ions range from 2.615 for the first (quadrupled\nprincipal) structure to 2.582 for the last (experimental) one i.e.are almost\nconstant. The significant variation of the total magnetic moment a long the\n”reaction path” observed in our numerical experiment is to be almos t com-\npletely attributed to that of the magnetic moments residing in the ”o rganic”\npart of the organometallic magnet. This is precisely what one should e xpect\nwithin the general picture including the formation of the [TCNE]2−\n2dimers.\n5On the other hand the stability of the d-bands along the path can be only\nunderstood if one assumes that the break of two V-N bonds at eac h atom\nis at least partially compensated by shortening (strengthening) of other two\nbonds extended in the b-direction.\nBycontrasttheprojectionsofDoStothecharacteristicorgano genicatoms\nsignificantly modify along the ”reaction path”. For example, the dom inant\npart of the N-DoS is concentrated in two broad bands at ca.−5 andca.\n−6÷ −6.5 eV. The lower of the two is almost equally contributed by the\nnitrogen atoms of all three types and its position is stable througho ut the\npath. The same applies to somewhat smaller contribution from the bo nding\nN-atoms extended in the b-direction to this band. Their contribution to the\nupper of the two mentioned bands is noticeably spin-polarized. The d ensity\nfrom the N atoms bearing the dangling lone pairs fairly manifests itself in\nthe same band. Weak features in the N-DoS can be observed in the 4 eV\nwide range right below the Fermi level. The peak right below the Fermi level\nin the ”right-spin” channel in the initial structure is equally contribu ted by\nthe ”bonding”, ”to be dangling”, and ”magnetic” nitrogens. (In ge neral the\nDoS projections for the ”bonding” and ”to be dangling” nitrogens c oincide\nin the initial structure). In the final structure this peak splits so t hat its\nupper part (”right-spin” channel right below the Fermi level) is con tributed\nby the ”magnetic” nitrogens, whereas two peaks next to it in the bo ttom\ndirection is contributed by the ”bonding” nitrogens. The contribut ion from\nthe ”dangling” nitrogens contributes to the upper part of the wide band at\nca.−3.5÷−5.5 eV.\nOfparticularinterest istheevolutionoftheC-projectionoftheDo S.That\non the ethylenic carbon atoms is most interesting. The ethylenic car bons\ncan be further subdivided in two categories: those in the planes ext ended\nin theaandchorizontal directions and those extended in the vertical ( b)\ndirection and further involved in the rotation yielding the [TCNE]2−\n2units.\nAt the initial stage both projected DoS are significantly polarized an d both\nnoticeably contribute to the ”right-spin” density right below the Fe rmi level\n(the DoS projection to the to be bonding atoms is not seen in the left most\ngraph of Fig. 6 since at this point they are degenerate with the ”mag netic”\nprojection and are masked by these latter). From further graph s of the DoS\nprojected to the atoms forming the emerging C-C bonds one can co nclude\nthat such bonds represented in our study by two spin sub-bands f or the\nleft and right spin channels completely develop at a pretty late stage of\n6the hypothetical transition: in the middle of the path the correspo nding\npeaks in the density projections can be yet clearly seen in both spin c hannels\nnear to the spin-polarized DoS of the ”magnetic” atoms, although t he peaks\ncorresponding to the emerging bonds are not polarized. By contra st the\nprojection of DoS on the magnetic C-atoms in the horizontal planes develop\ntwo sub bands right above and below the Fermi level of which the lowe r\none (right-spin) is completely occupied by electrons with the spin pro jection\nopposite to that of the electrons occupying the d-subband. This is in the\nfair agreement with the assumption concerning the nature of the s ubbands\nlocatedin thevicinity of the Fermi level madein our previous paper Re f. [10].\n3 Model Hamiltonians for V-TCNE system\n3.1 Model Orbital Hamiltonian\nFascinating properties of the V(TCNE) 2magnet call not only for numerical\nmodelling, but also for some qualitative picture. However, for the fin al (”ex-\nperimental”) structure all the DoS projections manifest themselv es as very\nnarrow bands. This indirectly indicates that the band picture used t hrough-\nout the calculations is not completely adequate and that an adequat e model\nmust begiven intermsof aneffective Hamiltonianrepresenting theele ctronic\nstructure of the HM V(TCNE) 2magnet (in its ”experimental” structure) in\nterms of some objects local in the direct space e.g.local spins similar to that\nproposed yet in Ref. [9]. As in the case of band models the most import ant\none-electron states to be included are these contributing to the e nergy bands\nin the vicinity of the Fermi level. Based on our analysis of projected D oS\nperformed in the previous Section we can conclude that the states of the\n[TCNE]2−\n2units contribute to the bands far away from the Fermi level. The\nDoS related to this unit goes away from the Fermi level along the ”re action\npath”. Thus the observed electronic structure is primary one of t he indi-\nvidual (ruffled) V-TCNE layer extended in the ac-plane. For constructing\nthe Hamiltonian for this layer one can employ the unit cell of the princip al\nmodel dropping from it the TCNE unit extended in the b-direction (and fi-\nnally engaged in formation of the [TCNE]2−\n2= C4(CN)2−\n8dimers). According\nto Ref. [9] on the vanadium sites it suffice to consider only the d-shells of the\nmetal ions which isalso confirmed by our current analysis. The overla p ofthe\n7d-shell of the metal ions with the σ-orbitals of the TCNE’s (including those\nimplied in the model) ensures the standard two-over-three splitting of thed-\nshell characteristic for the octahedral environment. In the cas e of vanadium\nthree unpaired electrons in the d-shell occupy respectively three orbitals in\nthet2g-manifold. The basis orbitals dxy,dxz, anddyzcan be characterized by\nthe normal to the plane in which each of the orbitals lays – ζ,η, andξ– are\nsubsequently used in the notation. For the donor sites in the ruffled planes\ntheb3g(π∗) LUMOs of the TCNE (singly occupied in the radical anion) are\nincluded. In each such a layer each metal ion is surrounded (coordin ated)\nby four TCNE units which are in their turn coordinated to (surround ed by)\nfour metal atoms. In this case the layer unit cell composition V:TCNE is\n1:1.\nThemodelHamiltonianfortheV(TCNE) 2magnet, formulatingtheabove\nideas, has the general form:\nH=/summationdisplay\nr(Hd(r)+Ha(r)+Hda(r)+Hdd(r)) (1)\nThe contributions to it are the following. Operator Ha(r) describes electrons\nin the acceptor orbital of the TCNE˙−radical-anion in the r-th unit cell:\nHa(r) =−αaˆnar+Uaaˆnar↓ˆnar↑\nˆnarσ=a+\nrσarσ;ˆnar=/summationtext\nσˆnarσ(2)\nSymbola+\nrσ(arσ) is the operator creating (annihilating) an electron with the\nspin projection σon the acceptor orbital of the TCNE molecules in the r-th\nunit cell. In eq. (2) the first term is the energy of attraction of an e lectron to\nthe core of TCNE – the orbital energy of the b3g(π∗) LUMO shifted by the\nelectrostaticfieldinducedbytheentirecrystalenvironment. The secondterm\nin eq. (2) is the Hubbard one, effectively describing the Coulomb repu lsion\nof electrons with opposite spin projections eventually occupying th e same\nacceptor orbital.\nThe operator Hd(r) describes electrons in the t2g-subshell of the d-shell\n8of the vanadium ion in the r-th unit cell:\nHd(r) = [−αd(ˆnζr+ ˆnηr+ ˆnξr) +\n+ (Udd+2Jdd)(ˆnζr↓ˆnζr↑+ ˆnηr↓ˆnηr↑+ ˆnξr↓ˆnξr↑)]+\n+(Udd+Jdd/2)\n2/summationdisplay\nσ,σ′(ˆnζrσˆnξrσ′+ ˆnζrσˆnηrσ′+ ˆnξrσˆnηrσ′)+ (3)\n−4Jdd(ˆSζrˆSξr+ˆSζrˆSηr+ˆSξrˆSηr)\nˆnγrσ=γ+\nrσγrσ,ˆnγr=/summationdisplay\nσˆnγrσ;γ=ξ, η,ζ.\nIn eq. (3) ˆ nγrσare the operators of the number of electrons with the spin\nprojection σon thedxy,dyz,anddxzorbitals of the vanadium ion in the r-th\nunit cell. The spin operators and spin-operator product terms are defined by\nthe well-known relations:\nˆSγrˆSγ′r= 1/2(ˆS+\nγrˆS−\nγ′r+ˆS+\nγ′rˆS−\nγr)+ˆSz\nγrˆSz\nγ′r\nˆS+\nγr=γ+\nr↑γr↓,ˆS−\nγr=γ+\nr↓γr↑,ˆSz\nγr= 1/2(ˆnγr↑−ˆnγr↓).\nwhere the symbols γ+\nrσ(γrσ) represent the operators creating (annihilating)\nan electron with the spin projection σon thedxy,dyz,anddxzorbitals of the\nvanadium ion in the r-th unit cell.\nThe first row in the above operator describes the attractionof ele ctrons in\nthed-orbitals to the cores of vanadium ions (shifted by the electrostat ic field\nof the rest of the crystal). Two further rows describe the spin-s ymmetric\npart of the Coulomb interaction of electrons in the d-shell. The last row\ndescribes the spin dependent part of the Coulomb interaction of ele ctrons\nin thed-shell (exchange). It is ultimately responsible for the Hund’s rule in\natoms and for the high spin of the ground state of electrons in the d-shell.\nThe contributions to the Hamiltonian eq. (1) described so far model isolated\nlocal states important for the crystal description. The magnetic order can\nonly be possible due to various interaction terms. Operator Hda(r) describes\nthe electron hopping between the d-states of vanadium ions and the acceptor\nstates. The dxy-state represented by the ζ+\nrσ(ζrσ) operators being of the\n(approximate) σ-symmetrywithrespect tothe acplane(theruffling oftheV-\nTCNE plane is neglected) has no overlap with the LUMO’s of TCNE’s which\nare (again approximately) of the π-symmetry with respect to the same plane.\nTwo others ( dxz- anddyz-states represented respectively by the η+\nrσ(ηrσ) and\n9ξ+\nrσ(ξrσ) operators) overlap with the LUMOs of two (different) neighbor\nTCNE units each. The phase relations between the orbitals involved in\nthe model lead to such a distribution of signs at the one-electron ho pping\nparameters that the hopping operator acquires the form:\nHda(r) =−tda/summationdisplay\nσ/bracketleftig\nξ+\nrσ(arσ+ar+a+cσ)−η+\nrσ(ar+aσ+ar+cσ)/bracketrightig\n+h.c.\n(4)\nwhere the parameter tda>0 describes the magnitude of the hopping between\nthe acceptor state and the neighbor d-state.\nThe sum of the above contributions to the effective Hamiltonian in fac t\nform that for an isolated V-TCNE layer. In the ”experimental” stru cture\nthe diamagnetic C 4(CN)2−\n8units seem to effectively isolate the V(TCNE)\nsheets fromeach other. Nevertheless, oneshould assume that c ertain indirect\ninteraction between the d-states in the b-direction is possible through the\nmediation of the [TCNE]2−\n2units. It was proposed in Ref. [10] to use an\neffectivehoppingsimilartoeq. (4). Sinceitisanywayaneffectiveinter action\nitcanbechoseninawaywhichfitsbettertothemethodthesystemis treated.\nFor this reason we postpone the discussion of this term.\nIn our previous paper Ref. [10] we considered the band model of th e V-\nTCNE organometallic magnet as derived from the orbital Hamiltonian e qs.\n(1) - (4) and employed them for analysis of results of our numerical experi-\nments performedwithuseoftheVASPpackage. These latterare, however, in\na kind of fundamental contradiction with the physics of the system at hand.\nThis manifests itself in the very narrow bands coming out of calculatio n, as\nwe already mentioned. The reasonis that thehopping parameter tdaentering\neq. (4) which are generally responsible for extension of one-electr on states\nover the crystal (band formation) and which are proportional to the overlap\nbetween the orbitals represent the smallest energy scale in the sys tem. Gen-\nerally it leads to a break of the delocalized (band) picture and makes a local\ndescription to be more adequate. The latter can be sequentially der ived by\ntreating perturbatively the hopping operator eq. (4). It yields th e effective\nHamiltonian of the Heisenberg form in terms of the spins of electrons occu-\npying the local states (orbitals) involved. Its parameters are est imated in\nAppendix A.\n10The overall result comes out as a spin Hamiltonian of the form:\nHlayer\nspin= −4Jdd/summationtext\nr(ˆSζrˆSξr+ˆSζrˆSηr+ˆSξrˆSηr)+\n+2Kda/summationtext\nr/bracketleftigˆSξr/parenleftigˆSar+ˆSar+a+c/parenrightig\n+ˆSηr/parenleftigˆSar+a+ˆSar+c/parenrightig/bracketrightig\n(5)\nwhich describes effective magnetic interactions in an isolated layer. I t must\nbe complemented by interlayer interactions. If the vanadium ions in a djacent\nlayers are coupled by an an effective hopping an analogous perturba tive pro-\ncedure results in the antiferromagnetic sign of the effective magne tic interac-\ntion. Thiscontradictstotheexistenceofthenonzerooverallmag netizationin\nthe V-TCNE magnets below the critical temperature (with the antif erromag-\nnetic interlayer coupling the magnetization of one layer would be canc elled\nby that of another). For that reason we have to supplement the H amiltonian\neq. (5) by an effective interlayer interaction with the ferromagnet ic sign of\nthe corresponding exchange parameter. It cannot directly come from any\nperturbative treatment of the hopping. By contrast some mecha nism of fer-\nromagnetic coupling described e.g.in Refs. [12] or [13] and implemented in\npapers [14,15] devoted to the exchange in metallocene based orga nometallic\nmagnets (Miller-Epstein magnets) acting through the [TCNE]2−\n2units might\nbe expected. Indeed as one can see from the Fig. 7 despite the fac t the\nthe states located in the [TCNE]2−\n2units are pulled up and down from the\nvicinity of the Fermi level, some spin polarization of these bands part icu-\nlarly of those which are contributed by the ”bonding” nitrogens indic ates\nthe involvement of the [TCNE]2−\n2units in transfer of magnetic interactions\nbetween the d-shells in the b-direction (between the layers).\nWith this caveatthe spin Hamiltonian written in terms of ”true” elec-\ntronic spins is the following:\nHspin= −4Jdd/summationtext\nr(ˆSζrˆSξr+ˆSζrˆSηr+ˆSξrˆSηr)+\n+ 2Kda/summationtext\nr/bracketleftigˆSξr/parenleftigˆSar+ˆSar+a+c/parenrightig\n+ˆSηr/parenleftigˆSar+a+ˆSar+c/parenrightig/bracketrightig\n+ 2Kdd/summationtext\nr/parenleftigˆSξr+ˆSηr+ˆSζr/parenrightig/parenleftigˆSξr+b+ˆSηr+b+ˆSζr+b/parenrightig(6)\nThe Hamiltonian eq. (6) is not a standard Heisenberg Hamiltonian usua lly\nused to describe magnetic properties of insulators. This latter is wr itten in\nterms of the effective local spins residing at each atomic magnetic ce nter.\n11In our case the vanadium ions represetn such nontrivial magnetic c enters\nbearing effective spins in the d-shells:\nˆSdr=/summationdisplay\nγˆSγr. (7)\nAccording to Ref. [16] using the effective spins is, however, an appr oximation\nsince the transition from the representation of the effective Hamilt onian in\nterms of the of individual electronic spins-1\n2eq. (6) which can be sequentially\nderived from the model orbital Hamiltonian eqs. (1) - (4) by pertur bative\ntreatment of the hopping term eq. (4) to the phenomenological Ha miltonian\neq. (8) operating with the effective spins eq. (7) is only possible if the\nexchange interactions of all individual spins in one magnetic center ( in our\ncase – the V ion) with those in the other magnetic center (in our case the\neffective spin in TCNE˙−coincides with the individual one) are equal. This is\nobviously not the case since the electronic spin ˆSζrin thed-shell to the first\napproximation does not interact with the spin residing in any of accep tor\norbitals.\nThis generally poses the problem since in the Hamiltonian eq. (6) at\nleast the exchange parameters JddandKdacan be independently determined\nrespectively by eq. (18) and atomic spectra, whereas the parame terJ⊥in\neq. (8) remains completely empirical quantity. The fact that Jdd≫Kdain\neq. (6) allows to approximately replace the spins of separate electr ons in the\nr-thd-shell by the operator of the total spin of the respective d-shell. Further\ndetails of this transition are given in Appendix B.\nThe phenomenological Hamiltonian written in terms of the effective sp ins\neq. (7) to be used for modeling the entire crystal is:\nHphen=J/bardbl/summationtext\nrˆSdr/bracketleftigˆSar+ˆSar+a+ˆSar+c+ˆSar+a+c/bracketrightig\n+\n+J⊥/summationtext\nrˆSdrˆSdr+b(8)\nIn the next Section we apply it to analysis of magnetic properties of t he HM\nV(TCNE) 2material.\n124 Magnetic properties of V(TCNE) 2as inter-\npretedwithuseofphenomenologicalHamil-\ntonian\nThe magnetic and thermodynamic properties of the HM V(TCNE) 2material\nmust be derived from the phenomenological Hamiltonian eq. (8). Two types\nof data will be of interest for us: the critical temperature of tran sition into\nmagnetically ordered state and the temperature dependence of s pontaneous\nmagnetization. The mean field estimates of the critical temperatur e used so\nfar in the literature lack the account of structural information. I t is impor-\ntant to realize that the quantities of interest are sensitive to thes e details as\nrepresented in the respective Hamiltonians.\nThe commonly used (see e.g.Ref. [6]) symmetric mean field formula:\nθMF\nN=zJeff\n3Sd(Sd+1)\n(whereT=θ/kB) ignores the acceptor spins and sets zto be the number\nof indirectly neighboring magnetic metal ions. It yields the quadratic de-\npendence of the critical temperature on the spin of the metal ion. On the\nother hand according to Ref. [8] the critical temperature for the magneti-\ncally ordered state of a material with two types of spins ( SdandSa(= 1/2))\nis described by the mean field formula:\nθMF\nN=/vextendsingle/vextendsingle/vextendsingleJMF\neff/vextendsingle/vextendsingle/vextendsingle\n3/radicalig\nZadZda/radicalig\nSd(Sd+1)Sa(Sa+1)\n(The factor of two is dropped here to get the formula to conform w ith the\nHamiltonian definition accepted in the present paper). This formula a ppears\nas a zero interlayer coupling limit of the mean field expression for the N ´ eel\ntemperature in a ferrimagnet eq. (35) derived in Appendix C. In Ref. [8]\nit had been applied to the Fe(TCNE) 2compound for which the structure\nmeasurements have been performed there. It, however, brings up two com-\nplications–onetheoreticalandanotherexperimental. Fromtheex perimental\npoint of view we notice that the above formula as well as formula eq. ( 35) is\neffectively linear in Sdrather than quadratic (in the high anisotropy limit).\nFor that reason even the mean field estimates of the exchange par ameters\n13as given in Table 1 must be reconsidered since the latter had been obt ained\nwith use of the quadratic dependence. As one can see from the str ucture the\nchoiceof Zad=Zda= 4yieldstheestimateofthemeanfieldexchangeparam-\neter forthe Fe(TCNE) 2compound of JMF\neff(Fe) = 43 K andforthe V(TCNE) 2\ncompound – JMF\neff(V) = 183 K. Both values are significantly larger than those\ngiven in Table 1, but it is remarkable that the difference between them (to\nbe explained) reduces from the factor of larger than five to that o f 4.25. We\nnotice that following the assumption of Ref. [24] and using Zdd= 6 for the\nV(TCNE) 2compound further reduces the difference and the value of the\nexchange parameter for the latter, but from our point of view it ca nnot be\nsubstantiated within the scope of the model considered in the pres ent paper.\nFrom the theoretical point of view, even the improved molecular field\nexpression eq. (35) has that disadvantage that it predicts a nonv anishing\nordering temperature for J⊥= 0. It is obviously wrong and such an es-\ntimate is not acceptable in the context where a strong anisotropy m ight be\nexpected on the structure basis. As it has been mentioned the mod el must be\ncomplemented by the interlayer interactions between the effective spins 3/2\non the vanadium sites mediated by the diamagnetic (closed shell) [TCNE ]2−\nunits. Remarkably enough the sign of this interaction must be ferromag-\nnetic(the magnetic moments residing in the layers must be pointing in the\nsame direction) to ensure the existence of the net spontaneous m agnetiza-\ntion in the three-dimensional sample, although in general one has to expect\nantiferromagnetic sign of such an interaction [12] (see below).\nIn a presumably rather anisotropic situation brought by tentative differ-\nence in mechanisms of the intralayer and interlayer interactions (re spectively\n”antiferromagnetic kinetic exchange” for the intralayer interact ion and the\n”ferromagnetic superexchange” for the interlayer one) the crit ical tempera-\nture has to be estimated from the spin-wave treatment taking an a dequate\ncare about the anisotropy of the effective spin-spin interaction an d at least\nproviding a correct asymptotic value of the critical temperature f or the van-\nishing interlayer coupling J⊥. This is done by the formula\nMs=M0\n1−/parenleftiggθ\nθN/parenrightigg3\n2\n (9)\nexpressing the Bloch T3\n2law for the temperature dependence of the spon-\ntaneous magnetization as derived in Appendix D with the critical (N´ e el)\n14temperature given by:\nθN=4π\nζ2\n3(3\n2)1\nSd−Sa3/radicalig\n(Sd+Sa)2S4\ndS2a/bracketleftig\nJ2\n/bardbl|J⊥|/bracketrightig1\n3(10)\n– the 3D structure-specific relation of the effective exchange inte raction to\nthe N´ eel temperature.\nTheassumptionsusedinAppendixDforderivingtheformulaeq. (10) are\nnot satisfied in strongly anisotropic systems where |J⊥| ≪J/bardbl. In this limit\nthe N´ eel temperature is to be determind from the transcendent al equation:\n1\n4πSd−Sa\nSdSaθN\nJ/bardbl1\nSd+Salog/parenleftiggSd−Sa\nS2\ndθN\n|J⊥|/parenrightigg\n= 1, (11)\nand the magnetization is given by:\nMs=M0/bracketleftigg\n1−1\n4πSd−Sa\nSdSa1\nSd+Saθ\nJ/bardbllog/parenleftiggSd−Sa\nS2\ndθ\n|J⊥|/parenrightigg/bracketrightigg\n(12)\nprovided θis close enough to θN.\nNeither three-dimensional (3D) or two-dimensional (2D) estimate s of the\nN´ eel temperature eqs. (10) and (11), respectively, permits to determine\nthe longitudinal and transversal interactions independently and t o estab-\nlish by this the amount of anisotropy. The effective exchange intera ction\nJSW\neff=3/radicalig\nJ2\n/bardblJ⊥as derived from eq. (10) and the experimental N´ eel tempera-\nture for the HM V(TCNE) 2compound amounts JSW\neff(V) = 36 K. This is due\nrather large numerical value of the transition coefficient in eq. (10) coupling\nthe effective exchange interaction with the N´ eel temperature (1 1.376 for the\nstructure depicted in Fig. 1 and Sd=3\n2;Sa=1\n2). This result is in a general\nagreement with the result of Ref. [18] which yields the correspondin g coeffi-\ncient to be 9.937 for a simple cubic ferrimagnet with the same values of the\neffective spins. (It is not clear how the estimate of ca.100 K for JSW\neff(V)\nis obtained in Ref. [23] since is also based on assumption of a simple cubic\nlattice magnetic structure, but apparently uses some different co efficient).\nIt also stresses the different character of averaging of intralaye r and inter-\nlayer exchange parameters in the mean field (arithmetic mean) and in the\nspin-wave (geometricmean) approximations. Atthehighanisotrop ies thege-\nometric mean provides much stronger dependence of the effective exchange\non the interlayer exchange than the arithmetic mean.\n15Whatever value of JSW\neffleaves a wide range of possibilities since each pair\nof values of J⊥andJ/bardblyielding the above value of JSW\neff(V) conforms with the\nexperimental data on magnetization. It has to be realized, howeve r, that us-\ning the Bloch T3\n2law for the magnetization in the entire temperature range\nbelowTNis an extrapolation of the data obtained at low temperatures. In\norder to check its validity in a wider temperature range we notice tha t ac-\ncording to it the magnetization depletion at the an intermediate temp erature\n(225 K) amounts the factor of 0.592 which looks out to be in an accep table\nagreement with experiment which shows the magnetization depletion by a\nfactor of ca.0.6 at this temperature as compared to that at T= 0.\nIn the HM V(TCNE) 2case the measured magnetization values are avail-\nable up to 300 K. The Bloch T3\n2-law for the temperature dependence of\nspontaneous magnetization results in a simple formula for the slope o f the\nmagnetization vs.temperature in the N´ eel point:\n−3\n2M0\nTN(13)\nAs one can derive from the magnetization data on the HM compound g iven\ninRef.[6]theslopeofthemagnetizationoftheHMmaterial intheN´ e el point\namounts −1.4M0\nTNwhich is a fair extrapolation of the last measured points.\nIt suggests the 3D regime for the HM material in the entire tempera ture\ninterval up to the N´ eel point. Nevertheless the possibility of tran sition to\nthe 2D regime at T >300 K cannot be a prioriexcluded. Then for the 2D\nregime the slope of magnetization vs.temperature in the N´ eel point is:\n−M0kB\n4πJ/bardblSd−Sa\nSdSa1\nSd+Sa/parenleftigg\n1+log/parenleftiggSd−Sa\nS2\ndkBTN\n|J⊥|/parenrightigg/parenrightigg\n(14)\nWhen combined with eq. (11) it yields:\n−M0/parenleftiggkB\n4πJ/bardblSd−Sa\nSdSa1\nSd+Sa+1\nTN/parenrightigg\n(15)\nEmploying the value of the slope extracted from Fig. 1 of Ref. [6] we d e-\nrivekBTN= 2.4πJ/bardbl, and inserting experimental (extrapolated) value of TN\nyields immediately J/bardbl= 54 K and together with the value of JSW\neffextracted\nfrom low temperature data allows to estimate anisotropy to be ( J/bardbl/J⊥≈3).\n16At the above intermediate temperature (225 K) the 2D estimate wit h this\nanisotropy shows the depletion of magnetization to be 0.595 of the m aximal\nvalue at 0 K as well in a perfect agreement with experiment, which sho ws\nthat the available data on the temperature dependence of magnet ization in\nHM V(TCNE) 2do not allow to distinguish between the 3D and 2D regimes.\nIt must be admitted that in general the above value of anisotropy is not large\nenough ( ∼3) for the 2D regime to install. This analysis, however, allows us\nto set bounds for the value J/bardblin the V(TCNE) 2compound as derived from\nthe spin-wave treatment: it appears that this compound resides in the 3D\nregimesothattheabovevalueofanisotropymust beconsidered as amaximal\npossible in thismaterial. Otherwise even higher N´ eel temperatures (although\nnot accessible expreimentally due to material’s decomposition) still co nform-\ning to the applicability conditions (2 J⊥S2\nd≪θ≪4J/bardblSdSa) Ref. [25] of the\nlogarithmic formula eq. (11) should have to be admitted.\nApplying analogous treatment to the Fe(TCNE) 2compound for which\nthe structure presented in Fig. 1 is experimentally established yields the\nfollowing: JSW\neff= 9.5 K (the coefficient of 12.914 coming from eq. (10) with\nSd= 2 is used). With the anisotropy of 2 .53= 15.625 the 2D estimate of the\nN´ eel temperature is 124 K again in a fair agreement with the experim ent.\nThe value of J/bardblis then 24 K. This turns out to be not that much different\nfromtheupper boundaryforthesamequantity fortheV(TCNE) 2compound\nyielding the ratio of the intralayer exchange parameters for the tw o materials\nofmaximumonlytwo, insteadof4 ÷5stipulatedbythemeanfieldestimates,\nand only 1.5 if the isotropic regime is accepted for the V(TCNE) 2compound.\nThis latter value can be fairly explained by addressing the formulae of\nAppendix A and the spectroscopic data. From eq. (18) it follows tha t the\nratioJ/bardbl(V)/J/bardbl(Fe) of the intralayer parameters for the vanadium and iron\ncompounds is that of the squared hopping parameters tda. (We assume here\nthat due to similarity of the environment in these two compounds the en-\nergy denominators in eq. (18) given by eq. (19) are the same for th e both\ncompounds since the values of ionisation potentials of the V2+and Fe2+ions\nwhich are respectively 29.55 and 30.90 eV as coming from Ref. [26] and sim-\nilarly close estimates for the electron affinities for these ions). Acco rding to\nsuggestionby [27]thoroughlytested numerically inRefs. [28–30]th eamounts\nof the crystal field splitting in the coordination compounds are prop ortional\nto analogousexpressions: squares of thehopping parameters div ided by some\n(other) energy denominators, which are, however, also approxim ately equal\n17in similar compounds. Thus the proportion holds:\nJ/bardbl(V)\nJ/bardbl(Fe)=10Dq(V)\n10Dq(Fe)\nforpairsofsimilar complexes ineachsideoftheproportion. Forther ightside\nof the proportion we find with the values of Refs. [31,32] 10 Dq(V) = 14700\ncm−1, 10Dq(Fe) = 10900 cm−1to be 1.35 for the hexacoordinate octahedral\ncomplexes with acetonitile, which basically explains the above ratio 1.5 o f\nthe intralayer exchange parameters. Of course, the significant d ifference of\nanisotropies in these two materials remains to be understood.\nThis can be tentatively done with use of the Goodenough-Kanamori\nrules [12]. Indeed, the difference in the interlayer interactions requ iring an\nexplanation is too large, so that probably a qualitative distinction bet ween\nthe two materials is responsible for it. As we mentioned above the simp lis-\ntic application of the Goodenough-Kanamori rules in the present sit uation\nyields an antiferromagnetic sign of the interlayer interaction (the situation\nfallsinto theGoodenough-Kanamorication-anion-cationcategor yinthe180◦\ngeometry). Thus the observed ferromagnetic sign of the interlayer interac-\ntion appears as a result of ferromagnetic contributions of the high er order.\nSuch contributions depend qualitatively on the possibility to take adv antage\nof the intrashell ferromagnetic interactions which in their turn dep end on the\noccupancies of the atomic orbitals in the d-shells of the interacting t ransition\nmetal cations. (Importance of such terms in the context of orga nometallic\nmagnets had been stressed in Refs. [14,15]). It can be easily under stood that\nthe conditions for appearance of the compensating ferromagnet ic terms are\nvery much different for the V2+and Fe2+ions. Indeed, in the case of the\nV2+ion twod-orbitals remain empty and can participate in the one-electron\ntransfer coupled with the intrashell exchange eq.(21) compensat ing other-\nwise dominating antiferromagnetic kinetic exchange. In the case of the Fe2+\nion only one doublyoccupied d-orbital can take part in a similar process, so\nthat one can expect that the compensating contribution will be sign ificantly\nweaker in the case of the Fe(TCNE) 2compound eventually leading to much\nweaker overall ferromagnetic interlayer interaction, than in the case of the\nV(TCNE) 2compound.\n185 Discussion\nIn the present Section we apply the models proposed above to analy sis of\nexperimantal data available for the HM V(TCNE) 2and Fe(TCNE) 2com-\npounds.\nFirst of all we notice that the spin polarization per unit cell (number\nof electrons with spin up minus that with spin down) which can be relate d\nwith observed magnetization per formula unit. We see that the calcu lation\nperformed for V(TCNE) 2at the experimental structure of Fe(TCNE) 2de-\npicted on Fig. 1 shows the spin polarization of ca.8 spins-1/2 per unit cell\ncorresponding to two netto unpaired electrons per formula unit wh ich is in a\nfair agreement with the magnetization measured in the HM V(TCNE) 2com-\npound. On the other hand the LM V(TCNE) 2material manifests a weaker\nsaturation magnetization, namely corresponding to ca.one netto unpaired\nelectron per formula unit. This allows to think about certain differenc es in\nthe structures of two materials. Nevertheless, both experiment ally observed\nvalues (ca.10·103emu·Oe·mol−1and 6·103emu·Oe·mol−1, respectively) both\ndeviate from the theoretical values of 11.2 and 5.6 giving the magnet ization\nproduced by the integer number of netto spin-polarized electrons in an as-\nsumption of the Land´ e factor being equal to 2.\nWhen trying to extend the model Ref. [8] of the Fe(TCNE) 2compound\nto analysis of the V(TCNE) 2compound the authors Ref. [8] argued that\nthe vanadium compound must have some structure different from t he iron\none since the saturation magnetization in it is lower and approximately cor-\nresponds to two spins 1 /2 compensating (interacting antiferromagnetically\nwith) one spin 3 /2 per formula unit. From this observation the authors\nof Ref. [8] conclude that the interlayer interactions must be mediat ed by\nµ4-TCNE radical-anions, as it has been suggested yet in [9], rather by t he\n[TCNE]2−\n2dimers. This argument applies of course only to the LM form\nof the V(TCNE) 2material since for its HM form our numerical experiment\nshows that for the experimental structure of the Fe(TCNE) 2compound the\ncalculated magnetization fairly corresponds to the experimental v alue ob-\ntained on the HM V(TCNE) 2material. Incidentally, the magnetization val-\nues obtained numerically at intermediate structures on the ”react ion path”\ndepicted on Fig. 3 allows us to assume that some similar structures ob tained\nfrom structures of Fig. 2 by rotations of some TCNE units may pres ent\nin the LM V-TCNE material. This view had found a recent support from\n19the computational side in Ref. [33] where a structure for the LM fo rm of\nV(TCNE) 2has been proposed. It would be fair to say (although it is not\nsaid in Ref. [33]) that this structure as well descends from the stru ctures of\nRefs. [9,10]. Specifically, in order to obtain the structure of Ref. [3 3] one has\ntorotatetheTCNE molecule laying inthe bc-faceofeither ofthese structures\ndepicted in Fig. 2 in each of the unit cells around the diagonal of the fa ce (or\naround an axis going through the pair of trans-nitrogen atoms of that TCNE\nunit) by ca.90◦so that two other N-atoms of each rotating TCNE unit go\nout of coordination with the V ions. Such a structure corresponds as that of\nRef. [9] and the principal one (see above) of Ref. [10] to two TCNE˙−units\nper unit cell each bearing one unpaired electron and thus expected ly yield\ntheoverall magnetizationcorresponding to oneunpairedelectron per formula\nunit. For such a structure the magnetic interaction parameters o btained in\nRef. [33] are almost isotropic ( J⊥= 720 K and J/bardbl= 690 K) which is also\nnot surprising since the character of interactions between the V d-shells and\nLUMO’s of the TCNE˙−units are fairly the same in either direction. The\nnumerical values of the exchange parameters obtained in Ref. [33] are for\nsure considerable overestimates of the true ones since the N´ eel temperature\nderived from them either by the mean field or spin-wave methods exc eeds the\nexperimental value by orders of magnitude. Nevertheless, applyin g the spin-\nwave theory similar to that described in Section D results in the estima te\nforkBTN= 9.14JSW\neffwhich yields the numerical value of JSW\nefffor the LM\nform of V(TCNE) 2material of 45 K in fair agreement with the similar above\nestimates for the intralayer effective exchange parameters. It m ust also be\nadmitted that the spin-wave treatment of the model of Ref. [33] le aves the\nquestion of the reason of complete disagreement of the temperat ure depen-\ndence of the magnetization in the LM compound as given in Ref. [6] with\nthe Bloch law which should be expected for almost isotropic ferrimagn et\nunanswered.\n6 Conclusion\nIn the present paper we performed detailed analysis of our numeric al results\nconcerning thinkable structure of room-temperature organome tallic magnet\nV(TCNE) 2as manifested in the corresponding projections of DoS. Similar\nanalysis of projected DoS for a sequence of structures leading to the ten-\n20tative experimental structure of V(TCNE) 2is performed as well. Model\nspin Hamiltonian is developed for analysis and interpretation of numer ical\nresults and experimental data. Analysis of magnetic data in terms o f the\napproximate models derived from the phenomenological Hamiltonian is per-\nformed. A remarkable correspondence between experimental (s tructural and\nmagnetic) data on V(TCNE) x·ysolvent and numerical model has been ob-\nserved previously: magnetization corresponding to two unpaired e lectrons\nper formula unit in fair agreement with experiment on HM V-TCNE ma-\nterial derived from V(CO) 6by CVD technique is obtained numerically for\nV(TCNE) 2taken intherelaxed experimental Fe(TCNE) 2geometry Ref. [10].\nNow it is complemented by the detailed analysis of the magnon spectru m of\nthis model. The possible transition between the low-temperature 3D and the\nhigh-temperature 2D regimes is discussed. Estimates of paramete rs of the\nproposed spin-Hamiltonian as treated in the spin-wave approximatio n are\nderived from the experimental data on the N´ eel temperature an d the tem-\nperature dependence of magnetization. The differences in magnet ic behavior\nof probably isostructural HM V(TCNE) 2and Fe(TCNE) 2are tentatively ex-\nplained.\nAcknowledgments\nThisworkisperformedwiththepartialsupportoftheRFBRgrantN o07-03-\n01128 extended to ALT. The generous support of the visit and sta y of ALT\nat RWTH – Aachen University by DFG through the grant No DR 342/20 -1\nis gratefully acknowledged. ALT is thankful to Dr B. Eck for his help in\nmastering the solid state electronic structure analysis software a t the IAC of\nthe RWTH andto Drs. A.M. Tokmachev, I.V. Pletnev, and J. von Appe n for\nvaluable discussions. Prof. Joel S. Miller is acknowledged for kindly dr awing\nthe authors’ attention to Ref. [33] and for sending a preprint of h is work [24]\nprior to publication.\n21A Spin Hamiltonian as derived from orbital\nmodel Hamiltonian\nSimple estimates of the parameters of the spin Hamiltonian eq, (6) ca n be\nbased on considerations dating back to Ref. [12] as specified for th e current\nsituation. FortheidealizedgeometryofM(TCNE) 2compoundswheretheM-\nTCNE layers are assumed to be planar the one-electron hopping par ameters\ntdaare those between the b3g(π∗) singly occupied orbital of the TCNE˙−\nradial-anion and one of the d-orbitals of the π-symmetry with respect to the\nabovementioned plane ( dxzanddyz). For a pair of the TCNE˙−and V2+ions\ndescribed by the Hamiltonians, eqs. (2) and (3), respectively, the energy of\nthe bare ground state reads:\nE0=−αa−3αd+3Udd−21/4Jdd (16)\nanddoesnot dependontheway thespinsinthese sitesarecoupled. Theone-\nelectron hopping couples them with two states with one electron tra nsferred\nbetween the two sites (from one of the d-states to the a-state and from the\na-state to one of the d-states). The energies of the charge transfer states are\nrespectively:\nEd→a=−2αa−2αd+Uaa+Udd−3Jdd/2\nEa→d=−4αd+6Udd+29/4Jdd\nThey both correspond to the lower spin of the state of the two ions which\nresults, as usual, to a Heisenberg-type interaction of two 1/2 elec tron spins:\n2KdaˆSγrˆSar (17)\noccupyingtheoverlappingorbitals(here γreferstothatofthethree d-orbitals\nwhich overlaps with the particular a-orbital) with the effective exchange con-\nstant given by:\nKda=t2\nda(∆E−1\nd→a+∆E−1\na→d)>0, (18)\nwhere\n∆Ed→a=αd−αa+Uaa−2Udd+15/4Jdd= ∆α+Uaa−2Udd+15/4Jdd>0,\n∆Ea→d=αa−αd+3Udd+50/4Jdd=−∆α+3Udd+50/4Jdd>0,\n∆α=αd−αa>0;∆Ea→d>∆Ed→a.\n22On the other hand the charge transfer energies can be expresse d through the\nspectral ionization potentials of the respective ions, their electro n affinities\nand the energy shifts CdandCaof these quantities induced by the Coulomb\nfield of the surrounding crystals already mentioned:\n∆Ed→a=Id−Aa−gad>0,,\n∆Ea→d=Ia−Ad−gad>0,\nId=I0\nd−Cd>0;I0\nd=α0\nd−(nd−1)Udd (19)\nA0\nd=I0\nd−Udd\nIa=α0\na−Ca>0.\n(here we omitted the intraatomic exchange parameters Jddknown to be by\norders of magnitude smaller than other quantities relevant here an d included\nthe electron-hole interaction energies gadpreviously absorbed in α’s).\nThe interaction eq. (17) must be repeated for each interacting pa ir of\nelectronic spins (pair of orbitals, coupled by the electron hopping op erator).\nFor the V ion in the crystal the terms appear for the ξ- andη-states on each\natom.\nFor the pair of metal ions interacting through the [TCNE]2−\n2unit the one\nelectron hopping is effectively possible not only between the states in thet2g-\nmanifolds of the ions involved, but also between other remaining d-orbitals.\nBy this the states admixed to the spin degenerate bare gound stat e of the\npair of ions may have either lower or higher spin which leads both to the\nantiferromagnetic contribution of the form:\n4t2\ndd\n∆Ed→d(20)\nand of the ferromagnetic contribution of the form:\n−/parenleftigg4t′\ndd\n∆Ed→d/parenrightigg2\nJdd (21)\nwhere the hopping parameters tddandt′\nddhave the same order of magnitude.\nThe terms of the antiferromagnetic sign appear for each pair of th ed-orbitals\ncoupled by the hopping, whereas the terms of the ferromegnetic s ign appear\nall singly occupied orbitals in both shells provided an empty or a doubly\noccupied in the given d-shell couples with a singly occupied orbital in other\nd-shell through the hopping.\n23B Phenomenological Hamiltonian for effec-\ntivespinsand itsrelationtothespinHamil-\ntonian\nThe phenomenological Hamiltonian written in terms of the effective sp ins eq.\n(7) to be used for modeling the entire crystal has the form of eq. ( 8)\nIn order to obtain it we follow the general recipes given in Ref. [16] fo r\nobtainingthespin-wave spectrum andwritethegeneralHeisenber g equations\nof motion for the spin-raising operators ˆS+\ndr,ˆS+\nar. They read:\ni/planckover2pi1∂\n∂tˆS+\ndr=−J⊥/parenleftigˆSz\ndrˆS+\ndr±b−ˆS+\ndrˆSz\ndr±b/parenrightig\n+J/bardbl/bracketleftigˆSz\ndr/parenleftigˆS+\nar+ˆS+\nar+a+c+ˆS+\nar+a+ˆS+\nar+c/parenrightig\n−ˆS+\ndr/parenleftigˆSz\nar+ˆSz\nar+a+c+ˆSz\nar+a+ˆSz\nar+c/parenrightig/bracketrightig\ni/planckover2pi1∂\n∂tˆS+\nar=J/bardbl/bracketleftigˆSz\nar/parenleftigˆS+\ndr+ˆS+\ndr−a−c+ˆS+\ndr−a+ˆS+\ndr−c/parenrightig\n−ˆS+\nar/parenleftigˆSz\ndr+ˆSz\ndr−a−c+ˆSz\ndr−a+ˆSz\ndr−c/parenrightig/bracketrightig(22)\nInthespin-wave approximationtheoperators ˆSz\ndandˆSz\naarereplaced bytheir\naverage values in the magnetic (ordered) phase:\nˆSz\ndr→/angbracketleftigˆSz\nd/angbracketrightig\n=3\n2;ˆSz\nar→/angbracketleftigˆSz\na/angbracketrightig\n=−1\n2(23)\nso that the equations of motion get the form:\ni/planckover2pi1∂\n∂tˆS+\ndr=−J⊥/parenleftig\n3\n2ˆS+\ndr±b−3ˆS+\ndr/parenrightig\n+J/bardbl/bracketleftig\n3\n2/parenleftigˆS+\nar+ˆS+\nar+a+c+ˆS+\nar+a+ˆS+\nar+c/parenrightig\n+2ˆS+\ndr/bracketrightig\ni/planckover2pi1∂\n∂tˆS+\nar=J/bardbl/bracketleftig\n−1\n2/parenleftigˆS+\ndr+ˆS+\ndr−a−c+ˆS+\ndr−a+ˆS+\ndr−c/parenrightig\n−6ˆS+\nar/bracketrightig\n.(24)\nGoing to the Fourier transforms of the raising operators we get:\ni/planckover2pi1∂\n∂tˆS+\ndk=−3J⊥(coskb−1)ˆS+\ndk+J/bardbl/bracketleftig\n3\n2ΩkˆS+\nak+2ˆS+\ndk/bracketrightig\ni/planckover2pi1∂\n∂tˆS+\nak=−J/bardbl/bracketleftigΩ∗\nk\n2ˆS+\ndk+2ˆS+\nar/bracketrightig\nΩk= 1+exp( ika)+exp(ikc)+exp(ika+ikc)(25)\nwhich must be complemented by analogous system of equations for t he spin\nlowering operators ˆS−\ndr,ˆS−\nar.\n24On the other hand the Heisenberg equations of motion for the oper ators\nˆS+\nζr,ˆS+\nηr,ˆS+\nξr,andˆS+\narasderived fromeq. (6)(aftertheFouriertransformation\nis performed) form a system of four equations of motion for the Fo urier\ncomponents of the spin-1\n2raising operators for each wave vector k. It can\nbe rewritten with use of k-dependent 4 ×4 matrices acting on the vectors ˆS+\nk\nwith the components ˆS+\nζk,ˆS+\nηk,ˆS+\nξk,ˆS+\nak:\ni/planckover2pi1∂\n∂tˆS+\nk=M(k)ˆS+\nk (26)\nwhere\nM(k) =A+T(k)+L(k) (27)\nand the matrix\nA=Jdd\n4−2−2 0\n−2 4−2 0\n−2−2 4 0\n0 0 0 0\n(28)\ndescribes the spin fluctuations in the d-shells, the matrix\nT(k) =Kdd\ncoskb\n1 1 1 0\n1 1 1 0\n1 1 1 0\n0 0 0 0\n−3\n1 0 0 0\n0 1 0 0\n0 0 1 0\n0 0 0 0\n\n(29)\ndescribes the spin wave propagation in the b-direction (transversal to the\nV-TCNE) planes and the matrix\nL(k) =Kda\n0 0 0 0\n0 1 01\n2Qck\n0 0 11\n2Qak\n0−1\n2Q∗\nck−1\n2Q∗\nak−2\n(30)\nwith\nQak= 1+e−ika−ikc\nQck=e−ika+e−ikc(31)\n25desribes the spin-wave propagation in the ac-planei.e.in the individual\nV-TCNE layer. Going to the linear combitations of the spin fluctuation\noperators in the d-shell:ˆS+\nζk+ˆS+\nηk+ˆS+\nξk;−ˆS+\nζk+ˆS+\nηk;−ˆS+\nζk+ˆS+\nξkintroduces\nthe fluctuation of the effective spin of the d-shell (the first combination)\nand incidentaly diagonalizes the sum of the first two matrix terms A+T(k)\nyielding the eigenvalues: 0; 3 Kdd(coskb−1); 6Jdd−3Kdd>0 of which the\nzero one refersto precession of the1\n2spin in theacceptor orbital, the next one\nrefers to precession of the total spin of the d-shell, and the doubly degenerate\nhighest eigenvalue corresponds to excitations changing the total spin of the\nd-shell. The interaction matrix L(k) in this basis acquires the form:\nKda\n6\n−12 Q∗\nck−2Q∗\nakQ∗\nak−2Q∗\nck−Q∗\nak−Q∗\nck\n3Qak 4 −2 2\n3Qck −2 4 2\n3(Qak+Qck) 2 2 4\n(32)\nThe states corresponding to the excitations of the d-shell (the above matrix\nis written so that they are the second and the third ones) are of mu ch higher\nenergythanthestatescorrespondingtoprecessionoftheeffec tivespinsintwo\ntypesofsitesofthemodel. Forthatreasontheformercanbeexc luded. Todo\nso we treat L(k) as a perturbation with the smallness parameterKda\nJddand in\nthe zero order we obtain an operator acting in the two-dimensional subspace\nspanned by the vector with the components ˆS+\nζk+ˆS+\nηk+ˆS+\nξk=ˆS+\ndk;ˆS+\nak:\n/parenleftigg\n−2Kda −1\n6(Q∗\nak+Q∗\nck)Kda\n1\n2(Qak+Qck)Kda2Kda\n3−3Kdd(1−coskb))/parenrightigg\n(33)\nComparing eq. (33)witheq. (25)andnoticing that Ω k=Q∗\nak+Q∗\nckwe arrive\nto the conclusion that they coincide after setting/vextendsingle/vextendsingle/vextendsingleSz\nξ/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingleSz\nη/vextendsingle/vextendsingle/vextendsingle=1\n2=1\n3|Sz\nd|\nandKdd=−J⊥;Kda= 3J/bardblwhich establishes the required relation between\nthe spin wave treatments of the Hamiltonians eq. (6) and eq. (8). T he first\norder correction to eq. (33) has the form:\nK2\nda\n9Jdd/parenleftigg1\n2(2−coska−coskc+2coskccoska)1\n12Ω∗\nk\n−1\n4Ωk1\n3/parenrightigg\n, (34)\nbut it is not used hereinafter.\n26C Mean Field treatment of the phenomeno-\nlogical Hamiltonian.\nIn order to obtain a structure dependent mean field estimate for t he N´ eel\ntemperature of the V-TCNE material we use the method described in Ref.\n[21]. Accordingly the N´ eel temperature for a system comprising tw o types\nof magnetic centers with effective spins SdandSaso that each center of\nthe with spin SdhasZddneighbours of the same type, Zdaneighbours with\nthe spin Saetc.with the interactions between the nearest neighbors of the\nspecific type given by Jdd,Jdaetc.satisfies the equation:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleJddZddSd(Sd+1)−3θMF\nN JdaZdaSd(Sd+1)\nJdaZadSa(Sa+1) JaaZaaSa(Sa+1)−3θMF\nN/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0\nwhere we used the fact that Jad=Jda. For the structure represented in\nFig. 1 and modelled by the Hamiltonian eq. (8) we set Sa=1\n2. Then\nZad=Zda= 4;Zdd= 2;Zaa= 0 (acceptors do not have acceptor neighbors\nin this model). In the notation of eq. (8) Jdd=J⊥;Jda=J/bardbl, so that we\nobtain:\nθMF\nN=2J⊥\n3Sd(Sd+1)+/radicaligg\nJ2\n⊥\n9S2\nd(Sd+1)2+4\n3J2\n/bardblSd(Sd+1)\nwhich in the limit of strong anisotropy yields the following mean field esti-\nmate:\nθMF\nN=2J⊥\n3Sd(Sd+1)+J/bardbl/radicaligg\n4\n3Sd(Sd+1) (35)\nwhich flows to the the limiting expression used in Ref. [8] for the single F e-\nTCNE layer with the variance of the factor of two which results from the\ndifferent definition of the Hamiltonian in Refs. [8,21].\nD Spin-wave model of magnetic properties of\nV(TCNE) 2\nNow we address the spin-wave theory of the M(TCNE) 2ferrimagnets de-\nscribed by the phenomenological Hamiltonian eq. (8) in order to deriv e its\n27magnetic properties. The temperature dependence of magnetiza tion and the\nN´ eeltemperature(thatateachthemagnetizationofeachsubla tticevanishes)\nare controlled by the spectrum of the lowest energy excitations: s pin waves\n(magnons). Calculation of these properties is customary perform ed with use\noftheHolstein-Primakoff Ref.[17]representation ofthemagnons . This latter\nhadbeen many times derived fortheferrimagnets with relatively simp le crys-\ntal lattice Refs. [18,19]. These derivations are based on the Green ’s function\ntechniques. The present case differs from those described there by a com-\nbination of the alternation of the interaction sign ( J/bardblandJ⊥have opposite\nsigns) and a relatively complex form of the structure factors which makes the\nGreen’s function too cumbersome and also prevents from using dire ctly the\nthe general formulae derived previously. We perform the required derivation\nfor the present structure for certainty using the equation of mo tion method.\nIt evolves as follows. Each term in the phenomenological Hamiltonian e q.\n(8) has the form:\nJ12ˆS1ˆS2=J12/bracketleftbigg\nˆSz\n1ˆSz\n2+1\n2/parenleftigˆS+\n1ˆS−\n2+ˆS−\n1ˆS+\n2/parenrightig/bracketrightbigg\n(36)\nIfthepairoftheabovespinscouplesferromagnetically( J12<0)theHolstein-\nPrimakoff operators bi(b+\ni) annihilating (crearing) an elementary excitation\n(magnon) at the i-th site are introduced by the relations:\nˆS+\ni=/radicalig\n2Si−ˆnibi\nˆS−\ni=b+\ni/radicalig\n2Si−ˆni (37)\nˆSz\ni=Si−ˆni;ˆni=b+\nibi\nThe operators bi(b+\ni) so defined obey the boson commutation relations:\n/bracketleftig\nb+\ni,b+\ni′/bracketrightig\n= [bi,bi′] = 0;/bracketleftig\nb+\ni,bi′/bracketrightig\n=δii′ (38)\nAfter being inserted in the above Hamiltonian term the square roots are\nexpandedandthetermsnonhigherofthesecondorderintheboso noperators\nbiare kept (linear spin-wave approximation) so that one gets:\nJ12ˆS1ˆS2=J12S1S2−J12(S1ˆn2+ ˆn1S2)+J12/radicalig\nS1S2/parenleftig\nb+\n1b2+b+\n2b1/parenrightig\n(39)\n28If the pair of effective spins couples antiferromagnetically ( J12>0) the\nHolstein-Primakoff bosons are obtained by a more complex trick which , first,\nrequires a transformation of the spins residing on the m-th site in the unit\ncell by the rotation matrices ωm:\n˜Si=ˆSiωi;ˆSi=ω†\ni˜Si (40)\nIn the assumption of the common quantization axis for all spins in the crys-\ntallographic unit cell one can select the rotations as follows:\nω1=\n1 0 0\n0 1 0\n0 0 1\n;ω2=\n1 0 0\n0−1 0\n0 0−1\n (41)\nin the basis of the coordinate spin components x,y,zor\nω1=\n1 0 0\n0 1 0\n0 0 1\n;ω2=\n0 1 0\n1 0 0\n0 0−1\n (42)\nin the basis of the tensor spin components + ,−,z. Then the interaction term\nrewrites:\nJ12ˆS1ˆS2=−J12ˆSz\n1˜Sz\n2+1\n2J12/parenleftigˆS+\n1˜S+\n2+ˆS−\n1˜S−\n2/parenrightig\n(43)\nThe boson operators for the transformed spins are then defined by the same\nrelations as the nontransformed ones which yields the following inter action\nterm:\nJ12ˆS1ˆS2=−J12S1S2+J12(S1ˆn2+ ˆn1S2)+J12/radicalig\nS1S2/parenleftig\nb+\n1b+\n2+b1b2/parenrightig\n(44)\nwhere the anomalous products b+\n1b+\n2andb1b2appear.\nDescribed procedures apply to each pair of the interacting spins ( J/bardbl>\n0;J⊥<0) in the Hamiltonian eq. (8) yielding the following:\nHSW=/summationdisplay\nr/parenleftig\n−4J/bardblSaSd+J⊥S2\nd+4J/bardbl(ˆnarSd+Saˆndr)−2J⊥Sdˆndr\n+J/bardbl/radicalig\nSaSd[Dr(Ar+Ar+a+Ar+c+Ar+a+c) (45)\n+D+\nr/parenleftig\nA+\nr+A+\nr+a+A+\nr+c+A+\nr+a+c/parenrightig/bracketrightig\n+J⊥Sd/parenleftig\nD+\nrDr+b+D+\nr+bDr/parenrightig/parenrightig\n29whereD+\nr(Dr) andA+\nr(Ar) are the magnon creation (annihilation) operators\nat the respective sites in the crystal and the ’numbers of magnons ’ operators\nnecessary to calculate the magnetization at each site are:\nˆnar=A+\nrAr;ˆndr=D+\nrDr (46)\nIntroducing the Fourier transforms of the magnon creation oper ators by the\nrelations:\nD+\nr=1√\nN/summationdisplay\nkexp(−ikr)D+\nk;\nA+\nr=1√\nN/summationdisplay\nkexp(−ikr)A+\nk (47)\nand by the and the hermitean conjugate ones for the annihilation op erators\nwe obtain the spin-wave Hamiltonian in the form:\nHSW=/summationdisplay\nk/parenleftig\n4J/bardbl/parenleftig\nSdA+\nkAk+SaD+\nkDk/parenrightig\n+J/bardbl/radicalig\nSaSd/bracketleftig\nΩ∗\nkDkA−k+ΩkD+\nkA+\n−k/bracketrightig\n(48)\n+ 2J⊥Sd(coskb−1)D+\nkDk/parenrightig\nwhere the unnecessary constant is omitted and the structure fa ctors Ω kco-\nincide with those defined by eq. (25).\nThe above Hamiltonian produces the Heisenberg equations of motion\ni/planckover2pi1˙b= [HSW,b]\nfor the annihilation operators and which are coupled with the analogo us ones\nfor the creation operators due to the presence of the anomalous terms:\ni/planckover2pi1˙Ak= 4J/bardblSdAk+J/bardbl/radicalig\nSaSdΩ−kD+\n−k\ni/planckover2pi1˙D+\nk=−4J/bardblSaD+\nk−J/bardbl/radicalig\nSaSdΩ∗\nkA−k+2J⊥Sd(coskb−1)D+\nk(49)\ni/planckover2pi1˙Dk= 4J/bardblSaDk+J/bardbl/radicalig\nSaSdΩkA+\n−k−2J⊥Sd(coskb−1)Dk\ni/planckover2pi1˙A+\nk=−4J/bardblSdA+\nk−J/bardbl/radicalig\nSaSdΩ∗\n−kD−k\nOne can easily see that in the above system the first one is coupled on ly to\nthe second whereas the third one is coupled only with the fourth. Th us eq.\n30(49) reduces to a pair of 2 ×2 matrix eigenvalue/eigenvector problems to be\nsolved for the stationary magnons:\n/parenleftigg\n4Sd−εk√SaSdΩ−k\n−√SaSdΩ∗\n−k−4Sa+2Sdak−εk/parenrightigg/parenleftigg\nuk\n−vk/parenrightigg\n= 0 (50)\n/parenleftigg\n4Sa−2Sdak−εk√SaSdΩk\n−√SaSdΩ∗\nk−4Sd−εk/parenrightigg/parenleftigg\nxk\n−yk/parenrightigg\n= 0 (51)\nwhere\nak=J⊥\nJ/bardbl(1−coskb) (52)\nThe matrix eigenvalue problems eqs. (50), (51) each yield two solutio ns for\nεkof which one is negative in either case and must be rejected (see Ref . [16]).\nThe nonegativesolutions ofbothproblems represent thespectru m ofmagnon\nexcitations having the form:\nε±\nk= Γk±(2(Sd−Sa)+Sdak) (53)\nΓk=/radicalig\n(2(Sd+Sa)−Sdak)2−SdSa|Ωk|2\nwhere the ” −” sign corresponds to the gapless band with the magnon anni-\nhilation operators Gk, whereas the ”+” sign in the expression for the energy\ncorresponds to the excitations with a gap and the magnon annihilatio n op-\neratorsFkgiven by the relations:\nFk=ukAk−vkD+\n−k (54)\nGk=xkDk−ykA+\n−k (55)\nThe site populations by the magnons are given by\n∝angbracketleftˆnar∝angbracketright=1\nN/summationdisplay\nk/angbracketleftig\nA+\nkAk/angbracketrightig\n;∝angbracketleftˆndr∝angbracketright=1\nN/summationdisplay\nk/angbracketleftig\nD+\nkDk/angbracketrightig\n(56)\nwhere\n/angbracketleftig\nA+\nkAk/angbracketrightig\n=|xk|2/angbracketleftig\nF+\nkFk/angbracketrightig\n+|yk|2/angbracketleftig\nGkG+\nk/angbracketrightig\n/angbracketleftig\nD+\nkDk/angbracketrightig\n=|vk|2/angbracketleftig\nFkF+\nk/angbracketrightig\n+|uk|2/angbracketleftig\nG+\nkGk/angbracketrightig\n(57)\n31andthe∝angbracketleft...∝angbracketrightaveragingisperformedovertheequilibiumstateofaferrimagnet,\nwhich can be rewritten in compliance with the commutation rules for th e\nboson operators:\n/angbracketleftig\nA+\nkAk/angbracketrightig\n=|xk|2∝angbracketleftˆnFk∝angbracketright+|yk|2(1−∝angbracketleftˆnGk∝angbracketright)\n/angbracketleftig\nD+\nkDk/angbracketrightig\n=|vk|2(1−∝angbracketleftˆnFk∝angbracketright)+|uk|2∝angbracketleftˆnGk∝angbracketright (58)\nThe average population of a magnon state at a temperature T=θ/kBex-\npresses through the energy of the corresponding magnon\n/planckover2pi1ω±\nk=J/bardblε±\nk (59)\nfollowing the boson statistics:\n∝angbracketleftˆnGk∝angbracketright=/bracketleftigg\nexp/parenleftigg/planckover2pi1ω−\nk\nθ/parenrightigg\n−1/bracketrightigg−1\n;∝angbracketleftˆnFk∝angbracketright=/bracketleftigg\nexp/parenleftigg/planckover2pi1ω+\nk\nθ/parenrightigg\n−1/bracketrightigg−1\n.\n(60)\nInthelow-temperaturelimit thecontributionofthegapmagnonsise xponen-\ntially small so that ∝angbracketleftˆnFk∝angbracketrightcan be set equal to zero. The expansion amplitudes\nuk,vk,xk,andykare themselves immaterial and the only required densities\nare|xk|2and|yk|2given by:\n|xk|2\n|yk|2/bracerightigg\n=1\nSd+Sa/braceleftbiggSd\nSa(61)\n(here we neglected the k-dependence of the magnon amplitudes).\nThe summation prescribed by eq. (56) replaces according to:\n1\nN/summationdisplay\nk→1\n8π3/integraldisplay\nBZd3k (62)\nwith the first Brillouin zone being a cube with the side of 2 π.\nEvaluating the integrals is based on the advantage of the long wave a p-\nproximation for the energy. Two versions of the latter are relevan t: in the\nfirst low-temperatureregime θ≪ |J⊥|< J/bardblthegaplessbranchofthemagnon\nenergy spectrum has the form:\n/planckover2pi1ω−\nk=/parenleftiggSdSa\nSd−SaJ/bardblk2\na−S2\nd\nSd−SaJ⊥k2\nb+SdSa\nSd−SaJ/bardblk2\nc/parenrightigg\n(63)\n32which by the standard moves (see e.g.Ref [22]) brings us to the estimates\n1\nN/summationdisplay\nk∝angbracketleftˆnGk∝angbracketright/braceleftigg|xk|2\n|yk|2/bracerightigg\n=1\nSd+Sa/braceleftbiggSd\nSa/bracerightbigg(Sd−Sa)3\n2\n8π3\n2S2\ndSaζ/parenleftbigg3\n2/parenrightbiggθ3\n2\nJ/bardbl|J⊥|1\n2(64)\n(hereζis the Riemann ζ-function). This is in some variance with Ref. [18]\nsince in the present case the structure factors for the magnetic interactions\ncorresponding to the realistic model of the material under study h ave been\nused rather the simple cubic ones and the definition of the exchange param-\neters in the Hamiltonian differs by a factor of two (each pair of intera cting\nspins counts once here so that the coefficient of the interaction pa rameter in\nthe diagonal matrix element of the equation of motion equals to the n um-\nber of neighbours of the correcponding type). Taking into accoun t quadratic\ncorrections to the densities eq. (61) yields the higher order corre ctions∝θ5\n2.\nNeglecting these corrections one gets the Bloch T3\n2law eq. 9 for the sponta-\nneous magnetization with the critical (N´ eel) temperature (one at which the\nspontaneous magnetization disappears and which in the present mo del coin-\ncides with the point where the sublattice magnetizations disappear e ither)\nto be found from the equiation:\n1\nSd+Sa(Sd−Sa)3\n2\n8π3\n2S2\ndSaζ/parenleftbigg3\n2/parenrightbiggθ3\n2\nN\nJ/bardbl|J⊥|1\n2= 1.\nExpanding the magnon energy upto the second order in all compone nts\nof the wave vector is possible only if the temperature is the smallest e nery\nscale. In the case of very high anisotropy the second low temperat ure regime\nis possible when |J⊥| ≪θ≪J/bardbl. In this case the gapless branch of the\nmagnon energy spectrum has the form:\n/planckover2pi1ω−\nk=/parenleftiggSdSa\nSd−SaJ/bardblk2\np−2S2\nd\nSd−SaJ⊥(1−coskb)/parenrightigg\n(65)\nk2\np=k2\na+k2\nc\nandtheintegrationof |xk|2or|yk|2divided by/bracketleftbigg\nexp/parenleftbigg\n/planckover2pi1ω−\nk\nθ/parenrightbigg\n−1/bracketrightbigg\nfirst performs\n(following Ref. [20]) over the planar projection kpof the wave vector k, which\n33reduces to\nab\n4π2∞/integraldisplay\n0kpdkp/bracketleftig\nexp/parenleftig\npk2\np/parenrightig\n−a/bracketrightig (66)\nwhere\na= exp/bracketleftigg2S2\nd\nSd−SaJ⊥\nθ(1−coskb)/bracketrightigg\n;p=SdSa\nSd−SaJ/bardbl\nθ;b=1\nSd+Sa/braceleftbiggSd\nSa/bracerightbigg\n.(67)\nAfter substituting z=k2\npthe integration yields:\n−b\n8π2plog(1−a) (68)\nProvided aisanexponential functionwithasmallnegativeargument, log(1 −\na) replaces by the logarithm of the absolute value of the argument:\nlog(1−a)≈log/parenleftigg2S2\nd\nSd−Sa|J⊥|\nθ(1−coskb)/parenrightigg\n, (69)\nso that the expression to be intrgrated over kbbecomes\n−1\n8π2Sd−Sa\nSdSaθ\nJ/bardbl1\nSd+Sa/braceleftbiggSd\nSa/bracerightbigg\nlog/parenleftigg2S2\nd\nSd−Sa|J⊥|\nθ(1−coskb)/parenrightigg\n(70)\nwhich immediately yields\n−1\n4πSd−Sa\nSdSaθ\nJ/bardbl1\nSd+Sa/braceleftbiggSd\nSa/bracerightbigg\nlog/parenleftiggS2\nd\nSd−Sa|J⊥|\nθ/parenrightigg\n. (71)\nThe sublattice magnetizations are:\n/braceleftbiggSd\nSa/bracerightbigg/parenleftigg\n1−1\n4πSd−Sa\nSdSa1\nSd+Saθ\nJ/bardbllog/parenleftiggSd−Sa\nS2\ndθ\n|J⊥|/parenrightigg/parenrightigg\nso that the N´ eel temperature is obtained by the condition of the e vanescence\nof the sublattice magnetizations which in the present approximation coincide\nwith that of the total spontateous magnetization.\n34References\n[1] J.M. Manriquez, G.T. Yee, S.R. McLean, A.J. Epstein, J.S. Miller. Sci-\nence (1991) 252, 1415.\n[2] E.B. Vickers, T.D. Selby, J.S. Miller. J. Am. Chem. Soc. (2004) 126,\n3716.\n[3] E.B. Vickers, T.D. Selby, M.S. Thorum, M.L. Taliaferro, J.S. Miller.\nInorg. Chem. (2004) 43, 6414.\n[4] M.L. Taliaferro, M.S. Thorum, J.S. Miller. Angew. Chem. Int. Ed.\n(2006) 45, 5326.\n[5] K.I. Pokhodnya, E.B. Vickers, M. Bonner, A.J. Epstein, J.S. Miller.\nChem. Mater. (2004) 16, 3218.\n[6] J.S. Miller, A.J. Epstein. Chem. Commun. (1998) 1319.\n[7] P. Zhou, S.M. Long, J.S. Miller, A.J. Epstein. Phys. Lett. A (1993) 181,\n71.\n[8] J.-H. Her, P.W. Stephens, K.I. Pokhodnya, M. Bonner, J.S. Miller.\nAngew. Chem. Int. Ed. (2007) 46, 1521 - 1524\n[9] A.L. Tchougr´ eeff, R. Hoffmann. J. Phys. Chem. (1993) 97, 350 .\n[10] A.L. Tchougr´ eeff, R. Dronskowski. J. Comp. Chem. (2008) 29 , 2220.\n[11] VASP the Guide. by G. Kresse and J. Furthm¨ uller, Institut f¨ u r Materi-\nalphysik, Universit¨ at Wien, (2007).\n[12] J.B. Goodenough. Magnetism and the Chemical Bond. Interscie nce-\nWiley, NY (1963).\n[13] O. Kahn. Molecular Magnetism. VCH (1993).\n[14] A.L. Tchougr´ eeff, I.A. Misurkin. Phys. Rev. B (1992) 46, 5357 .\n[15] A.L. Tchougr´ eeff. J. Chem. Phys. (1992) 96, 6026.\n35[16] S.V. Vonsovskii, Magnetism. Nauka, Moscow (1971) [in Russian]; S .V.\nVonsovsky, Magnetism. Wiley, NY (1974) in two volumes.\n[17] T. Holstein, H. Primakoff. Phys. Rev. (1940) 58, 1098.\n[18] G. Wei, R. Qiu, A. Du. Phys. Lett. A (1995) 205, 335.\n[19] R. Qiu, Z. Zhang. J. Phys. Cond. Matt. (2001) 13, 4165.\n[20] A. Singh, Z. Teˇ sanovi´ c. H. Tang, G. Xiao, C.L. Chien, J.C. Walk er.\nPhys. rev. Lett. (1990) 64, 2571.\n[21] K.H.J. Buschow, F.R. de Boer. Physics of Magnetism and Magnetic\nMaterials. Kluwer, NY (2004).\n[22] A. Auerbach. Interacting Electrons and Quantum Magnetism. Springer-\nVerlag, NY 1994.\n[23] K.I. Pokhodnya, D. Pejakovic, A.J. Epstein, J.S. Miller. Phys. Re v. B\n(2001) 63, 174408.\n[24] J. S. Miller. Polyhed. (2009) 28, 1596.\n[25] A.A. Katanin, V.Yu. Irkhin. Usp. Fiz. Nauk, 177 (2007) 639 - 662 [in\nRussian]; Physics – Uspekhi 50 (2007) No 6. [in English].\n[26] http://physics.nist.gov/PhysRefData/ASD/levels form.html\n[27] P.W. Anderson in Marnetism. Vol. 1. G.T. Rado, H. Suhl Eds. AP, N Y\n(1963).\n[28] A.V. Soudackov, A.L. Tchougr´ eeff, I.A. Misurkin. Theor. Chim. Acta\n(1992) 83, 389.\n[29] A.V. Soudackov, A.L. Tchougr´ eeff, I.A. Misurkin. Int. J. Quan t. Chem.\n(1996) 57, 663.\n[30] A.V. Soudackov, A.L. Tchougr´ eeff, I.A. Misurkin. Int. J. Quan t. Chem.\n(1996) 58, 161.\n36[31] P.B. Hitchcock, D.L. Hughes, G.J. Leigh, J.R. Sanders, J. De Sou za,\nC.J. McGarry, L.F. Larkworthy J. Chem. Soc., Dalton Trans. (1994 )\n3683.\n[32] A.B.P.Lever.InorganicElectronicSpectroscopy.2-ndEdit ion.Elsevier,\nNY (1984).\n[33] G.C. De Fusco, L. Pisani, B. Montanari, and N.M. Harrison. Phys. Rev.\nB (2009) 79, 085201.\n37Table 1: Saturation magnetization Msat 2 K; ordering temperature Tcand\neffective exchange energy JMF\neffbetween the metal residing effective spins\nin the assumption of the mean field connection between TcandJMF\nefffor\nM(TCNE) 2after [6].\nMetal Saturation magnetization MsTcin KJeffin K\nemu Oe mol-1 Spins per M atom\nVa10 300 2 ∼400 53\nVb6 100 1 ∼400 -\nMn 19 000 4 107 6.1\nFe 16 900 3 121 10\nCo 8 000 ∼1.5 44 5.9\nNi 15 800 3 44 11\naThe HM form.\nbThe LM form.\n38Figure 1: Structure of Fe(TCNE) 2as coming from the synchrotron radiation\nstudy [8]. Each unit cell contains four formula units. The solvent mole cules\nare omitted for clarity.\n39Figure 2: Hypothetical structures of V(TCNE) 2following [9] (left) and following [10] (right).\n4041Figure 3: Schematic representation of the intermediate structur es between\nthe quadrupled ”principal” (left upper corner) and the experiment al (right\nlower corner) ones.\n42M = 11.115 \nstatesEnergy (eV)\n–8–6–4–202468–5.0–4.5–4.0–3.5–3.0–2.5–2.0–1.5–1.0–0.50.00.51.01.52.02.53.03.54.04.55.0\nE_FM = 5.249 \nstatesEnergy (eV)\n–8–6–4–2024–5.0–4.5–4.0–3.5–3.0–2.5–2.0–1.5–1.0–0.50.00.51.01.52.02.53.03.54.04.55.0\nE_FM = 8.366 \nstatesEnergy (eV)\n–12–10–8–6–4–202–5.0–4.5–4.0–3.5–3.0–2.5–2.0–1.5–1.0–0.50.00.51.01.52.02.53.03.54.04.55.0\nE_F\nFigure 4: V-Densities of states in two spin channels for the limiting and an intermediate structures\nbetween the quadrupled ”principal” (left) and the experimental (r ight) ones. Observe different scale of\nthe abscissa for different pictures. The magnetization in units of nu mber of unpaired electrons per unit\ncell is indicated on top of each picture.\n43M = 11.115 \nstatesEnergy (eV)\n–4–2 024–10.0–9.5–9.0–8.5–8.0–7.5–7.0–6.5–6.0–5.5–5.0–4.5–4.0–3.5–3.0–2.5–2.0–1.5–1.0–0.50.00.51.01.52.0\nE_FM = 5.249 \nstatesEnergy (eV)\n–8–6–4–202468–10.0–9.5–9.0–8.5–8.0–7.5–7.0–6.5–6.0–5.5–5.0–4.5–4.0–3.5–3.0–2.5–2.0–1.5–1.0–0.50.00.51.01.52.0\nE_FM = 8.366 \nstatesEnergy (eV)\n–6–4–20246–10.0–9.5–9.0–8.5–8.0–7.5–7.0–6.5–6.0–5.5–5.0–4.5–4.0–3.5–3.0–2.5–2.0–1.5–1.0–0.50.00.51.01.52.0\nE_F\nFigure 5: N-Densities of states in two spin channels for the limiting and an intermediate structures\nbetween the quadrupled ”principal” (left) and the experimental (r ight) ones. Red/green is the DoS of\nthe N atoms in the V-TCNE layers, blue/yellow is that of the ”to be dan gling” ones; olive green/dark\nred is that of those involved in the V-V bonding through the [TCNE]2−\n2units. Observe different scale of\nthe abscissa for different pictures. The magnetization in units of nu mber of unpaired electrons per unit\ncell is indicated on top of each picture.\n44M = 11.115 \nstatesEnergy (eV)\n–2–10123–12–11–10–9–8–7–6–5–4–3–2–1012\nE_FM = 5.249 \nstatesEnergy (eV)\n–3–2–10123–12–11–10–9–8–7–6–5–4–3–2–1012\nE_FM = 8.366\nstatesEnergy (eV)\n–2–10123–12–11–10–9–8–7–6–5–4–3–2–1012\nE_F\nFigure 6: C-Densities of states in two spin channels for the limiting and an intermediate structures\nbetween the quadrupled ”principal” (left) and the experimental (r ight) ones. Red/green is the DoS of\nthe C atoms in the V-TCNE layers, blue/yellow is that of those forming the C-C bonds in the [TCNE]2−\n2\nunits. Observe different scale of the abscissa for different picture s. The magnetization in units of number\nof unpaired electrons per unit cell is indicated on top of each picture .\n45TCNE_2 \nstatesEnergy (eV)\n–8–6–4–202468–11–10–9–8–7–6–5–4–3–2–1012\nE_F\nFigure 7: The projection of the DoS in two spin channels to the [TCNE]2−\n2\nunits. in the ”experimental” structure [8]. Red/green is the DoS of t he N\natoms bonding the V-TCNE layers; blue/yellow is that of the ”dangling ”\nones; olive green/dark red is that of the C atoms.\n46"    },    {        "title": "1511.07923v2.Designing_a_fully_compensated_half_metallic_ferrimagnet.pdf",        "content": "arXiv:1511.07923v2  [cond-mat.mtrl-sci]  26 Nov 2015Designing a fully-compensated half-metallic ferrimagnet\nMarioˇZic,1Karsten Rode,2Naganivetha Thiyagarajah,2Yong-Chang Lau,2Davide Betto,2J.M.D. Coey,2\nStefano Sanvito,2Kerry J. O’Shea,3Ciaran A. Ferguson,3Donald A. MacLaren,3and Thomas Archer1,∗\n1CRANN and School of Physics, Trinity College Dublin, Dublin 2, Ireland\n2CRANN, AMBER and School of Physics, Trinity College Dublin, Dublin 2, Ireland\n3SUPA, School of Physics & Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom\n(Dated: November 30, 2015)\nRecent experimental work on Mn 2RuxGa demonstrates its potential as a compensated ferrimag-\nnetic half-metal (CFHM). Here we present a set of high-throu ghputab initio density functional\ntheory calculations and detailed experimental characteri sation, that enable us to correctly describe\nthe nominal Mn 2RuxGa thin films, in particular with regard to site-disorder and defects. We then\nconstruct models that accurately capture all the key featur es of the Mn-Ru-Ga system, including\nmagnetic compensation and the spin gap at the Fermi level. We find that electronic doping is nec-\ncessary, which is achieved with a Mn/Ga ratio smaller than tw o. Our study shows how composition\nand substrate-induced biaxial strain can be combined to des ign the first room-temperature CFHM.\nCompensated ferrimagnetic half metals (CFHM) have\nan ordered spin-state which results in half-metallicity,\nwithout any net magnetic moment. As the material cre-\nates no stray magnetic field it should have low Gilbert\ndamping, and offer numerous advantages compared to\nstandard ferromagnetic metals. These include higher fre-\nquency operation, higher packing density, reduced device\npowerrequirementanddevicesthat areimpervioustoex-\nternal magnetic fields. Although there is no net magnetic\nmoment, the highly-polarised spin-state allows switch-\ning of the magnetisation via spin-transfer torque. The\nclass of CFHMs was first envisaged by van Leuken and\nde Groot[1] in 1995, but despite significant effort[2–7] the\ngoal of a CFHM had proved elusive.[8]\nRecent experimental[9–11] and theoretical[12] efforts\ntowards creating a CFHM have concentrated on the\nHeusler alloy system Mn 2RuxGa (MRG). Heusler alloys\nare made of four interlaced fcclattices, which form a\nbcc-like structure. Mn 2RuGa has the full Heusler (L2 1)\nstructure, with Mn occupying the 4 aand 4csites, Ru the\n4dand Ga the 4 bsites.[9] Half-metallic Heuslers with the\nL21structure are expected to follow a modified Slater-\nPauling curve with the net magnetic moment mgiven by\nm=Nv−24 where Nvis the number of valence elec-\ntrons. For Mn 2RuGa,Nv=25, resulting in a net moment\nof 1µB. Mn 2Ga was expected to have a half-Heusler\n(C1b) structure,[13] with Mn on 4 a, 4csites and Ga on\n4bsites, thus leaving the 4 dsite empty. In half-metallic\nHeuslers with the C1bstructure, the magnetic moment\nmis given by m=Nv−18. For Mn 2Ga,Nv= 17 and\nhencem=−1µB. The idea behind the Mn 2RuxGa sys-\ntem, as proposed by Kurt et al.[9], was that by changing\nthe Ru content xa material can be formed mid-way be-\ntween Mn 2RuGa and Mn 2Ga, that is half-metallic, yet\npresents no net magnetic moment.\nCubic Mn 2RuxGa was stabilized in thin film form by\nKurt et al.[9], who showed that at x≈0.5 there is an\nordered spin state with a critical temperature of approx-\nimately 550K and a very small net magnetic moment. In\nFIG. 1: a) Spin resolved resistivity (green) and the corre-\nsponding density of states (DOS) (blue) calculated for the\nMn2Ru0.5Ga. The data corresponding to the spin down chan-\nnel is represented with negative values. b) Calculated tran s-\nport spin polarization.\nthe same work Andr´ eev reflection (PCAR) spectroscopy\nshowed a 54% spin polarization at the Fermi level. This\nislessthanthe100%expectedforanidealhalfmetal, but\nclose to the values measured in other Heusler half-metals\nin thin film form. Further evidence of half-metallicity\ncomes from the spontaneous Hall angle, more than an\norder of magnitude higher than that observed in the 3 d-\ntransitionmetals. This andthelinearvariationof mwith\nxprovided strong indications that Mn 2Ru0.5Ga, grown\nby Kurt et al. [9] was a CFHM.\nHowever, despite the experimental evidence, the mea-\nsurements do not match the theoretical understanding\nof Mn 2RuxGa provided by Galanakis et al.[12]. There\nare three main areas where disagreement between the-\nory and experiment exists, namely 1) the on-set of half-\nmetallicity; 2) the cell volume; and 3) the dependence of\nMonx.\nHalf-metallicity Density functional theory (DFT) in-\nvestigations of Mn 2Ru0.5Ga indicate that, although it\nmay be possible to engineer a phase with zero magnetic2\n# 4a4c4b4d∆H[eV] M [ µB]c/aVol [˚A3]\n1 Mn Ru Ga Ru -1.11 2.18 1.0 56.33\n2 Ga Mn Ga Ru -0.99 2.93 1.0 55.83\n3 Mn Mn Ga Ru -0.97 0.07a1.2 55.88\n4 Mn Mn Ga Ru -0.52 4.66b1.0 55.01\n...........................\n10 Ga Mn Ga – 0.02 3.14 1.0 52.01\n12 Ru Mn Ga – 0.27 0.19 1.0 44.70\n13 Ru Mn Ga Ru 0.29 4.44 1.0 59.20\n14 – Mn Ga Ru 0.52 4.50 1.0 49.60\n15 Mn Mn Ga – 0.54 0.47 1.0 46.54\n...........................\naMagnetization quenched by the tetragonal distortion.\nbFerromagnetic Mn 2RuGa phase.\nTABLE I: Calculated enthalpies of formation, ∆ H, for the\nmost stable competing Heusler phases of the 1221 structural\nand magnetic Mn-Ru-Ga cells investigated. The configura-\ntions investigated are limited to the primitive 4-atom Heus ler\ncell (3-atom for half-Heuslers).[15]\nmoment, this will not be half-metallic [3, 5, 12]. Our\ncalculations and those of others [12] find that the spin\ngap in the density of states (DOS) of Mn 2Ru0.5Ga lies\nabout 0.4eV above the Fermi level, EF. In Fig. 1 we plot\nthe calculated DOS andthe correspondingresistivity, ob-\ntained by solving the Boltzmann equations in the relax-\nation time approximation. Although the DOS exhibits\na large spin polarisation, ≈60%, this is not reflected in\nthatoftheresistivity,whichisonly30%. Sinceourtrans-\nport calculations are expected to overestimate the spin\npolarization, the presence of the ’pseudo-gap’[14] cannot\njustify the large experimentally observed spin polariza-\ntion of the current.\nCell volume X-ray diffraction (XRD) and transmis-\nsion electron microscopy (TEM) demonstrate that the\nfilms growepitaxially with the in-plane lattice parameter\nbeing dictated by the MgO substrate ( aMRG=√\n2aMgO)\nfor all film thicknesses grown, while the out-of-plane lat-\ntice parameter c, depends strongly on the film thick-\nness [10]. This results in cell volumes much larger than\nthose predicted by DFT; and their variation with film\nthickness cannot be explained.\nMagnetism The measured magnetic moment as a\nfunction of Ru concentrationis linear[9] for 0 .3≤x≤0.7\nwithslope dM/dx= 2, suggestingthatthereisaspingap\nat the Fermi level over an extended range of concentra-\ntions, in contradiction to the DFT results.\nHere we address and resolve the conflict between ex-\nperiment and theory. By applying a high-throughput ap-\nproach based on the VASP[16] implementation of DFT\nand the Perdew-Burke-Ernzerhof (PBE)[17] functional,\nwe have calculated the properties of 1221 Heusler phases\ncontaining Mn, Ru and Ga, that present different stoi-\nchiometry, magnetic order and site occupancy within theL21and C1 bsymmetries. Other symmetries were ex-\ncluded from the search. For each configuration we com-\npute the enthalpy of formation, ∆ H, with respect to the\nlowest energy phase of each of the constituent elements,\nallowing us to compare the relative stability of the var-\nious configurations. Our results for the lowest energy\nstructures are summarized in Table I.\nFor the Mn 2RuGa composition, we find that the low-\nest energy structure corresponds to Mn occupying the\ninequivalent 4 a, 4csites, Ga the 4 band Ru taking the\nremaining 4 dsite, consistent with literature[4, 9]. Cu-\nbic Mn 2Ga was found to have a positive enthalpy of\nformation of 0 .54eVf.u.−1, making the compound un-\nstable with respect to decomposition into its elementary\nphases. The symmetry of the stable D0 22structure was\nexcluded from our calculations. However, the most en-\nergetically favourable Mn 2Ga structure in the L2 1phase\n(# 15), places Mn on the inequivalent 4 aand 4csites\nwith Ga occupying the 4 bone. The structure remains\ncubic and the magnetic state is ferrimagnetic, consistent\nwith experimental characterisationpresented by Kurt et\nal. [9] for thin films stabilized on a suitable substrate or\na seed layer. We note that the formation of any half-\nHeusler in the Mn, Ru and Ga phase diagram is ener-\ngetically unfavourable, so that we would not expect pure\nhalf-Heuslers to be a significant constituent of the films.\nWe note that the energy of the system can be lowered\nby forming Mn-deficient MnRu 2Ga or MnGa 2Ru (# 1\nand 2 in table I). Given that the difference in formation\nenthalpy of these phases with respect to Mn 2RuGa (# 3\nin table I) is only 0 .14eVf.u.−1, we expect that actual\nsamples will not form distinct polycrystalline phases, but\ninstead display significant site disorder, particularly on\nthe4asite, withapreferencetowardsalowerMncontent.\nThis was confirmed by laser-assisted inductively coupled\nmassspectroscopy(ICPMS) measurementsofthe Mn-to-\nGa ratio for a series of samples with varying Ru concen-\ntration,x. The ratio was observed to be in the 1 .6-1.9\nrange, increasing with increasing film thickness.\nIn Fig. 2 (a) we show scanning transmission elec-\ntron microscopy (STEM) measurements of electron-\ntransparent lamellæ of Mn 2RuGa which indicate that\nthere is little variation in either the in-plane or out-of-\nplane lattice constants throughout the film. The corre-\nsponding electron energy loss (EELS) spectra and line\nprofiles are shown in Fig. 2 (b). Alternating light and\ndark bands in dark-field images indicate slight composi-\ntional variations, especially in the layers closest to the\nsurface. EELS measurements reveal that these bands\ncorrespond to layers of Mn enrichment and Ru depletion,\nsuggesting a degree of phase segregation during growth,\nbut notatthelevelofformationofhalf-Heuslers. Wealso\nnote that the Ru concentration, x, decreases by about\n20% from the interface with the substrate through the\nthickness of the film; to a lesser extent, the Mn concen-\ntration increases across the same range.3\n10 20 30 40 50 60 700.00.51.01.52.0\n00.81.0\nGaMn\nRu\nPosition [nm]Normalised Counts [Arb. Units]HAADF\n(a)\n(b)Cap Mn GaRu2 MgO\nFIG. 2: Analytical electron microscopy analysis of the thin\nfilm (x= 1) composition. (a) High angle annular dark field\n(HAADF) image of a typical sample cross-section, with the\nregion where EELS data were acquired indicated by a white\nrectangle. In HAADF images, heavy elements appear bright-\nest. (b) Elemental variations across the thin film stack, as\nmeasured by EELS. Data are normalised by setting the av-\nerage composition to Mn 2GaRu. The dashed vertical lines\nindicate regions where of Mn enrichment and Ru depletion,\ncorresponding to darker features in the HAADF image.\nIn order to investigate the properties of low-Mn-\ncontent films we have performed supercell calculations\nwhere 1/3 of the Mn atoms at the 4 asite are substi-\ntuted with Ga. The Mn-Ga substitution simultaneously\nchanges the lattice parameters, the magnetic properties\nand the electronic structure of the system. We find that\nthe ionic charges of Mn and Ga are +2 and +1, respec-\ntively. Hence a one-atom Mn-Ga substitution leaves the\nsystem with one unbound electron, thus creating elec-\ntronic doping. Below we describe in detail the properties\nof such Mn-deficient compounds.\nLattice Electronic doping provides an explanation for\nthe variation of the lattice parameter with film thick-\nness. From the volume difference between the relaxed\nDFT structure and the corresponding experimental one\nand by using the bulk modulus B0, we estimate the ex-\nperimental electronic doping. The bulk modulus is cal-\nculated for Mn 2RuxGa (x= 0.0, 0.33, 0.50, 0.66 and\n1.0) compounds by fitting the Murnaghan equation of\nFIG. 3: Comparison of the magnetic moment predicted by\nDFT with the experimental results. The calculated magnetic\nmoment for ideal MRG compounds is shown bythe green line.\nCorrection to the DFT magnetization due to the formation of\nGa defects is shown by the black points. The corresponding\nestimate oftheMn/Garatiois shownintheinset. Thedashed\nline equivalent to doping of 2 e/Ru is shown to guide the eye.\nFIG. 4: a) Spin resolved resistivity (green line) and the\ncorresponding DOS (blue shadowed area) for electron-doped\nMn2Ru0.5Ga. The doping was fixed at 0 .4 electrons per for-\nmula. The data corresponding to the spin down channel are\nrepresented by negative values. b) Calculated transport sp in\npolarization.\nstate[18, 19]. By using a simple model\nnel=B0\nS0/bracketleftBigg/parenleftBigc\na/parenrightBig\nexp·/parenleftbiggaexp\na0/parenrightbigg3\n−1/bracketrightBigg\n,(1)\nwe can relate the experimentally observed lattice param-\neters to the electron doping level, nel. In Eq.(1) S0is\nthe rate of change of the excess pressure with electron\ndoping, while aexpanda0correspond to the experimen-\ntal in-plane lattice constant and the relaxed theoretical\nlattice constant, respectively. This equation is easily de-\nrived under the assumption that the material is in me-\nchanicalequilibrium atthe experimentallattice constant,\ndue to the excess pressureprovidedby the Mn-Ga substi-\ntution, via the electron doping mechanism. Since we are\ncomparing pressure differences, we ignore constant pres-4\nsure terms. In order to stabilise the experimental lattice\nparameters, including the observed c/a >1, we find elec-\ntron doping in the range 0 .1ef.u.−1to 0.5ef.u.−1, cor-\nresponding to a Mn/Ga ratio in the interval 1.4 to 2.0.\nThe higher doping level occurs for the lower Ru concen-\ntrations, as shown in the inset of Fig. 3.\nTEM imaging and spectroscopy clearly indicate that\nthere are regions of high Mn content, and therefore re-\ngions of enhanced Ga content elsewhere in the sample. It\nis therefore reasonable to assume that films of different\nthicknesses will have a different electronic doping, which\nin turn alters the c-lattice parameter and does so uni-\nformly throughout the sample.\nMagnetism A key feature of a CFHM is the magneti-\ncally compensated ground state. In agreement with Ref.\n[12], we find that the magnetization calculated for MRG\ncompounds, as a function of the Ru doping, differs from\nthe experimental one (see Fig. 3). The discrepancy is\ntwo-fold: I) the slope of the magnetization with xdis-\nagrees by a factor of 2, and II) there is no compensation\nof the magnetization around x= 0.5. These discrep-\nancies are resolved if we take into account the effect of\nthe Mn-Ga substitution on the magnetic properties. Ga\ndefects introduce, in addition to the electronic doping,\na change in the net magnetic moment per unit cell, of\n−2µBper Mn substituted by Ga, which allows us to ex-\npress the expected moment MEXPas\nMEXP(x) =MDFT(x)−2·nel(x), (2)\nwhereMDFTis the theoretically calculated magnetic mo-\nment for a defect-free Mn 2RuxGa compound.\nThe corrections given by Eqns. (1) and (2) have been\napplied for each value of x, and the results are sum-\nmarised in Fig. 3. Notably, the presence of defects im-\nproves significantly the agreement between the experi-\nmental and theoretical magnetic moments, with the ex-\nception of concentrations around x= 1. A neutron\ndiffraction study by Hori et al. [4] has shown that the\nmagnetization of stoichiometric Mn 2RuGa (x= 1) is\n≈1µB, in good agreement with our calculations. This\nleads us to the conclusion that in the x= 1 limit there\nmay be a substantial content of the Ru 2MnGa phase,\nwhich is known to be antiferromagnetic [4].\nElectronic Structure Finally, we discuss the effect of\nthe electronic doping on the degree of transport spin\npolarisation. In figure 4 we show the DOS and corre-\nsponding Boltzmann resistivity for Mn 2RuGa with nel=\n0.4 extra electrons per formula unit, coresponding to a\nMn/Ga ratio of 1 .6 as observed by ICPMS. The addi-\ntional doping results in a transport spin polarization of\n≈60%, which is twice as large as the one calculated for\nthe original Mn 2RuxGa compound. At a doping level of\nnel= 1.0 the transport spin polarization becomes 100%.\nIt is important to note that the calculations presented\nhere do not take into account the effect of the disorderdue to the Mn-Ga substitution on the transport prop-\nerties. We anticipate that the presence of disorder may\nopen further the spin gap, resulting in a cumulative ef-\nfect where the disorder provides both the spin gap and\nthe electron doping necessary for reinstating the half-\nmetallicity. The improvement of spin-transport proper-\nties obtained by introducing disorder, has already been\ndiscussed by Chadov et al. [20].\nIn conclusion, a sustained dialogue between experi-\nmental measurements and theoretical calculations have\ndemonstrated that Mn 2RuxGa can form a true CFHM.\nAs a consequence we expect that it will become a cor-\nnerstone for future spintronics technology. By means of\nhigh-throughput calculations, we have shown that there\nare several competing phases in the Mn-Ru-Ga system;\nand that due to their small energy differences they ex-\nhibit a strong tendency towards site disorder, and a pref-\nerence for reduced Mn content. This has all been con-\nfirmedbyourexperimentalcharacterizationofMRG thin\nfilms. Furthermore the low Mn content provides an elec-\ntronic doping mechanism, pushing the system towards\nhalf-metalicity and improving the agreement between ex-\nperiment and theory regarding the structural and mag-\nnetic propertiesofthe system. Basedonourcalculations,\ncomplete transport spin polarisation can be achieved.\nWe have shown that chemical composition, c/aratio,\ntendency to site-disorder and cell volume are all corre-\nlated. To achieve transport half-metallicity and zero net\nmoment, a reduced Mn to Ga ratio of ≈1.4 is required,\nas well as a Ru concentration of ≈0.7. Fine-tuning of\nthe position of the Fermi level in the spin gap can then\nbe achievedthrough varyingthe c/aratio, which we have\nshown can be done by varying the film thickness.\nThe authors would like to thank Cora McKenna for\nthe ICPMS mesaurements, and H. Kurt for fruitful dis-\ncussions. Computational resources have been provided\nby the Trinity Center for High Performance Computing\n(TCHPC)and bythe IrishCentreforHigh-EndComput-\ning(ICHEC). The authorsacknowledgefinancialsupport\nfrom the FP7 project ’ROMEO’ (grant number 309729),\nScience Foundation Ireland through ’AMBER’, and from\ngrant 13/ERC/12561.\n∗archert@tcd.ie; www.materials-mine.com\n[1] H. Van Leuken and R. A. De Groot, Phys. Rev. Lett 74,\n1171 (1995).\n[2] M. Hakimi, M. Venkatesan, K. Rode, K. Ackland, and\nJ. M. D. Coey, J. Appl. Phys. 113, 17B101 (2013).\n[3] L. Yang, B. Liu, F. Meng, H. Liu, H. Luo, E. Liu,\nW. Wang, and G. Wu, J. Magn. Magn. Mater. 379, 1\n(2015).\n[4] T. Hori, M. Akimitsu, H. Miki, K. Ohoyoama, andY. Ya-\nmaguchi, Appl. Phys. A-Mater. 74, s737 (2002).\n[5] S. Wurmehl, H. C. Kandpal, G. H. Fecher, and C. Felser,5\nJ. Phys.-Condens. Mat. 18, 6171 (2006).\n[6] I. Galanakis, P. Mavropoulos, and P. H. Dederichs, J.\nPhys. D Appl. Phys. 39, 765 (2006).\n[7] E. S ¸a¸ sıo˘ glu, Phys. Rev. B 79, 100406 (2009).\n[8] X. Hu, Adv. Mater. 24, 294 (2012).\n[9] H. Kurt, K. Rode, P. Stamenov, M. Venkatesan, Y.-C.\nLau, E. Fonda, and J. M. D. Coey, Phys. Rev. Lett. 112,\n027201 (2014).\n[10] N. Thiyagarajah, Y.-C. Lau, D. Betto, K. Borisov,\nJ. M. D. Coey, P. Stamenov, and K. Rode, Appl. Phys.\nLett.106, 122402 (2015).\n[11] D. Betto, N. Thiyagarajah, Y.-C. Lau, C. Piamonteze,\nM.-A. Arrio, P. Stamenov, J. M. D. Coey, and K. Rode,\nPhys. Rev. B 91, 094410 (2015).\n[12] I. Galanakis, K. ¨Ozdo˘ gan, E. S ¸a¸ sio˘ glu, and S. Bl¨ ugel, J.Appl. Phys. 116, 033903 (2014).\n[13] I. Galanakis, P. H. Dederichs, and N. Papanikolaou,\nPhys. Rev. B 66, 134428 (2002).\n[14] T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999).\n[15] http://www.materials-mine.com.\n[16] G. Y. Sun, J. K¨ urti, P. Rajczy, M. Kertesz, J. Hafner,\nand G. Kresse, J. Mol. Struc.-Theochem 624, 37 (2003).\n[17] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.78, 1396 (1997).\n[18] C. L. Fu and K. M. Ho, Phys. Rev. B 28, 5480 (1983).\n[19] F. D. Murnaghan, 30, 244 (1944), PMC1078704.\n[20] S. Chadov, J. Kiss, and C. Felser, Adv. Funct. Mater.\n23, 832 (2013)."    },    {        "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\n<ij>σiSj+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)+<H−H 0>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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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": "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": "1912.10129v1.Picosecond_spin_orbit_torque_switching_of_ferrimagnets.pdf",        "content": "1 \n Picosecond spin -orbit torque switching of ferrimagnets  \nHao Wu1,*, Deniz Turan1, Quanjun Pan1, Guanjie Wu2, Seyed Armin Razavi1, Nezih Tolga \nYardimci1, Zongzhi Zhang2, Mona Jarrahi1, and Kang L. Wang1,† \n1Department of Electrical and Computer Engineering, University of California, Los Angeles, \nCalifornia 90095, USA  \n2Shanghai Engineering Research Center of Ultra -Precision Optical Manufacturing, Department \nof Optical Science and Engineering, Fudan University, Shanghai 200433, China  \n*Corresponding author : wuhaophysics@ucla.edu  \n†Corresponding author : wang@ee.ucla.edu  \n  2 \n Abstract:  \nSpintronic s provide s an efficient platform for realizing non -volatile memory and logic \ndevices. In these systems, data is stored in the magnetization of magnetic materials , and  \nmagnetization is switched in the writing process . In conventional spintronic devices, ferromagnetic \nmaterials are used which have a magnetization dynamics timescale of around the nanosecond s, \nsetting a limit for the switching speed . Increasing the magnetization switching speed has been one \nof the challenges  in spintronic research . In this work we take advantage of the ultrafast \nmagnetization dynamics in ferrimagnetic materials instead of ferrom agnets, and  we use \nfemtosecond laser pulses and a photoconductive Auston switch to create picosecond current pulses \nfor switching the ferrimagnet.  By anomalous Hall and magn eto-optic Kerr (MOKE) measurement, \nwe demonstrate the robust picosecond  SOT driven magnetization switching of ferrimagnetic \nGdFeCo. The time-resolved  MOKE shows more than 50 GHz magnetic resonance frequency of \nGdFeCo, indicating faster  than 20 ps spin dynamics and tens of picosecond SOT switching speed.  \nOur work provides a promisi ng route  to realize picosecond operation speed for non -volatile \nmagnetic memory and logic applications.  \nMain Text : \nElectrically manipulat ion of  the magnetization is both fundamentally interesting and \npractically important  for spintronic devices1,2. To date, two crucial issues need to be addressed : (1) \nreducing the energy consumption3 and (2) in creasing the switching speed.  There have been  several \napproaches to reduce the energy consumption, such as voltage -controlled magnetization4,5, the \ngiant spin -orbit torque in topological insulators6-11, and so on. As for the switching speed, up to \nnow, most spintronic devices are still operated in several nanoseconds’  timescale, which \ncorresponds to the GHz magnetic resonance frequency of ferromagnetic materials.  Ferrimagnets 3 \n have antiferromagnetically -coupled spin sublattices, as a result, the exchange coupling -induced \nmagnetic resonance frequency is on the order of 100 GHz near their compensation point12, which \ncan enable picosecond magnetization switching speed.  \nThere have been efforts  for realizing ultrafast all -optical switching of ferrimagnetic \nmaterials13, with circularly -polarized femtosecond laser pulses resulting in switching speeds on \nthe order o f tens of picoseconds14-17. Similarly, laser-induced hot-electron pulses have been \nexploited to induce ultrafast switching dynamics in ferrimagnetic materials18-20. However, these \nmethods mainly  rely on heating effects and have relat ively low energy efficiencies compared to \nelectrical methods . Therefore, picosecond electrical switching of ferrimagnets provides a \npromising route towards  ultrafast spintronic devices with high energy -efficiency , which has not \nbeen experimentally explored yet.  \nIn this work, a photoconductive Auston  switch is employed to generate the picosecond \nelectrical pulse from a femtosecond laser pulse21, which is used to provide  the picosecond spin-\norbit torque (SOT) switching22-24 in Ta/GdFeCo heterostructures . The magnetization switching is \ndetected by both the anomalous Hal l effect (AHE) and the magneto -optic optical Kerr effect \n(MOKE) , which demonstrate robust magnetization switching by the picosecond SOT  with a peak \ncurrent density of 107 A/cm2. Furthermore, 50 GHz spin dynamics of GdFeCo is detected by the  \ntime-resolved MOKE  (TR-MOKE) , further confirming the tens of picosecond  switching speed  of \nferrimagnets .  \nFigure 1a shows the schematic of the device structure and the measurement method, where a \nphotoconductive Auston switch is connected to the magnetic  strip of Ta/GdFeCo heterostructures. \nWhen the femtosecond laser pulse is focused on the gap of the Auston sw itch, the charge carriers \nwill be excited in the conduction band of low temperature grown GaAs (L T-GaAs), which are 4 \n subsequently accelerated  by the bias voltage . Based upon simulations in the Sentaurus software \npackages, the induced photocurrent response t ime is estimated  to be around 1 ps.  The \nmagnetization of GdFeCo is detected either by the AHE or the MOKE , where a n in-plane magnetic \nfield is applied to break the inversion symmetry between ±M during the SOT switching.  Figure \n1b shows the microscopic picture of the Auston switch+ magnetic  strip device , where the width of \nthe magnetic  strip is 20  μm. The Rxy-Hz curve in Figure 1c  demonstrates  the strong  perpendicular \nmagnetic anisotropy (PMA)  of GdFeCo.  \n \nFIG. 1. a, Schematic of the Auston switch+magnetic strip device and the measurement set up. \nThe THz lase r pulse is focused  on the gap of the voltage -biased photoconductive Auston switch \nto excite the picosecond current, which  then provide s a picosecond spin -orbit torque  (SOT) on \n5 \n the ferrimagnetic GdFeCo.  b, SEM picture of the patterned Auston switch+magnetic str ip \ndevice.  c, Rxy-Hz curve shows the strong perpendicular magnetic anisotropy (PMA) of GdFeCo.  \nTo test the device , we first perform all -electrical SOT switching of the Ta/GdFeCo \nheterostructure  using relatively long electrical pulses . Figure 2a,b show the current -driven SOT \nswitching, where the magnetization is detected by the AHE resistance. In this measurement scheme, \na 1-ms writing current pulse is applied to the switch the magnetization  firstly, and is followed one \nsecond later by another  1-ms reading current pulse to detect the AHE resistance.  Under ±50 Oe  \nin-plane magnetic field Hx, the current -induced switching polarity changes, which is the typical \nSOT characteristic.  The switching current density  Jc is around 1.1 ×107 A/cm2 for the 1 -ms writing \npulse, which is similar to th e previous report25. MOKE microscope is employed to detect the \nmagnetic domain directly , which also clearly show the robust SOT -driven magnetic domain \nswitching , as shown in Figure 2c . In order to figure out the current -induced Joule heating effect on \nthe SOT switching, we cha nge the writing pulse width from 0.5 s to 10 ns , as shown in Figure 2d . \nJc significantly increases for the writing pulse width from 0.5 s to 1  ms, and then slightly changes \nfrom 1 ms to 10 ns, indicat ing Joule heating is not a domina nt effect  with short er writing pulse s. 6 \n  \nFIG. 2. a and b show the current -driven SOT switching of the Ta/GdFeCo heterostructure  under \nHx = ±50 Oe . c shows the SOT -driven magnetic domain switching measured by the MOKE \nmicroscope . d, Switching current density Jc as a function of the pulse width of the writing \ncurrent.  \nThe magnetic precession period of the GdFeCo samples is more than 3 orders of magnitude \nshorter than the duration of the writing pulses in Fig ure 2d. It is therefore reasonable to expect that \nthe SOT switching of the GdFeCo could  be achieved in devices subjected to significantly shorter \nwriting pulses. Previous time resolved magnetization studies of GdFeCo structures report full \nmagnetization switching in structures subjected to ps and sub -ps length optical  excitation. \nHowever, t hese previous results are both rooted in thermal effects, therefore , the demonstration of \n7 \n deterministic, ultrafast, electrical switching of GdFeCo magnetization is highly desirable from a \ntechnological perspective.  \nTo test the temporal limitations of SOT  switching in the Ta/GdFeCo  heterostructure , we \nperform  the experiments on devices subjected to ps duration electrical pulses. These ultrashort \nelectrical excitations are generated by illuminating a DC biased photoconductive switch with the \nfs output of a mode locked Ti:Sapphire laser. A typical device used for these experiments is \ndepicted in Fig ure 1b. Given the duration of the laser pulse (~150  fs) and the known carrier  lifetime \nwithin the switch (~300  fs), electrical pulses on the order of 1  ps duration are predicted by \nsimulations in the Sentaurius software package. By tuning  the applied laser power (100 mW ) and \nthe bias voltage  (10 V) , average currents as high as 10  μA are observed in these devices, equating \nto a peak densit y of 107 A/cm2 for the picosecond current . When testing the viability of ultra fast \nSOT switching in our devices, we appl y an in -plane magnetic field to the device under test  to break \nthe inversion symmetry between ±M. Experiments are performed by illuminating the switch for \napproximately a minute, and then reading out the magnetization state through measurements of \nthe AHE  resistance (Fig ure 3a) and the Kerr rotation angle (Figure 3b). Significantly, we observe \nthat the magnetization state s of device are indeed changed when subject to this procedure. Unlike \nthe previous report of ultrafast electrical switching in GdFeCo, however, we observe that the final \nmagnetization state of our devices is dependent upon the polarity of the bias voltage across the \nswitc h as seen in Fig ure 3a,b. That the switching is dependent upon bias polarity indicates that this \nmagnetization switching  primarily comes from picosecond current -induced SOT, rather than  as a \nconsequence of thermal effects.  8 \n  \nFIG. 3. a, THz laser pulse is applied at the gap the photoconductive Auston switch to excite the \npicosecond current under the bias voltage, and then provides a SOT on the Ta/GdFeCo \nheterostructure, where the magnetization is detected by the AHE resistance. When the b ias \nvoltage polarity is switched from +10 V to -10 V, the sign change of the AHE resistance \ndemonstrate s the magnetization switching from up to down states, which indicates SOT \nswitching rather than thermal -induced switching. b, Picosecond -SOT driven magne tization \nswitching detected by the MOKE . \nWhile deterministic switching is observed in our ultrafast measurements, the measured Hall \nand Kerr signals provided in Fig ure 3a,b do not fully return to their initial state upon magnetization \nreversal, and appear to drift slightly over many switching cycles. The failure to recover the Hall \n9 \n resistivity or Kerr rotation observed at full magnetization is likely an indicator of only par tial SOT \nswitching in these devices, resulting in a multi -domain final state. That the Hall resistance and \nKerr rotation of the final state drift following several switching cycles stems from an asymmetry \nin the current generated in our Auston switch upon the reversal of the external bias.  \nWe have show n that the picosecond SOT can efficiently switch the ferrimagnetic GdFeCo, \nwhile another important question is how fast of the magnetization switching speed? In order to \nanswer this question, we perform the TR-MOKE  measurement26-28 to pick up the spin dynamics \nof the ferrimagnetic GdFeCo.  Figure 4a shows the typical TR -MOKE curve of the Ta/GdFeCo  \nheterostructure  under H = 15 kOe. To extract precession frequency f, the dynamic Kerr signals are \nfitted by using the following expression29: \n \n0 1 0 2 exp( / ) sin(2 )exp( / )= + + + −kc c t t c ft t      (1) \nHere c0 is the background signal. The second term is an exponential decaying signal representing \nthe slow recovery process, where c1 is the amplitude and t0 is the characteristic relaxation time. c2, \nf, φ, and τ in the third term represent the  magnetization preces sion parameters of amplitude, \nfrequency, initial phase, and decay time, respectively. The fitted frequencies under different H are \nshown in  Figure 4b , which is consistent with our expectations.  From the fitting, we can get as large \nas 52.4 GHz magnetic resonance frequency  of GdFeCo , which corresponds to a faster  than 20 ps \ntimescale of spin dynamics , and therefore, indicating the tens of picosecond speed  of SOT \nswitching . 10 \n  \nFIG. 4. a, Normalized dynamic TR-MOKE signals for  the Ta/GdFeCo heterostructure . b, \nMagnetic resonance frequency and damping factor as a function of the magnetic field.  \nIn summary, by combing the photoconductive Auston switch with the Ta/GdFeCo  \nheterostructure , a picosecond writing current pulse is applied to provide the SOT, where both the \nAHE and MOKE measurements demonstrate the robust picosecond SOT-induced magnetization \nswitching . The TR-MOKE  measurement demonstrate s faster  than 20 ps  spin dynamics, indicating \ntens of picosecond SOT switching speed of ferrimagnetic GdFeCo.  These results demonstrate the \npicosecond SOT switching of ferrimagnets, which can increase the speed of spintronic devices \nfrom current ns to ps  ranges . \nExperimental Section  \nPicosecond electrical pulses  are generated using a  photoconductive  Auston switch  based on \nthe low-temperature -grown gallium arsenide  (LT-GaAs) . LT-GaAs has a high mobility  and a  high \nintrinsic resistivity, and the  photogenerated carriers exhibit a short lifetime ( 300 fs), which leads \nto the short photocurrent response time (1 ps) . A mode locked Ti:Sapphire laser operating at 80 \nMHz is used to excite the Auston switch. In transport experiments, 800  nm pulses are focused on \nthe gap of Auston switch using a microscope objective. After illuminating the Auston switch for \napproximately one minute, the beam is blocked, and the AHE  resistance is read out using a four-\n11 \n contact  measu rement. All -optical experiments are conducted using the same protocol of minute -\nlong illumination of the Auston switch, followed by a measurement of the Kerr rotation  at the \nmagnetic strip . In these experiments, a n 820 nm-wavelength laser beam  is used to excite \nphotoconductive Auston switch, while the magnetization is extracted from the Kerr signal of a \nreflected 410  nm-wavelength laser  beam.  In both instances, the pulse duration is between 150 -\n200 fs. \nThe dynamic  TR-MOKE measurements are achieved by using a pulsed Ti:sapphire laser with \na central wavelength of 800 nm, a pulse duration of 150 fs, and a repetition rate of 1000  Hz. An \nintense pump pulse beam with a fluence of approximately 1.0 mJ/cm2 is used to excit e the dynamic \nmagnetization behaviors, and the transient MOKE signal is detected by a weak probe pulse of \napproximately 0.0 5 mJ/cm2, which is time delayed with respect to the pump beam. The pump and \nprobe laser beams are at almost perpendicular incidence, with spot diameters of about 1.0 and 0.2 \nmm, respectively. During the TR -MOKE measurement, an external field  H is applied in the x-z \nplane , with a polar angle of θH=71 to drive the magnetization orientation away from the \nperpendicular easy axis.  \nAcknowledgements  \nThis work is supported by the NSF Award No. 1611570, the  Nanosystems  Engineering \nResearch Center for Translational Applications  of Nanoscale Multiferroic Systems (TANMS), and \nthe Spins and Heat in  Nanoscale Electronic Systems (SHINES), an Energy Frontier Research  \nCenter funded by the US Department of Energy (DOE). The authors are  also grateful for the \nsupport from the Function Accelerated nanomaterial  Engineering (FAME) Center, and a \nSemiconductor Research Corporation  (SRC) program sponsored by Microelectronics Advanced \nResearch  Corporation (MARCO) and Defense Advanced Research Projects Agency  (DARPA).  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Laser -Induced Magnetization Dynamics in \nInterlayer -Coupled [Ni/Co] 4/Ru/[Co/Ni] 3 Perpendicular Magnetic Films for Information Storage. \nACS Applied Nano Materials  2, 5140 -5148 , doi:10.1021/acsanm.9b01028 (2019).  \n "    },    {        "title": "1302.5229v1.Large_zero_field_cooled_exchange_bias_in_bulk_Mn2PtGa.pdf",        "content": "arXiv:1302.5229v1  [cond-mat.mtrl-sci]  21 Feb 2013Large zero-field cooled exchange-bias in bulk Mn 2PtGa\nA. K. Nayak,1,∗M. Nicklas,1,†S. Chadov,1C. Shekhar,1Y. Skourski,2J. Winterlik,3and C. Felser1,3\n1Max Planck Institute for Chemical Physics of Solids, N¨ othn itzer Str. 40, 01187 Dresden, Germany\n2Dresden High Magnetic Field Laboratory (HLD),\nHelmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Ger many\n3Institut f¨ ur Anorganische und Analytische Chemie,\nJohannes Gutenberg-Universit¨ at, 55099 Mainz, Germany\n(Dated: August 21, 2018)\nWe report a large exchange-bias (EB) effect after zero-field c ooling the new tetragonal Heusler\ncompound Mn 2PtGa from the paramagnetic state. The first-principle calcu lation and the magnetic\nmeasurements reveal that Mn 2PtGa orders ferrimagnetically with some ferromagnetic (FM ) inclu-\nsions. We show that ferrimagnetic (FI) ordering is essentia l to isothermally induce the exchange\nanisotropy needed for the zero-field cooled (ZFC) EB during t he virgin magnetization process. The\ncomplex magnetic behavior at low temperatures is character ized by the coexistence of a field in-\nduced irreversible magnetic behavior and a spin-glass-lik e phase. The field induced irreversibility\noriginates from an unusual first-order FI to antiferromagne tic transition, whereas, the spin-glass like\nstate forms due to the existence of anti-site disorder intri nsic to the material.\nPACS numbers: 75.60.Ej, 75.30.Gw, 72.15.Jf, 75.50.Cc\nThe class of Ni 2Mn1+xZ1−xbased Heusler alloys ex-\nhibit a structural transition from a high temperature\ncubic austenite phase to a low temperature tetrago-\nnal/orthorhombic martensitic phase, whereas, a mag-\nnetic ordering transition takes place in the cubic phase.\nThe existence of a first-order structural transition with\nstrong magneto-structural coupling leads to the observa-\ntion of various functional properties [1–4]. In these off-\nstoichiometric Mn rich alloys the extra Mn replaces the\nZatoms. The Mn-Mn exchange interaction is ferromag-\nnetic (FM) within the regular Mn sublattices while it is\nantiferromagnetic (AFM) between the Mn atoms occu-\npying the regular Mn sublattice and Zsublattice [5, 6].\nThe tetragonal/orthorhombic distortion in the marten-\nsitic phase enhances this AFM interaction due to the de-\ncrease in the Mn-Mn distance, which also results in a\nlarge degree of magnetic frustration. Furthermore, many\nHeusler alloysalso display anti-site disorder[7]. All these\nfactors contribute to a complex magnetic state in the low\ntemperature regime. The observations of spin-glass be-\nhavior [8–10] and the exchange-bias (EB) phenomenon\n[8, 11] are direct evidence of this complex magnetic state.\nA new class of Mn 2YZbased binary and ternary\nHeusler compounds stabilize in the cubic/tetragonal\ncrystal structure with high Curie temperatures ( TC).\nThese materials are attractive candidates for spin-torque\ntransfer devices [12–15]. The magnetic interaction in\nmost of these compounds is found to be ferrimag-\nnetic (FI) in nature [13, 16]. The magnetic circular\ndichroism in x-ray absorption (XMCD) measurements in\nMn3−xCoxGa compounds show direct evidence of the FI\nordering [17]. In a recent work it is found that Mn 2PtIn,\nwhich crystallizes in a tetragonal structure with FI or-\ndering, exhibits an inhomogeneous magnetic state at low\ntemperature and shows a weak conventional EB effect\n[18]. In this Letter, we report a large unconventional EBeffect obtained after zero-field cooling the Heusler com-\npound Mn 2PtGa from its paramagnetic state. To further\nelucidate the microscopic origin of the EB behavior we\ncarried out a theoretical investigation on the magnetic\nstructure. Based on our experimental and theoretical re-\nsults we propose a phenomenological model for the large\nzero-field cooled (ZFC) EB.\nPolycrystalline ingots of Mn 2PtGa were prepared by\narc melting stoichiometric amounts of the constituent el-\nements and subsequent annealingfor one week at 1273K.\nThe samples were structurally characterized by X-ray\npowder diffraction. The details of the structural anal-\nysis is given in the supplementary material [19]. The\nphysical quantities were investigated utilizing Quantum\nDesign measurement systems. The pulsed magnetic field\nexperiments were performed at the Dresden High Mag-\nnetic Field Laboratory. We have also performed a cal-\nculation to map out the magnetic structure using the\nPYA-LMTO program package [20].\nMn2PtGa undergoes a paramagnetic (PM) to FM(FI)\ntransition at TC= 230K as exemplified in the magneti-\nzation,M(T), and ac-susceptibility, χ(T), data in Fig.1a\nand b. With further decreasing temperature M(T) ex-\nhibits a sudden drop at 150K. To probe the nature of\nthis transition we have measured field cooled (FC) and\nfield heated (FH) M(T) curves that show a significant\nhysteresis. The presence of such a thermal hysteresis be-\ntween the FC and FH curves is a strong evidence for the\nfirst-order nature of a phase transition [21]. Therefore,\nwe argue that Mn 2PtGa undergoes a first-order FM(FI)\nto AFM transition at 150K. However, the existence of\na small irreversibility between ZFC and FC curves sug-\ngests that the low temperature phase is not perfectly\nAFM in nature. This irreversibility mainly originates\nfrom the presence of anti-site disorder that gives rise to\na magnetically inhomogeneous state. We note that the2\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/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56/s72/s61/s48/s46/s49/s32/s84\n/s84/s32/s91/s75/s93\n/s32/s32/s90/s70/s67\n/s32/s70/s67\n/s32/s70/s72/s40/s97/s41 /s40/s98/s41\n/s84/s32/s91/s75/s93/s57/s57/s57/s55/s32/s72/s122\n/s51/s51/s55/s32/s72/s122/s49/s48/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s34\n/s97/s99\n/s39\n/s97/s99\n/s97/s99/s32/s91/s97/s46/s117/s46/s93\n/s40/s99/s41\n/s84/s61/s49/s46/s57/s32/s75\n/s32/s90/s70/s67\n/s32/s70/s67\n/s32/s32\n/s40/s100/s41\n/s84/s61/s49/s46/s57/s32/s75/s77\n/s66/s47/s102/s46/s117/s46/s93\n/s72 /s32/s91/s84/s93/s72 /s32/s91/s84/s93\n/s77\n/s66/s47/s102/s46/s117/s46/s93/s77\n/s66/s47/s102/s46/s117/s46/s93\n/s32/s32\n/s32 /s32\nFIG. 1. (Color online) (a) M(T) at 0.1T measured in ZFC,\nFC, andFH cycles. Inthe ZFCmode, the sample was initially\ncooled to 2K in 0 Tand data were takenupon increasing tem-\nperature in applied field. In the FC mode, data were collected\nwhile cooling in field and subsequently in FH mode data were\ncollected during heating. (b) Real ( χ′) and imaginary ( χ′′)\npart of the ac-susceptibility measured with different frequ en-\ncies and an amplitude of 5 Oe. (c) ZFC and FC M(H) loops\nat 1.9 K performed as 0 →7T→ −7T→7T(d) ZFC M(H)\nloop at 1.9 K performed as 0 → −7T→7T→ −7T. The\ninsets show a magnified view around H= 0.\ndegree of anti-site disorder can be tuned by the anneal-\ning/quenching procedure. The imaginary part of the ac-\nsusceptibility, χ′′(T) (Fig.1b), shows a maximum around\n120K which exhibits a weak frequency dependence indi-\ncating the presence of spin- or cluster-glass like states.\nMagnetization loops, M(H), have been measured us-\ning a ZFC or FC protocol, as indicated in Fig.1c. The\nmost fascinating behavior in the ZFC M(H) data is the\nshift of the hysteresis loop in negative field direction by\nabout 0.17 T. To verify this effect we have measured the\nM(H) loop also in the opposite direction (Fig.1d). As\nexpected, this shifts by 0.17T to the positive field di-\nrection. Therefore, we conclude that the observed EB-\nlike behavior is intrinsic to Mn 2PtGa. Thus, it is possi-\nble to induce the exchange anisotropy by zero-field cool-\ning from the paramagnetic state. The direction of the\nanisotropy field depends on the initial direction of the\nexternal field. In general, the EB effect in conventional\nexchange-coupled systems appears only after field cool-\ning from a temperature above the N´ eel temperature of\nthe AFM material. Thus, when the virgin M(H) curve\nis measured in positive field direction Mn 2PtGa behaves\nas cooled in the presence of positive field and vice versa.\nTheM(H) loop taken at 1.9K after field cooling in 7T\nshows approximately the same shifting as the ZFC loop\n(see Fig.1c). Therefore, in case of the ZFC EB, scanning\nthe virgin magnetization process provides the roll of the\ncooling field required in the conventional EB systems.\nWe note that the ZFC virgin M(H) curve at 1.9K dis-\nplays a sharp magnetization change at 4.8T, indicating\na field-induced first-order metamagnetic transition from/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s72 /s32/s91/s84/s93/s77/s32/s91 /s47/s102/s46/s117/s93\n/s32/s32\n/s84/s61/s52/s46/s50/s32/s75/s84/s61/s50/s55/s53/s32/s75\n/s72 /s32/s91/s84/s93/s77/s32/s91\n/s66/s47/s102/s46/s117/s46/s93\n/s32/s84/s61/s52/s46/s50/s32/s75/s40/s97/s41 /s40/s98/s41\n/s72 /s32/s91/s84/s93/s32/s72\n/s69/s66/s40/s90/s70/s67/s41\n/s32/s72\n/s69/s66/s40/s70/s67/s41\n/s32/s72\n/s67/s40/s70/s67/s41\n/s84/s32/s91/s75/s93/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53\n/s32/s32/s72\n/s67/s40/s90/s70/s67/s41\nFIG. 2. (Color online) (a) M(H)isotherms at 4.2Kand275K\nin fields up to 60T. Inset: data at 4.2K in the low field range.\n(b)HEB(T) andHC(T) taken from the ZFC (close symbols)\nand FC (open symbols) M(H) isotherms.\nan AFM to a FI phase, which will be addressed later.\nM(H) measured in pulsed fields does not show any\nsaturation up to 60T (see Fig.2a). The sample shows\na magnetization of 0 .8µB/f.u.in 7 T at 1.9 K and\n1.6µB/f.u.in 60T at 4.2K. The small value of the mag-\nnetization and the still increasing M(H) indicates the\npresence of FI ordering in Mn 2PtGa. At 275K, which\nis above TC, we find paramagnetic behavior. Interest-\ningly, the M(H) curve at 4.2K displays a similar shift\nas that observed in the low field ZFC M(H) data. This\nevidences that a field of 60T is too small to affect the\nexchange anisotropy produced in the sample during the\ninitial magnetization process. The temperature depen-\ndence of the EB field ( HEB) and the coercive field ( HC)\ndeduced fromthe ZFCandFC hysteresisloopsareshown\nin Fig.2b. In FC mode the sample wascooledin 7T from\n300K to the desired temperature. HEBandHCare cal-\nculatedusing HEB=|H1+H2|/2andHC=|H1−H2|/2,\nwhereH1andH2are the lower and upper cut-off fields.\nAt 1.9K,HEBpossesses a maximum value of around\n0.17T in ZFC mode and 0.16 T in FC mode. HEB(T)\nandHC(T)exhibit a monotonicdecreaseupon increasing\ntemperature, which is for HC(T) interrupted by a local\nmaximum around 40K. The origin of this anomaly will\nbe discussed later.\nWe will now turn our focus to the details of the mag-\nnetic phase diagram of Mn 2PtGa. Figure 3 displays the\nZFCM(H) loops at various temperatures. The 10K\nM(H) loop shows a similar behavior as that at 1.9K\n(Fig.1c) with a reduction of the field required to induce\nthe metamagnetic transition. Interestingly, the metam-\nagnetic transition is only observed in the virgin curve for\nT≤15K. However, at 17.5K the loop shows a signa-\nture of the metamagnetic transition in the negative field\npath. Upon increasing temperature this signature be-\ncomes more pronounced for negative as well as positive\nfields. The presence of the virgin curve outside the en-\nvelope loop and the incomplete metamagnetic transition\nresults in an irreversible hysteresis loop for T <40K.3\n/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s32/s32\n/s84/s61/s49/s48/s32/s75/s40/s97/s41\n/s40/s98/s41\n/s84/s61/s49/s55/s46/s53/s32/s75\n/s32/s32\n/s49/s50\n/s51\n/s52/s53/s40/s99/s41\n/s84/s61/s50/s48/s32/s75\n/s40/s100/s41\n/s72 /s32/s91/s84/s93/s84/s61/s51/s48/s32/s75/s40/s101/s41\n/s84/s61/s52/s48/s32/s75/s40/s102/s41\n/s84/s61/s55/s48/s32/s75\n/s40/s103/s41\n/s84/s61/s49/s48/s48/s32/s75/s40/s104/s41\n/s84/s61/s49/s54/s48/s32/s75/s40/s105/s41/s77\n/s66/s47/s102/s46/s117/s46/s93\n/s84/s61/s50/s54/s48/s32/s75\n/s32/s32\nFIG. 3. (Color online) ZFC M(H) loops taken at different T.\nTheM(H) loops for T≥40K are reversible in both the\nquadrants, also the virgin curve coincides with the corre-\nsponding part of the hysteresis loop. The critical field of\nthe metamagnetic transition decreases significantly upon\nincreasingtemperature. Thefeatureofthemetamagnetic\ntransition completely vanishes at 160K, where we find\nsoft FM behavior and finally a paramagnetic M(H) loop\nat 260 K. However, the loops remain shifted in the nega-\ntive field direction for T≤160K. From the above obser-\nvations we can divide the whole temperature range into\ndifferent intervals: 0 < T≤15K, the AFM phase, which\nwas converted to FI, cannot be recovered by any number\nof field cycling. This results in a large field induced irre-\nversibility in the M(H) loops. In the temperature range\n15K< T <40K the AFM phase can be partially re-\nstored. At higher temperatures (40K < T <150K), the\nAFM phasethat wasinitiallyconvertedtothe FIphaseis\nrecovered fully to its initial state by cycling the magnetic\nfield. Here the hysteresis loops are fully reversible and\nthe AFM phasereturnsto its initial state by reducingthe\nfield to zero. The peak in HC(T) around 40K, shown in\nFig.2b, originates from the irreversible hysteresis loops\nin the range of 17 .5K≤T≤40K. This results in a nom-\ninal increase of HC(T) upon increasing temperature for\n17.5K≤T≤40K followed by a reduction for reversible\nloops at higher temperatures.\nThe disorder influenced first-order magnetic to mag-\nnetic transition in Mn 2PtGa is found for the first time\nin any Heusler compound. The observation of different\nmagnitudes of field-induced irreversibility is a direct con-\nsequence of the competition between the potential and\nthermal energy. Though the AFM phase possesses a\nlower energy at H= 0 forT≤15K, the field induced FI\nphase forms a pinning potential, where disorder provides\nthe pinning centers. When the field is reduced to zero\nthe thermal energy cannot overcome this potential and\nthe AFM phase is not restored resulting in the observed\nirreversibilities. At higher temperatures the thermal en-\nergy becomes large enough to allow for a partial recovery\nFIG. 4. (Color online) Total energies for various magnetic\nconfigurations in case of inverse (black) and regular (red li ne)\nHeusler structures. The energy of the inverse Heusler struc -\nture in the nonmagnetic state is taken as reference. The\nchemical composition of each configuration is schematicall y\ndepicted by the 4 ×4-atom diagrams drawn closer to the cor-\nresponding energy points. Light green, yellow and hollow\nspheres mark Pt, Ga and Mn atoms, respectively. Arrows\non Mn spheres indicate the directions of the local magnetic\nmoments.\nof the AFM phase by field cycling. Only above 40K the\nthermalenergybecomeslargerthanthepinningpotential\nand a reversible hysteresis loop is observed. Similar phe-\nnomenahavebeenreportedinsystemsundergoingafirst-\norder magnetic to magnetic transition [22, 23]. Although\nsome of the Heusler systems with martensitic transition\nshow a similar type of field induced irreversibility, the\nunderlying physics for such a behavior categorically dif-\nfers from the case of Mn 2PtGa. In the former materials\nthe field induced structural change plays a major role in\ninducing the irreversibilities, which become stronger on\napproaching the martensitic transition upon increasing\ntemperature [24, 25]. This is in contrast to Mn 2PtGa\nwhich does not show any structural transition.\nTo map out the exact magnetic configuration in\nMn2PtGa, we have performed calculations using the\nfully-relativistic linearized muffin-tin orbitals band-\nstructure method within the PYA-LMTO program pack-\nage[20]. The unit cell parametersweretakenfromexper-\niment. The exchange-correlation potential was treated\nusing the Vosko-Wilk-Nusair form of the local density\napproximation [26]. The comparison of the total en-\nergycalculatedforthedifferentconfigurations(seeFig.4)\nindicates that the nonmagnetic configurations are ener-\ngetically the most unfavorable (regular and inverse vari-\nants). The energy is substantially reduced by switching\non the magnetism on the Mn atoms. For a ferromagnetic\nconfiguration this results into a large magnetic moment:\n7.2µB/f.u. in case of the regular and 7.12 µB/f.u. for\nthe inverse Heusler structure. In this fully ferromagnetic4\nsetup both regular and inverse structures possess rather\nsimilar total energies of about -15 meV/atom compared\nwith the nonmagnetic inverse structure. For the regu-\nlar Heusler variant the ferromagnetic state is the most\nstable. Enforcing any type of antiparallel alignment of\nMn moments within the regular structure always leads\nto a locally nonmagnetic state. On the other hand, by\nvarying the magnetic configurations further within the\ninverse structure, the total energy still can be lowered.\nBy reverting the directions of the magnetic moments in\neach second pair of layers, we arrive in the fully com-\npensated antiferromagneticstate, with anenergylowered\nby about 4 meV/atom compared with the ferromagnetic\nstate. The lowest energy configuration is achieved when\nthe Mn moment is reverted within each layer (FI state).\nSince the positions of the Mn atoms in the adjacent lay-\ners within the inverted Heusler structure are nonequiv-\nalent by symmetry, the total magnetic moment is not\ncompletely compensated: a small total moment of about\n0.55µB/f.u., which is the result of 3.65 µB/f.u.[Mn(I)]\n- 3.1µB/f.u.[Mn(II)], remains. Our results emphasize\nthat the formation of the magnetic state is driven by\ntwo mechanisms – ferromagnetic exchange between Mn\natoms at long and antiferromagnetic at short distances.\nThe observation of a magnetization of 0 .8µB/f.u.\nat 7 T is slightly larger than the calculated value\n(0.55µB/f.u.). This can be explained by ferromagnetic\nclustersembedded inthestrongly-compensatedferrimag-\nnetichost. Theferromagneticorderinginside theclusters\nis due to their regular Heusler structure, i.e. the inter-\nchange between Mn and Pt atoms (the so-called anti-\nsite disorder, which often takes place in Heusler materi-\nals). Samples prepared under different annealing and/or\nquenching conditions exhibit a slightly changed magne-\ntization. This is most likely connected to an atomic\nrearrangement toward the more stable, inverse Heusler\nstructure. The latter phase forms an almost compen-\nsated host, which surrounds the ferromagnetic clusters.\nBased on the first-principle calculations and experi-\nmental results we propose a simple model for the ZFC\nEB in Mn 2PtGa. We assume an exchange interaction\nbetween two dissimilar magnetically anisotropic phases.\nThe irreversibility between ZFC and FC M(T) curves in\nlow fields ( ≤0.1T), the frequency dependence of χ′′(T)\nbelowTC, and the theoretical calculations indicate the\npresence of magnetically inhomogeneous states originat-\ning from local FM clusters in a FI background. Upon\napplying field the virgin magnetization process will try\nto align the moments in the FI and FM phases along the\nfield direction, which implies that the interface spins of\nthe FI and FM phases will align in the same direction\nto minimize the energy. This sets up the exchange in-\nteraction between the magnetically soft FM phase and\nthe magnetically hard FI phase. Further, this implies an\nincrease of the coercive field in a perfectly ordered mate-\nrial, because the ferrimagnetically ordered moments col-lectively change their direction [27]. However, one would\nexpect a rough interface with disorder and uncompen-\nsated moments in the FI phase and additional defects in\nthe bulk [27–29]. All of these factors collectively con-\ntribute to the ZFC EB behavior. The fact that HEB\nobtained from the FC M(H) loops measured after field\ncooling in 7T, where the sample is only in the FI phase\nbelowTC, and the HEBderived from the ZFC loops al-\nmost match rule out any significance of the first-order\nFI to AFM transition in the formation of the ZFC EB.\nThis is furthermore supported by the observation of a\nsmall EB at 160 K, which is above the FI to AFM tran-\nsition. The pulsed field experiments up to 60T find only\naverysmallchangein HEB. Thisshowsthatoncetheex-\nchange interaction is set up the field does not change the\nFI phase significantlyto observea change in HEB. These\nfindings distinguish Mn 2PtGa fromthe off-stoichiometric\nNi-Mn-In Heusler alloys where recently a ZFC EB has\nbeen reported [8]. Mn 2PtGa exhibits a too weak fre-\nquency dependence of the peak in χ′′(T) to indicate su-\nper spin-glass behavior resulting in super FM clusters\nduring the initial magnetization process as in Ni-Mn-In\n[8]. One common factor, however, is the existence of FI\nordering in both systems. Therefore, we argue that the\npresence of FI ordering with embedded FM clusters is an\nimportant ingredient for the appearance of a ZFC EB.\nInconclusion,wehavesynthesizedandstudiedthenew\nHeusler compound Mn 2PtGa. Though, Mn 2PtGa orders\nferrimagnetically at TC= 230K, it undergoes an un-\nusual first order FI to AFM phase transition below TC.\nWe demonstrated, to our knowledge for the first time,\nthe presence of a ZFC EB effect in a stoichiometric bulk\nHeusler compound. We further show that the appear-\nance of this ZFC EB effect is related to the presence of\nFI ordering with embedded FM clusters. We ruled out\nany role of the first-order transition in the observation of\nthe ZFC EB.\nWe thank J.A.Mydosh for valuable discussions on the\npresentwork. Thisworkwasfinanciallysupportedbythe\nDeutsche Forschungsgemeinschaft DFG (Project No.TP\n2.3-A of Research Unit FOR 1464 ASPIMATT) and\nby the ERC Advanced Grant (291472) ”Idea Heusler”.\nThe experiments at the High Magnetic Field Laboratory\nDresden (HLD) were sponsored by Euro-MagNET II un-\nder the European Union contract 228043.\n∗nayak@cpfs.mpg.de\n†nicklas@cpfs.mpg.de\n[1] T. Krenke, E. Duman, M. Acet, E. F. Wassermann, X.\nMoya, L. Ma˜ nosa, and A. Planes, Nature Mater. 4, 450\n(2005).\n[2] R. Kainuma, Y. Imano, W. Ito, Y. Sutou, H. Morito,\nS. Okamoto, O. Kitakami, K. Oikawa, A. Fujita, T.\nKanomata, and K. Ishida, Nature 439, 957 (2006).5\n[3] J. Liu, T. Gottschall, K. P. Skokov, J. D. Moore, and O.\nGutfleisch, Nature Mater. 11, 620 (2012).\n[4] M. Pasquale, C. P. Sasso, L. H. Lewis, L. Giudici, T. Lo-\ngrasso, and D. 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Mater. 234, 584 (2001)."    },    {        "title": "1109.1709v1.Mixed_Ising_ferrimagnets_with_next_nearest_neighbour_couplings_on_square_lattices.pdf",        "content": "arXiv:1109.1709v1  [cond-mat.mtrl-sci]  8 Sep 2011Mixed Ising ferrimagnets with next-nearest\nneighbour couplings on square lattices\nW Selke1and C. Ekiz2\n1Institut f¨ ur Theoretische Physik, RWTH Aachen, 52056 Aachen, Germany\n2Department of Physics, Adnan Menderes University, 09010 Aydin, Turkey\nAbstract. We study Ising ferrimagnets on square lattices with antiferromagn etic\nexchange couplings between spins of values S=1/2 and S=1 on neighb ouring sites,\ncouplings between S=1 spins at next–nearest neighbour sites of th e lattice, and\na single–site anisotropy term for the S=1 spins. Using mainly ground s tate\nconsiderationsandextensiveMonteCarlosimulations, weinvestigat evariousaspectsof\nthephasediagram,includingcompensationpoints, criticalpropert ies,andtemperature\ndependent anomalies. In contrast to previous belief, the next–ne arest neighbour\ncouplings, when being of antiferromagnetic type, may lead to compe nsation points.\nPACS numbers: 75.10.-b, 75.10.Hk, 75.40.Mg, 75.50.Gg\nSubmitted to: Institute of Physics Publishing\nJ. Phys.: Condens. Matter\n1. Introduction\nMixed spin ferrimagnetic Ising models have been studied for some time , with a renewed\ninterest, especially, in connection with ’compensation points’, see, f or instance, [1-\n11]. These points occur at temperatures, below the critical tempe rature, at which the\nsublattice magnetizations cancel exactly, giving zero total momen t. As the temperature\nis tuned through such a point the total magnetization changes sign , which may be used\nin technological applications, most notably in magnetic recording.\nOne of the simplest such models consists of classical Ising spins S=1/ 2 and S=1\non a square or simple cubic lattice with the spins of different type being located on\nneighbouring sites of the lattice. Its Hamiltonian may be written in the form\nH=J1/summationdisplay\n/angbracketlefti,j/angbracketrightσiSj+D/summationdisplay\nj∈BSj2(1)\nwhereJ1>0 denotes the antiferromagnetic coupling between spins σi=±1/2 on\nthe sites of sublattice ’A’, and neighbouring spins Sj= 1,0,−1 on sites forming the\nsublattice ’B’. Then, spins on each sublattice tend to order ferroma gnetically, withMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 2\nopposite sign for the two types of spins. Dis the strength of a single–site anisotropy\n(or crystal–field) term acting only on the S=1 spins of sublattice B.\nOne may also choose σi=±1 rather than ±1/2. The change has to be taken\ninto account when calculating sublattice magnetizations and defining the compensation\npoint. Otherwise, themodifiedconvention simplyamountstoaresca lingoftheexchange\ncoupling [5, 10].\nThe mixed spin Ising model, eq. (1), is known, see, e.g., [3, 5, 10], to le ad to\ncompensation points for simple cubic lattices, in accordance with mea n–field theory [2],\nbut not for square lattices, in contrast to mean–field theory. As h as been observed\nalready some years ago by Buendia and Novotny [5, 12], compensat ion points may\noccur on square lattices, when adding a (ferromagnetic) coupling b etween next–nearest\nneighbouring (nnn) spins on the A sublattice. On the other hand, th e authors did not\nfind any evidence for compensation points, when considering nnn co uplings for B spins,\ninstead of the ones for A spins.\nIn the following article we shall challenge the latter suggestion, which seems to have\nbeen taken for granted by others, see, e.g., [7]. Indeed, based on extensive Monte Carlo\n(MC) simulations, we shall present clear evidence for compensation points due to nnn\nantiferromagnetic interactions between B, or S=1, spins on the sq uare lattice. In fact,\nthe effect is not easy to identify.\nThe outline ofthe article is as follows. Insection 2, to set the scene f or the following\nparts, we define the model with nnn couplings between S=1 spins, dis cuss its ground\nstates, and describe details of the simulations. The following section deals with the\ncompensation points. In section 4, MC results on critical propertie s and temperature\ndependent anomalies of the model will be presented. A brief summar y is given in the\nfinal section.\n2. Model, ground states, and simulations\nWe shall study the following mixed spin Ising model on a square lattice\nH=J1/summationdisplay\n/angbracketlefti,j/angbracketrightσiSj+D/summationdisplay\nj∈BSj2−J2/summationdisplay\n/angbracketlefti,j/angbracketright∈BSiSj (2)\nwith antiferromagnetic nearest neighbour couplings, J1>0 between A spins, σi=±1,\nand B spins, Sj= 0,±1, a crystal–field term of strength Dacting on the B spins, and\nnnn couplings, J2, between B spins. Note that we set A spins equal to ±1, in accordance\nwithprevious work [5, 10]. To identify possible compensation points, c are isthenneeded\nin defining the magnetization of the sublattice A, see above and below .\nTo determine the ground states of the model in the ( D/J1,J2/J1) plane, we first\nselect, like before [5, 12, 13], the structures which are stable amon g the ones described\nby 2×2 cells of spins on the square lattice. One readily observes the possib ility of\ndegenerate ground states, eventually with indefinitely large unit ce lls. Finally, we check\nthe tentative ground state structures by monitoring spin configu rations at very lowMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 3\ntemperatures obtained from careful cooling MC runs for lattices o f various sizes. The\nresulting ground state phase diagram is depicted in figure 1.\nThephasediagramatvanishingtemperature, T= 0,comprisesfourstructures. The\nantiferromagnetic(AF) structure isstableforferromagneticor weakly antiferromagnetic\nnnn couplings, J2, and negative or sufficiently small positive values of the crystal field ,\nD. The spins on each of the two sublattices order ferromagnetically, ±1, with opposite\nsign on the sublattices A and B.\nIn two of the remaining other three ground states, S=1 spins may b e in the state 0,\ndue to the single–site anisotropy term, D. Actually, for sufficiently large values of D, all\nspins of the sublattice B are in state 0. We abbreviate the resulting s tructure as ’0II’.\nThe structure is highly degenerate, with each A spin being either −1 or 1, leading to a\n2NA–fold degeneracy for NAsites on the sublattice A. In contrast, in the ’0I’ structure,\nsee figure 1, only half of the S=1 spins are in the state 0. They form, on the sublattice\nB, in the standard Wood’s notation for overlayers [14], a periodic c( 2×2) superlattice.\nThe other half of the spins on the B sites are ferromagnetically orde red,±1. The spins\non the sublattice A are ferromagnetically ordered as well, with oppos ite sign, compared\nto that of the B spins.\n-2-1012 3 4 56\nD/J1-3-2.5-2-1.5-1-0.500.51J2/J1AF\n0I 0II\nB-AF\nFigure 1. Ground state phase diagram of the mixed spin model, eq. (2), on a sq uare\nlattice, with solid lines separating the different phases. The dashed lin e in the 0I phase\nrefers to equation (7), see text.\nIn the fourth ground state structure, all spins on the sublattice B are\nantiferromagnetically aligned, due to the dominant antiferromagne tic nnn coupling, J 2.\nThe structure may beabbreviated asB–AF. Eachspin onthesublat tice Amay beeither\n−1 or 1, leading, as for 0II, to a large degeneracy.\nNote that there may be additional degeneracies at borderlines bet ween different\nground state structures [10].\nTo determine thermal properties of the model, (2), we mainly perfo rm MC\nsimulations, using the Metropolisalgorithmwithsingle–spin flips, provid ing, indeed, theMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 4\nrequired accuracy, so thatthereisno needtoapplyother techniq ues like cluster–updates\nor the Wang–Landau approach [15]. We consider lattices with LxLsites, employing full\nperiodic boundary conditions. Lranges from 4 to 120, to study finite–size effects.\nTypically, we do runs of 5 ×106to 107Monte Carlo steps per spin, where averages and\nerror bars may be obtained from evaluating a few of such runs, usin g different random\nnumbers. These quite long runs lead to reliable data, as before [10]. T he estimated\nerrors are usually smaller than the sizes of the symbols in the figures , and they are\nshown there only in a few cases.\nWe record the energy per site, E, the specific heat, C, both from the energy\nfluctuationsandfromdifferentiating Ewithrespecttothetemperature, andtheabsolute\nvalues of the sublattice magnetizations of the two sublattices\n|mA|=<|/summationdisplay\nAσi|> /(2(L2/2)) (3)\nand\n|mB|=<|/summationdisplay\nBSj|> /(L2/2) (4)\nas well as the absolute value of the staggered magnetization of sub lattice B, describing\nthe ordering for the B–AF structure,\n|mst\nB|=<|/summationdisplay\nB+Si−/summationdisplay\nB−Sj|> /(L2/2) (5)\nwith B+,−denoting an obvious bipartition of sublattice B. The brackets <>denote\nthe thermal average. Note the factor of 1/2 in the definition of |mA|, taking into\naccount the correct length of the S=1/2 spins, so that |mA(T= 0)|= 1/2 for the\nferromagnetic ground state of sublattice A. In addition, the corr esponding sublattice\n(staggered) susceptibilities, χA,χB, andχst\nB, have been computed from the fluctuations\nofthe(staggered)sublattice magnetizations. Wealsoanalyse the fourth–ordercumulant\nof various (sublattice) order parameters, the Binder cumulant [16 ], defined by\nU= 1−< m4> /(3< m2>2) (6)\nwith< m2>and< m4>being the second and fourth moment of (staggered)\nmagnetizationsofsublatticeAorB.Finally, wemonitortypicalequilibr iumMonteCarlo\nconfigurations, illustrating the microscopic behaviour of the syste m and providing, e.g.,\nevidence for the ground states structures, as mentioned above .\nTo test the accuracy of the simulations, we compared our MC data t o those of\nprevious accurate numerical work and to exact results by enumer ating all possible\nconfigurations for small lattices with L= 4 [5, 10].Mixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 5\n3. Compensation points\nCompensationpointsoccurattemperatures Tcomp,wherebothsublatticemagnetizations\ncancel each other, with a vanishing total magnetization. In finite s ystems, as one is\nstudying in MC simulations, a convenient and efficient way [5] to locate such points is to\nuse the crossing condition |mA|(Tcomp) =|mB|(Tcomp). At low temperatures sufficiently\nfar below the phase transition, finite–size effects are expected to be very weak, because\nthere the compensation effect is not related to critical phenomena .\n0 0.2 0.4 0.60.81 1.2 1.4 1.6\nkBT/J10.10.20.30.40.50.60.70.80.91|mA|, |mB|\nFigure 2. Sublattice magnetizations |mA|(open symbols, dotted lines) and |mB|(full\nsymbols,solidlines)atfixed D/J= 3.0andvarying J2/J1=0.25(circles), 0.0(triangles\ndown),−0.1 (squares), −0.24 (diamonds), and −0.26 (triangles left). Lattices of size\n602are simulated.\nIn the AF ground state, one has |mB|(T= 0)=1, while |mA|(T= 0)= 1/2. Now,\nat non–zero temperatures, |mB|(T) may fall off quite rapidly due to antiferromagnetic\ncouplings J2and due to a relatively large single–site anisotropy term, D, which favours\nflips of S=1 spins to state 0. Furthermore, at zero temperature, |mB|(T= 0) drops from\n1 to 1/2, when passing through the borderline between the AF and 0 I ground states.\nAccordingly, one might speculate that compensation points, at low t emperatures in the\nAF phase, may show up close to the AF–0I borderline at T=0. In fact, our preliminary\nmean–field calculations seem to suggest that lines of compensation p oints may spring\nfrom this AF–0I borderline.\nHowever, in our MC simulations, investigating carefully several case s, we find no\nevidence forsuch compensationpointsintheAFphase. Anexample is depicted infigure\n2. There, D/J1is fixed at 3.0, and J2/J1is varied, from +0.25 to −0.26, so that the\nAF–0I border at J2/J1=−0.25 is approached and crossed. One observes, that |mB|is\nalways larger than |mA|, albeit the difference between the two sublattice magnetizations\nmay get quite small when decreasing J2. MC data for lattices of fixed size, L= 60, areMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 6\ndisplayed. The observation on the absence of compensation points in the AF phase also\nholds for other lattice sizes as well as for other values of J2/J1andD/J1.\n0.10.2 0.3 0.4 0.50.60.7 0.8 0.9 1\nkBT/J10.10.20.30.40.50.60.70.80.91|mA|, |mB|\nTcomp\nFigure 3. Sublattice magnetizations |mA|(open symbols, dotted lines) and |mB|\n(full symbols, solid lines) at fixed J2/J1=−0.2, and varying D/J1= 3.0 (circles), 3.4\n(squares), 3.57 (diamonds), and 3.7 (triangles up), simulating lattic es of size L= 60.\nAtD/J1=3.57, a compensation point shows up, Tcomp.\nOn the other hand, compensation points will be argued below to occu r in the 0I\nphase, for J2/J1>−1.0, arising at vanishing temperature from the line\nJ2/J1=−2+D/(2J1) (7)\nwhich is depicted as the dashed line in the ground state phase diagram , figure 1.\nLet us first present numerical support for this claim, followed then by low\ntemperature energy considerations backing it up. A numerical exa mple is shown in\nfigure 3, where, for lattices with 602sites, the temperature dependences of the sublattice\nmagnetizations, |mA|and|mB|, are shown at fixed nnn coupling, J2/J1=−0.2, and\nvarying the strength of the single-site anisotropy term, D/J1, from 3.0 to 3.7. Note\nthat for this ratio of couplings, at T= 0, the AF–0I border is at 3.2, and the origin,\nat vanishing temperature, of the line of compensation points would b e, according to\nequation (7), at D/J= 3.6. As discussed in the context of figure 2, one observes\nthat|mB|(T) is always larger than |mA|(T) when being in the AF phase. However,\nin the 0I phase, here, at D/J1= 3.57, there seems to be a compensation point, with\nkBTcomp/J1≈0.26, clearly below the phase transition: At lower temperatures, |mB|is\nstill larger than |mA|, with reverse ordering at T > T comp. By further increase of D,\nD/J1= 3.7,|mA|(T) is larger than |mB|(T), at all temperatures up to Tc.\nTo establish numerically the presence of compensation points, care ful finite–size\nanalyses are required, as illustrated in figure 4. Here, MC data for v arious lattices sizes,\nwithLranging from 20 to 120, aredisplayed, at J2/J1=−0.2 andD/J1= 3.59, i.e. veryMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 7\n0.05 0.1 0.15 0.2 0.25 0.3\nkBT/J10.360.380.40.420.440.460.480.5|mA|, |mB|\nTcomp\nFigure 4. Sublattice magnetizations |mA|(open symbols, dotted lines) and |mB|\n(full symbols, solid lines) at J2/J1=−0.2 andD/J1= 3.59, with lattice sizes L= 20\n(squares), 40 (diamonds), 60 (triangles left), 80 (triangles down ), and 120 (circles).\nThe compensation point is located at kBT/J1≈0.16.\nclose to the point, from which, according to (7), the line of compens ation points may\narise. In fact, a compensation point may be located at kBTcomp/J1≈0.16. Below that\ntemperature, |mB|(T)supercedes |mA|(T), withfinite–size dependences being extremely\nsmall. Then, at Tcomp, a crossing of the two sublattice magnetizations occurs. The finite\nsize effects are still very weak up to kBT/J1≈0.25, providing clear evidence on the\nexistence of the compensation point in the thermodynamic limit.\nIndeed, similarfinite-sizeanalysesallowonetolocatetheratherste eplyrisinglineof\ncompensationpoints, atfixednnncouplings J2/J1=−0.2. Results onthecompensation\npoints,Tcomp, are summarized in figure 5, together with estimates for the trans ition line,\nTc, to the paramagnetic phase.\nThe critical line, Tc, has been determined by monitoring the size dependent\npositions of maxima in the specific heat C, in the susceptibility of the sublattice A,\nand in the intersection points of the Binder cumulant, especially, of s ublattice A. The\nintersections of the Binder cumulant U(L,T) for successive lattice sizes [16] turn out\nto have the smallest finite–size effects, thereby being most efficient in estimating the\ncritical temperature.\nThe line of compensation points seems to start at zero temperatur e atD/J1= 3.6\nforJ2/J1=−0.2, in agreement with equation (7). Furthermore, it extends only ov er\nquite a small region, with the compensation point coinciding with the cr itical point at\naboutD/J1= 3.52±0.02.\nLet us now turn to the low temperature considerations leading to (7 ). Specifically,\nwe consider, for the 0I structure, the one–spin flip excitations on sublattice B. For\nconcreteness, we analyse ground states with spins being in state −1 or 0 on sublatticeMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 8\n33.13.2 3.3 3.4 3.53.63.7 3.8 3.9 4\nD/J100.10.20.30.40.50.60.70.80.911.1kBT/J1\nAF 0ITc\nTcompparamagnetic\nFigure 5. Phase boundary to the paramagnetic phase, Tc(solid line), and\ncompensation points, Tcomp(dashed line), at J2/J1=−0.2, varying D/J1.\n22.12.2 2.3 2.4 2.52.62.7\nD/J100.10.20.30.40.50.60.70.80.9kBT/J1\n0Iparamagnetic\nTcompTc\nFigure 6. Phase boundary to the paramagnetic phase, Tc(solid line), and\ncompensation points, Tcomp(dashed line), at J2/J1=−0.8, varying D/J1.\nB, and in state 1 on sublattice A. Then there are four possible flips of S=1 spins,\nnamely flipping a spin from its state 0 to the state (i) −1 or (ii) 1, or flipping a spin\nfrom its state −1 to state (iii) 0 or (iv) +1. Obviously, only flip (i) will increase the\nsublattice magnetization |mB|above its value at zero temperature, |mB|(T= 0) = 1/2,\nwhile otherwise |mB|(T) will have a negative slope at low temperatures. Now, one\nmay readily calculate the energies of all four elementary flips. One ob tains that flip\n(i) costs the lowest energy either followed by flip (iii), when D/J1<2J2/J1+ 4,\nor followed by flip (ii), when J2/J1>−1.0. Equation (7) is then obtained underMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 9\nthe assumption, that an initial increase of |mB(T)|is needed to have a compensation\npoint, with |mB|>|mA|at sufficiently low temperatures. Otherwise, in the 0I phase,\n|mA|>|mB|at all temperatures 0 < T < T c, excluding a compensation point.\nWe further checked numerically the hypothesis, (7), in determining lines of\ncompensation points by fixing the nnn coupling J2/J1not only at −0.2, but also at\n−0.4 and−0.8, varying the strength of the single–site anisotropy term, D/J1. Indeed,\nwe observe compensation points in the 0I structure, which seem to arise, at zero\ntemperature, from D/J1= 3.2 at J2/J1=−0.4 and from D/J1= 2.4 at J2/J1=−0.8,\nin agreement with (7). The latter case is displayed in figure 6.\nNote that the range of values of D/J1, in which there is a line of compensation\npoints, shrinks appreciably as one decreases J2/J1. This may be seen by comparing\nfigures 5 and 6. Thence, it becomes increasingly difficult to locate com pensation points\nfor lower values of J2/J1.\n4. Phase transitions and anomalies\nWhile mixed spin Ising models are of much interest because of the pote ntial presence of\ncompensations points, they may exhibit other intriguing thermal pr operties as well.\nWe shall focus on two aspects, the characterization of the phase transitions\nassociatedwiththevariousgroundstates, andanomaliesinthetem peraturedependence,\nespecially, of the sublattice magnetization and the specifc heat C.\nOne expects that there is no phase transition associated with the h ighly degenerate\n0II ground state for the following reasons: At zero temperature , all spins of sublattice\nB are in state 0, while each spin of sublattice A may be either −1 or 1. The single–site\nanisotropy term, favouring the spin state 0 on sublattice B, does n ot support long–range\norder, in close analogy to the situation in the Blume–Capel model [17]. The spins on\nsublattice A act merely like a fluctuating random field, and may not lead to a phase\ntransition neither. In fact, our MC data, for various quantities, lik e the specific heat C\nor the probability to encounter a spin in state 0, give no indication for singular thermal\nbehaviour.\nIn case of the other three ground states, AF, 0I, and B–AF, we d etermine the\nuniversality class [18, 19] by analysing specific heat and susceptibilitie s. In particular,\nwemonitorthesize, L,dependenceofthatmaximumofthespecificheat, Cmax(L), which\ngoesover into a singularity inthe thermodynamic limit, L→ ∞(note that there may be\nother maxima as will be discussed below in the context of anomalies). F urthermore, we\nstudy the size dependence of the maxima in (sublattice) susceptibilit ies. For the AF and\n0I structures, we focus on χA,max(L). For the B–AF structure, we record the staggered\nsusceptibility of the antiferromagnetically ordered sublattice B, χst\nB,max(L). In all three\ncases, the critical behaviour is found to be consistent with having p hase transitions in\nthe universality class of the standard two–dimensional Ising model.\nTypical MC results on the size, L, dependence of maxima in the specific heat,\nCmax, are depicted in figure 7. For all three types of ground states, AF , 0I as wellMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 10\n2.2 2.4 2.62.8 3 3.2 3.4 3.63.8 4 4.2 4.4\nln(L)0.80.911.11.21.31.41.51.61.71.81.92Cmax\nFigure 7. Height of the critical maximum of the specific heat, Cmaxversus logarithm\nof the lattice size, ln L, forL= 10, 20, 40, 60, and 80 at (a) (circles) J2/J1=−0.3,\nD/J1=1.5,(b)(squares) J2/J1=−1.0,D/J1=1.5,and(c)(diamonds) J2/J1=−1.5,\nD/J1= 2.5. The three cases refer to the (a) AF, (b) 0I, and (c) B–AF st ructures.\nas B–AF, one approaches to a good degree, already for lattices of moderate sizes, the\nformCmax∝lnL, being characteristic for the two–dimensional Ising universality cla ss.\nNote that the prefactor in front of the logarithmic term depends s trongly on J2/J1and\nD/J1. Certainly, such Ising–like critical behaviour may be expected in the AF case. In\ncase of the B-AF phase, the spins on sublattice B order antiferrom agnetically, with the\nA spins providing effectively a randomly fluctuating field, which may be a rgued to be\nirrelevant for the universality class. Finally, in the 0I case, the Ising –like criticality may\nbe understood by the ferromagnetic order of the spins on sublatt ice A.\nThe analysis of the susceptibilies confirms the findings on the specific heat. Typical\nexamples, for the same model parameters as in figure 7, are shown in figure 8. The size\ndependent maxima in the susceptibilities are observed to follow the fo rmχmax∝L7/4,\nwith rather small corrections, for all three cases, AF, 0I, and B– AF. This behaviour\nmay be quantified by calculating the slope between successive points in the doubly\nlogarithmic plot of the MC data. The resulting effective local critical e xponent of the\nsusceptibility is near 7/4, even for moderate lattice sizes. Thence, in all three cases,\ncriticality seems to belong to the two–dimensional Ising universality c lass.\nCare is needed when attempting to determine the universality class f rom the value\nof the critical Binder cumulant U∗=U(Tc,L−→ ∞), because that value is known to\ndepend on boundary conditions, shape, and anisotropy of the cor relations [16, 20, 21].\nNotethatinthecases westudied, itsvalueseems tobeclose tothat ofthestandardtwo–\ndimensional Ising model with periodic boundary conditions for lattice s of square shape,\nU∗≈0.6107 [22]. However, the dependence, especially on anisotropy, may be very weak\n[20, 21], and a reliable analysis may require an exact or, at least, extr emely accurateMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 11\n2.2 2.4 2.62.8 3 3.2 3.4 3.63.8 4 4.2 4.4\nln(L)0123456ln(χA(B),max(st))\nFigure 8. Log–log plot for maxima of (staggered) sublattice susceptibilities χmax\nversus lattice size Lfor the AF, 0I, and B–AF cases, with the same parameters and\nnotation as in figure 7. In case of AF and 0I structures, χA,max, in case of the B–AF\nstructure, χst\nB,maxis recorded. For comparison, the solid line shows χmax∝L7/4.\ndetermination of the critical point. We therefore refrained from s uch an analysis here.\n0 0.2 0.4 0.60.81 1.2 1.4 1.6 1.8 2 2.2\nkBT/J100.10.20.30.40.5 C\nFigure 9. Specific heat Cas a function of temperature, kBT/J1, atJ2/J1=−0.2 and\nD/J= 3.4, simulating lattices of sizes L= 20 (circles), 40 (squares), 60 (diamonds),\nand 80 (triangles left).\nLet us now turn to discussing anomalies, exhibiting intriguing non–mon otonic\ntemperature dependences, which are not related to phase trans itions. Especially, we\nmonitor the magnetizations of sublattice B, |mB|, the specific heat C, and the thermallyMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 12\naveraged occupancy of B sites with spins in the state 0, pB(0).\nAn example for such anomalies is the overshooting of |mB|(T) at low temperatures\nin the vicinity of the 0I ground state, illustrated in figures 2 and 3. As discussed above,\nthe overshooting is related to decreasing the number of spins in sta te 0, by flipping such\nspins to, say, −1, costing minimal energy. Indeed, the anomaly in |mB|is monitored to\nbe accompanied by a pronounced decrease in pB(0).\nAnomalies may also occur in the specific heat, being caused either by a rapid\nincrease or decrease in the number of S=1 spins in state 0. For insta nce, such an\nanomaly in Chas been observed before [10] in the AF phase at J2/J1= 0, close to the\n0II structure. There, monitoring C(T), a three–peak–structure of the specific heat has\nbeen found, with a non-critical maximum at low temperatures, due t o easy flips of spins\nin the state 0, followed, at higher temperature, by a strongly size d ependent critical\npeak, and, at even larger temperature, another non–critical ma ximum, due to flipping\nof single spins for sublattices with rather large clusters of ’+’ or ’ −’ spins.\nWe observe a similar thermal behaviour of the specific heat in the 0I p hase, when\nturningonthennncoupling, J2<0. Anexample isdepicted infigure9, at J2/J1=−0.2\nandD/J1= 3.4. However, here the peak at lowest temperature, Tl, is due to a rather\ndrastic decrease of B spins in state 0, which has been discussed abo ve. Actually, by\nenhancing the strength of the single–site anisotropy term, D/J1= 3.6, the position of\nthat peak, at Tl, shifts first to somewhat higher temperatures. It still correspo nds to a\nlowering, with temperature, of the average number of 0’s for the s ublattice B, pB(0)(T).\nWhen moving even closer to the 0IIstructure, D/J1= 3.8 and 3.9, Tltends to be shifted\ntowards lower temperatures with increasing D. The peak position seems to follow the\nsame dependence as for vanishing J2[10], namely kBTl/J1∝4−D/J1. As may be seen\nfrom monitoring pB(0), the peak is then, as in the limit J2= 0, due to an increase in\nthe number of B spins in state 0 at low temperatures.\n5. Summary\nWehave studied a mixed spin Ising model with antiferromagneticcoup lings,J1, between\nspinsS=1/2andS=1onneighbouring sitesofasquarelattice, augme ntedbycouplings,\nJ2, between spins S= 1 on next–nearest neighbouring sites of the latt ice. An additional\nquadratic single–site anisotropy term, D, acts upon the S=1 spins. Based mainly on\ngroundstateconsiderationsandonextensive standardMonteCa rlosimulations, wehave\ndetermined the ground state phase diagram in the ( D/J1,J2/J1) plane and identified\ncompensation points, types of phase transitions corresponding t o different ground state\nstructures as well as anomalies for various physical quantities.\nIn particular, compensation points are found to exist, in contrast to previous belief.\nThey exist for antiferromagnetic nnn couplings, J2>−1.0, in the 0Iphase, springing, at\nzero temperature, from the line J2/J1=−2+D/(2J1). Across that line, different types\nof one–spin flips on the S=1 sublattice cost lowest energy, accompa nied by a change in\nthesignoftheslopeofthemagnetizationofthatsublatticeatlowte mperatures. AtfixedMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 13\nvalue ofJ2/J1, the range, in D/J1, of compensation points is quite narrow, becoming\nextremely small for decreasing nnn couplings, as we have shown by d ecreasing J2/J1\nfrom−0.2 to−0.8.\nThe transition to the paramagnetic phase appears to be always in th e universality\nclass of the two–dimensional Ising model, with the critical maxima of t he specific\nheat growing with lattice size, L, in a logarithmic fashion, and with the maxima\nof the appropriate (staggered) sublattice susceptibilities growing with lattice size\nproportionally to L7/4. These results hold in case of AF, 0I, and B–AF ground states of\nthe model. In case of the 0II ground state, there is no phase tran sition.\nThe model is observed to exhibit interesting anomalies by showing non –monotonic\ntemperature dependences for various quantities, including sublat tice magnetization and\nspecific heat. The anomalies are typically caused by energetically fav oured flips of S=\n1 spins between the state 0 and a non–zero state.\nIn conclusion, our study provides clear evidence for non–expecte d compensation\npoints in a rather simple mixed Ising model with antiferromagnetic nex t–nearest\nneighbour couplings between S=1 spins on a square lattice. Extensio ns to three\ndimensional lattices are well beyond the present study, and may be investigated in\nthe future.\n6. Acknowledgements\nC E thanks the Department of Physics at the RWTH Aachen for the k ind hospitality\nduring his stay there. We thank Prof. Mark Novotny for a very use ful discusssion as\nwell as Dr. Mukul Laad for interesting conversations.\nReferences\n[1] N´ eel L 1948 Ann. Phys. (Paris) 3137\n[2] Kaneyoshi T and Chen J C 1991 J. Magn. Magn. Mater. 98201\n[3] Zhang G-M and Yang C-Z 1993 Phys. Rev. B489452\n[4] Bob´ ak A and Jascur M 1995 Phys. Rev. B5111533\n[5] Buendia G M and Novotny M 1997 J. Phys.: Condens. Matter 95951\n[6] Oitmaa J and Zheng W-H 2003 PhysicaA328185\n[7] Godoy M, Leite V S, and Figueiredo W 2004 Phys. Rev. B69054428\n[8] Jascur M and Strecka J 2005 Condens. Matter Phys. 8869\n[9] Oitmaa J and Enting I G 2006 J. Phys.: Condens. Matter 1810931\n[10] Selke W and Oitmaa J 2010 J. Phys.: Condens. Matter 22076004\n[11] Zukovic M and Bob´ ak A 2010 PhysicaA3895402\n[12] Buendia G M and Novotny M A 1997 J. Phys. IV (C1) France 7175\n[13] Buendia G M and Liendo J A 1997 J. Phys.: Condens. Matter 95439\n[14] Oura K, Lifshits V G, Saranin A A, Zotov A V and Katayama M 2003 Surface Science: An\nIntroduction (Berlin: Springer)\n[15] Landau D P and Binder K 2005 A Guide to Monte Carlo Simulations in Statistical Physics\n(Cambridge: Cambridge University Press)Mixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 14\n[16] Binder K 1981 Z. Physik B43119\n[17] Blume M 1966 Phys. Rev. 141517; Capel H W 1966 Physica (Utr.) 32966\n[18] Fisher M E 1974 Rev. Mod. Phys. 46597\n[19] Pelissetto A and Vicari E 2002 Phys. Rep. 368549\n[20] Chen X S and Dohm V 2004 Phys. Rev. E70056136; Dohm V 2008 Phys. Rev. E77061128\n[21] Selke W and Shchur L N 2005 J. Phys. A: Math. Gen. 38L739; Selke W 2006 Eur. Phys. J. B\n51223\n[22] Kamieniarz G and Bl¨ ote H W J 1993 J. Phys. A: Math. Gen 26201"    },    {        "title": "1002.4544v1.Magnetic_Structure_of_CaBaCo4O7__Lifting_of_Geometrical_Frustration_towards_Ferrimagnetism.pdf",        "content": "1/19  Magnetic Structure of CaBaCo 4O7: \n \nLifting of Geometrical Frustr ation towards Ferrimagnetism \n \n \nV. Caignaert1,*, V. Pralong1, V. Hardy1, C. Ritter2 and B. Raveau1 \n \n(1)CRISMAT, UMR 6508, CNRS-ENSICAEN \n 6 Bd Marechal Juin, 14050 Caen, France \n(2) Institut Laue Langevin,  6 rue Jules Horowitz , 38042 Grenoble, France \n  \n  \nAbstract  \n CaBaCo\n4O7 represents a new class of ferrimagnets whose structure is built up of CoO 4 \ntetrahedra only, similarly to other members LnBaCo 4O7 of the “114” series,  forming an alternate \nstacking of kagomé and triangular layers. Neutr on powder diffraction reveals, that this compound \nexhibits the largest distortion within the “114” series, characterized by a strong buckling of the \nkagomé layers. Differently from all other members it shows charge ordering, with Co2+ sitting on \ntwo sites (Co2, Co3) and “m ixed valent” cobalt “Co3+/Co2+L” sitting on two other sites (Co1, \nCo4). The unique ferrimagnetic structure of this  cobaltite at 4 K can be described as the \nassemblage of ferrimagnetic triple chains (Co1  Co2 Co3) running perpendicular to the kagomé \nlayers, ferromagnetically coupled within the la yers, and antiferromagnetic ally coupled with a \nfourth cobalt species Co4. The lifting of the geometrical frustration towards ferrimagnetism, which \nappears in spite of the triangula r topology of the cobalt lattice, is  explained by the very large \nstructural distortion, charge ordering phenomena and large cobalt valence compared to other LnBaCo\n4O7 oxides. \n    \nKeywords \n \nCobalt oxides, CaBaCo\n4O7, 114, cobaltite, magnetic structure, ferrimagnetism, neutron. \n      * Corresponding author: Dr. Vincent Caignaert \ne-mail: vincent.caignaert@ensicaen.fr Fax: +33 2 31 95 16 00  Tel:  +33 2 31 45 26 32  \n  \nThe competition between geometrical frustration and magnetic ordering in transition metal \noxides leading to exotic magnetic properties and transitions has been the subject of many \ninvestigations these last few decades [1] that are still to date not completely understood. In this respect, oxides whose metallic subl attice exhibits vertex-sharing tr iangular or tetrahedral topology \nwith antiferromagnetic (AFM) interactions are of great interest. This is for example the case in \nA\n2B2O7 pyrochlores [2-3], whose metal fr amework consists of tetrahedral [A 4]∞ or [B 4]∞ \nsublattices. The spinels AB 2O4 may be also geometrically frustrated magnets when the A cation is \ndiamagnetic [4]. A second important example deal s with the “kagomé” oxides, where the magnetic \nfrustration originates from a 2D  triangular topology of their ma gnetic network. This class of \ncompounds is nicely illustrated by the Jarosite family [5], which display either a spin-glass \nbehavior or long-range magnetic order with propagation vectors k=0 and k=(1/3, 1/3, 0). \nThe discovery, some years ago of the cobaltites LnBaCo 4O7 [6-9] and very recently of the \nferrites CaBaFe 4O7 [10] and YBaFe 4O7 [11, 12] opened the route to the study of new \ngeometrically frustrated magnets. Structurally, th ese compounds are closely related to the spinels \nas far as they contain similar triangular subl attices of cobalt or iron atoms with the kagomé \ntopology, but they differ fundamentally from the la tter by the fact that their Co-O or Fe-O \nframework consists exclusively of CoO 4 or FeO 4 tetrahedra, forming two sorts of layers, called \ntriangular and kagomé respectivel y. As a consequence, the magnetic  properties of these oxides are \nvery different from those of other “kagomé” oxides and show very complex transitions. Such a behavior is exemplifie d by the cobaltite YBaCo\n4O7 for which a spin glass transition was first \nreported around T f ∼ 66 K [7], whereas long range ma gnetic order was evidenced below T N = \n110K [13], and a magnetic tran sition with short range correlations was revealed above T N in a \nsingle crystal of this phase [15]. Fr om the structural view point, this phase also exhibits a structural \ntransition from orthorhombic to he xagonal above room temperature at T S = 313 K [13], the \northorhombic low temperature form corresponding to a distortion of the high temperature \nhexagonal form. Quite remarkably, all th e magnetic transitions appear below T S and the recent \nstudies of YBaCo 4O7 by single crystal neutron scattering by Manuel et al. [15] demonstrate a very \nimportant feature: At 130 K, i.e. above T N, this phase exhibits a quasi 1D magnetic order along cr, \nwhereas in the kagomé layers (a-b plans) it di splays a strong degeneracy, so that it can be \ndescribed as a new class of 2D frustrated oxide. \nRecently, we synthesized the cobaltite CaBaCo 4O7 [16], which differently from all other \ncobaltites of this series, including Y 0.5Ca0.5BaCo 4O7 [17], exhibits ferrimagnetic properties below \nTC = 70 K. Bearing in mind that the orthorhombic dist ortion of this phase is much larger than any \nobserved for other cobaltites, we  have explored its nuclear a nd magnetic structure. We report \n2/19  herein on the crucial role  of the structural distortion, of the cobalt valence and of charge ordering \nin the appearance of ferrimagnetic ordering in this phase. We show that the AFM out of plane \ncoupling in the “Co1 Co2 Co3” triple chains running along cris a key parameter for weakening the \nin-plane magnetic frustration and consequent ly for inducing the 3 D magnetic ordering. \n \nThe sample was synthesized from a stoichiometric mixture of CaCO 3, BaCO 3 and Co 3O4 \nfirst heated at 900°C in air for 12 h for decarbona tion. The mixture was then heated in air at \n1100°C for 12 h, and quenched down to room temperature. Specific heat measurements were \ncarried out by means of a commercial Physical Properties Measurements System (PPMS, Quantum \nDesign) using a relaxation method with a 2 τ fitting procedure. Neutron powder diffraction (NPD) \nversus temperature was performed on the D2b diffractometer (ILL, Grenoble) at room temperature \n(λ = 1.59Å) and from 4 to 150 K ( λ = 2.428 Å). \n \nThe NPD patterns registered at various temperat ures reflect an excellent crystallization of \nthe compound, as illustrated for the pattern of CaBaCo 4O7 registered at 4 K (Fig.1). An impurity, \nidentified as CoO, was taken into account, due  to the presence of one magnetic reflection \n(2θ=28.2°, d=4.92Å) close to those of CaBaCo 4O7. The amount of CoO was estimated to 2% in \nweight and is probably related to Co 3O4 in excess during the synthesis.  Whatever the temperature, \nthe Rietveld refinements of CaBaCo 4O7 evidence the orthorhombic Pbn2 1 symmetry, in the whole \ntemperature range. This is in cont rast to most of the other LnBaCo 4O7 cobaltites [8-9] which \ngenerally show a structural transition from or thorhombic (O) to hexagonal (H) symmetry as the \ntemperature increases, corre sponding to the relations: a O ∼ aH ∼ 6.3 Å,  b O ∼ aH3 ∼ 11 Å  and  c O \n∼ cH ∼ 10.2 Å. The evolution of the cell parameters versus temperature (Fig.2a) shows that the ar \nand  parameters decrease with T, whereas the crbr\n parameter increases as T decreases, leading to \nan overall decrease of the cell volume (Fig .2b). The orthorhombic symmetry results from a \ndistortion of the hexagonal cell, wh ich can be quantified as D = (b/ 3-a)/a, corresponding to an \nexpansion of the hexagonal cell along one [110] H direction and a contraction along the [1 10]H \nperpendicular direction. The amplit ude of this distortion D in CaBaCo 4O7 is the largest ever \nobserved in the “114” series of cobaltites. It in creases linearly as T decreases from D=1.05% at \nroom temperature to D=1.8% at 50 K, and remain s constant below this temperature down to 4 K \n(Fig.2b). These values can be compared to th e D values previously observed for the LnBaCo 4O7 \ncompounds at low temperature. For the latter, they  are always much smaller, reaching a maximum \nvalue of 0.5% at 10 K for YbBaFe 4O7, i.e. still much smaller than the D value observed for \nCaBaCo 4O7 at room temperature. This high value of the distortion is quite remarkable, since it \ninduces a lifting of the geometrical frustration, wh ich is initially present in the cobalt network of \n3/19  the LnBaCo 4O7 series [6-9], due to its triangular topol ogy. It explains the absence of magnetic \nfrustration at low temperature, and the a ppearance of a ferrimagnetic state in CaBaCo 4O7. \n \nThe atomic coordinates of this oxide, determined  at 4 K (Table 1) and at room temperature \nby NPD are in agreement with the results obta ined from X-Ray powder diffraction refinements \n[16], but show a higher accuracy of the oxyge n positions. The crystal structure of CaBaCo 4O7 \nconsists of a 1:1 stacking of kagomé (K) and triangular (T ) layers of CoO 4 tetrahedra along cr. The \nprojections of the structure along  (Fig.3a) and arbr\n (Fig.3b) show that the triangular layers (T) \nformed by the Co1 tetrahedra are perfectly flat; in contrast, the kagomé layers (K) are strongly \ncorrugated, the apical oxygen of  the Co2, Co3 and Co4 tetrahed ra being located in a plane, \nwhereas the oxygen atoms of the basal planes of these tetrahedra form a waving layer. \nThe interatomic distances (Table 2) show that the four symmetry-indepe ndent cobalt tetrahedra \ncan be classified into two groups : the Co2 and the Co3 tetrahedra belonging to the K layers which \nexhibit larger Co-O distances, i.e. average dist ances ranging from 1.94-1.97 Å (RT) to 1.95 Å (4 \nK), and the Co1 and Co4 tetrahed ra belonging to the T and K la yers respectively, whose Co-O \ndistances are significantly smaller i.e. average distances ranging from 1.87-1.88 Å (RT) to 1.85-1.89 Å (4 K). Such a distribution of the Co-O  bond lengths between these four independent \ncrystallographic sites suggests that, whatever the temperature, CaBaCo\n4O7 exhibits charge \nordering in agreement with the stoichiometric formula CaBaCo 22+Co23+O7. Nevertheless, though \nthe bond lengths observed for Co2 and Co3 ar e in agreement with those observed for Co2+ in \ntetrahedral coordination, the Co-O bond lengths observed for Co1 and Co4 are significantly larger \nthan those expected for Co3+ in tetrahedral coordination, which were found typically close to 1.79 \nÅ in oxides. The bond valence sum calculations (BVS), performed accord ing to Alternatt and \nBrown [19], support this viewpoint (Table 3). On e indeed observes that Co2 and Co3 exhibit a \ncharge comprised between +1.9 and +2.1, whatev er the temperature and can be considered as \ndivalent, whereas a charge ranging from +2.4 to +2 .6 is observed for Co1 and Co4, suggesting that \nthese cations should be rather mixed valent, with an average value of +2.5. Such a mixed valence, \nsignificantly smaller than +3, is in contradiction with the stoichiometric formula. Indeed, from \nstructure refinements neither oxygen defi ciency leading to the formula CaBaCo 4O6.5, nor presence \nof protons according to the formula CaBaCo 4O6OH were detected. Bear ing in mind that the \ntetrahedral coordination of Co3+ is very rare, due to the crystal field effect which favors the \noctahedral coordination, the possibi lity of partial charge transfer from cobalt to one oxygen may be \nconsidered. This would lead to a ligand hole configuration ( L) similar to that observed for apical \noxygen in layered cuprates  [20], corresponding to a hybridizatio n of the bonds of the Co1 and Co4 \ntetrahedra according to the scheme Co3+-O = (Co3d6) Ù Co2+=O-(Co3d7L).  The significantly \n4/19  5/19  larger value of the apical Co-O 7 bond length, of the Co1 and C o4 tetrahedra, ranging from 1.91-\n1.95 Å at RT to 1.97 – 1.89 Å at 4 K, suggests that the hole appears on the O7 atom shared \nbetween one Co4 and one Co1 tetrahedron. A study of this compound by X-ray absorption \nspectroscopy would be necessary  to elucidate this point. \nThe geometry of the CoO 4 tetrahedra is significantly modi fied by temperature. The size of \nthe Co1 and Co2 tetrahedra increases as T d ecreases, whereas the si ze of the Co3 and Co4 \ntetrahedra decreases with temperature, as show n from the average <Co-O> bond lengths (Table2). \nThe evolution of the distortion is as well very  complex. The distortion of the Co1 and Co3 \ntetrahedra increases significantly as T decrease s, whereas the distortion of the Co2 tetrahedra \ndecreases slightly with T, and no significant va riation of the distorti on with temperature is \nobserved for the Co4 tetrahedra. Clearly, the stru ctural evolution vs T does not influence the \ncharge ordering of CaBaCo 4O7: it can be described as 1:1 charge  ordering between Co2/Co3 sites \noccupied by Co2+ and Co1/Co4 sites which are mixed or hybridized Co3+ (3d6) / Co2+ (3d7L). \nThe Ca-O distances of the CaO 6 octahedra, ranging from 2.25 to 2.40 Å at 4 K and from \n2.27 to 2.35 Å at RT are in agreement with th ose usually observed. The coordination of barium \nchanges from 5+7 at RT to 6+6 at 4 K. The 5 or 6 nearest oxygen neighbors are located at usual \ndistances from barium, ra nging from 2.80 to 2.90 Å, whereas the other seven or six oxygen atoms \nsit much further, at distances ranging from 3.36 to  3.58 Å. In spite of this reasonable coordination \nof barium, the computed valence of barium calcu lated from BVS is much smaller than expected \ni.e. +1.4 instead of +2, suggesting that Ba2+ is strongly underbonded. A similar behavior has been \nobserved previously for YbBaCo 4O7 [18] and was attributed to a structural instability, in \nagreement with the transition from hexagonal to orthorhombic symmetry at decreasing \ntemperature. Such a hypothesis does not hold here, since CaBaCo 4O7 does not exhibit any \nstructural transition in the whol e temperature range and moreover the so-obtained barium valence \nof 1.40 for CaBaCo 4O7 at 4K, which exhibits th e largest distortion, is st ill very close to that \nobtained for the hexagonal form of YbBaCo 4O7 (1.33). The origin of th is abnormally weak valence \nof barium remains unexplained.  \nThe NPD pattern registered at 4 K shows cl early the presence of magnetic Bragg peaks \n(Fig.1) with a propagation vector k=0, whose in tensity decreases as T increases. The latter \ndisappear above T\nC=70 K, in agreement with the specific h eat measurement (inset Fig.1) and with \nthe ferrimagnetic to paramagnetic transition previ ously observed for this oxi de [16]. To generate \nall the spin configurations comp atible with the crystal symmetry  we carried ou t a group theory \nanalysis using the program BasiReps. There are f our irreducible representa tions (IR) associated \nwith the Pbn2 1 space group and k=(0, 0, 0). Among the latter, only thre e IRs allow a ferrimagnetic \nalignment of Co sublattices. The ba sis vectors of these representati ons are listed in Table 4. The results of simulated annealing runs, to solve the ma gnetic structure, lead to a unique solution in the \nΓ4 representation. The final refinement shows that one of the 3 coefficients ( Ψ12 in Table 4), \nrepresenting the 3 basis v ectors, refines to zero leading to a magnetic structure where all the spins \nlie in the ( ab) plane. This solution of the magnetic structure at 4 K confirms its ferrimagnetic \nnature, characterized by a simultaneous ordering of Co spins in both triangular and kagomé layers. \nThus, the magnetic structure at 4 K (Fig.4) can be described in the Pbn2 1 space group, with all Co \nspins lying in the (a,b) plane, i.e. parallel to the kagomé and tr iangular layers. The ferrimagnetic \nstructure of this cobaltite is quite unique w ith respect to all other members of the LnBaCo 4O7 \nseries. In the kagomé layers, the Co2 and Co3 spins form zigzag chains running along b (Fig.4a \nand 7b). In those chains, one Co2 alternates wi th one Co3 species, and two successive Co2 (or \nCo3) spins are oriented at ∼60°, while “Co2Co3” ferromagnetic couples are formed. As a result, \nthe transverse magnetic compone nts of Co2 (and Co3) along ar are antiferromagnetically coupled \nand cancel each other, whereas the longitudinal magnetic components of Co2 and Co3 along br\n are \nall oriented in the same direction. Thus, C o2 and Co3 form ferromagnetic zigzag chains along br\n, \nwhich are themselves all ferromagnetically coupled  in the entire kagomé layer. Between those \nchains, the Co4 species have thei r spins oriented antiparallel to  the FM resultant of “Co2Co3” \nchains. Consequently, a ferrimagnetic order of the Co2, Co3 and Co4 sp ins takes place in the \nkagomé layers, whose facile ma gnetization is directed along br\n. The Co1 spins of the triangular \nlayers (Fig.4a-b) are practically antiparallel to Co2 and Co3 spins (Fig.4a and 7b), two successive \nCo1 spins, being oriented at 60°, similarly to “C o2 Co3” chains. Thus, their transverse components \nare antiparallel and cancel each other and their longitudinal components are parallel, forming \nferromagnetic chains Co1 running along br\n. The latter chains are antiferromagnetically coupled \nwith the “Co2 Co3” chains. In summary, the 3 D ferrimagnetic ordering of cobalt spins in \nCaBaCo 4O7, consists of “Co2 Co3” zigzag FM chains running along br\n, antiferromagnetically \ncoupled with Co4 and Co1 spins along ar and br\n respectively, the resultant magnetic moments of \ncobalt spins being parallel to the facile magnetization axis br\n. Bearing in mind the description of \nthe cationic framework of CaBaCo 4O7, previously proposed by Chapon et al [13], which consists \nof chains of corner-sharing “Co5” bipyramids running along c (Fig.4b) interconnected through \n“Co3” triangles in the (a,b) plane (Fig.4a), anothe r description of this ferrimagnetic structure can \nbe proposed. It consists of the assemb lage of ferrimagnetic chains running along cr, and \ncorresponding to the AFM coupling of  Co1, with Co2 and Co3. In th is description, each triple \n“Co1 Co2 Co3” chain is ferromagnetically c oupled to two other id entical chains, and \nantiferromagnetically coupled  to three Co4 species in the (a, b) plane. \n6/19  The evolution of the spontaneous magnetization versus temperature (Fig.5) corroborates the \nprevious magnetic measurements [16]. One ind eed observes a resultant  magnetic moment of ∼ 1μB \nper f.u. at low temperature, close to the value observed from M(T) data ( ∼ 0.6μB per f.u.), which \ndecreases abruptly at T C down to ∼0.2μB . \nThe evolution of the magnetic moments of the fo ur cobalt sites, versus temperature (Fig.6), \nshows also an abrupt  decrease around T C as expected. More importa ntly, the values of these \nmoments at low temperature supp ort strongly the existence of charge ordering. One indeed \nobserves significantly larger va lues of the magnetic moment s of Co1 and Co4 (2.5 - 3.0 μB per Co) \nin agreement with their valences deduced from BVS’s, i.e. Co2+L/Co3+ (theoretical moment ∼ \n4μB). These values of the magnetic moments, sma ller than the theoretical  ones, suggest that a \ndegree of frustration remains, inducing a sl ight disordering of the cobalt spins. \n \n \n7/19   This study shows that CaBaCo 4O7 exhibits, compared to all other “114” LnBaCo 4O7 \ncobaltites, a very unique ferrimagnetic structure, in spite of its close structural relationships with \nthese oxides. The origin of this complex magnetic structure is, most likely, reminiscent of the \ncompetition between geometric frustration that appears in the kagomé layers due to their triangular \ntopology and the antiferromagnetic inte ractions that take place along crof corner-shared tetrahedra \n(Co1, Co2 and Co3). The crucial role  of these triple AFM cobalt chai ns in the establishment of the \n3 D magnetic ordering in these compounds is clear ly shown by comparing th e magnetic structure \nof YBaCo 4O7 previously observed at 80 K by  Chapon et al [15] (Fig.7a)  with the structure of \nCaBaCo 4O7 at 4 K (Fig.7b). Both magnetic structures consist of very similar triple AFM chains \n“Co1 Co2 Co3” running along  and ferromagnetically coupled along , i.e. forming \nferrimagnetic layers parallel to (0 10). The two structures differ by the coupling of  those layers \nalong , which is antiferromagnetic for YBaCo 4O7 (Fig.7a) and ferromagnetic for CaBaCo 4O7 \n(Fig.7b). The spin orientation of Co4 is also di fferent in the two structures, the moment of Co4 \nbeing oriented at 90° from other spins in YBaCo 4O7 and at ∼ 30° in CaBaCo 4O7. The Co4 spins \nare antiferromagnetically  coupled in YBaCo 4O7 (Fig.7a) whereas they are ferromagnetically \ncoupled along b in CaBaCo 4O7 (Fig.7b), so that the magnetic structure of the latter phase remains \nferrimagnetic with b as facile magnetization axis. In both structures, the lifting of the magnetic \nfrustration is explained by the fact th at the in-plane exchange interaction J 1, which favors the 120° \ngeometry in the hexagonal structure, is reduc ed by the orthorhombic di stortion as T decreases. \nThis orthorhombic distortion is much larger for CaBaCo 4O7, as shown from the in-plane Co-Co \ndistances between Co2, Co3 and C o4 ranging from 3.07 to 3.33 Å at RT and from 3.00 to 3.31 Å \nat 4 K, so that the Co triangles are no more equilateral. The 3 D ma gnetic ordering is also crar\nbr\nr\nr8/19  supported by the out of plane J 2 interaction, which tends to favor the existence of AFM triple \nchains “Co1 Co2 Co3” in both structures. The fact  that the cobalt valence is larger in CaBaCo 4O7, \nbut especially that Co3+/or Co2+L are preferentially located on the Co1 and Co4 sites may change \nsignificantly J 2. Such a charge ordering may be at the origin of the FM coupling of the chains in \nCaBaCo 4O7, in contrast to the AFM coupling observed for YBaCo 4O7. Thus, the combination of \nthese different factors – structural  distortion, mixed valence of coba lt and charge ordering – appear \nas key factors for controlling the magnetic  ordering in the “114” cobaltite series. \n Finally, it is worth pointing out that  the ferrimagnetism observed for CaBaCo 4O7, is \nsignificantly different from what has been observed for CaBaFe 4O7 [10] where T C and magnetic \nmoments are much higher. The hexagonal symmetry of the latter suggests that th e iron spins in this \nphase might be lying out of plane. \n   9/19  References \n1. A. Ramirez, Ann. Rev. Mater. Sci. 24, 453 (1994)  \n2. J.E. Greedan, J. Alloys Compounds 408-412 , 444 (2006)  \n3. M. J. Harris and M. P. Zinkin, Modern Phys. Lett. B 10, 417 (1996) \n4. S.-H. Lee, C. Broholm, W. Ratcliff, G. Ga sparovic, Q. Huang, T. H. Kim and S.-W. \nCheong, Nature 418, 856 (2002) \n5. A. S. Wills, Can. J. Phys. 79, 1501 (2001) \n6. M. Valldor and M. Andersson, Solid State Sci. 4, 923 (2002)  \n7. M. Valldor, J. Phys. Condens. Matter 16, 9209 (2004)  \n8. A. Maignan, V. Caignaert, D. Pelloquin, S.  Hébert, V. Pralong, J. Hejtmanek, and D. \nKhomskii, Phys. Rev. B 74, 165110 (2006) \n9. V. Caignaert, A. Maignan, V. Pralong, S. Hé bert and D. Pelloquin, Solid State Sciences 8, \n1160-1163 (2006) \n10. B. Raveau, V. Caignaert, V. Pralong, D. Pelloquin and A. Maignan, Chem. Mater. 20, 6295 \n(2008) \n11. V. Caignaert, A.M. Abakumov, D. Pelloquin, V. Pralong, A. Maignan, G. Van Tendeloo \nand B. Raveau, Chem. Mater. 21, 1116 (2009) \n12. V. Pralong, V. Caignaert, A. Maigna n and B. Raveau, J. Mat. Chem., DOI: \n10.1039/B911611G (2009) \n13. L. C. Chapon, P.G. Radaelli, H. Zheng and J.F. Mitchell, Phys. Rev. B 74 172401 (2006) \n14. M. Soda, Y. Yosui, T. Moyoshi, M. Sato, N. Igawa and K. Kakurai, J. Phys. Soc. Japan 75, \n054707 (2006) \n15.  P. Manuel, L.C. Chapon, P.G. Radaelli, H.  Zheng and J.F. Mitche ll, Phys. Rev. Lett. 103, \n037202 (2009) \n16. V. Caignaert, V. Pralong, A. Maignan and B. Raveau, Solid State Comm. 149, 453 (2009)   \n17. M. Valldor, Solid State Sci. 8, 1272 (2006)   \n18. A. Hug, J.F. Mitchell, H. Zheng, L.C. Chapon, P.G. Radaelli, K.S. Knight and P.W. \nStephens, J. Solid State Chem. 179, 1125 (2006) \n19. I. D. Brown and D. Alternatt, Acta Cryst. B 41, 244 (1985)  \n20. A. Bianconi, J. Budnick, A.M. Flank, A. Fontai ne, P. Lagarde, A. Marcelli, H. Tolentino, \nB. Chamberland, G. Demazeau, C. Michel and B. Raveau, Phys. Lett. A 125, 285 (1988) \n \n 10/19  Table Captions: \n \nTable1: Atomic and magnetic parameters for CaBaCo 4O7 at 4K. All sites are fully occupied. \nSpace group: Pbn2 1. Cell parameters: a= 6.2613(1) Å, b=11.0399(2) Å, c=10.1642(2) Å. \n \nAtom x y z B \nCa 0.0066(21) 0.6729(13) 0.8708(13) 0.06(13) \nBa 0.0043(14) 0.6651(10) 0.5 0.06(13) \nCo1 0.0155(33) -0.0003(24) 0.9332(25) 0.09(13) \nCo2 -0.0067(26) 0.1688(20) 0.6902(28) 0.09(13) \nCo3 0.2527(33) 0.0955(16) 0.1904(21) 0.09(13) \nCo4 0.2673(25) 0.9209(15) 0.6841(21) 0.09(13) \nO1 -0.0155(27) 0.0044(11) 0.2460(16) 0.64(5) \nO2 -0.0045(20) 0.4914(11) 0.2285(16) 0.64(5) \nO3 0.7817(21) 0.2613(10) 0.7815(19) 0.64(5) \nO4 0.7319(18) 0.7418(12) 0.2141(17) 0.64(5) \nO5 -0.0549(12) 0.1522(8) 0.4997(16) 0.64(5) \nO6 0.2035(13) 0.1097(6) 0.0062(19) 0.64(5) \nO7 0.2645(15) 0.9471(6) 0.5002(22) 0.64(5) \n \nAtom M x M y M \nCo1 -1.78(16) -2.20(16) 2.83(16)\nCo2 1.01(14) 1.77(20) 2.04(19)\nCo3 0.72(22) 1.92(11) 2.05(15)\nCo4 -0.05(20) -2.44(12) 2.44(15)\n Table 2: \nInteratomic distances at 4K for CaBaCo 4O7. \nBond Distance (Å) Bond Distance (Å)\n Ca-O2 2.32(4) Co1-O1 1.90(6) \n    O3 2.24(4)     O5 1.82(5) \n    O4 2.32(4)     O6 1.85(5) \n    O5 2.35(4)     O7 1.98(5) \n    O6 2.38(4) Co2-O1 2.00(5) \n    O7 2.41(4)     O3 1.91(5) \n Ba-O2 3.36(3)     O4 2.00(4) \n    O2 2.89(3)     O5 1.97(6) \n    O3 3.54(4) Co3-O1 2.04(5) \n    O3 2.72(4)     O2 1.99(5) \n    O4 3.48(3)     O3 1.83(5) \n    O4 2.79(3)     O6 1.90(6) \n    O5 2.82(2) Co4-O1 1.89(5) \n    O5 3.45(2)     O2 1.78(4) \n    O6 3.57(3)     O4 1.84(4) \n    O6 2.81(3)     O7 1.89(6) \n    O7 3.51(3)   \n    O7 2.81(3)   \n \n Table 3 : Bond valence sum calcu lations (BVS) at 4K. \nAtom BVS \nCa 2.23(5)\nBa 1.48(3)\nCo1 2.39(8)\nCo2 1.89(7)\nCo3 2.08(7)\nCo4 2.63(9)\nO1 1.97(7)\nO2 1.82(6)\nO3 2.01(7)\nO4 1.74(5)\nO5 1.78(7)\nO6 1.79(6)\nO7 1.59(6)\n \nTable 4:  Basis vectors for the space group Pbn2 1 with k=(0,0,0). The decomposition of the \nmagnetic representation for the Co site is . The atoms are defined according \nto 1:( x,y,z), 2:( -x,-y,z+1/2 ), 3:( -x+1/2,y+1/2 ) and 4:( x+1/2,-y+1/2,z+1/2 ). 1\n41\n31\n21\n1 3 3 3 3Γ + Γ + Γ + Γ = Γmag\n \n  Γ 1   Γ2   Γ3   Γ4  \n Ψ1 Ψ2 Ψ3 Ψ4 Ψ5 Ψ6 Ψ7 Ψ8 Ψ9 Ψ10 Ψ11 Ψ12 \nAtom m //a m //b m //c m //a m //b m //c m //a m //b m //c m //a m //b m //c \nm1 1 1 1 1 1 1 1 1 1 1 1 1 \nm2 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 \nm3 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 \nm4 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 \n \n  \n11/19  Figure Captions \nFig. 1  (Color online) Rietveld  refinement of CaBaCo 4O7 at 4 K: The dots and solid line represent \nthe experimental data points and calculated di ffraction pattern, respectiv ely. The difference is \nshown at the bottom. The row of markers sh ows the positions of the nuclear and magnetic \nreflections for CaBaCo 4O7 and CoO. The thick solid line (blu e online) represents the contribution \nfrom magnetic scattering of CaBaCo 4O7 alone. The inset shows the sp ecific heat C versus T. \nFig. 2  (Color online) Evolution of the cell parameters (a) and of the cell volume and structural \ndistortion (D =  (b 3-a)/a)(b) versus temperature of CaBaCo 4O7  \nFig. 3  (Color online) Structure of CaBaCo 4O7 at 4 K showing the arrangement of the kagomé (K) \nand triangular (T) laye rs: (a) projection along a, (b) projection along b. \nFig. 4  (Color online) Magnetic structure of CaBaCo 4O7: projection along c (a) and projection \nalong  (b). The cobalt network, built up of “Co5” triangular bipyramids and “Co3” triangles is \ndrawn as a support r\nar\nFig. 5  Evolution of the spontaneous magnetization versus temperature for CaBaCo 4O7. The \nresulting moment is given in Bohr magneton per f.u. The line is a guide to the eye. \nFig. 6  (Color online) Evolution of the magnetic moments of the different cobalt ions versus \ntemperature for CaBaCo 4O7. The lines are guide to the eye. \nFig. 7  Comparison of the magnetic structures of YBaCo 4O7 at 80 K (a) and of CaBaCo 4O7 at 4K \n(b) view along . The ellipses show the similar triple ferrimagnetic chains “Co1 Co2 Co3” and \ntheir relative orientati ons in the two compounds. cr\n \n \n12/19  2-θ (degrees)Intensity\n20 40 60 80 100 120 140  0200400\n25 50 75 100  0 50100150\nT (K)C (J/Kmol)\n \n \n \nFigure 1 \n \n13/19   \n \n \n \nFigure 2 \n \n14/19   \n \nFigure 3 \n \n15/19   \n \n \n \nFigure 4 \n \n16/19   \n \nFigure 5 \n \n17/19   \n \n \nFigure 6 \n \n18/19   \n \n \n \nFigure 7 \n \n \n19/19  "    },    {        "title": "0805.3604v2.Frustration_Induced_Quantum_Phases_in_Mixed_Spin_Chain_with_Frustrated_Side_Chains.pdf",        "content": "arXiv:0805.3604v2  [cond-mat.str-el]  12 Aug 2008Frustration Induced Quantum Phases in Mixed Spin Chain with Frustrated Side\nChains\nKazuo Hida1and Ken’ichi Takano2\n1Division of Material Science, Graduate School of Science an d Engineering,\nSaitama University, Saitama, Saitama, 338-8570, Japan\n2Toyota Technological Institute, Tenpaku-ku, Nagoya 468-8 511, Japan\n(Dated: October 26, 2018)\nA mixed Heisenberg spin chain with frustrated side chains is investigated by numerical and per-\nturbational calculations. A frustration-induced quantum partially polarized ferrimagnetic phase\nand a nonmagnetic spin quadrupolar phase are found adjacent to the conventional Lieb-Mattis type\nferrimagnetic phase or the nonmagnetic singlet cluster sol id phases. The partially polarized ferri-\nmagnetic phase has an incommensurate spin structure. Simil ar structures are commonly found in\nother frustration-induced partially polarized ferrimagn etic phases. Numerical results also suggest a\nseries of almost critical nonmagnetic ground states in a hig hly frustrated regime if the side chain\nspins weakly couple to the main chain.\nPACS numbers: 75.10.Jm, 75.10.Pq, 75.30.Et, 75.30.Kz\nI. INTRODUCTION\nThe interplay of frustration and quantum fluctua-\ntion has been extensively studied in a variety of low-\ndimensional quantum magnets. Even in one-dimensional\ncases, various exotic quantum phenomena such as spon-\ntaneous dimerization, [1] 1/3-plateau with spontaneous\ntrimerization, [2] and transition between quantum and\nclassical plateaus [3] are reported. On the other hand,\nthe mixed quantum spin chains also have a variety of\nground states ranging from quantum ferrimagnetism [4]\nto spin gap phases [5–8].\nRecently, it has been reported that frustration induces\na partiallypolarizedferrimagnetic(PPF) phase [9–12] in\nadditiontoaconventionalLieb-Mattistypeferrimagnetic\n(LMF) phase. The PPF phase appear when both frus-\ntration and quantum fluctuations are fairly strong. It is\nan interesting issue how general such a ferrimagnetismis.\nHence it is important to investigate the features of PPF\nphasesin variousspin systems. We arethen motivated to\nfind a PPF phase in other models and investigate them\nin detail.\nWe have introduced a spin chain with side chains in a\nprevious paper [8]. This spin chain has frustration ow-\ning to the interaction among spins in the main chain and\nthose of side chains. The frustration varies in strength\nand in feature with the variation of parameters in the\nmodel. Sincewefocusedonthespin-gapphasesin [8], we\nhave investigated the parameter regimes where frustra-\ntion is not strong enough to destroy the spin-gap phases.\nWe then found two spin-gap phases and explained them\nby singlet cluster solid (SCS) pictures.\nIn the present work, we examine this model in a highly\nfrustrated regime. This regime, in which the frustration\nplays a central role, is of interest in its own right, since\nthe model exhibits features very different from those in\nthe weak frustration regimes. We actually find clear nu-\nmerical evidences not only for the above mentioned fas-\ncinating PPF phase but also for the spin quadrupolar(QP) phase [13–19]. These phases are totally different\nfromthe conventionalphasessuch asspin gapphasesand\nthe LMF phase which can be realized even in the unfrus-\ntrated case. They will be investigated in detail in the\npresent paper. Also numerical data are obtained which\nsuggest the possible existence of an exotic almost critical\nnonmagnetic ground state in the regime where the cou-\nplings between the side chain and main chain spins are\nweak but strongly frustrated.\nThe transition from a ferrimagnetic phase to a non-\nmagnetic phase in the present model takes place because\nthe quantum fluctuation in the side-chains destroys the\nferrimagnetic long range order in the main chain. The\nmechanism of quantum destruction of ferromagnetism\nand ferrimagnetism has been less studied than that of\nantiferromagnetism; the latter has been extensively stud-\nied in relation with the high- Tcsuperconductivity. Re-\ncently, however, experiments have been reported on the\nnonmagnetic ground states in one- and two-dimensional\nmaterials [23, 24] with ferromagnetic nearest neighbour\nand antiferromagnetic next nearest neighbour couplings.\nTheoretical investigation has also been carried out for\ncorresponding models [25–28].\nIn an unfrustrated ferrimagnet, the spontaneous mag-\nnetizationisuniquelydetermined bytheLieb-Mattisthe-\norem [29]. This type of quantum ferrimagnetism has\nbeen investigated in detail [4]. As far as the frustration\nis weak, the spontaneous magnetization remains locked\nto this value [30]. This phase is the LMF phase [10].\nThe spontaneous magnetization in this phase is a sim-\nple fraction of the saturated magnetization. In contrast,\nthe spontaneous magnetization in the PPF phase con-\ntinuously varies with the parameter characterizing the\nstrength of frustration and is not a simple fraction of\nthe saturated magnetization. The PPF phase appears\nbetween the LMF phase and the nonmagnetic spin gap\nphases. This type of phase is first predicted in the pio-\nneering work of Sachev and Senthil [21] in the quantum\nrotor model. Bartosch, Kollar and Kopietz [22] proposed2\na possibility of ferromagnetic Luttinger liquid in an itin-\nerant one-dimensional Fermi system. The first explicit\nexample of quantum PPF phase induced by frustration\nin one-dimensional quantum spin systems was proposed\nby Ivanov and Richter [9] in a frustrated mixed spin lad-\nder. Similar phases are also found by Yoshikawa and\nMiyashita [10] in a uniform spin chain and by one of the\npresent authors in a trimerized zigzag chain [11, 12]. We\nproposeanotherexampleofthe PPFphasein the present\nmixed spin chain. The present example is substantially\ndifferent from previous ones, because it is accompanied\nby the destruction of the ferrimagnetic order in the main\nchain by the frustrated coupling to the quantum fluctu-\nation in the side chains.\nThe QP phase is well known for the spin-1 bilinear-\nbiquadratic chain between the Haldane and ferromag-\nnetic phases [13–19]. The exact Bethe ansatz solution\nis available if the coefficients of the bilinear and the bi-\nquadratic terms coincide with each other [13–15]. Re-\ncently, similar phases are found in the frustrated two-\ndimensional Heisenberg model with ferromagnetic near-\nest neighbour interaction and antiferromagnetic next\nnearest neighbour interaction [28]. In this paper we\nexplicitly show that our model reduces to the bilinear-\nbiquadratic chain in appropriate limiting cases. It is also\nargued that the QP phase should appear in a wide class\nof complex spin models between ferrimagnetic and spin\ngap phases.\nThis paper is organized as follows. In the next section,\nthe model Hamiltonian is presented. Various limiting\ncases are discussed using the perturbational approxima-\ntion from the strong coupling limit in Sec. III. The nu-\nmerically obtained ground state phases are explained in\nSec. IV. The properties of PPF phase are described in\ndetail in Sec. V. Section VI is devoted to summary and\ndiscussion.\nT(p)\nS  (p)1S  (p)2J J 2K1K2\n1\nFIG. 1: The quantum spin chain with side chains which we\nstudy in this paper. S1(p),S2(p) andT(p) are spins in the\np-th unit cell. These magnitudes are S1= 1,S2=1\n2, and\nT=1\n2, respectively.II. HAMILTONIAN\nWeconsiderthemixedHeisenbergspinchaindescribed\nby the Hamiltonian\nH=N/3/summationdisplay\np=1[J1S1(p)S2(p)+J2S2(p)S1(p+1)\n+K1S1(p)T(p)+K2S2(p)T(p)]. (1)\nwhereS2(p) andT(p) are the spin-1 /2 operators and\nS1(p) is the spin-1 ones in the p-th unit cell, as shown\nin Fig. 1. In what follows, we use nondimensional pa-\nrameters j=J2/J1,k=K1/J1andr=K2/(2K1), and\nthe unit of J1= 1. The parameter rcharacterizes the\nstrength of frustration. The total number of spin sites is\ndenoted by N, and then the number of unit cells is N/3.\nIn regime 0 ≤r/lessorsimilar0.5, where frustration is not strong,\nwe have found two types of nonmagnetic ground states\nand have explained them by SCS pictures [8]. When\nfrustrationbecomesstrong,thephasediagramdrastically\nchanges. In the present paper, we will investigate the\nstrongly frustrated case r/greaterorsimilar1 in detail.\nIII. LIMITING CASES\nA. Small jregime\nForj= 0, the chain is decoupled into an assembly of\n3-spin clusters described by the Hamiltonian\nHA(p) =S1(p)S2(p)+kS1(p)T(p)+2krS2(p)T(p).\n(2)\nAs discussed in Ref. [8], the cluster ground state is a\nsinglet state with energy\nEs=−k−1+kr\n2(3)\nfork < kc≡2\nr−1, and a triplet state with energy\nEt=−(1+k+2kr)−/radicalbig\n(1+k−4kr)2+8(k−1)2\n4\n(4)\nfork > kc. Therefore, the ground state of the chain for\nsmalljis a gapped local 3-spin singlet phase for k <\nkc. We call this phase as Gap I phase following Ref. [8].\nBecause this cluster ground state is gapped, it cannot\ngainenergywithinthefirstorderin jeveninthepresence\nofj.\nIn the 3-spin triplet ground state for k > kc, we define\na composed spin with magnitude 1 as ˆS(p)≡S1(p) +\nS2(p) +T(p) to describe low energy phenomena. The\ntotal Hamiltonian is written as H=HA\n0+HA\nintwith an3\nunperturbed part HA\n0=/summationtext\npHA(p) and an interaction\npart\nHA\nint=N/3/summationdisplay\np=1jS2(p)S1(p+1). (5)\nThe perturbation calculation up to the second order in\nthe interaction part HA\nintyields the following effective\nbilinear-biquadratic Hamiltonian for the composed spin\nˆS(p):\nHA\neff=N/3/summationdisplay\np=1/bracketleftBig\nJA\neffˆS(p)ˆS(p+1)+DA\neff(ˆS(p)ˆS(p+1))2/bracketrightBig\n,\n(6)\nwhere the effective interaction parameters consist of the\nfirst and the second order perturbation terms as JA\neff=\nJA(1)\neff+JA(2)\neffandDA\neff=DA(1)\neff+DA(2)\neff.\nIn the first orderperturbation, the effective interaction\nparameters coming from JA(1)\neffandDA(1)\neffare\nJA\neff≃ −jX(α), (7)\nDA\neff≃0, (8)\nwhereX(α) is given by\nX(α) =α√\n2(1+α2)2/parenleftbigg\n1−α\n2√\n2/parenrightbigg/parenleftbigg\n1+α2\n2/parenrightbigg\n(9)\nwith\nα=2√\n2(1−k)/radicalbig\n(k+1−4kr)2+8(k−1)2−1−k+4kr.(10)\nFor small but finite j, the energy of the LMF phase is\ngiven by Et+JA\neffper unit cell while the energy of the\n3-spin singlet (Gap I) phase is given by Eswith no first\norder correction in j. Therefore, comparing the energies\nof these two ground states, we find that the phase tran-\nsition between the gapped 3-spin singlet phase and the\nLMF phase takes place at\nk=kTF≡kc+(1−r)(2−2r+r2)(3−r)j\n2r(2−r)(3−4r+2r2).(11)\nThe effective coupling JA(1)\neffvanishes for k= 1. There-\nfore, in the neighbourhood of k= 1, the higher order\nterms come into play. Within the second order pertur-\nbation with respect to 1 −kandj, the effective coupling\nconstants are\nJA\neff≃j2\n8(r−1)(1+2r)−j(1−k)\n2(2r−1),(12)\nDA\neff≃j2\n8(r−1)(4r2−1). (13)\nThe effective model (6) has a variety of phases [19].\nWithin the present parameter regime, we find the Hal-\ndane phase for 0 < DA\neff< JA\neffwhich corresponds to Gap0 0.5 100.10.2\nGap II\n(Haldane)\nGap I\n(3 spin singlet)LMFQP\nkj r=1.2\nFIG. 2: Phase diagram for small jwithr= 1.2.\nII phase in Ref. [8], the QP phase for 0 < JA\neff< DA\neff\nand the LMF phase for JA\neff<0. However, in the orig-\ninal Hamiltonian (1), the LMF phase is limited by the\ntransition to the 3-spin singlet phase at JA\neff=Es−Et\nas discussed above. For JA\neff> Es−Et, the ground state\nis the 3-spin singlet (Gap I) phase.\nThusthe conditionsforeachphasein terms ofthe orig-\ninal parameters are summarized as follows: The ground-\nstate phase of the present model is (i) the 3-spin singlet\n(Gap I) phase for\n0< k < k TF, (14)\n(ii) the LMF phase for\nkTF< k < k FQ≡1−(2r−1)j\n4(r−1)(2r+1),(15)\n(iii) the QP phase for\nkFQ< k < k QH≡1−j\n2(2r+1),(16)\nand (iv) the Haldane (Gap II) phase for\nk > kQH. (17)\nThese phase boundaries are plotted on the k-jplane in\nFig. 2 for r= 1.2. It should be remarked that the\nQP phase in the present model is realized without bi-\nquadratic interaction in the original Hamiltonian (1).\nB. Large jRegime\nIn the large jlimit,S1(p) andS2(p−1) form an effec-\ntiveS= 1/2spinσ(p)≡S1(p)+S2(p−1). The effective\nHamiltonian is given by\nH=N/2/summationdisplay\np=1(Keff\n1σ(p)T(p)−Jeff\nFσ(p)σ(p+1)\n−Keff\nFσ(p+1)T(p)), (18)4\n.\nJFeffKFeffKeff: S=1/2T(p)\nσ(p)AF\nFerro\nFIG. 3: ∆-chain realized in the limit j >>1.\nwhich form a ∆-chain structure depicted in Fig. 3. The\neffective interactions are given by\nJeff\nF=4\n9, Keff\nF=2kr\n3, Keff\n1=4k\n3,(19)\nas argued in Ref. [8]. The detailed analysis of this model\nis reported in a separate paper [20]. Therefore, we only\nquotetheresultsandrewritethemintermsoftheoriginal\nmodel (1).\nTheferromagneticphaseofthe model(18)corresponds\nto the LMF phase in the original model (1). This phase\nis stable for\n0< k < k FQ=r−2\n3r(20)\neven in the limit of large j.\nFork > kFQ, the ground state is nonmagnetic. Never-\ntheless,therearestillseveraldifferentphases. Forlarge r,\nthe model (18) reduces to the S= 1 bilinear-biquadratic\nmodel and the QP phase appear for\nkFQ< k < k QH≡3r−4\n9r, (21)\nand Haldane phase appear for k > kQH.\nThe results of the numerical diagonalization calcula-\ntion for the model (18) is summarized in Fig. 4. There\nare the QP, the Haldane and the LMF phases, which we\ndiscussed above. In addition, numerical results suggests\nthattherepossiblyexistanarrowPPFphasebetweenthe\nLMF phase and the QP phase, and almost critical non-\nmagnetic ground states for small values of k. The PPF\nphase is so narrow that it cannot be represented in Fig.\n4. We speculate that the almost critical nonmagnetic\nphases are spin gap phases with extremely small energy\ngap with large scale resonating singlet cluster solid struc-\nture. Corresponding phases are also found for finite jas\ndescribed in the next section.\nC. Large rRegime\nWe examine the case of K2≫K1,J1(r,kr≫1) in\nthis subsection. In the the limit of r,kr→ ∞, the chain\nis decoupled into an assembly of 3-spin clusters, each of\nwhich described by the Hamiltonian\nHB(p) =jS1(p+1)S2(p)+2krS2(p)T(p).(22)0 0.1 0.2 0.301020\nr\nGap II\n(Haldane)\nkLMF\nQP\nFIG. 4: Phase diagram in large jlimit. The shaded region is\nthe almost critical nonmagnetic phase.\nThe eigenvalues of this 3-spin Hamiltonian are given by\nE(S= 2) =kr\n2+j\n2, (23)\nE(S= 1;g) =−j−2kr−/radicalbig\n16k2r2−8krj+9j2\n4,\n(24)\nE(S= 1;e) =−j−2kr+/radicalbig\n16k2r2−8krj+9j2\n4,\n(25)\nE(S= 0) =kr\n2−j. (26)\nThe ground states are the triplet states with energy\nE(S= 1;g). Itshouldbenotedthatthelowestexcitation\nenergy is of the order of kreven ifjis small. Therefore\nthe perturbation calculation from this limit is valid even\nfor small j. As a result, each cluster has an effective spin-\n1 degree of freedom ˜S(p)≡S1(p+1)+S2(p)+T(p). The\ntotal Hamiltonian is written as H=HB\n0+HB\nintwith an\nunperturbed part HB\n0=/summationtext\npHB(p) and an interaction\npart\nHB\nint=N/3/summationdisplay\np=1(S2(p)+kT(p))S1(p).(27)\nWe can write down the effective Hamiltonian for ˜S(p) up\nto the second order in HB\nintin the form,\nHB\neff=N/3/summationdisplay\np=1/bracketleftBig\nJB\neff˜S(p)˜S(p+1)+DB\neff(˜S(p)˜S(p+1))2/bracketrightBig\n.\n(28)\nWithin the first order in HB\nint, the effective interaction\nparameters are given as\nJB\neff≃ −(X(α)+kX(−α)), (29)\nDB\neff≃0, (30)5\n0 5 10−0.6−0.4−0.20\nj/rGap II\n(Haldane)r∆k\nQP\nLMFr∆kQH\nr∆kFQ\nFIG. 5: Phase diagram on the j/r-r∆kplane where ∆ k=\nk−kc(j/r).\nwhereX(α) is given by Eq. (9) with\nα=j−4kr+/radicalbig\n16k2r2−8krj+9j2\n2√\n2j.(31)\nThe effective exchange constant JB\neffvanishes up to the\nfirst order in HB\nintifjandksatisfy the relation\nj\nr=8k(k2−1)\n(3k−1)(3k−5). (32)\nWe denote the value of kwhich satisfies this relation by\nkc(j/r) as a function of j/r. The terms of O(r−1) come\ninto play for k≃kc(j/r), and constitute the bilinear-\nbiquadratic form (28) with JB\neff∼O(k−kc,r−1) and\nDB\neff∼O(r−1).\nWe do not explicitly present the second order expres-\nsion for JB\neffandDB\neff, since we numerically carried out\nthe summation over the intermediate states in the sec-\nond order perturbation calculation. The LMF-QP phase\nboundary kFQis determined by setting JB\neff= 0. Be-\ncause the correction terms are of O(r−1), deviations\n∆kFQ≡kFQ−kc(j/r) and ∆ kQH≡kQH−kc(j/r)\nscale with 1 /rfor fixed j/r. Figure 5 shows the j/r-\ndependence of r∆kFQandr∆kQH. Thephaseboundaries\nforr= 10 determined by the present approximation are\nshown in Fig. 6.\nThecalculationin thissubsectionsuggeststhat theQP\nphase found in the small- jlimit (IIIA) and that in the\nlarge-jlimit (IIIB) form a single phase, although it is\nexplicitly demonstrated only in the large rlimit.\nIV. NUMERICAL GROUND STATE PHASE\nDIAGRAM\nForsmall r, therearetwotypesofnonmagneticground\nstates and the Gaussian transition occurs between them\nas described in Ref. [8]. The perturbational approaches\nin III, however, predict the presence of the LMF and\nthe QP phases in addition to the conventional spin gap\nphases.\nWe start with the case of r= 2, where the frustration\nisfairlystrong. Thegroundstatephasediagramisshown0 0.5 102040\nj\nkLMFGap II\n(Haldane)r=10\nQP\nFIG. 6: Phase diagram for r= 10 using the approximation in\nsubsection IIIC.\n0 0.5 10123\njr=2\nLMFPPF\nGap II\n(Haldane)\nkN=18\nN=24\nQP\nFIG. 7: Ground state phase diagram of (1) for r= 2.0. In\nthis and following figures, phase boundaries determined fro m\nthe numerical diagonalization data for N= 18 and 24 are\nshown. The solid lines are guide for eye. The broken lines are\nthe results of the small- japproximation in IIIA.\nin Fig. 7. The phase boundaries calculated by using the\nnumerical diagonalization data for N= 18 and 24 are\nshown. Between the Haldane-like Gap II phase and the\nLMFphasewithspontaneousmagnetization M0=Ms/2,\nthere appear a QP phase, which is identified by the low-\nest excitation with total spin 2, [16, 17] for 0 .66/lessorsimilark≤1\nas expected from the perturbational calculation. For\nk/greaterorsimilar0.5, only the data for N= 18 are shown consid-\nering the quasi-trimerized nature of the QP phase. In\nspite of the limited system size, the phase boundary co-\nincides well with the perturbational results for small j\nas depicted by the broken lines. However, the QP phase\nvanishes when kdecreasesand jincreases. Instead, there\nappears a PPF phase with the spontaneous magnetiza-\ntion of intermediate values between Ms/2 and 0. The\ndetailed properties of this phase is discussed separately\nin the next section.6\n0 0.5 10123\nGap II\n(Haldane)Gap\n      I\nGap ILMFPPF\nkjr=1.5\nN=18\nN=24\nQP\nFIG. 8: Ground state phase diagram of (1) for r= 1.5.\n0 0.5 10123\nGap IGap II\n(Haldane)\nPPF\nLMF Gap Ij\nkr=1.3\nN=18\nN=24\nQP\nFIG. 9: Ground state phase diagram of (1) for r= 1.3..\nForr <2, the 3-spin singlet Gap I phase appears for\n0< k < k c. As shown in the phase diagrams Fig. 8 for\nr= 1.5, Fig. 9 for r= 1.3 and Fig. 10 for r= 1.2,\nboth the QP phase and the PPF phase shrink to regions\naroundk∼1 with the decrease of r. Atr= 1, the 3-spin\nsinglet phase extends up to k= 1 for small j, and both\nthe QP and the PPF phases vanish.\nThe phase boundary between the Gap I and the\nGap II phases is determined by the twisted boundary\nmethod [31, 32]. In Fig. 8 for r= 1.5, the Gap I phase\nconsists of two separate regions which have the same\nparity under the twisted boundary condition. However,\nthese two regions merge with the decrease of ras shown\nin Fig. 10 for r= 1.2. Therefore we conclude these two\nregions belong to a single phase.0 0.50123\nGap II\n(Haldane)\nGap IPPF\nLMFQPN=18\nN=24r=1.2\nkj\nFIG. 10: Ground state phase diagram of (1) for r= 1.2.\nFor small k, we find the region in which the singlet-\ntriplet excitation gap ∆ Ebehaves almost critically. Fig-\nure11(a)showsthe j-dependence ofthe scaledgap N∆E\nforr= 1.3 andk= 0.12 forN= 12,18 and 24. Al-\nthough the boundary of such region cannot be precisely\ndetermined, they roughly correspond to the shaded re-\ngions in Figs. 8, 9 and 10. Applying the twist boundary\nmethod, [31, 32] we calculate the ground state energies\nE+andE−with spin inversion parities + and −, respec-\ntively. Then we find that the spin inversion parity of the\nground state, or the sign of E+−E−, changes several\ntimes with the variation of jas shown in Fig. 11(b). In\nthis region, the difference E+−E−is extremely small\n(typically less than O(10−3) forN= 24), and becomes\neven smaller with the decrease of k. This behavior is\nmost prominent for j∼1 as shown in Figs. 9 and 10,\nunless this regime is covered by the ferrimagnetic phase\nas in Figs. 7 and 8.\nThe critical values of jat which the parity changes\nare shown by the dotted lines in Figs. 8, 9 and 10 for\neach system size. They depend sensitively on the system\nsize. Due to the limitation of the system size, we cannot\nconclude whether these lines correspond to some phase\ntransitions in the thermodynamic limit. However, for\nlargej, these lines are expected to be continuously con-\nnected to the similar lines of the effective model (18) in\nthe corresponding regime (the shaded region in Fig. 4).\nThe numerically estimated values of the central charge\ncof the effective model (18) on these lines suggest that\nthey are Gaussian transition lines with c= 1 among spin\ngap phases with extremely small gap and large scale sin-\nglet clusters [20]. Therefore it is likely that these lines\nin the present model are also similar Gaussian transition\nlines. However, considering the large ambiguity in the\nestimation of cin Ref. [20], other possibilities cannot be\nruled out. The elucidation of the nature of the ground7\n.\n0 2 400.20.4\njN∆E r=1.3 k=0.12\nperiodic boundary\nN=12\nN=18\nN=24(a)\n0 2 4−0.00200.0020.004E+−E−\nN=12\nN=18r=1.3 k=0.12\ntwisted boundary\nN=24\nj(b)\nFIG. 11: (a) The j-dependence of thescaled singlet-triplet en-\nergy gap N∆Ewith periodic boundary condition and (b) the\nenergy difference between the different parity ground states\nE+−E−with twisted boundary condition. The parameters\narer= 1.3,k= 0.12 andN= 12,18 and 24. The triangles\nindicate the values of jwhere the ground state parity changes\nunder the twisted boundary condition for N= 24.\nstate in this regime is left for future studies.\nV. PARTIALLY POLARIZED\nFERRIMAGNETIC PHASE\nA. Numerical Results\nTo clarify properties of the PPF phase, we calculated\nthe spontaneous magnetization M0of the ground state\nby the density matrix renormalization group (DMRG)\nmethod. In Fig. 12, we show M0as a function of j\nforr= 1.3 andk= 0.5 withN= 72. For small j,\nthe ground state is in the Gap I phase with M0= 0.\nWhenjincreases, the LMF phase sets in where M0=\nMs/2. The slight deviation of M0/Msfrom 0.5 in this\nphaseisduetotheboundaryeffectinevitablefortheopen\nboundary DMRG. With further increase of j, the ground\nstate enters into the PPF phase where the spontaneous\nmagnetization gradually decreases down to zero.\nTypical magnetization curve calculated by the DMRG0 100.20.4M0/Ms\njr=1.3 k=0.5 N=72\nFIG. 12: The j-dependence of the spontaneous magnetization\nfork= 0.5 andr= 1.3. The results are calculated in the\nN= 72 system by the DMRG method. The magnetization is\nnormalized by the saturated magnetization Ms≡2N/3.\n0 2 400.51\nM/Ms\nHj=1.7 k=0.4 r=1.5\nN=192\n0 0.05 0.100.20.4M/Ms\nH\nFIG. 13: Magnetization curve in the PPF phase with k=\n0.4,j= 1.7 andr= 1.5 forN= 192 calculated by the DMRG\nmethod.\nis presented in Fig. 13 for r= 1.5,j= 1.7 andk= 0.4.\nThe magnetization increases continuously from the zero\nfield value in the PPF phase in contrast to the LMF\nphase where the magnetization is quantized to the zero\nfield value up to a finite critical field [4]. This implies\nthat the magnetic excitation is gapless in the PPF phase\nas in the previously reported systems [9, 21, 22].\nThe local magnetization profile /angbracketleftSz(p)/angbracketrightcalculated by\nthe DMRG is plotted against pin Fig. 14 for r= 1.5,j=\n1.7 andk= 0.4. In addition to the period 2 oscillation,\nan incommensurate modulation is clearly observed in the\nlocal magnetization.\nSimilar behaviors are found in other frustration in-\nduced PPF phases in the spin-1/2 period-3 chain with\nnext nearest neighbor interaction [11] and in the model\nof Ref. [10]. We expect these features are common as-\npects of the frustration induced quantum PPF phase.8\n0 100−0.500.51\nS1\nS2Tr=1.5 j=1.7 k=0.4<Sz(p)>\nN=192\np\nFIG. 14: Local magnetization profile in PPF phase with k=\n0.4,j= 1.7 andr= 1.5 forN= 192 calculated by the DMRG\nmethod.\nφ\nχ\nS1T\nS2θ\nFIG. 15: Classical planar spin configuration in a triangle.\nB. Classical Picture\nTo understand the physical picture of the PPF phase,\nwe consider the classical limit of the Hamiltonian (1).\nSince the J2-bonds are not frustrated, the relative angles\nbetween the spins S1(p),S2(p) andT(p), which form a\ntriangle, are not affected by jin the absence of magnetic\nfield. The ground state of the whole chain can be con-\nstructed by arranging the triangles so that S1(p) and\nS2(p−1) are antiparallel. The classical ground state en-\nergyE△\nGof a 3-spin cluster consisting of the spins S1(p),\n0 1 2024\nPPFLMFr\nk\nFIG. 16: Classical phase diagram on the k-rplane..\nS1(1)T(1)\nS2(1) S1(2) S2(2) S1(3)T(3)\nS2(3)T(2)(a)\nS1(1)T(1)\nS2(1) S1(2) S2(2) S1(3)T(3)\nS2(3)T(2)(b)\nFIG. 17: Examples of classical groundstate configurations ( a)\nwith finite magnetization and (b) with vanishing magnetiza-\ntion. The inner arrows indicate the direction of the rotatio n\nof spins along each triangle.\n0 5 1000.51r=1.5 j=1.7\nHM/Ms\nk=0.2\nk=1.0\nk=2.0\nFIG. 18: Classical magnetization curves for k= 0.2,1.0 and\n2.0 with j= 1.7 andr= 1.5.\nS2(p) andT(p) is given by\nE△\nG(φ,θ,χ) =k\n2cos(φ−θ)+kr\n2cos(φ−χ)+1\n2cos(χ−θ),\n(33)\nassuming a planar configuration. We confine ourselves\nto the planar configuration, because nonplanar configu-\nrations have higher energy. The angles θ,φandχdenote\nthe polar angle of S1(p),T(p) andS2(p) measured from\nz-axis as shown in Fig. 15, respectively. By minimiz-\ningE△\nG, we find that a stable noncollinear configuration\nwithin the triangle is realized in the region\n/vextendsingle/vextendsingle/vextendsingle/vextendsingler−1\nr/vextendsingle/vextendsingle/vextendsingle/vextendsingle< k <r+1\nr(34)\nwhich is indicated as PPF in Fig. 16. Outside this re-\ngion, three different collinear configurations are flavored9\ndepending on the values of kandras\n|Sz\n1(p)Tz(p)Sz\n2(p)/angbracketright=\n\n|⇑↓↑/angbracketright k >r+1\nr,\n|⇑↑↓/angbracketright0< k <r−1\nr,\n|⇑↓↓/angbracketright0< k <1−r\nr.(35)\nIf these are regularly arranged along the chain keeping\nthe spins S1(p) andS2(p−1) antiparallel, the LMF phase\nand two kinds of N´ eel phases are realized as indicated in\nFig. 16.\nIn the region (34), however, the energy is invariant un-\nder the simultaneous rotation of two of the spins (say\nS2(p) andT(p)) around the remaining one (say S1(p)).\nTherefore the ground state of the whole chain has a\nmacroscopic degeneracy. Among these highly degenerate\nground states, the states with various values of magneti-\nzation are included. For example, the state depicted in\nFig. 17(a) has finite magnetization, while the other one\ndepicted in Fig. 17(b) has a spiral structure and has no\nnet magnetization.\nIn the presence of magnetic field, the degeneracy is\nlifted and one of the ground states which has the largest\nmagnetization is selected by an infinitesimal magnetic\nfield. In this state, the spin configuration should be re-\nstricted in a single plane, because the tilt out ofthe plane\nreduces the net magnetization. Then each triangle in the\nspin chaintakesoneoftwofolddegeneratespinconfigura-\ntions with clockwise and counterclockwise spin rotations.\nThese two configurations are distinguished by chirality.\nInauniformarrayoftriangleswiththesamechirality,the\nnet magnetization tends to be averaged out. Therefore,\nthe magnetization per triangle decreases with increasing\nlength of the array. To maximize the magnetization, an\narray of the triangles with alternating chirality depicted\nin Fig. 17(a) is most favorable. This also explains the\nperiod 2 oscillation observed in Fig. 14. With the in-\ncrease of the magnetic field, the spin orientation gradu-\nally changes to the direction of the magnetic field.\nThe ground state in a finite magnetic field His ob-\ntained by numerically minimizing the classical energy\nEG(H) =N/3/summationdisplay\np=1/bracketleftbiggk\n2cos(φ(p)−θ(p))+kr\n2cos(φ(p)−χ(p))\n(36)\n+1\n2cos(χ(p)−θ(p))+j\n2cos(χ(p−1)−θ(p))\n(37)\n−H(cosθ(p)+1\n2cosφ(p)+1\n2cosχ(p))/bracketrightbigg\n.(38)\nwhere the magnetic field is in z-direction. Consider-\ning the argument in the preceding paragraph, we as-\nsume a period 2 structure in the minimization procedure.\nTypical magnetization curves are shown in Fig. 18 for\nr= 1.5 andj= 1.7 withk= 0.2,1.0 and 2.0. At firstsight, these three curves correspond to the LMF phase\n(k= 0.2), the PPF phase ( k= 1.0) and a nonmagnetic\nphase (k= 2.0). However, the classical phase including\nthe point of k= 2.0 correspond to a N´ eel ordered state\n|Sz\n1(1)Tz(1)Sz\n2(1)Sz\n1(2)Tz(2)Sz\n2(2)···/angbracketright=| ⇑↓↑⇓↑↓···/angbracketright\nas discussed above. Nevertheless, we may regard this\nN´ eel phase as a classical counterpart of the Gap II (Hal-\ndane) phase, since the short range antiferromagnetic cor-\nrelation between the total spins of the 3-spin clusters is\ncommon in the N´ eel state and in the Haldane-like state,\nand the latter is more stable in the quantum case. Simi-\nlarly, in the classical PPF ground state has noncollinear\nspin structure with broken U(1) symmetry around the\nz-axis, while in the quantum case, the U(1) symmetry is\nrestored due to quantum fluctuation. It should be also\nnoted that another ordered ground state with structure\n| ⇑↓↓⇑↓↓ .../angbracketrightcorresponding to the classical counterpart\nof the 3-spin singlet phase appears for small ras shown\nin Fig. 16.\nThus, all the phases in the quantum model (1) have\nthe classical counterparts except for the QP phase which\nis of essentially quantum origin. Therefore the classical\npicture appears to be satisfactory at least qualitatively.\nThe parameter regime for each phase is, however, largely\ndifferent from that of the original quantum model (1).\nFor example, the condition for the classical noncollinear\nspin configuration (34) is independent of jand does not\ncover the numerically obtained region of the PPF phase.\nIn addition, the incommensurate modulation of the spin\nprofile is not explained in the classical model. These\nfeatures are essentially quantum effect which is beyond\nthe classical interpretation.\nVI. SUMMARY AND DISCUSSION\nA mixed spin chain with frustrated side chains is in-\nvestigated, when the frustration is strong. Not only the\nLMF phase but also the PPF phase appears between\nthe nonmagnetic phase and the LMF phase. The PPF\nphase has continuously varying spontaneous magnetiza-\ntion, which is not a simple fraction of the saturated mag-\nnetization. The localmagnetizationprofilehasanincom-\nmensurate structure in the PPF phase. These features\nare common with other examples of PPF phases induced\nby frustration. Classical interpretation of the PPF phase\nis also presented. It is pointed out that an infinitesimal\nmagnetic field selects the PPF state in an appropriate\nparameter region. For the quantum model, however, the\nPPF state is realized in the absence of magnetic field.\nThis suggests that quantum fluctuation selects one of the\nclassical ground states.\nThe presence of QP phase is demonstrated by the per-\nturbational calculation as well as the numerical method.\nAlthough the present perturbational calculation is car-\nried out for our specific model (1), the derivation of the\neffective spin-1 bilinear-biquadratic chain is quite gen-\neral and is applicable to models which have an effective10\nspin-1 degree of freedom in each unit cell. The QP phase\nappears around the point where the first order effective\ncouplingvanishesduetofrustration. HencetheQPphase\nis expected to be commonly found in a wide variety of\nfrustrated quantum spin chains.\nForr/greaterorsimilar1 andj∼1 andk≪1, the main chain couples\nonly weakly with side spins but the frustration is strong.\nInthiscase, thereappearsanalmostcriticalnonmagnetic\nground state in which the spin inversion parity under the\ntwisted boundary condition changesmany times with the\nvariation of parameters. Considering the continuity to\nthe similar ground state in the effective model for large j,\nit is likely that a series of Gaussian transitions take place\namong gapped phases with an extremely small gap.\nAlthoughthenatureofthisalmostcriticalgroundstate\nremains unresolved, we may speculate its physical origin\nin the following way: In the absence of side spins T(p),\nthe ground state of the main chain is ferrimagnetically\nordered. For small k, side spins are coupled to this fer-\nrimagnetic moment antiferromagnetically via K1bond\nand ferromagnetically via K2bonds. For r∼1, the ef-\nfective coupling is even weakened due to frustration, and\nforj= 1, the ferrimagnetic state of the main chain has\nno local valence bond structure. Therefore the effective\ncoupling among the side spins, which is mediated by the\nfluctuation in the main chain, would be very long ranged\nforj∼1. This implies that the resultant nonmagneticstate should have a highly nonlocal character. Thus we\nmay speculate that the ground state has an extremely\nsmall energy gap and large scale singlet clusters. A sim-\nilar ground state with extremely small gap is known in a\nS= 1/2 zigzag chain with ferromagnetic nearest neigh-\nborinteractionandantiferromagneticnextnearestneigh-\nbor interaction [27]. However we do not find an explicit\nmapping of the present model onto the field theory of\nRef. [27].\nACKNOWLEDGMENTS\nThe numericaldiagonalizationprogramis basedon the\npackage TITPACK ver.2 coded by H. Nishimori. The\nnumerical 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 Su-\npercomputing Division, Information Technology Center,\nUniversity of Tokyo. This work is partly supported by\na Grant-in-Aid for Scientific Research on Priority Areas,\n”Novel States of Matter Induced by Frustration”, from\nthe Ministry of Education, Science, Sports and Culture\nof Japan, and Fund for Project Research in Toyota Tech-\nnological Institute.\n[1] C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10,\n1399 (1969).\n[2] K. Okunishi and T. Tonegawa, J. Phys. Soc. Jpn. 72, 479\n(2003); K. Okunishi and T. Tonegawa, Phys. Rev. B 68,\n224422 (2003); T. Tonegawa, K. Okamoto, K. Okunishi,\nK. Nomura and M. Kaburagi, Physica B 346-347 , 50\n(2004).\n[3] K.HidaandI.Affleck, J. Phys.Soc. Jpn. 74, 1849 (2005).\n[4] T. Kuramoto, J. Phys. Soc. Jpn. 67, 1762 (1998); K.\nMaisinger, U. Schollw¨ ock, S. Brehmer, H-J. Mikeska and\nS. Yamamoto, Phys. Rev. B58, R5908 (1998).\n[5] A. Koga, S. Kumada and N. Kawakami, J. Phys. Soc.\nJpn.68, 642 (1999).\n[6] K. Takano, Phys. Rev. Lett. 82, 5124 (1999)\n[7] K. Takano, Phys. Rev. B 61, 8863 (2000); Physica B\n284-288 , 1555 (2000); J. Phys. Chem. Solids 62, 377\n(2001).\n[8] K. Takano and K. Hida, Phys. Rev. B 77, 134412 (2008).\n[9] N. B. Ivanov and J. Richter, Phys. Rev. B 69, 214420\n(2004).\n[10] S. Yoshikawa and S. Miyashita, J. Phys. Soc. Jpn. Suppl.\n74, 71 (2005).\n[11] K. Hida, J. Phys. Condens. Matter, 19, 145225 (2007).\n[12] In Refs. [10, 11], this phase is called ’noncollinear fe rri-\nmagnetic phase’ which is based on the classical picture.\nHowever, this term gives the impression that the U(1)\nsymmetry around the z-axis is spontaneously broken in\nthis phase. Of course, this is not the case in the present\none-dimensional model. Therefore we use the term ’par-\ntially polarized ferrimagnetism’ following [9] to avoidconfusion.\n[13] G. V. Uimin, JETP Lett. 12, 225 (1970).\n[14] C. K. Lai, J. Math. Phys. 15, 1675 (1974).\n[15] B. Sutherland, Phys. Rev. B 12, 3795 (1975).\n[16] G. F´ ath and J. S´ olyom, Phys. Rev. B44, 11836 (1991).\n[17] G. F´ ath and J. S´ olyom, Phys. Rev. B47, 872 (1993).\n[18] C. Itoi and M-H. Kato, Phys. Rev. B 55, 8295 (1997).\n[19] A. Lauchli,G. Schmid and S. Trebst, Phys. Rev. B 74,\n144426 (2006).\n[20] K. Hida , J. Phys. Soc. Jpn. 77044707 (2008).\n[21] S. Sachdev and T. Senthil, Ann. Phys. 251, 76 (1996).\n[22] L. Bartosch, M. Kollar and P. Kopietz, Phys. Rev. B 67,\n092403 (2003).\n[23] M. Hase, H. Kuroe, K. Ozawa, O. Suzuki, H. Kitazawa,\nG. Kido and T. Sekine, Phys. Rev. B 70, 104426 (2004).\n[24] H. Kageyama, T. Kitano, N. Oba, M. Nishi, S. Nagai,\nK. Hirota, L. Viciu, J. B. Wiley, J. Yasuda, Y. Baba,\nY. Ajiro and K. Yoshimura, J. Phys. Soc. Jpn. 74, 1702\n(2005).\n[25] T. Hamada, J. Kane, S. Nakagawa and Y. Natsume, J.\nPhys. Soc. Jpn. 57, 1891 (1988).\n[26] T. Tonegawa and I. Harada, J. Phys. Soc. Jpn. 56, 2153\n(1987).\n[27] C. Itoi and S. Qin, Phys. Rev. B 63, 224423 (2001).\n[28] N. Shannon, T. Momoi and P. Sindzingre, Phys. Rev.\nLett.96, 027213 (2006).\n[29] E. Lieb and D. C.Mattis, J. Math. Phys. 3, 749 (1962).\n[30] N. B. Ivanov, J. Richter and U. Schollw¨ ock, Phys. Rev.\nB58, 14456 (1998).\n[31] A. Kitazawa, J. Phys. A Math. Gen. 30, L285 (1997).11\n[32] A. Kitazawa and K. Nomura, J. Phys. Soc. Jpn. 66, 3944\n(1997)."    },    {        "title": "2001.06918v5.Dephasing_of_Transverse_Spin_Current_in_Ferrimagnetic_Alloys.pdf",        "content": "1 \n Dephasing of Transverse Spin Current in Ferrimagnetic Alloys  \nYoungmin Lim1, Behrouz Khodadadi1, Jie-Fang Li2, Dwight Viehland2, Aurelien Manchon3,4, \nSatoru Emori1,* \n1. Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA  \n2. Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, \nUSA \n3. Physical Science and Engineering Division (PSE), King Abdullah University of Science and \nTechnology (KAUST), T huwal 23955 -6900, Saudi Arabia  \n4. Aix -Marseille Univ, CNRS, CINaM, Marseille, France  \n* email: semori@vt.edu  \n \nIt has been predicted that transverse spin current can propagate coherently (without \ndephasing) over a long distance in antiferromagnetically orde red metals. Here, we \nestimate  the dephasing length of transverse spin current in ferrimagnetic CoGd alloys by \nspin pumping measurements  across the compensation point . A modified drift -diffusion \nmodel, which accounts for spin -current transmission through the ferrimagnet, reveals \nthat the  dephasing length is about 4 -5 times longer in nearly compensated CoGd than in \nferromagnetic metals. This finding suggest s that antife rromagnetic order can mitigate \nspin dephasing – in a manner analogous to spin echo rephasing for nuclear and qubit \nspin systems – even in structurally disordered alloys at room temperature. We also  find \nevidence that  transverse spin current interacts  more strongly with the Co sublattice than \nthe Gd sublattice.   Our result s provide fundamental insight s into the interplay between \nspin current and antiferromagnetic order, which are crucial for engineering spin torque \neffects in ferrimagnetic and antiferromagnetic metals.  \n  2 \n I. INTRODUCTION  \nA spin current is said to be coherent when the spin polarization of its carriers, such as \nelectrons, is locked in a uniform orientation or precessional phase.  How far a spin current \npropagate s before decohering underpins various phenomena in solids  [1,2]. Spin decoherence \ncan generally arise from spin-flip scattering , where the carrier spin polarization is randomized \nvia momentum scattering  [3,4]. In magnetic materials, electronic spin current polarized \ntransverse to  the magnetization can also decohere by dephasing , where the total carrier spin \npolarization vanishes  due to the destructive interferences of precessing spins (i.e., upon \naveraging over the Fermi surface ) [5–9]. In typical ferromagnetic metals ( FMs), the dephas ing \nlength 𝜆𝑑𝑝 is only ≈1 nm  [5–7,10]  whereas the spin -flip (diffusion) length 𝜆𝑠𝑓 may be \nconsiderably longer (e.g., ≈10 nm)  [3,4], such that dephasing dominates the decoherence of \ntransverse spin current.   \nFigure 1(a) qualitatively illustrates  the dephasing of a coherent electronic spin current  in \na FM spin sink . In this particular illustration, a coherent ac transverse spin current is excited by \nferromagnetic resonance (FMR) spin pumping  [11], although in general a coherent transverse \nspin current may be generated by other means (e.g., dc electric current  spin-polarized \ntransverse to the magnetization  [5,6,12] ). This  spin current , carried by electrons,  then \npropagates coherently through the normal metal spacer  (e.g., Cu, where 𝜆𝑠𝑓~100 nm is much \ngreater than the typical spacer thickness ) [13]. However, this spin current enters  the FM spin \nsink with a wide distribution of incident wavevectors1, spa nned by the Fermi surface of the FM. \nElectronic spins with different wavevectors require different times  to reach a certain depth in the \nFM, thereby spending different times in the exchange field. Thus, even though these electronic \nspins enter the FM with the same phase, they  precess about the exchange field in the FM by \ndifferent amounts . Within a few atomic monolayers in the FM, the transverse spin  polarization \naverages to zero; the spin current dephases  within a short length scale 𝜆𝑑𝑝 ≈ 1 nm .   \n                                                           \n1 An insulating tunnel barrier is known to filter the incident wavevectors to a narrow distribution  [94]. This \nfiltering effect can reduce dephasing and thus extend 𝜆𝑑𝑝.  3 \n  \nFIG. 1.  Dephasing of a coherent transverse spin current excited by ferromagnetic resonance (FMR) in the \nspin source. The spin current carried by electrons is coherent in the normal metal (NM) spacer layer \n(indicated by the aligned black arrows), but ente rs the spin sink with different incident wavevectors \n(dashed gray lines). (a) In the ferromagnetic metal ( FM) spin sink, the propagating spins accumulate \ndifferent precessional phases in the ferromagnetic exchange field (red vertical arrows) and completely  \ndephase within a short distance. (b) In the ideal antiferromagnetic metal ( AFM ) or ferrimagnetic metal \n(FIM), the spin current does not dephase completely in the alternating antiferromagnetic exchange field \n(blue and green vertical arrows), as any precess ion at one sublattice is compensated by the opposite \nprecession at the other sublattice.  In the case of a FIM that is an alloy of a transition metal (TM , such as \nCo) and rare -earth metal (RE , such as Gd ), the TM constitutes one sublattice (e.g., blue arrow s) and the \nRE constitutes the other  sublattice  (e.g., green arrow s).  \n \nTransverse spin currents in antiferromagnetically ordered metals have been predicted to \nexhibit longer 𝜆𝑑𝑝 [14–17]. This prediction may apply not only to intrinsic antiferromagnetic \nmetals (AFMs) but also compensated ferrimagnetic metals (FIMs) , which  consist of transition -\nmetal (TM) and rare -earth -metal (RE) magnetic sublattices  that are antiferromagnetically \ncoupled to each other  [18]. In the ideal case as illustrated in Fig. 1(b), the spin current interacts \nwith the staggered antiferromagnetic exchange field whose direction alternates at the atomic \nlength scale. The propagating spins precess in alternating directions as they move from one \nmagnetic sublattice to the next, such that spin dephasing is suppressed over multiple \nmonolayers. This c ance llation of dephasing in AFMs and FIMs is analogous to spin rephasing \nNM spacer FM sink\nAFM/FIM sink\nsource\nNM spacer source(a)\n(b)4 \n by π-pulses (Hahn spin echo method) in nuclear magnetic resonance  [19], which has recently \ninspired several approaches of mitigating decoherence of qubit spin systems  [20–22].  \nThe above idealized picture for extended coherence in antiferromagnetically ordered \nmetals (Fig. 1(b)) assumes a spin current without any sca ttering and simple layer -by-layer \nalternating collinear magnetic order. Finite scattering, spin -orbit coupling, and complex \nmagnetization states in real materials  may disrupt transverse spin coherence  [23–25]. The \ntransverse spin coherence length 𝜆𝑐 accounting for both spin -flip scattering and spin dephasing \nis given by  [10,26] , \n1\n𝜆𝑐=Re[√1\n𝜆𝑠𝑓2−𝑖\n𝜆𝑑𝑝2].                          (1)      \nThus, in real AFMs and FIMs, a  shorter coherence length results from reduced 𝜆𝑠𝑓 due to \nincreased spin -flip rates, or reduced 𝜆𝑑𝑝 due to momentum scattering and non -collinear \nmagnetic order that prevents perfect cancellation of dephasing  [23,24] . Most experiments on \nAFMs (e.g., polycrystalline IrMn) indeed show short coherence lengths of  ≈1 nm  [7,27 –30].  \nNevertheless, a recent experim ental study utilizing a spin -galvanic detection \nmethod  [31–35] has reported a long coherence length of >10 nm  at room temperature in FIM \nCoTb  [18]. The report in Ref.  [18] is quite surprising considering the strong spin -orbit coupling of \nCoTb, primarily from RE Tb with a large orbital angular momentum, which can result in \nincreased spin -flip scattering  [36–38] and noncollinear sperimagnetic order  [39–41]. TM and RE \nelements also tend to form amorphous alloys  [39–42], whose structural disorder may result in \nfurther scattering and deviation from layer -by-layer antiferromagnetic order. It therefore remains \na critical issue to confirm whether the cancellation of  dephasing (as depicted in Fig. 1(b)) \nactually extends transverse spin coherence in antiferromagnetically  ordered metals, particularly  \nstructurally disordered FIMs.  \nHere, we test for the suppressed dephasing of transverse spin current in ferrimagnetic \nalloys. Our experimental test consists of FMR  spin pumping measurements  [7,28]  on a series of \namorphous FIM CoGd spin sinks, which exhibit significantly weaker spin -orbit coupling than \nCoTb due to the nominally zero orbital angular momentum of RE Gd.  Our experimental results \ncombined with  a modified drift -diffusion model  [9,10,43,44]  reveal that spin dephasing is indeed \npartially  cancelled  in nearly compensated CoGd , with 𝜆𝑑𝑝 extended by a factor of 4 -5 compared \nto that for FMs . Moreover, we find evidence  that transverse spin current interacts more strongly \nwith the TM Co sublattice  than the RE Gd sublattice . Our results  suggest  that, even in the \npresence of substantial structural disorder, the antiferromagnetic ally coupled sublattices  in FIMs 5 \n can mitigate the  decoherence of transverse spin current. On the other hand, the maximum 𝜆𝑑𝑝 \nof ≈5 nm in FIM CoGd found here is  quantitatively at odds with the report of 𝜆𝑑𝑝> 10 nm in FIM \nCoTb  [18]. Our study  sheds light on the interaction of  transverse spin current with \nantiferromagnetic order, which underpins spin  torque control of FIMs and AFMs for fast \nspintronic devices  [45–47].  \n \nII. FILM GROWTH AND STATIC MAGNETIC PROPERTIES  \nWe deposited spin -valve -like stacks of Ti(3)/Cu(3)/Ni 80Fe20(7)/Cu(4)/Co 100-xGd x(d)/Ti(3) \n(unit: nm) with x = 0, 20, 22, 23, 25, 28, and 30 by dc magnetron sputtering on Si/SiO 2 \nsubstrates. The base pressure in the deposition chamber was better than 8 10-8 Torr. The  Ar \nsputtering gas pressure was 3 mTorr. The Ti(3)/Cu(3) seed layer promotes the growth of \nNi80Fe20 with low Gilbert damping and minimal inhomogeneous linewidth broadening, whereas \nthe Ti(3) capping layer protects the stack from oxidation. FIM Co 100-xGd x films with various Gd \nconcentrations (x in atomic %) were deposited by co -sputtering Co and Gd targets at different \nGd sputtering powers (resulting in an uncertainty in composition of ≈±0.5 at.% Gd), except for \nCo80Gd 20 and Co 70Gd 30 films that were deposite d by sputtering compositional alloy targets.  The \ndeposition rate of each target  was calibrated by x -ray reflectivity and was set at 0.020 nm/s  for \nTi, 0.144 nm/s for Cu, 0.054 nm/s for NiFe, 0.011 or  0.020 nm/s for Co, 0.008 – 0.020 nm/s for \nGd, 0.01 4 nm/s for Co 80Gd 20, and 0.01 2 nm/s for Co 70Gd 30. \nWe performed vibrating sample magnetometry (with a Microsense EZ9 VSM) to identify \nthe magnetic compensation composition at room temperature for Co 100-xGd x films. We measured \nCo100-xGd x as single -layer films and as part of the spin -valve -like stacks (NiFe/Cu/CoGd), both \nseeded by Ti(3)/Cu(3) and capped by Ti(3). CoGd in both types of samples exhibited identical \nresults within experimental uncertainty. In the spin -valve -like stacks, the C u spacer layer \nsuppresses static exchange coupling between the NiFe and CoGd layers. This interlayer \nexchange decoupling is corroborated by magnetometry results (e.g., Fig. 2 (a)) that  indicate \nseparate switching for the NiFe and CoGd layers.  \nFigure 2(b -g) summarizes the composition dependence of the static magnetic properties \nof Co 100-xGd x. Our results  corroborate the well -known trend in ferrimagnetic alloys: the saturation \nmagnetization converges toward zero and coercivity diverges near the composition at  which the \nmagnetic moments of the Co and Gd sublattices compensate. Comparing the results for \ndifferent CoGd thicknesses, we find that the magnetization compensation composition shifts \ntoward higher Gd content with decreasing CoGd thickness. A similar thi ckness dependence of \nthe compensation composition in TM -RE FIM films has been seen in prior studies  [18]. Since 6 \n precise determination of the mag netic compensation composition wa s difficult for smaller Co 100-\nxGd x thicknesses, we tentatively identify t he magnetic compensation composition window to be \nx ≈ 22-25, which is what we find for 5 -nm-thick CoGd . We also remark that the angular \nmomentum compensation composition is expected to be ≈1 Gd at. % below the magnetic \ncompensation composition, considering  the g-factors of Co and Gd ( gCo ≈ 2.15, gGd = 2.0)  [48]. \nCoGd layers in our stack structures do not show perpendicular magnetic anisotropy  [18,49 –58], \ni.e., CoGd films here are in -plane magnetized  [59–62].  \n \n-50 -25 0 25 50\n0200400\n02040\n0100200\n04080\n0100200\n15 20 25 300816M (a.u.)\nm0H (mT)m0Hc (mT)    Ms (kA/m)\n    Ms (kA/m)CoGd 40 nm(b)\n(c)\n(d)\n(e)\n(f)\n(g)   Ms (kA/m) m0Hc (mT)(a)\n m0Hc (mT)\n  CoGd 10 nm\n   CoGd 5 nm\n \nGd at.%\n \nFIG. 2.   (a) Hysteresis loop of Ni 80Fe20(7)/Cu(4)/Co 75Gd25(13) (unit: nm) . The Ni80Fe20 and Co 75Gd25 \nmagnetization s switch separately at in -plane fields ≈ 0.2 mT and ≈13 mT, respectively.  (b,d,f) Saturation 7 \n magnetization Ms and (c,e,g) coercivity μ0Hc  of Co 100-xGdx with thicknesses of 40 nm (b,c), 10 nm (d,e), \nand 5 nm (f,g).  \n \nIII. CHARACTERIZATION OF SPIN TRANSPORT BY SPIN PUMPING  \nA. FMR Spin Pumping Experiment  \nThe multilayer stacks (described in Sec. II) in our study consist of a NiFe spin source \nand a Co100-xGd x spin sink, separated by a diamagnetic Cu spacer.  A coherent spin current \ngenerated by FMR  [11,13]  in NiFe propagates through the Cu spacer and decoheres in the \nCo100-xGd x spin sink, yielding nonlocal Gilbert damping  [7,28] . The Cu  spacer layer suppresses \nexchange coupling  – and hence direct magnon coupling – between the NiFe and CoGd \nlayers  [63]. The diamagnetic Cu spacer also accommodates spin transport mediated solely by \nconduction electrons, such that direct interlayer magnon coupling  [17,64 –66] does not play a \nrole here2.  \nIn our FMR spin pumping experiments performed at room temperature , the half -width -at-\nhalf-maximum linewidth ∆𝐻 of the NiFe spin source  is measured at microwave frequencies f = \n2-20 GHz . The d etails of the FMR measurement method are in Appendix A. The FMR response \nof the NiFe layer is readily separated  from that of pure Co (x = 0), and CoGd did not yield FMR \nsignals above our instrumental background  (see Fig. 10 in Appendix A) . Thus, as shown in Fig. \n3, the Gilbert damping parameter α for the NiFe layer is quantified from the f dependence of  ∆𝐻 \nthrough  the linear fit,  \n 𝜇0∆𝐻=𝜇0∆𝐻0+ℎ\n𝑔𝜇𝐵𝛼𝑓,  (2)  \nwhere 𝑔≈2.1 is the Landé g-factor of Ni 80Fe20, 𝜇0 is the permeability of free space, ℎ is \nPlanck’s constant, 𝜇𝐵 is the Bohr magneton, 𝜇0∆𝐻0 (< 0.2 mT) is the zero -frequency linewidth \nattributed to m agnetic inhomogeneity  [67]. For NiFe without a spin sink, we obtain αno-sink ≈ \n0.0067, similar to typically reported values for Ni 80Fe20 [68,69] .   \nA finite thickness 𝑑 of spin sink results in a damping parameter αw/sink that is greater than \nαno-sink. For examp le, the damping increases significantly with just 𝑑=1 nm of Co (Fig. 3(a)), \nsuggesting substantial spin absorption by the spin sink. By contrast, a stack structure that \nincludes an insulating layer of Ti -oxide before the spin sink does not s how the enhanc ed \ndamping (Fig. 3 (a)). This observation is consistent with the Ti -oxide layer blocking the spin \ncurrent  [70,71]  between the spin source a nd spin sink layers. Thus, the enhanced damping Δ α = \n                                                           \n2 We note that some of the electronic spin current might be converted into a magnonic spin current  [95] at \nthe Cu/CoGd interface. However, for the sake of simplicity, we assume the dominance of electronic spin \ntransport in CoGd here.  8 \n αw/sink – αno-sink is nonlocal in origin, i.e., due to the spin current propagating through the Cu \nspacer and decohering in the magnetic spin sink  [7,10,11,13,28,43] . The decoherence of \ntransverse spi n current in the spin sink is then directly related to Δ α. This spin pumping method \nbased on nonlocal damping enhancement provides an alternative to the spin-galvanic \nmethod  [18] that is known to contain parasitic voltage signals  unrelated to spin transport  [31–35], \nas discussed further in Sec. IV -C.  \n0 5 10 15 200246810\n0 5 10 15 20 0 5 10 15 20 NiFe/Cu/Co(3)\n NiFe/Cu/Co(1)\n NiFe/Cu/Co(0.6)\n NiFe/Cu/TiOx(3)/Co(1)\n NiFe/CuHWHM Linewidth (mT)\nFrequency (GHz)(a)\nFrequency (GHz)0 5 10 15 20\nFrequency (GHz) NiFe/Cu/Co75Gd25(9)\n NiFe/Cu/Co75Gd25(5)\n NiFe/Cu/Co75Gd25(2.5)\n NiFe/Cu/Co75Gd25(1.6)\n NiFe/Cu(c)\nFrequency (GHz) NiFe/Cu/Co77Gd23(9)\n NiFe/Cu/Co77Gd23(5)\n NiFe/Cu/Co77Gd23(2)\n NiFe/Cu/Co77Gd23(1)\n NiFe/Cu(b)  NiFe/Cu/Co72Gd28(8)\n NiFe/Cu/Co72Gd28(5)\n NiFe/Cu/Co72Gd28(2.1)\n NiFe/Cu/Co72Gd28(1.5)\n NiFe/Cu(d)\n \nFIG. 3.  (a) Half -width -at-half-maximum (HWHM) FMR linewidth versus frequency for stacks with a spin \nsink (NiFe/Cu/Co( d: nm)), a stack without a spin sink (NiFe/Cu), and a stack with an insulating Ti -oxide \nspin blocker before the spin sink (NiFe/Cu/TiO x/Co). (b -d) FMR linewidth versus frequency for stacks with \ndifferent thicknesses (d: nm) of Co 100-xGdx spin sink s, where x = 23 (b), 25 (c), and 28 (d).  The slope \n(proportional to the damping parameter  𝛼, see Eq. (2)) saturates at d ≈ 1 nm for the FM Co spin sink (a), \nwhereas the slope saturates at a much larger d for the FIM Co100-xGdx spin sinks (b -d).  \n \nIn contrast to the large Δ α with an ultrathin FM Co spin sink, the damping enhancement \nwith 𝑑 is more grad ual for FIM CoGd sinks. Figure 3 shows e xemplary linewidth versus \nfrequency results for the spin sink compositions of Co 77Gd 23 (Fig. 3(b)), Co 75Gd 25 (Fig. 3(c)), and \nCo72Gd 28 (Fig. 3(d)) . A damping enhancement similar in magnitude to that of the 1 -nm-thick FM \nCo spin sink is reached only when the CoGd  thickness is several nm. This suggests that \ntransverse spin -current decoherence takes place over a greater length scale in FIM spin sinks \nthan in FM spin sinks .  \nIn Fig. 4, we summarize our experimental results of transverse spin decoherence (i.e., \nΔα) as a function of spin sink thickness 𝑑. It can be seen that Δ α for each spin sink composition \nsatur ates above a sufficiently large  𝑑. This apparent saturation thickness – related to how far \nthe transverse spin current remains coherent  [7,10]  – changes markedly with the spin sink \ncomposition. With FM  Co as the spin sink , the saturation of Δ α occurs at 𝑑≈1 nm, in \nagreement with 𝜆𝑑𝑝 reported before for FMs  [7,10] . By contrast, Δ α saturates at 𝑑≫1 nm for 9 \n FIM CoGd sinks. This observation again implies that transverse spin current remains coherent \ndeeper within FIM sinks than within FM sinks.  \n \n   \n012\n012\n012\n012\n012\n012\n0 5 10012Da (10-3)\nPure Co\nCo80Gd20Da (10-3) Da (10-3)\nCo78Gd22\nCo77Gd23Da (10-3) Da (10-3)\nCo75Gd25\nCo72Gd28Da (10-3) Da (10-3)\nSpin sink thickness, d (nm)Co70Gd30  \nFIG. 4.  Nonlocal damping enhancement Δ α versus spin sink thickness d for CoGd spin sinks with \ndifferent compositions.  \n \nWe now  consider possible mechanisms  of the longer decoherence length s in FIM sinks. \nIncreasing the Gd concentration dilutes  the magnitude of the  exchange field in Co 100-xGd x alloys, \nas evidenced by the reduction of the Curie temperature  from ≈1400 K for pure Co to ≈700 K for  \nCo70Gd 30 [72]. The diluted  exchange field would lead to slower spin dephasing (i.e., longer 𝜆𝑑𝑝), \nthereby requiring a thicker spin sink  for complete spin-current decoherence. This mechanism  \nwould yield 𝜆𝑑𝑝 that is inversely proportional to the Curie temperature  [73], such  that the spin -\nsink thickness 𝑑 at which  Δα saturates would increase monotonically with Gd content. However, \nwe do not observe such a monotonic  trend in Fig. 4 . Rather , the saturation thickness appears to 10 \n plateau or peak at a Gd content of x ≈ 25, which is close to the magnetic compensation \ncomposition window.  \nWe therefore consider an alternative mechanism , where the antiferromagnetically \ncoupled Co and Gd sublattices in the FIM alloys mitigate dephasing, as qualitatively illustrated \nin Fig. 1(b). This mechanism would be expected to maximize  𝜆𝑑𝑝 near the compensation \ncomposition . In the following subsection, we describe and apply a modified spin drift -diffusion \nmodel to estimate 𝜆𝑑𝑝 and examine the possible role of the antiferromagnetic order in the \nmitigation of dephasing  in FIM CoGd .  \n \nB. Modified Drift -Diffusion Model  \nWe wish to model  our experimental results and quantify 𝜆𝑑𝑝 for the series of CoGd sinks. \nThe conventional drift -diffusion model captures spin -flip scattering in nonmagnetic  \nmetals  [11,74,75] , but not spin dephasing that is expected to be significant in magnetic metals. \nThis conventional model also predicts a monotonic increase of Δ α with 𝑑 [11,74,75] , whereas  \nwe observe in Fig. 4 a non-monotonic behavior where Δ α overshoot s before approaching \nsaturation  for some compositions of CoGd . For spin pumping studies with magnetic  spin sinks, \ntypical models assume  that the transverse spin current decoheres by dephasing as soon as it \nenters the FM spin sink, i.e., 𝜆𝑑𝑝=0 [13,75,76] . Others fit a linear increase of Δα with 𝑑 up to \napparent saturation , deriving  𝜆𝑑𝑝=1.2±0.1 nm for FMs  [7,28,73] . However,  it is questionable \nthat this linear cut -off model applies  in a physically meaningful way to our experimental results  \n(Fig. 4 ), in which the increase of Δα to saturation is not generally linear.   \nWe therefore apply  an alternative model that captures  the dephasing (i.e., precession \nand decay) of transmitted transverse spin current in the magnetic spin sinks  by invoking the \ntransmitted spin -mixing conductance  𝑔𝑡↑↓ [5,9,10,43,44,77,78] . As illustrated in Fig. 5, 𝑔𝑡↑↓ \naccounts for the spin current transmitted thro ugh the spin sink, in contrast to the conventional \nreflected  spin-mixing conductance that accounts for the spin current reflected at the spin sink \ninterface  [7,79] . 𝑔𝑡↑↓ is a function of the magnetic spin sink thickness 𝑑 that must vanishes in the \nlimit of 𝑑≫𝜆𝑑𝑝, i.e., when the transverse spin current completely dephases in  a sufficiently thick \nspin sink  [5,9,77,78] . In conventional FM spin sinks, it is often  assumed  that 𝑔𝑡↑↓=0, which is \nequivalent to assuming  𝜆𝑑𝑝=0 [79].  11 \n  \nFIG. 5.  (a) Cartoon schematic of transverse spin current transport  in the spin -source/spacer/spin -sink \nstack. The FMR -driven spin source pumps a coherent transverse spin current (polarized along the x-axis) \nthrough the Cu spacer. The spin current reflected at the Cu (spacer)/CoGd (spin sink) interface is \nparameterized by the reflected spin -mixing conductance 𝑔̃2↑↓, whereas the spin current transmitted through \nthe CoGd spin sink is parameterized by the transmi tted spin -mixing conductance 𝑔𝑡↑↓. (b) Illustration of the \ndephasing of the transverse spin current propagating in the CoGd spin sink. The shrinking circle s \nrepresent the decay of the transverse spin polarization due to dephasing while it precesses about the \neffective net exchange field 𝐻𝑒𝑥𝑛𝑒𝑡. (c) Illustration of the oscillatory decay of the transverse spin current. \nThe complex transmitted spin -mixing conductance 𝑔𝑡↑↓ captures the transmitted transverse spin \npolarization components sx and sy.  \n \nFurthermore, 𝑔𝑡↑↓(𝑑) is a complex value where the real and imaginary parts are \ncomparable in magnitude  [78]. Re[𝑔𝑡↑↓(𝑑)] and Im[𝑔𝑡↑↓(𝑑)] are related to the two orthogonal \ncomponents  of the spin current transmitted through the magnetic spin sink  [9]. As illustrated in \nFig. 5 (a,b), the incident spin polarization is along the x-axis, whereas  the y-axis is normal to \nboth the incident spin polarization and the magnetic order of the spin sink. Re[𝑔𝑡↑↓(𝑑)] and \nreflected \nxyspin source Cu spacer CoGd spin sink Ticapd\nxy\nsxsy\nxytransmitted(a)\n(b)\n(c)\n0 1 2 3 4\nd/ldp\nHexnetHexnet12 \n Im[𝑔𝑡↑↓(𝑑)] represent the x- and y-components, respectively, of the transverse  spin polarization  \n(Fig. 5) ; these components  oscillate and decay while the spin current dephases.  \nWe approximate 𝑔𝑡↑↓(𝑑) with an oscillatory decay function  [9],  \n𝑔𝑡↑↓(𝑑)=𝑔𝑡,0↑↓(𝜆𝑑𝑝\n𝜋𝑑sin𝜋𝑑\n𝜆𝑑𝑝±𝑖[(𝜆𝑑𝑝\n𝜋𝑑)2\nsin𝜋𝑑\n𝜆𝑑𝑝−𝜆𝑑𝑝\n𝜋𝑑cos𝜋𝑑\n𝜆𝑑𝑝])exp (−𝑑\n𝜆𝑠𝑓), (3) \nwhere  𝑔𝑡,0↑↓ represents the interfacial contribution to 𝑔𝑡↑↓. Equation (3)  is identical to the function \nproposed by Kim  [9], except that we incorporate an exponential decay factor (with 𝜆𝑠𝑓 = 10 nm \nas explained  in Appendix  B) that approximates incoherent spin scattering  as an additional \nsource of spin -current decoherence . The sign between the real and imaginary terms in Eq. (3) \nrepresents the net precession direction of the transverse spin polarization: the positive sign \nindicates precession about an  exchange field along  the net magnetization, whereas the \nnegative sign indicates precession about a n exchange  field opposing  the net magnetization.   \nAlthough  the effective exchange field is along the net magnetization in most cases, we discuss \nin Sec. IV -B a case where the exchange field opposes the net magnetization.  \nThe oscillatory decay of 𝑔𝑡↑↓(𝑑) modeled by Eq. (3) is illustrate d in Fig. 5(c).  Although  Fig. \n5(c) shows a case with 𝑔𝑡,0↑↓ as a positive real  quantity , 𝑔𝑡,0↑↓ is generally complex. In particular, \nRe[𝑔𝑡,0↑↓] and Im[𝑔𝑡,0↑↓] represent  the filtering and rotation  [5,6], respectively, of spin current at the \ninterface  of the Cu spacer and the Co 100-xGd x sink.  \nWe incorporate Eq. (3 ) into the drift -diffusion model  by Tanig uchi et al.  [10] that uses  the \nboundary conditions applicable to our multilayer systems. Accordingly, the nonlocal Gilbert \ndamping enhancement ∆𝛼 due to spin decoherence in the spin  sink is given by  \n∆𝛼=𝑔𝜇𝐵\n4𝜋𝑀𝑠𝑡𝐹(1\n𝑔̃1↑↓+1\n𝑔̃2↑↓)−1\n,     (4) \nwhere 𝜇𝐵 is the Bohr magneton; 𝑔=2.1 is the gyromagnetic ratio, 𝑀𝑠=800 kA/m is the \nsaturation magnetization, and 𝑡𝐹=7 nm is the thickness of the NiFe spin source. 𝑔̃1↑↓=16 nm-2 \nis the renormalized reflected spin -mixing conductance at the NiFe/Cu interface ( see Appendix  \nB). 𝑔̃2↑↓ is the renormalized reflected spin -mixing conductance at the Cu/Co 100-xGd x interface, \nwhich depends on the Co 100-xGd x spin sink thickness 𝑑 as \n𝑔̃2↑↓=(1+Re[𝑔𝑡↑↓]Re[𝜂]+Im[𝑔𝑡↑↓]Im[𝜂])\n(1+Re[𝑔𝑡↑↓]Re[𝜂]+Im[𝑔𝑡↑↓]Im[𝜂])2+(Im[𝑔𝑡↑↓]Re[𝜂]−Re[𝑔𝑡↑↓]Im[𝜂])2𝑔̃𝑟↑↓,     (5) 13 \n where 𝜂=(2𝑒2𝜌𝑙+/ℎ)coth  (𝑑/𝑙+) with 𝑙+=√1/𝜆𝑠𝑓2−𝑖/𝜆𝑑𝑝2, 𝜌 is the resistivity of the spin sink, \nand 𝑔̃𝑟↑↓ is the renormalized reflected mixing conductance a t the Cu/ Co 100-xGd x interface with \n𝑑≫𝜆𝑑𝑝.  \nGiven the rather large number of parameters in the model, some assumptions to \nconstrain the fitting are required , as outlined in Appendix  B. These assumptions leave us with \nthree  free fit parameters: 𝜆𝑑𝑝, Re[𝑔𝑡,0↑↓], and Im[𝑔𝑡,0↑↓]. It is also possible to further reduce the \nnumber of free parameters by fixing Re[𝑔𝑡,0↑↓], particularly near the compensation compositio n as \nexplained in  Appendix  C. Qualitatively similar results are obtained with Re[𝑔𝑡,0↑↓] free or fixed.   \n012\n-8-404\n-4-202\n012\n012\n-202\n012\n-202\n-101\n012\n012\n-101\n0 5 10 15-101\n0 5 10012Da (10-3)\nPure Co\nCo80Gd20 Co80Gd20Pure Co\ng↑↓\nt(nm-2)\nRe[g↑↓\nt]\nIm[g↑↓\nt]\ng↑↓\nt(nm-2) g↑↓\nt(nm-2)Da (10-3) Da (10-3)\nCo78Gd22 Co78Gd22Da (10-3)\nCo77Gd23 Co77Gd23\ng↑↓\nt(nm-2) g↑↓\nt(nm-2)Da (10-3)\nCo75Gd25\nCo72Gd28Co75Gd25\nCo72Gd28Da (10-3)\ng↑↓\nt(nm-2) g↑↓\nt(nm-2)Da (10-3)\nSpin sink thickness, d (nm)Co70Gd30 Co70Gd30\n \nFIG. 6.  Left column: spin sink thickness dependence of nonlocal damping enhancement Δ α. The solid \ncurve indicates the fit using the modified drift -diffusion model. Right column: the real and imaginary parts \nof the transmitted spin -mixing conductance 𝑔𝑡↑↓ derived from the fit. The vertical dashed line indicates the \nspin dephasing length 𝜆𝑑𝑝.  14 \n  \nThe fit results using the modified drift -diffusion model are  shown as solid curves in the \nleft column of Fig. 6. These curves adequately reproduce the 𝑑 dependence of Δα for all spin \nsink compositi ons. We also show in the right column of Fig. 6  the 𝑑 dependence of 𝑔𝑡↑↓, \nillustrating the net precession and decay of the transverse spin current.  \nWith the modeled results in Fig. 6 , the overshoot in Δα versus 𝑑 can now be attributed to \nthe precession of the transverse spin current  [9,44,77] . For a certain magnetic spin sink \nthickness, the polarization of the spin current leaving the sink is opposite to that of the spin \ncurrent entering the spin sink . Since the difference between the leaving and entering spin \ncurrents is related to the spin angular momentum transferred to the magnetic order, the spin \ntransfer – manifesting as Δα here – can be enhanced  [9,44,77]  compared to  when the spin \ncurrent is completely dephased for 𝑑≫𝜆𝑑𝑝.  \nWe acknowledge that our assumptions in the  modified drift -diffusion model  do not  \nnecessarily capture  all phenomena that could impact spin transport  in a quantitative manner . \nFor example , there remain unresolved questions regarding the relationship between  resistivity 𝜌 \nand spin -flip length 𝜆𝑠𝑓, as well as the role of the thickness dependent compensation \ncomposition on dephasing, which warrant further studies in the future . Nevertheless, as \ndiscussed in Sec. IV, our approach reveals sa lient features of spin dephasing  in compensated \nFIMs , as well as the upper  bound of the transverse spin coherence length in FIM CoGd . \n \nIV. DISCUSSION  \nA. Compositional Dependence of Spin Transport  \nFigure 7  summarizes the main finding  of our work, namely  the relationship between the \nmagnetic compensation (Fig. 7 (a)) and the spin transport parameters (Fig. 7 (b-d)) of  FIM Co 100-\nxGd x. We a lso refer the readers to Fig. 16  in Appendix C, which shows that qualitatively similar \nresults are obtained when Re[𝑔𝑡,0↑↓] is treated as a fixed parameter.  \nWe first discuss the composition dependence of 𝜆𝑑𝑝, summarized in Fig. 7(b), which is \nderived from the modified drift -diffusion  model (vertical dashed lines in Fig. 6).  The results in \nFig. 7 (b) show  a peak value of 𝜆𝑑𝑝 ≈ 5 nm at x ≈ 25-28, which is close to  the magnetic \ncompensation composition window  (x ≈ 22-25, the shaded region in Fi g. 7). It is worth noting \nthat 𝜆𝑑𝑝 is maximized on the somewhat Gd -rich side of the compensation composition  window . \nWe attribute this observation to the stronger contribution to spin dephasing  from the Co \nsublattice, as discussed more in detail in Sec. IV-B.  15 \n  \nFIG. 7.  (a) Static magnetic properties (saturation magnetization Ms and coercive field Hc) reproduced from \nFig. 2, (b) spin dephasing length 𝜆𝑑𝑝, and the (c) real and (d) imaginary parts of the interfacial contribution \nof the transmitted spin -mixing conductance 𝑔𝑡,0↑↓ versus Gd content in the spin sink. The shaded region \nindicates the window of composition corresponding to magnetic compensation. The error bars in (b -d) \nshow the 95% confidence interval.  \n \n Our results  (Fig. 7(b)) indicate  up to a factor of ≈5 enhancemen t in 𝜆𝑑𝑝 for nearly \ncomp ensated FIMs compared to FM Co. This observation is  qualitatively  consistent with the \ntheoretical prediction that antiferromagnetic order mitigates the decoherence of transverse spin \ncurrent  [14–18]. In a nearly compensated FIM CoGd spin sink, the alternating Co and Gd \n-12-8-404Re[gt,0] (nm-2)\n0246ldp (nm) \n0 20 25 30-2024Im[gt,0] (nm-2)\nGd at. %(a)\n(b)\n(c)\n(d)\n050100150Ms (kA/m)\n01020\nm0Hc (mT)\n16 \n moments of approximately equal magnitude partially cancel the dephasing of the propagating \nspins . This scenario, illustrated in Fig. 1(b), is corroborated by our tight-binding calculations \nassuming coherent ballistic transport  (see Appendix  D). Transverse spin current in \ncompensated CoGd is therefore able to remain coherent ove r a longer distance than in FMs. \nWe note , however,  that the spin current decohere s within a finite length scale in any real \nmaterials,  due to the imperfect suppression of dephasing and the presence of incoherent  \nscattering.  \nWe now comment on the compositional dependence of Re[𝑔𝑡,0↑↓] and Im[𝑔𝑡,0↑↓], particularly \nin the vicinity of magnetic compensation . Our calculations  in Appendix  C predict  Re[𝑔𝑡,0↑↓] to be \nonly weakly dependent on the net ex change splitting  𝑘Δ of the magnetic spin sink. Our \nexperimental results (Fig. 7(c)) are in qualitative agr eement with this prediction, as  Re[𝑔𝑡,0↑↓] \ntakes a roughly constant value near magnetic compensation.  \nBy contrast, the same calculations in Appendix C predict a quadrati c dependence of \nIm[𝑔𝑡,0↑↓] on 𝑘Δ. Specifically,  Im[𝑔𝑡,0↑↓] converges  to zero as the net exchange 𝑘Δ approaches zero \n(i.e., magnetic sublattices approaching compensation). Our experimental results indeed show a \nminimum in Im[𝑔𝑡,0↑↓] when  𝜆𝑑𝑝 is maximized . We remark that Im[𝑔𝑡,0↑↓] is related to  how much the \npolarization of the spin current rotates upon entering the magnetic spin sink  [5,6]. When the \nmagnetic sublattices are nearly compensated, the s pin current sees a nearly canceled net \nexchange field such that the spin rotation (precession) is suppressed. Therefore, both t he \nreduction of Im[𝑔𝑡,0↑↓] and the enhancement of 𝜆𝑑𝑝 arise naturally from the cancellation of the net \nexchange field. Our results in Figs. 6 and 7  consistently point to the suppression of spin -current \ndephasing enabled by antiferromagnetic order.  \nIt is important to note that FIM TM -RE alloys in general are amorphous with no long -\nrange structural order. Instead of the sim ple layer -by-layer alternating order illustrated in Fig. \n1(b), the TM and RE atoms are expected to be arranged in a rather disordered fashion. \nConsidering that disorder and electronic scattering tend to quench transverse spin \ncoherence  [18,23,24] , it is remarkable that such amorphous FIMs permit extended 𝜆𝑑𝑝 at all. We \nspeculate the observed enhancement of transverse spin coherence is enabled by short -range \nordering of Co and Gd atoms, e.g., finite TM -TM and RE -RE pair correlations in the film plane \n(and TM -RE pair correlation out of the film plane) as sugges ted by prior reports  [18,80] .  \n \n  17 \n B. Distinct Influence of the Sublattices on Spin Dephasing  \nOur results (Fig. 7 ) indicate  that magnetic compensation is achieved in the Co 100-xGd x \ncomposition range  of x ≈ 22-25, while 𝜆𝑑𝑝 is maximized (and Im[𝑔𝑡,0↑↓] is minimized) at x ≈ 25-28. \nThis observation deviates from  the simple expectation  (Fig. 1(b) ) that spin dephasing is \nsuppressed when the two sublattices are compensated. Here, w e discuss why 𝜆𝑑𝑝 should be  \nmaximized  at a more Gd -rich spin sink composition than the magnetic compensation \ncomposition.  \nFirst, it should  be recalled  that the magnetic compensation composition is dependent on \nthe FIM thickness . Our magnetometry data (Fig.  2(b-g)) indicate  that the magnetic \ncompensat ion composition becomes more Gd -rich with decreasing CoGd thickness  𝑑. However , \nthe CoGd thickness 𝑑= 5 nm shown in Fig. 2(f,g) , from which the compensation composition is \ndeduced,  is close to  the estimated maximum 𝜆𝑑𝑝 of ≈ 5 nm . Therefore, t he thickness \ndepende nce of the compensation composition alone does not  explain the maximum  𝜆𝑑𝑝 on the \nGd-rich side of magnetic compensation.  We consider an alternative  explanation below.   \nGenerally, it  might be expected that transverse spin current interacts more strongly with \nthe TM Co magnetization (from the spin -split itinerant 3 d bands near the Fermi level)  [50,81]  \nthan the RE Gd magnetization (primarily from the localized 4 f levels ≈7 -8 eV below the Fermi \nlevel [82,83] ). This is  analogous to magnetotransport effects dominated by itinerant 3 d band \nmagnetism in TM -RE FIMs  [60,84] . If the interaction of transverse spin current with the Co \nsublattice is stronger  [81], more Gd would be needed to compensate spin depha sing. Thus, t he \ngreater contribution  of the Co sublattice to dephasing could explain why 𝜆𝑑𝑝 is maximized at a \nmore Gd -rich composition than the magnetic compensation composition.  \n We observe additional evidence for the stronger interaction  with the TM Co sublattice  by \ninspecting the dependence of 𝑔𝑡↑↓ on spin sink thickness 𝑑. The relevant results can be seen in \nthe right column of Fig. 6, but for the sake of clarity, we highlight  𝑔𝑡↑↓ vs 𝑑 for a select ed few \nCoGd compositions near magnetic compensation in Fig. 8. For most spin sink compositions  \n(e.g., Fig. 8(a),(c)) , the results are consistent with the straightforward scenario : the net  \nexchange field , felt by the transverse spin  current,  points along the net magnetization . However, \nthe Co 75Gd 25 spin sink (Fig. 8(b)) , which is on the Gd -rich side of the magnetic compensation \ncomposition,  exhibits a qualitatively different  phase shift in Im[𝑔𝑡↑↓]. This noteworthy result is \nconsistent with the transverse spin  precessing about a  net exchange field  that opposes  the net 18 \n magnetization3. In other words , the net magnetization in Co 75Gd 25 is dominated by the Gd \nmagnetization  (from 4 f orbitals far below the Fermi level ), but the net exchange field points \nalong t he Co magnetization  (from 3d bands near the Fermi level ). The retrograde spin \nprecession in Co 75Gd 25 suggests that transverse electronic spin current interacts preferentially \nwith the itinerant 3 d TM magnetism.   \n \nFIG. 8.  Transmitted spin -mixing conductance 𝑔𝑡↑↓ versus spin sink thickness 𝑑 near the magnetic \ncompensation com position , derived from the modified drift -diffusion model fit (Fig. 6) . Note the phase shift \nin Im[𝑔𝑡↑↓] for Co 75Gd25 (b). The cartoons on the right illustrate the net precession direction of the \ntransverse spin polarization in each  CoGd spin sink. Co 75Gd25 exhibits retrograde precession, opposite to \nthe precession direction in the other spin sink compositions.   \n                                                           \n3 The modeled curves for Re[𝑔𝑡↑↓] and Im[𝑔𝑡↑↓] in Fig s. 6 and 8 for Co 75Gd25 showing retrograde \nprecession are obtained by taking the negative sign between the real and imaginary terms in Eq. (3). \nAdequate fits could also be obtained by fixing the sign between the real and imaginary terms to be \npositive, but th is would require the signs of  Re[𝑔𝑡,0↑↓] and Im[𝑔𝑡,0↑↓] for Co75Gd25 to be opposite to those of \nthe other CoGd compositions. Since a sign flip in Re[𝑔𝑡,0↑↓] or Im[𝑔𝑡,0↑↓] with respect to CoGd composition is \nnot expected (according to our calculations  in Appendix C), we  conclude that retrograde precession is the \nphysically reasonable scenario for the Co 75Gd25 sink.  \n-2-1012\n Re[gt ]\n Im[gt ]Co77Gd23gt (nm-2)\n0 5 10-1.0-0.50.00.5gt (nm-2)Co72Gd28\nspin sink thickness, d (nm)\n-2-101gt (nm-2)Co75Gd25CoGd\nCoGd\nCo\nGd\n  \n(a)\n(b)\n(c)19 \n We also comm ent on the possible role of  angular momentum compensation in FIM \nCoGd. As noted in Sec. II , angular momentum compensation composition is slightly more Co-\nrich than the magnetic compensation composition. While angular momentum compensation is \nkey to  fast antiferromagnetic -like dynamics in FIMs  [53,55] , it is evidently  unrelated to the \nmaximum  𝜆𝑑𝑝 on the slightly Gd -rich side of the magnetic compensation composition.  Rather, \nwe conclude that 𝜆𝑑𝑝 in FIMs is governed by the effective net exchange field , where the TM \nsublattice can play a greater role than the RE sublattice.   \n \nC. Comparison with a Prior Report of Long Transverse Spin Coherence in Ferrimagnetic \nMetals  \nOur results (e.g., Fig s. 6-8) point to  mitigation of spin decoherence  in nearly \ncompensated FIM CoGd. However,  we do not observe  evidence for  a transverse spin \ncoherence length  in excess of 10 nm , which was  recently reported for CoTb  [18]. Owing to the \nweaker spin -orbit coupling in CoGd than in CoTb, o ne might expect longer length scales for spin \ndephasing (more collinear antiferromagnetic order) and spin diffusion (less spin -flip scattering) \nin CoGd than in CoTb. We discuss possible reasons as to why the maximum coherence length \nin CoGd in our present study is significantly shorter than that reported in CoTb by Ref.  [18].  \nA plausible  factor  is the difference in experimental method for deducing the coherence \nlength . Yu et al.  in Ref. [18] utilize spin -galvanic measurements on Co/Cu/CoTb/Pt stacks: FMR \nin the Co layer pumps a spin current presumably through the CoTb spacer and generates a \nlateral dc voltage from the inver se spin -Hall effect in the Pt detector  [85]. A finite dc voltage is \ndetected for a range of CoTb alloy spacer thicknesses up to 12 nm, interpreted as evidence that \nthe spin current propagates from Co to Pt even with >10 nm of CoTb in between. However, \nthere could be coexisting voltage contributions besides the spin -to-charge conver sion in the Pt \ndetector . For example, spin scattering in the CoTb layer could yield an additional inverse spin -\nHall effect  [86], i.e., the  reciprocal of the strong spin -orbit torque reported in CoTb  [87]. \nFurthermore, the FMR -driven spin -galvanic measurement could  pick up spin rectification  [31–\n33] and thermoelectr ic voltages  [34,35]  from the dynamics of the FM Co layer, which might be \nchallenging to disentangle from the inver se spin -Hall effect in Pt. While  a number of control \nexperiments are performed in Ref.  [18] to rule out artifacts, it is possible that some spurious \neffects are not completely suppressed. Figures 4(d) and 5(c) in Ref.  [18] show that the spin -\ngalvanic signal drops abruptly  with the inclusion of a finite thickness of CoTb space r and \nremains constant up to CoTb thickness >10 nm . This trend , at odds with  the expected gradual 20 \n attenuation of spin current with CoTb  thickness , may arise from other mechanisms unrelated to \nspin transmission  through CoTb.  \nIn our present study , the spin pumping method measures the nonlocal damping Da that \nis attributed to spin-current decoherence in the spin sink ( see Sec. III -A). This method  does not \ninvolve any complicat ions from spin -galvanic signals. It still  might be argued , however,  that our \nmethod does not necessarily allow for precise, straightforward quantification of spin transport  \ndue to the large number of  parameters involved in modeling ( see Appendix B). Nevertheless, \neven if there are quantitative errors in our modeling, it is incontrovertible that Da saturates (i.e., \nthe transverse spin current pumped int o CoGd decoheres)  within a length scale well below 10 \nnm. Thus, our results indicate that the spin coherence length in CoGd does not exceed 10 nm.  \nThere may be other factors leading to the discrepancy  between our study and Ref.  [18]. \nThe CoTb films in Ref.  [18] may have a higher degree of layer -by-layer ordering than our CoGd \nfilms. This appears unlikely, since  CoTb “multilayers” (grown by alternately depositing Co and \nTb) and  CoTb “alloys” (grown by simultaneously depositing Co and Tb) exhibit essentially the \nsame CoTb thickness dependence of spin -galvanic signal  [18]. Another possibility is that \nmagnons, rather than spin -polarized conduction electrons, are responsible for the remarkably \nlong spin coherence through CoTb. If this were the case, it is yet unclear how CoTb might \nexhibit a longer magnon coherence length than CoGd, or how the spin -galvanic method in \nRef. [18] might be more sensitive to magnon spin transport than the nonlocal damping method \nin our present study. A future study  that directly compares  CoTb and CoGd spin sinks (e.g., with \nthe nonlocal damping method ) may verify  whether transverse spin current s survive over > 10 nm \nin ferrimagnet ic alloys with strong spin -orbit coupling .  \n \nD. Possible Implications for Spintronic Device Applications  \nThe dephasing length  𝜆𝑑𝑝 of transverse electronic spin current fundamentally impacts \nspin torque effects  in a magnetic metal  [5,6]. Specifically, the transverse spin angular \nmomentum lost by  the spin current is transferred to the magnetization , thereby giving rise to a \nspin torque within a depth  of order  𝜆𝑑𝑝 from the  surface of the magnetic metal.  Due to the  short \n𝜆𝑑𝑝 ≈ 1 nm, the spin torque is more efficient  in a thinner  FM layer, which comes  at the expense \nof reduced thermal stability of the stored magnetic information. The longer 𝜆𝑑𝑝 in compensated \nFIMs (or AFMs) may enable efficient spin torques in thicker, more thermally stable magnetic \nlayers  for high -density nonvola tile memory devices  [18,52] . Another practical benefit of \nextended 𝜆𝑑𝑝 is that  it facilitates  tuning the magnetic layer thickness to enhanc e spin 21 \n torques  [9,44,77] .  We further emphasize that the increase of 𝜆𝑑𝑝 is evident even in amorp hous  \nFIMs. This suggests that optimally engineered  FIMs or AFMs  (e.g., with high crystallinity)  could \nexhibit much longer 𝜆𝑑𝑝, potentially yielding spin torque effects that are qualitatively distinct from \nthose in FMs  [14].  \n In addition to estimating  the dephasing length scale in FIMs, our study highlights the \nroles of the chemically distinct sublattices on spin dephasing . From conventional spin  torque \nmeasurements, i t is generally difficult to deduce whether a spin current in a TM -RE FIM \ninteracts more strongly with a particular sublattice  [54]. Our results (as discussed in Sec. IV -B) \nimply that the TM sublattice contributes more strongly to the dephasing of transverse spin \ncurrent  – and hence to sp in torque s. The greater role of the TM sublattice over the RE sublattice \ncould be crucial for engineering spin torque effects in compensated TM -RE FIMs  – e.g., for \nenabling ultrafast (sub-THz-range ) spin torque oscillators  [88].  \n \nV. CONCLUSION  \nIn summary, we have utilized broadband FMR spin pumping to estimate  the dephasing \nlength 𝜆𝑑𝑝 of transverse spin current in ferrimagnetic CoGd alloys  across the compensation \npoint . We obtain a maximum of 𝜆𝑑𝑝≈5 nm in nearly compensated CoGd, consistent with the \nantiferromagnetic order mitigating the decoherence (dephasing) of transverse spin current. The \nobserved maximum 𝜆𝑑𝑝 constitutes a factor of ≈4 -5 enhancement compared to that for \nferromagnetic metals. On the other hand, we do not find evidence for 𝜆𝑑𝑝 in excess of 10 nm in \nferrimagnetic alloys reported  in a recent study  [18]. Despite this  quantitative difference, our \nresults suggest that partial spin rephasing by antiferromagnetic order – i.e., analogous to  the \nspin-echo  scheme to counter spin decoherence  – is indeed operative even in disordered \nferrimagnetic alloys at room temperature.  Moreover, o ur results suggest that transverse spin \ncurrent interacts more strongly with the itinerant Co sublattice than the localized Gd sublatt ice in \nnearly compensated CoGd. The spin rephasing effect and the sublattice dependent interaction  \ncould impact  spin torques in ferrimagnetic alloys , with possible applications in  fast spintronic \ndevices.  Our finding also points to the possibility of further extending transverse spin coherence \nin structurally pristine antiferromagnetic metals, thus opening a new avenue for fundamental \nstudies of spin transport in magnetic media.  \n \n  22 \n Acknowledgements  \nThis research was funded in part by 4 -VA, a collaborative partnership for advancing the \nCommonwealth of Virginia, as well as by the ICTAS Junior Faculty Award  and the NSF Grant \nNo. DMR -2003914 . We thank Jean J. Heremans  for helpful discussions .  \n  23 \n APPENDIX A: FMR MEASUREMENT METHOD  \n25 30 35 40 45 50 55 60dIFMR/dH\nm0H (mT)f = 6 GHz\nNiFe (7 nm)\n \nFIG. 9.  FMR spectrum  of Ni 80Fe20(7)/Cu(4) at 6 GHz. The red curve in the bottom panel represents the fit \nusing Eq. (A1).  \n \nFMR spectra are acquired using a broadband spectrometer, with each sample placed on \na coplanar waveguide (film side down) and magnetized in -plane (with a quasi -static magnetic \nfield, maximum value ~1 T, from an electromagnet). The quasi -static field is swe pt while fixing \nthe frequency of the microwave field (transverse to the quasi -static field) to acquire the \nresonance spectrum. A radio -frequency diode and lock -in amplifier (with 700 Hz modulation field \nas the reference) is used to detect the signal, which  is recorded as the derivative of the \nmicrowave power absorption with respect to the applied field, as shown in Fig.  9. To obtain  the \nhalf-width -at-half-maximum FMR linewidth Δ H, the measured signal is fit with the derivative of \nthe sum of symmetric and an tisymmetric Lorentzian functions  [89],  \n \n \nwhere Hres is the resonance field and the coefficients A and S are the proportionality factors for \nthe antisymmetric and symmetric terms, respectively.  𝑑𝐼𝐹𝑀𝑅\n𝑑𝐻=𝐴2(𝐻−𝐻𝑟𝑒𝑠)∆𝐻\n[(𝐻−𝐻𝑟𝑒𝑠)2+(∆𝐻)2]2+𝑆(𝐻−𝐻𝑟𝑒𝑠)2−(∆𝐻)2\n[(𝐻−𝐻𝑟𝑒𝑠)2+(∆𝐻)2]2, (A1)  24 \n  \n60 80 100 120 140dIFMR/dH (a.u.)\nm0H (mT)f = 10 GHz\nNiFe (7 nm)\nCo (4 nm)Ni80Fe20(7)/Cu(4)/Co100-xGdx(d) (Unit: nm)\nNo spin sink\n(x=0, d=4)\n(x=25, d=1.2)\n(x=25, d=9)\n(x=28, d=0.6)\n(x=28, d=8)  \nFIG. 10. FMR spectra for Ni80Fe20 (7 nm)/Cu (4 nm)/Co 100-xGdx (d nm) with no spin sink (magenta) and \nwith spin sink layer of pure Co (black), Gd 25% (red and green) and Gd 28% (blue and light blue). NiFe \nand Co exhibit well -separated FMR. The different thicknesses for Gd 25 and 28% was selected to show \ntwo different transp ort regimes, i.e., below/above the 𝜆𝑑𝑝 found from  Fig. 6 in the main text. No FMR \nsignal attributable to Co 100-xGdx in our samples (20 ≤ x ≤ 30) is detected.  \n \nAPPENDIX B: ASSUMPTIONS IN THE MODIFIED DRIFT -DIFFUSION MODEL  \nGiven the rather large number of  parameters in the model  (𝑔̃1↑↓, 𝑔̃𝑟↑↓, 𝜌, 𝜆𝑠𝑓, 𝜆𝑑𝑝, Re[𝑔𝑡,0↑↓], \nand Im[𝑔𝑡,0↑↓]), some assumptions are required to constrain the modified drift -diffusion model \n(Sec. III -B). Our assumptions are as follow:  \n(1)  The renormalized reflected spin-mixing  conductance (accounting for the Sharvin \nconductance of Cu  [79]) is set equal for the Ni 80Fe20/Cu and Cu/Co interfaces, i.e., \n𝑔̃1↑↓=𝑔̃𝑟↑↓. Here, 𝑔̃𝑟↑↓ is the renormalized reflected spin -mixing conductance for the \nCu/Co interface, 𝑔̃2↑↓, in the limit of 𝑑≫𝜆𝑑𝑝. We compute 𝑔̃1↑↓ from a modified form of Eq. \n(4),  25 \n ∆𝛼𝑠𝑎𝑡𝐶𝑜=𝑔𝜇𝐵\n4𝜋𝑀𝑠𝑡𝐹(2\n𝑔̃1↑↓)−1\n,     (B1) \nwhere ∆𝛼𝑠𝑎𝑡𝐶𝑜=0.0022  is the average of ∆𝛼 for Co spin sink thicknesses 𝑑≥3 nm (large \nenough that the transverse spin current is essentially  completely dephased). The \nrenormalized spin -mixing conductance 𝑔̃1↑↓ at the Ni 80Fe20/Cu interface is found to be 16 \nnm-2.  \n(2)  For each ferrimagnetic Co 100-xGd x spin sink composition, we compute 𝑔̃𝑟↑↓, shown in Eq. \n(5)) via \n∆𝛼𝑠𝑎𝑡𝐶𝑜𝐺𝑑=𝑔𝜇𝐵\n4𝜋𝑀𝑠𝑡𝐹(1\n𝑔̃1↑↓+1\n𝑔̃𝑟↑↓)−1\n,     (B2) \nwhere ∆𝛼𝑠𝑎𝑡𝐶𝑜𝐺𝑑 is the average of ∆𝛼 for the sa mples with the three largest spin sink \nthicknesses , where ∆𝛼 is essentially saturated . The values of 𝑔̃𝑟↑↓ are summarized in Fig. \n11.  \n \n0 20 25 30051015g\nr   \nGd at. %~(nm-2)  \nFIG. 1 1. Composition dependence of the renormalized reflected mixing conductance 𝑔̃𝑟↑↓ at the Cu/ Co 100-\nxGdx interface with the Co 100-xGdx spin sink thickness 𝑑≫𝜆𝑑𝑝.  \n \n(3) We set 𝜆𝑠𝑓=10 nm, which is of the same order as  the reported spin diffusion length in \nCo [4]. We further note that if the claim of 𝜆𝑐>10 nm for ferrimagnets in Ref.  [18] were \ncorrect, 𝜆𝑠𝑓 would necessarily need to be >10 nm. This relatively long 𝜆𝑠𝑓 is equivalent \nto assuming quasi -ballistic spin transport in the spin sink, such that spin dephasing \n(rather  than scattering) governs the transverse spin coherence length, 𝜆𝑑𝑝<𝜆𝑠𝑓. Our \nquantitative results of 𝜆𝑑𝑝 are essentially unaffected  if 𝜆𝑠𝑓 ≳ 4 nm (e.g., Fig. 12), while \nsome variation can be seen in  Im[𝑔𝑡,0↑↓] with 𝜆𝑠𝑓.  26 \n  \nFIG. 12.  Dependence of the spin dephasing length 𝜆𝑑𝑝 and interfacial transmitted spin -mixing \nconductance 𝑔𝑡,0↑↓ of Co 72Gd28 on the spin -flip length 𝜆𝑠𝑓 in the modified drift -diffusion model.  \n \n(4) For the sake of simplicity, we assume constant values of 𝜆𝑠𝑓 and 𝜌 for a given spin -sink \ncomposition (although in general, 𝜆𝑠𝑓 may depend on the resistivity 𝜌 of the spin sink, \nwhich in turn can depend on the spin -sink thickness). In particular, we set 𝜌 at the \nvalues obtained from four -point resistivity measurements of 10 -nm-thick Co 100-xGd x films \n(Fig. 13) ; the results are summarized in the figure below. (The exception was pure Co \nwhere 𝜌 was set at 1.0x10-6 m, since using the measured resistivity 𝜌 = 0.3x10-6 m \nyielded an unphysically large |𝑔𝑡,0↑↓| of ≫10 nm-2. The higher effective resistivity for Co is \npossibly justified considering that ∆𝛼 saturates at 𝑑≈1 nm, where the resistivity is likely \nmuch higher than 0.3x10-6 m due to surface scattering.) We remark that the \nuncertainty in the spin -sink thickness dependence of 𝜌 and 𝜆𝑠𝑓 may result in  a \nsystematic  error in quantifying 𝜆𝑑𝑝; further detailed studies are warranted to elucidate \nthe relationship between electronic and spin transport in ferrimagnets. Nevertheless, \nour approach is sufficie nt for semi -quantitative examination of 𝜆𝑑𝑝 as a function of CoGd \ncomposition. I t is incontrovertible that FIM CoGd has a significantly longer transverse \ncoherence length than FM Co and that this coherence length (albeit well below 10 nm)  \nshows a maximum  near the magnetic compensation .  \n0246ldp (nm)\n4 6 8 10-2-10123Im[gt,0]\nRe[gt,0]gt,0 (nm-2) \nlsf (nm)\n\n27 \n \n0 10 20 30 400.00.51.01.52.02.5Resistivity, r (10-6 m)\nGd at.% \nFIG. 1 3. Resistivity 𝜌 with varying Gd at.% for SiO x(thermally oxidized)/Co 100-xGd x(10)/TiO x(3) (unit: nm) . \nThe resistivi ties of intermediate compositions between x = 20 and 30 are derived via linear interpolation.  \n \n(5) We assume 𝜆𝑑𝑝 to be a constant parameter for each spin -sink composition, i.e., \nindependent of the spin sink thickness. We note that there could be a deviation from \nthis assumption, considering that the magnetic compensation composition (hence net \nexchange splitting) ev idently depends on the thickness of the ferrimagnetic spin sink, as \nseen in our magnetometry results (Fig. S3).  \n \nWe also add a few remarks regarding the details of our fitting protocol. In fitting the \nexperimental data ∆𝛼 versus 𝑑 using the modified dri ft-diffusion model  (Fig. 6, left column) , we \nassign a weight to each data point that is inversely proportional to the square of the error bar for \n∆𝛼 (from the linear fit of linewidth versus frequency). We fix Im[𝑔𝑡,0↑↓]=0.2𝑔̃𝑟↑↓ for pure Co, similar \nto what is suggested by Zwierzycki et al. [78]; for Co 80Gd 20, we impose the constraint 0.2𝑔̃𝑟↑↓<\n|𝑔𝑡,0↑↓|<0.3𝑔̃𝑟↑↓, since having the magnitudes of Re[𝑔𝑡,0↑↓] and Im[𝑔𝑡,0↑↓] as free parameters resulted \nin unphysically large |𝑔𝑡,0↑↓| with large error bars.   \n \nAPPENDIX C.CONSTANT 𝐑𝐞[𝒈𝒕,𝟎↑↓] NEAR MAGNETIC COMPENSATION  \nWhile the assumptions outlined in Appendix B results in three free fit parameters in the \ndrift-diffusion mode l (𝜆𝑑𝑝, Re[𝑔𝑡,0↑↓], and Im[𝑔𝑡,0↑↓]), it is possible to reduce the number of free \nparameters further by fixing Re[𝑔𝑡,0↑↓]. Here, we provide a physical justification for setting Re[𝑔𝑡,0↑↓] \nconstant, particularly near the magnetic compensation composition.  \nWe consider the simplest non -magnet and ferromagnet interface. In the nonmagnetic \nmetal, 𝑘𝑥=√𝑘𝐹2−𝜅2, and in the ferromagnet, 𝑘𝑥𝜎=√𝑘𝐹2+𝜎𝑘𝛥2−𝜅2. Here 𝜅 is the momentum 28 \n lying in  the plane of the interface and 𝑘𝛥=√2𝑚𝛥\nℏ quantifies the s -d exchange. The transmission \ncoefficient  𝑡𝜎 for spin 𝜎 reads  \n𝑡𝜎=2𝑘𝑥\n𝑘𝑥𝜎+𝑘𝑥 ,  (C1) \nand consequently, the transmitted mixing conductance at the interface  is \n𝑔𝑡,0↑↓=∫𝑑2𝜿\n4𝜋2𝑡↑(𝑡↓)∗ . (C2) \nFigure 14 displays the dependence of 𝑡↑(𝑡↓)∗ as a function of the in -plane momentum 𝜅2. \nThe real part is given in blue and the imaginary part is given in red. The first interesting feature \nis that all incident states contribute to Re[𝑔𝑡,0↑↓] whereas only states with incidence larger than \n𝜅2≥𝑘𝐹2−𝑘𝛥2 (rather close to grazing incidence) contribute to Im[𝑔𝑡,0↑↓].  \n \nFIG. 1 4. Momentum -resolved 𝑔𝑡,0↑↓ versus the in -plane momentum 𝜅2. Re[𝑔𝑡,0↑↓] in blue and Im[𝑔𝑡,0↑↓] in red.  \n \nUpon integration over the Fermi surface, the real and imaginary parts of the interfacial \ntransmitted mixing conductance reduce to Re[𝑔𝑡,0↑↓]≈𝑘𝐹2\n2𝜋 and Im[𝑔𝑡,0↑↓]=𝑘𝛥2\n4𝜋. In other words, the \nreal part is mostly independent of  the exchange  splitting , whereas the imaginary part is directly \nproportional to it (in fact, it is quadratic). This behavior is reported in Fig. 15 . When varying the \ncontent of Gd in the vicinity of  the magnetic compensation composition , it is therefore \nreasonable to assume that  Re[𝑔𝑡,0↑↓] is constant whereas Im[𝑔𝑡,0↑↓] varies.  \n29 \n  \nFIG. 15 . Interfacial transmitted mixing conductance versus  the exchange splitting  𝑘Δ. Re[𝑔𝑡,0↑↓] in blue and \nIm[𝑔𝑡,0↑↓] in red.  \n \nFigure 16  compares the fitting results of our experimental data with free Re[𝑔𝑡,0↑↓] (i.e., \nsame as Fig. 7) and with fixed Re[𝑔𝑡,0↑↓].  In Fig. 16(b), Re[𝑔𝑡,0↑↓] is fixed at a constant magnitude of \n1 nm-2 for all samples, except for Co and Co 80Gd 20 where larger Re[𝑔𝑡,0↑↓] (e.g., 8 and 4 nm-2, \nrespectively) is needed to obtain adequate fit curves. We observe that the qualitative conclusion \nis unaffected by whether or not Re[𝑔𝑡,0↑↓] is treated as a free parameter: 𝜆𝑑𝑝 is maximized (and \nIm[𝑔𝑡,0↑↓] is minimized) close to – or, more specificall y, on the slightly Gd -rich side of – the \nmagnetic compensation composition.   \n \n30 \n  \nFIG. 1 6. Comparison of modeling results with (a) free Re[𝑔𝑡,0↑↓] and (b) fixed Re[𝑔𝑡,0↑↓]. The shaded region \nindicates the window of composition corresponding to magnetic compensation.  \n \n \n  \n0246ldp (nm) \n-12-8-404Re[gt,0] (nm-2)\n0246ldp (nm) \n0 20 25 30-2024Im[gt,0] (nm-2)\nGd at. %(a) (b) \n-12-8-404Re[gt,0] (nm-2)\n0 20 25 30-2024Im[gt,0] (nm-2)\nGd at. %\n 31 \n APPENDIX D.  MODELING SPIN DEPHASING IN A FERRIMAGNETIC HETEROSTRUCTURE  \nTo understand the influence of the (collinear) magnetic order on the reflected and \ntransmitted mixing conductance s, we consider a magnetic trilayer composed of two equ ivalent \nnonmagnetic leads and a ferrimagnetic spacer, as illustrated in Fig. 17 . We compute the mixing \nconductances of this system assuming coherent ballistic transport.  \n \n \nFIG. 17.  Schematic of a ferrimagnetic trilayer as modeled below. The two magnetic sublattices are \ndenoted by red and blue arrows pointing in opposite directions.  \n \nSystem definition and boundary conditions:   Each layer is made of a three -dimensional \nsquare lattice with equal lattice parameter a for simplicity. The ferrimagnet is composed  of a \ntwo-atomic unit cell with sublattices A and B. In the {A,B} basis, the Hamiltonian for spin 𝜎 reads  \nℋ=(𝜀𝐴+𝜎𝛥𝐴−4𝑡𝐴𝜒𝑘 −2𝑡𝐴𝐵𝛾𝑘\n−2𝑡𝐴𝐵𝛾𝑘 𝜀𝐵+𝜎𝛥𝐵−4𝑡𝐵𝜒𝑘) \nwith \n𝜒𝑘=cos𝑘𝑥𝑎cos𝑘𝑦𝑎+cos𝑘𝑥𝑎cos𝑘𝑧𝑎+cos𝑘𝑧𝑎cos𝑘𝑦𝑎 \n𝛾𝑘=cos𝑘𝑥𝑎+cos𝑘𝑦𝑎+cos𝑘𝑧𝑎 \nIn this expression,  𝜀𝑖, 𝑡𝑖, 𝛥𝑖 are the onsite energy, hopping integral and magnetic exchange on \nsublattice i, and 𝑡𝐴𝐵 is the inter -sublattice hopping integral. The energy dispersion for spin 𝜎 and \nband 𝜂 is \n𝜀𝜂,𝜎=𝜀̅+𝜎𝛥̅−4𝑡̅𝜒𝑘+𝜂√(𝛿𝜀+𝜎𝛿𝛥 −4𝛿𝑡𝜒𝑘)2+4𝑡𝐴𝐵2𝛾𝑘2 \nand the associated eigenstate function reads  \n𝜙̂𝜂,𝜎=1\n√2√1+𝜂𝛽𝑘,𝜎|𝐴〉⊗|𝜎〉−𝜂\n√2√1−𝜂𝛽𝑘,𝜎|𝐵〉⊗|𝜎〉 \n𝛽𝑘,𝜎=𝛿𝜀+𝜎𝛿𝛥 −4𝛿𝑡𝜒𝑘\n√(𝛿𝜀+𝜎𝛿𝛥 −4𝛿𝑡𝜒𝑘)2+4𝑡𝐴𝐵2𝛾𝑘2 \n32 \n In the above  expressions, 𝛿𝜀=𝜀𝐴−𝜀𝐵\n2,𝜀̅=𝜀𝐴+𝜀𝐵\n2,𝛿𝛥=𝛥𝐴−𝛥𝐵\n2,𝛥̅=𝛥𝐴+𝛥𝐵\n2,𝛿𝑡=𝑡𝐴−𝑡𝐵\n2,𝑡̅=𝑡𝐴+𝑡𝐵\n2.  \nLet us now build the scattering wave function across the trilayer. The electron wave functions in \nthe left, central and right layers read  \n𝜓𝜎𝐿(𝒌)=(𝑒𝑖[𝑘𝑥𝐿(𝒌⊥)𝑥+𝒌⊥⋅𝝆]+∫𝑑2𝜿\n4𝜋2𝑟𝜎(𝜿)𝑒−𝑖[𝑘𝑥𝐿(𝜿)𝑥+𝜿⋅𝝆])|𝐶〉 \n𝜓𝜎𝐹(𝒌)=∫𝑑2𝜿\n4𝜋2∑𝑒𝑖𝜿⋅𝝆[𝐴𝜂,𝜎cos𝑘𝑥𝜂,𝜎(𝜿)𝑥+𝐵𝜂,𝜎cos𝑘𝑥𝜂,𝜎(𝜿)𝑥]𝜙̂𝜂,𝜎\n𝜂 \n𝜓𝜎𝑅(𝒌)=∫𝑑2𝜿\n4𝜋2𝑡𝜎(𝜿)𝑒𝑖[𝑘𝑥𝑅(𝜿)(𝑥−𝑑)+𝜿⋅𝝆]|𝐶〉 \nHere, 𝑘𝑥𝐿,𝑅(𝒌⊥) is a solution of the dispersion relation in the leads, 𝜀(𝒌)=𝜀𝑁−2𝑡𝑁𝛾𝑘=𝜀𝐹, 𝜀𝐹 \nbeing the Fermi energy, and 𝒌⊥is the in -plane component of the incoming wave vector. Similarly, \n𝑘𝑥𝜂,𝜎(𝜿) is determined by  the condition  𝜀𝜂,𝜎(𝒌)=𝜀𝐹. Now, let us determine the matching \nconditions at  𝑥=0 and  𝑥=𝑑. Because the leads and the ferr imagnetic layer possess a different \nfirst Brillouin zone, there will be Umklapp scattering  [90] that must be taken into account in the \nmatching procedure. In fact, the f irst Brillouin zone of the ferrimagnet is twice smaller than the \nfirst Brillouin zone of the leads, introducing an Umklapp momentum  𝑸=𝜋\n𝑎(𝒚+𝒛) [90–92]. We \nthen project the wave function of the ferrimagnetic layer on the normal metal orbital |𝐶〉 and use \nthe fact that  〈𝐶|𝐵〉=〈𝐶|𝐴〉𝑒𝑖(𝜅𝑦−𝑘⊥,𝑦)𝑎. In summary, the boundary conditions for normal \nscattering are  \n1+𝑟𝜎=∑[𝜂\n√2√1−𝜂𝜎𝛽 +1\n√2√1+𝜂𝜎𝛽 ]𝐴𝜂,𝜎\n𝜂 \n𝑖𝑘𝑥𝐿(1−𝑟𝜎)=∑[𝜂\n√2√1−𝜂𝜎𝛽 +1\n√2√1+𝜂𝜎𝛽 ]𝑘𝑥𝜂𝐵𝜂,𝜎\n𝜂 \n∑[𝜂\n√2√1−𝜂𝜎𝛽 +1\n√2√1+𝜂𝜎𝛽 ](𝐴𝜂,𝜎cos𝑘𝑥𝜂𝑑+𝐵𝜂,𝜎sin𝑘𝑥𝜂𝑑)\n𝜂 =𝑡𝜎 \n∑[𝜂\n√2√1−𝜂𝜎𝛽 +1\n√2√1+𝜂𝜎𝛽 ]𝑘𝑥𝜂(−𝐴𝜂,𝜎sin𝑘𝑥𝜂𝑑+𝐵𝜂,𝜎sin𝑘𝑥𝜂𝑑)\n𝜂 =𝑖𝑘𝑥𝑅𝑡𝜎\n  \nand for the Umklapp scattering, we obtain  \n𝑑𝜎=∑[𝜂\n√2√1−𝜂𝜎𝛽𝑄−1\n√2√1+𝜂𝜎𝛽𝑄]𝐴𝜂,𝜎\n𝜂 \n−𝑖𝑘𝑥,𝑄𝐿𝑑𝜎=∑[𝜂\n√2√1−𝜂𝜎𝛽𝑄−1\n√2√1+𝜂𝜎𝛽𝑄]𝑘𝑥,𝑄𝜂𝐵𝜂,𝜎\n𝜂 33 \n ∑[𝜂\n√2√1−𝜂𝜎𝛽𝑄−1\n√2√1+𝜂𝜎𝛽𝑄](𝐴𝜂,𝜎cos𝑘𝑥,𝑄𝜂𝑑+𝐵𝜂,𝜎sin𝑘𝑥,𝑄𝜂𝑑)\n𝜂 =𝑢𝜎 \n∑[𝜂\n√2√1−𝜂𝜎𝛽𝑄−1\n√2√1+𝜂𝜎𝛽𝑄]𝑘𝑥,𝑄𝜂(−𝐴𝜂,𝜎sin𝑘𝑥,𝑄𝜂𝑑+𝐵𝜂,𝜎sin𝑘𝑥,𝑄𝜂𝑑)\n𝜂 =𝑖𝑘𝑥,𝑄𝑅𝑢𝜎 \n \nIn order to make the above expressions easier to track, we  have  set \n𝑟𝜎=𝑟𝜎(𝒌⊥),𝑡𝜎=𝑡𝜎(𝒌⊥),𝛽=𝛽𝒌⊥,𝑘𝑥𝐿,𝑅=𝑘𝑥𝐿,𝑅(𝒌⊥),𝑘𝑥𝜂=𝑘𝑥𝜂(𝒌⊥) \n𝑑𝜎=𝑟𝜎(𝒌⊥+𝑸),𝑢𝜎=𝑡𝜎(𝒌⊥+𝑸),𝛽𝑄=𝛽𝒌⊥+𝑸,𝑘𝑥,𝑄𝐿,𝑅=𝑘𝑥𝐿,𝑅(𝒌⊥+𝑸),𝑘𝑥,𝑄𝜂=𝑘𝑥𝜂(𝒌⊥+𝑸) \n \nMixing Conductance s: The reflected and transmitted mixing conductances are \ndefined  [93] \n𝑔𝑟↑↓=(𝑒2\nℎ)∫𝑑2𝜿\n4𝜋2(1−𝑟↑𝑟↓∗−𝑑↑𝑑↓∗) \n𝑔𝑡↑↓=(𝑒2\nℎ)∫𝑑2𝜿\n4𝜋2(𝑡↑𝑡↓∗+𝑢↑𝑢↓∗) \nWe now compute these two quantities (both real and imaginary parts) as a function of the \nferrimagnetic layer thickness upon varying the magnetic exchange. For these calculations, we \nset 𝜀𝑁=𝜀̅=2.5 eV, 𝛿𝜀=0,𝑡𝑁=𝑡𝐴𝐵=1 eV, 𝑡̅=0. We also set 𝛥𝐴=1 eV and 𝛥𝐵 is varied \nbetween +1 eV (ferromagnet) and -1 eV (antiferromagnet). The results a re reported i n Fig . 18. \nThe reflected mixing conductance is weakly affected by the magnetic order, which is expected \nbased on simple free -electron arguments  [93], whereas the transmitted mixing conductance is \ndramatically modified when tuning the magnetic exchange. In the ferromagnetic limit ( 𝛥𝐵=\n+1 eV), it displays the expected damped osc illatory behavior already observed  [93] and \nattributed to the destructive interferences between precessing spins with different incidence (in \nother words, spin dephasing). Redu cing and then inverting the exchange leads to an overall \nreduction of the average exchange field, leading to an increase of the dephasing length, which \ndiverges in the antiferromagnetic limit ( 𝛥𝐵=−1 eV). 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/CS/CX/CP/D1/D3/D2/CS \r /CW/CP/CX/D2 /DB/CX/D8/CW /CU/D3/D9/D6/B9/D7/D4/CX/D2 \r/DD\r/D0/CX\r \r/D3/D9/D4/D0/CX/D2/CV/D7/B8 /CP/D2/CS /D7/D3/D1/CT /CQ/CP/D7/CX\r /D8 /DD/D4 /CT/D7 /D3/CU /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2/D0/CP/CS/CS/CT/D6/D7 /DB/CX/D8/CW /CV/CT/D3/D1/CT/D8/D6/CX\r /CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2/BA /CC/CW/CT /D0/CP/D7/D8 /CB/CT\r/D8/CX/D3/D2 \r/D3/D2 /D8/CP/CX/D2/D7 /D7/D3/D1/CT \r/D3/D2\r/D0/D9/D7/CX/D3/D2/D7/BA\nNOCNi\nCu\nab\nc❋ ❋ ❋\n❋ ❋\n❋ ❋❋\n❋❋J(a) (b)\nS1,nS2,nSS\n1,n+12,n+1/BY/CX/CV/D9/D6/CT /BD/BA /B4/CP/B5 /CB/D8/D6/D9\r/D8/D9/D6/CT /D3/CU /D8/CW/CT /D5/D9/CP/D7/CX/B9/D3/D2/CT/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /CQ/CX/D1/CT/D8/CP/D0/D0/CX\r \r/D3/D1/D4 /D3/D9/D2/CS/C6/CX/BV/D9/B4/D4/CQ/CP/B5/B4/C0 2\n/C7/B53· /BE/C02\n/C7 /DB/CX/D8/CW /CP/D0/D8/CT/D6/D2/CP/D8/CX/D2/CV /D7/CX/D8/CT /D7/D4/CX/D2/D7S1=SNi= 1 /CP/D2/CSS2=SCu=1\n2/CP/D0/D3/D2/CV /D8/CW/CT /CP/DC/CX/D7b /BA /C0/DD/CS/D6/D3/CV/CT/D2 /CP/D8/D3/D1/D7 /CP/D6/CT /D3/D1/CX/D8/D8/CT/CS /CU/D3/D6 \r/D0/CP/D6/CX/D8 /DD /CJ /BL ℄/BA /B4/CQ/B5 /CC/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /C6 /A1 /CT/CT/D0 /D7/D8/CP/D8/CT /D3/CU /D8/CW/CT/D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /BT/BY/C5 \r /CW/CP/CX/D2 /D1/D3 /CS/CT/D0 /B4/BD/B5 /CS/CT/D7\r/D6/CX/CQ/CX/D2/CV /D8/CW/CT /CQ/CX/D1/CT/D8/CP/D0/D0/CX\r \r/D3/D1/D4 /D3/D9/D2/CS /C6/CX/BV/D9/B4/D4/CQ/CP/B5/B4/C0 2\n/C7/B53· /BE/C02\n/C7/BA/BE/BA /CC/CW/CT /CP/D2/D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /C0/CT/CX/D7/CT/D2/CQ /CT/D6/CV \r/CW/CP/CX/D2/BT /CV/CT/D2/CT/D6/CX\r /D7/D4/CX/D2 /D1/D3 /CS/CT/D0 /D3/CU /D8/CW/CT /BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8 /CX/D7 /D8/CW/CT /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /D7/D4/CX/D2 \r /CW/CP/CX/D2 /DB/CX/D8/CW /BT/BY/C5/D2/CT/CP/D6/CT/D7/D8/B9/D2/CT/CX/CV/CW /CQ /D3/D6 /CT/DC\r /CW/CP/D2/CV/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /B4J >0 /B5 /CP/D2/CS /CP/D0/D8/CT/D6/D2/CP/D8/CX/D2/CV /D7/D4/CX/D2/D7S1\n/CP/D2/CSS2\n/B4S1> S2\n/B5/B8 /CS/CT/D7\r/D6/CX/CQ /CT/CS/CQ /DD /D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2\nH=JL/summationdisplay\nn=1(S1,n+S1,n+1)·S2,n−µBHL/summationdisplay\nn=1/parenleftbig\ng1Sz\n1,n+g2Sz\n2,n/parenrightbig\n. /B4/BD/B5/CC/CW/CT /CX/D2 /D8/CT/CV/CT/D6/D7n /D2 /D9/D1 /CQ /CT/D6 /D8/CW/CT /D9/D2/CX/D8 \r/CT/D0/D0/D7/B8 /CT/CP\r /CW \r/D3/D2 /D8/CP/CX/D2/CX/D2/CV /D8 /DB /D3 /D0/CP/D8/D8/CX\r/CT /D7/D4/CP\r/CX/D2/CV/D7 /CP/D2/CS /D8 /DB /D3 /CZ/CX/D2/CS/D7 /D3/CU /D5/D9/CP/D2 /D8/D9/D1/D7/D4/CX/D2 /D3/D4 /CT/D6/CP/D8/D3/D6/D7 S1,n\n/CP/D2/CSS2,n\n\r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS /CQ /DD /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D7/D4/CX/D2 /D2 /D9/D1 /CQ /CT/D6/D7 S1\n/CP/D2/CSS2\n/B4S1> S2\n/B5/B8/D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /BA g1\n/CP/D2/CSg2\n/CP/D6/CT /D8/CW/CTg /B9/CU/CP\r/D8/D3/D6/D7 /D3/CU /D8/CW/CT /D7/CX/D8/CT /D1/CP/CV/D2/CT/D8/CX\r /D1/D3/D1/CT/D2 /D8/D7/B8 µB\n/CX/D7 /D8/CW/CT /BU/D3/CW/D6 /D1/CP/CV/D2/CT/D8/D3/D2/B8/CP/D2/CSH /CX/D7 /CP/D2 /CT/DC/D8/CT/D6/D2/CP/D0 /D9/D2/CX/CU/D3/D6/D1 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /CP/D4/D4/D0/CX/CT/CS /CP/D0/D3/D2/CV /D8/CW/CTz /CS/CX/D6/CT\r/D8/CX/D3/D2/BA /BT/D7 /CP/D2 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT/CU/D3/D0/D0/D3 /DB/CX/D2/CV /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /B4/BD/B5 /CW/CP /DA /CT /CQ /CT/CT/D2 /CT/DC/D8/D6/CP\r/D8/CT/CS /CU/D6/D3/D1 /D1/CP/CV/D2/CT/D8/CX\r /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/D3/D2 /D8/CW/CT /D6/CT\r/CT/D2 /D8/D0/DD /D7/DD/D2 /D8/CW/CT/D7/CX/DE/CT/CS /D5/D9/CP/D7/CX/B9/BD/BW /CQ/CX/D1/CT/D8/CP/D0/D0/CX\r \r/D3/D1/D4 /D3/D9/D2/CS /C6/CX/BV/D9/B4/D4/CQ/CP/B5/B4/BW 2\n/C7/B53· /BE/BW2\n/C7 /CJ /BL ℄ /DB/CX/D8/CW /CP/D7/D8/D6/D9\r/D8/D9/D6/CT /DB/CW/CX\r /CW /CX/D7 /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /D8/CW/CT /D3/D2/CT /D7/CW/D3 /DB/D2 /CX/D2 /AS/CV/D9/D6/CT /BD /BM(S1,S2)≡(SNi,SCu) = (1,1/2) /B8J/kB=\n121K /B8g1≡gNi= 2.22 /B8g2≡gCu= 2.09 /BA /C0/CT/D2\r/CT/CU/D3/D6/D8/CW /DB /CT /D7/D9/CV/CV/CT/D7/D8g1=g2≡g /CP/D2/CS /D9/D7/CT /D8/CW/CT /C8/D0/CP/D2\r /CZ\r/D3/D2/D7/D8/CP/D2 /D8 ¯h /B8 /D8/CW/CT /BU/D3/D0/D8/DE/D1/CP/D2/D2 \r/D3/D2/D7/D8/CP/D2 /D8 kB\n/B8 /CP/D2/CS /D8/CW/CT /D0/CP/D8/D8/CX\r/CT /D7/D4/CP\r/CX/D2/CVa0\n/CP/D7 /D9/D2/CX/D8/CT/D7/BA/BE/BA/BD/BA /BZ/D6/D3/D9/D2/CS/B9/D7/D8/CP/D8/CT /D4 /D6/D3/D4 /CT/D6/D8/CX/CT/D7 /CP/D2/CS /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7/BE/BA/BD/BA/BD/BAH= 0/C4/CT/D8 /D9/D7 /D7/D8/CP/D6/D8 /DB/CX/D8/CW /D8/CW/CT \r/CP/D7/CTH= 0 /BA /BT \r\r/D3/D6/CS/CX/D2/CV /D8/D3 /C4/CX/CT/CQ/B9/C5/CP/D8/D8/CX/D7/B3 /D8/CW/CT/D3/D6/CT/D1 /CJ /BD/BI ℄/B8 /CU/D3/D6H= 0 /D8/CW/CT/CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /D3/CU /D8/CW/CT /CQ/CX/D4/CP/D6/D8/CX/D8/CT /D1/D3 /CS/CT/D0 /B4/BD/B5 /CW/CP/D7 /CP /D8/D3/D8/CP/D0 /D7/D4/CX/D2ST= (S1−S2)L /B8L /CQ /CT/CX/D2/CV /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU/BE/CB/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2/D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7/D9/D2/CX/D8 \r/CT/D0/D0/D7/B8 /D7/D3 /D8/CW/CP/D8 /CX/D8 /CX/D7 /D2/CT\r/CT/D7/D7/CP/D6/CX/D0/DD /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT /D3/D6/CS/CT/D6/CT/CS/BA /CB/D9\r /CW /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/D7 /D1/CP /DD /CP/D0/D7/D3/CQ /CT /D6/CT/CU/CT/D6/D6/CT/CS /D8/D3 /CP/D7 /D5/D9/CP/D2/D8/CX/DE/CT /CS /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT/D7 /D7/CX/D2\r/CT /D8/CW/CT /BY/C5 /D1/D3/D1/CT/D2 /D8 /CX/D7 /D5/D9/CP/D2 /D8/CX/DE/CT/CS /CX/D2 /CX/D2 /D8/CT/CV/D6/CP/D0 /B4/D3/D6/CW/CP/D0/CU/B9/CX/D2 /D8/CT/CV/D6/CP/D0/B5 /D1 /D9/D0/D8/CX/D4/D0/CT/D7 /D3/CU /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D9/D2/CX/D8 \r/CT/D0/D0/D7L /BA /BT/D7 /CT/DC/D4/D0/CX\r/CX/D8/CT/D0/DD /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/CS /CQ /CT/D0/D3 /DB /B4/CB/CT\r/D8/CX/D3/D2/BF/B5/B8 /D7/D8/D6/D3/D2/CV /CT/D2/D3/D9/CV/CW \r/D3/D1/D4 /CT/D8/CX/D2/CV /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 \r/CP/D2 /D7/D9/D7/D4 /CT/D2/CS /D8/CW/CX/D7 /D5/D9/CP/D2 /D8/CX/DE/CP/D8/CX/D3/D2 /D6/D9/D0/CT /CX/D2 /D7/D3/D1/CT /D6/CT/CV/CX/D3/D2/D7 /D3/CU/D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP\r/CT /DB/CW/CT/D6/CT /D8/CW/CT /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D3/D6/CS/CT/D6 /D7/D8/CX/D0/D0 /D7/D9/D6/DA/CX/DA /CT/D7/BA /CB/CX/D2\r/CT /D8/CW/CT /D1/D3 /CS/CT/D0 /CW/CP/D7 /CP/D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /D3/D6/CS/CT/D6/CT/CS /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/B8 /D8/CW/CX/D7 /D1/CP/CZ /CT/D7 /D8/CW/CT /BD/BW /D4/D6/D3/CQ/D0/CT/D1 /CP/D1/CT/D2/CP/CQ/D0/CT /D8/D3 /D8/CW/CT /D7/D4/CX/D2/B9/DB /CP /DA /CT /D8/CW/CT/D3/D6/DD/B4/CB/CF/CC/B5 /CP/D4/D4/D6/D3/CP\r /CW /CJ /BD/BJ ℄/BA/CE /CP/D0/D9/CP/CQ/D0/CT /D5/D9/CP/D0/CX/D8/CP/D8/CX/DA /CT /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CP/D2/CS /D0/D3 /DB/B9/D0/DD/CX/D2/CV /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 \r/CP/D2 /CQ /CT /CT/DC/B9/D8/D6/CP\r/D8/CT/CS /CP/D0/D6/CT/CP/CS/DD /CU/D6/D3/D1 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /D7/D4/CX/D2/B9/DB /CP /DA /CT /D8/CW/CT/D3/D6/DD /B4/C4/CB/CF/CC/B5 /CJ /BD/BK /B8 /BD/BL /B8 /BE/BC /B8 /BE/BD ℄/BA /CC/CW/CT /D3/D2/B9/D7/CX/D8/CT /D1/CP/CV/D2/CT/B9/D8/CX/DE/CP/D8/CX/D3/D2/D7 m1=∝an}bracketle{tSz\n1,n∝an}bracketri}ht /CP/D2/CSm2=−∝an}bracketle{tSz\n2,n∝an}bracketri}ht /B4m1−m2=S1−S2\n/B5 /CP/D6/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1/CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT /CZ /CT/CT/D4/CX/D2/CV /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT /D7/D4/CX/D2 \r/D3/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7/BA /C4/CB/CF/CC /CX/D1/D4/D0/CX/CT/D7/D8/CW/CP/D8 /CX/D2 /D8/CW/CT /CT/DC/D8/D6/CT/D1/CT /D5/D9/CP/D2 /D8/D9/D1 \r/CP/D7/CT(S1,S2) = (1,1/2) /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D7/D4/CX/D2 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /D6/CT/CS/D9\r/CT /D7/D9/CQ/B9/D7/D8/CP/D2 /D8/CX/CP/D0/D0/DD /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D3/D2/B9/D7/CX/D8/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/D7 /B4m2/S2≈0.39 /B5/BA /C6/D3/D8/CX\r/CT/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CP/D8 /CS/D9/CT /D8/D3 /D8/CW/CT/CQ/D6/D3/CZ /CT/D2 /D7/D9/CQ/D0/CP/D8/D8/CX\r/CT /D7/DD/D1/D1/CT/D8/D6/DD /B8 /CX/D2 /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D8/CW/CT/D6/CT /CP/D4/D4 /CT/CP/D6 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 \r/D3/D6/D6/CT\r/B9/D8/CX/D3/D2/D7 /D8/D3m1\n/CP/D2/CSm2\n/D6/CT/D7/D9/D0/D8/CX/D2/CV /CU/D6/D3/D1 /D8 /DB /D3/B9/CQ /D3/D7/D3/D2 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /CQ /D3/D7/D3/D2/CX\r /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CJ /BE/BE ℄/BA /CD/D4/D8/D3 /D7/CT\r/D3/D2/CS /D3/D6/CS/CT/D6 /CX/D21/S2\n/B4/CX/D2 /D6/CT/D7/D4 /CT\r/D8 /D8/D3 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B5/B8 /CB/CF/CC /D7/CT/D6/CX/CT/D7 /CV/CX/DA /CT/D7 /D8/CW/CT /D4/D6/CT\r/CX/D7/CT /D6/CT/B9/D7/D9/D0/D8m2= 0.29388 /CJ /BE/BF ℄/B8 /D8/D3 /CQ /CT \r/D3/D1/D4/CP/D6/CT/CS /DB/CX/D8/CW /D8/CW/CT /CS/CT/D2/D7/CX/D8 /DD/B9/D1/CP/D8/D6/CX/DC /D6/CT/D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2/B9/CV/D6/D3/D9/D4 /B4/BW/C5/CA /BZ/B5/CT/D7/D8/CX/D1/CP/D8/CT m2= 0.29248 /CJ /BE/BC ℄/BA /BT/D2/D3/D8/CW/CT/D6 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /D4 /CT\r/D9/D0/CX/CP/D6/CX/D8 /DD /D3/CU /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2 /CX/D7 /D8/CW/CT /CT/DC/B9/D8/D6/CT/D1/CT/D0/DD /D7/D1/CP/D0/D0 \r/D3/D6/D6/CT/D0/CP/D8/CX/D3/D2 /D0/CT/D2/CV/D8/CW /B4/D7/D1/CP/D0/D0/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D9/D2/CX/D8 \r/CT/D0/D0/B5 /D3/CU /D8/CW/CT /D7/CW/D3/D6/D8/B9/D6/CP/D2/CV/CT /D7/D4/CX/D2 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7/CJ /BD/BL /B8 /BE/BC ℄/BA /CC/CW/CX/D7 /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /CT/DC/D4/D0/CP/CX/D2/D7 /D8/CW/CT /CV/D3 /D3 /CS /D5/D9/CP/D2 /D8/CX/D8/CP/D8/CX/DA /CT /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/D3 /CS/CT/D0 /CP\r /CW/CX/CT/DA /CT/CS /DB/CX/D8/CW/D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2/CP/D0 /CP/D4/D4/D6/D3/CP\r /CW /D9/D7/CX/D2/CV /D1/CP/D8/D6/CX/DC/B9/D4/D6/D3 /CS/D9\r/D8 /D7/D8/CP/D8/CT/D7 /CJ /BE/BG /B8 /BE/BH ℄/BA\n00.511.522.53\n0 0.2 0.4 0.6 0.8 1E\nπkk+\n−\n2k/EExcitation energies\n+●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●/BY/CX/CV/D9/D6/CT /BE/BA /BW/CX/D7/D4 /CT/D6/D7/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D3/D2/CT/B9/D1/CP/CV/D2/D3/D2 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 /B4E±\nk\n/B5 /CX/D2 /D8/CW/CT /D7/DD/D7/D8/CT/D1(S1,S2) = (1,1/2)\r/CP/D0\r/D9/D0/CP/D8/CT/CS /D9/D4 /D8/D3 /D7/CT\r/D3/D2/CS /D3/D6/CS/CT/D6 /CX/D2 /D8/CW/CT1/S2\n/D6/CT/D0/CP/D8/CX/DA /CT /D8/D3 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CJ /BE/BJ ℄/BA /CC/CW/CT /D4 /D3/CX/D2 /D8/D7/CX/D2E±\nk\n/D7/CW/D3 /DB /D2 /D9/D1/CT/D6/CX\r/CP/D0 /BX/BW /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6 /D4 /CT/D6/CX/D3 /CS/CX\r \r /CW/CP/CX/D2/D7 /B4/CU/D3/D6E+\nk\n/B5 /CP/D2/CS /C9/C5/BV /D6/CT/D7/D9/D0/D8/D7 /B4/CU/D3/D6E−\nk\n/B5 /CJ /BE/BD ℄/BA/CC /D9/D6/D2/CX/D2/CV /D8/D3 /D8/CW/CT /CT/DC\r/CX/D8/CP/D8/CX/D3/D2 /D7/D4 /CT\r/D8/D6/D9/D1/B8 /CB/CF/CC /D4/D6/CT/CS/CX\r/D8/D7 /D8 /DB /D3 /D8 /DD/D4 /CT/D7 /D3/CU /D0/D3 /DB/B9/D0/DD/CX/D2/CV /D1/CP/CV/D2/D3/D2 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7\nE±\nk\n/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /B8 /CX/D2 /D8/CW/CT /D7/D9/CQ/D7/D4/CP\r/CT/D7 /DB/CX/D8/CW /DE \r/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /D8/D3/D8/CP/D0 /D7/D4/CX/D2Sz\nT=ST±1 /B4/D7/CT/CT /AS/CV/D9/D6/CT /BE/B5/BA/C1/D2 /D8/CW/CT /D0/D3/D2/CV /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /D0/CX/D1/CX/D8k≪1 /B8 /D8/CW/CT /BY/C5 /D1/D3 /CS/CTE−\nk\n/D8/CP/CZ /CT/D7 /D8/CW/CT /C4/CP/D2/CS/CP/D9/B9/C4/CX/CU/D7/CW/CX/D8/DE /CU/D3/D6/D1\nE−\nk=̺s\nM0k2+O(k4), /B4/BE/B5/DB/CW/CT/D6/CT̺s=JS1S2\n/CX/D7 /D8/CW/CT /D7/D4/CX/D2 /D7/D8/CX/AR/D2/CT/D7/D7 \r/D3/D2/D7/D8/CP/D2 /D8 /CJ /BE/BI ℄ /CP/D2/CSM0= (S1−S2)/2 /CX/D7 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CS/CT/D2/D7/CX/D8 /DD /D3/CU/D8/CW/CT /D2/CT/D8 /BY/C5 /D1/D3/D1/CT/D2 /D8/BA /CC/CW/CX/D7 /CU/D3/D6/D1 /D3/CU /D8/CW/CT /BZ/D3/D0/CS/D7/D8/D3/D2/CT /D1/D3 /CS/CT/D7 /CX/D7 /D8 /DD/D4/CX\r/CP/D0 /CU/D3/D6 /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/D7/B8 /CP/D2/CS/D6/CT/AT/CT\r/D8/D7 /D8/CW/CT /CU/CP\r/D8 /D8/CW/CP/D8 /D8/CW/CT /BY/C5 /D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /CX/D7 /CX/D8/D7/CT/D0/CU /CP \r/D3/D2/D7/D8/CP/D2 /D8 /D3/CU /D8/CW/CT /D1/D3/D8/CX/D3/D2/BA /BT/D7 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/CS/CQ /CT/D0/D3 /DB/B8 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 ̺s\n/CP/D2/CSM0\n/D4/D0/CP /DD /CP /CQ/CP/D7/CX\r /D6/D3/D0/CT /CX/D2 /D8/CW/CT /D0/D3 /DB/B9/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r/D7 /D3/CU/D8/CW/CT /D1/D3 /CS/CT/D0/BA /C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CW/CP/D2/CS/B8 /D8/CW/CT /BT/BY/C5 /CQ/D6/CP/D2\r /CWE+\nk\n/CX/D7 /CV/CP/D4/D4 /CT/CS/B8 /DB/CX/D8/CW /CP /D1/CX/D2/CX/D1 /D9/D1 /CP/D8k= 0 /CV/CX/DA /CT/D2/CQ /DD∆+= 2(S1−S1)J /CX/D2 /CP /C4/CB/CF/CC /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/BA /C6/D3/D8/CT/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CP/D8 /D8/CW/CT /C4/CB/CF/CC /CT/D7/D8/CX/D1/CP/D8/CT/D7 /CU/D3/D6\n̺s\n/CP/D2/CS /D8/CW/CT /BT/BY/C5 /CV/CP/D4∆+/CP/D6/CT /CU/D9/D6/D8/CW/CT/D6 /D6/CT/D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /CQ /DD /D8/CW/CT /CQ /D3/D7/D3/D2/B9/CQ /D3/D7/D3/D2 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CJ /BE/BJ ℄/BM /C1/D2 /D8/CW/CT/BF/C6/BA /BU/BA /C1/DA/CP/D2/D3/DA/CT/DC/D8/D6/CT/D1/CT /D5/D9/CP/D2 /D8/D9/D1 /D7/DD/D7/D8/CT/D1(S1,S2) = (1,1/2) /B8 /CP/D0/D6/CT/CP/CS/DD /D8/CW/CT /AS/D6/D7/D8 /D3/D6/CS/CT/D6 \r/D3/D6/D6/CT\r/D8/CX/D3/D2/D7 /CX/D21/S2\n/CV/CX/DA /CT /D8/CW/CT/D6/CT/D7/D9/D0/D8/D7̺s/(Js1s2) = 0.761 /CP/D2/CS∆(+)= 1.676J /BA /CC/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /C5/D3/D2 /D8/CT/B9/BV/CP/D6/D0/D3 /B4/C9/C5/BV/B5 /D6/CT/D7/D9/D0/D8 /CU/D3/D6 /D8/CW/CX/D7/D4/CP/D6/CP/D1/CT/D8/CT/D6 1.759 /CJ /BE/BD ℄ \r/D0/CT/CP/D6/D0/DD /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CT /CX/D1/D4 /D3/D6/D8/CP/D2\r/CT /D3/CU /D8/CW/CT1/S2\n\r/D3/D6/D6/CT\r/D8/CX/D3/D2/D7/BA /C8/D6/CT\r/CX/D7/CT /CT/D7/D8/CX/D1/CP/D8/CT/D7/CU/D3/D6 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT(1,1/2) \r /CW/CP/CX/D2 /CW/CP /DA /CT /CP/D0/D7/D3 /CQ /CT/CT/D2 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D9/D7/CX/D2/CV \r/D0/D9/D7/D8/CT/D6 /D7/CT/D6/CX/CT/D7 /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /CJ/BE/BK ℄/BMm2= 0.292487(6) /B8∆+/J= 1.7591(6) /CP/D2/CS̺s/(Js1s2) = 0.831(5) /BA/BE/BA/BD/BA/BE/BAH/negationslash= 0/CC/CW/CT /CX/D2 /D8/D6/D3 /CS/D9\r/D8/CX/D3/D2 /D3/CU /CP /CI/CT/CT/D1/CP/D2 /D8/CT/D6/D1 /CX/D2 /BX/D5/BA /B4/BD/B5 /D0/CT/CP /DA /CT/D7 /CP/D0/D0 /CT/CX/CV/CT/D2/D7/D8/CP/D8/CT/D7 /DB/CX/D8/CW /CP /CV/CX/DA /CT/D2 /DE \r/D3/D1/D4 /D3/D2/CT/D2 /D8/D3/CU /D8/CW/CT /D8/D3/D8/CP/D0 /D7/D4/CX/D2Sz\nT\n/CX/D2 /DA /CP/D6/CX/CP/D2 /D8/B8 /DB/CW/CX/D0/CT /D7/CW/CX/CU/D8/CX/D2/CV /D8/CW/CT /CT/CX/CV/CT/D2/CT/D2/CT/D6/CV/CX/CT/D7 /CQ /DDhSz\nT\n/B8 /DB/CW/CT/D6/CTh=gµBH /BA /C1/D2/D4/CP/D6/D8/CX\r/D9/D0/CP/D6/B8 /D8/CW/CT /D3/D2/CT/B9/D1/CP/CV/D2/D3/D2 /D7/D8/CP/D8/CT/D7E±\nk\n/B8 \r/CP/D6/D6/DD/CX/D2/CV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/D7 ±1 /B8 \r /CW/CP/D2/CV/CT /D8/D3E±\nk(h) =E±\nk∓h /B8/CX/BA/CT/B8 /D8/CW/CT /CI/CT/CT/D1/CP/D2 /D8/CT/D6/D1 /CX/D2 /D8/D6/D3 /CS/D9\r/CT/D7 /CP /CV/CP/D4∆−(h) =h /CU/D3/D6 /D8/CW/CT /BY/C5 /CQ/D6/CP/D2\r /CW /CP/D2/CS /D6/CT/CS/D9\r/CT/D7 /D8/CW/CT /CV/CP/D4 /D3/CU /D8/CW/CT/BT/BY/C5 /CQ/D6/CP/D2\r /CW/B8∆+(h) = ∆+−h /BA /C1/CU /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /CT/DC\r/CT/CT/CS/D7 /D8/CW/CT \r/D6/CX/D8/CX\r/CP/D0 /DA /CP/D0/D9/CTh=hc1= ∆+/B8 /D8/CW/CT/BT/BY/C5 /CV/CP/D4 \r/D0/D3/D7/CT/D7 /CP/D2/CS /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CT/D2 /D8/CT/D6/D7 /CP \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CT /DB/CW/CX\r /CW /CX/D7 /CT/DC/D4 /CT\r/D8/CT/CS /D8/D3 /CQ /CT /CP /CZ/CX/D2/CS /D3/CU /C4/D9/D8/D8/CX/D2/CV/CT/D6/D0/CX/D5/D9/CX/CS/B8 /CX/D2 /CP/D2/CP/D0/D3/CV/DD /DB/CX/D8/CW /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D3/D8/CW/CT/D6 /CV/CP/D4/D4 /CT/CS /D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D0/CX/CZ /CT /D8/CW/CT /D7/D4/CX/D2/B9/BD \r /CW/CP/CX/D2 /CP/D2/CS /D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /D0/CP/CS/CS/CT/D6 /CJ /BE/BL /B8 /BF/BC /B8 /BF/BD ℄/BA /CC/CW/CT \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CT /D8/CT/D6/D1/CX/D2/CP/D8/CT/D7 /CP/D8 /CP /D7/CT\r/D3/D2/CS \r/D6/CX/D8/CX\r/CP/D0 /AS/CT/D0/CSh=hc2= 3J /CP/D8/DB/CW/CX\r /CW /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CQ /CT\r/D3/D1/CT/D7 /CU/D9/D0/D0/DD /D4 /D3/D0/CP/D6/CX/DE/CT/CS /CJ /BF/BE ℄/BA /BT/D2 /CP\r\r/D9/D6/CP/D8/CT /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /B41,1/2 /B5/C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /B4H∝ne}ationslash= 0) /CX/D2 /D8/CW/CT \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CThc1< h < h c1\n/CW/CP/D7 /CQ /CT/CT/D2 /CP\r /CW/CX/CT/DA /CT/CS /CQ /DD /CP /D1/CP/D4/D4/CX/D2/CV /D8/D3/CP/D2 /CT/AR/CT\r/D8/CX/DA /CT /D7/D4/CX/D2/B9/BD/BB/BE /CG/CG/CI \r /CW/CP/CX/D2 /CX/D2 /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /DB/CW/CX\r /CW /D9/D7/CT/D7 /DA /CP/D6/CX/CP/D8/CX/D3/D2/CP/D0 /D1/CP/D8/D6/CX/DC/B9/D4/D6/D3 /CS/D9\r/D8/D7/D8/CP/D8/CT/D7 /CJ /BE/BH ℄/BA /C1/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV/D0/DD /B8 /D7/D9\r /CW /CP \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CT /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CU/D3/D6 /D8/CW/CT /CT/D2 /D8/CX/D6/CT \r/D0/CP/D7/D7 /D3/CU/D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 (S1,S2) /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX\r /CG/CG/CI /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2/D7 /DB/CX/D8/CW /CT/CP/D7/DD/B9/D4/D0/CP/D2/CT /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CP/D2/CS /D7/CW/D3/D6/D8/B9/D6/CP/D2/CV/CT/CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7/BA /BT/D2 /CT/DC/D8/CT/D2/D7/CX/DA /CT /D2 /D9/D1/CT/D6/CX\r/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7/B8 /D9/D7/CX/D2/CV /CT/DC/CP\r/D8/B9/CS/CX/CP/CV/D3/D2/CP/D0/CX/DE/CP/D8/CX/D3/D2 /B4/BX/BW/B5 /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6(1,1/2)/CP/D2/CS(3/2,1/2) /D4 /CT/D6/CX/D3 /CS/CX\r \r /CW/CP/CX/D2/D7/B8 /D7/D9/CV/CV/CT/D7/D8/D7 /D9/D2/CX/DA /CT/D6/D7/CP/D0 \r/D6/CX/D8/CX\r/CP/D0 /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /CU/D3/D6 /D8/CW/CT/D7/CT \r /CW/CP/CX/D2/D7 /CX/D2 /D8/CW/CT /CT/D2 /D8/CX/D6/CT/CX/D2 /D8/CT/D6/DA /CP/D0 /CU/D6/D3/D1 /D8/CW/CT /BY/C5 /D8/D3 /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /CX/D7/D3/D8/D6/D3/D4/CX\r /D4 /D3/CX/D2 /D8/D7 /CJ /BF/BF ℄/BA /BT/D0/D3/D2/CV /D8/CW/CX/D7 /D4/CW/CP/D7/CT/B8 /D8/CW/CT \r/D6/CX/D8/CX\r/CP/D0/AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D6/D9/D0/CT/CS /CQ /DD /CP \r/D3/D2/CU/D3/D6/D1/CP/D0 /AS/CT/D0/CS /D8/CW/CT/D3/D6/DD /D3/CU /BZ/CP/D9/D7/D7/CX/CP/D2 /D8 /DD/D4 /CT /DB/CX/D8/CW /CP /D8/D3/D4 /D3/D0/D3/CV/CX\r/CP/D0 \r /CW/CP/D6/CV/CTc= 1 /BA/BW/CX/D7\r/D9/D7/D7/CT/CS /D3/D2/CT/B9/D1/CP/CV/D2/D3/D2 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 \r/D3/D2 /D8/D6/D3/D0 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D4/D6/D3 \r/CT/D7/D7 /D3/CU /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2/BA/CB/CX/D2\r/CT /D8/CW/CT /D0/D3 /DB /CT/D7/D8 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2 /CX/D2\r/D6/CT/CP/D7/CX/D2/CV /D8/CW/CT /BY/C5 /D1/D3/D1/CT/D2 /D8 M=∝an}bracketle{tSz\n1,n+Sz\n2,n∝an}bracketri}ht /CT/DC/CW/CX/CQ/CX/D8/D7 /CP/D2 /CT/D2/CT/D6/CV/DD /CV/CP/D4/B8/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 \r/D9/D6/DA /CTM(H) /CW/CP/D7 /CP /D4/D0/CP/D8/CT/CP/D9 /CP/D8M=S1−S2\n/BA /CC/CW/CX/D7 /D4/D0/CP/D8/CT/CP/D9 /D4/CW/CP/D7/CT/B8 /CW/D3 /DB /CT/DA /CT/D6/B8/CP/D4/D4 /CT/CP/D6/D7 /CX/D2 /D8/CW/CT /D6/CT/D0/CP/D8/CT/CS \r/D0/CP/D7/D7/CX\r/CP/D0 /D7/DD/D7/D8/CT/D1/B8 /CP/D7 /DB /CT/D0/D0/BA /C1/D2 /D8/CW/CX/D7 \r/D3/D2/D2/CT\r/D8/CX/D3/D2/B8 /CX/D8 /CX/D7 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /D8/D3 /CT/DC/CP/D1/CX/D2/CT /D8/CW/CT/D6/D3/D0/CT /D3/CU /CS/CX/AR/CT/D6/CT/D2 /D8 /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX/CT/D7 /DB/CW/CX\r /CW /D1/CP /DD /CP/D4/D4 /CT/CP/D6 /CX/D2 /D6/CT/CP/D0 /D1/CP/D8/CT/D6/CX/CP/D0/D7/BA /BT/D7 /D8/D3 /D8/CW/CT /CT/DC\r /CW/CP/D2/CV/CT /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /B8/CX/D8 /D3 \r\r/D9/D6/D6/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /CT/AR/CT\r/D8/D7 /D7/CX/D1/D4/D0/DD /D7/D8/CP/CQ/CX/D0/CX/DE/CT /D8/CW/CT /D4/D0/CP/D8/CT/CP/D9 /CP/D8M=S1−S2\n/CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /CG/CH/CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /B8 /CX/BA/CT/BA/B8 /D8/CW/CT /CT/AR/CT\r/D8 /CX/D7 /D6/CT/CS/D9\r/CT/CS /D8/D3 /D5/D9/CP/D2 /D8/CX/D8/CP/D8/CX/DA /CT \r /CW/CP/D2/CV/CT/D7 /D3/CU /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D6/CT/D7/D9/D0/D8 /CJ /BF/BG ℄/BA /BT \r/D0/CT/CP/D6/CX/D2/CS/CX\r/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D3/D6/CX/CV/CX/D2 /D3/CU /D8/CW/CT /CS/CX/D7\r/D9/D7/D7/CT/CS /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D4/D0/CP/D8/CT/CP/D9 /DB /CP/D7 /D4/D6/D3 /DA/CX/CS/CT/CS /CU/D3/D6 /D8/CW/CT\n(1,1/2) \r /CW/CP/CX/D2 /DB/CX/D8/CW /CP /D7/CX/D2/CV/D0/CT/B9/CX/D3/D2 /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CU/D3/D6 /D7/D4/CX/D2/B9/BD /D7/CX/D8/CT/D7/B8D/parenleftbig\nSz\n1,n/parenrightbig2/B8 /DB/CW/CX\r /CW /CX/D7 /CT/DA /CT/D2 /D1/D3/D6/CT /D6/CT/CP/D0/CX/D7/D8/CX\r/CU/D3/D6 /D8/CW/CT /CT/DC/CX/D7/D8/CX/D2/CV /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D1/CP/D8/CT/D6/CX/CP/D0/D7 /CJ /BF/BH ℄/BA /C1/D8 /DB /CP/D7 /D6/CT/DA /CT/CP/D0/CT/CS/B8 /CX/D2 /D4/CP/D6/D8/CX\r/D9/D0/CP/D6/B8 /D8/CW/CP/D8 /D8/CW/CT /D1/CT\r /CW/CP/D2/CX/D7/D1 /D3/CU/D8/CW/CTM= 1/2 /D4/D0/CP/D8/CT/CP/D9 \r/CP/D2 \r /CW/CP/D2/CV/CT /CU/D6/D3/D1 /CP /C0/CP/D0/CS/CP/D2/CT /D8/D3 /CP /D0/CP/D6/CV/CT/B9D /D8 /DD/D4 /CT /D8/D6/D3/D9/CV/CW /CP /BZ/CP/D9/D7/D7/CX/CP/D2 /D5/D9/CP/D2 /D8/D9/D1\r/D6/CX/D8/CX\r/CP/D0 /D4 /D3/CX/D2 /D8/B8 /CT/D7/D8/CX/D1/CP/D8/CT/CS /CP/D7D/J= 1.114 /B8 /DB/CW/CX\r /CW /CX/D7 /CP /CZ/CX/D2/CS /D3/CU /CY/D9/D7/D8/CX/AS\r/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D3/D6/CX/CV/CX/D2/D3/CU /D8/CW/CTM= 1/2 /D4/D0/CP/D8/CT/CP/D9 /CX/D2 /D8/CW/CX/D7 /D7/DD/D7/D8/CT/D1/BA /CC /DB /D3 /D4/D0/CP/D8/CT/CP/D9 /D4/CW/CP/D7/CT/D7 /B4M= 1/2 /B5 /DB/CX/D8/CW /CS/CX/AR/CT/D6/CT/D2 /D8 /D4/CP/D6/CX/D8/CX/CT/D7/CW/CP /DA /CT /CP/D0/D7/D3 /CQ /CT/CT/D2 /D7/D8/D9/CS/CX/CT/CS /CX/D2 /CP /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2 /DB/CX/D8/CW /CS/CX/AR/CT/D6/CT/D2 /D8 /D2/CT/DC/D8/B9/D2/CT/CP/D6/CT/D7/D8/B9/D2/CT/CX/CV/CW /CQ /D3/D6 /BT/BY/C5/CQ /D3/D2/CS/D7 /B4/D7/CT/CT /AS/CV/D9/D6/CT /BG/CP/B5 /CJ /BF/BI ℄/BA/BE/BA/BE/BA /CC/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r/D7/C5/D3/D7/D8 /D3/CU /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D3/D2 /D5/D9/CP/D7/CX/B9/BD/BW /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D7/DD/D7/D8/CT/D1/D7 \r/CP/D6/D6/CX/CT/CS /D3/D9/D8 /CX/D2 /D8/CW/CT /D4/CP/D7/D8 \r/D3/D2\r/CT/D6/D2/CT/CS/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r \r /CW/CP/CX/D2/D7 /DB/CX/D8/CW /D6/CP/D8/CW/CT/D6 /CQ/CX/CVS1\n/D7/D4/CX/D2/D7 /B4/D8 /DD/D4/CX\r/CP/D0/D0/DD /B8 S1= 5/2) /B5/B8/D1/CP/CZ/CX/D2/CV /D8/CW/CT /D7/DD/D7/D8/CT/D1 /D6/CP/D8/CW/CT/D6 \r/D0/CP/D7/D7/CX\r/CP/D0 /CJ /BF/BJ ℄/BA /BT/D7 /CP /D1/CP/D8/D8/CT/D6 /D3/CU /CU/CP\r/D8/B8 /D5/D9/CP/D2 /D8/D9/D1 /CT/AR/CT\r/D8/D7 /CP/D6/CT /D1/D3/D7/D8 /D4/D6/D3/D2/D3/D9/D2\r/CT/CS/CX/D2 /D8/CW/CT /CT/DC/D8/D6/CT/D1/CT /D5/D9/CP/D2 /D8/D9/D1 \r/CP/D7/CT(S1,S2) = (1,1/2) /B8 /D6/CT/CP/D0/CX/DE/CT/CS/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /CX/D2 /D8/CW/CT /D5/D9/CP/D7/CX/B9/BD/BW /CQ/CX/D1/CT/D8/CP/D0/D0/CX\r/D1/CP/D8/CT/D6/CX/CP/D0 /C6/CX/BV/D9/B4/D4/CQ/CP/B5/BW 2\n/C7/B53· /BE/BW2\n/C7 /CJ /BL℄/BA /BT/D7 /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CP/CQ /D3 /DA /CT/B8 /D8/CW/CX/D7 /D1/CP/D8/CT/D6/CX/CP/D0 /CW/CP/D7 /CP/D2 /CT/DC\r /CW/CP/D2/CV/CT /CX/D2 /D8/CT/D6/CP\r/B9/D8/CX/D3/D2 /D3/CU /CP/CQ /D3/D9/D8 /BD/BE/BD /C3/B8 /CP/D2/CS /D7/CW/D3 /DB/D7 /CP /D8/CW/D6/CT/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /BT/BY/C5 /D3/D6/CS/CT/D6/CX/D2/CV /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /CP/D8 /BJ /C3 /B4/D7/CT/CT /AS/CV/D9/D6/CT /BF/B5/DB/CW/CX\r /CW /D7/D3/D1/CT/DB/CW/CP/D8 /D3/CQ/D7\r/D9/D6/CT/D7 /D8/CW/CT /DA /CT/D6/DD /D0/D3 /DB /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /B4T /B5 /CU/CT/CP/D8/D9/D6/CT/D7 /D3/CU /D8/CW/CT /BD/BW /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/BA /CB/D8/CX/D0/D0/B8 /D0/D3 /DB/B9/AS/CT/D0/CS /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /CP/D6/CT /CX/D2 /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0/D0/DD /CT/DC/D4 /CT\r/D8/CT/CS /D6/CT/D7/D9/D0/D8/D7/BA /C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6/CW/CP/D2/CS/B8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/D7 /D2/CT\r/CT/D7/D7/CP/D6/DD /D8/D3 /D6/CT/CP\r /CW /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0/D0/DD /D1/D3/D7/D8 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CT /DB /D3/D9/D0/CS/CQ /CT /CU/D3/D6 /D8/CW/CX/D7 \r/D3/D1/D4 /D3/D9/D2/CS /DB /CT/D0/D0 /CQ /CT/DD /D3/D2/CS /BD/BC/BC /CC/B8 /D1/CP/CZ/CX/D2/CV /D8/CW/CT /D7/CT/CP/D6\r /CW /CU/D3/D6 /D3/D8/CW/CT/D6 \r/D3/D1/D4 /D3/D9/D2/CS/D7 /DB/CX/D8/CW /DB /CT/CP/CZ /CT/D6/CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CP/D2 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV \r /CW/CP/D0/D0/CT/D2/CV/CT /CU/D3/D6 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/CX/D7/D8/D7/BA/BG/CB/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2/D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7/C4/D3 /DB/B9/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r/D7 /B4T≪̺sM0\n/B5 /D3/CU /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2/D7 /CX/D7 \r/D3/D2 /D8/D6/D3/D0/D0/CT/CS /CQ /DD /D8/CW/CT/BY/C5 /D1/CP/CV/D2/D3/D2/D7/B8 /D7/D3 /D8/CW/CP/D8 /CX/D2 /DE/CT/D6/D3 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /CX/D8 \r/CP/D2 /CQ /CT /CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /DD /D9/D7/CX/D2/CV /CC /CP/CZ /CP/CW/CP/D7/CW/CX/B3/D7 \r/D3/D2/D7/D8/D6/CP/CX/D2/CT/CS/CB/CF/CC /CU/D3/D6 /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/D7 /CJ /BG/BC /B8 /BG/BD ℄/BA /BT /CU/CT/DB /DA /CP/D6/CX/CP/D2 /D8/D7 /D3/CU /D8/CW/CX/D7 /D8/CW/CT/D3/D6/DD /CW/CP /DA /CT /CP/D0/D7/D3 /CQ /CT/CT/D2 /CP/D4/D4/D0/CX/CT/CS /D8/D3 /BD/BW /CU/CT/D6/D6/CX/B9/D1/CP/CV/D2/CT/D8/D7 /CJ /BF/BK /B8 /BG/BE /B8 /BG/BF ℄/BA /C1/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /CV/D6/D3/D9/D2/CS/B9/D7/D8/CP/D8/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 ̺s\n/CP/D2/CSM0\n/B8 /D8/CW/CT /CT/DC/D4/D0/CX\r/CX/D8/CT /CU/D3/D6/D1 /D3/CU/D8/CW/CT /D7/CT/D6/CX/CT/D7 /CX/D2 /D4 /D3 /DB /CT/D6/D7 /D3/CUt≡T/̺sM0\n/CU/D3/D6 /D8/CW/CT /D9/D2/CX/CU/D3/D6/D1 /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD χ /CP/D2/CS /D8/CW/CT /D7/D4 /CT\r/CX/AS\r /CW/CT/CP/D8Cv\n/D6/CT/CP/CS/CP/D7 /CJ /BG/BG ℄\nχT\nM0=2\n3t−1−ζ(1\n2)√πt−1\n2+ζ2(1\n2)\n2π+O/parenleftBig\nt1\n2/parenrightBig\n,\nCv\nM0=3\n8ζ(3\n2)√πt1\n2−t\n2+/bracketleftbigg\nconst−15ζ(1\n2)\n32√π/bracketrightbigg\nt3\n2+O/parenleftbig\nt2/parenrightbig\n. /B4/BF/B5/C0/CT/D6/CT/B8ζ(z) /CX/D7 /CA/CX/CT/D1/CP/D2/D2/B3/D7 /DE/CT/D8/CP /CU/D9/D2\r/D8/CX/D3/D2/B8 /CP/D2/CSconst= 15(S2\n1+S1S2+S2\n2)ζ(5\n2)/128√π /BA/BT/D7 /D1/CP /DD /CQ /CT /CT/DC/D4 /CT\r/D8/CT/CS/B8 /D8/CW/CT /CP/CQ /D3 /DA /CT /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /D6/CT/D4/D6/D3 /CS/D9\r/CT /CC /CP/CZ /CP/CW/CP/D7/CW/CX/B3/D7 /D3/D6/CX/CV/CX/D2/CP/D0 /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT/C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /BY/C5 \r /CW/CP/CX/D2 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS /CQ /DD /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 M0=S /CP/D2/CS̺s=JS2/BA /BT/D4/CP/D6/D8 /CU/D6/D3/D1 /D8/CW/CT\r/D3 /CTꜶ\r/CX/CT/D2 /D8 /D3/CUt3/2/CX/D2 /D8/CW/CT /CT/DC/D4/CP/D2/D7/CX/D3/D2 /CU/D3/D6Cv\n/B8 /D8/CW/CT /CP/CQ /D3 /DA /CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /D6/CT/D4/D6/D3 /CS/D9\r/CT /D4/D6/CT\r/CX/D7/CT/D0/DD /D8/CW/CT /D8/CW/CT/D6/D1/D3/B9/CS/DD/D2/CP/D1/CX\r /BU/CT/D8/CW/CT/B9/CP/D2/D7/CP/D8/DE \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /BY/C5 /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /CJ /BG/BH /B8 /BG/BI ℄/BA /C1/D8 /CX/D7 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV/D8/D3 /D2/D3/D8/CT /D8/CW/CP/D8 /DB/CX/D8/CW/D3/D9/D8 /D8/CW/CT /CU/CP\r/D8/D3/D6const /CX/D2 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6Cv\n/B8 /CQ /D3/D8/CW /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /CU/D9/D0/AS/D0/D0 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0/CW /DD/D4 /D3/D8/CW/CT/D7/CX/D7 /CP\r\r/D3/D6/CS/CX/D2/CV /D8/D3 /DB/CW/CX\r /CW /CX/D2 /BD/BW /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/D7 /CP/D0/D0 /D3/CQ/D7/CT/D6/DA /CP/CQ/D0/CT/D7 /D7/CW/D3/D9/D0/CS /CQ /CT /D9/D2/CX/DA /CT/D6/D7/CP/D0/CU/D9/D2\r/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CQ/CP/D6/CT \r/D3/D9/D4/D0/CX/D2/CV/D7 M0\n/B8̺s\n/B8 /CP/D2/CSh /B8 /D6/CT/CP/D0/CX/DE/CX/D2/CV /CP /D2/D3/B9/D7\r /CP/D0/CT/B9/CU/CP\r/D8/D3/D6 /D9/D2/CX/DA/CT/D6/D7/CP/D0/CX/D8/DD /CJ /BG/BJ /B8 /BG/BK ℄/BA /C1/D2/D4/CP/D6/D8/CX\r/D9/D0/CP/D6/B8 /D8/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /CU/D6/CT/CT/B9/CT/D2/CT/D6/CV/DD /CS/CT/D2/D7/CX/D8 /DD /D7/CW/D3/D9/D0/CS /CW/CP /DA /CT /D8/CW/CT /CV/CT/D2/CT/D6/CX\r /CU/D3/D6/D1\nF\nN=TM0ΦF/parenleftbiggρsM0\nT,h\nT/parenrightbigg\n, /B4/BG/B5/DB/CW/CT/D6/CTΦF(x,y) /CX/D7 /CP /D9/D2/CX/DA /CT/D6/D7/CP/D0 /D7\r/CP/D0/CX/D2/CV /CU/D9/D2\r/D8/CX/D3/D2 /DB/CX/D8/CW /D2/D3 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /D7\r/CP/D0/CT /CU/CP\r/D8/D3/D6/D7/BM /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT/D7/CX/D8/CT /D7/D4/CX/D2S /CT/D2 /D8/CT/D6/D7 /D3/D2/D0/DD /CX/D2/CS/CX/D6/CT\r/D8/D0/DD /B8 /D8/CW/D6/D3/D9/CV/CW ̺s\n/CP/D2/CSM0\n/BA /BT/D4/D4 /CT/CP/D6/CP/D2\r/CT /D3/CU /D8/CW/CT /CU/CP\r/D8/D3/D6 /DA/CX/D3/D0/CP/D8/CX/D2/CV /D8/CW/CT/CP/CQ /D3 /DA /CT /D7\r/CP/D0/CX/D2/CV /CW /DD/D4 /D3/D8/CW/CT/D7/CX/D7 /CX/D7 /D4/D6/D3/CQ/CP/CQ/D0/DD /CP/D2 /CP/D6/D8/CX/CU/CP\r/D8 /D3/CU /CC /CP/CZ /CP/CW/CP/D7/CW/CX/B3/D7 /CB/CF/CC/BA /BT/D7 /CU/CP/D6 /CP/D7 /DB /CT /CZ/D2/D3 /DB/B8 /D9/D4 /D8/D3/D2/D3 /DB /D9/D2/CX/DA /CT/D6/D7/CP/D0 /D0/D3 /DB/B9T /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7 /CW/CP /DA /CT /D2/D3/D8 /CQ /CT/CT/D2/D7/D8/D9/CS/CX/CT/CS/BA\n0.51.01.5\n0 50 100 150 200χT (emu K/mol)\nTemperature (K)AFMFM(a)\n0.0 0.5 1.0\nTemperature0.00.51.0Specific heat(b)\nAFM contribution\nspin−1/2 ferromagnet(1,1/2) ferrimagnet/BY/CX/CV/D9/D6/CT /BF/BA /B4/CP/B5 /BX/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /B4χ·T /DA/D7T /B5 /CU/D3/D6 /D8/CW/CT(1,1/2) /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2 /C6/CX/BV/D9/B4/D4/CQ/CP/B5/BW 2\n/C7/B53·/BE/BW2\n/C7 /CJ /BL℄/BA /B4/CQ/B5Cv\n/DA/D7T /D3/CU /D8/CW/CT(1/1/2) /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2 /B4/BW/C5/CA /BZ/B5 \r/D3/D1/D4/CP/D6/CT/CS /D8/D3Cv(T) /D3/CU /D8/CW/CT/D7/D4/CX/D2/B9/BD/BB/BE /BY/C5 \r /CW/CP/CX/D2 /B4/BW/C5/CA /BZ/B5 /CP/D8h= 0.05J /BA /CC/CW/CT /CS/CX/AR/CT/D6/CT/D2\r/CT /D3/CU /CQ /D3/D8/CW /D7/D4 /CT\r/CX/AS\r /CW/CT/CP/D8/D7 \r/CP/D2 /CQ /CT /CX/CS/CT/D2 /D8/CX/AS/CT/CS/CP/D7 /BT/BY/C5 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CJ /BF/BE ℄/BACv\n/CP/D2/CST /CP/D6/CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /D8/CW/CT /D9/D2/CX/D8/CT/D7 /D3/CUJ /BA/CF/CW/CT/D6/CT/CP/D7 /D0/D3 /DB/B9T /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r/D7 /D6/CT/AT/CT\r/D8/D7 /D8/CW/CT /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D6/CT/D0/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D0/D3 /DB /CT/D6 /CV/CP/D4/D0/CT/D7/D7 /BY/C5 /CQ/D6/CP/D2\r /CW\nE−\nk\n/B8 /D8/CW/CT /CX/D2 /D8/CT/D6/D1/CT/CS/CX/CP/D8/CT/B9 T /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D7 /CS/D3/D1/CX/D2/CP/D8/CT/CS /CQ /DD /D8/CW/CP/D8 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/BY/C5 /D2/CT/CP/D6/CT/D7/D8/B9/D2/CT/CX/CV/CW /CQ /D3/D6/B9 /B4J1<0 /B5 /CP/D2/CS /BT/BY/C5 /D2/CT/DC/D8/B9/D2/CT/CP/D6/CT/D7/D8/B9/D2/CT/CX/CV/CW /CQ /D3/D6 /B4J2>0 /B5 /CT/DC\r /CW/CP/D2/CV/CT /CQ /D3/D2/CS/D7/BA /CC/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS J1−J2\n/BY/C5 \r /CW/CP/CX/D2 /CW/CP/D7 /D6/CT\r/CT/D2 /D8/D0/DD /CP/D8/D8/D6/CP\r/D8/CT/CS/D1 /D9\r /CW /CP/D8/D8/CT/D2 /D8/CX/D3/D2/B8 /CP/D7 /CX/D8 /CX/D7 /D7/D9/D4/D4 /D3/D7/CT/CS /D8/D3 /CS/CT/D7\r/D6/CX/CQ /CT /CP /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /CT/CS/CV/CT/B9/D7/CW/CP/D6/CX/D2/CV \r/D9/D4/D6/CP/D8/CT/D7/B8 /D7/D9\r /CW /CP/D7/CA/CQ2\n/BV/D92\n/C5/D33\n/C712\n/CJ /BH/BJ ℄ /CP/D2/CS /C4/CX/BV/D9/CE /C7 4\n/CJ /BH/BK ℄/BA /CC/CW/CT /D0/CP/D8/D8/CT/D6 /D1/CP/D8/CT/D6/CX/CP/D0 /CT/DC/CW/CX/CQ/CX/D8/D7 /D1 /D9/D0/D8/CX/CU/CT/D6/D6/D3/CX\r /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /CJ /BH/BL ℄/CP/D7 /DB /CT/D0/D0 /CP/D7 /CP/D2 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /D7/D4 /CT\r/CX/AS\r /D4/CW/CP/D7/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /CX/D2 /CP /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /CU/D6/D3/D1 /CP/D2 /D3/D6/CS/CT/D6/CT/CS /D7/D4/CX/D6/CP/D0 /D8/D3 /CP/D2/D3/D6/CS/CT/D6/CT/CS /D1/D3 /CS/D9/D0/CP/D8/CT/CS/B9\r/D3/D0/D0/CX/D2/CT/CP/D6 /D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT /CJ /BI/BC ℄/BA /C7/D2 /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0 /D7/CX/CS/CT/B8 /D8/CW/CT /D0/CP/D8/D8/CT/D6 /BY/C5 /D1/D3 /CS/CT/D0/DB /CP/D7 /D7/CW/D3 /DB/D2 /D8/D3 /CT/DC/CW/CX/CQ/CX/D8 /CP /DA /CT\r/D8/D3/D6 \r /CW/CX/D6/CP/D0 /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT /D3 /CS/CT/D6 /B4/CX/D2 /CP /D1/D3 /CS/CT/D6/CP/D8/CT /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/B5 /CP/D7 /DB /CT/D0/D0 /CP/D7 /CP /D6/CX\r /CW/DA /CP/D6/CX/CT/D8 /DD /D3/CU /CT/DC/D3/D8/CX\r /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT/D7 /DB/CX/D8/CW /CS/CX/AR/CT/D6/CT/D2 /D8 /CZ/CX/D2/CS/D7 /D3/CU /D1 /D9/D0/D8/CX/D4 /D3/D0/CP/D6 /D7/D4/CX/D2 \r/D3/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/CX/D2 /CP /D0/CP/D6/CV/CT/D6/D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/B5 /CJ /BI/BD /B8 /BI/BE ℄/BA/CC /D9/D6/D2/CX/D2/CV /D8/D3 /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 J1−J2\n\r /CW/CP/CX/D2/B8 /D0/CT/D8 /D9/D7 /CQ /CT/CV/CX/D2 /DB/CX/D8/CW /CP \r/D3/D1/D1/CT/D2 /D8 /D3/D2 /CX/D8/D7 \r/D0/CP/D7/D7/CX\r/CP/D0 /D0/CX/D1/CX/D8/BA/CC/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /D3/CU /D8/CW/CT /D1/D3 /CS/CT/D0 \r/CP/D2 /CQ /CT /CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /DD /D8/CW/CT /CP/D2/D7/CP/D8/DESi,n=Si[ucos(Qn)+vsin(Qn)]/B4i= 1,2 /B5/B8 /DB/CW/CT/D6/CTu⊥v /CP/D6/CT /D9/D2/CX/D8 /DA /CT\r/D8/D3/D6/D7 /CX/D2 /D8/CW/CT /D7/D4/CX/D2 /D7/D4/CP\r/CT/BA /CC/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT/DB/CX/D8/CW /CP /D4/CX/D8\r /CW /CP/D2/CV/D0/CT /CQ /CT/D8 /DB /CT/CT/D2 /D2/CT/CX/CV/CW /CQ /D3/D6/CX/D2/CV /D7/D4/CX/D2/D7Q/2 =π /CX/D7 /D7/D8/CP/CQ/D0/CT /D9/D4 /D8/D3 /D8/CW/CT /D4/CW/CP/D7/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8\nJ2c=S1S2J1/[2(S2\n1+S2\n2)] /BA /C1/D2 /D8/CW/CT /D7/D8/D6/D3/D2/CV/D0/DD /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D6/CT/CV/CX/D3/D2J2> J2c\n/B8 /D8/CW/CT /D7/D8/CP/CQ/D0/CT /D7/D8/CP/D8/CT /CX/D7 /CP /D7/D4/CX/D6/CP/D0/DB/CX/D8/CW /CP/D2 /D3/D6/CS/CT/D6/CX/D2/CV /DB /CP /DA /CT /DA /CT\r/D8/D3/D6 /CV/CX/DA /CT/D2 /CQ /DDcos(Q/2) =−S1S2J1/[2J2(S2\n1+S2\n2)] /BA /C1/D2 /D8/CW/CT /D0/CX/D1/CX/D8J2→ ∞ /B8\nQ=π /CP/D2/CS /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 \r/D3/D1/D4 /D3/D7/CT/CS /D3/CU /D8 /DB /D3 /CS/CT\r/D3/D9/D4/D0/CT/CS /BT/BY/C5 \r /CW/CP/CX/D2/D7 /DB/CX/D8/CW /D7/CX/D8/CT /D7/D4/CX/D2/D7S1\n/CP/D2/CSS2\n/BA/C6/CT/DC/D8/B8 /D0/CT/D8 /D9/D7 /CS/CX/D7\r/D9/D7/D7 /D8/CW/CT /CT/DC/D8/D6/CT/D1/CT /D5/D9/CP/D2 /D8/D9/D1 \r/CP/D7/CT(S1,S2) = (1,1/2) /BA /BT/D0/D6/CT/CP/CS/DD /CP /D5/D9/CP/D0/CX/D8/CP/D8/CX/DA /CT /D7/CT/D1/CX/B9\r/D0/CP/D7/D7/CX\r/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7 /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CP/D8 /D2/CT/CP/D6 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CP/D8J2= 0.2J1\n/D8/CW/CT /BY/C5 /D1/CP/CV/D2/D3/D2/CQ/D6/CP/D2\r /CWE−\nk\n/B4/D7/CT/CT /AS/CV/D9/D6/CT /BE/B5 /CX/D7 /D7/D8/D6/D3/D2/CV/D0/DD /AT/CP/D8/D8/CT/D2/CT/CS/B8 /DB/CW/CT/D6/CT/CP/D7 /D8/CW/CT /D3/D4/D8/CX\r/CP/D0 /BT/BY/C5 /CQ/D6/CP/D2\r /CW /D7/CW/D3 /DB/D7 /D3/D2/D0/DD /CP/D7/D1/D3 /D3/D8/CW /CX/D2\r/D6/CT/CP/D7/CT /D3/CU /D8/CW/CT /BT/BY/C5 /CV/CP/D4∆+/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /D3/D4/D8/CX\r/CP/D0 /D1/CP/CV/D2/D3/D2/D7 /CS/D3 /D2/D3/D8 /D4/D0/CP /DD /CP/D2 /DD /CX/D1/B9/D4 /D3/D6/D8/CP/D2 /D8 /D6/D3/D0/CT /CX/D2 /D8/CW/CT /D1/CT\r /CW/CP/D2/CX/D7/D1 /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/BA /C7/D2 /CP/D4/D4/D6/D3/CP\r /CW/CX/D2/CV /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8/B8 /D8/CW/CT/CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /CP/D2/CS /D5/D9/CP/D2 /D8/D9/D1 /D7/D4/CX/D2 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /D4/D0/CP /DD /D7/D3/D1/CT/DB/CW/CP/D8 /CS/CX/AR/CT/D6/CT/D2 /D8 /D6/D3/D0/CT/D7/BM /CF/CW/CT/D6/CT/CP/D7 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r/CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /D7/D8/D6/D3/D2/CV/D0/DD /D6/CT/CS/D9\r/CT/D7 /D8/CW/CT /D7/CW/D3/D6/D8/B9/D6/CP/D2/CV/CT /D7/D4/CX/D2 \r/D3/D6/D6/CT/D0/CP/D8/CX/D3/D2 /D0/CT/D2/CV/D8/CW /CX/D2 /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT/B8 /D8/CW/CT/D5/D9/CP/D2 /D8/D9/D1 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /D7/D8/CP/CQ/CX/D0/CX/DE/CT /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D3/D6/CS/CT/D6 /D9/D4 /D8/D3 /D8/CW/CT /D4 /D3/CX/D2 /D8J2= 0.231J1\n/CQ /CT/DD /D3/D2/CS /D8/CW/CT\r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CP/D8J2= 0.2J1\n/BA /BT/D2/D3/D8/CW/CT/D6 /CT/AR/CT\r/D8 /D3/CU /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /CX/D7 /D6/CT/D0/CP/D8/CT/CS /D8/D3/D8/CW/CT \r /CW/CP/D2/CV/CT /D3/CU /D8/CW/CT \r /CW/CP/D6/CP\r/D8/CT/D6 /D3/CU /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/BM /BT /CS/CT/D8/CP/CX/D0/CT/CS /BW/C5/CA /BZ /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD/D0/CT/DA /CT/D0/D7 /CP/D6/D3/D9/D2/CS /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 \r/D0/CT/CP/D6/D0/DD /CX/D2/CS/CX\r/CP/D8/CT/D7 /CP /D0/CT/DA /CT/D0 \r/D6/D3/D7/D7/CX/D2/CV/B8 /CX/BA/CT/BA/B8 /CP /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /D5/D9/CP/D2 /D8/D9/D1/D4/CW/CP/D7/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D8/D3 /CP /D7/CX/D2/CV/D0/CT/D8 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CP/D8J2= 0.231J1\n/BA /BY /D9/D6/D8/CW/CT/D6/B8 /BW/C5/CA /BZ /CW/CP/D7 /CP/D0/D7/D3 /CX/D2/CS/CX\r/CP/D8/CT/CS /D8/CW/CP/D8/CP/D8 /D0/CT/CP/D7/D8 /CU/D6/D3/D1J2= 0.25J1\n/D9/D4 /DB /CP/D6/CS/D7 /D8/CW/CT /CS/CX/D7\r/D9/D7/D7/CT/CS /D1/D3 /CS/CT/D0 /CT/DC/CW/CX/CQ/CX/D8/D7 /CP /D7/CX/D2/CV/D0/CT/D8 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CJ /BE/BE ℄/BA/BT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D3/D4 /CT/D2 /CX/D7/D7/D9/CT/D7 /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 J1−J2\n/C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /D1/D3 /CS/CT/D0 \r/CP/D2 /CQ /CT /CX/D2/CS/CX\r/CP/D8/CT/CS/BA/C7/D2/CT /D3/CU /D8/CW/CT /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D3/D4 /CT/D2 /D4/D6/D3/CQ/D0/CT/D1/D7 \r/D3/D2\r/CT/D6/D2/D7 /D8/CW/CT /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT /D7/CX/D2/CV/D0/CT/D8 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/D7 /CT/D7/D8/CP/CQ/D0/CX/D7/CW/CT/CS/CQ /CT/DD /D3/D2/CS /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT/BA /CB/CX/D2\r/CT /CX/D2 /BD/BW /D7/DD/D7/D8/CT/D1/D7 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0/D0/DD /CQ/D6/D3/CZ /CT/D2 /CB/C7/B4/BF/B5 /D7/DD/D1/D1/CT/D8/D6/DD /CX/D2/D8/CW/CT /D7/D4/CX/D6/CP/D0 /D7/D8/CP/D8/CT /CX/D7 /CV/CT/D2/CT/D6/CP/D0/D0/DD /CT/DC/D4 /CT\r/D8/CT/CS /D8/D3 /CQ /CT /D6/CT/D7/D8/D3/D6/CT/CS /CQ /DD /D5/D9/CP/D2 /D8/D9/D1 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7/B8 /D3/D2/CT /CX/D7 /CT/D2/CU/D3/D6\r/CT/CS /D8/D3/D0/D3 /D3/CZ /CU/D3/D6 /D4 /D3/D7/D7/CX/CQ/D0/CT /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CS/CX/D7/D3/D6/CS/CT/D6/CT/CS /D7/D8/CP/D8/CT/D7/BA /BT /DA /CP/D0/D9/CP/CQ/D0/CT /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 \r/CP/D2 /CQ /CT /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1/D8/CW/CT /C4/CX/CT/CQ/B9/CB\r /CW /D9/D0/D8/DE/B9/C5/CP/D8/D8/CX/D7 /D8/CW/CT/D3/D6/CT/D1 /CJ /BI/BF ℄ /CP/CS/CP/D4/D8/CT/CS /D8/D3 /D1/CX/DC/CT/CS/B9\r /CW/CP/CX/D2 /D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /CJ /BI/BG ℄/BA /CC/CW/CT /D8/CW/CT/D3/D6/CT/D1 /CX/D7/CP/D4/D4/D0/CX\r/CP/CQ/D0/CT /D8/D3 /D7/DD/D7/D8/CT/D1/D7 /DB/CX/D8/CW /CP /CW/CP/D0/CU/B9/CX/D2 /D8/CT/CV/CT/D6 \r/CT/D0/D0 /D7/D4/CX/D2 /B4S1+S2\n/B5 /CP/D2/CS /D7/CP /DD/D7 /D8/CW/CP/D8 /D8/CW/CT /D1/D3 /CS/CT/D0 /CT/CX/D8/CW/CT/D6 /CW/CP/D7/CV/CP/D4/D0/CT/D7/D7 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 /D3/D6 /CT/D0/D7/CT /CW/CP/D7 /CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/D7/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D3/D2/CT /D1/CP /DD /D0/D3 /D3/CZ /CU/D3/D6 /D4/CW/CP/D7/CT/D7 /DB/CX/D8/CW/D4/D6/CT/D7/D9/D1/CP/CQ/D0/DD /D7/D3/D1/CT /CQ/D6/D3/CZ /CT/D2 /CS/CX/D7\r/D6/CT/D8/CT /D7/DD/D1/D1/CT/D8/D6/DD /BA /C1/D2 /D4/CP/D6/D8/CX\r/D9/D0/CP/D6/B8 /D8/CW/CT /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT/CS \r /CW/CX/D6/CP/D0 /D4/CW/CP/D7/CT /CU/D3/D9/D2/CS /CX/D2/D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D7/D4/CX/D2/B9/BD/BB/BE /BY/C5J1−J2\n/D1/D3 /CS/CT/D0 /CX/D2 /D8/CW/CT /D7/D1/CP/D0/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /D6/CT/CV/CX/D3/D2 \r/D3/D9/D0/CS /CQ /CT /CP /D4 /D3/D7/D7/CX/CQ/D0/CT\r/CP/D2/CS/CX/CS/CP/D8/CT /CJ /BI/BD ℄/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /DA /CP/D6/CX/CT/D8 /DD /D3/CU /D4 /D3/D7/D7/CX/CQ/D0/CT /D2/D3/D2/B9/D1/CP/CV/D2/CT/D8/CX\r /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT/D7 /D1/CP /DD /CQ /CT \r/D3/D2/D7/CX/CS/CT/D6/CP/CQ/D0/DD/CT/D2/D0/CP/D6/CV/CT/CS /CS/D9/CT /D8/D3 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2\r/CT /D3/CU /D8 /DB /D3 /CZ/CX/D2/CS/D7 /D3/CU /D7/CX/D8/CT /D7/D4/CX/D2/D7 /CX/D2 /D8/CW/CT /D1/CX/DC/CT/CS /DA /CP/D6/CX/CP/D2 /D8 /D3/CU /D8/CW/CT /D4/CP/D6/CT/D2 /D8 /D1/D3 /CS/CT/D0/BA/BF/BA/BE/BA /BY /D6/D9/D7/D8/D6/CP/D8/CT/CS /CS/CX/CP/D1/D3/D2/CS \r/CW/CP/CX/D2/D7/CC/CW/CT /CS/CX/CP/D1/D3/D2/CS /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /CX/D7 /CP/D2/D3/D8/CW/CT/D6 /CV/CT/D2/CT/D6/CX\r /D1/D3 /CS/CT/D0 /D3/CU /BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7 \r/D3/D2/B9/D7/D8/D6/D9\r/D8/CT/CS /CU/D6/D3/D1 /D3/D2/CT /CZ/CX/D2/CS /D3/CU /D7/D4/CX/D2/D7 /D0/CX/DA/CX/D2/CV /D3/D2 /D7/D9/CQ/D0/CP/D8/D8/CX\r/CT/D7 /DB/CX/D8/CW /CS/CX/AR/CT/D6/CT/D2 /D8 /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D7/CX/D8/CT/D7 /B4/D7/CT/CT /AS/CV/D9/D6/CT /BG/CQ/B5/BA/CC/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D7/DD/D1/D1/CT/D8/D6/CX\r /CS/CX/CP/D1/D3/D2/CS \r /CW/CP/CX/D2 /B4/CB/BW/BV/B5 /B4J1=J′\n1\n/B8J= 0 /B5 /DB/CX/D8/CW /BT/BY/C5 /DA /CT/D6/D8/CX\r/CP/D0 /CQ /D3/D2/CS/D7\nJ⊥>0 /CX/D7 /D4/D6/D3/CQ/CP/CQ/D0/DD /D8/CW/CT /AS/D6/D7/D8 /D7/D8/D9/CS/CX/CT/CS /D1/D3 /CS/CT/D0 /D3/CU /BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8 /DB/CX/D8/CW \r/D3/D1/D4 /CT/D8/CX/D2/CV /CX/D2 /D8/CT/D6/B9/BJ/C6/BA /BU/BA /C1/DA/CP/D2/D3/DA/CP\r/D8/CX/D3/D2/D7 /CJ /BI/BH /B8 /BI/BI ℄/BA /BT 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/D8/D9/D1/D4/CW/CP/D7/CT/D7 /CX/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP\r/CT /B4J1/J⊥,J′\n1/J⊥\n/B5 /CW/CP /DA /CT /CQ /CT/CT/D2 /CS/CX/D7\r/D9/D7/D7/CT/CS /CU/D3/D6 /D8/CW/CT /CS/CX/D7/D8/D3/D6/D8/CT/CS /D1/D3 /CS/CT/D0/BM /BY /D3/D6\nJ1/J⊥,J′\n1/J⊥≪1 /B8 /D8/CW/CT /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD /D7/CT\r/D8/D3/D6 /CX/D7 /CV/D3 /DA /CT/D6/D2/CT/CS /CQ /DD /CP/D2 /CT/AR/CT\r/D8/CX/DA /CT /D7/D4/CX/D2/B9/BD/BB/BE /BT/BY/C5 /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV/D1/D3 /CS/CT/D0/B8 /DB/CW/CX\r /CW /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CT /CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /CP /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /D4/CW/CP/D7/CT /B4/CB/BY/B5 /DB/CX/D8/CW /D7/D3/D1/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /CW/CX/CV/CW/B9/CT/D2/CT/D6/CV/DD /D1/D3 /CS/CT/D7 /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D0/D3 \r/CP/D0 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /DA /CT/D6/D8/CX\r/CP/D0 /CS/CX/D1/CT/D6/D7/BA /BY /D3/D6 /CX/D2 /D8/CT/D6/D1/CT/CS/CX/CP/D8/CT J1/J⊥\n/CP/D2/CSJ′\n1/J⊥\n/B8/D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CS/CX/D1/CT/D6/CX/DE/CT/D7/B8 /CU/D3/D6/D1/CX/D2/CV /CP /D8 /DB /D3/CU/D3/D0/CS /CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /D7/CT/D5/D9/CT/D2\r/CT /D3/CU /CP/D0/D8/CT/D6/D2/CP/D8/CX/D2/CV /D8/CT/D8/D6/CP/D1/CT/D6/D7 /CP/D2/CS/CS/CX/D1/CT/D6/D7 /B4/CC/BW1\n/D4/CW/CP/D7/CT/B5/BA /BY/CX/D2/CP/D0/D0/DD /B8 /CU/D3/D6 /CQ /D3/D8/CWJ1/J⊥\n/CP/D2/CSJ′\n1/J⊥\n/D7/D9Ꜷ\r/CX/CT/D2 /D8/D0/DD /D0/CP/D6/CV/CT/B8 /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CX/D7 /CP/BD/BW /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/BA /CC/CW/CT/D7/CT /D4/CW/CP/D7/CT/D7 \r/CP/D2 /CQ /CT \r/D0/CT/CP/D6/D0/DD /CX/CS/CT/D2 /D8/CX/AS/CT/CS /CP/D0/D6/CT/CP/CS/DD /CX/D2 /D8/CW/CT /CB/BW/BV/BM /CB/BY /B4J1<0.5J⊥\n/B5/B8 /CC/BW1/B4<0.5J⊥< J1<1.10J⊥\n/B5/B8 /CP/D2/CS /BD/BW /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8 /B4J1>1.10J⊥\n/B5/BA\n1.0 2.0 0.01.02.0\nFerrimagnet\nJ1’J1__\nSFTD1J =1(a)\n 1 2 3\n−2 −1  0 0\n 1  2  3L=8\nL=10\nL=12J\n1TD\nTD1\n2\nFerrimagnetFM\nFM1\n2 ?|\nKSF\nDM(b)/BY/CX/CV/D9/D6/CT /BH/BA /B4/CP/B5 /C9/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /B4T= 0 /B5 /D3/CU /D8/CW/CT /CS/CX/D7/D8/D3/D6/D8/CT/CS /CS/CX/CP/D1/D3/D2/CS \r /CW/CP/CX/D2 /CX/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/D4/CP\r/CTJ1\n/DA/D7/BAJ′\n1\n/B4J⊥= 1 /B5 /CJ /BI/BL℄/BA /CB/BY /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CT /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /D4/CW/CP/D7/CT/B8 /DB/CW/CT/D6/CT/CP/D7 /CC/BW1\n/D1/CP/D6/CZ/D7 /D8/CW/CT/D8/CT/D8/D6/CP/D1/CT/D6/B9/CS/CX/D1/CT/D6 /D4/CW/CP/D7/CT \r/D3/D2/D7/D8/D6/D9\r/D8/CT/CS /CU/D6/D3/D1 /CP/D0/D8/CT/D6/D2/CP/D8/CX/D2/CV /CS/CX/D1/CT/D6/D7 /CP/D2/CS /D8/CT/D8/D6/CP/D1/CT/D6/D7 /CX/D2 /D0/D3 \r/CP/D0 /D7/CX/D2/CV/D0/CT/D8 /D7/D8/CP/D8/CT/D7/BA/B4/CQ/B5 /C9/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /B4T= 0 /B5 /D3/CU /CP /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /CB/BW/BV /DB/CX/D8/CW \r/D3/D1/D4 /CT/D8/CX/D2/CV /CU/D3/D9/D6/B9/D7/D4/CX/D2 \r/DD\r/D0/CX\r/CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP\r/CTJ⊥\n/DA/D7/BAK /B4J′\n1=J′\n1= 1 /B5 /CJ /BJ/BD ℄/BA /C1/D2 /D8/CW/CX/D7 /D7/DD/D7/D8/CT/D1/B8 /D8/CW/CT/D6/CT /CP/D4/D4 /CT/CP/D6 /CP/D8/D0/CT/CP/D7/D8 /AS/DA /CT /D2/CT/DB /D4/CW/CP/D7/CT/D7 /CS/CT/D2/D3/D8/CT/CS /CP/D7 /CU/D3/D0/D0/D3 /DB/D7/BM /BY/C51\n/B4/CU/D9/D0/D0/DD/B9/D4 /D3/D0/CP/D6/CX/DE/CT/CS /BY/C5 /D4/CW/CP/D7/CT/B5/B8 /BY/C52\n/B4/CP/D2/D3/D8/CW/CT/D6 /D4/CW/CP/D7/CT/DB/CX/D8/CW /CP /D2/CT/D8 /BY/C5 /D1/D3/D1/CT/D2 /D8/B5/B8 /BW/C5 /B4/CU/D3/D9/D6/CU/D3/D0/CS /CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /D7/CX/D2/CV/D0/CT/D8 /D4/CW/CP/D7/CT/B5/B8 /CC/BW2\n/B4/CP/D2/D3/D8/CW/CT/D6 /D8/CT/D8/D6/CP/D1/CT/D6/B9/CS/CX/D1/CT/D6/D4/CW/CP/D7/CT /DB/CX/D8/CW /D8/CT/D8/D6/CP/D1/CT/D6/D7 /CX/D2 /D0/D3 \r/CP/D0 /D8/D6/CX/D4/D0/CT/D8 /D7/D8/CP/D8/CT/D7/B5/B8 /CP/D2/CS /CP/D2 /CT/DC/D3/D8/CX\r /D2/D3/D2/B9/C4/CX/CT/CQ/B9/C5/CP/D8/D8/CX/D7 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT/CS/CT/D2/D3/D8/CT/CS /CQ /DD /CP /D5/D9/CT/D7/D8/CX/D3/D2 /D1/CP/D6/CZ/BA /CC/CW/CT /CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/D7 /D8/D6/CP\r/CT /D8/CW/CT /D4/CW/CP/D7/CT /CQ /D3/D9/D2/CS/CP/D6/CX/CT/D7 /D3/CU /D8/CW/CT /D7/CX/D2/CV/D0/CT/B9/CS/CX/CP/D1/D3/D2/CS/D7/DD/D7/D8/CT/D1/BA/CC/CW/CT /CS/CX/CP/D1/D3/D2/CS /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /CX/D7 /CP/D0/D7/D3 /D3/D2/CT /D3/CU /D8/CW/CT /D7/CX/D1/D4/D0/CT/D7/D8 /D5/D9/CP/D2 /D8/D9/D1 /D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /CP/CS/D1/CX/D8/D8/CX/D2/CV /CP/CU/D3/D9/D6/B9/D7/D4/CX/D2 \r/DD\r/D0/CX\r /CT/DC\r /CW/CP/D2/CV/CT \r/D3/D9/D4/D0/CX/D2/CV/BA /BU/CT/D0/D3 /DB /DB /CT /CS/CX/D7\r/D9/D7/D7 /D8/CW/CT /CX/D1/D4/CP\r/D8 /D3/CU /D8/CW/CX/D7 \r/D3/D1/D4 /CT/D8/CX/D2/CV /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D3/D2/D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /B4T= 0 /B5 /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /CB/BW/BV /CJ /BJ/BD ℄/BA /CC/CW/CT /D7\r /CW/CT/D1/CP/D8/CX\r /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2/D3/CU /D8/CW/CT /D1/D3 /CS/CT/D0 /CX/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /AS/CV/D9/D6/CT /BG /CQ/B8 /D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS /CU/D3/D9/D6/B9/D7/D4/CX/D2 \r/DD\r/D0/CX\r /CT/DC\r /CW/CP/D2/CV/CT \r/D3/D9/D4/D0/CX/D2/CV /CJ /BJ/BE ℄ /CX/D2/CP /D7/CX/D2/CV/D0/CT /CS/CX/CP/D1/D3/D2/CS /CQ /CT/CX/D2/CV \r/D3/D2 /D8/D6/D3/D0/D0/CT/CS /CQ /DD /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 K /BA /C1/D8 /CX/D7 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D8/D3 /D2/D3/D8/CX\r/CT /D8/CW/CP/D8 /D8/CW/CT \r/DD\r/D0/CX\r\r/D3/D9/D4/D0/CX/D2/CV /CS/D3 /CT/D7 /D2/D3/D8 /DA/CX/D3/D0/CP/D8/CT /D8/CW/CT /D0/D3 \r/CP/D0 /D7/DD/D1/D1/CT/D8/D6/DD /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /B4J⊥>0 /B5 /CB/BW/BV /D1/D3 /CS/CT/D0 /D9/D2/CS/CT/D6 /D8/CW/CT/CT/DC\r /CW/CP/D2/CV/CT /D3/CU /D4/CP/CX/D6/D7 /D3/CU /D3/AR/B9\r /CW/CP/CX/D2 /D7/D4/CX/D2/D7S1a,n\n/CP/D2/CSS1b,n\n/CX/D2 /D8/CW/CT /CS/CX/CP/D1/D3/D2/CS/D7/BA /CC/CW /D9/D7/B8 /CX/D2 /D8/CW/CT /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 \r/CP/D7/CT/D3/CU /D7/D4/CX/D2/B91/2 /D3/AR/B9\r /CW/CP/CX/D2 /D3/D4 /CT/D6/CP/D8/D3/D6/D7/B8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS /B4/CP/D7 /CX/D2 /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D1/D3 /CS/CT/D0 /DB/CX/D8/CW/D3/D9/D8\r/DD\r/D0/CX\r /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7/B5 /CQ /DDL /D0/D3 \r/CP/D0 /CV/D3 /D3 /CS /D5/D9/CP/D2 /D8/D9/D1 /D2 /D9/D1 /CQ /CT/D6/D7 sn= 0,1 /B4n= 1,2,...,L /B5 /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT\r/D3/D1/D4 /D3/D7/CX/D8/CT /D3/AR/B9\r /CW/CP/CX/D2 /D7/D4/CX/D2/D7S1,n=S1a,n+S1b,n\n/BMS2\n1,n=sn(sn+1) /BA /CD/D7/CX/D2/CV /D8/CW/CX/D7 /D0/D3 \r/CP/D0 /D7/DD/D1/D1/CT/D8/D6/DD /B8 /D8/CW/CT/C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /D3/CU /D8/CW/CT /CB/BW/BV \r/CP/D2 /CQ /CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /D8/CW/CT \r/D3/D1/D4/CP\r/D8 /CU/D3/D6/D1\nHc=E0+L/summationdisplay\nn=1/bracketleftbigg\nJ1S1,n·(S2,n+S2,n+1)+JnS2,n·S2,n+1+K\n2{S1,n·S2,n,S1,n·S2,n+1}/bracketrightbigg\n. /B4/BH/B5/C0/CT/D6/CTE0=J⊥/summationtextL\nn=1[sn(sn+1)−3/4] /CX/D7 /CP /AS/DC/CT/CS /D2 /D9/D1 /CQ /CT/D6 /CU/D3/D6 /CT/DA /CT/D6/DD /D7/CT\r/D8/D3/D6 /CS/CT/AS/D2/CT/CS /CP/D7 /CP /D7/CT/D5/D9/CT/D2\r/CT /D3/CU /D8/CW/CT/BK/CB/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2/D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7/D0/D3 \r/CP/D0 /D5/D9/CP/D2 /D8/D9/D1 /D2 /D9/D1 /CQ /CT/D6/D7 [s1,s2,...,s L] /B8Jn=J+K/4−snK /B8 /CP/D2/CS{A,B} /CX/D7 /D8/CW/CT /CP/D2 /D8/CX\r/D3/D1/D1 /D9/D8/CP/D8/D3/D6/D3/CU /D8/CW/CT /D3/D4 /CT/D6/CP/D8/D3/D6/D7 A /CP/D2/CSB /BA/C4/CT/D8 /D9/D7 /CQ/D6/CX/CT/AT/DD \r/D3/D1/D1/CT/D2 /D8 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /CU/D3/D6 /D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /CB/BW/BV /B4/AS/CV/D9/D6/CT /BH/CQ/B5/B8 /CQ /DD /D9/D7/CX/D2/CV /D7/D3/D1/CT/D7/DD/D1/D1/CT/D8/D6/CX/CT/D7 /D3/CU /D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /B4/BH/B5/BA /C1/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP\r/CT /DB/CW/CT/D6/CT /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CX/D7 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS/CQ /DDsn= 0 /B4n= 1,2,...,L /B5/B8 /D8/CW/CT /AS/D6/D7/D8 /CP/D2/CS /D8/CW/CX/D6/CS /D8/CT/D6/D1/D7 /CX/D2 /D8/CW/CT /D7/D5/D9/CP/D6/CT /CQ/D6/CP\r /CZ /CT/D8/D7 /DA /CP/D2/CX/D7/CW /CP/D2/CS /D8/CW/CT /D1/D3 /CS/CT/D0/CX/D7 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D8/D3 /D8/CW/CT /D7/D4/CX/D2/B91/2 /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /DB/CX/D8/CW /CP/D2 /CT/AR/CT\r/D8/CX/DA /CT /CT/DC\r /CW/CP/D2/CV/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 J+K/4 /BA /CC/CW/CX/D7/CT/DC/D4/D0/CP/CX/D2/D7 /D8/CW/CT /D4/D6/CT/D7/CT/D2\r/CT /D3/CU /CP /CU/D9/D0/D0/DD /D4 /D3/D0/CP/D6/CX/DE/CT/CS /BY/C51\n/B4J+K/4<0 /B5 /CP/D2/CS /CP /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /B4J+K/4>0 /B5/D4/CW/CP/D7/CT/D7 /CU/D3/D6 /D0/CP/D6/CV/CT /CT/D2/D3/D9/CV/CWJ⊥/J1\n/BA /C6/D3/D8/CT /D8/CW/CP/D8 /CQ /CT/D7/CX/CS/CT/D7 /D8/CW/CT /DB /CT/D0/D0/B9/CS/D3 \r/D9/D1/CT/D2 /D8/CT/CS \r/D3/D0/D0/CT\r/D8/CX/DA /CT /D1/D3 /CS/CT/D7/B8 /D8/CW/CT/D7/CT/D4/CW/CP/D7/CT/D7 /CT/DC/CW/CX/CQ/CX/D8 /D7/D4 /CT\r/CX/AS\r /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 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/D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0 /D7/CX/CS/CT/B8 /D9/D2/CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /DA /CP/D6/CX/CP/D2 /D8/D7 /D3/CU /D8/CW/CT /D0/CP/CS/CS/CT/D6 /D1/D3 /CS/CT/D0/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /AS/CV/D9/D6/CT /BG /CW/CP /DA /CT/CP/D0/D6/CT/CP/CS/DD /CQ /CT/CT/D2 /CP/D2/CP/D0/DD/DE/CT/CS /CX/D2 /CP /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D4/D9/CQ/D0/CX\r/CP/D8/CX/D3/D2 /CJ /BJ/BI/B8 /BJ/BJ /B8 /BJ/BK /B8 /BJ/BL /B8 /BK/BC /B8 /BK/BD ℄/BA /CC/CW/CT /D9/D2/CU/D6/D9/D7/D8/D6/CP/D8/CT/CS\r /CW/CT\r /CZ /CT/D6/CQ /D3/CP/D6/CS /D0/CP/CS/CS/CT/D6 /B4 /AS/CV/D9/D6/CT /BG\r/B5 /DB/CX/D8/CW /BT/BY/C5 /CQ /D3/D2/CS/D7 /B4J,J1>0 /B5 /CT/DC/CW/CX/CQ/CX/D8/D7 /CP /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /CV/D6/D3/D9/D2/CS/D7/D8/CP/D8/CT/BA /C1/D8/D7 /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 \r/D0/D3/D7/CT/D0/DD /D6/CT/D4/D6/D3 /CS/D9\r/CT /D8/CW/CT /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /D8/CW/CT /CV/CT/D2/CT/D6/CX\r /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2/CS/CX/D7\r/D9/D7/D7/CT/CS /CX/D2 /CB/CT\r/D8/CX/D3/D2 /BE /CJ /BJ/BL ℄/BA /C6/D3/D8/CT /D8/CW/CP/D8 /CP /DA /CP/D6/CX/CP/D2 /D8 /D3/CU /D8/CW/CX/D7 /D7/D8/D6/D9\r/D8/D9/D6/CT /DB/CX/D8/CW /BY/C5 /D0/CT/CV/D7 /B4J1<0 /B5 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/D7/BL/C6/BA /BU/BA /C1/DA/CP/D2/D3/DA\nCB AM =3/2\nGapless spin fluid\nM =1/2\n012345h/J\n0.10.20.30.4 0.5 0.60.7\nJ /J211\nNon−Lieb−Mattisferrimagnet\nD(a)\n0\n0\nM\n00.20.40.6\n0.30 0.34 0.38 0.42J /J120(S ,S )=(1,1/2)\nh=01 2(b)/BY/CX/CV/D9/D6/CT /BI/BA /B4/CP/B5 /C8/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /D3/CU /D8/CW/CT \r /CW/CT\r /CZ /CT/D6/CQ /D3/CP/D6/CS /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 (1/1/2) /D0/CP/CS/CS/CT/D6 /B4/AS/CV/D9/D6/CT /BG \r/B5 /CX/D2 /D8/CW/CT\n(J2/J1,h/J1) /D4/D0/CP/D2/CT /CJ /BK/BE ℄/BA /CC/CW/CT /D4/CW/CP/D7/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/CU /D8/CW/CT /CU/D9/D0/D0/DD /D4 /D3/D0/CP/D6/CX/DE/CT/CS /D4/CW/CP/D7/CT /B4M0= 3/2 /B5 /CX/D7 /CT/DC/CP\r/D8/BA/C0/CT/D6/CTM0\n/CX/D7 /D8/CW/CT /BY/C5 /D1/D3/D1/CT/D2 /D8 /D4 /CT/D6 /D6/D9/D2/CV/BA /CC/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /B4/BT/BU/BV/BW/B5 /D3/CU /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r M0= 1/2/D4/D0/CP/D8/CT/CP/D9 /D4/CW/CP/D7/CT /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /BX/BW /D3/CU /D4 /CT/D6/CX/D3 /CS/CX\r \r/D0/D9/D7/D8/CT/D6/D7 /DB/CX/D8/CWL= 10 /B4/D7/D5/D9/CP/D6/CT/D7/B5 /CP/D2/CS\nL= 12 /B4\r/D6/D3/D7/D7/CT/D7/B5 /D6/D9/D2/CV/D7/BA /CC/CW/CT /D4 /D3/CX/D2 /D8/D7 /BU/B8 /BV/B8 /CP/D2/CS /BW /D1/CP/D6/CZ/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /B8 /D8/CW/CT \r /CW/CP/D2/CV/CT /D3/CU /D8/CW/CT /D0/D3 /DB /CT/D7/D8/CT/DC\r/CX/D8/CT/CS /D7/D8/CP/D8/CT/D7/B8 /D8/CW/CT /D8/CX/D4 /D3/CU /D8/CW/CT /D0/D3/CQ /CT/B8 /CP/D2/CS /D8/CW/CTh= 0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CP/D8J2/J1= 0.399 /CU/D6/D3/D1 /D8/CW/CT\nM0<1/2 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT /D8/D3 /CP /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /D4/CW/CP/D7/CT/D7/BA /CC/CW/CT /D0/CP/D8/D8/CT/D6 /D4/CW/CP/D7/CT /D3 \r\r/D9/D4/CX/CT/D7 /D8/CW/CT /D6/CT/D7/D8/D3/CU /D8/CW/CT /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1/BA /B4/CQ/B5M0\n/DA/D7J2/J1\n/CU/D3/D6 /D8/CW/CT /D2/D3/D2/B9/C4/CX/CT/CQ/B9/C5/CP/D8/D8/CX/D7 /B4/CX/BA/CT/BA/B8M0<1/2 /B5 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r/D4/CW/CP/D7/CT/B8 /CP/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /BX/BW /D3/CUL= 12 \r/D0/D9/D7/D8/CT/D6/D7 /B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/B5/BA /CC/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT \r/D3/D2/D2/CT\r/D8/D7/D8/CW/CT /D1/CX/CS/D4 /D3/CX/D2 /D8/D7 /D3/CU /D8/CW/CT /D7/D8/CT/D4/D7/CT /CJ /BK/BF ℄/BA\r/D3/D1/D4/D0/CT/D8/CT/D0/DD /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/CT/CP/D8/D9/D6/CT/D7/BM /CC/CW/CT /BY/C5 /D0/CT/CV \r/D3/D9/D4/D0/CX/D2/CV /CS/D6/CX/DA /CT/D7 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D2 /D8/D3 /CP /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /CV/D6/D3/D9/D2/CS/D7/D8/CP/D8/CT /DB/CW/CX\r /CW /CX/D7 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CU/D3/D6 /D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /BT/BY/C5 /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /CJ /BK/BD ℄/BA /CB/CX/D1/CX/D0/CP/D6 /CV/CP/D4/D0/CT/D7/D7 /D4/CW/CP/D7/CT/CP/D4/D4 /CT/CP/D6/D7 /CX/D2 /D8/CW/CT /D9/D2/CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /DA /CP/D6/CX/CP/D2 /D8 /D3/CU /D8/CW/CT /D1/D3 /CS/CT/D0 /CS/CX/D7/D4/D0/CP /DD /CT/CS /CX/D2 /AS/CV/D9/D6/CT /BG /CT /CJ /BJ/BJ /B8 /BJ/BK ℄/B8 /DB/CW/CT/D6/CT/CP/D7 /D8/CW/CT /D1/D3 /CS/CT/D0/D7/CW/D3 /DB/D2 /CX/D2 /AS/CV/D9/D6/CT /BG /CS /D4 /D3/D7/D7/CT/D7/D7/CT/D7 /CP /CV/CP/D4/D4 /CT/CS /D2/D3/D2/B9/CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /D6/D9/D2/CV/B9/D7/CX/D2/CV/D0/CT/D8 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CJ /BK/BC ℄/B8 /DB/CW/CX\r /CW /CX/D7 /D8/CW/CT\r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /D4/CW/CP/D7/CT /D3/CU /D8/CW/CT /D9/D2/CX/CU/D3/D6/D1 /D7/D4/CX/D2/B9/BD/BB/BE /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /D0/CP/CS/CS/CT/D6/BA/CC /D3 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS /D8/CW/CT /D6/D3/D0/CT /D3/CU /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX\r /CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /CX/D2 /D7/D9\r /CW /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7/B8 /D0/CT/D8 /D9/D7 /D8/D9/D6/D2/D8/D3 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /CX/D2 /AS/CV/D9/D6/CT /BI/CP /D4/D6/CT/D7/CT/D2 /D8/CX/D2/CV /D8/CW/CT /D4/CW/CP/D7/CT/D7 /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS (1,1/2) \r /CW/CT\r /CZ /CT/D6/CQ /D3/CP/D6/CS/D0/CP/CS/CS/CT/D6 /B4/AS/CV/D9/D6/CT /BG\r/B5 /CX/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP\r/CT(J2/J1,h/J1) /CJ /BK/BE ℄/BA /CF /CT /CP/D6/CT /D2/D3/D8 /CP /DB /CP/D6/CT /D3/CU /CP/D2 /DD /D4/D9/CQ/D0/CX\r/CP/D8/CX/D3/D2/D7/D7/D8/D9/CS/DD/CX/D2/CV /D8/CW/CT /D3/D8/CW/CT/D6 /D8 /DB /D3 /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D1/D3 /CS/CT/D0/D7 /CS/CX/D7/D4/D0/CP /DD /CT/CS /CX/D2 /AS/CV/D9/D6/CT/D7 /BG /CS/B8 /CP/D2/CS /CT/BA /BT/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT/CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 J2/J1\n/B8 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS \r /CW/CT\r /CZ /CT/D6/CQ /D3/CP/D6/CS /D1/D3 /CS/CT/D0/CT/DC/CW/CX/CQ/CX/D8/D7 /D8/CW/D6/CT/CT /D4/CW/CP/D7/CT/D7/B8 /DB/CW/CX\r /CW \r/CP/D2 /CQ /CT /CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /DD /D8/CW/CT /CP/D2/CV/D0/CT/D7(θ,φ) /AS/DC/CX/D2/CV /D8/CW/CT /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT\r/D0/CP/D7/D7/CX\r/CP/D0 /D7/D4/CX/D2/D7S1,n\n/CP/D2/CSS2,n\n/CX/D2 /D6/CT/D7/D4 /CT\r/D8 /D8/D3 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r \r/D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 (θ,φ) = (0,0)/DB/CX/D8/CW /D9/D4/B9S1,n\n/CP/D2/CS /CS/D3 /DB/D2/B9S2,n\n/D7/D4/CX/D2/D7/BA /C1/D2 /D8/CW/CT /D7/D4 /CT\r/CX/CP/D0 \r/CP/D7/CT(S1,S2) = (1,1/2) /B8 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 \r/CP/D2 /D8/CT/CS/D7/D8/CP/D8/CT /D7/CW/D3 /DB/D2 /CX/D2 /AS/CV/D9/D6/CT /BG\r /CX/D7 /D7/D8/CP/CQ/D0/CT /CX/D2 /D8/CW/CT /CX/D2 /D8/CT/D6/DA /CP/D0 0.3219J1< J2<0.4606J1\n/BA /BY /D3/D6 /D0/CP/D6/CV/CT/D6J2\n/B8 /CP\r/D3/D0/D0/CX/D2/CT/CP/D6 \r/D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 /DB/CX/D8/CW(θ,φ) = (π/2,π/2) /CX/D7 /D7/D8/CP/CQ/CX/D0/CX/DE/CT/CS/BA /C1/D2 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D0/CX/D1/CX/D8/B8 /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/D7/CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT \r/CP/D2 /D8/CT/CS /D7/D8/CP/D8/CT /CP/D2/CS /D8/CW/CT /D3/D8/CW/CT/D6 /D8 /DB /D3 /D4/CW/CP/D7/CT/D7 /CP/D6/CT \r/D3/D2 /D8/CX/D2 /D9/D3/D9/D7/BA /CC /D9/D6/D2/CX/D2/CV /D8/D3 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D1/D3 /CS/CT/D0/B8/D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV \r /CW/CP/D2/CV/CT/D7 /CX/D2 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /B4/D0/CX/D2/CTh= 0 /CX/D2 /AS/CV/D9/D6/CT /BI /CP/B5 \r/CP/D2 /CQ /CT /CX/D2/CS/CX\r/CP/D8/CT/CS/BA/CF/CW/CT/D6/CT/CP/D7 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT /D7/D9/D6/DA/CX/DA /CT/D7 /D5/D9/CP/D2 /D8/D9/D1 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7/B8 /D8/CW/CT \r/D3/D0/D0/CX/D2/CT/CP/D6 /D1/CP/CV/D2/CT/D8/CX\r/D7/D8/CP/D8/CT /CX/D7 \r/D3/D1/D4/D0/CT/D8/CT/D0/DD /CS/CT/D7/D8/D6/D3 /DD /CT/CS/BA /C1/D2/D7/D8/CT/CP/CS/B8 /CU/D3/D6J2>0.399J1\n/D8/CW/CT/D6/CT /CP/D4/D4 /CT/CP/D6/D7 /CP /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /D4/CW/CP/D7/CT/BA/C1/D2 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0 \r /CW/CP/D7/CT/B8 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT /CX/D7 /CT/CX/D8/CW/CT/D6 \r/D6/CX/D8/CX\r/CP/D0 /B4/CU/D3/D6 /CW/CP/D0/CU/B9/CX/D2 /D8/CT/CV/CT/D6 S1+S2\n/B5/B8/D3/D6 /CV/CP/D4/D4 /CT/CS /B4/CU/D3/D6 /CX/D2 /D8/CT/CV/CT/D6S1+S2\n/B5/BA/CC/CW/CT /D1/D3/D7/D8 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV \r /CW/CP/D2/CV/CT/D7 /CP/D4/D4 /CT/CP/D6 /CX/D2 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 \r/CP/D2 /D8/CT/CS /D7/D8/CP/D8/CT/BA /C1/D2 /AS/CV/D9/D6/CT /BI/CQ /DB /CT /D7/CW/D3 /DB /D2 /D9/D1/CT/D6/CX\r/CP/D0/BX/BW /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6 /D8/CW/CT /BY/C5 /D1/D3/D1/CT/D2 /D8 /D4 /CT/D6 /D6/D9/D2/CVM0\n/CP/D7 /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 J2/J1\n/BA /CF /CT /D7/CT/CT/D8/CW/CP/D8 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /DB/CW/CX\r /CW /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CT/D7 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 \r/CP/D2 /D8/CT/CS /D4/CW/CP/D7/CT /CX/D7 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS /CQ /DD /CP /AS/D2/CX/D8/CT/BY/C5 /D1/D3/D1/CT/D2 /D8 /DB/CW/CX\r /CW /CX/D7/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /D0/CT/D7/D7 /D8/CW/CP/D8 /D8/CW/CT /D5/D9/CP/D2 /D8/CX/DE/CT/CS /BY/C5 /D1/D3/D1/CT/D2 /D8 /D4 /CT/D6 /D6/D9/D2/CV /B4S1−S2= 1/2 /B5/D3/CU /CP /D7/D8/CP/D2/CS/CP/D6/CS /C4/CX/CT/CQ/B9/C5/CP/D8/D8/CX/D7 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/BA /C6/D3/D8/CX\r/CT /D8/CW/CP/D8 /D8/CW/CX/D7 /D4/CW/CP/D7/CT /CT/DC/CX/D7/D8/D7 /D3/D2/D0/DD /CP/D8h= 0 /BA /BY /D3/D6h >0/CX/D8 /D1/CT/D6/CV/CT/D7 /CX/D2 /D8/D3 /D8/CW/CT /C4/D9/D8/D8/CX/D2/CV/CT/D6 /D0/CX/D5/D9/CX/CS /D4/CW/CP/D7/CT /B4/D7/CT/CT /AS/CV/D9/D6/CT /BI/CP/B5/BA /BT/D2 /CT/DC/D8/D6/CP/D4 /D3/D0/CP/D8/CX/D3/D2 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/BU/B8 /BJ/BK /B8 /BC/BI/BG/BG/BC/BJ/BA/BK/BK/BA /CC /D3/D2/CT/CV/CP /DB /CP /CC/BA/B8 /C7/CZ /CP/D1/D3/D8/D3 /C3/BA/B8 /CB/CP/CZ /CP/CX /CC/BA/B8 /CP/D2/CS /C3/CP/CQ/D9/D6/CP/CV/CX /C5/BA/B8 /BE/BC/BC/BL/B8 /C2/BA /C8/CW /DD/D7/BA/BM /BV/D3/D2/CU/CT/D6/CT/D2\r/CT /CB/CT/D6/CX/CT/D7/B8 /BD/BG/BH /B8 /BC/BD/BE/BC/BI/BI/BA/BD/BF"    },    {        "title": "2401.08235v1.Spin_Waves_in_Ferrimagnets_near_the_Angular_Magnetization_Compensation_Temperature__A_Micromagnetic_Study.pdf",        "content": "SW in FiMs near the AMC Temperature AIP/123-QED\nSpin Waves in Ferrimagnets near the Angular Magnetization Compensation\nTemperature: A Micromagnetic Study\nLuis Sánchez-Tejerina,1David Osuna Ruiz,2Eduardo Martínez,3Luis López Díaz,3and\nÓscar Alejos1\n1)Department of Electricity and Electronics, University of Valladolid,\nSpain.\n2)Departament of Electrical, Electronical and Communications Engineering,\nPublic University of Navarra, Spain.\n3)Department of Applied Physics, University of Salamanca,\nSpain.\n(*Electronic mail: oscar.alejos@uva.es.)\n(Dated: 17 January 2024)\nSpin wave propagation along a ferrimagnetic strip with out-of-plane magnetization is stud-\nied by means of micromagnetic simulations. The ferrimagnetic material is considered as\nformed by two antiferromagnetically coupled sub-lattices. Two critical temperatures can\nbe defined for such systems: that of magnetization compensation and that of angular mo-\nmentum compensation, both different due to distinct Landé factors for each sub-lattice.\nSpin waves in the strip are excited by a spin current injected at one of its edges. The\nobtained dispersion diagrams show exchange-dominated forward volume spin waves. For\na given excitation frequency, Néel vector describes highly eccentric orbits, the eccentric-\nity depending on temperature, whose semi-major axis are oriented differently at distinct\nlocations on the FiM strip.\n1arXiv:2401.08235v1  [cond-mat.mtrl-sci]  16 Jan 2024SW in FiMs near the AMC Temperature\nMagnon propagation in antiferromagnetic (AFM) and/or ferrimagnetic (FiM) has drawn atten-\ntion recently and is the subject of very recent works1–9due to the variety of benefits as opposed to\nusing ferromagnetic (FM) materials, such as greater speeds, marginal sensitivity to external fields\nand the wider range of materials available, many being electrical insulators. Spin wave propaga-\ntion in insulating AFM materials connects magnonics with spintronics since spin current (a key\nconcept in spintronics) is mainly transported by magnons in such materials, due to the absence\nof conducting electrons as opposed to FM materials, where electrons are the main spin carriers.\nThe use of insulators also points to the importance of thermal effects due to local heating. The\ndynamics of spins in AFM materials have been widely investigated theoretically. For example,\ncharacterising the fundamental k=0 resonance modes2–4, propagating magnons5,6or AFM do-\nmain walls (DWs) and their interactions with magnons7,8, but always focused on their fundamental\nphysics, as AFM magnonics is a field still in its infancy2,9,10. Additionally, FiM materials at the an-\ngular momentum compensation point ( TA) can mimic the dynamical response of AFM11–13while\nbeing easier to manipulate and detect by electric and/or optical means14,15.\nIn this work, micromagnetic simulations have been used to investigate the propagating high-\nfrequency ( ≈THz) AFM modes in a FiM strip at different temperatures. Simulations consider\nFiMs as formed by two sub-lattices (indexes i=1,2 will be used to refer to each sub-lattice).\nAccordingly, a pair of coupled Landau-Lifshitz-Gilbert equations are used to model the material:16\n˙mi=−γimi×Heff,i+αimi×˙mi+τSOT ,i (i=1,2), (1)\nwhere mi,γiandαiare the local orientation of the magnetization, the gyromagnetic factor, and\nthe damping parameter for each sub-lattice, respectively. Heff,iis the effective field including all\nrelevant interactions within the system, and τSOT ,icorresponds to the spin-orbit torque (SOT) due\nto spin currents. In our model, the FiM is formed by computational elementary cells, in a finite\ndifference scheme. We have two magnetic moments within each cell, one for each sub-lattice, and\nboth sub-lattices are coupled by an interlattice exchange interaction (see details in Ref.16).\nSimulations have been carried out by means of a homemade code17implemented on graphic\nprocessing units. Details of the modeled device are depicted in Fig. 1. FiM strip parameters\nconsidered are those that can be found in the literature for a prototypical FiM as GdFeCo11. Sat-\nuration magnetization for each sub-lattice varies with temperature according to the law Ms,i=\nM0\ns,i\u0010\n1−T\nTC\u0011bi,TC=450 K being the Curie temperature, M0\ns,ibeing the saturation magnetization\nat 0K: M0\ns,1=1.71MA\nmM0\ns,2=1.4MA\nm, and b1=0.76, and b2=0.5, resulting in the tempera-\n2SW in FiMs near the AMC Temperature\nX YZ\n32 nm𝐽𝑡=𝐽0sinc 2𝜋𝑓𝑐𝑡−𝑡𝑐8 nm8 nmprobes\n𝑡=0𝐽0=1TA\nm2\n𝑡\n𝑡𝑐=0.5 ns\n𝑓𝑐=1 THz\n𝐉 =𝐽𝑡𝐮𝑦\n(a)\n(b)\n(c)\n𝐉s\nFIG. 1. Geometry and properties of the modeled device: (a) Temperature dependence of the saturation\nmagnetization for each sub-lattice, (b) dimensions of the FiM strip and of the exciting spin current injection\narea, (c) time dependence of the exciting current to obtain the dispersion diagrams.\nture dependence plotted in Fig. 1(a). According to these parameters, magnetization compensation\noccurs at TM=241 K. The rest of significant parameters are: intralattice exchange Ai=70pJ\nm, in-\nterlattice exchange Bi j=−9MJ\nm3, damping constant αi=0.001, anisotropy constant Ku,i=1.4MJ\nm3,\nDzyaloshinskii-Moriya constant Di=0.12mJ\nm2, and spin-Hall angle θSH=0.15. Finally, each sub-\nlattice is characterized by distinct Landé factors: g1=2,g2=2.05, which result in a temperature\nof angular momentum compensation TA=260K, slightly greater than TM.\nThe device consists of a FiM strip and a current line of a heavy metal (HM) running perpendicu-\nlar to the strip at one end as shown in Fig. 1(b). The FiM material is initially uniformly magnetized\nin the out-of-plane direction. The FiM strip area is 8192 nm ×32 nm, and it is 6-nm thick, while\nthe HM strip is 32-nm wide. This defines a 32 nm ×32 nm SW excitation area on the leftmost end\nof the FiM strip due to an electric current flowing through the HM line, then exciting spin waves\nvia SOT. Beside this area, a set of 254 probes are regularly spaced on the FiM strip to monitor the\nvalue of the local magnetization, i.e., Néel’s vector n=m1−m2. Probes are 3 nm ×8 nm in size\nand 5-nm distant from each other, then occupying a total length of 2032 nm.\nA spin current arising from an electric current carried by the HM strip is injected through the\nexcitation area. This spin current is polarized along the ±uxdirection depending on the instanta-\n3SW in FiMs near the AMC Temperature\nneous direction of the electric current along the HM strip. Figure 1(c) presents the time dependent\ncurrent considered to achieve a uniform excitation across a desired frequency range in the dis-\npersion diagram. In this first case, an (unnormalized) sinc-shaped electric current pulse has been\nused, J(t) =J0sinc[2πfc(t−tc)], where fcis the excitation cut-off frequency, which was set to\n1 THz, J0is the pulse amplitude, initially set to 1TA\nm2, and tcis a delay time which has been chosen\nas one half of the total simulation time, set to 1 ns. Using this activation, we ensure that each mode\nis equivalently fed. This current is sufficiently small to remain in the linear regime of activation of\ndamped, stable oscillations3and to avoid any static changes in the AFM phases of the sample. The\ndelay time also provides a reasonable offset to the peak of the pulse, allowing a gradual increase\nof the amplitude from the beginning of the simulation. However, when later analyzing the time\nevolution of the magnetic signal, a monochromatic wave (MW) excitation is to be applied with\na current at a specific frequency f0, i.e., J(t) =J0sin(2πft). Equivalently to the previous case,\nJ0is chosen to be 1TA\nm2to remain in the same modes regime and to obtain a good magnetization\ncontrast for m1,2(t). Finally, temperature is taken here as an additional parameter, i.e., a way to\nbalance the value of the saturation magnetization of both sub-lattices to achieve both magnetiza-\ntion compensation and angular momentum compensation. This means that only coherent magnons\nare considered in our simulations.\nSW spectra in AFM materials have revealed ultra-high frequency (in the THz regime), field-\ndependent modes that are related to each of the sub-lattices (strongly coupled by exchange) usu-\nally called ‘optic’ or higher frequency mode, and ‘acoustic’ or lower frequency mode2,3. For\neach mode, there are opposite senses of precession for each sub-lattice, either right-handed (RH)\nor left-handed (LH). The AFM resonance frequencies ( k=0 magnons) depend on the different\nequilibrium states of the coupled sub-lattices, or so-called AFM phases2, which can also be tuned\nby external fields. These modes can be locally excited via spin currents Js, as represented in\nFig. 1(b). The localized perturbation allows AFM magnons with k̸=0 to propagate in the material\nvia exchange interactions, consequently possessing very short wavelength. To investigate the SW\npropagation in the frequency domain, we analyze the Néel vector since it transports coherently\nexcited magnons, such as those generated by Js18.\nWe focus on the in-plane components of the Néel vector nxandnywhich account for the\ndynamic (differential) x-component and y-component respectively from the two sub-lattices at\nthe closest position to the excitation region. Figure 2(a) shows the SW spectra (FFT intensity of\nnx) and fundamental resonances at k=0 for different strip temperatures ranging from below TMto\n4SW in FiMs near the AMC Temperature\n(a)(b)\nFIG. 2. (a) SW spectra (FFT intensity of nx) and fundamental resonances at k=0 for different strip temper-\natures, and (b) dispersion relations obtained for nyat four different temperatures in the strip ranging below\nTMto above TA. The regularly spaced interruptions in the dispersion curves are related to the geometry of\nthe excitation zone (see text).\nabove TA, as a response to the sinc-shaped pulse current. The simulated (normalised) spectra are in\ngood agreement with recent work characterising such modes in FiMs19, where the two distinctive\npeaks overlap at TA. Similar results are obtained for ny(not shown), but with greater amplitude\nthan nxatTA, and generally wider peaks, suggesting an elliptical precession of m1,22. In contrast\ntony,nxis more attenuated at TAthan at any other temperature. In particular, Fig. 2(b) shows the\ndispersion relations obtained for nyat four different temperatures in the strip ranging below TM\nto above TA. The color scale is the same for all color plots, so intensities are directly comparable\nbetween different temperatures. The exchange-dominated character of the SWs is clear, in the\nform of a exchange-dominated k2-dependent mode frequency20. Solid curves show the fitting to\nthe data from Kalinikos and Slavin’s theory20, when each sub-lattice magnetization is modeled as\nof separate effective FM, i.e. as in a synthetic antiferromagnet (SAF), and MS,i(Tk). A Forward\nV olume SW (FVSW) configuration is assumed, which corresponds well to the initial out-of-plane\nconfiguration, where m1,2are perpendicular to both kand the strip plane at equilibrium. When\nthe AFM resonance frequencies ( k=0) for each sub-lattice (see Fig. 2(a)) are included as the cut-\nofffrequencies into the FVSW equations, the agreement with the simulated dispersion relations\nis remarkable (green curve for i=1 and red curve for i=2). Such dispersion relations can be\nanalytically described as2\nωi=γi\u0012\nHc,i+2Ai\nµ0MS,ik2\nx\u0013\n(i=1,2), (2)\nwith Hc,i=q\n2Hanis ,iHexch ,i+H2\nanis ,i,Hexch ,i=|Bi j|\nµ0MS,ibeing the interlattice exchange field, and\n5SW in FiMs near the AMC Temperature\nT<TM\nTM<T<TA\n0 0.5 1 1.5 2T≈TA\nnx,ny(a.u)nx\nnynx,ny(a.u)nx\nnynx,ny(a.u)\nx(µm)nx\nny\nFIG. 3. Typical instantaneous values of nxandnyat the location of each probe. The graphs shows the\nresults obtained at t=1 ns at different temperatures: T<TM,TM<T<TAandT≈TA.\nHanis ,i=2ku,i\nµ0MS,ibeing the anisotropy field. According to these expressions, the pair of dispersion\ncurves merges as the temperature approaches TA, as Fig. 2(b) confirms. It should be mentioned at\nthis point that the dispersion curves are regularly interrupted for the various multiples of a certain\nkx≈100rad\nµm. This modulation is caused by the spatially limited excitation zone 32 nm-wide. Note\nthat at the edges of the excitation zone, the spin accumulation vanishes giving a half-wavelength\nofλ\n2=32 nm corresponding to a k-vector with kx=2π\nλ.\nWith the purpose of analyzing the time evolution of the magnetic signal under a MW excitation,\nthe following results presented correspond to an excitation via a MW signal of frequency f=\n360 GHz. Although other frequency values have been considered, this one has been found to be\noptimal for revealing magnon propagation characteristics along the FiM strip. In particular, this\nvalue is slightly higher than the cut-off frequency ( k=0 magnons) for all of the temperatures\nconsidered. The intersection of this excitation frequency with the dispersion curves shows how\nthe wave vector is in general different for each of the two sub-lattices. Accordingly, SWs travel\nwith different phase velocities through each sub-lattice. This fact is illustrated in the graphs in\nFig. 3. The plots show projections along the x- and y-directions of the Néel vector orbits. Plots\nare obtained at the end of the excitation time to ensure that the stationary regime is achieved. The\ncoupled oscillations between both sublattices but with different kxgenerate patterns of nodes and\nanti-nodes of nxandnyalong the FiM strip. All patterns possess anti-nodes of nyat the closest\nzone to the excitation area in agreement with the injected spin current. In addition to the oscillatory\n6SW in FiMs near the AMC Temperature\nbehavior of the two components, their amplitudes are also modulated by the progressive vanishing\nof the amplitude of the oscillations as the SWs travel along the strip. To get a clearer idea of the\nmagnon propagation, a set of videos showing the evolution in time of the plots in Fig 3 has been\nadded as part of the supplementary material.\nWe can further analyze the consequences of the phase velocity difference between the two sub-\nlattices. Fig.4 represents the in-plane components of the Néel vector at different positions along\nthe strip for three different temperatures: below TM, in-between TMandTA, and at approximately\nTA. These trajectories are always elliptic, their eccentricity being determined by temperature and\nreaching a maximum at TA. Fig.4 also shows that the strip position plays no role in the eccentricity,\nthough the semi-major axis takes different orientations depending on location. The Néel vector\ntrajectory is composed of the two magnetization sub-lattice precessions. Because the phase ve-\nlocity of each sub-lattice differs, the relative phase between the two precessions is modified from\npoint to point, thus modifying the semi-major axis orientation. Consequently, different positions\nare subjected to different torques, not only by the amplitude fading but because of this reorienta-\ntion effect. Nevertheless, the reorientation vanishes as the temperature approaches TA, which can\nbe a touchstone to detect this temperature for FiM materials under test.\nIn conclusion, we have numerically explored the excitation and propagation of SWs along\nFiM strips as a function of temperature, below and above the angular and the magnetization com-\npensation points. The analysis of results could be extrapolated to FiM materials with different\ncompositions at room temperature. The fact that the eccentricity of the orbits can be controlled\nby temperature, in addition to the gradual rotation of the semi-major axis along the strip, provides\nan unprecedented local control of magnetization oscillations, which can be exploited to imple-\nment novel functionalities such as sensing in the emerging field of THz magnonics10. Our results\nmay also be relevant to spintronics for making very localized excitation positions given the sen-\nsitivity to distance of the different exerted torques along the magnetic strip. Besides, a potential\nreconfigurability can be envisaged by using controlled heat sources along the strip, such as laser\npulses.\nThis work was supported by Projects No. SA114P20 from Junta de Castilla y León, No.\nPID2020117024GB-C41 funded by MCIN/AEI/10.13039/501100011033, and Project MagnEFi,\nGrant Agreement No. 860060, (H2020-MSCA-ITN-2019) funded by the European Commission.\nThe data that support the findings of this study are available from the corresponding author\nupon reasonable request.\n7SW in FiMs near the AMC Temperature\n𝑇𝑀<𝑇<𝑇𝐴\n𝑇≈𝑇𝐴\n𝑇<𝑇𝑀\nSingle frequency  excitation  \n𝑓=360 GHz\n𝑡=0𝐽0=1TA\nm2\n𝑡\nFIG. 4. Trajectories of the in-plane component of the Néel vector in the stationary regime under a MW\nexcitation. Trajectories are obtained at different locations along the FiM strip and at different temperatures.\nREFERENCES\n1C. Chen, C. Zheng, J. Zhang, and Y . 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D.\nBeach, “Fast current-driven domain walls and small skyrmions in a compensated ferrimagnet,”\nNature Nanotechnology 3(2018).\n12S. A. Siddiqui, J. Han, J. T. Finley, C. A. Ross, and L. Liu, “Current-induced domain wall\nmotion in a compensated ferrimagnet,” Physical Review Letters 121, 057701 (2018).\n13F. Cutugno, L. Sanchez-Tejerina, R. Tomasello, M. Carpentieri, and G. Finocchio, “Micro-\nmagnetic understanding of switching and self-oscillations in ferrimagnetic materials,” Applied\nPhysics Letters 118, 052403 (2021).\n14K. Ueda, M. Mann, C.-F. Pai, A.-J. Tan, and G. S. D. Beach, “Spin-orbit torques in\nTa/TbxCo100-x ferrimagnetic alloy films with bulk perpendicular magnetic anisotropy,” Ap-\nplied Physics Letters 109, 232403 (2016).\n15K. Fleischer, N. Thiyagarajah, Y .-C. Lau, D. Betto, K. Borisov, C. C. Smith, I. V . Shvets, J. M. D.\nCoey, and K. Rode, “Magneto-optic kerr effect in a spin-polarized zero-moment ferrimagnet,”\n9SW in FiMs near the AMC Temperature\nPhys. Rev. B 98, 134445 (2018).\n16E. Martínez, V . Raposo, and Ó. Alejos, “Novel interpretation of recent experiments on the\ndynamics of domain walls along ferrimagnetic strips,” Journal of Physics: Condensed Matter\n32, 465803 (2020).\n17Ó. Alejos, V . Raposo, L. Sanchez-Tejerina, R. Tomasello, G. Finocchio, and E. Martinez,\n“Current-driven domain wall dynamics in ferromagnetic layers synthetically exchange-coupled\nby a spacer: A micromagnetic study,” Journal of Applied Physics 123, 013901 (2018).\n18R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati, J. Cramer, A. Brataas, R. A.\nDuine, and M. Kläui, “Tunable long-distance spin transport in a crystalline antiferromagnetic\niron oxide,” Nature 561, 222–225 (2018).\n19E. Haltz, J. Sampaio, S. Krishnia, L. Berges, R. Weil, A. Mougin, and A. Thiaville, “Quantitative\nanalysis of spin wave dynamics in ferrimagnets across compensation points,” Physical Review\nB105, 104414 (2022).\n20B. A. Kalinikos and A. N. Slavin, “Theory of dipole-exchange spin wave spectrum for ferro-\nmagnetic films with mixed exchange boundary conditions,” Journal of Physics C: Solid State\nPhysics 19, 7013–7033 (1986).\n10"    },    {        "title": "2007.11214v3.Nuclear_Magnetic_Relaxation_Time_near_Compensation_Temperature_in_Ferrimagnetic_Insulator.pdf",        "content": "Journal of the Physical Society of Japan FULL PAPERS\nNuclear Magnetic Relaxation Time near the Compensation Temperature in a\nFerrimagnetic Insulator\nMichiyasu Mori\nAdvanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 117-1195, Japan\nThe nuclear magnetic relaxation time T1in a ferrimagnetic insulators is calculated within the mean-field approxima-\ntion for the magnetic exchange interactions and the Raman process involving the hyperfine interaction. We find that the\nvalue of 1 /T1on one type of site increases rapidly near the compensation temperature T0, whereas that on the other type\nof site does not increase up to Curie temperature Tc. This is due to the fact that the soft-magnon bandwidth becomes\ncomparable to T0. An increase in 1 /T1below Tcis found also in another type ferrimagnet, which shows a hump structure\nin the temperature dependence of magnetization instead of compensation. Also in that case, we find the rapid increase in\n1/T1below Tc, even though the magnetization does not show compensation. The coexistence of soft and hard magnons\nleads to these remarkable properties of ferrimagnets.\n1. Introduction\nAferri- magnet is a kind of ferro- magnet, and it was\ntheoretically predicted by N ´eel.1–3)Soon afterward, the\nmagnetization compensation was observed in the LiFeCr\nspinel ferrite, for which the magnetization becomes zero at\nmagnetization-compensation temperature TMfar below the\nCurie temperature Tc.4)Such a ferrimagnet, called an N-type\nferrimagnet, also has been found in rare-earth iron garnets\n(RIGs).2, 5–11)The RIGs have been studied by many authors in\norder to apply their magnetization-compensation properites to\nmagneto-optical memories.12–14)\nThe dynamical aspects of ferrimagnetism were initially\nstudied using the electron spin resonance (ESR).15–23)The\nferrimagnetic resonance (FIR) di \u000bers from the ferromagnetic\nresonance (FMR) in having two branches. One gives the usual\nFMR, while the other, called the exchange frequency, is lo-\ncated higher in energy.21)It was di \u000ecult to measure the ex-\nchange frequency when it was first discovered, since its wave-\nlength is of the order of a tenth of a millimeter. However,\na singular behavior of the gyromagnetic ratio was observed\naround the angular momentum compensation temperature TA\nin a LiFeCr spinel ferrite.15, 16)On the lower branch, the e \u000bec-\ntive gyromagnetic ratio becomes small around TMand then\nincreases rapidly around TA.15–23)Theg-value of the upper\nbranch becomes small in a measurable range around TA.16)\nThe magnetization is the product of the Lande g-factor and\na total angular momentum. In general, hence, TMis di\u000berent\nfrom TA, when the orbital angular momentum is involved. In\ncontrast to the magnetization, the dynamics of a ferrimagnet\nbecome singular around TA.\nBecause the magnetization couples to a magnetic field,\nwhile the total angular momentum itself does not, it can be\ndi\u000ecult to measure TAdirectly using conventional methods.\nRecently, however, Imai et al. have successfully observed TA\nusing the Barnett e \u000bect.24–26)In a rotating frame, the rota-\ntion frequency couples to the angular momentum instead of\nto the magnetization, without any coupling constant. By spin-\nrotation coupling, a magnetization is induced through the an-\ngular momentum by mechanical rotation. This was originally\nstudied by Barnett,27)and it is now used to determine TAina RIG.24, 25)It has been reported that around TAthe magneti-\nzation reverse rapidly and that domain walls move fast.28–31)\nThose properties, which are advantageous for magnetic mem-\nories, are attributed to angular-momentum compensation.\nNuclear magnetic resonance (NMR) also is a powerful tool\nfor studying the magnetism of a broad range of materials.\nMagnetic excitations can be characterized by the nuclear mag-\nnetic relaxation time T1, which originates in the hyperfine in-\nteraction between electron and nucleus. For magnetic insu-\nlators, however, the origin of T1is not so obvious. If the sys-\ntem is isotropic and the nuclear and electron quantization axes\nare identical, the relaxation cannot be obtained within the lin-\nearized spin-wave approximation. Misalignment of the quan-\ntization axes–and /or the dipole-dipole interactions between an\nelectronic and a nuclear spins–induces relaxation through the\nRaman process.32–34)Interactions among magnons are also\nthe source of relaxation, e.g., through the three-magnon pro-\ncess.32–34)Those processes can be studied for both ferromag-\nnetic and antiferromagnetic insulators. Recently, Imai et al.\nhave reported an enhancement of the NMR signal around TA,\nwhich is closely related to domain wall motion.26)In contrast\nto ESR, NMR provides a site-selective measurement of mag-\nnetism. It is therefore interesting to study the dynamical as-\npects of magnetism site-by-site in a ferrimagnet. In addition,\na consistent understanding of ferrimagnetism among experi-\nmental methods–NMR, ESR, and neutron scattering–will be\nuseful.\nIn this paper, we study the nuclear magnetic relaxation\ntime in ferrimagnets. Section 2 explains the model Hamilto-\nnian and the approximation used. The nuclear magnetic relax-\nation time due to the Raman process is given in Sec. 3. Ad-\nditional changes due to orbital angular momentum are briefly\ndiscussed in Sec. 4. Below, Bohr magneton \u0016Band Planck\nconstant ~=h=2\u0019are set equal to 1 for brevity.\n2. Formalism: Magnons in Ferrimagnet\nWe will focus on a ferrimagnetic ”insulator,” which is sim-\nply called a ”ferrimagnet” below. The magnetic exchange in-\nteraction due to the Pauli principle and to the Coulomb in-\nteraction between electrons is the source of magnetism in a\nferrimagnet. Two sub-lattices with di \u000berent spin magnitudes\n1arXiv:2007.11214v3  [cond-mat.mtrl-sci]  29 Sep 2020J. Phys. Soc. Jpn. FULL PAPERS\nSA,SBcomprise the simplest model. The Hamiltonian is\ngiven by\nH=\u0000JAX\nhi;i0i~Si\u0001~Si0\u0000JBX\nhj;j0i~Sj\u0001~Sj0+JCX\nhi;ji~Si\u0001~Sj;(1)\nwith the spin operators ~Si(~Sj) on site i2A-sites ( j2B-sites).\nThe angular brackets h\u0001\u0001\u0001i denotes nearest neighbor sites. The\nmagnitudes of the magnetic exchange interactions JA,JB, and\nJCare assumed to be positive for brevity. First, we do not con-\nsider the orbital angular momentum ~Li. Hence, there is only\none compensation temperature T0; i.e., TM=TA\u0011T0. What\nis changed by ~Liwill be discussed in the last section. At T0,\nthe expectation values hSz\nAi\u0011MA>0 andhSz\nBi\u0011\u0000 MB<0\nsatisfy MA\u0000MB=0, where the bracket denotes the thermal\naverage. See also Appendix A. It is known that there are some\npossible cases of compensation. In the case considered above,\nboth sub-lattices have the same number of sites in a unit cell,\nas shown in the inset of Fig. 1. Another case has MAnA\u0000MBnB\n=0, for which the number nAof spins on sub-lattice Ais dif-\nferent from the number nBon sub-lattice B. As shown in Ap-\npendix B, those lattice structures have characteristic features\nin common. Hence, we consider the simplest case, shown in\nFig. 1 below.\nCompensation occurs at a finite temperature. To include the\ntemperature dependences of MAandMB, we adopt the mean-\nfield approximation and use the linearized spin-wave approx-\nimation around the mean-field solution. This is equivalent to\nTyablikov decoupling in the Green’s function method and is\na kind of random-phase approximation.35, 36)The mean-field\nsolution for JA=0.1,JB=1.0,JC=0.05, SA=1, and SB=1/2, is\nshown in Fig. 1, where T0=Tc\u00180.3 and the Curie temperature\nisTc\u00183.0.\nExpectation value of spin\nFig. 1. (Color online) The mean-field solution for JA=0.1, JB=1.0,\nJC=0.05, SA=1, and SB=1/2. The inset shows the lattice structure, which\nis three dimensional. The red (upward) and blue (downward) triangles denote\nMAandMB, respectively. The black circles are the sum of two expectation\nvalues, MA\u0000MB. The broken line indicates T0.\nThe Holstein-Primako \u000b(HP) bosons (magnons)–for which\nthe creation and annihilation operators are ay\ni;aionA-\nsublattice and by\nj;bjonB-sublattice–are given by, S\u0000\ni\u0018p2MAay\ni,S+\ni\u0018p2MAai,Sz\ni=MA\u0000ay\niai,S+\nj\u0018p2MBby\nj,S\u0000\nj\u0018p2MBbj,Sz\nj=by\njbj\u0000MB. Below, the spins are assumed to\nbe ordered in the z-direction. By the linearized approximation,\nthe action of the magnons is given by37)\nS=X\nq;i!n\by\" \u0000i!n 0\n0 i!n!\n+ \"1q\"\u0003\n3q\n\"3q\"2q!#\n\b;(2)\n\by\u0011\u0010\nay\nq(i!n);b\u0000q(\u0000i!n)\u0011\n; (3)\n\"1q\u0011zJCMB+zAJAMA(1\u0000\u0010Aq); (4)\n\"2q\u0011zJCMA+zBJBMB(1\u0000\u0010Bq); (5)\n\"3q\u0011JCp\nMAMBX\n\u0018eiq\u0001\u0018; (6)\n\u0010A(B)q\u00111\nzA(B)X\n\u0011cos(q\u0001\u0011); (7)\nWe use the boson operators aq(i!n) and b\u0000q(\u0000i!n) with mo-\nmentum q=(qx;qy;qz) and Matsubara frequency !n. The q-\nsummation is taken over the first Brillouin zone. The magnon\ndispersion relation depends on the connectivity of the sub-\nlattice, which gives the number of nearest-neighbor sites zA\nand zBon each sub-lattice, and the number zof nearest-\nneighbor sites between the two sub-lattices. In Eqs. (6) and\n(7),\u0018and\u0011mean the summation over the nearest-neighbor\nsites between the two sub-lattices and within each sublat-\ntice, respectively. From Eq. (2), the magnon Green’s functions\ng\u0017(q;i!n) are given by\ngA(q;i!n)\u0011haq(i!n)ay\nq(i!n)i\n=\u0000i!n+\"2q\u0010\ni!n\u0000E\u000bq\u0011\u0010\ni!n+E\fq\u0011; (8)\ngB(q;i!n)\u0011hbq(i!n)by\nq(i!n)i\n=\u0000i!n+\"1q\u0010\ni!n\u0000E\fq\u0011\u0010\ni!n+E\u000bq\u0011; (9)\nand the magnon dispersion relations E\u000bqandE\fqare given by\nE\u000bq=1\n22666664\u0010\n\"1q\u0000\"2q\u0011\n+r\u0010\n\"1q+\"2q\u00112\u00004\f\f\f\"3;q\f\f\f23777775; (10)\nE\fq=1\n22666664\u0000\u0010\n\"1q\u0000\"2q\u0011\n+r\u0010\n\"1q+\"2q\u00112\u00004\f\f\f\"3;q\f\f\f23777775:(11)\nThe small- Qapproximation for MA>MBleads to,\nE\u000bq\u0018CQ2; (12)\nE\fq\u001812JC(MA\u0000MB)+DQ2; (13)\nwhere CandDare constants given in Appendix B and Q\n\u0011q\nq2x+q2y+q2z. The mode E\u000bqis gapless, while E\fqhas an\n”optical gap,” Eg\u0011jE\u000bq=0\u0000E\fq=0j=12JC(MA\u0000MB), which\ndisappears at T0. The dispersion relations degenerate at the\ngamma point and increases linearly with Q, similar to an anti-\nferromagnet. Away from the gamma point, on the other hand,\nthe two dispersion relations deviate from each other:\nE\u000bq\u0018C1Q+C2Q2; (14)\nE\fq\u0018C1Q\u0000C2Q2; (15)\n2J. Phys. Soc. Jpn. FULL PAPERS\nwhere C1andC2are constants given in Appendix B. These\nQ-dependences are relevant to the temperature dependence of\nT1at low temperatures.\n3. Results: Nuclear Magnetic Relaxation due to the Ra-\nman Process\nIn this study, we consider the nuclear magnetic relaxation\ntime T1originating from the contact interaction between a nu-\ncleus and an electron\nHn\u0000el=1\n2X\n\u0017=A;B2666664\r\u0017X\nifi\u0017\u0010~Si\u0001~Ii\u00113777775; (16)\nwith the g-factor on\u0017-sites being given by \r\u0017and the nuclear\nspin~Iioni-site. If the system is isotropic and the quantiza-\ntion axes of the nucleus and electron are identical, we cannot\nobtain relaxation within the linearized approximation. Mis-\nalignment of the quantization axes–and /or the dipole-dipole\ninteractions between electronic and nuclear spins–will in-\nduce relaxation due to the Raman process.32–34)Interactions\namong magnons also cause relaxation, e.g., through the three\nmagnon-process.32–34)Below, we focus on the Raman process\ninduced by misalignment. This is su \u000ecient to enable us to\nfind some of the characteristics of T1near T0. The critical ex-\nponent of T1is beyond the scope of this study and will be\ndiscussed elsewhere. When the quantization axis of nucleus\ndeviates by an angle \u0012from that of the electron, Eq. (16) re-\nduces to the following component\nHz\nn\u0000el=1\n2X\n\u0017=A;B2666664\r\u0017X\nisin\u0012fi\u0017Sz\ni\u0000I+\ni+I\u0000\ni\u00013777775; (17)\nwhich are relevant for calculating T1. Assuming that \rA=\rB\n\u0011\rand the form factors fiAare constant, i.e., fiA=fiB\u0011f,\ntheT1on site\u0017=A;Bis given by\n1\nT1\u0017=FX\nqC\u0017(q;!0); (18)\nC\u0017(q;!0)=Z\ndt ei!0tD\nSz\n\u0017q(t)Sz\n\u0017;\u0000q(0)+Sz\n\u0017;\u0000q(t)Sz\n\u0017q(0)E\n;\n(19)\nwhereh\u0001\u0001\u0001i means the thermal average. The nuclear magnetic\nresonance energy is denoted by !0, and F=(\rfsin\u0012=2)2.\nUsing Eqs. (8) and (9), the spin-spin correlation function\nC\u0017(q;!0) is given by\nC\u0017(q;!0)=2\n1\u0000e!0=kBTIm\u0005R\n\u0017(q;!0); (20)\n\u0005\u0017(q;i!0)=kBTX\np;ng\u0017(p+q;i!n+i!0)g\u0017(p;i!n);(21)\nwhere Tis temperature, kBis the Boltzmann’s constant, and\nmomentum p=(px,py,pz) in the first Brillouin zone. The re-\ntarded function of \u0005\u0017(q;i!0) is denoted by \u0005R\n\u0017(q;!0). When\n!0is much smaller than kBT, the nuclear magnetic relaxation\ntime T1\u0017on site\u0017is given by\n1\nT1\u0017=2FX\nqlim\n!0!0kBT\n!0Im\u0005R\n\u0017(q;!0); (22)\n=2\u0019FX\np;q\u001a\nnB(E\u0017p)h\nnB(E\u0017p)+1i\u000bp\u000bq\n\u0001p\u0001q\u000e\u0010\nE\u0017p\u0000E\u0017q\u0011+nB(E\u0016p)h\nnB(E\u0016p)+1i\fp\fq\n\u0001p\u0001q\u000e\u0010\nE\u0016p\u0000E\u0016q\u0011\u001b\n; (23)\n\u000bp\n\u0001p=1\n2 \"1p+\"2p\n\u0001p+1!\n; (24)\n\fp\n\u0001p=1\n2 \"1p+\"2p\n\u0001p\u00001!\n; (25)\n\u0001p=r\u0010\n\"1p+\"2p\u00112\u00004\f\f\f\"3p\f\f\f2; (26)\nHere\u0016,\u0017, i.e.,\u0016=\u000bfor\u0017=\for\u0016=\ffor\u0017=\u000band the Bose\ndistribution function is denoted by nB(x)\u00111=[ex=(kBT)\u00001].\nNote that Eq. (23) can be checked by considering the ferro-\nand the antiferromagnet cases as discussed in Appendix C.\nUsing Eq. (23) and the mean-field solution shown in Fig. 1,\ntheT-dependence of 1 /T1\u0017can be calculated numerically as\nshown in Fig. 2 (a). Note that 1 =T1;Aincreases rapidly around\nFig. 2. (Color online) (a) The T-dependence of 1 =T1on the A-sites and B-\nsites (1 /T1Aand 1 /T1B) are plotted with upward triangles (red) and downward\ntriangles (blue), respectively. The mean-field solution for JA=0.1, JB=1.0,\nJC=0.05, SA=1, and SB=1/2 was used. The broken line indicates T0. See\nalso Fig. 1. (b) At low temperatures, T1Ais well fitted by T2(the dashed-\ndotted line) similar to the ferromagnet (See also Appendix C), whereas it is\ndeviated with increasing temperature as AT2+BT5(solid line). AandBare\nconstants.\nT=Tc\u00180.3, which corresponds to T0indicated by the bro-\nken line in Fig. 2. This contrasts sharply with 1 =T1B, which\ndiverges just below Tc. As shown in Fig. 2 (b), at low tem-\nperatures, T1Ais well fitted by T2, similar to the ferromagnet\n3J. Phys. Soc. Jpn. FULL PAPERS\n(See also Appendix C). With increasing temperature, on the\nother hand, it is fitted by AT2+BT5with constants AandB.\nThis is similar to the behavior of a ferromagnet, except for the\nfact that 1=T1Aincreases around T0instead of Tc.\nTo understand this behavior of 1 /T1around T0,E\u000bqandE\fq\nare plotted in Fig. 3 for (a) T=Tc=0.1 and (b) T=Tc=0.3 with\nqy=qz=0. At low temperatures, the T-dependence of 1 /T1\nFig. 3. (Color online) The dispersion relations E\u000bqandE\fqare plotted by\nupward triangles (red) and downward triangles (blue), respectively, for (a)\nT=Tc=0.1 and (b) T=Tc=0.3, with qy=qz=0. In the pre-set lattice struc-\nture, at q=(\u0019;\u0019= 2;0),E\u000bqis a maximum. This can be approximated by its\nvalue at q=(\u0019;0;0) due to the small value of JC. The low-energy region is\nenlarged and plotted in the insets.\nis determined by the Q2-dependences of E\u000bqandE\fqaround\nQ\u00180. See also the inset of Fig. 3 (a). At T=T0,Egbecomes\nzero, as shown in the inset of Fig. 3 (b), and both E\u000bqand\nE\fqbecome proportional to Qinstead of Q2around Q=0.\nNote that kBT=JBis shown by the broken line in Figs. 3 (a)\nand (b) as a measure of the temperature. In the pre-set lattice\nstructure, E\u000bqis a maximum at q=(\u0019;\u0019= 2;0). It can be ap-\nproximated by the value at q=(\u0019;0;0), since the small value\nofJC=0.05. We then find that at T=T0the bandwidth of\nE\u000bqbecomes comparable to kBT. This means that all states of\nE\u000bqcontribute to 1 /T1through nB(x) in the first term in Eq.\n(23), where the first term is dominant. This is the origin of\nthe rapid increase in 1 /T1AatT0. This is not accidental, due\nto the following points. The bandwidth of E\u000bqcan be very\nroughly estimated by \"1;q=(\u0019;0;0)\u0018(6JC+16JA)M0\u00188JA=0.8\nwith MA=MB\u0011M0\u00180.5. On the other hand, T0can be\nroughly estimated as T0\u0018zAJAXA=0.8. Since MAis soft and\ndecreases rapidly with T,T0is close to the Curie tempera-\nture of a system limited to the A-sublattice. Therefore, 1 /T1Arapidly increases around T0.\nSo far, we have discussed an N-type ferrimagnet.1, 3)An-\nother type of ferrimagnet–called P-type–shows a hump in the\ntemperature dependence of the magnetization instead of com-\npensation. Figure 4 is calculated from Eq. (1) for JA=0.5,\nJB=1.0,JC=0.2,SA=1/2, and SB=1. The lattice structure is\nthe same as the inset of Fig. 1. In a P-type ferrimagnet, the\nFig. 4. (Color online) The mean-field solution for a P-type ferrimagnet\nwith JA=0.5,JB=1.0,JC=0.2,SA=1/2, and SB=1.\nmagnetization does not show any singular behavior, such as\ncompensation, although it is composed of two di \u000berent sub-\nlattices, i.e., with soft and hard dispersion relations E\u000bqand\nE\fq. The value of 1 /T1\u0017in aP-type ferrimagnet is plotted in\nFig. 5 in the same way as for the N-type. We find that 1 /T1A\nFig. 5. (Color online) The relaxation rate 1 /T1in aP-type ferrimagnet. The\nT-dependences of 1 =T1on the A-sites and B-sites (1 /T1Aand 1 /T1B) are plot-\nted by upward triangles (red) and downward triangles (blue), respectively.\nThe mean-field solution for JA=0.5,JB=1.0,JC=0.2,SA=1/2, and SB=1 was\nused. The broken line is near the top of the hump structure.\nrapidly increases around T=Tc\u00180.4 close to the top of the\nhump structure, while 1 /T1Bincreases rapidly near Tc. It is\nnow straightforward to understand this behavior, since the A-\nsublattice is soft and the B-sublattice is hard. This is clearly\nshown by the dispersion relation in Figs. 6 for (a) T=Tc=0.1\n4J. Phys. Soc. Jpn. FULL PAPERS\nFig. 6. (Color online) The dispersion relations E\u000bqandE\fqare plotted by\nupward triangles (red) and downward triangles (blue), respectively, for (a)\nT=Tc=0.1 and (b) T=Tc=0.4, with qy=qz=0. The broken line indicates\nthe corresponding temperature kBT=JB.\nand (b) T=Tc=0.4. In each panel, kBT=JBis shown by the\nbroken line as measure of the temperature. At T=Tc=0.4, all\nofE\u000bqcontribute to 1 /T1A. Therefore, even in a P-type fer-\nrimagnet without compensation, we find a rapid increase of\n1/T1.\n4. Summary and Discussions\nWe have studied T1in a ferrimagnetic insulator that is in-\nduced by the Raman process involving the hyperfine interac-\ntion. To calculate 1 /T1, we adopted a Heisenberg model com-\nposed of two sublattices and used the linearized spin-wave ap-\nproximation around the mean-field solution. At T0<Tc, 1/T1\nincreases rapidly on one sublattice, whereas on the other site\nit does not increase up to Tc, as usual in a ferromagnet. This\nis due to the fact that an N-type ferrimagnet has two magnon\nexcitations. The soft magnon contributes to the increasing be-\nhavior of 1 /T1atT0, since the bandwidth of the soft magnon\nis less than T0in energy.\nAt low temperatures, the T-dependence of 1 /T1is well fit-\nted by T2, similar to the ferromagnetic case. With increas-\ning temperature, a T5-component is added, due to the mo-\nmentum dependence of the dispersion relation. In this paper,\nhowever, we have considered only the Raman process. When\nthe three-magnon process, magnon-magnon interactions, and\nother factors are involved, those T-dependences will be mod-\nified. Those are beyond the purpose of this paper and will be\nstudied elsewhere.\nThe increase in 1 /T1below Tcis found also in a P-type fer-\nrimagnet, which shows hump structure in the temperature de-pendence of the magnetization instead of compensation. Also\nin aP-type ferrimagnet, we find a rapid increase of 1 /T1be-\nlowTc, even though the magnetization does not show com-\npensation. This also can be explained by the fact that a P-type\nferrimagnet is composed of soft and hard magnons. Although\naP-type ferrimagnet does not show compensation, 1 /T1on\none sublattice still increases below Tc. We expect this to be\nexperimentally confirmed in the near future.\nSo far, we have not considered the orbital angular momen-\ntum~L. For example, in rare-earth (R) iron garnets, R 3Fe5O12\n(R=Ho, Er, Tb, etc.), the rare-earth magnetization is calcu-\nlated by using the total angular momentum ~J=~L+~S.38)The\nLand ´eg-factor on an R-site is di \u000berent from that on an iron\nsite, and TM,TAin general. Still, Eq. (1) is our starting point.\nThe expectation value of Sz\nAthen contains the extra factor ( \rA-\n1), so thathSz\nAi=(\rA-1)hJz\nAi,38)where Jz\n\u0017is the z-component of\n~Jon\u0017-site (\u0017=A, B). These factors can be renormalized into\nJA,JB, and JC:KA\u0011(\rA\u00001)2JA,KB\u0011(\rB\u00001)2JB, and\nKC\u0011(\rA\u00001)(\rB\u00001)JC.39, 40)Using KA,KB, and KC, the\nmagnon dispersion relations are obtained by substituting hJz\nAi\nandhJz\nBiforMAandMB, respectively. See also Appendix D.\nAround which temperature, TMorTA, does 1 /T1start to in-\ncrease? The magnon bandwidth is determined by the expec-\ntation value ofhSz\nAi=(\rA\u00001)hJz\nAiandhSz\nBi=(\rB\u00001)hJz\nBi\ninstead ofhJz\nAiandhJz\nBi. We recall that 1 /T1increases, when\nkBTis comparable to the bandwidth, and TAis determined by\nhJz\nAiandhJz\nBi. For example, in a case with \rA=5/4,\rB=2, such\nas for Ho 3Fe5O12, the factors ( \rA\u00001) and (\rB\u00001) are smaller\nthan 1. A rough estimate of the energy scale of the bandwidth\nis thus smaller than TA. This means that 1 /T1will start to in-\ncrease further blow TA. However, those energy scales are dif-\nferent depending on the materials involved. Thus, it is di \u000ecult\nto identify the temperature at which 1 /T1starts to increase.\nSuch a material dependence will be discussed in the near fu-\nture and will be clarified experimentally.\nOn the other hand, it is clear that Egbecomes zero at TA\ninstead of TM\nEg=12KCh\nhJz\nAi\u0000hJz\nBii\n: (27)\nMagnon excitations in RIGs have been reported from inelastic\nneutron scattering.41, 42)However, Eghas not yet been clar-\nified around the compensation temperature. The loss of Eg\nmust be associated with the increase of domain wall speed\natTA28–31)and the enhancement of NMR signals.26)Such re-\nmarkable changes of the domain walls will make ferrimagnets\nmore useful for spintronics. A consistent understanding of the\nNMR, ESR, and neutron-scattering results will be even more\nimportant and useful.\nThe author thanks S. Maekawa, H. Chudo, M. Imai, M.\nFujita, Y . Kawamoto, S. Kambe, Y . Tokunaga, and H. Sakai\nfor useful and helpful discussions. This work was supported\nby Grants-in-Aid for Scientific Research (Grant 18H04492\nand 20K03810) from JSPS and MEXT, and by the inter-\nuniversity cooperative research program of IMR Tohoku Uni-\nversity (20N0006). A part of the numerical calculation was\ndone with the supercomputer of JAEA.\n5J. Phys. Soc. Jpn. FULL PAPERS\nAppendix A: Mean-field Equation\nThe mean-field equation and its solution are straightfor-\nward. We define\nMA=fSA[(zAJAMA\u0000zJCMB)=(kBT)]; (A\u00011)\nMB=fSB[(zBJBMB\u0000zJCMA)=(kBT)]; (A\u00012)\nfS[x]\u0011(S+1=2) coth [x(S+1=2)]\u00001=2 coth (x=2):(A\u00013)\nNote that gand\u0016Bhave been omitted to make the equations\nclearer. Solving Eqs. (A \u00011) and (A\u00012) gives Tcin the form\nTc=1\n2\u0014\nXAzAJA+XBzBJB\n+q\n(XAzAJA\u0000XBzBJB)2+XAXB(2zJC)2\u0015\n; (A\u00014)\nXA\u0011(SA+1)SA=3; (A\u00015)\nXB\u0011(SB+1)SB=3; (A\u00016)\nFor the case shown in Fig. 1, T0can be approximated by the\nCurie temperature of the system limited to the A-sublattice, it\nis given by T0\u0018zAJAXA.\nAppendix B: Magnon Dispersion Relations in a Ferri-\nmagnet\nAt low energies away from the compensation temperature,\nthe magnon dispersion relations given by Eqs. (10) and (11)\ncan be expanded as\nE\u000bq\u0018CQ2; (B\u00011)\nE\fq\u001812JC(MA\u0000MB)+DQ2; (B\u00012)\nC=h\n3JAM2\nA\u0000(JA+JB\u00002JC)MAMB+3JBM2\nBi\nMA\u0000MB;\n(B\u00013)\nD=JAM2\nA\u0000(3JA+3JB+2JC)MAMB+JBM2\nB\nMA\u0000MB;\n(B\u00014)\n\"1q+\"2q=6JC(MA+MB)+4(JAMA+JBMB)Q2; (B\u00015)\n\u0001q=6JC(MA\u0000MB)\n+12JC(MB+MA) (JAMA+JBMB)+24J2\nCMAMB\n6JC(MA\u0000MB)Q2;\n(B\u00016)\nwhere Q\u0011q\nq2x+q2y+q2z. Note that (0,0,0) and ( \u0019;\u0019;\u0019 ) are\nequivalent for the case shown in Fig. 1. At the compensation\ntemperature MA=MB\u0011M, the excitation gap vanishes\nE\u000bq\u0018C2Q2+C1Q; (B\u00017)\nE\fq\u0018\u0000C2Q2+C1Q; (B\u00018)\nC1=2p\n3JC(JA+JB+JC)S; (B\u00019)\nC2=2(JA\u0000JB)S; (B\u000110)\n\"1q+\"2q=12JCS+4(JA+JB)S Q2; (B\u000111)\n\u0001q=4Sp\n3JC(JA+JB+JC)Q: (B\u000112)This does not depend on the lattice structure, as shown in Fig.\nB\u00011.\nFig. B \u00011.(Color online) (a) Schematic illustration of the compensation.\nTheA-sublattice has one site, whereas the B-sublattice two sites. (b) Magnon\ndispersion relation away from the compensation point for JA=JB=JC=1,\nSA=SB=1. (c) Magnon dispersion relation at the compensation point for\nJA=JB=JC=1,SA=1 and SB=1/2. In both (b) and (c), the parameters were\nchosen by hand to show the characteristics.\nAppendix C: Nuclear Magnetic Relaxation in a Ferro-\nmagnet and an Antiferromagnet\nThe ferromagnetic state is obtained by imposing the con-\nditions SB=0 and JC=JB=0 on the mean-field equation given\nby Eq. (1). Equation (23) then reduces to\n1\nT1=2\u0019FX\np;qnB(Ep)h\nnB(Ep)+1i\n\u0002\u000e\u0010\nEp\u0000Eq\u0011\n;(C\u00011)\nwith Eq=zAJAMA(1\u0000\u0010Aq). The temperature dependence of\n1=T1is shown in Fig. C\u00011. At low temperatures, it is well fitted\nbyCT2, because Eq/Q2.33, 34)On the other hand, it deviates\nfrom T2with increasing Tand is well fitted by AT2+BT5,\nsince a Q-linear component grows in Eq. Here, A,B, areCare\nconstants.\nThe antiferromagnetic state is given by SA=SBandJA=\nJB=0. In this case, E\u000bq=E\fq. Equation (23) then reduces to\n1\nT1;\u0017=2\u0019A\u0017X\np;qnB(E\u0017p)h\nnB(E\u0017p)+1i\"\"p\"q\n\u0001p\u0001q+1\n4#\n6J. Phys. Soc. Jpn. FULL PAPERS\nFig. C \u00011.(Color online) The T-dependence of 1 /T1in the ferromagnetic\nstate. The red dots were calculated numerically from Eq. (C \u00011) and the mean-\nfield solution. The thick and the broken lines are fitting results using AT2+\nBT5andCT2, respectively.\n\u0002\u000e\u0010\nE\u0017p\u0000E\u0017q\u0011\n; (C\u00012)\nwhere \u0001p=2q\n\"2p\u0000j\"3;pj2and\"p=\"1p=\"2p. Its temperature\ndependence is shown in Fig. C \u00012. At low temperatures, it is\nFig. C \u00012.(Color online) The T-dependence of 1 /T1in the antiferromag-\nnetic state. The red dots were calculated numerically from Eq. (C \u00012) and the\nmean-field solution. The thick and the broken lines are fitting results using\nAT3+BT5andCT3, respectively.\nwell fitted by CT3, because Eq/Qand (\"q=\u0001q)/Q\u00001.32, 34)\nOn the other hand, it deviates from T3with increasing Tand is\nwell fitted by AT3+BT5, since the curvature of Eqbecomes\nrelevant. Here, A,B, are Care constants.\nAppendix D: ESR Frequencies\nWhen we consider ~L,JA,JB, and JCare replaced by KA,KB,\nandKC, and further MAand\u0000MBare interpreted ashJz\nAiand\n\u0000hJz\nBiin Eqs. (10) and (11). In a magnetic field ~H=(0;0;H),\n\rAHand\u0000\rBHare added to Eqs. (4) and (5), respectively.\nThe magnon excitations at Q=0, which correspond to theESR frequencies \n\u000band\n\fare given by\n\n\u000b=1\n2\"\n(\"1\u0000\"2)+q\n(\"1+\"2)2\u00004j\"3j2#\n; (D\u00011)\n\n\f=1\n2\"\n\u0000(\"1\u0000\"2)+q\n(\"1+\"2)2\u00004j\"3j2#\n: (D\u00012)\n\"1=\u0015hJz\nBi+\rAH; (D\u00013)\n\"2=\u0015hJz\nAi\u0000\rBH; (D\u00014)\n\"3=\u0015q\nhJz\nAihJz\nBi; (D\u00015)\nwith\u0015=zKC. At low temperatures–below both TMandTA–\nand to first order in H, Eqs. 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Lett. 125, 027201\n(2020).\n8"    },    {        "title": "1805.02827v2.Spin_Hall_and_Anisotropic_Magnetoresistance_in_Ferrimagnetic_Co_Gd___Pt_layers.pdf",        "content": "Page 1    Spin-Hall and Anisotropic Magnetoresistance in Ferrimagnetic Co-Gd / Pt layers   W. Zhou,1† T. Seki,1,2,* T. Kubota,1,2 G. E. W. Bauer1-4, and K. Takanashi,1,2 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 3AIMR, Tohoku University, Sendai 980-8577, Japan 4Zernike Institute for Advanced Materials, University of Groningen, The Netherlands   * e-mail: go-sai@imr.tohoku.ac.jp †Present affiliation: National Institute for Materials Science, Tsukuba, Japan    Page 2 Abstract: We present the Co-Gd composition dependence of the spin-Hall magnetoresistance (SMR) and anisotropic magnetoresistance (AMR) for ferrimagnetic Co100-xGdx / Pt bilayers. With Gd concentration x, its magnetic moment increasingly competes with the Co moment in the net magnetization. We find a nearly compensated ferrimagnetic state at x = 24. The AMR changes sign from positive to negative with increasing x, vanishing near the magnetization compensation. On the other hand, the SMR does not vary significantly even where the AMR vanishes. These experimental results indicate that very different scattering mechanisms are responsible for AMR and SMR. We discuss a possible origin for the alloy composition dependence.   Page 3 1. Introduction  Antiferromagnetic spintronics [1-6] is an emerging research field that has attracted much attention because of the unique properties of antiferromagnets: zero net magnetization, small magnetic susceptibility [7], and magnetization dynamics characteristically different from ferromagnets [8-11]. Antiferromagnets have great potential for the development of novel spintronic devices such as crosstalk-free and ultrahigh-density non-volatile memories because they do not generate and are robust against magnetic stray fields [12]. However, several problems have to be solved before exploiting the aforementioned functionalities in practical devices. A major issue is the efficient control of the antiferromagnetic order. The small magnetic susceptibility of antiferromagnets renders magnetic field control difficult. Current-induced spin transfer phenomena may be a possible solution for this problem [13,14]. Recent studies demonstrate that spin current (Js) can be generated by an antiferromagnet [4,15] and also interacts with its magnetic moments [2,3,5,11,14,16-19]. However, the coupling phenomenon between antiferromagnetic order and spin currents has not yet been fully understood. It is a complex problem involving the spin-dependent scattering in the bulk and at interfaces to electric contacts. Here we focus on the latter by revealing details of the spin mixing at the interface between platinum and a ferrimagnet around the compensation point.  Co-Gd amorphous alloys are ferrimagnets, in which the Co and Gd moments (mCo and mGd) are coupled antiferromagnetically [20]. The net magnetic moment of Co-Gd (mCo-Gd) is given by | mCo - mGd |, Page 4 meaning that dominance of one of them in the mCo-Gd magnetization strongly depends on the alloy composition as shown in Figs. 1(a) and 1(b). The ferrimagnetic state with zero magnetization at the Co-Gd compensation point resembles an antiferromagnet. By exploiting ferrimagnetic materials such as Co-Gd and Co-Fe-Gd, several studies recently reported an interaction between Js and magnetizations near the compensation points. Ham et al. [21] found an enhanced damping-like component of the spin-orbit torque (SOT) near the compensated perpendicularly magnetized Gd25Fe65.6Co9.4 / Pt. This was explained by the reduction of the net magnetic moment. Although Mishra et al. [22] also observed a substantial increase of the SOT effective field and switching efficiency in perpendicularly magnetized Co-Gd / Pt, they conclude that the negative exchange interaction in the ferrimagnet enhances the SOT near compensation. Co-Gd alloys can also display angular momentum compensation, which is beneficial for ultra-fast magnetization dynamics: Kim et al. [23] demonstrated magnetic field-driven fast domain wall motion in a ferrimagnetic Co-Fe-Gd wire at the angular momentum compensation temperature. Even though Co-Gd is an important material class, both from fundamental and application point of view, the detailed mechanism of spin-dependent transport in this material is not well understood. Here we present a systematic study of magnetotransport of Co-Gd alloy / Pt thin films that accesses spin-dependent scattering parameters and sheds light on the interaction between Js and the ferrimagnetic order. Our analysis separates the contributions from the spin-Hall magnetoresistance (SMR) and Page 5 the anisotropic magnetoresistance (AMR) that occur simultaneously in all-metal magnetic bilayers, which should help to establish microscopic models for both effects.  We report the different composition dependences of SMR and AMR for the Co-Gd / Pt bilayers with in-plane magnetization. The sign of the AMR monotonically changes from positive to negative by increasing the Gd concentration and vanishes near the magnetization compensation composition. On the other hand, the SMR remains finite even when the AMR vanishes, which is a direct proof for different physics. We interpret the composition dependence of the SMR in terms of a spin mixing conductance that, in contrast to the conventional wisdom, depends on the magnet.   2. Experimental Procedure  Thin films were deposited on a thermally oxidized Si substrate using an ultrahigh vacuum compatible magnetron sputtering system with the base pressure below 2 × 10-7 Pa. First, a 4 nm-thick Cr buffer was deposited on the Si-O substrate. Then Co and Gd were co-deposited to form the Co100-xGdx layers with a thickness of 30 nm. Finally, a 4 nm-thick Pt layer was deposited. All the layers were deposited at room temperature. By tuning the sputtering powers of Co and Gd targets, the Gd concentration x (at. %) was widely varied from x = 0 to x = 45. Except for x = 0, i.e. pure cobalt, the Co-Gd layers were amorphous alloys, as confirmed by reflection high energy electron diffraction (RHEED) in Fig. 1(c). In contrast to the amorphous Page 6 phase of Co-Gd, the RHEED pattern of Fig. 1(d) indicates that the top Pt layer crystallizes on the amorphous Co-Gd. The top Pt layer serves as not only the capping layer to prevent the Co-Gd from oxidation, but also as converter of a charge current (Jc) to a transverse spin current Js by the spin-Hall effect (SHE) [25]. We also prepared reference samples consisting of Al / Co100-xGdx / Al, for which we anticipated negligible SMR because of the small spin-orbit coupling in Al. Magnetic properties were measured by a superconducting quantum interference magnetometer, a vibrating sample magnetometer and a longitudinal magneto-optical Kerr effect (L-MOKE) set-up with laser wavelength of 680 nm, all at room temperature.  The thin films were patterned into a 10 µm-wide Hall-cross by photolithography and Ar ion milling. In order to separate the contributions of AMR and SMR as depicted in Fig. 2, we measured the magnetoresistance as a function of direction of an applied magnetic field (H) in two configurations. In the g scan, H rotates in the x - z plane (Fig. 2(a)), anticipating an AMR since Jc flows along the x direction. The SMR is accessed by the b scan in which H rotates in the y - z plane (Fig. 2(b)) [26]. In these measurements we apply a large magnetic field |H|=70 kOe such that the magnetization and field are collinear to a good approximation. All measurements were carried out at room temperature.   3. Experimental Results and Discussion A. Composition Dependence of the Magnetic Properties Page 7  Figs. 3(a) - 3(d) show the magnetization versus magnetic field (M - H curves) of Cr / Co100-xGdx / Pt with (a) x = 12, (b) 24, (c) 25 and (d) 37 and Figs. 3(e) - 3(h) the corresponding L-MOKE loops. H was applied in the film plane. When x is increased from 12 to 37, the net magnetization is suppressed at x = 24 and 25 and accompanied by an increased coercivity (Hc). M is observed to increase again for x = 37. In contrast to the M - H curves, the L-MOKE loops show a gradual decrease in the magnitude of Kerr rotation angle (qK) with x. A remarkable feature of the L-MOKE loops is the sign reversal of qK  between x = 24 and 25.   The solid circles in Fig. 4 summarize (a) M, (b) the maximum qK and (c) its absolute value as a function of x for Cr / Co100-xGdx / Pt. As x is increased, M shows a local minimum at x = 24 and qK  reverses its sign at x = 25. |qK| slowly and monotonically decreases with x. We conclude that the room-temperature magnetization compensation point of Co-Gd lies between x = 24 and 25. This composition is close to the reported magnetization compensation point at room temperature [20]. We attribute the sign reversal of qK to the change of the dominant magnetic component from mCo to mGd. The present L-MOKE system is equipped with a 680-nm-wavelength semiconductor laser, which selectively probes mCo. At the Co-rich composition (x = 12), mCo dominates and is parallel to H, resulting in a positive qK. On the other hand, when mGd dominates, mCo is antiparallel to H and qK is negative. The results for the Al / Co100-xGdx / Al reference samples, denoted by open squares in Fig. 4, exhibited magnetic properties very similar to Cr / Co100-xGdx / Pt.  Page 8 B. Field Angular Dependence of SMR and AMR  Here we present results of the high field (70 kOe) SMR and AMR measurements for Cr / Co100-xGdx / Pt. The g scans of the longitudinal resistance (Rxx) are displayed in Figs. 5(a) - 5(d) ((a) x = 12, (b) 24, (c) 25 and (d) 45), and the corresponding b scans in Figs. 5(e) - 5(h). Jc was set at 0.1 mA or current density of 2.6 × 104 A cm-2. As shown in Fig. 2, the magnetization of the g (b) scan lies in the x - z (y – z) plane, which corresponds to the AMR (SMR), respectively. The AMR ratio AMR=Rxxγ=90º−Rxxγ=0º()Rxxγ=0º{}×100=ρ//−ρ⊥()ρ⊥{}×100 in terms of the longitudinal (r//) and transverse (r⊥) resistivities. The definition of SMR=Rxxβ=90º−Rxxβ=0º()Rxxβ=0º{}×100=Δρρ0{}×100, where r0 is the resistivity at the SMR maximum and Dr the resistivity modulation. This definition agrees with previous ones for metallic bilayers [27], but differs from that used for magnetic insulators / Pt [26]. Therefore, a negative SMR here corresponds to the “normal” situation in Ref. [28]. The longitudinal electric fields along Jc,x due to AMR (ExxAMR) and SMR (ExxSMR) are well described by [26, 27]: ExxAMR=ρ⊥+ρ//−ρ⊥()sin2γ{}Jx,   (1) and  ExxSMR=ρ0+Δρsin2β{}Jx.    (2) The solid lines in Fig. 5 denote fits by Eqs. (1) and (2), which is a strong evidence that the angular dependences indeed are caused by AMR and SMR.  Page 9  AMR is the dependence of the resistance on the angle between current and magnetization and defined to be positive when r// > r⊥. The SMR is generated by the spin-orbit interaction in the normal metal layer and the exchange interaction at the interface. Here, we find for Gd-Co alloys an AMR > 0 for x = 12, while AMR < 0 for x = 45, and AMR = 0 at x = 25. In contrast, the SMR ratio is negative regardless of x even at the composition for which the AMR vanishes. As mentioned above, a negative sign of SMR in the convention of Ref. [27] implies a net magnetization of Co-Gd is parallel to the external magnetic field.  Figure 6 summarizes (a) the composition dependence of the longitudinal resistance R (on this scale the dependence on magnetization direction is negligibly small), (b) AMR ratio and (c) SMR ratio for Cr / Co100-xGdx / Pt. The experimental values of R, AMR ratio and SMR ratio are also summarized in Table 1. Compared with pure Co, the alloy scattering of amorphous Co-Gd strongly increases the resistivity. Figure 6(b) clearly demonstrates the sign change of the AMR from positive to negative as x increases (see the inset of Fig. 6(b)). Pure Co shows a positive AMR [29] while a negative AMR has been reported for a Gd single crystal [30]. Our results suggest that the d-electrons of Co and the f-electrons of Gd contribute oppositely to the AMR phenomenon. Hence, the effect of local s-d (s-f) scattering appears to cancel (to the experimental accuracy) exactly at the compensation point. However, the microscopic mechanism of the AMR is much more complicated, being governed by the full electronic structure, see e.g. Ref. [31]. Nevertheless, the vanishing of the AMR at the compensation point is most likely not a coincidence and our results should help to develop better theoretical Page 10 models for spin and charge transport in magnetic metals. While the AMR changes sign at x = 25, the SMR is negative regardless of x, which implies that the scattering mechanisms of AMR and SMR are very different.   In order to shed light on this matter we carried out g and b scan magnetoresistance measurement for the Al / Co100-xGdx / Al reference samples as shown in Fig. 7. The Co-rich Co-Gd with x = 7 (Fig. 7a) exhibits a positive AMR whereas in the Gd-rich Co-Gd with x = 42 (Fig. 7b) AMR < 0. This sign change is consistent with the results for Cr / Co100-xGdx / Pt, and proves that the AMR is qualitatively not affected by the normal metal and the interfaces. The absence of a clear SMR for the reference samples in Figs. 7c and 7d can be attributed to the small spin-orbit coupling and negligibly small spin Hall effect in Al as anticipated. We therefore may conclude that (i) the sign change in the g scan with increasing x originates from the bulk scattering in the Co-Gd layer, and (ii) the SMR for the Cr / Co100-xGdx / Pt is dominantly caused by the direct and inverse spin Hall effects without a significant bulk contribution. A contribution from the transverse AMR [32] to the b scans of Rxx can be excluded because the Al / Co100-xGdx / Al sample resistance does not change in the b scans.  C. Discussion  We now discuss the composition dependence of SMR for the Cr / Co100-xGdx / Pt. In metallic bilayers of a nonmagnet (N) with large spin-orbit coupling and a ferromagnet (F) [27]  Page 11  Δρρ0~−θSH()2λNtNtanh2tN2λN()1+ξgR1+gRcothtNλN()−gF1+gFcothtNλN()⎧⎨⎪⎩⎪⎫⎬⎪⎭⎪,  (3) in terms of  ξ≡ρNtFρFtN,   (4)  gR≡2ρNλNReGMIX[],  (5)  gF≡1−P2()ρNλNρFλFcothtFλF(),  (6) where rN(F) ,lN(F) ,tN(F) are the resistivity, spin diffusion length, thickness of the N (F) layer, respectively, qSH is the spin-Hall angle of the N layer, P is the current spin polarization of the F layer, and GMIX is the spin mixing conductance of the interface. The first term in the curly bracket of Eq. (3) coincides with the expression for the SMR for a ferromagnetic insulator. The second term takes the absorption of the longitudinal spin current by the ferromagnetic metal into account (and is not to be confused with the unidirectional SMR [33]).   Calculating the SMR ratio Eq. (3), i.e. Δρρ0()×100, requires values of many parameters. The Gd concentration dependence of rF is displayed in Fig. 8a. rN for Pt was experimentally measured using the 4 nm-thick Pt single layer film, and was obtained to be 4 × 10-7 W m. We cannot simply measure lF in our Co-Gd, but it is a few nm at most and we assume lF ≈ 2 nm in the following. P = 0.3 has been derived from the tunneling spin polarization of Co-Gd [34]. GMIX is a measure of the transverse spin current absorption that we initially choose to not depend on x, e.g. GMIX = 1 × 1015 W-1 m-2. qSH ≈ 0.08 and lN ≈ 3 nm reported for the Pt [Ref. 35] are chosen. Figure 8b compares the observed absolute values of the experimental and model SMR Page 12 ratios for each alloy composition, where the small scatter of calculated SMR reflects that in the measured resistivities. Those agree quite well, but in contrast to the experimental observation, the absolute value of the calculated SMR ratios increases with increasing x. This calculated tendency as a function of composition does not agree with the experimental trend, suggesting an alloy concentration-dependence of material parameters that we assumed constant, such as lF and GMIX. P affects the SMR because a spin current can penetrate the metallic ferromagnets when polarized parallel the magnetization. However, because the alloy resistance is relatively high, the calculated SMR does not change much for P ≤ 0.4.  We can turn the table and calculate the parameter dependence on Gd concentration. The dependence would reproduce the experiments. Here, we focus on GMIX. The Gd concentration dependence of GMIX that results from inverting Eq. (3) is shown in Fig. 8c. GMIX is seen to decrease strongly with increasing x, implying that the ratio of mCo versus mGb at the interface to Pt plays an important role in the spin mixing. This is surprising, since theory predicts that spin mixing is mainly governed by the electron density [36] or the dynamical spin susceptibility [37] of the normal metal. However, these theories do not take the spin-orbit interaction into account, which might importantly modify the spin-mixing conductance of the CoGd / interface and cause its suppression with increasing Gd concentration. M. A. Schoen et al. [38] report a compositional dependence of GMIX for the NixCo1-x, NixFe1-x, and CoxFe1-x, even for such 3d transition-metal binary alloys. M. Tokaç et al. [39] report that GMIX depends on the crystal structure even for elemental Co. Page 13  D. In-plane Field Angular Dependence  The dependence of Rxx on the magnetic field angle a (left panel in Fig. 9) is plotted in Fig. 9 for x = 25 at which the AMR vanishes. The a scan should therefore give identical results with the b scan, as confirmed by comparison with Fig. 5(g). In a Co-Gd alloy with x = 25 the AMR and associated planar Hall effect vanish, which could be a technical advantage. For example, in attempts to measure the spin Hall angle by the spin pumping technique, spurious contribution from the planar Hall effect must be subtracted [40]. This technical difficulty can be overcome by choosing Co-Gd (or in fact any other ferrimagnet) at its compensation composition or temperature to accurately measure the spin-Hall effect, even in the case of metallic bilayer system.    4. Summary  We systematically investigated the Co-Gd composition dependence of the SMR and AMR of in-plane magnetized Co100-xGdx / Pt layers. As x increases, the dominant magnetization changes from the mCo to the mGd sublattices. We realized a nearly compensated ferrimagnetic structure at x = 24 (at room temperature). We find distinctly different composition dependences of the SMR and AMR. The AMR decreases monotonically with increasing x and changes sign near the compensation composition while SMR remain constant when the Page 14 AMR sign chances. The composition dependence of the AMR suggests a local picture of the AMR in which the magnetic moments of the Co and Gd contribute with opposite sign with canceling contributions at the compensation. The observed composition dependence of the SMR can be explained by an exchange interaction of the conduction electrons in Pt that is dominated by the Co magnetic moments or the spin-orbit interaction at the interface. Our findings contribute to a better understanding of an important material class.  Acknowledgement The authors thank J. Barker and S. Takahashi for valuable discussions. Y. M u r a k a m i a n d  I . N a r ita  provided technical support during the structural characterization. This work was supported by the Grant-in-Aid for Scientific Research B (16H04487) and Innovative Area “Nano Spin Conversion Science”(26103006）as well as the Research Grant from the TEPCO Memorial Foundation. The device fabrication was partly carried out at the Cooperative Research and Development Center for Advanced Materials, IMR, Tohoku University.  References [1] T. Jungwirth, X. Marti, P. Wadley and J. Wunderlich, Nature Nano. 11, 231 (2016). [2] Z. Wei, A. Sharma, A. S. Nunez, P. M. Haney, R. A. Duine, J. Bass, A. H. MacDonald, and M. Tsoi, Phys. Rev. Lett., 98, 116603 (2007). Page 15 [3] P. M. Haney and A. H. MacDonald, Phys. Rev. Lett., 100, 196801 (2008). [4] W. Zhang, M. B. Jungfleisch, W. Jiang, J. E. Pearson, A. Hoffmann, F. Freimuth, and Y. Mokrousov Phys. Rev. Lett., 113, 196602 (2014). [5] T. Moriyama, S. Takei, M. Nagata, Y. Yoshimura, N. Matsuzaki, T. Terashima, Y. Tserkovnyak, and T. Ono, Appl. Phys. Lett., 106, 162406 (2015). [6] T. Moriyama, N. Matsuzaki, K.-J. 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Page 19  Table 1 Experimental values of R, AMR ratio and SMR ratio for Cr / Co100-xGdx / Pt. x (at. %) R (W) AMR ratio (%) SMR ratio (%) 0 24.1 ± 0.1 1.5 ± 0.002 -0.200 ± 0.001 12 202.5 ± 0.1 0.050 ± 0.001 -0.045 ± 0.003 19 257.8 ± 0.7 0.022 ± 0.002 -0.043 ± 0.001 24 257.9 ± 0.1 0.012 ± 0.002 -0.035 ± 0.001 25 272.3 ± 0.6 0 ± 0.002 -0.036 ± 0.001 30 280.7 ± 3.3 0 ± 0.002 -0.020 ± 0.001 34 272.8 ± 0.3 -0.003 ± 0.002 -0.030 ± 0.002 37 274.6 ± 0.1 -0.007 ± 0.001 -0.025 ± 0.002 39 295.3 ± 0.1 -0.011 ± 0.003 -0.020 ± 0.001 45 301.0 ± 0.1 -0.012 ± 0.001 -0.019 ± 0.004       Page 20 Figure Captions Figure 1 (a,b) Relationship between net magnetization, Co magnetic moment, and Gd magnetic moment for (a) Co-rich Co-Gd and (b) Gd-rich Co-Gd. (c,d) Reflection high-energy electron diffraction patterns of the (c) Co-Gd surface (x = 12) and (d) Pt surface.  Figure 2 Measurement configurations for the angular dependence of (a) anisotropic magnetoresistance (AMR) and (b) spin-Hall magnetoresistance (SMR) of the Co100-xGdx / Pt bilayer. A charge current (Jc) was applied along the x direction. The AMR is observed when the external magnetic field (H) is rotated in the x-z plane by the angle of g. The SMR is the resistance change when H is rotated in the y-z plane by an angle b.  Figure 3 Magnetization curves for the Co100-xGdx / Pt bilayer films with (a) x = 12, (b) 24, (c) 25 and (d) 37, at room temperature and magnetic field H in the film plane. The longitudinal magneto-optical Kerr effect (L-MOKE) loops for (e) x = 12, (f) 24, (g) 25 and (h) 37.  Figure 4 (a) Gd concentration (x) dependence of magnetization (M), (b) the maximum Kerr rotation angle (qK), and (c) the absolute value of qK (|qK|). The solid circles represent the data for Cr / Co100-xGdx / Pt while the open squares are those of the reference samples Al / Co100-xGdx / Al.  Page 21  Figure 5 Angular (g) dependence of the AMR of the Cr / Co100-xGdx / Pt with (a) x = 12, (b) 24, (c) 25 and (d) 45, and angular (b) dependence of SMR for the Cr / Co100-xGdx / Pt with (e) x = 12, (f) 24, (g) 25 and (h) 45. Jc was set at 0.1 mA, and H = 70 kOe was applied. The solid curves are fits by Eqs. (1) and (2).    Figure 6 (a) Gd concentration (x) dependence of device resistance (R), (b) AMR and (c) SMR for the Cr / Co100-xGdx / Pt. The inset of (b) is a magnified plot of the AMR versus x.  Figure 7 Angular (g) dependence of AMR of the reference samples Al / Co100-xGdx / Al with (a) x = 7 and (b) 42, and angular (b) dependence of SMR for the Al / Co100-xGdx / Al with (c) x = 7 and (d) 42. Jc was set to 0.1 mA, and H = 70 kOe was applied. The solid curves are fits by Eq. (1).  Figure 8 (a) Gd concentration (x) dependence of resistivity (rF) of the present Co-Gd. (b) Comparison of the absolute values of the SMR ratios between the experiment (solid circles) and the model (solid line). (c) GMIX calculated from the SMR ratio as a function of x for the Cr / Co100-xGdx / Pt by inverting Eq. (3).  Figure 9 Illustration of the in-plane field angular (a) dependence of magnetoresistance, and a scan for the Cr / Page 22 Co100-xGdx / Pt with x = 25. Jc was set at 0.1 mA, and H = 70 kOe was applied. The solid curve is the fit by Eq. (2) with an offset of 90 degree for a.    Page 23    \n   Figure 1    \nPage 24    \n  Figure 2    \nPage 25    \n  Figure 3    \nPage 26   \n   Figure 4    \nPage 27   \n  Figure 5     \nPage 28    \n  Figure 6    \nPage 29     \n  Figure 7    \nPage 30   \n  Figure 8    \nPage 31      \n  Figure 9  \n"    },    {        "title": "2303.15191v4.Strain_effects_on_magnetic_compensation_and_spin_reorientation_transition_of_Co_Gd_synthetic_ferrimagnets.pdf",        "content": "Strain effects on magnetic compensation and spin reorientation transition of Co/Gd\nsynthetic ferrimagnets\nGiovanni Masciocchi,1, 2, a)Thomas J. Kools,3Pingzhi Li,3Adrien A. D. Petrillo,3Bert Koopmans,3Reinoud\nLavrijsen,3Andreas Kehlberger,2and Mathias Kläui1\n1)Institute of Physics, Johannes Gutenberg University Mainz, Staudingerweg 7, 55128 Mainz,\nGermany\n2)Sensitec GmbH, Walter-Hallstein-Straße 24, 55130 Mainz, Germany\n3)Department of Applied Physics, Eindhoven University of Technology, Eindhoven, 5612 AZ,\nNetherlands\n(Dated: 3 April 2023)\nSynthetic ferrimagnets are an attractive materials class for spintronics as they provide access to all-optical switching\nof magnetization and, at the same time, allow for ultrafast domain wall motion at angular momentum compensation.\nIn this work, we systematically study the effects of strain on the perpendicular magnetic anisotropy and magnetization\ncompensation of Co/Gd and Co/Gd/Co/Gd synthetic ferrimagnets. Firstly, the spin reorientation transition of a bilayer\nsystem is investigated in wedge type samples, where we report an increase in the perpendicular magnetic anisotropy\nin the presence of in-plane strain. Using a model for magnetostatics and spin reorientation transition in this type of\nsystem, we confirm that the observed changes in anisotropy field are mainly due to the Co magnetoelastic anisotropy.\nSecondly, the magnetization compensation of a quadlayer is studied. We find that magnetization compensation of this\nsynthetic ferrimagnetic system is not altered by external strain. This confirms the resilience of this material system\nagainst strain that may be induced during the integration process, making Co/Gd ferrimagnets suitable candidates for\nspintronics applications.\nI. INTRODUCTION\nRecent advances in spintronics have opened new possi-\nbilities for electronic applications beyond the CMOS stan-\ndard. New concepts of high density and ultrafast non-volatile\ndata storage have been proposed in magnetic memories1,2.\nThroughout the years, magnetic memories have evolved3,4\nexploiting different geometries5and new material platforms\nsuch as ferrimagnets6have been used to improve storage\ndensity7, reading and writing speed8and energy efficiency9,10.\nAt the same time, single-pulse optical-switching (AOS) of\nmagnetization has reduced the switching speed of the mag-\nnetization below ps timescale11–14. This bears promise for a\nnew generation of ultrafast data buffering, in a single chip that\nintegrates photonics with spintronics15–19.\nFerrimagnets are a class of magnets with unbalanced\nantiparallel-aligned sublattice moments. The compensation\nof the two inequivalent sublattices, combines the advantages\nof both antiferromagnets (antiparallel alignment of magnetic\nmoments) and ferromagnets (finite Zeeman coupling and spin\npolarization)16,20. Moreover, the drastic contrast between the\ntwo sublattices in non-adiabatic dynamics, could potentially\naccommodate AOS by a femtosecond laser pulse12,16. Single-\npulse AOS is typically observed in rare earth–transition metal\n(RE–TM) ferrimagnetic alloys like GdFeCo20or in multilayer\nsynthetic ferrimagnet, such as Co/Gd and [Co/Tb] n21,22. In\nparticular, the one based on multilayer of Co/Gd is a good can-\ndidate for integrated opto-spintronics devices as it shows AOS\n- without the constrains on the composition as imposed by al-\nloy system23,24- and at the same time exhibits magnetic and\nangular momentum compensation, allowing ultrafast domain\na)Electronic mail: gmascioc@uni-mainz.dewall motion25,26. For instance, the integration of Co/Gd syn-\nthetic ferrimagnets in an optically switchable magnetic tunnel\njunction has been recently reported27.\nWhen it comes to technological implementation, strain\ninduced effects must be considered, which could be in-\ncurred from processing steps such as packaging and layer\ndeposition28. Intrinsic stresses and strain could affect the\nmagnetic anisotropy via changes to the spin-orbit coupling\n(SOC)29or to the magnetization compensation of ferrimag-\nnets especially in RE–TM alloys30,31. However, in spite of\nbeing omnipresent in applications32–34, the effect of strain has\nnot yet been explored in these materials. In this work, we\npresent a systematic study of the effects of strain on Co/Gd\nsynthetic ferrimagnets. By the application of external strain,\nusing substrate bending, we investigate the impact of strain on\nthe perpendicular magnetic anisotropy (PMA) and the mag-\nnetization compensation of [Co/Gd] and [Co/Gd] 2multilay-\ners, respectively. Using wedge samples in a bilayer system of\nCo/Gd and polar magneto-optic Kerr effect (pMOKE) mea-\nsurements, we confirm that the PMA is increased by in-plane\ntensile strain and a negative magnetostriction is reported. By\nincluding the contribution of the strain-anisotropy for this sys-\ntem in a model for the magnetostatics, we show that the ef-\nfects of strain on the magnetization are mainly due to the\nmodification of the spin-orbit coupling within the magnetic\nlayer and at the the Pt/Co interface that increases the mag-\nnetic anisotropy via magnetoelastic coupling. Additionally,\nwe find that the magnetization compensation point is not af-\nfected significantly by strain, as the magnetoelastic coupling\naffects the anisotropy rather than the magnetization of the two\nsublattices. Our study explores the mechanisms that underlie\nthe influence of strain on the magnetic anisotropy of Co/Gd\nferrimagnets and contributes to a better understanding of the\nmagnetoelastic effects of ferrimagnetic multilayers. These re-\nsults could be employed for the optimization and developmentarXiv:2303.15191v4  [cond-mat.mtrl-sci]  31 Mar 20232\nof spintronics devices, as well as for potential applications in\nfields such as magnetic memory and sensing.\nII. METHODS AND SAMPLE FABRICATION\nThe samples were grown on a 1.5 µm thick, thermally oxi-\ndized SiOx on top of a 625 µm thick Si substrate by DC mag-\nnetron sputtering in a chamber with a typical base pressure\nof 5×10−9mBar. To obtain a variable thickness (wedge)\nalong the sample surface, a shutter in the close proximity of\nthe sample is gradually closed during deposition. This al-\nlows to study the compensation and spin reorientation tran-\nsition (SRT) within a single sample. Two types of sam-\nples are realized. Firstly, a bilayer of Ta(4 nm)/Pt(4)/ Co(0-\n2)/Gd(t Gd)/TaN(4) with a constant Gd layer on top of a Co\nwedge is considered to study the SRT. In addition, a quadlayer\nof Ta(4)/Pt(4)/Co(0.6)/Gd(0-2)/Co(0.6)/Gd(1.5)/TaN(4), this\ntime with a Gd wedge, is grown to study the magnetization\ncompensation.\nThe magnetic properties of these wedge samples were in-\nvestigated by pMOKE, where we are only sensitive to the\nout-of-plane (OOP) component of the Co magnetization at a\nwavelength of 658 nm. According to Fig. 1 (a), the surface\nof the sample is scanned along the y-direction using a focused\nlaser spot with a spot-size of /similarequal250µm diameter. Accord-\ningly, the local magnetic properties and hysteresis loops can\nbe measured as a function of layer thickness, with a negligi-\nble thickness gradient <0.025 nm within the used laser spot.\nAll the measurements are performed at room temperature. To\napply in-plane tensile strain to our multilayer, the substrate\nis mechanically bent using a three-point method35. A square\nsample of 1 by 1 cm is vertically constrained on two sides\nand pushed uniformly from below by a cylinder that has off-\ncentered rotation axis. The device generates a tensile strain in\nthe plane of the sample when the cylinder is rotated. As pre-\nviously reported, the tensile strain is uniaxial along xand uni-\nform in the measured area of the sample. The in-plane strain\nmagnitude is 0 .1% and has been measured with a strain gauge\n(RS PRO). More details about the strain generating device can\nbe found in section S2 of the supplementary information.\nIII. RESULTS AND DISCUSSION\nA. Spin reorientation transition in Co/Gd bilayer\nThe use of magnetic materials for high density data storage\nrequires magnetic systems that are OOP magnetized36,37. In\nthin films, an OOP magnetic easy axis can be obtained by\nmagnetocrystalline anisotropy induced at the interface with\nheavy metal38,39. In addition to that, strain has been shown\nto affect the magnetic easy axis direction in systems with\nPMA40. To understand the effect of external strain on Co/Gd\nsystems with PMA, we investigate bilayer samples consist-\ning of Ta(4 nm)/Pt(4)/ Co(0-2)/Gd( tGd)/TaN(4). Specifically,\nthe Co thickness is varied between 0 and 2 nm over a few\nmm along the ydirection, whereas tGdis constant (as in Fig.1 (a)). In this system, the balance between the interfacial\nanisotropy energy (magnetocrystalline anisotropy energy at\nthe Pt/Co interface) and demagnetization energy determines\nthe effective magnetic anisotropy. In such system, the demag-\nnetization energy increases with the thickness of the Co mag-\nnetic layer, and consequently, the magnetization will go from\nout-of-plane (OOP) to in-plane (IP). To probe the magnetiza-\ntion of our wedge sample, we record hysteresis loops from\nthe pMOKE signal. We repeat the measurement moving the\nlaser spot along the wedge in the y direction. Firstly, a sample\nwhere t Gd=0 is considered. This measurement can be seen in\nFigs. 1 (b) and (c). Fig. 1 (b) reports the magnetic response of\nthe Ta(4 nm)/Pt(4)/Co(0-2)/TaN(4) sample to an OOP mag-\nnetic film for different t Co. The effective anisotropy Ke f fwas\nestimated38recording hysteresis loops with magnetic field ap-\nplied OOP and IP and the corresponding anisotropy energy\nper unit area is Ks=1.7 mJ/m2. For tCo=1.35 nm the square-\nshaped loop indicates PMA with Ke f f= 1.5(2)×105J/m3. A\nvalue of MCo=1.3 MA/m was used in the calculation. As the\nthickness of Co is increased (moving the laser spot along the\nwedge direction - y) the remanence and squareness of the hys-\nteresis loop decreases together with the PMA of the system.\nFortCo=2.00 nm, the sample is IP magnetized and Ke f f=-\n0.8(2)×105J/m3is negative. The OOP to IP transition occurs\nattCo=1.85(2)nm in this system.\nTo investigate the effects of externally applied in-plane\nstrain, we repeat the measurement while the sample is me-\nchanically bent. The magnetization is coupled to the ex-\nternal strain and can be described by the expression for the\nanisotropy energy35:\nKME=−3\n2λsYε, (1)\nwhere λsis the saturation magnetostriction, Yis the\nYoung’s modulus and εis the strain. If the strain in the film\nis non-zero, the magneto-elastic coupling of Co contributes\nin principle to the effective anisotropy. Accordingly, the total\nanisotropy Ke f fof the magnetic stack is expected to change in\nthe presence of external strain. Fig. 1 (c) shows the OOP hys-\nteresis loops of Ta(4 nm)/Pt(4)/Co(1.85)/TaN(4) sample be-\nfore (blue) and after (red) the application of εxx=0.1%. We\nobserve that the anisotropy field is decreased after the appli-\ncation of in-plane strain. This happens because, in this sys-\ntem, the strain-induced magnetoelastic anisotropy KME=0.02\nmJ/m2is positive, as we expect from a material with negative\nmagnetostriction like Co40,41. More details about the calcu-\nlations of magnetoelastic anisotropy can be found in section\nS2 of the supplementary information. Accordingly, the PMA\nis increased by the applied strain, i.e. the system is expected\nto be OOP magnetized for thicker Co if compared to samples\nwithout strain.\nAfter this preliminary study on Pt/Co systems, we focused\nour attention on the magnetostriction of Co/Gd multilayers.\nIn Co-Gd alloys the magnetostriction has been reported to be\nstrongly dependent on the composition29,42due to the struc-\ntural modification occurring with different atomic content. In\ncontrast to this case, the effects of magnetostriction of a mul-3\nFIG. 1: (a) Sample sketch, red arrow indicates the direction of the applied strain. (b) Out of plane hysteresis loops of a Pt/Co/TaN stack for different Co\nthicknesses. (c) OOP hysteresis loops of Pt/Co(1.85 nm)/TaN before (blue) and after (red) application of 0 .1% in-plane strain. (d) MOKE intensity scan at\nremanence (no applied field) of Pt/Co/Gd/TaN films along the Co wedge.\ntilayer, are expected to be dependent on the magnetoelastic\ncoupling of the individual layers43.\nTo study the magnetostriction of a Co/Gd multilayer, a con-\nstant layer of Gd on top of the Co wedge is added. To perform\nthickness dependent studies, a thickness tGd=1 nm and 3 nm\nis considered. In the bilayer system, the magnetization in the\nGd layers is mainly induced at the interface with the Co layer,\nand couples anti-parallel the Co magnetization21. Accord-\ningly, tCorequired to reach SRT is expected to change with in-\ncreasing tGd44. To compare the SRT of Ta(4 nm)/Pt(4)/Co(0-\n2)/Gd( tGd)/TaN(4) samples with different tGdwe performed\nremanent intensity scan along our Co wedge, in addition\nto hysteresis loop measurements. After the sample is sat-\nurated with an OOP magnetic field of 1T, we determine\nthe thickness-dependent remanence from the pMOKE signal\nwithout external magnetic field. The remanent intensity scans\nare reported in Fig. 1 (d). As the pMOKE signal is mainly\nsensitive to the OOP component of Co magnetization, the nor-\nmalized remanent intensity will drop to zero at the SRT, when\nthe magnetization rotates IP. The SRT can be observed in Fig.\n1 (d) in samples with different thicknesses of Gd before and\nafter the application of strain. As previously reported44the\ncritical thickness tCo=tcat which SRT occurs, changes sig-\nnificantly in the presence of a Gd layer. For all the considered\nsamples, the in-plane strain shifts the OOP to IP transition\ntowards larger Co thickness. This suggests that the effective\nmagnetostriction of the Co/Gd bilayer is negative and its value\nλs=−10(5)×10−6is not significantly altered by the pres-ence of the Gd layer.\nTo obtain a quantitative understanding of the shape of the\nspin reorientation boundary, we employ an analytical model44\ndescribing the magnetostatic free energy of the anisotropy,\nwhich is zero at the SRT boundary. The first constituent ener-\ngies of the model are the demagnetization energies of the Co\nlayer\nEd,Co=1\n2µ0/integraldisplayy\n0M2\nCodq=1\n2µ0M2\nCoy (2)\nand of the Gd layer\nEd,Gd=1\n2µ0/integraldisplayx\n0M2\nGdexp(−2q/λGd)dq=\n1\n4µ0M2\nGdλGd/parenleftbigg\n1−exp/parenleftbigg−2x\nλGd/parenrightbigg/parenrightbigg (3)\nwhere λGdis the characteristic decay length of the Gd mag-\nnetization, which is induced at the Co/Gd interface, MCois the\nmagnetization of the Co layer, MGdis the effective Gd mag-\nnetization at the interface between Co and Gd and xandy\nare, respectively, the Gd and Co thicknesses in the diagram of\nFig.2 (a). The plot axes in Fig.2 (a) have been inverted for a\nbetter comparison with the other figures. The magnetocrys-\ntalline anisotropy is included with the term4\nEK=Ks−∆K/parenleftbigg\n1−exp/parenleftbigg−2x\nλK/parenrightbigg/parenrightbigg\n, (4)\nand it is also considered to decay with a characteristic de-\ncay length λKand magnitude ∆K. The second term in Eq.\n4 phenomenologically addressed the experimentally observed\ndecay in the effective anisotropy, which may be caused by\nsputter induced disordering of the Co45. Using a numeri-\ncal fit to the experimentally determined SRT, the parame-\ntersλK,λGdand∆Kfor our Co/Gd bilayer are determined.\nAll the other parameters were either experimentally measured\nor taken from literature and are reported in Table S.1 , sec-\ntion S1 of the supplementary information. In addition to the\nanisotropy term, and additional energy term Emixis included\nin the model. Emixtakes into account the mixing at the mag-\nnetic layer interfaces where the local net magnetization is\nzero. More details about the expression for this term and the\ndetermination of the fitting parameters can be found in the\nsupplementary information and in the work of Kools et al.44.\nIn this model, the expression of the total free energy density\nper unit area is, considering all the terms mentioned so far:\nEtot=−EK−Emix+Ed,Co+Ed,Gd. (5)\nThe magnetocrystalline anisotropy energy per unit area Ks,\ndue to the Pt/Co interface is assumed constant.\nEq. 5, describing the total energy of a\nTa(4nm)/Pt(4)/Co( tCo)/Gd( tGd)/TaN(4) sample, can be\nsolved for y (t Co) by imposing Etot=0 (spin reorientation\ntransition). The solution for the SRT obtained with the model\ndescribed above is reported in Fig. 2 (a) with a blue solid line\nin a phase diagram where tGd(x) and tCo(y) are continuously\nvaried from 0 to 3 nm and from 0 to 2 nm, respectively.\nTogether with the calculations, the SRT measured experimen-\ntally without externally applied strain is reported with blue\ndiamonds in Fig. 2 (a). The experimental data, follow well\nthe general trend of the calculations. Discrepancies between\nmodel and experimental values for tGd=0, might be due to\nadditional mixing between the layers.\nTo include the effects of strain, a magnetoelastic anisotropy\nKMEis added to Eq. 5 that becomes\nEtot=−EK−Emix−KME+Ed,Co+Ed,Gd. (6)\nIn our case KME=0.02 mJ/m2corresponds to the value of\nmagnetoelastic anisotropy induced with 0 .1% externally ap-\nplied in-plane strain in our experiments. As showed in Fig.\n1 (d), we do not observe significant changes to KMEwith in-\ncreasing tGd. Again considering the SRT-boundary to be at\nEtot=0, the solution of Eq. 6 (that includes the magnetoe-\nlastic term) is reported in Fig. 2 (a) with an orange solid line.\nAs expected from a material with negative magnetostriction,\nKMEsums to Ksand the PMA is enhanced by in-plane strain.\nThe SRT calculated including KMEto Eq. 6 is consequently\nshifted to larger values of tCo. This trend is in agreement with\nFIG. 2: (a) 2D phase diagram of the SRT of the a\nTa(4nm)/Pt(4)/Co( tCo)/Gd(t Gd)/TaN(4) stack as a function of tGd(x) and tCo\n(y). The axes have been inverted for a better comparison with other figures.\nBlue diamonds and red squares correspond to the experimental data,\nreported without and with strain applied, respectively. The solid lines\nindicate the calculated values using the model for the magnetostatics and Eq.\n6. A magnetoelastic anisotropy KME=0 and 0.02 mJ/m2is considered,\nrespectively, for the blue and orange curve. (b) Spin reorientation transition\nof a Ta(4)/Pt(4)/Co( tCo)/Gd(t Gd)/TaN(4) sample calculated for values of\ntGd=0, 1 and 3 nm and plotted as a function of tCo. The SRT is represented\nhere by a step function. Solid and dashed lines consider KME=0 and 0.02\nmJ/m2, respectively.\nthe experimentally determined SRT when and external strain\nεxx=0.1% is applied (orange squares in Fig.2 (a)).\nAnother way to visualize the SRT is solving Eq. 6 for fixed\nvalues of tGdand obtaining the critical thickness of tCosuch\nthatEtot=0. Then, the SRT can be represented as a step\nfunction in the diagram of Fig. 2 (b), analogue to the MOKE\nremanence scan shown in Fig. 1 (d). The values of Gd thick-\nnesses considered are tGd=0, 1 and 3 nm and are plotted in\nFig. 2 (b) with solid lines in black, blue and orange, in order.\nSolid lines consider KME=0 mJ/m2. Dashed lines consider\ninstead KME=0.02 mJ/m2in Fig. 2 (b). The information\ncontained here can be correlated to the experimental rema-\nnent intensity scan in Fig. 1 (d). Comparing Fig. 2 (b) with5\nFIG. 3: (a) Layerstack consisting of a Co/Gd quadlayer used to obtain magnetization compensation. (b) Coercivity and (c) remanent pMOKE intensity scan as\na function of tGd. Measurements before (blue) and after (orange) application of in-plane strain are reported. (d) Hysteresis loops in the Co dominated and (e)\nGd dominated state. Both curves with (orange) and without (blue) in-plane strain applied are shown.\nFig. 1 (d), a similar behavior can be observed. Firstly we can\nnote that the model predicts the SRT to shift when the thick-\nness of the Gd layer is tGd>0. Secondly, we observe a similar\nshift of the SRT point in Fig. 2 (b) and Fig. 1 (d) due to the\neffect of magnetoelastic anisotropy and of the external strain,\nrespectively. As we expect from a material with negative mag-\nnetostriction, Ksadds to KME, therefore the PMA is increased\nand the Co/Gd bilayer stays OOP magnetized for thicker Co\n(corresponding to larger Ed,Co). We confirm that the major ef-\nfect of strain on the Ta(4 nm)/Pt(4)/ Co(0-2)/Gd(t Gd)/TaN(4)\nsample is the alteration of the PMA. Moreover, the estimated\neffective magnetostriction of the stack - λs=−10(5)×10−6\n- is not significantly altered by the presence of the Gd layer in\nthe thickness range considered.\nIn this section, we examined the impact of in-plane strain\non the effective PMA of a Co/Gd ferrimagnetic bilayer. Our\nresults suggest negative magnetostriction of the stack for the\ninvestigated thickness values. We employ a recent model for\nthe magnetostatics of these type of systems, where we include\nthe effects of strain purely as magnetoelastic anisotropy. Our\nexperimental findings are in good agreement with the predic-\ntions made by this model, providing deeper understanding of\nthe response of this material platform to external strain.\nB. Magnetization compensation in quadlayer systems\nIn ferrimagnets, magnetization compensation can be\nachieved. This occurs when the net magnetization /vectorMtot=\n/vectorMGd+/vectorMCovanishes because the magnetization, coming from\nthe two sub-lattices, is equal in magnitude and opposite in\nsign.\nIn recent studies, changes to the saturation magnetization\nin the presence of strain were reported in epitaxial films31\nand rare earth free ferrimagnets30. To study the effectsof strain on magnetization compensation of synthetic fer-\nrimagnets, we consider a quadlayer sample44consisting of\nTa(4 nm)/Pt(4)/Co(0.6)/Gd(0-2)/Co(0.6)/Gd(1.5)/TaN(4) as\nschematically drawn in Fig. 3 (a). In this case, the thickness of\nthe bottom Gd layer is varied between 0 and 2 nm over a few\nmm, whereas all the other layers have constant thickness. The\nreason for this choice is that compared to the Co/Gd bilayer,\nthe magnetic volume of the Co is doubled while the number\nof Co/Gd interfaces where magnetization is induced in the Gd\nthrough direct exchange with the Co, is tripled. In this way\nmagnetization compensation can be more readily achieved.\nThe growing thickness of Gd, increases the contribution of\n/vectorMGdto/vectorMtot. For this reason, some areas of the wedge sam-\nple will be Co-dominated (for tGd<tcomp) and other will be\nGd-dominated (for tGd>tcomp) with /vectorMtot=0 attGd=tcomp.\nHere, tcomp is the thickness where magnetization compensa-\ntion is obtained. At magnetization compensation two effects\nare expected: a divergence of the coercivity and a sign change\nin the remanent pMOKE signal (Kerr rotation, normalized to\nits value in absence of Gd). The measurements for coercivity\nand intensity are reported in Figs. 3 (b) and (c), respectively.\nThe coercivity data were extracted from hysteresis loops mea-\nsured across the wedge direction (along y). The reason for\nthe sign change in the pMOKE signal, is the alignment of the\nGd magnetization along the field direction, in the Gd domi-\nnated regime. We report magnetization compensation in this\nquad-layer for tGd=1.25 nm.\nIn a similar fashion to what we have done investigating the\nPMA in the bilayer system, we repeat the experiment in the\npresence of εxx=0.1% in-plane strain. The results are reported\nin orange in Fig. 3 (b) and (c). Remarkably, the compensation\npoint of the Co/Gd quadlayer is unchanged by the application\nof this externally applied strain.\nFigs. 3 (d) and (e) contain OOP hysteresis loops\nof Ta(4 nm)/Pt(4)/Co(0.6)/Gd( tGd)/Co(0.6)/Gd(1.5)/TaN(4)6\nsamples for tGd=1.15 nm and tGd=1.35 nm, respectively,\nand further show the effects of magnetization compensation.\nThe sample is in this case OOP magnetized. As the thickness\nof Gd is increased, the magnetization of the sample goes from\nCo dominated (Fig. 3 (d)) to Gd dominated (Fig. 3 (e)). The\ninversion of hysteresis loops happens because for tGd>1.25\nnmthe Co-magnetization aligns antiparallel to the field, lead-\ning to the change in sign of the pMOKE signal. When the\nmeasurement is repeated in the presence of εxx=0.1% strain\n(orange line), no significant changes to the remanent intensity\nor coercivity are reported, if compared to the unstrained case\n(blue line). This suggests that magnetization compensation\ncan be achieved in these multilayer systems in the presence of\nexternal strain and, most importantly, that the magnetization\ncompensation point is unaffected.\nTo explain this, we can consider earlier studies about\nmagnetostatics of these types of systems. As previously\nreported25,44, magnetization compensation is due to the bal-\nance in Co magnetization and the Gd magnetization, induced\nin the Gd at the Co/Gd interfaces. In-plane strain in multilayer\nsamples with PMA modifies spin orbit coupling within one\nlayer46, thus altering the magnetocrystalline anisotropy en-\nergy of the system47. On the other hand the total magnetic mo-\nment per unit area /vectorMtotin synthetic ferrimagnets is obtained\nby integrating the magnetization of the Co and Gd sublattices\nover the respective layer thicknesses. Accordingly, in a mul-\ntilayer in-plane strain is not affecting the induced magnetic\nmoment from the Co onto the Gd, thus not altering magneti-\nzation compensation.\nIV. CONCLUSIONS\nThis work reveals the effect that external strain has on PMA\nand magnetization compensation of Co/Gd systems at room\ntemperature. Growing wedge samples, where the thickness of\none of the magnetic layers was varied, has allowed us to deter-\nmine thickness dependent transition in the magnetostatics of\nthis multilayer system. Deliberate in-plane strain was applied\nto the sample. In a bilayer Pt/Co/Gd system, we experimen-\ntally show that a sizable magnetoelastic coupling changes the\nSRT in the presence of strain. The contribution of the strain-\nanisotropy for this system has been included in a model for the\nmagnetostatics, describing the experimental observations well\nif an effective negative magnetostriction is considered. In a\nPt/Co/Gd/Co/Gd quadlayer we obtain magnetization compen-\nsation of the two sub-lattices by varying the thickness of the\nbottom Gd layer. Here, we find that the application of in-plane\nstrain does not affect the magnetization compensation. The\ninduced magnetic moment from the Co onto the Gd, being an\ninterface effect in a multilayer system, is not altered by such\nmechanical deformation. To conclude, this work provides a\nbroad understanding of the magnetoelastic properties of these\nmultilayer systems. As PMA and magnetic compensation are\nmaintained in the presence of externally applied strain, this\nmaterial system is a good candidate for technological imple-\nmentation of ferrimagnets.SUPPLEMENTARY MATERIAL\nSee supplementary material for magnetostatics model for\nthe spin reorientation transition and for more details about the\nsetup used for application of strain.\nACKNOWLEDGMENTS\nThis project has received funding from the European\nUnion’s Horizon 2020 research and innovation program un-\nder the Marie Skłodowska-Curie grant agreement No 860060\n“Magnetism and the effect of Electric Field” (MagnEFi), the\nDeutsche Forschungsgemeinschaft (DFG, German Research\nFoundation) - TRR 173 - 268565370 (project A01 and B02)\nand the Austrian Research Promotion Agency (FFG). The au-\nthors acknowledge support by the Max-Planck Graduate Cen-\ntre with Johannes Gutenberg University.\nAUTHOR DECLARATIONS\nConflict of interest\nThe authors have no conflicts to disclose.\nDATA SHARING POLICY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable reques.\n1T. Endoh, H. Honjo, K. Nishioka, and S. Ikeda, “Recent progresses in\nSTT-MRAM and SOT-MRAM for next generation MRAM,” in 2020 IEEE\nSymposium on VLSI Technology (IEEE, 2020) pp. 1–2.\n2S. S. Parkin, M. Hayashi, and L. Thomas, “Magnetic domain-wall race-\ntrack memory,” Science 320, 190–194 (2008).\n3S. Tehrani, “Status and outlook of MRAM memory technology,” in 2006\nInternational Electron Devices Meeting (IEEE, 2006) pp. 1–4.\n4K. Garello, F. Yasin, and G. S. 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Shull, “Strain-assisted magnetization reversal in Co/Ni multi-\nlayers with perpendicular magnetic anisotropy,” Scientific reports 6, 1–8\n(2016).Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nSupplementary material - Strain effects on magnetic compensation and spin\nreorientation transition of Co/Gd synthetic ferrimagnets\n(Dated: 3 April 2023)\n1arXiv:2303.15191v4  [cond-mat.mtrl-sci]  31 Mar 2023Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nS1 - Magnetostatics model for the Spin Reorientation Transition\nIn the expression of the free energy, a term describing the effect of intermixing at the multilayer\nsystem interfaces is added. The expression is\nEmix=1\n2µ0/integraltexta0x\n0M2\nCo+ (MGdexp(−q/λGd))2dq=\n1\n2µ0a0M2\nCox+1\n4µ0λGdM2\nGd/parenleftBig\n1−exp/parenleftBig\n−2a0x\nλGd/parenrightBig/parenrightBig\n.(S.1)\nTherefore, the total energy Etot=−EK−Emix−KME+Ed,Co+Ed,Gd, including all the terms\nwill be:\nEtot=−Ks+∆K/parenleftBig\n1−exp/parenleftBig\n−2x\nλK/parenrightBig/parenrightBig\n−KME−1\n2µ0a0M2\nCox\n−1\n4µ0λGdM2\nGd/parenleftBig\n1−exp/parenleftBig\n−2a0x\nλGd/parenrightBig/parenrightBig\n+1\n2µ0M2\nCoy\n+1\n4µ0M2\nGdλGd/parenleftBig\n1−exp/parenleftBig\n−2x\nλGd/parenrightBig/parenrightBig\n.(S.2)\nHere xandyare, respectively, the Gd and Co thicknesses in the phase diagram of Fig. S1. The\nvalue of the parameters used in our model are listed in Table S I .\nParameter Value Description\nKs 1.7 mJ/m2Interfacial anisotropy (from exp.)\nKME 0.02 mJ/m2Magnetoelastic anisotropy (from exp.)\nMCo 1.3 MA/m Cobalt magnetization (from exp.)\nMGd 1.4 MA/m Gadolinium magnetization at Co/Gd interface (from Ref.1)\na0 0.13 (-) Growth parameter of intermixing region (from exp.)\nλK 0.51(15) nm Change of PMA energy characteristic decay length (Fit parameter)\nλGd 0.59(22) nm Gd magnetization decay characteristic decay length (Fit parameter)\n∆K 3.96(41) ×10−4J/m2Change of PMA energy (Fit parameter)\nTABLE S I: Parameters used in the model for the magnetostatics of uncompensated Co/Gd synthetic ferrimagnets\nused for the calculations of the SRT. The term KMEis considered zero for when external strain is not applied to the\nsample.\nThe values of λK,λGdand∆Kare instead determined using a numerical fit and are reported\nin Table S I. To fit this equation to the phase diagram obtained experimentally, it is convenient to\n2Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nfind the Co-thickness (y) where the anisotropy energy ( Etot) is equal to zero (spin reorientation\ntransition, SRT). Solving Eq. S.2 for y gives:\ny0(x) =2\nM2\nCoµ0/parenleftBig\n−/parenleftBig\nKs−∆K/parenleftBig\n1−exp/parenleftBig\n−x\nλK/parenrightBig/parenrightBig/parenrightBig/parenrightBig\n−1\n2µ0a0M2\nCox\n−1\n4µ0λGdM2\nGd/parenleftBig\n1−exp/parenleftBig\n−2a0x\nλGd/parenrightBig/parenrightBig\n+1\n4µ0M2\nGdλGd/parenleftBig\n1−exp/parenleftBig\n−2x\nλGd/parenrightBig/parenrightBig\n.(S.3)\nNote that for determining the fit parameters, the measurements were taken without externally\napplied strain. Accordingly, the magnetoelastic energy term KMEis set to zero in Eq. S.3. The\nexperimental data used for the numerical fit are reported in Fig. S1. The sample used for the\nnumerical fit, explores a wide thickness range (Gd from 0 to 6 nm) in a double wedge fashion\nto improve accuracy. Consequently, the dimensions of this sample exceed the 1x1 cm size of\nthe bending device. For this reason, single wedge samples have been deposited for the strain-\ndependent study.\nFIG. S 1: Values for the SRT obtained experimentally on a Ta(4nm)/Pt(4)/Co(tCo)/Gd(tGd)/TaN (4)sample and\nused to extract the fitting parameters in Eq. S.3.\nThe feature around tGd=2 nm in Fig. S1 is not captured by out toy model, and might be due to\nthe additional intermixing caused during sputtering, not included in Eq. S.2.\n3Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nS2 -Application of strain\nTo obtain information about the magnetoelastic properties of the material, the substrate was\nbent mechanically with a 3 point bending sample holder, as shown schematically in Fig. S2 (a).\nA square sample of 1 by 1 cm is vertically constrained on two sides and pushed uniformly from\nbelow by a cylinder that has an off-centered rotation axis. The device generates a tensile strain\nin the plane of the sample up to 0 .1 % when the cylinder is rotated by 90◦. The strain is mostly\nuniaxial and has been measured with a strain gauge on the substrate surface.\nFIG. S 2: (a) schematic of the three point bending method used to externally strain the sample. The strain is mostly\nuniaxial along the xdirection. (b) hysteresis loops measured before (blue) and during (red) application of in-plane\nstrain for a sample of Pt/Co(1.85 nm)/Ta. The area highlighted in red corresponds to the magnetoelastic energy in the\nstrained system. The magnetic field was applied along the OOP direction ( z).\nMagnetic hysteresis loops are recorded before and after the application of the tensile strain\nand are used to estimate the magnetoelastic anisotropy. As previously reported2,3the magnetic\nanisotropy Ke f fis linked to the energy stored in the magnetization curves. For example the PMA\nenergy is given by the area enclosed between the magnetic loops measured with field along IP and\nOOP direction. If then the strain in the film is non-zero, the magneto-elastic coupling contributes\nin principle to the effective anisotropy. Two hysteresis loops measurements, before and after the\napplication of strain, are sufficient to estimate KME. Indeed the total anisotropy of the system\nisKe f f=KsandKe f f=Ks+KMEbefore and after the application of strain, respectively. The\nmagnetoelastic anisotropy KME=−3\n2λsYεis linked to reversible part of the hysteresis loops (close\nto the saturation) according to\n4Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nKME=Ms∆E=−3\n2λsYε (S.4)\nwhere ∆Eis the anisotropy energy measured by the difference in area below the strained and\nunstrained curves, εis the strain λsis the magnetostriction and Y is the Young’s mudulus of the\nmaterial. In our case ε=0.1% and Y=200 GPa. ∆Ecorresponds to the reversible part, i.e. the red\nmarked area in Fig. S2 (b). The value of magnetoelastic anisotropy was calculated using the value\nof saturation magnetization ( Ms) of the stack taken from literature and reported in Table S I.\n5Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nREFERENCES\n1Kools, T. J., van Gurp, M. C., Koopmans, B., and Lavrijsen, R. (2022). Magnetostatics of room\ntemperature compensated Co/Gd/Co/Gd-based synthetic ferrimagnets. Applied Physics Letters,\n121(24), 242405.\n2Johnson, M. T., Bloemen, P. J. H., Den Broeder, F. J. A., and De Vries, J. J. (1996). Magnetic\nanisotropy in metallic multilayers. Reports on Progress in Physics, 59(11), 1409.\n3Baril, L., Gurney, B., Wilhoit, D., Speriosu, V . (1999). Magnetostriction in spin valves. Journal\nof Applied Physics, 85(8), 5139-5141.\n6"    },    {        "title": "2111.08768v1.Ultrathin_ferrimagnetic_GdFeCo_films_with_very_low_damping.pdf",        "content": "Ultrathin ferrimagnetic GdFeCo \flms with very low damping\nLakhan Bainsla*,1,a)Akash Kumar,1Ahmad A. Awad,1Chunlei Wang,2Mohammad Zahedinejad,1Nilamani\nBehera,1Himanshu Fulara,1Roman Khymyn,1Afshin Houshang,1Jonas Weissenrieder,2and J. \u0017Akerman1,b)\n1)Physics Department, University of Gothenburg, 412 96 Gothenburg, Sweden.\n2)Department of Applied Physics, KTH Royal Institute of Technology, 106 91 Stockholm,\nSweden\nFerromagnetic materials dominate as the magnetically active element in spintronic devices, but come with\ndrawbacks such as large stray \felds, and low operational frequencies. Compensated ferrimagnets provide an\nalternative as they combine the ultrafast magnetization dynamics of antiferromagnets with a ferromagnet-like\nspin-orbit-torque (SOT) behavior. However to use ferrimagnets in spintronic devices their advantageous prop-\nerties must be retained also in ultrathin \flms ( t<10 nm). In this study, ferrimagnetic Gd x(Fe87:5Co12:5)1\u0000x\nthin \flms in the thickness range t= 2{20 nm were grown on high resistance Si(100) substrates and studied\nusing broadband ferromagnetic resonance measurements at room temperature. By tuning their stoichiometry,\na nearly compensated behavior is observed in 2 nm Gd x(Fe87:5Co12:5)1\u0000xultrathin \flms for the \frst time,\nwith an e\u000bective magnetization of Me\u000b= 0.02 T and a low e\u000bective Gilbert damping constant of \u000b= 0.0078,\ncomparable to the lowest values reported so far in 30 nm \flms. These results show great promise for the\ndevelopment of ultrafast and energy e\u000ecient ferrimagnetic spintronic devices.\nI. INTRODUCTION\nSpintronic devices utilize the spin degree of freedom for\ndata storage, information processing, and sensing1,2with\ncommercial applications such as hard drives, magnetic\nrandom access memories, and sensors. Besides conven-\ntional memory applications based on quasi-static opera-\ntion of magnetic tunnel junctions, high frequency spin-\ntronic oscillators3,4have recently been demonstrated for\nanalog computing applications such as bio-inspired neu-\nromorphic computing5,6, logic operations, energy har-\nvesting and Ising Machines.7For the \frst time, such oscil-\nlators are now used in commercial magnetic hard drives\nto facilitate writing to the disc.8The key challenges in\ndeveloping such devices is to \fnd material combinations\nwhich allow for fast operation, low-power consumption,\nnon-volatility, and high endurance. Due to their nat-\nural spin polarization and easy manipulation, ferromag-\nnetic materials (FM) dominate as active elements in these\ndevices.4However, FMs come with drawbacks such as:\n(i) large magnetic stray \felds a\u000becting the operation of\nneighbouring devices; (ii) limited scalability of magnetic\nbits in memory devices; (iii) the operating frequency of\nspin-based oscillators limited by ferromagnetic resonance\nfrequency, and (iv) slow synchronization of such oscilla-\ntors. These shortcomings drive researchers to \fnd more\nsuitable materials for future spintronic devices.\nVery recently, the interest in antiferromagnetic (AFM)\nspintronics9{11increased rapidly, as AFM materials have\nno stray \felds and can o\u000ber ultrafast spin dynamics, in-\ncluding AFM resonance frequencies in the THz region.\nIt was theoretically shown that such high-frequency ex-\ncitations are possible to achieve without any applied\nmagnetic \feld by injecting spin currents into AFM\na)Electronic mail: lakhan.bainsla@physics.gu.se\nb)Electronic mail: johan.akerman@physics.gu.sematerials.12{15Experiments have since demonstrated\npossible THz writing/reading capabilities.16However,\nthe absence of a net magnetic moment in AFMs leads\nto di\u000eculties in the read-out of the spin dynamics, in-\ncluding any microwave output signal from the AFM\noscillators.13{15\nA possible solution is presented by ferrimagnets\n(FiMs), which combine the properties of FMs and AFMs.\nFiMs posses magnetic sub-lattices in the same way as\nAFMs do, but their sub-lattices are inequivalent. The\nmagnetic sub-lattices in FiMs often consist of di\u000berent\nmagnetic ions, such as rare earth (e.g. Gd) and transi-\ntion metal (e.g. Fe, Co) alloys (RE-TM) such as CoGd,\nand as a result, a large residual magnetization remains\ndespite the two opposing sub-magnetizations. The tem-\nperature dependence of RE and TM sub-magnetizations\nin FiM can be quite di\u000berent which result in magneti-\nzations that can increase, and even change sign, with\ntemperature17,18, in stark contrast to the non-monotonic\ndecreasing temperature dependence for FMs and AFMs.\nSimilar e\u000bects could also be seen by varying the com-\nposition of ferrimagnetic alloys instead of changing the\ntemperature.19In addition, the di\u000berent properties of the\ntwo magnetic sub-lattices also results in two compen-\nsation points, namely the magnetization compensation\npointTmand the angular compensation point Ta. AtTm,\nthe two magnetic sub-lattices cancel each other, which re-\nsults in a zero net magnetic moment, while at T a, their\nnet angular momentum vanishes, as in AFMs. Therefore,\natTa, FiMs can have a near-THz resonance as in AFMs,\nwhile still having a net magnetic moment which can lead\nto strong read-out signals, including e\u000ecient microwave\nsignal output from FiM-based oscillators20, as well as ef-\n\fcient control and excitation. FiMs also show high spin\npolarization which also make them suitable candidate for\ne\u000ecient magnetic tunnel junctions.21\nDue to these unique properties, research in FiMs for\nspintronic applications is intensifying22, focusing mainly\non RE-TM based systems such as CoTb23, CoGd24, andarXiv:2111.08768v1  [cond-mat.mtrl-sci]  16 Nov 20212\nFigure 1. (a) Schematic illustration of the coplanar waveguide (CPW), the thin \flm sample and its orientation, the directions\nof the applied magnetic \feld H, the microwave \feld hrf, and the e\u000bective magnetic \feld He\u000bduring FMR measurements.\nInset shows the \flm stack. (b) FMR response (derivative of the FMR absorption) for a 10 nm Gd 12:5Fe76:1Co11:4\flm (S2)\nrecorded at di\u000berent frequencies and \ftted (solid lines) to Eq. 1. While FMR curves were recorded at 1 GHz frequency intervals\nthroughout this study, \fgure (b) only shows curves with \u0001 f= 2 GHz for clarity.\nGdFeCo25and Mn 3\u0000xPtxGa26,27based Heusler alloy.\nAmong these, GdFeCo has been studied the most with\ndemonstrations of fast domain wall motion28and ultra-\nfast spin dynamics17nearTa, large spin-orbit torques\nand their sign reversal,25,29low magnetic damping in\nthick 30 nm \flms,30and sub-picosecond magnetization\nreversal,31to name a few. What is missing, however, is a\ndemonstration that these unique material properties per-\nsist down to much thinner \flms, which will ultimately be\nneeded if FiMs are to be used in spin-Hall nano oscillators\n(SHNOs).4\nIn the present study, we systematically study the\ngrowth and functional properties of ultrathin ferrimag-\nnetic Gd x(Fe87:5Co12:5)1\u0000xthin \flms [referred to as\nGdx(FeCo) 1\u0000xhereafter]. GdFeCo thin \flms in the\nthickness range of 2{20 nm were grown on high resis-\ntance silicon (HR-Si) substrate. The atomic composi-\ntion of Gd x(FeCo) 1\u0000xwas controlled using co-sputtering\nand determined using inductively coupled plasma optical\nemission spectroscopy (ICP-OES). The magnetic prop-\nerties and Gilbert damping were studied using broad-\nband ferromagnetic resonance (FMR) measurements. We\nalso demonstrate ultra low Gilbert damping for 2 nm\nGdFeCo, near the compensation point of Gd x(FeCo) 1\u0000x.\nThese results paves the way for integration of FiMs into\nvarious spintronic devices and applications.\nII. RESULTS AND DISCUSSION\nThe growth conditions for GdFeCo were \frst optimized\nby growing four 10 nm thick Gd 12:5Fe76:1Co11:4\flms on\nHR-Si (100) substrates using di\u000berent MgO seed layer\nthicknesses: 0 nm (S1), 6 nm (S2), 10 nm (S3 & S4);in S4, the seed was annealed at 600C for 1 hour prior\nto GdFeCo deposition to check the e\u000bect of MgO crys-\ntallinity. MgO was chosen as seed since it is insulating\nand therefore will not contribute any spin sinking to the\nmagnetic damping.32\nA. Seed layer dependence on 10nm thick\nGd12:5Fe76:1Co11:4\flms\nFurther details of the growth conditions are given in\nthe experimental section. FMR measurements, on 6 \u00023\nmm2rectangular pieces cut from these \flms, were then\nperformed using a NanOsc PhaseFMR-40 FMR Spec-\ntrometer. The sample orientation on the coplanar waveg-\nuide (CPW), together with the directions of the applied\n\feld, the microwave excitation \feld hrf, and the e\u000bec-\ntive magnetic \feld He\u000b, are shown in Fig. 1(a). Typical\n(derivative) FMR absorption spectra obtained for S2 are\nshown in \fgure 1(b) together with \fts to a sum of sym-\nmetric and anti-symmetric Lorentzian derivatives:33\ndP\ndH(H) =\u00008C1\u0001H(H\u0000HR)\n[\u0001H2+ 4(H\u0000HR)2]2+2C2(\u0001H2\u00004(H\u0000HR)2)\n[\u0001H2+ 4(H\u0000HR)2]2\n(1)\nwhereHR, \u0001H,C1, andC2represent the resonance \feld,\nthe full width at half maximum (FWHM) of the FMR ab-\nsorption, and the symmetric and anti-symmetric \ftting\nparameters of the Lorentzian derivatives, respectively.\nThe extracted values of HRvs.fare shown in \fgure\n2 (b) together with \fts to Kittel's equation:34\nf=\r\u00160\n2\u0019q\n(HR\u0000Hk)(HR\u0000Hk+Meff) (2)3\nFigure 2. (a) Seed layer dependence of frequency vs resonance \feld of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid\nsymbols and solid lines are the experimental data points and \ftting with equation (2), respectively. (b) Resonance linewidth\n(\u0001H)vs.frequency of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid symbols and solid lines are the experimental data\npoints and \ftting with equation (3), respectively. The e\u000bective Gilbert damping constant values of all the samples are given in\n\fgure 2 (b). The black and violet dotted lines in \fgure 2(b) shows the \ftting of equation (3) in low and high frequency regions,\nrespectively.\nwhere,\r,HkandMe\u000bare the gyromagnetic ratio, the\nin-plane magnetic anisotropy \feld, and the e\u000bective mag-\nnetization of the sample, respectively, all allowed to be\nfree \ftting parameters. Values for \randHkonly showed\nminor variation between the four samples, with \r=2\u0019=\n29.4-30.0 GHz/T and Hk= 66-104 Oe. Me\u000bvaried more\nstrongly, with values of 0.79, 1.19, 0.71 and 0.76 T ob-\ntained for S1, S2, S3 and S4, respectively.\nThe e\u000bective Gilbert damping constant \u000bcan then be\nobtained from \fts of \u0001 Hvs.fto:35\n\u0001H= \u0001H0+4\u0019\u000bf\n\r\u00160(3)\nwhere the o\u000bset \u0001 H0represents the inhomogeneous\nbroadening. Equation (3) is well \ftted to the experimen-\ntal values, using \u0001 H0and\u000bas adjustable \ftting param-\neters for all the four samples, as shown in the \fgure 2(b).\n\u0001H0= 2{4 mT is essentially sample independent within\nthe measurement accuracy. In contrast, the obtained val-\nues of\u000bvary quite strongly and are given inside \fgure\n2(b). The GdFeCo grown with 6 nm MgO seed layer (S2)\nclearly shows the lowest value of \u000b= 0:0055, although\nthis might be a\u000bected by the slight non-linear behavior\naround 10 to 15 GHz. However, when only the high-\feld\ndata is \ftted, the extracted damping of \u000b= 0.0076 is\nstill the lowest and at all frequencies the linewidth of S2\nlies well below all the other samples. As damping is one\nof the most important parameters for spintronic devices,\nwe hence chose the growth conditions of S2 for all subse-\nquent \flms in this study.B. Thickness dependence on Gd 12:5Fe76:1Co11:4\flms\nAfter optimizing the growth conditions for\nGd12:5Fe76:1Co11:4, the thickness dependence of the\n\flms was studied with the same composition using the\ngrowth conditions of sample S2. The FMR linewidth\n\u0001Hvs. f is shown in \fgure 3(a) and exhibits a relatively\nstrong dependence on thickness. It is noteworthy\nthat the 4 nm \flm shows the narrowest linewidth at\nall frequencies, clearly demonstrating that very low\ndamping can be achieved also in ultra-thin GdFeCo.\nThe extracted Me\u000band\u000bare shown vs.thickness in\n\fgure 3(b), both showing a strong thickness dependence.\nDamping as low as \u000b= 0:0055 is obtained for the 10\nnm thick \flms. If only the high-\feld portion of the data\nis \ftted, the extracted damping increases to 0.0076,\nwhich is still about an order of magnitude lower than\nany literature value on 10 or 30 nm \flms.19,36Both\nthe 10 and 20 nm \flms showed a minor nonlinearity in\n\u0001Hvs.fdata and were therefore analysed by \ftting\nthe data in both the low and the high \feld regions\nseparately, as shown by the dotted lines in \fgure 3(a).\nThe\u000bvalue for the 20 nm \flm increased slightly from\n0.0098 to 0.0109 if only high \feld data is used for\nanalysis. The relatively higher damping for the 20 nm\n\flm might be due to the radiative damping mechanism\nwhich increases proportionally with magnetic layer\nthickness.37We conclude that 2 nm ultrathin \flms can\nindeed be grown with reasonably low damping. Since\nthe damping is strongly thickness dependent in this\nregime, the optimum thickness for devices may likely be\nfound in the 2{4 nm range.4\nFigure 3. (a) FMR linewidth \u0001 Hvs.ffor four Gd 12:5Fe76:1Co11:4\flms with di\u000berent thicknesses, together with linear \fts to\nequation (3). The dotted lines show \fts for the 20 nm \flm in its low and high frequency regions, respectively. (b) E\u000bective\nmagnetization and e\u000bective Gilbert damping constant vs.thickness; lines are guides to the eye.\nC. Composition dependence on 2nm thick \flms\nTo \fnally investigate whether we can achieve a com-\npensated ferrimagnetic behavior also in ultra-thin \flms,\nwe grew 2 nm Gd x(FeCo) 1\u0000x\flms in the composition\nrange 12{27 at.% Gd. The \flms were characterized using\nFMR spectrometry as described above and the extracted\nresults are shown in \fgure 4.\nThe extracted Me\u000band\u000bfollow a similar trend as re-\nported earlier for one order of magnitude thicker GdFeCo\n\flms characterized using an all-optical pump-probe tech-\nnique.17We \frst note that we can indeed reach an es-\nsentially fully compensated antiferromagnetic behavior in\ntwo \flms around a composition of 25 at.% Gd. We have\nmarked this compensation point with xmand a dashed\nline in \fgure 4 (c). Both \flms show very low damping of\n0.0078 and 0.009 respectively. However, just below this\ncomposition, the damping shows a peak, which is con-\nsistent with an angular compensation point, which we\ndenote byxa. It is noteworthy that the extracted damp-\ning value of \u000b= 0.0142 is still more than an order of\nmagnitude lower than \u000b= 0.45 of 30 nm \flms measured\nusing FMR spectrometry19and\u000b= 0.20 of 20 nm \flms\nmeasured using an optical pump-probe technique.17\nIII. CONCLUSION\nIn view of the potential application of compensated\nferrimagnets to spintronic devices, we prepared ferri-\nmagnetic thin \flms of Gd x(FeCo) 1\u0000xon high resistance\nSi(100) substrates and studied them using the FMR mea-\nsurements. Their growth conditions were optimized us-\ning 10 nm thick Gd 12:5Fe76:1Co11:4\flms, after which\nthickness dependent studies were done on the same com-\nposition in the thickness range of 2{20 nm. Composi-\ntion dependence studies were \fnally done on 2 nm thick\nGdx(FeCo) 1\u0000x\flms and an essentially compensated fer-rimagnetic behavior was observed for the \frst time in\nultrathin 2 nm \flms. The angular momentum compensa-\ntion and magnetic compensation points observed in this\nwork are very close to those reported earlier on much\nthicker \flms in the literature. A record low \u000bvalue of\nabout 0.0078 is obtained near the magnetic compensa-\ntion point, which is an order of magnitude lower than\nthe values reported in the literature using similar analysis\nmethods. The observation of compensated ferrimagnetic\nbehavior in ultrathin \flms together with very low value\nof\u000bare promising results for the future development of\nultrafast and energy e\u000ecient ferrimagnetic spintronic de-\nvices.\nEXPERIMENTAL SECTION\nA. Thin \flms growth and composition analysis\nAll the samples were prepared on high resistivity\nSi(100) substrates using a magnetron sputtering sys-\ntem with a base pressure of less than 2 \u000210\u00008torr.\nThin \flms of Gd x(FeCo) 1\u0000xwere deposited using the\nco-sputtering of high purity (more than 99.95%) Gd\nand Fe 87:5Co12:5targets, and composition analysis\nwas done using the inductively coupled plasma mass\nspectroscopy (ICP-MS). Thin \flms stacking structure\nof Si(100)/MgO(t)/Gd 12:5Fe76:1Co11:4(10)/SiO 2(4)\nwere used for seed layer dependence studies, here,\nthe number in the bracket is the thickness of the\nlayer in nm, where t=0, 6 and 10 nm. Four sam-\nples, namely S1 to S4 were prepared to obtain the\nbest conditions to grow Gd 12:5Fe76:1Co11:4(10) \flms.\nFor S1, Gd 12:5Fe76:1Co11:4(10) was grown directly\nover HR-Si (100) substrates, while in both S2 and\nS3 Gd 12:5Fe76:1Co11:4were grown with MgO seed\nlayer of 6 and 10 nm, respectively. All the lay-\ners in S1-S3 were grown at room temperature and5\nFigure 4. (a) Frequency vs.resonance \feld and (b) resonance linewidth vs.frequency, of 2 nm thick Gd x(FeCo) 1\u0000x\flms as\na function of Gd content in atomic %. (c) E\u000bective magnetization and e\u000bective Gilbert damping constant vs.Gd content.\nSolid symbols represent the values obtained by \ftting the experimental FMR data in (a) and (b) using the equation (2) and\n(3), respectively; solid lines in (c) are guides to the eye. xaand xmshow the angular and magnetic compensation points,\nrespectively, obtained from the literature17,19.\nno further heat treatment was given to them. In\nS4, 10 nm MgO seed layer were grown over HR\nSi(100) substrates at RT and followed by a in-situ\npost-annealing at 600C for 1 hour, and after that\nGd12:5Fe76:1Co11:4were deposited. The stacking struc-\nture of Si(100)/MgO(6)/Gd 12:5Fe76:1Co11:4(m)/SiO 2(4)\nwere used for thickness dependence studies, where\nm is the thickness of Gd 12:5Fe76:1Co11:4layer,\nand varied from 2 to 20 nm. For composi-\ntion dependence studies, stacking structure of\nSi(100)/MgO(6)/Gd x(FeCo) 1\u0000x(2)/SiO 2(4) were used,\nwhere xvaried from 12.5 to 26.7. The composition of\nGdx(FeCo) 1\u0000x\flms was varied by changing the sput-\ntering rate of Fe 87:5Co12:5target, while keeping the Gd\nsputtering rate \fxed for most \flms. All the samples for\nthickness dependence and composition dependence were\ngrown at room temperature and no post-annealing was\nused. Layer thicknesses were determined by estimating\nthe growth rate using the Dektak pro\fler on more than\n100 nm thick \flms.B. Inductively coupled plasma mass spectroscopy\n(ICP-MS) measurements\nThe elemental composition (Co, Fe, and Gd) of the\nthin \flm samples was determined by inductively coupled\nplasma optical emission spectroscopy (ICP-OES) using a\nThermo Fisher Scienti\fc iCAP 6000 Series spectrometer.\nEach thin \flm sample was exhaustively extracted in 5 mL\nHNO3 (65%, Supelco, Merck KgaA, Sigma-Aldrich) for\na duration of 30 min. 5 mL ultrapure MilliQ-water (18\nM\ncm) was added to the solution and the extract was al-\nlowed to rest for 30 minutes. The extract was transferred\nto a 100 mL volumetric \rask. The extracted sample was\nthen rinsed for several cycles in ultrapure water. The\nwater used for rinsing was transferred to the same volu-\nmetric \rask. The extract was diluted to 100 mL for ICP\nanalysis. ICP check standards were prepared from stan-\ndard solutions (Co and Fe: Merck, Germany; Ga: Accu-\nstandard, USA). The relative standard deviation (from\nthree individual injections) were within 1%.6\nTable I. The obtained values of e\u000bective Gilbert damping constant \u000bat room temperature (RT) in this work and comparison\nwith the lowest values reported so far in the literature at RT and also at their respective angular momentum compensation\n(Ta) and magnetic compensation (T m) points.\nFilm composition Film thickness \u000b Measurement technique Analysis method Reference\nGd23:5Fe68:9Co7:6 30 \u00180.45 (at RT) FMR Kittel's FMR19\n\u00180.35 (at RT) Pump-probe\nGd22Fe74:6Co3:4 20 \u00180.21 (at T a) Pump-probe -do-17\n\u00180.13 (at T m)\nGd25Fe65:6Co9:4 10 \u00180.07 (at RT) Spin torque FMR -do-36\n\u00190.01 (at RT) Spin torque FMR Ferrimagnetc resonance\nGd23:5Fe66:9Co9:6 30 0.0072 (at RT) Domain wall (DW) Field driven DW30\nmotion mobility\nGd12:5Fe76:1Co11:4 10 0.0055 Broadband FMR Kittel's FMR This work\n0.0076 (HF data) -do- -do- This work\nGd12:5Fe76:1Co11:4 4 0.0064 -do- -do- This work\nGd12:5Fe76:1Co11:4 2 0.0101 -do- -do- This work\nGd23:4Fe67:0Co9:6 2 0.0141 -do- -do- This work\nGd24:4Fe66:1Co9:5 2 0.0078 -do- -do- This work\nC. Ferromagnetic resonance (FMR) measurements\nRectangular pieces of about 6 \u00023 mm2were cut from\nthe blanket \flms and broadband FMR spectroscopy was\nperformed using a NanOsc Phase FMR (40 GHz) system\nwith a co-planar waveguide for microwave \feld excita-\ntion. Microwave excitation \felds hrfwith frequencies up\nto 30 GHz were applied in the \flm plane, and perpendic-\nular to the applied in-plane dc magnetic \feld H. All the\nFMR measurements were performed at the room tem-\nperature. The schematic of FMR measurement setup is\nshown in 1(a), and further details about the measure-\nments are given in Section 2 (results and discussions).\nSUPPORTING INFORMATION\nSupporting Information is available from the Wiley\nOnline Library or from the corresponding author.ACKNOWLEDGEMENTS\nLakhan Bainsla thanks MSCA - European Commission\nfor Marie Curie Individual Fellowship (MSCA-IF Grant\nNo. 896307). This work was also partially supported\nby the Swedish Research Council (VR Grant No. 2016-\n05980) and the Horizon 2020 research and innovation\nprogramme (ERC Advanced Grant No. 835068 \"TOP-\nSPIN\").\nCONFLICT OF INTEREST\nThe authors declare no con\rict of interest.\nAUTHOR CONTRIBUTIONS\nL.B. and J. \u0017A. planned the study. L.B. grew the \flms,\nperformed the FMR measurements and analysed the ob-\ntained FMR data. J.W. helped with ICP-MS measure-\nments and analysis. L.B. wrote the original draft of the\npaper. J. \u0017A. coordinated and supervised the work. All\nauthors contributed to the data analysis and co-wrote\nthe manuscript.7\nDATA AVAILABILITY STATEMENT\nThe data that support the \fndings of this study are\navailable from the corresponding author on reasonable\nrequest.\nREFERENCES\n1S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, v. S. von\nMoln\u0013 ar, M. Roukes, A. Y. Chtchelkanova, and D. 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Shaw, Nature Physics\n12, 839 (2016)."    },    {        "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 effect 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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Deniz Bozdag1*, Chia -Yi Chen2, P.A. Truitt1, A. J. Epstein1,3†and E. \nJohnston -Halperin1†\n \n1Department of Physics, The Ohio State Univers ity, Columbus, OH 43210 -1117, USA            \n2Chemical Physics Program, The Ohio State University, Columbus, OH, 43210 -1106, \nUSA   3Department of Chemistry, The Ohio State University, Columbus, OH 43210 -1173, \nUSA \nAbstract  \nWe report the successful extractio n of spin polarized current from the organic -based \nroom temperature ferrimagnetic semiconductor V[TCNE] x (x~2, TCNE: \ntetracyanoethylene; TC ~ 400 K, EG ~ 0.5 eV, ~ 10-2 S/cm ) and its subsequent injection \ninto a GaAs/AlGaAs light -emitting diode (LED). The spin current tracks the \nmagnetization of V[TCNE] x~2, is weakly temperature dependent, and exhibits heavy hole \n/ light hole asymmetry. This result has implications for room temperature spintronics and \nthe use of inorganic materials to probe spin physics in organic and molecular systems.  \n \nThe field of semiconductor spintronics promises the extension of spin -based \nelectronics beyond memory and magnetic sensing into active electronic components with \nimplications for next -generation computing [1] and quantum information  [2]. The \ndevelopment of organic -based magnets with room temperature magnetic ordering [3] and \nsemiconducting functionality [4] promises to further broaden this impact by providing a \nroute to all -organic spintronic devices [5-6] and hybrid organic/inorganic structures capable of exploiting the multifunctionality [7-8] and ease of production in organic \nsystems [9] as well  as the well established spintronic functionality of inorganic materials. \nOur work demonstrates electrical spin injection in a hybrid organic/inorganic spin -\nresolved light -emitting diode (spin -LED) [10] structure an d opens the door to a new class \nof active, hybrid spintronic devices with multifunctional behavior defined by the optical, \nelectronic and chemical sensitivity [11] of the organic layer. In addition, spin transport i n \nthese hybrid structures provides the opportunity to use well -characterized inorganic \nmaterials as a probe of spin physics in organic and molecular systems.  \nThe magnetic order in V[TCNE] x~2 (TC ~ 400 K) originates from direct \nantiferromagnetic exchange co upling between the unpaired spins of V2+ (t2g , S = 3/2) and \nTCNE- (*, S = 1/2). As shown in Fig. 1(a), the unpaired spin in the TCNE- anion is \ndistributed over the entire molecule [12] and occupies the * antibonding level.  The * \norbital can accept another electron with opposite  spin, which costs an additional Coulomb \nenergy [4, 13]  Uc ~ 2 eV. Therefore, the * band is split into two oppositely spin \npolarized subbands: occupied * and unoccupied *+U c. The t2g levels lie within the \nCoulomb  gap [14] and define the valence band while the * + U c levels define the \nconduction band with an 0.5 eV bandgap (  ~ 10-2 S/cm ). This proposed electronic \nstructure with ferromagnetically aligned conduction and valance bands is consistent with  \ntheoretical calculations [15-17] and experimental studies [5, 14, 18] . \nThe implications of the electronic and magnetic structure of V[TCNE] x~2 for a spin -\nLED device are depicted schematically in Fig. 1(b). Spin polarized carriers are extracted \nfrom a spin -polarized source (V[TCNE] x~2 here)  and injected into the conduction band of \nthe n-AlGaAs layer of an n-i-p diode with a GaAs quantum well ( QW) embedded in the intrinsic region. The sign and magnitude of the spin -polarized charge injection are \ndetermined by the magnetization of the V[TCNE] x~2 and can be analyzed through the \npolarization of the heavy -hole (HH) and light -hole (LH) electrolumines cence (EL) from \ncarrier s that relax into the quantum well. These optical polarizations have complementary \ncoupling to the spin polarization of the electron current [19], where the optical \npolarizat ion is defined as PEL = (IRCP – ILCP)/(IRCP + ILCP)  (IRCP and ILCP indicate the \nintensity of right and left circular polarization, respectively). For example, an S = + 1/2 \nelectron will yield an RCP photon on recombining with a HH and an LCP photon on \nrecombining with a LH. This asymmetry of HH and LH with a given magnetization of \nthe spin polarized source is shown schematically in Fig. 1(c) and can be resolved in EL \ndue to the energy difference between HH and LH states. Thus the optical polarization of \nthe EL is directly  proportional the spin polarized current present in the V[TCNE] x~2. \nThe organic and inorganic layers of the spin -LED are synthesized using chemical \nvapor deposition (CVD) and molecular beam epitaxy (MBE), respectively.  The \nAlGaAs/GaAs heter ostructure follows previous spin -LED devices . The MBE grown III -\nV material is grown on a p-doped ( p = 1× 1018 cm-3) GaAs (100) substrate with layer \nstructure as follows: 300 nm p-GaAs (1× 1017 cm-3)/ 200 nm p-Al0.1Ga0.9As (p = 1× 1016 \ncm-3)/ 25 nm i-Al0.1Ga0.9As/ 10 nm i-GaAs/ 25 nm i-Al0.1Ga0.9As/ 100 nm n-Al0.1Ga0.9As \n(n = 1× 1016 cm-3 )/ 15 nm nn+Al0.1Ga0.9As, n = 1× 1016 cm-3 to 5× 1018 cm-3/ 15 nm n+- \nAl0.1Ga0.9As ( n+ = 5× 1018 cm-3) [20]. Devices are fabricated using  standard \nphotolithography to define a mesa that is protected with photoresist except for a narrow \nwindow (100 m by 1  mm). The V[TCNE] x~2 is deposited uniformly across the sample at \n40 ° C and 35  Torr in an argon glovebox [9].  The top electrical contact consists of an optically transparent aluminium layer (7 nm) and a high conductivity gold layer (23 nm, \nFig. 2(b) inset). Due to the air -sensitivity of CVD -prepared V[TCNE] x~2 the samples are \ntransferred within the glovebox to a custom -designed air -free sample mount for transfer \nto a magneto -optical cryostat. See Auxiliary Mat erial 1 a for additional details.  \nThe electrical and optical properties of both V[TCNE] x~2 spin-LEDs and bare LEDs \nare studied and compared (Fig. 2(a); Auxiliary Material 1b). Both devices show modified \np-i-n diode behaviour with the V[TCNE] x~2 spin-LED sh owing an additional series \nimpedance  consistent with thermally activated transport in an intrinsic semiconductor [4] \nwith a bandgap of 0.5  eV  (Auxiliary Material  2a). Moreover a positive linear \nmagnetoresistance is  observed in V[TCNE] x~2 spin-LED devices with the same sign and \nmagnitude as isolated V[TCNE] x~2 films [4], confirming charge flow through the \nV[TCNE] x~2 layer ( Auxiliary Material  2b). Figure 2(b) shows an EL spectr um from a \nV[TCNE] x~2 spin-LED at a temperature of 60 K and a bias of +18.5 V collected using a \n0.3 m spectrometer and LN 2 cooled charge -coupled device (CCD). The peak at 1.533 eV \nrepresents the transition from the ground state of the conduction band of the  quantum \nwell to the HH states of the valence band. The higher energy peak at 1.541 eV is \nattributed to the corresponding LH transition, consistent with theoretical predictions for \nthe HH/LH splitting [21]. The broa d peak centred at 1.471 eV is due to recombination in \nthe p -doped GaAs substrate.  \nThe EL polarization, PEL, of the HH and LH transitions is measured and analysed \nas a function of the out of plane magnetic field at T = 60 K, with I  = 0.5 mA and V = \n+18.5 V. At each field a series of spectra with alternating helicity are acquired and \nanalyzed using a variable wave plate and a linear polarizer in the collection path [10]. IRCP and ILCP are determined individually by i ntegrating over the appropriate spectral \npeaks (highlighted regions in Fig. 2(b)) and PEL is calculated for the  HH and LH, \nrespectively . In these  spin-LED device s the total polarization  signal is composed of four \nindependent contributions:  the spin  injecti on EL, the intrinsic magnetic field response of \nthe AlGaAs/GaAs quantum well ( Zeeman effect)  [22], magnetic circular di chroism \n(MCD)  from the cryostat windows, and MCD [10, 2 3-24] in the V[TCNE] x~2 layer . \nFigure s 3 (a) – (c) show cartoon s explaining how  these contributions combine  to give the \ntotal measured EL polarization  while Figures 3 (d) – (f) show corresponding \nexperimental measurements of these effects.  \n Figure 3 (a) shows  the expected  linear response from  both the Zeeman effect  and \ndichroism in the  cryostat windows . In principle the HH should show a three times larger \nZeeman splitting than the LH due to their increased angular momentum ( mj = 3/2 and mj \n= 1/2, respectiv ely); however this difference is not observed (Fig. 3 (d)), suggesting  that \nthe dichroism of the cryostat windows is the dominant effect.  \nIn Fig. 3 (b) the expected MCD response is plotted as the dashed gray line, with the \ncontribution from  Fig. 3  (a) plotted as the dashed orange line . The  total of these two \neffects  is shown  as the green solid line, and qualitatively agrees with independent \nmeasurements of the MCD shown in Fig. 3 (e). This data is collected by optically  \nexciting unpolarized carriers in the quantum well using a linearly polarized pump  at \n1.771 eV and a power density of 100 W/cm2. The resulting photoluminescence (PL)  \nexhibits both  the linear window dichroism shown in Fig. 3(d) and saturation at ±  200 Oe, \nconsistent with the field dependence of  the magnetization of V[TCNE] x~2 (Fig. 4(a), \ngreen line ; Auxiliary Material 3)  as predicted in Fig. 3 (b) . Note that the since the HH and LH PL are so close in energy, there is no measureable difference between the HH and LH \nMCD response . \nFinally in Fig. 3  (c), the spin injection EL is added to the various background \nsignals shown in Figs. 3 (a) and (b) (green line). In contrast to these backgrounds, the \nspin injection EL should have opposite sign for the HH and LH (red and blue dashed \nlines, respectively).   The total EL signal  is given by  \nHH\nwindow ZeemanHH\nMCDHH\nspinHH\nEL P P P P  \nand \nLH\nwindow ZeemanLH\nMCDLH\nspinLH\nEL P P P P for the HH and LH, respectively. Note that in the \ncase where the magnitude of the spin injection signal is comparable to that of the MCD \nbackground the two terms that depe nd on the magnetization of the V[TCNE] x~2 cancel for \nthe LH, leaving a simple linear dependence due to Zeeman splitting and window \ndichroism. Figure 3 (f) shows the total EL signal from a full V[TCNE] x~2 spin-LED \ndevice. As expected, and in contrast to the  background measurements in Figs. 3 (d) and \n(e), a clear distinction between HH and LH polarization is observed and the linear \nresponse of the LH indicates that at this temperature the spin injection signal and MCD \nare of comparable magnitude.   \n The bare EL signal ( Pbare) for both HH and LH transitions can be removed by a \nlinear fit to the calibration data in Fig. 3( d), leaving the corrected polarization \nas\n/ / /HH LH HH LH HH LH\nEL MCD spinP P P . The background corrected polarization signals are plotted \nin Fig. 4(a) wh ere it can be seen that PEL of the HH peak reaches saturation at ~ 200 Oe, \nclosely tracking the out of plane magnetization of the V[TCNE] x~2 (Fig. 4(a) green line).  \nMoreover, subtracting \nLH\nELP from\nHH\nELP will result i n a cancellation of the MCD terms and \ngive twice the optical polarization (\n2spinP ).  This analysis yields a saturated spin polarization of \nsat\nspinP =0.098 ±  0.007% (the error is one standard deviation at saturation). \nThe saturated spin polarization is related to the injected electron spin polarization at the \nquantum well (inj) by\ninj sat\nspinP , where  \n(1 )r\ns  and τ r and τ s are the recombination \ntime and spin relaxation time of the injected electrons, respectively [19]. Independent \nphotoluminescence studies give a n η of 8.48 (Auxiliary Material 4), and thus a inj of \n0.83 ±  0.07% at 60 K. This number is comparable to the initial resul ts from GaMnAs and \nFe based spin -LED structures [10, 22] . \nThe dependence of PEL on bias and temperature is explored in Figs. 4(b) and 4(c), \nrespectively. The large impedance of the V[TCNE] x~2 layer, and the resultin g high turn \non voltage for the spin -LED, limits the range of applied bias over which measureable EL \ncan be obtained. Figure 4(b) shows a measurement of  PEL at a bias of +15.2 V and a \ncurrent of 0.08 mA (the minimum values that give measureable luminescence ), showing \nthe same PEL as at +18.5 V (Fig. 4a) to within the resolution of the measurement . This \ninsensitivity to bias is initially surprising given that the 3.3 V difference between \nmeasurements is roughly twice the bandgap of the III -V materials . Howeve r, comparison \nwith a bare LED device (Fig. 2 (b)) suggests that the voltage drop across the p-i-n \njunction is only 0.27 V and it is important to note that this measurement probes the \nground state of the quantum well and thus is sensitive only to carriers t hat have fully \nthermalized (see Auxiliary Material 5 for a full discussion).  \nFigure 4(c) shows PEL at a temperature of 140 K, the highest temperature at which \nEL from the quantum well can be clearly resolved. This data provides further support for \nthe simp le two -component model for the polarization presented above. At 140K the magnitude for both\nHH\nspinP  and \nLH\nspinP are reduced from 60K, consistent with an increased η of \n18.09  (see Auxiliary Material 4) and consistent with literature report s of the decrease in  \nspin relaxation time at high temperature  [25]. When this η is combined with \nsat\nspinP = 0.036 \n±  0.006%, a value for inj (140 K) of 0.66 ±  0.16% is determined. This relatively modest \ndependence on temperature ( inj(60 K) = 0.83 ± 0.07%) is consistent with spin injection \nfrom other  room temperature   ferromagnets  such as Fe [24][26][27].  \nIn conclusion,  optical detection of electrical spin injection across a \nV[TCNE] x~2/AlGaAs interface has been demonstrated using an active hybrid \norganic/inorganic spin -LED device. We report circular polarization of the \nelectroluminescence that tracks the magnetization of  the V[TCNE] x~2 layer and exhibits \nHH/LH asymmetry. These studies validate the spintronic functionality of organic -based \nmagnets,  laying the foundation for a new class of multifunctional hybrid spintronic \nstructures and representing the first all -semicondu ctor spintronic device with the potential \nfor room temperature operation. Moreover, we establish the technique of using \nextensively studied inorganic heterostructures as a sensitive probe of free carrier spin \nphysics in organic and molecular systems.  \nWe ac knowledge Dr. Mark Brenner and Semiconductor Epitaxial  and Analysis \nLaboratory (SEAL) of the Ohio State University for providing AlGaAs/GaAs quantum \nwell LED detector. We also acknowledge discussions and interactions with J. A. Gupta, \nR. Myers and J. -W. Y oo. This work was supported by  the NSF MRSEC program (DMR -\n0820414) , OSU -Institute of Material Research and DOE – spin (DE -FG02 -01ER45931) . \n References  \n* These authors contributed equally to this work  \n†To whom correspondence should be addressed. E -mail: epstein@mps.ohio -\nstate.edu and ejh@mps.ohio -state.edu .  \n [1] International technology roadmap for semiconductors. Executive summ ary for \n2009.  (2009).  \n[2]       D. P. Divincenzo , Science 270, 255 (1995).  \n[3] J. M. Manriquez et al. , Science 252, 1415 (1991).  \n[4] V. N. Prigodin et al. , Adv. Mater. 14, 1230 (2002).  \n[5] J.-W. Yoo  et al. , Nat. Mater. 9, 638 (2010) . \n[6]        J. -W. Yoo  et al. , Phys. Rev. B. 80, 205207 (2009).  \n[7] J.-W. Yoo  et al. , Phys. Rev. Lett. 97, 247205 (2006).  \n[8]        K. Deniz Bozdag et al., Phys. Rev. B. 82, 094449 (2010)  \n[9] K. I. Pokhodnya, A. J. Epstein, and J. S. Miller , Adv. Mater. 12, 410 (2000).  \n[10]      Y. Ohno  et al. , Nature 402, 790 (1999).  \n[11] J. S. Miller  and A. J. Epstein , Angew. Chem. Int. Edit. 33, 385 (1994).  \n[12] A. Zheludev  et al. , J. Am. Chem. Soc. 116, 7243 (1994).  \n[13] C. Tengstedt  et al. , Phys. Rev. B. 69, 165208 (2004).  \n[14] C. Tengsted t et al. , Phys. Rev. Lett. 96, 057209 (2006).  \n[15] A. L. Tchougreeff  and R. Dronskowski , J. Comput. Chem. 29, 2220 (2008).  \n[16] H. Matsuura, K. Miyake , and  H. Fukuyama , J. Phys. Soc. Jpn. 79, 034712 (2010).  \n[17] G. C. De Fusco et al. , Phys. Rev. B. 79, 085201 (2009).  \n[18] J. B. Kortright et al. ,  Phys. Rev. Lett. 100, 257204 (2008).  \n[19] F. Meier and B. P. Zakharchenya, Optical Orientation  Amsterdam, The \nNetherlands: North -Holland, vol. 8. (1984).  \n[20] C. Adelmann et al., Phys. Rev. B. 71, 121301(R) (2005)  \n[21] B. Jonker , Proc. IEEE. 91, 727 (2003).  \n[22] H. Zhu et al. , Phys. Rev. Lett. 87, 016601 (2001).  \n[23]     X. Y, Dong et al. , Appl. Phys. Lett. 86, 102107 (2005).  \n[24] A. T. Hanbicki et al. , Appl. Phys. Lett. 80, 1240 (2002).  \n[25]     J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80, 4313 (1998)    \n[26]    The spin injection efficiency tracks with the bulk magnetization, and is distinct \nfrom materials such as LSMO [27] where surface relaxation leads to a rapid \ndepolarization of the spin current at elevated temperature.  \n[27]      J. -H. Park et al. , Phys. Rev. Lett. 81,1953 (1998)  \n \n \n \n \n \n \n FIG. 1 (color online). (a) Schematic energy diagram for V[TCNE] x~2. (b) Schematic band \ndiagram of the spin -LED. The study of the V[TCNE] x~2/n-AlGaAs interface through \nwhich spin -polarized electrons are injected is complicated by the large and nonlinear \nresistance from bulk V[TCNE] x~2, which is represented by the dashed box. (c) The \nexpe cted polarization signal for HH/LH transitions (right) for a given magnetization of \nthe spin injector (left). Due to quantum selection rules, HH and LH show opposite \npolarization behavior.  \n \nFIG. 2 (color online). (a) I–V curve s for both bare LED (triangle s) and V[TCNE] x~2 spin-\nLED (diamond s) devices at T = 60 K .  (b) Typical EL spectrum of a V[TCNE] x~2 spin-\nLED device ( I = 0.5 mA, V = +18.5 V) at T = 60 K. The shaded areas in the spectrum \nindicate the region of polarization integration over the quantum well  heavy -hole (HH) \nand light -hole (LH) peaks, respectively. Inset: A schematic of the V[TCNE] x~2 spin-LED \ndevice structure.  \n \nFIG. 3  (color online).  (a) - (c) Cartoon showing the evolution of the detected \nluminescence signal from three different sources. (a) Linear background from the \nintrinsic magnetic field response of the AlGaAs/GaAs quantum well  and dichroism from \nthe cryostat window s. (b) Linear background from (a) plus the magnetic circular \ndichorism ( MCD ) response from the magnetic layer (c) Expected HH  (red) and LH  (blue) \nEL from spin injection  and background s from ( a and b). (d) -(e)  Experimental results \ncorresponding to cartoons (a) – (c), EL from a non -magnetic device, MCD from photoluminescence and EL from a spin -LED, respectively. Filled symbols ar e for field \nsweeping up and open symbols are for field sweeping down. Note that a field -\nindependent offset  is subtracted from all scans. Error bars are determined using the \nstandard deviation of multiple measurements at each field value.  \n \nFIG. 4(color onli ne).Circular polarization for HH and LH transitions in a V[TCNE] x~2 \nspin-LED device at (a) T = 60 K, I = 0.5 mA and V = +18.5 V, (b) T = 60 K, I = 0.08 mA \nand V = +15.2 V and (c) T = 140 K, I = 1.5 mA and V = +10.1 V  (red and blue symbols \nfor HH and LH, re spectively) . The solid green line in (a) represents out of plane \nmagnetization of the V[TCNE] x~2 layer.  Error bars are determined as in Fig. 3 except for \n(b), where the long measurement time required for low bias  measurements prevents \nstatistically signifi cant averaging.  As a result, the error is estimated from the scatter of the \nLH data.  \n \n \n \n \n \n \n Fig. 1.  \n \n \n \n \n \n \n \n Fig. 2  \n \n \n \n \n \n \n \n \n Fig. 3.  \n \n \n \n \n \n \n \n \n \n Fig. 4.  \n \n "    },    {        "title": "1703.08482v2.Electronic_structure_and_direct_observation_of_ferrimagnetism_in_multiferroic_hexagonal_YbFeO3.pdf",        "content": "1 \n Electronic structure  and direct observation of \nferrimagnetism in multiferroic hexagonal YbFeO 3 \n \nShi Cao ,1 Kishan Sinha,1 Xin Zhang,1 Xiaozhe Zhang,1,2 Xiao Wang,3 Yuewei Yin,1 Alpha T N'Diaye ,4 Jian \nWang,5 David J Keavney ,6 Tula R Paudel,1 Yaohua Liu,7 Xuemei Cheng,3 Evgeny Y Tsymbal ,1,8 Peter A \nDowben,1,8 Xiaoshan Xu1,8 \n1Depa rtment of Physics and Astronomy , University of  Nebraska, Lincoln, NE 68588, USA  \n2Department of Physics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China  \n3Department of Physics, Bryn Mawr College, Bryn Mawr, PA 19010, USA  \n4Advanced Light  Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  \n5Canadian Light Source, Saskatoon, SK S7N 2V3, Canada  \n6Advanced Photon Source, Argonne National Labora tory, Argonne, Illinois 60439, USA  \n7Quantum Condensed Matter Division, Oak Ridge National Lab, Oak Ridge, TN 37831, USA  \n8Nebraska Center for Materials and Nanoscience, University of  Nebraska, Lincoln, NE 68588, USA  \nAbstract  \nThe magnetic interaction between rare-earth and Fe ions in hexagonal rare -earth ferrites (h-REFeO 3), may \namplif y the weak ferromagnetic mo ment on Fe, making these materials more appealing as multiferroics. \nTo elucidate the interaction strength between the rare-earth and Fe  ions as well as the magnetic moment of \nthe rare -earth ions , element specific magnetic characterization is needed . Using  X-ray magnetic circular \ndichroism, we have s tudied the ferrimagnetism in h-YbFeO 3 by measuring the magnetization of Fe and Yb \nseparately.  The result s directly show  anti-alignment of magnetization of Yb and Fe  ions in h-YbFeO 3 at \nlow temperature , with an  exchange field on Yb  of about 17 kOe. The magnetic moment of Yb is about  1.6 \nµB at low -tempe rature,  significantly reduced compared with the 4.5 µ B moment of a free Yb3+. In addition, \nthe saturation magnetization of Fe in h -YbFeO 3 has a sizable enhancement compared with that in h -LuFeO 3. \nThese findings  directly demonstrate  that ferrimagnetic order exists in h -YbFeO 3; they also account for the \nenhancement of magnetization and the reduction of coercivity  in h-YbFeO 3 compared with those in h-\nLuFeO 3 at low temperature , suggesting an important role for the rare -earth ions in tuning the multiferroic  \nproperties of h-REFeO 3. \n  2 \n I. Introduction  \nThe divers e magnetic properties  of rare-earth (RE) transition -metal (TM) oxides  owe to the interplay \nbetween the distinct magnetism of rare-earth  and transition -metal  ions. For the transition -metal  ions, t he \nmagnetic moments come from d electrons  which are well exposed to the local environment . In contrast,  for \nrare-earth ions, the magnetic moments come from 4f electrons  which are close to the inner core  and have \nsignificant contributions from both spin and orbital angular moment um.1 While the stronger interaction \nbetween the transition -metal  ions determines the framework of the magnetic order  in the RE -TM oxides2–\n4, the weaker  interaction between the rare-earth  and transition -metal  ions,  on the other hand, generates \ninteresting phenomena such as spin reorientations  and moment compen sation .5–11 Despite the importance \nof the RE-TM interaction, a comprehensive understanding of its underpinnings  and implications is still \nlacking for many material systems.  \nIn this work, we study the magnetic interaction  between the rare-earth  and transition -metal  ions by \nmeasuring the magnetizati on of the rare-earth and transition -metal  ions separately using a n element -specific \nmethod.  In particular, we study hexagonal YbFeO 3, a member  of hexagonal rare -earth ferrites ( h-REFeO 3, \nRE=Ho-Lu, Y, and Sc) . Hexagonal R EFeO 3 have a layered crystal structure in which both RE and Fe atoms \nadopt a two -dimensional triangular lattice , as shown in Figure  1.12 Below about  1000 K, h -REFeO 3 crystal \nstructure undergoes a distortion corresponding to a rotation of the FeO 5 local structure and a buckling of \nthe rare -earth layer, which induces an improper ferroelectrici ty.13–17 The rotation of the FeO 5 also cants the \nmoment on Fe , via the Dzyaloshinskii -Moriya  interaction, generating weak ferr omagnetism on top of a \n120-degree antiferromagnetic order  below about 1 20 K, as illustrated  in Fig ure 1. 18–20 The spontaneous \nmagnetization is along the c axis. Recent work demonstrated that a super -lattice structure of hexagonal Lu -\nFe-O materials are promising for realizing room temperature mult iferroic materials with co -existing \nferroelectric ity and ferromagnetism,21 a property that has potential application in energy -efficient \ninformation processing and storage22. \nIn h-YbFeO 3, the Fe -Fe interaction is expected to dominate  the framework of the magnetic ordering , as \ncorroborated by the fact that  the ordering temperature of h -YbFeO 3 is almost the same as that of h -LuFeO 3 \n(noting that Lu3+ is non -magnetic)16,17,23,24. The Yb-Fe interaction is  weaker but enough to partially align \nthe moment on Yb and contribute to the total magnetization . Indeed, an enhancement of magnetization  of \nh-YbFeO 3, compared with that in h -LuFeO 3, has been ob served previously23,24, to be up to about  3 µB/f.u. \nat 3 K , in contrast to 0.0 18 µB/f.u. in h-LuFeO 3.13,16 The Yb -Fe interaction could , in principle , align or anti -\nalign the moment s of Fe and Yb. At the compensation temperature3,5, the magnetization of Fe and Yb \ncancel s, and an  indication of this was observed  previously  at about 80 K24. On the other hand , direct \nobservation of anti -alignment between the Fe and Yb magnetization is still lacking. In addition, t he \npreviously reported large magnetization (about 3 µ B/f.u.)24 at low temperature is more consistent with  a free \nYb3+, but unexpected when considering the effect of t he crystal field generated by the local environment ,25–\n29 which  could significantly change  the effective magnetic moment and the magnetic anisotropy  at low \ntemperature .28,30  \nTo elucidate the Yb-Fe interaction and the magnetic moment of  Yb, we have studied the electronic structure \n \nFigure 1  (color online) The c rystal structure of h -YbFeO 3 \nand schematic of the magnetic structure. The arrows on the \natoms indicate the atomic magnetic moments. 𝑀Fe and 𝑀Yb \nare the magnetization of Fe and Yb along the c axis \nrespectively, which are anti -aligned at low temperature.  The \nFe moments form a 120 -degree antiferromag netic order in \nthe basal plane, with only a very small component along  the \nc axis. The Yb moments are partially aligned by the Yb -Fe \nexchange field.  \n 3 \n of h-YbFeO 3 using X -ray absorption spectroscopy (XAS) and the X -ray photoemission spectroscopy (XPS)  \nand measured the magnetization of Fe and Yb separately using  X-ray magnetic circular dichroism (XMCD) . \nWe have found a large exchange field (1 7 kOe) on Yb, while the magnetic moment of Yb is significantly \nreduced from the value of a free ion.  Mixed valence of  Yb was investigated and found only at the surface \nof samples grown in reducing environment, suggesting minimal effect on the magnetism of h -YbFeO 3. \nII. Methods  \nHexagonal YbFeO 3 (001) films ( 20-50 nm) were deposited on yttrium stabilized zirconia (YSZ) (111) \nsubstrates  and on Fe 3O4 (111)/Al 2O3 (001) substrates using pu lsed laser (248 nm) deposition  in 5 mtorr \noxygen and argon environment , at 750 °C with a laser fluence of about 1 J cm−2 and a repetition rate of 2 \nHz.13,14,31 All the films studied  with X-ray absorption spectr oscopy and X -ray magnetic circu lar dichroism \nwere grown in oxygen enviro nment.  The film growth was monitored using reflection high energy electron \ndiffraction (RHEED).  The crystal structures  of the h -YbFeO 3 films were characterized by X-ray diffraction \n(XRD ) using a Rigaku D/Max -B diffractometer, with the Co K -α radiation (1.7903 Å). The linear X-ray \nabsorption spectroscopy  on the Fe L edge and O K edge was studied using X-ray photoemission electron \nmicroscope (X -PEEM) at the SM beamline of the Canadian Light Source with linearly polarized X -ray. \nThe circular X-ray absorption (fluorescence ) spectroscopy  of Yb M edge and Fe L edge  measurements were \nperformed at the bend magnet beamline 6.3.1 in the Advanced Light Source at Lawren ce Berkeley National \nLaboratory  and at the beamline 4IDC in the Advanced Photon Source at Argonne National Laboratory  \nrespectively.  The angle -resolved X -ray photoemission spectra (ARXPS) were obtained using SPECS \nPHOIBOS 150 energy analyzer. A non -monochromatized Al Kα x -ray source, with ph oton energy 1486.6 \neV, was used with various emission angles, as previously reported .32 \nIII. Results and analysis  \nA. Crystal structure and local environment of  Fe  \nTo verify  the structure and phases of the epitaxial films, we  carried out X -ray diffraction , electron \ndiffraction, and  X-ray spectroscopy measurements. Figure 2 (a) shows the X -ray diffractio n (θ-2θ scan) of \nh-YbFeO 3/YSZ films. No additional peak  other than those expected for h -YbFeO 3 and the substrate is \nvisible in this large -range scan, indicating no impurity phases.  As shown in Fig ure 2(b), RHEED images \nshow  diffraction streaks consistent with a flat surface and the structure of h -REFeO 3.13,31 \nThe X-ray absorption spectra provided further confirmation of  the local structure of Fe , from  the Fe L edge \n \nFigure 2  (color online) (a) θ -2θ X -ray \ndiffraction measurement of a n h-YbFeO 3 film \ngrown on yttrium stabilized zirconia (YSZ). (b) \nRHEED patter ns of an h-YbFeO 3 film with \nelectron beam along the <1 -10> and <100> \ndirections.  \n 4 \n spectra taken with linearly polarized X -ray. The local environment of Fe in h -YbFeO 3 is a trigonal \nbipyramid, with two apex O atoms ( top and bottom ) and three equator O atoms (in the Fe layer)  as shown \nin Fig ure 1 as well as in Fig ure 3(a) inset.  This structur e makes the out -of-plane  direction (along the c axis) \nand the in -plane direction (in the a-b plane) two distinct crystalline direction s. Using linearly polarized X -\nray, we measured the absorption spectra at the Fe L edge , as illustrated in Fig ure 3(b). As shown in Fig ure \n3(a), the spectrum with s-polarized X -ray ( E vector in the a-b plane) and that with p-polarized X -ray ( E \nvector along the c axis) show obvious contrast , consistent with the large structural anisotropy. The spectra \nand linear dichroism in Fig ure 3(a) match  those observed previously for  h-LuFeO 3,20,21,33,34 confirming that \nthe local environment of the FeO 5 moiety in the two materials are almost identical.  \nB. The Electronic structure of  Yb  \nWhile the electronic structur e of Fe in h -LuFeO 3 and h -YbFeO 3 are superficially similar, the electronic \nstructure of Yb3+ is expected to be different from that of Lu3+ by one less 4f electron. To probe the \nunoccupied states o f Yb, we measured the excitation of electrons from O 1s stat es to O 2p states (O K edge) \nusing X -ray. Nominally, O 2p states are fully occupied; the O 1s to O 2p excitation is forbidden by the Pauli \nexclusion principle. If, on the other hand, the O 2p states are hybridized with the Yb states, the O 2p states \nwill b e slightly unoccupied and give rise to observable O 1s to O 2p excitation; one can infer the energy of \nthe unoccupied Yb states using the excitation energies.20 As shown in Figure  4(a), with linearly polarized \nX-rays, several features can be observed in the absorption spectra. Previously, we carried out symmetry \nanalysis of the absorption spectra measured on h -LuFeO 3 and identified the origin of these features mainly \nas the 5d orbitals split in the crystal field: 𝑒𝜋, 𝑎1 and 𝑒𝜎 [see Figure  4(b)]20. Compared with the X -ray \nabsorption spectra of h -LuFeO 3, the spectra of h -YbFeO 3 show additional density of state s, as indicated in \nFigure 4(a), which is expected to be the unoccupied 4f state that is hybridized with the O 2p states.  \nThe 4f13 configuration  of Yb can also be probed by measuring the excitation  directly  to the  unoccupied  4f \nstates  (in the absence s -f hybridization, none exist with Lu3+). As shown in Figure  5(a), X -ray absorption \nspectra  at the Yb M edge  were measured at 18 K . Two peaks are observed in the absorption spectra at \napproximately  1513 and 1555 eV, which can be assigned to M 5 (initial state 3d 5/2) and M 4 (initial state 3d 3/2) \n \nFigure 3  (color online) (a) X -ray absorption \nspectra at the  Fe L edge measured using \nlinearly polarized X -ray. Inset: the FeO 5 \nlocal environment. (b) Schematic illustration \nof the L 2 and L 3 excitation.  \n \n  \nFigure 4  (color online) (a) X -ray \nabsorption spectra at the  O K edge of h -\nLuFeO 3 and h -YbFeO 3 measured using \nlinearly polarized X -ray. The arrow \nindicates the 4f state. (b) Schematic \nillustration of the O K edge excitation \nand the hybridization between the O and \nYb states.  \n 5 \n excitations respectively according to the photon energy35 [see Fig ure 5(b)] . The M 5 transition in  Yb, which \nis allowed by the angular -momentum selection rule,  can be described using the one-electron (hole) picture , \nwithout many -body interactions , due to the simple initial (full 3d 5/2, one hole in 4f 7/2) and final (one hole in \n3d5/2, full 4f 7/2) states, consistent with the observed  sharp, structureless  peak in Figure 5(a). The M 4 \nexcitation  (3d 3/2 to 4f 7/2), on the other hand , is not allowed by the angular -momentum selection rule. The \nnon-zero intensity of the M4 peak suggests that the crystal -field splitting and the Yb 4f -O 2p hybridization \nreduces the symmetry of the electronic states considerab ly, which is in line  with the observed contribution \nto the O K edge excitation by the Yb 4f state shown  in Fig ure 4(a). \nC. The Ferrimagnetism of h-YbFeO 3  \n1. Magnetization  of Yb and Fe   \nTo study the magnetization of Yb, we carried out X -ray magnetic circular dichroism measurements, by \ncomparing the absorption spectra using circular ly polarized X -ray in opposite  magnetic field s. As shown in \nFigure 5(a), the X -ray absorption spectra measured in 19 kOe and -19 kOe magnetic field s along the z \ndirection show a clear contrast . We define the XMCD contrast as 𝐼+−𝐼−\n(𝐼++𝐼−)/2, where 𝐼+ and 𝐼− are the M5 peak \nareas of the absorption spectra  in positive and negative magnetic fields respectively .  \nThe XMCD contrast measured at H = 19 kOe , for various temperatures  between 6.5 and 80 K, is displayed \nin Figure 6(a). The value of the XMCD signal decreases rapidly at low  temperature, inconsistent with \ntypical ferromagnetic dependence , which typically follows the Bloch’s law ( decrease slowly at low \ntemperature  but much faster close to the magnetic ordering temperature )36. Figure 6(b) shows the field \ndependence of the XMCD contrast of Yb at 18 K. A clear hysteresis is observed with a coercive field of \napproximately 3.5 kOe. The magnetization converted from the XMCD contrast (See Appendix A) is also \ndisplayed in Fig ure 6.  \n \nFigure 5  (color online) (a) X -ray \nabsorption spectra at the  Yb M edge \nmeasured using X -ray polarized \ncounter clockwise. XAS+ (XAS-) is the \nspectrum measured in magnetic field along \nthe + z (-z) direction.  (b) Schematic \nillustration of the Yb M edge excitation.  \nThe crystalline c axis of h-YbFeO 3 is along \nthe z direction.  \n \n \nFigure 6  (color online) XMCD contrast of Yb M 5 edge \nand the corresponding magnetization.  (a) Temperature \ndependence measured  in a 19 kOe magnetic field; the \nline is calculated using the parameters analyzed from \n(b). Inset: 𝐻𝑌𝑏 extracted  from the mean -field theory \n(see text in Section IV.B). (b) Magnetic field \ndependence measured at 18 K.  The magnetic field is \nalong t he c axis.  \n 6 \n Figure 7(a) shows the spectra of X -ray absorption of Fe L edge measured in circularly polarized X -ray in a \n10 kOe magnetic field at 6.5 K . A clear difference is observed between the spectra measured using  X-rays \nof different  polarization s, which can be used to estimate the magnetization of Fe .37 Figure 7(b) shows the \nmagnetic -field dependen ce of the Fe magnetization  calculate d from the XMCD contrast using the sum \nrule37–39. A hysteretic behavior is observed, with a coercive field of approximately 4 kOe, consistent with \nthe value found in  previous bulk magnetometry measurements .23,24 This coercive fields is also similar to  \nthat of Yb in Figure  6(b), indicative of the exchange field on Yb generated by Fe.  The saturation \nmagnetization of Fe is 0.05 +/ - 0.01 µB/f.u., which corresponds to a small projection of the Fe moment \nalong  the c axis. From  Figure 6 and 7 , we find that the magnetization of Fe is anti-parallel to  the magnetic \nfield and to that of the Yb magnetization  at low temperature, as also illustrated in Figure 1.  This provides  a \ndirect observation of ferrimagnetic order in h -YbFeO 3. \n2. The L ow temperature magnetic moment of Yb  \nAs shown in Fig ure 6(b),  the magnetization of Yb does not saturate in the measurement condition; instead, \nit shows a linear relation with magnetic field when the field is much larger than the coerci ve field , which is \nconsistent with a susceptibility behavior  and somewhat akin to paramagnetism for Yb. We can, nonetheless, \nfurther analyze  the magnetic moment  on Yb using  the mean -field theory36, which has been extensively \ndiscussed historically in orthorferrites and garnets10,11,40 –43. \nIn the mean -field theory , the exchange interactions are modeled using the molecular fields.  Assuming that \nthe saturation magne tization of Fe is 𝑀𝐹𝑒,𝑆 (in µB/f.u.), the magnetization of Fe is given by:  \n𝑀𝐹𝑒 =𝑀𝐹𝑒,𝑆𝐿(𝑥𝐹𝑒),     (1) \nwhere 𝐿(𝑥)=coth (𝑥)−1\n𝑥 is the Langevin function, 𝑥𝐹𝑒=(𝛾𝑌𝑏𝐹𝑒 𝑀𝑌𝑏+𝛾𝐹𝑒𝑀𝐹𝑒 +𝜇0𝐻)𝑀𝐹𝑒,𝑆\n𝑘𝐵𝑇, 𝑀𝑌𝑏 is the \nmagnetization of Yb, 𝛾𝑌𝑏𝐹𝑒 and 𝛾𝐹𝑒 are the molecular field parameters for the Yb -Fe and Fe -Fe interactions \nrespectively, 𝜇0 is the vacuum permittivity, 𝑘𝐵 is the Boltzmann constant , H is external magnetic field, and \nT is temperature . The magnetization  of Yb  is given by:  \n𝑀𝑌𝑏 =𝜇𝑌𝑏𝐿(𝑥𝑌𝑏),     (2) \nwhere 𝑥𝑌𝑏=(𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒+𝜇0𝐻)𝜇𝑌𝑏\n𝑘𝐵𝑇 and 𝜇𝑌𝑏 is the magnetic moment of Yb.  No Yb -Yb exchange interaction is \nincluded since such exchange interactions are  too weak to play a role in the temperature range investigated \nin this work.3,5 \nWhen the magnetic field is much  larger than the coer cive field and the temperature is much lower than the \n \nFigure 7 (color online) (a) Absorption spectra \nof Fe L edge measured with circularly \npolarized X -ray in a 10 kOe field at 6.5 K.  CW \nand CCW stand for clockwise and \ncounterclockwise polarization of the X -ray \nrespectively.  (b) Magnetic -field dependence of \nthe magnet ization of Fe  at 6.5 K, which \ncontains a soft and a hard component (see \ndiscussion in Section IV .D). The magnetic field \nis along the c axis.  \n 7 \n magnetic ordering temperature for the Fe (  120 K for h -YbFeO 3)24, one may treat  |𝑀𝐹𝑒|≈𝑀𝐹𝑒,𝑆 as a \nconstant. As shown in Figure  6(b), at T = 18 K , when H is between 6 and 19 kOe, the XMCD contrast \nshows a linear dependence wi th magnetic field, suggesting  that 𝑥𝑌𝑏 is small enough that  the Langevin \nfunction takes a linear form with respect to the magnetic field H: \n𝑀𝑌𝑏=𝜇𝑌𝑏2(𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒+𝜇0𝐻)\n3𝑘𝐵𝑇       (3). \nAccording to Eq. (3), the slope of the field dependence  of 𝑀𝑌𝑏 (susceptibility) is 𝜒𝑌𝑏=𝑑𝑀𝑌𝑏\n𝑑𝐻=𝜇𝑌𝑏2𝜇0\n3𝑘𝐵𝑇, \nwhich leads to  𝜇𝑌𝑏 = 1.6 +/- 0.1 µB, a value much smaller than the magnetic moment  of a free Yb  (4.5 \nµB/f.u.)44. \n3. Exchange field  on Yb  \nAccording to Eq. (3) , the remanent magnetization ( magnetization in zero H) is expected to be  \n𝑀𝑌𝑏,𝑅=𝜇𝑌𝑏2𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒\n3𝑘𝐵𝑇    (4). \nBecause  𝑀𝐹𝑒 and 𝑀𝑌𝑏 have different sign in zero H [see Figure 6(b) and Figure 7(b)], one finds  𝛾𝑌𝑏𝐹𝑒 < 0 \nfrom Eq. (4).  \nUsing the value 𝑀𝑌𝑏,𝑅 = 0.057 µB/f.u. at 18 K from Fig ure 6(b), one can calculate  the exchange field on Yb : \n𝐻𝑌𝑏=𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒\n𝜇0 = 17  kOe.  We also note that the exchange field on Yb generated by Fe in h -YbFeO 3 is \nabout an order of magnitude larger than  the value  1.6 kOe in orthorhombic YbFeO 3 and that in rare-earth \northoferrites in general3. This large difference may come from the dramatic differences between the bond \nlengths and bond angles in the hexagonal and orthorhombic YbFeO 3 structures.  \nD. The P ossible  mixed valence of Yb  \nMixed valence (Yb3+ and Yb2+) may play a role in  the magnetism of h -YbFeO 3 as well as the determination \nof the m agnetization on the Yb3+. In principle, there is a tendency to form Yb2+ due to the stability of the \n4f14 configuration. Although it will not affect the XMCD method discussed above  since Yb2+ does not \ncontribute to the Yb M 5 X-ray absorption in the first place (the excitations to the fully occupied 4f states \nare forbi dden in Yb2+), it will  be important for bulk magnetometry. We investigate d the possibility of mixed \nvalence in h -YbFeO 3 using ARXPS, by probing the core level electronic structure . \nFigure 8(a) shows  the Fe 2p X-ray photoemission spectra  for both h -LuFeO 3 and h -YbFeO 3. The good \n \nFigure 8  (Color online) (a) The X-ray \nphotoelectron spectra around Fe 2p edge of \nh-YbFeO 3 and h-LuFeO 3. (b) The X-ray \nphotoelectron spectra around Yb  5p edge of \nh-YbFeO 3 film samples grown in Ar and O 2 \nenvironment measured  at 0o and 70o take-off \nangle , corresponding to 2 nm and 0.7 nm \nprobing depth  respectively47,48. 8 \n match between the  Fe 2p 3/2 peaks  of h-LuFeO 3 and h -YbFeO 3 in Figure 8(a) indicates that Fe core level \nelectronic structure are similar in these  two ferrites . Previously, we have studied the X-ray photoemission \nspectra of Fe 2p using the Gupta and Sen (GS) multiplet fitting45,46 of Fe 2p 3/2 in h-LuFeO 3 and concluded \nthe Fe 2p and its satellite peaks are characteristic of a nominal Fe3+ valance32. The same analysis  applies \nhere in h-YbFeO 3 as well. These features also do not vary with emissio n angle  (data not shown). As a result , \nboth the surface and the bulk part of the h -YbFeO 3 are in the nominal Fe3+ valance state.   \nWe also did not find indication of Yb2+ in the film samples grown in oxygen environment (used for X-ray \nabsorption spectroscopy and X -ray magnetic circular dichroism in Figure 2 to Figure 7 ). To investigate the \npossible appearance of Yb2+, we studied ARXPS  on the h -YbFeO 3 films prepared in argon environment  A \ncomparison with two samples grown in oxygen and argon environments is displayed  in Fig ure 8(b). At the \nzero-degree  take-off angle  (perpendicular  to surface ), the XPS  spectra  of Yb are identical for both h-\nYbFeO 3 samp les. At the 70˚ take -off angle , which probes mostly the surface47,48, the XPS spectra of the \nsample grown in oxygen environment (lower panel) do not show clear  difference from that at zero degree , \nalso the surface appears to be slightly Yb rich . In contrast, for the sample grown in the argon environment, \nthe XPS spectra at the 70˚ take -off angle  exhibit additional intensity at the 5p peak, indicating a Yb2+ \nvalence49 at the surface  The correlation between the growth condition s indicates that the pr esence  of oxygen \nvacancy promotes the reduction of Yb3+. Although slightly YbO rich, t he mixed surface termination (both \niron oxide  and YbO  appear present at the surface) differs from the Fe–O termination seen for LuFeO 3.32 \nIV. Discussion  \nA. Origin of r educed moment of Yb  \nThe low -temperature magnetic moment of Yb is found to be 1.6 µB, a value significantly smaller than 4.5 \nµB for a free Yb 44. In h-YbFeO 3, Yb is surrounded by 7 oxygen atoms, a pproximately corresponding to a \nC3v symmetry. Analysis using double groups indicates that the 4f 7/2 states are split by the crystal field into \n4 levels: 3 𝐸1\n2 + 𝐸3\n2  (see Appendix B), where 𝐸1\n2 and 𝐸3\n2  are both  two dimensional44. The energy scale of the \ncrystal -field splitting is typically a few meV to a few tens meV,26–28 which cannot be resolved in the XAS \nspectra. This crystal field  splitting  means that , at low temperature , only the low -lying level (ground state) \nis populated and contribute s to the magnetization . The occupation of the  low-lying level , in turn, leads t o \nthe reduced value of 𝜇𝑌𝑏, and is the reason for the temperature -dependent magnetic momen ts and magnetic \nanisotropy observed previously in rare -earth -containing oxides28,30. \nB. Possible spin  reorientation  and magnetization compensation  \nOne can calculate the temperature dependence of Yb magnetization using Eq. (2). As shown in Fig ure 6(a), \nthe resu lt (with 𝜇𝑌𝑏=1.6 µ B, H=19 kOe,  and 𝐻𝑌𝑏= 17 kOe ) is compared with the measured values. The \nmeasured and the calculated magnetization  match well below  70 K, suggesting that mean -field theory can \ndescribe the temperature dependence of 𝑀𝑌𝑏 too.. The fact that the mean -field theory  can describe  both the \nmagnetic -field (Section III  C 2) and temperature dependence of 𝑀𝑌𝑏, indicates its validity in analyzing the \nmagnetic properties of h -YbFeO 3. \nOn the other hand,  at about 80  K, the calculated value is much larger than the measured value , suggesting \na reduction  of 𝐻𝑌𝑏 at higher temperature.  To reveal the temperature dependence of 𝐻𝑌𝑏, we calculate d 𝐻𝑌𝑏 \nfrom the measured magnetization value using Eq. (3) ( with 𝜇𝑌𝑏=1.6 µ B, H=19 kOe) ; the result is displayed \nin Fig.6(a) inset . Clearly, a sign change of 𝐻𝑌𝑏 occurs at about 80 K, indicating a possible realignment  \nbetween  the magnetization  𝑀𝑌𝑏 and 𝑀𝐹𝑒, which is discussed below.  \nIn principle, the alignment between 𝑀𝑌𝑏 and 𝑀𝐹𝑒 is determined by the minimization of total energy \n𝐸𝑡𝑜𝑡𝑎𝑙 =−1\n2𝜒𝑌𝑏(𝐻+𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒)2−𝑀𝐹𝑒𝐻, or the maximization of the total magnetization 𝑀𝑡𝑜𝑡𝑎𝑙 =\n𝑀𝐹𝑒(1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒 )+𝜒𝑌𝑏𝐻.11 Here the external field H is along the c axis and  𝑀𝐹𝑒 may point either along \nor opposite to H, corresponding to  the positive and negative  signs respectively.  9 \n Because  𝛾𝑌𝑏𝐹𝑒<0 and 𝜒𝑌𝑏=𝜇𝑌𝑏2𝜇0\n3𝑘𝐵𝑇 (see section  III C  3), the sign of 1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒 is expected to change \nwith temperature, possibly causing the reversal of  the direction of the magnetization 𝑀𝐹𝑒:  \n(1) At low temperature, 1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒 <0. In this case , 𝑀𝐹𝑒< 0 ( 𝑀𝐹𝑒 anti-parallel to H) is more \nfavorable  for maximizing 𝑀𝑡𝑜𝑡𝑎𝑙. this mean the exchange field 𝐻𝑌𝑏=𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒\n𝜇0>0 (parallel to the \nexternal field ).  \n(2) When temperature is increased and 1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒  > 0 is satisfied,  𝑀𝐹𝑒> 0 ( 𝑀𝐹𝑒 parallel to H) is \nmore favorable . In this case,  the exchange field 𝐻𝑌𝑏=𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒\n𝜇0<0 (anti parallel to the external \nfield);this could be the reason that at about 80  K 𝐻𝑌𝑏 becomes  negative  [Figure 6(a) inset ]. .  \n(3) At the compensation temperature, 1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒  = 0. Therefore, 𝑀𝑡𝑜𝑡𝑎𝑙 =𝜒𝑌𝑏𝐻, as if 𝑀𝐹𝑒 is \n“screened” by the part of Yb moment induced by the exchange field 𝐻𝑌𝑏. The magnetization \ncompensation can be understood as the cancellation of 𝑀𝐹𝑒 and 𝑀𝑌𝑏 at zero field . According to \nFig. 6(a) inset, the compensation temperature appears to be between 70 and 80 K, in fair agreement \nwith the previous estimation24. \nNonetheless, the magnetization of the Yb is largely a spectator to that of the Fe. The coercivity is the same \nas that observed for iron, with the essential observation [Figure 6(b)]  that the magnetization does not easily \nsaturate indicating that much of the magnetization depends on the magnetic susceptibility and possible \nalignment of the moments with external magnetic field H and with that of Fe (Figure 7).  \nC. Exchange field on Fe  \nThe exchange field may also have an effect on the Fe, which can be understood by combining Eq. (1) and \nEq. (2) to reach  𝑥𝐹𝑒=[𝛾𝑌𝑏𝐹𝑒 𝜇𝑌𝑏𝐿(𝛾𝑌𝑏𝐹𝑒𝑀𝐹𝑒𝜇𝑌𝑏\n𝑘𝐵𝑇)+𝛾𝐹𝑒𝑀𝐹𝑒]𝑀𝐹𝑒,𝑆\n𝑘𝐵𝑇, assuming H = 0. Since Fe moments in h -\nYbFeO 3 form weak ferromagnetic order, 𝛾𝐹𝑒 must be positive. Because of the properties of the Langevin \nfunction 𝐿(𝑥), 𝛾𝑌𝑏𝐹𝑒 𝜇𝑌𝑏𝐿(𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒𝜇𝑌𝑏\n𝑘𝐵𝑇) is always positive regardless of the sign of 𝛾𝑌𝑏𝐹𝑒. Therefore, the \nYb always enhance s the m olecular field on the Fe. That said, because  in general 𝛾𝐹𝑒≫|𝛾𝑌𝑏𝐹𝑒 |, the effect \nmay not be significant.   \nD. Comparison between magnetic properties of h-YbFeO 3 and h -LuFeO 3 \nHexagonal LuFeO 3 (h-LuFeO 3) is the most studied hexagonal rare -earth ferrites. Because Lu3+ is non -\nmagnetic, the magnetic properties of h -LuFeO 3 is less complex. By comparing h -LuFeO 3 and h -YbFeO 3, \none may gain insight on the effect of the rare earth  on the magnetism . \nOne dr amatic difference between h -YbFeO 3 and h -LuFeO 3 is in the co ercive field of magnetization. For h -\nYbFeO 3 at 18 K, the coercive field is about 4 kOe, which is much smaller than the value 25 kOe for h -\nLuFeO 3.16 For both h -LuFeO 3 and h -YbFeO 3, the magnetization -field loop s for Fe  have  a squared shape, \nsuggesting that t he magnetic coercive field is determined by the competition between the magnetic \nanisotropy energy and the Zeeman energy . Compared with h -LuFeO 3, h-YbFeO 3 has enhanced \nmagnetization due to the contribution of Yb. Therefore, a much smaller magnetic field is needed in h-\nYbFeO 3 to overcome the magnetic anisotropy, corresponding to a much smaller coerciv e field . \nAnother  differen ce between h -YbFeO 3 and h -LuFeO 3 is in the saturation magnetization of Fe. According \nto Fig ure 7, in h-YbFeO 3, 𝑀𝐹𝑒,𝑆 = 0.05 +/ - 0.01 µB/f.u., larger than that in h -LuFeO 3 ( 0.03 µB/f.u.)16. We \nnote that previously it was observed in h -LuFeO 3 that the magnetization contains a soft component and a \nhard component, in which only the  hard component  (0.018 µB/f.u.) is bel ieved to be  intrinsic to the weak \nferromagnetic ordering  because it disappears above the magnetic ordering temperature .16 In Fig ure 7, there \nis also one soft (coercive field  1 kOe) and one hard component (coercive field  4 kOe). If we only treat \nthe hard component to be intrinsic to the  canting of the Fe moment , the weak ferromagnetic  moment of Fe \nin h-YbFeO 3 is = 0.03 +/ - 0.01 µB/Fe according to Fig. 7(b) , still larger compared with  the value  0.018 10 \n µB/f.u. in h -LuFeO 3.16 Due to  the size difference of Lu3+ and Yb3+,14 the lattice constant  of the basal plane  \nof h-LuFeO 3 is smaller than th at of h-YbFeO 3: a = 5.963 Å  for h -LuFeO 3 and a = 6.021 Å  for h -YbFeO 3.31 \nOur recent work suggests that a compressive biaxial strain may reduce the canting of the Fe moments in h -\nREFeO 3,50 which is in line wi th the correlation between the lattice constant and weak ferromagnetic moment \non Fe observed here . \nV. Conclusion  \nWe have studied the electronic structure and magnetic ordering of  h-YbFeO 3 (001) thin films on YSZ (111) \nand on Fe3O4(111)/Al 2O3(001) substrates.  The magnetism of Yb in h -YbFeO 3 was studie d using the \nelement -specific  method X-ray magnetic circular  dichroism  based on X -ray absorption spectroscopy . From \nthe temperature and magnetic -field dependence of the  Yb magnetization, we found that the low temperature \nYb magnetic moment is significantly reduced compared with the value of free  Yb3+ ions, indicating the \neffect of crystal field.  The e xchange field on Yb , generated by the Fe moments, tends to anti -align the \nmagnetization of Fe and Yb  at low temperature.  We also investigated possible  valence mixing of  Yb and \nonly found indication of Yb2+ at the surface of samples grown in an Ar environment, suggesting an \ninsignificant e ffect to the magnetism of h -YbFeO 3. We expect  that future work, such as  optical spectroscopy \non probing Yb crystal field levels  and theoretical calculation s on Yb -Fe interaction strength , may provide \nmore insight on the ferrimagnetism of  h-YbFeO 3. \nAcknowledgements  \nThis project was primarily supported by the National Science Foundation through the Nebraska Materials \nResearch Science and Engineering Center (Grant No. DMR -1420645). Additional support was provided by \nthe Semiconductor Research Corporation through the Ce nter for Nanoferroic Devices and the SRC -NRI \nCenter under Task ID 2398.001. Work of Bryn Mawr College is supported by National Science Foundation \nCareer Award (Grant No. NSF DMR -1053854 ). This research used resources of the Advanced Photon \nSource, a U.S. D epartment of Energy (DOE) Office of Science User Facility operated for the DOE Office \nof Science by Argonne National Laboratory under Contract No. DE -AC02 -06CH11357 . Use of the \nAdvanced Light Source was supported by the U.S. Department of Energy, Office of  Science, Office of \nBasic Energy Sciences  under contract no. DE -AC02 -05CH11231 . The Canadian Light Source is funded by \nthe Canada Foundation for Innovation, the Natural Sciences and Engineering Research Council of Canada, \nthe National Research Council Canada, the Canadian Institutes of Health Research, the Government of \nSaskatchewan , Western Economic Diversification Canada, and the University of Saskatchewan.  The \nresearch was performed in part in the Nebraska Nanoscale Facility: National Nanotechnology Coordinated \nInfrastructure and the Nebraska Center for Materials and Nanoscience, which are supported by the National \nScience Foundation under Award ECCS: 1542182, and the Nebraska Research Initiative.  \n  11 \n Appendix  \nA. Converting XMCD contrast to magnetization of Yb  \nThe ca lculation of the magnetization of Yb from the  XMCD contrast  at the  M edge  is different from that of \n3d metals (e.g. Fe, Co, Ni)  at the  L edge , due to the strong spin -orbit coupling  in both initial and final states . \nWe hereby present a method based on  the XMCD contrast of  excitation s from the 3d 5/2 to the individual \n4f7/2 eigen states  Jz=-7/2 to 7/2, where  Jz is the projection of total angular moment J on the z axis; all possible \n4f7/2 states are superposition of these states.   \nExcited by an X -ray polarized clockwise, the transition from one 3d 5/2 state to one 4f 7/2 state needs to satisfy \nΔJz = 1. One can calculate the transition probabilities 𝑃 between individual states; the non -zero results are \ndisplayed in Fig ure 9(a). The projection of the magnetic moment of a Jz state on the z direction is µ z = gµBJz, \nwhere g = 1.14 is the Lande g-factor and µ B is the Bohr magneton.  Therefore,  one can calculate the XMCD \ncontrast, defined as 2(𝑃𝐽𝑧−𝑃−𝐽𝑧)\n𝑃𝐽𝑧+𝑃−𝐽𝑧 , with respect to µ z, where 𝑃𝐽𝑧 (𝑃−𝐽𝑧) is the transition probability for the final \nstate represented by Jz (-Jz); the result is  shown in Fig ure 9(b). Although f or large µz, the XMCD contrast \ndoes not  distinguish the | Jz|=7/2 and | Jz|=5/2 states,  for small µz, the relation between XMCD contrast and \nµz is approximately  linear . The measured XMCD contrast in this work falls in the small µz region (all values \nare less than 0.4). Therefore, we can use the relation in Fig. 9(b) to convert XMCD contrast to magnetization  \nas a fair  approximation.  \nB. Group  theory analysis of the crystal field splitting of Yb  states  \nIn h-YbFeO 3, the local environment of Yb has a symmetry that can be described using point group C3v [see \nFigure  9(b) inset].  The degenerate electronic state s in general are split according to the symmetry of the \nlocal environment. Because of the strong spin -orbit coupling, the angular momentum of the 4f state s takes \nhalf-integer 𝐽=5\n2 or 𝐽=7\n2, the analysis of which requires the double group.  TABLE 1  shows  the character \ntable for C 3v double group,  including irreducible representations 𝐴1, 𝐴2, 𝐸, 𝐸1\n2, 𝐸3\n2  (𝐸3\n2+ and 𝐸3\n2−). The \ncharacter of the representation with angular momentum 𝐽=5\n2 and 𝐽=7\n2 are also listed. Using these \ncharacters, one can reduce the 𝐽=5\n2 and 𝐽=7\n2 representations. The results are: 𝐽=5\n2→ 2𝐸1\n2+𝐸3\n2  and 𝐽=\n7\n2→ 3𝐸1\n2+𝐸3\n2 . \n \nFigure 9 (color online) (a) Transition \nprobability between individual 3𝑑5\n2 and \n4𝑓7\n2 states excited by a clockwise polarized \nX-ray. (b) XMCD contrast as a function of \n𝜇𝑧 (see text) calculated assuming a free \nYb3+ ion. 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B 95, 94110 (2017).  \n \n "    },    {        "title": "1704.07537v2.A_variety_of_elastic_anomalies_in_orbital_active_nearly_itinerant_cobalt_vanadate_spinel.pdf",        "content": "arXiv:1704.07537v2  [cond-mat.str-el]  18 Jul 2017Variety of elastic anomalies in an orbital-active nearly it inerant cobalt vanadate spinel\nTadataka Watanabe1,∗Shogo Yamada1, Rui Koborinai2, and Takuro Katsufuji2,3\n1Department of Physics, College of Science and Technology (C ST), Nihon University, Tokyo 101-8308, Japan\n2Department of Physics, Waseda University, Tokyo 169-8555, Japan and\n3Kagami Memorial Research Institute for Materials Science a nd Technology, Waseda University, Tokyo 169-0051, Japan\n(Dated: July 25, 2018)\nWe perform ultrasound velocity measurements on a single cry stal of nearly-metallic spinel\nCo1.21V1.79O4which exhibits a ferrimagnetic phase transition at TC∼165 K. The experiments\nreveal a variety of elastic anomalies in not only the paramag netic phase above TCbut also the\nferrimagnetic phase below TC, which should be driven by the nearly-itinerant character o f the\norbitally-degenerate V 3 delectrons. In the paramagnetic phase above TC, the elastic moduli ex-\nhibit elastic-mode-dependent unusual temperature variat ions, suggesting the existence of a dynamic\nspin-cluster state. Furthermore, above TC, the sensitive magnetic-field response of the elastic mod-\nuli suggests that, with the negative magnetoresistance, th e magnetic-field-enhanced nearly-itinerant\ncharacter of the V 3 delectrons emerges from the spin-cluster state. This should be triggered by\nthe inter-V-site interactions acting on the orbitally-deg enerate 3 delectrons. In the ferrimagnetic\nphase below TC, the elastic moduli exhibit distinct anomalies at T1∼95 K and T2∼50 K, with a\nsign change of the magnetoresistance at T1(positive below T1) and an enhancement of the positive\nmagnetoresistance below T2, respectively. These observations below TCsuggest the successive oc-\ncurrence of an orbital glassy order at T1and a structural phase transition at T2, where the rather\nlocalized character of the V 3 delectrons evolves below T1and is further enhanced below T2.\nPACS numbers: 62.20.de, 72.55.+s, 75.25.Dk, 75.50Gg\nI. INTRODUCTION\nVanadate spinels AV2O4with divalent A2+ions have\nattracted considerable attention owing to the interplay\nbetween the orbital degree of freedom and geometrical\nfrustration [1]. The trivalent magnetic V3+ions are char-\nacterized by double occupancy of the triply-degenerate\nt2gorbitals ( t2\n2g), and form a sublattice of corner-sharing\ntetrahedra (pyrochlore lattice). Upon cooling, AV2O4,\nwith nonmagnetic A= Zn, Mg, and Cd, undergoes a\nstructural phase transition followed by an antiferromag-\nnetic phase transition [2–10]. With magnetic A= Mn\nand Fe, owing to the presence of additional A2+-V3+\nexchange interactions, AV2O4exhibits a more complex\nstructural and magnetic behavior, with successive struc-\ntural and ferrimagnetic phase transitions [11–24].\nForAV2O4(A= Zn, Mg, Cd, Mn, and Fe), the struc-\ntural phase transition is considered to arise from a long-\nrangeorderingofthe V t2gorbitals, wherethe loweringof\nthe lattice symmetry should result in the release of frus-\ntration (magnetic ordering). The orbital order to explain\nthe structural and magnetic orders for AV2O4is still un-\nder debate both theoretically [25–31] and experimentally\n[4,7,12,15]; it is considered to be driven by the competition\namong Jahn-Teller coupling, Kugel-Khomskii exchange\ninteraction [32], and relativistic spin-orbit coupling.\nCoV2O4(Co2+:e4t3\n2, V3+:t2\n2g) is a unique vanadate\nspinel with its highest electrical conductivity among the\ninsulatingorsemiconducting AV2O4, whichisconsidered\nto arise from the nearly-itinerant character of the V 3 d\nelectrons [33–39]. CoV 2O4exhibits a ferrimagnetic order\nbelowTC∼150 K, which is similar to AV2O4with mag-\nneticA= Mn (TC= 57 K) and Fe ( TC= 110 K). How-\never, while AV2O4(A= Zn, Mg, Cd, Mn, and Fe) lowersthe cubic lattice symmetry by the structural phase tran-\nsition, CoV 2O4is unique in that the loweringofthe cubic\nlattice symmetry is absent down to low temperature. Al-\nthoughaveryrecentneutron scatteringstudy ofCoV 2O4\ndiscoveredavery-weakstructuralphasetransitionat ∼90\nK within the ferrimagnetic phase, this structural transi-\ntion was characterized as a short-range distortion of the\nO octahedra, which does not lower the global cubic sym-\nmetry of the crystal [40]. The very small magnitude of\nthe structural distortion in CoV 2O4is considered to be\nrelevant to the nearly-itinerant electron character.\nFor CoV 2O4, the previous reports of the experimen-\ntal studies in the single-crystalline and polycrystalline\nsamples suggested the presence of additional transition\nwithin the ferrimagnetic phase [34,35,38,39]. However, the\nreported transition temperatures, also including the fer-\nrimagnetic transition temperature TC, differ from study\nto study. This variation in the transition temperature\nvalues seems to arise from the off-stoichiometry of the\nsamples. In the crystal growth of the cobalt vanadate\nspinel, the Co : V ratio of the grown crystal is prone to\ndeviate from the stoichiometric ratio of 1 : 2 to 1+ x:\n2-xwith excess Co x. Very recently, the measurements\nof the magnetization, magnetostriction, neutron scatter-\ning, and dielectric constant for the single-crystalline and\npolycrystalline Co 1+xV2−xO4with various xextracted\nthe intrinsic magnetic and structural properties irrespec-\ntiveoftheoff-stoichiometry[39]. Itwassuggestedthat, in\naddition to the ferrimagnetic transition at TC, two other\npossible transitions occur within the ferrimagnetic phase\n[39].\nFig. 1(*) depicts the ferrimagnetic transition tempera-\ntureTCand two other transition temperatures T1andT2\nin Co1+xV2−xO4as functions of off-stoichiometry x(the2\nsolidlines), whichweresuggestedfromthemagnetization\nand magnetostriction measurements [39]. By increasing\nthe amounts of the off-stoichiometry x,TCincreases but\nT1andT2decrease. T1corresponds to a spin canting\ntemperature, where a collinear-to-noncollinear ferrimag-\nnetic ordering occurs upon cooling [39]. In Fig. 1(*), the\nstructural transition temperature of ∼90 K observed in\nthe neutron scattering experiments for the stoichiometric\nCoV2O4is also indicated as a circle [40], indicating the\noccurrence of the structural transition at T2.\nForAV2O4(A= Mn and Fe), the long-range or-\ndering of the V orbitals is considered to be accompa-\nnied by a canting of the V spins, which was observed\nas the appearance of a noncollinear ferrimagnetic order\n[15,21]. For Co 1+xV2−xO4, very recent neutron scatter-\ning experiments revealed the occurrence of a collinear-\nto-noncollinearferrimagneticordering (a spin canting) at\nT1[Fig. 1(*)] upon cooling to within the ferrimagnetic\nphase [39]. However, at the spin canting temperature T1,\nthe magnetostriction of Co 1+xV2−xO4exhibits a change,\nwhichissubtlerandmoregradualthanthatof AV2O4(A\n= Mn and Fe) [39]. Furthermore, the dielectric constant\nof Co1+xV2−xO4exhibits a frequency variation within\nthe noncollinear ferrimagnetic phase, indicating slow re-\nlaxation dynamics [39]. This dielectric behavior is simi-\nlar to that observed in the orbital glass state of FeCr 2S4\n[41]. From the results of the neutron scattering, magne-\ntostriction, anddielectric constantmeasurements, forthe\nnearly-itinerantCo 1+xV2−xO4, itissuggestedthatanor-\nbital glassy state exists in the noncollinear ferrimagnetic\nphase [39].\nHerein, we study the interplay of orbital, spin,\nand lattice degrees of freedom in the nearly-itinerant\nCo1+xV2−xO4by means of ultrasound velocity measure-\nments, which measure elastic moduli of this compound.\nThe elastic modulus of a crystal is a thermodynamic ten-\nsor quantity, and thus the ultrasound velocity measure-\nments in all the symmetrically-independent elastic mod-\nuli in a crystal can provide the symmetry-resolved ther-\nmodynamic information. Furthermore, since the ultra-\nsound velocity can be measured with a high precision of\n∼ppm, the ultrasound velocity measurements can sensi-\ntively probe elastic anomalies driven by phase transition\nand excitations. For the frustrated spinels, the ultra-\nsound velocity measurements have proven to be a useful\ntool for studying not only the ground state but also the\nexcited states [42–47].\nFor the vanadate spinels, the ultrasound velocity mea-\nsurementsinMgV 2O4[44]andMnV 2O4[45]wererecently\nreported. These measurements revealed the presence of\nanomalous elastic behavior in the paramagnetic phase\n(the magnetically disordered phase) and its absence in\nthe magnetically ordered phase. In contrast, the present\nstudy reveals that Co 1+xV2−xO4exhibits a variety of\nelastic anomalies in not only the paramagnetic phase\nbut also the magnetically ordered phase. Furthermore,\nthe present study also reveals that the anomalous elastic\nbehavior in the paramagnetic phase of Co 1+xV2−xO4isuniquely different from that of MgV 2O4and MnV 2O4.\nThe present study strongly suggests that the anoma-\nlous elastic behavior in Co 1+xV2−xO4is relevant to the\nnearly-itinerant character of the orbitally-degenerate V\n3delectrons, which causes the orbital glassiness within\nthe magnetically ordered phase.\nII. EXPERIMENTAL\nSingle crystals of Co 1+xV2−xO4(0≤x≤0.3) were\nprepared by the floating-zone method, where the chem-\nical compositions of the grown single crystals were esti-\nmated by the inductively coupled plasma analyses [39].\nFor the present experiments, we applied a large single\ncrystal of Co 1.21V1.79O4(x= 0.21), where the magne-\ntization, magnetostriction, and neutron scattering mea-\nsurements suggested the occurrence of magnetic and\nstructural transitions at TC∼165 K,T1∼100 K, and\nT2∼50 K [the solid lines in Fig. 1(*)] [39]. The ul-\ntrasound velocity measurements were performed by a\nhome-built apparatus, where the phase-comparison tech-\nnique was used with longitudinal and transverse sound\nwaves at a frequency of 30 MHz. The ultrasound waves\nwere generated and detected by LiNbO 3transducers\nglued on the parallel mirror surfaces of the crystal. We\nmeasured the sound velocities in all the symmetrically-\nindependent elastic moduli in the cubic crystal, specifi-\ncally, compression modulus C11, tetragonal shear modu-\nlusC11−C12\n2≡Ct, and trigonal shear modulus C44. From\nC11andCtdata, we also obtained the bulk modulus\nCB=C11+2C12\n3=C11−4\n3Ct. The respective measure-\nments of C11,Ct, andC44were performed using longi-\ntudinal sound waves with propagation k/bardbl[100] and po-\nlarization u/bardbl[100], transverse sound waves with k/bardbl[110]\nandu/bardbl[1¯10], and transverse sound waves with k/bardbl[110]\nandu/bardbl[001]. The sound velocities of Co 1.21V1.79O4mea-\nsured at room temperature (300K) are 6740m/s for C11,\n2750 m/s for Ct, and 3770 m/s for C44. In the present\nstudy, we also performed the electrical resistivity mea-\nsurements for the single-crystalline Co 1.21V1.79O4using\nthe conventional four-probe method.\nIII. RESULTS\nA. Elastic modulus\nFigures 1(a)-1(c) respectively present the temperature\n(T) dependence of the elastic moduli, CB(T),Ct(T),\nandC44(T), with zero magnetic field ( H= 0) in\nCo1.21V1.79O4. Here,CB(T) =C11(T)−4\n3Ct(T) is ob-\ntained from C11(T) [the inset in Fig. 1(a)] and Ct(T)\n[Fig. 1(b)]. From these experimental results, a variety of\nelastic anomalies are observed.\nFirst, the elastic moduli shown in Figs. 1(a)-1(c) ex-\nhibit anomalies at ∼165 K for all the elastic moduli,\nat∼95 K for Ct(T), and at ∼50 K for C44(T). As3\n37.6\n37.4\n37.2\n37.0\n36.8(b) CtCt (GPa)TC\nT1H = 0180\n179\n178CB (GPa)(a) CB\n177TCTC\n300250200150100500230\n229\n228\n227C11  (GPa)C11 \nH = 0H = 0\n72.0\n71.0\n70.0\n69.0\n300250200150100500\n T (K)(C) C44 H = 0C44  (GPa)TC T2150 \n100 \n50 TC\nT2(*)  T (K)\n xT1\n0 0.1 0.2 0.3 \nFIG. 1: (Color online) (*) Transition temperatures TC,T1,\nandT2for Co 1+xV2−xO4as functions of off-stoichiometry x,\nwhich were reported in Ref. [39] (the solid lines). The circle\nin (*) indicates a structural transition temperature for th e\nstoichiometric CoV 2O4reported in Ref. [40]. The vertical\ndashed line in (*) indicates x= 0.21, corresponding to the x\nvalue for the single-crystalline sample applied in the pres ent\nstudy. The crosses in (*) indicate transition temperatures TC,\nT1, andT2, which are obtained from the experimental results\nshown in (a)-(c). (a)-(c) Elastic moduli of Co 1.21V1.79O4as\nfunctions of TwithH= 0. (a) CB(T), (b)Ct(T), and (c)\nC44(T). The insetin(a) depicts C11(T)ofCo 1.21V1.79O4with\nH= 0. The arrows in (a)-(c) indicate TC,T1, andT2.\nindicated in Fig. 1(*), these temperatures [the crosses\nin Fig. 1(*)] respectively agree with TC,T1, andT2re-\nported in Ref. [39] [the solid lines in Fig. 1(*)]. At the\nferrimagnetic transition temperature TC, all the elastic\nmoduli exhibit a discontinuous anomaly, as marked by\nthe arrows in Figs. 1(a)-1(c). Additionally, within the\nferrimagnetic phase ( T < T C),Ct(T) andC44(T) respec-\ntively exhibit a rather gradualanomaly at T1∼95 K and\na discontinuous anomaly at T2∼50 K, as marked by the\narrows in Figs. 1(b) and 1(c). The gradual anomaly at\nT1∼95 K inCt(T) should be driven by the occurrence\nof the orbital glassy order accompanied by the spin cant-\ning, which was suggested in Ref. [39]. The discontinuous\nanomaly at T2inC44(T) indicates the occurrence of a\nphase transition at this temperature. It should be noted\nthat the neutron scattering experiments in the stoichio-\nmetric CoV 2O4observed a structural transition at ∼90\nK [the circle in Fig. 1(*)] [40]. Since this structural tran-\nsition temperature agrees with the temperature of T2forCo1+xV2−xO4withx= 0 [the solid line in Fig. 1(*)] [39],\nthe elastic anomaly at T2inC44(T) for Co 1.21V1.79O4\nshould be driven by the structural transition, which is\nidentical to that observed in the neutron scattering ex-\nperiments for the stoichiometric CoV 2O4[40]. Compar-\ning the elastic anomalies at TC,T1, andT2, the rather\ngradual observation of the anomaly at T1is compatible\nwith the occurrence of the orbital glassy order, which is\nin contrast to the discontinuous anomalies at TCandT2\ndriven by the ferrimagnetic and structural phase transi-\ntions, respectively.\nIn addition to the elastic anomalies at TC,T1, andT2,\nas shown in Figs. 1(a)-1(c), the elastic moduli exhibit\nelastic-mode-dependent unusual Tvariations in both the\nparamagnetic and ferrimagnetic phases. Upon cooling in\nthe paramagnetic phase ( T > T C),CB(T) exhibits ordi-\nnal hardening [48], butCt(T) andC44(T) exhibit anoma-\nlous softening. Here, the magnitudes of the softening in\nCt(T) andC44(T) are ∆Ct/Ct∼1.5 % and ∆ C44/C44∼\n0.5 %, respectively. In the ferrimagnetic phase ( T < T C),\nthe elastic moduli exhibit nonmonotonic Tvariations;\nCB(T) andCt(T) exhibit minima at ∼130 K and ∼\n30 K, and C44(T) exhibits a minimum at ∼130 K.\nIn the present study, we also investigated the magnetic\nfield effect on the elastic properties of Co 1.21V1.79O4. In\nthe ferrimagnetic phase ( T < T C), the ultrasound echo\nsignalsweretoostronglyattenuated bythe applicationof\na magnetic field to perform the ultrasound velocity mea-\nsurement, which should arise from the magnetostriction\n[39]. However, in the paramagnetic phase ( T > T C), we\nwere able to detect the ultrasound echo signals indepen-\ndent of whether the magnetic field was applied or not.\nThus, in the present study, the ultrasound velocity mea-\nsurements under magnetic field were performed only in\nthe paramagnetic phase ( T > T C).\nFigures 2(a)-2(c) respectively depict CB(T),Ct(T),\nandC44(T) with magnetic field H||[001] in the para-\nmagnetic phase ( T > T C) of Co 1.21V1.79O4. Here,\nCB(T) =C11(T)−4\n3Ct(T) is obtained from C11(T) [the\ninset in Fig. 2(a)] and Ct(T) [Fig. 2(b)]. In the zero-\nfield (H= 0) paramagnetic phase, as described above\nin conjunction with Figs. 1(a)-1(c), Ct(T) andC44(T)\nexhibit softening with decreasing T, whileCB(T) ex-\nhibits ordinal hardening [48]. Furthermore, in Figs. 2(a)-\n2(c),CΓ(T) exhibits Hvariation below ∼230 K;CB(T)\nbecomes softer with increasing H, but the softening in\nCt(T) andC44(T) is relaxed with H, as indicated by the\nsolid arrows.\nB. Magnetoresistance\nFigure 3(a) depicts the Tdependence of the magne-\ntoresistanceat 7 T, MR=ρ(7T)−ρ(0)\nρ(0), for Co 1.21V1.79O4\nwith the current I||[100] and the magnetic field H||[001]\n(I⊥H), which is obtained from the Tdependence of\nthe electrical resistivities with H= 0 (ρ(0)) and 7 T\n(ρ(7T)) depicted in Figs. 3(b) and 3(c). The Tdepen-4\n71.2\n71.1\n71.0\n70.9\n300 250 200175 \n T (K) 7 T\n 5 T\n 3 T\n H = 0(c) C44 C44  (GPa)HH || [001] \n       Eq. (1)  \n C44  = 71.4 GPa \n EJT  = 0.2 K \n θ = 192.9 K 0       Eq. (1)  \n Ct = 38.1 GPa \n EJT  = 1.4 K \n θ = 141.5 K 037.7\n37.6\n37.5\n37.4\n37.3 7 T\n 5 T\n 3 T\n H = 0(b) CtCt (GPa)\nHH || [001] \n300 250 200175 \n T (K)H = 0\n 3 T\n 5 T\n 7 T(a) CB179.5\n179.0\n178.5\n178.0\n177.5\n300 250 200175 \n T (K)H\nH || [001] 211.0\n210.5\n210.0\n209.5\n300 250 200175 H = 0\n 3 T\n 5 T\n 7 THH || [001] C11 C11  (GPa)CB (GPa)\nFIG. 2: (Color online) CΓ(T) of Co 1.21V1.79O4withH||[001] in the paramagnetic phase ( T > T C). (a)CB(T), (b)Ct(T), and\n(c)C44(T). The inset in (a) depicts C11(T) of Co 1.21V1.79O4withH||[001] in the paramagnetic phase ( T > T C). The solid\narrows in (a)-(c) are guides to the eye, indicating the varia tion ofCΓ(T) with increasing H. The dotted curves in (b) and (c)\nare, respectively, fits of the zero-field experimental Ct(T) andC44(T) to Eq. (1) in 250 K < T <300 K. The respective values\nof the fit parameters are also listed in (b) and (c).\ndence of the magnetoresistanceshown in Fig. 3(a) agrees\nwith the previous report on the stoichiometric CoV 2O4\n[37]; upon cooling, Co 1+xV2−xO4exhibits a large neg-\native magnetoresistance at temperatures around ∼TC,\na change to positive magnetoresistance at ∼T1, and a\nfurther enhancement of the positive magnetoresistance\nbelow∼T2.\nIV. DISCUSSION\nA. Paramagnetic phase ( T > T C)\nWe will discuss the origins of the elastic anomalies\nobserved in Co 1.21V1.79O4. First we address the ori-\ngins of the elastic anomalies in the paramagnetic phase\n(T > T C) [Fig. 2].\nIn zero magnetic field ( H= 0),Ct(T) andC44(T)\nsoften upon cooling in the paramagnetic phase, while\nC11(T) exhibits ordinalhardening [48]. One probable ori-\ngin for this softening in Ct(T) andC44(T) is a precursor\nto the Jahn-Teller (JT) structural transition, which has\nbeen observed in MgV 2O4[44] and MnV 2O4[45].\nFor the JT effect, a JT-active ion and its set of lig-\nands are considered to be a structural unit, where the\ncrystal-field striction mechanism leads to the JT distor-\ntion. In the JT magnets, the degenerate ground state\nis considered to couple strongly and selectively to the\nelastic modulus CΓ, which has the same symmetry as\nthe JT distortion. For such a JT-active elastic mode,\nTdependence of the elastic modulus CΓ(T) above the\nJT transition temperature is explained by assuming the\ncoupling of the ultrasound to the JT-active ions through\nthe crystal-field striction mechanism, and the presence of\ninter-JT-active-ion interactions. A mean-field expressionofCΓ(T) in the JT magnets is given as [49–51]\nCΓ(T) =C0,Γ(1−EJT\nT−θ), (1)\nwithC0,Γthe elastic constant without the JT effect, EJT\nthe JT coupling energy, and θthe inter-JT-active-ion in-\nteraction.\nAccording to Eq. (1), CΓ(T) of the JT system exhibits\na Curie-type ( ∼ −1/T-type) softening at temperatures\nabovethe JT transitiontemperature. ForCo 1.21V1.79O4,\nhowever, Ct(T) [Fig. 2(b)] and C44(T) [Fig. 2(c)] exhibit\nthe non-Curie-type softening in the paramagnetic phase\naboveTC. As a comparison, fits of the zero-field exper-\nimentalCt(T) andC44(T) to Eq. (1) in 250 K < T <\n300 K are presented in Figs. 2(b) and 2(c) as dotted\ncurves, respectively. It is evident in these figures that\nthe experimental Ct(T) andC44(T) both deviate from\nthe dotted curves below ∼250 K, which rules out the JT\neffect as a possible origin for the softening in Ct(T) and\nC44(T) in the paramagnetic phase above TC. Indeed,\nfor the stoichiometric CoV 2O4, the X-ray and neutron\ndiffraction experiments reported the absence of a struc-\ntural distortion at TC, indicating the absence of a JT\nstructural transition at TC[33,40].\nTheotherpossibleoriginforthesofteningin Ct(T)and\nC44(T) is the coupling between the correlated paramag-\nneticstateandtheacousticphonons. Itisnotedherethat\nthe inelastic neutron scattering experiments in the stoi-\nchiometric CoV 2O4observed quasielastic magnetic exci-\ntations in the paramagnetic phase [40]. A similar type of\nlocal spin excitations was also observed in AV2O4and\nACr2O4(A= Mg and Zn), which has been character-\nized as spin-cluster excitations on the V/Cr pyrochlore\nlattice [52–54]. Furthermore, for these compounds, the\nnon-Curie-type softening was observed in CΓ(T) in the\nparamagnetic phase, which is considered to be driven by5\nH = 0 \n7 T 0.4 \n0.3 \n0.2 \n0.1 \n30025020015010090 80 70 60 50 40 30 ρ (Ω cm)\n110100 H = 0 \n7 T (b) (c) \nT2\nT1TC\n I ||[100] \nH||[001]  I ||[100] \nH||[001] \nT1\n30 300 250 200 150 100 50\nT (K)MR  (%)10\n8\n6\n4\n2\n0\n−2\n−4\n−6TCT2\n I ||[100] \nH||[001] \n(a) (d) \neg\nt2g eg\na1g \nFIG. 3: (Color online) (a) Magnetoresistance ( MR) of\nCo1.21V1.79O4at 7 T for I||[100] and H||[001] (I⊥H) as a\nfunction of T. (b), (c) Electrical resistivities of Co 1.21V1.79O4\nwithH= 0 and 7 T ( I||[100] and H||[001]) as functions of T:\n(b) 30 K < T <100 K and (c) 100 K < T <300 K. The ar-\nrows in (a), (b), and (c) indicate the transition temperatur es\nTC,T1, andT2. The horizontal dashed line in (a) is a guide\nto the eye indicating MR= 0. (d) Energy-level scheme of the\nV 3dorbitals in Co 1+xV2−xO4. The arrows in (d) depict the\nHund-rule filling of 2 electrons/V.\nthe coupling of the lattice to the spin-cluster excitations\n[42,44]. Therefore, the non-Curie-type softening in Ct(T)\nandC44(T) in the paramagnetic phase of Co 1.21V1.79O4\nshould stem from the coupling of the lattice to the spin-\ncluster excitations on the V pyrochlore lattice. Taking\ninto account that the quasielastic magnetic excitations\nof CoV 2O4andAV2O4(A= Mg and Zn) were both ob-\nserved centered around a wave vector of Q∼1.3˚A−1\n[40,52], the shape of the spin cluster in these compounds\nmight be identical, which should be confirmed in a future\nstudy.\nThe softening in CΓ(T) driven by the spin-cluster ex-\ncitations is generally explained as the presence of a finite\ngapfortheexcitations, whichissensitivetostrain[42]. In\nthe mean-field approximation, CΓ(T) in the spin-cluster\nsystem is written as [42]\nCΓ(T) =C0,Γ−G2\nΓNχΓ(T)\n[1−KΓχΓ(T)],(2)\nwithC0,Γthe backgroundelastic constant, Nthe density\nofspin clusters, GΓ=|∂∆/∂ǫΓ|the couplingconstant for\nasinglespinclustermeasuringthestrain( ǫΓ)dependence\nofthe excitationgap∆, KΓthe inter-spin-clusterinterac-\ntion, and χΓ(T) the strain susceptibility of a single spincluster. From Eq. (2), when CΓ(T) strongly couples to\nthe excited state at ∆, this elastic mode exhibits soften-\ning upon cooling roughlydown to T∼∆, but recoveryof\nthe elasticity (hardening) roughly below T∼∆;CΓ(T)\nexhibits a minimum roughly at T∼∆. According to\nEq. (2), for Ct(T) andC44(T) of Co 1.21V1.79O4respec-\ntively shown in Figs. 2(b) and 2(c), the softening upon\ncooling down to TCindicates that this softening arises\nfrom the coupling of the lattice to the gapped excitations\nwith ∆< TC, which is compatible with the observation\nof the spin-cluster excitations at energies below ∼4 meV\n(< kBTC) in the paramagnetic phase of the stoichiomet-\nric CoV 2O4[40]. For the experimental results shown in\nFig. 2, the future identification of the shape of the spin\ncluster in Co 1+xV2−xO4by the inelastic neutron scat-\ntering study will enable the quantitative analyses using\nEq. (2).\nAs observed in Fig. 2, CB(T),Ct(T), andC44(T) in\ntheparamagneticphaseallexhibitthemagnetic-field( H)\nvariations. Notably, while the softening in Ct(T) and\nC44(T) is present already at H= 0 and relaxed with in-\ncreasing H, the softening in CB(T) is absent at H= 0\nbut induced by H. As described above, the softening in\nCt(T) andC44(T) withH= 0 is attributed to the cou-\npling of the lattice to the spin-cluster excitations. Thus\nthe relaxation of this softening by Hindicates that the\ncoupling of the lattice to the spin-cluster state is relaxed\nbyH. A similar type of this Heffect was also observed\nin MgV 2O4, which is also considered to be a result of\nthe relaxation of the spin-cluster-lattice coupling by H\n[44]. In contrast, the H-induced softening in CB(T) for\nCo1.21V1.79O4is a unique elastic anomaly, which was not\nobservedin other AV2O4(A=Mg [44] and Mn [45]) com-\npounds. Taking into account that, among the insulating\nor semiconducting AV2O4, Co1+xV2−xO4is closest to\nthe itinerant-electron limit [31,33], theH-induced soften-\ning inCB(T) unique to Co 1+xV2−xO4should be relevant\nto the nearly-itinerant character of the V 3 delectrons.\nAs shown in Fig. 3(a), Co 1.21V1.79O4exhibits the T-\ndependent magnetoresistance, which should arise from\nthe nearly-itinerant V 3 delectrons. Comparing Fig. 2(a)\nwith Fig. 3(a) in the paramagnetic phase ( T > T c), it is\nevident that the H-induced softening in CB(T) develops\nupon cooling below ∼230 K in accordance with the de-\nvelopment of the negative magnetoresistance. We note\nhere that, for the cubic crystal of Co 1+xV2−xO4,CB\nis a symmetry-conserving isotropic elastic mode, while\nCtandC44are symmetry-lowering anisotropic elastic\nmodes, and that the application of Hcauses the devel-\nopment of the softening for CB(T) and the relaxation\nof the softening for Ct(T) andC44(T). This elastic-\nmode-dependent Heffect indicates that the magnetoe-\nlastic coupling becomes rather isotropic with increasing\nH, whichshouldbe drivenbythe H-enhanceddelocaliza-\ntion of the strongly-directional V 3 delectrons. Thus, for\nCo1+xV2−xO4, theH-induced softening in CB(T) and\nthe negative magnetoresistance suggest the presence of\nthe intersite V 3 delectron interactions, which causes the6\nH-enhanced electron delocalization in the paramagnetic\nphase.\nAs is evident from the comparison between Fig. 2 and\nFig. 3(a), in the paramagnetic phase of Co 1.21V1.79O4,\nnot only the H-induced softening in CB(T) but also\ntheH-induced relaxation of the softening in Ct(T) and\nC44(T) coincide with the development of the negative\nmagnetoresistance below ∼230 K. As already described\nin conjunction with Figs. 2(b) and 2(c), the softening in\nCt(T) andC44(T) withH= 0 in the paramagneticphase\nis concluded to be driven by the coupling of the lattice\nto the spin-cluster state. Thus the H-induced relaxation\nof the softening in Ct(T) andC44(T) suggests the oc-\ncurrence of the H-induced ”bond dissolution” in the V\nspin clusters, which should be driven by the H-enhanced\ndelocalization of the V 3 delectrons.\nIn Co1+xV2−xO4, the O octahedra surrounding the V\natoms are trigonally distorted from the regular octahe-\ndral shape even in the cubic crystal structure [33,36,40].\nAdditionally, for the V 3 dorbitals, the small trigonal\ncrystal field splits the triply-degenerate t2gorbitals into\na localized a1gorbital and delocalized doubly-degenerate\ne′\ngorbitals [Fig. 3(d)]. Thus it is expected that the delo-\ncalizede′\ngorbitals are responsible for the nearly-itinerant\ncharacter of the V 3 delectrons [36]. Thee′\ngorbitals are\nalso expected to be responsible for the elastic anomalies\nin the paramagnetic phase [Fig. 2].\nFinally, we note that the softening in CΓ(T) driven by\nthe spin-cluster-lattice coupling is observed in not only\nthe orbital-degenerate Co 1+xV2−xO4and MgV 2O4[44]\nbut also the orbital-nondegenerate ACr2O4(A= Mg\nand Zn) [42] and ZnFe 2O4[43], which indicates that the\nspin-clusterstatecanuniversallyemergeinthefrustrated\nspinels. It is also noted that this type of elastic softening\nis relaxed by Hin Co1+xV2−xO4and MgV 2O4, but in-\nsensitive to HinACr2O4and ZnFe 2O4, indicating that\nthe orbital sector is responsible for the Heffect on the\nspin-cluster state. For Co 1+xV2−xO4and MgV 2O4, as\ndescribed above, it is suggested that the application of\nHresultsin the”bonddissolution”intheVspinclusters,\nwhich should be driven by the enhancement of the delo-\ncalized character for the V 3 delectrons. Here, compar-\ning Co 1+xV2−xO4and MgV 2O4, only MgV 2O4exhibits,\nin addition to the spin-cluster-driven elastic anomalies,\nthe Curie-type softening in CΓ(T), which is the JT ef-\nfect (a precursor to the structural transition) expressed\nby Eq. (1) [44]. Taking into account that the JT ef-\nfect is driven by the coupling of the elastic deformations\nto the localized magnetic moments, the presence of the\nJT effect in MgV 2O4but its absence in Co 1+xV2−xO4\nindicates that the localized character of the V 3 delec-\ntrons is stronger in MgV 2O4than Co 1+xV2−xO4, which\nis compatible with the poorer electrical conductivity of\nMgV2O4than Co 1+xV2−xO4[31,33].B. Ferrimagnetic phase ( T < T C)\nIn this section, we discuss the origins of the elastic\nanomalies observed in the ferrimagnetic phase ( T < T C)\nof Co1.21V1.79O4. As shown in Fig. 1, Co 1.21V1.79O4\nexhibits nonmonotonic Tdependence of the elastic mod-\nuli in the ferrimagnetic phase, which is in contrast to\nthe monotonic Tdependence observed in the magneti-\ncally ordered phase of other AV2O4(A= Mg [44] and\nMn [45]) andACr2O4(A= Mg and Zn [42]). This non-\nmonotonic elastic behavior in Co 1.21V1.79O4is expected\nto be relevant to the nearly-itinerant character of the V\n3delectrons.\nFirst, one of the remarkable elastic anomalies shown in\nFig. 1 is the elastic moduli minima at ∼30 K and ∼130\nK. Since there is no additional phase transition at these\ntemperatures, theelasticmoduli minimashouldoriginate\nfrom the coupling of the ultrasound to the magnetic ex-\ncitations, rather than to the static order. Indeed, in the\nferrimagneticphase,magnonexcitationswereobservedin\nthe inelastic neutron scattering experiments [40]. How-\never, in the ferrimagnetic phase of Co 1+xV2−xO4, there\nremains the possibility for the coexistence of the magnon\nexcitations and the spin-cluster excitations. Thus, at\npresent, we cannot identify which excitation is the ori-\ngin for the respective elastic moduli minima at ∼30 K\nand∼130 K.\nNext, asalreadymentionedinSec. IIIAinconjunction\nwith Fig. 1, the Tdependence of the elastic moduli sug-\ngests the successive occurrence of the orbital glassy order\natT1∼95 K and the structural phase transition at T2∼\n50 K. As shown in Figs. 1(a)-1(c), the elastic anomaly\natT1is observed only in the tetragonal Ct(T) withEg\nsymmetry. We note here again that, for Co 1+xV2−xO4,\nthe trigonal crystal field splitting of the V 3 dorbitals is\npresent even in the cubic crystal structure [Fig. 3(d)]\n[33,36,40]. Thus the elastic anomaly at T1only inCt(T)\nsuggests that the nearly-itinerant doubly-degenerate e′\ng\norbitals [Fig. 3(d)] play a dominant role for the occur-\nrence of the orbital glassy order. Further, as shown in\nFig. 3(a), the magnetoresistance exhibits a negative-to-\npositive sign change at T1, which indicates that, in the\norbital glassy state, the application of Henhances the\nlocalizedcharacterofthe V3 delectrons. While the nega-\ntive magnetoresistance in the paramagnetic phase is sug-\ngested to be driven by the H-induced ”bond dissolution”\nin the V spin cluster, the positive magnetoresistance in\nthe orbitalglassystatemight be drivenbythe H-induced\n”bond reinforcement” in the V spin clusters.\nFor the structural phase transition at T2, from Figs.\n1(a)-1(c), the selective observation of the elastic anomaly\nin the trigonal C44(T) suggests the occurrence of a trigo-\nnallatticedistortionat T2. Thusthestructuraltransition\natT2is expected to further enhance the trigonal crystal\nfieldsplittingoftheV3 dorbitals,whichispresentevenin\nthe cubic crystal structure for Co 1+xV2−xO4[Fig. 3(d)]\n[33,36,40]. Notably, for Co 1+xV2−xO4, Ref. [40] revealed\nthat the structural transition at T2causes a short-range7\n37.5\n37.4\n37.3179.8\n179.6\n179.4\n179.2\n70.6\n70.4\n70.2\n70.0\n69.8\n69.6\n90 80 70 60 50Ct (GPa) CB (GPa) C44  (GPa)H = 0\nH = 0\nH = 0\n T (K)(a) CB\n(b) Ct\n(C) C44        Eq. (1)  \n  Ct = 37.6 GPa \n  EJT  = 0.2 K \n  θ  = 30 K 09080706050229.8\n229.6\n229.4\n229.2C11  (GPa)C11 \nH = 0\nFIG. 4: (Color online) CΓ(T) of Co 1.21V1.79O4withH=\n0 inT2< T < T 1[from Figs. 1(a)-1(c)]. (a) CB(T), (b)\nCt(T), and (c) C44(T). The inset in (a) depicts C11(T) of\nCo1.21V1.79O4withH= 0 inT2< T < T 1[from the inset in\nFig. 1(a)]. The solid curve in (b) is a fit of the experimental\nCt(T) to Eq. (1) in 50 K < T <90 K. The values of the fit\nparameters are also listed in (b).\ndistortion of the O octahedra, but does not lower the\nglobal cubic symmetry of the crystal. Thus the C44(T)\nanomaly at T2should be driven by the short-range trig-\nonal lattice distortion. It should be noted that Refs. [39]\nand [40] suggestedthe existence ofthe orbitalglassystate\nattemperaturesofnotonly T2< T < T 1butalsoT < T 2.\nTaking into account that the localized electron character\nat low temperatures of T < T 2was revealedby the obser-\nvation of the spin gap in the inelastic neutron scattering\nexperiments [40], it is implied that the localized electron\ncharacter in the orbital glassy state is further enhanced\nbelowT2, which seems to give rise to the further en-\nhancement of the positive magnetoresistance below T2\n[Fig. 3(a)].\nFinally, we note that, in the orbital glassy phase in\nthe temperature range of T2< T < T 1,Ct(T) exhibits\nthe Curie-type softening, which is the JT effect expressed\nby Eq. (1). Figs. 4(a)-4(c) respectively depict CB(T),\nCt(T), andC44(T) inT2< T < T 1[from Figs. 1(a)-\n1(c)]. The Curie-type softening selectively observed in\nthetetragonal Ct(T)withEgsymmetryshouldarisefrom\nthe coupling of the ultrasound to the doubly-degenerate\ne′\ngorbitals [Fig. 3(d)]. Taking into account that the JT\neffect is driven by the coupling of the elastic deforma-\ntions to the localized magnetic moments, the presence\nof the Curie-type softening below T1is compatible with\nthe existence of the orbital glassy state below T1, wherethe V 3delectrons should become rather localized. In\nFig. 4(b), a fit of the experimental Ct(T) to Eq. (1)\nis depicted as a solid curve, which reproduces the ex-\nperimental data very well. Comparing the fit values of\nthe onsite JT energy EJTand the intersite orbital in-\nteraction θfor Co 1.21V1.79O4[Fig. 4(b)] with those for\nMgV2O4[44] and MnV 2O4[45], whileθforCo 1.21V1.79O4\n(30 K) is comparable to those for MgV 2O4(15 K) and\nMnV2O4(5 K),EJTfor Co 1.21V1.79O4(0.2 K) is much\nsmaller than those for MgV 2O4(10 K) and MnV 2O4(22\nK). Such a small value of EJTfor Co 1.21V1.79O4seems\nto be compatible with the rather delocalized character of\nthe V 3delectrons compared to MgV 2O4and MnV 2O4.\nBelowT2,CΓ(T) of Co 1.21V1.79O4exhibits the absence\nof the Curie-type softening, and instead exhibits an elas-\ntic moduli minimum at ∼30 K arising from the magnetic\nexcitations [Figs. 1(a)-1(c)]. For Co 1+xV2−xO4, the JT\nfluctuations might be suppressed below T2by the further\nstabilization of the orbital glassy state, which is accom-\npanied by the lattice distortion.\nAlthough the observation of the Curie-type soften-\ning inCt(T) inT2< T < T 1[Fig. 4(b)] indicates\nthe presence of the JT interaction, the e′\ngelectron [Fig.\n3(d)] is also expected to experience the onsite spin-\norbit interaction. The very small fit value of EJTfor\nCo1.21V1.79O4(0.2 K) [Fig. 4(b)] might reflect, in addi-\ntiontotheratherdelocalizedelectroncharacter,thecom-\npetition/coexistence of the onsite JT and spin-orbit in-\nteractions. Additionally, the disappearance of the Curie-\ntype softening in Ct(T) belowT2[Fig. 1(b)] might be\na result of the enhanced contribution of the spin-orbit\ninteraction below T2. For Co 1+xV2−xO4, the intricate\ninterplay of the JT, spin-orbit, and intersite spin-orbital\ninteractions is expected to play an important role for the\nemergence of the nearly-itinerant electron character and\nthe orbital glassy state, which remains to be understood.\nV. SUMMARY\nUltrasound velocity measurements of Co 1.21V1.79O4\nrevealed a variety of the elastic anomalies in both the\nparamagnetic phase ( T > T C) and the ferrimagnetic\nphase (T < T C). In the paramagnetic phase above TC,\nthe present study revealed the elastic-mode-dependent\nunusualtemperaturevariationsoftheelasticmoduli, sug-\ngesting the existence of the dynamic spin-cluster state.\nFurthermore, above TC, the present study revealed the\nsensitive magnetic-field response of the elastic moduli,\nsuggestingthat, withthenegativemagnetoresistance,the\nmagnetic-field-enhanced nearly-itinerant characterof the\nV 3delectrons emerges from the spin-cluster state, which\nshould be triggered by the inter-V-site interactions act-\ning on the orbitally-degenerate 3 delectrons. In the ferri-\nmagnetic phase below TC, the elastic anomalies at T1∼\n95 K and T2∼50 K were found to coincide, respec-\ntively, with the sign change of the magnetoresistance at\nT1(positive below T1) and the enhancement of the posi-8\ntive magnetoresistance below T2. These observations be-\nlowTCsuggest the successive occurrence of the orbital\nglassy order at T1and the structural phase transition\natT2, where the rather localized character of the V 3 d\nelectrons evolves below T1and is further enhanced be-\nlowT2. Further experimental and theoretical studies are\nindispensable if the spin and orbital states of the nearly-\nmetallic Co 1+xV2−xO4in both the magnetically disor-\ndered and ordered phases are to be understood.VI. ACKNOWLEDGMENTS\nThis work was partly supported by a Grant-in-Aid\nfor Scientific Research (C) (Grant No. 17K05520) from\nMEXT of Japan, and by Nihon University College of Sci-\nence and TechnologyGrants-in-Aidfor ProjectResearch.\n∗tadataka@phys.cst.nihon-u.ac.jp\n1P. G. Radaelli, New J. Phys. 7, 53 (2005).\n2Y. Ueda, N. Fujiwara, and H. Yasuoka, J. Phys. Soc. Jpn.\n66, 778 (1997).\n3M. Reehuis, A. Krimmel, N. Buttgen, A. Loidl, and A.\nProkofiev, Eur. Phys. J. B 35, 311 (2003).\n4S.-H. Lee, D. Louca, H. Ueda, S. Park, T. J. 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We survey the literature of two of the most commonly modelled ferrimagnets\nin the field of spintronics–the amorphous alloy GdFeCo and the magnetic insulator yttrium iron garnet. We\nlook at how generic models and material specific models have been applied to predict and understand spintronic\nexperiments, focusing on the fields of ultrafast magnetisation dynamics, spincaloritronics and magnetic textures\ndynamics and give an outlook for modelling in ferrimagnetic spintronics\nI. INTRODUCTION\nIn the field of spintronics1the goal is to use the spin de-\ngree of freedom to create new devices which are superior to\nelectronic devices in some aspect such as energy consump-\ntion. It also provides insights into the fundamental physics of\nspin conversion with electrons and phonons2. For example,\ncreating memory devices which can be controlled with elec-\ntric or spin currents but with better durability and lower power\nconsumption than electronic memories. There is also a push\nto develop new types of devices for specialised roles such as\nmachine learning where the additional degrees of freedom in\nmagnetic materials can be exploited3.\nFerromagnets provide us with a single magnetisation vec-\ntor which can be manipulated using external stimuli such as\nmagnetic fields and spin transfer from electrical currents. The\ninteraction of the magnetisation vector with the spin quantisa-\ntion axis of electrons gives rise to a range of physical effects\nsuch as giant magnetoresistance4–6, spin Hall7–11and Rashba-\nEdelstein effects12,13, which can be used to create devices or\nas fundamental probes in experiments. For quite some time\nferromagnetic behaviour has been the focus of studies in spin-\ntronics. Whilst ferrimagnetic materials have been used it is\ngenerally because of a desirable material property, such as\na low magnetic damping or high magneto-optical coupling.\nThe ferrimagnetic nature was incidental and often ignored in\ntheory and simulations due to their minor role in the macro-\nscopic magnetic properties of interest by the time. As re-\nsearch progressed it became clear that some results could only\nbe explained due to the unique characteristics of ferrimagnets\nwhich occur with two or more coupled but anti-aligned mag-\nnetisation vectors. It has long been known that ferrimagnets\nhave special points in their phase diagram such as the angu-\nlar momentum and magnetisation compensation points. For\na two sublattice ferrimagnet with antiparallel sublattice mag-\nnetisationsM1(T)andM2(T), the magnetisation compensa-\ntion point is when M1(T) =M2(T)(see Fig. 1(a)). There is\nzero net magnetisation because of the cancellation of the two\nantiparallel sublattices. If the two sublattices also have differ-\nent gyromagnetic ratios \r1and\r2, then the angular momen-\ntum compensation temperature is different from the magneti-\nsation compensation point, being at M1(T)=\r1=M2(T)=\r2as shown in Fig. 1(b).\nRecently, researchers have become interested in using\nthe compensation points to tune the macroscopic proper-\nties of ferrimagnets to impart some of the desirable quali-\nties of both ferromagnets–a measurable net magnetisation–\nand antiferromagnets–high frequency dynamics. The hope is\nthat the versatile spin couplings present in ferrimagnets can be\ntechnologically exploited in devices for ultrafast spintronics14,\nwith the potential to overcome the gigahertz frequency limi-\ntations of ferromagnetic based technologies and realise tera-\nhertz spintronics15. Ferrimagnetic insulators may also enable\nlow power devices which work by pure spin currents16. Con-\nsequently, a large and ever increasing range of ferrimagnetic\nmaterials are now used to investigate and push forward the\nfield of spintronics.\nComputer modelling of ferrimagnets aims to describe the\nmacroscopic magnetic behaviour, which can be observed in\nexperiments, by considering the microscopic or multiple sub-\nlattice variables which are difficult to isolate experimentally.\nModelling ferrimagnets involves many of the similar hurdles\nas modelling antiferromagnets. There are multiple connected\nsublattices and antiferromagnetic coupling between them can\nlead to complex phase diagrams and magnetisation dynamics.\nEspecially important in modelling ferrimagnets is the correct\naccounting of the thermodynamic properties as temperature\ntunes the system across the compensation points. Simulations\nare also frequently used in non-equilibrium physics where ex-\nperimental results can be noisy or of limited resolution and\nanalytic theory can be difficult. In this Review we focus on\nthe two major numerical modelling approaches; atomistic spin\ndynamics (ASD)17–21and the Landau-Lifshitz-Bloch (LLB)\nequation22–27which are able to produce accurate descriptions\nof both the dynamics and thermodynamics of ferrimagnets.\nThese computational methods are used for different length\nand timescale limits. Atomistic spin-dynamics simulations\nare restricted by computational costs to lateral dimensions of\n10-100 nm. Timescales are limited to tens of nanoseconds.\nThe simulation of mesoscopic spin textures such as magnetic\ndomains is extremely computationally expensive. For com-\nparison with experiments much larger sample sizes are often\nnecessary. This is usually tackled with micromagnetic ap-\nproaches. Conventional micromagnetics, rests on a general-arXiv:2102.11004v1  [cond-mat.mtrl-sci]  22 Feb 20212\nFIG. 1. Diagram of the (a) magnetisation compensation and (b) an-\ngular momentum compensation points in a two sublattice ferrimag-\nnet.M1andM2are the magnetisations of the antiparallel sublattices,\nwhich have different gyromagnetic ratios \r1and\r2respectively. \re\u000b\nis the effective gyromagnetic ratio defined in Eq. (6). If \r1=\r2\nthen the magnetisation and angular momentum compensation tem-\nperatures coincide.\nisation of the Landau-Lifshitz equation, which is a zero tem-\nperature approach and cannot be used when the temperature\nof the spin system varies in space and time. The primary fi-\nnite temperature micromagnetic approach is the LLB equation\nof motion.\nWe proceed in Section II by introducing the basic theory\nand principles behind the common methods and how they\napply to ferrimagnets. In Section III we explain how two\nmaterials–GdFeCo and yttrium iron garnet–have been mod-\nelled and why these two materials are so focused on in spin-\ntronics. In Section IV we survey the literature where these\nmethods and material models have played an important role\neither in elucidating the understanding of experiments or pre-\ndictions of ferrimagnetic behaviour. We conclude in Section V\nwith an outlook of the future directions for modelling in ferri-\nmagnetic spintronics.\nII. MODELLING TECHNIQUES\nAt a constant temperature and away from the magnetisa-\ntion or angular momentum compensation points ferrimagnets\nbehave like ferromagnets. Ferrimagnets show a net magneti-sation and the low frequency dynamics can be described with\nequations analogous to a ferromagnet. A considerable amount\nof modelling of ferrimagnetic materials has used the conven-\ntional micromagnetism approach which assumes smooth spa-\ntial variation of the net magnetisation. Either by considering\nthe material simply as a ferromagnet, an effective medium ap-\nproach which adjusts the parameters to mimic a given temper-\nature, or by coupling multiple magnetisation vectors in a cell\nor close proximity. These approaches describe many experi-\nmental situations well28,29, but when temperature becomes a\ndynamic variable or higher energy modes play a role in dy-\nnamics or thermodynamics, more sophisticated approaches\nare needed.\nA. Atomistic Spin Dynamics (ASD)\nAtomic spin models are a natural formalism for ferrimag-\nnetism because they can explicitly account for the different\nmagnetic moments at the atomic scale within the material.\nThey are often complemented with statistical physics meth-\nods because with large ensembles of spins, thermodynamic\nproperties can be calculated. A Heisenberg Hamiltonian de-\nscribes the energetics of the system owning to the interactions\nbetween the classical ‘spin’ vectors. The word spin is used\ncolloquially in this field and means the classical local mag-\nnetic moment ascribed to an atomic site. Additional terms of\nrelativistic origin or due to coupling to external stimulus can\nbe added to the Hamiltonian according to the material being\nstudied. A typical example Hamiltonian is\nH=\u00001\n2X\ni6=jJijSi\u0001Sj\u0000X\nidzS2\nz;i\u0000X\ni\u0016iB\u0001Si:(1)\nwhere the first term is the Heisenberg exchange with Jijis the\nelectronic exchange energy (the 1=2is for the double counting\nin the sum) and Siis a classical spin vector of unit length.\nThe second term is a uniaxial anisotropy in the z-direction\nof energydzrepresenting magneto-crystalline effects and the\nthird term is an external magnetic field Bmeasured in Tesla\nand\u0016iis the size of the magnetic moment of the i-th spin in\nBohr magneton.\nThe exchange parameters quantify the interaction energy\nbetween the spins upon small rotation from the ground state,\nas demonstrated by Liechestein et al.30. The exchange inter-\naction here can arise from direct exchange of the overlap of\nelectronic orbitals, typical in transition metals, or indirect such\nas super-exchange mediated by a third (non-magnetic) ion as\nis common in oxides. Ferrimagnets can be described by con-\nsidering this Hamiltonian with two or more distinct types of\nspin. Two possibilities are i) when there the spin moments\nof different atoms have different magnitudes, for example a\ncombination of transition metal and rare earth elements as in\nthe alloy GdFeCo ii) in a crystal where the same element ex-\nists in different environments which are antiferromagnetically\ncoupled. The magnetic garnets are a typical example of this,\nwhere Fe3+ions exist in both octahedral and tetrahedral en-\nvironments in a 2:3 ratio with antiferromagnetic coupling be-\ntween the two sublattices. In material specific models the ex-3\nchange parameters Jijare usually parametrised either from\nneutron scattering measurements of the magnon spectrum or\ndirect calculation of the electronic structure from first princi-\nples methods. Often the exchange interaction contains contri-\nbutions beyond the nearest neighbours which leads to complex\nmagnon band structures.\nClassical atomistic spin models are often solved using\nMonte Carlo techniques31or atomistic spin dynamics–which\nprovides dynamical quantities of interest. The equation of\nmotion for the spin dynamics is the Landau-Lifshitz-Gilbert\n(LLG) equation:\n@Si\n@t=\u0000j\rij\n(1 +\u00152\ni)[(Si\u0002Hi)\u0000\u0015i(Si\u0002(Si\u0002Hi))]:\n(2)\n\u0015iis the (dimensionless) local intrinsic damping assigned to\nthei-th site,\ri=gis the gyromagnetic ratio on the i-th site in\nradians per second per Tesla. Hi=\u0000(1=\u0016i)(@H=@Si)is the\neffective field local to a site iderived from the Hamiltonian.\nThe LLG equation is usually solved numerically for large\ncollections of coupled spins in a manner similar to molecu-\nlar dynamics for atomic motion. Besides the antiferromag-\nnetic interactions between different sublattices, ferrimagnets\nare modelled by using distinct parameters on different atomic\nlattice sites. Elements can have magnetic moments of differ-\ning values, different gyromagnetic ratios or exchange interac-\ntions. This atomic scale detail is something not possible in\nmicromagnetic approaches based on the continuum approxi-\nmation.\nTemperature is included in this dynamical model following\nBrown32by adding a stochastic field \u0010ito the effective field\nHi=\u0010i(t)\u0000(1=\u0016i)(@H=@Si). Equation (2) now becomes\na Langevin equation. Conventionally the classical fluctuation\ndissipation theorem is used (as this is a classical spin model)\nand the statistical properties of the stochastic field are there-\nfore a white noise with correlations33:\nh\u0010i;\u000b(t)i= 0 andh\u0010i;\u000b(0)\u0010j;\f(t)i=2\u0015ikBT\u000eij\u000e\u000b\f\u000e(t)\n\u0016i\ri:\n(3)\nwherekBTis the thermal energy provided by the heat bath\nand\u000b,\fare Cartesian components. Within ASD spin-spin\ninteractions occur naturally due to the coupled system of LLG\nequations and hence their effect is present without explicit\ninclusion–something which often limits analytic thermody-\nnamic models. The magnon-magnon interactions become sig-\nnificant for thermodynamics at temperatures above approxi-\nmately half the Curie temperature ( TC). Recently advances\nhave been made in the thermostating to use the quantum fluc-\ntuation dissipation theorem instead34,35. This is essentially the\nsemi-classical limit often used in analytic theory where spin\nmoments are considered in the classical limit but the quantum\n(Planck) distribution is used as the thermal distribution for the\nmagnons. The use of quantum statistics enables quantitative\ncalculations of thermodynamics even at low temperatures34,36.\nThe thermostated spin dynamics gives a canonical ensem-\nble from which thermodynamics can be calculated. The total\nmagnetisation at a given time is M(t) =P\ni\u0016s;iSi(t)and\na thermodynamic average can be calculated from a long timeseries in equilibrium. For ferrimagnets it is often useful to\ncalculate the sublattice magnetisation, something which only\na few experimental methods can probe.\nM\u0017(t) =X\nj\u0016s;jSj(t) (4)\nwherejare the indices belonging to the \u0017-th sublattice.\nResolving quantities by sublattice links into the Landau-\nLifshitz-Bloch method (described below in Section II D)\nwhich requires these thermodynamic quantities. The thermal\nspin fluctuations cause the system to sample broad regions of\nthe free energy and so although the anisotropy, dz, and ex-\nchange energy, Jij, in the Hamiltonian are independent of\ntemperature, the macroscopic anisotropy (often denoted K)\nand the macroscopic exchange stiffness (often A) are temper-\nature dependent and can be calculated with ASD37. These are\nalso key input parameters for macroscopic formalisms such\nas the micromagnetic and LLB methods. The statistical ther-\nmodynamic aspects of ASD means it can be used to calculate\nphase diagrams. The accuracy exceeds what is possible with\nmean field approaches which are known to significantly over\nestimate critical points due to the absence of magnon-magnon\ninteractions. For a set of material specific parameters, these\nmicroscopic details produce the thermodynamic angular mo-\nmentum and magnetisation compensation points which dis-\ntinguish ferrimagnets from ferro and antiferromagnets. Non-\ncollinear phases such as spin flop which occur in high mag-\nnetic fields can be modelled and even non-equilibrium and\ndynamical states38.\nThe damping parameter in ASD describes the rate of en-\nergy and angular momentum dissipation to the heat bath. It\nrepresents the electron and phonon systems. Detailed descrip-\ntions of the heat-bath dynamics are rarely39,40introduced or\nmodelled in detail. The coupling to electron and phonon sys-\ntems are “direct” damping pathways41involving charge carri-\ners or the lattice, with spin-orbit coupling being paramount\nfor the transfer of angular momentum from the spin space\nto the lattice. The atomic damping parameter appearing in\nEq. (2) is different from the macroscopic damping parameter–\nthe relaxation rate of the total magnetisation–as measured in\nan experiment. The macroscopic damping includes further\ncontributions from “indirect” mechanisms–magnon-magnon\ninteractions–which are internal to the spin system. This re-\nsults in an increase in the damping of the macroscopic mag-\nnetisation with increasing temperature.\nASD is used for both generic modelling as well as mate-\nrial specific modelling. Generic models include many of the\ncomplex effects which are heavily approximated in analytic\ntheory, such as magnon-magnon interactions and non-linear\ndynamics, but without being tied to a single material. This is\nwell exampled in the field of spincaloritronics (Section IV B)\nwhich studies the coupling between spin and heat–an ideal\nfield for the application of ASD.4\nB. Macrospin models\nMuch of the basic theory of ferrimagnets comes from a sim-\nple macrospin model of two coupled magnetisation vectors.\nThis provides simple equations for effective bulk properties\nwithin the limit that the two macrospins are rigidly coupled,\nassuming a strong inter-sublattice exchange interaction.\nIn ferrimagnets the macroscopic damping, \u000be\u000b–defined as\nthe rate of relaxation of transverse magnetisation oscillations–\ndiffers qualitatively from the microscopic damping parame-\nters. The reason is the coupled dynamics of both sublattices.\nThe collective oscillations, described as one effective ferro-\nmagnet, relax at a rate which is a combination of the damping\nparameters from different sublattices, weighted by their mag-\nnetisation. This rate of relaxation is what is experimentally\nobservable42. In the macrospin rigid coupling approximation\nand not too close to the magnetisation compensation point the\neffective damping of the net magnetisation is\n\u000be\u000b(T) =\u000b1(T)M1(T)=\r1+\u000b2(T)M2(T)=\r2\nM1(T)=\r1\u0000M2(T)=\r2: (5)\nAt low temperatures (much below the Curie temperature,\nbefore the magnon-magnon interactions become significant),\nin a first step, \u000b1(T) =\u00151and\u000b2(T) =\u00152can be taken\nas constant. The effective damping scales inversely to the\nnet magnetisation M1\u0000M2, which has a more pronounced\ntemperature dependence than in ferromagnets, with the strik-\ning difference that as the system approaches the magnetisation\ncompensation temperature, M1\u0000M2!0, the macroscopic\ndamping (Eq. (5)) starts to diverge. Divergence of the damp-\ning parameter is unphysical, and points to the failure of the\neffective macrospin approach to describe ferromagnetic mag-\nnetisation dynamics.\nThe local gyromagnetic ratio \rican also be site dependent\nin a ferrimagnet leading to an angular momentum compen-\nsation point where the sublattices no longer precess. For the\nmacrospin model this is\n\re\u000b(T) =M1(T)\u0000M2(T)\nM1(T)=\r1\u0000M2(T)=\r2: (6)\nThe frequency of the lowest energy mode (often called the\nferromagnetic mode), when neglecting damping is\n!(T) =\re\u000b(T)H: (7)\nApproaching the magnetisation compensation point !!0.\nAs demonstrated in Figure 1, at the angular momentum com-\npensation point M1(T)=\r1\u0000M2(T)=\r2= 0so\re\u000band!di-\nverge. When the inclusion of the damping contribution is used\nin the calculations, != (\re\u000b=1 +\u000b2\ne\u000b)H. This correction re-\nmoves the divergence of the frequency at the compensation\npoint.\nC. Conventional micromagnetism\nLength scales covered by atomistic spin dynamics simu-\nlations are limited to only several tens of nanometers of lat-eral size and macrospin approaches fail at describing mag-\nnetic textures, such as domain walls, vortices or skyrmions\n(Section IV C) which are a key aspect of many spintronic de-\nvice functionalities. To model spintronic phenomena happen-\ning at the micrometer scale, such as magnetic domain motion,\none needs to use the so-called micromagnetic models. Mi-\ncromagnetics has become an pivotal tool for not only for the\ninterpretation of experimental results but also for the design\nof spintronics devices and predicting novel effects. Origi-\nnally micromagnetics was developed as a purely theoretical\ntool for the study of magnetisation processes, however nowa-\ndays computer software using finite difference or finite ele-\nment methods to solve the equations for different geometries\nis common43–46.\nConventional micromagnetism was originally formulated\nwithin the ’ferromagnetic’ framework. The basic underlying\nassumption of micromagnetism is that neighbouring atomic\nspins can be considered parallel over a certain scale given\nby the ratio between the anisotropy and exchange interaction.\nThe magnetisation is therefore treated as a continuous field\nwhere at each point of the continuum space rthe magnetisa-\ntion is defined by one vector, m(r). Ferrimagnetism was first\nintroduced in micromagnetics by assuming that the opposing\nsublattices are locked exactly antiparallel and so the coupled\nequations of motion can be re-written with a single Landau-\nLifshitz-Gilbert equation of motion47using the effective pa-\nrameters for the damping (Eq. (5)) and gyromagnetic ratio\n(Eq. (6)). This is sometimes called an effective medium ap-\nproach. It is useful in ferrimagnets with very strong exchange\ncoupling between the sublattices and away from the compen-\nsation points which diverge in the effective parameter equa-\ntions. Micromagnetism for ferrimagnets with explicit consid-\neration of at least two sublattices is recent48–50. The assump-\ntion of a smoothly changing magnetisation only holds in the\nspin subspace corresponding to each magnetic sublattice. At\nthe local level, the spins are aligned antiparallel. Similarly to\nantiferromagnets, this causes an ‘exchange enhancement’ of\nthe spin dynamics. Exchange enhancement effects are impor-\ntant since allow for the speed up of the dynamics, for exam-\nple the motion of domain walls at relativistic velocities–near\nto the maximum magnon velocity of the medium51. Other\nmagnetic textures, such as skyrmions, in ferrimagnets are also\nof great fundamental and technological interest. Micromag-\nnetic models allow the investigation of low frequency dynam-\nics of such magnetic textures at the device scale. However,\nconventional micromagnetism is unable to describe thermal\neffects. These are crucial in fields such as ultrafast spin dy-\nnamics and spincaloritronics and possibly for realistic device\nmodelling due to contact heating. Solutions to the shortcom-\ning of thermodynamic considerations have been proposed for\nexample by rescaling the micromagnetic parameters to their\nvalues of at specific temperatures. Stochastic fields have been\nalso added to the equations of motion, however since the finite\nsize of the computational cells means that the full range of\nspin waves excitations can not be described. At intermediate-\nto-high temperatures the contribution of those excitations to\nthe thermodynamic properties becomes large and they cannot\nbe neglected. An elegant solution is the use of the Landau-5\nLifshitz-Bloch equation as a basis for micromagnetism, first\nderived for ferromagnets22,24and later on for ferrimagnets26.\nD. Landau-Lifshitz-Bloch (LLB) model\nTo include thermal effects into the equation of motion of\nmagnetisation one has to either include a noise term, simi-\nlar to atomistic spin dynamics methods (Section II A), or to\nexpand the equation of motion to take care of the effect of\ntemperature on a mesoscopic level. This leads to the so-\ncalled Landau-Lifshitz-Bloch equation. The LLB equation\ndescribes magnetisation dynamics (rather than spin dynam-\nics) at finite temperatures. It can be considered as an ex-\ntension of already established micromagnetic methods with a\ncomparable numerical effort. Standard micromagnetic mod-\nels are strictly zero temperature formalisms—the length of the\nmagnetisation is fixed. In contrast, the LLB equation admits\nthe relaxation of the magnetisation length. The higher order\nspin-spin interactions are accounted for in the derivation, pro-\nducing an equation in which parameters have a temperature\ndependence. These parameters can be calculated within the\nmean-field approximation (MFA), higher order methods using\nGreen’s functions52, atomistic spin dynamics, or parametrised\nfrom experimental measurements. This makes the LLB equa-\ntion well suited for modelling scenarios where temperature\nalso changes dynamically, such as the field of ultrafast mag-\nnetisation dynamics. Because of the increasing interest in an-\ntiferromagnetic and ferrimagnetic materials the LLB equation\nof motion for two sublattice magnets was recently derived26\nand we briefly introduce it here.\nThe LLB equation in a two sublattice ferrimagnet is ele-\nment specific, and for each sublattice averaged magnetisation\nat each site, mi;\u0017=hSi;\u0017i,(\u0017=1,2), hence, its value is lim-\nited to 1. Moreover, in a first step it is common to assume a\nmagnetically homogeneous state, mi;\u0017!m\u0017for all lattice\nsites at the same sublattice \u0017, the dynamics equation reads as\nfollows\n1\n\r\u0017dm\u0017\ndt=\u0000[m\u0017\u0002He\u000b;\u0017]\u0000\u000b\u0017\n?\nm2\u0017[m\u0017\u0002[m\u0017\u0002He\u000b;\u0017]]\n+\u000b\u0017\nk\nm2\u0017(m\u0017\u0001He\u000b;\u0017)m\u0017: (8)\nThe relaxation of the magnetisation within the LLB equation\ndepends on the longitudinal and transverse damping parame-\nters,\u000b\u0017\nkand\u000b\u0017\n?, and the effective field, He\u000b. The damping\nand other input parameters for the two sublattice LLB equa-\ntion are element specific. When the gyromagnetic ratio, \r\u0017,\nis different for each sublattice, owing to the particular relation\nbetween orbital and spin degrees of freedom at each sublat-\ntice, the total angular momentum compensation point differs\nfrom the magnetisation compensation point. Below TC, the\ndamping parameters \u000b\u0017\nkand\u000b\u0017\n?are\n\u000b\u0017\nk=2kBT\u0015\u0017\neJ0;\u0017; \u000b\u0017\n?=\u0015\u0017 \n1\u0000kBT\neJ0;\u0017!\n: (9)\nwhereeJ0;\u0017=J0;\u0017+jJ0;\u0017\u0014jme;\u0014=me;\u0017,J0;\u0017is the intra-\nsublattice exchange and J0;\u0017\u0014is the inter-sublattice exchange(note the sign of the second term does not depend on the\nsign of the inter-sublattice exchange interaction), me;\u0017is the\nlength of the magnetisation at equilibrium for a given tem-\nperature. Above TCthe longitudinal and transverse damp-\ning parameters are equal and coincide with the expression22\nfor the classical LLB equation of a ferromagnet above TC,\n\u000b\u0017\n?=\u000b\u0017\nk= 2\u0015\u0016T=3TC, and the equation of motion reduces\nto Bloch equation. In Eqs. (9), the intrinsic damping parame-\nters\u0015\u0017depend on the particularities of the spin dissipation at\nthe atomic level as discussed above in Section II A. They can\nbe the same or different for each sublattice. The effective field\nHe\u000b;\u0017for sublattice \u0017is defined as\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;(10)\nwhere H is an applied field, \u0005\u0017 =\n\u0000[m\u0014\u0002[m\u0014\u0002m\u0017]]=m2\n\u0014is transverse to m\u0014, and \u001c\u0017\nis the component of m\u0017parallel to m\u0014, in other words\n\u001c\u0017=m\u0014(m\u0017\u0001m\u0014)=m2\n\u0014, where\u00146=\u0017. The anisotropy\nfield,HA;\u0017is related to the zero-field transverse susceptibility\nor directly to the uniaxial anisotropy. The temperature depen-\ndence of the parameters defining the longitudinal dynamics in\nEq. (10) is\n\u0003\u0017\u0017=1\ne\u001fk;\u0017\u0012\n1 +J0;\u0017\u0014\n\u00160;\u0017e\u001fk;\u0014\u0013\n;\u0003\u0017\u0014=J0;\u0017\u0014\n\u00160;\u0014me;\u0014\nme;\u0017:\n(11)\nWithin the MFA, the equilibrium magnetisation\nof each sublattice can be obtained via the self-\nconsistent solution of the Curie-Weiss equations\nme;\u0017 = L ((J0;\u0017me;\u0017+jJ0;\u0017\u0014jme;\u0014)=kBT), (Lis the\nLangevin function) and the sublattice dependent longitudinal\nsusceptibilities derived directly from them, f\u001f\u0017\nk=@m\u0017=@H26.\nFor temperatures above TC, one can make use of the sym-\nmetry around TC,e\u001f(\u000f) = 2e\u001f(\u0000\u000f), where\u000f= 1\u0000T=TC\nis small26. These parameters can be calculated directly\nfrom MFA, gained from ASD simulations, or measured\nexperimentally.\nE. Comparisons of ASD and LLB approaches\nThere have been several studies comparing the ASD and\nferromagnetic LLB approaches to show consistency26,53,54.\nSince the LLB model describes both the longitudinal and\ntransverse relaxation of the magnetisation, two different types\nof comparison have been conducted so far. In the first case,\ntransverse relaxation, which can also be described by the con-\nventional micromagnetism, the LLB model provides also the\ntemperature dependence of the damping parameters, magneti-\nsation and effective fields. In the second case, longitudinal\nrelaxation, this is described by the last term in Eq. (8), and\naccounts for the dynamics of the magnetisation length. For\nexample, when the temperature of the heat-bath changes, the\nmagnetisation length also changes. This cannot occur in con-\nventional micromagnetic methods.6\nIn transverse relaxation numerical experiments, the main\ndifference between ferromagnets and ferrimagnets is found\naround the magnetisation and angular momentum compensa-\ntion temperatures. As already described by Eqs. (5) and (7),\neffective damping and frequency strongly depends on the net\nmagnetisation and angular momentum, which in turn depend\non temperature. Effective damping also depends on sublattice\ndamping parameters, which are assumed to be constant in con-\nventional micromagnetism, however within the LLB approach\nthey depend on temperature. As highlighted above, features\nsuch as the temperature dependence of damping should obey\nEq. (5). This emerges naturally in ASD but it is important\nthe LLB also reproduces this physics. Direct comparison be-\ntween ASD simulations and LLB predictions were made in\norder to validate the temperature dependence of the parame-\nters involved in the transverse dynamics. Schlickeiser et al.\nperformed this ASD-LLB comparison53. They showed good\nagreement between ASD and LLB results as shown in Fig. 2.\nTheir work makes it clear that the effective damping of ferri-\nmagnets is higher than predicted by macroscopic approxima-\ntions with a constant intrinsic damping parameter. As TCis\napproached, spin-spin interactions increase and so the intrin-\nsic damping increases. The simulations also provide access to\nnot only the effective Gilbert-like damping but also the damp-\ning of individual modes in the magnon spectrum. The ex-\nchange modes have a higher effective damping and a greater\ntemperature dependence. These modes can play an important\npart in non-equilibrium and dynamical processes55.\nIn longitudinal relaxation simulation, one aims to inves-\ntigate the time scales for the magnetisation length to reach\na new equilibrium when the temperature of the heat-bath is\nchanged suddenly. A longitudinal relaxation term was al-\nready proposed by Baryaktar as a direct generalisation of the\nLandau-Lifshitz equation56. However, the theory does not\nanswer what happens with phenomenological relaxation pa-\nrameters. Later on Garanin solved this issue with the LLB\nequation22, and Atxitia et al. extended Garanin’s ideas to\nferrimagnets26. In their work, Atxitia et al. already con-\nducted a systematic ASD-LLB comparison of the element-\nspecific longitudinal relaxation dynamics in a GdFeCo alloy.\nThis study was restricted to relatively large intrinsic damp-\ning parameters, \u0015\u0017= 0:1and temperatures below TC. Later,\nNieves et al.57extended the comparison to all temperatures.\nThe comparison between ASD simulations and LLB predic-\ntion is very good. This good agreement opens the door to use\nthe LLB model to derive analytical expression of the charac-\nteristic time scales of the longitudinal relaxation for situations\nwhere the deviations from equilibrium are relatively small. In\nferrimagnets, the main characteristic of the longitudinal relax-\nation is that the time scales are element specific. For example,\nin GdFeCo, Gd relaxes slower than Fe. This is rooted in their\ndifferent atomic magnetic moments. Interestingly, fluctua-\ntions act differently on Gd and Fe sublattices since the alloys\nare typical>70% Fe the Gd moments are like impurities in-\nside this Fe ferromagnet, therefore Gd behaves as a polarised\nparamagnet. As a consequence, whilst magnetic fluctuations\nmake Fe magnetisation dynamics experience critical slowing\ndown near the Curie temperature, in Gd they do not, and the\nFIG. 2. Comparison of ASD (points) and LLB (red and blue lines)\ncalculations of the effective frequency and damping of the ferromag-\nnetic (blue) and exchange (red) modes in a ferrimagnet. The black\ndashed line is the analytic solution when the temperature dependence\nof the microscopic damping parameters is not included. (Reprinted\nfigure with permission from [F. Schlickeiser et al., Phys. Rev. B, 86,\n214416 (2012)] Copyright (2012) by the American Physical Society.)\nGd sublattice relaxes faster than Fe at this point. This be-\nhaviour was predicted by the LLB model and it has recently\nbeen observed in femtosecond element-resolved experiments\nin GdFeCo58.\nIII. MATERIALS\nFerrimagnetic materials are everywhere, many of mag-\nnetic materials used in research are some kind of ferrimag-\nnet. Actually the majority of “ferromagnetic” insulators are\nferrimagnets59. There are many ferrimagnets with high poten-\ntial for technological applications. However two ferrimagnets\nhave been of sustained interest in spintronics ; GdFeCo metal-\nlic alloys for ultrafast spintronics applications, and YIG, for\ninsulator spintronics. From the computer modelling perspec-\ntive, ASD is ideal for the modelling of those two ferrimagnets.\nFirst, GdFeCo is most commonly grown as an amorphous al-\nloy, which makes it almost impossible for first principle cal-7\nculation of macroscopic properties but it is a problem well\nsuited for ASD and LLB models. Second, YIG presents a\ncomplex spin structure of the unit cell, and potential novel\nspintronic properties emerging from the atomic interactions\nbetween them can only be handled with atomistic models.\nA. Amorphous GdFeCo\nGdFeCo has long been used for magneto-optical based ex-\nperiments but is now finding use in other spintronics areas\ndue to its tuneable nature60–62. It is an amorphous alloy of\nGd, Fe and Co where the content of each can be adjusted in\nthe growth. In the Gd with partially filled 4f-shells a mag-\nnetic moment of\u00187\u0016Barises from the localised electrons\ndue to their coupling according to Hund’s rules. The valence\nband electrons are spin-polarised by the 4f electrons and give\nrise to an additional 5d6s atomic magnetic moment contribu-\ntion of around 0:5\u0016B. They are strongly exchange coupled to\nthe 4f electrons via a very strong intra-atomic exchange inter-\naction of the order of 200 meV , thus one can fairly consider\nboth kind of spins as one locked magnetic moment for most of\nthe physics studied here. The Gd moment is therefore much\nlarger than the Fe and Co moments, but in alloys with only\n15-30 % of Gd the net magnetisation can be balanced forming\nthe compensation points even though there are much fewer\nGd moments. Being able to continuously adjust the composi-\ntion allows fine tuning of the macroscopic quantities such as\nsaturation magnetisation and the temperature of the magneti-\nsation and angular momentum compensation points. In ex-\nperiments, the Gd or FeCo composition are often varied in a\nsystematic set of samples alongside the temperature or applied\nfield. This has lead to important discoveries such as field free\noptical switching38but makes the parameter space to model\nmuch larger55,63\nAlthough GdFeCo is amorphous it has generally been mod-\nelled on a lattice with random site occupancies64. The ex-\nchange interactions are parametrised by nearest neighbour\nvalues chosen to reproduce the magnetisation compensation\nand Curie temperatures63. While the number of neighbours of\neach site is constant (due to the lattice) the number of specific\nFe or Gd neighbours varies from site to site as would be ex-\npected in an amorphous system. Modelling the system as a\ntruly amorphous arrangement of atoms is eminently possible\nbut requires parameters such as the atomic radial distribution\nfunction and the distance dependent exchange which are un-\nknown and hard to obtain. The lattice based random site mod-\nelling is therefore a crossover between attempts at quantitative\nmodelling and toy models, where salient macroscopic features\nare reproduced using a somewhat generic base model.\nOstler et al. performed the first extensive ASD modelling\nof GdFeCo for different compositions, parametrised from a\nseries of experimental measurements63. Compensation points\nas well as critical temperatures were well reproduced by the\nmodel, showing an excellent agreement with both experimen-\ntal and mean field modelling (Fig. 3). They also parametri-\ncally change the Fe-Gd exchange coupling and calculate the\nthermal relaxation time of the element-specific magnetisation\nFIG. 3. The Cuire temperature ( TC- blue squares) and magnetisa-\ntion compensation temperature ( TM- blue circles) of the amorphous\nGdx(FeCo) 1\u0000xASD model as a function of the fractional Gd con-\ntent. Red triangles are experimental measurements of the magneti-\nsation compensation temperature (Reprinted figure with permission\nfrom [T.A. Ostler et al., Phys. Rev. B, 84, 024407 (2011)] Copyright\n(2011) by the American Physical Society.)\nlength, showing that stronger coupling leads to a faster ther-\nmalisation. Since then, the ASD model for GdFeCo proposed\nby Ostler et al. has been used for the description of ultrafast\nall-optical switching with great success, see Section IV A for\nfurther details.\nB. Garnets\nYttrium Iron Garnet (YIG)65is one of the main ferrimag-\nnetic materials used in insulator spintronics research. It’s very\nlow Gilbert damping, \u000b\u00181\u000210\u00005means that spin waves and\nmagnons have very long lifetimes. Recently it has become a\nplatform for spincaloritronics66research where magnon spin\nand heat currents are passed through the YIG16. Many other\ngarnets can be formed by substitution, for example the yttrium\nis easily substituted by other rare-earth elements such as Gd,\nTb or Er and the Fe can be substituted by gallium to reduce\nthe magnetisation and introduce compensation points.\nMacroscopically the magnetic properties of YIG appear\nsimilar to a ferromagnet. Most analytic theories approximates\nYIG as a simple one magnon band ferromagnet. In fact YIG\nis a complex ferrimagnet with the primitive cell containing\n20 Fe atoms. Much effort has been applied in building de-\ntailed, quantitative models of YIG and numerical modelling\nhas played a role in understanding spintronic effects in YIG\nand other garnets such as gadolinium iron garnet (GdIG) be-\nyond the ferromagnetic approximations.\nThe complexity of the crystal structures and magnetic spec-\ntra of ferrimagnetic insulators has meant that parametrising\natomistic models can be difficult. It is common to infer\nHeisenberg exchange parameters from inelastic neutron scat-\ntering measurements of the magnon spectrum. Linear spin8\nwave models are fitted to the measured dispersion67. In these\ncomplex magnets only a few of the magnon modes can be\nidentified in the neutron scattering making the fitting non-\nunique. For example YIG must contain 20 magnon modes but\nneutron measurements see only four or five modes clearly68.\nGd is also present in many ferrimagnets, particularly those\nused in magneto-optical based spintronics, as well as some\ninsulating garnets. But it provides a particular difficulty for\nneutron scattering because its natural isotope has one of the\nlargest neutron capture cross sections. Electronic structure\ncalculations are therefore being increasingly used to establish\nthe exchange interactions in material specific models.\nThe ferrimagnetic insulators like YIG have large electronic\nband gaps making the most common density functional the-\nory (DFT) methods poorly suited for calculations due to the\nsignificant Coulomb interaction. Attempts have been made\nto calculate the magnetic properties of ferrimagnetic garnets\nusing DFT+U but these could not simultaneously provide the\ncorrect electronic and magnetic properties69–72.\nRecently more advanced methods such as quasiparticle\nself-consistent GW73have shown great promise for these ma-\nterials with excellent results obtained for YIG74. This enables\na truly ‘ab initio’ parametrisation of atomic spin models, with-\nout even the choice of a Hubbard ‘U’. Progress in this area\nmay allow accurate modelling of specific ferrimagnetic insu-\nlators even where it is difficult to measure the exchange inter-\nactions experimentally.\nRecent advances in ASD calculations have introduced\nquantum statistics into the thermostat34,35. This means the\nthermal magnons obey Planck rather than Rayleigh-Jeans\nstatistics. The result is that quantitative calculations of ther-\nmodynamic properties can be made across almost the en-\ntire temperature range from zero Kelvin to the Curie point.\nThe method has been verified for complex ferrimagnets such\nas YIG with excellent agreement for the temperature depen-\ndence of magnetisation, magnon heat capacity34and magnon\nspectrum75. Excellent agreement was found. The magnon\nheat capacity in particular is an important parameter for ther-\nmal and spin transport76but can only be measured below\nabout 10 K in experiments due to the overwhelming contri-\nbution of the phonons77. Classical equipartition overestimates\nits value orders of magnitude and using quantum statistic al-\nlowed accurate calculations of the room temperature value34.\nThe ASD calculations showed that at room temperature the\npresence of the higher terahertz modes means the magnon heat\ncapacity is an order of magnitude larger than otherwise pre-\ndicted when assuming a simple single magnon band model78.\nIV . RESEARCH DOMAINS\nA. Ultrafast magnetisation dynamics\nThe field of ultrafast magnetisation dynamics began with\nthe discovery of the sub-picosecond magnetic response of\nNickel to femtosecond duration optical laser pulses79. The\nuse of femtosecond laser pulses has enabled the ultra-\nfast manipulation of the magnetic order, from femtoseconddemagnetisation79–81to ultrafast spin currents82,83and switch-\ning of magnetic polarity38,64,84–86. Later, alternative rapid ex-\ncitations have also been shown to drive magnetisation dynam-\nics, including recent demonstrations using picosecond electric\nand spin currents87,88.\nFerrimagnetic materials play a central role in this field.\nThe most prominent material is amorphous GdFeCo alloys\n(see Sec. III A) since it has long been the only material\nshowing the unusual deterministic heat-induced magnetisa-\ntion switching38. Recently, another class of ferrimagnet,\nMn2RuxGa89, has shown the same all thermal switching. The\ntheoretical description of laser induced all-optical switching\n(AOS)84of the magnetisation in GdFeCo ferrimagnetic alloys\nhas remained a challenge55,63,64,84,90–97Experimental findings\nare mostly compared or interpreted in terms of atomistic spin\ndynamics simulations55,63,98,99and the Landau-Lifshitz-Bloch\nequation26,93,100.\nLaser pulses can be as short as just a few femtoseconds,\nwhich can excite the electron system on timescales of the or-\nder of the exchange interaction allowing the investigation of\nthe fundamental physics governing switching. When a metal-\nlic ferrimagnetic thin film is subjected to a near infrared laser\npulse, only the electrons are accessible to the photon electric\nfield. Initially, the absorbed energy is barely transferred to the\nlattice and consequently the electron system heats up. The\nelectron and phonon temperatures, TelandTph, are decou-\npled for up to several picoseconds until the electron-phonon\ninteraction equilibrates the two heat-baths. This phenomenol-\nogy is well captured by the so-called two-temperature model\n(2TM)101,102which can be written as two coupled differential\nequations:\nCel@Tel\n@t=\u0000gep(Tel\u0000Tph) +Pl(t) (12)\nCph@Tph\n@t= +gep(Tel\u0000Tph): (13)\nThe parameters entering Eqs. (12) and (13) are material de-\npendent. For instance, for GdFeCo alloys, Cel=\relTelwhere\n\rel= 6\u0002102J/m3K2, andCph= 3:8\u0002106J/m3K rep-\nresent the specific heat of the electron- and phonon system.\nThe electron-phonon coupling, gep= 7\u00021017J/m3K. Here,\nPl(t) =P0exp(\u0000t2=t2\np)represents the absorbed energy by\nthe electron system, coming from the laser. The laser duration\nistp.\nASD simulations coupled to the 2TM predicted and ex-\nperiments subsequently confirmed, that the heat loaded by a\nfemtosecond laser pulse into these particular systems is suffi-\ncient to toggle switch the magnetisation polarity38as shown\nin the simulation results of Fig. 4. Despite the Gd and FeCo\nspin sublattices being coupled antiferromagnetically, during\nthe switching process, and for only a few picoseconds, the\nGd and FeCo spins show a parallel alignment, dubbed a ‘tran-\nsient ferromagnetic-like state’. These insights were only pos-\nsible thanks to element-specific time resolved femtosecond\nx-ray magnetic circular dichroism experiments conducted by\nRadu and co-workers64. This class of magnetic switching of-\nfers great promise for future applications as picosecond writ-\ning times have already been demonstrated in micro90and9\nFIG. 4. (a) Electron temperature in the two temperature model simu-\nlating the thermal laser pulses. (b) The sublattice magnetisation dy-\nnamics from ASD, Fe (dashed blue line) and Gd (solid red line). The\nsublattices switch deterministically with only a thermal pulse and no\napplied magnetic field. (c) Net magnetisation as a function of time\nthrough the switching events. (Reprinted figure from [T.A. Ostler et\nal., Nat. Commun., 3, 666 (2012)])\nnanoscale magnetic dots103. Recent experiments have brought\nthis closer to more conventional spintronics paradigms, show-\ning that besides using femtosecond laser pulses, GdFeCo\nnanostructures can be switched by the heat provided by pi-\ncosecond electric currents87and picosecond optical pulses104.\nThanks to ASD simulations it has been recently shown that\nswitching using femtosecond to picosecond heating follows\nthe same switching path.105\nFull micromagnetic models based in the LLB equation have\nbeen able to predict the ultrafast antiferromagnetic skyrmions\ncreation after ultrafast demagnetisation46. Micromagnetic\nsimulations based on the LLB have been also used for the\ndescription of all-optical switching of the magnetisation using\nmultiple-shot technique in cross devices made of synthetic fer-\nrimagnets composed of rare-earth free materials106. Switching\nbehaviours can be controlled by the Curie temperature of each\nlayer in the bilayer, which in turn is controlled by their thick-\nness.\nB. Spincaloritronics\nThe field of spincaloritronics studies the physical ef-\nfects coming out from the coupling between spin and\nheat66. Spincaloritronics phenomena include the spin Seebeck\neffect107–109, spin Peltier effect110,111, spin Nernst effect112, in\nmetallic, semiconductor, non-magnetic and magnetic insula-\ntors. Magnetic insulators are extensively used here because\nthey allow the study of purely magnon driven effects with-\nout charge transport convoluting results. Ideally, experiments\nwould study the magnon transport in ferromagnets before\nbranching into ferri- and antiferromagnets but, as mentioned\nbefore, most “ferromagnetic” magnetic insulators are in-fact\nferrimagnets. Many theoretical works in this field neglect thecomplications associated to the existence of at least two an-\ntiparallel sublattices and approximate the materials as a ferro-\nmagnet by assuming a single magnon band with a ~!\u0018k2\ndispersion. To explore what such approximations may miss\nin a true ferrimagnetic material ASD modelling has been used\nextensively.\nThe magnonic spin Seebeck effect in a ferrimagnet–the re-\nsponse of the magnetisation under a temperature gradient–has\nbeen studied using ASD simulations considering the simpli-\nfied situation of a temperature step instead of a gradient. In\nthis context, ASD modelling of a generic two sublattice fer-\nrimagnet in a temperature step provided information about\nthe spatial distribution of sublattice magnetisation113. In this\nnumerical experiment, a stationary non-equilibrium magnon\naccumulation arises and the shape corresponds to the deriva-\ntive of the equilibrium magnetisation @mz=@T in the absence\nof temperature step. In ferromagnets, @mz=@T < 0, but in\nferrimagnets showing compensation points, the sign of it can\nchange, hence, the existence of a temperature at which mag-\nnetisation accumulation vanishes at the step site.\nModelling in this field has also drawn attention to the\nmagnon polarisation in ferrimagnets68,75,114. Here polarisation\nis related to the clockwise versus anticlockwise rotation of\nthe spins or alternatively whether a magnon mode carries +~\nor\u0000~spin angular momentum. In a ferromagnet, magnons\nhave a single circular polarisation (or elliptical with dipolar\ninteractions) corresponding to the anticlockwise rotation of\nthe magnetic moments in a field. In a uniaxial antiferromag-\nnet there are both anticlockwise and clockwise polarisations\nbut the magnon modes degenerate so the polarisation cannot\neasily be measured. In a ferrimagnet the anticlockwise and\nclockwise polarisations also exist due to the opposing sublat-\ntices, but the exchange field between the sublattices splits the\nmodes so that clockwise magnons are higher in energy than\nanticlockwise.\nExperimental measurements and ASD modelling of the\nspin Seebeck effect in Gadolinium iron garnet (GdIG)\nshowed that magnon polarisation has observable effects in\nspintronics115. Two changes of sign in the spin Seebeck effect\nwere found as the temperature is increased. One is expected\nacross the magnetisation compensation point as the sublattices\nreverse in the applied field across this point. The change in\nsign of the spin Seebeck effect at low temperatures does not\ncorrespond to any macroscopic changes of the ferrimagnet.\nTheory and ASD modelling showed that the changing ther-\nmal occupation of magnon modes with different polarisation\ncauses the sign change at low temperature.\nAlthough the polarisation of magnons has been described\nby theory from the early understanding of magnons it had\nnever been directly measured, possibly due to the lack of ap-\nparent physical consequence until recently. Polarised inelas-\ntic neutron scattering was used to measure the polarisation of\nmagnons for the first time75, supported by quantitative ASD\nmodelling. The scattering cross section in the required exper-\nimental geometry is very small making the experiments dif-\nficult. The first attempt was only partially successful which\nwas identified because of the prior calculation of the polarised\nneutron scattering cross section using ASD modelling. A sec-10\nFIG. 5. Upper (c) experimental measurements of the magnetic chi-\nral part of polarised inelastic neutron scattering from YIG. Lower\n(d) ASD calculations of the polarised inelastic neutron scattering\ncross section from a parametrised model of YIG. The ASD results\nhave been convoluted with the experimental measurement resolution.\n(Reprinted figure with permission from [Y . Nambu et al., Phys. Rev.\nLett., 125, 027201 (2020)] Copyright (2020) by the American Phys-\nical Society.)\nond measurement was successful and very good agreement\nwas found between the ASD calculations and the experimen-\ntal measurements as shown in Fig. 5. The polarisation of dif-\nferent magnon modes may be useful in spintronics applica-\ntions as the magnon transport appears to be sensitive to the po-\nlarisation in experiments116. Theoretical suggestions of how\nto use the polarisation are also being made117.\nThe high frequency magnon modes of the YIG has\nmotivated experiments to excite these high frequency\nmodes118–121. Maehrlein et al.118resonantly excited long-\nwavelength THz phonons in YIG using THz laser pulses.\nChanges in the magnetisation were measured via the magneto-\noptical Faraday effect with an optical probe pulse. A reduction\nin the sublattice magnetisation was observed on a picosecond\ntimescale118. The change in magnetisation persists for mi-\ncroseconds before recovering. The excitation of the infra-red\nactive THz phonons predominantly makes the light oxygen\nions to move, displacing them from their equilibrium posi-\ntions. It is assumed to cause fluctuations in the superexchange\nbetween Fe atoms due to the changes in the bond distances\nand angles, shown pictorially in Fig. 6A. To confirm the ex-\nperimental findings ASD calculations were performed with\na modified Heisenberg exchange which included dynamical\nstochastic fluctuations of the intersublattice exchange param-\neters. From the model it was determined that the net magneti-\nsation remains constant, even though the sublattice magneti-\nFIG. 6. (A) Superexchange model of the inter-sublattice coupling in\nYIG. The oxygen displacement ( u(t)) caused by phonon excitations\nat 20 THz causes fluctuations in the superexchange Jad. (B) ASD\nmodel with stochastic exchange for 0.5 ps. The change in magneti-\nsation of each sublattice is shown in blue and green respectively. The\ntotal change in magnetisation (black line) is zero. (Reprinted figure\nfrom [Maehrlein et al., Sci. Adv., 4, eaar5164 (2018)])\nsation is decreasing (Fig. 6B). This is because the isotropic\nexchange interaction conserves the the spin angular momen-\ntum.\nThe fields of ultrafast magnetisation dynamics and\nspincaloritronics sometimes combine and effects such as\nthe spin Seebeck effect are probed on the sub-picosecond\ntimescale to understand the timescales of the fundamental pro-\ncesses. Seifert et al. used a femtosecond laser to suddenly\nheat the Pt of a YIG-Pt bilayer119. The YIG is transparent\nto the laser and so there is a significant temperature differ-\nence between the hot electrons in the Pt and the magnons and\nphonons of the YIG. The aim was to understand how quickly\nthe spin current across the YIG-Pt interface is established due\nto electron-magnon scattering. The spin current (measured\nfrom the THz emission due to the current formed by the in-\nverse spin Hall effect) emerges almost instantaneously and\npeaks within 500 femtoseconds of the laser excitation. A dy-\nnamical theory was created which required parameters such as\nthe frequency dependent spin susceptibility of the YIG. This11\nwas calculated using ASD with a detailed model of YIG and\nshowed that the magnetic system can respond instantly due to\nthe lack of inertia. The quantitative theory allowed an esti-\nmate of the interfacial sd-exchange coupling and spin mixing\nconductance in good agreement with other values in the liter-\nature.\nC. Magnetic textures\nThe motion of spin textures such as domain walls and\nskyrmions is a large area of study because of proposed de-\nvice concepts using the textures for information storage and\nprocessing. Much work has been done on ferromagnetic tex-\ntures but ferrimagnets are relatively unexplored beyond treat-\ning a ferrimagnet as a ferromagnet. The main difference be-\ntween ferro and ferrimagnets that can be expected is in how\nthe dynamics of textures changes due to the presence of the\nangular momentum and magnetisation compensation points.\nIt is expected that at the angular compensation point, fer-\nrimagnets behave like antiferromagnets. Antiferromagnetic\ntextures can move at high velocities due to the compensation\nof torques acting at each sublattice and responsible of the so-\ncalled Walker breakdown common for ferromagnets. Analyti-\ncal theory based on existing macrospin approaches have been\nused to predict and interpret domain wall dynamics under ex-\nternal fieldH122. The velocity, vDWand precession, \nDW\nof the ferrimagnetic domain wall of width \u0001DWcan be ex-\npressed in terms of effective damping (Eq. (5) and gyromag-\nnetic ratio (Eq. (6):\nvDW= \u0001 DW\u000be\u000b\n1 +\u000b2\ne\u000b\re\u000bH; \n DW=1\n1 +\u000b2\ne\u000b\re\u000bH:\n(14)\nStill, thermal effects, such as motion of magnetic textures\nunder thermal gradients require computational models. ASD\nmodelling has been useful in this area. Donges et al. per-\nformed an extensive study of how domain walls in ferrimag-\nnets move when thermal gradients are applied123. In ferro- and\nantiferromagnets, domain walls always move towards hotter\nregions124–126. In ferrimagnets they found the same behaviour\nbelow the Walker breakdown–where domain walls behave like\nan effective ferromagnetic DW under thermal gradient. But\nabove the Walker breakdown the domain walls can move to-\nwards colder regions if the temperature is less than the angular\nmomentum compensation point as shown in Fig. 7. Their re-\nsults explain anomalies in experimental measurements of do-\nmain wall motion in ultrafast laser experiments on GdFeCo\nwhere domain walls have been observed to move away from\nthe heated region127–the opposite of previous expectations.\nThis highlights the need for ASD modelling of ferrimagnets\nas these features cannot be seen in ferromagnetic models such\nas Eqs. (14) and are difficult to isolate in experiments from\nmany other possible effects. They also find a ‘torque com-\npensation point’ where the ferrimagnetic domain wall moves\nsimilarly to an antiferromagnet domain wall with inertia free\nmotion and no Walker breakdown.\nIn the magnetic skyrmion field there is an effort to remove\nor control the skyrmion Hall effect–the transverse motion of askyrmion when it is pushed by a directional stimulus. In an-\ntiferromagnets it is known that skyrmions have no hall effect\ndue to the exact cancellation of the Magnus force from the\ntwo sublattices114,128. Recent experiments in GdFeCo have\ndemonstrated that the skyrmion Hall effect can also be zero in\nferrimagnets at the angular momentum compensation point129\n. The experimental evidence comes from MOKE imaging and\nwere compared with ASD simulations showing a good agree-\nment confirming the absence of a skyrmion Hall effect when\nthe next spin density is zero.\nV . SUMMARY AND OUTLOOK\nThe modelling techniques presented in this review have al-\nready played a central role in uncovering material properties\nand physical phenomena in ferrimagnetic spintronics which\nare absent in the commonly used effective ‘ferromagnetic’\nmodels. Even with the rise of antiferromagntic spintronics130,\nwe believe ferrimagnetic spintronics will grow due to the\nmanipulations that can be induced by use of the compensa-\ntion points. One of the most important aspects of computer\nmodelling is the close collaboration with experimental part-\nners. This allows not only to interpret experimental results\nbut also to validate theories and predict new physical phe-\nnomena. Paradigmatic examples have been covered in this\nreview. For example, in the field of ultrafast dynamics, mod-\nelling techniques have been pivotal to the interpretation of the\ntransient ferromagnetic-like state in GdFeCo, validation of the\nfemtosecond formation of the Spin Seebeck effect in YIG,\nand the prediction of thermally-induced ultrafast magnetisa-\ntion switching in GdFeCo. One characteristic of ferrimagnets\nis that interesting physics appears when the energy input is\nof the same order as the antiferromagnetic coupling between\ndistinct sublattices. For example, by changing temperature\n(thermal energy input), magnetisation and angular momentum\ncompensation point can be crossed at which some dynami-\ncal properties are similar to antiferromagnets. Lately, models\nwhich include thermal effects have dominated this field. Im-\nprovements in the modelling of temperature by inclusion of\nquantum statistics are likely to become more prevalent and\nprovide more predictive weight to multiscale simulations. Ef-\nforts to directly model the heat bath subsystems are also likely\nto increase as there is much interest in the strong coupling be-\ntween systems such as phonons and magnons. As space and\ntimescales are pushed further models at the atomic level must\nbe further developed beyond the localised rigid spin approx-\nimation. There is also a notable absence of models for ele-\nments with significant orbital moments which may be required\nfor rare-earth elements which are found in many ferrimagnets.\nGoing forward we anticipate ferrimagnets to be ever more\ninvestigated for spintronic applications. Ferrimagnets are the\ndoor for fast control of magnetic functionalities as given by\ntheir antiferromagnetic character while conserving the advan-\ntage of ferromagnets in terms of measuring and controlling\nby conventional means such as magnetic fields. Recent ex-\namples, include the field of skyrmionics and spin-orbit torque\nswitching. Ferrimagnetic insulators in particular are likely to12\nFIG. 7. Velocity, precession and tilting of ferrimagnetic domain walls (DW) under thermal gradients from ASD simulations. (a) Biaxial\nanisotropy ferrimagnetic DWs dynamics can be described as an effective ferromagnet, velocity scales with temperature gradient. (b) Tilting\nof the DW vanishes at the torque compensation temperature Ttat which the DW dynamics shows similar characteristics to antiferromagnets,\nnamely, quasi-inertia-free motion and the absence of Walker breakdown. (c) Uniaxial anisotropy. Above Walker breakdown, the ferrimagnetic\nDW can show the opposite, counter intuitive behaviour of moving toward the cold end. Below the compensation temperature the wall moves\ntoward the cold end, whereas above it toward the hot end. This behaviour is driven by angular momentum transfer and therefore strongly\nrelated to the angular momentum compensation temperature. (e) The inset shows the temperature dependence of the non-adiabaticy parameter\n\fe\u000b, defined as the ratio between adiabatic and non-adiabatic torque created by a spin current. (Reprinted figure from [Donges et al., Phys.\nRev. Research, 2, 013293 (2020)])\ncontinue to be used widely in experiments as they provide an\nimportant spin source for fundamental and device inspired ex-\nperiments which attempt to modify spin current transmission.\nRecently, practical realisation of relativistic kinematics in iso-\nlated magnetic solitons has been demonstrated51in ferrimag-\nnets, establishing an experimental framework to study rela-\ntivistic solitonic physics.\nSo far ferrimagnets have hardly featured in some popular\ntopics such as topological magnetism where the additional\nsymmetries allowed by a two sublattice magnet with a net\nmagnetisation may be of interest. A similarly open avenue\nto be explored is the investigation of two-dimensional ferri-\nmagnets. A promising route to realise this class of 2D ferri-magnets could be by exploiting the highly tunable superlattice\npatterns in twisted (or moiré) bilayers of atomically thin ma-\nterials. 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Nan-\notechnol. 11, 231 (2016)."    },    {        "title": "1310.1782v1.Ferrimagnetic_Slater_Insulator_Phase_of_the_Sn_Ge_111__Surface.pdf",        "content": "arXiv:1310.1782v1  [cond-mat.mes-hall]  7 Oct 2013FerrimagneticSlater InsulatorPhaseoftheSn/Ge(111)Sur face\nJun-Ho Lee, Hyun-Jung Kim, and Jun-Hyung Cho∗\nDepartment of Physics and Research Institute for Natural Sc iences,\nHanyang University, 17 Haengdang-Dong, Seongdong-Ku, Seo ul 133-791, Korea\n(Dated: June 7, 2021)\nWe have performed the semilocal and hybrid density-functio nal theory (DFT) studies of the Sn/Ge(111)\nsurface to identify the origin of the observed insulating√\n3×√\n3 phase below ∼30 K. Contrasting with the\nsemilocal DFT calculation predicting a metallic 3 ×3 ground state, the hybrid DFT calculation including van\nder Waals interactions shows that the insulating ferrimagn etic structure with√\n3×√\n3 structural symmetry is\nenergetically favored over the metallic 3 ×3 structure. It is revealed that the correction of self-inte raction error\nwith a hybrid exchange-correlation functional gives rise t o a band-gap opening induced by a ferrimagnetic\norder. The results manifest that the observed insulatingph ase is attributedtothe Slatermechanism via itinerant\nmagnetic order ratherthan the hithertoaccepted Mott-Hubb ard mechanism via electron correlations.\nPACS numbers: 71.30.+h, 73.20.At, 75.50.Gg\nTwo-dimensional (2D) electronic systems formed at crys-\ntal surfaceshave attractedmuch attentionbecause of their in-\ntriguing physical phenomena such as charge density waves\n(CDW), magnetic order, Mott insulators, and 2D supercon-\nductivity [1–3]. One of the most popular quasi-2D systems\nis the√\n3×√\n3 phase formed by the 1/3-monolayer adsorp-\ntion of group IV metal atoms, Sn or Pb, on the Si(111) or\nGe(111) surface [4–12]. Here, adatoms locating at T4sites\n[Fig. 1(a)] saturate all the dangling bonds (DBs) of Si or\nGe surface atoms, leaving a single DB for each adatom [13].\nSincetheseDBelectronsareseparatedasfaras ∼7˚A,there-\nsulting narrow half-filled DB band is likely to invoke variou s\ninstabilitiesandstrongelectroncorrelations.\nWe here focus on a prototypical example of quasi-2D sys-\ntems,Sn/Ge(111). Below ∼220K,thissystemundergoesare-\nversible phase transition from a√\n3×√\n3R30◦(hereafter des-\nignated as√\n3×√\n3) structure to a 3 ×3 structure [14]. This\nphase transition was initially interpreted in terms of a sur -\nface CDW formation stabilized by electron correlation ef-\nfects in the√\n3×√\n3 structure [14, 15]. However, the high-\ntemperature√\n3×√\n3 phase was later explained as a dynam-\nical effect in which inequivalent Sn atoms interchange thei r\nverticalpositions[16–18]: i.e.,thethreeSnatoms[up(U) and\ndown (D) atoms in Fig. 1(b)] of two different heights within\nthe3×3structurefluctuatebetweentwopositionsastempera-\nture increases, apparentlyshowinga√\n3×√\n3 structural sym-\nmetry [18]. Nevertheless, the ground state of Sn/Ge(111)\nis still subject to much debate because, as the temperature\nfurther decreases to ∼30 K, various experimental techniques\nsuch as scanning tunneling microscopy (STM), low-energy\nelectron diffraction, and angle-resolved photoemission s pec-\ntroscopy (ARPES) explored a phase transition from the 3 ×3\nstructureto a new√\n3×√\n3 structure wherethree Sn atomsin\n3×3 unit cell have an equivalent height [19]. This structural\nphase transition was observed to accompany simultaneously\na metal-insulator transition (MIT) [19]. However, previou s\ndensity-functional theory (DFT) studies [20–25] with the l o-\ncaldensityapproximation(LDA)andgeneralizedgradienta p-\nproximation(GGA) predicted that the√\n3×√\n3 structure wasnot only metallic but also less stable than the 3 ×3 structure,\nthereby failing to describe the electronic and energetic pr op-\nerties of the observed insulating√\n3×√\n3 phase. To resolve\nthis problem of LDA, Profeta and Tosatti [4] took into ac-\ncountCoulombinteractions(Hubbard U)using the LDA+ U\nscheme. They found that electron correlations can stabiliz e\na magnetic insulator with√\n3×√\n3 structural symmetry,indi-\ncating a Mott-Hubbard insulating ground state. On the other\nhand,Flores et al.pointedoutthattheeffectofelectroncorre-\nlations cannotinduce a transition to a Mott insulating grou nd\nstate [22]. Thus, it remains elusive whether the formation o f\nthe insulating phase of Sn/Ge(111) is attributed to electro n\ncorrelations[26].\n(a) (b)\nD D\n(c)U\nFIG. 1: (Color on line) (a) Top and (b) side views of the metall ic\n3×3 structure of Sn/Ge(111). The side view of the insulating fe rri-\nmagneticstructureisgivenin(c). Thedarkandgraycircles represent\nSn and Ge atoms, respectively. U and D in (b) represent the up a nd\ndownSnatoms,respectively. TheSnatomsarelocatedatthe T4site,\ni.e.,asingle threefoldhollowsiteabove asecondlayerGea tom. For\ndistinction, Ge atoms in the subsurface layers are drawn wit h small\ncircles. In (a), the 3 ×3 and√\n3×√\n3 unit cells are indicated by the\nsolidand dashedlines, respectively. The xandzdirections in(b) are\n[112]and [111], respectively.\nIn the present work, we propose a new origin of the ob-\nserved insulating phase in Sn/Ge(111) due to itinerant mag-\nnetism. This magneticallydriveninsulatingphase via an it in-\nerant single-electron approach is characterized as a Slate r in-\nTypesetbyREVT EX2\nsulator[27]. ItisfoundthattheDBstatesofSnatomsexhibi t\nan itinerant character because of their significant hybridi za-\ntion with the Ge surface states, differing from the case of th e\npreviously proposed Mott-Hubbard insulator [4] where each\nDBelectronwastreatedtobelocalizedatSnadatomsite. We\nnotethat theLDAandGGA tendto stabilizeartificially delo-\ncalized electronic states due to their inherent self-inter action\nerror (SIE), because delocalization reduces the spurious s elf-\nrepulsion of electron [28, 29]. In this regard, previous LDA\nand GGA calculations for Sn/Ge(111) may overestimate the\nstabilityofthemetallic3 ×3structure[20–25]. Itis,therefore,\nvery interesting to examine if the observed insulating phas e\ncan be predicted by the correction of SIE with an exchange-\ncorrelationfunctionalbeyondtheLDAorGGA.\nIn this Letter, we present a new theoretical study for\nSn/Ge(111) based on the hybrid DFT scheme including\nvan der Waals (vdW) [30] interactions (termed DFT+vdW\nscheme). We find that the correction of SIE with the hybrid\nexchange-correlation functional of Heyd-Scuseria-Ernze rhof\n(HSE) [31] stabilizes the insulating ferrimagnetic (FI) st ruc-\nture where the band-gap opening occurs by a FI spin order-\ning of three Sn atoms within the 3 ×3 unit cell. Here, the\nbuckling of three Sn atoms is suppressed to show apparently\na√\n3×√\n3 structural symmetry, and the stability of the FI\nstructure is further enhanced by the inclusion of vdW inter-\nactions. The calculated magnetic moment of the FI structure\nis well distributed over Sn adatoms as well as Ge substrate\natoms, giving rise to a magnitude of 1 µBper 3×3 unit cell.\nIt is thus demonstrated that the observed insulating phase i n\nSn/Ge(111) can be represented as a Slater insulator through\nitinerantmagnetism,notasthepreviously[4]proposedMot t-\nHubbardinsulatorbyCoulombinteractions U.\nThe present semilocal and hybrid DFT calculations were\nperformed using the FHI-aims [32] code for an accurate,\nall-electron description based on numeric atom-centered o r-\nbitals, with “tight” computationalsettings. For the excha nge-\ncorrelation energy, we employed the GGA functional of\nPerdew-Burke-Ernzerhof (PBE) [33] as well as the hybrid\nfunctional of HSE [31]. The k-space integration was done\nwith the 15 ×15 and 9×9 uniformmeshesin the surface Bril-\nlouin zones of the√\n3×√\n3 and 3×3 unit cells, respectively.\nThe Ge(111) substrate was modeled by a 6-layer slab with\n∼34˚Aofvacuuminbetweentheslabs[34]. Here,weusedthe\noptimizedGelatticeconstants a0=5.783,5.718,and5.667 ˚A\nfor the PBE, HSE, and HSE+vdW calculations, respectively.\nThe HSE+vdWlattice constant [35, 36] agreesmost with the\nexperimentalvalueof5.658 ˚A[37]. EachGe atominthebot-\ntom layer was passivated by one H atom. All atoms except\nthe bottom layer were allowed to relax along the calculated\nforces until all the residual force components were less tha n\n0.02 eV/ ˚A. The employed HSE+vdW scheme was success-\nfully applied to determine the energy stability of the metal lic\nandinsulatingphasesinindiumnanowiresonSi(111)[38].\nWe begin to optimize the nonmagnetic (NM)√\n3×√\n3 and\n3×3 structures using the PBE functional. The optimized top\nand side views of the 3 ×3 structure are displayed in Fig.TABLEI:Calculatedtotalenergies(inmeVper√\n3×√\n3unitcell)of\nthe 1U2D, FM, and FI structures relative to the NM√\n3×√\n3 struc-\nture. For comparison, the previous LDA and GGA results are al so\ngiven.\n1U2D FM FI\nPBE −10.1 − −\nHSE+vdW −31.6 −18.7 −37.7\nLDA (Ref. [21]) −5 − −\nLDA (Ref. [24]) −7.5 − −\nLDA (Ref. [4]) −9 − −\nGGA (Ref. [17]) −5 − −\n-1-0.5 0 0.5 1\nΓ KM ΓEnergy (eV)\n-1-0.5 0 0.5 1\nΓ KM ΓEnergy (eV)\nS1S1S2ΓΜ\nΚ\nS3\n→S2S3\n→\nDUDDUD\n(b)(a)→\nFIG.2: (Coloronline)(a)Surfacebandstructureofthe1U2D struc-\nture computed using the PBE functional. The spin-polarized surface\nband structure of the FI structure computed using the HSE+vd W\nscheme is given in (b). The band dispersions are plotted alon g the\nsymmetrylinesoftheBrillouinzone ofthe3 ×3unitcell[seethein-\nset in(a)]. The Γ-M line corresponds to [11 2] direction. The charge\ncharacters of the DB states at the Γpoint are also displayed with an\nisosurfaceof0.006e/ ˚A3. TheenergyzerorepresentstheFermilevel.\nThe majority and minoritybands in (b) are drawn with the soli d and\ndashed lines,respectively.\n1(a) and 1(b), respectively. It is seen that the U atom posi-\ntions higher than the two D atoms by 0.35 ˚A, in good agree-\nment with those (ranging from 0.26 to 0.36 ˚A) of previous\nDFT studies [4, 20–25, 39]. This so-called 1U2D structure\nis more stable than the NM√\n3×√\n3 structure by 10.1 meV\nper√\n3×√\n3 unit cell, which is well comparable with previ-\nous DFT results (see Table I). As shown in Fig. 1S of the\nSupplemental Material [40] [Fig. 2(a)], the calculated ban d\nstructure for the NM√\n3×√\n3 (1U2D) structure exhibits the3\npresenceofoccupiedDBstate(s)attheFermilevel,indicat ing\na metallic feature. It is revealed that the charge character s of\nthe DB states in the 1U2D structure [see Fig. 2(a)] represent\na chargetransferfromtheD to the U atoms[24]. Thischarge\ntransfergivesrisetoareducedCoulombrepulsionbetweenS n\nDB electrons, resulting in a more stabilization over the NM√\n3×√\n3 structure. Ourspin-polarizedPBE calculationswere\nnotabletofindanyspinorderingwithinthe√\n3×√\n3and3×3\nunit cells. Thus, we can say that the semilocal DFT scheme\nwiththePBEfunctionalcannotpredicttheinsulating√\n3×√\n3\nphaseobservedbelow ∼30K [19].\nItisnoteworthythatarecenthigh-resolutionphotoemissi on\nstudy [5] for the Sn 4 dcore level of the metallic 3 ×3 struc-\nture resolved three components,which were assigned to each\nof the three Sn atoms within the 3 ×3 unit cell. On the basis\nofthisphotoemissiondatatogetherwith STMimages,Tejeda\net al.[5] concluded that the two D atoms position at slightly\ndifferentheights, formingan inequivalent-down-atoms(I DA)\nstructure. Our PBE calculation shows that the IDA struc-\nture with a height difference of ∼0.03˚A between the two D\natoms is almost degenerate in energy (less than 0.1 meV per\nSn atom)with the 1U2D structure,consistent with a previous\nLDA calculation [39]. For the 1U2D and IDA structures, we\ncalculate the Sn 4 dcore-levelshifts using initial-state theory,\nwhere the shift is defined by the difference of the eigenval-\nues of the Sn 4 dcore level at different sites. The results are\ndisplayedinFig. 3andcomparedtothephotoemissionexper-\niment[5]. WefindthateachSn4 dcorelevelissplitintothree\nsublevels, i.e., the degenerate Cd1(Cd2) sublevel arising from\nthedxyanddx2−y2(dyzanddxz) orbitals and the Cd3sublevel\nfrom the dz2orbital. Note that such a crystal field splitting\nis conspicuous for the two D atoms but negligible for the U\natom. On the basis of the calculated initial-state core leve ls,\nthe observed C1,C2, andC3components (see Fig. 3) can be\nassociated with the three sublevels of the U atom, Cd1of the\ntwo D atoms, and Cd2andCd3of the two D atoms, respec-\ntively. Thus, we can say that the observed two components\nof higher binding energy are attributed to the effect of crys -\ntal field splitting [41] on the Sn 4 dcore levels of the two D\natoms,ratherthanto differentcorelevelsforthetwo inequ iv-\nalentDatomsintheIDAstructure[5]. Theinitial-statethe ory\nfor the 1U2D (IDA) structure shows that the Cd1,Cd2, and\nCd3sublevels for the two D atoms shift to higher binding en-\nergy by 155 (157 ±1), 237 (238 ±2), and 256 (257 ±3) meV,\nrespectively, relative to the average value of three sublev els\nfor the up atom. These shifts are smaller than those (230 ±40\nand 390±40 meV) of C2andC3relative to C1, which were\nresolved from the high-resolution photoemission spectra [ 5].\nThis differenceof Sn 4 dcore-level shifts between the initial-\nstatetheoryandthephotoemissionexperimentmayreflectth e\nfinal-statescreeningeffects[42].\nIn order to provide an explanation for the observed insu-\nlating√\n3×√\n3 phase, Profeta and Tosatti [4] performed the\nLDA +Ucalculation to propose the Mott-Hubbardinsulator,\nwhere the inclusion of electron correlations strongly modi -\nfies the ground state of Sn/Ge(111) from a NM 3 ×3 metal0.40.30.20.1 00.40.30.20.1 0Experiment\nC3 C2 C1\nTheory-1U2Dz z\nd2z\ndC3dC2dC1 U\nD1\nD2\n0.40.30.20.1 0Theory-IDA\nBinding energy (eV) dC3dC2dC1\ndyz\nd2x2-y dxzdxy\nx y x y\nFIG. 3: (Color on line) Calculated Sn 4 dsurface core-level shifts of\nthe1U2DandIDAstructures,incomparisonwiththehigh-res olution\nphotoemission experiment [5]. The shifts for the two D atoms (D1\nand D2) are given with respect to the average of the three subl evels\nfor the U atom. The positive sign indicates a shift to higher b inding\nenergy. The five dorbitals of the D1 atom are displayed with an\nisosurface of ±0.2(e/˚A3)1/2.\ntoamagneticinsulatorwith√\n3×√\n3structuralsymmetry[4].\nHere, the FI spin order in the Mott-Hubbard insulating state\nwasstabilizedwithina localizedorHeisenbergpicturewhe re\nanunpairedelectronwaslocalizedateachSnadatomsitewit h\na spin moment of 1 µB. This localized picture for magnetism\ncontrastswiththepresentitinerantorSlatermagnetismwh ere\nthe magnetic moment is well delocalized over Sn adatoms\nas well as Ge substrate atoms, as discussed below. Since\nthe Sn DB state is largely delocalized through hybridizatio n\nwith the Ge surface states [see Fig. 2(a)], the SIE inherent\nto the PBE functional may cause the incorrect prediction of\nthe metallic 3 ×3 structure as a ground state. To circumvent\nsuch an over-delocalization of Sn DB electrons, we use the\nHSE+vdW scheme to optimize the NM and magnetic struc-\ntures of Sn/Ge(111). The calculated total energies of the\n1U2D, FI, and ferromagnetic (FM) structures relative to the\nNM-√\n3×√\n3 structure are given in Table I. We find that the\nFI structure is energetically favored over the 1U2D and FM\nonesby6.1and19.0meVper√\n3×√\n3unitcell,respectively.\nThe optimized geometry of the FI structure is shown in Fig.\n1(b), where the buckling of three Sn atoms within the 3 ×3\nunit cell is suppressed to become flat, leading to a√\n3×√\n3\nstructural symmetry [43]. In Fig. 2(b), the calculated band\nstructure of the FI structure shows a band-gap opening of 71\nmeV, in goodagreementwith the ARPES measurementof60\nmeV [19]. Therefore, the present HSE+vdW calculation pre-4\ndicts an insulating FI ground state with√\n3×√\n3 structural\nsymmetry, consistent with the observed insulating√\n3×√\n3\nphase[19].\nSince the total energyobtained from the HSE+vdW calcu-\nlation is composed of the HSE energy ( EHSE) and the vdW\nenergy(EvdW), the total energydifferencebetween the 1U2D\nand FI structures can be divided into the two components,\nΔEHSEandΔEvdW. For this decomposition,we obtain ΔEHSE\n=4.9meV[44],whichislargerthan ΔEvdW=1.2meV.Thus,\nwecansaythatthecorrectionofSIEwiththeHSEfunctional\ngivesa more dominantcontributionto the stabilization of t he\ninsulatingFIstructureoverthemetallic1U2Dstructure,c om-\nparedtothat fromvdWinteractions.\nFigure2(b)alsoshowsthechargecharactersofthespin-up\n(denotedas S1↑andS3↑)andspin-down( S2↓)DBstatesinthe\nFIstructure. Itisrevealedthat S1↑andS3↑representsomehy-\nbridization of two DB electrons. All of the three DB states\nstronglyhybridizewith the pzorbitalsofthesurfaceandsub-\nsurface Ge atoms. This strong hybridization gives rise to a\nlarge delocalization of spin moments up to deeper Ge atomic\nlayers (see Fig. 4). The sum ( m) of the spin moments of\nSn atoms or Ge atoms in each layer is also given in Fig. 4.\nHere, the spin moment of each atom is calculated by Mul-\nliken analysis. We find that the Sn layer has m= 0.22µB,\nwhile the first and third Ge layers have m= 0.40 and 0.22\nµB, respectively, which are significantly larger than those ob -\ntained from other Ge layers. Note that the total spin moment\nis1µBper3×3unitcell. Theresultofalargespindelocaliza-\ntion over Sn atoms and Ge substrate atoms contrasts with the\ncase of the previously proposed Mott-Hubbard insulator [4]\nwhereeachSn atom hasa localizedspinmomentof1 µBasa\nconsequenceofelectroncorrelations. Itisremarkabletha tthe\npresent FI order is determined by an itinerant single-elect ron\napproach with the correction of SIE, thereby representing a\nSlaterinsulatordrivenbyitinerantmagnetism.\n(µ  )Bm\nSn layer 0.22\n1st Ge layer 0.40\n2nd Ge layer 0.03\n3rd Ge layer 0.22\n4th Ge layer 0.04\n5th Ge layer 0.05\n6th Ge layer 0.03\nFIG.4: (Coloronline)SpindensityoftheFIstructure. Them ajority\n(minority)spindensityisdisplayedindark(bright)color withaniso-\nsurfaceof0.01( −0.01)e/˚A3. Thesum( m)ofthespinmomentsofSn\natoms or Ge atoms in each layer is also given. For the spin mome nt\nof eachGe atom, see Table ISof the Supplemental Material[40 ].\nWe note that the total energy difference ΔEbetween the\nmetallic 1U2D structure and the insulating FI structure is\n6.1 meV per Sn atom. Although the MIT temperature can\nbe predicted by the precise entropy-related free energy dif -\nference (TΔS) between the 1U2D and the FI structures, we\nroughly estimate it by considering only the electronic con-tribution to the entropy, as done by Profeta and Tosatti [4].\nAssuming that the FI structure has a lack of spin entropy\nand a charge gap, TΔSwas approximated to γT2whereγis\nthe electronicspecific heat coefficient(roughlyof order ∼0.1\nmeV/site K2) [4, 45]. We thus estimate the MIT temperature\nas∼8K,whichissomewhatbelowtheobservedMITtemper-\nature of∼30 K [19]. This deviation of the MIT temperature\nmayreflect that the 1U2Dand FI structureshavedifferentvi-\nbrationalcontributionstotheentropy,whicharenottaken into\naccountin TΔS.\nIn conclusion, our semilocal and hybrid DFT calculations\nshowed the different predictions for the ground state of the\nSn/Ge(111) surface. Contrasting with the PBE functional\npredicting a metallic 3 ×3 ground state, the HSE functional\nshowed that the correction of SIE cures the delocalization\nerror to predict an insulating FI ground state with√\n3×√\n3\nstructural symmetry. We found that the magnetic moment of\nthe FI structure is well distributed over Sn adatoms as well\nas Ge substrate atoms. It is thus demonstrated that the ob-\nserved insulating phase in Sn/Ge(111) can be represented as\na Slater insulator through itinerant magnetism rather than a\nMott-Hubbardinsulator driven by Coulomb interactions. We\nnotice that the Sn/Si(111) surface has also been much stud-\nied to determine its exact crystallographicarrangement,e lec-\ntronic structure, and ground state [9, 10]. 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Lett .77,\n3865 (1996); 78, 1396(E) (1997).\n[34] Calculations withthek-pointsamplingof20 ×20meshesinthe\nsurface Brillouin zone of the√\n3×√\n3 unit cell, the eight-layer\nslab of Ge(111), and the force criterion of 0.01 eV/ ˚A change\nthe energy difference between the 1U2D and FI structures by\nlessthan0.2meVper√\n3×√\n3unitcell.Itisalsofoundthatthe\ncalculatedspinmomentsofSnandGeatomsintheFIstructure\nchange littleby less than0.01 µB.\n[35] G.-X.Zhang, A.Tkatchenko, J.Paier,H.Appel,and M.Sc hef-\nfler,Phys. Rev. Lett. 107, 245501 (2011).\n[36] N. Marom, A. Tkatchenko, M. Rossi, V. V. Gobre, O. Hod, M.\nScheffler, and L. Kronik, J. Chem. Theory Comput. 7, 3944\n(2011).\n[37] C.Kittel,in IntroductiontoSolidStatePhysics ,8thed.(Wiley,\nNew York, 2005), p.20.\n[38] H.-J.Kim and J.-H.Cho, Phys. Rev. Lett. 110, 116801 (2013).\n[39] S.¨Ozkaya, M. C ¸akmak, and B. Alkan, Phys. Status Solidi B\n248, 2142 (2011).\n[40] See Supplemental Material at\nhttp://link.aps.org/supplemental/xxxx for the band stru c-\nture of the√\n3×√\n3 structure computed using the PBE\nfunctional and the spinmoments of Snand Geatoms.\n[41] F.A.Cotton, Chemical Applicationsof GroupTheory (Wiley,\nNew York, 1990).\n[42] E.Pehlke andM. Scheffler,Phys. Rev. Lett. 71, 2338 (1993).\n[43] Since the HSE calculation without vdW interactions pre dicts\nthe insulating FI ground state with the disappearance of Sn\nbuckling, we can say that the flat structure of Sn atoms is not\ndue to vdW interactions but tothe FIspin ordering between Sn\natoms. It is found that the Sn-Ge bond lengths and the height\ndifference between the up and down Sn atoms (in the 1U2D\nstructure) obtained usingHSE+vdWdecrease bylessthan0.0 2\n˚A, compared tothose obtained using HSE.\n[44] It is notable that ΔEHSEobtained using the HSE calcula-\ntion without vdW interactions is 2.0 meV. This HSE value\nis somewhat different from that (4.9 meV) obtained from the\nHSE+vdW calculation, possibly due to the use of two differen t\nGe lattice constants optimized usingHSEand HSE+vdW.\n[45] K. Kanoda, J.Phys.Soc. Jpn. 75, 051007 (2006)."    },    {        "title": "1410.8782v1.Enhanced_ferrimagnetism_in_auxetic_NiFe2O4_in_the_crossover_to_the_ultrathin_film_limit.pdf",        "content": "Enhanced ferrimagnetism in auxetic NiFe 2O4\nin the crossover to the ultrathin \flm limit\nMichael Hoppe,1Sven D oring,1Mihaela Gorgoi,2Stefan Cramm,1and Martina M uller1, 3,\u0003\n1Peter Gr unberg Institut (PGI-6) and JARA J ulich-Aachen Research Alliance,\nForschungszentrum J ulich GmbH, 52425 J ulich, Germany\n2Helmholtz-Zentrum Berlin f ur Materialien und Energie GmbH, 12489 Berlin, Germany\n3Fakult at f ur Physik, Universit at Duisburg-Essen, 47048 Duisburg, Germany\n(Dated: July 6, 2021)\nWe investigate the sensitive interplay between magnetic, electronic and structural properties in\nthe ferrimagnetic oxide NiFe 2O4. Emphasis is placed on the impact of reduced dimensionality in\nthe crossover from bulk-like to ultrathin \flms. We observed an enhanced saturation magnetization\nMSfor ultrathin NiFe 2O4\flms on Nb-SrTiO 3(001) substrates that co-occurs with a reduced out-\nof-plane lattice constant under compressive in-plane epitaxial strain. We found a bulk-like cationic\ncoordination of the inverse spinel lattice independent of the NiFe 2O4\flm thickness { thus ruling out\na cationic inversion that nominally could account for an enhanced MS. Our study instead uncovers\na reduction of the unit cell volume, i.e. an auxetic behavior in ultrathin NiFe 2O4\flms, which may\nresult in an enhanced magnetic exchange caused by an increased interatomic electronic localization.\nPACS numbers: 75.47.Lx, 75.50.Gg, 75.70.Ak, 79.60.Dp\nI. INTRODUCTION\nThe competition of charge, spin and orbital degrees\nof freedom in complex oxides leads to intriguing phys-\nical phenomena, including ferromagnetism, ferroelectri-\ncity or multiferroicity [1]. Fertilized by the continuously\nadvancing art of oxide growth, the controlled synthesis\nof high-quality oxide heterostructures now approaches a\nmonolayer-precision [2]. Designing electronic properties\nin ultrathin oxide \flms and interfaces thereby opens up\nroutes to explore novel nanoelectronic functionalities for\napplications.\nIn the context of spin-based electronics, oxides fea-\nturing both magnetic and insulating properties reveal\na highly e\u000bective spin \flter e\u000bect, where spin-polarized\nelectron currents are generated by a spin-dependent tun-\nnelling process. Up to 100 % spin \fltering has been\ndemonstrated in magnetic oxides with low Curie tem-\nperatureTc, such as the binary rare earth compounds\nEuO or EuS [3, 4], and hence their integration as model\nspin injection/detection contacts to silicon was explored\nrecently [5, 6]. Implementing the spin \flter functionality\nof magnetic insulators in all-oxide heterostructures can\nextend the scope of applications further towards a mul-\ntifunctional oxide-based spin electronics.\nIt is in this pursuit that ferrite materials are envisioned\nas high-Tcspin \flters with the ultimate goal to real-\nize e\u000ecient spin \fltering for application at room tem-\nperature. For example, NiFe 2O4shows ferrimagnetic\nordering up to TC= 865 K [7] and grows epitaxially\non Nb-doped SrTiO 3(001) perovskite electrodes [8, 9].\nIts inverse spinel lattice of the type Fe3+[Ni2+Fe3+]O4\nhowever, exhibits a high structural complexity: Ni2+-\n\u0003mart.mueller@fz-juelich.de\nFigure 1. Schematic representation of the inverse spinel lat-\ntice of NiFe 2O4: Fe3+-cations (red) are distributed equally\nacross tetra- ( Td) and octahedral ( Oh) lattice sites, while\nNi2+-cations (green) occupy Ohsites. An antiferromagnetic\ncoupling between the TdandOhsites compensates the mag-\nnetic moments of the Fe3+-cations, why only the Ni2+-cations\naccount for the net macroscopic magnetization of 2 \u0016B/f.u..\ncations are situated on octahedrally ( Oh) coordinated\nlattice sites, while Fe3+-cations are equally distributed\nacross both tetrahedral ( Td) andOhsites (Fig. 1). The\nelectronic and magnetic properties of spinel ferrites thus\nsensitively depend on the details of the interatomic co-\nordinations. In particular, magnetic ordering is domi-\nnated by superexchange interactions between TdandOh-\ncoordinated cations on two antiferromagnetically coupled\nsublattices.\nA structural inversion from the inverse to the normalarXiv:1410.8782v1  [cond-mat.mtrl-sci]  31 Oct 20142\nspinel lattice consequently alters the cationic coordina-\ntion, as quanti\fed by the inversion parameter \u0015. Hereby,\n\u0015is the fraction of A2+-cations occupying Ohsites, with\n\u0015= 0 denoting a normal (Ni2+[Fe3+Fe3+]O4) and\u0015= 1\nan inverse (Fe3+[Ni2+Fe3+]O4) spinel lattice. In pre-\nvious studies, an unexpected magnetic behaviour, i.e.\nan enhanced saturation magnetization was reported for\nNiFe 2O4\flms in the ultrathin \flm limit [10, 11]. The\norigin of this phenomenon was explained by a cationic\ninversion from an inverse to a partly normal spinel lat-\ntice (0< \u0015 < 1), since this structural redistribution of\nFe cations nominally accounts for an increased magnetic\nmoment. Theoretical considerations based on density\nfunctional theory calculations \fnd a partial cationic in-\nversion energetically favourable for NiFe 2O4\flms under\ntensile, but not under compressive strain [12], as is the\ncase for NiFe 2O4grown on Nb-SrTiO 3(001). The origin\nof the altered magnetic exchange interaction in ultrathin\nNiFe 2O4\flms thus still remains an open question.\nIn this work, we explore the details of the electronic\nand magnetic properties of single-crystalline NiFe 2O4\n\flms in the crossover from bulk-like to the ultrathin \flm\nlimit. The goal of our studies is to uncover modi\fca-\ntions of the structural, electronic and magnetic prop-\nerties with regard to the reduced \flm dimensionality.\nWe performed a complementing spectroscopic analysis\nemploying the bulk- and surface sensitive photon spec-\ntroscopy techniques HAXPES, XANES and XMCD, re-\nspectively, which allow for a precise quanti\fcation of the\nelement-speci\fc cationic valencies and spatial coordina-\ntions. From our throughout analysis, we can conclude on\nthe absence of a cationic inversion for all NiFe 2O4\flm\nthicknesses. Instead, we propose an auxetic behaviour of\nultrathin NiFe 2O4\flms as the mechanism that may lead\nto an enhanced interatomic electronic localization.\nII. EXPERIMENTAL DETAILS\nA series of NiFe 2O4thin \flms with varying thick-\nnesses between 2 and 20 nm has been deposited from\nstoichiometric targets on conductive 0 :1 % Nb-doped\nSrTiO 3(001) substrates by utilization of the pulsed laser\ndeposition technique. The substrates were previously\netched in bu\u000bered hydro\ruoric acid to provide a TiO 2-\nterminated terrace surface [13]. During growth, the laser\n\ruence was set to 1 :5 J/cm2and a repetition rate of 2 Hz.\nThe oxygen pressure was kept at 0 :04 mbar and the sub-\nstrate was heated to TS= 635\u000eC. After deposition, the\nsamples were post-annealed at TSfor 90 min in vacuum.\nThe thickness of the grown \flms was determined by\nX-ray re\rectivity measurements (XRR), while the struc-\ntural characterization was accomplished by X-ray di\u000brac-\ntion experiments (XRD). Both XRR and XRD experi-\nments were performed on a Philips XPert MRD using Cu-\nK\u000b-radiation. Bulk magnetic properties of the samples\nwere investigated on a Quantum Design MPMS SQUID.\nHysteresis loops were recorded at T= 5 K with a mag-netic \feld up to 3 :6 T, which was applied parallel to the\nin-plane [100]-axis of the \flms.\nHard X-ray photoelectron spectroscopy (HAXPES)\nwas conducted on the HIKE endstation of the KMC-1\nbeamline at the BESSY-II electron storage ring (HZB\nBerlin) [14]. In contrast to soft X-ray photoelectron\nspectroscopy, HAXPES experiments use high energy X-\nrays with photon energies EPranging from 2 to 15 keV.\nThus, the kinetic energy as well as the inelastic mean\nfree path of the emitted photoelectrons are strongly en-\nhanced, which gives HAXPES an information depth (ID)\nof several tens of nanometers, allowing one to probe the\nchemical properties of a multi-layered \flm structure with\ntrue bulk sensitivity [15]. All spectra shown in this work\nwere taken at EP= 4 keV and at room temperature. The\nX-ray beam was aligned at 3\u000egrazing incidence and the\nphotoelectron detector normal to the sample surface. By\nde\fning ID(95) as the probing depth from which 95 %\nof the photoelectrons originate, the given experimental\nparameters result in an ID(95) \u001917 nm.\nX-ray absorption near edge structure (XANES) exper-\niments have been performed at the HIKE endstation as\nwell. To record the XANES spectra, the X-ray absorp-\ntion of the samples was determined by the emitted \ruo-\nrescence, which was detected by an energy dispersive de-\ntector. In contrast, the absorption of the NiFe 2O4bulk\ntarget material was determined in total electron yield\nmode (TEY). All signals were normalized to the incident\nX-ray \rux, monitored by a ionization chamber in front\nof the sample. The angle between the incident X-ray\nbeam and the sample was optimized for every spectrum,\nin order to maximize the \ruorescence signal without sat-\nurating the detector.\nFor all photon energies used during the HAXPES and\nXANES experiments, photoemission spectra of the Au 4f\ncore level from an Au reference sample attached to the\nmanipulator have been recorded. The energy position of\nthe Au 4f lines was compared to standard values and all\nmeasured data corrected accordingly.\nX-ray magnetic circular dichroism (XMCD) data was\ndetermined by X-ray absorption spectroscopy (XAS) ex-\nperiments performed at the UE56-1 SGM beamline at\nBESSY-II. The sample surfaces were aligned in 20\u000egraz-\ning incidence. A magnetic \feld of 300 mT was applied\nparallel to the surface and in the plane spanned by the\nincident beam with the surface normal axis. The absorp-\ntion signal was taken in TEY mode and was normalized\nto the incident X-ray \rux. The XMCD asymmetry spec-\ntra were determined from the di\u000berence of two spectra\ncollected by changing either the magnetic \feld to the op-\nposite direction, or by two spectra recorded by changing\nthe polarization from left to right-handed. In total, at\nleast four absorption spectra were taken for every sam-\nple, each with a di\u000berent combination of polarization and\nmagnetization direction.\nFor XMCD data analysis, we have calculated model\nXMCD spectra using the program CTM4XAS 5.5 [16],\nwhich is based on crystal \feld multiplet calculations in-3\n101102103104105106107\nIntensity [a.u.]48474645444342412θ [°]8.358.308.258.208.15out of plane lattice constant [Å]20151050film thickness [nm]NiFe2O4 bulklattice constant\nd -> 0 d = 20.5 nm d = 10.2 nm d = 8.3 nm d = 6.3 nm d = 4.1 nm d = 2.0 nmSrTiO3 (002)NiFe2O4 (002)\nFigure 2.\u0012-2\u0012-scans of the NiFe 2O4(004) re\rection for vary-\ning \flm thickness. The out-of-plane lattice constant aoopde-\ncreases below the bulk value for ultrathin \flms (see inset).\ncluding charge transfer e\u000bects [17{19]. Using the parame-\nters from Ref. [20], the interatomic screening is taken into\naccount by reducing the Slater integrals F(dd), F(pd),\nand G(pd) with scaling factors F(dd) = 0.7, F(pd) = 0.8\nand G(pd) = 0.8. For octahedral (tetrahedral) symmetry,\nthe crystal \feld was set to 10 Dq = 1 :2 eV (\u00000:6 eV) and\nthe exchange \feld was set to M = 10 meV ( \u000010 meV).\nThe resulting spectra were broadened by a Lorentzian\nwidth with a half-width of 0.3 eV (0.5 eV) for the L3(L2)\nedge to respect the core-hole lifetime broadening, and by\na Gaussian width of 0.2 eV to account for instrumental\nbroadening.\nIII. RESULTS\nA. Structural and magnetic characterization\nFirst, the thickness-dependent structural properties of\nultrathin NiFe 2O4\flms on Nb-SrTiO 3(001) were in-\nvestigated by X-ray di\u000braction experiments. \u0012-2\u0012-scans\nranging from 2 \u0012= 20\u000eto 100\u000ewith a scattering vector\nparallel to the surface normal were used to analyze the\ncrystal structure of the \flms and to con\frm their epi-\ntaxial growth. All scans show the expected re\rections\nof a (001)-oriented SrTiO 3crystal, as well as two addi-\ntional re\rections at 2 \u0012\u001943\u000eand 2\u0012\u001995\u000e, which can\nbe attributed to the NiFe 2O4(004) and (008) re\rections.\nSince no other re\rections are observed, we conclude that\nthe NiFe 2O4\flms grow textured along the (001) direc-\ntion without any parasitic phases. \b-scans around the\nSrTiO 3(202) and NiFe 2O4(404) peaks both show a four-\nfold symmetry and provide evidence that the \flms grow\ncube-on-cube on the SrTiO 3substrate, despite the in-\nduced biaxial compressive strain of 6 :4 %. In Figure 2,\nthe details of the \u0012-2\u0012-scans around the NiFe 2O4(004)\n-4-3-2-101234\nM [µB/f.c.]\n-15-10-5051015H [kOe]4.03.02.01.00.0M [µB/f.c.]20151050film thickness [nm]MSat d = 20.5 nm d = 8.3 nm d = 6.3 nm d = 4.1 nm d = 2 nmMSat NiFe2O4 bulkFigure 3. In-plane M-Hhysteresis loops of NiFe 2O4on\nSrTiO 3(001) recorded at T= 5 K. The inset shows the\nsaturation magnetization MSas a function of NiFe 2O4\flm\nthickness.\nre\rection are shown, which reveal that for decreasing \flm\nthickness the center of the NiFe 2O4(004)-peak shifts to-\nwards larger angles, implying a decreasing out-of-plane\nlattice constant aoop. The broadening of the peaks for\nthinner \flms is due to the smaller amount of material\nthat contributes to coherent di\u000braction. For \flm thick-\nnesses above 6 nm, aoopis slightly larger than the bulk\nvalue (a bulk= 8:339\u0017A). In combination with the com-\npressive in-plane stress induced by the substrate, this\n\fnding reveals the tendency of the material to preserve\nits bulk unit cell volume. On the other hand, for lower\nthicknesses aoopdecreases, as compiled in the inset of\nFig. 2. This refers to a reduction of the unit cell vol-\nume for ultrathin \flms in comparison to the bulk value,\na result that also has been reported for CoFe 2O4\flms\non SrTiO 3[21]. In contrast to CoFe 2O4, however,aoop\nof NiFe 2O4even drops below its bulk value for ultrathin\n\flms, which implies that NiFe 2O4shows an auxetic be-\nhaviour, i.e. a negative Poisson ratio \u0017in the crossover\nto the monolayer regime.\nNext, the NiFe 2O4\flms were investigated with regard\nto their magnetic properties. Hereby, special attention\nis payed to changes dependent on their \flm thickness.\nHysteresis loops of all samples were recorded at T= 5 K,\nwhich are dominated by the diamagnetic contribution of\nthe SrTiO 3substrate. To extract the magnetic response\nof the NiFe 2O4\flms, a subtraction of the diamagnetic\nbackground is required. Therefore, linear slopes have\nbeen \ftted to the high-\feld tails of the raw signal and\nsubtracted afterwards.\nIn Figure 3, hysteresis loops after background correc-\ntion are depicted, which con\frm ferromagnetic behaviour\nfor all NiFe 2O4\flm thicknesses. The coercive \felds are\napproximately constant for thicknesses above d= 6 nm,\nbut dramatically decrease for d= 2 nm. For CoFe 2O4on\nMgO, this e\u000bect has been interpreted as a result of the\nreduction of the magnetic anisotropy for thin \flms [22].4\nNiFe 2O4\flms with thicknesses above 6 nm show a sat-\nuration magnetization of MS\u00191:3\u00001:5\u0016B/f.u., which\nis lower than the bulk value of 2 \u0016B/f.u. [7]. These de-\nviations are supposed to be related to structural disloca-\ntions, which form due to the strain incorporated by the\nsubstrate, and to the formation of anti-phase boundaries\nduring growth. The latter occur due to island forming at\ndi\u000berent positions on the substrate, which are shifted by\nhalf of a unit cell to each other and thus loose periodic-\nity upon merging [23]. This model is supported by the\nhigh external magnetic \felds required to drive the \flms\ninto saturation, which is even at 15 kOe not completely\naccomplished. More striking, when the \flm thickness\nscales below 6 nm, we \fnd the saturation magnetization\nenhancing up to 3 \u0016B/f.u. - thus signi\fcantly exceeding\nthe bulk value. This result is in agreement with previous\nstudies on NiFe 2O4/SrTiO 3[10] and CoFe 2O4/SrTiO 3\n[24]. So far, this phenomenon was explained in terms of\na cationic inversion, where the inverse spinel structure of\nthe bulk state partly changes to a normal spinel structure\nin the crossover to the ultrathin \flm limit. An experi-\nmental proof for this model is however still lacking.\nMoreover, for the thinner \flms, the contributions from\ncontaminations to the total signal increase. Foerster et\nal.[25] discussed the in\ruence of the substrate, for which\nin the case of NiFe 2O4\flms on MgAl 2O4, the observed\nincreased magnetization can be explained by a paramag-\nnetic contribution from the substrate, which even disap-\npears, if the magnetic response of the substrate is sub-\ntracted properly. Yet this cannot explain the \fndings for\nNiFe 2O4on SrTiO 3, since SrTiO 3shows a purely dia-\nmagnetic response and thus validates the applied back-\nground subtraction.\nIn order to evidence the existence or absence of a\ncationic inversion, we investigate the chemical proper-\nties and cationic distribution of NiFe 2O4as a function of\nthe \flm thickness in more detail.\nB. HAXPES\nIn a \frst step, we need to clarify whether the chemical\nproperties of NiFe 2O4di\u000ber for bulk-like and ultrathin\n\flms. HAXPES measurements have been performed to\nquantify the valence states of each cation species. In\ncontrast to soft X-ray photoemission, HAXPES allows us\nto identify these properties not only at the surface but\nwith bulk sensitivity. The increased information depth\neven allows us to record reference spectra of the pressed\nNiFe 2O4powder used as bulk-target for PLD deposition,\nwhich do not posses a \rat surface as typically required\nfor low-energy photoemission experiments.\nFigure 4 plots the Ni 2p and Fe 2p core level spectra\nfor NiFe 2O4\flms of 8 nm to 2 nm, and compares them to\nthe bulk reference. In Figure 4(a), all spectra of the Ni\n2p core level display the Ni 2p 3/2and Ni 2p 1/2peaks at\na binding energy of 855 :1 eV and 872 :4 eV respectively,\nwithout a chemical shift relative to the bulk material.\nIntensity [arb. units]890880870860850840Binding Energy [eV]Ni 2p core level d = 8.2 nmReference spectra: d = 6.3 nm NiO monolayer d = 4.1 nm NiO bulk d = ~2 nm NiFe2O4 bulk-target referenceNi 2p3/2Ni 2p1/2satellitesatelliteFe 2s(a)\nIntensity [arb. units]750740730720710700Binding Energy [eV]Fe 2p core level d = 8.2 nmModel spectra: d = 6.3 nm Fe 2p 2+ Oh d = 4.1 nm Fe 2p 3+ Oh d = ~2 nm Fe 2p 3+ Td NiFe2O4 bulk-target referenceFe 2p3/2Fe 2p1/2satellitesatellite(b)Figure 4. HAXPES spectra of NiFe 2O4\flms with varying\n\flm thickness recorded at a photon energy of h\u0017= 4 keV. (a)\nNi 2p core level spectra and references for one monolayer NiO\nand NiO bulk [26]. (b) Fe 2p core level spectra in comparison\nto model spectra taken from [27].\nThe two main peaks are both accompanied by satellite\npeaks at 7 eV above their binding energies and overlap\nwith the Fe 2s core level at lower energies. The shape\nof all spectra is comparable to that of a single mono-\nlayer of NiO [26], in particular there is no shoulder visible\nat the high energy side of Ni 2p 3/2. The occurrence of\nsuch a shoulder \u00181:5 eV above the 2p 3/2peak (see NiO\nbulk reference in Fig. 4(a) for comparison) has been the-\noretically described by a screening e\u000bect, that emerges\nfrom electrons not originating from the oxygen orbitals\naround the excited Ni cation, but from adjacent NiO 6\nclusters [28]. The HAXPES experiment thus con\frms,\nthat no NiO clusters have formed within the NiFe 2O4\n\flms. Moreover, the spectra do not show any contribu-\ntion of metallic Ni0, which would peak at around 852.8\neV. We therefore conclude, that the NiFe 2O4\flms con-\ntain completely oxidized and homogeneously distributed\nNi2+cations only without any NiO cluster formation.\nFigure 4(b) depicts the HAXPES data of the Fe 2p\ncore levels from all NiFe 2O4samples with d= 8\u00002 nm.5\nFor comparison, also model spectra of Fe cations in the\ninverse spinel structure of magnetite (Fe 3O4) are given\n(reproduced from Ref. [27]). These spectra have been\ncalculated individually for the di\u000berent possible Fe cation\nlattice site occupancies ( Oh,Td) and valencies (2+, 3+).\nThe Fe 2p 3/2and 2p 1/2core levels are peaking at bind-\ning energies of 710 :9 eV and 724 :4 eV, in agreement with\nthe spectrum of the NiFe 2O4bulk reference and consis-\ntent with literature [29]. The formation of under-oxidized\nFe2+ions during \flm growth would result in a charac-\nteristic shoulder at the low-energy side of the Fe 2p 3/2\npeak, due to a chemical energy shift (as observable in\nthe Fe2+Ohreference). All measured spectra coincide\nwith the bulk reference sample, thus con\frming that the\nNiFe 2O4\flms consist of fully oxidized Fe3+cations and\nthat the amount of underoxidized Fe2+cations is below\nthe detection limit.\nComparing the Fe2+Oh, Fe3+Ohand Fe3+Tdmodel\nspectra reveals, that the main peak binding energies are\nsensitive to the oxidation state, but not to the atomic site\noccupancy. In contrast, the Fe 2p 3/2satellite observable\nbetween the spin-orbit split Fe 2p peaks is caused by\na screening e\u000bect of the surrounding oxygen ions and\ndeviates signi\fcantly for TdandOhcation coordination.\nThus, its shape and binding energy position can serve as\na \fngerprint for the chemical state of di\u000berent iron oxides\nand the cationic lattice site occupancies [27]. A complete\ninversion to the normal spinel structure would shift the\nsatellite's spectral weight to lower binding energies by\nabout 0.8 eV. Since both shape and energy position of the\nthin \flm samples satellite peaks perfectly match that of\nthe NiFe 2O4bulk spectrum, we conclude, that the Fe3+\ncations occupy the bulk lattice sites { without any sign\nfor a cationic inversion from the inverse to the normal\nspinel structure in the binding energy resolution limit of\nthe performed HAXPES experiment.\nIn summary, both the Ni 2p and Fe 2p spectra are com-\nparable to the spectrum of bulk material for all \flm thick-\nnesses, and reveal that the chemical composition of the\nbulk material is well reproduced in the ultrathin NiFe 2O4\n\flms. The Fe 2p 3/2satellite gives no hint for a cationic\ninversion in ultrathin NiFe 2O4. In order to rule out also\nany smaller e\u000bect, we investigate the spatial cationic dis-\ntribution by further spectroscopic means.\nC. XANES\nTo gain precise information on the spatial cationic dis-\ntribution in the NiFe 2O4thin \flms, we recorded XANES\nspectra of the Fe and Ni K-edge. Since the \fne struc-\nture above the absorption edge is dominated by multiple\nscattering with the surrounding atoms of the investigated\ncation species, XANES is very sensitive to the distribu-\ntion of the oxygen anions around the cation. Thus, a\ncationic inversion - for which the local site occupancy\nchanges from tetrahedral to octahedral, or vice versa -\nconsiderably modi\fes the shape of the spectral \fne struc-\nNormalized Absortion [arb. units]8420840083808360834083208300Photon Energy [eV]Ni K-edgeXANES d = 10.3 nm d = 8.2 nm d = 6.3 nm d = 4.1 nm d = 2.0 nm NiFe2O4 bulk-target(a)\nNormalized Absortion [arb. units]718071607140712071007080Photon Energy [eV]0.160.120.080.04pre-edgeintensity [a.u.]108642thickness [nm]Fe K-edgeXANEStetrahedral Fe3+pre-edge feature d = 10.3 nm d = 8.2 nm d = 6.3 nm d = 4.1 nm d = 2.0 nm NiFe2O4 bulk-target ZnFe2O4 normal spinel(b)Figure 5. XANES spectra of (a) the the Ni K-edge and (b)\nthe Fe K-edge from NiFe 2O4\flms with varying \flm thick-\nness. For comparison a Fe K-edge spectrum of Zn ferrite,\nwhich exhibits the normal spinel structure, is plotted (repro-\nduced from [30]). The inset in Fig. 5(b) shows the integrated\nspectral weight of the Fe K-edge pre-edge feature in depen-\ndence of the \flm thickness, where the dotted line represents\nthe bulk value.\nture.\nThe absorption spectra of the NiFe 2O4\flm samples are\nrecorded by \ruorescence yield, thus the measured data\nprobes the bulk-like \flm properties. Figure 5(b) shows\nthe XANES spectra of the Fe K-edge for NiFe 2O4\flms\ndown to 2 nm and a bulk material reference spectrum.\nAll spectra show a pre-edge feature at 7111 eV, which in\ncase of the spinel structure is observable for cations in\naTdsymmetry only. While the main absorption line is\ncaused by a dipole transition from the 1s to the empty 4p\norbital, the pre-edge structures in transition metal oxides\nare assigned to quadrupole transitions to the empty 3d\nstates, and thus are only very weak [31]. If the inversion\nsymmetry of the transition metal cation is broken, the\nlocal 3d and 4p wavefunctions of the cation hybridize,\nand in turn dipole transitions into this orbital become\nallowed, leading to an increased weight of the pre-edge6\nfeature. In the spinel structure a broken symmetry is\ngiven for cations on Td, but not on Ohsites.\nXANES studies of the Fe K-edge of various spinels\nclearly show a sharp pre-edge for all materials exhibiting\nthe inverse spinel structure, where Fe cations are situated\nonTdsites. In contrast, the spectra of compounds fea-\nturing the normal spinel structure, in which Fe cations\nsolely occupy Ohsites, only show a weak broad feature\n[30]. In Fig. 5(b), this is exemplary shown by a XANES\nreference spectrum of the normal spinel ZnFe 2O4(repro-\nduced from [30]).\nThe normalized pre-edge intensity can be quantita-\ntively correlated to the local site symmetry of the inves-\ntigated cation species [32]. By monitoring the Fe K-edge\npre-edge intensity of the NiFe 2O4samples, no intensity\nchanges are resolvable between the various \flm thick-\nnesses and also not in comparison to the bulk reference\nsample. We thus can conclude once more, that the ultra-\nthin \flms do not undergo a cationic inversion, but remain\nin the bulk-like cationic distribution of the inverse spinel\nlattice. This is supported by the Ni K-edge spectra (Fig.\n5(a)), which also show no sign of an emerging pre-edge\nfeature, characteristic for Ni cations on Tdsites.\nFocussing on the main Fe K-edge in Fig. 5(b), a chem-\nical shift is expected for valency changes. A shift of about\n5 eV between Fe2+and Fe3+for octahedrally coordinated\niron oxides was observed previously [33]. A comparison\nof XANES spectra from bulk Fe 3O4with NiFe 2O4, for\nwhich Fe2+cations are replaced by Ni2+, reveals a chem-\nical shift of about 3 eV, that has been explained by the\nmissing Fe2+ions [34]. This energy shift has also been\nobserved in other ferrites, where the Fe2+cations are sub-\nstituted by a di\u000berent cation species [30]. In all cases, the\nFe-K-edge of the Fe2+-compounds was situated at lower\nbinding energies. In our case, we observe no chemical\nshift of the main-edge across all \flm thicknesses, thus\nagain supporting that no modi\fcation in the oxidation\nstate of the Fe cations occurs, fully consistent with our\nHAXPES results.\nThe results of this in-depth XANES pre-edge anal-\nysis clearly reveal that NiFe 2O4\flms grow in the in-\nverse spinel structure independent of their \flm thickness.\nMoreover, the position of the main K-edges con\frms,\nthat the Fe and Ni cations in all samples are present in\na bulk-like valency for all \flm thicknesses.\nD. XMCD\nIn a last step, we analyse the XMCD asymmetry sig-\nnal to quantitatively determine the cationic distribution\nacross the spinel lattice sites. The XMCD asymmetry\nspectra are element-speci\fc and sensitively in\ruenced by\nthe valency, the local lattice site symmetry and the mag-\nnetic ordering of the investigated cation species. The\nspectral details re\rect the superposition of cations oc-\ncupyingTdorOhsites with either divalent or trivalent\nvalency, respectively. Hereby, each con\fguration has its\n0.060.040.020.00-0.02-0.04MCD Intensity [a.u.]\n730720710700Photon Energy [eV]0.10.0-0.1MCD Intensity [a.u.] 0.70.60.50.4Ratio20151050d [nm] Fe3+ octahedral sites Fe3+  tetrahedral sites\n MCD signal of NiFe2O4 (d = 2nm) CTM4XAS model spectra fitFe L3Fe L2Fe L23-edgeXMCD\n Fe3+ Oh Fe2+ Oh Fe3+ Td Fe2+ Td    CTM4XAS model spectra fitCTM4XASmodelspectraIII\nIII(a)\n(b)Figure 6. Experimental XMCD spectrum from the Fe L2,3-\nedge of the 2 nm thick NiFe 2O4\flm and the corresponding\n\ft. The resulting lattice site occupancy for various \flm thick-\nnesses is depicted in the inset.\nown characteristic MCD spectrum, which serves as a \fn-\ngerprint for the certain atomic and geometric con\fgura-\ntion. We thus modelled those four XMCD spectra, which\nallows us to \ft them as a linear combination to the ex-\nperimental data and to quantify the fraction of each con-\n\fguration.\nWe investigated XMCD asymmetry spectra of the Fe\nL2,3-edge to identify any changes in the distribution of\nFe cations between TdandOhlattice sites for NiFe 2O4\n\flms with varying thickness. Site- and valency-speci\fc\nFeL2,3-edge XMCD spectra were computed by LFM cal-\nculations utilizing the software CTM4XAS [16], which\nare presented in Fig. 6(b). Due to the antiferromag-\nnetic alignment of the cation spins between TdandOh\nsites, their asymmetry signals are of opposite sign. Con-\nsequently, these signals mainly cancel out in the observ-\nable sum asymmetry signal, leaving the resulting di\u000ber-\nence spectrum extremely sensitive to subtle changes in\nthe cationic distribution.\nFig. 6(a) exemplary shows the MCD spectrum of the\nFeL2,3-edge for the 2 nm thick NiFe 2O4\flm with the\ncorresponding \ft. The L3-edge exhibits a pronounced\n-/+/- asymmetry structure, caused by the antiparallel\noriented Fe moments. The positive (+) peak at 709 :7 eV\n(II) is dominated by tetrahedral Fe3+and the high en-7\nergy negative (-) peak at 710 :5 eV (III) by octahedral\nFe3+cations. We note, that the \frst negative peak at\n708:5 eV (I) is enhanced in comparison to the model and\nto the reference data [20], thus indicating that a frac-\ntion of Fe2+cations of about \u00192 % is existent at the\n\flm surface. However, the result gives no indication for\na cationic inversion of the \flm, which would result in a\ndecrease of the positive peak (II) and strong enhance-\nment of the high energy negative peak (III). These re-\nsults are also observed for all other investigated NiFe 2O4\n\flm thicknesses, which give no clue for an increased oc-\ntahedral Fe3+fraction, as would be characteristic for a\ncationic inversion to the normal spinel structure. This\n\fnding is in perfect agreement with electronic structure\ncalculations, which \fnd the fully inverse spinel lattice to\nbe the ground state of bulk NiFe 2O4[35].\nSince the XMCD spectra are recorded in TEY mode,\nthe experiment probes the uppermost 2-3 nm of mate-\nrial. Complemented by the bulk-sensitive HAXPES and\nXANES techniques, the analysis yields a consistent pic-\nture of the stoichiometry, valency and cationic distribu-\ntion of the NiFe 2O4thin \flms. In particular, we \fnd\nthat the cationic site occupancy always belongs to that\nof an inverse spinel lattice { this result is found both at\nthe NiFe 2O4surface and in the bulk volume. This strik-\ning consistency provides clear evidence for the absence\nof a cationic inversion in NiFe 2O4in the crossover to the\nultrathin \flm limit, and thus rules out this mechanism\nas the origin of the observed enhanced MSin ultrathin\nNiFe 2O4\flms.\nIV. SUMMARY\nIn summary, we have investigated single-crystalline\nNiFe 2O4thin \flms grown cube-on-cube on Nb-dopedSrTiO 3(001) substrates, with thicknesses scaling down\nfrom 20 { 2 nm. In this crossover to the ultrathin \flm\nlimit, we focussed on the impact of reduced dimension-\nality on the structural, electronic and magnetic NiFe 2O4\nproperties. Foremost, we observed an enhanced satura-\ntion magnetization MSin ultrathin NiFe 2O4\flms. De-\nspite the substrate-induced compressive in-plane strain,\na reduced out-of-plane NiFe 2O4lattice constant is found,\nimplying that a reduction of the unit-cell volume is\nenergetically favourable. In order to investigate the\ncationic distribution in the NiFe 2O4thin \flms, comple-\nmenting bulk- and surface-sensitive analyses using HAX-\nPES, XANES and XMCD spectroscopy techniques have\nbeen performed, and special attention was paid to the\nelement-speci\fc cation valencies and -coordinations. We\n\fnd a bulk-like inverse spinel structure being present in\nall samples { independent of the NiFe 2O4\flm thickness.\nThereby, our results consistently reveal the absence of a\ncationic inversion from the inverse to the normal spinel\nstructure, as was so far held responsible for an enhanced\nMSin ultrathin spinels. From our experimental results\nwe thus propose an auxetic behavior, i.e. a structural\nunit cell reduction, being a possible mechanism for a\nstronger interatomic electronic localization in ultrathin\nNiFe 2O4\flms. Theoretical calculations that shall fur-\nther elucidate possible underlying physical mechanisms\nare currently underway.\nACKNOWLEDGMENTS\nWe acknowledge experimental support during beam-\ntimes by B. Zijlstra and C. Caspers. We thank R.\nDittmann for providing the PLD setup at FZJ. This\nwork has been funded by the Helmholtz Association un-\nder Grant HGF-NG-811.\n[1] H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Na-\ngaosa, and Y. Tokura, Nat. 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B 74,\n174431 (2006)."    },    {        "title": "2201.09067v1.Universal_criteria_for_single_femtosecond_pulse_ultrafast_magnetization_switching_in_ferrimagnets.pdf",        "content": "Universal criteria for single femtosecond pulse ultrafast magnetization switching in\nferrimagnets\nF. Jakobs and U. Atxitia\u0003\nDahlem Center for Complex Quantum Systems and Fachbereich Physik,\nFreie Universität Berlin, 14195 Berlin, Germany\nSingle-pulse switching has been experimentally demonstrated in ferrimagnetic GdFeCo and\nMn2RuxGa alloys. Complete understanding of single-pulse switching is missing due to the lack\nof an established theory accurately describing the transition to the non-equilibrium reversal path\ninduced by femtosecond laser photo-excitation. In this work we present general macroscopic theory\nfor the magnetization dynamics of ferrimagnetic materials upon femtosecond laser excitation. Our\ntheory reproduces quantitatively all stages of the switching process observed in experiments. We\ndirectlycompareourtheorytocomputersimulationsusingatomisticspindynamicsmethodsforboth\nGdFeCo and Mn 2RuxGa alloys. We provide explicit expressions for the magnetization relaxation\nrates in terms of microscopic parameters which allows us to propose universal criteria for switching\nin ferrimagnets.\nUltrafast magnetization switching induced by a single\nfemtosecond laser pulse has attracted a lot of attention\nas a promising solution for low energy, faster memory\napplications. [1–15]. Until recently only GdFeCo\nalloys and synthetic ferrimagnets [8], presented the\nability to switch under either optical femtosecond laser-\n[3, 4, 8] or electric picosecond current-pulses [16,\n17]. Although several micro- and macroscopic models\nhave reproduced single-pulse switching in GdFeCo\nferrimagnets [18–33], complete understanding of the\nrole of electrons, lattice and spin sublattices and their\nmutual interactions remains a challenge [34]. The\nexisting criteria for switching rely on the existence of\ntwo antiferromagnetically coupled magnetic sublattices\nshowingdistinctdynamicalresponsetofemtosecondlaser\nphoto-excitation. While in single species ferromagnets\nsuch as 3d transition metals, relaxation of angular\nmomentum occurs via dissipation into other degrees\nof freedom – relativistic relaxation – in two-sublattice\nmagnets, additionally, relaxation can occur via angular\nmomentum exchange between sublattices – exchange\nrelaxation. By driving the spin system into a non-\nequilibrium state where exchange relaxation dominates,\na non-equilibrium ultrafast reversal path opens. One\nof the most outstanding, open questions is about the\nconditions or criteria for the onset of the exchange\ndominated relaxation regime. Crucially, it is unclear\nhow previous understanding gained from observation\nin GdFeCo translates into the recent discovery of\nsingle-pulse switching in the ferrimagnetic Heusler alloy\nMn2RuxGa [35], where the two antiferromagnetically\ncoupled Mn atoms are of the same kind in comparison\nto Gd and FeCo.\nIn this work we present a general theoretical\nframework for the description of single pulse switching\nof ferrimagnets. We provide explicit expressions for\nthe relativistic and exchange relaxation parameters as a\nfunction of microscopic material parameters, including\ntheir dependence on temperature and non-equilibriumsublattice magnetization. This allows us to uncover\nthe criteria for the onset of the exchange-dominated\nrelaxationregimeandswitching. Weverifythevalidityof\nthe model by direct comparison to atomistic spin model\nsimulations of both GdFeCo and Mn 2RuxGa alloys.\nWe shall describe each magnetic atom at site ias a\nclassical spin vector sof a unit length. The magnetic and\nmechanical moments of each atom/element are given by\n\u0016=\u0016ssandS=\u0016ss=\rwhere\u0016sisthemagneticmoment\nand\ris the gyromagnetic ratio. We consider a classical\nHeisenberg spin Hamiltonian[36]:\nH=\u0000X\ni6=jhijiJijsi\u0001sj\u0000X\nidz\ni(sz\ni)2:(1)\nTo model a ferrimagnet, one needs to consider two\nsublattices with different and antiparallel magnetic\nmoments\u0016aand\u0016b, with three different exchange\ncouplingconstants. Twoferromagneticcouplingsforeach\nsublattice coupling with itself ( JaandJb>0) and a\nthirdfortheantiferromagneticinteractionbetweenthem,\nJab<0[37]. The second term in Eq. (1) describes the\ncontribution to the energy of on-site uniaxial anisotropy\nwith easy-axis in z-direction and anisotropy constant, dz\ni.\nThe macroscopic model we propose is derived from a\nmicroscopic spin model, where the equation of motion for\nthe spin dynamics of each atomic spin is the stochastic\nLandau-Lifshitz-Gilbert (LLG) equation [36]:\n@si\n@t=\u0000j\rj\n(1 +\u00152)\u0016i[(si\u0002Hi)\u0000\u0015(si\u0002(si\u0002Hi))]:\n(2)\n\u0015is the local intrinsic atomic damping, the effective field\nHi=\u0010i\u0000@H\n@si, wherethermalfluctuationsarerepresented\nby the stochastic field \u0010i.\nThe non-equilibrium macroscopic magnetization dy-\nnamics of the element-specific angular momentum Sa=\n\u0016ahsai=\r, wherema=hsaiis the first moment of the\nnon-equilibrium distribution function, can be describedarXiv:2201.09067v1  [cond-mat.mtrl-sci]  22 Jan 20222\nelectrons \nlattice \nFIG. 1. Degrees of freedom involved in single-pulse switching;\nelectrons, lattice and element-specific spin systems. Electrons\nact as heat-bath to the spin system with an element-specific\ncoupling strength given by \u000ba(b). Exchange relaxation rate\n\u000bex\u0018\u000ba=ma+\u000bb=mb, wherema(b)are the normalized\nsublattice magnetization magnitude.\nby [19]:\ndSa\ndt=\u000ba\u0016aHa+\u000bex(\u0016aHa\u0000\u0016bHb)(3)\nwherea6=b. The macroscopic relativistic damping\nparameter in Eq. (3) reads [38]\n\u000ba= 2\u0015aL(\u0018a)\n\u0018a: (4)\nHere,L(\u0018)stands for the Langevin function. The\nrelaxation parameter strongly depends on temperature\nand non-equilibrium magnetization state through the\nthermal field \u0018a=\f\u0016aHMFA\na. In the exchange\napproximation, the MFA field acting on sublattice ais:\n\u0016aHMFA\na =zaJaama+zabJabmb: (5)\nHere,zais the number of nearest neighbours (n.n.) of\nspins of type a, andzabis the amount of n.n. of type\nb. The macroscopic damping increases with temperature\nup to a value \u000ba= 2=3at the critical temperature [38].\nThe exchange relaxation parameter \u000bexis given by\n\u000bex=1\n2\u0012\u000ba\nzabma+\u000bb\nzabmb\u0013\n: (6)\nThisexpressionistheextensionofthenon-localexchange\nrelaxation in ferromagnets to local exchange relaxation\nin ferrimagnets. The role of \u000bex,\u000baand\u000bbas the\ncoupling between the sublattices and heat baths is\nvisualized in Fig. 1. In single species ferromagnets,\n0 0.2 0.4 0.6 0.8 1ma00.20.40.60.81mb0.1110\nFIG. 2. Normalized exchange relaxation parameter \u000bex=\u0015\nas function of the sublattice magnetization maandmb, at\nT= 600K. System parameters correspond to GdFeCo.\n\u0015a=\u0015b=\u0015= 0:01. The dotted black line corresponds\nto\u000bex=\u000ba. The white dot represents the starting\nsublattice magnetization ( ma;mb). The closed dashed orange\nline describes a trajectory meeting a no-switching criteria.\nThe open solid white line describes a trajectory meeting a\nswitching criteria.\nsublattices aandbrepresent the same spin lattice,\nhence\u000ba=\u000bb. Therefore, \u000bex=\u000ba=(zma), and\n\u0016aHa\u0000\u0016bHb=\u0016aHexa2\n0\u0001ma, witha0representing\nthe lattice constant. Hence, \u0000non\u0000loc:\nex =\u000bex(\u0016aHa\u0000\n\u0016bHb) =\u000ba(A=M a(T))\u0001ma, whereAis the so-\ncalled micromagnetic exchange stiffness [39]. Ma(T) =\n(\u0016a=\u001da)mais the magnetization density at temperature\nT, where\u001dais the unit cell volume. Non-local exchange\nrelaxation plays a minimal role in the field of ultrafast\nmagnetization dynamics since \u0001ma\u001c1.\nThe non-equilibrium fields ( \u0016aHa= 0at equilibrium)\nare given by\n\u0016aHa=(ma\u0000L(\u0018a))\n\fL0(\u0018a): (7)\nwhere,L0(\u0018) =dL=d\u0018. Notethattheyaredifferenttothe\nMFA fields in Eq. (5), and have been derived previously\n[38, 40]. As the magnetic system approaches thermal\nequilibrium, the non-equilibrium fields can be cast into\nLandau-like expressions [19, 38].\nEquation (3) has been proposed before based on\nsymmetry arguments [19, 23] as a direct generalization\nof the Landau-Lifshitz equation with longitudinal\nrelaxation terms. These models have introduced the\nrelaxation parameters at a purely phenomenological level\nand to some extent their values are arbitrary. Moreover,\nsince they were taken as constant values, most of the\nnon-equilibrium spin physics was not taken into account.\nOur model overcomes these assumptions by providing\nexpressions for the relativistic and exchange relaxation\nparameters as a function of the sublattice specific3\natomic relaxation parameter, \u0015a(b), and normalized\nmagnetization ma(b).\nThis insight is paramount to find the criteria for\nthe onset of a exchange relaxation dominated state.\nIn Fig. 2 we present a diagram of the normalized\nexchange relaxation parameter \u000bex=\u0015for GdFeCo alloy\nparameters as a function of maandmbat a fixed\ntemperature, T= 600K, which corresponds to the\nCurie temperature of the alloy. We observe that the\nexchange relaxation parameter strongly depends on the\nmagnitude of sub-lattice magnetization and its value\nranges over three orders of magnitude, \u000bex=\u0015= 0:1\u0000\n10. Importantly, only when magnetic states reduce\nsignificantly, does the exchange relaxation dominates\nover relativistic relaxation.\nSo far only two classes of ferrimagnets have\nshown single-pulse switching, Gd x(FeCo) 1\u0000xalloys and\nMn2RuxGa. Switching in GdFeCo has been thoroughly\nstudied both experimentally [1–5] and theoretically [3,\n11, 20–23, 31, 32], whereas in Mn 2RuxGa has been only\nrecently demonstrated for a range of Ru concentrations\n(x\u00150:9) [35]. In GdFeCo alloy, switching is\ncharacterised by a fast response of the Fe sublattice and\nslower response of the Gd sublattice to femtosecond laser\nphoto-excitation. This difference roots to their distinct\nmagnetic moment, \u0016Gd= 7:6\u0016Band\u0016Fe= 1:92\u0016B.\nDifferently to GdFeCo, antiferromagnetically coupled\nMn spins in Mn 2RuxGa have similar atomic magnetic\nmoments. We demonstrate the universality of our theory\nby direct comparison of the photo-induced magnetization\nswitching to computer simulations based on atomistic\nspin dynamics (Eq. (2)) for GdFeCo (disordered spin\nstructure) and Mn 2RuxGa (ordered spin structure).\nTheir magnetic properties such as magnetic moments\nand exchange interactions differ substantially. We\nconsider two typical compositions, that show switching,\nGd25(FeCo) 75alloysandMn 2Ru0:86Ga. Itisnoteworthy;\nthat while for GdFeCo atomistic spin models have been\nused thoroughly, in Mn 2Ru0:86Ga similar simulations are\nmissing. We derive the necessary material parameters\nbased on experimental measurements [41–45]. We find\nthat for Mn 2Ru0:86Ga the atomic magnetic moments,\n\u00164a= 2.88\u0016Band\u00164c=4.05\u0016B, and the exchange\nparameters, Ja= 1:28\u00021021J,Jb= 4:0\u00021022J\nandJab=\u00004:85\u00021022J describe the equilibrium\nmagnetization well as function of temperature. Using\nthese parameters the Curie temperature becomes Tc=\n600K and the compensation temperature TM\u0019300K.\nWe note that in the experimental samples, those\ntemperatures are sensitive to the growth conditions as\nwell as material composition. However our temperatures\nare within the reported range of experimentally found\ntemperatures [41–45]. We use the so-called two-\ntemperature model (TTM) to describe the dynamics\nof the electron and phonon systems[46, 47]. For both\nmaterials we use the same parameters in the TTM andthus the electron and phonon temperatures are the same\nfor the same fluence in both materials [45]. The heat-\nbath to which the spins are coupled is represented by the\nelectron system.\nFigures 3 (a) and (b) show excellent agreement\nbetween our macroscopic model and atomistic spin\ndynamics simulations for both alloys for all stages of\nthe magnetization dynamics leading to switching, from\nfast demagnetization, transient ferromagnetic-like state,\nto magnetization recovery. Interestingly, the switching\ndynamics in Mn 2Ru0:86Ga (Fig. 3 (a)) differs to GdFeCo\n(Fig. 3 (b)), both Mn sublattices demagnetize at\nsimilar rate and switch almost simultaneously. Although\ndemagnetization timescales are similar in Mn 2Ru0:86Ga\nand GdFeCo, the recovery of the magnetization in\nMn2Ru0:86Ga is significantly slower. The relaxation of\nthesublatticemagnetization(Eq. 3)canbesplitintotwo\ncontributions, the relativistic relaxation: \u0000r\na=\u000ba\u0016aHa,\nand exchange relaxation \u0000ex=\u000bex(\u0016aHa\u0000\u0016bHb)(see\nFig. 3(c) and (d)). For both ferrimagnetic alloys, \u0000r\na\ndrivesthedynamicsinthefirsthundredsoffemtoseconds,\nuntil sublattice magnetization reduces sufficiently to\nenter the exchange relaxation dominated region \u0000ex>\u0000r\na\n(Fig. 2). In this regime, the exchange relaxation steers\nthe systems towards an intermediate metastable state\ndefined by the condition \u0016aHa=\u0016bHb. Under some\nconditionsthisintermediatestateprecedeswitching. Itis\ncumbersome however to directly analyse Eqs. (7) due to\nitshighlynon-linearcharacter. Thereforeinthefollowing\nwe investigate some limiting scenarios.\nIn the high temperature limit, \u0018a=\f\u0016aHMFA\na!\n0,\u0000r\na= 2\u0015akBTma, which is the so-called thermal\nfluctuation field. The corresponding relaxation time,\n\u001ca=\u0016a=(\r2\u0015akBT), associated to relativistic relaxation,\nhas been discussed before [19, 48]. Similarly, we can\nestimate the high-temperature limit of the exchange\nrelaxation rate\n\u0000ex\n1(ma;mb) =\u0015kBT\nz(ma+mb)(mz\na\u0000mz\nb)\nmamb:(8)\nHigh temperature limits are valid when the temperature\nis larger than the exchange energy acting on the spins.\nOtherwise, intermediate-to-high temperature limit are\nnecessary. This limit adds corrections to previous\nestimations. For instance, \u0016aHa\u0000\u0016bHb= 3kB((T\u0000\nTa\nc)ma+ (T\u0000Tb\nc)mb), whereJ0;a+J0;ab= 3kBTa\nc. The\nexchange relaxation rate is\n\u0000ex\nT(ma;mb) = \u0000ex\n1\u0012\n1\u00001\nTTa\ncma+Tb\ncmb\nma+mb\u0013\n(9)\nFor element-specific critical temperature Ta\nc\u0019Tb\nc, the\ncorrection can be cast as \u0000ex\nT= \u0000ex\n1(T\u0000Ta\nc)=T.Since\nthe relativistic relaxation rate is also modified as \u0000r\na=\n\u0000r\na;1(T\u0000Ta\nc)=T, we can fairly investigate the crossover\nfrom relativistic to exchange dominated regime by4\n0123456FeGd\ntime [ps]−ΓrFeΓrGdΓexexchange relaxationexchange relaxationrelativistic relativistic relativistic −1−0.500.51\n0123\n0123456magnetizationMn4aMn4cΓ[µB/ps]time [ps]−Γr4aΓr4cΓex\n(a)(c)(d)(b)\nFIG. 3. Element-specific magnetization dynamics of single-pulse switching in ferrimagnets using an atomistic spin dynamics\nmodel (symbols) and a macroscopic model (solid lines) for Mn 2Ru0:86Ga (a) and Gd 25FeCo (b). The red area corresponds to\nthe critical values ( mc\na;mc\nb) to enter exchange relaxation regime. The blue area corresponds to the time lapse where relativistic\nrelaxation dominates. Different contributions to the element-specific magnetization relaxation for the single-pulse switching\ndynamics in Mn 2Ru0:86Ga (c) and Gd 25FeCo (d). \u0000r\na(b)stands for the relativistic relaxation rate of sublattice a= 4a(c)\nanda=Fe (d), and b= 4c(c) andb=Gd (d). \u0000exis the exchange relaxation rate, which is the same for both sublattices.\n\u0015a=\u0015b=\u0015= 0:01.\ncomparing \u0000ex\n1and\u0000r\na;1. Two cases of interest exist,\ni) one sublattice is faster than the other, and ii) both\nsublattices demagnetize at the same rate.\nIn the first case, \u001ca\u001c\u001cb, sublattice ademagnetizes\nfaster than sublattice b. This corresponds to GdFeCo\n(Fig. (3)(b)). Soon after the application of a fs\nlaser pulse, ma\u001cmb, and consequently, \u0000ex\u0019\n\u0015kBTmb=(zma)(cf. Eq.(8)). We estimate the conditions\nforthetransitionfromrelativistictoexchange-dominated\nregimes (see Fig. (3)(d)): From \u0000ex>\u0000r\na,ma<p\nmb=2zab\u00190:288pmb, and from \u0000ex>\u0000r\nbandma\u0014\n1=2z= 0:0833(red colored area in Fig. 3 (b)). A second\ncase,\u001ca\u0019\u001cb, might also arise when demagnetization\ntimes of both sublattices are similar. This is the case\nof Mn 2Ru0:86Ga alloys (Fig. (3)(c)). One can estimate\nthe conditions for the transition \u0000ex= \u0000r\na. This\nhappens for both sublattices when ma;b= (ma(0) +\nmb(0))2=(ma(0)mb(0))=2zab. Assuming a realistic value\nofmb(0)=ma(0) = 0:9, the exchange-dominate regime is\nreached when ma;b= 0:334(red colored area in Fig. 3\n(c)). Notably, this condition only depends on the initial\nvalues of the magnetization, ma(b)(0). These results are\nsimple and general, and one of the main result of the\npresent work. One can interpret the transition from\nrelativistic to exchange-dominated regime as follows:\nas the magnetization of one sublattice decreases, the\nphase space of available states for the spins of the other\nsublattice to switch by exchange of angular momentum\ndramaticallyincreases. Whileswitchingspinviacouplingto an external bath becomes increasingly difficult as the\nmagnetization reduces.\nOnce the system has entered the exchange dominated\nregime the dynamics follows a path where total angular\nmomentum is conserved towards a magnetic state where\n\u0016aHa=\u0016bHb. In the high temperature limit, this\ncondition reduces to mz\na=mz\nb, meaning that the\nexchange relaxation drives the magnetization of both\nsublattices into the same polarity, i.e. a ferromagnetic-\nlike state [2]. From this condition arises, that in order to\nhave a final mz\na<0the following condition is necessary:\nSex\na+Sex\nb<0. HereSex\na(b)standsfortheangularmagnetic\nmoment when the system enters the exchange dominated\nregime. For example, in Mn 2RuxGa, since Mn spins are\nassumed to demagnetize at the same rate, this condition\nreduces to Sex\na+Sex\nb\u0019(Sa(0) +Sb(0)) exp(\u0000t=\u001ca)<\n0. Namely, only for a starting temperature below the\ncompensation temperature, Sa(0)+Sb(0)<0, conditions\nfor switching are fulfilled, in complete agreement to\nexperimental observations [35]. Yet, the exchange\nrelaxation regime needs to be active for a significant time\nin order for the magnetization to switch. We compare\nthe time scales associated to both the relativistic and\nexchange relaxation. Relativistic relaxation rate follows\n\u0000r\na= 2\u0015akB(T\u0000Ta\nc)mz\na, which is strongly reduced by\nthe ultrafast dynamics of mz\na, in only a few hundred of\nfemtoseconds \u0000r\na!0. Whereas \u0000exrather follows the\ndynamics of the temperature T, and therefore decays\nslower than \u0000r\na(Fig. (3)(c) and (d)).5\nThe characteristic time scale of the electron and lattice\ntemperature is described by the TTM and for common\nparameter values in the range of 2 \u00003 ps. As the\ntemperature reduces, the exchange relaxation drives the\nsystem towards (T\u0000Ta\nc)mz\na= (T\u0000Tb\nc)mz\nb. For\nTb\nc< T < Ta\nc, the exchange relaxation drives the\nsystem back into an antiferromagnetic, but switched\nsate. As the magnetization builds up in the opposite\ndirection, \u0000ex, decreases and the relativistic relaxation\ntakes over. Interestingly, our theory predicts that for\nsystems where Ta\nc\u0019Tb\ncswitching would be unlikely,\nwhich has been recently demonstrated in rare-earth free\nsynthetic ferrimagnets [49]. Further, one can accelerate\nthe transition from exchange relaxation dominated to\nthe relativistic regime, and speed up complete switching\nby increasing difference between Ta\ncandTb\nc, an effect\nwhich has been observed by the substitution of Fe by\nCo, namely, GdFe by GdCo [50]. While GdFe recovers\nin tens of picoseconds, GdCo alloys only need of a couple\nof picoseconds.\nTo summarize, in this work we have proposed a general\nmacroscopic theory for the magnetization dynamics\nof ferrimagnetic materials driven by femtosecond laser\nphoto-excitation. Our theory reproduces quantitatively\nall stages of the switching process observed in ex-\nperiments. Notably, we have directly compared our\ntheory to computer simulations using atomistic spin\ndynamics methods for both GdFeCo and Mn 2RuxGa\nalloys. The magnetization dynamics transits from a\nrelativistic relaxation path to an exchange dominated\nregime due to the strong enhancement of the exchange\nrelaxation. 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Suzuki1\n1Department of Materials Science and Engineering,\nUniversity of California, Berkeley, CA 94720\n2School of Applied and Engineering Physics,\nCornell University, Ithaca, NY 14853\n3Advanced Light Source, Lawrence Berkeley National Laborat ory, Berkeley, CA 94720\n(Dated: November 3, 2018)\nAbstract\nWe investigate the magnetic properties of the isostructura l spinel-spinel interface of\nNiMn2O4(NMO)-Fe 3O4. Although the magnetic transition temperature of the NMO fil m is pre-\nserved, both bulk and interface sensitive measurements dem onstrate that the interface exhibits\nstrong interfacial magnetic coupling up to room temperatur e. While NMO thin films have a fer-\nrimagnetic transition temperature of 60K, both NiFe 2O4and MnFe 2O4are ferrimagnetic at room\ntemperature. Our experimental results suggest that these m agnetic properties arise from a thin\ninterdiffused region of (Fe,Mn,Ni) 3O4at the interface leading to Mn and Ni magnetic properties\nsimilar to MnFe 2O4and NiFe 2O4.\nPACS numbers:\n∗Electronic address: bbnelsonchee@berkeley.edu\n1The oxide spinel Fe 3O4is an ideal candidate for highly spin polarized electrode material\nto be used in spintronic applications. It has been theoretically predic ted to be half metallic,\nand is highly attractive for applications due to its high Curie temperat ure (T C) of 850K[1].\nExperimental studies of Fe 3O4in spintronic heterostructures, however, have exhibited much\nlower junction magnetoresistance (JMR) values than expected fr om a half-metallic electrode\nmaterial. Among the highest JMR values of Fe 3O4-based heterostructures are observed in\nlayered systems with epitaxially grown isostructural spinel barrier layers. Oxide spinels such\nas CoCr 2O4, MgTi 2O4, FeGa 2O4, and MnCr 2O4have been used as barrier layers in mag-\nnetic tunnel junctions with spinel Fe 3O4and half metallic perovskite electrodes[2, 3], while\nCoFe2O4has been used with Fe 3O4in spin-filter junctions[4, 5]. Recently, NiMn 2O4(NMO)\nhas also been identified as an effective spin filter barrier material in Fe 3O4-based magnetic\njunctions with perovskite counter electrodes[6]. Whereas perovs kite and spinel layers have\nbeen shown to be magnetically uncoupled in these structures[6], the magnetism near the\nisostructural spinel interfaces is a subject of interest. A more d etailed investigation of the\ninterfacial magnetic interactions between spinel structure mate rials is necessary in order to\nunderstand transport and magnetic interaction results attribut ed to these multilayers, as\nwell as to optimize the use of these heterostructures for spintro nic applications.\nIn this paper, we observe magnetic properties in NiMn 2O4thin film bilayers with Fe 3O4\nnot observed in either film alone. Although the NMO magnetic transitio n at 60K is pre-\nserved, interfacial element-specific magnetism measurements of NMO/Fe 3O4bilayers show\nstrong interfacial coupling oftheFe, MnandNi moments. Wesugge st these magnetic results\ncan be explained by a thin interdiffused layer at the interface.\nThin film heterostructures of NMO and Fe 3O4were grown by pulsed laser deposition\non (110)-oriented single crystal SrTiO 3(STO) substrates. The NMO was grown at 600◦C\nin 10mTorr of 99%N 2/1%O 2, while the Fe 3O4was grown at 400◦C in vacuum. The NMO\nfilm was grown first to minimize oxidation of the Fe 3O4film during deposition. The films\ngrow epitaxially on the STO substrates as confirmed by X-ray diffrac tion and Rutherford\nBackscattering mesasurements. The single NMO thin films have a T Cof 60K[7]. The\nbulk magnetism of the samples was probed by a superconducting qua ntum interference\ndevice (SQUID) magnetometer. The element specific magnetic prop erties of the inter-\nfacial Ni, Mn and Fe were investigated by X-ray magnetic circular dich roism (XMCD)\n(BL4.0.2 and BL6.3.1) in total electron yield at the Advanced Light Sou rce. Due to\n2FIG. 1: Moment as a function of temperature for (a) thick (40n m) NMO bilayer, and (b) thin\n(5nm) NMO bilayer.\nthe surface sensitive nature of this technique and in order to be int erface specific, all\nsamples had a 5nm Fe 3O4top layer. Additionally, because the NMO films have a low\nsaturation magnetization (0.8 µB) compared to Fe 3O4films (4.1 µB), two different thick-\nnesses of NMO film in the bilayer (40nm and 5nm) were utilized to elucidat e any effect\nof the bulk NMO film on the interface. Lastly, because these measur ements are relevant\nto spin polarized heterostructures, where the bottom spinel laye r is usually grown on a\nperovskite counter electrode, such a heterostructure was also investigated. Therefore, the\nNMO/Fe 3O4interface was investigated in the following three samples: a ’thick bilay er’ of\nSTO//NMO(40nm)/Fe 3O4(5nm), a ’thin bilayer’ of STO//NMO(5nm)/Fe 3O4(5nm), and a\n’trilayer’ of STO//La .7Sr.3MnO 3(LSMO)(40nm)/NMO(5nm)/Fe 3O4(5nm). All magnetic\nmeasurements were performed along the [100] in-plane direction.\nMoment versus temperature measurements taken at 10 Oe of the NMO/Fe 3O4bilayers\nare shown in Fig 1(a). The thick bilayer sample shows a Brillouin shape fo r the NMO\nTCof 60K; however, after reaching a minimum at 60K, the moment begin s to rise with\nincreasing temperature(Fig1a), uncharacteristic ofthemagnet icbehaviorobserved ineither\nindividual film. This behavior is largely absent in the thin bilayer sample, a lthough a\nslight discontinuity may be seen at 50K (Fig 1(b)). Such results prom pted more detailed\ninvestigation of the magnetic interactions at the interface.\nXMCD spectra and hysteresis loops were taken of the NMO/Fe 3O4interface in all het-\nerostructures at various temperatures. The thin NMO bilayer XAS and XMCD results are\ndisplayed in Fig 2. The NMO/Fe 3O4interface exhibits virtually identical Fe, Mn and Ni\nXAS and XMCD spectra for all temperatures between 30K-300K, a s seen in Fig 2. The\n3FIG. 2: Element specific magnetism of Fe 3O4/NMO interface in thin NMO/Fe 3O4bilayer. (a)\nXAS and XMCD spectra for Mn, Fe, and Ni as a function of tempera ture, (b) Mn, Fe and Ni\nnormalized XMCD hystersis loops at each temperature, (c) Fe normalized XMCD hysteresis loops\na function of temperature.\nthick NMO bilayer and trilayer samples also demonstrate this behavior . In addition, for a\ngiven temperature, the Fe, Mn and Ni XMCD hysteresis loops are ide ntical to one another.\nNevertheless, the shapeof the hysteresis loops changes distinctly as a function of temper-\nature, showing a dramatic increase in coercive field for all three elem ents below the NMO\nTC. Similar results are seen in the temperature dependent hysteresis loops of the trilayer\nsample. As shown in Fig 3, the coercive fields of the Fe are the same at 80K and 55K, but\nincrease at 30K and 15K. Furthermore, at 55K, the hysteresis loo p shows a slight decrease\nin remanent dichroism, which is consistent with the minimum moment in th e SQUID data.\nAs the normalized Fe, Mn and Ni hysteresis loops are identical for ea ch given temperature,\nthis hysteresis loop behavior is seen in the Mn and Ni as a function of t emperature also, but\nhas been omitted for clarity in Fig 3.\nOne can now discuss the apparent source of the bulk moment measu rements by utilizing\nthe element and interface specific XMCD information. First, it appea rs that there is signif-\nicant magnetic coupling at the interface as evidenced by identical Fe , Mn and Ni hysteresis\nloops. However, although they are identical for a given temperatu re, the magnetic nature\nof the hysteresis loops becomes increasingly harder as the temper ature is decreased through\n4FIG. 3: Fe XMCD hysteresis loops as a function of temperature , probing the top 5nm of the\ntrilayer sample.\n60K.This evolution suggests that thespecies at theinterface are coupled to themagnetically\nsoft Fe 3O4at temperatures above the NMO T C, but, as the NMO layer becomes ferrimag-\nnetic, the species couple to the magnetically hard NMO, as well. The de crease in remanent\nasymmetry observed in the trilayer around 60K could be due to a mag netic frustration of\nthe interfacial species as the NMO layer becomes ferrimagnetic. Th is can all be related to\ntheincrease in bulk moment observed above 60K in Fig 1a in the following way: just ab ove\nthe TCof the NMO film, the bulk hysteresis loop exhibits greater squarenes s than that at\nlower temperatures, which results in an effective increase in moment at small fields as the\ntemperature is increased.\nThe magnetic transition of the NMO film in the presence of this strong magnetic coupling\nat the interface is also of interest. It is apparent from the change in hysteresis loop shape\nand increase in coercive field that even the thin NMO layer undergoes a magnetic transition\naround 60K. Any depressed onset of the coupling to the NMO layer c ould be due to the\nrelatively low magnetization of the NMO compared to the Fe 3O4.\nNow that we have discussed how the interfacial species respond to the bulk of the NMO\nand Fe 3O4thin films, let us focus on how the magnetic species at the interface c an give rise\nto room temperature Mn and Ni magnetic circular dichroism. Two pos sible explanations\nare: (1) a thin interfacial layer of the NMO thin film is magnetized far a bove the NMO T C\nby the close proximity to the Fe 3O4layer, or (2) the presence of a mixed (Fe,Mn,Ni) 3O4\nspinel at theinterface thatis ferrimagnetic atroomtemperature . Such a solidsolutionat the\ninterface is reasonable as the cations of the spinel structure occ upy only a small fraction of\n5the available octahedral and tetrahedral sites of the oxygen fac e-centered-cubic sub-lattice,\nleaving ample opportunity for cation diffusion throughout the struc ture.\nThe XAS and XMCD data supports the presence of Mn and Ni in MnFe 2O4and NiFe 2O4\nenvironments at the interface, consistent with a mixed (Fe,Mn,Ni) 3O4spinel. The Ni XAS\nand XMCD spectra is characteristic of NiFe 2O4, while the Mn XAS and XMCD spectra is\ncharacteristic of MnFe 2O4[8]. Additionally, the alignment of the Ni and Mn moments with\nrespect to the Fe moments in the XMCD is consistent with MnFe 2O4and NiFe 2O4. As\nseen in Fig 2, the maximum Ni dichroism is parallel to the third peak of th e Fe dichroism,\nas in bulk NiFe 2O4, and the maximum Mn dichroism is antiparallel to the third peak of\nthe Fe dichroism, as in bulk MnFe 2O4[9]. Furthermore, the lack of a change in the Mn\nand Ni XMCD spectra below the NMO T Cmay result from probing the magnetism in an\ninterdiffused region, which would form the bulk of the XMCD probing de pth and, due to the\ncomparatively high saturation magnetization values of NiFe 2O4and MnFe 2O4with respect\nto NMO, overwhelm the NMO dichroism.\nIn conclusion, isostructural spinel interfaces of Fe 3O4and NiMn 2O4exhibit strong inter-\nfacial magnetic coupling, although the NMO T Cof 60K is preserved. Element and interface\nspecific magnetic analysis suggests that this behavior is due to limited interdiffusion of the\nFe, Mn and Ni cations at the interface, thus creating a spinel solid s olution of (Fe,Mn,Ni) 3O4\nthat exhibits the magnetic properties of NiFe 2O4and MnFe 2O4. Above the NMO T C, this\ninterdiffused region couples to the magnetic Fe 3O4layers; however, with the onset of ferri-\nmagnetism inthe NMO film, the interfacial region first becomes frust rated, and then couples\nto the magnetically hard NMO. This work is relevant for understandin g magnetic interfacial\ninteractions in spinel-spinel heterostructures.\nThis work was supported in full by the Director, Office of Science, Offi ce of Basic Energy\nSciences, Division of Materials Sciences and Engineering, of the U.S. D epartment of Energy\nunder Contract No. DE-AC02-05CH11231.\n[1] Z. Zhang and S. Satpatathy. Phys. Rev. B. 44, 13319 (1991).\n[2] G. Hu and Y. Suzuki. Phys. Rev. Lett. 89, 276601 (2002).\n[3] L.M.B. Alldredge, R.V. Chopdekar, B.B. Nelson-Cheesem an and Y. Suzuki. Appl. Phys. Lett.\n689, 182504 (2006).\n[4] M.G. Chapline and S.X. Wang. Phys. Rev. B 74, 014418 (2006).\n[5] A.V. Ramos, J.-B. Moussy, M.-J. Guittet, M. Gautier-Soy er, C. Gatel, P. Bayle-Guillemaud,\nB. Warot-Fonrose, and E. Snoeck. Phys. Rev. B 75, 224421 (2007).\n[6] B.B. Nelson-Cheeseman, R.V. Chopdekar, L.M.B. Alldred ge, J.S. Bettinger, E. Arenholz, and\nY. Suzuki. (arXiv.org/cond-mat/0706.2726.)\n[7] B.B. Nelson-Cheeseman, R.V. Chopdekar, M.F. Toney, J.S . Bettinger, E. Arenholtz, and Y.\nSuzuki. (Unpublished.)\n[8] R.A.D. Pattrick, G. van der Laan, C.M.B. Henderson, P. Ku iper, E. Dudzik, D.J. Vaughan.\nEur. J. Mineral. 14, 1095 (2002).\n[9] V.N. Antonov, B.N. Harmon, and A.N. Yaresko. Phys. Rev. B .67, 024417 (2003).\n7"    },    {        "title": "1303.5585v1.Linear_perturbation_renormalization_group_method_for_Ising_like_spin_systems.pdf",        "content": "arXiv:1303.5585v1  [cond-mat.stat-mech]  22 Mar 2013CondensedMatterPhysics,2013,Vol.16,No1,13704:1–8\nDOI:10.5488/CMP.16.13704\nhttp://www.icmp.lviv.ua/journal\nLinearperturbationrenormalizationgroupmethod\nforIsing-likespinsystems\nJ.Sznajd\nInstituteforLowTemperatureandStructureResearch,Poli shAcademyofSciences,Wroclaw\nReceivedJuly4,2012,infinalformJanuary3,2013\nThelinearperturbationgrouptransformation(LPRG)isuse dtostudythethermodynamicsoftheaxialnext-\nnearest-neighborIsingmodelwithfourspininteractions( extendedANNNI)inafield.TheLPRGforweaklyin-\nteractingIsingchainsispresented.Themethodisusedtost udyfinitefieldpara-ferrimagneticphasetransitions\nobservedinlayereduraniumcompounds,UAs 1−xSex,UPd 2Si2orUNi 2Si2.Theabove-mentionedsystemsare\nmadeofferromagneticlayersandthespinsfromthenearest- neighborandnext-nearest-neighborlayersare\ncoupledbytheantiferromagneticinteractions J1<0and J2<0,respectively.Eachofthesesystemsexhibitsa\ntriplepointinwhichtwoorderedphases(ferrimagneticand incommensurate)meettheparamagneticone,and\nallundergothehighfieldphasetransitionfrompara-toferr imagnetic( ++−)phase.However,ifinUAs 1−xSex\nthepara-ferriphasetransitionisofthefirstorderasexpec tedfromthesymmetryreason,inUT 2Si2(T=Pd,Ni)\nthistransitionseemstobeacontinuousone,atleastinthev icinityofthemulticriticalpoint.WithintheMFA,the\ncriticalcharacterofthefinitefieldpara-ferrimagnetictr ansitionatleastatoneisolatedpointcanbedescribed\nbytheANNNImodelsupplementedbyanadditional,e.g.,four -spininteraction.However,inLPRGapproxima-\ntionfortheratio κ=J2/J1around 0.5thereisacriticalvalueofthefieldforwhichanisolatedcri ticalpoint\nalsoexistsintheoriginalANNNImodel.Thepositivefour-s pininteractionshiftsthecriticalpointtowardshigher\nfieldsandchangestheshapeofthespecificheatcurve.Inthel attercaseforthefieldssmallenough,thespecific\nheatexhibitstwo-peakstructureintheparamagneticphase .\nKeywords:ANNNImodel,renormalizationgroup,isolatedcriticalpoi nt\nPACS:75.10.Hk,75.40.Cx\n1.Introduction\nTheLinearPerturbationRenormalizationGroup(LPRG)[1]m ethodusesasimpleone-dimensional\ndecimationtostudyuniversal(critical)andnon-universa lsuchasalocationofthecriticaltemperature\nandtemperatureorfielddependenceofthepropertiesofther modynamicquantitiesofseveralclassical\nandquantumhigher-dimensionalmodels.Forthefirsttime,t hiskindofmethodwasproposedbySuzuki\nandTakano(ST)[2].IntheSTapproachtheone-dimensionald ecimationiscombinedwiththeMigdal-\nKadanoff(MK)bondmovingapproximation.Thedisadvantage softhelattermethodespeciallyforthe\nquantumsystemswerediscussedbyBarmaetal.[3]andCastel lanietal.[4].Here,wewishonlytoremind\nthattheMKapproachgivesratherpoorquantitativeresults evenforthetwo-dimensionalIsingmodel\nandthereisnopossibilitytoconstructanysystematicappr oximationprocedurewithinthismethod.\nForexample,theMKproceduregivesfortheIsingmodelonthe squarelatticethevaluesoftheinverse\ncriticaltemperature kc≈0.61whereastheexactvalueis kc≈0.44andforthe s=1\n2XYmodel kc≈\n1.2[2]muchlargerthanthevalue kc≈0.64estimatedfromthehigh-temperatureseriesexpansion[5],\nkc≈0.71(67)foundfromMonteCarlosimulationsbyfittingtotheexponent iallaw[6,7]andpowerlaw\n[8],respectivelyorrotationallyinvariantnon-linear(b lock)transformation kc≈0.62[9,10].TheLPRG\nmethodhasbeenproposedfortheso-calledquasi-one-dimen sionalmagnetsmadeofspinchainswith\ntheintrachaincoupling kandmuchweakerinterchaincoupling k1<k.However,evenforthestandard\nIsingmodel k1=k,theLPRGapproximationgivestheresultswhichareinveryg oodagreementwiththe\n©J.Sznajd,2013 13704-1J.Sznajd\nexactones kc≈0.45.For k>k1>0.15 k,thedeviationfromthecriticaltemperatureexactvaluesi sless\nthan 2%andfor 0.5k>k1>0.15 kevenlessthan 1%[11].\nInthispaper,theLPRGisusedtostudythethermodynamicsan dtheexistenceofacriticalpointin\ntwo-dimensionalaxialnext-nearest-neighbourIsingmode lwithfourspininteractions(extendedANNNI)\ninafield.\n2.LPRG\nWefirstdescribeindetailtheLPRGapproachinthesimplestp ossiblecase,i.e.,theIsingchainswith\nintrachaininteraction Jcoupledbytheweakinterchaininteractions J1definedbytheHamiltonian\nH=k/summationdisplay\n〈i j〉Si,jSi,j+1+k1/summationdisplay\n〈i j〉Si,jSi+1,j, (2.1)\nwherethelabel ireferstorowsand jreferstocolumns,thefactor −1/kBThasalreadybeenabsorbed\nintheHamiltonian( k≡J/kBT,k1≡J1/kBT),and k1<k.TheLPRGapproachstartswithanexactdeci-\nmationforone-dimensionalsystem.Dividingthespinsofth echainintotwogroups,inthesimplestcase,\ns2i+1(oddspins—survived)and s2i+2(evenspin—decimated)onecanwritethedecimationtransfo r-\nmationintheform\neH(s2i+1)=Trs2i+2eH(s2i+1,s2i+2). (2.2)\nIneachstepofthetransformation(2.2),everyotherspinis decimated.Thesametransformationcanbe\nwrittenbyusingalinearweightoperator P(σi,si)\neH′(σ)=TrsP(σ,s)eH(s), (2.3)\nwheretheweightoperator P(σ,s)whichcouplestheoriginal (s)tothenewspins (σ)ischoseninalinear\nformasfollows:\nP(σ,s)=1\n2NN/productdisplay\ni=1(1+σi+1s2i+1). (2.4)\nTheweightoperator P(σ,s)projectstheoriginalspin sspaceontothespaceoftheeffectivespins\nσ.FortheonedimensionalIsingmodelwithinterchainintera ction k1=0,thetransformation(2.2)or\nequivalently(2.3)canbecarriedoutexactly.So,wecansep aratetheHamiltonian(2.1)inamanageably\nexactlyunperturbatedpart H0containingintrachaininteraction k[firsttermoftheHamiltonian(2.1)],\nandaremainder HIcontaininginterchaininteraction k1[secondterminequation(2.1)].Withthenota-\ntion\nz0=TrsP(σ,s)eH0(s)(2.5)\nand\n〈A〉0=1\nz0TrsAP(σ,s)eH0(s)(2.6)\nthetransformation(2.3)canbewrittenas\nH′(σ)=lnz0+ln〈eHI(s)〉, (2.7)\nwiththefollowingcumulantexpansionfor ln〈eHI(s)〉[12]\nln〈eHI(s)〉=〈HI〉0+1\n2!/parenleftbig\n〈HI2〉0−〈HI〉2\n0/parenrightbig\n. (2.8)\nTheideaofLPRGintwodimensionsispresentedinfigure1.The chainsaredividedintotwoblack\nandwhitegroups.Ineachrenormalizationstep,everyother spinfromablackchainisdecimatedandin\nawhitechainallspinsareremoved.Asaresult,onegetsthes ystemofeffectivespins σ.Thesinglechain\n“partial”partitionfunction z0canbeeasilyfoundas[13]\nz0=(cosh2 k+1)+(cosh 2 k−1)σi,jσi,j+1, (2.9)\n13704-2LPRGforIsing-likesystems\nFigure1.TheLPRGprocedureforweaklyinteractingchainsintwodime nsions.\nand\nlnz0=1\n2(ln2+ln cosh2 k)+1\n2σi,jσi,j+1ln cosh2 k. (2.10)\nToevaluatethecumulants(2.8),onehastoknowtheaverages 〈si,j〉and〈si,j...si,j+n〉fromtheblack\n(decimated)andwhite(removed)rows.Forthespinsfromthe blackrows(figure1)\n〈s1,1〉=σ1,1,〈s1,2〉=1\n2(σ1,1+σ1,2)tanh 2 k,〈s1,3〉=σ1,2,(2.11)\nand\n〈s1,1s1,2〉=1\n2(1+σ1,1σ1,2)tanh 2 k,〈s1,1s1,3〉=σ1,1σ1,2. (2.12)\nForthespinsfromthewhiterows\n〈s2,j〉=0,〈s2,js2,j+n〉=tanhnk. (2.13)\nNowitisrelativelyeasyto findtherenormalizedHamiltonian H′(σ)(2.7)intheform\nH′(σ)=G(ki)+k′/summationdisplay\nσi,jσi,j+1+k′\n1/summationdisplay\nσi,jσi+1,j+Ω(σ), (2.14)\nwhere G(ki)isaconstant(independentofeffectivespins σ)termwhichcanbeusedtocalculatethefree\nenergypersiteaccordingtotheformula\nf=∞/summationdisplay\nn=1G(k(n)\ni)\n3n, (2.15)\nand“n”numberstheLPRGsteps. Ω(σ)denotestheadditionalinteractionsgeneratedeventually byLPRG\ntransformation.\nTheapproachpresentedabovecanbealsousedtoconsideraqu antumspinmodeloraninteracting\nelectronmodel.However,inthesecasestheLPRGdoesnotsta rtwithanexactbutwithanapproximate\ndecimationforone-dimensionalsystems[2].Weshouldalso emphasizethatgenerally,theapproachis\nrelevantforhighertemperatures.UsingLPRGonecanshowth eexistenceofthecriticalpoint,andcan\nobtainthelocationofatransitionpoint,ifany,andthetem peratureorfielddependencesofthethermo-\ndynamicquantities.\n3.ExtendedANNNImodelinafield\nThereexistsaclassofcompoundswithalayeredstructurean dstrongc-axialanisotropywhichcan\nbedescribedbytheIsing-likeANNNImodeloritsextensions .Forexample,modelsofthiskindhave\nbeenproposedtodescribethephasediagramsoftheuraniumc ompounds:UAs 1−xSexwith x<0.1[14]\norUT 2Si2(T=Pd,Ni)[15,16]madeofferromagneticlayerswhichcomprise threeorderedphases:anti-\nferromagnetic (+−+− ),ferrimagnetic (++− )andincommensurate.Eachofthissystemexhibitsatriple\n13704-3J.Sznajd\npointinwhichtwoorderedphases(ferrimagneticandincomm ensurate)meettheparamagneticone,and\nallundergothehigh fieldphasetransitionfromparamagnetictoferrimagnetic (++− )phase.However,\nifinUAs 1−xSexthepara-ferriphasetransitionisofthe firstorder,inUT 2Si2thistransition,atleastin\nthevicinityofthemulticriticalpoint,seemstobeacontin uousone.Forthesymmetryreason,thepara-\nferrimagnetic (++− )phasetransitioninthepresenceofanexternal fieldshouldbediscontinuousanda\nquestionhasarisediftheANNNImodelcanexhibitinsuchaca seacriticaltransition.WithintheMFAit\nhasbeenshownthattheANNNImodelshouldbesupplementedby someadditional,e.g.,four-spininter-\nactiontoexhibitanisolatedcriticalpoint[17].Inordert oanswerthisquestion,wehaveusedtheLPRG\ntostudytheextendedANNNImodelina field\nH= k/summationdisplay\n〈i j〉Si,jSi,j+1+k1/summationdisplay\n〈i j〉Si,jSi+1,j+k2/summationdisplay\n〈i j〉Si,jSi+2,j (3.1)\n+k4/summationdisplay\n〈i j〉Si,jSi+1,jSi,j+1Si+1,j+1+h/summationdisplay\n〈i〉Si.\nInthegroundstateinzero field,thestandardANNNImodelwith k4=0,k1,k2<0andκ≡k2/k1=0.5\nexhibitsamulticriticalpointwhereantiferromagneticph ases (+−+− )and (++−− )meetaferrimagnetic\nphasewiththespinsoftheferromagneticplanesorderedint hesequence (++− )(〈2,1〉phaseinthe\nnotationofthepaper[16]).Theexternalmagnetic fieldextendsthe (++− )phaseinthe (κ,H)plane\n[19].AllmagneticstructuresofUPd 2Si2aswellasUNi 2Si2andUAs 1−xSex(0/leqslantx/leqslant0.1)inthefieldsalong\ncaxisexhibitferromagneticlayerswithmomentsperpendicu lartothelayers.So,onlythemagnetic\norderbetweentheferromagneticlayersisofinterest.Cons equently,itseemsthattounderstandthemain\nfeatureoftheparamagnetic-ferrimagnetic( ++−)phasetransitiononecancon fineoneselftoconsiderthe\ntwodimensional(2D)problem.Thus,inthepresentpaperweu setheLPRGmethodtostudytheextended\nANNNImodelintwodimensions.\nTheideaoftheLPRGforANNNI-typemodelin2Dwiththefollow ingprojectorforonedecimatedrow\nP(σ,s)=1\n4(1+σ1,1S1,3)(1+σ1,2S1,6) (3.2)\nispresentedinfigure2.Thefullcirclesrepresentthespinswhichsurvivein thedecimationprocedure.\nAccordingtofigure2,ineachstepoftheRGtransformationeveryotherrow( “evenrow”)isremoved,\nandfromoddrowseverythirdspinsurvives.Theclusterpres entedinfigure2canbeusedtostudysys-\ntemsmadeofferromagneticorantiferromagneticchains butitdoesnotpreservetheantiferromagnetic\nandmuchlessmodulatedorderbetweenthechains .However,thisclusterpreservestheferrimagnetic\n(++−)orderingwhichisamatterofinterestinthispaper.Inorde rtotakeintoaccountbothpossibilities\n(antiferromagneticandferrimagneticorders)oneshouldc onsideraclusterof 11chainsandforthree\nFigure2.Cluster(8-10-8-10-8)usedtogetrenormalizedHamiltonia noftheextendedANNNImodelinthe\nLPRGprocedure.\n13704-4LPRGforIsing-likesystems\npossibilities(antiferromagnetic +−+−,antiferromagnetic ++−−,andferrimagneticorders) 16chains.\nAlthoughcalculationwithsuchclustersisstraightforwar ditbecomesquiteinvolved.Itshouldbeem-\nphasizedoncemorethattheLPRGwiththeclusterpresentedi nfigure2issuitabletodescribe,exceptfor\npara-ferro,onlypara-ferrimagnetic( ++−)phasetransitions.Suchphasetransitionscanoccurinbot h2D\nand3Dsystems.Thus,wehavecon finedourselvestoconsider2Dproblemalthoughtheincommens urate\nphasesobservedinthementioneduraniumcompoundscanbede scribedby3DANNNImodel.\nAsusual,theRGprocedureleadsfromthesetoforiginalpara metersoftheHamiltonian(2.3)( h,k,\nk1,k2,k4)tothesetofrenormalizedparameters( h′,k′,k′\n1,k′\n2,k′\n4)\nh′σi\nj, k′σ(j)\n1σ(j)\n2, k′\n1σj\n1σ(j+1)\n1, k′\n2σ(j)\n1σ(j+2)\n1, k′\n4σ(j)\n1σ(j+1)\n1σ(j)\n2σ(j+1)\n2.(3.3)\nHowever,inthelowestnontrivialorderofthecumulantexpa nsion,whichinourcaseisthesecondorder,\nseveralnewoddandeveninteractionscomeintoplay.Forthe sakeofsimplicity,wewilltakeintoaccount\nthecontributionsuptothesecondorderonlyfromtheone-an dtwo-spininteractions,contributionsto\nthefirstorderfromthree-andfour-spininteractions,andnegle ctanycontributionsfromthehigherorder\nspinterms[21].Accordingly,weconsider17parameters,i. e.,fiveoriginal(3.3)and12generatedbythe\nRGtransformation.\nToevaluatethetransformation(2.8)onehastoknowtheaver agesofspins 〈Si,j〉andtwospinsprod-\nucts〈Si,jSi,j+n〉fromthedecimated( “odd”)andremoved(“even”)rows.Thechainaveragesforthespins\nfromdecimatedrowsoftheIsingmodelina fieldwiththeweightoperator(3.2)havebeenpresentedin\npaper[9],andforthespinsfromtheremovedrows,theseaver agesareknownexactly(seeforexample\n[18]andreferencestherein).Now,weareabletonumericall yevaluatetherenormalizationtransforma-\ntion(2.8)whichinourapproximationhasaformof17recursi onrelations.Asusual,inordertodeter-\nminethecriticaltemperature,onehasto findacriticalsurfacewhichseparatesintheparameterspace\ntheregionofattractionofthetwostable fixedpoints,zerotemperature, kα= ∞andinfinitetempera-\nture kα=0.Theexistenceofsuchasurfacemeansthatthesystemcanund ergoasecondorderphase\ntransition.Moreover,wecanalsonumericallycalculateth efreeenergyperspincollectingtheconstant\ntermsgeneratedineachstepoftheiterationprocess(2.15) .Inadisorderedphaseweobtainarapidly\nconvergentinfiniteseriesforthefreeenergywhichcanbeusedto findthespecificheatasafunctionof\nthereducedtemperature t=T/J.\nLetusstartwiththestandardANNNImodelwith k=1,k1=−0.6,k2=−0.3and k4=0[21].Thevalue\nofthecriticaltemperaturehasbeendeterminedonthegroun doftherecursionrelationanalysis.Using\ntheformula(2.15),thefreeenergypersiteandthespeci ficheatasfunctionsoftemperatureanda field\nhavebeenfound.In figure3,thetemperaturedependenceofthespeci ficheatforseveralvaluesofthe\nfieldispresented.Thespeci ficheatdivergencecorrespondingtoacriticalpointisvisib lefor h=0andfor\nharound 0.11.For 0<h<hc≈0.11,thespecificheatexhibitsamaximum.Unfortunately,theLPRGfails\n1.55 1.60 1.65 1.70 1.75 1.80 1.85t 050010001500ch/LParen1t/RParen1\nh/Equal0\nh/Equal0.01h/Equal0.06\nh/Equal0.08h/Equal0.1\nh/Equal0.108\nh/Equal0.15\nFigure3.(Coloronline)Thespecificheattemperaturedependencesof theANNNImodelwith k1=−0.6,\nk2=−0.3,k4=0forseveralvaluesofthefield.\n13704-5J.Sznajd\n1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74t 050001000015000ch/LParen1t/RParen1\nk4/Equal/Minus0.2k4/Equal/Minus0.1k4/Equal0k4/Equal0.1k4/Equal0.2\nFigure4.(Coloronline)Thespecificheattemperaturedependencesof theextendedANNNImodelwith\nk1=−0.6,k2=−0.3atzerofieldforseveralvaluesofthefourspincoupling k4.\n1.55 1.60 1.65 1.70 1.75 1.80t 0500100015002000ch/LParen1t/RParen1\nh/Equal0\nh/Equal0.03 h/Equal0.08h/Equal0.1\nh/Equal0.112h/Equal0.1214\nh/Equal0.13\nFigure5.(Coloronline)Thespecificheattemperaturedependencesof theextendedANNNImodelwith\nk1=−0.6,k2=−0.3,k4=0.2forseveralvaluesofthefield.\n1.56 1.58 1.60 1.62 1.64 1.66 1.68 1.70t 0200040006000800010000ch/LParen1t/RParen1\nk4/Equal0.0, h/Equal0.0 k4/Equal0.0, h/Equal0.11\nk4/Equal0.2, h/Equal0.121\nFigure6.(Coloronline)Comparisonofthespecificheatcurvesofthes tandardANNNImodelatfields\nh=0and h=0.11aroundthecriticalfieldwithsuchacurveforthemodelwith k4=0.2aroundthe\ncriticalfield h=0.121.\n13704-6LPRGforIsing-likesystems\n1.580 1.585 1.590 1.595t\n/Minus1.0/Minus0.50.00.51.0/Lesssisi/Plus1/Greater,/Lesssisi/Plus2/Greater\nFigure7.(Coloronline)Thetemperaturedependencesofthenearest- neighbor 〈sisi+1〉(bottomcurve)\nandnext-nearest-neighbor 〈sisi+2〉chainscorrelationfunctionsforextendedANNNI (k4=0.2)modelat\nh=0.121.\ntoproperlydescribethespeci ficheatbehaviorbelowthemaximumbecauseitisnotcapableof describing\ntheorderedphasewhichisexpectedbelowthemaximum,ifthe systemundergoesadiscontinuousphase\ntransition.Forsomevalueofthe fieldh>hc(h=0.15infigure3),specificheathasonlyabroadhump.\nThismeansthattheconsideredANNNIinthepresenceoftheex ternalfieldcanexhibitacontinuous\nphasetransitiontotheferrimagneticphase (++− )ath=0and h=hc≈0.11.Thus,withintheLPRG\nunlikeinMFAitisnotnecessarytosupplementtheoriginalA NNNImodelbyanadditionalfour-spin\ninteractiontoreachanisolatedcriticalpointfora finitefieldaround h=0.11[21].\nNow,weproceedtotheextendedANNNImodelwith k=1,k1=−0.6,k2=−0.3and k4/nequal0.Infigure4\nwepresentthetemperaturedependencesofthespeci ficheatinzerofieldforseveralvaluesofnegative\nandpositivevaluesof k4.Asseen,thefour-spininteractionsimplyshiftsthecriti calpointtowardshigher\ntemperaturefor k4>0andtowardslowertemperaturefor k4<0.Figure5showsthetemperaturede-\npendenceofthespeci ficheatforthemodelwith k4=0.2infinitefields.SimilarlytothestandardANNNI\nmodelcase,theapplicationoftheexternal fieldchangesthedivergenceofthespeci ficheatintoamaxi-\nmumfor 0<h<hc.For h≈0.1214,thedivergenceappearsagainindicatingacriticalpoint. Infigure6\nwecomparetheshapesofthespeci ficheatcurvesforthe fieldstrengthsclosetothecriticalvaluesfor\nstandardandextendedANNNImodels.Inthecaseofthe k4=0.2model,asmallpeakinspeci ficheat\nappearsabovethecriticaltemperature.In figure7,thetwo-sitecorrelationfunctionsforthespinfrom\nadjacent( G1=〈sisi+1〉)andnextnearestneighbor( G2=〈sisi+2〉)chainsarepresented.Asseen,akink\nofG2aroundthetemperatureofthespeci ficmaximumisobserved.Itcansuggestsomepreliminaryor-\nderingbetweenthespinsfromthenextnearestneighbourcha ins.Tosummarize,ithasbeenshownthat\nwithintheLPRGapproximation,boththeoriginalANNNImode landtheonesupplementedbythefour-\nspininteractionwith k2/k1=0.5and k1,k2<0undergothecontinuousphasetransitionat h=0andcan\nundergosuchatransitionforsome finitevalueofthe fieldh=hc.Inthecalculationspresentedinthis\npaper,asharpanomalyinspeci ficheatisevidentat fieldsaround hcwhichcouldexplaintheexperimen-\ntallyobservedanomaliesinUPd 2Si2.Thefour-spininteractionshiftsboththecriticaltemper atureand\nfield,andchangestheshapeofthespeci ficheatcurves.\nReferences\n1. SznajdJ.,Phys.Rev.B,2001, 63,184404;doi:10.1103/PhysRevB.63.184404.\n2. SuzukiM.,TakanoH.,Phys.Lett.A,1979, 69,426;doi:10.1016/0375-9601(79)90397-9.\n3. BarmaM.,KumarD.,PandeyR.B.,J.Phys.C:SolidStatePhy s.,1979,12,L909;doi:10.1088/0022-3719/12/23/008.\n4. CastellaniC.,DiCastroC.,RanningerJ.,Nucl.Phys.B,1 982,200,45;doi:10.1016/0550-3213(82)90057-8.\n5. RogiersJ.,GrundkeW.,BettsD.D.,Can.J.Phys.,1979, 57,1719;doi:10.1139/p79-237.\n13704-7J.Sznajd\n6. DingH.-Q.,MakivićM.S.,Phys.Rev.B,1990, 42,6827;doi:10.1103/PhysRevB.42.6827.\n7. DingH.-Q.,MakivićM.S.,Phys.Rev.B,1992, 45,230;doi:10.1103/PhysRevB.45.230.\n8. JonssonA.,MinnhagenP.,NylénM.,Phys.Rev.Lett.,1993 ,70,1327;doi:10.1103/PhysRevLett.70.1327.\n9. SznajdJ.,Z.Phys.B:Condens.Matter,1986, 62,349;doi:10.1007/BF01313458.\n10. DrzewińskiA.,SznajdJ.,PhysicaA,1991, 170,415;doi:10.1016/0378-4371(91)90052-E.\n11. SznajdJ.,Phys.Rev.B,2008, 78,214411;doi:10.1103/PhysRevB.78.214411.\n12. NiemeijerTh.,VanLeeuwenJ.M.J.,Physica,1974, 71,17;doi:10.1016/0031-8914(74)90044-5.\n13. NauenbergM.,J.Math.Phys.,1975, 16,703;doi:10.1063/1.522584.\n14. Rossat-MignodJ.,LanderG.H.,BurletP.,In:Handbooko nthePhysicsandChemistryoftheActinides,Vol.1,\nFreemanA.J.,LanderG.H.(Eds.),North-Holland,Amsterda m,1984,p.415.\n15. ShemiraniB.,LinH.,CollinsM.F.,StagerC.V.,Garrett J.D.,BuyersW.J.L.,Phys.Rev.B,1993, 47,8672;\ndoi:10.1103/PhysRevB.47.8672.\n16. HonmaT.,AmitsukaH.,YasunamiS.,TenyaK.,Sakakibara T.,MitamuraH.,GotoT.,KidoG.,KawarazakiD.,\nMiyakoY.,SugiyamaK.,DateM.,J.Phys.Soc.Jpn.,1998, 67,1017;doi:10.1143/JPSJ.67.1017.\n17. PlackowskiT.,KaczorowskiD.,SznajdJ.,Phys.Rev.B,2 011,83,174443;doi:10.1103/PhysRevB.83.174443.\n18. SelkeW.,Phys.Rep.,1988, 170,213;doi:10.1016/0370-1573(88)90140-8.\n19. RujanP.,SelkeW.,UiminG.V.,Z.Phys.B:Condens.Matte r,1983,53,221;doi:10.1007/BF01388543.\n20. ThompsonC.J.,In:PhaseTransitionandCriticalPhenom ena,DombC.,GreenM.S.(Eds.),AcademicPress,1972,\np.177.\n21. SznajdJ.,J.Phys.:Condens.Matter,2012, 24,436006;doi:10.1088/0953-8984/24/43/436006.\nМетодлiнiйноїпертурбативноїренормалiзацiйноїгрупи\nдляiзингоподiбнихспiновихсистем\nЙ.Шнайд\nIнститутнизькоїтемпературиiструктурнихдослiджень,По льськаакадемiянаук,Вроцлав,Польща\nЛiнiйнепертурбативнеперетворення(LPRG)використовуєт ьсядлявивченнятермодинамiкиаксiальної\nмоделiIзингазнаступнимидонайближчихсусiдамизчотирис пiновоювзаємодiєю(розширенамодель\nANNNI)уполi.ПредставленоLPRGдляслабовзаємодiючихлан цюжкiвIзинга.Методзастосованодови-\nвченняпара-феромагнiтнихфазовихпереходiвускiнченном уполi,щоспостерiгаютьсявшаруватихспо-\nлукахурануUAs 1−xSex,UPd 2Si2чиUNi 2Si2.Вищезгаданiсистемизробленiзферомагнiтнихшарiвiспi-\nнизнайближчихiнаступнихдонайближчихшарiвєзв’язанiан тиферомагнiтнимивзаємодiями J1<0\niJ2<0,вiдповiдно.Кожназцихсистемдемонструєпотрiйнуточку, вякiйдвiвпорядкованiфази(фе-\nромагнiтнаiнеспiвмiрна)зустрiчаютьсязпарамагнiтноюф азоюiвсiфазизазнаютьфазовогопереходу\nусильномуполiзпара-доферомагнiтної( ++−)фази.Проте,якщовUAs 1−xSexєпара-ферофазовий\nперехiдпершогороду,якочiкуєтьсязсиметрiйнихмiркуван ь,вUT 2Si2(T=Pd,Ni)цейперехiдвидається\nнеперервним,принаймнiвоколiмультикритичноїточки.Вра мкахнаближеннясередньогополя,критич-\nнийхарактерпара-феромагнiтногопереходувскiнченномуп олi,принаймнiводнiйiзольованiйточцi,\nможебутиописанийзадопомогоюмоделiANNNI,якадоповнена додатковою,наприклад,чотириспiно-\nвоювзаємодiєю.Проте,внаближеннiLPRGдлякоефiцiєнта κ=J2/J1поблизу 0.5єкритичнезначення\nполя,дляякогоiзольованакритичнаточкатакожiснуєвориг iнальнiймоделiANNNI.Позитивначотири-\nспiновавзаємодiязсуваєкритичнуточкудовищихполiвiзмi нюєформукривоїпитомоїтеплоємностi.В\nостанньомувипадкудлядостатньомалихполiвпитоматеплоє мнiстьдемонструєдвопiковуструктурув\nпарамагнiтнiйфазi.\nКлючовiслова:модельANNNI,ренормалiзацiйнагрупа,iзольованакритичн аточка\n13704-8"    },    {        "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. 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Sharp and Lynn E. \nGarn, J. Appl. Phys. 53, 8980 (1982). \n[26] Jeroen van den Brink, and Daniel I Khomskii, J. Phys. Condens. 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 <z <0.5.The different Co sit e s  a r e  l a b e l l e d  w i t h  s a m e  \nscheme as in Ref. [15].  \n \n13 \n  \nFIG.4. (Color online). (a) and (b)  Pic torial reperesentation of the c harge and magnetic \nordering mechanism of the appearence of polarization (green das hed arrows) in kagome \nlayers for CBCNO sample.  (c,d) Co2-Co3 and Co3-Co4 zig zag cha ins along b-direction in \nkagome layer ( z < 0.5).  \n \n \nSupplementary Information  \n \nSample preparation and experimental techniques \nThe samples of CaBaCo 4-xNixO7 (x=0, 0.05 and 0.10)  were prepared by standard solid state reaction \nmethod, using high purity CaCo 3, BaCo 3, NiO and Co 3O4. Stoichiometric mixtures of these \npowders were ground thoroughly, heated in air at 900 °C for 12 h, 1100 °C for 24 h and quenched \nto room temperature.  Phase purity of the samples is confirmed initially by x-ray powder diffraction \nat room temperature where all samples are assigned to orthorhom bic structure with Pbn2 1 space \n14 \n group. The neutron diffraction measurements on x=0 and 0.10 sam ples were carried out on the PD2 \npowder neutron diffractometer (λ=1.2443 Å) at Dhruva reactor, B habha Atomic Research Centre, \nMumbai, India. The powder samples of approximately 5g were pack ed in a cylindrical vanadium \ncontainer and attached to the cold finger of a closed cycle hel ium refrigerator. Rietveld refinement \nof the neutron diffraction pattern s were carried out using the FULLPROF program [4]. Magnetic \nmeasurements were carried in a commercial VSM SQUID magnetomete r (Quantum Design, \nUSA).Dielectric and pyroelectric current measurements were done  in parallel plate capacitor \ngeometry using Agilient 4291A and Keithley electrometer 6517A r espectively. For dielectric and \npyroeletric current ( Ip) measurements pellets of thickness 0.5 mm and diameter of 5 mm  were used.  \nPyroelectric behaviour of CBCO:  \nPyroelectric and ferroelectric materials have the same prerequi site symmetry and ferroelectric \nmaterials form a subset of pyroelectric materials, in which spo ntaneous electric polarization can be \nreversed by an external electri c field. Spontaneous polarizatio n of the pyroelectric materials is \nstrongly temperature dependent due to their crystallographic st ructure. A pyroelectric material \ngenerates a voltage (current) when they are heated or cooled, t hrough transition temperature even \nwithout electric field. The measured Ip in heating and cooling paths for different heating/cooling \nrates, in the absence of poling field is shown in the Fig.S1 (a ). It can be noticed from the Fig.S1 that \n(a) the sign of the  Ip is opposite for cooling and heating paths; this behavior is ty pical for \npyroelectric materials as depicted in Fig S1 (b); because Ip α dT/dt (T: temperature, t: time). \nPyroelectric materials possess permanent dipoles and thus induc e non-zero net polarization (Ps) and \nhence attract free charges to its surface. If there is no chang e in temperature, there is no change in \nPs and no current flows through the external circuit. When cool ing (heating) the sample across the \ntransition temperature the dipolar ordering attract the ambient  charges (randomization of dipoles \nrelease charge) to the surface of the electrodes resulting a fl ow of current (in opposite) direction in \nthe circuit. 15 \n \n \n \n \nSM \nFIG.S1  (a) Heating and cooling rate dependence of pyrocurrent ( Ip) for CBCO measured in absence of poling field. (b)  \nSchematic model for the observed phenomenon in (a). \nFerroelectric behaviour of CBCNO (Ni 5%) samples:  \n \nSM FIG.S2. Temperature dependent (a) magnetization (logarithmic scale) (b)  dielectric constant and (c) P of CBCNO \n(Ni 5%). Inset (a) shows the M-H at 5K and inset (c) shows the variation of the pyrocurrent and temperature with time \nin the temperature interval of 50-52 K.  \n16 \n Temperature dependence of magnetization for 5% Ni substituted s amples is shown in Fig.S2 (a); \ntwo magnetic transitions at 60 K and 82 K can be noticed simila r to 10% Ni sample in Fig.1 (b). \nThe temperature dependent dielectric permittivity of the 5% Ni substituted samples is shown in the \nFig.S2 (b). The 5% Ni substituted samples shows two peaks at 60  K  a n d  ~ 8 2  K ,  w h i c h  i s  a l s o  \nsimilar to the 10% Ni doped sample (Fig.2(c)). The switching of  the polarization for the opposite \npoling fields and same magnitude  of the polarization for variou s ramp rates suggests the \nferroelectric behaviour of the 5% Ni sample. The intrinsic ferr oelectric switching behavior in 5 % \nNi doped CBCO sample is confirmed by time vs temperature cycle measurements between the 50-\n52 K temperature intervals (done after waiting time of 40 minut es at 50 K).  \nThe neutron diffraction data on the CBCO and 10% Ni doped CBCO is given below.  \nTABLE Ia.   Atomic parameters for CBCO at 6 K.  \n Space group: Pbn2 1. Cell parameters: a = 6.2556(5) (Å), b = 11.0383(9)  (Å), c = 10.1563(8) (Å). \nV= 701.3(1) (Å) 3 \nAtom x y z    B reliability factors \nCa -0.005 (4) 0.674 (2) 0.87 3 (2) 0.3(1) Rp (%) = 2.77 \n \nRwp (%) = 3.53 \n \nχ2 (%)  = 2.33 Ba 0.003 (2) 0.667 (2) 0.50000 0.3(1) \nCo1 0.008 (5) -0.005 (4) 0.941 (3) 0.2(4)\nCo2 0.005 (5) 0.161 (3 ) 0.701 (4) 0.2(4) \nCo3 0.248 (5) 0.092 (3 ) 0.196 (4) 0.2(4) \nCo4 0.261 (5) 0.914 (2 ) 0.677 (3) 0.2(4) \nO1 -0.012 (4) -0.005 (3) 0.255 (3) 0.1(2) \nO2 -0.012 (4) 0.496 (2) 0.231 (3) 1.2 (3) \nO3 0.777 (5) 0.258 (2)  0.789 (3) 0.9 (4) \nO4 0.728 (4) 0.748 (2)  0.216 (3) 0.5 (3) \nO5 -0.047 (2) 0.157 (1) 0.501 (3) 1.0 (3) \nO6 0.206 (2) 0.108 (1) -0.002 (4) 0.2(2) \nO7 0.266 (2) 0.947 (1)  0.499 (4) 0.5 (2) \n                                                      Co site  magnetic moments at 6K  \n Mx My Mz  M total  \nCo1 -1.4 (1) -2.2 (1) 0 2.7(1)  \nCo2 1.3 (1) 1.4 (2) 1.4 (2) 2.2(2)  \nCo3 1.7 (2) 2.8 (1) 0.9 (2) 3.4(2)  \nCo4 -0.2(2) -2.0(1) 0 2.0(2)  \n \n \n  \n 17 \n TABLE Ib.   Atomic parameters for CBCO at 300 K. Space group: Pbn2 1. Cell parameters: a = \n6.2837(6) (Å), b = 11.0032(9) (Å), c = 10.1857(8) (Å). V= 704.2  (1) (Å) 3     \nAtom x y z    B reliability factors \nCa 0.003 (3)   0.676 (2)   0.87 2 (2) 0.4(1) Rp (%) = 2.65 \n Rwp (%) = 3.31 \n \nχ2 (%)  = 2.18 Ba -0.002 (2)   0.666 (2)   0.50000 0.4(1) \nCo1 -0.001 (4)  -0.007 (3)   0.947 (2) -0.4(1) \nCo2 0.007 (5)   0.177 (2)   0.698 (2) -0.4(1) \nCo3 0.225 (4)   0.100 (2)   0.202 (2) -0.4(1) \nCo4 0.242 (5)   0.908 (2)   0.677 (2) -0.4(1) \nO1 -0.001 (4)   0.001 (2)  0.263 (2) 0.6(2)\nO2 -0.005 (3)   0.493 (2)  0.240 (2) 1.0(2)\nO3 0.787 (2)   0.261 (1) 0.791 (2) 0.4(2) \nO4 0.746 (2)   0.746 (1)   0.219 (2) 1.0(2) \nO5 -0.052 (2)  0.161 (1) 0.502 (2) 0.9(2) \nO6 0.213 (2)   0.1020 (9) -0.000 (3) 0.7(2) \nO7 0.262 (2)   0.944 (1)  0.503 (3) 0.8(2)\n \n \nTABLE IIa.   Atomic parameters for CBCNO (Ni 10%) at 6 K. Space group: Pbn2 1. \nCell parameters: a = 6.2561 (6) (Å), b = 11.0246 (9) (Å), c = 1 0.1673 (9) (Å). V= 701.243(0.104) \n(Å) 3 \n \nAtom x y z   B Site Occupancy \nCa -0.007 (2)   0.676 (1)   0.877 (1) 0.05(8) 1.0\nBa 0.003 (1)   0.663 (1)  0.50000 0.05(8) 1.0 \nCo1 0.019 (3)   0.000 (3)   0.947 (2) -0.16(10) 1.0 \nCo2 \nNi2  0.013(3)   0.178 (1)   0.690 (2) -0.16(10) 0.96(1) \n0.04(1) \nCo3 \nNi3 0.250 (3)  0.094 (2)   0.200 (2) -0.16(10) 0.95(1) \n0.05(1) \nCo4 0.258 (4)  0.924 (1)   0.700 (2) -0.16(10) 1.0 \nO1 -0.019 (2) 0.007 (1)  0.258 (1) 0.59(16) 1.0 \nO2 -0.004 (2)   0.495 (1)   0.235 (1) 0.58(15) 1.0 \nO3 0.778 (2)   0.2603 (9)  0.794 (1) 0.42(15) 1.0 \nO4 0.717 (2)   0.7431 (9)   0.221 (1) 0.34(16) 1.0 \nO5 -0.050 (2)   0.162 (1)   0.511 (2) 0.96(17) 1.0 \nO6 0.207 (2)   0.1136 (8)   -0.002 (2) 1.13(17) 1.0 \nO7 0.266 (2)   0.9456 (7)   0.512 (2) 0.86(15) 1.0 \n                                                     Co  site  magnetic moments at 6K \n Mx My Mz  M total reliability factors  \nCo1 -2.03(4) 0 0 -2.03(4) Rp (%) = 1.77 \n \nRwp (%) = 2.24  \nχ2 (%) = 2.08 Co2 Ni2 1.84(8) 0 0 1.84(8) \nCo3 \nNi3 1.1(3) 0 0 1.1(3) \nCo4 1.4(2) 0 0 1.4(2) \n \n \n   \n \n 18 \n TABLE IIb .  Atomic parameters for CBCNO (Ni 10%) at 300 K. Space group: Pbn2 1.  Cell \nparameters: a = 6.2832(3) (Å), b = 10.9981(5) (Å), c = 10.1817( 4) (Å). V= 703.589 (0.056) (Å) 3 \n \nAtom x y z    B Site Occupancy \nCa 0.002 (2)   0.666 (1)   0.8754 (8) 0.41(6) 1.0 \nBa  0.004 (1)  0.6573 (8)   0.50000 0.41(6) 1.0 \nCo1 0.014 (3)  -0.002 (2)  0.948 (1) -0.13(7) 1.0\nCo2 \nNi2 0.0238 (2) 0.185(1)   0.691 (2) -0.13(7) 0.96(1) \n0.04(1) \nCo3 \nNi3 0.242 (3)   0.091(1)   0.195 (1) -0.13(7) 0.95(1) \n0.05(1) \nCo4 0.248 (3)  0.926 (1) 0.696 (1) -0.13(7) 1.0 \nO1 -0.005 (3)   -0.005 (1) 0.260 (1) 0.87(10) 1.0 \nO2 -0.003 (2)   0.4919 (93) 0.238 (1) 1.21(14) 1.0 \nO3 0.786 (1)  0.2566 (90) 0.789 (1) 0.84(13) 1.0 \nO4 0.739 (2)   0.7492 (83) 0.222 (1) 0.95(14) 1.0 \nO5 -0.060 (1)   0.1595 (87)   0.505 (1) 1.96(15) 1.0 \nO6 0.210 (1)   0.1022 (68)   -0.002 (2) 1.78(16) 1.0 \nO7 0.262 (1)   0.9415 (56)  0.513 (1) 0.77(10) 1.0\nreliability factors  Rp (%) = 1.24 \nRwp (%) = 1.61 \nχ2 (%) = 6.93 \n \n \nTABLE III.  Comparison of Interatomic distances at 300K and 6K for CBCO and  CBCNO (Ni 10 %) \nsamples. \n \nBOND                 CBCO CBCNO \n300K 6K 300K 6K \nCa-O2 \nO3 \nO4 \nO5 O6 O7 2.30(2) \n2.21(2) \n2.34(2) \n2.25(2) 2.35(3) 2.41(3) 2.37(3) \n2.12(4) \n2.33(4) \n2.30(3) 2.38(3) 2.34(3) 2.232(15) \n2.241(15) \n2.352(15) \n2.354(16) 2.310(16) 2.375(14) 2.375(18) \n2.112(18) \n2.290(18) \n2.275(19) 2.348(18) 2.394(19) \nBa-O2 O2 \nO3 \nO3 O4 \nO4 \nO5 O5 \nO6 \nO6 O7 \nO7 3.26(2)   \n3.01(2)   \n3.615(20) \n2.649(20) 3.391(19) \n2.884(19) \n3.482(18) 2.803(18) \n3.449(19) \n2.878(20) 3.48(2)   \n2.869(20) 3.32(3) \n2.96(3) \n3.56(3) \n2.68(3) 3.47(3) \n2.77(3) \n3.405(18) 2.855(18) \n3.56(2) \n2.80(2) 3.50(2) \n2.82(2) 3.228(12) \n2.927(12) \n3.633(12) \n2.690(12) 3.439(12) \n2.886(12) \n3.490(12) 2.795(12) \n3.399(12) \n2.970(12) 3.523(11) \n2.794(11) 3.268(14) \n2.952(14) \n3.635(15) \n2.640(15) 3.470(14) \n2.815(14) \n3.423(14) 2.837(14) \n3.561(14) \n2.797(14) 3.529(14) \n2.799(14) \nCo1-O1 \nO5 \nO6 \nO7 1.88(3) \n1.82(4) \n1.88(3) \n1.87(3) 1.89(4) \n1.80(5) \n1.85(4) \n1.92(4) 1.917(16) \n1.85(2) \n1.76(2) \n1.97(2) 1.92(2) \n1.91(3) \n1.80(3) \n1.99(2) 19 \n Co2-O1 \nO3 \nO4 \nO5 2.07(3) \n1.91(3) \n1.78(3) \n2.04(3) 1.81(5) \n1.99(4) \n1.95(4) \n2.06(5) 2.100(18) \n1.967(17) \n1.686(16) \n1.98(2) 2.158(20) \n2.02(2) \n1.93(2) \n1.87(3) \nCo3-O1 \nO2 \nO3 \nO6 1.89(3) \n2.15(3) \n1.82(3) \n2.06(4) 2.04(4) \n1.99(4) \n1.91(4) \n2.04(6) 1.99(2) \n2.01(2) \n1.958(17) \n2.01(2) 2.02(2) \n1.96(2) \n1.88(2) \n2.08(3) \nCo4-O1 O2 \nO4 \nO7 2.01(4) \n2.03(3) \n1.75(3) \n1.82(4) 2.02(4) \n1.82(4) \n1.84(3) \n1.85(5) 1.87(2) \n1.86(2) \n1.945(16) \n1.875(20) 1.78(3) \n1.77(3) \n1.871(19) \n1.92(3) \n \n \n \n "    },    {        "title": "1703.05220v1.Modeling_ultrafast_all_optical_switching_in_synthetic_ferrimagnets.pdf",        "content": "arXiv:1703.05220v1  [cond-mat.mes-hall]  15 Mar 2017Modeling ultrafast all-optical switching in synthetic fer rimagnets\nS. Gerlach1,∗L. Oroszlany2, D. Hinzke1, S. Sievering1, S. Wienholdt1, L. Szunyogh3, and U. Nowak1\n1Fachbereich Physik, Universit¨ at Konstanz, D-78457 Konst anz, Germany\n2Department of Physics of Complex Systems, E¨ otv¨ os Univers ity,\nH-1117 Budapest, P´ azm´ any P´ eter s´ et´ any 1/A, Hungary an d\n3MTA-BME Condensed Matter Research Group and Department of T heoretical Physics,\nBudapest University of Technology and Economics, Budafoki ´ ut 8., HU-1111 Budapest, Hungary\n(Dated: September 13, 2018)\nBased on numerical simulations, we demonstrate thermally i nduced magnetic switching in syn-\nthetic ferrimagnets composed of multilayers of rare-earth and transition metals. Our findings show\nthat deterministic magnetization reversal occurs above a c ertain threshold temperature if the ratio\nof transition metal atoms to rare-earth atoms is sufficiently large. Surprisingly, the total thickness\nof the multilayer system has little effect on the occurence of switching. We further provide a simple\nargument to explain the temperature dependence of the rever sal process.\nI. INTRODUCTION\nThe demonstration of helicity-dependent all-optical\nmagnetization switching1,2was one of the most sur-\nprising findings in ultrafast magnetization dynamics.\nThe experiments showed that magnetization switching\nis possible solely triggered by a single laser pulse in\nthe sub-picosecond range avoiding any externally applied\nmagnetic field. These experiments were performed on\nGdFeCo, a rare-earth-based ferrimagnet where the rare-\nearth (RE) sublattice is antiferromagnetically coupled to\nthe transition metal (TM) sublattice. First attempts to\ndescribe these unexpected processes were based on the\nassumption that the circularly polarized laser pulse in-\nduces a strong magnetic field via the inverse Faraday ef-\nfect which determines the direction of the switching2,3.\nOver all, this process takes place on a time scale orders\nof magnitude shorter than today’s writing procedures\nin hard discs. This calls for applications in magnetic\ndata storage and alternative materials including alloys4,\nheterostructures5and synthetic ferrimagnets6are cur-\nrently investigated.\nThe discovery of thermally-induced all-optical switch-\ning using linearly polarized light7,8cast a new light on\nall-optical switching and called for more sophisticated\nmodels since this switching works without any external\nor optically induced magnetic field, which could define\nthe magnetization direction during its recovery after the\nultrafast quenching.\nIn simulations this switching was observed in an atom-\nistic spin model developed by Ostler et al.7–9. An\nattempt to explain the occurrence of the transient\nferromagnetic-like state (TFMLS) was given by Mentink\net al. who identified angular momentum transfer driven\nby the inter-sublattice exchange as the crucial process10.\nLater on the thermally induced switching was more quan-\ntitatively described by means of an orbital-resolved spin\nmodel, where the magnetic moments stemming from d-\nelectrons of the TM, the d-electrons of the RE and the\nf-electrons of the RE are distinguished11. Here, it was\nshown that the initial laser excitation brings the sublat-\ntices into a strong non-equilibrium after 1 ps. This hap-pens due to the different demagnetization times of the in-\ndividual sublattices12. On that time scale, electron and\nphonon temperatures are nearly equilibrated below TC\nagain, but the Fe sublattice is already completely demag-\nnetized while the Gd sublattice remains still rather or-\ndered. The remagnetization dynamics of Fe taking place\nsubsequently leads to a state where the Gd spins and the\nFe spins are aligned — the transient ferromagnetic-like\nstate. The TFMLS arises naturally as a consequence of a\nredistribution of the energy and angular momentum be-\ntween the different sublattices due to the maximization of\nentropy under the constraint of energy and angular mo-\nmentum conservation. These processes, which are driven\nvia the precession term of the Landau-Lifshitz-Gilbert\nequation, dominate on shorter times scales. The follow-\ning relaxation back to a ferrimagnetic equilibrium state\nis on a longer time scale, where dissipative processes are\nresponsible, and does not necessarily lead to a switched\nstate4,13. The details of this relaxation process depend\non the material properties as well as the experimental\nspecifications and are still under investigation.\nIn the following, we will explore the possibility of\nthermally-induced switching in synthetic ferrimagnets\ncomprised of bilayers of Fe and Gd. We use ab-initio\nmethods to estimate spin model parameters for Fe-Gd bi-\nlayers. The dynamic simulations of the spin model allows\nfor an investigation of the preconditions for thermally\ninduced switching. We find that deterministic magne-\ntization reversal occurs only above a certain threshold\ntemperature and in bilayers where the ratio of transition\nmetal atoms to rare-earth atoms is sufficiently large. Fi-\nnally, we find a simple explanation why the compensation\ntemperature is so important.\nII. MODEL\nOur aim is to model a synthetic ferrimagnet as a bi-\nlayer of two ferromagnets, Fe and Gd, with a negative\ncoupling between the two layers. For that we consider an\natomistic spin model where localized spins are arranged\non a simple-cubic lattice structure. Spins experience ex-2\nchange interactions with their nearest neighbors only and\ndipole-dipole interaction is neglected. The Heisenberg\nHamiltonian of the system studied reads\nH=−/summationdisplay\nNNJijSiSj−/summationdisplay\nidzS2\ni,z. (1)\nThe first term represents the Heisenberg exchange en-\nergy, where the exchange interaction is either between\nspins of the Fe layer, spins of the Gd layer, or, across the\ninterface, between Fe and Gd spins. This term contains,\ntherefore, three interactions, JFe−Fe,JGd−GdandJFe−Gd.\nThe second term represents a uniaxial anisotropy with\nanisotropy constant dz. The lateral dimensions of the\nmodel are 150 ×150 atoms with the layers stacked along\nthezaxis and with periodic boundary conditions in\ntransverse directions. The thicknesses of the Fe and Gd\nlayers are varied.\nAb-initio calculations have been performed in terms of\nthe fully relativistic screened Korringa–Kohn–Rostoker\n(KKR) method, designed, in particular, for layered sys-\ntems and surfaces14. The LSDA parametrization from\nRef. 15 was used. The strong correlation of the localized\n4f-states of the Gd atoms was treated within the frame-\nwork of the LSDA+U approach16as implemented within\nthe KKR method17. The calculations were carried out\nwith the commonly used U= 6.7 eV and J= 0.7 eV\nvalues of the Coulomb and exchange integrals16and the\ndouble counting term derived in Refs. 18 and 19 satis-\nfying the atomic limit for the LSDA total energy. The\nexchange constants have been obtained by means of the\nrelativistic torque method20.\nThe geometry used in the ab-initio calculations were\nbased on a heterostructure of six Fe layers between two\nsemi-infinite bulk Gd regions. For the hcp structure of\nGd bulk we used the experimental c/aratio of 1.5904\nand an optimized lattice constant a= 3.450˚A21. The\ninterlayer distance in the Fe region was chosen such that\nthe volume per Fe atom was identical to that in bcc Fe\nwith the experimental lattice constant of 2.867 ˚A. The\ndistance of the Fe planes to the nearest Gd planes was\ntaken to be the average of the Fe-Fe and Gd-Gd layer dis-\ntances. The interlayer exchange interactions obtained by\ntheab-initio procedure were then mapped to the simple\ncubic structure with nearest-neighbor interactions used\nin the spin-dynamics simulations.\nThe calculations outlined above result in exchange con-\nstants of the spin model above with the ratios JFe-Fe :\nJGd-Gd :JFe-Gd = 1 : 0 .286 : −0.388 which we use in\nthe subsequent dynamic simulations. The magnetic mo-\nments of Fe and Gd have the values of µTM= 1.92µB\nandµRE= 7.63µB, refering to the values found for bulk\nFeGd in Ref. 11. These values are close to those obtained\nin our KKR LSDA+U calculations. In addition, we used\nan anisotropy constant of dz= 0.2 meV favoring magne-\ntization along the zaxis.\nThe dynamics of the system is governed by the stochas-0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4\nTemperature in JFe/kB−3.0−2.5−2.0−1.5−1.0−0.50.00.51.0Magnetization Mz(T)/MFe\nz(0)\nTGd\nCTcomp TFe\nCMFe\nMGd\nM\nFIG. 1. (Color online) Temperature dependent equilibrium\nmagnetization for a bilayer with 3 monolayers of Fe and 2\nmonolayers of Gd. The magnetization of the element with\nthe stronger intra-layer exchange interaction, in this cas e Fe,\nclosely follows the shape of a ferromagnet with its Curie tem -\nperature TFe\nCclose to the Curie temperature of the coupled\nsystem. The element with the weaker intra-layer exchange\ninteraction, in this case Gd, exhibits magnetic ordering ab ove\nits own bulk Curie temperature, TGd\nC, due to the interaction\nwith the other sublattice. The chosen ratio of layer thick-\nnesses leads to a larger magnetic moment of the Gd layer at\nlow temperatures and, consequently, to a magnetic compen-\nsation point at a temperature Tcomp which is slightly above\nthe bulk Curie temperature of the Gd.\ntic Landau-Lifshitz-Gilbert equation of motion,\n(1 + α2)µi\nγ˙Si=−Si×[Hi+α(Si×Hi)],(2)\nwith the gyromagnetic ratio γand a dimensionless\nGilbert damping constant αthat describes the coupling\nto the heat-bath. In our simulations the damping con-\nstant is set to α= 0.0211,22. Thermal fluctuations are\nincluded as an additional white noise term ζiin the in-\nternal fields Hi=−∂H\n∂Si+ζi(t) with\n/angbracketleftζi(t)/angbracketright= 0,/angbracketleftζiη(0)ζjθ(t)/angbracketright=2kBT αµ i\nγδijδηθδ(t),(3)\nwhere i, jdenotes lattice sites and η, θare Cartesian com-\nponents. All algorithms we use are described in detail in\nRef. 23.\nIn Fig. 1we present the equilibrium properties of a bi-\nlayer consisting of 3 monolayers of Fe and 2 monolayers of\nGd. This bilayer behaves as a synthetic ferrimagnet with\na magnetic compensation point Tcomp at a temperature\nof about 40% of the Curie temperature Tc.\nIt is well-known that laser-induced demagnetization in\ntransition metals is several times faster than in rare-eart h\nelements24,25. Several approaches have been proposed to\nexplain this behavior, including electron-phonon scatter -\ning processes of the Elliott-Yafet type26and intra-atomic3\nenergy transfer within the electronic subsystem11. How-\never, based on the different demagnetization time scales,\nwe may assume that heating a multilayer system with\nincident laser light can lead to a situation where the TM\nlayer is completely demagnetized while the RE layers still\nretains a substantial net magnetization. We use this fact\nin the following and do not calculate the action of the\nlaser pulse on the spin system explicitly. Instead we focus\non the relaxation of the magnetization at constant tem-\nperature starting our simulations with a spin configura-\ntion where the Fe sublattice is completely demagnetized\n(spins are randomly oriented) while we vary the degree\nof magnetization of the Gd layer. This initial Gd magne-\ntization and the temperature will turn out to be crucial\nquantities for the understanding of thermally triggered\nswitching.\nIII. RESULTS\nFor a fixed layer configuration, we treat the initial\nRE magnetization remaining after the laser excitation\nand the temperature Tas the relevant parameters which\ndetermine the magnetization dynamics triggered by the\nlaser pulse. In Fig. 2three different possible scenarios are\nshown, switching (top), switching followed by switching\nback to the initial state (center), and no-switching (bot-\ntom). The chosen values for Tand Gd magnetization are\nindicated in Fig. 3. The switching scenario corresponds\nto the work of Radu et al.7, while the back-switching\nwas measured by Khorsand et al.27. Note, that in all\nthree cases the magnetization of the transition metal\nstarts towards negative values (while the original sign\nbefore demagnetization by the effect of the laser heating\nwould have been positive) so that a TFMLS is obtained.\nNote also, that the transverse components of the mag-\nnetization are usually not small — apart from the case\nof switching — which indicates a linear mechanism for\nthe case of successful switching but a more precessional\nprocess for the case of no-switching and back-switching.\nA systematic variation of the two parameters, temper-\nature and initial Gd magnetization, allows for the con-\nstruction of a switching diagram as shown in Fig. 3. It\nis ternary in the way that it provides information about\nthe relaxation process, with the three scenarios above\n(switching, back-switching, or no-switching) as shown in\nFig. 2. The diagram is constructed by taking into ac-\ncount the magnetization dynamics of the two species in-\nvolved for a single run during a time interval of 40 ps. We\nidentify three distinct regions, a connected no-switching\nregion (/squaresolid), a connected switching region ( /squaresolid), and a back-\nswitching region along the boundary of the other two re-\ngions (/squaresolid).\nFor an interpretation of this diagram we first note that\nfor very low values of initial RE magnetization one would\nexpect the system to randomly pick one of the two pos-\nsible equilibrium configurations, either switched or not.\nThis sort of statistical behavior is indeed evident from−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0) Fe\nGd\n−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0) Fe\nGd\n0 5 10 15 20\nTimetin ps−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0) Fe\nGd\nFIG. 2. (Color online) The three possible time relaxation\nscenarios for the sublattice magnetizations of a RE-TM lay-\nered system. (top) Magnetization switches with respect to\nthe initial configuration, (center) both sublattice magnet iza-\ntions change their signs twice, ending up back in the initial\nstate, (bottom) no-switching, i.e. the rare-earth layer ma gne-\ntization does not change sign. Also shown are the transverse\ncomponents (dotted and dashed lines). For the case of back-\nswitching and no-switching these components are of the orde r\nof the magnitude of the magnetization.\nthe isolated data points at the very left hand side of the\ndiagram. Furthermore one would expect that high val-\nues of MREwould prevent the system from switching at\nlow temperatures because the level of order in the RE\nlayer is too high to become demagnetized. This is indeed\nwhat we find from Fig. 3. The no-switching region is\nfound in the bottom right corner, corresponding to low\ntemperature and high MRE.\nSo far, the switching diagram meets our intuitive ex-4\n20 40 60 80 100\nInitial Gd magnetization in %0.00.10.20.30.40.50.6Final temperature in JFe/kBTcomp\nTGd\nC\nno-switchingswitching\nFIG. 3. (Color online) Switching in a 3:2 layer sample after\n40 ps. Color coding: /squaresolidMagnetization switches, /squaresolidmagneti-\nzation does not switch and returns to the initial state, /squaresolidboth\nmagnetizations change their sign twice, ending up in the ini -\ntial state (back-switching). The compensation temperatur e\nTcomp is indicated by a solid black line, the Curie tempera-\nture TGd\nCof the isolated Gd layer by a black dashed line. The\nthree points indicate the chosen values for the scenarios in\nFig. 2.\npectations. It is less obvious, however, why high tem-\nperatures (above Tcomp) lead to switching regardless of\nhow strongly the RE layers are demagnetized by the heat\npulse. Qualitatively, this can be understood keeping\nin mind the linear switching mechanism, which avoids\ntransverse magnetization components (see Fig. 2and\nRefs.2,28). Linear switching needs a high degree of spin\ndisorder in the system and, consequently elevated tem-\nperatures.\nMore quantitatively, the role of temperature can be\nunderstood via the equilibrium layer magnetizations as\nshown in Fig. 1, since those mark the final values for\nthe relaxation process. Let us assume that following the\nexcitation of the laser pulse the Fe layer is completely de-\nmagnetized while the Gd is only demagnetized by 50%.\nThe bilayer is far from equilibrium and a relaxation pro-\ncess will set in of which the details depend on the tem-\nperature. Each layer will relax towards its individual\nequilibrium value. We can identify three important tem-\nperature ranges:\n•T < TRE\nC: Both layers tend to increase their net\nmagnetization magnitudes towards higher values.\n•TRE\nC< T < TTM\nC: To reach equilibrium, the mag-\nnitude of the TM magnetization must still increase,\nwhile that of the RE must decrease.\n•T > TTM\nC: The system will completely demagne-\ntize.\nFor most TM-RE layer ratio Tcomp (if it exists) is\nhigher than TRE\nCand only the temperature range be-−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0)\nFe\nGd\n0.0 0 .5 1 .0 1 .5 2 .0\nTimetin ps−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0)\nFe\nGd\nFIG. 4. (Color online) Strong dissipation ( α= 1) in a 3:2\nlayer sample, implying no conservation of angular momen-\ntum. The relaxation does not lead to a TFMLS, neither at\nlow temperature where the Gd magnetization increases nor at\nhigh temperature where it decreases. (top: T < TGd\nC, bottom:\nTGd\nC< T < TFe\nC). Also shown are the transverse components\n(dotted and dashed lines) which remain small for both cases.\ntween TRE\nCandTTM\nCsupports the dynamics necessary for\nswitching — decreasing Gd magnetization and increasing\nFe magnetization. Consequently, the temperature must\nbe at least above the critical temperature of bulk Gd.\nAbove the compensation temperature switching is always\npossible. That means, if Tcomp is very low, switching can\nalso be done below TRE\nC.\nThe next question is, why the Fe layer magnetization\nstarts recovering towards negative magnetization which\nresults in a TFMLS. This is a consequence of angular mo-\nmentum conservation, as was already pointed out by sev-\neral authors10,11. While the Gd still demagnetizes (since\nthe temperature is above the bulk critical temperature\nof Gd), the dynamics of the Fe layer magnetization must\nchange into the other direction keeping the angular mo-\nmentum constant. The argument also explains that one\nneeds a certain initial Gd magnetization to start with.\nWithout initial Gd magnetization there is no angular mo-\nmentum reservoir to drag the Fe magnetization towards\nnegative values.\nAngular momentum conservation is not strictly ful-5\n0 1 2 3 4 5 6 7\nTimetin ps−0.8−0.6−0.4−0.20.00.2Magnetization Mz(T)/Mz(0)\nFe\nGd\nFIG. 5. (Color online) Layer resolved magnetization dynam-\nics for switching in a 3:2 layer sample. The Gd layer is de-\nmagnetized to 50% of its zero temperature value. The bold\nlines corresponds to the layers at the Gd-Fe interface and th e\ndashed lines shows the average value for Gd or Fe. The tem-\nperature is T= 0.5JFe/kB.\nfilled in the spin system. The relaxation part of the equa-\ntion of motion breaks this conservation on time scales\nwhich are determined by the value of the damping con-\nstant α. For low values of α, the precessional part of the\nequation of motion is much larger leading to dynamics\nwhich keeps total energy and total angular momentum\nconserved on shorter time scales. This is different when\nconsidering larger values of α. For comparison, we inves-\ntigate in Fig. 4the regime of strong dissipation ( α= 1).\nHere, the time scales of precessional dynamics is of the\nsame order as the time scale of relaxation and dissipative\neffects counteract the conservation of angular momentum\nin the system. Only if the damping constant αis suffi-\nciently small, angular momentum is almost conserved on\nshort time scales, along with the total energy, leading\nto a TFMLS as seen in Fig. 2. The figure also illus-\ntrates that — depending on the temperature — the Gd\nmagnetization might relax either towards higher or lower\nequilibrium values. The transition from dissipationless\ndynamics to the regime where damping effects dominate\nthe dynamics has previously been investigated11.\nIn the following we turn to the peculiarities of the lay-\nered system. The layer resolved magnetization in Fig. 5\nshows the importance of the interface layers for switch-\ning. For low damping, the angular momentum conserva-\ntion leads to a relaxation dynamics with an exchange of\nangular momentum between the still demagnetizing Gd\nlayer and the Fe layer, leading to a negative Fe magne-tization and, consequently, to a TFMLS. Because of the\nantiferromagnetic coupling along the Fe-Gd interface, the\nFe interface monolayer lags behind the other layers. Af-\nter some ps, however, the Fe magnetization has reached\nits new, negative equilibrium value, pushing the Gd via\nthe negative interface coupling towards positive values.\nHere, the dynamics is quicker at the interface as for the\nother Gd layer which is lagging behind.\nWe also simulated other layer thicknesses and ratio.\nWhile Fe-Gd bilayer with 20-40% Gd (for example 4:1,\n3:1 or 3:2 layer) turned out to have the correct thickness\nratio for switching (and having a magnetization compen-\nsation temperature) we found that the over-all thickness\nis less relevant. Successful switching can also be seen in\nmuch bigger samples (although on longer time scales), for\ninstance in a 30:20 Fe-Gd layer. The antiferromagnetic\ncoupling of the interface layers finally leads to a switch-\ning of all layers when the temperature of the heat bath\nexceeds Tcomp (which is almost the same as TGd\nCfor big\nsamples). Changing the ratio of Fe and Gd layer does\nnot change the switching behavior as long as the sam-\nple maintains a magnetization compensation tempera-\nture. Only the temperature range for switching increases\nwith decreasing percentage of Gd.\nIV. SUMMARY\nWe explored thermally induced magnetic switching in\nsynthetic ferrimagnets composed of a bilayer of rare-earth\nand transition metal on the basis of spin model simula-\ntions where the model parameters were calculated from\nfirst principles. Varying the temperature and the de-\ngree of initial rare-earth magnetization directly after th e\nlaser pulse one may find either, back-switching, or no-\nswitching. Deterministic magnetization reversal occurs\nabove a certain threshold temperature which is above the\nbulk Curie temperature of the rare-earth since only then\nthe magnetization of the rare-earth sublattice relaxes to-\nwards lower magnitude. 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