[    {        "title": "1004.2786v1.How_to_recognize_a_nearly_flat_band_ferromagnet_by_means_of_thermodynamic_measurements_.pdf",        "content": "arXiv:1004.2786v1  [cond-mat.str-el]  16 Apr 2010How to recognize a nearly-ﬂat-band ferromagnet by means\nof thermodynamic measurements?\nVolodymyr Derzhko and Janusz Jedrzejewski∗\nInstitute of Theoretical Physics, University of Wroc/suppress law,\npl. Maksa Borna 9, 50–204 Wroc/suppress law, Poland\nFebruary 28, 2018\nAbstract\nWe make an attempt at unveiling the thermodynamic “signatur e” of a speciﬁc class of\nelectronic systems, the so called nearly-ﬂat-band paramag nets and ferromagnets that can\ntheoretically be described by appropriate versions of the H ubbard model.\n1 Introduction\nA nearly-ﬂat-band system is an electronic system whose one or mor e bands in the single-particle\nspectrum can be made arbitrarily narrow by continuously adjusting some parameters of the system,\nwithout destroying its structure. To make this phrase more explicit , let us consider a tight-binding\ndescription of an electronic system. In this description a single-par ticle system is given by a graph\nthat consists of a set of sites, with assigned values of external po tentials, and a set of bonds\nconnecting the sites, associated to non-vanishing hopping intensit ies. Such a graph is connected if\nany two sites are connected by a sequence of bonds.\nGraphs of nearly-ﬂat-band systems constitute a class of connec ted graphs, for which it is\nnecessary to tune the hopping intensities and on-site external po tentials, or only one of those sets\nof parameters, to get one or more degenerate energy levels, the so called non-dispersive bands\nor ﬂat bands , whose degeneracy is proportional to the number of lattice sites. As a result of\nthe tuning, the hopping intensities and on-site external potentials satisfy some relations, which\nconstitute suﬃcient conditions for ﬂat bands. Note, however, that there are connected graph s\n(bipartite graphs) whose spectrum does contain a ﬂat band, and n o tuning of hopping intensities\nis required [1].\nTo each site of a graph there corresponds its neighborhood , that is the set of all the sites\nconnected with it by bonds. The topology of a graph is speciﬁed by the set of all the neighborhoods.\nIt is the topology, and not the geometry, of the underlying connec ted graph, that decides whether\nﬂat bands can appear in a system. To the best of our knowledge, th ere is no characterization\nof the topology of graphs that admit ﬂat bands. Only examples, som etimes of classes, of graphs\nwhere nearly ﬂat bands do appear are known [2, 3, 4, 5, 6].\nBy perturbing continuously the suﬃcient conditions for the ﬂat ban d, that have to be satisﬁed\nby hopping intensities and on-site external potentials, we transfo rm a ﬂat-band system into a\n∗Corresponding author: jjed@ift.uni.wroc.pl\n1nearly-ﬂat-band system. This deﬁnition of a nearly-ﬂat-band sys tem, via a ﬂat-band system\nis a theoretical, mathematical one. The theoretical limit of non-disp ersive band constitutes a\nnonphysical system that cannot be realized in experiment. The res idual entropy of multi-electron\nfree ﬂat-band system, whose ﬂat band is partially ﬁlled, is ﬁnite, whic h violates the IIIrd law\nof thermodynamics, and there is no Fermi surface. The IIIrd law is violated also in interacting\nsystems; see [5, 6] for examples, where the ground-state degen eracy in an interacting case was\ncalculated exactly. Moreover, the density of states of those sys tems is singular. In experiments,\nwe can only attempt at constructing a nearly-ﬂat-band system, w here by suitably adjusting the\nparameters of the underlying system we can make one or more band s arbitrarily narrow, at least\nin principle.\nNevertheless, ﬂat-band systems are worth of theoretical stud ies, since they are more simple for\nanalysis than nearly-ﬂat-band systems, and moreover it might hap pen that some of their features\nare stable against perturbations that transform a ﬂat band into a nearly-ﬂat band. A prominent\nexample of such a feature is ferromagnetism, and quite naturally th e ﬁrst model studied was the\nparadigmatic Hubbard model:\nH=/summationdisplay\ni,j,σti,jc+\ni,σcj,σ+U\n2/summationdisplay\ni,σni,σni,−σ, (1)\nwhere the sums are over all the sites i,jof the underlying graph, and over projections, σ, of\nthe electron spin on some axis; ti,j– the matrix elements of a single-particle Hamiltonian between\nstates localized at sites iandjgive the hopping intensities and on-site external potentials; c+\ni,σ,ci,σ,\nstand for the electron creation and annihilation operators, respe ctively; the term proportional to\nU > 0 represents a strongly screened Coulomb repulsion. When the und erlying graph is bipartite,\nLieb [1] proved the existence of unsaturated ferromagnetism in the ground state (that is, for given\nnumber of electrons the total spin of the ground state is a fractio n of the maximal one), when it is\na line graph – a proof of saturated ferromagnetism in the ground state (the total spin of the ground\nstate is maximal) was given by Mielke [2], and when it belongs to Tasaki cla ss – the corresponding\nresult was obtained by Tasaki [3, 4]. In all those cases there is a ﬂat band in the single-electron\nspectrum, and on switching on the Hubbard repulsion the paramagn etic ground state of a free\nmulti-electron system turns into a ferromagnetic one, for a specia l value of electron density (or\na narrow interval of densities) and any nonzero value of the Hubba rd on-site repulsion U. The\nlatter statement means that paramagnetic ground state turns in to a ferromagnetic one without\nany competition between the kinetic and potential energies (the Hu bbard repulsion is needed only\nto lift the macroscopic degeneracy), which is another non-physica l feature of ﬂat-band systems.\nThe ﬂat-band ferromagnetism discovered by Mielke and Tasaki app eared to be robust against\nperturbations of the ﬂat band. The proofs of this fact for nearly -ﬂat band systems can be found\nin [7, 8].\nWhile there is a number of theoretical examples of nearly-ﬂat-band systems (see the papers\nquoted above), we do not know of any measurements performed o n real nearly-ﬂat-band systems.\nThere have been a few proposals of experimental realizations of ne arly-ﬂat-band systems such as:\natomic quantum wires [9], quantum-dot super-lattices [10] or organ ic polymers [11]. However,\nthe most promising seems to be a realization as cold atoms in optical lat tices [12]. Due to a\nvery good control of system parameters in the latter case, such a realization would open new\npossibilities of investigating the mechanism of nearly-ﬂat-band ferr omagnetism, inaccessible in\nother experimental realizations, and beyond the scope of presen t-day theoretical methods. In\nthis perspective, it is already interesting and challenging to determin e theoretically characteristic\nlow-temperature thermodynamic properties of the aforemention ed systems.\n22 The models and the goal\nIn this paper, we provide a resume of our investigations of the ques tion: how to recognize a nearly-\nﬂat-band ferromagnet by means of thermodynamic measurement s? In view of the high sensitivity\nof the considered systems to details of the underlying graph, it wou ld be naive to expect that some\ngeneral answer, good for any type of such systems, can be given . We have chosen to concentrate on\nnearly-ﬂat-band systems described by the Hubbard Hamiltonian, w ith the lowest band nearly-ﬂat\nand separated from the upper bands by a gap, which we call the principal gap , to distinguish it\nfrom any other gap in the spectrum of the system. Moreover, we s et the Hubbard Uto be small\ncompared to the value of the principal gap, and the electron densit y not to exceed nfb, which is\nthe density of those electrons that can be accommodated in the ne arly-ﬂat band. For instance,\nthe Tasaki models [7] and some related models [13] belong to this class .\nOne can argue, however, that our thermodynamic results, to be p resented below, hold as well\nfor a wider class of systems. This class includes systems where ther e are two groups of bands:\nlow-energy bands and high-energy bands with the nearly-ﬂat band being the highest band in\nthe group of the low-energy bands, and separated by a principal g ap from the upper bands [9].\nAnother group of systems that belongs to this class are the syste ms where the ﬂat band sticks\nto a dispersive one in a few points of the Brillouin zone, as in the Mielke cla ss of models (see a\nremark in [14] and numerical results in [6]).\nTo minimize the burden of large volume computer work we have perfor med calculations for the\nHubbard model whose graph is a one-dimensional lattice, decorate d with additional sites located\nin the middle between the sites of the lattice (known also as a ∆-chain [1 3] or a sawtooth chain\n[5]). For appropriate hopping intensities and external potentials th e lower band of the model is ﬂat\nand the ground state is ferromagnetic for any U > 0, provided the electron density is nfb/2. This\nmodel belongs to Tasaki class of models [4]. There are many ways of p erturbing the system to get\na nearly-ﬂat-band system, whose ground state is ferromagnetic only for suﬃciently large U. In\nour calculations we chose the Tasaki perturbation [7], whose advan tage is that the ferromagnetic\nground state and its energy are known explicitly. The real matrix ele mentsti,jof the perturbed\nmodel can be chosen as follows (with the lattice constant of the one -dimensional lattice set to\nunity):ti,i+1= 1,ti,i+1/2= 1 +s,ti−1/2,i+1/2=−s,ti,i= 2−s,ti+1/2,i+1/2= 1−2s, for an integer\ni, wheres >0 is the parameter of Tasaki perturbation; up to a factor it amoun ts to the width of\nthe nearly-ﬂat band. One ﬁnds that the width of the lower band is δ−= 4s, that of the upper\nband –δ+= 4, and the gap between the bands is ε= 1 +s.\nFor comparison, we consider also a ﬁctitious noninteracting electro n system, not born by a\nHamiltonian, whose single-particle spectrum consists of two bands, with some dispersion relations,\nseparated by a gap. The advantage of this model is that the width o f the lower band, δ−, the\nupper band δ+, and the gap between the bands, ε, can be varied independently, what facilitates\nobserving their impact on thermodynamic properties of the model.\n3 The isochoric heat capacity\nWe start our considerations of thermodynamic quantities with the e ntropy per particle, s(T,v), as\na function of temperature, T(throughout the paper Tis measured in energy units), and volume\nper particle, v– the inverse of electron density, n. Practically, since this entropy is not directly\nobservable, we consider a simply related and accessible to direct mea surements quantity – the\nisochoric heat capacity per particle (brieﬂy speciﬁc heat), cV(T), for speciﬁed values of electron\ndensityn. The heat capacity at constant volume, is ideally suited for our purp oses, since it\n3describes a response of a system to heating, exclusively due to the rmal excitations. In contrast\nto other heat capacities, the isochoric heat capacity contains no c ontribution of a mechanical or\nchemical work done by the system. Such an extra contributions blu r the response of the system\nto heating, making it more diﬃcult to identify the system as a nearly-ﬂ at-band system.\nTaking into account the variety of graphs and spaces of paramete rs of nearly-ﬂat-band systems,\nwe can claim that ground states of a nearly-ﬂat-band electronic sy stems are typically paramagnetic,\nthat is the total spin Stot=o(Ne), where Neis the number of electrons. This is true even if we\nrestrict the variety of graphs to those that admit ferromagnetic ground states, that is states whose\nStot=O(Ne), for suitable values of system parameters. The reason is that if t he underlying graph\nadmits ferromagnetism, rather limited values of electron densities, nearly-ﬂat band widths and\nHubbard repulsion Uare required to make the ground state ferromagnetic. Quite gene rally, there\nis no ferromagnetism, in the above sense, if the electron density an d/or Hubbard repulsion Uare\ntoo small, and/or the nearly-ﬂat band – too wide.\nA principal gap in a single-particle spectrum induces a gap in the spect rum of the corre-\nsponding many-electron noninteracting system and interacting sy stem described by the Hubbard\nHamiltonian. Consequently, the spectra of these many-electron s ystems can be split into a lower\npart and an upper part. As a result, plots of cV(T) consist of two, low-temperature and high-\ntemperature, humps. As long as our system is paramagnetic, the lo w-temperature hump is due\nto low-energy excitations, that is excitations whose energies do no t exceed the width of the lower\npart of energy levels (the width of the nearly-ﬂat band in the nonint eracting case). The high-\ntemperature hump is, in turn, due to high-energy excitations, who se energy does exceed the gap\nbetween the low-energy group and the high-energy group of ener gy levels (the principal-gap width\nin a free nearly-ﬂat-band system). Consequently, the position of the low-temperature hump is of\nthe order of the width of the nearly-ﬂat band, while that of the high -temperature one is shifted\nby an energy of the order of the principal gap, at least for not too largeU. The degree of overlap\nof the two humps, i.e the extent and the depth of the well between t hem, depends mainly on the\nratio of the principal gap and the width of the lower band. The larger this ratio is the better the\nseparation of the humps. For suﬃciently large gaps, the bottom of this well reaches zero and is\nﬂat. Of course, cV(T) tends to zero if T→0 orT→ ∞ . These features are well illustrated in\nFig. 1, where results for the two-band noninteracting system are shown. In particular it is clear\nthat this morphology of the cV(T) plot is no characteristic of a nearly-ﬂat-band system. It can be\nobserved in any system of the class described above, interacting o r noninteracting, where instead\nof a nearly-ﬂat band there is just a narrow band.\nIn contrast to a narrow-band system, in a nearly-ﬂat-band one w e can exploit the possibility of\nshrinking the lower band, while preserving the structure of our sys tem. On decreasing δ−, starting\nfrom suﬃciently small value, the low-temperature hump moves, of c ourse, towards zero and almost\nlinearly in δ−, its half-width shrinks, while its maximum remains essentially unchange d. These\neﬀects can be seen in Fig. 1 for the noninteracting two-band model and in Fig. 2 for the particular\nHubbard model – Tasaki model, deﬁned in previous section. In the t heoretical limit of ﬂat band,\nwe are left with one, high-temperature hump, which may be separat ed from zero temperature by\na visible plateau of zero value, provided the principal gap is large enou gh. This is because there\nis practically no low-temperature excitations from the ﬂat band.\nNow, suppose that the underlying graph, the electron density, an d the Hubbard repulsion U\nare such that for those widths of the nearly-ﬂat band that are sm aller than some threshold value,\nδp−f(which depends on U), the paramagnetic ground state changes into a ferromagnetic o ne.\nStarting measurements with the values of δ−somewhat larger than δp−f, and then repeating them\nfor a decreasing sequence of values above δp−f, one observes the above described “evolution” of the\nlow-temperature hump (see Fig. 2). However, below δp−fthe gapless low-temperature excitations\n4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 0.01  0.1  1  10cv(T)\nTδ-=0.00\nδ-=0.05\nδ-=0.10\nδ-=0.50\nδ-=1.00\nFigure 1: Speciﬁc heat versus temperature, cV(T), of the noninteracting two-band model, with\nδ+= 1 and ε= 6 (in arbitrary energy units), and partially ﬁlled lower band, for a de creasing\nsequence of lower-band widths δ−and for the ﬂat band. Temperature is in a logarithmic scale.\nof a paramagnet (of hole-particle type) turn into ferromagnetic e xcitations – the magnons. This\ntransition results in pinning the low-temperature hump at the tempe rature of the order of the\nwidth of the magnons band, which is determined by U. Further decrease of δ−, down to δ−=\n0, i.e. the limit of ﬂat-band ferromagnet, does not bring any signiﬁca nt changes to the low-\ntemperature hump, neither to its position nor to its shape. In Fig. 3 we see a transient region\nbetween the paramagnet and the ferromagnet, with the values of δ−greater than δp−f, but smaller\nthan the values of δ−for which a typical for a paramagnet, linear in δ−, “motion” of the low-\ntemperature hump towards zero is observed. In contradistinctio n to the overall shape of cV(T)\nplots in paramagnetic and ferromagnetic states, we expect that t he size (measured by a range of\nδ−) of the transient region and the shape of cV(T) plots in this region are rather sensitive to the size\nof the system used for calculations. Finally, in Fig. 4, we see the pinnin g of the low-temperature\nhump for δ−smaller than δp−f.\n4 Heat capacity at constant chemical potential\nIn the previous section we discussed the heat capacity per particle at constant volume, which is\nproportional to the second derivative with respect to temperatu re,T, of the fundamental ther-\nmodynamic function, the Helmholtz free energy per particle as a fun ction ofTand volume per\nparticle, v. An analog of this thermodynamic quantity, biased by ﬁnite-size eﬀe cts, can be calcu-\nlated numerically for small Hubbard systems, with ﬁxed volume and pa rticle number, by means of\nthe canonical ensemble. Formally, the heat capacity at constant c hemical potential, per unit vol-\nume,cµ, is another quantity of this kind. It is proportional to the second d erivative with respect to\ntemperature of the fundamental thermodynamic function, the g rand-canonical potential per unit\nvolume (i.e. minus the pressure) as a function of temperature and c hemical potential, µ. This is,\nhowever, an unusual heat capacity, since it refers to open syste ms that are in thermal equilibrium\n5 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0.001  0.01  0.1  1  10cv(T)\nTs=0.02\ns=0.03\ns=0.05\nFigure 2: Speciﬁc heat versus temperature, cV(T), of the particular Hubbard model – Tasaki\nmodel, deﬁned in the paper, that consists of ten sites with periodic b oundary conditions, and ﬁve\nelectrons (quarter ﬁlling), for U= 0.1 and for the range of lower-band widths δ−, where a typical\nparamagnetic behavior is observed. Temperature is in a logarithmic s cale.\nwith particle reservoir of given chemical potential. Consequently, in a plot of cµversusT, diﬀerent\npoints correspond, in general (depending on µ, the electron density at constant µ,nµ(T), can be\nmonotonic or not), to systems with diﬀerent amount of matter. An analog of this thermodynamic\nquantity, biased by ﬁnite-size eﬀects, can be calculated numerically for small Hubbard systems,\nwith ﬁxed volume and chemical potential, by means of the grand-can onical ensemble. Of course,\ncµ(T) can be related to ncV(T), which is the isochoric heat capacity per unit volume, by the\nthermodynamic identity:\ncµ(T) =nµ(T)cV(T,v(T,µ)) +T/parenleftbigg∂nµ(T)\n∂T/parenrightbigg2/parenleftbigg∂nµ(T)\n∂µ/parenrightbigg−1\n. (2)\nThe diﬀerence between those two heat capacities, cµ−nµcV, is born by the chemical work that\nan open system does exchanging matter with matter reservoir whe n heated.\nIn a free system, whose spectrum consists of a ﬂat band only, cV(T) vanishes identically, since\nthere are no excitations in this system. In contradistinction to cV(T),cµ(T) can be nonzero, due\nto the chemical work, the heated system does exchanging matter with a matter reservoir. In such\na system the electron density is a function of the activity ξ= exp(µ/T) only; we used here the\nfact that we can always set the energy of the ﬂat band to zero. He nce, the second term of identity\n(2) – the chemical work term can be written as\nξln2(ξ)dn(ξ)\ndξ≥0. (3)\nIn a free system, with only a ﬂat band, dn(ξ)/dξ > 0, therefore the chemical work term, is nonzero\nunless the chemical potential coincides with the energy of the ﬂat b and. Consequently, there is\njust one hump in a plot of cµ(T) forµ/negationslash= 0, located at low temperatures, if |µ|is small. The size\nand the position of this hump is sensitive to the value of the chemical p otential.\n6 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0.001  0.01  0.1  1  10cv(T)\nTs=0.010\ns=0.012\ns=0.015\nFigure 3: Speciﬁc heat versus temperature, cV(T), of the particular Hubbard model – Tasaki\nmodel, deﬁned in the paper, that consists of ten sites with periodic b oundary conditions, and ﬁve\nelectrons (quarter ﬁlling), for U= 0.1 and for the range of lower-band widths δ−, greater than\nδp−fbut smaller than the values of δ−in Fig. 2. Temperature is in a logarithmic scale.\nIf a ﬂat band is accompanied by some upper bands, separated by a g ap from the ﬂat one\n(like in the systems considered in the previous section), the above s tatements concerning low-\ntemperature hump remain qualitatively true at suﬃciently low temper atures and for suﬃciently\nsmall|µ|. Additionally, such a system contributes signiﬁcantly to cµ(T) at high-temperatures;\nthere is a high-temperature hump in the plot of cµ(T), which coincides essentially with ncV(T)\n– the isochoric heat capacity per unit volume. Small |µ|guarantees that the low- and high-\ntemperature humps are well separated.\nA weak perturbation that turns a ﬂat-band system into a nearly-ﬂ at-band one, could have\ninﬂuenced signiﬁcantly only the low-temperature hump. Unlike the low -temperature hump in\ncV(T) plot of a free nearly-ﬂat-band system, which is due to thermal ex citations from a nearly-\nﬂat-band, and therefore is sensitive to the width of this band (its p osition moves towards zero\ntemperature as the perturbation decreases), the low-tempera ture hump in cµ(T) is due to exchange\nof matter, and its position and shape are quite insensitive to the widt h of the nearly-ﬂat band.\nThus, a weak perturbation of the ﬂat band does not change essen tially the two-hump cµ(T) plot\nof the ﬂat-band case. The above observations are well illustrated in Figs. 5, 6, 7.\nNow, consider the Hubbard model (1) whose graph is speciﬁed in Sec tion 2. In the absence of\nperturbation ( s= 0), the lower band in the single-particle spectrum is ﬂat and is separ ated by a\ngap from the upper band. There is a basis of the ﬂat-band eigensub space that consists of localized\neigenstates. In terms of Slater determinants of those localized sin gle-particle eigenstates, one\ncan construct a basis of the ground-state eigensubspace of the Hubbard model, for any Hubbard\nrepulsion U > 0, provided the number of electrons does not exceed half-ﬁlling of t he ﬂat-band\n[4]. Derzhko et al [5] have demonstrated that those bases, as well as ground-state-eigensubspace\nbases of some similar ﬂat-band Hubbard models [6], can be mapped ont o a ﬁctitious hard-core\nlattice gas. The ground-state is paramagnetic, and the many-elec tron excited states are separated\nby a gap from the ground state, for any ﬁlling that is smaller than half -ﬁlling of the ﬂat-band.\n7 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0.001  0.01  0.1  1  10cv(T)\nTs=0.000\ns=0.005\ns=0.007\nFigure 4: Speciﬁc heat versus temperature, cV(T), of the particular Hubbard model – Tasaki\nmodel, deﬁned in the paper, that consists of ten sites with periodic b oundary conditions, and ﬁve\nelectrons (quarter ﬁlling), for U= 0.1 and for the range of lower-band widths δ−that are smaller\nthanδp−f, and for the ﬂat band. Temperature is in a logarithmic scale.\nConsequently, for such ﬁllings, at suﬃciently low temperatures tha t make excitations above the\nground state very unprobable, the response of the system to he ating amounts essentially to that\nof the hard-core gas corresponding to the macroscopically-dege nerate ground state. Since there\nare no excitations in a closed hard-core gas, cVvanishes identically. In contrast, the chemical\nwork term (3), which in the case under consideration and in many oth er cases can be calculated\nanalytically, is nonzero for µ/negationslash= 0. Therefore, there is just one hump in a plot of cµ(T) forµ/negationslash= 0,\nat low temperatures.\nA weak perturbation of the ﬂat-band ( s≪1) does not bring essential changes to this low-\ntemperature hump. It is given essentially by the chemical work term of the hard-core gas corre-\nsponding to the ground state of the ﬂat-band limit of the considere d nearly-ﬂat-band Hubbard\nmodel. Therefore, all the above described features of the low-te mperature hump of cµ(T) plot in\nfree nearly-ﬂat-band systems, in particular its insensitivity to the width of the nearly-ﬂat band,\nremain valid in the considered Hubbard model. These observations ap ply also to other Hubbard\nsystems mentioned in Section 2. It is the derivative dn(ξ)/dξthat diﬀerentiates between hard-core\ngases and/or ﬂat band systems and makes the shape of the low-te mperature hump speciﬁc for a\nsystem. However, those diﬀerences are not dramatic; they are h idden in ﬁne details of the shape\nof the low-temperature hump (see the plots in [6]).\nNaturally, at high temperatures, there develops a high-temperat ure hump due to excitations\nabove the ground state, much like in the case of a free systems con sidered above, and it amounts\nessentially to the speciﬁc heat per unit volume, ncV(T). The overall picture of cµ(T) plot in the\nconsidered Hubbard models is qualitatively the same as those in free n early-ﬂat-band systems con-\nsidered in the previous paragraph. The described above features ofcµ(T) plot are well illustrated\nby the plots displayed in [5], for the Hubbard model considered in this p aper, and in [6] for other\nHubbard models with ﬂat or nearly-ﬂat-bands. The data for all tho se plots were obtained from\nexact diagonalization of small Hubbard systems, with the number of sites in their graphs between\n8 0 0.05 0.1 0.15 0.2 0.25\n 0.001  0.01  0.1  1  10cµ(T)\nTδ-=0.00, µ=0.00\nδ-=0.00, µ=0.05\nδ-=0.01, µ=0.05\nFigure 5: Plot of cµ(T) for the noninteracting two-band model, with δ+= 1 andε= 6 (in arbitrary\nenergy units), with the ﬂat-band energy and the upper edge of th e corresponding nearly-ﬂat band\nset to zero. Temperature is in a logarithmic scale.\n10 and 20. Apparently, looking at a cµ(T) plot on can hardly infer whether the underlying system\nis a nearly-ﬂat-band (ﬂat-band) Hubbard system.\n5 Summary\nSumming up, we propose an answer to the question asked in the title o f our report. We argue\nthat the isochoric heat capacity per particle, cV(T), is a good candidate, a suﬃciently sensitive\nthermmodynamic quantity, to measure. At the heart of our answe r is the fact that in the case\nof nearly-ﬂat-band systems we can vary the width of the nearly-ﬂ at lower band, and make it as\nnarrow as we wish. Then, we can watch how cV(T) plot “evolves” with the decreasing width of the\nlower-band, at suﬃciently low temperatures. This low-temperatur e “evolution” provides a signa-\nture of a nearly-ﬂat band paramagnet, paramagnet-ferromagn et transition, and nearly-ﬂat-band\nferromagnet. A thermodynamic measurement performed only for a single value of a nearly-ﬂat-\nband width is not suﬃcient for this purpose. We demonstrate also th at the heat capacity at\nconstant chemical potential, per unit volume, versus T,cµ(T), is not a suitable quantity to mea-\nsure, since it depends weakly on the width of a nearly-ﬂat-band and is sensitive to the value of the\nchemical potential kept constant. The shape of cµ(T) plot is no characteristic of a nearly-ﬂat-band\n(ﬂat-band) Hubbard system. One can ask naturally, whether suc h thermodynamic characteristics\nof closed systems like the coeﬃcient of thermal expansion or the iso thermal compressibility can\nbe used to recognize nearly-ﬂat-band paramagnets and ferroma gnets. Our studies of the nearly-\nﬂat-band two-band model suggest that those quantities can be u sed to detect nearly-ﬂat-band\nHubbard paramagnets. However, those quantities are not well de ﬁned and cannot be calculated\nfor small lattice systems. Therefore, we have no data for Hubbar d systems with ferromagnetic\nground state to check if those quantities are suitable also for dete cting nearly-ﬂat-band ferromag-\nnets.\n9 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18\n 0.001  0.01  0.1  1  10cv(T)n(T)\nTδ-=0.00, µ=0.00\nδ-=0.00, µ=0.05\nδ-=0.01, µ=0.05\nFigure 6: Plot of nµ(T)cV(T,v(T,µ)) versus T, for the noninteracting two-band model, with\nδ+= 1 andε= 6 (in arbitrary energy units), with the ﬂat-band energy and the u pper edge of the\ncorresponding nearly-ﬂat band set to zero. The dashed line coincid es with the continuous one.\nTemperature is in a logarithmic scale.\nAcknowledgements\nWe thank Oleg Derzhko for discussions on ﬂat-band systems.\nReferences\n[1] E. H. Lieb, Two theorems on the Hubbard model , Phys. Rev. Lett. 62, 1201 (1989).\n[2] A. Mielke, Ferromagnetic ground states for the Hubbard model on line gr aphs, J. Phys. A:\nMath. Gen. 24, L73 (1991). Ferromagnetism in the Hubbard model and Hund’s rule , Physics\nLetters A 174, 443 (1993).\n[3] H. Tasaki, Ferromagnetism in the Hubbard models with degenerate singl e-electron ground\nstates , Phys. Rev. Lett. 69, 1608 (1992).\n[4] A. Mielke, H. Tasaki, Ferromagnetism in the Hubbard model , Commun. Math. Phys. 158,\n341 (1993).\n[5] O. Derzhko, A. Honecker, and J. Richter, Low-temperature thermodynamics for ﬂat-band\nferromagnet: Rigorous versus numerical results , Phys. Rev. B 76, 220402(R) (2007). A.\nHonecker, O. Derzhko, J. Richter, Ground-state degeneracy and low-temperature thermody-\nnamics of correlated electrons on highly frustrated lattic es, Physica B 404, 3316 (2009).\n[6] O. Derzhko, A. Honecker, and J. Richter, Exact low-temperature properties of a class of\nhighly frustrated Hubbard models , Phys. Rev. B 79, 054403 (2009). O. Derzhko, J. Richter,\n10 0 0.05 0.1 0.15 0.2 0.25\n 0.001  0.01  0.1  1  10cµ-ncv\nTδ-=0.00, µ=0.00\nδ-=0.00, µ=0.05\nδ-=0.01, µ=0.05\nFigure 7: Plot of cµ(T)−nµ(T)cV(T,v(T,µ)) versus T, for the noninteracting two-band model,\nwithδ+= 1 and ε= 6 (in arbitrary energy units), with the ﬂat-band energy and the u pper edge\nof the corresponding nearly-ﬂat band set to zero. Temperature is in a logarithmic scale.\nA. Honecker, M. Maksymenko, and R. Moessner, Low-temperature properties of the Hubbard\nmodel on higly frustrated one-dimensional lattices , Phys. Rev. B 81, 014421 (2010).\n[7] H. Tasaki, Ferromagnetism in Hubbard models , Phys. Rev. Lett. 75, 4678 (1995). Ferro-\nmagnetism in the Hubbard model: a constructive approach , Commun. Math. Phys. 242, 445\n(2003).\n[8] A. Tanaka and H. Ueda, Stability of ferromagnetism in the Hubbard model on the kago me\nlattice , Phys. Rev. Lett. 90, 067204-1 (2003).\n[9] R. Arita, K. Kuroki, H. Aoki, A. Yajima, and M. Tsukada, S. Watan abe, M. Ichimura, T.\nOnogi, and T. Hashizume, Ferromagnetism in a Hubbard model for an atomic quantum wire :\na realization of ﬂat-band magnetism from even-membered rin gs, Phys. Rev. B 57, R6854\n(1998).\n[10] H. Tamura, K. Shiraishi, T. Kimura, and H. Takayanagi, Flat-band ferromagnetism in quan-\ntum dot superlattices , Phys. Rev. B 65, 085324 (2002).\n[11] Y. Suwa, R. Arita, K. Kuroki, and H. Aoki, Flat-band ferromagnetism in organic polymers\ndesigned by a computer simulation , Phys. Rev. B 68, 174419 (2003).\n[12] I. Bloch, J. Dalibard, W. Zwerger, Many-body physics with ultracold gases , Rev. Mod. Phys.\n80, 885 (2008)\n[13] M. Ichimura, K. Kusakabe, S. Watanabe, T. Onogi, Flat-band ferromagnetism in extended\n∆-chain Hubbard models , Phys. Rev. B 58, 9595 (1998).\n[14] K. Kusakabe and H. Aoki, Ferromagnetic spin-wave theory in the multiband Hubbard mo del\nhaving a ﬂat band , Phys. Rev. Lett. 72, 144 (1994).\n11"    },    {        "title": "2305.04459v1.Enhanced_Itinerant_Ferromagnetism_in_Hole_doped_Transition_Metal_Oxides__Beyond_the_Canonical_Double_Exchange_Mechanism.pdf",        "content": "arXiv:2305.04459v1  [cond-mat.str-el]  8 May 2023Enhanced Itinerant Ferromagnetism in Hole-doped Transiti on Metal Oxides:\nBeyond the Canonical Double Exchange Mechanism\nZhao Liu1, 2,∗and Nikhil V. Medhekar1,2,†\n1Department of Materials Science and Engineering, Monash Un iversity, Victoria 3800, Australia\n2ARC Centre of Excellent in Future Low-Energy Electronics Te chnologies, Monash University, Victoria 3800, Australia\nHere we demonstrate the occurrence of robust itinerant ferr omagnetism in Mott-Hubbard systems\nat both low and high doping concentrations. Speciﬁcally, we study the eﬀect of hole doping on the\nexperimentally synthesized LaCrAsO via ﬁrst-principles c alculations and observe that the parent\nG-type antiferromagnetism vanishes quickly at low doping c oncentration ( x∼0.20) and the system\nbecomes ferromagnetic metal due to the canonical double exc hange (CDE) mechanism. As xcon-\ntinues to increase, the onsite energy diﬀerence between Cr 3 dand As 4 porbitals decreases and the\nsystem transitions to a ferromagnetic negative charge-tra nsfer energy metal. Therefore, the itinerant\nferromagnetism doesn’t terminate at intermediate xas CDE mechanism usually predicts. Further-\nmore, our calculations reveal that both nearest and next-ne arest ferromagnetic exchange coupling\nstrengths keep growing with x, showing that ferromagnetism caused by negative charge-tr ansfer\nenergy state is \"stronger\" than that of CDE picture. Our work not only unveils an alternative\nmechanism of itinerant ferromagnetism, but also has the pot ential to attract immediate interest\namong experimentalists.\nIntroduction. -One of the oldest but recurrent top-\nics in condensed matter physics is itinerant ferromag-\nnetism (FM) [1–10]. Discovered since ancient time\nin materials such as elemental iron and nickel, itiner-\nant FM now plays a crucial role in various technologi-\ncal applications, including modern-day data processing\nand storage [11]. Several mechanisms have been pro-\nposed for itinerant FM, including Stoner criterion [12],\nNagaoka’s theorem [13–15], ﬂat-band model [16–19],\nmulti-orbital Hubbard models [20–22], canonical dou-\nble exchange (CDE) [23, 24], etc. Among these mecha-\nnisms, CDE is a prominent one in both mixed valence\ntransition metal oxides (such as magnetites [25] and\nspinel ferrites [26]) and doped Mott-Hubbard insula-\ntors [27]. In the latter system, itinerant charge carri-\ners arise from cation’s broad dbands while the anion’s\nporbitals remain inactive. To maximize the kinetic\nenergy, an antiferromagnetism (AFM)-FM transition\noccurs at low doping concentration. In manganite per-\novskites, the occurrence of itinerant FM is marked by\na strong suppression of resistivity by magnetic ﬁeld\nduring the FM-paramagnetism phase transition [28].\nThis is now known as the colossal negative magnetore-\nsistance, a crucial concept in spintronics. Nevertheless,\na transition from FM to AFM always occurs at high\ndoping concentration, limiting the application of itin-\nerant FM in spintronics.\nThis work introduces a mechanism for persistent\nitinerant FM that can survive at high doping concen-\ntration in Mott-Hubbard systems. To set the stage,\nwe ﬁrst discuss the prototypical Mott-Hubbard sys-\ntem: LaCrAsO, which has been experimentally syn-\nthesized [29]. While LaCrAsO is metallic, several indi-\nrect observations suggest that it is in close proximity\nto a Mott-Hubbard insulator [29]. Firstly, the mea-\nsured room-temperature electrical resistivity ρ∼3.8\nmΩ·cm, which corresponds to a normalized mean free\npathkFl∼hc/e2ρ∼0.6 (where c = 8.98 Å is the lat-\ntice constant along the c direction). The fact that kFl\n< 1 strongly indicates LaCrAsO is a bad metal. Sec-ondly, a local magnetic momentum of 1.57 µB/Cr is\nreported at room temperature—the existence of local\nmagnetic momenta implies a strong correlation eﬀect.\nFinally, the ground state of LaCrAsO is G-type AFM,\nsuch a long-range order is the result of superexchange\nmechanism. Taking all of these experimental results\ntogether, even if LaCrAsO is not fully Mott-Hubbard\ninsulating, it should not be far away from it. Conse-\nquently, various theoretical studies have explored the\npossibility of high-temperature superconductivity in\nelectron doped LaCrAsO [30, 31] and BaCr 2As2[32],\naiming at constructing a similar 3 dorbital ﬁlling to\nFe2+in the well-known superconductor LaFeAsO [33].\nHere we investigate the other type of charge dop-\ning: hole doping (its concentration is labelled by x),\nwhere the driving force is to upgrade the valence state\nof Cr from 2+ to 3+. As described earlier, the CDE\nmechanism can induce an AFM-FM transition, but\nonly at small x, and at large xthe system often re-\nturns to AFM phase as the superexchange mecha-\nnism dominates. On the other hand, it is known that\nthe higher valence state Cr3+can lead to negative\ncharge-transfer energy states which greatly enhance\nFM [34, 35]. Based on these arguments, there will be\na competition between superexchange mechanism and\nnegative charge-transfer energy states in heavily hole-\ndoped LaCrAsO, which makes it intriguing to explore\nthe evolution of magnetic ground state as well as the\nexchange coupling strength ( J) with respect to x.\nIn this work, via ﬁrst-principles calculations, we\ndemonstrate that the parent G-type AFM quickly dis-\nappears at small xvalues, as predicted by the CDE\nmechanism. However, contrary to CDE mechanism,\nthe FM doesn’t vanish at an intermediate xvalue but\npersists up to x= 1.00, violating the superexchange\nmechanism. Additionally, our calculations reveal that\nthe nearest and next-nearest ferromagnetic exchange\ncoupling strengths (labelled as J1andJ2) continue to\ngrow with increasing x, indicating a stronger itinerant\nFM than that of the CDE mechanism. We attribute2\nFigure 1: (a) Perspective view of 2 ×2×1 (La1−xSrx)CrAsO supercell, here x= 0.125 is shown as an example. (b)\nLattice constants a, b, c (black color) and angles α,β,γ(blue color) of the ground state structures at diﬀerent xvalues.\n(c) Relative energies of three long-range magnetic orders a t diﬀerent xvalues.\nthis enhanced itinerant FM at large xvalues to the\nformation of negative charge-transfer energy states .\nFirst-principles calculations. -The ﬁrst-principles\ncalculations were performed using the Vienna ab-initio\nsimulation package (VASP) within the framework of\ndensity functional theory (DFT) [36]. For geometric\noptimization and electronic property calculations, a\nplane-wave cutoﬀ 600 eV was used. All our calcu-\nlations were converged within 10−5eV for energy\nand 0.01 eV/Å for Hellman-Feynman forces. For the\nLaCrAsO unitcell, the Brillouin zone integration was\ncarried out with 14 ×14×10 k-point sampling for\nself-consistency. The majority of calculations were\nbased on the non-empirical, strongly constrained and\nappropriately normed (SCAN) functional [37, 38],\nwhich is a parameter-free functional and can treat\ncharge, spin and lattice degrees of freedom on equal\nfooting. For the interlayer van der Waals (vdW)\ninteractions, the revised Vydrov-van Voorhis nonlocal\ncorrelation functional ( rVV10) was employed [39].\nIt is found that SCAN + rVV10 shows a good\nperformance on LaCrAsO by reproducing several key\nexperimental results (see Sec. A of Supplementary\nMaterials [40] for details). In addition, we also\nemployed the hybrid HSE06 functional [41, 42] to\ncorrect the magnetic band structure.\nThe hole doping in LaCrAsO is achieved by par-\ntially replacing La with Sr in a supercell. The ionic\nsize of Sr2+is almost identical to that of La2+, thus,\na true solid solution should be formed over a large x\nrange. This substitution method has been widely used\nin hole-doped materials, such as cuprates [43], iron-\nbased pnictides [44], inﬁnite-layer nickelate [45], and\nmagnanites [28]. Since charge/spin/orbital anomalies\nhave been reported at doping concentration x= n/8\n(n is an integer) for both square lattice based cuprates\n[46, 47] and perovskite based magnanites [28], a 2 ×2\n×1 supercell was adopted in this work (see Fig. 1(a) ).\nBy replacing n (n = 1–7) La atoms with Sr, xvalue\nfrom 0.00 to 1.00 can be simulated. To account for any\npossible distortion introduced by hole doping, both the\nsupercell and atomic coordinates were allowed to op-timize freely. For all xvalues, our calculations show\nthat (La 1−xSrx)CrAsO alloy is energetically preferred\n(see Sec. B of Supplementary Materials [40]). The\noptimized lattice constants and angles of the ground\nstates are presented in Fig. 1(b) , from which it is clear\nthat all the supercells have negligible distortions as the\nthree angles α,βandγare all close to 90◦. Such small\ndistortions can be traced back to the similar ionic ra-\ndius of La3+and Sr2+. To provide a better description\nof the electronic structures at x= 0.50, it was also sim-\nulated in a unitcell.\nResults and discussion. -First we explore the eﬀect of\nhole doping on the fundamental electronic structures\nand in particular, the charge-transfer energy. The\ncharge-transfer energy composes of two parts: single-\nparticle part εdpwhich is the onsite energy diﬀerence\nbetween Cr 3 dand As 4 porbitals, and interacting\npart E intwhich is a function of the interaction pa-\nrameters and electron ﬁllings. As shown in Fig. S3 ,\nasxincreases, the position of As 4 porbitals gradu-\nally shift to higher energy compared with Cr 3 d, lead-\ning to a reduced εdp. Such a phenomenon has also\nbeen reported in hole doped inﬁnite-layer nickelate re-\ncently [48]. Additionally, the charge-transfer energy\nof an electron from As 4 pto Cr 3dorbitals includes\nthe Hubbard repulsion of the transferred electron with\nthed-electrons already present on the Cr ions. There-\nfore, E intis reduced with a decrease in the number\nofd-electrons, following approximately E int(Cr3+) =\nEint(Cr2+) - Ud, where U dis the average Hubbard in-\nteraction of Cr 3 dorbitals [27]. Taken together, with\nhigh cationic valence states, charge-transfer energy is\nsigniﬁcantly reduced.\nTo investigate the magnetic ground state, three long-\nrange magnetic orders were considered: FM with mag-\nnetic ordering momentum q= (0, 0, 0), checkboard\nAFM (C-type AFM) with q= (π,π, 0) and strip\nAFM (S-type AFM) with q= (π, 0, 0). The rela-\ntive energies of these magnetic orders at diﬀerent x\nvalues are shown in Fig. 1(c) . Atx= 0.00, C-type\nAFM is the ground state, which is ascribed to the su-\nperexchange mechanism (see Sec. D of Supplementary3\nFigure 2: (a,b) Orbital-resolved magnetic band structure o f SrCrAsO for spin up and spin down channel. The high\nsymmetry k-path is R-A-Z- Γ-X-M-Γ: (0.5, 0.5, 0.0)-(0.5, 0.5, 0.5)-(0.0, 0.0, 0.5)-(0.0, 0.0 , 0.0)-(0.5, 0.0, 0.0)-(0.5, 0.5,\n0.0)-(0.0, 0.0, 0.0). (c) Magnetic band structure of (La 0.5Sr0.5)CrAsO (in a unitcell) in the spin up channel (black). Red:\nthe Wannier ﬁtted band structure. (d) Topview of the four max imally localized Wannier functions downfolded from the\ngrey color shaded bands in (c). The (La 0.5Sr0.5)O sublayer is omitted for a better view and the isovalue is ±0.60 Å−3/2.\nMaterials [40] for more details). This magnetic order\npersists up to x∼0.20, beyond which the FM order\nbecomes the lowest one. As xfurther increases, the\nenergy diﬀerence between FM and other AFM mag-\nnetic order grows larger, in accordance with the CDE\npicture. However, it is unexpected that the energy dif-\nference continues to grow without saturating even at x\n= 1.00. This behavior signiﬁcantly deviates from the\nCDE picture in the sense that there is no optimal dop-\ning concentration ( xc). In the CDE mechanism, the\ngain from kinetic energy reaches the most at xcwithTc\nthe highest. After xc, the gain from exchange energy\ngradually prevails and the superexchange mechanism\nshould make SrCrAsO AFM (see Sec. D of Supple-\nmentary Materials [40] for details). In order to explain\nthis unconventional doping behaviour, we now turn to\nthe electronic structures.\nFig. 2(a)-(b) presents the orbital- and spin-resolved\nFM band structures of SrCrAsO ( x= 1.00). We ob-serve that the spin up channel is metallic while spin\ndown channel is insulating, indicating that SrCrAsO is\na half metal (HSE06 functional also conﬁrms this, see\nFig. S7 ). Moreover, there is a signiﬁcant disparity in\norbital compositions for the two spin channels. Specif-\nically, in the spin down channel, the valence and con-\nduction bands around Fermi level (E f) are contributed\nby As 4p/O 2pand Cr 3 dorbitals respectively. Nev-\nertheless, in the spin up channel, there are states of\npredominantly As 4 pcharacter above E falong Z-Γ-\nX direction at around 2.60 eV. We stress that these\nAs 4pstates are not originated from d−phybridiza-\ntion. In the case here, the 4 pstates would be locating\nat Efeven with the hybridization switching oﬀ, indi-\ncating the negative charge-transfer energy nature [35].\nThese negative charge-transfer energy states make the\nmagnetic molecular orbitals as the underlying building\nblocks rather than the localized atomic orbitals, result-\ning in a large spreading of magnetic orbitals. This, in4\nFigure 3: (a) J1andJ2at diﬀerent xvalues. (b) Phase transition temperature at diﬀerent xvalues. The orange colored\nregion near x= 0.20 is the critical point region. (c) Total density of stat es atx> 0.20, red/blue represents spin dn/up\nchannel respectively. (d) O 2 porbital-resolved magnetic band structure of SrCrAsO in spi n up channel. (e) Evolution of\nmagnetic susceptibility with respect to temperature for di ﬀerent Heisenberg model in SrCrAsO.\nturn, strongly enhances the FM exchange interaction,\nas bothJ1andJ2will become FM [35].\nTo visualize the magnetic molecular orbitals, we fo-\ncus on the system at x= 0.50 in an unitcell instead of\nx= 1.00 for there is no local gap structure around E f\nin Fig. 2(a). In Fig. 2(c) , we observe four bands (grey\ncolor shaded) are isolated from the others, which can\nbe downfolded to obtain maximally localized Wannier\norbitals (MLWFs) [49]. The overall downfolding is sat-\nisfactory as evidenced by the good agreement between\nthe DFT and the Wannier ﬁtted bands (see Fig. 2(c) ).\nFig. 2(d) displays the four MLWFs, and it is clear that\nthey are composed of small CrAs clusters rather than\nlocalized atomic orbitals. This feature can be reﬂected\nin the magnetic form factor through inelastic neutron\nscattering, as observed in itinerant chiral magnet MnSi\nrecently [50]. Similar to the FM CrAs monolayer,\nthese MLWFs are of anti-bonding type, so in principle,\nwith further hole doping, the FM phase should become\nstronger as the occupation of anti-bonding orbitals de-\ncreases [35]. This actually explains the anomaly we\nobserve in Fig. 1(c) , where the FM order becomes in-\ncreasingly stable as xapproaches 1.00. Hence, at large\nx, (La1−xSrx)CrAsO becomes a FM negative charge-\ntransfer energy metal and it is noted that the itinerant\nFM here can’t be explained by other exchange mecha-\nnisms (see Sec. D of Supplementary Materials [40] for\ndetailed elaboration).\nAfter understanding the origin of FM in both smalland large xvalues, we next evaluate the phase transi-\ntion temperatures ( TN/Tcfor Neel/Curie temperature\nrespectively). By mapping the relative energies of FM,\nC-type AFM and S-type AFM to the Heisenberg model\nwithS= 3/2 (see [40] for detailed information), both\nJ1andJ2can be obtained as shown in Fig. 3(a) . Atx\n= 0.00 and 0.125, J1is AFM (positive value) while J2\nis FM (negative value), therefore C-type AFM is the\nground state as conﬁrmed by the DFT calculations.\nAtx∼0.20,J1changes its sign and becomes FM, and\nthen both J1andJ2are FM. As xincreases, the mag-\nnitude of both J1andJ2increase without saturation.\nAs mentioned before, such a feature stems from the\nfact that the system is closer to \"ideal ﬁlling\" as xap-\nproaches 1.00. To determine TN/Tc, classical Monte\nCarlo (MC) simulations were preformed on a 40 ×40\n×1 supercell based on Heisenberg Hamiltonian with\nJ1andJ2[51]. The phase transition temperatures ob-\ntained are presented in Fig. 3(b) . Initially, TNgradu-\nally decreases until a critical point region (labelled by\norange color). After that Tccontinuously increases to\n∼920 K at x= 1.00. The total density of states for x>\n0.20 are displayed in Fig. 3(c) . It is evident that most\nof these FM phases are half metal, except for x= 0.25.\nThe half-metallic gap reaches its maximum value of ∼\n1.88 eV at x= 0.375 and reduces to ∼0.80 eV at x=\n1.00. If HSE06 functional is further considered, there\nis a∼+0.80 eV correction to the half-metallic gap\n(seeFig. S7 ). The high Tcand the large half-metallic5\ngap makes (La 1−xSrx)CrAsO ( x> 0.25) a promising\ncandidate for half-metallic ferromagnets.\nIn the above discussions, we concentrated solely on\nthe intralayer coupling. However, we found that at\nhighxvalues, not only As 4 p, but also O 2 porbitals\nare polarized. In Fig. 3(d) , we present the O 2 p\norbital-resolved magnetic band structure in the spin\nup channel, and it is obvious that O 2 pbands cross\nEf. This means that these O 2 pstates can be regarded\nas electron or hole reservoirs between CrAs sublayers\nand promote a three-dimensional (3D) magnet behav-\nior. To investigate the 2D-3D crossover at x= 1.00,\nwe calculated the nearest out-of-plane exchange cou-\npling (J⊥) to be -6.40 meV and simulated the mag-\nnetic susceptibility with/without J⊥, as plotted in Fig.\n3(e). The introduction of J⊥not only pushes Tcup\nto a higher temperature ( ∼1080 K), but also gives a\nsharper peak with smaller full width at half maximum,\nsuggesting the 3D nature of SrCrAsO.\nBecause As 4 porbitals play a vital role in stabiliz-\ning the FM order, lastly we discuss how to track them\nexperimentally. The contribution of As 4 porbitals can\nbe identiﬁed in both energy space and real space. In\nenergy space, a \"shoulder\" or even a \"peak\" struc-\nture will occur at As L2/3edge in the electron energy\nloss spectroscopy as xincreases, just like the case in\ncuprates [52]. In real space, the valence charge den-\nsity around the As site should gradually reduce with\nincreasing x, which can be observed by synchrotron X-\nray diﬀraction. We also note that similar observations\nhave been made for the ligand hole in cubic perovskite\nSrFeO 3in a recent study [53].\nConclusions. - In summary, we propose an alterna-\ntive phase diagram for doped Mott-Hubbard system.\nAs the concentration of hole doping increases, the va-\nlence state of the metallic ions becomes higher and the\ncharge-transfer gap gradually turns to negative. The\nnegative charge-transfer energy states result in strong\nferromagnetism, causing the overdoped system to be\nferromagnetic rather than antiferromagnetic described\nby superexchange mechanism. Furthermore, the fer-\nromagnetism at large doping concentration is so ro-\nbust that it is even stronger than that of canonical\ndouble exchange mechanism at low doping. Based on\nthis proposed phase diagram, we expect that a colossal\nnegative magnetoresistance will occur at large doping\nconcentration, similar to that observed in double ex-\nchange systems. 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Arima, arXiv:\n2303.02571 ."    },    {        "title": "2011.13727v1.d0_Ferromagnetism_in_Ag_doped_Monoclinic_ZrO2_Compounds.pdf",        "content": "1 \n d0 Ferromagnetism in Ag -doped Monoclinic ZrO 2 \nCompounds  \n \nL. Chouhan1, G. Bouzerar2 and S. K. Srivastava1* \n \n1Department of Physics, Central Institute of Technology Kokrajhar, Kokrajhar -783370, India  \n2CNRS et Université Claude Bernard Lyon 1, F-69622, L yon, France  \n \n \n*Corresponding Author E -mail:  sk.srivastava@cit.ac.in  \nAbstract  \nRecently d0 or intrinsic ferromagnetism was believed to provide an alternative pathway to \ntransition metal induc ed ferromagnetism  in oxide . In pursuit  of augmenting  the area of d0 \nferromagnetism;  we have undertaken to  study  the crystal structure and magnetic properties of \nAg-doped ZrO 2 compounds. Polycrystalline samples of  Zr1-xAgxO2 (with x=0, 0.02, 0.04, \n0.06 and 0.08) were prepared by solid -state reaction route.  All t he prepared compounds are \nfound to crystallize in monoclinic  symmetry  of ZrO2. In our study,  pure ZrO 2 compound \nexhibit s paramagnetic behavior.  However, the Ag -doped ZrO 2 compounds exhibit \nferromagnetic  to paramagnetic  transition . The Curie temperature ( C) was found to increase \nfrom 28.7  K for x=0.02 to 173.2 K for x= 0.08 doped ZrO 2. Thus, the introduction of Ag in \nZrO 2 induces ferromagneti sm with a large  C. The measurements of hysteresis curves indicate  \nthat Ag doped Z rO2 compounds  exhibit hysteresis loops with a coercivity of around 1350 Oe. \nMoreover , increase in Ag concentration resulted increase in the  value of saturation \nmagnetization (M S); the maximum value of M S was recorded as  0.01 μB/Ag ion for x= 0.06 \nsample.  The sintering of sample  at high temperature (13500C) diminishes  the ferromagnetism \nand it leads to paramagnetic behaviour . \n \nKey Words:  Monoclinic ZrO 2; Ag-doping ; d0 Ferromagnetism ; Defect Induced Magnetism  \n \n \n \n 2 \n 1. Introduction:  \nOver last couple of decades, there  has been persistent effort by researchers to e xplore new \nmaterials which can be integrated for spintronics devices, such as giant magneto -resistance  \nsensors, magneto -resistive random -access  memories  and storage media  [1]. Spintronics  \ndevices , where both charge a s well as  spin are utilized , have been projected  to have plenty of \nadvantages over prevalent  semiconductor devices, such as faster data processing,  non-\nvolatility, low power consumption  and increased storage density  [1, 2]. Among the various \nexplored materials  for spintronics  devices , transition metal (TM) doped  semiconducting \noxide materials  were studied extensively. Many TM -doped oxides,  such as ZnO, SnO 2 and \nTiO 2 have been reported to exhibit room temperature ferromagnetism (RTFM)  [3-11]. The \nmagnetic coupling through excha nge interaction of delocalized 3d electrons via struct ural \ndefects (oxygen vacancies, Vo) has been put forward as  possible  explanation of the \nferromagnetism  observed  in TM-doped oxides [10]. Although, t he experimental search for \nferromagnetism  (FM)  in TM -doped oxide materials has been continued to be dogged , but it \nhas not yet resulted in reproducible and homogeneous magnetic materials  and there is a \ndebate whether RTFM is intrinsic or due to the magnetic ion cluster  [11-13].  \nIn order to get a clean  material exhibiting RTFM, several other types of  materials were \nstudied and explored. In addition to TM-doped oxide materials , unexpected ferromagnetism \nhas been reported or predicted in several pure oxides like HfO 2, CaO, ZnO, ZrO 2, TiO 2, MgO, \nSnO 2 and, even in CaB 6 [14-20]. Such unexpected ferromagnetism was termed as d0 \nferromagneti sm or intrinsic ferromagnets i.e. the materials that do not contain magnetic \nimpurities. Thus, the d0 or intrinsic ferromagnetism was believed to provide an alternative \npathway to TM -induced ferromagnetism.  The origin of the magnetism in these materials was \nconsidered due to  point defects such as cation vacancies , which  induces a local magnetic \nmoment  on the neighboring oxygen atoms  [15, 20] . To circumvent the difficulties of defect \ncontrol, an alternative way, which consists of  the substitution of non-magnetic elements in \ndioxides such as AO2 (A=Ti, Zr, or Hf), was proposed [ 21]. Following to th is idea, many ab-\ninitio  studies have  predicted ferromagnetism with  high Curie Temperature ( θC) in several \nnon-magnetic elements doped oxides , such as K -SnO 2 [22], Ag-SnO 2  [23], Mg–SnO 2 [24], \nanatase Li -TiO 2 [25], rutile K –TiO 2 [26], V-TiO 2 [27], K–ZrO 2 [16, 26]. Experimentally, d0 \nmagnetism  were  observed in severa l non-magnetic elements doped oxides  such as; alkali \nmetal doped ZnO [ 28-30]; Cu doped TiO 2 prepared in thin film form [ 31, 32], C-doped TiO 2 3 \n prepared by solid state route [ 33], K-SnO 2 [34], Li-SnO 2 [35], K-TiO 2 [36], Cu-ZnO [37] and  \nNa-SnO 2 [38]. \nRecently  Zirconium dioxide ( ZrO 2) was projected as one of the  promising candidates  \nexhibiting d0 ferromagnetism . ZrO 2 is a multipurpose  material used in various  scientific & \ntechnological applications  due to its high dielectric constant, ionic conductivity, wide optical \nband gap, high chemical and thermal stabilities, low optical loss and high transparency [ 39-\n41]. It can crystallize in three different forms i.e. monoclinic (s pace group: P2 1/c), tetragonal \n(space group: P4 2/nmc) and cubic (space group: Fm3m) structures [ 39-40]. The monoclinic \nphase of zirconia is usually thermodynamically stable up to 1400 K. The tetragonal and cubic \nphases of ZrO 2 can be stabilized either by heating at very high temperature ( 1480 <T <2650 \nK for tetragonal phase and , T >2650 K  for cubic phase) or by the addition of another cation \nsuch as Ca2+ or Y3+ [40]. In fact, several recent reports indicate that pure ZrO 2 exhibits room \ntemperature d0 ferromagnetism and it is related to the presence of oxygen vacancies or \nstructural defects. Some of these studies also show that the crystallographic phase is very \nimportant in this context, with reports of ferromagnetism more common for tetragonal ZrO 2 \nstructures  [42-44]. The theoretical studies  predicted high -temperature ferromagnetism in TM-\ndoped cubic zirconia [ 45] as well as non-magnetic element doped zirconia such as  K [16, 26] \nand V [46] in ZrO 2. However, it was  predicted that doping with Cu [ 47], Cr [ 45] or Ca [16] in \nZrO 2 can result in paramagnetism, antiferromagnetic  or non -magnetic ground states, \nrespectively.  \nAlthough, Ag -doped ZrO 2 was attempted by researchers for different kind of application \nsuch as in resistive switching, soot oxidation and opto -electronics [ 48-50] but, there exists no \nreport on study of magnetic properties of Ag -doped ZrO 2. In pursuit of augmenting  the \nresearch  area of d0 ferromagnetism, we have endeavoured  a study on the crystal structure and \nmagnetic properties of Ag -doped ZrO 2 compounds. We have chosen to prepare materials in \nbulk form at equilibrium  conditions to diminish the uncertainties in fabrications and any \ninaccuracies in characterization.  \n \n2. Experimental Details  \nThe polycrystalline samples of Zr1-xAgxO2 (x=0, 0.02, 0.04, 0.06 and 0.08) were prepared \nby solid -state reaction route. We used h igh-purity ZrO 2 and AgNO 3 as the starting materials  \nfor synthesis of our samples . The maximum amount of any kind of trace magnetic impurities \nin the starting materials was found to be less than 0.9 % ppm as mentioned by the supplier 4 \n ICP chemical analyses report.  Pre-sintering  of the  prepared samples was  performed  in \npowder form at various temperature s, i.e. 2000C and 3000C for about 20 hours at each \ntemperature. The  samples were further annealed  in pallet form at 500˚C for 30 hrs. One \nsample of Zr0.94Ag0.06O2 was prepared at 1 3500C to study the influence of sintering \ntemperature . The crystal structure of the prepared samples  was checked using  X-ray \ndiffractometer . The temperature  (T) variation of magnetization (M) and , magnetization versus \nmagnetic field (H) measurements were carried out using commercial  SQUI D magnetometer  \nand vibrating sample magnetometer (VSM) . The magnetic measurements were done with \nutmost care and repeated t wo times with different pieces of samples to guarantee the \nreproducibility of results  \n3. Results and Discussion  \nThe crystal structure and phase purity of all Ag-doped ZrO 2 compounds were checked \nby X -ray diffractometer and the  XRD patterns of these compounds  are shown in Figure 1.  All \nthe XRD peaks could be indexed to monoclinic  symmetry  of ZrO2. Within the instrumental \nlimit of t he X-ray diffractometer, the XRD patterns also indicate  that samples are formed in \nsingle phase and no secondary phase  is present. To gain insight into  various  crystal structur al \nparameters , the refinement  of the XRD patterns  was performed with the help of the Fullprof \nprogram by employing the Rietveld refinement technique  [51]. Figure 2 shows the typical \nRietveld  refinement of XRD patterns for ZrO 2 and Zr 0.94Ag0.06O2 compound s. It is seen that \nthe experimental XRD data matches perfectly  with the Rietveld  software calculated  XRD \ndata.  Figure 3 presents the variation of lattice parameters ‘a’ ‘b’, ‘c’  and cell volume  (V) for \nZr1-xAgxO2 (x=0, 0.02, 0.04, 0.06 and 0.08) compounds.  For pure ZrO2 compound , the lattice \nparameters are estimated to be  a=5.1468Å , b= 5.2041Å , c=5.3198  Å and they are  found to be  \ncomparable  with the values reported in  other work  [21]. The lattice parameter s and unit cell \nvolume of the all Ag -doped ZrO 2 compounds  are found to increase with the increase of Ag -\ndoping . Doping of bigger  Ag1+ ion (ionic radii of 1.15 Å) into the Zr4+ (ionic radii of 0.72 Å ) \nis the  likely  cause of  expansion of  the lattice parameters and cell volume.  5 \n  \nFigure 1 : XRD patterns of Zr1-xAgxO2 (x=0, 0.0 2, 0.04, 0.06 & 0.08) compounds.  \nZrO2\n   \nZr0.98Ag0.02O2\n  \nZr0.94Ag0.06O2Zr0.96Ag0.04O2\n  Intensity (a.u.)\n \n20 30 40 50 60 70 80Zr0.92Ag0.08O2\n2  (deg.)6 \n  \nFigure 2:  Refinement of XRD patterns for (a) ZrO 2 (b) Zr 0.94Ag0.06O2 compound s, obtained \nwith the help of the Fullprof program by employing the Rietveld refinement technique . \n7 \n  \nFigure 3:  Variation of lattice parameters ‘a’ ‘b’, ‘c’  and cell volume  (V) for Zr 1-xAgxO2 (x=0, \n0.02, 0.04, 0.06 and 0.08) compounds . \nIn order to explore the magnetic properties of these prepared compounds, the  zero-\nfield cooled (ZFC) magnetization curve s as a function of temperature for all Ag-doped ZrO 2 \ncompounds w ere measured under an applied field of 500 Oe  using SQUID magnetometer . \nThe temperature variation of magnetization as a function of temperature  i.e. M -T curve  for \npure ZrO 2 compound shows a paramagnetic behavior  of the sample , as depicted in Figure \n4(a). In addition , the magnetization versus applied field measurement performed  at 3K for \npure ZrO 2 again indicates  the paramag netic nature of the compound as shown in figure 4(b). \nThus, pure ZrO 2 compound is found to exhibit par amagneti c behavior . Figure 5 presents the \ntemperature variation of magnetization  curves for all Ag -doped ZrO 2 compounds. It is \nobserved that the magnetization decrease s with the increase of temperature throughout the \nmeasured temperature range for Zr0.98Ag0.02O2 compound . However, higher Ag -doped \ncompounds i.e. Zr 0.96Ag0.04O2 and Zr 0.94Ag0.06O2 compounds are found to exhibit a clear \nferromagnetic to paramagnetic transition. Moreover, Zr0.92Ag0.08O2 compound  exhibits \ninteresting feature. The measurement of ZFC M -T curve for this sample show that there is a \nlow temperature antiferro magnetic  (AFM)  transition at ~55 K, followed by weak \nferromagnetic to paramagnetic transition at ~165 K (as shown in the inset of Figure 5d ). It \nindicates that weak ferromagnetic phase is superimposed with the dominating \nantiferromagnetic phase. To get further insight into the magnetic property of Zr0.92Ag0.08O2 \n0.00 0.02 0.04 0.06 0.085.3205.3215.3225.323 c (Å)\nAg C oncentration (%)(Zr1-xAgx)O2\n0.00 0.02 0.04 0.06 0.085.2045.2055.2065.2075.208 b (Å)\nAg C oncentration (%)(Zr1-xAgx)O2\n0.00 0.02 0.04 0.06 0.085.1465.1485.1505.152 a (Å)\nAg C oncentration (%)(Zr1-xAgx)O2\n0.00 0.02 0.04 0.06 0.085.1465.1485.1505.152 V (Å3)\nAg C oncentration (%)(Zr1-xAgx)O28 \n compound , we have measured the M-T curve under field cooled (FC) condition along with \nZFC condition and the data are presen ted in Figure 5 (d). The measurements of ZFC and FC \nmagnetization data for this sample indicate that ZFC and FC M -T curves  coincide  at low \ntemperature and the magnitude  of magnetization at AFM  transition temperature  of under FC \ncondition  has increased , indicating an antiferromagnetic interaction in the matrix. Similar \nobservations have been reported in Fe-doped SnO 2 compounds [ 52]. To determine the \nferromagnetic ( FM) transition temperature  (TC), peak s observed in |dM/dT| versus \ntemperature plot  have been used . Here, T C value was taken as the minimum of |dM/dT | plot \nand fitting the curve with a Gaussian function. Typical plots of |dM/dT| versus temperature  \nfor x = 0.04 and 0. 08 samples are shown in Figu re 6. The T C values are obtained as  108.2, \n192.4 and 172.6 K for x  = 0.0 4, 0.06 and 0.08 samples respectively. Thus, the introduction of \nAg in ZrO 2 gives rise to  increase in ferromagnetic T C. Moreover, the magnitude of \nmagnetization is found to increase with Ag concentration in addition to increase in the FM \nTC. The decrease of magnetization for x=0.08 sample is due to presence of competing AFM \ninteraction . For the estimation of Curie -tempera ture ( C), the paramagnetic region of ZFC M -\nT curve was  analyzed using Curie -Weiss law, 𝜒=𝐶0𝑥/(𝑇−𝜃𝐶). Typical plots of 1/ dc \nversus temperature for x= 0.02, 0.04 and 0.08 samples are shown in Figure 7 along with \nCurie -Weiss law fitting and the estimated Curie -temperature ( C) values are listed in Table 1. \nThe value of Curie -temperature ( C) is found to be 28. 7, 126.8 and 163.2 K for x=0.02, 0.04 \nand 0.08 samples respectively. The positive values of C indicate the FM interaction. The \ndifference between T C and C are mainly due to the observed broad magnetic transition. We \ncould not fit data for x=0.06 sample due to non -availability of sufficient data in the \nparamagnetic region. For calculating effective paramagnet ic moment ( eff), the relation, \n𝜇𝑒𝑓𝑓=√3𝑘𝐵𝐶0𝑥/𝑁𝜇0𝜇𝐵2  was used and the values  were found to be 1.40 μB/Ag ion, 2.27 \nμB/Ag ion and0.16  μB/Ag ion for 2, 4 and 8 % Ag-doped samples respectively.  9 \n  \nFigure 4:(a) M-T curve for pure ZrO 2 under  a field of 0.05 T and (b) M -H loop obtained  at 3 \nK for the ZrO 2 sample.  \n \nFigure 5:(a) Temperature variation of magnetization of Ag-doped ZrO 2 samples measured \nunder an applied field of 0.05 T field.  \n-5-4-3-2-1012345-0.0010.0000.001(b)\nZrO2\n  Magnetization (B/f.u.)\nField (Tesla)At 3 K\n0 100 200 3000.000.010.020.03\n ZFCZrO2\n  Magnetization (emu/mol)\nTemperature(K)(a)\n0 100 200 3000.0000.0050.0100.015\n100 150 200 250 3000.0000.0010.0020.003\n ZFC\n FC\n  M (emu/mol)\nT (K)\n ZFC\n FC(d)\nZr0.92Ag0.08O2\n  Magnetization (emu/mol)\nTemperature (K)\n0 100 200 3000.00.30.60.91.21.51.8\nZr0.94Ag0.06O2\n  Magnetization (emu/mol)\nTemperature (K)(c)\n0 100 200 3000.000.050.100.150.200.25\nZr0.96Ag0.04O2\n  Magnetization (emu/mol)\nTemperature (K)(b)\n0 100 200 3000.000.010.020.030.040.05\n(a)\nZr0.98Ag0.02O2\n  Magnetization (emu/mol)\nTemperature (K)10 \n  \nFigure 6: Temperature v ariation of  |dM/dT|  for Zr0.96Ag0.04O2 and Zr 0.92Ag0.08O2. \n \nFigure 7: Temperature variation of inverse of susceptibility ( 1/dc) for Zr 1-xAgxO2 with (a) \nx=0.02 (b) x=0.04 and (c) x=0.08  compounds . Solid lines represent fit to the Curie -Weiss \nlaw.  \n0 100 200 300-0.003-0.002-0.0010.0000.001\nZr0.96Ag0.04O2(a)\n   \nTemperature (K)dM/dT\n100 200 300-0.00009-0.00006-0.000030.00000\nZr0.92Ag0.08O2\n  dM/dT\nTemperature (K)(b)\n0 50 100 150 200 2500.00.51.01.52.0\nTemperature (K) 1/dc [106 emu/mol-Oe ](a)\n0 100 200 30003691/ dc [106 emu/mol-Oe ]\nTemperature (K)(c)\n0 100 200 3000369121518\nTemperature (K)  dc [104 emu/mol-Oe ](b)11 \n      The field variation of magnetization measurements  i.e. M -H curves at 3 K were \nperformed for all Ag -doped ZrO 2 compounds  and they are  shown in Figure 8.  From the  \ncurves , it is observed that  the magnetization of these samples gets saturate d at relatively \nlarger applied field . One can see that  the samples show a clear ferromagnetic behaviour at 3 \nK. The value of saturation magnetization ( MS) is found to be 0.003, 0.006, 0.009 and 0.004  \nμB/Ag ion  for x=  0.02, 0.04, 0.06 and , 0.08 respectively.  The decrease in M s value for x=0.08 \nsample is possibly due to the presence of AFM interaction in the sample. Although, \nZr0.98Ag0.02O2 does not clearly exhibit any hysteresis loop but, higher Ag doped ZrO 2 \ncompounds  exhibit hysteresis loops  with a coercivity of around 1350 Oe.  \n \n \nFigure 8: Field variation of magnetization for Zr1-xAgxO2 (x=0.0 2, 0.04, 0.06 and 0.08) \ncompounds  measured at 3 K.  \n \n \n \n \n-5-4-3-2-1012345-0.003-0.002-0.0010.0000.0010.0020.003(a)\nZr0.98Ag0.02O2\n  Magnetization (B/Ag ion )\nField (T)At 3K\n0 1 2 30.0000.0010.0020.0030.004\n(d)\nZr0.92Ag0.08O2\n  Magnetization (B/ Ag ion )\nField (T)At 3K\n-3 -2 -1 0 1 2 3-0.012-0.008-0.0040.0000.0040.0080.012(c)\nZr0.94Ag0.06O2\n  Magnetization (B/Ag ion )\nField (T)At 3 K\n-3 -2 -1 0 1 2 3-0.008-0.006-0.004-0.0020.0000.0020.0040.006(b)\nZr0.96Ag0.04O2\n  Magnetization (B/Ag ion )\nField (T)At 3 K12 \n Table 1: Parameters obtained from ZFC magnetization measurements of Zr1-xAgxO2 (x=0, \n0.02, 0.04, 0.06, 0.08) compounds. Here , TC is ferromagnetic transition temperature.  C and \neff (B/ion) are Curie temperature  and experimental effective magnetic moment respectively, \nobtained from the Curie -Weiss law fit of susceptibility data.  \n \nSample/  \nParameters  x=0.02  x=0.04  x=0.06  x=0.08  \nTC (K) -- 108.6  192.4  172.6  \nc (K) 28.7 126.8  -- 163.2  \neff (B /Ag ion) 1.40 2.27 -- 0.16 \nMS (B/Ag ion)  0.003  0.006  0.009  0.004  \n \nTo study the influence of sintering temperature on the magnetic property , one sample \nof Zr0.94Ag0.06O2 was prepared by sintering  it at 1350 ºC. The crystal structure from XRD \npatterns indicates  that sample has been crystallized in monoclinic symmetry of ZrO 2 (not \nshown).  Figure 9 shows the M-T and M–H (at 3K) curves  measured  for th is sample.  The \nmeasurement of M-T curve , as shown in Figure 9(a)  indicate s that it exhibits  paramagnetic \nbehaviour and it is further corroborated by M -H curve , as shown in Figure 9 (b) . Moreover, \nfrom the M-H curve, it is observed that the value of magnetization decreases with increase in \nthe sintering  temperature . The s aturation magnetization i s two times larger for  the sample \nprepared  at 500 ºC, as compared with the sample  prepared  at 1350 ºC. These results suggest \nthat oxygen vacancies present  in the sample prepared at lower temperature lead to  \nferromagnetism, which get diminishe d in the sample prepared at  high-temperature. High-\ntemperature preparation  results  in the destruction of the ferromagnetic order ing and reduced \nmagnetic  moment , possibly due to decrease  in oxygen vacancies . Similar observation was \nmade for Cu doped TiO 2 [31-32]. 13 \n  \nFigure 9: (a) Temperature variation of magnetization and (b) Magnetization versus field \nvariation at 3K of Zr0.94Ag 0.06O2 compound , prepared at 1 3500C. \nLet us summarize  the magnetism  observed  in these samples. The measurement of \nmagnetic properties of these compounds indicates that pure ZrO 2 compound exhibit \nparamagnetic behavior, which is unlike few previous reports  where ferromagnetism was \nobserved in pur e ZrO 2 [42-43].  However, the Ag -doped ZrO 2 compounds exhibits \nferromagnetic to paramagnetic transition and FM transition temperature was found to \nincrease with the increase of Ag concentration.  Thus , the introduction of Ag in ZrO 2 gives \nrise to increase in ferromagnetic T C. Moreover, the magnitude of magnetization is found to \nincrease with Ag concentration in addition to increase in the FM T C. When we consider the \neffect of  sintering temperature on the magnetic property , the sample prepared at lower \ntemperature is  found to be ferromagnetic  and this is due to the vacancies present therein.   \nIt is an established fact in the realm of the physics and chemistry of solids  that that \nions substitute for one -another in structures if the charges and sizes are similar , it is usually \nevident from  a systematic cell parameter change . In the case of a direct cationic substitution, \ntheoretical model study [ 21] and first principle approach study in ZrO 2 [16, 26] and \n0 100 200 3000.000.040.080.12\nZr0.94Ag0.06O2\n  Magnetization(emu/mol)\nTemperature (K)(a)\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.0000.0020.0040.0060.008\nZr0.94Ag0.06O2\n  Magnetization (B/Ag /ion )\nField (T)At 3 K(b)14 \n observation of d0 magnetism in K doped SnO 2 [34] have demonstrated that three physical \nparam eters are essential to explain induced d0 magnetism: (i) the position of the induced \nimpurity band (ii) the density of carrier per defect and (ii) the electrons -electrons correlatio ns \n[21, 34]. In present study,  ZrO 2 has 6 -coordinate Zr4+ and t he replacement  of Zr4+ by Ag1+ \nwill produce  three holes. To retain  charge neutrality,  it is possible that  more defect species \nsuch as oxygen vacancies (V o) are created  in ZrO 2 structure as oxidation states of Ag1+ is \nlower in comparison to  Zr4+. A vacancy induces local magnetic moments on the neighboring \noxygen atoms which then interact with extended exchange couplings  they interact \nferromagnetically via O. Furthermore , it should be noted that we have not observed any \nsecondary phase from the crys tal structure and thus the observed ferromagneti sm has intrinsic \nnature.  It should be noted  that the tetragonal or cubic symmetry of ZrO 2 is not absolutely \nnecessary  to obtain d0 ferromagnetism . \n \n4. Conclusion  \nTo conclude, polycrystalline samples of  Zr1-xAgxO2 (x=0, 0.02, 0.04, 0.06 and 0.08) were \nprepared by solid -state reaction route.  All the prepared compounds are found to crystallize in \nmonoclinic  symmetry  of ZrO2 with typical lattice parameters of a=5.1468Å, b= 5.2041Å, \nc=5.3198 Å  for pure ZrO2 compound . The measurement of magnetic properties of these \ncompounds indicates that pure ZrO 2 compound exhibit s paramagnetic behavior.  However, \nthe Ag -doped ZrO 2compounds exhibit ferromagnetic  to paramagnetic  transition . The Curie \ntemperature ( C) was fou nd to increase from 28.7 K for x=0.02 to 173.2 K for x= 0.08 doped \nZrO 2. Thus, the introduction of Ag in ZrO 2 induces ferromagneti sm with a large C. The \nmeasurement of hysteresis curves indicates that Ag doped Z rO2 compounds  exhibit hysteresis \nloops with a coercivity of around 1350 Oe. In this study, increase in Ag concentration \nresulted increase in  the value of saturation magnetization (M S). The maximum value of M S \nwas found to be 0.00 9 μB/Ag ion for x= 0.06 sample. The study of in fluence of sintering \ntemperature suggests that ferromagnetism observed in the sample prepared at low \ntemperature (5000C) is possibly due to oxygen vacancies present  in the sample. The sintering \nof sample at high temperature (13500C) diminishe s the ferromagnetism and it leads to \nparamagnetic behaviour.  \nAcknowledgements  \nThis research did not receive any specific grant from funding agencies in the public, \ncommercial, or  not-for-profit sectors.  15 \n References:  \n \n[1] H. Ohno, Making Nonmagnetic Semiconductors Ferromagnetic, Science 281 (1998) \n951-956. https://doi.org/10.1126/science.281.5379.951 . \n[2] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. V. Molnar, M. L. \nRoukes, A. Y. Chtcheljanova and D. M. Treger, Spintronics: A Spin -Based \nElectronics Vision for the Future, Science 294 (2001) 1488 –1495. \nhttps://doi.org/10.1126/science.1065389 . \n[3] J. K. Furdyna, Diluted magnetic semiconductors, J. Appl. Phys. 64 (1988) R29 –R64.  \nhttps://doi.org/10.1063/1.341700 . \n[4] S.B. Ogale, R.J. Choudhary, J.P. Buban, S.E. Lofland, S.R. Shinde, S.N. Kale, V.N. \nKulkarni, J. Higgins, C. Lanci, J.R. Simpson, N.D. Browning, S. 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B 78 (2008). \nhttps://doi.org/10.1103/physrevb.78.024404 . \n "    },    {        "title": "2201.10722v1.Local_ferromagnetic_resonance_measurements_of_mesoscopically_patterned_ferromagnets_using_deterministically_placed_nanodiamonds.pdf",        "content": "Local ferromagnetic resonance measurements of mesoscopically\npatterned ferromagnets using deterministically placed\nnanodiamonds\nJe\u000brey Rable,1Benjamin Piazza,1,\u0003Jyotirmay Dwivedi,1and Nitin Samarth1,y\n1Department of Physics, Pennsylvania State University,\nUniversity Park, Pennsylvania 16802, USA\n(Dated: January 27, 2022)\n1arXiv:2201.10722v1  [cond-mat.mes-hall]  26 Jan 2022Abstract\nNitrogen-vacancy centers in diamond have recently been established as e\u000bective sensors of the\nmagnetization dynamics in vicinal ferromagnetic materials. We demonstrate sub-100 nm placement\naccuracy of nitrogen-vacancy-containing nanodiamonds and use these as local sensors that probe\noptically detected ferromagnetic resonance in mesoscopically patterned Permalloy islands. These\nmeasurements reveal variations in the ferromagnetic resonance signal at di\u000berent sites on these\nstructures with distinct behavior in the edge and the bulk of patterned features. These test\nmeasurements establish an easily implemented approach for spatially targeted measurements of\nspin dynamics in mesoscale ferromagnets. In principle, the methodology can also be extended to\nlocal studies of nanoscale ferromagnets such as single magnetic nanowires and nanoparticles.\nI. INTRODUCTION\nContemporary problems of interest in spintronics often require knowledge of the dy-\nnamical behavior of magnonic (spin wave) excitations in patterned ferromagnetic devices.\nFor example, this information is important in the creation and characterization of magnon\nquantum buses [1, 2] and other magnonic devices such as spin-based transistors [3]. The\nnitrogen-vacancy (NV) center in diamond has emerged as an e\u000bective non-perturbative local\nprobe for characterizing the magnetic properties of such systems [4]. NV center-based local\nmagnetometry has provided new insights into static spin con\fgurations of skyrmions [5, 6]\nand magnetic domain walls [7], as well as the dynamical behavior of magnons [8{11] and\nvortices [12]. In the latter context, it is important to develop techniques that allow local\nmeasurements of ferromagnetic resonance (FMR) at targeted locations in a ferromagnetic\nsample or device.\nContinuous wave NV center-based optically detected ferromagnetic resonance (ODFMR)\nmeasurements rely on the quenching of the NV center \ruorescence when a vicinal ferromag-\nnet meets the conditions for FMR [8]. This is attributed to an increased magnon density\nand an accompanying enhancement of the magnetic \feld noise sensed by the NV centers,\nthus leading to increased spin relaxation [9]. Local measurements of ODFMR have relied\n\u0003Current address: Network Science Institute, Northeastern University, Boston, MA 02115, USA\nynsamarth@psu.edu\n2on stochastic distribution of dropcast nanodiamonds [8, 13], stochastic distribution of NVs\nin a diamond \flm [12{15], proximate placement of a diamond nanobeam [9, 11], and chem-\nically patterned directed assembly of nanodiamonds [10]. In principle, scanned probe NV\ncenter magnetometry could provide a means of carrying out local ODFMR with imaging\ncapability. However, since NV detection of ODFMR rapidly decreases in sensitivity with\nincreasing sample-probe distance [15], NV scanning probe measurements of ODFMR may\nbe constrained by the tip-sample distances that are typically greater than about 100 nm\n[7, 16, 17]. Although one scanning NV technique allows for smaller tip-sample separation\n(\u001830 nm) [5], it would be technically challenging to engineer e\u000bective excitation of the FMR\nin a ferromagnetic sample in this geometry. This may account for the absence (as yet) of any\npublished reports of ODFMR using a scanning NV center probe. As an aside, we note that\nlocal magnetization dynamics of ferromagnets can also be e\u000bectively probed and imaged via\na completely di\u000berent method, namely scanning ferromagnetic resonance force microscopy\n(FMRFM) which uses a microscale force cantilever to detect FMR with \u0018100\u0000200 nm\nspatial resolution [18{21]. As with NV center based ODFMR, the FMRFM technique is\namenable to measurements at ambient temperature. However, in contrast with NV center\nODMFR, the FMRFM method requires more sophisticated instrumentation and it is more\ncumbersome to collect FMR data that densely spans frequency-magnetic \feld space. Thus,\na question of interest is whether one can develop a simpler approach for locally measuring\nFMR at targeted sites on a mesoscopic or nanoscale ferromagnetic structure without having\nto resort to the sophisticated instrumentation required by scanning microscopy techniques.\nIn this paper, we demonstrate the use of an atomic force microscope (AFM) to achieve\nwell-controlled positioning of NV-containing nanodiamonds as FMR sensors, with sizes be-\ntween 40 nm - 100 nm and with sub-100 nm accuracy. We use these deterministically placed\nnanodiamonds to perform local ODFMR measurements of mesoscale (5 \u000010\u0016m lateral size)\nfeatures patterned in ferromagnetic (Permalloy, Py) thin \flms. Although similar to a 'pick-\nand-place' technique previously reported for assembling nanodiamonds at desired locations\n[22, 23], this approach has not yet been exploited to probe the localized magnetization\ndynamics of magnetic materials. We note that chemically patterned directed assembly of\nnanodiamonds has been e\u000bectively used for probing ODFMR in YIG devices [10]; however,\nthe measured locations are constrained in advance by lithographic patterning and subject\nto overlay error. We seek a more \rexible approach that allows the measured locations to\n3FIG. 1. Demonstration of the nanodiamond placement process with a 40 nm diameter nanodi-\namond (1). After lifting the nanodiamond and con\frming it is no longer on the sample (2), we\nmove over to the deposition site, a 100 nm wide nanowire (3). Then, we can deposit via ramp or\nlift mode and con\frm placement via another scan (4).\nbe varied at will. The placement precision in our proof-of-concept demonstration can, in\nprinciple, also allow for the targeted measurement of smaller nanoscale structures, such as\nnanowires, or of localized modes in larger structures, such as edge modes and defect modes.\nII. METHODS\nA. Nanodiamond Placement and ODFMR\nWe \frst describe the experimental approach for deterministic placement. We begin with\na sample containing patterned permalloy structures in the center and drop cast an aqueous\nsolution of 100 nm diameter nanodiamonds with 3 ppm NV centers onto the edge of the\nsample. This is done to avoid directly depositing nanodiamonds onto the features. Next,\nwe scan the area where the nanodiamonds were drop cast using a Veeco Nanoscope IIIA\nMultimode AFM with a gold coated silicon tip, which increases the probability of pickup over\na standard silicon tip, likely because of stronger van der Waals forces or malleability of the\n4gold. When we \fnd a particle that matches the dimensions of the dispersed nanodiamonds,\nwe zoom in on the particle (Fig. 1, step 1) and enter ramp mode. Then, we ramp into the\nnanodiamond and re-scan the area where it was in the prior scan to con\frm pickup (Fig. 1,\nstep 2). If the nanodiamond remains at the site, we ramp in again until it is picked up or\nattempt pickup on a di\u000berent particle.\nIf the nanodiamond does not appear on the post-ramp scan, indicating that it was picked\nup, we can begin the placement process by moving the AFM tip to our patterned features.\nWith the nanodiamond still attached to the tip, we can scan the sample looking for the\nfeature that we want to deposit the particle on (Fig. 1, step 3). When we \fnd the device,\nwe either repeat the process used in pickup, ramping into the feature until the nanodiamond\ndislodges, or we scan across the sample at a constant height below the surface using the\nAFM's lift mode, scraping the nanodiamond-coated tip along until it dislodges. While this\nsecond method works more consistently, it risks scratching the sample if the tip is dug into\nthe feature. Finally, with the diamond dislodged, we perform a \fnal scan to con\frm that\nthe nanodiamond is in the desired location (Fig. 1, step 4). We caution that we do not\nyet have a systematic measure of the success rate of the method. The repeatability of the\ntechnique appears to depend on factors that are not completely understood and varies with\nthe details of the drop casting and the substrate.\nFor optical polarization and readout of the NV center \ruorescence, we used a 1 mW, 532\nnm continuous wave laser and an ID Quantique ID100 avalanche photodiode. A scanning\nmirror scans across the surface for imaging and allows focusing on a site for ODFMR mea-\nsurements. A static magnetic \feld is applied using a permanent N52 magnet mounted on\na highly repeatable stepper motor linear stage. The applied \feld is calibrated in the plane\nof the sample using a single crystal diamond \flm containing NV centers; this is achieved\nusing the known orientations and the Zeeman splitting of the NV electronic ground state\nspin transition. During the ODFMR measurements, a microwave magnetic \feld is applied\nvia a 25 µm diameter gold wire run across the sample. This microwave \feld both drives\nFMR in the magnetic features and the NV spin state transitions. Additionally, when FMR\nis driven in the sample, new, higher frequency magnons are generated via scattering and\nthermal mechanisms. The dipolar \feld noise generated by these incoherent magnons also\na\u000bects the NV spin state transitions; this e\u000bect is believed to be responsible for the detection\nof FMR via an optical contrast even though the FMR is driven at frequencies o\u000b-resonant\n5from those that drive the NV spin transitions [9].\nB. Micromagnetic Simulations\nPrior to our measurements, we performed micromagnetic simulations of the FMR modes\nin patterned Py features identical to those measured experimentally. The Py \flm thickness\nin all these simulations is 10 nm. The simulations were performed with the Mumax3 software\npackage, which uses the Landau-Lifshitz-Gilbert equation:\n@~M\n@t=\rLL1\n1 +\u000b2(~ m\u0002~Be\u000b+\u000b(~ m\u0002(~ m\u0002~Be\u000b)) (1)\nto calculate the evolution of the magnetization ~Mof \fnite ferromagnetic cells. In Eq. 1, \u000b\nis the Gilbert damping of the material, \rLLis the gyromagnetic ratio of the material, and\nBeffis the e\u000bective magnetic \feld at that cell, which includes contributions from external,\ndemagnetization, exchange, and anisotropy \felds [24].\nThe simulations were performed using 5 nm x 5 nm x 10 nm cells and the geometries\nconsisted of permalloy features with the parameters in table I.\nParameter Value\nMs 8 x 105A/m\nAex 1.3 x 10\u000011J/m\n\u000b 0.0063\nTABLE I. Permalloy material parameters used in micromagnetic simulations\nAfter de\fning the sample geometry, the system was given an initial magnetization point-\ning along the (0,0,1) direction out of the plane of the \flm and allowed to relax to the\nminimum energy state in the applied bias \feld, which ranged from 1 to 30 mT in our\nsimulations.\nTo excite the system, we applied a Gaussian pulse with a 20 ps full width half maximum\nand a 0.5 mT amplitude. The system was then allowed to freely evolve in time for 20 ns\nand average magnetization was sampled every 5 ps. Finally, we used a discrete fast Fourier\ntransform to analyze the data in the frequency domain. Using the above sampling rates and\nsimulation lengths, we obtain a resolution of 50 MHz.\n6FIG. 2. AFM images of the measured permalloy features with superimposed scanning confocal\n\ruorescence images showing where NV-containing nanodiamonds are located. (a) A 10 µm x 5\nµm x 10 nm permalloy rectangle with nanodiamonds located near the edges and in the middle.\n(b) Two 6 µm diameter, 10 nm thick permalloy circles connected at the edges. Nanodiamonds are\nlocated at the periphery of the left disk and near the junction between the disks.\nIII. RESULTS\nTo begin, we placed multiple nanodiamonds on the 10 µm x 5 µm x 10 nm rectangle in\nFig. 2 (a) to con\frm that we could replicate previous macroscale FMR measurements carried\nout on arrays of such rectangular islands. These studies showed the presence of an easy-axis\nand hard-axis resonance [25, 26] which can be modelled using a geometry-dependent form\nof the Kittel equation:\nf=\r\n2\u0019q\n(Happ\u0000(Nx\u0000Ny)4\u0019\u00160Ms)(Happ\u0000(Nx\u0000Nz)4\u0019\u00160Ms): (2)\nHere,\ris the gyromagnetic ratio of the material, Happis the applied \feld, Msis the\nsaturation magnetization of the material, and Niare the three geometric demagnetization\nfactors which sum to 1.[27] We use NxandNyas in-plane demagnetization factors and Nzas\nthe out-of-plane demagnetization factor. We can further simplify these three geometric pa-\nrameters and the saturation magnetization into two quantities that represent the anisotropy\n\felds of the features - BkandB?. This yields:\n7FIG. 3. Measurement of ODFMR using a nanodiamond in the bulk of a rectangular Py island\nwith the applied magnetic \feld oriented along (a) the longitudinal easy axis and (b) along the\ntransverse hard axis. The corresponding micromagnetic simulations of the FMR with the applied\nmagnetic \feld oriented along (c) the longitudinal easy axis and (d) along the transverse hard axis.\nf=\r\n2\u0019q\n(Happ+Bk)(Happ+B?): (3)\nBecause of Py's low intrinsic anisotropy, we can also assume that these anisotropy \felds\nsolely result from the shape anisotropy of our features. In the case of thin, rectangular\nfeatures like the one measured, NxandNyin eqn. 2 will be small and unequal, while Nz\nwill still be close to 1, its value in an in\fnite thin \flm. This results in B?being close to the\n4\u0019\u00160Msand inBkhaving a comparatively small magnitude. Furthermore, when rotated 90\u000e\nin plane,NxandNyswap positions in eqn. 2, leading to Bkretaining its magnitude while\nswitching signs in 3. This results in a positive Bkalong the easy axis, but negative Bkalong\nthe hard axis, leading to a divergence as Happapproaches it.\nFigure 3 (a) and (b) show the measurements of ODFMR obtained using a dim nanodia-\nmond located near the center of the rectangle. When a magnetic \feld is applied along the\n8easy axis (long edge of the rectangle), we detect a single mode that increases approximately\nlinearly starting at 2 GHz (Fig. 3 (a)), as expected from eqn. 3. Fitting eqn. 3 to this\nresult, we \fnd the in plane anisotropy \feld Bkof this feature to be 3.3 mT, and the out\nof plane anisotropy \feld B?to be 1.04 T, close to the accepted 1 T Msof Py as expected.\nWhen a magnetic \feld is applied along the hard axis (short edge of the rectangle), we see a\nV-shaped dispersion, which reaches a minimum at approximately 3 mT, near the divergence\npoint expected from our previously measured Bk(Fig. 3 (b)). The micromagnetic simula-\ntions shown in Fig. 3 (c) and (d) largely match these results, though the frequencies are\nhigher, most likely a result of our applied microwave power, which can result in NV contrast\nabove the FMR frequency [28].\nHowever, along the edges of the rectangle, we detected new features in addition to the\nones seen by the nanodiamond located in the middle (or bulk) of the ferromagnetic rectangle\n(Fig. 4). For a nanodiamond located on the long edge, when the magnetic \feld is applied\nalong the easy axis (long edge of the rectangle), we observe an incomplete switching with\na faint signal from the V-shaped resonance (Fig. 4 (a). We believe this to be the result of\na small magnetic \feld misalignment. When the \feld is applied along the hard axis (Fig.\n4 (b)), we see a faint easy-axis signal as well as faint traces of additional signals between\nthe easy and hard axis FMR signals. For a nanodiamond placed on the short edge of the\nrectangle and with the \feld applied along the easy axis Fig. 4 (c)), we observe a strong\neasy axis signal and a faint hard axis signal, similar to Fig. 4 (a). However, we can also\nsee faint traces of additional signals above the easy-axis resonance. We speculate that these\nadditional signals in Fig. 4(b) and (c) are higher order magnon modes.\nWe now discuss measurements on a feature composed of two connected 6 µm diameter, 10\nnm thick circular Py disks (Fig. 2 (b)). These features had a slight fabrication error in one of\nthe circles near the constriction where they meet, leading to a sharp edge slightly protruding,\nwhich could cause a local distortion of the ferromagnetic resonance. We positioned and\nmeasured nanodiamonds at three di\u000berent sites - one on the edge approximately 4 µm from\nthe constriction, one on the pristine side of the constriction approximately 2 µm from the\nfabrication error, and one on the misfabricated side of the constriction, approximately 750\nnm away from the error. Far away from the constriction, the data shows streaking at\napproximately 2 mT, and a signal that begins suddenly at approximately 4 mT (Fig. 5\n(a)). Near the constriction (Fig. 5 (b)), we see additional noise broadband noise at lower\n9FIG. 4. ODFMR measurements using a nanodiamond located on the long edge of a rectangular\nisland, with \feld oriented along (a) the hard axis (long edge), (b) along the easy axis (short edge).\n(c) ODFMR measurements using a nanodiamond located on the short edge of a rectangular island\nwith \feld oriented along the hard axis (long edge). We attribute the anomalous high \feld behavior\nof the NV center resonance lines to fringe \feld e\u000bects that occur at the edge of the patterned\nfeature.\n\felds (between 2 mT - 4 mT), but the signal largely matches that in Fig. 5 (a). On the\nopposite end of the constriction (Fig. 5 (c)), close to the misfabricated edge, the primary\nresonance detected at higher \felds matches the measurements at the other two sites, but\nan additional high frequency resonance emerges, leaving our detection range between 5 and\n10 mT. At this site, the signal up to 5 mT appears to be broadband noise, with greater\ncontrast as the \feld increases. The line width also increases dramatically, from 156 MHz at\n10 mT at the opposite, pristine side of the constriction to 278 MHz at 10 mT. Micromagnetic\nsimulations of FMR in the double circle feature (5 (d)) match our experimental results well\nfor \felds stronger than 4 mT. We speculate that the disagreement between measurements\n10FIG. 5. ODFMR measurements using a nanodiamond located at di\u000berent sites along the edge\nof a circular double disk feature where the disks meet to form a constriction. Measurements are\nmade at three distinct sites: (a) Far away from the constriction;(b) Near the constriction; (c) On\nthe opposite end of the constriction, close to a misfabricated edge. (d) Micromagnetic simulations\nof FMR in the double circle feature.\nand simulations at lower \felds is caused by di\u000berences in the magnetic texture of the material,\nas the applied \feld will not saturate the ferromagnet and our simulation setup procedure\ndoes not perfectly replicate the history of the features.\nIV. SUMMARY\nWe have demonstrated a straightforward method to deterministically place NV-containing\nnanodiamonds at desired locations on lithographically patterned ferromagnetic thin \flms.\nUsing this placement, we performed local ODFMR measurements on single mesoscopic Py\nislands, revealing position-dependent variations in spin dynamical behavior that cannot be\n11detected using conventional FMR measurements of ensembles of patterned islands. After\ncon\frming that this technique worked on a simple rectangular island, we then applied it to a\nmore complex feature composed of two circles with a subtle fabrication error near the point\nof closest approach. In the vicinity of this fabrication error, we measured both a larger line\nwidth and an additional signal that did not appear in the control measurement away from the\nerror or in our micromagnetic simulations of pristine patterns. This \fnding shows that local\nODMFR measurements using targeted placement of nanodiamonds can provide information\nabout the in\ruence of defects on the spin dynamical behavior of patterned ferromagnets.\nMoving forward, we see this technique being used as a more general method of measuring\nlocalized magnetization dynamics in various patterned ferromagnetic structures. For exam-\nple, the technique could provide new insights into the properties of spin wave edge modes\npreviously detected using scanning FMRFM [20]. It could also be used for probing the\nlocal magnetization dynamics of arti\fcial spin ice arrays [29, 30]. 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Lett. 127, 117203 (2021).\n15"    },    {        "title": "0811.4252v2.Spin_Valve_Effect_of_the_Spin_Accumulation_Resistance_in_a_Double_Ferromagnet___Superconductor_Junction.pdf",        "content": "arXiv:0811.4252v2  [cond-mat.supr-con]  3 May 2009Spin-valve eﬀect of the spin accumulation resistance in a do uble ferromagnet -\nsuperconductor junction\nP. S. Luo,1T. Crozes,1B. Gilles,2S. Rajauria,1B. Pannetier,1and H. Courtois1,∗\n1Institut N´ eel, C.N.R.S. and Universit ´eJoseph Fourier,\n25 Avenue des Martyrs, 38042 Grenoble, France.\n2SIMAP, C.N.R.S., Universit ´eJoseph Fourier and Grenoble I.N.P.,\nDomaine Universitaire, 1130 rue de la Piscine, 38402 Saint M artin d’H` eres, France\n(Dated: October 29, 2018)\nWe have measured the transport properties of Ferromagnet - S uperconductor nanostructures,\nwhere two superconducting aluminum (Al) electrodes are con nected through two ferromagnetic iron\n(Fe) ellipsoids in parallel. We ﬁnd that, below the supercon ducting critical temperature of Al, the\nresistance depends on the relative alignment of the ferroma gnets’ magnetization. This spin-valve\neﬀect is analyzed in terms of spin accumulation in the superc onducting electrode submitted to\ninverse proximity eﬀect.\nPACS numbers: 74.45.+c\nAt a Normal metal-Superconductor (N-S) junction bi-\nased at a voltage below the superconducting gap ∆ /e,\nAndreev reﬂection is the dominant contribution to trans-\nport. Here, onespin-upelectronpenetratesthesupercon-\nductor with a spin-down electron so that a Cooper pair\nis formed. This can also be viewed as the reﬂection of\nan electron into a hole.1Since the two electron spin pop-\nulations are involved, Andreev reﬂection is reduced in a\nF-S junction based on a Ferromagnetic (F) metal, and\nsuppressed in the case of a full spin-polarization.2\nElectron transport in F-S hybrid structures exhibits\nseveral remarkable phenomena. Crossed Andreev Reﬂec-\ntion (CAR) is predicted to occur when two normal metal\nleads contacting a superconductor are separated by at\nmost the superconducting coherence length ξs:3one elec-\ntron from one lead is reﬂected as a hole in the other\nlead. Similarly, an electron can travel from one lead to\nthe other by elastic co-tunneling (EC).4Experiments on\nmulti-terminal F-S structures showed a non-local signal\nsensitive to the relative magnetization alignment.5This\nwas attributed to CAR, which is enhanced in a anti-\nparallel state (AP) and inhibited in a parallel (P) state.\nA similar bias-dependent non-local signal was observed\nin N-I-S-I-N planar junctions (I stands for Insulator)6\nand N-S multerminal devices.7Spin switches made ofone\nsuperconductor sandwiched by two ferromagnetic layers\nhave been studied close to the superconductivity criti-\ncal temperature Tc, which was found to be larger in the\nAP state than in the P state8in accordance to the ex-\npected proximity eﬀect.9Opposite behaviors were also\nobserved10and explained in terms of stray ﬁeld eﬀects.11\nAt a F-S junction, a spin-polarized current is converted\ninto a spinless current.12This occurs in the ferromag-\nnetic metal on a characteristic length scale given by the\nspin relaxation length λsf. Electrons with the minority\nspinthenaccumulateclosetotheinterface,whichinduces\nan extra resistance of amplitude determined by a length\nλsf. In a F-I-S tunnel junction, quasi-particules can be\ninjected only at an energy above the gap, generating a\nlarge spin-accumulation resistance.13,14Although non-local mechanisms, spin switch and spin\naccumulation eﬀects can coexist in hybrid nanostruc-\ntures,theirrelativecontributiontoelectrontransporthas\nbeen little studied. In this paper, we address the spin-\ndependent transport at the junction between two ferro-\nmagnetic leads and a superconductor, in the regime of a\nmetallic contact. We observeabias-dependent spin-valve\neﬀect, which we analyze in terms of spin accumulation in\nthe superconducting electrode submitted to inverse prox-\nimity eﬀect.\nFig. 1 a, b show the two sample geometries that we\nhave investigated. In every case, two superconducting Al\nreservoirs or wires are connected through two Fe ellip-\nsoids in parallel. We have chosen an ellipsoidal shape in\norder to ensure a single magnetic domain regime within\none ellipsoid. The spacing between the Fe ellipsoids was\nvaried between 100 and 500 nm. The separation between\nthe two Al reservoirs is 100 nm, which is much larger\nthan the proximity eﬀect decay length in a ferromagnetic\nmetal. This means that the two superconducting inter-\nfaces of the same ellipsoid are decoupled. Geometry a)\nis designed to have bulk Al contacts with voltage probes\nclose to the interface, while geometry b) reduces signiﬁ-\ncantly the inﬂuence on the Al electrodes of the stray ﬁeld\ninduced at the Fe ellipsoids ends. The fabrication pro-\ncedure starts from epitaxial Fe ﬁlms that were grown on\na MgO substrate at room temperature under a residual\npressure below 10−9mbar and annealed at 600◦C for 3\nh. The ﬁlms are 40 nm thick and protected by a 3 nm\nlayer of Pt or Au. First, the Fe ellipsoids are patterned\nby e-beam lithography and Ar ion-etching. After a sec-\nond e-beam lithography, a 70 nm Al ﬁlm is deposited on\na resist mask and lifted-oﬀ. Prior to the deposition of Al,\nthe protection layer is removed by a soft ion-milling.\nIn a given sample, the two ellipsoids have been made\nwith diﬀerent dimensions (900 ×100 nm2and 500 ×150\nnm2) in order to obtain diﬀerent coercive ﬁelds. Fig. 1c\ndisplays the topographical and the magnetic images of\nthe same area of a test sample featuring a large number\nofFe ellipsoidspairs. Themagneticimageswereacquired2\nFIG. 1: (Color online) Top: Micrographs of the two sample\ngeometries based on a Fe ellipsoids pair together with the\nmeasurement connections. (a) The two wide Al pads have\nvoltage probes close to the interface. (b) The two Al wires\ndo not overlap the ends of the ellipsoids, where the largest\nstray ﬁeld is induced. Bottom: (c) Topographical (left) and\nmagnetic (right) images of the same area of a test sample\nmade of a large number of Fe ellipsoids pairs. The magnetic\nimage was taken at a magnetic ﬁeld of 30 mT after having\npolarized the sample in the opposite direction at - 200 mT.\nThe pairs indicated by a circle show an anti-parallel (AP)\nmagnetization state.\nwith a Magnetic Force Microscope (MFM) and give ac-\ncess to the perpendicular to the surface component of\nthe magnetic ﬁeld gradient. Fig. 1c data was acquired\nat a moderate in-plane magnetic ﬁeld of 30 mT after\nfull polarization of the sample in the opposite direction\nat - 200 mT. For every ellipsoid, the magnetic image is\ncompatible with a single magnetic domain conﬁguration.\nAlthough all ellipsoids pairs were made identically, part\nof them show an AP magnetization conﬁguration, mean-\ning that the short ellipsoid has switched, while the long\none remains pinned. This scattering is presumably due\nto slight changes in the precise ellipsoids geometry. For\ninstance, it is expected that the ellipsoid edge roughness\nplays a signiﬁcant role in the exact value of the coercive\nﬁeld. Based on the full series of measurements, we ﬁnd\nthat the switching ﬁelds of the ellipsoids are about 30\nmT for the short one and 50 mT for the long one, with\nsigniﬁcant variations from one sample to the other.\nWe have measured the electron transport properties of\na series of samples at very low temperature down to 260\nmK. We used a lock-in technique with an a.c. current of\namplitude 100 to 200 nA superposed to a d.c. bias cur-\nrent. Herewepresentexperimentaldatafromonesample\n(Sample 5) out of 8 samples showing a similar behavior.\nIt is of geometry a) and was measured in a 2-wire conﬁg-\nuration, unless otherwise speciﬁed. The data displayed\nFIG. 2: (Color online) Probe magnetic ﬁeld dependence of\nSample 5 zero-bias diﬀerential resistance at 310 mK. Red\nsquare (blue circular) dots: the Fe ellipsoids are ﬁrst pola r-\nized with + (-) 300 mT magnetic ﬁeld and the diﬀerential\nresistance is measured at zero external magnetic ﬁeld with\ndecreasing (increasing) probe ﬁelds.\nhere was acquired at zero applied magnetic ﬁeld.\nLet us ﬁrst discuss the spin-valve eﬀects in our sam-\nples. First we polarizedthe two ellipsoidsmagnetizations\nwith a magnetic ﬁeld of absolute value 300 mT. After-\nwards and for every data point, we applied for about 1\ns a probe magnetic ﬁeld of a varying value. The ﬁeld\nis then ramped back to zero and the resistance is mea-\nsured. This procedure is repeated at a series of values\nfor the probe magnetic ﬁeld, starting from the polariza-\ntion ﬁeld and until a ﬁeld opposite in sign is reached. In\nthis way, we systematically measure the resistance of the\ndevice in diﬀerent magnetization conﬁgurations, without\nthe parasitic eﬀects of a non-zero magnetic ﬁeld.\nFig. 2showsanexampleofsuchameasurement, where\nthehorizontalaxisindicatestheprobemagneticﬁeldthat\nwas applied just before the measurement. We observe\nsharp stepwise changes of the resistance, with two sym-\nmetric domains featuring a lower value. The values of\nthe probe magnetic ﬁeld at the resistance changes are\ncompatible with the switching ﬁelds of the two diﬀerent\nellipsoids. In this respect, we ascribe the low-resistance\ndomains to the regime of AP magnetizations ( ↑↓or↓↑).\nBoth at smaller ﬁeld and at higher ﬁeld, the resistance\nis higher and constant within the measurement accu-\nracy. We ascribe these states to a P conﬁguration ( ↓↓\nor↑↑). The resistance diﬀerence between the AP and\nthe P states is about 40 mΩ or 3 %. This spin-valve\neﬀect is the central result of this paper. Let us now de-\nscribe our further experimental study and data analysis\naimed at identifying the involved physical eﬀect.\nFig. 3 inset showsSample 5resistancetemperaturede-\npendence, when prepared in a P or AP state. The mag-\nnetic state has a small eﬀect (about 3 mK) on the critical\ntemperature of the Al electrodes. The sign of the shift\n(at the sharpest resistance drop) suggests an eﬀect of the3\nFIG. 3: (Color online) Temperature dependence of Sample\n5 zero-bias resistance in parallel (P) and anti-parallel (A P)\nmagnetization states. The inset shows a zoom close to the Al\nsuperconducting critical temperature.\ndipolar magnetic ﬁeld arising from the Fe ellipsoids.11\nSome other samples showed an opposite eﬀect but with\na similar amplitude, compatible with a dominating prox-\nimity eﬀect.9At low temperature, this spin switch eﬀect\nmay modify the superconducting gap and hence inﬂu-\nence the transport properties. Nevertheless, all samples\nshowed a similar spin-valve behavior although they ex-\nhibit a diﬀerent spin switch eﬀect. Moreover, the voltage\nacross the device is well below the gap in the discussed\ndata. Thus the spin switch eﬀect does not explain the\nspin-valve behavior observed at very low temperature.\nFig. 3 main panel displays the Sample 5 resistance\ntemperature dependence in both P and AP states. Be-\nlowTc, the resistances in the two states show a non-\nmonotonous behaviour. The spin-valve eﬀect amplitude\nhas alsoa non-monotonousbehavior, with a maximum at\nabout 0.9 K. At lower temperature, it decreases steadily\ntowards zero. The eﬀect is absent within a 1 mΩ res-\nolution above the critical temperature Tc= 1.18 K of\nAl. This conﬁrms that the observed spin-valve eﬀect is\nrelated to superconductivity. Fig. 4 left shows Sample\n5 diﬀerential resistance in diﬀerent magnetic states. In\nthe AP states, we ﬁnd a zero-bias resistance peak, which\nis suppressed in the P states. The spin valve eﬀect is\nlarger for ﬁnite voltage bias, which is consistent with the\nabove observation that it is larger at ﬁnite temperature.\nThe two P states on one side and the two AP states on\nthe other side behave very similarly. This conﬁrms that\nelectron transport depends on the relative magnetization\nalignment of the two ellipsoids, not on the direction of a\ngiven magnetization.\nTable 1 lists the main properties of the 8 investigated\nsamples, showing a spin-valve eﬀect as discussed above,\nat 275 mK. The spin-valve eﬀect amplitude ∆ Rvaries\nquite little, between 18 to 43 mΩ. The sample resis-\ntanceRis small and quite constant for Samples 2-5,\nwhere all F-S interfaces are very transparent. Samples 1\nand 6-8 with a larger resistance feature presumably less\ntransparent F-S interfaces at one of the superconducting\nFIG. 4: (Color online) Left: Voltage dependence of Sample\n5 diﬀerential resistance in the two P (top curves, ↓↓and↑↑)\nand the two AP states (bottom, ↑↓and↓↑). Right: Schemat-\nics of the sample geometry a) outlining the region S’ of the\nsuperconducting electrodes S submitted to inverse proximi ty\neﬀect and where spin accumulation is expected to occur.\nelectrodes, bringing to almost zero the spin-valve eﬀect\nin that electrode. In the geometry a), we have probed\nthe resistance either in a 2 wires geometry by using the\nwide Al electrodes for both current bias and voltagemea-\nsurement, or in a 4 wires geometry by using the voltage\nprobes for the measurement. We observed an identical\nlow temperature behavior in both cases. In a 4-wire con-\nﬁguration, a resistance peak appears close to the critical\ntemperature, due to charge-imbalance in the Al pads.15\nLet us now turn to the interpretation of the observed\nspin-valve eﬀect. As for non-local eﬀects, EC would have\nno contribution here, since we current-bias the two el-\nlipsoids in parallel. The sign and amplitude of the mea-\nsured eﬀect are compatible with CAR. Nevertheless, the\nabsence of a signiﬁcant inﬂuence of the ellipsoids separa-\ntion in the 100-500 nm range investigated here, whereas\nthe coherence length ξsis estimated to be about 100 nm,\ndiscards an interpretation in terms of CAR.\nA signiﬁcant inverse proximity eﬀect is expected in a\nsuperconductor in metallic contact with a ferromagnetic\nmetal. The sub-gap electronic density of states is non-\nzero in a region S’ extending in the superconductor over\na few times the coherence length ξs,16see Fig. 4 right\npart. Thisallowsfortheinjection, evenatasub-gapbias,\nof spin-polarized quasi-particules from the two Fe ellip-\nsoids into every Al electrode. In a P state, the current\nthroughboth ellipsoids injects the same majorityofspins\nand a signiﬁcant spin accumulation builds up in S’. In a\nAP state, one ellipsoidinjects spin-up quasi-particlesand\nthe other one injects spin-down quasi-particles. The two\nspin populations are then balanced and little spin accu-\nmulation is expected in S’. Thus a AP state is expected\nto have a lower resistance than a P state, as observed in\nthe experiment.\nThe consideredspin-accumulationbuilds upwithin the\nAl electrode in the region S’, while the Andreev reﬂec-4\nSample Geometry Separation (nm) R(Ω) ∆R(Ω)\n1 a 150 9.21 0.022\n2 a 150 1.94 0.041\n3 a 150 2.16 0.023\n4 b 100 2.76 0.018\n5 a 150 1.54 0.043\n6 b 150 8.37 0.024\n7 b 150 18.45 0.018\n8 b 500 5.62 0.023\nTABLE I: Sample parameters including the geometry type,\nthe ellipsoids separation, the P state resistance Rand the\nspin-valve eﬀect amplitude ∆ Rboth at 275 mK.\ntion occurs at the S’-S interface. No spin-valve eﬀect is\nexpected in the absence of inverse proximity eﬀect, in\nwhich case spin accumulation would occur separately in\nthe two ellipsoids. The observation of a spin-valve ef-\nfect in Sample 8 with a ellipsoids separation of 500 nm\nindicates that the region S’ extends over about 250 nm\nfrom a F-S interface, which is less than 3 times the coher-\nence length ξs. This is in agreement with Ref. 16, which\nshows that a signiﬁcant sub-gap density of states level\nremains at such a distance. The inverse proximity eﬀect\nand the relatedspin accumulationthus decayslowerthan\nthe CAR amplitude5,6when the separation between the\ntwo ellipsoids is increased in the range of a few times the\nsuperconducting coherence length.\nAt an F-Sinterface, the magnitude ofthe spin accumu-\nlation induced resistance is ∆ R=Rsq.(λsf/w).(α2/(1−α2)), where Rsqisthe squareresistance, wthewirewidth\nandαis the spin polarization.12Applying this analysis\nto the S’-S interface, with a square resistance Rsqof 1\nΩ, a wire width wof 400 nm, a spin relaxation length\nλsfof 400 nm in Al17and a polarization αof 40 % close\nto the one of bulk Fe,18one obtains ∆ R= 0.02 Ω per\ninterface, in fair agreement with the resistance change\namplitude at zero bias. The spin polarization in the Al\nelectrodeS’ regionis presumablysmaller than in bulk Fe,\nwhich would decrease the amplitude of the eﬀect. Even-\ntually, the dependence on bias and temperature of the\nspin-valve eﬀect can be related to changes of the size of\nthe region S’. As the current bias or the temperature in-\ncreases, the inverse proximity eﬀect extends over a larger\ndistance and the spin accumulation eﬀects increase.\nIn conclusion, we have investigated the sub-gap trans-\nport properties in double F-S hybrid structures with two\nF elements. Below the critical temperature of the super-\nconductor, the resistance depends on the relative magne-\ntization alignment. This spin-valve behavior is related to\nthe spin accumulation in the superconducting electrode\nsubmitted to inverse proximity eﬀect. This approach is\nsimilar to considering a out-of-equilibrium region in the\nvicinity of the interface19and may hold for previous ex-\nperiments in similar hybrid structures.20\nThesampleshavebeenfabricatedatNanofab-C.N.R.S.\nGrenoble plat-form. We thank M. Giroud for contribut-\ning at the early stage of this project, I. L. Prejbeanu for\nthe MFM measurements, D. Beckmann and R. M´ elin for\ndiscussions. We acknowledge support from ”Elec-EPR”\nANR contract and STREP ”SFINx” EU project.\n∗Also at Institut Universitaire de France\n1A. F. Andreev, Zh. Ekesp. Teor. Fiz. 46, 1823 (1963) [Sov.\nPhys. JETP 19, 1228 (1964)]; D. Saint-James, J. Phys.\n(Paris)25, 899 (1964).\n2M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev. Lett.\n74, 1657 (1995).\n3J. M. Byers and M. E. Flatt´ e, Phys. Rev. Lett. 74, 306\n(1995); G. Deutscher and D. Feinberg, Appl. Phys. Lett.\n76, 487 (2000).\n4G. Falci, D. Feinberg and F. W. J. Hekking, Europhys.\nLett.54, 255(2001); R.M´ elin, andD.Feinberg, Eur.Phys.\nJ. B26, 101 (2002); J. P. Morten, A. Brataas, and W.\nBelzig, Phys. Rev. B 74, 214510 (2006).\n5D. Beckmann, H. B. Weber, and H. v. L¨ ohneysen, Phys.\nRev. Lett. 93, 197003 (2004).\n6S. Russo, M. Kroug, T. M. Klapwijk, and A. F. Morpurgo,\nPhys. Rev. Lett. 95, 027002 (2005); A. L. Yeyati, F. S.\nBergeret, A. Martin-Rodero, and T. M. Klapwijk, Nature\nPhys.3, 455 (2007).\n7P. Cadden-Zimansky and V. Chandrasekhar, Phys. Rev.\nLett.97, 237003 (2006).\n8J. Y.Gu, C.-Y.You, J. S.Jiang, J. Pearson, Ya.B. Bazaliy,\nand S. D. Bader, Phys. Rev. Lett. 89, 267001 (2002); I. C.\nMoraru, W. P. Pratt, and N. O. Birge, Phys. Rev. B 74,\n220507(R) (2006).9P. G. de Gennes, Phys. Lett. 23, 10 (1966); L. R. Tagirov,\nPhys. Rev. Lett. 83, 2058 (1999).\n10A. Yu. Rusanov, S. Habraken, and J. Aarts, Phys. Rev. B\n73, 060505(R) (2006).\n11R. Steiner and P. Ziemann, Phys. Rev. B 74, 094504\n(2006).\n12F. J. Jedema, B. J. van Wees, B. H. Hoving, A. T. Filip,\nand T. M. Klapwijk, Phys. Rev. B 60, 16549 (1999).\n13M. Johnson, Appl. Phys. Lett. 65, 1460 (1994).\n14N. Poli, J. P. Morten, M. Urech, A. Brataas, D. B. Hav-\niland, and V. Korenivski, Phys. Rev. Lett. 100, 136601\n(2008); S. Takahashi and S. Maekawa, Phys. Rev. B 67,\n052409 (2003).\n15J. Clarke, Phys. Rev. Lett. 28, 1363 (1972); A. Schmid\nand P. Martinoli, J. of Low Temp. Phys. 20, 207 (1975).\n16M. A. Sillanp¨ a¨ a, T. T. Heikkil¨ a, R. K. Lindell, and P. J.\nHakonen, Europhys. Lett. 56, 590 (2001).\n17M. V. Costache, M. Zaﬀalon, and B. J. van Wees, Phys.\nRev. B74, 012412 (2006).\n18I. I. Mazin, Phys. Rev. Lett. 83, 1427 (1999).\n19R. M´ elin, Phys. Rev. B 72, 054503 (2005).\n20M. Giroud, K. Hasselbach, H. Courtois, D. Mailly, and B.\nPannetier, Eur. Phys. J. B 31, 103 (2003)."    },    {        "title": "0803.3174v1.Induced_Triplet_Pairing_in_clean_s_wave_Superconductor_Ferromagnet_layered_structures.pdf",        "content": "arXiv:0803.3174v1  [cond-mat.supr-con]  21 Mar 2008Induced Triplet Pairing in clean s-wave Superconductor/Fe rromagnet layered\nstructures\nKlaus Halterman,1,∗Oriol T. Valls,2,†and Paul H. Barsic2,‡\n1Physics and Computational Sciences, Research and Engineer ing Sciences Department,\nNaval Air Warfare Center, China Lake, California 93555\n2School of Physics and Astronomy, University of Minnesota, M inneapolis, Minnesota 55455\n(Dated: March 1, 2022)\nWe study induced triplet pairing correlations in clean ferr omagnet/superconductor/ferromagnet\nheterostructures. The pairing state in the superconductor is the conventional singlet s-wave, and\nthe angle αbetween the magnetizations of the two ferromagnetic layers is arbitrary. We use a\nnumerical fully self-consistent solution of the microscop ic equations and obtain the time-dependent\ntriplet correlations via the Heisenberg equations of motio n. We ﬁnd that in addition to the usual\nsinglet correlations, triplet correlations, odd in time as required by the Pauli principle, are induced\nin both the ferromagnets and the superconductor. These time -dependent correlations are largest\nat times of order of the inverse of the Debye cutoﬀ frequency, ωD, and we ﬁnd that within that\ntime scale they are often spatially very long ranged. We disc uss the behavior of the characteristic\npenetration lengths that describe these triplet correlati ons. We also ﬁnd that the ferromagnets\ncan locally magnetize the superconductor near the interfac e, and that the local magnetization then\nundergoes strongly damped oscillations. The local density of states exhibits a variety of energy\nsignatures, which we discuss, as a function of ferromagneti c strength and α.\nPACS numbers: 74.45.+c, 74.25.Bt, 74.78.Fk\nI. INTRODUCTION\nTriplet Cooper pairing is no new phenomenon: it has\nlong been recognized to be responsible for superﬂuidity1\nin3He as well as for superconductivity in some electronic\nmaterials. This occurs when the pairing interaction is in\na partial wave with odd ℓ. However, recent observations\nhave raised the possibility of induced triplet pairing cor-\nrelations in s-wave superconductors. It is a matter of\nelementary physics that the Cooper pair wavefunction\nmust be antisymmetric under exchange of the two elec-\ntronstosatisfythePauliprinciple. Forspatiallysymmet-\nric s-wave superconductors with decoupled spatial and\nspin degrees of freedom, the spin singlet pair is the only\npossible antisymmetric state of an electron pair. Triplet\npairing states on the other hand, where the spin state\nis symmetric, are obviously allowed when the pairing is\nspatially antisymmetric, such as in p-wave superconduc-\ntors. Triplet states in systems with s-wave pairing, even\nin momentum or coordinate space, would naively appear\nto violate the Pauli principle. However, many years ago\nBerezinskii proposed2a triplet state in superﬂuid3He,\nwhichinvolvedspatiallysymmetriccorrelations. Berezin-\nskii’s triplet pairing correlations, involving diﬀerent-time\npairing, did not violate the Pauli principle by virtue of\nbeing odd in time, thus allowing a triplet state in a sys-\ntem with s-wave interactions. While such a state did not\nturn out to be appropriate to describe superﬂuidity in\n3He, its consideration has led the way to the study of\ncases where some sort of time-reversal symmetry break-\ning mechanism may allow an odd time triplet state to be\ninducedin systemswith spatiallysymmetricinteractions.\nInterestin exotictripletpairingarisesfrommanyquar-\nters. In the case of two component cold atomic gases3with short-range s-wave interactions, in which the two\nspecies have the same mass but diﬀerent chemical po-\ntentials, it may be possible to induce a triplet pairing\nthatbreakstime reversalsymmetry. In electronicmateri-\nals, an important issue is the possible existence of a long\nrange proximity eﬀect in Superconductor/Ferromagnet\n(SF) heterostructures. Interest in these heterostructures\narises in turn from their possible4applications. Of par-\nticular interest is that the thermodynamic and transport\nproperties of FSF trilayers are found to depend strongly\non the relative orientation of the magnetization in the\ntwo F layers.5,6,7,8This rather well-understood9,10,11,12\nfact makes these structures candidates as spin valves.\nThere have been no unambiguous observations of in-\nduced triplet correlations in SF structures involving s-\nwave superconductors. However, there have been some\nenticing experimental hints in the form of long ranged\nproximity induced superconducting behavior in SF mul-\ntilayers with strong exchange ﬁelds. The observed ef-\nfects are over length scales much larger than those of\nthe usual SF proximity eﬀect and more like the much\nlonger length scales associated with the standard prox-\nimity eﬀect between a superconductor and a normal non-\nmagnetic metal. These observations include measure-\nments in superlattices13with ferromagnetic spacers and\nSQUIDs14withferromagneticinterlayers. Superconduct-\ning characteristics, such as a critical temperature and en-\nhancedsub-gapconductance,havebeenobservedinpoint\ncontact conductance measurements on s-wave supercon-\nductor/half metallic systems.15Perhaps most compelling\nistheobservationofaJosephsoncurrentthroughastrong\nalmost half-metallic, ferromagnet.16All of these experi-\nments indicate long range superconducting correlations\nthat are not destroyed by a strong exchange ﬁeld: a2\ntriplet state would obviously be consistent with these ex-\nperiments. To fully understand the behavior of these\nsystems, further studies involving, for example, quanti-\nties sensitive to the gap such as the local density of states\n(local DOS), are needed. Until both better theoretical\nmodels and more varied experimental observations are\nmade available, one cannot conclusively say that there is\nindeed an induced triplet state in these systems, but at\nthis point the facts ﬁt this explanation and no better one\nhas been proposed.17\nMany theoretical studies agree that it is pos-\nsible to induce this exotic state in certain SF\nsystems17,18,19,20,21,22,23,24,25,26,27,28,29(perhaps even in\nthe nonmagnetic30case) with ordinary singlet pairing in\nS. Some studies use an SFF′S arrangement in which the\nFandF′layershavediﬀerentmagnetizationorientations.\nOthersassumethat adomainstructurein asingleF layer\nis responsible for the symmetry breaking. Yet others as-\nsume an FSF system with diﬀerent in-plane magnetiza-\ntion orientations in the F layers. Whatever the mecha-\nnism for the symmetry breaking, such arrangements can\ninduce, via proximity eﬀects, triplet correlations of dif-\nferent kinds. Recently it was shown that a Josephson\nsupercurrent can exist in a half metal by virtue of equal\nspin triplet pairs and spin ﬂip scattering events at the\ninterfaces.28To understand and probe the underlying\ntriplet state, investigations have been done on conduc-\ntance spectra in simpler FS structures31with arbitrary\nmagnetization alignment and with spin active interfaces,\nas well as in diﬀusive29SF junctions, through character-\nizing the possible superconductor symmetry classes con-\nsistent with Pauli’s principle. With the exception of our\nearly work18on SFS trilayers and some recent work on\nSFS Josephson junctions,28the studies above are done\nin the dirty limit through linearized Usadel-type or other\nquasiclassicalequations. Thedisadvantageofaquasiclas-\nsical approachis that it is unsuitable formagnets with an\nexchange ﬁeld on the order of the Fermi energy. Thus, it\ncannot properly model a strong ferromagnet, and it does\nnot allow for atomic scale oscillations in the pair ampli-\ntude. A good quantitative explanation requires a self-\nconsistent treatment of a fully microscopic model. Thus,\nas pointed out in a recent review,27the very existence of\ntriplet correlations in clean FS structures was until very\nrecently generally doubted, and these doubts have only\nvery recently18been dispelled.\nIn this paper we explore the phenomenon of induced\ntriplet correlations, odd in time, of clean FSF structures,\nwhere S is an ordinary s-wave superconductor and the\nmagnetizationsin thetwoFlayersarerotatedbyanarbi-\ntrary angle α. We assume strong ferromagnets(up to the\nhalf-metallic limit), and smooth, sharp interfaces. In this\ngeometry, triplet correlations with total spin projection\nm= 0 on the axis of quantization of the Cooper pairs are\nin general possible and, when the relative magnetizations\n(which we assume as usual are both parallel to the inter-\nfaces) are not aligned, triplet components with m=±1are allowed also. To satisfy the Pauli principle, these\nspatially symmetric triplet pairing correlations must be\nodd in frequency or time.2That such correlations are al-\nlowed, does not mean that they must exist, nor that they\nmust exist over an extended spatial range. We ﬁnd, how-\never,viaafullyself-consistentsolutiontothemicroscopic\nBogoliubov de-Gennes32(BdG) equations that such cor-\nrelations do indeed exist, and that the penetration depth\nassociated with them can be very long. Our use of the\nBdG equations allows us to study strong ferromagnets.\nSelf-consistency is fundamental: non-self-consistent solu-\ntions are found to violate the Pauli principle. Thus, the\ntime consuming step of calculating fully self-consistent\nsolutions is necessaryto properly model the proximityef-\nfects which allow for the mixing of superconducting and\nferromagnetic orderings that causes these induced corre-\nlations.\nIn Sec. II of this paper, we discuss the basic equations\nand our method for numerical self-consistent solution.\nThere, the extraction of the all-important time depen-\ndence via solution of the Heisenberg equations of motion\nfor the relevant operators is explained in detail. Expres-\nsions for all of the time-dependent triplet correlationsare\nalso derived. The equations for the local density of states\n(local DOS) and the local magnetic moment (which we\nuse to discuss the reverse proximity eﬀect, that is, the\npenetration of the magnetism into the superconductor)\nare also presented. In the next section (Sec. III) we be-\ngin by presenting an extensive discussion of the triplet\ncorrelations as a function of position, α, and magnet\nstrength. The appropriate penetration depths are ex-\ntractedanddiscussed. ResultsfortheDOS,the magnetic\nmoment and the temperature dependence of both triplet\nand ordinary singlet correlations are also given. Finally,\nin Sec. IV, a brief conclusion and summary is given.\nII. METHODS\nThe geometry we consider consists of a planar FSF\njunction as depicted in Fig. 1. The thickness of the su-\nperconducting layer is dSand the F layers have thick-\nnessesdF1anddF2. The system is assumed to be inﬁnite\nin the plane perpendicular to the layers, which we label\nas ourx−zplane. The magnetizations of the F layers,\nwhich are in this plane, form an angle ±α/2 with the z\naxis, which is that of the direction of quantization of the\nspins.\nOur starting point is the Bogoliubovde-Gennes (BdG)\nequations32for the system under consideration. The\nderivation of the BdG equations for the case of interest\nrequires some care with the conventions for all operator\nphase factors, which are not universally agreed upon in\nthe literature, and which may give rise to diﬀerent signs\nin some of the equations below. We write the eﬀective\nBCS Hamiltonian, Heﬀ, as3\nFIG. 1: (Color online) Schematic of the FSF junction. The\nyaxis is normal to the interfaces. The left ferromagnet layer\ndenoted F 1has a magnetization oriented at an angle −α/2 in\nthex−zplane, while the other magnet F 2, has a magnetiza-\ntion orientation at an angle α/2 in the x−zplane. All layer\nwidths are labeled.\nHeﬀ=/integraldisplay\nd3r/braceleftig/summationdisplay\nαψ†\nα(r)Heψα(r)+1\n2[/summationdisplay\nα,β(iσy)αβ∆(r)ψ†\nα(r)ψ†\nβ(r)+h.c.]−/summationdisplay\nα,βψ†\nα(r)(h·σ)αβψβ(r)/bracerightig\n,(2.1)\nwhereHe=−1/(2m)∇2−EF+U(r),σare the set\nof Pauli matrices, spin is denoted by Greek indices, and\nas usual, we represent the magnetism of the F layers by\nan eﬀective exchange Stoner energy h(r) which will in\ngeneral have components in both the transverse ( x,z)\ndirections. The spin independent scattering potential is\ndenotedU(r), and ∆( r) is the usual pair potential.\nTo diagonalize the eﬀective Hamiltonian, the ﬁeld op-\neratorsψ†\nαandψαare expanded by means of a Bogoli-\nubov transformation, which, for our phase convention,\nwe write33as:\nψ↑(r) =/summationdisplay\nn/parenleftbig\nun↑(r)γn−vn↑(r)γ†\nn/parenrightbig\n,(2.2a)\nψ↓(r) =/summationdisplay\nn/parenleftbig\nun↓(r)γn+vn↓(r)γ†\nn/parenrightbig\n,(2.2b)whereunαandvnαare the quasiparticle and quasihole\namplitudes, and γnandγ†\nnare the Bogoliubov quasipar-\nticle annihilation and creation operators, respectively.\nWe require that the transformations in Eqs. (2.2) di-\nagonalize Heﬀ,\n[Heﬀ,γn] =−ǫnγn, (2.3a)\n[Heﬀ,γ†\nn] =ǫnγ†\nn. (2.3b)\nOne can also take the commutator [ ψα(r),Heﬀ]. With\nthe magnetizations in the x−zplane as explained above,\nthis gives the following,\n[ψ↑(r),Heﬀ] = (He−hz)ψ↑(r)−hxψ↓(r)+∆(r)ψ†\n↓(r), (2.4a)\n[ψ↓(r),Heﬀ] = (He+hz)ψ↓(r)−hxψ↑(r)−∆(r)ψ†\n↑(r). (2.4b)\nInserting (2.2) into (2.4) and using Eqs. (2.3) yields the general spin -dependent BdG equations,\n\nH0−hz(y)−hx(y) 0 ∆( y)\n−hx(y)H0+hz(y) ∆(y) 0\n0 ∆( y)−(H0−hz(y))−hx(y)\n∆(y) 0 −hx(y)−(H0+hz(y))\n\nun↑(y)\nun↓(y)\nvn↑(y)\nvn↓(y)\n=ǫn\nun↑(y)\nun↓(y)\nvn↑(y)\nvn↓(y)\n, (2.5)\nwhere the single particle Hamiltonian H0is deﬁned as,\nH0≡/hatwidep2\ny\n2m+ε⊥−EF+U(y). (2.6)A plane wave factor eik⊥·rhas been canceled in both4\nsides of Eq. (2.5). The longitudinal momentum opera-\ntor,/hatwidepy, is given by, /hatwidepy=−i∂/∂y,ε⊥is the kinetic en-\nergy of the transverse modes, ∆( y) is the self-consistent\npair potential, and U(y) is a scalar potential represent-\ning interface scattering characterized by a delta func-\ntion of strength HB. The ferromagnetic exchange ﬁeld\nh(y) = (hx(y),0,hz(y)), vanishes in the S layers. We\nhavehx(y) =h0sin(−α/2) andhz(y) =h0cos(−α/2)\nin the F 1layer, where h0is the magnitude of the ex-\nchange ﬁeld, while in F 2,hx(y) =h0sin(α/2), and\nhz(y) =h0cos(α/2). We refer to Fig. 1 for details. The\ndimensionless parameter I≡h0/EFconveniently char-\nacterizes the strength of the magnetism. One thus has\nI= 1 in the half metallic limit. If we take α= 0 orπ,\ni.e. the magnetizations of both layers lie along the same\ndirection, or if there is only one F layer, then hx= 0\nand we recover the simpler form of the BdG equations\nused9,34in other contexts. We can ﬁnd the quasiparticle\namplitudes on a diﬀerent quantization axis in the x−z\nplane forming an angle α′withz, by performing a spin\nrotation Φ n→/hatwideU(α′)Φnwith,\n/hatwideU(α′) = cos(α′/2)ˆ1⊗ˆ1−isin(α′/2)ρz⊗σy,(2.7)\nwherewehaveintroduced ρasaset ofPauli-likematrices\nin particle-holespace. It isconvenienttouse thematricesρin conjunction with the ordinary Pauli matrices σin\nspin space to rewrite Eqs. (2.5) in the more compact but\nperhaps less transparent way:\n/bracketleftbig\nρz⊗/parenleftbig\nH0ˆ1−hzσz/parenrightbig\n+/parenleftbig\n∆(y)ρx−hxˆ1/parenrightbig\n⊗σx/bracketrightbig\nΦn=ǫnΦn,\n(2.8)\nwhere Φ n≡(un↑(y),un↓(y),vn↑(y),vn↓(y))T, with the\nsuperindex denoting transposition.\nThe usual self consistency condition relates the spec-\ntrum obtained from Eq. (2.5) to the inhomogeneous pair\npotential ∆( y) by an appropriate sum over states:\n∆(y) =g(y)\n2/summationdisplay\nn′/bracketleftbig\nu↑\nn(y)v↓\nn(y)+u↓\nn(y)v↑\nn(y)/bracketrightbig\ntanh(ǫn/2T),\n(2.9)\nwhere the prime on the sum indicates that only those\npositive energy states with energy less than the pairing\ninteraction energy cutoﬀ, ωD, are included, and Tis the\ntemperature. The function g(y) vanishes in the F layers\nwhile in the S layers it takes the value of the usual BCS\nsingletcoupling constant in the S material.\nWith an appropriatechoice ofbasis,34,35Eqs.(2.5) can\nbe cast into a ﬁnite 4 N×4Ndimensional matrix eigen-\nvalue system. In dimensionless form, it reads,\n\nH0−Hz−Hx 0 D\n−HxH0+HzD 0\n0D−(H0−Hz)−Hx\nD 0 −Hx−(H0+Hz)\nΨn=/tildewideǫnΨn, (2.10)\nwhere/tildewideǫn≡ǫn/EF, and Ψ n, the transpose of\nΨT\nn= (u↑\nn1,...,u↑\nnN,u↓\nn1,...,u↓\nnN,v↑\nn1,...,v↑\nnN,v↓\nn1,...,v↓\nnN), (2.11)\ncontains the expansion coeﬃcients associated with the set of orth onormal basis functions. We write uα\nn(z) =/radicalbig\n2/d/summationtextN\nq=1uα\nnqsin(qπz/d), andvα\nn(z) =/radicalbig\n2/d/summationtextN\nq=1vα\nnqsin(qπz/d),forα=↑,↓. The necessary matrix elements\nanalogous to Eqn. (2.5) for diﬀerent πjunction geometries and for strictly collinear magnetization orienta tions have\nbeen calculated in previous work.34,35The situation is more complicated for the case of a FSF trilayer consid ered5\nhere, with the magnetization angles of the two F layers forming an an gleα. The matrix elements are then written as,\n(H0)mn=/bracketleftigg/parenleftbiggmπ\nkFd/parenrightbigg2\n+ε⊥\nEF−1/bracketrightigg\nδmn+ZB[Um−n(dF1)+Um−n(dF1+dS)−Um+n(dF1)\n−Um+n(dF1+dS)], (2.12a)\n(Hz)mn=h0\nEFcos(α/2)/bracketleftbig\nKm−n(dF1)−Km+n(dF1)+Km+n(dF1+dS)\n−Km−n(dF1+dS)/bracketrightbig\n, m/ne}ationslash=n, (2.12b)\n=h0\nEFcos(α/2)/bracketleftbiggdF1+dF2\nd+K2m(dF1+dS)−K2m(dF1)/bracketrightbigg\n, m=n, (2.12c)\n(Hx)mn=h0\nEFsin(α/2)/bracketleftbig\nKm+n(dF1)−Km−n(dF1)+Km+n(dF1+dS)\n−Km−n(dF1+dS)/bracketrightbig\n, m/ne}ationslash=n, (2.12d)\n=h0\nEFsin(α/2)/bracketleftbiggdF2−dF1\nd+K2m(dF1+dS)+K2m(dF1)/bracketrightbigg\n,m=n, (2.12e)\n(D)mn=2\nEFd/integraldisplaydF1+dS\ndF1dysin/bracketleftigmπy\nd/bracketrightig\n∆(y)sin/bracketleftignπy\nd/bracketrightig\n, (2.12f)\nwhereZB≡2HB/(kFd) is a convenient dimensionless\nmeasure of interfacial scattering. We have also deﬁned:\nKn(y)≡sin/parenleftbignπy\nd/parenrightbig\nnπ,Un(y)≡cos/parenleftignπy\nd/parenrightig\n.(2.13)\nWe now consider the appropriate quantities that char-\nacterize the induced triplet correlations. To do this, we\ndeﬁne the following tripletpair amplitude functions in\nterms of the ﬁeld operators,\nf0(r,t) =1\n2[/an}bracketle{tψ↑(r,t)ψ↓(r,0)/an}bracketri}ht+/an}bracketle{tψ↓(r,t)ψ↑(r,0)/an}bracketri}ht],\n(2.14a)\nf1(r,t) =1\n2[/an}bracketle{tψ↑(r,t)ψ↑(r,0)/an}bracketri}ht−/an}bracketle{tψ↓(r,t)ψ↓(r,0)/an}bracketri}ht].\n(2.14b)\nWe will later demonstrate that these amplitudes vanish\natt= 0, as required by the Pauli principle.\nTo make use of these expressions, it is most convenient\nto use the Heisenberg picture. Thus we write ψςin the\nHeisenberg representation:\nψς(t) =e(iHefft)ψςe(−iHefft). (2.15)\nTo put this in terms of the quasiparticle amplitudes, we\napplyEqns.(2.2)andthetransformationEqns.(2.3). We\ncan then immediately write down the Heisenberg equa-\ntions of motion for the γ’s as\ni∂γn\n∂t= [γn,Heﬀ] (2.16)\nand\ni∂γ†\nn\n∂t= [γ†\nn,Heﬀ]. (2.17)These equations of motion, given Eqns. (2.3), have the\nsolutionsγn(t) =γne−iǫntandγ†\nn(t) =γ†\nneiǫnt. When\nwe substitute these results into the above equations for\nf0andf1, taking into account Eqns. (2.2) we obtain:\nf0(y,t) =1\n2/summationdisplay\nn[un↑(y)vn↓(y)−un↓(y)vn↑(y)]ζn(t),\n(2.18a)\nf1(y,t) =−1\n2/summationdisplay\nn[un↑(y)vn↑(y)+un↓(y)vn↓(y)]ζn(t),\n(2.18b)\nwhereζn(t)≡cos(ǫnt)−isin(ǫnt)tanh(ǫn/2T). The spa-\ntial dependence of the complex quantities f0(y,t) and\nf1(y,t) is, in our geometry, on the ycoordinate only.\nThey vanish identically at t= 0.\nWe will focus on in our study of the induced triplet\ncorrelationsonthe time dependent quantities f0(y,t) and\nf1(y,t). Their existence at t>0 is allowed by the Pauli\nprinciple. It is also important to sort out when it is\nallowed by the spin symmetries: when the axis of quanti-\nzation ofthe Cooperpairsis the only axis ofquantization\nin the system (i.e., when α= 0) then it is not hard to see\nthat the total spin operator Sof the Cooper pairs does\nnot commute with the Hamiltonian. This is best seen\ndirectly from the matrix expression on the left side of\nEqn. (2.5). On the other hand, Szand the Hamiltonian\ndocommuteinthiscase. However,when α(andtherefore\nhx) is nonzero, then no component of Scommutes with\nthe eﬀective Hamiltonian. From this spin symmetry ar-\ngumentitfollowsthatthe inducedamplitude f1(y,t)may\nexist (at ﬁnite times) only at nonzero α, whilef0(y,t) is\nallowed ant any α. Forα=π, when the magnetizations\nare antiparallel and along the xaxis, no triplet ampli-\ntudes with nonzero component along that axis can exist.6\nThe matrix /hatwideU(α) in Eq. (2.7) can be used to verify this\nby performing the corresponding spin rotations. That\nthe existence of certain quantities is consistent with all\nsymmetry properties does not mean that these quanti-\nties will indeed be nonvanishing, and it certainly tells us\nnothing about the possible range and behavior in space\nand time of these amplitudes. To determine this requires\ndetailed calculations.\nAlso of considerable interest in F/S structures is the\nreverse proximity eﬀect: the leakage of magnetism outof the magnets and into the superconductor. This can\nbe characterized by the local magnetization m(y). It is\ndeﬁned as,\nm=−µB/an}bracketle{t/summationdisplay\nσψ†\nσσψσ/an}bracketri}ht, (2.19)\nwhereµBis the Bohr magneton. The vector mhas two\ncomponents in the FSF geometry discussed. Both com-\nponents depend on y. They are:\nmz(y) =−µB/summationdisplay\nn/braceleftbig/bracketleftbig\n|un↑(y)|2−|un↓(y)|2/bracketrightbig\nfn+/bracketleftbig\n|vn↑(y)|2−|vn↓(y)|2/bracketrightbig\n(1−fn)/bracerightbig\n, (2.20)\nand\nmx(y) =−2µB/summationdisplay\nn{[un↑(y)un↓(y)]fn+[vn↑(y)vn↓(y)](1−fn)}. (2.21)\nIt is convenient to normalize these components to −µB(N↑+N↓), whereN↑=k3\nF(1 +I)3/2/(6π2), andN↓=\nk3\nF(1−I)3/2/(6π2).\nThe proximity eﬀects can also be examined through the local DOS, N(y,ǫ), given by,\nN(y,ǫ) =−/summationdisplay\nn{[u2\nn↑(y)+u2\nn↓(y)]f′(ǫ−ǫn)+[v2\nn↑(y)+v2\nn↓(y)]f′(ǫ+ǫn)}, (2.22)\nwheref′=∂f/∂ǫ. We will be concerned mainly with\nthe DOS normalized to the DOS of a bulk (unpolarized)\nnormal metal, DN(0) =k3\nF/(2π2EF).\nIII. RESULTS\nInthissectionwepresentourresults, obtainedselfcon-\nsistently as explained above and in previous18,34,35work.\nWe have assumed a coherence length kFξ0= 100. We\nwill choose a geometry in which the layers are relatively\nthick:kFds≡DS= 200 (that is, two coherence lengths)\nandDF1=DF2= 250. These values ensure that the\nsample will be overall superconducting at temperatures\nup to about 1 /3 of the transition temperature Tc, of a\npure S bulk sample. This was not the case for the smaller\nvalues used in Ref. 18 where the condensationenergywas\nquite small (see e.g. ﬁgure 6 in Ref. 35). This allows us\nto study the temperature dependence of the quantities\ninvolved over a broad range. The most important pa-\nrameters are the angle αand the magnet strength I. We\nwill varyαin its full range between 0 and πand give re-\nsults for the values of Iof 0.25, 0.5, and unity. No triplet\namplitudes arise at I= 0 (when magnetism is absent).\nIn the results presented we have ZB= 0, when proximity\neﬀects are in general maximized.\nIn Fig. 2 we present comprehensive results for the real\nparts of f0(y,t) andf1(y,t), which we denote simply asf0(y,t) andf1(y,t) respectively. These are plotted in\nterms of the dimensionless variable Y≡kFy. The am-\nplitudes are normalized to the value of the usual singlet\namplitude in a pure bulk S sample. The temperature\nis set to zero in this ﬁgure. In the main plots, half of\nthe S region and a portion (three ﬁfths) of the left ( F1)\nregion are included. The corresponding portion on the\nF2side can be inferred from the geometry and symme-\ntry considerations. Results are plotted at three values\nofIand at a number of ﬁnite times τ≡ωDtbetween\n0.4 and 8 as indicated in the legends. We have veriﬁed\nthat att= 0 the computed triplet amplitudes vanish\nidentically, in agreement with the Pauli principle. This\nis true, however, only when the calculation is performed\nto self-consistency: non-self consistent results invariably\nviolate the Pauli principle near the interface. The results\nforf0are given at an angle α= 0 while those for f1are\natα=π/2. Atα= 0,f1vanishes identically since the\nzcomponent of the total spin is then a good quantum\nnumber. At α=π/2, and short time scales, the spatial\ndependences of the two triplet components coincide, al-\nbeit with diﬀerent signs in the two magnet regions, due\nto the magnetization vectors having equal projections on\nthexandzaxes. At longer times, when the triplet am-\nplitudes extend throughout the S layer and couple the\ntwo magnets, f0andf1deviate from one another. The\ninsets in each panel amplify and clarify the region near\nthe interfaces.7\n-0.06-0.04-0.020.000.020.040.06\n0.4\n2\n4\n6\n8I=0.25\n-108 -106 -104 -102 -100-0.020.000.020.040.06Normalized f0\n-0.06-0.04-0.020.000.020.040.06\nI=0.5\n-108 -106 -104 -102 -100-0.020.000.020.040.06\nY-250 -200 -150 -100 -50 0-0.06-0.04-0.020.000.020.040.06\nI=1\n-108 -106 -104 -102 -100-0.06-0.04-0.020.000.020.040.06-0.06-0.04-0.020.000.020.040.06\n0.4\n2\n4\n6\n8I=0.25\n-108 -106 -104 -102 -100-0.020.000.020.040.06Normalized f1\n-0.06-0.04-0.020.000.020.040.06\nI=0.5\n-108 -106 -104 -102 -100-0.020.000.020.040.06\nY-250 -200 -150 -100 -50 0-0.06-0.04-0.020.000.020.040.06\nI=1\n-108 -106 -104 -102 -100-0.06-0.04-0.020.000.020.040.06\nFIG. 2: The real parts, f0andf1, of the triplet pair ampli-\ntudesf0andf1(Eqns.(2.18)), plottedasafunctionofposition\n(in terms of Y≡kFy) for three values of Iat diﬀerent times\nτ≡ωDtindicated in the legends of the top panels. These\nquantities are normalized to the value of the singlet pairin g\namplitude in a bulk S material. The main plots show half of\nthe S region (right side of the vertical dashed line), and par t\nof the F 1region. The insets are blow-ups of the region near\nthe interface. The angle αis zero in the left panel and π/2 in\nthe right panel.\nFIG. 3: Penetration depths for the triplet amplitudes (see\nEq.(3.1)), plottedasafunctionof τ, as calculated from f0and\nf1in both the S and F regions, for the values of Iindicated\nand the same angles as in Fig. 2.FIG. 4: Maximum absolute values of f0andf1(see text) as a\nfunction of dimensionless time τatI= 1. In the top panel we\nconsider f0at bothα= 0 and α=π/2 while in the bottom\npanel we consider f1atα=π/2.\nOn the F side, both amplitudes peak very near the\ninterface and then decay in an oscillatory manner, rem-\niniscent of the behavior of the usual pair amplitude.\nAlthough the height of the ﬁrst peak does not depend\nstrongly on I, the subsequent decay in the F material is\nfaster for larger values of I. This can be attributed to a\ndecreased overall proximity eﬀect: here we have assumed\nthat atI= 0 there would be no mismatch between the\nFermi surface wavevectorsof the two materials, implying\nthat asIincreases the mismatch between either the up\nor the down Fermi wavevectors k↑andk↓, on the F side,\nand that in the S side increases. The location of this ﬁrst\npeak depends very clearly on I, its distance to the inter-\nface decreasing as approximately 1 /Iconsistent with the\ngeneral rule that the oscillatory spatial dependences on\nthe F side are determined by the inverse of k↑−k↓. The\nheight of the ﬁrst peak depends strongly on time and is\nmaximum at times τof about 2π. It is quite obviousthat\nat intermediate values of Ithe penetration of the triplet\ncorrelation into the F material is rather long ranged.\nOn the superconducting side the behavior is quite dif-\nferent: the triplet correlationspenetrate into the S mate-\nrial overa distance that rather quickly reaches two corre-\nlation lengths and then of course saturates at the sample\nsize, without signs of decaying in time at these length8Normalized f0\n-0.06-0.04-0.020.000.020.040.06\n0.4\n2\n4\n6\n8I=1\n-104 -103 -102 -101 -100-0.04-0.03-0.02-0.010.000.01\n-100 -98 -96 -94 -92 -90-0.04-0.03-0.02-0.010.000.010.02~\nF region\nS region\nY-250 -200 -150 -100 -50 0Normalized f1\n-0.06-0.04-0.020.000.020.040.06\n-104 -103 -102 -101 -100-0.03-0.02-0.010.000.01\n-100 -98 -96 -94 -92 -90-0.03-0.02-0.010.000.010.020.030.04~F region\nS region\nFIG. 5: Imaginary parts, ˜f0(y,t) and˜f1(y,t) of the complex\ntriplet amplitudes f0(y,t) andf1(y,t). These quantities are\nnormalized and plotted exactly as their corresponding real\nparts are in Fig. 2, except that here we consider only the\nI= 1 case, and both plots are for α=π/2. As in Fig. 2\nthe main plot shows the behavior over an extended region.\nThere are now two insets to each main plot, each showing the\ndetailed behavior near the interface itself, on either the S or\nF side.\nscales. Furthermore this eﬀect now increases sharply\nwithIand is maximal in the half metallic case. Thus,\nthe magnets act as sources, so to speak, of triplet correla-\ntions that enter the S material and this eﬀect is stronger\nwhenIis larger.\nIt is instructive to extract characteristic penetration\nlengthsℓifrom the above data using the deﬁnition,\nℓi=/integraltext\ndy|fi(y,t)|\nmax|fi(y,t)|, i= 0,1, (3.1)\nwhere the integration is either over the S or the F re-\ngion. In Fig. 3, the top two panels show the penetration\nlengths for the F material, at three values of Iand the\nsame values of αas for Fig. 2. The results are very sim-\nilar whether they are calculated from the results for f0\nor from those for f1. The penetration length at constant\ntime decreases with Ias already noted and shows signs\nof saturating with time at a value which for I= 0.25 ap-\nproaches that of the superconducting coherence length.On the S side (bottom panels) the situation is very dif-\nferent: the results for f0andf1arenow clearlydissimilar\nwith the penetrationlength forthe former quantitybeing\n(forcasesshownhere)thelargerone. Thisarisesfromthe\ngeometry and magnetization projections of each F layer\non thexaxis, which are in opposite directions, forcing\nthe tripletf1to possess a node at the center of the tri-\nlayer. No such requirement exists for f0, as it is spatially\nsymmetric. Except for the case of f1atI= 1, we see\nno sign of saturation. In fact, the maximum value of τ\ndisplayedherecorrespondsto the casein which the entire\nintrinsically singlet superconductor layer, two coherence\nlengths thick and sandwiched between two magnets, is\nwholly pervaded by induced triplet correlations.\nIt is alsoofinterestto considerthe variationofthe spa-\ntial maximum values of f0(y,t) andf1(y,t) with time. In\nFig. 4 we show, for each time, the largest value of these\nquantities, which typically is attained near the interface,\nin either the F or S regions. By “maximum” value we\nmean the maximum of |f0|and|f1|, not to be confused\nwith the absolute value of the complex quantities |f0|\nor|f1|. In this ﬁgure, the magnets are half metallic,\nI= 1. In the top panel, we plot the results for f0at\nbothα= 0 andα=π/2. We see that at earlier times,\nthe valuef0(y,t) at its peak just inside the F region (see\nFig. 2, bottom left panel) exceeds the maximum value of\nthis quantity in S. At longer times, however, there is a\ncrossover as the size of the peaks in F decreases rather\nsharply, as explained above, while the size of the ampli-\ntude in S decreases only slowly, as the triplet correlations\nﬁll the S layer. It is apparent from careful examination of\ntheI <1 panelsin Fig.2, that this crossoverdoesnot oc-\ncur for smaller values of Iexcept possibly on a time scale\nmuch longer than that considered here. In the bottom\npanel, wepresentasimilarstudy of f1, this timeofcourse\nonly atα=π/2 since this quantity vanishes identically\nfor collinear magnetizations. The results are clearly very\nsimilar except that the results in the F region appear\nto saturate and do not decrease at long times. This is\nconsistent with the earlier discussion where we saw that\n|f0|and|f1|overlap atα=π/2, except at suﬃciently\nlong times. In all cases the maximum value of the quan-\ntity plotted crests near τ= 6 in agreement with previous\nremarks.\nAll of the above results have been given in terms of the\nreal parts, f0andf1of the complex amplitudes f0and\nf1. The behavior of the corresponding imaginary parts is\nqualitatively very similar and thus we will present only\ntwoexamples, inFig.5. We denotethese imaginaryparts\nby˜f0(y,t) and˜f1(y,t) respectively. In Fig. 5 we consider\nonly the case I= 1 (compare with Fig. 2) and α=π/2.\nAs in Fig. 2 the main plots include an extended region\nnear one of the interfaces and the insets are close views\nof the interface itself, in this case one of the insets shows\na more detailed view of the S side. On the F side, the\nbehavior is reminiscent to that of the real parts, except\nthat the very prominent peak seen in the real parts right\nat the interface is absent for the imaginary parts. On the9\nFIG. 6: Magnetic moment component mx(see Eq. (2.21)),\nnormalized as explained in the text, plotted vs position (at\nI= 1) for several values of α. The main plot shows the\nbehavior near the interface (vertical dashed line), while t he\ninset covers the whole sample.\nS side, the sign is now initially negative and it changes to\npositive atτoforderunity. No such changewasobserved\nfor the real parts. At longer times, the imaginary part of\nthe triplet correlations also eventually penetrates several\ncorrelationlengthsintothe S sample, just asthe realpart\ndoes.\nIn the next two ﬁgures, we explore the reverse prox-\nimity eﬀect (the spreading of the magnetism into the S\nlayer) as a function of α. This is best done by consider-\ning separately the two components of the local magnetic\nmoment vector. First, in Fig. 6 we consider the xcom-\nponentmx(see Eq. (2.21)) normalized to the absolute\nvalue of its bulk value in a pure F material. The results\nin this ﬁgure are for half metallic magnets, ( I= 1). In\nthe main plot of the ﬁgure we display the value of mxin\nthe region very near an interface for several values of α.\nOf course, mxvanishes at α= 0. At other values of α\nit is large in the F material and it penetrates into S in\nan oscillatory way that is quite reminiscent of the corre-\nspondingpenetrationofthe superconductingcorrelations\nintoS. We seethatthe periodofthe spatialoscillationsof\nthe magnetization is independent of the angle αbetween\nthe two F layer magnetizations. Another discernible fea-\nture is that mxdampens out over relatively short length\nscales, consistent with past work.34The inset shows the\noverall behavior of mxin the entire sample, demonstrat-\ning also the opposite signs between in the two magnets\nin accordance with Fig. 1.\nSimilarly, in Fig. 7 we display the z-component of mz\n(see Eq. (2.20)), normalized in the same way as mx, and\nfor the same values of αbut including now three diﬀerent\nvalues ofI. Again, the main plots display the behavior\nnear the interface while the insets are for the entire sam-\nple. One can see here that the reverse proximity eﬀect\nis very weak at small Iand largest in the half metallicFIG. 7: Normalized (see text) magnetic moment component\nmz(Eq. (2.20)) plotted vs. dimensionless position for three\nvalues of I. Again, the main plot is the behavior near the\ninterface, whiletheinsetscoverthewhole sample. Itisevi dent\nthatmzpoints in the same direction for both magnets\ncase. The magnetic moment oscillates in the supercon-\nductor with a period that is independent of the direc-\ntion and magnitude of the mutual magnetization in the\nF layers. The observed trends in mzhold also for the mx\ncomponent.\nWe next study the energy dependence of the single\nparticle quasiparticle spectrum by considering the lo-\ncal density of states (local DOS), N(y,ǫ), as deﬁned in\nEqn. (2.22). In Fig. 8 we consider the local DOS, in-\ntegrated either over either the entire S or the entire F\nregion, and normalized to its bulk value on a sample of10\nFIG. 8: Normalized local DOSfrom Eq. (2.22)integrated over\n(see text) over either the F region (left panels) or the S regi on\n(right panels) for three values of Iand several relative mag-\nnetization orientations. The temperature is at T= 0.01Tc.\nthe S materialin its normal (non-superconducting) state.\nThe results are displayed for three values of Iand sev-\neral values of α. The results reﬂect and conﬁrm what we\nalready have found out from analyzing the triplet ampli-\ntudes. On the F side, the proximity eﬀect increases the\ncorrelationssomewhatwith α, asareductioninquasipar-\nticle states emerges for low energies due to increased cor-\nrelations as the relative magnetizations become increas-\ningly antiparallel. As Iincreases, the proximity eﬀects\nweaken. Indeed, at I= 1 the DOS is nearly ﬂat except\nat the larger values of α, suggesting that in the absence\nof a down spin band there may be a contribution from a\npossible triplet presence. On the superconducting side,\nthe situation is somewhat diﬀerent: the results never ap-\nproach a limit where the local DOS would look similar\nto that of a bulk superconductor. Even at I= 1, there is\nnever a gap and even when that situation is approached\nat largerα, when the magnetizations in the two F lay-\ners are antiparallel, the shape of the DOS curve does not\nresemble the signature bulk superconductor result. The\nprofound diﬀerence between the parallel ( α= 0) and an-\ntiparallel(α=π) casesis consistentin everyrespect with\nthat previously9found.\nThe above results were all obtained in the low tem-\nperature limit. In the remaining ﬁgures we consider the\nNormalized f0\n-0.02-0.010.000.010.020.030.040.050.06\n0\n0.225\n0.270\n0.300\n0.305\n0.306T/Tc\n-104 -102 -100 -98 -96-0.02-0.010.000.010.020.030.040.050.06\nY-200 -150 -100 -50 0Normalized f1\n-0.02-0.010.000.010.020.030.040.05\n0 \n0.225 \n0.270 \n0.300 \n0.305\n0.315\n-104 -102 -100 -98 -96-0.02-0.010.000.010.020.030.040.050.06\nT/Tc\nFIG. 9: Temperature and position dependence of f0andf1at\nI= 0.5 andτ= 4. In the top panel α= 0 and in the bottom\npanelα=π/2. The values of T/Tc, whereTcis the transition\ntemperature of bulk S, are indicated. The main plots show\na rather wide region near the interface (vertical dashed lin e)\nand the insets focus on the region very near the interface.\nFIG. 10: The maximum values of the real parts of the triplet\namplitudes at I= 0.5 andτ= 4, as a function of temperature\nand the same values of αas in Fig. 9. The inset shows the\ncorresponding peak values of the ordinary singlet amplitud e,\nf3≡∆(y)/g(Eq. (2.9)).11\ntemperature dependence. In Fig. 9 we plot directly the\nspatial behavior of f0andf1over a broad range of tem-\nperatures. As in previous ﬁgures for the triplet ampli-\ntudes, the main plot shows the behavior over a relatively\nextended region of the sample and the insets magnify the\nregionnearinterface. Inbothcaseswehavetaken I= 0.5\nandτ= 4 whileα= 0 forf0andα=π/2 forf1. It can\nbe inferred that at low T the temperature dependence\nof the triplet amplitudes is weak, while as Tincreases,\nthecloserspacingintemperaturesshownsuggestthatthe\ncorrelationsbecomedestroyedatamuchmorerapidrate.\nIn comparing the two panels, it is seen that for much of\nthe temperature range, |f0|>|f1|, but at higher tem-\nperatures (T/greaterorsimilar0.3Tc),f1becomes the larger of the two.\nThe temperature dependence of the characteristic pene-\ntration depths is weak. These observations are further\nexempliﬁed in Fig. 10, where the peak values of |f0|and\n|f1|(also atτ= 4) are shown as a function of T. These\nquantities are determined by calculating max {f0(y,T)}\nand max {f1(y,T)}throughout the structure for a given\ntemperature. The inset depicts the corresponding peak\nvalues of the ordinary self- consistent equal-time singlet\namplitudef3(y)≡∆(y)/g, (see Eq. (2.9)). For both the\ntriplet and singlet behavior, there is a strong dependence\nonTas the temperature approaches the system’s transi-\ntion temperature, which is about 0 .32Tcfor our system.\nTechnically, the determination of the self consistent am-\nplitudes is more diﬃcult at higher T, when the number\nof iterations is in principle much higher. This increase in\ncomputational time can be reduced by up to an order of\nmagnitude by taking as the initial spatial pair potential\nat a givenTthe result for the previously obtained next\nlower temperature times a Tdependent factor derived\nfrom the linearized Ginzburg-Landau theory32.\nIn accordancewith what wehavejust seen, it is anatu-\nral extension to study how thermal eﬀects might destroy\nthe spatial characteristicsof singlet correlationsthrough-\nout the structure. In Fig. 11, we therefore display f3as\na function of Y, for several temperatures, and in which\nthe relative magnetizations are collinear ( α= 0) and at\nright angles ( α=π/2). Remarkably, one can see that\nthe temperature dependence of the triplet components is\nsomewhatweakerthat thatofthestandardsingletampli-\ntude. One can also clearly see that, as indicated above,\nthe penetration of the triplet amplitudes into the F ma-\nterial over a length scale that is clearly much longer than\nthat of the singlet amplitude. This is again a strong in-\ndication that experimental tunneling results indicating\nlong ranged penetration eﬀects in F/S structures are in-\ndeed evidence for induced triplet correlations.\nIV. CONCLUSIONS\nIn this paper, we have presented a detailed study of\ninduced time dependent (odd in time) triplet pairing\ncorrelations in clean planar FSF junctions consisting of\nan ordinary s-wave superconductor sandwiched between0.00.10.20.30.4\n0\n0.105\n0.180\n0.225\n0.255\n0.278\n0.293\n0.3\n0.3047\n0.3064\n-106 -105 -104 -103 -102 -101-0.04-0.020.000.02\nY-200 -150 -100 -50 0Normalized f3\n0.00.10.20.30.4\n0\n0.105\n0.180\n0.225\n0.255\n0.278\n0.3\n0.3095\n0.3148\n-106 -105 -104 -103 -102 -101-0.06-0.04-0.020.000.02\nFIG. 11: Temperature and position dependence of the or-\ndinary singlet pair amplitude (or ∆( y)/g)), normalized to its\nvalue in bulk S material, for moderate magnetic strength, I=\n0.5. Top panel: the magnetizations are parallel ( α= 0). Bot-\ntom panel: the magnetizations are perpendicular ( α=π/2).\ntwo relatively thick ferromagnets whose magnetizations\nare misoriented with respect to each other by an angle\nα. Our microscopic formalism allowed us to investigate\ncases involving strong magnets, as well as atomic scale\nphenomenon, two things not possible in the widely used\nquasiclassical approaches. We have obtained results as a\nfunction of α, time, the strength Iof the ferromagnets,\nand the temperature. We have presented results for the\nspatialbehaviorofthe time-dependent triplet pairampli-\ntudes,f0andf1, and for the corresponding penetration\nlengths extracted from them. We have found that these\ntriplet correlations are indeed induced via the proxim-\nity eﬀect, that they completely pervade even a super-\nconductor several coherence lengths thick, and also sub-\nstantially penetrate the ferromagnetic layers. These re-\nsults haveclear implications for the experimental workin\nwhich long range proximity eﬀects in SF nanostructures\nhave been reported, eﬀects that have been speculated to\nbe due to the existence of some kind of triplet pairing.\nOur calculations, in which the time dependence is stud-\nied from the Heisenberg picture, emphasize the need for\nfull self-consistency of the solutions, without which we\nﬁnd that the Pauli principle is violated.\nWe have also considered the reverse proximity eﬀect,\nwhich is ofparticularinterest in this case due to the pres-\nenceoftwocomponentsofthemagnetization,andwealso\nhave given results for the experimentally measurable lo-12\ncal density of states, which revealed clear subgap energy\nsignatures as a function of αandI. We have studied\nthe temperature dependence of the triplet amplitudes (as\nwell as the ordinary singlet amplitude) and found that\nthe temperature dependence of the penetration depths\nassociated with these triplet amplitudes is weak: these\nlengths remain large all the way up to the vicinity of the\ntransition temperature of the system. 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B 70, 104516\n(2004)."    },    {        "title": "2202.04286v1.Ferromagnetism_induced_by_hybridization_of_Fe_3d_orbitals_with_ligand_InSb_bands_in_n_type_ferromagnetic_semiconductor__In_Fe_Sb.pdf",        "content": " 1 Ferromagnetism induced by hybridization of Fe 3d orbitals with ligand  \nInSb bands  in n-type ferromagnetic semiconductor (In,Fe)Sb  \n \nRyo Okano1, Tomoki Hotta1 Takahito Takeda1, Kohsei Araki1, \nKengo Takase1, Le Duc Anh1, Shoya Sakamoto2, Yukiharu Takeda3, Atsushi Fujimori4,5,  \nMasaaki Tanaka1,6, and Masaki Kobayashi1,6,* \n1Department of Electrical Engineering and Information Systems, The University of Tokyo, 7 -3-1 Hongo, \nBunkyo -ku, Tokyo 113 -8656, Japan  \n2The Institute for Solid State Physics, The University of Tokyo, 5 -1-5 Kashiwanoha, Kashiwa, Chiba 277 -\n8581, Japan  \n3Materials Sciences Research Center, Japan Atomic Energy Agency, Sayo -gun, Hyogo 679 -5148, Japan  \n4Department of Physics, The University of Tokyo, 7 -3-1 Hongo, Bunkyo -ku, Tokyo 113 -0033, Japan  \n5Department of Applied Physics, Waseda University, Okubo, Shinjuku, Tokyo 169 -8555, Japan  \n6Center for Spintronics Research Network, The University of Tokyo, 7 -3-1 Hongo, Bunkyo -ku, Tokyo 113 -\n8656, Japan  \n(Date: 8th Feb., 202 2) \n*Author to whom all correspondence should be addressed: masaki.kobayashi@ee.t.u -\ntokyo.ac.jp  \n \nABSTRACT  \nFe-doped III -V ferromagnetic semiconductor (FMS) (In,Fe)Sb is a promising material for \nspintronic device applications because of the n -type carrier conduction and the  \nferro magnetism with high Curie temperature ( TC > 300 K) . To clarify the mechanism of \nthe high-TC ferromagnetism, we have investigated the electronic structure and magnetic \npropert ies of an (In0.94,Fe0.06)Sb thin film by performing x-ray absorption spectroscopy  \n(XAS) and x -ray magnetic circular dichroism (XMCD) measurements  at the Fe L2,3 edges . \nThe magnetic -field (μ0H) dependence of the XMCD spectra reveal s that there are \nferromagnetic -like Fe and paramagnetic -like Fe components in the (In,Fe)Sb thin film.  2 The XAS and XMCD spectra of the ferrom agnetic -like and paramagnetic -like Fe \ncomponents resemble those of other Fe -doped FMSs and extrinsic oxides, respectively.  \nThe finite value of the ratio between the orbital and spin magnetic moments estimated by \napplying t he XMCD sum rules indicate s that the valence state of the Fe ions substituting \nfor the In sites in (In,Fe)Sb is not purely  ionic Fe3+, but intermediate between Fe3+ and \nFe2+. The qualitative correspondence between the 0H dependence of the visible -light \nmagnetic circular dichroism intensity  and that of  the XMCD intensity  demonstrates that  \nthe Zeeman splitting of the InSb band  is proportional to the net magnetization of the \ndoped Fe . These results suggest that  the ferromag netism of (In,Fe)Sb originates fro m the \nFe 3d orbital s hybridized with the host InSb bands.   \n \n \nI. INTRODUCTION  \nFerromagnetic  semiconduct ors (FMSs) exhibit both the properties of semiconduct ors \nand ferromagnet s simultaneously and thus are promising materials for spintronics  devices  \n[1-3], which exploits both the charge and spin degrees of freedom of electrons. In III -V \nFMSs, magnetic elements such as Mn and Fe partially  replace the group III sites. The Fe -\ndoped III -V FMSs, (In,Fe)As [4-6], (In,Fe)Sb [7,8] , (Ga,Fe)Sb [9-11] and (Al,Fe)Sb [12], \nhave been successfully grown by molecular beam epitaxy (MBE) in the last deca de. Since \nthe doped Fe ions substitute for group  III elements as Fe3+, it has been believed  that the \nFe ions  in a III-V semiconductor  matri x do not simply act as donors o r acceptors. In fact,  3 various carrier types of Fe -doped III -V FMSs have been realized by  co-doping or defect \ncontrol:  (In,Fe)As:Be [4] and (In,Fe)Sb [7] are n -type, (Ga,Fe)Sb [9] is p-type, and \n(Al,Fe)Sb [12] is insul ating. Moreover, (Ga,Fe)Sb and (In,Fe)S b show ferromagnetism \nwhose  Curie temperature ( TC) is higher than  room temperature. The highest TC of (Ga 1-\nx,Fex)Sb is about 400 K at Fe co ncentration x = 0.20 [13], and that of (In 1-x,Fex)Sb is about \n385K at x = 0.35 [14]. Therefore, Fe -doped III -V FMSs a re promising materials for the \nrealization of spintronics devices with pn junctions [15] operating at room temperature.  \nVarious studies of the physical properties of n -type (In,Fe)Sb have been carried out to \nclarify the origin of the fe rromagnetism [7,8,1 6]. X-ray diffraction and scanning \ntransmission electron microscopy measurements indicated that (In 1-x,Fex)Sb maintains the \nzinc-blende -type crystal structure up to at least x ≦ 0.16 [7]. Magnetic circular dichroism \n(MCD) in a visible -light range (1 – 5 eV) and anomalous Hall effect measurements have \nconfirmed intrinsic ferromagnetism in (In,Fe)Sb, in which ferromagnetic order appears \nin the zinc -blende semiconductor phase (there is no visible  evidence for secondary \nphases) [7]. A first -principles calc ulation [16] for (In,Fe)Sb has predicted that the \nisoelectronic Fe dopants induce antiferromagnetic interaction between the Fe ions \nthrough the super -exchange mechanism and the transition from the antiferromagnetic to \nferromagnetic states is induced by add itional carrier doping. This behavior can be well  4 understood in terms of the Alexander -Anderson -Moriya mechanism [17,18]. The \nelectrical control of ferromagnetism in (In,Fe)Sb by applying a gate voltage indicated that \nboth the electron -carrier -induced ferr omagnetic interaction and the super -exchange \nmechanism contribute to the emergence of the ferromagnetism in (In,Fe)Sb [8]. \nFor further understanding of the o rigin of magnetism in (In,Fe)Sb, it is necessary to \ncharacterize and reveal the electronic states o f the doped Fe ions related to the \nferromagnetism in detail. To address this issue, we investigate the relationship between \nthe local electronic states of the Fe io ns in (In,Fe)Sb and  the ferromagnetic behavior b y \nx-ray absorption spectroscopy (XAS) and x -ray magnetic circular dichroism (XMCD). \nSynchrotron radiation based XMCD is an element -specific magnetic probe  and a \npowerful tool to study the electronic state of the doped magnetic ions in FMSs [19-23]. \nThe experimental findings based on the XMCD measurements suggest that the \nferromagnetism in (In,Fe)S b is intrinsic and origin ates from the Fe 3 d orbitals hybridized \nwith the ligand bands of the host InSb.  \n \nII. EXPERIMENTAL  \nAn (In 0.94,Fe0.06)Sb thin film with a thickness of 15 nm was grown on a p -type \nGaAs(001) substrate by MBE. In order to avoid surface oxidation, the sample was  5 covered with a thin amorphous As capping la yer after the MBE growth of the \n(In0.94,Fe0.06)Sb layer. The sample str ucture is, from top to bottom, As capping layer ~1 \nnm/(In 0.94,Fe0.06)Sb 15 nm/AlSb 100 nm/AlAs 6 nm/GaAs:Be 100 nm  grown on a p+ \nGaAs  (001)  substrate. The TC of the sample was about 100 K estimated  by the Arrott plot \nof visible -light MCD  intensity - perpen dicular magnetic field ( μ0H) characteristics .  \nXAS and XMCD measurements  were performed at beamline BL23 -SU of SPring -8. \nThe measurements were conducted under an ultrahigh vacuum below 1.0×10-8 Pa at a \ntemperature ( T) of 10 K. Circularly polarized x -rays in the energy range of 690  – 740 eV \nnear the Fe L2,3 absorption edges were used for the measurements. The absorption spectra \nfor circularly polarized x -rays were obtained by reversing photon helicity at each photon \nenergy ( ℎ𝜈 or ℏ𝜔) and were taken in the  total electron yield mode. Here, XAS spectra  \ntaken with left and right circularly polarized x -rays are defin ed as 𝜇+  and 𝜇− , \nrespectively, and then the XAS spectrum 𝜇(𝜔) and the XMCD s pectrum ∆𝜇𝐻(𝜔) are \nrepresented as 𝜇(𝜔)=(𝜇++𝜇−)2⁄  and ∆𝜇(𝜔)=(𝜇+−𝜇−) , where ℏ𝜔  is the \nphoton energy . Magnetic fields were varied from -7 T to 7 T and applied parallel to the \nincident x -rays corresponding to the surface -normal direction. Th e sample was divided \ninto two pieces and one of them was etched by HCl to obtain the clean surface. We etched \nthe sample with HCl (2.4 mol/L) for 5 s to remove the capping layer and subsequently  6 rinsed it with water just before loading the sample in the vacuum chamber of the \nspectro mete r, followi ng the same procedure reported elsewhere [19].  \n \nIII. RESULTS AND DISCUSSION  \nFigure 1(a) shows XAS spectra of the as -grown and H Cl-etched (In 0.94,Fe0.06)Sb thin \nfilms at the Fe L2,3 absorption edges. The XAS spectrum of the as -grown sample shows \ntwo peaks at  hν ∼ 707.7 eV and ∼ 709.7 eV in t he Fe L3 edge, while the XAS spectrum \nof the HCl -etched sample shows a single peak at hν ∼ 707.7 eV . This spectral line -shape \ndifference indicates that there are two Fe components in the film: the component having \na peak at 709.7 eV , which disappears by the  HCl etching, and the component having a \npeak at 707.7 eV , which remains after the HC l etching.  \nFigure 1(b) shows the XMCD spectra of the as -grown film with varying μ0H. At the \nFe L3 edge, only the peak around 707.7 eV is observed in the XMCD spectrum taken with \nμ0H = 0.1 T, while the peak around 709.7 eV increases with increasing μ0H. This suggests \nthat there are mainly two kinds of magnetic components in the (In 0.94,Fe0.06)Sb thin film, \nand the components having the peaks at 707.7 eV and 709.7 eV a re ferromagnetic -like \n(FM-like) and paramagnetic -like (PM -like), respe ctively. The peak positi ons of the two \npeaks in the XMCD spectra are almost the same as those in the XAS spectra, as shown  7 by the vertical dashed lines in Fi g. 1. The XAS and XMCD peaks around 707.7 eV are \nalso observed in other Fe -doped FMSs , such as (Ga,Fe)Sb [19] and (In,Fe) As:Be [22], \nwhile the peak position a t ∼ 709.7 eV coincides with that of γ-Fe2O3 [24].  \nFigure 2 shows the μ0H dependence of the XMCD intensities (XMCD  - H curve) \nmeasured at hν = 707.7 eV and 709.7 eV . The XMCD  - H curve measured at 707.7 eV \nsteeply increases near the zero magnetic field (0 T < μ0H < 1 T) and gradually increases \nabove μ0H ~1 T. In contrast, the steep increase is almost absent for the XMCD  - H curve \nmeasured at 709.7 eV , but the linear gradual increase is dominant. The se results indicate \nthat the two components having peaks at 707.7 eV and 709.7 eV in the measured thin film \nare predominantly FM an d PM, respectively.  \nSince the μ0H dependence of  the X MCD intensit ies indicates  that there are two \ncomponents in (In,Fe)Sb, it is necessary to clarify the origin of each component. We \ndenote the FM -like and PM -like componen ts by α and β. The XMCD spectrum and the \nXAS spectrum were  decomposed into the two components  respectively  by the following \nprocedure: Firstly, assuming that the magnetization of the β component responds linearly \nto the magnetic field, the XMCD spectra of the α and β components  (∆𝜇𝛼 and ∆𝜇𝛽) were \nobtain ed by the following equations:  \n∆𝜇𝛽=7\n3(∆𝜇7T−∆𝜇4T), (1)  8 ∆𝜇𝛼=∆𝜇7T−∆𝜇𝛽. (2) \nHere, ∆𝜇nT  is the XMCD spectrum at μ0H = n T. Secondly, we conducted the \ndecomposition of the XAS spec tra. Since XA S spectral line shapes are usually insensitive \nto μ0H, the XAS spectra cannot be decomposed from the μ0H dependence as done for the \nXMCD spectra. Therefore, we have extracted those Fe components by comparing the \nXAS spectra before and after the HCl etching as in the previously reported XAS spectra \nof (Ga,Fe )Sb [19]. Since the peak position of 707.7 eV in the XAS spectrum after the HCl \netching corresponds to that of t he FM -like component in the XMCD spectra, it is likely \nthat the α component is predominant in the XAS spectrum after the HCl etching.  \nTherefore, we have adopted the XAS spectrum after the HCl etching as the XAS spectrum \nfor the α component ( μα). The  XAS spectrum for the β component (μβ) is then obtained \nby subtracting a fraction of μα spectrum from the XAS spectrum of the as -grown sample \nsuch that the α-component shoulder at 707.7 eV  disappears in the μβ spectrum.  \nFigure 3 shows the XAS and XMCD spectra decomposed into the α and β components  \nat μ0H = 7 T . The XAS spectrum of the α component shows a peak only at 707.7 eV in \nthe Fe L3 edge. The spectral line shapes of the XAS and XMCD spectra of the α \ncomponent resemble those o f other FMSs such as (Ga,Fe)Sb [19], (In,Fe)As [22], \n(Al,Fe)Sb [20]. This indicates that the electronic state of the α component is close to those  9 of the substitutional Fe ions in the other Fe -doped F MSs and that the α component is \nintrinsic to (In,Fe)Sb. On the other hand, the spectral line shape s of the XAS and XMCD \nspectra of the β component having peaks at 708 eV and 709.7 eV in the Fe L3 edge are \nsimilar to those of γ-Fe2O3 [24]. Since the HCl etching is considered to remove the layers \nnear the surface, the β component removed by the HCl etching is most likely an extrinsic \ncomponent such as surface oxides.  \nTo identify the electronic states of the α and β comp onents in more detail, w e have \nestimated the magnetic moments using the XMCD sum rules [25,26]. By appl ying the \nXMCD sum rules to the obtained XMCD spectra ∆𝜇(𝜔), the spin and orbital magnetic \nmoments of the doped Fe ions in units of μB/atom are estimated separately. The XMCD \nsum rules are as follows:  \n𝑀𝑜𝑟𝑏=−2𝑞\n3𝑟(10−𝑁𝑑), \n𝑀𝑠𝑝𝑖𝑛 ≈−3𝑝−2𝑞\n𝑟(10−𝑁𝑑). (3) \n(4) \nHere, 𝑝=∫∆𝜇(𝜔)𝑑𝜔𝐿3 , 𝑞=∫ ∆𝜇(𝜔)𝑑𝜔𝐿2,3 , 𝑟=∫ 𝜇(𝜔)𝑑𝜔𝐿2,3  and Nd is the number \nof 3d electrons. It should be noted here that the ratio between Morb and Mspin can be \nestimated from the XMCD spectra without the values of r and Nd. The estimated values \nof Morb/Mspin for the α and β compo nents at μ0H = 7 T and T = 10 K  are listed in Table I. \nThe Morb/Mspin value of the α component, which appears to be intrinsic to (In,F e)Sb, has  10 a finite positive value of 0.06 ± 0.01. This value is close to 0.065  ± 0.014 reported for \n(In0.95,Fe0.05)As:Be [21] and larger than  0.043  ± 0.001 for Fe (bcc) [27]. Therefore, the \nelectronic state of the substitutional Fe ions in (In,Fe)S b is sim ilar to that of (In,Fe)As:Be  \nbut is different from that of Fe (bcc). Since the orbital magnetic moment of an ionic Fe3+ \nion ( d5 high-spin state) is zero, the finite Morb/Mspin suggests that the valence of Fe in \n(In,Fe)Sb is different from the purely ionic Fe3+. The valenc e state of the doped Fe ions \nis likely an intermediate state between trivalent and divalent due to a charge transfer from \nthe ligand Sb to the Fe 3 d orbitals via the hybridization with the surrounding ligand Sb \nbands. Assuming that C 1|d5> + C 2|d6L> as the Fe 3 d state, which is a superposition of the \nstates d5 and d6L (L is the hole in the ligand), t he values of C 12 and C 22 for the α component \nare estimated to be 0.94 and 0.06, respectively, from Morb/Mspin. Therefore, the number of \nelectrons Nd is estimated to be 5.06. The values of Morb and Mspin estimated  by applying \nNd = 5.06 are shown in Table I. Here, the applied correction factor is 0.685 ( d5) [28]. Since \nthe Morb/Mspin value for the β component is almost zero, the β component probably \noriginates from trivalent Fe3+ (d5) oxides like Fe 2O3.  \nTo further clarify the magnetic behavior, the μ0H dependences of the magnetizations \n(M - H curves) are studied. Th e inset of Fig. 3(b) shows the M - H curves. Here, the \nmagnetizat ion values estimated fr om the XMCD sum rules are in units of μB per Fe atom.  11 While the magnetization of the β component is linearly proportional to μ0H, the \nmagnetization of the α component shows FM -like behavior with μ0Hs. It should be noted \nhere that the magnetization of the α component is not saturated even  at the highest μ0H, \nand the magnetization at 7 T of ~2.8 μB/Fe is smaller than the full moment of 5 μB/Fe for \nFe3+ (d5). This suggests that the magnetic behavior of the doped Fe ions in (In,Fe)Sb \ninvolves not only a FM component, but also a PM component. The details of the magnetic \ncomponents are discussed later.  \nFigure 4(a) shows comparison between the M - H curve of the intrinsic Fe component \n(α) and th e MCD  - H curve from  visible -light MCD  measurements . Since the MCD is the \ndifference between the intensities of the transition from the valence band to the \nconduction band with circularly polarized lights under an applied μ0H, the MCD signal \nreflects the Zeeman splitting of the host InSb bands. As shown in Fig. 4(a), th e MCD  - H \ncurve is almost identical to the M - H curve estimated from the XMCD spectra, indicating \nthat the magnetic behavior of the doped Fe ions in (In,Fe)Sb is proportional to the Zeeman \nsplitting of the InSb  bands. It should be mentioned here that the net magnetization of the \nintrinsic Fe component contributes to the magnitude of the Zeeman splitting, although the \nintrinsic Fe of (In,Fe)Sb contains not only a FM component but also a PM component. \nConsideri ng the finite Morb discussed above, the substitutional Fe ions in (In,Fe)Sb induce  12 the Zeeman splitting of the host InSb via the hybridization between the Fe 3 d orbital with \nthe ligand sp bands. Based on these findings, we conclude that the ferromagnetism in \n(In,Fe)Sb originates from the h ybridization between the Fe 3d orbitals and the valence or \nconduction bands of InSb, but not from oxidized Fe or precipitated Fe metal.  \nAs we noted above, the intrinsic component involves both the FM -like and PM -like \ncomponen ts. To further investigate this magnetization process of the intrinsic component, \nwe have analyzed the shape of the M - H curve. Note that the local electronic state of the \ndoped Fe ions is common for these FM -like and PM -like components within the \nexperimental accuracy. This suggests that th e different magnetic behavior of the doped \nFe ions comes from the inhomogeneous distribution of the Fe ions in (In,Fe)Sb [20,22]. \nWhile the PM component likely originat es from  isolated Fe ions  in regions with low  Fe \ndensity , regions with hig h Fe density may contribute to the FM and/or superparamagnetic \n(SPM) magnetic behavior depending on the size of the regions (Fe domain). In order to \ndetermine the ratio of the FM, SPM, and PM components, the M - H curve has been fitted \nusing the following functions :  \n𝑀=𝑠𝑚sat𝐿(𝜇FM𝜇0𝐻\n𝑘B𝑇)+𝑡𝑚sat𝐿(𝜇SPM𝜇0𝐻\n𝑘B𝑇)+(1−𝑠−𝑡)𝐶𝜇0𝐻\n𝑇+𝑇𝐴 . (5) \nHere, the FM and the SPM components are assumed to be represented by Langevin \nfunctions, and the PM component is represe nted by a linear function. 𝑠 (𝑡) is the ratio of  13 Fe atoms participating in the FM (SPM) , 𝑚sat is the total magnetic moment of the Fe \natom , 𝜇FM (𝜇SPM) is the magnitude of the magnetic moment per FM (SPM) region where \nthe magnetic moments are aligned, 𝐶 is the Curie constant, 𝑇𝐴 is the Weiss temperature, \nand kB is the Boltzmann constant. We have assumed that 𝑚sat=5 𝜇B, which is the total \nmagnetic moment of Fe3+. As shown in Fig. 4(b), the fitting well reprod uces the M - H \ncurve, suggesting that all the three magnetic components contribute to the intrinsic \nmagnetic behavior. Table II lists the fitted results of the parameters. The percentages of \nFM, SPM, and PM are about 27 %, 13 %, and 60 %, respectively. Note that w e have  also \ntried fitting with a single Langevin function and a linear function, but the M - H curve \nobtained by the expe riment is not reproduce d well.  𝑇𝐴 is about 28 K, which is close to \nthe Weiss temperature of PM (Ga 0.96,Fe0.04)As without carrier doping (32  K) [29]. This \nimplies that the magnetic interaction between the isolated PM Fe ion in (In,Fe)Sb is \nantiferromagnetic, consistent wit h previous theoretical calculations  [16]. The magnetic \nmoment per FM domain 𝜇FM  is deduced to be abo ut 840  μB – 1025  μB, which \ncorresponds to 168  – 205 Fe atoms i n each FM domain on average, and 𝜇SPM is about \n80 μB – 125 μB, which corresponds to 16  – 25 Fe atoms in each SPM domain on average. \nThe formation of Fe -rich domains is consistent with the attractive interaction of Fe atoms \nin (In,Fe)Sb at the second nearest neighbor site, as predicted by theoretical calculations  14 [16]. Although the majority of the Fe io ns seems isolated in (In,Fe)Sb and shows PM \nbehavior, a significant fraction of the Fe ions are located close to eac h other and exhibit \nthe FM or SPM behavior. As the Fe dopin g concentration is increased, the fraction of Fe \natoms involved in the FM region is expected to increase. This model is consistent with \nthe experimental results that TC increases with increasing Fe doping concentration [7,14] . \nTo elucidate the magnetic behavior of (In,Fe)S b in more detail, the  XMCD studies on \n(In,Fe)Sb with different Fe concentrations are desirable.  \n \nIV . CONCLUSION  \nIn conclusion, we have performed XMCD measurements on an (In 0.94,Fe0.06)Sb thin \nfilm to investigate the electronic states of the doped Fe ions related to the ferromagnetism. \nThe XAS and XMCD spectra taken at the Fe L2,3 edges have been decomposed into the \nsubstitutional Fe ions of (In,Fe)Sb and extrinsic Fe oxide s formed near the surface. The \nfinite ratio of the orbital to spin magnetic moments estimated by applying the XMCD \nsum rules indicates that the valence state of the substitutional Fe ions in (In,Fe)Sb is not \npurely ioni c Fe3+, but int ermediate between Fe3+ and Fe2+. The q ualitative correspondence \nbetwee n the visible -light MCD  - H and XMCD  - H curves demonstrates that the Zeeman \nsplitting of the InSb band is proportional to the net magnetization of the doped Fe atoms .  15 Based on these findings, we conclude that the ferromagnetism in (In,Fe)Sb originates \nfrom the Fe 3 d orbitals hybridized with the host InSb bands.  In addition, t he fitting result \nof the M - H curve suggests that the magnetism of (In,Fe )Sb consists of FM, PM, and \nSPM components, and that all of the magnetic components are derived from  the orbital \nhybridization of the Fe 3d orbitals  with the host InSb bands. These results are important \nfor understanding the physical properties of (In,Fe)S b, such as the mechanism of \nferromagnetism, and will be useful for applications such as spintronics devices using \n(In,Fe)Sb in the future.  \n \nACKNOWLEDGEMENT  \nThis work was supported by a G rants-in-Aid (Nos. 20H05650, 18H05345), CREST \n(No. JPMJCR1777), and PRESTO Programs (Grant No. JPMJPR19LB) of the Japan \nScience and Technology Agency . Thi s work was partially supported the Spintronics \nResearch Network of Japan (Spin -RNJ). This work was performed under the Shared Use \nProgram of Japan Atomic Energy Agency (J AEA) Facilities (Proposal No. 2018B -E23, \n2020A -E18 and 2021A -E24) supported by JAEA Advanced Characterization \nNanotechnology Platform as a program of  “Nanotechnology Platform ” of the Ministry of \nEducation, Culture, Sports, Science and Technology (MEXT) (Pr oposal No. A-18-AE- 16 0041 , JPMXP09A20AE0018  and JPMXP09A21AE0022 ). 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The red dashed line is the peak position at μ0H = 0.1 T near 70 7.7eV . \nThe blue dashed line is the energy of the XMCD peak position at μ0H = 7 T near 7 09.7 \neV . The inset shows an enlarged plot of XMCD spectra at Fe L3 absorption edge . \n 20  \nFIG. 2. Magnetic -field dependence of the XMCD intensity  (XMCD  - H curve) of the as-\ngrown (In 0.94,Fe0.06)Sb thin fi lm. T he red circle and blue rhombic  markers  are XMCD  - H \ncurves taken at  hν ∼ 707.7 eV  and hν ∼ 709.7 eV, respectively . \n  \n 21 TABLE I. Spin and orbital magnetic moments of the α and β in the (In0.94Fe0.06)Sb thin \nfilm at μ0H = 7 T and T = 10 K. Here, the nu mber of d electrons is assumed  to be 5 .06. \nThe correction factor for the Fe  3d ion (0.685) is employed  [28]. \n Morb/Mspin Morb (B/Fe) Mspin (B/Fe) \nα 0.06 ± 0.01 0.16 2.60 \nβ 0.016 ± 0.003  0.044 2.75 \n \n   22   \nFIG. 3. Decompos ition analysis for XAS and XMCD spectra  of the as -grown \n(In0.94,Fe0.06)Sb thin film  at Fe L2,3 edges. (a) XAS s pectra  and (b) XMCD spectra at μ0H \n= 7 T. The black line is the raw XMCD spectrum. The inset is the magnetic field \ndependence of the magnetization ( M - H curve). Based on the XMCD sum rules  [25,26], \nthe vertical axis was con verted from the XMCD intensity to the magn etization per Fe \natom. The circle  and rhombic  markers are the α and β component s, respectively .  \n 23  \nFIG. 4. M - H curves of the doped Fe ions in (In,Fe)Sb . (a) M - H curve of the α component \nand visible -light MC D - H curve of the (In 0.94,Fe0.06)Sb thin films. (b) M - H curve of the \nα component  and the fitting result by Langevin functions and liner function. Dotted, dash -\ndotted, and  dashed lines are the FM, SPM and PM component s, respectively. Solid line \nis the fitting result corresponding to the sum of all the component s.  \n 24 TABLE II. Fitting parameters using Eq.  (5) for the M - H curve.  \n𝑠 𝜇FM 𝑡 𝜇SPM 𝑇𝐴 \n0.272 ± 0.017  933 ± 92  0.129  ± 0.016 102 ± 22 27.7 ± 1.7 \n "    },    {        "title": "2310.05514v1.The_influence_of_metallic_overlayers_on_ferromagnetism_in_LaMnO__3_.pdf",        "content": "The influence of metallic overlayers on ferromagnetism in LaMnO 3\nBart Folkers,1,∗Thies Jansen,1,∗Thijs J. Roskamp,1,∗Pim Reith,1Andr ´e Timmermans,1Daen\nJannis,2Nicolas Gauquelin,2, 3Johan Verbeeck,2Hans Hilgenkamp,1,†and Carlos M. M. Ros ´ario1, 4\n1MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands\n2Electron Microscopy for Materials Research (EMAT),\nDepartment of Physics, University of Antwerp, BE-2020 Antwerpen, Belgium\n3NANOlab Center of Excellence, University of Antwerp, BE-2020 Antwerpen, Belgium\n4INL - International Iberian Nanotechnology Laboratory, 4715-330 Braga, Portugal\n(Dated: October 10, 2023)\nLaMnO 3(LMO) thin films epitaxially grown on SrTiO 3(STO) usually exhibit ferromagnetism above a crit-\nical layer thickness. We report the use of scanning SQUID microscopy (SSM) to study the suppression of the\nferromagnetism in STO/LMO/metal structures. By partially covering the LMO surface with a metallic layer,\nboth covered and uncovered LMO regions can be studied simultaneously. While Au does not significantly in-\nfluence the ferromagnetic order of the underlying LMO film, a thin Ti layer induces a strong suppression of the\nferromagnetism, over tens of nanometers, and a large change in the out-of-plane lattice parameter. We relate the\nsuppression of the ferromagnetism to the scavenging of oxygen and diffusion of Ti approximately 5 nanometers\ndeep into the film, which takes place at timescales of days. Furthermore, we demonstrate that by patterning\nTi/Au overlayers, we can define ferromagnetic structures down to sub-micrometer scales.\nComplex oxide thin films exhibit remarkable versatility,\nmaking them suitable for many potential applications in elec-\ntronics, spintronics and catalysis, among other fields [1–4].\nTheir tunability allows for precise control and manipulation\nof their properties, for example by the background pressure\nduring growth [5–7], the incorporation of the oxide in a het-\nerostructure [8–11], the modulation of strain [12, 13], dop-\ning, or selecting the thickness [14, 15]. LaMnO 3(LMO) is a\ngood example of such a complex oxide with versatile proper-\nties, in particular its magnetic properties. LMO is an antifer-\nromagnetic insulator in the bulk, but becomes ferromagnetic\nwhen grown epitaxially on SrTiO 3(STO) under the right oxy-\ngen background conditions [6]. However, the ferromagnetism\nonly arises above a critical thickness of 6 unit cells [15, 16].\nFor a thin film of a complex oxide, such as LMO, to be in-\ncorporated into an electronic device, it is imperative to contact\nit with conventional metals. When the electronic properties\nare studied, it is common to contact the complex oxides with\na Ti/Au layer, where Ti serves as an adhesion layer [17, 18].\nHowever, it is widely known that Ti has a high oxygen affinity,\neasily oxidizing in contact with oxygen from the environment\n[17, 19] or with oxygen-rich materials, in the right thermo-\ndynamic conditions [20, 21]. In spite of the importance, a\nsystematic study about the interface between metals and com-\nplex oxides lacks. Which is quite surprising, considering that\none may expect significant changes in oxide properties due to\ntheir tuneability and the chemical reactivity of elements like\nTi, for instance.\nIn this work, we investigate the influence of the metal-\nlic overlayers Ti and Au on the magnetic, structural and\nchemical properties of LMO thin films grown on STO sub-\nstrates. The technique of scanning SQUID microscopy (SSM)\nis used, as it is a powerful tool to image ferromagnetism with\na micrometer-scale resolution [22, 23]. We find that the Ti\nlayer suppresses the ferromagnetism in LMO and demonstrate\na clear time dependency. Moreover, we report a significantchange in the out-of-plane lattice parameters and in the sto-\nichiometry for several nanometers deep at the interface. Fi-\nnally, we show that by patterning the Ti layer we can lo-\ncally modulate the magnetic properties and observe dipolar-\nlike magnetic signals in sub-micrometer ferromagnetic struc-\ntures.\nFor the experiments, metal layers of Ti and Au were sput-\ntered in an inert Ar atmosphere on top of pulsed-laser de-\nposited LMO on SrTiO 3substrates. Details on the sample\nfabrication are provided in the supplemental materials [24].\nIn order to have a clear measurement of the influence of the\nmetallic overlayers on LMO, Ti and Au were deposited by\nmeans of photolithography and lift-off processing, and cover\nonly one half of the LMO film. In this way it is possible to\nmeasure both covered and uncovered regions in the same SSM\nscan. The Au layer impedes the oxidation of the Ti film upon\natmospheric exposure. Fig. 1(a) shows a scan over the border\nregion, where in the right side of the picture, a 20 unit cell\n(u.c.) thick LMO film (approximately 7.9 nm) is covered by\n4 nm of Ti and 60 nm of Au. On the left (uncovered) side, the\nmagnetic field exhibits a pattern in line with previous SSM ex-\nperiments on ferromagnetic LMO thin films performed in our\ngroup [15, 23]. It is presumed that the magnetism in the LMO\nis predominantly in-plane, with stray out-of-plane magnetic\nfields arising at domains walls. However, on the right side the\nsignal is heavily suppressed, showing the effect of covering\nthe LMO with the metal layers. Additional scans were per-\nformed entirely on the right side of the sample to show that no\nsignal has actually been measured above the noise level of the\nSQUID. The SSM setup [15, 22, 25] has a sensitivity in the\norder of 10µΦ0Hz−1/2with a bandwidth of 1000 Hz, where\nΦ0= 2×10−15T m2is the magnetic flux quantum. For more\ndetails on the setup and SSM measurements see supplemental\nmaterial Sec. S3 [24].\nA control sample with no Ti layer, where only Au was sput-\ntered using the exact same processing conditions as the Ti/AuarXiv:2310.05514v1  [cond-mat.mtrl-sci]  9 Oct 20232\nsamples, showed no significant decrease in the measured mag-\nnetic signal strength with respect to the uncovered LMO film,\nsee Fig. S3(a). Therefore, it is concluded that the presence of\nthe overlaying Au layer does not significantly affect the ferro-\nmagnetism in LMO, nor should the sputtering process itself be\nresponsible for the suppression observed. On the other hand,\na sample with only a Ti layer did show a significant suppres-\nsion, see Fig. S3(b), so the Ti layer does play a dominant role\nin the effect.\n(a)\nSQUID Signal ( μT)18 -18 9 -9 050 μm50 u.c. LaMnO3tTi = 4 nm\ntTi = 24 nm(b)\nSQUID Signal (μT)\n-5.0 5.0 0.0 -2.5 2.550 μm\nLaMnO320 u.c.\nTi 4 nm\nAu60 nm\nTime (days)Ratio (%)304050\n20\n10\n06070\n0 4 8 12 16\nTime (days)\n(c)\nFIG. 1. Scanning SQUID microscopy of LMO thin films partially\ncovered by Ti and Au layers showing the suppression of the ferro-\nmagnetism on the Ti-covered side. (a) SSM scan of a 20 u.c. LMO\nthin film partially covered by Ti(4 nm)/Au(60 nm) layers, as illus-\ntrated by the stack on top of the graph. The shaded red area is an\nindication of the scan area during the measurement. (b) Panels of\nmultiple SSM scans performed on the same 50 u.c. LMO thin film\npartially covered by Ti(4 nm)/Au(60 nm) over a time period of sev-\neral days (0.2, 1, 3, 8 and 13 days from top to bottom respectively).\n(c) Ratio of the SQUID voltages on the covered- and uncovered sides\nas a function of time after deposition of the metallic layers for two\n50 u.c. LMO samples with different thicknesses of Ti: 4 and 24 nm.\nThe ratio between the thickness of the LMO and Ti layers\nis important for the level of suppression of the ferromagnetic\nbehavior. Experiments with a thicker LMO layer of 50 u.c.\nshow that it is still possible to measure a significant stray mag-\nnetic field originating from the Ti-covered LMO for the same\nthickness of the Ti layer of 4 nm, although the suppression\neffect remains very clear. Again, a Au capping layer is used\nto impede oxidation of the Ti from the ambient environment.\nFig. 1(b) shows a panel comprised of several scans on a sam-\nple of 50 u.c. thick LMO partially covered with 4 nm of Ti,performed at different moments in time after the deposition\nof the metal layers. Between the measurements the sample\nwas stored at room temperature in a nitrogen flushed desic-\ncator. Again, the left side of the scans shows the uncovered\nregion, while the right side shows the Ti/Au-covered film. To\nenable a quantitative comparison of the ferromagnetic signal\nin both regions of the scans, the variance (root mean square\nvalue) of the SQUID voltage signal ( V) in a defined area on\neach side of the border was calculated. The SQUID voltage is\nlinearly dependent on the magnetic field as measured by the\nSQUID. This quantity can be used to estimate the variance of\nthe strength of the measured magnetic field and can be used\nto compare between samples and when scanning parameters\nare changed [23]. Then, for each scan, a ratio Ris defined\nbyR=VRMS,uncov. /VRMS,Ti . The panels in Fig. 1(b) show that\nthe suppression of the ferromagnetism in the Ti-covered LMO\nfilm increases with time in a timescale of several days. After\n13 days, Ris reduced to approximately 12 % compared to the\nuncovered region. It is also noteworthy that there must be at\nleast two different rates of the decrease of R, because there\nis an abrupt decrease immediately after the deposition of the\nTi layer, as after a few hours the ratio is already down to ap-\nproximately 64 %. Previous experimental studies dedicated to\nthe oxidation of Ti thin films show that the oxidation process\ncan be divided into two stages: a fast initial oxygen absorp-\ntion step, taking place in a few minutes, followed by a slower\nprocess that can take hours to days [26, 27]. The occurrence\nof these different oxidation regimes is dependent on the Ti\nfilm thickness [28]. Our experiments corroborate this. Thus,\nthis leads us to believe that after depositing Ti/Au, oxygen va-\ncancies are created rapidly, which is sufficient to completely\nsuppress the magnetism in 20 u.c. thick films but not in the 50\nu.c. films. Here diffusion of the oxygen vacancies takes place\non a longer time scale after the initial deposition, suppress-\ning the ferromagnetic order further. Data on the diffusion of\noxygen species in LMO is scarce, but recently a study gave\nindications for a high mobility of oxygen vacancies in LMO\nat moderate temperatures [29].\nTo study the dependence of the ferromagnetism suppres-\nsion on the thickness of the Ti layer, several samples were\nfabricated while keeping the thickness of LMO to 50 u.c. and\nchoosing different values of the Ti layer thickness. Fig. 1(c)\nshows the ratio Ras a function of time for two different thick-\nness values of the Ti layer: 4 and 24 nm. The comparison of\nthe time evolution of the ratio for these two samples shows\nthat for the same time period since the deposition of Ti, the\nlevel of suppression is different, i.e.the Ti thickness seems to\ndetermine the rate of suppression of the ferromagnetism and\npossibly the final level of suppression.\nTheoretically, the STO substrate induces the correct amount\nof strain in the LMO film to stabilize the ferromagnetic ground\nstate via strain-induced orbital ordering [30]. In this model\nthe two main prerequisites for ferromagnetism are: 1) crys-\ntallinity and 2) good stoichiometry, which is strongly tied to\nthe oxygen content and Mn valency. This is also reflected by\nstudies showing the dependence of the magnetic order on the3\nFIG. 2. Structural characterization of the Ti/Au - LMO interface. (a) HAADF image of the STO/LMO/Ti/Au structure. (b) Corresponding\nEELS linecut, showing the presence of the Mn2+and Mn3+valence states. (c) Corresponding EELS linecut, showing the presence of O, Mn\nand Ti. (d) 2θ−ωscan of sample covered with and without Ti/Au and Experimental and simulated 2θ−ωscan of LMO covered with Ti/Au.\n(e) Extracted clattice parameter from the HAADF image as function of sample depth (blue circles). The caxis lattice parameter as function\nof sample depth used in the model shown in (d) (red solid line).\noxygen background pressure during growth [6, 31, 32]. To\nget further insight into the possible mechanisms of the sup-\npression, these two prerequisites are investigated by means\nof scanning transmission electron microscopy (STEM), elec-\ntron energy loss spectroscopy (EELS) and X-ray diffraction\n(XRD).\nFig. 2(a) shows a high-angle annular dark field (HAADF)\nSTEM image of the STO/LMO/Ti/Au structure, where the\nLMO layer is approximately 50 u.c. thick, and the Ti and\nAu layers have a thickness of 12 and 60 nm, respectively. The\nimage shows that the LMO film is grown epitaxially and crys-\ntalline on STO without any noticeable defects. Fig. 2(b) and\n(c) show a line cut of the corresponding EELS signal for the\npresence of Mn2+and Mn3+valency and the elements Mn,\nTi and O as function of sample depth. The interface between\nthe Ti and LMO is indicated with a black dashed line, which\nis estimated to be at half of the maximum Ti signal. Far above\nthe interface there is mainly pure Ti present, but in a region of\napproximately 3 nm above the interface oxygen starts to be-\ncome more present. This is accompanied by an evolution of\nthe Ti valence state from Ti0via Ti1+/Ti2+to Ti3+and finally\nto Ti4+below the interface, see S5 for more details. In the re-\ngion of 3 nm above the interface Mn is present with a valency\nof Mn2+. At the interface the presence of Ti drops rapidly,\naccompanied by a sudden increase of Mn. Interestingly, the\noxygen content also drops for over a distance of 2 nm, after\nwhich it saturates. In the same region the Mn is predominately\nMn2+, which reflects the presence of oxygen vacancies. After\nthe steep drop of the Ti signal, it decays slowly over a length\nof 5 nm to eventually vanish. Simultaneously, the valency ofthe Mn becomes completely Mn3+. This shows that the inter-\nface region extends over approximately 8 nm and consist of\nMn, Ti and O, with different valence states present. Focusing\non the LMO, it is clear that for approximately 5 nm into the\nfilm from the interface the LMO is not stoichiometric due to a\ncombination of oxygen scavenging and Ti interdiffusion.\nSuch a change of the LMO film on these length scales\nshould also be observable in XRD. Fig. 2(d) shows 2 θ−ω\nscans on a sample with and without a 24 nm-thick Ti layer\n(capped with Au). A clear STO 002 peak is visible in both\nscans accompanied by a LMO 002 peak with the correspond-\ning Laue fringes. There is indeed a significant difference in\nthe LMO 002 peak position between the Ti-covered and un-\ncovered samples. This indicates a (local) change of the out-\nof-plane lattice parameter cin the LMO film, in line with pre-\nvious reports where the amount of oxygen vacancies in LMO\nwas controlled by a difference in oxygen pressure during the\ngrowth [31, 32]. To quantify this change, the XRD pattern of\na LMO film on a STO substrate was simulated, allowing for\nboth the thickness of the LMO layer and the lattice parameter\ncof the LMO to vary as function of film depth. This was per-\nformed with interactive XRD fitting [33]. Fig. 2(d) shows the\nexperimental XRD data of the Ti/Au covered area together\nwith the XRD simulated pattern where cvaries as function\nof the sample depth with an exponential decay towards the\nSTO/LMO interface. The red line in Fig. 2(e) shows the vari-\nation of the lattice parameter that is used in the simulation.\nThe best fit was obtained using an exponential dependence of\nthe lattice parameter over a thickness of 45 LMO unit cells.\nOther models were also considered, where the lattice param-4\netercis constant as function of sample depth or it has a step\nfunction-like behaviour, but these simulations deviated more\nfrom the experimental data. Fig. 2(e) also shows the extracted\ndifference between the lattice planes dyas a function of sam-\nple depth from the HAADF image shown in Fig. 2(a). Here\na sample depth of 0 corresponds to the STO/LMO interface.\nThe STO substrate and the LMO film can be distinguished by\ntheir different lattice parameters. A small upturn is observed\ntowards the LMO/Ti interface, which suggest an increase of\nthe out of plane lattice parameter towards the LMO/Ti inter-\nface.\nHere, the STEM image shows that the LMO film is still\nfully crystalline, but the addition of a Ti scavenging layer\nleads to the active removal of the oxygen from the LMO layer,\nseveral nanometers in depth. This is shown by the presence of\nTi4+(indicating the presence of Titanium on the Mn site in the\nLaMnO 3perovskite structure) and Mn2+at the interface. The\noxygen deficiency in LMO also leads to significant elongation\nof the lattice as observed by XRD, with an increase of the lat-\ntice parameter c. The simulation in Fig. 2(e) further shows\nthat this elongation occurs mostly at the interface, where the\nhighest concentration of oxygen vacancies is expected. This\nall suggests that the change in stoichiometry is likely the cause\nof the ferromagnetic suppression.\nThe suppression of the ferromagnetism in LMO achieved\nby the coverage with a Ti layer can be exploited to selectively\npattern ferromagnetic structures in LMO thin films. To show-\ncase this possibility, micrometer-sized areas of the LMO thin\nfilms were left uncovered by the Ti/Au bilayer, by means of\nlithography and lift-off. Four examples of such patterns can be\nseen in Fig. 3. Areas where the SSM measures a distinct mag-\nnetic signal correspond to uncovered patches of the 20 u.c.\nLMO film, while in the regions covered by the metal layers\nthe ferromagnetic signal is significantly suppressed.\nFig. 3(a) shows the SQUID signal of a 200×200µm2\nsquare. The magnitude of the measured SQUID signal is sim-\nilar to that of an uncovered film, see for example Fig. 1(a). So,\nfor patterns of this size, the size restrictions have no notable\ninfluence on the magnetism of the LMO underneath. Fig. 3(b)\nshows that as the size of the structure decreases, the measured\nSQUID signal is reduced. The smaller SQUID signal can be\nexplained by the fact that the SQUID resolution limits the sig-\nnal and that by decreasing the structure size there are less fer-\nromagnetic domains under the sensor area that contribute to\nthe measured flux. The difference in the magnitude between\nthe SQUID signal in Figs. 3(a) and (b) is due to a different\nSQUID chip used in the measurements and a varying angle,\nfor a more comprehensive explanation see the supplemental\nmaterial [24].\nInterestingly, Fig. 3(b) also reveals that as the structures get\nsmaller, from squares with a side of 25µmto1µm, the mag-\nnetic behaviour changes and becomes reminiscent of magnetic\nmultipoles and eventually dipoles. This can also be seen in\nFig. 3(c), where only squares with sides of 5µmare imaged.\nThis is the expected behaviour of a single ferromagnetic do-\nmain with in-plane magnetization [23]. To lift the geometric\n(d)\n50 μmSQUID Signal (μ T)\n0 -5 5(c)\n50 μmSQUID Signal (μ T)\n0 -1 1(b)\n50 μmSQUID Signal (μ T)\n0 -20 20(a)\n100 μmSQUID Signal (μ T)\n0 -6 6FIG. 3. Scanning SQUID microscopy images of the patterned mi-\ncrostructures on 20 u.c. LMO by selectively covering with Ti and Au\nlayers. In the top left of each subfigure an illustration is shown about\nthe structures measured. (a) A 200×200µm2uncovered square.\n(b) Uncovered squares with descending size from top left to bottom\nright with a long side of respectively 25,20,15,10,5,2.5, and1µm.\n(c) Array of 16 uncovered 5×5µm2squares. (d) 23 bar structures\ndisposed on a circumference, each with size 10×0.5µm2.\nsymmetry of the squares, bars of 10×0.5µmwere created\nand positioned radially inward on a circumference, as shown\nin Fig. 3(d). Here, the dipole-like field distribution almost per-\nfectly aligns radially, evidencing a favoured orientation of the\nmagnetic moments due to the lateral constriction of the bars.\nThese results raise the question whether the small patches\nof LMO are uniformly polarized and represent real dipoles,\nor that the magnetic structure in the LMO is more complex\nand the magnetic field at the height of the pickup loop only\nemerges as that of a dipole. Domain sizes in LMO have\nbeen reported to be in the order of 200 nm [34, 35], which\nis an order smaller than our patterns. Our pickup loop scans\nat a height of approximately 3µm, because this is an order\nof magnitude larger than the expected size of the domains,\nwe don’t expect to distinguish domains of these sizes indi-\nvidually. Finally, a uniform magnetically polarized patch of\n5×5µm2LMO would result in a magnetic field, at the pickup\nloop height, several orders of magnitude higher than we mea-\nsure. So, although our small ferromagnetic patterns emerge\nas dipoles, we don’t think that they are uniformly polarized.\nRather we expect these LMO patches to posses more complex\nmagnetic structures at length scales we can not resolve with\nour current experimental setup.5\nIn summary, scanning SQUID microscopy studies of LMO\nthin films partially covered by a thin Ti layer show the sup-\npression of the ferromagnetism in LMO. For a combination of\n20 u.c. of LMO and 4 nm of Ti, there is no stray magnetic\nfield originating from the LMO film underneath. For thicker\nLMO films, the same thickness of Ti suppresses only partially\nthe ferromagnetic phase in LMO and the rate of suppression\nis on the order of several days. STEM, EELS and XRD re-\nveal the creation of oxygen vacancies in the LMO layer due\nto oxygen scavenging from the Ti layer and Ti interdiffusion\nfor approximately 5 nm deep into the film. This process first\nhappens rapidly, resulting in immediate suppression in the 20\nu.c. thick LMO samples. Further diffusion of oxygen va-\ncancies deeper in the LMO films happens on timescales of\ndays, resulting in a further suppression of ferromagnetism af-\nter initial Ti/Au deposition in 50 u.c. thick LMO samples.\nThis suppression effect can be used to pattern the ferromag-\nnetism in LMO thin films, as showcased by ferromagnetic\n(sub-)microstructures that were successfully fabricated in this\nway. Small structures show dipolar-like magnetic signatures,\nwhich raises questions about the effective domain size of the\nLMO and provides motivation for further research to study\nthese structures with SQUID microscopes having higher spa-\ntial resolution.\nThe authors acknowledge support from the project ”TOP-\nCORE” (project number OCENW.GROOT.2019.048) which\nis financed by the Dutch Research Council (NWO). The au-\nthors acknowledge the research program “Materials for the\nQuantum Age” (QuMat) for financial support. This program\n(registration number 024.005.006) is part of the Gravitation\nprogram financed by the Dutch Ministry of Education, Culture\nand Science (OCW). JV acknowledges The eBEAM project\nwhich is supported by the European Union’s Horizon 2020\nresearch and innovation programme FETPROACT-EIC-07-\n2020: emerging paradigms and communities. 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Hilgenkamp, Review of\nScientific Instruments 88, 123706 (2017).\n[24] See Supplemental Material at insert link for details on the sam-\nple fabrication, structural characterization of LaMnO 3, scan-\nning SQUID microscopy setup and technique, extra SSM scans\non covered LaMnO 3, and the Ti valence state evolution.\n[25] J. R. Kirtley, M. B. Ketchen, K. G. Stawiasz, J. Z. Sun, W. J.\nGallagher, S. H. Blanton, and S. J. Wind, Applied Physics Let-\nters66, 1138 (1995).\n[26] M. Martin, W. Mader, and E. Fromm, Thin Solid Films 250, 61\n(1994).\n[27] M. C. Passeggi, L. I. Vergara, S. M. Mendoza, and J. Ferr ´on,\nSurface Science 507, 825 (2002).6\n[28] I. Vaquila, M. Passeggi, and J. Ferr ´on, Physical Review B -\nCondensed Matter and Materials Physics 55, 13925 (1997).\n[29] R. Rodriguez-Lamas, C. Pirovano, A. Stangl, D. Pla,\nR. J´onsson, L. Rapenne, E. Sarigiannidou, N. Nuns, H. Rous-\nsel, O. Chaix-Pluchery, M. Boudard, C. Jim ´enez, R.-N. Vannier,\nand M. Burriel, J. Mater. Chem. A 9, 12721 (2021).\n[30] Y . S. Hou, H. J. Xiang, and X. G. Gong, Physical Review B 89,\n064415 (2014).\n[31] J. Roqueta, A. Pomar, L. Balcells, C. Frontera, S. Valencia,\nR. Abrudan, B. Bozzo, Z. Konstantinovi ´c, J. Santiso, andB. Mart ´ınez, Crystal Growth and Design 15, 5332 (2015).\n[32] Y . K. Liu, H. F. Wong, K. K. Lam, C. L. Mak, and C. W. Leung,\nJournal of Magnetism and Magnetic Materials 481, 85 (2019).\n[33] C. Lichtensteiger, Journal of Applied Crystallography 51, 1745\n(2018).\n[34] Y . Anahory, L. Embon, C. J. Li, S. Banerjee, A. Meltzer, H. R.\nNaren, A. Yakovenko, J. Cuppens, Y . Myasoedov, M. L. Rap-\npaport, et al. , Nature Communications 7, 12566 (2016).\n[35] T. Gao, S. Cao, Y . Liu, Y . Zhang, and J. Zhang, Rare Metals\n30, 359 (2011).Supplementary material: The influence of metallic overlayers on ferromagnetism in LaMnO 3\nBart Folkers,1,∗Thies Jansen,1,∗Thijs J. Roskamp,1,∗Pim Reith,1Andr ´e Timmermans,1Daen\nJannis,2Nicolas Gauquelin,2, 3Johan Verbeeck,2Hans Hilgenkamp,1,†and Carlos M. M. Ros ´ario1, 4\n1MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands\n2Electron Microscopy for Materials Research (EMAT),\nDepartment of Physics, University of Antwerp, BE-2020 Antwerpen, Belgium\n3NANOlab Center of Excellence, University of Antwerp, BE-2020 Antwerpen, Belgium\n4INL - International Iberian Nanotechnology Laboratory, 4715-330 Braga, Portugal\n(Dated: October 10, 2023)\n∗These authors contributed equally to this work\n†j.w.m.hilgenkamp@utwente.nlarXiv:2310.05514v1  [cond-mat.mtrl-sci]  9 Oct 20232\nS1. SAMPLE FABRICATION\nThe LaMnO 3samples were fabricated using pulsed laser deposition in a vacuum system with a base pressure of 10−8mbar on\nTiO 2-terminated SrTiO 3(001) substrates with dimensions of 5.0×5.0×5.0 mm . The deposition was performed at a substrate\ntemperature of 750◦Cand in an oxygen background pressure of 10−2mbar . The laser fluence used was 1.8 J cm−2with a\nrepetition rate of 1 Hz. Reflection high energy electron diffraction (RHEED) is used to monitor the thickness during the growth.\nOscillations of the RHEED spots are observed during deposition, indicating layer by layer growth which can be seen in Fig. S1.\nAfter deposition, the samples are cooled down to room temperature under the deposition pressure.\nThe thin films of Ti and Au are deposited at room temperature on top of the LaMnO 3using RF sputtering (vacuum base\npressure 10−7mbar ) in an inert argon gas. Ti and Au were sputtered with a sputter power of 150 W, under a deposition pressure\nof2×10−2mbar . To pattern the Ti/Au film, either photo- or electron-beam lithography was used in combination with standard\nlift-off in acetone to create sub-micrometer structures.\nS2. STRUCTURAL CHARACTERIZATION OF BARE AND COVERED LaMnO 3FILMS\nA. RHEED\nFig. S1(a) shows the RHEED intensity of the first order diffraction spot monitored during the deposition of LaMnO 3on a\nSrTiO 3(001) substrate. Clear oscillations are visible, indicating layer by layer growth. This enables us to control the thickness\nof the LaMnO 3films with unit cell precision. Fig. S1(b) shows the RHEED diffraction pattern after the deposition of 20 unit\ncells of LaMnO 3on SrTiO 3. The clear diffraction peaks indicate a high crystalline film.\nFIG. S1. Characterization of LaMnO 3film by RHEED during the deposition. (a) shows the intensity of the first order diffraction peak\nmonitored during the growth, the oscillations indicate layer-by-layer growth. (b) RHEED diffraction pattern after the deposition of 20 unit\ncells of LaMnO 3on a SrTiO 3substrate.\nB. XRD\nThe XRD data are taken using the Bruker D8 Discover X-ray diffractometer with a monochromated Cu K α1source. Simula-\ntions of the XRD data are performed with InteractiveXRDFit [1].\nC. STEM and EELS\nTransmission electron microscopy studies were performed at room temperature using the QuAntEM microscope, which is an\naberration-corrected scanning transmission electron microscope (STEM), FEI Titan operated at an acceleration voltage of 3003\nkV , equipped with a high-brightness field-emission electron source (X-FEG) and a monochromator giving a resolution of less\nthan 150 meV with less than 1 ˚A spatial resolution. Cross-sectional cuts of the sample were prepared using a dual-beam focused\nion beam (FIB) instrument. Electron energy-loss spectroscopy (EELS) was performed in STEM on a Gatan K2 direct electron\ncamera, the collection semi-angle was set to 61 mrad. Spectrum images (SI) were acquired with a beam current of 80 pA and an\nacquisition time of 200 ms/pixel and a pixel size of 0.1 ˚A with a dispersion of 0.1 eV/pixel.\nMulti linear least squares fitting is used to determine the valence state of Manganese where reference standards for the 2+ and\n3+ EEL spectra. The elemental distribution of Ti, O and Mn is determined via a model based fitting procedure. The HAADF\nsignal is used as a marker for the heavy atom La compositional profile since this signal could not be acquired simultaneously\ndue to the limiting field of view in energy.\nTo visualize the change in fine structure of Titanium, a local peak fitting routine is applied on each Ti L edge where the\nposition and amplitude of the peak is stored. This shows the splitting and abundance of the Titanium signal as function of probe\nposition.\nAtomic positions based on the STEM image were obtained with STATSTem [2]. Subsequently the out-of-plane coordinate\nwas obtained by averaging over all atom positions in the same plane. The lattice parameter, i.e. dy, was then obtained by simply\ntaking the difference between neighbouring lattice planes. Finally a moving average is applied to show the trend.\nS3. PRINCIPLE OF SCANNING SQUID MICROSCOPY\nThe most essential part of any scanning SQUID microscope (SSM) is the sensor chip containing the SQUID itself. In\nFig. S2(a) optical microscope images of the chip containing the dc SQUID chip can be seen. The SQUID washer, bottom\ninset, contains the dc SQUID made of layers of superconducting Nb and insulating SiO 2and Josephson junctions made of a\nNb/AlO x/Al sandwich. The SQUID washer is extended using (magnetically shielded) superconducting Nb leads which end in\na small (unshielded) tip, forming the pickup loop area. The upper inset shows the unshielded pickup loop, which has inner\ndimensions of approximately 3 x 3 µ m2. However, these dimensions and especially the shape vary from chip-to-chip, which is\nshown in Fig. S2(b) where the pickup loop structures of the two main SQUIDs used in this research are shown.\nThe SQUID chip is mounted on a flexible PCB, known as the cantilever, which is placed at an angle of approximately 15owith\nrespect to the sample, see Fig. S2(c). Scanning is performed when the edge of the SQUID chip is in contact with the sample, the\npoint of contact is detected using two strain gauges connected in a Wheatstone bridge which are present on the cantilever. By\nmeasuring under a contact angle the pickup loop is located as close to the sample as possible, to further minimize the height of\nthe pickup loop with respect to the sample the edge of the SQUID chip is ground away as can be seen in Fig. S2(b). This ensures\nthat the distance between the contact point and center of the pickup loop is approximately 8µm, meaning that under a contact\nangle of 15othe height of the center of the pickup loop with respect to the sample surface is approximately between 2 and 3µm.\nThe sensor is mounted on a stationary holder at the bottom of a movable tube, see the inset of Fig. S2(d) for a schematic\nillustration. The sample of interest is pasted on the bottom of a movable lever inside the outer tube. [3, 4] The tube descends\ninto the cryostat such that the measurement stage is submersed into the liquid Helium bath at 4.2 K. The sensor is kept stationary\nwhilst the sample is moved in three spatial axes using propulsion provided ex-situ by three dc motors. Furthermore, the cryostat\nis equipped with a µ-metal shield, which has a shielding factor of approximately 25. Moreover, a superconducting Nb shield is\nused to further shield the measurement stage from external magnetic fields. As a result the typical flux sensitivities are in the\norder of 10µΦ0Hz−1/2, where Φ0is the magnetic flux quantum equal to 2.067×10−15T m2.\nIn order to maximize the SQUID sensitivity the SQUID in our SSM is operated using a flux-locked loop (FLL) feedback\nsystem [4, 5]. The FLL amplifies the SQUID signal and supplies an opposite field Φmback to the SQUID washer using a\nmodulation coil, which can be seen in the lower inset of Fig. S2(a). A change in external flux, ∆Φ e, is fed back through the\nFLL to the modulation coil which creates an opposite signal Φm=−∆Φ e. Therefore, the working point is locked and the\nSQUID remains equally sensitive. The system is thus a ”zero detector”, the flux through the SQUID loop is constant and the\nsystem is only sensitive to variations in magnetic flux. The measured voltage from the FLL is related to the external flux by the\nflux-to-voltage ratio F:\nVSQUID =F·Φe. (S1)\nThe value of Fis measurement-dependent and is generally between 16.6and17.4 V/Φ0. Conversion from the measured SQUID\nvoltage to an average magnetic field strength can be done using the following equation:\nB= Φ e·1\nA=VSQUID\nF·Φ0·1\nAeff, (S2)4\n≈9μm\n(a) (b)\n(d)Pickup loop\nMagnetic shielding\nModulation loop\nJosepson junctions\nHeliumMeasurement stage\nCryostatOuter tubedc motors\nSampleLever (inner tube)\nFlexible PCBSQUID chipDC motors\nOuter tube\nMeasurement stage\nCryostatLever (inner tube)\nSample\nSQUID chip\nFlexible PCB(c)\nθ≈14.9º5 μm\nFIG. S2. (a) Optical microscope images of the SQUID chip including SQUID washer with modulation loop and pickup loop. (b) Optical\nmicroscope images of the pickup loop areas of the two mainly used polished SQUID chips. (c) Contact angle of the SQUID chip on a flexible\nPCB with the sample. (d) Schematic illustration of the scanning SQUID microscope, including the enlarged measurement stage. The SQUID\nchip is glued on a flexible PCB and is placed in contact with the sample which is glued on the inner tube. The dc linear actuators drive the\ninner tube in order to scan the sample across the SQUID chip. The outer tube including measurement stage is fully immersed in the liquid\nHelium in the cryostat during measurements.\nwhere Bis the magnetic field strength in T,Ais the area through which the flux threads in m2,Φ0is the magnetic flux quantum,\nandAeffis the effective area of the pickup loop in m2.\nThe effective area of the pickup loop is influenced by several factors of the experiment and determines the spatial resolution\nof the microscope. The area of the loop is enlarged slightly by the Meissner effect of the superconducting Nb, this is called flux\nfocusing. Furthermore, the height and angle of the pickup loop with respect to the sample also influence the effective area and\nspatial resolution. By basing the spatial resolution of the microscope on the separation between the two extrema of a point-dipole\n[6] we can estimate that our spatial resolution is between 5and10µm. Furthermore, the effective area of the pickup loop can\nbe obtained by performing a calibration measurement for the specific sensor using Abrikosov vortices. This leads to an effective\narea of approximately 18µm2for sensors used in this work. It is important to note that due to the angle of the pickup loop with\nrespect to the sample the measured magnetic field values are not purely the perpendicular component of the field emanating from\nthe sample, but also consist of in-plane components.\nS4. EXTRA SSM MEASUREMENTS ON LaMnO 3\nIn Fig. S3 a comparison is shown of the influence of the individual metallic layers on the ferromagnetism of 50 u.c. thick\nLMO. In Fig. S3(a) a layer of 60 nm Au partially covers the LMO film. On the other hand, in Fig. S3(b) only a 4 nm thin layer of5\nTi covers the film. The SSM images in the figure are both taken on top of the covered LMO and in both cases the ferromagnetic\ndomain landscape is still visible, as expected for thicker 50 u.c. films.\nHowever, comparing the magnitude of the measured SQUID signals in Fig. S3 one can clearly see an order of magnitude\ndifference. Furthermore, also in comparison to SSM images on uncovered LMO (see Fig. 1), the image in Fig. S3(b) has a\nremarkably low signal, whereas the signal measured in Fig. S3(a) is similar to measurements on uncovered LMO. The differences\nare too large to be attributed to variations in the SQUID pickup loop or angle and can thus be attributed to the effect of the Ti\nlayer. Clearly it is the Ti layer which suppresses the ferromagnetic order in the STO/LMO/Ti/Au structures, whereas the Au\nlayer has no noticeable influence. The residual signal measured in Fig. S3(b) is both a result from the fact that the LaMnO 3film\nis thick, 50 u.c., and thus not fully suppressed and due to the fact that the sputtered Ti film is not capped with a gold layer and is\nthus exposed to the ambient environment. As a result of this the Ti film is also oxidised from the ambient environment and the\neffective thickness of the Ti film that can be oxidised by the LMO film is reduced.\n(a)\n50 μmSQUID Signal (μ T)\n11\n-5.5\n-1105.5\n50 μm(b) SQUID Signal (μ T)\n1\n-0.5\n-100.5\nLaMnO350 u.c.\nAu60 nm\nLaMnO350 u.c.\nTi4 nm\nFIG. S3. Comparison between the effect of the two metallic layers, Au and Ti, on 50 u.c. thick LMO. (a) SSM scan of a 60 nm Au covered\nregion of LMO. (b) SSM scan of a 4 nm Ti covered region of LMO.\nIn Fig. S4 two SSM images can be seen which show the time-dependent effect of the suppression of ferromagnetism in 50\nu.c. thick LMO by a Ti layer of 24 nm at the same position on the film. As shown by the schematic illustration two depositions\nof Ti(24 nm)/Au(60 nm) were performed on the same 50 u.c. thick LMO film partially covering perpendicular sides of the film.\nIn the SSM scans the intersection of these two layers, A and B, is shown. It is important to note that layer B was deposited a\nweek after the deposition of layer A.\nClearly visible in the bottom left of Fig. S4(a) is the ferromagnetic signal from the uncovered LMO, which is comparable\nto previous measurements on uncovered LMO. Using a similar approach as in the main text, one finds that the, older, area\nunderneath layer A has a signal strength of 16% in comparison to the uncovered signal. Whereas the area underneath layer B\nhas a relative signal strength of 35%. The SSM image in Fig. S4(b) was taken at the same position as in Fig. S4(a), but seven\ndays later. After a week the relative signal strengths of layers A and B are 14% and 17% respectively. Clearly showing that the\nsuppression of the ferromagnetism is time-dependent with a time span of days.\nS5. Ti V ALENCE STATE EVOLUTION AT THE LaMnO 3/Ti INTERFACE\nExtra information about the evolution of the valence state of Ti across the interface can be obtained by looking at the Ti\nL2,3edge. Fig. S5 shows the EELS data as function of sample depth around the Ti L 2,3edge together with the data of the\nrelative composition as shown in the main text. Also the linecut of the ADF signal is shown, together with the ADF image of\nthe interface. From the peak positions at the Ti L 2,3edge the valency can be deduced. Fig. S5 shows that the valency of the\nTi evolves from elemental Ti, to Ti1+/Ti2+, to Ti3+and finally to Ti4+. The indicated composition reflects only the elements\npresent in each layer where xrepresents the minority substitutional element, zthe oxygen content of the amorphous transition\nmetal oxide layer and dthe possible presence of oxygen vacancies. From top to bottom the interface consist of elemental Ti,\nsome form of Ti xOz, with a small amount of Mn doping, Ti-rich La(Ti,Mn)O 3−d, Mn-rich La(Ti,Mn)O 3−dand finally LaMnO 3.6\n(a)\n50 μmSQUID Signal (μ T)\n14\n+7 days\n-7\n-1407\n100 μm(b) SQUID Signal (μ T)\n18\n-9\n-1809A A\nB\nLaMnO350 u.c.\nTi24 nm\nAu60 nmA\nBA\nB\nFIG. S4. SSM scans of the same border region between an uncovered region of 50 u.c. thick LMO, and 24 nm Ti and 60 nm Au covered\nregions. Scan (b) was made 7 days later then scan (a) at the same position. The grey dashed line indicates the observed edge between layer A\nand B in the SSM scans.\nFIG. S5. EELS data at the Ti L 2,3edge with the corresponding ADF signal as function of sample thickness across the Ti/LaMnO 3interface.\nThe left 2-panels show the elemental composition as showed in the main text. The right panel the ADF image. From top to bottom the Ti\nvalency evolves from elemental Ti, to Ti1+/Ti2+, to Ti3+and finally to Ti4+when the perovskite structure starts (it shares then the Mn B site\nof LaMnO 3). Also the tentative composition in each layer is indicated, where xrepresents the minority substitutional element, zthe oxygen\ncontent of the amorphous transition metal oxide layer and dthe possible presence of oxygen vacancies.\n[1] C. Lichtensteiger, InteractiveXRDFit : a new tool to simulate and fit X-ray diffractograms of oxide thin films and heterostructures, Journal\nof Applied Crystallography 51, 1745 (2018).\n[2] A. De Backer, K. van den Bos, W. Van den Broek, J. Sijbers, and S. Van Aert, StatSTEM: An efficient approach for accurate and precise\nmodel-based quantification of atomic resolution electron microscopy images, Ultramicroscopy 171, 104 (2016).\n[3] J. R. Kirtley, M. B. Ketchen, K. G. Stawiasz, J. Z. Sun, W. J. Gallagher, S. H. Blanton, and S. J. Wind, High-resolution scanning SQUID\nmicroscope, Applied Physics Letters 66, 1138 (1995).\n[4] J. R. Kirtley and J. P. Wikswo, Scanning SQUID microscopy, Annual Review of Materials Science 29, 117 (1999).\n[5] P. J. Kung, R. R. Bracht, E. R. Flynn, and P. S. Lewis, A direct current superconducting quantum interference device gradiometer with\na digital signal processor controlled flux-locked loop and comparison with a conventional analog feedback scheme, Review of Scientific\nInstruments 67, 222 (1996).\n[6] P. Reith, X. Renshaw Wang, and H. Hilgenkamp, Analysing magnetism using scanning SQUID microscopy, Review of Scientific Instru-\nments 88, 123706 (2017)."    },    {        "title": "2207.14196v2.Evolution_of_short_range_magnetic_correlations_in_ferromagnetic_Ni_V_alloys.pdf",        "content": "Evolution of short-range magnetic correlations in ferromagnetic Ni-V alloys\nShiva Bhattarai,1Hind Adawi,1, 2Jean-Guy Lussier,1Adane Gebretsadik,1,\u0003\nMaxim Dzero,1Kathryn L. Krycka,3and Almut Schroeder1,y\n1Physics Department, Kent State University, Kent OH 44242, USA\n2Department of Physics, Jazan University, Jazan 45142, Kingdom of Saudi-Arabia\n3NIST Center of Neutron Research, National Institute of Standards and Technology, Gaithersburg MD 20899, USA\nWe experimentally study how the magnetic correlations develop in a binary alloy close to the fer-\nromagnetic quantum critical point with small-angle neutron scattering (SANS). Upon alloying the\nitinerant ferromagnet nickel with vanadium, the ferromagnetic order is continuously suppressed. The\ncritical temperature Tcvanishes when vanadium concentrations reach the critical value of xc= 0:116\nindicating a quantum critical point separating the ferromagnetic and paramagnetic phases. Earlier\nmagnetization and \u0016SR data have indicated the presence of magnetic inhomogeneities in Ni 1\u0000xVx\nand, in particular, recognize the magnetic clusters close to xc, on the paramagnetic and on the ferro-\nmagnetic sides with nontrivial dynamical properties [R. Wang et al. , Phys. Rev. Lett. 118, 267202\n(2017)]. We present the results of SANS study with full polarization analysis of polycrystalline\nNi1\u0000xVxsamples with x= 0:10 andx= 0:11 with low critical temperatures Tc<50 K. For both\nNi-V samples close to xcwe \fnd isotropic magnetic short-range correlations on the nanometer-scale\npersisting at low temperatures. They are suppressed gradually in higher magnetic \felds. In addi-\ntion, signatures of long-range ordered magnetic domains are present below Tc. The fraction of these\nmagnetic clusters embedded in the ferromagnetic ordered phase grows toward xcand agrees well\nwith the cluster fraction estimate from the magnetization and \u0016SR data. Our SANS studies provide\nnew insights into the nature of the inhomogeneities in a ferromagnetic alloy close to a quantum\ncritical point.\nI. INTRODUCTION\nFerromagnetic order emerging as a result of electron-\nelectron interactions in metals has been studied exten-\nsively during the last several decades. In clean metals if\nthe spin-exchange interactions between the itinerant elec-\ntrons are strong enough, then the ferromagnetic order de-\nvelops at some \fnite temperature.1{3For example, for Ni\nthe critical temperature of the ferromagnetic transition\nisTc\u0019630 K. Introduction of disorder by alloying is a\nway to suppress the ferromagnetic transition, so that the\ncritical temperature becomes zero at some \fnite concen-\ntration of impurity atoms xc. For example, in Ni 1\u0000xVx\nalloys the critical concentration is xc\u00190:116.4\nWhenx=xca quantum phase transition (QPT) be-\ntween the ferromagnetic and paramagnetic states is ex-\npected to occur. Since the symmetry of the ground state\nchanges at the quantum critical point (QCP), the evolv-\ning quantum critical \ructuations a\u000bect the vicinity of\nthe QCP, including \fnite temperatures that lead to un-\nusual thermal properties.5Since the transition is driven\nby spin-exchange interactions, which for each realization\nof disorder become nonlocal and random, QPTs are ex-\npected to be a\u000bected by disorder more likely than thermal\nphase transitions.6\nUnder the proper conditions random quenched disor-\nder in a metallic system can produce a quantum Gri\u000eths\nphase7,8that is usually recognized by anomalous ther-\nmodynamic properties8close to the QCP. In this case, a\ndistribution of magnetic \ructuations with di\u000berent time\nscales including very slow rare regions7dominates the\nmagnetic response. Speci\fcally, experimental signatures\nof the Gri\u000eths phase are power laws with non-universalexponents9in thermodynamic responses. These power\nlaws are observed in many systems4,10, but the responsi-\nble \ructuations with di\u000berent time and and length scales\nhave not been demonstrated directly. In general, disor-\ndered alloys close to a ferromagnetic QCP are recognized\nby unusual scaling behavior11{13. Speci\fc e\u000bective ex-\nponents are predicted for disordered FMs.14,15Note that\nQCPs in FMs are rare; more often new phases emerge\nwhenTcgets suppressed.15Typically, clean FMs rather\npresent a \frst-order transition15without critical \ructu-\nations. So remarkably, introducing disorder is a unique\nroute for a QCP in itinerant FM such as Ni. The char-\nacterization of the critical \ructuation spectrum should\nreveal the nature of this special point.\nLastly, we note that the conclusion that only disorder\nallows to the QCP in itinerant ferromagnets is a sim-\npli\fcation as alternative mechanisms have been recently\nproposed to explain a ferromagnetic QCP16with critical\n\ructuations. For example, the experimental signatures of\nthe quantum critical \ructuations were reported in chem-\nically tuned Ni-Rh17and pressure-tuned CeRh 6Ge418,19\nfrom ferromagnetic systems where disorder is considered\nnegligible.\nAs we have mentioned above, Ni 1\u0000xVxis an example\nof a disordered metallic system which exhibits the signa-\ntures of a ferromagnetic QCP indicating the presence of\nquantum critical \ructuations. In this paper, we present\nthe results of the measurements which directly probe\nquantum critical magnetic \ructuations in the vicinity of\nthe ferromagnetic QPT driven by disorder. In particular,\nwe use small-angle neutron scattering (SANS) to probe\nthe magnetic microstructure and magnetic inhomogene-\nity on the mesoscopic length scale of interest from a few toarXiv:2207.14196v2  [cond-mat.str-el]  24 Feb 20232\n(b)\nFIG. 1. (a) Magnetic phase diagram of Ni 1\u0000xVxdisplay-\ning the critical temperature Tcvs. vanadium concentra-\ntionxseparating the ferromagnetic ordered phase (FM) from\nthe paramagnetic phase (PM) leading to a quantum critical\npoint.26The red markers indicate the samples for this in-\nvestigation. The right panel (b) demonstrates the magnetic\ninhomogeneities in Ni 1\u0000xVxwithx= 0:10 in an fcc lattice\nplane: it shows how randomly placed V atoms (in blue) sepa-\nrate Ni-rich regions (in red) responsible for magnetism. Since\nV a\u000bects more the moment of the local Ni neighbors, we dis-\ntinguish these close Ni-sites (in white) from the rest, the more\ndistant Ni atoms (in red).\na hundred nanometers.20We are motivated by the fact\nthat SANS successfully revealed the correlation length\nchange of critical \ructuations close to a phase transi-\ntion in several magnetic alloys.21{25We present the ex-\nperimental evidence of magnetic correlations at various\nlength scales in a ferromagnetic Ni alloy introduced by\nrandom atomic substitution of Ni by V. These \fndings\nare compatible with a quantum Gri\u000eths phase.\nII. EXPERIMENTAL METHODS\nA. Ni-V\nThe alloys of Ni 1\u0000xVxprovide a good platform to\nstudy a random disordered QPT not only because of their\nlow magnetic anisotropy and the \\large defects\" caused\nby vanadium, but also because they form indeed good\nsolid solutions with random atomic disorder. In addition,\nwhen the vanadium concentration reaches the value of xc\n= 0:116 these binary alloys exhibit a quantum phase tran-\nsition (QPT) from the ferromagnetic (FM) to a paramag-\nnetic (PM) phase with a quantum critical point (QCP)4.\nFig. 1(a) shows the phase diagram. It is known that V\na\u000bects the electronic state of the Ni neighbors and causes\ntherefore large magnetic inhomogeneities27and an e\u000bec-\ntive reduction of the magnetic moment with low xc28. In\nFig. 1(b) we present a schematic view of a lattice plane\nwith V on random lattice sites. Assuming that V causes\na moment reduction of neighboring Ni, the undisturbed\nmagnetic Ni network becomes inhomogeneous.\nPrevious studies reveal some signatures of a quantum\nGri\u000eths phase. Speci\fcally, for x\u0019xcthe magne-tization data M(B) show the non-universal power law\ndependencies.4,26This is further supported by \u0016SR data\nrecognizing a \feld distribution in the samples29, and\nthe ferromagnetic alloys with values of xclose toxcin-\nclude a dynamic contribution to M(B) besides the static\ncontribution.26\nAll samples keep the simple fcc crystal structure. The\npolycrystalline samples were arc-melted and annealed at\nhigh temperatures and cooled down fast to maintain the\nrandom atomic placement of the V on the fcc lattice sites\nat low temperatures. A structural study (PDF analysis\nfrom wide-angle neutron di\u000braction data26,30) con\frmed\nthe random atomic distribution of V and the otherwise\nunchanged crystalline fcc lattice at low temperatures.\nB. Experimental Details\nWe use the same polycrystalline samples prepared for\noptimized random atomic distribution as studied before\nby di\u000berent methods.4,26,30For the small-angle neutron\nscattering (SANS) study we chose the concentrations\nx= 0:10,x= 0:11 andx= 0. The samples with x= 0:11\nare made with58Ni, while the others contain natural Ni\nisotope mixtures, which yield di\u000berent nuclear neutron\ncross sections of Ni. The SANS experiments were per-\nformed at the instruments NG7SANS31and VSANS32,\nat the NIST Center for Neutron Research (NCNR) at\nthe National Institute of Standards and Technology and\nat the instrument GPSANS, at the high \rux isotope re-\nactor (HFIR) at Oak Ridge National Laboratory. We fo-\ncus here most on the SANS experiments that allow a full\npolarization analysis from NG7SANS. Several 3 mm di-\nameter pellets of each concentration were wrapped in Al\nfoil and placed on a Cd-mask framed Al-sample holder\nattached to the cold plate of the cryostat. To cover a\nwave vector range of Q= (0.06 - 1) nm\u00001with neutron\nwavelengths of 0.55 nm and 0.75 nm, the SANS inten-\nsity was collected in the xyplane on a 2D detector at\ndi\u000berent sample to detector distances (from 2m to 11m).\nWe obtained the di\u000berent polarized cross sections, e.g.\nthe non-spin \rip (NSF) scattering with unchanged po-\nlarization state of the neutrons (DD and UU) and spin\n\rip (SF) scattering with reverse polarization state (DU\nand UD) from the sample, using the super mirror polar-\nizer and3He-cell as a spin analyzer as stated in detail in\nRefs. 33 and 34. See set up in Fig. 2. U, D refer to\nthe up, down aligned neutron spins with regard to the\naxis of neutron polarization determined by the external\nmagnetic \feld. The magnetic \feld was applied in the x\ndirection (Bmin= 7 mT,Bmax= 1.5 T) perpendicular\nto the beam (kz).\u0012indicates the azimuthal angle within\nthexyplane, with \u0012= 0\u000ein the horizontal x-direction\nand\u0012= 90\u000ein verticalydirection of the detector. Most\nof the data were reduced and analyzed with the IGOR\nsoftware35.3\nFIG. 2. (a) Polarized SANS set up with polarizing supermir-\nror, spin \ripper, sample in a cryostat with a magnetic \feld\nperpendicular to the neutron beam, a3He analyzer, and a\nposition sensitive detector. Arrows indicate neutron polar-\nization direction. (b) Coordinate axes with beam direction ^ n\nalongzde\fning angle \u0012in thexyplane of the detector. (c)\nPresent set up of \feld and polarization direction ^ palongx\n(from Ref. 36)\nC. Polarized SANS Analysis\nThe full polarized SANS (PASANS) technique traces\nthe neutron spin before it enters and after it leaves the\nsample separately. The four di\u000berent cross sections (DD,\nUU, DU, UD) o\u000ber the advantage of separating the small\nmagnetic scattering from overwhelming nuclear scatter-\ning in our samples.37Here we use in particular the angle\ndependence36of the signal in the xyplane of the detec-\ntor to resolve magnetic components M2\nx,M2\ny,M2\nx, with\na magnetic \feld Bxperpendicular to the beam. We ap-\nply two methods to best trace the magnetic signal. First\nwe collect the pure SF signal (DU+UD) that ideally se-\nlects \\magnetic\" scattering. But it contains other \\back-\nground\" (BG) contributions due to incoherent scattering\nfrom the sample and other sources. A small SF contri-\nbution of V nuclear spins is also included that does not\nshow any angle dependence.37,38\nThe NSF signal (DD+UU) is only used in limited cases\nto extract some strong longitudinal component M2\nxwhen\nthe nuclear contribution N2and its sample variation are\nnot dominating. Otherwise, we use the \ripper contrast\nof NSF data called DIF of fully polarized (DD-UU) or\nhalf polarized data (D-U) to recognize the anisotropic\nMxcomponent of the mixed term 2 NMxthat contains\na strong nuclear contribution. For our geometry we do\nnot expect and do not see any indication of anisotropy\nin the transverse direction of the \feld, so that we keep\nMy=Mz= 0, whileMx>0.\nNote that the intensity shown is not calibrated, it is\nconsistent for each sample. Di\u000berent number and size\nof pellets and the di\u000berent cross sections of58Ni and of\nnatural Ni lead to di\u000berent responses for both samples\nwithx= 0:11 andx= 0:10, respectively. We omit alsoany proportionality factor in the equation (more details\nand general forms are given in Ref. 36).\nThe angular dependence for the SF response is there-\nfore\nSF(\u0012) =M2\nz+M2\nycos4\u0012+M2\nxcos2\u0012sin2\u0012+BGSF:(1)\nFor the case of isotropic magnetic correlations where\nM2\nx=M2\ny=M2\nz=1\n3M2\ntot, we expect a simple cosine-\nsquare response with amplitude of1\n3M2\ntot:\nSF(\u0012) =M2\ny(1 + cos2\u0012) +BGSF: (2)\nIn particular, the SF contrast, the di\u000berence between SF\ndata in a horizontal sector ( \u0012= 0\u000eand 180\u000e) called SFH\nand a vertical sector ( \u0012=\u000690\u000e) called SFV, each col-\nlected typically within \u000e\u000630\u000e, produces the transverse\ncomponent M2\nywithout any BG SF. Ideally BG SFcan-\ncels out in the di\u000berence assuming the other contributions\nto SF(\u0012) are not angle dependent:\nSF(\u0012= 0\u000e)\u0000SF(\u0012= 90\u000e) =M2\ny(Q): (3)\nThis SF contrast can be evaluated over the accessible\nQ-regime to study the magnetic correlation lengths.\nIn principle the total non-spin \rip data NSF serve to\nextract the longitudinal component of the magnetic scat-\ntering,M2\nx, from the angular dependence. But the ex-\ntra constant is not small and not angle independent due\nto the non homogeneous sample arrangement. The nu-\nclear scattering of the sample N2dominates typically the\nresponse compared to other external BG and magnetic\nsignals.\nNSF tot(\u0012) =N2+M2\nxsin4\u0012+M2\nycos2\u0012sin2\u0012+BGNSF:\n(4)\nFor isotropic correlations we get a simple sine-square vari-\nation with amplitude M2\nx=1\n3M2\ntot:\nNSF (\u0012) =N2+M2\nxsin2\u0012+BGNSF (5)\nThe di\u000berent response between the two polarization\ndirections (without registering spin \rip), the NSF asym-\nmetry or DIF (DD-UU or D-U) yields an interference\nterm of nuclear and magnetic origin. It signals a weak\ncontribution from a center with a net magnetic compo-\nnent along the xdirectionMxin the presence of a strong\nnuclear contribution from the same center:\nDIF (\u0012) = 2NMxsin2\u0012 (6)\nThe vertical sector cut DIFV (collected at \u0012=\u000690\u000e\nwithin\u000e\u000630\u000e) gives the maximum signal 2 NMx. We\ncan trace this DIFV term over a large Q-regime resolving\nan anisotropic magnetic response Mxfrom 0.06 nm\u00001to\n1 nm\u00001in the Ni 1\u0000xVxsamples.\nIII. RESULTS OF SANS STUDY\nWe collected indication of disorder and clusters in\nNi1\u0000xVxalloys26related to a quantum Gri\u000eth phase.4\nBut any information of length and time scales of these\nunusual magnetic clusters is still lacking. Initial SANS\nexperiments demonstrated that some magnetic signal can\nbe detected for samples with small vanadium concentra-\ntionx < xc, while the magnetic intensity in the para-\nmagnetic regime ( x>xc) seems too small to be noticed\nwith an averaged high-\feld magnetic moment of less than\n0.01\u0016B/atom4. More recent (unpolarized) SANS data\ncollected at GPSANS show promising temperature and\n\feld dependent signals, and \fnally polarized SANS data\ncan identify consistently a magnetic response. Some data\nwith preliminary analysis for the compound Ni 0:9V0:10\nwith the higher critical temperature Tc\u001950 K are shown\nin Ref. 38. Here we present the PASANS results of the\nsamples closest to the critical point with the lowest Tcin\ncontrast to pure Ni. We show in detail the magnetic re-\nsponses of Ni 0:89V0:11withTc= 7 K and compare them\nwith Ni 0:9V0:10withTc\u001950 K to \fgure out the rele-\nvant magnetic correlations, their correlation length, and\nevolution toward xc.\nA. Short-Range Correlations in Ni 0:90V0:10\nThe \frst challenge is to extract the small magnetic\nscattering from the large nuclear contribution of Ni. The\nbackground of \\non magnetic scattering\" is high because\nof the high nuclear cross section of Ni compared to the\nsmall magnetic cross section due to the reduced average\nmagnetic moment ( <0:1\u0016B=Ni) in these alloys. The ad-\nditional grain boundary scattering observed in our poly-\ncrystalline samples dominates toward lower scattering\nvectorsQ. We use the (azimuthal) angle dependence (in\nthexyplane of the detector) of selected cross sections\nto resolve the magnetic components M2\nx,M2\ny,M2\nzwith a\nmagnetic \feld Bxperpendicular to the beam (along z).\nSection II C presents the relevant expressions for our set\nup shown in Fig. 2(b,c).\nFirst, we focus on the spin \rip (SF) signal to recog-\nnize the magnetic scattering. The angle dependence of\nSF(\u0012) as predicted by Eq. (1) further separates magnetic\nscattering from the background contribution and recog-\nnizes spin anisotropies. We distinguish a longitudinal\nmagnetic component M2\nxfrom transverse magnetic com-\nponentsM2\nyandM2\nz(with a wave vector Qperpendicular\ntoz). Fig. 3 presents SF(\u0012) and the non-spin \rip signal\nNSF (\u0012) of Ni 0:9V0:10in a medium Q-range (0:45\u00060:15)\nnm\u00001. Both show signi\fcant variations in the angle \u0012\nmost obvious for a temperature close to Tc, which sig-\nnals a magnetic response according to Eqs (1) and (4).\nThe \ft looks even like a simple cosine square function for\nSF(\u0012) and a sine square for NSF( \u0012) with the same am-\nplitude, which signals M2\ny=M2\nx(see Eqs. (2) and (5)).\nThe \ft parameters M2\nyandM2\nxevaluated at a speci\fc Q\nrange from SF(\u0012) andNSF (\u0012), are presented in Fig. 4\nfor di\u000berent temperatures and \felds. It is obvious in the\nleft panels (a) and (b) that both, the transverse M2\nyand\nlongitudinal M2\nxcomponent, are the same at Tcand stay\nFIG. 3. Neutron scattering intensity of Ni 0:90V0:10vs az-\nimuthal angle \u0012collected in medium Q-range of (0 :3\u00000:6)\nnm\u00001at di\u000berent temperatures Tand magnetic \felds B. The\nupper panel (a) presents the SF data, the lower panel (b) the\nNSF data with \ft as solid line using Eq. (1) in (a) and Eq.\n(4) in (b). An angle independent background BG has been\nsubtracted from the data, which varies with spin \flter and B.\nthe same for all Tin small \felds.\nFor low \felds (of 7 mT) the magnetic response is\nisotropic in this medium Q-regime for high temperatures\nT > Tclike expected in the PM regime. The signal is\nstrongest at the critical temperature Tcand decreases for\ntemperatures above and below Tcas seen in Fig. 3 and\nFig. 4. Note that at the lowest temperature, T= 3 K, the\nmagnetic signal does not completely vanish. In the FM\nregime below Tcwe need to apply a slightly higher \feld\nof 50 mT to keep the neutron beam su\u000eciently polarized\nwithPS>85% for a polarization analysis (see Appendix\nA). In this small \feld of 50 mT the response looks still\nisotropic. The simple cosine-square or sine-square form\nwith reduced but \fnite amplitude indicates that isotropic\nmagnetic \ructuations remain in the FM state as observed\nin the PM state. These magnetic \ructuations change in\nhigher magnetic \felds. The M2\nycomponent in the SF re-\nsponse is reduced while the longitudinal component M2\nx\nseems less a\u000bected. The M2\nxcomponent is more di\u000ecult\nto determine, it is a small disturbance to the SF(\u0012) shape\nor is the main parameter from the NSF (\u0012) data that in-\nclude a high nuclear response. Through these angle de-\npendent data we are able to con\frm magnetic scattering5\nFIG. 4. Temperature Tand magnetic \feld Bdependence of\nmagnetic transverse components M2\nyand longitudinal compo-\nnentM2\nxof Ni 0:90V0:10. In the upper panel (a,c) M2\nyandM2\nx\nare extracted from \ft (Eq. (1)) of SF( \u0012) atQ-range (0:2\u00000:5)\nnm\u00001. The lower panel (b,d) presents the main components\nfrom the data shown in Fig. 3 at Q-range (0:3\u00000:6) nm\u00001:\nM2\nycomes from \ft of SF( \u0012) andM2\nxfrom \ft (Eq. (4)) of\nNSF(\u0012).\nat \fniteQin Ni 0:90V0:10at lowT. Since they display the\nsame isotropy as in the PM state, it is likely that they\nare \ructuations with su\u000ecient long time scales to be ob-\nserved as static by the neutrons. This magnetic signal\nlooks promising to uncover the \\magnetic clusters\", the\ndistribution of clusters with di\u000berent sizes and \ructua-\ntion rates that evolve at a disordered QCP.\nTheQ-dependence of SANS reveals some direct infor-\nmation about the relevant magnetic correlation length of\nthese clusters. Fig. 5 displays the Q-dependence of the\nSF signal, i.e. the M2\nycomponent from the SF contrast\n(SFH-SFV). Since the signal is isotropic (in low \felds)\nM2\nyrepresents 1/3 of the total magnetic signal M2\ntot.M2\ny\ncan be resolved in a limited Q-range above 0.1 nm\u00001.\nToward lower Q, the dominating nuclear NSF contribu-\ntion increases and makes the extraction of a small SF\ncontribution with polarization corrections less reliable.\nThe even smaller SF contrast cannot be resolved. The\nstrongest response is found at 47 K close to Tc.M2\nyis\nclearlyQdependent, steeper at Tcthan at higher tem-\nperaturesT=70 K. At the lowest temperature of T= 3 K\nM2\ny(Q) can be barely resolved beyond the middle- Qre-\ngion but its Qdependence looks similar to the T= 47 K\ndata. Within this limited Qregion of (0.1-1) nm\u00001the\nSF contrast can be approximated well by a Lorentzian\nfunction as expected for paramagnetic critical scattering\nof the Ornstein-Zernike form with a correlation length\nFIG. 5. Magnetic neutron scattering intensity vs. wave vector\nQfor Ni 0:90V0:10. The spin \rip contrast SFH-SFV is shown\nin small magnetic \felds ( B\u001450 mT) representing M2\nyor\n1/3 of the total isotropic magnetic response M2\ntotfor di\u000berent\ntemperatures T. Solid lines are Lorentzian \fts using Eq. (7).\n\u0018= 1=\u0014:\nM2\ny=AL\u0002\u00142=(\u00142+Q2): (7)\nWe estimate a correlation length using Eq. (7) as \u0018\u0019\n(7\u00062) nm for T\u0019Tc= 47 K, and we \fnd the\ncomparable value of \u0018= (10\u00066) nm in a more re-\nstrictedQ-regime for T= 3 K. About 10% of the am-\nplitudeALof the Lorentzian form remains at low T:\nAL(T= 3 K)\u0019(1=10)AL(T= 47 K). That corresponds\nto the same fraction of M2\nyat medium Q. These data\nclearly show some leftover short-range magnetic correla-\ntions in the FM state at T < Tcthat do not contribute\nto the long-range order. The remaining magnetic re-\nsponse at low temperatures is similar to the PM scat-\ntering. The cluster sizes or range of correlation length at\nlowTlook like those estimates in the PM regime close to\nTc. The non-polarized data estimates for x= 0:10 using\nhigh \feld data as reference agree with the main PASANS\n\fndings.38\nB. Short-Range Correlations in Ni 0:89V0:11\nThe discovery of a remaining magnetic signal in\nNi0:90V0:10motivates us to investigate the more diluted\nNi0:89V0:11that is closer to critical concentration xcwith\na lowerTc= 7 K. As expected the magnetic response in\nNi0:89V0:11is even smaller than in Ni 0:90V0:10and more\nchallenging to resolve. Note that the data shown are not\ncalibrated and a direct comparison of the scattering in-\ntensities of x= 0:11 andx= 0:10 cannot be made. We\ncannot extract any clear magnetic signal from the NSF\ndata due to the strong nuclear contribution in x= 0:11.\nBut the SF data reveal successfully a magnetic response\nfrom the distinct angle dependence in a limited Q-regime.\nThe strongest SF response in Ni 0:89V0:11close toTcat6\nFIG. 6. Angle dependence of SF signal collected in medium\nQrange (0:35\u00060:15) nm\u00001for Ni 0:89V0:11. Panel (a) shows\ndata in small magnetic \felds Bat di\u000berent temperatures T,\nand panel (b) data at low Tin di\u000berent B. Fit is shown as\nsolid line using Eq. (1). A constant BG has been subtracted\nfrom the data that depends on polarization \flter and mag-\nnetic \feld.\nT= 8 K follows a simple cosine square function as seen\nin Fig. 6(a). All SF(Q) data can be described well by\nEq. (1) con\frming an isotropic magnetic response with\nM2\ny=M2\nxin low \felds ( B\u001450 mT). The Tdependence\nof these transverse and longitudinal components is shown\nin Fig. 8(a). Similarly to Ni 0:90V0:10the strongest PM\nresponse of \ructuating short-range correlations develops\nupon cooling toward Tc, decays for T <Tcbut does not\nvanish for the lowest T. Also theQdependence of the SF\ncontrast (SFH-SFV) that measures the transverse com-\nponentM2\nyof the isotropic \ructuations looks similar for\nboth concentrations as displayed in Fig. 5 and 7. For\nNi0:89V0:11theQrange and precision of the data are\nmore limited. M2\ny(Q) can be described by a Lorentzian\nline shape with Eq. (7). Cooling toward Tcincreases the\ncorrelation length \u0018= 1=\u0014, but it remains \fnite at Tc.\nM2\ny(Q) shows about the same Qdependence at the low-\nestTof 4 K compared to Tc. Only the amplitude AL\nis reduced. Fitting the SF contrast at 8 K and 4 K in-\ncluding data up to 1nm\u00001(not all data shown) yields\n\u0018= (4\u00062) nm and\u0018= (6\u00063) nm, respectively. Within\nthis large uncertainty we conclude that the typical cor-\nrelation length at Tcand at low Tis quite similar in the\nrange of 5-10 nm and not very distinct from Ni 0:90V0:10.\nThese simple estimates show some indication that the\nFIG. 7. Magnetic neutron scattering intensity vs. wave vector\nQfor Ni 0:89V0:11. The spin \rip contrast SFH-SFV is shown\nin small magnetic \felds ( B\u001450 mT) representing M2\nyor\n1/3 of the total isotropic magnetic response M2\ntotfor di\u000berent\ntemperatures T. Solid lines are Lorentzian \fts using Eq. (7).\nFIG. 8. Magnetic components M2\nyandM2\nxextracted from \ft\n(Eq. (1)) of SF( \u0012) data (shown in Fig. 6) of Ni 0:89V0:11with\nTc\u00197 K at medium Qrange of (0.2-0.5) nm\u00001. Panel (a)\nshowsTdependence of isotropic \ructuations in small \felds\n(B\u001450 mT) and (b) shows Bdependence of short-range\ncorrelations at low T= 4 K, which are di\u000berent for transverse\nand longitudinal components. Lines just connect data points.\nmagnetic length scale describing the cluster distribution\nmight become smaller towards the critical point, but only\nbetter precision estimates from a larger Qregime with\nimproved statistics of the low intensity x= 0:11 data can\nclarify that. Most importantly, these polarized data are\nable to con\frm that magnetic clusters are still present at\nlowT, which look similar to PM scattering with a typi-\ncal scale of a few nm. In Ni 0:89V0:11the magnetic cluster\nresponseM2\nyat 4 K is about half of M2\nyatTcfor this\nmediumQrange. While the remaining cluster contribu-\ntion is about of similar magnitude in both samples, the\nfraction of remaining clusters is much higher in x= 0:11\nthan inx= 0:10. The present data (not shown in the\nsame scale) do not allow to trace a detailed concentra-\ntion dependence of the remaining clusters. However, we\ncan rule out similar short-range correlations in polycrys-7\nFIG. 9. Field dependence of magnetic component M2\nyof\nNi0:89V0:11at lowT= 4 K.M2\nyvalues estimated from SF( \u0012)\nusing Eq. (1) at two di\u000berent Qranges as indicated. Lines\njust connect data.\ntalline Ni samples (see in Section IV A).\nThe signature of a quantum Gri\u000eths phase with a clus-\nter distribution in FM Ni 1\u0000xVxcame from the \feld de-\npendence of the bulk magnetization M.26The charac-\nteristicBdependence of M(B) in form of a power law\nindicates that these \ructuations of di\u000berent sizes and\nrates are gradually freezing out toward higher B. The\nSANS data at low \felds con\frm already the existence\nof isotropic magnetic \ructuations at base T, SF data in\nhigher magnetic \felds should reveal more detailed infor-\nmation about the distribution. We notice in Fig. 6(b)\nalready that the SF response and SF contrast M2\nyat 4\nK decrease in higher \felds. The angle dependence also\nchanges, it shows deviations from a pure sine-square form\nin higherBthat indicates an anisotropic magnetic signal.\nThe \ft of Eq. (1) reveals di\u000berent M2\nyandM2\nxcompo-\nnents as presented in Fig. 8(b). The transverse com-\nponentM2\nyis decreasing with Bwhile the longitudinal\ncomponent M2\nxdoes not change within large error bars.\nThe Ni 0:90V0:10data (Fig. 3 (a,b), Fig. 4 (c,d)) support\nthe same trend: M2\nydecreases with BwhileM2\nxremains\n\fnite. Both SF( \u0012) and NSF( \u0012) yield consistently a dom-\ninantM2\nxcomponent in high \feld. We did not expect to\nsee a remaining magnetic longitudinal component M2\nxat\n\fniteQin strong magnetic \felds. Since these SF data\ndo not allow to explore further the Qdependence of this\nterm we postpone the discussion to the the next sections.\nIn Secs. III C and III D we employ another method DIF\nthat is better suited to resolve this anisotropic term.\nWe do understand and expect that M2\nydecreases with\nB. That signals that the \ructuations indeed freeze out\nin higher \feld, more moments from short-range correla-\ntions do not contribute to this higher Qregion since they\nmight join larger domains. Unfortunately, we cannot re-\nsolve the detailed Qdependence and Bdependence to\n\fnd a speci\fc cluster distribution. The statistics are too\npoor to even extract a Lorentzian \ft with two param-\neters fromM2\ny(Q) in high \felds that is further reduced\ncompared to low \felds. Fitting the angle dependence of\nSF data collected in wide Q-ranges of \u0001 Q= 0:15 nm\u00001\nFIG. 10. Angle dependence of neutron response DIF( \u0012) of\nNi0:90V0:10for di\u000berent Qranges at low T= 3 K< Tccon-\n\frming magnetic response 2 NM x, solid lines follow Eq. (6).\n(a) LowQrange covers (0 :06\u00000:14) nm\u00001, (b) highQrange\nincludes (0:4\u00000:8) nm\u00001. HP denotes half polarized data,\nwhere DIF=D-U.\nprovides su\u000ecient statistics to trace M2\ny(B) at two dif-\nferentQlocations. Fig. 9 presents M2\ny(B) for Ni 0:89V0:11\nobtained from SF(\u0012) at di\u000berent Q, comparing a lower Q\nrange (centered at 0 :25 nm\u00001) and a higher Qrange (cen-\ntered at 0:45 nm\u00001). We notice that M2\nyshows a steeper\nB-dependence at lower Qthan at higher Q. This trend is\nconsistent with the idea that larger slower clusters freeze\nout in smaller \felds than smaller clusters with higher\n\ructuations rates. Our data might support a distribu-\ntion of \ructuating cluster sizes consistent with quantum\nGri\u000eths singularities. Better resolution is certainly nec-\nessary to see and con\frm more details.\nC. Search for Long-Range Order in Ni 0:90V0:10\nAnother way to extract the longitudinal magnetic scat-\ntering is to consider the di\u000berence response \\DIF\" be-\ntween the two initial polarization directions without an-\nalyzing the neutron polarization after the sample. The\nNSF asymmetry or \ripper di\u000berence is collected from\nfully polarized data (DD-UU) or half polarized (HP) data\n(D-U) as introduced before in Section II C, where Eq. (6)\npresents the characteristic angle dependence of DIF( \u0012).\nThis interference term of nuclear and magnetic origin,\n2NMx, more easily con\frms a weak magnetic signal with\na strong nuclear contribution. It is therefore accessible in\na largerQregime in Ni 1\u0000xVxproviding further informa-\ntion on di\u000berent length scales than the \ructuations from\nthe SF signal.8\nFIG. 11.Qdependence of DIFV=2 NM xof Ni 0:90V0:10at low\nT= 3 K< Tcin magnetic \felds B. Solid line indicates \ft\naccording to Eq. (8).\nFIG. 12. Magnetic response 2 NM xof Ni 0:90V0:10for lowQ\n(upper panels (a,b)) and high Q(lower panels (c,d) derived\nfrom DIF(\u0012). The left panels (a,c) compare the temperature\ndependence in small magnetic \felds as indicated ( Blow\u001450\nmT), the right panels (b,d) the \feld dependence at T= 3 K\n< TcandT= 47 K =Tc. While 2NM xatQlowrepresents\nthe long-range magnetic domain contribution, 2 NM xatQhigh\nsignals local magnetic impurities. The solid lines connect data\npoints in (a,c,d).\nFirst we check the angular dependence of the DIF data\nin Ni 0:90V0:10for traces of a magnetic response in di\u000ber-\nentQregimes. Fig. 10 shows DIF( \u0012) at lowT= 3 K in\nB= 0:2 T for a low Qregion (centered at Qlow= 0:10\nnm\u00001) in panel (a) and a high Qregion (centered at\nQhigh= 0:6 nm\u00001) in panel (b). The DIF( \u0012) data fol-\nlow well the expected response of a sine square function\n(see Eq. (6)) that con\frms a longitudinal magnetic con-tributionMxin the magnetized state. The maximum\namplitude observed at DIF (\u0012= 90\u000e)=DIFV=2 NMx\nis the only \ft parameter. Fig. 11 presents the de-\ntailedQ-dependence of DIFV in a wide Qrange from\nQ= (0:06\u00001) nm\u00001for Ni 0:90V10at baseT= 3 K well\nbelowTc. As long as the applied \feld is large enough\n(B\u001550 mT) that the neutron beam does not become\ndepolarized ( Ps>85%) in the FM state of the sample,\nthe DIFV(Q) response looks similar for di\u000berent B. To-\nwards low Q, the signal presents a steep 1 =Qnupturn\nthat follows a \\Porod\" term39with power n= 4 . To-\nward higher Qthe intensity remains constant. DIF( Q)\ncan be represented by\nDIFV (Q) =AD=Q4+CD (8)\nWe recognize two di\u000berent responses of aligned magnetic\nmomentsMxin theBdirection dominant at lower Q\nand remaining at higher Qin the FM state at 3K. To\ndistinguish their origin we study the evolution with T\nandB.\nThe upper panels of Fig. 12(a,b) focus on the inter-\nference term 2 NMxatQlowrepresentative of the low Q\nupturn with amplitude AD. The 2NMx(Qlow) signal is\npresent at the lowest temperature in \felds B\u001550mT\nthat can be analyzed in the FM state. As shown in panel\n(a) a clear positive response of 2 NMxappears only in\nthe FM phase. 2 NMxbecomes very small close to zero\ntoward higher Twhen crossing the critical temperature\nTc= 47 K into the PM state. The minor deviation from\nzero at higher Tis not signi\fcant. The signal at low T\nis further increasing by applying a higher Bas shown\nin panel (b). The \feld also changes the response in the\nPM regime at T= 47 K\u0019Tcbut much less than in the\nordered state below Tc. Further higher precision data at\nseveralTbelow and above Tccould trace and reveal the\nonset of a FM response in more detail. However, these\nchanges in 2 NMx(T;B) are clearly related to the \\mag-\nnetic\" contribution Mxthat appears below and vanishes\naboveTc.\nAs this mixed term 2 NMxincludes the nuclear contri-\nbution, we check the pure nuclear contribution N2sepa-\nrately for any anomalies in T,Band in particular in Q\nto get a better estimate of the pure magnetic response.\nThe pure nuclear signal from coherent scattering from Ni,\nN2, is estimated from the total NSF contribution in a sec-\ntor along the \feld direction ( NSFH =N2+BG) after\nother BG subtraction of mainly sample holder. N2(Q)\nshows a 1=Q4dependence toward low Q(see Ref. 38)\nwithout any obvious TorBdependence. This strong\nnuclear response in this lower Qrange stems from grain\nboundaries of crystallites on the order of \u0016min these\npolycrystalline samples. The grain size is too large to be\ndetermined from these SANS data in this Qregime (1\u0016m\n>1=Qmin= 1=0:05 nm\u00001). Since the nuclear N2(Q) and\nthe cross term 2 NMx(Q) exhibit a 1 =Q4dependence in\nthe observed Qrange, the magnetic term ( Mx)2(Q) fol-\nlows then also a 1 =Q4dependence from the simple es-\ntimate of ( Mx)2= (2NMx)2=(4N2)\u0018Q\u00004\u00022=Q\u00004=9\nQ\u00004. The precision of 2 NMxandMxis far less than\nN2but deviations from 1 =Q4are not obvious. There-\nfore theQdependence of 2 NMx(Q) does not contradict\nand rather might support large-scale magnetic domains\nof similar order to the grain sizes. This is in agreement\nwith the simple depolarization estimates (see Appendix\nA). The fact that the sample depolarizes the neutron\nbeam strongly indicates FM order of large-scale domains\non the order of \u0016m. LowerQdata with better preci-\nsion might increase the chance to notice deviations from\npowern= 4. This could reveal indication of a frac-\ntal nature of perforated domains40or reduced domain\nsizes. Recognizing saturation e\u000bects at lower Qmight\nimprove estimates of a domain size, e.g. assuming a sim-\nple Lorentzian square \ft41. So far, the lower limit is\nonly 50nm (see Ref. 38). Note that e.g. nanocrystalline\nNi with average crystallite size of about 50nm presents a\ndi\u000berent response42. DIF(Q) is weakly Q-dependent42, it\ndoes not display the steep 1 =Q4upturn, or the constant\nQterm toward higher Q.\nThis DIFV signal at low Qin Ni 0:90V0:10shows the\nevolution of magnetic domains, long-range ordered re-\ngions, that develop below Tc. As we saw in the previous\nSection III A some short-range \ructuations remain at low\nT, but most of the Ni moments seem to form still a long-\nrange ordered network in the FM state below Tc. We do\nnot have evidence that the overall macroscopic domain\nsize is reduced to short-range order or cluster freezing by\nthe introduction of disorder through V in this alloy. As\nwe see in Fig. 11 and Fig. 12 the upturn 2 NMx(Qlow) still\nincreases with Bat lowT= 3 K. At the same \felds we\nnotice that M2\nyat higherQrange from SF is decreasing.\nWhile the short-range cluster \ructuations are freezing\nout, we expect that the contribution to the long-range\nmagnetic domains are growing. Indeed, the data support\nconsistently that the magnetic clusters freeze out grad-\nually to join the ordered net moments that increase in\n\feld direction. Since the nuclear response does not show\nany anomaly at a magnetic transition, the response of\n2NMx(T;Q low) at lowBmarks the onset of FM order.\nThe second magnetic term in 2 NMx(Q) that becomes\ndominant at higher Q > 0:2 nm\u00001is ratherQindepen-\ndent within the investigated Q-regime up to 1nm\u00001.\nTheTandBdependence is di\u000berent for this \\high\"\nQterm and points to a di\u000berent origin than the low\nQterm. This constant CDor 2NMx(Qhigh) gradually\ndecreases with increasing Twithout vanishing at Tc\nas shown in panel (c) of Fig. 12. Also, 2 NMx(Qhigh)\ndoes not change much with higher magnetic \felds (see\npanel (d)). It rather saturates at 2 NMmax\nx already\ninB\u001550 mT for low and higher T=Tc, di\u000berent\nthan the low Qupturn that was related to long-range\nmagnetic domains. The Q-independence supports that\nthis highQ2NMxcontribution stems from rather local\ndefects in this Ni 0:90V0:10alloy. An extended Q-range\nto largerQis essential to probe the e\u000bective length\nscale of these defects. They become visible when the\nmaterial is magnetized through entering the FM stateforming domains or through an external magnetic \feld.\nWe believe that this is not an instrumental artifact but\na signature of defects introduced by the vanadium as we\nwill show later in Section IV A through comparison with\npure Ni data.\nD. Search for Long-Range Order in Ni 0:89V0:11\nNext, we explore the DIF response for the other con-\ncentrationx= 0:11 closer to xc. Also here we \fnd signs\nof a longitudinal magnetic response MxbelowTcfrom\nthe cross term 2 NMxin a large Qregime similar to\nx= 0:10. Fig. 13 presents the characteristic angle vari-\nation of sin2(\u0012) inDIF (\u0012) (see Eq.(6)) at extreme Q\nregions, at Qlow= 0:10\u00060:04 nm\u00001in panel (a) and at\nQhigh= 0:6\u00060:2 nm\u00001in panel (b) at the lowest tem-\nperatureT= 4 K for B\u00150:05 T. Fig. 14 shows then\ntheQdependence of the maximum response at DIFV at\nT= 4 K forB\u001550 mT. It can be described by Eq. (8)\nby a 1=Q4upturn and a remaining constant CDsimilar\nto DIFV(Q) of Ni 0:90V0:10. We see that this low Qscat-\ntering related to magnetic domains is increasing slightly\nwith the magnetic \feld B, while the constant at high Q\ndoes not show any variation with the \feld. This extra\nDIFV=2NMxcontribution at high Qcon\frms clearly a\nnet magnetic Mxterm that backs up the anisotropic com-\nponentM2\nx>M2\nyfrom SF that persists in high \felds as\npresented earlier in Fig. 8 in Section III B.\nThis cross term DIFV that evaluates 2 NMx(derived\nfrom the di\u000berence of dominant nuclear responses) is of\nsimilar scale in x= 0:11 but shows more overall scatter\nthan forx= 0:10. This is not surprising, we expect a\nsmallerMx(comparing M(B) data26) and a larger N2\nfor Ni 0:89V0:11than for Ni 0:90V0:10. Thex= 0:10 sam-\nple contains \\natural\" Ni, while the x= 0:11 sample\nis made from pure isotope58Ni with twice the coherent\ncross section of natural Ni. For x= 0:11 we take the max-\nimum variation, the contrast between DIFV and DIFH,\nas the best 2 NMxestimate from DIF (\u0012). We accepted\nconsistently for all runs a negative DIFH = -0.18/-0.12\n(pol/HP) instead of ideally 0. This small deviation of\n<1% of the large NSF signal seems within the reso-\nlution of the instrument and analysis. The parameter\n2NMxis evaluated for x= 0:11 for several temperatures\nand \felds as shown in Fig. 15 for Qlow(upper panels)\nandQhigh(lower panels), similar to Fig. 12 for x= 0:10.\nThe lowQupturn in 2 NMxis represented through\n2NMx(Qlow) in Fig. 15(a,b). It is present in the FM\nstate at low Tin magnetic \felds ( B\u00150:05 T) that\nkeep the beam su\u000eciently polarized. Panel (a) con\frms\nthat it develops only below Tcand marks the onset of\nFM order with long-range domains. The \\precise\" Q-\ndependence for only ( Mx)2(Q) cannot be revealed from\nDIFV(Q), but it does not contradict a 1 =Q4dependence\nwithout deviations due to long-range magnetic domains10\nFIG. 13. Angle dependence of DIF( \u0012) of Ni 0:89V0:11for dif-\nferentQ-ranges at low T= 4 K< Tccon\frming magnetic\nresponse 2NM x; solid lines follow Eq. (6) with a small o\u000bset.\n(a) LowQrange covers (0.06-0.14) nm\u00001, (b) highQrange in-\ncludes (0.4-0.8) nm\u00001. HP denotes half polarized data, where\nDIF=D-U.\nFIG. 14. Qdependence of DIFV=2 NM xof Ni 0:89V0:11at\nlowT= 4 K< Tcin magnetic \felds B. Solid line indicates\n\ft according to Eq. (8).\nas explained for Ni 0:90V0:10above. Additional support\nfor long-range ferromagnetic order comes from the fact\nthat this Ni 0:89V0:11sample depolarizes the beam below\nTc(see Appendix A). A simple estimate from neutron\ndepolarization suggests a domain size of few \u0016m. Data\ntoward lower Qwith better statistics are necessary to\nclarify more details.\nThe amplitude of the low Qupturn recorded at low\nT, 2NMx(Qlow), is gradually growing in higher \felds as\nshown in Fig. 15(b) and Fig. 14. Previously in Section\nFIG. 15. Magnetic response 2 NM xof Ni 0:89V0:11for lowQ\n(upper panels (a,b)) and high Q(lower panels (c,d) derived\nfrom DIF(\u0012). The left panels (a,c) compare the temperature\ndependence in small magnetic \felds as indicated ( Blow\u001450\nmT), the right panels (b,d) the \feld dependence at T= 4 K\n<TcandT= 8 K \u0019Tc. While 2NM x(Qlow) represents the\nordered domain contribution, 2 NM x(Qhigh) stems from local\nmagnetic impurities.\nIII B we saw that the \ructuations M2\nyat highQdecrease\nwithB. This consistently supports the picture of short-\nrange clusters joining larger domains. Ni 0:89V0:11shows\nsimilar magnetic signatures to Ni 0:90V0:10, mainly that\nlarge-scale magnetic domains still form in the FM state\nalthough short-range magnetic \ructuations are becom-\ning more dominant. The high QDIF signal or remaining\nconstantCDrepresented by 2 NMx(Qhigh) is also present\nin Ni 0:89V0:11. The lower panels (c,d) of Fig. 15 show\nthat it vanishes gradually with increasing temperature\nand saturates already in medium \felds at any T. Simi-\nlarly to Ni 0:90V0:10any external \feld or FM state is suf-\n\fcient to magnetize the sample to notice these defects in\nNi0:89V0:11.\nThe more diluted sample Ni 0:89V0:11follows the same\ntrend of Ni 0:90V0:10, the magnetic 2 NMxsignal is essen-\ntially zero at temperatures above T\u0015Tcbut appears be-\nlowTc. This indicates FM order within macroscopic do-\nmains even very close to the critical point. We collected\nthree di\u000berent magnetic responses in Ni 0:90V0:10and\nNi0:89V0:11through SF and DIF in di\u000berent Qranges.\nFor both samples, we could extract short-range \ructua-\ntions remaining at low Tbut also \fnd signatures of long-\nrange domain scattering. To further check the origin and\nthe importance for a disordered alloy we compare the\nsignals with pure Ni and discuss the evolution with the\nvanadium dilution.11\nIV. DISCUSSION\nA. SANS Response of Ni vs. Ni-V Alloy\nTo clarify which magnetic e\u000bects are induced by vana-\ndium, we compare the magnetic responses of the alloy\nNi0:90V0:10with pure Ni. Since x= 0:11 was made with\n58Ni, this sample cannot be used for this comparison. We\nperformed a polarized SANS measurement at a di\u000berent\ninstrument, VSANS, for a Ni sample that was prepared\nwith the same annealing protocol as the alloys. We cov-\nered a large Qrange at room temperature, 300 K, where\nNi is in the ferromagnetic phase ( Tc=630 K). A stronger\nmagnetic \feld of 0.5 T is needed for the Ni samples to\navoid neutron beam depolarization. No clear net mag-\nnetic intensity results from the spin-\rip contrast (SFH-\nSFV) with su\u000ecient resolution that compares to the data\nofx= 0:10. The upper limit is well below the cluster\ncontribution of the alloys. However, we resolve a signif-\nicant magnetic response from the DIF term, the \ripper\ncontrast. To better compare the observed intensities of\nthe alloy Ni 0:90V0:10and of Ni, measured under di\u000ber-\nent conditions, we calibrate the signal with the nuclear\nscattering term at low Q. In both samples we expect the\nnuclear scattering to be dominated by the same (natu-\nral) Ni material with the same Ni coherent scattering.\nSo we calibrate the observed intensity DIFV=2 NMxby\ndividing through the observed nuclear response N2. This\nsignal is measured by NSFH (after BG correction using\nthe empty sample holder). We use here the value of N2\natQlow= 0:1 nm\u00001that is part of the 1 =Q4Porod term\nas calibration constant nc=N2(Qlow). The calibrated\n2NMx(Q)=ncshould then be independent of the spec-\ntrometer con\fguration and sample size.\nFig. 16 compares the Qdependence of the calibrated\nDIF term, 2 NMx=nc, of Ni 0:90V0:10and Ni at the lowest\npossible \felds. We see clearly for both samples at low\nQa dominating term AD=Q4, the amplitude is higher in\npure Ni than in Ni 0:90V0:10. At highQ, 2NMx=ncin Ni\ndips below the signal in the alloy, the extra constant CD\nterm of the alloy (in Eq. (8)) seems missing in Ni. That\nsupports that these local magnetic defects are caused by\nvanadium in Ni 0:90V0:10and not already present in Ni\ndue to other structural defects or domain wall bound-\naries.\nFig. 17 summarizes the magnetic cross term 2 NMx=nc\natQlowat di\u000berent V-concentration x. For better com-\nparison ratios are shown normalized to x= 0. We see\nthat the two SANS data points follow well the bulk mag-\nnetization data from previous study4.M(5T) represents\nthe magnetization at high \felds and M(0T) the sponta-\nneous magnetization, the extrapolated value for 0T. The\nslight magnetic \feld dependence does not a\u000bect much the\nratio. (The ratio of the bulk moment estimate of x= 0\nandx= 0:1 increases from 0.13 in B= 0 T to 0.14 in\nB= 5 T, while the 2 NMx=ncratio estimate is 0 :12 in\n0.2T and 0:13\u00060:02 in 1.5 T).\nIt is remarkable that the magnetic moment estimates\nFIG. 16. Calibrated DIF response 2 NM x=ncfor Ni and\nNi0:90V0:10vs. wave vector Q. In Ni only the long-range\nordered domain term is present, the local defects appear in\nthe V alloy. The solid line is \ft according to Eq. (8).\nFIG. 17. Relative moment contribution vs. vanadium con-\ncentrationxfrom SANS results and other methods from Ref.\n26. 2NM x(Qlow)=ncand bulk magnetization data M(B) nor-\nmalized to the value of Ni are shown. The long-range ordered\ndomain term follows the trend of the linear reduction of the\naverage magnetic moment with x.\nfrom bulk data M(B) and from SANS, extracting the\nPorod term 2 NMx(Qlow), agree so well with each other.\nThis DIF term does not only signal a magnetic response\nat lowQ, it represents long-range ferromagnetic domain\nscattering. It even seems to measure the magnetic mo-\nment density and to function as an order parameter in\nthis alloy. This cross term signal is so dominant since it is\nenhanced through the strong nuclear scattering of grain\nsizes in these polycrystalline samples. On one hand this\nis a huge \\background\" obstructing the SF scattering at\nthe lowQrange (that limits the precision of the mag-\nnetic short-range \ructuations), on the other hand this\nlowQupturn does provide useful information. This sim-\nple scaling and successful calibration also imply that the\nmagnetic structure and crystalline structure of Ni do not12\nchange very much in the alloy Ni-V, at least in these sam-\nples with low V concentrations prepared with the same\nprotocol. Not only the cross sections but also the do-\nmain and crystallite structure determine the small angle\nscattering response. The Porod term with amplitude AD\ntypically depends on the contrast ( b1\u0000b2)2of the scat-\ntering length densities b1andb2but also on the surface\nproperties and area Sin between the di\u000berent scatter-\ning centers. Proper calibration with the nuclear term\n(nc=N2) assumes that the grain boundaries do not\nchange much in the alloy compared to Ni. This agrees\nwith a structural PDF analysis that notices minor defects\nin Ni but does not \fnd changes in crystalline quality\nupon alloying26,30. The fact that the 2 NMx(Qlow)=nc\nterms scale as the spontaneous magnetization M(0T)\nterm suggests that also the FM domain sizes and bound-\naries between FM domains and potential PM inclusion\ndo not di\u000ber much in the diluted alloys and in pure Ni.\nSuch structural and magnetic characterization is possi-\nble to extract from a simple cross term DIFV( Q) that\npresents only one \ft parameter: the amplitude of the\nPorod term AD. It works because Ni and Ni 0:90V0:10\ncontain the same nuclear scattering centers (natural Ni)\nand magnetic scattering centers. The present data from\nthe Ni 0:89V0:11samples made with the58Ni isotope can-\nnot be easily used for this simple comparison. Further\nstudies become necessary to study the domain scattering\nat higher vanadium concentration closer to xcwhen the\n\ructuations become more dominant and the remaining\nordered network weakens.\nB. Evolution of Magnetic Clusters with Disorder\nThe contribution of FM ordered moments measured\nby the domain term decreases with introducing vana-\ndium into the alloy Ni 1\u0000xVx. At the same time some\nshort-range magnetic \ructuations seem to develop in the\nFM state toward the critical V concentration xcbefore\nthe long-range order breaks down. We measure the mag-\nnetic cluster contribution by the left over SF contrast at\nlowT. To assess a cluster fraction we take the ratio of\nthe signal intensity at lowest Tand atT\u0019Tcwhere all\nmoments should be still in the PM phase. Fig. 18 shows\nthe cluster fraction of the SANS measurement comparing\nwith previous estimates from \u0016SR and magnetization26\nas a function of x. The magnetization data de\fne the\npower law increase in \feld up to 5 T as cluster contri-\nbution \u0001M. The\u0016SR analysis recognizes an additional\ndynamic term as paramagnetic cluster contribution be-\nsides a static term describing the FM response. All these\nestimates agree well within their precision con\frming the\nevolution of increasing magnetic \ructuations breaking\nthe FM order at xc= 0:116. All data were taken at\nthe lowest temperature of about 2 K. The present SANS\ndata include data at 3 K and 4 K and non polarized data\ntaken at 1.5 K. The cluster ratio for x=0.11 is slightly\nT-dependent, but not vanishing. The error bar includes\nFIG. 18. Magnetic cluster fraction vs. vanadium concentra-\ntionxin Ni 1\u0000xVxobtained from SANS compared to other\nmethods: magnetization M(B),\u0016SR from Ref. 26. The\nSANS estimate is the ratio of the SF contrast at lowest T\nandTc, taken in medium Qrange.\npossible uncertainties due to low Textrapolations. A\ntwo component picture of PM and FM contribution is\ncertainly oversimpli\fed, but the present new SANS data\nrectify that extreme scales of long-range domains and a\nshort-range scale of few nm are relevant in this alloy.\nGenerally, it is clear that disorder renders the exchange\ninteraction which ultimately leads to the emergence of\nthe itinerant ferromagnetic state to become spatially in-\nhomogeneous. As a result, one may expect that in a\nregion of a sample, the local value of the exchange cou-\npling would exceed the critical value of the averaged in-\nteraction constant leading to the formation of the ferro-\nmagnetic droplet. In two spatial dimensions, the size of\nthe droplet does not depend on a size of the magnetic\nmoment43, while in the three dimensional case this is not\nso. In either case, the dynamics of these droplets will\nbe determined by the correlators of the randomly dis-\ntributed coe\u000ecients in the Ginzburg-Landau functional.\nWe leave the analysis of the magnetic susceptibility and\nheat capacity for future studies.\nV. CONCLUSION\nAlthough the average magnetic moment in Ni 1\u0000xVx\nbecomes very small close to the critical point where\nFM order vanishes, this SANS study identi\fes success-\nfully magnetic scattering. The polarized SANS technique\nhelps in particular to extract the small magnetic signal\nfrom the huge nuclear scattering. The pure spin \rip (SF)\nresponse and the \ripper di\u000berence (DIF) reveal comple-\nmentary magnetic contributions at various length scales\nin Ni 1\u0000xVxclose to critical concentration xcwhere the\nonset of ferromagnetism at Tcextrapolates to zero. The\ntwo samples with vanadium concentration of x= 0:10\nandx= 0:11 present similar responses in the accessi-\nbleQ-regime of (0.06-0.1) nm\u00001and are analyzed using13\nsimple forms. The result of a pure Ni sample serves as\nreference for a polycrystalline FM without atomic disor-\nder.\nFirst, we con\frm the presence of magnetic clusters at\nthe lowest temperatures far below Tc. The angular and\nQdependence (of the SF response) reveal isotropic mag-\nnetic short-range correlations similar to the paramagnetic\n\ructuations seen close to Tc. The scale of these cor-\nrelations is 5-10 nm estimated by a simple Lorentzian\nfunction. These \ructuations get suppressed gradually\nin higher \felds and follow qualitatively the expectations\nof the quantum Gri\u000eths phase where magnetic clusters\nfreeze with di\u000berent \ructuations rates. The SF data give\nan estimate of the fraction of these magnetic clusters that\ndo not contribute to the magnetic order. The cluster ra-\ntio is determined through the remaining few clusters at\nlowTin the FM phase compared to the PM signal close\ntoTc. This cluster contribution increases signi\fcantly\nfromx= 0:10 tox= 0:11, demonstrating that these\nshort-range correlated \ructuations become more impor-\ntant toward xc. The new SANS data agree well with\nprevious cluster estimates from magnetization and \u0016SR\ndata26supporting further the di\u000berent dynamics of these\ncontributions.\nIn addition, we \fnd signatures of long-range domain\nscattering, which supports macroscopic long-range or-\nder for samples even very close to xc. Within this Q-\nrange we cannot measure the macroscopic length scale,\nbut only set lower limits and collect supporting evidence.\nThe neutron beam gets depolarized below Tcfor both\nsamples. The magnetic cross term DIF=2 NMxshows a\nlowQupturn due to long-range domain scattering with\naligned moments Mxsimilar to the pure nuclear scat-\ntering term N2due to grain boundary scattering. This\nlowQmagnetic domain term only appears below Tcand\nis \feld dependent. In higher magnetic \felds, the low Q\ndomain term increases while the medium Q\ructuation\nterm decreases, indicating that the short-range \ructua-\ntions freeze to join the larger domains with aligned mo-\nments. A comparison with Ni even shows that this low Q\ncontribution scales with the magnetic moment estimates\nfrom bulk magnetization measurements. So, this low Q\ndomain term not only con\frms long-range order but also\nreveals the ordered moment of the compound.\nThese SANS data provide remarkable information al-\nthough the individual data lack precision in resolving de-\ntails of a system with moments of less than 0 :1\u0016B/Ni.\nWe see clear evidence of various magnetic correlations in\nthis alloy produced by random atomic dilution. We no-\ntice correlation lengths from nm scale to \u0016m employing\nsimple tools. These various contributions con\frm that\nthe FM order does not get homogeneously destroyed. On\nthe other hand we do not notice any signs of a spin glass\nor cluster glass with only frozen short-range correlations.\nWe do not see macroscopic phase separation either of dif-\nferent ordered phases. Note that a SANS response with a\nPorod or Lorentzian square term indicating longer-range\ndomains and a shorter-range Lorentzian term is often ob-served in alloys and transition metals oxides (see e.g. in\nRef 20). Typically, the size of the long-range order is al-\nready limited41or gets reduced by chemical substitution\nbefore the magnetic order breaks down44. The two coex-\nisting phases are expected both to be magnetic ordered\nphases with relevant coercivity45. We come to a di\u000berent\nconclusion in our simple dmetal alloy Ni 1\u0000xVxwith sus-\ntained long-range order and coexisting \ructuations close\ntoxc. Our prepared samples remain a soft metal and\ndo not show any hysteresis in M(B)4. The SANS data\nreveal macroscopic domains that are unavoidable in a \f-\nnite sized soft FM. We \ft only two contributions of FM\ndomains and PM clusters to facilitate a description of a\nvariety of length and time scales present in these samples.\nIt is likely that through quenched disorder the correlated\nregions vary throughout the alloy, leading to a network\nof longer-range domains that are perforated by shorter-\nrange \ructuations (provoked by the random location of\nthe V atoms). This disorder can lead to a quantum crit-\nical point in a FM metal. Here we tried to measure a\ntypical scale of short-range correlations that stems from\na distribution. We cannot see the local arrangement or a\npattern formation. A complimentary method with spa-\ntial resolution in nm would be of high interest to perform\nto reveal the location, distribution and shape of clusters\nand of coexisting long-range domains. This simple model\nsystem Ni 1\u0000xVxwith controlled random atomic distri-\nbution o\u000bers a great opportunity to study the complex\ninteraction of a dmetal with short and long-range inter-\naction including local random defects. The present SANS\ndata allow insight in the mechanism of how \ructuations\nevolve in the FM ordered state that lead to a quantum\ncritical point in an itinerant FM.\nVI. ACKNOWLEDGEMENT\nWe thank J. Kryzwon, T. Dax, S. Watson and T. Has-\nsan for their support with NG7SANS, cryogenics and3He\nspin \flters preparation at NIST. We thank Lisa DeBeer-\nSchmitt for her support at GPSANS, ORNL. This re-\nsearch is funded in part by a QuantEmX grant from\nICAM and the Gordon and Betty Moore Foundation\nthrough Grant GBMF5305 to Hector D. Rosales. H.A.\nacknowledges support from Jazan University. M.D. ac-\nknowledges the \fnancial support by the National Science\nFoundation grant NSF- DMR-2002795. We acknowledge\nthe support of the National Institute of Standards and\nTechnology, U.S. Department of Commerce, in providing\nthe neutron research facilities used in this work. Support\nfor usage of the3He spin polarizer on the NG7 SANS\ninstrument was provided by the Center for High Resolu-\ntion Neutron Scattering, a partnership between the Na-\ntional Institute of Standards and Technology and the Na-\ntional Science Foundation under Agreement No. DMR-\n1508249. A portion of this research used resources at\nthe High Flux Isotope Reactor, which are DOE O\u000ece of\nScience User Facilities operated by Oak Ridge National14\nFIG. 19. Transmitted neutron beam polarization ratio Psof\nthe sample (Ni 0:90V0:10and Ni 0:89V0:11) at di\u000berent environ-\nments. Panel (a) presents the temperature Tdependence in\na small \feld, both samples depolarize the beam for T < T c.\nPanel (b) shows how the polarization recovers in applied mag-\nnetic \felds Bat a low temperature.\nLaboratory.\nS.B. and H.A. contributed equally to this work.\nAppendix A: Neutron depolarization\nThe main study concentrates on the \\scattered\" neu-\ntron intensities from the magnetic sample using the dif-\nferent cross sections with selected neutron polarization.\nThe polarizing devices such as the supermirrors, spin\n\ripper and3He spin \flter, where the polarizing e\u000e-\nciency decays with time, are certainly not perfect and\ndo not produce a 100% polarized beam. Collecting the\ntransmitted direct beam intensity for all 4 spin combi-\nnations gives typically su\u000ecient information for correc-\ntion of such spin leakage. Another source for the de-\npolarization of the neutron beam is the magnetic sam-\nple. On one hand depolarization measurements pro-\nvide important information46{48, they can help to diag-\nnose the magnetic state of the sample in particular for\nferromagnets49,50. On the other hand a strong depolar-\nization mixes and a\u000bects the di\u000berent scattered intensi-\nties of the 4 cross sections that might make a proper po-\nlarization analysis unresolvable51. The polarization state\nPSof the transmitted neutron beam due to the sample\nis essentially measured by the contrast of transmitted di-\nrect beam without spin \rip TNSFand with spin \rip TSF\n(for perfect polarizing devices with initial P0= 1).\nPs= [TNSF\u0000TSF]=[TNSF+TSF] (A1)Psremains 1, when no change in the initial spin direction\noccurs. Conversely, Psgoes down to 0, when the neutron\nspin looses completely its original orientation that the\nintensity of both initial and reversed direction are equal.\nTo check if the Ni-V samples depolarize the neutron\nbeam and to \fnd optimized working conditions to col-\nlect polarized scattering data we recorded the transmit-\nted neutron beam polarization Psfrom the (e\u000eciency\ncorrected) transmission data at di\u000berent temperature T\nand magnetic \felds B. Fig. 19 presents Psfor alloys\nwithx= 0:10 andx= 0:11.Psis close to 1 for high\ntemperatures in the paramagnetic region, but starts de-\nclining below the critical temperature Tc. Both samples\nare clearly depolarizing the beam in a small \feld of 7 mT.\nForx= 0:10Pssaturates at low Tat a small value of\nPs= 0:3, forx= 0:11Psreaches 0.7 at 3 K. As expected\nthe strong depolarization changes in higher \felds. At a\nmagnetic \feld of 1 T, the maximum value close to 1 is\nrecovered, at 50 mT Psincreases to 0.85 for x= 0:10.\nWe conclude that low Tscattering data taken at 50 mT\nshow su\u000ecient polarization for Ni 1\u0000xVxwithx\u00150:10.\nNeutron depolarization is caused by strong internal\n\felds that change on a mesoscopic scale. The di\u000berent\n\feld directions make the neutron spin precess and devi-\nate from the main polarization direction. Local defects\nor fast \ructuations in the sample do not a\u000bect the spin,\nbut di\u000berently oriented long-range domains in a soft fer-\nromagnet is the typical case that leads to depolarization\nas we are suspecting here. When these magnetic domains\nalign in higher magnetic \felds, the neutron polarization\ngets restored. The observed values of Psdepend on the\nstrength of the inner \felds and the neutron time spend\nprecessing to reach an e\u000bective precession angle \u001eand\ntherefore on the dimensions, the domain size \u000eand the\nsample thickness d. We use a simple model proposed in\nRef. 49 to estimate a typical domain size from Ps:\nPs=Pf=Pi= exp\u0012\n\u0000d\n3\u000e\u001e2\n\u000e\u0013\n; (A2)\nwherePfandPiare the \fnal and initial neutron\nbeam polarization. The precession angle \u001e\u000e= (4:63\u0002\n10\u000010G\u00001\u0017A\u00002)\u0015B\u000e depends on neutron wave length \u0015,\nmagnetic \feld Band the domain size \u000e.B= 4\u0019Ms\u001acan\nbe estimated from spontaneous magnetization Mswith\ndensity\u001a(all expressions in cgs units49). Forx= 0:10\nandx= 0:11 we \fndB= 810 G and B= 190 G from the\nmagnetization data26, which yield domain sizes of 3 \u0016m\nand 2\u0016m, respectively. 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Spin-spin i nteractions near the interf ace ferro magnet/sem iconductor \nplay crucial role in the o ptical readout and the m anipulation of ferr omagnetism .  \n \nE-mail: koren ev@ori ent.ioffe.rssi.ru\nPACS num bers: 71.70 .-d, 72.25.-b, 75.75.+a \n \n1. Introduction \n Nowaday s large efforts are directed to th e integra tion of m agnetism  into sem iconducto r \narchitecture of m odern c omputers. Th e first appro ach is to construct a universal object combining \nferro magneti c (FM) and sem iconducting (SC) properti es. Following pioneering research of the 60-s of  \nthe last century on  Eu-bas ed chalcogenid es and Cd- Cr spinels, Ohno ( 1998) and Dietl et al  (200 0) \ninitiated the recent active studies of new III-V-b ased ferro magnetic sem iconductors. The second \napproach – the so-called “semiconductor spintr onics” – is based on generation and m anipulation of \nspin and spi n currents entirely  with in non-m agnetic semiconductors (Zutic et al  200 4). T he third \napproach (Pr inz 199 0) ex ploits ferromagnet/sem iconductor (FM /SC) h ybrid systems. One of the \nadvantages is an additional degree of freedom  consisting in the choi ce of desirable pair FM/SC am ong \nparam agnet sem iconductors and a larg e num ber of ferromagnetic materials. As dem onstrated b y \nZakharchenya and Korenev (2005) , anot her advantage is the possibilit y both to read and to control the \nmagnetizatio n of FM  with  the help of semiconductor. Here I focus on the physics of spin int eractions \nin FM/SC h ybrids and thei r effect on the optical orient ation of spins  in sem iconductor. \n 1Optical or ientation in fe rromagnet/sem iconductor  hybrids \nElectron spin polarization is sensitive m onitor of  various spin i nteractions. For exam ple, a \nferro magnetic film  creates a static magnetic fiel d. Although re latively  weak, this magnetic field \npenetrates deep into sem iconductor . The electron spin interacts with the stray  magnetic field by means \nof Zeeman i nteraction. Random ly distributed st ray fields bring about the Lar mor precession of  \nsemiconductor electron spins, thus leadi ng to their spin  relaxation. This effect (section 2) can be used \nto distinguish between magnetized and de magnetiz ed states of the f erromagnet.  \nThe exchange interaction (Coulom b interaction rest ricted by  the Pauli exclusion principle) of \ncharge carriers is the strongest spin interaction in hybrids. It is sh ort-range depending strong ly on the \nwavefunction overlapping. The exchange interac tion causes ferr omagnetism  and splits the energy \nlevels of the  ferromagnet itself by a large va lue (~1 eV). T he wavefunction overlap  between  \nferro magnetic ato ms and sem iconductor charge carriers in a FM/SC hybrid leads in equilibri um to the \nlocal spin pol arization of sem iconductor carriers – the so-called proxim ity effect. In non-eq uilibrium \nconditions the FM-induced spin polari zation is tr ansferred into sem iconductor over the macroscopic \ndistance (~10 µm) leading to the d ynam ical proxim ity effect (section 3).  \nThe FM-SC interfacial exc hange coupling form s a unified spin s ystem whose properties differ  \nsubstantially  from  those of  an isolated FM film . At first glance this sounds strange because the density \nof charge (and spin) carriers in param agnetic se miconductors is m uch less than the density  of magneti c \natoms in ty pical ferro magnets. Therefore, “co mmo n sense” suggests that a semiconducto r is not \ncapable to manipulate the powerful ferromagnetic spin  system . However, the contact of FM  film  and \nSC leads to t he band bending (Schottky barrier) and accumulation  of a fair qua ntity of charge carri ers \n(electrons or  holes) near  the interface.  The strong exchange interaction of S C charge carriers and  \nmagnetic ato ms of FM lea ds to a strongly  coupled  spin sy stem . The uniqueness of the FM/SC sy stem \nlies in the electrical  and optical tunabilit y of elect rical and magnetic properties of the param agneti c \nSC. This m eans that the  ferromagnetism  (for exam ple, magnetic hysteresis loop)  of the unified s ystem \ncan be tuned  as well. As a result, SC is not onl y a substrate for a FM to lie upon but pa rticipates \nactively  in information processing (se ction 4).   \n \n 2Optical or ientation in fe rromagnet/sem iconductor  hybrids \n2. Zeeman interaction in FM/SC hybrid  \nAn electron with spin sr and magnetic moment sgBr rµµ−=  interacts wit h an externa l \nmagnetic field Hr\n. The Ha miltonian of Zee man interacti on \nHsg H HB Zrrrr⋅=⋅−= µµ ˆ , \nwhere  TeVcme\neB / 602µ µ ≈=h is the Bohr magnetron, and  g is  the electron’ s g-factor. The  \ninteraction leads to spin precession around Hr\n with a Larm or frequency , hrr\nHgBµ=Ω  accord ing to \nthe equation of m otion ] [ s srr&r×Ω= . The spin precession of particles is also affected by  local magneti c \nfields of othe r sources, and in general th e tota l field is the vector sum  of the field s.   \n \n2.1. Magnet ostatic stray fields created by the ferromagnetic film i n semiconductor \nLocal stray  magnetic fields generated by  ferro magnetic film  affect the sem iconductor electron  \nspin sim ilarly to the external magnetic fi eld. The distribution of str ay fields in s pace depends strongly  \non the m agnetic state of the FM film . The magnetic field outside the uniform ly m agnetized laterally  \ninfinite film  is zero becau se the magnetic  \nlines of force close at infinity . The film of  \nfinite lateral  size L creates at point A \n(Fig.1a) a magnetic field  LdM HArr\nπ4≈  \n(d is the  film  thickness, Mr\n is \nmagnetizatio n). It penetra tes deep ~L into \nSC. For a Ni fil m wit h , \n nm and L=0.1 cm the fiel d  Oe. The stray  field near the FM/SC  interface is the  \nstrongest when the film  is dem agnetized, i.e. br oken int o a larg e num ber of dom ains with  different  \norientations o f magnetization.  In t his case both th e direction and  strength of magnetic field var y in \nspace with c haracteristi c scale de termined by  the domain size  (Fig.1b). For Oe M 510=\n10=d 06.0≈AH\nl µ10=l  the field \n is hundred ti mes larger t han that of the uniform ly magnetized FM. However it decay s \ndepthward m uch faster – over the length  . Oe HA6~\nl≥SC\naSC\nllbAAd\nL\nMr\nFigure 1. Stray magnetic field creat ed by the FM film \n \n 3Optical or ientation in fe rromagnet/sem iconductor  hybrids \n2.2 Effect of s tray fields on the optical orientation of semiconductor electrons. \nDzhioev et al (1994, 19 95) observed the spin precessi on of SC electrons in static stray  fields in  \na Ni/n-GaAs hybrid b y optical orientation m ethod. The essence of optical orientation (Meier an d \nZakharchenya Eds, 1984) is the gene ration of spin-p olarized conduction band e lectrons by  circularly \npolarized lig ht transferring the p hoton angular \nmomentu m into the  spin s ystem of \nsemiconducto r electrons. In its turn, \nannihilation of the electrons em its circularly \npolarized light. The degree ρ of the circular  \npolarization of ph otoluminescence (PL) is \nequal to the projection of the electron \nensem ble-averaged spin cSr\n onto registration  \ndirection z, usually  coi nciding with  the \nnorm al to the structure plane (top of Figure \n2). An exte rnal magneti c field induces  \nprecession of each electron spin about Hr\n with Larm or frequency Ωr\n. In steady  state conditions,  \nhowever, the average spin does not c hange with time but defl ects fro m the initial dire ction (z) \ndecreasing in absolute value (Fig.2 ). This leads to the Hanle ef fect – depolarization of PL with \nmagnetic field. The halfwi dth  of the magnetic depol arization curve is determ ined by  the equality  \nof frequency 2/1H\nh2/1gHBµ  and reciprocal non-equilibrium  spin lifetim e sT1 . The longer th e spin  \nlifeti me Ts, the s maller magnetic field neces sary to rotate spin and depolarize PL. As a result, we have  \nthe Hanle effect-based optical magnetomete r whose sensitivity is determ ined by the H 1/2 value. As we \ndiscussed in subsection 2. 1, the am plitude of the st ray fields  is pretty sm all (~ 1 G), so that the \nhalfwidth should be narr ow enough. Such a narrow halfwidth is realized in n-GaAs samples (see \nChapter b y Kavokin an d Dzhioev). Fig ure 3a com pares the Hanle effects in n-GaAs when the film  is \npreviousl y magnetized (filled circles) and demagne tized (open circles).  It is seen that the \ndemagnetizat ion decrease s the degree ρ with the maximum difference being achieved in the zero  \nexternal magnetic field.  \nFigure 2. Optical orientation ex periment and t he Ha nle \neffect of sem iconductor electr ons.  Arrows show the stea dy \nstate mean electron s pin in transversal magnetic field of \ndifferent val ues. ( Adapted from Zakh arche nya an d \nKorenev 200 5). \n 4Optical or ientation in fe rromagnet/sem iconductor  hybrids \nThe magnetic field strength hc making the magnetic moment of th e sam ple van ish, is referred \nto as the co ercive force.  It can be  \nestim ated by measuring the zero-\nfield polarization ( 0=H)ρ  after  \nflipping a previousl y magnetized  \nFM film  by an external magnetic \nfield of strength H*. When the \nswitching field H* is equal to the \nfield hc the film is demagnetized and  \nthe polarizati on value () 0=Hρ  is \nminimal. Th e data point s on the \nupper depen dence in Fig .3b were \nmeasured after switchin g in the  \ndark. The sharp minimum \ncorresponds to hc=90 Oe . Remarkably , that the coercive force va lue appeared to be 2.5 tim es larger  \nthan that m easured in t he sam e structure with th e use of a supercon ductor q uantum interference device \n(SQUID). Th e difference between the two technique s is that SQUID regist ers the total  magnetic \nmoment of the structure. It is mainly the magnetic moment of the nickel film  whose thickness ( 40 nm ) \nis much larger than that of the interface NiGaAs (a few nm). However, the contributio n of stray  fields \nof nickel and the interfac e to the Hanle  effect  is not reduced to the su m of their magnetic mom ents. \nIndeed, Dzhioev et al (1997) have shown that the non-equilibri um electron spin diffuses into n-GaAs \nover the distance µ10≈sL  that is 1 0 times longer than the µ1 size (Bochi et al 1995) of ni ckel film  \ndomain. Therefore the nickel stray  fields decay  quickly  near the surface, a nd the basic mass of  \nelectrons does not “feel” the m. On the contrary , the interface st ray fields penetrate deep  into \nGaAs and dephase electron spins  inside. Therefor e, the Hanle effect magnetometer de tects the  \nferro magneti c interface Ni GaAs rather t han the Ni fi lm itself  due t o the space  selection of their stray  \nfields. sL≥-6 -4 -2 0 2 4 612\n(b) (a) \nSwitching field H* (Oersted)Zero-field PL polarization ρ(0) (%)PL Polarization ρ (%)\nExternal field H (Oersted)0 50 100 150 2001,52,02,5\n  \ncoercivity = 90  Oe\n0 50 100 150 2001,52,02,5 \ncoercivity = 45  Oe\nFigure 3. (a) T he Ha nle effec t in a Ni/n-Ga As struct ure th at was \npreviously magnetized  (filled circles) in  plane and  demagnetized  \n(open circles). T=4. 2 K. \n(b) The de gree of zero -field value of circul ar pola rization ()0ρ  \nversus swi tching field H*. Top  panel: switch ing in the d ark, \nBottom panel: illu mination by He-Ne laser with  power d ensity  \n5 W/cm2 was  on during switching process ( Adapted from \nDzhioev et al 1995). \n 5Optical or ientation in fe rromagnet/sem iconductor  hybrids \nThe bottom  curve in Fig.3b was measured under illumination by  He-Ne laser. The coercivity \nof interface d ecreased twi ce. This experiment appeare d to be the first exa mple of  the optical control of \nferro magnetism  in FM/SC hybrids. It  is discussed in detail in section 4. \nTo detect the  small stray  field ~1 Oe  Dzhioev et al ( 1995)  used n -type GaAs where at low \ntemperature  (T=4.2 K ) the halfwidth  corresponds to a very  long optical orientation  \nlifeti me . However the next attem pt (Crowell et al  1997) usin g semimagnetic \nsemiconductor ZnCdMnSe (instead of GaAs) with giant g-factor g~100 was unsuccessful because the \nlifeti me  was too short , so that . Recently  Meier et al (2006) have used  \nelectrons with longer spin  memory  in InGaAs S C quantum  well to succes sfully  observe  \nthe stray  fields fro m Fe patterned stripes, wh ere the str ay fields were enhanced up to ~ 200 Oe . Oe H 22/1=\nns Ts130=\nps Ts15= kOe H 1~2/1\nns Ts 5.1≈\n \n3. Exchange interaction in FM/SC hybrids \n \n3.1. Exchan ge interaction of magnetic atoms w ith semiconductor charge carriers at the FM /SC \ninterface.  \nThe exchange interaction between magnetic ato ms and sem iconductor charge carriers is very  \nimportant in FM/SC hy brids. Tunneling of SC ca rriers through the interface i nduces in equilibrium  \nproxim ity effect – spontaneous ordering  \nof electron spins localized near interface  \nfrom  the sem iconductor side.  \nEquilibrium spin polarization decay s \ninto the sem iconductor ov er the length \nof the order of de Brogli e wavelength  \nBrλ. Equilibrium proxim ity effect is \nconsidered in  subsection 3. 2. A num ber \nof new phenomena out of equilibrium \nare discussed in subsection 3.3. J\nSemico nductorFM insu lator\nMr\ntSr\n(a)J\nSemiconductorFM metal\ntSr\n(b)Mr\nFig.4. Pr oximity effect  in (a) FM -insulator/SC and (b) FM -\nmetal/SC h ybrids. Sp in sp litting of SC el ectron s (t-electron s) \nlocalized nea r interface indu ces their polarization in \nequilibriu m. In case (b) sp in sublevels are b roadened due to \nescape i nto the FM m etal.  \n \n 6Optical or ientation in fe rromagnet/sem iconductor  hybrids \n3.2. Equilibrium proximity effect. \nThe proxim ity effect in the insulating FM/SC sy stem  was considered theoretically  by Korenev  \n(1996, 1997).  Figure 4a shows a contact between FM  insulator and sem iconductor with electrons (t-\nelectrons) trapped on deep isolated centers near heterojunctio n. Tu nneling into t he ferro magnet lead s \nto their exchange interaction with FM atoms and splits the states of t-electrons with spins along ()↑ \nand opp osite ()↓ to the m agnetization Mr\n. The energy  (per unit area) of exchange interaction betwe en \nt-electrons with surface  concentration  and m ean spin tntSr\n and t he unif ormly magnetized \nferro magnet (Appendix) is:  \nωrr\nhrr\n⋅−=⋅−=tt tt ex Sn mSJn E ,                                                   ( 3.1) \nwhere J is the exchange constant determ ined by  the FM/SC wav efunction ov erlap1, mr is the uni t \nvector along Mr\n. Exchange interaction (3.1) is equivalent to the interaction of each t-electro n spin \nwith an effective magnetic field colline ar to mr, and ve ctor hrrmJ=ω  means the Lar mor preces sion \nfrequency  of the t-electron spin in this fi eld. In equ ilibrium  the mean spin polarization of t-electrons at  \ntemperature T \n T t STth Srhrr\n≡⎟⎠⎞⎜⎝⎛=2 21ω\nωω                                                  (3.2) \nis directed along ωr. \n Wavefunctions of t-electrons penetrate deep int o the FM in case of t he conta ct with m etallic \nFM fil m (fig .4b). This induces the broadening of the split ↑ ()↓ states by  the +γ (−γ) value,  \nrespectively .  McGuire et al  (2004) calculated param eters J, +γ and −γ for t he two-dim ensional \nelectron gas near the FM metal for diff erent barrier s at the interface. They  found that the proxim ity \neffect is ba sically  determ ined by  spin- dependent lifetim e broadening ±γ, whereas the spin splitt ing J \nis sizeable on ly for a very  thin barrier thickness. \n Exchange coupling takes place for the contact with p-type sem iconductor whe re the holes are \nmajority  charge carriers. Korenev (2003) considered  the interaction of the two-di mensional hole gas in \nquantum  well with magnetic ato ms of the nearby  insulating FM fi lm. The problem  is specifi c in that  \n                                                 \n1 Constant J can have any sign in spite of the ferromagnet ic ordering inside of FM.  \n 7Optical or ientation in fe rromagnet/sem iconductor  hybrids \nthe size quantization induces the splitting of the  heavy  and light hole states with an gular \nmomentum projections hl∆\n23± and 21±  onto  growth direction   with the heavy  holes being the  \ngroun d state. Merkulov an d Kavokin  (1995) f ound that the exchange interaction becom es anisotropi c \nif the splitting  is larger than the exchange constant : . Within this \napproxim ation onl y the out-of-plane magnetizatio n com ponent  induces the z\nhl∆hJzh\nzhh ex mSnJ E−=\n0≠zm 23±  states  \nsplitting. The  mean effecti ve spin of heavy  holes hSr\n is collinear to the z-axis. Its equilibrium  value is \ngiven b y the equation sim ilar to Eq. (3 .2) where the vector ωr has only z-co mponent hzh z mJ=ω . \n Myers et al ( 2004)  observ ed the hole s pin polarization in duced b y the pro ximity effect. The \nstructure consisted of GaAs quantum  well between AlGaAs barri ers. A ferro magnetic MnAs lay er was  \ninserted into one of the barriers. Gate b ias contro ls the overlap of the hole wavefunctions with MnAs. \nHole polarization was m easured b y the degree ρ of the circular polarization of lum inescenc e excite d \nby the linearly  polarized light. In the absen ce of exte rnal magnetic  field the ma gnetization Mr\n lies in  \nthe film  plane due to dem agnetizing field. Deviation of Mr\n out of pla ne induces equilibrium  hole spin  \npolarization . In wide range of bias the hole spin was opposite to t he Mn spin and along %6≤hS Mr\n \nin accordance  with anticipated antif erromagnetic sig n of Mn-hole excha nge interaction (Mn g-factor is  \npositive henc e magnetizati on Mr\n is antiparallel to the Mn spin and pa rallel ).  hSr\n \n3.3. Optically ind uced spin pol arization of  semi conductor electrons near FM /SC interface. \n3.3.1 . Dynamical proximity effect  in insul ating FM/SC hybrids. \nOptical orient ation of t-electrons in the i nsulati ng FM/SC (Fig.4a) was considered theoretical ly by \nKorenev (19 97). T he illu mination of t he hybrid by circularly  polarized light with the energ y νh of \nquanta greater than the SC energy  gap  leads to the optical orientation  of sem iconductor \nconduction band electrons. It is assu med that the band carriers do not interact with ferro magnet, so that \nonly the t-electrons, localized near interface, under go the exchange interaction with magnetic ato ms. \nUnder these conditions the mean spin gE\ncSr\n of the el ectrons is parall el to the exciting beam  and is \ndeterm ined b y its helicit y. The vale nce band holes are depolarized due to their fast spin relaxation. \n 8Optical or ientation in fe rromagnet/sem iconductor  hybrids \nThey recom bine with t-electrons. The o ptically orien ted conductio n band  electrons are trapp ed by the \nsurface states, thus replacing the reco mbined t-electr ons.  As a r esult, the non-e quilibrium  polarization  \nof t-electrons appears. The steady  state mean spin tSr\n obeys the Bloch  equation \n0=×+−+−ωττrrrrrr\nt\nst T\ntt cSS S S S                                                 (3. 3) \nThe first term  in Eq.(3.3 ) describes the genera tion of spin com pensated b y recom bination with  \ncharacte ristic time tτ. The second ter m takes into account the spin relaxation during tim e sτ to \nequilibrium , whereas the last ter m gives the precessi on with Larm or frequency ωr around ex change \nfield of m agnetic ato ms. The solution  of Eq.(3.3)  is \n[]\n222\n0\n220\n01 1ss\nss\nss\nT tTTS\nTT SSTS S\nωωω\nωω\nτ +××+\n+×++=rrrrrrrr\n                                    (3.4) \nwhere \nts\ncTS Sτrr\n=0 ; the optical orientation lifetim e  is determ ined by  the shorte st of the lifetime and \nthe spin relaxation tim e: sT\ns t sT ττ1 1 1 += . It follows from  Eq. (3.4) that even the non-polarize d \nexcitation  decreas es the mean spin  due to the replacement of polarized equilibrium t-\nelectrons with non- polarized ones. Thus the non-p olarized excitation decouples FM and SC. In case of \nthe circularly polarized excitation the mean spin of conductio n band electrons ( 00==cS S)tS\ncSr\n is not colli near to  \nmagnetizatio n Mr\n. Then new co mponents of the t-electron m ean spin tSr\n appear: (i) co mponent 0Sr\n \nparallel to the conduction band electron spin cSr\n, (ii) com ponent ωrr\n×0S  perpendicular to both cSr\n and \nMr\n, (iii) com ponent []0Srrr××ωω  directed from  0Sr\n to Mr\n. The latter two com ponents result from  the \nprecession of non-equilibrium  spin of t-electrons in the exchange field of the ferromagnet.  \n The i mportant role of optically  oriente d surface st ates has be en de monstrated  by Prins et al \n(1996) in the ir experim ents on the spin-dependent tunneling of s pin polarized electrons fro m GaAs \ninto a ferromagnetic Pt/Co m ultiplay er sam ple. \n \n3.3.2 . Dynamical proximity effect  in metallic FM/SC hybrids.  \nIn previous subsection we assu med that the proxi mity effect is absent for the conduction band \nelectrons. However, if the Schottky  barrier is transp arent (for exam ple due to the  intentional d oping of \n 9Optical or ientation in fe rromagnet/sem iconductor  hybrids \nsemiconducto r) the free electrons reach the interfa ce and undergo the exchange interaction with \nmagnetic ato ms. In this c ase ther e is noticeable  \nspin-depende nt am plitude of transm ission of  the \nsemiconducto r electron into the ferrom agnet.  \nSimilar to l ocalized stat es the fre e electrons  \nacquire the equilibri um spin polarization over the \nlength scale ~Brλ due to the proxim ity effect  \nwith the FM fil m. However out of equilibri um \nthe conducti on band  electrons are able to transfer \nthe non-equil ibrium  spin over the macroscopic \ndistance into sem iconducto r via drift-diffusion proce sses. Aronov and Pikus (19 76) suggested to inject  \nspin from  FM into SC  by externall y applied bias. The difference in electroche mical potentials of FM  \n()fµ and SC ()eµ induces both t he particle flux eqr and the spin flux (cu rrent) due to the spin-\ndependent tu nneling thr ough the heteroj unction (Fig .5а). Electrical spin injection was first realized by  \nAlvarado and Reneaud (1992). Recently this res earch has beco me very popular (Zutic et al 2004).  eµ fµeqr\nFM SCeµ\nfµeqr\nFM SC\nhqrhµ\n(a) (b)\nFigure 5. Band diagram of the FM /SC hybrid out of \nequilibriu m due to: (a) electric cu rrent; (b) pho to-\nexcitatio n \nThe electroche mical potential difference  can be cr eated optically , too. For example, absorption of \nnon-p olarized light  creates SC electron-hole pairs who se distributio n is determ ined by the q uasi-Ferm i \nlevels for electrons eµ and h oles hµ (Fig.5b). The holes are attract ed to the FM/SC  interface, thus \ndragging the electrons. Part of the electrons tran smits into FM; the other  part reflects from  the \nboun dary. Bulk non-equ ilibrium  spin p olarization of SC  electrons develops du e to the spin-dependent \nextraction  into FM with the polarization sign being opposite  to that in the case of injection. The spin  \ndensity  ccSnSrr\n=  of conductio n band electrons with concentration  in n-type SC is purely  non-\nequilibrium  because  in the bulk of non -magnetic SC. The spin dynam ics can b e described b y \nBloch equations taking i nto account diffusion, dri ft, spin relaxation and Larm or precession in an  \nexternal m agnetic field cn\n0=TS\nSS\nxJ\ntS\nsS rrrrr\n×+−=∂∂+∂∂Ωταα                                                        (3. 5а) \nwhere the vec tor of spin current in the -direction ( )  αx zyx x ,,=α\n 10Optical or ientation in fe rromagnet/sem iconductor  hybrids \nαααxSDSV JS\n∂∂−=rrr\n                                                        (3. 5b) \nαV is the −αcomponent of dr ift velocity , D – diffusion coefficient . The effici ency of spin  \ninjection/extr action is deter mined by  the processes at the FM/SC interface and is taken into account in \nthe boundar y condition for the norm al co mponent nr of spin current as  prop osed b y Aronov an d Pikus  \n(1976) \nma JS\nnrr\n⋅=                                                           (3.6) \nwhere the pa rameter  is deter mined by the spin-dependent transmission/refle ction at the interface.  \nParameter a depends on the applied bias in th e case of el ectrical injection (Fig.5 а) and b y the \ngeneration/reco mbination processes for the optical injection (Fig.5b). In any case the SC e lectrons  \nacquire the non-equilibri um spin density a\nmSrr\n∝ , which can be transferred  into sem iconductor.  \n Optically  induced spin pol arizatio n of SC electrons was observed by Kawaka mi et al (2001) in  \nMnAs/n-GaAs and GaMnAs/n-GaAs heterostructures. Under illumination by l inearly  polari zed light  \nelectrons acquire a non-equilibrium  spin density  Sr\n, whose direction is collinear to the magnetization  \nMr\nof nearby  FM. Vector Mr\n lies in the plane due to the dem agnetization field, wh ich excludes the \npossible role of the m agnetic circular di chroism  (MCD) effect. The spin density  inside a  \nsemiconducto r undergoes Larm or precession  around an external magnetic field Sr\nHr\n. Another evidence  \nof the accu mulated non-equilibrium  spin results from the dy namical nuclear p olarization. It is known  \n(Meier and Zakharchenya Eds. 1984) that out of equilibrium  electrons tr ansfer their angular \nmomentu m into the nuclear spin-sy stem due to the hy perfine interac tion. In turn, polarized nuclei  \ncreate an eff ective hy perfine magnetic  field chan ging the precession frequency  of electro ns in an \nexternal field. Kawakam i et al (2001) o bserved this  effect referred to as “ferro magnetic im printing of  \nnuclear spins”. Epstein et al (2003) stu died an optic ally induced dynamical proxim ity effect com bined \nwith an externally applied  bias. The y found t hat the value of the o ptically  induce d spin densit y mSrr\n∝  \nvaries strongly with b ias: it is m aximal at a forw ard bias. This result can be u nderstood fro m Fig.5b.  \nForward bias (+ at ferro magnet) rectifies the band bending, th us increasing the electron spin current \nthrough the FM/SC interface. It results  in the increase d spin density. \n 11Optical or ientation in fe rromagnet/sem iconductor  hybrids \n \n3.3.3 . Dynamic proximity effect for the non-collinea r spins of SC and F M at the interface \nConsider the case  when the spin density  Sr\n near the FM/SC boundar y is non-collinear to \nmagnetizatio n Mr\n. It appears, for exam ple, under opt ical orientation conditi ons, i.e. excitation b y \ncircular polarized light (Fig.1). Optically  oriented  electrons reach  the FM/SC boundar y, partly going  \nthrough (reflecting from ) it with the change of thei r initial spin ori entation. This  problem  reminds the \nwell-known problem  of scattering of the spin-polar ized electron  beam  by magnetic target (Kessler \n1985).  Besides the co mponents  and mr()Smmrrr⋅ along m agnetization, the reflected electrons acquire \ntransverse co mponents  and Smrr× ()Smmrrr×× , which describe the rotation of spin Sr\n and its \nrelaxation toward Mr\n, respec tively . It is re asonable to a pply the old methods exploiting the m atrix of  \ndensity  to the FM/SC prob lem. Ciuti et al (2002) cal culated the spin-dependen t reflection coefficients \nfor the FM/SC heterojunction within t he effective mass approach. A little b it earlier Brataas et al \n(2001)  consi dered spin the transport t hrough th e param agnet metal-FM junction for  non- collinear \npolarizations. They deduce d the bo undar y conditi on for the spin cu rrent  \n()[][][]SmmrSmcmSmbma JS\nnrrrrrrrrrr\n××+×+⋅+⋅=                                            (3.7) \nwhere the pa rameters  and b are determ ined b y the spin-dependent conductance  ( ) for the \nelectron spins along ( oppose) to the FM  magnetization, param eter (a↑g↓g\ncr) is given by im aginary  (real ) \npart of the so-called mixing conductanc e ( ). The third ter m descri bes the spin preces sion around \nan effe ctive exchange magnetic field, wherea s the f orth ter m gives the  relaxat ion of  toward ↑↓g\nSr\nMr\n. \nThey are si milar to the la st two terms in E q.(3.4) for the insu lating FM/SC  hybrid (subs ubsection \n3.3.1).   \n Strictly speaking not only the spin current but also the charge current through the FM/SC  \nboun dary is spin-dependen t. Johnso n and Silsbee (19 85) argued that the electron current at t he FM-\nparam agnetic metal interface \n() ()[ ]Sm g g jerr⋅+⋅+=↓↑αξ∆                                                        (3.8) \n 12Optical or ientation in fe rromagnet/sem iconductor  hybrids \nis determ ined not onl y by the electroche mical potential difference ξ∆ but also b y the projection o f Sr\n \nonto Mr\n.  \n An experim ental study  of the effects given b y the first term in Eq. (3.7) was discussed in \nsubsubsection 3. 3.2. To t he best of m y knowledge,  the experim ental observation of the other three \nterms in Eq.(3.7) has not been reported y et. Therefore, the experim ental proof of the Eq. ( 3.7) (an d \nEq.(3.4) in insulating hybri ds) for the non-collinear  spins rem ains a challenge for experi mentalists.  \nThe spin-dependent charge current of optically orie nted electrons Eq.(3.8) was studied by Prins et \nal (1996), Isakovic et al (2 001) and Hirohata et al (2001). The authors modulated spin density  Sr\n of \nSC electrons via modulation of the helicity  of phot ons. A magnetic field perpendicular to the structure \nplane broug ht the m agnetization out of the film  plane and induced t he non-zero p rojection ()0≠⋅mSrr\n. \nThe signal of  photocurrent through the interface was detected. At the sam e time Prins et al (1996), \nIsakovic et al (2001) stated the strong influence of the MCD effect, i.e. the difference in absorption \ncoefficients ()mSrr\n⋅∝α∆  of the left and right circul ar pola rizations. Hence care should be t aken fo r \nthe quantitati ve interpretation of t he results.   \n   \n3.3.4 . Dynamics of dynamic proximity effect \nEpstein et al (2002) observed the two-step dy namics of the non-equilibrium  spin polarization of  \nSC electrons due to t he dynam ic proxim ity effect . The electron spin polarization appears during the  \nfirst 50 ps aft er excitation b y linearl y polarized laser pulse – the fi rst step. The second step is the long-\nterm spin relaxation of the order of nano seconds. Bau er et al (2004) carried out calculation of the two-\nstep dynam ics for the case of uniform  spatial distri bution of electron spin insi de of SC, when drift-\ndiffusion length is larger than the SC t hickness. Th ey found that the first  step  results from the fast  \nwithdrawal o f photo-holes toward the i nterface under the action of a built-up electric fi eld (F ig.5b). In  \nturn, this induces the spin-dependent  transfer of  electrons int o the ferro magnet (spin extraction) \naccompanied  by the accum ulation of non-equilibriu m spin in sem iconductor. The non-equilibrium \nelectron spin survives tens of nan oseconds at lo w temperature in n-t ype GaAs based sem iconductors \n(Dzhioev et al 1997). Th is explains the long-term  stage of the spin d ynamics. Gridnev (200 7) \n 13Optical or ientation in fe rromagnet/sem iconductor  hybrids \ncalculated the nanosecond dynamics with the use of Eqs.(3.5, 3.7) for the case of thick sem iconductor \nwhen the spin densit y  is spatially  non-uniform . He found the non-u niform  distributio n of spin \ndensity due to the Larm or precession and spin relaxati on on its way into the inte rior of sem iconductor . Sr\n \n4. Optical control of ferromagnetism in FM/SC hybrid. \n \nIt is well known that light,  besides trivial heati ng, exerts non-therm al influence upon  the m agnetic  \nproperties of ferro magnetic sem iconductors (Nagaev 198 8), such as Curie tem perature, magnetic \nhysteresis loop, m agnetic anisotropy e tc. Recently  similar phenomena have been reported in new \nGaAs based ferro magnetic sem iconductors (Wang et  al 2005). This section considers the ph ysics \nunderl ying the optical cont rol of  ferrom agnetism  specifically  for the FM/SC h ybrids: m agnetism  of the \nhybrid (m agnetic hy steresis loop and the orientat ion of magnetization vector in space)  is tuned \nopticall y with the help of sem iconductor. Th e electrical manipulation of ferro magnetism  was \nconsidered b y Zakharchenya  and Korenev (200 5).  \n \n4.1. Increase of the exchange s tiffness coefficient of t he FM/SC with electron accumulation l ayer. \nIn case of an  electron ac cumulation la yer the exchange interacti on of m agnetic ato ms with SC \nelectrons is isotropic. Kor enev (199 6) considered  the exchange interaction between magnetic atoms  \nand SC elec trons localized at deep centers n ear the FM/SC interface (s ubsection 3.2, Fig.4a). \nExchange favors collinear orientation of  t-electron sp in and the m agnetic atom  spins located within the  \nlocalization a rea of the t- electron. The local enha ncem ent of the ferro magnetic ordering increases the \nexchange rigidit y coefficient of the hybrid in equilibri um.  \nQuantitative analy sis is based on the Eq.(3.1) t hat should be m odified in t he non-unif orm case \n(Korenev 1996, 19 97). Assu me that magnetization ()xmr changes orientation in on e direction x over  \nthe distance δ (dom ain wall thickness) m uch larger then the loca lization radius . Within t he \nmacros copic approach (see  Appendix) the exchange e nergy per unit surface ar ea  ta\n()()()()x xSxn xEt t ex ωrr\nh ⋅ −=                                                      (4.1) \nwhere the fre quenc y  \n 14Optical or ientation in fe rromagnet/sem iconductor  hybrids \n()()()\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n+ = xmaxmJxxt ''2\n2r r\nhrω                                                    (4.2) \ncontains in c omparison with Eq.( 3.1) an additi onal term with the second space derivative. It results \nfrom  the smooth variatio n of  within the localization area  and represents the first \nnontrivial term of expansion of  the exchange in seri es in powers ()xmr2~taπ\n() 12<<δta . As a result the mean  \nspin of t-electrons also depends on x: Eq.(3.2) gives its equilibrium  value with vector ()xωr being  \ntaken from  Eq.(4.2). At low te mperatur e the mean s pin()()()x x xSt ωω 2rr\n= . Substitut ing the ()xStr\n \nand Eq. (4 .2) into Eq.  (4.1)  one finds \n()2\n1\n02⎟⎠⎞⎜⎝⎛\n∂∂+=xm AxEexr\nε                                                     ( 4.3) \nwhere 2/0 tnJ−=ε  does not depend on the distribution of m agnetization in space; the ex change  \nstiffness coef ficient 2/2\n1 ttanJ A= ; the i dentity ()2m mm ′−=′′⋅rrr was used. Eq.(4.3 ) shows that the \nexchange interaction incr eases the exchange stiffness of the hy brid by A1 value in agree ment with \nqualitative di scussion.  \nThe increase of the exchange stiffn ess coefficient gives an additiona l contribution to the coercivit y \nof the hybrid. However let us recall first the origin of coercivity . \n \n4.2. Origin of coercivity (Chikazumi 198 4). \nThe width of  the magnetic h ysteresis loop is ofte n determ ined by t he displacement of the domain  \nwalls separat ing adjacent domains. Space fluc tuations of the anisotropy  constant  or the \nexchange stiffness coeffici ent  about t heir average values  and  induce the fluctuations of  \nthe dom ain wall energy  ()xK∆\n()xA∆0K0A\n() ()[] ()[ ]xK KxA A x ∆+⋅∆+0 0 ~γ . The potent ial relief ()xγ  fixes the \ndomain walls giving the source of coercivit y. Coercivit y  is determ ined by  both the am plitude of \nfluctuations ch\nγ∆ and their characteristi c length . An external magnetic field H exerts press ure Λ\nMH P2=  on 1800-domain wall. It is co mpensated by  the pressure ()\nΛ∆\n∂∂γγ~xx due to the \nfluctuations. The coercivity  can be es timated by  equating the pressure at  to the maximum ch H=\n 15Optical or ientation in fe rromagnet/sem iconductor  hybrids \nrestoring for ce ()\nδγ γ ∆\n∂∂= ~ 2\nmaxxxMhc . Here we have taken into accoun t that the maxim um is \nreached whe n Λ is close to the dom ain wall thickness 0 0KA=δ :  pinni ng is negligi ble at \nδ>>Λ , whereas the effect of fluctuations is averaged out over the wall thickness at  δ<<Λ . We \nconclude that one should decrease the amplitude of  potential relief to decrease coercivity . For sm all \nfluctuations ()γγ<<∆  we esti mate MK hc∆~  for the fl uctuations of the anisotrop y constant and  \n()2~ δ⋅∆ MA hc  for the fluctuations of the exchange stiffness coeffici ent.  \n \n4.3. Optical c ontrol of  coercivity of the FM/SC with electron accumulation layer. \nIn case of a two-dimensional electron accu mulation layer the exchange of magnetic ato ms \nwith SC ele ctrons is isotropic and affe cts the exchan ge stiffness  coefficient  (subsection 4.1 ). The \nfluctuations  of the surface density   of t-electrons cr eate fluctuati ons 1A\ntn∆ ()xnt 2/2\n1 ttanJ A∆=∆ . If \n then the coercivity  is enhanced in the dark by  the  value \n(  is the m agnetic moment per unit surface area, d is the width of the FM film ). Under \nillum ination of the hybri d by light  with power density W and p hoton  energ y t tn n~∆ ()2\n1/ ~0 δ δsurf c MA h\ndM Msurf⋅=\ngE h>ν  the photo-\nexcited holes m ove to the interface (Fig.5b) and recom bine with t-electrons, thus decreasing both  \nand coercivity (Korenev, 1 996)  tn\n   ()()\n01/ 10\nWWhh Whc\nc++=δ                                                                (4.4) \nwhere W0 is the characteristic power density depen ding on the absorption coeffi cient and the  capture \nefficiencie s of electrons and holes by t-c enters, para meter  takes into account ot her contributio ns to \ncoercivity . 1h\nThe optical control of the coercivity  of FM/SC hybrid has been d emonstrated experim entally b y \nDzhioev et al (1994, 19 95) in a Ni/n-GaAs structu re. We have already  con sidered this system in \nsubsection 2. 2 and fou nd that the optic ally oriented electrons rep resent a s ensitive detector of weak  \nstray  fields of the Ni-Ga As interface r ather than Ni film itself. The  important feature of the N i/n-GaA s \nhybrid consis ts in the optical tunability  of th e inte rface ferro magnetism . Figure 6 shows that th e \n 16Optical or ientation in fe rromagnet/sem iconductor  hybrids \nillum ination by He-Ne laser ( eV96.1=νh ) decreas es the interf ace coer civity  by half (co mpare al so \nthe lower dependence on  Fig.3b with  the \nupper one). However illumination doe s not \naffect the coercive force of  nickel fil m. The \neffect  we  called photocoercivity  is not \nsensitive to t he light polarization and takes \nplace only in the illum inated region. \nPhotocoercivity takes place at a low powe r \ndensity  (a few mW/cm2) and is not related  \nto the heating of sam ple: the heating of the \nhybrid by pa ssing the dc electric curre nt across the FM/SC heterojunction, w ith dissipated power \nbeing 1 0 times larger than the power of li ght, remained the coercive forc e unchanged. Spectral \nmeasurements have sho wn that the phot ocoercivit y is due to the effect of sem iconductor o n \nferro magneti c interface: it diminishes if the photon en ergy is below  the energy  gap of GaAs. The solid  \nline in Fig.6 i s calculated with the use of Eq.(4.4) for () Oe h hc 4501 ==δ ,  showin g \ngood  agreement. 2\n0 / 8 cm mW W=\nhc, Oe \nW, mW/cm2\nFig.6. De pende nce of the interface c oercivity on light \nintensity. Adapted from Dzhi oev et al 1995. \n The question about t he origin of  the interf ace ferr omagnetism  remains for future studie s. \nLahav et al (1986) have shown that already  at , Ni diffuses i nto GaAs to  form an \ninterm ediate layer. This lay er may be ferro magnetic and form  what we understand by  “int erface”. \nHowever, I have no infor mation on the  research in magnetism  of NiGaAs co mpoun ds. C T0100≥\n \n4.4. Surface a nisotropy in t he FM /SC with hole accum ulation layer. \n Another situation arises from the contact of  the FM/ SC with a hole accu mulation lay er. For \nexample, it ta kes place in inversion la yers, with stro ng band bending (Fig.7), or in case of the p-ty pe \nquantum  well near FM. Size quantization leads to anisotropic exchange interaction (subsection 3. 2) \nwith the energy  per unit area  \nzh\nzh zh\nzhh ex Sn mSnJ E ωh−= −=                                                (4.5) \n 17Optical or ientation in fe rromagnet/sem iconductor  hybrids \nand frequency hzh z mJ=ω . At low te mperature t he holes are com pletely spin-polarized due to \nproxim ity effect (subsection 3.2) wit h mean spin z zh\nzS ωω2= . Then the exchange energy \nhzh\nex nmJE2−= . One can see that the exchange interaction can be  considered as a peculiar magnetic \nsurface anisotropy  with easy  axis be ing directed alo ng z-axis (pe rpendi cular anisotropy ). This is in  \nstriking contrast with the case of an el ectron accumulation la yer (subsection 4.1). If the exchange  \nconstant and hole concentration are strong enough , then t he orientational  transition is  possible \n(Korenev 200 3): the m agnetic moment leaves th e plane and becom es oriented along the n ormal. \n  \n4.5. P hotocoe rcivity in the FM/SC with the hole accumulation l ayer. \nSimilar to su bsections 4.2, 4.3 the fluctuations  of exchange coupling between magnetic atom s and \nSC holes induce fluctuations of the an isotrop y cons tant hhnJ K~δ  increa sing the coerci vity in the  \ndark b y ()surf hh c MnJ h ~0δ .  The fluctuations Kδ \ncan be much less than  the averag ed anisotropy \nconstant , and no visible change of the average \nmagnetic anisotropy  takes place. How ever they  are \ncrucial for t he coercivity. The illu mination of the \nhybrid m ay decrease the exchange constant  due to \nthe band flatt ening, th us decreasing the overlap of th e \nhole wavefunctions with m agnetic ato ms (Fig.7). 0K\nhJFerromag net Semiconductor\nFermi level in\nequil ibrium\nHole accumulation layer\nFig.7 Band diagram of the FM/SC hybrid with \nhole inversion layer \nOiwa et al (2 001) dem onstrated the optical contro l of the coercive force of p-InMnAs FM grown  \non no n-magnetic GaSb sem iconductor.  The authors observed that the photo-ex citation of th e hybrid \ndecreased the coercive force. Th is effect was the strongest unde r the illum ination by  light with the \nphoto n energ y larger than  the energ y gap of n onmagnetic sem iconductor  GaSb, th us poi nting t o the \nimportant role of SC. Strong bend bending in GaSb (si milar to the Fig.7) takes place near the FM/SC  \ninterface in the dark. This  leads to accum ulation of a substantial num ber of holes in the vicinity  of \ninterface. Aut hors explained the photocoercivity  as due to the change of ferro magnetism  by the holes \nexcited in the GaSb and transferred in to the In MnAs lay er. Another explan ation follows from  the \n 18Optical or ientation in fe rromagnet/sem iconductor  hybrids \nabove discussion: o ne cou ld take int o account the w eakening of t he FM/SC ex change coupl ing near \ninterface. \n \n4.6. P hoto-i nduced change  of the Curie t emperature.  \nLight can aff ect the Curie temperature  , too. For exam ple, the photoexcitated carriers may \nchange the exchange constants of ferrom agnets. The light-induced increase in Curie te mperature  \nis well known in ferrom agnetic sem iconductors (N agaev 1988). A sim ple estim ation shows that this \neffect is pretty  sm all: CT\nCT∆\n() K T Nn TC fm C 1.0≈ ≈∆  if the concentration of m agnetic ato ms \n,  and the con centration of photocarriers .  3 2210~−cm Nfm K TC1000=3 1810−= cm n\nIt is reasonable to expect a si milar s mall change in the FM/SC hy brids. Ho wever, Koshihara et al  \n(1997)  claim ed the ph oto-induced ferrom agnetic orde r in p-InM nAs/GaSb hybrid persistent up  to \n35 K. The authors m easured the magnetization co mponent n ormal to the sam ple plane vs m agnetic \nfield with th e use of SQUID and an omalous Hall effect. They  found no n-hysteretic behavior of \nmagnetizatio n in the dark. In contrast, excitation w ith ph oton en ergy larger t han the energ y gap of  \nGaSb induced clear hy steresis. The authors inte rpreted this effect as a p aramagnet-ferromagnet  \ntransition of magnetic semiconductor In MnAs due to the transfer of SC photo- holes into InM nAs FM. \nIn other wor ds, the change K TC35≈∆  takes place under the change  in hole  \nconcentration inside of I nMnAs (from  3.76*103 1810−≈∆ cm nh\n19 cm-3 in the dark up to  3.90*1019 cm-3 on the light). It \nis not clear whether the existing theo ries (for ex ample, Dietl et al 2000, Kam inski and Das Sarm a \n2003) are abl e to explain t his unusual ef fect. \n \n4.7. Effect of circularly polarized light o n the magneti zation.  \n The circularly  polarized light transfers the angular m omentum  per photo n. Hence its \nabsorption m agnetizes the sam ple. For exam ple, it ex cites spin oriented charge carriers ( Meier and \nZakharchenya 1984). Ordi narily  their concentration is much sm aller than that of m agnetic atoms, so  \nthat the direct magnetizatio n by light is i nefficient.  Much stronger effect consists in the appearance o f \nan effective magnetic field pro portio nal to the degree  of circular polarization of light . If the field  \nvalue is larger than coercivity, then the s ample is magnetized even in the ab sence of external magneti c h±\ncP\n 19Optical or ientation in fe rromagnet/sem iconductor  hybrids \nfield. Van der Ziel et al (1 965) pro posed inverse Faraday  effect: the circularly  polarized light acts on \natoms in non-absorbing  media as a magnetic field lifting Kram er’s degeneracy . This field is relatively  \nweak: ~0.01 Oe at power density 10effH7 W/cm2 for Er+2:CaF 2. Therefore optical pulses heating the \nmagnetic sy stem  close to TC shoul d perform  the m agnetizatio n. Stanciu  et al (20 07) re ported t he \noptical m agnetization of th e ferrim agnetic GdFeCo.  \nEffective magnetic fields can be create d under adso rption of the circular polarized light. In this \ncase optically  oriented carriers undergo a strong exch ange interaction with m agnetic ato ms. Merkulo v \nand Sam sonidze (1980) considered theoretically  the do main wall motion un der the actio n of the  \ncircularly  polarized light exciting the fe rrom agnetic  semiconductor with perpend icular anisotrop y. The \nmagnetic circ ular dichrois m (MC D) effect leads to th e optical orientation of carriers: a large r electron \nconcentration is excited in dom ains with one orienta tion of Mr\n with respect to the dom ains with the \nopposite Mr\n. Then the size of one t ype of domains increas es at the expense of the  others.  Mer kulov  \nand Sam sonidze (1980) solved the d ynamic Landau-Li fshitz equation and fou nd that the do main wall \nmoves as if it was an effective magnetic field whos e value and direction is de termined by  the helicity \nof light.  Nagaev (1988)  reviewed early  experim ents in ferro magnetic sem iconductors on this t opic.  \n \n4.7.1 . Effective magnetic fields in FM /SC with electron accumulat ion layer.  \nKorenev (199 7) considered theoretically  the action of circularly  polarized light on the dom ain \nwall in the F M/SC hy brids. He argued that the effective field effHr\n appears as a result of a pres sure \nexerted by  the optically  oriented sem iconductor el ectrons on the dom ain wall. Both the usual optical \norientation of  carriers in non-m agnetic semiconducto r and the MCD effect resu lt in the appearance of \nthe effective field whose value and directi on are determ ined b y the angular m omentum cPr\nh⋅ of the \nphoto n. Imagine a thin FM film  with perpendicu lar anisotrop y grown on the sem iconducto r surfac e \nwith distribut ion ()xMr\n in a 1800-wall vary ing in one direction x (Fig.8). Below we shall assu me the \nusual distribu tion of  within the dom ain wall ()rmrr),( cos)( x xmzθ=  ),(sin)( x xmyθ=   ,0)(=xmx\n)/( )( cos δ θ xth x−=  (Chikazum i 1984).  Let t he exchange in teraction between magnetic atom s and SC \nt-electrons b e small enough not to affect both the dom ain wall distributio n and thickness δ. The \n 20Optical or ientation in fe rromagnet/sem iconductor  hybrids \ndomain wall centered at the position  undergoes th e pressure due to the exch ange interaction  \nbetween magnetic ato ms and SC electrons 0xx=\n()()∫ ∫+∞\n∞−+∞\n∞−∂−∂=\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n−∂∂−= = dxxxxSn dxxxExH M Ptt ex eff surf\n00\n0\n02ωrr\nh                                  (4.6) \nwhere the exchange energy per unit surface ar ea  and ()xEex ()xωr are give n by Eq. (4.1) and  Eq. \n(4.2), respectively. Space derivation in Eq.(4.6) \nshould be perform ed under fixed ttSnr\n in spit e of \nthe fact that it may (and really  do) depends on  \n()xmr distribution. This is becau se the cal culation \nof force in open s ystem s should be perform ed \nunder fixed external conditions (Landau and \nLifshits 1979 ). In our case the spin s ystem  of t-\nelectrons is a n external sy stem with resp ect to the ferromagnetic one.  Fig.8 Bold arr ows show the magnetization profile \nwithin the  180-wall. Thin solid arrows show t he \nprojection of the initial spin (up) onto vectorSemicondu ctorFerromagnetz\ny\nxθ\nωr\nIn equilibrium the force will be absent. I ndeed, the substitution of equilibr ium value TSr\n \nEq.(3.2)  into Eq.(4.6)  gives zero. Hence the n onequil brium  spin d ensity of t-el ectrons ttSnr\n shou ld be \ncalculated. In the MCD case the mean electron spin i s in equilibri um and com pletely  spin- polarized at  \nlow te mperature, whereas t he concentration of t-electrons is different in different dom ains \n))( 1( )( xmP nxnc t trr\n⋅+=γ , ()()()x x xSt ωω 2/vr\n=                                             (4.7) \nParameter γ characterizes both t he value and sign of dichroism . Substituting Eq.(4.7) i nto Eq.(4.6) we  \nreproduce Merkulov and  Sam sonidze (1980) result deduced from  the Lan dau-Lifshitz d ynamic \nequation \n21\n3δγ\nsurfc effMAP H=                                                     ( 4.8a) \nAlternatively  one m ay consider the case of opticall y oriented electrons when the non-equili brium spin \n is governed by  the equation (3.4) b ut the concentration  is the same in different \ndomains (no MCD effect). Taking  into account that only the pr ojection tSr()x constnt=\n()2ωωωrrr\n⋅tS  of the  spin tSr\n \nof t-electrons is conserved in the case of strong exchan ge 1>>sTω  we obtain from  Eqs.(3.4, 4. 6) \n 21Optical or ientation in fe rromagnet/sem iconductor  hybrids \n21\n32\nδτsurf ts\nc effMA TS H=                                                     ( 4.8b) \n Note that the expressions (4.8) for the field effHr\n contain a sm all param eter \n3 2 2 210 10~− −− δta  (for t ypical values of localization radius , and domain wall thickness nm at1~\nnm30~δ ), so that for the point defect . This is the resul t of the flexibilit y of t he \nsemiconducto r electron spin sy stem  adjusti ng the dire ction of the m ean spin to 0=effH\nMr\n.  \n \n4.7.1 . Effective magnetic fields in FM /SC with hole a ccumulation l ayer. \nThe effective  field value increas es dras tically  if th e direction of t -electron spin is fixed, thu s the \nspin precessi on in the ferromagnet exchange field is  absent. Suppose that a p-type accu mulation la yer \nwith hole concentration  is form ed in sem iconductor (or quant um well filled with holes) near th e \nFM/SC interface. Due to t he reduced sy mmetry the hol e spin states are split into heavy  and light hole  \ndoublets as discussed in subsections 3.2,  4.4. Then the hole spin hn\n2z hm S=  is fix ed along  z-\ndirection. Taking i nto account t he expr ession for th e Larm or frequenc y ()hxmJzh z=ω  and using \nEq.(4.6) we get for the MCD cas e \nsurfhh\nch\neffMnJP H6γ=                                                    (4.10a) \nand for the case of the heavy hole o ptical orientation \n \nsurfhh s\nhh\neffMnJTS Hτ=                                                    (4.10b) \nOne can see  that the  field is indeed greatly  (effH2 2\ntaδ  times) enhanced due to the rigidity  of the \nhole spin s ystem.  \nThe effective field will shift the magnetic hy steresis loop M(H)  by the Heff value over the H \naxis. Let us esti mate the  field value for  the FM/SC couplin g const ant  and t he hole \nsurface conce ntration . We take  Bohr magnetons per unit effH eV Jh 1.0=\n2 1210−= cm nh15102 2 ⋅≈ ≈ dN Mfm B surfµ\n 22Optical or ientation in fe rromagnet/sem iconductor  hybrids \narea for the FM fil m thickness  and the concentrati on of FM atoms . For \nthe MCD cas e we find from  the Eq.(10a) that =1400  Oe under favorable condi tion nm d1=3 2210−≈ cm Nfm\neffH 1=γ   and \n for the 100 % circularly  polarized light. For the case of the optic al orientation  of holes we get \nfor 1=cP\n1 2=t scTSτ  a value 4200 Oe. We conclude that the ex change coupling with holes loo ks very  \nprom ising for the optical control of ferrom agnetism  of the h ybrid.  \nOiwa et al (2002) observed the m agnetization of GaMnAs FM fil m by the circ ularly po larized  \nlight in the GaMnAs/GaAs hy brid. T he authors e xplain it b y the photo-creation inside GaMnAs of \nspin-oriented holes, which dynam ically polarize th e Mn spins . However, a very  sharp spectral  \ndependence correlates with the excitatio n of parama gnetic GaAs rather than  ferrom agnetic GaMnAs \nwhose spectrum  is very  smooth due to a strong disorder. This fact provides a strong evide nce of the \ncrucial role of GaAs excit ation in the optical magnetization of GaMnAs. Th erefore, one could also \nconsider the possible role of opt ically o riented hol es in GaAs and their exchange with m agnetic atoms  \nas discussed in this section.  \n \n5. Summa ry \nSpin-spin i nteractions in the FM/SC h ybrid lead to a strongl y coupled  spin s ystem  of \nferro magnet and sem iconductor. On t he one hand t hey in duce the proxim ity effect – spin polarization \nof sem iconductor electrons. Hence semiconductor electrons monitor the magnetic state of the \nferro magnet. On the ot her hand the  magnetic propert ies of the unified sy stem  differ drasticall y from \nthose of the F M film  alone. The m agnetism  of the en tire sy stem  can be controlled opticall y. As a result \nthe hy brid constitutes an elem entary  magnetic st orage with the sem iconductor being not onl y a \nsubstrate but an active participant in inform ation processing. An additional degree of freedo m \nconsisting in  the choice of desirable FM/SC pa ir am ong param agnet sem iconductors a nd a large  \nnumber of ferromagnetic materi als provides m any possi bilities. The ulti mate goal is the discovery  of \nFM/SC h ybrids with the o ptical control  of m agnetism  at room  temperature. The m ost prom ising for  \nthis purpose seems to be the FM/SC sy stem operati ng on the o ptically tunable pr oximity effect.    \nAuthor greatly  appreciates I.A. Merkulo v for va luable rem arks. The paper is supported in pa rt \nby RFBR, Russian Science Supp ort Fou ndation an d program s of Russian Academ y of Sciences. \n 23Optical or ientation in fe rromagnet/sem iconductor  hybrids \nAppendix \nEquations (3 .1, 4. 1) can be derived from  the Ham iltonian ∑ −⋅ =\njij i j i\nfmRrIsINJH\n,) (ˆˆ ˆrrrrδ  \ndescribing th e isotropic exchange interaction be tween t-electrons and m agnetic atoms with spin I and \nconcentration . Here fmNisˆr and jIˆr\n are the operators of spi ns of the i-th t-electron and j-t h magnetic \natom located at irr and jRr\n, respe ctively . Deep centers ar e assumed to be isolated ( ).  12<<ttan\nAssu me that  the localizat ion radius of t-electron  is much large r than the distance  \nbetween magnetic ato ms. In this case ferro magnet ca n be consid ered as a co ntinuo us m edium with  \nmacroscopic m agnetizatio n ta0a\n()rMrr\n. Replacing the spin density operator ∑− =\njj j RrI rI ) (ˆ)(ˆrrrrr\nδ  with \nits clas sical v alue )( )( rmIN rIfmrr rr\n−=  (the sign is negative because the spin of the m agnetic ato m is \nusually  ant iparallel to its magnetic moment) we arrive  at the mean-field H amiltonian  \n. It describes the interaction of each el ectron spin with an effective non-unif orm \nmagnetic field ∑⋅−=\nii i rmsJ H )(ˆ ˆrrr\n()rmrr∝ . If the exchange interaction acts as perturbation we obt ain the spin  \nHam iltonian  \n∑∫−=\nii i S rdrmr sJ Hrrrrr)()(ˆ ˆ2Φ                                                 (A1) \nby averaging the mean-fi eld Ha miltonian w ith the groun d state orbital wavefunction ()rirΦ  of t-\nelectron with coordinate rr located near t he i-th center. It is reasonable to consider the magnetization  \nto be a slowly  varying function, because the char acteristic distan ce nm30≈δ  is much larger than  \n. Then the integral in Eq.(A1) can be expanded in series in  powers nm at1≈ ()δta . If the  \nmagnetizatio n ()xmr varies only along x-direction then  we have up to  the second order term s \n() () ∑⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛′′+⋅−=\niit\ni i S xmaxmsJ Hr rr\n2ˆ ˆ2\n                                                (A2) \nHere magnetization and its second de rivative are taken at the position  of the i-th center.  \nLocalization radius is de termined by  the relation ix\n() () ∫Φ−≡ rdr xx ai i trr22 2. Equation (A2) has a form  \n 24Optical or ientation in fe rromagnet/sem iconductor  hybrids \n∑⋅=\nii i S s H ωrrhˆ ˆ , where the vector is giv en by Eq.(4 .2) and  sum ming goes ove r all param agnetic \ncenters. Introducing the sp in densit y operator of t-electrons ()()i\nii rrs rSrrrrr\n− ≡∑δˆ ˆ we can present it in \nequivalent for m \n()()∫⋅= rdrSr HSrrrrrh2 ˆ ˆω                                             (A3) \nWe can further treat the p roblem  macroscopically  if the num ber of centers wi thin the characteristi c \narea  is very  large: . Replacing in this cas e the spi n density  operator 2δ≥ 12>>δtn ()rSrrˆ by its \nclassi cal value ()()rSrnt trrr we arrive a t the quasiclassical expre ssion for the total interaction energy  \n() ()()() ∫ ∫⋅ = = rdr rSrn rdrE Et t exc Totalrrrrrrhrr2 2ω                               (A4) \nThis justifies the Eq.(4.1) f or the exchange energy  per unit area ()r Eexcr, which reduces to the Eq.(3.1) \nfor the unif orm Mr\n.  \n The lim its of applicabil ity of t he macros copic description  bring ab out the inequa lity \n2 2 2\n0 1 1 /1 δ>>>>>>t t n a a . For ,  and  it is satisfied i n \nthe wide concentration rang e cm a8\n0 103−⋅= cm at710−= cm6103−⋅=δ\n[]2 14 1110...10−∈ cm nt .  \n 25Optical or ientation in fe rromagnet/sem iconductor  hybrids \nRefe renc es: \nAlvarado S F  and Renaud  P 199 2 Observation of sp in-polarized-electron tunneling from  a ferromagnet  \ninto GaAs Phys. Rev. Lett.  68 1387-9 0 \nAronov A G and Pikus G E 197 6 Spin  injection in sem iconductors Sov. 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Phys.  \n76 323-41 0 \n 29"    },    {        "title": "1208.1580v1.Magnetic_properties_of_charged_spin_1_Bose_gases_with_ferromagnetic_coupling.pdf",        "content": "arXiv:1208.1580v1  [cond-mat.stat-mech]  8 Aug 2012Magnetic properties of charged spin-1 Bose gases with ferro magnetic coupling\nJihong Qin∗, Xiaoling Jian and Qiang Gu\nDepartment of Physics, University of Science and Technolog y Beijing, Beijing 100083, China\nMagnetic properties of a charged spin-1 Bose gas with ferrom agnetic interactions is investigated\nwithin mean-ﬁeld theory. It is shown that a competition betw een paramagnetism, diamagnetism\nand ferromagnetism exists in this system. It is shown that di amagnetism, being concerned with\nspontaneous magnetization, cannot exceed ferromagnetism in very weak magnetic ﬁeld. The critical\nvalue of reduced ferromagnetic coupling of paramagnetic ph ase to ferromagnetic phase transition Ic\nincreases with increasing temperature. The Lande-factor gis introduced to describe the strength\nof paramagnetic eﬀect which comes from the spin degree of fre edom. The magnetization density\nMincreases monotonically with gfor ﬁxed reduced ferromagnetic coupling IasI >Ic. In a weak\nmagnetic ﬁeld, ferromagnetism makes immense contribution to the magnetization density. While\nat a high magnetic ﬁeld, the diamagnetism inclines to satura te. Evidence for condensation can be\nseen in the magnetization density at weak magnetic ﬁeld.\nPACS numbers: 05.30.Jp, 75.20.-g, 75.10.Lp, 74.20.Mn\nI. INTRODUCTION\nThemagnetismofFermigaseshasalwaysreceivedcon-\nsiderable attention in solid-state physics, such as local-\nized and itinerant electrons. While the magnetic prop-\nerties of Bose gases has been less studied. But since the\nrealization of Bose-Einstein condensation (BEC) in ul-\ntracold atomic gases1, more interests have been cast to\nthis system. The Bose gases plays an important role\nin understanding some exotic quantum phenomena, such\nas superconductivity and superﬂuid. The ideal charged\nbosons were used originally to describe the supercon-\nductivity. It has been shown by Schafroth2, Blatt and\nButler3that an ideal gas of charged bosons exhibits\nthe essential equilibrium features of superconductor. Al-\nthough the Bardeen-Cooper-Schrieﬀer(BCS) theory4ex-\nplained the microscopic nature of conventional supercon-\nductivity, the charged Bose gas exhibits strong diamag-\nnetism at low temperature, which can be attributed to\nMeissner eﬀect. In recent years, the normal-state dia-\nmagnetism of high-temperature cuprate superconductors\nhas been explained by real-space charged bosons5. This\nalso recasts new research interest in charged Bose gases.\nExperimentally, since the realization of spinorBEC in\noptical traps6,7the magnetic properties of spinorBose\ngases has received considerable attention. Moreover, an\nultracold plasma can be created by photoionization of\nlaser-cooled neutral atoms8. The temperatures of elec-\ntrons and ions can reach as low as 100 mK and 10 µK,\nrespectively. The ions can be regardedas charged bosons\nif their spins are integers. The Lande-factor for diﬀerent\nmagnetic ions could also be diﬀerent.\nItisknownthatparamagnetismisfromthespindegree\noffreedomofparticles. WhilechargedspinlessBosegases\ncan exhibit strong diamagnetism, similar to Meissner ef-\nfect, which comes from the orbital motion of charge de-\n∗Corresponding author, E-mail: jhqin@sas.ustb.edu.cngree offreedom in magnetic ﬁeld. Theoretically, both the\nparamagnetism9,10in neutral spin-1 Bose gases and the\ndiamagnetismofthechargedspinlessBosegases11,12have\nbeen studied. Moreover, we13have discussed the compe-\ntition of paramagnetism and diamagnetism in charged\nspin-1 Bose gases in external magnetic ﬁeld, using the\nLande-factor gto evaluate the strength of paramagnetic\n(PM) eﬀect. It is shown that the gasexhibits a shift from\ndiamagnetism to paramagnetism as gincreases.\nThe ferromagnetism and superconductivity are not\ncompatible in conventional physical models. The\nMeissner-Ochsenfeldeﬀect shows the conventional super-\nconductor cancels all magnetic ﬁeld inside when the tem-\nperature below the superconducting transition tempera-\nture, which means they become perfectly diamagnetic.\nThe discovery of several ferromagnetic (FM) supercon-\nductors in experiments14–16stimulates the research in-\nterest in the exotic magnetic properties of FM supercon-\nductors. The state of the Cooper pairs in the FM su-\nperconductors has been wildly studied14–18. A stronger\nspin-orbit interaction in UGe 2results in an abnormal\nhuge magnetocrystalline anisotropy14–16. Monthoux et\nal.18indicates that the favorite superconducting pairing\ntype of this anisotropy is triplet. Although the exact\nsymmetry of the paired state has not yet been identi-\nﬁed, a spin-triplet pairing is more likely than the spin-\nsinglet pairing in these superconductors14–16. These be-\nhaviors are somewhat like charged spin-1 bosons. Thus\nthe charged spin-1 boson model helps to understand the\nexotic magnetic properties observed in such materials.\nAlthough the ferromagnetism19–24in a chargeless\nspinor Bose gas has also been involved in theory, it is\nlittle discussed when FM interaction exists in a charged\nspin system. Accordingly the magnetic behavior will be-\ncome more complex in charged spin systems with FM\ninteractions, where diamagnetism, paramagnetism and\nferromagnetism compete with each other in such case.\nIn this paper, the magnetic properties of a charged\nspin-1 Bose gas with FM interactions are studied via\nmean-ﬁeld theory. Alexandrov et al. found that the2\nCoulomb or any other scattering may make chargedBose\ngases superconducting below a critical ﬁeld25with a spe-\nciﬁc vortex matter26. Superconducting is not obtained\nin our paper, probably because we used the mean-ﬁeld\napproximation to deal with the FM interaction. In de-\nspite of this, mean-ﬁeld theory is still eﬀective to point\nout the main physics of the magnetism, especially the\nferromagnetic transition21. The remainder of this pa-\nper is structured as follows. In Section 2, we construct\na model including Landau diamagnetism, Pauli param-\nagnetism and FM eﬀect. The magnetization density is\nobtained through the analytical derivation. In Section 3,\nthe results is obtained and the discussions of our results\nis presented. A summary is given in Section 4.\nII. THE MODEL\nThe spin-1 Bose gas with FM couplings is described by\nthe following Hamiltonian:\nH−µN=DL/summationdisplay\nj,kz,σ/parenleftbig\nǫl\njkz+ǫze\nσ+ǫm\nσ−µ/parenrightbig\nnjkzσ,(1)\nwhereµis the chemical potential and the Landau levels\nof bosons with charge qand mass m∗in the eﬀective\nmagnetic ﬁeld Bis\nǫl\njkz= (j+1\n2)/planckover2pi1ω+/planckover2pi12k2\nz\n2m∗, (2)\nwherej= 0,1,2,...labels diﬀerent Landau levels and\nω=qB/(m∗c) is the gyromagnetic frequency. The en-\nergy level is degenerate with degeneracy\nDL=qBLxLy\n2π/planckover2pi1c, (3)\nwhereLxandLyare the length in xandydirections of\nthe system, respectively. The intrinsic magnetic moment\nassociated with the spin degree of freedom leads to the\nZeeman energy levels split in the magnetic ﬁeld,\nǫze\nσ=−g/planckover2pi1q\nm∗cσB, (4)\nwheregis the Lande-factor and σdenotes the spin-z\nindex of Zeeman state |F= 1,mF=σ/an}bracketri}ht(σ= 1,0,−1).\nThe contribution to the eﬀective Hamiltonian from the\nFM couplings is\nǫm\nσ=−2Iσ(m+σnσ), (5)\nwhereIdenotes FM coupling and spin polarization m=\nn1−n−1. The grand thermodynamic potential is ex-\npressed as\nΩT/negationslash=0=−1\nβlnTre−β(H−µN)\n=1\nβDL/summationdisplay\nj,kz,σln[1−e−β(ǫl\njkz+ǫze\nσ+ǫm\nσ−µ)],(6)whereβ= (kBT)−1. Through converting the sum over\nkzto continuum integral, we obtain\nΩT/negationslash=0=ωm∗V\n(2π)2/planckover2pi1β∞/summationdisplay\nj=0/summationdisplay\nσ/integraldisplay\ndkz\n×ln{1−e−β[(j+1\n2)/planckover2pi1ω+/planckover2pi12k2z\n2m∗−g/planckover2pi1q\nm∗cσB−2Iσ(m+σnσ)−µ]},\n(7)\nwhereVis the volume of the system. Eq. (7) can be\nevaluated by Taylor expansion, and then performing the\nintegral over kz. We get\nΩT/negationslash=0=−ωV\n/planckover2pi12/parenleftbiggm∗\n2πβ/parenrightbigg3/2\n×∞/summationdisplay\nl=1/summationdisplay\nσl−3\n2e−lβ[/planckover2pi1ω\n2−g/planckover2pi1q\nm∗cσB−2Iσ(m+σnσ)−µ]\n1−e−lβ/planckover2pi1ω.\n(8)\nFor convenience’s sake, we introduce some compact no-\ntation for the class of sums. It can be deﬁned as\nΣκσ[α,δ] =∞/summationdisplay\nl=1lα/2e−lx(ε+δ)\n(1−e−lx)κ, (9)\nwherex=β/planckover2pi1ωandµ−ǫze\nσ−ǫm\nσ= (1\n2−ε)/planckover2pi1ω. Within\nthis notation, Eq. (8) can be rewritten as\nΩT/negationslash=0=−ωV\n/planckover2pi12/parenleftbiggm∗\n2πβ/parenrightbigg3/2/summationdisplay\nσΣ1σ[−D,0].(10)\nwithD= 3. The particle density n=N/Vcan be\nexpressed as\nnT/negationslash=0 =−1\nV/parenleftBig\n∂ΩT/negationslash=0\n∂µ/parenrightBig\nT,V\n=x/parenleftBig\nm∗\n2πβ/planckover2pi12/parenrightBig3/2/summationtext\nσΣ1σ[2−D,0].(11)\nThe magnetization density Mcan be obtained from the\ngrand thermodynamic potential,\nMT/negationslash=0=−1\nV/parenleftbigg∂ΩT/negationslash=0\n∂B/parenrightbigg\nT,V\n=/planckover2pi1q\nm∗c/parenleftbiggm∗\n2πβ/planckover2pi12/parenrightbigg3/2/summationdisplay\nσ/braceleftbigg\nΣ1σ[−D,0]\n+x(gσ−1\n2)Σ1σ[2−D,0]−xΣ2σ[2−D,1]/bracerightbigg\n.\n(12)\nThe relation among eﬀective magnetic ﬁeld B, external\nmagnetic ﬁeld Hand magnetization density Mis for-\nmally expressed as\nB=H+4πM, (13)3\n0.00.51.0M(a)\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.0(b)\n||m\nI_\nFIG. 1: (a) The total magnetization density M, (b)m=n1−\nn−1versusIat reduced temperature t= 0.6 and magnetic\nﬁeldh= 0.00001. The Lande-factor g is chosen as: g=\n0.1(solid line), 0.3(dashed line), 0.5(dotted line).\nFor computational convenience, some dimensionless pa-\nrameters are introduced below. t=T/T∗,M=\nm∗cM/(n/planckover2pi1q),ω=/planckover2pi1ω/(kBT∗),I=In/(kBT∗),µ=\nµ/(kBT∗),m=m/n,nσ=nσ/nandh=\n/planckover2pi1qH/(m∗ckBT∗), and then x=ω/t, where T∗is the\ncharacteristic temperature of the system, which is given\nbykBT∗= 2π/planckover2pi12n2\n3/m∗. The mean-ﬁeld self-consistent\nequations are derived,\nn1=ωt1/2Σ′\n1,σ=1[2−D,0], (14a)\n1 =ωt1/2/summationdisplay\nσ=1,0,−1Σ′\n1σ[2−D,0], (14b)\nMT/negationslash=0=t3/2/summationdisplay\nσ/braceleftbigg\nΣ′\n1σ[−D,0]+x(gσ−1\n2)Σ′\n1σ[2−D,0]\n−xΣ′\n2σ[2−D,1]/bracerightbigg\n, (14c)\nω=h+4πγM, (14d)\nwhereγ=q2n1/3/(2πm∗c2), and\nΣ′\nκσ[α,δ] =∞/summationdisplay\nl=1lα/2e−lx(ε+δ)\n(1−e−lx)κ, (15)\nwithµ+gσω+2Iσ(m+σnσ) = (1\n2−ε)ω.\nSimilar method has been used to study the diamag-\nnetism of the charged spinless Bose gas12. Furthermore,\nwe have extended it to investigate the magnetic proper-\nties of charged spin-1 Bose gas13.\nIII. RESULTS AND DISCUSSIONS\nIn the following calculations from Fig. 1 to Fig. 6, the\ncharacteristic parameter γhas been set as 10−10, which\nis estimated for a system with the charge and mass of\n4He, and the particle density being set as (1 nm)−3. Fig.0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.4\n|\nIC\nt\nFIG. 2:Icvs reducedtemperature tphase diagram of charged\nspin-1 Bose gases at magnetic ﬁeld h= 0.00001.\n1 is plotted in a very weak magnetic ﬁeld h= 0.00001.\nAs shown in Fig. 1(a), the value of total magnetization\ndensityMpresents a turning point from zero to nonzero.\nIt is shown that the zero-ﬁeldspontaneousmagnetization\nexists in this system with increasing I, whereIis the re-\nduced FM coupling of charged spin-1 Bose gases. The\ncurves of mversusIin Fig. 1(b) are superposed for dif-\nferent Lande-factors ( g= 0.1, 0.3 and 0.5). It suggests\nthatm=n1−n−1is independent with the Lande-factor,\nsoIcat a certain temperature are equal for any Lande-\nfactor. Here Icis the critical value of reduced FM cou-\npling of PM phase to FM phase transition. Ic≈0.19 in\nthis situation. When I <Icmequals 0, and the value of\nmincreases with increasing IwhileI >Icuntil saturate.\nIn the region of I >Ic, the magnetization density Min-\ncreases with Lande-factor for ﬁxed I, which is attributed\nto the PM eﬀect13. Diamagnetism, paramagnetism and\nferromagnetism compete with each other in such system.\nThe diamagnetism of charged Bose gases, which is due\nto the internal ﬁeld induced by the spontaneous magne-\ntization, cannot overcome ferromagnetism in very weak\nmagnetic ﬁeld. While the competition between param-\nagnetism and diamagnetism has been discussed in Ref.\n13.\nFig. 2 plots the Icdependence of temperature at mag-\nneticﬁeld h= 0.00001. Theregionbelow IcisPMphase,\nwhile the region above it is FM phase. As the temper-\nature increases, Icincreases monotonically. It is shown\nthat spontaneous magnetization is hard to occur at high\ntemperature, when the Bose statistics reduces to Boltz-\nmann statistics.\nIt is supposed that mwill reach to a nonzero equiva-\nlenceatI= 0.2forarbitraryvalueofLande-factorforthe\nsituation of Fig. 1. To further study the inﬂuence of FM\ncoupling to spontaneousmagnetization, Fig. 3is plotted.\nIt is shown when I <Ic(≈0.19), the value of mwill be\nzero for any Lande-factor values. So the evolution of m\nwith Lande-factor gare superposed and keeps zero for\nI= 0 and I= 0.1. For ﬁxed IwhenI >Ic(≈0.19), the4\n0.00.51.0\n|(a)M\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.0\n|(b)m\ng\nFIG. 3: (a) The total magnetization density M, (b)m=n1−\nn−1as a function of Lande-factor gof charged spin-1 Bose\ngases at reduced temperature t= 0.6 and magnetic ﬁeld h=\n0.00001. The reduced FM coupling Iis chosen as: I= 0(solid\nline), 0.1(dashed line), 0.2(dotted line), 0.3(dash dotte d line),\nand 0.5(dash dot dotted line).\n-0.50.00.5\n(a)M\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.0|\n|(b)m\ng\nFIG. 4: (a) The total magnetization density M, (b)m=\nn1−n−1as a function of Lande-factor gof charged spin-\n1 Bose gases at reduced temperature t= 0.1 and magnetic\nﬁeldh= 0.1. The reduced FM coupling Iis chosen as:\nI= 0(solid line), 0.01(dashed line), 0.1(dotted line), 0.3( dash\ndotted line), and 0.5(dash dot dotted line).\nmagnetizationdensity Mincreasesmonotonicallywith g.\nWhilemmaintainsaconstantin despiteof g. Ourresults\nalso show that diamagnetism gives little contribution to\nthe magnetism in the weak magnetic ﬁeld, while param-\nagnetism and ferromagnetismplay signiﬁcant roles in the\nmagnetization density in the region for I >Ic. The in-\nteraction between paramagnetism and ferromagnetism is\nintricate. Theincreaseof mduetoincreasingthereduced\nFM coupling Iwill contribute to the paramagnetism.\nAbove we have discussed the very weak magnetic ﬁeld\nsituation, now we turn to investigate the magnetic prop-\nertiesofchargedspin-1Bosegasesatﬁnitemagneticﬁeld,\nwhere diamagnetism will emerges clearly. The result of\nthe dependenceofthetotalmagnetizationdensity Mand\nm=n1−n−1with Lande-factor gat a deﬁnite magnetic0.00.20.40.6\n||(a)M\n0 2 4 6 8 10 0.00.51.0\n(b)m\nh\nFIG. 5: (a) The total magnetization density M, (b)m=n1−\nn−1as a function of magnetic ﬁeld hof charged spin-1 Bose\ngases at reduced temperature t= 0.6 with Lande-factor g=\n0.5. The reduced FM coupling Iis chosen as: I= 0(solid\nline), 0.1(dashed line), 0.3(dotted line), and 0.5(dash do tted\nline).\nﬁeldh= 0.1 at reduced temperature t= 0.1 is shown\nin Fig. 4. At low temperature in the deﬁnite magnetic\nﬁeld, there is a competition among the paramagnetism,\ndiamagnetism and ferromagnetism. It is shown that dia-\nmagnetismdominatesin the small gregion, andtherefore\nthe magnetization density exhibits negative value. When\ng >0.45, thesystem presentsparamagnetismwhichisin-\ndependent of reduced FM coupling I. As seen from Fig.\n4, the curves of I= 0.1,I= 0.3 andI= 0.5 match to-\ngether. It means that mtends to saturate if Iis greater\nthan a critical value. The increase of Iafter this critical\nvalue does not contribute to the magnetization density.\nThen the system exhibits similar magnetization density\natI= 0.1,I= 0.3 andI= 0.5.\nThe discussions above all focused on ﬁxed magnetic\nﬁeld. Next we study the inﬂuence of magnetic ﬁeld on\nmagnetism. The evolution of the total magnetization\ndensityMandm=n1−n−1with magnetic ﬁeld at re-\nduced temperature t= 0.6 withg= 0.5 is shown in Fig.\n5. The gas always manifests paramagnetism no matter\nwhat the values of Iare. It indicates that in the case of\ng= 0.5, diamagnetism can not overcome paramagnetism\nno matter how strong the magnetic ﬁeld is. This behav-\nior is qualitatively consistent with the result of charged\nspin-1 Bose gases13. In this region, the stronger ferro-\nmagnetism induce larger m, which will enhance param-\nagnetism. With increasing the magnetic ﬁeld, diamag-\nnetism also increases. While this will not change the\nparamagnetism of this system. Whether diamagnetism\ncan increase inﬁnitely with magnetic ﬁeld is an impor-\ntant issue.\nIn order to manifest the paramagnetism and diamag-\nnetism in detail, in Fig. 6 we study the dependence of\nthe total magnetization density M, the paramagnetiza-\ntion density Mpand the diamagnetization density Md\nwith magnetic ﬁeld in reduced temperature t= 0.6 with5\n0.00.30.6\n||\n|(a)M\n0.30.6 (b)MP\n0 2 4 6 8 10 -0.6-0.30.0(c)Md\nh\nFIG. 6: (a) The total magnetization density M, (b) the para-\nmagnetization density Mp, and (c) the diamagnetization den-\nsityMdas a function of magnetic ﬁeld hof charged spin-1\nBose gases with g= 0.5 andI= 0.5, at reduced temperature\nt= 0.6.\ng= 0.5 andI= 0.5.Mpholds a constant since FM\ncoupling is larger. Mdtends to saturate with magnetic\nﬁeld. It indicates that diamagnetism will not increase\ninﬁnitely with magnetic ﬁeld. This is why in Fig. 5 the\ngas preserves paramagnetism even though the magnetic\nﬁeld is large.\nIt is signiﬁcant to evaluate the diamagnetic behavior\nat high magnetic ﬁeld limit. Without consideration of\nspin, the diamagnetization density,\nMd=t3/2∞/summationdisplay\nl=1l−3/2e−l(ω/2−µ)/t\n(1−e−lω/t)\n×[1+lω(−1\n2−e−lω/t\n1−e−lω/t)/t],(16)\nwhenω→ ∞,Mdcan be reduced to,\nMω→∞\nd=−1\n2ωt1/2∞/summationdisplay\nl=1l−1/2elµ/t\nelω/(2t), (17)\nfrom equation (14b), we can obtain,\n1 =ωt1/2∞/summationdisplay\nl=1l−1/2elµ/t\nelω/(2t), (18)\nSubstituting equation (18) into (17), Mω→∞\nd=−1/2\ncan be obtained. This analytical result illustrate the dia-\nmagnetization density Mdtends to a ﬁnite value at high\nmagnetic ﬁeld.\nIn order to investigate the magnetic properties of the\ncharged spin-1 Bose gas in low temperature, we suppose\nγ= 0.1. The evolution of the total magnetization den-\nsityMandm=n1−n−1with reduced temperature\nath= 0.00001 and g= 1 is shown in Fig. 7. It is\nshownthat Mincreaseswithincreasingtemperature, and\nreaches a maximum, then decreases at high temperature\nregion. The upward trend at low temperature reﬂects0.00.51.0\n(a)M\n0.0 0.5 1.0 1.5 2.00.00.51.0\n||\n(b)m\nt\nFIG. 7: (a) The total magnetization density M, (b)m=n1−\nn−1versusreducedtemperature tofchargedspin-1Bose gases\nwithγ= 0.1 andg= 1, at magnetic ﬁeld h= 0.00001.\nThe reduced FM coupling Iis chosen as: I= 0(solid line),\n0.1(dashed line), 0.3(dotted line), 0.5(dash dotted line) .\nthe diamagnetism, comparing with our results in Ref.\n21, which shows a ﬂat trend at the same temperature re-\ngion. A sharp decline can be seen when Mclose to zero.\nThis suggests that there is a pseudo-condensate temper-\nature in the transition from ferromagnetism to paramag-\nnetism. Although condensation has not been considered,\nthe magnetic ﬁeld is faint in such a case. It is reason-\nable that the pseudo-critical temperature increases with\nincreasing reduced FM coupling I. Therefore, the tem-\nperature region of ferromagnetism enlarges from I= 0\ntoI= 0.5 in turn.\nIV. SUMMARY\nIn summary, we study the interplay among param-\nagnetism, diamagnetism and ferromagnetism of charged\nspin-1 Bose gas with FM coupling within the mean-\nﬁeld theory. In very weak magnetic ﬁeld, it is shown\nthat the ferromagnetism is stronger than the diamag-\nnetism, where the diamagnetism is related with sponta-\nneous magnetization. The critical value of reduced FM\ncoupling IcofPMphasetoFMphasetransitionincreases\nwith increasing temperature. The Lande-factor gis sup-\nposed as a variable to evaluate the strength of the PM ef-\nfect. The gasexhibits a shift from diamagnetism to para-\nmagnetismas gincreasesataﬁnitemagneticﬁeld. Ferro-\nmagnetism plays an important role in the magnetization\ndensity in the weak magnetic ﬁeld. Diamagnetism can\nnot increase inﬁnitely with magnetic ﬁeld at high mag-\nnetic ﬁeld. Condensation is predicted to occur through\nstudying the low-temperature magnetic properties in a\nweak magnetic ﬁeld.6\nAcknowledgments\nJQ would like to thank Professor Huaiming Guo for\nthe helpful discussions. This work was supported by the\nNational Natural Science Foundation of China (Grant\nNo. 11004006), and the Fundamental Research Fundsfor the Central Universities of China.\nReferences\n1Anderson M H, Ensher J R, Matthews M R, Wieman C E\nand Cornell E A 1995 Science269198; Davis K B, Mewes\nM-O, Andrews M R, Druten N J V, Durfee D S, Kurn D\nM and Ketterle W 1995 Phys. Rev. Lett. 753969; Bradley\nC C, Sackett C A, Tollett J J and Hulet R G 1995 Phys.\nRev. Lett. 751687\n2Schafroth M R 1955 Phys. Rev. 100463\n3Blatt J M and Butler S T 1955 Phys. Rev. 100476\n4Bardeen J, Cooper L N and Schrieﬀer J R 1957 Phys. Rev.\n1081175\n5Alexandrov A S 2006 Phys. Rev. Lett. 96147003 ; Alexan-\ndrov A S 2010 J. Phys.: Condens. Matter 22426004;\nAlexandrov A S 2011 J. Supercond. Nov. 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Phys.: Condens.\nMatter23026003\n14Saxena S S, Agarwal P, Ahilan K, Grosche F M, Hasel-\nwimmer R K W, Steiner M J, Pugh E, Walker I R, Julian\nS R, Monthoux P, Lonzarich G G, Huxley A, Sheikin I,\nBraithwaite D and Flouquet J 2000 Nature406587\n15Aoki D, Huxley A, Ressouche E, Braithwaite D, Flouquet\nJ, Brison J -P, Lhotel E and Paulsen C 2001 Nature413\n613\n16Slooten E, Naka T, Gasparini A, Huang Y K and Visser A\nde 2009 Phys. Rev. Lett. 103097003\n17Machida K and Ohmi T 2001 Phys. Rev. Lett. 86850\n18Monthoux P and Lonzarich G G 1999 Phys. Rev. B59\n14598\n19Ho T -L 1998 Phys. Rev. Lett. 81742\n20Ohmi T and Machida K 1998 J. Phys. Soc. Jpn. 671822\n21Gu Q and Klemm R A 2003 Phys. Rev. A68031604(R)\n22Tao C J, Wang P L, Qin J H and Gu Q 2008 Phys. Rev.\nB78134403\n23Kis-Szab´ o K, Sz´ epfalusy P. and Szirmai G 2005 Phys. Rev.\nA72023617\n24Ashhab S 2005 J. Low Temp. Phys. 14051\n25Alexandrov A S 1993 Phys. Rev. B4810571; Alexandrov\nA S, Beere W H and Kabanov V V 1996 Phys. Rev. B54\n15363\n26Alexandrov A S 1999 Phys. Rev. B6014573"    },    {        "title": "0707.3265v1.Magnetic_State_Modification_Induced_by_Superconducting_Response_in_Ferromagnet_Superconductor_Hybrids.pdf",        "content": "arXiv:0707.3265v1  [cond-mat.supr-con]  22 Jul 2007Submitted to Phys. Rev. Lett. July 12th2007\nMagnetic State Modiﬁcation Induced by Superconducting Res ponse in\nFerromagnet/Superconductor Hybrids\nC. Monton, F. de la Cruz, and J. Guimpel\nCentro At´ omico Bariloche & Instituto Balseiro,\nComisi´ on Nacional de Energ´ ıa At´ omica & Universidad Naci onal de Cuyo,\n(8400) S.C. de Bariloche, Argentina\nMagnetization measurements in superconductor/ferromagn etNb/Cosuperlattices show a com-\nplex behavior as a function of temperature, applied ﬁeld and sample history. In base to a simple\nmodel it is shown that this behavior is due to an interplay bet ween the superconductor magne-\ntization temperature dependence, the ferromagnet magneti zation time dependence, and the stray\nﬁelds of both materials. It is also shown that the magnetic st ate of the Colayers is modiﬁed by\ntheNbsuperconducting response, implying that the problem of a su perconductor/ferromagnetic\nheterogeneous sample has to be solved in a self-consistent m anner.\nThe interaction between a superconductor, SC, and a\nferromagnet, FM, in close contact at an interface, as in a\nsuperlattice, has attracted attention in the last years due\nto the possibility offabricatingSC/FM hybriddevices.[1]\nThese engineered materials originate the appearance of\ninterestingphysicalphenomenaduetothediﬀerentscales\nand mechanisms of interaction, like SC pair breaking ef-\nfects related to exchange interaction at the interface,[1]\nor electromagnetic interaction with the stray ﬁelds of the\nFM both at the mesoscopic and macroscopic level.[2, 3]\nMost of the research has focused on the ways in\nwhich the FM aﬀects the SC response. For example,\nin the Domain Wall Superconductivity eﬀect, observed\nin SC/FM bi-layers,[2, 3] superconductivity nucleates\nin those places where the perpendicular component of\nthe inhomogeneous FM domain structure´s stray ﬁeld\nis close to zero. In the Spin Switch eﬀect, observed in\nFM/SC/FM trilayers,[1] the Cooper pair, due to its ﬁ-\nnite size, experiments diﬀerent average values of the ex-\nchange ﬁeld when the FM layers are ferro- or antiferro-\nmagnetically oriented. As a consequence, the SC order\nparameter is more depressed when the FM layers are\nferro-magnetically oriented. This allows the control of\ntheSCorderparametervaluethroughanexternalmacro-\nscopic parameter.\nIn contrast, very little work has been done in explor-\ning in which way the SC aﬀects the magnetic state of\nthe FM layer.[4] Recently,[5] we have shown the impor-\ntance of the FM stray ﬁelds in the overall magnetic re-\nsponse of Nb/Cosuperlattices. In that work we also\nhinted to the possibility that the SC response may mod-\nify the magnetic state of the FM layers. In this letter we\nshowthat, indeed, theSCresponsemodiﬁesthemagnetic\nstate of the FM layers. The system global electromag-\nnetic response is determined by an interplay between the\nSC magnetization temperature dependence and the FM\nmagnetization time evolution.\nWe present data on the temperature, T, dependence\nof the magnetic ﬂux expulsion, ∆ φ, directly propor-/s45/s52/s53/s48 /s45/s51/s48/s48 /s45/s49/s53/s48 /s48 /s49/s53/s48 /s51/s48/s48 /s52/s53/s48/s45/s49/s48/s48/s48/s45/s53/s48/s48/s48/s53/s48/s48/s49/s48/s48/s48\n/s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s45/s56/s48/s48/s48\n/s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s56/s48/s48\n/s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s56/s48/s48/s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s51/s48/s48/s54/s48/s48\n/s84/s47/s84\n/s67/s83\n/s84/s47/s84\n/s67/s83/s84/s47/s84\n/s67/s83/s48/s48/s48\n/s48/s32/s77/s32/s32/s91/s101/s109/s117/s47/s99/s109/s51\n/s93\n/s72\n/s97/s32/s91/s79/s101/s93/s84/s47/s84\n/s67/s83/s32/s32/s40/s97/s41\n/s40/s99/s41\n/s32/s32\n/s40/s100/s41\n/s32/s32/s40/s98/s41/s32\n/s32\nFIG. 1: Main panel: Magnetization, M, as a function of ap-\nplied ﬁeld, Ha, for the [ Nb(44nm)/Co(10nm)]x19 superlat-\ntice atT= 7K > T CS= 6.2K. Panels (a) through (d)\nshow the temperature, T, dependence of the superconducting\nmagnetic ﬂux response, ∆ φ, in units of the superconducting\nﬂuxquantum, φo, at diﬀerentapplied ﬁelds andferromagnetic\nlayers initial state, as indicated by gray dots and connecti ng\narrows in the main panel. Arrows in panels indicate the direc -\ntion of the Tsweeps. Panel (a): Ha=−22Oe, +FC. Panel\n(b):Ha= 75Oe, -FC. Panel (c): Ha= 38Oe, -FC. Panel\n(d):Ha= 22Oe, -FC.\ntional to the SC magnetization, for Nb/Cosuperlattices.\nData is presented for a [ Nb(44nm)/Co(10nm)]x19 and\na [Nb(44nm)/Co(7.5nm)]x19 superlattice. The results\non these samples are representative of the measurements\nwe performed on a collection of Nb/CoFM/SC super-\nlattices. Since our experimental setup measures mag-\nnetic ﬂux variations,[5] all ∆ φvalues are measured with\nrespect to the ﬁrst data point, always above the super-\nconducting critical temperature, TCS. Sample prepara-\ntion, characterization method and measurement details\nare described in reference 5. For all ﬂux expulsion data,2\nthe applied ﬁeld, Ha, is parallel to the sample surface.\nIn the normal state, both superlattices present FM be-\nhavior with Curie temperatures, TC, above 300 K. Flux\nexpulsioninthe SCstatewasmeasuredasafunction of T\nin ﬁeld cooling experiments, for two diﬀerent Colayer´s\ninitial FM states, -FC, with the Colayers initially sat-\nurated in the negative Hadirection, and +FC, with the\nColayers initially saturated in the positive Hadirection.\nA detailed explanation of these measurement protocols\nis also included in reference 5.\nFigure 1 is a composite that summarizes the experi-\nmental results. In the main panel we show the FM hys-\nteresis loop of the Colayers at T= 7K, close but above\nTCSof theNblayers. In the superimposed panels, we\nshow the Tdependence of ∆ φfor +FC and -FC mea-\nsurements. For each initial state, several Tcycles were\nmeasured, each cycle sweeping Tdown from 7 Kto 5.5K\nand up to 7 Kagain. The solid dots in the hysteresis\ncurve connected with arrows to the panels indicate the\ninitial magnetic state for each experiment.\nThe data for the ﬁrst Tdown-sweep in each panel\nshows the behavior already discussed in our previous\nwork.[5]. As discussed there, the SC response is pro-\nportional to the eﬀective ﬁeld, Heff, originated by the\nsuperposition of the applied ﬁeld, Ha, and the Colay-\ners´ stray ﬁeld, Hs. For the -FC measurements (lower\nbranch of the Cohysteresis loop), at low Haand neg-\nativeComagnetization, the SC layers sense a positive\nHeffdue to the Co´sHs, see panel (d). At higher Ha,\ntheComagnetizationbecomespositive, Hsbecomesneg-\native and larger than Ha, and the SC senses a negative\nHeff, see panel (b). Panel (c) shows an intermediate\ncase, where the magnetization is already reversed, but\nHsis smaller than Haand the SC still senses a positive\nHeff. Panel (a), +FC initial state (upper branch of the\nCohysteresis loop) is the mirror experiment from panel\n(d).\nThe novel feature observed in these data is present in\nthe dependence of the Comagnetization in the normal\nstate with the number of Tsweeps, i.e. cycles. This de-\npendence is observedasa non-repeatabilityofthe normal\nstatemagnetizationvalueafteracycleiscompleted. This\nbehavior is not due to an experimental artifact related to\nan instrumental drift, since this instrumental drift has\nbeen substracted from the data. A systematic behavior\nisobservedinspiteoftheseeminglycomplexdependence.\nThe direction of the variation follows the sign of the ap-\npliedﬁeld, and is independent of the Comagnetization\ndirection, i.e. the strayﬁeld, compare data in panel (b)\nand (d), for example. The diﬀerence between the ﬁrst\nand second normal state ∆ φvalues in panel (d) of 360\nsuperconducting ﬂux quantums is equivalent to a change\nof 1.1emucm−3in theColayer´s magnetization, which\nshows that this eﬀect is small but not negligible.\nIn order to understand this behavior, we have con-\nstructed a simple “toy model” to qualitatively simulateNb Co Nb CoT< Tcs\nNb CoT >Tcs\nCo Nb\n/c40 /c41a /c40/c32/c32/c41b\nFIG. 2: Schematics of the “toy model” behavior. Panel (a):\natT > T CSthe normal Nbellipsoid experiences an eﬀective\nﬁeld due to the applied ﬁeld (straight lines) and the ferroma g-\nneticCoellipsoid stray ﬁeld (dipole like lines). Panel (b): As\nTis reduced below TCSthe magnetic ﬂux expulsion from the\nNbellipsoid modiﬁes the eﬀective ﬁeld over the Coellipsoid.\nthe experimental data. Although the model is very sim-\nple, a careful consideration of its hypothesis should be\nmade to fully understand the implications of the results.\nThe ﬁrst requirement is that an electromagnetic stray-\nﬁeld mediated interaction should exist between the FM\nand the SC components. This is not achievable if the\nmaterials are modelled as nearly inﬁnite slabs parallel to\nthe applied ﬁeld, since the stray ﬁeld of this geometry\nis negligible. Consequently, both materials are modelled\nas ellipsoids with one of the principal axis parallel to the\nﬁeld. The “toy model” sample consists, then, of a FM\nand a SC ellipsoids, located side by side. Figure 2 de-\npicts the main ideas of the model. Panel (a) shows the\nsituation at T > T CSwhere the Heffsensed by the SC\nellipsoid is composed by Ha(straight lines) and the FM\nHs(dipolar lines arisingfrom the FM ellipsoid). When T\nis reduced below TCS, as depicted in panel (b), the ﬂux\nexpulsion from the SC ellipsoid modiﬁes the Heffsensed\nby the FM ellipsoid and consequently its magnetization.\nThe solution of the problem has now to be found in a\nself-consistent way.\nThe ellipsoid shape or eccentricity, ǫ, was selected as\nto maximize the stray ﬁeld eﬀects. That an optimum\nvalue exists is clear from the fact that in the ǫ→ ∞\n“needle” limit, the stray ﬁelds approach zero due to the\nnegligible demagnetizing eﬀects, and that in the ǫ→0\n“disk” limit, the stray ﬁelds also approach zero since the\nellipsoid is being magnetized along the shape anisotropy\n“hardaxis”. The optimal ǫvalue actually depends on the\nmaterial´s magnetization, but since it is weakly depen-\ndent on it, a value of 10 was found to maximize the stray\nﬁeld eﬀects in nearly all the T-Harange. Also, since an\nexact three dimensional spatial solution of this electro-\nmagnetic problem is beyond the scope of this work, and\nwould only obscure the results of the model, the spatial\ndependence of the stray ﬁelds is neglected, and Hsdue to3\neach ellipsoid is evaluated only at the center of the other\nellipsoid.\nThe second ingredient in the model is a “time” de-\npendence. This dependence cannot be ascribed to the\nsuperconducting material since we have shown that no\nvortices are present in the T−Harange of these exper-\niments. Consequently, it must be arising from the creep\nin the FM material. Following this idea, the magnetiza-\ntion ofthe SC Nbellipsoid is modelled by a Tdependent,\ntime independent Meissner state. As a further simpliﬁ-\ncation of the model the Tdependence is forced to follow\nthat of a parallel slab with a two ﬂuid Tlaw.[5] On the\nother hand, the FM Coellipsoid magnetization does not\npresent a Tdependence since its TCis much higher than\nthe measurementrange. It onlyshowsatime dependence\nwhich must be numerically simulated, as described in the\nnext paragraph.\nTo simulate the Tsweeps at constant Ha, the self-\nconsistent equilibrium state of the magnetized ellipsoids\nis solved at a given T. After this, the magnetization\nchange for the Coellipsoid, is reduced by a given per-\ncentage, and the magnetization of the Nbellipsoid is\nrecalculated for this, now ﬁxed, value of the Coellip-\nsoid´s magnetization, i.e. stray ﬁeld. This algorithm re-\nsults in an eﬀective exponential time dependence for the\nComagnetization. The sample´s magnetization, MT,\nis deﬁned as the total magnetic moment divided by the\ntotal sample volume. In order to compare the results\nto the experiments, the simulation data is presented as\n∆MT=MT−Mo, whereMois the value for the ﬁrst\nsimulated point, always at T > T CS.\nPanels (a) and (b) in ﬁgure 3 show the prediction of\nthe model for situations similar to panels (c) and (a) in\nﬁgure 1, i.e. opposite direction of Haand same value of\nMo. It is clear that the principal features of the exper-\nimental data are qualitatively reproduced. First, there\nis a dependence of the normal state magnetization with\nthe number of cycles. This dependence follows the sign\nof the applied ﬁeld and is not correlated to the mag-\nnetization direction. Second, there is an irreversibility\nbetween cooling-down and warming-up sweeps. Third,\na non-monotonic Tdependence is observed for cooling-\ndown sweeps.\nAn interesting feature not actually observable in the\ndata in ﬁg.1, but presented in reference 5 is a non-\nmonotonic T dependence that develops for applied ﬁelds\nnearthecoerciveﬁeldofthe Colayers. Themainpanelin\nﬁgure 4 shows a comparison between experimental and\nsimulated data, where the simulation parameters have\nbeen selected as to maximize this non-monotonic Tde-\npendence. The origin of this behavior becomes clear\nwhen examining separately the NbandComagnetiza-\ntion response in the simulated data. The inset shows the\nSC ellipsoid magnetization, MSC, the FM ellipsoid mag-\nnetization, MFM, and the sample´s magnetization, MT,\nas a function of simulated data point number, i.e.“time”,/s45/s48/s46/s53/s48/s46/s48\n/s48/s46/s52 /s48/s46/s56 /s49/s46/s50/s48/s46/s48/s48/s46/s53/s77\n/s84/s32/s91/s101/s109/s117/s47/s99/s109/s51\n/s93\n/s32/s32\n/s84/s47/s84\n/s67/s83/s40/s97/s41\n/s32/s32\n/s32\n/s40/s98/s41\nFIG. 3: Model prediction for the temperature, T, dependence\nof the sample magnetization change, ∆ MT=MT(T)−Mo,\nwhereMois the magnetization for the initial simulated data\npoint at T > T CS. Panels (a) and (b) show the results\nfor two sets of parameters qualitatively equivalent to the e x-\nperimental data in panels (c) and (a) of ﬁgure 1. Panel\n(a):Ha= 22.85Oe,Mo= 5.12emucm−3. Panel (b):\nHa=−17.15Oe,Mo= 5.12emucm−3\n.\nwhileTis swept down from above TCS. TheTsweep is\nlinear with this “simulated time”. The time dependence\nofMSCis that arising from the Tsweep, given that the\nMeissner state does not present an intrinsic time depen-\ndence. The time dependence of MFM, on the otherhand,\nhas a twofold origin. First, the ﬂux expulsion in the SC\noriginates an increase of local magnetic ﬁeld in the FM\nmaterial, as schematized in panel (b) of ﬁgure 2. Second,\ntheMFMpresents an intrinsic time dependence in its re-\nsponse to the magnetic ﬁeld changes. In this light, the\noriginofthenon-monotonic Tdependencebecomesclear.\nAsTis swept down from above TCS, theTdependence\nof the SC ellipsoid magnetization produces a ﬂux expul-\nsion in the sample. This originates a ﬁeld increase in the\nFMmaterialincreasingits MFM. Atlowertemperatures,\ntheTdependence of the SC material is relatively weak,\nand the time dependence of the FM material emerges as\na “paramagnetic” like signal, resembling a paramagnetic\nMeissner eﬀect.[6, 7, 8].\nThe results and the toy model presented here clarify\nthe response of SC/FM hybrid structures and, at the\nsametime, raiseaninterestingquestion. Wehavedemon-\nstrated that the electrodynamic response of these hybrid4\n/s48/s46/s52 /s48/s46/s56 /s49/s46/s50/s45/s50/s53/s48/s50/s53\n/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52\n/s45/s50/s48\n/s49/s56/s46/s48/s49/s56/s46/s54\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s56/s46/s52/s56/s46/s56\n/s84/s47/s84\n/s67/s83/s77/s32 /s91 /s101/s109/s117/s47/s99/s109/s51 \n/s93\n/s116/s105/s109/s101/s32/s91/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s93\n/s32\n/s77\n/s84 /s32/s91/s101/s109/s117/s47/s99/s109/s51\n/s93/s32\n/s32/s32\n/s77\n/s83/s67/s32\n/s32/s77\n/s70/s77/s32\n/s32/s77\n/s84\nFIG. 4: Magnetic ﬂux expulsion, ∆ φfor a\n[Nb(44nm)/Co(7.5nm)]x19 superlattice with TCS= 5.9K,\nand model prediction, ∆ MT, forHa= 15Oeand\nMo= 9emucm−3. Inset shows the dependence on\nsimulated data point number, i.e.“time”, for the supercon-\nducting ellipsoid magnetization, MSC, the ferromagnetic\nellipsoid magnetization, MFM, and the sample´s total\nmagnetization, MT.\nsystems involves a combination of two separate phenom-\nena. In the ﬁrst place, the diamagnetic response of the\nSC layers expels the magnetic ﬂux into the FM layers.\nAs a consequence, the FM material responds with a time\ndependence, clearly in the direction of the applied ﬁeld.\nBoth materials aﬀect each other with their respective\nstray ﬁelds. In this process, the magnetic domain struc-\nture of the FM seems to play an important role, since\nthe stray ﬁelds of an inﬁnite slab are negligible. Clearly,\nin order to observe stray ﬁeld eﬀects, a non slab geome-\ntry has to be present in the samples. In this picture, aninteresting point arises. Given that the response of the\nhybrid material is aﬀected, and in some TandHrange,\ndominated, by the intrinsic time dependence, the eﬀects\ndescribed here may be important if the device operation\nis based on magnetization changes and designed to work\nat frequencies similar to the creep of the FM.\nIn summary, we have demonstrated that the electro-\ndynamic response of SC/FM hybrid materials is deter-\nmined by an interplay between the temperature depen-\ndence of the SC magnetization, the time dependence of\nthe FM magnetization, and the eﬀective interaction be-\ntween them mediated by the stray ﬁelds.\nWork partially supported by ANPCyT PICT2003-03-\n13511,ANPCyT PICT2003-03-13297andFundaci´ onAn-\ntorchas. CMacknowledgesﬁnancialsupportfromandJG\nis a member of CONICET, Argentina.\n[1] J.Y. Gu, C.-Y. You, J.S. Jiang, J. Pearson, Ya.B. Bazaliy\nand S.D. Bader, Phys. Rev. Lett. 89, 267001 (2002).\n[2] A.Yu. Rusanov, M. Hesselberth, J. Aarts and A.I. Buzdin,\nPhys. Rev. Lett. 93, 057002 (2004); Z. Yang et al., Nat.\nMater.3, 793 (2004).\n[3] D. Stamopoulos and M. Pissas, Phys. Rev. B 73, 132502\n(2006).\n[4] L.N. Bulaevskii and E.M. Chudnovsky, Phys. Rev. B 63,\n012502 (2000).\n[5] C. Monton, F. de la Cruz and J. Guimpel, Phys. Rev. B\n75, 64508 (2007).\n[6] M. A. L´ opez de la Torre, V. Pe˜ na, Z. Sefrioui, D. Arias,\nC. Leon, J. Santamaria and J. L. Martinez, Phys. Rev. B\n73, 052503 (2006).\n[7] W. Braunisch, N. Knauf, V. Kataev, S. Neuhausen,\nA. Gr¨ utz, A. Kock, B. Roden, D. Khomskii, and D.\nWohlleben, Phys. Rev. Lett. 68, 1908 (1992).\n[8] D. J. Thompson, M. S. M. Minhaj, L. E. Wenger and J.\nT. Chen, Phys. Rev. Lett. 75, 529 (1995)."    },    {        "title": "0711.0758v2.Low_density_ferromagnetism_in_biased_bilayer_graphene.pdf",        "content": "arXiv:0711.0758v2  [cond-mat.mes-hall]  5 May 2008Low density ferromagnetism in biased bilayer graphene\nEduardo V. Castro,1N. M. R. Peres,2T. Stauber,2,3and N. A. P. Silva2\n1CFP and Departamento de F´ ısica, Faculdade de Ciˆ encias Uni versidade do Porto, P-4169-007 Porto, Portugal\n2Centro de F´ ısica e Departamento de F´ ısica, Universidade d o Minho, P-4710-057, Braga, Portugal and\n3Instituto de Ciencia de Materiales de Madrid. CSIC. Cantobl anco. E-28049 Madrid, Spain\n(Dated: June 11, 2018)\nWe compute the phase diagram of a biased graphene bilayer. Th e existence of a ferromagnetic\nphase is discussed with respect both to carrier density and t emperature. We ﬁnd that the ferro-\nmagnetic transition is ﬁrst order, lowering the value of Urelatively to the usual Stoner criterion.\nWe show that in the ferromagnetic phase the two planes have un equal magnetization and that the\nelectronic density is hole like in one plane and electron lik e in the other.\nPACS numbers: 73.20.Hb,81.05.Uw,73.20.-r, 73.23.-b\nIntroduction.— Graphene, a two-dimensional hexago-\nnal lattice of carbon atoms, has attracted considerable\nattention due to its unusual electronic properties, char-\nacterized by massless Dirac fermions [1, 2]. It was ﬁrst\nproduced by micromechanical cleavage of graphite and\nits hallmark is the half integer quantum Hall eﬀect [3].\nInadditiontographene,few-layergraphenecanalsobe\nproduced. Of particular interest to us is bilayer graphene\n(BLG), where two carbon planes lay on top of each other\naccordingto AB-Bernalstacking. InBLGitispossibleto\nhave the two planes at diﬀerent electrostatic potentials.\nAs a consequence, a gap opens at the Dirac point and\nthe low energy band acquires a Mexican hat dispersion\n[4]. This system is called a biased BLG, and provides\nthe ﬁrst semiconductor with a gap that can be tuned\nexternally [5, 6, 7]. Due to the Mexican hat dispersion\nthe density of states (DOS) close to the gap diverges as\nthe square root of the energy. The possibility of hav-\ning an arbitrary large DOS at the Fermi energy poses\nthe question whether this system can be unstable toward\na ferromagnetic ground state – a question we want to\naddress in this Letter. From the point of view of the\nexchange instability, BLG was found to be always unsta-\nble toward a ferromagnetic ground state for low enough\ndensities [8, 9].\nThe question of magnetism in carbon based systems\nhas already a long history. Even before the discovery of\ngraphene, graphite has attracted a broad interest due to\nthe observation of anomalous properties, such as mag-\nnetism and insulating behavior in the direction perpen-\ndicular to the planes [10, 11, 12, 13, 14]. The research\nofs−pbased magnetism [15, 16, 17] was especially\nmotivated by the technological use of nanosized parti-\ncles of graphite, which show interesting features depend-\ning on their shape, edges, and applied ﬁeld of pressure\n[18]. Microscopic theoretical models of bulk carbon mag-\nnetism include nitrogen-carbon compositions where fer-\nromagnetic ordering of spins could exist in πdelocalized\nsystems due to a lone electron pair on a trivalent element\n[19] or intermediate graphite-diamond structures where\nthe alternating sp2andsp3carbon atoms play the roleof diﬀerent valence elements [20]. More general models\nfocus on the interplay between disorder and interaction\n[21, 22]. Further, midgap states due to zigzag edges play\na predominant role in the formation of magnetic mo-\nments [23, 24] which support ﬂat-band ferromagnetism\n[25, 26, 27]. Magnetism is also found in fullerene based\nmetal-free systems [28]. For a recent overview on metal-\nfree carbon based magnetism see Ref. [29].\nModel and mean ﬁeld treatment.— Due to the electro-\nstatically invoked band-gap, there is a large DOS for\nlow carrier density and thus eﬀective screening of the\nCoulomb interaction. Coulomb interaction shall thus be\ntreated using a Hubbard on-site interaction.\nThe Hamiltonian of a biased BLG Hubbard model is\nthe sum of two pieces H=HTB+HU, whereHTBis\nthe tight-binding part and HUis the Coulomb on-site\ninteraction part. The term HTBis a sum of four terms:\nthe tight-binding Hamiltonian of each plane, the hopping\nterm between planes, and the applied electrostatic bias.\nWe therefore have HTB=/summationtext2\nι=1HTB,ι+H⊥+HV,with\nHTB,ι=−t/summationtext\nr,σa†\nισ(r)[bισ(r) +bισ(r−a1) +bισ(r−\na2)] +h.c.,H⊥=−t⊥/summationtext\nr,σ[a†\n1σ(r)b2σ(r) +h.c.], and\nHV=V\n2/summationtext\nr,x,σ[nx1σ(r)−nx2σ(r)].The term HUis\ngivenby HU=U/summationtext\nr,x[nx1↑(r)nx1↓(r)+nx2↑(r)nx2↓(r)].\nWe used nxισ(r) =x†\nισ(r)xισ(r),x=a(b), as the\nnumber operator at position rand sublattice Aι(Bι)\nof layer ι= 1,2, for spin σ=↑,↓;a1=a(1,0) and\na2=a(1,−√\n3)/2 are the basis vectors and a≈2.46˚A\nthe lattice constant. Unless stated otherwise, we use\nt= 2.7eV,t⊥= 0.2t, andV= 0.05eV[30].\nThe problem deﬁned by HTB+HUcannot be solved\nexactly. We adopt a mean ﬁeld approach, recently ap-\nplied to describe magnetic properties of graphene nanois-\nlands [31]. Since the two planes of the BLG are at dif-\nferent electrostatic potentials, we expect an asymmetry\nbetween layers for the charge density nand the magneti-\nzationm=n↑−n↓(per unit cell). Accordingly, we pro-\npose the following broken symmetry ground state, which\nalso deﬁnes the mean ﬁeld parameters: ∝angbracketleftnx1σ(r)∝angbracketright=\nn+∆n\n8+σm+∆m\n8and∝angbracketleftnx2σ(r)∝angbracketright=n−∆n\n8+σm−∆m\n8,2\n-4 -2 0 2 4\n    /t00.20.40.60.81ρ(   ) (1/eV)-0.02 0 0.0200.51\n0.00924 0.00928\n    /t0102030\n0.05.0×10-51.0×10-41.5×10-4\nδn   (e- / unit cell)010203040Uc (eV)\nnumeric\nanalyticgap edge(a) (b)µ∼\nµ∼µ∼\nFIG. 1: (Color online) (a) Bilayer graphene DOS for U= 0.\nInset: Zoom near the gap region. (b) Ucvsδnin the low\ndoping regime. Inset: The same as a function of ˜ µ.\nwhere ∆ nand ∆mrepresent the charge density and\nthe spin polarization diﬀerence between the two lay-\ners, respectively [32]. This leads to an eﬀective bias\nVσ=V+U∆n/4−σU∆m/4.\nIf one assumes the ferromagnetic transition to be sec-\nond order, with m= 0 and ∆ m= 0 at the transition, we\nare lead to a U−criticalUcgiven by,\nUc= 1/ρb(˜µ,Uc), (1)\nwhereρb(˜µ,Uc) is the DOS per spin per lattice point and\n˜µ=µ−nUc/8, with chemical potential µ. Although\nEq. (1) looks like the usual Stoner criterion, the eﬀective\nbiasVσdepends on Udue to ∆ n. This makes Eq. (1)\nnon-linear, and Uchas to be found numerically in a self-\nconsistent way.\nSimple results.— We start with the zero temperature\n(T= 0) phase diagram in the plane Uvsδn, where\nδnis the doping relatively to the half ﬁlled case. An\napproximate analytic treatment is possible in this limit,\nwhich is used to check our numerical results.\nIn Fig. 1 (a) we represent the DOS of a biased BLG\nwithU= 0. As seen in the inset, the DOS diverges at\nthe edges of the gap. As a consequence, the closer the\nchemical potential to the gap edges, the lower the critical\nUcvalue. The low doping Ucvalue – given by Eq. (1) in\nthe limit U∆n≪V– is shown in Fig. 1 (b), both as\na function of δnand ˜µ(inset). The lowest represented\nvalue of Ucis about Uc≃2.7eVto which corresponds\nδn≃2.5×10−5electrons per unit cell. The step like\ndiscontinuity shown in panel (b) for Ucoccurs when the\nFermi energy equals V/2, signaling the top of the Mexi-\ncan hat dispersion relation.\nIt is clear from Fig. 1 (b) that in the low doping limit\nUcisalinearfunctionof δn. Tounderstandthisbehavior,\nﬁrstwenotethatforverylowdopingtheDOSclosetothe\ngapedgesbehavesas ρb(˜µ)∝(|˜µ|−∆g/2)−1/2,where∆ g\nis the size of the gap. Using this approximate expression\nto compute the doping, δn∝sign(˜µ)×/integraltext|˜µ|\n∆g/2dx ρb(x), we\nimmediately get δn∝sign(˜µ)/ρb(˜µ) and thus Uc∝ |δn|.\nIn Fig. 1 (b) both the numerical result of Eq. (1) and the\napproximated analytical result just derived are shown.\nThe agreement is excellent.0 5 10 15 200.02.5×10-55.0×10-57.5×10-51.0×10-4m   (e- / unit cell)\n-15-10-505101520-0.0027-0.0026-0.0025-0.0024-0.0023-0.0022-0.0021∆n   (e- / unit cell)δn = 0.00003\nδn = 0.00004\nδn = 0.00005\nδn = 0.00006\nδn = 0.00007\nδn = 0.00008\nδn = 0.00009\nδn = 0.00010\nδn = 0.00011\nδn = 0.00012\n0 5 10 15 20\nU (eV)0.02.5×10-55.0×10-57.5×10-51.0×10-41.2×10-4∆m   (e- / unit cell)\n0 5 10 15 20\nU (eV)05×10-51×10-4δn   (e- / unit cell)1st order\n2nd order (numeric)\n2nd order (analytic)\nV = 0.01 eVV = 0.05 eV(a)\n(b)(c)\n(d) P\nFV = 0.1 eV\nVV = 0.05 eV\nFIG. 2: (Color online) Panels (a), (b), and (c) show the\nT= 0 solution for m, ∆m, and ∆n, respectively. Panel (d)\nshows the Uvsδnphase diagram at T= 0: symbols are in-\nferred from panel (a) and signal a ﬁrst-order transition; lines\nstand for the second-order one given by Eq. (1). Labels: P-\nparamagnetic, F-ferromagnetic.\nSelf-consistent solution.— In order to obtain the T= 0\nphase diagram of the biased BLG, we study how m, ∆m,\nand ∆ndepend on the interaction U, for given values of\nthe electronic doping δn.\nIn Fig. 2 (a) it is shown how mdepends on Ufor\ndiﬀerentvaluesof δn. Thechosenvaluesof δncorrespond\nto the chemical potential being located at the divergence\nofthelowenergyDOS,whichexplainsthesmallercritical\nUcvalue for smaller δn. It is interesting to note that the\nsaturation values of the magnetization correspond to full\npolarization of the doping charge density with m=δn,\nalso found within a one-band model [9] . In Fig. 2 (b)\nwe plot the ∆ mvsU. Interestingly, the value of ∆ m\nvanishes at the same Ucasm. For ﬁnite values of m\nwe have ∆ m > m, which means that the magnetization\nof the two layers is opposite and unequal. In Fig. 2 (c)\nwe show ∆ nvsU. It is clear that |δn|<|∆n|, which\nimplies that the density of charge carriers is above the\nDirac point in one plane and below it in the other plane.\nThis means that the charge carriers are electron like in\none plane and hole like in the other. As Uis increased\n∆nis suppressed in order to reduce the system Coulomb\nenergy.\nIn Fig. 2 (d) we show the T= 0 phase diagram in the\nUvsδnplane. Here we concentrate on the V= 0.05eV\ncase. Symbols are inferred from the magnetization in\npanel (a). They signal a ﬁrst-order transition when m\nincreases from zero to a ﬁnite value [see panel (a)]. The\nfull (red) line is the numerical self-consistent result of\nEq.(1), andthe dashed(blue) line isthe approximatean-\nalytic result described above. The discrepancy between\nlines and symbols hasa clear meaning. In orderto obtain\nEq. (1) we assumed that a second-order transition would\ntake place. This is not the case, and the system under-3\n~ \nKM Γ KM Γ KM Γµ\nµµU > UcU < UcU >> Uc E\nFIG. 3: (Color online) Hartree-Fock bands for ↑(full lines)\nand↓(dashed lines) spin polarizations.\ngoes a ﬁrst-order transition for smaller Uvalues. There\nare clearly two diﬀerent regimes: one for δn/lessorsimilar10−4,\nwhere the dependence of δnonUcis linear, and another\nforδn >10−4, where a plateau like behavior develops.\nThis plateau has the same physical origin as the step like\ndiscontinuity we have seen in Fig. 1 (b). In the limit\nδn→0 we have not only Uc→0, but also m→0 and\n∆m→0 [see panels (a) and (b) of Fig. 2], implying a\nparamagnetic ground state for the undoped biased BLG.\nFigure 2 (d) shows also the eﬀect of Von theT= 0\nphase diagram (the eﬀect of t⊥being similar). Raising\neitherVort⊥leads to a decreaseofthe critical- Uneeded\nto establish the ferromagnetic phase for a given δn. The\norder of the transition, however, remains ﬁrst-order . We\nhave observed that decreasing t⊥leads to a decrease in\n∆m, and below some t⊥we can have ∆ m < m. A sim-\nilar eﬀect has been seen when Vis increased. It should\nbe noted, however, that mand ∆mareU-dependent,\nmeaning that, depending on Vandt⊥, we can go from\n∆m < m to ∆m > m just by increasing U. Irrespec-\ntive ofVandt⊥we have always observed |δn|<|∆n|:\nelectron like carriers in one plane and hole like in the\nother.\nUnderstanding the asymmetry between planes.— The\nasymmetry between planes regarding both charge and\nspin polarization densities can be understood based on\nthe Hartree-Fock bands shown in Fig. 3. Additionally,\nwe note that in the biased BLG the weight of the wave\nfunctions in each layer for near-gap states is strongly de-\npendent on their valence band or conduction band char-\nacter [6, 33, 34]. Valence band states near the gap are\nmostly localized on layer 2, due to the lower electrostatic\npotential −V/2. On the otherhand, near-gapconduction\nband states have their highest amplitude on layer 1, due\nto the higher electrostatic potential + V/2.\nThe case U < U cshown in Fig. 3 (left) stands for the\nparamagnetic phase. The values m= 0 and ∆ m= 0\nare an immediate consequence of the degeneracy of ↑\nand↓spin polarized bands. The presence of a ﬁnite\ngap, however, leads to the abovementioned asymmetry\nbetween near-gap valence and conduction states. As a\nconsequence, a half-ﬁlled BLG would have n2= (4 +0 5 10 15 200.01.0×10-52.0×10-53.0×10-54.0×10-55.0×10-5m   (e- / unit cell)\n0 5 10 15 20-0.0027-0.0026-0.0025-0.0024-0.0023-0.0022-0.0021∆n   (e- / unit cell)T = 0.1\nT = 0.3\nT = 0.5\nT = 0.7\nT = 0.9\nT = 1.1\n0 5 10 15 20\nU (eV)0.01.0×10-52.0×10-53.0×10-54.0×10-55.0×10-56.0×10-5∆m   (e- / unit cell)\n0 1 2 3 4 5 678 9 10 11 12\nU (eV)0.00.20.40.60.81.0T (K)paramagnetic\nferromagnetic(a)\n(b)(c)\n(d)\nFIG. 4: (Color online) Panels (a), (b), and (c) show the ﬁnite\nTsolution for m, ∆m, and ∆n, respectively, with Tmeasured\ninK. Panel (d) shows the UvsTphase diagram.\n∆n)/2e−/unit cell on layer 2 (electron like carriers)\nandn1= (4−∆n)/2e−/unit cell on layer 1 (hole like\ncarriers), with ∆ n∝negationslash= 0. Even though the system is not at\nhalf-ﬁlling, as long as |δn|<|∆n|the carriers on layers 1\nand 2 will still be hole and electron like, respectively.\nLet us now consider the case U/greaterorsimilarUcshown in Fig. 3\n(center). The degeneracy lifting of spin polarized bands\ngives rise to a ﬁnite magnetization, m∝negationslash= 0. Interest-\ningly enough, the degeneracy lifting is only appreciable\nfor conduction bands, as long as Uis not much higher\nthanUc. This explains why we have m≈∆m, as\nshown in panels (a) and (b) of Fig. 2 – as only conduc-\ntion bands are contributing to ∆ m, the spin polarization\ndensity is almost completely localized in layer 1, where\nm1= (m+ ∆m)/2≈m, while the spin polarization in\nlayer 2 is negligible, m2= (m−∆m)/2≈0.\nIt is only when U≫Ucthat valence bands become\nnon-degenerate, as seen in Fig. 3 (right). This implies\nthat near-gap valence states with ↑and↓spin polariza-\ntion have diﬀerent amplitudes in layer 2. As the valence\nband for ↓spin polarization has a lower energy the near-\ngap valence states with spin ↓have higher amplitude in\nlayer 2 than their spin ↑counterparts. Consequently, the\nmagnetization in layer 2 is eﬀectively opposite to that in\nlayer 1, i.e., ∆ m > m, as can be observed in panels (a)\nand (b) of Fig. 2.\nWenote thatthe cases U/greaterorsimilarUcandU≫Ucareparam-\neterdependent. Thevalencebandscanshowanapprecia-\nble degeneracy lifting already for U/greaterorsimilarUc, especially for\nsmall values of the t⊥parameter. In this case the mag-\nnetization of the two layers is no longer opposite, with\n∆m < m. This can be understood as due to the fact that\nast⊥is decreased the weight of near-gap wave functions\nbecomes more evenly distributed between layers, leading\nnot only to a decrease in ∆ nbut also in ∆ m.\nFinite temperature.— Now we describe the phase dia-\ngram of the biased BLG in the TvsUplane. This is\ndone in Fig. 4 for δn= 5×10−5e−/unit cell . For4\nT= 0−1.1Kwe studied the dependence of m, ∆mand\n∆non the interaction U. First we note that the min-\nimum critical- Uis not realized at T= 0. There is a\nreentrant behavior which is signaled by the smallest Uc\nforT= 0.06±0.02K. For temperatures above T≈0.1K\nwe have larger Ucvalues for the larger temperatures, as\ncan be seen in panel (a). The same is true for ∆ min\npanel (b). As in the case of Fig. 2, the value of ∆ m, at\na givenTandU, is larger than m. Also the value of\n∆n, shown in panel (c), is larger than δn. Therefore we\nhave the two planes presenting opposite magnetization\nand the charge carriers being hole like in one graphene\nplane and electron like in the other. In panel (d) of Fig. 4\nwe present the phase diagram in the UvsT. Except at\nvery low temperatures, there is a linear dependence of Uc\nonT. It is clear that at low temperatures, T≃0.2K,\nthe value of Ucis smaller than the estimated values of U\nfor carbon compounds [35, 36].\nDisorder.— Crucial prerequisite in order to ﬁnd ferro-\nmagnetism is a high DOS at the Fermi energy. The pres-\nence of disorder will certainly cause a smoothing of the\nsingularity in the DOS and the band gap renormaliza-\ntion, andcanevenleadtotheclosingofthegap. Wenote,\nhowever,thatforsmallvaluesofthedisorderstrengththe\nDOS still shows an enhanced behavior at the band gap\nedges [37]. The strong suppression of electrical noise in\nBLG [38] further suggests that in addition to a high crys-\ntal quality – leading to remarkably high mobilities [39] –\nan eﬀective screening of random potentials is at work.\nDisorder should thus not be a limiting factor in the pre-\ndicted low density ferromagnetic state, as long as stan-\ndard high quality BLG samples are concerned.\nLet us also comment on the next-nearest interlayer-\ncoupling γ3, which in the unbiased case breaks the spec-\ntrumintofourpocketsforlowdensities[40]. Inthebiased\ncase,γ3still breaks the cylindrical symmetry, leading to\nthe trigonal distortion of the bands, but the divergence\nin the density of states at the edges of the band gap is\npreserved[37]. Therefore, the addition of γ3to the model\ndoes not qualitatively change our result.\nConclusion.— We have found that in the ferromagnetic\nphase the two layers in general have opposite magneti-\nzation and that the electronic density is hole like in one\nplane and electron like in the other. We have also found\nthat at zero temperature, where the transition can be\ndriven by doping, the phase transition between param-\nagnetic and ferromagnetic phases is ﬁrst-order .\nEVC, NMRP and TS acknowledge the ﬁ-\nnancial support from POCI 2010 via project\nPTDC/FIS/64404/2006, the ESF Science ProgramINSTANS. This work has also been supported by MEC\n(Spain) through Grant No. FIS2004-06490-C03-00, by\nthe European Union, through contract 12881 (NEST),\nand the Juan de la Cierva Program (MEC, Spain).\n[1] A. H. Castro Neto et al., arXiv:0709.1163 (to appear in\nRev. Mod. Phys.).\n[2] M. I. Katsnelson, Mater. Today 10, 20 (2007).\n[3] A. K. Geim et al., Nat. Mater. 6, 183 (2007).\n[4] F. Guinea et al., Phys. Rev. B 73, 245426 (2006).\n[5] T. Ohta et al., Science 312, 951 (2006).\n[6] E. V. Castro et al., Phys. Rev. Lett 99, 216802 (2007).\n[7] J. B. Oostinga et al., Nature Mater. 7, 151 (2007).\n[8] J. Nilsson et al., Phys. Rev. B 73, 214418 (2006).\n[9] T. Stauber et al., Phys. Rev. B 75, 115425 (2007).\n[10] P. Esquinazi et al., Phys. Rev. B 66, 024429 (2002).\n[11] H. Kempa et al., Phys. Rev. B 65, 241101(R) (2002).\n[12] H. Kempa et al., Solid State Commun. 125, 1 (2003).\n[13] Y. Kopelevich et al., Phys. Rev. Lett. 90, 156402 (2003).\n[14] H. Ohldag et al., Phys. Rev. Lett. 98, 187204 (2007).\n[15] A. V. Rode et al., Phys. Rev. B 70, 054407 (2004).\n[16] P. Turek et al., Chem. Phys. Lett. 180, 327 (1991).\n[17] V. I. Srdanov et al., Phys. Rev. Lett. 80, 2449 (1998).\n[18] T. Enoki et al., J. Mater. Chem. 15, 3999 (2005).\n[19] A. A. Ovchinnikov, Theor. Chem. Acta 47, 297 (1978).\n[20] A. A. Ovchinnikov and I. L. Shamovsky, J. Mol. Struct.\n(Theochem) 251, 133 (1991).\n[21] T. Stauber et al., Phys. Rev. B 71, 041406(R) (2005).\n[22] M. A. H. Vozmediano et al., Phys. Rev. B 72, 155121\n(2005).\n[23] M. Fujita et al., J. Phys. Soc. Jpn. 65, 1920 (1996).\n[24] L. Pisani et al., Phys. Rev. B 75, 064418 (2007).\n[25] A. Mielke, J. Phys. A 24, L73 (1991).\n[26] H. Tasaki, Prog. Theor. Phys. 99, 489 (1998).\n[27] K. Kusakabe et al., Phys. Rev. B 67, 092406 (2003).\n[28] J. A. Chan et al., Phys. Rev. B 70, 041403(R) (2004).\n[29]Carbon Based Magnetism , edited by T. Makarova and F.\nPalacio (Elsevier, Amsterdam, 2006).\n[30] We assume that Vand the charge density are indepen-\ndent and can be externally controlled [7].\n[31] J. Fernandez-Rossier and J. J. Palacios, Phys. Rev. Let t\n99, 177204 (2007).\n[32] Assuming equal spin densities in sublattices AandBof\nthe same layer is a reasonable approximation for t⊥≪t.\n[33] E. McCann, Phys. Rev. B 74, 161403(R) (2006).\n[34] H. Min et al., Phys. Rev. B 75, 155115 (2007).\n[35] R. G. Parr et al., J. Chem. Phys. 18, 1561 (1950).\n[36] D. Baeriswyl et al., Phys. Rev. Lett. 56, 1509 (1986).\n[37] J. Nilsson et al., Phys. Rev. Lett. 98, 126801 (2007);\narXiv:0712.3259v2.\n[38] Y.-M. Lin and P. Avouris, arXiv:0801.4576v1.\n[39] S. V. Morozov et al., Phys. Rev. Lett. 100, 016602\n(2008).\n[40] E. McCann et al., Phys. Rev. Lett. 96, 086805 (2006)."    },    {        "title": "2103.09456v1.Tunable_electronic_structure_and_magnetic_anisotropy_in_bilayer_ferromagnetic_semiconductor_Cr2Ge2Te6.pdf",        "content": "Tunable electronic structure and magnetic anisotropy in \nbilayer ferromagnetic semiconductor Cr 2Ge2Te6 \nWen-ning Ren1,2, Kui-juan Jin1,2,3,*, Jie-su Wang1, Chen Ge1,2, Er-Jia Guo1,2, \nCheng Ma1,2, Can Wang1,2,3 & Xiulai Xu1,2,3 \n1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, \nChinese Academy of Sciences, Beijing 100190, China  \n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing \n100049, China \n3Songshan Lake Materials Laboratory, Dongguan 523808, China \n*Correspondence and requests for materials should be addressed to Kuijuan Jin: \nkjjin@iphy.ac.cn  \nThe emergence of ferromagnetism in two-dimensional van der Waals materials \nhas aroused broad interest. However, the ferromagnetic instability has been a \nproblem remained. In this work, by using the first-principles calculations, we \nidentified the critical ranges of strain and doping for the bilayer Cr 2Ge2Te6 \nwithin which the ferromagnetic stability can be enhanced. Beyond the critical \nrange, the tensile strain can induce the phase transition from the ferromagnetic \nto the antiferromagnetic, and the direction of magnetic easy axis can be \nconverted from out-of-plane to in-plane due to the increase of compressive strain, \nor electrostatic doping. We also predicted an electron doping range, within which \nthe ferromagnetism can be enhanced, while the ferromagnetic stability was \nmaintained. Moreover, we found that the compressive strain can reverse the spin \npolarization of electrons at the conduction band minimum, so that two categories \nof half-metal can be induced by controlling electrostatic doping in the bilayer \nCr2Ge2Te6. These results should shed a light on achieving ferromagnetic stability \nfor low-dimensional materials. \nSince 2004 when the graphene was exfoliated by Geim and Novoselov 1, researchers \nhave revealed many unique physical properties from various two-dimensional (2D) \nmaterials, e.g. quantum spin Hall candidate monolayer WTe 2 2, stanine 3 with topological band inversion, and high-mobility black phosphorus 4,5. Nevertheless, the \nabsence of intrinsic ferromagnetism limits their application in spintronic devices. \nRecently, the intriguing intrinsic ferromagnetism has been proved both theoretically \n6-9 and experimentally 10 in Cr 2Ge2Te6, one of the layered transition metal \ntrichalcogenide’s family, which broke the long-established Mermin-Wagner theorem \n11 and greatly enriched the versatility of 2D materials. Some applications have been \nproposed in new-generation magnetic memory storage devices 12 and nanoelectronic \ndevices 13. \nAs the existence of ferromagnetism is one of the most charming features in 2D \nlayered materials, it is important to enhance the stability and realize the tunability of \nthe long-range magnetic ground state. An effective avenue is to increase the magnetic \nanisotropy energy (MAE), which is based on the energy difference between the \nin-plane and out-of-plane magnetization direction. Large MAE in van der Waals (vdW) \nmagnets would lift Mermin-Wagner restriction 11,14, for that the out-of-plane magnetic \nanisotropy would open a spin-wave gap and counteract magnetic fluctuations, \nresulting in the stabilization of the long-range ferromagnetic order 9,15,16. The \nmodulation of the magnetic properties based on the band engineering is highly desired \nin 2D layered ferromagnets, and the applications of external electric field 17-21, \npressure 22,23, electrostatic doping 24-26, and strain engineering 7,27-31 offer some valid \napproaches to tuning electronic structures, as well as the physical properties. Although \nrevealing the transformation of MAE under external factors is imperative, so far, few \nresearches has focused on the tunability of MAE with strain engineering or \nelectrostatic doping for bilayer Cr 2Ge2Te6. \nIn this letter, the first-principles calculations were carried out to study the \ntunability of electronic structures and the magnetism in bilayer Cr 2Ge2Te6 with biaxial \nstrain or electrostatic doping. We determined a critical range of strain or doping in \nwhich the MAE is increased, in other words, the ferromagnetic stability is enhanced. \nA range of electron doping is also predicted, within which the ferromagnetism and the \n(Curie temperature) 𝑇 can be raised. We also showed that two types of half-metal \nwere induced based on the external regulation. We further explored the possible mechanism involved in the variations of MAE, which would provide deeper \nunderstanding of 2D ferromagnetic materials.  \nResults and discussion  \nThe crystal structures from top and side views of bilayer Cr 2Ge2Te6 are shown in \nFig. 1a and b, respectively. The unit cell of bilayer Cr 2Ge2Te6 is denoted by the gray \nshaded region, which contains four Cr atoms, and each of them is bonded to six \nnearest-neighboring Te anions and locates at the center of an octahedron (denoted by \nlight green) formed by these Te atoms. Each layer consists of a honeycomb network \nof Cr atoms similar to graphene and comprises a Ge -Ge metal bond 32, which is a \ndimer lying perpendicularly at the central position of the CrTe 6 nets, forming \nethane-like groups of Ge 2Te6. The bilayer Cr 2Ge2Te6 contains two layers placed in AB \nstacking sequence. According to the structure optimization, the vdW gap in the bilayer \nis 3.437 Å, and the determined lattice parameters is a = b = 6.838(2) Å, as listed in \nTable 1. \n \n \nFigure 1. (a) Top and  (b) side views for pristine bilayer Cr 2Ge2Te6 in AB stacking. Blue, purple, \nand yellow balls represent Cr, Ge, and Te atoms, respectively. The gray shaded region denotes the \nunit cell. The red shaded areas indicate the Cr -Te-Cr angle α and the bond between the nearest \nneighbor Cr atoms are connected by red solid lines.  \nWe firstly explore the lattice distortion with the biaxial strain and electrostatic \ndoping. Here, the strain is denoted by 𝜂 =൫𝑎𝑎ൗ−1൯×100%, where 𝑎 and 𝑎 \ncorrespond to the strained and pristine lattice constants (without any strain or doping), \nrespectively. So, the positive sign of 𝜂 represents tensile strain, and the negative sign \nof it represents compressive strain. The electrostatic doping concentration is tuned by \naltering the total number of electrons in a unit cell. The positive and negative signs of \nconcentration represent hole and electron doping, respectively. Figures 2a and b \nexhibit how the Cr-Te-Cr angles 𝛼 and Cr-Cr bond 𝑑 change versus biaxial strains. \nAccording to the previous theoretical predictions 7,8,22,33-35, since the Cr-Te-Cr angle is \nclose to 90°, the super-exchange interaction favors ferromagnetic (FM). The direct \nexchange interaction favors antiferromagnetic (AFM), which is inversely proportional \nto 𝑑 36,37. The competition between super-exchange interaction and direct exchange \ninteraction can be effectively tuned by controlling Cr-Te-Cr angle and the \nnearest-neighbor Cr-Cr bond, which affects the magnetic ground state directly 22,34. As \nshown in Fig. 2b, the 𝛼 and 𝑑 decrease (increase) with the increase of compressive \n(tensile) strain. As shown in Fig. 2c, we define the lengths of bonds in Cr-Te as 𝑙 \nand 𝑙 of octahedra a and b, respectively. 𝑙̅ and 𝑙̅ are the average bonds of six \nCr-Te bonds in a and b octahedra, respectively. As shown in Fig. 2d, the 𝑙̅ and 𝑙̅ \ndecrease (increase) with the increase of compressive (tensile) strain. The increase \n(decrease) in average bonds 𝑙̅ and 𝑙̅ means the octahedra expanding (shrinking) \nwith the increase in the tensile (compressive) strain which is closely related to the \nmagnetic and electronic structure, and the relations will be discussed below. \n \n  \nFigure 2. (a) Schematic illustration of Cr -Te-Cr angles 𝛼 and Cr-Cr bond 𝑑. (b) Strain \ndependence of Cr -Te-Cr angles (left axis) and Cr -Cr bond lengths (right axis) for bilayer \nCr2Ge2Te6. The arrows point to the axises for each curve in the corresponding color. ( c) The bond \nlength of Cr octahedron a and b: 𝑙 and 𝑙, respectively. 𝑙̅ and 𝑙̅ denote the average bonds. \n(d) The average bonds 𝑙̅ and 𝑙̅ versus strain.  \n \n \n \n \n \nFigure 3. Electron localization function (ELF). ( a) The isosurface demonstration of pristine (0) \nbilayer Cr 2Ge2Te6. (b) The two -dimensional contour map in the direction of [001], the cut surface \nis shown in blue.  \nTo explore the electronic structures of the bilayer Cr 2Ge2Te6, the electron \nlocalization function (ELF) is simulated. Figure 3a shows the isosurface \ndemonstration of pristine bilayer Cr 2Ge2Te6. In order to explain the electron \nlocalization distribution more clearly, the two -dimensional contour map in the \ndirection of [001] is shown in Fig 3b, from which we can see the electron distribution \naround Te and Ge atoms is highly localized. The ELF value around Cr atom is 0, \nindicating the highly delocalization of electrons around Cr. The boundary between the \nlocalized and delocalized electron distribution is green, which means that the ELF \nvalue is about 0.5 and the ionic bond exists between Cr and Te atoms. We also studied \nthe effects on the electron localization distribution around different kinds of atoms \nunder biaxial strains or electrostatic doping concentrations, and found no obvious \ndifference in the electron localization distribution.  \nThe MAE is defined as MAE = 𝐸 [ଵ]−𝐸[ଵ] for each unit cell (four Cr atoms), \nwhere 𝐸[ଵ] and 𝐸[ଵ] denote the total energy for the magnetic moments oriented \nalong in-plane and out-of-plane, respectively. From this definition, we can determine \nthat increased 𝑀𝐴𝐸 means the enhanced ferromagnetic stability 14. The detailed \nresults are listed in Table 1, from which we can see the positive MAE for pristine \nbilayer Cr 2Ge2Te6, illustrating that the out-of-plane ( [001]) direction is the easy axis \nfor the magnetization. Also, we can see that the spin magnetic moment of Cr increases \nas the electron doping concentration increases from 0 to 0.2 e/u.c. Furthermore, from \nour results, the negative spin magnetic moment of Te relative to Cr is obtained, which \nis crucial for the stability of ferromagnetic ordering of Cr ions 38. \n \nTable 1. The lattice constant 𝑎 and interlayer distance 𝑙 (vdW gap) of pristine (0) bilayer \nCr2Ge2Te6 used in the present calculations. Eg and 𝑀𝐴𝐸 stand for the band gap and magnetic \nanisotropy energy, respectively. Spin ( msCr, msGe, msTe) and orbital ( moCr, moGe, moTe) moments of \nthe structures calculated by GGA+U with the spin-orbit coupling included. The results under \nspecific compressive strain (-2%), tensile strain (1%), electron doping (-0.1 e/u.c.) and hole \ndoping (0.1 e/u.c.) are also included. \n \nStructure  a l Eg 𝑀𝐴𝐸 msCr (moCr) msGe (moGe) msTe (moTe) \n (Å) (Å) (eV)  (meV/u.c.)  (μB/at) (μB/at) ( μB/at) \nBilayer 0 6.838  3.437 0.374  0.162 3.233 (0.002)  0.034 (0.001) −0.115 (–0.002)  \n-2% 6.701  3.454 0.174  0.300 3.206 (0.006)  0.027 (0.001) −0.109 (−0.002)  \n1% 6.907  3.369 0.446  0.222 3.248 (0.001)  0.037 (0.001) −0.118 (−0.002)  \n−0.1 6.853 3.443 − 0.126 3.245 (0.002)  0.035 (0.001) −0.114 (−0.002)  \n-0.2 6.867  3.447 − 0.062 3.255 (0.002)  0.035 (0.001) -0.114 (-0.002) \n0.1 6.853  3.425 − 0.302 3.229 (0.002)  0.033 (0.001) −0.115 (-0.002)  \n \nWe then investigated the spin-polarized band structures of pristine bilayer \nCr2Ge2Te6 just for comparing, as shown in Fig. 4. From Fig. 4a, we can see that both \nconduction band minimum (CBM) and valence band maximum (VBM) are of purely \nspin-up character, possessing indirect band gap. The spin-polarized band structures of \nbilayer Cr 2Ge2Te6 with 𝜂= −2% are plotted in Fig. 4b. Compared that of the \npristine one [Fig. 4a], the band gap is significantly reduced due to the increased \nbandwidth. Attractively, the spin-polarized character at the CBM has even changed from spin-up [Fig. 4a] to spin-down [Fig. 4b], indicating that the electrons at the \nCBM and VBM are with opposite spin. As shown in Fig. 4c, the larger compressive \nstrain (𝜂= −5%) further increases the bandwidth due to the stronger interatomic \ncoupling in the few-layer Cr 2Ge2Te6, inducing a semiconductor-metal phase transition. \nOn the contrary, the tensile strain (1%) increases the band gap, and the spin \npolarization at the CBM is enhanced compared with the pristine Cr 2Ge2Te6, as shown \nin Fig. 4d. Interestingly, in the vicinity of the Fermi level, the compressive strain \ninduces a degeneracy of the spin-down energy band along the K-M direction, which is \nmarked in the dashed blue circle. The half-metallic state (conduction electrons being \nspin-up) is induced with electron doping (-0.1 e/u.c.) due to the obvious spin \npolarization, as shown in Fig. 4e. While with the hole doping (0.1 e/u.c.), the Fermi \nlevel shift down into the valence band, which makes the material metallic, as shown \nin Fig. 4f. \n \n \n \n \n \n \n \n \n  \nFigure 4. The spin -polarized electronic band structures of bilayer Cr 2Ge2Te6: (a) pristine, ( b) \ncompressive strain of 2%, ( c) compressive strain of 5%, ( d) tensile strain of 1%, ( e) electron \ndoping of 0.1 e/u.c, and ( f) hole doping of 0.1 e/u.c. To achieve the visualizations of band gaps, \nthe CBM and VBM are denoted by olive horizontal dashed lines, which are connected by \ndouble-headed olive arrows. The gray horizontal dashed lines denote the Fermi level. The \ndegenerate spin -down energy band is marked in the dashed blue circle.  \nWe further studied the band structure of bilayer Cr 2Ge2Te6 under dual regulations, \ndemonstrating that the half -metallic state (conduction electrons being spin -down) can \nalso be induced in bilayer Cr 2Ge2Te6 by combining electron doping ( -0.1 e/u.c.) and \ncompressive strain ( 𝜂 = −2% ), as shown in Fig. 5a. These results indicate the \npotential usage of few -layer Cr 2Ge2Te6 in spintronic devices. In Fig. 5b, the schematic \nof the electronic structures restructuring illustrates the regulation mechanism more \nvisually. \n \n \n \nFigure 5. (a) The band structure of bilayer Cr 2Ge2Te6 under dual regulations (compressive \nstrain of 2% and electron doping of 0.1 e/u.c.). Horizontal dashed lines denote the Fermi level. ( b) \nThe schematic of the electronic structures restructuring models under the electron doping ( -0.1 \ne/u.c.) or dual regulations. Spin -up and spin -down states are marked by red and blue, respectively. \nThe white plane denotes the Fermi surface.  \n \nFigure 6 shows the atomic projected density of states (PDOS) of the bilayer \nCr2Ge2Te6 under specific strains or electrostatic doping concentrations, as well as the \nschematic electron configurations for pristine or doping of -0.1 e/u.c. Here, only the \nPDOS of Cr d orbitals are presented, which makes a major contribution to MAE \nbased on the perturbation theory analysis. As shown in Fig. 6b, the spin polarization at \nthe CBM of bilayer Cr 2Ge2Te6 is weakened under the strain of 𝜂 = −2% . Moreover, \nthe states at CBM is transformed into spin -down, which originates mainly from Cr \n𝑑௬௭/𝑑௫௭ orbitals. As the compressive strain further increases ( 𝜂 = −5% ), the band \ngap disappears and the orbital overlaps. With tensile strain of 𝜂 = 1%, the \ncontribution of Cr 𝑑௫௬/𝑑௫మି௬మ orbitals near the Fermi level is reduced compared to \nthe pristine one, as shown in Fig. 6d. As seen in Fig. 6e, the half -metallic state is \ninduced due to the obvious spin polarization at the doping concentration of -0.1 e/u.c \nin the bilayer Cr 2Ge2Te6, and the spin -up states near the Fermi level  are mainly from \nCr 𝑑௬௭/𝑑௫௭ orbitals. To understand the increased magnetic moment for Cr atoms \nwith electron doping (listed in Table 1), the diagrammatic electronic configurations \nare shown in Fig. 6g for the pristine or doped ( -0.1 e/u.c.) bilayer Cr 2Ge2Te6. \nCompared with the pristine one, the Cr -𝑑௬௭ state is occupied by doped electrons, so \nthat the net magnetic moment increases at the doping concentration of -0.1 e/u.c., \nindicating the enhanced ferromagnetism of the system. It is worth mentioning that Cr \n𝑑௬௭ and 𝑑௫௭, as well as 𝑑௫௬ and 𝑑௫మି௬మ orbitals are degenerate because of the \ncrystal symmetry. The shift of the Fermi level due to the strain or electrostatic doping \nchanges the 𝑑 projected orbitals near the Fermi level, which further changes the \nMAE of bilayer Cr 2Ge2Te6 according to the following discussion.  \n \n   \n \nFigure 6. The spin-polarized projected density of states (PDOS) of Cr 2Ge2Te6-Cr for bilayer \nCr2Ge2Te6: (a) pristine, ( b) compressive strain of 2%, ( c) compressive strain of 5%, ( d) tensile \nstrain of 1%, ( e) electron doping of 0.1 e/u.c, and ( f) hole doping of 0.1 e/u.c. ( g) The electronic \nconfiguration for pristine (left) and doping of -0.1 e/u.c. (right). The red and blue arrows indicate \nspin-up and spin-down states, respectively. The occupied states are presented by colorfully filled \nareas. \nThe MAE plays a crucial role in the stability of the long-range magnetic ground \nstates 11,39. We therefore investigated the variations of MAE and magnetic ground \nstate with various strains or electrostatic doping concentrations in the bilayer \nCr2Ge2Te6. As shown in Fig. 7a, the ferromagnetic to antiferromagnetic transition can \nbe induced by applying tensile strain more than 1%. Although larger 𝛼 than 90o and \nlonger Cr-Cr bond caused by the tensile strain indicating a weaker super-exchange \ninteraction and a weaker direct interaction, combining the results in Fig. 3 and the \nferromagnetic to antiferromagnetic transition shown in Fig. 7a, we can deduce that the \ndirect exchange interaction dominates for tensile strain more than 1%. The ground \nstate of the bilayer Cr 2Ge2Te6 is ferromagnetic when the applied strain is in the range \nof -4% to 1% while the magnetization direction remains out-of-plane. More \nmeaningfully, we determine the critical strain range of -3% to 1%, within which MAE \ncan be effectively enhanced comparing to the pristine one, and the ferromagnetic \nground state maintains as well. From Fig. 7a, we can also see that large compressive \nstrain (-5%) makes the direction of the easy axis change from out-of-plane to in-plane, \nwhich reduces the magnetic stability of the material. In the same way, we determined \nthe critical doping range about 0 to 0.2 e/u.c, as shown in Fig. 7b. Beyond this range, \nthe out-of-plane magnetization transformed into in-plane magnetization as \nelectrostatic doping concentrations increase. Similarly, a range of electron doping \nfrom -0.25 e/u.c. to 0 is also predicted, within which the ferromagnetic stability can be \nmaintained and the 𝑇 should be increased as the electron doping concentration \nincreases due to the enlarged absolute value of ∆𝐸ிெିிெ according to the mean \nfield theory 40. While the hole doping concentration has little effect on the magnetic \nground state. In order to see the insight of physical mechanisms beneath the variations of MAE, \nthe second-order perturbation theory is engaged, which indicates that only the \noccupied and unoccupied Cr d states near the Fermi level make major contributions to \nMAE in 2D magnetic systems 41. Depending on the different spin channels, the \ncontributions to MAE can be divided into two parts 41,42, including the same spin \npolarization and different spin polarizations, namely, 𝑀𝐴𝐸=𝐸±,±+𝐸±,∓ 40, which \nare expressed by \n \n𝐸±,±= (𝜉)ଶ| <𝑜±|𝐿௭|𝑢±> |ଶ−| <𝑜±|𝐿௫|𝑢±> |ଶ\n𝜀௨−𝜀±,௨±,                   (1) \n𝐸±,∓= (𝜉)ଶ| <𝑜±|𝐿௫|𝑢∓> |ଶ−| <𝑜±|𝐿௭|𝑢∓> |ଶ\n𝜀௨−𝜀±,௨∓,                   (2) \nwhere 𝑜 and 𝑢 denote the occupied and unoccupied states, respectively. The \nmagnetization directions are denoted by 𝑥 and 𝑧. Positive sign represents spin-up \nstates, and reversely, negative sign represents spin-down states. 𝜀௨ (𝜀) stands for the \nenergy of unoccupied (occupied) states, and the spin-orbit coupling constant is \nrepresented by 𝜉. Herein, five angular momentum matrix elements between two Cr 𝑑 \norbitals are nonvanishing 41 : <𝑑௫௬|𝐿௫|𝑑௫௭>, <𝑑௭మ|𝐿௫|𝑑௬௭>, <\n𝑑௫మି௬మ|𝐿௫|𝑑௬௭> , <𝑑௫௭|𝐿௭|𝑑௬௭>, and <𝑑௫మି௬మ|𝐿௭|𝑑௫௬>. It is seen from \nEquation (1) that the out-of-plane spin polarization is favored for the occupied and \nunoccupied degenerate states due to the nonvanishing <𝐿௭>ଶ and vanishing \n<𝐿௫>ଶ, while the in-plane spin polarization is favored with the nonvanishing \n<𝐿௫>ଶ for nondegenerate states. In contrary to Equation (1), for Equation (2), the \nout-of-plane spin polarization is favored for the occupied and unoccupied \nnondegenerate states due to the nonvanishing <𝐿௫>ଶ and vanishing <𝐿௭>ଶ , \nwhile the in-plane spin polarization is favored with the vanishing <𝐿௫>ଶ and \nnonvanishing <𝐿௭>ଶ for degenerate states. \nBased on above analyses, we can further understand the variations of MAE under \ndifferent strains or electrostatic doping concentrations. For pristine bilayer Cr 2Ge2Te6, the spin-up Cr 𝑑௫௬/𝑑௫మି௬మ orbitals contribute peaks in both VBM and CBM as \nshown in Fig. 6a, so the <𝑑௫మି௬మ|𝐿௭|𝑑௫௬> in Equation (1) remained, leading to the \npositive MAE, which means the out-of-plane anisotropy is favored. With strain \nengineering ( 𝜂= −2%), we can see from the states around the Fermi level in Fig 6b \nthat MAE is mainly contributed from 𝑑௬௭/𝑑௫௭/𝑑௫௬/𝑑௫మି௬మ orbitals with different \nspin states, and the <𝑑௫௬|𝐿௫|𝑑௫௭> and  <𝑑௫మି௬మ|𝐿௫|𝑑௬௭> remained in \nEquation (2), so that we can obtain positive value of MAE, or in another words, the \nout-of-plane anisotropy is favored. Also, the reduced energy bandgap in Fig. 6b offers \nthe decreased energy difference between the unoccupied and occupied states in \nEquation (2), which results in the enhanced ferromagnetic stability, identical with our \nresults in Fig. 7a. With  𝜂= −5% shown in Fig. 6c, MAE are mainly derived from \nthe inverse spin-polarized 𝑑௫௭/𝑑௬௭, the <𝑑௫௭|𝐿௭|𝑑௬௭> in Equation (2) maintains, \nimplying the in-plane anisotropy is favored, also identical with the results shown in \nFig. 7a. From the results in  Fig. 6d, the complex competition between orbital \ninteractions may be the reason for maintaining out-of-plane anisotropy. At small \nelectron doping concentrations (not larger than -0.1 e/u.c.), the Fermi level can only \ncross the spin-up states due to the obvious spin polarization in bilayer Cr 2Ge2Te6, the \nspin-up Cr 𝑑௬௭/𝑑௫௭/𝑑௫௬/𝑑௫మି௬మ orbitals contribute peaks near the Fermi level in \nFig. 6e, the out-of-plane anisotropy is favored because of the nonvanishing <\n𝑑௫௭|𝐿௭|𝑑௬௭>and <𝑑௫మି௬మ|𝐿௭|𝑑௫௬> in Equation (2) [see Fig. 7b]. In Fig. 6f, the \nspin-up Cr 𝑑௫௬/𝑑௫మି௬మ orbitals and spin-down Cr 𝑑௫௭/𝑑௬௭ orbitals contribute \npeaks near the Fermi level, the <𝑑௫మି௬మ|𝐿௭|𝑑௫௬> and <𝑑௫௭|𝐿௭|𝑑௬௭> in \nEquation (1) results in the positive MAE, i.e. the out-of-plane anisotropy [see Fig. \n7b]. \n \n  \n \n \n  Figure 7. (a) Strains and ( b) electrostatic doping concentration dependence of the magnetic \nanisotropy energy (MAE) and magnetic ground state for bilayer Cr 2Ge2Te6. The arrows point to \nthe axis for each curve in the corresponding color. The gray horizontal dashed lines denote the \nboundary of spin polarization or magnetic ground state. The critical range is marked over the \nmagenta dashed line in ( a) and (b). \nConclusion \nIn summary, we have studied the variations of electronic structures and magnetic \nproperties of bilayer Cr 2Ge2Te6 with different strains or electrostatic doping \nconcentrations. We proposed a critical strain ranges of -3% ~ 1% for bilayer \nCr2Ge2Te6, within which the ferromagnetic stability can be enhanced, as well as the \ncritical doping range of 0 ~ 0.2 e/u.c. While beyond the critical range, the tensile \nstrain induces a phase transition from the ferromagnetic to the antiferromagnetic, \nwhich is attributed to the competition between exchange interactions. Moreover, \nbeyond the critical range, the compressive strain or electrostatic doping induced the \nmagnetization direction to change from out-of-plane to in-plane. We also identified a \nrange of electron doping from -0.25 e/u.c. to 0, within which the magnetic moment \nand 𝑇 can be increased, while the ferromagnetic stability was maintained. We have \nshown two ways for inducing half-metal in the bilayer Cr 2Ge2Te6. The compressive \nstrain induced the reversed electron spin state at the conduction band minimum and \nthe transition from semiconductor to metallic state. The second-order perturbation \ntheory was applied to explain these variations of MAE. These results illustrated the \ntunability of electronic structures and magnetic properties by strain and electrostatic \ndoping in the bilayer Cr 2Ge2Te6 and hopefully shed a light on achieving ferromagnetic \nstability for low-dimensional materials. \nMethods \nAb initio calculations were performed based on density functional theory. The \nexchange-correlation interaction was treated with the scheme of generalized gradient \napproximation (GGA) parametrized by the Perdew-Burke-Ernzerhof revised for solids \n(PBEsol) 43 as implemented in the Vienna ab initio Simulation Package (VASP) 44,45. \nThe accurate projector augmented wave method (PAW) 46 was employed for the \nfollowing electronic configurations: 2 p63d54s1 (Cr), 4s24p2 (Ge), and 5 s25p4 (Te). A \n500 eV kinetic energy cutoff of the plane-wave basis set was used for all calculations. \nThe GGA+U was adopted for improving the description of on-site Coulomb \ninteractions to the Cr d orbital 47. Different effective on-site Coulomb energy value \nUeff =U-J were conducted for magnetism, optimized lattice constants, and electronic \nstructures with bilayer Cr 2Ge2Te6 (as shown in Supplementary Figs. S1, S2, and S3), \nwhich indicated that the results of Ueff =1.7 eV were consistent with previous \nexperiments and theoretical calculations 9,17,48. 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Journal of Applied Crystallography  44, \n1272-1276, (2011).  \nAcknowledgments \nThis work was supported by the National Key Basic Research Program of China \n(Grant 2019YFA0308500), the National Natural Science Foundation of China (Grant Nos. 11721404, 51761145104, 11974390 and 11674385), the Key Research Program \nof Frontier Sciences of the Chinese Academy of Sciences (Grant No. \nQYZDJ-SSW-SLH020), the Youth Innovation Promotion Association of CAS (Grant \nNo. 2018008). We acknowledge Zhicheng Zhong and Peiheng Jiang for the helpful \ndiscussions. \nAuthor contributions \nW.-N.R. and K.-J.J. contributed the whole idea and designed the research. W.-N.R., \nK.-J.J., and J.-S.W. wrote up the paper. W.-N.R. performed the theoretical calculations. \nC. M. prepared figure 5. K.-J.J. supervised the overall project. All authors discussed \nthe results and commented on the manuscript. \nCompeting interests \nThe authors declare no competing interests. \nAdditional information \nCorrespondence  and requests for materials should be addressed to K.-J.J. "    },    {        "title": "1206.3468v1.Comment_on__Anisotropic_Critical_Magnetic_Fluctuations_in_the_Ferromagnetic_Superconductor_UCoGe_.pdf",        "content": "arXiv:1206.3468v1  [cond-mat.str-el]  15 Jun 2012Comment on ”Anisotropic Critical Magnetic Fluctuations in the Ferromagnetic\nSuperconductor UCoGe”\nV.P.Mineev and V.P.Michal\nService de Physique Statistique, Magn´ etisme et Supracond uctivit´ e,\nInstitut Nanosciences et Cryog´ enie, UMR-E CEA/UJF-Greno ble1, F-38054 Grenoble, France\n(Dated: August 16, 2018)\nPACS numbers: 75.40.Gb, 74.70.Tx, 75.50.Cc\nThe results of neutron scattering measurements of\nmagnetic ﬂuctuations in weakly ferromagnetic supercon-\nductor UCoGe have been reported in a recent Letter1.\nThere was observed ﬁnite attenuation of excitations at\nzero wave vector earlier found also in another related fer-\nromagnet UGe 22that has been interpreted by the au-\nthors as ”strong non-Landau damping of excitations” .\nHere we point out that revealed phenomenon can be\ntreated as the Landau damping corresponding to the in-\ntersection of Fermi surfaces relating to diﬀerent bands.\nThe intensity of neutron scattering is proportional to\nthe imaginary part of susceptibility (see for instance2).\nIn the isotropic case the susceptibility is scalar and its\nimaginary part is given by\nχ′′(q,ω)\nω=χ(q)Γq\nω2+Γ2q, (1)\nχ(q)∝χpk2\nF\nξ−2+q2, (2)\nHere,χpis the Pauli susceptibility and the line width is\ndetermined by equality\nΓqχ(q) =χpω(q) (3)\nwhereω(q) is the Landau damping frequency.\nTakinginto accountthe possibilityofbandintersection\none can deﬁne the Landau damping frequency through\nthe imaginary part of bubble diagram with one electron\nGreen’s functions\nN0ω\nωνν′(q)∝ImT/summationdisplay\nk,ωnGν(k,ωn)Gν′(k+q,ωn+νm)|iνm→ω+i0,\n(4)whereνandν′are the band indices and N0is average\ndensity of states at the Fermi surface. For the intraband\ncaseν=ν′the Landau damping frequency ω(q)≈vFq\nvanishes linearly at q→0. For the case when the Fermi\nsurfaces of two bands intersect each other along a line l\nthe Landau damping at q= 0 acquires ﬁnite value3\nN0\nω12(q= 0)≈/contintegraldisplaydl\n(2π)3|v1×v2|. (5)\nHere vectors v1andv2are the Fermi velocities on the\nFermi sheets 1 and 2 at point klat linel.\nAtq= 0 one can estimate the product (3) as fol-\nlows Γ qχ(q)|q=0=χpω12(q)|q=0≈χpεF. Numerically\nthis value is of the order of 10−2Kthat is in correspon-\ndence with the experimentally found values of 0 .7µeV\nand 0.4µeV in UGe 22and UCoGe1correspondingly.\nThe intersection of the diﬀerent band Fermi surfaces\ncan be established from the ab initio calculations. The\nlatter have been already performed4–7although without\nspecial attention to this problem.\nWe have presented the potential explanation of nonva-\nnishing at q= 0 Landau damping measured experimen-\ntally in ferromagnetic compounds UGe 22and UCoGe1\nbased on possible intersection of the Fermi sheets cor-\nresponding diﬀerent bands. Quite large and nonvanish-\ning atq= 0 value of the Landau damping means that\nthe amplitude of pairing interaction is determined by fre-\nquency independent susceptibility. The latter of course\nshould not be taken as it is in the isotropic case given by\nequation (2).\n1C. Stock, D. A. Sokolov, P. Bourges, P. H. Tobash, K. God-\nfryk, F. Ronning, E. D. Bauer, K. C. Rule, and A. D. Hux-\nley, Phys. Rev. Lett. 107, 187202 (2011).\n2A.D. Huxley, S. Raymond, and E. Ressouche, Phys. Rev.\nLett.91, 207201 (2003).\n3M. F. Smith, Phys. Rev. B 74, 172403 (2006).\n4M. Biasini, R. Troc, Phys. Rev. B 68, 245118 (2003).5M. Divis Physica B 403, 2505 (2008).\n6P. de la Mora and O. Navarro, J. Phys.: Condens. Matter\n20, 285221 (2008).\n7M. Samsel-Czekala, S. Elgazzar, P.M. Oppeneer, E. Tallik,\nW. Walerczyk and R.Troc, J. Phys.: Condens. Matter 22,\n015503 (2010)."    },    {        "title": "1106.3795v1.Ferromagnetic_state_and_phase_transitions.pdf",        "content": "1 \n Ferromagnetic state  and phase transitions  \n \nYuri Mnyukh  \n76 Peggy Lane, Farmington, CT, USA,  e -mail: yuri@mnyukh.com  \n(Dated: June 20, 2011)  \n \n   Evidence is summariz ed attesting that the standard exchange field theory of ferromagnetism by \nHeisenberg has not bee n successful. It is replaced by the crystal field and a simple assumption that \nspin orientation is inexorably associated  with the orientation of its carrier. It follows at once that \nboth ferromagnetic phase transitions and magnetization must involve a stru ctural rearrangement. \nThe mechanism of structural rearrangements in solids is nucleation and interface propagation. The \nnew approach accounts coherently for ferromagnetic state and its manifestations.  \n \n \n1. Weiss ' molecular and Heisenberg's electron \nexchang e fields   \n \nGenerally, ferromagnetic s are  spin-containing  material s \nthat are (or can be) magnetized and remai n magnetized \nin the absence of magnetic field. This definition also \nincludes ferrimagnetics , antiferromagnetics , and \npractically unlimited variet y of magnetic structures.  \nThe classical Weiss / Heisenberg theory of \nferromagnetism , taught  in the universities  and presented \nin many textbooks ( e. g., [1-4]), deals basically with the \nspecial case of a collinear (parallel and antiparallel)  \nspin arrangeme nt. \n \n   The logic  behind the theory  in question is as follows.  \nThere is a spontaneously  magnetized crystal ( e. g., of \nFe or Ni ) due to a parallel alignment of the elementary \nmagnetic dipoles . It remains stable up to  its critical \n(Curie) temperature  point  when  the thermal agitation \nsuddenly  destroys that align ment. It need ed to be \nexplained how the ferromagnetic  state can be \nthermodynamically stable  up to the really observed \ntemperatures so high as  1042 K  in Fe.  It seemed \nunavoidable  to suggest that the for ce holding the \ndipoles in parallel  is the  dipole  interaction. Setting aside \nthe probability that  such interaction in Fe would rather \ncause mutual dipole repulsion than attraction , how \nstrong must this interaction be? It followed from  the  \nWeiss'  theory that it had to be about 104 times  stronger \nthan the magnetic dipole interaction alone.  The \nconclusion  seemed undeniable : besides the magnetic \ndipole interaction,  there is also interaction due to a \nmuch more powerful \"molecular field\" of unknown \nphysical nature.  \n  \n   Heisenberg [ 5] accepted  the Weiss'  theory  and \ndeveloped  its quantum -mechanical i nterpretation.   His \ntheory  maintains  that overlap ping of the electron shells \nresults in extremely strong electron exchange \ninteraction  responsible for  collinear ori entation of the \nmagnetic moments.  The main parameter in the \nquantum -mechanical formula was exchange integral .  Its positive sign led to a collinear ferromagnetism, and \nnegative to a collinear antiferromagnetism.  Since then it \nhas become accepted that Heis enberg gave a quantum -\nmechanical explanation  for Weiss' \"molecular field\" : \n\"Only  quantum mechanics has brought about \nexplanation of the true nature of ferromagnetism\" \n(Tamm [ 2]). \"Heisenberg has shown that the Weiss' \ntheory of molecular field can get a sim ple and \nstraightforward  explanation in terms of quantum \nmechanics\" (Seitz [ 1]).  \n \n \n2. Inconsistence  with the reality   \n \n   General acceptance of the Heisenberg's theory  of \nferromagnetism  remains unshakable to the present \ndays. Judging from the textbooks on  physics , one may \nconclude that it is rather successful  [6]. In these books \nand other concise presentations every effort was made \nto portray it as basically valid and a great achievement, \nwhile contradictions, blank areas, and vast \ndisagreements with exper iment are either omitted as \n\"details\" or only vaguely mentioned.  As a result, a new \nstudent gets wrong impression about the real status of \nthe theory. In general, t he theory  remains basically \nunchallenged . But the more detailed the source is, the \nmore drawbacks are exposed. There are experts who \npointed out to  its essential shortcomings .  \n    \n   Bleaney & Bleaney [ 7]: \"There is no doubt that \nferromagnetism is due to the exchange forces first \ndiscovered by Heisenberg, but the quantitative theory of \nferromag netism contains many difficulties \".   \n   \"We have a broad understanding of the outlines of \nferromagnetic theory, but not of the details.  The \nexchange interaction between two electrons cannot be \ncalculated a priori ... We cannot even be certain of its \nsign.\" \n \n   Belov [ 8]: \"...Many important questions connected \nwith the behavior of materials in the region [of \nferromagnetic transition] remain unsettled or in dispute 2 \n to the present time.  These include ...the actual \ntemperature behavior of the spontaneous mag netization \nnear the Curie point, the causes of the 'smearing out' of \nthe magnetic transition... the existence of 'residual' \nspontaneous magnetization above the Curie \ntemperature, and the nature of the temperature \ndependence of elastic, electric, thermal, a nd other \nproperties near the Curie point.  It even remains \nunsettled what we should take to be the Curie \ntemperature, and how to determine it\".  \n   \"The theory of Weiss and Heisenberg cannot be \napplied to the quantitative description of phenomena in \nthe nei ghborhood of the Curie point... Even for such a \n'simple' ferromagnetic substance as nickel it is not \npossible to 'squeeze' the experimental results into the \nWeiss -Heisenberg theory\" . \n \n   Bozorth [3]: \"The data for iron and for nickel [at low \ntemperatures] show that the Weiss theory in either its \noriginal or modified form is quite inadequate\".  \n   \"The Curie point is not always defined in accordance \nwith the Weiss theory but in other more empirical \nways...\"  \n \n   Crangle  [9]: \"It seems difficult to be convinced  that \ndirect exchange between localized electrons can be the \nmain origin of the ferromagnetism in metals of the iron \ngroup\".  \n \n   Kittel [ 6]: \"The Neel temperatures T N often vary \nconsiderably between samples, and in some cases there \nis large thermal hystere sis\". \n \n   Feynman [ 10]: \"Even the quantum theory deviates \nfrom the observed behavior at both high and low \ntemperatures\".  \n   \"The exact behavior near the Curie point has never \nbeen thoroughly figured out\".  \n   \"The theory of the sudden transition at the Curi e point \nstill needs to be completed.\"  \n   \"We still have the question: why is a piece of \nlodestone in the ground magnetized?\"  \n   \"To the theoretical physicists, ferromagnetism \npresents a number of very interesting, unsolved, and \nbeautiful challenges. One ch allenge is to understand \nwhy it exists at all\".  \n    \n   The last statement is especially indicative, \nconsidering that  it was the primary  purpose of the \nWeiss' and Heisenberg's theories to explain why \nferromagnetism exists at all.  Moreover, it  turned out  \nthat the exchange forces , as powerful as they assumed \nto be , do not physically  participate  in the actual \nferroma gnetic phenomena . Thus, Seitz [1] maintained \nthat the \"Heisenberg's model…is too simple to be used \nfor quantitative  investigation of the real ferro magnetic materials\".  Tamm [2] noted that \"it is the usual \nmagnetic interaction of atoms [rather than exchange \ninteraction] that is responsible for such, for example, \nphenomena as magnetic anisotropy and \nmagnetostriction\".  In this respect many  other \nphenome na could also be mentioned: domain structure, \nmagnetic hysteresis, magnetocaloric effect, Barkhausen \neffect, first -order magnetic phase transitions, \nmagnetization kinetics, and more. Remarkably , the \nquestion  why the exchange forces do no t exhibit \nthemselve s in those  phenomena has never  been raised .  \n \n      There are also other  phenomena and facts the \nexchange interaction offer s no reasonable explanation , \nif at all. Among them:   \n   (A) The value of the exchange integral for Ni was \nfound lower by about two or ders of magnitude needed \nto account for its Curie temperature.  \n   (B) A collinear order of the atomic magnetic \nmoments in ferro -, antiferro - and ferrimagnetics \nrepresents only particular cases , while  there is, in fact, a \ngreat variety of non -collinear magn etic structures as \nwell. The exchange field was unable to provide a \nparallel alignment in those innumerable magnetic \nstructures.  \n   (C) There are  materials where magnetic moments are \ntoo far apart to make any direct exchange possible. The \nappropriate elect ron shells in the ferromagnetic \nrare-earth metals  do not overlap. The „exchange field‟ \ntheory was expanded to those cases anyway, to become \n\"superexchange\".  \n   (D) The actual speed of magnetization is well below \nof the theoretically expected.  \n   (E) The ex change forces have the wrong sign.  \n \n \n3. The sign problem  \n \n   Even the initial verifications of the Heisenberg's \ntheory had to preven t its acceptance. The verifications \nhave produced a wrong sign  of the exchange forces. \nFeynman  [10] was skeptical at least,  as seen from these \nstatements: \"When it was clear that quantum mechanics \ncould supply a tremendous spin -oriented force - even if, \napparently, of the wrong sign - it was suggested that \nferromagnetism might have its origin in this same \nforce\", and \"The most  recent calculations of the energy \nbetween the two electron spins in iron still give the \nwrong  sign\", and even \"This physics of ours is a lot of \nfakery.\" The sign problem was later carefully examined \nin a special review [11] and found fundamentally \nunavoid able in the Heisenberg model. It was suggested \nthat the \"neglect of the sign may hide important \nphysics.\"  \n \n 3 \n 4. Ferromagnetic phase transitions: from \ncooperative to magnetostructural  \n \n   In order to presen t a coherent picture of \nferromagneti sm, which is the  purpose of this article, the \nmolecular mechanism of ferromagnetic phase transition \nshould be established.  With this in mind, i t will be \nhelpful to trace the evolvement  of views on \nferromagnetic phase transitions.  Initially it was \neveryone's belief tha t they are of the second order - a \ncooperative phenomenon with a fixed (Curie) \ntemperature of phase transition. Kittel [6] used Ni as an \nexample to state: \"This behavior classifies the usual \nferromagnetic/paramagnetic transition as second order\". \nIn 1965 Belov  wrote in his monograph \"Magnetic \nTransitions\" [ 8] that ferromagnetic and \nantiferromagnetic transitions are \"concrete examples\" \nof second -order phase transitions.  His work was \ndevoted to the investigation of spontaneous \nmagnetization and other properties in the vicinity of the \nCurie points.  The problem was, however, how to \nextract these \"points\" from the experimental  data which \nwere always \"smeared out\" and had \"tails\" on the \ntemperature scale, even in single crystals.   \n  \n   Vonsovskii [4]  was still on th at initial stage when \nstated that the theory of second -order phase transitions \nprovided an \"impetus\" to studies  of magnetic phase \ntransitions.  But he already entered the second sta ge of \nthe \"evol vement \" by recognizing that there are a \nnumber of  the first -order ferromagnetic phase \ntransitions . In his book about 25 such phase transitions \nwere listed , still as rather \"exotic\" . They were \ninterpreted in the usual narrow -formal manner as those \nexhibiting abrupt  changes and/or hysteresis of the \nmagnetization and o ther properties. Some of these \nfirst-order ferromagnetic transitions Vonsovskii \nerroneously described as \"apparent\" , where  structural \ntransition s occur before the ferromagnetic -to-\nparamagnetic transition s, but existence of genuine \nfirst-order ferromagnetic  transitions was also \nrecognized.  The puzzling fact of their existence led to \nthe numerous theoretical and experimental studies \nsurveyed in th e book. The conventional theory was in a \npredicament: the Curie point was not a point any more, \nand was rather a range of points and, even worse, was a \nsubject to temperature hysteresis. A ttempts were made , \nwith no success,  to complicate the theory by making  the \nexchange field dependent on the lattice deformation, \ninteratomic parameters, energy of magnetic anisotropy , \netc. The first -order ferromagnetic phase tr ansitions, so \nalien to the conventional theory, had to be accepted \nsimply as an undeniable reality. It was not realized that \na first -order phase transition meant nucleation and \ngrowth, and not a critical phenome non.  \n     The number of recognized first-order ferromagnetic \nphase transitions continued growing. They  were found  \nto be of the fist order even in the basic ferromagnetics - \nFe, Ni and Co [ 12-14]. This process was accompanied \nby the increasing re aliza tion of structural changes \ninvolved. A new term \"magnetostructural \" transitions \nhas come into use  to dis tinguish  them from not  being \n\"structural\".  At the prese nt time the quantitative ratio \n\"magnetostructural / second order\"  is dramatically \nshifting in favor o f the \"magnetostructural\" phase \ntransitions . The search with Google in June 8, 2011 \nproduced  \n'second order ferromagnetic '.…286,000 hits , \n'first order ferromagnetic'...…...926,000 hits , \n'magnetostructural transition'…718,000 hits . \n  \n \n5. The assumptions  \n \n   The above trend is obvious, address ing us toward the \nconclusion that all ferromagnetic phase transitions are \n\"structural\", meaning they are always realized by \nnucleation and crystal r earrangements at the interfaces , \nrather than cooperatively . While this conclusion will \nformally remain our assumption , it is destined to be \naccepted as a fact. Designation s of phase transitions  as \nsecond order are always superficial . Not a single \nsufficiently documented example , ferromagnetic or \notherwise , exists. This is bec ause a nucleation -growth \nphase transition represents  the most energy -efficient  \nmechanism,  considering that it needs energy to relocate \nonly one molecule at a time, and not the myriads of \nmolecules at a time as a cooperative process requires.  \nRefer  to [15 ]. \n \n   The other assumption is: the orientation of a spin is \ndetermined by the orientation  of its atomic carrier . \nConsidering that the atomic carrier is an asymmetri c \nentity, this simple assumption  is more probable  than \nability of a spin to acquire different  orientation s in the \nsame atom.  These two assumptions represent the new \nfundamentals allowing to  coherently account for \nferromagnetic state and the numerous ferroma gnetic \nphenomena . Knowledge of  the actual molecular \nmechanism of nucleation -and-growth phase  transitions \nwill be necessary. Importantly, this will not require \nintroduction of a \"molecular field\" of any kind in \naddition to the already existing chemical crystal \nbonding and magnetic dipole interaction. . \n \n \n6. The crucial part of crystal structure  \n   \n   Two opposing factors were considered by the Weiss' \ntheory: the \"molecular field\" causing a parallel \nalignment of the ensemble of elementary magnets and 4 \n the thermal agitation destroying this alignment.  There \nthe role of a crystal structure was implicitly  reduced \nonly to providing a positional, but not orientational, \norder to its magnetic dipoles. A system of atomic \nmagnetic dipoles was a dipole system only. The objects \nof thermal agitation were the elementary magnets, and \nnot the atoms carrying them. The crystal field was \noverlooked . There are powerful bonding forces \ncombining molecules, ions, atoms, magnetic or not, into \na crystal 3 -D long -range order, both positional and \norientational .  It is the crystal field that imposes one or \nanother magnetic  order by packing spin carriers in \naccordance with the structural requirements .   \n \n \n7. The m echanism of nucleation -and-growth phase \ntransitions  \n \n   The following is a synopsis of the general mechanism \nof solid -state phase transitions and other structural  \nrearrange ments, deduced from the studies presented by \nthe sequence of journal articles [ 16-29] and summarized \nin the book [13].  \n FIG. 1. The edgewise  mechanism of phase transitions and \nany other rearrangements in solid state , such as at domain \nboundaries.  The sketch illustrates the mode of advancement \nof inte rface in the n direction by shuttle -like strokes of small \nsteps (kinks), filled by molecule -by-molecule, in the  \ndirection ; i and r – are initial  and resultant crystals, \nrespectively.  (A crystal growth from liquids  is realized by the \nsame manner).  The kin ks may consist of a single molecular \nlayer or be a ladder -like conglomeration of smaller steps. \nRefer to [24,13] for m ore detailed description .    \n \n  Rearrangements in a solid state are a crystal growth \nby nucleation and propagation of interfaces. Neither \nferromagnetic and ferroele ctric phase transitions , nor \nphase transitions involving the orientation -disorder \ncrystal (ODC) phase  are excluded from this rule. Not a \nsingle sufficiently documented example exists of a \ntransition  being homogeneous (cooperative).  \n   The nuclei are locat ed in specific crystal defects -  \nmicrocavities of a certain optimum size. These defects \ncontain information on the condition ( e.g., temperature) \nof their activation and orientation of the resultant \ncrystal lattice. The nucleation can be epitaxial, in whic h case a certain orientation relationship between the \ninitial and resultant structures is observed.   \n   The interface is a rational crystallographic plane of \nthe resultant crystal lattice. It is named \"contact \ninterface\" owing to a direct molecular contac t between \nthe two lattices without any intermediate layer. The \nmolecular rearrangement proceeds according to \nedgewise  (or stepwise ) mechanism (Fig.1) involving \nformation of \"kinks\" (steps) at the flat interface and \nfilling them, molecule -by-molecule, until  the layer is \ncomplete, and building successive layers in this \nmanner.  \n \n8. Accounting for ferromagnetism and its \nmanifestations  (including the problems cited by \nFeynman ) \n \n   This will be done below within reasonable limits of a \nsingle article  - mostly in a synopsis form . \n \n    Some  problems are eliminated automatically : \n- There are t wo types of ferromagnetic phase transitions \n- second order and first order . (Only one exists).    \n- Application of the statistical  mechanics to first -order \nferromagnetic phase transitions . (Not applicable ). \n- The Curie point is blurred  and subjected to hysteresis . \n(Phase transition temperature is not a Curie point) . \n- Magnetocrystalline (anisotropy) energ y. (The partial \nimpact of the crystal on s pin directions  is replaced by \nour premise that spin orientation is bound to the \norientation of its carrier) . \n \n   Stability of a fe rromagnetic state . (Feynman : \"why \nferromagnetism exist s at all ?\"). Ferromagnetic st ate is a \n\"slave\"  of crystal structure . A particular spin alignment \n(\"magnetic structure\") is determined by the \nrequirements of crystal packing.  The magnetic \nstructure is an element of that 3 -D packing, contributing \na small positive or negative addition to  the total crystal \nfree energy.  Ferromagnetism materializ es in those \ncases when minimum free energy of the crystal packing  \nrequires placing spin carriers in the positions w ith their \nspins not mutually compensated.  Despite of the \npossible destabilizing ef fect of the magnetic interaction , \nit is too weak  to make any alternative crystal structure \npreferable . In brief: c ontribution of the magnetic \ninteraction to the total crystal free energy is small as \ncompared to that of crystal bonding; a ferromagnetic \ncrystal is stable due to its low total free energy in spite  \nof the destabilizing effect of the magnetic interaction.  \n \n   \"Why is a piece of lodestone in the ground \nmagnetized? \"  By razing  this question, Feynman meant  \nthat, besides the stability  problem , there  must be a n \noriginal  cause turning non -ferromagnetic lodestone to \n5 \n ferromagnetic. Answer: it became ferromagnetic in the \nprehistoric  times during its crystallization from liquid \nphase . The ferromagnetic  state of  lodestone is an \ninherent  element of its cryst al structure.  \n \n   Existence of a great variety of non -collinear \nmagnetic structures.  These are some types of magnetic \nstructures in crystals: “simple ferromagnetic\", “simple \nantiferromagnetic\", \"ferrimagnetic\", \"weakly \nferromagnetic\", \"weakly non -collinea r \nantiferromagnetic\", \"triangle\", “simple helical\", \n\"ferromagnetic helical\", and more.  Only in the heavy \nmetallic rare earths the following magnetic structures \nwere listed [ 9]: \"ferromagnet\", \"helix\", “cone\", \n\"antiphase cone\", \"sinusoidally modulated\", \n\"square -wave modulated\". The diversity in the mutual \npositions and orientations of spins  can only be matched \nby the diversity in the world of crystal structures . This \nis not accidental : a magnetic structu  re is imposed  by \nthe crystal, being secondary to the requirements of the \ncrystal geometry.  \n \n   Paramagnetic state.   It is usually assumed, as Weiss \ndid, that the magnetic dipoles  of the high-temperature \nphase  of a ferromagnet lost their ferromagnetic  \nalignment due to thermal rotation . The Weiss ' view  is \nunderstandable , for in his times the orientation -\ndisordered crystals (ODC)  were not yet discovered. T he \natoms and molecules , and not their spins alone, in the \nODC state are  engaged in a hindered thermal rotation . \nA zero magnetic moment of the high -temperature  phase \nin question can also be not owing to the ODC state, but \nresult from mutual compensation of its spins in the \ncentrocymmetrical structure.  \n \n   Ferromagnetic phase transitions .  A reorientation \nof spins involved in th ese phase transition s requires \nchanging  the orientation of spin carriers. The only way \nto achieve that is replac ing the crystal structure. This \noccurs by  nucleation and interface propagation . It \nfollows  that all ferromagnetic phase transitions are \n\"magnetostructural\" . The term, however,  is defective in \nthe sens e that it suggests existence of ferromagnetic \nphase transitions without  structural change . \n \n   Magnetization by interface propaga tion. The \nconventional theory does not explain why \nmagnetization occurs in this manner  rather than \ncoope ratively  in the bulk.  Once again: m agnetizatio n is \nnot a spin reorientation in the same crystal structure , but \nrequires turning the spin carrier s. The only way to turn \nthe carriers  is by crystal rearrangement.  The mechanism \nof crystal rearrangements is nuc leation and interface \npropagation. The possibility of a cooperative \nmagnetization \"by rotation\" is thus ruled out. Refer to \n[31].   \n   Magnetization  \"switching\" and \"reversal\" . Their \nexperimentally estimated ultimate speed in single -\ndomain particles turn ed out three orders of magnitude \nlower than theoretically predicted [ 30]. The cause: \nwhether they are activated by temperature, pressure, or \nexternal magnetic field, they always materialize by a \nrelatively slow process of nucleatio n and propagation of \ninterfaces. Refer to [ 31,32].   \n \n   Origin of magnetic hysteresis . The current theory \nwas powerless to deal with magnetic hysteresis  other \nthan in a phenomenological manner, while its physical \ncause remained a question mark. Solution : Magnetic \nhysteresis is a  reflection of the structural  hysteresis \nboth in ferromagnetic phase transitions and in \nmagnetization of domain systems. The y require 3 -D \nnucleation to begin and 2 -D nucleation to proceed. The \nnucleation is heterogeneous, localized in specific \ndefects – microcavities – where nucleation lags are \nencoded. These nucleation lags are the cause of  \nmagnetic hysteresis . Refer to [32]. \n \n   Formation of magnetic hysteresis  loops . The \n\"sigmoid\" shape of the hysteresis loops is due to the \nbalance between the increase  in nucleation sites and the \ndecrease in the amount of the original phase . Refer to \n[32].  \n \n   Specific heat near the Curie transition .  (Feynman: \n\"One of the challenges of theoretical physics today is to \nfind an exact theoretical description of the chara cter of \nthe specific heat near the Curie transition - an intriguing \nproblem which has not yet been solved.  Naturally, this \nproblem is very closely related to the shape of the \nmagnetization curve in the same region\"). Solution : The \ncooperative \"Curie trans ition\" does not exist. Solid -\nstate phase transitions occur by nucleation and growth \n(Section 6). What believed to be a specific heat \nanomaly (called -anomaly) is not anomaly at all . It is \nthe latent heat  of a first-order phase transition (Fig. 2). \nRefer t o [33] and Chapter 3 in [13] . \n \n   Ferromagnetic  domain structure . An essential fact \nregarding ferroma gnetic domain structure is that it is \nnot specifically rooted in a ferromagnetic state , as \nLandau and Lifshitz [34] assumed.  Domain structures \nare found also in antiferromagnetic s, ferroelectrics, \nsuperconductors, organic crystals, etc. Their origin is \nstructural . A ferromagnetic domain structure originates \nby multiple nucleation of the ferromagnetic phase in \nseveral equivalent structural orientations within the \nparamagnetic matrix.  Growth of these nuclei and \nsubsequent \"magnetic aging\" proceed toward 6 \n minimizing the magnetic energy.  Refer to [ 13], Sec. \n2.8.6, 4.5 and 4.9.  \n \n \nFIG. 2. The \"anomalous\" peaks of a physical property P, \nbelieve to be a heat capacity or magnetization,  reside in the \nranges of transition (actually, ranges of nucleation). The \n“critical  (Curie)  point T c” at the -peak  top (the common \nchoice) is a subject of hysteresis, for there are two non-\noverlapp ing transition ranges, one above T o - for heating, and \nthe other below  To - for cooling.  In the adiabatic calorimetr y \nthese peaks are not a specific heat, but  the latent hea t of first-\norder (nucleation and growth) phase transitions . A differential  \nscanning calorimetry would reveal the peak in a cooling run \nactually looking downward , being  exothermic.  \n \n.   Barkhausen effect  - short advances and stops \nduring  magnetization by magnetic field - is foreign  to \nthe traditional theory. The exchange field theory did not \nassume it. The domain the ory may account only for the \nlargest magnetization jumps, but they always consist of \nmuch smaller steps. The recent  scientific work was \ndevoted only to the phenomenological description of \nthe effect , shedding no light on its nature  [35]. But the \neffect is a direct manifestation of the crystal growth.  In \norder to lower the crystal free energy in the applied \nmagnetic field  H, the spins of the ferromagnetic crystal \nhave to turn toward the H direction, causing the \nstructural rearrangement at the interfaces  as shown in  \nFig. 1 . Quick re crystallization  of a whole  layer at the \ndomain boundary produces a magnetic \"jump\". The \nrearrangement of every successive layer is delayed by \navailability of next nucleus . The layers can be as thin as \none lattice space, or they can be conglomerat ions of \nnumerous elementary layers. In the latter case  larger \nsteps (“avalanches”) appear on the magnetization curve.  A quick restructuring of a whole domain would  produce \nthe largest  step, but it will inevitably consist of many  \nsmall er ones. Refer to [13 ], Sec. 4.10  and Addendum H . \n \n   Magnetostriction  of Fe. The phenomenon is not a \nkind of d eformation, as usually believed. The α-Fe has \na tetragonal rather than a cubic crystal structure. The \nmagnetostriction results from the structural \nrearrangement, induced b y application of magnetic \nfield,  that makes the direction of the longer \ncrystallographic axis of the participated domains \ncoincide with, or become closer to  the direction of the \napplied magnetic field . Refer to [ 36]. \n \n   Magnetocaloric effect.    It was acknowledged  [37] \nthat the \"underlying physics behind the magnetocaloric \neffect is not yet completely understood\". Now the \nphysical nature of a \"giant\" magnetocaloric effect is \nexplained in terms of the new fundamentals of phase \ntransitions, ferromagnetism and ferroelectricity  [13]. It \nis the latent heat  of structural (nucleation -and-growth) \nphase transitions from a normal crys tal state to the \norientation -disordered crystal (ODC) state where the \nconstituent particles are engaged in thermal rotation. \nThe ferromagnetism of the material provides the \ncapability to trigger  the structural phase transition by \napplication  of magnetic fi eld. Refer to [ 38]. \n \n   Disparity with ferroelectricity . Ferromagnetism and \nferroelectricity are very similar phenomena with \nanalogous set of manifestations. The standard theory \nwas unable to find a unified approach to them since the \nWeiss/Heisenberg mole cular field was applied only to \nferromagnetism. No analog to it was found (or even \nneeded) for ferroelectricity. Solution:  This profound \ninconsistency disappears after the Weiss/Heisenberg \nmolecular field is eliminated from consideration. Now \nthe two pheno mena have quite parallel explanations.  \nRefer to [13].  \n \n \nReferences  \n \n[1]  F. Seitz, The Modern Theory of Solids, Mc Grow -\nHill (1940), and the numerous subsequent editions.  \n[2]  I.E. Tamm, Fundamentals of the Theory of \nElectricity , Mir Publications, Moscow  (1979).  \n[3]  R.M. Bozorth, Ferromagnetism , D. Van Nostrand \nCo., New York  (1951).  \n[4]  S.V. Vonsovskii, Magnetism, vol. 2, Wiley (1974).  \n[5]  W. Heisenberg, Z. Physik  49, 619 (1928).  \n[6]  C. Kittel, Introduction to Solid State  Physics , 4th \nEd., Wiley, (1971) . \n[7]  B.I. Bleaney , B. Bleaney, Electricity and \nMagnetism , Oxford , Clarendon Press (1963).  \n7 \n [8]  K.P. Belov, Magnetic Transitions , Boston  Tech. \nPubl. (1965).  \n[9]  J. Crangle, The Magnetic Properties of Solids , \nEdward Arnold, London (1977).  \n[10]  R. P. Feyn man, R. B. Leighton, M. Sands , The \nFeynman Lectures on physics , v.2, Addison -Wesley \n(1964).  \n[11]  J.H. Samson, Phys. Rev . B 51 , 223 (1995).  \n[12]  R.S. Preston, Phys. Rev. Let . 19, 75 (1967).  \n[13] Y. Mnyukh, Fundamentals of Solid -State Phase \nTransitions, Fe rromagnetism and Ferroelectricity , \nAuthorhouse, 2001 [or 2nd (2010) Edition].  \n[14]   Sen Yang et al. , Phis. Rev . B 78 ,174427 (2008).  \n[15]   Y. Mnyukh, http://arxiv.org/abs/1102.1085 . \n[16]   Y. Mnyukh, J. Phys. Chem. Solids , 24 (1963) \n631. \n[17]   A.I. Kitai gorodskii, Y. Mnyukh, Y. Asadov,  \nSoviet Physics - Doclady 8 (1963) 127.  \n[18]   A.I. Kitaigorodskii, Y. Mnyukh, Y. Asadov,  J. \nPhys. Chem. Solids  26 (1965) 463.  \n[19]   Y. Mnyukh, N.N. Petropavlov, A.I. \nKitaigorodskii, Soviet Physics - Doclady 11 (1966) 4.  \n[20]   Y. Mnyukh, N.I. Musaev, A.I. Kitaigorodskii, \nSoviet Physics - Doclady 12 (1967) 409.  \n[21]   Y. Mnyukh, N.I. Musaev, Soviet Physics - \nDoclady 13 (1969) 630.  [22]   Y. Mnyukh, Soviet Physics - Doclady  16 (1972) \n977. \n[23]   Y. Mnyukh, N.N. Petropavlov,  J. Phys. Chem. \nSolids  33 (1972) 2079.  \n[24]   Y. Mnyukh, N.A. Panfilova, J. Phys. Chem. \nSolids 34 (1973) 159.  \n[25]   Y. Mnyukh, N.A. Panfilova, Soviet Physics - \nDoclady 20 (1975) 344.  \n[26]   Y. Mnyukh et al. , J. Phys. Chem. Solids 36 \n(1975) 127.  \n[27]   Y. Mnyukh, J. Crystal Growth  32 (1976) 371.  \n[28]   Y. Mnyukh, Mol. Cryst. Liq. Cryst.  52 (1979) \n163. \n[29]   Y. Mnyukh, Mol. Cryst. Liq. Cryst.  52 (1979) \n201. \n[30]   I. Tudosa et al., Nature  428, 831 (2004).  \n[31]  Y. Mnyukh, http://arxiv.org/abs/1101.1249.  \n[32]  Y. Mnyukh, http://arxiv.org/abs/1103.2194.  \n[33]  Y. Mnyukh, http://arxiv.org/abs/1104.4637.  \n[34]   Collicted Papers of L.D. Landau, Gordon & \nBreach  (1967).  \n[35]   G. Durin ,  S. Zapperi, cond -mat/0404512.  \n[36]   ]  Y. Mnyukh, http://arxiv.org/abs/1103.45 27. \n[37]   N.A. de Oliveira, P.J. von Ranke, Phys. Rep.   \n489, 89-159 (2010).  \n[38]  V.J. Vodyanoy , Y. Mnyukh, \nhttp://arxiv.org/abs/1012.0967..  \n "    },    {        "title": "1407.4906v1.Local_characterization_of_ferromagnetic_properties_in_ferromagnet_superconductor_bilayer_by_Point_Contact_Andreev_Reflection_Spectroscopy.pdf",        "content": "Local  characterization of ferromagnetic properties \nin ferromagnet/superconductor bilayer by Point \nContact Andreev Reflection Spectroscopy  \nFilippo  Giubileo *,a,b,‡, Francesco  Romeob,‡, Roberta  Citrob,a, Antonio  Di Bartolomeob, \nCarmine  Attanasiob,a, Carla Cirilloa,b, Albino  Polcaric, Paola Romanoc,a \na CNR -SPIN Salerno, via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy  \nb Dipartimento di Fisica “E.R. Caianiello”, Università degli Studi di Salerno, via Giovanni \nPaolo II 132, 84084 Fisciano (SA), Italy  \nc Dipartimento di Scienze e Tecnologie, Università degli Studi del Sannio, via Port’Arsa 11, \nBenevento, Italy  \n \n \n \n \n \n \n \n \n  \nABSTRACT  \n \nWe realized point contact spectroscopy experiment on ferromagnet/superconductor bilayers. \nDifferential conductance curves show several features that we explained  within Bogoliubov -\nde Gennes formalism  considering  the presence of two interfaces in the normal -metal -\ntip/ferromagnet /superconductor device . We demonstrate that such configura tion is suitable as \nlocal probe of  the spin polarization and thickness of ferromagnetic layer , directly on bilayer \nareas . This is due to the high sensitivity of the Andreev surface states to the physical properties \nof the ferromagnetic interlayer.  \n \n \nKEYWOR DS:  \nAndreev reflection, spin polarization, point contact spectroscopy, \nferromagnet/superconductor interface, scattering theory , Andreev bound states . \n \n \n \n \n \n  \nSpin polarization (P) represents an intrinsic parameter that characterizes a ferromagnet \nmeasur ing the spin imbalance for the occupied electronic states. Experimentally, P can be \ndetermined by photoemission spectroscopy1 (PS) as well as by spin-dependent spectroscopy \non magnetic tunnel junctions2 (MTJ) , but both methods have important drawbacks: PS has  \nlimited energy resolution  (few meV) and spatial sensitivity (few Angstrom of the surface)  \nwhile MTJs need high quality fabrication process to get planar structures with uniform thin \ninsulating barrier  and a setup  to apply high magnetic fields . Less than twenty years ago, De \nJong and Beenakker3 proposed the possibility to measure P by means of Point Contact Andreev \nReflection (PCAR) spectroscopy  exploiting the Andreev reﬂection  (AR)  process at the \nmetal/superconductor interface for which an incoming electr on with energy less than the \nsuperconducting energy gap is retroreflected  in the metal as a hole with opposite spin, while a \nCooper pair enters the superconductor4. If the metal is a ferromagnet, the probability for AR is \nreduced and (transport) polarization can be obtained from the study of the differential \nconductance spectra G(V) (conductance vs voltage) . The main advantage is clearly related to \nthe simplicity to realize a superconductor/ferromagnet micro -constrictio n by pushing a tip on \na sample, avoiding fabrication process of tunnel junctions and without application of external \nmagnetic field. On this idea, first experimental evidence s of measuring P in ferromagnet \nmaterials by PCAR have been independently reported  in 1998 by R.J. Soulen et al.5 and by \nS.K. Upadhyay et al.6. Later, this technique has been used to characterize a large number of \nferromagnetic metals5-8 (Fe, Co, Ni ), alloys5,9 (permalloy NixFe1-x), manganites5,10-11 (La1-\nxSrxMnO3), ruthenates12-13 (SrRuO 3) and half metals5,14-15 (CrO 2). Nowdays, PCAR experiment \nis well established as a valid method to measure P.  \nFrom a theoretical point  of view, a  simple approach by Strijkers et al.7 gives a generalization \nof the BTK model16 to spin polarized materials by considering the current flowing in a ferromagnet/superconductor (F/S) contact as  𝐼=(1−𝑃)∙𝐼𝑢+𝑃∙𝐼𝑝, with 𝑃 the spin \npolarization in F and 𝐼𝑝and 𝐼𝑢 the fully polarized and fully not  polarized current, respectively \n(the AR process being zero for the polarized case).  Moreover, by considering the presence of \na weak superconducting layer at the interface due to proximity effect, this model succeeded in \nsome cases to fit  conductance dips often experimentally observed at en ergies close to the gap \nenergy. Another model to describe AR at the F/S interface by F. Peréz -Willard et al.17 takes \ninto account two spin -dependent transmission coefficients for the majority and minority \ncarriers in the ferromagnet. Both models have been widely applied in order to extract spin \npolarization in several experiments in which point contact is realized between a ferromagnetic \nmaterial and a superconductor.  \nIn this Letter we extend the use of point contact technique to characteriz e F/S bilayers and \nextract direct local  information about spin polarization and thickness of the ferromagnetic \nlayer. We have developed a theoretical model within a Bogoliubov -de Gennes (BdG) \nformalism18 taking into account the presence of two interfaces, tip/sample (N/F ) and F/S (in \nthe bilayer).  We then applied such model to analyse experimental results obtained in PCAR \nexperiment by pushing a gold tip on the ferromagnetic side of a PdNi/Nb  bilayer. We \ndemonstrate that in this configuration, PCAR can give extremely precise estimation of transport \nspin polarization as well as of the local ferromagnet thickness, the high sensitivity being due \nto the strong dependence of the surface (Andreev) b ound states on such physical properties.  \nModel . We adopt a Bogoliubov -de Gennes formalism to describe the PCAR setup. \nAccordingly, the wave function (𝑟), describing an excitation of energy 𝐸 in the tip, in the \nferromagnetic layer or in  the superconducting substrate , is derived by solving the eigenvalues \nproblem given by ( 𝑥≠0,𝑑) \n[𝐻(𝑟)(𝑟)\n+(𝑟)−𝐻∗(𝑟)](𝑟)=𝐸 (𝑟).                                                                 (1)  \n \nFigure 1. Schematic repr esentation of theoretical model: transport current  𝐼𝑡 flows from \nmetallic tip through N/F (parameterized by 𝑍1) and then F/S ( 𝑍2) interfaces , barrier strengths \ndepending on the scattering potential 𝑉(𝑟). Ferromagnetic region ( 0 < 𝑥 < 𝑑) has \nmagnetization 𝑀 perpendicular to transport direction. Superconducting pairing potential, \nexisting in S, causes AR and normal reflection only at F/S interface.  \n \nThe tip region, characterized by 𝑥<0, does not present superconducting correlations and thus  \nwe set (𝑟)=0 (see Figure 1), while  the  tip quasi -particle Hamiltonian is assumed to be of \nfree-particle form 𝐻(𝑟)=−2∇2\n2𝑚−𝐸𝐹≡𝐻0(𝑟). The thin ferromagnetic layer ( 0<𝑥<𝑑) is \nmodeled by setting  (𝑟)=0 and by adding to the free -particle Hamiltonian 𝐻0(𝑟) a Zeeman \nenergy term, −𝑔𝐵𝑀z≡−𝐸𝐹ℎ𝜎𝑧, describing a magnetization  𝑀 belonging to the 𝑦−𝑧 \nplane (i.e. the magnetic easy plane) orthogonal to the transport direct ion, i.e. the  𝑥−direction. \nThe superconducting region ( 𝑥>𝑑) is described by a homogeneous pairing potential (𝑟)=\n𝑖𝑦, while the quasi -particle Hamiltonian 𝐻0(𝑟) is assumed. The Fermi velocities mismatch \namong the different regions and the non -ideality of the interfaces are modeled by using a \nscattering potential 𝑉(𝑟)=𝑈1(𝑥)+𝑈2(𝑥−𝑑) to be added to the single particle \nHamiltonian 𝐻0(𝑟),  (𝑥) being the Dir ac delta function.  We assume the translational \ninvariance of the problem along any direction belonging to the 𝑦−𝑧 plane which implies the \nconservation of the linear momentum 𝐤≡(0,𝑘𝑦,𝑘𝑧) parallel to the interface. The wave \nfunction in each region can be written in the form  (𝑟)=𝑒𝑖(𝑘𝑦𝑦+𝑘𝑧𝑧)ψ(𝑥|𝐸,𝐤) leading to \nan effective one -dimensional problem for ψ(𝑥|𝐸,𝐤), being the energy 𝐸 and 𝐤 conserved \nquantum numbers during a sc attering event. Once the wave functions ψ𝑡(𝑥|𝐸,𝐤), \nψ𝑓(𝑥|𝐸,𝐤) and ψ𝑠(𝑥|𝐸,𝐤) describing, respectively,  the tip, the magnetic layer  and  the \nsuperconducting substrate have been expressed in terms of eigenfunctions associated to  the \neigenvalues problem given in Equation  1, the scattering coefficients  are determined by \nimposing the boundary conditions:  (i) ψ𝑡(𝑥=0|𝐸,𝐤)=ψ𝑓(𝑥=0|𝐸,𝐤), (ii) \nψ𝑠(𝑥=𝑑|𝐸,𝐤)=ψ𝑓(𝑥=𝑑|𝐸,𝐤), (iii)  ∂𝑥ψ𝑓(𝑥|𝐸,𝐤)|𝑥=0−∂𝑥ψ𝑡(𝑥|𝐸,𝐤)|𝑥=0=\nkF 𝑍1ψ𝑡(𝑥=0|𝐸,𝐤), (iv)  ∂𝑥ψ𝑠(𝑥|𝐸,𝐤)|𝑥=𝑑−∂𝑥ψ𝑓(𝑥|𝐸,𝐤)|𝑥=𝑑=\nkF 𝑍2ψ𝑓(𝑥=𝑑|𝐸,𝐤), where kF indicates the Fermi wave vector, while 𝑍1/2=2𝑚𝑈1/2/\n(2kF) represents the BTK parameter describing the interface properties. The current 𝐼𝑡 \nflowing through the constriction can be expressed via the AR coefficients  a′(𝐸,𝐤) and the \nnormal reflection coefficients b′(𝐸,𝐤) defining the tip wave function ψ𝑡=ψ𝑒in+\n∑b′′ ψ𝑒′out+∑a′′ ψℎ′out .  Here ψ𝑡 is decomposed into incoming ( in) or outgoing ( out) \nelectron -like ( ψ𝑒in/out) and hole -like ( ψℎin/out) modes having spin projection /2, with =\n±1. The experimentally measured differential conductance has to be compared with G(𝑉)=\n𝑑𝐼𝑡\n𝑑𝑉  where:  \n𝐼𝑡(𝑉)∝𝐴\n(2)2∫𝑑𝐸 𝑑2𝐤[2+∑ |a′(𝐸,𝐤)|2 ′ −\n                ∑ |b′(𝐸,𝐤)|2 ′ ](𝑓(𝐸−𝑒𝑉)−𝑓(𝐸)), \n 𝑓(𝐸) is the Fermi -Dirac distribution and 𝐴 represents the junction cross section. The above \nexpression can be rewritten in terms of angular integration over the incidence angles (𝜃,𝜑) by \nchanging the double integral variables taking the modulus  k(𝐸)=√2𝑚𝐸/ of the wave vector as fixed, i.e.  ∫ 𝑑2𝐤→∫ k2(𝐸) (sin𝜑)2cos𝜃𝑑𝜃𝑑𝜑 . In the PCAR experiment the voltage \nbias  𝑒𝑉 ranges from zero to few times the superconducting gap , as a consequence, in the \nrelevant energy window [𝜇,𝜇+𝑒𝑉] around the chemical potential  𝜇, the wave vector k(𝐸) is \nwell approximated by the constant value  kF, being the corrections to the leading term of order \nof  𝑒𝑉\n𝜇~\n𝜇≈10−3. Thus the central qu antity of our analysis can be written as:  \n𝐼𝑡(𝑉)∝kF2𝐴\n(2)2∫𝑑𝐸 𝑑[2+∑ |a′(𝐸,𝜃,𝜑)|2 ′ −\n             ∑ |b′(𝐸,𝜃,𝜑)|2 ′ ](𝑓(𝐸−𝑒𝑉)−𝑓(𝐸)), \nwhere we introduced the notation  𝑑≡(sin𝜑)2cos𝜃𝑑𝜃𝑑𝜑 , while the angular integration is \nperformed over 𝜃∈[−𝜋/2,𝜋/2] and 𝜑∈[0,𝜋]. We notice that the factor  kF2𝐴\n(2)2 is related to \nthe number of transverse modes which participate in the charge transport. Once the differential \nconductance 𝐺(𝑉) is determined using the above relation, it is normalized with respe ct to the \ndifferential conductance of the junction at high bias, i.e. using the value 𝐺𝑁𝑁=𝐺(𝑉)|𝑒𝑉≫∆. \nThe quantity 𝐺(𝑉)/𝐺𝑁𝑁 is directly compared with the experimental data.  \nExperiment . The bilayers measured in this study were grown in -situ by a three -target ultra high \nvacuum  dc magnetron sputtering on Al 2O3 substrates in Argon pressure (few µbar) depositing \nfirst a 40 nm thick Nb layer and then a 4 nm thick Pd0.84Ni0.16 layer.  The critical temperature \nof the bilayer 𝑇𝑐𝑏𝑖𝑙𝑎𝑦𝑒𝑟 was checked by resistive transition measurement and it has been \ncompared with the same parameter of a twin Nb film ( 𝑇𝑐𝑁𝑏=8.2 𝐾, without  the ferromagnetic \nlayer on top) resulting about 1K low er. \nPCAR experiments have been performed by pushing a mechanically etched gold tip on the \nferromagnetic side of the PdNi/Nb bilayer. The tip is installed on a screw driven chariot in \norder to allow gentle approach to the sample surface. The measuring inset  is directly introduced in a liquid  helium cryostat, with the device exposed to helium atmosphere. Conventional four -\nprobe technique (see inset of Figure  1) has been applied in order to measure current -voltage (I -\nV) characteristics in the temperature range  between 4.2K and 10K (i.e., above the Nb critical \ntemperature). Differential conductance spectra (G -V) are obtained by numerical derivative of \nthe I-V curves.  Several different contact resistance s have been obtained  in the range 2-10 \nby simply varying the position and the pressure of the tip on the sample.  \nThe transport regime for such contacts can be easily estimated to be ballistic or diffusive by \nusing Wexler’s formula19   \n𝑅𝑃𝐶=4𝜌𝑙\n3𝜋𝑎2+𝜌\n2𝑎 \nin which the first term gives the  Sharvin resistance20 describing the ballistic regime and the \nsecond one is the Maxwell21 resistance describing the diffusive regime. The dominating term \nwill depend on the contact dimension 𝑎, the resistivity 𝜌 of the sample and the mean free path \n𝑙 of the charge carriers. For 𝜌=13 𝜇Ω 𝑐𝑚 (as resulting by direct measurements) and \nconsidering that 𝜌𝑙=3.72×10−6𝜇Ω 𝑐𝑚2 for niobium22-23, it comes out that the minimum \ncontact dimension in our junctions is 𝑎≈ 8 nm to be compared with the mean free pat h 𝑙≈ 3 \nnm. The ratio 𝑙/𝑎<1 gives indication for diffusive transport. However, the extention of BTK \ntheory to diffusive regime has been proven24 to be successful in correctly identify the effect of \nspin polarization on the conductance spectra with respec t the effects due to the diffusive \ntransport. Moreover, it has been demonstrated25 that the application of ballistic model to \nanalyze PCAR spectra obtained in the diffusive regime will allow an estimation of the spin \npolarization with an error below 3%. At  the same time, the barrier parameter Z evaluated in a \nballistic model will be systematically larger  than what obtained in a diffusive model (with a \nvariation of about 0.5 -0.6) due to the fact that the parameter Z should include either the ballistic \nbarrie r strength (as expected in the BTK theory) and other physical effect s (diffusion, velocity and/or mass mismatch) .  In Figure  2, we show a variety of normalized conductance spectra \n(conductance is expressed as 𝐺(𝑉)/𝐺𝑁𝑁 while energy scale, eV/ ΔNb, is normalized to the Nb \nenergy gap ) obtained at T=4.2 K. At a first qualitative analysis, these data can appear quite \npuzzling : Zero Bias Conductance Peak (ZBCP) higher than 2 (i.e., 𝐺(𝑉=0)/𝐺𝑁𝑁 > 2, where \n𝐺(𝑉=0) is the conductance at zero bias and  𝐺𝑁𝑁 is the conductance at high bias)  appears in \nmany spectra ; two conductance  dips, one fixed at the niobium gap energy, another at lower  \n(not fixed)  energy , are always present;  conductance maxima within gap energy  appear with \ndifferent intensity in th e various spectra.  To quantitatively analyze the conductance curve \nreported in Figure  2a-2d, experimental data (empty circles) are compared to theoretically \ncalculated spectra (solid lines) according to the model introduced in the previous section : the \nresult is satisfactory with  all features properly reproduced. We used as fitting parameters the \nbarrier strength 𝑍1 (describing the N/F interface), the barrier strength 𝑍2 (describing the F/S \ninterface), the thickness parameter 𝑟 (describing the thickness  of the ferromagnetic layer  \naccording to the formula 𝑟=𝑘𝐹𝑁𝑏∙𝑑, where 𝑘𝐹𝑁𝑏≈11.8 𝑛𝑚−1 is the Fermi momentum26 and \n𝑑 is the real thickness), and the spin polarization ℎ; we do not consider as fitting parameter the \nniobium superconducting energy gap and the effective temperature that we fixed at the values \n∆𝑁𝑏=1.5 meV and 𝑇𝑒𝑓𝑓=0.7 K. We notice that the effective temperature has been fixed at a \nvalue sensibly lower t han the bath temperature ( 𝑇𝑏𝑎𝑡 ℎ=4.2K): this discrepancy will be \ndiscussed in the next section.  Figure 2. Differential conductance spectra measured at low temperature (T = 4.2 K) on \ncontacts realized by pushing an  Au tip on a PdNi/Nb bilayer . Inset in (b) shows a scheme of \nthe setup. Different contacts are classified by 𝑅𝑁𝑁, i.e. the high bias resistance of the device. \nExperimental data (empty circles) are normalized to 𝐺𝑁𝑁=1 (where 𝐺𝑁𝑁=1/𝑅𝑁𝑁 is the high \nbias conductance) and compar ed to curves (solid lines) resulting from the theoretical model \ndiscussed above. In each plot are listed the parameters used in the model to reproduce the data: \n𝑍1 and 𝑍2 are the barrier height of the tip (Au) /ferromagnet (PdNi)  interface  and of the \nferrom agnet(PdNi)/superconductor(Nb) interface, respectively; 𝑟 is related to the ferromagnet \nthickness  𝑑  via the equation 𝑟=𝑘𝐹𝑁𝑏∙𝑑; ℎ is the polarization of the ferromagnetic layer. All \nfits are performed by considering a temperature value 𝑇𝑒𝑓𝑓= 0.7 K well below the bath \ntemperature of 4.2  K. (d) The asymmetry of the spectrum is reproduced by simply assuming a \nhigher temperature for the positive energy side  𝑇𝑒𝑓𝑓= 1.1 K. \n \nFitting parameters used to reproduce experimental data are reported in F igure 2 for each plot. \nAll conductance spectra are characterized by a large 𝑍1 value ( 2.3<Z1<5.1) indicating a \nlow transparency of the  contact between the gold tip ant the PdNi layer; at the same time, \nsignificantly lower 𝑍2 values are always found ( 0.26<Z2<0.50) as expected for in -situ \nfabricated interface Nb/PdNi. More interestingly, the other two parameters involved in the \nfitting procedure take on values in narrow intervals, 51.2<𝑟<55.1 and 0.140 <ℎ<0.146 \nfrom which it is possible to give an extimation of the ferromagnet thickness 4.3nm <𝑑<\n4.6nm  in the various sample positions and of the corresponding spin polarization 14.0% <\nP<14.6%. We will discuss in the following section on the precision of such estimation . \n \nFigure 3.  Temperature dependenc e of the conductance spectra measured for the contact of \nFigure 2a. (a) Spectra have been shifted for clarity. Black arrows identify the position of the \nconductance dip at the gap edge. (b) Temperature evolution of the relative amplitude of the \nzero bias c onductance 𝐺(𝑉=0)/𝐺𝑁𝑁 is compared with the theoretical BCS behavior of Δ(T). \n(c) Temperature dependence of the Nb energy gap as extracted by the experimental data.  \n \nIn order to verify that conductance features are strictly related to the superconductivity of Nb, \nwe performed complete temperature dependence of the conductance spectra. We show in \nFigure  3 that for T  = 7.7 K the device is not anymore superconducting and all conductance \nfeatures are washed out. Moreover, the conductance dip position (black arrows in Figure  3a) \nand the amplitude of the zero bias peak both correctly follow the expected BCS behaviour for \nΔ(T).   \nDiscussion . The N -F/S configuration of the poin t contact experiment allowed to measure \nseveral different conductance spectra, as reported in Figure  2. The use of a theoretical model \nbased on  BdG formalism give s a complete explanation of all conductance features measured \nat low temperatures. In Figure 4 we show the effect of the variation of the model parameters \non the conductance curves. In all plots of Figure  4 the solid (black ) spectrum corresponds to \nthe calculated curve of Figure  2a whose parameters are 𝑍1=3.85, 𝑍2=0.35, 𝑟=51.25, ℎ=0.1403. \nThe f our plots of Figure  4 are then obtained by keeping fixed three parameters and allowing \nonly one parameter to vary in a range . Thus,  Figure  4a for 0< 𝑍1<8, Figure  4b for 0< 𝑍2<8, \nFigure  4c for 48< 𝑟<58, Figure  4d for 0.135< ℎ<0.145 are obtained.  \n \nFigure 4. Evolution of the differential conductance spectra calculated within our theoretical \nmodel. The black spectrum evidenced in each plot corresponds to Figure  2a (𝑍1=3.85, 𝑍2=0.35, \n𝑟=51.25, ℎ=0.1403). Different plots are obt ained by varying only one parameter (a) 𝑍1, (b) 𝑍2, \n(c) 𝑟, (d) ℎ and keeping the other three parameters fixed at the value of Figure  2a. \n \nA completely transparent contact ( 𝑍1 = 0) results in a simple spectrum with two maxima close \nto the gap energy. For increasing 𝑍1, there is a slow evolution of the spectra and two \nconductance minima appear exactly at 𝛥𝑁𝑏 and a ZBCP  arises as well as other two maxima \nwithin energy gap. For l argest 𝑍1 values , the zero bias peak further splits. Differently, by \nvarying 𝑍2 there is a fast variation of the conductance spectra that become fully gapped as soon \nas 𝑍2 approaches about 1, suggesting that the wide variety of the observed spectra are favored \nby the high transparency of the F/S interface in the bilayer.  In the latter case, t he formation of \nsurface (Andreev) bound states is expected27. \nFrom Figure  4c we notice a strong dependence of the conductance features on the parameter 𝑟. \nThe range  48<𝑟<58 corresponds to a thickness 𝑑 of the ferromagnetic layer of less than  1nm. \nAccording to this, the precise fitting of the conductance spectrum in such N -F/S configuration \nis suitable for direct local measurement of the F -layer thickness. As an ex ample, in Figure  5a \nwe show the different spectra obtained for 𝑟 = 51.25 (i.e. 𝑑≈4.3nm), 𝑟 = 52.25 (i.e. 𝑑≈4.4nm) \nand 𝑟 = 53.25 (i.e. 𝑑≈4.5nm).  To complete the analysis of the conductance spectra dependence \non the various parameters, we show in Figure  4d the result obtained by varying ℎ in the range \n0.135 <ℎ<0.145. This parameter gives direct information about the spin polarization of the \nferromagnetic layer. Also in this case, the fast modification of the spectra for small variations \nof the param eter allows precise estimation of P (see Figure  5b). \n  \nFigure 5. Comparison of conductance spectra calculated within our theoretical model for small \nvariations of the parameter (a) 𝑟  and (b) ℎ. The lower (black) spectrum in both panels is the \nconductance spectrum of Figure  2a calculated with the parameters 𝑍1=3.85, 𝑍2=0.35, 𝑟=51.25, \nℎ=0.1403.   \n \nWe notice that the values of the polarization obtained from the various contacts (and \ncorrespondin g fittings) of Figure  2 span on a larger range 14.0% <𝑃<14.6%. Due to the high \nsensibility of the conductance curve on the ℎ value , we can conclude that we are probing small \nspatial variation of the polarization probed locally by the PCAR setup with the di fferent \ncontacts that have been realized, since we demonstrate that the technique has a resolution better \nthan 0.001 variations on the ℎ values.  \nEffective temperature.  As mentioned in the previous section, the numerical simulation s need  \nfour fitting parameters, namely 𝑍1, 𝑍2, 𝑟, ℎ, while we fixed 𝛥𝑁𝑏 = 1.5 meV and 𝑇𝑒𝑓𝑓 = 0.7 K. \nAccording to our experimental setup, in which the point contact inset is directly placed in the \nliquid helium bath, the temperature of the device is nat urally expected to be stabilized at 𝑇=\n4.2 K. On the other hand, the reported conductance features are clearly very sharp in contrast \nwith the thermal smearing expected at T  = 4.2 K. Such phenomenon has been already observed \nfor Cu/Nb contacts7 and it has been ascribed to possible non -equilibrium transport processes in \npresence of proximity effect at the interface.  \nWe also have to take into account  that our experiment (pushing a normal tip on F/S bilayer) \nrealizes a N -F-S device. It has been reported28 that in such case the  ferromagnetic interlayer \nreduces the Andreev Joule heating favoring an efficient cooling, the cooling power depending \non the spin polarization. Moreover, maximum cooling power is expected29 at 𝑇≈0.5 𝑇𝑐 and \nfor bias voltages of about eV /Δ≈1. This process can reasonably affect the local temperature of \nthe N -F-S device under investigation in this work, explaining a discrepancy between 𝑇𝑏𝑎𝑡 ℎ and \n𝑇𝑒𝑓𝑓. However, the cooling power generally acting on this systems seems to be not enough  to \ncause a temperature reduction of few Kelvin, suggesting that further physical effects should \ncontribute to obtain such experimental observation.  \nFinally, we also consider the possibility that such small effective temperature could arise from \nthe predi cted “extraordinary ” temperature dependence  of the resonant Andreev reflection peak \nexpected in N/quantum -dot/S systems30, the tunneling mediated by discrete energy levels  being \nresponsible for an anomalous broadening of the conductance peak with respect the thermal one . \nExperimentally, such behavior has been reported for N/semiconductor/S systems31 and in \nN/High -Tc-Superconductors constrictions32, where the discrete  levels  could be due to the \nexistence of surface (Andreev) bound states.  \nIn conclusion, we used Bogoliubov -de Gennes formalism to describe N -F-S systems. \nExperimentally, such configuration has been realized by means of point contact Andreev \nreflection spectroscop y by pushing a metallic tip on the ferromagnetic side of PdNi/Nb bilayer. \nDifferential conductance spectra for several contacts have been measured at low temperature, \nshowing several different features, all consistently explained within our theoretical mod el. \nMoreover, we demonstrated that this setup configuration is suitable to locally measure the \nferromagnetic properties (polarization and thickness) of the F -layer  with high precision.  \n AUTHOR INFORMATION  \nCorresponding Author  \n*Dr. Filippo Giubileo,  \nAddres s: CNR -SPIN, via Giovanni Paolo II, 132 , 84084 Fisciano (SA), Italy  \nTel: +39.089.969329  \nFax: +39. 089.96 8817   \nE-mail: filippo.giubileo@spin.cnr.it  \n \nAuthor Contributions  \nF.G. designed the experiment, performed PCAR measurements and numerical fittings,  did data \nanalysis  and wrote the paper; F.R. developed theoretical model , performed numerical fittings  \nand contributed to data analysis  and paper writing ; P.R. contributed to the e xperiment design, \nmeasurements and data analysis; C.C . and C.A. provided the bilayers  and their electrical \ncharacterization ; R.C. contributed to develop the theoretical model; A. DB. and A.P. helped \nwith the data analysis.  The manuscript was revise d by all  authors. All authors have given \napproval to the final version of the manuscript.  ‡These authors contributed equally.  \n \n \n \n \n \n REFERENCES  \n1. Johnson , P. D. Rep. Prog. Phys.  1997 , 60, 1217 -1304 . \n2. Meservey,  R.; Tedrow,  P. M.  Phys. Rep.  1994 , 283, 173-243. \n3. de Jong M. J. M. ; Beenakker  C. W. J.  Phys. Rev. Lett.  1995 , 74, 1657 -1660.  \n4. Andreev  A. F.  Sov. Phys. JETP 1964 , 19, 1228 -1231.   \n5. Soulen,  R. J.,  Jr.; Byers,  J. M.; Osofosky,  M. S.; Nadgorny , B.; Ambrose,  T.; Cheng,  S. F.; \nBroussard,  P. R.; Tanaka,  C. T.; Nowak,  J.; Moodera,  J. S.; Barry,  A.; Coey,  J. M. D.  \nScience  1998 , 282, 85-88. \n6. Upadhyay, S. K.; Palanisami, A.; Louie, R. N.; Buhrman, R. A.  Phys. Rev. Lett. 1998 , 81, \n3247 -3250 . \n7. Strijkers, G.; Ji, Y.; Yang, F.; Chien, C.; Byers, J.  Phys . Rev. B 2001 , 63, 104510.  \n8. Kant, C. H. ; Kurnosikov, O. ; Filip, A. 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Ponmurugan∗\nSchool of Basic and Applied Sciences, Central University of Tamil Nadu,\nThiruvarur - 610 005, Tamil Nadu, India.\nAbstract\nThe Monte Carlo analysis for the magnetic response of a single-walled nanotube using the\nMetropolis and Wang Landau algorithms is reported in the present paper. The nanotube\narchitecture used in the present study utilizes the spin half Ising model with nearest neigh-\nbors interaction and obtained various magnetic orderings namely, ferromagnetic, G-type anti-\nferromagnetic, A-type anti-ferromagnetic, and C-type anti-ferromagnetic. It is also found that\nthe phase changes from ferromagnetic/anti-ferromagnetic to paramagnetic with the modiﬁci-\nation of system’s control parameters. The transition temperatures is determined for various\ninteraction strength in the absence of magnetic ﬁeld and for ﬁxed interaction strength with the\ninclusion of external magnetic ﬁeld. The present study conﬁrms the transition from ferromag-\nnetic /anti-ferromagnetic to paramagnetic is a second order transition.\nKey words: Nanotube, Ising model, Monte-Carlo techniques, Phase changes, transition tem-\nperature.\n1 Introduction\nThe study of magnetic properties in low-dimensional systems has attracted a lot of theoretical\nand experimental interest recent years [1,2]. The low dimensional systems such as the magnetic\nnano-materials have wider range of technical applications, viz., sensors [3], drug delivery [4],\nbio-medical applications [5], magnetic resonance imaging [6], magnetic recording media [7],\nmemorydevices[8]andpermanentmagnets[6]. Thenano-structuredmaterialscanbeproduced\nin a variety of shapes, such as nano ﬁlms [9], nano rings [10], nanotubes [11], nano wires [12]\nand nano particles [13]. Among other shapes, the magnetic nanotube has interesting uses in\nhigh performance battery [14], humidity sensors [15], tactile sensors [16] and energy storage\napplications [17]. Thus, analyzing the mechanical and thermal properties of a nanotube is\nimportant in enhancing its applications in various ﬁleds. The simple interaction model known\nas the Ising model [18–20] is well-suited to study the magnetic behavior of nanotubes using\nvarious theoretical techniques.\nThe phase transition of a hexagonal cylindrical magnetic super-lattice nanotube is inves-\ntigated using the molecular ﬁeld theory approximation [21]. The eﬀective ﬁeld theory with\n∗email correspondence: ponphy@cutn.ac.in\n1arXiv:2112.12393v2  [cond-mat.stat-mech]  25 Apr 2022correlations is used to analyze the phase diagram and magnetization of a cylindrical nanotube\nas a transverse Ising model [22]. Using the diﬀerential operator technique and eﬀective ﬁeld\ntheory, the magnetic properties of a six-legged spin half and spin one nanotubes are studied\nin the presence of an applied magnetic ﬁeld [23,24]. The phase transition and magnetiza-\ntion of Ising nanotubes are analyzed by cellular automata approach with ferromagnetic and\nanti-ferromagnetic interactions [25]. Magnon thermodynamic properties of nanotubes are in-\nvestigated using the many-body green function approach and studied the eﬀect of the spin\ncorrelation on the magnetic properties of the single walled nanotubes [26]. The thermal and\nmagnetic properties of low dimensional spin related systems are also reported in detail in\nRefs. [28–35]. The Monte-Carlo algorithm is used to study the dielectric properties along with\nthe polarization and hysteresis behavior of mixed spin in a nanotube system [27]. Several stud-\nies also discussed the magnetic responses of Ising modeled nanotubes [36–39] using Metropolis\nMonte Carlo algorithm. In the present paper, both the Boltzmann (Metropolis) and non-\nBoltzmann (Wang Landau) Monte-Carlo algorithms are used to analyze the phase changes and\nthe order of the transition of the single-walled nanotube which is modeled as spin half Ising\nsystem.\nThe Metropolis algorithm [40] is particularly useful to compute the mechanical properties\nand the hysteresis behavior by generating the micro-states that correspond to the Boltzmann\ndistribution [41]. However, the Metropolis algorithm is not suitable for obtaining the thermal\nproperties of nanotube by estimating its density of states. This is due to the fact that the\nindividualmicro-statescannotbeassignedanentropyvaluesinceentropyisacollectiveproperty\n[42]. Hence, in the present study, the Wang Landau (WL) algorithm is used to calculate the\ndensity of states and thereby the temperature-dependent observables can be determined with\nun-weighting and re-weighting techniques [43,47].\nIn this paper, a systematic analysis of the magnetic properties of a single walled Ising\nnanotube is explored in detail. The inﬂuence of the control parameters (the interaction strength\nand the external magnetic ﬁeld) on the properties of the Ising nanotube are also investigated.\nThe manuscript is organised as follows: In section. 2, the detailed description of the model\nand the simulation techniques are explained. The results and its discussions are eloborated\nin section.3. The observations and its conclusions are given in the conclusion section of the\nmanuscript.\n2 Model description and the Monte Carlo simulation tech-\nniques\n2.1 Model\nIn the present work, the spin half Ising model is examined and decorated as a single wall nan-\notube which consists of a layered graphite-like structure with magnetic moments Si, occupied\nin its respective lattice sites. Each layer would be composed of 6lattice sites Si. Every lattice\nsite can take a spin value of ( +1)/(\u00001) for spin up/spin down, respectively. The Hamiltonian\nof the conﬁguration can be described as,\nH=\u0000J1X\n<i;j>SiSj\u0000J2X\n<i;k>SiSk\u0000BX\niSi: (1)\nWhere< i;j >,< i;k > represents the nearest neighbour sites, J1stands for the exchange\ninteraction strength between the ﬁrst neighbouring sites within every layer, J2stands for the\nexchange interaction strength between the layers and Bis the external magnetic ﬁeld. These\ninteractions ( J1&J2) will favour the spin to align either parallel (Ferro Magnetic: FM) or\n2anti-parallel (Anti Ferro Magnetic: AFM). If J1andJ2are greater than zero, then the system\nfollows ferromagnetic order, whereas the system follows anti-ferromagnetic order if either of the\ninteraction ( J1orJ2) is less than zero. The detailed picture of the Ising nanotube is depicted\nin Fig. 1. In the ﬁgure, each site (green ball) is connected with 4neighbouring sites. The\nJ1, i.e., the intralayer interaction where each site is connected with two neighbouring sites are\nshown in red lines and the blue line that connects the two neighbouring sites of diﬀerent layers\nrepresents the inter-layer interaction ( J2). The periodic boundary condition is applied in the\naxis of the system’s length (z)whereas the closed boundary conditions are given in the xand\nyaxes.\nFigure 1: Illustration of Ising nanotube. Green balls represents lattice points, red lines represents J1\ninteractions and blue line represents J2interactions.\nThe following sub section reviews the Metropolis algorithm for canonical ensemble by con-\nsidering the system in equilibrium with the reservoir and further considers WL algorithm with\nun-weighting and re-weighting to calculate the thermal properties.\n2.2 Simulation technique 1: Boltzmann technique\nIsing nanotube can be analyzed in simulation by using the temperature dependent Metropolis\nalgorithm [40,44], called the Boltzmann technique. The spin conﬁgurations (Markov chain) are\ngenerated according to the Boltzmann distribution ( C0!C1!C2!:::Ci!Ci+1!\n:::CN). The simulation begins with the initial random conﬁguration (C0)followed by the\ntrial conﬁgurations (Ct)which are generated by ﬂipping a randomly selected spin. This trial\nconﬁguration is accepted with transition probability p=minf1;exp(\u0000\f\u0001E)g, where \u0001E=\nEt\u0000Eiis the energy diﬀerence between the trial and the initial spin conﬁgurations, \f=1\nkBT\n(kBis the Boltzmann constant and it is set to unity for the whole simulation) is the inverse\ntemperature and Tis the temperature of the reservoir. About 106Monte Carlo Sweeps (MCS)\nis performed for each temperature in which the ﬁrst 105MCS are discarded for the system\nequilibration. The remaining MCS part is taken into account for calculating the average values.\nThe canonical ensemble average of a desired physical quantity Ois given by,\nhOiT=1\nMsX\nCO(C); (2)\nwhereMsis the total number of MCS after equilibration. The average energy and average\nmagnetization can be obtained from the equation 2. The speciﬁc heat capacity (CV)and\nmagnetic susceptibility (\u001f)were calculated using the following expressions,\nCV(T) =1\nkBT2(hE2iT\u0000hEi2\nT); (3)\n\u001f(T) =1\nkBT(hM2iT\u0000hMi2\nT); (4)\n3wherehEiis the average energy and hMiis the average magnetization of the system. The\nMagnetization per spin ( M) is calculated by summing all magnetic moments Siand is given by\nM=1\nNP\niSi, where,N= 6Lis the total number of spins and Lrepresents the total number\nof layers (each layer consist of six spins). In general (otherwise speciﬁed), N= 180spins with\nL= 30layers are used in the entire simulation. As mentioned earlier, the Metropolis algorithm\nis not suitable for analyzing the thermal properties in terms of the system density of states,\nhence the WL algorithm is used to simulate the Ising nanotube for obtaining the other thermal\nproperties in addition to the above properties (calculated from the estimated density of states).\n2.3 Simulation technique 2: Non-Boltzmann technique\nThe conventional density of states g(E)is calculated using the temperature-independent WL\nalgorithm, called non-Boltzmann Monte-Carlo technique [45,46]. This approach aims to per-\nform a random walk on the energy space by randomly choosing a lattice site and changing its\nmagnetic moment with an appropriate probability, which is proportional to the inverse of the\ndensity of states g(E), i.e.P(E)/1=g(E). The density of states (DOS) is not known as\na priori. Thus, the algorithm is initiated by assuming g(E) = 1and the energy histogram,\nH(E) = 0for all energy levels. The simulation is started with a random conﬁguration of energy\nEi. The trial conﬁguration is made by ﬂipping a randomly chosen spin and its energy is given\nbyEt. The trial conﬁguration is accepted with the transition probability,\nP(Ei!Et) =min(\n1;g(Ei)\ng(Et))\n: (5)\nThen, the corresponding g(E)is updated by multiplying with a modiﬁcation factor f(f >1)\nand the histogram is updated with unity as,\ng(E) =g(E)\u0002f; (6)\nH(E) =H(E) + 1: (7)\nThe logarithm of DOS is updated as,\nln [g(E)] = ln [g(E)] + lnf: (8)\nIf the conﬁguration gets rejected, the initial conﬁguration is taken as a trial conﬁguration\nand update it accordingly. Once the histogram is ﬂat, the modiﬁcation factor is reduced\nmonotonically by multiplying with 0:5 (lnf(lnf\u00020:5)and the histogram resets to H(E) = 0\nfor all energies. Typically, the ﬂatness (80%)condition for the energy histogram is checked for\nevery 10;000MCS. The algorithm repeats the iteration with a new lnfuntil it reaches a small\nenough value (e.g., lnf < 10\u00008). The entire process results in the converged DOS of the\nsystem. The thermo-dynamical quantities can be calculated by ﬁnding the partition function\nof the system from the estimated g(E). The canonical partition function Z(\f)at any ﬁnite\ntemperature can be obtained by, Z(\f) =P\nEg(E) exp[\u0000\fE]. The production run is performed\nwith the WL acceptance rule as in the equation (5) with converged DOS. It begins with an\narbitrary conﬁguration C0and generates the micro-state sequence that constitutes the Markov\nchain,\nC0!C1!C2!::: Ci!Ci+1!::: CM:\nThe average of any macroscopic observable Ocan be computed for the canonical distribution\nusing the estimated DOS by un-weighting and re-weighting with the corresponding weight\n4factor. Un-weighting is the process of dividing the value O(C)by the probability weight factor\n[g(E(C))]\u00001for each conﬁguration while, re-weighting is the process of multiplying the value\nO(C)with the Boltzmann weight factor exp[\u0000\fE]for each conﬁguration. Thus, the weight\nfactor [43,47],\nW(C;\f) =g(E(C)) exp[\u0000\fE(C)]; (9)\nsuch that, the canonical ensemble average of Ocan be calculated as,\nhOi\f=P\nCO(C)W(C;\f)\nP\nCW(C;\f); (10)\nhOi\f=P\nCO(C)g(E(C)) exp[\u0000\fE(C)]\nP\nCg(E(C)) exp[\u0000\fE(C)]; (11)\nwhereCrepresents the conﬁguration generated by the Monte-Carlo production run. The aver-\nage energy and average magnetization are calculated using the above equation (11). Further,\nthe equations (3) and (4) are used to calculate the speciﬁc heat capacity CV(T)and suscepti-\nbility\u001f(T)of the system. In addition to that, the Gibbs free energy (F)and Canonical entropy\n(S)are calculated as,\nF(T) =\u0000kBTln(Z) =\u0000kBTln X\nEg(E) exp[\u0000\fE]!\n; (12)\nS(T) =U(T)\u0000F(T)\nT(13)\nwhereU(T) =hEiTis the internal energy. The Monte Carlo simulation of Ising nanotube using\nthe above explained techniques is carried out and the results are discussed in the following\nsection.\n3 Simulation results and its discussions\nThe simulations are carried out for the Ising nanotube with diﬀerent control parameters over\na ﬁnite range of temperatures. The Fig. 2 shows the ground state spin orientation for various\nmagnetic orderings namely, FM (Fig. 2a), G-AFM (Fig. 2b), A-AFM (Fig. 2c) and C-AFM\n(Fig. 2d) ordering [48,49]. In FM ordering, all the magnetic spins are aligned parallel to each\nother by the ferromagnetic interaction with the interaction strength J1= +1andJ2= +1.\nWhereas in G-type AFM, all the spins are aligned anti-parallel among the nearest neighbouring\nsites with the interaction strength J1=\u00001andJ2=\u00001. While the other two, in which A-type\nAFM magnetic ordering is characterized with parallel alignment due to FM interaction and\nthe spins between the layers are anti-parallel due to AFM interaction which is viceversa for\nC-type. The interaction strengths are (J1;J2) = (+1;\u00001)and(J1;J2) = (\u00001;+1), respectively,\nfor A-type and C-type ordereing.\n5(a)\n (b)\n (c)\n (d)\nFigure 2: Graphical representation of ground state ferromagnetic (FM) and anti-ferromagnetic (G-\nAFM, A-AFM and C-AFM) spin order for various interaction strength (a) FM: J1= 1;J2= 1, (b)\nG-AFM: J1=\u00001;J2=\u00001, (c) A-AFM: J1= +1; J2=\u00001and (d) C-AFM: J1=\u00001;J2= +1in\nthe absence of external magnetic ﬁeld B= 0.\nThe present simulation is performed by varying either of the control parmeters J1and B\nthereby keeping the other to be constant. While the control parameter J2is ﬁxed for every\nset of simulations. The result analysis of the above mentioned framework is discussed in the\nsubsequent sections.\n3.1 Metropolis results\nIn the absence of magnetic ﬁeld\nThe system evolves under the Metropolis algorithm without the external magnetic ﬁeld (B= 0)\nin this part of the simulation. Initially, the interaction strength between the adjacent layers,\nJ2, is set to 1:0andJ1, the interaction strength within the layers is varied from 0:0to1:0to\nsimulate the ferromagnetic system. The temperature-dependent magnetization and magnetic\nsusceptibility graphs are plotted in Fig. 3. Fig. 3a shows that the system exhibits spontaneous\nmagnetizationatlowertemperatures. Thesaturationofmagnetizationoccursduetotheparallel\norderingofthespins. Thespontaneousmagnetizationbreaksatacertaintemperatureandturns\ninto random ordering. Thus, the net (absolute) magnetization curve smoothly settles down to\nthe minimum magnetization at higher temperatures. When J1= 0, the interaction of spins\nwithin the layer is zero and hence only the interactions of adjacent layer spins will contribute\nto the FM ordering. So the total magnetization is reduced (inset Fig. 3a). The Monte Carlo\nerrors for all simulation data are calculated and it is found to be smaller than the size of the\ndata points in the entire analysis. The transition temperature (TC)is positioned by identifying\nthe peak value of the susceptibility curve (Fig. 3b). The susceptibility curve for diﬀerent\nvalues of interaction strength are plotted and the peak value of the susceptibility is found to\nbe decreasing (as a result of the decrease in the ﬂuctuations) and shifting towards the higher\ntemperatures with increasing interaction strength J1. The maximum value of susceptibility is\nfound to be higher for J1= 0(inset Fig. 3b).\n6(a)\n (b)\nFigure 3: Temperature dependence of observables for ferromagnetic interactions in the absence of\nmagnetic ﬁeld B= 0, (a) average magnetization and (b) susceptibility. The interactions are 0:0\u0014\nJ1\u00141:0andJ2= 1:0. The inset plot shows the observables for J1= 0:0.\nIn a similar way, the interaction strength between the layers, J2, is ﬁxed as\u00001:0and the\ninteractionstrengthwithinthelayer, J1, isvariedfrom\u00001:0to0:0, whichbringsthesysteminto\nG-type AFM ordering. The corresponding average magnetization and magnetic susceptibility\nplots are shown in Fig. 4. The temperature-dependent magnetization for G-AFM in Fig. 4a is\nzero at the lower temperature region, as all the spins are aligned in the anti-parallel pattern.\nThe average magnetization curve increases steadily and reaches its saturation point, evidencing\nthe transition to a random order state. The transition point is determined from the maximum\nvalue of susceptibility (Fig. 4b). The transition temperature is shifted to the lower values as\nthe interaction strength, J1, increases. The maximum value of susceptibility also increases with\nincrease in J1. Though the interaction strength, J1= 0, the system follows the G-type AFM\norder.\n(a)\n (b)\nFigure 4: Temperature dependence of observables for G-type anti-ferromagnetic interactions exclud-\ning the magnetic ﬁeld, i.e., B= 0, (a) average magnetization and (b) susceptibility. The interactions\nare\u00001:0\u0014J1\u00140:0andJ2=\u00001:0.\nThe system evolving with opposite interaction (A type: J1= +1,J2=\u00001) and (C type:\nJ1=\u00001,J2= +1) are respectively shown in Figs. 5 and 6. The average magnetic observables\n7for the A-type interaction namely, the average magnetization and susceptibility are shown in\nFigs. 5a and 5b. The magnetization curve (Fig. 5a) clearly shows that the system exhibits\nAFM dominance despite the presence of FM interaction within the layers. The susceptibility\nplot (Fig. 5b) shows a transition from the anti-ferromagnetic to the paramagnetic phase though\nthe FM interaction is present. The transition temperature rises as J1interaction increases.\n(a)\n (b)\nFigure 5: Temperature dependence of observables in the absence of magnetic ﬁeld B= 0, (a) average\nmagnetization and (b) susceptibility for the interactions 0:0\u0014J1\u00141:0andJ2=\u00001:0.\nDespite having a high FM interaction of J2= +1, the magnetization plot in Fig. 6a\nshows that the system follows C-AFM ordering. While for the interaction J1= 0:0,J2= +1,\nsystem follows FM ordering which is already shown earlier in the inset plot of Fig. 3a. The\nsusceptibility curves are shown in Fig. 6b. As the interaction J1is reduced towards \u00001:0, the\nvalue ofTCchanges towards higher temperature. The susceptibility peak is higher for J1= 0:0,\nJ2= 1:0and it is already highlighted in the inset plot of Fig. 3b.\n(a)\n (b)\nFigure 6: Temperature dependence of observables in the absence of magnetic ﬁeld B= 0, (a) average\nmagnetization and (b) susceptibility for the interactions \u00001:0\u0014J1\u0014\u0000 0:2andJ2= +1:0.\n8(a)\n (b)\nFigure 7: Phasediagram: J1vsTC. (a)J2= +1;\u00001:0\u0014J1\u00141:0, and(b) J2=\u00001;\u00001:0\u0014J1\u00141:0.\nThe phase diagram is plotted for TCwith various values of interaction strength, J1and is\nshown in Fig. 7. The interaction J2is set to +1and the interaction J1is raised from\u00001:0to1:0\nas in Fig. 7a. Therein, the phase line separates the phases into C-AFM, FM and paramagnetic\nregimes. At J1= 0:0, the C-type AFM turns in to FM ordering due to the ﬁxed value of\nJ2= 1:0. The transition temperature for the interactions J2=\u00001and\u00001:0\u0014J1\u0014+1\nare plotted and shown in Fig. 7b. TCis found to decreases with increase in J1tillJ1= 0:0\nand above that it increases. The phase line separates the region of G-AFM, A-AFM and the\nParamagnetic order. The system will follow the G-AFM ordering when J1= 0:0andTC= 2:0\ndue to the ﬁxed value of J2=\u00001:0. The observations from these results suggest that though\nthe system has a ferromagnetic interaction, a small anti-ferromagnetic interaction would lead\nthe system to possess the anti-ferromagnetic ordering.\nThe following part explores the system with the inclusion of magnetic ﬁeld of various ﬁeld\nstrengths for diﬀerent types of interaction.\nIn the presence of magnetic ﬁeld\nThe magnetic ﬁeld is applied along the Zaxis. The interaction strengths, J1= +1andJ2= +1\nare applied to set the system’s state in the ferromagnetic order. The external magnetic ﬁeld is\nvaried from 0:5to3:0and the average observables are plotted in Fig. 8. Fig. 8a depicts the\naverage magnetization and the inﬂuence of ﬁeld in the spin ordering. At lower temperature,\nthe average magnetization attains its saturation value and the system is found to maintain\nFM ordering at those lower temperatures. Due to the presence of magnetic ﬁeld, the spins\nare favoured to orient in the direction of the ﬁeld. Increasing the temperature breaks that\norientationintoarandomspinorder. Thisbreakingpoint(thepointatwhichrandomspinorder\noccurs)increaseswhileincreasingtheﬁeld,since,thesystemisfavouredtosustainferromagnetic\nordering at the higher magnetic ﬁeld and it needs more temperature to break this orientation.\nField dependent magnetic susceptibility is shown in Fig. 8b and the transition temperature is\nobtained from the peak value of susceptibility, which is observed to be shifting and decreasing\ntowards the higher temperatures with increasing magnetic ﬁeld.\n9(a)\n (b)\nFigure 8: Field dependence of observables for ferromagnetic interactions with magnetic ﬁeld, (a)\naverage magnetization and (b) susceptibility. The interactions are J1= +1:0andJ2= +1:0.\nThe magnetic observables for anti-ferromagnetic interactions with the inclusion of magnetic\nﬁeld is plotted in Fig. 9. The average magnetization at lower temperatures is zero due to the\nanti-parallel alignment of the spins that cancels each other. The value of average magnetization\nincreases as the ﬁeld increases (see the Fig. 9a). The critical temperature is found from the\nsusceptibility plot Fig. 9b. The peak value of susceptibility increases and its position is shifted\ntowards the lower temperatures while increasing the magnetic ﬁeld value.\n(a)\n (b)\nFigure 9: Field dependence of observables for G-type anti-ferromagnetic interactions with magnetic\nﬁeld, (a) average magnetization and (b) susceptibility for the interactions J1=\u00001:0andJ2=\u00001:0.\nThe variation of transition temperature with ﬁeld intensity for FM and G-AFM interactions\nare plotted as phase diagram and is shown in Fig. 10. It is observed that the transition\ntemperature shifted towards higher temperature for FM interactions (Fig. 10a) and towards\nlower temperature for G-AFM interactions with the increase in the ﬁeld intensity (Fig. 10b).\nFrom the ﬁgure (Fig. 10), it is also observed that the system transits from FM/G-AFM to\nparamagnetic phase with respect to the values of interactions J1andJ2.\n10(a)\n (b)\nFigure 10: Phase diagram: BvsTC. (a)J1= 1:0&J2= 1:0;0:5\u0014B\u00143:0, and (b) J1=\u00001:0&\nJ2=\u00001:0;0:5\u0014B\u00143:0.\nThe observables for A-type AFM ordered interaction are depicted in Fig. 11. As discussed\nin the preceding section, a small AFM interaction is suﬃcient to produce an AFM order in\na FM system. Hence the system is maintained in A-AFM order even in the presence of FM\ninteraction. However, as the external ﬁeld is increased, the system loses its AFM behavior\nand changes to FM behavior. The magnetization plot for various values of Bis shown in Fig.\n11a. The system retains in the A-AFM order when the external ﬁeld Bis less than 2:0and it\nexperiences FM order when the external ﬁeld, Bis greater than 2. The transition temperature\nis observed from the susceptibility plot (Fig. 11b). As the external magnetic ﬁeld is increased,\ntheTCdecreases and the peak value of susceptibility increases. When it is increased beyond\nB= 2:0, theTCincreases but the peak value of susceptibility decreases with increase in ﬁeld\nstrength. The susceptibility plot for B= 2:0is highlighted in inset of the Fig. 11b.\n(a)\n (b)\nFigure 11: Field dependence of observables with increasing magnetic ﬁeld, (a) average magnetization\nand (b) susceptibility for the interactions J1= +1:0andJ2=\u00001:0. The inset plot represents the\nsusceptibility at B= 2:0.\nSimilarly, for the next set of simulations, interaction strengths of J1=\u00001andJ2= +1is\napplied to the system, which has a C-type magnetic ordering. The results of this interaction\nare depicted in Fig. 12. From the results as shown in Fig. 12a, it is clear that, if the magnetic\n11ﬁeld is less than 2:0, the system is maintained in C-AFM order and the system turns to FM\nordering when the magentic ﬁled is greater than 2:0. As the ﬁeld increases (Fig. 12b), the\nTCdecreases and the maximum value of susceptibility increases until B= 2:0. Beyond that,\ntheTCincreases and the maximum value of susceptibility decreases with increasing ﬁeld. The\ninset in Fig. 12b shows the susceptibility curve for B= 2:0.\n(a)\n (b)\nFigure 12: Field dependence of observables with increasing magnetic ﬁeld, (a) average magnetization\nand (b) susceptibility. The interactions J1=\u00001:0andJ2= +1 :0. The inset plot represents the\nsusceptibility at B= 2:0.\n(a) 0\n (b)\nFigure 13: Phase diagram: BvsTC. (a)J1= +1:0&J2=\u00001:0;0:5\u0014B\u00143:0, and (b) J1=\u00001:0\n&J2= +1:0;0:5\u0014B\u00143:0.\nThe transition temperatures for the interactions (A-AFM and C-AFM) against an external\nmagnetic ﬁeld are plotted in Fig. 13. The phases are well separated in the phase diagram for\ninteraction J1= +1; J 2=\u00001(Fig. 13a). The A-AFM dominates the system below B= 2:0\nand above that FM interaction will follow. Similarly, the C-AFM dominates the system below\nB= 2:0, beyond that the FM dominates the system for the interaction J1=\u00001; J 2= +1\n(Fig. 13b). The evolution of both the systems with interactions J1= +1;J2=\u00001and\nJ1=\u00001;J2= +1possess AFM ordering in low magnetic ﬁeld ( B < 2:0). As the magnetic ﬁeld\nincreases, the system changes to FM ordering.\n12In the subsequent section, the analysis of hysteresis behavior of the system with the appli-\ncation of the external magnetic ﬁeld is given in detail.\nHysteresis process\nThe magnetization is calculated by increasing and decreasing the external magnetic ﬁelds. The\nsaturation limit M=\u00061, is achieved as the system spins are favoured to align with the ﬁeld.\nThe net magnetization increases with increasing magnetic ﬁeld, further it decreases in a slightly\ndiﬀerent path with the decrease in the ﬁeld strength to form a loop. The hysteresis loops for\ndiﬀerent temperatures are plotted in Fig. 14. The magnetization shows a sudden shift to\nits saturation value at lower temperatures. As the temperature increases, the magnetization\nfollows a curved path and gradually reaches the saturation value. The remanent magnetization\nis observed at zero ﬁeld and the coercive force is observed within the ﬁeld values for which the\nnet magnetization is zero. With the increase in temperature, the hysteresis loop area reduces\nand the coercive force is also found to decrease. At the transition temperature ( TC= 2:2), the\nremanent magnetization gets reduced and the coercive force becomes zero. The forward and\nreverse paths merge to give a zero hysteresis area at TC= 2:2. After the transition temperature,\nthe system is no longer ferromagnetic but transits to a paramagnetic state.\nFigure 14: Hysteresis loop for FM interactions at diﬀerent temperatures.\nFig. 15 shows the hysteresis process for the G-type anti-ferromagnetic ordering. The system\nexhibits double loop hysteresis for T= 0:01(Fig. 15a) and T= 0:05(Fig. 15b). The coercive\nforce is bounded between B=\u00064:5and is found to decrease with increasing temperature. The\ndouble loop gradually vanishes and forms a step hysteresis as shown in Fig. 15c and 15d. As\nthe temperature increases, the cycle forms a single line and beyond the transition temperature,\nTC= 3:5, the system is found to be in the paramagnetic phase (15f). There is no remanent\nmagnetization for all the hysteresis loops.\n13(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFigure 15: Hysteresis loop for G-AFM order of interaction at diﬀerent temperatures.\nFig. 16 shows the hysteresis behavior of the system with A-type AFM ordering for various\ntemperatures. The system has a double loop hysteresis at lower temperatures ( T= 0:01)\n(Fig. 16a). The minimum remanent magnetization and coercive force are observed for the\ntemperatures T= 0:05(Figs. 16b), T= 0:1(Figs. 16c) and T= 0:2(Figs. 16d). The\nassociated coercive force is bounded by B=\u00064. While increasing the temperature, the width\nof the coercive ﬁeld is reduced as shown in Figs. 16b, 16c and 16d. Nearly at the temperature,\nT= 0:4, the coercive force vanishes and the double loop turns into a single line and is shown\nin Fig. 16e. There is no magnetization and no coercive force above T= 2:0and the system\nenters a paramagnetic phase at TC= 3:2(Fig. 16f).\n14(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFigure 16: Hysteresis loop for A-AFM order of interaction at diﬀerent temperatures.\nFigure 17 depicts the hysteresis curve for the C-type AFM system ( J1=\u00001;J2= +1). A\nstep behaviour is observed at lower temperature ( T= 0:01) and is shown in Fig. 17a. The\nwidth of coercive force in the double loop hysteresis decreases as the temperature increases\n(please see Fig. 17b, 17c and 17d). At T= 0:3, the loop becomes a single line and two\nstep hysteresis occurs (Fig. 17e). There is a minimum remanent magnetization found at that\ntemperature ( T= 0:3). The Fig. 17f shows that the system transits from C-type AFM phase\nto paramagnetic phase at the TC= 3:2.\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFigure 17: Hysteresis loop for C-AFM order of interaction at diﬀerent temperatures.\n15Inthepresentwork, thecanonicalensemblefromtheMetropolisalgorithmaresimulatedand\nthe mechanical and thermal properties are determined. However, this method is not suitable\nfor calculating the system’s entropy, free energy and the order of the transition [42]. Hence, to\ndiscuss the additional thermal properties and the order of phase transition of the Ising nano\ntube, WL algorithm is utilized and the following section discusses about them in detail.\n3.2 Wang-Landau results\nIn this section, the Wang-Landau technique is applied to investigate the system with FM and\nG-type AFM order of interaction. The logarithm of the density of states is calculated for\nvarious control parameters and are shown in Fig.18. In the absence of magnetic ﬁeld, the\nspan of energy increases for FM interactions (Fig.18a) and decreases for G-AFM interactions\n(Fig.18b) with increasing interaction strength J1. The shape of DOS is symmetric about the\nzero-energy for FM and AFM interactions without any external ﬁeld and the shape becomes\nasymmetric while applying an external magnetic ﬁeld. The logarithm of DOS for FM and G-\nAFM interactions in the presence of external magnetic ﬁeld are shown respectively in Fig.18c\nand Fig.18d. Irrespective of the change in the applied magnetic ﬁeld, the highest possible value\nof DOS remains same in both the cases. However, the DOS plot for FM interactions show an\nincrease in the range of negative energy values with an increase in the applied magnetic ﬁeld,\nwhereas the G-AFM interactions show an increase in the range of positive energy values with\nan increasing magnetic ﬁeld. The maximum value of the DOS is same ( 122:2959) for the energy\nE= 0for all the plots in Fig.18.\n(a)\n (b)\n(c)\n (d)\nFigure 18: Logarithm of density of states versus energy. (a) J1>0withB= 0, (b)J1<0with\nB= 0, (c)B > 0for FM interaction, and (d) B > 0for G-AFM interaction.\nThe DOS of FM and G-AFM are similar to each other in the absence of magnetic ﬁeld.\nThe DOS shows interesting characteristics in the presence of the magnetic ﬁeld. Thus, the\nfree energy and entropy for ferromagnetic and anti-ferromagnetic interactions in the presence\n16of the magnetic ﬁeld are analyzed in detail and are depicted in Fig.19. From the Fig.19a,\nit is observed that the free energy of the ground state for FM interaction decreases with the\nincrease in the external magnetic ﬁeld which reﬂects the characteristics of the DOS as given in\nFig.18c. The entropy plot for FM system (Fig.19b) starts from zero entropy for all the external\nﬁeld strengths and it shows the convergence of the system to a single microstate for the lowest\nenergy (ground state). As the ﬁeld strength increases, the plots follows a decreased trend in\nentropy values with increasing temperature.\n(a)\n (b)\n(c)\n (d)\nFigure 19: Thermodynamic quantities for the Ising nanotube were calculated from DOS with varying\nmagnetic ﬁelds. (a) free energy and (b) entropy for FM interaction; (c) free energy, and (d) entropy\nfor G-AFM interaction.\nFig. 19c shows the ground state free energy plot for anti-ferromagnetic interation under\ndiﬀerent magnetic ﬁelds and is found to approached at a particular value ( \u00002:0). The free\nenergy plot for G-AFM interaction shows the decreasing behavior as in the FM interaction with\nthe increasing temperature. The Fig.19d shows a crossing point for all the plots with diﬀerent\nmagnetic ﬁelds at kBT= 2:8. This crossing point (inset of Fig.19d) suggests the isentropic\nbehavior of the system with the variation in the magnetic ﬁeld at a particular temperature,\nkBT= 2:8. The entropy of the system increases for the increase in the magnetic ﬁeld upto\nthe crossing point, beyond that the entropy decreases. The trend of decreasing free energy and\nincreasing entropy for all the plots in Fig.19 shows the phase change from FM (or G-AFM) to\nparamagnetic order. The phase transition is conﬁrmed to be second-order as the entropy plots\nare continuous.\n17Figure 20: DOS for diﬀerent number of layer L(J1= +1:0;J2= +1:0) atB= 0:0.\nThe logarithm of density of states for ferromagnetic interaction in the absence of magnetic\nﬁeld is shown in Fig. 20 for diﬀerent number of layers L. The DOS is computed by increasing\nthe number of layers Nfrom 30to100. The energy range is maintained between \u00002and+2.\nThemaximumvalueofDOSincreasesinproportiontothenumberoflayers. Theaverageenergy\nis plotted against the temperature (Fig. 21a) and the system’s stability can be assessed from\nits proﬁle. The average energy begins in a stable ground state. Then, it rapidly rises near the\ntransition temperature and eventually reaches its maximum value with increasing temperature.\nThis conﬁrms the transition from ferromagnetic to paramagnetic phase. The system under the\npresent study clearly follows a second-order transition since the average energy curve behaves\ncontinuously without any jumping. The speciﬁc heat capacity as a function of temperature\nfor various number of layers is shown in Fig. 21b. The speciﬁc heat peak value reduces with\nincreasing the number of layers. For diﬀerent number of layers, the TCremained to be the same\nand the approximate value of kBTC(L)is around 2:2.\n(a)\n (b)\nFigure 21: Thermodynamic observables (a) average energy and (b) speciﬁc heat capacity for diﬀerent\nlayerLwithout applying magnetic ﬁeld.\nIn order to further conﬁrm the second-order transition, the canonical distribution for FM\nand G-AFM interaction at the transition temperature is analyzed and is depicted in Fig. 22.\nAll the micro-states are chosen based on their weight ( g(E) exp[\u0000\fE]) and the system shows\na single peak for the canonical distribution. Temperatures below and above the vicinity of\nthe transition temperature also show single peak in the distribution. The single peaked form\n18of the probability distribution at the transition temperature is a common indication for the\nexistence of a second-order transition [50]. It is also conﬁrmed that both the FM and G-AFM\ninteractions show the second-order phase transition.\n(a)\n (b)\nFigure 22: Canonicaldistributionfor(a)FM( J1= +1;J2= +1)and(b)G-AFM( J1=\u00001;J2=\u00001)\ninteraction.\n4 Conclusion\nThe Metropolis and WL algorithms are used to explore the magnetic response of the Ising\nnanotube. The obtained magnetic spin orientations of the ground state conﬁrmed the FM\nand AFM (G, A and C type) ordering about the zaxis. The variations of TCfor diﬀerent\ninteraction strengths are calculated in the absence of an external magnetic ﬁeld. At J1= 0:0,\nthe C-AFM order will turn to FM ordering and similarly, the A-AFM order changes to G-AFM\nordering. A small disturbance (hump) is found in the magnetization curve for the interaction\nJ1= 0:0; J 2= +1:0at lower temperatures and it can be given a special attention in future\nstudies. It is found that a small anti-ferromagnetic interaction is enough to bring the spins in\nthe anti-ferromagnetic ordering, even though a FM interaction is present in the system.\nThe magnetic response of FM and diﬀerent AFM interactions is analyzed by including\nthe magnetic ﬁeld for unit interaction strength. It is found that the transition temperature\nincreases for the FM system and decreases for the G-AFM system when the magnetic ﬁeld is\nincreased. Both the alternative interactions, i.e., A and C type AFMs, evolve similarly. In\nwhich the transition temperature TCﬁrst decreases then increases with the increasing external\nmagnetic ﬁeld. When the magnetic ﬁeld, Bis less than 2.0, the AFM order dominates the\nsystem and the FM order dominates when B\u00152:0. AtB= 2:0, the A-AFM and C-AFM\nsystems changes to FM ordering. In general, it is found that the Ising nanotube transits from\nferromagnetic (or anti-ferromagnetic) to paramagnetic phase over the ﬁnite temperatures. The\nhysteresis plot shows that the coercive ﬁeld and the remanent magnetization reduces to zero\nwith an increase in temperature. As a result, the system exhibits a narrow hysteresis loop for\nhigher temperatures. The G, A, and C type AFM interactions exhibit the double step loop\nhysteresis at lower temperatures and which disappears after the transition temperature.\nThe DOS is determined using the WL algorithm to obtain the thermal properties of Ising\nnanotube. The symmetric DOS is observed for FM and G-AFM interaction without an external\nmagnetic ﬁeld. By applying a magnetic ﬁeld, the system exhibits the asymmetric DOS pattern.\nThefreeenergyandentropyarecalculatedinthepresenceofamagneticﬁeldfromtheconverged\nDOS. This study also shows that in the absence of magnetic ﬁled, the diﬀerence in the number\nof layers does not alter the transition temperature. Using Wang Landau algorithm, it is also\n19found that there is no discontinuity in canonical entropy and a single peak in the canonical\ndistribution conﬁrms that the system follows the second-order transition.\nAcknowledgement\nThe authors would like to thank the anonymous referees for their insightful comments and\nvaluable suggestions which helped to improve the manuscript. One of the author A. Arul Anne\nElden would like to thank M. Suman Kalyan, S. Siva Nasarayya Chari and Asweel Ahmed A.\nJaleel for fruitful discussions, and also thank T. Kiran for useful discussions and critical reading\nof the manuscript.\nReferences\n[1] W. E. Parente, J. T. M. Pacobahyba, M. A. Neto, I. G. Araújo and J. A. Plascak, “Spin-\n1/2anisotropicHeisenbergantiferromagnetmodelwithDzyaloshinskii-Moriyainteraction\nvia mean-ﬁeld approximation”, Journal of Magnetism and Magnetic Materials , Vol. 492,\npp. 8-12, 2008.\n[2] N. Hachem, I. A. Badrour, A. El Antari, A. Lafhal, M. Madani and M. 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Dagotto, “Nanoscale phase separation and colossal magnetoresistance: the physics of\nmanganites and related compounds”, Springer Science & Business Media , 2003.\n[50] D. P. Landau, S. H. Tsai and M. Exler, “A new approach to Monte Carlo simulations in\nstatistical physics: Wang-Landau sampling”, American Journal of Physics , Vol. 72, no.\n10, pp. 1294-1302, 2004.\n23"    },    {        "title": "0705.3299v1.Role_of_long_range_ferromagnetic_order_in_the_electronic_structure_of_Sr___1_x__Ca__x_RuO__3_.pdf",        "content": "arXiv:0705.3299v1  [cond-mat.str-el]  23 May 2007Role of long range ferromagnetic order in the electronic str ucture\nof Sr1−xCaxRuO3\nRavi Shankar Singh, V.R.R. Medicherla, and Kalobaran Maiti∗\nDepartment of Condensed Matter Physics and Materials Scien ce,\nTata Institute of Fundamental Research,\nHomi Bhabha Road, Colaba, Mumbai 400005, India\n(Dated: September 12, 2021)\nAbstract\nWe investigate the role of long range ferromagnetic order in the electronic structure of\nSr1−xCaxRuO3usinghighresolution photoemission spectroscopy. SrRuO 3is aferromagnetic metal\nbut isostructural, isoelectronic CaRuO 3is an enhanced paramagnet. Surface spectra of CaRuO 3\nexhibit temperature induced modiﬁcations. This is not sign iﬁcant in other compositions. This\nmay be attributed to the structural changes observed in prev ious studies. Interestingly, the bulk\nspectra reveal unusual spectral changes exhibiting large d ecrease in the coherent feature intensity\ncorresponding to only ferromagnetic samples, although the Ru moment is very similar in all the\ncompositions.\nPACS numbers: 75.30.Kz, 75.50.Cc, 71.45.Gm, 71.20.-b\n∗Author to whom correspondence should be addressed; electronic mail: kbmaiti@tifr.res.in\n1Ruthenium based perovskite oxides have attracted a great deal o f attention due to possi-\nbilitiesinsigniﬁcanttechnologicalapplicationsinadditiontovariousinte restingfundamental\nissues. In particular, SrRuO 3, the only itinerant ferromagnet among the 4 dtransition metal\noxides (Curie Temperature, T c∼165 K) [1], is a promising candidate for several techno-\nlogical applications due to its metallic character, high magnetic momen t (1.4µB/Ru), high\nchemical stability, etc. [1, 2, 3] Ultrathin ﬁlms of SrRuO 3have already been used for normal\nmetal layers in Josephson junctions [4], spintronic devices based on spin polarized ferromag-\nnetic tunnel junctions with ferromagnetic metal as an electrode [5 ], etc. Thus, ferromagnetic\nmaterials form basis for technological advances and microscopic un derstanding of the origin\nof such eﬀect is crucial to design new materials for future applicatio ns.\nHere, we investigate the role of ferromagnetic transition in the elec tronic structure of\northorhombically distorted perovskites (space group Pbnm), Sr1−xCaxRuO3. The average\nRu-O-Rubondanglegraduallydecreasesfrom ∼165◦inSrRuO 3toabout150◦inCaRuO 3[1,\n3, 6]. Magnetic measurements exhibit ferromagnetic ground state forx <0.8 and enhanced\nparamagnetic phase for higher xvalues [2]. Such diﬀerent magnetic ground states have also\nbeen observed in ab initio calculations based on local spin density approximations (LSDA)\n[7,8]. Variousphotoemissionstudies[9,10,11]suggestthattheele ctroncorrelationstrength,\nU/W(U= Coulomb interaction strength, W= bandwidth) is signiﬁcantly weak and similar\nin all the compositions. Transport measurements, on the other ha nd, indicate Fermi liquid\nbehavior in SrRuO 3while CaRuO 3is non-Fermi liquid [12]. It is thus, clear that these\nsystems exhibit varieties of interesting ground state properties, which cannot be attributed\nsolely to the electron correlation eﬀect. Despite numerous studies , the origin of such widely\ndiﬀerent ground state properties is still unclear.\nIn this letter, we report our results on the modiﬁcation of the elect ronic structure across\nthe magnetic phase transition in this system using state of the art h igh resolution photoe-\nmission spectroscopy. Experimental spectra exhibit qualitatively d iﬀerent bulk and surface\nelectronic structures in all the samples and interesting evolutions w ith the change in tem-\nperature.\nSamples were prepared by solid state reaction route followed by sint ering in the pellet\nform for about 72 hours at 1523 K to achieve large grain size. Sharp features in the x-ray\ndiﬀraction (XRD) patterns with lattice parameters similar to those in single crystalline sam-\nples [2, 6] and no signature of impurity feature indicate high quality of the samples. DC\n2magnetic susceptibility, measured using high sensitivity vibrating sam ple magnetometer,\nshow sharp ferromagnetic transition at 165 K in SrRuO 3. The sharpness of the transition\ngradually reduces and becomes insigniﬁcant for x≥0.8.µefffor all the samples in the\nparamagnetic region has been estimated to be 2.8 ±0.2µB, which is close to the theoretical\nspin only value of 2.83 µBcorresponding to t3\n2g↑t1\n2g↓conﬁguration of Ru4+[2, 9]. Photoe-\nmission measurements were performed on in situ(base pressure ∼3×10−11torr) scraped\nsample surfaces using Gammadata Scienta analyzer, SES2002 with a n energy resolution set\nto 4 meV, 900 meV and 300 meV for the measurements with monochro matic He II(40.8 eV),\nAlKα(1486.6 eV) (twin source) and Al Kα(monochromatic source) respectively. Clean-\nliness of the sample surface was ensured by minimizing the higher bindin g energy feature\nin O 1sspectra [13] and the absence of C 1 ssignal. A polycrystalline silver was mounted\non the same sample holder in electrical contact with other samples to determine the Fermi\nedge,ǫF.\nInFig.1(a),weplotRu4 dHeIIspectraaftersubtractingtheO2 pcontributionsappearing\nat higher binding energies. All the spectra exhibit an intense, broad feature at 1.2 eV along\nwith ﬁnite intensity at ǫF. While the intensities at ǫFcorrespond well to the band structure\nresults (termed as coherent feature ), signature of the dominant contributions at 1.2 eV is\nnot present in these results [7, 8]. The 300 K spectra for all the valu es ofxare very similar\nexhibiting weak coherent feature intensity suggesting metallic phas e in these materials. Ru\n4dHeIIspectra at 20 K exhibit signiﬁcant decrease in intensity of the coher ent peak, when\nit is normalized by the intensity at 1.2 eV of 300 K spectrum. The diﬀere nce in intensity\nbetween the spectra at room temperature and 20 K spectra grad ually increases with the\nincrease in x.\nSince, the surface contribution is signiﬁcantly large ( ∼80%) in the He IIspectra, we probe\nthe inﬂuence of the temperature on the Ru 4 dcontributions in the Al Kαspectra of the\nvalence band, where the surface sensitivity is reduced to about ∼40%. All the spectra are\nshown in Fig. 1(b) after normalizing by the intensity at 1.5 eV. The lines hape of the Ru\n4dspectra is signiﬁcantly diﬀerent from the He IIspectra shown in Fig. 1(a). The 300 K\nspectrumofSrRuO 3exhibitsintensecoherentpeakaround0.5eVandanasymmetrytow ards\nhigher binding energies. Interestingly, corresponding 20 K spectr um exhibits signiﬁcant\nlowering in intensity compared to the intensity at 1.5 eV. The diﬀerenc e in coherent feature\nintensity is signiﬁcantly large in SrRuO 3and becomes almost insigniﬁcant in CaRuO 3. This\n3temperature induced modiﬁcation is strikingly diﬀerent from that ob served in the He II\nspectra. This is veriﬁed in the high resolution spectra of end member s shown in Fig. 1(b).\nThe high resolution spectra of SrRuO 3exhibits a large decrease in the coherent feature\nintensity with the decrease in temperature, while CaRuO 3spectra remain unchanged.\nIn order to understand the contrasting spectral changes in the HeIIand AlKαspectra,\nwe extract the surface and bulk spectral functions from these t wo sets of spectra at all\nthe temperatures for all the samples. Photoemission intensity can be expressed as I(ǫ) =\n[1−e−d/λ]Fs(ǫ) +e−d/λ.Fb(ǫ), where dis the thickness of the surface layer and λis the\nescape depth of the photoelectrons. Fs(ǫ) andFb(ǫ) represent the surface and bulk spectra,\nrespectively. Using the values of d/λfrom Ref. [8], we have extracted the surface and the\nbulk contributions as shown in Fig. 2(a) and Fig. 2(b), respectively. The surface spectra\nexhibit dominant contributions at 1.2 eV binding energy. The threefo ld degeneracy of the\nRu 4d t2gband is already lifted in the bulk electronic structure due to the disto rtion of the\nRuO6octahedra [8]. The absence of periodicity along the surface normal will further reduce\nthe crystal symmetry from Ohsymmetry towards D4hsymmetry at the surface. Thus, the\nfeature at 1.2 eV is often attributed to the egband derived from the t2gband due to such\nsymmetry breaking [9, 14].\nThe coherent feature intensity is signiﬁcantly weak in the surface s pectra of all the com-\npositions. The decrease in temperature down to 20 K does not lead t o signiﬁcant change in\nthesurface spectra of SrRuO 3andSr 0.7Ca0.3RuO3. This is alsoevident inthe highresolution\nspectra of SrRuO 3. Small change in lineshape is observed for higher xvalues, which is most\nsigniﬁcant in CaRuO 3exhibiting a large reduction in coherent feature intensity with the\ndecrease in temperature as clearly visible in the high resolution spect ra of CaRuO 3. Various\ncore level studies [13, 15] indicate signiﬁcant change in the lineshape suggesting temperature\ninduced modiﬁcation in structural parameters. It is already clear t hat the two dimensional\nnature, defects, reconstructions at surface play key roles in de termining the surface elec-\ntronic structure. Thus, the spectral modiﬁcations observed in t he surface spectra may be\nattributed to such temperature induced changes of the surface structure.\nInFig. 2(b), weshow thebulkspectra forallthe xvalues. Theroomtemperature spectra,\nnormalizedbyintegratedintensity under thecurvearealmostsimilar inall thecompositions.\nThe coherent peak appears at about 0.5 eV with the contribution of incoherent peak (the\nlower Hubbard band) appearing at 2 eV. This suggests that the cha nge in Ru-O-Ru bond\n4angle across the series does not introduce signiﬁcant change in the electronic structure\nof this system. Bulk spectra at 20 K are shown in the same ﬁgure by n ormalizing the\nintensity of the incoherent feature. Intensity of the coherent f eature in SrRuO 3is found to\ndecrease signiﬁcantly with the decrease in temperature across th e magnetic phase transition.\nInterestingly, such lowering incoherent features intensity is clear ly visible inthe bulk spectra\nofallthecompositionsexhibiting longrangeferromagneticorder. T hespectracorresponding\ntox≥0.8 remain unchanged down to the lowest temperature studied.\nBand structure calculations [8] for various magnetic and non-magn etic solutions suggest\nthat in the ferromagnetic groundstate, the contribution fromth e down spin density of states\nmoves above the Fermi level due to the exchange coupling between the 4delectronic states.\nSince, the coherent feature represents the density of states o bserved in the band structure\nresults, the lowering of coherent feature intensity across the ma gnetic phase transition may\nbeattributedtotheshiftofdownspinspectralintensityabove ǫF. Thisshiftofthedownspin\ndensity of states depends on the exchange splitting, which is also re ﬂected in the magnetic\nmoment. This appears to explain the change in the electronic struct ure in ferromagnetic\ncompositions. However, various magnetic measurements suggest similar Ru 4 dmoment\nacross the entire series [2, 9]. Thus, no change in the bulk spectra o f paramagnetic samples\niscuriousandopensupaninteresting question inmicroscopic unders tanding oftheevolution\nof magnetism.\nThe ferromagnetism is often described within two models. (a) The St oner description\n[16]: the exchange splitting gradually decreases with the increase in t emperature and be-\ncomes zero at the Curie temperature leading to zero magnetic mome nt. (b) On the other\nhand, a spin mixing behavior [17] leads to a reduction in spin polarization with the increase\nin temperature keeping the magnetic moment unchanged. The pure ly Stoner behavior can\nbe ruled out since the magnetic moment exists even in the paramagne tic phase in all the\ncompositions. If the second case is active, the spin integrated spe ctra should be identi-\ncal in all the compositions. Thus, signiﬁcant change only in the spect ra of ferromagnetic\ncompositions in this study is curious and opens up a new dimension in understanding ferro-\nmagnetism. Since, all the compounds are essentially identical excep t the diﬀerence in long\nrange order, the signiﬁcant modiﬁcation observed in the samples ha ving long range order\nnaturally suggests a relation among themselves. This suggests tha t in addition to the in-\ntrasite exchange interactions (responsible for local magnetic mom ents), intersite exchange\n5correlations, which give rise to long range order presumably play a ke y role in determining\nthe spectral functions observed by photoemission spectroscop y. We hope, this study will\nhelp to initiate further eﬀorts in this direction to understand this eﬀ ect in ferromagnetic\nmaterials.\nIn summary, we investigate the change in the electronic structure across the magnetic\nphase transition in a series of compounds exhibiting magnetic ground states ranging from\nferromagnetic to enhanced paramagnetic. Although the intrasite exchange interactions are\nsimilar in all the compositions, the bulk spectra exhibit signiﬁcant modiﬁ cation in the line-\nshape across the Curie temperature in ferromagnetic materials, w hile the spectra in param-\nagnetic samples remain unchanged down to the lowest temperature studied. This suggests\nthat intersite exchange interactions responsible for long range or der presumably play an\nimportant role in determining the electronic structure of these inte resting materials.\n6[1] J. J. Randall and R. Ward, J. Amer. Chem. Soc. 81, 2629 (1959); A. Callaghan, C. W.\nMoeller, and R. Ward, Inorg. Chem. 5, 1572 (1966); J. M. Longo, P. M. Raccah, and J. B.\nGoodenough, J. Appl. Phys. 39, 1327 (1968).\n[2] G. Cao, S. McCall, M. Shepard, J. E. Crow, and R. P. Guertin , Phys. Rev. B 56, 321 (1997).\n[3] R. S. Singh, P. L. Paulose, and K. Maiti, Solid State Physi cs (India) 49, 876 (2004).\n[4] S. C. Gausepohl, M. Lee, L. Antognazza, and K. Char, Appl. Phys. Lett. 67, 1313 (1995).\n[5] K. S. Takahashi, A. Sawa, Y. Ishii, H. Akoh, M. Kawasaki, a nd Y. Tokura, Phys. Rev. B 67,\n094413 (2003).\n[6] Kobayashi, H., M. Nagata, R. Kanno, and Y. Kawamoto, Mate r. Res. Bull. 29, 1271 (1994).\n[7] D.J. Singh, J. Appl. Phys. 78, 4818 (1996); I.I. Mazin and D.J. Singh, Phys. Rev. B 56, 2556\n(1997).\n[8] K. Maiti, Phys. Rev. B 73, 235110 (2006).\n[9] K. Maiti and R. S. Singh, Phys. Rev. B 71, 161102(R) (2005). In this paper, CaRuO 3was\ndescribed to be antiferromagnetic due to (-ve) θPderived from the susceptibility in the para-\nmagnetic region. However, recent studies suggest an enhanc ed paramagnetic phase in this\ncompound.\n[10] K. Maiti, R. S. Singh, and V. R. R. Medicherla, Europhys. Lett.78, 17002 (2007).\n[11] M.Takizawa, D.Toyota, H.Wadati, A.Chikamatsu, H.Kum igashira,A.Fujimori, M.Oshima,\nZ. Fang, M. Lippmaa, M. Kawasaki, and H. Koinuma, Phys. Rev. B 72, 060404(B) (2005)\n[12] L. Klein, L. Antognazza, T. H. Geballe, M. R. Beasley and A. Kapitulnik, Phys. Rev. B 60,\n1448 (1999); P. Khalifah, I. Ohkubo, H. Christen and D. Mandr us, Phys. Rev. B 70, 134426\n(2004); Y. S. Lee et al., Phys. Rev. B 66, 041104(R) (2002).\n[13] R. S. Singh and K. Maiti, Solid State Comm. 140, 188 (2006).\n[14] K. Maiti, A. Kumar, D. D. Sarma, E. Weschke, and G. Kaindl , Phys. Rev. B 70, 195112\n(2004).\n[15] R. S. Singh and K. Maiti, cond-mat/0605552\n[16] E.C. Stoner, Proc. R. Soc. London A 154, 656 (1936).\n[17] V. Korenman et al., Phys. Rev. B 16, 4032 (1977); 16, 4048 (1977); H. Capellman, Z. Phys.\nB34, 29 (1979); A.J. Pindor et al., J. Phys. F 13, 979 (1983); H. Hasegawa, J. Phys. Soc.\n7Jpn.46, 1504 (1979).\n8FIGURE CAPTIONS\nFig. 1: Photoemission spectra of Ru 4 dvalence band for diﬀerent values of xin\nSr1−xCaxRuO3at 300 K (closed circle) and 20 K (open circle) using (a) He IIand (b)\nAlKαradiations. The top and bottom sets in (b) are the high resolution sp ectra of SrRuO 3\nand CaRuO 3using monochromatic Al Kαsource.\nFig. 2: (a) Surface and (b) Bulk spectra of Sr 1−xCaxRuO3at 300 K (closed circle) and\n20 K (open circle). Top and bottom sets are the high resolution spec tra of SrRuO 3and\nCaRuO 3, respectively.\n91.0Al Kα\n0.80.50.3\n300 K\n  20 K0.0 \n3 2 1 εF-1Mono\n Binding Energy (eV)1.0(b)\nMonox = 0.0 \n2.5 2.0 1.5 1.0 0.5 εF\n Binding Energy (eV)1.00.8Intensity (arb. units)0.50.3(a) He \u0001 \u0000\n300 K\n  20 Kx = 0.02 1 εFMono 1.0\nBinding Energy (eV)1.00.80.50.3\n Intensity (arb. units)0.0\n (a) Surface\nMono x = 0.0\n3 2 1 εFMono\n1.0\nBinding Energy (eV)1.00.80.50.3\n 300 K\n  20 K\n0.0\n (b) Bulk\nMono x = 0.0"    },    {        "title": "0908.3048v1.Spin_transfer_torque_in_disordered_weak_ferromagnets.pdf",        "content": "arXiv:0908.3048v1  [cond-mat.mes-hall]  21 Aug 2009Spin-transfer torque in disordered weak ferromagnets\nYoshisuke Ban∗, Gen Tatara\nDepartment of Physics,\nTokyo Metropolitan University,\nHachioji, Tokyo 192-0397, Japan\n(Dated: December 4, 2018)\nAbstract\nWe study theoretically the spin transfer eﬀect on a domain wal l in disordered weak ferromagnets.\nWe have identiﬁed the adiabatic condition for the disordere d case as λ≫λD≡/radicalbig\n/planckover2pi1D/∆sd, where\nDand ∆ sdare the diﬀusion constant and the spin splitting energy due to thes-dtype exchange\ninteraction, respectively, and found out that perfect spin -transfer eﬀect occurs even in weak fer-\nromagnets as long as this condition is satisﬁed. The eﬀective βterm arising from the force turns\nout to govern the wall dynamics, and therefore, the wall moti on can be as eﬃcient as in strong\nferromagnets even if ∆ sdis small.\n∗Present address: Department of Electrical Engineeringand Info rmation Systems,The University of Tokyo,\n7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.\n1The spin-transfer eﬀect has been intensively studied recently as a novel eﬃcient magne-\ntization switching mechanism without magnetic ﬁeld. The eﬀect arises from the transfer of\nthe spin angular momentum between the conduction electron and th e localized spins (mag-\nnetization), as ﬁrst pointed out by Berger in the case of domain wall [1]. In conventional 3 d\nferromagnets, domain walls are thick (with thickness λ∼100nm) and the coupling between\nthe electron spin and the localized spin ( s-dtype interaction) is strong. This is the adia-\nbatic regime where the electron spin can follow the localized spin textu re when traversing\nthe domain wall. In the ballistic case, the adiabatic condition is given by [ 2]\nλ≫vF\n∆sd, (1)\nwhereλis the wall thickness, vFis the Fermi velocity and ∆ sdis the spin splitting of the\nconduction electron. In this limit, the motion of the wall under curre nt is dominated by the\nspin-transfer eﬀect and the eﬃciency is governed by the angular m omentum conservation\nlaw [1]. The microscopic justiﬁcation for the series of pionnering work s by Berger [1, 3] was\ndone in Refs. [4, 5].\nThe eﬀect of spin-relaxation, which acts as an eﬀective non-adiaba ticity, was studied by\nZhang and Li and Thiaville et al. [6, 7] and was shown to signiﬁcantly aﬀ ect the current-\ndriven wall motion close to the adiabatic limit. The eﬀect of non-adiaba ticity (electron\nreﬂection by the wall) studied by Berger [3] and one of the present a uthors [4] was then\nidentiﬁed as another crucial factor even in the case close to the ad iabatic limit.\nWhile diﬀusive electron transport has been considered in the contex t of domain wall re-\nsistance [8, 9] and tunnel junctions [10], only ballistic case has been c onsidered in discussing\nthe spin-transfer torque. The reason would be that normal (spin -conserving) impurity scat-\ntering has been believed not to aﬀect the spin transfer processes . In contrast, spin relaxation\ndue to the spin-dependent impurity scattering and the spin-orbit in teraction was shown to\ngive rise to the eﬀective force (called βterm) and thus modiﬁes the domain wall motion\ngreatly [11, 12, 13].\nThe aim of the present paper is to investigate the eﬀect of normal im purities on the spin\ntransfer eﬀects, including the diﬀusion ladder. We demonstrate th at the diﬀusive electron\nmotion results in the modiﬁcation of the adiabatic condition to be\nλ≫/radicalbigg\n/planckover2pi1D\n∆sd≡λD, (2)\n2whereD=/planckover2pi12kF2\n3m2τisthediﬀusionconstant with kF,mandτbeingtheFermienergy, electron\nmass and elastic lifetime, respectively. Here, λDis the distance that a electron can reach\ndiﬀusively within a time of spin precession caused by the s-dinteraction, ∆ sd. Therefore\nif disordered adiabatic condition is satisﬁed, the electron spin can fo llow the localized spin\nproﬁle while going through the wall if λ≫λD(Fig. 1). As a result, even in a weak\nferromagnets with ∆ sd≪ǫF(ǫFis the Fermi energy), perfect spin transfer eﬀect is realized\nif the system is disordered. The adiabatic condition (2) was already p ointed out in Ref. [5],\nλ\nλD\nFIG. 1: The diﬀusive adiabatic limit we consider. λD=/radicalbig\nD/planckover2pi1/∆sdis the lenth scale the diﬀusive\nelectron can reach within the precession period of its spin, /planckover2pi1/∆sd. If the wall thickness is larger\nthanλD, the electron spin can follow the localized spin proﬁle and a diabatic spin transfer occurs.\nbut the eﬀect of diﬀusion ladder was not discussed there since only t he strong spin splitting\ncase was considered there.\nThe Lagrangian we consider is given by [5]\nL0\ne≡1\n/planckover2pi1/integraldisplay\nd3x/bracketleftbigg\ni/planckover2pi1c†˙c−/parenleftbigg/planckover2pi12\n2m|∇c|2−ǫFc†c/parenrightbigg\n+∆sdn·(c†σc)/bracketrightbigg\n+Himp, (3)\nwheren(x) represents the direction of the lozalized spin. In this paper, we co nsider the\ncase of a planar wall, given by nz(x) = tanhz\nλ,nx(x)±iny(x) =e±iφ1\ncoshz\nλ, wherezis the\ndirection perpendicular to the wall plane and φis a constant representing the angle out of\nthe easy plane. Scattering by normal impurities is described by Himp. Treating the impurity\npotential as an on-site type, it is given by\nHimp=Nimp/summationdisplay\ni=1/summationdisplay\nkk′vimp\nNei(k−k′)·Ric†\nk′ck, (4)\nwherevimprepresents the strength of the impurity potential, Rirepresents the position of\nrandom impurities, Nimpis the number of impurities, and N≡V/a3is number of sites. To\n3estimate physical quantities, we take the random average over imp urity positions. The self-\nenergy type processes due to the impurity scattering results in th e electron Green’s function\nwith lifetime τ, e.g.,gr\nk(ω) =1\nω−ǫk+i\n2τ, where the inverse lifetime is given as\n1\nτ=2π\n/planckover2pi1nimpv2\nimpν. (5)\nHereνis the density of states per site at the Fermi level and nimp≡Nimp\nNis the impurity\nconcentration. In this paper, we consider a weak ferromagnet an d thus neglect the spin-\ndependence of νandτ.\nThe force and torque due to applied current are given as [5]\nF=−∆sd/integraldisplay\nd3x∇zn·s, (6)\nτ=−∆sd/integraldisplay\nd3x(n×s), (7)\nwheresis the electron spin density induced by the current and domain wall.\nThe calculation of the elecron spin density is carried out by use of the spin gauge trans-\nformation, a≡Uc, whereUis a 2×2 unitary matrix and ais the electron operator in the\ngauge transformed frame. The marix Uis chosen to diagonalize the s-dtype interaction\nasU≡m·σ, wherem≡(sinθ\n2cosφ,sinθ\n2sinφ,cosθ\n2) (θandφare the polar coordinates\nof the localized spin direction, n). This approach is justiﬁed if the adiabatic condition (2)\nholds [5].\nThe spin-transfer torque and the force acting on a planar wall (wit h the wall plane\nperpendicular to the direction z) is expressed by the transverse spin densities, sθ≡s·eθ\nandsφ≡s·eφ(eθ≡(sinθcosφ,sinθsinφ,cosθ) andeφ≡(−sinφ,cosφ,0)) as\nF=−∆sd/integraldisplay\nd3x(∇zθ)sθ\nτz=−∆sd/integraldisplay\nd3xsinθsφ. (8)\nEach component is expressed as\nsθ=−1\n2/summationdisplay\n±e∓iφ˜s±\nsφ=−1\n2/summationdisplay\n±(∓)ie∓iφ˜s±, (9)\nwhere\n˜s±(x,t)≡/angbracketleftbig\na†σ±a/angbracketrightbig\n, (10)\n4are the spin densities in the gauge-transformed frame, calculate b y the standard diagramat-\nical expansion.\nThe spin density induced by the applied electric ﬁeld Ewas calculated in Ref. [5] without\nincluding the vertex correction. The result (denoted by ˜ s±(1)) is\n˜s±(1)\nq=−e\nπma3/summationdisplay\nijEiA±\nj(q)I±\nij(q), (11)\nwherea3is the unit volume and\nI±\nij(q)≡1\nN/summationdisplay\nk/bracketleftbigg\ngr\nk−q\n2,∓ga\nk+q\n2,±δij+/planckover2pi12kikj\nmgr\nk−q\n2,∓/parenleftBig\ngr\nk+q\n2,±+ga\nk−q\n2,∓/parenrightBig\nga\nk+q\n2,±/bracketrightbigg\n,(12)\nwhereN≡V/a3is the number of sites and gr\nk,∓are the Green’s function at zero frequency.\nHereA±\nµis the gauge ﬁeld ( µand±are the spatial and spin index, respectively). (Diagrams\nare shown in Fig. 2 (a)). In the adiabatic limit, kFλ≫1 and thus I±\nij(q) can be approxi-\nmated by the value at q= 0, i.e., I±\nij(0), since the transfer of the linear momentum between\nthe gauge ﬁeld and the electron can be neglected.\nThe aim of the present paper is to evaluate the vertex corrections , diagramatically shown\nin Fig. 2 (b), which were not addressed to in Ref. [5]. The vertex cor rection contribution\nEs    =+−~(1)σ+−\nσ+−σ+− σ+−σ+−σ+−A\nEs    =+−~(V) σ+−\nσ+−σ+− σ+−σ+−σ+−A(a)\n(b)\nΓ+−\nFIG. 2: (a) Diagrammatic representation of the spin density without vertex correction considered\nin Ref. [5]. (b) Vertex corrections to the spin density. Hatc hed square represents the diﬀusive\nladder, Γ±, arising from successive electron scattering by the normal impurities.\nto the spin density is easily calculated as\n˜s±(V)\nq=−e\nπma3/summationdisplay\nijEiA±\nj(q)I±\nij(q)Γ±(q), (13)\n5where\nΓ±(q)≡∞/summationdisplay\nn=1(nimpv2\nimpI±\n0(q))n. (14)\nHere\nI±\n0(q)≡1\nN/summationdisplay\nkgr\nk−q\n2,∓ga\nk+q\n2,±. (15)\nis written also as\nI±\n0=−1\nN/summationdisplay\nk(gr\nk−q\n2,∓−ga\nk+q\n2,±)1\n±2∆sd+/planckover2pi12k·q\nm+i/planckover2pi1\nτ. (16)\nIt was noted in Ref. [5] that ˜ s±(V)is negligiblly small in ballistic 3 dferromagnets,\ndue to the strong spin splitting, ∆ sdτ//planckover2pi1≫1. This fact is easily checked by noting that\nnimpv2\nimpI±\n0(q) =±iπ\n2nimpv2\nimpν++ν−\n∆sd+O(q2) =O/parenleftBig\n/planckover2pi1\n∆sdτ/parenrightBig\n≪1 in this limit, and thus Γ±≪1.\nIn this paper, we are considering the opposite limit, ∆ sdτ//planckover2pi1≪1, namely, a dirty weak\nferromagnet. We now demonstrate that the spin-transfer eﬀec t exists even in this case.\nMathematically, the dominant spin-transfer eﬀect in this limit is include d in the vertex\ncorrection, ˜ s±(V). Expanding eq. (16) with respect to qand ∆ sdτ//planckover2pi1≪1, we obtain\nnimpv2\nimpI±\n0(q) =nimpv2\nimp2πντ\n/planckover2pi1/parenleftbig\n1±2i∆sdτ//planckover2pi1−Dq2τ/parenrightbig\n=/parenleftbig\n1±2i∆sdτ//planckover2pi1−Dq2τ/parenrightbig\n.(17)\nThe summation in Eq. (14) is then carried out as\nΓ±(q) =1\nDq2τ∓2i∆sdτ//planckover2pi1−1. (18)\nTherefore the total spin density including the vertex correction, ˜s±≡˜s±(1)+ ˜s±(V), is\nobtained by use of Eqs. (11)(13)(18) as\n˜s±\nq=−e\nπma3/summationdisplay\nijEiA±\nj(q)I±\nij(q)1\nDq2τ∓2i∆sdτ//planckover2pi1. (19)\nThe last factor describes the long-range correlation of the torqu e induced by the diﬀusive\nelectron motion. I±\nij(q) in the limit of ∆ sdτ//planckover2pi1≪1 is calculated as\nI±\nij(q) =δij\nN(±∆sd)/summationdisplay\nk/bracketleftbig\n(gr\nk)2ga\nk−gr\nk(ga\nk)2/bracketrightbig\n=∓4πiν∆sdτ2\n/planckover2pi12δij+o(q2,∆sdτ//planckover2pi1), (20)\n6where we have neglected the contribution containing higher order o fqand ∆ sdτ. The\nexpression for the gauge ﬁeld in the case of a planar wall is given as [5]\nA±\nz(q) =∓iπ\n2Le±iφ1\ncoshπ\n2qλδq⊥,0, (21)\nwhereqrepresents the momentum transfer in the direction zandq⊥represents that in the\ntransverse direction and Lis the system length. From Eqs. (9)(19)(20)(21), the result of\nthe spin polarization is obtained as\nsφ(q) =3\n2πδq⊥,0j\ne∆sdτ\nǫF/planckover2pi1L1\ncoshπ\n2qλ/summationdisplay\n±(±i)1\nDq2τ∓2i∆sdτ//planckover2pi1\n=−6πδq⊥,0j\ne(∆sdτ)2\nǫF/planckover2pi12L1\ncoshπ\n2qλ1\n(Dq2τ)2+4(∆ sdτ//planckover2pi1)2(22)\nsθ(q) = 3πδq⊥,0j\ne∆sdτ\nǫF/planckover2pi1L1\ncoshπ\n2qλDq2τ\n(Dq2τ)2+4(∆ sdτ//planckover2pi1)2, (23)\nwherej≡(e2nτ/m)Eis the current density. The ﬁnal result of the torque and the forc e\nthen becomes (using z≡π\n2qλ)\nτ=π4/planckover2pi1\n8I\ne∆sd\nǫF/parenleftbiggλ\nλD/parenrightbigg4/integraldisplay∞\n−∞dz1\ncosh2z1\nz4+π4\n4/parenleftBig\nλ\nλD/parenrightBig4(24)\nF=3π4/planckover2pi1\n4I\ne∆sd\nǫFλ/parenleftbiggλ\nλD/parenrightbigg2/integraldisplay∞\n−∞dz1\ncosh2zz2\nz4+π4\n4/parenleftBig\nλ\nλD/parenrightBig4. (25)\nWe are interested in the adiabatic limit, λ≫λD, and the integrals are estimated in this\nlimit as\n/integraldisplay∞\n−∞dz1\ncosh2z1\nz4+π4\n4/parenleftBig\nλ\nλD/parenrightBig4∼4\nπ4/parenleftbiggλD\nλ/parenrightbigg4/integraldisplay∞\n−∞dz1\ncosh2z=8\nπ4/parenleftbiggλD\nλ/parenrightbigg4\n(26)\n/integraldisplay∞\n−∞dz1\ncosh2zz2\nz4+π4\n4/parenleftBig\nλ\nλD/parenrightBig4∼4\nπ4/parenleftbiggλD\nλ/parenrightbigg4/integraldisplay∞\n−∞dzz2\ncosh2z=2\n3π2/parenleftbiggλD\nλ/parenrightbigg4\n(27)\nThe torque and the force in the disordered adiabatic limit is ﬁnally obta ined as\nτ=/planckover2pi1I\ne∆sd\nǫF(28)\nF=π2\n2/planckover2pi1I\ne∆sd\nǫFλ/parenleftbiggλD\nλ/parenrightbigg2\n(29)\n7What is signiﬁcant is that the result of the torque indicates that the transfer of the spin\nangular momentum is essentially 100% in the disordered adiabatic limit. I n fact, if we deﬁne\nthe spin polarization in the disordered case as\nPD≡∆sd\nǫF, (30)\nwe see that the torque is simply given as\nτ=/planckover2pi1I\nePD. (31)\nIn the ballistic adiabatic limit, on the other hand, we know that the spin -transfer torque\nis given by τ=/planckover2pi1I\neP, where the polarization is deﬁned as P≡n+−n−\nn++n−(n±is the density of\nthe electron with spin ±) [5]. Since PD∼Pin most cases, we see that Eq. (31) indicates\nthat the perfect spin transfer occurs even in the disordered wea k ferromagnets as long as\nthe disordered adiabatic condition λ≫λDis satisiﬁed. One should note that the actual\nmagnitude of the spin transfered in a weak ferromagnet is small, pro portional to the small\npolarization factor, PD. Nevertheless, the eﬃcienty of the wall motion is as high as in\nstrongly polarized ferromagnets, as we will demonstrate below.\nThe force on domain wall can be measured by a dimensionless paramet erβdeﬁned as\nβ≡eSλ\n/planckover2pi1IF[5], where Sis the magnitude of localized spin. From Eq. (29), the parameter β\narising from the difussive electron motion is given by\nβ=π2\n2SPD/parenleftbiggλD\nλ/parenrightbigg2\n. (32)\nWe see here that/parenleftbigλD\nλ/parenrightbig2is a measure of the non-adiabaticity. If PD∼0.1 andλD/λ∼0.2\nwithS∼1, we obtain β= 0.02, which is suﬃciently large to improve the wall motion\ngreatly [5, 7].\nLet us study the wall dynamics in the present diﬀusive regime. We neg lect the extrinsic\npinning. The equation of motion of a planar wall is given by [5]\n˙φ0+α˙X\nλ=a3\n2eSλβj\n˙X−αλ˙φ0=vcsin2φ0+a3\n2eSPDj, (33)\nwherevc≡K⊥λS\n2/planckover2pi1is a critical velocity corresponding to the hard axis anisotropy ener gyK⊥.\nFrom Eq. (33), we immediately notice that the intrinsic threshold, giv en as2eSvc\nPDa3is very\n8high for a small spin polarization. Nevertheless, the wall motion occu rs due to the βterm\ninduced by the diﬀusive motion. The velocity calculated as function of the applied current\ndensity for diﬀerent values of PDandγ≡/parenleftbigλD\nλ/parenrightbig\nis plotted by solid lines in Fig. 3. It is seen\nthat the wall motion indeed occurs at lower current density even co mpared with the the\nintrinsic threshold at 100% spin polarization ( ji\nc=2eSvc\na3). We also see that the wall motion\nis governed by the parameter βand not much by the polarization PDin the present diﬀusive\nregime. Therefore, strogly polarized ferromagnet is not the nece ssary condition for eﬃcient\ndomain wall motion, but disordered weak ferromagnet is another op tion as promising as\nstrong ferromagnets.\n 0 0.5 1 1.5 2\n 0  0.5  1  1.5  2v/vcPD=0.2\nγ=0.2\nβ=0.04PD=0.1\nγ=0.2\nβ=0.02PD=0.2\nγ=0.1\nβ=0.01 PD=0.1\nγ=0.1\nβ=0.005PD=1, β=0\nj / j      ci\nFIG. 3: The wall velocity as function of the applied current d ensity for diﬀerent values of PDand\nγ≡/parenleftBig\nλD\nλ/parenrightBig\n(solid lines). The case of the ballistic adiabatic limit wit hβ= 0 is plotted by a dotted\nline for comparison. The velocity and the current density ar e normalized by vcand the intrinsic\nthreshold current ( ji\nc=2eSvc\na3) atPD= 1, respectively.\nTo conclude, we have derived the adiabatic condition for the spin tra nsfer eﬀect in disor-\ndered weak ferromagnets (Eq. (2)), and showed that the perfe ct spin transfer eﬀect occurs if\nthat condition is fullﬁlled even in the case of weak s-dtype coupling. We have also derived\nthe force acting on the wall in the diﬀusive limit and estimated the corr esponding β, which\nturned out to govern the wall motion in the diﬀusive weak ferromagn ets. By solving the\nequation of motion of a planar wall under current, we found that wa ll motion as eﬃcient as\nstrong ferromagnets can be realized in the present system, due t o a large value of force or\n9β.\nOur result also serves as a proof that the gauge ﬁeld expansion is ju stiﬁed as long as\ndisordered adiabatic condition (Eq. (2)) is satisﬁed.\nThis work was supported by a Grant-in-Aid for Scientiﬁc Research in Priority Areas,\n”Creation and control of spin current” (1948027), the Kurata M emorial Hitachi Science and\nTechnology Foundation and the Sumitomo Foundation.\n[1] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[2] X. Waintal and M. Viret, Europhys. Lett. 65, 427 (2004).\n[3] L. Berger, J. Appl. Phys. 49, 2156 (1978).\n[4] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004).\n[5] G. Tatara, H. Kohno, and J. Shibata, Physcs Reports 468, 213 (2008).\n[6] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[7] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Euro phys. Lett. 69, 990 (2005).\n[8] P. E. Falloon, R. A. Jalabert, D. Weinmann, and R. L. Stamp s, Phys. Rev. B 74, 144425\n(2006).\n[9] G. Tatara and H. Fukuyama, Phys. Rev. Lett. 78, 3773 (1997).\n[10] N. Theodoropoulou, A. Sharma, J. W. P. Pratt, J. Bass, M. D. Stiles, and J. Xiao, Phys. Rev.\nB76, 220408 (2007).\n[11] Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W. Ba uer, Phys. Rev. B 74, 144405\n(2006).\n[12] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, 113706 (2006).\n[13] G. Tatara and P. Entel, Phys. Rev. B 78, 064429 (2008).\n10"    },    {        "title": "1412.1521v1.Proximity_induced_ferromagnetism_in_graphene_revealed_by_anomalous_Hall_effect.pdf",        "content": "1 \n Proximity -induced f erromagnetism in  grap hene  revealed by anomalous Hall effect  \nZhiyong Wang, Chi Tang, Ray mond  Sachs, Yafis Barlas , and Jing Shi  \nDepartment of Physics and Astronomy, University of California, Riverside, CA 92521  \n \n \nWe demonstrate the anomalous Hall effect (AHE) in single -layer graphene \nexchange -coupled to  an atomically flat yttrium iron garnet (YIG)  ferromagnetic thin film . \nThe anomalous Hall conductance has magnitude  of ~0.09(2e2/h) at low temperatures and is \nmeasurable  up to ~ 300 K. Our observations indicate  not only proximity -induced \nferromagnetism in graphene /YIG  with large exchange interaction , but also  enhanced \nspin-orbit coupling  which is believed to be inheren tly weak in ideal graphene. T he \nproximity -induced ferromagnetic order in graphe ne can lead to novel transport phenomena \nsuch as the quantized AHE  which are  potentially useful for spintronics.  \n  2 \n Although pristine graphene sheets only exhibit  Laudau  orbital diamagnetism, l ocal \nmagnetic moments can be introduced in a variety of forms, e.g.  along  the edges of \nnanoribbons  [1] around  vacancies  [2] and adatoms  [3]. However,  a long-range ferromagnetic \norder in graphene  does not occur without exchange coupling between the local moments. In \ngeneral, i ntroducing  local moments and the exchange interaction  in bulk material s can be  \nsimultaneously  accomplished by doping  atoms with unfilled d- or f-shells  [4]. For graphene , \nscattering caused by random impurities  could be detrimental to  its high  carrier mobility , a \nunique electronic property  that should be preserved . By coupling the single atomi c sheet of \ncarbons with a magnetic insulator  film, e.g. YIG, we may introduce ferromagnetism in \ngraphene without sacrificing its excellent transport properties. T he hybridization between the \n-orbitals in graphene and the nearby  spin-polarized d-orbitals in magnetic insulators  gives \nrise to the exchange interaction  required for long -range ferromagnetic ordering . On the other \nhand, such proximity coupling does not bring unnecessary disorder to graphene. In addition, \nunlike ferromagnetic metals that could in principle mediate proximity exchang e coupling , the \ninsulating material does not shunt current away from graphene.  In this work, we demonstrate \nferromagneti c graphene via the proximity effect  and directly probe the ferromagnetism by \nmeasuring the anomalous Hall effect (AHE) .  \nTo bring  graphene in contact with YIG substrates , we apply  a previously developed \ntransfer technique (see SM) that is capable of transferring pre -fabricated functional graphene \ndevices to any target substrates  [5]. We first fabricate exfoliated single -layer graphene \ndevices on 290 nm -thick SiO 2 atop highly doped Si substrates using standard electron -beam \nlithography and Au electron -beam evaporation. Both longitudinal and Hall resistivities are 3 \n measured at room temperature to characterize the state of the pre -transferred devices. To \ntransfer selec ted devices, we spin -coat the chip with poly-methyl  methacrylate  (PMMA) \nfollowed by a hard bake at 170 °C for 10 minutes. The entire chip is then s oaked in 1 M \nNaOH solution for two  days to etch away SiO 2 so that the device /PMMA layer is released \nfrom the substrate. The PMMA layer attached with the fully nano -fabricated graphene \ndevices is then placed on the target substrate. Finally, the PMMA is dissolved with acetone \nfollowed by careful rinsing and dryi ng, and the device is  ready for electrical transport and/or \nRaman measurements. This technique was previously applied to fabricate  graphene devices \non SrTiO 3, a high nominal dielectric constant pervoskite material  [5,6]. The transfer steps are \nschematically shown in Fig . S-1 in SM.  \nFor this study, ~ 20 nm thick atomically flat  YIG films are grown epitaxially on 0.5 \nnm-thick gadolinium gallium garnet (GGG)  substrates by pulsed lase r deposition  as described \nelsewhere  [7], which  are then subsequently annealed in an  oxygen -flow furnace at 85 0 °C for \n6 hours  to minimize  oxygen deficiency . Magnetic hysteresis loop measurements and atomic \nforce microscop y (AFM)  are performed  to characterize the magnetic properties  and the \nmorphology  of YIG films , respectively . The hysteresis loop s of a representative YIG/GGG \nsample  are displayed in Fig. 1(a). The YIG film clearly shows  in-plane magnetic anisotropy . \nThe in-plane  coercive field and saturation field are  both small (~ a few G and <  20 G,  \nrespectively) , and t he out-of-plane loop indicate s a typical  hard-axis behavior with a \nsaturation field ~2000 G , which can  vary from 1500  to 2500 G  in different YIG samples . Fig. \n1(a) inset shows the  AFM topographic image  of a typical YIG film. The nearly parallel lines \nare terraces  separated by steps with the atomic height and the roughness on the terrace is ~ 4 \n 0.06 nm . The smooth ness of the  YIG surface  is not only critical to a strong  induced proximity \neffect  in graphene , but also favorable for maintaining high carrier mobility  [8].  \nIn order t o effectively tune the carrier density in graphene/YIG,  we fabricate a thin methyl \nmethacrylate ( MMA ) or PMMA top gate.  Fig. 1(b) shows  a false -colored  optical image of a \ngraphene device on YIG/GGG  before the  top gate  is fabricated . Room -temperature Raman \nspectroscopy i s performed at different stages of the device fabrication.  Representative spectra \nare shown in F ig. 1(c) for the same graphene device on SiO 2 (before transfer ) and YIG (after \ntransfer), and for YIG/GGG only . Graphene/YIG  show s both the characteristic  E2g (~1580  \ncm-1) and 2D peaks  (~2700  cm-1) of single -layer graphene as well as  YIG’ s own peaks, \nsuggesting success ful transfer. We also note that the transfer process does not produce  any \nmeasurable  D peak ( ~1350  cm-1) associated with defects  [9]. Fig. 1(d) is a schematic drawing \nof a top-gated transferred  device on YIG/GGG .  \nLow-temperature transport measurements are performed in Quantum Design’s Physical \nProperty Measurement System . Fig. 2(a) is a plot of the gate  voltage  dependence of the \nfour-terminal electrical conductivity scaled by the effective capacitance per unit area, Cs. \nSince different gate dielectric s are used in the back - and top-gated  graphene devices, Cs is \ncalculated based on the quantum Hall data which agrees with the calculated value usi ng the \nnominal dielectric constant and the  measured  dielectric film thickness . Before transfer, the \nDirac point is  at ~ -9 V and the field-effect mobility is ~ 6000  cm2/V∙s. After transfer,  the \nDirac point is shifted to ~ -18 V . The slope  of the σxx/Cs vs. Vg curve increases somewhat , \nindicating slight ly higher mobility , which  suggest s that the transfer process, the  YIG substrate, \nand the top -gate dielectric do  not cause  any adverse effect on graphene mobility . At 2 K, the 5 \n mobility improves further , exceeding  10000 cm2/V∙s on the electron side. Well-defined \nlongitudinal resistance  peaks  and quantum Hall plateau s are both present  at 8 T as shown  in \nFig. 2(b), another indication of uncompromised device quality after transfer . In approximately \n8 devices studied,  we find that the mobility of graphene/YIG is  either compar able with or \nbetter than that of  graphene/SiO 2.  \nTo s tudy the proximity -induced magnetism  in graphene, we  perform the  Hall effect  \nmeasurements  in the field range where the magnetization of YIG rotates out of  plane over a \nwide range of temperatures . Nearly  all graphene/YIG devices exhibit  similar  nonline ar \nbehavior  at low temperature s as shown in Fig. 2(c). Fig. 2(d) only  shows  the Hall data after \nthe linear ordinary Hall background  (the straight line in Fig. 2(c))  is subtracted . In \nferromagnets, the Hall resistivity generally consists of two parts  [10]: from the ordinary Hall \neffect and the anomalous Hall  effect (AHE) , i.e. 𝑅𝑥𝑦=𝑅𝐻(𝐵)+𝑅𝐴𝐻𝐸(𝑀)=𝛼𝐵+𝛽𝑀, \nhere B being  the external magnetic field,  M being the magnetization component in the \nperpendicular direction , and and  are two B- and M-independent parameters  respectively . \nThe B-linear  term results from the Lorentz force  on one type of carriers. Higher order terms \ncan appear if there are two or more types of carriers present.  The M-linear  term is due to the \nspin-orbit coupling  in ferromagnets  [10]. The observed non-linearity  in Rxy suggests the \nfollowing three  possible  scenarios: the ordinary  Hall effect  arising  from more than one type  \nof carriers  in response to the external magnetic field, the same Lorentz force related ordinary \nHall effect  but due to the stray magnetic field from the underlying YIG film, and AHE  from \nspin-polarized carriers . The  nonlinear Hall curves saturate  at Bs ~ 230 0 G, which is \napproximately  correlated with  the saturation of the  YIG magnetization  in Fig . 1(a). This 6 \n behavior is  characteristic of AHE , i.e. RAHE ∝ MG, where MG is the  induced magnetization of  \ngraphene . Since  MG result s from the proximity  coupl ing with the magnetization  of YIG , MYIG, \nboth MG and MYIG should saturate when the external field exceeds some value . The saturation \nfield of YIG is primarily determined by its  shape anisotropy , i.e. 4πM YIG, which  should not  \nchange significantly  far below the Curie  temperature  (550 K)  of YIG . On the other hand, if it  \nis caused by the  Lorentz force  on two type s of carriers , the nonlinear feature  would  not have  \nany correlation  with MYIG. These experimental facts do not support  the first scenario . To \nfurther exclude the ordinary Hall effect due to the Lorentz force from stray fields from  YIG, \nwe fabricate graphene devices on Al 2O3/YIG, in which the 5 nm thick continuous Al2O3 layer \nshould have  little effect on the strength of the stray  field but effectively cut off the proximity \ncoupling.  We do not observe any measurable nonlinear Hall signal similar to those in \ncompanion graphene/YIG devices  (Figs. S -6 and  S-7 in SM). It  excludes the effect of the \nstray field. Therefore, we attribute the non linear Hall signal in graphene/YIG to AHE which \nis due to spin -polarized carriers in ferromagnetic graphene. Further evidence will be  \npresent ed when the  gate voltage dependence  is discussed below . \nFig. 3(a) shows  the AHE resistance , RAHE, vs. the positive out-of-plane magnetic field \ntaken from 5 to 250 K . All linear background has been removed. Fig. 3 (b) is the extracted  \ntemperature dependence of  the saturated AHE resistance . The AHE signal decre ases as the \ntemperature is increase d, but it stays finite  up to nearly 300  K. We note that t he AHE \nmagnitude changes  sharply  in the temperature ran ge of 2 – 80 K , and then stays relatively \nconstant above 80 K before it approaches  ~ 300 K, which defines the Curie temperature of \nMG. In conducting ferromagnets , the AHE resistance , RAHE, scales with the longitudinal 7 \n resistance , Rxx, in the power -law fashion  [10], i.e. 𝑅𝐴𝐻𝐸∝𝑀𝐺𝑅𝑥𝑥𝑛. Thus t he temperature \ndependence of RAHE could originate from MG and/or Rxx. Here MG should be a slow -varying \nfunction of  the temperature below 80 K; however, the temperature dependence of Rxx in 1T \nfield (inset of Fig . 3(b)) cannot account for the steep temperature dependence of RAHE either. \nTherefore, we attribute the  discrepancy  to possible  physical  distance  change  between the \ngraphene sheet and YIG either due to an increase in the vibration al amplitude or different \nthermal expansion coefficients  between the top -gate dielectric and YIG/GGG . We have  \nobserve d variations  in both the Curie temperature  Tc for MG and the maximum RAHE (see Fig . \nS-2 and S -3). Among all 8 devices studied, the highest Tc is ~ 300 K and the largest RAHE at 2 \nK is ~ 200 Ω.  \nWith a top gate, we can control the  position of the Fermi level in graphene at a fixed \ntemperature , not possible in ferromagnetic metals. By sweeping  the top -gate voltage,  Vtg, we \nsystematically vary both RAHE and Rxx and keep  the induced magnetization and  exchange \ncoupling strength  unchanged . More importantly , by changing the carrier type, a sign reversal \noccurs in  the ordinary Hall , i.e. the  slope of the  linear background signal . We remove this \ncarrier density dependent linear background for each gate voltage and obtain the AHE signal. \nFig. 4(a) is the AHE resistivity  of a device  measured at 20 K  for several Vtg’s: 60 V (red \nsquares), 0  V (green circle s), and -20 V (blue triangle s), respectively. The inset shows the \nVtg-dependence  of the resistivity . The Dirac point is at ~35 V; therefore,  carriers are \npredominately electrons at 60 V  with a density ~ 2.5x1011 cm-2, but predominately  holes at \nboth 0  and -20 V . We deliberately avoid the region close to the Dirac point where both \nelectrons and holes coexist and the ordinary Hall signal acquires high -order terms in B. In the 8 \n gate dependence data, i t is important to note that the AHE sign remains unchanged regardless \nof the carrier type.  This is strong evidence that the observ ed non linear Hall signal is not due \nto the ordinary Hall effect from two types of carriers , either from the external or stray field,  \nbut due to the AHE contribution from spin -polarized carriers  in ferromagnetic sampl e. In \naddition, t he resistance at 60  V is the highest among the  three, followed by that at  0 V, and \nthen -20 V , and t he corresponding RAHE magnitude follows the same order.  \nTo further reveal  the physical origin of  AHE, we now focus on  the relationship between  \nRAHE and Rxx as Vtg is tuned. Fig. 4(b) shows  more gate -tuned AHE data in another top -gated \ndevice  measured at 2 K . We also exclude the data close to the Dirac point  (-14 V for this \ndevice)  for the reason mentioned above . Starting from -10 V , RAHE is the largest . As Vtg is \nincreased , the electron density increases, and  Rxx decreases  accordingly, which is \naccompanied by a steady decrease in  RAHE. Due to the negatively biased Dirac point, we \ncannot reach the completely hole-dominated region within the safe Vtg range  (gate leakage \ncurrent < 10 nA) . On the hole side where the background is still influenced by the two -band \ntransport, we do not observe any evidence of  a sign change  in RAHE. In the inset we plot RAHE \nvs. Rxx as Vtg is varied . From the slope of the straight line in the log-log plot, we obtain the \nexponent of  the power -law: n =1.9 ± 0.2 . The same exponent is also obtained in a  different  \ngate-tuned device  (see Fig . S-4 and S -5). As in many ferromagnetic conductors, t he quadratic \nrelationship indicates a scattering -independent AHE mechanism, which is different from the \nskew scattering induced AHE[10] . \nIt is understood that a necessary ingredient for AHE is the presence of SOC along with  \nbroken  time reversal symmetry  [10]. AHE can result from  either intrinsic  (band structure 9 \n effect) or extrinsic (impurity scattering)  mechanisms . Haldane  showed that for a honeycomb \nlattice (graphene) the presence of intrinsic SOC (which breaks time reversal symmetry) can \nlead to quantized  AHE ( QAHE ) for spin -less electrons  [11]. Since intrinsic SOC in graphen e \nis very weak (~10  μeV) [12], this effect has not been observed experimentally.  \nHowever, an enhanced Rashba SOC is possible when graphene is placed on substrates  \n[13,14] or subjected to hydrogenation  [15] due to broken inversion symmetry.  Recently, Qiao \net al. predicted that ferromagnetic graphene with Rashba SOC should exhibit QAHE  [16,17]. \nIn this case , the Dirac spectrum opens up a topological gap with  magnitude  smaller than \ntwice the minimum of exchange and SOC  energy scale (see SM). As the Fermi level is turned \ninto the gap, a decrease in the four-terminal resistance  is expected  along with a  simultaneous  \nquantization of the AHE  conductivity approaching 2e2/h. In devices exhibiting AHE, the \nlargest AHE at 2 K is ~ 200 Ω. Using the  corresponding Rxx of 5230 Ω , we calculate the AH E \ncontribution and obtain  σAHE ≈ 7 μS ≈ 0.09(2e2/h), nearly one order of magnitude smaller than \nthe predicted QAHE conductivity 2e2/h. Clearly we have not reached the QAHE regime  due \nto the intrinsic band structure effect , indicating that the Rashba SOC strength  λR is smaller \nthan the  disorder  energy scale . From the minimum conductivity plateau , we estimate  the \nenergy scale associated with the  disorder Δdis = ħ/τ ≈ 12 meV , assuming  long-ranged \nCoulomb scattering  [18]. Therefore  our experimental results suggest that  λR < 12 meV .  To \nobserve QAHE , it is important to further improve the quality of the devices or to strengthen \nthe Rashba SOC  to fulfill  λR > Δdis, both of which are highly possible.  \nIn order to understand the physical origin  of the observed unquantized AHE in our devices, \nwe calculate  the intrinsic  AHE (see SM ) at the relevant densities for λR < 12 meV .  Our results 10 \n show that the intrinsic AHE conductivity  at these densities  is an order of magnitude  smaller  \nthan the observed value , which argues against the intrinsic mechanism . Since  charged \nimpurity screening in graphene becomes extremely weak  as the Dirac point is approached , it \nis likely that the ex trinsic mechanisms play a more important role  here. We would like to \npoint out that  gate tunability in ferromagnetic graphene allow s for the observation of Fermi \nenergy dependen ce of the AHE conductivity , which  cannot be achieved  in ordinary \nferromagnet metals . If the carrier density can be modulated by gating, b esides the exponent, \nthe Fermi energy dependence of the AHE conductivity can be experimentally determined \nover a broad range of energy  [19]. This additional information can help further  pinpoint the \nphysical origin of AHE  in 2D Dirac fermion systems .  \nWe thank Z.S. Lin, T. Lin, B. Barrios, Q. Niu, and W. Beyerman for their help and useful \ndiscussions. ZYW and JS were supported by the DOE BES award #DE-FG02 -07ER46351 , \nCT was supported by NSF/ECCS, and RS was supported by NSF/NEB.   \n \n  11 \n FIG. 1. (a) Magnetic hysteresis loop s in perpendicular and in -plane magnetic field s. Inset is \nthe AFM topographic i mage of YIG thin film surface. (b) Optical image  (without top gate)  \nand (d) schematic drawing  (with top gate)  of the devices  after transferred to YIG/GGG \nsubstrate (false color). (c) Room  temperature Raman spectra of graphene/YIG  (purple), \ngraphene/SiO 2 (red), and YIG/GGG substrate only (blue).  \n \nFIG. 2. (a) The gate voltage dependence of the device conductivity scaled by the c apacitance  \nper unit area  for the pre -transfer  (293 K, black) and transferred devices  (300 K, red; 2 K, \ngreen) with the same graphene sheet . (b) Quantum Hall effect of transferred graphene/YIG \ndevice in an 8 T perpendicular magnetic field at 2 K.  (c) The measured total Hall resistivity \ndata at 2 K with a straight line indicating the ordina ry Hall background. (d) The non linear \nHall resistivity after the linear background is removed from the data in (c).  \n \nFIG. 3. (a) AHE resistance at different temperatures.  (b) The temperature dependence of AHE \nresistance. 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Lett. 96, 086805 (2006).  \n \n "    },    {        "title": "1908.03226v1.Ferromagnetism_in_Quantum_Dot_Plaquettes.pdf",        "content": "Ferromagnetism in Quantum Dot Plaquettes\nDonovan Buterakos1and Sankar Das Sarma1\n1Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics,\nUniversity of Maryland, College Park, Maryland 20742-4111 USA\n(Dated: August 12, 2019)\nFollowing the recent claimed obsevation of Nagaoka ferromagnetism in \fnite size quantum dot\nplaquettes,1a general theoretical analysis is warranted in order to ascertain in rather generic terms\nwhich arrangements of a small number of quantum dots can produce saturated ferromagnetic ground\nstates and under which constraints on interaction and inter-dot tunneling in the plaquette. This is\nparticularly necessary since Nagaoka ferromagnetism is fragile and arises only under rather special\nconditions. We test the robustness of ground state ferromagnetism in the presence of a long-\nrange Coulomb interaction and long-range as well as short-range interdot hopping by modeling\na wide range of di\u000berent plaquette geometries accessible by arranging a few ( \u00184) quantum dots\nin a controlled manner. We \fnd that ferromagnetism is robust to the presence of long range\nCoulomb interactions, and we develop conditions constraining the tunneling strength such that the\nground state is ferromagnetic. Additionally, we predict the presence of a partially spin-polarized\nferromagnetic state for 4 electrons in a Y-shaped 4-quantum dot plaquette. Finally, we consider\n4 electrons in a ring of 5 dots. This does not satisfy the Nagaoka condition, however, we show\nthat the ground state is spin one for strong, but not in\fnite, onsite interaction. Thus, even though\nNagaoka's theorem does not apply, the ground state for the \fnite system with one hole in a ring\nof 5 dots is partially ferromagnetic. We provide detailed fully analytical results for the existence\nor not of ferromagnetic ground states in several quantum dot geometries which can be studied in\ncurrently available coupled quantum dot systems.\nI. INTRODUCTION\nJohn Hubbard introduced the celebrated Hubbard\nmodel2as a minimal model to study ferromagnetism\nin narrow band itinerant electron systems such as Fe,\nNi, and Co. The hope was that the minimal Hubbard\nmodel, with just one dimensionless interaction parame-\nterU=twhereUis the on-site interaction (arising from\nCoulomb repulsion) between two electrons with unlike\nspins andtis the nearest-neighbor tunneling associated\nwith kinetic energy, would make the di\u000ecult problem\nof itinerant electron metallic ferromagnetism tractable\nand perhaps even exactly solvable. This early hope of\nthe Hubbard model leading perhaps to an understanding\nof narrow band metallic ferromagnetism was echoed in\nother early publications also.3,4After almost 60 years of\nextensive research, we still do not have a general solu-\ntion to the Hubbard model (except under very restricted\nconditions, e.g., one dimensional, 1D, systems) and the\nHubbard model has become the archetype underlying the\nwhole subject of strongly correlated materials. In fact,\nlarge teams of computational physicists work on large\ncomputers with the single goal of trying to understand\nnumerically the implications of the Hubbard model in\nvarious situations, and no clear signatures for ferromag-\nnetic ground states in the Hubbard model have emerged\nfrom these extensive numerical calculations.5Perhaps the\nmost ironic aspect of the Hubbard model is that it is\nnow universally accepted to be an excellent model to\nstudy antiferromagnetism, local moment formation, and\nMott metal-insulator transition in narrow band lattice\nsystems rather than as a model for metallic ferromag-\nnetism as Hubbard originally dreamed of. Any ferromag-netism arising within the Hubbard model is fragile and\nis certainly limited to very narrow parameter ranges (i.e.\nband \flling and the interaction strength U=t), and it is\nentirely possible that generic 2D and 3D ferromagnetic\nsystems cannot be described by the Hubbard model at\nall.\nOne important early result in this context is the con-\ncept of Nagaoka ferromagnetism6which arises naturally\nin the 2D Hubbard model on square (and other bipartite)\nlattices under rather nongeneric and highly restrictive\nconditions (see, e.g. Refs. 7, 8, and references therein).\nThis is an exact result which asserts that the 2D Hubbard\nmodel doped by precisely one hole (i.e. one missing elec-\ntron) away from the half-\flling has full ferromagnetism\nof the whole system in the thermodynamic limit provided\nUis in\fnite. Since the half-\flled 2D Hubbard model is\nsurely not a ferromagnet at any interaction strength, the\nNagaoka theorem appears pretty amazing in the sense\nthat removing just one electron from the system drives\nthe whole ground state completely ferromagnetic. The\ntheorem derives from the kinetic constraint on the mo-\ntion of a hole in the half-\flled system in the in\fnite U\nlimit, leading to the lowest energy state being the state\nof all the electrons becoming spin-polarized in order to\nminimize the kinetic energy in the strongly interacting\nlimit (where double occupancy is not allowed). While Na-\ngaoka ferromagnetism is of some theoretical signi\fcance\nbecause it is an exact result, it is of no consequence for\nany experimental situation since creating precisely one\nhole in a thermodynamic system is obviously an impos-\nsible constraint (and the in\fnite interaction limit is un-\nphysical as well). The very fragile nature of the proof\nunderlying this theorem does not allow its generalizationarXiv:1908.03226v1  [cond-mat.mes-hall]  8 Aug 20192\nto a dilute density of holes around half-\flling, and Na-\ngaoka ferromagnetism in its original form6is unlikely to\nbe observable experimentally in spite of its theoretical\nvalidity.\nThe question we address in the current work is the rel-\nevance of Nagaoka ferromagnetism in small \fnite 2D sys-\ntems, which can be constructed by using semiconductor\nquantum dots with a few electrons in it. In such a sys-\ntem, withNelectrons in Mdots, the e\u000bective \fnite-size\nNagaoka situation is easily achieved by tuning the sys-\ntem to having N=M\u00001, assuming each dot to have one\ne\u000bective orbital energy level with two spin states. Such\na scenario was recently achieved experimentally in Ref.\n1, and signatures for ferromagnetism were observed. Our\ngoal in the current work is to ask a general theoretical\nquestion on the existence or not of ferromagnetic ground\nstates in small 2D plaquettes made of tunable semicon-\nductor quantum dots: What experimentally accessible\narrangements of a few coupled quantum dots ( \u00184) with a\nfew electrons would manifest stable ferromagnetic ground\nstates? It turns out that this question can be answered\nanalytically for several interesting quantum dot struc-\ntures which are currently experimentally viable because\nof recent advances in control, engineering, and fabrication\nof coupled semiconductor quantum dots in the context of\ndeveloping spin qubits.1,9{13\nIt was pointed out 25 years ago14{16that semiconduc-\ntor quantum dot arrays may be capable of simulating the\nHubbard model in \fnite solid state systems searching for\nMott transition and related strong correlation phenom-\nena. Advances in materials growth and nanofabrication\ntechniques \fnally made this idea practical in laboratory\nsettings only in 2017 when Mott physics in the form of the\npredicted collective Coulomb blockade14was observed in\na small linear array of coupled GaAs quantum dots emu-\nlating the Hubbard model.9There has been rapid recent\ndevelopment in controlling small coupled quantum dot\narrays in several laboratories1,9{13, and experimentalists\ncan now study up to 4-8 dots with variable numbers of\nelectrons per dot along with precise control of coherent\nelectron tunneling between the dots. Our work, although\npurely theoretical, is inspired by these developments in\nthe precise experimental control over small systems of\ncoupled quantum dots. In particular, the recent exper-\nimental work from Delft1reporting the observation of\nNagaoka ferromagnetism in a 2D square array of quan-\ntum dots has directly motivated our work although our\nemphasis is on the generality of the possible emergence\nof Nagaoka-type ferromagnetism in quantum dot arrays,\nnot describing the observations in Ref. 1 which require a\ndetailed numerical approach.17\nElectrons in quantum dots interact via the long-range\nCoulomb interaction, and hence our model is a gener-\nalized or extended Hubbard model which includes both\non-site and inter-site Coulomb interaction. In addition,\nelectrons in quantum dots could, in general, have dis-\ntant neighbor hopping, not just nearest-neighbor hop-\nping as in the minimal Hubbard model. We thereforeinclude both nearest-neighbor and next-nearest-neighbor\nhopping in the theory. One other possible practical com-\nplication, which may be relevant to the experimental\nquantum dot arrays, is that each dot may have more\nthan one relevant orbital level, making the system akin\nto an SU(2n) Hubbard model where nis the number of\norbitals (\\quantum dot energy levels\") playing a role in\neach dot.18In such a situation, the inter-site hopping pro-\ncess could involve inter-orbital hopping also. We neglect\nthis complication and consider a purely SU(2) system\nwith each dot having just two spin states, assuming the\nhigher orbital levels in each dot to be reasonably high in\nenergy. This is not an essential approximation, and is\ndone to enable us to carry out our work completely ana-\nlytically. In any case, the neglect of higher orbital levels is\na well-de\fned and well-controlled theoretical approxima-\ntion since this can always be achieved experimentally by\nmaking each dot con\fnement potential su\u000eciently deep\n(and keeping the temperature su\u000eciently low) so that\nonly the lowest orbital state in each dot is operational in\nthe physics of the system. The \fnite size Hubbard model\nwe consider is therefore a generalization of the minimal\nHubbard model, and includes both distant neighbor hop-\nping and inter-site Coulomb interaction, but no higher\norbital physics.\nWe also should mention here that although the quan-\ntum ferromagnetism discussed in our work is adiabati-\ncally connected to the Nagaoka ferromagnetism in the\nhalf-\flled in\fnite- UHubbard model with one hole, there\nare important di\u000berences to keep in mind in order to\navoid confusion and misunderstanding. First, our sys-\ntem is a \fnite 2D plaquette (Fig. 1) with 4 dots and 3-5\nelectrons whereas Nagaoka ferromagnetism is obviously a\nthermodynamic result. Second, in our system the inter-\naction could be large, but never in\fnite, since the in\fnite-\nUlimit is unphysical for actual quantum dots. Third,\nour model being semi-realistic includes distant neighbor\nhopping and interaction, so we are considering a general-\nized and extended Hubbard model. Fourth, our inter-site\ntunneling (i.e. the hopping parameter t) matrix element\nis negative, not positive as in the original work of Na-\ngaoka. Fifth, because of the small size of our system, one\nmissing electron (i.e. a hole) corresponds to a \fnite hole\ndensity in contrast to the Nagaoka situation where the\nhole density is by de\fnition zero (e.g. 3 electrons in a\n2D square with 4 dots at the corners correspond to one\nhole in the system, but the hole density is 25%!). Thus,\nthe ferromagnetism we consider should perhaps be better\ncalled \\Nagaoka-type ferromagnetism\" rather than just\nNagaoka ferromagnetism. The really important point is,\nhowever, the fact that the quantum ferromagnetism we\npredict can be observed experimentally in already exist-\ning semiconductor quantum dot arrays.\nThe rest of this manuscript is organized as follows.\nIn sec. II, we investigate Nagaoka-type ferromagnetism\nby \fnding the ground states of three electrons in 4-dot\nplaquettes of various geometries. In sec. III, we repeat\nthe calculations for a half-\flled band (4 electrons) for the3\nsame geometries. In sec. IV, we look at the case of one\nhole in a 5-dot ring, and we summarize our results in sec.\nV.\nII. THREE ELECTRONS IN FOUR DOTS\nA. General Model And Method\n1. Hamiltonian\nWe consider a single-band Hubbard model with onsite\ninteraction energy U0, long-range Coulomb interaction\ntermsVijand hopping terms tij. Thus the Hamiltonian\nis given by:\nH=X\ni6=j;\u000btijcy\ni;\u000bcj;\u000b+X\niU0ni\"ni#+X\ni6=jVij\n2ninj(1)\nNagaoka's theorem predicts ferromagnetism in systems\nwith one hole in a half-\flled band with certain geometries\nwhere Nagaoka's condition holds. The simplest of these\nsystems are a triangle or square plaquette of three or four\nsites. However, of particular importance is the sign of the\nproduct of hopping elements around loops t12t23t31. In\norder for the Nagaoka condition to hold, quantities of\nthis form must be positive; however, in reality, this sign\nis determined by the number of sites in the loop, and is\nnegative for an odd number of sites. Thus a triangular\nplaquette with two electrons does not satisfy the Nagaoka\ncondition, as must be the case since it is well known that\nthe ground state of two electrons in any potential must\nnecessarily be a singlet. Thus the addition of next near-\nest neighbor hopping terms (the dashed lines in \fg. 1)\nbreak the Nagaoka condition and can potentially destroy\nferromagnetism if strong enough. It is interesting to de-\nrive a condition on the relative strengths of the hopping\nterms that determines whether ferromagnetism exists.\n \na \na \nd \na \nb \nd \na \n a \n a \na \na \n a \nd 1, 2:  3, 4: \n5: 6, 7: \nFIG. 1: A depiction of di\u000berent 4-dot geometries, numbered as\nthey appear in this work. Solid lines depict nearest-neighbor\nhopping terms, and dashed lines next nearest neighbor hop-\nping terms, which we consider in some cases. In all cases\nlong-range Coulomb interactions are included.We consider four di\u000berent geometries with 4 quantum\ndots: a square, a rectangle, a linear array, and Y-shaped\nplaquette, all with and without diagonal hopping terms\nwhere applicable. We note that only the \frst two satisfy\nthe Nagaoka condition, and only in the absence of the\ndiagonal hopping, as discussed above. We de\fne ato be\nthe distance between nearest neighbors, along with b>a\nin the case of the rectangle, and we de\fne dto be the dis-\ntance between next nearest neighbors in each respective\ngeometry. We de\fne Vrto be the Coulomb interaction\nenergy between electrons separated by a distance r, and\ntrbe the magnitude of the hopping strength between\ndots separated by a distance r.U0will be the onsite in-\nteraction energy as de\fned above. The bare parameters\nVrandU0are not important by themselves, but rather\ntheir di\u000berences are what a\u000bect the dynamics of the sys-\ntem, as a uniform shift in all these values will simply\ncause a constant shift in total energy, since the number\nof particles is conserved. Thus we will de\fne new pa-\nrametersUandVcorresponding to the relevant energy\ndi\u000berences, which vary for each geometry. We will also\nshift the total energy of the Hamiltonian by a constant\nsuch that the lowest energy con\fguration of electrons in\nthe absence of tunneling is 0.\n2. Spin 3/2 States\nA system of three electrons can have either spin 1/2\nor 3/2. To investigate the spin 3/2 states, we merely\nconsider the case where all electrons are spin up, as all\nother states in the spin 3/2 quartet will be identical, aside\nfrom the value of Sz. We de\fne the notation jd1d2d3d4i\nto be the state where the electron \flling of dot iis given\nbydi, wheredi2f0;\";#;\u0018 \u0017g. Since the Pauli exclusion\nprinciple forbids two spin up electrons from occupying\nthe same orbital state, there are four possible spin 3/2\nstates for each value of Sz. ForSz= 3=2, these are:\nj\"\"\" 0i;j\"\"0\"i;j\"0\"\"i;j0\"\"\"i (2)\nThe Hamiltonian is then constructed in this basis and\ndiagonalized to \fnd the eigenstates and energies. The\nlowest energy spin 3/2 state is compared to the lowest\nenergy spin 1/2 state to detrmine whether the ground\nstate is ferromagnetic. Additionally, for comparison, we\ncalculate the spin gap \u0001, de\fned to be the energy di\u000ber-\nence between the two lowest energy spin 3/2 states.\n3. Spin 1/2 States\nFor the spin 1/2 state, we consider the case where two\nelectrons are spin up and one is spin down. For con\fgu-\nrations with at most one electron per site, this gives three\nstates, one of which is part of the spin 3/2 quartet, and4\nthe other two of which have spin 1/2, as follows:\nj 3=2i=1p\n3\u0000\nj\"\"#i +j\"#\"i +j#\"\"i\u0001\nj +\n1=2i=1p\n3\u0000\ne2\u0019i\n3j\"\"#i +j\"#\"i +e\u00002\u0019i\n3j#\"\"i\u0001\nj \u0000\n1=2i=1p\n3\u0000\ne\u00002\u0019i\n3j\"\"#i +j\"#\"i +e2\u0019i\n3j#\"\"i\u0001\n(3)\nDe\fne a matrix Msuch that\n \nj +\n1=2i\nj \u0000\n1=2i!\n=M0\n@j\"\"#i\nj\"#\"i\nj#\"\"i1\nA (4)\nwhich can be obtained simply by reading o\u000b the coef-\n\fcients of eq. (3). Then we have a total of 8 low-energy\nspin 1/2 states with Sz= 1=2:\nj +\n1 +\n2 +\n30i;j +\n1 +\n20 +\n3i;j +\n10 +\n2 +\n3i;j0 +\n1 +\n2 +\n3i;\nj \u0000\n1 \u0000\n2 \u0000\n30i;j \u0000\n1 \u0000\n20 \u0000\n3i;j \u0000\n10 \u0000\n2 \u0000\n3i;j0 \u0000\n1 \u0000\n2 \u0000\n3i\n(5)\nHere i\njrefers to the state of the jth spin ofj i\n1=2ide-\n\fned as in eq. (3). For example, the state j +\n10 +\n2 +\n3i=\n1p\n3(e2\u0019i\n3cy\n1\"cy\n3\"cy\n4#+cy\n1\"cy\n3#cy\n4\"+e\u00002\u0019i\n3cy\n1#cy\n3\"cy\n4\")j0i. There\nare also 12 high energy states, corresponding to all per-\nmutations ofj\u0018 \u0017\"0 0i. These states only a\u000bect the ener-\ngies to order t2=U. Since Nagaoka's theorem applies only\nin the in\fnite Ulimit, we will initially consider only the\nlow energy states, and afterward calculate corrections to\nordert2=U.\nNagaoka ferromagnetism occurs because as a hole tun-\nnels around a loop, it causes the other electron spins in\nthe loop to be cyclically shifted one position. In the ferro-\nmagnetic state, all spins point in the same direction, and\nthus cycling them does not change the spin con\fguration.\nAt a lower total spin, however, there is a mixture of up\nand down spins, and thus cycling them will have some ef-\nfect such as rotating one spin con\fguration into another\nor adding a phase, which can potentially increase the en-\nergy of the state with lower total spin. In our calculation,\nwe see this e\u000bect when calculating the matrix elements of\nHbetween states where one electron has tunneled. If the\ntwo dots where the tunneling occurred are in consecutive\norder, then the spins remain in the same order, and the\nmatrix element is given by the corresponding term in the\nHamiltonian, as in the following example:\nhs1s2s30jHjs0\n1s0\n20s0\n3i=\u0000t\u000es1s0\n1\u000es2s0\n2\u000es3s0\n3(6)\nand thus matrix elements between  ican be found via:\nh i\n1 i\n2 i\n30jHj j\n1 j\n20 j\n3i=\u0010\nM\u0003(\u0000t)MT\u0011\nij=\u0000t\u000eij\n(7)\nand similarly for all other states of this form. How-\never, if the dots are not in consecutive order, such as forexample hopping between dots 1 and 4, then the spins\ncan potentially be rearranged:\nhs1s2s30jHj0s0\n1s0\n2s0\n3i=\u0000t\u000es2s0\n1\u000es3s0\n2\u000es1s0\n3(8)\nand therefore:\nh i\n1 i\n2 i\n30jHj0 j\n1 j\n2 j\n3i=\u0000t\"\nM\u00030\n@0 1 0\n0 0 1\n1 0 01\nAMT#\nij\n=\u0012\n\u0000te\u00002\u0019i\n3 0\n0\u0000te2\u0019i\n3\u0013\nij(9)\n4. Finite UCorrections\nFor several of the geometries, we also determine the\nleading order corrections to E1=2forU\u001dtbut not in-\n\fnite. This is done using perturbation theory, but is\ncomplicated by the fact that the spin 0 states are often\ndegenerate. We determine the matrix elements of Hbe-\ntween the lowest energy spin 0 states, which we denote\nj\ti\n1=2iand the high energy (2 ;1;0;0) states, which we\ndenotej\biiand order as follows:\nj\u0018 \u00170\"0i;j0\u0018 \u00170\"i;j\"0\u0018 \u00170i;j0\"0\u0018 \u0017i;\nj\u0018 \u0017\"0 0i;j\u0018 \u00170 0\"i;j0\u0018 \u0017\"0i;j\"\u0018 \u00170 0i;\nj0 0\u0018 \u0017\"i;j0\"\u0018 \u00170i;j\"0 0\u0018 \u0017i;j0 0\"\u0018 \u0017i;(10)\nWe de\fne the matrices Tand \u0003 as follows:\nTij=h\bijHj\tj\n1=2i (11)\n\u0003ij=h\bijHj\bji (12)\nNote that \u0003 is diagonal to leading order in t=U, and is\ngiven simply by the energies of j\bii. Then the corrections\nto the singlet state energies to order t2=Uare given by\nthe eigenvalues of the matrix \u0000Ty\u0003\u00001T.\nB. Ground State Calculations\n1. Square with no Diagonal Hopping\nWe initially consider a system of four dots in a square,\nwheretijandVijare given as follows:\ntij=(\n\u0000taifi\u0000j=\u00061 mod 4\n0 otherwise(13)\nVij=(\nVaifi\u0000j=\u00061 mod 4\nVdifi\u0000j= 2 mod 4(14)5\nUp to symmetry, three di\u000berent electron con\fgurations\nare possible:\n(1;1;1;0) with energy: 2 Va+Vd\n(2;0;1;0) with energy: U0+ 2Vd\n(2;1;0;0) with energy: U0+ 2Va (15)\nWe shift the total energy of the Hamiltonian by a con-\nstant amount 2 Va+Vd, and de\fne UandVas:\nU\u0011U0\u00002Va+Vd\nV\u0011Va\u0000Vd (16)\nso that the energies of the three electron con\fgurations\nin eq. (15) become 0, U, andU+ 2Vrespectively. Then\nthe spin 3/2 Hamiltonian in the basis given by eq. (2) is:\nH3=2=\u0000ta0\nB@0 1 0 1\n1 0 1 0\n0 1 0 1\n1 0 1 01\nCA (17)\nwhich has ground state \t 3=2=1\n2(1 1 1 1)Tand energy\nE3=2=\u00002ta. The \frst excited spin 3/2 state has energy\n0, so the spin gap is \u0001 = 2 ta.\nWe now \fnd the spin 1/2 Hamiltonian. From eq. (9),\na phase is introduced when tunneling the hole around the\nloop. Thus the spin 1/2 Hamiltonian is given by a block\ndiagonal matrix consisting of two blocks, corresponding\nto \u0006\n1=2as de\fned in eq. (3):\nH\u0006\n1=2=\u0000ta0\nBB@0 1 0e\u00072\u0019i\n3\n1 0 1 0\n0 1 0 1\ne\u00062\u0019i\n30 1 01\nCCA(18)\nwhich has ground states given by:\n\t\u0006\n1=2=1\n2\u0014\nj \u0006\n1 \u0006\n2 \u0006\n30i+e\u0006\u0019i\n6j \u0006\n1 \u0006\n20 \u0006\n3i\n+e\u0006\u0019i\n3j \u0006\n10 \u0006\n2 \u0006\n3i\u0006ij0 \u0006\n1 \u0006\n2 \u0006\n3i\u0015\n(19)\nwith energy E\u0006\n1=2=\u0000p\n3ta. Thus in the in\fnite U\nlimit, the system exhibits ferromagnetism, since the spin\n3/2 state has lower energy.\nWe also determine the \fnite Ucorrections to E1=2.\nSince there are two degenerate spin 1/2 states, \u0000Ty\u0003\u00001T\nis a 2\u00022 matrix, given by:\n\u0000Ty\u0003\u00001T=h\n\u00003t2\na\nU\u00002t2\na\nU+ 2Vi\u0012\n1 0\n0 1\u0013\n(20)\nHence we \fnd that the \t\u0006\n1=2degeneracy remains un-\nbroken, and the spin 1/2 ground state energy is given\nby:\nE1=2=\u0000p\n3ta\u00003t2\na\nU\u00002t2\na\nU+ 2V+O\u0010t3\na\nU2\u0011\n(21)\n2468101214V/ta1415161718Ucrit/taFIG. 2: Ucritversus Vfor three electrons in a four-dot square\ncon\fguration. Here Uand 2 Vare de\fned as in eq. (16).\n10 15 20 25 30U/ta-2.2-2.1-2.0-1.9-1.8E/ta\nE3/2\nE1/2(V=0)\nE1/2(V=2t a)\nE1/2(V=5t a)\nFIG. 3: E3=2andE1=2versus Ufor di\u000berent values of Vfor\nthree electrons in a four-dot square con\fguration. The point\nwhere E3=2andE1=2cross is Ucrit.\nThen forV!0, we recover a correction of \u00005t2\na=U,\nagreeing with the result given in Ref. 1. Using this re-\nsult, we can derive the value Ucrit(to \frst order in ta=U)\nwhich marks the transition between the ferromagnetic\nand antiferromagnetic phases:\nUcrit=1\n2(2\u0000p\n3)\"\n\u00002(2\u0000p\n3)V+ 5ta+\nq\n(2(2\u0000p\n3)V\u00005ta)2+ 24(2\u0000p\n3)Vta#\n(22)\nForV!0, this gives Ucrit= 5ta=(2\u0000p\n3)\u001918:7ta.\n2. Square with Diagonal Hopping\nWe now investigate how diagonal hopping terms ef-\nfect the system. We use the same square con\fgura-\ntion of four dots, but now add extra hopping terms\nt13=t31=t42=t24=\u0000td. We again de\fne UandVas\nin equation (16). The analysis for the spin 3/2 states is\nsimilar to above, except there are now extra matrix ele-\nments corresponding to td. These will be positive rather6\nthan negative as an extra minus sign is introduced due\nto Fermi statistics, since diagonal tunneling essentially\nexchanges two electrons. Then the spin 3/2 Hamiltonian\nis given as follows:\nH3=2=0\nB@0\u0000tatd\u0000ta\n\u0000ta0\u0000tatd\ntd\u0000ta0\u0000ta\n\u0000tatd\u0000ta01\nCA (23)\nwhich has ground state \t 3=2=1\n2(1 1 1 1)Tand energy\nE3=2=\u00002ta+td. The \frst excited state has energy \u0000td,\nso the spin gap is \u0001 = 2 ta\u00002td.\n0.1 0.2 0.3 0.4 0.5td/ta\n-2.0-1.5-1.0-0.5E/ta\nE3/2\nE1/2\nFIG. 4: Plot of E3=2andE1=2versus td=tafor three electrons\nin a four-dot square con\fguration with diagonal hopping in\nthe in\fnite Ulimit. We see that ferromagnetism is only pos-\nsible for td< ta=4.The analysis for the spin 1/2 states is also similar to\nthe square model, with again the only di\u000berence in the in-\n\fniteUlimit being the diagonal hopping terms td. Then\na calculation similar to eq. (9) yields:\nh i\n1 i\n2 i\n30jHj j\n10 j\n2 j\n3i=\u0012\n0tde\u00002\u0019i\n3\ntde2\u0019i\n3 0\u0013\nij\nh i\n1 i\n20 i\n3jHj0 j\n1 j\n2 j\n3i=\u00120tde2\u0019i\n3\ntde\u00002\u0019i\n3 0\u0013\nij(24)\nThus diagonal hopping rotates j +\n1=2iintoj \u0000\n1=2iand\nvice versa. Then H1=2is no longer block-diagonal, and is\ngiven by:\nH1=2=0\nBBBBBBBBBBB@0\u0000ta 0\u0000tae\u00002\u0019i\n3 0 0 tde\u00002\u0019i\n3 0\n\u0000ta 0\u0000ta 0 0 0 0 tde2\u0019i\n3\n0\u0000ta 0\u0000tatde\u00002\u0019i\n3 0 0 0\n\u0000tae2\u0019i\n3 0\u0000ta 0 0 tde2\u0019i\n3 0 0\n0 0 tde2\u0019i\n3 0 0 \u0000ta 0\u0000tae2\u0019i\n3\n0 0 0 tde\u00002\u0019i\n3\u0000ta 0\u0000ta 0\ntde2\u0019i\n3 0 0 0 0 \u0000ta 0\u0000ta\n0tde\u00002\u0019i\n3 0 0\u0000tae\u00002\u0019i\n3 0\u0000ta 01\nCCCCCCCCCCCA(25)\nwhich has two degenerate ground states with energy\nE1=2=\u0000p\n3t2a+t2\nd. Thus, in the in\fnite Ulimit,E3=2<\nE1=2as long astd<ta=4, and thus ferromagnetism only\nexists fortd<ta=4.\n3. Rectangle with no Diagonal Hopping\nWe now model a rectangular con\fguration of four dots.\nThis will be similar to the square model, except tijandVijare given by:\ntij=8\n><\n>:\u0000taiffi;jg=f1;2gorf3;4g\n\u0000tbiffi;jg=f2;3gorf1;4g\n0 otherwise(26)\nVij=8\n><\n>:Vaiffi;jg=f1;2gorf3;4g\nVbiffi;jg=f2;3gorf1;4g\nVdifi\u0000j=\u00062(27)\nWithout loss of generality, we will assume b > a , and\nthusta>tbandVa>Vb. We note that up to symmetry7\nthe following four electron con\fgurations are possible:\n(1;1;1;0) with energy: Va+Vb+Vd\n(2;0;1;0) with energy: U0+ 2Vd\n(2;0;0;1) with energy: U0+ 2Vb\n(2;1;0;0) with energy: U0+ 2Va (28)\nWe shift the total energy by Va+Vb+Vd, and de\fne\nU,V, andWas:\nU\u0011U0\u0000Va\u0000Vb+Vd\nV\u0011Va\u0000Vd\nW\u0011Vb\u0000Vd (29)\nso that the energies of the electron con\fgurations in\neq. (28) become 0, U,U+2W,U+2Vrespectively. The\nanalysis for the spin 3/2 states is identical to the square\nmodel, except that care must be taken to distinguish be-\ntweentaandtb. Thus we construct the Hamiltonian:\nH3=2=0\nB@0\u0000ta0\u0000tb\n\u0000ta0\u0000tb0\n0\u0000tb0\u0000ta\n\u0000tb0\u0000ta01\nCA (30)\nwhich has ground state \t 3=2=1\n2(1 1 1 1)Tand energy\nE3=2=\u0000ta\u0000tb. The \frst excited state has energy \u0000ta+\ntb, so the spin gap is \u0001 = 2 tb.\n0.2 0.4 0.6 0.8 1.0tb/ta\n-2.0-1.5-1.0-0.5E/ta\nE3/2\nE1/2\nFIG. 5: Plot of E3=2andE1=2versus tb=tafor three electrons\nin a four-dot rectangular con\fguration with no diagonal hop-\nping in the in\fnite Ulimit.\nThe analysis for the spin 1/2 states is also similar to\nthe square model, with again the only di\u000berence in the\nin\fniteUlimit being the the second hopping strength tb.\nThen the spin 1/2 Hamiltonian is given by:\nH\u0006\n1=2=0\nBB@0\u0000ta0\u0000tbe\u00072\u0019i\n3\n\u0000ta 0\u0000tb 0\n0\u0000tb0\u0000ta\n\u0000tbe\u00062\u0019i\n30\u0000ta 01\nCCA(31)\nwhich has energy E\u0006\n1=2=\u0000p\nt2a+tatb+t2\nb, andground state given by:\n\t\u0006\n1=2=1\n2\u0014\nj \u0006\n1 \u0006\n2 \u0006\n30i+e\u0006i'j \u0006\n1 \u0006\n20 \u0006\n3i\n+e\u0006i\u0019\n3j \u0006\n10 \u0006\n2 \u0006\n3i+e\u0006i('+\u0019\n3)j0 \u0006\n1 \u0006\n2 \u0006\n3i\u0015\n(32)\nwhere'\u0011arctanp\n3tb\n2ta+tb. Thus, three electrons in four\ndots arranged in a rectangular con\fguration will exhibit\nferromagnetism for large U, regardless of the ratio of ta\nandtb. This is assuming that there is no diagonal hop-\nping, an assumption that may break down for extreme\nratios oftatotb.\n0.0 0.2 0.4 0.6 0.8 1.0tb/ta20406080Ucrit/ta\nFIG. 6: Plot of Ucritversus tb=tafor three electrons in a four-\ndot rectangular con\fguration with V=W= 0.\nThe procedure for calculating the \fnite Ucorrections\ntoE\u0006\n1=2is also similar to the square model. We calculate\n\u0000Ty\u0003\u00001Tlike before, obtaining:\n\u0000Ty\u0003\u00001T\n=\u0000t2\na\u0000tatb\u0000t2\nb\nU\u0012\n1e\u0000i\u0019\n3\u0000i'cos 3'\nei\u0019\n3+i'cos 3' 1\u0013\n\u0000t2\na\nU+ 2W\u0012\n1e\u0000i\u0019\n3\u0000i'cos'\nei\u0019\n3+i'cos' 1\u0013\n\u0000t2\nb\nU+ 2V\u0012\n1\u0000e\u0000i\u0019\n3\u0000i'cos('\u0000\u0019\n3)\n\u0000ei\u0019\n3+i'cos('\u0000\u0019\n3) 1\u0013\n(33)\nThe o\u000b-diagonal terms break the j\t+\n1=2i;j\t\u0000\n1=2idegen-\neracy, with the lower energy state given by:\nj\t1=2i=1p\n2h\nj\t+\n1=2i+ei\u0019\n3+i'j\t\u0000\n1=2ii\n(34)8\n20 40 60 80100U/ta\n-2.0-1.5-1.0-0.5E/ta\nE3/2\nE1/2\nE1/2ex\n20 40 60 80100U/ta\n-2.0-1.5-1.0-0.5E/ta\nE3/2\nE1/2\nE1/2ex\nFIG. 7: Plot of E3=2,E1=2, and the nearly-degenerate ex-\ncited state energy Eex\n1=2versus Ufor three electrons in a four-\ndot rectangular con\fguration with no diagonal hopping with\ntb=ta=:8 (Top) and tb=ta=:2 (Bottom). Here V=W= 0.\nand thus, the energy of the lowest energy state is:\nE1=2=\u0000q\nt2a+tatb+t2\nb\u0000t2\na+tatb+t2\nb\nU(1 + cos 3')\n\u0000t2\na\nU+ 2W(1 + cos')\u0000t2\nb\nU+ 2V(1\u0000cos('\u0000\u0019\n3))\n(35)4. Rectangle with Diagonal Hopping\nWe now address the case of diagonal hopping in a rect-\nangular system. We de\fne taandtbas in eq. (26), and\nlet the diagonal hopping term be given by td. We assume\nta>tb>td. We shift the total energy by Va+Vb+Vd,\nas in the rectangular case, and de\fne U,V,Was in\nequation (29).\nThe analysis for the spin 3/2 states is similar to above.\nThus we construct the Hamiltonian:\nH3=2=0\nB@0\u0000tatd\u0000tb\n\u0000ta0\u0000tbtd\ntd\u0000tb0\u0000ta\n\u0000tbtd\u0000ta01\nCA (36)\nwhich has ground state \t 3=2=1\n2(1 1 1 1)Tand energy\nE3=2=\u0000ta\u0000tb+td. The \frst excited state has energy\n\u0000ta+tb\u0000td, so the spin gap is \u0001 = 2 tb\u00002td.\nThe analysis for the spin 1/2 states is also similar to\nabove. Then H1=2is given by:\nH1=2=0\nBBBBBBBBBBB@0\u0000ta 0\u0000tbe\u00002\u0019i\n3 0 0 tde\u00002\u0019i\n3 0\n\u0000ta 0\u0000tb 0 0 0 0 tde2\u0019i\n3\n0\u0000tb 0\u0000tatde\u00002\u0019i\n3 0 0 0\n\u0000tbe2\u0019i\n3 0\u0000ta 0 0 tde2\u0019i\n3 0 0\n0 0 tde2\u0019i\n3 0 0 \u0000ta 0\u0000tbe2\u0019i\n3\n0 0 0 tde\u00002\u0019i\n3\u0000ta 0\u0000tb 0\ntde2\u0019i\n3 0 0 0 0 \u0000tb 0\u0000ta\n0tde\u00002\u0019i\n3 0 0\u0000tbe\u00002\u0019i\n3 0\u0000ta 01\nCCCCCCCCCCCA(37)\nwhich has a nondegenerate ground state with energy\nE1=2=\u0000p\nt2a+t2\nb+t2\nd+tatb+tatd\u0000tbtd. From this, it\nis easy to show that in the in\fnite Ulimit,E3=2<E 1=2\nas long astd<tatb=(3ta+tb).5. Linear Array of Four Dots\nWe also model a linear array of four dots. This will be\nsimilar to the square model, except t14=t41= 0, and9\nVijis given by:\nVij=8\n><\n>:Vaifi\u0000j=\u00061\nV2aifi\u0000j=\u00062\nV3aifi\u0000j=\u00063(38)\nWe note that up to symmetry, the following electron\ncon\fgurations are possible:\n(1;1;0;1) with energy: Va+V2a+V3a\n(1;1;1;0) with energy: 2 Va+V2a\n(2;0;0;1) with energy: U0+ 2V3a\n(2;0;1;0) with energy: U0+ 2V2a\n(2;1;0;0) with energy: U0+ 2Va (39)\nWe shift the total energy by Va+V2a+V3a, and de\fne\nU,V, andWas:\nU\u0011U0\u0000Va\u0000V2a+V3a\nV\u0011Va\u0000V3a\nW\u0011V2a\u0000V3a (40)\nso that the energies of the electron con\fgurations in eq.\n(39) become 0, V,U,U+ 2W,U+ 2Vrespectively. The\nanalysis for the spin 3/2 states is identical to the square\nmodel, except that some states have an extra energy V,\nand no hopping is permitted between dots 1 and 4. Thus\nwe construct the Hamiltonian:\nH3=2=0\nB@V\u0000ta0 0\n\u0000ta0\u0000ta0\n0\u0000ta0\u0000ta\n0 0\u0000taV1\nCA (41)\nwhich has a nondegenerate ground state with energy:\nE3=2= (V\u0000ta\u0000p\n(V+ta)2+ 4t2a)=2 (42)\nand ground state given by:\n\t3=2=1\np\n2q\n1 +(V\u0000E3=2)2\nt2\na0\nB@1\n(V\u0000E3=2)=ta\n(V\u0000E3=2)=ta\n11\nCA (43)\nFor convenience, we de\fne A(V;t) andB(V;t) from\neq. (43) above such that \t 3=2= (AB B A )T. The \frst\nexcited state has energy ( V+ta\u0000p\n(V\u0000ta)2+ 4t2a)=2,\nand so the spin gap is given by the di\u000berence of this\nenergy and E3=2.\nIn the square model without diagonal hopping, the\nonly di\u000berence between the spin 3/2 and spin 1/2 sub-\nspaces in the in\fnite Ulimit is in the hopping term be-\ntween dots 1 and 4. Since this term no longer exists\nin the linear model, we \fnd that the spin 1/2 Hamilto-\nnian is simply two exact copies of the spin 3/2 Hamil-\ntonian,H\u0006\n1=2=H3=2, and thus the ground state energy\n20 40 60 80100U/ta\n-2.0-1.5-1.0-0.5E/ta\nE3/2\nE1/2\nE1/2ex\n20 40 60 80100U/ta\n-2.0-1.5-1.0-0.5E/ta\nE3/2\nE1/2\nE1/2exFIG. 8: Plot of E3=2,E1=2andEex\n1=2versus Ufor three elec-\ntrons in a four-dot linear array for V= 0 (Top) and V= 5ta\n(Bottom). Here W=V=4.\nE\u0006\n1=2=E3=2, as well. Thus for \fnite U, the system can-\nnot exhibit ferromagnetism, since the \fnite Ucorrections\nwill lower the energy of the spin 1/2 states.\nWe repeat the procedure discussed above to calculate\nthe \fniteUcorrections to E\u0006\n1=2. Then\u0000Ty\u0003\u00001Tis given\nby:\n\u0000Ty\u0003\u00001T\n=\u0014\u0000B2t2\na\nU\u0000((A+B)2+A2)t2\na\nU+ 2W\u00002A2t2\na\nU+ 2V\u0015\u0012\n2 1\n1 2\u0013\n(44)\nThe o\u000b-diagonal terms break the j\t+\n1=2i;j\t\u0000\n1=2idegen-\neracy, with the lower energy state given by:\nj\t1=2i=1p\n2h\nj\t+\n1=2i+j\t\u0000\n1=2ii\n(45)\nwhich corresponds to the spin con\fguration:\n1p\n6\u0002\n\u0000j\"\"#i + 2j\"#\"i\u0000j#\"\"i\u0003\n(46)\nThis spin con\fguration is the spin 1/2 state which\nmaximizes overlap with the alternating spin con\fgura-\ntionj\"#\"i , and so the ground state of 3 electrons in a\nlinear array of 4 dots is an antiferromagnet. The ground\nstate energy is given by:\nE1=2=V\u0000ta\u0000p\n(V+ta)2+ 4t2a\n2\n\u00003t2\na\u0014B2\nU+((A+B)2+A2)\nU+ 2W+2A2\nU+ 2V\u0015\n(47)10\n6. Y-Shaped Con\fguration\nWe now model a Y-shaped con\fguration of four dots.\nWe will let dots 2 through 4 be positioned at the corners\nof an equilateral triangle, and dot 1 be at the center, with\nhopping terms only between a corner dot and the center\ndot. ThentijandVijare given by:\ntij=(\n\u0000taifiorj= 1\n0 otherwise(48)\nVij=(\nVaifiorj= 1\nVdotherwise(49)\nThen up to symmetry, the following electron con\fgu-\nrations are possible:\n(0;1;1;1) with energy: 3 Vd\n(1;1;1;0) with energy: 2 Va+Vd\n(0;2;1;0) with energy: U0+ 2Vd\n(2;1;0;0) with energy: U0+ 2Va\n(1;2;0;0) with energy: U0+ 2Va (50)\nWe shift the total energy by 3 Vd, and de\fne UandV\nas:\nU\u0011U0\u0000Vd\nV\u0011Va\u0000Vd (51)\nso that the energies of the electron con\fgurations in\neq. (50) become 0, 2 V,U,U+ 2V,U+ 2Vrespectively.\nUsing the same methods as above, we construct the spin\n3/2 Hamiltonian:\nH3=2=0\nB@2V0 0\u0000ta\n0 2V0ta\n0 0 2V\u0000ta\n\u0000tata\u0000ta01\nCA (52)\nwhich has a nondegenerate ground state with energy:\nE3=2=V\u0000p\nV2+ 3t2a (53)\ngiven by:\n\t3=2=1r\n3 +9t2a\nE2\n3=20\nB@1\n\u00001\n1\n3ta=(\u0000E3=2)1\nCA (54)The \frst excited state has energy 2 V, and so the spin\ngap is given by the di\u000berence 2 V\u0000E3=2.\nFor the spin 1/2 case, in the in\fnite Ulimit, the Hamil-\ntonian separates into a block-diagonal matrix with two\nblocks, where the basis for each block is given by:\nj \u0006\n1 \u0006\n2 \u0006\n30i;j \u0007\n1 \u0007\n20 \u0007\n3i;j \u0006\n10 \u0006\n2 \u0006\n3i;j0 \u0006\n1 \u0006\n2 \u0006\n3i\n(55)\nIn this basis, the two blocks of the spin 1/2 Hamilto-\nnianH\u0006\n1=2are given by:\nH\u0006\n1=2=0\nBB@2V 0 0\u0000tae\u00072\u0019i\n3\n0 2V 0tae\u00072\u0019i\n3\n0 0 2 V\u0000ta\n\u0000tae\u00062\u0019i\n3tae\u00062\u0019i\n3\u0000ta 01\nCCA(56)\nwhich is identical to H3=2up to a phase rede\fnition\nof some of the states. Therefore in the in\fnite Ulimit,\nE\u0006\n1=2=E3=2, and thus for \fnite U, the system cannot\nexhibit ferromagnetism, since the \fnite Ucorrections will\nlower the energy of the spin 1/2 states.\n7. Y-Shaped Con\fguration With N.N.N. Hopping\nWe now add a next nearest neighbor hopping term td\nbetween the outer corners of the Y-shaped con\fguration.\nThentijis given by:\ntij=(\n\u0000taifiorj= 1\n\u0000tdotherwise(57)\nThe same electron con\fgurations as in eq. (50) above\nare possible. We again shift the total energy by 3 V2, and\nde\fneUandVas in eq. (51). Using the same methods\nas above, we construct the Hamiltonian:\nH3=2=0\nB@2V\u0000tdtd\u0000ta\n\u0000td2V\u0000tdta\ntd\u0000td2V\u0000ta\n\u0000tata\u0000ta01\nCA (58)\nwhich has a nondegenerate ground state with energy:\nE3=2=V+td\u0000p\n(V+td)2+ 3t2a (59)\nThe \frst excited state has energy 2 V\u0000td.\nWe construct the spin 1/2 Hamiltonian in the basis\ngiven by eq. (5) as follows:11\nH1=2=0\nBBBBBBBBBBB@2V\u0000td 0\u0000tae\u00002\u0019i\n3 0 0 tde\u00002\u0019i\n3 0\n\u0000td 2V\u0000td 0 0 0 0 tae2\u0019i\n3\n0\u0000td 2V\u0000tatde\u00002\u0019i\n3 0 0 0\n\u0000tae2\u0019i\n3 0\u0000ta 0 0 tae2\u0019i\n3 0 0\n0 0 tde2\u0019i\n3 0 2V\u0000td 0\u0000tae2\u0019i\n3\n0 0 0 tae\u00002\u0019i\n3\u0000td 2V\u0000td 0\ntde2\u0019i\n3 0 0 0 0 \u0000td 2V\u0000ta\n0tae\u00002\u0019i\n3 0 0\u0000tae\u00002\u0019i\n3 0\u0000ta 01\nCCCCCCCCCCCA(60)\nThis matrix has two degenerate ground states with en-\nergy given by the smallest root of a cubic polynomial\nP(E1=2) = 0, where P(E) is given by:\nP(E) =E3\u00004VE2+ (\u00003t2\na\u0000t2\nd+ 4V2)E+ 6t2\naV\n(61)\nTo compare E1=2withE3=2, one can show that\nP(E3=2)>0 for 0< td< ta. This implies that there\nmust be a root of P(E) which lies to the left of E3=2,\nand thusE1=2<E 3=2. Therefore the ground state is not\nferromagnetic.\nC. Summary\nWe have explored many di\u000berent plaquette geometries\nin the presence of long-range Coulomb interactions, withand without next nearest neighbor hopping. We have\nfound that in these systems, Nagaoka ferromagnetism is\nrobust to the presence of long-range Coulomb interac-\ntions, and is present even if the plaquette is rectangular\nrather than square. We argued that next nearest neigh-\nbor hopping destroys Nagaoka ferromagnetism, and de-\nrived conditions for the value of tdwhere this transition\noccurs for both the square and rectangular geometries.\nfor completeness, we showed that other geometries such\nas a linear array and Y-shaped con\fguration have an an-\ntiferrromagnetic ground state. We present these \fndings\nin a table below:\nSec. dot n.n.n. E3=2 E1=2 Spin Ferro-\nnum. con\fg. hopping forU!1 gap magnetism?\n1 square no\u00002ta \u0000p\n3ta 2ta yes\n2 square yes\u00002ta+td \u0000p\n3t2a+t2\nd2ta\u00002tdiftd<ta=4\n3rectangle no\u0000ta\u0000tb \u0000p\nt2a+tatb+t2\nb2tb yes\n4rectangle yes\u0000ta\u0000tb+td\u0000p\nt2a+t2\nb+t2\nd+tatb+tatd\u0000tbtd2tb\u00002tdiftd<tatb\n3ta+tb\n5 linear no1\n2 \nV\u0000ta\n\u0000p\n(V+ta)2+ 4t2a!\n1\n2 \nV\u0000ta\n\u0000p\n(V+ta)2+ 4t2a!\n\u0001lin no\n6Y-shaped noV\u0000p\nV2+ 3t2a V\u0000p\nV2+ 3t2a 2V\u0000E3=2 no\n7Y-shaped yesV+td\n\u0000p\n(V+td)2+ 3t2agiven byP(E1=2) = 02V\u0000td\n\u0000E3=2no\n\u0001lin=ta+1\n2\u0010p\n(V+ta)2+ 4t2a\u0000p\n(V\u0000ta)2+ 4t2a\u0011\nP(E1=2) =E3\n1=2\u00004VE2\n1=2+ (\u00003t2\na\u0000t2\nd+ 4V2)E1=2+ 6t2\naV\nV\u0011(\nVa\u0000V3afor sec. 5\nVa\u0000Vdfor sec. 6 & 7\nIII. FOUR ELECTRONS IN FOUR DOTS\nA. General Method\nWe now consider a half-\flled band consisting of four\nelectrons and four dots in an arbitrary con\fguration, forlargeU0. It is well-known that for large systems, the12\nground state of a half-\flled band is antiferromagnetic;\nhowever, we show that a four dot plaquette can have a\npartially ferromagnetic spin-1 ground state for certain\ngeometries.\nThe lowest energy states will be in the (1 ;1;1;1) con-\n\fguration, and we will shift the energy of our Hamilto-\nnian to account for the Coulomb interaction energy in\nthis con\fguration. Thus, by de\fnition, the spin 2 states,\nwhich are not a\u000bected by tunneling, have energy E2= 0,\nand in the in\fnite Ulimit, the spin 0 and 1 states have\n0 energy as well. We will de\fne Ui(j)to be the Coulomb\ninteraction energy of the (2 ;1;1;0) con\fguration states\nwith two electrons in dot iand no electrons in dot j,\nagain o\u000bset by the energy of the (1 ;1;1;1) state. We ig-\nnore the (2 ;2;0;0) states, as they are not connected to\nthe (1;1;1;1) states by a single tunneling operation, and\nthus will have no e\u000bect on the ground state energies to\nordert2=U.\nOur strategy is the same as when \fnding the \fnite U\ncorrections in the previous section. We list all low energy\nstates with a given spin. Since these will all be in the\n(1;1;1;1) con\fguration, they will be degenerate to lead-\ning order. We then list the relevant high energy states,\nand let \u0003 be the diagonal matrix with entries given by\nthe energies of the high energy states, and let the entries\nofTbe given by the matrix elements of Hbetween a low\nand a high energy state. Then the \frst order corrections\nint2=Uto the energies of the low energy states are given\nby diagonalizing the matrix \u0000Ty\u0003\u00001T. This will poten-\ntially also break the degeneracy, as long as \u0000Ty\u0003\u00001Tis\nnot proportional to the identity matrix.\n1. Spin 0 States\nThere are two states with total spin 0 for electrons in\nthe (1;1;1;1) con\fguration:\nj\t\u0006\n0i=1p\n6\u0014\ne\u00062\u0019i\n3j\"\"##i +j\"#\"#i +e\u00072\u0019i\n3j\"##\"i\n+e\u00072\u0019i\n3j#\"\"#i +j#\"#\"i +e\u00062\u0019i\n3j##\"\"i\u0015\n(62)\nThere are 24 high energy states connected to j\t\u0006\n0iby a\nsingle tunneling operation, corresponding to all permuta-\ntions ofj\u0018 \u0017\"#0i. However, due to conservation of spin,\nonly states where the two single electrons form a spin\nsinglet will contribute, and thus we need only consider\n12 states. We calculating matrix elements between these\nstates andj\t\u0006\n0i, we obtain the matrix \u0000Ty\u0003\u00001T:\n\u0000Ty\u0003\u00001T=\u0000X\ni6=jt2\nij\nUi(j) \n1e\u0000i'ij\nei'ij 1!\n(63)where'ijis given by:\n'ij=8\n><\n>:\u0019\n3iffi;jg=f1;2gorf3;4g\n\u0019 iffi;jg=f1;3gorf2;4g\n5\u0019\n3iffi;jg=f1;4gorf2;3g(64)\nThus, to order t2=U, the total energy of the spin 0\nground state is:\nE0=\u0000X\ni6=jt2\nij\nUi(j)\u0000\f\f\f\f\fX\ni6=jt2\nij\nUi(j)ei'ij\f\f\f\f\f(65)\n2. Spin 1 States\nTo investigate the spin 1 states, we consider the sub-\nspace where Sz= 1. There are three states with total\nspin 1 for electrons in the (1 ;1;1;1) con\fguration:\nj\t1\n1i=1\n2h\nj\"\"\"#i +j\"\"#\"i\u0000j\"#\"\"i\u0000j#\"\"\"ii\nj\t2\n1i=1\n2h\nj\"\"\"#i\u0000j\"\"#\"i +j\"#\"\"i\u0000j#\"\"\"ii\nj\t3\n1i=1\n2h\nj\"\"\"#i\u0000j\"\"#\"i\u0000j\"#\"\"i +j#\"\"\"ii\n(66)\nThere are 12 high energy states connected to j\ti\n1i,\ngiven by all permutations of j\u0018 \u0017\"\"0i. Calculating ma-\ntrix elements between these states and j\ti\n1i, we \fnd that\n\u0000Ty\u0003\u00001Tis given by:\n\u0000Ty\u0003\u00001T=\u0000X\ni6=jt2\nij\nUi(j)1+\n0\nB@A12+A34A23\u0000A14A13\u0000A24\nA23\u0000A14A13+A24A12\u0000A34\nA13\u0000A24A12\u0000A34A14+A231\nCA(67)\nwhereAijis given by:\nAij=t2\nij\u00121\nUi(j)+1\nUj(i)\u0013\n(68)\nB. Ground State Calculations\n1. Square with no Diagonal Hopping\nFor four dots in a square, with no diagonal hopping,\nwe have for spin 0,\n(\u0000Ty\u0003\u00001T)0=\u0000t2\na\nU \n8 4\n4 8!\n(69)\nwhereU\u0011U0\u0000Va. The o\u000b-diagonal terms break the\ndegeneracy, and the ground state and energy is given by:13\nj\t0i=1\n2p\n3\u0014\n\u0000j\"\"##i + 2j\"#\"#i\u0000j\"##\"i\n\u0000j#\"\"#i + 2j#\"#\"i\u0000j##\"\"i\u0015\n(70)\nE0=\u000012t2\na\nU(71)\nWe note that as expected, this is the spin 0 state which\nmaximizes overlap with the antiferromagnetic con\fgura-\ntionsj\"#\"#i andj#\"#\"i . For spin 1, we have\n(\u0000Ty\u0003\u00001T)1=\u0000t2\na\nU0\nB@4 0 0\n0 8 0\n0 0 41\nCA (72)\nE1=\u00008t2\na\nU(73)\nHere the degeneracy is also broken, and the ground\nstate is given byj\t2\n1ias de\fned in eq. (66).\n2. Square With Diagonal Hopping\nFor four dots in a square, with diagonal hopping, we\nhave\n(\u0000Ty\u0003\u00001T)0=\u0000t2\na\nU \n8 4\n4 8!\n\u0000t2\nd\nU+V \n4\u00004\n\u00004 4!\n(74)\nE0=\u000012t2\na\nU(75)\nwhereU\u0011U0\u0000VaandV\u0011Va\u0000Vd. For spin 1,\n(\u0000Ty\u0003\u00001T)1=\u0000t2\na\nU0\nB@4 0 0\n0 8 0\n0 0 41\nCA\u0000t2\nd\nU+V0\nB@4 0 0\n0 0 0\n0 0 41\nCA\n(76)\nE1=\u00008t2\na\nU(77)\nInterestingly, diagonal hopping for a square does not\na\u000bect the ground state energies E0orE1, and only serves\nto decrease the energy of the excited states. This can be\nunderstood by noticing that in each of the ground states,\nspins at opposite corners of the square (dots 1 and 3 or\ndots 2 and 4) only occur in a triplet con\fguration. This\nis necessary to allow adjacent spins to anti-align as much\nas possible.3. Rectangle\nFor four dots in a rectangle, with no diagonal hopping,\nwe have\n(\u0000Ty\u0003\u00001T)0=\n\u00004 t2\na\nU+t2\nb\nU+Vt2\na\nUe\u0000\u0019i\n3+t2\nb\nU+Ve\u0019i\n3\nt2\na\nUe\u0019i\n3+t2\nb\nU+Ve\u0000\u0019i\n3t2\na\nU+t2\nb\nU+V!\n(78)\nE0=\u00004\"\nt2\na\nU+t2\nb\nU+V+s\nt4a\nU2+t4\nb\n(U+V)2\u0000t2at2\nb\nU(U+V)#\n(79)\nwhereU\u0011U0\u0000Va, andV\u0011Va\u0000Vb. For spin 1,\n(\u0000Ty\u0003\u00001T)1=\u00004\u0014t2\na\nU+t2\nb\nU+V\u0015\n1+ 40\nB@t2\na\nU0 0\n0 0 0\n0 0t2\nb\nU+V1\nCA\n(80)\nE1=\u00004\"\nt2\na\nU+t2\nb\nU+V#\n(81)\nWe note that the spin 1 ground state remains the same\nas in the square case, while the spin 0 ground state ro-\ntates, essentially in such a way as to include a greater\nweight to singlets across the shorter edge of the rectan-\ngle than the longer edge. This must be the case, as when\ntb!0, the ground state must become two spin singlets.\n4. Rectangle With Diagonal Hopping\nFor four dots in a rectangle, with diagonal hopping, we\nhave\n(\u0000Ty\u0003\u00001T)0=\n\u00004 t2\na\nU+t2\nb\nU+V+t2\nd\nU+Wt2\na\nUe\u0000\u0019i\n3+t2\nb\nU+Ve\u0019i\n3\u0000t2\nd\nU+W\nt2\na\nUe\u0019i\n3+t2\nb\nU+Ve\u0000\u0019i\n3\u0000t2\nd\nU+Wt2\na\nU+t2\nb\nU+V+t2\nd\nU+W!\n(82)\nE0=\u00004\"\nt2\na\nU+t2\nb\nU+V+t2\nd\nU+W\n+\u0012t4\na\nU2+t4\nb\n(U+V)2+t4\nd\n(U+W)2\n\u0000t2\nat2\nb\nU(U+V)\u0000t2\nat2\nd\nU(U+W)\u0000t2\nbt2\nd\n(U+V)(U+W)\u00131=2#\n(83)\nwhereU\u0011U0\u0000Va,V\u0011Va\u0000Vb, andW=Va\u0000Vd.14\nFor spin 1,\n(\u0000Ty\u0003\u00001T)1=\n\u00004\u0014t2\na\nU+t2\nb\nU+V+t2\nd\nU+W\u0015\n1+ 40\nB@t2\na\nU0 0\n0t2\nd\nU+W0\n0 0t2\nb\nU+V1\nCA\n(84)\nE1=\u00004\"\nt2\na\nU+t2\nb\nU+V#\n(85)\nAgain the the spin 1 ground state is una\u000bected by the\npresence of diagonal hopping, for the same reason dis-\ncussed above. However, the diagonal hopping terms do\na\u000bect the spin 0 state, since the imbalance between ta\nandtbcauses opposite spins to no longer only appear in\ntriplets.\n5. Linear Array\nFor four dots in a line, we have\n(\u0000Ty\u0003\u00001T)0=\n\u00002t2\na0\n@1\nU+1\nU+2V+1\nU+V(1\nU+1\nU+2V)e\u0000\u0019i\n3+e\u0019i\n3\nU+V\n(1\nU+1\nU+2V)e\u0019i\n3+e\u0000\u0019i\n3\nU+V1\nU+1\nU+2V+1\nU+V1\nA\n(86)\nE0=\u00002t2\na\"\n1\nU+1\nU+2V+1\nU+V\n+s\n\u00001\nU+1\nU+2V\u00012+1\n(U+V)2\u00001\nU+V\u00001\nU+1\nU+2V\u0001#\n(87)\nwhereU\u0011U0\u00002Va+V3a, andV\u0011Va\u0000V3a. For spin\n1,\n(\u0000Ty\u0003\u00001T)1=\u00002t2\nah1\nU+1\nU+ 2V+1\nU+Vi\n1\n+ 2t2\na0\nB@1\nU+1\nU+2V1\nU+V0\n1\nU+V0 0\n0 01\nU+V1\nCA(88)\nE1=\u0000t2\na\nU\u0000t2\na\nU+ 2V\u00002t2\na\nU+V\n\u0000t2\nas\n\u00001\nU+1\nU+ 2V\u00012+4\n(U+V)2(89)\nIn the limit where V!0, this reduces to E0=\u0000(6 +\n2p\n3)t2\na=UandE1=\u0000(4 + 2p\n2)t2\na=U.6. Y-Shaped Con\fguration\nFor four dots in a Y-shaped con\fguration, we have\n(\u0000Ty\u0003\u00001T)0=\u0000\u0010t2\na\nU+t2\na\nU+ 4V\u0011 \n3 0\n0 3!\n(90)\nE0=\u00003\u0010t2\na\nU+t2\na\nU+ 4V\u0011\n(91)\nwhereU\u0011U0\u00003Va+2VdandV\u0011Va\u0000Vd. Thus, the\nj\t\u0006\n0idegeneracy remains unbroken, due to the three-fold\nrotational symmetry of the system. For spin 1,\n(\u0000Ty\u0003\u00001T)1=\u0010t2\na\nU+t2\na\nU+ 4V\u00110\nB@\u00002\u00001 1\n\u00001\u00002 1\n1 1\u000021\nCA(92)\nE1=\u00004\u0010t2\na\nU+t2\na\nU+ 4V\u0011\n(93)\nInterestingly, the ground state is the spin 1 state rather\nthan the spin 0 state. This state is given by:\nj\t1i=1\n2p\n3h\nj\"\"\"#i +j\"\"#\"i +j\"#\"\"i\u0000 3j#\"\"\"ii\n(94)\nwhich is the state maximizes the weight of the spin\ncon\fguration where the center electron has opposite spin\nas the three corner electrons. Thus, the ground state can\nbe thought of as antiferromagnetic in the sense that ad-\njacent spins are anti-aligned; however, since there is an\nimbalance in the number of sites in the odd and even sub-\nlattices, assigning alternating spins to these sites causes\na total spin of 1 rather than 0.\n7. Y-Shaped Con\fguration with N.N.N. Hopping\nFor four dots in a Y-shaped con\fguration, with next\nnearest neighbor hopping (that is hopping between the\nouter corners), we have\n(\u0000Ty\u0003\u00001T)0=\u0000\u0010t2\na\nU+t2\na\nU+ 4V+2t2\nd\nU+ 3V\u0011 \n3 0\n0 3!\n(95)\nE0=\u00003\u0010t2\na\nU+t2\na\nU+ 4V+2t2\nd\nU+ 3V\u0011\n(96)\nwhereU\u0011U0\u00003Va+ 2VdandV\u0011Va\u0000Vd. For spin\n1,\n(\u0000Ty\u0003\u00001T)1=\u00002\u0010t2\na\nU+t2\na\nU+ 4V+2t2\nd\nU+ 3V\u0011\n1\n+\u0010t2\na\nU+t2\na\nU+ 4V\u00002t2\nd\nU+ 3V\u00110\nB@0\u00001 1\n\u00001 0 1\n1 1 01\nCA (97)\nE1=\u00004\u0010t2\na\nU+t2\na\nU+ 4V\u0011\n(98)15\nHere the presence of next nearest neighbor hopping re-\nduces the energy of the spin 0 states, while still maintain-\ning thej\t\u0006\n0idegeneracy, as the three-fold symmetry of\nthe system remains unbroken. The next nearest neighbor\nhopping terms do not a\u000bect the spin 1 ground state, how-\never, as the spins in any of the two corners only appear\nin triplet con\fgurations. Thus, as tdis increased there\nexists a crossover point between E0andE1.E1<E 0as\nlong as6t2\nd\nU+3V<t2\na\nU+t2\na\nU+4V.\nC. Summary\nWe have calculated the energies of the lowest energy\nspin 0 and spin 1 state for half-\flled band in severaldi\u000berent four-dot geometries to \frst order in t2=U. In\neach case, the ground state is antiferromagnetic; how-\never, with the Y-shaped con\fguration, alternating spins\non each cite causes a total spin of 1 rather than 0, because\nthere are 3 corner dots and only 1 center dot. Adding\nnext nearest neighbor interactions reduces the energy dif-\nference between the two states, up to a critical strength\nat which point the spin 0 state becomes the ground state.\nWe summarize these \fndings in a table:\nSec. dot nnn. E0 E1 E2E1<E 0?\nnum. con\fg. hop. to ordert2=U to ordert2=U\n1 square no \u000012t2\na=U \u00008t2\na=U 0 no\n2 square yes \u000012t2\na=U \u00008t2\na=U 0 no\n3rectangle no\u00004t2\na\nU\u00004t2\nb\nU+V\n\u00004q\nt4a\nU2+t4\nb\n(U+V)2\u0000t2at2\nb\nU(U+V)\u00004t2\na\nU\u00004t2\nb\nU+V0 no\n4rectangle yes\u00004t2\na\nU\u00004t2\nb\nU+V\u00004t2\nd\nU+W\n\u00004 t4\na\nU2+t4\nb\n(U+V)2+t4\nd\n(U+W)2\n\u0000t2\nat2\nb\nU(U+V)\u0000t2\nat2\nd\nU(U+W)\u0000t2\nbt2\nd\n(U+V)(U+W)!1\n2\u00004t2\na\nU\u00004t2\nb\nU+V0 no\n5 linear no\u00002t2\na\nU\u00002t2\na\nU+2V\u00002t2\na\nU+V\n\u00002t2\na \u00001\nU+1\nU+2V\u00012+1\n(U+V)2\n\u00001\nU+V\u00001\nU+1\nU+2V\u0001!1=2\u0000t2\na\nU\u0000t2\na\nU+2V\u00002t2\na\nU+V\n\u0000t2\naq\u00001\nU+1\nU+2V\u00012+4\n(U+V)20 no\n6Y-shaped no \u00003\u0010\nt2\na\nU+t2\na\nU+4V\u0011\n\u00004\u0010\nt2\na\nU+t2\na\nU+4V\u0011\n0 yes\n7Y-shaped yes\u00003\u0010\nt2\na\nU+t2\na\nU+4V+2t2\nd\nU+3V\u0011\n\u00004\u0010\nt2\na\nU+t2\na\nU+4V\u0011\n0if6t2\nd\nU+3V<\nt2\na\nU+t2\na\nU+4V\nNote we de\fne UandVslightly di\u000berently each time:\nU\u00118\n><\n>:U0\u0000Va for sec. 1, 2, 3 & 4\nU0\u00002Va+V3afor sec. 5\nU0\u00003Va+ 2Vdfor sec. 6 & 7\nV\u00118\n><\n>:Va\u0000Vbfor sec. 3 & 4\nVa\u0000V3afor sec. 5\nVa\u0000Vdfor sec. 6 & 7\nW\u0011Va\u0000VdIV. FOUR ELECTRONS IN FIVE DOTS\nA. Model\nWe now consider a ring of \fve dots with four electrons.\nThis does not satisfy the Nagaoka condition, and thus we\ndo not predict the ground state to be ferromagnetic. The\nHamiltonian is given by eq. (1), with tijandVijgiven\nas follows:\ntij=(\n\u0000taifi\u0000j=\u00061 mod 5\n0 otherwise(99)\nVij=(\nVaifi\u0000j=\u00061 mod 5\nVdifi\u0000j=\u00062 mod 5(100)16\n \nd \na \nFIG. 9: A depiction of a ring of 5 dots. Solid lines depict\nnearest-neighbor hopping terms and Coulomb interactions,\nand dashed lines long-range Coulomb interactions.\nUp to symmetry, only one low energy electron con\fgu-\nration is possible. There are also three high energy con-\n\fgurations that are connected to the low energy states\nby a single tunneling operation. These are:\n(1;1;1;1;0) with energy: 3 Va+ 3Vd\n(2;0;1;1;0) with energy: U0+Va+ 4Vd\n(2;1;0;1;0) with energy: U0+ 2Va+ 3Vd\n(2;1;1;0;0) with energy: U0+ 3Va+ 2Vd (101)\nWe shift the total energy of the Hamiltonian by 3 Va+3Vd,\nand de\fneUandVas:\nU\u0011U0\u00002Va+Vd\nV\u0011Va\u0000Vd (102)\nso that the energies of the electrons con\fgurations in eq.\n(101) become 0, U,U+V, andU+ 2Vrespectively.\nB. Ground State Calculation\n1. Spin 2\nWe proceed in a similar fashion as above. For spin 2,\nthere are \fve states for each value of Szcorresponding\nto the position of the hole, since there is only one spin\ncon\fguration for a given value of Szthat has spin 2. For\nSz= 2, these states are:\nj\"\"\"\" 0i;j\"\"\" 0\"i;j\"\"0\"\"i;j\"0\"\"\"i;j0\"\"\"\"i\n(103)\nIn this basis, the spin 2 Hamiltonian is given as follows:\nH2=\u0000ta0\nBBBBB@0 1 0 0\u00001\n1 0 1 0 0\n0 1 0 1 0\n0 0 1 0 1\n\u00001 0 0 1 01\nCCCCCA(104)\nHere the sign in the (1,5) elements is due to Fermi\nexchange statistics. There are two degenerate groundstates to this Hamiltonian given by:\nj\t\u0006\n2i=1p\n5\u0010\n1e\u0006\u0019i\n5e\u00062\u0019i\n5e\u00063\u0019i\n5e\u00064\u0019i\n5\u0011T\n(105)\nwith energy:\nE2=\u00001 +p\n5\n2ta (106)\n2. Spin 1\nWe now consider the spin 1 subspace. We de\fne the\nfollowing spin con\fgurations:\nj j\n1i=1\n2h\nj\"\"\"#i +ej\u0019i\n2j\"\"#\"i\n+e2j\u0019i\n2j\"#\"\"i +e3j\u0019i\n2j#\"\"\"ii\n(107)\nforjbetween 1 and 3. We see that cycling the spins\nwill return the same state with an extra phase ej\u0019i\n2. The\norbital part will be similar to the spin 2 case discussed\nabove, and thus the spin 1 Hamiltonian will be given by\na block-diagonal matrix, with blocks given as follows:\nHj\n1=\u0000ta0\nBBBBB@0 1 0 0\u0000ej\u0019i\n2\n1 0 1 0 0\n0 1 0 1 0\n0 0 1 0 1\n\u0000e\u0000j\u0019i\n20 0 1 01\nCCCCCA(108)\nThis has a nondegenerate ground state with energy\nE1=\u00002ta. The ground state has spin con\fguration\ngiven byj 2\n1i, and orbital part1p\n5(1 1 1 1 1)T.\n3. Spin 0\nFinally, we examine the spin 0 subspace. There are\ntwo spin con\fgurations, which we de\fne as follows:\nj 0\n0i=1\n2p\n3\u0014\n\u0000j\"\"##i + 2j\"#\"#i\u0000j\"##\"i\n\u0000j#\"\"#i + 2j#\"#\"i\u0000j##\"\"i\u0015\n(109)\nj 1\n0i=1\n2\u0014\nj\"\"##i\u0000j\"##\"i\u0000j#\"\"#i +j##\"\"i\u0015\n(110)\nWe note that cycling the spins of j j\n0ireturns the same\nstate with an additional phase ( \u00001)jj j\n0i. The the spin 0\nHamiltonian will be a block-diagonal matrix with blocks:17\nHj\n0=\u0000ta0\nBBBBB@0 1 0 0 (\u00001)j+1\n1 0 1 0 0\n0 1 0 1 0\n0 0 1 0 1\n(\u00001)j+10 0 1 01\nCCCCCA(111)\nThis also has a nondegenerate ground state with en-\nergyE0=\u00002ta. This state has spin con\fguration given\nbyj 1\n0i, and orbital part1p\n5(1 1 1 1 1)T.\n4. Finite UCorrections\nAs before, the spin 2 energy is exact for \fnite U, since\nthe Pauli exclusion principle forbids any other states than\nthe \fve examined. Additionally, since neither the spin 1\nnor spin 0 ground states are degenerate with other states\nof the same spin, we simply use nondegenerate perturba-\ntion theory to calculate the leading order correction to\nthe energy. We \fnd that to order t2=U, the energy of the\nlowest energy spin 1 state is given by:\nE1=\u00002ta\u00004t2\na\nU\u00002t2\na\nU+V\u00002t2\na\nU+ 2V+O\u0010t3\na\nU2\u0011\n(112)\nand the energy of the lowest energy spin 0 state is given\nby:\nE0=\u00002ta\u00002t2\na\nU\u0000t2\na\nU+V\u0000t2\na\nU+ 2V+O\u0010t3\na\nU2\u0011\n(113)\nThus, for \fnite U, the ground state of the system is\nthe spin 1 state. This means the ground state is partially\nferromagnetic.\nV. CONCLUSION\nWe have theoretically considered 4-dot quantum ar-\nrays in several di\u000berent geometries investigating ana-\nlytically within a simple, but semi-realistic, model the\nexistence or not of Nagaoka-type ferromagnetic ground\nstates. Our work includes distant-neighbor hopping and\ndistant-neighbor Coulomb coupling within a one orbital\n(with two spins) per dot model. Although the interac-\ntion is always \fnite in our system we \fnd several situa-\ntions where Nagaoka-type ferromagnetism should emergeprovided the kinetic and potential energies obey certain\nconstraints (which we derive). We calculate the spin gap\nfor our system, and obtain the di\u000berence in energies be-\ntween the ferromagnetic ground state and other nearby\nground states. We also provide results for a 5-dot ring\nwith 4 electrons, \fnding a partially ferromagnetic ground\nstate. We believe that our predictions are experimentally\ntestable in currently available quantum dot arrays as long\nas there is su\u000ecient control over the system (i.e. hopping\nmatrix elements, number of electrons in the system) and\nthe temperature is low. In principle, one can try to nu-\nmerically calculate the hopping and the interaction ma-\ntrix elements for a given system of coupled dots to make\nthe prediction quantitative. We, however, do not be-\nlieve that such an endeavor, which would be numerically\nvery demanding involving large con\fguration interaction\ncalculations19{21for the coupled dot system, is particu-\nlarly useful since the necessary information for the quan-\ntum con\fnement in each dot is unknown and therefore,\nthe results would be numerically unreliable. Since all the\nmatrix elements of hopping and interaction entering the\nmodel are likely to be exponentially sensitive to the un-\nknown dot con\fnement potential, our phenomenological\napproach using model parameters based on a delta func-\ntion con\fnement model is likely to have reasonable quali-\ntative accuracy. In particular, our speci\fc predictions on\nwhich geometry would lead to ferromagnetism and which\nwould not and the conditions necessary for obtaining full\nor partial ferromagetism in the ground states of di\u000berent\narrays should motivate experiments in current semicon-\nductor dot based qubit structures where the observation\nof di\u000berent types of nontrivial magnetic ground states\ncould be construed as quantum emulation of interacting\nHamiltonians in small systems. We think that the ex-\nperimental control already achieved in the laboratory for\nsemiconductor qubit systems should enable the commu-\nnity to see various magnetic ground states in quantum\ndot plaquettes as predicted in our theory.\nAcknowledgments\nThis work is supported by the Laboratory for Physical\nSciences.\n1J. P. Dehollain, U. Mukhopadhyay, V. P. Michal, Y.\nWang, B. Wunsch, C. Reichl, W. Wegscheider, M. Rud-\nner, E. Demler, & L. M. K. Vandersypen. arXiv:1904.05680\n(2019).\n2J. Hubbard, Proc. R. Soc. Lond. A 276, 1365, p. 238-257(1963).\n3J. Kanamori, Prog. Theor. Phys. 30, 3, p. 275-289 (1963).\n4M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963).\n5J. P. F. LeBlanc et al. , Phys. Rev. X 5, 041041 (2015).\n6Y. Nagaoka, Phys. Rev. 147, 392 (1966).18\n7H. Tasaki, Prog. Theor. Phys. 99, 4 (1998).\n8E. Bobrow, K. Stubis, & Y. Li. Phys. Rev. B 98, 180101\n(2018).\n9T. Hensgens, T. Fujita, L. Janssen, X. Li, C. J. Van\nDiepen, C. Reichl, W. Wegscheider, S. Das Sarma & L.\nM. K. Vandersypen. Nature 548, p. 70-73 (2017).\n10C. J. Van Diepen, P. T. Eendebak, B. T. Buijtendorp, U.\nMukhopadhyay, T. Fujita, C. Reichl, W. Wegscheider, & L.\nM. K. Vandersypen. Appl. Phys. Lett. 113, 033101 (2018).\n11H. G. J. Eenink, L. Petit, W. I. L. Lawrie, J. S. Clarke,\nL. M. K. Vandersypen & M. Veldhorst. arXiv:1907.08523\n(2019).\n12A. R. Mills, D. M. Zajac, M. J. Gullans, F. J. Schupp, T.\nM. Hazard & J. R. Petta. Nat. Comm. 10, 1063 (2019).\n13A. R. Mills, M. M. Feldman, C. Monical, P. J. Lewis, K.\nW. Larson, A. M. Mounce & J. R. Petta. arXiv:1907.10775\n(2019).14C. A. Sta\u000bord & S. Das Sarma. Phys. Rev. Lett. 72, 3590\n(1994).\n15R. Kotlyar, C. A. Sta\u000bord, & S. Das Sarma. Phys. Rev. B\n58, 3989 (1998).\n16R. Kotlyar, C. A. Sta\u000bord, & S. Das Sarma. Phys. Rev. B\n58, R1746 (1998).\n17Y. Wang, J. P. Dehollain, F. Liu, U. Mukhopadhyay,\nM. S. Rudner, L. M. K. Vandersypen & E. Demler.\narXiv:1907.01658 (2019).\n18A. V. Onufriev & J. B. Marston. Phys. Rev. B 59, 12573\n(1999).\n19X. Hu & S. Das Sarma. Phys. Rev. A 61, 062301 (2000).\n20X. Hu & S. Das Sarma. Phys. Rev. A 64, 042312 (2001).\n21E. Nielsen, E. Barnes, J. P. Kestner & S. Das Sarma. Phys.\nRev. B 88, 195131 (2013)."    },    {        "title": "0812.1703v1.Spin_Wave_Relaxation_in_a_Quantum_Hall_Ferromagnet.pdf",        "content": "arXiv:0812.1703v1  [cond-mat.mes-hall]  9 Dec 2008Spin-Wave Relaxation in a Quantum Hall Ferromagnet\nS. Dickmann1and S.L. Artyukhin1,2\n1Institute for Solid State Physics of RAS,\nChernogolovka 142432, Moscow District, Russia.\n2University of Groningen, Broerstraat 5, 9712 CP Groningen, N etherlands\n(Dated: February 5, 2020)\nWe study spin wave relaxation in quantum Hall ferromagnet re gimes. Spin-orbit\ncoupling is considered as a factor determining spin noncons ervation, and external\nrandom potential as a cause of energy dissipation making spi n-ﬂip processes irre-\nversible. We compare this relaxation mechanism with other r elaxation channels\nexisting in a quantum Hall ferromagnet.\nPACS numbers 73.21.Fg, 73.43.Lp, 78.67.De\n1.Last years are characterized by growing interest in spin relaxation (SR) in low-\ndimension systems — ﬁrst of all, in the relaxation in quantum dots stud ied within the\nprojects aimed at development of a computer employing spin memory . Yet, the relaxation\nof an electron spin in lateral quantum dots manufactured on the ba sis of two-dimensional\n(2D) heterostructures, should be in many respects similar to the S R of electrons localized in\nthe 2D layer in minima of a smooth random potential (SRP). In high mag netic ﬁelds this\nsingle-electron relaxation corresponds to the situation occurring at low Landau level (LL)\nﬁlling:ν≪1 or|ν−2n| ≪1 (nis an integer).1\nThe SR at diﬀerent ﬁling factors, ν>∼1, has quite diﬀerent nature representing in this\ncase a many-electron process. In particular, in a quantum Hall fer romagnet (QHF), i.e. at\nν= 1,3,...orν= 1/3,1/5,..., the SR reduces to the relaxation of lowest collective ex-\ncitations, i.e. spin waves.2,3The SR observation would thereby be a good tool to study\nfundamental collective properties of a strongly correlated 2D elec tron gas (2DEG). However,\nin spite of much recent interest in the SR in a 2DEG, up to now only a han dful of exper-\niments relevant to the SR in a QHF were performed: these are indirec t results based on\nthe linewidth measurements in the electron spin resonance,4and a direct observation where\nthe photoluminescence dynamics of spin-up and spin-down states w as studied.5Meanwhile,\navailability of the new time-resolved technique of photon counting allo ws us to believe that\nnew direct experiments on observation of excitations’ relaxation in a 2DEG, in particular of\nthe spin wave relaxation (SWR), will become available in the near futur e.6\nTheoretically the SWR in a QHF was studied in works 7,8. It is worth notin g here that2\ntheSWRrepresents actuallynotspindephasing buttheenergyrela xationduetothespin-ﬂip\nprocess. Indeed, any spin-ﬂip means at least dissipation of the Zee man energy ǫZ=|g|µBB\n(g≈ −0.44 in a GaAs structure). The latter is a part of the spin-wave (spin e xciton, SE)\nenergy\nEsw=ǫZ+Eq, (1)\nwhereEqis the SE correlation energy depending on the 2D wave vector q.2,3At variance with\nthe relaxation channel of Ref. 7 where electron-phonon interact ion was considered as the\nmechanism making the relaxation irreversible, and contrary to the c ase of Ref. 8 where the\nirreversibility was provided by an inter-spin-exciton interaction mec hanism, we now study\nsmooth disorder ﬁeld as the reason causing the energy transform . The SRP thereby deter-\nmines an alternative relaxation channel competing with the ones stu died earlier. Another\ndistinction of the present work from Refs. 7,8 consists in the study of not only the integer\nQHF (atν=1,3,...) but also of the fractional one ( ν=1/3,1/5,...) as well. At the same\ntime we again consider the spin-orbit coupling (SO) as the cause mixing diﬀerent spin states\nand therefore providing the spin nonconservation. Actually, vario us SWR channels coexist\nin parallel. We consider the total rate and ﬁnd crossover regions of external parameters\n(magnetic ﬁeld, temperature, etc.) where one relaxation channel ceases to be dominant and\nchanges into another.\nThe SR channel due to SRP was already considered in the integer qua ntum Hall ferro-\nmagnetic case.1,9However, studied in these works instead of the SWR was a speciﬁc SR\nwhen initially the total macroscopic spin /vectorSof the system as a whole is turned away from\nthe equilibrium direction parallel to /vectorB. (Relaxation of this Goldstone mode microscopically\nreduces to annihilation processes of the so-called zero SEs, having exactly zero momenta.)\nContrary to this case, the spin perturbation determined by excita tion of the spin waves\n(non-zero SEs) represents an initial deviation where ∆ S=∆Sz, so that/vectorSis kept parallel to\n/vectorBand the total symmetry of system remains unchanged.\nConcerning the origin of SRP, one should note that it has in the 2D laye r the “direct”\ncomponent and the eﬀective one. The former is the SRP determined by charged donors\nlocated outside the spacer. The latter is essential in some kinds of q uantum wells, being\ndetermined by spatial ﬂuctuations (in the plane of the layer) of qua ntum well width. These\nﬂuctuations lead to ﬂuctuations of the size-quantization energy a nd may be presented as an\nSRP term in the single electron Hamiltonian. Both SRP components hav e approximately\nthe same amplitude ∆ ∼10K and correlation length Λ ∼30−50nm.\n2.The total Hamiltonian has form Htot=/summationtext\njH(j)\n1+Hint, wherejenumerates electrons,3\nHintis thee-einteraction, and the single-electron operator is\nH1= ¯h2ˆq2/2m∗\ne−ǫZˆσz/2+HSO+ϕ(r). (2)\nIn this equation ϕ(r) is the SRP ﬁeld; the SO Hamiltonian is speciﬁed for the (001) GaAs\nplane,\nHSO=α(ˆq×ˆσ)z+β(ˆqyˆσy−ˆqxˆσx), (3)\npresenting a combination of the Rashba term and the crystalline anis otropy term10(ˆq=\n−i∇+eA/c¯his a 2D operator, σx,y,zare the Pauli matrices). If the SRP is assumed to be\nGaussian, then it is deﬁned by the correlator K(r) =/angb∇acketleftϕ(r)ϕ(0)/angb∇acket∇ight. By choosing /angb∇acketleftϕ(r)/angb∇acket∇ight= 0,\nin terms of the correlation length Λ and the LL width ∆ the correlator is\nK(r) = ∆2exp(−r2/Λ2). (4)\nWe ﬁrst ﬁnd the bare single-electron basis diagonalizing the Hamiltonia n (2) without the\nSRP ﬁeld. To within the leading order in the HSOterms we obtain\nΨpa=/parenleftBig\nψnp\nv√n+1ψn+1p+iu√nψn−1p/parenrightBig\n,\nΨpb=/parenleftBig\n−v√nψn−1p+iu√n+1ψn+1p\nψnp/parenrightBig (5)\nHereψnpis the electron wave function in the Landau gauge, nis the number of the half-ﬁlled\nLL in the odd-integer quantum Hall regime, i.e. in the ν=2n+1 case. Otherwise, if ν≤1, we\nsetn=0.uandvare small dimensionless parameters: u=β√\n2/lB¯hωcandv=α√\n2/lB¯hωc\n(ωcandlBare the cyclotron frequency and the magnetic length, respective ly). The single-\nelectron states thus cease to be purely spin states but acquire a c hiralityaorb. The spin\nﬂip corresponds thereby to the a→bprocess now.\nBy analogywith previous works1,7,8,9(see alsoRef. 11)we deﬁne the SEcreationoperator\nQ†\nabq=1/radicalbig\nNφ/summationdisplay\npe−iqxpb†\np+qy\n2ap−qy\n2, (6)\nwhereapandbpare the Fermi annihilation operators corresponding to states (5) ,Nφis\nthe LL degeneracy number. In Eq. (6) and everywhere below we me asure wave vector q\nin the 1/lBunits. If the ratio rc= (αe2/κlB)/¯hωcis considered to be small ( α<1 is the\naveraged formfactor which appears due to ﬁniteness of the layer thickness), and the SRP\nand SO terms in Eq. (2) are ignored, then the operator (6) acting o n the ground state in\ntheodd-integer quantum Hall regime yields the eigen state of the total Hamiltonian: namely,\n[Htot,Q†\nabq]|0/angb∇acket∇ight=(ǫZ+Eq)Q†\nabq|0/angb∇acket∇ight, where|0/angb∇acket∇ight=|Nφ/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n↑,↑,...↑/angb∇acket∇ight. This basic property of the exciton\nstate,Q†\nabq|0/angb∇acket∇ight, is the asymptotically exact one to the ﬁrst order in rc.4\nNow consider corrections arising due to the HSOterms. When presented in terms of basis\nstates(5),spinoperators/integraltext\nΨ†ˆS2Ψd2rand/integraltext\nΨ†ˆSzΨd2r[whereΨ=/summationtext\np(apΨpa+bpΨpb)]preserve\ninvariant form up to the second order in uandv. However, the interaction Hamiltonian\nHint=1\n2/integraltext\ndr1dr2Ψ†(r2)Ψ†(r1)U(r1−r2)Ψ(r1)Ψ(r2) acquires proportional to uandvterms\nwhich correspond to creation and annihilation of SEs in the system. I t is exactly these terms\nthat lead to the “coalescence” channel of the SWR.8In the present work we study another\nrelaxation channel. Therefore, neglecting this SO corrections to ˆHint, we focus on the SRP\nterm. Calculating/integraltext\nΨ†ϕ(r)Ψd2r, we get the terms responsible for a spin-ﬂip:\nˆϕ=N1/2\nφlB/summationdisplay\nqϕ(q)(iuq+−vq−)Qq+H.c. (7)\n(it is assumed here that q≪1).ϕ(q) is the Fourier component [i.e. ϕ=/summationtext\nqϕ(q)eiqr], and\nq±=∓i(qx±iqy)/√\n2.\nAt variance with integer QHF, the use of the excitonic basis Q†\nabq|0/angb∇acket∇ightpresents only a\nmodel approach in the case of fractional quantum Hall regime . Generally, spin-ﬂip excitations\nwithin the same Landau level might be many-particle rather than two -particle excitations\nat fractional ﬁlling because the same change of the spin numbers δS=δSz=−1 may be\nachieved with participation of arbitrary number of intra-spin-suble vel excitations (charge-\ndensity waves). These waves are generated by the operator A†\nq=N−1/2\nφQ†\naaqacting on the\nground state |0/angb∇acket∇ight=|νNφ/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n↑,..↑,..↑/angb∇acket∇ight.12It is trivial in the case of integer ν(A†\nq|0/angb∇acket∇ight=δq,0|0/angb∇acket∇ight);\nhowever, states of the Q†\nabq1A†\nq2A†\nq3...|0/angb∇acket∇ighttype might constitute a basis set if one studies a\nspin-ﬂip at fractional ν. On the other hand, a comprehensive phenomenological analysis3,12\nsuggests that even the spin-ﬂip basis reduced to single-mode (sing le-exciton) states would be\nquite appropriate, at least for lowest-energy excitations in the ca se of fractional QHF. This\nsingle-modeapproachis indirectly substantiated bythefact thatt hecharge-density wave has\na Coulomb gap12which is well larger than the Zeeman gap ǫZ. Hence for a fractional QHF,\njust as in Ref. 3, we will consider the only state Q†\nabq|0/angb∇acket∇ightto describe the spin-ﬂip excitation.\nThe commutation algebra for operators Q†\nabq,A†\nq′andB†\nq′′=N−1/2\nφQ†\nbbq′′is certainly the\nsame as for integer ﬁlling,7,8,9. However, a diﬀerence arises in the calculation of expectation\n/angb∇acketleft0|AqA†\nq′|0/angb∇acket∇ightwhich is needful for the following. This value is simply δq,0δq′,0at integer ﬁlling,\nbut atν<1 it is expressed in terms of the two-particle correlation function g(r) calculated\nfor the ground state:\n/angb∇acketleft0|AqA†\nq′|0/angb∇acket∇ight=ν\nNφ/bracketleftBig\n2πνg(q)eq2/2+1/bracketrightBig\nδq′,q. (8)\nHereg(q)=1\n(2π)2/integraltext\ng(r)e−iqrd2ristheFouriercomponent. Function g(r)iswell known, e.g., in\nthe case of Laughlin’s state.12,13If the ground state is presented in terms of the Hartree-Fock5\nmodel, we get the expression 2 πg=/parenleftBig\nNφδq,0−e−q2/2/parenrightBig\nwhich does not depend on ν. Besides,\nat odd-integer ﬁlling factors this Hartree-Fock expression becom es Fourier component of the\nexactcorrelation function. In the latter case one should also make the su bstitutionν→ν−2n\nin Eq. (8), i.e. formally set ν=1 there.\n3.The operator (7) obviously does not conserve the number of SEs. However, if the\nSWR is governed by this operator, the corresponding problem can n ot be solved in terms\nof a single-exciton study. Indeed, the SE interaction with the SRP in corporates the energy\nUx-SRP∼qlB∆/Λ (the SE possesses the dipole momentum elB[q×ˆz])2. The SE momentum\nis estimated from the condition Eq<∼T, and we therefore ﬁnd that Ux-SRP≪ǫZ, T. Due\nto this inequality, the energy of annihilating exciton can not be trans formed to anywhere.\nBy analogy with Ref. 8, we study a coalescence process where initial double-exciton state\n|i/angb∇acket∇ight=Q†\nabq1Q†\nabq2|0/angb∇acket∇ighttransforms to ﬁnal single-exciton state |f/angb∇acket∇ight=Q†\nabq′|0/angb∇acket∇ighthaving the combined\nenergy:\nǫZ+Eq′= 2ǫZ+Eq1+Eq2 (9)\n(c.f. also the Auger magnetoplasma relaxation considered in Ref. 14 ). At the same time,\ncontraryto Ref. 8, there is nomomentum conservation inthis SWR c hannel. Thus thephase\nvolume where the Xq1+Xq2→Xq′transition is possible turns out to be much larger than\nthat in the coalescence process of Ref. 8. This transition is govern ed by the Fermi golden\nrule probability: wfi= (2π/¯h)|Mfi|2δ(Ef−Ei), and our immediate task is to calculate the\nmatrix element Mfi=ν−3/2/angb∇acketleftf|ˆϕ|i/angb∇acket∇ight. (The factor ν−3/2appears due to the normalization since\nnorms of the |i/angb∇acket∇ightand|f/angb∇acket∇ightstates areν2andν, respectively.)\nWe perform the calculation for relevant values of momenta q1,q2,q′≪1 which satisfy\nthe conditions Eq1,Eq2<∼T<∼1K. (These inequalities correspond to q1,q2,q′≪1/lBin usual\ndimensional units). By employing exciton-operators’ commutation rules7and evident iden-\ntitiesQabq|0/angb∇acket∇ight≡Bq|0/angb∇acket∇ight≡0 and/angb∇acketleft0|Aq|0/angb∇acket∇ight≡ν, we obtain with the help of Eqs. (7)-(8) that\nMfi(q1,q2,q′)=2πν1/2\nN1/2\nφ/bracketleftBigg2/summationdisplay\nj=1g(|qj−q′|)e(qj−q′)2/2/bracketrightBigg/summationdisplay\nqϕ(q)(iuq+−vq−)δq1+q2,q+q′.(10)\nBesides, within our approximation, g(q)eq2/2should be replaced with g(q)eq2/2/vextendsingle/vextendsingle/vextendsingle\nq→0. The\nlatter quantity is equal to −1/2πin the Hartree-Fock approach or −1/2πνwhen calculated\ninthecaseofLaughlin’sgroundstatedescribing thefractionalQHF .So, forν=1,1/3,1/5,...,\nreplacing the terms in square brackets with −1/πν, we obtain a simple result:\n|Mfi(q1,q2,q′)|2=4πK(q)q2(u2+v2)\nνN2\nφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=q1+q2−q′. (11)6\nIt is used that the squared modulus of ϕ(q) may be expressed in terms of Fourier component\nof the correlator (4): |ϕ(q)|2=2πK(q)/Nφ. In the Hartree-Fock model the expression (11)\nshould be multiplied by ν2; therefore the calculated relaxation rate would be by a factor of\nν2slower. Notice also that if ν=3,5,..., one should formally set ν=1 in Eqs. (10) and (11).\nThe SWR rate is deﬁned as the diﬀerence between the ﬂuxes of annih ilating and cre-\nated SEs. We assume that the thermodynamic equilibrium in the syste m of spin waves is\nestablished much faster than the spin-ﬂip processes occur so tha t the rate is\nR=1\n2/summationdisplay\nq1,q2,q′2π\n¯h|Mfi(q1,q2,q′)|2δ(E1+E2−E′)[n1n2(1+n′)−n′(1+n1)(1+n2)].(12)\nThe notations used here are Ei=ǫZ+Eqi, ni=n(Ei) (i= 1,2) andE′=ǫZ+Eq′, n′=n(E′),\nwhere the Bose distribution function is n(E) = 1/(e(E−µ)/T−1). The rate Ris completely\ndetermined by Eqs. (11)-(12) and is a function of parameters B,T, and of the total number\nof SWs in the system: Nx=/summationtext\nqn(ǫZ+Eq). In our case, when temperature is rather low, we\ncan certainly use quadratic approximation for the “kinetic” exciton energy:Eq≈q2/2Mx.\nChemical potential µis determined by the ratio of the exciton number and the total spin:\nNx(µ) =νNφ/2−S. Calculating the quantity N(0)\nx=Nx/vextendsingle/vextendsingle/vextendsingle\nµ=0, one obtains the equilibrium\nnumber of excitons. We will ﬁnd the rate at the ﬁnal stage of the re laxation process where\nNx−N(0)\nx≪N(0)\nx. So,byemployingthequadraticapproximationfortheSEkineticene rgy, and\nchanging in Eqs. (11)-(12) from summations to integrations we obt ainR=/parenleftBig\nNx−N(0)\nx/parenrightBig\n/τsrp,\nwhere\n1/τsrp=(u2+v2)M3\nx\n2νπ¯h/parenleftbigg∆ΛT\nlB/parenrightbigg2/parenleftbig\ne−ǫZ/T−e−2ǫZ/T/parenrightbig\nFSRP(Λ2MxT/l2\nB, ǫZ/T).(13)\nHereFSRP(α, β) is a dimensionless function arising as a result of integrations over q1andq2\nand averaging over angles θ1=q1∧q′andθ2=q2∧q′:\nFSRP(α, β) =/integraldisplay∞\n0/integraldisplay∞\n0e−x−ydxdy\n(1−e−x−β)(1−e−y−β)(1−e−x−y−2β)\n×/integraldisplayπ\n−πdθ1/integraldisplayπ\n−πdθ2r(x,y,θ1,θ2)exp[−αr(x,y,θ1,θ2)],\nwherer(x,y,θ1,θ2)=x+y+β/2−√x+y+β(√xcosθ1+√ycosθ2)+√xycos(θ1−θ2).\n4.Now we calculate the numerical value of 1 /τsrpat typical SRP parameters and com-\npare it with inverse relaxation times 1 /τe−eand 1/τphgoverned by the inter-SEs’ interaction\nmechanism8and the SE-acoustic-phonon coupling.7We carry out this analysis for the ν=1\nQHF assuming that ∆ = 10K and Λ = 40nm. The Zeeman splitting at g=−0.44 is\nǫZ=0.295BK (Bis everywhere in Teslas), and the combination of SO parameters is es ti-\nmated asu2+v2= 10−3/B. The SE mass Mxmight be calculated theoretically by using7\ngeneral expressions for Eq.2,3Yet, the result depends on speciﬁc formfactor inherent in a\ngiven heterostructure due to ﬁnite thickness and it is therefore m ore convenient to extract\nMximmediately from experiments. According to recent data available fo r currently used\nwide quantum wells,15,16we estimate that 1 /Mx= 9.24√\nBK. Using Eq. (13), we thus\ncalculate 1/τsrpas a function of temperature Tat given ﬁeld B. The results are presented\nin Fig. 1 by dash curves. The dot and dash-dot curves correspond to the 1/τe−eand 1/τph\nvalues given by formulas17\n1/τe−e=2\n¯h(u2+v2)T/parenleftbig\ne−ǫZ/T−e−2ǫZ/T/parenrightbig\nFe−e(ǫZ/T), (14)\nwhere\nFe−e(β) =/integraldisplay/integraldisplay\nxy>β2/4dxdy(x+y+β)e−x−y\n(xy−β2/4)1/2(1−e−β−x)(1−e−β−y)(1−e−2β−x−β);\nand\nτ−1\nph=MTǫZ(u2+v2)\n¯hcsp3\n0l2\nB/bracketleftBigg\nγ1(ǫZ/T)\nτD+10MT\nτP/parenleftbigg¯hcs\nǫZ/parenrightbigg4/parenleftbiggp0\nlB/parenrightbigg2\nγ2(ǫZ/T)/bracketrightBigg\n,(15)\nwhere\nγk(β) = (e2β−eβ)/integraldisplay∞\n0exxkdx\n(eβ+x−1)2, k=1,2.\n(See Ref. 7; the used material parameters characterizing the ele ctron-phonon coupling are\ncs=5.14·105cm/s,τD=0.8·10−12s−1,τP=35·10−12s−1, andp0= 2.52·106cm−1; both kinds\nofe-phinteraction, deformation and polarization ones, are taken into acc ount.)\nIt is seen from Fig. 1 that the SRP relaxation channel actually compe tes with other\nmechanisms in the experimentally relevant range of parameters: na mely, at ﬁelds B≤5\nand temperatures T∼0.3−0.5K. We have indicated above that the basic advantage of the\nSRP channel, as compared to the e-eone, consist in the absence of momentum conservation\nin the coalescence process. On the other hand, the SRP mechanism s is also determined by\neﬀective SE-SE collisions. Therefore the inverse relaxation time is pr oportional to the SE\nconcentration and drops exponentially as ∼exp(−ǫZ/T) with vanishing T[rather than as ∼\nexp(−2ǫZ/T) which occurs for the e-emechanism due to the SEs momentum conservation!].\nThe phonon mechanism of SWR dominates at low temperatures due to its weak temperature\ndependence ( ∼T), in spite of small value of the electron-phonon coupling constant in GaAs.\nThe dependence on the ﬁlling factor in the case of integer QHF is only d etermined by the\nSE massMxbecauseνin Eq. (13) is formally set equal to unit. For fractional QHF there\nare both direct and indirect (through the mass Mx) dependences on ν.\nFinally we calculate the combined inverse relaxation time determined by the SO interac-\ntion:\n1/τtot= 1/τsrp+1/τe−e+1/τph (16)8\nTheresultispresentedbysolidcurvesinFig. 1. Itisworthmentioning thatitdemonstratesa\ngoodagreementwiththemeasuredvalue τtot≃10nsofRef. 5whencalculatedforparameters\nBandTcorresponding to the experiment.\nThe authors acknowledge support of the RFBR and hospitality of th e Max Planck Insti-\ntute for Physics of Complex Systems (Dresden) where this work wa s partly carried out. The\nauthors also thank S.V. Iordanskii and L.V. Kulik for discussion.\n1S. Dickmann, JETP Lett. 78, 452 (2003).\n2Yu.A. Bychkov, S.V. Iordanskii, and G.M. Eliashberg, JETP L ett.33, 143 (1981); C. Kallin\nand B.I. Halperin, Phys. Rev. B 30, 5655 (1984).\n3J. P. Longo and C. Kallin, Phys. Rev. B 47, 4429 (1993).\n4M. Dobers, K.v. Klitzing, and G. Weimann, Phys. Rev. B 38, 5453 (1988); M. Dobers et al.,\nPhys. Rev. Lett. 61, 1650 (1988).\n5V.E. Zhitomirskii et al., JETP Lett. 58, 439 (1993).\n6L.V. Kulik, private communication.\n7S. Dickmann and S.V. Iordanskii, JETP 83, 128 (1996).\n8S. Dickmann and S.V. Iordanskii, JETP Lett. 70, 543 (1999).\n9S. Dickmann, Phys. Rev. Lett. 93, 206804 (2004).\n10Yu.A. Bychkov and E.I. Rashba, JETP Lett. 39, 78 (1984); M.I. D’yakonov and V.Yu. Ka-\nchorovskii, Sov. Phys. Semicond. 20, 110 (1986).\n11A.B. Dzyubenko and Yu.E. Lozovik, Sov. Phys. Solid State 25, 874 (1983) [ ibid.26, 938 (1984)].\n12S.M. Girvin, A.H. MacDonald, and P.M. Platzman, Phys. Rev. B 33, 2481 (1986).\n13S.M. Girvin, Phys. Rev. B 29, 6012 (1984).\n14S. Dickmann and Y. Levinson, Phys. Rev. B 60, 7760 (1999).\n15Y. Gallais et al., Phys. Rev. Lett. 100, 086806 (2008).\n16I.V. Kukushkin et al., Phys. Rev. Lett. 96, 126807 (2006).\n17The corresponding formula for inverse relaxation time in Re f. 8 contains a misprint. Now we\npresent the corrected result in Eq. (14).9\nInverse   relaxation   times   [1/s]\nFIG. 1: Inverse SWR times against Tcalculated by using formulas (13)-(15) at B= 3,5,10T.\nSpeciﬁc material parameters are given in the text. Dash, dot , and dash-dot lines are for 1 /τsrp,\n1/τe−eand 1/τph, respectively. Solid lines present the result of calculati on of the combined inverse\ntime (16)."    },    {        "title": "2007.01700v1.Unsaturated_bipartite_entanglement_of_a_spin_1_2_Ising_Heisenberg_model_on_a_triangulated_Husimi_lattice.pdf",        "content": "arXiv:2007.01700v1  [cond-mat.stat-mech]  3 Jul 2020Vol. XXX (20XX) CSMAG’19 No. X\nUnsaturated bipartite entanglement of a spin-1/2\nIsing-Heisenberg model on a triangulated Husimi lattice\nC. EKIZ∗1and J. STRE ˇCKA2\n1Faculty of Science and Art, Aydın Adnan Menderes University, 0901 0 Aydın, Turkey\n2Faculty of Science, P. J. ˇSaf´ arik University, Park Angelinum 9, 04001 Koˇ sice, Slovakia\nA bipartite entanglement between two nearest-neighbor Heisenbe rg spins of a spin-1/2 Ising-\nHeisenberg model on a triangulated Husimi lattice is quantiﬁed using a concurrence. It is shown\nthat the concurrence equals zero in a classical ferromagnetic and a quantum disordered phase,\nwhile it becomes sizable though unsaturated in a quantum ferromagn etic phase. A thermally-\nassisted reentrance of the concurrence is found above a classica l ferromagnetic phase, whereas a\nquantumferromagneticphasedisplaysastrikingcuspoftheconcu rrenceatacriticaltemperature.\nKeywords: Ising-Heisenberg model, Husimi lattice, geometric frustration, en tanglement\n1.Introduction\nThepolymericcompoundCu 9Cl2(cpa)6·nH2O(cpa=carboxypentonicacid)hasrecentlyattracted\na lot of attention, because it does not order down to the lowest exp erimentally reached temper-\natures due to a geometric spin frustration of the underlying triang ulated kagom´ e lattice [1, 2].\nBecause of intractability of the respective spin-1/2 Heisenberg mo del on a triangulated kagom´ e\nlattice we have proposed and exactly solved a simpler spin-1/2 Ising- Heisenberg model on related\ntriangulated (triangles-in-triangles) structures with the aim to br ing insight into unconventional\nmagnetism of this highly frustrated magnetic material [3, 4]. From th is perspective, the spin-1/2\nIsing-Heisenbergmodel on a triangulated kagom´ elattice [3] and it s related recursivetriangulated\nHusimi counterpart [4] aﬀords a long sought-after playground fo r a theoretical investigation of\nthe quantum entanglement, which is eligible also for an experimental t esting.\n∗Corresponding author: cekiz@adu.edu.tr2 Template for Calculation of Length of Manuscript ...\nJH\nJ JI IR R\nR\nFig. 1: The spin-1/2 Ising-Heisenberg model on a triangulated Husim i lattice (left-hand-side)\nand its rigorous mapping to the eﬀective spin-1/2 Ising model on a tr iangular Husimi lattice\n(right-hand-side). Filled (empty) circles denote lattice positions of the Heisenberg (Ising) spins.\n2.Ising-Heisenberg model on a triangulated Husimi lattice\nThe spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattic e schematically illustrated\non the left-hand-side of Fig. 1 can be deﬁned through the Hamiltonia n\nˆH=−JH2N/summationdisplay\n/angbracketleftk,l/angbracketright[∆(ˆSx\nkˆSx\nl+ˆSy\nkˆSy\nl)+ˆSz\nkˆSz\nl]−JI4N/summationdisplay\n/angbracketleftk,j/angbracketrightˆSz\nkˆσz\nj,\nwhere ˆσz\njandˆSα\nk(α=x,y,z) label spatial components of the usual spin-1/2 operator assign ed\nto the Ising and Heisenberg spins, respectively, Ndenotes the total number of the Ising spins,\nthe parameter JHis the XXZ interaction between the nearest-neighbor Heisenberg s pins, ∆ is\nan exchange anisotropy, and the parameter JIlabels the Ising interaction between the nearest-\nneighbor Heisenberg and Ising spins. The overall magnetic structu re of the triangulated Husimi\nlattice form smaller triangles of the Heisenberg spins (trimers), whic h are embedded into larger\ntriangles of the triangular Husimi lattice involving in its nodal lattice sit es the Ising spins.\nIt has been previously proved [4] that the generalized star-triang le transformation provides\nan exact mapping correspondence between the partition function s and associated free energies of\nthe spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattic e and the eﬀective spin-1/2\nIsing model on a triangular Husimi lattice shown on right-hand-side o f Fig. 1\nFIHM(β,JH,JI,∆) =FIM(βR)−2\n3lnA. (1)\nHere,β= 1/(kBT),kBis Boltzmann’s constant, Tis absolute temperature and two mapping3 Template for Calculation of Length of Manuscript ...\nparameters AandβRare given by Eqs. (6)-(9) of Ref. [4]. The free energy of the eﬀec tive spin-\n1/2 Ising model on a triangular Husimi lattice can be found through e xact recursive relations\nFIM=β−1[2ln(eβR+2x+x2)−ln(1+x2)−βR/2]. (2)\nTheparameter xcanbeobtainedbysolvingtherecursiverelation(Eq. (13)inRef. [4]) iteratively\nor by solving the polynomial equation x3+(2−eβR)x2+(eβR−2)x−1 = 0 with the roots\nx1,2=1\n2/bracketleftbigg\neβR−3±/radicalBig\n(eβR−5)(eβR−1)/bracketrightbigg\n, x3= 1. (3)\nIt is noteworthy that the ﬁrst two solutions x1,2correspond to a spontaneously ordered phase\nwith two opposite signs of the spontaneous magnetization, while the third solution x3= 1\ncorresponds to a disordered paramagnetic phase without any long -range order. Hence, it follows\nthat the critical temperature of the eﬀective spin-1/2 Ising mode l on a triangular Husimi lattice\nis given by the condition βcR= ln5 being consistent with a coalescence of all three roots\nx1=x2=x3= 1, which also represents the critical condition for the spin-1/2 Is ing-Heisenberg\nmodel on a triangulated Husimi lattice due to the mapping relation (1) between the free energies.\nThe main goal of this work lies in a rigorous analysis of entanglement. A bipartite entangle-\nment between two nearest-neighbor Heisenberg spins can be quan tiﬁed via concurrence [5]\nC= max\n\n0,4|Cxx\nHH|−2/radicalBigg/parenleftbigg1\n4+Czz\nHH/parenrightbigg2\n−m2\nH\n\n, (4)\nwhich can be expressed in terms of three local observables, namely , two spatial components\nof the pair correlation function Cxx\nHH=/angbracketleftˆSx\nk,iˆSx\nk,i+1/angbracketright,Czz\nHH=/angbracketleftˆSz\nk,iˆSz\nk,i+1/angbracketrightand the sublattice\nmagnetizationofthe Heisenbergspins mH=/angbracketleft(ˆSz\nk,i+ˆSz\nk,i+1)/2/angbracketright. An exactresultforthesublattice\nmagnetization mHwas already reported in Ref. [4] [see Eq. (19)], while both spatial com ponents\nof the pair correlation function can be calculated from Eq. (1) acco rding to the formulas\nCxx\nHH=−1\n4∂FIHM\n∂JH∆, Czz\nHH=−1\n2∂FIHM\n∂JH. (5)4 Template for Calculation of Length of Manuscript ...\nThe ﬁnal formulas for Cxx\nHHandCzz\nHHare too complex to write them down here explicitly.\n3.Results and Discussion\nLet us explore the bipartite entanglement of the spin-1/2 Ising-He isenberg model on a trian-\ngulated Husimi lattice, whereas our further attention will be restr icted to the model with the\nferromagnetic coupling constant JI>0 because the antiferromagnetic counterpart JI<0 causes\na mere ﬂip of all Ising spins. First, we will brieﬂy comment on all possible ground states of\nthe spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattic e, which have been already\nreported in Ref. [4] and can be classiﬁed either as the classical ferr omagnetic (CF) phase\n|CF/angbracketright=N/productdisplay\ni=1|↑/angbracketrightσz\ni2N/3/productdisplay\nk=1|↑↑↑/angbracketrightSz\nk1,Sz\nk2,Sz\nk3, (6)\nor the quantum ferromagnetic (QF) phase\n|QF/angbracketright=N/productdisplay\ni=1|↑/angbracketrightσz\ni2N/3/productdisplay\nk=11√\n3(|↑↑↓/angbracketright+|↑↓↑/angbracketright+|↓↑↑/angbracketright)Sz\nk1,Sz\nk2,Sz\nk3, (7)\nor the highly degenerate ground-state manifold further referre d to as the quantum disordered\n(QD) phase. The QD phase emerges in the frustrated parameter s paceJH/JI<−2/(2+∆), the\nQF phase is realized whenever ∆ >1 andJH/JI>1/(∆−1), while the CF phase is allocated in\nthe parameter region bounded by the inequalities JH/JI>−2/(2+∆) and JH/JI<1/(∆−1).\nThe sublattice magnetizations of the Ising and Heisenberg spins ( mI,mH), the correlation\nfunctions ( Cxx\nHH,Czz\nHH) and the concurrence Care plotted against temperature in Fig. 2(a)-(d).\nThermal variations displayed in Fig. 2(a)-(b) show qualitative similarit ies, since the selected\nparameters coincide with the CF ground state. It is evident that th e concurrence may display\na striking thermally-assisted reentrance, which is much more prono unced for the ferromagnetic\nHeisenberg coupling [ JH>0, Fig. 2(b)] than the antiferromagnetic one [ JH<0, Fig. 2(a)].\nFig. 2(c) depicts typical behavior at a coexistence point of the CF a nd QF ground states, while\nFig. 2(d) displays typical temperature dependences when startin g from the QF ground state.\nUnder this condition, the concurrence decreases upon increasing temperature until it reaches an5 Template for Calculation of Length of Manuscript ...\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s67/s32\n/s32\n/s67/s32/s120/s120\n/s72/s72/s32/s67/s32/s122/s122\n/s72/s72/s32/s109\n/s72/s32/s109\n/s73/s32\n/s32/s32\n/s32/s40/s97/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s74\n/s73/s109\n/s73/s32/s44/s32/s109\n/s72/s32/s44/s32 /s67/s32/s122/s122 /s72/s72/s32/s44/s32 /s67/s32/s120/s120 /s72/s72/s32/s44/s32 /s67 /s32\n/s32/s61/s32/s49/s74\n/s72/s32/s47/s32 /s74\n/s73/s32/s61/s32/s45/s48/s46/s54 /s32/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52/s48/s46/s48/s48/s54\n/s32/s32/s32\n/s32\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s67/s32\n/s32\n/s67/s32/s120/s120\n/s72/s72/s32/s67/s32/s122/s122\n/s72/s72/s32/s109\n/s72/s32/s109\n/s73/s32\n/s32/s32\n/s32/s40/s98/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s74\n/s73/s109\n/s73/s32/s44/s32/s109\n/s72/s32/s44/s32 /s67/s32/s122/s122 /s72/s72\n/s32/s44/s32 /s67/s32/s120/s120 /s72/s72/s32/s44/s32 /s67 /s32/s32/s61/s32/s50/s74\n/s72/s32/s47/s32 /s74\n/s73/s32/s61/s32/s48/s46/s57\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s67/s32\n/s32\n/s67/s32/s120/s120\n/s72/s72/s32\n/s67/s32/s122/s122\n/s72/s72/s32/s109\n/s72/s32/s109\n/s73/s32\n/s32/s32\n/s32/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/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s74\n/s73/s109\n/s73/s32/s44/s32/s109\n/s72/s32/s44/s32 /s67/s32/s122/s122 /s72/s72\n/s32/s44/s32 /s67/s32/s120/s120 /s72/s72/s32/s44/s32 /s67 /s32/s32/s61/s32/s50/s74\n/s72/s32/s47/s32 /s74\n/s73/s32/s61/s32/s49/s46/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s67/s32\n/s32\n/s67/s32/s120/s120\n/s72/s72/s32\n/s67/s32/s122/s122\n/s72/s72/s32/s109\n/s72/s32/s109\n/s73/s32\n/s32/s32\n/s32/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/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s74\n/s73/s109\n/s73/s32/s44/s32/s109\n/s72/s32/s44/s32 /s67/s32/s122/s122 /s72/s72\n/s32/s44/s32 /s67/s32/s120/s120 /s72/s72/s32/s44/s32 /s67 /s32/s32/s61/s32/s50/s74\n/s72/s32/s47/s32 /s74\n/s73/s32/s61/s32/s49/s46/s50\nFig. 2:(Color online) Typical thermal variations of the sublattic e magnetizations of the Ising ( mI) and\nHeisenberg ( mH) spins, the correlation functions ( Cxx\nHH,Czz\nHH) and the concurrence Cfor a few selected\nsets of the interaction parameters. The inset in Fig. 2(a) sh ows concurrence in enlarged scale.\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s67/s80/s32/s61/s32/s49/s46/s48\n/s67\n/s81/s68 /s81/s70/s67/s70\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/s74\n/s72/s32/s47/s32 /s74\n/s73/s107\n/s66/s32/s84 /s32/s47/s32 /s74\n/s73\n/s48/s46/s48/s48/s48/s48/s46/s48/s48/s49/s48/s46/s48/s48/s50/s48/s46/s48/s48/s51/s48/s46/s48/s48/s52/s48/s46/s48/s48/s53\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s81/s68/s67/s80/s32/s61/s32/s50/s46/s48\n/s67\n/s81/s80\n/s81/s70/s67/s70\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/s74\n/s72/s32/s47/s32 /s74\n/s73/s107\n/s66/s32/s84 /s32/s47/s32 /s74\n/s73\n/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\nFig. 3:(Color online) A density plot of the concurrence in the JH/JI−kBT/JIplane for two diﬀerent\nvalues of the exchange anisotropy: (a) ∆ = 1; (b) ∆ = 2. Dotted l ines separate the entangled region\n(C >0) from the disentangled one ( C= 0), while solid lines display a critical temperature assoc iated\nwith a breakdown of the spontaneous long-range order.\noutstanding cusp at a critical temperature associated with break down of the spontaneous long-\nrangeorder,whichissuccessivelyfollowedbyagradualdeclineendin gatathresholdtemperature.\nLet us summarize our ﬁndings by constructing global phase diagram s of the spin-1/2 Ising-\nHeisenberg model on a triangulated Husimi lattice for two values of t he exchangeanisotropy ∆ =\n1and2. Tothisend, adensityplot ofthe concurrenceisshownin Fig. 3(a)-(b)in JH/JI−kBT/JI\nplane along with a critical temperature connected with a breakdown of the spontaneous order. It\nfollows from Fig. 3(a)-(b) that a weak entanglement can be found w ithin the CF phase close to a\nphase boundary either with the QD or QF phase. However, the prep onderant entanglement can\nbe detected in the QF phase, which exhibits a sizable drop of the conc urrence around the critical\ntemperature successivelyfollowed by a more gradualthermally-as sisteddecline. The concurrence6 Template for Calculation of Length of Manuscript ...\nthus survives far above the critical temperature of the QF phase .\n4.Conclusions\nIn the present work, the quantum entanglement of the spin-1/2 I sing-Heisenberg model on a\ntriangulated Husimi lattice has been examined in detail. Exact results for the sublattice magne-\ntization and two spatial components of the pair correlation functio n were employed in order to\ncalculate the quantum concurrence, which serves as a measure of the bipartite entanglement be-\ntween the nearest-neighbor Heisenberg spins. It has been found that the bipartite entanglement\nis totally absent within the CF and QD ground states, while it becomes s izable though unsatu-\nrated within the QF groundstate. Strikingly, a thermally-assistedr eentrance ofa relativelyweak\nbipartite entanglement (concurrence) can be detected above th e CF ground state in a vicinity\nof phase boundary either with the QF or QD ground states. In addit ion, it turns out that the\nthreshold temperature, above which the bipartite entanglement v anishes, may seemingly exceed\nthe critical temperature of the QF phase accompanied with a cusp in the relevant temperature\ndependence of the concurrence.\n5.Acknowledgement\nThis work was supported under the grant Nos. VEGA 1/0531/19 an d APVV-16-0186.\nReferences\n[1] H.S.C. Hamilton, W.M. Farmer, S.F. Skinner, L.W. ter Haar, AIP Adv. 8, 055802 (2018).\nDOI: 10.1063/1.5006791.\n[2] W.M. Farmer, S.F. Skinner, L.W. ter Haar, AIP Adv. 8, 101404 (2018). DOI:\n10.1063/1.5042777\n[3] J. Streˇ cka, L. ˇCanov´ a, M. Jaˇ sˇ cur, M. Hagiwara, Phys. Rev. B 78, 024427 (2008). DOI:\n10.1103/PhysRevB.78.024427\n[4] J. Streˇ cka, C. Ekiz, Phys. Rev. E 91, 052143 (2015). DOI: 10.1103/PhysRevE.91.0521437 Template for Calculation of Length of Manuscript ...\n[5] L. Amico, A. Osterloh, F. Plastina, R. Fazio, G.M. Palma, Phys. Rev. A 69, 022304 (2004).\nDOI: 10.1103/PhysRevA.69.022304"    },    {        "title": "2204.01551v2.Hole_doping_induced_ferromagnetism_in_2D_materials.pdf",        "content": "Hole\n-\ndop\ning\n \ninduced ferromagnetism in 2D materials\n \n \nR. Meng\n1\n*\n, L.\nM.C. \nPereira\n1\n, J.P. Locquet\n1\n, V .V . Afanas’ev\n1\n, G. Pourtois\n2\n, and M. Houss\na\n1\n,2\n*\n \n \n1\nDepartment of Physics and Astronomy, KU Leuven, Celestijnenlaan 200D, Leuven B\n-\n3001, Belgium\n \n2\nimec, Kapeldreef 75, B\n-\n3001 Leuven, Belgium\n \n*\nEmail: ruishen.meng@kuleuven.be; michel.houssa@kuleuven.be\n \n \n \nAbstract\n \n \nT\nwo\n-\ndimensional (2D) ferromagnetic materials \nare considered\n \nas \npromising\n \ncandidates for the \nfuture \ngenerations of \nspintronic \ndevices\n. Yet, 2D materials with intrinsic ferromagnetism are scarce.\n \nHigh\n-\n \nthroughput first\n-\nprinciples simulations are performed in \norder to screen \n2D materials that \npresent\n \na \nnon\n-\nmagnetic to \na \nferromagnetic transition\n \nupon hole doping. \nA \nglobal evolutionary search is\n \nsubsequently \nperformed, in order to identify\n \nalternative \npossible atomic structures \nof \nthe eligible candidates\n, and \n122\n \nmaterials\n \nexhibiting a \nhole\n-\ndoping induced \nferromagnetism\n \nare\n \nidentified. \nT\nheir energetic and dynamic \nstability, \nas well as their \nmagnetic properties under hole doping \nare\n \ninvestigated \nsystematically\n.\n \nHalf of these \n2D materials are metal halides, \nfollowed by chalcogenides, oxides and nitrides, some of them having predicted \nCurie temperature\ns\n \nabove 300 K. The exchange interactions\n \nresponsible for the ferromagnetic order in these \n2D materials are also discussed. This work not only provides theoretica\nl insight\ns\n \ninto hole\n-\ndoped 2D \nferromagnetic materials, but also enriches the family of\n \n2D magnetic materials for possible spintronic \napplications. \n \n \nIntroduction\n \n \nTwo\n-\ndimensional \nmagnetic materials \nare \nserving as \na \npowerful platform for the investigation and \nunderstandin\ng\n \nof magnetism at \nthe \nultima\nte 2D limit\n.\n[1]\n \nO\nwing to the atomic thickness and controllable \nelectron\n-\nspin degree of freedom\n, t\nhey are also considered to be \npromising\n \ncandidates \nfor the next\n-\ngeneration \nspintronic\n \ndevices\n.\n \nEspecially\n, \n2D\n \nferromagnetic materials\n \nwith semiconducting \nor \nhalf\n-\nmetallic properties \n(\nmetallic \non one spin channel and \nsemiconducting \non the other spin channel)\n \ndraw \nparticular interest\n.\n[2]\n \nBy \nmeans of theoretical calculations, a variety of 2D \nferromagnetic\n \nmaterials\n \nhave been discovered.\n[3]\n \nSome of \nthese \n2D\n \nmaterials \nhave bee\nn successfully synthesized and \ntheir fascinating magnetic properties have been explored subsequently\n.\n[4]\n \nFor instance, 2D \nCrI\n3\n \nand Cr\n2\nGe\n2\nTe\n6\n \nwere \npredicted to be intrinsic ferromagnetic \nsemiconductors\n,\n \na prediction \nwhich w\nas later \nconfirmed \nexperiment\nally\n.\n[5]\n \nFurther investigation sho\nws that\n \n2D \nCrI\n3\n \nused as \na \nspin\n-\nfilt\ner tunnel barrier sandwiched between \ngraphene contacts can have ultra\n-\nhigh \ntunneling magnetoresistance.\n[6]\n \nHowever, \nthe\n \nscarcity\n \nof\n \n2D \nf\nerromagnetic\n \nsemiconductors\n/\nhalf\n-\nmetals\n \nand \nthe\nir\n \nrather \nlow \nCurie temperature (\nT\nc\n)\n \nm\nay \nhamper\n \npractical application\ns\n \nas well as \nthe \nfurther investigation \nof 2D magnetism. \nContinuous\n \nseeking\n \nfor more 2D \nferromagnetic\n \nsemiconducting\n \nor \nhalf\n-\nmetallic materials \nwith relatively high \nCurie temperature\n \nis \nnecessary\n.\n \nIn addit\nion to these \nintrinsic \n2D \nmagnetic \nm\naterials,\n \nthere \nar\ne also\n \nstudies\n \nwhich focus \non inducing \nmagnetism \nin\nto\n \nnon\n-\nmagnetic \n2D \nma\nterials. Quite a\n \nfew theoretical \nreports\n \nhave sho\nwn that 2D III\n-\nVI (gallium \noxides/chalcogenides and indium oxides/chalcogenides)\n[7]\n \nand IV\n-\nVI (tin oxides/chalcogenides and lead \noxides)\n[8]\n \nsemiconductors, and \nsome\n \nother 2D materials such as InP\n3\n, etc.\n[9]\n \nwould exhibit non\n-\nmagnetic\n \nto \nferroma\ngnetic\n \nproperties \nupon\n \nhole \ndoping.\n \nF\nerromagn\netism in \nthe\nse\n \nhole\n-\ndoped 2D\n \nmaterials aris\nes from an \nexchange \nsplitting of electronic states at the top of the valence band, where t\nhe\n \ndensity of states (\nDOS\n)\n \nexhibits \na sharp \nV\nan\n \nHove\n \nsingula\nrity\n, \nt\nhat would lead\n \nto\n \nan electronic\n \ninsta\nbility\n.\n \nControlling the magnetic state of \nthese materials by \ne.g.,\n \na gate bias has potential interesting applications for novel spintronic devices.  \n \nUnlike most of intrinsic magnetic mater\nials\n, \nwhose magnetic moments\n \nare mainly origin\nated from\n \nunsaturated \nd\n \norbitals of\n \ntransition metal\ns\n, \nthe \nmagnetic moments of these hole\n-\ndoped 2D ferromagnetic \nmaterials are mainly contributed by the delocalized \np\n \norbitals of anions, which could be \nadvantageous\n \nfor \nthe \nlong\n-\nrange \nferromagnetic\n \norder.\n \nHowever, a \nsystematic investigation of the magnetic properties of these 2D \nmaterials (spin\n-\npolarization energies, magnetic exchange coupling strength, magnetic anisotropy, and Curie \ntemperature) is lacking. Besides, the mechanism responsible for the exchange coupling\n \nand delocalization of \nthe spin\n-\npolarized holes has not been discussed, to our knowledge. In addition, \nit is also \nof significance to find \nout whether there are\n \nother 2D materials which become ferromagnetic upon hole doping.\n \n \nI\nn this study, \nwe screen thousands of 2D \nnon\n-\nmagnetic\n \nsemiconductor\ns\n/insulator\ns\n \nf\nrom three databases,\n[3b, \n3c, 10]\n \nfollowed by\n \na high\n-\nthroughput density functional theory calculation to identify potential 2D \nferromagnetic \nmateria\nls \ninduced by \nhole doping. \nW\ne then verify the stability of the potential candidates \nwith \nrespect to co\nmpeting \natomic \nstructures\n,\n \no\nbtained by g\nlobal struc\nture search\ning\n \nbased on evolutionary \nalgorithm\n[11]\n, via \nexploring energy convex hulls\n \nand \nphonon spectrum\n. \nAfterwards, \nthe magnetic behaviors of \nthe stable candidates at different doping densities are \ninvestigated systematically. \nEventually, \n122 \nmaterials \nare recognized as stable 2D \nferromagnetic\n \nmateri\nals\n \nupon hole doping\n, some of them having a computed Curie \ntemperature near or above room temperature\n. The\n \nexchange \ninteraction mechanisms \nresponsible for the\n \nferromagnetic coupling in these 2D materials \nare\n \nalso discussed.\n \n \nRESULTS AND DISCUSSION\n \nT\nhe general \nworkflow\n \nof\n \nthis research is \nillustrated \nby \na funnel plot\n \non the left panel \nof\n \nFigure 1\n, w\nh\nich \nis\n \ncategorized into five processes\n; \nthe number of materials\n \nused for screening and the screening criteria\n \nin \neach \nprocess is also given\n. \nAs a starting point, \nto include as many 2D materials as possible, we gather about 8000 2D \ncrystal structures from three databases, \n2DMatPedia\n,\n[10]\n \ncomputational 2D materials database (C2BD)\n[3b]\n \nand \nMaterials Cloud two\n-\ndimensional crystals database\n \n(MC2D)\n[3c]\n \nthat contains exfoliable 2D materials\n. \nIn \nthe prescreening process, the magnetic as well as the metallic materials are excluded since we \nar\ne only \ninterested in\n \nthe \nnon\n-\nmagnetic\n \nsemiconductors.\n \nBesides, \nthe repeated structures and the structures with low \nthermodynamic \nstability \n(\nE\nc\n \n< 0.2 eV/atom, \nE\nc\n \nis the \ncohesive energy\n \ndefined\n \nby the total energy of the \ncompound \nminus the total energies of\n \nits constituent \natoms\n)\n \nare also\n \nscreened out\n. As a result, \nabout \n3\n000 \nnon\n-\nmagnetic\n \n2D \nsemiconducting materials\n, \nwith moderate stability\n, \nare \npick\ned\n \nout\n \nfor the\n \nsubsequent \nhole doping \ns\nimulation\ns\n. \n \nThe \ntypical \ndoping densities,\n \nranging\n \nfrom 5\n×\n10\n12\n \ncm\n-\n2\n \nto 8×10\n1\n4\n \ncm\n-\n2\n \nare used in the hole doping \nsimulat\nions\n, \nand\n \nstructural relaxation\ns\n \nare\n \ncarried out \nfor\n \neach\n \ndopi\nng density. \nThe \nspin\n-\npolarization energy\n \nis \nobtained by\n \nE\nspin\n \n=\n \nE\nNM\n \n−\n \nE\nFM\n, where \nE\nFM\n \nand \nE\nNM\n \nare the energy of the ferromagnetic and non\n-\nmagnetic \nstates, respectively\n \nat PBE level\n. \nIn this case\n, a positive \nE\nspin\n \nsuggests that the ferromagnetic state is more \nenergetically \nstable\n. \nTypically, f\nor the Ga\nS\n,\n \nGaSe\n \nand InS\ne\n \nmonolayer, \nthe maximum \nE\nspin\n \nis \nabout 10\n \nmeV\n/hole\n,\n[7a, 7b, 7e]\n \nwhereas other materials, like GaO, SnO, etc.\n[7d, 8b]\n \nhave relatively larger \nE\nspin\n.\n \nHerein, we\n \nuse \nthis \n10\n \nmeV\n/\nhole as \na \nstandard\n \nvalue\n, and materials with \nE\nspin\n \nsmaller than\n \n10 mev/hole are discarded.\n \nThe \ninduced magnetic moment is also compared with the number of holes used in the simulation. If the ratio of \nmagnetic moment to hole number is ~1 (1 \nμ\nB\n/\nhole), it indicates that the injected holes can be fully spin \npolarized. Materials with induced magneti\nc moment much smaller than 1 \nμ\nB\n/\nhole are excluded. Finally, \nmaterials \nexhibit\ning\n \nferromagnetism\n \nonly \nat \na \nhole density below 10\n13\n \ncm\n-\n2 \nor only after a high hole density \nof 6 × 10\n14\n \ncm\n-\n2\n \nare \nalso discarded. \n \n \n \nF\nigure 1. General prescreening and \nhigh\n-\nthroughput screening strategies applied to 2D structures from 2DMatpedia Database, \nComputatio\nnal 2D Materials Database (C2DB) and Materials Cloud two\n-\ndimensional crystals database (MC2D) to identify \npotential 2D ferromagnetic materials upon hole dopin\ng (left panel). The number of candidate materials and the screening \ncriteria used for each screening step are provided. Polar histogram showing the number of structures belonging to different \nchemical compositions after hole doping screening (top right pan\nel); Polar histogram displaying the number of final stable \n2D ferromagnetic materials in terms of chemical compositions and their crystal space groups (bottom right panel).\n \n \nA \nglobally \nevolutionary \nstructur\ne\n \nsearching is\n \nsubsequently \ncarried out to \nexplore\n \natomic structures with \ndifferent stoichiometr\nies\n \nfor a given \nmaterial \nthat shows promising \nferromagnetic\n \nproperties \nupon \nhole doping\n.\n \nIn this step, \nthe \nconvex hull of \na\n \nspecific chemical composi\ntion \nis\n \nprovided\n \nwith the apex determined by the \nmost \nstable \nmaterials in energy\n.\n \nIf a new structure/phase on the \nconvex hull\n \nis found,\n \nit \nis\n \nsent back \nto the hole \ndoping simulations. \nSubsequently\n, \nthe number of \nc\nandidate materials \nis\n \nnarrowed down to 73\n6\n.\n \nAccording to \ntheir \nchemical \nformula\n, these candidate materials are classified into \ndifferent \nprototypes\n, as shown \nby\n \nthe\n \np\nolar \nhistogram\n \non the top right \npanel \nof Figure 1\n.\n \nThe most prevalent chemical compos\nition \ncorresponds to metal\n \nhalides\n \n(~47%)\n, followed by \nother materials such as ternary compounds, phosphorides, carbides, etc. (\n~\n26%, \ngrouped as “Others”).\n \nO\nxides \n(~12%) \nand chalcogenides \n(~11%) \nalso account for a part of \nthe \ncandidate \nmaterials. \nThere \nis\n \nalso a fraction of \nhydroxides \n(~3%) \nand nitrides\n \n(~3\n%)\n.\n \nPhonons calculations on these \ncandidates are next performed, \nto\n \nveri\nfy their \ndynamic\n \nstab\nility.\n \nIn this \nprocedure\n, \nmaterials with imaginary \nfrequency\n \nonly appears \naround\n \nthe \nГ \npoint in the first Brillouin zone and less than 50 cm\n-\n1 \nare regarded as \ndynamically stable. \nAfterwards, 14\n6\n \ncandidates\n \nare\n \nselected\n \nfor further screening.\n \nT\nhe\n \nnext step \nconsists in performing \ndetail\ned\n \ncalculations \nof \nthe magnetic \nproperties\n \nof these selected 2D \nmaterials\n,\n \ni.e.,\n \nmagnetic \nexchange \ncoupling strength, magnetic\n \nanisotropy energy\n \n(MAE)\n, and Curie \ntemperat\nure\n.\n \nGenerally, the magnetic moments of \n2DHDFM\n \nare well localized on the \np\n-\norbitals of the anion \nsites\n, \nwhich\n \nis also confirmed by the HSE\n06 calculation\ns in Figure S1\n \nthat the spin density localized almost \nsolely on the anion site.\n \nThe\nrefore, their \nFM configuration\n \nas well as a few \nantiferromagnetic (AFM) \nconfigurations can be constructed and\n \ntheir \nenergies \nobtained from first\n-\nprinciples calculations \ncan be mapped \nto the\n \nspin Hamiltonian\n:\n \n\u0000\n=\n−\n\u0000\n\u0000\n\u0000\u0000\nS\n\u0000\n⋅\nS\n\u0000\n\u0000\n\u0000\n\u0000\n−\n\u0000\n\u0000\n(\nS\n\u0000\n\u0000\n)\n\u0000\n\u0000\n \nwhere \n\u0000\n\u0000\u0000\n \ni\ns the exchange interaction between \n\u0000\n \nand \n\u0000\n \natomic sites, \nS\n\u0000\n \nis a unit vector denoting the local \nspin \ndirection\n \nof atom \ni\n, \nand \n\u0000\n\u0000\n \nis the unit vector of the spin moment direction of\n \natom\n \nj\n.\n \nFor ferromagnetic \nmaterials\n \nwhere neighbo\nring spins align in paralle\nl\n,\n \n\u0000\n\u0000\u0000\n \n>\n \n0, while\n \nfor\n \nantiferromagnetic materials where the \nspins prefer to align\n \nanti\n-\nparallel\n,\n \n\u0000\n\u0000\u0000\n \n< 0. \n\u0000\n \nis the\n \nMAE\n \nper \nmagnetic ion\n, \nwhere\n \npositive va\nlue\ns\n \ncorrespond to\n \na preferred \nalignmen\nt along the z\n-\naxis, while negative val\nues \ncorrespond to a \nprefer\nred\n \nalignment \nalong\n \nthe\n \nx\n \n−\ny\n \nplane\n.\n \nThe magnetic exchange interaction strength can then be obtained by \ncomparing the total energ\nies\n \nof \ndifferent magnetic configuration\ns\n.\n[12]\n \nNote that \nmaterials with negative \nmagnetic \nexchange \ncouplings\n, i.e., \nwhose antiferromagnetic state is more ener\ngetically stable than the \nferromagnetic one\n, \nor materials with weak \nFM coupling (\n\u0000\n\u0000\u0000\n<\n \n1\n \nm\neV) \nare \nnot consi\ndered\n \nfurther\n. \n \nThrough the screening\n \nprocess described above, \n1\n22\n \ncandidate\ns\n \nare\n \nidentified as stable\n \n2D\n \nhole\n-\ndoped\n \nf\nerromagnetic materials\n \nupon hole doping\n,\n \nwhich are denoted as \n2D\nHD\nF\nM\n \nhereafter\n.\n \nThese materials become \nhalf\n-\nmetals after hole doping, with a metallic DOS at the Fermi level (\nE\nF\n) for the spin\n-\ndown channels and a simultaneous band gap for the spin\n-\nup channels. The magnetic moment of \n2DHDFM\n \nis primarily originating \nfrom the \np\n-\norbitals of the anions.\n \nIn the following, we\n \nfocus \non the analysis of \ntheir\n \nstructural and \nmagnetic \nproperties. \nMore\n \ndetailed information\n, \ninc\nluding their fo\nrmation energies in the convex hull, atomic st\nructure\ns\n \nand \natomic coordinates, phonon dispersion spectra, band structures\n \nand\n \nelectronic\n \ndensity of states\n, \nmagnetic \nmoment and spin polarization \nenergy\n \nas a function of hole density, \nm\nagnetic configurations and the \ncorresponding spin Hamiltonian\n,\n \nthe magnetic exchange coupling strength, MAE \nand \ntemperature\n-\ndependent \nnormalized magnetization curves \nat\n \nspecific hole density \nare provided in the supplementary\n \nmaterials.\n \n \nT\nhe\n \nclassification of \n2DHDFM\n \nbased on\n \ntheir \nchem\nical composition \nand \nspace group\n \nis shown \nby\n \nthe\n \np\nolar histogram\n \non \nthe bottom right \npanel\n \nof Figure 1\n. Clearly\n, 6\n5\n \nout of 1\n22\n \ncandidates \nbelong\n \nto\n \nmetal halides\n, \nincluding \n20\n \nfluorides, 1\n7\n \nchlorides, 1\n7\n \nbromides and 1\n1\n \niodides. \nThe rest covers 21 chalcogenides, 13 oxides, \n9 nitrides, 6 hydroxides and 8 \nother materials\n.\n \nAmong those materials,\n \nP3m1 is the\n \nprimary \nspace group\n, \nfollowed by the P4m2 and P6m2.\n \nThe representative atomic structures of \nthese materials \nin different space \ngroups \nare given in Figure 2\n.\n \n \nT\nhe metal\n \nelement \nin the 2D \ndi\nhalides wit\nh\n \nP3m1 and P4m2\n \nspace groups\n \nare\n \nmainly from the \ngroup IIA\n \nBe, Mg, Ca, Sr and Ba, \nand\n \ngroup IIB\n \nZn\n \nCd and \nHg\n. \nThe a\ntomic structure of th\ne \nP3m1\n-\ndihalides\n \nis \ncommonly \nknown as the 1T structure, with one hal\nogen\n \nato\nm bond\ned\n \nto three neighboring metal atoms. \nT\nhe \nones with \nP4m2\n \nspace group\n \nhav\ne a \ntetragonal lattice with on\ne hal\nogen\n \natom connecting to only two neighboring metal \natoms.\n \nAccording to their convex hulls, \nP3m1\n-\ndi\nhalide\ns\n \nare usually more stable than the \nP4m2\n-\nones, except \nfor ZnBr\n2\n. PbCl\n2\n \nand PbBr\n2\n \nare the only two halides \nwith \nP6m2\n \nspace group\n,\n \nand\n \nthe\nir\n \nstructure \nis commonly \nknown \nas \nthe \n2H hexagonal st\nructure. Nevertheless, they are less stable than the P3m1\n \ncounterparts\n. In \naddition, there are two different halide structures with t\netragonal\n \nP4/\nm\nmm space group. They contain\n \nmetal\n \nelements from \ngroup IA\n \nand \ngroup IV\nA\n,\n \nrespectively. The former form metal monohalid\nes,\n \nsuch as \nLiCl\n \na\nnd \nNaBr,\n \nwith bo\nth \nhalogen\n \nand metal atoms being fourfold coordinated.\n \nThe latt\ner \nconsist in \nte\ntrahalide\ns\n, XF\n4\n \n(X=Si, Ge, Sn, Pb),\n \nwith \nan \natomic structure similar to the P4m2 \ndi\nhalides,\n \nthe \ntwo extra \nfluorine\n \natoms \nforming \nbond\ns\n \nto one X atom \nalong\n \nthe out\n-\nof\n-\nplane direction\n \nin the unit\n-\ncell\n.  \n \nThe majority of chalcogenide\ns \nbelong\ns\n \nto\n \nthe \nP3m1 and P6m2\n \nspace groups\n, while a few of them belong \nto the \nP4m2 and P4/mmm space groups. \nThe chalcogenides mainly con\nsist\n \nof\n \nelements from the \ngroup IIIA\n \nAl, Ga, In, Tl\n,\n \ngroup IV A\n \nSi, Sn, Pb and \ngroup IIB\n \nZn\n \nand \nCd. \nThe \nmonochalcogenides,\n \nMX\n \n(M=Al, Ga, In, \nTl; X=S, Se)\n, \nwhich contain two metal atoms and two chalcogenide atoms in the unit cell,\n \nhave \nboth P3m1 \nand P6m2 space groups. \nE\nach \nX atom is bonded to three \nneighboring M atoms, \nand\n \nevery M atom is fourfold \ncoordina\nted, forming bonds with three adjacent X atoms and one\n \nM atom, with a characteristic X\n-\nM\n-\nM\n-\nX \nvertical stacking.\n \nThe only difference\n \nbetween the two space groups is that the X atoms in the upper and \nbottom layer overlap with each other from the top view of the atomic plane for the P6m2 structu\nres, while \nthey h\nave stagger position in th\ne P3m1 structures.\n \nFor a given material, t\nhe \ntotal \ne\nnergy difference \nbetween\n \nthese two space groups is very small\n \n(\ntypically \nless than 50\n \nmeV/unit cell)\n. \nNote that \nthese \nmono\nchalcogenides \ngradually become non\n-\nmagnetic\n \nwhen the \nhole doping density is\n \ntypically \nla\nrger than 2 \n× 10\n1\n4 \ncm\n-\n2\n.\n \nThere are \ntwo \nother\n \nP3m1\n \natomic struct\nures\n \nfor\n \nthe chalcogenides. \nThe first one\n \ncorresponds to the \ndichalcogenide\ns,\n \nincluding SnS\n2\n, PbS\n2\n \nand PdS\n2\n, ha\nving\n \nthe 1T hexagonal structure\n.\n \nThe second one corresponds to \nthe monochalcogenide\ns\n \nSiS, SnS, PbS, PbSe and ZnSe, \nholding\n \nthe buckled hexagonal structure. The space group \nof CdS and ZnS, however, belongs to P6m2, because of the planar hexagonal atomic plane. \n \n \n \nF\nigure 2. The representative atomic structures of the \n2DHDFM\n, sorted by chemical composit\nions \nand space groups. The \nchemical formula and the corresponding constituent elements of each atomic structure\ns\n \nare provided. The unit cells are marked \nout by the black dotted lines. \n \n \nConcerning the \n2DHDFM\n \noxides, \nGe\nO\n2\n, SnO\n2\n, TiO\n2\n \nand ZrO\n2 \nbelong to the P3m1 space group with 1T \nstructure, while the space grou\np of BeO, CdO \nis \nP6m2\n \n(\nsimilar to CdS and ZnS\n)\n,\n \nwith\n \ntheir atomic planes \nbeing\n \nalso \np\nlanar. Another space group of the oxides is P4/nmm, whi\nch \ninclude\ns GeO, SnO and PbO\n. In these \nthree materials, \neach cation \n(\nanion\n)\n \nis bonded to four neighboring anions\n \n(\ncations\n)\n \nin the\n \ntetragonal structure\n. \nSrO\n2\n \nis the\n \nonly oxide \nwith\n \nP4/mmm space group, and each O atom bonds to four neighboring Sr atoms, while \neach Sr atom bonds to eight O atoms. \n \nAlN, GaN, InN and TlN are the four \n2DHDFM\n \nnitrides\n \nwith planar hexagonal P6m2 structure. \nThe rest\n \nof \nthe\n \n2D \nnitrides \ninclud\ne\n \nplanar t\netragonal\n \nAl\n4\nN\n4\n \nand Ga\n4\nN\n4\n \nwith\n \nP4/mbm, two\n \nAl\n2\nN\n2\n \nstructures \nwith\n \neither\n \nP3m1 \nor\n \nP4/nmm\n \nspace group\n,\n \nand C\n3\nN\n4\n \nwith\n \nC2 space group.\n \nIn addition, t\nhere are only si\nx\n \n2DHDFM\n \nhydroxides\n, \nincluding Ca(OH)\n2\n, Mg(OH)\n2\n, Cd(OH)\n2\n \nand Zn(OH)\n2\n \nwith\n \nP3m1\n \nspace group, \nand Li\n2\n(OH)\n2\n, \nNa\n2\n(OH)\n2\n \nwith\n \nP4/nmm\n \nspace group\n. The P3m1 and P4/nmm\n-\nhydroxi\ndes have\n \nstructures simila\nr to the 1T \nstructure, and the ones of GeO, SnO and PbO, respectively, with the two O atoms terminated by two H atoms \nin the unit cell.\n \nNote that\n \nt\nhese six hydroxides\n \nhave\n \nall\n \nparent 3D bulk structure\ns\n \nand can be easily exfoliated.\n \n[3c]\n \nT\nh\ne re\nmaining\n \n2DHDFM\n \nmainly \ninvolve \nternary compounds, \nsuch as \nPb\n2\nB\nr\n2\nF\n2\n, Sc\n2\nBr\n2\nO\n2\n \nand Sr\n2\nBr\n2\nH\n2\n \nwith\n \nP4/nmm, In\n2\nBr\n2\nO\n2\n \nand In\n2\nCl\n2\nO\n2\n \nwith\n \nPmmn, \nand \nTiPbO\n3\n \nwith\n \nP4mm\n \nspace group\n.\n \n \n \n \nF\nigure 3. \nBox plots of the \ns\npin\n-\npo\nlarization energy (a) magnetic anisotropic energy (MAE) per magnetic ion (\nb\n) nearest\n-\nneighbor exchange interaction parameter, \nJ\n1\n \n(\nc\n) \nand \nnext\n-\nnearest\n-\nneighbor exchange interaction parameter \nJ\n2\n \n(d) \nat specific \nhole doping density of the 122 \n2DHDFM\n,\n \nclassified by space group and chemical composition. The red lines inside the boxes \nrepresent the median values and the horizontal\n \nbars denote the low/high boundaries of the spreads.\n \n \nT\nh\ne\n \nspin\n-\npolarization energies\n \nE\nspin\n \nof \n2DHDFM\n \nat different doping densities are \npresented\n \nin Figure \n3\n \n(a)\n. \nGenerally, \nE\nspin\n \nincreases\n \nmonotonically\n \nas the hole doping density increases\n.\n \nThere is a clear\n \ndist\ninction\n \nin the \nE\nspin\n \ndistribution\n \namong\n \ndifferent compounds \nwith different\n \nspace groups.\n \nF\nor the\n \nP3m1, P4m2 and \nP4/mmm\n \nspace groups\n, \nhalide\ns \ntypically \nhave \nhigh \nmedian\n \nas well as \nlarge\n \nE\nspin\n \nat \na given\n \nhole \ndoping density\n.\n \nBesides, \nthe\n \nox\nides\n \nand nitrides \nwith P6m2\n \nalso have \nmoderate\n \nto high\n \nE\nspin\n,\n \nwith\n \nmedians of ~200 to ~250 \nmeV/hole at 6 × 10\n14\n \ncm\n-\n2\n.\n \nOn the other hand, \nthe \n2DHDFM\n \nwith\n \nP4/nmm\n \nspace group\n \nhave rel\natively\n \nsmall\n \nE\nspin\n,\n \ntypically \nless than 100 meV/hole at 6 × 10\n14\n \ncm\n-\n2\n.\n \n \nE\nspin\n \nha\ns the \ntendency\n \nof being relatively large\nr\n \nfor the binary compounds with lighter anions in the same \ncrystal structure. This\n \nis in agreement with values reported in the\n \nliterature\n[7a, 7b, 7d, 7e]\n \nfor the 2D gallium or \nindium oxides and chalcogenides, with \nE\nspin\n \nfollowing the order of Ga\nO > GaS > GaSe, and InO>InS>InSe. \nFrom our calculations, \nwe fi\nnd that for\n \nhalides \nwith\n \nthe same space group, fluorides or chlorides usually have \nthe largest \nE\nspin\n, followed by bromides and iodides at the same hole density. As can be seen from \nTable 1\n, \nMgF\n2\n \nhas larger \nE\nspin\n \nthan MgCl\n2\n, and for the calcium halides, their \nE\nspin\n \nfollow the order\n \nCaF\n2\n \n> CaCl\n2\n \n> \nCaBr\n2\n \n> CaI\n2\n.\n \nA\n \ns\nim\nilar trend is also clear for the stronti\num, barium, cadmium and zinc halides.\n \nNote, however, \nthat t\nhe fluorides of barium, cadmium and mercury \ndo not follow the same trend, \ntheir \nE\nspin\n \nbeing \nsmaller t\nhan \nthe ones of their chloride counterparts.\n \nIn fact, f\nor materials with \na \nlarge\nr\n \ndifference in electronegativity\n \n(\nΔχ\n) \nb\netween the\n \nanions and cations\n, \none \ne\nxpect\ns\n \na\n \nlarger coulomb interaction and thus a larger exchange splitting\n, \nand subsequently, \na larger \nE\nspin\n.\n \nThis may explain why the metal halides have generally larger \nE\nspin\n, since their \nΔχ\n \nis large. In particular, the \nP4/mmm\n-\nmonohalides have the largest \nE\nspin\n, \nbe\ncause the \ngroup IA\n \nmetals\n \nhave \nsmall electronegativities\n, \nwhile the\n \nhalides\n \n(\nespecially \nfluoride\n) have large electroneg\nativities\n.\n \nFor instance, \nNaBr and KBr have\n \nquite\n \nhigh\n \nE\nspin\n,\n \nover 400 meV/hole at 6 \n× 10\n14\n \ncm\n-\n2\n.\n \nThe trend mentioned above for Ga \nand In oxides, sulfides, and selenides is also consistent with the decrease of the electronegat\nivity from O to \nSe.  \n \nT\nhe \nmagnetic anisotropy energy (\nMAE\n)\n \nof \n2DHDFM\n \nvaries from materials to materials, and generally \ngrows as the hole doping density increases\n, as can be seen from Figu\nre \n3\n(b)\n.\n \nT\nhe \nMAE of the \nmajority of \n2DHDFM\n \nare below 1 meV\n \nper magnetic ion\n.\n \nParticularly, \nmetal halides have relatively large \nabso\nlute \nMAE\n, \nas compared to \nother compounds, especially \nthe ones\n \nwith\n \nP4/mmm and P4m2 space group\ns\n.\n \nS\nome of them \neven have \nabsolute \nMAE larger than\n \n10 meV\n \nper magnetic ion\n \nat 4 and \n6 \n× 10\n14\n \ncm\n-\n2\n. \nThis \nis likely related\n \nto \nthe large spin\n-\norbit coupling \nstrength \nof\n \nbromi\nne\n \nand iodi\nne\n.\n \nOther materials\n \nwith lighter elements (and thus \nsmaller spin\n-\norbit coupling), \nlike\n \nfluorides and oxi\ndes\n, \nhave much smaller \nabsolute \nMAE, usually \nless\n \nthan \n100 µeV/magnetic\n \nion.\n \nInterestingly, \nthe median of MAE of most \n2DHDFM\n \nare\n \nnegative, which reveals that \nthe spins\n \nprefer to align along the in\n-\nplane \ndirection. \n \nThe magnetic exchange coupling parameters of the \n2DHDFM\n \nat different doping densities are summarized \nin Figure 3 (c) and (d), where \nJ\n1\n \nand \nJ\n2 \nrepresents the \nnearest and next\n-\nnearest neighbor coupling, respectively. \nThese parameters can be obtained from the total energy difference of various possible ferromagnetic and \nantiferromagnetic configurations.\n[12]\n \nOverall, the magnetic exchange coupling parameters also increase by \nincreasing the hole doping density. \nJ\n1 \nis positive for most \n2DHDFM\n, which is characteristic of a ferro\nmagnetic \nfirst\n-\nneighbor interaction. On the other hand, \nJ\n2\n \nis typically smaller than \nJ\n1\n \n(as expected), and \nJ\n2\n \nis negative \nfor a couple of \n2DHDFM\n, suggesting competing ferromagnetic and antiferromagnetic interactions in these \ncompounds, as \ndiscussed further below. T\nhe computed Curie temperature \nT\nc\n \nof \n2DHDFM\n \nis summarized in \nFigure 4. Increasing the hole doping density also gives rise to a higher \nT\nc \nfor most of \n2DHDFM\n. Upon a hole doping density of 4 \n× 10\n14 \ncm\n-\n2\n, the medians \nT\nc \nof various \n2DH\nDFM\n \nare in the range of ~50 K to ~170 K, and \nfurther increase to ~70 K to ~270 K at 6 \n× 10\n14 \ncm\n-\n2\n. As expected, the Curie temperature is correlated to the \nvalues of the magnetic exchange coupling parameters. Very interestingly, the sign of \nJ\n2\n \nis found to p\nlay a very \nimportant role on \nT\nc\n. Consider the typical case of CaF\n2\n \n(\nT\nc\n=70 K) and HgF\n2\n \n(\nT\nc\n=333 K), which have the same \nP3m1 crystal structure (see Table 1). While these two materials have similar \nJ\n1\n \n(about 23 meV), CaF\n2\n \nhas a \nnegative \nJ\n2\n \n(about \n-\n2 meV) and HgF\n2\n \nhas a positive \nJ\n2\n \n(about 12 meV). A ferromagnetic next\n-\nnearest neighbor \ninteraction \nthus favors a large \nT\nc\n \nin these 2D materials, such as in \nBaF\n2 \n(337 K), CdF\n2\n \n(320 K), HgF\n2 \n(333 K), \nPbCl\n2\n \n(394 K) and PbBr\n2 \n(301 K) at a hole doping density of \n6 \n× 10\n14 \ncm\n-\n2\n.\n \n \n \nF\nigure \n4\n. \nBox plots of the \nCurie temperatures \n(\nT\nc\n) \nobtained from Monte Carlo simulations for \n2DHDFM\n \nat different hole \ndoping densities. The red lines inside the boxes represent the median values and the horizontal bar denotes the low/high \nboundar\nies\n \nof the spread\ns\n.\n \n \nIn general, the\n \nmagnetic moment\ns of \n2DHDFM\n \nare \nmainly \ncontributed by\n \nthe \nanion \np\n-\norbitals,\n \nas shown \nin Figure \n5\n \n(a)\n. \nSince a large portion of these 2D materials consists of \nP3m1\n-\ndihalides, the magnetic properties \nof these materials are \nfirst\n \ndiscussed below\n. \nIn the\nse compounds\n, \nJ\n1\n \ncorresponds to the out\n-\nof\n-\nplane exchange \ncoupling between two halogens from the top and bottom atomic planes, as shown in Figure 6 (a); \na\n \nlocal \ncoordinate system is used for the\nse\n \nP3m1\n-\ndihalides, where the metal \nion\n \nis \nin\n \nthe center \nof a distorted octahed\nra \nwith\n \nsix \nligan\nd\ns\n \nat the vertices.\n \nN\note that the \nshorter \nhalogen\n-\nhalogen distance \nis about \n2.3 Å \nin \nfluorides\n,\n \nand \ngradually increase\ns\n \nto \nabout \n5\n \nÅ from chlorides to iodides.\n \nConsidering the delocalized nature of the \np\n \norbitals\n, \nt\nhe direct out\n-\nof\n-\nplane magnetic interaction mainly arises from the exchange coupling between the \np\nz\n \norbitals \nof the halogens, as illustrated in Figure 6 (b);\n \nthis direct exchange coupling is ferromagne\ntic, leading to a \npositive value of \nJ\n1\n, which lies between \nabout \n5\n \nand \n39 \nmeV (see Table 1).\n \nNote also that \nJ\n1\n \nis smaller in \niodi\nd\nes, due to the larger out\n-\nof\n-\nplane halogen\n-\nhalogen distance in these compounds.\n \n \n \nFigure 5. Spin density \nplot\ns\n \n(a) and o\nrbital projected density of states\n \n(b) and (c) for P3m1 strontium dihalides and cadmium \ndihalides monolayers at hole density of 6 × 10\n14\n \ncm\n-\n2\n. Positive (negative) values refer to up (down) spins. The Fermi level is \nset at zero energy.\n \n \n \nOn the other hand, \nJ\n2\n \ncorresponds to the in\n-\nplane exchange coupling between two halogens from the same \natomic plane, see Figure 6 (a). In this case, \nthe direct exchange \ninteraction between the\n \nanion \np\n-\norbitals\n \nis \nweaker\n \nsince the in\n-\nplane halogen\n-\nhalogen distan\nce is \nlarger.\n \nConsequently, t\nhe indirect exchange interaction, \ninvolving the coupling between the halogen\n-\np\n \norbitals with the metal orbitals, should \nalso \nplay \nan\n \nimportant \nrole\n \nin the next\n-\nnearest neighbor interaction. \nGroup IIA and IIB metals \nare transferring their two outmost \nvalence electrons to the \nhalogens \np\n-\norbitals, to form the dihalide compounds. In the indirect exchange \nmechanisms, it is assumed that some covalent mixing between the cation and anion orbitals is energetically \nfavorable. \nFrom the p\nrojected\n \nelectronic density of states (PDOS) of \nstrontium and c\nadmium dihalides, shown \nin Figure 5 (b) and (c),\n \nrespectively,\n \non\ne can indeed visualize the possible hybridization between the (spin\n-\ndown) \np\n \norbitals of the halogen and the (spin\n-\ndown\n) \np\n \nor \nd\n \norbitals of the metals near the Fermi level; note \nthat for group IIB metal dihalides, the degeneracy between the metal \nd\n \norbitals is partially lifted by the \ndistorted octahedral crystal field, resulting in the formation of e\n1\n(\nd\nxz\n \nand \nd\ny\nz\n), e\n2\n(\nd\nx\n2\n-\ny\n2\n \nand \nd\nxy\n) and a\n1\n(\nd\nz\n2\n) \nstates.\n \nSpecifically, the in\n-\nplane halogen\n-\np\nx\n/\np\ny\n \norbitals \ncan \ncouple t\no metal\n-\np\nx\n/\np\ny\n \norbitals\n \nand halogen\n-\np\nz\n \norbitals couple to metal\n-\np\nz\n \norbitals \nfor strontium halides\n. For cadmium dihalides, the halogen\n-\np\nx\n/\np\ny\n \norbitals \ncouple\n \nto metal\n-\nd\nx\n2\n-\ny\n2\n/\nd\nxy\n \nand \nd\nxz\n/\nd\ny\nz\n \norbitals\n, and \nthe halogen\n-\np\nz\n \norbitals also hybridize with the \nd\nxz\n/\nd\nyz\n \norbitals. \nFrom Table 1, \nJ\n2\n \nof group IIA metal dihalides can be positive or negative, indicating the competition between \ndifferent indirect exchange mechanisms, such as the super\n-\nexchange coupling between orbitals pointing in the \nsame direction (antiferromagnetic\n, e.g., hybridizatio\nn between halogen\n-\np\nz\n \norbitals with metal\n-\np\nz\n \norbitals,\n \nas \nillustrated\n \nin Figure 6 (c)\n), the super\n-\nexchange mechanism involving the coupling between orthogonal orbitals \n(ferromagnetic\n, e.g., hybridization between halogen\n-\np\nx\n/\np\ny\n \norbitals with metal\n-\np\nx\n/\np\ny\n \norbitals, \nas depicted\n \nin \nFigure 6 (c)\n) and the double exchange mechanism (ferromagnetic). This latter mechanism involves the \ninteraction between anions orbitals with different charged (or valence) states, e.g., between partially filled \nhalogen orbitals (w\nhere holes are localized) with filled halogen orbitals.\n \nVery interestingly, \nJ\n2\n \nis found to be \npositive for all group IIB metal halides (see Table 1), the coupling between the halogen \np\n \norbitals with the \nmetal \ne\n1\n \nand e\n2\n \nstates most likely favoring the ferromagnetic exchange mechanisms\n, as shown in Figure 6 (d)\n-\n(f)\n; note that the cont\nribution of the metal \nd\n-\nstates to the \nspin\n \ndensity, observed on Figure 5\n \n(a)\n, also points \ntowards the possible hybridization between the cation and anion o\nrbitals near the Fermi level. \nAlthough GGA \nfunctional \ntends \nto over\n-\ndelocalize electrons\n, the contribution of the \nmetal \nd\n-\nstates in CdBr\n2\n \ncan still be \nconfirmed\n \nin the spin density obtained from HSE06 functional\n.\n \n \n \nFigure 6.  (a) Exchange interaction paths in P3m1 metal dihalides. \nJ\n1\n \nis the exchange coupling parameter originated from the \nhalide ions on the upper layer (\na\n1\n) and the bottom layer (\na\n2\n). \nJ\n2\n \nis the exchange coupling parameter originated from the halide \nion\ns on the same atomic plane (\na\n2\n \nand \na\n3\n)\n, \nm\n \nis the metal ions\n. \nEdge\n-\nsharing \ndistorted \nma\n6\n \noctahedra with the local xyz\n \ncoordinates\n \nof the \nP3m1 \nstructure are also given\n \non the right\n.\n \nSchematic diagrams of \n(\nb\n)\n \ndirect exchange between\n \nhalogen\n-\np\nz\n \norbitals\n,\n \n(c) \nthe super\n-\nexchange \ninteractions \nbetween \nhalogen\n-\np\nx\n/\np\ny\n \norbitals with \nmetal\n-\np\nx\n/\np\ny\n \norbitals \nand \nhalogen\n-\np\nz\n \nor\nbitals \nwith metal\n-\np\nz\n \norbitals\n, \n(d) \nhalogen\n-\np\nx\n/\np\ny\n \norbitals with \nmetal\n-\nd\nx\n2\n-\ny\n2\n/\nd\nxy\n \norbitals\n,\n \n(e) \nhalogen\n-\np\nx\n/\np\ny\n \norbitals with \nmetal\n-\nd\nxz\n/\nd\nyz\n \norbitals\n, \nand (f) \nhalogen\n-\np\nz\n \norbitals with \nmetal\n-\nd\nxz\n/\nd\nyz\n \norbitals\n. \n \n \nWe now \nconsider \nthe possible magnetic exchange interactions in planar \n2DHDFM\n, namely without\n \nout\n-\nof\n-\nplane \nanion \ninteractions. \nSimil\nar to the case of P3m1\n-\ndihalides, \nJ\n1\n \nof the P4/mmm\n-\nlithium monohalides \ndecrease\ns\n \nfrom LiCl to LiI, \nas can be seen in Table S1.\n \nIn this case, as shown in Fig\nure S\n2\n, \nJ\n1\n \nis mainly due to \nthe direct exchange between the halogen \np\nz\n-\norbitals. \nIn addition, from the PDOS shown \nin Figure S\n3\n,\n \none\n \ncan \nsee that there is\n \na very \nlimited \npossible \nhybridization between the Li\n-\ns\n \norbitals with the halogen\n-\np\n \norbitals,\n \npointing out to a \nrelatively weak\n \nindirect exchange interaction in these materials.\n \nOn the other hand, i\nn the \ncase of\n \nhole\n-\ndoped 2D planar\n \nP6m2\n-\nsulfides (such as ZnS and CdS) and nitrides (such as AlN and GaN), \nthe \nferromagnetic order \nis\n \nessentially \nmedia\nted by the indirect exchange interaction between the\n \nsulfur or\n \nnitrogen \np\nz\n-\nstates and\n \nthe metal\n-\nd\nxz\n/\nd\nyz\n \nand \np\nz\n \nstates, as evidence by the PDOS in \nFigure S\n3\n.\n \nConsidering the possible \nuse\n \nof these \n2DHDFM\n \nin spintronic devices, the \nmetal \ndihalides\n \n(like BaF\n2\n \nand \nCdF\n2\n) \nappear as interesting candidates for high\n-\ntemperature (above room temperature)\n \napplications. However, \nthese materials might have some stability issues, possibly interacting\n/reacting\n \nwith the \nambient\n \nand being \noxidized in air\n. On the other hand,\n \npotential \nmore stable \n2DHDFM\n \nhave also been identified for low \ntemperature applications. For example, AlN\n, \nGaN\n \nand ZnS \ncrystallize\ns\n \nin the Wurtzite phase. In ultra\n-\nthin \nfilms (and at the 2D limit)\n, such\n \nmaterials could be grown in a h\n-\nBN like phase\n[13]\n. \nIn addition, these 2D \nmaterials should be much less reactive with the ambient. Other materials of interest are e\n.g.\n,\n \n2D ZrO\n2\n \nand \nTiO\n2\n, which are predicted to be \n2DHDFM\n \nwith moderate Curie temperatures of 176 and \n228 \nK, respectively. \n \n \n \n   \n \n \n \nT\nable \n1\n. \nSpin polarization energy\n \nE\nspin\n, electronegativity difference \nΔχ,  \nmagnetic\n \nanisotropic energy MAE, nearest\n-\nneighbor \nexchange interaction parameter \nJ\n1\n, next\n-\nnearest\n-\nneighbor exchange interaction parameter \nJ\n2\n, and Curie temperature \nT\nc\n \nat hole \ndoping density of 6 \n× 10\n14 \ncm\n-\n2\n \nfor the P3m1\n-\nhalides.\n \n \n \nE\nspin\n \n(meV)\n \nΔχ\n \nMAE\n \n(\nμ\neV)\n \nJ\n1\n \n(\nmeV)\n \nJ\n2\n \n(meV)\n \nT\nc\n \n(K)\n \nBeF\n2\n \n1\n20 \n \n1\n.94\n \n-\n9\n \n21.440\n \n-\n3.343 \n \n2\n6\n \nMgF\n2\n \n1\n52\n \n2\n.67\n \n-\n15\n \n2\n8.624\n \n-\n3.931)\n \n6\n3\n \nMg\nCl\n2\n \n10\n1\n \n1\n.85\n \n-\n163\n \n34.845\n \n3.081\n \n224\n \nCaF\n2\n \n1\n84\n \n2\n.98\n \n-\n62\n \n2\n3.212\n \n-\n1.996\n \n7\n0\n \nC\na\nCl\n2\n \n8\n0\n \n2\n.16\n \n-\n11\n \n3\n6.036\n \n-\n5.636\n \n4\n9\n \nCaBr\n2\n \n4\n0\n \n1\n.96\n \n-\n613\n \n2\n7.296\n \n-\n1.552\n \n1\n04\n \nCaI\n2\n \n1\n4\n \n1.66\n \n-\n620\n \n5\n.415\n \n0\n.663\n \n4\n2\n \nSrF\n2\n \n1\n95\n \n3\n.03\n \n-\n125\n \n1\n6.900\n \n-\n0.740\n \n6\n3\n \nSr\nCl\n2\n \n11\n8\n \n2\n.21\n \n-\n295\n \n32.319\n \n-\n4.562\n \n83\n \nSrBr\n2\n \n7\n1\n \n2\n.01\n \n1\n26\n \n3\n8.474\n \n-\n2.717\n \n1\n26\n \nSrI\n2\n \n3\n5\n \n1\n.71\n \n-\n1956\n \n9\n.024 \n \n3\n.092\n \n1\n07\n \nBaF\n2\n \n9\n6\n \n3\n.09\n \n9\n46\n \n3\n8.685\n \n7\n.868\n \n3\n37\n \nBa\nCl\n2\n \n16\n7\n \n2\n.27\n \n195\n \n20.222\n \n-\n0.71\n0\n \n90\n \nBaBr\n2\n \n1\n10\n \n2\n.07\n \n3\n20\n \n3\n2.384\n \n1\n.431\n \n1\n81\n \nBaI\n2\n \n7\n3\n \n1\n.77\n \n-\n1776\n \n1\n1.161\n \n5\n.215\n \n1\n62\n \nZnF\n2\n \n7\n6\n \n2\n.33\n \n3\n14\n \n1\n4.890\n \n4\n.560\n \n1\n55\n \nZn\nCl\n2\n \n65\n \n1\n.51\n \n-\n314\n \n20.431\n \n2.748\n \n1\n61\n \nZnBr\n2\n \n4\n2\n \n1\n.31\n \n-\n3800\n \n1\n3.446\n \n5\n.452\n \n1\n79\n \nCdF\n2\n \n107\n \n2\n.29\n \n8\n16\n \n2\n9.730\n \n9\n.200\n \n3\n20\n \nC\ndCl\n2\n \n1\n14\n \n1\n.47\n \n-\n171\n \n31.241\n \n0.785\n \n1\n62\n  \n \n \n \n \nConclusion\ns\n \nUsing \nhigh\n-\nthroughput density functional\n \ntheory\n \ncalculations\n, we have identified \n122\n \nstable \n2DHDFM\n \nthat \nexhibit \na \nnon\n-\nmagnetic to \na \nferromagnetic\n \nphase\n \ntransition \nupon hole doping\n. \nIn these\n \n2DHDFM\n, \nmetal halides \ntake up the biggest proportion, followed by the sulfides, oxides\n \nand nitrides\n.\n \nThe magnetic properties of these \n2D materials, such as \ntheir \nspin polarization energy, magnetic anisotropic energy, magnetic exchange coupling \nparameters a\nnd Curie temperature typically increase with the hole doping density. Among these materials, \nmetal dihalides with a P3m1 phase, like BaF\n2\n, CdF\n2\n,\n \nPbCl\n2\n \nand \nPbBr\n2\n \nare predicted to have Curie temperatures \nabove \n300 K \nat a typical hole density of \n6 \n× 10\n14 \ncm\n-\n2\n, these materials being potentially interesting for \nspintronic devices operating above room temperature. \nIn general, the ferromagnetic interaction in these \nmaterials is mediated by a direct exchange interaction bet\nween \np\nz\n-\norbitals of anions in different atomic planes \n(out\n-\nof\n-\nplane coupling) as well as the indirect exchange interaction between the \np\nz\n-\nanion states and the metal \np\n \nor \nd\n \nstates (in\n-\nplane coupling). On the other hand, 2D planar materials with moderate Cur\nie temperature\ns\n, \ntypically ranging between 110 K and 230 K, have also been identified for possible \nspintronic \napplications at \nlow temperature. Such materials are \ne.g.\n,\n \nP3m1 sulfides\n \nand \nnitrides,\n \nlike\n \nZnS, AlN, and GaN. \nIn these \nmaterials, the ferromagnetic coupling mainly arises from the indirect exchange interaction between the anion \np\nz\n-\norbitals and the \np\n \nor \nd\n \nmetal states. \nThese materials should be more stable than their halogen counterparts,\n \nespecially considering the\nir \npossible interaction \nwith the ambient\n \n(oxidation)\n, \nand could potentially \nbe \ngrown \nin a planar 2D form\n, \nlike h\n-\nBN.\n \n \n \n \nAcknowledgments\n \nPart of this work has been financially supported by \nthe FLAG\n-\nERA grant DIMAG, by the Research \nFoundation \n–\n \nFlanders (FWO)\n \nas well as\n \nthe KU Leuven Research Fund, project C14/17/080\n \nand \nC14/21/083\n.\n \nPart of the computational resources and services used in this work have been provided by the VSC (Flemish \nSupercomputer Center), funded by the \nFWO\n \nand the Flemish Government \n–\n \ndepartment EWI.\n \n \nCdBr\n2\n \n7\n8\n \n1\n.27\n \n-\n5005\n \n2\n0.870\n \n3\n.980\n \n1\n90\n \nCdI\n2\n \n3\n7\n \n0\n.97\n \n-\n7917\n \n8.155\n \n4.530\n \n141\n \nHgF\n2\n \n7\n1\n \n1\n.98\n \n-\n1316\n \n2\n3.307\n \n1\n2.027\n \n3\n33\n \nH\ngCl\n2\n \n8\n4\n \n1\n.16\n \n366\n \n30.671\n \n6.515\n \n2\n70\n \nHgBr\n2\n \n7\n7\n \n0\n.96\n \n-\n3391\n \n2\n7.471\n \n6\n.280\n \n2\n64\n \nH\ngI\n2\n \n5\n2\n \n0\n.66\n \n-\n7939\n \n1\n2.72\n \n5\n.672\n \n1\n85\n \nPbCl\n2\n \n5\n3\n \n0\n.83\n \n0\n \n2\n7.041\n \n1\n4.43\n \n3\n94\n \nPbBr\n2\n \n2\n3\n \n0\n.63\n \n-\n1013\n \n1\n8.215\n \n1\n1.75\n \n3\n01\n References\n \n[1]\n \na) M. 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Lett.\n \n2013\n, 103, 251605.\n \n Supplementary Materials  for \nHole -doping induced ferromagnetism in 2D materials  \n \nR. Meng1*, L.M.C. Pereira1, J.P. Locquet1, V .V . Afanas’ev1, G. Pourtois2, and M. Houssa1,2* \n \n1Department of Physics and Astronomy, KU Leuven, Celestijnenlaan 200D, Leuven B -3001, Belgium  \n2imec, Kapeldreef 75, B -3001 Leuven, Belgium  \n*Email: ruishen.meng@kuleuven.be; michel.houssa@kuleuven.be  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \nFigure  S1. Spin density plots calculated by HSE06 functional for P3m1 strontium dihalides and cadmium dihalides \nmonolayers at hole density of 6 ×  1014 cm-2. \n \n \nFigure  S2. (a) L ocal xyz coordinates of the lithium monohalide with P4/mmm space group , a1 and a2 are the two nearest \nhalide ions . (b) Schematic diagrams of the direct exchange  coupling J1 between  pz orbitals  of the nearest hal ide ions .  \n \n \nFigure  S3. Orbital projected density of states  for planar 2DHDFM  under hole doping. The doping densities are 4 ×  1014 cm-2 \nfor lithium monohalides  with P4/mmm space group , ZnS, CdO and CdS  with P6m2 space group  and 6 × 1014 cm-2 for the \nnitrides  with P6m2 space group . Positive (negative) values refer to up (down) spins. The Fermi level is set at zero energy.  \n \n \n \nTable S1. Spin polarization energy  Espin, electronegativity difference Δχ, magnetic anisotropic energy MAE, nearest -neighbor \nexchange interaction parameter J1, next -nearest -neighbor exchange interaction parameter J2, and Curie temperature Tc. unless \notherwise stated, the hole doping density is 6  × 1014 cm-2. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n    Espin (meV)  MAE  (μeV) J1 (meV)  J2 (meV)  Tc (K) \n  LiCl \n(@4× 1014) 74 -703 17.186  -4.158  81 \n LiBr \n(@4× 1014) 70 -5684  14.856  -3.304  87 \n LiI \n(@4× 1014) 67 -10596  11.105  -2.303  76 \n NaBr \n(@4× 1014) 216 -4569  5.733 -1.239  37 \n NaI \n(@4× 1014) 204 -13476  1.998  -0.447  16 \n KBr 435 8932 4.438 1.330 80 \n SiF4 104 86 60.935  13.560  181 \n GeF4 136 262 66.101  17.979  239 \n SnF4 204 1254 41.674  16.566  224 \n PbF 4 221 1777 47.632  14.948  214 \n  Al2S2 30 0  1.198 6.646 26 \n  Al2Se2 19 -13 2.478  11.954  35 \n-160  Ga2S2 7 -2 1.403 6.086 29 \n  In2S2 18 -7 2.088 10.015  43 \n  Tl2S2 2 -19 2.155 10.024  48 \n  CdS 327 -14 1.513 -0.709  2 \n  ZnS 230 2 4.198 4.633 111 \n  SnS2 45 -160 15.621  2.803 136 \n  PbS 2 56 -83 16.746  5.295 176 \n  PdS 2 \n(@4× 1014) 5 163 1.965 0.940 28 \n  SiS 100 11 25.120  -0.052  114 \n  SnS 30 -81 32.824  -3.198  84 \n  PbS \n(@4× 1014) 9 -921 12.006  -0.356  148 \n  PbSe \n(@2× 1014) 23 -961 4.179 -0.892  41 \n  ZnSe 158 -1593  8.749 -1.448  92 \n  GeO2 58 0 6.709 -0.855  22 \n  SnO 2 160 6 22.170  -1.288  78 \n  TiO 2 72 -2 28.475  4.176 228 \n  ZrO 2 91 -7 29.899  1.660 176 \n  Ge2O2 35 -2 4.906 -1.399  40 \n  Sn2O2 41 -16 8.415 -3.919  61 \n  Pb2O2 19 -95 2.874 2.166 46 \n  BeO 203 0 10.600  1.074 149 \n CdO 250 -4 5.751 2.175 102 \n  SrO 2 125 99 171.382  20.580  282 \n  AlN 215 2 13.287  1.178 186 \n  GaN 223 -1 15.387  0.784 205 \n  InN 313 -18 21.218  -5.844  173 \n  TlN 345 -269 20.763  -7.956  90 \n  Al4N4 253 -3 109.369  7.384 101 \n  Ga4N4 245 -1 89.792  12.179  164 \n  Al2N2 46 -2 24.138  -6.191  101 \n  Al2N2 97 0 30.102  -2.211  97 \n  C2 C3N4 91 -4 26.360  5.933 213 \n  Mg(OH) 2 41 -6 8.349 4.370 117 \n  Ca(OH) 2 65 2 15.051  5.868 179 \n  Zn(OH) 2 63 -2 17.292  8.873 245 \n  Cd(OH) 2 85 -12 23.053  12.691  341 \n  Li2 (OH) 2 91 7 17.512  -0.214  132 \n  Na2 (OH) 2 78 8 18.275  1.077 175 \nHalides  \nP4/mmm  \nP6m2  \nP3m1 \nChalcogenides  \nP3m1 \nP4/nmm \nP4/mmm  \nP6m2 \nChalcogenides  \nP6m2 \nP4/mbm  \nP4/nmm\nm \nP3m1 \nNitrides  Hydroxides  \nP3m1 \nP4/nmm\nm Computational Method s \nIn this work, all the density functional theory calculations were performed using the Vienna ab initio \nsimulation package (V ASP)  package1,2, with electron -ion interaction described by projector augmented wave \n(PAW) pseudopotentials . The generalized gradient approximation (GGA), parameterized by the Perdew -\nBurke -Ernzerhof (PBE)3 approach  was used as the exchange correlation functional. The energy cutoff of 550 \neV and  k-point meshes of 0.03 ×2π and 0.02 ×2π Å-1 were used for structural optimizations and self -consistent \ncalculations.  Total energy convergence criterion of 10-6 eV/cell and force convergence criterion of 0.005 eV/Å \nwere chosen for complete relaxations of lattice constants as well as atomic positions.  The \nHeyd−Scuseria−Ernzerhof functional (HSE06),4 which mixes 25% nonlocal exchange with the PBE \nfunctional, was also used for  testing purpose. In the doping calculations, the hole densit y was tuned by \nremoving electrons from the cell, with a jellium background with  opposite charge added to maintain charge \nneutrality  and the atomic positions are reoptimized at different hole densities . \nPhonon dispersion curves were calculated by the PHONOP Y package5 on the basis of Density Functional \nPerturbation Theory (DFPT).  Curie temperatures  were estimated using Monte Carlo simulations, as \nimplemented in the V AMPIRE package6. The simulated system s for all materials consist  of a platelet with  at \nleast 10000 spins with a rectangular supercell . The spins were thermalized for 10000 equilibrium steps, \nfollowed by 20000 averaging steps for the calculation of the thermal equilibrium magnetization at every \ntemperature.  \nThe ab initio evolutionary algorithm was performed by USPEX7,8, interfaced with VASP, was used to find \nthe potential structures that are ferromagnetic under ho le doping. The variable -composition searching was \nchosen with the total atom numbers of the 2D crystals set to be 2 -8, and their thicknesses and vacuum spaces \nrestricted to be 4 Å and 20 Å, respectively. 150 groups of symmetry were used to produce random s ymmetric \nstructure generator for initial population. Then, the full structure relaxations were performed, and the most \nstable and metastable structures were screened and inherited into the next generation by comparing their \nformation enthalpy. The number o f generation is set to 40.   \n \n \n \n \n \n \n List of Structures  \nMore details about the  122 2DHDFM  are summarized in the tables and they are provided  in term of  their \nmolecular formula, and classified by  their space group s. The 2D materials are selected if their spin polarization \nenergies are larger than 10 meV/hole. In the tables, the 2dmat -ID and MC2D -ID are the identification number  \nof the 2D materials from the 2DMatPedia  database9 and the supplementary information of the article10, \nrespectively. Meanwhile, if the materials exist in the Computational 2D Materials Database (C2DB)11, they \nwill be marked with ✓. At the same time, if the material is solely found by the USPEX and not existed in the \ndatabases mentioned above, this material is also marked with ✓.  \nThe convex hulls are defined as the enthalpy of formation versus the composition of  different elements \nwith all possible stoichiometries , which are searched by USPEX . The formation enthalpy  in the convex hull \nfigure is obtained by subtracting the total energy of the total energy of the compound by the total energies of \nthe 2D structures of its constituent element s found by USPEX. The corresponding atomic structures with the \nunit-cell marked out by the black dashed line, as well as the lattice geometry and the atomic  positions  in \nfractional coordinates are also provided. The spin polarization energy is defined as the total energy of the non -\nmagnetic state minus the one of the ferromagnetic sta te. Thus, a positive spin polarization energy indicates \nthat the ferromagnetic state is more stable.  The magnetic configurations, i.e., ferromagnetic (FM), and \ndifferent antiferromagnetic (AFM) structures and their corresponding spin Hamiltonian are given to extract \nthe nearest -neighbor and second nearest -neighbor and/or third nearest -neighbor exchange coupling parameter s. \nThe energy of the FM structure is calculated using the unit -cell while the supercell used to obtain the energies \nfor the AFM structures are marked by the black dashed lines. Magnetic anisotropic energies (MAE) are \ndetermined by obtaining the energy difference by orienting the spins in the out -of-plane and the in -plane (from \n0° to 360°) direction. Finally, the Curie temperatures (Tc) are evaluated using Monte Carlo calculations. The \ntemperature dependent magnetization is  fitted using the Curie -Bloch equation in the classical limit:    \nm(𝑇)=(1−𝑇\n𝑇𝑐)𝛽 \nwhere T is the temperature and β is a critical exponent.  \n \n \n  P3m1  \n1. BeF2 2. MgF 2 3. MgCl 2 4. CaF 2 5. CaCl 2 6. CaBr 2 7. CaI 2 \n8. SrF 2 9. SrCl 2 10.SrBr 2 11. SrI2 12. BaF 2 13. BaCl 2 14. BaBr 2 \n15. BaI 2 16. ZnF 2 17. ZnCl 2 18. ZnBr 2 19. CdF 2 20. CdCl 2 21. CdBr 2 \n22. CdI 2 23. HgF 2 24. HgCl 2 25. HgBr 2 26. HgI 2 27. PbCl 2 28. PbBr 2 \n29. Al2S2 30. Al2Se2 31. In2S2 32. SiS 33. SnS 34. PbS 35. PbSe  \n36. ZnSe  37. PdS 2 38. SnS 2 39. PbS 2 40. Al2O3 41. GeO2 42. SnO 2 \n43. TiO 2 44. ZrO 2 45. Al2N2 46. Mg(OH) 2 47. Ca(OH) 2 48. Zn(OH) 2 49. Cd(OH) 2 \n50. TeC       \nP6m2  \n51. PbCl 2 52. PbBr 2 53. Al2S2 54. Al2Se2 55. Ga2S2 56. In2S2 57. Tl2S2 \n58. CdS 59. ZnS 60. BeO 61. CdO  62. TeO 3 63. AlN 64. GaN  \n65. InN 66. TlN      \nP4m2  \n67. BeF 2 68. BeCl 2 69. MgF 2 70. MgCl 2 71. MgBr 2 72. CaF 2 73. CaCl 2 \n74. CaBr 2 75. CaI 2 76. SrF 2 77. SrCl 2 78. SrBr 2 79. SrI2 80. BaF 2 \n81. BaCl 2 82. BaBr 2 83. BaI 2 84. ZnF 2 85. ZnCl 2 86. ZnBr 2 87. CdF 2 \n88. CdCl 2 89. CdBr 2 90. CdI 2 91. SnS 2 92. PbS 2   \nP4/mmm  \n93. SiF 4 94. GeF 4 95. SnF 4 96. PbF 4 97. LiCl 98. LiBr  99. LiI \n100. NaBr  101. NaI 102. KBr 103. SrO 2    \nP4/nmm  \n104. Cd2S2 105. Ge2O2 106. Pb2O2 107. Sn2O2 108. Al2N2 109. Li2(OH) 2 \n110. Na2(OH) 2   111. Pb2Br2F2   112. Sc2Br2O2   113. Sr2Br2H2 \nThe rest  \n114. AlF 3   115. Al4N4   116. Ga4N4   117. B2O3   118. C3N4   119. In2Cl2O2   120. In2Br2O2 \n121. Mg 2Ge2O6   122. TiPbO 3 \n \n \n \n 1. BeF 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-43 - P3m1 9.28 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Be1F2                                  \n   1.00000000000000      \n   2.6145080605214228  0.0000000000000000  0.0000000000000000  \n  -1.3072541701761635  2.2642303665544037  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  23.5298450000000017  \n   Be   F  \n   1     2  \nDirect  \n  0.0000000084641769  0.9999999978600869  0.9999999098079044  \n  0.6666681049509435  0.3333319357115911  0.0384319141545930  \n  0.3333318865848796  0.6666680664283149  0.9615681760375026  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 2 K 2×1014 cm-2: 3 K \n 4×1014 cm-2: 15 K 6×1014 cm-2: 26 K \n2. MgF 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-1016  - P3m1 7.43 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Mg1F2                                  \n   1.00000000000000      \n   3.1052391845157428    0.0000000000000000    0.0000000000000000  \n  -1.5526285592966793    2.6892109035038221    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.5298450000000017  \n   Mg   F  \n   1     2  \nDirect  \n  0.9999999999447198  0.9999999999375007  0.0000000006161187  \n  0.6666669873034792  0.3333329995349033  0.0399737101175219  \n  0.3333330127518010  0.6666670005275961  0.9600262892663594  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 4 K 2×1014 cm-2: 12 K \n 4×1014 cm-2: 40 K 6×1014 cm-2: 63 K \n3. MgCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n118 ✓ 2dm-3734  - P3m1 6.00 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Mg1Cl2                                  \n   1.00000000000000      \n   3.6665835364464066    0.0000000000000000    0.0000000000000000  \n  -1.8332905255061329    3.1753551780468938    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.7262264346999991  \n   Mg   Cl  \n     1     2  \nDirect  \n  0.6666666683536562  0.3333333367887548  0.4999999819558667  \n  0.0000000213911591  0.0000000059911471  0.5603683136885778  \n  0.3333333102710228  0.6666666572517741  0.4396317043577582  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 8 K 2×1014 cm-2: 26 K \n 4×1014 cm-2:158 K 6×1014 cm-2: 224 K \n4. CaF 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-253 - P3m1 7.05 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ca1F2                                  \n   1.00000000000000      \n   3.6146954836474197  0.0000000000000000  0.0000000000000000  \n  -1.8073574282525919  3.1304124666103319  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  23.0162860000000009  \n   Ca   F  \n   1     2  \nDirect  \n  0.9999999944597491  0.9999999764067269  0.0000000056559131  \n  0.6666661430052940  0.3333340805736000  0.0416578000119898  \n  0.3333338625349569  0.6666659430196731  0.9583421943320971  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 7 K 2×1014 cm-2: 18 K \n 4×1014 cm-2: 67 K 6×1014 cm-2: 70 K \n5. CaCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-290 - P3m1  5.80 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ca1Cl2                                 \n   1.00000000000000      \n   4.1328322659408245  0.0000000000000000  0.0000000000000000  \n  -2.0664142106727197  3.5791387990563859  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  23.0162860000000009  \n   Ca   Cl  \n   1     2  \nDirect  \n  0.0000000041059991  0.0000000017533566  0.9999999944880287  \n  0.6666666435974022  0.3333333674675671  0.0609587123737469  \n  0.3333333522965987  0.6666666307790834  0.9390412931382244  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 20 K 2×1014 cm-2: 52 K \n 4×1014 cm-2: 92 K 6×1014 cm-2: 49 K \n6. CaBr 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-618 - P3m1 5.02 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ca1Br2                                 \n   1.00000000000000      \n   4.2897231099517441  0.0000000000000000  0.0000000000000000  \n  -2.1448649768165291  3.7150068475578428  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000   22.7427169999999990  \n   Ca   Br  \n    1     2  \nDirect  \n  0.9999999673048805  0.0000000562025804  0.9999999967490183  \n  0.6666668435005079  0.3333332155573601  0.9312910311879961  \n  0.3333331891946187  0.6666667282400596  0.0687089720629928  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 23 K 2×1014 cm-2: 62 K \n 4×1014 cm-2: 89 K 6×1014 cm-2: 104 K \n7. CaI 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n29 ✓ - - P3m1 3.91 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ca1I2                                 \n   1.00000000000000      \n     4.5415685628997693  0.0000000000000000    0.0000000000000000  \n    -2.2707073727749525  3.9331250512874485    0.0000000000000000  \n     0.0000000000000000  0.0000000000000000   18.5370006561279332  \n   Ca   I  \n    1     2  \nDirect  \n  0.9999770747201140  0.9999822193259575  0.5000122657880297  \n  0.6665529080537027  0.3332915086209738  0.5953312750419357  \n  0.3332904005901298  0.6665868203645928  0.4046833791565660  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 5 K 2×1014 cm-2: 25 K \n 4×1014 cm-2:48 K 6×1014 cm-2: 42 K \n8. SrF 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-325 - P3m1 6.54 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sr1F2                                  \n   1.00000000000000      \n   3.9100925209269715    0.0000000000000000    0.0000000000000000  \n  -1.9550419704333364    3.3862411587467638    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.5298450000000017  \n   Sr   F  \n   1     2  \nDirect  \n  0.9999999919565923  0.9999999886292770  0.0000000036421426  \n  0.6666661064332828  0.3333339471161452  0.0412259079451118  \n  0.3333339016101320  0.6666660642545779  0.9587740884127456  \n \nProjected band structure  \nand density of states  Magnetic moment  and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 9 K 2×1014 cm-2: 22 K \n 4×1014 cm-2: 50 K 6×1014 cm-2: 63 K \n9. SrCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-491 - P3m1 5.76 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sr1Cl2                                 \n   1.00000000000000      \n   4.4521103983943764    0.0000000000000000    0.0000000000000000  \n  -2.2260537991918081    3.8556412623226648    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.52984500000 00017  \n   Sr   Cl  \n   1     2  \nDirect  \n  0.0000000015717418  0.0000000177476167  0.0000000092065093  \n  0.6666665225228101  0.3333334107469312  0.0596025144725019  \n  0.3333334759054551  0.6666665715054592  0.9403974763209888  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 16 K 2×1014 cm-2: 47 K \n 4×1014 cm-2: 83 K 6×1014 cm-2: 48 K \n10. SrBr 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-349 - P3m1 5.00 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sn1Te1                                 \n   1.00000000000000      \n   4.1801374371344826    0.0000000000000000    0.0000000000000000  \n  -2.0900848226707285    3.6201327930791019    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.3597700000000010  \n   Sn   Te  \n   1     1  \nDirect  \n  0.9999798074185620  0.9999593082243052  0.5382850831818260  \n  0.3333351925814441  0.6666706917756997  0.4626419168181712  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping co ncentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 29 K 2×1014 cm-2: 75 K \n 4×1014 cm-2: 101 K 6×1014 cm-2: 126 K \n11. SrI2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-980 - P3m1 4.33 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sr1I2                                  \n   1.00000000000000      \n   4.8374045014613749    0.0000000000000000    0.0000000000000000  \n  -2.4187021135054843    4.1893150489074769    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.74271699999 99990  \n   Sr   I  \n   1     2  \nDirect  \n  0.9999999763343652  0.9999999463252820  0.0000000200908161  \n  0.6666667330449343  0.3333332401574580  0.9211748513637446  \n  0.3333332906207005  0.6666668135172600  0.0788251285454393  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 19 K 2×1014 cm-2: 59 K \n 4×1014 cm-2: 81 K 6×1014 cm-2: 107 K \n12. BaF 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-519 - P3m1 6.75 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ba1F2 \n   1.00000000000000      \n     3.9612593115589059    0.0000000000000000    0.0000000000000000  \n    -1.9806442192289506    3.4306998803184459    0.0000000000000000  \n     0.0000000000000000    0.0000000000000000   23.5298450000000017  \n   Ba   F  \n     1     2  \nDirect  \n  0.0000012602392090  0.0000002237310426  0.9999981429803526  \n  0.6666713792017092  0.3333413211465981  0.0500095127199103  \n  0.3333273605590890  0.6666584551223664  0.9499923442997513  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 33 K 2×1014 cm-2: 91 K  \n 4×1014 cm-2: 211 K 6×1014 cm-2: 337 \nK  \n13. BaCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-375 - P3m1 5.62 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ba Cl2                                 \n   1.00000000000000      \n   4.8143921693898761  0.0000000000000000  0.0000000000000000  \n  -2.4071905294981946  4.1693892796060368  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  23.5298450000000017  \n   Ba   Cl  \n   1     2  \nDirect  \n  0.0000000216941487  0.0000000156677160  0.0000000113663816  \n  0.6666662872184119  0.3333337859237773  0.0591696875755261  \n  0.3333336910874394  0.6666661984085067  0.9408303010580994  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis . +  \n Tc: 1×1014 cm-2: 15 K 2×1014 cm-2: 44 K  \n 4×1014 cm-2: 88 K 6×1014 cm-2: 90 K  \n14. BaBr 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-606 - P3m1 4.95 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ba1Br2                                 \n   1.00000000000000      \n   4.9850549887801128  0.0000000000000000  0.0000000000000000  \n  -2.4925186364639837  4.3171932506893418  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  23.5298450000000017  \n   Ba   Br  \n   1     2  \nDirect  \n  0.9999999932224100  0.9999999868052072  0.0000000047005386  \n  0.6666665621886381  0.3333334368866261  0.0655372747501346  \n  0.3333334445889591  0.6666665763081667  0.9344627205493268  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \n Tc: 1×1014 cm-2: 27 K 2×1014 cm-2: 81 K  \n 4×1014 cm-2: 139 K 6×1014 cm-2: 181 \nK  \n15. BaI 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-804 - P3m1 4.15 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ba1I2                                  \n   1.00000000000000      \n   5.2073983008725655  0.0000000000000000  0.0000000000000000  \n  -2.6037054740318659   .5097379194447873  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  22.7427169999999990  \n   Ba   I  \n   1     2  \nDirect  \n  0.0000000031733549  0.0000000081148528  0.9999999910277495  \n  0.6666664477720730  0.3333334854018304  0.9217489752268904  \n  0.3333335490545721  0.6666665064833168  0.0782510337453601  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \n Tc: 1×1014 cm-2: 32 K 2×1014 cm-2: 76 K  \n 4×1014 cm-2: 122 K 6×1014 cm-2: 162 \nK  \n16. ZnF2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - - ✓ P3m1 4.47 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zn1F2                                 \n1.00000000000000      \n3.1800763879302671    0.0000000000000000    0.0000000000000000  \n-1.5901079974308439    2.7539876665380705    0.0000000000000000  \n0.0000000000000014    0.0000000000000024   22.7725935562999986  \n   Zn   F  \n   1     2  \nDirect  \n  0.6666707971307986  0.3333293903718726  0.5000079107596136  \n  0.3333324786651346  0.6666674978137763  0.5431399079587464  \n  0.9999967242521777  0.0000031119105515  0.4568600920412536  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 8 K 2×1014 cm-2: 32 K \n 4×1014 cm-2: 95 K 6×1014 cm-2: 155 K \n17. ZnCl2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n238 ✓ 2dm-4402  - P3m1 4.46 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zn1Cl2                                 \n   1.00000000000000      \n   3.6006943636000002    0.0000000000000000    0.0000000000000000  \n  -1.8003471818000001    3.1182927901999999    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.7725935562999986  \n   Zn   Cl  \n   1    2  \nDirect  \n  0.6666666666880445  0.3333333333760891  0.4999999818575773  \n  0.3333333333440223  0.6666666666880445  0.5601819124713145  \n  0.0000000000160369  0.0000000000320668  0.4398181056733037  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2:40 K \n 4×1014 cm-2: 97 K 6×1014 cm-2: 161 K  \n18. ZnBr 2` \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n237 ✓ 2dm-5204  - P3m1 3.47 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zn1Br2                                 \n   1.00000000000000      \n   3.8002466363168996    0.0000000000000000    0.0000000000000000  \n   1.9001385057259335    3.2911013689901698    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.0561270000000000  \n   Zn   Br  \n   1     2  \nDirect  \n  0.0000000025429330  0.9999999899984786  0.4999999992549746  \n  0.6666660428341586  0.6666660416888561  0.4355832714836581  \n  0.3333339546229013  0.3333339683126582  0.5644167292613744  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 102 K 6×1014 cm-2: 179 K \n19. CdF2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-766 - P3m1 3.83 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1F2                                  \n   1.00000000000000      \n   3.5610764138459019    0.0000000000000000    0.0000000000000000  \n  -1.7808839336124016    3.0837829331036968    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.95605199999 99997  \n   Cd   F  \n   1     2  \nDirect  \n  0.0000000010281257  0.0000000007603228  0.9999999996078505  \n  0.3333344642687417  0.6666654662189303  0.0436518055167469  \n  0.6666655347031326  0.3333345330207468  0.9563481948754026  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 18 K 2×1014 cm-2: 64 K \n 4×1014 cm-2: 186 K 6×1014 cm-2: 320 K \n20. CdCl2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n32 - 2dm-3485  - P3m1 3.89 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1Cl2                                 \n   1.00000000000000      \n   3.9076173896460085    0.0000000000000000    0.0000000000000000  \n   1.9538089758997854    3.3840958059197010    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.86522799999 99983  \n   Cd   Cl  \n   1     2  \nDirect  \n  0.9999999951674496  0.0000000068514439  0.4999999994666027  \n  0.6666667349110043  0.6666666788783928  0.4370233396324821  \n  0.3333332699215532  0.3333333142701633  0.5629766609009152  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 70 K 2×1014 cm-2: 41 K \n 4×1014 cm-2: 117 K 6×1014 cm-2: 162 K \n21. CdBr2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n31 - 2dm-3696  - P3m1  3.22 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1Br2                                 \n   1.00000000000000      \n   4.0706621156352378    0.0000000000000000    0.0000000000000000  \n  -2.0353408016766656    3.5252915802523042    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.24151600000 00007  \n   Cd   Br  \n   1     2  \nDirect  \n  0.9999999992555857  0.0000000008387815  0.5000121988331898  \n  0.3333332777723470  0.6666666985006202  0.5674896742607487  \n  0.6666667229720673  0.3333333006605983  0.4324991269060590  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 43 K \n 4×1014 cm-2: 107 K 6×1014 cm-2: 190 K \n22. CdI2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n33 - 2dm-4402  - P3m1 2.49 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1I2                                  \n   1.00000000000000      \n   4.3316988530728873    0.0000000000000000    0.0000000000000000  \n  -2.1658519942253878    3.7513592953485837    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.4358480000000000  \n   Cd   I  \n   1     2  \nDirect  \n  0.0000000000086544  0.9999999991882902  0.5000000011422969  \n  0.3333332763331356  0.6666666941107380  0.5732266571983189  \n  0.6666667236582100  0.3333333067009789  0.4267733416593913  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 98 K 6×1014 cm-2: 141 K \n23. HgF 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-1267  - P3m1 1.83 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Hg1F2                                  \n   1.00000000000000      \n   3.6961332543963730    0.0000000000000000    0.0000000000000000  \n  -1.8480904290674907    3.2009317551727383    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.9560519999999997  \n   Hg   F  \n     1     2  \nDirect  \n  0.9999999999731912  0.0000000000039009  0.0000000000325286  \n  0.3333340419202813  0.6666660076113686  0.0438198432261103  \n  0.6666659581065275  0.3333339923847305  0.9561801567413610  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 150 K 6×1014 cm-2: 333 K \n24. HgCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-704 - P3m1 2.43 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Hg1Cl2                                 \n   1.00000000000000      \n   3.9910470260351625    0.0000000000000000    0.0000000000000000  \n  -1.9955138134886130    3.4563539216977830    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.9560519999999997  \n   Hg   Cl  \n   1     2  \nDirect  \n  0.0000000003411529  0.0000000009434515  0.0000000002844232  \n  0.3333329915070777  0.6666669812547212  0.0636426794320073  \n  0.6666670081517623  0.3333330178018272  0.9363573202835695  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 16 K 2×1014 cm-2: 62 K \n 4×1014 cm-2: 176 K 6×1014 cm-2: 270 K \n25. HgBr2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-5582  - P3m1 2.03 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Hg1Br2                                 \n1.00000000000000      \n4.1414124754368009    0.0000000000000000    0.0000000000000000  \n-2.0707091579513657    3.5865667601677167    0.0000000000000000  \n0.0000000000000014    0.0000000000000024   22.9560519999999997  \nHg   Br  \n1     2  \nDirect  \n  0.0000000013484254  0.0000000062100511  0.4999999727847353  \n  0.3333329961901370  0.6666670063158548  0.5691642222798095  \n  0.6666670024614234  0.3333329874740940  0.4308357777201905  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 63 K \n 4×1014 cm-2: 173 K 6×1014 cm-2: 264 K \n26. HgI2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-1787  - P3m1 1.42 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Hg1I2                                  \n1.00000000000000      \n4.3801552151490224    0.0000000000000000    0.0000000000000000  \n-2.1900890609255428    3.7933179985396399    0.0000000000000000  \n 0.0000000000000014    0.0000000000000024   22.9560519999999997  \n   Hg   I  \n     1     2  \nDirect  \n  0.9999999734892455  0.9999999764134557  0.4999999925911638  \n  0.3333333609336862  0.6666664630857753  0.5758827326418228  \n  0.6666666655770612  0.3333335605007690  0.4241172673581772  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 120 K 6×1014 cm-2: 185 K \n27. PbCl2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-1488  - P3m1 3.07 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  PbCl2                                 \n1.00000000000000      \n4.3970613879130758    0.0000000000000000    0.0000000000000000  \n -2.1985156727696702    3.8079873908668000    0.0000000000000000  \n0.0000000000000014    0.0000000000000025   23.1145000457999998  \n   Pb   Cl  \n   1     2  \nDirect  \n  0.9999999958593691  0.9999999535866948  0.5000068738244536  \n  0.3333332074778212  0.6666674573040368  0.4357256432128835  \n  0.6666667966628239  0.3333325891092471  0.5642743567871165  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n \n \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 209 K 6×1014 cm-2: 394 K \n28. PbBr 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ - ✓ P3m1 2.75 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb1Br2                                \n   1.00000000000000      \n   4.4853000640999996    0.0000000000000000    0.0000000000000000  \n  -2.2426500319999998    3.8843837991000001    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.1145000457999998  \n   Pb   Br  \n   1     2  \nDirect  \n  0.0000000000000000  0.0000000000000000  0.5000000000000000  \n  0.3333333730000021  0.6666666870000029  0.4278699759999967  \n  0.6666666269999979  0.3333333129999971  0.5721300240000033  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n \n \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 49 K \n 4×1014 cm-2: 172 K 6×1014 cm-2: 301 K \n29. Al2S2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- ✓ 2dm-1791  - P3m1 2.15 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Al2S2                                  \n   1.00000000000000      \n   3.5921342265381568  0.0000000000000000   0.0000000000000000  \n  -1.7960664300799600  3.1108798811072038  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  24.6231550000000006  \n   Al   S  \n   2     2  \nDirect  \n  0.0000001739833166  0.9999997859377459  0.0073385258748786  \n  0.9999998212522385  0.0000002233968743  0.1125265962129731  \n  0.6666667216689461  0.3333331687641419  0.9642231695601424  \n  0.3333332830954916  0.6666668219012379  0.1556407083519957  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−6𝐽1𝑆2+3𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \n Tc: 0.5×1014 cm-2: 2 K 1×1014 cm-2: 6 K  \n 2×1014 cm-2: 21 K 4×1014 cm-2: 66 K  \n30. Al2Se2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- ✓ 2dm-835 - P3m1 2.14 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Al2Se2                                 \n   1.00000000000000      \n   3.7890287407636372  0.0000000000000000  0.0000000000000000  \n  -1.8945134127802470  3.2813958126517888  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000   24.6231550000000006  \n   Al   Se \n   2     2  \nDirect  \n  0.0000001475127291  0.9999998252788131  0.0076291995589983  \n  0.9999998536617909  0.0000001758314951  0.1122351700220108  \n  0.6666668171954555  0.3333331551883560  0.9605922183525735  \n 0.3333331816300174  0.6666668437013357  0.1592724120664144  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−3𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−6𝐽1𝑆2+3𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 2 K 0.8×1014 cm-2: 5 K  \n 1×1014 cm-2: 7 K 2×1014 cm-2: 25 K  \n31. In2S2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-1655  - P3m1 1.59 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  In2S2                                  \n   1.00000000000000      \n   3.9255056553423078    0.0000000000000000    0.0000000000000000  \n  -1.9627590902974188    3.3995840234451506    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   24.62315500000 00006  \n   In   S  \n   2     2  \nDirect  \n  0.0000002887760360  0.9999998500440341  0.0030620583553400  \n  0.9999997100388427  0.0000001837059500  0.1167939975452299  \n  0.6666671347126112  0.3333330130271150  0.9556137015571053  \n  0.3333328664725101  0.6666669532229008  0.1642592425423217  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−3𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−6𝐽1𝑆2+3𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 2 K 0.8×1014 cm-2: 4 K \n 1×1014 cm-2: 9 K 2×1014 cm-2: 30 K \n32. SiS \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-776 - P3m1 2.20 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Si1S1                                  \n   1.00000000000000      \n   3.3033376893731692    0.0000000000000000    0.0000000000000000  \n  -1.6516683655338755    2.8607747078457786    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   21.4786740000000016  \n   Si   S  \n   1     1  \nDirect  \n  0.3333332618974936  0.6666667344429342  0.1629379298674891  \n  0.6666667381025064  0.3333332655570658  0.1012060701325126  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n \n \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−𝐽1𝑆2+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 6 K 2×1014 cm-2: 17 K \n 4×1014 cm-2: 58 K 6×1014 cm-2: 114 K \n33. SnS \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-280 - P3m1 2.32 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sn1S1                                  \n   1.00000000000000      \n   3.7550081475776218    0.0000000000000000    0.0000000000000000  \n  -1.8775079252228946    3.2519302232273097    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   21.4786740000000016  \n   Sn   S  \n   1     1  \nDirect  \n  0.3333332934743112  0.6666667044025658  0.1660752026681962  \n  0.6666667065256888  0.3333332955974342  0.0980687973318055  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n \n \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−𝐽1𝑆2+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 6 K 2×1014 cm-2: 26 K \n 4×1014 cm-2: 88 K 6×1014 cm-2: 84 K \n34. PbS \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-3013  - P3m1 2.05 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb1S1                                  \n   1.00000000000000      \n   3.9574464751196028    0.0000000000000000    0.0000000000000000  \n   1.9787250178403992    3.4272481510099406    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   21.5987730000000013  \n   Pb   S  \n   1     1  \nDirect  \n  0.9999998641125032  0.9999998528727545  0.5332669634183915  \n  0.3333331358875000  0.3333331471272487  0.4667330365816085  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 6 K 1×1014 cm-2: 25 K \n 2×1014 cm-2: 78 K 4×1014 cm-2: 148 K \n35. PbSe \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-458 - P3m1 1.87 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb1Se1                                 \n   1.00000000000000      \n   4.0889834380812813    0.0000000000000000    0.0000000000000000  \n  -2.0444941902647757    3.5411621784365397    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   21.47867400000 00016  \n   Pb   Se  \n   1     1  \nDirect  \n  0.3333333170308492  0.6666666697625487  0.1680887920243137  \n  0.6666666829691508  0.3333333302374513  0.0960552079756880  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: - K 0.8×1014 cm-2: 7 K \n 1×1014 cm-2: 11 K 2×1014 cm-2: 41 K \n36. ZnSe  \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-6047  - P3m1 1.75 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zn1Se1                                 \n   1.00000000000000      \n   4.0978712846507142    0.0000000000000000    0.0000000000000000  \n  -2.0488732777045082    3.5488967196092260    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.16693000000 00007  \n   Zn   Se  \n   1     1  \nDirect  \n  0.0000053148122987  0.9999947018611266  0.4953029702306182  \n  0.6666616851876981  0.3333382981388766  0.5046970297693818  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 25 K 2×1014 cm-2: 62 K \n 4×1014 cm-2: 104 K  6×1014 cm-2: 92 K \n37. PdS2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-71 - P3m1 1.26 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pd1S2                                  \n   1.00000000000000      \n   3.5495563932969119    0.0000000000000000    0.0000000000000000  \n  -1.7747775327287703    3.0740067111284404    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.61340099999 99996  \n   Pd   S  \n   1     2  \nDirect  \n  0.0000000094028891  0.9999999939163047  0.0000001727369536  \n  0.3333334999086475  0.6666665007981649  0.9451683186251998  \n  0.6666664906884634  0.3333335052855233  0.0548315086378466  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 2 K 2×1014 cm-2: 5 K \n 4×1014 cm-2: 9 K 6×1014 cm-2: 28 K \n38. SnS2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n184 ✓ 2dm-3203  - P3m1 1.58 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sn1S2                                   \n   1.00000000000000      \n   3.6986775778313943    0.0000000000000000    0.0000000000000000  \n  -1.8493394205076867    3.2031503712734675    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.9654742180 999989  \n   Sn   S  \n   1     2  \nDirect  \n  0.0000000020600694  0.9999999885573345  0.5000000109287441  \n  0.3333333536954370  0.6666666786313584  0.4357795055875542  \n  0.6666666442172726  0.3333333328113000  0.5642204834815274  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 64 K 6×1014 cm-2: 136 K  \n39. PbS2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-773 - P3m1 0.72 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb1S2                                  \n   1.00000000000000      \n   3.8539737416739341    0.0000000000000000    0.0000000000000000  \n  -1.9269871390013520    3.3376389717959429    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.1928920000000005  \n   Pb   S  \n   1     2  \nDirect  \n  0.0000000094714636  0.0000000083801268  0.9999999986081747  \n  0.3333331807494631  0.6666667919214007  0.0665891499451021  \n  0.6666668097790733  0.3333331996984725  0.9334108514467232  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in-plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 26 K \n 4×1014 cm-2: 88 K 6×1014 cm-2: 176 K  \n40. Al2O3 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-2031  - P3m1 5.06 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Al2 O3                                  \n   1.00000000000000      \n   2.8880760942522561  0.0000000000000000  0.0000000000000000  \n  -1.4440323919937297  2.5011505816539961  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  24.6292340000000003  \n   Al   O  \n   2     3  \nDirect  \n  0.9999991221962148  0.0000009400556067  0.6275054727579317  \n  0.3333338851025545  0.6666660406841700  0.5156122354868558  \n  0.3333333154995515  0.6666667389266365  0.6550782369124093  \n  0.9999997577686770  0.0000001839094494  0.4880375566537793  \n  0.66666691 94329984  0.3333330964241341  0.5715634981890148  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−9𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2−5𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \n \n \n \n \n Tc: 1×1014 cm-2: 5 K 2×1014 cm-2: 11 K  \n 4×1014 cm-2: 26 K 6×1014 cm-2: 34 K  \n41. GeO2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-6440  ✓ P3m1 3.59 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ge1O2  \n   1.00000000000000      \n   1.4533940999104427    2.5173465541177111    0.0000000000000000  \n  -1.4533932615547911    2.5173466481221562    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.0000000000000000  \n   Ge   O  \n   1     2  \nDirect  \n  0.9999999915906770  0.9999999726070286  0.4999999723726987  \n  0.3333330089489692  0.3333331709755214  0.5488004338288164  \n  0.6666669994603538  0.6666668564174500  0.4511995937984778  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 2 K 2×1014 cm-2: 5 K \n 4×1014 cm-2: 11 K 6×1014 cm-2: 14 K \n42. SnO 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-5126  - P3m1 2.58 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sn1O2                                  \n   1.00000000000000      \n   3.2314916001454064    0.0000000000000000    0.0000000000000000  \n   1.6157408761073357    2.7985563250660106    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.0306889999999989  \n   Sn   O  \n   1     2  \nDirect  \n  0.6666666176169969  0.6666665950884152  0.4999997058801355  \n  0.9999999018715400  0.9999999166747600  0.5460274559632978  \n  0.3333334805114632  0.3333334882368177  0.4539728381565666  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 4 K 2×1014 cm-2: 25 K \n 4×1014 cm-2: 50 K 6×1014 cm-2: 78 K \n43. TiO 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n-  2dm-3816  - P3m1 2.72 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ti1O2                                  \n   1.00000000000000      \n   2.9924450441581008    0.0000000000000000    0.0000000000000000  \n   1.4962176577629793    2.5915362450782649    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   21.93526999999 99992  \n   Ti   O  \n   1     2  \nDirect  \n  0.9999999976287199  0.0000000025867664  0.5000000167018683  \n  0.6666668161742066  0.6666668110108134  0.5440614346086221  \n  0.3333331861970734  0.3333331864024203  0.4559385486895025  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 10 K 2×1014 cm-2: 34 K \n 4×1014 cm-2: 122 K  6×1014 cm-2: 228 K  \n44. ZrO 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-2760  - P3m1 4.41 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zr1O2                                  \n   1.00000000000000      \n   3.2782740240424788    0.0000000000000000    0.0000000000000000  \n  -1.6391299774822352    2.8390725995268471    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   21.9372839999999982  \n   Zr   O  \n   1     2  \nDirect  \n  0.0000000003210161  0.0000000002877698  0.0000000040734349  \n  0.3333335748378232  0.6666664039804999  0.0439843153940842  \n  0.6666664248411678  0.3333335957317303  0.9560156805324809  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 8 K 2×1014 cm-2: 27 K \n 4×1014 cm-2: 95 K 6×1014 cm-2: 176 K  \n45. Al2N2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - - ✓ P3m1 3.50 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Al2N2  \n   1.00000000000000      \n 3.1569069837959960  0.5321837153485067  0.0000000000000000  \n   1.1181249120853491  3.0003695011230347  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  20.0000000000000000  \n   Al   N  \n   2     2  \nDirect  \n  0.3676633334244457  0.4537661869794078  0.4494394326520350  \n  0.0349973019316394  0.1202368071170525  0.5505605699356337  \n  0.3682612689215006  0.4535880144369600  0.5563400399759786  \n  0.0343940957224120  0.1204139914665703  0.4436599574363527  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \n Tc: 1×1014 cm-2: 5 K 2×1014 cm-2: 16 K  \n 4×1014 cm-2: 59K 6×1014 cm-2: 97 K  \n46. Mg(OH) 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n116 ✓ 2dm-5739  - P3m1 3.30 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Mg1 H2O2                               \n   1.00000000000000      \n   3.1751312257357447    0.0000000000000000    0.0000000000000000  \n  -1.5876568617590670    2.7496917417189422    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   24.0290010000000009  \n   Mg   H    O  \n   1     2     2  \nDirect  \n  0.0000000000649081  0.0000000000122924  0.4999999989443822  \n  0.6666665528492572  0.3333333508677612  0.5833277350975337  \n  0.3333334469813565  0.6666666491729387  0.4166722634303568  \n  0.6666689772502252  0.3333311958666556  0.5431118051139165  \n  0.33333102 28542530  0.6666688040803521  0.4568881974138108  \n \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 8 K 2×1014 cm-2: 23 K \n 4×1014 cm-2: 53 K 6×1014 cm-2: 117 K \n47. Ca(OH) 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n26 - 2dm-3595  - P3m1 3.68 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ca1H2O2                               \n   1.00000000000000      \n   3.6323417965460472    0.0000000000000000    0.0000000000000000  \n  -1.8161768156685321    3.1456965616529682    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   24.2520099999999985  \n   Ca   H    O  \n   1     2     2  \nDirect  \n  0.0000000001817000  0.9999999996367777  0.4999999993959605  \n  0.6666669284561024  0.3333330562961052  0.5869570112582636  \n  0.3333330701477379  0.6666669444442306  0.4130429870705470  \n  0.6666667311249554  0.3333334392343161  0.5469845406018976  \n  0.3333332700895042  0.6666665603885633  0.4530154616733242  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 13 K 2×1014 cm-2: 40 K \n 4×1014 cm-2: 102 K  6×1014 cm-2: 179 K  \n48. Zn(OH) 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-3835  - P3m1 2.31 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zn1H2O2                               \n   1.00000000000000      \n   3.2353808825138515    0.0000000000000000    0.0000000000000000  \n  -1.6180078068454582    2.8017390107423221    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   24.0492699999999999  \n   Zn   H    O  \n  1     2     2  \nDirect  \n  0.0000000029575205  0.9999999973844496  0.4999999581771419  \n  0.6666625554332768  0.3333374095621267  0.5832637052899159  \n  0.3333374438495511  0.6666625913620479  0.4167362088128357  \n  0.6666674842100448  0.3333328930466024  0.5429472369259187  \n  0.3333325135496068  0.6666671086447735  0.4570528907941878  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 10 K 2×1014 cm-2: 32 K \n 4×1014 cm-2: 122 K  6×1014 cm-2: 245 K \n49. Cd(OH) 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n30 - 2dm-4273  - P3m1 2.27 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1H2O2                               \n   1.00000000000000      \n   3.5799420594205853    0.0000000000000000    0.0000000000000000  \n  -1.7900444418957295    3.1002784089441624    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   24.1669234368999994  \n     Cd   H    O  \n      1     2     2 \nDirect  \n  0.9999999998872511  0.0000000000548539  0.5000000004885621  \n  0.3333334950215061  0.6666664932644011  0.5857309909998492  \n0.6666665049997746  0.3333335067164072  0.4142690100591722  \n  0.3333333550734849  0.6666666401951815  0.5455607873114161  \n  0.6666666449 619711  0.33333 33597691635  0.4544392111389257   \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 13 K 2×1014 cm-2: 51 K \n 4×1014 cm-2: 182 K  6×1014 cm-2: 341 K  \n50. TeC \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-6385  - P3m1  1.29 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Te1C1                                  \n   1.00000000000000      \n   3.3479315019084788    0.0000000000000000    0.0000000000000000  \n  -1.6739654406742093    2.8993885712899390    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   19.07737699999 99985  \n   Te   C  \n   1     1  \nDirect  \n  0.3333397911702534  0.6666799631394653  0.4712258947687573  \n  0.9999752088297456  0.9999500368605325  0.5297011052312470  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 2 K 1×1014 cm-2: 13 K \n 2×1014 cm-2: 59 K 4×1014 cm-2: 155 K  \n51. PbCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-2389  - P6m2 3.51 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb1Cl2                                 \n   1.00000000000000      \n   4.1792382900134513    0.0000000000000000    0.0000000000000000  \n  -2.0896228122357243    3.6193252552410371    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.83800000000 00010  \n   Pb   Cl  \n   1     2  \nDirect  \n  0.0000000478759219  0.9999999537932069  0.0000000242073526  \n  0.6666669767475994  0.3333329992222076  0.0713952306129286  \n  0.6666669753764722  0.3333330469845919  0.9286047451797188  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n \n \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 18 K 2×1014 cm-2: 60 K \n 4×1014 cm-2: 178 K 6×1014 cm-2: 300 K \n52. PbBr 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-2251  - P6m2 3.20 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb1Br2                                 \n   1.00000000000000      \n   4.3075062621630185    0.0000000000000000    0.0000000000000000  \n  -2.1537530794790190    3.7304106452124888    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.83800000000 00010  \n   Pb   Br  \n   1     2  \nDirect  \n  0.0000003932458341  0.9999995832829853  0.0000000056039582  \n  0.6666668147712116  0.3333332242805724  0.0773418257317360  \n  0.6666667919829479  0.3333331924364487  0.9226581686643058  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n \n \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 23 K 2×1014 cm-2: 56 K \n 4×1014 cm-2: 177 K  6×1014 cm-2: 253 K  \n53. Al2S2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- ✓ 2dm-289 - P6m2 2.10 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Al2S2                                  \n   1.00000000000000      \n   3.5798922574179861   0.0000000000000000  0.0000000000000000  \n  -1.7899470606356469  3.1002771729430374  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  25.0192729999999983  \n   Al   S  \n   2     2  \nDirect  \n  0.0000001465943669  0.9999998397079537  0.6845740298811052  \n  0.0000001467123454  0.9999998475117806  0.7883356698716995  \n  0.6666668464147989  0.3333331535408490  0.8309130204768067  \n  0.6666668602784895  0.3333331592394160  0.6419972797703934  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−6𝐽1𝑆2+𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \n Tc: 0.5×1014 cm-2: 3 K 1×1014 cm-2: 7 K  \n 2×1014 cm-2: 26 K 4×1014 cm-2: 36 K  \n54. Al2Se2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- ✓ 2dm-868 - P6m2 2.00 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Al2S e2                                  \n1.00000000000000      \n3.7778825182436613    0.0000000000000000    0.0000000000000000  \n-1.8889414578327082    3.2717419256532256    0.0000000000000000  \n0.0000000000000000    0.0000000000000000   25.0192729999999983   \nAl   S e  \n 2     2  \nDirect  \n  0.0000001429421630  0.9999998645452521  0.6848207534099870  \n  0.0000001493541149  0.9999998696059151  0.7880892077043526  \n  0.6666668562835198  0.3333331334181153  0.8346121006108405  \n0.6666668514202030  0.3333331324307238  0.6382979382748175  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentrati on  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−6𝐽1𝑆2+𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 2 K 0.8×1014 cm-2: 6K   \n 1×1014 cm-2: 10 K 2×1014 cm-2: 35K  \n55. Ga 2S2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-3608  - P6m2 2.37 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ga2S2                                  \n   1.00000000000000      \n   3.6284444130234483    0.0000000000000000    0.0000000000000000  \n  -1.8142235231982928    3.1423242251638892    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   24.6403500000000015  \n   Ga   S  \n   2     2  \nDirect  \n  0.9999998311713725  0.0000003277878093  0.4498288151669172  \n  0.9999996742776460  0.0000001562129910  0.5501698700550293  \n  0.3333333133584020  0.6666668247670984  0.4058138022918598  \n  0.3333331811925930  0.6666666912320949  0.5941875124862008  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−6𝐽1𝑆2+𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 2 K 0.8×1014 cm-2: 6 K \n 1×1014 cm-2: 10 K 2×1014 cm-2: 29 K \n56. In2S2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n92 ✓ 2dm-341 - P6m2 1.67 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  In2S2                                  \n   1.00000000000000      \n   3.9185666080547219    0.0000000000000000    0.0000000000000000  \n  -1.9592897715604429    3.3935744799206717    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   25.0192729999999983  \n   In   S  \n     2     2  \nDirect  \n  0.0000001819039639  0.9999998993744370  0.6799257115131994  \n  0.0000000934144708  0.9999998335445142  0.7929862264336265  \n  0.6666668458848051  0.3333331219688347  0.8401302385281824  \n  0.6666668787967538  0.3333331451122206  0.6327778235249895  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−6𝐽1𝑆2+𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 4 K 0.8×1014 cm-2: 10 K \n 1×1014 cm-2: 14 K 2×1014 cm-2: 43 K \n57. Tl2S2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-59 - P6m2 0.64 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Tl2S2                                  \n   1.00000000000000      \n   4.0686518437142709    0.0000000000000000    0.0000000000000000  \n  -2.0343271845599413    3.5235546946074092    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   25.0192729999999983  \n   Tl   S  \n   2     2  \nDirect  \n  0.0000002362939000  0.9999998127287384  0.6785093255514667  \n  0.0000001279460093  0.9999998324554156  0.7944036021698508  \n  0.6666667720654900  0.3333331170876477  0.8423251117015980  \n  0.6666668636945943  0.3333332377282048  0.6305819605770822  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−6𝐽1𝑆2+𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 5 K 0.8×1014 cm-2: 12 K \n 1×1014 cm-2: 18 K 2×1014 cm-2: 48 K \n58. CdS \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- ✓ 2dm-3242  - P6m2  1.63 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1S1                                  \n   1.00000000000000      \n  4.2580819629675677  0.0000000000000000    0.0000000000000000  \n -2.1290382627071618  3.6876071496348870    0.0000000000000000  \n  0.0000000000000000  0.0000000000000000   20.1669300000000007  \n   Cd   S  \n   1     1  \nDirect  \n  0.0000005926146045  0.9999996012260013  0.4990068239281555  \n  0.6666664073853923  0.3333333987740019  0.5009931760718445  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: - K 0.8×1014 cm-2: 60 \nK   \n 1×1014 cm-2: 111 K 2×1014 cm-2: - K  \n59. ZnS \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-6003  - P6m2 2.54 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zn1S1                                  \n   1.00000000000000      \n 3.8920917828964909  0.0000000000000000    0.0000000000000000  \n  -1.9460098501003000   3.3706717941884605    0.0000000000000000  \n  0.0000000000000000  0.0000000000000000   20.1669300000000007  \n   Zn   S  \n   1     1  \nDirect  \n  0.0000055430588475  0.9999947329815484  0.5000067577056129  \n  0.6666614569411493  0.3333382670184548  0.4999932422943871  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off-plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 27 \nK 0.8×1014 cm-2: 65 \nK   \n 1×1014 cm-2: 129 K 2×1014 cm-2: 111 K  \n60. BeO  \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - - ✓ P6m2 5.64 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Be1O1  \n   1.00000000000000      \n   2.6794513013437462    0.0000000000000000    0.0000000000000000  \n   1.3385932197899217    2.3222466752701587    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.0000000000000000  \n   Be   O  \n   1     1  \nDirect  \n  0.2477499428836225  0.9866502521319376  0.4999999576193730  \n  0.9144080571163826  0.6531267478680647  0.5000000423806270  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 45 K \n 4×1014 cm-2: 104 K  6×1014 cm-2: 149 K  \n61. CdO  \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- ✓ 2dm-6249  - P6m2  0.82 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1O1                                  \n   1.00000000000000      \n 3.6825886440488960  0.0000000000000000  0.0000000000000000  \n -1.8412959146795791  3.1892143383270488  0.0000000000000000  \n 0.0000000000000000  0.0000000000000000  20.6349209999999985  \n   Cd   O  \n   1     1  \nDirect  \n  0.9999998982562843  0.0000000992136862  0.5000065496344277  \n  0.3333331017437189  0.6666669007863106  0.4999934503655723  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K  \n 4×1014 cm-2: 45 K 6×1014 cm-2: 102 K  \n62. TeO 3 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-4882  - P6m2  0.86 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Te1O3                                  \n   1.00000000000000      \n   3.7689421325705479    0.0000000000000000    0.0000000000000000  \n  -1.8844707233302402    3.2639992105142546    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.5916649999999990  \n   Te   O  \n   1     3  \nDirect  \n  0.9999999967672437  0.0000000340517587  0.5000001130785705  \n  0.9999999778634034  0.9999999944631028  0.4238697578018247  \n  0.9999999783810836  0.9999999934514747  0.5761301269029389  \n  0.3333330469882725  0.6666669780336605  0.5000000022166660  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−𝐽2𝑆2−9𝐽3𝑆2  \n𝐸𝐴𝐹𝑀1=𝐸0+𝐽2𝑆2−5𝐽3𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−2𝐽1𝑆2−𝐽2𝑆2+3𝐽3𝑆2 \n𝐸𝐴𝐹𝑀3=𝐸0+2𝐽1𝑆2−𝐽2𝑆2+3𝐽3𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 15 K 2×1014 cm-2: 54 K \n 4×1014 cm-2: 77 K 6×1014 cm-2: 111 K  \n63. AlN \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- ✓ 2dm-3085  - P6m2  2.91 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Al N                                   \n1.00000000000000      \n  3.1261529833522546   0.0000000000000000    0.0000000000000000  \n -1.5630816990343190  2.7073252116660003    0.0000000000000000  \n  0.0000000000000000  0.0000000000000000   20.0000000000000000  \n   Al   N  \n    1     1  \nDirect  \n  0.3333330967632762  0.6666668749795690  0.5000000011677059  \n  0.9999999032367271  0.0000001250204278  0.4999999988322941  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off-plane (in -plane) easy axis .  \n Tc: 1×1014 cm-2: 16 K 2×1014 cm-2: 63 K  \n 4×1014 cm-2: 129 K 6×1014 cm-2: 186 K  \n64. GaN  \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-2992  - P6m2 2.16 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ga1N1                                  \n   1.00000000000000      \n   3.2095416635794085    0.0000000000000000    0.0000000000000000  \n  -1.6047941441788967    2.7795964982737638    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.0053009999999993  \n   Ga   N  \n   1     1  \nDirect  \n  0.3333558881396570  0.6667181924672363  0.4999606694951524  \n  0.0000061118603440  0.0000058075327587  0.5000393305048476  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 15 K 2×1014 cm-2: 66 K \n 4×1014 cm-2: 145 K  6×1014 cm-2: 205 K  \n65. InN \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-6424  - P6m2 0.57 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  In1N1                                  \n   1.00000000000000      \n   3.5856211591877236    0.0000000000000000    0.0000000000000000  \n  -1.7928138495758739    3.1052374082318037    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.00000000000 00000  \n   In   N  \n   1     1  \nDirect  \n  0.3333329143395574  0.6666670480444168  0.3853517022048223  \n  0.0000000856604458  0.9999999519555800  0.3853742977951811  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 29 K 2×1014 cm-2: 75 K \n 4×1014 cm-2: 151 K  6×1014 cm-2: 173 K  \n66. TlN \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n-  2dm-6412  - P6m2 0.002  \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Tl1N1                                  \n   1.00000000000000      \n   3.7270386939767035    0.0000000000000000    0.0000000000000000  \n  -1.8635196930302853    3.2277107372986182    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.00000000000 00000  \n   Tl   N  \n   1     1  \nDirect  \n  0.3333330275056099  0.6666669015254953  0.3853632018469071  \n  0.9999999724943933  0.0000000984745014  0.3853627981530892  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−12𝐽1𝑆2−12𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 30 K 2×1014 cm-2: 75 K \n 4×1014 cm-2: 145 K  6×1014 cm-2: 90 K \n67. BeF 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-951 - P4m2 8.57 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Be1 F2                                  \n   1.00000000000000      \n   2.7287100377127125  0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  2.7287099143497966  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  22.6561909999999997  \n   Be   F  \n   1     2  \nDirect  \n  0.9999999995816324  0.0000000004229719  0.0000000105880460  \n  0.0000000005111787  0.4999999998487610  0.9641986322388192  \n  0.4999999999071889  0.9999999997282671  0.0358013571731419  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 8 K 2×1014 cm-2: 25 K \n 4×1014 cm-2: 77 K 6×1014 cm-2: 146 K  \n68. BeCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-1681  - P4m2 5.40 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Be1Cl2                                 \n   1.00000000000000      \n   3.3430584071816685  0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  3.3430589003723514  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  22.6561909999999997  \n   Be   Cl  \n   1     2  \nDirect  \n  0.0000000210738023  0.9999999953296026  0.9999999744550081  \n  0.9999999542964204  0.5000000206131787  0.9471076325499652  \n  0.5000000246297702  0.9999999840572187  0.0528923929950338  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 9 K 2×1014 cm-2: 39 K  \n 4×1014 cm-2: 132 K 6×1014 cm-2: 236 K  \n69. MgF 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-1799  - P4m2 6.63 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n \n Mg1F2                                  \n   1.00000000000000      \n   3.3964268531687636    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.3966978452804462    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.65619099999 99997  \n   Mg   F  \n   1     2  \nDirect  \n  0.9999950786037601  0.0000018575789156  0.0000100712674111  \n  0.9999979803661745  0.5000084214424518  0.9628688135455903  \n  0.5000069410300654  0.9999907209786372  0.0371211151869986  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 12 K 2×1014 cm-2: 36 K \n 4×1014 cm-2: 91 K 6×1014 cm-2: 147 K  \n70. MgCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- ✓ 2dm-4009  - P4m2 5.30 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Mg1Cl2                                 \n   1.00000000000000      \n   3.8860546408558880    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.8860549545644787    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.6561909999999997  \n   Mg   Cl  \n   1     2  \nDirect  \n  0.4999999972785929  0.5000000026059581  0.4999999975733260  \n  0.5000000116682841  0.9999999847521011  0.4402447806052976  \n  0.9999999910531159  0.5000000126419408  0.5597552218213693  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 12 K 2×1014 cm-2: 41 K \n 4×1014 cm-2: 116 K 6×1014 cm-2: 175 K \n71. MgBr 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-1719  - P4m2 4.29 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Mg1Br2                                 \n   1.00000000000000      \n   4.0734703651003379    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.0734704136992779    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.6561909999999997  \n   Mg   Br  \n   1     2  \nDirect  \n  0.9999999865177287  0.0000000583975179  0.9999999995217266  \n  0.9999999528752852  0.5000000356119969  0.9335674807916021  \n  0.5000000606069790  0.9999999059904923  0.0664325196866713  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 45 K \n 4×1014 cm-2: 123 K  6×1014 cm-2: 172 K  \n72. CaF 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-690 - P4m2 6.35 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ca1F2                                  \n   1.00000000000000      \n   3.9545539778349492  0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  3.9545558625799551  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  22.6561909999999997  \n   Ca   F  \n   1     2  \nDirect  \n  0.9999999378009363  0.0000000158025415  0.9999999761766531  \n  0.0000001156571017  0.4999999831472124  0.9603490458055433  \n  0.4999999465419620  0.0000000010502461  0.0396509780178036  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 12 K 2×1014 cm-2: 30 K \n 4×1014 cm-2: 59 K 6×1014 cm-2: 88 K \n73. CaCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-1617  - P4m2 5.57 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ca1Cl2                                 \n   1.00000000000000      \n   4.5059930501459551  0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  4.5059939016548638  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  22.6561909999999997  \n   Ca   Cl  \n   1     2  \nDirect  \n  0.0000000006244889  0.9999999674575051  0.9999999883601873  \n  0.0000000040950212  0.5000000350374876  0.9398917786764045  \n 0.4999999952804899  0.9999999975050073  0.0601082329634011  \n \nProjected band structure  \nand density of states  Magnetic moment and spi n polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 5 K 2×1014 cm-2: 18 K \n 4×1014 cm-2: 60 K 6×1014 cm-2: 111 K  \n74. CaBr 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-265 - P4m2 4.83 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ca1 Br2                                 \n   1.00000000000000      \n   4.6464431803270090  0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  4.6464438158227459  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  22.6561909999999997  \n   Ca   Br  \n   1     2  \nDirect  \n  0.0000000070004447  0.0000000073439566  0.0000000149294976  \n  0.9999999878355510  0.4999999976954754  0.9320430549211309  \n  0.5000000051640043  0.9999999949605680  0.0679569301493785  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 24 K \n 4×1014 cm-2: 85 K 6×1014 cm-2: 117 K  \n75. CaI 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n29 ✓ - - P3m1 3.91 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ca1I2                                 \n   1.00000000000000      \n     4.5415685628997693  0.0000000000000000    0.0000000000000000  \n    -2.2707073727749525  3.9331250512874485    0.0000000000000000  \n     0.0000000000000000  0.0000000000000000   18.5370006561279332  \n   Ca   I  \n    1     2  \nDirect  \n  0.9999770747201140  0.9999822193259575  0.5000122657880297  \n  0.6665529080537027  0.3332915086209738  0.5953312750419357  \n  0.3332904005901298  0.6665868203645928  0.4046833791565660  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+3𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in-plane) easy axis .  \nTc: 1×1014 cm-2: 11 K 2×1014 cm-2: 37 K \n 4×1014 cm-2: 103 K  6×1014 cm-2: 105 K  \n76. SrF 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-736 - P4m2 5.95 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sr1F2                                  \n   1.00000000000000      \n   4.2629461998541327    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.2629736922051258    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.6561909999999997  \n   Sr   F  \n   1     2  \nDirect  \n  0.0000003237238957  0.0000003252872460  0.9999991898838232  \n  0.9999998753638977  0.4999997930998674  0.9587276808113430  \n  0.4999998009122066  0.9999998816128794  0.0412731293048409  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 13 K 2×1014 cm-2: 24 K \n 4×1014 cm-2: 54 K 6×1014 cm-2: - K \n77. SrCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-1453  - P4m2 5.31 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sr1Cl2                                 \n   1.00000000000000      \n   4.8543299045047013    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.8543300630774118    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.65619099999 99997  \n   Sr   Cl  \n   1     2  \nDirect  \n  0.0000000025609310  0.9999999744235950  0.9999999912219621  \n  0.9999999772453876  0.4999999599091751  0.9392024719621190  \n  0.5000000201936814  0.0000000656672441  0.0607975368159188  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 21 K 6×1014 cm-2: 67 K \n78. SrBr 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-470 - P4m2 4.67 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sr1Br2                                 \n   1.00000000000000      \n   5.0131425786126185    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    5.0131451710533899    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.6561909999999997  \n   Sr   Br  \n   1     2  \nDirect  \n  0.9999999712321994  0.9999999511638364  0.9999999691335972  \n  0.0000000168529155  0.4999999431819333  0.9317467581101084  \n  0.5000000119148922  0.0000001056542231  0.0682532727562943  \n \nProjected band structure  \nand density of states  Magnetic moment  and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 13 K \n 4×1014 cm-2: 71 K 6×1014 cm-2: 61 K \n79. SrI 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-1251  - P4m2 4.15 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sr1I2                                  \n   1.00000000000000      \n   5.2456463437361229    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    5.2456598323804133    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.65619099999 99997  \n   Sr   I  \n   1     2  \nDirect  \n  0.0000000830145623  0.0000000068729662  0.9999998411190703  \n  0.9999998019985270  0.4999999113557791  0.9208822368191676  \n  0.5000001149869178  0.0000000817712547  0.0791179220617622  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 33 K \n 4×1014 cm-2: 87 K 6×1014 cm-2: 65 K \n80. BaF 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-880 - P4m2 5.65 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ba1F2 \n1.00000000000000  \n4.5690708279870531  0.0000000000000000  0.0000000000000000  \n0.0000000000000000  4.5690607374883241  0.0000000000000000  \n\\  0.0000000000000000  0.0000000000000000  22.6561909999999997  \nBa   F  \n1     2  \nDirect  \n0.0000000045197126  0.0000000149795838  0.0000000784861740  \n0.9999999917091813  0.5000000175521677  0.9556433456282321  \n0.5000000037711061  0.9999999674682556  0.0443565758855939  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole d oping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \n Tc: 0.5×1014 cm-2: 35 K 1×1014 cm-2: 72 K  \n 2×1014 cm-2: 159 K  4×1014 cm-2: 303 K   \n81. BaCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-1724  - P4m2 5.19 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ba1Cl2                                 \n   1.00000000000000      \n   5.2204455449080065 0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  5.2204455608024087  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  22.6561909999999997  \n   Ba   Cl  \n   1     2  \nDirect  \n  0.9999999895210436  0.9999998649340824  0.0000000120736772  \n  0.9999999136415667  0.4999997978403741  0.9379403053198416  \n  0.5000000968373897  0.0000003372255364  0.0620596826064812  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentra tion  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 3 K 1×1014 cm-2: 5 K  \n 2×1014 cm-2: 15 K 4×1014 cm-2: 53 K  \n82. BaBr 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-940 - P4m2 4.65 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ba1Br2                                 \n   1.00000000000000      \n   5.3962707755781327  0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  5.3962708320850812  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  22.6561909999999997  \n   Ba   Br  \n   1     2  \nDirect  \n  0.9999999445990824  0.0000000188573068  0.9999999890720801  \n  0.0000000144325156  0.4999999835966094  0.9309651109781285  \n 0.5000000409684091  0.9999999975460838  0.0690348999497914  \n \nProjected band structure  \nand density of states  Magnetic moment and spi n polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \n Tc: 0.5×1014 cm-2: - K 1×1014 cm-2: - K  \n 2×1014 cm-2: 13 K 4×1014 cm-2: 68 K  \n83. BaI 2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-578 - P4m2 4.17 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ba1I2                                  \n   1.00000000000000      \n   5.6312877034679518  0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  5.6312855784550733  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  22.6561909999999997  \n   Ba   I  \n   1     2  \nDirect  \n  0.0000000097016439  0.0000000261087223  0.0000000364624881  \n  0.9999999760593496  0.5000000066512271  0.9202877037178823  \n  0.5000000142390064  0.9999999672400506  0.0797122598196296  \n \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: - K 1×1014 cm-2: 1 K  \n 2×1014 cm-2: 27 K 4×1014 cm-2: 69 K  \n84. ZnF2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-5027  - P4m2 4.40 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zn1F2                                 \n1.00000000000000      \n3.4021277956138194    0.0000000000000000    0.0000000000000000  \n-0.0000000067903241    3.4021231044657156    0.0000000000000000  \n 0.0000000000000014    0.0000000000000014   22.7593417019000022  \n   Zn   F  \n    1     2  \nDirect  \n  0.0000000118231895  0.0000000385394259  0.4999999886403614  \n  0.0000000122277015  0.4999999796982166  0.4590367377307842  \n  0.4999999759356513  0.9999999817488643   0.5409632622692158  \n0.4999999946864051  0.0000000006091483  0.5599368665589637  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n \n \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 10 K 2×1014 cm-2: 37 K \n 4×1014 cm-2: 119 K 6×1014 cm-2: 203 K \n85. ZnCl 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n239 ✓ 2dm-4713  - P4m2 4.25 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zn1Cl2                                 \n   1.00000000000000      \n   3.7221357404532283    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.7221358331008698    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.7501749999999987  \n   Zn   Cl  \n   1     2  \nDirect  \n  0.9999999979183585  0.0000000005683205  0.5000000000157669  \n  0.0000000073952364  0.4999999988225241  0.4400631334252694  \n  0.4999999946864051  0.0000000006091483  0.5599368665589637  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 13 K 2×1014 cm-2: 48 K \n 4×1014 cm-2: 142 K  6×1014 cm-2: 225 K  \n86. ZnBr 2` \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- ✓ 2dm-186 - P4m2 3.37 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Zn1Br2                                 \n   1.00000000000000      \n     3.9153705009491917  0.0000000000000000    0.0000000000000000  \n     0.0000000000000000  3.9153710571021980    0.0000000000000000  \n     0.0000000000000000  0.0000000000000000   22.7501749999999987  \n   Zn   Br  \n    1     2  \nDirect  \n  0.9999999799870594  0.0000000040239740  0.4999999955561520  \n  0.0000000202587032  0.5000000005451213  0.4347980916009320  \n  0.4999999997542375  0.9999999954308976  0.5652019128429160  \n \nProjected band structure  \nand density of states  Magnetic moment and sp in polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 14 K 2×1014 cm-2: 57 K \n 4×1014 cm-2: 140 K  6×1014 cm-2: 208 K  \n87. CdF2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-1124  - P4m2 3.78 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1F2                                  \n   1.00000000000000      \n   3.8660617979077605    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.8660617644304405    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.74136899999 99988  \n   Cd   F  \n   1     2  \nDirect  \n  0.5000000002372786  0.4999999990947259  0.0000000089976382  \n  0.0000000011284911  0.5000000001739124  0.9582099986260815  \n  0.4999999986342374  0.0000000007313616  0.0417899923762732  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 3 K 2×1014 cm-2: 5 K \n 4×1014 cm-2: 14 K 6×1014 cm-2: 42 K \n88. CdCl2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-5027  - P4m2 3.63 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1Cl2                                 \n   1.00000000000000      \n   4.0781723446932565    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.0724475101274082    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.7505990000000011  \n   Cd   Cl  \n   1     2  \nDirect  \n  0.8381951176880946  0.1710117550868731  0.5000431008569990  \n  0.8369366399360061  0.6711498327356509  0.4352580040546670  \n  0.3380302423758934  0.1725564121774710  0.5646998950883315  \n \nProjected band structure  \nand density of states  Magnetic moment  and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 16 K 2×1014 cm-2: 47 K \n 4×1014 cm-2: 125 K  6×1014 cm-2: 167 K  \n89. CdBr2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-24 - P4m2 3.04 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1Br2                                 \n  1.00000000000000      \n   4.2145244047588637    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.2145256921027006    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.741368999999 9988  \n   Cd   Br  \n    1     2  \nDirect  \n  0.5000000009435723  0.4999999953282668  0.9999999850019776  \n  0.0000000012902532  0.4999999993134665  0.9291315382588436  \n  0.4999999977661815  0.0000000053582667  0.0708684767391787  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 16 K 2×1014 cm-2: 53 K \n 4×1014 cm-2: 123 K  6×1014 cm-2: 161 K  \n90. CdI2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-498 - P4m2 2.59 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd1I2                                  \n   1.00000000000000      \n   4.4830774580651989    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.4830733220360397    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.74136899999 99988  \n   Cd   I  \n   1     2  \nDirect  \n  0.4999999740505032  0.5000000072614341  0.0000000437352909  \n  0.0000000222610765  0.4999999900661791  0.9230432821464731  \n  0.5000000036884202  0.0000000026723868  0.0769566741182359  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 42 K \n 4×1014 cm-2: 94 K 6×1014 cm-2: 102 K  \n91. SnS2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-6028  - P4m2  1.45 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sn1S2                                  \n   1.00000000000000      \n   3.7843209366167074    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.7843210308975257    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.8554570000000012  \n   Sn   S  \n   1     2  \nDirect  \n  0.9999999970905904  0.0000000007089014  0.5000000052448712  \n  0.5000000031336782  0.9999999969897146  0.5672176036748198  \n  0.9999999997757243  0.5000000023013840  0.4327823910803161  \n \nProjected band structure  \nand density of states  Magnetic moment  and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 59 K 6×1014 cm-2: 96 K \n92. PbS2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-6220  - P4m2 0.65 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb1S2                                  \n   1.00000000000000      \n   3.9362004097085279    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.9362002933386218    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.85545700000 00012  \n   Pb   S  \n   1     2  \nDirect  \n  0.9999999987829113  0.0000000019596129  0.4999999970302511  \n  0.4999999983130792  0.9999999968400317  0.5699955865671171  \n  0.0000000029040095  0.5000000012003554  0.4300044164026389  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 6 K 2×1014 cm-2: 25 K \n 4×1014 cm-2: 81 K 6×1014 cm-2: 122 K  \n93. SiF 4 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-1452  - P4/mmm  6.66 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Si1F4                                  \n   1.00000000000000      \n   3.6037410880278817    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.6037413636842635    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.8182620000000007  \n   Si   F  \n   1     4  \nDirect  \n  0.0000000001529230  0.0000000001174740  0.0000000008728449  \n  0.9996137853834597  0.9999999996778186  0.9327723585694088  \n  0.4999999997505071  0.0000000005697984  0.9999999998743334  \n  0.0000000006462031  0.4999999997206146  0.9999999998793285  \n  0.00038621 40669071  0.9999999999142943  0.0672276408040844  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 20 K 2×1014 cm-2: 54 K \n 4×1014 cm-2: 142 K 6×1014 cm-2: 181 K \n94. GeF 4 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-1570  - P4/mmm  4.26 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ge1F4                                  \n   1.00000000000000      \n   3.8411487860685023    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.8411460687975425    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.8182620000000007  \n   Ge   F  \n   1     4  \nDirect  \n  0.9999999963869257  0.0000000020198172  0.0000000002135891  \n  0.9995969344907110  0.0000000015022934  0.9279121530585428  \n  0.5000000035436329  0.0000000000330260  0.0000000006181509  \n  0.9999999995331805  0.4999999991252650  0.0000000006424727  \n  0.00040306 60455498  0.9999999973195983  0.0720878454672373  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 23 K 2×1014 cm-2: 68 K \n 4×1014 cm-2: 170 K  6×1014 cm-2: 239 K \n95. SnF 4 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n181 - 2dm-3719  - P4/mmm  3.85 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sn1F4                                  \n   1.00000000000000      \n   4.1703085515655678    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.1703085104161621    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.8182429999999989  \n   Sn   F  \n   1     4  \nDirect  \n  0.9999999943846589  0.9999999915730413  0.4999999996232276  \n  0.9999544901801372  0.0000000014527046  0.5797577264576859  \n  0.5000000055319589  0.9999999858996134  0.4999999991057820  \n  0.0000000061119465  0.5000000211374811  0.4999999987110613  \n  0.0000455037912985  0.9999999999371596  0.4202422761022433  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 36 K 2×1014 cm-2: 88 K \n 4×1014 cm-2: 177 K  6×1014 cm-2: 224 K  \n96. PbF 4 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n150 - 2dm-4359  - P4/mmm  2.48 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb1F4                                  \n   1.00000000000000      \n   4.3854459551064870    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.3854457720421047    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.9935650000000003  \n   Pb   F  \n   1     4  \nDirect  \n  0.9999999991334647  0.9999999973692013  0.5000000065320052  \n  0.0000000001832490  0.9999999998796909  0.5827573533406962  \n  0.0000000007667609  0.5000000032074752  0.5000000007589378  \n  0.5000000022286315  0.0000000008707701  0.5000000001661036  \n  0.99999999 76879010  0.9999999986728767  0.4172426392022643  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 40 K 2×1014 cm-2: 100 K \n 4×1014 cm-2: 217 K 6×1014 cm-2: 214 K \n97. LiCl  \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-6230  - P4/mmm  5.35 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Li1Cl1                                 \n   1.00000000000000      \n   3.4326989999999999    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.4326989999999999    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.00000000000 00000  \n   Li   Cl  \n   1     1  \nDirect  \n  0.5000000000000000  0.5000000000000000  0.5000000000000000  \n  0.0000000000000000  0.0000000000000000  0.5000000000000000  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2−2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 60 K \n 4×1014 cm-2: 81 K 6×1014 cm-2: 77 K \n98. LiBr  \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-6152  - P4/mmm  4.64 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Li1Br1                                 \n   1.00000000000000      \n   3.6852198488649122    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.6852197352056466    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.00000000000 00000  \n   Li   Br  \n   1     1  \nDirect  \n  0.4999999994756266  0.4999999998885158  0.4999999989455830  \n  0.0000000005243734  0.0000000001114842  0.5000000010544170  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2−2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 64 K \n 4×1014 cm-2: 87 K 6×1014 cm-2: 71 K \n99. LiI \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-6194  - P4/mmm  4.05 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Li1I1                                  \n   1.00000000000000      \n   4.0410291989920726    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.0410296835011872    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.00000000000 00000  \n   Li   I  \n   1     1  \nDirect  \n  0.5000000009804637  0.4999999989172252  0.5000000001712479  \n  0.9999999990195363  0.0000000010827748  0.4999999998287521  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2−2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 44 K \n 4×1014 cm-2: 76 K 6×1014 cm-2: - K \n100. NaBr \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-6127  - P4/mmm  4.36 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Na1Br1                                 \n   1.00000000000000      \n   4.0826855104855753    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.0826855851440875    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.00000000000 00000  \n   Na   Br  \n   1     1  \nDirect  \n  0.5000000000311715  0.4999999999283915  0.5000000000226166  \n  0.9999999999688285  0.0000000000716085  0.4999999999773834  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2−2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 25 K 2×1014 cm-2: 23 K \n 4×1014 cm-2: 37 K 6×1014 cm-2: - K \n101. NaI \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-3215  - P4/mmm  3.84 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Na1I1                                  \n   1.00000000000000      \n   4.4235681542498817    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.4235689204567565    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.00000000000 00000  \n   Na   I  \n   1     1  \nDirect  \n  0.4999999996511733  0.5000000022245175  0.4999999994690754  \n  0.0000000003488267  0.9999999977754825  0.5000000005309246  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2−2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 12 K 2×1014 cm-2: - K \n 4×1014 cm-2: 16 K 6×1014 cm-2: 44 K \n102. KBr \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-6264  - P4/mmm  4.37 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  K1Br1                                  \n   1.00000000000000      \n   4.5696917733565918    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.5696918158573636    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.00000000000 00000  \n   K    Br  \n   1     1  \nDirect  \n  0.4999999999747899  0.5000000000775557  0.4999999989178505  \n  0.0000000000252101  0.9999999999224443  0.5000000010821495  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2−2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 24 K 2×1014 cm-2: 19 K \n 4×1014 cm-2: - K 6×1014 cm-2: 80 K \n \n103. SrO 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-11 - P4/mmm  2.94 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sr1O2                                  \n   1.00000000000000      \n   3.4794399348957756    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.4794400036870723    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   21.51940099999 99984  \n   Sr   O  \n   1     2  \nDirect  \n  0.4999999999049933  0.4999999998392468  0.0000000076882714  \n  0.9999999977551539  0.9999999955740932  0.9643033953589608  \n  0.0000000023398599  0.0000000045866528  0.0356965969527749  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 23 K 2×1014 cm-2: 49 K \n 4×1014 cm-2: 163 K  6×1014 cm-2: 282 K  \n104. Cd 2S2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-2392  - P4/nmm  2.26 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Cd2S2                                  \n   1.00000000000000      \n   4.5311467153995970    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.5311449053809385    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.9900120000000001  \n   Cd   S  \n   2     2  \nDirect  \n  0.5000000090778087  0.4999999708515048  0.0000000259332467  \n  0.9999999888466249  0.0000000341068613  0.9999999734893308  \n  0.9999999556635117  0.5000000067003185  0.0583492951533486  \n  0.5000000464120546  0.9999999883413224  0.9416507054240739  \n \n \nProjecte d band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 78 K 2×1014 cm-2: 77 K \n 4×1014 cm-2: 128 K  6×1014 cm-2: 129 K  \n105. Ge2O2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-831 - P4/nmm  2.13 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ge2O2                                  \n   1.00000000000000      \n   3.6230918133660310    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.6230921794361621    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.3791590000000014  \n   Ge   O  \n     2     2  \nDirect  \n  0.9999999480668578  0.5000000375351021  0.9526917621287510  \n  0.5000000158606497  0.0000000044229722  0.0473088295918700  \n  0.5000000191862455  0.4999999800669102  0.9999997135117695  \n  0.0000000168862542  0.9999999779750084  0.9999996947676095  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−8𝐽1𝑆2−4𝐽2𝑆2−4𝐽3𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽2𝑆2+4𝐽3𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀3=𝐸0+2𝐽1𝑆2+4𝐽3𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: - K 0.8×1014 cm-2: 3 K \n 1×1014 cm-2: 11 K 2×1014 cm-2: 40 K \n106. Sn2O2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n182 - 2dm-3629  - P4/nmm  3.01 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sn2O2                                  \n   1.00000000000000      \n   3.8480553455705597    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.8480551394167248    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.3242170000000009  \n   Sn   O  \n   2     2  \nDirect  \n  0.0000000026401210  0.4999999951072667  0.4471324917123454  \n  0.4999999976294731  0.0000000023310065  0.5528675071911024  \n  0.0000000048212740  0.9999999976863023  0.4999999994032223  \n  0.4999999949091318  0.5000000048754174  0.5000000016933299  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−8𝐽1𝑆2−4𝐽2𝑆2−4𝐽3𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽2𝑆2+4𝐽3𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀3=𝐸0+2𝐽1𝑆2+4𝐽3𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 5 K 1×1014 cm-2: 17 K \n 2×1014 cm-2: 61 K 4×1014 cm-2: - K \n107. Pb2O2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n153 ✓ 2dm-3561  - P4/nmm  2.49 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb2O2                                  \n   1.00000000000000      \n   4.0492641769418585    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.0492642957581655    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   22.3791699999999985  \n   Pb   O  \n   2     2  \nDirect  \n  0.0000000109132117  0.4999999827697224  0.4464929309991490  \n  0.4999999860015976  0.9999999991165467  0.5535070723400963  \n  0.5000000029956198  0.5000000161566831  0.5000000014446400  \n  0.0000000000895710  0.0000000019570550  0.4999999952161147  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n \n \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 5 K 0.8×1014 cm-2: 10 K \n 1×1014 cm-2: 15 K 2×1014 cm-2: 46 K \n108. Al2N2 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-1029  - P4/nmm  3.54 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Al2N2                                  \n   1.00000000000000      \n   3.6327475818023207  0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  3.6327475947125496  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000 23.9423190000000012  \n   Al   N  \n   2     2  \nDirect  \n  0.0000000002115428  0.9999999991086526  0.9999999984051726  \n  0.4999999994827178  0.5000000007099743  0.9999999984286347  \n  0.5000000007493099  0.9999999975236946  0.9765657033252495  \n  0.9999999995564295  0.5000000026576856  0.0234342998409502  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \n Tc: 1×1014 cm-2: 8 K 2×1014 cm-2: 25 K  \n 4×1014 cm-2: 69 K 6×1014 cm-2: 101 K  \n109. Li2(OH) 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n111 ✓ 2dm-3650  - P4/nmm  3.92 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Li2H2O2                               \n   1.00000000000000      \n   3.5713938126153830    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.5713935372583849    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.6428140000000013  \n   Li   H    O  \n   2     2     2  \nDirect  \n  0.9999999940638915  0.0000000017331061  0.5000000001397638  \n  0.4999999962725070  0.5000000333215198  0.5000000065823755  \n  0.4999999832262390  0.9999999969004563  0.5765971880356275  \n  0.9999999969126421  0.5000000019982949  0.4234028051547583  \n  0.00000008 47418349  0.5000000288072073  0.4643706448608924  \n  0.4999999447828856  0.9999999372394157  0.5356293552265896   \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 11 K 2×1014 cm-2: 35 K \n 4×1014 cm-2: 81 K 6×1014 cm-2: 132 K  \n110. Na2(OH) 2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n131 ✓ 2dm-5304  - P4/nmm  2.77 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Na2H2O2                               \n   1.00000000000000      \n   3.4173975040401374    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.4202648848351274    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   24.497505000000 0003  \n   Na   H    O  \n   2     2     2  \nDirect  \n  0.8328904903480279  0.6813309661559188  0.5410498809907835  \n  0.3328756579871879  0.1817173217661505  0.4589499776824937  \n  0.8334356528785563  0.6819885831771160  0.4060455645891565  \n  0.3334398469365425  0.1807410466727646  0.5939543914272249  \n  0.8336403829060757  0.6818801626118756  0.4456891253749831  \n  0.3336399689436220  0.1808349196161743  0.5543110599353582 8 \n \nProject ed band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−4𝐽1𝑆2+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 3 K 1×1014 cm-2: 11 K \n 2×1014 cm-2: 40 K 4×1014 cm-2: 175 K  \n111. Pb2Br2F2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n148 - 2dm-3625  - P4/nmm  3.04 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Pb2Br2F2                                \n   1.00000000000000      \n   4.1039841614003132    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.1039841659102736    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   25.4031884695 999999  \n   Pb   Br   F  \n   2     2     2  \nDirect  \n  0.0000000029635459  0.5000000040512447  0.5603851148380130  \n  0.4999999961748287  0.9999999984268086  0.4396148853497124  \n  0.4999999998419469  0.0000000001062332  0.6147554852933865  \n  0.0000000002298322  0.5000000000353140  0.3852445149141914  \n  0.9999999998432116  0.9999999975649132  0.4999999998302727  \n  0.5000000009466348  0.4999999998154863  0.4999999997783604  \n \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 20 K 2×1014 cm-2: 51 K \n 4×1014 cm-2: 199 K  6×1014 cm-2: 238 K  \n112. Sc2Br2O2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n176 ✓ 2dm-4564  - P4/nmm  2.72 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n \n Sc2Br2O2                              \n   1.00000000000000      \n   3.5745649225414975    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.9886947265690655    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   25.8493729999999999  \n   Sc   Br   O  \n   2     2     2  \nDirect  \n  0.9999998932254002  0.5000000070878201  0.5358870875154977  \n  0.5000001052406517  0.9999999966401631  0.4641129213449062  \n  0.4999999522509384  0.5000000653313492  0.6130380560660740  \n  0.0000000403175449  0.9999999274374076  0.3869618954813916  \n  0.49999998 75516323  0.4999999459086979  0.4892888382271607  \n  0.0000000214138325  0.0000000575945620  0.5107112013649697  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 103 K  6×1014 cm-2: 206 K  \n113. Sr2Br2H2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n188 ✓ - - P4/nmm  4.34 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Sr2Br2H2                                \n   1.00000000000000      \n   4.1041140556000002    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.1041140556000002    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   25.0283088683 999999  \n   Sr   Br   H  \n   2     2     2  \nDirect  \n  0.0000000000000000  0.5000000000000000  0.5562024709999989  \n  0.5000000000000000  0.0000000000000000  0.4437975589999965  \n  0.5000000000000000  0.0000000000000000  0.6102313399999986  \n  0.0000000000000000  0.5000000000000000  0.3897686600000014  \n  0.0000000000000000  0.0000000000000000  0.5000000000000000  \n  0.5000000000000000  0.5000000000000000  0.5000000000000000   \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n \n \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.5×1014 cm-2: 5 K 0.8×1014 cm-2: 11 K \n 1×1014 cm-2: 14 K 2×1014 cm-2: 26 K \n114. AlF 3  \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - - ✓ Pmmm  7.73 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  AlF3 \n   1.00000000000000      \n   2.8217698873637747  0.0000000000000000  0.0000000000000000  \n   0.0000000000000000  3.6404754518615086  0.0000000000000000  \n   0.0000000000000000  0.0000000000000000  20.0000000000000000  \n   Al   F  \n   1     3  \nDirect  \n  0.5011636113421850  0.5617777439934213  0.3749814757054186  \n  0.0011379468021243  0.5617020862032926  0.3155663691980434  \n  0.5011793557123678  0.0617761716906813  0.3750111511360803  \n  0.0011721077792330  0.5618489820967347  0.4344402663104455  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−𝐽1𝑆2−8𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+𝐽1𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−𝐽1𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis.   \n Tc: 1×1014 cm-2: - K 2×1014 cm-2: 39 K  \n 4×1014 cm-2: 97 K 6×1014 cm-2: 140 K  \n115. Al4N4 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-2590 - P4/mbm  2.86 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Al4 N4                                  \n   1.00000000000000      \n   6.156056481602916  0.0000000000000000   0.0000000000000000  \n   0.000000000000000  6.1560563947606166   0.0000000000000000  \n   0.000000000000000  0.0000000000000000   20.0000000000000000  \n   Al   N  \n   4     4  \nDirect  \n  0.3561749438853923  0.1438250552105913  0.6067160362650981  \n  0.1438250167987931  0.6438250168000081  0.6067160376887344  \n  0.8561749845307816  0.3561749838010471  0.6067160340132887  \n  0.6438250553214573  0.8561749445924391  0.6067160357455847  \n  0.1536257579985119  0.3463742399625076  0.6067159645347076  \n  0.8463742404299310  0.6536257593352417  0.6067159638015340  \n  0.65362579 48346247  0.1536257898751501  0.6067159680978946  \n  0.3463742062005082  0.8463742104230079  0.6067159598531475   \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy \nas a functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−8𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−2𝐽1𝑆2+8𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE indicates \nthe off -plane (in -plane) easy axis .  \n Tc: 1×1014 cm-2: 9 K 2×1014 cm-2: 26 K  \n 4×1014 cm-2: 51 K 6×1014 cm-2: 101 K  \n116. Ga 4N4 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- - 2dm-2582  - P4/mbm  2.05 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ga4N4                                  \n   1.00000000000000      \n   6.3251827589122840    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    6.3251971034337258    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   20.0000000000000000  \n   Ga   N  \n     4     4  \nDirect  \n  0.3503042731550039  0.1496967050798901  0.6067164713012545  \n  0.1496947757320584  0.6496939522736795  0.6067161534779828  \n  0.8503036384762837  0.3503025847410797  0.6067167725671752  \n  0.6496944167056071  0.8503064182194251  0.6067164587253018  \n  0.14909584 82973710  0.3509043025197585  0.6067155413565217  \n  0.8509058038166657  0.6490947246571181  0.6067155360478154  \n  0.6490955845930699  0.1490957611569925  0.6067154072219907  \n  0.3509056592239332  0.8509055513520494  0.6067156593019476   \nProjected band struct ure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−8𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0−2𝐽1𝑆2+8𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off-plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 17 K 2×1014 cm-2: 47 K \n 4×1014 cm-2: 102 K  6×1014 cm-2: 164 K  \n117. B2O3 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-33 - P62m  5.22 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  B2O3                                   \n 1.00000000000000      \n 4.4159446740658321  0.0000000000000000  0.0000000000000000  \n -2.2079736202051823  3.8243201010166517  0.0000000000000000  \n 0.0000000000000000  0.0000000000000000  21.6555759999999999  \n   B    O  \n   2     3  \nDirect  \n  0.3333336116611321  0.6666666779422172  0.6291477653214699  \n  0.6666666758417463  0.3333335782488263  0.6291710961926356  \n  0.6166768581259490  0.0000003034604745  0.6291587479318892  \n  0.0000002949047229  0.6166768751406337  0.6291587490202772  \n  0.38332255 94664498  0.3833225652078482  0.6291586415337420  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−3𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−2𝐽1𝑆2+𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2+𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \n Tc: 1×1014 cm-2: 7K 2×1014 cm-2: 41 K  \n 4×1014 cm-2: 177 K  6×1014 cm-2: 234 K   \n118. C3N4 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-4847  - C2 2.25 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  C3N4                                   \n   1.00000000000000      \n     4.7148016012078937  0.0000000000000000    0.0000000000000000  \n     2.3515188622351082  4.0710647115704353    0.0000000000000000  \n     0.0000000000000000  0.0000000000000000   20.7846499999999992  \n   C    N  \n   3     4  \nDirect  \n  0.8292154477102325  0.8656598237157468  0.4938500673793084  \n  0.3583144737507584  0.8640682525088920  0.4998956424374154  \n  0.8294419568887541  0.3911639484662840  0.5062309821806974  \n  0.5091577933182134  0.5520897669134094  0.5204099538574705  \n  0.9971507890925141  0.5444678953312492  0.5002984702008131  \n  0.5090099178357270  0.0251595474506033  0.4793819486384657  \n  0.0033326214037945  0.0414337656138267  0.4999349353058247   \nProjected band structure  \nand density of states  Magnetic moment and spin po larization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−6𝐽1𝑆2−6𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0−2𝐽1𝑆2+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2+2𝐽2𝑆2 \n \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: - K \n 4×1014 cm-2: 117 K  6×1014 cm-2: 213 K  \n119. In2Cl2O2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n- ✓ 2dm-3585  - Pmmn  2.56 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  In2Cl2O2                              \n   1.00000000000000      \n   3.5731505241839181    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.1589447996631739    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   25.663499999999 9991  \n   In   Cl   O  \n   2     2     2  \nDirect  \n  0.9999999987798702  0.0000000176380510  0.5393361404053039  \n  0.4999999997657127  0.5000000162422111  0.4606638625722042  \n  0.9999999998758398  0.4999999998113651  0.3895169892393682  \n  0.4999999980386818  0.9999999911125528  0.6104830086990489  \n  0.5000000012555006  0.9999999910346347  0.4855965158030031  \n  0.0000000022843949  0.4999999841611782  0.5144034832810718   \nProjected  band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+4𝐽1𝑆2−4𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+4𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 0.8×1014 cm-2: 10 K 1×1014 cm-2: 92 K \n 2×1014 cm-2: 199 K  4×1014 cm-2: 214 K \n120. In2Br2O2 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n91 ✓ 2dm-3667  - Pmmn  2.29 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  I In2Br2O2                              \n   1.00000000000000      \n   3.6777167334034697    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    4.1352408658591528    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   25.9786509999999993  \n   In   Br   O  \n   2     2     2  \nDirect  \n  0.4999999980116812  0.4999999961174453  0.4612293562923924  \n  0.0000000048975579  0.0000000030597818  0.5387706370112966  \n  0.9999999991153103  0.4999999835151883  0.3850082613192214  \n  0.5000000025816149  0.0000000170015824  0.6149917430419549  \n  0.99999999 87333652  0.5000000040808814  0.5125732056066994  \n  0.4999999966604634  0.9999999962251280  0.4874267967284354   \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2+2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 68 K \n 4×1014 cm-2: 112 K  6×1014 cm-2: 249 K  \n121. Mg 2Ge2O6 \nMC2D -ID C2DB  2dmat -ID USPEX  Space group  Band gap (eV)  \n- - 2dm-4548  - Pmma  1.83 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n \n Mg2Ge2O6                              \n   1.00000000000000      \n   3.0059731683423987    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    8.0018786807623012    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   24.5345280000000017  \n   Mg   Ge   O  \n   2     2     6  \nDirect  \n  0.9999999854917405  0.7500000228972823  0.4428084404274983  \n  0.0000000032055354  0.2499999931306931  0.5571915758536861  \n  0.5000000212605755  0.5000000543986332  0.4999999655723784  \n  0.4999999934689470  0.9999999389468712  0.4999999701877158  \n  0.00000005 83089417  0.9950844787043565  0.4481664650504484  \n  0.9999999378169875  0.0049156040384446  0.5518335574013022  \n  0.0000000420817869  0.4950843724664722  0.5518335522298301  \n  0.9999999689551089  0.5049155438453283  0.4481664516883015  \n  0.5000000016533050  0. 2499999920652627  0.5089925584737145  \n  0.4999999877570787  0.7499999995066702  0.4910074631151176   \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−4𝐽2𝑆2−4𝐽3𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽1𝑆2−4𝐽2𝑆2−4𝐽3𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2−4𝐽2𝑆2+4𝐽3𝑆2 \n𝐸𝐴𝐹𝑀3=𝐸0−2𝐽1𝑆2+4𝐽2𝑆2−4𝐽3𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in-plane) easy axis .  \nTc: 1×1014 cm-2: - K 2×1014 cm-2: 33 K \n 4×1014 cm-2: 97 K 6×1014 cm-2: 171 K  \n122. TiPbO 3 \nMC2D -ID C2DB  2dmat -ID USPEX Space group  Band gap (eV)  \n-  2dm-5482  - P4mm  2.65 \nConvex hull  Atomic structure  Atomic coordinates  Phonon dispersion curve  \n  Ti1Pb1O3                              \n   1.00000000000000      \n   3.9178046187640239    0.0000000000000000    0.0000000000000000  \n   0.0000000000000000    3.9178042947442169    0.0000000000000000  \n   0.0000000000000000    0.0000000000000000   23.8189570000000010  \n   Ti   Pb   O  \n   1     1     3  \nDirect  \n  0.4999999814381937  0.5000000060258714  0.4842658842307088  \n  0.0000000042042529  0.0000000083075875  0.5681751813756151  \n  0.5000000131192692  0.9999999961594170  0.5155089190967530  \n  0.0000000026095179  0.5000000029248710  0.5155089310102241  \n  0.49999999 86287591  0.4999999865822531  0.4165420842866965  \n \nProjected band structure  \nand density of states  Magnetic moment and spin polarization energy as \na functio n of hole doping concentration  \n  \nMagnetic conf igurations and spin Hamiltonian  Magnetic exchange coupling parameters  \n \n𝐸𝐹𝑀=𝐸0−2𝐽1𝑆2−2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀1=𝐸0+2𝐽2𝑆2 \n𝐸𝐴𝐹𝑀2=𝐸0+2𝐽1𝑆2−2𝐽2𝑆2  \nMagnetic anisotropy energy  (MAE, μeV)  \nper magnetic atom  Monte Carlo simulations of the normalized \nmagnetization of as a function of temperature  \n \nMAE = E‖ − E⊥, a positive (negative) value of MAE \nindicates the off -plane (in -plane) easy axis .  \nTc: 1×1014 cm-2: 7 K 2×1014 cm-2: 17 K \n 4×1014 cm-2: 66 K 6×1014 cm-2: 105 K  \nSupplementary Reference  \n1. Kresse, G. (1999). From ultrasoft pseudopotentials to the projector augmented -wave method. Phys. Rev. B 59, 1758 -1775.  \n2. Kresse, G., and J, F. (1996). Efficient iterative schemes for ab initio total -energy calculations using a plane -wave basis set. \nPhys. Rev. B 54, 11169.  \n3. Perdew, J.P., Burke, K., and Ernzerhof, M. (1996). Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, \n3865.  \n4. Heyd, J., Scuseria, G.E., and Ernzerhof, M. (2003). Hybrid functionals based on a screened Coulomb potenti al. J Chem \nPhys 118, 8207 -8215.  \n5. Togo, A., and Tanaka, I. (2015). First principles phonon calculations in materials science. Scr. Mater. 108, 1-5. \n6. Evans, R.F., Fan, W.J., Chureemart, P., Ostler, T.A., Ellis, M.O., and Chantrell, R.W. (2014). Atomistic  spin model \nsimulations of magnetic nanomaterials. J. Phys. Condens. Matter 26, 103202.  \n7. Lyakhov, A.O., Oganov, A.R., Stokes, H.T., and Zhu, Q. (2013). New developments in evolutionary structure prediction \nalgorithm USPEX. Comput. Phys. Commun. 184, 1172 -1182.  \n8. Oganov, A.R., and Glass, C.W. (2006). Crystal structure prediction using ab initio evolutionary techniques: principles and \napplications. J Chem Phys 124, 244704.  \n9. Zhou, J., Shen, L., Costa, M.D., Persson, K.A., Ong, S.P., Huck, P., Lu, Y ., Ma, X., Chen, Y ., Tang, H., and Feng, Y .P. \n(2019). 2DMatPedia, an open computational database of two -dimensional materials from top -down and bottom -up \napproaches. Sci Data 6, 1-10. \n10. Mounet, N., Gibertini, M., Schwaller, P., Campi, D., Merkys, A., Marrazzo, A., Sohier, T., Castelli, I.E., Cepellotti, A., \nPizzi, G., and Marzari, N. (2018). Two -dimensional materials from high -throughput computational exfoliation of \nexperimentally known compounds. Nat. Nanotechnol. 13, 246 -252. \n11. Haastrup, S., Strange, M., Pan dey, M., Deilmann, T., Schmidt, P.S., Hinsche, N.F., Gjerding, M.N., Torelli, D., Larsen, \nP.M., Riis -Jensen, A.C., et al. (2018). The Computational 2D Materials Database: high -throughput modeling and discovery \nof atomically thin crystals. 2D Mater. 5, 0420 02. \n "    },    {        "title": "2003.12270v1.Spin_polarized_Current_driven_Ferromagnetic_Domain_Wall_Motion_with_a_Skyrmion_Building_Block.pdf",        "content": "Spin -polarized Current -driven Ferromagnetic Domain Wall Motion  with  \na Skyrmion Building Block  \n \nO. Gorobets1,2*, Yu. Gorobets1,2 , I. Tiukavkina1, R. Gerasimenko1  \n \n1National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic \nInstitute», 37 Peremohy Ave., 03056 Kyiv, Ukraine  \n2Institute of Magnetism of NAS and MES of Ukraine, 36b Acad. Vernadskoho \nBlvd., 03142 Kyiv, Ukraine  \n \n*Correspondent author e -mail: gorobets.oksana@gmail.com , National Technical \nUniversity of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», 37 Peremohy \nAve., 03056 Kyiv, Ukraine  \n \nAbstract  \nThe purpose of the research is the construction of the analytical model for \ndescription of spin-polarized current -driven ferromagnetic domain wall m otion  with \na skyrmion building block. The dependence of velocity of ferromagnetic domain \nwall motion  with a skyrmion building block is found as a function of driving torques \nand an external magnetic field strength.  \n  \nKeywords ferromagnet, domain wall,  skyrmion , spin-polarized current  \n \n1. Introduction  \nRecently, domain walls in ferromagnetic nanosized samples have been an \nurgent object of research as promising carriers of information bits for applications in magnetic memory devices [1]. Moreover, the domain wall in a ferromagnet can \nhave the simplest defect -free structure, such as a “transverse” wall, or include \nvortices and other topological defects, such as a “vortex” wall. Among the wide \nvariety of magnetic topological objects, the following are most distinguished as \n“building blocks” in the internal structure of domain walls: vortex, antivortex, \nbimeron, Bloch lin e, Bloch point.  \nIn magnets, the Bloch lines divide the surface of a domain wall into two \nsubdomains and significantly affect the properties of domain walls. Numerous \nstudies have been devoted to the construction of magnetic memory devices based on \nBloch lines [2,3] . To date, the B loch lines are the most studied in ferromagnets with \nhigh uniaxial anisotropy [4]. The corresponding studies began much earlier in \nweakly anisotropic films [5], and modern achievements in thi s direction are \ndescribed in [6]. The Bloch lines are obse rved regardless of the sign of the magnetic \nanisotropy constant in cubic ferromagnets [7,8] . A theoretical model of Bloch lines \nin weak ferromagnets was proposed in [9]. Local bends were observed in the Bloch \nlines moving at high speeds in yttrium orthoferrite, and they were associated with \nthe movement of vortices along the domain wall  [10]. \nThe Bloch point is one of the examples of point topological defects in domain \nwalls and was first proposed in [11,12] . The defining property of the Bloch point is \nthat it represents the topological singularity of the magnetization field, and one can \nfind all possible directions of the magnetization vector on the sphere of infinitely \nsmal l radius centered at the Bloch point. Unlike other topological spin textures, such \nas magnetic skyrmions and vortices, the Bloch points have a unique feature – the \nlocal magnetization at the Bloch point completely disappears. This was \nexperimentally confir med in yttrium ferrite garnet crystals, micron thick garnet \nfilms, and magnetic cylindrical wire based on static measurements [13–15]. \nIt was shown in [11] that the structure of the Bloch point is mainly determined \nby the exchange energy. Later in [12], the specific energy was calculated and it was \nshown that its value is topolo gically invariant. A family of magnetization textures with a local rotation angle γ (in the azimuthal direction) was considered in [12] and \nit was found that minimizing the magnetostatic energy sele cts a specific angle γ ≈ \n112 ° [12]. In order to  study the region near the singular point, the Landau magnetic \nenergy [16] was included in [17] and the neglecting of magnetostatic energy was \njustified, and as a result, it wa s shown that the magnitude of the magnetization vector \nincreases linearly with the radial distance from the center. The magnetization field \nof the Bloch point was calculated taking into account the exchange energy, Landau \nmagnetic energy, and magnetostatic  energy in [18]. The Bloch point in the domain \nwall of a ferromagnet is characterized by a topological (skyrmionic) charge q = ± 1 \n[19]. There are an infinite number of Bloch point configurations. However, there are \nthree main possible configurations of the B loch point, namely, a hedgehog  \nconfiguration in which the magnetization distribution around the Bloch point is \nspherically symmetric, and the magnetization vector is directed away from the Bloch \npoint (diverging Bloch point q = +1) or to the Bloch point ( converging Bloch point \nq = −1), vortex  or antivortex  (q = +1 or −1) and spiral (q = + 1 or −1) configurations, \nwhich are obtained by 90 ° and 180 ° rotation of the magnetization of the similar \nconfiguration, respectively [20–22]. Direct observation of the stabilized structures \nof Bloch points with a skyrmion charge q = + 1, namely, hedgehog -like, vortex  and \nspiral configurations, is reported in [19]. The  in-plane and the out of plane \nmagnetization components were observed using magnetic transmission soft X -ray \nmicroscopy MTXM [19] and the corresponding structures were determined based \non numerical micromagnetic simulation [19]. \n“Vortex” or “topological” domain walls with integer topological charges, as \nwell as integer or fractional winding numbers of volume vortices and edge defects \nare observed, for example, in ferromagnetic nanowires and nanorings [1]. The \ndynamics of a domain wall in a ferromagnet depends on the topological charge of \nthe v ortices in its structure. The movement of the domain wall leads to the creation, \npropagation, and annihilation of such defects.  The interest to the dynamics of magnetic vortices and Bloch points is also \nassociated with the discovery of the fast magnetizati on reversal of the core of a \nmagnetic vortex by alternating external influences (magnetic field [23] or spin \ncurrent [24]). Numerical micromagnetic simulation of the ma gnetization reversal of \nthe vortex core [25] showed that the annihilation mechanism of the vortex – \nantivortex pairs [26] requires mediation of the magnetization singularity: the \n“magnetic monopole” or, in other words, of the Bloch point [21]. \nAt the same time, the vast majority of theoretical studies on the internal \nstructure of domain walls in ferro - and antiferro magnets are based on numerical \nmicromagnetic modeling. However, the results of the a nalysis of exact analytical \nsolutions of the Landau -Lifshitz equations in ferro - and antiferromagnets have \nshown [27] that  there can exist an infinite number of magnetic textures under the \nsame boundary conditions for the magnetization vector  (as well as the \nantiferromagnetism vector) , which obviously represents problem for numerical \nmicromagnetic simulat ion. \nTherefore, in this work, we will obtain the exact dynamic solution of the \nLandau –Lifshitz –Gilbert –Slonczewski equation in a ferromagnet with uniaxial \nmagnetic anisotropy, which describes the motion of a domain wall with a skyrmion \nin the internal stru cture under the influence of an external magnetic field and spin \ncurrent.  \n \n2. Theory and calculation  \nLet us consider a ferromagnet with uniaxial magnetic anisotropy  and \nmagnetization  \nM\n, \n0 MM=\n  where absolute value of magnetization  is equal to \n0M const=\n. The expression for the magnetic energy of a ferromagnet and an \nequation of magnetization dynamics can be written though the angular variables that \nare introduces by the standard way:  \n0 0 0sin cos , sin sin , cosx y zM M M M M M     = = =\n,                  (1) where  \n  and \n  are the polar and azimuth angles for the magnetization, \nxM , \nyM , \nzM\n are the Cartesian coordinates of the magnetization vector.  \nThe magnetic energy of a ferromagnet has the form  \n22\n2 2 2 0\n0\n0sin sin cos22ex\niiHW M drx x M         = + + −      \n,            (2) \nwhere \n  is the nonuniform exchange constant (\n0 ), \n is the constant of uniaxial \nmagnetic anisotropy, \n0H\n  is an external magnetic field strength, integration in (2) is \ntaken over the volume of a ferromagnet.  \nThe Landau –Lifshitz –Gilbert –Slonczewski equation  for a ferromagnet ha s the \nform  \n0eff G MMg M H M g Tt M t =−  +  +   \n,                       (3) \nwhere \nT T T⊥=+\n  is the spin -transfer torque, i.e. the torque induced upon the \nmagnetization by spin -polarized current flowing through the ferromagnet, \n0J\npaT M M mM =−  \n, \nJp T b M m⊥=\n , \ng  is the gyromagnetic ratio, \nG  is \nthe damping factor, \neffH\n  is the effective field , \neff WHM\n=−\n , \nJa and  \nJb are the \ndriving torques, and  \npm\n is the unit vector along the polarization of the current . \nThe Landau –Lifshitz –Gilbert –Slonczewski equation for a  ferromagnet can be \nwritten though the angular variables  ( )\n( )\n( )\n( )0\n0\n22sin sin sin cos cos cos\nsin cos\nsin sin cos sin\nsin cos cos sin cos sin sin sinp p p p\nJ z z y x\npp\nJ x y G\npp\nJ y x\np p p\nJ x y z GgWg b m m m mtM\ng a m mt\ngWg b m mtM\ng a m m mt     \n  \n   \n        = + − + + −\n − − + − = + − + − + − +       (4) \nThe Landau –Lifshitz –Gilbert –Slonczewski equation can be simplified \nconsidering spin -torque polarization along OZ axis \n0, 1p p p\nx y zm m m= = =  \n0\n22\n0sin sin\nsin sin sinp\nJ z G\np\nJ z GgWg b mt M t\ngWg a mt M t    \n     = − +=− − −\n.                   (5) \nIt is possible to obtain the following equations for the magnetization dynamics \nsubstituting the energy of a ferromagnet with uniaxial magnetic anisotropy  into the \nset of Landau –Lifshitz –Gilbert –Slonczewski equations  \n \n() \n 2\n0\n0\n0\n0sin sin cos\nsin cos sin sin\ncos sin sin sinex\np\nJ z G\np\nex J z GMgt\nHg b mMt\nM g g a mtt     \n     \n         = − +  +  \n  + + − +  \n=   +  − −\n.       (6) \n \n3. Results and Discussion  \nThe equations for the magnetization dynamics have the following exact dynamic \nsolution  0\n0exp ln2n v t\nz vt rtg nr   \n\n= + +\n  −=+  ,                                         (7) \nwhere  \n()\n()()()0\n22\n0 2211\n11p\nz\nG J J\nGG\np\nz G\nJ G J\nGG\nexg H g mv a b\ngmv g H a b \n\n\n= − +++\n= + −++\n\n=\n,                          (8) \nn\n is an arbitrary integer number, \n0  is an arbitrary initial phase, \n0 02 , \n is \nan arbitrary constant with the dimension of length.  \nThis solution (7), (8) describes movement of a ferromagnetic domain wall of \nwidth \n  with built in skyrmion  with a constant velocity along OZ axis . The \nskyrmion as a domain wall building block can be of an arbitrary topological charge \nn\n (\nn is equal to both the skyrmion charge and the skyrmion winding number ), of \narbitrary size \n  and of arbitrary initial helicity \n0\nn= . The analytical model \ndescribes the temporary oscillations of skyrmion helicity from zero to \n2\nn= . It \nmeans that during the period of such oscillations \n2Tv=  the skyrmion type is \ntransforming from the Neel type at\n00= , \n0= , \n02=  to the intermediate type at \narbitrary \n0 0 0 00, , , 22         , then  to the Blo ch type \n02= , and then \nagain to the Neel type . \n \n4. Conclusions  The exact dynamic solution (8), (9) of Landau –Lifshitz –Gilbert –Slonczewski \nequation in a ferromagnet with uniaxial magnetic anisotropy, obtained in the present \npaper, describe the spin -polarized current -driven ferromagnetic domain wall motion  \nwith a skyrmi on building block. The re is a linear  dependence of velocity of \nferromagnetic domain wall motion  with a skyrmion building block \nv  as a function \nof driving torques and an external magnetic field strength  according to the \nexpression ( 8). The temporary oscillation of skyrmion type from Neel to Bloch one \nis predicted according to the formula (7) during the domain wall motion, the period \nof the oscillation is \n2Tv= . \n \nReferences  \n[1] A. Pushp, T. Phung, C. Rettner, B.P. Hughes, S.H. Yang, L. Thomas, S.S.P. \nParkin, Domain wall trajectory determined by its fractional topological edge \ndefects, Nat. Phys. (2013). https://doi.org/10.1038/nphy s2669.  \n[2] S. Konishi, A new ultra -high-density solid state memory: Bloch line \nmemory, IEEE Trans. Magn. (1983). \nhttps://doi.org/10.1109/TMAG.1983.1062715.  \n[3] L.J. Schwee, H.R. Irons, W.E. Anderson, The crosstie memory, IEEE Trans. \nMagn. (1976). https://d oi.org/10.1109/TMAG.1976.1059098.  \n[4] A. In, Magnetic Domain Walls in Bubble Materials, 1979. \nhttps://doi.org/10.1016/c2013 -0-06998 -8. \n[5] E.E. Huber, D.O. Smith, J.B. Goodenough, Domain -wall structure in \npermalloy films, J. Appl. Phys. (1958). https://doi .org/10.1063/1.1723105.  \n[6] A. Hubert, R. Schäfer, 1.1 What are Magnetic Domains?, 2009. \nhttps://doi.org/10.1007/978 -3-540-85054 -0. \n[7] Theorie der Domänenwände in geordneten Medien, 1974. https://doi.org/10.1007/3 -540-06680 -2. \n[8] U. Hartmann, H.H. Mende,  Observation of subdivided 180°Bloch wall \nconfigurations on iron whiskers, J. Appl. Phys. (1986). \nhttps://doi.org/10.1063/1.336670.  \n[9] FARZTDINOV MM, MAL’GINOVA SD, DOMAIN STRUCTURE OF \nRARE -EARTH ORTHOFERRITES, Fiz Tverd Tela. (1970).  \n[10] M. V. Chetkin, Y.N. Kurbatova, T.B. Shapaeva, O.A. Borshchegovskii, \nGyroscopic quasi -relativistic dynamics of antiferromagnetic vortex in \ndomain boundary of yttrium orthoferrite, J. Exp. Theor. Phys. Lett. (2004). \nhttps://doi.org/10.1134/1.1776235.  \n[11] E. Feldt keller, Mikromagnetisch stetige und unstetige \nMagnetisierungskonfigurationen, Z. Angew. Phys. 19 (1965) 530 –536. \n[12] W. Döring, Point singularities in micromagnetism, J. Appl. Phys. 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Verga, Magnetization structure of a Bloch point singularity, \nEur. Phys. J. B. 82 (2011) 159 –166. https://doi.org/10.1140/epjb/e2011 -\n20146 -6. \n[19] M.Y. Im, H.S. Han, M.S. Jung, Y.S. Yu, S. Lee, S. Yoon, W. Chao, P. \nFischer, J. Il Hong, K.S. Lee, Dynamics of the Bloch point in an asymmetric \npermalloy disk, Nat. Commun. (2019). https://doi.org/10.1038/s41467 -019-\n08327 -6. \n[20] O. V. Pylypovskyi, D.D. Sheka, Y. G aididei, Bloch point structure in a \nmagnetic nanosphere, Phys. Rev. B - Condens. Matter Mater. Phys. (2012). \nhttps://doi.org/10.1103/PhysRevB.85.224401.  \n[21] A. Thiaville, J.M. García, R. Dittrich, J. Miltat, T. Schrefl, Micromagnetic \nstudy of Bloch -point -mediated vortex core reversal, Phys. Rev. B - Condens. \nMatter Mater. Phys. (2003). https://doi.org/10.1103/PhysRevB.67.094410.  \n[22] S.K. Kim, O. Tchernyshyov, Pinning of a Bloch point by an atomic lattice, \nPhys. Rev. B - Condens. Matter Mater. Phys. (2013) . \nhttps://doi.org/10.1103/PhysRevB.88.174402.  \n[23] B. Van Waeyenberge, A. Puzic, H. Stoll, K.W. Chou, T. Tyliszczak, R. \nHertel, M. Fähnle, H. Brückl, K. Rott, G. Reiss, I. Neudecker, D. Weiss, C.H. \nBack, G. Schütz, Magnetic vortex core reversal by excitati on with short \nbursts of an alternating field, Nature. (2006). \nhttps://doi.org/10.1038/nature05240.  \n[24] K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, T. \nOno, Electrical switching of the vortex core in a magnetic disk, Nat. Mater. \n(2007). https://doi.org/10.1038/nmat1867.  \n[25] R. Hertel, S. Gliga, M. Fähnle, C.M. Schneider, Ultrafast nanomagnetic toggle switching of vortex cores, Phys. Rev. Lett. (2007). \nhttps://doi.org/10.1103/PhysRevLett.98.117201.  \n[26] R. Hertel, C.M. Schneider, Exchange explosions: Magnetization dynamics \nduring vortex -antivortex annihilation, Phys. Rev. Lett. (2006). \nhttps://doi.org/10.1103/PhysRevLett.97.177202.  \n[27] O.Y. Gorobets, Degeneration of magnetic states of the order parameter \nrelative to the boundary c onditions and discrete energy spectrum in \nferromagnetic and antiferromagnetic nanotubes, Chaos, Solitons and Fractals. \n36 (2008). https://doi.org/10.1016/j.chaos.2006.06.106.  \n "    },    {        "title": "1301.0678v1.Ferromagnetic_bubble_clusters_in_Y___0_67__Ca___0_33__MnO__3__thin_films.pdf",        "content": " \nFerromagnetic bubble clusters in Y 0.67Ca 0.33MnO 3 thin films \n \nJeehoon Kim,1 N. Haberkorn,2 L. Civale,1 P. C. Dowden1 and R. Movshovich.1 \n \n1 Los Alamos National Laboratory, Los Alamos, NM 87545. \n2 Centro Atómico Bariloche, 8400 Bariloche, Argentina. \n(Dated: 3 January 2013) \nWe studied the ferromagnetic topology in a Y 0.67Ca0.33MnO 3 thin film with a combination of \nmagnetic force microscopy and magnetization measur ements. Our results show that the spin-glass \nlike behavior, reported previously for this system, could be attributed to frustrated interfaces of the \nferromagnetic clusters embedded in a non-ferromagn etic matrix. We found temperature dependent \nchanges of the magnetic topology at low temperatures, which suggests a non-static Mn3+/Mn4+ ratio.  \n \n The coexistence of distinct magnetic phases within a single sample of a perovskite \nmanganite compound ( A1-xBxMnO 3: A and B represent rare-earth and alkaline-earth elements) has \nbeen intensely studied for decades because of both technological applications and fascinating \nphysics.1,2,3,4 The electronic and magnetic properties of ma nganites can be tuned by substitution of \ncations and/or by the modification of the oxygen co ntent. A certain range of doping levels results in \na drastic change of resistance, i.e., a colossal magnetoresistance (CMR) effect.5 These properties are \nstrongly related to a structural distortion, which can be analyzed by the tolerance factor t, defined as \nݐൌሺ ൏ݎ /ݎ ைሻ/ሺ൏ ݎ ெݎ ைሻ, where ݎ/, rO, and rMn are the radii of the rare-earth/alkaline-\nearth, the oxide, and the manganese, respectively.6 The structure of perovskite materials is, in \ngeneral, stable in the range of 0.8 < t < 1. The perovskite structure tends to distort when t deviates \nfrom 1 and, in particular, manganites do not u ndergo a metal-insulator transition (MIT) when t < \n0.91. In this extreme regime Gd 2/3Ca1/3MnO 3 (GCMO, t ≈0.89) and Y 2/3Ca1/3MnO 3 (YCMO, t≈0.88) \nappear. Both systems display a ferromagnetic (FM)  ordering at around 80 K, associated with the \nMn ions, and none of them exhibit a MIT.7,8 Recently we reported magnetic phase coexistence and \nmagnetization reversal in ferrimagnetic GCMO thin films,9 which is consistent with those \npreviously reported in single crystals.10 The magnetic response of YCMO is governed by short-\nrange FM correlations below the Curie temperature ( TC), and its magnetization shows magnetic \nhistory dependence.11 The YCMO system shows a spin glass-like behavior with a freezing \ntemperature ( Tf) of about 30 K. The dynamics above Tf is attributed to a thermally activated \nredistribution of FM-ordered clusters and a random  dipolar interaction of their magnetic moments.8  \n Although magnetic properties might be expect ed to be similar for GCMO and YCMO \nowing to similar structural distortions and tolera nce factor, the resulting magnetic properties are in \nfact different. In GCMO the magnetic coupling via a d-f exchange interaction between Gd and Mn \nplays an important role and leads to the presence of a compensation temperature ( Tcomp) as a result of a competing ferrimagnetic order and a giant magnetostriction.12  YCMO, on the other hand, \ndisplays a spin-glass behavior, which can be interpre ted in terms of FM clusters with an associated \nlattice distortion and magnetic inhomogeneity of the system.13,14 In this Letter we report the FM \nphase topology of a YCMO thin film studied by magnetic force microscopy (MFM). We observe \nFM nanoclusters embedded in a non-FM matrix, in  support of the previously reported data and \ninterpretations.8,11,13 Images of the FM nanoclusters as a function of temperature demonstrate that \nthe nanoclusters exhibit fluctuations under specific conditions. \n The Y 0.67Ca0.33MnO 3 (YCMO) thin film was grown by pulsed-laser deposition (PLD) on a \nSrTiO 3 (100) substrate using a commercial target  with the same chem ical composition. The \nsubstrate temperature was kept at 790 °C in an oxyg en atmosphere at a pressure of 200 mTorr. After \ndeposition, the O 2 pressure was increased up to 200 Torr, and the temperature was decreased down \nto room temperature at a rate of 30 °C/min. Bulk  YCMO is an orthorhombic perovskite with lattice \nparameters of /ܽ√2 = 0.392 nm, /ܾ2  = 0.375 nm, /ܿ√2 = 0.372 nm.8 The YCMO film was \nexamined by x-ray diffractometry, and was found to be single phase with a (0 l0) orientation. The \nlattice parameters of the film [ /ܽ√2 = 0.392 (1), /ܾ2 =0.378 (1), /ܿ√2 =0.374(1)] were determined \nusing  ( 0l0), (200), and (002) reflections from a four-circle diffractometer/goniometer. No \nadditional peaks due to secondary phases or differe nt crystalline orientations were observed (see \nfigure 1). The rocking curve FWHM of the (040) peak  of the film was 0.24°. Furthermore, the four \npeaks at 90° intervals in the φ scan make evident the existence of  in-plane order of the film. The \nfilm thickness of 33(2) nm was determined by a low-angle x -ray reflectivity measurement with an \nangular resolution of 0.005°. \n \n A Quantum Design MPMS superconducting quantum interference device (SQUID) \nmagnetometer was used for measurements of th e global magnetization with the magnetic field \nperpendicular to the film surface. All MFM measurem ents described in this paper were carried out \nin a home-built low-temperature MFM apparatus.15 MFM images were obtained in high vacuum of \n1x10-6 Torr in a frequency-modulated mode. Commerc ially available cantilevers with a Co/Cr \ncoating layer16 were used for MFM measurements. The MFM tip was magnetized along the tip axis \nin a field of 3 T prior to MFM measurements. The external magnetic field ( H) was always applied \nperpendicular to the film surface. The negative fre quency shift of the tip results from the attractive \ninteraction between the tip and the sample magneti zation. Therefore, the dark features in the MFM \nimage, displaying a negative frequency shift of the tip, indicate that the sample magnetization is \nparallel to the tip magnetization. \n In figure 2(a) we present magnetization ( M) vs Temperature ( T) at μ0H= 0.1 T for H \nperpendicular to the surface. The global magnetic  measurements were performed in the same \nconfiguration as the local measurements in MFM. The M-T curve shows an inflection at \napproximately 75 K, which corresponds to the FM  order reported previously for a bulk sample.8 \nFigure 2(b) shows the coercive field ( Hc) vs T obtained from magnetic hysteresis loops at each \ntemperature (see inset). The data show an increase of Hc below 30 K, which corresponds to the \nfreezing temperature, signaling that the system  possibly undergoes a spin-glass transition.8 \nAdditionally, the saturation magnetization ( Ms), obtained from the subtraction of the paramagnetic background, is always smaller than the theoretical value of Ms ≈ 560 emu/cm3. The Ms value at 5 K \nis 170 ± 30 emu/cm3, indicating the presence of non-FM regions or frustrated magnetism. \n \n Figures 3(a)-3(e) display MFM images obtained sequentially at 4 K along an upper branch \nof the magnetic hysteresis loop after saturation at μ0H=1 T. The MFM image obtained at μ0H=0.5 T \n[see Fig. 3( a)] shows coexistence of isolated round and elongated domains. The bright domains are \nantiparallel to the tip magnetization, and thei r size is around 200 nm. The remanent state ( H=0), \nshown in Fig. 3(b), is characterized by dark spots (bubble domains parallel to the tip field), \nappearing in the matrix of a homogeneous magne tization. The size of the ferromagnetic bubble \ndomains is around 100 nm, smaller than that for μ0H=0.5 T. The shape of the bubbles persists up to \nμ0H= -0.1 T, and changes back to large domains at μ0H= -0.5 T, showing similar shapes to those in \nFig. 3(a): the cross correlation map, shown in Fi g. 3(f), shows strong positive correlations. This \nindicates that magnetization reversal takes place via rotation of the magnetic domains instead of the \nnucleation of the reversed domai ns that expand with increasing H. This type of magnetization \nreversal via domain rotation is a typical signature  of phase separated magnetic materials. Data taken \nat μ0H= -3 T (not shown) are similar to those at μ0H=-1 T, indicating the saturation of the sample \nand the presence of non-FM regions in the film. Th e rapid change of the domain features at low \nfield is related to the stiffness of the magnetic domains due to the dominant shape anisotropy. The \nout-of plane magnetic saturation field ( Hs) can be estimated by considering the theoretical \nexpression for an isolated bubble domain, assumi ng a disk with a diameter of 200 nm and a \nthickness of 33 nm, 4ߨሺ1െܦ ሻܯ௦ൎ 2000 Oe , where D is the demagnetization factor17 and Ms is \nthe saturation magnetization. The experimental values of Hs [see the inset in Fig 2(b)] are around \n3600 Oe, larger than the theoretical value of 2000 Oe, indicating that the system presents large \ndomains produced by interconnected bubbles with common boundaries, which is in good agreement \nwith the experimental data [see Fig. 3(e)]. Th e magnetic topology of YCMO shows isolated bubble \ndomains, different from the topology of the GCMO film,9, which showed non-symmetric and larger \nmagnetic regions. The presence of the unconnected bubble domains at low field in YCMO can be \nunderstood by the inhomogeneous Mn3+/Mn4+ ratio as a mechanism for stress relaxation.18 Figure 4 \nshows MFM images, obtained sequentially from th e same place at 4 K and 10 K, respectively. \nThermal drift was negligible from 4 K to 15 K due to the rigid design of the microscope.9, 15 The \nimages were obtained at μ0H = -0.1 T after the sample was saturated at μ0H = 1 T, as in Fig. 3(c). \nThe features between 4 K and 10 K have no spatial correlation,10 indicating that the bubble domains \nchange drastically with the temperature under these conditions.   \n \n Having both the MFM images in Figs. 3 and 4 and those discussed in Ref. [8] in GCMO \nfilms allows us to discuss similarities and di fferences between GCMO and YCMO. GCMO thin \nfilms exhibit phase coexistence between ferrima gnetic domains and non-ferrimagnetic regions9 and \nshow larger domains than do YCMO films. The main  contrast between the two films arises from the \nGd-Mn  interaction in GCMO, which modifi es the magnetism and results in the Tcomp, where the \nmagnetizations from Gd and Mn sublattices are antip arallel and equal to each other. The large \nchanges and the complex behavior of the magnetism due to the Gd-Mn  antiferromagnetic coupling \nmake the analysis of the evolution of the magnetic domains difficult.12 There are several possible \nmechanisms of the magnetic interaction w ithin an assembly of magnetic particles.19 In general, the \ndipole-dipole interaction between particles is of primary importance for such systems. A direct exchange interaction via the surface of the bub ble domains should be taken into account as well \nwhen the clusters are in close contact with each other. Another possible explanation is the presence \nof frustrated interfaces between FM and non-FM regi ons, which is also consistent with the spin \nglass-like behavior reported in the bulk samples.8 Unconventional glass-like behaviors appear either \nin bulk manganites with phase separation or in films and multilayers with strained \ninterfaces.20,21,22,23 No correlation was detected between the FM domains at 4 K and 10 K in this \nstudy, which is consistent with random nucleation, and suggests that the intrinsic distortion, due to \nthe low tolerance factor, plays a salient role in  the magnetic topology. Alth ough magnetic properties \nin thin films could be strongly affected by stress and strain,24 the drastic change of the Hc (T) and the \nmagnetic topology of isolated bubbles at low temperatures (see Fig. 4) suggest a non-static \nMn3+/Mn4+ ratio as a mechanism for strain relaxation. We believe that the freezing temperature \ncould be associated with changes of the non-FM ma trix, which prevent mobility of the FM cluster \nand produce changes of the dynamics of the material.8 This hypothesis is in agreement with the fact \nthat Hs does not change significantly between 5 and 30 K (not shown), which indicates no \nsignificant change of the demagnetization factor due to coupling between the bubbles.  \n In conclusion, we studied topology of the magnetic domains in a high quality epitaxial \nYCMO thin film. Our results show the phase coex istence between FM and non-FM domains and a \nspin-glass behavior below T\nC, which is supported by a strong suppression of the saturation \nmagnetization.  We found that at low magnetic fi eld the unusually small size of the isolated \nbubbles. Our observations are consistent  with the fluctuation of the Mn3+/Mn4+ ratio as the \nmechanism of strain relaxation in YCMO films. Our temperature-dependent studies show a direct \nevidence of the spin glass-like behavior and magne tism reported previously in bulk samples. The \nsmaller size of the round shape domains in YC MO, compared to those in GCMO, suggests a \npotential application for a magnetic memory devi ce, and magnetic templa te of magnetic pinning \ncenters in superconductors.  \n \nWe thank J. O. Willis for providing useful comments. This work was supported by the US \nDepartment of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and \nEngineering and the Los Alamos National Labor atory’s Laboratory Directed Research and \nDevelopment Program, Project No. 20130285ER. N. H. is a member of CONICET (Argentina). \n \n \n  \n \n \n \n  \n \n \n \n \n                                                             \n1 M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998). \n2 Jin, S. et al. Science 264, 413–415 (1994). \n3 D. Niebieskikwiat and M. B. Salamon, Phys. Rev. B 72, 17422  (2005). \n4 T. Kimura, Y. Tomioka, R. Kumai, Y. Okimoto, and Y. Tokura, Phys. Rev. Lett. 83, 3940 (1999). \n5 L. P. Gor'kov and V. Z. Kresin, Physics Reports 400, 149 (2004). \n6 J. P. Attfield, Crystal Engineering 5, 427 (2002). \n7 G. Y. Snyder, C. H. Booth, F. Bridges, R. Hiskes, S. DiCarolis, M. R. Beasley, and T. H. Geballe, Phys. \nRev. B 55, 6453 (1997). \n8 R. Mathieu, P. Nordblad, D. N. H. Nam, N. X. Phuc, and N. V. Khiem. Phys. Rev. B 63, 174405 (2001). \n9 J. Kim, N. Haberkorn, L. Civale, E. Nazaretski,  P. Dowden, A. Saxena, J. D. Thompson, and R. \nMovshovich, Appl. Phys. Lett. 100, 022407 (2012).   \n10 N. Haberkorn, S. Larregola, D. Franco, and G. Nieva, J. Magn. Magn. Mater. 321, 1133 (2009). \n11 X. L. Wang, J. Horvat, H. K. Liu, and S. X. Dou, J. Magn. Magn. Mater. 182, L1 (1998). \n12 V. F. Correa, N. Haberkorn, G. Nieva, D. J. García, and B. Alascio, EPL 98, 37003 (2012). \n13 D. E. Cox, P. G. Radaelli, M. Ma razio, and S-W. Cheong, Phys. Rev. B 57, 3305 (1998). \n14 I. G. Deac, J. F. Mitchell, and P. Schiffer, Phys. Rev B 63, 1724081(2001). \n15 E. Nazaretski, K. S. Graham, J. D. Thompson, J.  A. Wright, D. V. Pelekhov , P. C. Hammel, and R. \nMovshovich, Rev. Sci. Instrum. 80, 083074 (2009). \n16 A SSS-QMFMR cantilever,  Nanosensors, Inc. \n17 J. A. Osborn,  Phys. Rev 67, 351 (1945). \n18 J. Burgy, A. Moreo, and E. Dagotto, Phys. Rev. Lett. 92, 097202 (2004). \n19 X. Batlle and A. Labarta, J. Phys. D 35, R15 (2002).  \n20 M. Bibes, LI Balcells, S. Valencia, J. Fontcuberta, M. Wojcik, E. Jedryka, and S. Nadolski, Phys. Rev. Lett. \n87, 067210 (2001). \n21 N. Haberkorn, F. Lovey, A. M. Condó, G. Nieva, and J. Guimpel, Phys. Rev B 75, 024427 (2007). \n22 D. Niebieskikwiat, J. Tao, J. M. Zuo, and M. B. Salamon, Phys. Rev. B 78, 014434 (2008). \n23 N. Haberkorn, J. Guimpel, M. Sirena, L. B. Steren, E.  Baca, W. Saldarriaga, and M. E. Gómez, Appl. Phys. \nLett. 84, 3927 (2004).  \n24 I. C. Infante, F. Sánchez, J. Fontcuberta, M. Wojcik, E. Jedryka, S. Estradé, F. Peiró, S. Estradé, F. Peiró, J. \nArbiol, V. Laukhin, and J. P. Espinós, Phys. Rev. B 76, 224415 (2007).     \n \n \n \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n                                                                                                                                                                                      \nFIG. 1. X-ray diffractogram (logarithmic intensity scale) of the YCMO film at room temperature. \nThe inset shows the rocking curve for the (040) reflection. \n \n \nFIG. 2. (Color online) (a) Magnetization vs temperature at μ0H= 0.1 T. (b) Coercive field vs \ntemperature obtained from magnetic hysteresis loops. Inset: typical hysteresis loop at 15 K. All the \nmeasurements were performed with H ⊥ to the surface.  \n \n \nFIG. 3. (Color online) (a)-(e) MFM images in YCMO taken sequentially in different fields at 4 K. \n(f) Cross correlation images between (a) and (d). Th e bright spot marked by the white arrow shows \na strong positive correlation and indicates a simila r domain structure between (a) and (d).  The \nposition of the spot in the correlation map is off-cen tered, indicating a small field drift is present. \nThe tip lift height was 100 nm from the surface. \n \n \nFIG. 4. (Color online) MFM images obtained at different temperatures. (a) The image was obtained \nin μ0H =-0.1 T after the sample was saturated in th e field of 1 T. (b) The MFM image taken and \nafter warming the sample of (a) in the same field of -0.1 T. The tip lift height was 100 nm above the \nsample surface. No spatial correlation was resolved between 4 K and 10 K. \n \n \n \t\r 20304050607080-2-1012YCMO (040)\nYCMO (060)STO(003)STO(002)I [arb. units]2 Θ [°]STO(001)YCMO (020)\nI [arb. units]ω [°]FWHM ~ 0.24 °\t\r 0102030405003006009001200-1.0-0.50.00.51.0-150015003060901200306090\nb) T [K]Hc[Oe]a)\nHcMs 15 KM [emu/cm3]µoH [T]Hs µ0H = 0.1 T T [K]M [emu/cm3]Tc1 µm (a) (b) (c) (d) (e) YCMO 0.5 T 0 T -0.1 T -0.5 T -1 T \n(f) \n1 µm (a) (b) 4 K 10 K "    },    {        "title": "0910.2442v1.Resonant_coupling_of_coplanar_waveguides_with_ferromagnetic_tubes.pdf",        "content": "A. Kozhanov et al. October 2009 1\nResonant coupling of coplanar wave guides with ferromagnetic tubes. \n \nA. Kozhanov1, D. Ouellette1, M. Rodwell1, D. W. Lee2, S. X. Wang2 and S. J. Allen1 \n1California Nanosystems Institute, University of Ca lifornia at Santa Barbara, Santa Barbara, CA, 93106 \n2Department of Materials Science and Engineer ing, Sanford Univers ity, Stanford, CA, 94305 \n \n(Received      ) \n \nResonant coupling of coplanar waveguides is explored by wrapping proximate shorted ends of the waveguides with micron \nsize ferromagnetic Co 90Ta5Zr5 tubes.  Ferromagnetic resonance and up to 7 outer surface modes are identified. Experimental \nresults for these contorted rectangular tubes are in good agreem ent with micromagnetic simula tions and model calculations \nof magnetostatic modes for an elliptical fe rromagnetic tube.   These results indicate that the modes are largely determined by \ntube topology and dimensions but less so by the detailed shape. (PACS: 76.50.+g) \n \nMonolithic micro and nanoscale filters, delay lines and \nresonators are potentially important for high frequency \nelectronics.  Here we explor e metallic ferromagnetic micron \nscale magnetostatic wave structures in the form of tubes \nwrapped around coplanar wave guides.  Microwave devices \nbased on magneto-static spin waves in insulating ferrimagnetic \nmaterials like yttrium-iron garnet (YIG)1 have long been \nexplored and developed.  Howeve r, future micro and nano scale \nspin wave based devices ma y benefit from exploiting \nferromagnetic metals that are more easily deposited, processed \nand nanofabricated than ferrim agnetic oxides.  Further, \nferromagnetic metals like CoTaZr, CoFe and CoFeB have nearly \nan order of magnitude larger  saturation magnetization than \ntypical ferrimagnets.2  As a result, they will support higher, \nshape defined, zero magnetic fiel d resonances and consequently \nintrinsically faster response. \nRecent theoretical3,4,5 and experimental6,7,8 work has \nfocused on the magnetization dynamics in ferromagnetic \nnanotubes. Ferromagnetic nanot ubes could serve as magnetic \ncores in nano scale transformers as well as active elements of \ntunable high frequency filters. Several theoretical models predict \nexistence of quantized surface modes of magnetostatic \noscillations in the ferromagnetic nano tubes magnetized along \nthe axis of the tube4,5. The experiments of Mendach et al. \nobserved ferromagnetic resonance in a Permalloy tube, the lowest order and essentially spa tially uniform mode.  They did \nnot report on the rich spectrum of  standing magneto-static waves \nthat circulate around the tube.\n8. \nThis letter describes excitati on and detection of quantized \nsurface magnetostatic oscillations in the ferromagnetic tubes \nformed by wrapping metallic fe rromagnetic film  around exciting \nand detecting coupling loops at e nds of co-planar waveguides.  \nResonances are displayed that are defined by the magnetostatic \nmodes indexed by periodic boundary conditions around the tube \ndespite their contorted geometry. Such a structure could form \nthe building block for filtering elements at microwave \nfrequencies.    \nThe ferromagnetic tube coupler was fabricated in the \nfollowing manner.  200nm thick ferromagnetic Co 90Ta5Zr5 films \nwere sputtered onto Si/SiO 2 wafers, lithographically patterned \ninto two xxx x xxx micron rectangles and covered with an \ninsulating SiO 2 layer. (A saturation magnetization of M s=1.2T \nand a coercive field H c~2 Oe were measured on an unpatterned \nCo90Ta5Zr5 film using a vibrating sa mple magnetometer.)   \nCoupling loops formed by the shorte d ends of a pair of coplanar \nwaveguides were positioned over Co 90Ta5Zr5 rectangles.  The structure was then covered with a 100 nm thick SiO 2 insulating \nlayer and holes etched to allow a subsequent top Co 90Ta5Zr5 \nlayer to complete the magnetic circuit. The top Co 90Ta5Zr5 layer \nwas sputtered on the resist covered structure and lifted-off. This \nprocess resulted in two shorte d coplanar waveguides wrapped \ntogether by a Co 90Ta5Zr5 film (Fig.1). The “tubes” thus formed \nare of course not circular but  topologically equivalent to a \ncylinder and provide a closed ma gnetic circuit (Fig. 1b,c) that \ncaptures the magnetic fields produced by the high frequency \ncurrents flowing in the shorted ends of the coplanar waveguides.  \nFocused ion beam etching (Fig.1 c) exposes the cross section \nimaged in the SEM micrograph.  \nThe closed magnetic circuit effectively couples the two \nshorted ends of the coplanar waveguides only when the tube \nmagnetization is oriented along the axis of the tubes, along the \nshorting lines of the coplanar waveguides.  Only then can the \nmicrowave magnetic fields induce changes in the magnetization \nand couple to the magnetostatic oscillations in the tubes. \n \n \n \nFig. 1.  Fabricated structure SEM micrograph (a), profile scheme(b), \nSEM micrograph of structure cross section.(c) A. Kozhanov et al. October 2009 2\nS-parameters were measured at room temperature using \nAgilent 8720ES vector network analyzer operating from 0.05 to \n20 GHz.  Only S 21, the ratio of high frequency voltage at \nterminals 2 to the input high freque ncy voltage at terminals 1, is \nanalyzed in the following discu ssion.  The test devices were \npositioned on the narrow gap of  small electro-magnet that \nprovided magnetic field bias up to 1000 Oe. By comparing the \nS-parameters at disparate bias  magnetic fields, the magnetic \nfield independent instrument response can be effectively \nremoved to expose the S-parameters related to the magneto-\nstatic mode coupling of exciting a nd detecting wires. See Fig. 2.  \n \n \nIn the absence of external magnetic field, () 21Sf  has a \nnumber of irreproducible peaks that are strongly dependent on \nthe history of the bias magnetic field. With increase of the \nexternal magnetic field directed along the axis of the tube ( Hx), \nthe magnitude of these peaks decreases and related magnetic \ncoupling disappears at Hx~100 Oe.  \nAt Hx ≥ 200Oe we detect a strong peak followed by a series \nof smaller peaks. These peaks shift towards higher frequencies \nand grow in magnitude with increase of Hx. Transmission at the \nlowest frequency peak reaches | S21|~0.012  at Hx=988 Oe. \nThese results suggest that at ~ 100-200 Oe there is a change in \nthe magnetization distribution.  Micromagnetic simulations were \ncarried out to investigate the magnetization alignment within our \nstructures at magnetic field values |H| ≤1kOe. We used a \nrectangular Co 90Ta5Zr5 tube as the model with dimensions \nsimilar to the dimensions of the fabricated ferromagnetic tubes.  \nMicromagnetic structure was si mulated by solving the Landau-\nLivshitz-Gilbert equation using LLG Micromagnetics Simulator. \nThe results of the simulations are shown on Figures 3 \nand 4. The ground state at H=0 is described by a double vortex7 \nstate. The magnetization is circ ularly oriented around the tube \nperimeter pointing in a clockwise direction on one end, counter-\nclock wise direction on the other end, with a domain wall in the \ncenter. This is similar to what was found by Lee et al. for \ncircular ferromagnetic nanotubes7.  For the rectangular tube \nsome of the quasi stable states apparent in the in  circular tube at H=0 are not observed  (The “ferromagnetic” state7 with \nmagnetization aligned along the tube is not stable at H=0 in the \nparticular rectangular tube used  in these experiments.)      \n \n \n \n \nIn our experiment the paired vortex state of our \nferromagnetic tube will result in a complex configuration in \nwhich most of the magnetic moments are parallel to the exciting \nhigh frequency magnetic fields produced by the RF currents in \nthe waveguide. Only the very narrow area of the domain wall \nbetween the paired vortex states magnetic moments will have \nsome alignment along the tube axis and couple to the exciting \nmagnetic fields.  We speculate th at they are responsible for the \nirreproducible low amplitude peaks in () 21Sf  at H=0.  \nThe simulation indicates that with sufficient field, Hx, the \ntube is magnetized along the axis ; this is indicated by the \nsaturation in the hysteresis curve (Fig.4). This proceeds either by \ndomain wall widening or formati on of in plane magnetization \nvortex.  The details are not impo rtant for the present discussion \nbut following the jump of the M(H)  curve at H≈200 Oe we \nassume the magnetic moments are largely pointing along the \ntube axis. Then the high frequency magnetic field is \nperpendicular to th e magnetic moments and can excite the \nvarious standing magnetostatic waves.  \nFig. 3.  Schematic of magnetization alignment in the tube at H x=0. \nFig. 4.  Results of micromagnetic simulations: hysteresis curve of the \nrectangular ferromagnetic tube. \nFig. 2.  Frequency and magnetic field dependence of |S21| measured\nwith the fabricated structure. A. Kozhanov et al. October 2009 3\nStarting from magnetic field valu es of ~200 Oe we detect a \nseries of resonant peaks in () 21Sf  whose frequency increases \nwith increasing Hx (Fig.6). The development of the systematic \nstructure above 200 Oe shown in Fig. 6 confirms the results of \nthe simulation: above 200Oe, the magnetization of the \nrectangular tube is well al igned with the tube axis. \nWe identify up to 8 peaks in () 21Sf  and display them \nversus external magnetic field in Figure 6. We use model by \nPopov and Zavislyak5 to analyze the experimental data. This \nmodel calculates standing magne tostatic wave modes in a \nferromagnetic tube with elliptic al profile magnetized along the \ntube axis. Two classes of solution appear: standing spin waves \non the inner and on the outer surfaces of the tube. We assign the \nobserved peaks to the 7 outer  standing magnetost atic modes plus \nthe lowest, and essentially unif orm, ferromagnetic resonance. \nThe frequency for the latter uniform mode of the elliptical \nnanotubes coincides with the fr equency of the ferromagnetic \nresonance8 of an infinite ferromagnetic film and described \nby1/2~( 4 )fHMγπ+ . The other modes describe standing \nmagnetostatic waves with a whole number of wavelengths \nwithin the outer perimeter of the tube. The inner modes shown \nwith the dashed lines in the Fig. 6 do not seem to appear in the \nexperimental data.  We can offer no explanation. \n \n \nThe profile of the fabricated ferromagnetic tubes is not at \nall elliptical but has gross distortions where the CoTaZr film \ngoes over the coupling loops.  Re markably, we conclude that \nresponse is controlled by the to pology and scale: the detailed \nshape is not critical.  \nIn summary, we fabricated and measured microwave \ntransmission through coplanar waveguides coupled by novel \nrectangular ferromagnetic tubes.  We identified ferromagnetic \nresonance and up to 7 outer surf ace magnetostatic oscillation \nmodes guided by models of the magnetostatic oscillations in \nferromagnetic ellipti cal nanotubes. \nThese structures are potentially  important as high frequency \ntunable filters. The frequency of th e lowest and strongest peak is \nthe ferromagnetic resonance as described earlier. The frequency \nof the higher modes are defined by  the tube geometry (diameter, \nwall widths). As interesting as th e surface modes are, the strong \nferromagnetic mode is probably the most useful.    In order to make the filter work efficiently we will increase \nthe input inductance of the filter either by increasing the number \nof exciting wires inside the magnetic tube (winding) or more \nconveniently by increasing the t ube length while keeping other \ndimensions the same. Lengthening the tubes will introduce \nstrong shape anisotropy.  Th e ground state will have the \nmagnetization along the tube without external bias and a strong \nresonance transmission at H=0 .  Resonance could be fine tuned \nby “on circuit board” fields or self fields produced by DC \ncurrents flowing in the wires encased by the tubes. \nThe authors are grateful to Andrew Cleland for the use \nof the vector network analyzer and hosting aspects of this work \nin his laboratory.  This work is supported by NERC via the \nNanoelectronics Research Initiative (NRI), by Intel Corp. and \nUC Discovery at the Western Institute of Nanoelectronics \n(WIN) Center. \n                                                                 \n1 A review: J.D. Adam, L.E. Davis,  G.F. Dionne, E.F. Schloemann and \nS.N. Stitzer, IEEE. Trans.  Microwave Theory Tech. 50, 721 (2002). \n2 B.Kuanr, I.R. Harward, D.L. Marvin, T, Fal, R.E. Camley, D.L. Mills, \nand Z. Celinski, IEEE Trans. Magnetics, 41, 3538 (2005). \n3 H.Leblond, V.Veerakumar, Phys. Rev.B 70, 134413 (2004) \n4 T.M. Nguyen, M.G. Cottam Surface Science 600, 4151 (2006) \n5 M.A. Popov, I.V. Zavislyak ISSN 0503-1265, Ukr. J. Phys., 53, 7, 702 \n(2008) \n6 F.S.Li, D. Zhou, T. Wang, L.J. Song, and C.T. Xu, J. Appl. Phys, 101, \n014309 (2007) \n7 J. Lee, D. Suess, T. Schrefl, K. Oh, J. Fidler JMMM 310 (2007) 2445-\n2447. \n8 S. Mendach, J. Podbielski, J. Topp,  W. Hansen and D. Heitmann, Appl. \nPhys. Lett. 93, 262501 (2008) \n0 200 400 600 800 100002468101214161820f, GHz\nH, Oeinnerouter\nFig. 6.  |S 21| peaks frequency dependence on the applied magnetic field: \nexperiment (circles), theory for t ubes with elliptical profile (lines) 5. "    },    {        "title": "1410.5080v2.Coexistence_and_competition_of_ferromagnetism_and_p_wave_superconductivity_in_holographic_model.pdf",        "content": "arXiv:1410.5080v2  [hep-th]  3 Nov 2014Coexistence and competition of ferromagnetism and\np-wave superconductivity in holographic model\nRong-Gen Cai∗, Run-Qiu Yang†\nState Key Laboratory of Theoretical Physics,\nInstitute of Theoretical Physics, Chinese Academy of Scien ces,\nBeijing 100190, China\nAugust 27, 2018\nAbstract\nBy combining a holographic p-wave superconductor model and a holographic fer-\nromagnetismmodel, westudythecoexistence andcompetitio nofferromagnetismand\np-wave superconductivity. It is found that the results depe nd on the self-interaction\nof magnetic moment of the complex vector ﬁeld and which phase appears ﬁrst. In the\ncase that the ferromagnetic phase appears ﬁrst, if the inter action is attractive, the\nsystem shows the ferromagnetism and superconductivity can coexist in low tempera-\ntures. If the interaction is repulsive, the system will only be in a pure ferromagnetic\nstate. In the case that the superconducting phase appears ﬁr st, the attractive inter-\naction will leads to a magnetic p-wave superconducting phas e in low temperatures.\nIf the interaction is repulsive, the system will be in a pure p -wave superconducting\nphase or ferromagnetic phase when the temperature is lowere d.\n∗E-mail: cairg@itp.ac.cn\n†E-mail: aqiu@itp.ac.cn\n1Contents\n1 Introduction 2\n2 The model 5\n3 EoMs and free energy density 7\n3.1 Ansatz and EoMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3.2 Free energy and magnetic moment . . . . . . . . . . . . . . . . . . . . . . 9\n4 Coexistence and competition 11\n4.1 Superconducting ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . .11\n4.2 Ferromagnetic superconductor . . . . . . . . . . . . . . . . . . . . . . . . .15\n5 Summary and discussions 18\nA The method to compute αc 20\n1 Introduction\nIn condensed matter physics, there are two kinds of critical phen omena which have at-\ntracted a lot of attention for a long time. One is ferromagnetism whe re the electron spins\nalign to produce a net magnetization, which breaks the time reversa l symmetry sponta-\nneously and happens in the ferromagnets at the Curie temperatur e,TC(sometimes it is\neven higher than the indoor temperature). The other is supercon ductivity where electrons\ncondenseintoCooperpairsinthemomentumspace, whichbreaksth eU(1)symmetryspon-\ntaneously and usually happens at a much low temperature Tsc. Though ferromagnetism\nhas been found for more than thousands years and the supercon ductivity was found about\n100 years ago, their physical explanations was provided after the complete quantum theory\non materials was built.\nIn a very long time, it was thought these two phenomenons are incom patible with\neach other. This is rooted in the microscopic theory of supercondu ctivity by Bardeen,\nCooper, and Schrieﬀer (BCS) [ 1]. In the framework of BCS theory, the superconductivity\nappears when the electrons were bounded with antiparallel spins in s inglet Cooper pairs\ndue to the eﬀective attractive force coming from the lattice vibrat ions. When magnetic\nimpurityatomsareplacedinaconventionalsuperconductor, they arecapableofﬂippingthe\nelectron’s spin. Hence, impurity will suppresses the singlet Cooper p air formation, which\ncauses a rapid depression of the superconducting transition temp eratureTsc. Likewise,\nthe superconductivity can screen oﬀ the magnetic ﬁeld, which leads to the long-range\nmagnetic order is accompanied by the expulsion of superconductivit y. These features\nlead to the competition of such two orders. However, around 1980 s, it was recognized\nthat under special conditions superconductivity may coexist with a ntiferromagnetic order,\nwhere neighboring electron spins arrange in an antiparallel conﬁgur ation. For instance, in\n2Figure 1: The experimental results on the p-wave superconductin g ferromagnetic material\nURhGe. (a)Temperature dependence of the spontaneous magnetization M(T) and the in-\nverse of the magnetic susceptibility χ−1(T) under the normal pressure [ 6].(b)Temperature\ndependence of the resistivity of URhGe for diﬀerent pressures at low temperature. The su-\nperconducting critical temperature is deﬁned by the zero resistiv ity point [ 7].(c)Pressure-\ntemperature phase diagram of URhGe [ 7].\nheavyfermionantiferromagnets, theitinerantmagneticmoments havealmostnode-pairing\neﬀect on singlet Cooper pairs, because the average exchange inte raction is zero.\nThe discovery of the ﬁrst superconducting ferromagnet1UGe2in the year 2000 came as\na big surprise [ 2]. In this material, superconductivity is realized well below the Curie te m-\nperature, withoutexpellingtheferromagneticorder. Sincethen, othertwosuperconducting\nferromagnets have been discovered, such as URhGe [ 3] and UCoGe [ 4], which display in-\ntrinsic coexistence of ferromagnetism and superconductivity. Ev idence of superconducting\nferromagnetic phase was also reported for ZrZn 2in 2001 [5].\nFigure1shows the experimental results of a typical superconducting fer romagnetic\nmaterial URhGe. The ferromagnetic order is observed near TC≃9.5K, where the spon-\ntaneous magnetization begins appearing and the magnetic suscept ibility diverges. Below\nthis temperature, the initial magnetic susceptibility is positive and th e spontaneous mag-\nnetization increases with decreasing the temperature. In the fer romagnetic phase, when\ntemperature is decreased to near Tsc≃0.25K(at atmospheric pressure), the resistivity\napproaches to zero, which shows that the material is in a supercon ducting phase. In the\nregion of T < T sc, spontaneous magnetization and zero resistivity coexist, which sh ows\na superconducting ferromagnetism. The P−Tphase diagram shows that this kind co-\nexistence can appear in a very wide pressure region (up to near 30 k bar, where the Tsc\ndisappears.).\n1In this paper, we will use “superconducting ferromagnet” to deno te the materials whose Curie tem-\nperature is higher than superconducting transition temperature and “ferromagnetic superconductor” to\ndenote the opposite case.\n3The nature of superconducting state in ferromagnetic materials is currently under de-\nbate. Early investigations [ 8] studied the coexistence of conventional s-wave superconduc-\ntivity with itinerant ferromagnetism. However, the scenario of spin -triplet pairing soon\ngained the upper hand [ 9]. For a review of phenomenological theory of ferromagnetic un-\nconventional superconductors with spin-triplet Cooper pairing of electrons, one can see\nRef. [10]. Ref.[11] presents a general thermodynamic theory that describes phas es and\nphase transitions of ferromagnetic superconductors with spin-t riplet electron Cooper pair-\ning, based on an extended Ginzburg-Landau theory. Generally spe aking, the coexistence\nof ferromagnetism and superconducting is discovered in spin triplet rather than the usual\nspin singlet superconducting materials, because that p-wave pairin g allows parallel spin\norientation of the fermion Cooper pairs in unconventional superco nductors [ 12]. This\nunconventional paired manner makes p-wave superconductivity b eing robust under the in-\nﬂuences of external magnetic ﬁeld and spontaneous magnetizatio n. So they may coexist\nwith each other.\nA mean-ﬁeld model for coexistence of spin-triplet pairing and ferro magnetism was de-\nveloped in [ 9,13]. The model considers a uniform coexistence of ferromagnetism an d\nsuperconductivity, i.e., the same electrons play the role for the fer romagnetism and su-\nperconductivity at the same time. Another scenario where there is an interplay between\nmagnetic and superconducting orders in the same material is super conductors with spiral\norhelical magneticorder. Examples of such includeErRh 4B4andHoMo 6S8. Inthesecases,\nthe superconducting and magnetic order parameters entwine eac h other in a spatially mod-\nulated pattern, which allows for their mutual coexistence, althoug h it is no longer uniform.\nEven spin-singlet pairing may coexist with ferromagnetism in this mann er.\nUp to now, the theoretical investigation have been concentrated on the weak coupling\ncase, where some approximations and conception of free ﬁeld are s till valid. The investiga-\ntions on strong correlated system in theoretical and experimenta l aspects have challenged\nthe pictures about the materials. A crucial feature of these syst ems is the nonzero mag-\nnetic moment of the spin-triplet Cooper pairs. However, the micros copic theory about the\ncoexistence of magnetism and superconductivity in strongly intera cting heavy electrons is\neither too complex or insuﬃciently developed to describe the complica ted behavior. So it\nis still a fascinating thing to ﬁnd a suitable theory to describe the coe xistence and com-\npetition of the ferromagnetism and superconductivity in strong co rrelated system, such as\nthe heavy fermion system [ 14], Iron-based superconductor [ 15] and unconventional super-\nconductor [ 16].\nIn the strong coupling case, the usual methods developed in conde nsed matter theory\n(CMT) are considered losing their eﬃcacy. A new method, named gau ge/gravity duality\nor AdS/CFT correspondence, is considered as a promising approac h [17,18,19,20]. This\nmethodrelatesaweak couplinggravitationaltheoryina( d+1)-dimensionalasymptotically\nanti de-Sitter (AdS) space-time to a d-dimensional strong coupling conformal ﬁeld theory\n(CFT) in the AdS boundary. In recent years, this duality has been e xtensively applied into\ncondensed matter systems. Some remarkable progresses have b een made in this direction.\nFor example, some gravitational dual models of superﬂuid/superc onductor [ 21,22,23],\n(non-)Fermi liquid [ 24,25,26], Josephson junctions [ 27,28,29], superconducting quantum\n4interference device [ 30] and magnetic properties in superconductors [ 31,32,33,34,35]\nhave been constructed and intensively studied. The models in the Ad S/CFT frame for fer-\nromagnetism/paramagnetism and anti-ferromagnetism/paramag netism phase transitions\nhave also been proposed in [ 36,37].\nIn this paper, we will explore the coexistence and competition betwe en ferromag-\nnetism and p-wave superconductivity in the strong coupling case. O ur tool is just the\ngauge/gravity duality. We noted that the similar topic appeared in Re f. [38], where the\nauthors use two U(1) ﬁelds condense simultaneously from SU(2) mo del to present the\nsuperconducting and spontaneous magnetic orders. Our model is diﬀerent from theirs.\nInordertoconstructaholographicmodeltoinvestigatethecoex istenceandcompetition\nbetween ferromagnetism and superconductivity, we need a model where ferromagnetic\nproperties (such as spontaneous magnetic moment, time reversa l symmetry broken and\ndivergedmagneticsusceptibility atCurietemperatureandsoon)ca nappearindependently\nfrom the superconductivity. This is just supplied by the model in Ref . [36]. By combining\nthe complex vector ﬁeld model for the holographic p-wave superco nductor [ 39,40,41] and\nthe real antisymmetric tensor ﬁeld model for the holographic ferr omagnetism in Ref. [ 36],\nwe study the coexistence and competition of ferromagnetism and s uperconductivity. It\nturns out that the results depend on magnetic moment self-intera ction of complex vector\nﬁeld and which phase appears ﬁrst. In the case that the ferromag netic phase appears ﬁrst\nand the interaction is attractive, the p-wave superconductivity c an still appear and the\nsystem can show ferromagnetism and superconductivity both whe n temperature is lower\nthan a critical value. But if the interaction is repulsive, the p-wave s uperconductivity\ncan not appear and the system will only be in a pure ferromagnetic st ate. In the case\nthat the superconducting phase appears ﬁrst, the system will sh ow a magnetic p-wave\nsuperconducting phase with decreasing temperature if the intera ction is attractive. If\nthe interaction is repulsive, the system is in a pure p-wave supercon ducting phase or\nferromagnetic phase when the temperature is lowered.\nThe paper is organized as follows. In section 2, we will ﬁrst describe the holographic\nmodel. We will give our ansatz for matter ﬁelds, the equations of mot ion (EoMs) and\nthe expression of free energy density in section 3. In section 4, we will investigate the\npossible phases and the coexistence and competition between ferr omagnetism and super-\nconductivity with diﬀerent parameters. A brief summary and discus sion will be given in\nsection5.\n2 The model\nIn this paper, the model we are considering is the combination of the holographic p-wave\nsuperconductor model described by a complex vector ﬁeld [ 39,40,41] and holographic\nferromagnetism model described by a real antisymmetric tensor ﬁ eld [36]. The action is\nS=1\n2κ2/integraldisplay\nd4x√−g/bracketleftbigg\nR+6\nL2−FµνFµν+Lρ+LM+LρM/bracketrightbigg\n, (1)\n5with\nLρ=−1\n2ρ†\nµνρµν−m2\n1ρ†\nµρµ+iqγρµρ†\nνFµν−Vρ,\nLM=−1\n4∇µMντ∇µMντ−m2\n2\n4MµνMµν−λ\n2MµνFµν−VM,\nLρM=−iαρµρ†\nνMµν,(2)\nwhereLis the AdS radius which will be set to be unity and κ2≡8πGis related to the\ngravitational constant in the bulk. In the following, we will set 2 κ2= 1 for simplicity. λ,γ\nandJare three constants with J <0 [36],γ >0.m1,m2are the masses of the complex\nvector ﬁeld ρµandreal tensor ﬁeld Mµν.gis thedeterminant ofthe bulkmetric gµνandqis\nthe charge of complex vector ﬁeld. We deﬁne Fµν=∇µAν−∇νAµandρµν=Dµρν−Dνρµ\nwith the covariant derivative Dµ=∇µ−iqAµ. Theγ’s term characterizes the magnetic\nmoment of the vector ﬁeld ρµ[39,42]. The antisymmetric tensor Mµνis the eﬀective\npolarization tensor of the U(1) gauge ﬁeld strength Fµν.VMdescribes the self-interaction\nof the polarization tensor. Following Ref. [ 36], we take\nVM=J\n8MµνMντMτδMδµ(3)\nforsimplicity. Theterm Vρdescribe theself-interactionofmagneticmoment ofthecomplex\nvector ﬁeld, a simple form is\nVρ=−Θ\n2ρ[µρ†\nν]ρµρ†ν. (4)\nHereΘisaconstant, itcharacterizesthefeatureofmagneticmom entinteractionofcomplex\nvector ﬁeld. From the free energy density given later, we can see t hat a positive Θ gives an\nattractive interaction between the magnetic moment of complex ve ctor ﬁeld itself, while a\nnegative value of Θ gives an repulsive interaction. In Refs. [ 39,40,41,42], the term Vρis\nnot considered as there the aim is to study the superconductivity o f the model. However,\nwe can see that this term will play an important role in the following when we consider\nthe magnetic properties of the model.\nOne may ﬁnd that the Lagrangian LMhas a little diﬀerence from the form in [ 36]. This\nis just for convenience. In order to simplify our discussion in the pro be limit, we make\nfollowing transformations\nMµν→λMµν, J→λ−2J, ρµ→λρµ,α→α/λ,Θ→Θλ−2. (5)\nUnder these transformations, action ( 1) can be rewritten as\nS=/integraldisplay\nd4x√−g/bracketleftbigg\nR+6\nL2−FµνFµν+λ2(Lρ+LM+LρM)/bracketrightbigg\n, (6)\nwith\nLM=−1\n4∇µMντ∇µMντ−m2\n2\n4MµνMµν−1\n2MµνFµν−VM (7)\n6and the others are kept the same as the forms in ( 2). The probe limit corresponds to\nkeeping all the quantities ﬁnite with λ→0. In this limit, we can ﬁx the background\ngeometry and Maxwell ﬁeld and only consider the dynamics of complex vector ﬁeld and\npolarization ﬁeld.\n3 EoMs and free energy density\n3.1 Ansatz and EoMs\nAs in Ref. [ 36], we will work in the probe limit with λ→0, by which we can ﬁx the\ngeometry background and neglect the back reaction of matter ﬁe lds on the background\ngeometric and Maxwell ﬁeld. The equations for complex vector ﬁeld a nd polarization ﬁeld\nread\nDνρνµ−m2\n1ρµ+iqγρνFνµ+Θρ[νρ†\nµ]ρν−iαρνMνµ= 0.\n∇2Mµν−m2\n2Mµν−JMµδMδτMτν−2iαρ[µρ†\nν]−Fµν= 0.(8)\nWe take the AdS Reissner-Nordstr¨ om (RN) black hole with a planar h orizon as the back-\nground metric [ 43]\nds2=r2(−f(r)dt2+dx2+dy2)+dr2\nr2f(r),\nf(r) = 1−1+µ2\nr3+µ2\nr4, Aµ=µ(1−1/r)dt.(9)\nHere the horizon radius has been set to rh= 1 and µcan be identiﬁed with the chemical\npotential in the dual ﬁeld theory. The temperature of the bounda ry theory is\nT=1\n4π(3−µ2). (10)\nA self-consistent ansatz for complex vector ﬁeld and polarization ﬁ eld is\nMµν=−p(r)dt∧dr+h(r)dx∧dy, ρ µ=ρxdx+ieiθ(r)ρydy. (11)\nAs in Ref. [ 39], we can take ρx(r) to be real by some suitable U(1) gauge. However, the\ny-component of ρµthen can not be set to be real. So we have to assume that ρyandθ\nare real functions depending on r. Taking the ansatz into equations ( 8), we ﬁnd that only\nwheneiθ(r)=±1 for∀r∈(rn,∞) can we ﬁnd a self-consistent solution. Without loss\ngenerality, we can assume θ(r) = 0. Considering the symmetry between ρxandρy, we\nassume that\nρy=c(r)ρx (12)\n7in what follows. Thus we can reach the following equations for the com ponents of matter\nﬁelds\nh′′+f′\nfh′+/parenleftbiggJh2\nr6f−2f′\nrf−4\nr2−m2\n2\nfr2/parenrightbigg\nh−2cαρ2\nx\nr2f= 0,\nρ′′\nx+(f′\nf+2\nr)ρ′\nx+/parenleftbiggq2φ2\nr4f2−Θc2ρ2\nx\nr4f−m2\n1\nfr2−chα\nfr4/parenrightbigg\nρx= 0,\nc′′+/parenleftbiggf′\nf+2\nr+2ρ′\nx\nρx/parenrightbigg\nc′−(1−c2)(cΘρ2\nx+αh)\nfr4= 0.(13)\nThe equation for p(r) decouples from the above equations, so we will not write down it\nhere.\nNear the AdS boundary, the linearized equations give following asymp totic solutions2\nρx=ρx+r(δ1−1)/2+ρx−r−(δ1+1)/2, c=c+rδ1+c−,\nh(r) =h+r(1+δ2)/2+h−r(1−δ)/2,(14)\nwhereδ1=/radicalbig\n1+4m2\n1andδ2=/radicalbig\n17+4m2\n2. According to the AdS/CFT dictionary,\nρx+,c+andh+are sources terms for corresponding operators, while ρx−,c−andh−are\nvacuum expectation values, respectively. As in Ref. [ 36], we need impose the condition\nh+= 0 for the polarization ﬁeld and ρx+=c+= 0 for the complex vector ﬁeld. This is\nconsistent with the spirit of AdS/CFT correspondence: one requir es that the condensation\nand magnetization happen spontaneously.\nAt the horizon, we impose regular conditions for ρx, c(r) andh, which give following\nrelationships for the initial values at r=rh\nρ′\nx=1\n4πTρx(m2\n1+cαh−cΘρ2\nx),\nh′= 2h−h(Jh2−m2\n2)+2cαρ2\nx\n4πT,\nc′=1\n4πT(1−c2)(cΘρ2\nx+αh).(15)\nNote that there exists a symmetry as {ρx→ρy, ρy→ρx}, by which we can set that\n|ρy(rh)| ≤ |ρx(rh)|, i.e.,\n−1≤c(rh)≤1. (16)\nThus once given the value of parameters {Θ, α, J, q, m2\n1, m2\n2, T}and some suitable\ninitial values {h(rh), ρx(rh), c(rh)}, we can integrate equations ( 13) to obtain the whole\nsolutions matching the boundary conditions h+=ρx+=c+= 0 at the AdS boundary.\nOur numerical results show that3, except for case that c(rh) =h(rh) = 0 orc(rh) =±1,\nthe integration will meet a divergency at somewhere in rh< r <∞and the equations ( 13)\n2The asymptotic solution of c(r) depends on the source free condition of ρx. Whenρx+/ne}ationslash= 0, asymptotic\nsolution of c(r) isc=c++c−r−δ1.\n3We use the function ode45 in MATLAB R2012b with the relative and abs olute errors 10−13to solve\nequations ( 13) numerically.\n8do not have physical solutions. When c(rh) =h(rh) = 0, according to the relationship\nof initial values in ( 15), we can ﬁnd that c(r) =h(r) = 0, which is just the p-wave\nsuperconductor solution found in Ref. [ 39]. When c(rh) =±1, the solution for cisc(r) =\n±1, i.e.,ρx=±ρy, which leads that the nontrivial solutions are either pure ferromag netic\nphase or p-wave superconductivity phase with nonzero magnetiza tion ifα/ne}ationslash= 0. From the\nequations ( 13), we can see that the transformation α→ −αis equivalent to ﬁx αbut make\nthe transformation of c(r)→ −c(r). So the case of α <0 is equivalent to the case of\nα >0 with exchanging the results of c(r) =±1. In the special case that α= 0, from the\nequations ( 13), we can see that the equations for handρxare decoupled with each other.\nThe ferromagnetic phase and p-wave superconducting phase hav e no interaction with each\nother4. We are here not interested in that case. In the following sections, therefore we\nwill setα >0.\n3.2 Free energy and magnetic moment\nLet us now compute the on-shell action and give the thermodynamic of the dual boundary\ntheory. Here, we use grand canonical ensemble by ﬁxing the bound ary chemical potential.\nFor convenience, we can set µ= 1 in the numerics. The results for other chemical potential\nvalues can be obtained by scaling relations. We will ﬁrst ﬁx the horizon radiusrhto\nsolve the equations ( 13) and compute the free energy density, then we use the scaling\ntransformation to obtain the results in the grand canonical ensem ble. In gauge/gravity\nduality, the Gibbs free energy Fcan be obtained by temperature timing the on-shell bulk\naction with Euclidean signature. Since we work in the probe approxima tion, we can ignore\nthe gravity part. Given that the system is stationary, the Euclidea n action is related to the\nMinkowskian one by a total minus. Using the equations of motion ( 8) and the source free\nconditions, we can get the free energy density contributed by vec tor ﬁeld and polarization\nﬁeld as\nF\nV=λ2/integraldisplay∞\nrhdr√−g/parenleftbigg\n−Vρ+1\n4MµνFµν−VM−iα\n2Mµνρµρν†/parenrightbigg\n, (17)\nwhereVis the area spanned by coordinates xandyon the boundary. Taking the\nansatz (11) into the free energy density, it turns out to be\nF\nV=λ2/integraldisplay∞\nrhdr/parenleftbiggJh4\n4r6−c2Θρ4\nx\nr2−cαρ2\nxh\nr2/parenrightbigg\n. (18)\nNote that the contribution to the free energy density from p(r) is not relevant to our\ndiscussions, therefore here we have neglected that part in ( 18). The integration is ﬁnite\nwhenρx+=h+= 0 and m2\n1>−1/4, m2\n2>−4. In addition, note that there are two\nsymmetries in the free energy and EoMs ( 13) such as\n{ρx→ −ρx, ρy→ −ρy},{c→ −c, h→ −h} (19)\n4This is the consequence of the probe limit. Once the back reaction is t aken into account, even when\nα= 0,ρµandMµνwill interact each other through gravity background.\n9which make we can specify ρx(rh)≥0 andh(rh)≤0. It is a general requirement that the\nfunctions of ρxandhdo not have zero points in the region of [ rh,∞). Thus we can assume\nρx(r)>0 andh(r)≤0 in the region [ rh,∞). According to the deﬁnition of magnetic\nmoment density and using the expression ( 17), we have\nN=−1\nVlim\nB→0/parenleftbigg∂F\n∂B/parenrightbigg\nρµ,p=−/integraldisplay∞\nrhh\n2r2dr, (20)\nwhich is the same as in Ref. [ 36]. By the dictionary of AdS/CFT, the expectation value\nof p-wave superconducting order parameter is a complex vector− →P, whose mode is P=√\n1+c2|ρx−|. Here it is worthwhile to mention that though the complex vector ﬁeld does\nnot appear in the ﬁnal expression of magnetic moment density, it ma kes contribution\nthrough the mixture terms in equations ( 13): onceρxandcdo not vanish, hwill not\nvanish.\nIn the pure p-wave model, when nontrivial solutions of ρµappear, the global U(1) and\nspatial rotation symmetries are broken spontaneously. In this mo del, it is also true. In\naddition, there is an another possible symmetry breaking in this mode l. If one notes the\nfact that external magnetic ﬁeld Bwill be transformed in −Bunder the time reversal\ntransformation, by the expression of free energy density in ( 17), we have following rules\nfor time reversal transformation,\nh→ −h, ρy→ −ρy. (21)\nSo when h/ne}ationslash= 0 orρy=±ρx/ne}ationslash= 0 (they both lead to nonzero magnetic moment), the time\nreversal symmetry is broken spontaneously. This agrees with the fact that a spontaneously\nmagnetized phase will break time reversal symmetry spontaneous ly.\nFrom Refs. [ 39,36], we can see that the complex vector ﬁeld and polarization ﬁeld can\nboth condense in low temperatures in an AdS RN black hole backgroun d. We take Tsc0\nandTC0as the critical temperatures of ρxandh, whenα= 0, in the AdS RN black hole\nbackground, respectively. With decreasing the temperature of t he system, three interesting\nquestions appear immediately:\n(1) IfTC0> Tsc0, then the system will enter into the ferromagnetic phase ﬁrst. Ca n\na p-wave superconducting phase still appear at lower temperatur e? If yes, is the critical\ntemperature still Tsc0?\n(2) IfTsc0> TC0, then the system will enter into the p-wave superconducting phas e\nﬁrst. Can a ferromagnetic phase appear at some lower temperatu re? If yes, is the critical\ntemperature still TC0?\n(3) In an enough low temperature, for example, near the zero tem perature limit, can\nthe p-wave superconductivity and ferromagnetism coexist in this m odel?\nIn what follows, we will consider all these questions. We will ﬁrst solve equations ( 13)\nwith the regular condition ( 15) at the horizon and source free conditions at the AdS\nboundary. In general, depending on parameters, the system has four kinds of solutions:\none is a trivial solution without vector and tensor hairs, one is a solut ion with vector hair,\nbut no tensor hair (which describes the pure p-wave superconduc tivity phase), one is the\n10solutionwithtensorhair, butnovectorhair(whichdescribesthepu referromagneticphase),\nand ﬁnal one is the solution with both hairs (which describes both the superconductivity\nand ferromagnetism coexistence phase). The phase is physical fa vored if it has the lowest\nfree energy.\n4 Coexistence and competition\n4.1 Superconducting ferromagnet\nLet us ﬁrst consider the case with TC0> Tsc0, i.e., the ferromagnetic phase appears ﬁrst.\nAccording to the equation for cin equations ( 13), we ﬁnd c/ne}ationslash= 0 when h/ne}ationslash= 0. So there is not\na phase such that {h <0, ρx/ne}ationslash= 0, ρy= 0}. When we decrease the temperature to be lower\nthanTC0, ﬁvekindsofsolutionsmayappear. TheyarephaseA {h=ρx=ρy= 0},phaseB\n{h <0, ρx=ρy= 0}, phase C {h=ρy= 0, ρx/ne}ationslash= 0}, phaseD1{h <0, ρx=ρy/ne}ationslash= 0}and\nphaseD2{h <0, ρx=−ρy/ne}ationslash= 0}, which corresponds to normal phase, pure ferromagnetic\nphase, pure p-wave superconducting phaseandtwo kinds ofsupe rconducting ferromagnetic\nphases, respectively.\nFor small ρx, we can get its eﬀective mass square from equations ( 13) as\nm2\n1eﬀ=m2\n1−q2φ2\nr2f+cαh\nr2. (22)\nIn general, h(r) does not have zero point in the region of rh< r <∞in the condensed\nphase. With the choice h≤0,α >0, we have αh≤0. Ifc= 1, the eﬀective mass square of\ncomplex vector ﬁeld will decrease by the condensate of h. Ifc=−1, instead the eﬀective\nmass square of complex vector ﬁeld will increase by the condensate ofh, which leads the\ncritical temperature of complex vector ﬁeld to decrease. In this c ase, if−hαis enough\nlarge, the complex vector ﬁeld can not condense even in zero tempe rature. These imply\nthat when we decrease the temperature, the instability of complex vector ﬁeld will appear\nin the manner of ρy=ρxorρy=−ρxfor small αand in the manner of ρy=ρxfor enough\nlargeα. This instability tells us that in the ferromagnetic phase, the margina lly stable\nmode for complex vector ﬁeld can still appear at some temperature less than TC0. The\nanalysis can also be conﬁrmed directly by solving equations ( 13) numerically under the\ncase of neglecting the ρ2\nxterms, i.e.,\nh′′+f′\nfh′+/parenleftbiggJh2\nr6f−2f′\nrf−4\nr2−m2\n2\nfr2/parenrightbigg\nh= 0,\nρ′′\nx+(f′\nf+2\nr)ρ′\nx+/parenleftbiggq2φ2\nr4f2−m2\n1\nfr2−chα\nfr4/parenrightbigg\nρx= 0,(23)\nwith the initial value of ρx(rh) = 1. As a typical example, we choose parameters as\nm2\n1=−3/16,m2\n2=−3,J=−1 andq= 1.3 (for other parameter values, the results are\nqualitatively similar) . The results are shown in ﬁgure 2. In the left plot, we show ρ+as a\n110 0.2 0.4 0.6 0.8 1 1.2−0.1−0.0500.05\nT/TC0ρ+\n  \nTsc0\nTC0Tsc\nTC0\nTsc′\nTC0c=1\nc=−1\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\nα  \nαc\nTsc\nTC0Tsc′\nTC0\nFigure 2: Left:ρ+as a function of temperature Twhenh <0. The green line and blue\nline stand for c=−1 andc= 1, respectively. The red dashed line stands for the case in\nthe pure AdS RN background with h= 0.Right:Tsc/TC0andTsc′/TC0as functions of α.\nWe take m2\n1=−3/16,m2\n2=−3,J=−1,q= 1.3. HereTC0≃0.00925µ, Tsc0≃0.8135TC0\nfunction of temperature Tforc=±1 in the case of α= 0.1 whenh <0. The green line\nand blue line stand for c=−1 andc= 1, respectively. The red dashed line stands for the\ncase in the pure AdS RN background with h= 0. In the case of c= 1, there is a critical\ntemperature Tscless than TC0but higher than Tsc0to makeρ+= 0. In the case of c=−1,\nthere is a critical temperature Tsc′less than Tsc0to make ρ+= 0. In the right plot, we\nshowTsc/TC0andTsc′/TC0as functions of α. When we increase α,Tscwill increase but\nTsc′decreases. Numerical results show that there is a critical value at α=αc≃0.57649,\nless than which Tsc′will be less than zero. In appendix A, we give the method to compute\nthe critical value αc.\nNote that equations ( 23) and ﬁgure 2only show that there is a marginally stable mode\nfor the complex vector ﬁeld at T=TscorT=Tsc′. When temperature is below these\ncritical temperatures, whether the complex vector ﬁeld can cond ense and which one of two\nphases{h <0,ρx=−ρy/ne}ationslash= 0}and{h <0,ρx=ρy/ne}ationslash= 0}can appear in the physical phase\nspace are determined by their free energy density. In order to ﬁn d the phase diagram, we\nhave to solve equations ( 13) numerically to compute the free energy of possible solutions.\nIt turns out that the results depend on the sign of Θ. The possible p hases and the physical\nfavored phase in diﬀerent temperature regions are summarized in t able1.\nIn the case of Θ >0, we ﬁnd that the the phase D1is physical favored when T < T sc.\nThis means that the system displays ferromagnetism and supercon ductivity both in low\ntemperatures. In addition, by computing the ferromagnetic orde r parameters and p-wave\nsuperconducting order parameter, we see they are both continu ous at two critical temper-\natures. With lowering the temperature, the system will ﬁrst trans it into the ferromagnetic\nphase at TC0and then into the superconducting ferromagnetic phase at Tscthrough two\nsecond order phase transitions. The critical temperature of com plex vector ﬁeld grows\nwhenαgets increased. This shows that, in the ferromagnetic state, the interaction be-\n12Phases in the case of α < α c\nTemperature T > T C0Tsc< T < T C0Tsc′< T < T scT < T sc′\nPossible A A,BA,D1A, B,D1\nC(ifT < T sc0)D2, C\nPhysical(Θ >0)A B D1\nPhysical(Θ <0)A B\nPhases in the case of α > α c\ntemperature T > T C0Tsc< T < T C0 T < T sc\nPossible A A,B A, B,D1, C(ifT < T sc0)\nPhysical(Θ >0)A B D1\nPhysical(Θ <0)A B\nTable 1: The possible and physical phases in the case of TC0> Tsc0. Phase A is {h=ρx=\nρy= 0}. Phase B is {h <0, ρx=ρy= 0}. Phase C is {h=ρy= 0, ρx/ne}ationslash= 0}. PhaseD1is\n{h <0, ρx=ρy/ne}ationslash= 0}. PhaseD2is{h <0, ρx=−ρy/ne}ationslash= 0}.\ntween p-wave pairs and spontaneous magnetic moment will promote the appearance of\np-wave superconductivity. On the other hand, in the case of Θ <0, though the solutions\nofρx/ne}ationslash= 0 exist, they are not physical favored because they have higher free energy than\nthe solution of ρx= 0. As a result, in this case, there is only a ferromagnetic phase (ph ase\nB) when T < T C0.\nHere we only show the example for the case of α < α c. The case of α > α cis similar\nexcept for the diﬀerence that the solution for the phase D2does not occur. In the left plot\nof ﬁgure3, we show the free energy density of phases D1andD2in the case of Θ = ±1 and\nphaseBwith respect to temperature. Note that here the on-shell free e nergy density of\nphasesAandCis zero in the probe approximation. One can see that phase D1and phase\nBare physical favored when Θ = ±1 respectively when T < T sc. Thus we see that the\nphysical favored phase depends on the sign of Θ. In the case of Θ >0, with decreasing the\ntemperature, thesystem willﬁrstgoesintotheferromagneticph asewhen T < T C0andthen\ninto the p-wave superconducting ferromagnetic phase when T < T sc. However, there are\ntwo superconducting ferromagneticphases D1({ρx=ρy})andD2({ρx=−ρy}). Which has\na lower free energy than the pure ferromagnetic phase when T < T sc? It turns out that the\nphysical favored one is phase D1. The numerical results show that the ferromagnetism and\nsuperconductivity can coexist in the whole region of T < T sc. It seemingly indicates that\nthe ferromagnetism and superconductivity can coexist even in the zero temperature limit.\nHowever, in the case of Θ <0, we see that although the superconducting ferromagnetic\nphases exist (phase D1andD2), these two phases have higher free energy density than the\npure ferromagnetic phase. In this case, therefore the physical favored phase is the pure\nferromagnetic phase and the p-wave superconductivity will not oc cur.\nIn the right plot of ﬁgure 3, we show the superconducting order parameter Pand\nspontaneous magnetic moment density Nas functions of temperature Tin the case of\n130.20.40.60.8 1−0.05−0.04−0.03−0.02−0.0100.01\nT/TC02κ2F\nV\n  \nPhaseD1,Θ= 1\nPhaseD2,Θ= 1\nPhaseD1,Θ=−1\nPhaseD2,Θ=−1\nPhase B\n0.20.40.60.8 100.050.10.150.20.25\nT/TC0  \nP/√\n2µ∆−,Θ= 1\n3N/µ,Θ= 1\nP/√\n2µ∆−,Θ=−1\n3N/µ,Θ=−1\nFigure 3: Left:The free energy density with respect to temperature in the case o f Θ =\n±1.Right:The p-wave superconducting order parameter Pand spontaneous magnetic\nmomentdensity NasfunctionsoftemperatureinthecaseofΘ = ±1inthephysical favored\nphase. Here m2\n1=−3/16,m2\n2=−3,J=−1,q= 1.3 andα= 0.1. ∆−= 1+(1+ δ1)/2.\n00.020.040.060.080.100.511.522.5x 10−3\n1−T/TC0N2\nµ2\n  \nNumerical results\nFitting curve\n00.01 0.02 0.03 0.04012345x 10−3\n1−T/TscP2\nµ2∆−\n  \nNumerical results\nFitting curve\nFigure 4: The p-wave superconducting order parameter Pand spontaneous magnetic mo-\nment density Nwith respect to temperature near the critical temperature in the PhaseD1.\nNumerical ﬁttings give that Pµ−∆−≃0.3721/radicalbig\n1−T/TscandN/µ≃0.1588/radicalbig\n1−T/TC0.\nHerem2\n1=−3/16,m2\n2=−3,J=−1,q= 1.3 andα= 0.1. ∆−= 1+(1+ δ1)/2.\n14Θ =±1 in the physical favored phase. We see that there is a second orde r phase transition\natT=TC0, below which the system enters into a ferromagnetic phase. When t emperature\ndecreases below Tsc, the situations depend on the sign of Θ. In the case of Θ >0, there is a\nsecond order phase transitions at T=Tscsuch that the system will transit into the p-wave\nsuperconducting ferromagnetic phase. From the curve of Nin the right plot of ﬁgure 3,\none can see that the condensation of p-wave order will increase th e magnetic moment. This\nimplies that the p-wave pair carries a nonzero magnetic moment, whic h contributes to the\ntotal magnetic moment of the system. In ﬁgure 4we plot the behaviors of the magnetic\nmoment and p-wave order parameter with respect to temperatur e in the phase D1near\nthe critical temperature, which clearly shows a square root behav ior for both quantities.\nLet us make a brief summary for this subsection. In the case of TC0> Tsc0, i.e., the\ncasewiththeferromagneticphaseappearingﬁrstwithdecreasing thetemperature, whether\nthe p-wave superconductivity can appear depends on the sign of Θ . If Θ>0, there is a\ncritical temperature Tscwhich is lower than TC0but higher than Tsc0. WhenT < T sc, the\np-wave superconductivity can appear and the system will show the ferromagnetism and\nsuperconductivity both. Even in the near zero temperature limit, t hey can coexist. On\nthe other hand, if Θ <0, the p-wave superconductivity can not appear and the system\nwill only be in a pure ferromagnetic phase. These results are summar ized in table 1.\n4.2 Ferromagnetic superconductor\nLet us now consider the other case that Tsc0> TC0, i.e., the case where the p-wave super-\nconducting phase appears ﬁrst. When TC0< T < T sc0, according to the equations ( 13),\nthere may exist three kinds of p-wave superconducting phase. On e is just the usual p-wave\nsuperconducting phase C( {h=ρy= 0,ρx/ne}ationslash= 0}), the other two are new superconducting\nphaseE1with{h <0,ρx=ρy/ne}ationslash= 0}andE2with{h <0,ρx=−ρy/ne}ationslash= 0}. Thought phase\nE1andE2have nonzero magnetic moment, they are diﬀerent with phases D1andD2,\nbecause the appearance of nonzero hin the former two is induced by the p-wave pair but\nin the latter two is spontaneously produced. Thus we have two ques tions as follows. Can\nthese three solutions exist ? And which one is physical favored when temperature is in the\nregionTC0< T < T sc0?\nOur numerical results show that the answers depend on the sign of Θ. If Θ >0,\nthe phases A,CandE1can exist and the phase E1is physical favored which has the\nlowest free energy. Therefore the system will show magnetism onc e it goes into the p-wave\nsuperconducting phase. This case is very similar to the Anderson-B rinkman-Morel (ABM)\nphase in3He superﬂuid [ 44], where the superﬂuid phase is also of magnetism. Though\nthe magnetism and superconductivity appear together, it has an e ssential diﬀerence from\nphasesD1andD2just as we mentioned before. So the phase E1(andE2) should be called\n“magnetic superconducting” phase rather thanferromagnetic s uperconducting phase. This\ndiﬀerence can also be shown in the magnetic moment density near the phase transition\npoint. In the phases D1andD2, we see that the Nshows a square root behavior with\nrespect to1 −T/TC0(seeﬁgure 4), whileinphase E1, weseethat Nhasalinearrelationship\nwith respect to 1 −T/Tsc0(see ﬁgure 6). On the other hand, if Θ <0, we ﬁnd that the\n150.5 1 1.5−0.1−0.08−0.06−0.04−0.020\nT/TC02κ2F\nV\n  \nPhaseE1Phase B\n0 0.5 1 1.5 200.10.20.30.4\nT/TC0  \nP/√\n2µ∆−\n3N/µ\nFigure 5: Left:The free energy density with respect to temperature in phase E1and\nphaseB.Right:The p-wave superconducting order parameter Pand induced magnetic\nmoment density Nwith respect to temperature in phase E1. Herem2\n1=−3/16,m2\n2=\n−3,J=−1,q= 1.4,Θ = 1 and α= 0.1. ∆−= 1+(1+ δ1)/2.\nphaseE1andE2do not appear and the equations ( 13) only have the trivial solution and\nthe pure p-wave superconductivity solution. As a result, in this cas e, the system is in a\npure p-wave superconducting phase without magnetism in the regio n ofTC0< T < T sc0.\nAs an example, let us consider the parameters as m2\n1=−3/16,m2\n2=−3,J=−1,q=\n1.4 (again, the results are qualitatively similar for other parameter va lues, the only re-\nquirement is to have Tsc0> TC0). In this case, we have Tsc0≃1.8383TC0. The free energy\ndensity, the p-wave superconducting order parameter Pand induced magnetic moment\ndensityNwith respect to temperature in the phase E1are shown in ﬁgure 5. Because the\non-shell free energy for phases A and C is zero in the probe approx imation, we see that\nphaseE1has the lowest free energy. Furthermore from ﬁgure 6, we see that the p-wave\nsuperconducting order parameter Phave a square root behavior with respect to 1 −T/Tsc0\nbut the induced magnetic moment density has a linear relationship with 1−T/Tsc0when\nT→T−\nsc0.\nWhen temperature is lower than TC0and Θ>0, phases B can also appear. The\nnumerical results show that the phase E1is still the lowest free energy phase (see ﬁgure 5\nas an example). So there is no phase transition at T=TC0. If Θ<0, we have known that\nwhenT > T C0, this system is in a pure p-wave superconducting phase C with {h=ρy=\n0,ρx/ne}ationslash= 0}. AtT=TC0, because the equation for his a homogeneous one ( ρy=cρx, in\nphase C we have c= 0), there is critical point for h. However, the equation for c(r) in (13)\nrestricts c(r) to be 0 or ±1. When h/ne}ationslash= 0, the only solution for c(r) isc(r) =±1. This\nindicates that in this case the solutions for ( 13) are either just phase E1(orE2) or phase\nB, i.e., phases {h <0,ρx=±ρy/ne}ationslash= 0}or{h <0,ρx=ρy= 0}. However, our numerical\ncalculationsshowthatsuchsolutions {h <0,ρx=±ρy/ne}ationslash= 0}donotexist. Thusthepossible\nphases are only phase A, phase B and phase C. In this model, the on s hell free energy is\nzero for phase A and C but is negative for phase B. So the phase {h <0,ρx=ρy= 0}\n1600.005 0.01 0.015 0.0200.511.522.53x 10−3\n1−T/Tsc0P2\nµ2∆−\n  \nNumerical results\nFitting curve\n00.005 0.01 0.015 0.020123456x 10−4\n1−T/Tsc0N\nµ\n  \nNumerical results\nFitting curve\nFigure 6: The behaviors of NandPand near the critical temperature Tsc0in the phase E1.\nNumerical ﬁttings show that Pµ−∆−≃0.3699/radicalbig\n1−T/Tsc0andN/µ≃0.0294(1−T/Tsc0).\nHerem2\n1=−3/16,m2\n2=−3,J=−1,q= 1.4,Θ = 1 and α= 0.1. ∆−= 1+(1+ δ1)/2.\nPhases in the case of Θ >0\nTemperature T > T sc0TC0< T < T sc0T < T C0\nPossible A A,E1, CA,E1, C, B\nPhysical A E1\nPhases in the case of Θ <0\nTemperature T > T sc0TC0< T < T sc0T < T C0\nPossible A A, C A, C, B\nPhysical A C B\nTable 2: The possible and physical phases in the case of Tsc0> TC0. Phase A is {h=ρx=\nρy= 0}. Phase B is {h <0, ρx=ρy= 0}. Phase C is {h=ρy= 0, ρx/ne}ationslash= 0}. PhaseE1is\n{h <0, ρx=ρy/ne}ationslash= 0}. Phases E2is{h <0, ρx=−ρy/ne}ationslash= 0}.\nis physical favored in the case of Θ <0 whenT < T C0. The ferromagnetism can still\nappear from the p-wave superconducting phase at the same critic al temperature TC0, but\nthe p-wave superconducting phase will disappear. In other words , the superconductivity\nand ferromagnetism can not coexist in the case of Θ <0. In ﬁgure 7, we plot the diﬀerence\nof free energy between phase C and phase B and magnetic moment d ensityNwith respect\nto temperature in phase B, whereδF=Fphase B−Fphase C, from which we can see that\nthe pure ferromagnetic phase has lower free energy than the pur e p-wave superconducting\nphase.\nLet us make abrief summary forthissubsection. Inthis caseof TC0< Tsc0, i.e., thecase\nthat the superconducting phase will appear ﬁrst, the results dep end on the sign of Θ. If\nΘ>0, the ferromagnetic phase can not appear but the magnetic p-wa ve superconducting\nphase will appear in region of T < T sc0. If Θ<0, the system will be in a pure p-wave\n170.2 0.4 0.6 0.8 1−8−6−4−20x 10−3\nT/TC02κ2δF\nV\n0.2 0.4 0.6 0.8 100.020.040.06\nT/TC0N\nµ\nFigure 7: Left:The diﬀerence of free energy between phase C and phase B. Here δF=\nFphase B−Fphase C.Right:The magnetic moment density Nwith respect to temperature\nin phase B. Herem2\n1=−3/16,m2\n2=−3,J=−1,q= 1.4,Θ =−1 andα= 0.1.\nsuperconducting phase in the region TC0< T < T sc0and a pure ferromagnetic phase when\nT < T C0. All the results are summarized in table 2.\n5 Summary and discussions\nIn this paper, by combining the complex vector ﬁeld model for the ho lographic p-wave\nsuperconductor and the real antisymmetric tensor ﬁeld model fo r the holographic ferro-\nmagnetism, we have investigated the coexistence and competition o f ferromagnetism and\nsuperconductivity in the holographic setup. Depending on model pa rameters, we found\nthat the model shows rich phases in low temperatures. The study is done in the probe\nlimit, the background geometry is taken to be anAdS RNblack hole with a planar horizon.\nIn the case of TC0> Tsc0, i.e., the case where the ferromagnetic phase appears ﬁrst,\nwhether the p-wave superconductivity can appear depends on th e sign of Θ, the interac-\ntion strength of magnetic moment of the complex vector ﬁeld. If Θ >0, there is a critical\ntemperature Tscwhich is lower than TC0but higher than Tsc0. When temperature is higher\nthanTsc, the system only shows the ferromagnetism. When T < T sc, the p-wave super-\nconductivity can appear and the system will show ferromagnetism a nd superconductivity\nboth. Because of the the spontaneous magnetization, the critica l temperature of p-wave\ncondensation is higher than the critical temperature without the f erromagnetic phase, and\nincreases with the increasing of interaction strength between com plex vector ﬁeld and an-\ntisymmetric tensor ﬁeld. Even in the near zero temperature limit, th e magnetism and\nsuperconductivity can coexist. But if Θ <0, the p-wave superconducting state can still\nexist but it is not the lowest free energy state. So the supercondu ctivity can not appear\nand the system will only be in a pure ferromagnetic state.\nIn the case of TC0< Tsc0, i.e., the case where the superconducting phase appears ﬁrst,\nthe results also depend on the sign of Θ. If Θ >0, in the region of TC0< T < T sc0, the\n18system will show the p-wave superconductivity and a kind of induced magnetism. The\nsuperconductivity and magnetism appear both, however, it is a mag netic superconducting\nphase rather than a ferromagnetic superconducting phase, bec ause the magnetic moment\nis not spontaneously produced. The magnetic moment is proportion al toTsc0−Trather\nthan√Tsc0−Tnear the critical temperature. When temperature is lower than TC0, the\nferromagnetic phase B can exist, but it has higher free energy tha n phaseE1. So in the\nwhole region of T < T sc0, the physical favored phase is magnetic p-wave superconducting\nphaseE1. On the other hand, if Θ <0, when temperature is less than Tsc0, the system\nwill be in the pure p-wave superconducting phase without magnetism . If temperature is\nlower than TC0, the system will transit into the pure ferromagnetic phase from th e pure\np-wave superconducting phase. Therefore the ferromagnetism and superconductivity can\nnot coexist in the case of Θ <0.\nNow let us discuss some implications of our results. We have seen that the sign of\nΘ plays a crucial role in this model. This phenomenological parameter in (4) describes\nthe self-interaction between the magnetic moments of complex vec tor ﬁeld. A positive Θ\nmeans that the magnetic moments with same direction are attractiv e, while a negative\nΘ indicates that the magnetic moments with same direction are repuls ive. From tables 1\nand2, we can see that the ground state in the near zero temperature lim it only depends\non the sign of Θ. If we translate these attraction and repulsion into the boundary theory,\nthen our results can be understood well. Since p-wave pair is spin trip let, it can be in the\nstate of spin-up or spin-down. So every p-wave pair carries magne tic moment of ±2µB.\nIf Θ>0, which means that the p-wave pair will attract the pair which has th e same\nmagnetic moment direction and repulse the one which has opposite ma gnetic moment.\nSo under the inﬂuence of spontaneous magnetization, the p-wave pair will be enhanced\nand survive. In addition, the magnetic moment of p-wave pair will ten d to align along\nthe direction of spontaneous magnetization, which increases the t otal magnetic moment\nof the system. As a result we indeed see the ground state is the pha se where the p-wave\nsuperconductivity and ferromagnetism coexist. However, if Θ <0, the p-wave pair will\nrepulse the pair which has the same magnetic moment direction. So in t he region where\nsuperconductivity dominates, the p-wave pair will align without net m agnetism and the\nsystem is in a pure p-wave superconducting phase. When T < T C0, the ferromagnetism\nwill appear. Under the inﬂuence of spontaneous magnetization, th e magnetic moment of\np-wave pair will be compelled to align the same or opposite (depends on the value of α)\ndirection of spontaneous magnetization, which leads to the magnet ic moment of p-wave\npairhassamedirection. Butthep-wavepairswhichhavesamemagne ticmomentdirections\nwill repulse each other, so the p-wave pair is not stable and will be de- paired. Thus the\nsystem can only be in the ferromagnetic phase.\nFinally let us make some additional comments on superconducting fer romagnetic ma-\nterials, since the Curie temperature is higher than superconductin g critical temperature\nin general. Although superconductivity in ferromagnets was predic ted more than 30 years\nago, it took many years before the ﬁrst material UGe 2was discovered and the research\nin superconducting ferromagnets has just begun recently. The m ain reasons why we say\nsuperconducting state in superconducting ferromagnets is unco nventional are (i) Cooper\n19pairing carry magnetism and (ii) the gap structure of superconduc ting has a lower symme-\ntry than the crystal lattices [ 45]. Let G represent the point-group symmetry of the lattice,\nT denote time reversal symmetry, and U(1) be the gauge symmetr y. In the paramagnetic\nstate (T > T C0> Tsc) the symmetry group is given by G×T×U(1). In the ferromagnetic\nphase (T < T C0) time-reversal symmetry is broken, and in the superconducting p hase\n(T < T sc< TC0) gauge symmetry is broken as well. The coexistence of such two crit ical\nphenomena oﬀers an attractive playground for the investigation o f new phenomena, like\nthe elusive spontaneous vortex lattice, the inﬂuence of spin-triple t superconductivity on\nthe ferromagnetic domain size, control of tunneling currents by m agnetization and so on.\nAlso, it isa central issue in the understanding of superconductivity itself by the interplay of\nmagnetism and superconductivity. Research on superconducting ferromagnetic materials\nwill help us to expound how magnetic ﬂuctuations can arouse superc onductivity. This fun-\ndamental insight might turn out to be crucial in designing new superc onducting materials\nwith high transition temperatures. We hope that the holographic mo del in this paper or\nthe correspondence of AdS/CMT can give some helpful guidance in f uture.\nAcknowledgements\nThis work was supported in part by the National Natural Science Fo undation of China\nwith grants No.11035008, No.11375247 and No.11435006.\nA The method to compute αc\nIn this appendix, we will give the method to compute the value of αc. We need to solve\nfollowing equations in the zero temperature case\nh′′+f′\nfh′+/parenleftbiggJh2\nr6f−2f′\nrf−4\nr2−m2\n2\nfr2/parenrightbigg\nh= 0,\nρ′′\nx+(f′\nf+2\nr)ρ′\nx+/parenleftbiggq2φ2\nr4f2−m2\n1\nfr2+hα\nfr4/parenrightbigg\nρx= 0.(24)\nHere\nf(r) = 1−4/r3+3/r4, φ(r) =√\n3(1−1/r). (25)\nBy analyzing the behavior of handρx(r) near the horizon, we ﬁnd the regular form for\nthem are,\nh(r) =h0+h1(r−1)β1+···, ρx= (r−1)β2[1+ρx1(r−1)+···].(26)\nSubstituting them into equations ( 24), the leading orders of the solutions give the coeﬃ-\ncients in ( 26) as\nh0=−/radicalBig\nm2\n2/J, β 1=−1\n2+1\n6/radicalBig\n9−12m2\n2\nβ2=−1\n2+1\n6/radicalBig\n9−6αh0+6m2\n1−3q2, ρx1=−β2\nβ2+1.(27)\n20Taking (27) and (26) , we can integrate equations ( 24) from the horizon to the AdS bound-\nary and get the solutions outside the horizon. When the values of m2\n1, m2\n2, qandJare\ngiven, we can treat h1andαas shooting parameters to match the boundary conditions\nh+=ρx+= 0. Since the equation for his dependent of ρx, we can solve hﬁrst. Numerical\nresult shows that h1≃15.40714197. Then we put the solution hinto the equation of ρx\nand treat αas a shooting parameter to match the boundary condition ρx+= 0. Finally\nwe ﬁnd that there is solution with αc≃0.57649.\nReferences\n[1] J. 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Bhatt3, 4\n1Department of Physics, Princeton University, Princeton, NJ 08544, USA\n2Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA\n3Department of Physics, Princeton University, Princeton, NJ, 08544, USA\n4Department of Electrical and Computer Engineering,\nPrinceton University, Princeton, NJ 08544, USA\n(Dated: November 17, 2023)\nThe search for elusive Nagaoka-type ferromagnetism in the Hubbard model has recently enjoyed\nrenewed attention with the advent of a variety of experimental platforms enabling its realization, in-\ncluding moir´ e materials, quantum dots, and ultracold atoms in optical lattices. Here, we demonstrate\na universal mechanism for Nagaoka ferromagnetism (that applies to both bipartite and nonbipar-\ntite lattices) based on the formation of ferromagnetic polarons consisting of a dopant dressed with\npolarized spins. Using large-scale density-matrix renormalization group calculations, we present a\ncomprehensive study of the ferromagnetic polaron in an electron-doped Hubbard model, establish-\ning various polaronic properties such as its size and energetics. Moreover, we systematically probe\nthe internal structure of the magnetic state—through the use of pinning fields and three-point spin-\ncharge-spin correlation functions—for both the single-polaron limit and the high-density regime of\ninteracting polarons. Our results highlight the crucial role of mobile polarons in the birth of global\nferromagnetic order from local ferromagnetism and provide a unified framework to understand the\ndevelopment and demise of the Nagaoka-type ferromagnetic state across dopings.\nI. INTRODUCTION\nThe Hubbard model [1–3], a veritable workhorse for\nmuch of our modern understanding of strongly correlated\nquantum matter, is believed to underlie the physics of a\nwide variety of complex materials [4]. In its simplest\nform, the model describes a system of itinerant spin-1 /2\nelectrons hopping on a lattice of Nsites with a tunneling\namplitude twhile interacting via a local onsite potential\nof strength U. The corresponding fermionic Hamiltonian\ncan be written as\nH0=−tX\n⟨i,j⟩,σ\u0010\nc†\niσcjσ+ h.c.\u0011\n+UX\nini↑ni↓,(1)\nwhere c†\ni,σ, ci,σare the creation and annihilation opera-\ntors, respectively, for an electron with spin σ={| ↑⟩,| ↓⟩}\non site i,ni,σ≡c†\ni,σci,σdenotes the associated number op-\nerator, and the sum on ⟨i, j⟩runs over all pairs of nearest-\nneighbor (hereafter, NN) sites. In the sixty years since\nits proposal, the Hubbard model and its variants have\nbeen found to host a fascinatingly diverse set of quan-\ntum phases that run the gamut from magnetic states,\nsuch as antiferromagnets and topological spin liquids, to\ncharge density waves and superconductivity [5].\nGiven the inherent complexity of the correlated elec-\ntron problem, it is perhaps unsurprising that although\nremarkable progress has been made with numerical stud-\nies of the Hubbard model [6, 7], to date, only a few ex-\nact analytical results are known [8–10]. One such result\nis the rather striking Nagaoka theorem [11], which as-\nserts that for U=∞and nonnegative t, the ground state\nof the Hubbard model on a bipartite lattice with peri-\nodic boundary conditions (in D≥2 spatial dimensions)\ndoped with a single hole away from half filling is fer-romagnetic [12–14], as opposed to the antiferromagnetic\nground state of the half-filled system [15]. Intuitively, this\nfollows from very general kinetic considerations, depicted\nin Fig. 1. The hopping of dopants, either holes or dou-\nblons, necessarily scrambles an antiferromagnetic spin\ntexture [16, 17], leaving behind energetically unfavorable\n“strings” of displaced spins. However, such charge mo-\ntion does not disrupt a ferromagnetic configuration, thus\nallowing carriers to be less confined, whereupon the ki-\nnetic energy gain from delocalization wins over the com-\npeting antiferromagnetic superexchange.\nWhile mathematically rigorous, the Nagaoka theorem\nis of limited practical utility since any realistic system\ncan only ever be at finite U/t, which introduces its own\nsubtleties [18–20]. Moreover, the stringent requirement\nof exactly one dopant is not generalizable to the ther-\nmodynamic limit; the situation with a finite density of\ncarriers is also far from clear-cut, with arguments both\nfor [21–29] and against [30–38] the existence of high-spin\nground states under certain conditions.\nNonetheless, a few years ago, signatures of this elu-\nsive itinerant ferromagnetism were observed experimen-\ntally for the first time in small (four-site) quantum dot\nplaquettes [39]. Another especially promising platform\nfor the quantum simulation of Fermi-Hubbard models is\nproferred by ultracold atoms trapped in optical lattices\n[40–42], with recent experiments on these systems also\ndemonstrating ferromagnetism [43], albeit in a frustrated\ntriangular-lattice geometry. On such triangular lattices,\nmagnetism is invariably intertwined with kinetic frustra-\ntion [44–46] as follows. As pointed out by Haerter and\nShastry [47], the motion of a single hole (doublon) in a\nspin-polarized background leads to destructive (construc-\ntive) quantum interference between different paths on a\nnonbipartite lattice. To maximally lower their kineticarXiv:2311.09279v1  [cond-mat.str-el]  15 Nov 20232\n<latexit sha1_base64=\"2G3gr0FoY9J486ENLC7eF67/iBE=\">AAAB+nicbVDLSgMxFL3js9bXVJdugkVwIWVGfG2EohuXFewD2qFk0kwbmkmGJGMptZ/ixoUibv0Sd/6NaTsLbT0QOJxzD/fmhAln2njet7O0vLK6tp7byG9ube/suoW9mpapIrRKJJeqEWJNORO0apjhtJEoiuOQ03rYv5349UeqNJPiwQwTGsS4K1jECDZWarsFdN06QajVkQOBlZID1HaLXsmbAi0SPyNFyFBpu182TdKYCkM41rrpe4kJRlgZRjgd51uppgkmfdylTUsFjqkORtPTx+jIKh0USWWfMGiq/k6McKz1MA7tZIxNT897E/E/r5ma6CoYMZGkhgoyWxSlHBmJJj2gDlOUGD60BBPF7K2I9LDCxNi28rYEf/7Li6R2WvIvSuf3Z8XyTVZHDg7gEI7Bh0sowx1UoAoEBvAMr/DmPDkvzrvzMRtdcrLMPvyB8/kDfemS4w==</latexit>=#\n<latexit sha1_base64=\"+yRr4KFqXP7Xh39U9oa3eYMmCUc=\">AAAB+HicbVDLSsNAFJ3UV62PRl26GSyCCymJ+NoIRTcuK9gHNKFMppN26GQmzEOpoV/ixoUibv0Ud/6N0zYLbT1w4XDOvdx7T5QyqrTnfTuFpeWV1bXiemljc2u77O7sNpUwEpMGFkzIdoQUYZSThqaakXYqCUoiRlrR8Gbitx6IVFTwez1KSZigPqcxxUhbqeuW4VVwDGFgUiSleIRdt+JVvSngIvFzUgE56l33K+gJbBLCNWZIqY7vpTrMkNQUMzIuBUaRFOEh6pOOpRwlRIXZ9PAxPLRKD8ZC2uIaTtXfExlKlBolke1MkB6oeW8i/ud1jI4vw4zy1GjC8WxRbBjUAk5SgD0qCdZsZAnCktpbIR4gibC2WZVsCP78y4ukeVL1z6tnd6eV2nUeRxHsgwNwBHxwAWrgFtRBA2BgwDN4BW/Ok/PivDsfs9aCk8/sgT9wPn8A6b+R/A==</latexit>=\"\nUxx(a)(b)(c)<latexit sha1_base64=\"Cflx2J0B47wjyVRNEmM3baf0nhU=\">AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARXEiZEV8boejGZQX7gM5QMmmmDc0kQ5KxlLELf8WNC0Xc+hvu/BvT6Sy09UDgcM493JsTxIwq7TjfVmFhcWl5pbhaWlvf2Nyyt3caSiQSkzoWTMhWgBRhlJO6ppqRViwJigJGmsHgZuI3H4hUVPB7PYqJH6EepyHFSBupY+9decdeEiMpxRB6XTHkGe3YZafiZIDzxM1JGeSodewvE8ZJRLjGDCnVdp1Y+ymSmmJGxiUvUSRGeIB6pG0oRxFRfprdP4aHRunCUEjzuIaZ+juRokipURSYyQjpvpr1JuJ/XjvR4aWfUh4nmnA8XRQmDGoBJ2XALpUEazYyBGFJza0Q95FEWJvKSqYEd/bL86RxUnHPK2d3p+XqdV5HEeyDA3AEXHABquAW1EAdYPAInsEreLOerBfr3fqYjhasPLML/sD6/AEEOJYh</latexit>=\"#\nt\nFigure 1. Schematic illustration of how ferromagnetism can\nbe kinetically favored. (a) In a ferromagnetic state, the hop-\nping of a down spin (red) in a background of up spins (blue)\nallows the doublon (green) to move freely. However, for an an-\ntiferromagnetic configuration (b), the motion of the doublon\ncreates defects in the underlying spin texture, as sketched in\n(c) for a doublon moving two steps to the right. Due to the bi-\npartite nature of the square lattice, the same argument holds\nirregardless of whether the relevant charge carriers are dou-\nblons or holes.\nenergy, propagating holes therefore prefer to promote\nantiferromagnetic spin correlations around themselves\n(thereby releasing the frustration) [48–50] whereas dou-\nblons induce a local ferromagnetic environment [23, 51].\nThis phenomenon of kinetic ferromagnetism has only re-\ncently been observed in cold-atom experiments [52, 53],\nwhich demonstrated the development of ferromagnetic\npolarons: bound states consisting of a dopant dressed\nwith polarized spins. A natural question to then ask,\nwhich we address in our work, is whether this mechanism\nof polaron formation holds even without kinetic frustra-\ntion.\nSuch magnetic polarons have been extensively docu-\nmented for quantum antiferromagnets in which the move-\nment of a hole distorts the underlying N´ eel order [17, 54–\n58]. However, ferromagnetic polarons (henceforth re-\nferred to as “Nagaoka polarons”) have been less well char-\nacterized, with nearly all theoretical studies [59–63] fo-\ncusing on the so-called t-Jmodel [Eq. (5) below], which\nrepresents an approximation to the Hubbard model in\nthe limit of large U/t. Here, we present a comprehensive\ninvestigation of the Nagaoka polaron problem in a Hub-\nbard model, without simplification to the aforementioned\nt-Jlimit, using large-scale density-matrix renormaliza-\ntion group (DMRG) calculations [64–67]. In particular,we will consider an extended version of the doped Hub-\nbard model [68, 69], in which the second electron on any\nsite of the lattice is much more weakly bound than the\nfirst, and accordingly, the hopping depends on the occu-\npation of the site. The main advantage afforded by this\nmodel is that it greatly reduces the critical U/trequired\nfor ferromagnetism on the square lattice, which is bene-\nficial for the numerical stability of variational algorithms\nlike DMRG. Crucially, the ferromagnetic ground states\nof both the extended and the regular Hubbard models\nbelong to the same quantum phase and can be smoothly\nconnected by varying the microscopic parameters of the\ntheory; hence, they share the same physics.\nTo begin, in Sec. III, we first consider square clusters\nwith open boundary conditions and substantiate the for-\nmation of Nagaoka polarons as a route to itinerant fer-\nromagnetism at large U/t. Strictly speaking, such fer-\nromagnetism arises without all the conditions for Na-\ngaoka’s theorem being met but for the rest of this work,\nwe adopt the nomenclature “Nagaoka ferromagnetism”\nto label this phenomenon, even though, more accurately,\nit is only Nagaoka- type. We then systematically establish\nthe properties of individual polarons—including their en-\nergetics, size, and mobility—and discuss their extension\nto the higher-density regime of interacting polarons. Mo-\ntivated by these observations, we turn thereafter to the\nstudy of square-lattice geometries compactified on long\ncylinders in Sec. IV. With such cylindrical boundary con-\nditions (open along the cylinder axis and periodic in the\ntransverse direction), in addition to the fully saturated\nNagaoka state, we find striped configurations compris-\ning ferromagnetic domains interrupted by domain walls.\nHowever, irrespective of the global or local natures of\nthe ferromagnetic order, we show the emergence of po-\nlaronic quasiparticles with various techniques, including\nthe judicious application of pinning fields and examining\nspecially tailored three-point spin-charge-spin correlation\nfunctions. Our main findings are highlighted in Sec. V\nand also briefly summarized in Fig. 2, which depicts the\ndifferent magnetic ground states of the system as the dop-\ning concentration is varied. Some additional calculations\non the one-dimensional version of this model and smaller\n(square) cylinders are detailed in Appendices A and B,\nrespectively.\nII. MODELS AND METHODS\nThe extended Hubbard model that we investigate was\noriginally introduced to study hydrogenic donors in semi-\nconductors [68, 69; see also 70–72]. In an isolated hydro-\ngen atom, the one-electron 1 sbound state has a binding\nenergy of 1 Ry, but the two-electron state (H−) is bound\nby only 0.055 Ry, i.e., it is much more weakly bound\nthan the single-electron state. Consequently, H−is much\nlarger in size than the neutral H atom. A Hubbard-like\nmodel for an array of hydrogenic centers thus naturally\nneeds to be generalized to one where the hopping param-3\nPartially polarized FMExtended (global) FMParamagnetEffective single particle description\nStrongly interacting polaronsAFM / local FM<latexit sha1_base64=\"G8bqSPj9q+UC2B8GxtQyPeKxTLo=\">AAAB7XicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cI5gHJEmZnO8mY2ZllZlYIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9oapkqig0quVTtkGjkTGDDMMOxnSgkccixFY5up37rCZVmUjyYcYJBTAaC9RklxkrNboTckF654lW9Gdxl4uekAjnqvfJXN5I0jVEYyonWHd9LTJARZRjlOCl1U40JoSMywI6lgsSog2x27cQ9sUrk9qWyJYw7U39PZCTWehyHtjMmZqgXvan4n9dJTf86yJhIUoOCzhf1U+4a6U5fdyOmkBo+toRQxeytLh0SRaixAZVsCP7iy8ukeVb1L6sX9+eV2k0eRxGO4BhOwYcrqMEd1KEBFB7hGV7hzZHOi/PufMxbC04+cwh/4Hz+AJTMjyY=</latexit>\u0000<latexit sha1_base64=\"yqxL/LeN5Drp9Rf7ULunYgj27fw=\">AAACDHicbVC7TsMwFHXKq5RXgZHFokJiKknFa6xgYSyIPqQmjRzHaa3aSWQ7SFWUD2DhV1gYQIiVD2Djb3DaDNByJEtH55wr33u8mFGpTPPbKC0tr6yuldcrG5tb2zvV3b2OjBKBSRtHLBI9D0nCaEjaiipGerEgiHuMdL3xde53H4iQNArv1SQmDkfDkAYUI6Ult1qzfcIUGqSZi6EtKYfWyZ1rc6RGgqdxxLJBQ6fMujkFXCRWQWqgQMutftl+hBNOQoUZkrJvmbFyUiQUxYxkFTuRJEZ4jIakr2mIOJFOOj0mg0da8WEQCf1CBafq74kUcSkn3NPJfEs57+Xif14/UcGlk9IwThQJ8eyjIGFQRTBvBvpUEKzYRBOEBdW7QjxCAmGl+6voEqz5kxdJp1G3zutnt6e15lVRRxkcgENwDCxwAZrgBrRAG2DwCJ7BK3gznowX4934mEVLRjGzD/7A+PwBb4ybPw==</latexit>\u0000c⇠1/R2pol\nFigure 2. As a function of the electron doping concentration\naway from half filling, δ >0, we find four distinct magnetic\nregimes as illustrated pictorially here, with blue (red) sites\ndenoting up (down) spins. For small δ, the dopants form iso-\nlated polarons with local ferromagnetic (FM) ordering around\neach doublon core and antiferromagnetic (AFM) order further\naway. The radius of this polaronic cloud, Rpol, grows as U/t\nis increased. When doped with more electrons, the system\ncrosses over to a multipolaron regime forming ferromagnetic\ndomains (due to the kinetic energy gain from electron de-\nlocalization) but with different polarizations across domains\n(as favored by the superexchange). This—and the previous—\nregime may be viewed as a dilute gas of polarons, which is\nwell described by an effective noninteracting single-particle\npicture. As the doping is increased even further, one transi-\ntions to a regime of strongly interacting polarons at a critical\nδc∼R−2\npol; the individual mobile polarons now overlap, causing\nthe corresponding magnetic domains to be polarized homoge-\nneously. The system, which can be regarded as correlated\npolaronic fluid in this regime, thus becomes fully ferromag-\nnetic. Finally, at very large dopings, this global ferromag-\nnetism is progressively destroyed due to the reduced availabil-\nity of singly occupied sites, which suppresses electron hopping\nand fragments the extended domains.\neter is dependent on the occupation. This is captured by\nthe extended Hubbard model\nH=−X\n⟨i,j⟩,σ\u0010\nt(ni, nj)c†\niσcjσ+ h.c.\u0011\n+UX\nini↑ni↓,(2)\nwhere ni=P\nσc†\ni,σci,σis the total occupation of site iand\nthe correlated hopping alluded to above is of the form\nt(ni, nj) =(\n˜t ifni= 1 and nj= 2\nt otherwise. (3)\nNote that the choice of the barehopping tfor the second\ncase of Eq. (3) is essential to recover the exact asymptoticspatial dependence [73] of the effective exchange inter-\naction ∼e−2r/a b(∼t2/Ufort∼e−r/a b), where abis the\neffective Bohr radius of the hydrogenic centers. By con-\nstruction, this model is patently electron-hole asymmet-\nric and previously, high-spin ground states were found\nto be attained at much lower U/tfor electron doping\nthan hole doping [69]. Therefore, throughout this work,\nwe will focus exclusively on the electron-doped case. On\nsetting ˜t=t,Hjust reduces to the conventional Hub-\nbard model H0. However, a larger value of ˜t/texpands\nthe regions where the ground state attains its maximum\npossible spin [68] (since an enhanced hopping amplitude\nincreases the kinetic benefit of electron delocalization).\nAccordingly, we will work at a fixed ˜t/t= 4 unless men-\ntioned otherwise, but, as stressed earlier, all our conclu-\nsions about the Nagaoka polaron should apply to the case\nof˜t/t= 1 as well.\nIn the regime of large U/t≫1 and at above half-filling,\none can construct a low-energy theory of the extended\nHubbard model (2) by projecting out the unoccupied\nstate. In this reduced Hilbert space, defined by retain-\ning the states {| ↑⟩,| ↓⟩,| ↑↓⟩} on each site, the effective\nHamiltonian is\neH=−˜tX\n⟨i,j⟩,σ\u0010\n¯c†\niσ¯cjσ+ h.c.\u0011\n+4t2\nUX\n⟨i,j⟩\u0012\nSi·Sj−1\n4ninj\u0013\n,\nSα\ni≡X\nµ,νc†\niµτα\nµνciν;α=x, y, z, (4)\nwhere ¯ ciσ=ciσni¯σis the projected electron operator and\nταis a Pauli matrix in spin space. Importantly, the spin\nexchange is independent of ˜tand equals 4 t2/U(on the\nsquare lattice), which is the same as that for the regular\nHubbard model (1). Defining the conventional t-Jmodel\n[74, 75] as\nHtJ=−tX\n⟨i,j⟩,σ\u0010\n¯c†\niσ¯cjσ+ h.c.\u0011\n+JX\n⟨i,j⟩\u0012\nSi·Sj−1\n4ninj\u0013\n,\n(5)\nit easy to observe that the effective Hamiltonian (4) de-\nrived above is simply a rescaled version of Eq. (5), i.e.,\neH≡(˜t/t)HtJwith J= 4t3/(˜t U). Although we do not\ndirectly study the t-Jmodel in our numerical investiga-\ntions, we will see that it serves as a useful descriptor of\npolaronic properties in certain limits.\nWe analyze the extended Hubbard model (2) using\nDMRG, which provides an optimized matrix product\nstate representation of a target wavefunction. Through-\nout our calculations, we maintain a truncation error of\n<10−6by adaptively increasing the bond dimension as\nrequired, up to χ= 6000. Employing both open and\ncylindrical (unidirectionally periodic) boundary condi-\ntions, we explore the possible ground states for a broad\nrange of U/tand dopings. In particular, we find a variety\nof magnetically ordered states (including fully polarized\nhigh-spin ones) at moderate to large U/t.4\nPolaron formation[DMRG, open boundaries]\nDoping = 1eDoping = 2eDoping = 3eDoping = 4e\nFigure 3. Ground states, with open boundary conditions, on 5 ×5 (top panel) and 6 ×6 (bottom panel) square arrays doped\nwith one to four electrons, for ˜t/t= 4,U= 10 ˜t. The diameter of each circle is proportional to the local excess electron density\n⟨ni⟩ −1. The length of the arrows on each site indicates the magnitude of ⟨S0·Si⟩, with the central (white, unmarked) site\nchosen as the reference spin S0. The color of the circles as well as the orientation of the arrows conveys the sign of the spin\ncorrelations, with blue (red) denoting positive (negative) correlations.\nIII. SQUARE CLUSTERS\nMost numerical studies of Nagaoka ferromagnetism\nin the square-lattice Hubbard model have focused on\nthe infinite- Ulimit, which has been investigated for\nsmall clusters [76–78], with Lanczos techniques, as well\nas extended domains, using dynamical mean-field the-\nory [79, 80], variational quantum Monte Carlo (QMC)\n[26, 38], or DMRG [81, 82]. However, in order to under-\nstand the magnetic interactions in the spin sector, it is\nimportant to consider the (more generalizable) Hubbard\nmodel at finite U. The question of ferromagnetism in\nthis case poses a much more challenging problem, and\nthe system sizes probed thus far have been rather lim-\nited, ranging from plaquettes of ∼5–16 sites (amenable\nto exact diagonalization) [68, 69, 83] to ∼20 sites in\nmore recent works on full configuration interaction QMC\n[84, 85].\nHere, we start by studying the full extended Hubbard\nmodel (2) on L×Lsquare arrays for L= 5,6; our re-\nsults are tabulated for clusters with open boundary con-\nditions in Fig. 3. The corresponding results with cylindri-\ncal boundaries will be discussed in Appendix B, Fig. 13.\nWhile some ground-state properties can depend on the\nmicroscopics for these finite system sizes, let us highlight\nthe salient features observable in Fig. 3 that underscore\na few general trends. First, on doping the system with\na single electron, we can visually identify the formation\nof a ferromagnetic bubble residing near the center of the\nlattice (owing to the boundary conditions) for both L= 5and 6. While the spins are polarized within this bubble,\nfar away from it, the spin-spin correlations turn anti-\nferromagnetic. By virtue of the reasoning presented in\nFig. 1, the doublon can move around freely only inside\nthis bubble whereas its longer-range motion would nec-\nessarily disrupt the antiferromagnetic background. We\nrefer to this combination of the doublon and the polar-\nization cloud in its vicinity as a polaron. Now, if we add\nan extra electron, the two clouds of polarized spins sur-\nrounding each doublon can either be of the same polar-\nization ( L= 5) or the opposite ( L= 6). Which situation\nprevails is decided by the delicate interplay between the\ngain in kinetic energy from delocalization, which is aided\nby enlarging ferromagnetic domains, and the antiferro-\nmagnetic exchange energy that prefers to maximize do-\nmain walls, thereby favoring smaller domains. However,\naway from the bipolaron, the correlations still continue\nto be antiferromagnetic. Increasing the doping further,\nto three electrons, leads to the onset of long-range global\nferromagnetic order that extends across the entire sys-\ntem (in distinction to the local ferromagnetism observed\nin the previous two cases) as the polaronic wavefunctions\nstart to overlap. For both the doped 5 ×5 and 6 ×6 ar-\nrays, we find that the spins are now all fully polarized\nin the quantum ground state, forming a saturated Na-\ngaoka ferromagnet. Lastly, we observe that proceeding to\neven higher dopant concentrations (four electrons) actu-\nally impedes ferromagnetism and the system transitions\nto a paramagnetic phase [26], which can be understood\nas follows. Recall, per Fig. 1, that the very origin of fer-5\nromagnetism is due to the enhancement in the kinetic\nenergy gained by a delocalized electron in a spin-aligned\nbackground relative to the case of the background spins\nbeing in an antiferromagnetic (or random) configuration.\nHowever, such favorable hopping processes are hindered\nat large doublon concentrations because electrons cannot\nmove between two sites which are both doubly occupied.\nIn fact, at high densities and large U/t, the doublons\nshould have correlations resembling those of free hard-\ncore bosons [78, 86] and the collective charge motion is\ngoverned by a reduced hopping probability that depends\non the spin part of the wavefunction.\nThe ground states with cylindrical boundary condi-\ntions, shown in Fig. 13, are qualitatively similar, with\nthe key difference being the development of stripes, for\ncertain dopings, which compete with extended ferromag-\nnetic ordering. In this section, however, we focus on ar-\nrays with open boundary conditions, deferring a detailed\ndiscussion of clusters with cylindrical boundaries to Ap-\npendix B.\nA. Properties of the Nagaoka polaron\nHaving identified the formation of magnetic polarons,\nwe now characterize their size and energetics. To specif-\nically study the properties of individual noninteracting\npolarons, we consider the case of a single electron doped\ninto a 5 ×5 or 6 ×6 square cluster.\nFirst, we compute the polaron’s energy, defined as\nEpol=E1−(E0+U), (6)\nwhere E1is the energy of the system doped with one\nexcess electron and E0is that of the undoped system.\nEpoltherefore represents the energy gained by ferromag-\nnetically polarizing some subset of the spins (i.e., by the\ncreation of the polaron) relative to the N´ eel-ordered anti-\nferromagnetic ground state of the doublon-free undoped\nsystem. Note that in the definition of Epolin Eq. (6), we\nhave subtracted out a trivial shift of the energy due to the\ninteraction Uso as to isolate the magnetic contribution to\nthe polaron’s energy. For the L= 5,6 clusters, Fig. 4 dis-\nplays that Epolis lowered—and correspondingly, polaron\nformation is favored—with growing U/t. This behavior\nis in accordance with the intuition that the antiferro-\nmagnetic exchange interaction, which competes against\nferromagnetism, is diminished as U/tis increased.\nIn the t-Jmodel, a straightforward analysis balancing\nthe kinetic energy of a doublon propagating freely within\na ferromagnetic droplet against the magnetic energy of\nthe bubble (vis-` a-vis the N´ eel state) shows that the en-\nergy of the polaron should scale as√\nJ[61]. Subsequent\nwork [62] has since shown that a better numerical fit of\nthe polaronic energy in the t-Jmodel is given by (in units\nwhere t= 1)\nEtJ=−4 + 4 .6J0.42. (7)\n10 20 30 40 50 60-13-12-11-10-9Figure 4. Energy of an isolated Nagaoka polaron as a function\nofU/tat˜t/t= 4, as determined from Eq. (6) for a 5 ×5 (green\ncircles) or 6 ×6 (blue triangles) square lattice doped with one\nelectron above half filling. The dashed and dash-dotted lines\nmark the rescaled (by ˜t/t) predictions from the t-J[Eq. (7)]\nandt-Jz[Eq. (8)] models, respectively.\nThis curve is plotted in Fig. 4 for comparison to our data,\nand the reasonable agreement of the numerically deter-\nmined Epolwith this theoretical scaling further confirms\nour picture of polaron formation. Furthermore, in the\nsmall- Jlimit, the motion of a doublon is confined to\nits associated ferromagnetic polaron cloud, so spin-flip\n(S+\niS−\nj+S−\niS+\nj) processes are strongly suppressed. This\nmotivates the consideration of an Ising version of the t-J\nmodel [87, 88]\nHtJz=−tX\n⟨i,j⟩,σ\u0010\n¯c†\niσ¯cjσ+ h.c.\u0011\n+JzX\n⟨i,j⟩\u0012\nSz\niSz\nj−1\n4ninj\u0013\n,\nwhich drops the spin-flip part of the Heisenberg interac-\ntion in (5), thereby lifting the SU(2) spin-rotation sym-\nmetry inherent to the regular Hubbard and t-Jmodels.\nIn this case, the energy of the polaron is roughly given\nby [61]\nEtJz=−4 + 6 .03p\nJz, (8)\nwhich is also compared against our data in Fig. 4.\nWhile the values of Epolobtained for the extended\nHubbard model are broadly consistent with the predic-\ntions for both EtJandEtJz(after rescaling by the factor\nof˜t/t), Fig. 4 does exhibit noticeable deviations even\nfor large U/t, where the t-Jmodels are supposed to be\ngood approximations. This difference between the Hub-\nbard and t-Jbehaviors can be understood by examin-\ning the higher-order magnetic interactions, which arise\nin a perturbative expansion of the Hubbard model. The\nleading correction is a biquadratic ring exchange [89, 90]\ndescribed by\nH□=J□X\n⟨i,j,k,l⟩\u0002\u0000\nSi·Sj\u0001\n(Sk·Sl) + (Si·Sl)\u0000\nSj·Sk\u0001\n−(Si·Sk)\u0000\nSj·Sl\u0001\u0003\n, (9)6\nSize of the polaron\n(a)(b)\n-0.7-0.6-0.5-0.4-0.3-0.2-0.1\nFigure 5. The antiferromagnetic exchange energy ⟨Si·Sj−\nninj/4⟩for nearest-neighboring i,jon a 6 ×6 lattice doped\nwith one excess electron at (a) U= 5˜t, and (b) U= 10˜t(˜t/t=\n4 in both cases). The color of each bond as well as its thickness\nis scaled according to the value of ⟨Si·Sj−ninj/4⟩. The\nvanishing correlations at the center of the lattice delineate\nthe extent of the ferromagnetic polaron.\nwhere i, j, k, l label the four spins located around a square\nplaquette and J□∼O(t4/U3)>0 can be as large as 20%\nofJ[91] depending on the bandwidth. In a ferromagnetic\nbackground, as occurring for large U/t, this term thus has\na positive contribution, wherefore the energies of the t-J\nandt-Jzmodels underestimate the Hubbard Epol.\nAs the energy Epoldecreases with increasing U/t, the\npolaron also grows in size (as J−1/4for the t-Jmodel\n[61]), eventually expanding to fill the whole system be-\nlow some threshold J∼O(1/N2). This theoretically ex-\npected growth of the polaron can be observed in Fig. 5,\nwhere we plot the exchange energy ⟨Si·Sj−ninj/4⟩for\nnearest-neighboring i, jon the square lattice [61]. From\nEq. (5), one can infer that this expectation value is a di-\nrect measure of the disturbance of an antiferromagnetic\nspin texture by a ferromagnetic polaron. Note that al-\nthough ⟨Si·Sj−ninj/4⟩is indeed seen to be enhanced\naround a doublon in Fig. 5, it never exactly attains its\nmaximal value of zero due to a combination of finite-size\neffects and the fact that we calculate this quantity for a\nHubbard, rather than a t-J, model.\nB. Role of polaronic mobility\nOur previous calculations pertain to the limit where\nthe density of doublons is low enough such that the\nsystem is well-described by a dilute gas of isolated po-\nlarons coupled to a spin background via the kinetic term.\nHowever, as the doping concentration is increased, in-\nteractions between these polaronic quasiparticles become\nmore important. In this regime (see, e.g., the three- and\nfour-electron-doped cases in Fig. 3), extended ferromag-\nnetic order can arise from the spatial overlap between\nthe wavefunctions of different (mobile) polarons, which\nprompts the spins around their respective doublons tobe polarized similarly. This is because if two like po-\nlarons are positioned adjacent to each other, the doublon\ncores of each can now collusively delocalize over twice as\nlarge a ferromagnetic region [5].\nCentral to this mechanism therefore is the mobility of\nthe Nagaoka polaron. To corroborate this hypothesis, we\nengineer its contrapositive by explicitly pinning the dou-\nblons to certain sites of the lattice using an attractive\nlocal potential [92, 93], which disfavors their delocaliza-\ntion. Specifically, we consider a 6 ×6 array doped with\nNd= 4 electrons and apply a pinning potential, −V ns,\non four sites schosen so as to respect the rotational and\nreflection symmetries of the underlying square lattice.\nTreating the Nddoublons as spinless noninteracting\nfermions that fill a quadratic band, the total energy of\nsuch a multipolaron system can be easily approximated in\nthet-Jmodel, along the same lines as the single-polaron\ncalculation. For the optimal polaron size, this evaluates\nto [62]\nEtJ(Nd) = 2 Nd\u0010√\n2πJ−2\u0011\n, (10)\nin units where t= 1. Equation (10) yields an initial esti-\nmate for the threshold value of the pinning potential per\npolaron, Vth= (˜t/t)|EtJ(Nd)|/Nd, that must be applied\nin order for the energy gain from the pinning to disrupt\nthe ferromagnetic state.\nThe spin correlations of the 6 ×6 cluster (doped with\nfour electrons) at large U/tin the absence of any onsite\npinning field ( V= 0) are plotted at the bottom right in\nFig. 3. These are to be contrasted with the situation\nfor nonzero Vshown in Fig. 6. First, upon the appli-\ncation of a pinning field of strength V=U/2< V th\n[Fig. 6(a)], we see that the system forms four ferromag-\nnetic patches, one centered around each doublon. Hence,\nlocal ferromagnetic order still persists but the domains\nthus formed are of smaller size than in the field-free case\ndue to the reduced doublon mobility. On the other hand,\nfor a potential V= 3U/8> Vth[Fig. 6(b)], this phase be-\ncomes unstable to the creation of a predominantly anti-\nferromagnetic state but with weak ferromagnetic corre-\nPinning fields(a)(b)\nFigure 6. Charge densities and spin correlations of the ground\nstates of a 6 ×6 square lattice with open boundaries, doped\nwith four excess electrons, for ˜t/t= 4, U= 10 ˜t, and pinning\nfields of strength (a) V=U/4, and (b) V= 3U/8 applied on\nthe four sites at the center of each 3 ×3 corner of the array\n(marked by green dots).7\n(a)(b)(c)\nFigure 7. Ground states of the extended Hubbard model (2) on a 30 ×4 cylinder at δ= 1/12 electron doping with ˜t/t= 4,\nU= 10 ˜t, and local pinning potentials (a) V= 0, (b) V=U/4, and (c) V=U/2 applied to the ten lattice sites marked by the\ngreen dots in (b,c). The effect of the pinning can be visually discerned from the growth of the charge density (the diameter of\nthe circle) on these sites as Vis increased. The spin-spin correlations are plotted using the same conventions as in Fig. 3.\nlations on only the NN bonds next to the tightly pinned\npolarons.\nIV. POLARON FORMATION IN EXTENDED\nSYSTEMS\nHaving demonstrated the origin of Nagaoka ferromag-\nnetism via polaron formation in relatively small square\narrays, we now proceed to investigate this mechanism in\nextended systems. To this end, we study long cylinders\nof width four and length up to 30 sites; these dimensions\nare close to the current limit of state-of-the-art ground-\nstate DMRG numerics [94, 95]. We will further focus on\nthe optimal doping fraction (for ferromagnetism) of δ≡\nNd/N= 1/12 suggested by the results of Fig. 3.\nThe distinctive new feature that emerges on such cylin-\ndrical geometries, as identified by Ref. 46, is the existence\nof competing magnetically ordered ground states with\nstripes, i.e., unidirectional charge- and spin-density mod-\nulations [96–98]. Such an inhomogeneous striped ground\nstate (Fig. 13), which breaks both rotational and trans-\nlational symmetries, arises due to the competition be-\ntween the domain walls favored by the antiferromagnetic\nexchange and the lack thereof preferred for kinetic delo-\ncalization. However, increasing U/tweakens the spin ex-\nchange and eventually, the system undergoes a first-order\nquantum phase transition to a fully saturated ferromag-\nnetic state at U/t∼30 [46].\nThe polaronic nature of this ferromagnetism can be\ndemonstrated once again by using local pinning poten-\ntials, which we now apply in a staggered fashion along\nthe length of the cylinder. The number of pinned sites ischosen to be the same as the number of excess electrons.\nWhen the pinning fields are absent, as mentioned above,\nthe system exhibits long-range ferromagnetic order that\nspreads across the entire lattice without any domain walls\n[Fig. 7(a)]. On applying a potential of strength V=U/4\n[Fig. 7(b)], this global order fractures into smaller stripes,\neach of a width such that it accommodates exactly one\ndoublon on average. The natural interpretation here is\nthat while polarons still continue to form, their extent is\nlimited. As before, this effect of the pinning can be at-\ntributed to the reduced mobility of the doublons which,\nin turn, lowers the kinetic energy gain driving ferromag-\nnetism. Upon increasing Veven further, to U/2, we ob-\nserve that the polaron’s radius shrinks to now encompass\nonly the NN sites of a doublon [Fig. 7(c)] and the corre-\nlations in the ground state are mostly antiferromagnetic.\nTo gain further insights into the ferromagnetic po-\nlarons that we have seen develop, it is useful to probe\nthe spin environment around the doublons at a micro-\nscopic level. The polarization of the spins in the vicinity\nof a dopant electron can be quantified by a three-point\nfunction\nG(r0;r1,r2) =\n\u0000\nnr0−1\u0001\nSr1·Sr2\u000b\n, (11)\nwhich measures the correlations between two spins posi-\ntioned at lattice sites r1andr2given some excess charge\ndensity at site r0.G(r0;r1,r2) can equivalently be ex-\npressed in terms of the displacement between the spins,\nd=r2−r1, and the vector to the location of the doublon,\nr= (r1+r2)/2−r0, as\nG(r0;r,d) =D\u0000\nnr0−1\u0001\nSr0+r−d/2·Sr0+r+d/2E\n.(12)8\n0 2 4 6 8 10 12 140.2020.2040.2060.2080.2100.212\nFigure 8. Decay of the (normalized) three-point correlator\nG(¯r, d)/p[Eq. (11)] in the Nagaoka ferromagnetic state of\nFig. 7(a), as a function of the coarse-grained distance ¯ r(with\n∆r= 0.75) of a NN ( d= 1), 2NN ( d=√\n2), or 3NN ( d= 2)\npair of spins from the excess charge. The error bars on the\ny-axis represent the standard deviation of G(r, d)/p∀r∈[¯r−\n∆r,¯r+ ∆r]. The small dynamic range of the variation in G\nas a function of ¯ ris indicative of the presence of long-range\nferromagnetic order.\nWe define G(r, d) as this three-point correlator spatially\naveraged over r0as well as radially averaged over randd\n(with r≡|r|,d≡|d|). Working in units where the lattice\nspacing ais set to unity, we analyze G(r,1),G(r,√\n2),\nandG(r,2)—which correspond to first- (NN), second-\n(2NN), and third-nearest-neighboring (3NN) pairs of\nspins, respectively—for the ferromagnetic state sketched\nin Fig. 7(a). To avoid edge effects due to the open bound-\naries at the ends of the cylinder, we restrict r1andr2\nto the ten central columns of the 30 ×4 lattice. Since\nthe discrete nature of the lattice results in a set of often\nclosely spaced distances r, we coarse-grain the data in\n(nonoverlapping) windows [¯ r−∆r,¯r+ ∆r] to separate\nout the features of the Nagaoka state, which has a long\ncorrelation length, from nonuniversal short-wavelength\n(lattice-scale) fluctuations. Figure 8 plots G(r, d) as a\nfunction of the distance to the doublon rfor three bond\nlengths d= 1,√\n2,2. The decrease in G(r, d) with increas-\ningrconveyed by Fig. 8 indicates that spins are more\nlikely to be aligned closer to a doublon—in consistency\nwith our polaronic picture—while the slow nature of the\ndecay points to the presence of long-range ferromagnetic\norder.\nWhile G(r, d) characterizes the local distortion and re-\norganization of magnetic correlations in the proximity\nof a doublon, it also includes contributions from virtual\ndoublon-hole quantum fluctuations. Holes contribute\nwith an opposite sign to ( nr0−1) than doublons, and\ntheir effects may thus be difficult to disentangle in an\naveraged correlator ` a la Eq. (12). To circumvent this\ncomplication, we sample the ground-state DMRG wave-\nfunction in the ˆ zbasis, {|0⟩,| ↑⟩,| ↓⟩,| ↑↓⟩} , and generate\n100,000 snapshots; this is analogous to performing pro-\njective measurements in experiments. Using these sam-ples, we then compute the modified three-point correlator\nG(r0;r,d) =D\nSz\nr0+r−d/2·Sz\nr0+r+d/2E\f\f\f\f\n••r0,(13)\nwhich tracks the correlations between two spins separated\nbydconditioned on the presence of a doublon at r0[93].\nNote that the quantum expectation value indicated by\nthe angular brackets now reduces to an average over the\nindividual sampled configurations. To differentiate be-\ntween actual dopants and naturally occurring doublon-\nhole fluctuations, we exclude any doubly occupied site\nthat has a hole as its nearest neighbor. For each of the\nthree states depicted in Fig. 7 (ferromagnetic, striped,\nand antiferromagnetic), we evaluate G(r,d)—defined as\n(a)\n(b)\n(c)ryrx\n-0.5 0. 0.51.0\nFigure 9. NN ( d= 1, left) and 2NN ( d=√\n2, right) conditional\nspin correlations G(r, d), plotted as a function of r= (rx, ry),\nfor the (a) ferromagnetic, (b) striped, and (c) antiferromag-\nnetic states of Fig. 7. The correlations are represented by the\nbonds connecting two lattice sites (white dots) and are sorted\naccording to their distance from a doublon (black circle at\ncenter). The thickness of each bond is scaled in proportion to\n|G(r, d)|.9\nG(r0;r,d) averaged over all doublon positions r0—for\nvectors dcorresponding to the NN and 2NN bonds. This\nspin-charge-spin correlator allows us to directly exam-\nine the internal structure of the Nagaoka polaron. For\ninstance, in the ferromagnet [Fig. 9(a)], we observe, on\nboth NN and 2NN bonds, that the spin-spin correlations,\nwhile all positive, are strongest closest to the doublon and\ndecay with increasing distance therefrom. Likewise, in\nthe striped phase [Fig. 9(b)], the spins immediately next\nto the doublon remain positively correlated. On the con-\ntrary, we find that the NN bonds situated at rx=±1.5\nare antiferromagnetic, implying that the stripes are of\nwidth three in the ˆ xdirection. The anticorrelations vis-\nible in the 2NN ( |d|=√\n2) links also owe their origin\nto the same effect. We emphasize here that the spa-\ntial resolution of the vector rinto rxandrycompo-\nnents proves essential for distinguishing between the fer-\nromagnetic and striped states as the distinction between\nglobal and local ferromagnetic order can be washed out\nupon radially averaging r. Finally, in the antiferromag-\nnet [Fig. 9(c)], we see that the NN spin-spin correlations\nare negative while the 2NN ones are positive. However,\nthese (anti)correlations weaken for distances |rx|>2, re-\nflecting the influence of another doublon further away\nfrom the one at the origin.\nGoing beyond the properties of the individual polarons\nestablished above, we can additionally probe their inter-\nplay in a multiple-dopant system by studying the interac-\ntions between doublons. To do so, we define the doublon-\ndoublon correlation function [93]\nCd(r1,r2) =\nnd\nr1nd\nr2\u000b\n\nndr1\u000b\nndr2\u000b−1, (14)\nwhere nd\nr= 1 if there is a doublon on site rand 0 other-\nwise. Figure 10 shows that the doublons appear anticor-\nrelated at short distances (with an exchange-correlation\nhole approximately three lattice spacings in size) and\nuncorrelated beyond this length scale, as expected for\nfermionic particles.\nV. DISCUSSION AND OUTLOOK\nTo summarize, in this work, we have presented exten-\nsive numerical evidence coupled with theoretical analysis\nto demonstrate that the formation of magnetic polarons\nlies at the heart of Nagaoka ferromagnetism in the Hub-\nbard model. Our analysis illustrates that Nagaoka ferro-\nmagnetism is fundamentally a cooperative phenomenon\nin that the interaction of individual polarons, each pos-\nsessing only local ferromagnetic correlations around a\ndopant, can engender global ferromagnetism in a macro-\nscopic system. This also implies that the ferromagnetism\ncan be tuned by modifying the properties of the under-\nlying polarons. For instance, we have seen that starting\nfrom a predominantly antiferromagnetic state, one can\ninduce—or increase the extent of—ferromagnetic correla-\n1 2 3 4 5 6-0.8-0.6-0.4-0.20.0Figure 10. The radially averaged doublon-doublon correla-\ntion function of the δ= 1/12 electron-doped extended Hub-\nbard model on a 30 ×4 cylinder at ˜t/t= 4, for U= 6˜t(blue\ncircles) and U= 10˜t(red triangles), corresponding to striped\nand ferromagnetic ground states, respectively. The dashed\nline marks the average distance between doublons ≃3.20, as\ncomputed from the statistics of 100,000 projective samples of\nthe wavefunctions.\ntions by increasing U/t; at the microscopic level, this cor-\nresponds to enlarging the Nagaoka polaron. Conversely,\ngiven a state that isferromagnetic to begin with, one can\ndestroy the long-range magnetic order by preventing the\ndelocalization of doublons, such as via pinning potentials.\nThis underscores the vital importance of the mobility of\nthe polarons, which coalesce to form an extended ferro-\nmagnetic state. All these considerations taken together\nlead to the schematic phase diagram of Fig. 2, which out-\nlines the correspondence between the magnetic phases of\nthe Hubbard model and their associated polaronic inter-\npretations developed in our study.\nWithin the broader theoretical landscape, our results\nshed new light on the possibility and origin of itinerant\nferromagnetism in the Hubbard model, a long-standing\nproblem that has been tackled with diverse approaches\nover the years. Perhaps the simplest starting point in\nthis regard is Hartree-Fock theory, which yields ferro-\nmagnetic ground states whenever the Stoner criterion is\nsatisfied, i.e., D(EF)U >1, where D(EF) is the density\nof states at the Fermi energy. Such a theory does pre-\ndict ferromagnetism in extended regions of the Hubbard\nmodel’s phase diagram, but the validity of this purely\nstatic mean-field picture expectedly breaks down in the\nintermediate- to strong-coupling regime where one antic-\nipates ferromagnetism [99]. A proper treatment of the\nHubbard model, accounting for correlation effects, shows\nthat the behavior of the Nagaoka ferromagnet is highly\nlattice-dependent [100]. In particular, certain routes to\nferromagnetism are often specific to nonbipartite lattices:\nthese include the Haerter-Shastry mechanism [47], which\nresults from frustration due to three-site loops, as well\nas the so-called “low-density” or M¨ uller-Hartmann ferro-\nmagnetism [101] that arises due to a large and asymmet-\nric density of states at the band edge. At first glance, this\nsuggests that the microscopic details of the system can-10\nnot be neglected when it comes to understanding ferro-\nmagnetism, which would preclude a universal description\nof the physics. However, by studying the simple bipar-\ntitesquare lattice here, we establish that the polaronic\nmechanism driving ferromagnetism is a universal and ro-\nbust property of the Nagaoka state which does not rely\non kinetic frustration or other lattice-specific considera-\ntions. Hence, our general conclusions regarding polaron\nformation should also apply to triangular lattices, which\nhave been recently investigated in ultracold-atom exper-\niments [43, 52, 53] and semiconductor moir´ e superlattice\nsystems such as WSe 2/WS 2bilayers [44, 102, 103].\nLooking ahead, other interesting directions in which\nour calculations can be extended include exploring the in-\nfluence of disorder, finite temperatures, and long-ranged\nCoulomb interactions on ferromagnetism. Incorporation\nof these effects would be both useful and important for\ndescribing arrays of gate-defined semiconductor quan-\ntum dots [39], which have recently emerged as another\npromising platform for quantum simulation of the Hub-\nbard model [104, 105] and potentially, Nagaoka ferromag-\nnetism.\nACKNOWLEDGMENTS\nWe thank W. S. Bakr, I. Bloch, J. Dieplinger, A. Kale,\nL. H. Kendrick, M. Lebrat, and M. L. Prichard for useful\ndiscussions. R.S. is supported by the Princeton Quantum\nInitiative Fellowship. R.N.B. acknowledges support from\nthe UK Foundation at Princeton University. This work\nwas performed in part at the Aspen Center for Physics,\nwhich is supported by National Science Foundation grant\nPHY-2210452. The participation of R.S. at the Aspen\nCenter for Physics was supported by the Simons Founda-\ntion. The calculations presented in this paper were per-\nformed using the ITensor library [106] on computational\nresources managed and supported by Princeton Research\nComputing, a consortium of groups including the Prince-\nton Institute for Computational Science and Engineering\n(PICSciE) and the Office of Information Technology’s\nHigh Performance Computing Center and Visualization\nLaboratory at Princeton University.\nAppendix A: One-dimensional model\nWhile the primary focus of our work has been on two\nspatial dimensions, the problem of ferromagnetism in the\none-dimensional Hubbard model [107] also has a long and\nrich history. Here, we briefly note some salient results\nand direct the reader to Refs. 108 and 109 for more de-\ntailed reviews.\nFor 1D, Lieb and Mattis [110] rigorously proved that\nthe ground state of the single-band Hubbard model—\nwith only nearest-neighbor hoppings and onsite density-\ndensity interactions—is a singlet. Therefore, obtain-\ning ferromagnetic ground states requires circumventionof the assumptions underlying the Lieb-Mattis theorem.\nBroadly speaking, four different routes towards this end\nhave been investigated. One such way is to introduce or-\nbital degeneracy by considering a multiband extension\nof the Hubbard model. Then, the local exchange in-\nteractions between electrons in different orbitals on the\nsame site (which align unpaired electrons on each atom\nby Hund’s rule) may lead to ferromagnetism, i.e., the\nhopping of holes or electrons can yield a bulk order-\ning of preformed atomic moments [99, 111–113]. An-\nother option is to add in interactions such as the nearest-\nneighbour Coulomb repulsion terms [19, 20, 114], which\nare always present in the underlying electronic system\nbut are abstracted away in the Hubbard model. Simi-\nlarly, the inclusion of longer-range hopping terms such as\n−t2P\ni(c†\ni,σci+2,σ+ h.c.) has also been tied to the emer-\ngence of ferromagnetism both analytically (in certain lim-\nits) [101, 115, 116] as well as numerically [117–121]. This\nis because the proof of the Lieb-Mattis theorem relies on\na definite ordering of the particles, which is no longer\nenforced when t2̸= 0. Lastly, it is possible to assemble\nseveral (identical) copies of such long-range models to\nobtain models with only short-range hoppings that still\nexhibit ferromagnetism [108]. This opens up the direc-\ntion of inducing “flat band” ferromagnetism [122–127] by\nmodifying the Hubbard model such that the lowest bands\n(in the single-particle spectrum) are altered to be either\nexactly or nearly dispersionless.\nGiven this backdrop, it is thus only natural to ask\nabout the physics of the extended Hubbard model in 1D.\nRewriting the Hamiltonian (2) as\nH=−(˜t−t)X\n⟨i,j⟩,σ\u0010\nc†\niσcjσni(nj−1) +c†\njσciσnj(ni−1)\u0011\n−tX\n⟨i,j⟩,σ\u0010\nc†\niσcjσ+c†\njσciσ\u0011\n+UX\nini↑ni↓, (A1)\nwe observe that the hopping in the first line gets dressed\nby the occupation factors resulting in a four-operator\nterm, which describes a correlated hopping process [128]\njiij(a)(b)\n51015205101520\n-0.63 0. 0.74\n51015205101520\n-0.40 0. 0.67\nFigure 11. Two-point correlation function Cijof a 24-site\nchain at ˜t/t= 4,U= 10˜t, doped away from half filling with (a)\ntwo electrons and (b) six electrons, corresponding to dopant\nconcentrations of 1 /12 and 1 /4, respectively.11\n(a)(b)(c)12345678-0.04-0.020\n12345678-0.08-0.040\n12345678-0.1-0.050\nFigure 12. Absence of polaron formation in the 1D extended\nHubbard model (2). Here, we plot the three-point correlator\nGi(r) [Eq. (12)], averaged over the central L/2 lattice sites\ni, as a function of rfor five system sizes and electron doping\nconcentrations (a) 1 /12, (b) 1 /6, and (c) 1 /4, at ˜t/t= 4,\nU= 10 ˜t.\nwith no counterpart in the conventional Hubbard model\n(1). Consequently, determining the ground state of this\nmodel and its spin properties is a nontrivial task that is\nnot immediately addressed by the Lieb-Mattis theorem.\nHere, we study the one-dimensional system numeri-\ncally on long chains of up to 96 sites using DMRG. The\nfirst quantity that we examine is the connected two-point\ncorrelation function,\nCij=⟨Si·Sj⟩ − ⟨Si⟩ · ⟨Sj⟩, (A2)\nwhich is plotted in Fig. 11 for a chain of length L= 24\ndoped away from half filling with electrons at two differ-\nent doping concentrations. We observe that the domi-\nnant NN correlations are actually antiferromagnetic and\nupon increasing the doping fraction, the 2NN correla-\ntions also become antiferromagnetic. This antiferromag-\nnetic character is found to hold for a wide variety of chain\nlengths ( L= 12ℓ, ℓ= 2,3, . . . , 8), doping concentrations\n(1/12, 1/6, 1/4), and model parameters ( U/˜t∈[5,50]),\nand is numerically robust in that it persists even when\nthe system is explicitly initialized with a ferromagnetic\nconfiguration. This is in stark contrast to the behavior\nin two dimensions. The difference between the two cases\ncan be understood per the intuition outlined in Fig. 1:\nthe hopping of a dopant does not scramble an antiferro-\nmagnetic background in 1D since the associated domain\nwall is a point-like (as opposed to line-like in 2D) object.\nTo microscopically probe the origin of the antiferro-\nmagnetic correlations seen in Fig. 11, it is also useful toquantify the polarization of the spins in the vicinity of a\ndopant electron. This is achieved by the one-dimensional\nversion of the three-point function in Eq. (11),\nGi(r) =\n(ni−1)Si−r·Si+r\u000b\n, (A3)\nwhich should show the development of a ferromagnetic\npolaron, if any. In Fig. 12, we plot the correlation func-\ntion Gi(r) averaged over i, denoted as G(r), for sev-\neral different lattice sizes and doping concentrations in\nthe regime of large U/twhere one might expect ferro-\nmagnetism. In order to avoid trivial boundary effects,\nwe exclude L/4 sites from each end of the chain, so\nthat the computed three-point function accurately re-\nflects the bulk behavior. We find that for moderately\nlarge U(≥5˜t),G(r) is always negative for r= 1, inde-\npendent of system size, and never becomes appreciably\npositive for distances of up to r= 8. This reveals that\nthe spins tend to be partially antialigned near an ex-\ncess electron, and the magnitude of this anticorrelation\nincreases with doping. Hence, a ferromagnetic polaron\nnever forms.\nSince the site-averaged correlation function G(r) could\npotentially suffer from cancellations between contribu-\ntions from electron-rich and hole-rich spatial regions, due\nto the factor of ( ni−1) in Eq. (12), we also compute the\nmodified three-point function\n˜Gi(r) =D\n(ni−1)2Si−r·Si+rE\n. (A4)\nWhile not explicitly shown in Fig. 12, the site-averaged\nGi(r) and ˜Gi(r) are found to be virtually identical, indi-\ncating that for the doped system, the dominant contri-\nbution to G(r) is from the majority carriers.\nAppendix B: Square arrays on cylinders\nThe ground states of the extended Hubbard model on\nsquare clusters with cylindrical boundary conditions are\narrayed in Fig. 13. While these states exhibit some sim-\nilarities to the ones with open boundaries, displayed in\nFig. 3, a new feature, for certain dopings, is the develop-\nment of stripe ordering, characterized by unidirectional\nspin and charge density modulations [46]. Such stripes\nare well exemplified, for instance, by the two-electron-\ndoped systems, which convey that it can sometimes be\nenergetically favorable to form two smaller ferromag-\nnetic domains (thus optimizing the antiferromagnetic ex-\nchange contribution along the long domain wall) at the\nexpense of a single larger one. This is because the peri-\nodic boundaries along the circumference of the cylinder\nincrease the kinetic energy gain from delocalization over\na given area, relative to a system with open boundaries,\nand together with the superexchange, this can offset the\nenergetic cost of confining the doublon to a smaller spa-\ntial region. Under certain circumstances, such as for the\n6×6 system doped with four electrons, the spin texture\ncan also form square domains, as opposed to elongated12\nDoping = 1eDoping = 2eDoping = 3eDoping = 4ePolaron formation[DMRG, cylindrical boundaries]\nFigure 13. DMRG ground states on 5 ×5 (top) and 6 ×6 (bottom) square clusters with cylindrical boundary conditions, doped\nwith one to four electrons, for ˜t/t= 4,U= 10 ˜t. All the plots follow the same conventions as used in Fig. 3.\nstripes. 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China  \n3 Beijing Key Laboratory for Magnetoele ctric Materials and Devices ，Beijing \n100871, P.  R. China  \n4. Department of Materials Science and Metallurgy, University of Cambridge, \nCambridge, CB3 0FS, United Kingdom  \n \nAbstract:  The spin configuration in the ferromagnetic part during the magnetization \nreversal plays a crucial role in the exchange bias  effect . Through  Monte Carlo \nsimulation, the exchange bias effect in ferromagnetic -antiferromagnetic core-shell \nnanoparticles  is investigated. Magnetization reversal s in the ferromagnetic core were  \ncontrolled b etween the coherent rotation and the domain wall motion by modulati ng \nferromagnetic domain wall width  with parameters of uniaxial anisotropy constant  and \nexchange coupling strength. An anomalous monotonic dependence of exchange bias on \nthe uniaxial anisotropy constant  is found  in systems with small exchange coupling , \nshowing an obvious violation of classic Meiklejohn -Bean model, while domain wall s \nare found to form  close to the interface  and propagate in the ferromagnetic  core with 2 \n large r uniaxial anisotropy in both branch es of the hysteresis. The asymmetric \nmagnetization reversal with  the formation  of a spherical domain wall  dramatically  \nreduce s the coerci ve field  in the ascending  branch , leading to the enhancement of the \nexchange bias.  The results provide a nother degree of freedom to optimize  the magnetic \nproperties of magnetic nanoparticles  for applications .  \nKeywords:  exchange bias, Monte Carlo, domain wall, core -shell nanoparticle, \nmagnetic anisotropy  \nIntroduction  \nExchange bias has been found in magnetic materials  containing exchange -coupled \ninterfaces between  two different magnetic phases . Being  of great interest  both for the \napplications in spintronic devices and fund amental physics of condensed matter physics, \nexchange bias has been extensively  studied  since its first discovering  more than half a \ncentury ago1. Exchange bias has been found in a wide range  of materials including  low \ndimensional composites  with ferri-/ferro magnet  (FM)/antiferromagnet (AFM) \ncombinations2-4, single -phase  bulk materials  with spin glass (SG) or super spin glass \n(SSG)5-7. Recently, giant exchange bias effect has also  been  reported in magnetic single \nphases with inter -sublattice interaction s8-10. Typical systems such as FM/AFM bilayer, \nand FM/AFM core -shell nanoparticles usually serves as prototype models for the study \nof exchange bias, due to the well -defined FM and AFM parts as well as the interfaces.  \nMagnetic nanoparticles are  of increasing appeal  and have found  numerous applications \nin engineering (magnetic recording media or magnetic seals) and biomedical \napplications (magnetic resonance imaging, drug delivery, or thermotherapy)11. With the 3 \n first exchange bias reported in Co/CoO core-shell nanoparticles , in recent years, the \nstudy of exchange bias in nanoparticles and nanostructures has gained renewed interest \nsince it has been shown that control of the core/shell in teractions or of the exchange \ncoupling between the particle surface and the embedding matrix can increase the \nsuperparamagnetic limit for their use as magnetic recording media12 . Advances in  \ntechniques for synthesis of nanomaterials13-15 allow the magnetic properties in both the \ncore and the shell to be continuously  controlled  with morphology16 -19  and \ncomposition20 -23  tailoring. A number of  factors have show n, in both experimental \nstudies and Monte Carlo  (MC)  simulations,  strong effect s in the observed  exchange \nbias or magnetic properties  in the core -shell structures , including the  core/shell \nthickness es24 -27 , particle shape/morphology28 ,29 , cooling field30 ,31 , dipol ar \ninteractio ns32 ,33  /interparticle exchange interactions11, and interface lattice/magnetic \ndisorder /mismatch34,35.  \nIn the Meiklejohn -Bean ( M-B) model, by assuming a collinear magnetization reversal \nin both FM and uncompensated AFM parts, the exchange bias field was predicted to be  \nex\nE\nFM FMhtM\n   (1) \nwhere σex, tFM, and MFM stand for the interfacial exchange coupling energy, the FM \nthickness, and the FM magnetization , respectively.  Thus, a n inversely  linear \ndependence on the thickness of FM layer36,37 and no dependence on intrinsic properties \nof FM part, including the magnetic anisotropy and the exchange coupling strength  were \nindicated  in the model . Since the M -B model  work s very well in many systems, the \neffect of inner magnetic structure in the FM part has been overlooked  to some extent 4 \n for quite a long time  while the most effort ha s been devoted to  the magnetic structures \nin AFM part s and interface s. However, recent experimental and theoretical results \nindicate that this rule can be violated while a partial domain wall parallel  to FM -AFM \ninterface forms in FM layer  during the magnetization reversal process38,39. Although \nnonuniform  magnetization configurations  have  been reported in  magnetic nanoparticles  \nvia small -angle neutron scattering40,41 magnetic force microscopy42, magnetic electron \nholography43  and MC simulation s44 -47 , its effect on the exchange bias of FM -AFM \ncore-shell structures  and how it can be controlled remain  unknown .  \nIn this paper, it is shown , through MC simulations based on a simple model  of single  \ncore/shell nanoparticle, how the formation of a spherical domain wall  in the FM core is \nrelated to exchange bias in this system.  The spherical domain wall  is induced  or \nsuppressed  in the FM core by tuning the domain wall  width by varying the anisotropy \nconstant and the exchange coupling strength . This result is confirmed by inspection of \nmagnetic config urations and  curls of magnetic  configuration s in the core along the \nhysteresis loops . It is further  demonstrated that the formation of a spherical domain wall  \nin the core while magnetization reversal significantly reduce s the coercive field  in the \nascending branch , and consequently enhance s the exchange bias field.  \nModel  \nThe considered nanoparticles have  a spherical  shape with  a total radius of R = 12 a, \nrespectively, with a being the unit cell size. All the particles are made of an FM core \nsurrounded by an AF shell of a constant thickness RSh = 3a with magnetic properties \ndifferent from the core as well as from the spins at the interface between core and shell 5 \n spins. Taking a = 0.3 nm, such a particle corresponds to typical real dimensions R ≈ \n4 nm with a fixed shell thickness of RSh ≈ 1 nm and contains 5575 spins , with 307 1 \nspins in the FM core and 2504 spins in the AFM shell . The interface is defined to be \nthe atoms in the AFM shell which have direct exchange coupling  with the FM core  and \ncontains 918 spins . The anisotropic Heisenberg spin model is adopted in the \ncalculations with a Hamiltonian given by \n, , ,\n22\n1/B FM i j AFM i j INT i j\ni j FM i j AFM i FM j AFM\nFM iz AFM iz\ni FM i AFM\nN\ni\niH k J S S J S S J S S\nK S K S\nhS   \n\n     \n\n  \n\n\n    (2) \nwhere \niS\n  are classical Heisenberg spins of unit magnitude placed at the nodes of a \nsimple cubic lattice.  The first row gives the exchange energy between spins located in \nFM core, AFM shell and FM -AFM interface with exchange coupling constants denoted \nby JFM, JAFM and JINT, respectively. The second row gives the local anisotropic energy \nfor each spin in FM core and AFM shell with the anisotropy constant represented by \nKFM and KAFM, respectively. The local anisotropy axes are set to be the z-direction  for \nall spins to impart a uniaxial anisotropy to the simulated systems. The last term \ndescribes the Zeeman coupling to an external field H applied along the easy -axis \ndirection, which in reduced units reads \n=/B h H k\n (with μ the magnetic moment of the \nspin) and will be denoted in temperature units48. \nTo calculate the magnetic properties, the MC method with a standard Metropolis \nalgorithm  is employed49. As for the spin updates,  an attempt to change the spin  at a \nrandomly picked site  i from \niS\n  to \niS\n  is made in a Monte Carlo trial step with the 6 \n acceptance rate given by  \n ( ) min 1,exp /i i BP S S E k T   \n   (3) \nwhere \nE  denotes the change in free energy of the system if \niS\n  is accepted. To get \nan optimum efficiency for the Heisenberg system with finite uniaxial anisotropies , a \ncombination of three kinds of trial steps, a uniform movement, a small movement,  and \na reflection, with a ratio of 3:1:1, is adopted50. In the uniform movement,  the direction \nof \niS\n is selected by  random sampling on a sphere with Marsaglia method51. In the small \nmovement, the direction of \niS\n   is selected by rando m sampling in a cone centered \nabout  \niS\n. A reflection movement,  where  the direction of \niS\n  is selected to be \niS\n , is \nincluded to simulate nucleation processes even more efficiently in the limit of very large \nanisotropy . \nAn MC step (MCS) is finished while every spin in the whole system has undergone a \ntrial step for once. To get the equilibrium state, at each field (or temperature) point, \n10000 MCSs are performed with 9800 MCSs for configuration relaxation and the \nremaining 200 MCSs for averaging the quantities , which is enough to minimize  the \nfluctuation in the data, especially  at low temperature.  To get more detailed \nmagnetization reversals around the coercive field, smaller field steps are used for the \nspin configuration calculatio n with keeping the total MCSs.  \nResults and discussion  \n1. The KFM dependence of exchange bias  \nSystems with different ferromagnetic anisotropy constants, KFM, are field cooled (FC)  \nfrom a high -temperature ( far above N éel temperature of AFM shell, TN) disordered 7 \n phase in a constant step down to the measur ing temperature T = 0.1 K in the presence \nof a cooling  field hFC = 0.4J0 applied along the easy -axis direction , with J0 = 10 K as a \nreference parameter . All the other parameters, the exchange coupling in FM core JFM = \nJ0, exchange coupling at the FM -AFM interface JINT = -0.5J0, exchange coupling in \nAFM shell JAFM = -0.5J0, anisotropy constant in AFM shell  KAFM = J0, were kept the \nsame within all systems, which was  targeted to give a larger Curie temperature  TC of \nFM core  than TN and a relatively large anisotropy of AFM part due to the ultrathin \nthickness  of the AFM shell28. The KFM/J0 is varied from 0 to 0.1, which is  in the  \nreasonable  range  for real ferromagnetic systems52. The temperature dependence of the \nnormalized  magnetizations  M/M S (with MS being the total number of spins in the \nnanoparticle) in core , shell , and interface in a system with KFM/J0 = 0.1 is given in FIG. \n1(a), where a paramagnetic to ferromagnetic transition is observed when temperature \ndecreases across  the Curie temperature ( TC ≈ 15 K) of the FM core and a \nparamagnetic to antiferromagnetic transition is observed when temperature decreases \nacross the Néel temperature ( TN ≈ 6.5 K) of the AFM shell. Due to the \nantiferromagnetic exchange coupling at the interface between the FM core and the \nAFM shell, the uncompensated interfacial  spins  give a negative net magnetization. \nFrom FIG. 1(b), the interfacial net magnetization  MINT remains nearly invariant with \nincreasi ng KFM at all temperatures, indicating that the spin configuration in the FM core \nis dominated by the exchange coupling JFM and the cooling field hFC.  \n 8 \n  \nFIG.  1. (a) The FC M-T curves of different parts of the core -shell structure  with \nKFM/J0 = 0.1. (b) The FC M-T curves of interfacial spins , (c) hysteresis loops after \nFC, and (d) extracted hE and hC obtained in core -shell structures with 0  ≤ KFM/J0 \n≤ 0.1. All the data in (d) are averaged with three independent calculations with \nerror bars coming from the calculated standard deviations.  \n   \nAfter the FC, hysteresis loop calculations are undertaken  for each system  with different \nKFM using the starting configuration  obtained with the FC process and by cycling the \nmagnetic field from h = 0.4J0 to h = −0.4J0 in steps h = −0.005J0. Integration of the \nmagnetization is carried out over the whole system. As shown in FIG. 1(c), the \nhysteresis loops change significantly with the increasing uniaxial anisotropy constant \n9 \n of the FM core. As expected, a larger KFM unambiguously gives a larger coercivity in \nthe hysteresis loop where nearly zero coercive fields were obtained with KFM = 0 with \nthe hard-axis switching characteristi cs presented , showing a progressive approach to \nboth positive and negative saturation , due to the spin -flop coupling between  FM spins  \nand those compensated AFM spins  at the interface53,54,55. As the KFM increases, the \ninduced anisotropy perpendicular to z-axis is overwhelmed by the uniaxial anisotropy \nof the FM core itself, showing a sharper magnetization switching in both sides and a \nsignificantly enhanced coercivity. However, as shown FIG. 1(d), it is found  that the \ndependence of the coercivity hC [defined as  hC = (hCR-hCL)/2 where hCR and hCL are the \nleft coercive field and right coercive field , respectively] on the KFM is not linear.  \nMoreover, the exchange bias fields hE [defined as hE = (hCR+hCL)/2] also show s a \nmonotonic increase with increasing KFM which violates the result predicted by M -B \nmodel  where the exchange bias field only depends on the interfacial exchange coupling \nenergy σex ~ JINTMINT and the total magnetization of the FM part tFMMFM. Since  both \nJINT and tFM are invariant with KFM, to reveal the underlying origin  of this effect, \nconstrained  MC calculations  are undertaken , in which the AFM spins are fixed  in the \nhysteresis loop calculation s after the same FC process with the non-constrained  MC \ncalculations. Thus,  the effect of the FM core behavior on the exchange bias can be \nstudied separately .  10 \n  \nFIG. 2 . (a) The h ysteresis loops calculated with constrained MC  and (b) the \nextracted hE, hC in core -shell structures  with 0≤KFM/J0≤0.1 after FC. All the data \nin (b) are averaged with three independent calculations with error bars coming \nfrom the calculated standard deviations.  \n \nAs shown in FIG. 2 , the hyste resis loops calculated with the constrain ed MC show  \nsimilar KFM dependence with those obtained with non -constrained MC method \nespecially when the KFM is small, where the hysteresis also shows a hard -axis like \nmagnetization switching coming from the spin -flop coupling.  However, with higher \nKFM, the hysteresis shows hig her asymmetry with sharper magnetization switching  in \nthe descending  branch  than the one o btained with non -constrained MC ; this effect being  \nascribed  to the rigidness of the interfacial AFM spins  in the constrained MC. Meanwhile, \nboth hC and hE given  in FIG. 2(b) are larger than those obtained with non -constrained \nMC, indicating stronger pinning effect of the constrained AFM magnetic moments. \nFurther , it is worth noting that t he KFM dependence of hE and hC shows similar behavior \nwith those obtained from non -constrained MC calculation with monotonic dependence \nwith KFM. An increment of 64.6% in hE is obtained in the hysteresis loop with KFM/J0 = \n11 \n 0.1 compared to that with KFM/J0 = 0, which is even a little larger than the result  51.4%  \nobtained in non -constrained MC.  \nSince the AFM spins  are fixed in the constrained MC, it is demonstrate d that the \nmonotonic increase of hE and the non -linear increase of hC with the increasing KFM is \ncontributed by  the FM core. This can be corroborated by direct inspection  of the spin \nconfigurations along the loops, as presented in the main panel of Fig. 3 for KFM/J0 = 0. \nAs it is evidenced by the sequence of snapshots, the reversal proceeds by quasi -uniform \nrotation along both descending and ascending branches at magnetic fields around left \nand right coercive fields , respectively . The hysteresis loop shows different app roaching \nbehaviors  to the two saturation directions, although both  are reversible. T he progressive \napproac hing to negative saturation has been proven  to originate from a planar  domain \nwall formed parallel to the FM/AFM interface56,57. As shown in FIG. 3(c) and (d), t his \ndomain wall is also observed with a spherical shape in the core-shell nanoparticle  where  \nthe spins  close to core center  reverse before those close to the core-shell interface in the \ndescending  branch (FIG. 3(c)).  While in the approaching to positive  saturation  (FIG. \n3(a) and (f)) , all the spins in the co re rotate coherently without  formation of the domain  \nwall.  12 \n  \nFIG. 3 . Snapshots of s pin configurations during magnetization reversals around \nthe left coercive field (a -c) and right coercive field (d -f) in the system with JFM/J0 \n= 1 and KFM/J0 = 0, calculated with constrained MC . The color  of the arrow \nindicat es the magnitude of  the z component  of each spin.   \n \nFor comparison, spin configurations of the nanoparticle with FM anisotropic constant \nof KFM/J0 = 0.1 are inspected . As shown in FIG. 4, the magnetization  reversal along the \ndescending branch proceeds first with  quasi -uniform rotation  and then with a fast \npropagation of planar domain wall nucleat ed at on e point of the interface , while the \nnucleation of reversed domains at the whole interface and its subsequent slow shrink  \nacross  the core center is the major rever sal process  along the ascending branch, \nresulting in an asymmetric characteristic in the hysteresis loop . Similar asymmetry in \nhysteresis loops also has been observed experimentally in discontinuous \nnanostructure41,2. The asymmetric magnetization reversals here have  similar features \nSz1\n0\n-1\nxz\nR. Wu et al. Figure 313 \n but different mechanisms from those obtained in continuous films , where the domain \nwall motion occurs in the descending branch while domain rotation occurs in ascending \nbranch58,59, originating from a biaxial magnetic anisotropy in the AFM part60.  \n \nFIG. 4 . Snapshots of s pin configurations during magnetization reversals around \nthe left coercive field (a -c) and right coercive field (d -f) in the system with JFM/J0 \n= 1 and KFM/J0 = 0.1, calculated with constrained MC . The color of the arrow \nindicat es the magnitude of  the z component  of each spin.  \n \nHere,  it is demonstrated that  the exchange bias field is strongly  correlat ed with  special \nreversal mechanism  in the core-shell nanoparticle . First,  \nR\n, the curl  of the spin  vector \nfield \n ,,x y zS S S S\n , is used to describe the  non-colline arity of  spin configuration  in the \nFM core, which reads   \nSz1\n0\n-1\nxz\nR. Wu et al. Figure 414 \n \n=x y z\nyy xx zzR S R i R j R k\nSS SS SSi j ky z z x x y  \n                         \n    (4) \nwith Rx, Ry and Rz represent ing three components of  local curls . The differentials are \ncalculated with finite difference method.  The vortex -like local spin configurations will \nyield  non-zero local curls while the collinear spin configurations will give zero local \ncurls. The overall  magnitude  of the microscopic (local) curls can be given as  \n2 2 2\nmicro ix iy iz\ni FMC R R R\n  \n(5) \nwhich enable us to get an insight in to the noncollinearity of  FM core while t he \nmagnitude of  macroscopic  (global) curls can be represented as  \n2 2 2\n222macro ix iy iz\ni FM i FM i FM\nx y zC R R R\nCCC                   \n    \n(6) \nwhich  enable s us to investigate the evolution of  macroscopic curling while the \norientation of  macroscopic  curling can be obtained with its components  \nxC, \nyCand\nzC\n.  \nAs shown in FIG. 5, within a core with KFM/J0 = 0, both  overall microscopic curling \nmicroC\n and macroscopic curling \nmacroC  show very small deviations at all fields from  \nsaturation state s, which confirms  magnetization reversals in t he core are nearly coherent \nin both branches. However, it is worth noting that th ere is a significant shoulder at  the \nleft side  of each coercive field in\nmicroC , which is absent in\nmacroC , while both \nmicroC  and \nmacroC\n show two peaks at  coercive fields . From  spin configurations given in FIG. 3  (c) \nand (d) , the shoulders in \nmicroC  are related to  the formati on of spherical domain wall s \nin these  field region s. In the spherical domain wall , the local curl at one point is opposite 15 \n to that at its symmetric point , giving zero contribution to th e macroscopic curl. Thus, \nin macroscopic curls, two peaks  without shoulders  around coercive field s are obtained , \nwhich are also present in microscopic curl s. As KFM/J0 increases, the right peak shows \na monotonic increase while the left peak nearly does not change. From FIG. 4 (b) and \n(f), it can be  infer red that peak s at the left coercive field s and the right coercive field s \nare relate d to a planar domain wall and an incomplete spherical domain wall , \nrespectively . The peak shoulder  in \nmicroC  , which is related to a complete spherical \ndomain wall , maintains  in ascendi ng branches but decreases in  descending branch es \nand finally  disappears in the system with KFM/J0 = 0.1, showing asymmetric \nmagnetization reversal in the two branches . Three components of  macroscopic curl s \nshow similar dependence to  KFM/J0. Cx and Cy always follow  each other  due to the \nrotation symmetry of the considered systems in the x-y plane . With increasing KFM/J0, \nboth Cx and Cy peaks increase monotonically in descending branches but  keep nearly \ninvariant  in ascending branches . Differently, Cz only occurs in  ascending branch es in \nsystems with KFM/J0 > 0.06, indicating the emerging of a curling in the x-y plane .  16 \n  \nFIG. 5.  The overall microscopic curls (left column), the macroscopic curls (middle \ncolumn) and three different components of macroscopic curls (right column) of \nsystems  with JFM/J0 = 1 and 0  ≤ KFM/J0 ≤ 0.1. \n \n2. Effect of exchange strength JFM \nThe anomalous dependence of hE and hC on KFM may originate  from  this asymmetric \nmagnetization reversal behavior, which is related to different  domain structures formed \nin descending and ascending branches . The formation of a spherical domain wall  in the \nascending branch of the hysteresis loop can effectively reduce the increment of right \ncoercive field caused by increasing KFM and consequ ently increases  hE (as shown in \nFIG. 2).  For a classic approximation , the domain wall width of a ferromagnetic material \nis determined by the competition between exchange and effective anisotropic energy, \nR. Wu et al. Figure 517 \n which is given by  \nFM\nw\nFMJ A\nKK  \n  (7) \nwhere A = nS2JFM /a = J FM (with n=1 for simple cubic structure , S = 1, a = 1 for \nconsidered  system s) is the exchange stiffness constant and K = KFM is the  anisotropy \nconstant of the material.  \nThe KFM and JFM dependences of \nw   are plotted in FIG. 6. For a given JFM, \nw\ndecreases sharply at the beginning and  then gradually in  the end with the increasing \nKFM. For a given KFM, a smaller \nw  is obtained with a small  JFM than that obtained \nwith a large  JFM. Consequently , given a smaller JFM and a larger KFM, the \nw  will be \nsmall enough to enable domain wall  formation  in the FM core  with a diameter of 18 a. \nAlso, the domain wall in FM core  will be suppressed  with larger JFM and smaller KFM. \n \nFIG. 6 . The dependence  of domain wall width on KFM and JFM calculated from \nEqn. ( 7) with  a grey dashed line indicating the diameter of the FM core . \n18 \n  \nTo verify this hypothesis, KFM dependence of exchange bias in this system with varying \nJFM is studied . In all the calculations, AFM spins are constrained. As sho wn in FIG. 7(a) , \nthe exchange bias field hE shows a very sharp increase with increasing KFM with a small \nferromagnetic exchange coupling JFM/J0 = 0.5. This monotonic dependence of hE on \nKFM maintains with increasing JFM up to JFM/J0 = 2 and finally disappears in the system \nwith JFM/J0 = 4, where hE shows no obvious dependence on KFM. The effect of  JFM is \nmore prominent in the relative increment of exchange bias field , δhE/hE0, where hE0 and \nδhE are the hE at KFM/J0 = 0 and the increment of hE relative to hE0 at KFM/J0 ≠ 0. As \nshown in FIG. 7(c) , δhE/hE0 shows a very sharp increase with increasing KFM in a system \nwith JFM/J0 = 0.5. The increase is largely reduced in systems with larger JFM. Finally, a \nnearly zero increment  in hE is obtained with increasing KFM in the  system with JFM/J0 = \n4.  \n Meanwhile, hC also show s a strong  dependence on  both KFM and JFM. As shown in \nFIG. 7(b), with a small JFM, the system shows superparamagnetic characteristic with \nnearly zero  hC0 (the hC at KFM/J0 = 0). As the JFM increase s, hC0 shows a monotonic \nincrease , indicating  an increasing  magnetic anisotropy given  by the exchange coupling \nat the core -shell interface. Consequently, the relative increment of coercivity, δhC/hC0, \nwhere δhC is the increment of hC relative to hC0 at KFM/J0 ≠ 0, increases with KFM but \ndecreases with JFM, as shown in FIG. 7(d). Moreo ver, it is found that the way in which  \nhC depends on KFM varies with JFM significantly. When  JFM is small,  hC show s a \nnonlinear dependence on increasing KFM, with a gradual increase at lower KFM and a 19 \n steeper increase at higher KFM. However, when JFM increases, the nonlinearity of the \ndependence is reduced and , finally , becomes  a linear dependence in the system with \nJFM/J0 = 4.  \n \n \nFIG. 7.  The KFM dependence of (a) hE, (b) hC, (c) relative change of hE and ( d) \nrelative change of hC plots with different JFM. All the data are averaged with three \nindependent calculations  with error bars coming from the calculated standard  \ndeviations . \n \nAn invariant hE and a linear dependent hC on KFM is exactly the results predicted by M -\nB single spin model, which is absent in system with small JFM and presents in system \n20 \n with large JFM, indicating  an evolution of the spin configuration from non -collinear to \ncollinear during  magnetization reversals as JFM increases , which is verified by an \ninspection of spin configurations and overall microscopic curls of the systems with \ndifferent JFM during  magnetization reversal s. \n \nIt can be seen from  the first column of FIG. 8 , the planar domain wall at the left coercive \nfield shows strong  dependence with the increasing JFM. The planar domain wall with a \nsmall width is very significant in a system with small JFM, and becomes  weaker with a \nlarger domain wall width as JFM increases . A collinear alignment of core spins and  \ndecreased contrast in color map of Sz are observed in the system with JFM/J0 = 4.0, as \nshown in FIG. 8(j) . The spherical domain wall  at the right coercive field shows similar \nJFM dependence as the planar domain wall, as shown in middle column of FIG. 8, which \nbecome s weaker and broader with increasing JFM and nearly disappears in the system \nwith JFM/J0 = 4.0. The evolution of  domain structure with JFM is also reflected in overall  \nmicroscopic curls (FIG. 8, right c olumn). As shown in  FIG. 8 (c), overall microscopic \ncurls in the system with JFM/J0 = 0.5 are very large with contributions  including a large \nbackground coming from the random thermal fluctuation, two peaks from planar \ndomain wall and incomplete spherical domain wall  at left coercive field and right \ncoercive field, respectively, and broad shoulders from the complete spherical domain \nwalls. With an increased JFM/J0, as shown in FIG. 8 (f) and 8(i) , overall microscopic \ncurls are lowered significantly, which is in good agreement with the spin configurations. \nMeanwhile, the background is also reduced largely, which is ascribed to th e effectively 21 \n suppressed thermal fluctuations by the large exchange coupling. Finally,  as shown in \nFIG. 8(l),  with the largest JFM/J0 of 4.0, both the peaks and the background are largely \nreduced  corresponding to nearly collinear spin configurations  during the magnetization \nreversals.  \n \nFIG. 8. Spin configuration snapshots of systems with the same KFM/J0 = 0.1 but \nwith 0.5 ≤ JFM /J0 ≤ 4.0 taken at left coercive fields ( left column ) and right \ncoercive fields (middle column) and overall microscopic curls as functions of  the \nmagnetic  field in systems with different JFM, calculated with constrained MC . The \ncolor of the arrow s in the spin configuration snapshots indicates the magnitude of \nthe z -component of each spin.   \nSz\n1\n0\n-1\nxz\nR. Wu et al. Figure 822 \n  \nFor a realistic material consideration, typical domain wall widths of  Fe, Co, and Ni \nnanoparticles are around 138 nm, 36 nm, and 285 nm, respectively61 . However, the  \nmagnetic vortex state has b een observed in Fe nanoparticle  with a size of 26  nm43, \nindicating the noncollinear magnetic configuration can be obtained in magnetic \nnanoparticles much smaller than the bulk domain wall width. In harder magnetic \nmaterials,  much  smaller domain wall width can be obtained. For instance, domain wall \nwidths  for CoFe 2O4 and Nd 2Fe14B are about 8  nm62  and 5 nm63 , respectively . An \nincomplete spherical domain wall  can exist in a nanoparticle  around  this length scale , \nwhich  can be easily manipulated with size controlling and composition tailoring  to give \noptimized exchange bias effect and other magnetic properties .  \nConclusions  \nTo conclude, the effect of FM spin configuration on the exchange bias eff ect of \nFM/AFM core -shell nanoparticles has been studied with MC method. A significant \nenhancement of the exchange bias effect accompanied by a nonlinear behavior of \ncoercivity with increasing magnetic anisotropy constant KFM has been observed , \nshowing a violation of classic M -B model . This anomalous effect  is ascribed  to the \nasymmetr ic magnetization reversal in  the FM core with a spherical domain wall  \nformation in the ascending branch of the hysteresis loop, which largely reduces the right \ncoercive field and enhances the exchange bias field . This is demonstrated by adjust ing \nthe domain wall width in the FM core  with varying JFM and KFM. Finally, the anomalous \ndependence of hE and hC on KFM disappears when the domain wall in the core is 23 \n suppressed . The  results provide another freedom to tailor the exchange bias in the \nFM/AFM systems.  \nAcknowledgements  \nThis work is supported by the National Key Research and Development Program \nof China ( No. 2017YFA0206303, 2016YFB0700901 and 2017YFA0401502 ) and \nNational N atural Science Foundation of China (Grant Nos. 51731001 , 51371009, \n11504348, 11675006), the Ph.D. Programs Foundation of Ministry of Education of \nChina (No. 20130001110002).  \n* Corresponding author: jbyang@pku.edu.cn and rw556@cam.ac.uk  \nReferences  \n[1] W. H. Meiklejohn, and C. P. Bean, Phys. Rev.  102, 1413  (1956 ). \n[2] K. Liu, S. M. Baker, M.  Tuominen, T. P. Russell , and  I. K. Schuller, Phys. Rev. B  \n63, 060403  (2001 ). \n[3] W. Zhang , A. Chen,  J. Jian, Y. Zhu, L. 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"    },    {        "title": "2301.11027v3.Anisotropic_spin_current_spectroscopy_of_ferromagnetic_superconducting_gap_symmetries.pdf",        "content": "Anisotropic Spin-Current Spectroscopy of Ferromagnetic Superconducting Gap\nSymmetries\nHiroshi Funaki,1Ai Yamakage,2and Mamoru Matsuo1, 3, 4, 5\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n4RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n(Dated: May 9, 2023)\nWe develop a microscopic theory of tunneling spin transport at the magnetic interface between\na ferromagnetic insulator (FI) and a ferromagnetic superconductor (FSC) driven by ferromagnetic\nresonance. We show that the spin susceptibilities of the FSC can be extracted from the spin currents\nby tuning the easy axis of the FI, and thus the spin currents can be a probe for the symmetries of\nthe spin-triplet Cooper pairing. Our results will o\u000ber a route to exploiting the synergy of magnetism\nand superconductivities for spin devices.\nI. INTRODUCTION\nTunneling spin current in magnetic heterostructures\ndriven by magnetization dynamics using ferromagnetic\nresonance (FMR) has been studied intensively in spin-\ntronics. It is widely known as the spin pumping e\u000bect[1],\na versatile way to generate the spin current in nanohy-\nbrid systems from a ferromagnet into various conducting\nmaterials. Recently, the spin pumping has been recog-\nnized as a quantum probe to detect magnetic proper-\nties of thin \flms[2] because the generated spin current\ncan be measured very sensitively even for nano-scale thin\n\flms[3]. From a theoretical point of view, the spin cur-\nrent re\rects the spin susceptibility of adjacent materi-\nals [4]. This property has a signi\fcant impact on su-\nperconducting spintronics research[5], where a variety of\nconversions between Cooper pair supercurrents and spin\ncurrents have been intensively studied including triplet\nCooper pair currents[6]. In particular, the tunneling spin\ncurrent can be utilized as a direct probe of spin excita-\ntions in the Cooper pair symmetries of conventional SCs\n[7{20] and unconventional SCs [21{24].\nSymmetry of the Cooper pair characterizes the na-\nture of superconductors [25]. In particular, spin-triplet\nCooper pairs o\u000ber a fascinating state from the viewpoint\nof superconducting spintronics, because they can carry\nspin one as a supercurrent without dissipation.3He is\na well-established spin-triplet super\ruid with (breaking)\ntime-reversal symmetry in the B (A) phase. On the\nother hand, almost all existing superconductors are spin-\nsinglet rather than spin-triplet superconductors. Estab-\nlishing spin-triplet candidates in superconductors is an\nessential issue in condensed matter physics. Indeed, the\nsearch for spin-triplet superconductors continues vigor-\nously [26{30]. Among superconductors, it is also believed\nthat the spin-triplet pair is likely formed in the ferro-\nmagnetic superconducting state in uranium compounds\n[31{33], UGe 2[34], UIr [35], URhGe [36], and UCoGe\n[37]. However, despite many years of research, no de\fni-\n－ +＋\n－FIG. 1. (a) Schematic diagram of the spin pumping e\u000bect\nat a junction system of a ferromagnetic insulator (FI) and a\nferromagnetic superconductor (FSC). The tunneling spin cur-\nrentISis generated at the interface driven by magnetization\ndynamics due to microwave irradiation in FI. (b) The spin\npolarization of the generated spin current can be controlled\nby tuning the easy axis of the magnetization in FI. The z-\npolarized spin current Iz\nSand thex-polarized one Ix\nSre\rect\nthe magnetic properties of the FSC characterized by the spin\nsusceptibilities \u001f?and\u001fk. The transverse spin susceptibil-\nity\u001f?can be extracted from the z-polarized spin current Iz\nS\nwhile the longitudinal one \u001fkfrom 2Ix\nS\u0000Iz\nS, where\u001f?is the\ncorrelation between the majority spin and the minority spin\nand\u001fkconsists of the spin susceptibility of the majority spins\n\u001f\"and that of the minority spins \u001f#. (c) The superconducting\ngaps we consider in this paper.\ntive conclusions have been reached on its superconduct-\ning symmetry, such as the gap node and d-vector con-\n\fguration. A complete characterization of the properties\nof ferromagnetic superconductivity will provide the basis\nfor further understanding the physics of superconductiv-\nity and its development into anisotropic superconductingarXiv:2301.11027v3  [cond-mat.supr-con]  6 May 20232\nspintronics. Moreover, it has been proposed that these\nferromagnetic superconductors may exhibit gapless sur-\nface states characterized by the Z4topological invariant\nunder high pressure if a speci\fc superconducting symme-\ntry is realized [38]. Determining the symmetry of their\nparent state, ferromagnetic superconductivity, is one of\nthe most fundamental issues for studying topological ma-\nterials.\nIn this paper, we propose a spectroscopy of ferromag-\nnetic superconducting gap symmetries of FSC thin \flms\nby using the tunnel spin current at a magnetic interface\nexcited by FMR. To this end, we consider the tunnel-\ning spin transport at magnetic interface between a fer-\nromagnetic insulator (FI) and a ferromagnetic supercon-\nductor (FSC) driven by FMR as shown in Fig. 1. We\ndevelop a microscopic theory of the spin current gener-\nation, caused by the di\u000berences of the nonequilibrium\ndistribution functions of the FI and the FSC. We show\nthat we can generate several types of tunneling spin cur-\nrent, including Iz\nSandIx\nSby tuning the relative angle\nbetween the easy axis of the magnetization of the FI and\nthat of the FSC. We \fnd that the spin susceptibilities\nof the FSC can be extracted from the spin currents and\npropose a method to determine the symmetries of the\nspin-triplet Cooper pairing by using the inverse spin Hall\nvoltage measurements. Our results will o\u000ber a route to\nexploiting the synergy of magnetism and superconduc-\ntivities for spin devices.\nII. MODEL\nLet us consider a model of tunneling spin transport at\nthe magnetic interface consisting of a FI and a FSC aim-\ning at extracting the magnetic properties of the FSC from\nthe generated spin currents as shown in Fig. 1. The gen-\nerated spin current can be calculated by the spin tunnel-\ning Hamiltonian method [4, 8, 23, 24]. The total Hamil-\ntonianHconsists of the three terms:\nH=HFSC+HFI+Hex: (1)\nThe \frst termHFSCis the mean \feld Hamiltonian of the\nbulk FSC:\nHFSC=1\n2X\nkcy\nkHBdGck; (2)\nwhere the fermion operator is de\fned by ck=\n(ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)Tin the Nambu space for k=\n(kx;ky;kz). The Bogoliubov-de Gennes (BdG) Hamil-\ntonianHBdG for a spin-triplet superconductor with an\nequal-spin pairing is given by\nHBdG=0\nBB@\u0018k\" 0 \u0001 k\"\" 0\n0\u0018k# 0 \u0001 k##\n\u0000\u0001\u0003\n\u0000k\"\" 0\u0000\u0018\u0000k\" 0\n0\u0000\u0001\u0003\n\u0000k## 0\u0000\u0018\u0000k#1\nCCA;(3)where\u0018k\u001b=~2k2\n2m\u0000\"F\u0000\u001b\u0001FMis the energy dispersion\nof spin up [ \u001b=\"(+)] and down [ \u001b=#(\u0000)] electron in a\nferromagnet in the normal state, and \"Fand \u0001 FMare the\nFermi energy and the spin-splitting energy, respectively.\nHere we focus on a single-band superconductor in order\nto elucidate properties intrinsic to ferromagnetic super-\nconductivity, especially to gapless excitations of quasi-\nparticles and nonunitarity of breaking time-reversal sym-\nmetry. For simplicity we here neglect the collective ex-\ncitations in the FSC and assume that the spin-splitting\nenergy \u0001 FMis the constant and su\u000eciently larger than\nthe superconducting gap, where only the spin-up band\nis superconducting, i.e., \u0001 k##= 0. This assumption is\nappropriate for examining the key characteristics of non-\nunitary superconductivity, a leading candidate for the\norder parameter in FSC. In particular, we consider two\ntypes of the pair potential\n\u0001k\"\"=\u0000\u00010sin\u0012ke\u0000i\u001ek;for axial; (4)\n\u0001k\"\"=\u0000\u00010cos\u0012k;for polar; (5)\nas shown in Fig. 1 (c). Note that the above pair potentials\nare of generic form within the single-band p-wave equal-\nspin-pairing superconductivity. The axial gap, Auirrep\nof the magnetic point group 4 =mm0m0[39{43], has point\nnodes on the north and south poles while the polar gap,\nEuirrep, has a line node on the equator. This di\u000berence\nin the excitation gap is re\rected in the spin excitations\ninvolved in FMR.\nThe second term HFIdescribes the bulk FI given by\nHFI=\u0000JX\nhi;jiSi\u0001Sj\n\u0000~\rX\nih\nhdcSz\ni+hac(cos\ntSx\ni\u0000sin\ntSy\ni)i\n;(6)\nwhere Siis the magnetization at the site iin the FI,\nJis the exchange interaction, hdcis a static magnetic\n\feld,hacand \n are the amplitude and frequency of the\napplied microwave radiation, respectively, and \ris the\ngyromagnetic ratio.\nThe third termHexis the interfacial exchange coupling\nwhich describes the spin transfer between the FI and the\nFSC:\nHex=TX\nk;qSk\u0001sq; (7)\nwhereTis the tunneling amplitude between the magne-\ntization in the FI Skand the conduction electron spin in\nthe FSC sq. We assumed a constant tunneling amplitude\ncorresponding to a rough interface limit.\nIII. TUNNELING SPIN CURRENTS\nThe tunneling spin current at the interface driven by\nFMRh^Ii\nSiis calculated by the statistical average of the3\nV V\nFIG. 2. Schematic diagram of measuring spin currents Iz\nS\n(a) andIx\nS(b) by the inverse spin Hall e\u000bect (ISHE). The\nspin current is converted to a voltage and measured. The\nconversion results from the ISHE in a heavy metal (HM) with\nstrong spin-orbit interaction. The spin current at the interface\nbetween the FI and the FSC can be measured almost directly,\nespecially when the FSC is thin.\nspin current operator ^Ii\nSde\fned by\n^Ii\nS=\u0000~@t(si\ntot) =i[si\ntot;H] =\u0000X\nk;q\u000fijkTSj\nksk\nq;(8)\nwheresi\ntot=si\nq=0. We calculate the statistical average\nh^Ii\nSiusing the Schwinger-Keldysh approach. By taking\ninto account the second-order perturbation of the interfa-\ncial exchange coupling Hexand assuming that the Fermi\nenergy is su\u000eciently larger than the spin splitting energy\nin FSC, we obtain the relations between the generated\nspin currents and the dynamic spin susceptibilities of the\nFSC (see the Appendices A and B for the detailed deriva-\ntion):\nIz\nS:=h^Iz\nSi=T2\nSIm\u001fR;?\nloc;\n\u0001n\nImGR\n0;\n; (9)\nIx\nS:=h^Ix\nSi=T2\n2Sh\nIm\u001fR;?\nloc;\n+ Im\u001fR;k\nloc;\ni\n\u0001n\nImGR\n0;\n;\n(10)\nwhere\u001fR;?\nloc;\nand\u001fR;k\nloc;\nare transverse and longitudi-\nnal components of the local spin susceptibility in the\nFSC, respectively, and Im GR\n0;\nis spin susceptibility at\nk=0in the FI, and \u0001n\nis a change of magnon num-\nber due to microwave irradiation. Using sxand spin-\ndependent electron number n\u001b,\u001fR;?\nloc;\nand\u001fR;k\nloc;\nare\ndescribed as \u001fR;?\nloc;\n=iR\ndtP\nq\n[sx\nq(t);sx\n\u0000q(0)]\u000b\n\u0012(t)ei\nt\nand\u001fR;k\nloc;\n= (\u001fR;\"\nloc;\n+\u001fR;#\nloc;\n)=4 with\u001fR;\u001b\nloc;\n=\niR\ndtP\nqh[nq\u001b(t);n\u0000q\u001b(0)]i\u0012(t)ei\nt, respectively. Equa-\ntions (9) and (10) indicate that the local spin suscep-\ntibilities can be extracted from the spin currents as\nIm\u001fR;?\nloc/Iz\nSand Im\u001fR;k\nloc/2Ix\nS\u0000Iz\nS(see Fig.1 (b)). The\nspin polarization of the spin currents can be controlled\nby tuning the relative angle between the easy axis of the\nmagnetization of the FI and that of the FSC. Therefore,\nwe can systematically identify the spin susceptibilities of\nthe FSCs by measuring the spin currents and combining\ntheir frequency dependencies, as shown in Sec. IV.\nWe consider using the inversion spin Hall e\u000bect (ISHE)\nto measure the generated spin currents at the magneticinterface. Our measurement setup is shown in Fig. 2,\nwhere the heavy metal (HM) with strong spin-orbit in-\nteraction (SOI), such as Pt, is attached to the FSC thin\n\flm as the spin-current detector. Here we assume the\nFSC is su\u000eciently thin to avoid bulk spin scattering pro-\ncesses in the FSC for simplicity. The generated spin cur-\nrent between the FI and the FSC \rows into the HM and\nis converted into an inverse spin Hall voltage due to the\nstrong SOI. In particular, the z-polarized spin current Iz\nS\ngenerated when the easy axis of the FI is parallel to the\nz-axis can be measured by the setup (a) in Fig. 2, while\nthex-polarized spin current Ix\nSby (b). Note that we can\nobtain su\u000ecient information to identify the pairing sym-\nmetries of the FSC from Iz\nSandIx\nSas discussed below.\nNamely, the y-polarized spin current, which cannot be\nmeasured in our setup, is unnecessary to determine the\nsymmetries.\nIV. FREQUENCY DEPENDENCIES OF THE\nSPIN SUSCEPTIBILITIES\nFigure 3 shows the numerical results of the spin suscep-\ntibilities. We focus on the frequency dependence of the\nspin susceptibility when the temperature is lower than\nthe frequency of the applied microwave radiation, thus\nthe temperature can be approximated as zero. The fre-\nquency dependencies of the imaginary part of the local\nspin susceptibilities in the axial type FSC Im \u001fR;?\nloc(a) and\nIm\u001fR;k\nloc(b) and those in the polar type FSC (c, d) at\nzero temperature represented on a log-log scale, where\nthe spin susceptibilities are normalized as Im\u0016 \u001fR;\u000b\nloc=\nIm\u001fR;\u000b\nloc;\n=Im\u001fR;?\nloc;N;\n=\u0001 0=~(\u000b=?;k;\";#). Here Im\u001fR;?\nloc;N\nis the spin susceptibility in the normal state, and the fre-\nquency is normalized by \u0001 0. The susceptibility Im\u0016 \u001fR;\u000b\nloc\ndoes not depend on the spin-splitting energy \u0001 FMap-\nproximately because the Fermi energy \"Fis su\u000eciently\nlarger than \u0001 FM. Thus, the normalized spin susceptibil-\nity has only one parameter, ~\n=\u00010. It is remarkable that\nthe spin susceptibilities show the characteristic power-law\nfrequency dependencies. The transverse spin susceptibil-\nity in the axial superconducting state Im \u001fR;?\nlocis propor-\ntional to \n3while that in the polar state to \n2in the low\nfrequency region ~\n.\u00010, as indicated in the blue area\nin Figs. 2 (a) and (c). Such frequency dependencies can\nbe obtained from the analytical power expansions of the\nspin susceptibility (see the Appendix E for the detailed\nderivation). For instance, the power expansion of Im \u001fR;?\nloc\nin the axial superconducting state when ~\n.\u00010is given\nby\nIm\u0016\u001fR;?\nloc;\n\u0019(~\n)3\n3\u00013\n0: (11)\nIt should be noted that the dependence on \n3appears\nwhen~\n.\u00010, whereas it is proportional to \n when the\nfrequency becomes larger than the superconducting gap,\nwhere it no longer di\u000bers from the normal state.4\n－ +\n＋\n－\nFIG. 3. The frequency dependencies of the imaginary part of the local spin susceptibilities in the axial type FSC (a,b) and\nthose in the polar type FSC (c,d) at zero temperature. represented on a log-log scale. The characteristic power-law frequency\ndependencies (indicated in the blue area) originate from the symmetries of the superconducting gaps. Together with the fact\nthat Im\u001fR;?\nloc(a,c) is extracted from Iz\nSand Im\u001fR;k\nloc(b,d) from 2 Ix\nS\u0000Iz\nS(see also Fig. 1), we can identify the ferromagnetic\nsuperconducting gap symmetries from the tunneling spin currents.\nTABLE I. The characteristic power-law frequency dependen-\ncies of the transverse (Im \u001fR;?\nloc) and longitudinal (Im \u001fR;k\nloc=\n[Im\u001fR;\"\nloc+ Im\u001fR;#\nloc]=4) components of the imaginary part of\nthe local spin susceptibilities in various states. The suscepti-\nbilities in the anisotropic superconducting states indicate the\npower-law frequency dependencies in sharp contrast to the\nexponential dependence in the s-wave spin-singlet SC. In ad-\ndition to the axial and polar properties, the non-unitarity of\nthe FSC can be identi\fed from the frequency dependencies of\nthe susceptibilities.\nSC state Im \u001fR;?\nlocIm\u001fR;\"\nlocIm\u001fR;#\nloc\nFM p-wave (axial) non-unitary 3 5 1\nFM p-wave (polar) non-unitary 2 3 1\nFM p-wave (axial) unitary 5 5 5\nFM p-wave (polar) unitary 3 3 3\nd-wave (2D polar) singlet 3 3 3\ns-wave singlet exp. exp. exp.\nIn addition to the axial and polar features, the fre-\nquency dependencies in the low frequency region provide\ninformation on the non-unitarity of the FSCs. Due to\nthe anisotropy of the non-unitary pair, \u0001 \"\"6= 0 and\n\u0001##= 0, the spin excitations are also anisotropic, mak-\ning\u001fR;\"\nlocand\u001fR;#\nlocdi\u000berent. Moreover, \u001fR;?\nlocdi\u000bers from\n\u001fR;\"\nlocand\u001fR;#\nlocbecause it depends on both spin-up and\ndown bands. In particular, their exponents are also dif-ferent. They are given by Im \u001fR;?\nloc/\n3, Im\u001fR;\"\nloc/\n5,\nand Im\u001fR;#\nloc/\n for the axial non-unitary pair. On the\nother hand, no such anisotropy is observed for the uni-\ntary pair with \u0001 \"\"= \u0001##, and then all components of\nsusceptibility are proportional to the \ffth power of fre-\nquency. Thus, the measurement of spin excitations via\nspin currents by FMR is also helpful in measuring the\nnon-unitary nature of the Cooper pair.\nWe mention two points concerning the scope of our\nmodel. Firstly, our model does not account for the multi-\nband of FSC. Even when the multiband is considered,\nthe exponent of the frequency dependence remains un-\nchanged for intraband pairings because the node struc-\nture of the superconducting gap, which are the same as\nfor the single-band FSCs, primarily governs the expo-\nnent. When the interband paring is dominant, a di\u000ber-\nent exponent is expected to be observed. Secondly, our\nmodel does not consider the Andreev bound states be-\ncause our objective is to study the spin excitation in the\nbulk and not the Andreev bound states emerging on the\ninterface between the FM and FSC. They are gapless sur-\nface excitations in superconductors with gap nodes and\nhence can change the exponents discussed above. In the\ncase of line nodes, Andreev bound states form in the\nxy-plane and do not impact the exponents. In contrast,\npoint nodes create Andreev bound states along the xz\nplane, which in\ruences the exponents. In fact, a previ-\nous study has demonstrated that Andreev bound states5\nsigni\fcantly contribute to spin pumping in d-wave su-\nperconductors [44]. These issues will be left for future\nstudies.\nV. CONCLUSION\nIn this paper, we have developed a microscopic theory\nof the tunneling spin current at the magnetic interface\nof a FI and a FSC by microwave irradiation, aiming to\nidentify the ferromagnetic superconducting gap symme-\ntries. We obtained the relations between the tunneling\nspin currents and the dynamic spin susceptibilities of the\nFSCs, and found that the spin susceptibilities can be\nextracted from the spin currents by tuning the relative\nangle between the easy axis of the magnetization of the\nFI and that of the FSC. We revealed that the spin sus-\nceptibilities of the FSC indicate the characteristic power-\nlaw frequency dependencies re\recting the axial and polar\nproperties as well as the non-unitarity and unitarity of\nthe FSCs. Accordingly, the tunneling spin currents in our\nsetups can be a probe of the ferromagnetic superconduct-\ning gap symmetries by combining the tunability of their\nspin polarization and their frequency dependencies. Our\ntheory paves the way for ferromagnetic superconducting\nspintornics, where the synergy of magnetism and super-\nconductivities are exploited.\nACKNOWLEDGMENTS\nThe authors are grateful to Yuya Ominato for valu-\nable comments. This work was supported by the Pri-\nority Program of the Chinese Academy of Sciences un-\nder Grant No. XDB28000000, and by JSPS KAK-\nENHI for Grants (Nos. 20K03835, 20H04635, 20H01863,\n21H01800, 21H04565, and 23H01839) from MEXT,\nJapan.\nAppendix A: Spin currents at the interface of\nferromagnetic junctions\nIn this Section, we describe spin currents generated\nby spin pumping at the interface of ferromagnetic junc-\ntions. We derive the explicit expression of the spin\ncurrent of the second-order perturbation of the interfa-\ncial exchange coupling between a ferromagnetic insulator\n(FI) and a ferromagnetic superconductor (FSC) using the\nSchwinger-Keldysh approach.\nWe de\fne the spin operator in the FI, Sk, and that in\nthe FSC, sq, as\nSk=Sr\nker+S\u0012\nke\u0012+S\u001e\nke\u001e; (A1)\nS\u0006\nk=S\u0012\nk\u0006iS\u001e\nk; (A2)\nsq=sz\nqez+sx\nqex+sy\nqey; (A3)\ns\u0006\nq=sx\nq\u0006isy\nq; (A4)whereS\u0006\nkands\u0006\nq are ladder opera-\ntors, er = (sin\u0012cos\u001e;sin\u0012sin\u001e),e\u0012 =\n(cos\u0012cos\u001e;cos\u0012sin\u001e;\u0000sin\u0012),e\u001e= (\u0000sin\u001e;cos\u001e;0),\nez= (0;0;1),ex= (1;0;0), and ey= (0;1;0). Here,\neach quantization axis is chosen along each magnetic\neasy axis er= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012) for the FI\nandez= (0;0;1) for the FSC, respectively.\nThe interfacial exchange coupling between the FI and\nthe FSC is\nHex=TX\nk;qSk\u0001sq=TX\nk;qB\u000b\fS\u000b\nks\f\nq; (A5)\nwhereTis the strength of the interfacial exchange cou-\npling andB\u000b\f=e\u000b\u0001e\f. The spin-current operator is\nde\fned by\n^Ii\nS=\u0000~@t(si\ntot) =i[si\ntot;H] =\u0000X\nk;q\u000fijkTSj\nksk\nq\n=\u0000TX\nk;qAi\n\u000b\fS\u000b\nks\f\nq; (A6)\nwheresi\ntot=si\nq=0andAi\n\u000b\f=ei\u0001(e\u000b\u0002e\f).\nUsing the Schwinger-Keldysh approach and taking into\naccount the second-order perturbation of T, the statisti-\ncal average of the spin current is\nh^Ii\nSi=\u0000T\n2X\nk;qAi\n\u000b\fhTC[s\f\nqS\u000b\nk]Ki\n=\u0000T\n2X\nk;qh\nAi\n\u000b\fhS\u000b\nkihs\f\nqi\n+iTZd!\n2\u0019Ai\n\u000b\fB\u000b0\f0[\u001f\f\f0\nFSC;q;!\u001f\u000b\u000b0\nFI;k;!]Ki\n;(A7)\nwhereTCis the time-ordering operator on the Keldysh\ncontour and spin susceptibilities are de\fned as\n\u001f\u000b\u000b0\nFI;k;!=iZ\nCd(\u001c1\u0000\u001c2)\u0002\nhTC[S\u000b\nk(\u001c1)S\u000b0\n\u0000k(\u001c2)]i\n\u0000ihS\u000b\n0(\u001c1)ihS\u000b\n0(\u001c2)i\u0003\nei!(\u001c1\u0000\u001c2); (A8)\n\u001f\f\f0\nFSC;q;!=iZ\nCd(\u001c1\u0000\u001c2)\u0002\nhTC[s\f\nq(\u001c1)s\f0\n\u0000q(\u001c2)]i\n\u0000ihs\f\n0(\u001c1)ihs\f\n0(\u001c2)i\u0003\nei!(\u001c1\u0000\u001c2); (A9)\nwhereR\nCmeans integral on the Keldysh contour. The\nspin susceptibilities are detailed in Appendices C and D.\nThe integrand of the last term in Eq. (A7) can be ex-\npanded with the Langreth rule as\n[\u001f\f\f0\nFSC;q;!\u001f\u000b\u000b0\nFI;k;!]K=\u001fR;\f\f0\nFSC;q;!\u001fK;\u000b\u000b0\nFI;k;!\n+\u001fK;\f\f0\nFSC;q;!\u001fA;\u000b\u000b0\nFI;k;!; (A10)6\nwhere each component is de\fned as\n\u001fK;\f\f0\nFSC;q;!=iZ1\n\u00001dt\u0002\nhs\u000b\nq(t)s\u000b0\n\u0000q(0) +s\u000b0\n\u0000q(0)s\u000b\nq(t)i\n\u00002hs\f\n0(t)ihs\f0\n0(0)i\u0003\nei!t; (A11)\n\u001fR;\f\f0\nFSC;q;!=iZ1\n\u00001dth[s\u000b\nq(t);s\u000b0\n\u0000q(0)]i\u0012(t)ei!t; (A12)\n\u001fK;\u000b\u000b0\nFI;k;!=iZ1\n\u00001dt\u0002\nhS\u000b\nk(t)S\u000b0\n\u0000k(0) +S\u000b0\n\u0000k(0)S\u000b\nk(t)i\n\u00002hS\u000b\n0(t)ihS\u000b0\n0(0)i\u0003\nei!t; (A13)\n\u001fA;\u000b\u000b0\nFI;k;!=\u0000iZ1\n\u00001dth[S\u000b\nk(t);S\u000b0\n\u0000k(0)]i\u0012(\u0000t)ei!t:(A14)\nHere, the Keldysh component of the Green's function is\nde\fned by\nGK\nA;B(q;!) =\niZ1\n\u00001dt\u0002\nhAq(t)B\u0000q(0) +B\u0000q(0)Aq(t)i\u0003\nei!t;(A15)\nwhereAandBare arbitrary bosonic operators. Fur-\nthermore, the Green's function becomes pure imag-\ninary after integrating over k;q, and!because\n([\u001f\f\f0\nFSC;q;!\u001f\u000b\u000b0\nFI;k;!]K)\u0003=\u0000[\u001f\f\f0\nFSC;\u0000q;\u0000!\u001f\u000b\u000b0\nFI;\u0000k;\u0000!]K. Ac-\ncordingly, we can rewrite the spin current as\nh^ISi=\u0000TX\nk;qh\n(er\u0002ez)hSr\nkihsz\nqi\n\u0000T\n2Zd!\n2\u0019n\n(er\u0002ez)h\n(er\u0001ez)\nIm[(\u001fzz\nFSC;q;!\u0000\u001fxx\nFSC;q;!)(\u001frr\nFI\u0000\u001f\u0012\u0012\nFI;k;!)]K\n+ Im[\u001fxy\nFSC;q;!\u001f\u0012\u001e\nFI;k;!]Ki\n+fer\u0000(er\u0001ez)ezg\nIm[(\u001fzz\nFSC;q;!+\u001fxx\nFSC;q;!)\u001f\u0012\u001e\nFI;k;!]K\n+ 2(er\u0001ez)ezIm[\u001fxx\nFSC;q;!\u001f\u0012\u001e\nFI;k;!]K\n\u0000fez\u0000(ez\u0001er)erg\nIm[\u001fxy\nFSC;q;!(\u001frr\nFI;k;!+\u001f\u0012\u0012\nFI;k;!)]K\n\u00002(ez\u0001er)erIm[\u001fxy\nFSC;q;!\u001f\u0012\u0012\nFI;k;!]Koi\n;(A16)\nwhere we use equations originating from the spin conser-\nvation law:\n\u001fzy\nFSC;q;!=\u001fyz\nFSC;q;!=\u001fzx\nFSC;q;!=\u001fxz\nFSC;q;!= 0;(A17)\n\u001fzz\nFSC;q;!6=\u001fxx\nFSC;q;!=\u001fyy\nFSC;q;!; (A18)\n\u001fxy\nFSC;q;!=\u0000\u001fyx\nFSC;q;!; (A19)\n\u001fzy\nFI;k;!=\u001fyz\nFI;k;!=\u001fzx\nFI;k;!=\u001fxz\nFI;k;!= 0; (A20)\n\u001fzz\nFI;k;!6=\u001fxx\nFI;k;!=\u001fyy\nFI;k;!; (A21)\n\u001fxy\nFI;k;!=\u0000\u001fyx\nFI;k;!; (A22)In addition, we can approximate \u001fxy\nFSC\u00190 because the\nFermi energy in the FSC is usually su\u000eciently larger than\nthe considered frequency. Therefore, we obtain\nh^ISi\u0019\u0000TX\nk;qh\n(er\u0002ez)hSr\nkihsz\nqi\n\u0000T\n2Zd!\n2\u0019n\n[er\u0000(er\u0001ez)ez]\nIm[(\u001fzz\nFSC;q;!+\u001fxx\nFSC;q;!)\u001f\u0012\u001e\nFI;k;!]K\n+ 2(er\u0001ez)ezIm[\u001fxx\nFSC;q;!\u001f\u0012\u001e\nFI;k;!]Koi\n;(A23)\nwhere we retain only the lowest order of Tin each direc-\ntion component of the spin current.\nAppendix B: Spin susceptibilities extracted from the\nmeasured spin currents\nIn this section, we discuss how to extract the spin\nsusceptibilities of FSC from the measured spin current.\nFrom Eq. (A23), the z- andx-polarized spin currents are\nIz\nS=T2X\nk;qZd!\n2\u0019h\nIm\u001fR;xx\nFSCRe\u001fK;\u0012\u001e\nFI+ Im\u001fK;xx\nFSCRe\u001fA;\u0012\u001e\nFIi\n;\n(B1)\nIx\nS=T2X\nk;qZd!\n2\u0019h1\n2[Im\u001fR;xx\nFSC+ Im\u001fR;zz\nFSC]Re\u001fK;\u0012\u001e\nFI\n+1\n2[Im\u001fK;xx\nFSC+ Im\u001fK;zz\nFSC]Re\u001fA;\u0012\u001e\nFIi\n; (B2)\nwhere we use that \u001fK;\u0012\u001e\nFI is a real number as shown in\nAppendix C. Furthermore, rewriting the spin suscepti-\nbility of the FI in terms of magnon Green's function (See\nAppendix C for detail), we obtain\nIz\nS=T21\nS[Im\u001fR;?\nloc;\n]\u0001n\nImGR\n0;\n; (B3)\nIx\nS=T21\n2S[Im\u001fR;?\nloc;\n+ Im\u001fR;k\nloc;\n]\u0001n\nImGR\n0;\n;(B4)\nwhereGR\n0;\nand\u0001n\nare the retarded component of\nmagnon Green's function and the deviation in the num-\nber of magnons under the microwave radiation with fre-\nquency \n, respectively, and \u001fR;\f\f\nloc;\n=P\nq\u001fR;\f\f\nFSC;q;\nis the\nlocal spin susceptibility of the FSC. The transverse ?\nand the longitudinal kcomponents are equal to xxand\nzzcomponents, respectively. The spin susceptibilities of\nthe FSC normalized by the spin susceptibility of the nor-\nmal state are\nIm\u001fR;?\nloc;\nIm\u001fR;?\nloc;N;\n=Iz\nS\nIz\nS;N; (B5)\nIm\u001fR;k\nloc;\nIm\u001fR;?\nloc;N;\n=2Ix\nS\u0000Iz\nS\nIz\nS;N; (B6)7\nwhere Im\u001fR;?\nloc;N;\nandIS;Nare the spin susceptibility and\nthe spin current in the normal state, respectively. The\nspin susceptibility of FI disappears by the normalization\nbecause it is equal in the superconducting and normal\nstates. These equations tell us the relationships between\nthe spin susceptibilities of the FSC and the measured\nspin currents.\nAppendix C: Spin susceptibilities in ferromagnetic\ninsulators\nIn this section, we derive the spin susceptibilities\n\u001f\u0012\u001e\nFI;k;!in the FI under microwave radiation from the\nmagnon Green's function, and show the detailed calcula-\ntions for the spin currents in Eqs. (B3) and (B4).\nWe use the Holstein-Primako\u000b transformation to the\nspin operators in the FI and employ the spin-wave ap-\nproximation:\nSr\nk= (S\u0000ay\nkak); (C1)\nS+\nk=S\u0012\nk+iS\u001e\nk\u0019p\n2Sak; (C2)\nS\u0000\nk=S\u0012\nk\u0000iS\u001e\nk\u0019p\n2Say\nk; (C3)\nwhereay\nkandakare the boson creation and annihilation\noperators, respectively. The Hamiltonian in the FI is\nrepresented in terms of the boson operators as\nHFI=\u0000JX\nhi;jiSi\u0001Sj\n\u0000~\rX\nih\nhdcSz\ni+hac(cos\ntSx\ni\u0000sin\ntSy\ni)i\n\u0019X\nk(!k+!0)ay\nkak\u0000Vac(e\u0000i\ntay\nk=0+ei\ntak=0);\n(C4)\nwherehdcis static magnetic \feld, hacand \n are ampli-\ntude and frequency of applied microwave, respectively,\nandJis the exchange coupling constant, hi;jirepresents\nsummation over all nearest-neighbor sites, \ris the gyro-\nmagnetic ratio, and !0=~\rhdc; Vac=~\rhacq\nSN\n2with\nthe number of sites N. Here, we assume the parabolic\nmagnon dispersion: !k/k2. The magnon Green's func-\ntions are de\fned as\nGk;!:=\u0000iZ\nCd(\u001c1\u0000\u001c2)hTC[ak(\u001c1)ay\nk(\u001c2)]iei!(\u001c1\u0000\u001c2);\n(C5)\nGR\nk;!:=\u0000iZ1\n\u00001dth[ak(t);ay\nk(0)]i\u0012(t)ei!t; (C6)\nGK\nk;!:=\u0000iZ1\n\u00001dthak(t)ay\nk(0) +ay\nk(0)ak(t)iei!t:(C7)\nIntroducing a phenomenological lifetime with the Gilbert\ndamping constant \u000band considering the second-orderterm ofVacas self-energy, the magnon Green's functions\nare written as\nGR\nk;!=1\n~!\u0000(!k+!0) +i\u000b~!; (C8)\nGK\nk;!=GK\n0;k;!+ 2GR\nk;!h\n\u0000i\n~V2\nac\u000ek;02\u0019\u000e(!\u0000\n)i\nGA\nk;!\n=\u00002iImGA\nk;!h\n2n!+ 2\u000enk;!+ 1i\n; (C9)\nwhereGK\n0is the non-perturbative Keldysh component,\nn!is the Bose distribution function, and \u000enk;!is the\ndeviation in the Bose distribution function due to the\noscillating magnetic \feld. Note that the self-energy does\nnot a\u000bect the retarded and advanced components since\nVacis a c-number. Here, \u000enk;!is given by\n\u000enk;!=\u0001n!\u000ek;02\u0019\u000e(!\u0000\n); (C10)\nwhere\u0001n!=V2\nac=(2\u000b~2!). The spin susceptibility\n\u001f+\u0000\nFI;k;!is represented by\n\u001f+\u0000\nFI;k;!=\u00001\n2SGk;!; (C11)\nwhere\n\u001f+\u0000\nFI;k;!:=iZ\nCd(\u001c1\u0000\u001c2)hTC[s+\nk(\u001c1)s\u0000\n\u0000k(\u001c2)]iei!(\u001c1\u0000\u001c2):\n(C12)\nThe\u0012\u0012,\u001e\u001e,\u0012\u001eand\u001e\u0012components of the spin suscepti-\nbility are rewritten as\n\u001f\u0012\u0012\nFI;k;!=\u001f\u001e\u001e\nFI;k;!=1\n4(\u001f+\u0000\nFI;k;!+\u001f\u0000+\nFI;k;!); (C13)\n\u001f\u0012\u001e\nFI;k;!=\u0000\u001f\u001e\u0012\nFI;k;!=i\n4(\u001f+\u0000\nFI;k;!\u0000\u001f\u0000+\nFI;k;!); (C14)\nwhere\n\u001f\u0000+\nFI;k;!:=iZ\nCd(\u001c1\u0000\u001c2)hTC[s\u0000\nk(\u001c1)s+\n\u0000k(\u001c2)iei!(\u001c1\u0000\u001c2):\n(C15)\nThe +\u0000and\u0000+ components have relationships:\n\u001fR;\u0000+\nFI;k;!=\u001fA;+\u0000\nFI;\u0000k;\u0000!; (C16)\n\u001fK;\u0000+\nFI;k;!=\u001fK;+\u0000\nFI;k;!: (C17)\nTherefore, the \u0012\u001ecomponents of the spin susceptibility\nare\n\u001fR;\u0012\u001e\nFI;k;!=\u0000i1\n8S[GR\nk;!\u0000GA\n\u0000k;\u0000!]; (C18)\n\u001fK;\u0012\u001e\nFI;k;!=\u0000i1\n8S[GK\nk;!\u0000GK\n\u0000k;\u0000!]: (C19)\nThe Keldysh component \u001fK;\u0012\u001e\nFI;k;!is real because GK\nk;!is\npure imaginary. From Eqs. (C18), (C19) and (C9), we\nobtain\n\u001fK;\u0012\u001e\nFI;k;!=\u00002(2n!+ 1)Re\u001fA;\u0012\u001e\nFI;k;!\n\u00001\n2S[\u000enk;!\u0000\u000en\u0000k;\u0000!]ImGA\n0;\n: (C20)\nWith this equation, we can reproduce the spin currents\nin Eqs. (B3) and (B4).8\nAppendix D: Spin susceptibilities in ferromagnetic\nsuperconductors\nIn this section, we derive the spin susceptibilities in\nFSC, and obtain the \",#, +\u0000and\u0000+ components,\ntransform them into the ?andkcomponents required\nto calculate the spin currents in Eqs. (B3) and (B4).\nThe components \u001b(=\";#),?andkare de\fned as\n\u001f\u001b\nFSC;q;!=iZ\nCd(\u001c1\u0000\u001c2)hTC[nq\u001b(\u001c1)n\u0000q\u001b(\u001c2)]iei!(\u001c1\u0000\u001c2)\n(D1)\n\u001fk\nFSC;q;!=\u001fzz\nFSC;q;!; (D2)\n\u001f?\nFSC;q;!=\u001fxx\nFSC;q;!=\u001fyy\nFSC;q;!; (D3)\nwherenq\u001b=P\nkcy\nk\u001bck+q\u001bandck\u001bis the annihilation\noperator of the electron with spin \u001b.\nThe mean \feld Hamiltonian in the FSC is\nHFSC=1\n2X\nkcy\nkHBdGck; (D4)\nwhere ck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T, andHBdGis\nHBdG=0\nBBB@\u0018k\" 0 \u0001 k\"\" 0\n0\u0018k# 0 \u0001 k##\n\u0000\u0001\u0003\n\u0000k\"\" 0\u0000\u0018\u0000k\" 0\n0\u0000\u0001\u0003\n\u0000k## 0\u0000\u0018\u0000k#1\nCCCA;(D5)\nwhere\u0018k\u001b=\"k\u0000\"F\u0000\u001b\u0001FM, with kinetic energy\n\"k=~2\n2mk2, the Fermi energy \"F, the spin-splitting en-\nergy \u0001 FM, and \u0001 k\u001b\u001bis the superconducting gap. We\nconsider that the superconducting gap opens only in the\nspin-up band (i. e. \u0001 k##= 0). The retarded Green's\nfunctions are given by\n\u001fR;\"\nFSC;q;\n=i1\nN2X\nkk0Z1\n\u00001dt ei\nt\nD\n[cy\nk\"ck+q\"(t);cy\nk0\"ck0\u0000q\"(0)]E\n\u0012(t)\n=1\nNX\nkX\n\u0015=\u0006X\n\u00150=\u0006\n1\n4h\n1 +\u0018\"\u00180\n\"+E\"\u0015\u00180\n\"+E0\n\"\u0015\u0018\"+ \u0001k\"\"\u0001k+q\"\"\nE\"\u0015E0\n\"\u00150i\nf(E0\n\"\u00150)\u0000f(E\"\u0015)\n~\n +i\u000e\u0000E0\n\"\u00150+E\"\u0015; (D6)\n\u001fR;#\nFSC;q;\n=i1\nN2X\nkk0Z1\n\u00001dt ei\nt\nD\n[cy\nk#ck+q#(t);cy\nk0#ck0\u0000q#(0)]E\n\u0012(t)\n=1\nNX\nkf(\u0018k+q#)\u0000f(\u0018k#)\n~\n +i\u000e+\"k\u0000\"k+q; (D7)\u001fR;+\u0000\nFSC;q;\n=i1\nN2X\nkk0Z1\n\u00001dt ei\nt\nD\n[cy\nk\"ck+q#(t);cy\nk0#ck0\u0000q\"(0)]E\n\u0012(t)\n=1\nNX\nkX\n\u0015=\u00061\n2\u0010\n1 +\u00180\n\"\nE0\n\"\u0015\u0011f(E0\n\"\u0015)\u0000f(\u0018#)\n~\n +i\u000e\u0000E0\n\"\u0015+\u0018#;\n(D8)\n\u001fR;\u0000+\nFSC;q;\n=i1\nN2X\nkk0Z1\n\u00001dt ei\nt\nD\n[cy\nk#ck+q\"(t);cy\nk0\"ck0\u0000q#(0)]E\n\u0012(t)\n=1\nNX\nkX\n\u0015=\u00061\n2\u0010\n1 +\u0018\"\nE\"\u0015\u0011f(\u00180\n#)\u0000f(E\"\u0015)\n~\n +i\u000e\u0000\u00180\n#+E\"\u0015\n= [\u001fR;+\u0000\nFSC;\u0000q;\u0000\n]\u0003; (D9)\nwhere\u0018=\u0018k\u001b,\u00180=\u0018k+q\u001b,E\"\u0015=\u0015q\n\u00182\nk\"+ \u00012\nk\"\", and\nE0\n\"\u0015=\u0015q\n\u00182\nk+q\"+ \u00012\nk\"\", respectively. To calculate the\nspin currents in Eqs. (B3) and (B4), we only need the\nimaginary part of the local spin susceptibilities. Inte-\ngrating the wave number k,qwith the axial \u0001 k\"\"=\n\u0000\u00010sin\u0012ke\u0000i\u001ekor the polar \u0001 k\"\"=\u0000\u00010cos\u0012ktype\nsuperconducting gap and using1\nx+i\u000e=\u0000i\u0019\u000e(x) + P1\nx,\nthe imaginary part of the local spin susceptibilities are\nIm\u001fR;\"\nloc;\n\u0019\u0000\u0019Z1\n\u00001d\"DS;\"DS;\"+~\n[f\"+~\n\u0000f\"];(D10)\nIm\u001fR;#\nloc;\n\u0019\u0000\u0019Z1\n\u00001d\"D#\nFD#\nF[f\"+~\n\u0000f\"]\n\u0019\u0019D#\nF2~\n; (D11)\nIm\u001fR;+\u0000\nloc;\n\u0019\u0000\u0019Z1\n\u00001d\"D#\nFDS;\"+~\n[f\"+~\n\u0000f\"];(D12)\nIm\u001fR;\u0000+\nloc;\n\u0019\u0019Z1\n\u00001d\"D#\nFDS;\"\u0000~\n[f\"\u0000~\n\u0000f\"]\n=\u0000Im\u001fR+\u0000\nloc;\u0000\n= Im\u001fR+\u0000\nloc;\n: (D13)\nHere, we replaced the wave number summation with the\nenergy integral as\n1\nNX\nk(\u0001\u0001\u0001)!Z1\n\u00001d\"D#\nF(\u0001\u0001\u0001); (D14)\nfor the normal state and\n1\nNX\nk(\u0001\u0001\u0001)!D\"\nF\n4\u0019Z1\n\u00001dEZ\u0019\n0d\u0012Z2\u0019\n0d\u001e\njEjsin\u0012p\nE2\u0000\u00012(\u0012;\u001e)(\u0001\u0001\u0001)\n=Z1\n\u00001d\"DS;\"(\u0001\u0001\u0001); (D15)9\nfor the superconducting state, where D\u001b\nFis the spin-\ndependent density of states in the normal state at the\nFermi energy.\nThe?component of the spin susceptibility is rewritten\nas\n\u001f?\nFSC;q;!=1\n4(\u001f+\u0000\nFSC;q;!+\u001f\u0000+\nFSC;q;!); (D16)\nIn itinerant electron systems, the kcomponent of the spin\nsusceptibility can be rewritten as\n\u001fk\nFSC;q;!=1\n4(\u001f\"\nFSC;q;!+\u001f#\nFSC;q;!): (D17)\nTherefore, we obtain\nIm\u001fR;k\nloc;\n=\u0019\n4~\nD#\nF2\u0000\u0019\n4D\"\nF2\n\u0002Z1\n\u00001d\"\u0016DS;\"\u0016DS;\"+~\n[f\"+~\n\u0000f\"];(D18)\nIm\u001fR;?\nloc;\n=\u0000\u0019\n2D\"\nFD#\nFZ1\n\u00001d\"\u0016DS;\"+~\n[f\"+~\n\u0000f\"];\n(D19)\nwhere \u0016DS;\"=DS;\"=D\"\nF. Thus, we can calculate the spin\ncurrents in Eqs. (B3) and (B4) with these longitudinal\nand transverse components of the local spin susceptibil-\nity.\nAppendix E: Power series expansion for spin\nsusceptibilities of FSC\nIn this section, we show the power expansion of the\nspin susceptibility of the FSC. The spin susceptibilities\nof the superconducting state are normalized by those of\nthe normal state with the frequency \fxed to be equal to\n\u00010as\nIm\u0016\u001fR;?\nloc;\n=Im\u001fR;?\nloc;\nIm\u001fR;?\nloc;\n=\u0001 0=~;T>T c\n=\u00001\n\u00010Z1\n\u00001d\"\u0016DS;\"+~\n[f\"+~\n\u0000f\"];(E1)\nIm\u0016\u001fR;k\nloc;\n=1\n2\u00101\n2Im\u0016\u001fR;\"\nloc;\n+1\n2Im\u0016\u001fR;#\nloc;\n\u0011\n; (E2)\n1\n2Im\u0016\u001fR;\"\nloc;\n=1\n2Im\u001fR;\"\nloc;\nIm\u001fR;?\nloc;\n=\u0001 0=~;T>T c\n\u0019\u00001\n\u00010Z1\n\u00001d\"\u0016DS;\"\u0016DS;\"+~\n[f\"+~\n\u0000f\"];\n(E3)\n1\n2Im\u0016\u001fR;#\nloc;\n=1\n2Im\u001fR;#\nloc;\nIm\u001fR;?\nloc;\n=\u0001 0=~;T>T c\u0019~\n\u00010; (E4)where Im\u001fR;?\nloc;\n=\u0001 0=~;T>T c=\u0019\n2\u00010D\"\nFD#\nFand we use the\napproximation D\"\nF\u0019D#\nF, assuming that the spin split-\nting energy \u0001 FMis su\u000eciently smaller than the Fermi\nenergy\"F. In the axial superconductor, which has point\nnodes, the density of states is\n\u0016DS;\"=\"\n2\u00010ln\f\f\f\"+ \u0001 0\n\"\u0000\u00010\f\f\f: (E5)\nWhen\"\u001c\u00010, it becomes \u0016DS;\"\u0019\"2\n\u00012\n0. In the polar su-\nperconductor, which has line nodes, the density of states\nis\n\u0016DS;\"=(\n\"\n\u00010arcsin\u00010\n\"\u0000\u00010\n\"<1\u0001\n\u0019j\"j\n2\u00010\u0000\u00010\n\">1\u0001\n:(E6)\nWhen \n<\u00010, we can expand the spin susceptibility to\npowers of frequency. In the case of axial type, they are\n1\n2Im\u0016\u001fR;\"\nloc;\n\u00191\n30(~\n)5\n\u00015\n0; (E7)\nIm\u0016\u001fR;?\nloc;\n\u00191\n3(~\n)3\n\u00013\n0: (E8)\nIn the case of polar type, they are\n1\n2Im\u0016\u001fR;\"\nloc;\n\u0019\u00192\n24(~\n)3\n\u00013\n0; (E9)\nIm\u0016\u001fR;?\nloc;\n\u0019\u0019\n4j~\nj(~\n)\n\u00012\n0: (E10)\nHere, we expand the integral required to calculate the\nspin susceptibility as\nZ1\n\u00001d\"DA;\"+~\n=2DB;\"\u0000~\n=2[f\"+~\n=2\u0000f\"\u0000~\n=2]\n=Z1\n\u00001d\"DA;\"+~\n=2DB;\"\u0000~\n=21X\nn=0f(2n+1) \n2n+1\n22n(2n+ 1)!\n\u0019\u00001X\nn=0h@2n\n@\"2n(DA;\"+~\n=2DB;\"\u0000~\n=2)i\n\"=0\n2n+1\n22n(2n+ 1)!;\n(E11)\nwhereDA;Bis the density of states in the superconduct-\ning or normal state, and we use f0(\")\u0019\u0000\u000e(\") assuming\nzero temperature. Thus, we can analytically obtain the\npower exponent of the spin susceptibility of superconduc-\ntors.10\n[1] Y. Tserkovnyak, A. Brataas, G. E. Bauer, and B. 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B 107, 144504 (2023)."    },    {        "title": "1703.08422v1.Probing_the_degree_of_spin_polarization_of_a_ferromagnet_with_ferromagnet_superconductor_proximity_effect.pdf",        "content": "Probing the degree of spin polarization of a ferromagnet \nwith ferromagnet/superconductor proximity effect  \nPavel V. Leksin1,2 , Andrey A. Kamashev2, Joachim Schumann1, Vladislav Kataev1, Jürgen Thomas1, Thomas \nGemming1, Bernd Büchner1,3, and Ilgiz  A. Garifullin2  \n \n \nAbstract  \n \n Superconductor/ferromagnet proximity effect has been studied for Pb/Co 2Cr1-xFexAl \nbilayers. Different substrate temperatures allowed us to prepare the Heusler alloy Co 2Cr1-xFexAl \nfilms with different degree of spin polarization of conduction band of the Heusler layer. We \nobtain a strong correlation between the dependence of the superconducting transition \ntemperature on the Pb -layer thickness at fixed thickness of the Heusler lay er and the degree of \nspin polarization in the ferromagnetic layer.  \n \n \n The effectiveness of magnetoelectronic devices depends on the extent to which a current \nis spin -polarized [1]. A typical transition -metal ferromagnet has two components to its \nelectronic  structure: narrow d band that may be fully or partially spin polarized and broad s \nband with a lesser degree of spin polarization (DSP) (due to hybridization with the d band). The \nknowledge of DSP gives an opportunity to distinguish between an ordinary ferromagnet and a \nhalf-metal (ferromagnet with 100 % DSP). Half -metallic ferromagnets represent the class of \nmaterials which have recently attracted a considerable interest due to their possible applications \nin spin electronics. In these materials the two spin bands show a completely different behavior. \nUsually the majority band (spin -up) shows a typical metallic behavior with a nonzero density of \nstates (DOS) at the Fermi level EF  and the minority  (spin -down) band exhibits a semiconducting \nbehavior with a  gap at EF. Therefore, such half -metals (HM) can be considered as hybrides \nbetween metals and semiconductors.   \nSome Heusler alloys of Co 2YZ composition (where Y is a 3 d transition metal and Z is an s-p \nmetal) are ferromagnets and are expected to exhibit 10 0 % spin polarization due to the energy -\nband gap for the minority –spin electrons at the Fermi level ( EF) (see, e.g., [2]). DSP of these \nalloys and their films strongly depends on the preparation conditions [3, 4]. In order to get a \nferromagnetic film with DSP approaching the meaning of half -metal usually it is necessary to hold substrate during the film growth at temperatures well above the room temperature.  \nFor both scientific and technological reasons it is important to be able to directly and easily \nmeas ure DSP of a ferromagnet. The most natural and common definition of DSP of a \nconduction band of ferromagnet is [5]:   \n                 \n                     \nwhere           is the density of electronic states at the Fermi level with corresponding spin \ndirection (    ).  \nThe following direct methods are usually involved for investigating DSP . Typical \nexperiments that can probe P are spin resolved photoemission spectroscopy [6] and hard X -ray \nphotoemission spectroscopy [7, 8]. However, their resolution is hundreds of meV, which is \nsignificantly less than the necessary one (~1 meV). They also imply complicated apparatus such \nas syn chrotron radiation, and very stringent surface preparation. A very useful method is the \ntunneling spectroscopy . The pioneering experiments by Tedrow and Meservey [5, 9] have \nshown, that the tunnel junction for ferromagnet/ insulator/superconductor (F/I/S) can be used to \nmeasure the DSP . One can also measure tunneling currents separately for both spin polarization \nchannels with superconducting point contacts. However, to get precise results, the barrier and \ninterface quality requirements are severe, so the m easured DSP correlates with the junction \nstructural quality . Another technique, based on the combination of tunneling magnetoresistance \n(TMR) and F/I/F junction -Julliere's model, is widely used for estimating the DSP [10]. It is clear, \nhowever, that the re sults depend on the structural quality of the tunnel junction and the choice \nof the tunneling barrier . It was also suggested [11, 12] that Andreev’s reflection at the interface \nbetween ferromagnet and a superconductor can be used for direct probing of DSP .  The \nadvantage of such relatively novel device -independent technique is its experimental simplicity . \nUnlike other methods, which have stringent requirement for the surface atomic cleanness \nand/or an ultrathin uniform oxide layer, and thus may make the stud y of some interesting \nmaterials difficult, the superconducting point contact method requires no magnetic field and \nhas no special constraints on a sample.  The S/F point contact is usually organized between the \nsurface of the sample and a superconducting pr obing element, a sharp needle, for instance. At \nsmall voltages the differential conductance of such a contact decreases with the rise of the DSP \nvalue, which makes it advantageous for a routine optimization of DSP for the material. The \nresults can be model ed by using different models, such as the Blonder -Tinkham -Klapwijk \nmodel [13] and its modifications (see, eg., [14 -16]). However the theoretical calculations sometimes fit results unsatisfactorily [17] or produce unrealistic fitting parameters.  The data \nwhich come from all the described methods were analyzed by Mazin [18]. Most of the above \ndescribed methods of measuring DSP require clean and well -ordered surfaces, which for \nHeusler compounds is difficult to obtain by surface cleaning procedures of ex situ  prepared \nsamples.  \n In this paper we show that the dependence of the superconducting transition \ntemperature on the superconducting layer thickness in a superconductor/ferromagnet (S/F) \nbilayer dramatically changes with changing DSP . This suggests a simple a lternative method for \nestimation of DSP of the ferromagnetic layer. In contrast to other techniques such method does \nnot involve the top clean surface or junction effects and thus may provide the information about \nDSP in simple bilayer geometry . In S/F bil ayers the singlet Cooper pair wave function penetrates \nfrom superconductor to ferromagnet over a certain distance, which is usually associated with \nthe penetration depth for ferromagnet depending on the exchange splitting of conduction band \nof ferromagnet.  This process is usually accompanied by a pair breaking effect, which decreases \nTc of the S layer and leads to a complete suppression of the superconductivity at a certain critical \nsuperconductor layer thickness        . \nWe show the efficiency of our  method using the Heusler alloy with the nominal \ncomposition of the target Co 2Cr0. 4Fe0. 4Al1.2. This choice is determined by the possibility to change \neasily DSP of conduction band by changing the substrate temperature during the growth of the \nHeusler  alloy layer [3, 4].  \nIt is known [3] that the alloy Co 2Cr1-xFexAl forms the film with high DSP if during the film \ngrowth the substrate temperature is hold at Tsub ≥ 600 K. The films prepared at lower Tsub appear \nto be weak ferromagnets  due to disordered structure [3]. According to our data on the point -\nconctact spectroscopy for the studied samples, DSP reaches 70 %. Bearing this in mind we \nprepared two sets of MgO/Heusler(12nm)/Cu(1.5nm)/Pb( dPb) samples with variable Pb -layer \nthickness and with the substrate temperature Tsub = 300 K (Set 1) and 600 K (Set 2) when \nevaporating the Heusler alloy by sputtering technique. Here we used Cu(1.5nm) as \nantidiffusion layer. To optimize the growth of the Cu/Pb fragment after deposition of the \nHeusle r layer we decreased the temperature of the substrate Tsub down to 150 K. An advantage \nof the low Tsub was shown in our previous papers [19 -20]. Finally all samples were covered by \nSi3N4 protective layer against oxidation. We used the following deposition rates: 0.37 Å/s for \nHeuser alloy, 0.5 Å/s for Cu layers and 12 Å/s for Pb films.  \nThe deposition of layers was performed using a combination of the sputtering technique (for Heusler alloy) and an e -gun in ultra -high vacuum (UHV) with pressure 10-9 mbar (for  Cu \nand Pb). The deposition setup had a load lock station with vacuum shutters, allowing changing \nthe sample holder without breaking the UHV in the main deposition chamber. First, the \nsubstrates were fixed on a sample holder and transferred into the sputte ring chamber for \ndeposition of the Heusler alloy using the dc sputtering technique. Then the sample holder was \nmoved to the main deposition chamber through the load lock station. We used a rotating wheel \nsample holder in order to prepare a set of samples w ith different Pb -layer thickness in a single \nvacuum cycle.   \nTo inspect the layer stacks regarding the thickness of the layers as well as the interface \nroughness and the morphology of the Pb layer, cross sections of the samples were investigated \nwith a tran smission electron microscope FEI TEM/STEM Tecnai F30 working at an acceleration \nvoltage of 300 kV. The electron -transparent lamellas were prepared by the focused ion beam  \n \n \n \n \n \n \n \n \n \n \nFigure 1  Microscopic cross -sectional image of the Heusler/Cu/Pb  sample with Tsub = 300 K. This image is \ntypical also for Heusler(12)/Cu(1.5)/Pb(20) with Tsub = 600 K.  \n(FIB) technique using a Zeiss 1540XB cross beam machine. After inspection of imaging a C/Pt -O-\nH protection layer (technical layer in Fig. 1) was deposit ed at the position of interest. The \nlamella was cut by a focused 30 keV Ga ion beam and after its lifting out welded on an electron \nmicroscopic girder. The protection layer reduces the Ga implantation in the sample region close \nto the surface. The cross se ctions were analyzed in the TEM by means of conventional fixed \nbeam imaging including its high resolution option as well as in the scanning transmission \nelectron microscopic mode (STEM) using a high angle annular dark field detector (HAADF). \nAdditionally , by energy dispersive X -ray spectroscopy (EDX S) in analytical mode the \ncomposition Co 61Cr13Fe11Al15 for the samples prepared at Tsub = 300 K and Co 55Cr15Fe13Al17 at Tsub = \n600 K was determined. The composition in both cases is nearly the same with some deficiency \nof Al. The interfaces between the single layers could clearly be seen in the TEM micrographs as \nwell as in the STEM -HAADF images (Fig. 1). In both structures prepar ed at Tsub = 300 K and 600 \nK all thicknesses are nearly the same and the interfaces are flat. The EDX S linescan intensity \nprofiles for the cross -sections revealed no chemical differences of the interfaces Heusler/Cu and \nCu/Pb for both types of Heusler film s, evidencing the absence of the oxide layers in both cases. \nThis fact was checked especially carefully .  \nBoth structures were magnetically characterized using a standard 7 T VSM SQUID \nmagnetometer . Magnetic hysteresis loops were measured at T = 10 K with magnetic film in the \nfilm plane (see Fig 2). One can see that the saturation magnetization of 850 emu/cm3 for the \nsample from Set 2 prepared at Tsub = 600 K is larger than that for sample from Set 1 (570 \nemu/cm3). Note, that for pure iron this value is equ al to 1730 emu/cm3. According to the data by \nMiura et al. [21] Co   Cr disorder (i. e. the appearance of Cr atoms at the Co sites and vice \nversa) significantly reduces the total magnetic moment and DSP . Thus, our data are in a good \nagreement with the data by Miura et al.  \n \n \n \n \n \n \n \n \nFigure 2  Magnetic hysteresis curves for Heusler(12) with Tsub = 300 K (open circles) and Heusler(12) \nwith Tsub = 600 K (closed circles). The measurements were carried out at T = 10K. Arrows indicate the \ndirection of the magnetic field sweeps.  \nThe transport properties of the samples were studied using a 4 -contact resistivity \nmeasurement method using the B2902A precision source/measure unit from Keysight Techn. \nThe temperature of  the sample was controlled with the 230 Ω Allen -Bradley thermometer which \nis particular sensitive in the temperature range of interest. We found that the residual resistivity \nratio RRR  = ρ(300 K)/ρ(10 K) of the studied samples lies in the interval 10 < RRR  < 17. Using \nρ(300 K) = 21 ·cm [22] we obtain ρs = ρ(10 K) = 1.2 - 2.1 ·cm for the residual resistivity . The \nBCS coherence length for Pb amounts to ξ0 = 83 nm [22] and the mean -free path of conduction \nelectrons obtained using the Pippard’s  relations [23] is about ls ~ 17 nm. The comparison of ls \nwith ξ0 shows that ls  << ξ0 implying the “dirty” limit for the superconducting part of the system. \nTherefore, we calculate the superconducting coherence length as ξs =            = 41 nm. We also  \nperformed the measurements of the temperature dependence of the resistivity of the single \nfilms of the Heusler alloy prepared at Tsub = 300 K and 600 K. For the films prepared at Tsub = 300 \nK we obtain that the resistivity does not depend on temperature and amounts to ρf = 143·cm \n(cf. 220·cm in Ref. [3] and 170 ·cm in Ref. [4]). For the film prepared at Tsub = 600 K this \nvalue is also independent of temperature and amounts to ρf = 130·cm (cf. 330 ·cm in Ref. \n[3] and 170 ·cm in Ref. [4]).  \nWe have measured Tc(dPb) for all prepared samples. The obtained results are shown in Fig. \n3(a). The shape of the dependence of Tc on the Pb -layer thickness dPb at fixed thickness of the \nHeusler layer thickness is conventional: with decreasing Pb thickness, Tc decreases slowly at \nlarge dPb and then drops sharply to zero when dPb approaches the critical thickness          Our \nprevious results for Fe/Cu/Pb structure [24] are also show n for comparison. One can see that the \nlargest Tc-suppression as function of dPb is observed for Fe/Cu/Pb structures. For Set 1 prepared \nat Tsub = 300 K the suppression is weaker [closed circles in Fig. 3(a)].  \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3  (a) The Tc dependence on the superconducting Pb layer thickness dPb for: \nFe(5nm)/Cu(2nm)/Pb( dPb ) [24] (half -opened circles); Heusler(12nm)/Cu(1.5nm)/Pb( dPb) with Tsub = 300 \nK (closed circles); two independent sets of samples Heusler(12nm)/Cu(1.5nm)/ Pb( dPb) with Tsub = 600 K \n(opened circles and black stars). (b) Superconducting transition curves for samples \nHeusler(12nm)/Cu(1.5nm)/ Pb(30) with Tsub = 300 K (i) and Tsub = 600 K (ii). The sketch visualizes the \nprocess at the F/S interface for samples with small (left)  and large (right) DSP, respectively.  \n \n \nFinally , for Set 2 prepared at Tsub = 600 K the  Tc(dPb)-curve [open circles in Fig. 3(a)] is \nshifted further to the left from the data for Set 1. To confirm the latter result we prepared the \ncontrol set of samples at Tsub = 600 K [Set 3 black stars on Fig.3 (a) ]. The obtained results coincide \nperfectly with the data for Set 2. Fig. 3(b) demonstrates the shift of the superconducting \ntransitions curves for two samples from Set 1 (i) and Set 2 (ii). Both have the same layer stack \ncomposition, but the Tc values differ dramatically [cf. Fig. 3 (a)]. The Tc differe nce between (i) \nand (ii) is about 2.1 K. When increasing Tsub, the shift of Tc(dPb)-curve occurs not due to the \nchange of the saturation magnetization (Fig. 2). Since                    (Fig. 2), the Tc(dPb)-curve for \nSet 2 should be shifted then to higher thicknesses dPb, as it indeed the case for the Fe/Cu/Pb \nstructure [Fig. 3(a)], which has the largest saturation magnetization among the studied samples. \nFurthermore, the composition of the Heusler layer and the chemical composition of the \nHeusler/Cu and Cu/Pb interfaces are the same for both types of the Heusler -based structures \nand thus could not be the reason for the significant shift of the Tc (dPb) dependence.  \nTherefore, we argue that a substantial offset of the Tc(dPb)-curves may be caused by \ndifferent DSP of the Heusler part in samples Set 1 and Set 2. The larger DSP of the F - layer in \nthe S/F - heterostructure should inhibit the penetration of Cooper electron pairs with opposite \nspins from the S - layer into the F - layer because of the different density of states of the up -spin \nand down -spin electrons in the F - layer. Thus, the suppression of Tc should take place at \nsmaller thicknesses of the S - layer. To verify experimentally this conjecture we have studied our \nHeusler films, prepared at Tsub = 300 K and 600 K, with Andreev’s reflection point contact \nspectroscopy . Similar to Soulen et al.  [12] the tunnel junction technique has been successfully \nused to compare the spin polarization DSP for the studied Heusler films . The general drawback \nof this technique is the constraint on the fabrication of a device consisting of the oxidized \nHeusler film. We have formed a metallic point contact between the sample and a \nsuperconducting needle using a simple mechanical adjustment.  The needle was driven by a \nmicrometer piezo mechanism. A metallic contact allows coherent two -particle transfer at the \ninterface between the normal metal and a superconductor. The electronic transport properties at \nthe point contact measures the conversio n between superconducting pairs and the single -\nparticle charge carriers of the metal. We used Nb tips as the probing needles. Special care was \ntaken to prevent Nb oxidation during the needle preparation in air. The transport \nmeasurements were performed usi ng the standard four -contact technique while the point \ncontact and sample were immersed in liquid helium. The dI/dU  data in this study was obtained \nby a standard dc technique. The resistance of the measured Nb/Heusler contacts was around 60 Ohms for both t ypes of the samples. The contacts were tested first at temperature T = 15 K \nwhich is higher, than the Tc = 9.25 K for Nb and have shown a constant differential resistance at \nany U value. The results measured at T = 4.2 K are plotted in Fig. 4 (a). The cond uctance dI/dU  of \nsample from Set 1 ( Tsub = 300 K) is maximum at the voltage U ≈ ± 1 V and continuously decreases \nwith either increasing or decreasing U. In contrast, dI/dU  of sample from Set 2 ( Tsub = 600 K) is \npractically constant at | U| ≥ 3.5 V and exhib its a minimum at  U = 0. \n \n \n \n \n \n \n \n \n \nFigure 4  (a) Andreev's reflection point contact spectra for: (a) Heusler with Tsub = 300 K (open circles) \nand Heusler with Tsub = 600 K (closed circles). The thicknesses of the films were 200 nm; (b) theory by G. \nI. Strijkers et at.  [14]. \nIf P = 100 % near EF, then there are no spin -down states in the Heusler -film to provide the \nspin-down electron of the superconducting pair for Andreev reflection. Supercurrent \nconversion via Andreev reflection at the interface is eff ectively blocked, allowing only single -\nparticle excitation to contribute to the conductance. These single -particle states necessarily see \nthe gap in the energy spectrum of the superconductor, thus suppressing the conductance for \nenergies less than exchange  splitting of the conduction band of the Heusler -film.  \nThe calculated dI/dU  dependences for different P values of DSP using a classical theoretical \nmodel by Strijkers et al.  [14] are plotted in Fig.4 (b). As can be seen from Fig. 4 (b), the curves at \nsmall voltages exhibit the tendency to rise up as the P rises from 30 to 70 %. Here we set the \ntemperature for the system T = 4.2 K, S/F interface transparency parameter Z = 0.4 and the \nsuperconducting energy gap Δ = 1.5 meV. The experimental curves on Fig. 4 (a) exhibit exactly \nthe same tendency as the calculated ones, evidencing that the DSP value for the sample from Set \n2 is higher , than the one for the sample from Set 1. First , according to theory [see Fig.4 (b)] the \ndI/dU for DSP = 30 %  monotonically rises up as the voltage amplitude is varied from 8 mV to 2 \nmV .  In contrast, the dI/dU curve for DSP = 70 % is almost constant within this voltage range.  \nSecond, the curve for DS P= 30 %   at |U| < 1.5 mV has two maxima and one minimum. At high \npolarization values about DSP = 70 % the curve has one pronounced minimum with the \nstrongly damped maxima. The experimental curves in Fig. 4 (a) show qualitatively the same \ncharacteristic cha nges, suggesting the larger value of DSP ~ 70 % for Heusler film, prepared at \nTsub = 600 K, as compared with DSP ~ 30 % of the Heusler film prepared at Tsub = 300 K.  \nWith this knowledge let us discuss our results on the Tc(dPb) dependences  (Fig. 3). In \naccordance with the theory by Fominov et al. [25], the explicit result for the critical thickness of \nthe S - layer        can be obtained in the limit ( γ/γ b )(ds/ξs) << 1 as  \n      \n        \n                 \n                 \n              \nHere γE ≈ 1.78 is the Euler constant,  ρS and ρF are the normal -state resistivity of the S - and F - \nlayer; Rb is the normal -state resistance of the S /F boundary and A is its area. The value of γb = 0 \ncorresponds to the fully transparent S /F interface.         is smaller for larger values of γb.  \nThe Tc curve in Fig. 3(a) show that the critical thickness         below which \nsuperconductivity vanishes amounts to        ~ 42 nm for Fe/Cu/Pb,        ~ 23 nm for Set 1 and \n       ~ 12 nm for Set 2. The DSP value for iron has been found by Soulen et al.  [12] to be 42 %. \nOur data on the point -contact spectroscopy with Nb tip enable to estimate DSP as P = 30 % for \nSet 1 and as P = 70 % for Set 2.  \nThe main parameters of the theory by Fominov et al.  [25] are the transparency parameter \nγb (Eq. 4) and the exchange splitting of conduction band of the ferromagnet h. To estimate γb \nfrom Fig. 3 (a) we use the values ρs , ξs, ρf  presented above and ξf  = 14 nm which we roughly \nestimate basing on our data. From Eq. (3) we obtain γ = 0.034 for both sets of samples. Finally , \nfrom Eq. (2) we get γb = 0.15 for Set 1 and γb = 0.35 for Set 2. With these parameters we have \ncalculated theoretical curves Tc(dPb) in Fi g. 3(a). As can be seen there the theory and experiment \nagree reasonable well. From Eq. (3) one can conclude that the resistance of the S/F boundary Rb \nfor Set 2 is larger by a factor of 2 than the one for Set 1. The question arises if the difference \nbetwe en two sample sets could be due to a more oxidized F/S interface in the sample from the \nSet 2. As it has been established from the EDX S line scan intensity profiles for the cross -section \nthere are no chemical differences in Heusler/Cu and Cu/Pb interfaces,  evidencing the absence of \nthe oxide layer for both types of structures.  Hence the additional oxidation or any another \nchemical modification of the interfaces cannot be the reason for the reduction of the \ntransmission coefficient for electrons moving thro ugh the Heusler/Cu/Pb interfaces.  \nThere are two points of view concerning the processes taking place at the S/F interface with high DSP value of the ferromagnetic layer. The first one [26] claims that if a ferromagnet  is \nin the half -metallic regime with only one occupied spin -band, the Cooper pairs with zero spin \nprojection reaching the interface should cease at the interface because one electron of the pair \npenetrates through the interface into the half -metal while an other one with the opposite spin is \nnormally reflected back to the superconductor. Based on our results we conclude that γb \nincreases with increasing DSP , i. e. the S/F interface becomes less transparent. This means that \nthe Cooper pairs are mostly reflect ed from the S/F interface without being destroyed [see sketch \nin Fig. 3 (b)].  \nThe starting point for the second approach is that the theory by Mironov and Buzdin [26] \ndoes not account for the possible relative shift between the spin -majority energy band i n the \nhalf-metal and the electron bands in the superconductor, which may renormalize the \nprobabilities of the electron transmission through the S/F interface. Such kind of effects have \nbeen analyzed by Takahashi et al. [27]. Following them, we assume a nea rly full polarization of \nconduction band of the Heusler layer in our samples. In that case the resistance of the \nferromagnet for the electrons in the Cooper pair is different for spin -up and spin -down \nelectrons. The larger the exchange splitting of the con duction band of ferromagnet is, the larger \nresistance of the S/F interface is expected. Hence, for larger h values the larger values of γb are \nexpected. This spin imbalance plays a key role in the processes taking place at the interface. The \ngeneralization  of the theory by Fominov et al. [25] for strong ferromagnets requires taking into \naccount the fact that the penetration of electrons through the S/F interface occurs with different \nprobabilities for the electrons with different spin projections as it was assumed in the boundary \nconditions by Eschrig et al.  [28]. The development of the theory in this direction is appealing for \na better, quantitative determination of DSP based on the measurements of the superconducting \ncritical temperature in S/F bilayers.  \nIn summary , we have demonstrated that the dependence of the superconducting critical \ntemperature Tc of the S/F bilayer Pb/Co 2Cr1-xFexAl on the thickness of the S - layer is sensitive to \nthe degree of the spin polarization of the F -layer. Our findings may pave the way to establishing \na new advantageous method of determination of DSP in ferromagnetic films beyond already \nknown techniques.  \n \n \n Acknowledgements  \n We gratefully acknowledge S. V . Mironov and Y a. V. Fominov for stimulating and \nfruitful discussions.  This work was financially supported by the Deutsche \nForschungsgemeinschaft through project No. LE 3270/1 -2. It was also partially supported by \nRussian Foundation for Basic Research through project No. 17 -02-00229 -a and the Program of \nthe Russian Academy o f Sciences.  \n \nReferences  \n[1]  Prinz, G. A. Magnetoelectronics. Science  1998,  282, 1660-1663 . \n[2]  Irkhin, V. Yu.; Katsel’son, M. I. Half -metallic magnets. Uspekhi Fizicheskih Nauk 1994,  164, 705 -\n724 [ Physics -Uspekhi  1994,  37, 659-676)].  \n[3]  Kudryavtsev , Y. V.; Uvarov, V. N.; Oksenenko, V. A.; Lee, Y. P.; Kim, J. B.; Hyun, Y. H.; Kim, K. W.; \nRhee, J. Y .; Dubowik, J . 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Spin Imbalance and Magnetoresistance in \nFerromagnet/Superconductor/ Ferromag net Double Tunnel Junctions. Phys. Rev. Lett. 1999 , 82 3911-\n3914.  [28] Eschrig, M.; Cottet, A.; Belzig, W.; Linder, J. General boundary conditions for quasiclassical theory \nof superconductivity in the diffusive limit: application to trongly spin -polarized systems. New J. \nPhys . 2015 , 17, 083037.  \n \n \n \n \n \n \n "    },    {        "title": "0904.0412v2.Probing_interaction_induced_ferromagnetism_in_optical_superlattices.pdf",        "content": "Probing Interaction-Induced Ferromagnetism in\nOptical Superlattices\nJ. von Stecher,1E. Demler,2;3M. D. Lukin,2;3and A. M. Rey1\n1JILA, University of Colorado and National Institute of Standards and Technology,\nBoulder, Colorado 80309-0440,\n2Physics Department, Harvard University, Cambridge, Massachusetts, 20138,\n3Institute for Theoretical Atomic, Molecular and Optical Physics, Cambridge,\nMassachusetts 02138, USA\nAbstract. We propose a method for controllable preparation and detection of\ninteraction-induced ferromagnetism in ultracold fermionic atoms loaded in optical\nsuperlattices. First, we discuss how to probe and control Nagaoka ferromagnetism in\nan array of isolated plaquettes (four lattice sites arranged in a square). Next, we allow\nfor weak interplaquette tunneling. Since ferromagnetism is unstable in the presence\nof weak interplaquette couplings, we propose to mediate long-range ferromagnetic\ncorrelations via double-exchange processes by exciting atoms to the \frst vibrational\nband. We calculate the phase diagram of the two-band plaquette array and discuss\nconditions for the stability and robustness of the ferromagnetic phases in this system.\nExperimental implementations of the proposed schemes are discussed.arXiv:0904.0412v2  [cond-mat.quant-gas]  22 Mar 2010Probing Interaction-Induced Ferromagnetism in Optical Superlattices 2\n1. Introduction\nThe origin of ferromagnetism in itinerant electron systems remains an important open\nproblem in condensed matter physics. Mean \feld approaches, such as the Hartree-\nFock approximation [1] and the Stoner criterion [2] for ferromagnetic instabilities,\nare extremely unreliable since they overestimate the stability of the magnetic-ordered\nphases [3]. The only rigorous example of ferromagnetism in the generic Hubbard\nmodel [4], predicted by Nagaoka in 1965 [5], was proven for a system with one fewer\nelectron than half \flling (i.e., one hole) in the limit of in\fnite interactions. Even though\nsuch a ferromagnetic state is an iconic example of a strongly correlated many-body state,\nit is highly unstable and counter examples indicating the absence of ferromagnetism with\ntwo or more holes have been found [6].\nThe experimental observation of Nagaoka ferromagnetism (NF) is a challenging\ntask, as it requires a system with a \fnite and controllable number of holes. Even\nthough there have been recent attempts to explore Nagaoka ferromagnetism using\narrays of quantum dots [7], the exponential sensitivity of the tunneling rates to the\ninterdot distance and the random magnetic \feld \ructuations induced by the nuclear\nspin background have prevented its direct experimental observation.\nHere, we propose to use cold fermionic atoms in optical superlattices for the\ncontrollable observation of interaction-induced ferromagnetism. First, we show how\nto probe the onset of NF in an array of isolated plaquettes (four lattice sites in a square\ngeometry). Next, we discuss how to engineer long-range ferromagnetic correlations,\nstarting from the isolated plaquette arrays. Since weak coupling of the plaquettes\ndestroys ferromagnetism [8], we instead propose to use additional atoms loaded in\nexcited bands. The underlying idea is to use Hund's rule couplings [9] to favor local\nferromagnetic alignment among the atoms in di\u000berent bands and a double-exchange\nmechanism (tunneling-induced alignment of the spins) [10] to stabilize ferromagnetic\ncorrelations between adjacent plaquettes. Exact numerical calculations for an array of\nweakly coupled plaquettes con\frm the existence of stable ferromagnetic order in this\ntwo-band setup. Finally, we discuss methods for experimental preparation and detection\nof the ferromagnetic correlations.\n2. Probing Nagaoka ferromagnetism in superlattices.\nThe low-energy physics of fermionic atoms loaded in the lowest vibrational band of an\noptical lattice is well described by the Hubbard Hamiltonian:\n^H=\u0000X\nhr;r0i;\u001bJr;r0^cy\nr\u001b^cr0\u001b+UX\nr^n\"r^n#;r; (1)\nwhereJr;r0=Jis the tunneling energy, and Uis the onsite interaction energy. In\nEq. (1), ^cr\u001bare fermionic annihilation operators, ^ nr\u001b= ^cy\nr\u001b^cr\u001bare number operators,\nr= 1;:::L labels the lattice sites, and hr;r0iin the summation indicates that the sum\nis restricted to nearest neighbors.Probing Interaction-Induced Ferromagnetism in Optical Superlattices 3\nBefore starting let us explicitly state Nagaoka theorem [5]: \\Let the tunneling matrix\nelement between lattice sites randr0be negative, Jrr0<0, for anyr6=r0andU=1\nand let the number of fermions be N=L\u00001, withLthe total number of sites. If\nthe lattice satis\fes certain connectivity condition, then the ground state has total spin\nS=N=2and it is unique, apart from a trivial (N+ 1)-fold spin degeneracy\" .\nThe notion of \\connectivity\" requires that each site in the lattice is contained in\na loop (of non-vanishing Jrr0) and furthermore that the loops should pass through no\nmore than four sites [11]. The requirement Jrr0<0 is just the opposite of that assumed\nin most discussions of the Hubbard model. However, in bipartite lattices, there is always\na canonical transformation connecting Jrr0<0 andJrr0>0.\nFrom the conditions of the theorem it is clear that the minimal geometry to observe\nNagaoka ferromagnetism is a triangle, however a triangle is a trivial example since in\nthis case either the ground state is always a singlet ( case Jr;r0=J >0 ) or it is always\na triplet (J < 0)z. The \frst non trivial example of Nagaoka crossing takes place in a\nplaquette loaded with three fermions(Fig. 1(a)).\nHbL\n0510152025-4.0-3.5-3.0-2.5-2.0-1.5-1.0\nUJEJ\n121824-2.2-2.1-2-1.9\nFigure 1. (a) Schematic representation of a plaquette. (b) Low energy spectrum of\nthree fermions. Solid lines and dash-dotted are S= 1=2 while dashed curve corresponds\ntoS= 3=2. Inset: Zoom in of the spectrum at the Nagaoka crossing. Solid lines are the\nspectrum for zero gradient \feld while dashed lines are the spectrum at \fnite gradient\n\feld.\nThe energy levels of a plaquette loaded with three fermions can be classi\fed\naccording to the total spin Sand the symmetries of the wave function. It is known (e.g.,\nRef. [8]) that for U < Ut\u001918:6Jthe ground state is a degenerate doublet S= 1=2\nstate with\u001c=px\u0006ipysymmetry (the wave function changes phase by \u0006\u0019=2 upon\u0019=2\nrotation). For U > Ut\u001918:6J, the ground state becomes a ferromagmetic S= 3=2\nstate, in agreement with the Nagaoka theorem (Fig. 1(b)). We denote these eigenstates\nasjS= 1=2;Sz;\u001c=\u0006iandjS= 3=2;SziwithSz=\u0000S;:::S and recall that the\nenergies are independent of the Szvalue. The onset of Nagaoka ferromagnetism can be\nunderstood as competition between the kinetic energy and superexchange interactions.\nIn theU!1 limit, double occupancies are energetically suppressed, and the low-energy\nzSince a triangular lattice is not a bipartite lattice, Nagaoka theorem only holds for J <0.Probing Interaction-Induced Ferromagnetism in Optical Superlattices 4\nstates are singly occupied with an energy spectrum given by E=\u00062J;\u0006p\n3J;\u0006J;0. The\nrelevant low-lying eigenstates are the ones with ES=3=2=\u00002JandES=1=2=\u0000p\n3J.\nAsUbecome \fnite, while the fully polarized states remain eigenstates for any Uand\ntheir energy is una\u000bected by interactions, the jS= 1=2;Sz;\u0006istates acquire some\nadmixture of double occupancies, which tend to lower their energy. The energy shift in\ntheS= 1=2 states can be calculated by using second order perturbation theory, yielding\nES=1=2=\u0000p\n3J\u00005J2\nU. The Nagaoka crossing occurs at the Ut=Jvalue when the two\nenergies become equal, Ut= 5=(2\u0000p\n3)J\u001818:66J, in very good agreement with the\nexact diagonalization.\nAn array of plaquettes can be created by superimposing two orthogonal optical\nsuperlattices formed by two independent sinusoidal potentials that di\u000ber in periodicity\nby a factor of two, i.e., V(x) =Vs=2 cos(4\u0019x=\u0015s)\u0000Vl=8 cos(2\u0019x=\u0015s), whereVlis the\nlong lattice depth, Vsis the short lattice depth, and \u0015sis the short lattice wavelength.\nBy controlling the lattice intensities, it is possible to tune the intra- and interplaquette\ntunneling and, in particular, to make the plaquettes independent. Here the axial optical\nlattice is assumed to be deep enough to freeze any axial dynamics. To load the plaquettes\nwith three atoms, one can start by preparing a Mott insulator with \flling factor three in\na 3D lattice and then slowly split the wells along x and y. Since only two fermions with\nopposite spin can occupy the lowest vibrational level, loading three fermions per site\nrequires populating the \frst excited vibrational state before splitting the wells. This\nloading procedure creates plaquettes with S= 1=2.\nSinceSandSzare conserved quantum numbers in clean cold atom set-ups, to\nprobe the Nagaoka transition we require the presence of a weak magnetic-\feld gradient.\nWe choose for this case a \feld pointing along zwith a constant gradient along the x\ndirection B(x) =\u000eEB\n\u0016Bg2x\n\u0015s^ z, where\u0016Bis the Bohr magneton gis the gyromagnetic factor.\nThe magnetic-\feld gradient couples the j3=2istate with some linear combination of\nj1=2;1=2;\u0006iwhich we denote as j1=2;1=2;1i, through a Hamiltonian matrix element\nH3=2;1=2=\u00002=3(1 +p\n3)\u000eEBand leaves another linear combination of j1=2;1=2;\u0006i,\nwhich we denote as j1=2;1=2;2i, uncoupled. It consequently transforms the crossing at\nUtinto an avoided crossing [see inset of Fig. 1 (b)] which can be used to adiabatically\ntransform thej1=2;1=2;1i, ground state for U <Ut, intoj3=2iasUis slowly increased.\nUcan be tuned by a Feshbach resonance in the presence of an external magnetic \feld.\nThe variation ofhSiin the ground state below and above the Nagaoka point can\nbe inferred from a band mapping analysis [12] after collapsing the plaquettes into single\nwells: The states with S= 3=2 andS= 1=2 will occupy three and two bands respectively\nwhen a plaquette is transformed into a single site. Since the state j1=2;1=2;2iremains\ndecoupled, one can improve the band mapping signal by adiabatically turning on the\nmagnetic \feld gradient at the same time the plaquettes are prepared. In this way the\ninitial plaquette con\fguration will be almost in a pure j1=2;1=2;1istate and we will be\nable to fully transform the state into the j3=2iatU >Ut.\nThe energy di\u000berence between the j3=2iandj1=2;1=2;\u0006istates can also be probed\ndynamically. After preparing the state j (0)i= cos\u000bj3=2;1=2i+ sin\u000bj1=2;1=2;1iProbing Interaction-Induced Ferromagnetism in Optical Superlattices 5\none can suddenly turn o\u000b the magnetic-\feld gradient. By measuring the Neel order\nparameter or spin imbalance along the xdirection [NS(t) = 1=2(P\nr=1;2n\"r\u0000n#r\u0000P\nr=3;4n\"r\u0000n#r)] as a function of time, one can track the Nagaoka point by the oscillation\nperiod ofhNS(t)i/cos[(ES=3=2\u0000ES=1=2)t=~]. AsU=J approaches Ut=J, the period\nwill become very long, indicating that the character of ground state has changed. This\nsimple treatment ignores the admixture of states with double occupied sites in the\nj1=2;1=2;1istate. When included, the excitations introduce fast but small oscillations\nof frequency J. Comparisons between the exact and analytic solutions are shown in\nFig. 2. The spin imbalance NS(t) can be experimentally probed by \frst splitting the\nplaquettes into two double wells and then following the same experimental methods used\nfor measuring superexchange interactions [13] that rely on band-mapping techniques and\na Stern-Gerlach \fltering.\n0 20 40 60 80-1.5-1.0-0.50.00.51.01.5\ntimeHmsLSpinImbalanceUJ=13.6\n0 20 40 60 80-1.5-1.0-0.50.00.51.01.5\ntimeHmsLUJ=18.6\n0 20 40 60 80-1.5-1.0-0.50.00.51.01.5\ntimeHmsLUJ=23.5\nFigure 2. Normalized spin population imbalance. At the Nagaoka crossing, the\nenvelope frequency becomes very long, indicating zero-energy splitting between the\nj3=2iandj1=2ilevels.\nThe constant magnetic \feld needed for tuning a Feshbach resonance does not a\u000bect\nthe dynamics since the relative energy spacing of the various levels within a plaquette is\ninsensitive to such magnetic \felds. The big advantage of this probing method is that it\ndoes not require \fxing the same magnetization for the various plaquettes. Consequently,\nwe can relax the temperature constraint for preparing the Mott insulator used for the\ninitial loading. The insensitivity of this probing method to the initial magnetization can\nbe understood by the fact that the dynamic taking place in a plaquette initially loaded\nwithSz=\u00001=2 is identical to that described for the Sz= 1=2 case. Furthermore,\npreparation of plaquettes with Sz=\u00063=2 is energetically suppressed due to the large\nvibrational energy spacing. It should be stressed that \fnite temperature e\u000bects only\ndetermines the \fdelity of the initial preparation of a Mott insulator. Once the Mott\ninsulator is achieved, the observation of ferromagnetism depends on the e\u000eciency of\nimplementing the adiabatic manipulation discussed above.Probing Interaction-Induced Ferromagnetism in Optical Superlattices 6\n3. Engineering long-range ferromagnetic correlations.\nWe now study the more general case in which one allows a weak interplaquette\ntunneling, J0, by lowering the long lattice depth along both the xandydirections\n(or along only x). This procedure generates a 2D (1D) array of plaquettes. In the\nNagaoka regime ( U=J > 18:6) to zero order in J0, the many-body ground state has\na degeneracy of 4N(Nis the number of plaquettes) and is spanned by states of the\nformj\biSz1;:::SzN=Q\nijS= 3=2;Szii. A \fniteJ0breaks the degeneracy between the\nstates, but as long as J0\u001cJ, the occupation of states with Si<3=2 is energetically\nsuppressed. These states can only be populated \\virtually,\" leading to an e\u000bective\nHeisenberg interaction between the various e\u000bective S= 3=2 states at each plaquette [8],\ni.e.,\nHeff=GX\nhi;ji~^Si\u0001~^Sj: (2)\nHere,~^Si= (^Sxi;^Syi;^Szi) are spin 3 =2 operators acting on the pseudospin states\njS= 3=2;Szii, and we have set ~= 1. The interaction coe\u000ecient can be written\nasG=gJ02=J, whereg > 0 is an antiferromagnetic-coupling constant that slowly\nvaries as a function of J=U. Equation (2) explicitly shows the fragility of Nagaoka\nferromagnetism, since a weak coupling among the plaquettes leads to a many-body\nground state with antiferromagnetic correlations.\nTo overcome this limitation, we consider a di\u000berent initial con\fguration. Starting\nwith four atoms per plaquette in the lowest orbital, we excite one of the atoms to a\nnondegenerate vibrational level [see Fig. 3]. This system is described by a two-band\nHubbard Hamiltonian of the form\n^H=\u0000X\nhr;r0i;\u001b;nJn^cy\nrn\u001b^cr0n\u001b+X\nrnUn;n^nrn\"^nrn#+VX\nr^nr1^nr2\n\u0000JexX\nr\u001b\u001b0^cy\n1r\u001b^c1r\u001b0^cy\n2r\u001b0^c2r\u001b (3)\nwhich is characterized by on-site interactions between particles in the ground ( U11\u0011U)\nand excited ( U22\u0011Ue) bands, tunneling in the lower ( J1\u0011J) and upper ( J2\u0011Je)\nbands, and direct ( V) and exchange ( Jex) interactions between the two bands. In the\npresent implementation, V=Jex. In Eq. (3), we have neglected terms that transfer\natoms between bands, since they are energetically suppressed. The energy splitting\nbetween them has been omitted in our rotating frame.\nA single plaquette with three atoms in the lowest band and the fourth in the second\nband exhibits a crossing between an S=1 and an S=2 state (ferromagnetic state) at a\nvalue of ~Uthat is smaller than Ut. Figure 4(a) shows the low energy behavior of this\nsystem. For the parameters used in Fig. 4(a), the crossing occurs at ~U\u00196J. In general,\n~Udepends on J,Je,U, andJex. Figure 4(b) analyzes the existence of a ferromagnetic\nground state as a function of U=J and\u000bJ=Je=J. For this analysis, we assume that the\ninteraction terms are proportional, i.e., Jex=\u000bJexU. Di\u000berent shaded regions displayProbing Interaction-Induced Ferromagnetism in Optical Superlattices 7\nFigure 3. Schematic representation of two coupled double-band plaquettes. Solid\ncircles represent occupied orbitals.\nthe ferromagnetic regions for di\u000berent \u000bJex, and the dashed curves their corresponding\ncritical values ~U. As both \u000bJand\u000bexincrease, the ~Uvalues decrease, extending the\nferromagnetic region. This behavior is consistent with our physical picture that both\ndouble exchange processes and the Hund's rule coupling stabilize ferromagnetism.\n5.766.3-14.5-14.4(a)\n(a)\n02468-16.0-15.5-15.0-14.5-14.00 100 200 300\nUJEJaa0\n012344681012\naJUJ(b)\nFigure 4. (a) Energies of a plaquette as a function of U=J and the scattering length\nin Bohr radii a0. The parameters that characterize the Hamiltonian [Eq. (3)] are\nobtained for a superlattice constructed with a short-wavelength laser of \u0015s= 765 nm\nthat characterizes the short-lattice recoil energy Er=h2=(2m\u00152\ns). The energies of the\n\fgure corresponds to Vl= 20ErandVs= 7:5Er. Inset: Zoom in of the spectrum at\nthe ferromagnetic crossing. Solid lines are the spectrum for zero gradient \feld while\ndashed lines are the spectrum at \fnite gradient \feld. (b) Ferromagnetism in a two-\nband plaquette. Shaded areas correspond to a ferromagnetic ground state. Dashed\ncurves correspond (from top to bottom) to \u000bJex= 0:2;0:4;0:6, and 0.8.\nThe mobile atoms in the excited band are also expected to stabilize, via double-\nexchange processes, the ferromagnetic phase when a weak tunneling between plaquettes\n(J0andJ0\ne) is allowed. The stabilization mechanism relies on the preservation of the\nspin when an atom hops, and on the energy penalty of 2 Jexwhen ground and excited\natoms form a singlet instead of a triplet at a given site. Only when the spins of\nadjacent plaquettes are fully aligned, the mobile atoms are free to hop. We con\frm theProbing Interaction-Induced Ferromagnetism in Optical Superlattices 8\nstabilization of the ferromagnetic correlations by studying the weakly coupled regime,\nwhich can be described again by an e\u000bective Heisenberg Hamiltonian as in Eq. (2), but\nnow between the S= 2 states at each plaquette.\n012345-1.5-1.0-0.50.0\naJg¥,ge¥H10-2LHaL\n501001502000.00.10.20.30.40.50.60.7\nUJaUe(b)\nFigure 5. (a) Asymptotic coe\u000ecients g1andg1\neas functions of \u000bJ. The solid\ncurve corresponds to g, and the dotted curve corresponds to ge. The dashed curve\ncorresponds to gewhen\u000bUe= 0. (b) Phase diagram for J0= 0. The shaded regions\ncorrespond to the ferromagnetic regime for di\u000berent \u000bJvalues. From left to right,\n\u000bJ= 0:5,\u000bJ= 1:1,\u000bJ= 2.\nThe coupling coe\u000ecient Gcan be obtained by considering virtual processes in\nwhich one atom from the ground (excited) band hops from one plaquette to the other\nand returns to the original con\fguration. In practice, we extract Gfrom the analysis\nof the low-energy spectrum of two weakly coupled plaquettes [see Fig. 3]. Gdepends\nin a nontrivial way on the interaction and kinetic energy parameters and it is given\nbyG=gJ02=J+geJ02\ne=JeforJ0;J0\ne\u001cJ;Je. Heregandgeare proportional to\nGwhen the interplaquette tunneling is allowed only in the ground or excited band,\nrespectively. Both depend on J,Je,U, andJex; andgealso depends on Ue. In the\ne\u000bective model, the existence of ferromagnetism can be directly deduced from the sign\nofG. To further understand the robustness of the ferromagnetic phase, we consider\nthe general case for which all the interaction terms are proportional to each other,\nUe=\u000bUeU,V=Jex=\u000bJexU, and the kinetic terms are related as Je=\u000bJJ. For large\ninteraction values ( U!1 ), the coe\u000ecients gandgeapproach to their asymptotic\nvaluesg1andg1\nethat only depend on \u000bJ(independent of \u000bJexand\u000bUeso long as\n\u000bJex>0 and\u000bUe>0). Figure 5 (a) shows that both g1andg1\nebecome negative\nfor a large parameter regime, con\frming the robustness of the ferromagnetic phase. If\nUe= 0 [dashed curve in Fig. 5 (a)], g1\nebecomes positive for an important region of \u000bJ\nvalues. This \fnding is consistent with Ref. [14] where it is pointed out that a nonzero\ninteraction between atoms in the excited band ( Ue>0) can be crucial for the transition\nto a ferromagnetic ground state.\nFigure 5 (b) shows the phase diagram as a function of \u000bUeandU=J for three\ndi\u000berent\u000bJ= 0:5, 1.1, 2 values, with the assumption that J0= 0. For this study, we\nset\u000bJex= 0:4, which is a typical value for optical superlattices. In general, we observe\nthat as\u000bJexincreases, ferromagnetism becomes more favorable. The transition to aProbing Interaction-Induced Ferromagnetism in Optical Superlattices 9\nferromagnetic ground state depends strongly on \u000bJand, for the J0= 0 case, low values\nof\u000bJenhance ferromagnetism. For J0\ne= 0, the phase diagram is independent of \u000bUe,\nand higher mobility of the excited atoms, large \u000bJ, favors ferromagnetic correlations.\nFor example, for \u000bJ= 2 the critical value Ucis\u0019147J. It increases to Uc\u0019488Jfor\n\u000bJ= 1:1, and for\u000bJ= 0:5, no ferromagnetic phase is observed.\n50100150200-20246\nUJGJH10-3L\n50 100 150 200-6-4-20220003000400050006000\nUJEJH10-2Laa0\nS=0\nS=3\nS=4S=1\nS=2\nFigure 6. Lowest energies as a function of U=J and the scattering length of two\nweakly coupled plaquettes ( Vl= 4ErandVs= 5:5Er) with six particles in the lowest\nband and two in the excited band and a total Sz= 0. Circles correspond to exact\nnumerical calculations, and lines correspond to the e\u000bective Hamiltonian [Eq. (2)]\ndescription. Top grid: scattering length values in Bohr radii for6Li parameters. Inset:\nTheGcoe\u000ecient as a function of U/J for a plaquette with Vl= 4ErandVs= 5:5Er\n(dashed curve). For this case, the tunneling is J\u00190:085Er. The transition occurs at\nU=J\u0019100.\nHowever, in realistic experimental setups it is not possible to explore the complete\nparameter space. For standard superlattice geometries, in which only the ratio Vl=Vs\ncontrols the relation between the di\u000berent tunneling parameters, we \fnd that a favorable\nferromagnetic scenario takes place in the regime when J=JeandJ0=J0\ne. We\ncan achieve this scenario by loading the atoms in the lowest vibrational state of a\n2D plaquette array (tight con\fnement along z) creating a 2D lattice in the x\u0000y\ndirection and and then exciting one atom per plaquette to the \frst vibrational\norbital along the zdirection. The atoms are initially all in the ground-state orbital\n\bg(r) =\u001e0(x)\u001e0(y)e\u001e0(z), and one of the atoms in each plaquette is excited to the\n\be(r) =\u001e0(x)\u001e0(y)e\u001e1(z) vibrational state. The tunneling along the xandydirections\nis independent of the z-orbital ( e\u001e0(z) ore\u001e1(z)) and, therefore, J=JeandJ0=J0\ne.\nThe tunneling in the zdirection depends on the z-orbital but it is negligible in the 2D\ngeometry in consideration.\nFigure 6 presents this case for realistic6Li experimental parameters ( \u000bJex= 0:5 and\n\u000bUe= 0:7) where we observe a change in the sign of the coupling coe\u000ecient G(see inset)\nfrom positive to negative at U=J\u0018100, a value of interaction accessible with a FeshbachProbing Interaction-Induced Ferromagnetism in Optical Superlattices 10\nresonance. Even though for the perturbative regime under consideration, where the\ne\u000bective Hamiltonian is valid, Gis small ( of the order of few Hz), it is measurable with\nstate-of-the-art technology, as demonstrated in recent experiments where superexchange\ninteractions as low as 5 Hz have been resolved [13].\nFor preparing and probing ferromagnetism in the isolated two-band plaquette array,\nwe propose to start by loading atoms into a band insulator in a deep 3D lattice[15, 16].\nBy applying a double-well superlattice along the z-direction and manipulating the\ndouble-well bias, one can excite one of the four atoms in each double well to the \frst\nvibrational state via interaction blockade [17]. Then, by slowly merging the double wells\nalong thez-direction and splitting them along the x\u0000ydirections, one can load the\ndesired plaquette con\fguration with three atoms in the lowest and the fourth in the \frst-\nexcited vibrational state along z. Alternatively, spatially selective two-photon Raman\npulses can be applied to excite one atom in each plaquette to the desired vibrational\nstate [18, 19, 20]. This procedure will lead to plaquettes with S= 0. The state\nS= 0 can be adiabatically converted into S= 1 by applying a nonlinear magnetic \feld\ngradientxalongzand tuning the interactions from U < 0 toU > 0. The transition\nbetweenS= 1 andS= 2 states in a single two-band plaquette can be probed using the\nsame experimental techniques proposed for the single band plaquette. A magnetic \feld\ngradient couples the S= 1 andS= 2 states [see inset of Fig. 4(a)] and this coupling\ncan be used to probe the transition with spin-imbalance measurements or band mapping\ntechniques.\nInteraction induced ferromagnetism in a few coupled plaquette array could also be\nobserved by varying U=J in the presence of a magnetic \feld gradient. Since our initially\nprepared state is a band insulator with S= 0 the magnetic \feld gradient has to be\nlarge enough to couple S= 0 withS=Smaxacross the transition. After this procedure\nis applied the ferromagnetic nature of the ground state can be inferred in the applied\nmagnetic-\feld gradient by measuring the local magnetization of the system [21, 22, 23].\nA linear magnetic-\feld gradient produces a perturbation in the e\u000bective Hamiltonian of\nthe formHp=P\nii\u000eEp^Szi=~, where\u000eEpis the average energy shift between consecutive\nplaquettes. In the ferromagnetic phase, the formation of a domain wall is expected. The\ndomain-wall width will be determined by the dimensionless parameter zGS=\u000eE p, wherez\nis the number of nearest-neighbor plaquettes, and S= 2. The measurement of this width\ncan be used to extract Gin the ferromagnetic regime. In the antiferromagnetic phase on\nthe contrary, no domain wall will be formed, and the local Neel order parameter should\nvary smoothly. The onset of ferromagnetic correlations as the system is driven through\nthe critical point should be signaled by a suppression in inelastic collisions, a minimum\nin kinetic energy, and a maximum in the size of the cloud. The latter signatures have\nbeen demonstrated to be useful smoking guns of a ferromagnetic transition in recent\nexperiments carried on in fermionic gases without a lattice (See Ref. [21] and references\ntherein).\nxOur calculations show that a simple linear gradient does not couple the S= 1 andS= 0 states.Probing Interaction-Induced Ferromagnetism in Optical Superlattices 11\n4. Conclusions\nIn summary, we have proposed a controllable and experimentally realizable scheme to\nstudy interaction-induced ferromagnetism in ultracold atoms. Our method exploits the\nadvantage o\u000bered by these systems to divide the full lattice into plaquettes.\nWe showed that a plaquette loaded with three fermionic atoms is a promising\nset-up to experimentally observe Nagaoka ferromagnetism for the \frst time. We also\nused the plaquettes as the fundamental building blocks to create interaction induced\nferromagnetism at longer length scales. We analyzed the system of weakly coupled\ntwo-band plaquettes and demonstrated that in the limit where the interplaquette\ncoupling can be treated perturbatively, the system maps out into an e\u000bective Heisenberg\nHamiltonian with a coupling constant Gwhich changes sign from positive to negative as\ninteractions are increased. The change in sign is a manifestation of an antiferromagnetic\nto ferromagnetic transition.\nIn the perturbative regime, where our analysis is valid, the coupling parameter\nGis small. Consequently, it will be experimentally challenging to adiabatically reach\nthe ferromagnetic ground state for large number of coupled plaquettes by increasing\ninteractions. In this situation, the required temperature would be smaller than\nG\u001810\u00002J. Nevertheless, exact diagonalization in a two-plaquette array con\frmed the\npersistence of itinerant ferromagnetism beyond the weakly coupling regime. Actually,\nwe found excellent agreement between our perturbative Hamiltonian and the many-\nbody spectrum even at values of J0=Jas high as 1 =4, as shown in the low energy\nspectrum presented in Fig. 6. Our two-plaquette results seem to be consistent with\nvariational Monte Carlo [24] and dynamical mean-\feld [25] predictions that have found\nferromagnetic phases in the two-band generic square-lattice Hubbard model. The\nactual stability of the ferromagnetic phase in larger plaquette arrays and stronger\ninterplaquette couplings will need however to be resolved ultimately by experiments.\nAcknowledgments\nThis work was supported by NSF, ITAMP, CUA and DARPA.\nReferences\n[1] D. R. Penn, Phys. Rev. 142, 350 (1966).\n[2] E. Stoner, Proc. R. Soc. London 165, 372 (1938).\n[3] P. Fazekas, B. Menge, and E. M uller-Hartmann, Z. Phys. B 78, 69 (1990). A. N. Tahvildar-Zadeh,\nJ. K. Freericks, and M. Jarrell, Phys. Rev. B 55, 942 (1997).\n[4] Some examples of ferromagnetism in Hubbard models with complex lattice geometries have also\nbeen predicted: H. Tasaki, Phys. Rev. Lett. 75, 4678 (1995), S. Zhang, H. Hung and C. Wu,\narXiv:0805.3031, .\n[5] Y. Nagaoka, Phys. Rev. 147, 392 (1966).\n[6] M. Takahashi, J. Phys. Soc. Japan 51, 3475 (1982). Y. Fang et al. , Phys. Rev. B 40, 7406 (1989);\nB. Doucot and X. G. Wen, Phys. Rev. B 40, 2719 (1989).Probing Interaction-Induced Ferromagnetism in Optical Superlattices 12\n[7] E. Nielsen and R. N. Bhatt, Phys. Rev. B 76, 161202(R) (2007).\n[8] H. Yao, W. F. Tsai, and S. A. Kivelson, Phys. Rev. B 76, 161104 (2007).\n[9] A. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge,\n1993). H. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod. Phys 69, 809 (1997); M. Gulacsi,\nPhil. Mag. 86, 1907 (2006).\n[10] C. Zener, Phys. Rev. 81, 440 (1951).\n[11] H. Tasaki, Progr. Theor. Phys. 99, 489 (1998).\n[12] M. Greiner, I. Bloch, O. Mandel, T. W. Hansch, and T. Esslinger, Phys. Rev. Lett. 87, 160405\n(2001).\n[13] S. Trotzky, P. Cheinet, S. Folling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A.\nDemler, M. D. Lukin, and I. Bloch, Science 319, 295 (2008).\n[14] P. Simon and D. Loss, Phys. Rev. Lett. 98, 156401 (2007).\n[15] R. J ordens, N. Strohmaier, K. G unter, H. Moritz, T. Esslinger, Nature (London) 455, 204-207\n(2008)\n[16] U. Schneider, L. Hackermuller, S. Will, T. Best, I. Bloch, T. A. Costi, R. W. Helmes, D. Rasch,\nA. Rosch, Science 322, 1520 (2008).\n[17] P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger, M. Moreno-Cardoner, S. Foelling, and I. Bloch,\nPhys. Rev. Lett. 101, 090404 (2008).\n[18] T. M uller, S. F olling, A. Widera, I. Bloch, Phys. Rev. Lett. 99, 200405 (2007)\n[19] B. Paredes and I. Bloch, Phys. Rev. A 77, 023603 (2008).\n[20] A. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. Lukin, Phys. Rev. Lett. 100, 093005 (2008).\n[21] G. B. Jo, Y. R. Lee, J. H. Choi, C. A. Christensen, T. H. Kim, J. H. Thywissen, D. E. Pritchard,\nW. Ketterle, Science 325, 1521 (2009)\n[22] D. Weld, P. Medley, H. Miyake, D. Hucul, D. Pritchard, and W. Ketterle, Phys. Rev. Lett. 103,\n245301 (2009)\n[23] M. Babadi, D. Pekker, R. Sensarma, A. Georges, E. Demler, arXiv:0908.3483 (2009).\n[24] K. Kubo, Phys. Rev. B 79, 020407 (2009).\n[25] K. Held and D. Vollhardt, Eur. Phys. J. B 5, 473 (1998)."    },    {        "title": "1903.06841v1.Novel_critical_behavior_of_magnetization_in_URhSi_Similarities_to_uranium_ferromagnetic_superconductors_UGe__2__and_URhGe.pdf",        "content": "arXiv:1903.06841v1  [cond-mat.str-el]  15 Mar 2019APS/123-QED\nNovel critical behavior of magnetization in URhSi:\nSimilarities to uranium ferromagnetic superconductors UG e2and URhGe∗\nNaoyuki Tateiwa1,†Yoshinori Haga1, and Etsuji Yamamoto1\n1Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai,\nNaka, Ibaraki 319-1195, Japan\n(Dated: March 19, 2019)\nWe study the critical behavior of dc magnetization in the ura nium ferromagnet URhSi around\nthe paramagnetic to ferromagnetic phase transition at TC∼10 K with a modiﬁed Arrott plot, a\nKouvel-Fisher plot, the critical isotherm analysis and the scaling analysis. URhSi is isostructural to\nuranium ferromagnetic superconductors URhGe and UCoGe. Th e critical exponent βfor the tem-\nperature dependence of the spontaneous magnetization belo wTC,γfor the magnetic susceptibility,\nandδfor the magnetic isotherm at TCin URhSi have been determined as β= 0.300 ±0.002,γ=\n1.00±0.02, and δ= 4.38±0.04 by the scaling analysis and the critical isotherm analy sis. These\ncritical exponents fulﬁll the Widom scaling law δ= 1 +γ/β. Magnetization has strong uniaxial\nmagnetic anisotropy in the ferromagnetic state of URhSi. Ho wever, the universality class of the\nferromagnetic transition does not belong to the 3D Ising sys tem with short-range exchange interac-\ntions between magnetic moments ( β= 0.325, γ= 1.241, and δ= 4.82). The obtained exponents in\nURhSi are similar to those in the uranium ferromagnetic supe rconductors UGe 2and URhGe, and\nuranium ferromagnets UIr and U(Co 0.98Os0.02)Al. We have previously reported the unconventional\ncritical behavior of magnetization in the uranium ferromag netic superconductors [N. Tateiwa et al.\nPhys. Rev. B 89, 064420 (2014)]. The universality class of the ferromagnet ic transition in URhSi\nmay belong to the same one in the uranium ferromagnetic super conductors and the uranium fer-\nromagnets. The unconventional critical behavior of the mag netization in the uranium compounds\ncannot be understood with previous theoretical interpreta tions of critical phenomena. The absence\nof the superconductivity in URhSi is discussed from several viewpoints. The improvement of the\nsample quality in URhSi could provide a good opportunity to g ain a deeper understanding of the\nferromagnetic superconductivity in the uranium ferromagn ets.\nPACS numbers:\nI. INTRODUCTION\nA. General introduction\nMany experimental and theoretical studies have been\ndone for intriguing physical properties in uranium com-\npounds with 5 felectrons such as mysterious “hidden\norder” in URu 2Si2, unconventional superconductivity in\nUPt3or UBe 13, and the coexistence of the superconduc-\ntivity and antiferromagnetismin UPd 2Al3or UNi 2Al3[1–\n3]. The most unique feature ofuranium 5 fsystems is the\ncoexistence of the superconductivity and ferromagnetism\nboth carried by the same 5 felectrons in UGe 2, URhGe,\nand UCoGe[3–8]. Novel physical phenomena associated\nwith a quantum phase transition between ferromagnetic\nand paramagnetic states have been the subjects of exten-\nsive researches from both experimental and theoretical\nsides[9].\nIt is important to understand detailed ferromagnetic\npropertiesin the uranium ferromagneticsuperconductors\nUGe2, URhGe, and UCoGe for a better understanding\n∗Phys. Rev. B 99, 094417 (2019).\n†Electronic address: tateiwa.naoyuki@jaea.go.jpof the superconductivity. This is because ferromagnetic\ninteractions between the 5 felectrons may play an im-\nportant role for the appearance of the superconductivity\nin the ferromagnetic state as theoretically shown[10–13].\nThe ferromagnetic states in UGe 2, URhGe, and UCoGe\naremagneticallyuniaxial[14–16]. Theuraniumferromag-\nnetic superconductors have been regarded as a three-\ndimensional (3D) Ising system. We focus on a classical\ncriticalbehaviorofthe magnetizationaroundaferromag-\nnetic transition temperature from which the type of the\nmagnetic phase transition and the nature of magnetic\ninteractions can be studied[17]. We have previously re-\nported that the universality class of the critical phenom-\nena in UGe 2and URhGe does not belong to any known\nuniversality classes of critical phenomena[18]. The ferro-\nmagnetism of the uranium ferromagnetic superconduc-\ntors may not be described only with the 3D Ising model.\nIn this paper, we report the novel critical behavior of\nthe magnetization in URhSi. The compound crystalizes\nin the same orthorhombic TiNiSi-type crystal structure\n(space group Pnma) to those of the uranium ferromag-\nnetic superconductors URhGe and UCoGe[19]. URhSi\nshows a ferromagnetic transition at the Curie tempera-\ntureTC∼10 K. The superconductivity has not been ob-\nserved down to 40 mK[20]. The ferromagnetic state in\nURhSi has uniaxial magnetic anisotropy with the mag-2\nnetic easy axis parallel to the caxis in the orthorhombic\ncrystal structure[20], which is similar to the uranium fer-\nromagnetic superconductors URhGe[15] and UCoGe[16].\nWe ﬁnd that the universality class of the critical phe-\nnomenon in URhSi does not belong to the 3D Ising\nmodel. The values of the critical exponents in URhSi\nare similar to those in URhGe and UGe 2. The univer-\nsality class of the ferromagnetic transition in URhSi may\nbelong to the same one in URhGe and UGe 2. We discuss\nthe static and dynamical magnetic properties of URhSi\nin comparison with those of URhGe, UCoGe, and UGe 2.\nPossible reasons for the absence of the superconductivity\nin URhSi are discussed.\nB. Physical properties in URhSi and comparison\nwith URhGe, UCoGe, and UGe 2\nWe summarize the crystal structure and basic physical\nproperties of URhSi in this subsection. Figure 1 shows\nthe orthorhombic TiNiSi-type crystal structure and Ta-\nble I shows the structural parameters of URhSi. Here,\nBeqis the equivalent isotropic atomic displacement pa-\nrameter. Lattice parameters at room temperature are\ndetermined as a= 0.69970(4) nm, b= 0.42109(2) nm,\nandc= 0.74458(4) nm by single-crystal x-ray diﬀraction\ntechniques using an imagingplate areadetector (Rigaku)\nwith Mo K αradiation. The distances between the ura-\nnium atoms are d1= 0.3638 nm and d2= 0.3408 nm\nalong the aandbaxes, respectively. The structure can\nbe regarded as coupled chains of the nearest-neighbor\nuranium atoms (“zigzag chain”) running along the crys-\ntallographic baxis. Meanwhile, the distance d1is shorter\nthand2in URhGe and UCoGe[3]. The crystal structure\nof both compounds can be viewed as the coupled zigzag\nchains along the aaxis.\nTableII tabulatesthe valuesof TC,peﬀ,ps,T0,TA, and\nTC/T0for URhSi, URhGe, UCoGe, and UGe 2. Here,\npeﬀandpsare the eﬀective and the spontaneous mag-\nnetic moments, respectively. The deﬁnitions of T0and\nTAwill be explained later. The values of the parameters\nfor URhSi are determined in this study. UGe 2orders\nferromagnetically at the relatively high Curie tempera-\ntureTCof 52.6 K with the large spontaneous magnetic\nmoment ps= 1.41µB/U[18]. Meanwhile, UCoGe shows\nthe ferromagnetictransitionat TC=2.4Kwith the small\nspontaneousmagneticmoment ps=0.0039 µB/U[7]. The\nvalues of TCandpsin URhSi are similar to those in\nURhGe[18]. Neutron diﬀraction studies on URhSi have\nshown a collinear ferromagnetic structure with an ura-\nnium magnetic moment of 0.50 - 0.55 µB/U oriented\nalong the caxis[21, 22]. The ferromagnetic structure is\nthe same as those in URhGe and UCoGe[6, 23]. The lin-\near speciﬁc heat coeﬃcient γin URhSi was determined\nasγ= 164.2 mJ/(mol ·K2) in the ferromagnetic ordered\nstate[20]. This value is almost the same as that [ ∼160\nmJ/(mol ·K2)] in URhGe estimated from the C/Tvalue\njust above the superconducting transition temperature\n(a)\n(b) (c)\nacabc\nba\ncbd2\nd1U\nRh \nSi\nd2d1\nd2\nd1\nFIG. 1: (a) Representation of the orthorhombic TiNiSi-type\ncrystal structure of URhSi. Projections of atoms on (b) the\nbcand (c) the acplanes.\nTABLE I: Crystallographic parameters for URhSi at room\ntemperature in the orthorhombic setting (space group Pnma)\nwith lattice parameters a= 0.69970(4) nm, b= 0.42109(2)\nnm, and c= 0.74458(4) nm. The conventional unweighted\nand weighted agreement factors of R1andwR2are 3.47 and\n9.01%, respectively.\nAtom Site x y z B eq(nm2)\nU 4(c) 0.00257(6) 1/4 0.18536(7) 5 .1(2)×10−3\nRh 4(c) 0.15004(17) 1/4 0.57177(14) 7 .7(3)×10−3\nSi 4(c) 0.7870(7) 1/4 0.6056(5) 5 .7(6)×10−3\nTsc[24]. The γvalue in UCoGe [= 57 mJ/(mol ·K2)] is\nabout one-third of those in URhSi and UCoGe[7]. There\nare several similarities in the basic physical properties\nbetween URhSi and UCoGe.\nNext, we compare the dynamical magnetic property\nin URhSi with those in the uranium ferromagnetic su-\nperconductors UGe 2, URhGe, and UCoGe. Recently, we\nhave studied the applicability of Takahashi’s spin ﬂuctu-\nation theory to the actinide 5 fsystems[25–28]. We ana-\nlyzed the magnetic data of 80 actinide ferromagnets and\ndetermined spin ﬂuctuation parameters T0andTA: the\nwidths ofthe spin ﬂuctuation spectrum in the energyand3\n/s49/s49/s48/s49/s48/s48 /s112/s101/s102/s102/s32/s47/s112/s115\n/s48/s46/s48/s49 /s48/s46/s49 /s49\n/s84/s67/s47/s84/s48/s32/s85/s114/s97/s110/s105/s117/s109/s32/s99/s112/s100/s115\n/s32/s78/s101/s112/s116/s117/s110/s105/s117/s109\n/s32/s80/s108/s117/s116/s111/s110/s105/s117/s109\n/s32/s51/s100/s32/s115/s121/s115/s116/s101/s109\n/s32\n/s85/s82/s104/s71/s101/s85/s67/s111/s71/s101\n/s85/s71/s101/s50/s78/s112/s79/s115/s50\n/s78/s112/s65/s108/s50/s80/s117/s65/s115/s90/s114/s90/s110/s50\n/s70/s101/s78/s105/s51/s65/s108/s49/s45/s120/s71/s97/s120/s85/s82/s104/s83/s105\nFIG. 2: Generalized Rhodes-Wohlfarth plot for uranium, nep -\ntunium and plutonium ferromagnets, and the 3 dmetals and\ntheir intermetallic ferromagnetic compounds shown as clos ed\ncircles, squares, triangles and anti-triangles, respecti vely[25].\nThe data for UCoGe and the 3 dsystems are cited from the\nliterature[26–32]. Solid line shows a theoretical relatio n be-\ntweenTC/T0andpeﬀ/psin the Takahashi’s spin ﬂuctuation\ntheory[26–28].\nTABLE II: Basic magnetic and spin ﬂuctuation parame-\nters for URhSi, and uranium ferromagnetic superconductors\nURhGe[18, 25], UCoGe[7, 29] and UGe 2[18, 25].\nTCpeﬀ psT0TATC/T0\n(K) (µB/U) (µB/U) (K) (K)\nURhSi 10.5 2.94 0.571 64.5 354 0.163\nURhGe 9.47 1.75 0.407 78.4 568 0.121\nUCoGe 2.4 1.93 0.039 362 5.92 ×1030.0065\nUGe252.6 3.00 1.41 92.2 442 0.571\nmomentum spaces, respectively. Figure 2 shows the plot\nofpeﬀ/psandTC/T0(the generalized Rhodes-Wohlfarth\nplot) for the actinide ferromagnets, and the 3 dmet-\nals and their intermetallic ferromagnetic compounds[25].\nThe data for uranium, neptunium, and plutonium com-\npounds, and the 3 dsystems are plotted as closed cir-\ncles, squares, triangles, and anti-triangles, respectively.\nThe data for URhSi, URhGe, UCoGe, and UGe 2are\nhighlighted. A solid line represents a theoretical rela-\ntion between TC/T0andpeﬀ/psin the Takahashi’s spin\nﬂuctuation theory. The data for UCoGe and the 3 dsys-\ntems are cited from the literature[26–32]. The parame-\nters of the other actinide compounds were determined by\nus[25]. The data of the actinide ferromagnets follow the\ntheoretical relation for TC/T0<1.0. This suggests the\napplicability of the theory to most of the actinide ferro-\nmagnets. Several data points deviate from the relationnearTC/T0= 1, which may be due to some eﬀects aris-\ning from the localized character of the 5 felectrons not\nincluded in the theory. In the spin ﬂuctuation theory,\nthe degree of the itinerancy of magnetic electrons can be\ndiscussed from the parameter TC/T0[26–28]. The strong\nitinerant character of the magnetic electrons is suggested\natTC/T0≪1 and a relation TC/T0= 1 indicates the lo-\ncal moment ferromagnetism. The value of TC/T0= 0.571\nfor UGe 2suggests that it is located comparably close to\nthe local moment system. Meanwhile, the small values of\nTC/T0= 0.0065 and psfor UCoGe suggest the weak fer-\nromagnetism, similar to those in Y(Co 1−xAlx)2[30] and\nNi3Al1−xGax[31]. URhSi and URhGe are located in an\nintermediate region between the two limiting cases. The\ntwo uranium ferromagnets share several similarities in\nterms of the basic physical and the dynamical magnetic\nproperties.\nII. EXPERIMENT AND ANALYSIS\nA single-crystal sample of URhSi was grown by\nCzochralski pulling in a tetra arc furnace. The value\nof the residual resistivity ratio (RRR) ( = ρRT/ρ0) is\nabout 2.5. Here, ρRTandρ0represent the resistivity\nvalue at room temperature and the residual resistivity at\nlowtemperatures, respectively. Impuritiesormisoriented\ngrains were not detected in the x-ray diﬀraction experi-\nment on the single-crystal sample for this study. Magne-\ntization was measured in a commercial superconducting\nquantum interference (SQUID) magnetometer (MPMS,\nQuantum Design). The internal magnetic ﬁeld µ0Hwas\nobtained by subtracting the demagnetization ﬁeld DM\nfrom the applied magnetic ﬁeld µ0Hext:µ0H=µ0Hext\n-DM. The demagnetizing factor D(= 0.22) was cal-\nculated from the macroscopic dimensions of the sample.\nWe determine the critical exponents using a modiﬁed Ar-\nrott plot, critical isotherm analysis, a Kouvel-Fisherplot,\nand scaling analysis.\nIII. RESULTS\nWe show the temperature dependencies of the mag-\nnetic susceptibility χand its inverse 1 /χin a magnetic\nﬁeld of 0.1 T applied along the magnetic easy caxis\nof URhSi in Fig. 3(a). The magnetic susceptibility\nχwas analyzed using a modiﬁed Curie-Weiss law χ=\nC/(T−θ) +χ0shown as solid line. Here, Candθare\nthe Curie constant and the paramagnetic Curie temper-\nature, respectively. χ0is the temperature-independent\ncomponent of the magnetic susceptibility from the den-\nsity of states at the Fermi energy from other than the\n5felectrons. The eﬀective magnetic moment peﬀis de-\ntermined as peﬀ= 2.94µB/U per uranium atom from C\n=NAµ2\nBp2\neﬀ/3kB. Here, NAis the Avogadro constant.\nThe smaller value of peﬀthan those expected for 5 f2\n(U4+,peﬀ= 3.58µB/U)and 5 f3(U3+,peﬀ=3.62µB/U)4\n/s52/s48/s48\n/s51/s48/s48\n/s50/s48/s48\n/s49/s48/s48\n/s48/s99/s45/s49/s32/s91/s101/s109/s117/s47/s109/s111/s108/s93/s45/s49\n/s32\n/s51/s48/s48 /s50/s48/s48 /s49/s48/s48 /s48\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s91/s75/s93/s51\n/s50\n/s49\n/s48/s99/s32/s91/s101/s109/s117/s47/s109/s111/s108/s93\n/s32\n/s109/s48/s72/s32/s47/s47/s32/s99\n/s32/s32/s32/s32/s32/s32/s32/s61/s32/s48/s46/s49/s32/s84/s85/s82/s104/s83/s105\n/s84/s67/s40/s97/s41\n/s48/s46/s56\n/s48/s46/s54\n/s48/s46/s52\n/s48/s46/s50\n/s48/s46/s48/s77/s32/s91/s109/s66/s47/s85/s93\n/s32\n/s54 /s52 /s50 /s48\n/s109/s48/s72/s32/s91/s84/s93/s109/s48/s72/s32/s47/s47/s32/s99\n/s32/s109/s48/s72/s32/s47/s47/s32/s98\n/s32/s32/s32/s32/s32/s32/s32/s47/s47/s32 /s97/s85/s82/s104/s83/s105\n/s84/s32/s61/s32/s50/s46/s48/s32/s75/s40/s98/s41\n/s84/s32/s61/s32/s50/s46/s48/s32/s75/s68/s84/s32/s61\n/s50/s46/s48/s32/s75/s32/s54/s46/s48/s32/s75\n/s32\n/s32/s50/s48/s32/s75\nFIG. 3: (a)Temperature dependencies of the magnetic sus-\nceptibility χand its inverse 1 /χin a magnetic ﬁeld of 0.1\nT applied along the magnetic easy caxis in URhSi. Solid\nline represents the result of the ﬁt to the inverse of the mag-\nnetic susceptibility 1 /χusing a modiﬁed Curie-Weiss law. (b)\nMagnetic ﬁeld dependencies of the magnetization at several\ntemperatures in magnetic ﬁeld applied along the caxis, and\nthe magnetization at 2.0 K in ﬁelds along the magnetic hard\nbandaaxes in URhSi.\nconﬁgurations suggests the itinerant character of the 5 f\nelectrons in URhSi. We show the magnetic ﬁeld depen-\ndencies of the magnetization at several temperatures in\nmagnetic ﬁeld applied along the magnetic easy caxis of\nURhSi in Fig. 3 (b). The spontaneous magnetic moment\npsis determined as ps= 0.571 µB/U from the magneti-\nzation curve at 2.0 K. The value of psis consistent with\nthose (0.50−0.55µB/U) determined by the elastic neu-\ntron scattering studies[21, 22]. The value is smaller than\nthemagneticmoment µ[=µU+µRh=0.66(2)+0.05(2)=\n0.71(4)µB] determined at 2 K with magnetic ﬁeld of 6 T\nby the polarized neutron scattering experiment[33]. The\nreason for this discrepancy is not clear. The magnetiza-\ntion curves in ﬁelds along the magnetic hard aandbaxes\nat 2.0 K are also shown in Fig. 3 (b). Clearly, the fer-romagnetic ordered state has large magnetic anisotropy,\nsimilar to those in the uranium ferromagnetic supercon-\nductors UGe 2, URhGe, and UCoGe[14–16]. This uniax-\nial magnetic anisotropy is consistent with the collinear\nferromagnetic structure with the uranium magnetic mo-\nmentsorientedalongthe caxisdeterminedintheneutron\nscattering studies[21, 22].\nThere are diﬀerences in the magnetization curves be-\ntween the presentand the previous studies[20, 22]. In the\nprevious studies, the value of psis less than 0.5 µB/U in\nthe magnetization curve along the caxis and the sponta-\nneous magnetic moment above 0.1 µB/U occurs in mag-\nnetic ﬁelds applied along the magnetic hard aandbaxes.\nThe magnetization along the aaxis is slightly largerthan\nthat along the baxis at 2.0 K[20, 22]. These features are\nnot consistent with the present data shown in Fig. 3 (b).\nThe magnetization curves in the previous studies suggest\nthe tilt ofthe magnetic moment from the caxisto the a-b\nplane. However,thisisinconsistentwiththesimpleferro-\nmagnetic structure with the magnetic moments aligned\nalong the caxis determined by the neutron scattering\nexperiments[21, 22]. The authors of Ref. 22 proposed\nseveral possible reasons for this discrepancy such as the\nexistenceofgrainsintheirsingle-crystalsample. It seems\nthat a ﬁnal conclusionhasnot been made. We stressthat\nthemagnetizationcurvesinFig. 3(b)areconsistentwith\nthe magnetic structure determined in the neutron scat-\ntering studies[21, 22].\nIn the mean-ﬁeld theory, the free energy of a ferromag-\nnet in the vicinity of TCcan be expressed as a power-\nseries expansion in the order parameter M:\nF(M) =F(0)+1\n2aM2+1\n4bM4+....−HM.(1)\nThe following equation of state is derived from the\nequilibrium condition by minimizing the free energy\n∂F(M)/∂M= 0:\nH=aM+bM3. (2)\nThe mean-ﬁeld theory fails in the asymptotic critical\nregion whose extent can be estimated by the Ginzburg\ncriterion[34]. The correlation length ξ=ξ0|1−T/TC|−ν\ndiverges in the critical region, which leads to univer-\nsal scaling laws for the spontaneous magnetization Ms,\nthe initial susceptibility χ, and the magnetization at TC.\nHere,νisthe criticalexponent. Fromthescalinghypoth-\nesis, the spontaneous magnetization Ms(T) below TC,\nthe inverse of the initial magnetic susceptibility χ(T) be-\nlowand above TC, andthe magnetization M(µ0H) atTC\narecharacterizedasetofcriticalexponentsasfollows[17]:\nMs(T)∝ |t|β(T < T C), (3)\nχ(T)−1∝ |t|γ′\n(T < T C),|t|γ(TC< T),(4)\nM(µ0H)∝(µ0H)1/δ(T=TC). (5)\nHere,tis the reduced temperature t= 1−T/TC.β,γ,\nγ′, andδare the critical exponents.5\n/s49/s46/s53/s120/s49/s48/s49/s48\n/s48/s46/s48/s77/s49/s47/s98/s32/s91/s65/s47/s109/s93/s49/s47/s98\n/s52/s48 /s50/s48 /s48/s40/s72/s47/s77/s41/s49/s47/s103/s77/s101/s97/s110/s32/s102/s105/s101/s108/s100\n/s98/s32/s32/s61/s32/s48/s46/s53\n/s103/s32/s32/s61/s32/s49/s46/s48/s40/s97/s41\n/s68/s84/s32\n/s61/s32/s48/s46/s51/s32/s75/s32/s84/s32/s61/s32/s57/s46/s50/s32/s75/s32\n/s84/s32/s61/s32/s49/s49/s46/s48/s32/s75/s32/s85/s82/s104/s83/s105\n/s56/s120/s49/s48/s49/s54\n/s48/s77/s49/s47/s98/s32/s91/s65/s47/s109/s93/s49/s47/s98\n/s52/s48 /s50/s48 /s48\n/s40/s72/s47/s77/s41/s49/s47/s103/s77/s65/s80\n/s98/s32/s32/s61/s32/s48/s46/s51/s48/s49\n/s103/s32/s32/s61/s32/s49/s46/s48/s48/s40/s99/s41/s52/s120/s49/s48/s49/s53\n/s48/s77/s49/s47/s98/s32/s91/s65/s47/s109/s93/s49/s47/s98\n/s50/s48 /s49/s48 /s48/s40/s72/s47/s77/s41/s49/s47/s103/s83/s82/s32/s51/s68/s32/s73/s115/s105/s110/s103\n/s98/s32/s32/s61/s32/s48/s46/s51/s50/s53\n/s103/s32/s32/s61/s32/s49/s46/s50/s52/s49/s40/s98/s41\nFIG. 4: Magnetization isotherms in the forms of M1/βvs.\n(H/M)1/γin the temperature range 9.2 K ≤T≤11.0 K,\nwith (a) the mean-ﬁeld theory, (b) the short-range (SR) 3D-\nIsing model, and (c) the modiﬁed Arrott plot (MAP) with\nβ= 0.301 and γ= 1.00 in URhSi. Bold circles indicate the\nisotherms at 10.1 K. Solid lines in (c) show ﬁts to the data\nwith Eq. (6).\nUsually, the Arrott plots technique has been used to\ndetermine the phase transition temperature TC. In the\nmean-ﬁeld theory, isotherms plotted in the form of M2\nvs.H/Mshould be a series of parallel straight lines and\nthe isotherm at TCshould pass through the origin[17].\nThe critical exponents with β= 0.5,γ= 1.0, and δ=/s49/s120/s49/s48/s53\n/s48/s77/s32/s91/s65/s47/s109/s93\n/s54 /s52 /s50 /s48\n/s109/s48/s72/s32/s91/s84/s93/s100/s32/s61/s32/s52/s46/s51/s56/s32/s32/s32/s84/s32/s61/s32/s49/s48/s46/s49/s32/s75/s32/s40/s97/s41\n/s85/s82/s104/s83/s105\n/s53/s54/s55/s56/s57/s49/s48/s53/s77/s32/s91/s65/s47/s109/s93\n/s48/s46/s49 /s49\n/s109/s48/s72/s32/s91/s84/s93/s85/s82/s104/s83/s105\n/s32/s32/s32/s84/s32/s61/s32\n/s32/s49/s48/s46/s49/s32/s75/s32/s40/s98/s41\n/s84/s32/s61/s32/s57/s46/s48/s32/s75/s32\n/s84/s32/s61/s32/s49/s49/s46/s48/s32/s75\n/s68/s84/s32/s32\n/s61/s32/s48/s46/s49/s32/s75/s100/s32/s61/s32/s52/s46/s51/s56\nFIG. 5: Magnetic ﬁeld dependencies of the magnetization (a)\nat 10.1 K and (b) from 9.0 to 11.0 K in URhSi. Bold cir-\ncles indicate the critical isotherm data at 10.1 K. Solid lin es\nrepresent ﬁts to the critical isotherm with Eq. (5).\n3.0 in the mean-ﬁeld theory are assumed in the Arrott\nplot.\nFigures4(a)and(b)showthemagnetizationisotherms\nin the forms of M1/βvs. (H/M)1/γwith (a) the mean-\nﬁeld theory ( β= 0.5 and γ= 1.0) and (b) the 3D-Ising\nmodel with short-range (SR) exchange interactions ( β\n= 0.325 and γ= 1.241), respectively. The isotherms\ndo not form straight lines in the two plots. Therefore,\nthe Arrott-Noakes equation of state has been used to re-\nanalyze the magnetization isotherms[35]. The following\nequation should hold in the asymptotic critical region.\n(H/M)1/γ= (T−TC)/T1+(M/M1)1/β(6)6\n/s52\n/s50\n/s48/s99/s32/s45/s49/s54/s120/s49/s48/s52\n/s52\n/s50\n/s48/s77/s115/s32/s91/s65/s47/s109/s93/s85/s82/s104/s83/s105\n/s70/s77 /s80/s77/s84/s67/s32/s40/s97/s41\n/s45/s52\n/s45/s50\n/s48/s77/s115/s32/s40/s100/s77/s115/s47/s100/s84/s41/s45/s49/s32/s91/s75/s93\n/s32\n/s49/s49/s46/s48 /s49/s48/s46/s53 /s49/s48/s46/s48 /s57/s46/s53 /s57/s46/s48\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s91/s75/s93/s48/s46/s53\n/s48/s46/s48/s99/s45/s49/s32/s40/s100/s99/s45/s49/s47/s100/s84/s41/s45/s49/s32/s91/s75/s93\n/s32\n/s84/s67/s32/s61/s32\n/s49/s48/s46/s49/s53/s32/s75\n/s103/s32/s32/s61/s32/s49/s46/s48/s48 /s98/s32/s32/s61/s32/s48/s46/s50/s57/s54/s40/s98/s41\nFIG. 6: (a) Temperature dependencies of the spontaneous\nmagnetization Ms(T) belowTC(left) and the inverse of the\ninitial magnetic susceptibility χ−1aboveTC(right) deter-\nmined from the modiﬁed Arrott plot. (b)Kouvel-Fisher plots\nofMs(T)[dMs(T)/dT]−1(left) and χ−1(T)[dχ−1(T)/dT]−1\n(right) in URhSi.\nHere,T1andM1are material constants. The data for\nURhSi are plotted in the form of M1/βvs. (H/M)1/γin\nthe modiﬁed Arrott plots. The isotherms exhibit a linear\nbehavior when the appropriate values of TC,β, andγare\nchosen as shown in Fig. 4 (c). The values are determined\nasTC= 10.12±0.02 K,β= 0.301±0.002, and γ= 1.00\n±0.04 from a best ﬁt of Eq. (6) to the data for 9.2 K ≤\nT≤11.0 K and 0.1 T ≤µ0H≤7.0 T in URhSi.\nThe third critical exponent δis determined as δ= 4.38\n±0.04 for URhSi from ﬁts to the critical isotherm at\n10.1 K with Eq. (5) as shown in Figure 5. The value is\nlower than that in the 3D Ising model with short-range\nexchange interactions ( δ= 4.82). The exponents β,γ,\nandδshouldfulﬁlltheWidomscalinglaw δ=1+γ/β[36].\nWe estimate the value of δas 4.32±0.10 from the βand\nγvalues determined in the modiﬁed Arrott plots using\nthe law. This value is consistent with that determined\nfrom the critical isotherm.\nThe data are analyzed using the Kouvel-Fisher (KF)/s49/s46/s49\n/s49/s46/s48\n/s48/s46/s57/s103/s101/s102/s102\n/s48/s46/s48/s49 /s48/s46/s49\n/s124/s116/s124/s32/s61/s32/s124/s40 /s84/s45/s84/s67/s41/s47/s84/s67/s124/s40/s98/s41/s48/s46/s51/s52\n/s48/s46/s51/s50\n/s48/s46/s51/s48\n/s48/s46/s50/s56/s98/s101/s102/s102/s40/s97/s41/s85/s82/s104/s83/s105\nFIG. 7: Eﬀective exponents (a) βeﬀfor the spontaneous mag-\nnetization Ms(T) belowTCand (b)γeﬀfor the magnetic sus-\nceptibility χaboveTCas a function of the reduced tempera-\nture|t|[=|(T−TC)/TC|] in URhSi.\nmethod by which the critical exponents βandγcan be\ndetermined more accurately[37]. At ﬁrst, we determine\nthe temperature dependencies of the spontaneous mag-\nnetization Msand the initial magnetic susceptibility χ\nfrom the modiﬁed Arrott plots as follows. The ﬁtted\nstraight lines in the plots intersect with the vertical axis\natM1/β=Ms1/βforT < T Cand with the transverse\naxis at ( H/M)1/γ= (1/χ)1/γforTC< T[38]. Next,\nthe temperature dependencies of Msandχ−1(T) are ob-\ntained by inserting the values of the exponents βandγ\ndetermined in the modiﬁed Arrott plots. Figure 6 (a)\nshows the temperature dependencies of Msandχ−1(T)\nin URhSi. Solid lines show the ﬁts to the data using Eqs.\n(3) and (4) for Ms(T) andχ−1(T), respectively. In the\nKF method, temperature-dependent exponents β(T) and\nγ(T) are deﬁned as follows:\nMs(T)[dMs(T)/dT]−1= (T−TC−)/β(T),(7)\nχ−1(T)[dχ−1(T)/dT]−1= (T−TC+)/γ.(T) (8)\nEquations (7) and (8) can be obtained from Eq. (6)\nin the limit H→0 forT <and> TC, respectively.\nWe determine the values of βandγfrom the slope\nofMs(T)[dMs(T)/dT]−1andχ−1(T)[dχ−1(T)/dT]−1-\nplots, respectively, at TCas shown in Fig. 6 (b). Note\nthat the quantities β(T) andγ(T) in the limit T→TC7\n/s55/s56/s57/s49/s48/s53/s50/s51/s52/s77/s47/s124/s116/s124/s32/s98/s32/s91/s65/s47/s109/s93\n/s49 /s49/s48 /s49/s48/s48 /s49/s48/s48/s48\n/s40/s109/s48/s72/s41/s47/s124/s116/s124/s40/s32/s98/s43/s103/s41/s32/s91/s84/s93/s84/s67/s32/s61/s32/s49/s48/s46/s49/s50/s32/s75\n/s98/s32/s32/s61/s48/s46/s51/s48/s48\n/s103/s39/s32/s32/s61/s32/s49/s46/s48/s48 /s32/s40/s84/s60/s84/s67/s41\n/s103/s32/s32/s61/s32/s49/s46/s48/s51/s32/s40 /s84/s67/s60/s84/s41/s84/s32/s60/s32/s84/s67\n/s84/s67/s32/s60/s32/s84/s85/s82/s104/s83/s105/s40/s98/s41/s52/s120/s49/s48/s53\n/s48/s77/s47/s124/s116/s124/s32/s98/s32/s91/s65/s47/s109/s93\n/s50/s48/s48/s48 /s49/s53/s48/s48 /s49/s48/s48/s48 /s53/s48/s48 /s48\n/s40/s109/s48/s72/s41/s47/s124/s116/s124/s40/s32/s98/s43/s103/s41/s32/s91/s84/s93/s84/s67/s32/s61/s32/s49/s48/s46/s49/s50/s32/s75\n/s98/s32/s32/s61/s48/s46/s51/s48/s48\n/s103/s39/s32/s32/s61/s32/s49/s46/s48/s48 /s32/s40/s84/s60/s84/s67/s41\n/s103/s32/s32/s61/s32/s49/s46/s48/s51/s32/s40 /s84/s67/s60/s84/s41/s84/s32/s60/s32/s84/s67\n/s84/s67/s32/s60/s32/s84/s85/s82/s104/s83/s105/s40/s97/s41\nFIG. 8: Renormalized magnetization m(≡ |t|−βM(µ0H,t))\nof URhSi as a function of renormalized ﬁeld h[≡H|t|−(β+γ)]\nfollowing Eq. (11) below and above TCwithTC,β,γ, andγ′\nvalues mentioned in the main text. Solid lines represent bes t-\nﬁt polynomials. The magnetization data in the temperature\nranget=|(T−TC)/TC|<0.1 are plotted.\ncorrespond to the critical exponents βandγ, respec-\ntively. Solid lines in Fig. 6 (b) show the ﬁts to the\ndata using Eqs. (7) and (8). The determined values of\nthe exponents βandγareβ= 0.296 ±0.002 and γ=\n1.00±0.02 with TC= (TC++TC−)/2 = 10.15 ±0.01\nK. The determined exponents are consistent with those\ndetermined in the modiﬁed Arrott plot.\nIf there are various competing interactions or disor-\nders, crossover phenomena could occur in the critical ex-\nponentsonapproaching TCasobservedinNi 3Al[39]. The\nconvergence of the critical exponents should be checked.\nEﬀective exponents βeﬀandγeﬀare useful to examinethis possibility.\nβeﬀ(t) =d[lnMs(t)]/d(lnt), (9)\nγeﬀ(t) =d[lnχ−1(t)]/d(lnt). (10)\nWe show the eﬀective exponents βeﬀandγeﬀas a func-\ntion of the reduced temperature tin Figs 7 (a) and (b),\nrespectively. A monotonic |t|dependence is observed in\nbothβeﬀandγeﬀfor|t|≥5.12×10−3and 4.73 ×10−3,\nrespectively. The crossover phenomenon in the critical\nbehavior between two universality classes can be ruled\nout.\nItisnecessarytoexaminethepossibilityofthestrongly\nasymmetric critical region or the change of the universal-\nity class across TC. The values of γ’ forT < T Candγfor\nTC< Tcan be determined separately with the scaling\ntheory where a reduced equation of state close to TCis\nexpressed as follows[17]:\nm=f±(h). (11)\nHere,f+forTC< Tandf−forT < T Care regular\nanalytical functions. The renormalized magnetization m\nand ﬁeld hare deﬁned as m≡ |t|−βM(µ0H,t) andh≡\nH|t|−(β+γ), respectively. This relation implies that the\ndata ofmversushwith the correctvalues of β,γ’,γ, and\ntfall on two universal curves, one for T < T Cand the\nother for TC< T. We show the renormalized magneti-\nzationmas a function of the renormalized ﬁeld hbelow\nand above TCin Figs. 8 (a) and (b). The magnetization\ndata in the temperature range t=|(T−TC)/TC|<0.1\nare plotted. All data points collapse onto two indepen-\ndent curves. The values of TCand the critical exponents\nare determined as TC= 10.12 ±0.02 K,β= 0.300 ±\n0.002,γ′= 1.00±0.02 forT < T C, andγ= 1.03±0.02\nforTC< Tin URhSi. The scaling analysis suggests that\nthe set of the critical exponents are the same below and\naboveTCin URhSi. We can rule out the strongly asym-\nmetric critical region and the change of the universality\nclass across TC.\nTable III shows the critical exponents β,γ,γ′, and\nδin various theoretical models[17, 40, 41] and those in\nURhSi. We also show the exponents in the uranium fer-\nromagnetic superconductors UGe 2[18], URhGe[18], and\nUCoGe[7], and uranium ferromagnets UIr with TCof 46\nK[42, 43] and U(Co 0.98Os0.02)Al with TCof 25 K[44, 45].\nThe strong uniaxial magnetization in the ferromagnetic\nstate of URhSi suggests the universality class of the 3D\nIsing model. However, the obtained critical exponents in\nURhSi are diﬀerent from those of the 3D Heisenberg ( d\n= 3,n=3), 3D XY ( d= 3,n=2), 3D Ising ( d= 3,n\n=1), and 2D Ising ( d= 2,n=1) models where magnetic\nmoments are interacted via short-range (SR) exchange\ninteractions of a form J(r)∼e−r/b. Here,bis the corre-\nlation length. The value of βin URhSi is close to those in\nthe 3D models but the γvalue is close to unity, expected\none in the mean-ﬁeld theory. While the magnetization\nMsshows the critical behavior around TC, the magnetic8\nTABLE III: Comparison of critical exponents β,γ,γ′, andδof various theoretical models[17, 40, 41] with those in\nURhSi, uranium ferromagnetic superconductors UGe 2[18], URhGe[18] and UCoGe[7], and uranium ferromagnets UIr [42] and\nU(Co0.98Os0.02)Al[44]. Abbreviations: RG- φ4, renormalization group φ4ﬁeld theory; SR, short-range; LR, long-range.\nMethod TC(K) β γ′γ δ Reference\n(T < T C) (TC< T)\n(Theory)\nMean-ﬁeld 0.5 1.0 3.0\nSR exchange: J(r)∼e−r/b\nd= 2,n=1 Onsager solution 0.125 1.75 15.0 [17, 40]\nd= 3,n=1 RG- φ40.325 1.241 4.82 [41]\nd= 3,n=2 RG- φ40.346 1.316 4.81 [41]\nd= 3,n=3 RG- φ40.365 1.386 4.80 [41]\nURhSi This work\nModiﬁed Arrott 10.12 ±0.02 0.301 ±0.002 1.00 ±0.04\nKouvel-Fisher 10.15 ±0.01 0.296 ±0.002 1.00 ±0.02\nScaling 10.12 ±0.02 0.300 ±0.002 1.00 ±0.02 1.03 ±0.02\nCritical isotherm 4.38±0.04\nUGe2 [18]\nModiﬁed Arrott 52.6 ±0.1 0.334 ±0.002 1.05 ±0.05\nKouvel-Fisher 52.60 ±0.02 0.331 ±0.002 1.03 ±0.02\nScaling 52.79 ±0.02 0.329 ±0.002 1.00 ±0.02 1.02 ±0.02\nCritical isotherm 4.16±0.02\nURhGe [18]\nModiﬁed Arrott 9.44 ±0.02 0.303 ±0.002 1.02 ±0.03\nKouvel-Fisher 9.47 ±0.01 0.303 ±0.002 1.01 ±0.02\nScaling 9.47 ±0.01 0.302 ±0.001 1.00 ±0.01 1.02 ±0.01\nCritical isotherm 4.41±0.02\nUCoGe 2.5 ∼mean-ﬁeld type ∼ [7]\nUIr [42]\nModiﬁed Arrott 45.15 0.355(50) 1.07(10)\nCritical isotherm 4.01(5)\nU(Co0.98Os0.02)Al [44]\nModiﬁed Arrott 25 0.33 1.0\nCritical isotherm 4.18\nsusceptibility χfollows the mean-ﬁeld theory in URhSi.\nThe universality class of the ferromagnetic transition in\nURhSi may not belong to any known universality class.\nThe obtained critical exponents in URhSi are similar\nto those in UGe 2, URhGe, UIr, and U(Co 0.98Os0.02)Al\nwhere the ferromagnetic state has strong uniaxial mag-\nnetic anisotropy. The universality class of the ferromag-\nnetic transitions in URhSi may belong to the same one\nof the uranium ferromagnets. Note that we previously\nreported the unconventional critical scaling of magneti-\nzation in UGe 2and URhGe[18]. The values of the ex-\nponentβslightly diﬀer, depending on each ferromagnet.\nMeanwhile, the γvalues of the uranium ferromagnetsare\nclose to unity. This almost mean-ﬁeld behavior of the\nmagnetic susceptibility χmay be a characteristic feature\nof the unconventional critical behavior of the magneti-\nzation in the uranium ferromagnets. The ferromagnetic\ncorrelation in the uranium ferromagnetsmay be diﬀerent\nfrom that in the 3D Ising system. This unusual critical\nbehaviorof the magnetizationmay be inherent in the fer-\nromagnetismof5 felectronswherethe superconductivity\ncould appear.\nWe discuss the extent of the asymptotic critical regionwhere magnetic data for the determination of the critical\nexponents should be collected. The extent of the region\ncan be estimated by the Ginzburg criterion[34, 46, 47]:\n∆TG/TC=k2\nB/[32π2(∆C)2ξ06].(12)\nHere, ∆Cis the jump of the speciﬁc heat at TCandξ0\nis the correlation length ξ(1/ξ2= 1/ξ2\n0|1−T/TC|) at\n2TC. It is possible to estimate the temperature region\nwhere the mean-ﬁeld theory fails by the Ginzburg crite-\nrion. Previously, we determined the value of ∆ TGas∼\n100 K for UGe 2in Ref. 18 using the neutron scatter-\ning and speciﬁc heat data[5, 48] and concluded that the\ndata used for the determination of the critical exponents\nin UGe 2were collected inside the asymptotic critical re-\ngion. Thelargevalueof∆ TGmaybeduetoexperimental\nerrors in the values of ∆ Candξ0. ∆TGis very sensitive\ntoξ0. Unfortunately, it is impossible to estimate ∆ TG\nfor URhSi since there has been no report for the correla-\ntion length ξ0. The critical exponents in URhSi are de-\ntermined using the data in the temperature region from\n9.0 K to 11.0 K (0 < t <0.1). The present analyses sug-\ngestthat this temperature regionis inside the asymptotic\ncritical region. The T-linear dependence of the magnetic9\nsusceptibility χ−1does not indicate that the analysis is\ndone using the data taken outside the asymptotic critical\nregion. We also rule out the possibility of the strongly\nasymmetric critical region or the change of the universal-\nity class across TCas mentioned before.\nThe mean-ﬁeld behavior of the magnetization in\nUCoGe is brieﬂy discussed[7]. As shown in Table II, the\nvalues of the spontaneous magnetic moment ( ps= 0.039\nµB/U) and the parameter TC/T0( = 0.0065) are very\nsmall compared with those in UGe 2and URhGe. These\nresults suggest the strong itinerant characters of the 5 f\nelectrons. We previously estimated the value of ∆ TGas\nless than 1 mK using the speciﬁc heat and the neutron\ndata[7, 18, 49]. This value suggestsaverynarrowasymp-\ntotic critical region. The strongitinerant characterofthe\n5felectrons masks the critical behavior in UCoGe. The\nmean-ﬁeld behavior of the magnetization is expected to\nappear since most of the magnetic data around TCmight\nbe collected outside the very narrow asymptotic critical\nregion.\nWe estimate the critical exponent αfor the speciﬁc\nheat [C(T)∝|t|α] as∼0.4 using the Rushbrooke scaling\nrelation ( α+ 2β+γ= 2)[50]. In the mean ﬁeld theory\n(α= 0), the speciﬁc heat does not exhibit divergence at\nTCand there is no contribution from the magnetic criti-\ncal ﬂuctuations to the speciﬁc heat ( Cmag= 0) above TC.\nTheαvalue in URhSi suggests the signiﬁcant contribu-\ntion toCmag(>0) from the critical ﬂuctuations above\nTC. This is consistent with a speciﬁc heat tail in the tem-\nperature range 0 < t[=|(T−TC)/TC|]<∼0.1 above\nTC[20].\nThe critical exponents in URhSi were previously re-\nported as β= 0.36±0.02 and γ= 1.14±0.06 from\nthe analysis of the magnetization with the scaling the-\nory in Ref. 20. The values of βandγare larger than\nthose in the present study. In the magnetization data\nreported in Refs. 20 and 22, the spontaneous magnetic\nmoment occurs in magnetic ﬁelds applied along the mag-\nnetic hard aandbaxes as mentioned before. The mag-\nnetization curves in the previous studies are not com-\npatible with the simple ferromagnetic structure with the\nmagnetic moments oriented along the caxis determined\nin the neutron scattering studies[21, 22]. Meanwhile, the\nmagnetization curves in this study shown in Fig. 3 (b)\nare consistent with the magnetic structure. In this study,\nthe critical exponents βandγare determined by several\ndiﬀerent methods: the modiﬁed Arrott plot, the Kouvel-\nFisher plot, and the scaling analysis. The values of β,γ,\nandδsatisfy the Widom scaling law ( δ= 1+γ/β).\nIV. DISCUSSIONS\nA. Unconventional critical behavior of\nmagnetization\nWe discuss the unconventional critical behavior of\nthe magnetization in URhSi, UGe 2, URhGe, UIr, andU(Co0.98Os0.02)Al with previous theoretical approaches\nto critical phenomena.\n(1) The universalityclass of the magnetic phase transi-\ntion is aﬀected by the long-range nature of the magnetic\nexchange interaction. The strength of the magnetic ex-\nchange interaction J(r) decreases rapidly with distance\nin the theoretical models with short-range (SR) interac-\ntions. The exchanged interaction can be long-ranged for\nthe itinerant electron system. When the range of the in-\nteraction becomes longer, the critical exponents of each\nuniversality class are shifted towards those in the mean-\nﬁeld theory. Fischer et al.analyzed systems with the\nexchange interaction of a form J(r)∼1/rd+σby a renor-\nmalization group approach[51]. Here, dis the dimension\nof the system and σis the range of exchange interaction.\nTheyshowedthe validity ofsuchamodel with long-range\ninteractions for σ <2 and derived a theoretical formula\nfor the exponent γ= Γ{σ,d,n}. Here,nis the dimen-\nsion of the order parameter and the function Γ is given\nin Ref. 51. Recently, we studied the critical behavior of\nthe magnetization in URhAl around TC= 26.02 K[52].\nThe critical exponents in URhAl were explained with the\nresult of this renormalization group approach for the 2D\nIsing model coupled with long-range interactions decay-\ning asJ(r)∼1/r2+σwithσ= 1.44. We try to reproduce\nthe critical exponents in URhSi, UGe 2, URhGe, UIr, and\nU(Co0.98Os0.02)Al using the formula for diﬀerent sets of\n{d:n}(d,n= 1, 2, 3). However, no reasonable solution\nofσis found.\n(2) Next, we discuss the eﬀect of classical dipole-\ndipole interaction on the critical phenomenon. The ef-\nfect on the critical behavior of the magnetization in\ngadolinium ( TC= 292.7 K, ps= 7.12µB/Gd) has been\nstudied[53]. This scenario seems not applicable to the\nuranium ferromagnets since the strength of the eﬀect de-\npends on the square of the spontaneous magnetic mo-\nmentps[54]. The theoretical values of the critical ex-\nponents for the critical phenomena associated with the\nisotropic or anisotropic dipole-dipole interaction are not\nconsistent with those in URhSi, UGe 2, URhGe, UIr, and\nU(Co0.98Os0.02)Al[55, 56].\n(3) Spin ﬂuctuation theories have been developed\nto explain the ﬁnite temperature magnetic properties\nin itinerant ferromagnets of the 3 dmetals and their\nintermetallics[57]. For example, the nearly T-linear de-\npendence of χ−1aboveTCobserved in the 3 delec-\ntrons systems has been reproduced in numerical calcu-\nlations based on Moriya’s self-consistent renormalization\n(SCR) theory[58, 59] and the Takahashi’s spin ﬂuctua-\ntion theory[26–28]. It is diﬃcult to discuss the exact\ntemperature dependencies of χ−1in the temperature re-\ngion close to TC. This is because the behavior of χ−1\nnearTCdepends on the values of parameters in the the-\nories. Calculated χ−1-Tcurves for certain parameter re-\ngions are concave upward near TC[58, 59]. The T4/3-\ndependence of p2\nswas derived in the weak coupling limit\nby the SCR theory[60] and the dependence was roughly\nreproduced numerically at certain parameter regions in10\nthe Takahashi’s spin ﬂuctuation theory[26–28]. These re-\nsults are not consistent with the experimentally observed\ncritical exponents of the uranium ferromagnets. Further-\nmore, we note that the critical exponents are determined\nin the asymptotic critical region where the spin ﬂuctua-\ntions theories cannot be applied to.\n(4) We discuss the critical exponents in the ura-\nnium ferromagnets from the viewpoint of the local mo-\nment magnetism. The orthorhombic TiNiSi-type crys-\ntal structure of URhSi, URhGe, and UCoGe can be re-\ngarded as the coupled zigzag chains of the nearest neigh-\nbor uranium atoms as mentioned in the introduction.\nUGe2and UIr crystalize in the orthorhombic ZrGa 2-\ntype (space group Cmmm) and the monoclinic PbBi-\ntype (space group P21) structures, respectively[5, 61].\nThe crystal structures also can be regarded as the cou-\npled zigzag chains along the crystallographic aaxis for\nUGe2and thebaxis for UIr. The magnetic structures of\nthese ferromagnetscould be mapped ontothe anisotropic\n3D Ising model or the anisotropic next-nearest-neighbor\n3D Ising (ANNNI) model. However, the critical expo-\nnents of the uranium ferromagnets are not consistent\nwith those for the two models obtained by numerical\ncalculations[62, 63].\n(5) Recently, Singh, Dutta, and Nandy discussed the\nunconventional critical behavior of the magnetization in\nUGe2and URhGe with a non-local Ginzburg-Landau\nmodel focusing on magnetoelastic interactions that give\na nonlocal quartic interaction[64]. The authors claimed\nthat the calculated critical exponents are comparable\nwith the experimentally observed critical exponents in\nUGe2, URhGe, and UIr. We hope that the almost mean-\nﬁeld behavior of the magnetic susceptibility χin the ura-\nnium ferromagnets is completely reproduced.\nIt is diﬃcult to explain the critical exponents in the\nuranium ferromagnets with previous approaches to crit-\nical phenomena as discussed in points 1 −5. Here,\nwe introduce several interesting experimental studies on\nUGe2and suggest the relevance of the dual nature of\nthe 5felectrons between itinerant and localized char-\nacters to the critical behaviors of the magnetization in\nthe uranium ferromagnets[65–67]. The long correlation\nlength of ξ0=48˚A with a magnetic moment of 0.02\nµB/U was detected in UGe 2by the Muon spin rotation\nspectroscopy[65]. The value of ξ0is more than two times\nlarger than that ( ∼22˚A) determined by the inelastic\nneutron scattering experiment[48]. A main contribution\nto the magnetic scattering intensity in the neutron scat-\ntering experiment comes from the localized component\nof the 5felectrons in UGe 2since the intensity is propor-\ntional to the square of the magnetic moment. The mag-\nnetic moment on the uranium site was determined as µU\n= 1.45−1.46µB/U at 6 K by the polarized neutron scat-\ntering experiment[68]. The longer magnetic correlation\nwiththesmallermagneticmomenthasbeenattributedto\nthe itinerant component of the 5 felectrons[65, 66]. Very\nrecently, Haslbeck et al.have reported the results of the\nultrahigh-resolution neutron scattering experiment[67].According to the authors, their results suggest the dual\nnature of spin ﬂuctuations in UGe 2; local spin ﬂuctu-\nations described by the 3D Ising universality class and\nitinerant spin ﬂuctuations. The concept of the duality of\nthe 5felectrons has been employed in theoretical models\nfor the superconductivity in the ferromagnetic state of\nUGe2and URhGe[69, 70], and in the antiferromagnetic\nstate of UPd 2Al3[71, 72]. There might be a Hund-type\ncoupling between the itinerant and localized components\nof the 5felectrons. A novel critical phenomenon could\nappear due to the diﬀerent nature of the two correlations\nand the coupling of the two components.\nIn Fig. 2, weshowthe resultsofthe analysesonthe ac-\ntinide ferromagnetswith the Takahashi’sspin ﬂuctuation\ntheory[25]. The applicability of the theory to actinide 5 f\nsystems is discussed. Huxley et al.reported from the\ninelastic neutron scattering experiment that χ(q)Γqre-\nmains large for q→0 from the data of χ(q)Γqmeasured\nforq≥0.03˚A−1[48]. Here, Γ qis the relaxation rate for\nthe magnetization density. This non-Landau damping of\nmagnetic excitations suggests that the uniform magne-\ntization density is not a conserved quantity. This fact\nmay raise doubts about the applicability of spin ﬂuctua-\ntion theories to the actinide 5 felectrons systems. Phe-\nnomenological and microscopic theories were proposed\nto explain this non-zero Γ(0) focusing on the duality of\nthe 5felectrons[73, 74]. In the recent experiment by\nHaslbeck et al.[67], the qdependence of χ(q)Γqwas de-\ntermined down to q∼0.02˚A−1, lower than the low limit\nofq(= 0.03˚A−1) in the previous study[48]. χ(q)Γqis al-\nmostconstantfor q0(=0.038 ˚A−1)<qbut it approaches\nto zero [χ(q)Γq→0] forq→0 belowq0. This latest result\nin turn implies that the magnetization density is a con-\nserved quantity. This is favorable to the application of\nthe theory to the actinide 5 fsystems, though this result\ndoes not completely justify it. It is important to under-\nstand how the result reconciles with the two theoretical\nstudies. Haslbeck et al.also reported that the critical\nexponents [ β= 0.32(1), γ= 1.23(3), and ν= 0.63(2)] de-\ntermined by them are similar to those ( β= 0.325, γ=\n1.241, and ν= 0.630) in the 3D Ising model[41]. The\nvalue ofγ(= 1.23) is diﬀerent from that ( γ= 1.0) deter-\nmined in our previous study[18]. Our result is consistent\nwith that in the previous neutron scattering study[48].\nThe reason for this discrepancy is not clear.\nB. Possible reasons for absence of the\nsuperconductivity in URhSi\nWe discuss the absence of the superconductivity in\nURhSi and future prospects for the study of uranium fer-\nromagnets. At ﬁrst, we introduce ourstudy on UGe 2[75].\nWe measured the dc magnetization under high pressure\nand determined the pressure dependence of the charac-\nteristic energy of the longitudinal spin ﬂuctuations T0.\nWe have found a clear correlation between TscandT0\nin UGe 2. Our result suggests that the superconductiv-11\nity is mediated by the spin ﬂuctuations developed at the\nphase boundary of FM1 and FM2 in UGe 2. The corre-\nlation between the two quantities has been discussed in\nhigh-Tccuprate and heavy-fermion superconductors[77–\n79]. The importance of the longitudinal spin ﬂuctuations\nfor thep-wave superconductivity in the ferromagnetic\nstate has been theoretically pointed out as mentioned in\nthe Introduction[10–13]. The values of T0in URhGe and\nURhSi are similar as shown in Table II. Furthermore,\nthis study suggests the similarity in the critical behav-\nior of the magnetization between URhSi, URhGe, and\nUGe2[18]. Then, one could expect the superconductiv-\nity in URhSi. However, it has not been observed at low\ntemperatures down to 40 mK in URhSi according to Ref.\n22. We discuss the absence of the superconductivity in\nURhSi from two viewpoints.\n(i) Generally, it is diﬃcult to grow the high-quality\nsingle crystal of UTX ferromagnets with the orthorhom-\nbic TiNiSi-type crystal structure. Here, Tis a transition\ndmetal.Xis ap-block element[19]. The unconven-\ntional non s-wave superconductivity in strongly corre-\nlated electron systems is sensitive to a small amount of\nnonmagnetic impurities[3, 80]. The high sensitivity of\nTscin URhGe to the electronic mean-free path was an-\nalyzed with the Abrikosov-Gorkov model assuming that\nthe mean free path is proportional to the inverse of the\nresidual resistivity ratio (1/RRR)[8, 81]. One possibility\nis that the quality of samples of URhSi studied so far are\nnot enough for the appearance of the superconductivity.\nIt is necessary to grow the high-quality single crystal of\nURhSi in order to check the appearance of the supercon-\nductivity at low temperatures.\n(ii) We discussthis issuefrom the recentuniaxialstress\nexperiment on URhGe[76]. The uniaxial stress σapplied\nalongthemagnetichard baxisenhances Tsc, withamerg-\ning of the low- and high-ﬁeld superconducting states in\nURhGe[76]. The value of Tscin URhGe increases from\n0.4Katambientpressureto0.8Kat σ∼0.6GPaapplied\nalong the baxis. Meanwhile, the ferromagnetic transi-\ntion temperature TCshows only a 10 % decrease. There\nis no report for the dc magnetization under the uniax-\nial pressure. We speculate that the value of T0does not\nchangesigniﬁcantly. The relationbetween TscandT0un-\nder uniaxial stress in URhGe may be diﬀerent from ap-\nproximately linear relations in the high- Tccuprate and\nheavy-fermion superconductors[77–79]. It is interesting\nto note that the relation in FM1 of UGe 2is expressed as\nTsc∝(T0)αwithα=2.3±0.1[75].Tscisverysensitiveto\nT0. We suggest one possibility: T0is not a sole param-\neter that determines Tscin the uranium ferromagnetic\nsuperconductors.\nThe magnetic susceptibility χbin a magnetic ﬁeld\nalong the magnetic hard baxis increases with increas-\ning uniaxial stress applied along the same direction in\nURhGe[76]. The increase in Tscof the compound may\nbe related to enhanced transverse magnetic ﬂuctuations\nalong the hard baxis as theoretically shown[82]. Mineev\nderived a theoretical expression of Tscin a BCS-type for-mulaTsc=εexp(−1/g) for a two-band superconduct-\ning state in orthorhombic systems where the coupling\nbetween the two components of the equal spin-triplet\np-wave order parameter is taken into account[82]. The\ncoupling constant gis expressed as follows:\ng=(g↑\n1x+g↓\n1x)\n2+/radicalBigg\n(g↑\n1x−g↓\n1x)2\n4+g↑\n2xg↓\n2x(13)\nHere,g↑\n1xandg↓\n1xare coupling constants for intraband\npairing, and g↑\n2xandg↓\n2xfor interband pairing. g↑\n1xand\ng↓\n1xare proportional to the magnetic susceptibility along\nthemagneticeasy caxis. Theintra-bandpairingisdriven\nby the longitudinal spin ﬂuctuations whose energy scale\ncan be estimated from T0.g↑\n2xandg↓\n2xare determined\nby the diﬀerence of the magnetic susceptibilities in the\nmagnetic hard bandaaxes (χb−χa). Thus, g↑\n2xg↓\n2x\nis proportional to ( χb−χa)2. Note that g↑\n2xg↓\n2x= 0 in\ntetragonalsystems. The strengthofthe anisotropyin the\nmagnetic susceptibilities along the magnetic hard axes\nincreases Tsc.χbis about 5 times larger than χain the\nferromagnetic state of URhGe at ambient pressure[15].\nThe enhancement of Tscin the uniaxial stress applied\nalong the baxis can be understood as the result of the in-\ncreasing transverse ﬂuctuations along the magnetic hard\nbaxis. Onthe otherhand, the degreeofthe anisotropyin\nthe magnetic susceptibilities χbandχais at most two in\nthe ferromagnetic state of URhSi as shown in Fig. 3(b).\nThe contribution to raise Tscby this mechanism may be\nsmaller than that in URhSi.\nIt is not clear which pairing mechanism (the intraband\nor interband one) takes a dominant role for the appear-\nance of the superconductivity at ambient pressure in this\nsystem. It is diﬃcult to make quantitative estimates of\ncontributions from the two mechanisms to Tsc. If the in-\ntrabandpairing mediated by the longitudinal spin ﬂuctu-\nations is dominant, the superconductivity with the sim-\nilar value of Tscto that in URhGe could appear in the\nhigh-quality sample of URhSi since the values of T0in\nURhSi and URhGe are similar. Meanwhile, if the in-\nterband pairing mediated by the transverse ﬂuctuations\nplaysacertainroleinthesuperconductivity,the Tscvalue\nin URhSi could be smaller than that of URhGe even\nif the quality of samples is improved. This is because\nthe strength of the transverse ﬂuctuations in URhSi is\nsmaller than that in URhGe as discussed above. It is\nnecessary to grow the high-quality single-crystal sample\nof URhSi to settle this problem.\nIt haslongbeen thoughtthat the p-waveferromagnetic\nsuperconductivity driven by the longitudinal spin ﬂuctu-\nations appears in the uranium ferromagnetic supercon-\nductors where the ferromagnetism can be described with\nthe 3D Ising model. The importance of the longitudinal\nspin ﬂuctuations for the ferromagnetic superconductiv-\nity has been stressed in the theoretical studies[10–13].\nOur study for the critical behavior of the magnetization\nin UGe 2and URhGe suggests that the 3D Ising uni-12\nversality class is not appropriate to describe the ferro-\nmagnetic phase transition in the uranium ferromagnetic\nsuperconductors[18]. The pairing mechanism other than\nthat driven by the longitudinal spin ﬂuctuations might\ntake an important role for the uranium ferromagnetic\nsuperconductors. The improvement of the sample qual-\nity in URhSi would provide good opportunity to make\nfurther progress toward a complete understanding of the\nuranium ferromagnetic superconductors.\nV. SUMMARY\nThe critical behavior of the magnetization in uranium\nferromagnet URhSi has been studied around its ferro-\nmagnetic transition temperature TC∼10 K. We have an-\nalyzed the magnetic data with a modiﬁed Arrott plot, a\nKouvel-Fisher plot, the critical isotherm analysis, and\nthe scaling analysis in order to determine the critical ex-\nponentβfor the temperature dependence of the spon-\ntaneous magnetization MsbelowTC,γfor the magnetic\nsusceptibility χ, andδfor the magnetization isotherm at\nTC. We determine the values of the critical exponents as\nβ= 0.300 ±0.002,γ′= 1.00±0.02 for T < T C,γ=\n1.03±0.02 forTC< T, andδ= 4.38±0.04 from the\nscaling analysis and the critical isotherm analysis. The\nWidom scaling law δ= 1 +γ/βis fulﬁlled with the\nobtained critical exponents. The ferromagnetic state in\nURhSi has strong uniaxial magnetic anisotropy. How-\never, the obtained critical exponents diﬀer from those inthe 3D Ising model ( β= 0.325, γ= 1.241, and δ= 4.82)\nwithshort-rangeexchangeinteractions. Thevaluesofthe\nexponents βandγin URhSi are similar to those in ura-\nnium ferromagnetic superconductors UGe 2and URhGe,\nand uranium ferromagnets UIr and U(Co 0.98Os0.02)Al.\nWe suggest that the universality class of the ferromag-\nnetic transition in URhSi belong to the same one for the\nuraniumferromagneticsuperconductorsandtheuranium\nferromagnets. In these uranium ferromagnets, the mag-\nnetic susceptibility χshows a mean-ﬁeld-theory-like be-\nhavior (χ−1∝T) around TCbut the critical exponents\nβare close to those of 3D ferromagnets. The anomalous\ncritical exponents in the uranium ferromagnets cannot\nbe explained with previous theoretical studies for critical\nphenomena. This conventional critical behavior of the\nmagnetization may reﬂect peculiar features in the ferro-\nmagnetism of the 5 felectrons where the superconductiv-\nity could appear. The absence of the superconductivity\nin URhSi is discussed from several viewpoints. 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B 96, 104501 (2017)."    },    {        "title": "1302.2166v1.Reply_to_the_Comment_on__Unified_Formulism_of_Andreev_Reflection_at_a_Ferromagnetic_Superconductor_Interface__by_Eschrig_et_al.pdf",        "content": "arXiv:1302.2166v1  [cond-mat.supr-con]  8 Feb 20131\nReply to the Comment on “Uniﬁed For-\nmalism of Andreev Reﬂection at a Ferromag-\nnet/Superconductor Interface” by Eschrig et al.\nAfter the publication of the paper by Chen, Tesanovic,\nand Chien [1] (hereafter noted as the CTC model) on the\nuniﬁed formalism of Andreev reﬂection, Eschrig et al.[2]\n(hereafter noted as the Comment), assert that a number\nof works [2-7 in Ref.2] have already solved the problem\nand that the CTC model violates basic physics. Both\nassertions are false.\nIn metals, the spin polarization P, where 0 ≤P≤1, is\na quantity intrinsic to the material and uniquely deﬁned\nby its band structure and carrier density. The Andreev\nreﬂection spectroscopy (ARS) at the interface between a\nnormal metal (N) and a superconductor (S), can mea-\nsurePof the normal metal and the gap of the super-\nconductor through proper analyses, therefore is of great\nimportance. Over the last three decades, there have been\nmany theoretical models pertaining to ARS, but only\nsome are relevant to, and conﬁrmed by, the ARS exper-\niments. Numerous experimental ARS measurements us-\ning point contact Andreev reﬂection (PCAR) have ﬁrmly\nestablishedthattheexperimentalresultsofnon-magnetic\nmetals (P= 0, e.g., Cu and Au), and a true half metal\n(P= 1 in CrO 2) [5] arequantitatively close to those de-\nscribed by the BTK [6] model and the Mazin model [7]\nrespectively. A linear model with a linear combination\nof the BTK ( P= 0) and the Mazin( P= 1) results has\nbeen widely used but not rigorously justiﬁed. The Dynes\nmodel [8] for tunneling has been experimentally estab-\nlished even earlier. The CTC model combines the BTK,\nthe Mazin, and the Dynes models, the three experimen-\ntally established models.\nThe key physics that has eluted previous theories but\nincludedintheCTCmodelisthecrucialeventattheN/S\ninterface. A single electron cannot go through the N/S\ninterface to have AR because it always requires another\nelectron with the proper spin orientation. On the N side,\nthere is a polarized current with an imbalance of spins,\nwhereas on the S side, the supercurrent carries no spin\nangular momentum. In ARS, charge is alwaysconserved,\nas guaranteed by the AR, but spin angular momentum is\nnot. The spin angular momentum carried by the redun-\ndant majority must become evanescent. The AR wave-\nfunction in the CTC model is ψAR=aeαq−zeiq−x, where\nzin the ﬁrst exponentialis the coordinateofthe spin cur-\nrent, generallyunrelatedtothechargecurrentcoordinate\nxin the second exponential. This is the same situation\nencountered in optical total reﬂection on a 2D surface\nwith the spin current and the charge current as the two\neﬀective dimensions. However, in 1D ARS, the evanes-\ncent spin current is also along the xdirection, thus we\nhave the wave function ae(α+i)q−xand only the deriva-\ntive boundary condition needs a factor of 2, which has\nbeen attributed to the dimensionless interfacial factor Z.\nThe paper [1] clearly emphasizes this essential physics,starting from the abstract.\nUnfortunately, theCommentfailstorecognizethiscru-\ncial novelty and erroneously treats the spin current as\ncharge current and calculates the current of the evanes-\ncent wave function, which we have emphasized clearly\nthatitdoesNOTcontributetochargecurrent. TheCom-\nment further incorrectly assumes the evanescent wave is\nin the superconductor ( x >0 ) while it is actually on\nthe normal metal side with x <0. Since it occurs only\nnear the interface, the calculated jAR∼e2αq−xby the\nComment is zero at x<<0 as expected for the evanes-\ncent spin current. In the CTC model, charge conser-\nvation is guaranteed by the conservation of probability.\nThe charge current density in the normal metal is always\njC=ejP, wherejPis the probability current. Speciﬁ-\ncallyfor the AR term, the chargecurrentis eA, withAas\ntheARprobability. The CTC model violates no physical\nlaws. Should the CTC model violate any, so would the\nthree models (BTK, Mazin, and Dynes) it encompasses,\nnot to mention disagreement with experiments.\nWithout the explicit inclusion of spin into the wave-\nfunction, the theoretical ARS analyses cannot determine\nP,whichisdeﬁnedbythenumberofconductionelectrons\ninthetwospindirections. TheAR isnotasingleelectron\nevent since it always needs another electron with proper\nspin orientation. The availability of the other electron\nwith proper spin should notdepend on the dimensions of\nthe interface or the wavevectors of the conduction elec-\ntrons, as the models [3, 4] suggested by the Comment,\nbut rather onlythe spin polarization as represented by\nαin the CTC model. Therefore, αisunrelated to any\ncharacteristic length of charge decay as the Comment\nspeculated. It rather describes the characteristics of the\nredundant majority spins at the interface. Any model\nof ARS must be consistent with the two obvious facts:\nall electrons can go through the interface for P= 0\nbut no electrons go through the interface for P= 1,\nand there should be no variation in spin current on the\nnormal metal side for either case. Indeed, in the CTC\nmodel the characteristic length, ∼1/α, is zero for P= 1\nand inﬁnite with zero magnitude for P= 0, so the CTC\nmodeldoescapture these crucial features. Furthermore,\nfor 0< P <1, the CTC model indicates that the char-\nacteristic length is ﬁnite but depends on the Pvalue, as\nexpected because the eﬀect of the superconductor should\nbe localized near the interface. These eﬀects have never\nbeen addressed before.\nBecause of the constraints from experimental ARS re-\nsults, we now know any viable theoretical model must\nprovidequantitative results that are close to those of the\nBTK and the Mazin models at the two limits, or will not\nbeconﬁrmedexperimentally. Thereareindeedmanythe-\noreticalmodels but some haveserious shortcomings. The\nComment goes into great length describing the salient\nfeatures and merit of some theoretical models [3, 4], but\nneglects to mention that these models do notgenerate2\nresults that are even close to those of experiments (or\nthe BTK and the Mazin models), and some predicted\nfeatures that have never been observed. For example,\nthe Comment cites the theoretical model of Zutic et al.\n[3]. While in search of a viable model for analyses dur-\ning the earlier times of ARS, we have extensively tested\nthat model, which treats the AR events in analogy to\nthe Snells law in optics with the suppression of conduc-\ntance depending on the injection angle. However, after\nnumerous measurements using tips of various materials,\nshapes, and contact angles, we could not obtain experi-\nmental results in any contacts at any temperature that\nagree with the theory. We now know since that theory\ndoesnotproduceresultsclosetothoseat the P= 1limit,\nany experimental attempt for conﬁrmation would be fu-\ntile. The Comment mentions another more recent model\nby Grein et al., [4]. Among other features, this complex\nmodel predicts subgap Andreev bound states for spin-\npolarized N/S interfaces. Numerous ARS measurements\nduring the last two decades on magnetic materials, in-\ncluding half metals, at various temperatures, some down\nto less than 1 K, have never uncovered such subgap fea-\ntures.\nThe CTC model provides in concise analytical forms\nall the probabilities involved in AR with any Pvalue and\nthe results encompass those of the BTK and the Mazin\nmodels in appropriate limits as shown in Table 1 of Ref.\n1. One immediately knows that the charge current is\nconserved since for α= 100 (P∼=1), the AR is zero,\nA= 0, but the normal reﬂection is 1 ( B= 1), as shown\nin Fig. 1 of Ref. 1. Other eﬀects, such as the inelastic\nscattering in the Dynes model are now included for each\nprobability of any Pvalue. It veriﬁes the validity of\nthe widely used linear polarization model but only under\nappropriate conditions. These are the key results that no\nprevious theoretical models have provided.In summary, there are indeed many theoretical models\nfor ARS, some elaborate, complex, and intricate. How-\never, only those that can be experimentally conﬁrmed\nare relevant and useful in extracting physically impor-\ntant quantities in ARS. The CTC model with analytic\nsolutions, unifying the BTK, the Mazin, and the Dynes\nmodels, has been experimentally and quantitatively es-\ntablished, and revealednew physics about the evanescent\nspin currentat the N/S interface. The alleged unphysical\nresults are entirely due to the mistreatment of the spin\ncurrent at the N/S interface and careless calculations in\nthe Comment.\nT. Y. Chen1and C. L. Chien2\n1Department of Physics, Arizona State University,\nTempe, AZ 85287,2Department of Physics and Astron-\nomy, Johns Hopkins University, Baltimore MD 21218\nWe are conﬁdent that Z. Tesanovic, who passed away\nrecently, would concur with the content of the Response.\n[1] T. Y. Chen, Z. Tesanovic, and C. L. Chien, Phys. Rev.\nLett.109, 146602 (2012).\n[2] M. Schrig, A. A. Golubuv, I. I. Mazin, B. Nadgorny, Y.\nTanaka, O. T. Valls, and I. Zutic, arXiv.1301.3511.\n[3] I. Zutic and O. T. Valls, Phys. Rev. B 61, 1555 (2000).\n[4] R. Grein, T. Lofwander, G. Metalidis, and M. Eschrig,\nPhys. Rev. B 81, 094508 (2010).\n[5] Y. Ji, G. J. Strijkers, F. Y. Yang, C. L. Chien, J. M.\nByers, A. Anguelouch, G. Xiao, and A. Gupta, Phys.\nRev. Lett. 86, 5585 (2001).\n[6] G. E. Blonder and M. Tinkham and T. M. Klapwijk,\nPhys. Rev. B 25, 4515 (1982).\n[7] I. I. Mazin, A. A. Golubov, and B. Nadgorny, J. Appl.\nPhys.89, 7576 (2001).\n[8] R. C. Dynes, J. P. Garno, G. B. Hertel, and T. P.\nOrlando, Phys. Rev. Lett. 53, 2437 (1984)."    },    {        "title": "1007.4536v1.Thermodynamics_of_classical_frustrated_spin_chain_at_the_ferromagnet_helimagnet_transition_point.pdf",        "content": "arXiv:1007.4536v1  [cond-mat.stat-mech]  26 Jul 2010Thermodynamics of classical frustrated spin chain at the fe rromagnet-helimagnet\ntransition point\nD. V. Dmitriev∗and V. Ya. Krivnov\nJoint Institute of Chemical Physics, RAS, Kosygin str. 4, 11 9334, Moscow, Russia.\n(Dated:)\nLow-temperature thermodynamics of the classical frustrat ed ferromagnetic spin chain is studied.\nUsing transfer-matrix method we found the behavior of the co rrelation function and zero-ﬁeld sus-\nceptibility at the ferromagnetic-helical transition poin t. It is shown that the critical exponent for\nthe susceptibility is changed from 2 to 4/3 at the transition point.\nLately, there has been considerable interest in low-dimensional spin models that exhibit frustration. One of them\nis the spin chain with the ferromagnetic interaction J1of nearest neighbor (NN) spins and the antiferromagnetic\nnext-nearest-neighbor (NNN) interaction J2, so called the 1D F-AF model. Its Hamiltonian has a form\nH=J1/summationdisplay\nSn·Sn+1+J2/summationdisplay\nSn·Sn+2 (1)\nwhereJ1<0 andJ2>0.\nThis model is characterized by a frustration parameter α=J2/|J1|. The ground state properties of the quantum\ns= 1/2 F-AF chain have been intensively studied last years [1–4]. It is know n that the ground state of the model is\nferromagnetic for α<1/4. Atα= 1/4 the ground state phase transition to the incommensurate singlet phase with\nhelical spin correlations takes place. Remarkably, this transition po int does not depend on a spin value, including the\nclassical limit s=∞.\nInteresting question is the inﬂuence of the frustration on the low- temperature thermodynamics of the model es-\npecially near the transition point α= 1/4. We study this problem for the classical version of model (1). At z ero\ntemperature the classicalmodel has long range-order(LRO) for all values of α: the ferromagneticLRO at α≤1/4and\nthe helical one at α>1/4. At ﬁnite temperature the LRO is destroyed by thermal ﬂuctuat ions and thermodynamic\nquantities have a singular behavior at T→0. In particular, the zero-ﬁeld susceptibility χdiverges. For the 1D\nHeisenberg ferromagnet ( α= 0)χ= 2|J1|/3T2[5]. At 0< α <1/4 the susceptibility is χ= 2(1−4α)|J1|/3T2.\nThis behavior of χis similar to that for the quantum s= 1/2 F-AF model [6]. The prefactor in χvanishes at the\ntransition point indicating the change of the critical exponent. We f ocus our attention on the behavior of χat the\ntransition point.\nThe partition function Zof model (1) at α= 1/4 is\nZ=N/productdisplay\nn=1/integraldisplay\ndΩnexp/braceleftbigg1\nT/summationdisplay\n(/vectorSn·/vectorSn+1−1\n4/vectorSn·/vectorSn+2)/bracerightbigg\n(2)\nwhere/vectorSnis unit vector, dΩnis the volume element of the solid angle for n-th site, we put |J1|= 1 and the periodic\nboundary conditions are proposed.\nOur further calculations are based on the transfer matrix method and we use a version of this method adapted to\nthe model with NNN interactions by Harada and Mikeska in [7].\nFollowing Ref.[7] we represent Zin a form\nZ=N/productdisplay\nn=1/integraldisplay\ndΩnK(θn−1,θn;ϕn) (3)\nwhere\nK(θn−1,θn;ϕn) = exp/parenleftbiggcosθn−1+cosθn\n2T−(cosθn−1cosθn+sinθn−1sinθncosϕn)\n4T/parenrightbigg\n(4)\nwhereθnis the angle between /vectorSnand/vectorSn+1andϕnis the angle between components of /vectorSn−1and/vectorSn+1projected\nonto (Xn,Yn) plane of the n-th local coordinate system with the Znaxis parallel to /vectorSn.\n∗Electronic address: dmitriev@deom.chph.ras.ru2\nIntegrating Eq.(4) over ϕnwe obtainZin a form\nZ=/productdisplay\nn/integraldisplayπ\n0dθnsinθnA(θn−1,θn) (5)\nwhere\nA(θn−1,θn) =1\n2I0(−z)exp/parenleftbiggcosθn−1+cosθn\n2T−cosθn−1cosθn\n4T−3\n4T/parenrightbigg\n(6)\nandI0(−z) is the modiﬁed Bessel function of\nz=sinθn−1sinθn\n4T(7)\nLet us consider an integral equation\n/integraldisplayπ\n0A(θ1,θ2)ψα(θ2)sin(θ2)dθ2=λαψα(θ1) (8)\nwhereψα(θ) satisfy normalization condition\n/integraldisplayπ\n0ψα(θ)ψβ(θ)sinθdθ=δα,β (9)\nEigenfunctions ψαand eigenvalues λαcan be chosen as real, since the kernel A(θ1,θ2) is real and symmetric. Then,\nA(θ1,θ2) =/summationdisplay\nαλαψα(θ1)ψα(θ2) (10)\nSubstituting Eq.(10) into Eq/(5) we obtain in the thermodynamic limit\nZ=λN\n0 (11)\nwhereλ0is the largest eigenvalue of Eq.(8).\nIn the low-temperature limit the angles θnare small and we can use the asymptotic expansion of the modiﬁed\nBessel function\nI0(−z) =ez\n√\n2πz/parenleftbigg\n1+1\n8z+O(z−2)/parenrightbigg\n(12)\nThen, we expand the expression in the exponent of the transfer m atrix to the fourth order in θito obtain\nA(θ1,θ2) =/radicalbigg\nT\n2πθ1θ2/parenleftbigg\n1+T\n2θ1θ2/parenrightbigg\nexp/parenleftbigg\n−(θ1−θ2)2\n8T−θ2\n1θ2\n2\n8T+(θ1−θ2)4\n96T/parenrightbigg\n(13)\nWe can neglect the term ( θ1−θ2)4/96Tas will be seen below.\nAs a result integral equation (8) reduces to\n/integraldisplayπ\n0/radicalbigg\nTθ2\n2πθ1/parenleftbigg\n1+T\n2θ1θ2/parenrightbigg\ne−(θ1−θ2)2\n8T−θ2\n1θ2\n2\n8Tψα(θ2)dθ2=λαψα(θ1) (14)\nThe maximum of the expression in the exponent (saddle point) is at θ2=θ1(more exactly θ2=θ1−θ3\n1+..., but\nit suﬃces to put θ2=θ1). Near this saddle point we expand ψα(θ2) as follows\nψα(θ2) =ψα(θ1)+(θ2−θ1)ψ′\nα(θ1)+(θ2−θ1)2\n2ψ′′\nα(θ1)+... (15)\nand\n/radicalbigg\nθ2\nθ1=/radicalbigg\n1+θ2−θ1\nθ1= 1+θ2−θ1\n2θ1−(θ2−θ1)2\n8θ2\n1+... (16)3\nLet us introduce new scaled variables\nθ2−θ1=T1/2x\nθ1=T1/3r (17)\nNowψα(θ)→ψα(r) and\nψα(θ2)→ψα(r)+T1/6xψ′\nα(r)+T1/3x2\n2ψ′′\nα(r)+... (18)\n/radicalbigg\nθ2\nθ1→1+T1/6x\n2r−T1/3x2\n8r2+O(T1/2) (19)\nexp/parenleftbigg\n−(θ1−θ2)2\n8T−θ2\n1θ2\n2\n8T/parenrightbigg\n→exp/parenleftbigg\n−x2\n8−T1/3r4\n8/parenrightbigg\n(20)\nSummarizing all above we arrive at\n/integraldisplayπ/√\nT\n−r/T1/6/parenleftbigg\n1+T1/6x\n2r−T1/3x2\n8r2/parenrightbigg/parenleftbigg\n1+T1/3\n2r2/parenrightbigg/parenleftbigg\nψα(r)+T1/6xψ′\nα(r)+T1/3x2\n2ψ′′\nα(r)/parenrightbigg\ne−x2/8−T1/3r4/8Tdx√\n2π=λαψα(r)\n(21)\nAtT→0, we can change the limits in the integral to [ −∞,∞], then only even powers in xgives contribution, so\ntaking into account only terms up to T1/3we obtain\n/integraldisplay∞\n−∞/bracketleftbigg/parenleftbigg\n1−T1/3r4\n8+T1/3\n2r2/parenrightbigg\nψα+T1/3x2\n2/parenleftbigg\nψ′′\nα+1\nrψ′\nα(r)−1\n4r2ψα/parenrightbigg/bracketrightbigg\ne−x2/8Tdx√\n2π=λαψα (22)\nAfter integration over xwe obtain a linear diﬀerential equation\n2T/parenleftbigg\n1−T1/3r4\n8+T1/3\n2r2/parenrightbigg\nψα+4T4/3/parenleftbigg\nψ′′\nα+1\nrψ′\nα(r)−1\n4r2ψα/parenrightbigg\n=λαψα (23)\nand, ﬁnally,\n−ψ′′\nα−1\nrψ′\nα+r4\n16ψα=εαψα (24)\nwith\nεα=2T−λα\n4T4/3(25)\nThus, we have got a Schr¨ odinger equation for a particle with Zcomponent of the angular momentum lz= 0 in 2D\npotential well U(r) =r4/16. Normalization condition for ψα(r) is\n/integraldisplay∞\n0ψα(r)ψβ(r)2πrdr=δα,β (26)\nNumerical solution of Eq.(24) gives the following lowest eigenvalues (c orresponding to the largest λ):\nεα= 0.9305;3.78;7.44... (27)\nAs was shown in Ref.[7] the two-spin correlation function can be expr essed by the following integral\n/angbracketleftBig\n/vectorS1·/vectorS1+n/angbracketrightBig\n=1\nλn−1\n0/integraldisplayπ\n0dθnsinθnn−1/productdisplay\nl=1dθlsinθlψ0(θl)ψ0(θn)/parenleftbig0 1/parenrightbig\nB(θ1)H(θl,θl+1)B(θn)/parenleftbigg\n0\n1/parenrightbigg\n(28)\nwhere\nB(θ) =/parenleftbigg\ncosθ/2 sinθ/2\n−sinθ/2 cosθ/2/parenrightbigg\n(29)4\nH(θ1,θ2) =B(θ1)/parenleftbigg\n−/tildewideA(θ1,θ2) 0\n0A(θ1,θ2)/parenrightbigg\nB(θ2) (30)\nand/tildewideA(θ1,θ2) is given by Eq.(6) with I0(−z) replaced by I1(−z).\nUsing the asymptotic expansion of the Bessel function\nI1(−z) =−ez\n√\n2πz/parenleftbigg\n1−3\n8z+O(z−2)/parenrightbigg\n(31)\nwe obtain\nH(θ1,θ2) =A0(θ1,θ2)/parenleftBigg\n1−3T\n2θ1θ2θ1+θ2\n2\n−θ1+θ2\n21+T\n2θ1θ2/parenrightBigg\n(32)\nwhere\nA0(θ1,θ2) =/radicalbigg\nT\n2πθ1θ2exp/parenleftbigg\n−(θ1−θ2)2\n8T−θ2\n1θ2\n2\n8T/parenrightbigg\n(33)\nThe matrix H(θ1,θ2) is not symmetric. Therefore, to calculate/angbracketleftBig\n/vectorS1·/vectorS1+n/angbracketrightBig\nit is necessary to solve a pair of the\nintegral equations\n/integraldisplayπ\n0H(θ1,θ2)/vector uα(θ2)sin(θ2)dθ2=ηα/vector uα(θ1) (34)\n/integraldisplayπ\n0HT(θ1,θ2)/vector vα(θ2)sin(θ2)dθ2=ηα/vector vα(θ1) (35)\nwhereHT(θ1,θ2) is transposed matrix H(θ1,θ2) and two-component vectors /vector uαand/vector vα\n/vector uα=/parenleftbigg\nu1,α\nu2,α/parenrightbigg\n, /vector v α=/parenleftbigg\nv1,α\nv2,α/parenrightbigg\n(36)\nsatisfy orthonormality relations,\n/integraldisplayπ\n0/vector uT\nα(θ)/vector vβ(θ)sin(θ)dθ=/integraldisplayπ\n0/vector vT\nα(θ)/vector uβ(θ)sin(θ)dθ=δα,β (37)\nThen, the matrix H(θ1,θ2) can be represented as\nH(θ1,θ2) =/summationdisplay\nαηα/vector uα(θ1)/vector vT\nα(θ2) (38)\nAt smallθ1,θ2Eqs.(34) and (35) reduce to\nπ/integraldisplay\n0A0(θ1,θ2)/bracketleftbigg/parenleftbigg\n1−3T\n2θ1θ2/parenrightbigg\nu1,α(θ2)+θ1u2,α(θ2)/bracketrightbigg\nsin(θ2)dθ2=ηαu1,α(θ1) (39)\nπ/integraldisplay\n0A0(θ1,θ2)/bracketleftbigg\n−θ1u1,α(θ2)+/parenleftbigg\n1+T\n2θ1θ2/parenrightbigg\nu2,α(θ2)/bracketrightbigg\nsin(θ2)dθ2=ηαu2,α(θ1) (40)\nIntegratingthese equationsnearthe saddlepoint similarto Eqs.(21 ),(22), weget apairoflineardiﬀerential equation\n2T/parenleftbigg\n1−T1/3r4\n8−3T1/3\n2r2/parenrightbigg\nu1,α+4T4/3/parenleftbigg\nu′′\n1,α+1\nru′\n1,α−1\n4r2u1,α/parenrightbigg\n+2T4/3ru2,α=ηαu1,α (41)\n2T/parenleftbigg\n1−T1/3r4\n8+T1/3\n2r2/parenrightbigg\nu2,α+4T4/3/parenleftbigg\nu′′\n2,α+1\nru′\n2,α−1\n4r2u2,α/parenrightbigg\n−2T4/3ru1,α=ηαu2,α (42)5\nand, ﬁnally,\n−u′′\n1,α−1\nru′\n1,α+1\nr2u1,α+r4\n16u1,α+r\n2u2,α=µαu1,α (43)\n−u′′\n2,α−1\nru′\n2,α+r4\n16u2,α−r\n2u1,α=µαu2,α (44)\nwhere\nµα=2T−ηα\n4T4/3(45)\nA few lowest eigenvalues of Eqs.(43) and (44) are\nµα= 1.4113;1.83;3.98... (46)\nFor/vector vαsimilar procedure gives\n−v′′\n1,α−1\nrv′\n1,α+1\nr2v1,α+r4\n16v1,α−r\n2v2,α=µαv1,α (47)\n−v′′\n2,α−1\nrv′\n2,α+r4\n16v2,α+r\n2v1,α=µαv2,α (48)\nIt follows from Eqs.(43)-(44) and (47)-(48) that the functions /vector vαis connected with /vector uαby the relations v1,α=−u1,α,\nv2,α=u2,αand, therefore, normalization condition (37) transforms to\nT2/3/integraldisplay∞\n0(u2,αu2,β−u1,αu1,β)rdr=δα,β (49)\nUsing Eqs.(38) and (37) we obtain the correlation function (28) in a f orm\n/angbracketleftBig\n/vectorS1·/vectorS1+n/angbracketrightBig\n=/summationdisplay\nαyn−1\nαf2\nα (50)\nwhereyα=ηα/λ0and\nfα=/integraldisplayπ\n0ψ0(θ)u2,α(θ)sin(θ)dθ=T2/3/integraldisplay∞\n0ψ0(r)u2,α(r)rdr (51)\nAtT→0\nyα=2T−4T4/3µα\n2T−4T4/3ε0≈1−2T1/3(µα−ε0) (52)\nand the correlation function becomes\n/angbracketleftBig\n/vectorS1·/vectorS1+n/angbracketrightBig\n=/summationdisplay\nαf2\nαexp[−2T1/3(µα−ε0)(n−1)] (53)\nAccording to Eq.(53) the correlation length ξatT→0 is\nξ=1\n2(µ0−ε0)T1/3=1.04\nT1/3(54)\nNow we are ready to calculate the magnetic susceptibility at T→0, which is\nχ=1\n3TN/summationdisplay\nn/angbracketleftBig\n/vectorS1·/vectorS1+n/angbracketrightBig\n=1\n3T(1+2/summationdisplay\nαf2\nα\n1−yα) =1\n3T+1\n3T4/3/summationdisplay\nαf2\nα\nµα−ε0(55)\nNow we see that f2\nαand (µα−ε0) depends on the solutions of diﬀerential equations which are indepe ndent ofT.\nSo, the sum in χgives numerical constant\n/summationdisplay\nαf2\nα\nµα−ε0= 3C (56)6\nTherefore, the low-temperature susceptibility behaves as\nχ=C/vextendsingle/vextendsingle/vextendsingleJ1/3\n1/vextendsingle/vextendsingle/vextendsingle\nT4/3(57)\nNumerical calculationsgivesfor the constant Cthe valueC≈1.07. Thus, the criticalexponent for the susceptibility\nat the transition point is 4 /3 and that for the correlation length is 1 /3.\n[1] A. V. Chubukov, Phys. Rev. B 44, 4693 (1991).\n[2] D. V. Dmitriev and V. Ya. Krivnov, Phys. Rev. B 73, 024402 (2006).\n[3] F. Heidrich-Meisner, A. Honecker, and T. Vekua, Phys. Re v. B74, 020403(R) (2006).\n[4] T. Hikihara, L. Kecke, T. Momoi, and A. Furusaki, Phys. Re v. B78, 144404 (2008).\n[5] M. E. Fisher, Am. J. Phys. 32, 343 (1964).\n[6] M. Hartel, J. Richter, D. Ihle, and S.-L. Drechsler, Phys . Rev. B 78, 174412 (2008).\n[7] I. Harada and H. J. Mikeska, Z. Phys. B: Condens. Matter 72, 391 (1988)."    },    {        "title": "0902.0308v1.Evolution_of_the_local_superconducting_density_of_states_in_ErRh__4_B___4___close_to_the_ferromagnetic_transition.pdf",        "content": "arXiv:0902.0308v1  [cond-mat.supr-con]  2 Feb 2009Evolution of the local superconducting density of states in ErRh 4B4close to the\nferromagnetic transition\nV. Crespo,1J.G. Rodrigo,1H. Suderow,1S. Vieira,1D. Hinks,2and I.K. Schuller3\n1Laboratorio de Bajas Temperaturas, Departamento de F´ ısic a de la Materia Condensada,\nInstituto de Ciencia de Materiales Nicol´ as Cabrera, Facul tad de Ciencias,\nUniversidad Aut´ onoma de Madrid, 28049 Madrid, Spain\n2Materials Science Division, Argonne National Laboratorie s, Argonne, Illinois 60439\n3Physics Department, University of California-San Diego, L a Jolla California 92093-0319, USA\n(Dated: March 13, 2022)\nWe present local tunneling spectroscopy experiments in the superconducting and ferromagnetic\nphases of the reentrant superconductor ErRh 4B4. The tunneling conductance curves jump from\nshowing normal to superconducting features within a few mK c lose to the ferromagnetic transition\ntemperature, with a clear hysteretic behavior. Within the f erromagnetic phase, we do not detect\nany superconducting correlations. Within the superconduc ting phase we ﬁnd a peculiar V-shaped\ndensity of states at low energies, which is produced by the ma gnetically modulated phase that\ncoexists with superconductivity just before ferromagneti sm sets in.\nPACS numbers: 74.70.Dd, 74.25.Jb, 74.25.Dw\nThe physics of competing orders has been the subject\nof much research over the years. A particularly interest-\ning and extensively studied area is the competition and\ncoexistence between superconductivity and magnetism\n[1, 2, 3, 4, 5]. This interest has been reemphasized by\nadvances in highly correlated superconductors, such as\nthe cuprates [6, 7], and, very recently, the Fe pnictide\n[8] and Ni [9] and Fe [10] phosphide superconductors.\nIn all these materials there is evidence of some sort of\nmagnetism and of superconductivity perhaps at diﬀerent\ntemperatures or even coexisting. It may even be that\nfor the existence of high temperature superconductiv-\nity proximity to a magnetic boundary is important[11].\nThus investigating the form in which these two orders\ninteract is not only unusual and interesting, but it may\nhold information regarding the mechanism of supercon-\nductivity in highly correlated systems. A classical ex-\nample of a superconductor which also exhibits magnetic\norder is ErRh 4B4[12]. This is a reentrant superconduc-\ntor in which, with decreasing temperature, ﬁrst a super-\nconducting transition occurs, at T c1≈8 K, and then\nat Tc2≈0.7 K, where local Er moments order ferro-\nmagnetically, the material becomes normal again. Un-\ntil now this material was only probed with macroscopic\nprobes, such as thermal studies, resistivity and suscepti-\nbility, neutron scattering or tunneling spectroscopy using\nthing ﬁlms [1, 2, 3, 13, 14, 15, 16, 17, 18]. The super-\nconducting properties far from the magnetic transition\nremain unknown to a large extent. For example, the\njump in the speciﬁc heat at T c1is sizable. However,\nalready shortly below, magnetic contributions are over-\nwhelming. Even more mysterious is the behavior close\nto ferromagnetism, where always a strong hysteresis is\nfound. When heating, the transition to the full super-\nconductingstateoccursT c2↑,whichisconsiderablyabove\n(around 100 mK) the transition temperature found whencooling, T c2↓. There are no pronounced eﬀects associ-\nated with the velocity of the temperature ramps, so it\nseems that superconductivity truly hinders the outcome\nof ferromagnetism and viceversa [3, 17, 18]. Remarkably,\nneutron scattering experiments by Sinha et al. [13] have\nshown that in the superconducting phase, when cooling,\na new magnetically modulated state appears around 1\nK, and disappears below T c2↓. When heating, the same\nstate appears at T c2↑and disappears again around 1 K\n[1, 13]. This peculiarmagnetic state is supposed to be in-\nduced by superconductivity[3], as proposed by Anderson\nand Suhl[3, 19]. More recently, it has been shown that\nthemodulatedstatemostlikelycoexistswithinsupercon-\nducting domainsformingclosetoferromagnetism[17, 18].\nTherearemanyimportantopenquestionsraisedbythese\nand other studies. Among them, we may ask: What\nis the shape of the superconducting density of states?\nWhat changes are produced in the coexistence region\nby the magnetically modulated state? Are there some\nsuperconducting correlations in the electronic density of\nstates below T c2↓or Tc2↑? Here we present atomic res-\nolution scanning tunneling microscopy and spectroscopy\n(STM/S)experimentsonaErRh 4B4singlecrystal,which\nprovide new insight into these problems.\nWe use a STM/S system in a3He refrigerator,\nequipped with a Pb tip and an in-situ system, described\npreviously, to obtain clean and sharp tips[20, 21]. The\nsuperconducting density ofstates ofPb tips has been dis-\ncussed in previous work[20, 21, 22]. As it is well known,\ntunneling spectroscopy on a superconducting sample us-\ning a superconducting counter electrode (S-S’ tunnel-\ning) is superior to tunneling spectroscopy with a nor-\nmal electrode, because the sharp density of states of\nthe counter electrode allows for a better determination\nof ﬁne structure in the density of states of the sample,\neven at temperatures where thermal smearing is impor-2\n-8 -6 -4 -2 0 2 4 6 80246\n2 nm \n Normalized conductance \nVoltage (mV) Pb - ErRh 4B4\nT = 1.1 K a) \n-8 -6 -4 -2 0 2 4 6 801234\n  NPb  (E) \nEnergy (meV) Pb ErRh 4B4 b) \n-4 -3 -2 -1 0 1 2 3 40.0 0.5 1.0 1.5 2.0  \n N ErRh 4B4 (E)  \nEnergy (meV) c) 2ΔErRh 4B4\nFIG. 1: (Color online) In (a) we show local tunneling spec-\ntroscopy curves obtained in the superconducting phase of\nErRh4B4at 1.1 K with a Pb tip, and in the inset atomic\nresolution topography. Red line is the calculated tunnelin g\nconductance using NPb(E) andNErRh 4B4(E) shown, respec-\ntively, in (b) and (c). Arrows in (a) and (b) mark Pb phonon\nmode features.\ntant. The tunneling current between a superconduct-\ning tip with density of states NPb(E) and a sample with\nNErRh 4B4(E)canbewrittenas I(V)∝/integraltext\ndE[f(E−eV)−\nf(E)]NPb(E−eV)NErRh 4B4(E), wheref(E)istheFermi\nfunction. At all temperatures I(V), and its derivative\nthe tunneling conductance σ(V), present sharp features\nat|V|= ∆Pb+∆ErRh 4B4, andattemperaturesaboveap-\nproximately Tc/3 at ∆ Pb−∆ErRh 4B4[20, 21, 23]. With\nthe accurately previously determined density of states of\nthe Pb tip as a function of temperature, NPb(T;E), it\nis possible to obtain the density of states of the sample\nNErRh 4B4(T;E) from the previous expression[23].\nThe sample is a single crystal in the primitive tetrago-\nnal phase grown from an Er-Rh-B melt. The ingot con-\ntained two large single crystals ( ≈0.3 g), the larger of\nwhich was used for the present studies and is the same\nearlier used [13] for combined neutron, transport and\nmagnetic studies. We measured the single crystal on dif-\nferentlyorientedsurfaces,in andoutofplaneofthe prim-\nitive tetragonal crystal structure. In all cases, tunneling\ncharacteristics found on as-grown surfaces, correspond\nto good vacuum tunnel junction, with reproducible to-pographic images and spectra, independent on the bias\nvoltage, and work functions of the order of some eV. The\ntopographyofthesampleisirregular,anditappearsdiﬃ-\ncult to ﬁnd extended ﬂat regions. Many regions show de-\npressed superconducting properties, and even no super-\nconductivityatall, possiblydue tochangesinthe compo-\nsition of the surface, surface reconstructions, or strongly\nenhanced magnetic scattering. However, we were able\nto ﬁnd some (around twenty) locations with ﬂat areas of\nsome 100 nm x 100 nm, where we often obtain atomic\nresolution images (inset of Fig.1). These areas show the\nspectroscopic features discussed in the following. These\nfeatures are much sharper than those found on macro-\nscopic measurements[1, 2, 3, 13, 14, 15, 16, 17, 18], which\nshould, even in high quality samples, give an averaged\nbehavior.\n-4 0 4\n0 1 2 3 4 5 6 7 80.0 0.5 1.0 1.5 1Normalized conductance \n  \nVoltage (mV) 0 01\n-4 -2 0 2 4x 10 \nx 5 \n b) \n N ErRh 4B4 (E)  \nEnergy (meV) a) 7.2 K \n6.0 K \n5.5 K \n5.0 K \n4.0 K \n2.0 K \n1.1 K 6.5 K \n  \n Temperature (K)  Δ ErRh 4B4 (meV) c) \nFIG. 2: (Color online) In (a) we show the temperature depen-\ndence of the tunneling conductance curves measured (points )\nand those calculated (lines) using NErRh 4B4(E;T) shown in\n(b)(curvesare shifted for clarity; scale inthe toptwocurv esis\nenlarged). In (c) we show the temperature dependence of the\nvoltage position of themaximumin NErRh 4B4(E), ∆ErRh 4B4,\ntogether with the BCS curve (line).\nAt the lowest temperatures, T=0.3 K, only the super-\nconducting features due to the Pb tip are observed in the\ntunneling conductance curves, and NErRh 4B4(E) is ﬂat\nand featureless. Well within the superconducting phase\nof ErRh 4B4we measure the expected curves characteris-\ntic for tunneling between two superconductors [Fig.1(a)].\nUsing N Pb(E) (Fig.1(b) and Ref.[24]), we obtain the\ndensity of states of ErRh 4B4,NErRh 4B4(E), shown in\nFig.1(c). NErRh 4B4(E) has a well opened superconduct-\ning gap, with a zerodensity ofstatesup to 0.75meV, and3\na rounded quasiparticle peak whose maximum is located\nat ∆ErRh 4B4=1.2 meV. As shown in Fig.2, when heat-\ning, the local tunneling conductance curves show S-S’\nfeatures, with peaks appearing at ∆ Pb−∆ErRh 4B4from\nthermal excitations. The rounded shape of NErRh 4B4(E)\nis maintained as a function of temperature. ∆ ErRh 4B4\npresents a temperature evolution (Fig.2) which is very\nclose to the expected behavior from weak coupling BCS\ntheory (∆ BCS= 1.76 k BTc1= 1.2 meV).\nThe rounded quasiparticle peaks in NErRh 4B4(E) are\nvery diﬀerent from the divergency expected within sin-\ngle band s-wave BCS theory. Most probably, this shows\npeculiar, temperature independent, magnetic pair break-\ning eﬀects from disorder in the Er paramagnetic sublat-\ntice [25, 26, 27, 28]. However, we cannot exclude an in-\ntricate dependence of the superconducting gap over the\nFermi surface. Band structure calculations show that\nthe largest contribution to the Fermi level density of\nstates comes from the 4d electrons, with smaller addi-\ntional p and s character contributions, of the Rh atoms,\nand a contribution from Er 5d electrons[29]. In any case,\nthe close agreement between experiment and BCS theory\nshowninFig.2(c), andthefactthatthePbphononmodes\nare displaced in the tunneling conductance curves by ex-\nactly∆ ErRh 4B4/e(Fig.1), pinpoint that themainopened\ngap feature over the Fermi surface is that of ∆ ErRh 4B4.\n-3 -2 -1 0 1 2 30123456\nVoltage (mV) \n Normalized conductance 0.91 K \n0.86 K \n1.11 K \n0.91 K \n0.86 K a) \n-3 -2 -1 0 1 2 3\n Voltage (mV) \n 1.11 K \n0.83 K \n0.77 K 1.11 K \n0.89 K b) \n1.11 K \n0.89 K \n0.83 K \n0.77 K Cooling \n-3 -2 -1 0 1 2 30.0 0.5 1.0 1.5 \nErRh 4B4ErRh 4B4\n  NErRh 4B4 (E) \nEnergy (meV) d) Heating Heating \n-3 -2 -1 0 1 2 3 \nEnergy (meV) c) Cooling \nFIG. 3: (Color online). A close-up view of experiments made\nnear the ferromagnetic transition. In the left panels, beha vior\nwhen heating, from 0.86 K to 1.1 K, and in right panels, when\ncooling from 1.1 K to 0.77 K. In (a) and (b) we show the\ntunneling spectroscopy curves obtained, together with tho se\ncalculated using NErRh 4B4(E) shown in (c) and (d).\nMost remarkable features in NErRh 4B4(E) are found\nclose to the ferromagnetic transition, where the tunnel-\ning conductance curves strongly change its shape. InFig.3 we show a close-up view of typical heating and\ncooling experiments. When heating, superconductivity\nin the sample appears abruptly at T c2↑, and the tun-\nneling spectroscopy curves show, within a few mK, the\nS-S’ behavior representedby the lowestcurve in Fig.3(a).\nThe resulting temperature dependence of NErRh 4B4(E)\nis shown in Fig.3(c). Close to T c2↑, at T = 0.86 K in\nFig.3 (a) and (c), the curves are signiﬁcantly diﬀerent\nthan well within the superconducting phase, at T = 1.1\nK in Fig.3 (a) and (c). The position of the maximum\nof the quasiparticle peaks is still at ∆ ErRh 4B4. However,\nthe quasiparticle peaks are smeared and the energy in-\nterval with zero NErRh 4B4(E) is of about 0.3 meV, i.e.\nsmaller than well within the superconducting phase (0.75\nmeV, see Figs.1 and 4).\nWhen cooling again into the ferromagnetic phase, we\nﬁnd identical tunneling conductance curves, at, however,\nlowertemperaturesthan in the heating process, as shown\nin Fig.3(b), and the disappearance of superconducting\nfeatures when cooling occurs at T c2↓, which is 90 mK\nbelow T c2↑. The transition to the normal state occurs\nagain abruptly, within a few mK. Just before the tran-\nsition, at a few mK above T c2↓, we ﬁnd tunneling con-\nductance curves with most strongly smeared supercon-\nducting features [see curves at 0.77 K in Figs.3(b) and\n(d)]. Remarkably, the interval with zero NErRh 4B4(E) is\nfully lost close to T c2↓, where we get truly gapless super-\nconductivity with V shaped increase of NErRh 4B4(E) at\nlow energies. Such a behavior is never observed close to\nTc2↑when heating, as highlighted in Fig.4, where we plot\nthe temperature dependence of the energy interval with\nzeroNErRh 4B4(E). Note that this intervalis signiﬁcantly\ninﬂuenced by magnetism up to about 0.2K abovethe ap-\npearance of the ferromagnetic state, whereas ∆ ErRh 4B4\n(inset of Fig.4) remains constant.\nThese results have been reproduced in the locations\nwhere we focus on here, making measurements at con-\nstanttemperaturewithstepsassmallas2mK.Atagiven\ntemperature and location, the tunneling conductance\ncurves are spatially homogeneous, and do not change as\na function of time. There are small diﬀerences in the\ntransition temperatures and height and position of the\nquasiparticle peaks in diﬀerent locations, possibly due to\ninternal stress. Nevertheless, the diﬀerence in transition\ntemperatures, T c2↑-Tc2↓, the abrupt nature of the transi-\ntion, the fact that the position of the quasiparticle peaks\nremains at ∆ ErRh 4B4, and the gapless V shaped increase\nofNErRh 4B4(E) close to T c2are always found.\nThe possible coexistence between long rangeferromag-\nnetic order and superconductivity has been discussed\nby comparing diﬀerent macroscopic experiments[1, 2, 3].\nAs shown in Refs.[13], the height of the ferromagnetic\nBragg peaks jumps at temperatures close to T c2↑and\nTc2↓, showing the same hysteretic behavior as we ﬁnd\nhere. Moreover, it does not saturate but has a continu-\nous increase below these temperatures[13]. The full dis-4\n0.8 0.9 1.0 1.1 0.0 0.2 0.4 0.6 0.8  Size of  zero N ErRh 4B4 interval (meV) \nTemperature (K) Tc2  ↓ Tc2  ↑ \nTc2 ↑Tc2 ↓0.7 0.8 0.9 1.0 1.1 \n0.0 0.4 0.8 1.2  Temperature (K) ΔErRh 4B4  (meV) \nFIG. 4: (Color online). The width of thezero density ofstate s\ninterval is shown as a function of temperature when cooling\nand heating (respectively, blue squares and red circles). T he\ninset shows ∆ ErRh 4B4(T) close to the magnetic transition.\nappearance of any localsuperconducting signal in the\ntunneling spectroscopydataat T c2↓, and the correspond-\ning appearance at T c2↑, shows that there is no evidence\nfor coexistence between long range ferromagnetic order\nand superconductivity. If there would be superconduct-\ning correlations in extended ferromagnetic regions, these\nshould lead to some signal in the local tunneling conduc-\ntance curves below T c2↑and T c2↓, such as a decrease of\nNErRh 4B4(E) close to the Fermi level[3], which we do not\nobserve here.\nThe presence of the magnetically modulated state dis-\ncovered in Ref.[13] best explains the observed smearing\nin the tunneling density of states. The peaks satel-\nlite to some of the main ferromagnetic Bragg reﬂec-\ntions, which correspond to a modulated magnetic mo-\nment at (0.042a,0.055c), exist in the superconducting\nphase on the same temperature range where we observe\nthe changes in the width of the zero NErRh 4B4(E) inter-\nval (Fig.4). Moreover, the satellite Bragg peaks grow to\nmuch higher values on cooling at T c2↓than on heating at\nTc2↑, and disappear abruptly when entering long range\nferromagneticorder[13]. Clearly,thesmearingoftheden-\nsity of states found here (Fig.4) must directly show the\neﬀect of this kind of magnetic order on the supercon-\nducting density of states. This is strongest when the mo-\nments associated with the magnetically modulated state\nare highest (at T c2↓) and where we observe the peculiar\ngapless regime.\nThe superconducting density of states in the magnet-\nically modulated phase has been qualitatively calculated\npreviously[3, 30]. The predictions coincide with our ob-\nservations. Within this scenario, while the main gap\nparameter of the superconducting phase ∆ ErRh 4B4re-\nmains unchanged, the presence of a ﬁnite magnetization\nin some directions leads to selective pair breaking eﬀects\nwhichproducestronglyanisotropicgapstructuressimilar\nto those observed in d-wave superconductors. In partic-\nular, there are zero gap regions along the lines formedby equally oriented magnetic moments, although no sign\nchanges of the phase of the Cooper pair wavefunction,\nand no concomitant zero energy bound states (charac-\nteristic of d-wave superconductivity, see e.g. Ref.[31]),\nare found.\nIn summary, local tunneling spectroscopy experiments\nhave allowed us to ﬁnd locations in the surface of\nErRh4B4with areas showing clear-cut superconducting\nfeatures in the density of states and a superconducting\ngap parameter close to BCS expectations. The temper-\nature evolution of the superconducting density of states\nfollows the temperature dependence of magnetic signals\nfound in previous neutron scattering experiments. Fer-\nromagnetism seems to totally cancel superconductivity,\nandthe magneticallymodulated phasehasastrongeﬀect\non the superconducting density of states, which leads to\nfully gapless superconductivity.\nWe acknowledge conversations with A.I. Buzdin, F.\nGuinea, H. Suhl and S.K. Sinha. The Laboratorio de\nBajas Temperaturas is associated to the ICMM of the\nCSIC. This work was supported by the Spanish MICINN\n(Consolider Ingenio Molecular Nanoscience CSD2007-\n00010 program and FIS2008-00454), by the Comunidad\nde Madrid through program ”Science and Technology at\nMillikelvin”, by NES and ECOM programs of the ESF,\nand by the US Department of Energy.\n[1]Superconductivity in Ternary Compounds II, Supercon-\nductivity and Magnetism (M B Maple and φFischer,\nBerlin, 1982).\n[2]Proceedings of the International Conference on Ternary\nSuperconductors (G.K. Shenoy and B.D. Dunlap and\nF.Y. Fradin, North-Holland, New York, 1981).\n[3] L. N. Bulaevski, et al., Adv. Phys. 34, 175 (1985).\n[4] J. Flouquet and A. I. Buzdin, Phys. World 15, 9 (2002).\n[5] A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n[6] D. C. Johnston, S. K. Sinha, and A. J. Jacobson, Physica\nC153, 572 (1988).\n[7] J. M. Tranquada, et al., Phys. Rev. Lett. 60, 156 (1988).\n[8] Y. Kamihara, etal., J. Am.Chem. Soc. 130, 3296(2008).\n[9] T. Watanabe, et al., Inorg. Chem. 46, 7719 (2007).\n[10] Y. Kamihara, et al., J. Am. Chem. Soc. 128, 10012\n(2006).\n[11] J. Orenstein and A. Millis, Science 288, 468 (2000).\n[12] W. A. Fertig, et al., Phys. Rev. Lett. 38, 987 (1977).\n[13] S. K. Sinha, et al., Phys. Rev. Lett. 48, 950 (1982).\n[14] C. Umbach, et al., Physica B 108, 803 (1981).\n[15] U. Poppe, Physica B 108, 805 (1981).\n[16] J. M. DePuydt, E. D. Dahlberg, and D. G. Hinks, Phys.\nRev. Lett. 56, 165 (1986).\n[17] R. Prozorov, et al., Phys. Rev. B 77, 100503(R) (2008).\n[18] R. Prozorov, et al., cond-mat p. 0806.2479 (2008).\n[19] P. W. Andersonand H.Suhl, Phys.Rev. 116, 898(1959).\n[20] J. G. Rodrigo, H. Suderow, and S. Vieira, Eur. Phys. J.\nB40, 483 (2004).\n[21] J. G. Rodrigo and S. Vieira, Physica C 404, 306 (2004).\n[22] H. Suderow, et al., Phys. Rev. B 65, 100519(R) (2002).5\n[23] I. Guillamon, et al., Physica C 468, 537 (2008).\n[24] J. G. Rodrigo, et al., J. Phys.: Condens. Matter 16,\nR1151 (2004).\n[25] J. P. Brison, et al., J. Magn. Magn. Mater. 272, 158\n(2004).\n[26] M. Crespo, et al., Phys. Rev. Lett. 96, 027003 (2006).\n[27] D. Gusakova, et al., Pis’ma V ZhETF 83, 385 (2006).\n[28] L. Coﬀey, K. Levin, and G. S. Grest, Phys. Rev. B 27,2740 (1983).\n[29] T. Jarlborg, A. J. Freeman, and T. J. Watson-Yang,\nPhys. Rev. Lett. 39, 1032 (1977).\n[30] L. N. Bulaevski and S. V. Panjukov, J. Low Temp. Phys.\n52, 137 (1983).\n[31] S. Pan, et al., Nature 403, 746 (2000)."    },    {        "title": "1410.4651v1.Coexistence_of_ferromagnetism_and_superconductivity_in_iron_based_pnictides__a_time_resolved_magnetooptical_study.pdf",        "content": "arXiv:1410.4651v1  [cond-mat.supr-con]  17 Oct 2014Coexistence of ferromagnetism and superconductivity in ir on based\npnictides: a time resolved magnetooptical study\nA. Pogrebna,1, 2T. Mertelj,1,∗N.Vujičić,1, 3G. Cao,4Z. A. Xu,4and D. Mihailovic1, 5\n1Complex Matter Dept., Jozef Stefan Institute,\nJamova 39, SI-1000 Ljubljana, Slovenia\n2Jožef Stefan International Postgraduate School,\nJamova 39, SI-1000 Ljubljana, Slovenia\n3Institute of Physics, Bijenička 46, HR-10000 Zagreb, Croat ia\n4Department of Physics, Zhejiang University,\nHangzhou 310027, People’s Republic of China\n5CENN Nanocenter, Jamova 39, SI-1000 Ljubljana, Slovenia\n(Dated: 7th June 2021)\nAbstract\nFerromagnetism and superconductivity are antagonistic ph enomena. Their coexistence implies\neither a modulated ferromagnetic order parameter on a lengt hscale shorter than the superconduct-\ning coherence length or a weak exchange coupling between the itinerant superconducting electrons\nand the localized ordered spins. In some iron based pnictide superconductors the coexistence of fer-\nromagnetism and superconductivity has been clearly demons trated. The nature of the coexistence,\nhowever, remains elusive since no clear understanding of th e spin structure in the superconducting\nstate has been reached and the reports on the coupling streng th are controversial. We show, by a\ndirect optical pump-probe experiment, that the coupling is weak, since the transfer of the excess en-\nergy from the itinerant electrons to ordered localized spin s is much slower than the electron-phonon\nrelaxation, implying the coexistence without the short-le ngthscale ferromagnetic order parameter\nmodulation. Remarkably, the polarization analysis of the c oherently excited spin wave response\npoints towards a simple ferromagnetic ordering of spins wit h two distinct types of ferromagnetic\ndomains.\n∗Correspondence to tomaz.mertelj@ijs.si\n1In the iron-based superconductors family[1, 2] EuFe 2(As,P) 2[3] and Eu(Fe,Co) 2As2[4] oﬀer\nan interesting experimental possibility to study the compe tition between the ferromagnetic\n(FM) and superconducting (SC) order parameters that can lea d to nonuniform magnetic\nand SC states[4–7] since the optimal superconducting criti cal temperature Tc∼28K[8] is\ncomparable to the FM Eu2+-spin ordering temperatures TC∼18K.[3, 9]\nThe strength and nature of the coupling between the carriers in the FeAs planes, respon-\nsible for superconductivity, and localized Eu2+f-orbitals spins, responsible for ferromag-\nnetism, is expected to inﬂuence strongly any possible magne tic as well as SC modulated\nstate.[6] To enable coexistence of the singlet superconduc tivity with ferromagnetism in the\ncase of strong exchange-interaction-dominated coupling t he magnetization modulation pe-\nriod should be short on the lengthscale below the SC coherenc e length[5, 6], which is a\nfew[10, 11] tens of nm in 122 iron based compounds. On the othe r hand, in the case of\nweaker long-range magnetic-dipole dominated coupling a lo nger lengthscale FM domain\nstructure can eﬀectively minimize the internal magnetic ﬁe ld enabling coexistence of the\nsinglet superconductivity and FM state.[12] Alternativel y a spontaneous SC vortex state[6]\nmight form as proposed recently[13] for EuFe 2(As,P) 2.\nIn the literature opposing claims regarding the coupling be tween the carriers in the FeAs\nplanes and localized Eu2+spins exist. A weak coupling between Fe and Eu magnetic order s\nwas initially suggested by Xiao et al.[14], while recently a strong coupling was suggested\nfrom the in-plane magnetoresistance[15] and NMR[16].\nThe strength of the coupling between the carriers in the FeAs planes and localized Eu2+\nf-orbitals spins should be reﬂected also in the energy transf er speed between the two subsys-\ntems upon photoexcitation. We therefore systematically in vestigated the ultrafast transient\nreﬂectivity ( ∆R/R) dynamics and time resolved magneto-optical Kerr eﬀect (TR -MOKE)\nin EuFe 2(As1−xPx)2in both, the undoped spin-density wave (SDW) and doped SC sta te.\nIn addition to the relaxation components, that were observe d earlier in related non-FM\nBa(Fe,Co) 2As2,[17, 18] we found another slow-relaxation component assoc iated with Eu2+-\nmagnetization dynamics. The relatively slow 0.1-1 nanosec ond-timescale response of the\nEu2+spins to the optical excitation of the FeAs itinerant carrie rsindicates a rather weak\ncoupling between the two subsystems suggesting the magnetic-dipole dominated coupling\nbetween SC and FM order parameters.\nMoreover, the antiferromagnetic (AFM) Eu2+-spin order in the undoped SDW EuFe 2As2,\n2where the spins are aligned ferromagnetically in the abplane with the A-type AFM order\nof the adjacent Eu2+planes along the caxis, is rather well understood.[9, 14] Contrary,\nno coherent picture of Eu2+-spin ordering upon P or Co doping exists. In addition to\nthe proposal of a SC induced helimagnetic ordering[4] in Eu( Fe,Co) 2As2a canted AFM\nwas proposed by Zapf et al. [19] in superconducting EuFe 2(As1−xPx)2, while a pure FM\nordering[13] at x= 0.15coexisting with superconductivity was reported by Nandi et al.\n[13]. A spin-glass state over the all P doping range was also s uggested by Zapf et al. [20]\nrecently.\nThe observed time-resolved magnetooptical transients in t he presence of an in-plane mag-\nnetic ﬁeld reveal an additional coherent magnon response in the superconducting sample.\nThe polarization dependence of the coherent magnon oscilla tionspoints towards a FM do-\nmain state consistent with results of Nandi et al. [13].\nRESULTS\nTemperature dependence of photoinduced reﬂectivity. In Fig. 1 c)-f) we show\ntemperature dependence of the transient reﬂectivity ( ∆R/R) measured with the probe pulses\npolarized in the ab-plane in undoped nonsuperconducting EuFe 2As2(Eu-122) and doped\nsuperconducting EuFe 2(As0.81P0.19)2(EuP-122). The transient reﬂectivity is anisotropic\nin theab-plane, consistent with the orthorhombic crystal structur e. We indicate the two\northogonal polarizations P+andP−according to the sign of the subpicosecond transient\nreﬂectivity. In addition to the anisortopic fast component associated with the SDW order\ndiscussed elsewhere[21] we observe in both samples, concur rently with emergence of the\nEu2+-spin ordering,[14, 19] appearance of another much slower r elaxation component [see\nFig. 1 a) and b)] with a risetime of ∼1ns in Eu-122 and ∼100ps in EuP-122 (at\nT= 1.5K) and the decay time beyond the experimental delay range. In the vicinity of the\nEu2+magnetic ordering temperatures a marked increase of the ris etime is observed in both\nsamples. In Eu-122 the slow component is rather anisotropic , while in EuP-122 it appears\nalmost isotropic.\nThe probe-photon-energy dependence of the transients in Eu -122 is shown in Fig. 2. The\ndispersion of the fast component[21] is much broader than th at of the slow one, which shows\na relatively narrow resonance around ∼1.7 eV.\n3Metamagnetic transitions. Upon application of magnetic ﬁeld lying in the ab-plane\nthe Eu2+AFM order in Eu-122 is destroyed above µ0H∼0.8T in favor of an in-plane\nﬁeld-aligned FM state.[22–24] In EuP-122 a similar ﬁeld-in duced spin reorientation from the\nout-of-plane FM into the in-plane ﬁeld-aligned FM state was observed around µ0H∼0.6\nT.[13] These metamagnetic transitions have remarkable inﬂ uence on the transient reﬂectivity\nas shown in Fig. 3. While the fast picosecond response associ ated with the SDW state[21]\nshows virtually no dependence on the magnetic ﬁeld, the slow response shows a marked\nchange in the ﬁeld-induced FM state[13, 22–24].\nIn undoped Eu-122 the P+-polarization slow response is suppressed above the metama g-\nnetic transition [Fig. 3 (a)] and is magnetic-ﬁeld independ ent above 2 T. Concurrently, for\ntheP−polarization, which is parallel to the magnetic ﬁeld, [Fig. 3 (b)] the slow response\nis ﬁrst enhanced at low magnetic ﬁeld above the transition, r esembling a rotation of the\nanisotropy by π/2, and then slightly suppressed upon increasing the ﬁeld to 7 T .\nIn EuP-122 the initially positive rather isotropic slow res ponse [Fig. 3 (c), (d)] switches to\na negative anisotropic one along the P+polarization, parallel to the magnetic ﬁeld. Similar\nto Eu-122 the slow response is slightly suppressed at the hig hest ﬁeld with a faster relaxation.\nCoherent spin waves. In EuP-122 at low magnetic ﬁelds below ∼0.5T additional\ndamped oscillations appear on top of the slow relaxation in ∆R/R[see Fig. 4 (a), (b)].\nThese oscillations appear rather isotropic. The amplitude of the oscillations, shown in Fig.\n5 (d), is strongly peaked around ∼0.25T and vanishes at 0.5 T. The frequency of the\noscillations, as determined by a damped oscillator ﬁt shown in Fig. 5 (a), is Hindependent\nat low ﬁelds and starts to decrease with increasing ﬁeld abov eµ0H∼0.3T. The damping,\non the other hand, is magnetic-ﬁeld independent at τ−1∼10GHz.\nAnother oscillation with a higher frequency ( ∼14.5GHz at 0.3 T) is revealed by the\ntransient magnetooptical Kerr eﬀect (TR-MOKE) shown in Fig . 4 (c)-(f). The oscillatory\npart of the transient rotation and ellipticity is polarizat ion independent and almost even\nwith respect to the reversal of magnetic ﬁeld.\nDISCUSSION\nEu2+ions have [Xe] 4f76s2(8S7/2) electronic conﬁguration. The lowest excited states of a\nfree Eu2+ion are∼3.5eV above the ground state.[26] In oxides, however, this spli tting can\n4be reduced down to ∼1eV.[26, 27] In Eu-122 the position of f-derived states was calculated\nto be∼2eV below the Fermi energy,[28] close to the observed Eu2+-spin ordering related\nslow-component resonance around 1.7 eV [see Fig. 2 (b)]. It i s therefore plausible that the\ncoupling of the Eu2+magnetism to the dielectric constant at the probe photon ene rgy of\n1.55 eV is through the resonant magneto-optical Cotton-Mou ton eﬀect with the location of\nthe Eu2+-4fstates∼1.7eV below the Fermi level.\nOn the other hand, a large magnetostriction is indicated fro m the realignment of the\ncrystal twin domain structure in magnetic ﬁeld,[23] sugges ting a possibility of the indirect\ncontribution to the optical dielectric function through th e magnetoelastic eﬀect. The rather\nnarrow probe-photon-energy resonance of the slow componen t does not support this mech-\nanism.\nWe should also note that the realignment of the twin domain st ructure[23] was not ob-\nserved in our experiment, since the anisotropy of the fast co mponent, which is associated\nwith the structural twin domains,[18] shows no dependence o n magnetic ﬁeld in both sam-\nples (see Fig. 3) up to µ0H= 7T. Moreover, the realignment of the twin domain structure\nobserved in Ref. [23] might be related to the Fe spin ordering as indicated by observation\nof a partial magnetic ﬁeld detwinning also in non-ferromagn etic Ba(Fe 1−xCox)2As2.[29]\nThe strong in-plane anisotropy of the slow component in Eu-1 22 indicates that the re-\nsponse corresponds to the dynamics of the in-plane componen t of the sublattice Eu2+magne-\ntizations. The presence of qualitatively same response in t he in-plane ﬁeld-aligned FM state\nsuggests that the observed slow dynamics is not the dynamics of the AFM order parameter,\nbut rather the dynamics of the individual AFM sublattice mag netizations. The response\ncan therefore be associated with a decrease of the Eu2+magnetization upon photoexcitation\nin both, the zero-ﬁeld AFM and the ﬁeld-induced in-plane FM s tate.\nTo understand the change of the anisotropy between weak and s trong magnetic ﬁelds in\nEuP-122 let us look at the symmetric part of the in-plane diel ectric tensor components ǫii.\nWithin the orthorhombic point symmetry ǫiican be expanded in terms of magnetization to\nthe lowest order as:\nǫii=ǫ0,ii+aiizzM2\nz+aiixxM2\nx+aiiyyM2\ny, (1)\nwithi∈ {x,y}. HereMwould correspond to the Eu2+sublattice magnetization in the case\nof a canted AFM ordering, or the total Eu2+magnetization in the case of FM ordering. In\nEuP-122 in low magnetic ﬁelds Mis predominantly oriented along the c-axis[13, 30] leading\n5to the nearly isotropic response, since axxzz∼ayyzzdue to the small orthorhombic lattice\ndistortion. In the ﬁeld-induced FM state and the zero-ﬁeld A FM state of Eu-122 Mlies in\ntheab-plane leading to an anisotropic response since it is quite u nlikely that aiiii∼ajjii,\nwithi/ne}ationslash=j.\nThe photoinduced Eu2+demagnetization is therefore slow, on a nanosecond timesca le\nin Eu-122 and a ∼100ps timescale in EuP-122. It can not be due to a direct emission\nof incoherent Eu2+magnons by the eV-energy photoexcited Fe- d-bands electron-hole pairs\nsince it has been shown, that in the case of iron-based pnicti des in the SDW state the Fe- d-\nbands quasiparticle relaxation occurs on a picosecond time scale[21, 31, 32] and goes through\nemission of Fe- d-spin magnons[21] followed by relaxation to phonons. It can therefore be as-\nsumed that the Fe- d-bands quasiparticle and lattice degrees of freedom are ful ly thermalized\nbeyond∼10ps when the slow component starts to emerge. This suggests th atthe energy\ntransfer from the excited quasiparticles in the Fe- dbands to the Eu2+magnons is rather in-\neﬃcient . The incoherent Eu2+magnons are therefore excited indirectly via the spin-latt ice\ncoupling only after the initial excitation energy was therm ally distributed between the Fe-\nd-bands quasiparticles and phonons. The Eu2+spins therefore appear only weakly coupled\nto the Fe- d-bands quasiparticles with the coupling increasing with th e P doping. The rather\nlarge in-plane magnetoresistance observed by Xiao et al. [15] in Eu-122 can therefore be\nattributed to slow magnetostriction eﬀects modifying the l attice twin domain structure.\nThe light penetration depth at the probe-photon energy of /planckover2pi1ωpr= 1.55eV is∼27\nnm,[33] while the beam diameters are in a 100 µm range. Irrespective of the excitation\nmechanism, which can be either nonthermal impulsive[34] in verse Cotton-Mouton eﬀect or\nthermal displacive non-Raman[35] like, it can be assumed th at the relevant wavevectors are\nq/lessorsimilar1/30nm−1and dominantly a uniform coherent magnetization precessio n is excited and\ndetected. (In the case of helical magnetic order with the pro pagation vector q0, spin waves\natq=±mq0,m∈Z, also need to be considered.[36])\nThe low frequency mode observed in the transient reﬂectivit y response softens with in-\ncreasing temperature and vanishes in the ﬁeld induced in-pl ane FM state so it can deﬁnitely\nbe assigned to a magnetic mode. The high frequency mode has al so a magnetic origin since\nit appears in the TR-MOKE conﬁguration only.\nAnalyzing contributions of the magnetization displacemen ts,δMi, to the symmetric part\nof the optical response it follows from (1),\n6δǫii= 2aiizzMzδMz+2aiixxMxδMx+2aiiyyMyδMy. (2)\nThe low-frequency mode is very strong in ∆R/Rand rather isotropic in the ab-plane in-\ndicating that is either associated with the out of-plane ter ms (i)2aiizzMzδMzor (ii) both,\nMxandMy, are ﬁnite such as in the case of a helimagnetic ordering. In t he latter case\nthe local magnetization needs to be considered since the ave rage of the terms < MiδMi>,\ni∈ {x,y}, over Eu2+planes is ﬁnite despite < Mi>= 0. Concurrently, it is weak in the\nTR-MOKE conﬁguration, which is sensitive to δMz. SinceδMz/ne}ationslash= 0for both (i) and (ii) (see\nSupplemental information for case (ii)) this indicates tha t the measured volume is composed\nfrom the “up” and “down” magnetic domains magnetized along t hec-axis. The sign of δMz\nvaries in diﬀerent magnetic domains leading to a vanishing T R-MOKE response averaged\nover many magnetic domains, while the sign of MzδMzdoes not depend on the domain\norientation and averages to a ﬁnite value.\nFor the high-frequency mode observed in the TR-MOKE conﬁgur ation, on the other\nhand, the averaged δMzis ﬁnite while the averaged MzδMzis rather small in comparison to\nthe low frequency mode. This indicates that in addition to th ec-axis magnetized domains\nin-plane magnetized regions exist with Mz∼0. The in-plane magnetization leads to an out\nof plane magnetization displacement with δMz/ne}ationslash= 0andMzδMz∼0, consistent with the\nobserved magnetic ﬁeld dependence of the mode frequency. Th e invariance of the oscillatory\nTR-MOKE response [see. Fig. 4 (c)-(f)] with respect to the in version of the magnetic ﬁeld\nis also consistent with the in-plane magnetization orienta tion.\nA ﬁt of the frequency magnetic-ﬁeld dependence [see Fig. 5 (b )] using the standard\nuniaxial ferromagnet formula for a parallel magnetic ﬁeld[ 25] ignoring demagnetization fac-\ntors,ω=γab(Hab+H), yieldsµ0Hab= 0.3T andγab/µ0= 182 GHz/T. The obtained\ngyromagnetic ratio gab= 2.06is consistent with8S7/2state of Eu2+ions. The absence of\ndemagnetization factors suggests that the response does no t originate from the domain walls\nbetween the c-axis oriented domains but rather from planar shaped domain s. Due to surface\nsensitivity ( ∼30nm) of the optical probe these are very likely surface domain s, however,\nthe bulk nature of these domains can not be entirely excluded .\nThe observed behaviour is compatible with the simple ferrom agnetic order (within the\ndomains) proposed by Nandi et al.[13]. In the absence of the in-plane magnetic ﬁeld the\nstatic magnetization in the c-axis domains is along the c-axis and δMz= 0, consistent with\n7the vanishing amplitude of the low frequency mode near the ze ro ﬁeld. Upon application of\nthe in-plane magnetic ﬁeld the magnetization is tilted away from the c-axis (see inset to Fig.\n5) leading to a ﬁnite δMzand the observed decrease of the mode frequency.[25] The dec rease\nof the transient-reﬂectivity amplitude, when approaching to the metamagnetic transition,\ncan be associated with the vanishing Mz.\nA ﬁt of the frequency magnetic-ﬁeld dependence using the sta ndard uniaxial ferromag-\nnet formula for the perpendicular magnetic ﬁeld[25] ignori ng demagnetization factors, ω=\nγc/radicalbig\nH2c−H2, results in µ0Hc= 0.52T andγc/µ0= 119 GHz/T. The small value of γc\nleading to a small gyromagnetic ratio ( gc= 1.35) can be attributed to the ignored unknown\ndemagnetization factors of the c-axis magnetized domains. Moreover, it suggests that the\nc-axis magnetized domains have a ﬂat shape with the normal per pendicular to the caxis.\nOn the other hand, the presence of two distinct modes and the m agnetic ﬁeld depen-\ndence of the mode frequencies [see Fig. 5 (b)] resembles the s tandard uniaxial AFM cases\nwith the magnetic ﬁeld perpendicular to the easy/hard axis[ 25, 37] indicating a possible\ncanted AFM[19] (CAFM) order. The polarization dependence o f the modes is, however, not\ncompatible with the CAFM picture since both, the quasi-AFM mode[38] and the quasi-FM\nmode, contribute to δMz(see Supplemental) and should, contrary to the observation s, con-\ntribute concurrently to the transient reﬂectivity and the T R-MOKE with identical relative\namplitudes.\nIn the case of the conical helimagnetic ordering[4, 30, 39] t he in-plane isotropy naturally\nappears for certain modes (see Supplemental). However, sin ce, as in the case of the CAFM\nstate, contributions of more than one magnetic mode to δMzare expected, our data do not\nsupport the conical helimagnetic ordering.\nIn conclusion, our data point towards the simple FM Eu2+-spin order in superconducting\nEuFe2(As,P) 2proposed by Nandi et al. [13]. The observed weak coupling between the FeAs-\nplane quasiparticles and Eu2+spins indicates a weak magnetic-dipole dominated coupling\nbetween the SC and FM order parameters. This indicates that t he coexistence of the singlet\nsuperconductivity with ferromagnetism in EuFe 2(As1−xPx)2is possible without necessity of\nthe magnetic structure modulation on the lengthscale short er than the SC coherence-length.\nThe presence of the FM domain structure on longer lengthscal es, which is inferred from\nthe coherent-spin-wave response, might additionally cont ribute to stability of the coexisting\nstate.\n8METHODS\nSample preparation. Single crystals of EuFe 2(As1−xPx)2were grown by a ﬂux method,\nsimilar to a previous reports[21, 40] The out-of-plane magn etic susceptibilities shown in\nFig. 1 are consistent with previous results. [20, 22] From th e susceptibility we infer Eu2+\nspin ordering temperatures TN= 19 K andTCur= 17.6K in EuFe 2As2(Eu-122) and\nEuFe2(As0.81P0.19)2(EuP-122), respectively. EuP-122 also shows the onset of su perconduc-\ntivity at Tc=22.7K.\nOptical measurements. Measurements of the photoinduced transient reﬂectivity,\n∆R/R, fromabfacets of freshly cleaved samples at nearly normal incidenc e were performed\nusing a standard pump-probe technique, with 50 fs optical pu lses from a 250-kHz Ti:Al 2O3\nregenerative ampliﬁer seeded with an Ti:Al 2O3oscillator.[18] We used the pump photons\nwith both, the laser fundamental ( /planckover2pi1ωP= 1.55eV) and the doubled ( /planckover2pi1ωP= 3.1eV) photon\nenergy, and the probe photons with the laser fundamental ( /planckover2pi1ωpr= 1.55eV) photon energy.\nMagnetooptical measurements. Transient Kerr rotation, ∆φK, was also measured\nonabfacets of freshly cleaved samples at nearly normal incidenc e by means of a balanced\ndetector scheme using a Wollaston prism and a standard homod yne modulation technique\nin a 7-T split-coil optical superconducting magnet. To meas ure the transient Kerr ellipticity,\n∆ηK, aλ/4-waveplate was inserted in front of the Wollaston prism. 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Doping dependence of\nfemtosecond quasiparticle relaxation dynamics in Ba(Fe, Co )2As2single crystals: Evidence for\nnormal-state nematic ﬂuctuations. Physical Review B 86, 024519 (2012).\n[19] Zapf, S. et al. Varying Eu2+magnetic order by chemical pressure in EuFe 2(As1−xPx)2.Phys.\n10Rev. B 84, 140503 (2011).\n[20] Zapf, S. et al. EuFe 2(As1−xPx)2: Reentrant spin glass and superconductivity. Physical Review\nLetters 110, 237002 (2013).\n[21] Pogrebna, A. et al. Spectrally resolved femtosecond reﬂectivity relaxation dynamics in undoped\nspin-density wave 122-structure iron-based pnictides. Phys. Rev. B 89, 165131 (2014).\n[22] Jiang, S. et al. Metamagnetic transition in EuFe 2As2single crystals. New Journal of Physics\n11, 025007 (2009).\n[23] Xiao, Y. et al. Field-induced spin reorientation and giant spin-lattice coupling in EuFe 2As2.\nPhysical Review B 81, 220406 (2010).\n[24] Guguchia, Z. et al. 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Impulsive excitation of coherent magnons and phonons by su bpi-\ncosecond laser pulses in the weak ferromagnet FeBO 3.Phys. Rev. B 78, 104301 (2008).\n[36] Cooper, B. R. & Elliott, R. J. Spin-wave theory of magneti c resonance in spiral spin structures:\nEﬀect of an applied ﬁeld. Phys. Rev. 131, 1043–1056 (1963).\n[37] Subkhangulov, R. et al. All-optical manipulation and probing of the d−fexchange interaction\nin EuTe. Scientiﬁc reports 4, 4368 (2014).\n[38] Iida, R. et al. Spectral dependence of photoinduced spin precession in Dy FeO3.Phys. Rev. B\n84, 064402 (2011).\n[39] Nowik, I., Felner, I., Ren, Z., Cao, G. H., & Xu, Z. A.57Fe and151Eu Mössbauer spectroscopy\nand magnetization studies of Eu(Fe 0.89Co0.11)2As2and Eu(Fe 0.9Ni0.1)2As2.New Journal of\nPhysics 13, 023033 (2011).\n[40] Jiao, W.-H. et al. Anisotropic superconductivity in Eu(Fe 0.75Ru0.25)2As2ferromagnetic super-\nconductor. EPL (Europhysics Letters) 95, 67007 (2011).\nACKNOWLEDGMENTS\nWork at Jozef Stefan Institute was supported by ARRS (Grant N o. P1-0040). Work\ndone in Zhejiang University was supported by NSFC (Grant No. 11190023)\nAUTHOR CONTRIBUTIONS\nT. M. conceived the idea and experiment, A. P. performed opti cal measurements, N. V.\nbuilt broadband optical setup, G. C. and Z.-A. X. grew and cha racterized single crystals, T.\nM. and A. P. analyzed the data, T. M., A. P. and D. M. wrote the pa per.\nADDITIONAL INFORMATION\nCompeting ﬁnancial interests: The authors declare no competing ﬁnancial interests.\n12Figure 1. Temperature dependence of the ab-plane transient reﬂectivity. The amplitude of the\nphotoinduced reﬂectivity transients at long delays as a fun ction of temperature in EuFe 2As2, (a),\nand EuFe 2(As0.81P0.19)2, (b), compared to the magnetic moment along the c-axis .P+andP−\ncorrespond to two orthogonal in-plane probe-photon polari zations while ZFC and FC correspond\nto cooling in the presence and absence of magnetic ﬁeld, resp ectively. Photoinduced reﬂectivity\ntransients at low- Tin EuFe 2As2(c), (e) and EuFe 2(As0.81P0.19)2(d), (f) for the two probe-photon\npolarizations. Inset to ( e) represents a schematic of the probe beam conﬁguration.\n13Figure 2. Photoinduced reﬂectivity transients at low- Tin EuFe 2As2as a function of probe photon\nenergy for the P+polarization. The spectral dependencies of the fast and slo w response amplitude\nare shown as red and dark-grey lines, respectively.\n14Figure 3. In-plane magnetic ﬁeld dependence of the transien t reﬂectivity. ( a), (b) The reﬂectivity\ntransients in EuFe 2As2with the magnetic ﬁeld ﬁeld paralel to the P−polarization. ( c), (d) The\nreﬂectivity transients in EuFe 2(As0.81P0.19)2with the magnetic ﬁeld ﬁeld paralel to the P+polar-\nization. All transient were measured at T= 2K,F ∼3µJ/cm2and 1.55-eV pump-photon energy.\nInsets show shematically magnetization reorientation in m agnetic ﬁeld.\n15Figure 4. ( a), (b) The reﬂectivity transients in EuFe 2(As0.81P0.19)2in low magnetic ﬁelds at T= 2\nK,F= 3µJ/cm2and 1.55-eV pump photon energy. Transient Kerr ellipticity , (c), (d), and\nrotation, ( e), (f), upon reversal of the magnetic ﬁeld at T= 1.5K andF= 10µJ/cm2. Odd and\neven part of the responses correspond to the diﬀerence an the sum of the responses measured at\ndiﬀerent signs of the magnetic ﬁeld, respectively.\n16Figure 5. ( a) The oscillatory part of the isotropic ∆R/Rcomponent in EuFe 2(As0.81P0.19)2at\nlow magnetic ﬁelds, F= 3µJ/cm2and 1.5-eV pump photon energy. Thin lines represent the\ndamped oscillator ﬁts discussed in text. The frequency (b), decay time (c) and amplitude ( e)\nof the oscillations as functions of the magnetic ﬁeld. The po ints (open symbols) at B= 0.15T\nwere obtained from the P−polarization ﬁt due to the lack of data at the P+polarization. The\nred squares were obtained from TR-MOKE ﬁts. The lines in ( b) are uniaxial ferromagnet[25] ﬁts\ndiscussed in text. The inset to ( a) shematically shows magnetization precession in small mag netic\nﬁelds with corresponding projections onto the z-axis.\n17"    },    {        "title": "2303.07993v1.The_unusual_distribution_of_spin_triplet_supercurrents_in_disk_shaped_Josephson_junctions.pdf",        "content": "The unusual distribution of spin-triplet\nsupercurrents in disk-shaped Josephson junctions\nRemko Fermin, Junxiang Yao, Kaveh Lahabi, and Jan Aarts\nHuygens-Kamerlingh Onnes Laboratory\nLeiden University\nP.O. Box 9504, 2300RA Leiden, The Netherlands\nE-mail: aarts@physics.leidenuniv.nl\nAbstract. The phenomenon of s-wave spin triplet Cooper pairs induced in\nferromagnetic metals has been researched now for more than a decade, and its\nmain aspects are well understood. Crucial in converting s-wave singlet pairs in the\nsuperconductor to s-wave triplets in the ferromagnet is the engineering of well-de\fned\nmagnetic inhomogeneity (the 'generator') at the interface with the superconductor.\nVertical layer stacks are typically used as such, where two separate thin ferromagnetic\nlayers with homogeneous but non-collinear magnetizations, provide the inhomogeneity.\nAlternatively, magnetic textures, like ferromagnetic domain walls and vortices, are\npossible triplet generators, although they are far less studied. In this paper we review\nour experiments on lateral disk-shaped Josephson junctions where a ferromagnetic\nbottom layer provides a weak link with a vortex magnetization imposed by the shape\nof the disk. We present three di\u000berent junction con\fgurations, exhibiting their own\ngenerator mechanism. In the \frst, we utilize the non-collinearity with a second\nferromagnetic layer to produce the triplet correlations. The second con\fguration\nconsists of only the bottom ferromagnet and the superconducting contacts; it relies\non the vortex magnetization itself to generate the spin-polarized supercurrents. In the\nthird case we exploit an intrinsic generator by combining a conventional superconductor\n(NbTi) and a half-metallic ferromagnetic oxide (La 0:7Sr0:3MnO 3). We \fnd strong\nsupercurrents in all cases. A particularly interesting \fnding is that the supercurrents\nare strongly con\fned at the rims of the device, independent of the generating\nmechanism, but directly related to their triplet nature. What causes these rim currents\nremains an open question.\n1. Introduction\nSupercurrents of s-wave nature can be induced in ferromagnets by exploiting their anti-\nsymmetrical pairing in the time domain. Speci\fcally, the Pauli exclusion principle is\nnot violated when the wave function describing the Cooper pair obtains a minus sign\nunder exchange of time variables. Therefore, such pairing (usually called odd-frequency)\ncan be of equal-spin and s-wave nature simultaneously. These supercurrents, carried by\nspinfull triplet Cooper pairs, are therefore spin-polarized. A triplet pair in a ferromagnet\nis typically broken by the temperature (similar to a singlet in a normal metal), or byarXiv:2303.07993v1  [cond-mat.supr-con]  14 Mar 2023The unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 2\nthe spin lifetime, giving rise to a signi\fcantly longer coherence lengths than a spinless\nsinglet pair in the same ferromagnet: the singlet Cooper pair is broken by the exchange\nenergy, leading to a coherence length of the order of only nanometers. Due to their long\ncoherence lengths in ferromagnets, triplet pairs form the building block for supercon-\nducting spintronics, a \feld that strives towards dissipationless information processing\nusing spin [1{3]. The possibility of robustly generating a long-range proximity e\u000bect\nwas discussed in a seminal paper by Bergeret et al. [4], who \frst suggested a mechanism\nfor formation of the triplet pairs in S/F-hybrids, using a controlled local inhomogeneity\nof the magnetization of the F layer. Speci\fcally, they considered a changing direction\nof the magnetization, as can be found in domain walls [5, 6]. Generally, magnetic in-\nhomogeneity is key to generating triplets, as elucidated for instance in Refs. [7{9]: a\nspin singlet Cooper pair has to undergo spin-selective scattering to produce the spinless\ncomponent of a triplet, and its quantization axis has to be rotated to become a spin-\nfull triplet. These processes are referred to as spin mixing and spin rotation, respectively.\nExperimentally, rather than using domain walls, magnetic non-collinearity can be\nengineered in a controlled way by using layer stacks with di\u000berent magnetic materi-\nals and magnetization directions. Such stacks of type F 1/F2/F1were used early on to\nstudy the long range triplet (LRT) proximity e\u000bect. This was done by using a Ho/Co/Ho\nstack, which bene\fted from the helical nature of the ferromagnetic holmium [10], but\nalso with stacks of homogeneous ferromagnets such as weakly ferromagnetic PdNi or\nstrongly ferromagnetic Ni for the F 1layer [11, 12]. Most of the subsequent work relied\non such stacks as triplet generator, although LRT supercurrents were also reported in\nsingular ferromagnets such as Co [13] or the Heusler alloy Cu 2MnAl [14].\nSpecial mentioning deserves the case of the ferromagnetic metallic oxides CrO 2\nand La 0:7Sr0:3MnO 3(LSMO), that are both half metals, meaning they are fully spin-\npolarized. In that case, the range of the proximity e\u000bect can be expected to be par-\nticularly long. An early report of a CrO 2\flm with NbTi contacts grown on a TiO 2\nsubstrate, but without obvious triplet generator, reported supercurrents \rowing over\na distance of almost 0.5 µm [15]. Similar results were found for CrO 2\flms grown on\nAl2O3substrates [16]. Later work showed that a generator stack could be fruitfully used\nto yield large spin valve e\u000bects [17,18] and large supercurrent densities. [19,20] Triplet\ngeneration in CrO 2without such a stack is currently believed to be due to intrinsic\nstrain-induced magnetic inhomogeneity (for \flms on TiO 2) or grain boundary disor-\nder (for \flms on Al 2O3). [21] For LSMO, the picture is intriguing: supercurrents were\nreported in hybrid structures with the high-T csuperconductor YBa 2Cu3O7, both for\nvertically stacked structures [22], and recently for a lateral junction [23]. Since there are\nno established sources of magnetic inhomogeneities in these structures, the mechanism\nfor triplet generation is not yet clear.\nMore recently, alternative methods of generating LRT currents were discussed.The unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 3\nFigure 1. (a) Schematic of the disk-shaped Josephson junctions. The\nsuperconducting electrodes are separated by a trench, forming a ferromagnetic weak\nlink. The pattern on the ferromagnetic layer corresponds to micromagnetic simulations\nof a micron-size disk. The arrows correspond to the in-plane magnetization, while the\nout-of-plane component is represented by color, which only appears at the vortex\ncore (blue region; less than 5 nm in diameter). (b)False colored scanning electron\nmicrograph of a structured bilayer. The 20 nm gap indicates the weak link at the\nbottom of the trench. The scale bar is equivalent to 400 nm. The inset shows a\nzoom of a notch formed by the geometry dependent milling rate. The image is from a\ndi\u000berent disk, fabricated in the same way. Figure adapted after one in Ref. [24].\nSpeci\fcally, spin textures, such as domain walls [5, 25{27] or magnetic vortices [28, 29]\nwere predicted as generators. Besides, spin mixing can also be achieved by spin-\norbit coupling (SOC), leading to various theoretical considerations [30{37]. Although\nthis spawned various experiments on Josephson junctions with heavy metal interlay-\ners [38{40], experiments on junctions with controlled spin textures remain scarce.\nIn this paper, we review the research we have carried out on mesoscopic planar\nJosephson junctions, featuring a well-de\fned spin texture induced by the disk shape of\nthe device. Here the disk geometry brings two advantages with respect to classic stacked\njunctions: stray \felds are (almost) absent and the disk geometry allows us to directly\nstudy the role of a vortex magnetization on the generation of triplet supercurrents. The\nconcept is shown in Figure 1a. The device consists of a disk-shaped bilayer or layer\nstack. Here the bottom ferromagnetic layer exhibits a ferromagnetic vortex magnetiza-\ntion pattern, whereas the top layer is split into two superconducting electrodes. This is\ndone by locally removing the superconductor in the middle of the disk. Since we exclu-\nsively remove the superconductor, the current path is forced through the ferromagnet\nbelow, which creates an S/F junction.\nThe paper is organized as follows. First we present our basic study object, the\nplanar disk-shaped planar Josephson junction; we describe the device fabrication, and\nwe show the results on singlet junctions made from a superconductor and a normal (N)\nmetal. By analyzing the dependence of the critical current on an out-of-plane magnetic\n\feldIc(B?), we demonstrate that in such S/N junctions the current distribution is fullyThe unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 4\nhomogeneous. In contrast, when the bottom layer is replaced by a ferromagnet, we\n\fnd the triplets supercurrents to \row in highly localized channels at the rims of the\ndevice. We review the similarities and di\u000berences between three types of ferromagnetic\njunctions. The \frst consists of a stack Nb/Ni/Cu/Co, where the design philosophy\nwas that of magnetic non-collinearity of stacked junctions. The second type is based\non a Nb/Co bilayer, and makes full and only use of the spin texture to generate LRT\ncurrents. In the third we use LSMO as the ferromagnet, which is a special case due to\nits fully spin-polarized charge carriers.\n2. Planar disk junctions\n2.1. Fabication of disk-shaped junctions\nCentral to the fabrication of the junctions is the focused Ga+-ion beam (FIB) milling\nprocedure. After lithographically de\fning a four-probe geometry and sputter depositing\nthe bilayer or layer stack, we use FIB milling to de\fne the shape of the junction. By\napplying an ultra-low beam current of 1.5 pA, the weak link is formed by a line cut in the\nsuperconducting layer along the diameter of the device. Generally, forming this trench\nseparates the superconducting electrodes by a roughly 20 nm weak link, allowing for\nJosephson coupling. In Figure 1b we show a false colored scanning electron micrograph\nof a typical disk-junction. The cut becomes slightly deeper at the edge of the structure\ndue to a geometrical dependence of the milling rate, leading to a small notch of order\n50 nm on the sides of the disk, making the diameter of the weak link slightly smaller\nthan the disk diameter. The notches, on the side of the disk, can be seen in Figure 1b.\n2.2. S/N disk junctions: MoGe/Ag\nTo con\frm that we can reliably fabricate disk-shaped Josephson junctions and rule out\nany secondary e\u000bect induced by the shape of the superconducting electrodes, we fab-\nricated junctions with a normal metal weak link from a MoGe (55 nm)/Ag (20 nm)\nbilayer. We use Ic(B?) measurements to con\frm the uniformity of the critical current\nin these S/N devices. We determine Ic(B?) of the device, with \u00160Hz=B?an out-of-\nplane magnetic \feld (perpendicular to the current \row), by means of current ( I) versus\nvoltage (V) measurements. A color plot of the resulting Ic(B?) pattern is shown in\nFigure 2a. For the normal metal weak link junctions, we observe a typical Fraunhofer\ninterference pattern corresponding to a single Josephson junction with a uniform critical\ncurrent distribution.\nTo analyze the Ic(B?) patterns in more detail, we perform a Fourier analysis to\nextract the critical current density distribution using a technique originally proposed by\nDynes and Fulton [42]. In order to do so, we \frst extract Icfrom theIc(B?) pattern\nby a voltage threshold criterion. Next we use a complex inverse Fourier transform to\nextract the critical current density distribution from Ic(B?). The Fourier analysis isThe unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 5\nFigure 2. (a) Ic(B?) measurement of a disk-shaped S/N Josephson junction as a\ndV=dIcolor map. As expected for an S \u0000N\u0000S junction, we observe a typical Fraunhofer\ninterference pattern. in (b)we plot the critical current distribution, obtained from\nFourier analysis of the data shown in (a). We indicate the sides of the electrodes by\nsolid vertical reference lines; the dashed lines indicate the sides of the actual weak link.\nFigure adapted after Ref. [41].\nbased on knowledge of the shielding currents in the electrodes. For junctions between\nmacroscopic superconducting electrodes, these are given by the Meissner e\u000bect. As a\nconsequence, the e\u000bective length of such junctions is given by d+ 2\u0015L, wheredis the\nlength of the junction (i.e., distance between the superconducting electrodes) and \u0015L\nthe London penetration depth. However, since our junctions are in the thin \flm limit\n(i.e., the superconducting layer thickness is smaller than \u0015L) and our junctions are lat-\nerally constricted in a disk geometry, the shielding currents are given by a non-local\nelectrodynamic relation, which alters the e\u000bective junction length in a signi\fcant way.\nWe recently developed a method to evaluate the shielding currents using simulations\nof the gauge-invariant phase gradient in our junctions. A detailed discussion of these\nsimulations and the technical details of the Fourier analysis can be found elsewhere [41].\nFigure 2b shows the critical current density distribution corresponding to the Ic(B?)\npattern of Figure 2a. Here we indicate the width of the electrodes by solid vertical\nreference lines. By dashed lines we indicate the width of the weak link itself, which is\nslightly smaller due to the aforementioned notches. As expected for a single junction, we\nobserve a uniform distribution of critical current throughout the weak link. This means\nwe can reliably fabricate disk-shaped Josephson junctions using our FIB fabrication\ntechnique.The unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 6\n3. Superconductor-ferromagnet junctions\n3.1. Magnetic non-collinearity: Nb/Ni/Cu/Co disk junctions\nThe \frst type of ferromagnetic junction consists of a layer stack of Co (60 nm)/Cu (5\nnm)/Ni (1.5 nm)/Nb (45 nm), where we utilize the non-collinearity of the two magnetic\nlayers to generate triplet Cooper pairs. We reported results on this device in Ref. [43]\nbut re-analyzed Ic(B?) according to our more recent insights [41]. Here we form the\n20 nm Co barrier by a trench that cuts through the layers above (see Fig. 3a). Micro-\nmagnetic simulations con\frmed that the bottom (Co) layer has a vortex magnetization,\nwhile shape anisotropy would force the Ni magnetization along the edges of the trench,\nas schematically shown in Fig. 3b. Note that the Cu layer decouples the two magnetic\nlayers. The calculated amount of non-collinearity in terms of the angular di\u000berence\nbetween the local magnetization directions in the Co and the Ni is shown in Fig. 3c.\nThe values are high along the trench, except for a small region at the center, where the\nFigure 3. (a) Schematic of the Nb/Ni/Cu/Co device layout, showing the di\u000berent\nlayers and their thickness. The Nb layer contains singlet Cooper pairs, the current\nin the Co is carried by triplets. (b) The magnetic texture in the Co and Ni layers,\ncalculated by 3D simulations using the OOMMF package. Note that the magnetic\nmoments in Ni tend to align with the gap and therefore perpendicular to the magnetic\nmoments in the Co. (c) The magnetic non-collinearity pro\fle in terms of the calculated\nangle\u0012between the magnetization vectors in the Co and the Ni. The red regions are\nareas of large non-collinearity. Figure adapted after one in Ref. [43].The unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 7\nFigure 4. (a) The dependence of the resistance Rof the Nb/Ni/Cu/Co device on\ntemperature Tfor the virgin (pink) state and the conditioned (blue) state, measured\nusing a current of 10 µA. The inset show two I-Vtraces (current-voltage), taken\nat 2.08 K for the virgin (pink) and the conditions (blue) case. (b) Critical current\nvariation and simulations of the magnetic non-collinearity as function of an in-plane\n\feldBxdirected perpendicular to the trench. Note the motion of the vortex core\nis directed along the trench towards the side of the disk. The vortex state in Co\ndisappears around 35 mT and the Ni layer becomes magnetized antiparallel to the Co.\nAbove 45 mT, the \feld starts to align the Ni magnetization against the stray \felds\nfrom the Co layer, leading again to non-collinearity. The alignment is complete above\n60 mT, and superconductivity is suppressed. Figure adapted after one in Ref. [43].\nout-of-plane \feld from the vortex core (about 20 nm in diameter) locally couples to the\nNi magnetization.\nTypical to the Nb/Ni/Cu/Co devices is that magnetic conditioning of the nickel\nlayer is required to achieve the non-collinearity, as is also evidenced by our transport\nexperiments. Fig. 4a compares the resistance Rversus temperature Tof the device and\nIV measurements taken before and after conditioning the device with an out-of-plane\n\feld of 2.5 T. In the virgin state, hardly any supercurrent is observed below the initial\nsuperconducting transition (5.5 K), but after conditioning, Rgoes fully superconducting\nbelow 3 K and the I-Vcharacteristics (inset Fig. 4a) show a well-de\fned zero-voltage\nstate with an enhanced Ic. Next we examined the precise in\ruence of the spin texture in\nthe Co layer by applying an in-plane \feld Bxalong the x-direction of the device, which\nis perpendicular to the trench, thereby moving the vortex core along the trench. The\nresults, complemented with non-collinearity maps from the micromagnetic simulations,\nare given in Fig. 4b. The \frst thing to note, is a deep dip in Icaround 8 mT. This is a\nsurprisingly robust feature, which cannot be explained by the magnetic non-collinearity,\nalso occurs in Co disks without the Ni layer. As we discuss in the next section, this dip\nappears to signal a 0 to \u0019phase shift in the junction, which results from the asymmetric\nspin texture of a displaced vortex. Next we observe Icto go to zero around 34 mT. At\nthis \feld the vortex core leaves the disk, and stray \felds that emerge from the Co-layer\nalign the magnetization in the Ni layer antiparallel. A further increase of the \feld startsThe unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 8\nFigure 5. (a) The critical current Icof the Nb/Ni/Cu/Co device as function of\nan out-of-plane magnetic \feld \u00160Hztaken at 2.8 K. (b) The current density pro\fle\nconstructed from the Fourier analysis. The sides of the electrodes are shown by the\nsolid lines. The critical supercurrent density is predominantly distributed on the sides\nof the junction.\nto align the Ni and Co magnetizations, but in the process non-collinearity occurs again,\nand superconductivity reenters. Only around 70 mT, when both magnetizations are\naligned, superconductivity disappears.\nNext we consider the critical current distribution in the junction by measuring\nIc(B?) (at zero in-plane magnetic \feld). The result, given in Fig. 5a is surprising, and\nqualitatively di\u000berent from the S/N case. The \frst lobe in the pattern is wider than\nthe next ones, but less than expected for a Fraunhofer pattern. It rather tends to a\ntwo-channel interference pattern, where all lobes have equal width. The Fourier anal-\nysis of the pattern con\frms the presence of two parallel channels. The supercurrent\ndistribution pro\fle presented in Fig. 5b, shows that the current is inhomogeneous, with\nclear peaks at the rims of the disk. Apparently, the current \rows primarily around the\nrim, although appreciable transport still takes place via the center of the disk. Note\nthat we re-analyzed the data with the nonlocal electrodynamics approach, removing the\nambiguity about the actual diameter of the sample [43].\nReiterating the most salient points, this device shows long range triplet\nsupercurrents through the Co weak link. The triplet supercurrents are generated by\nthe amount of magnetization non-collinearity of the two F-layers and the experimental\nresponse of the device can be almost completely determined by this magnetic non-\ncollinearity. However, the critical current density at the rims of the device is\nsubstantially larger than in the center. Besides, there is a robust dip in the critical\ncurrent as function of in-plane magnetic \feld, which cannot be explained by the magnetic\nnon-collinearity of the Co and Ni. These two observations will come back in the nextThe unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 9\ntwo paragraphs.\n3.2. LRT generation using a ferromagnetic vortex: Nb/Co disk junctions\nAs discussed in the previous section, we can capture much of the characteristics of the\nNb/Ni/Cu/Co junctions by the non-collinearity of the Co and Ni layers. However, there\nremain a couple of open questions. One is: why a portion of the supercurrent \rows along\nthe sides of the junction; another is why is there a sudden decrease of Icat an in-plane\nmagnetic \feld of around 10 mT. Besides these questions raised by our experiments, there\nexist multiple theoretical proposals that discuss the generation of long-range triplet cor-\nrelations using the vortex magnetization of the weak link itself [28,29]. This motivated\nthe study of Nb/Co devices that lack the Ni and Cu layer [24]. Any superconducting\ncorrelations in these devices must be solely generated by the spin texture of the ferro-\nmagnetic layer.\nIndeed, we \fnd that the Nb/Co devices a show long-range proximity e\u000bect, with\nsimilar critical current values as those measured in the Nb/Ni/Cu/Co junctions. We\nhave veri\fed that the naturally present vortex spin texture is responsible for the gen-\neration of long-range triplet correlations by showing that the critical current of the\nNb/Co junctions vanishes when the magnetization is uniform (either by magnetizing\nthe Co-layer or by various control experiments). Besides, the Nb/Co junctions show\nno di\u000berence between the virgin or \feld conditioned states, as is the case with the\nNb/Ni/Cu/Co devices. This di\u000berence is caused by the fact that, unlike the 1.5 nm Ni\nlayer in the electrodes, the Co disk has a highly stable vortex ground state, which does\nnot require magnetic conditioning.\nWe investigated the critical current distributions in the Nb/Co devices by Ic(B?)\nmeasurements; the results are plotted in Figure 6, along with the critical current distri-\nbutions evaluated using inverse Fourier transform. Speci\fcally, we show the interference\npatterns for two junctions with di\u000berent diameters (1.62 and 1.05 µm) in Figure 6a and\nb. Note that the period of the oscillations scales inversely with the area of the junction,\nwhich is determined by the radius of the disk. This implies that we consistently \fnd the\nsupercurrent to be highly localized in 70 nm wide channels at the rims of the sample,\nregardless of the sample area. We therefore call these currents rim currents . As a con-\ntrol experiment, we also prepared a disk junction with a relative shallow trench. This\nprovides a non-magnetic weak link for singlet correlations. In that case we observed\na typical Fraunhofer-like interference pattern with a two times wider central lobe (not\nshown here). In reference [24], we suggested a model to explain the appearance of the\nrim currents in terms of a spin accumulation at the edges due to an e\u000bective orbit cou-\npling generated by the vortex magnetization. We will further detail this model in the\ndiscussion in section 4.The unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 10\nFigure 6. (a) and(b)ShowIc(B?) patterns obtained on two Nb/Co disk junctions\nwith di\u000berent diameters. The disk diameters in (a) and (b) are 1.62 µm and 1.05 µm,\nrespectively. The period of the oscillations scales inversely with the junction area. In\nboth cases, the junctions show a clear two-channel interference pattern. (c) and (d)\nshow the critical current density distributions obtained by the Fourier analysis of the\npatterns in (a) and (b) respectively. The vertical lines indicate the boundaries of the\nelectrodes of the device, whereas the dashed lines indicate the sides of the actual weak\nlink. Figure adapted after one in Ref. [24] z.\nComparing the critical current distributions in Figure 6b and d to those obtained\non the Nb/Ni/Cu/Co devices (Figure 5b), we observe that both devices share a ten-\ndency for supercurrent to \row along the rims of the device. The di\u000berence, however,\nis the relative distribution of rim currents: the Nb/Co junctions show a far stronger\nconcentration of critical current on the rims than the Nb/Ni/Cu/Co devices. Finally,\nnote that the two-channel behavior is absent in all our control samples that are either\nsinglet dominated (e.g., those presented in section 2.2) or lack any spin texture (by fully\nmagnetizing the Nb/Co devices).\nTo further investigate the interplay between spin texture and the generation of long-\nrange triplet supercurrents, we modify the vortex magnetization pattern using in-plane\nzReference [24] can be found at: https://pubs.acs.org/doi/10.1021/acs.nanolett.1c04051. Further\npermission related to the reuse of these \fgures should be directed to the ACS.The unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 11\nFigure 7. Ic(B?) patterns (left column) obtained on Nb/Co junctions, measured at\ndi\u000berent in-plane \felds and the corresponding simulated spin textures (right column).\nIn the simulations, the gray line represents the position of the weak link, and the red\ndot indicates the location of the vortex core. (a) and (b) are obtained at zero in-plane\n\feld: the vortex is at the center of the disk, and a two channel pattern is observed, i.e.,\nlobes of equal width and slow decay of peak height. (c) and (d) Applying \u00160Hy= 10\nmT breaks the axial symmetry of the vortex magnetization, by e\u000bectively displacing\nit away from the junction. This results in the suppression of the middle peak in the\ninterference pattern, characteristic of a 0 \u0000\u0019SQUID. Figure adapted after one in\nRef. [24]z.\nmagnetic \felds and study the result on the critical current distribution. In this case, we\napply a magnetic \feld along the trench direction (the y-direction) and e\u000bectively move\nthe vortex away from the junction (towards the contacts). This displacement is linear\nfor small in-plane \felds. While maintaining a static in-plane \feld, we can sweep the\nout-of-plane magnetic \feld by the use of a vector magnet. The Ic(B?) patterns for a\nNb/Co disk-device at zero in-plane magnetic \feld and 10 mT in-plane \feld are compared\nin Figure 7. At zero in-plane \feld, we observe a clear two-channel interference pattern,\ncharacterized by lobes of equal width and a maximum of Icat zero out-of-plane \feld.\nSimilarly, at \u00160Hy= 10 mT we observe a pattern featuring equal-width lobes, but now\nwith a strong suppression of Icat zero out-of-plane \feld. The resulting Ic(B?) pattern\ntherefore bears resemblance of a 0 \u0000\u0019SQUID. Such a 0 to \u0019transition is also observed in\nour Nb/Ni/Cu/Co devices, where a \u001810 mT in-plane \feld perpendicular to the trench\nresults in a sharp drop in critical current (see Fig. 4b). Note that in the Nb/Co case\nthe in-plane \feld is along the trench and in the Nb/Ni/Cu/Co case the in-plane \feld isThe unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 12\nperpendicular to the trench. Still, we conclude that the 0 to \u0019transition is a universal\nfeature of the vortex magnetization of the Co-layer and the relative displacement of the\nvortex core from the center of the disk.\n3.3. Half metallic ferromagnet: NbTi/LSMO disk junctions\nThe third system we studied is similar to the Nb/Co system: it does not rely on the\nrelative orientation of multiple magnetic layers but consists out of a single ferromag-\nnet. However, instead of a 3 dferromagnetic transition metal, we use the halfmetallic\nferromagnetic oxide La 0:7Sr0:3MnO 3(LSMO). The halfmetallic nature of LSMO means\nthat only one type of triplet pair can exist, either spin-up or spin-down. We have also\nreplaced the Nb for NbTi superconducting electrodes.\nFigure 8. (a) Atomic Force Microscopy image of a 40 nm thick LSMO \flm grown\non an LSAT substrate. Terraces can be seen. The inset shows an xray difraction\nmeasurement (XRD). Intensity oscillations around the main Bragg re\rection at 47\u000e\nsignals a high \ratness of the \flm. (b) Resistivity \u001aversus temperature Tof a 40\nnm thick LSMO \flm. The Curie temperature can be read o\u000b from the peak of the\nderivative shown in the inset (362\u000eC). (c) Resistance RversusTof a NbTi/LSMO\ndisk junction. The inset shows the temperature dependence of the critical current\ndensityJc.\nUsing the method described above, a disk-shaped NbTi/LSMO junction was made,\nwith similar dimensions as the Co ones. We \frst grow a thin (40 nm) LSMO \flm on\nan (La 0:18Sr0:82)(Al 0:59Ta0:41)O3(LSAT) substrate by o\u000b-axis sputtering (growth pres-\nsure 0.7 mbar in an Ar:O (3:2) atmosphere; background pressure about 10\u00007mbar) at\na substrate temperature of 700oC. The substrate is chosen to minimize the mismatch\nbetween \flm and substrate. The LSMO is of high quality, as can be seen from the\nx-ray di\u000braction and atomic force microscopy data in Fig. 8a. Fig. 8b shows the R(T)\ndata of a typical \flm. The ferromagnetic ordering temperature (the Curie temperature)\nis at about 360 K, and the resistivity drops by almost two orders of magnitude upon\ndecreasing the temperature, with a value of 40 µ\ncm at 10 K. In the same system, we\nsputter-deposit a NbTi \flm of about 60 nm with a superconducting transition temper-\nature of about 7.5 K.The unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 13\nFig. 8c shows R(T) of the device, with the typical signature of a proximity e\u000bect:\n\frst the contacts go superconducting, then R(T) shows a plateau where the weak link is\nstill in the normal state, followed by a second transition and zero resistance reached just\nbelow 5.5 K.The inset shows the critical current density Jcas function of temperature.\nJcsteadily increases, reaching 1.8 \u00021010A/m2at the lowest temperature of 1.5 K. This\nis an even higher value than found in the disk junctions discussed before. For example,\nthe Nb/Co disk of the previous section, we estimate the current density in the channels\nto be around 8\u0002109A/m2. High values are consistent with the earlier works on long-\nrange triplet proximity in half-metals (e.g., CrO 2junctions and spin valves). As far as\nwe are aware, there are no previous reports on the proximity e\u000bect in LSMO or similar\nmetallic oxides with a conventional superconductor.\nTheIc(B?) pattern of the device, taken at 4.1 K, is shown in Fig.9(a). We again\nobserve a clear two channel interference pattern, similar to what is seen in the Nb/Co\nsystem. The current distribution as calculated by Fourier analysis (not shown) con\frms\nthat also in the NbTi/LSMO disk junctions the supercurrent distribution strongly\npeaks at the rims of the device. However, the behavior upon applying an in-plane\n\feld is very di\u000berent, as seen Fig.9(b). Under the application of in-plane \felds (either\nparallel or perpendicular to the trench), the critical current is quite robust: there is\nonly a negligible decrease of critical current upon increasing the \feld to 200 mT. Note\nthat 200 mT is su\u000ecient to remove any spin texture in the disk-shaped NbTi/LSMO\njunction. Besides, the 0 \u0000\u0019transition observed in the Co-based systems when moving\nthe vortex core is absent. However, the aforementioned rim currents persist regardless\nof the magnetization state. Therefore, it appears that the triplet supercurrents arise\nfrom an intrinsic mechanism that is not changed by even strong in-plane \felds.\n4. Discussion and Conclusions\nIn this section we will review the similarities and di\u000berences of the three di\u000berent mag-\nnetic disk junctions. The Nb/Ni/Cu/Co system is, at \frst sight, mostly understood.\nThe non-collinearity of the Ni/Cu/Co stack appears to act as the generator, and the\nsupercurrent disappears when an in-plane \feld has homogeneized the magnetizations.\nThere is also an e\u000bect of the vortex magnetization of the Co-disk, however, since chang-\ning that spin texture by moving the vortex core with a small in-plane \feld leads to a 0- \u0019\ntransition. However, the non-collinearity cannot explain why the current distribution\nwould peak at the rims of the disk.\nThat brings us to the Nb/Co system. Here, the current distribution peaks more\nsharply at the rims of the device. In Ref. [24] we suggested a model to explain the rim\ncurrents. Within the framework of the linearized Usadel equations, the vortex spin tex-\nture is equivalent to an e\u000bective spin-orbit coupling (SOC). Such an e\u000bective SOC wouldThe unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 14\nFigure 9. (a)Ic(B?) pattern recorded at 4.1 K on a disk-shaped NbTi/LSMO\njunction. (b) IV-characteristics under the application of constant in-plane \felds\nparallel or perpendicular to the junction of the device. The illustrations in the legend\ndemonstrate the corresponding magnetization states. Note that the critical current is\nhardly suppressed although no ferromagnetic vortex is present in the LSMO layer.\ngive rise to a spin current directed along the local magnetization direction x. When this\nspin current encounters a vacuum boundary (i.e., the rims of the disk), the spin cur-\nrent accumulates. Since the spin current across that vacuum boundary has to be zero, a\nsource of long range triplets emerges to generate an opposing spin current. Looking back\nat the Nb/Ni/Cu/Co system, that appears to show a combination of the conventional\nmechanism of magnetic non-collinearity coming from two F layers, andthe mechanism\nwe just described for the Nb/Co disks. The end result is a current distribution peaked\nat the rim, but \fnite in the center.\nShifting attention to the third ferromagnetic system: the observation that LSMO\ncan be proximized with a conventional superconductor is an important result in its own\nregard. It might appear that all the ingredients from the Nb/Co case are also at play\nin the NbTi/LSMO system, since the supercurrent peaks at the rim. That cannot ex-\nplain, however, the fact that these supercurrents are fully insensitive to in-plane \felds,\nboth directed along the trench or perpendicular to the trench. Once the magnetiza-\ntion becomes homogeneous, which we certainly expect for a 200 mT \feld, the vortex\nmagnetization along with its e\u000bective SOC disappears, but the supercurrent is almost\nunchanged. Apparently, the LRT correlations in the NbTi/LSMO devices do not result\nfrom the vortex magnetization of the disk. Rather, the observations point to a magnet-\nically disordered layer at the NbTi/LSMO interface, which can have various origins but\nis insensitive to the in-plane \felds. This picture is attractive in the sense that it could\nequally well explain the generation of LRT correlations in the reports mentioned ear-\nxThe spin current is carried by the spinless component of a triplet condensate naturally present at\nthe interface between a ferromagnet and a superconductor.The unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 15\nlier [22,23]. What is not easy to understand from this picture is why the current peaks\nat the rim, if that is not due to an e\u000bective SOC. This requires better understanding of\nthe magnetic landscape in the LSMO disk.\nIn summary, we have researched the behavior of lateral disk-type Josephson junc-\ntions for several di\u000berent types of superconductor-ferromagnet con\fgurations. Our most\nimportant \fnding is that triplet currents are generated in such junctions. The spin tex-\nture of the ferromagnetic disk plays an important role here, as evidenced by the strong\ndependence of the critical currents and their distribution over the junctions on the micro-\nmagnetic texture of the devices. In the Nb/Ni/Cu/Co devices, the generator is provided\nby the non-collinearity of the magnetic layers. However, by studying the Nb/Co devices,\nwe found that the ferromagnetic vortex itself can generate triplet correlations as well.\nLooking back, the behavior of the Nb/Ni/Cu/Co devices can be completely described\nby a combination of the vortex-related e\u000bects observed in the Nb/Co case and magnetic\nnon-collinearity. What is still less clear is why the critical supercurrent density peaks\nat the rim of the device: although our model based on an e\u000bective SOC captures the\nbehavior well in the Nb/Co case, it cannot explain the appearance of rim currents in\nthe LSMO-based devices. In these devices, the triplet correlations seem to be generated\nby an intrinsic mechanism at the NbTi/LSMO interface. What causes the unusual con-\ncentration of the triplet currents at the rims of the device remains an open question.\nWe want to end by itemizing some of the most salient observations to facilitate further\nunderstanding:\ni Rim currents only emerge when transport is carried out by spin-polarized triplet\nCooper pairs. The two-channel interference pattern corresponding to rim currents,\nappear in all the disk junctions with a strong ferromagnetic barrier (Co or LSMO),\nwhere only long-range triplet correlations can survive. In contrast, the disk\njunctions with a non-magnetic barrier (singlet transport) all show a standard\nFraunhofer di\u000braction pattern corresponding to a single transport channel.\nii The rim currents seem to emerge independently of the mechanism behind the\ngeneration of long-range triplet correlations. The di\u000berent types of disk systems\ndiscussed above suggest that, while long-range triplet correlations are crucial,\nthe mechanism that generates the triplets may not be relevant for formation of\nrim channels. Whether the long-range triplets are generated by magnetic non-\ncollinearity between separate F layers (Nb/Ni/Cu/Co devices), the spin texture of\na single ferromagnet (Nb/Co disks), or an intrinsic magnetic inhomogeneity at the\nS/F interface (LSMO-based disks), the rim currents appear as long as the long-\nrange triplets are present.\niii The size of the rim channels does not scale with the size of the disk; it appears to\nvary according to the material. As shown in Fig. 6, the width of the rim channels\n(\u001970 nm) is the same for Nb/Co devices, despite the di\u000berence in their diameters.The unusual distribution of spin-triplet supercurrents in disk-shaped Josephson junctions 16\nHowever, when a 5 nm Cu layer is placed at the S/F interface, the channels become\nconsiderably wider (see Fig. 5).\niv We can induce supercurrents in halfmetallic LSMO disks, with very high current\ndensities. The disk shows rim currents, similar to what we observe in Co disks.\nPuzzling is that, unlike in Co disks, the long-range triplet correlations are not\nsuppressed when the disk is uniformly magnetized by a large in-plane \feld. Both\nthe halfmetallicity and the oxide magnetism make this a di\u000berent system, that\ninvites further study.\n5. Acknowledgements\nThis work was supported by the project `Spin texture Josephson junctions' (project\nnumber 680-91-128) and by the Frontiers of Nanoscience (NanoFront) program, which\nare both (partly) \fnanced by the Dutch Research Council (NWO). J. Y. is funded by\nthe China Scholarship Council (No. 201808440424).The work was further supported\nby EU Cost actions CA16218 (NANOCOHYBRI) and CA21144 (SUPERQMAP). It\nbene\ftted from access to the Netherlands Centre for Electron Nanoscopy (NeCEN) at\nLeiden University.\nReferences\n[1] J. Linder and J. W. A. Robinson. Superconducting spintronics. NatPhys, 11:307{315, 2015.\n[2] M. Eschrig. Spin-polarized supercurrents for spintronics: a review of current progress. Rep. Prog.\nPhys., 78:104501, 2015.\n[3] G. Yang, C. Ciccarelli, and J. W A Robinson. Boosting spintronics with superconductivity. APL\nMater., 9:050703, 2021.\n[4] F. S. Bergeret, A. F. Volkov, and K. B. Efetov. 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Uemura2and Fanlong Ning1;5\u0003\n1Department of Physics, Zhejiang University, Hangzhou 310027, China\n2Department of Physics, Columbia University, New York, New York 10027, USA\n3Department of Physics, Hangzhou Normal University, Hangzhou 310016, China\n4Beijing National Laboratory for Condensed Matter Physics,\nand Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China and\n5Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China\nDiluted ferromagnetic semiconductors (DMSs) that combine the properties of semiconductors\nwith ferromagnetism have potential application in spin-sensitive electronics (spintronics) devices.\nThe search for DMS materials exploded after the observation of ferromagnetic ordering in III-V\n(Ga,Mn)As ﬁlms. Recently, a series of DMS compounds isostructural to iron-based superconduc-\ntors have been reported. Among them, the highest Curie temperature TCof 230 K has been achieved\nin (Ba,K)(Zn,Mn) 2As2. However, most DMSs, including (Ga,Mn)As, are p-type, i.e., the carriers\nthat mediate ferromagnetism are holes. For practical applications, DMS with n-type carriers are also\nadvantageous. Here we report the successful synthesis of a II-II-V diluted ferromagnetic semiconduc-\ntor with n-type carriers, Ba(Zn,Co) 2As2. Magnetization measurements show that the ferromagnetic\ntransition occurs up to TC\u001845 K. Hall eﬀect and Seebeck eﬀect measurements jointly conﬁrm\nthat the dominant carriers are electrons. Through muon spin relaxation ( \u0016SR), a volume sensitive\nmagnetic probe, we have also conﬁrmed that the ferromagnetism in Ba(Zn,Co) 2As2is intrinsic and\nthe internal ﬁeld is static.\nThe combination of spin and charge degrees of\nfreedom in diluted magnetic semiconductors (DMSs)\nmakes them promising materials in spintronics. The\nobservation of ferromagnetism in Mn doped III-V\nGaAs has therefore attracted extensive attention in\nlast two decades [1–4]. (Ga,Mn)As ﬁlms are typically\nfabricated via low-temperature molecular beam epitaxy\n(LT-MBE), where Mn2+substitution for Ga3+intro-\nduces both spins and holes simultaneously. Despite\nthe controversy about the origin of ferromagnetism in\n(Ga,Mn)As[5], it has been widely accepted that the\nitinerant carriers mediate the ferromagnetic interaction\nbetween spatially separated magnetic ions. To date,\nthe Curie temperature TCin (Ga,Mn)As has reached a\nmaximum of \u0018190-200 K [6–8], which is still far below\nroom temperature and therefore limits the possibilities\nfor practical applications. Recently, a series of DMS\nmaterials that are structural derivatives of iron-based\nsuperconductors have been synthesized, including I-II-\nV Li(Zn,Mn)As [9], “1111” (La,Ba)(Zn,Mn)AsO[10]\nand II-II-V (Ba,K)(Zn,Mn) 2As2[11]. Of these,\n(Ba,K)(Zn,Mn) 2As2has the highest TCof\u0018230\nK[12], exceeding the highest record of (Ga,Mn)As.\n(Ba,K)(Zn,Mn) 2As2was synthesized through the doping\nof K and Mn into the parent semiconductor BaZn 2As2,\nwhere the substitution of Mn for Zn introduces magnetic\nmoments and the substitution of K for Ba introduces\ncarriers. Considering that the end member BaMn 2As2is\nan antiferromagnet with Neel temperature of 625 K [13],\n\u0003Electronic address: ningﬂ@zju.edu.cnit seems promising that TCmay reach room temperature\nin II-II-V systems when the synthesis conditions and the\nselection of elements are optimzed[14].\nThe above-mentioned DMSs are all p-type, i.e., the\ndominant carriers are holes. N-type DMSs with elec-\ntron carriers are still exceptionally rare. In practical\napplications, both p- and n-type DMSs are required to\nfabricate junctions and devices. Furthermore, n-type\nDMSs may shed light on the general mechanism for fer-\nromagnetic ordering in DMSs. In the past, Co:ZnO\nﬁlms have been proposed to be a candidate for n-type\nDMS [15–17]. However, the underlying mechanism is\nstill in debate. For example, one careful investiga-\ntion showed that the ferromagnetism may arise from a\nhydrogen-facilitated interaction[18]. Co:TiO 2ﬁlms are\nalsoreportedtopossessferromagnetismaboveroomtem-\nperature, with electrons provided by defects or electric\nﬁelds acting as carriers[19, 20]. Recently, Hai et: al:\nreported the observation of electron-mediated ferromag-\nnetism in (In,Fe)As ﬁlms where interstitial Be provides\nelectrons[21–23]. Similar fabrication routes have also\nbeen tried in (In,Co)As ﬁlms, but no ferromagnetic or-\ndering has been observed[24]. Theoretically, Gu et: al:\npredicted that n-type DMSs may be realized in narrow-\nband-gap semiconductors[25].\nIn this paper, we demonstrate the successful synthe-\nsis of a high-quality n-type ferromagnetic semiconductor\nby doping Co onto the Zn sites of the narrow-band-gap\n(0.2 eV) semiconductor BaZn 2As2[26]. The highest TC\nof Ba(Zn 1\u0000xCox)2As2reaches \u001845 K forx= 0:04. Us-\ning muon spin relaxation ( \u0016SR) measurements, we havearXiv:1712.06764v1  [cond-mat.mtrl-sci]  19 Dec 20172\n0.000 .020 .044.1164.1204.12413.57213.57613.580(c)c (Å)a  (Å)x\n 0.00\nZn/Co \n0.01Ba(Zn1-xCox)2As2x\n = (a)B aA\ns \n0.02 \n0.03 \n0.041\n02 03 04 05 06 07 08 09 01 00 0.052\nθ (degree)Intensity (a. u.)\n204 06 08 01 00120Intensity (a. u.)2\nθ (degree) Observed \nCalculated \nDifference \nTetragonal structure(b)\nFigure 1: Structural and X-ray diﬀraction results. (a) X-ray diﬀraction patterns of Ba(Zn 1\u0000xCox)2As2with diﬀerent\ndoping levels. (b) Rietveld reﬁnement proﬁle for x= 0:04. (c) Lattice parameters for Ba(Zn 1\u0000xCox)2As2.\nconﬁrmed the homogeneous and intrinsic nature of the\nferromagnetic ordering in Ba(Zn,Co) 2As2.\nResults\nX-ray diﬀraction. In Fig. 1(a), we show the X-\nray diﬀraction patterns for Ba(Zn 1\u0000xCox)2As2with dif-\nferent doping levels. In general, BaZn 2As2is polymor-\nphic, typically crystallizing into either an orthorhom-\nbic structure (space group Pnma) or a tetragonal struc-\nture (space group I4/mmm )[11]. The tetragonal struc-\nture results in a semiconductor with a band gap of\n\u00180.2 eV[26] that forms the parent compound of the\n(Ba,K)(Zn,Mn) 2As2DMS system. In this structure, lay-\ners of {ZnAs 4} tetrahedra stack alternately with Ba lay-\ners along the c axis. We note that if the Zn atoms are\nreplaced by Fe, BaFe 2As2is the parent compound for\nmany iron based superconductors[27]. The X-ray diﬀrac-\ntion peaks in Fig. 1(a) can be well indexed with a tetrag-\nonal structure (space group I4/mmm ) with no sign of the\northorhombic phase or other impurities. In Fig. 1(b), weshow the Reitveld reﬁnement proﬁle for the x= 0:04\nsample using the GSAS-II package[28]. No obvious im-\npurity peak was observed, and the resulting weighted re-\nliability factor Rwpis\u00189:87%, indicating high sample\nquality. In Fig. 1(c), we show the lattice parameters\nfor diﬀerent doping levels. With increasing Co concen-\ntrationaincreases and cdecreases monotonically. The\nmonotonic behavior of lattice parameters indicates the\nsuccessful doping of Co up to x= 0:05.\nMagnetic properties. In Fig. 2(a), we\nshow the temperature-dependent magnetization of\nBa(Zn 1\u0000xCox)2As2(x= 0:01;0:02;0:03;0:04;0:05) in an\nappliedmagneticﬁeldof100Oe. Zeroﬁeldcooling(ZFC)\nand ﬁeld cooling (FC) data are represented by open and\nﬁlled symbols, respectively. For x= 0:01, no magnetic\ntransition was observed down to the base temperature\nof 2 K, and the magnetic moment at 2 K in H= 100\nOe is only 0.005 \u0016B/Co. However, for Co concentrations\nexceeding 1%, a sudden increase of the magnetization\ndevelops around 35-45 K, indicative of a ferromagnetic\ntransition.\nWe used the Arrott plot method for precise determina-\ntion of the Curie temperature TCfor Ba(Zn 1\u0000xCox)2As23\n02 00004 00006 00000.00.10.20.30.4Ba(Zn0.95Co0.05)2As2Magnetic Moment (µB/Co)M\nagnetic Field (Oe)   2 K  30 K \n32 K  34 K \n36 K  38 K \n40 K  42 K \n44 K  46 K(d)\n05 00001000001500002000000.00.10.2 \n  2 K \n30 K \n32 K \n34 K \n36 K \n38 K \n40 K \n42 K \n44 K \n46 KBa(Zn0.95Co0.05)2As2M2 ((µB/Co)2)H\n/M (Oe/(µB/Co))(e)T\nC = 41 K\n-1000-800-600-400-20002004006008001000-0.4-0.3-0.2-0.10.00.10.20.30.4B\na(Zn0.96Co0.04)2As2Magnetic Moment (µB/Co)M\nagnetic Field (Oe)   2 K \n30 K \n40 K \n50 K(c)\n-200-150-100-50050100150200-0.3-0.2-0.10.00.10.20.3(b)B\na(Zn1-xCox)2As2Magnetic Moment (µB/Co)M\nagnetic Field (Oe) 0.01 \n0.02 \n0.03 \n0.04 \n0.05T = 2 K\n02 04 06 08 01 001 201 401 601 802 000.000.020.040.060.080.100.120.140.160.180.200.220.240.260.28Magnetic Moment (µB/Co)T\nemperature (K)   x = \n0.01 (ZFC) \n0.01 (FC) \n0.02 (ZFC) \n0.02 (FC) \n0.03 (ZFC) \n0.03 (FC) \n0.04 (ZFC) \n0.04 (FC) \n0.05 (ZFC) \n0.05 (FC)H = 100 OeBa(Zn1-xCox)2As2(a)0\n102030405060708090100050100150200250300 \n0.02 \n0.03 \n0.04 \n0.051/(χ−χ0) (mol Co*Oe/emu)T\nemperature (K)\nFigure 2: Magnetization results. (a) The temperature-dependent magnetization for Ba(Zn 1\u0000xCox)2As2(x=\n0:01;0:02;0:03;0:04;0:05) in a magnetic ﬁeld of 100 Oe. The open and ﬁlled symbols represent the zero-ﬁeld-cooled and\nﬁeld-cooled data, respectively. Inset: Plot of 1=(\u001f\u0000\u001f0)versusT. Straight lines represent a Curie-Weiss ﬁt. (b) The isothermal\nmagnetization for Ba(Zn 1\u0000xCox)2As2(x= 0:01;0:02;0:03;0:04;0:05) at 2 K in an applied magnetic ﬁeld ranging from -200\nOe to 200 Oe. (c) Evolution of the hysteresis loop for Ba(Zn 0:96Co0:04)2As2with increasing temperature. (d) Isothermal\nmagnetization for Ba(Zn 0:95Co0:05)2As2at diﬀerent temperatures. (e) The Arrott plot for Ba(Zn 0:95Co0:05)2As2at diﬀerent\ntemperatures. Lines show the best linear ﬁt.\n[29]. According to the Ginzburg-Landau mean ﬁeld the-\nory for magnetism, the free energy close to the phase\ntransition can be written as F(M) =\u0000HM +a(T\u0000\nTC)M2+bM4+\u0001\u0001\u0001. For a stable state, the derivative\nofF(M)with respect to Mshould be 0. After omit-\nting the high order items, the function is rewritten as:\nM2=1\n2bH\nM\u0000a\n2b(T\u0000TC). Therefore, around TC, the\nplot ofM2versusH=Mshould be an array of paral-\nlel lines. The intercept is positive below TCand neg-ative above TC, respectively. The temperature where\nthe line passes through the origin is TC. In Fig. 2(e),\nwe show the Arrott plot for x= 0:05. AroundTC, the\npoints at high magnetic ﬁeld fall approximately on a se-\nries of parallel lines. The solid lines displayed on the\nplot are the linear ﬁts at high magnetic ﬁeld, and the\nnonlinear behavior at low ﬁeld is ascribed to the higher-\norder terms we omitted in the analysis or other devia-\ntions from mean ﬁeld theory. We identify TCas 41 K,\nthe temperature at which the parallel line would pass4\nTable 1: Curie Temperature ( TC), Weiss Temperature and\nEﬀective Moment ( \u0012and\u0016eff, derived from Curie-Weiss ﬁt-\nting), Saturation Moment ( \u0016s, the value measured at T= 2\nK andH= 200Oe), Coercive Field ( Hc)\nCo concentration ( x)TC\u0012 \u0016 eff\u0016sHc\n(K) (K) (\u0016B/Co) (\u0016B/Co) (Oe)\n0.01 =0.03 2.0 = =\n0.02 35 53 1.1 0.18 16\n0.03 37 54 1.7 0.20 22\n0.04 45 57 1.4 0.24 6\n0.05 41 51 1.4 0.22 11\nTable 2: Comparison of selected properties of (Ga,Mn)As,\n(Ba,K)(Zn,Mn) 2As2and Ba(Zn,Co) 2As2.\n(Ga,Mn)As (Ba,K)(Zn,Mn) 2As2Ba(Zn,Co) 2As2\nValence before doping III-V II-II-V II-II-V\nCarrier type holes holes electrons\nMaximum TC190 K[6] 230 K[12] 45 K\nSaturation moment 5\u0016B/Mn 2\u0016B/Mn 0.2 \u0016B/Co\nSample form thin ﬁlm bulk form bulk form\nthrough the origin. TCfor other doping levels was also\ndetermined by this method (see Supplement). We list\nTCfor Ba(Zn 1\u0000xCox)2As2in Table 1. We can also ob-\ntain the eﬀective moments \u0016effby ﬁtting the tempera-\nture dependent magnetization above TCwith a modiﬁed\nCurie-Weiss law: \u001f=\u001f0+C=(T\u0000\u0012), where\u001f0is the\ntemperature independent component, Cis the Curie con-\nstant and\u0012is the Weiss temperature. \u0016effis\u00181.1-1.7\n\u0016B/Co. According to \u0016eff=\u0016Bgp\nS(S+ 1), where\u0016B\nis the Bohr magneton and g= 2is the Lande factor for\nelectrons, we estimate the average spin state of Co to be\nclose to S = 1=2.\nIn Fig. 2(b), we show the isothermal magnetiza-\ntion at 2 K. Clear hysteresis loops are observed for all\ndoping levels except the paramagnetic x= 0:01sam-\nple. The coercive ﬁeld of Ba(Zn,Co) 2As2is on the or-\nder of \u001810 Oe, which is much smaller than 1 T in\n(Ba,K)(Zn,Mn) 2As2[11]. The small coercive ﬁeld is con-\nsistent with the minimal bifurcation of ZFC and FC\ncurves at 100 Oe shown in Fig. 1(a). In Fig. 2(c), we\nshow the temperature dependence of the hysteresis loop\nforx= 0:04. With increasing the temperature, the mo-\nment become smaller and the hysteresis loop eventually\ndisappears above 50 K. The saturation moment ( \u0016s) is\u0018\n0.2-0.3\u0016B/CoforBa(Zn,Co) 2As2whichismuchsmaller\nthan 2\u0016B/Mn for (Ba,K)(Zn,Mn) 2As2and5\u0016B/Mn for\n(Ga,Mn)As[2, 11].\nHall eﬀect, Seebeck eﬀect and transport. We\njointly utilized measurements of the Hall eﬀect and See-\nbeck eﬀect (see Supplement) to investigate the properties\nof the carriers. Since RHall =B=(ne), whereBis the\n05 01 001 502 002 503 004.0x10176.0x10178.0x10171.0x10181.2x10181.4x1018 \nCarrier ConcentrationCarrier Concentration (-cm-3)T\nemperature (K)-10-50510-0.010.000.01 \n25 K \nLinear FittingHall Resistivity (Ω*cm)M\nagnetic Field (×10000 Oe)Figure 3: Results of Hall eﬀect measurements. Carrier\nconcentration for Ba(Zn 0:96Co0:04)2As2calculated from Hall\nresistivity curves. Inset is the Hall resistivity at 25 K, with a\nlinear ﬁt shown as the green line.\nexternal ﬁeld perpendicular to the current and eis the\nelementary charge, we obtained the carrier concentration\nfromRHallversusBcurves. InFig. 3, weshowtherepre-\nsentative Hall resistivity ( RH) at 25 K and the variation\nof carrier density ( n) versus temperature ( T). The neg-\native slope of Hall resistivity curve indicates that domi-\nnantcarriersinBa(Zn,Co) 2As2areelectrons. Thecarrier\nconcentration is about 1017\u00181018/cm\u00003depending on\nthe measuring temperature, which is much smaller than\n1020/cm\u00003of (Ba,K)(Zn,Mn) 2As2[11], but comparable\nto that of Li(Zn,Mn)P[30]. The carrier density decreases\ngradually with decreasing temperature. Seebeck eﬀect\nmeasurements at room temperature were also conducted\nto investigate the carrier type. The Seebeck coeﬃcient\nisS=\u0000\u0001U=\u0001T, where \u0001Uis the voltage diﬀerence\nbetween two electrodes and \u0001Tis the temperature dif-\nference. The sign of the Seebeck coeﬃcient is related to\nthe carrier type, positive for p-type carriers and negative\nfor n-type carriers. The room temperature Seebeck coef-\nﬁcient is \u0018-15.86\u0016V/K forx= 0:04and\u0018-6.95\u0016V/K\nforx= 0:05. The negative Seebeck coeﬃcient (see Sup-\nplement) conﬁrms our conclusion of n-type carriers.\nIn Fig. 4(a), we show the electrical transport proper-\nties for diﬀerent doping levels. With Co doping, the re-\nsistivity retains its semiconductor behavior but the mag-\nnitude decreases, indicating the successful introduction\nof carriers by Co substitution for Zn. In Fig. 4(b), we\nshow the magneto-resistivity (MR) for the x= 0:04sam-\nple under an applied magnetic ﬁeld of 1 Tesla. The MR\ncurve decreases clearly below TC, which is due to the\nsuppression of magnetic scattering by the external ﬁeld.\nAtT= 2K, the negative MR saturates at H= 6000\nOe with (\u001a\u0000\u001a(0))=\u001a(0)reaching \u0018 \u000017%, as shown in\nthe inset of Fig. 4(b). This value is much larger than the\nvalue of \u00187:5%for (Ba,K)(Zn,Mn) 2As2at 7 T[12]. After5\n02040608010012014016018020022024026028030010-1100101102103104105106Ba(Zn1-xCox)2As2Resistivity (Ω∗mm)T\nemperature (K) 0.00  0.01  \n0.02  0.03 \n0.04  0.05(a)\n0102030405060708090100152025303540Ba(Zn0.96Co0.04)2As2 \nH = 0 T \nH = 1 TResistivity (Ω*mm)T\nemperature (K)TC(b)-\n10-50 5 1032343638Resistivity (Ω*mm)M\nagnetic field (×10000 Oe)\nFigure4:Resultsofresistivityandmagneto-resistivity.\n(a) The resistivity as a function of temperature for diﬀerent\ndoping levels. (b) The magneto-resistivity for the x= 0:04\nsample. The red arrow marks the position of TC\u001845K.\nInset is the ﬁeld-dependent resistivity, with a ﬁeld interval of\n2000 Oe.\nsaturation at 6000 Oe, the MR displays a slight increase\nwith increasing external ﬁeld. Usually, depending on the\nconﬁgurationofthemutuallyperpendicularﬁeldandcur-\nrent, the Lorentz force can aﬀect the electrons’ path and\ntherefore increase the resistance. Nonetheless, the rela-\ntively large negative magneto-resistivity and smaller sat-\nuration ﬁeld indicate that the electrical transport prop-\nerties in Ba(Zn,Co) 2As2can be be easily controlled by\nexternal magnetic ﬁeld.\nZF- and LF- \u0016SR.Generally speaking, a small\namount of magnetic impurities such as Co nanoparticles\nor unknown Co compounds can give rise to magnetic sig-\nnals that may obscure the intrinsic magnetic properties.\nTo rule out such scenario, we applied \u0016SR, a volume-\nsensitive magnetic probe, to investigate Ba(Zn,Co) 2As2.\nIn Fig. 5(a), we show the zero ﬁeld (ZF-) \u0016SR time\nspectra for Ba(Zn 0:95Co0:05)2As2. A fast-relaxing com-\nponent clearly arises below TC, consistent with the for-\n010203040500.0000.0020.0040.006B\na(Zn0.95Co0.05)2As2Relaxation Rate (µs-1)T\nemperature (K)(f)T\nC = 41 KTf = 5.5 K\n02 4 6 8 0.160.180.200.220.24Ba(Zn0.95Co0.05)2As2AsymmetryT\nime (µs) 145 K \n40 K \n30 K \n10 K \n5.5 K \n2.1 K(e)\n010203040500.0000.0020.0040.0060.008B a(Zn1-xCox)2As2λd (µs-1)T\nemperature (K)x= \n0.03 \n0.05(d)\n02 4 6 8 0.000.050.100.150.200.25B a(Zn0.95Co0.05)2As2( c)AsymmetryT\nime (µs) 200 Oe \n100 Oe \n50 Oe \n25 Oe \n10 Oe \n0 Oe\n010203040500.00.10.20.30.40.5B\na(Zn1-xCox)2As2as (µs-1)T\nemperature (K)x= \n0.03 \n0.05(b)\n02 4 6 8 0.000.050.100.150.200.25B a(Zn0.95Co0.05)2As2AsymmetryT\nime (µs)100 K4\n0 K3\n0 K2\n0 K1\n0 K5\n K2\n K(a)Figure 5: Results of \u0016SR characterization. (a) ZF-\u0016SR\ntime spectra of Ba(Zn 0:95Co0:05)2As2. The solid lines show\nthe best ﬁt to the dynamic-static relaxation function with\nthe static local ﬁeld amplitude parameter asshown in (b)\nand the dynamic relaxation rate parameter \u0015dshown in (d).\nThe LF-\u0016SR time spectra are shown in (c), exhibiting full\ndecoupling at 200 Oe. (e) The time spectra of LF- \u0016SR in\nBa(Zn 0:95Co0:05)2As2with an external ﬁeld of 100 Oe at dif-\nferent temperatures. (f)The muon spin relaxation rate 1/T 1.\nmation of ferromagnetic ordering. Similar to the case\nof p-type “1111” DMS systems (La,Ba)(Zn,Mn)AsO[10],\nwe use a dynamic spin freezing model to ﬁt the ZF- \u0016SR\ndata. As shown by the solid curves in Fig. 5(a), the\ntime spectra can be well ﬁtted by the dynamic-static re-\nlaxation function (Eq. (26) of Ref. 31). This indicates\nthatBa(Zn 0:95Co0:05)2As2achievesstaticmagneticorder\nthroughout the entire volume at low temperatures, con-\nﬁrming that the previous magnetization measurements\nare intrinsic to the samples and not due to a small im-\npurity phase. The static local ﬁeld amplitude asand the\ndynamic relaxation rate \u0015ddetermined from the ﬁts are\ndisplayed in Fig. 5(b) and (d). The parameter asis pro-\nportional to the individual ordered moment size multi-\nplied by the moment concentration. asis zero above TC,\nandstartstoincreasebelow TC, indicatingtheemergence\nof a static ﬁeld in the ferromagnetic state.\nWe used longitudinal ﬁeld (LF-) \u0016SR to investigate the\nspin dynamics in Ba(Zn,Co) 2As2. In Fig. 5(c), we show\nthe ﬁeld dependence LF- \u0016SR spectra measured at 2 K.\nAn external ﬁeld of \u0018100 Oe fully decouples the LF-\n\u0016SR time spectra, indicating that the internal magnetic\nﬁeld at the muon stopping sites is fully static and has a\nmagnitude about 10 times less than the decoupling ﬁeld,\ni.e.\u001810 Oe. In Fig.5(e), we show the temperature-\ndependent LF- \u0016SR spectra conducted under a constant6\n02 04 06 08 0100120140020406080as (µs-1)T\nC (K) (Ga,Mn)As \nLi(Zn,Mn)As \nLi(Zn,Mn)P \n(Ba,K)(Zn,Mn)2As2 \n(La,Ba)(Zn,Mn)AsO \nBa(Zn,Co)2As20204060801000.00.51.0as (µs-1)T\nC (K)\nFigure 6: Plot ofasversusTC.Correlation between\nthe static internal ﬁeld parameter asdetermined at T= 2\nK by ZF-\u0016SR versus the ferromagnetic Curie temperature\nTCobserved in (Ga,Mn)As (Ref. 32), Li(Zn,Mn)As (Ref.\n9), Li(Zn,Mn)P (Ref. 33), (La,Ba)(Zn,Mn)AsO (Ref. 10),\n(Ba,K)(Zn,Mn) 2As2(Ref. 11), and Ba(Zn,Co) 2As2(current\nstudy).\nexternal ﬁeld of 100 Oe and plot the extracted relaxation\nrate 1/T 1in Fig. 5(f). 1/T 1displays similar behavior\nas\u0015dobtained from ZF- \u0016SR(Fig. 5(d)). The dynamic\nrelaxation exhibits two peaks, one corresponding to TC\narising from the critical slowing down of spin ﬂuctuations\nnearTC, and the other arising at the temperature where\nZFC and FC curves start to bifurcate as shown in Fig.\n2(a). This temperature should be related to the freezing\nof magnetic domains.\nWhen we plot the internal ﬁeld strength asversusTC\nin Fig. 6, the point for the present n-type system lies\nat a location very diﬀerent from the linear trend shown\nby many other p-type DMS systems [9–11, 32, 33]. Since\nthe static internal ﬁeld parameter asis proportional to\nthe concentration multiplied by the average static mo-\nment size in dilute spin systems, the trend for the n-\ntype system implies that TCis relatively high for a given\nsize and density of the static ordered moments. Hence\nthe ferromagnetic exchange coupling is much larger in\nthe n-type system compared to the p-type systems. (A\npreliminary estimation of the s-d exchange interaction\nin Ba(Zn 1\u0000xCox)2As2is much larger than 1.2 eV of\n(Ga,Mn)As[34].) Thistendencycanbepartlyascribedto\nthe diﬀerence between the present Co-doped system and\nMn-doped p-type 122 DMS systems, which involve frus-\ntration because the nearest-neighbor Mn pairs are cou-\npled antiferromagnetically, as can be seen in BaMn 2As2\nbeing a strong antiferromagnet with TN\u0018625K[13].\nIn contrast, BaCo 2As2is a paramagnet showing a ten-\ndency towards ferromagnetic correlation[35, 36]. There-\nfore, there is no frustration between neighboring Co spins\nin the Co-doped 122 system. This could lead to the\nsmallercoerciveﬁeldandstrongerferromagneticcoupling\nin the n-type system compared to the p-type Mn doped\nDMS system. This feature may be helpful in obtaining\nhigherTCin n-type DMS systems.Discussion\nWe have successfully synthesized the ferromagnetic\nsemiconductor Ba(Zn,Co) 2As2through the solid state\nreaction method. Hall resistivity and Seebeck coeﬃ-\ncient measurements jointly conﬁrmed that the carriers\nin Ba(Zn 1\u0000xCox)2As2are electrons. Magnetization mea-\nsurements showed that the highest TCis\u001845K for the\n4%doping level and the coercive ﬁeld is on the order\nof 10 Oe. ZF- and LF- \u0016SR measurements show that a\nstatic ﬁeld arises throughout the full sample volume be-\nlowTC, with a magnitude of about 10 Oe at the muon\nstopping sites. In the temperature-dependent LF- \u0016SR\nmeasurements, we observed critical slowing down of spin\nﬂuctuations around TCand the freezing of magnetic do-\nmains at lower temperature. Combining the ZF- and\nLF-\u0016SR time spectra, we can conclude that the present\nn-type DMS system exhibits characteristic signatures of\ndynamic slowing down followed by static magnetic order,\nwith a magnetically ordered state in the entire volume.\nThen-typeDMSBa(Zn,Co) 2As2(TC=45K)fromthe\ncurrent study joins several related compounds including\nthe p-type DMS (Ba,K)(Zn,Mn) 2As2(TC= 230 K)[12],\nthe Fe-based superconductor Ba(Fe,Co) 2As2(Tc= 25\nK)[27], theantiferromagneticinsulatorBaMn 2As2(TN=\n625 K)[13], and the paramagnetic metal BaCo 2As2[36].\nThey all share a common tetragonal crystal structure\nwith a lattice mismatch of less than 5%. Superconduct-\ning ﬁlms of Ba(Fe 1\u0000xCox)2As2have been fabricated suc-\ncessfully with pulsed laser deposition methods by many\ngroups in past [37]. Recently, Xiao et alhave success-\nfully grown high-quality epitaxial ﬁlms of the tetrago-\nnal\f-BaZn 2As2[26], and Cao et alare working on the\ngrowth of Ba(Zn,Co) 2As2ﬁlms [38]. With the progress\nof thin ﬁlm growth, it is conceivable that various junc-\ntions and devices can be fabricated to combine n-type\nDMS Ba(Zn,Co) 2As2, p-type DMS (Ba,K)(Zn,Mn) 2As2,\nand the superconductor Ba(Fe,Co) 2As2through the As\nlayers.\nAcknowledgments\nThe work at Zhejiang was supported by MOST\n(No. 2016YFA0300402 and No. 2014CB921203), NSF\nof China (11574265), NSF of Zhejiang Province (No.\nLR15A040001 and No. LY14A040007) and the Funda-\nmental Research Funds for the Central Universities; at\nColumbia by NSF (DMR 1610633 and DMREF DMR-\n1436095) and thank JAEA Reimei project; at IOPCAS\nby NSF & MOST through research projects. F.L. Ning\nacknowledge helpful discussions with B. Gu, S. Maekawa,\nKaiyou Wang, Hanoh Lee, Igor Mazin, Igor Zutic and\nJianhua Zhao.7\nMethod\nMaterial synthesis. Polycrystalline samples of\nBa(Zn,Co) 2As2were synthesized through solid state re-\naction of high purity elements ( \u001599.9%) Ba, Zn, Co and\nAs. Mixed ingredients were placed in aluminum crucibles\nand sealed in evacuated silica tubes. All handling of the\nelements was conducted in a glove box ﬁlled with high\npurity Ar (the content of H 2O and O 2is less than 0.1\nppm) except the sealing of the silica tubes. The mixture\nwasheatedto900\u000eCfor10h, thenheldat1150\u000eCfor24\nh followed by furnace cooling. Next, the products were\nground, pressed into pellets, sealed in evacuated silica\ntubes, then heated to 1150\u000eC and held for over 24 hours\nfollowed by fast cooling to keep the tetragonal phase.\nProperty characterization. Powder X-ray diﬀrac-tion was performed at room temperature using a PANa-\nlytical X-ray diﬀractometer (Model EMPYREAN) with\nmonochromatic Cu-K \u000b1radiation. The DC magnetiza-\ntion measurements were conducted on a Quantum De-\nsign Magnetic Property Measurement System (MPMS3).\nThe Hall eﬀect and magneto-resistivity were measured\non a Quantum Design Physical Property Measurement\nSystem (PPMS). The Seebeck coeﬃcient was measured\nat room temperature using a commercial thermopower\nmeasurement apparatus. The zero ﬁeld resistivity was\nmeasured by the typical four-probe method with a Keith-\nley 6221 DC and AC current source and Keithley 2182A\nnanovoltmeter. \u0016SR measurements were performed with\nLAMPF spectrometeron theM20beamline atTRIUMF,\nCanada, and \u0016SR data were analyzed using the musrﬁt\npackage[39].\n[1] Ohno, H. (Ga,Mn)As: A new diluted magnetic semicon-\nductor based on GaAs. Appl. Phys. Lett. 69, 363 (1996).\n[2] Ohno, H. Making Nonmagnetic Semiconductors Ferro-\nmagnetic. 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B79, 21 (2009).\n[37] Haidnl. S. et al.Thin ﬁlm growth of Fe-based supercon-\nductors: from fundamental properties to functional de-\nvices.Acomparativereview.Rep.Prog.Phys. 77, 046502\n(2014).\n[38] Cao. L. X. at Institute of Physics, Chinese Academy of\nScience. Private communications.[39] Suter, A., Wojek B. M. Musrﬁt: A Free Platform-\nIndependent Framework for \u0016SR Data Analysis Physics\nProcedia30, 69 (2012).\nAuthor contributions\nF.L.N. conceived and proposed the present study and\norganized the research project with Y.J.U.; S.L.G. grew\nthe materials and conducted transport and magneti-\nzation measurement with F.L.N, H.Y.M., C.D., Y.Z.,\nL.C.F., Y.L.G, G.X.Z., B.C. and H.D.W.; Z.D. and\nC.Q.J.measuredtheSeebackeﬀect; S.L.G,F.L.N,B.A.F,\nS.C.C, K.Y. and Y.J.U. worked on \u0016SR data acquisition\nat TRIUMF, and S.L.G, Z.G., F.L.N. and Y.J.U. anal-\nysed the\u0016SR spectra. The main text was drafted by\nF.L.N. after input from S.L.G. and Y.J.U.; Supplemen-\ntary Information was drafted by S.L.G. and F.L.N.; All\nauthorssubsequentlycontributedtorevisionsofthemain\ntext and Supplementary Information."    },    {        "title": "1302.4398v2.Ferromagnetic_planar_Josephson_junction_with_transparent_interfaces__a_φ_junction_proposal.pdf",        "content": "arXiv:1302.4398v2  [cond-mat.supr-con]  8 May 2013Ferromagnetic planar Josephson junction with\ntransparent interfaces: a ϕjunction proposal\nD M Heim1, N G Pugach2,3, M Yu Kupriyanov2, E Goldobin4,\nD Koelle4and R Kleiner4\n1Institut f¨ ur Quantenphysik and Center for Integrated Quantu m Science and\nTechnology (IQST), Universit¨ at Ulm, D-89069 Ulm, Germany\n2Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State\nUniversity, 119991 Leninskie Gory, Moscow, Russia\n3Faculty of Physics, M. V. Lomonosov Moscow State University, 119 991 Leninskie\nGory, Moscow, Russia\n4Physikalisches Institut and Center for Collective Quantum Phenome na in LISA+,\nUniversit¨ at T¨ ubingen, D-72076 T¨ ubingen, Germany\nE-mail:dennis.heim@uni-ulm.de\nAbstract. We calculate the current phase relation of a planar Josephson\njunction with a ferromagnetic weak link located on top of a thin norma l metal\nﬁlm. Following experimental observations we assume transparent s uperconductor-\nferromagnetinterfaces. Thisprovidesthebestinterlayercouplin gandalowsuppression\nof the superconducting correlations penetrating from the super conducting electrodes\ninto the ferromagnetic layer. We show that this Josephson junctio n is a promising\ncandidate for an experimental ϕjunction realisation.\nPACS numbers: 85.25.Cp, 74.78.Fk, 74.45.+c, 74.50.+rFerromagnetic planar Josephson junction with transparent interfaces 2\n1. Introduction\nAϕjunction [ 1,2] is a Josephson junction with a doubly degenerate ground state, in\nwhich the Josephson phase takes the values + ϕor−ϕ(0< ϕ < π) [3]. This junction\nbeing closed into a ring is able to self-generate a fractional ﬂux Φ 0ϕ/(2π), where Φ 0is\nthe magnetic ﬂux quantum.\nIn this sense the ϕjunction is a generalisation of the πjunction [ 4] which has a\nJosephson phase + πor−πin itsgroundstate. It hasbeen experimentally demonstrated\nthat the πjunction improves the performance and simpliﬁes the design of class ical and\nquantum circuits [ 5,6,7]. Since the ϕjunction oﬀers the possibility to choose a special\nvalue of the phase in the ground state it may further optimize these circuits.\nThe initial ϕjunction proposal [ 1] investigated grain-boundary junctions, which\nwere analysed experimentally in [ 8]. From then on ϕjunctions were studied more\nand more intensively and many other systems appeared as possible c andidates for the\nrealisation of ϕjunctions, e.g. [ 2,3,9,10,11,12,13,14]. Only recently, anexperimental\nevidence of a ϕjunction made of 0 and πparts [2,11,12] was reported [ 15]. One half\nof the junction had the Josephson phase 0 in its ground state and t he other half the\nphaseπ. Thiswasrealised[ 15]byconnectingtwosuperconductor-insulator-ferromagnet-\nsuperconductor (SIFS) junctions in parallel. The advantage of th is concept is that it is\nbased on the technology already developed for the fabrication of 0 -πjunctions [ 16,17].\nOn the other hand this ϕjunction concept is diﬃcult to realise experimentally\nbecause, e.g., a step in the thickness of the F layer must be realised w ith very high\naccuracy [ 11,12,15]. A completely other method, the “ramp-type overlap” (RTO) ϕ\njunction, was proposed by Bakurskiy et al. [ 18]. It only requires one small SFS junction\nlocated on a thin normal (N) metal layer, see ﬁgure 1. This basic setup provides a\nminiaturized ϕjunction. Moreover, this type of junction has already been realise d\nexperimentally for the analysis of the double proximity eﬀect [ 19].\nA simple model [ 3] to show that the RTO junction can be used as a ϕjunction\nFigure 1. The geometry of the considered system. The Josephson junction consists\nof two superconducting (S) electrodes separated by a ferromag netic (F) weak link of\nthickness dFand length L. It is located on top of a thin normal (N) metal ﬁlm of\nthickness dN.Ferromagnetic planar Josephson junction with transparent interfaces 3\nrequires its current-phase relation (CPR). By writing it in terms of a sine series\nI(φ) =Asin(φ)+Bsin(2φ), (1)\nwhereφis the Josephson phase, the amplitudes have to obey the conditions [3]\n|B|>|A|/2 and B <0. (2)\nThe RTO junction, schematically shown in ﬁgure 1, can fulﬁl these conditions\nbecause the current ﬂows between the S electrodes through the F metal andthe N\nlayer. In this way the properties of an SFS and SNS junction are com bined. The SFS\njunction can have a negative [ 20,21] amplitude AFin (1), while the SNS junction has\na positive [ 20,22] amplitude ANin (1). By adding both the total amplitude Acan\nbe minimized and a dominant negative amplitude Bfrom the SNS part is obtained to\nfulﬁl conditions ( 2). Since supercurrents in SFS junctions are rather small, the SNS\ncontribution has to be reduced. This is done by using only a thin norma l metal ﬁlm.\nIn the present paper we investigate anRTO junction which has, diﬀe rently from the\none proposed in [ 18], transparent SF interfaces in order to amplify the SFS contributio n\nto the total current. This assumption has already successfully be en used to describe\nvarious experiments [ 19,23,24]. As a result, we obtain slightly smaller system sizes\nfor theϕjunction realisation than [ 18], where weakly transparent interfaces were\nassumed. Moreover, our approach provides a better penetratio n of the superconducting\ncorrelationsintotheFlayerwhichmayincreasetheJosephsoncurr ent. Intheframework\nof transparent SF interfaces we cannot use linearised equations f or the SFS part, as it\nwas done in [ 18]. Therefore, we use non-linearised equations in the SFS andSNS part\nfor our analytical approach.\nWe derive the CPR in the “dirty” limit. For this purpose, we combine the solution\nof the Usadel equations in the N ﬁlm [ 18] with the solution of the Usadel equations in\nthe SFS layer [ 25]. The resulting current phase relation consists of three parts: (i) a\ncontribution fromthe SFSlayer, (ii) a contribution fromthe Nﬁlm and (iii) a composite\nSNFS term.\nThe paper is organized as follows. In section 2we introduce the model of the\nconsideredJosephsonjunctionintermsofUsadelequations. The analyticalexpression of\nthe CPR of our system is based onthis model and presented in sectio n3. In section 4we\nuse this expression together with realistic system parameters to d iscuss its applicability\nasϕjunction. Finally, an appendix provides a detailed derivation of the co mposite\nSNFS current.\n2. Model\nThe considered Josephson junction is sketched in ﬁgure 1. It consists of an SFS junction\nlocated on a normal metal ﬁlm. The F layer has a thickness dFand a length Lwhile the\nN layer has a thickness dNandis considered as inﬁnitely long. We have chosen the xand\nzaxis in directions parallel and perpendicular to the plane of the N ﬁlm, r espectively.Ferromagnetic planar Josephson junction with transparent interfaces 4\nFor the calculation of the current I(φ) ﬂowing from one superconducting electrode\nto the other we determine the Green’s functions describing our sys tem. We consider the\n“dirty” limit [ 19,23,24], in which the elastic scattering length is much smaller than the\ncharacteristic decay length, we can use the Usadel equations [ 26] to model our system.\nWe write them in the form [ 20]\nξ2\nj\nGj/bracketleftbigg∂\n∂x/parenleftbigg\nG2\nj∂\n∂xΦj/parenrightbigg\n+∂\n∂z/parenleftbigg\nG2\nj∂\n∂zΦj/parenrightbigg/bracketrightbigg\n−˜ω\nπTcΦj= 0,\nGj=˜ω/radicalbig˜ω2+ΦjΦ∗\nj, j∈ {N,F} (3)\nin the N and F layer, respectively. Here, Φ jandGjare the Usadel Green’s functions\nin the Φ parametrization [ 27]. The frequencies ˜ ω=ω+ iHcontain the Matsubara\nfrequencies ω=πT(2n+1) at temperature T, wheren= 0,1,2,..., and the exchange\nﬁeldHof the ferromagnetic material which is assumed to be zero in the N lay er. The\ndecay lengths\nξN=/radicalbigg\nDN\n2πTc, ξF=/radicalbigg\nDF\n2πTc(4)\nof the superconducting correlations are deﬁned via the critical te mperature Tcof the\nsuperconductor (we use /planckover2pi1=kB= 1) and the diﬀusion coeﬃcients DNandDFin the\nnormal and ferromagnetic metal, respectively.\nWe assume that superconductivity in the S electrodes is not suppre ssed by the\nneighbouring N and F layers. This assumption is valid in our case of tran sparent SF\ninterfaces with the conditions for the suppression parameters\nγBSF=RBSFABSF\nρFξF≪1, γSF=ρSξS\nρFξF≪1, (5)\nγBSN=RBSNABSN\nρNξN≫γSN=ρSξS\nρNξN. (6)\nHere,RBSN,BSFandABSN,BSFare the resistances and areas of the SN and SF interfaces.\nThe values of ρN,F,Sdescribe the resistivity of the N, F, and S metals.\nThis allows us to use the rigid boundary conditions [ 20]\nΦS(±L/2) = ∆exp( ±iφ/2), GS=ω√\nω2+∆2, (7)\nwhere ∆ is the absolute value of the order parameter in the superco nductor.\nThe boundary conditions [ 27,28,20] at the free interfaces are\n∂\n∂zΦj= 0, j∈ {N,F}, (8)\nand at the interfaces of the superconductor they are\nγBSNξN∂ΦN\n∂z=GS\nGN[ΦS(±L/2)−ΦN] (9)\nand\nΦF=˜ω\nωΦS(±L/2). (10)Ferromagnetic planar Josephson junction with transparent interfaces 5\nAdditionally we use\nγBNFξF∂ΦF\n∂z=GN\nGF/parenleftbigg˜ω\nωΦN−ΦF/parenrightbigg\n(11)\nat the NF interfaces, where\nγBNF=RBNFABNF\nρFξF(12)\nis deﬁned analogous to ( 6).\nFinally we calculate the total current\nI(φ) =IN(φ)+IF(φ) (13)\nby integrating the standard expressions [ 20] for the current densities of the N and F\npart over the junction cross section along the zaxis. This leads us to\nIN(φ) = iπTW\n2eρN∞/summationdisplay\nω=−∞/integraldisplaydN\n0dzG2\nN\nω2\n×/bracketleftbigg\nΦN(ω)∂\n∂xΦ∗\nN(−ω)−Φ∗\nN(−ω)∂\n∂xΦN(ω)/bracketrightbigg\nx=0(14)\nand\nIF(φ) = iπTW\n2eρF∞/summationdisplay\nω=−∞/integraldisplaydN+dF\ndNdzG2\nF\n˜ω2\n×/bracketleftbigg\nΦF(ω)∂\n∂xΦ∗\nF(−ω)−Φ∗\nF(−ω)∂\n∂xΦF(ω)/bracketrightbigg\nx=0. (15)\nThe width Wof the junction along the yaxis is supposed to be small compared to the\nJosephson penetration depth. We have chosen the position x= 0 for the integration\nover the junction cross section since the zcomponent of the current densities vanishes\nthere because of the symmetry of the considered junction geome try.\n3. Currents\nIn order to calculate the current I(φ) from (13) we cannot simply add the current\nthrough the N layer calculated by Bakurskiy et al. [ 18] to the SFS current calculated\nby Buzdin et al. [ 25] because we have to take into account a composite SNFS current\nwhich appears due to a penetration of superconductivity from the N layer into the F\nlayer. Therefore, we split the current IF(φ) into a contribution IF,dir(φ) due to a direct\npenetration of superconductivity into the F layer and the additiona l partINF(φ). This\nleads us to\nI(φ) =IN(φ)+IF,dir(φ)+INF(φ). (16)\nIn the following three sections we derive the expressions of these t hree currents using\nthe scaling\n/tildewideIj(φ) =I(φ)eρj\nW∆. (17)Ferromagnetic planar Josephson junction with transparent interfaces 6\n3.1. Current in the N layer\nIn this layer we adopt the current\n/tildewideIN(φ) = 2dNT\nξNTc/summationdisplay\nω>0Γ(φ)\nµ(φ)rsin(φ) (18)\nwith the deﬁnitions\nΓ(φ) =rδ√γBMΩ+GS/radicalBig\n2γBMΩ(√\nΩ2+δ2r2+µ(φ)), (19)\nδ=∆\nπTc, γBM=γBSNdN\nξN,Ω =ω\nπTc, (20)\nr=/parenleftbiggγBM\nπTc√\nω2+∆2+1/parenrightbigg−1\n, (21)\nµ(φ) =/radicalbig\nΩ2+r2δ2cos2(φ/2) (22)\nfrom [18]. Its derivation is based on the assumption L≪ξNand an inﬁnitely long N\nlayer. It is calculated with the help of the solution Φ N(x) (A9) of the non-linear Usadel\nequations which depends only on the coordinate xbecause the thickness dN≪ξNis\nassumed to be small.\n3.2. Current in the F layer\nThe current\nIF,dir(φ) =√\n264dFκe−2κLFsin/parenleftBig\n2κL+π\n4/parenrightBig\nsinφ, (23)\nwith\nκ=√\nh√\n2ξF, h=H\nπTc,F=πT/summationdisplay\nω>0Θ2\n∆, (24)\nΘ =∆\nη+|ω|+/radicalbig\n2η(η+|ω|), η=√\nω2+∆2, (25)\nis a result of [ 25]. It also has been calculated with the help of a solution of the non-linea r\nUsadel equations because γBSF= 0 is assumed. Additionally the condition ξF≪Lis\nrequired.\n3.3. Composite NF current\nWedeterminethecurrent INF(φ)bycombining thetwonon-linearsolutionsΦ F,dir(x)and\nΦN(x)of(A6)and(A9)inAppendix A .Themainideaistodecomposetheferromagnetic\nGreen’s function\nΦF(x,z) = ΦF,dir(x)+ΦNF(x,z) (26)\ninto a function Φ F,dir(x), which corresponds to currents only ﬂowing in the F layer, and\na function Φ NF(x,z), which corresponds to currents ﬂowing through the N layer into t heFerromagnetic planar Josephson junction with transparent interfaces 7\nF layer. The second function is obtained by linearising the Usadel equ ations (3) in the\nF layer. Then we connect it to the N layer solution Φ N(x) via the boundary conditions.\nThe superposition ( 26) of the solution Φ F,dirof the non-linear Usadel equation with\nthe solution Φ NFof the linearised Usadel equation is valid because we distinguish in the\nF part between two cases: (i) at x≈ ±L/2 near the boundaries to the S regions the\nGreen’s function Φ F,diris very dominant |ΦF,dir| ≫ |ΦNF|due to a transparent boundary\nbetween the S and the F part, that is γBSF= 0; (ii) at x≈0, that is away from the\nboundaries the contribution of Φ Fdecays exponentially. Therefore, the contribution\nfrom the N part is dominant |ΦNF| ≫ |ΦF,dir|.\nAs a result ( A12) we obtain the current\n/tildewideINF(φ) =16cos(φ/2)ξF\nγBNFh∆ξNe−κL/2\n×/bracketleftbigg\nsinκL\n2+κL√\n2e−κL/2cos/parenleftBig\nκL+π\n4/parenrightBig/bracketrightbigg\n×2πT/summationdisplay\nω>0ΘΓ(φ)sinφ\n2, (27)\nwith the deﬁnitions of Γ( φ) from (19),κfrom (24) and Θ together with ηfrom (25).\n4. Discussion\nIn this section we estimate the geometrical parameters dN,dFandL, see ﬁgure 1, for\nwhich the considered Josephson junction obeys the ϕjunction conditions ( 2). We use\nthe analysing scheme of [ 18] and ﬁnally compare our results with the ones obtained\nin [18].\nWe split the sine series amplitudes\nA=AN+AF,dir+ANF, (28)\nB=BN+BNF (29)\nof thetotal current ( 16), scaled according to( 17), into partsoriginating fromthe current\nof the N layer ( 18), the F layer ( 23) and the composite NF current ( 27). There is no\namplitude BF,dirbecause we have a pure sinusoidal CPR ( 23) in the F layer.\nIn our calculations we chose the temperature T= 0.1Tc. We make this choice\nbecause far away from the critical temperature the CPR has large r deviations from the\nsinφform [22] which results in a larger second harmonic B. As S electrode material we\nchose Nb with Tc= 9.2K because it is commonly used in superconducting circuits.\nOur ﬁrst step is to ﬁnd suitable parameters dF. For this purpose we analyse the\namplitudes ( 28) and (29) as a function of Lfor diﬀerent values of dFfor the same\nparameters as in [ 18]:dN= 0.64ξN,ξF= 0.1ξN,H= 10Tc, ∆ = 1.76Tc,ρF=ρN=ρ\nandγBNF= 1. Figure 2shows threetypical examples: (a) dF= 0.15ξN, (b)dF= 0.31ξN\nand (c)dF= 0.35ξN. The ﬁrst (a) and last (c) examples correspond to limiting cases\nwhere it is diﬃcult to realize a ϕjunction because the intervals of Lwhere conditions ( 2)Ferromagnetic planar Josephson junction with transparent interfaces 8\nhold are not large. These intervals of Lare highlighted by bold lines. In between the\ntwo limiting values for dFthis line becomes longer. Figure 2(b) shows an optimum\nsituation because there is a wide range of Lwhich yields a ϕjunction conﬁguration.\n0.1 0.2 0.310−210−1100 dN=0.64 ξN,\ndF=0.15 ξN\nLCPR-Amplitudes ×eρ\nW∆(a)\n  \n|A|/2\n|B|\n0.1 0.2 0.3dN=0.64 ξN,\ndF=0.31 ξN\nL(b)\n0.1 0.2 0.3dN=0.64 ξN,\ndF=0.35 ξN\nL(c)\nFigure 2. The functions |A|/2 and|B|, based on ( 28) and (29), as functions of Lfor\ndN= 0.64ξNand three characteristic values of dF. The bold lines correspond to values\nofLwhere the conditions ( 2) for the ϕjunction realization are fulﬁlled.\nFor the optimum value dF= 0.31ξNwe calculate the magnitudes\nΥA=ANW∆/(eρ) = 0.534 and Υ B=BNW∆/(eρ) =−0.106. Inserting them together\nwith the amplitude AF,dirfrom (23) into (2) and neglecting the small NF contributions\nleads us to the condition/vextendsingle/vextendsingle/vextendsingle/vextendsingleΥA+1\nεΨ(L)/vextendsingle/vextendsingle/vextendsingle/vextendsingle<2|ΥB|. (30)\nHere, we use the constant ε=ξF/(64F√\nhdF) withdF= 0.31ξN,F= 0.0691 and\nΨ(L) = exp(−2κL)sin(2κL+π/4). (31)\nFrom (30) we ﬁnd the minimum value 0 .10ξNand maximum values 0 .17ξNofL.\nFor summarising our suggestion of the geometrical conﬁguration o f aϕjunction we\nusethevalue ξN= 100nmforCuasNlayer, astronglydilutedferromagnetsuchasFe Pd\northeCuNialloywith ξN= 10nmand H= 10TcasFmetal. Oursetofparametersthen\nbecomedN/greaterorsimilar50nm, 15nm /lessorsimilardF/lessorsimilar35nm and 10nm /lessorsimilarL/lessorsimilar17nm, which we compare\nto the values dN/greaterorsimilar50nm, 19nm /lessorsimilardF/lessorsimilar48nm and 7nm /lessorsimilarL/lessorsimilar22nm of [ 29].\nSince we use the same N layer conﬁguration, the value for dNis the same. But the\nsuggested regime for dFdiﬀers. A change in this direction was expected because we only\nneed a thin F layer since the transparency of our interfaces alread y ampliﬁes our SFS\ncurrent contribution. The possible range for the length Lof the F part is smaller in our\ncase but the whole junction conﬁguration is still experimentally feas ible.\n5. Conclusion\nWe have shown that the considered Josephson junction with a ferr omagnetic weak link\nlocated on a thin normal metal ﬁlm is a good candidate for a ϕjunction realisation. ByFerromagnetic planar Josephson junction with transparent interfaces 9\nchoosing transparent SF interfaces we obtained slightly diﬀerent s ystem sizes for the ϕ\njunction existence compared to a junction with weakly transparen t interfaces.\nThe current was split into a contribution through the N layer, the F la yer and a\ncomposite term which described the current ﬂowing through the N a nd F parts of the\njunction simultaneously. We performed our calculations in the “dirty ” limit, that is, the\ncurrents are obtained from solutions of the non-linear Usadel equ ations.\nSince our case of a large interface transparency corresponds be tter to the\nexperimental situation[ 23,24,19]thanweakly transparent interfaces [ 29] itisimportant\nto note that a smaller thickness and length of the F layer have to be c hosen than\npredicted in [ 29]. We are looking forward to experiments realising this ϕjunction and\nits application in classical and quantum devices.\nAcknowledgments\nWethank SVBakurskiy forfruitful andstimulating discussions. DMH thanks Professor\nWPSchleich andK Vogelfor giving himthepossibility to workattheM. V. Lomonosov\nMoscow State University. Financial support by the DFG (Project N o. SFB/TRR-21),\nthe Russian Foundation for Basic Researches (RFBR grants No. 11 -02-12065-oﬁ-m, 13-\n02-01452-a)and the Ministry of Education and Science of the Russ ian Federation (grant\n8641) is gratefully acknowledged.\n*\nAppendix A. NF current derivation\nIn this appendix we derive the current ( 27) which ﬂows through the N and F part of the\njunction, sketched in ﬁgure 1, simultaneously. We ﬁrst linearise the Usadel equations ( 3)\nand then combine the solution with the Green’s functions from [ 18] and [25].\nFor the linearisation of ( 3) we assume the superconducting correlation coming from\nthe N part into the F part as rather small. Then, the Green’s functio n ΦNF(x,z) can\nalso be assumed to be small. Using GNF= sign(ω) we obtain the linearised Usadel\nequation [ 21]\nξ2\nF/parenleftbigg∂2\n∂x2+∂2\n∂z2/parenrightbigg\nΦNF=/tildewideΩ ΦNF, (A1)\nwith the deﬁnitions\n/tildewideΩ =|Ω|+i sign(Ω) h,Ω =ω\nπTc, h=H\nπTc. (A2)\nIts solution in the form of a series\nΦNF(x,z) =∞/summationdisplay\nn=1bnsin/parenleftbigg2π\nLnx/parenrightbigg\ncosh[κn(z−dN−dF)], (A3)\nwith\nκ2\nn=/parenleftbigg2πn\nL/parenrightbigg2\n+/tildewideΩ\nξ2\nF(A4)Ferromagnetic planar Josephson junction with transparent interfaces 10\nand a Fourier coeﬃcient bn, already obeys the boundary condition ( 8) at the upper\nborder (z=dN+dF).\nTheboundaryconditionsattheleftandrightendoftheFpartat x=±L/2arealso\nalready fulﬁlled. They follow from ( 10) withγBSF= 0. Using here the deﬁnition ( 26) of\nΦFleads us to the condition\nΦF,dir(±L/2)+Φ NF=/tildewideΩ\nΩΦS(±L/2). (A5)\nThis equation is already fulﬁlled by the solution\nΦF,dir(x) =/tildewideΩ\nGF,dir/parenleftbig\ne−iφ/2sinα−+e+iφ/2sinα+/parenrightbig\n(A6)\nwith\nα±= 4arctan/bracketleftbigg\nΘexp/parenleftbigg\n±/radicalbig\n/tildewideΩx∓L/2\nξF/parenrightbigg/bracketrightbigg\n, (A7)\nΘ =∆\nη+|ω|+/radicalbig\n2η(η+|ω|), (A8)\nfrom[25]alone. Therefore, theNFGreen’sfunction( A3)onlyhastoobeytheconditions\nΦNF= 0 atx=±L/2. Note that we do not need the expression for GF,dirto ﬁnally\ncalculate the current.\nIn order to obtain the Fourier coeﬃcient bnwhich ﬁxes the solution Φ NFfrom (A3)\nwe use the boundary condition ( 11) where we neglect the term Φ NFassuming |ΦNF| ≪\n|ΦN|. Now, we replace the Green’s function GFbyGNF= sign(ω) and insert\nΦN(x) =r∆cosφ\n2+2iµ(φ)Γ(φ)sin/parenleftbiggφ\n2/parenrightbiggx\nξN, GN=Ω\nµ(φ), (A9)\nfrom [18], where we use the deﬁnitions ( 19), (21) and (22) for Γ(φ),randµ(φ),\nrespectively. By neglecting the real part of Φ Nwe obtain the Fourier coeﬃcient\nbn=2i/tildewideΩL(−1)nΓ(φ)sin(φ/2)\nκnsinh(κndF)ξFγBNFπnξN. (A10)\nOur last step is to calculate the current INF. Therefore, we insert the Green’s\nfunction ( 26), which contains the Green’s functions from ( A3) and (A6), into the\ndeﬁnition ( 15) of the F layer current. Due to the condition x= 0 it reduces to a\nsumIF(φ) =INF(φ)+IF,dir(φ), where the NF current is deﬁned by\nINF(φ) = iπTW\n2eρF∞/summationdisplay\nω=−∞/integraldisplaydN+dF\ndNdzGNFGF,dir\n˜ω2\n×/bracketleftbigg\nΦF,dir(ω)∂\n∂xΦ∗\nNF(−ω)−Φ∗\nF,dir(−ω)∂\n∂xΦNF(ω)/bracketrightbigg\nx=0(A11)\nandIF,dir(φ) is the current ﬂowing only through the F layer [ 25] summarized in ( 23).\nWe insert the Green’s functions Φ NFand Φ F,dirfrom (A3) and (A6) into (A11).\nThen, by ﬁnally using the the approximation /tildewideΩ≈ih, which holds for the conditionFerromagnetic planar Josephson junction with transparent interfaces 11\nπTc≪H, we obtain the scaled current\n/tildewideINF(φ) =16cos(φ/2)ξF\nγBNFh∆ξNe−κL/2\n×/bracketleftbigg\nsinκL\n2+κL√\n2e−κL/2cos/parenleftBig\nκL+π\n4/parenrightBig/bracketrightbigg\n×2πT/summationdisplay\nω>0ΘΓ(φ)sinφ\n2. (A12)\nReferences\n[1] Mints R G 1998 Phys. Rev. B 57R3221–R3224\n[2] Buzdin A and Koshelev A E 2003 Phys. Rev. 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Phys. 77935–977\n[22] Likharev K K 1979 Rev. Mod. Phys. 51101–159\n[23] Oboznov V A, Bol’ginov V V, Feofanov A K, Ryazanov V V and Buzdin A I 2006 Phys. Rev. Lett.\n96197003\n[24] Bannykh A A, Pfeiﬀer J, Stolyarov V S, Batov I E, Ryazanov V V a nd Weides M 2009 Phys. Rev.\nB79054501Ferromagnetic planar Josephson junction with transparent interfaces 12\n[25] Buzdin A I and Kupriyanov M Y 1991 Pis’ma Zh. Eksp. Teor. Phys. 53308–312[1991 JETP Lett.\n53321–326]\n[26] Usadel K D 1970 Phys. Rev. Lett. 25507–509\n[27] Kuprianov M Y and Lukichev V F 1988 Zh. Eksp. Teor. Fiz. 94139–149 [1988 Sov. Phys. JETP\n671163–1168]\n[28] Koshina E A and Krivoruchko V N 2000 Low Temp. Phys. 26115–120\n[29] The estimations for the thickness dFof an RTO ϕjunction with weakly transparent SF interfaces\nare taken from [ 18] and divided by a missing factor π."    },    {        "title": "2303.11604v1.Unique_Itinerant_Ferromagnetism_in_4d_electron_System_Ca2RuO4.pdf",        "content": "Unique Itinerant Ferromagnetism in 4 d-electron \nSystem Ca 2RuO 4 \n \nFumihiko NAKAMURA1*, Hiroki OGURA1, Tatsuhiro SAKAMI1, and Takashi \nSUZUKI2 \n \n1Kurume Institute of Technology, 2228 -66 Kamitsu, Kurume 830 -0052, Japan  \n2Department of Quantum Matter, AdSE, Hiroshima University, Higashi -Hiroshima 739 -8530, \nJapan  \n \n*E-mail: fumihiko@kurume -it.ac.jp   \n(Received July 15 , 2022) \n \nWe have studied the magnetic properties of pressure-induced ferromagnet Ca 2RuO4 to reveal \nthe uniqueness of the 4 d-electron ferromagnetism in the quasi-two-dimensional conductor. \nThe magnetic parameters have been estimated from the paramagnetic susceptibility and the \nmagnetisation process under pressure up to ~2 GPa. The parameters can well be interpreted \non the basis of the self-consistent renormalization theory of spin fluctuation for “three-\ndimensional” itinerant ferromagnet. Nevertheless, the metallic Ca2RuO4 shows quite strong \nanisotropy not only in the conductivity but also in the magnetisation process. Such the strong \nanisotropy is rare for an itinerant ferromagnet and is a unique characteristic of the 4 d electron \nsystem Ca2RuO4.  \nKEYWORDS:   Mott transition, Itinerant ferromagnet, 4 d electron, Pressure study, \nMagnetisation process, Arrott's plot,  \n \n \n1.  Introduction  \n \nRecently, the \"quantum phase transition\" has been proposed as a key concept for the \nunconventional superconductivity in the vicinity of magnetic ordered states [1,  2]. In \nparticular, much attention has recently been gained on the unconventional super-\nconductivity in the vicinity of a ferromagnetic (FM) state while it is generally known \nferromagnetism (FM) plays as a competitive factor for superconductivity. Moreover, w e \ncan fully expect unconventional superconductivity due to FM fluctuations from analogy \nwith the triplet pairing in the superfluid 3He [3].  \nAnother interest is a role of two-dimensional (2D) electronic state in a FM magnetic \nordering. Although low dimensionality is generally known as a destructive factor for \nlong-range order , exotic quantum phenomena have often been found in quasi two-\ndimensional (Q2D) systems such as high- Tc cuprates. As seen in previous report , SC-Tc \nin cuprate superconductors can be risen by releasing frustration due to the 2D electronic \nstates [4]. Now, our concern has been focused on h ow about ferromagnetism.  There is, \nhowever, a great lack of experimental studies of the itinerant ferromagnetism in a Q2D \nmetal. To our know ledge, the pressurised Ca 2RuO4 (CRO) is one of the most suitable systems to investigate the relation between itinerant FM and unconventional super-\nconductivity in a quasi-two-dimensional (Q2D) metal.  \nA single-layered ruthenate (Ru4+-4d4) CRO is a Mott insulator and a compound \nisostructural with the exotic superconductor Sr2RuO4 [5]. The pressurised CRO is a rare \nsystem that the FM order actually occurs in a Q2D metal with a strongly anisotropic \nconduction [6 ]. Application of pressure ( P) to CRO induces versatile quantum \nphenomena, ranging from the insulator to a superconduct or (above ~9 GPa with \nmaximum SC-Tc ~0.4 K) via a Q2D metal with a FM ground state (in the P range from \n0.5 to 8 GPa with maximum FM-TC ~25 K) [6, 7].  The FM in the pressurised CRO is, \nthus, attractive system; however, the quantitative characteristic of the itinerant FM \nremains unknown so far.  \nI\nn this report, w e mention the magnetisation process and the paramagnetic suscepti-\nbil\nity of a Q2D metal of CRO under P up to ~2 GPa in order to grasp the FM \nparameters such as the anisotropy and the itinerancy. In particular, the itinerant FM \nparameters are quantitatively evaluated on the basis of the self-consistent \nrenormalization (SCR) theory of spin fluctuation for itinerant ferromagnet [8 ].  \n \n2. Experiments \n \nIn order to evaluate the FM parameters in pressurised CRO, temperature ( T) and \nmagnetic field ( 0H) dependences of the magnetisation ( M) under P up to ~2 GPa were \nmeasured by using our developed \ncell [9] equipped with a commercial \nS\nQUID magnetometer (Quantum \nDesign, model MPMS). We used the \nDaphne oil 7243 (Idemitsu Kosan \nCo., Ltd.) as a pressure transmitting \nmedium. Pressure at low \ntemperatures was estimated by the \nsuppression of SC-Tc of tin loaded \nwithin the P cell. Our measurement \nwas performed by using several \npieces of single-crystalline CRO \n(total masses ~20 mg). Our crystals \ngrown by a floating-zone method are \nof single phase of K 2NiF4 structure \nwith the c axis of 11.915 Å. Thus, \nour single crystals are high-purit y \nwith essentially stoichiometric \noxygen content.  \nFigure 1(a) shows the comparison \namong the magnetisation curves at 2 \nK and 1.8 GPa as a function of the \nfields along the a, b and c axes up to \n0H =5.5 T. It can clearly be seen \nthat the P-induced FM in CRO is  \nFig. 1. Magnetic field variations of (a) magnetisation \ncurves in the fields along the a, b and c and (b) \ntransverse magnetoresistance in the fields along the \nab and the c axes. Linearly extrapolated M||a, M|| b and \nM||c cross at the field of ~9.5 T. They were measured \nat 2 K and ~2 GPa.     \n \n \n \n \n \n \n \n \n \n0 5 100.00.51.00.00.20.40.6\n(b)(a)\nCa2RuO4\n1.8 GPa\n1.9 GPacbaM (B/Ru-ion)\n  ab/ab(0)\n 0H (T)0H||ab0H||c\n  strongly anisotropic. We note here that the a axis is the easy direction of the mag-\nnetisation and the c axis is the hard direction.  Moreover, we deduce that application of \n0H||c above the anisotropy field ( 0HA) forces the direction of the spin orientation from \nthe a to the c axis. Indeed, the linearly extrapolated magnetisations of M||a, M||b and M||c \ncross at the field of ~9.5 T, at which the transverse magnetoresistance in 0H||c peaks as \nshown in Fig. 1 (b) (The magnetoresistance in 0H||c is partly taken from Ref. 10 ). From \nthe extrapolation, w e obtain the anisotropy field to  be 0HA ~9.5 T. \nIn order to evaluate the itinerant FM parameters, it is important to measure the \nmagnetisation in the field along the easy direction (namely, the a-axis). Figure. 2 (a) and \n(b) shows the remnant FM moment ( Mrem)  at 2 K and the effective paramagnetic (PM) \nmoment ( peff) as a function of P, respectively. The Mrem was obtained from the magneti-\nsation curve in the field of 0H||a at 2 K. The FM moment was induced by pressurising \nover 0.5 GPa. The value of Mrem increases rapidly from ~0.1  μB/Ru-ion at 0.6 GPa but \nsaturates toward ~0.44 μB/Ru-ion pressuring over ~1.7 GPa. The inset of (b) repre-\nsentatively shows inverse PM susceptibility ( 0H||a/M) at 2.0 GPa as a function of T. We \nobserved a linear relation obeying the Curie-Weiss law M/0H||a= C/(T–p). We \nestimated the Curie temperature p ~ +10 K and the effective PM moment peff ~1.85 μB. \nMoreover, these parameters are almost constant in the P range above  1.5 GPa. \nNext, we mention the itinerancy of the P-induced FM in CRO, focusing on  the \nfollowing four viewp oints: First, the FM magnetisation linearly increases without satu-\nration even in strong fields. Second,  Mrem~0.44 μB/Ru-ion is much smaller than the satu-\nrated moment of 2 μB/Ru-ion as a \nlocalised system of S=1. Third, \nalthough a Curie-Weiss type \nsusceptibility has been observed \nabove FM- TC ~ +10 K, the value of \npeff ~1.85  μB/Ru-ion is quite smaller \nthan 2.73 μB estimated as a localised \nsystem of S=1. Last, a ratio of \npeff/Mrem ~4.2 is much larger than 1 \nof localised spins systems. It can, \ntherefore, be seen that the P-induced \nFM nature in CRO is quantitatively \ninterpreted in terms of an itinerant \nFM.  \nMoreover,  we have observed \nthat the c-axis susceptibility also \nobeys the Curie-Weiss law with the \nparameters of p ~ −15 K and peff \n~2.1 μB at 2 GPa.  Thus, the PM \nsusceptibility is quantitatively \nanisotropic between the a and the c \naxes.  \nIn order to quantitatively \nevaluate the itinerant parameters, the \nmagnetisation process has been \ninterpreted on the basis of the SCR  \nFig. 2. Pressure variation of the FM parameters \nestimated from the magnetisation in the pressurised \nCRO. (a) Remnant magnetisation  at 2 K. (b) Effective \nPM moment estimated from a Curie constant. Inverse \nsusceptibility at 2.0 GPa is representatively plotted as a \nfunction of temperature in inset (b).   \ntheory for three-dimensional (3D) \nspin fluctuations. We, in itially, refer \nto the Arrott's plot criterion [8 ]. \nThat is, the square of M at 2.0 GPa \nfor several-fixed T is plotted as a \nfunction of 0H/M as shown in Fig. \n3(a). The magnetisation process at \nthe lowest temperature of 2 K \n(namely ground state) shows good \nlinearity in the Arrott's plot. From \nits slope, we have estimated two \nenergy scales of T0 ~80 K and TA \n~550 K, where T0 and TA \ncharacterize the spectral dist ribution \nof the spin-fluctuation spectrum in \nthe frequency and wave-vector \nspaces, respectively.  \nA number of experimental and \ntheoretical works indicate that a \ntypical magnetisation process is \ncharacterised by a linear relation in \nthe Arrott's plot, especially the \nproportional relation at FM- TC. The pressurised CRO, indeed, shows the good linear \nrelation at 2 K. However, the magnetisation process in the vicinity of FM- TC shows a \nproportional relation not in the  M 2–0H/M plot but in the M 4 –0H/M plot, as shown in \nFig. 3(b). Similar behaviour has been reported in the system with the specific-range \nparameters of peff/Mrem and TC/T0 [8]. This is due to that thermal spin fluctuations cannot \nbe neglect at the critical temperature in such the system. We expect that itinerancy of the \nP-induced FM in CRO is close to that in MnSi [8].       \nLet us compare the itinerancy of the FM in CRO against other ferromagnets \nreported so far. The obtained parameters are additionally plotted in the generalized \nRhodes-Wohlfarth plot for itinerant ferromagnets [8 , 11, 12] as shown in Fig. 4 , where \npeff/Mrem is plotted as a function of TC/T0. The itinerant parameters of the FM in CRO in \nthe P range from  1.0 to 2.0 GPa are well located on the theoretical curve (see also in the \ninset). It can be seen that the pressurised CRO is an itinerant ferromagnet. Moreover, \nitinerancy of the CRO is very close to that of the above-mentioned system “MnSi”.   \nThus, itinerant nature of the P-induced FM in CRO can well be interpreted on the \nbasis of the SCR theory for “three-dimensional” (3D) spin fluctuations although the \nconduction shows essentially 2D metal.  Now, we consider how about role of “2D” in \nitinerant ferromagnet of the pressurised CRO. As shown in our previous report, the in-\nplane resistance shows metallic behaviour whereas the c-axis resistance has a negative \nslope indicating non-metallic conduction  [4]. That is, the system should be interpreted \nas not a 3D metal with strong anisotropy but an essentially 2D metal. On the other hand, \nwe indicate the FM parameters in pressurised CRO can be explained by the SCR theory \nfor a “3D” spin system rather than \"2D\" one. It may be difficult to realize a 2D spin sys-\ntem comparing with theoretical predictions even in an ideal 2D conductor such as \npressurised CRO. This might be because the ferromagnetic interlayer interaction is hard  \nFig. 3. Representative  magnetisation curves at 2.0 \nGPa analysed by the Arrott -plot criterion.  (a) The \nsquare of the magnetisation (from Ref. 12) is plotted \nas a function of  0H/M. (b) The fourth power of the \nmagnetisation TC ~10 K is replotted as a function of  \n0H/M.  \n0 5 100.000.050.100.150.200.250.30\n(b)12K\n13K10K\n8K\n2K\n  M 2 (B/Ru-ion)2\n0H/M[ T/ (B/Ru-ion)]0 5 100.000.010.020.030.040.050.060.07\n(a)10K\nM 4 (B/Ru-ion)4 \n0H/M[ T/ (B/Ru-ion)]to suppress even if the conduction \namong RuO 2 planes can \nelectrically be disturbed.  \nMove on last topic of the FM \nanisotropy in the pressurised CRO, \nnamely a role of spin-orbit \ncoupling in the magnetism. As \nshown in previous reports, spin-\norbit coupling is a key to \nunderstand the attractive \nphenomena,  such as the orbital \nordering [1 3, 14], the giant \nmagnetoresistance [10] and  the \ninsulator-metal transition accom-\npanied by a bulk structural tran-\nsition [15-17 ]. We note here that \nFM anisotropy is known as an \nindicator of the strongness of spin-\norbit coupling. Nevertheless, there \nhave been few discussions of a role \nof spin-orbit coupling in a FM \nordering. This is because many of \nstudies for itinerant FM have been \nconcentrated to a 3 d-electron metallic system where the orbital angular momentum is \nquenched, and then there have been few reports on a role of spin-orbital coupling in a d-\nelectron FM system. It is actually known that the FM anisotropy is quite small in many \nof 3d-electron systems, except for the cobalt metal, where the orbital quenching is most \nlikely incomplete [18, 19].  \nOn the other hand,  strong anisotropy has been reported in some uranium FMs (100-\n1000 time larger than in 3 d FMs) [20 ]. For example, the itinerancy of uranium \ncompounds URhSi and URhGe is comparable to that of the pressurised CRO although \ntheir FM anisotropy is quite stronger than that of the pressurised CRO. Moreover,  the \nmagnetic and conductive properties in uranium compounds are mainly governed by 5 f \nelectrons having intermediate natures between localized 4 f and itinerant 3 d electrons. \nWe note that strong spin-orbit coupling is a key character of 5 f electrons because the \norbital angular momentum of 5 f electrons is not quenched.  \nAs mentioned above, the P-induced FM in 4 d-electron system CRO is characterised \nby strong anisotropy as a d-electron FM. The anisotropic energy of EA~2.1×105 J/m3 \nwas obtained by using the experimental values of 0HA ~9.5 T and Mrem ~0.44 μB/Ru-\nion [19]. This result indicates that quenching the orbital angular momentum in 4 d \nelectron system of CRO is not completely.  \nThis deduction is also indicated by some phenomena such as giant magneto-\nresistance [10 ], the orbital-selective Mott transition [2 1, 22], orbital ordering [1 3, 14], \nand the Mott insulator-metal switching induced by a quite small electric-field [2 3]. All \nof them strongly suggest that the FM properties in CRO are qui te sensitive to strong \nspin-orbit coupling. Namely, many of interesting phenomena in CRO can be interpreted \nin terms that the orbital angular momentum is not entirely quenching in 4 d electron.  \nFig. 4. The itinerant FM parameters of pressurised  CRO \nare plotted in the generalized Rhodes -Wohlfarth plot \nafter Ref. 8, 11 and 12.  \n \n3.  Conclusion  \n \nWe have demonstrated the uniqueness of the P-induced FM in CRO. First, the \nitinerancy of the FM is comparable to that in the typical 3D itinerant-ferromagnet MnSi. \nSecond, it is well interpreted in terms of the SCR theory for a \"3D\" spin system \nalthough the pressurised CRO is an essentially 2D metal in electric conduction. Last, we \nhave observed quite strongly anisotropy in the itinerant FM as a d-electron system. Such \na strong anisotropy in FM is difficult to understand regardless of considering the strong \nspin-orbit coupling. We, thus, expect to build up a theory of itinerant FM including the \nstrong spin-orbit coupling. \n \nAcknowledgment  \n \nWe acknowledge T. Takemoto, R. Nakai and Y. Kimura for their experimental helps. \nA part of this work was supported by Grant- in-Aid for Scientific Research (Grant Nos. \n26247060, 26287083, 17H06136, 22K03485 and 22H01166) by  JSPS. \n \nReferences  \n \n[1] For a review, see G. G. Lonzarich , Nature Physics 1, 11 - 12 (2005); P. Monthoux and G.G. \nLonzarich. Phys. Rev. B 63, 054529 (2001).  \n[2] S. S. Saxena et al., Nature 406, 587 (2000).  \n[3] A. J. Leggett, Rev. Mod. Phys.  47, 331 (1975).  \n[4] F. Nakamura, T. Goko, J. Hori, Y. Uno, N. Kikugawa, and T. Fujita, Phys. Rev. B 61, 107 (2000).  \n[5] S. Nakatsuji and Y. Maeno, J. Phys. Soc. Jpn.  66, 1868 (1997). \n[6] F. Nakamura  et al.,  Phys. Rev. B 65, 220402R (2002). \n[7] P. L. Alireza  et al., Journal of Physics: Condensed Matter, 22, 052202 (2010). \n[8] Y. Takahashi, J. Phys. Soc. Jpn. 55, 3553 (1986). \n[9] R. Nakai, F. Nakamura and T. Suzuki, J. Phys. Soc. Jpn. 76 (2007) Suppl.A, 219.  \n[10] F. Nakamura et al., Phys. Rev. B 80, 193103 (2009). \n[11] S. Ikeda, Y. Maeno, T. Fujita, Phys. Rev. B  57, 978 (1998). \n[12] Y. Yamauchi  et al., Physica C 470, S740 (2010).  \n[13] M. Kubota et al., Phys. Rev. Lett. 95, 26401 (2005). \n[14] I. Zegkinoglou  et al., Phys. Rev. Lett. 95, 136401 (2005). \n[15] C. S. Alexander  et al., Phys. Rev. B 60, R8422 (1999).  \n[16] O. Friedt  et al., Phys. Rev. B 63, 174432 (2001). \n[17] P. Steffens  et al., Phys. Rev. B 72  094104 (2005). \n[18] For example, see J. Stohr, J. Mag. Mag. Mater. 200, 470 (1999). \n[19] For detailed estimation, see K. Ohta: Jikikougaku no Kiso (II)-Jiki no Butsuri (Kyoritsu Shuppan, \nTokyo, 1973) in Japanese. \n[20] Naoyuki Tateiwa,  Yoshinori Haga, and Etsuji Yamamoto, Phys. Rev. B  99, 094417 (2019).  \n[21] V. I. Anisimov  et al., Eur. Phys. J. B 25 (2002) 191.  \n[\n22] J. S. Lee, S. J. Moon, T. W. Noh, S. Nakatsuji, and Y. Maeno, Phys. Rev. Lett. 96, 057401(2006).  \n[23] F. Nakamura, M. Sakaki, Y. Yamanaka, S. Tamaru, T. Suzuki and Y. Maeno, Sci. Rep. 3, 2536 (2013) .  \n \n "    },    {        "title": "1404.5456v1.Superconductivity_induced_Magnetic_Modulation_in_a_Ferromagnet_Through_an_Insulator_in_LaCaMnO3_SrTiO3_YBa2Cu3O7_δ_Hybrid_Heterostructures.pdf",        "content": "1 \n Superconductivity -induced Magnetic Modulation in a Ferromagnet Through  \nan Insulator in La2/3Ca 1/3MnO 3/SrTiO 3/YBa 2Cu 3O7-δ Hybrid Heterostructures  \n \nC. L. Prajapat1, Surendra Singh2, Amitesh  Paul3,  D. Bhattacharya2, M. R. Singh1,                      \nG. Ravikumar1 and S. Basu2,* \n1Technical Physics Division, Bhabha Atomic Research Centre, Mumbai -400085.  \n2Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai -400085.  \n3Technische Universität München , Physik Department, Lehrstuhl für Neutronenstreuung, James -\nFranck -Straße 1,D-85748 Garching, Germany  \n*email: sbasu@barc.gov.in  \n \n \nCoexistence of ferromagnetic and superconducting orders and their interplay in \nferromagnet -superconductor heterostructures1-5 is a topic of intense research. While it is \nwell known that proximity of a ferromagnet suppresses superconducting order in the \nsuperconductor, there exist few studies indicating the proximity of a superconductor \nsuppressing ferromagnetic order in a ferromagnet.6-11 Here we demonstrate a rare \nobservation of the suppression of ferromagnetic order in a La2/3Ca1/3MnO 3 layer separated \nfrom a YBa 2Cu 3O7-δ layer by a thin insulator (SrTiO 3). Polarized neutron reflectivity \nmeasurements on La2/3Ca1/3MnO 3/SrTiO 3/YBa 2Cu 3O7-δ trilayer deposited on [ 001] SrTiO 3 \nsingle crystal  substrates  shows  the emergence of a thin magnetic “dead” layer in \nLa2/3Ca1/3MnO 3 adjacent to the insulating layer  below its superconducting transition \ntemperature  of YBa 2Cu 3O7-δ. Further, the magnetic dead layer grows in thickness when the 2 \n insulating layer is made thinner. This indicates a possible tunneling of the superconducting \norder -parameter through the insulating SrTiO 3 inducing modulation of magnetization in \nLa2/3Ca1/3MnO 3.   \nSuperconductors (SC)  and ferromagnetic (FM) heterostructure like SC/FM/SC, are \nknown to exhibit  coupling between the SC  through a thin intervening FM layer producing what \nis known as π -state, essentially due to the presence of SC order -parameter in the FM layer, albeit \nover an extremely short range.12  However , in thin film hetrostructure s of SC cuprates and FM \nmanganites, there are sufficient evidence s which suggest that the SC order persists over much \nlonger length scales extend ing  up to 100 Å,12,13 which  is very intrigu ing by itself . YBa 2Cu3O7-δ \n(YBCO) and La 2/3Ca1/3MnO 3 (LCMO) are ideal candidates for growth of epitaxial thin films on a \nvariety of oxide substrate s like SrTiO 3 (STO), LaAlO 3 and others . Heterostructures of these \nmaterials grown with high interface quality14, 15 are ideal candidates for investigating the \ninteraction between mutually antagonistic SC and FM order ings. The properties of \nLCMO/YBCO heterostructures are strongly influenced by a coupling phenomena at the \ninterface,1-3 which can lead to complex beh aviors in these heterostructures such as giant \nmagnetoresistance4, transient photo -induced superconductivity5 and magnetic proximity effect s.6-\n11 While the competition between the two ordered ground states1-3,16 leads to suppression of both \nsuperconducting  and magnetic transition temperatures ,17, 18 a variety of exotic phenomenon have \nbeen seen in these heterostrures.19  \n Hoppler et al .,9 inferred a giant modulation of the in -plane magnetization in LCMO \nlayers below the superconducting transition of LCMO/YB CO multilayer. It was observed that \nthe magnetization in alternate LCMO layers are strongly suppressed and enhanced, doubling the \nperiodicity of the magnetic lattice. There also exist experimental studies which indicated  the 3 \n depletion of magnetization or a magnetic dead (MD) layer in the adjacent region of LCMO at the  \nLCMO/YBCO interface.6-11 Magnetic dead layers are known to result from chemical inter -\ndiffusion/alloying or interface roughness20-22. However , scattering techniques6-11 and electron \nspectroscopy23 have ruled out these factors in LCMO/YBCO heterostructures. While magnetic \nmodulation in LCMO layers , induced by superconductivity in the adjacent YBCO layer in \nLCMO/YBCO multilayers  have been  extensively studied5-9, to the b est of our knowledge no \nstudy involving superconducting order -parameter influencing a ferromagnet through an \ninsulating (I) barrier in FM/I/SC hetro structures  has been reported so far. Here  we observed \ntunneling effect across an intermediate insulator laye r and found that thickness of the insulating \nlayer play an important role for deciding the behavior of such hybrid system.   \nTwo trilayer  samples  labeled as S1: STO[substrate]/YBCO (300 Å) / STO (25 Å)/ \nLCMO (300 Å)  and S2: STO[substrate]/YBCO (200 Å) / STO (50 Å)/ LCMO (200 Å)  were  \ngrown with  STO (1 00) as substrate by pulse laser deposition (PLD). Using polarized neutron \nreflectivity (PNR), we present direct evidence of magnetic modulation in LCMO layer across \ninsulating S TO layer below superconducting transition temperature ( TSC). PNR data reveals  that \nthe magnetization in LCMO was suppressed to zero (magnetic “dead” layer) near the \nLCMO/STO interface below TSC. The thickness of the magnetic “dead” layer is estimated to be  \nabout 100  Å in sample S1. The magnetic dead layer thickness is reduced to ~ 40 Å in sample S2 \nwhere  the thickness of insulator was increased to  50 Å. This clearly signifies the tunneling of the \nSC order -parameter through an insulator into a FM.   \nThe structural characterization of the samples was done by  using X -ray diffraction (Fig. \nS1 in Supplementary), showing high quality epitaxial growth of the films. X -ray reflectivity \n(XRR) measurements were performed to determine the depth dependent layer structure of the se 4 \n hetrostructures  (Supplementary  Fig. S2 ).  Fig. 1 depicts SQUID data for  the d. c. magnetization \nmeasurements on S1 under  field cooled  (FC) condition  (cooling field HFC = 300 Oe). The  zero \nfield cooled  (ZFC)  SQUID data are shown  in the inset  of Fig.1 . The ZFC data shows TSC ~ 60 K , \nsuggesting the YBCO is under -doped  and FC data shows the LCMO layer has a Curie \ntemperature  ~ 150 K.  Similar behaviors for SQUID data from S2 was also observed.  \nPNR measurements were carried out  to obtain  depth dependent magnetization profile  in \nthe sample s. PNR involves specular reflection of polarized neutron from magnetic film as a \nfunction of wave vector transfer, Q (= 4πsinθ/λ, where, θ is angle of incidence and λ is neutron \nwavelength).22,24,25 ,26 Specular reflection of neutron beam with polarization parallel (+) and anti -\nparallel ( -) to sample magnetization corresponds to reflectivities, R±(Q). We have measured the \nPNR data for S1 and S2 at 10 K, 50 K, 100 K and 300 K , with an applied in -plane field of 300 \nOe after cooling the sample in the same field from 300 K.  Our aim was to obtain the \nmagnetization depth  profile from the fits to these data sets  and look for any possible modulations  \nin magnetization across  TSC.  Fig. 2a shows the PNR measurements from S1 at 300 K and 10  K. \nThe normalized s pin asymmetry  (NSA ) plots at 300 K and 10 K shown in the bottom  panel of \nFig. 2a  is given by (R+ - R-)/R F, where R F = 16π2/Q4 is Fresnel reflectivity .24 At 300 K, R+ and R- \nare same ( NSA  = 0), indicating no net magnetization of the sample at this temperature , consistent \nwith the macroscopic magnetization (SQUID) measurements.  \nIn order to extract the magneti zation profile from the PNR measurements,25 we first \noptimized the n uclear scattering length density (NSLD) profile at 300 K by constraining the layer \nthicknesses , density  and interface roughness to be within the error estimated on  parameters \nobtained from XRR. Keeping  the NSLD fixed, the magnetization depth profile [ M(z)] was \noptimized to fit the PNR data  at lower  temperatures . Since the effect that we  observed is more 5 \n enhanced for thinner intervening insulator (STO) layer, we will first discuss the results obtained \nfrom S1, followed by the results  from S2.  The solid lines in Fig. 2a represent  the fit to \nexperimental data at  300 K  and 10 K , from S1. Top panel of Fig. 2 b shows the NSLD obtained  \nfrom PNR  data at 300 K .   \nThe PNR data from  S1 at 10 K (Fig. 2a) shows a clear separation between R+ and R- \nindicating a ferromagnetic state of the LCMO layer  (contrast with the data at 300 K in the same \nfigure) . To fit the PNR data at 10 K , we have optimized  several models for the magneti zation \ndepth  profile in the LCMO layer and a detailed comparison of the fits corresponding to different \nmodels is  shown in the Supplementary Fig. S3. Uniform magnetization profile (dash blue line in \nthe bottom  panel  of Fig. 2 b) for the entire LCMO layer clearly does not agree  with the NSA  data \nat 10 K as shown by the dash (blue) lines in Fig. 2 a (bottom panel) . The fit  (black and green \ncurves in top panel of Fig. 2 a) to PNR data at 10 K  with minimum χ2 was obtained for a \nmagneti zation  depth profile shown as a solid  (black)  curve in  the bottom  panel of Fig. 2 b, \nindicating a magnetic “dead” layer ( shaded area in bottom  panel of Fig. 2 b) at LCMO/STO \ninterface with a thickness ∆ ~ 100 Å  and a non-uniform magnetization in the rest of the LCMO \nlayer . The magnetization in the rest of the LCMO layer is seen to be gradually increasin g from  \nzero to a maximum value ~205 emu/cm3 near the film-air interface .  \nTo confirm the role of superconductivity on the modulation of the magneti zation depth  \nprofile of LCMO layer in S1 we carried out  PNR  (NSA ) measurements at 100 K (well above the \nTSC) and 50 K ( marginally below TSC~60 K ) as shown in top and bottom panel of Fig.  3a, \nrespectively.  We attempted to fit PNR data from  S1 at 100 K with similar magneti zation depth  \nprofile as obtained at 10 K . The fit is shown in F ig. 3b (top panel)  as solid curve . Also a fit with \nuniform magnetization in the LCMO layer without any dead -layer , is shown in the top panel of 6 \n Fig. 3b as blue dashed line . Comparison of these two fits clearly suggest s that a small but \nuniform magnetization ( dash line) of ~17 emu/cm3 for the entire LCMO  layer  fits the NSA  data \nbetter tha n the model with MD layer (solid curve) . However t he PNR data at 50 K ( bottom panel \nof Fig. 3a and 3b) is consistent with the magnetization  (with reduced magnetization)  model \nobtained at 10 K and fits the PNR data with the same profile as obtained from the PNR data at 10 \nK. A comparison of fitted NSA  data and magnetization depth profile model  at 50 K is given in \nthe bottom  panel of Fig. 3a and b. Overall  decrease  in the magnetization  at 50 K as compared to \n10 K  indicates the  change in magnetization of a ferromagnetic material with  temperature.  These \nresults clearly suggest that the MD layer emerges only below the superconducting transition  \ntemperature  of YBCO (~ 60 K) .   \nFurther, to study the effect of insulator thickness on SC induced magnetization \nmodulati on in LCMO layer, we now focus  on the result of the PNR experiments at 10 K under \nsimilar conditions on S2  (with STO layer thickness 50 Å) . The sam ple was also characterized \nusing XRR for depth dependent scattering length density profile and  the details are given in the \nSupplementary information (Fig. S2). Fig. 4 shows the PNR ( NSA ) data from  S2 at 300 K and \n10 K. At 300  K we did not observe any difference between R+ and R-, indicating that the LCMO \nlayer was non-magnetic  (similar to S1  at 300 K ).  The PNR data at 300 K was analyzed to get a \ndetail ed NSLD profile of the sample ( top panel of Fig. 4 a -b). PNR data at 10 K and a \ncomparison of fit ass uming different magnetization model is shown in Fig. S 5 of supplementary \ninformation. The PNR ( NSA ) data at 10 K ( bottom  panel of Fig. 4 a -b) confirms the modulation \nof magnetization with occurrence of magnetic dead layer at LCMO/STO interface. We obtained  \na magnetic dead layer of thickness ~ 40 Å at the LCMO/STO interface of S2. The magnetization \nprofile in the rest of the LCMO layer was uniform with M = 100 emu/cm3.   7 \n At this point it is relevant to mention that the length scale of the MD layer at LCMO/STO \ninterface is much higher (≈ 40 -100 Å) than the interface roughness (≈ 5 Å) as obtained from the \nXRR and PNR  data. The interface roughness in these samples ranged between 13 Å and 4 Å, the \nhighest being at the air film interface  (13 ± 3 Å) and the lowest at the LCMO on STO interface \n(4 ± 1 Å). This  fact overrules interface mixing as  a possible cause of the observed dead -layer .   \nIn addition, the average magnetization of S1 was estimated around 120 ± 10 emu/cm3 at \n10 K  using  PNR  measuremen ts, which is comparable with earlier measurements7 and well below \nthe saturation magnetization ( Ms) ≈ 400 emu/cm3 observed for single layer LCMO thin films.27 \nThis is also in agreement with the value of 11 2 emu/cm3, obtained from SQUID by considering a \nmagnetic dead layer of thickness ~ 100 Å. It has been reported that magnetization in LCMO \ndepend  on the thickness of YBCO layers.17 We believe  that lower value of TSC, Curie \ntemperature and magnetic moment in this system is an important issue  for observed  results,  \nbecause these  may well provide an additional energy and length scale that must be considered in \ndescribing the competing SC and magnetic interactions.28 \nOur PNR results from LCMO/STO /YBCO  systems  clearly indicate  that the existence of a \nMD layer is related to the superconducting state of YBCO  layer . It is a distinct possibility that \nthe depletion of magnetization in the LCMO layer is caused by the tunneling of SC order -\nparameter into the LCMO layer. Both FM and the SC state s derive their existence from the local \ndensity of states. Possibility of long range coherence length  of Cooper pairs in FM has been \ntheoretically  discussed by Bobkova and Bobkov .29 We argue that the system prepared under field \ncooled condition is in a non -equilibrium state which comprises a nanoscopic phase -coexistence \nof FM , AFM  and charge ordered states and FM domain walls. We suspect  that the LCMO layer \nwhich show coexistence30, 31,32 of different phases mentioned above will provide a spatially 8 \n varying characteristic length scales for SC order -parameter and thus play a fundamental  role for \nobservation of such SC -induced modulation in magnetization. The decay of such \nsuperconducting wave functions  in LCMO layer after tunneling through STO is  depic ted in Fig. \n5a.  This decay of SC order in the FM layer , in our samples, is indicated by the gradual increase  \nof the magnetic moment density profile.  In view of this we fitted  (Fig. 5 b)  the magnetization \ndepth profile of S1  at 10 K obtained from PNR measurements to the expression:        [  \n      \n    , where x is distance of surface (air/LCMO interface) from YBCO/STO interface, ξ \nis coherence length scale, M0 magnetization at surface and α is an exponential coefficient, which \nwill dictate the possible order of length scale  involved in LCMO  layer. From the fitted line \n(Solid line in Fig. 5  b), the exponential ( α) for the system is estimated at around 18:  a \nsurprisingly large value ! This indicates  the presence of a large number of length scales in the \nsystem.  \nIn conclusion, we have shown unambiguously that superconductivity induced  modulation \nin magnetization depth  profile in LCMO layer across  an insulating  STO layer  in two \nLCMO/STO/YBCO hybrid structure s. We observed that a magnetic dead layer formed at the \nLCMO/STO (LCMO on STO) interface below superconducting  transition temperature of YBCO.  \nWe conjecture that this happens probably due to tunneling of superconducting order -parameter in  \nto the FM layer. The length scale for the magnetic dead layer depends on the thickness of the \ninsulator layer. Thinner the insulator layer , thicker was the magnetic dead layer at LCMO/STO \ninterface. We believe  the presence of phase coexistence over many length scales in LCMO layer \nis responsible for superconductivity -induced magnetic modulation in these hetrostructures. Our \nresults open s a way to explore the fundamental study of tunneling of superconducting order \nparameter in FM/I/SC system. Nevertheless  future experiments using advanced local imaging 9 \n techniques in combination of scattering techniques may provide further insight for \nsuperconducting induced phenomena in FM/I/SC systems.  \n \nMethods:  \nTwo trilayer  samples labeled as S1: STO[substrate]/YBCO (300 Å) / STO (25 Å)/ LCMO \n(300 Å)   and S2:  STO[substrate]/YBCO (200 Å) / STO (50 Å)/ LCMO (200 Å)   were grown on \nSTO ( 001) substrate using  pulse  KrF laser (248 nm) laser  deposition ( PLD ). During growth, the \nsubstrate temperature was 770°C, O 2 partial pressure was 0.5 mbar, laser fluence was 2.5 J/cm2, \nand the pulse repetition rate was 2 Hz.  \nMagnetization measurements were performed using superconducting quantum interface \ndevice (SQUID) magnetometry under field cooled (FC) and zero field cooled (ZFC) conditions . \nPolarized neutron reflectivity (PNR) measurements of the samples were carried out using the \npolarized reflectometer MARIA at the FRM II research reactor in Garching , Munich . In PNR the \nintensity of the specularly reflected neutron beam was measured as  a function of wave vector \ntransfer, Q (= 4πsinθ/λ, where, θ is angle of incidence and λ is neutron wavelength), and for \nneutron beam polarization parallel (+) and anti -parallel ( -) to sample magnetization. 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Structure \nand microstructure of La 1-xCaxMnO 3-δ thin films prepared by pulsed laser deposition. Phys. \nRev. B 58, 8065 (1998).  \n32. Dhital, C. et al.  Neutron scattering study of magnetic phase separation in nanocrystalline \nLa5/8Ca3/8MnO 3 Phys. Rev. B  84, 144401 (2011)  \n \n \n \nAcknowledgements:  \nWe are thankful to S. Mattauch for assisting in the PNR measurements and P. Böni for his \nencouragements during the course of the work  \n \n \nAuthor Contributions:  \n \n \nC. L. P., M. R. S. and G. R. grew samples by pulsed laser deposition and carried out SQUID \nmeasu rements. S. S., D. B. and S. B. designed the XRR and neutron scattering experiments as \nwell as analyzed the results. A. P. proposed for the beamtime and carried out the neutron \nreflectivity measurements  and discussed the results .  All the authors contribut ed in writing the \npaper. Authors C.L.P. and S.S.  have equal contributions of the work.  \n \nCompeting Interests:  \nThe authors declare that they have no competing financial interests.  \n \n \n 14 \n Figure Captions:  \n \nFig. 1: Temperature dependent Magnetization data . Magnetization data of the YBa 2Cu3O7-δ \n(300 Å) /SrTiO 3 (25 Å) /La2/3Ca1/3MnO 3 (300 Å) sample in f ield cooled (FC) condition in a field \nof ~300 Oe showing the FM transition ( TC) ~ 150 K.   Inset show the zero field cooled (ZFC) \ndata suggesting a superconducting transition temperature ( TSC) ~ 60 K.  \n \nFig. 2: Polarized neutron reflectivity (PNR) measurements and their modeling. a, PNR (spin \nup, R+ and spin down, R- ) data from the YBa 2Cu3O7-δ (300 Å) /SrTiO 3 (25 Å) /La2/3Ca1/3MnO 3 \n(300 Å) sample  at 300 K  and 10 K , with an applied in -plane field of 300 Oe  after cooling the \nsample in the same field from 300 K.  Reflectivity data at 300 K and 10 K  are shifted by a factor \nof 20 for the sake of clarity. Normalized s pin asymmetry  (NSA ) data, defined a s (R+ - R-)/R F, \nwhere R F = 16π2/Q4 is Fresnel reflectivity , at 300 K and  10 K ( bottom  panel  of a).  b, Nuclear \nscattering length density ( NSLD) and magnetization ( M) depth profile  extracted from fitting \nPNR data at 300 K  and 10 K . Two magnetization models,  with and without magnetic dead (MD) \nlayer at LCMO/STO interface,  at 10 K are  also depicted in  b (bottom panel)  and the \ncorresponding fits to PNR data are shown in a (bottom  panel).  \n  \nFig. 3: Polarized neutron reflectivity (PNR) measurements and their modeling across \nsuperconducting transition temperature. a, normalized  Spin asymmetry  (NSA ) data from the \nYBa 2Cu3O7-δ (300 Å) /SrTiO 3 (25 Å) /La2/3Ca1/3MnO 3 (300 Å) sample  at 100 K ( top panel) and 50 \nK (bottom  panel) , with an applied in -plane field of 300 Oe  after cooling the sample in the same \nfield from 300 K. b, magnetization ( M) depth profile  at 100 K ( top panel) and 50 K ( bottom  15 \n panel) which are fitted to NSA  data shown in a. Comparison of t wo magnetization models at 100 \nK and 50 K  are depicted in b and the corresponding fits to PNR data are shown in a. \n \n \nFig. 4: Polarized neutron reflectivity (PNR) measurements and their modeling from S2 with \ninsulator layer of double thicknes s. a, Normalized  spin asymmetry  (NSA ) data from the \nYBa 2Cu3O7-δ (220 Å) /SrTiO 3 (50 Å) /La2/3Ca1/3MnO 3 (190 Å) sample  (S2) at 300 K ( top panel) \nand 10 K ( bottom  panel) , with an applied in -plane field of 300 Oe  after cooling the sample in the \nsame field from 300 K. . b, Nuclear scattering length density (NSLD) ( top panel)   and \nmagnetization (bottom  panel) depth profile extracted from fitting PNR data at 300 K and 10 K as \nshown in a. Two magnetization mod els at 10 K are depicted in  b (bottom panel)  and the \ncorresponding fits to PNR data are shown in a (bottom  panel ). \n \nFig. 5: Schematic of tunneling of Cooper pair across insulator . a, Schematic showing \nrepresentation of LCMO/STO/YBCO system with phase coexistence of different length in \nLCMO layer whic h is deciding the perturbation of superconducting wave functions tunneled \nthrough STO (insulator).  b, Fitting of magnetization depth profile of YBa 2Cu3O7-δ (300 \nÅ)/SrTiO 3 (25 Å) /La2/3Ca1/3MnO 3 (300 Å) sample  at 10 K obtained from PNR data using \nexpression:        [        \n    , where x is distance of surface (air/LCMO interface) \nfrom YBCO/STO interface, ξ is coherence length scale, M0 magnetization at surface and α is an \nexponential coefficient, which will dictate the possible order of length scale in LCMO  layer.  \n \n \n 16 \n  \n \n \n  \nFig. 1  \n \n \n \n30 60 90 120 15001020304050M (10-6 emu )\nT (K)FC @300 Oe40 80 120-90-60-300M (10-6 emu )\nT (K)TSC ~ 60 K ZFC17 \n  \n \n Fig. 2  \n \n \n \n \n \n  \n \n \n \n \n \n10-410-310-210-1100\n0510\n0.02 0.03 0.0405103.54.04.5\n0 200 400 6000100200b\n 10 KNeutron reflectivity R+\n R-300 Ka\n NSA (10-11)T = 300 K  NSA data\n M at 10 K with MD layer\n M at 10 K with no MD layer \nT = 10 K\n Q (Å-1)\nNSLD (10-6 Å-2 )\nYBCOSTOLCMO NSLD\n Depth (Å)M (emu/cm3) M at 10 K with MD Layer\n M at 10 K with no MD layer\n M at 300 K\nSubstrate\n18 \n   \n \n \n \n \nFig. 3  \n \n \n \n \n012\n0.02 0.03 0.0402401020\n0 200 400 600055110  NSA (10-11)a\n fit with no MD layer\n   fit with MD layer\n   NSA Data at 100 K \n Q (Å-1)b\n fit with MD layer\n fit with no MD layer\n NSA Data at 50 K\n \nLCMO M with no MD layer\n M with MD layer\n                  T = 100 KSTOYBCO\nSubstrate  \n Depth (Å)M (emu/cm3)\n M with no MD layer  \n M with MD layer  \nT = 50 K19 \n  \n  \nFig. 4  \n \n \n-202\n0.02 0.03 0.04-2024.04.5\n0 200 400050100  T = 300 Ka b\nT = 10 K\n Q (Å-1 )NSA (10-11)\n NSA data\n with MD layer\n no MD layer NSLD (10-6 Å-2 )LCMO\nSTOYBCO\nSubstrate\n Depth (Å)M (emu/cm3)        M at 10 K\n with MD layer\n no MD layer\n      M at 300 K20 \n  \n \n \nFig 5  \n \n \n \n \n \n \n0 100 200 300 4000100200\nb\nMagnetization (emu/cm3)\nDepth (Å)LCMODead\nlayer\nSTOYBCO  Sample S1\nM at 10 K from PNR\n fita LCMO STO YBCO21 \n Supplementary information for  \nSuperconductivity -induced Magnetic Modulation in a Ferromagnet Through  \nan Insulator in La2/3Ca 1/3MnO 3/SrTiO 3/YBa 2Cu 3O7-δ Hybrid Heterostructures  \nC. L. Prajapat1, Surendra Singh2, Amitesh  Paul3, D. Bhattacharya2, M. R. Singh1, G. \nRavikumar1 and         S. Basu2,* \n1Technical Ph ysics Division , Bhabha Atomic Research Centre, Mumbai -400085.  \n2Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai -400085.  \n3Technische Universitat Munchen, Physik Department E21, Lehrstuhl  fur Neutronenstreuung, \nJames -Franck -Strasse 1,D -85748 Garching b. M ünchen, Germany  \n*email: sbasu@barc.gov.in  \n \n \nPulsed laser (KrF) deposition was used to grow La2/3Ca1/3MnO 3 (LCMO)/SrTiO 3 (STO)/  \nYBa 2Cu3O7-δ (YBCO) hetrostructure on single crystalline STO ( 001) substrates. The deposition \nrate was controlled through appropriate focus of laser beam on the target. The substrate \ntemperature during film growth was initially optimized and was maintained at 770 °C. The \noxygen pressure during deposition was 0.5 m bar. The laser fluence was 2.5 J/cm2, and the pulse \nrepetition rate was 2 Hz.  \nThe degree of crystallinity of the hetrostructure was evaluated by X -ray Diffraction \n(XRD) measurements. Before deposition of hetrostructure we optimized the growth of LCMO \nand YBCO  on single crystalline STO ( 001) substrate. Fig. S1 represents the XRD data from the \nsample. Upper and middle panel s of the Fig. S1 show the XRD pattern from LCMO and YBCO 22 \n layer grown on STO substrate. Lower panel of the Fig. S1 show the XRD pattern from the \nLCMO ( 300 Å)/  STO  (25 Å)/YBCO (300  Å) hetrostructure sample. These results are evidence \nfor a high degree of perfection of atomic structure along the growth direction .  \n \nFig. S1:  X-ray diffraction data from LCMO/STO/YBCO system.  \n101102103104102103104105106\n10 20 30 40 50101102103104105106107108109 \nY(004)Y(007)Y(006)/STO(002)\nY(005)\nY(004)Y(003)/STO(001)\nY(002)Intensity (arb. unit)YBCO on STO (001)\nY(001)LCMO(002)STO(002)STO(100) \n LCMO on STO (001)Y(001) Y(002) Y(005)\nY(007)LCMO(001)LCMO(001)\nLCMO(002)STO(002) /\n     Y(006)STO(001)/\n     Y(003) \n 2 (deg.)LCMO/STO/YBCO on STO (001)\n6 7 8 9\n2 (deg.)23 \n  \nFig. S2 shows the X -ray reflectivity (XRR) pattern from two hetrostructures , LCMO (300 \nÅ)/STO (25 Å)/YBCO (300 Å)  and LCMO (200 Å)/STO (5 0 Å)/YBCO ( 200 Å) , samples. The \nspecular reflectivity  (R) was measured as a function of wave vector transfer, Q = 4π sinθ/λ \n(where, θ is angle of incidence and λ is x-ray). The reflectivity is qualitatively related to the \nFourier transform of the scattering length density (SLD) depth profile     1,2, averaged over \nwhole sample area .  For XRR ,     , is proportional to electron density1,2. Thus the chemical \ndepth  profiles were inferred  from the  data by fitting a model ρ (z) whose reflectivity best fit the \ndata. The reflectivities were calculated using the dynamical formalism of Parratt3, and \nparameters of the model were adjusted to minimize the  value of reduced χ2 –a weig hted measure \nof goodness of fit .4 A model consisted of a layer(s) representing regions with different electron \nSLD. The parameters of the model included layer thickness, interface (or surface) roughness and \nelectron SLD.  \nXRR pattern from LCMO (300 Å)/STO (25 Å)/YBCO (300 Å)  and LCMO (200 Å)/STO \n(50 Å)/YBCO ( 200 Å)  hetrostructure samples are shown in upper and lower panel of Fig. S2. \nInset show the corresponding electron scattering length density (ESLD) profile  which gave best \nfit to XRR data. The parameters obtained from the analysis of the XRR data are shown i n Table \n1.  \nTable 1 : parameters obtained from XRR measurements  \n \n LCMO (300 Å)/STO (25 Å)/YBCO (300 Å)   \nhetrostructure  LCMO (200 Å)/STO (5 0 Å)/YBCO ( 200 Å)  \nhetrostructure  24 \n layer  Thickness  \n(Å) Electron SLD \n(10-5 Å-2) Roughness \n(Å) Thickness  \n(Å) Electron SLD \n(10-5 Å-2) Roughness \n(Å) \nLCMO  320±15  4.97±0.06  13±3  187±12  4.88±0.07  10±3  \nSTO  23±2  4.30±0.05  4±1 50±3  4.25±0.05  5±1 \nYBCO  285±15  4.76±0.04  12±4  185±11  4.74±0.05  14±4  \n \n10-410-310-210-1100\n0.05 0.10 0.1510-410-310-210-1100X-ray reflectivity\nQ (Å-1)0 200 400 6004.24.54.8ESLD (10-5 Å-2)\nDepth (Å)\n0 200 4004.24.54.8ESLD (10-5 Å-2)\nDepth (Å)25 \n Fig. S2: X -ray reflectivity (XRR) pattern from LCMO/STO/YBCO hetrostructures.  Inset show \nthe corresponding electron scattering length density (ESLD) depth profile which gave best fit to \nXRR data.  \n \nFig. S3 show the polarized neutron reflectivity (PNR) data from LCMO (300 Å)/STO (25 \nÅ)/YBCO (300 Å)  hetrostructure at 10 K. W e first optimized the nuclear scattering length \ndensity (or NSLD) profile from PNR data at 300 K  (where there is no magnetism)  by \nconstraining layer thicknesses and interface roughness to be within the 95% confidence limit,4 \ni.e., 2 -σ error, established from the a nalysis of the XRR data . To fit PNR data at 10 K we \noptimized magnetization depth profile only and NSLD profile was fixed. Fig. S3 a show the \nPNR data at 10 K. upper panels  show the spin difference ( R+ - R-) data.  \n 26 \n  \nFig. S3: a, PNR  (spin up, R+ and spin down, R-) data from the YBa 2Cu3O7-δ (300Å) /SrTiO 3 \n(25Å)/ La2/3Ca1/3MnO 3 (300 Å) sample  at 10 K. upper panel show the normalized spin \nasymmetry (NSA)  [ =(R+ - R-)/R F, where R F is Fresnel reflectivity]  data at 10 K.  b, shows the \ncorresponding magnetization  (M) depth profiles which fitted PNR data at 10 K.  \n \nWe used different models  for the magnetization depth profile  by considering uniform and \nnon uniform magnetization across LCMO layer. A comparison of three models which gave better \nfit (with smaller χ2) to PNR data at 10 K are shown in Fig. S3b. These three models are  (a) \n0.02 0.03 0.0410-310-210-1100\n0.02 0.03 0.04 0.02 0.03 0.040510\n0 250 5000100200\n0 250 500 0 250 5000.02 0.03 0.04\n0 250 500Neutron reflectivity\nQ (Å-1) R+\n R-\n R+ Fit\n R- Fit\n \nQ (Å-1)\n \nQ (Å-1) \nb\nNSA (10-11)a\n  NSA data\n  fit\n \n \n \n \nYBCOM (emu/cm3)\nDepth (Å)LCMO\nSTO\n \nDepth (Å)\n \nDepth (Å) \n  \nQ (Å-1) \nDepth (Å)27 \n Where  the magnetization is homogeneous throughout  LCMO layer  (left panel) , (b) \nmagnetization is  suppressed (or formation of magnetic dead layer) at LCMO/STO  interface but \nuniform magneti zation in the rest of LCMO layer (middle panel) and ( c) formation of magnetic \ndead layer at LCMO/STO  interface and non uniform magnetization in the rest of LCMO layer \n(right panel) . Fig S3 clearly depicts that model (c) best fit  (with smallest χ2 ) the PNR data at 10 \nK, suggesting modulation in magnetization depth profile LCMO layer.  \n \nFig. S4:  Variation of magnetization (M) as a function of temperature for field cooled condition \nin a magnetic field of 300 Oe.  \n30 60 90 1200255075100125 M (emu/cm3)\nT (K)   SQUID measurements\n M with magnetic dead layer \n M with no magnetic dead layer\nM from PNR measurements28 \n  \nFig. S5:  a, PNR  (spin up, R+ and spin down, R- ) data from the YBa 2Cu3O7-δ (200 Å) /SrTiO 3 (50 \nÅ)/La2/3Ca1/3MnO 3 (200 Å) sample  at 10 K. upper panel show the normalized spin asymmetry \n(NSA)  [= (R+ - R-)/RF, where R F is Fresnel reflectivity]  data at 10 K. b, shows the corresponding \nmagnetization depth profiles which fitted PNR data at 10 K.  \n \n0.02 0.03 0.0410-310-210-1100\n0.02 0.03 0.04-2024\n0 200 400050100\n0 200 400Neutron Reflectivity\nQ (Å-1)\n \nQ (Å-1) NSA (10-11)\n  \nb\n Depth (Å)M (emu/cm3)a\n Depth (Å) LCMO YBCOSTO29 \n The variation of magnetization (M in emu/c m3) as a function of temperature is \ndetermined from SQUID measurements are shown in Fig. S4. Using thickness of LCMO layer as \nmeasured by scattering tech niques (XRR and PNR) we obtained magnetization in emu/cc and are \nplotted in Fig. S4 assuming two cases: with (open circle with line) and without (open triangle \nwith line) magnetic dead layer in LCMO layer. The magnetization obtained from SQUID on \nassuming a magnetic dead layer of thickness ~ 100 Å matches well with the ones obtained from \nthe best fit of PNR data (shown by open star).  \nFig. S5 show PNR data analysis assuming different magnetization profile for sample S2 \nat 10 K. It is clear from the Fig. S5 t hat magnetic dead layer model best fit the PNR data.  \n \nReferences:  \n1. Singh, S. et al. Growth kinetics of intermetallic alloy phase at the interfaces of a Ni/Al \nmultilayer using polarized neutron and x -ray reflectometry.  Phys. Rev. B  81 \n235413(2010).  \n2. Fitzsimmons, M. R. & Majkrzak, C. Modern Techniques for Characterizing Magnetic \nMaterials (Springer, New York, 2005), Chap. 3, pp. 107 –155. \n3. Parratt, L. G. Surface studies of solids by total reflection of x -rays. Phys. Rev . 95, 359 -\n369 (1954).  \n4. Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. Numerical Recipes in \nFortran: The Art of Scientific Computation , 2nd ed. (Cambridge University Press, \nCambridge, 1992).  \n \n \n "    },    {        "title": "1006.5564v1.Ferromagnetic_phase_diagram_of_neutron_matter.pdf",        "content": "arXiv:1006.5564v1  [nucl-th]  29 Jun 2010Ferromagnetic phasediagram ofneutronmatter\nJ.P.W. Dienera,∗,F.G. Scholtza,b\naInsitute ofTheoretical Physics, Stellenbosch University , P.O.BoxX1,Matieland, 7602, South Africa\nbNational Insitute ofTheoretical Physics, P.O.BoxX1,Mati eland, 7602, South Africa\nAbstract\nThe magnetic properties of matter under extreme conditions are of particular importance to un-\nderstanding the neutron star interior. One contributing fa ctor to the magnetic ﬁeld of a neutron\nstarcouldbetheferromagneticphaseofnuclearmatter. Int hisletterwepresentaself-consistent,\nrelativisticdescriptionofferromagnetismindensematte r,withinwhichtheferromagneticphase\ndiagramforneutronmatteriscalculated.\nKeywords: ferromagnetism,relativisticmean-ﬁeld,neutronmatter\nThepropertiesofdensenuclearmatterareofinteresttovar iousbranchesoftopicalresearch.\nOne area of interest is the description of the constituent ma tter of neutron stars. Neutron stars\nare extreme laboratories: extremes in pressure, density an d magnetic ﬁeld are just some of the\nfactorsthatneedtobeconsideredwhenstudyingneutronsta rmatter. Theneutronstarinterioris\nbelieved, at least in part, to consist of nuclear (baryonic) matter. Since charge neutrality of the\nstar is required (a nett charge would result in a Coulomb forc e that will rip the star apart) the\nmajorityofthe baryonicmattershouldbeneutrons[1].\nTheoriginoftheverystrongmagneticﬁeldofneutronstarsh asbeenatopicofdiscussionfor\nsometime. BrownellandCallaway[2]aswellasSilverstein[ 3]proposedthattheferromagnetic\nphase of nuclear matter can make a signiﬁcant contribution t o the magnetic ﬁeld of a neutron\nstar. This idea was investigated by various authors, using b oth non-relativistic and relativistic\nmodels, most recently Modarresand Pourmirjafari[4], whic h also providesreferencesto previ-\nous investigations. The outcome of these investigationsse ems to be inconclusive regarding the\nexistenceofaferromagneticphaseofneutronmatterandapp earto beverymodel-dependent.\nDue to the extreme densities encountered in the neutron star interior relativistic e ﬀects be-\ncome more pronounced and thus it has been argued [1, 5] that a r elativistic description for the\ninterior of the star may be necessary. Thus, to investigate f erromagnetic eﬀects in the baryonic\npart of the neutron star interior a relativistic descriptio n of ferromagnetism in neutron and nu-\nclear matter needs to be considered. Such a description of fe rromagnetism was investigated by\nMaruyama and Tatsumi [6], which based their work on that of Ni embro et al (see [7] and ref-\nerences therein). We aim to further this work by presenting a self-consistent calculation of the\nmagneticﬁeld. Webelievethataself-consistentapproachi swell-suitedtothisproblemsincethe\noriginofthemagneticﬁeld intheferromagneticphaseisthe nucleonmagneticdipolemoments,\nwhichinturnreactsto thepresenceofamagneticﬁeld.\n∗Corresponding author\nEmailaddress: jpwd@sun.ac.za (J.P.W.Diener)\nPreprint submitted to Physics Letters B November 7, 2018Weperformthisself-consistentcalculationintherelativ isticmean-ﬁeldapproximation. This\nis done by coupling the magnetic dipole moment of the neutron to the magnetic ﬁeld of the\nferromagneticphasewithintherelativisticdescriptionp ioneeredbyWalecka[5]calledQuantum\nHadrodynamics(QHD). QHD is an e ﬀective ﬁeld theory model for nuclear matter with mesons\nas degrees of freedom. It had undergone various extensions a nd modiﬁcations, as reviewed in\n[8] and has been used extensively to study the properties of n uclei, nuclear matter and neutron\nstars; see for instance [1, 9, 10, 11]. We employ the coupling between the dipole moment and\nthe magnetic ﬁeld which was introduced by Broderick et al [12 ] to investigate the interaction\nbetween the magnetic dipole moment of nucleons and an extern al magnetic ﬁeld. Within the\nrelativistic mean-ﬁeld approximationwe use the Euler-Lag rangeequation to derive an equation\nofmotionfortheferromagneticmagneticﬁeld(4),whichwe t hensolveself-consistently. Asfar\naswecanestablishsuchaself-consistentcalculationusin gthiscouplinghasnotbeenperformed\npreviously.\nTo describe the interaction between the neutrons we couple t he nucleon ﬁeld ψto a scalar\n(sigma)φand vector (omega) ωµmeson ﬁeld and to the photon ﬁeld tensor Fµν. Theφmeson\ndescribes the long-range attraction of the NN-potential, w hile theωmesons accounts for the\nshort-range repulsion [5]. This description is valid for co ld, neutral neutron matter, since the\nneutronmatter is consideredto be at zero temperatureand th e includedmesonshave no charge.\nTheLagrangianis\nL=LDirac+LKG+LProca+LEM−gv¯ψγµωµψ−gb¯ψFµνσµνψ+gs¯ψφψ. (1)\nHereLDiracis the free ﬁeld component of the Lagrangian for the neutrons ,LKGis the Klein-\nGordon Lagrangian for the scalar mesons and likewise LProcais the Proca Lagrangian for the\nmassive vector mesons. LEMis the free-ﬁeld component of the electromagnetic Lagrangi an.\nThe coupling of the mesons to the di ﬀerent nucleon densities is the standard coupling as in [5].\nσµνare the generators of the Lorentz group. If A0=0 there is no electric ﬁeld and Fµνσµν\nreduces to−2BiΣiin a co-moving frame of reference, with Bithe components of the magnetic\nﬁeld andΣithethreespatialfour-componentspinmatrices.\nFor the coupling strengths gvandgsthe values of the QHD1 parameter set given in [5]\nwere used. These couplingconstantsare chosento ﬁt various propertiesof nuclearmatterat the\nnuclear saturation point ( 0 .16fm−3). The strength of the nucleonand electromagneticcoupling\ngbisleftasa freeparameterwith theunitsofthenuclearmagne ticdipolemoment( Cfm).\nThe magnetic ﬁeld is solved for in a co-moving frame of refere nce chosen, without loss of\ngenerality,so thatthemagneticﬁeldliesinthe z-direction.\nIn the relativistic mean-ﬁeld (RMF) approximation the nucl eon and meson operators are\nreplaced by their groundstate expectation values [5]. The e quations of motion of the various\nﬁeldsarethenfoundto be\nφ=gs\nm2σ/angbracketleftbig¯ψψ/angbracketrightbig(2)\nω0=gv\nm2ω/angbracketleftBig\nψ†ψ/angbracketrightBig\n(3)\nBz=−gb\ne2/angbracketleftbig¯ψΣzψ/angbracketrightbig(4)\n0=/bracketleftbigg\ni/∂−gvγ0ω0−gbBzΣz−/parenleftBig\nm−gsφ/parenrightBig/bracketrightbigg\nψ. (5)\nFor the equation of motion of the ωmeson ﬁeld in (3), the RMF approximation results in the\nspatial componentsbeingzeroandthusonlythezerothcompo nentisconsidered[8].\n2From (5) it can be seen that the equation of motion of the nucle ons is a modiﬁed version of\nthe Diracequation:\n/bracketleftBig\ni∂t−gvω0/bracketrightBig\nψ=/bracketleftBig\n−iα·∇+gbBzβΣz+βm∗/bracketrightBig\nψ, (6)\nwithm∗=m−gsφtheeﬀectivemassandαandβtheDiracmatrices.\nThus the nucleon (Dirac) ﬁeld ψcan be constructed in analogy to the free particle solutions\noftheDirac equation[5]. Therefore ψisassumedto havethe form\nψk,λ(x)∝ψ(k,λ)eik·x−ie(k,λ)t(7)\nwithe(k,λ)thesingleparticleenergy. Sinceanon-zeromagneticﬁeld wouldbreakthespherical\nsymmetryofthegroundstate it isconvenienttoexpress kincylindricalcoordinatesas\nk=(k⊥,kz)=/parenleftBig/radicalBig\nk2x+k2y,kz/parenrightBig\n. (8)\nAspointedoutbypreviousauthors,[6]andreferencesthere in,carehastobetakenwhenconsid-\nering the magnetic dipole moment (spin) of nucleonsin the co ntext of the Dirac equation. This\nisduetothefactthatthespinoperatordoesnotcommutewith theDiracHamiltonian,thematrix\non the right hand side of (6), and thereforespin is not a good q uantum number. Using the form\nofthewavefunctionin (7),thesingleparticleenergiesare foundto be\ne(k,λ)−gvω0=±/radicalBigg/parenleftBigg/radicalBig\nk2\n⊥+m∗2+λgbBz/parenrightBigg2\n+k2z. (9)\nHeree(k,λ) is labeled by λ=±1, which refers to the contribution of the magnetic ﬁeld to th e\nsingleparticleenergies. λwillservetodiﬀerentiatebetweenthetwospeciesofneutrons,butthis\nshould not be mistaken for a spin label. As in the case of the un modiﬁed Dirac equation both\npositive(particle)andnegative(antiparticle)energies arefound.\nUsingthenotationof(7) theunnormalisedDirac spinorsfor particleare\nψ(k,λ)=k2\n⊥+[e(k,λ)+gbBz+m∗][m∗+λ√\nk2\n⊥+m∗2]\n[kx+iky][e(k,λ)+gbBz+λ√\nk2\n⊥+m∗2]\n−kz\ne(k,λ)+gbBz+λ√\nk2\n⊥+m∗2\nkz[m∗+λ√\nk2\n⊥+m∗2]\n[kx+iky][e(k,λ)+gbBz+λ√\nk2\n⊥+m∗2]\n1. (10)\nThespinorsarenormalisedsothat\n/integraldisplay\nd3xψ†\nk′,λ′(x)ψk,λ(x)=δ(k′−k)δλ,λ′. (11)\nBy inspectionof the single particle energies(9) it clear th at when the magneticﬁeld is zero,\nthe normal dispersion relation is recovered and the lowest s ingle particle energy state that a\n3Figure 1: Illustration of the single particle energy levels that neutrons can populated in the presence of a magnetic ﬁel d.\nTheunpaired neutrons areof course the source of the magneti c ﬁeld.\nfermion can occupy is the e ﬀective mass, m∗. In the presence of a magnetic ﬁeld the lowest\nsingle particle energy would be m∗±gbBzforλ=±1. The magnetic ﬁeld therefore lifts the\ndegeneracybetween the states of the two neutronspecies. Th is splitting is illustrated in Fig.(1)\nforpositiveenergystates,butalsoappliestothenegative energystates. Howeverthegroundstate\nweconsideronlyhaspositiveenergynucleonstates,whichi sﬁlledtoacertainenergy,theFermi\nenergy.\nUsing(7)to calculatethe nucleondensitiesonthe righthan dsideof equations(2 -4), it can\nbe shown that these equations are equivalent to the minimiza tion of the energy density of the\nsystem,ǫ,\nǫ=/summationdisplay\nλ/integraldisplaydk\n(2π)3e(k,λ)Θ/bracketleftBig\nµ−e(k,λ)/bracketrightBig\n+1\n2m2\nσφ2+1\n2m2\nωω2\n0+1\n2e2B2\nz, (12)\nwithµthechemicalpotentialand Θastep functiontoensurethatonlyenergiesbelowtheFermi\nenergyare considered.\nWe used the equations of motion to calculate the values of the diﬀerent ﬁelds at a given\ndensity and value of the coupling gb. The coupling constant was increased at a ﬁxed density\nuntil a non-zerovalueof themagneticﬁeld was found. The res ultsare shownin Fig.(2). Inthis\nﬁgurethesolidlinerepresentsthecouplingstrength gbataspeciﬁcdensitywhichwoulde ﬀecta\nferromagneticphasetransitionwhennomesonsare present. The dashedline representsthecase\nwhenmesonsareincluded.\nWe notethatthe vectormesonsdonotinﬂuencethephasetrans itionsince theycoupleto the\nnucleondensityandmerelyshifttheenergyspectrum. Thisc anbeseenintheexpressionforthe\nsingleparticleenergies(9).\nTheeﬀectofthescalarmesonistoreducethemassofnucleonresult inginaneﬀectivemass\nm∗. This reductionin the mass implies that for the same density the couplingconstant gbhas to\nbestrongerthaninthecasewhennomesonsarepresent. Fig.( 2)clearlyshowsthattheinclusion\n40 2 4 6 8 10Ρ/Slash1Ρ0 0.00.51.01.52.02.53.0gb/LBracket1Cfm/RBracket1\n/MultiplyΣ,Ωmesonsno mesons\nMagnetic field /NotEqual 0\nMagnetic field = 0\nFigure2: Ferromagneticphasediagramindensityandcoupli ngconstant, gb. Thesolidlinerepresentsthephaseboundary\nwhen no mesons are included in the description and thus the ne utrons only interact with the magnetic ﬁeld. The dotted\nline represents the phase boundary when mesons are also incl uded. The×indicates the point where gbis at the value of\nthe magnetic dipole momentof the neutron at adensity which i s equal to nuclear saturation density ρ0=0.16fm−3.\nofthescalar mesonsresultinthephasetransitionoccurrin gconsistentlyat largervaluesof gb.\nTheshapeofthecurveinFig.(2)atlowdensitiesandwithmes onsincluded,resultsfromthe\ninterplaybetweenthecontributionsfromthescalar andnuc leonﬁeldstotheenergydensity.\nFrom Fig.(2) it is apparentthat for the ferromagneticphase to present itself in cold neutron\nmatter the coupling between the nucleon ﬁeld and the magneti c ﬁeld has to have some den-\nsity dependence. The coupling at normal nuclear density (th e magnetic dipole moment of the\nneutron) is just not strong enough to a ﬀect a phase transition even at very high densities. The\ndensity dependenceof gbmight be calculated by includingrenormalizatione ﬀects, but this was\nnotinvestigatedin thecurrentwork.\nIt is possible that the densities at which the ferromagnetic phase of neutron matter is stable\ncanbereachedinneutronstars. Inthecalculationdonehere suchastablephasegeneratesamag-\nneticﬁeldoftheorderof1010Gauss. Thismagnitudeoftheﬁeldinitselfisnotenoughtoex plain\nthe origin of the magnetic ﬁeld of neutron stars, which are ca lculated to have surface magnetic\nﬁelds of the order of 1011−1013G[13]. The model presented here is a ﬁrst approximation for\nneutronstarmatter,sinceonlyneutronsareincludedandon lyinﬁnitematterconsidered. Further\nworkwouldentailstudyingthee ﬀectsofincludingotherhadronsandleptonsinthemodeltoob -\ntain a more realistic description for neutron star matter. W ith such a model it might be possible\ntoinvestigatetheeﬀectsofaferromagneticphaseontheequationofstate ofneut ronstarmatter.\n1. Acknowledgements\nThis research is supported by the South African SKA project a nd a grant from the National\nResearchFoundationofSouthAfrica.\n5References\n[1] N.K.Glendenning, Compact Stars,Nuclear Physics,Part icle Physics and General Relativity, 2ndedition, Springer,\nNew York 2000.\n[2] D.H.Brownell, J.Callaway, Nuovo Cimento 60B (1969) 169 .\n[3] S.D.Silverstein, Physical Review Letters 23 (1969) 139 .\n[4] M. Modarres, T.Pourmirjafari, Nuclear Physics A836 (20 10) 91.\n[5] B.D.Serot, J.D.Walecka, TheRelativistic Nuclear Many -Body Problem,in: J.W.Negele, E.Vogt(Eds.)Advances\nin Nuclear Physics 16 (1986) 1.\n[6] T.Maruyama, T.Tatsumi, Nuclear Physics A693 (2001) 710 .\n[7] R. Niembro, S.Marcos, M.L.Quelle and J.Navarro, Physic s Letters B249 (1990) 373.\n[8] B.D. Serot, J.D.Walecka, International Journal of Mode rn Physics E6 (1997) 515.\n[9] C.J.Horowitz, J.Piekarewicz, Physical Review Letters 86 (2001) 5647.\n[10] J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long and L.S . Geng, Progress in Particle and Nuclear Physics 57\n(2000) 470.\n[11] J.P.W. Diener, Relativistic mean-ﬁeld applied to the s tudy of neutron star properties, MSc thesis, Stellenbosch\nUniversity 2008, arXiv:0806.0747.\n[12] A. Broderick, M. Prakash and J.M.Lattimer, TheAstroph ysical Journal 537 (2000) 351.\n[13] A. Harding, D. Lai,Reports on Progress in Physics 69 (20 06) 2631.\n6"    },    {        "title": "1803.03079v1.Hole_doping_induced_half_metallic_ferromagnetism_in_highly_air_stable_PdSe2_monolayer_under_uniaxial_stress.pdf",        "content": "arXiv:1803.03079v1  [cond-mat.mes-hall]  8 Mar 2018Hole-doping-induced half-metallic ferromagnetism in hig hly-air-stable PdSe 2\nmonolayer under uniaxial stress\nShi-Hao Zhang1,2and Bang-Gui Liu1,2,∗\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China\n(Dated: June 18, 2021)\nTwo-dimensional (2D) high-temperature ferromagnetic mat erials are important for spintronic ap-\nplication. Fortunately, a highly-air-stable PdSe 2monolayer semiconductor has been made through\nexfoliation from the layered bulk material. It is very highl y desirable to realize robust ferromag-\nnetism, even half-metallic ferromagnetism (100% spin pola rization), in such excellent nonmagnetic\nmonolayer semiconductors. Here, the ﬁrst-principles inve stigation shows that the PdSe 2monolayer\ncanbe madetoattain Stonerferromagnetism withthemaximal Curie temperature reachingto800K,\nand the hole concentration threshold for ferromagnetism de creases with applied uniaxial stress. Fur-\nthermore, half-metallicity can be achieved in some hole con centration regions. For the strain of 10%\n(uniaxial tensile stress of 4.4 N/m), the monolayer can atta in half-metallic ferromagnetism up to\n150 K. The magnetic anisotropic energy is suitable to not onl y stabilizing the 2D ferromagnetism\nbut also realizing fast magnetization reversal. The magnet ization can be also controlled by applying\na transverse uniaxial stress. The highly-air-stable PdSe 2monolayer, with these advantages, should\nbe promising for spintronic applications.\nI. INTRODUCTION\nRecent experimental discovery of exfoliated two-\ndimensional ferromagnetic materials CrI 3[1] and\nCr2Ge2Te6[2] inspires people to seek more two-\ndimensional realizable magnetic materials for device ap-\nplications. Besides exfoliating two-dimensional mag-\nnetic van der Waals crystals, there are many methods\nto obtain the two-dimensional magnetism, such as elec-\ntric ﬁeld modulation [3, 4], strain engineering [5, 6],\nnanoribbon edge modiﬁcation [7], surface adsorption [8–\n10], transition-metal atom doping [11–13], defect engi-\nneering [14, 15], and so on. Carrier doping is always\neﬀective when there is high density of states near the\nFermi level and the doped carrier can create itinerant\nferromagnetism obeying Stoner’s criterion N(EF)I >1,\nwhereN(EF) is the density of states at the Fermi energy\nin the nonmagnetic state and Iis Stoner parameter de-\nﬁnedasI= ∆/M(∆isthespinsplittingenergyand Mis\nthe spin moment). The Stoner ferromagnetism has been\npredicted in the nonmagnetic two-dimensional materials\nGaSe [16], InP 3[17], PtSe 2[18], C 2N [19] monolayer,\nphosphorene and arsenene [20].\nRecently, PdSe 2monolayer as a two-dimensional ma-\nterial has been synthesised by exfoliating from bulk\nPdSe2crystals [21]. Very importantly, the semiconduct-\ning PdSe 2monolayer has high air stability under am-\nbient conditions, in contrast to the fast degradation of\nblack phosphorus in air [22, 23], which makes the PdSe 2\nmonolayer promising for electronic devices. Furthermore\nultrathin PdSe 2ﬁeld-eﬀect transistor with high mobility\nhas been reported [24]. Previous theoretical calculations\n∗bgliu@iphy.ac.cnshowed that the monolayer has large Seebeck coeﬃcients\nfor both p- and n-type carrierswhen doping level is lower\nthan 2×1013cm−2[25], but there is no magnetic explo-\nrationinthePdSe 2monolayer,althoughitishighlydesir-\nableforspintronicapplicationsbasedonhighly-air-stable\n2D materials.\nThe sharp peak in the density of states near the va-\nlence band maximum, always coexisting with large ther-\nmoelectric Seebeck coeﬃcients [26], implies that doping\nsome carriers can move the Fermi level to the peaked\ndensity of states, and induce Stoner instability and then\nitinerantferromagnetisminthe PdSe 2monolayer. In this\nwork, our ﬁrst-principles calculations show that doping\nhole into the PdSe 2monolayer can create Stoner ferro-\nmagnetism and the Curie transition temperature can be\nfar beyond room temperature. The magnetization, spin\npolarization energy, magnetic anisotropic energy, and\nCurie temperature are systematically studied with dif-\nferent hole doping concentration under diﬀerent uniaxial\ntensile stress. Stable half-metallic ferromagnetism, im-\nplying 100% spin polarization at the Fermi level[27–29],\ncan be achieved by tuning the doped hole concentration\nandthe applieduniaxialstress. Moredetailed resultswill\nbe presented in the following.\nII. COMPUTATIONAL METHODS\nOurﬁrst-principlescalculationsareperformedwiththe\nVienna Ab initio Simulation Package (VASP) [30] with\nthe projector-augmented wave (PAW) method [31]. The\ngeneralized gradient approximation (GGA) by Perdew,\nBurke, andErnzerhof(PBE)[32] is usedasthe exchange-\ncorrelation potential because the band gap calculated\nwith PBE is in good agreement with the experimental\nvaule [21]. The thickness of vacuum region is set as2\n/s40/s97/s41 \n/s40/s98/s41 /s97/s98\n/s97/s99\nFIG. 1. The top view (a) and side view (b) of PdSe 2mono-\nlayer. The red and blue balls represent Pd and Se atoms,\nrespectively. The a, b, and c directions correspond to the y,\nx, and z axes, respectively.\n20˚A to avoid any artiﬁcial interaction in the computa-\ntional model. The cutoﬀ energy is set to 500 eV, and\nthe convergence standard is that the total energy diﬀer-\nence between two successive steps is smaller than 10−6\neV. The structures are fully optimized to ensure all the\nHellmann-Feynman forces on each atom are less than\n0.01 eV/ ˚A. The hole doping is achieved by changing the\ntotal number of electrons of the unit cell, with a com-\npensating jellium background of opposite charge added.\nBecause the hole-doped magnetism is very sensitive to\nthe sampling of densities of states (DOS), we carry out\nthe Brillouin zone integration with a dense Γ-centered\n(41×41×1) Monkhorst-Pack grid [33]. When calculating\nthe small magnetic anisotropic energy (MAE), the en-\nergy convergence criterion is promoted to 10−8eV for\nachieving high accuracy.\nIII. RESULTS AND DISCUSSION\nA. Intrinsic electronic structure\nThe crystal structure ofPdSe 2monolayer[21] is shown\nin Fig. 1. The unit cell of PdSe 2monolayer (P2 1/c space\ngroup) has two palladium and four selenium atoms, with\nthe lattice parameters a= 5.74˚A (y-axis) and b= 5.92\n˚A (x-axis). The monolayer shows puckering pentagonal\nring which is like Cairo pentagonal tiling from the top\nview [21]. The energy bands and the density of states\nare showed in Fig. 2. The monolayer shows semicon-\nducting feature and has an indirect band gap of 1.33 eVwhichisagreementwiththeexperiment[21]. Thevalence\nband maximumis located at(0.34, 0)and the conduction\nband minimum is situated on the (0.34, 0.44) point. In\nFig. 2(c), the energy values of the highest valence band\nover the entire Brillouin zone are presented. There are\nfour maximum in the energy distribution, and there is\na ﬂat band (within the blue contour) and a sharp DOS\npeak at -0.15eV (below the Fermi level). This DOS peak\nindicates a great probability for Stoner ferromagnetism\ninduced by hole doping.\nB. Hole doping and ferromagnetism\nWe perform ﬁrst-principles calculations about spin\nmagnetic moment and spin polarization energy Epol, the\nenergy diﬀerence between the nonmagnetic and ferro-\nmagnetic states Epol=Enon−Efer). The calculated\nresults are shown with solid lines in Fig. 3 (a,b). With\nintroducing hole hoping, the Fermi level can touch the\nhigh density of states and then makes the strong on-site\ninteractions between the opposite spins which induces\nthe spin splitting. The calculated results shows that the\nsystem favors ferromagnetism under appropriate hole-\ndoping levelsbecause the energiesof ferromagneticstates\nare lower than those of nonmagnetic cases. For PdSe 2\nmonolayer, ferromagnetism begins to occur when hole\nconcentration is largerthan 1.5 ×1014cm−2(0.25 hole per\nformula unit) and magnetic moment per hole becomes\npeaked (0.83 µBper hole) at the 1.8 ×1014cm−2(0.3 hole\nper formula unit) hole level. The positive spin polariza-\ntion energy means stable magnetization and it reaches\nthe maximum (7.0 meV per hole) at the 2.1 ×1014cm−2\nhole level, which is comparable to those of GaSe (3 meV\nper carrier) [16] and C 2N monolayer (8.5 meV per car-\nrier) [19].\nMagnetic anisotropic energy (MAE) plays an impor-\ntant role in the two-dimensional stable long-range fer-\nromagnetism [1]. Deﬁned as the total energy diﬀerence\nbetween the ferromagnetic conﬁgures along the out-of-\nplane(z)andthelowestin-plane(xorb)directions,MAE\nmainly originates from electronic contribution. The\nshapeanisotropycaused bythe dipole-dipoleinteractions\ncan be neglected [34, 35]. The calculated results are de-\nscribed with solid line in Fig. 3(c). The maximum of\nMAE reaches at 32 µeV per hole for the hole concentra-\ntion 1.8×1014cm−2, which is comparable to that of hole-\ndopedphosporene[20]andmuchlargerthanthoseofcon-\nventional transition metals: Fe, Co, and Ni (several µeV\nper atom) [36]. Compared to other two-dimensional ma-\nterials with intrinsic magnetism, the hole-doped PdSe 2\nmonolayer has smaller MAE than CrXTe 3(X=Si,Ge,Sn,\n0.069-0.419 meV) [37], CoBr 2(2.6meV) [38], and Fe 2Si\n(0.55–0.57meV) [39] monolayers. This comparison indi-\ncates that the hole-doped PdSe 2monolayers can be used\nasair-stableandappealingtwo-dimensionalmagneticdy-\nnamic layers for fast spin dynamics.\nThe Stoner magnetism originates from the strong ex-3\n/s45/s50 /s45/s49 /s48/s49/s50/s51/s69/s32/s45/s32/s69 \n/s70/s40/s101/s86/s41 \n/s77/s88 /s83 /s89 /s83/s49/s46/s51/s51/s32/s101/s86 /s88/s83 /s89/s77\n/s45/s48/s46/s55 /s45/s48/s46/s54 /s45/s48/s46/s53 /s45/s48/s46/s52 /s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s69/s32/s45/s32/s69 \n/s70/s40/s101/s86/s41 \n/s88/s89\n/s48 /s49/s48 /s50/s48 /s51/s48 \n/s68/s79/s83 /s32/s84/s111/s116/s97/s108 \n/s32/s80/s100/s45/s100 \n/s32/s83/s101/s45/s112 /s40/s97/s41 /s40/s98/s41 /s40/s99/s41 \nFIG. 2. The energy bands (a) and density of states (DOS) (b) of PdSe2monolayer, where the M point represents the (0.39,\n0.5) point. (c) The energy distribution of the highest valen ce bands over the ﬁrst Brillouin zone.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s48/s46/s48 /s49/s48/s46/s48 /s50/s48/s46/s48 /s51/s48/s46/s48 /s52/s48/s46/s48 /s53/s48/s46/s48 /s77/s65/s69/s32/s40 /s101/s86/s47/s104/s111/s108/s101/s41 \n/s72/s111/s108/s101/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s49/s48 /s49/s52 \n/s47/s99/s109 /s50\n/s41/s32/s48/s37 \n/s32/s51/s37 \n/s32/s54/s37 \n/s32/s56/s37 \n/s32/s49/s48/s37 \n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s48/s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s67/s117/s114/s105/s101/s32/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41 \n/s72/s111/s108/s101/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s49/s48 /s49/s52 \n/s47/s99/s109 /s50\n/s41/s32/s48/s37 \n/s32/s51/s37 \n/s32/s54/s37 \n/s32/s56/s37 \n/s32/s49/s48/s37 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s77/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32/s40 \n/s66/s47/s104/s111/s108/s101/s41 \n/s72/s111/s108/s101/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s49/s48 /s49/s52 \n/s47/s99/s109 /s50\n/s41/s32/s48/s37 \n/s32/s51/s37 \n/s32/s54/s37 \n/s32/s56/s37 \n/s32/s49/s48/s37 \n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s48/s46/s48 /s50/s46/s48 /s52/s46/s48 /s54/s46/s48 /s56/s46/s48 /s83/s112/s105/s110/s32/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s104/s111/s108/s101/s41 \n/s72/s111/s108/s101/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s49/s48 /s49/s52 \n/s47/s99/s109 /s50\n/s41/s32/s48/s37 \n/s32/s51/s37 \n/s32/s54/s37 \n/s32/s56/s37 \n/s32/s49/s48/s37 /s40/s97/s41 /s40/s98/s41 \n/s40/s99/s41 /s40/s100/s41 \nFIG. 3. The magnetic moment(a), spinpolarization energy(b ), magnetic anisotropic energy(MAE)(c), andCurie tempera ture\n(d) versus hole doping concentration under diﬀerent x-axis strain.\nchange ﬁeld in the system. When hole doping level is\n1.8×1014cm−2, the spin splitting energy at the Γ point\nreaches 144 meV which corresponds to an eﬀective Zee-\nman splitting from an external magnetic ﬁeld of 1243 T.The exchange correlation induces the spontaneous mag-\nnetization obeying\nm=1\nN/summationdisplay\nk(/angbracketleftnkσ/angbracketright−/angbracketleftnk˜σ/angbracketright) (1)4\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s49/s48 /s53/s48/s53/s49/s48 /s68/s79/s83 \n/s69/s32/s45/s32/s69 \n/s70/s32/s40/s101/s86/s41 /s32/s84/s111/s116/s97/s108 \n/s32/s80/s100/s45/s100 \n/s32/s83/s101/s45/s112 \n/s45/s50 /s45/s49 /s48/s49/s50/s51/s69/s32/s45/s32/s69 \n/s70/s32/s40/s101/s86/s41 \n/s77/s88 /s83 /s89 /s83/s45/s50 /s45/s49 /s48/s49/s50/s51/s69/s32/s45/s32/s69 \n/s70/s32/s40/s101/s86/s41 \n/s77/s88 /s83 /s89 /s83\n/s48/s37 /s50/s37 /s52/s37 /s54/s37 /s56/s37 /s49/s48/s37 /s48/s49/s50/s51/s52/s53/s83/s116/s114/s101/s115/s115/s32/s40/s78/s47/s109/s41 \n/s83/s116/s114/s97/s105/s110 /s40/s97/s41 \n/s40/s99/s41 /s40/s98/s41 \n/s40/s100/s41 \nFIG. 4. (a) The uniaxial stress on the monolayer as a function of x-axis strain. (b) The energy bands of PdSe 2monolayer\nunder the x-axis strain of 10%. The spin-polarized energy ba nds (c) and density of states (DOS) (d) of the monolayer with 0 .2\nholes per formula unit (1 .08×1014cm−2) under the same strain.\n=1\nN/summationdisplay\nk/braceleftbigg1\neβ(Ek−∆−µ)+1−1\neβ(Ek+∆−µ)+1/bracerightbigg\n(2)\nwhereβis (kBT)−1,µis the chemical potential, and 2∆\nis the spin splitting energy between two spin channels, σ\nand ˜σ.\nHere we discuss how to estimate the Curie transition\ntemperature of the itinerant ferromagnetism. We slowly\nincrease the Gauss smearing factor to simulate the ef-\nfect of increasing temperature kBT. This will reduce the\ndiﬀerence between /angbracketleftnkσ/angbracketrightand/angbracketleftnk˜σ/angbracketrightin Eq. (1) and en-\nlarge thermal excitations which weaken the magnetism\nand ﬁnally cause the Curie transition. By minimizing\nthe free energy of the system at given temperatures,\nwe can obtain the Curie temperature at the mean ﬁeld\nlevel [16]. Calculated Curie transition temperature as\na function of hole concentration is presented with solid\nline in Fig. 3(d). The Curie temperatures are higher\nthan 600 K when the hole concentration is larger than\n1.4×1014cm−2, which are much higher than those of\nGaSe monolayer (about 90 K) [16]. These results reveal\nthat the stable Stoner ferromagnetism can be observedbeyond the room temperature. The magnetization M\nversus temperature Tfor 1.8×1014cm−2hole level shows\na critical exponent of 1/2, ∆ M(T)∼(Tc−T)1/2, where\nTcis Curie temperature. This is in agreement with the\nmean ﬁeld model.\nC. Uniaxial stress and half-metallic ferromagnetism\nTensile stress can always make the energy bands be-\ncome more narrow (larger DOS) and thus make the\nStoner ferromagnetism easier to occur. Actually, it was\nfound that tensile strain decreases the critical value of\nhole concentration in the arsenene [20] and PtSe 2mono-\nlayer [18]. Due to the anisotropy between the x and y\ndirections, uniaxial tensile stress is more accessible than\nbiaxial stress. We apply a tensile stress along x-axis. As\na result, it will cause a tensile strain in the x-axis and a\ncompressive strain in the y-axis. The stress as a function\nof the x-axis strain is presented in Fig. 4(a). The strain\ncan be applied by stretching a ﬂexible substrate on which5\nthe 2Dmaterialis attaching, orelectricallycontrollingby\nusing a piezoelectric substrate [40–42]. It only needs 4.4\nN/m to realize the x-axis strain of 10% in the case of the\nPdSe2monolayer.\nThe calculated magnetic moment, spin polarization\nenergy, magnetic anisotropic energy (MAE), and Curie\ntemperature as functions of hole concentration for dif-\nferent x-axis strains are presented in Fig. 3. It is noted\nthat the hole concentration threshold for ferromagnetism\nsubstantially decreases with increasing the x-axis strain,\nwith the threshold being 1.5 ×1014cm−2for zero strain\nagainst 1.9 ×1013cm−2for 10%. It should be noted that\ntheholeconcentration2.0 ×1013cm−2correspondsto0.04\nholes per formula unit.\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 \n/s54/s52/s48 /s54/s54/s48 /s54/s56/s48 /s55/s48/s48 /s55/s50/s48 /s48/s46/s48/s48 /s48/s46/s48/s54 /s48/s46/s49/s50 /s48/s46/s49/s56 /s48/s46/s50/s52 /s48/s46/s51/s48 /s124/s77/s124 /s50\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41 /s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40 \n/s66/s47/s104/s111/s108/s101/s41 \n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 \n/s51/s52/s48 /s51/s54/s48 /s51/s56/s48 /s52/s48/s48 /s52/s50/s48 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s124/s77/s124 /s50\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41 /s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40 \n/s66/s47/s104/s111/s108/s101/s41 \n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41 /s40/s97/s41 \n/s40/s98/s41 \nFIG. 5. The magnetizations versus temperature of PdSe 2\nmonolayer under x-axis strain 10%, with the two hole concen-\ntrations of 5.4 ×1013cm−2(a) and 1.08 ×1014cm−2(b). Insets:\nThe magnetization squared as a function of temperature near\nthe Curie transition temperature for both of the cases.\nEspecially, half-metallic ferromagnetism begins to oc-\ncur when the x-axis strain reaches to 6%. It accompaniesthe saturated magnetic moment: 1 µBper hole. Keeping\non increasingthe strain, the doped hole concentrationfor\nhalf-metallicity develops into a plateau region. For ex-\nample, it becomes 0 .2∼1.6×1014cm−2when the strain\nreaches 10%. Because doped carrier concentrations were\nachieved to the order of 1013cm−2in transition metal\ndichalcogenide monolayers by back-gate gating [43, 44]\nandto1014cm−2in graphenebyionliquid gating[45,46],\nthe doped hole concentrations for half-metallic ferromag-\nnetism in the PdSe 2monolayer should be accessible ex-\nperimentally.\nFor comparison, we also present the energy bands of\nPdSe2monolayer under the x-axis strain of 10% without\nhole doping in Fig. 4(b). The band width of the high-\nest valence bands is 0.57 eV, smaller than that of the\nequilibrium state (0.72 eV). The energy bands and DOS\nof the monolayer under 10% with 0.2 holes per formula\nunit are showed in Fig. 4(c,d). The DOS at the Fermi\nlevel,N(EF), for the nonmagnetic state with 0.2 holes\nper formula unit reaches 7.8/eV and the spin splitting\nenergy near the Fermi level is 0.12 eV. Thus the Stoner\nparameter Iis 0.3 eV and the product N(EF)Iequals\n2.34, which satisﬁes the Stoner’s criterion. It is clear\nthat the energy bands and DOS show half-metallic ferro-\nmagnetism.\nWe alsostudythe temperaturedependence ofthe mag-\nnetization. For the strain of 10%, we present the mag-\nnetization as a function of temperature for two hole con-\ncentrations of 5.4 ×1013cm−2and 1.08×1014cm−2in Fig.\n5. These two hole concentrations are equivalent to 0.1\nand 0.2 holes per formula unit, and the two Curie tem-\nperatures are higher than 400 K and 700 K, respectively.\nMore importantly, the half-metallic ferromagnetism can\npersist up to 70K and 150 K, respectively. These imply\nthat nearly 100% spin polarization can be achieved and\ntherefore the half-metallic PdSe 2monolayer can be used\nfor high-performance spintronic devices.\nD. Further discussions\nAs for the spin polarization energy, the maximum is\nstill approximately7meV perhole when the x-axisstrain\nincreases to 10%, which proves the spin-polarized conﬁg-\nures are still stable. The maximal Curie temperatures\nstill remain approximately 800 K under diﬀerent strains.\nIn contrast, the maximum ofMAE increasesfrom 32 µeV\nper hole (0%) to 45 µeV per hole (10%), which implies\nthat the uniaxial tensile stress can enhance the stability\nof the magnetization direction.\nWhen a tensile stress is applied along the y-axis (the\na direction), there will be a tensile strain in the y-axis\nand a compressive strain along the x-axis (the b direc-\ntion). With 0.25 holes per formula unit doped into the\nmonolayer, the magnetization along the x-axis can per-\nsist when a tensile y-axis strain up to 6% is applied. If\nthe y-axisstrainislargerthan 6%, themagnetizationwill\nswitch from the x-axis to the y-axis. This is like ferro-6\nelectric polarization switching in GaTeCl monolayer[47].\nTherefore, the stress can change the MAE of the mono-\nlayer, and can be used to control the magnetization di-\nrection of the hole-doped PdSe 2monolayer.\nIV. CONCLUSION\nThe ﬁrst-principles investigation has shown that when\nhole carriers rae doped into PdSe 2monolayer, Stoner\nferromagnetism can be induced and the maximal Curie\ntemperature can reach to 800K. The hole concentra-\ntion threshold for ferromagnetism decreases with applied\nstress (x-axis strain), reducing to 1.9 ×1013cm−2at the\nstrain of 10%. More importantly, half-metallicity can\nbe formed in some hole concentration regions, in addi-\ntion to the ferromagnetism. For the strain of 10%, es-\npecially, when the hole doping concentration is in the\nrange of 0 .2∼1.6×1014cm−2, the monolayer can at-\ntain half-metallic ferromagnetism up to 150 K. These im-\nply that 100% spin polarization can be achieved in these\nhole concentration regions. A uniaxial tensile stress 4.4N/m can produce this large x-axis strain of 10%. The\nmagnetic anisotropic energy is suitable to stabilizing the\ntwo-dimensional ferromagnetism and ensuring fast mag-\nnetization reversal. 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Liu,\nNanoscale (2018), 10.1039/C7NR09588K."    },    {        "title": "1612.06216v1.Spontaneous_Distortion_and_Ferromagnetism_Induced_by_Quantum_well_States_in_Pd_100__Ultrathin_Films.pdf",        "content": "arXiv:1612.06216v1  [cond-mat.mtrl-sci]  19 Dec 2016Spontaneous Distortion and Ferromagnetism Induced by Quan tum-well States in\nPd(100) Ultrathin Films\nShunsuke Sakuragi,1,∗Hiroo Tajiri,2Hiroyuki Kageshima,3and Tetsuya Sato1\n1Department of Applied Physics and Physico-Informatics,\nKeio University, Hiyoshi, Yokohama 223-0061, Japan\n2Japan Synchrotron Radiation Research Institute/SPring-8 , Kouto, Sayo 679-5198, Japan\n3Interdisciplinary Graduate School of Science and Engineer ing,\nShimane University, Nishikawatsu-cho, Matsue 690-8504, J apan\n(Dated: September 20, 2018)\nWe study the crystal structure of Pd(100) ultrathin ﬁlms, wh ich show ferromagnetism induced\nby the quantum conﬁnement eﬀect, using in-situ X-ray crysta l truncation rod measurement and\ndensity functional calculation. The energy gain due to the a ppearance of ferromagnetism in Pd\nresults in ﬂatter and uniform ﬁlm growth of ferromagnetic Pd ﬁlms compared with paramagnetic\nPd. In addition, ferromagnetic Pd ﬁlms expand the lattice co nstant in order to suppress the increase\nin kinetic energy of electrons accompanied by the occurrenc e of exchange splitting. Although the\ntraditional theory of magnetism in metals indicates that th e increase in density of states that induces\nferromagnetism (Stoner criterion), our present ﬁnding rev eals a mechanism of modulation in the\ndensity of states via the appearance of ferromagnetism, i.e ., the inverse mechanism of Stoner’s\ntheory.\nPACS numbers: 75.50.-Cc, 75.30.Kz, 75.70.-Ak\nRecently, the importance of magnetic materials in in-\ndustry has increased due to the development of novel\nelectronic devices based on the spin degree of freedom of\nelectrons [1, 2]. However, in bulk form elements, ferro-\nmagnetism appears in a small fraction of the 3 dtransi-\ntion metals, only Fe, Co and Ni. Over the past decade,\nthe appearance of ferromagnetism in several metal par-\nticles, such as Pd, Pt and Au, has been reported [3–6].\nThe physical properties of metals are determined by elec-\ntronic states near the Fermi energy ǫF. Thus, it has been\nunderstood that the conﬁnement of electrons, surface ef-\nfect, and/or distortion in the crystal is accompanied by\nnano-scaling-induced ferromagnetism in the noble met-\nals. Although the origin of ferromagnetism in nanoparti-\ncles remains to be solved, this phenomenon suggests the\npossibility of fabricating fascinating materials with novel\nproperties not realized in bulk form.\nRecently, it was shown that Pd(100) ultrathin ﬁlms\nshowed ferromagnetism in an oscillatory manner, depen-\ndent on ﬁlm thickness [7–10]. From the standpoint of\nStoner’s theory for the appearance of ferromagnetism\n[11],\nID(ǫF)>1, (1)\nwhereIis the exchange integral and D(ǫF) is the den-\nsity of states at the Fermi energy, the ferromagnetism\nin Pd(100) is interpreted based on the increase in D(ǫF)\ndue tod-electron quantum-well states. The magnetiza-\ntion and Curie temperature of epitaxial Pd(100) ultra-\nthin ﬁlms on SrTiO 3(100) substrates are comparable to\nthose of Ni [10], where the magnetization is larger than\none digit from the theoretical expectation [9]. Elucidat-\ning the origin of the enhancement of ferromagnetic mo-\nment in Pd(100) would contribute to our understandingof the expression mechanism of ferromagnetism in tran-\nsition metals, because the magnetism of this system can\nbesystematicallycontrolledthroughquantum-wellstates\ndepending on ﬁlm thickness, according to Stoner’s the-\nory. Because of its potential for the development of novel\nmagnetic nanoscale materials whose magnetism can be\ncontrolled, this matter is of great technological interest\n[12–15].\nIn this Letter, we demonstrate the relationship be-\ntween the crystal structure and magnetism in Pd(100)\nultrathin ﬁlms using surface X-ray diﬀraction and den-\nsity functional calculations. The ﬂat growth of Pd(100)\nﬁlms is attributed to the energy gain in the system ac-\ncompanied by the appearance of ferromagnetism induced\nby quantum-well states, and 0.8% lattice expansion was\nneeded in order to suppress energy loss via the occur-\nrence of exchange splitting. Our results suggest that\nthe spontaneous stabilization in the magnetic state is\nbrought about by lattice expansion via the appearance\nof ferromagnetism in the transition metal.\nAll experiments were performed at SPring-8 BL13XU\n[16]. We used atomically ﬂat SrTiO 3(100) substrates\ntreated with buﬀered hydroﬂuoric acid (SHINKOSHA\nCo., Ltd.) [17], and the following three-step growth\nmethodtoprepareatomicallyﬂatPd(100)ultrathinﬁlms\n[10, 18, 19]. First, we deposited 1/5 of total thickness of\nPd ﬁlm at 300 C, and cooled it to room temperature.\nWe then deposited 4/5 of total thickness of Pd. After\ndeposition, post annealing was performed to ∼250 C to\nimprove the crystallinity. Finally, X-ray reﬂectivity and\nX-raycrystal truncation rod (CTR) scattering were mea-\nsured in-situ, at room temperature using synchrotron X-\nrays at 15 keV. The pressure of the chamber was kept2\n!\"#$!%#$\n&'()*+ $\n&',)*+ $\n-'.)*+ $\n-'-)*+ $-'/)*+ $\n,'0)*+ $\n,'1 $&'1 $2'1 $\n,3)!456755#$!\"# \n!\"$ \n!\"% \n!\"& \n!\"! \n'()*+,-./010+*,/2µ34(,105\n$6%7&\n89-.:*+;;/2*05<! <<! &<! 7<! %\n=+(:4>(??+@/2(AB\"/C*-,;5/'()*+,-./010+*,\n/=+(:4>(??+@ \n!\"#$\"%&#'()*+,-(.\"&#%/\n0-1 2-1 3-1 \nFIG. 1. (a) X-ray reﬂectivities of Pd(100) ultrathin ﬁlms of\nvarious thicknesses on SrTiO 3; (b) thickness dependence of\npeak/valley ratio of X-ray reﬂectivity. The thickness depe n-\ndence of magnetic moments in Ref. [10] is also shown for\ncomparison.\nlower than 1 ×10−7Pa.\nFig. 1(a)showsX-rayreﬂectivitiesforseveralPdﬁlms,\nwhose thicknesses were evaluated from the fringe peri-\nods. It was found that visibility of reﬂectivity changes\ndepended on ﬁlm thickness. To make these trends\nclearer, we plotted ratios of peak-to-valley values in the\nfringes (i.e., peak/valley), as shown in Fig. 1(b). The\npeak/valley values oscillate as ﬁlm thickness increases.\nSince visibility reﬂects ﬁlm roughness in general, high\npeak/valleyvaluesindicate the ﬁlm is ﬂat. Here, wecom-\npare the present reﬂectivity results with the thickness-\ndependent magnetization in Pd(100) ﬁlms [10] prepared\nunder the same conditions as those shown in Fig. 1(b).\nIt is apparent that both peak/valley values and the mag-\nnetic moment in Pd(100) show the same oscillatory be-\nhavior. This indicates that the improved uniformity of\nthe ﬁlm structure occurs due to the appearance of ferro-\nmagnetism.\nThe thickness dependence of 00 rod proﬁles of the X-\nray CTR scatterings are shown in Fig. 2(a), which al-\nlows us to investigate the details of the ﬁlm structures.\nThe surface normal components Lof the Miller indices\nare expressed in the reciprocal lattice unit (r.l.u.) in the\nlattice constant of the SrTiO 3substrate. All ﬁlms ex-\nhibited Laue-function-like thickness fringes. The ﬁlms of\n3.3- and 4.2-nm thickness (i.e., the ferromagnetic ﬁlms),\nwhere the peak/valley ratios of the reﬂectivities are high-\nest, show regular forms of Laue-function oscillation. In\nother samples (namely near paramagnetic ﬁlms), on the\notherhand, disturbed Laue-functionoscillationswereob-\nserved. These disturbances are attributed to spread in\nﬁlm thickness and lattice constants in the ﬁlm, as re-\nvealed by the following analysis.\nFor the quantitative discussion, we analyzed the pro-\nﬁles of X-ray CTR scatterings by least-squares ﬁt. We\nassumed that the Pd(100) ﬁlms had certain distributions\n!\"#$\"%&#'()*+,-(.\"&#%/\n0-0 0-1 2-3 \n!\")+-4-.-/!\"# \n!\"$ \n!\"% \n!\"& \n!\"! \n%! &' &( &# \n)*+,-./01-23.455/67*.*8,94:5;!\"#$%& '\n!\"($%& '\n)\"*$%& '\n)\")$%& ')\"+$%& '\n(\",$%& '!\"($%&'\n)\"*$%&'!\"# \n!\"$ \n!\"% \n!\"! &''()*+',-$. $$ $! %/ %0 !\"# \n!\"$ \n!\"% \n!\"& \n!\"! \n&$ &% && &! '# \n)\")$%&')\"*,$Å'\n)\"*-$Å')\"*-$Å'./0' .10'\n)\"*-$Å' )\"*#$Å'\nFIG.2. (a)Thickness-dependent00rodproﬁles ofX-rayCTR\nscattering of Pd(100) ultrathin ﬁlms on SrTiO 3. Dashed lines\nshow the simulation data. (b) Thickness distribution of 4.2 -,\n3.8-, and 3.3-nm ﬁlms. The lattice constant of each domain\nis shown for the 3.3-nm ﬁlm.\nin both ﬁlm thickness and lattice constant on the Ti-\nO terminated SrTiO 3substrate. The ﬁtting results are\nshown in Fig. 2(a). The obtained ﬁlm thickness distri-\nbutions of samples are shown in Fig.2 (b); thicknesses es-\ntimated from reﬂectivity are 4.2 nm (ferromagnetic), 3.8\nnm (paramagnetic), and 3.3 nm (ferromagnetic). A wide\nand anomalous thickness distribution, which indicates a\ndisturbed ﬁlm structure, was observed in the paramag-\nnetic ﬁlm with a thickness of 3.8 nm. On the other hand,\nferromagnetic ﬁlms had narrow thickness distributions\nsignifying uniformity of ﬁlm structure. In addition, the\nlattice constant of each domain in the ferromagnetic 3.3-\nnm sample was evaluated from the ﬁtting, and the 17-\nmonolayers thick domain, corresponding to the peak of\nthe thickness distribution, showed ∼0.8% lattice expan-\nsion, as may be seen in Fig. 2(b). This indicates that\nthe appearance of ferromagnetism due to quantum-well\nstates in Pd(100) ultrathin ﬁlms promoted the atomic\ngrowth of ﬂat ﬁlm, which was accompanied by sponta-\nneous lattice expansion.\nThis experiment shows the signiﬁcant correlation be-\ntween the appearance of ferromagnetism and the crystal\nstructure of the ﬁlm. In order to clarify the origin of\nthis correlation, we performed a density function calcu-\nlation. The PHASE/0 program [20] using the projector\naugmented wave-type (PAW) pseudopotential [21] to the\nspin-polarized local density approximation reported by\nPerdew and Wang [22] was used. The values of the lat-\ntice constant converge to 3.84 ˚Afor fcc bulk Pd, and we\nused this value for ﬁlm-shaped Pd(100). To evaluate the\nmagnetismofPd(100)ultrathinﬁlms, aslabofVacuum(2\nmonolayers)/Pd(N monolayers)/Vacuum(3 monolayers),\n56×56×1k-points, and 36 Ry of cut-oﬀ energy was\nused. Based on this, we calculated the diﬀerence of total3\n!\"# \n!\"$ \n!\"! %&'()*+,- ./0*&'1\n#! $2 $! 2\n!\"#$ \n!\"#% \n!\"#& \n!'#( )*+,-./0/1/+230\n45%& 6! 007-+8+//9\n$& %' %& '0:-+-;-21/8<. \n0=/++>;-21/8<. \n!\"#$ !%$ !&$ $\n'()**+ ,!,' -.*.,\n/,0\"$ !1 ,,2.*3,4\n#$ \"5 \"$ 5\n6789:;)<<,/=><4!\"#$# \n!\"#$$ \n!\"#$% \n!\"#!& \n'())*+,-+./0)(/)-1Å2\n%\"3 %\"% \n4.5,/)-16 78().52!\" \n!# \n!$ \n%\n&'()*+,-./0.*,1,(*.2*,1,(*3(45\n6$7\" 6$7# $7$ $7# \n8)(9:-.2(45.;$7<.= \n..$7$.= \n.6!7$.= \n.6!7><.= \n!\"#$%&%'()*+#,-./(0.\n/10.\n/20./-0.\n/*0.\nFIG. 3. (a) Calculation prediction of thickness-dependent\nmagnetic moment in freestanding Pd(100) slabs. (b)\nThickness-dependent surface energy of freestanding Pd(10 0)\nslabs with and without ferromagnetism. (c) The diﬀerence\nin surface energy between ferromagnetic states and paramag -\nnetic states of Pd(100) slabs. (d) The total density of state s of\n10-monolayers Pd(100) slab near the Fermi energy as a func-\ntion of out-of-plane lattice expansion, where 0.0 eV means\nthe Fermi energy. (e) The out-of-plane lattice constant of t he\n15-monolayers Pd(100) slab with the appearance of quantum-\nwell-induced ferromagnetism as a function of magnetic mo-\nment.\nenergies between paramagnetic and ferromagnetic states,\nwhere the spin polarization was ﬁxed to a curtain value\nin freestanding Pd(100)[14]. We calculated the magnetic\nstates of freestanding Pd(100) from 2-23 monolayers, as\nshowninFig. 3(a). FerromagnetismappearedinPd(100)\ndepending on ﬁlm thickness, in an oscillatory manner.\nThe period of oscillation was consistent with previous\ncalculations [9] and experiments [10], although the cal-\nculated ferromagnetic moment was smaller compared to\nprevious ﬁndings. Despite the underestimation of the\nferromagnetic moment of Pd, the present calculation is\nconsistent with quantum-well induced ferromagnetism.\nIn order to investigate the stability of the ﬁlm struc-\nture, the thickness dependence of the surface energy of\nfreestanding Pd(100) was studied, as shown in Fig. 3(b),\nwherethesurfaceenergyofparamagneticPd(100),which\nis calculated by assuming no spin polarization, is also\nshown for comparison. The diﬀerence of surface energies\nbetween ferromagnetic and paramagnetic ﬁlms, shown in\nFig. 3(c), indicates that there is a gain of surface en-\nergy due to the appearance of ferromagnetism. For fer-\nromagnetic ﬁlms with a large magnetic moment of Pd\n(monolayers 4 and 9), the surface energy decreased with\nthe appearanceofferromagnetismcomparedto paramag-netic ﬁlm of equivalent thickness. This indicates that the\nferromagnetic domain is more suitable for growing, com-\npared to paramagnetic domains of equivalent ﬁlm thick-\nness. This result can explain our experimental ﬁnding\nthat ferromagnetic Pd ﬁlms have a ﬂat and uniform ﬁlm\nstructure.\nStoner’s theory means that the ferromagnetic transi-\ntion occurs if the energy gain due to exchange splitting\ncansurpasstheincreaseinthekineticenergyofelectrons.\nThis condition is satisﬁed when the system has a large\nD(ǫF). Pd has a sharp shape of density of states orig-\ninating from 4 dbands near the Fermi energy, and thus\nitsD(ǫF) is large compared to the other transition met-\nals, even in bulk. When an increase in D(ǫF) occurs by\nquantum-well states, Stoner’s criterion eq. (1) is satisﬁed\nin Pd(100), and there is an energy gain due to the ap-\npearance of spin polarization. Our results indicate that\nthe energy gain due to spin polarization reduces the sur-\nface energy and makes ferromagnetic ﬁlm ﬂat. Crystal\ngrowth that depends on such a change in surface energy\nby nano-scaling is known as “electronic growth”, which\nis the formation mechanism of ﬂat metal ﬁlms depend-\ning on ﬁlm thickness [23]. Our present ﬁnding, in other\nwords, describes magnetic-induced electronic growth.\nThe experimentally observed spontaneous growth of\nﬂat ﬁlm in ferromagnetic Pd(100) can be explained by\nthe usual Stoner theory, as mentioned above, although\nthe 0.8% of lattice expansion due to the appearance of\nferromagnetism is not explained. The lattice distortion\nbrings changes to the D(ǫF) of transition metals as well\nas quantum-well states. The lattice expansion narrows\nthe width of the band, and the shape of the total den-\nsity of states becomes sharp. Fig. 3(d) shows total den-\nsity of states near the Fermi energy of the Pd(100) ul-\ntrathin ﬁlm, calculated with various out-of-plane lattice\nconstants, where the 0.0% means the lattice constant of\nbulk fcc Pd. The D(ǫF) increases with lattice expansion.\nThis calculation result suggests that the D(ǫF) increases\ndue to the appearance of ferromagnetism. Stoner’s the-\noryshowsthe mechanism bywhich the D(ǫF) determines\nthe magnetic state. In contrast, our results indicate the\nmechanism of modulation in D(ǫF) due to spin polariza-\ntion, i.e., the inverse mechanism of Stoner’s theory.\nHere, we calculate the magnetic moment as a function\nof the out-of-plane lattice constant of Pd(100) ﬁlms in\norder to discuss the relationship between the appearance\nof ferromagnetism and spontaneous distortion [Fig. 3\n(e)]. There is no change in the lattice constant of Pd be-\ntween nonmagnetic states and weak ferromagnetic states\nwith a spontaneous moment of 0.1 µB/atom. On the\nother hand, the 0.3% lattice expansion is observed at the\nspontaneous moment of the 0.6 µB/atom, which is the\nexperimentally obtained value of the magnetic moment.\nThis result supports the existence of a mechanism by\nwhich the system is stabilized through the spontaneous\nchange in electronic states accompanied by the occur-4\n!\"#!\"# !\"#Paramagnetic # Ferromagnetic #Ferromagnetic \nLattice expansion \nHigh D(εF)#High kinetic energy #\nStabilization #\nFIG. 4. A schematic image of spontaneous lattice distortion\ninduced by the appearance of ferromagnetism.\nrence of the huge exchange splitting; that is, the inverse\neﬀect of Stoner’s theory.\nWhen the exchangesplitting occurs, the kinetic energy\nof the electrons with the majority spin increases. The\nband narrowing, caused by lattice expansion, reduces the\nnumber of electrons with high kinetic energy, and thus\nsuppresses the increase of total energy in the system by\nexchange splitting. Therefore, this mechanism explains\nthe spontaneous lattice expansion in Pd(100) in transi-\ntioning to a ferromagnetic state, as shown schematically\nin Fig. 4.\nThe Stoner criterion means that the increase in D(ǫF)\nstabilizes the ferromagnetic state more than the param-\nagnetic state. Therefore, spontaneous lattice expansion\ninferromagneticPd(100)ultrathinﬁlms canoccurtosta-\nbilize ferromagnetic states of Pd.\nThe stabilization of the magnetic states by sponta-\nneous lattice distortion was previously predicted for Fe,\nwhich is a typical ferromagnetic transition metal [24–\n26]. In previous density functional calculations, it was\nreported that the 2% lattice expansion in bulk bcc Fe\nvia the appearance of ferromagnetism altered electronic\nstates to spontaneously stabilize the magnetic states. In\nour present study, this theoretical prediction was veriﬁed\nby systematic experiments using quantum-well induced\nferromagnetism in Pd.\nIn conclusion, we investigated the thickness-dependent\nstructuralchangeinPd(100)ultrathinﬁlmsaccompanied\nby a change in magnetic state in terms of quantum-well\ninduced ferromagnetism. When spontaneous magnetiza-\ntion is induced in Pd(100) by quantum-well states, stabi-\nlization of ﬁlm structure by reduced surface energy was\nobserved. In addition, a reduction of the total energy\nof the system by spontaneous lattice distortion occurred,\naccompanied by the appearance of ferromagnetism. Our\nﬁndings suggest a mechanism for determining the mag-\nnetic states in transition metals by the mutual relationsbetweenmagnetism, quantum-wellstates, andlatticedis-\ntortion. This mechanism can be extended to other mag-\nnetic materials, opening the possibility of tailoring mag-\nnetic states and/or magnetization by appropriate struc-\ntural electronic engineering.\nWe thank Y. Watanabe for technical support with\nrespect to the X-ray diﬀraction measurements and R.\nItotani, S. Urasaki, and K. Okada for fruitful advice on\ntheoretical calculation. The synchrotron radiation ex-\nperiments were performed at the BL13XU of SPring-8\nwith the approval of the Japan Synchrotron Radiation\nResearch Institute (JASRI) (Proposal No. 2014A1675,\n2015A1775, 2015B1689). This work was supported\nby JSPS KAKENHI Grant Numbers #15J00298 and\n15H01998. One of the authors (S.S.) also acknowledges\na fellowship from the JSPS.\n∗e-mail: sakuragi@az.appi.keio.ac.jp\n[1] I.ˇZuti´ c, J. Fabian, and S. Das Sarma,\nRev. Mod. Phys. 76, 323 (2004).\n[2] S. Bader and S. Parkin,\nAnnual Review of Condensed Matter Physics 1, 71 (2010).\n[3] T. Taniyama, E. Ohta, and T. 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Shimamura, N. Ishiwata,\nK. Kobayashi, and T. Ono, Nat Mater 10, 853 (2011).\n[13] Y. Sun, J. D. Burton, and E. Y. Tsymbal,\nPhys. Rev. B 81, 064413 (2010).\n[14] S. Aihara, H. Kageshima, T. Sakai, and T. Sato,\nJ. Appl. Phys. 112, 073910 (2012).\n[15] S. Shimizu, K. S. Takahashi, T. Hatano,\nM. Kawasaki, Y. Tokura, and Y. Iwasa,\nPhys. Rev. Lett. 111, 216803 (2013).\n[16] O. Sakata, Y. Furukawa, S. Goto, T. Mochizuki,\nT. Uruga, K. Takeshita, H. Ohashi, T. Ohata,\nT. Matsushita, S. Takahashi, H. Tajiri, T. Ishikawa,\nM. Nakamura, M. Ito, K. Sumitani, T. Taka-\nhashi, T. Shimura, A. Saito, and M. Takahasi,\nSurf. Rev. Lett. 10, 543 (2003).5\n[17] M. Kawasaki, K. Takahashi, T. Maeda, R. Tsuchiya,\nM. Shinohara, O. Ishiyama, T. Yonezawa, M. Yoshimoto,\nand H. Koinuma, Science 266, 1540 (1994).\n[18] T. Wagner, G. Richter, and M. Ruhle,\nJ. Appl. Phys. 89, 2606 (2001).\n[19] S. Sakuragi, H. Tajiri, and T. Sato,\nPhysics Procedia 75, 1167 (2015).\n[20]https://azuma.nims.go.jp/software/phase .\n[21] P. E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994).\n[22] J. P. Perdew and Y. Wang,Phys. Rev. B 45, 13244 (1992).\n[23] Z. Zhang, Q. Niu, and C.-K. Shih,\nPhys. Rev. Lett. 80, 5381 (1998).\n[24] M. Shiga and Y. Nakamura,\nJournal of the Physical Society of Japan 26, 24 (1969).\n[25] M. Ekman, B. Sadigh, K. Einarsdotter, and P. Blaha,\nPhys. Rev. B 58, 5296 (1998).\n[26] S. Mankovsky, S. Polesya, H. Ebert, W. Ben-\nsch, O. Mathon, S. Pascarelli, and J. Min´ ar,\nPhys. Rev. B 88, 184108 (2013)."    },    {        "title": "2105.11647v1.Spin_State_of_Single_Molecular_Magnet__SMM__Creating_Long_Range_Ordering_on_Ferromagnetic_Layers_of_Magnetic_Tunnel_Junction__A_Monte_Carlo_Study.pdf",        "content": "Spin State of Single Molecular Magnet (SMM) Creating Long Range Ordering on \nFerromagnetic Layers of Magnetic Tunnel Junction -A Monte Carlo Study  \n \nAndrew Grizzle, Christopher D’Angelo, Pawan Tyagi * \nCenter for Nanotec hnology Research and Education, Mechanical Engineering, University of the \nDistrict of Columbia, Washington DC -20008,  USA  \nEmail of corresponding author: ptyagi@udc.edu  \n \nABSTRACT  Single molecular magnet (SMM) like paramagnetic molecules interacting with the \nferromagnetic electro des of a magnetic tunnel junction (MTJ) produce a new system that differs \ndramatically from the properties of isolated  molecules and ferromagnets. However, it is unknown \nhow far deep in the ferromagnetic electrode the impact of the paramagnetic molecule an d \nferromagnet interactions can travel for various levels of molecular spin states. Our prior \nexperimental studies showed two types of paramagnetic SMM s, the hexanuclear Mn6 and \noctanuclear Fe -Ni molecular complex es, covalently bonded to ferromagnets produced \nunprecedented strong antiferromagnetic coupling between two ferromagnets at room temperature \nleading to a number of intriguing observations. In this paper, we report Monte Carlo Simulations \n(MCS) study focusing on the impact of the molecular spin state on cross junction shaped MTJ \nbased molecular spintronics device (MTJMSD). Our MCS study focused on the Heisenberg model \nof MTJMSD and investigated the impact of various molecular coupling strengths, thermal energy,  \nand molecular spin states. To gauge the impact of the molecular spin state on the region of \nferromagnetic electrodes, we examined the spatial distribution of molecule -ferromagnet correlated \nphases. Our MCS study shows that under a strong coupling regime m olecular spin state should be \n~30% of the ferromagnetic electrode’s atomic spins to create long -range correlated phases . \n \n I.  INTRODUCTION  \n Molecules are the only mass -producible nanostructures with customizable chemical, \nelectrical, optical, and magnetic properties  that can be prod with sub -angstrom sc ale precision .  \nMolecules are extremely ve rsatile and practically billions of types are possible and so are the \nmolecule -based  devices1-3. Several molecules  such as single molecular magnets (SMM)4, \nporphyrin5, DNA6, organometallic molecules7 have a high potential to be included as the device \nelement in futuristic molecular spintronics devices (MSD). MSD fabrication requires a molecule \nof interest to be si multaneously connected with at least a source and drain -type metal electrode8. \nThe intensity of interaction can be weak if it is physically separated from the two -metal electrode \nor connected by weak bonds9. However, a molecule with functional group s like sulfur can form \ncovalent and ionic bonds leading to very strong coupling10, 11. In the strong coupling regime, \nmolecules and metal electrodes near the interface show strong hybridization of energy levels12. \nSuch strong hybridization has been observed to create novel properties on both metal electrodes \nand molecules. For example, the interaction of thiolate molecule produced magnetism in non -\nmagnetic electrode13 and further enhanced the degree of spin polarization on ferromagnets. It is \nalso well known that a molecule connected to metal electrodes cannot exhibit the properties \nmeasured in its isolated st ate. Therefore, t he combined system of metal electrodes and molecules \nbecomes a new composite system altogether13, 14. Understanding this system is extremely \nimportant to progress the field of MSD where SMM -like molecules possess  a wide range of  spin \nstates interact ing with magnetic electrodes13. Magnetic electrodes, such as nickel (Ni), cobalt  (Co), \niron (Fe), exhibit strong long -range ordering. This long -range ordering can further transport the effect of molecule -ferromagnet interaction over microscopic range. Our previous experimental \nstudies showed that M n hexanuclear15 and Fe -Ni octa nuclear  molecular c omplexes (OMC)14 based \nSMM produced long -range impacts on ferromagnetic electrodes leading to room temperature \nobservations of several orders current suppression, spin photovoltaic effects, and several  orders of \nmagni tude magnetoresistance15, 16. Other groups have also observed strong coupling between C 60 \nmolecules and ferromagnetism of the nickel e lectrodes leading to the Kondo splitting phenomenon \nwithout applying the estimated ~50 T field needed for this observation17. However, experimentally \ndetermining the spin state of a paramagnetic molecule after forming a complete MSD is extremely \nchallenging. Additionally, Density Function Theory (DFT) study is extremely challenging to \nsimulate SMM -connected to realistic large -scale MSD s with long ferromagnetic electrodes18. This \npaper inv estigat es the effect of molecular spin state on the experimentally studied cross junction \nshaped MTJMSDs15, 16 with extended ferromagnetic electrodes beyond the  molecular junction \narea. For this study, we have employed the Heisenberg Model19 of M TJMSD that showed \npromising results in our prior MCS20. This paper provid es new insights into the effect of molecular \nspin state and evaluates the properties of whole MSD. In addition, w e have varied the molecular \nspin state from low to high to observe its impact on long -range ordering on the MTJMSD.  \n \n II. METHOD   \n \n We utilized a \ncontinuous spin model  to allow \nspin vector s of the \nferromagnets ’ atoms  and \nmolecules to assume any \ndirections in a spherical \ncoordinate system21. To \nunderstand the property of \nexperimentally studied \nMTJMSD via this MC S study , \nwe focused on the Heisenberg \nmodel (Fig.1e) as a  3D analog \nof an MTJMSD (Fig.  1b)14. This \nMCS study represented a tunnel \nbarrier with empty space within \na square -shaped molecular \nperimeter (Fig.1f). The \nmolecular perimeter was a 5x5 \nsquare with 16 molecular \nanalogs; in Fig.1f, we showed a \n4x4 molecular square to \nproduce a vivid illustration.  \nParamagnetic  SMM  molecules \nof MTJMSD  (Fig.1d) were \nrepresented by the atomic scale \nanalog with adjustable spin ( Sm) \nparameter . The coupling between two FM electrode s were only caused by the  paramagnetic \nmolecules (Fig. 1f). The molecule -mediated exchange coupling between the left and right FM \nFIG. 1. MSD formed by utilizing exposed edges of (a) Bare MTJ \nto attach (b) paramagnetic molecules between two ferromagnets. \n(c) SMM and (d) OMC paramagnetic molecules connected to \nferromagnets via sulfur atom. (e) 3D Heisenberg  Model of \nMolecular device. ( f)) Exc hange coupling parameters associated \nwith molecule -ferromagnet  interactions.  \nelectrodes is governed by the molecule's coupling  with left electrode (JmL) and molecule coupling \nwith the right elec trode (JmR), respectively . To simulate the effect of change in ambient temperature \nwe varied t hermal energy (kT) of the MTJMSD Heisenberg model.  The MTJMSD energy was \ncalculated using equation (1). A new state was selected or rejected according to the Metropolis \nalgorithm21. \n \n𝑈=−𝐽𝐿(∑𝑆→\n𝑖𝑆→\n𝑖+1 𝑖∈𝐿)−𝐽𝑅(∑𝑆→\n𝑖𝑆→\n𝑖+1 𝑖∈𝑅)−𝐽𝑚𝐿(∑ 𝑆→\n𝑖 𝑖∈𝐿,𝑖+1∈𝑚𝑜𝑙𝑆→\n𝑚𝑖+1)−\n𝐽𝑚𝑅(∑ 𝑆→\n𝑚𝑖−1 𝑖−1∈𝑚𝑜𝑙,𝑖∈𝑅𝑆→\n𝑖) (1) \n  \n In this study, S is a 3D vector that represents the discrete atomic spin of FM electrodes . Smi \nvectors  represent the Sm of molecules  at ith position . Sm was varied over the 0 to 4 range. However, \nmain discussion is around the critical Sm values for which transition in the molecular device was \nobserved. JL, and JR, are the Heisenberg exchange coupling strengths for the left and right FM \nelectrodes (Fig. 1 b). The molecule’s spin was coupled with ferromagnetic electrode spin via the \nexchange coupling parameters and create a correlated system. In our MCSs, the atoms beyond the \nboundary of the MTJMSD model (Fig. 1b) were set with zero spin state21. Energy  (U), described \nin equation (1), of the whole system , was minimized by running the Markov process. Markov \nprocess generate d a new state  after 200-2000 Million iterations  to reach  a stable low energy state . \nFurther details of MCS are published elsewhere14. The units of total energy U and ex change coupling \nparameters are the same as kT. In this study, the exchange coupling parameters and kT are referred to as the \nunitless parameters. The overall magnetic moment of the MTJMSD is the sum of the magnetic moment of \nthe molecules, Left FM and Righ t FM electrodes. We have mainly focused on the molecule -induced strong \nantiferromagnetic coupling where JmL= -1 and JmR =1. The reason for the emphasis on the molecule -induced \nantiferromagnetic coupling is the observation of molecule -induced strong exchang e coupling in our prior \nexperimental work14. To make this study generic , we also varied molecular coupling strength, thermal \nenergy, molecular spin state, and MTJMSD dimensions.  \n \n III. RESULTS AND DISCUSSIONS  \n \n To investigate the impact of Sm, we recorded the magnetic moment of the FM electrodes \nand the MTJMSD as a function o f iterations steps. We generally ran a MCS o ver ~200 Million \niterations  and recorded the magnetic moment of the FM electrodes, molecules, and whole \nMTJMSD at the interval of 50,000 steps. According to our previous study , OMC induced strong \nantiferromagnetic coupling14. Since we experimentally observed molecule -induced strong \nantiferromagnetic coupling well above room temperature14, we have investigated stabilization \nstudy at kT=0.1. To represent molecule -induced strong antiferromagnetic exchange coupling , we \nfixed  JmL = -1 and JmR = 1. We varied Sm from 0 to 4 range. However, we observed that the nature \nof MTJMSD stabilization dramatically changed around Sm = ~0.2 ( Fig. 2 a). For Sm ≤ 0.1, the left \nferromagnet (Left -FM) and right ferromagnet (Right -FM) stabilized around 1200 magnetic \nmoment  (Supplementar y Material -Fig.S1). However,  MTJMSD stabilized near 2000 (Fig. 2a) . For \nSm ≥ 0.3, Left-FM and Right-FM both still stabilized around 1000. However, MTJMSD’s total \nmagnetic moment, which is the sum of the magnetic moment of Left -FM, Right -FM, and \nmolecules, started settling below the individual electrode magnetic moment around 600. This \nresult suggest s that even  though  the molecule made  the same level of strong coupling with two \nelectrodes but , Sm dictate the MTJMSD stabilization dynamics. We also explored the effect of a \nwider range of Sm (Fig. 2c) on MTJMSD and left and right FM electrodes. The Left-FM and Right FM ele ctrodes settled  around  1100, i.e., close to their maximum possible magnetic moment of FM \nelectrodes, i.e., 1250 for Sm range from 0 to 1 (Fig. 2c). Interestingly, around Sm = 0.2, molecule \nstarted forcing Left -FM and Right -FM to settle in the antiparallel state due to the molecule induced \nantiferromagnetic coupling (Fig. 2c). This result suggests that strong exchange coupling between \nmolecule and FM electrodes can only impact MTJMSD when the molecular spin magnitude is \nabove a critical value. For Sm = 4, we saw FM electrode and MTJMSD stabilization patte rn like \nthat of Sm = 1 (Supplementar y Material -Fig.S2).  However, major difference was that from a very \nearly stage the MTJMSD magnetic moment became lower than  that of  Left-FM and Right -FM \nelectrodes. It means increasing Sm promoted early stabilization of MTJMSD into an \nantiferromagnetic state.  \n  \n  \n Understanding the spatial range of  Sm on the electrode is critical in understanding how far molecule \ninfluence can penetrate FM electrodes. For the calculation of spatial correlation between molecular spin \nstate and the ma gnetic electrode’s spin state, we calculated the dot product between the average magnetic \nmoment of the molecules with each atom’s magnetic moment in Left -FM and Right -FM and termed this \nproduct as correlation factor. To produce a 2D spatial correlation fa ctor graph, we averaged the data along \nthe width of the FM electrode; here, width is the shorter dimension parallel to the molecular plane (Fig.1e).   \n We also investigated the spatial magnetic susceptibility of MTJMSD. For the magnetic \nsusceptibility  calculation , the magnetic moment of 16 molecules were utilized . However, for the \ncalculation of spatial magnetic susceptibility of the FM electrodes the magnetic moment (m) of \neach atom present along the width dimensions , shorter dimension parallel to the molecular plane,  \nof each FM electrodes were utilized (Equation 2)21  \n \n𝜒=𝑘𝑇.N(⟨𝑚2⟩−⟨𝑚⟩2)  -------------- (2) \n \n For the case of Sm = 0.1, molecules ’ magnetic susceptibility (ꭓ) was very high as compared \nto the two FM electrodes (Fig.  4a). A higher ꭓ for molecule suggest s that for Sm = 0.1 external \nmagnetic  field can align the molecular spin vector selectively. However, for Sm = 0.3 case , the \nmagnitude of for  molecule were around 4 and 0, respectively (Fig. 4b). For Sm = 1, this difference \nbetween the ꭓ for molecules and magnetic electrode were ~1 and 0, respectively (Fig. 4c). \nUltimately, for Sm = 4, the value of ꭓ for molecul es and FM electrodes was almost the same and \nnear 0 (Fig.4d). This study suggests that if an MTJMSD posses ses strongly exchange -coupled high \nFIG .2. Iteration count vs magnetic moment of MTJMSD, left FM, and right FM for (a) Sm = 0.1, (b) Sm \n= 0.3, (c) MTJMSD and Fm electode  magnetic moment for molecular spin ranging 0 to 1. For all the \ncases kT = 0.1, JmL = -1 and JmR = 1.  \nspin molecular magnets , then realizing selective switching of molecules will be extremely \nchallenging.   \n   \n In the data discussed in figures 2 -4 we only \ndiscussed kT = 0.1 and  JmL = -1 and JmR = 1. To make \nthis study applicable for a wide range of possibilities , we \ninvestigated the effect of thermal energy and molecular \ncoupling strengths on MTJMSDs for different  Sm. \nTo investigate the effect of thermal energy , we \nvaried kT from 0.01 to 1.1. The molecular coupling \nstrength was varied by ensuring that the modulus of \nJmL and JmR were  equal. To consider both cases, \nmolecule inducing ferromagnetic and antiferromagnetic  \ncoupling, JmL was varied from -1 to 1 range. In this case  \nJmR was equal to | JmL |. The c ontour plot for Sm = 0 shows \nthat MTJMSD’s magnetic moment settled in high and \nlow magnitude state irrespective of the sign and \nmagnitude of JmL and JmR (Supplementary Material -Fig. \nS4). Increasing kT settled  MTJMSD into a highly \ndisordered state producing  a low MTJMSD magnetic \nmoment. Contour plot for Sm = 1 and  kT<0.2  the \nMTJMSD ’s magnetic moment remained close to 300 -\n900 for negative  JmL and JmR=| JmL | (Fig. 5a). \nInterestingly, MTJMSD low magnetic moment state was \nmore prevalent on both side s of the JmL = -0.5. As kT \nincreased , the MTJMSD  started attaining the higher \nmagnetic moment and finally settled into a low magnetic \nmoment state due to thermal energy induced disordering \n(Fig. 5a). Contour plot for Sm = 1 and  kT<0.2  the \nMTJMSD’s magnetic moment was as high as ~2400 for \npositive JmL and JmR= JmL  (Fig. 5a). For a positive sign \nof JmL and JmR, as kT increased , the MTJMSD’s magnetic \nmoment started attaining the lower higher magnetic \nmoment and finally settled into a low magnetic moment state due to thermal energy induced disordering \nFIG. 5. Contour plots of magnetic moment \nof MTJMSD as a function of kT and JmL & \n|JmR| for (a) Sm = 1, and (b) Sm = 4.   \nFIG . 4. Magnetic susceptibility(ꭓ) of FM electrodes and molecular layers of MTJMSD  for (a) Sm = \n0.1, (b) Sm = 0.3, (c) Sm = 1, and (d)  Sm = 4. For all the case s kT = 0.1, JmL = -1 and JmR = 1.  (Fig. 5a). The contour plot for Sm = 4 was somewhat similar to that of Sm = 1 (Fig. 5b). However, for Sm =4 \nand kT<0.2 , the MTJMSD’s magnetic moment persisted around ~500 for weaker molecular coupling. For \ninstance, MTJMSD magnetic moment state that was seen for Sm = 1 around kT = 0.1-0.2 for JmL ≤-0.9 was \nseen for Sm = 4 around kT = 0.1 -0.2 for JmL ≤-0.6 (Fig. 5 b). Also, MTJMSD magnetic moment state that was \nseen for Sm = 1 around kT=0.1 -0.2 for over very tight space for positive JmL (0.6-1) was seen over a broad \nrange for Sm = 4 around kT = 0.1 -0.2 for 0.2  ≤ JmL ≤1 (Fig. 5b).  Sm  played an important role in deciding the \noverall MTJMSD magnetic moment.  \n We also investigated the effect of  Sm and thermal energy on various parts of the MTJMSDs (Fig. \n6). For this study , we focused on  Sm ranging from 0 to 0.4 and kT ranging from 0.01 to 0.5  for JmL =-1 and \nJmR =1. The ranges of  Sm and kT is selected to focus on major transitions observed in Fig. 2 and Fig. 5. In \nthe contour plot of MTJMSD’s magnetic moment was ~2000 for Sm <0.2 and kT <0.1 (Fig. 6a). However, \nas Sm goes beyond 0.2 , MTJMSD started to settle in the low magnetic moment state due  to molecule -induced \nstrong antiferromagnetic coupling (Fig.  6a). This result is congruent with the data shown in Fig.  2c. It is \nimportant to note that with increasing kT, for Sm <0.2, MTJMSD lo ses a high magnetic  moment state very \nrapidly as compared to the variations observed for Sm >0.2 (Fig.  6a). It is apparent that MTJMSD magnetic \nmoment start s to get coupled with molecular spin state for Sm >0.2, which  remain s stable for higher thermal \nenergy. The molecule’s cumulative magnetic moment also get s impacted due to kT (Fig. 6b). The n et \nmagnetic moment of the molecule g ot disturbed with a slight increase in kT (Fig. 6b). However, as  kT \nincreases the  molecular magnetic moment persisted more for the higher magnitude of Sm. However, Left -\nFM (Fig.  6c) and Right -FM (Fig.  6d) both showed high magnetic moment for kT <0.2 over 0 -0.4 \nmolecular spin magnitude. Electrode  finally settled into a thermally induced di sturbed low \nmagnetic moment state (Fig.  6c-d). The m ain message this study suggest s is that uniform \nmolecular magnetic moment existed around linear boundaries on Sm vs kT graph  (Fig.  6b).  \n We also investiga ted the effect of MTJMSD’s along with Sm. For this study, we changed \nthe length of the Left -FM and Right -FM electrodes from 50 to 200, keeping the width and height \nto 5. The quick analysis of the spatial \ncorrelation factor indicated that for \nMTJMSD of 50 a tom length, the \nmolecules were strongly correlated \nwith the magnetic moment of the Left -\nFM and Right -FM (Fig. 7a); however, \nfor 200 atomic length MTJMSDs, \nmolecules were only correlated to the \nFM electrodes near the junction area \n(Fig. 7b). Similarly, we a lso increase \nthe thickness of each FM electrode \nfrom 5 to 25, while the length and \nwidth were fixed to 50 and 5, \nrespectively.  Spatial correlation data \nfor the extreme case of thickness = 25 \nsuggest that Left -FM and Right –FM \nelectrodes were weakly correl ated \nwith the molecules’ magnetic \nmoment. However, unlike 200 atomic \nlength MTJMSD, the spatial \ncorrelation factor was relatively \nuniform over the whole MTJMSD for \n50 atoms thick MTJMSD (Fig. 7c). \nFIG .6 Contour plot showing m agnetic moment for  \nthermal energy (kT)  and  Sm for (a) full MTJMSD , (b)  \nmolecular layer , (c) Left-FM, and (d)  Right -FM. For all \nthe case s molecular coupling was  JmL = -1 and JmR = 1.  For further investigation, we plotted the magnetic moment o f the MTJMSD and two FM electrodes \nas a function of the electrode length (Fig. 7d).   The effect of molecule -induced strong exchange \ncoupling could force the large area of MTJMSD only for short lengths (Fig. 7d). As length doubled, \nMTMSD’s Left -FM and Righ t FM electrode stop aligning perfectly antiparallel to each other, and \nmany metastable phases started becoming possible. As length increased to 150, the MTJMSD \nmagnetic moment was in between the Left -FM and Right -FM electrodes (Fig. 2c). It is apparent \nthat as the length of the electrode increases to 150 or more, FM electrodes appear to have multiple \nphases leading to lowered magnetic moment (Fig. 7d). Since increasing length did not allow the \nantiparallel alignment of the two FM electrodes over the full le ngth hence MTJMSD’s net \nmagnetic moment was significantly high. The increase in thickness of the FM electrode was more \ninfluential in determining the Sm effect on MTJMSD (Fig. 7e). Generally, increasing thickness \nforced MTJMSD to settle in a higher magneti zation state above the individual FM electrode’s \nmagnetic moment (Fig. 7e). Interestingly, for the 20 -atom thick FM electrode thickness, the \nMTJMSD’s magnetic moment was consistently below the FM electrode magnetic moment. Each \ndata point in Fig. 7d -e was repeated five times, and simulations were conducted for 2 billion \niterations to ensure we reach an equilibrium state. We hypothesized that changing the dimensions \nof the FM electrode impacted the stabilization dynamics; for the 20 -atom thick FM electrode \nthickness, the equilibrium magnetization state was akin to 5 atoms thick FM electrode (Fig. 7e). \nThe size effect data shown in Figure 7 provide direct insights into the consequences of varying the \nFM electrode dimensions.            \n  \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFIG . 7. Effect of FM electrode length and thickness: Molecule -FM correlated \nphases for FM electrodes with (a)length=50, width=5, thickness=5, (b) \nlength=200, width=5, thickness=5, and (c)length=50, width=5, thickness=25.  \nMTJMSD magnetization as a function of FM e lectrode’s (d) length and (e). For \nall the case s Sm = 0.3, kT = 0.1, JmL = -1 and JmR = 1.  \nFIG . 7 Spatial correlation factor for FM electrodes with (a) Length  = 50, \nWidth=5, and Height =5  (b) Length  = 200, Width=5, and Height =5 , (c) Length  \n= 50, Width= 25, and Height =5 . (d) Magnetization vs FM Electrode length (e) \nMagnetization vs. FM electrode thickness .  For all the case s Sm = 0.2. kT = 0.1, \nJmL = -1 and JmR = 1. Here , width =thickness of FM electrodes.  IV. CONCLUSION  \n  \nWe conducted a Monte Carlo simulation to study the impact of the molecular spin state ( Sm) on \nthe MTJMSD and ferromagnetic electrodes. This research produced a number of lessons that may \nhelp in understanding and designing futuristic molecular spintronics devices. (1)  In the strong \ncoupling regime, the molecular spin state must be above 0.2 to create a random state to antiparallel \nFM electrodes in an MTJMSD. (2) Switchabl e MTJMSD is only possible for a low molecular spin \nstate. (3) In a strong ferromagnetic coupling regime, increasing molecular spin state to S=4 enabled \nfast equilibration and enhanced the thermal stability molecule induced high magnetic moment. (4) \nMagneti c electrode thickness and length are critical in determining the molecular spin state effect. \nWe will focus on studying the effect of spin fluctuations on MTJMSD with different molecular \nspin states in future work.  \nACKNOWLEDGEMENT  \nThis research is supported by National Science Foundation -CREST Award (Contract # HRD - 1914751), \nDepartment of Energy/ National Nuclear Security Agency (DE -FOA -0003945). Author contributions: \nAndrew Grizzle  conducted simulations studies . Andrew Grizzle developed analysis software to analyze the \ndata and Christopher D’Angelo wrote C++ program under supervision of Pawan Tyagi. and Andrew Grizzle \nwrote the manuscript and an alyzed the data.  \n \nDATA AVAILABILITY STATEMENT  \nThe data that support the findings o f this study are available from the corresponding author upon \nreasonable request.  \nREFERENCES  \n1. L. Bogani and W. Wernsdorfer, Nat. Mater. 7 (3), 179 -186 (2008).  \n2. D. F. Li, S. Parkin, G. B. Wang, G. T. Yee, R. Clerac, W. Wernsdorfer and S. M. Holmes, J. Am. \nChem. Soc. 128 (13), 4214 -4215 (2006).  \n3. M. Fonin, S. Voss, S. Herr, G. de Loubens, A. D. Kent, M. Burgert, U. Groth and U. Rudiger, \nPolyhedron 28 (9-10), 19 77-1981 (2009).  \n4. A. J. Epstein, MRS Bull. 28 (7), 492 -499 (2003).  \n5. M. Jurow, A. E. Schuckman, J. D. Batteas and C. M. Drain, Coord. Chem. Rev. 254 (19-20), \n2297 -2310 (2010).  \n6. M. 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(Clarendon \nPress, Oxford, 1999).  \n \n \n Supplementar y Material  \nSpin State of Single Molecular Magnet (SMM) Creating Long Range Ordering on \nFerromagnetic Layers of Magnetic Tunnel Junction -A Monte Carlo Study  \n \nAndrew Grizzle, Christopher D’Angelo, Pawan Tyagi * \nCenter for Nanotec hnology Research and Education, Mechanical Engineering, University of the \nDistrict of Columbia, Washington DC -20008,  USA  \nEmail of corresponding author: ptyagi@udc.edu  \n \n \n \nFig. S1: M vs. time for Sm=0.  \n \n \nFig.S2: Magnetizatio n vs. Sm for 0 -4 Sm range  \n \nFig. S3: Spatial correlation fa ctor for Sm=0.4  \n \nFig. S4: kT vs Jm L for Sm=0. JmR=|JmL|.  \n"    },    {        "title": "0909.4917v2.Correlated_versus_Ferromagnetic_State_in_Repulsively_Interacting_Two_Component_Fermi_Gases.pdf",        "content": "arXiv:0909.4917v2  [cond-mat.quant-gas]  14 Dec 2009Correlated vs Ferromagnetic State in Repulsively Interact ing Two-Component Fermi\nGases\nHui Zhai\nInstitute for Advanced Study, Tsinghua University, Beijin g, China, 100084\n(Dated: September 15, 2021)\nWhether a spin-1 /2 Fermi gas will become ferromagnetic as the strength of repu lsive interaction\nincreases is a long-standing controversial issue. Recentl y this problem has been studied experimen-\ntally by Jo et al, Science, 325, 1521 (2009) in which the authors claim a ferro magnetic transition is\nobserved. This work is to point out the results of this experi ment can not distinguish whether the\nsystem is in a ferromagnetic state or in a non-magnetic but st rongly short-range correlated state. A\nconclusive experimental demonstration of ferromagnetism relies on the observation of ferromagnetic\ndomains.\nItinerant ferromagnetism is a common phenomenon in\nnature, but not yet well understood. Rigorous examples\nof itinerant ferromagnetic ground state have only been\nobtained for a few speciﬁc cases. For instance, Nagaoka\nshows that for inﬁnite strong repulsive interaction, in a\nbipartite lattice the ground state is ferromagnetic if one\nhole is doped into a half-ﬁlled system [1]. Lieb shows for\na half ﬁlled bipartite lattice, the ground state of repul-\nsively interacting fermions has non-zero spin if the num-\nber of total lattice site of each sub-lattice is not equal\n[2]. Mielke [3] and Tasaki [4] propose a class of mod-\nels whose single particle ground states have degeneracy,\nand show they become ferromagnetic with repulsive in-\nteractions. However, there is no conclusive results for a\ngeneric dispersion and ﬁlling number.\nStoner considered spin-1 /2 fermions with short range\ninteractions, spin polarization can lower the interaction\nenergy since two spin align fermions will not interact due\nto the Pauli exclusion principle, while it costs the ki-\nnetic energy. With Hatree-Fock approximation, one can\nconclude that when UN(EF)>1 there exists a second-\norderferromagneticphasetransition[5, 6], where Uisthe\ninteraction strength and N(EF) is the density-of-state\nnearby the Fermi surface. This is known as Stoner crite-\nria. Fors-wave scattering, this condition corresponds to\nkFas> π/2, where kFis the Fermi momentum, and asis\nthe s-wave scattering length. Higher order perturbation\nof interactions will lower the critical value of kFas, and\nmay change the transition to ﬁrst order [7].\nMany authors have proposed to study itinerant ferro-\nmagnetism transition using two-component Fermi gases\nwhereascan be tuned by Feshbach resonance [8]. Based\non the physical picture above, in a trapped system one\nshould observe non-monotonic dependence of the kinetic\nenergy with the increase of as, namely, the kinetic energy\nshall ﬁrst decrease before ferromagnetic transition due to\nthe expansion of the cloud, and then increase after the\ntransition. The inelastic collision rate shall ﬁrst increase\nand then decrease as diﬀerent components begin to sep-\narate spatially [9]. Recently, a beautiful experiment by\nJoet al[10] have observed all these monotonic features,\nand the agreement between experiment and ferromag-netic theory [9, 11] leads to the claim that this has shown\nexperimentally a ferromagnetic transition in continuum\nwithout particularrequirement of lattice and band struc-\nture [10].\nHowever, the itinerant ferromagnetic issue is in fact\nmore complicated than this. The question is, whether\nspin polarizationisthe only wayto reduceinteractionen-\nergy. Theansweris no. InthecontentofHubbardmodel,\nGutzwiller constructed his famous projected wave func-\ntion as/producttext\ni(1−ηni↑ni↓)|Ψ0/angb∇acket∇ight, where|Ψ0/angb∇acket∇ightis free fermion\nFermi sea, and iis the index of the lattice site. The pro-\njection operator/producttext\ni(1−ηni↑ni↓) (η >0) suppresses the\nprobability of having two fermions at the same lattice\nsite, and consequently reduces on-site interaction energy\n[12]. This state is non-magnetic if |Ψ0/angb∇acket∇ightis chosen as non-\nmagnetic state. Hereafter we shall call this state “corre-\nlated state” to distinguish it from “ferromagnetic state”.\nNevertheless, we shall note this state is not an exotic\nstate but still a Fermi liquid state, we use the term “cor-\nrelated state” in the sense that the projection operator\nintroduces strong short-range correlation into this state.\nIn continuum, a Jastrow factor can play the role of the\nprojection operator.\nIn the Hubbard model, using the projected wave func-\ntionasavariationalwavefunction, Gutzwillershowsthat\nat low-density, the correlated state has lowerenergy than\na ferromagnetic state [12]. An alternative view is that\nthe short-range correlation, which has been ignored in\nthe Hatree-Fock and perturbation treatment, will signif-\nicantly renormalize down the interaction. Kanamori ar-\nguedthattheup-boundoftheeﬀectiveinteractionshould\ncorrespond to the kinetic cost to put a node in the wave-\nfunction where two fermions overlap, which should alway\nbe ﬁnite even when bare interaction goes to inﬁnite, and\nhe also argued that the renormalized interaction is not\nsuﬃcient for ferromagnetic transition at low density [13],\nwhich is supported by some later calculations [14].\nIn short, the key of the itinerant ferromagnetism prob-\nlem is whether the system will choose spin polarization\nor building up short-range correlation to reduce interac-\ntion energy as the strength of interaction increases. The\nadvantage of cold atom is to provide an opportunity for2\na direct quantum simulation of the Stoner model, and\nhopefully can settle the issue of itinerant ferromagnetism\nexperimentally. Sothe question comestowhether the ex-\nperiment of Ref. [10] has conclusively settled the issue.\nThe answer is no. The purpose of this Rapid Communi-\ncation is to point out a non-magnetic “correlated” state\ncan explain the main observation of Ref. [10] equally\nwell as a ferromagnetic state, in another word, from the\nexisting experimental results, it is very hard to distin-\nguish whether the system is in a “correlated” state or in\na ferromagnetic state. Further experimental eﬀorts are\nrequired to distinguish them.\nEquation-of-state for a “correlated” state. Let us ﬁrst\nconsider two-component fermions in free space (without\noptical lattice and harmonic trap), the Hamiltonian is\ngiven by\nH=/summationdisplay\nrσ\ni,σ=↑,↓−/planckover2pi12∇2\ni\n2m+/summationdisplay\nr↑\ni,r↓\njv(r↑\ni−r↓\nj) (1)\nwherev(r↑\ni−r↓\nj) is a short-range pairwise interacting\npotential. For a non-polarized free Fermi sea |Ψ0/angb∇acket∇ight=\nDet(eikir↑\ni)Det(eikjr↓\nj), the kinetic energy of each compo-\nnent is given by E0\nkin= 3EFn/5, where EF=/planckover2pi12k2\nF/(2m),\nnis the density of each component, and n=k3\nF/(6π2).\nFor a Fermi sea, the interaction energy is proportional to\nthek= 0 Fourier component of v(r) (denoted by v0), i.e.\nE0\nint=/angb∇acketleftΨ0|/summationdisplay\nr↑\ni,r↓\njv(r↑\ni−r↓\nj)|Ψ0/angb∇acket∇ight=v0n2.(2)\nAway from a Feshbach resonance, E0\nint= 4π/planckover2pi12asn2/m.\nNow we consider Gutzwiller’s projected wave function\nin continuum |Ψ/angb∇acket∇ight=P|Ψ0/angb∇acket∇ightas a class of varational states.\nWith the projection operator, the probability of having\ntwo spin-opposite fermions closely changes from n2to\n(1−g)n2, and the interaction energy decreases if g >0\nand increases if g <0, thus the interaction energy shall\nlinearly depend on the “projection strength” gas\nEint=4π/planckover2pi12as\nmn2(1−g) (3)\nBy dimension analysis the kinetic energy shall be of the\nform\nEkin=/angb∇acketleftΨ|/summationdisplay\nrσ\ni,σ−/planckover2pi12∇2\ni\n2m|Ψ/angb∇acket∇ight=3\n5EFnw(g),(4)\nwherew(g) is a dimensionless function of g. There are\nsome simple properties of w(g) one can make use of. For\ng= 0, there is no projection and the free Fermi sea is the\nstate that minimizes the kinetic energy, thus w(0) = 2.\nIfg/negationslash= 0, both positive and negative gwill lead to the\nincrease of the kinetic energy, thus g= 0 is the minimumofw(g), namely, ∂w(g)/∂g|g=0= 0. Hence, up to the\nsecond order of g, one has the form\nEkin=3\n5EFn(2+αg2) =4π/planckover2pi12\nma0n5/3(2+αg2) (5)\nwhereα >0, anda0= 3(6π2)2/3/(40π) = 0.36. We shall\nnow stress that the purpose of this work is neither to\nrigorously derive this equation-of-state and calculate the\nnumber of α, nor to prove theoretically that this state\ncan energetically do better than a ferromagnetic state.\nInstead, we shall take Eq. 3 and Eq. 5 together as a sim-\nple “phenomenological ” equation-of-state for this class\ncorrelated state, and the key of work is to point out the\ngeneral behavior of this correlated state in trap, which\ndoes not depend on the speciﬁc value of α, and hereafter\nwe shall use αas an unspeciﬁed parameter.\nFor a given density nandas, one shall ﬁrst minimize\nthe free energy with respect to g. Forasn1/3≤2αa0,\ng=asn1/3/(2αa0). In this regime,\nEkin=4π/planckover2pi12\nm/bracketleftbigg\na0n5/3/parenleftbigg\n2+a2\nsn2/3\n4a2\n0α/parenrightbigg/bracketrightbigg\n(6)\nEint=4π/planckover2pi12\nm/bracketleftbigg\nasn2/parenleftbigg\n1−asn1/3\n2a0α/parenrightbigg/bracketrightbigg\n(7)\nthe total energy\nEtot=4π/planckover2pi12\nm/bracketleftbigg\n2a0n5/3+asn2−a2\nsn7/3\n4a0α/bracketrightbigg\n(8)\nand the chemical potential\nµ=4π/planckover2pi12\nm/bracketleftbigg5a0n2/3\n3+asn−7a2\nsn4/3\n24a0α/bracketrightbigg\n.(9)\nForasn1/3>2αa0,g= 1. In this regime,\nEkin=4π/planckover2pi12\nm/bracketleftBig\na0n5/3(2+α)/bracketrightBig\n, (10)\n12345askF0.51.01.52.02.53.0E/LParen1Ekin0/RParen1\nFIG. 1: (Color online) The total energy (blue solid line), th e\nkinetic energy (red dashed line) and the interaction energy\n(black dotted line) of the “correlated state” as a function o f\naskF. For this plot we set α= 1.E0\nkinis the kinetic energy of\na free Fermi sea /planckover2pi12k2\nF/(2m).3\nandEint= 0, the total energy\nEtot=4π/planckover2pi12\nm/bracketleftBig\na0n5/3(2+α)/bracketrightBig\n(11)\nand the chemical potential\nµ=4π/planckover2pi12\nm/bracketleftbigg5a0n2/3\n6(2+α)/bracketrightbigg\n. (12)\nThe kinetic, interaction and total energy (in unit of E0\nkin)\nas a function of askFare illustrated in Fig. 1. When\ng= 1 at very large kFas, the energy of a correlated state\nis lower than a fully polarized ferromagnetic state if α <\n25/3−2∼1.17.\nTrapped System. From the discussion above, we have\nobtained the relation µ(n,as). For a given as, one can\ninvert this relation to obtain n(µ,as). Considering the\nharmonic trapping potential Vtrap(r) = (mω⊥(x2+y2)+\nmωzz2)/2, weshalluselocaldensityapproximationtore-\nplaceµwithµ0−Vtrap(r) andbysolvingthetotalnumber\nof particle constraint/integraltext\nd3rn(µ0−V(r),as) =N, one can\nobtainµ0(N,as). Then the local fermion density is given\nbyn(r) =n(µ0(N,as)−V(r),as). Using the expres-\nsions for kinetic and interaction energy density discussed\n0123450.30 0.35 0.40 0.45 \nask0Ek/LParen1Ek0/RParen1\n0123450.00 0.02 0.04 0.06 \nask0Ei/LParen1Ek0/RParen1\n0123451.00 1.05 1.10 1.15 1.20 \nask0Μ0/LParen1Ek0/RParen10123450.38 0.40 0.42 0.44 0.46 \nask0Ep/LParen1Ek0/RParen1\n0123450.75 0.80 0.85 0.90 \na\"k0Et/LParen1Ek0/RParen1\n0123450.00 0.05 0.10 0.15 0.20 0.25 \na\u0000k0/CapGamma/LParen1/CapGamma0/RParen1(a) (b)\n(c) (d)\n(e) (f)×5×10 \nFIG. 2: (Color online) From (a)-(e) is the kinetic energy\nper particle Ek=Ekin/(2N), the interaction energy per\nparticle Ei=Eint/(2N), the potential energy per particle\nEp=Epot/(2N), the total energy per particle Et=Etot/(2N)\nand the chemical potential µ0as a function of ask0.k0is the\nFermi momentum for free Fermi gas at the center of the trap,\nand the energy unit is taken as E0\nk=/planckover2pi12k2\n0/(2m). (f) the\nthree-body loss rate as a function of ask0.α= 1 for blue\nsolid line, α= 0.75 for red dashed line and α= 0.5 for black\ndotted line.above, one can compute the total kinetic and interac-\ntion energy as Ekin/int=/integraltext\nd3rEkin/int(n(r),as), and the\npotential energy is given by Epot=/integraltext\nd3rVtrap(r)n(r),\nand the total energy is Etot=Ekin+Eint+Epot. The\nloss rate is computed in a very phenomenological way as\nΓ = 2Γ 0(ask0)6/integraltext\nd3rn3(r)(1−g(n(r)) [15].\nAs shown in Fig. 1, for a uniform system the kinetic\nenergy for a correlated state monotonically increases for\nanyas>0. To show whether for small asthe kinetic en-\nergywillﬁrstdecreasewiththeincreaseof asinatrapped\nsystem, we shall note\n∂Ekin\n∂as=/integraldisplay\nd3r/parenleftbigg∂Ekin\n∂n∂n\n∂µ∂µ0\n∂as+∂Ekin\n∂as/parenrightbigg\n.(13)\nThe ﬁrst term is negative and the second is positive. It is\nimportant to note that when as→0 the ﬁrst term does\nnot vanish while the second term does, since µ0linearly\ndepends on aswhileEkinquadratically depends on as,\ntherefore the ﬁrst term is always dominative in small as,\nwhich gives ∂Ekin/∂as<0, and leads to a non-monotonic\nbehavior of kinetic energy.\nWe consider the experimental condition as Ref. [10],\ni.e.N= 6.5×105andωz/ω⊥= 7/30. The results\nare shown in Fig. 2. Comparing them with the predic-\ntion of a ferromagnetic state, for instance, Fig 1 and 2\nof Ref. [9] and Fig. 2 of Ref. [11], they display simi-\nlar non-monotonic behavior and also qualitatively agree\nwith the observation of Ref. [10]. This leads to the main\npoint of this work, that is, the non-monotonic behavior\nobserved in Ref. [10] is not suﬃcient to distinguish a fer-\nromagnetic state from a non-magnetic correlated state,\nandthusnotconclusiveformakingtheclaimofferromag-\nnetictransition. Weemphasizethat despiteofthesimilar\nnon-monotonic behavior, there is no phase transition in\nthis scenario. In fact, the suppression of interaction en-\nergy and the inelastic collision rate due to correlation is\nnot surprising in strongly interacting systems. Quantum\nHall eﬀect and the Tonk gas of one-dimensional bosons\nare two of the examples. Suppression of the three-body\nrecombinationrate has been observed in one-dimensional\nBose gas as it approaches the Tonk gas regime [16].\nDiscussions. There are a few points we would like to\ncomment on before ending. First, there are some quanti-\ntative diﬀerences between the results of Fig. 2 and that\nfrom a ferromagnetic theory (for instance, Fig 1 of Ref.\n[9]). In Fig. 2, the extreme of kinetic energy, potential\nenergy and the loss rate are not very close, while they are\nvery close in the ferromagnetic theory prediction. And\nthere is nomaximum in the chemicalpotential (i.e. cloud\nsize) plot of Fig 2(e). However, both calculation above\nand the theoretical work of Ref. [8, 9, 11] are not quan-\ntitatively correct. The important eﬀect of Feshbach res-\nonance and unitary limit of the repulsive interaction is\nnot taken into account. For instance, the Hatree-Fock\nenergy of a free-Fermi gas is taken as linearly increasing\nwithas, while the accurate Hatree-Fock energy should4\nbe smaller and saturates at large as. The resonance\nphysics has to be taken into account seriously for mak-\ning a quantitative comparison between theory and ex-\nperiments, for instance, the value kFasof kinetic energy\nturning point, and for constructing a correct microscopic\nFermiliquidtheory. And forthe correlatedstate, the cor-\nrection should be treated more seriously rather than the\nphenomenological way presented above, for instance, by\nquantum Monte Carlo simulation. It remains to be seen\nwhether these quantitative diﬀerence between the pre-\ndiction of two scenarios can be used to distinguish these\ntwo states, when a more careful analysis in the theory is\ndone. We leave this for follow up works.\nSecondly, a conclusive experimental evidence of ferro-\nmagnetism is the observation of ferromagnetic domains.\nRef. [10] fails to observe the ferromagnetic domains.\nThey attribute this reason to short lifetime that prevents\nthesystemtoreachequilibrium. However,oneshouldno-\ntice that this systemis the sameaswhat hasbeen used to\nstudy BEC-BCS crossover before. Maybe there is some\nparticular physics reason to believe the relaxation time\nis particularly long in this case than in the case of BEC-\nBCS time. If it is the case, the dynamics remains to be\nexplored.\nAcknowledgment: The author would like to thank Tin-\nLun Ho for helpful comments on the manuscript.\n[1] Y. Nagaoka, Phys. Rev. 147, 392 (1966)\n[2] E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989)[3] A. Mielke, J. Phys. A 24, L73 (1991); ibid,25, 3311\n(1991) and ibid,25, 4335 (1992)\n[4] H. Tasaki, Phys. Rev. Lett. 69, 1608 (1992); ibid,75,\n4678 (1995) and A. Tanaka and H. Tasaki, Phys. Rev.\nLett.98, 116402 (2007)\n[5] E. Stoner, Phil. Mag. 15, 1018 (1933)\n[6] C. J. Pethick and H. Smith, Bose-Einstein Condensation\n(Cambridge University Press, Cambridge 2002) Chapter\n14.2.,\n[7] D. Belitz, T. R. Kirkpatrick, and T. Vojta, Phys. Rev.\nLett.82, 4707(1999) andD.Belitz andT.R.Kirkpatrick,\nPhys. Rev. Letts. 89, 247202 (2002)\n[8] M. Houbiers, R. Ferwerda, H. T. C. Stoof, W. I.\nMaAlexander, C. A. Sackett and R. G. Hulet, Phys. Rev.\nA56, 4864 (1997); T. Sogo, H. Yabu, Phys. Rev. A 66,\n043611 (2002) and R. A. Duine and A. H. MacDonald,\nPhys. Rev. Lett. 95, 230403 (2005)\n[9] L. J. LeBlanc, J. H. Thywissen, A. A. Burkov, and A.\nParamekanti, Phys. Rev. A 80, 013607 (2009)\n[10] G. B. Jo, Y. R. Lee, J. H. Choi, C. A. Christensen, T.\nH. Kim, J. H. Thywissen, D. E. Pritchard, W. Ketterle,\nScience, 325, 1521 (2009)\n[11] G. J. Conduit and B. D. Simons, arXiv: 0907.3725\n[12] M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963) and\nPhys. Rev. 137, A1726 (1965)\n[13] J. Kanamori, Prog. Theor. Phys. 30, 275 (1963)\n[14] Y. M. Vilk, L. Chen, and A.-M. S. Tremblay, Phys. Rev.\nB49, 13267 (1994); L. Chen, C. Bourbonnais, T. Li,\nand A. M. S. Tremblay, Phys. Rev. Lett. 66, 369 (1991);\nvon der Linden and D. M. Edwards, J. Phys: Condens.\nMatter3, 4917 (1991)\n[15] D. S. Petrov, Phys. Rev. A 67, 010703(R) (2003)\n[16] B. L. Tolra, K. M. OHara, J. H. Huckans, W. D. Phillips,\nS. L. Rolston, and J. V. Porto, Phys. Rev. Lett. 92,\n190401 (2004)"    },    {        "title": "0812.2171v2.Coexistence_of_triplet_superconductivity_and_itinerant_ferromagnetism.pdf",        "content": "Coexistence of triplet superconductivity and itinerant\nferromagnetism\nV .P.Mineev\nCommissariat a l’Energie Atomique, INAC/SPSMS, 38054 Grenoble, France\nAbstract. The triplet superconductivity in UGe 2andURhGe coexists with itinerant ferromagnetism such that in the pressure-\ntemperature phase diagram the whole region occupied by the superconducting state is situated inside a more vast ferromagnetic\nregion. In the same family metal UCoGe the pressure dependent critical lines TCurie(P)andTsc(P)of the ferromagnet and\nthe superconducting phase transitions intersect each other. The two-band multidomain superconducting ferromagnet state\narises at temperatures below both of these lines. Here I describe the symmetry and the order parameters of the paramagnet\nas well of the multidomain ferromagnet superconducting states. The Josephson coupling between two adjacent ferromagnet\nsuperconducting domains is discussed.\nKeywords: ferromagnetic superconductor\nPACS: 74.20.De,74.20.Rp\nINTRODUCTION\nA phase transition of the second order breaks some sym-\nmetry such that below the critical temperature the or-\ndered phase of lower symmetry in comparison with the\ninitial state is formed. As it was ﬁrst pointed out by\nL.D.Landau [1] an intersection of critical lines on the\nphase diagram leads to formation of an ordered phase\nwith symmetry lower than the symmetries of both initial\nordered states existing below of each critical lines sepa-\nrately. Here we study the symmetries of ordered phases\narising at intersection of critical lines of ferromagnet and\nsuperconducting phase transitions.\nThe co-existence of superconductivity and ferromag-\nnetism in several uranium compounds UGe 2, [2]URhGe ,\n[3] and the recently revealed UCoGe . [4] is found to arise\nas a co-operative phenomena rather than as the overlap of\ntwo-mutually competing orders. In all these compounds\nthe substantial reduction of the ordered moment as com-\npared with the Curie-Weiss moment provides clear ev-\nidence of 5 fitineracy. In the ﬁrst two compounds the\nCurie temperatures TCurie is more than the order of mag-\nnitude higher than their critical temperatures for super-\nconductivity. In UCoGe the ratio TCurie=Tscat ambient\npressure is about four. The large exchange ﬁeld and also\nhigh upper critical ﬁeld at low temperatures strongly ex-\nceeding the paramagnetic limiting ﬁeld [5, 6, 7] indicate\nthat here we deal with Cooper pairing in the triplet state.\nThe singlet superconductivity coexists with ferromag-\nnetism in a form known as the Anderson-Suhl or crypto-\nferromagnetic superconducting state (for review see [8])\ncharacterized by the formation of a transverse domain-\nlike magnetic structure. The structure period or domainsize is larger than interatomic distance and smaller than\nthe superconducting coherence length that weakens the\ndepairing effect of the exchange ﬁeld. The latter is irrele-\nvant in the case of triplet superconductivity. Hence, there\nis no reason for the formation of a cryptomagnetic state.\nIndeed, no traces of space modulation of magnetic mo-\nments directions on the scale smaller than the coherence\nlength has been revealed.[3, 9, 10, 11] On the other hand\nthe neutron depolarization measurements on UGe 2down\nto 4.2 K (that is in the ferromagnet but not superconduct-\ning region) establish, that the magnetic moment strictly\naligned along a-axis, with a typical domain size in the bc-\nplane of the order 4 :4\u000210\u00004cm [12] that is about two\norders of magnitude larger than the largest superconduct-\ning coherence length in b-direction xb\u00197\u000210\u00006cm.\nArising at temperatures far below the corresponding\nCurie temperature the superconductivity in UGe 2and\nURhGe coexists with ferromagnetism in some pressure\ninterval such that in the (P;T)phase diagram the whole\nregion occupied by the superconducting state is situ-\nated inside a more vast ferromagnetic region.[13, 14] In\nURhGe the Curie temperature increases up to the high-\nest pressure achieved (130kbar). The superconducting\ncritical temperature decreases slowly up to 20 kbar. It is\nmore peculiar the behavior of UGe 2, where at low tem-\nperatures the ferromagnetism and the superconductivity\nabruptly ( by means the ﬁrst-order-type transition) disap-\npears at the same critical pressure Pc\u001915kbar.\nThe observation that the superconductivity in UGe 2\nis conﬁned to the ferromagnet state can be trivially ex-\nplained by an assumption that the ferromagnetism in this\ncompound is formed by f-electrons with half-metallic\nbands ﬁlling. Namely, the band with the spin-down elec-\ntrons is completely ﬁlled, whereas the band with spin-uparXiv:0812.2171v2  [cond-mat.supr-con]  15 Apr 2009T\nPP*NSFSFFIGURE 1. The schematic pressure-temperature phase diagram of superconducting UCoGe . Here, Nis the normal paramagnet\nphase, Fis the ferromagnet phase, Sis the paramagnet superconducting phase, FSis the multi-domain ferromagnet superconducting\nphase. All the lines are the lines of the second-order phase transitions.\nelectrons ﬁlled up to the Fermi level. The triplet spin-\nup superconducting state formed in this band persists\nso long the Fermi level intersects this band. The pres-\nsure induced the Fermi level lifting above the band upper\nboundary kills both the itinerant ferromagnetism and the\nsuperconductivity.\nThe particular one-band superconducting state was\nchosen [6] for successful explanation of temperature de-\npendences of the upper critical ﬁeld in URhGe in dif-\nferent crystallographic directions. This state is also ap-\npropriate for the description [15] of the transition driven\nby the change of orientation of the ordered magnetic\nmoment in this compound by the application of mag-\nnetic ﬁeld in perpendicular direction accompanying by\nthe arising a reentrant superconducting state [16]. The\none band superconductivity, of course, does not exclude\nthe existence of the other conducting but not supercon-\nducting bands, or, more exactly, the bands with negligi-\nbly small superconducting gaps. The latter is in corre-\nspondence with reduced speciﬁc heat jump in compari-\nson with BCS value, and the ﬁnite residual zero temper-\nature ratio (C(T)=T)T!0comparable with its magnitudein the normal state found in all uranium superconducting\nferromagnets.\nThe phase diagram of the new ferromagnetic super-\nconductor UCoGe is qualitatively different (see Fig-\nure 1).[17] At ambient pressure, the ferromagnetism\n(TCurie\u00193K) coexists with superconducting state ( Tsc\u0019\n0:7K). Then at applied pressure, the Curie temperature\ndecreases such that no indication of ferromagnetic order\nis observed above P\u0003\u001910kbar. The resistive supercon-\nducting transition is, however, quite stable with changes\nin temperature and persists up to the highest measured\npressure of about 24 kbar. Thus, the pressure dependent\ntransition lines TCurie(P)andTsc(P)apparently intersect\neach other and the superconductivity exists both in the\nparamagnet and in the ferromagnet state.\nThe ferromagnet superconducting state in an or-\nthorhombic metal is similar to the superﬂuid3He\u0000Ain\nan external magnetic ﬁeld known as A2state. The su-\nperﬂuid3He\u0000Ais the spin nonpolarized state formed\nby the spin-up and the spin-down Cooper pairs in equal\namounts. There is also the spin-polarized A1state where\nthe pairing only spin-up particles occurs.[18] The3He\u0000A1arises from the normal Fermi liquid in an external\nmagnetic ﬁeld. Then, at lower temperature, the liquid\npasses to the A2state where the paired spin-up and spin-\ndown states are almost equally populated. The presence\nof spin-orbital coupling admixes some amount of the\nspin-down Cooper pairs to pure A1state [19], such that\ntheA1andA2states are in fact qualitatively indistin-\nguishable. The phase transition between these two states\nis a crossover, looking as a phase transition due to the\nsmallness of the spin-orbital coupling in superﬂuid3He.\nThe nonunitary two-band superconducting state arising\nin the ferromagnet state of UCoGe can be considered as\nan analog of superﬂuid A2phase arising from the normal\nliquid3Heunder magnetic ﬁeld.\nThe increasing pressure causes the decrease of the ex-\nchange ﬁeld that suppresses the spin-up and spin-down\nband difference. The restoration of the time reversal sym-\nmetry occurs at recreation of spin-up and spin-down\nband degeneracy by the phase transition from the ferro-\nmagnet axiplanar superconducting state to the paramag-\nnet superconducting state similar to the planar state of\nthe superﬂuid3He(for the superﬂuid3Hephase deﬁni-\ntions see for instance [20]). Thus, the ferromagnet super-\nconducting state is separated from the normal state by the\nmore symmetric, paramagnet planar-like state. We see,\nthat the (P;T)phase diagram in UCoGe is quite naturally\nexplained in terms of two band superconducting state in\nthis material. The observation of the upward curvature in\nthe temperature dependence of the upper critical ﬁeld in\nUCoGe [7] adds the additional argument in support of\nthis point.\nThe symmetries and the order parameters of uncon-\nventional superconducting states arising from the normal\nstate with a ferromagnetic order in orthorhombic crys-\ntals with strong spin-orbital coupling have been found\nin the paper. [21] Then it was pointed out that super-\nconducting states in triplet ferromagnet superconduc-\ntors represent a special type of two band superconduct-\ning states. [22, 23]. There were obtained several results\nbased on phenomenlogical (Ginzburg-Landau) and mi-\ncroscopic descriptions of two-band superconductivity. It\nwas proved, however, that the superconducting ferro-\nmagnet classes pointed there have been found improp-\nerly. Although, it leaves untouched the main results of\n[21, 22, 23], the based on these papers description of\npossible (P;T)phase diagrams for two band supercon-\nducting states in an orthorhombic itinerant ferromagnet\nis incorrect.[24] To make it correctly we return to the\ndeﬁnition of the superconducting ferromagnet classes.\nIt will be proven that TCurie(P)andTsc(P)can intersect\neach other as the critical lines of the phase transition of\nthe second order. The symmetry and the order parameters\nof the multidomain ferromagnet as well of the paramag-\nnet superconducting states are established. The Joseph-\nson coupling between neighboring superconducting do-mains is also discussed.\nTWO-BAND SUPERCONDUCTING\nFERROMAGNET PHASE DIAGRAM\nAll uranium ferromagnetic superconductors are or-\nthorhombic metals. The symmetry of its normal para-\nmagnetic state is determined by the elements of the\ngroup\nGN=D2\u0002U(1)\u0002R; (1)\nwhere D2= (E;Cz\n2;Cx\n2;Cy\n2)is the point symmetry group\nincluding the operations Cx\n2;Cy\n2;Cz\n2of rotation on the\nangle pabout the x;y;z- axes correspondingly, U(1)is\nthe group of gauge transformations, and Ris the time\nreversal operation.\nIf the pressure dependent transition lines TCurie(P)and\nTsc(P)intersect each other at P=P\u0003then the region of\nthe coexistence of superconductivity and ferromagnetism\nis separated from the normal state by the region of ferro-\nmagnet normal state at P<P\u0003and by the region of the\nsuperconducting state at P>P\u0003(see Figure 1).\nIn the transition from the normal paramagnet state to\nthe normal ferromagnet state the magnetic moment di-\nrected along one crystallographic axis appears. We chose\nthis direction as the ˆ zaxis. Hence, in the ferromagnet\nstate the symmetry reduces to the\nGF=D2(Cz\n2)\u0002U(1); (2)\nwhere\nD2(Cz\n2) = ( E;Cz\n2;RCx\n2;RCy\n2) (3)\nis the so called magnetic class [25] or the point symme-\ntry group of the ferromagnet. The rotations on the angle\npabout the x- and y- directions are accompanied by the\ntime inversion Rthat changes the direction of magnetiza-\ntion to the opposite one.\nIn the transition from the normal paramagnet state to\nthe superconducting paramagnet state the gauge symme-\ntry is broken, such that the symmetry of this, so called\nconventional superconducting state is\nGS= (E;Cz\n2;Cx\n2;Cy\n2)\u0002R: (4)\nThere is another possibility related to the formation of\nnonconventional superconducting state where, in the ad-\ndition to the gauge symmetry, the point symmetry is also\nbroken. We shall not discuss it here.\nNow, we shall consequently describe the phase transi-\ntions from the normal ferromagnet state (F) to the super-\nconducting ferromagnet state (FS) taking place at P<P\u0003\nand from the paramagnet superconducting state (S) to the\nsuperconducting ferromagnet state (FS) at P>P\u0003.F to FS phase transition\nAs it was remarked in [22] superconducting state in\nan itinerant ferromagnet represents the special type of\ntwo band superconducting state consisting of pairing\nstates formed by spin-up electrons from one band and\nby spin-down electrons from another band. Hence, a\nsuperconducting state characterizes by two component\norder parameter\nd1(k) =D\"(k)(ˆx+iˆy);d2(k) =D#(k)(ˆx\u0000iˆy):(5)\nHere, ˆ xand ˆyare the unit vectors of the spin coordinate\nsystem pinned to the crystal axes.\nThe unconventional superconducting states arising\nfrom the normal state with a ferromagnetic order in or-\nthorhombic crystals with strong spin-orbital coupling be-\nlong to the two different corepresentations AandB.[21]\nAll the states relating to the given corepresentation obey\nthe same critical temperature. The order parameter am-\nplitudes for AandBstates correspondingly are given by\nDA\n\"(k) =h1(kxu1+ikyu2);\nDA\n#(k) =h2(kxu3+ikyu4); (6)\nDB\n\"(k) =h1(kzv1+ikxkykzv2);\nDB\n#(k) =h2(kzv3+ikxkykzv4): (7)\nThey are odd functions of the momentum directions of\npairing particles on the Fermi surface. The functions\nui=ui(k2\nx;k2\ny;k2\nz)andvi=vi(k2\nx;k2\ny;k2\nz)are invariant in\nrespect of all transformations of orthorhombic group. For\nthe brevity, in that follows, we shall discuss only the A\nstate. This state is related to the family of nonunitary\naxiplanar states.\nThe complex order parameter amplitudes h1=\njh1jeij1andh2=jh2jeij2are not completely indepen-\ndent. The relative phase difference j1\u0000j2is chosen\nsuch that the quadratic in the order parameter part of\nthe Ginzburg-Landau free energy density should be\nminimal. In an ordinary two-band superconductor it is\nF=a1jh1j2+a2jh2j2+g(h\u0003\n1h2+h1h\u0003\n2); (8)\nandj1\u0000j2=pforg>0 and j1\u0000j2=0 for g<\n0. In the case of ferromagnetic normal state the time\nreversal symmetry is broken and the quadratic in the\norder parameter components free energy density has the\nform\nF=a1jh1j2+a2jh2j2+g(h\u0003\n1h2+h1h\u0003\n2)\n+id(h\u0003\n1h2\u0000h1h\u0003\n2): (9)\nHere, all the coefﬁcients are the functions of the ex-\nchange ﬁeld h. The last term breaks the time reversalsymmetry. In the absence of exchange ﬁeld d=0. Min-\nimization of free energy (9) ﬁxes the order parameter\ncomponents phase difference tan (j1\u0000j2) =d=g. Af-\nter substitution of this value back to (9) we come to the\nexpression\nF=a1jh1j2+a2jh2j2\u0000p\ng2+d2(h\u0003\n1h2+h1h\u0003\n2):\n(10)\nHere ai=ai0(T\u0000Tci),i=1;2 are the band indices, Tci\nare the critical temperatures in each band in the absence\nof band mixing. Unlike eqn. (9) the complex amplitudes\nh1=jh1jeiq,h2=jh2jeiqin the eqn. (10) have common\nphase factors with q= (j1+j2)=2. This form of free en-\nergy valid near the phase transition from the ferromagnet\nstate to the ferromagnet superconducting state has been\nused in the papers. [22, 23] The common for the each\nband superconductivity critical temperature is given by\nTsc=Tc1+Tc2\n2+s\u0012Tc1\u0000Tc2\n2\u00132\n+g2+d2\na10a20(11)\nIn the superconducting A-state the gauge symmetry\nis broken. Acting on the order parameters (5), (6) by\nthe elements gofD2(Cz\n2) = ( E;Cz\n2;RCx\n2;RCy\n2)group we\nobtain the following coefﬁcients of transformation, or\nmatrices of corepresentation\nG1= (1;1;e\u00002ij1;e\u00002ij1);G2= (1;1;e\u00002ij2;e\u00002ij2);\n(12)\ncorrespondingly. Corepresentations G1andG2are equiv-\nalent or they are transformed each other by an uni-\ntary matrix UasG1(g) =U\u00001G2(g)Uif the element\ngdoes not include the time inversion, and as G1(g) =\nU\u00001G2(g)U\u0003if the element gincludes the time inversion.\nIt is easy to check that here the matrix of transformation\nisU=ei(j2\u0000j1).\nThe order parameter component d1(k)relating to the\nspin-up band is invariant in respect to the following\ngroup of transformations\nGFS= (E;Cz\n2;RCx\n2;RCy\n2) =D2(Cz\n2): (13)\nAction of the time reversal operation Ron superconduct-\ning order parameter implies also the multiplication of it\nby the square of its phase factor: R!e2ij1R. The second\ncomponent d2(k)possess the same symmetry. So, the\ngroup of symmetry of superconducting ferromagnet state\nAcalled also by the superconductiing magnetic class is\nD2(Cz\n2). This group is the subgroup of the group of sym-\nmetry of the ferromagnet state (2).\nSuperconducting ferromagnet domains\nThe Cooper pairing changes the magnitude of sponta-\nneous magnetization in respect to its value in normal fer-romagnet state. Namely, the superconducting spin mag-\nnetic moment density is\nMs=mB\u0002\nN0\n0\"hjD\"(k)j2i\u0000N0\n0#hjD#(k)j2i\u0003\n: (14)\nHere, in the ﬁrst term, N0\n0\"is the derivative of the density\nof states at the Fermi surface of the spin-up band , and the\nangular brackets means the averaging over it. The second\nterm presents the corresponding input of the spin-down\nband. One can write also the orbital magnetic moment\ndensity. [20]\nAlong with the introduced state A, there is its time\nreversed state A\u0003characterized by the complex conjugate\norder parameter components\nd\u0003\n1(k) =z1(ˆx\u0000iˆy)(kxu1\u0000ikyu2);\nd\u0003\n2(k) =z2(ˆx+iˆy)(kxu3\u0000ikyu4): (15)\nThe states AandA\u0003occupy neighboring domains with\nthe opposite direction of magnetization. The state A\u0003\norder parameter amplitudes are z1=jz1jeif1andz2=\njz2jeif2. The phase difference is ﬁxed by tan (f1\u0000f2) =\nd(\u0000h)=g.\nThe matrices of corepresentations for the state A\u0003are\nobtained from (12) by the substitution j1;2!f1;2. So,\nthey transformed each other by means the matrices Ui=\nei(ji\u0000fi). It means, that the corepresentations for the state\nA\u0003are equivalent to the corepresentations for the state\nA. Hence, the superconducting states in the neighboring\ndomains obey the same critical temperature.\nThe symmetry of the time reversed states A\u0003belong to\nthe same superconducting ferromagnet class D 2(Cz\n2)as\ntheA-states.\nS to FS phase transition\nInP>P\u0003region at temperature decrease UCoGe pass\nto the nonmagnetic superconducting state. Let us assume\nthe simplest and quite natural situation that it is the\nsuperconducting state with the order parameter\nd(k) =2h(kxw1ˆx+kyw2ˆy); (16)\ntransforming according to the unit representation of the\nnormal state point symmetry group D2. Here h=jhjeij\nand the functions w1;2=w1;2(k2\nx;k2\ny;k2\nz)are invariant in\nrespect of all transformations of orthorhombic group.\nThis state reminds planar phase of superﬂuid3He. The\nparamagnet superconducting state is invariant in respect\nto the group (4) which can be rewritten as\nGS=D2(Cz\n2)+R\u0002D2(Cz\n2): (17)\nBy further decrease the temperature we approach to\nTCurie(P). At this temperature the exchange ﬁeld appears,and the Kramers degeneracy between spin-up and spin-\ndown electron states is lifted accompanied by arising of\ndeviation from of the order parameter (16)\nd(k) =2h(kxw1ˆx+kyw2ˆy)\n=h(kxw1\u0000ikyw2)(ˆx+iˆy)+h(kxw1+ikyw2)(ˆx\u0000iˆy)\n!˜d(k) =h1(kxw1\u0000ikyw2)(ˆx+iˆy)\n+h2(kxw1+ikyw2)(ˆx\u0000iˆy) (18)\nThe order parameter ˜d(k)transforms according to corep-\nresentation of the symmetry group (17) of the paramag-\nnet superconducting state.\nAlong with increase of the band splitting the two\ncomponent of the order parameter ˜d(k)are transformed\nto the order parameters of spin-up and spin-down bands\ngiven by eqns. (5), (6). The ferromagnet superconducting\nstate determined by eqn. (18) as well by the eqns. (5) and\n(6) is invariant in respect to the group\nGFS=D2(Cz\n2): (19)\nThe latter is the subgroup of the group of symmetry of\nferromagnet state GF(2) as well as of the symmetry\ngroup of paramagnet superconducting state GS(17). So,\nthe lines of the ferromagnet and the superconducting\nphase transitions can intersect each other as the critical\nlines of the phase transitions of the second order.\nINTERDOMAIN JOSEPHSON\nCOUPLING\nLet us consider a ﬂat domain wall dividing magnetic\nmoment-up and -down domains in single band ferromag-\nnet. This case, the localized at x=0 domain wall con-\ntribution to the superconducting free energy density is\ngiven by [26]\nFDW=\u0002\ng1(jhj2+jzj2)+g2(h\u0003z+hz\u0003)\n+ig3(h\u0003z\u0000hz\u0003)]d(x): (20)\nHere h=jhjeijandz=jzjeifare the superconducting\norder parameters in the left (magnetic moment-up) do-\nmain and in the right (magnetic moment-down) domain,\ncorrespondingly. The boundary conditions at x=0 are\nderived by the minimization of the sum of domain wall\n(20) and the gradient free energies.[20]\nK¶z\n¶x=g1z+(g2+ig3)h\n\u0000K¶h\n¶x=g1h+(g2\u0000ig3)z: (21)\nHere, the rigidity coefﬁcients K\u0018¯h2=m. The solutions of\nleft and right domain nonlinear Ginzburg-Landau equa-\ntions supplemented by these boundary conditions deter-\nmine the order parameter distribution of two domain su-\nperconducting structure. The solution of correspondinglinear problem is physically relevant only in the case\nof stimulation of superconductivity by the domain wall\nwhen the localized near domain wall superconducting\nstate arises at temperatures higher than the temperature\nof superconducting phase transition in single domain ge-\nometry.\nThe situation for two band superconductivity is much\nmore complicated. This case the two-band domain wall\nfree energy density is obtained by the addition to the\neqn.(20) the corresponding terms for the second band\norder parameters h2andz2and also the interband terms\nsymmetric in respect to the substitutions hibyziand vice\nversa.\nSubstituting the boundary conditions (21) in the sum\nof the left domain and the right domain current through\nthe domain wall (see for instance [20]) we obtain the\ndensity of the interdomain Josephson current:\nj=8eK\n¯hjzjjhj[g2sin(f\u0000j)\u0000g3cos(f\u0000j)] (22)\nThus, due to the time reversal breaking ( g36=0) the ex-\npression for the Josephson current between the adjacent\nsuperconducting domains with spin-up and spin-down\nmagnetization differs from the usual weak link Joseph-\nson current formula. In the equilibrium, the phase differ-\nence between domains is ﬁxed: tan (f\u0000j) =g3=g2, and\na spontaneous interdomain current is absent.\nIn conclusion of this section it is worth to be noted\nthat the existence of the interdomain Josephson coupling\nbilinear in respect of jhjandjzjis typical for the Asuper-\nconducting states. The order paramer for the Bstates is\nvanishing in the equatorial plane kz=0. This case, there\nis only the higher order Josephson coupling between the\ndomains divided by a ﬂat domain wall parallel to the\nmagnetization direction.\nCONCLUSION\nThe superconducting state in the itinerant ferromagnet\nuranium compound UCoGe manifests the properties nat-\nurally explained in terms of two band superconductiv-\nity with triplet pairing. We discussed the symmetry and\nthe order parameters of such a state. It was proven that\nthe pressure dependent critical lines of the ferromag-\nnetTCurie(P)and the superconducting phase transition\nTsc(P)can intersect each other in the correspondence\nwith the experimental observations.[17] The Josephson\ncoupling in between the adjacent superconducting ferro-\nmagnet domains is found to be different in comparison\nwith the usual Josephson coupling between two super-\nconductors.REFERENCES\n1. L. D. Landau, ZhETF 7, 19 (1937) [ Phys. Zs. Sowjet. 11,\n26 (1937)].\n2. S. S. Saxena, P. Agarval, K. Ahilan, F. M. Grosche,\nR. K. W. Hasselwimmer, M. J. Steiner, E. Pugh,\nI. R. Walker, S. R. Julian, P. Monthoux, G. G. Lonzarich,\nA. Huxley, I. Sheikin, D. Braithwaite, and J. Flouquet,\nNature 406, 587 (2000).\n3. D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite,\nJ. Flouquet, J.-P. Brison, E. Lhotel and C. Paulsen, Nature\n413, 613 (2001).\n4. N. T. Huy, A. Gasparini, D. E. de Nijs, Y . Huang,\nJ. C. P. Klaasse, T. Gortenmulder, A. de Visser,\nA. Hamann, T. Gorlach, and H. v. Lohneysen, Phys.\nRev. Lett. 99, 067006, (2007).\n5. A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois,\nD. Braithwaite, R. Calemzuk, and J. Flouquet, Phys. Rev.\nB63, 144519 (2001).\n6. F. Hardy and A. D. Huxley, Phys. Rev. Lett. 94, 247006\n(2005).\n7. N. T. Huy, D. E. de Nijs, Y . Huang, and A. de Visser,\nPhys.Rev.Lett. 100, 007002 (2008).\n8. L. N. Bulaevskii, A. I. Buzdin, M. L. Kulic, and\nS. V . Panjukov, Adv. Phys. 34, 175 (1985).\n9. N. Aso, H. Nakane, G. Motoyama, N. K. Sato,\nY . Uwatoko, T. Takeuchi, Y . Homma, Y .Shiokawa,\nK. Hirota, Physica B 359-361 , 1051 (2005).\n10. H. Kotegawa, A. Harada, S. Kawasaki, Y . Kawasaki,\nY . Kitaoka, Y . Haga, E. Yamamoto, Y . Onuki, K. M. Itoh,\nE. E. Haller, and H. Harima, J. Phys. Soc. Jpn. 74, 705\n(2005).\n11. T. Ohta, Y . Nakai, Y . Ihara, K. Ishida, K. Deguchi,\nN. K. Sato, and I. Satoh, J. Phys. Soc. Jpn. 77, 023707\n(2008).\n12. S. Sakarya and N. H. van Dijk, E. Bruck, Phys. Rev.B 71,\n174417 (2005).\n13. A. Huxley, E. Ressouche, B. Grenier, D. Aoki, J. Flouquet,\nand C. Pﬂeiderer, J. Phys.: Condens. Matter 15, 1945\n(2003).\n14. F. Hardy, A. Huxley, J. Flouquet, B. Salce, G. Knebel,\nD. Braithwaite, D. Aoki, M. Uhlarz, C. Pﬂeiderer, Physica\nB359-361 , 1111 (2005).\n15. V . P. Mineev, C. R. Physique 7, 35 (2006).\n16. F. Levy, F. Hardi, I. Sheikin, B. Grenier, A. D. Huxley,\nScience 309, 1343 (2005).\n17. E. Hassinger, D. Aoki, G. Knebel, and J. Flouquet, J.\nPhys. Soc. Jap. 77, 073703 (2008).\n18. A. Leggett, Rev. Mod. Phys 47, 331 (1975).\n19. A Yamaguchi, S. Kobayashi, H. Ishimoto and H. Kojima,\nNature 444, 909 (2006).\n20. V . P. Mineev, K. V . Samokhin, Introduction to\nUnconventional Superconductivity , Gordon and Breach\nScience Publishers, Amsterdam, 1999.\n21. V . P. Mineev, Phys. Rev. B 66, 134504 (2002).\n22. V . P. Mineev and T. Champel, Phys. Rev. B 69, 144521\n(2004).\n23. V . P. Mineev, Int. J. Mod. Phys. 18, 2963 (2004).\n24. V . P. Mineev, J. Phys. Soc. Jpn. 77, 103702 (2008).\n25. L. D. Landau and E. M. Lifshitz, Electrodynamics of\nContinuous Media , Pergamon Press, Oxford, 1984.\n26. K. V . Samokhin and D. Shirokoff, Phys. Rev. B 71,\n104527 (2005)."    },    {        "title": "2302.00553v1.Ferromagnetism_of_sputtered_Fe3GeTe2_ultrathin_films_in_the_absence_of_two_dimensional_crystalline_order.pdf",        "content": "1 \n Ferromagnetism  of sputtered Fe3GeTe 2 ultrathin films in the absence of  \ntwo-dimensional crystalline order  \nQianwen Zhao1, 2, ChaoChao Xia1, 3, Hanying Zhang1, 2, Baiqing Jiang1, 2, Tunan  Xie1, 2, Kaihua \nLou1, 2, and Chong Bi1, 2, 3* \n1State Key Lab of Fabrication Technologies for Integrated Circuits, Institute of Microelectronics, \nChinese Academy of Sciences, Beijing 100029, China  \n2University of Chinese Academy of Sciences, Beijing 100049,  China  \n3School of Microelectronics, University of Science and Technology of China, Hefei 230026, \nChina  \n \n*bichong@ime.ac.cn  \n  2 \n Abstract : \nThe discovery of ferromagnetism in two -dimensional (2D) monolayers has stimulated growing \nresearch interest in both spintronics and material science. However , these 2D ferromagnetic \nlayers are mainly prepared th rough an incompatible approach for large -scale fabrication and  \nintegration, and moreover, a fundamental question whether the observed ferromagneti sm actually \ncorrelates with the 2D crystalline order has not been explored. Here, we choose a typical 2D \nferromagnetic material, Fe 3GeTe 2, to address these two issues by  investigating its \nferromagnetism in a n amorphous state.  We have fabricated nanometer -thick amorphous \nFe3GeTe 2 films approaching  the monolayer thickness limit of crystallized Fe 3GeTe 2 (0.8 nm) \nthrough magnetron sputtering.  Compared to crystallized Fe 3GeTe 2, we found that the basic \nferromagnetic attributes, such as the Curie temperature  that directly reflects magnetic exchange \ninteractions and local anisotropic energy, do not change significantly in the  amorphous states. \nThis is attributed to that the short -range atomic order, as  confirmed by valence state analysis , is \nalmost the same for both phases . The persistence of ferromagnetism in the ultrathin amorphous \ncounterpart has also been confirmed through magnetoresistance measurements , where two \nunconventional switching dips arising from  electrical transport within domain walls are clearly \nobserved in the amorphous  Fe3GeTe 2 single layer . These results indicate that the long -range \nferromagnetic order of crystallized Fe 3GeTe 2 may not correlate to the 2D crystalline order and \nthe corresponding ferromagnetic attributes can be  utilized in an amorphous state which suit s \nlarge -scale fabrication  in a semiconductor technology -compatible manner  for spintronics \napplications . \nKeywords: two-dimensional ferromagnetism, low -dimensional materials, spintronics, \namorphous ferromagnetic materials, amorphous two -dimensional materials  \n  3 \n INTRODUCTION  \nLong -range f erromagneti c order which cannot persist down to two -dimensional (2D) regime s at \na finite temperature has been proved in the well -known Mermin -Wagner theorem for several \ndecades1. However, in recent years, many groups report that this restriction, arising from thermal \nfluctuations, can be counteracted by magnetocrystalline  anisotropic fields (or external magnetic \nfields) that may open an energy gap in the dispersion of thermally excited magnons for \nstabilizing long -range magnetic order in the 2D regimes at a nonzero temperature2–5. \nExperimentally, ferromagnetism has been discovered in plenty of van der Waals materials  down \nto a few layers or even the monolayer l imit, such as CrI 3, Cr 2Ge2Te6 and Fe 3GeTe 26–8. Like other \nconventional 2D materials, the typical approach to obtain these ferromagnetic low-dimensional \nmaterials is by using  mechanical exfoliation of their bulk counterparts, which, nonetheless, is \nfacile neither for large -scale production, nor for integration with other materials, especially in \nspintronic applications where the spin transp ort between two functional layers is extremely \nsensitive to the quality of interfaces9 (the same situations for molecular beam epitaxy  \nfabrications ). More importantly, practical spintronic products, for example, magnetic sensors10,11, \nhard disk  drive s12–14, and magnetic random -access memory (MRAM)15, do n ot require single \ncrystalline or even polycrystalline magnetic materials, but instead, uniform amorphous magnetic \nmaterials that can be fabricated through the complementary metal oxide semiconductor (CMOS) -\ncompatible techniques  are preferred10–16. This is because, b esides the compatibility of \nfabrication, the ferromagnetic properties  of most conventional magnetic materials keep well in \nan amorphous state17–21 (except crystal orientation -dependent anisotropy) , and furthermore, no  \nelectrical noises or  random domain -wall pinning sites  formed at grain boundaries or along the \ncrystal directions will benefit device reliabilities during magnetization manipulation10–15. \nTherefore, from the point of view of spintronic applications, it will be interesting t o verify if the \nobserved ferromagnetism in low -dimensional materials is actually related to the 2D crystalline \norder, which will provide a technology benchmark for utilizing their ferromagnetic properties in \nCMOS -compatible processes.  \nTheoretically, unlike  electrical properties determined by the electronic band structure that \nstrongly correlates with long -range crystalline order , magnetic properties mainly rely on short -4 \n range  atomic  order16,18,19,21. A simple yet successful model to describe magnetism is using  a \nHeisenberg Hamiltonian  \n𝐻=∑ 𝐽𝑖𝑗𝑺𝑖 𝑖,𝑗 ∙𝑺𝑗+∑𝐷(𝒏𝑖∙𝑺𝑖)2\n𝑖 +𝑯∙∑𝑺𝑖𝑖,          (1) \nwhere 𝑺𝑖 is the spin operator on site i, 𝒏𝑖 is a unit vector of the local atomic anisotropy axis, \n𝐽𝑖𝑗 is the exchange coupling strength between sites i and j, and H is the applied magnetic field. \nThe first term describes the sum of isotropic Heisenberg exchange interactions between spin i \nand neighbored spin j, and the second term describes the contribution from uniaxial magnetic \nanisotropy with an anisotropy constant D. Considering a material with fixed atomic ratios, in a \nsolid crystalline state, both terms can be well calculated by using detailed lattice structures, while \nin an amorphous state, they can be evaluated through a disorder -modified 𝐽𝑖𝑗 and D, where the \nmodification reflects the change of relative spatial position between two neighbored spin \nsites18,22. 𝐽𝑖𝑗 is usually dominated by the short -range exchange interaction arising from \nelectrons’ antisymmetric wave function governed by the C oulombic interaction18,22,23 and can be \nestimated through the so -called Bethe -Slater curve22,24 based on the interatomic spacing that does \nnot change largely in a stable equilibrium state . Therefore, 𝐽𝑖𝑗 can be treated as 𝐽𝑖𝑗= 〈𝐽𝑖𝑗〉+\n∆𝐽𝑖𝑗 with an average exchange strength  〈𝐽𝑖𝑗〉 and an exchange fluctuation  ∆𝐽𝑖𝑗 in amorphous \nstates22. Indirect exchange inter action s like super - or double -exchange mediated by non -\nmagnetic atoms may be treated in the same  way.  Meanwhile, D can also be represented by a \n“local field” in amorphous materials with a correlation length of several angstroms, where the \nsign and strength of D are determined by spin -orbit interactions16,18,22,23,25 –28. In some cases, \nstress, shape, or interface may also contribute D18,22. Given the fact that the third term describin g \nthe Zeeman interaction does not depend on crystallinity, the basic magnetic properties such as \nthe Curie temperature  (Tc) and saturation magnetization  do not change remarkably  in the \ncrystalline and amorphous states  for plenty of  ferromagnets17–22,25, and the co rresponding \nexperimental results can also be well explained by using Eq. (1) in both states16,22,28. Therefore, \nin pri nciple, the magnetic attributes of layered van der Waals materials could also not correlate \ntheir crystalline order strongly, even more weakly than 3D ferromagnetic materials, by \nconsidering that the nearest -neighboring spin sites are limited in the 2D lay er. So far, intensive \nefforts have been focused on the exfoliation  and characterization  of 2D magnetic materials6–8, \nbut the correlation between magnetism and 2D crystalline order has not been discussed.  5 \n In this work, we choose a widely investigated 2D ferromagnet, F e3GeTe 2 (FGT)4,5, to explore the \npossible ferromagnetism in an amorphous state. By comparing to the reported ferromagnetic \nproperties of crystallized FGT (c -FGT), deep insights into the role of 2D crystalline order on the \nobserved magnetic properties can be gained. C -FGT belongs to the hexagonal crystal system, \nwhere the Te -Fe3Ge-Te slabs lying in the ab plane stack along the c axis, coupled via vdW \ninteraction. The T c of bulk c -FGT is about 220 K, which reduces to 130 K (may vary with \nsubstrates) when the thickness is down to the monolayer limit (0.8 nm)4,5. The ferromagnetic \norder persisting in a monolayer was attributed to the sustaining perpendicular magnetic \nanisotro py (PMA) that suppresses the thermally excited magnons2–5,7. These clear ferromagnetic \nbehaviors of c -FGT will be helpful for comparison with its amorphous counterp art in this work. \nWe prepared the amorphous FGT (a -FGT) thin films through magnetron sputtering from a FGT \ntarget under high vacuum conditions. The sputtered a-FGT was controlled between 1 nm and 120 \nnm in thickness (shortened for FGT(t) with t being the t hickness in nm) and capped with a 2.5 \nnm TaO x or 10 nm Si 3N4 layer  unless otherwise specified , which was then patterned into a Hall \nbar structure for Hall and magnetoresistance measurements.  \nRESULTS AND DISCUSSION  \nStructural Characterization . We first characterized structural and compositional properties of \nsputtered FGT by using high -resolution transmission electron microscopy (HRTEM), scanning \ntransmission electron microscopy (STEM) equipped with energy dispersive x -ray spec troscopy \n(EDS), and X-ray photoelectron spectroscopy (XPS). No capping layer was deposited for these \nsamples so that the degree of natural oxidation under ambient conditions can also be evaluated. \nFigure 1a shows the HRTEM images of 30 nm FGT, in which a n aturally oxidized surface up to \n4.1 nm can be observed clearly, highlighting the essentials of TaO x or Si 3N4 capping layers for \nultrathin FGT samples. The roughness of surfaces or interfaces is less than 0.4 nm even for the \n30 nm FGT, evidencing that the s puttered FGT can be continuous thin films when the thickness \nis close to the c -FGT monolayer limit (0.8 nm). The surface roughness of several nanometer -\nthick FGT has also been confirmed by  atomic force microscopy (AFM), where the average \nsurface roughness is about 0.21 nm for 1 nm FGT (see Supporting Information Figure S1). In a \nsimilar amorphous ferromagnetic layer, we have also demonstrated that the thickness of \ncontinuous thin films can be safely controlled down to 0.6 nm through the magnetron 6 \n sputtering29. The structural and compositional distributions have also been examined in a region \nof 30 nm × 1 μm, and no crystalline lattices or element aggregates were found. The typical \nstructural and elemental mapping images shown in Figure 1a,b provide direct evid ences that the \nsputtered FGT thin films are amorphous  (see Supporting Information Figure S2 for X -ray \ndiffraction results)  and uniform in structure and composition.  \n \nFigure 1 . (a) HRTEM images of sputtered FGT thin films with the thickness of 30 nm. The enlarged part shows that the  naturally  \noxidized layer is about 4.1 nm near the top surface. (b) Corresponding Fe, O, Ge, and Te maps of the 30 nm FGT. (c -e) XPS \nspectra of Fe 2p, Ge 3d and Te 3d.  The filled peaks with different colors represent different v alence states of each element. The \nisolated dots and connected lines are experimental data and fitting results, respectively.  \nThe most important feature of sputtered FGT that can show similar magnetic properties as c -FGT \nis the valence states of each elem ent, which directly reflect the chemical bond and short -range \natomic order  and determine the strength of direct exchange as well as spin -orbit coupling18,22. To \nanalyze the valence bond states, we performed XPS measurements, as shown in Figure 1c -e. \nBefore the XPS data was collected, the top oxidized surface had been etched. Generally, the XPS \nspectra of each element in c -FGT can be well reproduced in the sputtered FGT30,31. As shown in \nFigure 1c, the Fe 2p spectrum can be deconvoluted into six peaks, where the peaks at 706.6 \neV/719.8 eV attributed to Fe0 and 710.9 eV/724.6 eV attributed  to Fe3+ are exactly the same as \n710 720 730 740Intensity (a.u.)  Fe3+\n Fe0\n28 30 32 34\nBinding energy (eV)Ge 3d Fe 2p\n Ge4+\n Ge0\n570 575 580 585Te 3d\n Te4+\n Te0\n (a) (b)\n(c) (d)\n10 nm\nFGT FGT+O\nFe O\nGe Te\n20 nm\n(e)7 \n those of crystallized bulk FGT. Two  other peaks at 713.2 eV/ 726.2 eV can be attributed to the \nsatellites of 710.9 eV/724.6 eV peaks. For the Ge 3d and Te 3d spectra shown in Figure 1d,e, the \ndeconvoluted  peaks are also the same as those of c -FGT31. These XPS results provide strong \nevidences that the valence bond states and local positions of each el emental atoms in c -FGT, at \nleast in the scale of nearest -neighbors, are also sustained in the sputtered a-FGT . \nCorrespondingly,  those attributes mainly relying on the nearest -neighboring interaction , such as \nmagnetism , may also maintain in  the sputtered  a-FGT. It should be noted that the detailed \nchemical states of each element determined through XPS spectra have not been well understood \neven in c -FGT. For instance, the peak around 724.6 eV was also attributed to Fe2+ in some \nworks30, while the Fe3+ signals were explained as originating from surface oxidation31 but still \nappear in the samples afte r removing oxidized top surfaces30. Regardless of t hese debates on the \norigin of detailed peaks, the identical XPS results between c -FGT and sputtered a -FGT \ndemonstrate that the same chemical states of each element are sustained in both phases. The \nestimated atomic percentages of Fe, Ge, and Te from XPS sp ectra are 42.1%, 24.6%, and 33.3%, \nrespectively, in which the Fe concentration is less while Ge is higher than their corresponding \nconcentrations in FGT sputtering targets with the same components as c -FGT (Fe decreases from \n50.0% to 42.1%, and Ge increase s from 16.7% to 24.6%). The low Fe concentration was also \nconfirmed by using energy dispersive X -ray spectrometry ( EDX; see Supporting Information \nTable S1). Compared to c -FGT, the lower Fe concentration of sputtered c-FGT indicates that \nthere are enough G e and Te to form Ge -Fe or Te -Fe bonds and probably no extra isolated Fe \natoms contributing to  the ferromagnetism as discussed below.  8 \n  \nFigure 2. (a) R xx of sputtered a -FGT as a function of temperature with two typical thicknesses. Inset schematically shows the \nexperimental configuration for electrical measurements. (b -e) R xy curves (b, d) and corresponding Arrott plots (c, e) of 5 nm (b, \nc) and 3 nm (d, e ) a-FGT measured at different temperatures under a perpendicular magnetic field. (f) The thickness dependence \nof extracted T c for sputtered a -FGT. The T c data of c -FGT is from other references5 for comparison.  \n \nDetermination of T c through Hall Measurem ents. The longitudinal resistance (R xx) and \nanomalous Hall resistance (R xy) of sputtered FGT were examined by using a four -point \nmeasurement configuration, as schematically shown in the inset of Figure 2a. The temperature \ndependences of R xx shown in Figure  2a illustrate that the sputtered FGT shows semiconducting \nbehaviors for all thicknesses up to 60 nm, in sharp contrast to c -FGT that is metallic in bulk and \nbecomes semiconducting down to trilayer (2.4 nm). This can be understood that the electrical \ntrans port properties of both a -FGT and c -FGT thinner than trilayer are dominated by disorder, \nmainly interlayer (or interfacial) structural disorder  in the trilayer or thinner c -FGT , although in \nwhich the intralayer 2D crystalline order still keeps. Figure 2b a nd Figure 2d present R xy curves \nunder a perpendicular magnetic field (H z) for 5 nm and 1 nm FGT, respectively. At 300 K, only \nthe linear response contributed from the normal Hall effects can be observed, indicating that \nthere is no ferromagnetism in the sp uttered FGT at room temperature. With decreasing \ntemperature, the typical ferromagnetic hysteresis appears for both samples and finally dominate s \n0 100 200 3003.03.5Rxx (kW)\nT (K) FGT(10)3060\n FGT(3)\n-5 0 5-303Rxy (W)\nH (T)FGT(5)300 K5 K\n0 5 100510R2\nxy(W2)\nH/Rxy (T/W)FGT(5) 250 K200 K150 K80 K\n-5 0 5-10-50510Rxy (W)\nH (T)FGT(1)150 K2 K\n0 1 20204060R2\nxy(W2)\nH/Rxy (T/W)FGT(1)\n80 K50 K30 K20 K2 K\n1 10100200 Tc (K)\nThickness (nm) Crystal\n Amorphous(a) (b) (c)\n(d) (e) (f)RxxRxy\nyz\nx9 \n Rxy signals at low temperatures below 100 K. The transition temperature to ferromagnetism \nstrongly depends on the thickness of sputtered FGT like c -FGT and other ferromagnetic \nmaterials.  \nTo determine T c accurately, the Arrott plots32 of R xy curves for 5 nm and 1 nm a -FGT, 𝑅𝑥𝑦2 as a \nfunction of H/R xy, are replotted in Figure 2c and Figure 2e, respectively, where the positive \nintercept of linear part in the high -field range can be thought as ferromagnetic states. T c is \ndetermined when the intercept of linear part approaches zero.  For the thickness larger than 5 nm, \nthe determined T c is about 225 K, which is close to that of c -FGT by considering around 25 K \nmisestimation due to the interference of normal Hall contribution. As shown in Figure 2e, for 1 \nnm a -FGT, the temperature at w hich the intercept approaches zero is apparently lower than that \nof 5 nm FGT because of a reducing T c. The thickness dependences of T c for c-FGT4,5 and \nsputtered a -FGT are shown in Figure 2f, in which similar T c values  for both phases  emerge at \neach thicknes s. These results indicate that the basic exchange interactions responsible for the \nobserved ferromagnetism, which are directly reflected through T c, are not changed \nextraordinarily  in both phases . Therefore, the fundamental magnetic attributes of c -FGT \ndominated by the exchange interactions may not be related to the long -range crystalline order, \nbut instead, are mainly determined by the short -range atomic order like electron transport \nbehaviors down to trilayer  (as discussed above in Figure 2a) . Remarkably,  for the 1 nm a -FGT \napproaching the thickness limit of c -FGT monolayer, ferromagnetic characteristics still keep well \nbelow 100 K as shown in Figure 2d, indicating the existence of ferromagnetism.    \nDemonstration of Ferromagnetism through Magnetoresistance Measurements . Previous studies \nhave attributed the ferromagnetism of c -FGT in a few layers to the appearance of PMA \nsuppressing magnon excitation4,5. As shown in Figure 2b,d, the sputtered a -FGT does not show \nPMA , but does show hysteresis b ehaviors down to 1 nm at low temperatures. One possible \nreason for stabilizing ferromagnetism in the sputtered a -FGT can be the in -plane magnetic \nanisotropy (IMA)26. To reveal IMA and further confirm ferromagnetism, we performed \nanisotropic magnetoresistance (AMR) measurements33 by detecting angle dependence of R xx \nwithin different planes, as schematically shown in the inset of Figure 3a. Moreover, AMR \nmeasurements can also provide clear evidences to distinguish possible superparamagnetism \ninduced by magnetic nanoparticles or aggregates, which is always isotropic as  demonstrated in 10 \n granular films34,35. The applied current is along the x direction and the strength of applied \nmagnetic field (H) was fixed at 6 T , which is large enough to saturate magnetization in all \ndirection s as demonstrated in Figure 2b,c. Figure 3 and Figure 4 present the typica l AMR results \nof 3 nm a -FGT at several representative temperatures. The absence of magnetic field and angle \ndependences of R xx at 300 K are consistent with R xy results (Figure 2b), further confirming no \nferromagnetism at room temperature.  \n \nFigure 3. (a-c) The angle dependences of R xx for 3 nm a -FGT at 300 K (a), 50 K (b), and 3 K (c). The solid lines are  sin(2𝜑) or \ncos(2𝜑) fitting results, where φ represents α, β, or  γ in each scan plane as defined in the inset of (a). The applied external field is \n6 T. \nWhen the temperature drops to 50 K, as shown in Figure 3b, R xx shows clear angle dependences. \nAccording to the AMR theory, R xx depends on the angle (φ) between electrical current and \nmagnetization, that is, 𝑅𝑥𝑥∝𝑐𝑜𝑠2𝜑 and shows maximum when magnetization is along to the \ncurrent direction. For the α and γ scans shown in Figure 2b, Rxx can be explained through the \n21.74 aFGT(3) @ 300K\n b\n gH = 6 T\n33.68Rxx (kW)           @ 50K\n03\n0 90 180 270 36057.2857.30\na, b, g (°)            @ 3K\n03\nxy\nαzβ\nγ(a)\n(b)\n(c)\nMR ( ×10-4)11 \n AMR mechanism. However, the β scan results contradict AMR theory since R xx should not \nchange when magnetizati on is perpendicular to current and the angle dependence in the β scan is \nnot expected33. The β dependent R xx is usually observed in a ferromagnetic multilayer (usually a \nbilayer) involving a spin Hall layer or a Rashba interface due to spin Hall magnetoresistance \n(SMR)36–39, or in a thin ferromagnet due to geometrical size effects (GSE)40,41. SMR appears \nwhen the spin polarization generated through the spin Hall or Rashba effects (along the y \ndirection) modulates the spin absorption at the spin Hall or Rashba interfaces, which induces a \nresistance change when the spin polarization is parallel or perpendicular to magnetization. In this \ncase, the minimum of R xx corresponds to H along the y direc tion. This does not agree with \nexperimental data with a maximum value around β = 90º even though we consider  that a spin \ncurrent  may be  generated in a single ferromagnet with broken inversion symmetry42–44. \nTherefore, we attribute the β scan results to the GSE -related magnetoresistance, in which the β \ndependence of R xx is also consistent with that observed in most ferromagnetic materials40. As \nshown in Figure 3c, the AMR effects become more pronounced at 3 K and the corresponding \nmagnetoresistance ratio (MR) of γ scan increases about two times, as expected originating from  \nferromagnetic behaviors which are usually enhanced with decreasing temperature.  12 \n  \nFigure 4. (a-c) R xx of 3 nm a -FGT as a function of applied external fields measured at 3 K. Inset schematically shows field \ndirections. The enlarged parts in the low and high field ranges are shown in (b) and (c), respectively. The arrows in (b) ind icate \nfield sweep directions. (d) The corresponding R xx versus applied magnetic field at 300 K.  (e) Schematic of transverse  and vortex  \nDWs.  \nTo further examine the detailed magnetization switching driven by an applied magnetic field, R xx \nas a function of H along different directions was also recorded. Figure 4 shows the field-\ndependent R xx curves at two t ypical temperatures showing ferromagnetic or nonmagnetic \ncharacteristics  respectively . At 3 K, a negative MR that R xx decreases with increasing H can be \nobserved for all directional fields, as shown in Figure 4a. The negative MR has been \ndemonstrated in ba tch of 2D materials with strong disorder45–47, 2D electron systems48,49, and \nultrathin ferromagnetic layers50–52. The decreased resistance at high magnetic fields can be \nattributed to the reduced scattering pr obabilities of electrons from thermally excited magnons or \n-4 0 421.6821.72\nH (T)FGT(3) @ 300 K\n-4 0 457.558.0Rxx (kW) Hx\n Hy\n Hz\nFGT(3) @ 3 K\nxyz\n-0.2 0.0 0.225 W\n5 6100 W(a) (b)\n(c)\n(d)\nxyz (e)\nTransverse Vortex13 \n short -range disorder such as ionized impurities40. In the sputtered a -FGT showing \nferromagnetism, both the magnon and disorder  scatterings occur and thus may contribute the \nnegative MR simultaneously. Figure 4b and 4c show the enlarged R xx curves in low and high \nfield ranges, respectively. At high fields, R xx curves do not overlap under different directional \nmagnetic fields, refl ecting the AMR effects show n in Figure 3c.  \nMore interesting results are shown in Figure 4b, in which there are two dips, instead of two sharp \npeaks like conventional ferromagnets33, appearing in the R xx curves around zero field. The two \ndips are the typical giant magnetoresistan ce (GMR) phenomena in spin valve s or domain wall \n(DW) magnetoresistance in narrow ferromagnetic wires53. The former requires  at least two \nferromagnetic layers9, which probably does not happen in a single ferromagnetic layer . The latter \nhappens because the magnetization in DWs rotates to other directions and results in a resistance \nchange. For example, as schematically shown in Figure 4e, for an in -plane magnetized  \nferromagnet  along the x direction , the magnetization in a transverse  DW will rotate to the y \ndirection  (partially to the z direction for vortex DWs)54. Since R xx (H//y) < R xx (H//x) according \nto the AMR theory, the resistance of DW regions and thus the total R xx reduces when \nmultidomain states are formed during magnetization switching53. To verify the possible DW -\nrelated resistance change  as the mechanism of two resistance  dips in Figu re 4b, we first inspect \nthe field range where the two dips appear. As shown in Figure 4b, the two dips happen to appear \nin the field range where the multidom ain state appears (180 Oe ≤|𝐻|≤ 750 Oe, also \nconfirmed by direct magnetometry measurements shown below  in Figure 5a), and thus, the \nreduced R xx can only be explained by considering the electrical transport within DWs.  Second, \nthe dips at positive and negative fields almost overlap for t he field along x and y directions and \nthe resistance drops about 28 Ω. The relative resistance change is about 4.81 × 10-4, which is \nlarger than the MR value of α and β scans but smaller than γ scans as shown in Figure 3c. This \nindicates that the magnetiza tion of DWs in the multidomain state mostly rotates  from the x to z \ndirection  (vortex DWs ) and the resistance decrease is also consistent with Figure 3c  where R xx \n(H//z) < R xx (H//x). For the R xx versus H z curve, the resistance drop is about 31 Ω and the relative \nresistance change (about 5.32 ×  10-4) is still smaller than the MR value of γ scan , which can also \nbe understood as the DW -induced resistance decrease. Third, since the two dips in the three \ndirect ional Rxx curves can only be explained as the magnetization rotat ing from  the x to z \ndirection , it indicates a strong IMA along the x direction with the strength larger than  180 Oe  14 \n (domain nucleation field) . This is because, if the in -plane anisotropic  field is smaller than 180 \nOe, the magnetization should be aligned to the external field direction before domain nucleation \nand the dips in the H y and H z curves cannot arise from the x to z magnetization rotation.  The \nIMA along x direction  is also consistent with the expected  shape anisotropy  along the length of \nHall bar.  \nIt should be noted that, the two dips are in general the same as two switching peaks in the AMR \ncurves of conventional fe rromagnets except that the multidomain states survive in a large field \nrange and the switching dips are negative for all directional fields. The negative dips can be \nunderstood that R xx in the multidomain states is dominated by electrical transport within the DW \nregions that result in a decreasing R xx as explained above. In conventional ferromagnets, R xx of \nthe multidomain states is usually dominated by electrical transport within the domain  (rather than \nDW) regions , where R xx, and thus the sign of two swit ching peaks, is determined by the relative \norientation between applied current and the total magnetization of all domains as predicted by \nthe AMR theory33. The DW transport -dominated R xx in a single a -FGT layer not only confirm s \nthe persistence of ferromagnetism but also indicate s the possible unconventional DW behaviors \nthat may be interesting in theory and application.  The direct observation of DWs requires low -\ntemperature magneto -optical Kerr -effect (MOKE) microscope  with high -spatial  resolution, \nwhich is beyond  the scope of current work.  At 300 K, no any field -dependent R xx signals are \ndetected due to the lack of ferromagnetism, as shown in Figure 4d.  \nMagnetization Characterization through VSM  Measurements . The magnetization of sputtered a -\nFGT was also directly characterized by using  vibrating sample magnetometer (VSM). To gain \nclear magnetic signals, a 10 nm a -FGT film was adopted for the VSM measurements. Figure 5a \nshows magnetization as a function of appl ied in -plane H at 50 K and 3 K, in which typical \nhysteresis loops due to ferromagnetism can be observed. Both the saturation magnetization and \ncoercivity increase with decreasing temperature from 50 K to 3 K, in consistent with \nferromagnetic behaviors in m ost conventional ferromagnets. At 3 K, the magnetization gradually \nswitches to a reversed direction around zero field due to domain formation and the coercivity is \nabout 310 Oe. The magnetization switching process and switching fields agree well with that \nrevealed by AMR measurements shown in Figure 4b. Figure 5b shows the temperature \ndependence of magnetization under a 500 Oe in -plane H. The increase of magnetization with 15 \n decreasing temperature further confirms the ferromagnetic behaviors. The temperature at which \nmagnetization drops to zero is about 240 K, very close to T c = 220 K determined through Hall \nmeasurements (Figure 2f). These VSM results provide direct magnetometry evidences for the \nappearance of ferromagnetism in the sputtered a -FGT.  \n \nFigure 5. (a) In -plane magnetized hysteresis loops measured at 50 K and 3 K. A diamagnetic linear background due to substrates \nhas been subtracted. (b) The temperature dependence of magnetization under a 500 Oe in -plane field.  \nDiscussion . Ferromagnetism in 2D materials has attracted growing attention in both material and \nspintronic research areas. In fact, ferromagnetism in atomically thin conventional ferromagnets, \nsuch as Fe, Co , and Ni, has been investigated for several decades55. These materials are \nepitaxially grown on single crystal substrates to form special crystal orientations, and similar to \nferromagnetic 2D materials, magnetocrystalline anisotropy is thought to be a key to removing the \nrestriction of  Mermin -Wagner theorem. However, as a fundamental question, the correlation \nbetween observed ferromagnetism and 2D crystalline order has never been explored. As \nmentioned above, clarifying this question is not only helpful for understanding the mechanism of \n-3 0 3-2000200M (emu/cm3)\nH (kOe) 50 K\n 3 K(a)\n100 20001530M (emu/cm3)\nT (K) 500 Oe\n(b)16 \n ferromagnetism in 2D materials but also important in application to evaluate if they can be \ndeposited through CMOS -compatible technologies. Through structural, electrical, and \nmagnetization characterizations, our results  incontrovertibly demon strate that the ferromagnetism \ncan also persist in a-FGT down to 1 nm in the absence of crystalline order.  \nFirst, as shown in Figure 1a,b, there are no visible crystalline regions, nanoparticles, or Fe \naggregates in the sputtered FGT, which was examined by HRTEM with a spatial resolution less \nthan 1 nm. Moreover, the Fe concentration estimated from XPS spectra (at. 42.1%) is also much \nless than that of c -FGT (at. 50%), indicating that there are probably no extra isolated Fe atoms. \nEven though the re are some Fe aggregates with the size less than 1 nm that cannot be \ndistinguished by HRTEM, they should show superparamagnetic behaviors like magnetic ion -\ndoped granular films. The superpar amagnetic features are isotropic34,35 and not consistent with \nAMR results shown in Figure 3b,c.  In addition, the clear magnetoresistance switching signals \naround zero field driven by applied H (Figure 4a,b ) cannot arise from superparamagnetism.  \nSecond, XPS spectra show that the chemical valence states of each element are exactly the same \nas c-FGT an d all valence bond states in c -FGT can also be found in the sputtered a-FGT. These \nvalence state results demonstrate that the chemical bond and thus the relative position between \ntwo atomic sites remains at least in the length of next -nearest neighbors in the sputtered a -FGT. \nRemarkably, the T c of a- and c -FGT that is mainly determined by the strength of exchange \ncoupling between two neighbored sites18,22,23 is also correspondingly the same  with the thickness \nlarger than 5 nm, indicating that the long -range crystallize order, both in -plane and out -of-plane, \nmay not be the main factors determining  the exchange interactions and local magnetic \nanisotropic fields in the layered c -FGT. As shown i n Figure 2f, for the FGT layer less than 5 nm, \nTc shows almost the same reduction for both phases, also confirming similar exchange \ninteractions and local magnetic anisotropic energies in c - and a -FGT , although the former usually \nshows PMA while the latter  shows IMA. It should be noted that PMA of sputtered FGT can also \nbe expected by introducing an interfacial PMA like amorphous CoFeB with selected buffer and \ncapping layers29.  \nThird, the ferromagnetism of amorphous FGT persists down to 1 nm, approaching the thickness \nof c-FGT monolayer. According to the Hall measurements, T c of the sputtered 1 nm FGT is \nabout 100 K, which is very close to the reported T c values of c -FGT monolayer5, demonstratin g 17 \n that the similar exchange interactions and local magnetic anisotropic energies in a -FGT as c -FGT \nmaintain even below 1 nm. As demonstrated by AMR (Figure 4b) and VSM (Figure 5a) \nmeasurements independently, the sputtered ultrathin a -FGT films show IMA and  gradually \nmagnetization switching around zero field due to domain formation. The IMA may be the source \nto counteract restriction of the Mermin -Wagner theorem and stabilize ferromagnetism in \nultrathin a -FGT films26.  \nCONCLUSIONS  \nThe ferromagnetism of a -FGT  thin films down to 1 nm  has been demonstrated  by using electrical \nand magnetometry measurements. Similar to c -FGT monolayers with PMA, our results show that \nferromagnetism can also persist in the in -plane magnetized a -FGT with the thickness close to the \nc-FGT monolayer limit, where IMA may take a cri tical role in the stabilization of long -range \nferromagnetic order by creating a magnon energy gap. The T c of a-FGT is the same as that of c -\nFGT when the thickness is larger than 5 nm and also shows close valves for a thinner FGT below \n3 nm, indicating that  the two fundamental factors contributing ferromagnetism, exchange \ninteraction and local magnetic anisotropy, are similar for both amorphous and crystallized phases \nand may not relate to 2D crystalline order. The clear ferromagnetic switching of a -FGT with  \ndomain formation is also revealed by using magnetoresistance and VSM measurements \nindependently.  In addition, the DW-dominated magnetoresistance  that usually appears in very \nnarrow ferromagnetic wires  is also observed in the single a -FGT layer  and can onl y be explained \nby considering vortex -type DWs , indicating the possible unconventional magnetic domain \nbehaviors in the a -FGT . \nFrom the viewpoint of spintronic applications, similar magnetic attributes in crystallized and \namorphous FGT indicate that a -FGT t hat can be fabricated facilely in large scales may be used \nfor replacing c -FGT in most devices. For instance, it will be interesting to verify large voltage \ncontrol and GMR effects by employing a -FGT5,56,57. Moreover, as mentioned above, fabrication \nof perpendicularly magnetized a -FGT by using selected adjacent layers and exploration of \npossible special domain or skyrmion structures58–60 and corresponding spin -orbit torque \nswitching61,62 are also interesting in applications. Like c -FGT, Tc of a-FGT may also be \nincreased up to room temperature by modulating Fe concentrations63. \n 18 \n EXPERIMENTAL SECTION  \nThe a -FGT thin films were sputtered from a Fe 3GeTe 2 target with the purity higher than 99.9% \nthrough DC magnetron sputtering. The base vacuum before sputtering was pumped down to 8 ×  \n10-9 Torr and the Ar pressure during sputtering was set to 2 mTorr. The thickness of sputtered \nFGT was controlled between 1 nm and 120 nm by using a deposition rate about 0. 17 Å/s with the \nDC power of 15 W. A 2.5 nm Ta or 10 nm Si 3N4 layer as the capping layer was then \nsubsequently deposited, in which the 2.5 nm Ta would be naturally oxidized when the samples \nwere transferred o ut from the vacuum chamber. All samples were deposited on silicon wafers \nwith a 300 nm thermally oxidized SiO 2 layer. By using the standard photolithography and ion \nmilling processes,  the deposited thin films were then patterned into a Hall bar structure ( as \nschematically shown in the inset of Figure 2a) with the width of 10 μm and length of 50 μm for \nthe Hall (R xy) and resistance (R xx) measurements. A 30 nm FGT without capping layers were \nalso deposited for structural and compositional characterization by using commercial XPS  \n(ESCALAB 250Xi)  and HRTEM (FEI Titan Themis 200 TEM) measurements. R xx and R xy were \nmeasured by using Physical Properties Measurement System (PPMS, Quantum Design) in the \ntemperature range of 3 - 300 K.   \nASSOCIATED CONTENT  \nThe Supporting Information is available free of charge at  \nSupplementary Note, Figure S1 -3, and Table S1 present structure and composition of sputtered \nFGT analyzed by  AFM, X -ray diffraction (XRD), and  scanning electron microscopy -energy \ndispersive X -ray spectr ometry (SEM -EDX).  \nACKNOWLEDGMENTS  \n  This work is supported by the National Key R&D Program of China (Grant No. \n2019YFB2005800 and 2018YFA0701500), the National Natural Science Foundation of China \n(Grant No. 61974160, 61821091, and 61888102), and the Strate gic Priority Research Program of \nthe Chinese Academy of Sciences (Grant No. XDB44000000).  \n \nREFERENCES  19 \n (1) Mermin, N. D.; Wagner, H. Absence of Ferromagnetism or Antiferromagnetism in One - \nor Two -Dimensional Isotropic Heisenberg Models. Phys. Rev. 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B  \n2016 , 93 (1), 014411. https://doi.org/10.1103/PhysRevB.93.014411.  \n \n  25 \n Supporting Information  \nFerromagnetism  of Nanometer -Thick Sputtered Fe3GeTe 2 Films in the \nAbsence of  Two -Dimensional Crystalline Order:  Implications for \nSpintronics Applications  \nQianwen Zhao1, 2, ChaoChao Xia1, 3, Hanying Zhang1, 2, Baiqing Jiang1, 2, Tunan Xie1, 2, Kaihua \nLou1, 2, and Chong Bi1, 2, 3* \n1State Key Lab of Fabrication Technologies for Integrated Circuits, Institute of Microelectronics, \nChinese Academy of Sciences, Beijing 100029, China  \n2University of Chinese Academy of Sciences, Beijing 100049, China  \n3School of Microelectronics, University of Science and Technology of China, Hefei 230026, China  \n*E-mail: bichong@ime.ac.cn  \n  26 \n Supplementa ry Note  1: Surface roughness of sputtered FGT  \nThe surface roughness of sputtered FGT has also been confirmed by atomic force microscopy \n(AFM) . Figure S1 shows AFM images of sputtered 1 nm FGT without capping layer, from which \nthe calculated average roughness is about 0.21 nm. Please note that the 1 nm FGT has been \noxidized according to TEM analyses as mentioned in main text.  \n \nFigure S1. S urface morphology of 1 nm  oxidized a -FGT thin films  scanned by AFM . \nSupplementa ry Note  2: XRD results of sputtered FGT  \nX-ray diffraction  (XRD) measurements were performed by using a sputtered 120 nm FGT thin \nfilm without capping layer s. No extra XRD peaks due to FGT can be detected, indicating no \ncrystalline order formed in the sputtered FGT.  \n \n30 40 500300 FGT (120 nm)\n2θ (degree)Intensity (a. u.) Sub.a27 \n Figure S2. XRD results of sputtered 120 nm FGT (no capping layers). XRD signals from Si/SiO 2 \nsubstrates are also presented for comparison.  \nSupplementa ry Note  3: Fe concentration of sputtered FGT analyzed by EDX  \nExcessive Fe may induce ferromagnetism  in the sputtered FGT. We have also measure d the Fe \nconcentration by using scanning electron micros copy -energy dispersive X -ray spectrometry \n(SEM -EDX) . Figure S 3 shows a typical c ross-sectional SEM image  of 60 nm sputtered FGT (no \ncapping layer s). EDX data was collected at two different positions, P 1 and P 2. The corresponding  \nFe, Ge, and Te  concentrations are calculated and listed in Table S1,  in which  the measured  Fe \nconcentration approach es that acquired by XPS  and is also much less than that of crystallized \nFGT.   \n \nFigure S 3. Cross -sectional SEM images of 60 nm sputtered FGT. P 1 and P 2 indicate two positions \nfor acquiring EDX data.  \nTable S1. Composition of sputtered FGT acquired through SEM -EDX.  \nPosition  Fe (% , at.) Ge (%, at.) Te (%, at.) \nP1 38.09  28.62  33.29  \nP2 38.24  28.77  32.99  \n \n \n"    },    {        "title": "1302.1003v2.Reflection_and_refraction_process_of_spinwave_in_a_ferromagnet_frustrated_ferromagnet_junction_system.pdf",        "content": "arXiv:1302.1003v2  [cond-mat.mes-hall]  3 Jul 2013Journal of thePhysical Society of Japan DRAFT\nReﬂection and refraction process ofspinwaveinaferromagn et/frustrated\nferromagnet junction system\nYutaSasakiandHiroakiT.Ueda\nDepartment of Physics, Tokyo MetropolitanUniversity, Hac hioji, Tokyo192-0397, Japan\nFrustration introduces a nontrivial dispersion relation o f spinwave even in a ferromagnetic phase in a spin system.\nWe study the reﬂection and refraction process of spinwaves i n the ferromagnet/frustrated ferromagnet junction system\nby using the Holstein-Primako ﬀspinwave expansion and taking the large- Slimit. We discuss the relation between\nthe incident angle and the refraction angle of spinwave, nam ely, the Snell’s law of spinwave. As concrete examples\nof frustrated ferromagnets, we study the fully polarized fe rromagnet phases in J1-J2chains and the J1-J2models on\nthe square lattice. The interesting refraction processes, such as the splitting of the incident spinwave and the negati ve\nrefraction, are discussed. Wealsostudy the transmittance and reﬂectance inthese concrete models.\nKEYWORDS: spinwave, magnonics, junction system, frustrat ion, negative refraction\n1. Introduction\nSpinwave is a collective excitation of spins in magnets.\nSincespinwavecancarryinformationonthenano /microscale,\nrecently,potentialapplicationofspinwavetonewdevices has\nattracted much attention to researchers.1)In many circum-\nstances, spinwave behaves like a sound or light wave; reﬂec-\ntionandrefractionprocessofspinwavecanbeconsidered.\nUnderstanding of the reﬂection and refraction process is\nimportanttocontrolspinwave.Inthejunctionsystemofusu al\nferromagnets in which spinwave has a ‘trivial’2)dispersion\nrelation, the reﬂection and refraction process is theoreti cally\nunderstood by using geometrical-opticsapproximation3–6)or\nLandau-Lifshitz-Gilbert equation.7)One of the important re-\nsults of these studies is that the direction of the propagati ng\nspinwave depends on the strength of applied magnetic ﬁeld.\nThe experimental control of propagation of spinwave is in-\ndeedrealized bytuningthe internaldemagnetizingﬁelds in a\npermalloy waveguid.8)If the dispersion relation of spinwave\nis nontrivial, an exotic refraction process is expected. In the\ncase of light, for example, the negative refraction occurs d ue\nto the nontrivial dispersion relation.9,10)Recently, it is theo-\nreticallyproposedthattheanisotropicnatureofthedispe rsion\nrelationof dipole-exchangespinwavescan result in the neg a-\ntiverefraction.7)\nRecent experimental advance has been exhibiting a wide\nvariety of frustrated magnets. Magnetic frustration provi des\nus with an exotic magnetic behavior.11)Even in a ferromag-\nnet,in whichall spinsalign,thefrustrationcan induceanu n-\nusual dispersion relation of spinwaves. Hence, it will be na t-\nuraltoexpectinterestingrefractionprocessesofspinwav esin\nfrustratedferromagnets.In this paper,we study the reﬂect ion\nand refraction process of spinwave in ferromagnet /frustrated\nferromagnetjunctionsystem.\nAsa concreteexampleof frustratedferromagnets,we con-\nsider the spin systems which are already realized experi-\nmentally. For example, a frustrated ferromagnet can be pre-\npared by applying high magnetic ﬁeld on every frustrated\nmagnet. One-dimensional (1D) J1-J2Heisenberg chain with\na nearest-neighbor exchange coupling J1and next-nearest-\nneighbor coupling J2is one of the famous frustrated mag-\nnets.12–16)In the fully polarized phase under high magneticﬁeld, the dispersion of spinwave has two minima at12,17–19)\nthe wavevector Qs of cosQ=−J1/4J2for|J1|/J2<4 and\nJ2>0. There are many compounds consisting of the J1-J2\nchains (see Table. I in Ref. 13). The saturated ferromagneti c\nphases under high magnetic ﬁeld are experimentallyrealize d\nin13)Rb2Cu2Mo3O12,20–22)(N2H5)CuCl3and23–25)LiCuVO 4.\nAs another example of the frustrated magnets, the var-\nious compounds of the square-lattice J1-J2model, e.g.,\nBaCdVO(PO 4)2,arereported.26,27)Withoutexternalﬁeld,the\ncollinear antiferromagnet phase appears in BaCdVO(PO 4)2;\nthe saturation ﬁeld is about 4 ∼6T. In the saturated ferromag-\nnetic phase of this compound, theoretically, the dispersio n\nrelation is expected to be nontrivial; it has two minima at\nQ=(0,π)andQ=(π,0).26–28)\nThe organization of the present paper is as follows. In\nSec.2,webrieﬂyreviewthedispersionrelationsofspinwav es\nin (frustrated) ferromagnets by using the Holstein-Primak oﬀ\nspiwave expansion.InSec. 3,we generallydiscussrefracti on\nangles of transmission spinwaves in the ferromagnets junc-\ntion systems satisfying the given conditions. Then, we appl y\nthis discussion to the several junction systems including t he\nfrustrated J1-J2chains or the J1-J2model on the square lat-\ntice. We shall explicitly see the splitting of the incident s pin-\nwave and the negative refraction in these models. In Sec. 4,\nwestudythereﬂection-andthetransmissionrates(reﬂecta nce\nand transmittance)in part of the junction systems consider ed\ninSec.3bysolvingthesimpleSchr¨ oedingerequationswith in\nthe large Slimit. Our approach naturally treats the lattice\nstructureofferromagnetswithoutthecoarsegraining.\n2. Dispersion RelationofFrustratedFerromagnets\nAs a brief review, let us discuss the dispersion relation\nof spinwaves in the fully-polarized ferromagnetic phase of\n(frustrated) spin systems. For simplicity, we consider the lat-\ntice systems with one magnetic ion per unit cell, and assume\nthe rotational symmetry around the zdirection in spin space.\nThisassumptionleadstotheconservationlawofthetotalan -\ngular momentum along the zdirection, namely,/summationtext\ni/angb∇acketleftSz\ni/angb∇acket∇ight. We\nstudytheHamiltonianwiththegenericexchangeinteractio ns\nJij=Ji−j=Jj−i=Jl, the on-site anisotropicinteraction term\n1J.Phys. Soc. Jpn. DRAFT\nK,andtheexternalmagneticﬁeld H:\nH=/summationdisplay\n/angb∇acketlefti,j/angb∇acket∇ightJijSi·Sj+/summationdisplay\ni(−K(Sz\ni)2+HSz\ni),(1)\nwhere we use the coordinate ( x,y,z) in spin space, and i=\n(ai,bi,ci) in lattice space. Let us rewrite the spin operator\nby using the Holstein-Primako ﬀtransformation on the fully-\npolarizedferromagneticphase:\nSz\ni=−S+α†\niαi,\nS+\ni=√\n2Sα†\ni/radicalBigg\n1−α†\niαi\n2S≈√\n2Sα†\ni,\nS−\ni=√\n2S/radicalBigg\n1−α†\niαi\n2Sαi≈√\n2Sαi.(2)\nByusingthe1/Sexpansion,weapproximatelyobtainthefree\nbosonicHamiltonianintheleadingorderin 1 /S:\nH=/summationdisplay\nk(ω(k)−µ)α†\nkαk, (3)\nwhere\nǫ(k)=1\n2/summationdisplay\nlJlcosk·l, ω(k)=2S(ǫ(k)−ǫmin),\nµ=2S(ǫ(0)−ǫmin)−(H+2SK),(4)\nandǫminis the minimumof ǫ(k). The conservationlaw ofthe\nangular momentum along the zdirection assures the conser-\nvationofthetotalmagnonnumber.Foranyexchangeinterac-\ntions, a suﬃciently large H or Kcan induce the gap ( µ≤0)\nin the magnon dispersion. If µ≤0, the ferromagnetic phase\nisstable.Throughoutthispaper,wefocusonthisfreeboson ic\nHamiltonianbyassumingthelarge Slimit.\nWehaveseenthatadispersionrelationtakesvariousforms\nduetofrustration.Thisdispersionrelationcanleadtothe non-\ntrivialgroupvelocitynotparalleltothephasevelocity,w here\nthegroupvelocityisgivenby\nvg(k)=∇kω(k). (5)\nThe group velocity has an important physical meaning: it\ncarry the angular momentum as reviewed in Appendix. Not\nthe phase-but the groupvelocitydeterminesthe traveling d i-\nrection of spinwave. Next, let us study the variousdispersi on\nrelationsintheconcretemodels.\n2.1 simplecase\nFirst, we considerthe simple Heisenbergmodel on the cu-\nbic lattice with the nearest neighbor ferromagnetic coupli ng\nJ<0asshownin Fig.1. ǫ(k) isgivenby\nǫs(k)=J(coska+coskb+coskc),\nǫsmin=ǫs(k0=(0,0,0))=3J.(6)\nThis leads to the dispersion relation ωs(k) shown in Fig. 2.\nIn the long wavelength limit |k|→0, this dispersion relation\nbecomesisotropicandisgivenby\nωs(k)≈k2\n2ms,ms=−1\nJ. (7)\nInthefollowingdiscussion,weusethismodelastheconcret e\nexampleof‘usual’ferromagnets.\nFig. 1. (Color online) The cubic lattice with the nearest neighbor H eisen-\nberg exchange coupling J. The dots represent spins. The lattice constant\na0=1is assumed.\nFig. 2. (Coloronline)Thedispersionrelation ωs(k)ofthetrivialferromag-\nnet on the cubic lattice for J<0.kc=0is assumed.\n2.2 J 1-J2chain\nNext, let us discuss the dispersion relation of the 1D J1-J2\nchains with the ferromagnetic interchain coupling J3<0 on\nthecubiclattice17–19,29)asillustratedin Fig.3:\nǫ1(k)=J1coskc+J2cos2kc+J3(coska+coskb)\n=2J2(coskc+J1\n4J2)2−J2\n1\n8J2−J2+J3(coska+coskb),\n(8)\nwherethe J1-J2chainsareassumedtolieparalleltothe caxis.\nFor|J1|/J2≤4,J2>0andJ3<0,thedispersionrelationhas\ntwo minimaat Q=(0,0,±Qc)where\ncosQc=−J1\n4J2, ǫmin=−J2\n1\n8J2−J2+2J3.(9)\nThe dispersion relation ω1(k) is graphically shown in\nFigs. 4,5.\nFig. 3. (Color online) J1-J2chains with the interchain coupling J3on the\ncubic lattice. Thedots represent spins.\n2J.Phys. Soc. Jpn. DRAFT\nFig. 4. (Color online) The dispersion relation ω1(k) of the 1D J1-J2chain\nforJ1/J2=0.5 andJ2>0.J3=0orka=kb=0 is assumed.\nFig. 5. (Coloronline) Thedispersion relation ω1(k)ofthe1D J1-J2chains\non the cubic lattice for J1/J2=0.5,J2>0 andJ3/J2=−2.kc=0 is\nassumed.\n2.3 J 1-J2modelonthesquarelattice\nThe fully saturated phase in the J1-J2model on the square\nlattice (seeFig.6)also hasthenontrivialdispersionrela tion:\nǫ2(k)=J1(coska+coskb)+J2(cos(ka+kb)+cos(ka−kb)).\n(10)\nFor−2<J1/J2<2 andJ2>0,the dispersionrelation ω2(k)\nhastwo minimaat Q(2)\n1=(0,π)andQ(2)\n2=(π,0)asshownin\nFig.7.\nFig. 6. (Color online) Two dimensional square lattice. The dots rep resent\nspins connected by the nearest neighbor Heisenberg exchang e coupling J1\nand the next nearest neighbor coupling J2.\nFig. 7. (Color online) The dispersion relation ω2(k) forJ1/J2=−0.5 and\nJ2>0 in theJ1-J2model on the two-dimensional square lattice.\n3. TheSnell’sLawintheFerromagnetsJunctionSystem\nIn this section, we study the relation of the angles of\nincident-, reﬂected- and transmission- spinwaves passing\nthrougha boundary(the Snell’slaw).Let usconsidera ferro -\nmagnet/ferromagnet junction system whose boundary plane\nisﬂatandisperpendicularto cdirection.Ontheboundarythe\nproximityeﬀect mayinducea spin-exchangecouplingwhich\nmagnetically relates two ferromagnets. This proximity e ﬀect\nleadstotransmissionofincidentspinwave.Theschematicﬁ g-\nure of the reﬂection and refractionprocessin the usual ferr o-\nmagnetsjunctionsystemisshowninFig.8.\nFig. 8. (Color online) Schematic ﬁgureofrefraction and reﬂection process\nof spinwave in the usual ferromagnets junction system in the case ofkb=0.\nThetravelling direction of spinwave is determined by the gr oup velocity.\nInthecaseoflight,theSnell’slawisunderstoodbythecon-\nservationlawoftheenergyandthatofthemomentumparallel\ntothesurface.Thesamecontextcanbeappliedtodetermines\nthe Snell’s law of spinwave in the ferromagnetsjunctionsys -\ntemifthefollowingconditionsaresatisﬁed:(i)Theexchan ge\ninteractions in each ferromagnet and on the boundary have\na rotational symmetry around the zdirection in spin space.\n(ii) Each ferromagnethas a translationalsymmetry (far awa y\nfromtheboundary).(iii)Bothferromagnetshavethesamela t-\ntice vectors in the a-bplane, and the translational symmetry\nin thea-bplane exists in the whole junction system.30)The\nassumption (i) leads to the conservation law of a total num-\nberofmagnons,and(ii),(iii)leadtotheconservationlawo fa\ntotal momentum of the aandbdirection. Of course, the mo-\nmentum conservationlaw of the cdirection does not hold by\ntheboundaryeﬀect.\nSince the above discussion may be formal, let us consider\nthe simple junction system consists of ferromagnetswith th e\n3J.Phys. Soc. Jpn. DRAFT\nnearest neighbor interaction J(J′) on the cubic lattice (see\nSec. 2.1)withthe same lattice constant.Thisjunctionsyst em\nis schematicallyillustrated in Fig. 9. It may be appropriat eto\nconsidertheexchangeinteractionontheboundaryas\n∆/summationdisplay\na,bSi=(a,b,c1)·Sj=(a,b,c2), (11)\nwhere∆is the strength of the exchange interaction on the\nboundary,and c1andc2isthe positionofthe boundaryinthe\nc direction. This boundary condition satisﬁes the conditio ns\n(i),(iii).31)\nFig. 9. (Color online) Junction system of usual ferromagnets on the cubic\nlattice.Thedotsrepresentspins.Theexchangecoupling ∆duetothequantum\nproximity eﬀect is considered on the boundary.\nLet us consider the case that the incident spinwave in the\nmedium(1)hasthe momentum ki.The incidentspinwavear-\nriving at a boundaryis divided into the transmission wave of\nktandthereﬂectedwaveof kr.\nTheassumption(i)leadsto theenergyconservation\nΩ=ω(1)(ki)−µ(1)=ω(1)(kr)−µ(1)=ω(2)(kt)−µ(2).(12)\nAs discussed in Appendix,the group velocity correspondsto\nthecurrentofmagnons.Hence,itisappropriateinoursetti ng\nto assume that the group velocity of the transmission (reﬂec -\ntion) wave has the same (opposite) direction along the c axis\nasthatoftheincidentwave.Namely,\nv(1)c\ng(ki)/|v(1)c\ng(ki)|=−v(1)c\ng(kr)/|v(1)c\ng(kr)|=v(2)c\ng(kr)/|v(2)c\ng(kr)|.\n(13)\nInaddition,themomentumconservationlawparalleltothe\nboundaryleadsto\nka\ni=ka\nr=ka\nt,kb\ni=kb\nr=kb\nt. (14)\nTheseequationsdetermine krandkt,whichleadtotheSnell’s\nlaw.Ingeneral,thetransmission-andreﬂectedspinwavesd e-\ntermined by eqs. (12), (13), (14) are not necessarily single\nvalued.\nFinally, let us discuss the general relation between the re-\nﬂectance and transmittance if the conditions (i) (ii) (ii) a re\nsatisﬁed. By labeling spinwaves as k(1),k(2),k(3)···, this re-\nﬂectionandrefractionprocesscanbewrittenastheket\n(Aα(1)†\nki+/summationdisplay\njBjα(1)†\nk(j)\nr+/summationdisplay\njCjα(2)†\nk(j)\nt)|0/angb∇acket∇ight,(15)\nwhere|0/angb∇acket∇ightis vacuum state and a(1,2)(k) approaches the freeboson in the media (1,2) with the momentum ksuﬃ-\nciently away from the boundary. |v(1)c\ng(k(j)\nr)/v(1)c\ng(ki)||Bj/A|2\n(|v(2)c\ng(k(j)\nt)/v(1)c\ng(ki)||Cj/A|2)isreﬂectance(transmittance).By\nconsideringthe cuboiddiscussed in Appendix,we obtain the\nmagnonnumberconservationlaw:\n|A|2v(1)c\ng(ki)+/summationdisplay\nj|Bj|2v(1)c\ng(k(j)\nr)=/summationdisplay\nj|Cj|2v(2)c\ng(k(j)\nt).(16)\nThe concrete expression of BjandCjwill be discussed in\nSec. 4byexplicitlyconsideringtheboundarycondition.\nSince we have discussed the procedure to ﬁnd the Snell’s\nlaw, let us study the refractionprocesses in the variousfer ro-\nmagnetsjunctionsystems.\n3.1 usualcase\nFirst, let us further study the Snell’s law of the junction\nsystemoftheusualferromagnetsonthecubiclatticewithth e\nnearest neighbor exchange coupling J<0 (J′<0) as illus-\ntrated in Fig. 9. The dispersion relation on each ferromagne t\nisrespectivelygivenby\nωs1(k)=2S(J(coska+coskb+coskc)−3J),(17a)\nωs2(k)=2S(J′(coska+coskb+coskc)−3J′),(17b)\nwhereµ1<0 andµ2<0.µ1,2are given in eq. (4). In these\nusualferromagnets,thegroupvelocity vg,whichcarrythean-\ngularmomentum,isgivenby\nv(s1)\ng(k)=2S∇kǫ(k)=−2SJ(sinka,sinkb,sinkc),(18a)\nv(s2)\ng(k)=−2SJ′(sinka,sinkb,sinkc). (18b)\nThe sign of the group velocity v(s1)a,b,c\ngis always the same\nas the sign of ka,b,cfor−π<ka,b,c<π. If we consider the\nincidentwaveofwavevector ki, eqs.(12),(13),(14)uniquely\ndeterminethereﬂectedwaveof krandthetransmissionwave\nofkt.We obtain\ncoskc\nt=1\nJ′(Jcoskc\ni+(J−J′)(coska\ni+coskb\ni)−3(J−J′)\n+µ2−µ1\n2S),\n(19)\nwherekc\nt/|kc\nt|=kc\ni/|kc\ni|for−π<kc\nt<π. Thisimpliesthat, the\nlarger the chemical potential µ2is, the slower the group ve-\nlocity of the transmission wave becomes. Since µ1,2depends\nontheexternalmagneticﬁeldH 1,2ineachcompound,weex-\nplicitly see that the propagation of the spinwave can be con-\ntrolled by the (local) external magnetic ﬁeld as discussed i n\nRefs. 3–6.\nIf we consider the long-wavelength limit |k| →0, the\nSnell’s law in this system becomes simple. The group veloc-\nityisproportionaltothephasevelocityinthelongwavelen gth\nlimitk→0.Uptotheorderof k2,therefractiveindexisgiven\nby\nn=sinθi\nsinθt=|kt|\n|kc|=/radicalBigg\nΩ−J′+µ2\nΩ−J+µ1,(20)\nwhereθi(θt)istheangleofincidence(refraction)asshownin\nFig.8,andΩisgivenineq.(12).Eq.(20)isexactlythesame\nastheresult obtainedin Ref.4.\n4J.Phys. Soc. Jpn. DRAFT\n3.2 J 1-J2chainsparallelto theboundary\nNext, let us consider the reﬂection and refraction process\nof spinwavesin the usual ferromagnet /frustratedferromagnet\njunction system. In this subsection, as a concrete example o f\nfrustrated ferromagnets, we consider the J1-J2chains on the\ncubic lattice discussed in Sec. 2.2. We assume that the J1-J2\nchainsareparalleltotheboundaryplaneasshownin Fig.10.\nThe dispersion relation and the group velocity in the J1-J2\nchainsonthecubiclattice aregivenby\nω11(k)=2S(J1coska+J2cos2ka\n+J′(coskb+coskc−2)−J2\n1\n8J2−J2),\nv(11)\ng(k)=−2S(J1sinka+2J2sin2ka,J′sinkb,J′sinkc).\n(21)\nIfweconsidertheincidentwaveofwavevector kiintheusual\nferromagnet,thetransmissionwaveof ktisuniquelygivenby\neqs.(12),(13),(14):\ncoskc\nt=1\nJ′/parenleftbigΩ+µ2\n2S−J1coska\ni−J2cos2ka\ni\n−J′(coskb\ni−1)−J2\n1\n8J2−J2/parenrightbig,(22)\nwherekc\nt/|kc\nt|=kc\ni/|kc\ni|for−π <kc\nt< π. As discussed in\nSec 2.2, if−4<J1/|J2|<4 andJ2>0, the equation\nJ1coska\ni+J2cos2ka\nihas two minima at cos ka\ni=−J1\n4J2. If\ncoska\ni>−J1\n4J2, theacomponentof the groupvelocitiesofthe\nincident spinwave is of the opposite sign to that of the trans -\nmissionspinwave:therefractiveindexbecomesnegative32)as\nillustratedinFig.11.\nFig. 10. (Color online) Junction system of the usual ferromagnet (le ft,\ngreen)/ferromagnet consisting of J1-J2chains which lie along the aaxis\n(right, red). Theexchange coupling ∆due to the quantum proximity e ﬀect is\nconsidered on the boundary.\n3.3 J 1-J2chainsperpendiculartothe boundary\nInthissubsection,westudythecasethat thefrustratedfer -\nromagnetconsistsof J1-J2chainsperpendiculartothebound-\nary plane as shown in Fig. 12. The dispersion relation in the\nJ1-J2chainsisgivenby\nω12(k)=2S(J1coskc+J2cos2kc\n+J′(coska+coskb−2)+J2\n1\n8J2+J2).(23)\nFig. 11. (Color online) Schematic ﬁgure of refraction and reﬂection pro-\ncess of spinwave in the case of the negative refraction index andkc=0. The\ntravelling direction of spinwave is determined by the group velocity.\nIf the spinwaveof kiin the usual ferromagnetis injectedinto\ntheJ1-J2chains, the transmission waves of k(±)\ntare given by\neqs.(12),(13),(14):\ncosk(±)c\nt=±/radicalBigg\nΩ−2SJ′(coska+coskb−2)+µ2\n4SJ2−J1\n4J2,\n(24)\nwhere−J1sink(±)c\nt+2J2sin2k(±)c\nt\n|J1sink(±)c\nt+2J2sin2k(±)c\nt|=kc\ni\n|kc\ni|. Theappearanceof ±in the\nright-hand side of (24) is because the dispersion relation h as\ntwo minima in the direction of c axis. Hence, when both k±\narepermitted,twospeciesoftransmissionspinwaveappear as\nshowninFig.13.\nFig. 12. (Color online) Junction system of the usual ferromagnet (le ft,\ngreen)/ferromagnet consisting of the J1-J2chains which lie along the caxis\n(right, red). Theexchange coupling ∆due to the quantum proximity e ﬀect is\nconsidered on theboundary.\nFig. 13. (Color online) Schematic ﬁgure of refraction and reﬂection pro-\ncess of spinwave in the case of kb=0. The incident spinwave can split into\ntwo transmission waves. The travelling direction of spinwa ve is determined\nby the group velocity.\n5J.Phys. Soc. Jpn. DRAFT\n3.4 J 1-J2modelonthesquarelattice\nFinally, let us brieﬂy consider the fully polarized phase in\ntheJ1-J2model on the square lattice as an example of frus-\ntratedferromagnets.Westudythe2-dimensionaljunctions ys-\ntem in the a-bplane shown in Fig. 14. The dispersion rela-\ntion in the J1-J2model is given by eq. (10). In the case that\n−2<J1/J2<2 andJ2>0, by injecting the spinwave of ki\nfrom the usual ferromagnet, the transmission spinwave of kt\nisgivenby\nka\nt=ka\ni,\ncoskb\nt=1\nJ1+2J2coska\ni(Ω+µ2\n2S−J1coska\ni−2J2),(25)\nwhere−J1sinkb\nt+2J2coska\ntsinkb\nt\n|J1sinkb\nt+2J2coska\ntsinkb\nt|=−sinkb\ni\n|sinkb\ni|. The negative refrac-\ntion can be realized as well as the junction system discussed\ninSec. 3.2asillustratedinFig. 11.\nFig. 14. (Color online) Junction system of the usual ferromagnet (le ft,\ngreen)/ferromagnetconsistingofthe J1-J2modelonthesquarelattice (right,\nred). The exchange coupling ∆due to the quantum proximity e ﬀect is con-\nsidered on the boundary.\n4. ReﬂectanceandTransmittance\nWe have studied the relation between the refractionangles\noftransmissionspinwavesandtheangleofincidentspinwav e,\nnamely,theSnell’slawofspinwavesinferromagnetsjuncti on\nsystem. In this section, let us discuss the reﬂectance and th e\ntransmittance of spinwaves. Although in this section we fo-\ncus on the concrete models discussed in Secs. 3.1, 3.2, 3.3,\nour approach will be easily applied to other frustrated ferr o-\nmagnets junction systems satisfying the conditions (i), (i i),\n(iii)discussedin Sec.3.\n4.1 usualcase\nLet us study the reﬂectance and the transmittance of spin-\nwaves in the usual ferromagnet /ferromagnetjunction system\non the cubic lattice with the boundary condition (11) (see\nFig.9).The Hamiltonianofthe totaljunctionsystemis give n\nby\nHtot1=/summationdisplay\n/angb∇acketlefti,j/angb∇acket∇ightforic,jc≤0JSi·Sj+/summationdisplay\nic≤0(−K1(Sz\ni)2+H1Sz\ni)\n+/summationdisplay\n/angb∇acketlefti,j/angb∇acket∇ightforic,jc≥1J′Si·Sj+/summationdisplay\nic≥1(−K2(Sz\ni)2+H2Sz\ni)\n+∆/summationdisplay\na,bSi=(a,b,c=0)·Sj=(a,b,c=1),(26)where we assume that the boundaryis located between c=0\nand 1, and/angb∇acketlefti,j/angb∇acket∇ightrepresents the pairs of the nearest neighbor\ncoupling. The reﬂection and refraction process with the inc i-\ndentspinwaveof kimaybewrittenasthe ket\n|RR1/angb∇acket∇ight=/summationdisplay\na,bei(ka\nia+kb\nib)(A0/summationdisplay\nc=−∞eikc\nicα†\na,b,c+B0/summationdisplay\nc=−∞e−ikc\nrcα†\na,b,c\n+C∞/summationdisplay\nc=1eikc\nt(c−1)α†\na,b,c)|0/angb∇acket∇ight,\n(27)\nwherekc\ntis given by eq. (19). Then, A,B,Care determined\nbytheSchr¨ oedingerequation:\nHtot1|RR1/angb∇acket∇ight=Ω|RR1/angb∇acket∇ight. (28)\nIf we consider the transition matrix element of Htot1|RR1/angb∇acket∇ightto\na lattice position, (28) is always satisﬁed except at c=0,1.\nHence,weconsider\n/angb∇acketleft0|aa,b,c=0Htot1|RR1/angb∇acket∇ight=Ω(A+B)ei(ka\nia+kb\nib),\n/angb∇acketleft0|aa,b,c=1Htot1|RR1/angb∇acket∇ight=ΩCei(ka\nia+kb\nib).(29)\nEquivalently,\n(ξ1(kc\ni)−Je−ikc\ni)A+(ξ1(kc\ni)−Jeikc\ni)B=∆C,\n∆(A+B)=(ξ2(kc\nt)−J′eikc\nt)C,(30)\nwhere\nξ1(kc\ni)=2J(coskc\ni−1)−J+∆,\nξ2(kc\nt)=2J′(coskc\nt−1)−J′+∆.(31)\nHence,weobtain\nB\nA=−(ξ1(kc\ni)−Je−ikc\ni)(ξ2(kc\nt)−J′eikc\nt)−∆2\n(ξ1(kc\ni)−Jeikc\ni)(ξ2(kc\nt)−J′eikc\nt)−∆2,\nC\nA=−∆2iJsinkc\ni\n(ξ1(kc\ni)−Jeikc\ni)(ξ2(kc\nt)−J′eikc\nt)−∆2.(32)\nBy usingthe relation\n|(ξ1(kc\ni)−Je−ikc\ni)(ξ2(kc\nt)−J′eikc\nt)−∆2|2\n−|(ξ1(kc\ni)−Jeikc\ni)(ξ2(kc\nt)−J′eikc\nt)−∆2|2\n=4JJ′∆2sinkc\nisinkc\nt=∆2v(s1)\ng(ki)v(s2)\ng(kt)/S2,(33)\nthemagnon-number-conservationlawgivenbyeq.(16)iseas -\nily conﬁrmedas\nv(s1)c\ng(ki)(1−|B\nA|2)=v(s2)c\ng(kt)|C\nA|2, (34)\nregardless of any kiandkt. From eq. (32), we explicitly\nﬁnd the dependence of the transmittance and the phase shift\nof spinwave on the boundary condition ∆. For example,\nthe transmittance |v(s2)c\ng(kt)/v(s1)c\ng(ki)||C/A|2, the reﬂectance\n|B/A|2andtherefractionindexsin θi/sinθtinthespeciﬁccase\nare shown in Fig. 15. Since the travelling direction of spin-\nwaveisgivenbythegroupvelocity,incidentspinwaveofdif -\nferent frequencies could have the same incident angle. For\n∆→0,thetransmittanceleads: |C/A|2=O(∆2/J′2).\n6J.Phys. Soc. Jpn. DRAFT\nFig. 15. (Color online) The incident angle θi, the transmittance, the re-\nﬂectance (rate), and the refraction index n=sinθi/sinθtin the usual fer-\nromagnet/ferromagnet junction system for J=−1,J′=−1.5,∆=−1.5,\nµ1/2S=−1.6,µ2/2S=−1.7. The incident spinwave with ka=π/4 and\nkb=0 in the usual ferromagnet is assumed. I,R,Trespectively denotes the\nincident wave, the reﬂection wave and the transmission wave . The incident\nangle and the refraction index are determined by the directi on of the group\nvelocity.\n4.2 J 1-J2chainsparallelto theboundary\nNext,letusstudythecasethatonesideoftheferromagnets-\njunction system consists of the J1-J2chains parallel to the\nboundary as shown in Fig. 10. The reﬂection and refraction\nprocessisdescribedbythe followingket:\n|RR2/angb∇acket∇ight=/summationdisplay\na,bei(ka\nia+kb\nib)(A0/summationdisplay\nc=−∞eikc\nicα†\na,b,c+B0/summationdisplay\nc=−∞e−ikc\nrcα†\na,b,c\n+C∞/summationdisplay\nc=1eikc\nt(c−1)α†\na,b,c)|0/angb∇acket∇ight,\n(35)\nwherekc\ntisgivenbyeq.(22).Theproceduretodetermine B,C\nis exactly the same as that of the previous section. Hence,\nBandCare given by (32). For exapmle, the transmittance\n|v(11)c\ng(kt)/v(s1)c\ng(ki)||C/A|2, the reﬂectance|B/A|2and the re-\nfractionindexinthespeciﬁccaseareshowninFig.16,where\nthe occurrence of the negative refraction index is explicit ly\nseen.\n4.3 J 1-J2chainsperpendiculartothe boundary\nFinally,weconsiderthejunctionsystemofthe J1-J2chains\nperpendicular to the boundary as shown in Fig. 12. In this\ncase,theket isgivenby\n|RR3/angb∇acket∇ight=/summationdisplay\na,bei(ka\nia+kb\nib)(A0/summationdisplay\nc=−∞eikc\nicα†\na,b,c+B0/summationdisplay\nc=−∞e−ikc\nrcα†\na,b,c\n+C1∞/summationdisplay\nc=1eik(+)c\nt(c−1)α†\na,b,c+C2∞/summationdisplay\nc=1eik(−)c\nt(c−1)α†\na,b,c)|0/angb∇acket∇ight,\n(36)\nFig. 16. (Color online) The incident angle θi, the transmittance, the re-\nﬂectance (rate), and the refraction index nin the usual ferromagnet /frus-\ntrated ferromagnet junction system consisting of the J1-J2chains parallel to\nthe boundary plane for J=−1,J1=−2,J2=1,J3=−0.5,∆=−1.0,\nµ1/2S=−1.6,µ2/2S=−1.5. The incident spinwave with ka=π/4 and\nkb=0 in the usual ferromagnet is assumed. I,R,Trespectively denotes the\nincident wave, the reﬂection wave and the transmission wave . The incident\nangle and the refraction index are determined by the directi on of the group\nvelocity.\nwherek(±)c\ntare givenbyeq. (24).We see that (28)is satisﬁed\nat anylattice cite exceptat c=0,1,2.Hence,we consider\n/angb∇acketleft0|aa,b,c=0Htot3|RR1/angb∇acket∇ight=Ω(A+B)ei(ka\nia+kb\nib),\n/angb∇acketleft0|aa,b,c=1Htot3|RR1/angb∇acket∇ight=Ω(C1+C2)ei(ka\nia+kb\nib),\n/angb∇acketleft0|aa,b,c=2Htot3|RR1/angb∇acket∇ight=Ω(C1eik(+)c\nt+C2eik(−)c\nt)ei(ka\nia+kb\nib),(37)\nwhereHtot3is the Hamiltonianof the total system considered\nin thissubsection.Equivalently,\n(ξ′\n1(kc\ni)−Je−ikc\ni)A+(ξ′\n1(kc\ni)−Jeikc\ni)B=∆(C1+C2),\n(ξ′\n2(k(+)c\nt)−J1eik(+)c\nt−J2ei2k(+)c\nt)C1\n+(ξ′\n2(k(+)c\nt)−J1eik(−)c\nt−J2ei2k(−)c\nt)C2=∆(A+B),\n(ξ′\n3(k(+)c\nt)−2J1cosk(+)c\nt−J2ei2k(+)c\nt)C1eik(+)c\nt\n+(ξ′\n3(k(+)c\nt)−2J1cosk(−)c\nt−J2ei2k(−)c\nt)C2eik(−)c\nt=0,(38)\nwhere\nξ′\n1(kc\ni)=2J(coskc\ni−1)−J+∆,\nξ′\n2(k(+)c\nt)=ξ′\n2(k(−)c\nt)=2(J1cosk(+)c\nt+J2cos2k(+)c\nt+J2\n1\n8J2)\n−J1+J2+∆,\nξ′\n3(k(+)c\nt)=ξ′\n2(k(+)c\nt)+J1−∆.\n(39)\nB/A,C1/A,C2/Aaregivenbysolvingtheseequations.Wenu-\nmericallyconﬁrmedthefollowingmagnonnumberconserva-\ntionlawin thevariousconcreteparameters:\nv(s1)c\ng(ki)(1−|B\nA|2)=v(12)c\ng(k(+)\nt)|C1\nA|2+v(12)c\ng(k(−)\nt)|C2\nA|2,(40)\n7J.Phys. Soc. Jpn. DRAFT\nwherev(12)c\ng(k)=−2S(J1sinkc+J2sin2kc). For ex-\nample, the transmittance |v(12)c\ng(k+\nt)/v(s1)c\ng(ki)||C1/A|2,\n|v(12)c\ng(k−\nt)/v(s1)c\ng(ki)||C2/A|2, the reﬂectance|B/A|2and the\nrefraction index in the speciﬁc case are shown in Fig. 17,\nwhere two species of transmission wave appear when both\nk(±)c\ntsineq.(24)haverealvalues.\nFig. 17. (Color online) The incident angle θi, the transmittance, the re-\nﬂectance(rate),andtherefraction index nintheusualferromagnet /frustrated\nferromagnet junction system consisting of the J1-J2chains perpendicular to\nthe boundary plane for J=−1,J1=−1.5,J2=1,J3=−0.5,∆=−1.0,\nµ1/2S=−1.6,µ2/2S=−1.5. The incident spinwave with ka=π/4 and\nkb=0 in the usual ferromagnet is assumed. I,R,T±respectively denotes\nthe incident wave, the reﬂection wave and the transmission w ave with the\nwavevector k(±)c\ntin eq. (24). The incident angle and the refraction index are\ndetermined by the direction of the group velocity.\n5. Conclusion\nWe studied the reﬂection and refraction process of spin-\nwavesintheferromagnet /frustratedferromagnetjunctionsys-\ntem whose ferromagnets are described by the spin Hamilto-\nnianwiththegenericHeisenberg-exchangecoupling,theun i-\naxial anisotropic interaction and the external magnetic ﬁe ld.\nByusingtheHolstein-Primako ﬀspinwaveexpansionandtak-\ning the large Slimit, the ferromagnetic phase is given by\nthe free bosonic Hamiltonian, which describes the dynam-\nics of magnons. If frustration exists, the dispersion relat ion\nofmagnons(spinwave)canbecomenontrivial.\nIn Sec. 3, we discussed the Snell’s law of the spinwave in\nthecasethatthefollowingconditionsaresatisﬁedintheju nc-\ntion system: (i) The exchange interactions in each ferromag -\nnet and on the boundary have a rotational symmetry around\nthezdirectioninspinspace.(ii)Eachferromagnethasatrans-\nlational symmetry (far away from the boundary). (iii) Both\nferromagnets have the same lattice vectors in the a-bplane,\nand the translational symmetry in the a-bplane exists in the\nwhole junction system.30)By studying the various junction\nsystems satisfying these conditions, we found the nontrivi al\nrefraction process, e.g., the splitting of the spinwave and the\nnegativerefraction.\nIn Sec. 4, we studied the ‘reﬂectance’, ‘transmittance’and\n‘phase shift at the boundary’in the concrete examples of the\njunction systems. Within the large Slimit, we exactly ob-\ntainthesequantities,whichexplicitlydependonthebound arycondition.Throughoutthispaperourresultsdoesnotneedt he\nlongwavelengthapproximation.\nWe thank G. Tatara, S. Murakami and A. Yamaguchi for\nusefuldiscussions.\nAppendix: Currentofangularmomentum\nIn this appendix,we brieﬂyreview the correspondencebe-\ntween the group velocity of spinwave and the current of an-\ngular momentum.We study the spin system describedby the\nHamiltonian (1). Now, the Hamiltonian (1) commutes with\nthe totalspin Sz\ntot=/summationtext\nlSz\nj. Hence,Sz\ntotisthe conservedquan-\ntity. This implies that the total magnon number/summationtext\nlα†\nlαlcon-\nserves, and the conserved current of magnons (related to the\nU(1)symmetry)exists.\nThemagnoncurrentisunderstoodbystudyingin-andout-\nﬂowofthemagnonnumberinagivenregion.Letusconsider\nthe cuboid C1of which two face-to-face surfaces S1andS2\nareinﬁnitelylargeasshowninFig.A ·1.\nFig. A·1.Magnon current of the group velocity vgpasses through the\ncuboidC1.\nThe spatially-averaged ﬂow of magnons is determined by\nthecurrentpassingthrough S1andS2.By usingtherelation\n[α†\niαi+α†\njαj,Si·Sj]=0, (A·1)\nwe obtain\nid(/summationtext\nl∈C1α†\nlαl)\ndt=[/summationdisplay\nl∈C1α†\nlαl,H]=S/summationdisplay\ni∈C1,j∈/C1Jij(α†\niαj−αiα†\nj),\n(A·2)\nwhereweuseeq.(2).Weassumethat Jij=Jlisshortrangeso\nthatJl=0 for a large|l|, and that S1is suﬃciently separated\nfromS2.Then,ifweconsiderthestate |k/angb∇acket∇ight=α†\nk|0/angb∇acket∇ight,weobtain\n/angb∇acketleftk|d(/summationtext\nl∈C1α†\nlαl)\ndt|k/angb∇acket∇ight=2S\nN/summationdisplay\ni∈C1,j∈/C1Jijsink·(i−j)\n=1\nN(AS1n1·vg(k)+AS2n2·vg(k)),\n(A·3)\nwhereASiistheareaof Siand\nvg(k)·n=2S∂ǫ(k)\n∂k·n=−2S/summationdisplay\nlJlsin(k·l)l·n.(A·4)\nThese equations imply that the group velocity can be viewed\nas the magnon current which carry the angular momen-\ntum. 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In addition, the translation al invariance\nexists within the least common multiple lattice vectors in t hea-bplane.\nHence, even ifthedi ﬀerence ofthelattice vectors is large, insomecases\nour simple discussion may predict the reasonable result, e. g.,in the case\nof the long wavelength limit.\n31) As will be discussed, the refraction angles are independ ent of the con-\ncrete form of the boundary condition as long as the assumptio ns (i), (iii)\nhold: the Snell’s law is independent of ∆. Only the transmission rate\nand the phase shift at the boundary depend on ∆as will be discussed\nin Sec. 4.1.\n32) S.Murakami pointed outthe possibility of the negative r efraction in the\nprivate communication.\n9"    },    {        "title": "2308.15372v1.Theory_of_Fractionally_magnetized_Quantum_Ferromagnet.pdf",        "content": "Theory of Fractionally-magnetized Quantum Ferromagnet\nIsao Maruyama∗\nDepartment of Information and Systems Engineering,\nFukuoka Institute of Technology, 3-30-1 Wajiro-higashi,\nHigashi-ku, Fukuoka 811-0295, Japan\nShin Miyahara\nDepartment of Applied Physics, Fukuoka University,\n8-19-1 Nanakuma, Jonan-ku, Fukuoka 814-0180, Japan\n(Dated: August 30, 2023)\n1arXiv:2308.15372v1  [cond-mat.str-el]  29 Aug 2023Abstract\nWe present a theory to realize entangled quantum spin states with fractional magnetization. The\norigin of magnetization reduction is partly emergent antiferromagnetism, that is, spin-liquefaction\nof ferromagnetism. We study a ferromagnetic bilinear coupling region of the spin- S(≧1) bilinear-\nbiquadratic spin chain based on (i) a rigorous eigenstate correspondence between the spin- Smodel\nand spin-1\n2model and (ii) a numerical exact-diagonalization calculation up to S= 3. As a result,\nwe obtain a fractional magnetized M= 1−1/(2S) phase, where ground states have quantum\nentanglement-reflecting corresponding spin-1\n2antiferromagnetic ground states in a ferromagnetic\nbackground. This spin-liquefaction theory of ferromagnets can be generalized to any-dimensional\nlattices even under a magnetic field. This fractional ferromagnetism opens the new research field\nof quantum ferromagnets.\nEntangled quantum states have been attracting not only researchers in physics but also\ndevelopers in quantum computer science. In condensed-matter physics, antiferromagnets\ninvolve many interesting topics, including entangled gapped quantum spin-liquid states [1]\nin integer spin- Schain with a Haldane gap [2], and fractionalized S/2 spins that form an\nentangled spin singlet on a bond in the valence-bond-solid picture of the Affleck–Kennedy–\nLieb–Tasaki (AKLT) model [3]. On the other hand, ferromagnetically ordered states in\nquantum systems can be approximated as “classical” states in the sense that fully polarized\nlocal spins have no quantum entanglement. Is there any ferromagnet with an entangled\nquantum state?\nA key to realizing an entangled ferromagnetic state is to partly create an antiferromag-\nnetic quantum state in a ferromagnetic classical background, that is, “spin liquefaction” of a\nferromagnet. When the total spin of a partly emergent “spin-liquid” (a phase with nonmag-\nnetic long-range N´ eel order) is zero, the coexistent states are fractionally magnetized. In\nthis Letter, we propose a simple procedure to construct a quantum spin- SHamiltonian that\nleads to the property of a phase transition from fully magnetized ground states to fraction-\nally magnetized ground states under zero magnetic field. This transition is accomplished\nby flat-band one-magnon instability and magnetization changes from M= 1 to a fraction\nM < 1. Note that this is not a magnetization-plateau state under an external magnetic field\nbut macroscopically degenerate ferromagnetic ground states with fractional magnetization\nunder zero magnetic field, that is, a “fractional ferromagnet.”\n2FIG. 1: Known phase diagram of S= 1 BLBQ chain. The high-symmetry points αr=π/4\nandαc=π/2 are generalized to higher- Sin Eqs. (2) and (3). ˆP(s)\nijis a projection operator\ndefined later in Eq. (4).\nThe realization of the spin-liquefaction is supported by a rigorous correspondence between\na subset of eigenstates in the spin- Smodel and whole eigenstates in the spin-1\n2antiferro-\nmagnetic model. In other words, the rigorous correspondence is “eigensystem embedding.”\nThus, it might be interesting even in the context of quantum many-body scars[4–8]. As an\nexample, we consider a spin- S(S≧1) bilinear-biquadratic (BLBQ) chain described by the\nHamiltonian\nˆH(S)\nα= cos αNX\ni=1ˆSi·ˆSi+1+ sin αNX\ni=1\u0010\nˆSi·ˆSi+1\u00112\n(1)\nwith the periodic boundary condition ˆSN+1=ˆS1. The phase diagram for the S= 1 case,\nshown in Fig. 1, has been massively studied [9] and includes the AKLT point at α= arctan1\n3\n[3], the SU(3) point at α=π\n4[10–12], and the other high-symmetry points at5π\n4[13],3π\n2[14–\n17], and7π\n4[18–21]. As explained later, for any S, the rigorous eigenstate correspondence\nbetween eigenstates consisting of SandS−1 spin states in the BLBQ chain and eigenstates\nin the spin-1\n2Heisenberg chain (i.e., spin-1\n2liquefaction) is realized at α=αrandα=αr+π,\nwhere\nαr=\n\n−arctan\u0010\n1\n2S(S−2)+1\u0011\n, S≦3/2\nπ−arctan\u0010\n1\n2S(S−2)+1\u0011\n, S≧2.(2)\nForS= 1,αr=π\n4corresponds to the SU(3) point. In other words, αris a generalization of\ntheS= 1 SU(3) point via preservation of partial SU(2) symmetry for the spin-1\n2liquefaction.\nNote that, because this correspondence at αris for eigenstates, numerical evidence is required\n3FIG. 2: (a) Phase transition from ferromagnetic phase to antiferromagnetic phase in S=1\n2\nHamiltonian JPN\ni=1ˆsi·ˆsi+1. (b) Phase transition at αcfrom ferromagnetic M= 1 phase\nto fractionally magnetized M= 1−1\n2Sphase in the higher- SBLBQ Hamiltonian ˆH(S)\nα\ndescribed by Eq. (1). Rigorous ground-state correspondence with spin-1\n2antiferromagnetic\nchain realized at αrforS≧2.\nto obtain the ground-state properties.\nAs a result of numerical calculation of the BLBQ chain, we find that the ground state\nof the S≧2 BLBQ model at αris equivalent to that of the S=1\n2antiferromagnetic\nchain, and the fractionally magnetized state is stabilized in a finite parameter region for\nS≧3/2. The spin-liquefaction transition from the fully magnetized M= 1 phase around\nthe ferromagnetic Heisenberg point α=πto the fractionally magnetized M= 1−1\n2Sphase\noccurs at\nαc=π−arctan\u00121\n2S(S−1)\u0013\n(3)\nforS≧3/2, as shown schematically in Fig. 2(b). This spin-1\n2liquefaction of the spin- S\nsystem can be considered as a generalization of the “entire” spin-liquefaction from the spin-\n1\n2ferromagnetic-ordered phase to the antiferromagnetic quantum-disordered phase of the\nS=1\n2Hamiltonian JPN\ni=1ˆsi·ˆsi+1atJ= 0, as shown in Fig. 2(a).\nTo explain the theoretical detail, let us start with a spin-projection Hamiltonian of a\nspin-Smodel on any lattice with general coefficients J(s)\nijdefined as ˆH=P\nijP2S\ns=0J(s)\nijˆP(s)\nij,\nwhere ˆP(s)\nijis a projection operator onto the subspace with total spin s∈[0,2S] for two\nspins at sites iandj. There is a general relation [22]\nˆP(s)\nij=2SY\nn=0\nn̸=sˆSi·ˆSj−qn\nqs−qn,(ˆSi·ˆSj)n=2SX\ns=0qsnˆP(s)\nij, (4)\nwith qs=s(s+ 1)/2−S(S+ 1). GivenP2S\ns=0ˆP(s)\nij= 1, the (2 S+ 1)-dimensional parameter\nspace of J(0)\nij, J(1)\nij, . . . , J(2S)\nijis reduced to 2 Sdimensions. By ignoring the positive energy\nscale factor, the intrinsic parameter space becomes a 2 S-dimensional sphere: for S= 1,\n4a two-dimensional sphere is a circle parameterized by α, that is, the BLBQ Hamiltonian.\nMoreover, the spin-projection Hamiltonian can simply express high-symmetry points of the\nS= 1 BLBQ chain, as summarized in Fig. 1, by ignoring the positive energy scale factor and\nenergy shift. In previous studies for S= 2, 2 S= 4 independent parameters are assumed to\nbeJ(1)\nij=J(3)\nij= 0 [23–25] and J(0)\nij=J(1)\nij= 0 [26, 27].\nIn this Letter, we consider the condition J(2S)\nij=J(2S−2)\nij for spin-liquefaction, which gives\nˆH(S)\nr=X\nij2SX\ns=0J(s)\nijˆP(s)\nij\f\f\f\f\f\nJ(2S)\nij=J(2S−2)\nij, (5)\nwhere a subset of eigensystem has a rigorous correspondence with whole eigensystem in\nthe spin-1\n2Heisenberg model ˆH(1/2)=P\nij(J(2S)\nij−J(2S−1)\nij )ˆsi·ˆsj+ε0with the S=1\n2\noperator ˆsiand energy shift ε0=P\nij(3J(2S)\nij+J(2S−1)\nij )/4. In short, for any eigenstate |ψ⟩\nofˆH(1/2), corresponding eigenstates of ˆH(S)\nrare rigorously written as |Ψ0⟩=ˆC|ψ⟩with\nan intertwiner[28, 29] ˆC=QN\ni=1(|S⟩i⟨↑|+|S−1⟩i⟨↓|), which is a mapping operator from\nthe spin-1\n2Hilbert space spanned by |↑⟩and|↓⟩, to the spin- SHilbert space spaned by\n|S⟩,|S−1⟩, . . . ,|−S⟩. The degeneracy in ˆH(S)\nris greater than that in ˆH(1/2)\nr because of\na ferromagnetic moment in |Ψ0⟩. The additional degenerate states are |Ψs⟩= (ˆS−\ntot)s|Ψ0⟩,\nwhere ˆSα\ntot=P\niˆSα\niis a total spin operator. This rigorous eigenstate correspondence is easily\nproved [30]. Note also that a numerical calculation is required to confirm that a ground state\nofˆH(S)\nrmay also be written as |Ψs⟩. For eigenstates, however, the correspondence is valid\nfor a general lattice in any dimension, and even under a magnetic field.\nThe BLBQ chain, Eq. (1), is rewritten as ˆH(S)\nα=P\niP2S\ns=0J(s)\nii+1(α)ˆP(s)\nii+1, where J(s)\nii+1(α) =\nqscosα+qs2sinαbased on Eq. (4) [31]. At the two point α=αrandαr+π, given by\nEq. (2), the BLBQ chain satisfies the condition J(2S)\nii+1(α) = J(2S−2)\nii+1(α). As a result, a\nsubset of eigensystem in H(S)\nαrcorreponds to whole eigensystem in spin-1\n2antiferromagnetic\nHeisenberg chain. In addition, ˆH(S)\nαatαrcan be considered as a higher- Sgeneralization\nofˆH(1)\nr=−P\niˆP(1)\nii+1atαr=π/4 in Fig. 1, which leads us to the spin-1\n2SU(2) model.\nThis generalization is not the usual SU(2 S+ 1) generalization with J(2S)\nij=J(2S−2)\nij =···=\nJ(0)\nij, J(2S+1)\nij =J(2S−1)\nij =···=J(1)\nij[10–12, 24].\nSimilarly, as a higher- Sgeneralization of ˆH(1)\nc=P\niˆP(0)\nijatαcin Fig. 1, let us introduce\n5another limitation J(2S)\nij=J(2S−1)\nij for the spin-projection Hamiltonian\nˆH(S)\nc=X\nij2SX\ns=0J(s)\nijˆP(s)\nij\f\f\f\f\f\nJ(2S)\nij=J(2S−1)\nij<J(s)\nij,(s≦2S−2), (6)\nwhich gives a phase boundary of the ferromagnetic phase ( J(2S)\nij< J(s)\nij). For the BLBQ\nchain ˆH(S)\nαc, at the phase transition point αcgiven by Eq. (3), the condition J(2S)\nii+1(α) =\nJ(2S−1)\nii+1(α)< J(s)\nii+1(α) is satisfied. For S=1\n2, this is a quantum phase transition between\nferromagnetic and antiferromagnetic phases via a trivial Hamiltonian ˆH(1/2)\nc= 0, as shown\nin Fig. 2(a). In general, it is easy to check that the ferromagnetic state |0⟩=Q\ni|S⟩iand\nthe one-magnon excited state ˆS−\ni|0⟩are ground states of ˆH(S)\ncwith eigenenergyP\nijJ(2S)\nij:\nthat is, the one-magnon flat band degenerate at the ground-state energy. In addition, other\nground states are multi-sublattice N´ eel-like states, defined as |m⟩= (Q\ni∈LmˆS−\ni)|0⟩for any\nmth sublattice Lm, where |m⟩hasSz\ntot=NS−NmandNm=|Lm|is number of mth\nsublattice sites. If a ground state for J(2S−1)\nij < J(2S)\nijoverlaps with |m⟩, the ground state\nhasSz\ntot=NS−NmandStot≧Sz\ntot, which becomes Stot≧Sz\ntot=N(S−1\n2) for the\nBLBQ bipartite chain ( Nm=N/2). It is naively expected that the magnetization jumps\ntoM=Stot/(NS) = (S−1\n2)/Sfrom M= 1 at αc, whereas numerical evidence is required\nbecause other states can be more stable.\nTo observe fractional magnetization for the spin- SBLBQ N-site chain Hamiltonian ˆH(S)\nα\n[Eq. (1)], we perform an exact diagonalization with the Lanczos method in the region π/2<\nα < π up to S= 3 by using translational symmetry. Figure 3 shows the magnetization M\nof the ground state and energy gap ∆ Eof the first excited state in the subspace of Sz\ntot= 0\nand wave number q= 0 or πforS= 3/2 and S= 2 and N= 8,10,12, and 14, which shows\nclear transitions at αc[Eq. (3)]. The magnetization is fractionalized as M= 1−1/(2S) in\na certain region α < α c. Here, M=Stot/(NS) is calculated from Stot=f(⟨ˆStot·ˆStot⟩) via\nf(x) = (√1 + 4 x−1)/2.\nIn most of the M= (S−1\n2)/Sphases in Fig. 3, wave-vector qof the ground state\ndepends on the system-size Nbecause q= 0 ( π) for even (odd) N/2, while q= 0 for\nα > α c. This even-odd effect of N/2 is consistent with that in the spin-1\n2Heisenberg\nchain [32]. In detail, in the vicinity of α≲αcfor large system size N≧14, a state with\nM= 1−1/(2S) + 1/(NS) has slightly lower energy than that with M= 1−1/(2S), and\ntwo states are almost degenerate, which reflects doubly degenerate q= 0 and q=πmodes\n6 0 1/3  2/3 1\n/g83/2 /g68c/g83 0  1  2 M\n/g39E\n/g68N=8, Stot  / N\nN=10, Stot  / N\nN=12, Stot  / N\nN=14, Stot  / NN = 8, /g39E\nN = 10, /g39E\nN = 12, /g39E\nN = 14, /g39E(a)S= 3/2\n 0 1/4  2/4 3/4  1 \n/g83/2 /g68r/g68c/g83 0  1  2 M\n/g39E\n/g68N = 8, Stot / N\nN = 10, Stot / N\nN = 12, Stot / N\nN = 14, Stot / NN = 8, /g39E\nN = 10, /g39E\nN = 12, /g39E\nN = 14, /g39E (b)S= 2\nFIG. 3: Magnetization M=Stot\nNSand the energy gap ∆ Efor the spin- SBLBQ N-site chain\nHamiltonian ˆH(S)\nαEq. (1) in the Sz\ntot= 0 and q= 0, πsubspace. Phase transition from\nM= 1 to M= 1−1/(2S) occurs at αccorresponding to Eq. (3). (a) S= 3/2 and (b)\nS= 2.\nin the thermodynamic limit ( N→ ∞ ) [30].\nAt the rigorous point αr, magnetization of the ground states becomes M= 1−1/(2S)\nonly for S≧2 while M̸= 1−1/(2S) for S≦3/2. A main difference is whether the\nbilinear term in Eq. (1) is ferromagnetic ( S≧2) or antiferromagnetic ( S≦3/2). Since\nthe eigenstate correspondence is rigorous for any S, the eigenstate of spin-1\n2liquefaction for\nS≦3/2 can become stable under a magnetic field. For S= 1, the magnetization is M= 0\natαr=π/4, which is the critical point between the trimer and the Haldane phase [33], as\nshown in Fig. 1. However, a magnetic field induces a phase transition to the magnetized\nHaldane phase [34], which is known to have exact correspondence to the spin-1\n2model [35].\nFor general S, a rigorous correspondence between the ground state of the BLBQ model and\nthat of the spin-1\n2antiferromagnetic model can be realized under an external magnetic field.\nForS= 3/2, magnetic-field-induced spin liquefaction occurs at αr= arctan(2) ≃0.35π.\nHowever, this is left as a future problem.\nThe transition point αcis at least the phase boundary of fully magnetized ferromagnetic\nphase M= 1. The proof is simple because ground states and one-magnon excitation are\n7FIG. 4: Energy gap ∆ Ein the Sz\ntot=N(S−1\n2) sector as a function of phase-twist angle θ\nand bond-alternation δatα0=αc−2π×0.04 (αr< α 0≲αc) and system-size N= 8 for\n(a)S= 3/2 and (b) S= 2.\nwritten exactly [36]. As an exact result, the one-magnon band becomes flat at αc, as is\nalready known from spin-wave theory [37]. Note that the continuous one-magnon excitation\nis not depicted in Fig. 3 because the energy gap ∆ Eis restricted in the sector q= 0 and π.\nTo confirm the thermodynamic limit under the existence of a finite-size gap, we adopt the\ntwisted boundary condition [38] or quantized Berry phase [39], introducing ˆHα,δ,θwith bond-\nalternation δand boundary twist angle θby using the δ-dependent coefficient J(s)\ni,i+1(α, δ) =\n[1+(−1)δi]J(s)\ni,i+1(α) and the θ-dependent boundary condition ˆS±\nN+1=e±ıθˆS±\n1andˆSz\nN+1=ˆSz\n1.\nThe energy gap ∆ Ein the sector for Sz\ntot=N(S−1\n2) opens due to finite system size N= 8\neven in the uniform case ( δ= 0), while the finite gap closes under the twisted boundary\ncondition ( θ=π) only at δ= 0, as shown in Fig. 4 at α0=αc−2π×0.04 (αr< α 0≲αc).\nThe result of Fig. 4 is identical to that of dimer singlets in a spin-1\n2dimerized Heisenberg\nchain. In the dimerized limit ( δ= 1), the unique ground state in the subspace for Sz\ntot=\nN(S−1\n2) is given as a direct-product state of two-site dimerQN/2\ni=1(ˆS−\n2i−ˆS−\n2i+1)|0⟩forα < α c\nexactly [40]. Twist-angle θdependence appears as ˆS−\nN−ˆS−\nN+1=ˆS−\nN−e−ıθˆS−\n1in the boundary\ndimer, while in the other dimerized limit ( δ=−1) the ground stateQN/2\ni=1(ˆS−\n2i−1−ˆS−\n2i)|0⟩\ndoes not depend on θdue to the absence of the boundary dimer. This difference of θ\ndependence results in the difference in Berry phase γ. the change of quantized value γ= 0, π\nis accompanied by the Dirac cone shown in Fig. 4. The two-fold degenerate states at the\nDirac point ( θ=π) adiabatically connect to two states separated by the finite-size gap\nat the periodic boundary condition ( θ= 0). These two states have q= 0 and πfor the\nuniform case δ= 0 depending on the even-odd parity of N/2. The scenario of the finite-\nsize effect directly corresponds to the S=1\n2case, which is for the dimer-singlet state\n|↑↓⟩ − |↓↑⟩ = (ˆS−\ni−ˆS−\ni+1)|↑↑⟩= (ˆS−\ni−ˆS−\ni+1)|0⟩i,i+1existing in the Sz\ntot=N(S−1\n2) = 0\n8subspace; the finite-size gap disappears in the thermodynamic limit [41]. The Dirac point\nis observed in most of M= (S−1\n2)/Sphases. However, an interesting discrepancy from\ntheS=1\n2case occurs in the vicinity of α≲αc, where the additional Dirac cone appears at\nθ= 0 and δ=±δc.\nApart from our numerical results on the chain, the general theory can be applied to\nprevious studies on other lattices. On a square lattice [42], magnetic-field-induced spin-1\n2\nliquefaction of the S= 1 BLBQ model is realized. Moreover, on a S= 1 BLBQ triangular\nlattice [43], exact correspondence at αr=π/4 exists for M≥2/3; for example, the M= 2/3-\nplateau state must be regarded as the 1 /3-plateau state of the spin-1\n2model and the ↑↑↓\nstate with spin-1\n2fully polarized in the ↑↑↑background.\nGeneralizing ˆH(S)\ncfor the spin-1\n2liquefaction, it is naively expected that the spin- sliq-\nuefaction point is given by J(2S)\nij=J(2S−1)\nij =···=J(2S−2s)\nij < J(m)\nij, (m < 2S−2s) and\nperturbation from the point toward the other 2 s+ 1 parameter space generates several\nphases, including the ferromagnetic phase ( M= 1) and a fractionally magnetized phase\n(M= 1−s\nS).\nIn summary, we present herein the theory of entangled fractionally-magnetized quantum\nstates providing the viewpoint of spin liquefaction on a d-dimensional lattice. In the general\ndiscussion, the entangled states turn out to be antiferromagnetic entangled states in ferro-\nmagnetic background. To address this fractional ferromagnet, the ferromagnetic region of\nthe spin- SBLBQ chain was studied numerically. The fractional magnetization was revealed\nto have M= 1−1/(2S) even under zero magnetic field; for example, M= 2/3 for S= 3/2,\nandM= 3/4 for S= 2. Numerous future problems remain. From a theoretical viewpoint,\nfurther calculations (using other numerical or analytical techniques) in the one-dimensional\nS≧3/2 BLBQ model are required to clarify the magnetization curve as a function of\nexternal magnetic field, the boundary edge-spin problem (especially under open boundary\nconditions), the excitation spectrum as a function of q, and the entanglement entropy and\nspectrum. A more generic theoretical task is to establish the origin of the interaction in real\nmaterials or by optical-lattice experiments.\nThe spin-1\n2liquefaction at αropens up further discussion, for example, a comparison with\nferrimagnetism [44]. In a ferrimagnet, spin- sand the SHamiltonian break one-site transla-\ntion symmetry because s̸=S, whereas a fractional ferromagnet holds that symmetry. The\ndifference can induce anomalous low-energy excitations in the BLBQ model. In particu-\n9lar, the fractional ferromagnet at αrexhibits linear magnon excitation, which reflects the\ntwo-fold degeneracy of N´ eel-like states in the uniform Hamiltonian (i.e., the des Cloizeaux–\nPearson mode [41]), and its existence is guaranteed thanks to the rigorous correspondence\nto spin-1\n2antiferromagnetic chain.\nAs mentioned above, fractional ferromagnets are not conventional ferrimagnets. In ad-\ndition, the fractional ferromagnetic state is not the classical ferromagnetic state near the\nquantum critical point [45]. Even after spontaneous magnetization, the ground state of\na fractional ferromagnet has quantum entanglement corresponding to the spin-1\n2antiferro-\nmagnetic state. For the quantum entanglement in a fractional ferromagnet, the external\nmagnetic field has the potential to be a tool to manipulate an entangled quantum state,\nwhich can be useful in the context of quantum computer science. From the viewpoint of\ncondensed-matter physics, the key word “quantum magnet” has been used and accepted for\nantiferromagnets. Given that the present theory abolishes the prejudice that ferromagnetism\nis classical, quantum magnets will also be used for fractional ferromagnets.\nTo summarize, this Letter develops the new frontier of quantum spin states (i.e., “quan-\ntum ferromagnet”), which opens new field not only in fundamental physics but also in\nquantum computer science.\nThe authors thank Hosho Katsura for stimulating discussions. 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B 338, 602 (1990).\n[29] P. A. Pearce and Y.-K. Zhou, Intertwiners and a-d-e lattice models, Int. J. Mod. Phys. B 07,\n3649 (1993).\n[30] The basic idea is that triplet (singlet) projection of spin-1\n2model corresponds to spin-\nSoperator ˆP(2S)\nij +ˆP(2S−2)\nij (ˆP(2S−1)\nij ). The proof is based on the following formula\n\u0010\nˆP(2S)\nij+ˆP(2S−2)\nij\u0011\n|Tk⟩ij=|Tk⟩ij, ˆP(2S−1)\nij |S0⟩ij=|S0⟩ij,\u0010\nˆP(2S)\nij+ˆP(2S−2)\nij\u0011\n|S0⟩ij=\n0,ˆP(2S−1)\nij |Tk⟩ij= 0,ˆP(s)\nij|Tk⟩ij=ˆP(s)\nij|S0⟩ij= 0,(s≤2S−3),for polarized triplet\n|T1⟩ij=|S⟩i|S⟩j=ˆC|↑⟩i|↑⟩j,|T0⟩ij=ˆC|↑⟩i|↓⟩j+|↓⟩i|↑⟩j√\n2,|T−1⟩ij=ˆC|↓⟩i|↓⟩jand polarized\nsiglet|S0⟩ij=ˆC|↑⟩i|↓⟩j−|↓⟩i|↑⟩j√\n2. The formula is valid not only for S≥1 but also for S=1\n2if we\nputˆP(−1)\nij= 0. Using the above formula, one can prove ˆH(S)\nrˆC=ˆCˆH(1/2). IfˆH(1/2)|ψ⟩=ϵ|ψ⟩,\n12one finds ˆH(S)\nr\u0010\nˆC|ψ⟩\u0011\n=ˆCˆH(1/2)|ψ⟩=ϵ\u0010\nˆC|ψ⟩\u0011\n. More details of the proof and demonstra-\ntion are provided in §S.2 in the supplement.\n[31] The coefficients J(s)\nijas a function of one-parameter αare depicted in §S.1 of the supplement.\nIn addition, J(s)\nij(αr) and J(s)\nij(αc) are demonstrated for S= 3/2, . . . , 3.\n[32] Ref. [41] shown that in spin-1\n2antiferromagnetic Heisenberg chain q= arg( ⟨ˆT⟩) and total spin\nStotbecomes ground state has ( q, Stot) = (0 ,0) and excited state has ( q, Stot) = (π,1) for even\nN/2 and ground state has ( q, Stot) = ( π,0) and excited state has ( q, Stot) = (0 ,1) for odd\nN/2. In the thermodynamic limit, these two states become degenerated, following the fact\nthat energy spectrumπ|J|\n2|sinq|, that is, des Cloizeaux-Pearson mode becomes zero at q= 0\nandπ.\n[33] A. Lauchli, G. Schmid, and S. Trebst, Spin nematics correlations in bilinear-biquadratic s=1\nspin chains, Phys. Rev. B 74, 144426 (2006).\n[34] S. R. Manmana, A. M. L¨ auchli, F. H. L. Essler, and F. Mila, Phase diagram and continuous\npair-unbinding transition of the bilinear-biquadratic s = 1 heisenberg chain in a magnetic\nfield, Phys. Rev. B 83, 184433 (2011).\n[35] G. F´ ath and P. B. Littlewood, Massless phases of haldane-gap antiferromagnets in a magnetic\nfield, Phys. Rev. B 58, 014709(R) (1998).\n[36] For α > α c, the ground-states are\u0010\nˆS−\ntot\u0011s\n|0⟩, (s∈[0,2NS]), with eigenenergy Eα=\nNS2(cosα+S2sinα) =⟨0|ˆH(S)\nα|0⟩,which corresponds to classical energy of spin-vector\nSi·Sj=S2cosθfor ferromagnetism θ= 0. One magnon exact excitation has eigenenergy\nEα+Wα(1−cosq) with qand the hopping element Wα=−2S[cosα+ 2S(S−1) sin α]. Ex-\ncitation energy is positive due to Wα>0 for α > α c. At αc,Wαc= 0 means one-magnon\nflat band. For α < α c, flipped band is realized due to Wα<0 and fully ferromagnetic states\nbecome excited states: that is, end of fully magnetized ferromagnetic phase M= 1.\n[37] R. A. Muniz, Y. Kato, and C. D. Batista, Generalized spin-wave theory: application to the\nbilinear-biquadratic model, Prog. Theor. Exp. Phys. 2014 , 083101.\n[38] A. Kitazawa, Twisted boundary conditions of quantum spin chains near the gaussian fixed\npoints, J. Phys. A: Math. Gen. 30, 285 (1997).\n[39] Y. Hatsugai, Quantized berry phases as local order parameters of quantum liquids, J. Phys.\nSoc. Jpn. 75, 123601 (2006).\n13[40] Degenerate eigenstates of two-site Hamiltonian at δ= 1 written by the projections ˆP(s)\n2i,2i+1\nare\f\f\fS2i,2i+1=s, Sz\n2i,2i+1=mE\n2i,2i+1with the eigen energy J(s)\n2i,2i+1and 2 s+ 1-fold de-\ngeneracy. For α < α c, the minimum coefficient is J(2S−1)\n2i,2i+1and the ground states are\n\f\f\fS2i,2i+1= 2S−1, Sz\n2i,2i+1=mE\n2i,2i+1, (m=−2S+1, m=−2S+2, . . . , m = 2S−1) with the\nenergy-gap J(2S)\n2i,2i+1−J(2S−1)\n2i,2i+1>0. The Sz\ntot=N(S−1\n2) subspace considered in Fig. 4 has unique\nground stateQN/2\ni=1\f\f\fS2i,2i+1= 2S−1, Sz\n2i,2i+1= 2S−1E\n2i,2i+1=CQN/2\ni=1(ˆS−\n2i−ˆS−\n2i+1)|0⟩, with\nnormalization constant C.\n[41] J. Cloizeaux and J. J. Pearson, Spin-wave spectrum of the antiferromagnetic linear chain,\nPhys. Rev. 128, 2131 (1962).\n[42] T. A. T´ oth, A. M. L¨ auchli, F. Mila, and K. Penc, Competition between two- and three-\nsublattice ordering for s=1 spins on the square lattice, Phys. Rev. B 85, 140403(R) (2012).\n[43] D. Yamamoto, C. Suzuki, G. Marmorini, S. Okazaki, and N. Furukawa, Quantum and thermal\nphase transitions of the triangular su(3) heisenberg model under magnetic, Phys. Rev. Lett.\n125, 057204 (2020).\n[44] L. N´ eel, Propri´ et´ es magn´ etiques des ferrites; ferrimagn´ etisme et antiferromagn´ etisme, Annales\nDe Physique, 12, 137 (1948).\n[45] M. Brando, D. Belitz, F. M. Grosche, and T. R. Kirkpatrick, Metallic quantum ferromagnets,\nRev. Mod. Phys. 88, 025006 (2016).\n14"    },    {        "title": "1304.5103v2.Spontaneous_ferromagnetism_in_the_spinor_Bose_gas_with_Rashba_spin_orbit_coupling.pdf",        "content": "Spontaneous ferromagnetism in the spinor Bose gas with Rashba spin-orbit coupling\nKira Riedl,1Casper Drukier,1Peter Zalom,1,2and Peter Kopietz1\n1Institut für Theoretische Physik, Universität Frankfurt,\nMax-von-Laue Strasse 1, 60438 Frankfurt, Germany\n2Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 040 01 Košice, Slovakia\n(Dated: April 18, 2013)\nWe show that in the two-component Bose gas with Rashba spin-orbit coupling an arbitrarily small\nattractive interaction between bosons with opposite spin induces spontaneous ferromagnetism below\na ﬁnite critical temperature Tc. In the ferromagnetic phase the single-particle spectrum exhibits a\nunique minimum in momentum space in the direction of the magnetization. For suﬃciently small\ntemperatures below Tcthe bosons eventually condense into the unique state at the bottom of the\nspectrum, forming a ferromagnetic Bose-Einstein condensate.\nPACS numbers: 67.85.Fg, 03.75.Mn, 05.30.Jp\nI. INTRODUCTION\nDue to recent progress in the ﬁeld of ultracold gases,\nspinor Bose-Einstein condensates with various types of\nspin-orbit coupling can now be realized experimentally1,2\nby using spatially varying laser ﬁelds to couple in-\nternal pseudo-spin degrees of freedom to the momen-\ntum. These experiments have motivated many theoreti-\ncal investigations of spin-orbit coupled multi-component\nBose gases.3–15Of particular interest have been two-\ncomponent bosons with isotropic Rashba-type16spin-\norbit coupling, where in the absence of interactions the\nenergy dispersion assumes a minimum on a circle in mo-\nmentum space.3If the bosons do not condense, such a\nsurface in momentum space can be called a Bose sur-\nface, in analogy with the Fermi surface of an electronic\nsystem.17,18Of course, bosons do not obey the Pauli ex-\nclusion principle, so that the Bose surface cannot be the\nboundary between occupied and unoccupied states; how-\never, the Bose surface deﬁnes the location of the low-\nenergy excitations in the system, similar to the Fermi\nsurface of an electronic system.\nThe fact that Bose-Einstein condensation (BEC) in\nBose systems where the dispersion has degenerate min-\nima on a surface in momentum space diﬀers qualitatively\nfrom conventional BEC has been pointed out a long time\nago by Yukalov.19He studied BEC in an interacting Bose\nsystem whose energy dispersion is minimal on a sphere in\nmomentum space. Assuming that the bosons condense\nwithequalweightintoallstatesonthissphere, heshowed\nthat the condensed state does not exhibit oﬀ-diagonal\nlong-range order and is also not superﬂuid.\nDue to the degeneracy of the single-particle energy in\nthe spinor Bose gas with Rashba-type spin-orbit cou-\npling, in the non-interacting limit BEC is prohibited at\nany ﬁnite temperature. Hence, ﬁnite temperature BEC\nin this system must be an interaction eﬀect.7,10Interac-\ntions are expected to remove the ground state degener-\nacy and various types of exotic ground states have been\nproposed.3–15Which phase is realized experimentally de-\npends on the speciﬁc properties of the interaction. Atthis point a generally accepted agreement on the nature\nof the ground state in the spinor Bose gas with Rashba-\ntype spin-orbit coupling has not been reached.\nPhase transitions in systems whose ﬂuctuation spec-\ntrum exhibits minima on a surface in momentum space\nform their own universality class, the so-called Brazovskii\nuniversality class;20for example, the critical behavior in\ncholesteric liquid crystals belongs to this class.21Because\nscaling transformations and mode elimination in renor-\nmalization group calculations should be deﬁned relative\nto the low-energy manifold, the classiﬁcation of interac-\ntion vertices in systems belonging to the Brazovskii uni-\nversality class is diﬀerent from the corresponding classi-\nﬁcation in systems where the low-energy manifold con-\nsists of a single point; in particular, all two-body scat-\ntering processes where the momenta of the incoming and\nthe outgoing particles lie on the low-energy manifold are\nmarginal, so that in renormalization group calculations\none should keep track of inﬁnitely many marginal cou-\nplings. Note that also normal fermions belong to the\nBrazovskii universality class, because the Fermi surface\ncan be identiﬁed with the low-energy manifold relative\nto which scaling transformations should be deﬁned.22–24\nTwo-component bosons with Rashba-type spin-orbit cou-\npling are therefore another example for a quantum sys-\ntem which belongs to the Brazovskii universality class.\nIn this work we shall further investigate interaction ef-\nfects on spinor Bose gases with Rashba-type spin-orbit\ncoupling. We shall consider the speciﬁc case where the\ninteraction g?between two bosons with opposite pseudo-\nspin is attractive. We ﬁnd that in this case for any ﬁnite\ndensity an arbitrarily small interaction g?<0leads to\nspontaneous ferromagnetism below some ﬁnite tempera-\ntureTc. In the ferromagnetic phase, the single-particle\ndispersion has a unique minimum in momentum space,\nso that the bosons eventually condense at some temper-\natureTBEC<Tcinto the unique single-particle state at\nthe bottom of the spectrum.arXiv:1304.5103v2  [cond-mat.quant-gas]  24 Jun 20132\nII. SPIN-ORBIT COUPLED BOSONS\nWe consider a two-component Bose gas with Rashba\nspin-orbit coupling and a two-body interaction which is\ninvariant under rotations around the z-axis in spin-space.\nThe Hamiltonian is given by H=H0+Hint, with\nH0=X\nk(ay\nk\";ay\nk#)\u0014k2\u00002k0k?\u0001\u001b\n2m\u0015\u0012\nak\"\nak#\u0013\n;(2.1)\nHint=1\n2VX\nk0\n1k0\n2k2k1X\n\u001b1\u001b2\u000ek0\n1+k0\n2;k2+k1\n\u0002U\u001b1\u001b2(k0\n1;k0\n2;k2;k1)ay\nk0\n1\u001b1ay\nk0\n2\u001b2ak2\u001b2ak1\u001b1;(2.2)\nwherek?=kx^x+ky^yistheprojectionofthewave-vector\nkonto thexy-plane, and the components of the vector\noperator\u001baretheusualPaulimatrices. Thewave-vector\nk0measures the strength of the spin-orbit coupling. The\ninvariance of the interaction under spin-rotations around\nthez-axis implies that the function U\u001b1\u001b2(k0\n1;k0\n2;k2;k1)\nis symmetric with respect to the simultaneous permuta-\ntion of its incoming and outgoing labels23\nU\u001b1\u001b2(k0\n1;k0\n2;k2;k1) =U\u001b2\u001b1(k0\n2;k0\n1;k1;k2):(2.3)\nFor simplicity, we assume that the bare interaction is\nmomentum independent, so that the interaction is char-\nacterized by three diﬀerent coupling constants\ng\"=U\"\"(0);g#=U##(0);g?=U\"#(0) =U#\"(0);\n(2.4)\nandHintsimpliﬁes to\nHint=1\n2VX\nk0\n1k0\n2k2k1\u000ek0\n1+k0\n2;k2+k1h\ng\"ay\nk0\n1\"ay\nk0\n2\"ak2\"ak1\"\n+g#ay\nk0\n1#ay\nk0\n2#ak2#ak1#+ 2g?ay\nk0\n1\"ay\nk0\n2#ak2#ak1\"i\n:(2.5)\nIntroducing the usual ﬁeld operators\n^ \u001b(r) =1p\nVX\nkeik\u0001rak\u001b; (2.6)\nand the corresponding density operators ^\u001a\u001b(r) =\n^ y\n\u001b(r)^ \u001b(r)the interaction can be written as follows,\nHint=1\n2Z\ndDrh\ng\": ^\u001a2\n\"(r) : +g#: ^\u001a2\n#(r) :\n+2g?: ^\u001a\"(r)^\u001a#(r) :i\n;(2.7)\nwhere :::::denotes normal ordering. Assuming for sim-\nplicityg\"=g#=gk, we may write the interaction as\nHint=1\n2Z\ndDrh\ng\u001a: ^\u001a2(r) : +g\u001b: ^\u001b2(r) :i\n;(2.8)\nwhere ^\u001a= ^\u001a\"+ ^\u001a#and^\u001b= ^\u001a\"\u0000^\u001a#represent the density\nand the spin density, and the associated couplings are\ng\u001a=1\n2(gk+g?); g\u001b=1\n2(gk\u0000g?):(2.9)From Eq. (2.8) it is clear that for g\u001b<0(i.e.,g?>gk)\nstateswithaﬁnitespin-densityarefavored; anexampleis\nthe standing wave spin-striped state proposed in Ref. [4],\nwhere the bosons condense simultaneously into two mo-\nmentum states with opposite momenta on the low-energy\nmanifoldinmomentumspace. Ontheotherhand, g\u001b>0\nfavors states with vanishing spin density, such as plane\nwave condensate where the bosons condense into a single\nmomentumstate,orthechargestripedstatesdiscussedin\nRefs. [6 and 13]. In this work, we focus on the case where\ng?is negative. We show below that even for inﬁnitesi-\nmally small g?<0the system exhibits spontaneous fer-\nromagnetism in the plane of the spin-orbit coupling at\nsuﬃciently low temperatures.\nTo set up our notation, let us review the diagonaliza-\ntion ofH0. Performing a momentum-dependent rotation\nin spin space around an axis \u0012k=j\u0012kjwith anglej\u0012kj,\n\u0012\nak\"\nak#\u0013\n=e\u0000i\n2\u001b\u0001\u0012k\u0012\nak\u0000\nak+\u0013\n;(2.10)\nand using the fact that25\nei\n2\u001b\u0001\u0012k\u001be\u0000i\n2\u001b\u0001\u0012k=e\u0012k\u0002\u001b; (2.11)\nwe obtain\nH0=X\nk(ay\nk\u0000;ay\nk+)\u0014k2\u00002k0k?\u0001(e\u0012k\u0002\u001b)\n2m\u0015\u0012\nak\u0000\nak+\u0013\n:\n(2.12)\nWe now choose the rotation matrix e\u0012k\u0002such that it\nrotates the z-axis into the direction ^k?=k?=jk?j, i.e.,\n^k?=e\u0012k\u0002^z; (2.13)\nwhich can be achieved by setting\n\u0012k=\u0019\n2^z\u0002^k?: (2.14)\nDue to the rotational invariance of the scalar product,\nwe may hence write\n^k?\u0001(e\u0012k\u0002\u001b) = (e\u0012k\u0002^z)\u0001(e\u0012k\u0002\u001b) =^z\u0001\u001b=\u001bz:(2.15)\nOur rotation matrix in spin space is then explicitly given\nby\ne\u0000i\n2\u001b\u0001\u0012k= cos\u0012\u0012k\n2\u0013\u0012\n1 0\n0 1\u0013\n\u0000isin\u0012\u0012k\n2\u0013\n\u001b\u0001^\u0012k\n=1p\n2\u0014\u0012\n1 0\n0 1\u0013\n\u0000i\u001b\u0001(^z\u0002^k?)\u0015\n=1p\n2\u0012\n1\u0000^kx+i^ky\n^kx+i^ky 1\u0013\n=1p\n2\u0012\n1\u0000e\u0000i'k\nei'k 1\u0013\n; (2.16)\nwhere in the last line we have set ^kx= cos'kand^ky=\nsin'k. In the new basis (which we shall call the helicity\nbasis) the non-interacting part of the Hamiltonian is\nH0=X\nkX\n\u0015=\u0006Ek\u0015ay\nk\u0015ak\u0015; (2.17)3\nk2\n0\n2m\n0\n−k2\n0\n2m\n−k0 −k0\nk0 k00 0E(k)\nkx ky\nFIG. 1. (Color online) Graph of the energy dispersions (2.18)\nof the spinor Bose gas with Rashba-type spin-orbit coupling.\nThe minimum of the lower helicity branch is a circle in the\nplanekz= 0with radius k0. The spacing between contours\nscales quartically.\nwith energy dispersions\nEk\u0015=k2\n2m+\u0015v0jk?j=(jk?j+\u0015k0)2+k2\nz\u0000k2\n0\n2m:(2.18)\nHerev0=k0=mand the helicity index \u0015=\u0006labels the\ntwo branches of the dispersion. A graph of these dis-\npersions is shown in Fig. 1. Note that the energy Ek;\u0000\nassumes its minimum \u0000k2\n0=(2m)on a circle of radius k0\nin thexy-plane, while Ek;+is non-negative and vanishes\nonlyatk= 0. Duetothemomentum-dependentrotation\nin spin-space, the interaction vertices in the helicity ba-\nsis acquire a momentum-dependence. For completeness\nwe give the properly symmetrized expressions for these\nvertices in Appendix A. For our mean-ﬁeld calculation it\nis more convenient to work in the original spin basis.\nIII. MEAN-FIELD THEORY FOR\nTRANSVERSE FERROMAGETISM\nA. Derivation of the mean-ﬁeld equations\nTo study transverse ferromagnetism we add a uni-\nform magnetic ﬁeld h?in thexy-plane, so that the non-\ninteracting part of our Hamiltonian is now given by\nH0=X\nk(ay\nk\";ay\nk#)\u0014k2\u00002k0k?\u0001\u001b\n2m\u0000h?\u0001\u001b\u0015\u0012\nak\"\nak#\u0013\n:\n(3.1)\nFor convenience we measure the magnetic ﬁeld in units\nof energy. The system exhibits spontaneous ferromag-\nnetism if the magnetization remains ﬁnite when h?!0.\nThe spin-rotational invariance with respect to rotations\naround the z-axis is then spontaneously broken. While in\nthe symmetric phase the self-energies \u0006\u001b\u001b0are diagonal\nin the spin-labels, in the symmetry broken phase there\nare ﬁnite oﬀ-diagonal components \u0006\"#and\u0006#\". Withinthe self-consistent Hartree-Fock approximation the self-\nenergies are independent of momentum and frequency if\nwestartfromamomentum-independentbareinteraction.\nThe mean-ﬁeld Hamiltonian is therefore of the form\nHMF=H0+X\nk(ay\nk\";ay\nk#)\u0012\n\u0006\"\"\u0006\"#\n\u0006#\"\u0006##\u0013\u0012\nak\"\nak#\u0013\n:(3.2)\nWithin the self-consistent Hartree-Fock approximation\nthe self-energies are\n\u0006\"\"= 2g\"\u001a\"+g?\u001a#; (3.3a)\n\u0006##= 2g#\u001a#+g?\u001a\"; (3.3b)\n\u0006\"#=g?\u001a#\"; (3.3c)\n\u0006#\"=g?\u001a\"#; (3.3d)\nwhere we have introduced the densities\n\u001a\u001b=1\nVX\nkhay\nk\u001bak\u001bi; (3.4)\n\u001a\"#=\u001a\u0003\n#\"=1\nVX\nkhay\nk\"ak#i: (3.5)\nHere the expectation values should be evaluated with\nthe grand canonical density matrix associated with the\nmean-ﬁeld Hamiltonian (3.2). Note that our mean-ﬁeld\ndecoupling excludes states with broken translational in-\nvariance. This will be justiﬁed a posteriori from the fact\nthat the irreducible ferromagnetic susceptibility is expo-\nnentially large at low temperatures [see Eq. (3.54)], such\nthat, at least at weak coupling, the ferromagnetic insta-\nbility is dominant. Keeping in mindthat \u001a\"#=Mx+iMy\ncan be expressed in terms of the components of the trans-\nverse magnetization M?=Mx^x+My^y, we see that\ntheCartesiancomponentsoftheoﬀ-diagonalself-energies\nare proportional to the corresponding components of the\nmagnetization,\n\u0006x=1\n2(\u0006\"#+ \u0006#\") =g?Mx; (3.6)\n\u0006y=i\n2(\u0006\"#\u0000\u0006#\") =g?My: (3.7)\nIt is convenient to deﬁne in addition the self-energies\n\u0006z=1\n2(\u0006\"\"\u0000\u0006##) =g\"\u001a\"\u0000g#\u001a#\u0000g?\n2(\u001a\"\u0000\u001a#);(3.8)\n\u00060=1\n2(\u0006\"\"+ \u0006##) =g\"\u001a\"+g#\u001a#+g?\n2(\u001a\"+\u001a#);(3.9)\nand the wave-vector\np= (h?\u0000\u0006)=v0; (3.10)\nwhere\u0006= \u0006x^x+ \u0006y^y+ \u0006z^zis proportional to the in-\nternal magnetic ﬁeld induced by the interaction. The\nHamiltonian can now be diagonalized via a momentum-\ndependent rotation in spin space of the form (2.10). The\nrotation matrix can be written as\ne\u0000i\n2\u001b\u0001\u0012k=\u0012\ncos(\u0012k=2)\u0000sin(\u0012k=2)e\u0000i'k\nsin(\u0012k=2)ei'k cos(\u0012k=2)\u0013\n;\n(3.11)4\nk2\n0\n2m\n0\n−k2\n0\n2m\n−k0 −k0\nk0 k00 0E(k)\nkx ky\nFIG. 2. (Color online) Graph of the energy dispersions (3.13)\nof the spinor Bose gas with Rashba-type spin-orbit coupling\nand an eﬀective magnetic ﬁeld h?\u0000\u0006=v0ppointing in the\ndirection of the positive x-axis. The lower helicity branch has\na unique minimum at k0=k0p=jpj. The spacing between\ncontours scales quartically.\nwhere the rotation angles in the presence of an eﬀective\nmagnetic ﬁeld h?\u0000\u0006=v0pare now given by\ncos\u0012k=pz\njk?+pj; (3.12a)\nsin\u0012k=s\n1\u0000p2z\njk?+pj2; (3.12b)\ncos'k=kx+pxp\n(kx+px)2+ (ky+py)2;(3.12c)\nsin'k=ky+pyp\n(kx+px)2+ (ky+py)2:(3.12d)\nThe energy dispersions of the eigenmodes are\nEk\u0015=k2\n2m+ \u0006 0+\u0015v0jk?+pj;(3.13)\nwhere\u0015=\u0006labelsagainthehelicityofthemodes. These\ndispersions are shown graphically in Fig. 2. Obviously,\nfor any ﬁnite pthe degeneracy of the corresponding dis-\npersion without magnetic ﬁeld shown in Fig. 1 is com-\npletelyremoved, sothat Ek;\u0000nowhasauniqueminimum\natk0=k0p=jpj.\nTo derive a self-consistency equation for the transverse\nmagnetization, we simply evaluate the oﬀ-diagonal den-\nsity (3.5), using the grand canonical density matrix asso-\nciatedwiththemean-ﬁeldHamiltonianontheright-hand\nside. We thus obtain\n\u001a\"#=Mx+iMy=\u00001\n2VX\nk\u0015\u0015sin\u0012kei'knk\u0015;(3.14)\nwhere\nnk\u0015=1\ne\f(Ek\u0015\u0000\u0016)\u00001(3.15)\nis the average occupation of the mode with energy Ek\u0015.\nBelow we shall work at constant density, so that weshould eliminate the chemical potential \u0016in favor of \u001a.\nTherefore we need the diagonal densities (3.4),\n\u001a\u001b=1\n2VX\nkX\n\u0015\u0014\n1\u0000\u001b\u0015pz\njk?+pj\u0015\nnk\u0015;(3.16)\nimplying that the total density is\n\u001a=X\n\u001b\u001a\u001b=1\nVX\nk;\u0015nk\u0015: (3.17)\nTo show that the phase transition to the ferromagnetic\nstate is continuous, it is useful to calculate the grand\ncanonical potential, which in a mean-ﬁeld approximation\nis given by\n\n(T;\u0016;h ) =TX\nkX\n\u0015=\u0006lnh\n1\u0000e\u0000\f(Ek\u0015\u0000\u0016)i\n\u0000hH inti;\n(3.18)\nwhere the expectation value of the interaction part of the\nHamiltonian is\nhHinti=V=g\"\u001a2\n\"+g#\u001a2\n#+g?(\u001a\"\u001a#+M2\n?)\n=\u0010gk\n2+g?\n4\u0011\n\u001a2+g?M2\n?;(3.19)\nand the second line holds for g\"=g#=gk. For simplic-\nity, let us now choose h?=h^xso thatM?=M^x. To\nexplore the possibility of spontaneous transverse magne-\ntizationatconstantdensity, weshouldconsidertheGibbs\npotential\nG(T;\u001a;M ) = \n(T;\u0016;h ) +V(\u0016\u001a+hM);(3.20)\nwhere\u0016=\u0016(\u001a;M)andh=h(\u001a;M)should be deter-\nmined by inverting the equations\n\u001a=\u00001\nV@\n@\u0016; M =\u00001\nV@\n@h:(3.21)\nTo study spontaneous transverse ferromagnetism, we\nshalllatertakethelimit h!0. Forsimplicity, weassume\nthatg\"=g#=gk, so that\u001a\"=\u001a#=\u001a=2and\u00060=\n(gk+g?=2)\u001a. As a consequence \u0006z= 0andpz= 0.\nOur self-consistency equation (3.14) then reduces to the\nfollowing equation for the transverse magnetization,\nM?=\u00001\n2VX\nk\u0015\u0015k?+p\njk?+pjnk\u0015:(3.22)\nNote thatpin the right-hand side depends again on M?\nvia Eqs. (3.10) and (3.6,3.7); for g\"=g#andh!0the\nrelation between pandM?is simplyp=\u0000g?M?=v0.\nTo determine the order of the phase transition for h!0,\nit is suﬃcient to consider the change in the free energy\nF(T;\u001a) = \n +V\u0016\u001afor arbitrary magnetization M,\n\u0001F(T;\u001aM ) =F(T;\u001a;h = 0)M6=0\u0000F(T;\u001a;h = 0)M=0:\n(3.23)\nThe physical state of the system at vanishing external\nﬁeld is determined by @\u0001F(T;\u001a;M)=@M = 0, which\nis another way of deriving the self-consistency equation\n(3.22) for the order parameter M.5\nB. Spectral densities\nTo evaluate the integrals appearing in Eqs. (3.17) and\n(3.18), it is useful to introduce the density of states\n\u0017\u0015(\u000f;p) =1\nVX\nk\u000e(\u000f\u0000Ek\u0015) =1\nVX\nk0\u000e(\u000f\u0000Ek0\u0000p;\u0015);\n(3.24)\nwherep= (h\u0000g?M)=v0and we have shifted k0=k+p\non the right-hand side. The density equation (3.17) can\nthen be written as\n\u001a=Z1\n\u00001d\u000fX\n\u0015\u0017\u0015(\u000f;p)1\ne\f(\u000f\u0000\u0016)\u00001;(3.25)\nwhile our expression (3.18) for the grand canonical po-\ntential per volume becomes\n\n(T;\u0016;h )\nV=TZ1\n\u00001d\u000fX\n\u0015\u0017\u0015(\u000f;p) ln[1\u0000e\u0000\f(\u000f\u0000\u0016)]\n\u0000gk\n2\u001a2\u0000g?\u0012\u001a2\n4+M2\u0013\n: (3.26)\nIt is also useful to rewrite the self-consistency equation\nforM?=M^xin terms of a generalized susceptibility as\nfollows. Shifting k0=k+pin Eq. (3.22) we obtain\nM=\u00001\n2VX\nk0\u0015\u0015k0\nx\njk0\n?jnk0\u0000p;\u0015:(3.27)\nWe now introduce cylindrical coordinates in k0-space and\nperform a partial integration in the angular part. Rear-\nranging terms we ﬁnd that the self-consistency equation\n(3.27) can be written as\nh\nM=1\n\u001f?(M)+g?; (3.28)\nwhere the irreducible susceptibility \u001f?(M)is deﬁned by\n\u001f?(M) =\u0000\f\n2VX\nk0\u0015k02\ny\njk0\n?jk0\u0015nk0\u0000p;\u0015[nk0\u0000p;\u0015+1]:(3.29)\nIt is easy to see that \u001f?(M)>0, implying that, only\nforg?<0, the magnetization Mcan remain ﬁnite for\nh!0. From now on we shall therefore assume that g?\nis negative. In this case the magnetization in the broken\nsymmetry phase satisﬁes the self-consistency equation\n\u001f?(M) =\u00001\ng?: (3.30)\nNote that the physical susceptibility M=hdiverges at\nthe critical point. To evaluate \u001f?(M), we introduce the\nweighted density of states,\n\u001b\u0015(\u000f;p) =1\nVX\nk0k02\ny\njk0\n?jk0\u000e(\u000f\u0000Ek0\u0000p;\u0015):(3.31)Then we may write\n\u001f?(M) =\u0000\f\n2Z1\n\u00001d\u000fX\n\u0015\u0015\u001b\u0015(\u000f;p)e\f(\u000f\u0000\u0016)\n[e\f(\u000f\u0000\u0016)\u00001]2:(3.32)\nIntroducing cylindrical coordinates in k0-space in the\nabove integrals deﬁning \u0017\u0015(\u000f;p)and\u001b\u0015(\u000f;p), the inte-\ngrations over k0\nzandjk0\n?jcan be carried out exactly so\nthat we can write these functions as one-dimensional an-\ngular integrals. The results can be written in the scaling\nform\n\u0017\u0015(\u000f;p)\n\u00170= ~\u0017\u0015 \n\u000f\u0000\u00060\u0000p2\n2m\n\u000f0;p\nk0!\n;(3.33)\n\u001b\u0015(\u000f;p)\n\u00170= ~\u001b\u0015 \n\u000f\u0000\u00060\u0000p2\n2m\n\u000f0;p\nk0!\n;(3.34)\nwhere\n\u00170=mk0\n2\u0019; \u000f 0=k2\n0\n2m: (3.35)\nIn Appendix B we give explicit expressions for the di-\nmensionless scaling functions ~\u0017\u0015(~\u000f;~p)and~\u001b\u0015(~\u000f;~p)as one-\ndimensional integrals. In fact, for negative ~\u000fthe remain-\ning angular integration in the expressions for ~\u0017\u0000(~\u000f;~p)and\n~\u001b\u0000(~\u000f;~p)can also be done analytically, see Eqs. (B3) and\n(B5). Graphs of the scaling functions are shown in Fig. 3.\nThe behavior of the spectral densities \u0017\u0000(\u000f;p)and\n\u001b\u0000(\u000f;p)associated with the negative energy branch is\nrather interesting. Both functions vanish if \u000fis smaller\nthan the lower threshold energy\n\u000f\u0000=\u0000\u000f0+ \u0006 0\u0000v0p: (3.36)\nRecall that in the absence of an external magnetic ﬁeld\np=\u0000\u0006x=v0= (\u0000g?)M=v 0. For energies slightly above\nthe lower threshold we obtain from Eqs. (B6, B7),\n\u0017\u0000(\u000f;p)\n\u00170\u0018p\n2(1 + ~p)\n\u0019r\u000f\u0000\u000f\u0000\nv0p;(3.37)\n\u001b\u0000(\u000f;p)\n\u00170\u0018p1 + ~p\n3\u0019\u0014\u000f\u0000\u000f\u0000\nv0p\u00153=2\n;(3.38)\nwhere ~p=p=k0. Eqs. (3.37, 3.38) can also be derived by\nexpandingtheenergydispersion Ek;\u0000ofthelowerbranch\naround the minimum k0=k0^xto quadratic order,\nEk0+q;\u0000\u0019\u000f\u0000+q2\nx+q2\nz\n2m+p\nk0+pq2\ny\n2m:(3.39)\nThis approximation is only accurate for jEk0+q;\u0000\u0000\u000f\u0000j\u001c\n2v0p; the energy surface Ek;\u0000=\u000fcan then be approxi-\nmated by an ellipsoid. However, for higher energies the\ntopology of the energy surface changes, as illustrated in\nFig. 4. With increasing energy the ellipsoid distorts into\nabean-shapedsurfaceuntilthetwoendsofthebeanmeet6\n(a)\n0\n0123\n5 10 15 200.00.51.01.5\n−0.5 −1.0 ˜ǫ− ˜ǫ∗\n˜ǫ˜ν±˜ν−\n˜ν+\n(b)\n0\n012345\n5 10 15 200.00.51.0\n−0.5 −1.0 ˜ǫ− ˜ǫ∗\n˜ǫ˜σ±˜σ−\n˜σ+\nFIG. 3. (Color online) (a) Graph of the scaling functions\n~\u0017\u0006(~\u000f;~p)of the density of states deﬁned via Eq. (3.33) for\n~p= 0:1, see Eqs. (B1, B3). (b) Graph of the scaling func-\ntions ~\u001b\u0006(~\u000f;~p)of the weighted density of states deﬁned via\nEq. (3.34) for ~p= 0:1, see Eqs. (B4, B5). The solid lines cor-\nrespond to the scaling functions in the \u0015=\u00001branch while\nthe dashed lines are the scaling functions in the \u0015= 1branch.\nThe insets show a closeup of the negative energy part of the\nspectral functions in the lower helicity branch. We use ~\u000f\u0000=\u0000\n\u000f\u0000\u0000\u00060\u0000\u0000\np2=2m\u0001\u0001\n=\u000f0and~\u000f\u0003=\u0000\n\u000f\u0003\u0000\u00060\u0000\u0000\np2=2m\u0001\u0001\n=\u000f0.\nat a critical energy. For higher energies, a hole emerges\nin the energy surface so that it assumes the topology of\na torus. At the critical energy \u000f\u0003where the topology of\nthe energy surface changes the spectral densities have a\ncusp. From the exact expressions for the spectral densi-\nties given in Appendix B it is easy to see that the critical\nenergy is\n\u000f\u0003=\u0000\u000f0+ \u0006 0+v0p=\u000f\u0000+ 2v0p: (3.40)\nA similar transition in the topology of the Fermi sur-\nface of metals as a function of external pressure has been\ndiscussed a long time ago by Lifshitz.26Close to such a(a)\nkx\nk0ky\nk0kz\nk02\n2\n2−2\n−2−2˜ǫ= ˜ǫ−+ 0.02 (b)\nkx\nk0ky\nk0kz\nk0\n1.3\n0.70.3\n−0.30.3\n−0.3˜ǫ= ˜ǫ−+ 0.02\n(7×enlarged)\n(c)\n˜ǫ/˜ǫ∗= 1.37˜ǫ/˜ǫ∗= 1.25˜ǫ/˜ǫ∗= 1.12\n˜ǫ/˜ǫ∗= 1˜ǫ/˜ǫ∗= 0.88˜ǫ/˜ǫ∗= 0.75\nkx\nk0kx\nk0kx\nk0kx\nk0kx\nk0kx\nk0\nky\nk0ky\nk0ky\nk0ky\nk0ky\nk0ky\nk0\nkz\nk0kz\nk0kz\nk0kz\nk0kz\nk0kz\nk0\nFIG.4. (Coloronline)(a)Surfaceofconstantenergy Ek;\u0000=\u000f\nof the lower helicity branch in the ferromagnetic phase just\nabove the lower energy threshold ~\u000f\u0000=\u0000(1 + ~p)2. (b) Same\nas (a) but enlarged by a factor of seven. (c) Evolution of the\nconstant energy surface for diﬀerent energies. The scales are\nthe same as in (a). All plots are for ~p= 0:1.\ntransition, Lifshitz predicted anomalies in the thermody-\nnamics and the kinetics of the electrons. Therefore we\nexpect that in phases with spontaneous transverse ferro-\nmagnetism the kinetics of spin-orbit coupled bosons with\nenergies close to \u000f\u0003is rather unusual.\nC. Solution of the mean-ﬁeld equations\nFor the numerical solution of the above mean-ﬁeld\nequations it is useful to introduce the dimensionless den-\nsity, magnetization, susceptibility, and interaction as fol-7\nlows,\n~\u001a=\u001a\n\u00170\u000f0=4\u0019\u001a\nk3\n0; (3.41a)\n~M=M\n\u00170\u000f0=4\u0019M\nk3\n0; (3.41b)\n~\u001f?=\u001f?\n\u00170; (3.41c)\n~g?=\u00170g?: (3.41d)\nWe also introduce the dimensionless energy != (\u000f\u0000\n\u000f\u0000)=\u000f0which is measured relative to the bottom of the\nlower helicity branch, and deﬁne\n\u0016\u0017\u0015(!;~p) =\u0017\u0015(\u000f\u0000+\u000f0!;k0~p)\n\u00170= ~\u0017\u0015\u0000\n\u0000(1 + ~p)2+!;~p\u0001\n;\n(3.42)\n\u0016\u001b\u0015(!;~p) =\u001b\u0015(\u000f\u0000+\u000f0!;k0~p)\n\u00170= ~\u001b\u0015\u0000\n\u0000(1 + ~p)2+!;~p\u0001\n:\n(3.43)\nFinally, we introduce the dimensionless temperature\n\u001c=T=\u000f0; (3.44)\nand the fugacity\nz=e(\u0016\u0000\u000f\u0000)=T: (3.45)\nWith this notation the density equation (3.17) can be\nwritten as\n~\u001a=Z1\n0d!X\n\u0015\u0016\u0017\u0015(!;~p)z\ne!=\u001c\u0000z;(3.46)\nwhile the self-consistency equation (3.30) for the dimen-\nsionless order parameter ~p=p=k0=\u0000~g?~M=2becomes\n~\u001f?(\u001c;z;~p) =\u00001=~g?; (3.47)\nwith\n~\u001f?=\u00001\n2\u001cZ1\n0d!X\n\u0015\u0015\u0016\u001b\u0015(!;~p)ze!=\u001c\n[e!=\u001c\u0000z]2:(3.48)\nFinally, the dimensionless free energy f(\u001c;~\u001a;~p) = (\n +\n\u0016N)=(V\u00170\u000f2\n0)can be written as\nf(\u001c;~\u001a;~p) =\u001cZ1\n0d!X\n\u0015\u0016\u0017\u0015(!;~p) lnh\n1\u0000ze\u0000!=\u001ci\n\u0000~gk\n2~\u001a2\u0000~g?\u0012~\u001a2\n4+~M2\u0013\n+ ~\u0016~\u001a;(3.49)\nwhere ~\u0016=\u0016=\u000f0. Note that at constant density we should\ndetermine\u0016as a function of ~\u001aand~p.\nLet us ﬁrst discuss the critical temperature \u001cc=Tc=\u000f0\nbelow which the system exhibits spontaneous transverse\nferromagnetism. AccordingtoEq.(3.47), foragivenden-\nsity~\u001athe critical temperature is determined by\n~\u001f?(\u001cc;zc(\u001cc;~\u001a);~p= 0) =\u00001=~g?;(3.50)(a)\n0.00.51.01.52.02.5\n0 1 2 3 4 5\n|˜g⊥|τc˜ρ= 2.0\n˜ρ= 1.0\n˜ρ= 0.5\n(b)\n0 1 2 3 4 50.00.51.01.52.02.5\n˜ρτc|˜g⊥|= 2.0\n|˜g⊥|= 1.0\n|˜g⊥|= 0.5\nFIG. 5. (Color online) (a) Critical temperature for trans-\nverse ferromagnetism as a function of the dimensionless cou-\npling constantj~g?jfor three diﬀerent densities. The dots\nhave been obtained from the numerical solution of the mean-\nﬁeld equations (3.50) without further approximation, while\nthe solid lines represent the low-temperature approximation\n(3.55). (b) Critical temperature as a function of density for\ndiﬀerent values of the interaction.\nwhere the fugacity zc(\u001cc;~\u001a)at the critical point is deter-\nmined by\n~\u001a=Z1\n0d!X\n\u0015\u0016\u0017\u0015(!;0)zc\ne!=\u001cc\u0000zc: (3.51)\nNumerical results for \u001ccas a function ofj~g?jfor diﬀer-\nent densities are shown in Fig. 5 (a), while in Fig. 5 (b)\nwe show the critical temperature as a function of den-\nsity for diﬀerent values of j~g?j. The numerical results in\ngeneral are obtained without approximating the spectral\ndensities and choosing gksuch that \u00060vanishes. Note\nthat for smallj~g?jthe critical temperature approaches\nzero with inﬁnite slope. In this regime it is easy to ob-\ntain an analytic expression for the critical temperature.\nAssuming that the temperature is small compared with\n\u000f0(corresponding to \u001c\u001c1) we may neglect the contri-8\nbution of the upper helicity branch and approximate the\nspectral densities by their leading asymptotics for fre-\nquencies close to the bottom of the lower helicity branch\ngiven in Eqs. (B14) and (B15). In the symmetric phase\nwhere ~p= 0the density equation (3.46) then reduces to\n~\u001a=Z1\n0d!z\ne!=\u001c\u0000z=\u0000\u001cln(1\u0000z):(3.52)\nWe conclude that in the regime where the critical tem-\nperature\u001ccis small compared with unity, the critical fu-\ngacity is given by\nzc= 1\u0000e\u0000~\u001a=\u001cc: (3.53)\nFor\u001cc\u001c~\u001athis is exponentially close to unity, which\nis a consequence of the ﬁnite density of states of the\nspin-orbit coupled Bose gas close to the bottom of the\nlower energy branch. To determine the critical tem-\nperature as a function of the density, we calculate the\ntransverse susceptibility from Eq. (3.48) with ~p= 0,\nusing the approximation (B15) for the spectral density\n\u0016\u001b\u0000(!;0) = ~\u001b\u0000(\u00001 +!;~p= 0)for frequencies close to the\nbottom of the lower helicity branch,\n~\u001f?=1\n4Z1\n0d!ze!=\u001c\n[e!=\u001c\u0000z]2=1\n4z\n1\u0000z=1\n4h\ne~\u001a=\u001c\u00001i\n:\n(3.54)\nHence, in the paramagnetic phase the transverse suscep-\ntibility becomes exponentially large at low temperatures.\nAs a consequence, for any ﬁnite attractive interaction\ng?<0, we can ﬁnd a solution of the self-consistency\nequation (3.47) at suﬃciently low temperatures. Com-\nbining Eqs. (3.50) and (3.54) we obtain for the critical\ntemperature\n\u001cc=~\u001a\nln (1 + 4=j~g?j): (3.55)\nFrom the derivation of this expression it is clear that\nEq. (3.55) is valid as long as \u001cc\u001c1, which can always\nbe satisﬁed for suﬃciently small densities. The approxi-\nmation (3.55) corresponds to the solid lines in Fig. 5.\nNext, let us discuss the low-temperature phase \u001c <\u001cc\nwith spontaneous transverse ferromagnetism. In Fig. 6\nwe show the change\n\u0001f=f(\u001c;~\u001a;~p)\u0000f(\u001c;~\u001a;0) (3.56)\nin the dimensionless free energy deﬁned in Eq. (3.49) as\na function of the order parameter ~M= 2~p=j~g?jfor three\ndiﬀerenttemperatures. For \u001c <\u001ccthefreeenergycontin-\nuously develops two degenerate minima, corresponding\nto the Hartree-Fock solutions \u0006~M. The phase transi-\ntion to the magnetic state is therefore second order. The\ntransversemagnetizationasafunctionoftemperaturefor\nthree diﬀerent densities, obtained numerically, is shown\nin Fig. 7. In the regime where \u001cc\u001c1the behavior of\nthe magnetization for temperatures close to the critical\n/Minus/Minus/Minus0.000.050.10\n0.0 0 .2 0.2 0 .4 0.4 0.6 0.6\n˜M∆fFIG. 6. (Color online) Dimensionless free energy \u0001f, deﬁned\nin equation (3.56), as a function of the dimensionless magneti-\nzation ~Mfor three diﬀerent temperatures. \u001c=\u001cc= 0:9(dotted\nline);\u001c=\u001cc= 0:95(dashed line); \u001c=\u001cc= 1:1(solid line). For\nall plots we have used j~g?j= 5and~\u001a= 1.\n0.00.10.20.3\n0.30.4\n0.40.5\n0.5 0 .6 0 .7 0 .8 0 .9 1 .0˜M\nτ/τc˜ρ= 2.0\n˜ρ= 1.0\n˜ρ= 0.5\nFIG. 7. (Color online) Transversemagnetization as a function\nof temperature. The plots are for j~g?j= 1. The dots are\nobtained numerically, while the solid lines correspond to the\nanalytic result (3.57) which is only valid for \u001cc\u0000\u001c\u001c\u001cc\u001c1.\ntemperature can be calculated analytically, as shown in\nAppendix C. We ﬁnd\n~M\u0018~\u001aq\u0000\n8 +3\n2j~g?j\u0001\nln[1 +4\nj~g?j]r\u001cc\u0000\u001c\n\u001cc:(3.57)\nThe reason we obtain the usual mean-ﬁeld exponent \f=\n1=2is of course related to the fact that our calculation is\nbased on the Hartree-Fock approximation.\nIV. SUMMARY AND CONCLUSIONS\nIn summary, we have shown that in the spinor Bose\ngas with Rashba-type spin-orbit coupling an arbitrarily\nweakattractiveinteractionbetweenbosonswithopposite\nspin triggers a ferromagnetic instability for temperatures9\nbelow some ﬁnite temperature Tcif the density of the\nbosons is ﬁxed. Note that spontaneous ferromagnetism\nin electronic systems usually appears only if the relevant\ninteraction exceeds a ﬁnite threshold.27The fact that in\nthe spinor Bose gas such a threshold does not exist is\nrelated to the singularity of the Bose function for small\nenergies in combination with the ﬁnite density of states\nin the non-magnetic phase due to the spin-orbit coupling\nof the Rashba-type. Because for T < Tc, the density of\nstates exhibits the usualp!-behavior at low energies, at\nsome temperature below Tcthe bosons eventually con-\ndense into the single-particle state with the lowest mo-\nmentumk0, which is unique in the ferromagnetic phase.\nIn the weak coupling regime, we may estimate the crit-\nical temperature for BEC by using the critical tempera-\nture for free bosons with anisotropic dispersion given by\nEq. (3.39),\nTBEC=2\u0019\nm\u0012\u001a\n\u0010(3=2)\u00132=3\u0012p\nk0+p\u00131=3\n=2\u0019\nm\u0012\u001a\n\u0010(3=2)\u00132=3\u0012jg?jM\nk2\n0=m+jg?jM\u00131=3\n:(4.1)\nForconsistency, weshouldrequirethat TBEC<Tc, which\nis satisﬁed at weak coupling where p\u001ck0. We thus\nconclude that the ground state of the spinor Bose gas\nwith Rashba spin-orbit coupling and attractive interac-\ntiong?<0is a ferromagnetic Bose-Einstein condensate.\nIn principle, an attractive interaction g?<0could\nalso trigger an instability in the particle-particle channel,\nleading to a pair condensate at low temperatures. How-\never, as shown in Appendix D, at least for suﬃciently low\ndensities the ferromagnetic instability has a higher crit-\nical temperature if the relevant coupling constants have\nthe same order of magnitude.\nLet us also point out that the constant g?in our mean-\nﬁeld calculation should be considered as an eﬀective\nlow-energy interaction in the opposite-spin particle-hole\nchannel. Thisinteractionincludesrenormalizationeﬀects\nfrom high-energy ﬂuctuations in all channels, so that it\nis in principle possible that the eﬀective low-energy cou-\npling is attractive even if we start from a repulsive bare\ninteraction. Note that recently Gopalakrishnan et al.5\nperformed a momentum shell renormalization group cal-\nculation for the spin-orbit coupled Bose gas whose dis-\npersion exhibits a minimum on a circle in momentumspace; at ﬁnite temperature they found evidence for an\nattractive renormalized coupling in the particle-particle\nchannel, although the corresponding bare interaction is\nrepulsive, suggesting an instability towards pair conden-\nsation. However, a proper renormalization group calcu-\nlation of the eﬀective low-energy coupling of the spinor\nBose gas with Rashba-type spin-orbit coupling, taking\nhigh-energy ﬂuctuations in all channels and all relevant\nand marginal couplings consistently into account, still re-\nmains to be done. Given the fact that the system belongs\nto the Brazovskii universality class, the rescaling step in\nthe renormalization group transformation is non-trivial\nand all two-body scattering processes describing particles\nwith momenta on the low-energy manifold are marginal\nand should be retained. It should be advantageous to use\nthe well developed machinery of the functional renormal-\nization group23,24,28to carry out such a calculation.\nACKNOWLEDGMENTS\nThis work was ﬁnancially supported by SFB/TRR49.\nP. Z. would like to thank M. Zentková and A. Lencsesová\nfor ﬁnancial support from project EDUFYCE (ITMS\ncode 26110230034).\nAPPENDIX A: INTERACTIONS IN THE\nHELICITY BASIS\nIn this appendix we explicitly give the interaction part\nof the boson Hamiltonian (2.2) in the helicity basis. For\nsimplicity, we assume momentum independent bare in-\nteractions, see Eq. (2.4). To transform the interaction\nfrom the spin basis to the helicity basis, it is useful to\nintroduce the notation ak=ak\u0000andbk=e\u0000i'kak+, so\nthat our transformation (2.10) to the helicity basis can\nbe written as\nak\"=1p\n2\u0002\nak\u0000\u0000e\u0000i'kak+\u0003\n=1p\n2[ak\u0000bk];(A1)\nak#=1p\n2\u0002\nei'kak\u0000+ak+\u0003\n=ei'k\np\n2[ak+bk]:(A2)\nDeﬁning the coupling constants g\",g#andg?as in\nEq. (2.4), we ﬁnd that in the helicity basis the interaction\npart (2.5) of our Hamiltonian can be written as10\nHint=1\nVX\nk0\n1k0\n2k2k1\u000ek0\n1+k0\n2;k2+k1n1\n(2!)2h\n\u0000\u0016a\u0016aaa\n0(10;20; 2;1)ay\n10ay\n20a2a1+ \u0000\u0016b\u0016bbb\n0(10;20; 2;1)by\n10by\n20b2b1\n+\u0000\u0016a\u0016abb\n0(10;20; 2;1)ay\n10ay\n20b2b1+ \u0000\u0016b\u0016baa\n0(10;20; 2;1)by\n10by\n20a2a1i\n+1\n2!h\n\u0000\u0016a\u0016aab\n0(10;20; 2;1)ay\n10ay\n20a2b1+ \u0000\u0016b\u0016bba\n0(10;20; 2;1)by\n10by\n20b2a1\n+\u0000\u0016b\u0016aaa\n0(10;20; 2;1)by\n10ay\n20a2a1+ \u0000\u0016a\u0016bbb\n0(10;20; 2;1)ay\n10by\n20b2b1i\n+\u0000\u0016a\u0016bba\n0(10;20; 2;1)ay\n10by\n20b2a1o\n=1\nVX\nk0\n1k0\n2k2k1\u000ek0\n1+k0\n2;k2+k1n1\n(2!)2U1(k0\n1;k0\n2;k2;k1)[ay\n10ay\n20a2a1+by\n10by\n20b2b1]\n+1\n(2!)2U2(k0\n1;k0\n2;k2;k1)[ay\n10ay\n20b2b1+by\n10by\n20a2a1]\n+1\n2!U3(k0\n1;k0\n2;k2;k1)[ay\n10ay\n20a2b1+by\n10by\n20b2a1]\n+1\n2!U4(k0\n1;k0\n2;k2;k1)[by\n10ay\n20a2a1+ay\n10by\n20b2b1]\n+U5(k0\n1;k0\n2;k2;k1)ay\n10by\n20b2a1o\n: (A3)\nThe properly symmetrized interaction vertices are\n\u0000\u0016a\u0016aaa\n0(10;20; 2;1) = \u0000\u0016b\u0016bbb\n0(10;20; 2;1)\u0011U1(k0\n1;k0\n2;k2;k1)\n=g\"\n2+g#\n2e\u0000i('10+'20\u0000'2\u0000'1)+g?\n4(e\u0000i'10+e\u0000i'20)(ei'1+ei'2);(A4a)\n\u0000\u0016a\u0016abb\n0(10;20; 2;1) = \u0000\u0016b\u0016baa\n0(10;20; 2;1)\u0011U2(k0\n1;k0\n2;k2;k1)\n=g\"\n2+g#\n2e\u0000i('10+'20\u0000'2\u0000'1)\u0000g?\n4(e\u0000i'10+e\u0000i'20)(ei'1+ei'2);(A4b)\n\u0000\u0016a\u0016aab\n0(10;20; 2;1) = \u0000\u0016b\u0016bba\n0(10;20; 2;1)\u0011U3(k0\n1;k0\n2;k2;k1)\n=\u0000g\"\n2+g#\n2e\u0000i('10+'20\u0000'2\u0000'1)+g?\n4(e\u0000i'10+e\u0000i'20)(ei'1\u0000ei'2);(A4c)\n\u0000\u0016b\u0016aaa\n0(10;20; 2;1) = \u0000\u0016a\u0016bbb\n0(10;20; 2;1)\u0011U4(k0\n1;k0\n2;k2;k1)\n=\u0000g\"\n2+g#\n2e\u0000i('10+'20\u0000'2\u0000'1)+g?\n4(e\u0000i'10\u0000e\u0000i'20)(ei'1+ei'2);(A4d)\n\u0000\u0016a\u0016bba\n0(10;20; 2;1)\u0011U5(k0\n1;k0\n2;k2;k1)\n=g\"\n2+g#\n2e\u0000i('10+'20\u0000'2\u0000'1)+g?\n4(e\u0000i'10\u0000e\u0000i'20)(ei'1\u0000ei'2):(A4e)\nThe parametrization of the vertices in terms of ﬁve func-\ntionsU1;:::;U 5is similar to the parametrization used\nby Ozawa and Baym;7however, we ﬁnd it convenient\nto introduce slightly diﬀerent numerical prefactors in the\nsecond line of Eq. (A3) in order to simplify the combi-\nnatorial factors due the permutation symmetries of the\nvertices in higher order calculations. Note that the in-\nteraction vertices are invariant under arbitrary rotations\naround the zaxis, corresponding to a shift 'k!'k+\u000bin all angles.\nAPPENDIX B: SPECTRAL DENSITIES\nIn this appendix we discuss the dimensionless scaling\nfunctions ~\u0017\u0015(~\u000f;~p)and~\u001b\u0015(~\u000f;~p)which are deﬁned by writ-\ning the density of states and the weighted density of\nstates in the scaling form (3.33, 3.34). The scaling func-\ntion ~\u0017\u0015(~\u000f;~p)for the density of states can be written as\n~\u0017\u0015(~\u000f;~p) = \u0002(~\u000f)(p\n~\u000f\n\u0019+Z2\u0019\n0d'\n2\u0019(~pcos'\u0000\u0015)\u00141\n2+1\n\u0019arctan\u0012~pcos'\u0000\u0015p\n~\u000f\u0013\u0015)\n+ \u0002(\u0000~\u000f)Z2\u0019\n0d'\n2\u0019(~pcos'\u0000\u0015)\u0002(~pcos'\u0000\u0015\u0000p\n\u0000~\u000f): (B1)11\nRecall that in Eqs. (3.33) and (3.34) the dimensionless variables ~\u000fand~prepresent\n~\u000f=\u000f\u0000\u00060\u0000p2\n2m\n\u000f0; ~p=p\nk0: (B2)\nFor negative ~\u000fthe angular integration in Eq. (B1) can be done analytically. For the lower energy branch ( \u0015=\u00001)\nwe obtain\n~\u0017\u0000(~\u000f;~p) = \u0002(1\u0000p\n\u0000~\u000f\u0000~p) + \u0002(~p\u0000j1\u0000p\n\u0000~\u000fj)1\n\u0019\u0014\narccos\u0012p\u0000~\u000f\u00001\n~p\u0013\n+q\n~p2\u0000(1\u0000p\n\u0000~\u000f)2\u0015\n;for~\u000f<0:(B3)\nThe scaling function for the weighted density of states ~\u001b\u0015(~\u000f;~p), can be written as\n~\u001b\u0015(~\u000f;~p) = \u0002(~\u000f)Z2\u0019\n0d'\n2\u0019sin2'\n2(\n\u0002\n3(~pcos'\u0000\u0015)2+ ~\u000f\u0003\u00141\n2+1\n\u0019arctan\u0012~pcos'\u0000\u0015p\n~\u000f\u0013\u0015\n+3p\n~\u000f\n\u0019(~pcos'\u0000\u0015))\n+ \u0002(\u0000~\u000f)Z2\u0019\n0d'\n2\u0019sin2'\n2\u0002\n3(~pcos'\u0000\u0015)2+ ~\u000f\u0003\n\u0002(~pcos'\u0000\u0015\u0000p\n\u0000~\u000f): (B4)\nFor negative ~\u000fthe angular integration can again be performed analytically. For the lower energy branch ( \u0015=\u00001) we\nobtain\n~\u001b\u0000(~\u000f;~p) = \u0002(1\u0000p\n\u0000~\u000f\u0000~p)1\n4\u0012\n3 + ~\u000f+3\n4~p2\u0013\n+ \u0002(~p\u0000j1\u0000p\n\u0000~\u000fj)1\n4\u0019\"\u0012\n3 + ~\u000f+3\n4~p2\u0013\narccos\u0012p\u0000~\u000f\u00001\n~p\u0013\n+q\n~p2\u0000(1\u0000p\n\u0000~\u000f)2\u0012(1 + ~\u000f)(1 +p\u0000~\u000f)\n2~p2+13 + 3p\u0000~\u000f\n4\u0013#\n;for~\u000f<0: (B5)\nPlots of the above scaling functions are shown in Fig. 3.\nThe asymptotic behavior slightly above the lower thresh-\nold~\u000f\u0000=\u0000(1 + ~p)2is,\n~\u0017\u0000(~\u000f;~p)\u0018p1 + ~p\n\u0019s\n~\u000f+ (1 + ~p)2\n~p; (B6)\n~\u001b\u0000(~\u000f;~p)\u0018p1 + ~p\n3\u0019\u0014~\u000f+ (1 + ~p)2\n~p\u00153=2\n:(B7)\nTodiscussthethermodynamicsclosetothecriticalpoint,\nwe need the expansion of the density of states and the\nweighted density of states for small ~p=p=k0to order ~p2.\nWe obtain\n\u0017\u0015(\u000f;p)\n\u00170= ~\u0017(0)\n\u0015\u0012\u000f\u0000\u00060\n\u000f0\u0013\n+ ~p2~\u0017(2)\n\u0015\u0012\u000f\u0000\u00060\n\u000f0\u0013\n+O(~p4);\n(B8)\nwhere\n~\u0017(0)\n\u0015(~\u000f) = \u0002(~\u000f)\"p\n~\u000f\n\u0019\u0000\u0015\n2+1\n\u0019arctan\u00121p\n~\u000f\u0013#\n+ \u0002(\u0000~\u000f)\u000e\u0015;\u00001\u0002(1\u0000p\n\u0000~\u000f); (B9)\n~\u0017(2)\n\u0015(~\u000f) =\u0000\u0002(~\u000f)\n2p\n~\u000f\n\u0019(1 + ~\u000f)2\n+\u0002(\u0000~\u000f)\n4\u000e\u0015;\u00001\u000e0(1\u0000p\n\u0000~\u000f); (B10)\nand\u000e0(x) =d\ndx\u000e(x)is the derivative of the \u000e-function\nwith respect to its argument. The order parameter ex-pansion of the weighted density of states is\n\u001b\u0015(\u000f;p)\n\u00170= ~\u001b(0)\n\u0015\u0012\u000f\u0000\u00060\n\u000f0\u0013\n+ ~p2~\u001b(2)\n\u0015\u0012\u000f\u0000\u00060\n\u000f0\u0013\n+O(~p4);\n(B11)\nwith\n~\u001b(0)\n\u0015(~\u000f) =\u0002(~\u000f)\n4(\n(3 + ~\u000f)\u00141\n2\u0000\u0015\n\u0019arctan\u00121p\n~\u000f\u0013\u0015\n\u0000\u00153p\n~\u000f\n\u0019)\n+\u0002(\u0000~\u000f)\n4\u000e\u0015;\u00001\u0002(1\u0000p\n\u0000~\u000f)(3 + ~\u000f); (B12)\n~\u001b(2)\n\u0015(~\u000f) =\u0002(~\u000f)\n16\"\n\u0015p\n~\u000f(1\u0000~\u000f)\n\u0019(1 + ~\u000f)2\u00001\n2+\u0015\n\u0019arctan\u00121p\n~\u000f\u0013#\n+\u0002(\u0000~\u000f)\n16\u000e\u0015;\u00001h\n\u0000\u0002(1\u0000p\n\u0000~\u000f) + 2\u000e(1\u0000p\n\u0000~\u000f)\n+3 + ~\u000f\n2\u000e0(1\u0000p\n\u0000~\u000f)i\n: (B13)\nAt low temperatures only the leading asymptotics of\nthese functions close to the bottom of the lower energy\nbranch is relevant. Shifting the dimensionless energy as12\n~\u000f=\u00001 +!, we may approximate for j!j\u001c1,\n~\u0017(0)\n\u0000(\u00001 +!)\u0019\u0002(!); (B14)\n~\u001b(0)\n\u0000(\u00001 +!)\u0019\u0002(!)\n2; (B15)\n~\u0017(2)\n\u0000(\u00001 +!)\u00191\n4\u000e0(1\u0000p\n1\u0000!)\u0019\u000e0(!);(B16)\n~\u001b(2)\n\u0000(\u00001 +!)\u00191\n16[\u0000\u0002(!) + 2\u000e(!) + 4\u000e0(!)];(B17)\nwhere we have used \u000e0(a!) =a\u00002\u000e0(!)and!\u000e0(!) =\n\u0000\u000e(!).\nAPPENDIX C: ORDER PARAMETER\nEXPANSION\nFor temperatures slightly below the critical tempera-\nture the order parameter ~p=j~g?j~M=2is small so that\nwe may expand all quantities to second order in pow-\ners of ~p. Having expressed all quantities in terms of the\nspectral densities \u0017\u0015(\u000f;p)and\u001b\u0015(\u000f;p), we use the order\nparameter expansion of these, given in Eqs. (B8, B11), to\ngenerate the corresponding expansion of any other quan-\ntity. Using the density equation (3.46) we obtain for the\nchemical potential\n\u0016=\u00160+\u00162~p2+O(~p4); (C1)\nwhere the chemical potential for vanishing order param-\neter is determined by\n~\u001a=Z1\n\u00001d~\u000fX\n\u0015~\u0017(0)\n\u0015(~\u000f)n0(~\u000f); (C2)\nwith\nn0(~\u000f) =1\ne(~\u000f+~\u00060\u0000~\u00160)=\u001c\u00001: (C3)\nHere ~\u00060= \u0006 0=\u000f0and~\u00160=\u00160=\u000f0. The leading correction\nto the chemical potential in the symmetry broken phase\nis\n\u00162\nT=~\u00162\n\u001c=R1\n\u00001d~\u000fP\n\u0015~\u0017(2)\n\u0015(~\u000f)n0(~\u000f)\nR1\n\u00001d~\u000fP\n\u0015~\u0017(0)\n\u0015(~\u000f)n0(~\u000f)[n0(~\u000f) + 1]:(C4)\nThe corresponding expansion of the susceptibility ~\u001f?de-\nﬁned in Eq. (3.48) is\n~\u001f?= ~\u001f(0)\n?+ ~p2~\u001f(2)\n?+O(~p4); (C5)\nwith\n~\u001f(0)\n?=\u00001\n2\u001cZ1\n\u00001d~\u000fX\n\u0015\u0015~\u001b(0)\n\u0015(~\u000f)n0(~\u000f)[n0(~\u000f) + 1];(C6)\nand\n~\u001f(2)\n?=\u00001\n2\u001cZ1\n\u00001d~\u000fX\n\u0015\u0015\u001a\n~\u001b(2)\n\u0015(~\u000f)n0(~\u000f)[n0(~\u000f) + 1]\n\u0000~\u00162\n\u001c~\u001b(0)\n\u0015(~\u000f)n0(~\u000f)[n0(~\u000f) + 1][2n0(~\u000f) + 1]\u001b\n:(C7)And ﬁnally, the expansion of the dimensionless free en-\nergy deﬁned in Eq. (3.49) is\nf\u0011\n +\u0016N\nV\u00170\u000f2\n0=f0+ ~p2f2+O(~p4);(C8)\nwith\nf0=\u001cZ1\n\u00001d~\u000fX\n\u0015~\u0017(0)\n\u0015(~\u000f) lnh\n1\u0000e\u0000(~\u000f+~\u00060\u0000~\u00160)=\u001ci\n\u00001\n2\u0014\n~gk+~g?\n2\u0015\n~\u001a2+ ~\u00160~\u001a; (C9)\nf2=4\n\u0000g?+\u001cZ1\n\u00001d~\u000fX\n\u0015~\u0017(2)\n\u0015(~\u000f) lnh\n1\u0000e\u0000(~\u000f+~\u00060\u0000~\u00160)=\u001ci\n:\n(C10)\nIn the low-temperature regime \u001c\u001c1the above ex-\npressions can be evaluated analytically because it is then\nallowed to substitute the leading asymptotics of the spec-\ntral densities for energies close to the bottom of the lower\nbranch given in Eqs. (B14–B17). The zeroth order chem-\nical potential and the corresponding fugacity z0are then\ngiven by\nz0=e(~\u00160\u0000~\u00060+1)=\u001c= 1\u0000e\u0000~\u001a=\u001c;(C11)\nwhile the correction to the chemical potential is\n~\u00162=1\n\u001c(1\u0000z0)=e~\u001a=\u001c\n\u001c: (C12)\nThe zeroth order susceptibility ~\u001f(0)\n?is given by the right-\nhand side of Eq. (3.54), while the leading correction is\n~\u001f(2)\n?\u0019\u0000z0(3 +z0)\n8\u001c2(1\u0000z0)3: (C13)\nThe order parameter close to the critical point shows the\nusual mean-ﬁeld behavior,\n~p\u0018(1\u0000z0)s\n2~\u001a\u001cc\nz0(3 +z0)r\u001cc\u0000\u001c\n\u001cc: (C14)\nAt the critical temperature we may write\n1\u0000z0=e\u0000~\u001a=\u001cc=1\n1 +4\nj~g?j; (C15)\nso that we arrive at Eq. (3.57) for the transverse magne-\ntization ~M= 2~p=j~g?j. Finally, the leading coeﬃcient in\nthe expansion of the free energy is\nf0=\u0000\u001c2Li2(z0)\u00001\n2\u0014\n~gk+~g?\n2\u0015\n~\u001a2+ ~\u00160~\u001a;(C16)\nwhere Li2(z0)is the polylogarithm. The coeﬃcient of the\nterm proportional to ~p2is\nf2=\u00004\n~g?\u0000z0\n1\u0000z0= 4\u00141\nj~g?j\u0000~\u001f(0)\n?\u0015\n:(C17)\nNotethatfor \u001c <\u001cctheright-handsideofthisexpression\nis negative, so that in the ferromagnetic phase the system\nindeed gains energy.13\nAPPENDIX D: PAIR CONDENSATION\nFor attractive g?the system can also exhibit a pairing\ninstability which competes with the ferromagnetic insta-\nbility discussed in the main text. In this appendix we\nshow however that, at least for suﬃciently low densities,\nthe ferromagnetic instability is dominant.\nIf we decouple the interaction in the particle-particle\nchannel, we obtain the mean-ﬁeld Hamiltonian\nHMF\u0000\u0016N=X\nkh\n(Ek\u0000\u0000\u0016)ay\nkak+ (Ek+\u0000\u0016)by\nkbk\n+\u0016\u001f0ei'ka\u0000kbk+\u001f0e\u0000i'kby\nkay\n\u0000ki\n+V\njg?jj\u001f0j2:(D1)\nIn general, the boson-pairing order parameter \u001f0=\nj\u001f0jei\u000bis complex, but all phases can be eliminated by\nsetting ~ak=ei'k\u0000i\u000bak. The Hamiltonian can then be\ndiagonalized by means of a Bogoliubov transformation,\n\u0012~ak\nby\nk\u0013\n=\u0012\nu\u0003\nk\u0000v\u0003\nk\n\u0000vkuk\u0013\u0012\u000bk\n\fy\nk\u0013\n;(D2)\nwith\nuk=r\n\u0018k+!k\n2!k; vk=r\n\u0018k\u0000!k\n2!k;(D3)\nwhere\n\u0018k=k2\n2m\u0000\u0016; !k=q\n\u00182\nk\u0000j\u001f0j2:(D4)\nThe corresponding grand canonical potential is\n\nMF=TX\nk\u0015lnh\n1\u0000e\u0000\f(!k+\u0015v0jk?j)i\n+X\nk(!k\u0000\u0018k) +V\njg?jj\u001f0j2: (D5)\nThe self-consistency equation for the order parameter is\n1\njg?j=1\n2VX\nk\u00151\n!k\u00141\ne\f(!k+\u0015v0jk?j)\u00001+1\n2\u0015\n;(D6)\nwhile the density equation is\n\u001a=1\nVX\nk\u0015\u0018k\n!k1\ne\f(!k+\u0015v0jk?j)\u00001+1\nVX\nk\u0014\u0018k\n!k\u00001\u0015\n:\n(D7)\nThe integrals are ultraviolet divergent and must be reg-\nularized. One possibility is to eliminate the bare interac-\ntion in favor of the two-body t-matrix t?, which can be\ndeﬁned via\n1\nt?=1\ng?+1\nVX\nk1\n2\u000fk; (D8)where\u000fk=k2=2m. The regularized gap equation is then\n1\n\u0000t?+1\n2VX\nk\u00121\n\u000fk\u00001\n!k\u0013\n=1\n2VX\nk\u00151\n!k1\ne\f(!k+\u0015v0jk?j)\u00001:(D9)\n4.04.55.05.5\n0 5 10 15 20\nν0|t⊥|˜ρc\nno pairingτ= 1.0τ= 0.5τ= 0.05\nFIG. 8. (Color online) Critical density ~\u001acfor pair conden-\nsation as a function of the two-body t-matrixjt?j(note that\nt?<0, so\u0000t?=jt?j)forﬁxedtemperatures \u001c. Thesolidline\ndepicts ~\u001acfor vanishing temperature as given in Eq. (D10).\nThe dots represent numerical solutions of the regularized gap\nequation (D9) for the indicated temperatures. The shaded\narea underneath the solid line represents the regime where\npairing cannot occur at any temperature.\nAt the critical point we set \u001f0= 0. At low temperatures,\nwe may replace \u0018k\u0019\u000f0\u0000\u0016= 2\u000f0in the prefactor on the\nright-hand side. Then we obtain for the critical density\n~\u001ac=\u001ac=(\u000f0\u00170)at zero temperature\n~\u001ac= 4\u00141\n\u0000(\u00170t?)+ 1\u0015\n: (D10)\nFor densities smaller than ~\u001acthere is no pairing insta-\nbility at zero temperature. For ﬁnite temperatures we\nhave solved the regularized gap equation (D9) numeri-\ncally, see Fig. 8. Because the zero-temperature result\n(D10) for the critical density forms a lower bound for the\ncritical density at ﬁnite temperatures, we conclude that\nfor suﬃciently small densities, pairing cannot occur at\nany temperature. For densities smaller than this thresh-\nold the ferromagnetic instability discussed in the main\ntext is dominant.14\n1Y.-J. Lin, R. L. Compton, A. R.Perry, W. D. Phillips, J.\nV.Porto, and I.B.Spielman, Phys. Rev.Lett. 102, 130401\n(2009); Y.-J.Lin, R.L.Compton, K.Jiménez-García, J.V.\nPorto, and I. B. 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The cobalt-doped rutile TiO 2 epitaxial thin films with different \nelectron densities and cobalt contents were grown on r-sapphire substrates with laser molecular beam epitaxy. \nResults of magnetization, magnetic circular dichroism, and anomalous Hall effect measurements were examined \nfor samples with systematically varied electron dens ities and cobalt contents. The samples with high electron \ndensities and cobalt contents show the high temperature ferromagnetism, suggesting that charge carriers induce \nthe ferromagnetism.   \n PACS number(s): 75.50.Pp, 78.20.Ls, 73.50.Jt, 72.25.Dc, 75.30.Gw \n 11. Introduction \nCobalt-doped TiO 2 is one of the wide gap ferroma gnetic semiconductors exhibiting high \ntemperature ferromagnetism [ 1,2]. Many magnetic oxide semiconductors have been reported so far \n[3,4,5,6,7,8], but cobalt-doped TiO 2 is the most pronounced compound because its high temperature \nferromagnetism was evidenced also via various types of magnetic characterizations such as \nmagneto-optical and anomalous Hall effects and because the ferromagnetism varies systematically \nwith the sample parameters reporte d by different research groups [ 9,10,11,12,13,14,15,16,17]. \nFurthermore, tunneling magnetoresistance effect was successfully observed in a magnetic tunneling \njunction up to 200 K with use of cobalt-doped TiO 2 as a ferromagnetic electrode [ 18]. Various \nspintronic device applications at room temperatur e will be expected by improving device fabrication \nprocess.  \nThe cobalt-doped TiO 2 has several crystal structures. Among them, both rutile and anatase phases \nshow high temperature ferromagne tism, but various magnetic and electronic properties are different \neach other in a quantitative sense [10,15-17]. Accordingl y, comparison of such properties is interesting \nto investigate the origin of high temperature ferromagnetism. In this article, however, we mainly focus on results of cobalt-doped rutile TiO\n2 since its various experimental data are available with a \nsystematic series of samples. We will discuss a role of charge carriers for appearance of the \nferromagnetism by showing systematic results of magnetization, magnetic circular dichroism, and \nanomalous Hall effect measurements. In this article, we denote cobalt-doped TiO 2 as Ti 1-xCoxO2-δ since \noxygen vacancies ( δ) are present in order to make the samples electrically conductive.  \n \n2. Film growth  \nRutile Ti 1-xCoxO2-δ epitaxial thin films were fabricated by using laser molecular beam epitaxy. \nPrescribed amount of TiO 2 and CoO powders were mixed and sint ered for making ceramic targets. The \ntargets were ablated by pulsed KrF excimer laser. The films with (101) orientation were epitaxially \n 2grown on r-sapphire substrates. All the sapphire substr ates were subject to ultrasonic cleaning in \norganic solvents followed by annea ling in a furnace to have atomica lly flat step and terrace surface \nprior to the film growth. TiO 2 buffer layers with 5 nm thick were grown on the substrates at 400 ºC in 1 \n× 10-4 Torr of oxygen followed by post-annealing at 600 ºC in 1 × 10-2 Torr of oxygen for an hour. \nOxygen pressures during growth ( PO2) of Ti 1-xCoxO2-δ films grown at 400 ºC on the buffer layers were \nvaried from 1 × 10-7 Torr to 1 × 10-4 Torr in order to control the amount of δ. The reflection high \nenergy electron diffraction (RHEED) intensity was in-situ  monitored during the film growth. The film \nthicknesses were typically 70-120 nm. From X- ray diffraction, scanning electron microscopy, \ntransmission electron microscopy, RHEED, and atomic force microscopy, neither impurity phase nor \nsegregation of secondary phase was observed. Phot olithographically patterned Hall bars (220 μm long \n× 60 μm wide) were used for electronic transport measurements. Electron density ne was evaluated \nfrom linear slope of normal part of Hall resistance vs. magnetic field curve. Table 1 shows correspondence between x, P\nO2, and ne at 300 K for Ti 1-xCoxO2-δ in this study. \nFigure 1 shows typical in-situ  RHEED oscillation during growth and RHEED images of the \nbuffer TiO 2 layer and Ti 0.95Co0.05O2-δ film. The RHEED oscillation is kept up to an initial growth of \nTi0.95Co0.05O2-δ thin film. A sizable decrease in the RHEED intensity for growth of the Ti 0.95Co0.05O2-δ \nfilm is due to the relatively low growth temperature for the Ti 0.95Co0.05O2-δ film. The clear RHEED \nstreak pattern, as well as atomically flat atomic forc e microscope image, reflects high crystallinity of \nthe films without any surface segregation. As described  in section 5, various spectroscopies were also \nperformed to represent that Co ions in TiO 2 are Co2+ (d7) state with high spin configuration, thus \npossible segregation of Co metal is ruled out again.  \nIt is emphasized that optimization of various growth conditions such as pre-treatment of the \nsubstrate, deposition rate, and growth temperatur e was indispensable to obtain high quality films \nreproducibly. Such films show systematic properties as described in this article hereafter.  \n \n 33. Electric properties \nIn TiO 2-δ, δ yields in charge carriers. Varying PO2 enables to control an amount of δ leading to a \nsystematic change in resistivity with several decades. Figure 2 shows temperature dependence of \nresistivity for Ti 0.97Co0.03O2-δ grown in different PO2. By decreasing PO2, the resistivity decreases \nmonotonically due to increase in δ. The decrease in resistivity is s ynchronized with the increase in the \nlattice constant as shown in inset of Figure 2.  \nThe mobility at 300 K is kept to be ~10-1 cm2/Vs for different x  as shown in Figure 3, being \ncomparable with that of bulk rutile TiO 2, implying that negligible effect of x on the film quality. The \nsmall dependence of the mobility on ne in inset of Figure 3 represents that the change in resistivity \nwith δ is mainly caused by the change in ne. Hereafter, we examine various experimental data as a \nfunctions of ne and x.   \n \n4. Magnetic properties \n4.1. Magnetization \nFigure 4 shows magnetization curves at 300 K for Ti 1-xCoxO2-δ with different x and ne. In Figure \n4(a), the magnetization of Ti 0.97Co0.03O2-δ is negligible for n e = 7 × 1018 cm-3, and a tiny magnetization \nis seen for ne = 4 × 1019 cm-3. For higher ne ≥ 2 × 1020 cm-3, clear ferromagnetic behaviour is observed \nwith saturation magnetization of ~1.0 μB/Co. Figures 4(b) and (c) show the magnetization curves for \nTi0.95Co0.05O2-δ and Ti 0.90Co0.10O2-δ, respectively. Ti 0.95Co0.05O2-δ shows the saturation magnetization of \n1.1 μB/Co for ne = 2 × 1020 cm-3 and 7 × 1021 cm-3 with small difference in onset of the magnetization \nat low magnetic field. On the other hand, Ti 0.90Co0.10O2-δ shows the saturation magnetization that \ndepends on ne (ne = 2 × 1020 cm-3 and 4 × 1021 cm-3): the lower ne shows the larger saturation \nmagnetization of 1.5 μB/Co than the higher ne (1.1 μB/Co). The onset of the magnetization is different \nas well as that in Figure 4(b). Therefore, the magnitude of saturation magnetization and the magnetic \nanisotropy depends on both ne and x.  \n 4 \n4.2. Magnetic circular dichroism \nMagnetization measurements  provide indispensable information of magnetic properties such as \nthe magnetization value and the magnetic anisotro py. However, magnetometer is not able to \ndistinguish extrinsic signals such as a magnetic segregation from intrinsic magnetization. Hence, \ncomprehensive measurements are necessary in orde r to investigate magnetism of new compounds like \nTi1-xCoxO2-δ. For this purpose, magnetic circular dichro ism is a useful method since intrinsic and \nextrinsic origins can be distinguished as well as a nomalous Hall effect [7]. In several ferromagnetic \nsemiconductors, the absorption and MCD spectra were observed to have an intimate relation, where \nenergy derivative of absorption coefficient is proportional to MCD [ 19]. Also, effect of substrate, that \nis usually not negligible due to its much la rger volume than ferromagnetic film in case of \nmagnetization measurement, can be completely excluded when using magneto-optically inactive \nsubstrate.  \nFigure 5 shows the absorption and magnetic circular dichroism spectra at 300 K for \nTi0.97Co0.03O2-δ with different ne. The absorption spectra represent almo st transparency against visible \nlight, and the absorption increases over 3 eV . For ne ≤ 4 × 1019 cm-3, MCD is negligible, and MCD \ndevelops with increasing ne. Below the absorption edge, MCD is negative with broad spectral feature \nprobably originated from d-d transition [ 20]. At the higher photon energy, the MCD changes its sign. \nThese ferromagnetic MCD spectra have the same magnetic field dependence at any photon energy \n(not shown), thus the MCD is originated from Ti 1-xCoxO2-δ without any magnetic sources [ 21]. For ne ≥ \n4 × 1022 cm-3, the absorption edge shifts toward higher energy side with small increase in the \nabsorption below 2 eV , that was observed in a reduced TiO 2 [22]. It is noted that nearly the same \namount of blue shift is seen in the MCD spectrum as well as absorption spectrum for ne = 4 × 1022 cm-3, \nreflecting a close relation between the absorption and MCD spectra.  \nFor MCD measurement, effect of the substrate on MC D signal is negligible, thus it is useful to \n 5study magnetic anisotropy in spite of a lack of the quantitative magnetization value. Figure 6 shows \nmagnetic field dependence of normalized MCD for different ne and x. It is noted that the dependence \nof the MCD magnitude on ne and x is consistent with that of the magnetization in Fig. 4. For ne = 2 × \n1020 cm-3 (Figure 6(a)), onset of MCD at low magnetic field is steeper for the higher x . For n e ≥ 4 × \n1021 cm-3 (Figure 6(b)), on the other hand, the onset of MCD is nearly independent on x and the \nsaturation needs higher magnetic field. These results suggest that the moderate ne and higher x lead to \nthe steeper and larger out-of-plane magnetization taking result of Figure 4 into account, although \nfurther compilation of data is needed to investigat e detail behaviours of the magnetization properties.  \n \n4.3. Anomalous Hall effect \nFor thin film of dilute ferromagnetic systems,  anomalous Hall effect is a good measure of the \nferromagnetism because of its high sensitivity to th e magnetization. Also, the anomalous Hall effect \nwill not be affected by small amount of segregation, hence bulk magnetic property can be probed.  \nFigure 7 shows magnetic field dependence of Hall resistivity at different temperatures for \nTi0.97Co0.03O2-δ with different ne (300 K). In Figure 7(a), Hall r esistivity steeply increases with \nmagnetic field up to 0.2 T, and decreases almost lin early with further increasing magnetic field. This \nbehaviour clearly represents an empirical relation of Hall effect of ferromagnetic metals and \nsemiconductors: Hall resistivity is a sum of normal a nd anomalous Hall resistivities, where the former \nand the latter are proportional to the magnetic fi eld and the magnetization, respectively. With \ndecreasing temperature, the negative slope correspo nding to the normal Hall effect becomes steeper \ndue to the decrease in n e. For the lower ne (Figure 7(b)), normal Hall effect is dominant, hence \nanomalous Hall effect is difficult to be observed. Nevertheless, anomalous Hall effect can be seen by \nsubtracting the linear part with respect to magnetic field. For Ti 0.97Co0.03O2-δ with further lower ne (ne \n(300 K) ≤ 4 × 1019 cm-3), the anomalous Hall effect was negligible, representing absence of \nferromagnetism.  \n 6Figure 8 shows magnetic field dependence of Hall resistivity at different temperatures for \nTi0.90Co0.10O2-δ with different ne (300 K). The anomalous Hall effect becomes dominant in comparison \nwith Figure 7 because of the higher x leading to increase in the volume magnetization. In this case, the \ndecrease in ne also leads to steeper slope of the normal Hall effect. Figure 9 shows magnetic field \ndependence of Hall resistivity for different x at 300 K. With increasing x, anomalous Hall resistivity \nmonotonously increases mainly due to the increase in  the volume magnetization as described above. \nFor Ti 0.99Co0.01O2-δ with ne (300 K) ≥ 2 × 1020 cm-3, anomalous Hall effect was negligible.  \nThe Hall effect of ferromagnetic Ti 1-xCoxO2-δ can be simply regarded as the sum of normal and \nanomalous Hall effects, hence it is straightforward to extract the anomalous Hall part. Study on the \nanomalous Hall effect is recently revived because of  its quantal nature and relevance to spin Hall \neffect [ 23]. From the viewpoint of semiconductor spintronics, it is important to confirm whether the \nanomalous Hall effect can be controlled by external perturbation such as electric field or not, some of \nwhich have been demonstrated in (Ga,Mn)As [ 24]. Figure 10 shows magnitude of anomalous Hall \nconductivity vs. conductivity plots for various  ferromagnetic compounds having metallic or \nsemiconducting conduction. It can be seen that each class of compounds such as Mn-doped III-V \nsemiconductors, transition metal oxides, and Ti 1-xCoxO2-δ shows a scaling relation: magnitude of \nanomalous Hall conductivity is proportional to ( σxx)1.6, where σxx is conductivity [10,16, 25], where \nunderlying physics is described elsewhere [ 26,27]. This relation represents that the magnitude of \nanomalous Hall conductivity can be evaluated fro m the conductivity. In case of ferromagnetic \nsemiconductors, the conductivity in single substance can be varied via field effect and so on, thus \nleading to control of anomalous Hall effect, so that  it will be useful for implementation of spintronics \ndevices.   \n5. Discussion   \nHall resistivity, magnetic circular dichroism, and magnetization for Ti\n1-xCoxO2-δ with different ne \n 7and x show the same magnetic field dependence, except unimportant deviation in the Hall resistivity at \nhigh magnetic field due to the normal Hall effect as shown in Figure 11. This result rules out a \npossibility that magneto-optically inactive and/or  insulating substance produces the ferromagnetism. \nThe resultant magnetic phase diagram shows that the higher ne and x induce the ferromagnetic phase \n(Figure 12).  \nIn addition to the above magnetic measuremen ts, various types of characterization were \nperformed using the systematic series of our samp les. From x-ray photoemission spectroscopy (XPS) \nand x-ray magnetic circular dichroism (XMCD), electronic state of Co ions was Co2+ (d7) high spin \nstate although the magnetization values in Figure 4 is smaller than the d7 state ideally that yields in 3 \nμB/Co [ 28,29]. In order to keep charge neutrality in Ti 1-xCoxO2-δ, δ is equal to x assuming that valences \nof Ti and Co ions are 4+ and 2+, respectively. For electrically conductive Ti 1-xCoxO2-δ, 3d state of Ti \nions should be partially filled so th at the valence is less than 4+, hence δ > x. Such significant amount \nof oxygen vacancies will be energetically favourable for the high spin state of Co2+ ions due to the \ndecrease in the crystal field assuming that oxygen vacancies are adjacent to Co2+ ions. The presence of \nCo2+ ions and oxygen vacancies in the TiO 2 may cause local charge imbalance and lattice distortion. \nThe local lattice distortion was evidenced by x-ra y anomalous scattering of both rutile and anatase \nTi1-xCoxO2-δ, where Co ions deviate from exact location of Ti sites [ 30]. Nevertheless, this result does \nnot rule out the substitutional occupation of Co i ons since the x-ray absorption and XMCD spectra \nsuggest Co2+ high spin state in the D2h-symmetry crystal field at Ti site [29]. The local charge \nimbalance and the lattice distortion as well as high spin  state of Co ions might cause large exchange \ncoupling between localized spins and overlapping electron wave function leading to the high Curie temperature [ 31].  \n \n6. Summary \nVarious types of magnetic and electronic characterizations using a systematic series of high \n 8quality samples gradually unveil origin of the ferromagnetism in Ti 1-xCoxO2-δ. The present results \nsuggest an important role of charge carriers to induce the high temperature ferromagnetism. \nNevertheless, high temperature ferromagnetism was also observed for insulating cobalt-doped TiO 2 \n[32]. It is worth investigating whether such ferroma gnetism in the insulating sp ecimens is caused by a \nlocal exchange mechanism without  presence of charge carriers. \n \nAcknowledgements \nThe authors gratefully acknowledge T. Chikyow, J. M. Dong, A. Fujimori, T. Hasegawa, T. Hitosugi, Y . \nKawazoe, H. Koinuma, F. Matsukura, Y . Matsumoto,  T. Matsumura, T. Mizokawa, Y . Murakami, K. \nNakajima, H. Ohno, and H. Weng for discussions. This work was partly supported by the New \nEnergy and Industrial Technology Developm ent Organization, the Industrial Technology \nResearch Grant Program (05A24020d), MEXT fo r Scientific Research on Priority Areas \n(16076205) and for Young Scientists (A19686021) , and the Tokyo Ohka Foundation for \nPromotion of Science and Technology.  \n 9References \n \n[1] Matsumoto Y , Murakami M, Shono T, Hasegawa T, Fukumura T, Kawasaki M, Ahmet P, \nChikyow T, Koshihara S and Koinuma H 2001 Science  291 854  \n[2] Matsumoto Y , Takahashi R, Murakami M, Koida T, Fan X-J Hasegawa T, Fukumura T, Kawasaki \nM, Koshihara S and Koinuma H 2001 Jpn. J. Appl. Phys.  40 L1204 \n[3] Pearton S J, Abernathy C R, Overberg M E, Thaler G T, Norton D P, Theodoropoulou N, Hebard \nA F, Park Y D, Ren F, Kim J and Boatner L A 2003 J. Appl. Phys.  93 1 \n[4] Prellier W, Fouchet A and Mercey B 2003 J. Phys.: Condens. 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Mater. 4 173 \n[32] Griffin K A, Pakhomov A B, Wang C M, Heald S M and Krishnan K M 2005 Phys. Rev. Lett.  94 \n157204 \n 11Table 1. Correspondence between x, PO2, and ne (300 K) for Ti 1-xCoxO2-δ in this study. \nx PO2 [Torr] ne (300 K) [cm-3]\n0 1 × 10-6 4 × 1020 \n0 1 × 10-7 3 × 1022 \n0.01 1 × 10-6 2 × 1020 \n0.01 1 × 10-7 2 × 1021 \n0.03 1 × 10-4 7 × 1018 \n0.03 1 × 10-5 4 × 1019 \n0.03 1 × 10-6 2 × 1020 \n0.03 1 × 10-7 4 × 1022 \n0.05 1 × 10-6 2 × 1020 \n0.05 1 × 10-7 7 × 1021 \n0.10 1 × 10-6 2 × 1020 \n0.10 1 × 10-7 4 × 1021 \n \n 12Figure captions \n \nFigure 1. (a) RHEED oscillations during growth of rutile TiO 2 buffer layer and rutile Ti 0.95Co0.05O2-δ \nfilm. RHEED images of (b) annealed buffer layer and (c) Ti 0.95Co0.05O2-δ film.  \n Figure 2. Temperature dependence of resistivity for Ti\n0.97Co0.03O2-δ with different PO2. Inset shows P O2 \ndependence of conductivity at 300 K (solid circle) and lattice constant along (101) direction ( d (101)) \n(open circle) for these films.  \n \nFigure 3. x dependence of mobility at 300 K for Ti 1-xCoxO2-δ with different PO2. PO2 = 1 × 10-7 (solid \ncircle), 10-6 Torr (solid square), 1 × 10-5 (solid triangle), and 10-4 Torr (solid diamond). Inset shows ne \ndependence of mobility at 300 K for Ti 1-xCoxO2-δ with different x. x = 0 (open diamond), 0.01 (open \ntriangle), 0.03 (open square), 0.05 (open circle), and 0.10 (open inverted triangle).  \n \nFigure 4. Magnetic field dependence of magnetization at 300 K for Ti 1-xCoxO2-δ with different ne. (a) x \n= 0.03. Each data is shifted vertically. (b) x = 0.05. (c) x  = 0.10.  \n Figure 5. (a) Absorption and (b) magnetic ci rcular dichroism spectra at 300 K for Ti\n0.97Co0.03O2-δ with \ndifferent n e.  \n \nFigure 6. Magnetic field dependence of normaliz ed magnetic circular dichroism at 300 K for \nTi1-xCoxO2-δ with different x. Data for (a) the lower ne and (b) the higher n e samples. \n \nFigure 7. Magnetic field dependence of Hall resistivity at different temperatures for Ti 0.97Co0.03O2-δ. \nData for (a) the higher ne and (b) the lower ne samples. Inset of (b) shows magnetic field dependence \n 13of anomalous Hall resistivity obtained by subtr acting magnetic-field-linear component in (b).  \n \nFigure 8. Magnetic field dependence of Hall resistivity at different temperatures for Ti 0.90Co0.10O2-δ. \nData for (a) the higher ne and (b) the lower ne samples. \n Figure 9. Magnetic field dependence of Hall resistivity at 300 K for Ti\n1-xCoxO2-δ with different x. Data \nfor (a) the higher ne and (b) lower ne samples. \n \nFigure 10. Magnitude of anomalous Hall conductivity vs. conductivity for various ferromagnetic compounds with metallic or semiconducting conduction. Detail of each data is . \ndescribed elsewhere [25].   \nFigure 11. Magnetic field dependence of Hall resistivity (blue curve), magnetization (green open \ncircle), and MCD (red circle) for Ti\n0.90Co0.10O2−δ.  \n Figure 12. Magnetic phase diagram at 300 K for Ti\n1-xCoxO2-δ as functions of n e and x deduced from \nanomalous Hall effect and MCD measurements. Solid and open symbols denote ferromagnetic and \nparamagnetic phases, respectively. Circle, squa re, triangle, and diamond symbols correspond to PO2 = \n10-7, 10-6, 10-5, and 10-4 Torr, respectively.   \n 14Fig. 1  Fukumura et al. Deposition time [s]Intensity [arb. units]\n600 400 200 0TiO 2 buffer\nTi0.95Co0.05O2-δ\n(a)\nfilm\nannealed buffer(b)\n(c)Fig.2  Fukumura et al.300 200 100Resistivity [ ΩΩcm]x=0.03\n10-7Torr10-6Torr10-5TorrPO2=1×10-4Torr\nTemperature [K]0102\n100\n10-2101\n10-1\n10-30.254\n0.250\n0.246\nd(101) [nm]10-710-610-510-4\n10-410-2100102104PO2 [Torr]conductivity [ Ω-1cm-1]\n300KFig.3  Fukumura et al.101\n100\n10-1\n10-2\n10-3Mobility [cm2V-1s-1]\n0 0.02 0.04 0.06 0.08 0.1\nx in Ti 1-xCoxO2-δ300K\n10-7Torr10-4TorrPO2=10-5Torr\n10-6Torr0.100.01\n0x=0.03 0.05100\n10-2\n101810201022\nne [cm-3]Fig.4  Fukumura et al.H⊥⊥plane4\n3210\n-1-2-3-2 -1 0 1 2\n300Kx=0.034×1022cm-3\nμ0H [T]Magnetization [ μB/Co](a)\n2×1020cm-3\n4×1019cm-3\n7×1018cm-32\n10\n-1\n-22\n1\n0\n-1\n-2 -1 0 1 2\nμ0H [T]x=0.05\nx=0.10(b)\n(c)2×1020cm-3\n7×1021cm-3\n4×1021cm-32×1020cm-3Fig.5  Fukumura et al.(a)Absorption coeff.\n[103 cm-1]\n4×1019cm-3\n7×1018cm-32×1020cm-3\n4×1022cm-3800\n400\n0300K x=0.03MCD [103 deg./cm](b)\n-201\n4 3 2\nPhoton energy [eV]1T7×1018cm-3\n4×1019cm-3\n2×1020cm-34×1022cm-3\n-1\nH⊥plane1.5\n1\n0.5\n0\n-0.5\n-1\n-1.5-1 -0.5 0 0.5 1MCD/MCD (1.3T)\nμ0H [T]300KE=2.50eV\n0.03\n0.05x=0.10\nH⊥plane(a)\n-1 -0.5 0 0.5 1\nμ0H [T]E=3.05eV\n4×1022cm-30.03\n0.050.10(b)\nFig.6  Fukumura et al.2×1020cm-3\n2×1020cm-32×1020cm-34×1021cm-3\n7×1021cm-3Fig.7  Fukumura et al.Hall resistivity [ μΩμΩcm]\n-2 -1 0 1 21\n0.5\n0\n-0.5\n-1\nμ0H [T]x=0.03(a)\n300K\n4×1022cm-3100K3×10\n20cm-3\n200K5×10\n21cm-3\n-2 -1 0 1 2100\n50\n0\n-50\n-100\nμ0H [T]x=0.03(b)\n300K\n2×1020cm-3100K3×10\n18cm-3\n200K4×10\n19cm-33\n0\n-32 0 -2-2 -1 0 1 210\n50\n-5\n-10\nμμ0H [T]x=0.10300K\n4×1021cm-3100K1×10\n20cm-3\n200K4×10\n20cm-3Hall resistivity [ μΩμΩcm](a)\nx=0.10\n-2 -1 0 1 2\nμ0H [T]300K\n2×1020cm-3100K2×10\n19cm-3\n200K3×10\n19cm-3(b)\nFig.8  Fukumura et al.Fig.9 Fukumura et al.Hall resistivity [ μΩμΩcm]\n0.03\n4×1022cm-3 \n-2 -1 0 1 2\nμ0H [T](a)2\n10\n-1-2x=0.10\n4×10\n21cm-3\n0.057×10\n21cm-3 \n300K\n-2 -1 0 1 2\nμ0H [T](b)\n5\n0\n-50.03\n2×1020cm-3 0.10\n2×1020cm-3 \n0.05\n2×1020cm-3\n300KFigure 10  Fukumura et al.10-810-610-410-2100102104106\n10-2100102104106\nConductivityConductivity [Ω-1cm-1]|Anomalous Hall conductivityAnomalous Hall conductivity| [Ω-1cm-1]\nSr2FeMoO 6SrRuO 3\n(FeMn)Si(FeCo)Si\nr-(TiCo)O 2r-(TiCo)O 2a-(TiCo)O 2a-(TiCo)O 2\n (LaCa)MnO 3(LaCaPb)MnO 3\n(LaSr)MnO 3Nd2Mo2O7(LaSr)CoO 3\n(LaSr)CoO 3(GaMn)As\n(GaMn)As(GaMn)As\n(GaMn)As(InMn)As\n(InMn)As(GaMn)Sb(InMn)Sb\n(LaCa)MnO 3(SrCa)RuO 3300K-2\n-2-101\n-2 -1 0 1 2-2-1012Hall resistivity [ μΩμΩcm]\nμ0H[T]\nMagnetization [ μB/Co]x=0.10\n2×1021cm-3\n2kdeg./cm-MCD\nFig. 11  Fukumura et al.H⊥planeFig. 12  Fukumura et al.10181019102010211022\n0.1 0.08 0.06 0.04 0.02 0\nx in Ti 1-xCoxO2-δ300Kferromagnetic\nparamagneticElectron density [cm-3]"    },    {        "title": "1303.4016v2.Designing_ferromagnetism_in_vanadium_oxide_based_superlattices.pdf",        "content": "Designing ferromagnetism in vanadium-oxide based superlattices\nHung T. Dang1and Andrew J. Millis1\n1Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027, USA\n(Dated: September 24, 2018)\nMotivated by recent reports (Phys. Rev. B 80, 241102) of room-temperature ferromagnetism in\nvanadium-oxide based superlattices, a single-site dynamical mean \feld study of the dependence of\nthe paramagnetic-ferromagnetic phase boundary on superlattice geometry was performed. An exam-\nination of variants of the experimentally determined crystal structure indicate that ferromagnetism\nis found only in a small and probably inaccessible region of the phase diagram. Design criteria for\nincreasing the range over which ferromagnetism might exist are proposed.\nPACS numbers: 73.21.Cd,71.28.+d,75.10.Lp\nI. INTRODUCTION\n\\Materials by design\", the ability to design and create\na material with speci\fed correlated electron properties, is\na long-standing goal of condensed matter physics. Super-\nlattices, in which one or more component is a transition\nmetal oxide with a partially \flled d-shell, are of great cur-\nrent interest in this regard because they o\u000ber the possi-\nbility of enhancing and controlling the correlated electron\nphenomena known1to occur in bulk materials as well as\nthe possibility of creating electronic phases not observed\nin bulk.2Following the pioneering work of Ohtomo and\nHwang,3heterostructures and heterointerfaces of tran-\nsition metal oxides have been studied extensively. Ex-\nperimental \fndings include metal-insulator transitions,4\nsuperconductivity,5magnetism6,7and coexistence of fer-\nromagnetic and superconducting phases.8,9\n1.5 1.6 1.7 1.8 1.9 2.0\nfilling81012141618 tilt angle (Degree)\n(La/Sr)VO3 solid solutionPMFM\nFIG. 1: Ferromagnetic-paramagnetic phase diagram for\nLa/SrVO 3solid solution in plane of carrier concentration\n(changed by Sr concentration) and tilt angle in Pnma struc-\nture but with all three Glazer's angles nearly equal. Dashed\nline indicates relation between carrier concentration and ro-\ntation amplitude in physically occurring bulk solid solution.\nFrom Ref. 10.In this paper we consider the possibility that ap-\npropriately designed superlattices might exhibit ferro-\nmagnetism. Our work is partly motivated by a recent\nreport6of room-temperature ferromagnetism in super-\nlattices composed of some number mof layers of LaVO 3\n(LVO) separated by one layer of SrVO 3(SVO), even\nthough ferromagnetism is not found at any xin the bulk\nsolid solution La 1\u0000xSrxVO3. Our study is based on a pre-\nvious analysis10of the possibility of obtaining ferromag-\nnetism in variants of the crystal structure of bulk solid\nsolutions of the form La 1\u0000xSrxVO3. A key result of the\nprevious work was that ferromagnetism is favored by a\ncombination of large octahedral rotations and large dop-\ning away from the Mott insulating LaVO 3composition.\nA schematic phase diagram is shown in Fig. 1. How-\never, as indicated by the dashed line in the \fgure, in the\nphysical bulk solid solution, doping away from the Mott\ninsulating concentration reduces the amplitude of the oc-\ntahedral rotations so that the physical materials remain\nfar from the magnetic phase boundary. The motivating\nidea of this paper is that in the superlattice geometry, oc-\ntahedral rotation amplitude may be decoupled from car-\nrier concentration. The rotations can be controlled by\nchoice of substrate while the carrier concentration can\nbe controlled by choice of chemical composition and may\nvary from layer to layer of a superlattice. In e\u000bect, an\nappropriately designed superlattice could enable the ex-\nploration of di\u000berent paths in Fig. 1.\nIn this study, we combine single-site dynamical mean\n\feld approximation11with realistic band structure cal-\nculations including the e\u000bects of the octahedral rotations\nto determine the ferromagnetic-paramagnetic phase dia-\ngram in superlattices with the crystal structures believed\nrelevant12,13to the experiments of Ref. 6. Unfortunately\nwe \fnd that the experimentally determined crystal struc-\nture is in fact less favorable to ferromagnetism than the\none found in the bulk solid solution, but we indicate\nstructures that may be more favorable.\nThe paper has following structure. The model and\nmethods are described in Sec. II. Sec. III establishes the\nmethods via a detailed analysis of the phase diagram of\nsuperlattices with no rotations or tilts. In Sec. IV we\npresent the magnetic properties of superlattices with oc-arXiv:1303.4016v2  [cond-mat.str-el]  2 Jun 20132\ntahedral rotations similar to those observed experimen-\ntally. Section V is a summary and conclusion.\nII. MODEL AND METHODS\nA. Overview\nThis paper builds on a previous study of the magnetic\nphase diagram of bulk vanadates.10The new features rel-\nevant for the superlattices studied here are (i) the change\nin geometrical structure, including the di\u000berences from\nthe bulk solid solution in the pattern of octahedral tilts\nand rotations and (ii) the variation of electronic density\narising from superlattice structure. In the rest of this sec-\ntion we brie\ry summarize the basic theoretical method-\nology (referring the reader to Ref. 10 for details), de\fne\nthe crystal structures more precisely, explain the conse-\nquences for the electronic structure and explain how the\nvariation of density appears in the formalism.\nB. Geometrical structure\nWe study superlattices composed of layers of SrVO 3\n(SVO) alternating with layers of LaVO 3(LVO). If we ide-\nalize the structures as cubic perovskites, then the layers\nalternate along the [001] direction. In bulk, SVO crystal-\nlizes in the ideal cubic perovskite structure,14while LVO\ncrystallizes in a lower symmetry Pnma structure derived\nfrom the cubic perovskite via a four unit-cell pattern of\noctahedral tilts.15The crystal structure of bulk solid so-\nlutions La 1\u0000xSrxVO3interpolates between that of the\ntwo end-members with the rotation amplitude decreas-\ning asxincreases. In the superlattice, the presence of a\nsubstrate and the breaking of translation symmetry can\nlead to di\u000berent rotational distortions of the basic per-\novskite structure and also to a di\u000berence between lattice\nconstants parallel and perpendicular to the growth direc-\ntion.\nOctahedral rotations in perovskites can be described\nusing Glazer's notation.16In the coordinate system de-\n\fned by the three V-O bond directions of the original\ncubic perovskite, there are 3 tilt angles \u000b;\fand\rwith\ncorresponding rotation axes [100] ;[010] and [001]. The\ntilt is in-phase if successive octahedra rotate in the same\ndirection, and anti-phase if they rotate in opposite di-\nrections. Rotational distortions of the cubic perovskite\nABO3structure may be denoted by aibjckwherei;j;k\ncan be +;\u0000or 0 denoting in-phase, anti-phase or no tilt-\ning, respectively and a=\u000b;b=\f;c=\r.16{18Bulk LVO\n(Pnma ) is of the type a\u0000b+a\u0000with\u000b=\r= 8:7\u000eand\n\f= 7:9\u000e.13,15\nFor superlattices, substrate-induced strain may change\nthe situation in a way which depends on the growth direc-\ntion. Experiments12,13con\frm that the growth direction\nfor the experimentally relevant superlattices is [001] (in\nthe ideal cubic perovskite notation) and we focus on thiscase here. Recent experimental studies of superlattices12\nand of LaVO 3thin \flms, which apparently have the same\ngrowth direction,13suggest that the rotations are of the\ntypea\u0000a+c\u000017,18and indicate that the dominant rota-\ntion is around the axis de\fned by the growth direction:\n\u000b=\f\u00193\u000eand\r\u001911:5\u000e. This distortion pattern is\ndi\u000berent from that occurring in bulk. To explore its ef-\nfects we set \u000b=\f= 3\u000eand consider the consequences\nof varying\r.\nIn bulk La 1\u0000xSrxVO3, while the 4-sublattice Pnma\nstructure implies a di\u000berence in lattice constants, all V-\nO bond lengths are the same.15The di\u000berence in lat-\ntice constants arises from a di\u000berence in tilting pattern.\nSuperlattices are typically grown on a substrate, and in\nepitaxial growth conditions the lattice constants perpen-\ndicular to the growth direction (which we denote here by\na) are \fxed by the substrate, while the lattice parameter\nalong the growth direction ( c) is free to relax. The result\nis ac=aratio typically6= 1 contributed by both tilting\nand anisotropy in V-O bond lengths and possibly varying\nfrom layer to layer of the superlattice. For the experi-\nmentally studied superlattices, c=a\u00181:02.6,12The V-O\nbond lengths have not been determined but, as discussed\nin more detail in the Appendix, our studies indicate that\nall V-O bonds have essentially the same length. Further\nwe show that a few percent di\u000berences have no signi\f-\ncant e\u000bect on our study of ferromagnetism. In the rest\nof the paper we therefore ignore these distortions, setting\nall V-O bond lengths to be equal.\nC. Electronic structure\nWe study superlattices designed to be similar to the\nsystem studied in Ref. 6. In these superlattices, units of\nmlayers of LaVO 3are separated by one layer of SrVO 3.\nTo de\fne the superlattice, we begin from LVO in the\nappropriate bulk structure, then break translation in-\nvariance along the [001] ( z-direction) by replacing every\n(m+1)thLaO plane with an SrO plane. Fig. 2a shows\nsuch a superlattice with m= 3.\nWe assume that the superlattice is grown epitaxially so\nthat in-plane bond lengths and other aspects of the local\nstructure including rotations are the same for all layers.\nWe therefore take the electron transfer integrals which\nde\fne the band structure to the be same for all layers.\nIn this case the electronic structure of a superlattice is\nde\fned by adding the electrostatic potentials of the Sr\nand La ions to the basic translationally invariant hopping\nHamiltonian describing the bulk materials.\nIn our calculations we follow the common practice in\nstudies of early transition metal oxides by assuming that\nthe energy splitting between transition metal d-bands\nand oxygen p-bands is large enough to justify the use\nof a \\frontier orbital\" model focusing on the p-danti-\nbonding bands which are mainly composed of vanadium\nt2g-symmetry d-states.3\nLaSr\nLa\nSrttt'ttt'\n...z(a)\nLaLa\n12\n03\nxy\ndxy pypxdxy\ntt(b)\nFIG. 2: (Color online) (a) Schematic of superlattice lattice\nstructure (LaVO 3)m(SrVO 3)1withm= 3. Vanadium sites\nindicated as circles with charge density indicated by shading:\nheavy shading (black online) indicating higher charge density\nand light shading (yellow online) indicating lower charge den-\nsity. LaO and SrO planes are shown as solid and dashed lines\nrespectively. Nearest neighbor ( t) and next-nearest neighbor\n(t0) hoppings between vanadium sites indicated by arrows.\nThe numbers on the right are VO 2layer indices. (b) Inset:\npd\u0019hopping between t2gorbital and p-orbital. Main panel:\ntwo-dimensional nearest neighbor hopping tmade of two pd\u0019\nhoppings from xyorbital of one vanadium site to oxygen px\norpyorbital, then to xyorbital of another vanadium site.\nThe Hamiltonian for the superlattice is thus\nH=Hkin+Honsite +Hcoulomb; (1)\nwhereHcoulomb describes the electron-ion interaction and\nelectron-electron interaction between di\u000berent sites and\nHonsite describes the d-dinteractions, which we take\nto be on-site. Hkinis a tight binding model, derived\nby using maximally-localized Wannier function (MLWF)\ntechniques19to \ft thet2g-derived antibonding bands.\nThe detailed procedure is described in our previous\nwork.10\nThe kinetic Hamiltonian has the quadratic form\nHkin=X\nk;\u000b;\f;\u001bH\u000b\f\nband(k)cy\nk\u000b\u001bck\f\u001b; (2)\nwherecy\nk\u000b\u001bandck\f\u001bare electron creation and annihi-\nlation operators in reciprocal space with wavevector k.\n\u000band\fare orbital and layer indices, and \u001bis the spin\nindex.\nWe assume that the interaction takes the standard\nSlater-Kanamori form20{22which following Ref. 10 we\nwrite asHonsite =UX\ni\u000bni\u000b\"ni\u000b#+ (U\u00002J)X\ni\u000b6=\fni\u000b\"nj\f#+\n+ (U\u00003J)X\ni;\u000b>\f;\u001bni\u000b\u001bni\f\u001b+\n+JX\ni\u000b6=\f y\ni\u000b\" i\f\" y\ni\f# i\u000b#+\n+JX\ni\u000b6=\f y\ni\u000b\" i\f\" y\ni\u000b# i\f#;\n(3)\nwhere the values of the on-site interaction Uand the\nHund's coupling JareU= 6eV\u001822tandJ= 1eV so\nthat LVO is an insulator in bulk while SVO is a metal.\nIn the approximation employed here, the superlattice\nis de\fned by the Coulomb interaction between the La/Sr\nions and electrons. This, and the o\u000b-site part of the\nelectron-electron interaction is contained23in\nHcoulomb =Hel\u0000ion+Hel\u0000el: (4)\nTo construct Hel\u0000ion, we assume that the whole ion\ncharge of SVO or LVO unit cell comes into the Sr or\nLa site. Consider SrVO 3, the valence of V is +4 ( d1).\nIf this one d-electron is removed, the SVO unit cell will\nhave charge +1, hence, in our model, Sr site has charge\n+1. Similarly, LaVO 3has V+3(d2), thus La site has\ncharge +2. As a result, Hel\u0000ionhas the form\nHel\u0000ion=X\ni;RSr\u0000e2^ni\n4\u0019\u000f\u000f0jRi\u0000RSrj+\n+X\ni;RLa\u00002e2^ni\n4\u0019\u000f\u000f0jRi\u0000RLaj:(5)\nwhereniis electron-occupation operator at V-site i,\u000fis\nthe relative dielectric constant. The part Hel\u0000elis the\ninter-site Coulomb interaction of vanadium d-electrons\nHel\u0000el=1\n2X\ni;j\ni6=je2^ninj\n4\u0019\u000f\u000f0jRi\u0000Rjj: (6)\nHel\u0000elis treated in the Hartree approximation. Note\nthat in Eq. (6), ^ niis the operator giving the total d-\nelectron occupation of site i, whilenj=h^njiis the ex-\npectation value of d-electron occupancy at site j, which\nis determined self consistently. From Hcoulomb , the\nCoulomb potential Vifor siteiis calculated using Ewald\nsummation.24\nThe dielectric constant \u000fis an important parameter in\nEqs. (5, 6). It accounts for screening on the scale of a lat-\ntice constant so bulk measurements are not directly rele-\nvant and an appropriate value has not been determined.\nValues ranging from 4 to 15 have been reported in the\nliterature for similar systems.25,26Because the appropri-\nate value of \u000fhas not been determined, we have studied\nseveral cases and present results mainly for \u000f= 8;15.4\nD. Methods\nWe treat the on-site interaction terms using single-site\ndynamical mean \feld theory (DMFT)11with the hy-\nbridization expansion continuous time quantum Monte\nCarlo (CTQMC) solver.27The superlattice e\u000bect is taken\ninto account by the Coulomb potential ^V. We use the\nsuperlattice dynamical mean \feld theory introduced by\nPottho\u000b and Nolting28,29in the form given in Ref. 23.\nHere each V site ihas a self energy (site local but de-\npendent on site) determined from the solution of a quan-\ntum impurity model which has parameters \fxed by the\nDMFT self-consistency equation linking the site local\nterm of the lattice Green function fgiito the quantum\nimpurity model Green function30\n^Gi\nimp(!) =\u001ah\n(!+\u0016)1\u0000^Hband\u0000^V\u0000^\u0006(!)i\u00001\u001b\nii;\n(7)\nwhere\nVi=X\nj;j6=ie2nj\n4\u0019\u000f\u000f0jRi\u0000Rjj\u0000X\nRSre2\n4\u0019\u000f\u000f0jRi\u0000RSrj\u0000\n\u0000X\nRLa2e2\n4\u0019\u000f\u000f0jRi\u0000RLaj(8)\nis a site dependent quantity, diagonal in spin and or-\nbital indices but linking di\u000berent sites, derived from\nEqs. (5, 6). The layers are coupled by a self-consistency\ncondition which as discussed in Refs. 23,28,29 \fxes\nboth the hybridization function of the quantum impu-\nrity model and the layer-to-layer variation in the charge\ndensity.\nAs described in Ref. 10, it is advantageous to perform a\nsite-local rotation to align the orbital basis to the local V-\nO bond directions of each octahedron before solving the\nimpurity model. This reduces the sign problem in the\nCTQMC impurity solver and restores in-plane transla-\ntion invariance in the sense of making the self-consistency\nequations the same for all sites in a given plane.\nIn a superlattice composed of Nlayers, it is in princi-\nple necessary to solve Ndynamical mean \feld problems,\ncoupled by the self-consistency condition. However, we\n\fnd (see section III) that the susceptibility for a given\nlayer of the superlattice may be determined from a bulk\ncomputation at the same local density and crystal struc-\nture. Because the layer dependent density has no signif-\nicant dependence on the temperature or the many-body\nphysics, it may be determined once from a band structure\ncalculation and then bulk results with the appropriate\ndensity for a wide range of temperature may be used to\ninfer the Curie temperature, substantially reducing the\ncomputational burden.\nThe Curie temperature for ferromagnetism is deter-\nmined by extrapolating the inverse susceptibility \u001f\u00001(T)\nto 0 based on Curie-Weiss law \u001f\u00001\u0018T\u0000Tc. The test\nfor the reliability of this method for Tchas been done in\nRef. 10. A similar approach can be found in literature.31III. RELATION BETWEEN SUPERLATTICE\nAND BULK SYSTEM CALCULATIONS\nIn this section, we demonstrate that the magnetic\nphase diagrams of superlattice systems may be inferred,\nto reasonable accuracy, from the study of appropriately\nchosen bulk systems. This enables a considerable reduc-\ntion in the computation resources required.\n(a) Bulk DOS\n0\n2\n4\n6\n8\n(b) m=3 layer-DOS\nLayer 3\n0\n2\n4\n6\n8\nε\n = 15\nε\n = 5\nLayer 2density of states\n0\n2\n4\n6\n8\nLayer 1\n0\n2\n4\n6\n8\nLayer 0\n0\n2\n4\n6\n8\nenergy (eV)\n−1.5\n−1\n−0.5\n0\n0.5\n1\n1.5\nFIG. 3: (Color online) Panel (a): Non-interacting density\nof states for bulk system at carrier density n= 1. Panels\n(b): Non-interacting density of states for di\u000berent layers of\n(LVO) 3(SVO) 1superlattice for two di\u000berent values of dielec-\ntric constant \u000f= 5 (solid) and \u000f= 15 (dashed) with hopping\nparameters t= 0:264eV and t0= 0:084eV. SrO plane is be-\ntween layers 0 and 1 (the index is de\fned in Fig. 2). The\nFermi energy is at 0.\nWe begin with a study of \\untilted\" or \\cubic\" su-\nperlattices: those in which all V-O-V bond angles are\n180\u000e. We focus speci\fcally on [001] superlattices in which\nthe unit cell contains mlayers LVO and one layer SVO,\nwherem= 3;4;5. For orientation, we present the density\nof states (DOS) of the non-interacting system in Fig. 3.\nIn obtaining these densities of states we used the simple\ntight binding parametrization. The DOS for the bulk sys-\ntem is shown in panel (a). One sees the typical three-fold\ndegenerate DOS for t2gband, the Van Hove singularity\nis visible as a peak near the upper band edge. It is at\nhigh energy because the next-nearest neighbor hopping\nt0>0. The remaining panels show the layer-resolved\ndensities of states for the m= 3 superlattice. The upper\ntwo panels show layers sandwiched by La on both sides;\nthe lower two panels show the layers adjacent to the SrO\nplane. The superlattice-induced changes in the density\nof states are seen to be relatively minor: the main e\u000bects5\nare a weak splitting of the van Hove peaks re\recting the\nbreaking of translational invariance in the z-direction,\nand a relative shift in the positions of the van Hove peaks\narising from band bending associated with the di\u000berent\ncharges of the Sr and La ions.\nFig. 4 shows the layer-resolved charge density and in-\nverse susceptibilityH\nm(H)plotted against temperature\nfor three di\u000berent superlattice structures corresponding\ntom= 3;4;5. As expected from electrostatic consid-\nerations, the charge is lower for the VO 2planes nearer\nthe SrO layer and the charge variation between layers is\ncontrolled by the dielectric constant.\nThe magnetization mat the V sites on each layer was\ncomputed at \feld H= 0:01eV=\u0016Band the inverse sus-\nceptibility was obtained as H=m . Linearity was veri-\n\fed by repeating the computation using H= 0:02eV=\u0016B\n(not shown). For the m= 3;\u000f= 15 case (Fig. 4a),\nwe extended the computation to the lower temperature\nT= 0:03eV; for the other two cases T= 0:06eV was\nthe lowest temperature studied. The inverse susceptibil-\nities are approximately linear in temperature at higher\ntemperatures and in all cases, extrapolation to \u001f\u00001= 0\nrevealsTc<0, implying absence of ferromagnetism.\nEspecially for the layer nearest the SrO plane the \u001f\u00001\ncurves exhibit weak upward curvature at the lowest tem-\nperatures studied. As shown in Ref. 10, the curvature\nis a signature that the system is entering a Fermi-liquid\ncoherence regime. The Fermi liquid coherence tempera-\nture is highest for the layers nearest the SrO because the\ncharge in these planes is farther from the n= 2 Mott\ninsulating state. To verify this we followed Ref. 10 and\ncomputed the Wilson ratio RWfor each layer of the su-\nperlattice for the case m= 3;\u000f= 15, \fnding (not shown)\nthat for each layer the RWextrapolates to 2 at low tem-\nperature. The approach to the low temperature value\nis faster for layers with low density (near SrO planes)\nthan for layers with high density (far from SrO planes).\nRW= 2 is the value for a Kondo lattice, while ferromag-\nnetism is characterized by an RW>2.10We therefore\nbelieve that for \\untilted\" superlattices, the di\u000berences\nin\u001f\u00001among layers arise from di\u000berences in quasipar-\nticle coherence scale, there is no evidence for ferromag-\nnetism in this system, consistent with the solution of the\ncorresponding bulk problem.\nTo gain insight into the physics underlying the layer\ndependence of \u001f\u00001we have computed \u001f\u00001(T) for the\ncubic bulk system ( Hcoulomb = 0,Hkinis constructed\nfrom the two-dimensional dispersion \u000f(k) =\u00002t(coskx+\ncosky)\u00004t0coskxcosky) for carrier densities equal to\nthose on the di\u000berent VO 2layers. In Fig. 4, we present\nbulk calculations for n= 1:62 andn= 1:88 correspond-\ning to the densities calculated for layer 0 and 2 of the\nsuperlattice for all cases m= 3;4;5. Forn= 1:88, bulk\n\u001f\u00001atT= 0:06;0:10 and 0:14eV are very close to those\nofL= 2 layer of m= 3 superlattice, which has the\nsame density. For m= 4;5 superlattices, bulk n= 1:88,\n\u001f\u00001(T) (not shown) almost coincides with those of L= 2\n1.5\n1.7\n1.9\n2\n0\n1\n2\n3\n4\n5\n1.5\n1.7\n1.9\n2\n0\n1\n2\n3\n4\n1.5\n1.7\n1.9\n2\n0\n1\n2\n3\n(a) m = 3\nBulk L0\nBulk L2\nLayer 0\nLayer 2χ-1 (arb. unit)\n0\n0.1\n0.2\n0.3\n0.4\nT (eV)\n0\n0.05\n0.1\n0.15\nBulk L0\nLayer 0\nLayer 2\nLayer 3\n(b) m = 4\nT (eV)\n0\n0.05\n0.1\nBulk L0\nLayer 0\nLayer 2\nLayer 3\n(c) m = 5\nT (eV)\n0\n0.05\n0.1\n0.15FIG. 4: (Color online) Temperature-dependent layer-resolved\ninverse magnetic susceptibilities for symmetry-inequivalent\nlayers of untilted (LVO) m(SVO) 1superlattice structures with\ndi\u000berent numbers of LVO layers m= 3;4 and 5. Layer 0 is\nadjacent to SrO and layers 2 and 3 are between two LaO lay-\ners. The relative dielectric constant is \u000f= 15, magnetic \feld\nH= 0:01eV=\u0016B. The\u001f\u00001(T) obtained from solution of bulk\ncubic systems with charge density set to the density on the\ngiven layer are also shown. \\Bulk L0\" (\\BulkL2\") denotes a\ncalculation performed for a bulk system with density the same\nas forL= 0 (L= 2) layer density. Inset: the electron layer\ndensity distribution corresponding to the susceptibility plot,\nx-axis is the layer index, y-axis is the layer density. On-site\ninteractions U= 6eV,J= 1eV.\nlayer. For bulk n= 1:62, the di\u000berence between bulk\nand superlattice L= 0 layer is small. These calculations\ndemonstrate a general rule: within the single-site DMFT\napproximation, the layer-resolved properties of a super-\nlattice correspond closely to those of the corresponding\nbulk system at a density equal to that of the superlattice.\nThe superlattices of experimental relevance have crys-\ntal structures which are distortions of the \\untilted\" one,\ninvolving in particular a P21=mstructure characterized\nby a rotational distortion of the a\u0000a+c\u0000type17,18involv-\ning a large rotation about an axis approximately paral-\nlel to the growth direction and much smaller rotations\nabout the two perpendicular axes. Fig. 5 compares the\nnon-interacting DOS of bulk and (LVO) 3(SVO) 1super-\nlattice systems (both with the same P21=mstructure)\ncalculated using DFT and a MLWF parametrization of\nthe frontier bands. The DOS of bulk system is shifted so\nthat it has the same carrier density as layers of the su-\nperlattices near SrO plane. For three di\u000berent structures\n(untilted structure and P21=mstructure with \r= 11:5\u000e\nand 16\u000e), the basic features of the partial DOS are simi-6\n1.5\n0.01.5untilted structure\nyz zx xy\n1.5\n0.01.5gamma = 11.5\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5\nenergy (eV)1.5\n0.01.5gamma = 16\nFIG. 5: (Color online) Comparison between bulk LVO partial\nDOS (positive curves) and (LaVO 3)3(SrVO 3)1superlattice\nlayer DOS of layers near SrO (negative curves) derived from\nband structure calculations (DFT+MLWF). Both systems\nhave the same lattice structure for each case: untilted struc-\nture for the top panel and P21=mstructure (Glazer's notation\na\u0000a+c\u0000) with\u000b=\f=a= 3\u000eand\r=c= 11:5\u000eand 16\u000e\nfor other panels. The DOS of bulk system is shifted towards\nhigher energy so that bulk carrier density is the same as layer\ndensity of superlattices for the layers near SrO ( n\u00191:55).\nThe vertical dashed line marks the Fermi level.\nlar between bulk and superlattice. The translation sym-\nmetry breaking in z-direction leads to small extra peaks\nin the superlattice DOS. These di\u000berences are smoothed\nout by the large imaginary part of the DMFT self en-\nergy. Because the DMFT equations depend only on the\ndensity of states it is reasonable to expect that, as in the\nuntilted case, they will therefore give the same results in\nthe superlattice as in the bulk material with correspond-\ning density of states.\nTo verify that this is the case we have also compared\nbulk and superlattice susceptibilities for tilted structures.\nThe four VO 6octahedra in a unit cell are related by ro-\ntation, so an appropriate choice of local basis means that\nonly one calculation needs to be carried out for a given\nlayer. Fig. 6 compares the inverse susceptibilities for an\nm= 3 superlattice to calculations performed on a bulk\nsystem with the same P21=mstructure. In these calcu-\nlations, we choose \r= 11:5\u000e;16\u000e;18\u000eand dielectric con-\nstant\u000f= 8. We see that in this case, as in the \\untilted\"\ncase, the superlattice inverse susceptibilities \u001f\u00001(T) are\nalmost the same as those for bulk system calculated at\nthe same density, with di\u000berences only resolvable in the\nexpanded view for the largest tilt angles.\nFIG. 6: (Color online) Comparison in temperature depen-\ndent inverse susceptibility between bulk LVO (solid lines) and\n(LaVO 3)3(SrVO 3)1superlattice (dashed lines). Both have the\nsame lattice structure P21=mwith tilt angle \r= 11:5;16 and\n18\u000e. Bulk system has the same densities as those of layers\nof superlattice near and far from SrO planes ( n= 1:55;1:95).\nLeft column: the plots in wide temperature range. Right col-\numn: the expanded views near zero temperature.\nIV. SUPERLATTICES WITH GdFeO 3-TYPE\nROTATION\nIn this section we present and explain our results for\nthe magnetic phase diagram of (LVO) m(SVO) 1super-\nlattices with the P21=mstructure (Glazer's notation\na\u0000a+c\u0000) reported for the experimental systems.12,13In\nthese structures in-plane rotation along the growth di-\nrection ^z= [001] is large \r=c= 11:5\u000e(presumably\nbecause of the strain imposed by the substrate), while\nthe out-of-plane rotation is small ( \u000b=\f=a= 3\u000e) per-\nhaps because the system is free to relax along the growth\ndirection. We concentrate on the e\u000bect of the large rota-\ntion by \fxing the in-plane angles to 3\u000ewhile varying the\nout-of-plane angles over a wide range from 10\u000e!18\u000e.\nBased on the results of Section III we generate a phase\ndiagram for the superlattice from calculations for a bulk\nsystem which is a P21=mdistortion of the ideal cubic per-\novskite structure of chemical composition LaVO 3. The\nbulk system results are presented as a phase diagram in\nthe plane of carrier concentration and \r-rotation. Spe-\nci\fc layers of the superlattice will correspond to particu-\nlar points on the phase diagram, with the layer dependent\ndensity \fxed by number of LVO layers mand the dielec-\ntric constant \u000fand the rotation \fxed by the substrate\nlattice parameter.7\n012\ngamma=10xy yz zx\n012\ngamma=11.5\n012\ngamma=13\n012\ngamma=14\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5\nenergy (eV)012\ngamma=16\nFIG. 7: (Color online) Partial DOS derived from\nDFT+MLWF for \\bulk\" P21=mstructure (Glazer's notation\na\u0000a+c\u0000) with\u000b=\f=a= 3\u000eand\r=cchanging from\n10\u000eto 16\u000e. Onlyt2gbands are plotted because egbands are\nnegligible in this range of energy.\nWe use DFT+MLWF methods to obtain the frontier\norbital band structure for the t2g-derived antibonding\nbands. Fig. 7 presents representative results for the or-\nbitally resolved local density of states. In this \fgure the\norbitals are de\fned with respect to the local basis de\fned\nby the 3 V-O bonds of a given VO 6octahedron. We de-\n\fne ^z= [001] as the axis (approximately parallel to the\ngrowth direction) about which the large rotation occurs.\nFig. 7 shows that yzandzxorbitals are almost degen-\nerate, while xyorbital is strikingly di\u000berent. The DOS\nofxyorbital maintains the shape of a two-dimensional\nenergy dispersion with a van Hove peak well above the\nchemical potential, similar to the bulk cubic structure\n(see e.g. Fig. 3a). There are noticeable di\u000berences only\nat very high rotation angles. On the other hand, yzand\nzxorbitals are spread out with two small peaks, because\nhoppings along xorydirections (more distorted) are\ndi\u000berent from those along z-direction (less distorted).\nWhen the distortion gets larger, the d-bandwidth be-\ncomes smaller, the xypeak gets larger and slightly closer\nto the Fermi level, and yzandzxpeaks near the Fermi\nlevel also develop.\nBased on ^Hband(k) generated by DFT+MLWF, we\ncarry out DMFT calculations for in-plane rotation an-\ngle\rto get\u001f\u00001curves whose extrapolations de\fne the\nCurie temperatures Tc. Fig. 8 shows how Tcevolves when\nthe rotation angle \rincreases from 10 to 18\u000e. In this \fg-\nure, we consider two di\u000berent carrier densities n= 1:55\nand 1:95, corresponding to the band structure prediction\nfor the layer densities of layers near and far from SrO\nplanes in the superlattice. Tcforn= 1:95 is a slow func-\ntion of rotation and is always negative for the range of\n\runder consideration, while Tcforn= 1:55 increasesfaster, so that the system becomes ferromagnetic when\n\ris between 14 and 16\u000e. Ferromagnetism is therefore\nexpected only in superlattices with very large rotations,\nand then only in the layers with large hole doping (i.e.\nthe layers closest to the SrO planes).\nFIG. 8: (Color online) Inverse susceptibility \u001f\u00001vs. tem-\nperatureTfor bulkP21=mstructure of LaVO 3at densities\nn= 1:55 (black circle solid lines) and n= 1:95 (red diamond\ndashed lines) for rotation angle \rincreasing from 10 !18\u000e.\nOn-site interaction U= 6eV and J= 1eV. Left column: the\ncircles and diamonds are data points, the solid and dashed\nlines are \ftted from these data points. Right column: ex-\npanded view at small \u001f\u00001region. The vertical dashed line\nmarks zero temperature.\nFrom a range of calculations such as those shown in\nFig. 8 we have constructed the superlattice magnetic\nphase diagram shown in Fig. 9. Similar to Ref. 10, there\nare uncertainties in our extrapolation for Curie tempera-\nture, we consider 0 :004eV as the error bar for positions on\nthe phase diagram. Thus, Tc<0:004eV is considered as\nTc= 0 within the error bar. We see that ferromagnetism\nis favored only for very large rotations, much larger than\nthe 11\u000edetermined experimentally, and only for carrier\nconcentrations far removed from n= 2. We may locate\nthe experimentally studied superlattices on this phase\ndiagram. For an m= 3 superlattice, band structure cal-\nculations indicate layer densities 1 :55 for layers near SrO8\nplane and 1 :95 for the other layers. The experimentally\ndetermined rotation angle is \u001811:5\u000e. These two points\nare indicated by squares in Fig. 9.\n1.5 1.6 1.7 1.8 1.9 2.0\nfilling81012141618 tilt angle (Degree)Bulk LVO boundary\nPMFM\n(La/Sr)VO3 solid solution\nPossible superlattice points\n0.0040.0080.0120.0160.0200.0240.0280.032\nFIG. 9: (Color online) The magnetic phase diagram with x-\naxis carrier density nandy-axis tilt and rotation angle along\n^z= [001] direction \rfor bulk system LVO with the same\ntype of distortion as for (LVO) m(SVO) 1superlattices ( P21=m\nstructure), in-plane tilt angles \u000b;\f\u00193\u000e. On-site interactions\nU= 6eV,J= 1eV. The white regime indicates absence of\nferromagnetism ( Tc<0:004eV), the colored regime indicates\nferromagnetism with Tcindicated by the color bar. Also in-\ndicated are results for bulk La 1\u0000xSrxVO 3in thePnma struc-\nture, from Ref. 10. Note that in the calculations for the Pnma\nstructure all three tilt angles are almost the same.\nIt is interesting to compare our results to those previ-\nously obtained10for the bulk solid solution La 1\u0000xSrxVO3\n(Pnma structure). The dashed line in Fig. 9 shows the\ntheoretically estimated phase diagram for the bulk solid\nsolution. We see that the bulk structure is more favorable\nfor ferromagnetism than the superlattice structure. An\nimportant di\u000berence between the Pnma structure and\ntheP21=mof the superlattice is that in the former case\nall three tilt angles are of comparable magnitude whereas\nin theP21=mstructure only one rotation is large. We be-\nlieve that this di\u000berence is responsible for the di\u000berence\nin phase boundary.\nV. CONCLUSIONS\nIn this paper, we have studied the possibility of ferro-\nmagnetism in superlattice structures of vanadium oxides\nderived from LaVO 3and SrVO 3. Our investigation was\nbased on the idea that ferromagnetism depends on an in-\nterplay between carrier density and octahedral rotation,\nand while these are coupled in bulk (see the solid solution\ncurve in Fig. 9) they may be decoupled in the superlat-\ntice. In particular, the charge density varies across the\nsuperlattice, being lowest near the SrO planes, while therotation angle is controlled by the substrate. Thus in\nan appropriately designed superlattice at least some por-\ntions of the system might be moved closer to (or perhaps\ninto) the ferromagnetic region. In several important as-\npects this idea is consistent with calculations. We \fnd\nthat the local carrier density determines the local mag-\nnetic susceptibility (see section III) and the density/tilt\nangle relationship may be signi\fcantly altered (see solid\nline and square points in Fig. 9).\nHowever, we \fnd that the P21=moctahedral rota-\ntion pattern characteristic of experimentally discovered\nsuperlattices is in fact less favorable to ferromagnetism\nthan thePnma pattern characteristic of bulk materials\n(compare the phase boundaries in Fig. 9). Thus while\nthe general idea that an appropriately designed super-\nlattice might provide conditions favorable for ferromag-\nnetism thereby providing a potential explanation for the\nremarkable experimental report of room-temperature fer-\nromagnetism in (LaVO 3)m(SrVO 3)1superlattices with\nm= 3;4;5;6 by L uders et. al.,6(even though there is no\nferromagnetism in the bulk solid solution), our detailed\n\fndings are not consistent with the experimental result.\nOur results indicate that designing ferromagnetism\ninto a vanadate superlattice will require both large ampli-\ntude rotations about the growth axis and also substantial\nrotations about the other two axes. Rotations about the\ngrowth axis arise from substrate-induced strain, so choos-\ning substrates with smaller lattice parameter would be\ndesirable. Introduction of rotations about the orthogo-\nnal axes may be done by replacing the La with a smaller\ncounterion such as Y.\nOur study has certain limitations. The calculations\nemploy a frontier orbital model which includes only the\nt2g-derived antibonding bands. DFT+DMFT calcula-\ntions based on correlated atomic-like d-states embedded\nin the manifold of non-correlated oxygen states provide\na more fundamental description. Our previous work10\nindicates that the two models give very similar results\nif both calculations are tuned so that bulk LaVO 3is\na Mott insulator, but the implications of the full (but\ncomputationally very heavy) DFT+DMFT procedure for\nthe superlattice problem remain an open problem for fu-\nture research. Further, our calculations are based on the\nsingle-site DMFT approximation, which includes all local\ne\u000bects but misses inter-site correlations. While it is gen-\nerally accepted that these calculations give the correct\ntrends and qualitative behavior, the quantitative accu-\nracy of the methods is not known. Unfortunately, as yet\ncluster extensions of DMFT are prohibitively expensive\nfor the multiband models considered here.\nThe experimental results of L uders et. al.6there-\nfore provide an interesting challenge to materials the-\nory. They indicate that superlattices display ferromag-\nnetism when the corresponding bulk solid solutions do\nnot, whereas the present state of the art of real materials\ndynamical mean \feld calculations suggests that super-\nlattices should be less likely to display magnetism than\nthe corresponding bulk solid solutions. This discrepancy9\nrequires further investigation.\nAcknowledgements\nWe thank U. L uders and J. Okamoto for helpful conver-\nsations. We acknowledge support from DOE-ER046169.\nHTD acknowledges partial support from Vietnam Educa-\ntion Foundation (VEF). We acknowledge travel support\nfrom the Columbia-Sorbonne-Science-Po Ecole Polytech-\nnique Alliance Program and thank Ecole Polytechnique\n(HTD and AJM) and J ulich Forschungszentrum (HTD)\nfor hospitality while portions of this work were con-\nducted. A portion of this research was conducted at the\nCenter for Nanophase Materials Sciences, which is spon-\nsored at Oak Ridge National Laboratory by the Scienti\fc\nUser Facilities Division, O\u000ece of Basic Energy Sciences,\nU.S. Department of Energy. We use the code for CT-\nHYB solver27written by P. Werner and E. Gull, based\non the ALPS library.32\nAppendix: Lattice constant and V-O bond length\nratio\nIn this appendix, we present a more complete discus-\nsion of the strain-induced lattice distortions. The in-\nplane lattice constant of a superlattice epitaxially grown\non a substrate matches that of the substrate and may\ntherefore be di\u000berent from the lattice constant preferred\nin a free-standing \flm or bulk material. The out-of-plane\nlattice constant is typically free to relax, and in the pres-\nence of an in-plane strain may also be di\u000berent from that\nfound in bulk materials.\nA di\u000berence in V-V distance may arise from a change\nin V-O bond length or from a di\u000berence in buckling of\nV-O bonds. We consider both possibilities here, but \frst\nremark that the main di\u000berences in structure between\nbulk and experimentally studied superlattices arise from\ndi\u000berences in octahedral rotation. In the experimentally-\nstudied superlattices, the in-plane V-V distance is in fact\nslightly less than the V-V distance in LVO. The V-O bond\nlengths have not been measured for the superlattice, but\nto a high degree of accuracy we are able to reconstruct\nthe measured superlattice using the measured tilt an-\ngles given from experiments13structure, assuming that\nall V-O bond lengths are equal. Assuming the P21=m\nstructure, we varied the in-plane and out-of-plane V-O\nbond lengths to \ft the experimental data and found that\nc=a\u00191:02 only when the mean bond length dis found\nin the range from 1 :983 to 2 \u0017A depending on which ex-\nperimental result is \ft but in all cases the V-O bond\nlengths are found to be equal to within an accuracy\nof 0:3%. Therefore, we believe that all the V-O bond\nlengths should, to a good approximation, be the same.\nThe structure used in our calculations is presented in Ta-\nble. I. Although there are slight mismatches in in-plane\nangle and lattice constants, the c=a\u00191:02 ratio andbond angles are compatible with the experiment.\n0.00.51.01.5\n(a)xy zx (yz)\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5\nenergy (eV)012\n(b)\nFIG. 10: (Color online) Partial DOS for bulk LVO with c=a=\n1:02 with the c=aratio due to a change in V-O bonds (panel\n(a)) and toP21=mlattice structure with \u000b=\f= 0;\r= 11:5\u000e\n(similar to superlattice structure)(panel (b)). The dashed\nblue curve is the xyorbital, the solid red curve is the de-\ngenerateyz(orzx) orbital. The dashed vertical line marks\nthe Fermi level.\nChanging the amount of rotation has a di\u000berent e\u000bect\non the electronic structure than does changing the ra-\ntio of V-O bond lengths. Fig. 10 compares the partial\nDOS for the two cases, using as example a hypotheti-\ncal LaVO 3crystal with c=a= 1:02. The upper panel\npresents the DOS for the untilted structure with straight\nV-O-V bonds and the c=aratio induced by a di\u000berence in\nin-plane and out-of-plane V-O bond lengths. The lower\npanel presents the case of all equal V-O bonds, with the\nc=aratio produced by octahedral rotations about the z\naxis. The densities of states are quite di\u000berent, but can\nbe understood from the simple energy dispersion\n\u000fxy(k) = 2tk(2\u0000coskx\u0000cosky)+\n+ 4t0\nk(1\u0000coskxcosky);\n\u000fxz(k) = 2tk(1\u0000coskx)+\n+ 2t?(1\u0000coskz) + 4t0\n?(1\u0000coskxcoskz);\n(A.1)\nwheretkandt?are the in-plane and out-of-plane near-\nest neighbor hopping integrals and t0\nk;?are the second\nneighbor hoppings. The lower band edge is assumed to\nbe the same for all orbitals but we assume that the lat-\ntice distortions lead to di\u000berent values for the in-plane\nand out of plane hoppings.\nThe lower band edge is de\fned to be zero and is inde-\npendent of the distortion. The energy of the upper edge\nof thexyband is\u000fxy(\u0019;\u0019) = 8tkand of thexz=yz bands\nis\u000fxz(\u0019;\u0019) = 4tk+ 4t?. The positions of the van Hove\nsingularities are at k= (0;\u0019) or (\u0019;0). Forxyband there\nis only one van Hove peak, at \u000fV= 4tk+ 8t0; while for10\nTABLE I: Wycko\u000b positions and lattice constants for LaVO 3withP21=mstructure from our calculation based on tilt angles\ntaken from the experiment (Ref. 13). La positions are \fxed manually but do not a\u000bect lattice constants or c=aratio. In\nour notation, dis the V-O bond length, a0;b0andc0are lattice constants, \f0is the angle between a0andc0,aandcare\npseudocubic lattice constants (growth direction is along cdirection,ais perpendicular to c).\nAtom x y z Atom x y z\nLa(1)0 0.25 0 La(2)0.5 0.25 0.5\nV(1)0.5 0 0 V(2)0 0 0.5\nO(1)\n1 0.4662 0.25 0.0660 O(2)\n1 0.0392 0.25 0.4392\nO(1)\n2 0.7638 -0.0138 0.2362 O(2)\n2 0.2652 -0.0493 0.2652\na0(\u0017A)b0(\u0017A)c0(\u0017A) \f0 a(\u0017A)c(\u0017A)c=aratio\nexp. LVO thin \flm135.55 7.82 5.55 89 :489\u000e3.91 3.945 1.008\nexp. superlattice6,12NA NA NA NA 3.88 3.95 1.018\ncalculated with d= 2\u0017A 5.5988 7.8290 5.5821 88 :9732\u000e3.915 3.988 1.019\ncalculated with d= 1:983\u0017A 5.5512 7.7623 5.5346 88 :9732\u000e3.881 3.954 1.019\nχ-1\n0\n0.01\n0.02\n0.03\nT (eV)\n−0.15\n−0.1\n−0.05\n0\nc/a=0.98, N\nd\n=1.55\nc/a=0.98, N\nd\n=1.95\nc/a=1.00, N\nd\n=1.55\nc/a=1.00, N\nd\n=1.95\nc/a=1.02, N\nd\n=1.55\nc/a=1.02, N\nd\n=1.95inverse susceptibility χ -1\n0\n0.05\n0.1\n0.15\ntemperature T (eV)\n−0.2\n−0.1\n0\n0.1\n0.2\n0.3\n0.4\nFIG. 11: (Color online) Inverse susceptibility vs. tempera-\nture for cubic structure of bulk hole-doped LVO. The in-plane\nand out-of-plane bondlengths are changed so that the octahe-\ndral volume is unchanged: tensile strain ( c=a= 0:98 - black\nlines), no strain ( c=a= 1:00 - red lines) and compressive strain\n(c=a= 1:02 - blue lines). Two levels of hole doping are con-\nsidered:n= 1:55 (solid lines) and n= 1:95 (dashed lines).\nThese lines are linear \fts for the data points.\nxzband, there are two van Hove peaks at \u000fV\n1=\u000fVand\n\u000fV\n2= 4t?+ 8t0. Whent?is di\u000berent from tk, the di\u000ber-\nence in bandwidth of xyandzxorbitals is 4 tk\u00004t?,\nwhich is also the distance between the two van Hovepeaks ofzxband\u000fV\n1\u0000\u000fV\n2.\nWith these de\fnitions, we are in a position to under-\nstand the changes in the band structure. When the V-\nO bond lengths change (Fig. 10a) so that the z-bond is\nlonger and the in-plane bond is shorter but the octahe-\ndral volume is unchanged, the band structure calculation\nindicates that t?decreases but tkincreases slightly. The\ndi\u000berence between the bandwidth of the xyandxzband-\nwidths is 4 tk\u00004t?which is the same as the splitting\nbetween the van Hove peaks in the xz=yz bands. On\nthe other hand, if the c=aratio is produced by rotation,\n(Fig. 10b), the change is opposite. The in-plane hop-\npingtkdecreases because of the buckled in-plane V-O-V\nbonds, while the out-of-plane hopping t?is unchanged.\nThexyband therefore narrows substantially relative to\nthexz=yz bands. In addition the splitting of the van\nHove peaks is greater. From the bandwidth of xyand\nzxbands (Fig. 10b), tk\u00190:225eV,t?\u00190:35eV, the van\nHove peak distance is \u00190:5eV, which is compatible with\nthe peak positions shown in Fig. 10b.\nWe tested with DMFT calculations for the Curie tem-\nperatures with the V-O bondlength changed. Fig. 11 is\nthe temperature-dependent inverse susceptibility derived\nfrom DMFT for the bulk cubic structure with the c=a\nratio changing from 0.98 (tensile strain) to 1.02 (com-\npressive strain). For all the levels of hole doping under\nconsideration, the results are nearly the same for every\ncase ofc=aratio. We conclude that even when the V-O\nbondlength changes within the physical range, the ferro-\nmagnetism is not a\u000bected. However, we also found that\nwhen the V-O bondlength is such that c=a\u00151:06 or\n\u00140:90, there is large orbital polarization and the fer-\nromagnetism can be largely a\u000bected. But that range is\nunphysical and can be neglected in the context of this\nwork.11\n1M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys.\n70, 1039 (1998).\n2A. J. Millis, Nature Physics 7, 749 (2011).\n3A. Ohtomo and H. Y. Hwang, Nature 427, 423 (2004),\nISSN 0028-0836.\n4K. Yoshimatsu, T. Okabe, H. Kumigashira, S. Okamoto,\nS. Aizaki, A. Fujimori, and M. Oshima, Phys. Rev. Lett.\n104, 147601 (2010).\n5N. Reyren, S. Thiel, A. D. Caviglia, L. F. Kourkoutis,\nG. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S.\nRuetschi, D. Jaccard, et al., Science 317, 1196 (2007).\n6U. L uders, W. C. Sheets, A. David, W. Prellier, and\nR. Fr\u0013 esard, Phys. Rev. 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B\n67, 165408 (2003).\n30Our codes for solving the DMFT self consistent equa-\ntion can be downloaded from http://phys.columbia.edu/\n~hungdt/codes/ .\n31U. Yu, K. Byczuk, and D. Vollhardt, Phys. Rev. B 78,\n205118 (2008).\n32A. Albuquerque, F. Alet, P. Corboz, P. Dayal, A. Feiguin,\nS. Fuchs, L. Gamper, E. Gull, S. G urtler, A. Honecker,\net al., J. Magn. Magn. Mater. 310, 1187 (2007)."    },    {        "title": "0704.3125v1.Theory_of_the_tunneling_spectroscopy_of_ferromagnetic_superconductors.pdf",        "content": "arXiv:0704.3125v1  [cond-mat.supr-con]  24 Apr 2007Theory of the tunneling spectroscopy of ferromagnetic supe rconductors\nT. Yokoyama and Y. Tanaka\nDepartment of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan\nand CREST, Japan Science and Technology Corporation (JST) N agoya, 464-8603, Japan\n(Dated: August 15, 2021)\nWe study tunneling conductance in normal metal / insulator / ferromagnetic superconductor\njunctions. The tunneling spectra show a clear diﬀerence bet ween spin-singlet s-wave pairing, spin-\ntriplet opposite spin pairing and spin-triplet equal spin p airing: These pairings exhibit, respectively,\ngap struture, double peak structure and zero bias peak in the spectra. The obtained result may\nserve as a tool for determining the pairing symmetry of ferro magnetic superconductors.\nMagnetism and superconductivity have been under in-\ntensive pursuit in the ﬁeld of low temperature physics.\nRecently the interplay of them has also attracted much\nattention because nontrivial phenomena are predicted or\nfound experimentally. Such phenomena are expected to\noccur in ferromanget/superconductor junctions1,2,3and\nalso in ferromangnetic superconductors (FS). Up to now,\nseveralbulk materials, e.g., UGe 24, ZrZn 25and URhGe6,\nare identiﬁed as FS. How Cooper pairs are formed in\nFS or under the coexistence of ferromagnetism and su-\nperconductivity is an interesting problem. However the\npairing symmetries of FS are still controversial.\nFerromagnetic superconductors seem to be triplet su-\nperconductors because singlet pairing and ferromag-\nnetism are antagonist while triplet pairing have a uni-\nform magnetic moment. However the possibility of s-\nwave pairing cannot be excluded.7,8,9,10,11,12,13,14For ex-\nample it is predicted that UGe 2can haves-wave super-\nconductivity mediated by local ferromagnetic spins.10,11\nThe study of the nuclear relaxation rate cannot rule out\nthe possibility of s-wave pairing in UGe 2.12,13A weak\nferromagnetic Fermi liquid theory also suggests the pos-\nsibility ofs-wave superconductivity.14Therefore detailed\ncomparison between theoretical predictions and exper-\nimantal data is required to settle this problem. Then\nthe properties of thermodynamic quantities should be\nnoted: For example equilibrium thermodynamic quanti-\nties for Balian-Werthamer state of p-wave pairing, which\nis realized in B phase of3He, are expected to show s-\nwave property because its gap is constant.15In this way,\nequilibrium thermodynamic quantities for p-wavepairing\ncould not be clearly distinguished from those of s-wave\npairing. Therefore nonequilibrium quantities are more\ndesirable to compare with experimental data. Although\nsome predictions are made on the properties of junctions\nwith equal spin pairing(ESP) FS,16,17,18,19the study of\ntunneling spectra for possible candidate pairings of FS is\ninsuﬃcient.\nTunneling spectroscopy provides an important infor-\nmation on the superconducting gap and its pairing sym-\nmetry. In normal metal / supercunductor junctions,\nAndreev reﬂection (AR)20is a key concept for low en-\nergy transport. Blonder, Tinkham and Klapwijk (BTK)\nformulated the tunneling conductance where the AR is\ntaken into account21. This enables us to study the en-\nergy gap of superconductors. The generalization of theBTK formula for normal metal / unconventional super-\nconductor junctions are also useful to study the prop-\nerties of unconventional superconductors22,23,24because\nthetunneling conductanceissensitivetothe pairingsym-\nmetry due to the formation of midgap Andreev resonant\nstates22,23.\nIn the present paper we study the tunneling conduc-\ntance in normal metal / insulator / ferromangnetic su-\nperconductor (N/FS) junctions. The tunneling spectra\nshow a clear diﬀerence between spin-singlet s-wave pair-\ning, spin-triplet opposite spin pairing(OSP) and spin-\ntriplet equal spin pairing(ESP). This result may be use-\nful in determining the pairing symmetry of ferromagnetic\nsuperconductors.\nLet us start with an eﬀective Hamiltonian for the\nBogoliubov-de Gennes (BdG) equation. The Hamilto-\nnian reads\nˇH=/parenleftbiggˆH(k)ˆ∆(k)\n−ˆ∆∗(−k)−ˆH∗(−k)/parenrightbigg\n(1)\nwithˆH(k) =ξk+h·σ, andˆ∆(k) =i∆σyfor singlet\npairing or ˆ∆(k) = (d(k)·σ)iσyfor triplet pairing. Here\nξk,k,handσdenote electron band energy measured\nfromtheFermienergy,electronmomentum, appliedmag-\nnetic ﬁeld and Pauli matrices respectively. In this paper\nwe consider three types of pairings: spin-singlet s-wave\npairing, spin-triplet OSP and spin-triplet ESP. OSP and\nESP are characterized by the relations h×d(k) = 0 and\nh·d(k) = 0 respectively.25\nWeconsideratwodimensionalballisticN/FSjunctions\nat zero temperature. The N/FS interface located at x=\n0 (along the y-axis) has an inﬁnitely narrow insulating\nbarrier described by the delta function U(x) =Uδ(x).\nWe ﬁrst consider OSP. The BdG equation reads\nˇH/parenleftbigg\nˆu±\nˆv±/parenrightbigg\n=E±/parenleftbigg\nˆu±\nˆv±/parenrightbigg\n(2)\nfor electron-like quasiparticles, and\nˇH/parenleftbigg\nσyˆv±σy\nσyˆu±σy/parenrightbigg\n=−E±/parenleftbigg\nσyˆv±σy\nσyˆu±σy/parenrightbigg\n(3)\nfor hole-like quasiparticles, with\nE±=/radicalig\n(ξk)2+|∆|2±|h|, (4)2\nˆu±=u±\n0/parenleftig\n1±ˆh·σ/parenrightig\n/2, (5)\nˆv±=v±\n0ˆ∆†\n|∆|/parenleftig\n1±ˆh·σ/parenrightig\n/2, (6)\nu±\n0=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\n1+/radicalig\nE2\n±∓h−|∆2|\nE±∓h\n,(7)\nv±\n0=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\n1−/radicalig\nE2\n±∓h−|∆2|\nE±∓h\n,(8)\nˆh=h/|h|and|∆|2=1\n2Trˆ∆ˆ∆†.We assume ∆ < h\nbecause otherwise the gap vanishes for the ”-” state as\ncan be seen in Eq. (4). The solution ofthe BdG equation\nfors-wavepairinghavethesameformasthat ofOSPand\nobtained by choosing ˆ∆(k) =i∆σy. Below we consider\nunitary state for triplet superconductors and choose, as a\nmodel calculation, h=−hˆ z,d(k) = ∆(kx+iky)/kˆ zfor\nOSP, and d(k) = ∆(kx+iky)/kˆ xfor ESP. Here ˆ xand\nˆ zare unit vectors oriented to x- andz-axis respectively.For ESP, eigenfunctions for the Hamiltonian are given by\n\nu−\n0\n0\n−v−\n0e−iθ\n0\n,\nv−\n0\n0\n−u−\n0e−iθ\n0\n,\n0\nu+\n0\n0\nv+\n0e−iθ\n,\n0\nv+\n0\n0\nu+\n0e−iθ\n,\n(9)\nu±\n0=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\n1+/radicalig\nE2\n±−|∆2|\nE±\n, (10)\nv±\n0=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\n1−/radicalig\nE2\n±−|∆2|\nE±\n. (11)\nE±=/radicalig\n(ξk±|h|)2+|∆|2(12)\nwhereθis an angle with respect to the interface normal\nin the N region. Note that the magnitude of hcan be\nlarger than that of ∆ for ESP because Cooper pairs are\ninsensitive to the exchange ﬁeld.\nWe will calculate the tunnling condunctance, following\nthe BTK method.21,22Wave function ψ(x) forx≤0 (N\nregion) is represented as\nψ(x≤0) =\n\n1\n0\n0\n0\neikFcosθx+a\n0\n0\n0\n1\neikFcosθx+b\n1\n0\n0\n0\ne−ikFcosθx\neikFsinθx(13)\nfor an injection wave in up spin state, for s-wave pairing\nand OSP.ais AR coeﬃcient and bis normal reﬂection\n(NR) coeﬃcient. Foraninjectionwaveindownspinstate\nand the junction with ESP, wave functions are given in\na similar form.\nSimilarly for x≥0 (FS region) ψ(x) is given by the\nlinear conbination of the eigenfunctions. Note that since\nthe translational symmetry holds for the y-direction, the\nmomenta parallel to the interface are conserved.\nThe wave function follows the boundary conditions:\nψ(+0) =ψ(−0),(14)\n∂\n∂xψ(+0)−∂\n∂xψ(−0) =2mU\n¯h2ψ(+0).(15)\nApplying BTK theory with AR and NR coeﬃcients for\nelectron injections with up and down spin states, we can\ncalculate the angle-resolved dimensionless conductance\nfor OSP represented in the form:\nσSσ=4(4+Z2\nθ)+16|Γp\nσ|4−4Z2\nθ|Γp\nσΓm\nσ|2\n|4+Z2\nθ−Z2\nθΓp\nσΓmσ|2,(16)Γp\nσ= Γσe−iθ,Γm\nσ=−Γσe−iθ,(17)\nΓσ=∆\nE+σh+/radicalig\n(E+σh)2−|∆|2,(18)\nσ=±,Zθ=Z\ncosθ,Z=2mU\n¯h2kFwith quasiparticle energy\nE≡E+=E−, eﬀective mass m, Fermi wavenumber\nkFand Fermi energy EF. Fors-wave pairing, the con-\nductance is given by just replacing Γp\nσand Γp\nσwith Γ σ\nin Eq.(16). We deﬁne σNσas the conductance in the\nnormal state which is given by\nσNσ=4\n4+Z2\nθ. (19)\nThe normalized conductance is represented as\nσT=/integraltextπ\n2\n−π\n2dθcosθ(σS++σS−)\n/integraltextπ\n2\n−π\n2dθcosθ(σN++σN−).(20)\nFor ESP, the conductances are given by\nσSσ= 4λσ×3\n/braceleftbig\nZ2\nθ+(λσ+1)2/bracerightbig\n+4λσ|Γp|2−/braceleftbig\nZ2\nθ+(λσ−1)2/bracerightbig\n|ΓpΓm|2\n|(λσ+1)2+Z2\nθ−{Z2\nθ+(λσ−1)2}ΓpΓm|2,\n(21)\nΓp= Γe−iθ,Γm=−Γe−iθ, (22)\nΓ =∆\nE+/radicalig\nE2−|∆|2, (23)\nσNσ=4λσ\n(1+λσ)2+Z2\nθ,λσ=/radicalbigg\n1−σh\nEFcos2θ.(24)\nNote that Θ( θC−|θ|) have to be multiplied for σ= +\n(minorityspin)inEq.(20)with θC= cos−1/radicalig\nU\nEFbecause\nof the mismatch of Fermi surfaces of majorityand minor-\nity spins.26Here Θ(x) is the Heaviside step function.\nIn the above we choose the same eﬀective mass in N\nand FS. In most cases the eﬀective mass in N is much\nsmaller than that in FS. However it is expected that\nthis eﬀect does not change the results qualitatively for\nlargeZ(Z >1).27Thereforewechoosethe sameeﬀective\nmass. The inclusion of the diﬀerence of eﬀective masses\nis straightforward.27,28Although it is known that other\ncharacteristics, e.g., the shape of Fermi surfaces should\nbe taken into account in some phase-sensitive tests,29we\nuse a cylindrical Fermi surface in this paper for simplic-\nity because Fermi surface of FS has very complicated\nstructure.30,31,32\nWe study the normalized tunneling conduntace σTas\na function of bias voltage V. The conductances with\nZ= 10 are shown in Figs. 1(a), 1(b), and 1(c) for s-\nwave pairing, OSP and ESP respectively. For s-wave\npairing, a gap-like sturcture appears at h= 0.21With\nthe increase of h, the magnitude of the gap is reduced\nfrom 2∆ to 2∆ −2h(Fig. 1 (a)). For OSP, a zero bias\npeakappearsat h= 0asshownin Fig. 1(b), whichstems\nfrom the formation of midgap Andreev resonantstates.22\nWe ﬁnd a splitting of peak for OSP as hincreases. These\nshifted structures are attributed to the hdependence of\nwave function in Eqs.(7) and (8), and hence expected to\nemerge for all OSP (not restricted to the present choice\nofd(k)). On the other hand, the tunneling conductance\nhas a zero bias peak and is almost independent of the\nexchange ﬁeld for ESP as shown in Fig.1 (c). This is\nbecause there is no energy shift in the eigenfunctions as\nshown in Eqs.(10) and (11). We also ﬁnd that the ab-\nsence of the shifted structure is expected for all ESP by\ncalculating the eigenfunctions of the Hamiltonian with\nESP. Therefore a clear diﬀerence between three types of\nparings can be seen. Especially when the magnitude of\nthe gap ∆ is comparable to h, the tunneling spectra are\ncharacterized by gap struture, double peak structure and\nzero bias peak for s-wave pairing, OSP and ESP respec-\ntively.\nA corresponding plot for Z= 1 is shown in Fig. 2.\nAs shown in Fig. 2(a), the reduced dip structure ap-\npears fors-wave pairing, the width of which is given by\n\u0001 \u0000 \u0002 \u0003\n\u0004\u0005\n\u0006\u0007\n\b \t\n\n\u000b \f\n\r\u000e \u000f\n\u0010\n\u0011 \u0012 \u0013\u0014 \u0015 \u0016 \u0017 \u0018 \u0019 \u001a\n\u001b \u001c \u001d \u001e \u001f  !\"\n#\n$%\n&\n'(\n)*\n+ ,-\n./\n0 1 2 34\n56\n7 8 9 :;\n<=\n> ? @ ABC D E\nF\nG\nHI\nJ KL\nM NO\nP QR\nS T U VW\nX YZ\n[ \\ ] ^\n_\n`\nab\nc\nde\nf\ngh\ni\njk\nlm\nn op\nqr\ns t u vw\nxy\nz { | }~\n\n   \n\n\n\n\n\n \n\n   \n\n   \n\n¡ ¢ £ ¤¥¦ § ¨\n©\nª\n«¬\n ®¯\n° ±²\n³ ´µ\n¶ · ¸ ¹º\n» ¼½\n¾ ¿ À ÁÂÃ Ä Å\nÆ\nÇ\nÈÉ\nÊ ËÌ\nÍ ÎÏ\nÐ ÑÒ\nÓ Ô Õ Ö×\nØ ÙÚ\nÛ Ü Ý Þ\nFIG.1: (color online) Normalized tunnelingconductancewi th\nZ= 10 for (a) s-wave pairing, (b)OSP and (c)ESP.\n2∆−2h. When ∆ ∼h, the dip transforms into a single\npeak. As for OSP, a zero bias peak is formed and its\nwidth is reduced by the increase of h(see Fig. 2(b)). A\nzero bias peak remains with the increase of hfor ESP as\nshown in Fig. 2(c). Thus there is no qualitative diﬀer-\nence between OSP and ESP. This is because the eﬀect of\nmidgap Andreev resonant states becomes weak for small\nZand hence the zero bias anomaly is smeared for small\nZ. Therefore we ﬁnd that the diﬀerence between for s-\nwave pairing, OSP and ESP becomes clear for large Z.\nIn summary we have studied the tunneling conduc-\ntance in normal metal / insulator / ferromagnetic super-\nconductor junctions. We have found a clear diﬀerence\nin tunneling spectra between spin-singlet s-wave pairing,\nspin-triplet OSP and spin-triplet ESP. The diﬀerence is\nclearforlargebarrierparameter Z. Thisresultmayserve\nas a tool for determining the pairing symmetry of ferro-\nmagnetic superconductors.\nT. Y. acknowledges support by the JSPS. This work\nwas supported by NAREGI Nanoscience Project, the\nMinistry of Education, Culture, Sports, Science and\nTechnology, Japan, the Core Research for Evolutional\nScience and Technology (CREST) of the Japan Science\nand Technology Corporation (JST) and a Grant-in-Aid4ß\nà\ná\nâã\nä åæ\nçè\né ê ë ìí\nîï\nð ñ ò óô õ ö\n÷ ø ù\nú û ü\ný þ ÿ\u0001\n\u0003\u0002\n\u0004 \u0000\u0005\n\u0006\u0007\n\b \t \n \u000b\f\n\r\u000e\n\u000f \u0010 \u0011 \u0012\u0013\n\u0014\u0015\n\u0016 \u0017 \u0018 \u0019\u001a \u001b \u001c \u001d\n\u001e\n\u001f  !\n\" # $\n% & '\n( ) *\n+,\n- ./\n0 12\n3 45\n6 7 8 9:\n; <=\n> ? @ A\nB\nC\nD\nEF\nG HI\nJK\nL M N OP\nQR\nS T U VW\nX\nY\nZ[\n\\ ]^\n_`\na b c de\nfg\nh i j kl m n\no p q\nr s t\nu v wx\nyz\n{ |}\n~\n   \n\n   \n\n   \n  \n  \n  \n    \n¡ ¢ £\n¤ ¥ ¦\n§ ¨ ©ª\n«¬\n ®¯\n°±\n² ³ ´ µ¶\n·¸\n¹ º » ¼½\n¾¿\nÀ Á Â ÃÄ Å Æ Ç\nÈ\nÉ Ê Ë\nÌ Í Î\nÏ Ð Ñ\nÒ Ó Ô\nÕÖ\n× ØÙ\nÚ ÛÜ\nÝ Þß\nà á â ãä\nå æç\nè é ê ëì í î ï\nð\nñ ò ó\nô õ ö\n÷ ø ù\nú û ü\nýþ\nÿ \u0003\u000f\n\u0004 \u0000\u0001\n\u0002 \u0005\u0006\n\u0007 \b \t \n\u000b\n\f \r\u000e\n\u0010 \u0011 \u0012 \u0013\n\u0014\u0015\n\u0016\u0017\u0018 \u0019 \u001a \u001b\n\u001c \u001d\n\u001e\u001f  \n!\" #\n$\nFIG. 2: (color online) Normalized tunnelingconductancewi th\nZ= 1 for (a) s-wave pairing, (b)OSP and (c)ESP.for the 21st Century COE ”Frontiers of Computational\nScience” . 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Successive layers in the bulk compound are weakly bound by van der Waals forces so\nthat individual layers can be easily exfoliated. A monolayer of CrCTe 3is also an anti-ferromagnetic\nsemiconductor. The monolayer is structurally stable over a large range of compressive and tensile\nstrains, and the anti-ferromagnetic state is robust over this strain range. Band gap of the monolayer\ncan be tuned by as much as 50% by applying strain in this range.\nPACS numbers: 31.15.A-, 75.50.EearXiv:1707.00878v1  [cond-mat.mtrl-sci]  4 Jul 20172\nI. INTRODUCTION\nSince the exfoliation of graphene sheets from graphite [1] little more than a decade ago, two dimensional (2D)\nmaterials have attracted a lot of research attention. A major motivation for this is use of these materials in electronic\napplications which may open up new avenues not available to 3D materials. Many interesting 2D materials have been\nproposed in the past few years which include compounds such as hexagonal boron nitride (h-BN), transition metal\ndichalcogenides, and elemental 2D materials such as phosphorene, germanene, silicene, stanene, arsenene, borophene\nand aluminene [2{14]. Many of these have been realized in experiments in 2D single layer form (transition metal\ndichalcogenides, for example), for some, multilayers have been created (arsenene) [15], while some others remain\nas theoretical proposals as yet (aluminine) [14]. Some of these are promising for applications such as \feld e\u000bect\ntransistors (FET). MoS 2and phosphorene are good examples of this. Even though there are reports on intrinsic\nmagnetism in MoS 2and WS 2arising out of zig-zag edges in the grain boundaries and defects [16, 17], most of these\n2D materials are non-magnetic, and hence are not suitable for spintronics or storage applications. There can be two\nways of making 2D materials magnetic: functionalizing them by incorporating magnetic elements (transition metal\n(TM) atoms, for example), or identifying novel 2D materials that are inherently magnetic.\nSearch for magnetic 2D materials has generated new interest in the class of TM tri-chalcogenides with the general\nformula MAX 3(M= 3dTM atom, A=group 14 or 15 element, and X=chalcogen atom). Bulk compounds in this\nfamily have been known for quite some time [18]. But it is only recently that mono- and few layers of these compounds\nhave been successfully exfoliated [19]. One major advantage of the MAX 3family is that it has a very wide range of\nchemical diversity as well as structural complexity. Compounds in the MAX 3family display interesting properties\nsuch as half-metallicity, or semiconducting coupled with ferromagnetic (FM) and anti-ferromagnetic (AFM) order.\nThough MAX 3compounds have been studied extensively in the past [18, 20, 21], recently the Si and Ge compounds\nCrSiTe 3and CrGeTe 3have attracted a lot of renewed attention [22{26].\nFM materials are widely used in spintronics and their behavior is also well understood. AFM materials, on the\nother hand, are less explored due to their complex magnetic structures. In spite of their complexity, AFM materials\nhave generated a lot of interest recently [27{32]. AFM-ordered nano-structures consisting of small amounts of Fe-\natoms have been demonstrated to encode and store information using a spin-polarized tunneling current [31]. A\nnovel phenomenon related to AFM materials is the exchange bias e\u000bect [33], which is being used for development of\nmagnetic read heads in hard disks and magnetic memory devices. FMs can create parasitic magnetic \felds that can\ninterfere with each other. AFMs do not create such redundant magnetic \felds and are largely insensitive to the e\u000bect\nof external magnetic \feld due to their zero (or nearly zero) total magnetization. Another advantage of AFMs over\nFMs is that switching between di\u000berent states of an AFM is much faster than that in FM [32]. So, AFM materials are\npotential candidates for developing ultrafast and ultrahigh density spintronic and storage devices. Recent experiments\nshow that bimetallic antiferromagnets like Mn 2Au [34] and MnIr [35] are suitable for application in tunnel anisotropic\nmagnetoresistance (TAMR) devices due to their high magnetic ordering temperatures and strong magnetic anisotropy.\nSo, it is interesting and necessary to explore new potential AFM materials that can be used in nano-scale devices.\nBoth CrSiTe 3and CrGeTe 3are FM semiconductors. FM ordering in these compounds arises due to Cr-Te-Cr\nsuperexchange [36], and has been rationalized within the Goodenough-Kanamori-Anderson (GKA) rules [37]. The Cr\natom in these compounds are at the center of a Te 6trigonal anti-prism with D3dpoint group symmetry. The Te 6\ntrigonal anti-prism cages are arranged in an edge-sharing manner in each layer of the crystal. The Cr-Te-Cr angle is\nnearly, but not exactly, 900, and the Cr-Cr distances are relatively large ( \u00184\u0017A compared to 2.49 \u0017A in bcc Cr),\nmaking their direct exchange weak. The 900cation-anion-cation superexchange for a d3con\fguration of the TM\ncations in compounds with Ohsymmetry is quite complex. There are competing FM and AFM couplings. Generally\nTM oxides end up having AFM ground states, while the chlorides are FM [37]. The situation is more complicated\nin the tri-chalcogenides due to two factors: \frst, the crystal-\feld splitting of the Cr 3 dlevels is slightly di\u000berent in\ntheD3dcrystal-\feld as compared to the Ohcrystal-\feld assumed in the GKA analyses; second, the Cr-Te-Cr angle is\nnot exactly 900, as already stated. Therefore, whether the resultant superexchange is FM or AFM depends on a \fne\nbalance of di\u000berent factors, and it may be possible to a\u000bect the magnetic ground state by small structural changes.\nIn fact Casto et al [36] have shown that there is a strong spin-lattice coupling in CrSiTe 3.\nCompressive in-plane strain decreases the distance between the Cr atoms and increases overlap between the occupied\ndorbitals on neighboring atoms. Increase in direct overlap between atomic orbitals may drive the direct exchange it to\nbe antiferromagnetic beyond a point [38]. In bulk bcc Cr, the direct exchange is AFM. Interestingly, the Si compound\nis almost at the verge of a FM-AFM transition. A mere 0.5% compressive strain drives 2D layers of CrSiTe 3AFM [22].\nGe having a bigger atomic radius, the Cr-Cr distance increases from 3.90 \u0017A in CrSiTe 3to 3.94 \u0017A in CrGeTe 3, and the\nFM phase is stabilized. The Curie temperature (T C) increases from 33 K in CrSiTe 3[23, 39] to 61 K in CrGeTe 3[21].\nIn the theoretically proposed CrSnTe 3[40], the Cr-Cr distance increases even further (4.05 \u0017A in HSE06) [40] and the\nTCis estimated to be greater than 170 K.\nThus if one could design a similar tri-chalcogenide compound with smaller Cr-Cr distance, it is possible that the3\nground state would be AFM. An obvious option is to replace Si by C. Chemically, C would behave the same way\nas far as crystal binding is concerned. However, being smaller in size, would lead to a shorter Cr-Cr distance. The\nquestions are, is the compound CrCTe 3dynamically and mechanically stable in the R \u00163 crystal structure? If it is, is\nthe Cr-Cr distance short enough to give rise to an e\u000bective AFM superexchange? And \fnally, can 2D layers of this\ncompound be exfoliated? What are the properties of such a novel 2D material?\nThe rest of the paper is organized as follows. In Section II we describe the theoretical methods used for our\ncalculations. In Section III we discuss stability and electronic properties of bulk CrCTe 3(CCT). In various subsections\nof Section IV we discuss stability and electronic properties of a monolayer CCT. Section V contains our results on\nstrain engineering of a monolayer CCT. Finally, we draw our conclusions in Section VI.\nII. COMPUTATIONAL DETAILS\nSpin polarized density functional theory (DFT), as implemented in Vienna Ab-initio Simulation Package (VASP) [41{\n46], was used for all electronic structure calculations. Electronic wave functions were expressed in a plane wave basis\nwith an energy cuto\u000b of 500 eV. Electron-ion interactions were treated using the projector augmented wave (PAW)\nmethod. The gradient corrected Perdew-Burke-Ernzerhof (PBE) [47, 48] functional was employed for electronic min-\nimization, and structure relaxation. It is well known that the local and semi-local exchange-correlation functionals\nunderestimate band gaps. Therefore, for a more accurate estimation of the band gaps in monolayer CCT, the hybrid\nfunctional proposed by Heyd, Scuseria and Ernzerhof (HSE06) [49, 50] was used. Because HSE06 calculations are\nconsiderably more expensive than those with local/semi-local functionals, we did not employ it in any other case. To\naccount for dispersion interactions, a non-local correlation functional [51{53] (vdW-DF2) was used. Earlier works\non graphite have shown that the best description of dispersion interactions can be obtained by the vdW-DF2 func-\ntional [54, 55]. The convergence criteria for electronic minimization and structure optimization were set to 10\u00006eV\nand 0.001 eV/ \u0017A respectively. The Brillouin zone (BZ) integration was performed within the Monkhorst-Pack scheme\nusing 9 \u00029\u00023 and 11 \u000211\u00021 meshes for bulk and monolayer respectively. A unit cell was used for bulk electronic\nstructure calculations. Bigger supercells were used for calculating phonon band structure and cleavage energy. For\nthe monolayer, the usual repeated slab geometry was used with a vacuum space of 20 \u0017A. Phonon band structures were\ncalculated using the inter-atomic force constants obtained from VASP within the density functional perturbation\ntheory (DFPT) framework, with the PHONOPY code [56] as the post-processor. Energy and force convergence\ncriteria were set to 10\u00008eV and 0.00001 eV/ \u0017A for these calculations. A (3 \u00023\u00022) supercell was used for bulk\nphonon calculations. For phonon calculations of monolayers, a (3 \u00023) planar supercell was used. Visualization\npackages VESTA [57] and VSim [58] have been used to reproduce the crystal structures and vibration modes shown\nin this work.\nFIG. 1. Optimized lattice structure of CrCTe 3(a) top view (b) side view.4\nFIG. 2. Phonon band structure of bulk CrCTe 3\nIII. BULK CCT\nA. Structure and magnetic ordering\nSince CrSiTe 3, CrGeTe 3[21, 23], and the theoretically proposed compound CrSnTe 3[40] all have stable R \u00163 structure,\n. we assume that CCT also has the same structure. Fig. 1 shows the structure of CCT. Each Cr atom is six-fold\ncoordinated to Te-atoms arranged in a trigonal anti-prism structure with D3dpoint group symmetry. Each C atom\nis three-fold coordinated to Te-atoms. The formal valence of the atoms are Cr+3, C3+and Te\u00002, which gives a 3 d3\nelectronic con\fguration on Cr. Hund's rule would then give spin S= 3=2 on each Cr atom. We considered FM and\nfour di\u000berent AFM order of the Cr spins. These AFM spin arrangements are shown in Fig S1(a) in Supplementary\nInformation (SI) [59], and their energies are given in Table S1. The AFM state in which nearest neighbor Cr spins are\noriented in opposite directions within each CrCTe 3layer and between layers (3D N\u0013 eel order, called AFM1 here) turns\nout to have the lowest energy. The energy di\u000berence \u0001 E=EFM\u0000EAFM1 = 13:6 meV/atom. The magnetic moment\ncomes mostly from the Cr atoms. In the FM state, each Cr atom contributes 3.03 \u0016B, while in the AFM1 state the\ntwo Cr atoms in the unit cell contribute \u00062:85\u0016B. Further details about magnetic moment can be found in Table S3\nin SI [59]. This con\frms that bulk CCT has an AFM order in its ground state. The PBE-vdW-DF2 optimized lattice\nparameters for antiferromagnetic bulk CCT are a = b = 6.64 \u0017A and c = 21.41 \u0017A with an interlayer separation of\n3.85 \u0017A. The Cr-Te bond length is 2.80 \u0017A, and Cr-Cr bond length is 3.83 \u0017A , with a Cr-Te-Cr bond angle of 86.24\u000e.\nB. Stability and electronic structure\nWhile studying a new material theoretically, it is essential to check whether it is dynamically and mechanically\nstable. In order to ensure that the optimized R \u00163 lattice structure for CCT is dynamically stable, we calculated its\nphonon band structure. Any structural instability would show up as soft phonon modes with imaginary frequencies.\nAs can be seen in Fig. 2, all phonon branches have real frequencies signifying dynamical stability of the R \u00163 crystal\nstructure. We also calculated the elastic constants C11andC12to test for mechanical stability of CCT. These elastic\nconstants are de\fned as,\nC11=1\nV0:@2E\n@\u000f2\n11andC12=1\nV0:@2E\n@\u000f11@\u000f12: (1)\nHereEis the total energy of CCT, V0is its equilibrium volume, and \u000fare the components of strain. C11andC12\nwere found to be 112 GPa and 23.50 GPa respectively. The elastic constants satisfy the Born stability criterion\nC11\u0000C12>0 indicating mechanical stability of bulk CCT.\nAfter having established that bulk CCT is structurally stable, we now calculate its electronic structure. Electronic\nband structure calculated using PBE-vdW-DF2 is shown in Fig. 3(a). The valence band maximum (VBM) occurs at5\nFIG. 3. (a) Band structure of bulk CrCTe 3(b) Total and orbital resolved DOS of bulk CrCTe 3.\nthe \u0000-point while the conduction band minimum (CBM) occurs between \u0000 and M-points (\u0001-point henceforth). The\nsmallest direct gap appears at the \u0001-point and is 1.28 eV in PBE. But the fundamental band gap is an indirect one\nfrom \u0000 to \u0001, and is 1.12 eV. Fig. 3(b) shows the total and orbital resolved density of states (DOS) of bulk CCT.\nC states have very little contribution near the band edges, so contributions of only the Cr and Te atoms have been\nshown. In the D3dcrystal \feld of the Te anions, the Cr 3 dstates are split into three levels: E g(dx2\u0000y2,dxy); Eg(dyz,\ndzx); and A 1g(dz2) as seen in the bottom three panels of Fig. 3(b). The states near the VBM region have contributions\nfrom both Te- pand Cr-dorbitals while the states near the CBM have major contributions only from the Cr- dstates.\nTo conclude this section we note that our guess that a shorter Cr-Cr distance in CrCTe 3may lead to an AFM\nground state for the material is indeed borne out by DFT calculations. The material is found to be dynamically and\nmechanically stable, so it should be possible to synthesize it in the laboratory.6\nFIG. 4. Increase in energy as function of distance (relative to the equilibrium separation) between two half crystals of CrCTe 3.\nIV. MONOLAYER CCT\nWe now study properties of 2D monolayers of CCT (MCCT). Before studying properties MCCT, it is important to\n\fnd out how easy or di\u000ecult it is to exfoliate such layers from the bulk. We estimate the ease of exfoliation by the\nso-called cleavage energy. Cleavage energy is de\fned as the energy required to separate the crystal into two halves\nalong the gap between two successive CCT layers, and is calculated as follows. The increase in energy is calculated as\nthe distance between two halves of bulk CCT is increased compared to the equilibrium separation. We have calculated\nthis quantity with three and six layers in the simulation cell, and we get the same exfoliation energy in both cases\nindicating that our estimation does not su\u000ber from \fnite size e\u000bects. Energy increase for a six-layer supercell is\nshown in Fig. 4. The energy increases sharply at \frst, but saturates at larger separations. The saturation value of\nthe energy, 0.24 J/m2, is the cleavage energy. Interestingly, cleavage energy of CCT is smaller than that of graphite,\n0.37 J/m2[60]. It may be noted that the cleavage energies of other members of the family, CrSiTe 3, CrGeTe 3, are also\nhigher, 0.35 J/m2and 0.38 J/m2[61] respectively. Since graphene, and CrSiTe 3and CrGeTe 3layers can be easily\nexfoliated using simple mechanical means, one expects that the same procedure would work for CCT.\nA. Stability\nBefore calculating its electronic properties, we re-optimized the in-plane lattice parameters for MCCT should they\nchange compared to the bulk values. The PBE-vdW-DF2 optimized lattice parameters for monolayer CCT turn out\nto bea=b= 6:63\u0017A, very close to the bulk values. Such an insigni\fcant change in the lattice constants is perhaps\nbecause of a relatively weak inter-layer binding. In monolayer CCT we calculated energies of FM and three di\u000berent\nAFM ordering of the Cr spins: the 2D N\u0013 eel, zigzag and stripe phases (Fig. S1(b) in the SI [59]). Similar to its bulk\nform, the 2D N\u0013 eel state (AFM1) turns out to have the lowest energy with \u0001 E=14.6 meV/atom. Energies of all the\nmagnetic states are given in Table S2 in the SI [59]. The moment on each Cr atom in the FM state in MCCT is\npractically same as what we found in the bulk, 2.98 \u0016B. The moment on the two Cr atoms in the AFM1 state are\n\u00062:76\u0016B, slightly lower than the bulk value. The energy di\u000berence between the FM and AFM1 states in a monolayer\nis thus very close to that in the bulk, perhaps because the coupling between successive layers is rather weak.\nFrom our calculations of the FM and AFM1 states, we estimated the N\u0013 eel temperature ( TN) of the monolayer by\ntreating the Cr spins as S= 3=2 Ising spins. The energy di\u000berence between the two spin ordered states, gives the\nnearest neighbor exchange interaction energy as 10.8 meV. Using the expression for the transition temperature for\nan Ising model on a honeycomb lattice [62], ( T\u0003\u0018JS2=1:3kB) we \fndTN\u0018217K. We have used the fact that the7\nFIG. 5. (a) Phonon band structure of monolayer CrCTe 3; calculated (b) infrared and (c) Raman spectra of monolayer and\nbulk CCT. Insets in (b) and (c) are enlarged views of the low-frequency region.\ntransition temperature in the nearest-neighbor Ising model is independent of the sign of the exchange interaction.\nIt is important to check the dynamical and mechanical stability of a monolayer as well. As in the case of bulk,\nphonon band structure is calculated for the 2D layer and is shown in Fig. 5(a). All phonon branches turn out to have\npositive frequencies signifying dynamical stability of the structure. Te, being the heaviest of all the constituent atoms,\ndominates the lower frequency region, which is followed by Cr and C. Similar to CrSiTe 3[36], \rat optical phonon\nbranches is a feature of CrCTe 3also.\nTo check for mechanical stability of the 2D monolayer, we calculated its elastic constants. Elastic constants of a\nmonolayer are de\fned as\nC11=1\nA0:@2E\n@\u000f2\n11andC12=1\nA0:@2E\n@\u000f11@\u000f12: (2)8\nHereEis the total energy of MCCT, A0is its equilibrium area, and \u000f's are the components of strain. The calculated\nvalues of elastic constants for MCCT are: C 11= 81:70 N/m, C 12= 17 N/m which are slightly lower than that of\nbulk CCT. MCCT also satis\fes Born's criterion for mechanical stability C11\u0000C12>0. To avoid curling during\nthe exfoliation process of a 2D crystal, a high in-plane sti\u000bness is necessary. To estimate the in-plane sti\u000bness, we\ncalculate the in-plane Young's modulus of monolayer CCT using the formula: Ys= (C2\n11\u0000C2\n12)=C11. For monolayer\nCCT,Ys= 78 N/m. This is about 23% of that of graphene (341 N/m) [63], one of the strongest materials. Thus, it\ncan be assumed that MCCT can keep its free-standing structure. The in-plane sti\u000bness of monolayer CCT is higher\nthan that of monolayer CrSnTe 3withYs= 55 N/m [C 11= 60 N/m, C 12= 17 N/m] [40]. Therefore we conclude that\nmonolayers of CCT are dynamically and mechanically stable, and they can be mechanically exfoliated for further\nstudies in their free-standing structures.\n2D materials are often characterized by their infra-red (IR) and Raman spectra. Therefore, we have calculated the\nIR and Raman spectra for both bulk and a monolayer CCT. O\u000b-resonant Raman activity of a mode was calculated\nby computing the derivative of macroscopic dielectric tensor with respect to normal mode coordinates [64]. For this,\nphonons at Gamma, and macroscopic dielectric tensors were calculated using DFPT as implemented in VASP. The\nderivatives were calculated using the script developed by Fonari and Stau\u000ber [65].\nFor infrared modes, the tensor of the Born e\u000bective charges (the \frst derivative of the polarization with respect to\nthe ionic coordinates) was calculated using DFPT. Within the dipole approximation, the infrared intensity ( I) of an\neigenmode can be expressed in terms of the Born e\u000bective charges Z\u0003\n\u000b\fand the eigenvectors e\f(l), where\u000band\fare\ncartesian polarizations, and llabels the atoms of the system [66, 67].\nI=X\n\u000b2\n4X\nl;\fZ\u0003\n\u000b;\fe\f(l)3\n52\n(3)\nBulk CCT has six infra-red (IR) active modes. These are at 442.9 cm\u00001, 210.5 cm\u00001and 116.3 cm\u00001. All\nthese frequencies are doubly degenerate. These are shifted to slightly higher energies, 447.8, 212.3 and 116.5 cm\u00001\nrespectively in MCCT. It may be noted that out of the six IR-active modes, the \frst peak is the most prominent one\nin both bulk (442.9 cm\u00001) and monolayer CCT (447.8 cm\u00001) (Fig. 5(b)). Analyzing the eigenvector of these phonon\nmodes at \u0000, it is seen that these involve motion of the C atoms. The second peak at 210.5 cm\u00001in bulk (212.3 cm\u00001\nin monolayer) comes from Te-Cr-Te bond bending. The third one at 116.3 (116.5) cm\u00001is from both C-Te bond\nstretching and Te-Cr-Te bond bending. These are very similar to what was found in CrSiTe 3[36].\nEight Raman active modes have been found in MCCT. Three of these, at 992.2, 563.8 and 193.8 cm\u00001, are most\nintense. These modes are found in bulk CCT too but at 994.9, 556.7 and 192.3 cm\u00001. (Fig. 5(c)). Atomic motions\ncorresponding to the IR and Raman active modes are shown in Figs. S2 and S3 in the SI [59]. Thus the \frst Raman\nactive mode shifts to a lower energy in the monolayer while the other two shift to higher energies like the IR-active\nmodes.\nB. Electronic structure\nWe now proceed to study the electronic structure of MCCT. The electronic band structure of MCCT calculated\nusing PBE-vdW-DF2 is shown in Fig. 6. The band structure near the gap looks very similar to the bulk except\nthat there are fewer bands in the monolayer, as there are a smaller number of atoms in the unit cell. Again, as the\ninter-layer coupling is weak, band dispersion arises mainly due to in-plane bonding. Thus there is little di\u000berence\nbetween the band structure of the bulk and the monolayer. As a consequence, the energy gaps in a monolayer are also\nnearly the same as those in the bulk. The smallest direct PBE gap is 1.29 eV at the CBM which is at the \u0001-point.\nThe fundamental band gap is indirect and is equal to 1.15 eV from \u0000 to \u0001. Band structure of MCCT calculated with\nthe HSE06 functional is given in Fig. S4 of the SI [59]. We get qualitatively the same band structure with larger band\ngaps. In HSE06, the direct (at \u0001) and indirect (\u0000 to \u0001) gaps are nearly the same, and are 1.87 eV. Since the bulk\nand the monolayer have almost the same band gap at the PBE-vdW-DF2 level, it is reasonable to believe that the\nHSE06 functional would produce a similar band gap for the bulk material. Thus a more accurate estimate of band\ngap in the bulk would be around 1.87 eV.\nTo understand contributions of di\u000berent atomic states to the electronic structure of monolayer CCT, we have plotted\nthe total and atom and orbital resolved DOS in Fig. 7. As in the bulk, states near the VBM has contributions from\nboth Te-pand Cr-d(mostlyx2\u0000y2,xy,yzandzx) states. CBM has contributions mostly from the Cr- dstates.9\n-2-1012Energy (eV)\nK M Γ M∆=1.29 ∆=1.15 \nFIG. 6. Electronic band structure of monolayer CrCTe 3. Direct and indirect band gaps are shown by arrows.\n060DOS (arb. units)Cr (dx2-y2+dxy)060\nTe (p)\n060\nCr (dyz+dxz)060\nTotal\n-6 -5 -4 -3 -2 -1 0 1 2 3\nEnergy (eV)60060\nCr (dz2)4X\n4X\n4X\n4X\nFIG. 7. Total and orbital resolved DOS of monolayer CrCTe 3.\nV. STRAIN ENGINEERING OF MONOLAYER CCT\nNext we explore how electronic properties of MCCT can be engineered using in-plane strain. In this work we\nconsider application of biaxial strain only. The \frst point to check is the range of strain over which the R \u00163 structure is\ndynamically stable. For this, phonon band structure is calculated at di\u000berent applied strain. Based on our results, we\nclaim, with a caveat, that the R \u00163 structure for MCCT is stable from 4% compressive to 16% tensile strain as all phonon\nbranches have positive frequencies in this range. Fig. 8 shows the calculated phonon spectra of MCCT at di\u000berent10\nFIG. 8. Phonon band structures of monolayer CrCTe 3under di\u000berent compressive and tensile strain.\ncompressive (negative) and tensile (positive) strains. Phonons are calculated within a harmonic approximation for\nthe interatomic forces. Whether this approximation remains valid up to 16% strain is a pertinent question. How to\napply such large tensile strain in practice is also not obvious. Therefore, while our results are de\fnitely valid up to\nfew percent strain, they should be taken as indicative at larger strains.\nA long wavelength acoustic phonon branch becomes unstable at \u00005% strain. This soft phonon mode is clearly\nseen in at Fig. 8(a) at \u00006% strain. The eigenvector of this mode indicates that this is a \rexural mode (Fig. S5 in\nSI) [59]. This instability thus indicates a buckling transition in MCCT. It is interesting that the buckling instability\nsets in beyond 4% compressive strain in CCT whereas graphene develops this instability at a much smaller strain of\n0.75% [68]. Beyond 16% tensile strain an acoustic branch becomes soft but now at the M-point. This can be seen in\nFig. 8(d) for +16 :5% strain. This is an in-plane mode, and thus indicates a structural transition in the 2D monolayer.\nAn optical mode also becomes soft over the entire 2D BZ. Eigenvectors of these two modes are shown in Fig. S6 in\nSI [59].\nWe calculate the band structures of MCCT by applying biaxial strain in steps of 2 % on both the compressive\nand tensile sides. Fig. 9 shows the PBE-vdW-DF2 band structure of MCCT under di\u000berent strain conditions. In the\nunstrained case, MCCT is an indirect band gap semiconductor with a gap of 1.15 eV as mentioned earlier. Under\ncompressive strain, the indirect nature of the band gap is maintained with VBM and CBM still at the \u0000 and \u0001 points\nrespectively. However, the band gap drops marginally down to 1.13 eV at \u00004% strain. Di\u000berence between the direct\nand indirect gaps is quite small as seen in panels (a) and (b) in Fig. 9.\nWith tensile strain, the fundamental gap remains indirect between \u0000 and \u0001 up to 2% strain (Fig. 9-(c)). However,\nat 4% strain, while CBM still remains at the \u0001-point, the VBM moves to the K-point. With further increase in tensile\nstrain from 4% to 10 %, the band gap decreases from 1.15 eV to 0.80 eV with the VBM still at K-point shown in\npanels (d)-(g) in Fig. 9. At 12 % strain, the VBM shifts to the M-point. The band gap reduces further to 0.69 eV\nat 12 %, and to 0.55 eV at 15 % strain. This can be seen in panels (h)-(j) in Fig. 9. The energy di\u000berence between\nthe valence band states at M and K points is always small. Similarly, the energy di\u000berence between the conduction11\nFIG. 9. PBE electronic band structure of monolayer CrCTe 3under di\u000berent strains (a) \u00004% (b) \u00002% (c)+2% (d)+4% (e)+6%\n(f)+8% (g)+10% (h)+12% (i)+14% (j)+15% (direct and indirect gaps are shown by arrows).\n-4 -2 0 2 4 6 8 10 12 14 16\nStrain (%)0.60.811.2Bandgap (eV)PBE-vdW-DF2\nFIG. 10. Variation of band gap of monolayer CCT with biaxial strain.\nband states at these two points is also small. As a consequence, the di\u000berence between the direct and indirect gaps\nis small at all strains. In fact, the valence and conduction bands being rather \rat between M and K points, there are\nlot of states available for electron-hole excitations in a narrow energy range. This should make monolayer CCT an\nattractive material for photovoltaic applications.\nVariation of the band gap of MCCT with strain is shown in Fig. 10. The band gap changes very little under\ncompressive strain. It increases marginally between 0 \u00002% strain, but then starts decreasing. Beyond 4% strain, it\ndecreases sharply all the way up to 16%. Therefore, if our conclusion about structural stability of MCCT is valid in\nthe high strain regime, band gap of MCCT can be tuned by as much as 50% by applying tensile strain.\nIn order to get more insights into the electronic structure of MCCT at di\u000berent strains, we calculated atom and\norbital resolved DOS at various strains. These quantities at \u00004% and +4% strain are shown in Figs. 11(a) & (b). As\nin the pristine monolayer, the VBM originates primarily from the Te pstates, while the CBM has major contribution\nfrom the Cr dstates. A comparison of Figs. 7 and 11 also show that as the lattice constants of monolayer CCT12\nFIG. 11. Total, and orbital resolved DOS of monolayer CCT at (a)-4% strain and (b)+4% strain.\nincrease (tensile strain starting from its \u00004% structure), the contribution of the Cr dx2\u0000y2,dxy,dyzanddzxorbitals\nto the states in the range VBM to \u00002 eV decreases. It is the maximum at \u00004%, and is much lower at +4%. This\ncan be due to increasing Cr-Te distance with increasing tensile strain. The peak of the Te partial DOS in this energy\nrange shifts to higher energies. A decrease in the mixing with the Cr dstates destabilizes the Te pstates somewhat.\nAt still higher tensile strains (SI) [59], new states with contributions from both Cr dand Teporbitals appear \u00180:5 eV\nabove the VBM (Fig. S7) forming the new conduction band edge. These states reduce the band gap as seen Fig. 10.\nAn important question in the context of strain engineering is how robust the AFM order is with respect to applied\nstrain. Speci\fcally, it is possible that under tensile strain, as the Cr-Cr distance increases, the AFM state may become\nunstable towards a FM ground state. In order to check this we calculated energies of both FM and AFM1 ordered\nstates of the Cr spins under all strain conditions from \u00004% to +16%. It is interesting to note that \u0001 Eis positive\nat all strains indicating greater stability of the AFM1 state, and in fact, beyond \u00182% tensile strain the AFM1 state\ngains more stability relative to the FM state. In pristine MCCT, the Cr-Te bond length is 2.80 \u0017A, and Cr-Cr bond\nlength is 3.83 \u0017A , with a Cr-Te-Cr bond angle of 86.24\u000e, as stated earlier. With tensile strain, Cr-Te and Cr-Cr bond\nlengths increase monotonically and reach 3.11 \u0017A and 4.44 \u0017A at +16% strain. The bond angle also increases to 90 :89\u000e\nat +16% strain. With compressive strain, Cr-Te and Cr-Cr bond lengths decrease monotonically to reach 2.74 \u0017A and\n3.67 \u0017A at \u00004% strain. The bond angle also reduces to 84.13\u000eat\u00004% strain. Variation of bond lengths and bond\nangles are given in Table S4 in the SI [59]. For comparison, we optimized the structure of monolayer CrSiTe 3with the\nPBE-vdW-DF2 functional. Cr-Cr bond distance in this compound is 4.06 \u0017Aand the Cr-Te-Cr bond angle is 89 :97\u000e.\nCr-Te bond length is found to be 2.87 \u0017A. What is interesting is that the Si compound has a FM ground state with a\nCr-Cr distance of 4.06 \u0017A, but MCCT remains AFM up to a Cr-Cr distance of 4.44 \u0017A at +16% strain.\nIt is puzzling that the AFM1 state becomes more stable with larger tensile strain. Tensile strain monotonically\nincreases Cr-Cr distance. Therefore, any direct AFM exchange between neighboring Cr atoms must become weaker.\nHowever, as we discussed earlier, the superexchange in the TM tri-chalcogenides are quite complex. Therefore, one\ncan only speculate that perhaps in CCT it has an AFM character. This issue requires further careful study.13\nVI. CONCLUSION\nUsing DFT and DFPT calculations we have established that CrCTe 3is a structurally stable compound in the R \u00163\nstructure. It turns out to be an AFM semiconductor, a welcome addition to this family. The fundamental band\ngap is 1.12 eV and an indirect one as found in our PBE calculations. The successive layers of the bulk material are\nrather weakly bound by van der Waals forces. The cleavage energy is estimated to be 0.24 J/m2, smaller than that of\ngraphene, CrSiTe 3and CrGeTe 3. Monolayers of CCT are also AFM semiconductors with nearly the same indirect gap\nas the bulk. MCCT remains structurally stable between 4% compressive and 16% tensile biaxial strain. 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B 82, 115411 (2010)."    },    {        "title": "1912.01331v1.Nanoscale_Tantalum_Layer_Controlling_the_Magnetic_Coupling_between_Two_Ferromagnetic_Electrodes_via_Insulator_of_a_Magnetic_Tunnel_Junction.pdf",        "content": "1 \n Nanoscale Tantalum Layer Controlling the  Magnetic Coupling between \nTwo F erromagnetic Electrodes via Insulator  of a Magnetic Tunnel \nJunction  \nPawan Tyagi1,2* and Tobias Goulet1 \n1Mechanical Engineering, University of the District of Columbia , Washington DC-20008. USA  \n2Chemical and Materials Engineering, University of Kentucky, Lexington, KY -40566, USA  \n*Corresponding Author: ptyagi@udc.edu  \n \nABSTRACT:  Ability to tailor the nature of the magnetic coupling betw een two ferromagnetic electrodes  \ncan enable the realization of new spintronics device systems . This paper discusses our finding that \ndeposition of an ultrathin tantalum (Ta) on the  NiFe top electrode revers ed the nature of inter -\nferromagnetic electrode coupling. W e observed that the deposition of ~ 5 nm Ta on the top of a magnetic \ntunnel junction with Ta( 2 nm)/Co(5 nm )/NiFe (5 nm)/AlOx( 2  nm)/NiFe (10 -15 nm) configuration \nchanged the magnetic coupling between two ferromagnetic electrodes from antiferromagnetic to \nferromagnetic. We investigated Ta effect using multiple magnetic characterizations like ferromagnetic \nresonance, magnetometry, and polarized neutron reflectometry. Ferromagnetic resonance characterization \nwas very sensitive for detecting the changes in mag netic coupling via the insulating spacer. This simple  \napproach of adding Ta film to alter the magnetic coupling can impact the other burgeoning areas like \nmolecular spintronics. We found that preexisting magnetic coupling between two ferromagnetic \nelectrodes impacted the resultant magnetic properties of magnetic tunnel junctions based molecular \nspintronics devices.  \nKey words:  Magnetic tunnel junctions; molecular spintronics; tantalum; exchange  coupling;  \nI INTRODUCTION:   2 \n Tailoring the  nature of the magnetic coupling between two ferromagnetic electrodes has been the \ntopic of intense interest  [1, 2].  Ability to change the inter -ferromagnetic electrode coupling can lead the \ndevelopment of new device forms  and materials [2]. For instance, nanoscale spintronics  devices focus on \nmaneuvering the nature and strength of the inter-ferromagnetic electrode coupling  (IFMEC) [2]. To date, \nthree key approache s have been employed t o tailor the IFMEC. The first approach  involves inserting the \nnanoparticles between two ferromagnetic electrodes  [1]. The second method involves  changing the \nthickness  of nonmagnetic spacers between two ferromagnetic electrodes  [3]. The third method  requires  \nchanging of the nonmagnetic spacer material between two ferromagnetic electrodes . However, these \napproaches are very challenging to implement.  For instance, controlling the distribution and sizes of \nnanoclusters between two ferromagnetic electrodes  is very challenging to exercise and difficult to \nreproduce  [1]. Similarly, tailoring the spacer thickness to sub -nm scale and even changing the spacer \nmaterial altogether requires intensive device optimization [2, 4].  An approach that do es not physically \naffect the spacer between two ferromagnetic electrodes and easy to implement can lead to new \nopportunities. Recently , magnetic tunnel junction based molecular spintronics devices  (MTJMSDs)  were \ndeveloped [5-7]. One can alter the IFMEC by adding Ta top layer before transforming a magnetic tunnel \njunction into a MTJMSD. In this paper , we first discuss the role of Ta on IFMEC. We also discussed the \nimpact  of preexisting IFMEC on the magnetic properties of the MTJMSD .  \nII EXPERIMENTAL DETAILS:  \nTo investigate Ta effect on IFMEC we employed various physical property measurement  \ntechniques  such as Ferromagnetic resonance (FMR) , magnetometry, and polarized neutron reflectivity \n(PNR) .  For the  PNR measurements, unpatterned MTJ and MTJ -Ta samples were employed. The reason \nfor utilizing unpatterned samples was based on the strong effect from t he uncovered substrate that \nstrongly impact ed the measurement and modeling accuracy. The FMR and magnetometry study utilized \npatterned tunnel junctions. The FMR and magnetometry methods  are sensitive towards the magnetic \nproperty of the materials.  Unlike P NR the FMR and magnetometry do not get influenced by the 3 \n nonmagnetic su bstrate . Also, utilizing patterned tunnel junction was necessary for making molecular \ndevices for the FMR and magnetometry study. It is notworthy that e dge effects become prominent for \nsubmicron or nm scale magnetic features  [8]. To avoid the undesirable impact of edges we produced the \nMTJ and MT -Ta with several tens of micron area. We fabricated an array of ~7000  patterned magnetic \ntunnel junctions per sample. Every magnetic tunnel \njunction w as ~5 µm in diameter and ~10 µm distance \nfrom the neighboring magnetic tunnel junctions . We \nutilized an oxidized silicon substrate and performed \nphotolithography to produce a photoresist layer with \nan array of mi cro-cavities. For this study, Shipley®  \n(S1813 ) photoresist was spin coated on an oxidized \nsilicon wafer piece at 3000 rpm speed. This  spin \ncoated  photoresist film was baked at 90 ⁰C for one \nminute. Subsequently, the photoresist was exposed  to \nUV light thro ugh a photomask containing an array pattern. We developed the photoresist film in MF 319 \nMicroposit developer to produce an array of microc avities in the photoresist film . These microcavities \nwere filled with multiple thin  films to create a n array of magnetic tunnel  junctions. Tantalum (Ta), cobalt \n(Co), alumina(AlOx), and the alloy of NiFe  alloy with 80% nickel w ere sputter  deposited with AJA \nInternational sputtering machine. We first sputter depos ited Ta(2 nm)/Co(5 nm)/NiFe (5 nm)/AlOx( 2  \nnm)/NiFe (10 -15 nm) thin film configuration . This magnetic tunnel junction configuration is named as \nMTJ. To prepare the sample for the study of Ta impact on IFMEC MTJ -Ta samples were produced. To \ncreate MTJ -Ta samples , we additionally sputter deposited 5 nm Ta on the top of MTJ configuration . A \nresultant MTJ -Ta sample had Ta(2 nm)/Co(5 nm)/NiFe (5 nm)/AlOx( 2nm)/NiFe (10 -15 nm)/Ta (5 nm) \nconfiguration. The ~ 5 nm Ta top layer thickness ensured a conformal film o n the top NiFe electrode. We \nnoted that deposition of NiFe on the alumina (AlOx)  generally produced as high as 3-5 nm RMS \nroughness.  The high roughness is inevitable due to the amorphous nature of AlOx [8]. With such high \nFig. 1. FMR spectra recorded on MTJ and \nMTJ -Ta samples.  \n4 \n roughness Ta  < 3 nm was unable to form conformal films.  Hence, Ta with thickness around 5 nm was a  \ngood choice. On the other hand increasing Ta thickness also reduced device yield, presumably due to \nhigher mechanical stresses. Mechanical st resses have been found to create tunnel barrier failures [9]. \nThese MTJ and MTJ -Ta samples were characterized by the X -band Bruker EMX300 FMR and  NanoOsc \nPhase FMR over 2 -17 G Hz frequency range at room temperature. Magnetization studies were performed \nwith Quantum Design PPMS SQUID m agnetometer  at 150 K . Low temperature was chosen to avoid \nnoise in the magnetization study. PNR study w as performe d at 150 K and at 150 mT magnetic field at \nNational Institute of Standards and Technology, Gaithersb urg USA. The l ight reflectivity studies were \nperformed with Semiconsoft ® Mprobe thin film measurement sy stem at room temperature .      \nIII RESULTS AND  DISCUSSIONS :  \nWe first studied the IFMEC on MTJ and MTJ -Ta. It is noteworthy that t wo magnetic structu res \nseparated by nm gap exhibit  ferromagnetic or antiferromagnetic couplings  [10]. FMR is a powerful tool to \nstudy the charact eristics of magnetic coupling between two magnetic structures, especially two \nferromagnetic films  [11]. Our FMR studies revealed a striking difference between MTJ and MTJ -Ta (Fig. \n1). Under the identical experimental conditions , both samples showed t wo distinct FMR modes . For both \nsamples, an in -plane DC magnetic field up to 4000 Oe and 9.75G Hz microwave was applied to study the \nresonance modes. Before every measurement , the cavity’s spectra w as checked for the background signal \nat fivefold higher gain than that use d for the MTJ and MTJ -Ta samples. We also investigated  if the \nmanual err or in the sample alignment with respect to  the direction of the magnetic field could impact the \nintensity of FMR modes. The FMR spectra of a sample did not change noticeably within the  ±10º \nvariation on the magnetic field direction. We also utilized the FMR response from the graphite tape  as a \ncontrol sample  to ensure that experimental conditions were identical for the MTJ and MTJ -Ta (Fig. 1). \nDuring the FMR study of MTJ and MTJ -Ta, the  graphite tape produced a delta function type resonance \npeak at 3367±4 Oe (Fig. 1). In our study , we utilized the same graphite tape to mount the MTJ and MTJ -\nTa samples for the FMR study. This sharp resonance peak from the graphite tape was statistically 5 \n identical for the MTJ and MTJ -Ta samples. Invariance of the graphite tape’s signal suggests that \nexperimental conditions for measuring MTJ and MTJ -Ta were identical. The reproducibility of the \ngraphite tape’s resonance signal ensured the robustness and rep roducibility of microwave power, DC \nmagnetic field , and losses due to the measurement system, etc.  \nThe MTJ sample showed acoustic mode (higher intensity resonance peak) before the optical \nmode (lower intensity peak). According to FMR theory [12],[13] two ferromagnetic electrodes  of the MTJ \nsample are antiferromagnetic ally coupl ed. On the other hand, t he two resonance peaks from MTJ -Ta \nsample were significantly different as compared to the FMR peaks from MTJ. It appears that presence of \nTa on the top of NiFe ferromagnetic electrode reduced the intensity of the first resonance peak. As a \nresult , MTJ -Ta exhibited smaller intensity resonance mode (optical mode ) appear ing before the higher \nintensity resonance ( acoustic mode ). This particular form of the FMR spectra from MTJ -Ta is indicative \nof ferromagnetic coupling between the two ferromagnetic electrodes [12], [13].  \nWe estimated the strength of exchange coupling between two ferromagnetic electrodes of the \nMTJ and MTJ -Ta to be of the similar magnitude . \nWe estimated the order of magnitude of the \nmagnetic coupling by  two ways. (a) First , we \nevaluated the slop e of the lines joining the two \nresonance modes of the MTJ and MTJ -Ta. It is \nnoteworthy that zero slop means no interaction \nbetween the two ferromagnetic electrodes. The FMR \nmodes recorded on the isolated top and bottom \nferromagnetic electrodes are uncoupled (Fig. 2). \nHowever, for MTJ and MTJ -Ta the slop e of the line  \nbetween two modes is roughly the same. (b)  The \nestimation of magnetic coupling strength is also possible from the difference in tunnel junction’s mode \nFig. 2. FMR spectra of MTJ and MTJ -Ta samples  with \nrespect to FMR peaks from the isolated Ta/Co/NiFe and \nNiFe electrodes . \n6 \n positions with respect to the mode positions from the isolated ferromagnetic electrodes. It is noteworth y \nthat tunnel junction ’s mode positions shift as a function of the magnetic coupling strength between two \nelectrodes [10, 14]. We noticed that resonance position for the MTJ and MTJ -Ta was only ~45  Oe less as \ncompared to the resonance magnetic field of the Ta/Co/NiFe bottom electrode  grown in isolation. We \nsurmise that addition of Ta only affected the nature of IFMEC, not its magnitude. It is also noteworthy \nthat adding Ta app ears to reduce the intensity of acoustic mode, which appeared close to the top NiFe \nelectrode’s resonance position (Fig. 2).   \nWe also investigated the difference \nin magnetization data obtained from the \nMTJ and MTJ -Ta. We found that both \nsamples produced almost identical \nmagnetization loop s (Fig. 3 ). However, \nMTJ -Ta was relatively less sloped in the \nunsaturation state ( Inset of Fig. 3). This Ta \ninduced subtle difference in the \nmagnetization loop affirms two important \npoints : (i) For MTJ the antiferromagnetic \nIFMEC is not strong otherwise there could be a significant changes in  the magnetization loop  [15]; (ii) \nmagnetization data for MTJ -Ta show ed moderate  increase in the magnetic moment between saturation \nstates as compared t o that of MTJ  (Fig. 3 a). We surmise that t his moderate  increase in magnetization is \ndue to the  emergence of ferromagnetic coupling. If the addition of Ta did not affect the coupling, then \nmagnetization loop should have been the same for the MTJ and MTJ -Ta sample s. On the other hand , if \nthe addition of Ta enhanced the antiferromagnetic coupling, the n the magnetic moment in the \nunsaturation region should have been reduced. Hence, the current form of the magnetization data assert s \nFig.3: .Magnetic field  vs. normalized moment for ±2000 Oe \nrange.  Inset image show zoomed in image of magnetization \ncurve for ± 200 Oe . \n7 \n with the FMR data (Fig. 1) , which indicates that the addition of Ta produced ferromagnetic coupling \nbetween two ferromagnetic electrodes .    \nTo investigate the mechanism \nbehind the Ta effect, we conduc ted FMR \nstudy on tunnel junctions where the \nferromagnetic  electrodes on both sides of \nthe AlOx tunneling barrier were the same. \nWe studied the FMR response from a \ntunnel junction comprising of Ta(2 \nnm)/Co(5 nm)/NiFe (5 nm)/AlOx( 2 \nnm)/NiFe (10 nm)  (Fig. 4 ). This \nconfiguration essentially showed that \nbottom ferromagnetic electrode of the \nMTJ and MTJ -Ta is present on the both \nsides of AlOx  insulator. The FMR spectra \nfor this tunnel junction configuration revealed antiferromagnetic coupling between the two ferromagnetic \nelectrodes via AlOx insulator. The existence of this antiferromagnetic coupling is evident from the \nlocations of acoustic a nd optical modes  [13]. In this case, a coustic mode appeared at 1025 Oe, and optical \nmode existed at 3125 Oe  (Fig. 4) . The resonance peaks for this  configuration are ~2000 Oe apart, whereas \ntwo resonance modes for the MTJ and MTJ -Ta were positioned at a gap of ~340 Oe only.  \nWe also studied FMR on the tunnel junction with NiFe (10 nm)/AlOx (2 nm)/NiFe (1 0 nm) \nconfiguration  (Fig. 4) . It is noteworthy that the top and bottom ferromagnetic electrodes of this tunnel \njunction are similar to the ferromagnetic film utilized in the MTJ and MTJ -Ta. This tunnel junction also \nexhibited antiferromagneti c coupling between two NiFe ferromagnetic electrodes. For this configuration , \nacoustic mode appeared before the optical mode. Acoustic mode appeared at ~1130 Oe, and optical mode \nFig. 4: FMR of magnetic tunnel junctions with  \nTa(2 nm)/Co(5 nm)/NiFe (5 nm)/AlOx( 2 nm)/NiFe (10 nm) and  \nNiFe(10 nm)/AlOx (2 nm)/NiFe (10 nm) configurations.    \n8 \n \nFig. 5: Polarized beam neutron reflectivity study of (a) MTJ and (b) MTJ -Ta samples. Solid line \nshow fitted curve on the experimental d ata. R -- correspond to reflectivity from nuclear profile minus \nreflectivity from magnetic profile. R++ corresponds to reflectivity from nuclear profile plus \nreflectivity from magnetic profile. (c) Nuclear and magnetic scattering length density (ρ) vs. MTJ \nand MTJ -Ta thickness. (d) Comparison of modeled and expected nuclear and magnetic scattering \nlength density data for the MTJ.  \nappeared at 1230 Oe. Hence, the difference between two modes for this tunnel junction was only ~100 Oe \nthat is significantly smaller than that for MTJ and MTJ -Ta, i.e. 340 Oe . This tunnel junction  with similar \nferromagnetic electrodes also possessed antiferromagnetic coupling  [13]. Adding Ta on the top of this \ntunnel junction did not produce the change in the IFMEC.  It appears that impact of Ta is pronounced for \nthe magnetic tunnel junction with the dissimilar magnetic electrodes.  \nTo understand the impact of Ta along the depth of the MTJ , we conducted PNR studies. We \nhypothesized that  magnetic attributes of the MTJ’s top electrode should  be affected by the Ta layer. We \nemployed Polarized Neutron Reflectometry (PNR) to investigate the difference in magnetic attributes of \nthe MTJ and MTJ -Ta. Under this experiment , a polarized beam of neutron s interacted with th e MTJ and \nMTJ -Ta. Subsequently, t he spin and angle of the neutron beam reflected by the samples were analyzed. 9 \n We utilized nuclear reflectivity and n on- spin flip reflectivity  to calculate R -- and R++. Here, R -- \ncorresponds to the difference in reflectivity  from nuclear and magnetic profiles (Fig. 5). Whereas, the \nR++ correspond to the sum of reflectivity  due to nucle ar and magnetic profiles (Fig. 5 ).  The R -- and R++ \nvs. wave  vector graphs for the M TJ and MTJ -Ta are shown in Fig 5a and Fig 5 b, respectively. The \nreflectivity profile for the MTJ (Fig. 5a) is diffe rent than that of MTJ -Ta (Fig. 5 b). We fitted the \nexperimental data  with the scattering length density model to record the depth -wise changes in MTJ  and \nMTJ -Ta magnetization  (Fig.  5c). For MTJ -Ta m agnetic signal in the 220 to 230 nm thickness range \nsuggests that Ta gained magnetic moment. However, the magnetization in the adj acent NiFe’s region \ndecreased ; the NiFe/Ta region for the magnetic sig nal in Fig. 5 c indicates this possibility . Howe ver, for \nthe MTJ sample magnetic signal for the top NiFe electrode has expected profile in the 22 0-230 nm \nthickness range (Fig. 5 c). This PNR observation is in agreement with the prior studie s which reported that \na Ta film deposited on the NiFe gained magn etic moment  [16, 17]. The reduction in magnetic  moment  of \ntop NiFe electrode is a lso in agreement with the decrease  in the intensity of the acoustic mode of MTJ as \ncompared to MTJ -Ta (Fig. 2). Although  Fig 5 c represents the best fit for the nuclear and magnetic data , \nwe do not believe the model ed data is perfectly accurate. To estimate  the degree of deviation we \ncompared the modeled data for the MTJ sampl e with the expected data (Fig. 5 d). Expected data was \ncalculated for the ideal MTJ with perfect interfaces and atomically smoot h films. We attribute the \ndifference between modeled and e xpected data to \nthe significantly high roughness and diffusive \ninterfaces  between AlOx/NiFe (top) and \nNiFe(top)/Ta . Prior study  [8] and our AFM study \nshowed that growth of AlOx induced high \nroughness.  \nIn the quest of getting additional insights, \nwe attempted to fit PNR data by fixing the thin fil m thickness in the  scattering length density  model (Fig. \nFig. 6: Reflectance vs. wavelength graph for \nMTJ and MTJ -Ta. \n10 \n 1S, Supplementary material). The regions of nuclear and magnetic scattering length density for the \nbottom electrode (Ta/Co/NiFe) were in agreement with the expected thickness regions for the individual \nfilms. However, the nuclear and magnetic scattering length density did not show the  good fit  in sections \ncorresponding to top NiFe  and Ta films. This PNR study agrees with our hypothesis that issues  mainly \nstart after the deposition of AlOx. The PNR data provided in  the Fig. 1S of the s upplementary material  \nindicate d that magnetic moment was also prese nt in the Ta region of MTJ -Ta. Future study may \nemphasize on  producing smoothe r top AlOx and improved PNR modeling .    \nEven though roughness in the magnetic tun nel junctions  impacted PNR mode ling, but we do not \nbelieve this roughness level affected  the integrity of magnetic tunnel junctions.  The most delicate part of \nthe magnetic tunnel junction is the AlOx  tunneling barrier. We  found that 3 -5 nm level roughness is not \ndetrim ental to the integrity of tunnel barrier that separates the two ferromagnetic layers. For the \nvalidation, we produced tunnel junctions for the transport study by following the method described \nelsewhere  [7]. The transport study conducted on \ntunnel junctions exhibited excellent tun neling \nbehavior (Supplementary m aterial -Fig.2S). The \npresence of tunneling response confirms that \ntunneling barrier is in good condition and \nunaffected by the level of roughness observed \nin our sample.  \nWe conducted additional experiment to \njudge the quality of MTJ and MTJ -Ta. To make \nsure that top NiFe and Ta films were  continuous \nwe conducted light reflectivity study  on MTJ and MTJ -Ta. We hypothesized that a continuous Ta film on  \nthe top of MTJ -Ta must produce clearly noticeable effects.  The reflectivity data for MTJ and MTJ -Ta \nfollowed the similar trend (Fig. 7). The reflectivity data was consistent with the reflectivity profile from a \nFig. 7 : Magnetic moment of MTJ and MTJ -Ta \nbefore and after treating with OMCs.   \n11 \n continuous NiFe film [18].  However, the reflectivity data below ~450 nm was higher for the MTJ -Ta as \ncompared to the MTJ sample  (Fig. 6). It appears that Ta on top has a  higher reflectivity for light radiation \nbelow ~450 nm only. We also noted that peaks of reflectivity  data for MTJ -Ta were  ~10 nm ahead of the \nMTJ samples  for 200 to 450 nm range. S uch a shift in the position of the reflectivity peaks for short \nwavelength is observed when the top NiFe film is cover ed with a conformal film of different composition \n[18]. This study suggests that although NiFe may be rough , but Ta has formed a conformal film on the \ntop.         \nTo demonstrate the application of maneuvering IFMEC with Ta, w e studied the impact of Ta top \nlayer on the magnetic properties of the magnetic tunnel junction based molecular devices  (MTJMSD) . \nThe MTJMSD  were produced by bridging the paramagnetic molecules between the ferroma gnetic \nelectrodes of the MTJ and MTJ -Ta [19]. These paramagnetic molecules are essentially organometallic \nmolecular clusters (OMCs). For this study, the MTJ and MTJ -Ta samples discussed in Fig. 3, were \nutilized. The method of molecule attachment was described elsewhere [6, 7] . The attributes of  OMC \nparamagnetic molecules have also been published  elsewhere  [20]. An OMC molecule contained an octa -\nnuclear cubic cage with a net spin state. Every corner of the OMC’s cub ic cage possessed an alkane \ntether. At  the end of each alkane tether, a thiol functional group was provided. The thiol functional groups \nhelped to bridge the molecules across the ~ 2 nm AlOx tunnel barrier along the exposed edges of the \ntunnel junctions. Each thiol group had a strong affinity towards the NiFe ferromagnetic electrode of the \nMTJ and MTJ -Ta. The magnetic study showed that OMCs created remarkably stronger exchange \ncoupling as compared to AlOx  tunnel barrier.  It is noteworthy that MTJ and MTJ -Ta samples contained \nseveral thousand tunnel junctions to yield the high signal to noise ratio during magnetic studies. To study \nthe paramagnetic  molecule effect the magnetic moment were  measured for  ±2000 Oe field range at 150 K \ntemperature. The magnetization study showed that OMC channels across the AlOx insulator on the \nexposed sides produced the opposite effect s on MTJ and MTJ -Ta (Fig. 7). The magnetic mom ent of MTJ \ndropped nearly by ~84 % (Fig. 7 ); it must be noted that MTJ had preexisting antiferromagnetic coupling 12 \n between the ferromagnetic electrodes. On the other hand, OMCs increased the magnetic moment of the \nMTJ -Ta by ~116%; it is noteworthy that MTJ -Ta possessed pre -existing ferromagnetic coupling  between \nthe two ferromagnetic electrodes (Fig. 7). This study indicates that preexisting IFMEC  is important in \ndetermining the magnetic properties of the  molecular spintronics devices.  We have conducted Monte \nCarlo study [19] to get qualitative understanding; however, simulation studies with continuous spin \nmodels are recommended to investigate the impact on preexisting IFMEC.     \nHere w e propose  the potential mechanism behind the Ta induced changes in the IFMEC of the \nMTJ. The previous observations of the reversal of IFMEC are re ported due to variation in the property of \nthe tunneling barriers [1]. However, it is noteworthy that in our study the tunneling barrier was grown with \nthe same procedure for MTJ and MT J-Ta. Hence, tunneling barrier is not expected to play a role in \nimpacting IFMEC after the addition of Ta on MTJ. In our case , the magnetic interaction via the tunneling \nbarrier is expected to be governed by the exchange coupling. Prior research showed tha t in the case of \nmagnetic tunn el junctions exchange coupling was the most dominant [21] . The exchange coupling \nstrength decreases exponentially [22] with the tunneling barrier thickness . IFMEC  is also sensitive \ntowards the crystallinity of the tunneling barrier [21, 23]. Our AlOx tunneling barrier growth method is \nbased on prior work that leads to amorphous tunneling barrier [8]. In fact , AlOx tunneling barrier is by \ndefault  amorphous [4, 23] and it is extremely challenging to get crystalline AlOx  tunneling barrier  [8].   \nBased on the experimental studies and prior literature we hypothesized the following mechanism \nbehind the Ta effect on IFMEC. The reduced intensity of NiFe after the deposition of Ta may be due t o \nthe increased damping factor. We surmise that increase in damping may be associated with the creation of \na dead layer at NiFe/Ta interface [16, 17, 24]. In this paper, we also proposed a mechanism based on the \npresences of a dead layer at NiFe/Ta interface  [16, 17, 24]. Ta is found to  acqui re ~0.34 -0.56 µ B magnetic \nmoment when deposited on NiFe surface  14,15. These Ta atoms established antiferromagnetic  coupling \nwith the Ni atoms near the NiFe surface region. This antiferromagnetic coupling between the acquired \nmagnetic moment in Ta  and Ni atoms near the NiFe surface yielded a dead layer  of ~ 2 nm thickness  that 13 \n does not possess any net magnetic moment  [17]. Based on the  prior study14,15 the following mechanism is \nhypothesized  about the Ta effect on IFMEC .   \nOn a MTJ -Ta sample , the addition of Ta layer \nappears to gain magnetic moment by diminishing the spin \ndensity of NiFe  (Fig. 8). Since, Ta layer acquire s a net \nmagnetic moment14,15 hence it is imperative that Ta layer \ncan only pick majority or minority spin density  from NiFe , \nnot both. If it picks both type s of spins , then there may not \nbe any net magnetic moment as reported by previous \nstudies 14,15 and also seen i n our PNR study (Fig. 5c).   Furthermore, we conjecture that FMR  resonance \npeaks for the ferromagnets mainly depend  on the majority spins  only. In fact , FMR theoretical studies  \nhave mainly account ed for majority  types of spin [25, 26] and ignor ed minority spin population . It is also \nwell established that when two ferromagnets are coupled antiferromagnetically , then it means that the \nmajority spins of the two ferromagnets are antiparallel  to each other  (Fig. 8a). For a MTJ sample , with \nantiferromagnetic IFMEC, the majority spin density  (minority spin density) of the top NiFe electrode was \nantiparallel (parallel) to the majority spins of the bottom ferromagnetic electrode  (Fig. 8a). We \nhypothesize that after addition of Ta, a fraction of the NiFe’s majority spins move d into the Ta layer  (Fig. \n8b). As a result, Ni Fe’s major spin density depleted  and became lower than t he minor spin density (Fig. \n8b). Subsequently, the mi nor spin density of NiFe became  new major spin density due to the presence of \nTa (Fig.  8b). It is noteworthy that  NiFe’s  new majority spin after Ta addition is parallel to the majority \nspins of the bottom electrode of the resultant MTJ -Ta. This new configuration is tantamount to \nferromagnetic IFMEC on MTJ -Ta. Hence, the ferromagnetic coupling observed on MTJ -Ta is due to Ta \ninduced rearrange ment of the majority spin density of states  on the top NiFe electrode  (Fig.  8b). One can \nsee that Ta effect is not possible if minority spins from the top NiFe enter in the Ta layer. In that case , \nNiFe’s original majority spin remains unc hanged before and after the ad dition of Ta layer. As a result,  the \nFig.8: Ta effect on spin density of top \nferromagnetic (FM) electrode an d on the \nchange in IFMEC.  \n14 \n nature of IFMEC will also not change. Conceptually , Ta induced IFMEC reversal is only possible when \nmajority spins from the NiFe enter in the Ta layer.  This hypothesis is  in agreement with  the difference in \nthe intensities of the first peak for MTJ and MTJ -Ta. For MTJ -Ta the maximum intensity of the first peak \nis ~15191 around 598 Oe  (Fig. 2) . However, for MTJ  the intensity of the first peak is 62720 at 620 Oe  \n(Fig. 2). The addition of Ta appears to influence the population of spins responsible for producing the \nacoustic mode of the MTJ sample. According to FMR theory peak intensity is directly associated with the \nmagnetic moment of the ferromagnetic  electrode s[10, 14].  \nIV CONCLUSIONS:    \nThis paper discussed the effectiveness of Ta layer in changing the inter -ferromagnetic electrode \nexchange coupling. FMR was found to be us eful in recording the subtle changes due to the addition of Ta \nlayer. The magnetization study was only able to register very small change due to Ta. We also conducted \nneutron scattering studies on MTJ and MTJ -Ta sample s. These neutron studies observed  the moderate \nchange in the magnetic attributes in the top Ta layer and neighboring NiFe region.  We found that \nferromagnetic resonance is extremely sensitive for studying the effect of the change in electrode \ncomposition on the inter -electrode exchange coupling . We found that the ability to change the nature o f \ninter-electrode coupling of a  magnetic tunnel junction  can impact the resulting properties of the molecular \nspintronics devices. We observed that paramagnetic molecules decreased the magnetic moment of th e \nMTJ with pre -existing antiferromagnetic coupling. However, the same paramagnetic molecules increase d \nthe magnetic moment of the MTJ -Ta with preexisting ferromagnetic exchange coupling.  At present w e \nare unsure about the mechanism by which the  nature of Ta and NiFe interaction influences the IFMEC. \nFirst principle calculations are expected to shine light about the underlying mechanism.   \nSUPPLEMENTARY MATERIAL  \nFigure 1S showing the additional PNR results and Figure 2S showing the tunneling type transport via the \nAlOx tunnel barrier is provided in the supplementary material file.  15 \n ACKNOWLEDGEMENTS:   \nPawan Tyagi thank Dr. Bruce Hinds and Department of Chemical and  Materials engineering at the \nUniversity of Kentucky for facilitating experimental work on molecular spintronics during his Ph.D. \nOMC was produced Dr. Stephen Holmes’s group. The preparation of this paper and complementary \nexperiments were in part supporte d by National Science Foundation -Research Initiation Award (Contract \n# HRD -1238802), Department of Energy/ National Nuclear Security Agency (Subaward No. 0007701 -\n1000043016), and  Air Force Office of Sponsored Research (Award #FA9550 -13-1-0152). We also t hank \nCentre of Nanoscience and T echnology, NIST Gaithersburg for allowing the use of microscopy resou rces. \nWe also acknowledge Dr. Brian Kirby of NIST Center of Neutron Reflectivity for the polarized beam \nreflectivity study.  We also thank STEM C enter at UD C for providing the partial funding. Any opinions, \nfindings, and conclusions expressed in this material are those of the author(s) and do not necessarily \nreflect the views of any funding agency and corresponding author’s affiliations and collaborators.    \nREFERENCES:  \n[1] J.J.I. Wong, L. Ramirez, A.G. Swartz, A. Hoff, W. Han, Y. Li, R.K. 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Tsymbal, T. Katayama, S. Yuasa, \nDefect -mediated properties of magnetic tunnel junctions, IEEE  Trans. Magn., 43 (2007) 2770 -2775.  \n[24] M. Kim, W.T. Geng, A.J. Freeman, L.P. Zhong, J. Fernandez -de-Castro, First -principles calculations for \nthe structural and magnetic properties of ordered NiFe(001) thin films with and without a Ta overlayer, \nJ. App. Phys., 87 (2000) 5735 -5737.  \n[25] A. Layadi, Ferromagnetic resonance modes in coupled layers with cubic magnetocrystalline \nanisotropy, J. App. Phys., 83 (1998) 3738 -3743.  \n[26] J. Lindner, K. Baberschke, Ferromagnetic resonance in coupled ultrathin films, Jo urnal of Physics -\nCondensed Matter, 15 (2003) S465 -S478.  \n \n "    },    {        "title": "0708.3323v1.Enhancement_of_the_Gilbert_damping_constant_due_to_spin_pumping_in_noncollinear_ferromagnet_nonmagnet_ferromagnet_trilayer_systems.pdf",        "content": "arXiv:0708.3323v1  [cond-mat.mes-hall]  24 Aug 2007Enhancement of the Gilbert damping constant due to spin pump ing in non-collinear\nferromagnet / non-magnet / ferromagnet trilayer systems\nTomohiro Taniguchi1,2, Hiroshi Imamura2\n1Institute for Materials Research, Tohoku University, Send ai 980-8577,\n2Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology,\n1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan\n(Dated: October 29, 2018)\nWe analyzed the enhancement of the Gilbert damping constant due to spin pumping in non-\ncollinear ferromagnet / non-magnet / ferromagnet trilayer systems. We show that the Gilbert\ndamping constant depends both on the precession angle of the magnetization of the free layer and\non the direction of the magntization of the ﬁxed layer. We ﬁnd the condition to be satisﬁed to\nrealize strong enhancement of the Gilbert damping constant .\nPACS numbers: 72.25.Mk, 75.70.Cn, 76.50.+g, 76.60.Es\nThere is currently great interest in the dynamics of\nmagnetic multilayers because of their potential applica-\ntions in non-volatile magnetic random access memory\n(MRAM) and microwave devices. In the ﬁeld of MRAM,\nmuch eﬀort has been devoted to decreasing power con-\nsumption through the use of current-induced magnetiza-\ntion reversal (CIMR) [1, 2, 3, 4, 5, 6, 7]. Experimentally,\nCIMR is observed as the current perpendicular to plane-\ntype giant magnetoresistivity (CPP-GMR) of a nano pil-\nlar, in which the spin-polarized current injected from the\nﬁxed layer exerts a torque on the magnetization of the\nfree layer. The torque induced by the spin current is\nutilized to generate microwaves.\nThe dynamics of the magnetization Min a ferromag-\nnet under an eﬀective magnetic ﬁeld Beﬀis described by\nthe Landau-Lifshitz-Gilbert (LLG) equation\ndM\ndt=−γM×Beﬀ+α0M\n|M|×dM\ndt,(1)\nwhereγandα0are the gyromagnetic ratio and the\nGilbert damping constant intrinsic to the ferromagnet,\nrespectively. The Gilbert damping constant is an im-\nportant parameter for spin electronics since the critical\ncurrent density of CIMR is proportional to the Gilbert\ndamping constant [8, 9] and fast-switching time magne-\ntization reversal is achieved for a large Gilbert damp-\ning constant [10]. Several mechanisms intrinsic to ferro-\nmagnetic materials, such as phonon drag [11] and spin-\norbit coupling [12], have been proposed to account for\nthe origin of the Gilbert damping constant. In addition\nto these intrinsic mechanisms, Mizukami et al.[13, 14]\nand Tserkovnyak et al.[15, 16] showed that the Gilbert\ndampingconstantinanon-magnet(N) /ferromagnet(F)\n/ non-magnet(N) trilayersystem is enhanced due to spin\npumping. Tserkovnyak et al.[17] also studied spin pump-\ning in a collinear F/N/F trilayer system and showed that\nenhancement of the Gilbert damping constant depends\non the precession angle of the magnetization of the free\nlayer.\nOn the other hand, several groups who studied CIMR\nin a non-collinear F/N/F trilayer system in which theFIG. 1: (Color online) The F/N/F trilayer system is schemat-\nically shown. The magnetization of the F 1layer (m1) pre-\ncesses around the z-axis with angle θand angular velocity ω.\nThe magnetization of the F 2layer (m2) is ﬁxed with tilted\nangleρ. The precession of the magnetization in the F 1layer\npumpsspin current Ipump\nsintotheNandF 2layer, andcreates\nthe spin accumulation µNin the N layer. The spin accumu-\nlation induces the backﬂow spin current Iback(i)\ns(i= 1,2).\nmagnetization of the free layer is aligned to be perpen-\ndicular to that of the ﬁxed layer have reported the reduc-\ntion of the critical current density [5, 6, 7]. Therefore, it\nis intriguing to ask how the Gilbert damping constant is\naﬀected by spin pumping in non-collinear F/N/F trilayer\nsystems.\nIn this paper, we analyze the enhancement of the\nGilbert damping constant due to spin pumping in non-\ncollinear F/N/F trilayer systems such as that shown in\nFig. 1. Following Refs. [15, 16, 17, 18], we calculate the\nspin current induced by the precession of the magnetiza-\ntion of the free layer and the enhancement of the Gilbert\ndamping constant. We show that the Gilbert damping\nconstant depends not only on the precession angle θof\nthe magnetization of a free layer but also on the angle ρ\nbetweenthemagnetizationsoftheﬁxedlayerandthepre-\ncession axis. The Gilbert damping constant is strongly\nenhanced if angles θandρsatisfy the condition θ=ρor\nθ=π−ρ.\nThe system we consider is schematically shown in Fig.\n1. A non-magnetic layer is sandwiched between two fer-\nromagnetic layers, F 1and F 2. We introduce the unit2\nvectormito represent the direction of the magnetiza-\ntion of the i-th ferromagnetic layer. The equilibrium\ndirection of the magnetization m1of the left free fer-\nromagnetic layer F 1is taken to exist along the z-axis.\nWhen an oscillatingmagnetic ﬁeld is applied, the magne-\ntization of the F 1layer precesses around the z-axis with\nangleθ. The precession of the vector m1is expressed\nasm1= (sinθcosωt,sinθsinωt,cosθ), whereωis the\nangular velocity of the magnetization. The direction of\nthe magnetization of the F 2layer,m2, is assumed to be\nﬁxed and the angle between m2and thez-axis is repre-\nsented byρ. The collinear alignment discussed in Ref.\n[17] corresponds to the case of ρ= 0,π.\nBefore studying spin pumping in non-collinear sys-\ntems, we shall give a brief review of the theory of\nspin pumping in a collinear F/N/F trilayer system [17].\nSpin pumping is the inverse process of CIMR where the\nspin current induces the precession of the magnetization.\nContrary to CIMR, spin pumping is the generation of\nthe spin current induced by the precession of the mag-\nnetization. The spin current due to the precession of the\nmagnetization in the F 1layer is given by\nIpump\ns=/planckover2pi1\n4πg↑↓m1×dm1\ndt, (2)\nwhereg↑↓is a mixing conductance [18, 19] and /planckover2pi1is the\nDirac constant. Spins are pumped from the F 1layer\ninto the N layer and the spin accumulation µNis cre-\nated in the N layer. Spins also accumulate in the F 1\nand F 2layers. In the ferromagnetic layers the trans-\nverse component of the spin accumulation is assumed to\nbe absorbed within the spin coherence length deﬁned as\nλtra=π/|k↑\nFi−k↓\nFi|, wherek↑,↓\nFiis the spin-dependent\nFermi wave number of the i-th ferromagnet. For fer-\nromagnetic metals such as Fe, Co and Ni, the spin co-\nherence length is a few angstroms [20]. Hence, the spin\naccumulation in the i-th ferromagnetic layer is aligned to\nbe parallel to the magnetization, i.e., µFi=µFimi. The\nlongitudinal component of the spin accumulation decays\non the scale of spin diﬀusion length, λFi\nsd, which is of the\norder of 10 nm for typical ferromagnetic metals [21].\nThe diﬀerence in the spin accumulation of ferromag-\nnetic and non-magnetic layers, ∆ µi=µN−µFimi(i=\n1,2), induces a backﬂow spin current, Iback(i)\ns, ﬂowing\ninto both the F 1and F 2layers. The backﬂow spin cur-\nrentIback(i)\nsis obtained using circuit theory [18] as\nIback(i)\ns=1\n4π/braceleftbigg2g↑↑g↓↓\ng↑↑+g↓↓(mi·∆µi)mi\n+g↑↓mi×(∆µi×mi)/bracerightbig\n,(3)\nwhereg↑↑andg↓↓are the spin-up and spin-down con-\nductances, respectively. The total spin current ﬂowing\nout of the F 1layer is given by Iexch\ns=Ipump\ns−Iback(1)\ns\n[17]. The spin accumulation µFiin the F ilayer is ob-\ntained by solving the diﬀusion equation. We assume\nthat spin-ﬂip scattering in the N layer is so weak thatwe can neglect the spatial variation of the spin current\nwithin the N layer, Iexch\ns=Iback(2)\ns. The torque τ1\nacting on the magnetization of the F 1layer is given by\nτ1=Iexch\ns−(m1·Iexch\ns)m1=m1×(Iexch\ns×m1). For\nthe collinear system, we have\nτ1=g↑↓\n8π/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\nm1×dm1\ndt,(4)\nwhereν= (g↑↓−g∗)/(g↑↓+g∗) is the dimensionless\nparameter introduced in Ref. [17]. The Gilbert damping\nconstant in the LLG equation is enhanced due to the\ntorqueτ1asα0→α0+α′with\nα′=gLµBg↑↓\n8πM1dF1S/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\n,(5)\nwheregLis the Land´ e g-factor,µBis the Bohr magneton,\ndF1is the thickness of the F 1layer andSis the cross-\nsection of the F 1layer.\nNext, we move on to the non-collinear F/N/F trilayer\nsystem with ρ=π/2, in which the magnetization of the\nF2layer is aligned to be perpendicular to the z-axis. Fol-\nlowing a similar procedure, the LLG equation for the\nmagnetization M1in the F 1layer is expressed as\ndM1\ndt=−γeﬀM1×Beﬀ+γeﬀ\nγ(α0+α′)M1\n|M1|×dM1\ndt,(6)\nwhereγeﬀandα′are the eﬀective gyromagnetic ratio\nand the enhancement of the Gilbert damping constant,\nrespectively. The eﬀective gyromagnetic ratio is given by\nγeﬀ=γ/parenleftbigg\n1−gLµBg↑↓νcotθcosψsinωt\n8πMdF1Sǫ/parenrightbigg−1\n,(7)\nwhere cosψ= sinθcosωt=m1·m2and\nǫ= 1−ν2cos2ψ−ν(cot2θcos2ψ−sin2ψ+sin2ωt).(8)\nThe enhancement of the Gilbert damping constant is ex-\npressed as\nα′=gLµBg↑↓\n8πMdF1S/parenleftbigg\n1−νcot2θcos2ψ\nǫ/parenrightbigg\n.(9)\nItshouldbenotedthat, fornon-collinearsystems, both\nthegyromagneticratioandtheGilbert dampingconstant\nare modiﬁed by spin pumping, contrary to what occurs\nin collinear systems. The modiﬁcation of the gyromag-\nnetic ratio and the Gilbert damping constant due to spin\npumping can be explained by considering the pumping\nspincurrentandthe backﬂowspincurrent[SeeFigs. 2(a)\nand 2(b)]. The direction of the magnetic moment car-\nried by the pumping spin current Ipump\nsis parallel to\nthe torque of the Gilbert damping for both collinear and\nnon-collinear systems. The Gilbert damping constant is\nenhanced by the pumping spin current Ipump\ns. On the\notherhand, the directionofthe magneticmoment carried3\nFIG. 2: (Color online) (a) Top view of Fig. 1. The dotted\ncircle in F 1represents the precession of magnetization M1\nand the arrow pointing to the center of this circle represent s\nthe torque of the Gilbert damping. The arrows in Ipump\nsand\nIback(1)\nsrepresent the magnetic moment of spin currents. (b)\nThe back ﬂow Iback(1)\nshas components aligned with the di-\nrection of the precession and the Gilbert damping.\nby the backﬂow spin current Iback(1)\nsdepends on the di-\nrection of the magnetization of the F 2layer. As shown in\nEq. (3), the backﬂowspin current in the F 2layerIback(2)\ns\nhas a projection on m2. Since we assume that the spin\ncurrent is constant within the N layer, the backﬂow spin\ncurrent in the F 1layerIback(1)\nsalso has a projection on\nm2. For the collinear system, both Ipump\nsandIback(1)\ns\nare perpendicular to the precession torque because m2\nis parallel to the precession axis. However, for the non-\ncollinear system, the vector Iback(1)\nshas a projection on\nthe precession torque, as shown in Fig. 2(b). Therefore,\nthe angular momentum injected by Iback(1)\nsmodiﬁes the\ngyromagnetic ratio as well as the Gilbert damping in the\nnon-collinear system.\nLet us estimate the eﬀective gyromagnetic ratio using\nrealistic parameters. According to Ref. [17], the con-\nductancesg↑↓andg∗for a Py/Cu interface are given\nbyg↑↓/S= 15[nm−2] andν≃0.33, respectively. The\nLand´ eg-factor is taken to be gL= 2.1, magnetization is\n4πM= 8000[Oe] and thickness dF1= 5[nm]. Substitut-\ning these parameters into Eqs. (7) and (8), one can see\nthat|γeﬀ/γ−1| ≃0.001. Therefore, the LLG equation\ncan be rewritten as\ndM1\ndt≃ −γM1×Beﬀ+(α0+α′)M1\n|M1|×dM1\ndt.(10)\nThe estimated value of α′is of the order of 0.001. How-\never, we cannot neglect α′since it is of the same order\nas the intrinsic Gilbert damping constant α0[22, 23].\nExperimentally, the Gilbert damping constant is mea-\nsuredasthe width ofthe ferromagneticresonance(FMR)\nabsorptionspectrum. LetusassumethattheF 1layerhas\nno anisotropy and that an external ﬁeld Bext=B0ˆzisapplied along the z-axis. We also assume that the small-\nangle precession of the magnetization around the z-axis\nis excited by the oscillating magnetic ﬁeld B1applied in\nthexy-plane. The FMR absorption spectrum is obtained\nas follows [24]:\nP=1\nT/integraldisplayT\n0dtαγMΩ2B2\n1\n(γB0−Ω)2+(αγB0)2,(11)\nwhere Ω is the angular velocity of the oscillating mag-\nnetic ﬁeld, T= 2π/Ω andα=α0+α′. Sinceαis very\nsmall, the absorption spectrum can be approximately ex-\npressedasP∝α0+∝an}bracketle{tα′∝an}bracketri}htandthehighestpointofthepeak\nproportional to ∝an}bracketle{t1/(α0+α′)∝an}bracketri}ht, where∝an}bracketle{tα′∝an}bracketri}htrepresents the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant. In Fig. 3(a), the time-averaged value\n∝an}bracketle{tα′∝an}bracketri}htfor a non-collinear system in which ρ=π/2 is plot-\nted by the solid line as a function of the precession an-\ngleθ. The dotted line represents the enhancement of\nthe Gilbert damping constant α′for the collinear system\ngiven by Eq. (5). The time-averaged value of the en-\nhancement of the Gilbert damping constant ∝an}bracketle{tα′∝an}bracketri}httakes\nits maximum value at θ= 0,πfor the collinear system\n(ρ= 0,π). Contrary to the collinear system, ∝an}bracketle{tα′∝an}bracketri}htof the\nnon-collinear system in which ρ=π/2 takes its maxi-\nmum value at θ=π/2.\nAs shown in Fig. 2(b), the backﬂow spin current\ngives a negative contribution to the enhancement of the\nGilbert damping constant. This contribution is given by\nthe projection of the vector Iback(1)\nsonto the direction\nof the torque of the Gilbert damping, which is repre-\nsented by the vector m1×˙m1. Therefore, the condition\nto realize the maximum value of the enhancement of the\nGilbert damping is satisﬁed if the projection of Iback(1)\ns\nontom1×˙m1takes the minimum value; i.e., θ=ρor\nθ=π−ρ.\nWe can extend the above analysis to the non-collinear\nsystemwith anarbitraryvalue of ρ. After performingthe\nappropriate algebra, one can easily show that the LLG\nequation for the magnetization of the F 1layer is given\nby Eq. (6) with\nγeﬀ=γ/bracketleftBigg\n1−gLµBg↑↓νsinρsinωt(cotθcos˜ψ−cscθcosρ)\n8πMdS˜ǫ/bracketrightBigg−1\n(12)\nα′=gLµBg↑↓\n8πMdS/braceleftBigg\n1−ν(cotθcos˜ψ−cscθcosρ)2\n˜ǫ/bracerightBigg\n,\n(13)\nwhere cos ˜ψ= sinθsinρcosωt+ cosθcosρ=m1·m2\nand\n˜ǫ=1−ν2cos2˜ψ\n−ν{(cotθcos˜ψ−cscθcosρ)2−sin2˜ψ+sin2ρsin2ωt}.\n(14)\nSubstituting the realistic parameters into Eqs. (12) and\n(14), we can show that the eﬀective gyromagnetic ratio4\n\u0013\n\u0003\u0013\u0011\u0013\u0013\u0015\u001a\u0003\u0013\u0011\u0013\u0013\u0016\u0015\u0003\u0013\u0011\u0013\u0013\u0016\u001a\n\u000bD\f\n\u000bE\f\u0013\u0011\u0013\u0013\u0016\u001a\n\u0013\u0011\u0013\u0013\u0016\u0015\n\u0013\u0011\u0013\u0013\u0015\u001a/c50/c0f/c12 /c50\n/c51/c1c/c41\n/c1e\n\u0013 /c50/c0f/c12 /c50\n/c52\u0013/c50/c0f/c12/c50\n/c51/c1c/c41\n/c1e\nFIG. 3: (Color online) (a) The time-averaged value of the en-\nhancement of the Gilbert damping constant α′is plotted as a\nfunction of the precession angle θ. The solid line corresponds\nto the collinear system derived from Eq. (9). The dashed\nline corresponds to the non-collinear system derived from E q.\n(5). (b) The time-averaged value of the enhancement of the\nGilbert damping constant α′of the non-collinear system is\nplotted as a function of the precession angle θand the an-\ngleρbetween the magnetizations of the ﬁxed layer and the\nprecession axis.γeﬀcan be replaced by γin Eq. (6) and that the LLG\nequation reduces to Eq. (10). Figure 3(b) shows the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant ∝an}bracketle{tα′∝an}bracketri}htof Eq. (13). Again, the Gilbert\ndamping constant is strongly enhanced if angles θandρ\nsatisfy the condition that θ=ρorθ=π−ρ.\nIn summary, we have examined the eﬀect of spin\npumping on the dynamics of the magnetization of mag-\nnetic multilayers and calculated the enhancement of the\nGilbertdampingconstantofnon-collinearF/N/Ftrilayer\nsystems due to spin pumping. The enhancement of the\nGilbert damping constant depends not only on the pre-\ncession angle θof the magnetization of a free layer but\nalso on the angle ρbetween the magnetizations of the\nﬁxed layerand the precession axis, as shown in Fig. 3(b).\nWe have shown that the θ- andρ-dependence of the en-\nhancement of the Gilbert damping constant can be ex-\nplained by analyzing the backﬂow spin current. The con-\ndition to be satisﬁed to realizestrongenhancement of the\nGilbert damping constant is θ=ρorθ=π−ρ.\nThe authors would like to acknowledge the valuable\ndiscussions we had with Y. Tserkovnyak, S. Yakata, Y.\nAndo, S. Maekawa, S. Takahashi and J. Ieda. This work\nwas supported by CREST and by a NEDO Grant.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] 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).\n[4] A. Deac, K. J. Lee, Y. Liu, O. Redon, M. Li, P. Wang,\nJ. P. Nozi´ eres, and B. Dieny, J. Magn. Magn. Mater.\n290-291 , 42 (2005).\n[5] A. D. Kent, B. Ozyilmaz, and E. del Barco, Appl. Phys.\nLett.84, 3897 (2004).\n[6] K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86,\n022505 (2005).\n[7] T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl.\nPhys. Lett. 89, 172504 (2005).\n[8] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[9] J. Grollier, V. Cros, H. Jaﬀres, A. Hamzic, J. M. George,\nG. Faini, J. B. Youssef, H. L. LeGall, and A. Fert, Phys.\nRev. B67, 174402 (2003).\n[10] R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett.\n92, 088302 (2004).\n[11] H. Suhl, IEEE Trans. Magn. 34, 1834 (1998).\n[12] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[13] S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn.\nMater.239, 42 (2002).[14] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n[15] Y. Tserkovnyak and A. Brataas and G. E. W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[16] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n[17] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404(R) (2003).\n[18] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur.\nPhys. J. B 22, 99 (2001).\n[19] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n[20] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[21] J. Bass and W. P. Jr., J. Phys.: Condens. Matter 19,\n183201 (2007).\n[22] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[23] F. Schreiber, J. Pﬂaum, Th. M¨ uhge, and J. Pelzl, Solid\nState Commun. 93, 965 (1995).\n[24] S. V. Vonsovskii, ed., FERROMAGNETIC RESO-\nNANCE (Israel Program for Scientiﬁc Translations Ltd.,\nJersalem, 1964)."    },    {        "title": "1805.05630v1.Ferromagnetic_to_paramagnetic_transition_in_spherical_spin_glass.pdf",        "content": "Ferromagnetic to paramagnetic transition in spherical spin glass\nJinho Baik\u0003Ji Oon LeeyHao Wuz\nOctober 18, 2018\nAbstract\nWe consider the spherical spin glass model de\fned by a combination of the pure 2-spin spher-\nical Sherrington-Kirkpatrick Hamiltonian and the ferromagnetic Curie-Weiss Hamiltonian. In\nthe large system limit, there is a two-dimensional phase diagram with respect to the temperature\nand the coupling strength. The phase diagram is divided into three regimes; ferromagnetic, para-\nmagnetic, and spin glass regimes. The \ructuations of the free energy are known in each regime.\nIn this paper, we study the transition between the ferromagnetic regime and the paramagnetic\nregime in a critical scale.\n1 Introduction\nWe consider a disordered system de\fned by random Gibbs measures whose Hamiltonian is the sum\nof a spin glass Hamiltonian and a ferromagnetic Hamiltonian. Depending on the strength of the\ncoupling constant and the temperature, the system may exhibit several phases in the large system\nlimit. The paper is concerned with the \ructuations of the free energy near the boundary between\ntwo phases known as ferromagnetic and paramagnetic regimes.\nConsider the sum of the pure 2-spin spherical Sherrington-Kirkpatrick (SSK) Hamiltonian and\nthe Curie-Weiss (CW) Hamiltonian. We call this sum the SSK+CW Hamiltonian. We denote the\ncoupling constant by Jand the inverse temperature by \f. We consider the random Gibbs measure\nwith the SSK+CW Hamiltonian. The focus of this paper is on the free energy.\nThe limiting free energy was obtained non-rigorously by Kosterlitz, Thouless, and Jones [21] in\n1976. When J= 0, this formula is the explicit evaluation of the Crisanti{Sommers formula [15]\n(which was proved rigorously by Talagrand [26]) in the case of the pure 2-spin SSK. The Crisanti{\nSommers formula is the spherical version of the Parisi formula [24, 27]. The formula of Kosterlitz,\nThouless, and Jones shows a two-dimensional phase transition: see Figure 1. The three regimes are\ndetermined by the condition that max f1;1\n2\f;Jgis equal to 1 (spin glass regime),1\n2\f(paramagnetic\n\u0003Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA\nemail: baik@umich.edu\nyDepartment of Mathematical Sciences, KAIST, Daejeon, 34141, Korea\nemail: jioon.lee@kaist.edu\nzDepartment of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA\nemail: lingluan@umich.edu\n1arXiv:1805.05630v1  [math.PR]  15 May 2018-6\nJ1\n2\f\n01\n1spin glass\u0000\u0000\u0000\u0000\u0000\nparamagnetic\nferromagnetic\nFigure 1: Phase diagram for SSK+CW model. Here, \fis the inverse temperature and Jis the\ncoupling constant.\nregime) or J(ferromagnetic regime). The limiting free energy is analytic with respect to both \f\nandJin each regime, but not on the boundary.\nRecently, the authors of [6] showed that the result of Kosterlitz, Thouless, and Jones is rigorous.\nFurthermore, the authors also evaluated the distribution of the \ructuations of the free energy in\neach regime. (The case when J= 0 was obtained earlier in [4].) The order of the \ructuations\nareN\u00002=3;N\u00001;N\u00001=2and the limiting distributions are Tracy-Widom, Gaussian, and Gaussian\nin the spin glass, paramagnetic regime, ferromagnetic regime, respectively. In the same paper, the\ntransition between the spin glass regime and the ferromagnetic regime was also studied. However,\nthe other two transitions and the triple point were left open. The goal of this paper is to describe\nthe transition between the paramagnetic regime and and the ferromagnetic regime.\nAnother system which combines a spin glass and a ferromagnetic model is the SSK with an\nexternal \feld. The di\u000berence between the CW Hamiltonian and an external \feld is that one is a\nquadratic function and the other is a linear function of the spin variables. These two models are\nrelated; see [12] for a one-sided inequality. For the spin glass with external \feld, the \ructuations\nof the free energy were computed recently in [13, 14] when the coupling constant is positive (for\nboth SSK and SK (Sherrington-Kirkpatrick) cases with general spin interactions). However, the\ntransitions are not obtained except for certain large deviation results [18, 16]. One of the interests of\nthe SSK+CW model is that it is an easier model which can be analyzed in detail in the transitional\nregimes.\n1.1 Model\nLet\nSN\u00001=f\u001b= (\u001b1;\u0001\u0001\u0001;\u001bN)2RN:\u001b2\n1+\u0001\u0001\u0001+\u001b2\nN=Ng (1.1)\nbe a sphere in RNof radiusp\nN. De\fne the SSK+CW Hamiltonian by\nHN(\u001b) =HSSK\nN(\u001b) +HCW\nN(\u001b); \u001b2SN\u00001 (1.2)\n2where\nHSSK\nN(\u001b) =1p\nNNX\ni;j=1Aij\u001bi\u001bj; HCW\nN(\u001b) =J\nNNX\ni;j=1\u001bi\u001bj=J\nN NX\ni=1\u001bi!2\n: (1.3)\nHereJis the coupling constant. The random coe\u000ecients AijsatisfyAij=AjiandAij,i\u0014j, are\nindependent centered random variables. We call Aijdisorder variables. The precise conditions are\ngiven in De\fnition 1.1 below. Note that as a function of \u001b,HCW\nN(\u001b) is large when the coordinates\nof\u001bhave same sign. On the other hand, the maximizers \u001bofHSSK\nN(\u001b) depend highly on fAijg.\nWith\f > 0 representing the inverse temperature, the free energy and the partition function\nare de\fned by\nFN=1\nNlogZN; ZN=Z\nSN\u00001e\fHN(\u001b)d!N(\u001b) (1.4)\nwhere!Nis the normalized uniform measure on SN\u00001. Note that FNandZNare random variables\nsince they depend on the disorder variables Aij. The free energy and the partition function depend\non the parameters \fandJ,\nFN=FN(\f;J); ZN=ZN(\f;J): (1.5)\nSince the Curie-Weiss Hamiltonian is a quadratic function of the spin variable, we can write\nthe SSK+CW Hamiltonian as HN(\u001b) =PN\ni;j=1Mij\u001bi\u001bjwhereMij=1p\nNAij+J\nNare non-centered\nrandom variables. In terms of matrix notations,\nHN(\u001b) =\u001bTM\u001b; M =1p\nNA+J\nN11T(1.6)\nwithA= (Aij)1\u0014i;j\u0014N,1= (1;\u0001\u0001\u0001;1)T,M= (Mij)1\u0014i;j\u0014N, and\u001b= (\u001b1;\u0001\u0001\u0001;\u001bN)T. The non-\ncentered random symmetric matrix Mis an example of a real Wigner matrix perturbed by a\ndeterministic \fnite rank matrix. Such matrices are often called spiked random matrices. We will\nuse the eigenvalues of spiked random matrices in our analysis of the free energy.\nWe assume the following conditions on the disorder variables.\nDe\fnition 1.1 (Assumptions on disorder variables) .LetAij,i\u0014j, be independent real random\nvariables satisfying the following conditions:\n\u000eAll moments of Aijare \fnite and E[Aij] = 0 for alli\u0014j.\n\u000eFor alli < j ,E[A2\nij] = 1 ,E[A3\nij] =W3, and E[A4\nij] =W4for some constants W32Rand\nW4\u00150.\n\u000eFor alli,E[A2\nii] =w2for a constant w2\u00150.\nSetAij=Ajifori>j . LetA= (Aij)N\ni;j=1and we call it a Wigner matrix (of zero mean).\n3De\fnition 1.2 (Eigenvalues of non-zero mean Wigner matrices) .LetMbe theN\u0002Nsymmetric\nmatrix de\fned in (1.6) . We call it a Wigner matrix of non-zero mean1. Its eigenvalues are denoted\nby\n\u00151\u0015\u00152\u0015\u0001\u0001\u0001\u0015\u0015N: (1.7)\nWe introduce the following terminology.\nDe\fnition 1.3 (High probability event) .We say that an N-dependent event \nNholds with high\nprobability if, for any given D> 0, there exists N0>0such that\nP(\nc\nN)\u0014N\u0000D\nfor anyN\u0015N0.\n1.2 Previous results in each regime\nWe review the results on the \ructuations in each regime obtained in [6]. We state two types of\nresults: one in terms of the eigenvalues of Mand the other in terms of limiting distributions.\nSet\n~J:= maxfJ;1g: (1.8)\nIt was shown in [6] that the following holds with high probability. In both ferromagnetic and the\nspin glass regimes (given by ~J >1\n2\f), with any \u000f>0,\nFN=~FN+\u0012\n\f\u00001\n2~J\u0013\u0012\n\u00151\u0000~J\u00001\n~J\u0013\n+o(N\u00001+\u000f): (1.9)\nIn the paramagnetic regime (given by ~J <1\n2\f),\nFN=~FN\u00001\n2NNX\ni=1log\u0012\n2\f+1\n2\f\u0000\u0015i\u0013\n+o(N\u00001): (1.10)\nHere, ~FNis a deterministic function of N;\f;J . The above results show that the \ructuations of FN\nare determined, to the leading order, by the top eigenvalue \u00151in the ferromagnetic and spin glass\nregimes, while they are determined by all eigenvalues in the paramagnetic regime.\nA limit theorem for FNfollows if we use limit theorems for the eigenvalues of random matrices.\nThe relevant random matrices are Wigner matrices of non-zero mean in (1.6). For such random\nmatrices, the following is known [25, 11] (see [3] for complex matrices):\n(\nN2=3(\u00151\u00002))TW 1 ifJ <1,\nN1=2\u0000\n\u00151\u0000J\u00001\nJ\u0001\n)N (W3(J\u00002\u0000J\u00004);2(1\u0000J\u00002)) ifJ >1;(1.11)\n1In [6], we consider the case when the diagonal entries of Mhave meanJ0\nNand the o\u000b-diagonal entries have mean\nJ\nNwhereJandJ0are allowed to be di\u000berent. However, in this case, M=1p\nN+J\nN11T+J0\u0000J\nNIwhereIis the\nidentity matrix. This only shifts all eigenvalues by a deterministic small number. As we will see in Remark 2.2, it is\nnot more general than the case with J0=J.\n4where the convergences are in distribution. Here TW 1denotes the GOE Tracy-Widom distribution\nandN(a;b) denotes the Gaussian distribution of mean aand variance b. The dichotomy is due\nto the e\u000bect of the non-zero mean; if Jis not large enough (i.e. J < 1), then the in\ruence of\nthe non-zero mean is negligible to contribute to the \ructuations of the top eigenvalue. For J <1,\nthe top eigenvalue is close to the second eigenvalue with order O(N\u00002=3+\u000f). But for J > 1, the\ndi\u000berence of the top eigenvalue and the second eigenvalue is of order O(1).\nOn the other hand, the following is also known (see Theorem 1.6 of [6]): if a function 'is\nsmooth in an open interval containing the interval [ \u00002;~J+~J\u00001], then\nNX\ni=1'(\u0015i)\u0000NZ2\n\u00002'(x)d\u001bscl(x))N (f;a); d\u001bscl(x) :=p\n4\u0000x2\n2\u0019dx; (1.12)\nfor some explicit constants f;a. This result is applicable to the paramagnetic regime.\nTogether, we have the following asymptotic results obtained in Theorem 1.4 of [6] (with a small\ncorrection in [5]):\n(i) (Spin glass regime) If \f >1\n2andJ <1, then\n1\n\f\u00001\n2N2=3(FN\u0000F))TW 1: (1.13)\n(ii) (Paramagnetic regime) If \f <1\n2and\f <1\n2J, then\nN(FN\u0000F))N (f1;\u000b1): (1.14)\n(iii) (Ferromagnetic regime) If J >1 and\f >1\n2J, then\np\nN(FN\u0000F))N\u0000\nf0\n2;\u000b0\n2\u0001\n: (1.15)\nfor some deterministic function F=F(\f;J) and some explicit constants f1;\u000b1,f0\n2and\u000b0\n2depending\non\fandJ.\n1.3 Results\nWe state the results on the transition between the paramagnetic regime and the ferromagnetic\nregime. The boundary between these two regimes is given by the equation1\n2\f=JwithJ >1. In\nthe transitional regime, the correct scaling turns out to be the following: let J >1 be \fxed and let\n\f=\fNbe given by\n2\f=1\nJ+Bp\nN(1.16)\nwith \fxedB2R. The following is the \frst main result of this paper. This relates the free energy\nwith the eigenvalues of M.\n5Theorem 1.4. Let\fbe given by (1.16) . Then, for every 0<\u000f<1\n8,\nFN=~FN\u00001\n2NNX\ni=2g(\u0015i) +1\nNQ(\u001fN) +O(N\u00003=2+4\u000f); \u001fN:=p\nN(\u00151\u0000J+J\u00001);(1.17)\nwith high probability as N!1 , where\n~FN=\f(J+J\u00001)\u00001\n2\u00001\n2log(2\f) +1\nN\u00121\n4logN+ log\fp\u0019\u0013\n; g(z) := log(J+J\u00001\u0000z):(1.18)\nAlso,\nQ(x) =s(x)\n2(s(x)\u0000x)\u0000s(x)2\n4(J2\u00001)+log(s(x)\u0000x)\n2+ log I\u0012(s(x)\u0000x)2\nJ2\u00001\u0013\n(1.19)\nwith\ns(x) =x\u0000B(J2\u00001) +p\n(x+B(J2\u00001))2+ 4(J2\u00001)\n2(1.20)\nand\nI(\u000b) =Z1\n\u00001e\u0000\u000b\n4t2+it\n2p1 + itdt; (1.21)\nwhere the square root denotes the principal branch.\nThe formula (1.17) shows a combined contribution from \u00152;\u0001\u0001\u0001;\u0015Nand a distinguished contri-\nbution from \u00151. Compare the formula with (1.9) and (1.10).\nNow we state a result analogous to (1.14) and (1.15). This follows if we have limit theorems for\nQ(\u001fN) andPN\ni=2g(\u0015i). From the second part of (1.11), Q(\u001fN) converges to an explicit function of a\nGaussian random variable. On the other hand,PN\ni=2g(\u0015i) is di\u000berent fromPN\ni=1g(\u0015i) by one term.\nIt is not di\u000ecult to show that removing one term does not a\u000bect the \ructuations much and the\n\ructuations are still given by a Gaussian random variable similar to (1.12); see Theorem 2.1 in the\nnext section. In random matrix theory, these sums are known as partial linear statistic and linear\nstatistic, respectively. The main technical part of this paper is to evaluate the joint distribution of\nQ(\u001fN) andPN\ni=2g(\u0015i). We show that jointly they converge in distribution to a bivariate Gaussian\nvariable with an explicit covariance. See the next section for the precise statement. These results\nare interesting on their own in random matrix theory. Putting together, we obtain the following\nresult.\nTheorem 1.5. We have\nN\u0012\nFN\u00001\n4J2\u0000B\n2Jp\nN\u0000logN\n4N\u0000B2J2\n4N\u0013\n)G 1+Q(G2) (1.22)\nin distribution as N!1 whereG1andG2are bivariate Gaussian random variables with\nE[G1] =1\n4log(J2\u00001) +w2\u00002\n4J2+W4\u00003\n8J4+ log1\n2p\u0019J; (1.23)\nVar[G1] =\u00001\n2log(1\u0000J\u00002) +w2\u00002\n4J2+W4\u00003\n8J4; (1.24)\n6(a) pdf ofQ(G2)\n (b) pdf of normalized Q(G2)\nFigure 2: (a) Probability density function of Q(G2) forB=\u00001;0;1, (b) Probability density function\nof normalized Q(G2) resembles a Gaussian density as B!+1.\nE[G2] =W3(J\u00002\u0000J\u00004); Var[G2] = 2(1\u0000J\u00002); (1.25)\nand\nCov(G1;G2) =W3(J\u00002\u0000J\u00004)\n2: (1.26)\nNote thatG1andG2do not depend on B. The function Qis de\fned in (1.19) .\nNote that if the third moment W3ofAijwithi6=jis zero, thenG1andG2are independent\nGaussians.\nThe above result is consistent with the results on ferromagnetic and paramgnatic regimes if we\nlet formally B!+1andB!\u00001 , respectively. One can show that when B!+1,Q(G2)\ndominatesG1. Furthermore, while Q(G2) is not Gaussian, upon proper normalization, it converges\nto a Gaussian as B!+1. See Figure 2. On the other hand, when B!\u00001 , the leading two\nterms ofQ(G2) are constants and the random part is smaller than G1. See Section 6 for details.\nLet us comment on the other transitions in the phase digram in Figure 1. As mentioned before,\nthe transition between the spin glass and ferromagnetic regimes was discussed in [6]. Note that (1.9)\nis valid in both regimes. It was shown that if we let \f >1=2 be \fxed and consider N-dependent\nJ= 1 +wN\u00001=3, then for each w2R, (1.9) still holds. Now, for such J, it was shown in [9] that\nN2=3(\u00151\u00002))TW 1;wwhere TW 1;wis a one-parameter family of random variables interpolating\nTW and Gaussian distributions. Hence, we obtain the \ructuations for the transitional regime.\nOn the other hand, the transition between the spin glass and paramagnetic regimes is an open\nquestion. By matching the \ructuation scales in both regimes, we expect that the critical scale is\n\f=1\n2+O(plogN\nN1=3).\n1.4 Organization\nThe rest of the paper is organized as follows. In Section 2, we \frst state new results on random\nmatrices. They are given in Theorem 2.1 (partial linear statistics) and Theorem 2.3 (joint conver-\ngence). Using them, we derive Theorem 1.5 from Theorem 1.4. In Section 3, we prove Theorem\n71.4. In the next two sections, we prove the random matrix results stated in Section 2; Theorem 2.1\nin Section 4 and Theorem 2.3 in Section 5. In Section 6, we show that Theorem 1.5 is consistent\nwith the previous results on ferromagnetic and paramagnetic regimes.\nAcknowledgments\nThe work of Jinho Baik was supported in part by NSF grant DMS-1664692 and the Simons Fellows\nprogram. The work of Ji Oon Lee was supported in part by Samsung Science and Technology\nFoundation project number SSTF-BA1402-04.\n2 Results on Wigner matrices with non-zero mean\nIn order to prove Theorem 1.5 from Theorem 1.4, we need some new results on random matrices.\nWe need (i) a limit theorem for partial linear statisticsPN\ni=2g(\u0015i) and (ii) a joint convergence of the\nlarge eigenvalue and partial linear statistics. These results are interesting on their own in random\nmatrix theory. We state them here and prove them in Section 4 and Section 5 below. Using these\nresults, we prove Theorem 1.5 in Subsection 2.3.\nRecall that the N\u0002Nsymmetric matrix Mis given byM=1p\nNA+J\nN11TwhereA= (Aij) is a\nsymmetric matrix with independent entries for i\u0014jsatisfying the conditions given in De\fnition 1.1\nand1= (1;\u0001\u0001\u0001;1)T. The matrix Mis called a Wigner matrix with a non-zero meanJ\nN. Recall\nthat we assume\nJ >1: (2.1)\nThe eigenvalues of Mare denoted by \u00151\u0015\u0001\u0001\u0001\u0015\u0015N.\nIt is known that \u00151is close toJ+J\u00001with high probability and \u00152;\u0001\u0001\u0001;\u0015Nare in a neighborhood\nof [\u00002;2] with high probability. See Lemma 3.2 below for the precise statement.\n2.1 Partial linear statistics\nA linear statistic is the sum of a function of the eigenvalues. The \ructuations of linear statistics for\nWigner matrices and other random matrix ensembles are of central interest in the random matrix\ntheory; see, for example, [19, 2, 22]. For Wigner matrices with non-zero mean, the following result\nwas obtained in Theorem 1.6 and Remark 1.7 of [6]. Set\n^J=J+J\u00001: (2.2)\nLet':R!Rbe a function which is analytic in an open neighborhood of [ \u00002;^J] and has compact\nsupport. Then, as N!1 , the random variable\nNN(') :=NX\ni=1'(\u0015i)\u0000NZ2\n\u00002'(x)d\u001bscl(x))N (M(');V(')) (2.3)\n8where\nM(') =1\n4('(2) +'(\u00002))\u00003\n2\u001c0(')\u0000J\u00001\u001c1(') + (w2\u00002)\u001c2(')\n+ (W4\u00003)\u001c4(') +'(^J)\u00001X\n`=2J\u0000`\u001c`(');\nV(') =(w2\u00002)\u001c1(')2+ (W4\u00003)\u001c2(')2+ 21X\n`=1`\u001c`(')2:(2.4)\nHere,W4=E[A4\n12],w2=E[A2\n11], and\n\u001c`(') =1\n\u0019Z2\n\u00002'(x)T`(x=2)p\n4\u0000x2dx=1\n2\u0019Z\u0019\n\u0000\u0019'(2 cos(\u0012)) cos(`\u0012)d\u0012; (2.5)\nwhereT`(t) are the Chebyshev polynomials of the \frst kind.\nWe are interested in a partial linear statistic,PN\ni=2'(\u0015i). See [7, 23] for other types of partial\nlinear statistics. The partial linear staticPN\ni=2'(\u0015i) is the linear statistic minus one term '(\u00151).\nSince\u00151!^Jin probability (see the second part of (1.11)), by (2.3), Slutsky's theorem implies\nthat\nNX\ni=2'(\u0015i)\u0000NZ2\n\u00002'(x)d\u001bscl(x))N (M(')\u0000'(^J);V(')):\nSince this follows from (2.3), this is true assuming that 'is analytic in an open neighborhood of\n[\u00002;^J]. However, we are interested in the test function '(x) =g(x) = log( ^J\u0000x) (see (1.17)). Since\nthis function is not analytic at x=^J, the above simple argument does not apply. Nonetheless,\nif we adapt the proof of (2.3), one can show that it is enough to assume that the test function is\nanalytic in a neighborhood of the interval [ \u00002;2], not of [\u00002;^J].\nTheorem 2.1. LetJ >1. Then for every test function 'which is analytic in a neighborhood of\n[\u00002;2],\nN(2)\nN(') :=NX\ni=2'(\u0015i)\u0000NZ2\n\u00002'(x)d\u001bscl(x))N (M(2)(');V(2)(')) (2.6)\nasN!1 with\nM(2)(') =1\n4('(2) +'(\u00002))\u00003\n2\u001c0(')\u0000J\u00001\u001c1(') + (w2\u00002)\u001c2(')\n+ (W4\u00003)\u001c4(')\u00001X\n`=2J\u0000`\u001c`(');(2.7)\nandV(2)(') =V(')whereV(')is de\fned in (2.4) .\nNote that\nM(2)(') =M(')\u0000'(^J) (2.8)\nfor'analytic in a neighborhood of [ \u00002;^J].\n9Remark 2.2.We comment on a case when the test function depends on N. Consider the function\n'Nde\fned by\n'N(x) ='(x) +\u001e(x)\nN+O(N\u00002)\nuniformly for xin a neighborhood of [ \u00002;2] for analytic functions 'and\u001e. De\fne the corresponding\nlinear statisticN(2)\nN('N) =PN\ni=2'N(\u0015i)\u0000NR2\n\u00002'N(x)d\u001bscl(x), then\nN(2)\nN('N) =NX\ni=2'N(\u0015i)\u0000NZ2\n\u00002'N(x)d\u001bscl(x)\n=N(2)\nN(') +1\nN NX\ni=2\u001e(\u0015i)\u0000NZ\n\u001e(x)d\u001bscl(x)!\n+O(1\nN):(2.9)\nBy Theorem 2.1, the second order term converges to zero in probability. Thus, N(2)\nN('N) and\nN(2)\nN(') converge to the same Gaussian distribution. The same argument also applies to full linear\nstatistics; this is used in Remark 5.4 below. Now, the claim in footnote1is veri\fed by noting that\n'(x+J0\u0000J\nN) ='(x) +'0(x)(J0\u0000J)\nN+O(N\u00002).\n2.2 Joint convergence of the largest eigenvalue and linear statistics\nBy Theorem 2.1 and the second part of (1.11), the partial linear statistic and the largest eigen-\nvalue each converge to Gaussian distributions individually. The following theorem shows that they\nconverge jointly to a bivariate Gaussian with an explicit covariance.\nTheorem 2.3. LetJ >1. Then for'(x)which is analytic in a neighborhood of [\u00002;2],N(2)\nN(') :=PN\ni=2'(\u0015i)\u0000NR2\n\u00002'(x)d\u001bscl(x)and\u001fN:=p\nN(\u00151\u0000^J)converges jointly in distribution to a\nbivariate Gaussian variable with mean\n(M(2)(');W3(J\u00002\u0000J\u00004)) (2.10)\nand covariance  \nV(2)(') 2W3\u001c2(')(1\u0000J\u00002)\n2W3\u001c2(')(1\u0000J\u00002) 2(1\u0000J\u00002)!\n: (2.11)\nThe proof of this theorem, given in Section 5, is the main technical part of this paper. We prove\nthe theorem \frst for the Gaussian case, and then use an interpolation argument.\n2.3 Proof of Theorem 1.5\nWe now derive Theorem 1.5 from Theorem 1.4 using the results on the eigenvalues stated in the\nprevious two subsections. The term Q(\u001fN) converges to Q(G2) in distribution from Theorem 2.3.\nConsider the rest. It was shown in (A.5) of [4] that for g(z) = log(J+J\u00001\u0000z),\nZ\ng(z)d\u001bscl(x) =1\n2J2+ logJ: (2.12)\n10Inserting 2\f=J\u00001+BN\u00001=2and using the Taylor expansion log(1+BJp\nN) =BJp\nN\u0000B2J2\n2N+O(N\u00003=2),\n~FN\u00001\n2Z\ng(z)d\u001bscl(x) =1\n4J2+B\n2Jp\nN+logN\n4N+1\nN\u0014B2J2\n4+ log1\n2p\u0019J\u0015\n+O(N\u00003=2):(2.13)\nWe can evaluate M(2)(g) using (2.7) of [6] which evaluated the M(h) withh(x) = log(2\f+1\n2\f\u0000x):\n(note thatJ0=Jhere)\nM(2)(g) = lim\n\f!1\n2J\u0012\nM(h)\u0000log(2\f+1\n2\f\u0000J\u0000J\u00001)\u0013\n=\u00001\n2log(J2\u00001)\u0000w2\u00002\n2J2\u0000W4\u00003\n4J4:(2.14)\nThe variance V(2)(g) =V(g), which is independent of J, is given by 4 times (3.13) of [4] if we\nreplace 2\fbyJ\u00001:\nV(2)(g) =\u00002 log(1\u0000J\u00002) +1\nJ2(w2\u00002) +1\n2J4(W4\u00003): (2.15)\nFor the covariance term, we have \u001c2(g) =\u00001\n2J2from (A.17) of [4]. Hence, from Theorem 2.1 and\n2.3, we obtain the result.\n3 Proof of Theorem 1.4\nThe proof follows the steps for the proof of the Theorem 1.5 of [6] for paramagnetic and ferro-\nmagnetic regimes with necessary adjustments. The analysis is based on applying a method of\nsteepest-descent to a random integral. The location of the critical point is important. In the\ntransitional regime, the critical point is close to the largest eigenvalue but not as close as the fer-\nromagnetic case. On the other hand, the critical point is away from the largest eigenvalue in the\nparamagnetic case. See Subsection 3.2 below for details.\n3.1 Preliminaries\nThe following formula is a simple result in [21].\nLemma 3.1 ([21]; also Lemma 1.3 of [4]) .LetMbe a realN\u0002Nsymmetric matrix with eigenvalue\n\u00151\u0015\u00152\u0015\u0001\u0001\u0001\u0015\u0015N. Then for \fxed \f >0,\nZ\nSN\u00001e\f\u001bTM\u001bdwN(\u001b) =CNZ\r+i1\n\r\u0000i1eN\n2G(z)dz; G (z) = 2\fz\u00001\nNNX\ni=1log(z\u0000\u0015i); (3.1)\nwhere\ris any constant satisfying \r >\u0015 1, the integration contour is the vertical line from \r\u0000i1\nto\r+ i1, the logfunction is de\fned in the principal branch, and\nCN=\u0000(N=2)\n2\u0019i(N\f)N=2\u00001: (3.2)\nHere \u0000(z)denotes the Gamma function.\n11LetMbe a Wigner matrix with non-zero mean as in (1.6). Then its eigenvalues \u0015iare random\nvariables, and hence the above result gives a random integral representation of the partition func-\ntion. In [6, 4], the above random integral was evaluated using the method of steepest-descent for\ndi\u000berent choices of random matrices. The key ingredient in controlling the error term is a precise\nestimate for the eigenvalues which are obtained in the random matrix theory.\nLemma 3.2 (Rigidity of eigenvalues: Theorem 2.13 of [17] and Theorem 6.3 of [20]) .For each\npositive integer k2[1;N], set ^k:= minfk;N + 1\u0000kg. Let\rkbe the classical location de\fned by\nZ1\n\rkd\u001bscl(x) =1\nN\u0012\nk\u00001\n2\u0013\n: (3.3)\nThen, for every 0<\u000f<1\n2,\nj\u0015k\u0000\rkj\u0014^k\u00001=3N\u00002=3+\u000f(3.4)\nfor allk= 2;3;\u0001\u0001\u0001;Nwith high probability. Furthermore, for \fxed J >1, recall ^J=J+J\u00001,\nj\u00151\u0000^Jj\u0014N\u00001=2+\u000f(3.5)\nholds with high probability.\nFrom the rigidity, it is easy to obtain the following law of large numbers for eigenvalues.\nCorollary 3.3 (c.f. Lemma 5.1 of [4]) .Fix\u000e >0, letff\u000bg\u000b2I\u001aC1[\u00002\u0000\u000e;2 +\u000e]be a family of\nmonotonic increasing functions satisfying sup\u000b2Imaxxjf\u000b(x)j\u0014C0andsup\u000b2Imaxxjf0\n\u000b(x)j\u0014C1.\nThen, for every 0<\u000f< 1,\nsup\n\u000b2I\f\f\f\f\f1\nNNX\ni=2f\u000b(\u0015i)\u0000Z2\n\u00002f\u000b(x)d\u001bscl(x)\f\f\f\f\f=O(N\u00001+\u000f) (3.6)\nwith high probability.\nProof. Letf=f\u000bfor some\u000b2I. The absolute value on the left hand-side is bounded above by\n\f\f\f\f\f1\nNNX\ni=2f(\u0015i)\u00001\nNNX\ni=2f(\ri)\f\f\f\f\f+\f\f\f\f\f1\nNNX\ni=2f(\ri)\u0000Z2\n\u00002f(x)d\u001bscl(x)\f\f\f\f\f: (3.7)\nBy Lemma 3.2,\n\f\f\f\f\f1\nNNX\ni=2(f(\u0015i)\u0000f(\ri))\f\f\f\f\f\u0014maxjf0(x)j\nNNX\ni=2j\u0015i\u0000\rij\u0014C0\nN1\u0000\u000f(3.8)\nwith high probability. On the other hand, set ^ \rjby\nZ2\n^\rjd\u001bscl(x) =j\nN; j = 1;2;\u0001\u0001\u0001;N; (3.9)\n12and by convention ^ \r0= 2. Asf(x) is a monotonic increasing function, for i= 2;3;\u0001\u0001\u0001;N\u00001,\nZ^\ri\n^\ri+1f(x)d\u001bscl(x)\u00141\nNf(\ri)\u0014Z^\ri\u00002\n^\ri\u00001f(x)d\u001bscl(x): (3.10)\nThus,\f\f\f\f\f1\nNNX\ni=2f(\ri)\u0000Z2\n\u00002f(x)d\u001bscl(x)\f\f\f\f\f\u00143 maxjf(x)j\nN\u00143C1\nN: (3.11)\nSince the upper bounds are independent of f, we obtain the result.\n3.2 Steepest-descent analysis\nWe now apply steepest descent analysis to the integral in Lemma 3.1. We deform the contour\nto pass a critical point and show that the main contribution to the integral comes from a small\nneighborhood of the critical point. For G(z) given in (3.1), it is easy to check that all solutions of\nG0(z) = 0 are real-valued, and there is a unique critical point \rwhich lies in the interval ( \u00151;1)\n(see Lemma 4.1 of [6]).\nNote that since Gis random, the critical point is also random. For the paramagnetic regime, it\nwas shown in [6] that \r\u0000\u00151=O(1) with high probability. In the same paper, it was also shown\nthat in the ferromagnetic regime, \r\u0000\u00151=O(N\u00001+\u000f) with high probability. The following lemma\nestablishes a corresponding result for the transitional regime; it shows that \r\u0000\u00151=O(N\u00001\n2+\u000f)\nwith high probability.\nLemma 3.4 (Critical point) .Recall that (see (1.16) )J >1is \fxed and 2\f= 2\fN=1\nJ+Bp\nNwith\n\fxedB2R. Then, for every 0<\u000f<1\n4,\n\r=\u00151+1\n2p\nN\u0010\n\u0000\u001fN\u0000B(J2\u00001) +p\n(\u001fN+ (J2\u00001)B)2+ 4(J2\u00001)\u0011\n+O(N\u00001+\u000f) (3.12)\nwith high probability, where we set \u001fN:=p\nN(\u00151\u0000^J).\nNote that\rgiven above is larger than \u00151with high probability since the term in the big\nparenthesis is positive.\nProof. Set\n\u0012:=\u0000\u001fN\u0000B(J2\u00001) +p\n(\u001fN+ (J2\u00001)B)2+ 4(J2\u00001)\n2: (3.13)\nNote that\u0012>0. By the rigidity of \u00151, we havej\u001fNj\u0014N\u000f\n4and hence,\u0012\u0014N\u000f\n3with high probability.\nOn the other hand, using \u0000a+p\na2+b2=b2p\na2+b2+a,\n\u0012=2(J2\u00001)p\n(J2\u00001)B+\u001fN)2+ 4(J2\u00001) + ((J2\u00001)B+\u001fN);\nand hence\u0012\u0015CN\u0000\u000f\n4for some constant C > 0 with high probability. Hence,\nN\u0000\u000f\n3\u0014\u0012\u0014N\u000f\n3 (3.14)\n13with high probability. Set\n\r\u0006:=\u00151+\u0012p\nN\u0006N\u00001+\u000f: (3.15)\nBy the above properties of \u0012, we have\r\u0006>\u0015 1with high probability. We will show that G0(\r\u0000)<0\nandG0(\r+)>0 with high probability. Since G0(z) is a monotone increasing function for real zin\nthe interval ( \u00151;1), this shows that \r\u0000<\r <\r +with high probability, proving the lemma.\nRecall that \u00151!^Jin probability. Let us write\n\r\u0006=J+1\nJ+\u001ep\nN\u0006N\u00001+\u000f; \u001e :=\u0012+\u001fN (3.16)\nwhere\u001fN=p\nN(\u00151\u0000^J). Note that \u001e=O(N\u000f\n3) with high probability. Now, notice that\nG0(z) = 2\f\u00001\nNNX\ni=21\nz\u0000\u0015i\u00001\nN(z\u0000\u00151): (3.17)\nWe apply Corollary 3.3 to the family of the function f1\nz\u0000xgz>2+cfor some constant c > 0 and\nobtain\nG0(\r\u0006) = 2\f\u0000\r\u0006\u0000q\n\r2\n\u0006\u00004\n2+O(N\u00001+\u000f\n3)\u00001\nN(\r\u0006\u0000\u00151)\nwith high probability. By (3.16),\n\r\u0006\u0000q\n\r2\n\u0006\u00004\n2=1\nJ\u00001\nJ2\u00001\u0012\u001ep\nN\u0006N\u00001+\u000f\u0013\n+O(N\u00001+2\u000f\n3):\nBy (3.15),\n1\nN(\r\u0006\u0000\u00151)=1\n\u0012p\nN \n1\u0007N\u00001\n2+\u000f\n\u0012+O\u0012N\u00001+2\u000f\n\u00122\u0013!\n:\nUsing the formula of 2 \fand the estimate (3.14) for1\n\u0012, we \fnd that\nG0(\r\u0006) =1p\nN\u0012\nB+\u001e\nJ2\u00001\u00001\n\u0012\u0013\n\u0006\u00121\nJ2\u00001+1\n\u00122\u0013\nN\u00001+\u000f+O(N\u00001+2\u000f\n3) (3.18)\nwith high probability since 0 < \u000f <1\n4. By the de\fnition of \u0012, the leading term is zero. The\ncoe\u000ecient of the second term is positive. Hence we \fnd that G0(\r\u0000)<0 andG0(\r+)>0, and we\nobtain the lemma.\nThen we have the following lemma.\nLemma 3.5. Set\ns=sN:=p\nN(\r\u0000J\u0000J\u00001)and\u0001 = \u0001N:=p\nN(\r\u0000\u00151) =sN\u0000\u001fN. (3.19)\nThen, for every \u000f>0,\ns=\u001fN\u0000B(J2\u00001) +p\n(\u001fN+ (J2\u00001)B)2+ 4(J2\u00001)\n2+O(N\u00001\n2+\u000f) (3.20)\n14with high probability. We also have\njsj\u0014N\u000fandN\u0000\u000f\u0014\u0001\u0014N\u000f(3.21)\nwith high probability.\nProof. The previous lemma implies (3.20). The \frst part of (3.21) follows from the fact that\n\u001fN=O(N\u000f) with high probability. The second part is the estimate (3.14) in the proof of the\nprevious lemma.\nWe also need the following lemma.\nLemma 3.6. For every 0<\u000f< 1,\n1\nNNX\ni=21\n(\r\u0000\u0015i)2=1\nJ2\u00001+O(N\u00001+\u000f) (3.22)\nwith high probability.\nProof. This follows from Corollary 3.3 applied to f(x) =1\n(\r\u0000x)2.\nThe following auxiliary lemma is used to estimate an error in the steepest descent analysis.\nLemma 3.7. De\fne\nIm(\u000b) :=Z1\n\u00001tm\np1 + ite\u0000\u000b\n4t2+it\n2dt (3.23)\nfor non-negative integers mand\u000b>0, where the square root is the de\fned on the principal branch.\nWe set I(\u000b) :=I0(\u000b); see (1.21) . Then,\nI(\u000b) =r\n4\u0019\n\u000b(1 +O(\u000b\u00001))as\u000b!+1, (3.24)\nI(\u000b) =r\n8\u0019\ne(1 +O(\u000b))as\u000b!0+, (3.25)\nand for every m\u00150,\nIm(\u000b)is uniformly bounded for \u000b2(0;1). (3.26)\nA particular consequence is that the derivative I0(\u000b) =\u00001\n4I2(\u000b)is uniformly bounded for \u000b > 0.\nFurthermore, I(\u000b)>0for all\u000b>0.\nProof. Consider (3.24). Applying the method of steepest-descent to I(\u000b) =R1\n\u00001g(t)e\u000bh(t)dtwith\nh(z) =\u0000z2\n4andg(z) =1p1+izeiz\n2, we \fnd that\nI(\u000b) =e\u000bh(zc)\np\u000b\"s\n2\u0019\njh00(zc)jg(zc) +O(\u000b\u00001)#\n=r\n4\u0019\n\u000b(1 +O(\u000b\u00001)) (3.27)\n15as\u000b!+1. For Im(\u000b), usingR1\n\u00001yme\u0000\u000by2dy=O(\u000b\u0000(m+1)=2), we \fnd that\nIm(\u000b) =O(\u000b\u0000m+1\n2) as\u000b!+1. (3.28)\nConsider the limit \u000b!0+. After the change of the variables t=z=\u000b,\nI(\u000b) =e\u00001\n4\u000bp\u000bZ1\n\u00001e\u0000(z\u0000i)2\n4\u000bp\u000b+ izdz: (3.29)\nThe integrand is analytic in the complex plane minus the vertical line from i \u000bto i1. Note that the\nsaddle point is i and it is on the branch cut. We show that the main contribution to the integral\ncomes from the branch point z= i\u000b. We deform the contour so that it consists of the following\nfour line segments: L1from i\u00001 to i on the left half-plane, L2from i to i\u000blying on the left of the\nbranch cut, L3from i\u000bto i lying on the right of the branch cut, and L4from i to i +1lying on\nthe right-half plane. On L4, settingz= i +p\u000bx,\nZ\nL4e\u0000(z\u0000i)2\n4\u000bp\u000b+ izdz=p\u000bZ1\n0e\u0000x2\n4p\n\u000b\u00001 + ip\u000bxdx=O(p\u000b) (3.30)\nas\u000b!0. Similarly, the integral over L1is also of the same order. On the other hand, setting\nz= i\u000b+ iy,\nZ\nL2[L3e\u0000(z\u0000i)2\n4\u000bp\u000b+ izdz= 2Z1\u0000\u000b\n0e(\u000b+y\u00001)2\n4\u000b\npydy= 2e(\u000b\u00001)2\n4\u000bZ1\u0000\u000b\n0ey\n2+y2\u00002y\n4\u000b\npydy: (3.31)\nThe function y2\u00002ydecreases as yincreases from y= 0 toy= 1. Hence the main contribution to\nthe integral comes near the point y= 0. Using Watson's lemma,\nZ1\u0000\u000b\n0ey\n2+y2\u00002y\n4\u000b\npydy= \u0000(1=2)p\n2\u000b(1 +O(\u000b)): (3.32)\nCombining together and using \u0000(1 =2) =p\u0019, we obtain (3.25). For Im(\u000b), the analysis is same\nexcept that we use\nZ1\u0000\u000b\n0(i\u000b+ iy)mey\n2+y2\u00002y\n4\u000b\npydy=O(\u000bm+1=2): (3.33)\nHence, we \fnd that for m\u00150,Im(\u000b) =O(1) as\u000b!0+. Together with (3.28), this implies the\nuniform boundness of Im(\u000b).\nFor the positiveness of I(\u000b), we \frst write it as\nI(\u000b) =Z1\n\u00001e\u0000\u000b\n4t2+i\n2(t\u0000arctant)\n(1 +t2)1=4dt= 2Z1\n0e\u0000\u000b\n4t2\n(1 +t2)1=4cos\u00121\n2(t\u0000arctant)\u0013\ndt: (3.34)\nThe function \u0012(t) =t\u0000arctantis monotone increasing. We use the inverse function, t=t(\u0012), to\nchange the variables and \fnd that\nI(\u000b) = 2Z1\n0e\u0000\u000b\n4t2(1 +t2)3=4\nt2cos\u0012\u0012\n2\u0013\nd\u0012; t =t(\u0012): (3.35)\n16Sincee\u0000\u000b\n4t(\u0012)2is positive and monotone decreasing in \u0012, we obtain I(\u000b)>0 for every \u000b>0 if we\nshow that (i)Z\u0019\n0(1 +t2)3=4\nt2cos\u0012\u0012\n2\u0013\nd\u0012\u0015\u0000Z3\u0019\n\u0019(1 +t2)3=4\nt2cos\u0012\u0012\n2\u0013\nd\u0012; (3.36)\nand (ii)\n(\u00001)kZ(2k+1)\u0019\n(2k\u00001)\u0019(1 +t2)3=4\nt2cos\u0012\u0012\n2\u0013\nd\u0012; k = 1;2;3;\u0001\u0001\u0001; (3.37)\nis decreasing in k. (i) can be veri\fed numerically. On the other hand, (ii) follows immediately from\nthe fact (1 + t2)3=4=t2is a decreasing function of t. This completes the proof.\nWe now evaluate the integral in (3.1) using the steepest descent analysis.\nLemma 3.8. FixJ >1and let 2\f=J\u00001+BN\u00001=2. Consider G(z) de\fned in (3.1) . Then, for\nevery 0<\u000f<1\n8,\nZ\r+i1\n\r\u0000i1eN\n2G(z)dz=i\u0001eN\n2G(\r)\np\nNI(F00(\r)\u00012)\u0010\n1 +O(N\u00001\n2+4\u000f)\u0011\n(3.38)\nwith high probability, where\nF(z) = 2\fz\u00001\nNNX\ni=2log(z\u0000\u0015i)\u00001\nNlog(\r\u0000\u00151)\u0000z\u0000\r\nN(\r\u0000\u00151)(3.39)\nandI(\u000b)is de\fned in (1.21) . Recall that \u0001 =p\nN(\r\u0000\u00151)(see Lemma 3.5.)\nProof. We choose the \r, which de\fnes the contour, as the critical point of G(z). The path of\nsteepest-descent is locally a vertical line near the critical point. It turns out that, instead of using\nthe path of steepest-descent, it is enough to proceed the analysis using the straight line \r+ iR\nglobally. This choice was also made for the analysis in the paramagnetic regime in [6].\nWe \frst write, using the function F(z),\nZ\r+i1\n\r\u0000i1eN\n2G(z)dz=eN\n2G(\r)Z\r+i1\n\r\u0000i1eN\n2(G(z)\u0000F(z))+N\n2(F(z)\u0000G(\r))dz: (3.40)\nFrom the de\fnitions of G(z) andF(z),\neN\n2(G(z)\u0000F(z))=r\n\r\u0000\u00151\nz\u0000\u00151ez\u0000\r\n2(\r\u0000\u00151): (3.41)\nChanging the variables z=\r+ itN\u00001=2and using the notation \u0001 =p\nN(\r\u0000\u00151),\nZ\r+i1\n\r\u0000i1eN\n2G(z)dz=ieN\n2G(\r)\np\nNZ1\n\u00001eit\n2\u0001q\n1 +it\n\u0001eN\n2(F(\r+itN\u00001=2)\u0000G(\r))dt: (3.42)\n17It is easy to check that the part of the integral with jtj\u0015N\u000fis small. To show this, we \frst\nnote that\n<\u0012\nN\u0012\nF(\r+itp\nN)\u0000G(\r)\u0013\u0013\n=\u0000<NX\ni=2log \n\r\u0000\u0015i+ itN\u00001=2\n\r\u0000\u0015i!\n\u0014\u0000N\u00001\n2log\u0012\n1 +c2t2\nN\u0013\nwith high probability for some constant c>0, since there is a constant c>0 such that c\u0014\r\u0000\u0015i\u00141\nc\nfor alli= 2;\u0001\u0001\u0001;N, with high probability. Hence,\n\f\f\f\f\f\fZ1\nN\u000feit\n2\u0001q\n1 +it\n\u0001eN\n2(F(\r+itN\u00001=2)\u0000G(\r)dt\f\f\f\f\f\f\u0014Z1\nN\u000fe\u0000N\u00001\n2log\u0010\n1+c2t2\nN\u0011\ndt\n\u0014ZN\nN\u000fe\u0000c2\n8N2\u000fdt+Z1\nN1\n(c2N\u00001t2)N=4dt=O(e\u0000N\u000f) +O(N\u0000N=8)(3.43)\nwith high probability.\nConsider the part jtj\u0014N\u000f. Note that F(z) satis\fesF(\r) =G(\r),F0(\r) =G0(\r) = 0, and for\neachm\u00152,F(m)(z) =O(1) uniformly for zin a small neighborhood of \r(by Corollary 3.3). For\nm= 2, by Lemma 3.6,\nc1\u0014F00(\r)\u0014c2 (3.44)\nfor some constants 0 <c1<c2, uniformly in N. By Taylor expansion, for jtj\u0014N\u000f,\nF(\r+ itN\u00001=2)\u0000G(\r) =\u0000F00(\r)t2\n2N\u0000iF000(\r)t3\n6N3=2+O(N\u00002+4\u000f) (3.45)\nand hence,\neN\n2(F(\r+itN\u00001=2)\u0000G(\r))=e\u0000F00(\r)t2\n4\u0012\n1\u0000iF000(\r)t3\n12N1=2+O(N\u00001+6\u000f)\u0013\n: (3.46)\nTherefore,\nZN\u000f\n\u0000N\u000feit\n2\u0001q\n1 +it\n\u0001eN\n2(F(\r+itN\u00001=2)\u0000G(\r))dt\n=Z1\n\u00001eit\n2\u0001q\n1 +it\n\u0001e\u0000F00(\r)\n4t2dt\u0000iF000(\r)\n12N1=2Z1\n\u00001t3eit\n2\u0001q\n1 +it\n\u0001e\u0000F00(\r)\n4t2dt+O(N\u00001+6\u000f)\n= \u0001I(F00(\r)\u00012)\u0000iF000(\r)\u00014\n12N1=2I3(F00(\r)\u00012) +O(N\u00001+6\u000f):(3.47)\nBy (3.44) and Lemma 3.5, c1N\u0000\u000f\u0014F00(\r)\u00012\u0014c2N\u000f. Hence, Lemma 3.7 implies that\nI(F00(\r)\u00012)\u0015cN\u0000\u000f(3.48)\nfor some constant c >0. Hence, using Lemma 3.5, Lemma 3.7, and the uniform boundedness of\nF000(\r), we \fnd that (3.47) is equal to\n\u0001I(F00(\r)\u00012)(1 +O(N\u00001\n2+4\u000f)) (3.49)\n18if 0<\u000f<1\n8. Thus, using (3.48) and Lemma 3.5 again, we conclude that\nZ\r+i1\n\r\u0000i1eN\n2G(z)dz=i\u0001eN\n2G(\r)\np\nNI(F00(\r)\u00012)(1 +O(N\u00001=2+4\u000f)): (3.50)\n3.3 Proof of Theorem 1.4\nProof of Theorem 1.4. From Lemma 3.1 and Lemma 3.8, for every 0 <\u000f<1\n8,\nZN=CNi\u0001eN\n2G(\r)\np\nNI(F00(\r)\u00012)(1 +O(N\u00001\n2+4\u000f)) (3.51)\nwith high probability. Using Stirling's formula,\nCN=\u0000(N=2)\n2\u0019i(N\f)N=2\u00001=p\nN\f\nip\u0019(2\fe)N=2(1 +O(N\u00001)); (3.52)\nthus we \fnd that FN=1\nNlogZNsatis\fes\nFN=1\n2(G(\r)\u00001\u0000log(2\f)) +1\nN\u0012\nlog\u0012\f\u0001p\u0019\u0013\n+ log I(F00(\r)\u00012)\u0013\n+O(N\u00003\n2+4\u000f) (3.53)\nwith high probability.\nLet us consider G(\r). Since\rand ^J=J+J\u00001are away from \u00152;\u0001\u0001\u0001;\u0015Nwith high probability,\nlog(\r\u0000\u0015i) = log( ^J\u0000\u0015i)\u0000log \n1\u0000\r\u0000^J\n\r\u0000\u0015i!\n= log( ^J\u0000\u0015i) +\r\u0000^J\n\r\u0000\u0015i+(\r\u0000^J)2\n2(\r\u0000\u0015i)2+O(j\r\u0000^Jj3)(3.54)\nfori= 2;\u0001\u0001\u0001;N, where we also use that \r\u0000^J=O(N\u00001\n2+\u000f) with high probability (see Lemma 3.5).\nThen, using Lemma 3.6 and the fact that G0(\r) = 2\f\u00001\nNPN\ni=11\n\r\u0000\u0015i= 0,\n1\nNNX\ni=2log(\r\u0000\u0015i) =1\nNNX\ni=2log(^J\u0000\u0015i) + 2\f(\r\u0000^J)\u0000\r\u0000^J\nN(\r\u0000\u00151)+(\r\u0000^J)2\n2(J2\u00001)+O(N\u00003\n2+3\u000f)\nwith high probability. Hence, from the formula of G(z) in (3.1),\nG(\r) = 2\f^J\u00001\nNNX\ni=2log(^J\u0000\u0015i)\u00001\nNlog(\r\u0000\u00151) +\r\u0000^J\nN(\r\u0000\u00151)\u0000(\r\u0000^J)2\n2(J2\u00001)+O(N\u00003\n2+3\u000f)\n= 2\f^J\u00001\nNNX\ni=2log(^J\u0000\u0015i)\u00001\nNlog\u0012\u0001p\nN\u0013\n+sN\nN\u0001\u0000s2\nN\n2N(J2\u00001)+O(N\u00003\n2+3\u000f)\n19using the notations sN=p\nN(\r\u0000^J) and \u0001 =p\nN(\r\u0000\u00151) in Lemma 3.5. Thus,\nFN=\f^J\u00001\n2\u00001\n2log(2\f)\u00001\n2NNX\ni=2log(^J\u0000\u0015i) +1\n4NlogN\n+1\nN\u0012sN\n2\u0001\u0000s2\nN\n4(J2\u00001)+1\n2log \u0001 + log\fp\u0019+ log I(F00(\r)\u00012)\u0013\n+O(N\u00003\n2+4\u000f):(3.55)\nTo conclude Theorem 1.4, we use (i) the fact that \u0001 = sN\u0000\u001fN, (ii) the asymptotic (3.20) of sN\nin terms of \u001fN, (iii) the fact that F00(\r) =1\nJ2\u00001+O(N\u00001+\u000f) which follows from Lemma 3.6, and\n(iv) the fact that I0(\u000b) is uniformly bounded for \u000b>0 (see Lemma 3.7).\n4 Partial linear statistics\nThis section is devoted to a proof of Theorem 2.1 on partial linear statistics. The proof is a simple\nmodi\fcation of [6] for the linear statistics of all eigenvalues, which, in turn, follows the proof of\n[2, 1] for the case when the random matrix has zero mean.\n4.1 Proof of Theorem 2.1\nRecall ^J:=J+J\u00001denotes the classical location of the largest eigenvalue of a Wigner matrix of\nnon-zero mean. Fix ( N-independent) constants a\u0000<\u00002 and 2<a+<^J. Let \u0000 be the rectangular\ncontour whose vertices are ( a\u0000\u0006iv0) and (a+\u0006iv0) for somev02(0;1]. The contour is oriented\ncounter-clockwise. For a test function '(x) which is analytic in a neighborhood of [ \u00002;2], we\nconsider\nN(2)\nN(') :=NX\ni=2'(\u0015i)\u0000NZ\nR'(x)d\u001bscl(x)\n=NX\ni=21\n2\u0019iI\n\u0000'(z)\nz\u0000\u0015idz\u0000N\n2\u0019iZ\nRI\n\u0000'(z)\nz\u0000xdzd\u001bscl(x) =\u00001\n2\u0019iI\n\u0000'(z)\u0018(2)\nNdz;(4.1)\nwhere\n\u0018(2)\nN(z) :=NX\ni=21\n\u0015i\u0000z\u0000NZ\nR1\nx\u0000zd\u001bscl(x): (4.2)\nDecompose \u0000 into \u0000 u[\u0000d[\u0000l[\u0000r[\u00000, where\n\u0000u=fz=x+ iv0:a\u0000\u0014x\u0014a+g; (4.3)\n\u0000d=fz=x\u0000iv0:a\u0000\u0014x\u0014a+g; (4.4)\n\u0000l=fz=a\u0000+ iy:N\u0000\u000e\u0014jyj\u0014v0g; (4.5)\n\u0000r=fz=a++ iy:N\u0000\u000e\u0014jyj\u0014v0g; (4.6)\n\u00000=fz=a\u0000+ iy:jyj<N\u0000\u000eg[fz=a++ iy:jyj<N\u0000\u000eg (4.7)\n20for some\u000e>0. In the proof of Theorem 1.6 in [6], the authors showed that\n\u0018N(z) :=NX\ni=11\n\u0015i\u0000z\u0000NZ\nR1\nx\u0000zd\u001bscl(x) =\u0018(2)\nN(z) +1\n\u00151\u0000z(4.8)\nconverges weakly to a Gaussian process with mean b(z) =b(2)(z) +1\n^J\u0000zand covariance \u0000( zi;zj) =\n\u0000(2)(zi;zj) whereb(2)(z) and \u0000(2)(zi;zj) are given in the proposition below. Since for each \fxed\nz2C+,1\n\u00151\u0000z!1\n^J\u0000zin probability (by Lemma 3.2), it is natural to expect the following result for\na partial sum.\nProposition 4.1. Let\ns(z) =Z1\nx\u0000zd\u001bscl(x) =\u0000z+p\nz2\u00004\n2(4.9)\nbe the Stieltjes transform of the semicircle measure. Fix a constant c>0and a pathK\u001aC+such\nthat=z >c forz2K. Then the process f\u0018(2)\nN(z) :z2Kg converges weakly to a Gaussian process\nwith the mean\nb(2)(z) =s(z)2\n1\u0000s(z)2\u0012\n\u0000J\n1 +Js(z)+ (w2\u00001)s(z) +s0(z)s(z) + (W4\u00003)s(z)3\u0013\n\u00001\n^J\u0000z(4.10)\nand the covariance matrix\n\u0000(2)(zi;zj) =s0(zi)s0(zj)\u0012\n(w2\u00002) + 2(W4\u00003)s(zi)s(zj) +2\n(1\u0000s(zi)s(zj))2\u0013\n: (4.11)\nRemark 4.2.Note that as z!^J,\ns(z)2\n1\u0000s(z)2J\n1 +Js(z)=s0(z)\n1\nJ+s(z)=1\nz\u0000^J+s00(^J)\ns0(^J)+O(z\u0000^J): (4.12)\nHence,b(2)(z) is analytic near ^Jand thus analytic for z2Cn[\u00002;2].\nIn order to complete the proof of Theorem 2.1, we will prove the following lemma.\nLemma 4.3. De\fne the events\n\nN:=f\u00151\u0015^J\u0000N\u00001=3;\u00152\u00142 +N\u00001=3g (4.13)\nwhich satis\fes P(\nc\nN)<N\u0000Dfor any \fxed (large) D> 0. Then for some \u000e>0,\nlim\nv0!0+lim sup\nN!1Z\n\u0000#Ej\u0018(2)\nN(z) 1\nNj2dz= 0; (4.14)\nwhere \u0000#can be \u0000r,\u0000lor\u00000.\nFrom the explicit formulas (4.10) and (4.11), it is easy to check that\nlim\nv0!0+Z\n\u0000#Ej\u0018(2)(z)j2dz= 0: (4.15)\n21Proposition 4.1, Lemma 4.3 and (4.15) imply that N(2)\nN(') converges in distribution to a Gaussian\nrandom variable with the following mean and variance:\n\u00001\n2\u0019iI\n\u0000'(z)b(2)(z)dz;1\n(2\u0019i)2I\n\u0000I\n\u0000'(z1)'(z2)\u0000(z1;z2)dz1dz2: (4.16)\nIt is direct to check that these are equal to M(2)(') andV(2)(') (see Section 4.2 in [6]). We thus\nobtain Theorem 2.1.\n4.2 Proof of Proposition 4.1\nFrom Theorem 7.1 of [8], we need to show (i) the \fnite-dimensional convergence of \u0018(2)\nN(z) to a\nGaussian vector with desired mean and variance, and (ii) the tightness of \u0018(2)\nN(z). We will base our\nproof on the corresponding properties of \u0018N(z) obtained in [6]. Let us \frst recall the limit theorem\nfor\u0018N(z).\nLemma 4.4 (Proposition 4.1 in [6]) .Lets(z)andKde\fned in the same way as in Proposition\n4.1. Then, the process f\u0018N(z) :z2Kg converges weakly to a Gaussian process f\u0018(z) :z2Kg with\nthe mean\nb(z) =s(z)2\n1\u0000s(z)2\u0012\n\u0000J\n1 +Js(z)+ (w2\u00001)s(z) +s0(z)s(z) + (W4\u00003)s(z)3\u0013\n(4.17)\nand the covariance matrix\n\u0000(zi;zj) =s0(zi)s0(zj)\u0012\n(w2\u00002) + 2(W4\u00003)s(zi)s(zj) +2\n(1\u0000s(zi)s(zj))2\u0013\n: (4.18)\nLetz1;z2;\u0001\u0001\u0001;zparepdistinct points in K. The above lemma implies that the random vector\n(\u0018N(zi))p\ni=1converges weakly to a p-dimensional Gaussian distribution with the mean ( b(zi))p\ni=1and\nthe covariance matrix \u0000( zi;zj). Since the distance between Kand\u00151is bounded below,1\n\u00151\u0000zi!\n1\n^J\u0000ziin probability for i= 1;\u0001\u0001\u0001;p. Hence, by Slutsky's theorem, ( \u0018(2)\nN(zi))p\ni=1converges weakly to\nap-dimensional Gaussian distribution vector with the mean ( b(2)(zi))p\ni=1and the covariance matrix\n\u0000(2)(zi;zj), where\nb(2)(z) =b(z)\u00001\n^J\u0000z; (4.19)\nand \u0000(2)(zi;zj) = \u0000(zi;zj).\nFrom Theorem 12.3 of [8], in order to show the tightness of a random process ( \u0010N(z))z2K, it is\nsu\u000ecient to show that (i) ( \u0010N(z))Nis tight for a \fxed z, and (ii) the following H older condition\nholds: for some N-independent constant K > 0,\nEj\u0010N(z1)\u0000\u0010N(z2)j2\u0014Kjz1\u0000z2j2; z 1;z22K: (4.20)\nIn [6], the authors considered the random process \u0010N(z) :=\u0018N(z)\u0000E[\u0018N(z)], and proved that\nit satis\fes conditions (i) and (ii). Now, we consider \u0018(2)\nN(z) :=\u0010(2)\nN+E[\u0018N(z)], where\u0010(2)\nN(z) :=\n\u0010N(z)\u00001\n\u00151\u0000z. Since E[\u0018N(z)] converges, it is enough to check that ( \u0010(2)\nN(z))Nsatis\fes conditions\n22(i) and (ii). Now for a \fxed z, the tightness of ( \u0010N(z))Nand the boundedness of1\n\u00151\u0000zimply that\n(\u0010(2)\nN(z))Nis tight. On the other hand, since \u0010N(z) satis\fes the H older condition and =z\u0015cfor\nz2K,\nEj\u0010(2)\nN(z1)\u0000\u0010(2)\nN(z2)j2\u00142Ej\u0010N(z1)\u0000\u0010N(z2)j2+ 2E\f\f\f\f1\n\u00151\u0000z1\u00001\n\u00151\u0000z2\f\f\f\f2\n\u00142Kjz1\u0000z2j2+2jz1\u0000z2j2\nc4=\u0012\nK+2\nc4\u0013\njz1\u0000z2j2:(4.21)\nThusf\u0018(2)\nN(z);z2Kg is tight. This completes the proof of Proposition 4.1.\n4.3 Proof of Lemma 4.3\nForz2\u00000, we notice thatj\u0018(2)\nN1\nNj\u0014CNand then\nZ\n\u00000Ej\u0018(2)\nN1\nNj2\u0014CN2\u0000\u000e: (4.22)\nThus (4.14) holds for \u0000 0with\u000e > 2. For \u0000 rand \u0000l, it is su\u000ecient to show Ej\u0018(2)\nNj2< K for\nsomeN-independent constant K > 0. The authors in [6] showed2thatEj\u0018N(z)j2<K. Hence, for\nz2\u0000r,\nj\u0018(2)\nN(z) 1\nNj2\u00142j\u0018N(z) 1\nNj2+ 2\f\f\f\f1\n\u00151\u0000z1\nN\f\f\f\f2\n: (4.23)\nThe lemma then follows from the fact that j1\n\u00151\u0000z1\nNjis bounded.\n5 Joint Distribution of \u001fNandN(2)\nN(')\nAs before, let Abe a random symmetric matrix of size Nwhose entries are (up to the symmetry\ncondition) independent centered random variables satisfying De\fnition 1.1. Let M=1p\nNA+J\nN11T\nwhereJ >1. Let\u00151\u0015\u0001\u0001\u0001\u0015\u0015Nbe the eigenvalues of M.\nLet\u001fN=p\nN(\u00151\u0000^J) denoting the rescaled largest eigenvalue. Given an analytic function\n'(x), recall the partial linear statistics N(2)\nN(') =PN\ni=2'(\u0015i)\u0000NR2\n\u00002'(x)d\u001bscl(x). We saw in the\nprevious sections that \u001fNandN(2)\nN(') converge individually to Gaussian random variables. In this\nsection, we consider the joint distribution and prove Theorem 2.3. In Subsection 5.1, we \frst prove\nTheorem 2.3 assuming that the disorder variables are Gaussian random variables. In Subsection\n5.2, the general disorder variables are considered using an interpolation trick.\n5.1 Asymptotic Independence for the GOE case\nLet the o\u000b-diagonal entries of Abe Gaussian random variables of variance 1 and the diagonal entries\nbe Gaussian random variables of variance 2. In random matrix theory, the random symmetric\n2Even though it is stated in Lemma 4.2 of [6] that the lemma holds for su\u000eciently small \u000e>0, the proof of it is\nvalid for any \u000e>0, and we use \u000e>2 for our purpose.\n23matrixH=1p\nNAis said to belong to the Gaussian orthogonal ensemble (GOE). A special property\nof GOE, compared with general random symmetric matrices, is that the probability measure of\nGOE is invariant under orthogonal conjugations.\nThe following result is basically in [10].\nLemma 5.1. Let(1p\n2Aii;Aij;yi)1\u0014i<j\u0014Nbe i.i.d. standard Gaussian random variables. Let H=\n1p\nNAwithA= (Aij)1\u0014i;j\u0014Nand letY=1p\nN(y1;\u0001\u0001\u0001;yN)T. De\fneG(z) = (H\u0000zI)\u00001for\nz2Cn[\u00002\u0000\u000e;2 +\u000e], which is well de\fned with high probability for \fxed \u000e > 0. Then, for\nz2Rn[\u00002\u0000\u000e;2 +\u000e],\nnN(z) :=p\nN(Y\u0003G(z)Y\u00001\nNTr(G(z))))n(z) (5.1)\nwheren=n(z) :=N\u0010\n0;2Rd\u001bscl(x)\n(x\u0000z)2\u0011\nis a Gaussian random variable.\nProof of Lemma 5.1. We follow the idea presented in [10]. By Theorem 5.2 of [10], it is enough\nto check the following three conditions for G: (i) There exists an N-independent constant asuch\nthatkGk\u0014awith high probability, (ii)1\nNTrG2converges to a constant in probability, and (iii)\n1\nNPN\ni=1G2\niiconverges to a constant in probability. They follow from rigidity of eigenvalue (Lemma\n3.2), law of large numbers (Corollary 3.3), and local law (Theorem 2.9 of [17]), respectively.\nWe are now ready to prove the following property of GOE matrices.\nProposition 5.2. ForHde\fned in Lemma 5.1, denote its eigenvalues by \u001a1\u0015\u001a2\u0015\u0001\u0001\u0001\u0015\u001aN:For\n\fxedk, consider a random vector (X1\nN;X2\nN;\u0001\u0001\u0001;Xk\nN)whose entries are real measurable functions\nof those eigenvalues, i.e., Xi\nN=Xi\nN(\u001a1;\u001a2;\u0001\u0001\u0001;\u001aN)fori= 1;2;\u0001\u0001\u0001;k. Suppose there is a random\nvector (Xi)k\ni=1such that (Xi\nN)k\ni=1)(Xi)k\ni=1asN!1 . Then for nNandnde\fned as in (5.1) ,\n(X1\nN;X2\nN;\u0001\u0001\u0001;Xk\nN;nN))(X1;X2\u0001\u0001\u0001;Xk;n), wherenis independent from (X1;X2;\u0001\u0001\u0001;Xk).\nProof. For the convergence, it is enough to show (i) ( X1\nN;X2\nN\u0001\u0001\u0001;Xk\nN;nN) is tight, and (ii) con-\nvergence of characteristic function. The tightness follows from the tightness of individual random\nvector (variable), which is a consequence of individual convergence.\nFor (ii), consider the eigenvalue decomposition H=OPOT, whereP= diag(\u001a1;\u001a2;\u0001\u0001\u0001;\u001aN)\nandOis an orthogonal matrix. Since the His orthogonal invariant, PandOare independent. Set\nX=OTY. ThenX=1p\nN(x1;\u0001\u0001\u0001;xN) wherex1;\u0001\u0001\u0001;xNare i.i.d standard Gaussian ( Xis also\nindependent with P).\nNow,nN=Y\u0003G(z)Y\u00001\nNTrG(z) =1\nNPN\ni=1x2\ni\u00001\n\u001ai\u0000z. Since E[etx2\n1] =1p1\u00002t, we \fnd that for any\nt2iR, the conditional expectation over XgivenPsatis\fes\nEX\u0002\netnN\f\fP\u0003\n=EX\u0002\netp\nNPN\ni=1x2\ni\u00001\n\u001ai\u0000z\f\fP\u0003\n=NY\ni=1e\u00001\n2log(1\u00002tp\nN(\u001ai\u0000z))\u0000tp\nN(\u001ai\u0000z):\nNote that ( X1\nN;X2\nN;\u0001\u0001\u0001;Xk\nN) only depends on the eigenvalues, and hence it is independent of X.\nThus, for any u1;u2;\u0001\u0001\u0001;uk;t2iR,\nEh\nePk\nj=1ujXj\nN+tnNi\n=E\"\nePk\nj=1ujXj\nNNY\ni=1e\u00001\n2log(1\u00002tp\nN(\u001ai\u0000z))\u0000tp\nN(\u001ai\u0000z2)#\n: (5.2)\n24Since\u00001\n2log(1\u00002z)\u0000z=z2+O(z3) asz!0, using Corollary 3.3,\nNY\ni=1e\u00001\n2log(1\u00002tp\nN(\u001ai\u0000z))\u0000tp\nN(\u001ai\u0000z)=e1\nNPN\ni=1t2\n(\u001ai\u0000z)2+O(N\u00001\n2)=et2R1\n(x\u0000z)2d\u001bscl(x)+O(N\u00001\n2)\n=Eh\netn(z)i\neO(N\u00001\n2)(5.3)\nwith high probability. Denote this high probability event by \n N. Then,\nlim\nN!1Eh\nePk\nj=1ujXj\nN+tnNi\n= lim\nN!1\u0010\nEh\nePk\nj=1ujXj\nN+tnN\f\f\nNi\nP(\nN) +Eh\nePk\nj=1ujXj\nN+tnN\f\f\nc\nNi\nP(\nc\nN)\u0011\n=Eh\nePk\nj=1ujXji\nEh\netn(z)i\n;\n(5.4)\nsincet;u1;u2;\u0001\u0001\u0001;uk2iRand hence all exponents are pure imaginary. Note that the characteristic\nfunction of ( X1;\u0001\u0001\u0001;Xk;n) is equal to the product of the characteristic functions of individual\nrandom vector (variable). Thus n(z) is independent from ( X1;\u0001\u0001\u0001;Xk). This completes the proof.\nCorollary 5.3. Fix\u000e > 0, considerz12CnRandz22Rn[\u00002\u0000\u000e;2 +\u000e]. Recalls(z)de\fned\nin(4.9) . Then (Tr(G(z1))\u0000Ns(z1);nN(z2))converges in distribution to independent Gaussian\nrandom variables.\nProof. Note that Tr( G(z1))\u0000Ns(z1) is complex, we consider the random vector ( <(Tr(G(z1))\u0000\nNs(z1));=(Tr(G(z1))\u0000Ns(z1))). By Proposition 5.2, it is enough to show that ( <(Tr(G(z1))\u0000\nN(s1));=(Tr(G(z1))\u0000Ns(z1)) converges to a Gaussian random vector. Consider the expression\nz1=E+i\u0011for\u000f;\u00112Rand\u00116= 0. Recalling the de\fnition of linear statistics NN(') de\fned in\n(2.3), we have\n<(Tr(G(z1)\u0000Ns(z1))) =NN('r); 'r(x) =x\u0000E\n(x\u0000E)2+\u00112;\nand\n=(Tr(G(z1)\u0000Ns(z1)) =NN('i); 'i(x) =\u0011\n(x\u0000E)2+\u00112:\nThat is, they are both linear statistics. Then Corollary then follows from Theorem 1.1 of [2].\nRemark 5.4.When we prove Theorem 2.3 for GOE, we use Proposition 5.2 and Corollary 5.3 with\nN-dependent zi. First, for a \fxed z22Rn[\u00002\u0000\u000e;2+\u000e] for some\u000e>0, let ~z2= ~z2(N) :=q\nN+1\nNz2.\nUsing the exactly same argument in the proof of Lemma 5.1, one can show nN(~z2))n(z2). Since\nthe (5.3) still holds for ~ z2andn(z2), the asymptotic independence in Proposition 5.2 is still valid,\ni.e.\n(X1\nN;X2\nN;\u0001\u0001\u0001;Xk\nN;nN(~z2)))(X1;X2\u0001\u0001\u0001;Xk;n(z2));\nwheren(z2) is independent from ( X1;X2;\u0001\u0001\u0001;Xk). Second, for z12CnR, consider ~z1= ~z1(N) :=q\nN+1\nNz1. Notice that\n1\nx\u0000~z1=1\nx\u0000z1+z1\n2N(x\u0000z1)2+O(N\u00002):\n25Then, by the discussion in Remark 2.2, Tr( G(~z1))\u0000Ns(~z1) =NN(1\nx\u0000~z1) converges to a Gaus-\nsian random variable. Now, putting together, for ~ z1and ~z2de\fned as above, (Tr( G(~z1))\u0000\nNs(~z1);nN(~z2)) converge jointly to independent Gaussian random variables.\nWe now prove Theorem 2.3 for the case where the disorder belongs to GOE.\nProof of Theorem 2.3 when Abelongs to GOE. Recall that \u0015iare the eigenvalues of M=1p\nNA+\nJ\nN11TwithAfrom the GOE. Since the means and variances follow from [10] and Theorem 2.1, it\nis enough to prove the asymptotic independence of \u001fNandN(2)\nN('). (Notice that that W3= 0 for\nGaussianAij.) Now, for any analytic test function ', the partial linear statistics can be expressed\nas (see (4.2)) an integral of\n\u0018(2)\nN(z) =NX\ni=21\n\u0015i\u0000z\u0000NZ\nR1\nx\u0000zd\u001bscl(x); z2CnR: (5.5)\nThen according to Lemma 4.3 and what follows, it is enough to prove that \u001fNand\u0018(2)\nN(z) are\nasymptotically independent for \fxed z2CnR. Let\n\u0018N(z) =\u0018(2)\nN(z) +1\n\u00151\u0000z= Tr(M\u0000zI)\u00001\u0000Ns(z):\nSince1\n\u00151\u0000z!1\n^J\u0000zin probability, it is enough to prove that \u001fNand\u0018N(z) are asymptotically\nindependent.\nSince the GOE is orthogonal invariant, for every deterministic matrix U, the eigenvalues of\nA+Uhave the same distribution as A+OUOTfor any orthogonal matrix O. Thus, we may\nconsider the following equivalent model:\nM=1p\nNA+ diag(J;0;\u0001\u0001\u0001;0): (5.6)\nFollowing the proof of Theorem 2.2 in [10], we write\nM=\"A11p\nN+J Y\u0003\nY ^M#\n: (5.7)\nSince det(M\u0000zI) = det( ^M\u0000zI)\u0010\nA11p\nN+J\u0000z\u0000Y\u0003^G(z)Y\u0011\nwith\n^G(z) := ( ^M\u0000zIN\u00001)\u00001= (1p\nN^A\u0000zIN\u00001)\u00001; (5.8)\nthe largest eigenvalue of Msatis\fes\n\u00151=J+A11p\nN\u0000Y\u0003^G(\u00151)Y (5.9)\nif\u00151is not an eigenvalue of ^M, which holds with high probability. Using the resolvent formula\ntwice, we write\n^G(\u00151) =^G(^J) + ( ^G(\u00151)\u0000^G(^J)) = ^G(^J) + (\u00151\u0000^J)^G(\u00151)^G(^J)\n=^G(^J) + (\u00151\u0000^J)^G(^J)2+ (\u00151\u0000^J)2^G(\u00151)^G(^J)2:\n26Hence,\n\u00151\u0000^J=A11p\nN\u00001\nJ\u0000Y\u0003^G(\u00151)Y\n=A11p\nN\u00001\nJ\u0000Y\u0003^G(^J)Y+ (\u00151\u0000^J)Y\u0003^G(^J)2Y+ (\u00151\u0000^J)2Y\u0003^G(\u00151)^G(^J)2Y\nwith high probability. Moving all terms with factor \u00151\u0000^Jto the left and taking it out as a common\nfactor, we arrive at\n\u001fN=p\nN(\u00151\u0000^J) =A11\u0000p\nN(1\nJ+Y\u0003^G(^J)Y)\n1 +Y\u0003^G(^J)2Y+ (\u00151\u0000^J)Y\u0003^G(\u00151)^G(^J)2Y(5.10)\nwith high probability.\nNote that ^MandYsatisfy the setting of Corollary 5.3 up to the scaling factorq\nN\nN\u00001. Set\n~Y=r\nN\nN\u00001Y; ~G(z) =\u0000r\nN\nN\u00001^M\u0000zIN\u00001\u0001\u00001(5.11)\nThen, ~Yand ~Gsatisfy the setting of Corollary 5.3, and\nY\u0003^G(^J)Y=r\nN\u00001\nN~Y\u0003~G(~J)~Y; ~J:=r\nN\nN\u00001^J: (5.12)\nNow, by Corollary 3.3,\n1\nN\u00001Tr(~G(~J)) =s(^J) +O(N\u00001+\u000f) =\u00001\nJ+O(N\u00001+\u000f) (5.13)\nwith high probability. By Lemma 5.1, Corollary 3.3 and Lemma 3.2,\nY\u0003^G(^J)2Y!1\nJ2\u00001; (\u00151\u0000^J)Y\u0003^G(\u00151)^G(^J)2Y!0 (5.14)\nin probability. Using (5.14), (5.13) and denoting the denominator in (5.10) by D1, we write\n\u001fN=D\u00001\n1\u0010\nA11\u0000~nN\u00001(~J) +O(N\u00001\n2+\u000f)\u0011\n; (5.15)\nwherenN\u00001(~J) =p\nN\u00001(~Y\u0003~G(~J)~Y\u00001\nN\u00001Tr(~G(~J))) (see (5.1)) and D1!J2\nJ2\u00001in probability.\nNote thatA11andnN\u00001(~J) are independent, the distribution of \u001fNis governed by their convolution.\nWe now turn to the linear statistic \u0018N(z). Using Schur complement of Mwith block structure\nin (5.7), for any z2CnR,\nTr(M\u0000zI)\u00001=(J+A11p\nN\u0000z\u0000Y\u0003^G(z)Y)\u00001(1 +Y\u0003^G(z)2Y) + Tr( ^G(z)) (5.16)\nUsing Lemma 5.1 and Lemma 3.3,\nD2=D2(N) :=1 +Y\u0003^G(z)2Y\nJ+A11p\nN\u0000z\u0000Y\u0003^G(z)Y!1 +s0(z)\nJ\u0000z\u0000s(z)\n27in probability. Then, by setting ~ z:= ~z(N) =q\nN\nN\u00001z, we write\n\u0018N(z) =Tr(M\u0000zI)\u00001\u0000Ns(z) =D2+ Tr ^G(z)\u0000Ns(z) +O(N\u00001\n2+\u000f)\n=D2\u0000s(z)\n2+zs0(z)\n2+r\nN\nN\u00001\u0010\nTr~G(~z)\u0000(N\u00001)s(~z)\u0011\n+O(N\u00001\n2+\u000f):(5.17)\nThat is, the \ructuation of \u0018N(z) is govern by Tr ~G(~z)\u0000(N\u00001)s(~z). Now using Corollary 5.3\nand Remark 5.4, one can conclude that (Tr ~G(~z)\u0000(N\u00001)s(~z);nN\u00001(~J)) converge to independent\nGaussian random variables. Furthermore, A11is independent of both Yand ^M. Thus by (5.15)\nand (5.17), ( \u0018N(z);\u001fN) converge to independent random variables. Theorem 2.3 then follows.\n5.2 Proof of Theorem 2.3 for general case\nWe prove Theorem 2.3 for general disorders, where the disorder matrix Ais a Wigner matrix and\nsatis\fes De\fnition 1.1. Unlike the GOE, Wigner matrices are not orthogonal invariant, hence we\ncannot apply (5.6) where we replaced the rank-1 perturbation in Mby a diagonal matrix. To\novercome the di\u000eculty, we use an interpolation method. It has been successfully applied in many\nworks in random matrix theory, where a given matrix and a reference matrix such as GOE are\ninterpolated. We refer to [22] for its application in the analysis of linear eigenvalue statistics.\nLetV=1p\nNAbe a (normalized) Wigner matrix and VGbe a (normalized) GOE matrix\nindependent from V. De\fne\nH(t) =Vcost+VGsint (5.18)\nso thatH(0) =VandH(\u0019\n2) =VG. Note that E[H2\nij] =1\nNfori6=j. Let\ne=1p\nN1T=1p\nN(1;1;:::; 1)T2RN(5.19)\nand\nM(t) =H(t) +JeeT; (5.20)\nwhose eigenvalues are denoted by \u00151\u0015\u00152\u0015\u0001\u0001\u0001\u0015\u0015N. De\fne the resolvents\nG(z) = (M\u0000zI)\u00001; ^G(z) = (H\u0000zI)\u00001: (5.21)\nHere, we omit the dependence on tfor the ease of notation. We note that Gand ^Gare symmetric\n(not Hermitian). For any (small) \fxed \u000e >0,^G(z) is well-de\fned for z2Cn[\u00002\u0000\u000e;2 +\u000e] with\nhigh probability.\nFor\u001fN=p\nN(\u00151\u0000^J), we notice that\n^Gee(\u00151) :=he;^G(\u00151)ei=\u00001\nJ(5.22)\nwith high probability. The claim holds since\n0 = det(M\u0000\u00151I) = det(H\u0000\u00151I) det(I+J^G(\u00151)eeT)\n= det(H\u0000\u00151I) det(I+JeT^G(\u00151)e) = det(H\u0000\u00151I)\u0010\n1 +J^Gee(\u00151)\u0011(5.23)\n28and\u00151is not an eigenvalue of Hwith high probability (See Lemma 6.1 of [20]). Furthermore, by\nTaylor expansion,\n\u00001\nJ=^Gee(\u00151) =^Gee(^J) +^G0\nee(^J)(\u00151\u0000^J) +O(N\u00001+\u000f) (5.24)\nwith high probability, since j\u00151\u0000^Jj=O(N\u00001\n2+\u000f) andk^G00(z)k=O(1) with high probability. From\nthe isotropic local law, Theorem 2.2 of [20], we \fnd that\n^Gee(^J) =s(^J) +O(N\u00001\n2+\u000f); ^G0\nee(^J) =s0(^J) +O(N\u00001\n2+\u000f) (5.25)\nwith high probability. Thus, using Lemma 5.1,\n\u001fN=p\nN(\u00151\u0000^J) =\u0000p\nN(J\u00001+^Gee(^J))\ns0(^J)+O(N\u00001\n2+2\u000f) (5.26)\nwith high probability. That is, the behavior of \u001fNis governed by the \ructuation of ^Gee(^J).\nTo prove the Theorem 2.3, as in the Gaussian disorder case, it is enough to show the convergence\nof the joint distribution of \u001fNand the full linear statistics \u0018N(z) = Tr(G(z))\u0000Ns(z) for \fxed\nz2CnR. Under the light of (5.26), we set out to calculate the following characteristic function\ninvolving\u0018N(z) and ^Gee(^J). Explicitly, for t1;t2;t32iRandz=E+ i\u0011withE2Rand\u0011 >0,\nwe de\fne\nEh\neP(t)i\n:=Eh\net1<\u0018N+t2=\u0018N+t3nNi\n; P (t) :=t1<\u0018N(z) +t2=\u0018N(z) +t3nN; (5.27)\nwhere\nnN=p\nN\u0012\n^Gee(^J) +1\nJ\u0013\n: (5.28)\nNote thatnNis real, the exponent P(t) is pure imaginary and thus jeP(t)j\u00141. For our purpose,\nit is desired to estimate E[eP(0)]. Att=\u0019\n2, the disorder H(\u0019\n2) reduces to the GOE case. From\nSubsection 5.1, \u001fNand\u0018Nare asymptotically independent in the GOE case, then\nlim\nN!1Eh\neP(\u0019\n2)i\n=Eh\net1<\u0018+t2=\u0018i\n\u0001E\u0002\net3n\u0003\n(5.29)\nfor some Gaussian random variables \u0018;nwith known mean and variance. Thus, it only remains to\nestimate the t-derivative of E[eP(t)]. Here, we recall the following identity for the derivative of the\nresolventG. Fori;j;a;b = 1;2;\u0001\u0001\u0001;N,\n@\n@MijGab=\u0000\fjk(GajGkb+GakGjb) (5.30)\nwith\n\fjk=(\n1j6=k;\n1=2j=k:(5.31)\n29We note that the above identity also holds if one replace Gby^G. Thus for any \fxed event \n,\nd\ndtEh\neP(t)j\ni\n=E2\n4X\ni\u0014jdMij\ndt@\n@MijeP(t)\f\f\f\f\f\n3\n5\n=X\ni;jE\"\n\u0000\nVijsint\u0000VG\nijcost\u0001 \nt1<\u0000\nG2\u0001\nij+t2=\u0000\nG2\u0001\nij+t3p\nNX\np;q^Gpi^Gjq!\neP(t)\f\f\f\n#\n:\n(5.32)\nThe reason for the introduction of \n will be revealed in a minute. The right hand side of (5.32)\nmotivates us to apply the generalized Stein's lemma. More precisely, we will use Proposition 3.1 of\n[22] with a small modi\fcation as follows:\nProposition 5.5. Given an event \n, letXbe a random variable such that E[jXjp+2j\n]<1\nfor a certain non-negative integer p. Denote the conditional cumulants of Xby\u0014l:=\u0014l(\n),\nl= 1;:::;p + 1. Then for any function \b :R!Cof the class Cp+1with bounded derivatives\n\b(l);l= 1;:::;p + 1, we have\nE[X\b(X)j\n] =pX\nl=0\u0014l+1\nl!E[\b(l)(X)j\n] +\u000fp; (5.33)\nwhere the remainder term \u000fpadmits the bound\nj\u000fpj\u0014CpE2\n4jXjp+20\n@1 + max\n1\u0014j\u0014p+1\u0012Z1\n0j\b(p+1)(vX)jdv\u0013p+2\nj1\nA\f\f\f\f\f\n3\n5 (5.34)\nfor some constant Cpthat depends only on p.\nProof. We basically follow the proof of Proposition 3.1 of [22]. Let \u0019pbe the degree pTaylor\npolynomial of \b and let rp= \b\u0000\u0019p. Then, as in the proof of Proposition 3.1 of [22],\nE[X\u0019p(X)j\n] =pX\nj=0\u0014j+1\nj!E[\u0019(j)\np(X)j\n]: (5.35)\nThus\f\f\f\f\fE[X\b(X)j\n]\u0000pX\nl=0\u0014l+1\nl!E[\b(l)(X)j\n]\f\f\f\f\f\u0014jE[Xrp(X)j\n]j+pX\nl=0j\u0014l+1j\nl!\f\f\fEh\nr(l)\np(X)j\ni\f\f\f:(5.36)\nSince\nrp(X) =Xp+1\np!Z1\n0\b(p+1)(vX)(1\u0000v)pdv; (5.37)\nby the estimatej\u0014jj\u0014(2j)jE[jXjjj\n] and H older's inequality,\npX\nl=0j\u0014l+1j\nl!\f\f\fEh\nr(l)\np(X)j\ni\f\f\f\u0014pX\nl=0\u0014l+1\nl!(p\u0000l)!E\u0014\njXjp+1\u0000lZ1\n0j\b(p+1)(vX)jdv\f\f\f\n\u0015\n\u0014pX\nl=0(2l+ 2)l+1\nl!(p\u0000l)!E2\n4jXjp+20\n@1 +\u0012Z1\n0j\b(p+1)(vX)jdv\u0013p+2\np+1\u0000l1\nA\f\f\f\f\f\n3\n5:\n(5.38)\n30AsjE[Xrpj\n]jcan also be bounded by the right hand side of (5.38), the proof is complete.\nIn order to apply Proposition 5.5 to (5.32), we need prior bounds of P(t) and its derivatives\nto bound\u000fpin (5.33). As we will see later, it is enough to bound Gij, (G2)ij,^GijandP\np^Gip. In\nthe following, we are going to introduce a high probability event \n, on which we have the desired\nbounds.\nWith the trivial bound kGk\u00141\n\u0011(recall that z=E+i\u0011), we have thatjGijj\u00141\n\u0011and\f\f(G2)ij\f\f\u0014\r\rG2\r\r\u00141\n\u00112:For^Gij, we introduce the high probability event \n 1=f\u00151\u0014(2 + ^J)=2g. It is easy to\ncheck thatk^Gk 1\n1\u00141\n^J\u00002and thus\nj^Gij 1\n1j\u00141\n^J\u00002; (5.39)\nForP\np^Gip, we recall the following concentration theorem for the quadratic function of ^G:\nProposition 5.6 (Theorem 2.3 and Remark 2.4 of [20]) .Fix\u0006\u00153. Set'= (logN)log logN. Then\nthere exist constants C1andC2such that for any\nE2[\u0006;\u00002\u0000'C1N\u00002\n3][[2 +'C1N\u00002\n3;\u0006];\nand any\u00112(0;\u0006], and any deterministic v;w2CN,\njhv;^G(z)wi\u0000s(z)hv;wij\u0014'C2s\n=s(z)\nN\u0011kvkkwk (5.40)\nwith high probability, uniformly on z=E+ i\u0011.\nLetei:= (0;\u0001\u0001\u0001;1;\u0001\u0001\u0001;0). Noting thatPN\np=1^Gpi=p\nNhe;^Geii, we can derive a prior bound\nforPN\np=1^Gpi, which is summarized in the following Corollary.\nCorollary 5.7. For any \fxed E2Rn[\u00002;2], the tail bound\njX\np(^G(E))pij\u0014N\u000f(5.41)\nholds simultaneously for i= 1;\u0001\u0001\u0001;Nwith high probability. We also have that\njhv;^G(E)wi\u0000s(E)hv;wij\u0014kvkkwkN\u00001\n2+\u000f(5.42)\nwith high probability.\nProof. We \frst prove (5.42). Consider z=E+ iN\u00001=2. Using Proposition 5.6, we \fnd there exists\nsomeC > 0 such that\nhv;^G(E)wi\u0000s(E)hv;wij\u0014jhv;(^G(z)\u0000^G(E)wij+jhv;^G(z)wi\u0000s(z)hv;wij+js(z)\u0000s(E)jjhv;wij\n\u0014CN\u00001=2kvkkwk+C'CN\u00001\n2kvkkwk+CN\u00001=2:\n(5.43)\nHere we also use the fact that \n 1holds with high probability. Since '\u001cN\u000f, (5.42) then follows.\nThe tail bound (5.41) can be obtained from (5.42) by setting v=p\nNeandw=ei.\n31Based on our discussion above, we are ready to introduce the high probability event as promised.\nSets1:=s(z),s0\n1:=s0(z) ands2:=s(^J) =\u0000J\u00001, the desired high probability event \n is the\nintersection of \n 1and the following events:\n\n2=fjX\np(^G(^J))pij\u0014N\u000f;8i= 1;\u0001\u0001\u0001;Ng\\fj ^Gee(^J)\u0000s2j\u0014N\u00001\n2+\u000fg; (5.44)\n\n3=fj^Gij\u0000\u000eijs2j\u0014N\u00001\n2+\u000f;8i;j= 1;\u0001\u0001\u0001;Ng; (5.45)\n\n4=fjGij\u0000\u000eijs1j;j(G2)ij\u0000\u000eijs0\n1j\u0014N\u00001\n2+\u000f;8i;j= 1;\u0001\u0001\u0001;Ng; (5.46)\n\n5=fjVijj;jVG\nijj;jMijj\u0014N\u00001\n2+\u000f;8i;j= 1;\u0001\u0001\u0001;Ng: (5.47)\nHere, by Corollary 5.7, \n 2is a high probability event. The fact that \n 3and \n 4are high probability\nevents can be checked from Theorem 2.8 and Theorem 2.9 of [17]. It is easy to check that \n 5is a\nhigh probability event from the existence of all moments. Furthermore, by the Lipshitz continuity\nof the resolvents, we also \fnd that \n holds uniformly on twith high probability.\nApplying Proposition 5.5 to Equation (5.32) conditioning on \n, we claim\nX\ni;jE\"\nVij \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\neP\f\f\f\n#\n=3X\nl=1cosltX\ni;j\u0014Vij\nl+1\nl!E\"\u0012@\n@Mij\u0013l  \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\neP!\f\f\f\n#\n+O(N\u00001\n2+\u000f)\n(5.48)\nwhere\u0014Vij\nldenotes the l-th cumulant of Vij. Here, it is legal to replace the conditional cumulants\nby\u0014Vij\nl, since \n is a high probability event.\nTo prove the claim, we begin by controlling the remainder term \u000fpin (5.33). On \n, Gij;^Gij\nand (G2)ijareO(1), and\nN\u00001\n2X\np;q^Gpi^Gjq=N\u00001\n2 X\np^Gpi! X\nq^Gqj!\n=O(N\u00001\n2+\u000f):\nThus,@\n@MijP=O(1) on \n. From the resolvent identity and the de\fnition of event \n, we \fnd\nkG(z;vVij)\u0000G(z;Vij)k=O(N\u00001\n2+\u000f) for 0\u0014v\u00141. Thus on \n,@\n@MijP(t;vVij) =O(1) for\n0\u0014v\u00141. Furthermore, we notice that\n@\n@Mij(G2)ij=@\n@MijX\nkGkiGjk=\u0000\fij\u0000\n2Gij(G2)ij+Gii(G2)jj+Gjj(G2)ii\u0001\n; (5.49)\nand\n@\n@MijX\np^Gpi=\u0000\fij \n^GjiX\np^Gpi+^GiiX\np^Gpj!\n: (5.50)\nThus we can obtain similar estimates for higher derivatives of P. SinceV5\nij=O(N\u00005\n2+5\u000f) on \n 5,\nwe \fnd that\njVijj50\n@1 + max\n1\u0014j\u00145 Z1\n0\f\f\f\f\f\u0012@\n@Mij\u00135\nP\f\f\f\f\fdv!5\nj1\nA\u0014CN\u00005\n2+C\u000f(5.51)\n32on \n. That is, \u000f3\u0014CN\u00005\n2+C\u000f, and after summing over i;j, the claim (5.48) is proved.\nWe next consider the term in (5.32) containing VG. Noting that the cumulants of order higher\nthan 2 vanish for Gaussian random variables, it reduces to\nX\ni;jE\"\nVG\nij \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\neP\f\f\f\n#\n=(sint)X\ni;j\u0014VG\nij\n2E\"\n@\n@Mij  \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\neP!\f\f\f\n#\n+O(N\u00001\n2+\u000f);\n(5.52)\nwhere\u0014VG\nij\n2denotes the second cumulant of VG\nij. We now put (5.48) and (5.52) into (5.32) condi-\ntioning on \n. This yields\nd\ndtEh\neP(t)\f\f\f\n] = (sint)3X\nl=1(coslt)Il\u0000(costsint)IG\n1+O(N\u00001\n2+\u000f); (5.53)\nwhere we de\fne\nIl=X\ni;j\u0014Vij\nl+1\nl!E\"\u0012@\n@Mij\u0013l  \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\neP!\f\f\f\n#\n(5.54)\nand\nIG\n1=X\ni;j\u0014VG\nij\n2E\"\n@\n@Mij  \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\neP!\f\f\f\n#\n: (5.55)\nIn the following, we will evaluate Ilforl= 1;2;3 separately. We may omit the conditioning on \nfor the ease of notation.\n5.2.1 Estimate for I1\u0000IG\n1\nSince\u0014VG\nij\n2=\u0014Vij\n2=1\nNfori6=j, we only need to consider the contribution from the diagonal entries\ntoI1\u0000IG\n1. By the de\fnition of I1andIG\n1,\nI1\u0000IG\n1=X\ni(\u0014Vii\n2\u0000\u0014VG\nii\n2)E\"\n@\n@Mii  \nt1<(G2)ii+t2=(G2)ii+t3p\nNX\np;q^Gpi^Giq!\neP!#\n:(5.56)\nFrom (5.50), we \fnd that\nt3p\nN@\n@MiiX\np;q^Gpi^Giq=O(N\u00001\n2+\u000f):\nSimilarly, it can be checked that all terms in the right-hand side of (5.56) involving ^GareO(N\u00001\n2+\u000f).\nCollecting the terms of order 1 only, we obtain that\nI1\u0000IG\n1=1\nNX\ni(w2\u00002)E\u0002\u0000\n2t1<\u0000\n(G2)iiGii\u0001\n+ 2t2=\u0000\n(G2)iiGii\u0001\n+ (t1<(G2)ii+t2=(G2)ii)2\u0001\neP\u0003\n+O(N\u00001\n2+\u000f):\n(5.57)\n33Using the estimate jGij\u0000\u000eijs1j;j(G2)ij\u0000\u000eijs0\n1j\u0014N\u00001\n2+\u000fon \n 4, we conclude that\nI1\u0000IG\n1= (w2\u00002)\u0000\n2t1<(s0\n1s1) + 2t2=(s0\n1s1) + (t1<(s0\n1) +t2=(s0\n1))2\u0001\nE\u0002\neP\u0003\n+O(N\u00001\n2+\u000f):(5.58)\n5.2.2 Estimate for I2\nWe decompose I2into\nI2=X\ni;jW3\n2N3\n2E\"\u0012@\n@Mij\u00132  \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\neP!#\n:=I2;0+ 2I1;1+I0;2;(5.59)\nwhere\nIr;2\u0000r:=X\ni;jW3\n2N3\n2E\"\u0012@\n@Mij\u0013r \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\n\u0001\u0012@\n@Mij\u00132\u0000r\neP#\n:\n(5.60)\nWe \frst consider the case i6=jin the summand of Ir;2\u0000rforr= 0;1;2. Recall that all terms\nofO(N\u00001\n2+\u000f) are negligible in the sense that they can be absorbed into the error term in the\nright-hand side of (5.53).\n\u000fForI2;0, we note that the terms arising from the derivatives of the G2are negligible, which\ncan be checked by following the argument in the proof of Theorem 3.3 in [22], especially the\nestimate of T3in (3.53) of [22]. For example, one of such terms is bounded by\n\f\f\f\f\f\fN\u00003\n2X\ni;jW3\n2E\u0002\nt1<(GiiGjj(G2)ij)eP\u0003\f\f\f\f\f\f\u0014C\n\u00114p\nN: (5.61)\nTo prove it, we consider a vector u= (G11;G22;:::;GNN) and proceed as\n\f\f\f\f\f\fX\ni;jGiiGjj(G2)ij\f\f\f\f\f\f=\f\fhu;G2ui\f\f\u0014kG2kkuk2\u0014NkG2kkGk2\u0014N\n\u00114:\nOn the other hand,\n\u0012@\n@Mij\u00132\n^Gpi^Gjq= 6( ^Gpi^G2\nji^Gjq+^Gpj^Gii^Gji^Gjq+^Gpi^Gji^Gjj^Giq)\n+^Gii^Gjj(4^Gpi^Gjq+ 2^Gpj^Giq):(5.62)\nFrom the estimate j^Gij\u0000\u000eijs2j\u0014N\u00001\n2+\u000fon \n 3the concentration of ^Geeon \n 2, we then\nclaim that\nI2;0=W3t3\n2N2X\ni;jE\"\n6^Gii^Gjj(X\np^Gpi)(X\nq^Gqj)eP#\n+O(N\u00001\n2+\u000f)\n=3W3t3s2\n2Eh\n^G2\neeePi\n+O(N\u00001\n2+\u000f) = 3W3t3s4\n2E[eP] +O(N\u00001\n2+\u000f):(5.63)\n34All the other terms in I2;0arising from\u0010\n@\n@Mij\u00112P\np;q^Gpi^Gjqare negligible. One of such terms\nis bounded by\n\f\f\f\f\f\fW3t3\n2N2X\ni;jE\"\n(X\np^Gpj)^Gii^Gji(X\nq^Gjq)eP#\f\f\f\f\f\f\u00142jW3jjt3j\n(^J\u00002)N5\n2\u00003\u000fX\ni;jE\u0002\njePj\u0003\n=O(N\u00001\n2+3\u000f)\n(5.64)\nwhere we use the de\fnitions of \n 1, \n2and \n 3.\n\u000fForI1;1, the estimates for the negligible terms can be done by using the argument similar to\n(5.64) and (5.61). The remaining O(1)-terms are\nW3t3\nN2X\ni;jE\"X\np;q^Gpi^Gjq\u0000\nt1<\u0000\nGii(G2)jj+Gjj(G2)ii\u0001\n+t2=\u0000\nGii(G2)jj+Gjj(G2)ii\u0001\u0001\neP#\n:\nUsing the de\fnitions of \n 2and \n 4, we write\nI1;1= 2W3t3\u0000\nt1<(s1s0\n1) +t2=(s1s0\n1)\u0001\nEh\n^G2\neeePi\n+O(N\u00001\n2+\u000f)\n= 2W3t3\u0000\nt1<(s1s0\n1) +t2=(s1s0\n1)\u0001\ns2\n2E\u0002\neP\u0003\n+O(N\u00001\n2+\u000f):(5.65)\n\u000fForI0;2, from the same analysis as for I1;1,\nI0;2= 2W3t3\u0000\nt1<(s1s0\n1) +t2=(s1s0\n1)\u0001\ns2\n2E\u0002\neP\u0003\n+O(N\u00001\n2+\u000f): (5.66)\nAgain, the estimate can be done in a similar manner.\nFor the case i=j, since there are only Nterms in the summation in I2, all terms are negligible\ndue to the priori bounds on kGkandP\np^Gpi.\nCollecting the terms in (5.63), (5.65), and (5.66), we get\nX\ni;j\u0014Vij\n3\n2!E\"\u0012@\n@Mij\u00132  \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\neP!#\n=W3\u0002\n3t3s4\n2+ 6t1t3<(s1s0\n1)(s2)2+ 6t2t3=(s1s0\n1)(s2)2\u0003\nE\u0002\neP\u0003\n+O(N\u00001\n2+\u000f):(5.67)\n5.2.3 Estimate for I3\nNote that any term in I3involving ^Gis negligible due to the extra N\u00001\n2factor. Estimating as in\nthe previous subsection, we obtain that\nI3=X\u0014Vij\n4\n3!E\"\u0012@\n@Mij\u00133  \nt1<(G2)ij+t2=(G2)ij+t3p\nNX\np;q^Gpi^Gjq!\neP!#\n=\u00004(W4\u00003)\"\nt1<(s3\n1s0\n1) +t2=(s3\n1s0\n1) +\u0000\nt1<(s1s0\n1) +t2=(s1s0\n1)\u00012#\nE\u0002\neP\u0003\n+O(N\u00001\n2+\u000f)(5.68)\nWe remark that O(1)-terms in I3contribute only to the corrections of linear statistics.\n355.2.4 Proof of Theorem 2.3 for general case\nLet\n~P(t) =P(t)\u0000(W2\u00002)(cost)2\u0012\nt1<(s0\n1s1) +t2=(s0\n1s1) +1\n2\u0000\nt1<(s0\n1) +t2=(s0\n1)\u00012\u0013\n+W3(cost)3\u0000\nt3s4\n2+ 2t1t3<(s1s0\n1)s2\n2+ 2t2t3=(s1s0\n1)s2\n2\u0001\n\u0000(W4\u00003)(cost)4\u0000\nt1<(s3\n1s0\n1) +t2=(s3\n1s0\n1) + (t1<(s1s0\n1) +t2=(s1s0\n1))2\u0001\n:(5.69)\nThen, plugging (5.58), (5.67), and (5.68) into (5.48), we \fnd that\nd\ndtE[e~Pj\n] =O(N\u00001\n2+\u000f); (5.70)\nwhich implies that\nE[e~P(0)j\n] =E[e~P(\u0019\n2)j\n] +O(N\u00001\n2+\u000f): (5.71)\nThus,\nlim\nN!1E\u0002\neP(0)\u0003\n= lim\nN!1\u0010\nEh\neP(0)j\ni\nP(\n) + Eh\neP(0)j\nci\nP(\nc)\u0011\n=eP(0)\u0000~P(0)lim\nN!1E[e~P(0)j\n] =eP(0)\u0000~P(0)lim\nN!1E[eP(\u0019\n2)]:(5.72)\nHere we use the fact that \n holds with high probability and ~P(\u0019\n2) =P(\u0019\n2). We can now conclude\nthat (<\u0018N(z);=\u0018N(z);nN) converges to a multivariate Gaussian vector in distribution as N!1 .\nBy direct calculation, we also \fnd that\n \n\u0018N(z)\nnN!\n)N  \nb(z)\n\u0000W3s4\n2!\n; \nV(z1)\u00002W3s1s0\n1s2\n2\n\u00002W3s1s0\n1s2\n22\nJ2(J2\u00001)!!\n(5.73)\nwithb(z) andV(z) are de\fned in Lemma 4.4. Now, using (5.26), we arrive at\n \n\u0018N(z)\n\u001fN!\n)N  \nb(z)\nW3\nJ2(1\u00001\nJ2)!\n; \nV(z1) 2W3s1s0\n1(1\u00001\nJ2)\n2W3s1s0\n1(1\u00001\nJ2) 2(1\u00001\nJ2)!!\n: (5.74)\nHence, the asymptotic Gaussianity of ( N(2)\nN(');\u001fN) follows. For (2.10) and (2.11), the mean and\nthe variance ofN(2)['] is given in Theorem 2.1. The limiting covariance is given by\n\u00002W3(1\u00001\nJ2)I\n\u0000'(z)s(z)s0(z)dz\n2\u0019i= 2W3(1\u00001\nJ2)\u001c1('): (5.75)\nwhere we use the change of variables z7!smapping Cn[\u00002;2] to the diskjsj<1 withs+1\ns=\u0000z\nand (4.16) in [6]. This completes the proof of Theorem 2.3 for general case.\n6 Matching\nIn the transitional regime, we took 2 \f=1\nJ+Bp\nN. The ferromagnetic regime and the paramagnetic\nregime correspond to the limiting cases 2 \f > J and 2\f < J , respectively. In this section, we will\n36consider formal limits B!\u00061 of the formula given in the main result, Theorem 1.5, and check\nthe consistency with the results for ferromagnetic and paramagnetic regimes obtained in [6].\nTheorem 1.5 states that the free energy FNis close to the random variable\nFtran\nN:=1\n4J2+B\n2Jp\nN+logN\n4N+B2J2\n4N+1\nNG1+1\nNQ(G2) (6.1)\nin an appropriate sense. Here, ( G1;G2) is a Gaussian vector independent of B. The function Q(x)\nis given by (1.19). In ferromagnetic and paramagnetic regimes, [6] shows that the free energy is\nclose to\nFferro\nN:=\f\u0000\nJ+1\nJ\u0001\n\u00001\n2log(2\fJ)\u00001\n4J2\u00001\n2+\f\u00001\n2Jp\nNN(f0\n2;\u000b0\n2) (6.2)\nand\nFpara\nN:=\f2+1\nNN(f1;\u000b1); (6.3)\nrespectively, where N(f;\u000b) denotes a Gaussian distribution of mean fand variance \u000b. The pa-\nrameters for the Gaussians are (see (4) of [5] which corrected an error in [6])\nf0\n2=W3(J\u00002\u0000J\u00004);\n\u000b0\n2= 2(1\u0000J\u00002)(6.4)\nand (see (1.11) and (1.12) of [6]; we set J0=J)\nf1=1\n4log(1\u00004\f2) +\f2(w2\u00002) + 2\f4(W4\u00003)\u00001\n2log(1\u00002\fJ);\n\u000b1=\u00001\n2log(1\u00004\f2) +\f2(w2\u00002) + 2\f4(W4\u00003):(6.5)\nThe function Q(x) in (6.1) is given by\nQ(x) =s(x)\n2(s(x)\u0000x)\u0000s(x)2\n4(J2\u00001)+log(s(x)\u0000x)\n2+ log I\u0012(s(x)\u0000x)2\nJ2\u00001\u0013\n(6.6)\nwhere (recall the formula (1.20))\ns(x) =x\u0000B(J2\u00001) +p\n(x+B(J2\u00001))2+ 4(J2\u00001)\n2: (6.7)\nFrom the formula, for x=O(1),\ns(x) =(\nx+1\nB+O(B\u00002) as B!+1,\n\u0000B(J2\u00001)\u00001\nB+O(B\u00002) asB!\u00001 .(6.8)\nNote that since we set 2 \f=1\nJ+Bp\nNin the transitional regime, we regard B=O(p\nN) when\nwe takeB!\u00061 .\n376.1B!+1\nUsing (6.8), we \fnd that for x=O(1),\nQ(x) =Bx\n2+O(logB): (6.9)\nHence, sinceG1does not depend on B, we see that as B=O(p\nN) withB > 0,\nFtran\nN=1\n4J2+B\n2Jp\nN+B2J2\n4N+B\n2NG2+O\u0012logB\nN\u0013\n+O\u0012logN\nN\u0013\n: (6.10)\nwhereO(f(B;N )) represents a random variable Xsuch that the moments ofX\nf(B;N)are all bounded\nby constants independent of BandN.\nWe compare the above formula with the ferromagnetic case (6.2). If we set 2 \f=1\nJ+Bp\nN, then\nFferro\nN=1\n4J2+B\n2Jp\nN+B2J2\n4N+B\n2NN(f0\n2;\u000b0\n2) +O(N\u00003=2): (6.11)\nWe note that (see (6.4) and (1.25)) the mean and variance are f0\n2=E[G2] and\u000b0\n2= Var[G2]. The\nabove formula of Ftran\nNis thus consistent with Fferro\nN.\n6.2B!\u00001\nConsider (6.6). Recall that I(\u000b) =q\n4\u0019\n\u000b(1 +O(\u000b\u00001)) as\u000b!+1from (3.24). Hence, if x=O(1)\nands(x)!1 , then\nQ(x) =\u0000s(x)2\n4(J2\u00001)+ logs\n4\u0019(J2\u00001)\ns(x)+1\n2+O\u00121\ns(x)\u0013\n: (6.12)\nUsing (6.8), we \fnd that for x=O(1),\nQ(x) =\u0000B2(J2\u00001)\n4+ logs\n4\u0019\njBj+O(B\u00001): (6.13)\nHence, the two leading terms of Q(G2) do not depend on G2. Therefore, for B=O(p\nN) with\nB < 0,\nFtran\nN=1\n4\u00121\nJ+Bp\nN\u00132\n+1\n2Nlog \n4\u0019p\nN\njBj!\n+1\nNG1+O\u00121\nNB\u0013\n: (6.14)\nOn the other hand, in the paramagnetic regime, if we set 2 \f=1\nJ+Bp\nNwithB < 0, then the\nparameters in (6.5) satisfy (see (1.23))\nf1=1\n4log(1\u0000J\u00002) +1\n4J2(w2\u00002) +1\n8J4(W4\u00003)\u00001\n2log\u0012jBjJp\nN\u0013\n+O(N\u00001=2)\n=E[G1] +1\n2log \n4\u0019p\nN\njBj!\n+O(N\u00001=2)(6.15)\n38and\n\u000b1=\u00001\n2log(1\u0000J\u00002) +w2\u00002\n4J2+W4\u00003\n8J4+O(N\u00001=2) = Var[G1] +O(N\u00001=2) (6.16)\nThus, if we set 2 \f=1\nJ+Bp\nNwithB < 0, then\nFpara\nN=1\n4\u00121\nJ+Bp\nN\u00132\n+1\n2Nlog \n4\u0019p\nN\njBj!\n+1\nNN(E[G1];Var[G1]) +O(N\u00003=2): (6.17)\nThis is consistent with the formula of Ftran\nN.\nReferences\n[1] Z. Bai and J. W. Silverstein. Spectral analysis of large dimensional random matrices . Springer\nSeries in Statistics. Springer, New York, second edition, 2010.\n[2] Z. Bai and J. Yao. 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Stat. , 49(1):64{94, 2013.\n[26] M. Talagrand. Free energy of the spherical mean \feld model. Probab. Theory Related Fields ,\n134(3):339{382, 2006.\n40[27] M. Talagrand. The Parisi formula. Ann. of Math. (2) , 163(1):221{263, 2006.\n41"    },    {        "title": "2303.15820v2.Flat_band_ferromagnetism_in_the_SU__N___Hubbard_and_Kondo_lattice_models.pdf",        "content": "Flat-band ferromagnetism in the SU( N) Hubbard\nand Kondo lattice models\nKensuke Tamura\nDepartment of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo\n113-0033, Japan\nE-mail: tamura-kensuke265@g.ecc.u-tokyo.ac.jp\nHosho Katsura\nDepartment of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo\n113-0033, Japan\nInstitute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo,\nBunkyo-ku, Tokyo 113-0033, Japan\nTrans-scale Quantum Science Institute, The University of Tokyo, 7-3-1 Hongo,\nBunkyo-ku, Tokyo 113-0033, Japan\nE-mail: katsura@phys.s.u-tokyo.ac.jp\n12 September 2023\nAbstract. We develop a general theory of flat-band ferromagnetism in the SU( N)\nFermi-Hubbard model, which describes the behavior of N-component fermions with\nSU(N) symmetric interactions. We focus on the case where the single-particle spectrum\nhas a flat band at the bottom and establish a necessary and sufficient condition for\nthe SU( N) Hubbard model to exhibit ferromagnetism when the number of particles\nis the same as the degeneracy. We show that the occurrence of ferromagnetism is\nequivalent to the irreducibility of the projection matrix onto the space of single-particle\nground states. We also demonstrate that this result can be exploited to establish a\nrigorous result for the ferromagnetic SU( N) Kondo lattice model with a flat band.\nSpecifically, we prove that when the SU( N) Hubbard model is ferromagnetic, the\nferromagnetic SU( N) Kondo lattice model with the same hopping matrix also exhibits\nSU(N) ferromagnetism.\nKeywords : SU( N) Hubbard model, SU( N) Kondo lattice model, flat-band\nferromagnetism. Submitted to: J. Phys. A: Math. Theor.arXiv:2303.15820v2  [cond-mat.str-el]  11 Sep 2023Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 2\n1. Introduction\nIn recent years, advances in experimental techniques with ultracold atoms have allowed\nfor the simulation of various quantum systems in optical lattices [1–5]. With the\nability to precisely control lattice potentials and interaction strengths, ultracold atomic\nsystems are expected to be a versatile tool for investigating many-body physics in\nstrongly correlated systems. It is worth noting that ultracold atoms are not restricted to\nsimulating known models describing conventional quantum systems but can also realize\nnovel quantum systems with no counterpart in conventional materials.\nOne example of such a novel quantum system is fermionic systems with SU( N)\nsymmetry realized with alkaline-earth-like atoms. These atoms trapped in an optical\nlattice are described by the SU( N) Fermi-Hubbard model [6], which generalizes the\nstandard Hubbard model with SU(2) symmetry [7–11]. In conventional condensed-\nmatter physics, the SU( N) Hubbard model has mainly been explored with the large- N\napproach [12,13]. This approach is primarily concerned with the behavior of the model\nwith infinitely large N, and little attention has been paid to the properties of the model\nfor finite N(N > 2). However, recent experimental realizations of the SU( N) Hubbard\nmodel with ultracold atoms have inspired theoretical studies on the properties of the\nSU(N) Hubbard model with finite N[14–16]. Recent studies have shown that the\nSU(N) Hubbard model can exhibit phases, and interest in this model has continued to\ngrow.\nMoreover, in a specific limit, the system of alkaline-earth-like atoms can be\ndescribed by the SU( N) Kondo lattice model [6, 17], in which itinerant fermions and\nlocalized SU( N) spins interact with SU( N) symmetric exchange interaction. Efforts\nhave also been made to realize such systems described by the SU( N) Kondo lattice\nmodel using two-orbital alkaline-earth-like atoms [18–20]. The SU( N) Kondo lattice\nmodel was also introduced in the large- Napproach to study the SU(2) Kondo lattice\nmodel [21–23]. However, in this approach, the main focus was on the case with infinitely\nlarge N, and the properties of the models at finite Nhave been less studied except for\nthe case of N= 2.\nWhile the SU( N) Hubbard and Kondo lattice models have attracted much\ninterest both theoretically and experimentally, such models are notoriously difficult to\nsolve analytically. Nevertheless, obtaining mathematically rigorous results in special\nsituations would be possible. Although the model in such a situation may be unrealistic,\nit can serve as a basis for other theoretical studies. Here we review rigorous results for\nSU(N) symmetric models, mostly for the SU( N) Hubbard model (including the case\nwith N= 2).\nThe Nagaoka ferromagnetism is the first rigorous result for the SU(2) Hubbard\nmodel [24, 25]. It was proved that when the Coulomb repulsions are infinitely large,\nand there is exactly one hole, the ground state of the Hubbard model is ferromagnetic\nand unique, provided the lattice satisfies a certain connectivity condition. Recently,\nRefs. [26, 27] have reported that the Nagaoka ferromagnetism can be extended to theFlat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 3\nSU(N) Hubbard model with general N. In the multiorbital Hubbard model, theorems\nregarding ferromagnetism have been rigorously proved in the Refs. [28, 29]. These\ntheorems have also been extended to the SU( N) case, as discussed in [28].\nIn Ref. [30], Lieb established rigorous results for both attractive and repulsive\nSU(2) Hubbard model. In particular, for the repulsive case, it was proved that Lieb’s\nferrimagnetism is exhibited in a wide range of models, which can also be considered\nas the case with a flat band. Subsequently, Mielke [31] and Tasaki [32] independently\nestablished new rigorous results for the SU(2) Hubbard model, known as flat-band\nferromagnetism. The term flat band refers to the structure of the single-particle energy\nspectrum with macroscopic degeneracy. They constructed tight-binding models that\nproduce a flat band at the bottom of the single-particle spectrum and then showed\nthat the ground states of the Hubbard model are ferromagnetic and unique when the\nnumber of particles equals the multiplicity of the single-particle ground states. There\nare systematic methods for constructing tight-binding models with flat bands. For\nexample, Mielke proposed a method based on line graphs [31]. Another method called\ncell construction was introduced by Tasaki [32–35]. Other methods of constructing\nvarious classes of flat bands have also been proposed [36–41]. Based on these methods,\nvarious types of flat-band ferromagnetism have been studied so far [42–46]. Furthermore,\nextensions of flat-band ferromagnetism to the SU( N) case have recently been discussed,\nand rigorous results were proved in Refs. [47–49].\nAlthough there are various tight-binding models that have a flat band at the bottom\nof their energy spectrum, we should note that it is not always guaranteed that the ground\nstate of the Hubbard model, which is formed by adding the on-site interaction term to\nthe tight-binding model, is uniquely ferromagnetic. In the case of the SU(2) Hubbard\nmodel, a general theory of flat-band ferromagnetism has been developed [35, 50, 51].\nThis theory provides a necessary and sufficient condition to determine whether an SU(2)\nHubbard model with a bottom flat band exhibits ferromagnetism in the ground state.\nHowever, the corresponding general theory of flat-band ferromagnetism in the SU( N)\nHubbard model has not yet been established.\nWe also comment on rigorous results for the Kondo lattice model. As for the SU(2)\nKondo lattice model, a few rigorous results are known. In Ref. [52], the equivalence\nbetween the antiferromagnetic SU(2) Kondo lattice model in the strong-coupling limit\nand the SU(2) Hubbard model with infinitely large Coulomb repulsion was found. It was\nshown that there exist some ferromagnetic regions. In Ref. [53], the antiferromagnetic\nSU(2) Kondo lattice model with one electron was investigated, and it was rigorously\nproved that the ground state exhibits a ferromagnetic order. The SU(2) Kondo lattice\nmodel with a flat band was also discussed in Ref. [54]. For the SU( N) Kondo lattice\nmodel, the one-dimensional SU( N) Kondo lattice model has recently been discussed\nin Ref. [17]. In the strong-coupling limit, the effective Hamiltonian of the model was\nderived, and rigorous results for the ground states were proved, which can be seen as a\ngeneralization of the result in Ref. [55] to the SU( N) case.\nThis paper presents a general theory of flat-band ferromagnetism in the SU( N)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 4\nHubbard model. This is a natural extension of the general theory in the SU(2)\nHubbard model presented in Refs. [50,51]. We consider a hopping matrix whose ground\nstates are degenerate. Then we study the SU( N) Hubbard model with the hopping\nmatrix. We give a necessary and sufficient condition for the model to exhibit SU( N)\nferromagnetism. It is proved that the emergence of SU( N) ferromagnetism is equivalent\nto the irreducibility of the orthogonal projection matrix onto the space spanned by the\nlowest energy states of the single-particle spectrum.\nIn addition, we find an application of the result to the ferromagnetic SU( N)\nKondo lattice model and prove a rigorous result for flat-band ferromagnetism in this\nmodel. The standard alkali-earth-like atoms, such as87Sr and173Yb, exhibit the\nferromagnetic Kondo coupling rather than antiferromagnetic Kondo coupling [17,18,56,\n57]. Consequently, in the context of ultracold atomic experiments, it is more physically\nnatural to consider the ferromagnetic SU( N) Kondo lattice model. Supposing that the\nSU(N) Hubbard model exhibits SU( N) ferromagnetism, it is rigorously proved that the\nferromagnetic SU( N) Kondo lattice model with the same hopping matrix also exhibits\nSU(N) ferromagnetism in its ground states.\nThe present paper is organized as follows. In Sec. 2, we consider the SU( N) Hubbard\nmodel with degenerate single-particle ground states. We then discuss the necessary and\nsufficient condition for the SU( N) Hubbard model to exhibit ferromagnetism when the\nnumber of particles is the same as the degeneracy and prove that the irreducibility\nof the projection matrix onto the space of single-particle ground states is equivalent\nto the occurrence of ferromagnetism. In Sec. 3, we further discuss the ferromagnetic\nSU(N) Kondo lattice model with a flat band. By exploiting the general theory for the\nHubbard model, we also establish a rigorous result for flat-band ferromagnetism in the\nferromagnetic SU( N) Kondo lattice model. Finally, in Sec. 4, we give a summary and\npresent some remarks on the theorem concerning the SU( N) Kondo lattice model.\n2. The SU( N) Hubbard model and main result\n2.1. The SU( N) Hubbard model\nLet Λ be a finite lattice. We denote by ˆ c†\nx,αand ˆcx,α, respectively, the fermionic creation\nand the annihilation operators at site x∈Λ with color α= 1, . . . , N . They satisfy the\nanticommutation relations\n{ˆcx,α,ˆcy,β}={ˆc†\nx,α,ˆc†\ny,β}= 0, (1)\n{ˆcx,α,ˆc†\ny,β}=δα,βδx,y. (2)\nThe number operator of fermion at site xwith color αis defined by ˆ nx,α= ˆc†\nx,αˆcx,α, and\nthe total fermion number is ˆNc=P\nx∈Λˆnx, where ˆ nx=PN\nα=1ˆnx,α. The Fock space of\nthe fermionic operators is denoted by H(Λ). The Hamiltonian of the SU( N) HubbardFlat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 5\nmodel is given by\nˆHHub=ˆHhop+ˆHint, (3)\nˆHhop=NX\nα=1X\nx,y∈Λtx,yˆc†\nx,αˆcy,α, (4)\nˆHint=UX\nα<βX\nx∈Λˆnx,αˆnx,β, (5)\nwhere T= (tx,y)x,y∈Λis the hopping matrix on the lattice Λ, and the parameter Uis\nassumed to be positive.\nIn the SU( N) Hubbard model, the total number of fermions is trivially conserved,\nwhich can be seen as\n[ˆHHub,ˆNc] = 0. (6)\nWe define color raising and lowering operators by\nˆFα,β=X\nx∈Λˆc†\nx,αˆcx,βforα̸=β, (7)\nand the total number operator of fermions with color αby\nˆFα,α=X\nx∈Λˆc†\nx,αˆcx,αforα= 1, . . . , N. (8)\nDue to the SU( N) symmetry, one can see that the operators ˆFα,βcommutes with ˆHHub.\nTogether with the conservation of the total number of fermions, the Hamiltonian (3)\npossesses U( N) = U(1) ×SU(N) symmetry. In what follows, we denote the eigenvalues\nofˆFα,αbyMα, and the eigenvalue of ˆNcbyNc.\nLet us introduce some subspaces of H(Λ). We define a subspace HNc(Λ) by\nHNc(Λ) = {|Φ⟩ ∈ H (Λ)|ˆNc|Φ⟩=Nc|Φ⟩}. (9)\nWe also define a subspace HM1,...,M N(Λ) by\nHM1,...,M N(Λ) = {|Φ⟩ ∈ H (Λ)|ˆFα,α|Φ⟩=Mα|Φ⟩for all α= 1, . . . , N }. (10)\nTo define SU( N) ferromagnetism, we introduce the quadratic Casimir operator ˆC2\nof the SU( N) group, which is defined by [58]\nˆC2=1\n2 NX\nα,β=1ˆFα,βˆFβ,α−ˆN2\nc\nN!\n. (11)\nWhen N= 2, in the standard notation, one may write ˆS+\ntot=ˆF1,2,ˆS−\ntot=ˆF2,1, and\nˆSz\ntot=\u0010\nˆF1,1−ˆF2,2\u0011\n/2. With the notation, the operator ˆC2is written as\nˆC2=1\n2\u0010\nˆS+\ntotˆS−\ntot+ˆS−\ntotˆS+\ntot\u0011\n+\u0010\nˆSz\ntot\u00112\nforN= 2. (12)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 6\nThis operator is the square of the magnitude of the total spin operator defined by\u0010\nˆStot\u00112\n=\u0010\nˆSx\ntot\u00112\n+\u0010\nˆSy\ntot\u00112\n+\u0010\nˆSz\ntot\u00112\n. Therefore, the operator ˆC2can be seen as a\ngeneralization of the operator\u0010\nˆStot\u00112\n.\nNow we are ready to state the definition of SU( N) ferromagnetism.\nDefinition 1. Consider the Hamiltonian (3)with the total fermion number Nc. We\nsay that the model exhibits SU( N) ferromagnetism if any ground state |ΦGS⟩has the\nmaximum eigenvalue of ˆC2inHNc(Λ), i.e.,\nˆC2|ΦGS⟩=Nc(N−1)\n2\u0012Nc\nN+ 1\u0013\n|ΦGS⟩. (13)\nNote that the above definition of ferromagnetism is the strongest form of\nferromagnetism, which should be referred to as complete ferromagnetism or saturated\nferromagnetism. In the case N= 2, let Stot(Stot+ 1) be the eigenvalue of\u0010\nˆStot\u00112\n.\nEven if Stotof the ground states is macroscopically large but not the maximum value,\nit is commonly considered that ferromagnetism is manifested. In this paper, however,\nwe only study complete ferromagnetism and refer to it simply as ferromagnetism.\nWe call a state satisfying Eq. (13) as a fully polarized state. The eigenvalue equation\n(13) is satisfied if a state has no double occupancy and is fully symmetrized with respect\nto the color degrees of freedom. Conversely, a state satisfying Eq. (13) is such a fully\nsymmetrized state.\n2.2. Main theorem\nHere we state our main theorem. First, we introduce some notation and make\nassumptions about the model. The single-particle Hilbert space is denoted by h∼=C|Λ|,\nand we write a |Λ|-dimensional vector in hasϕ= (ϕ(x))x∈Λ. The inner product of two\nvectors, ϕandψ, is defined by\n⟨ϕ,ψ⟩=X\nx∈Λϕ(x)∗ψ(x). (14)\nWe now assume that\nT≥0, (15)\nand denote the kernel of Tbyh0= ker T. We also assume that h0is not empty and\nwrite D0= dim h0. Let P0be the orthogonal projection matrix onto the subspace h0,\nand we define Λ 0={x∈Λ|(P0)x,x̸= 0}. We say the |Λ0|×|Λ0|matrix\u0010\n(P0)x,y\u0011\nx,y∈Λ0is\nreducible if and only if Λ 0can be decomposed as Λ 0= Λ 1∪Λ2with Λ 1∩Λ2=∅, Λ0̸=∅,\nand Λ 2̸=∅so that ( P0)x,y= 0 for any x∈Λ1andy∈Λ2. The matrix\u0010\n(P0)x,y\u0011\nx,y∈Λ0\nis said to be irreducible if it is not reducible. If Thas translation symmetry, we have\nenergy bands as a function of wave vectors. Moreover, if D0is proportional to theFlat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 7\nFigure 1. The lattice structure of the delta chain with hopping amplitude t1=t/√\n2\nandt2=t/2, where we impose the periodic boundary conditions. All the sites have\nthe uniform on-site potentials t.\n−π 0 π\nk0.00.51.01.52.02.53.0ε(k)\nFigure 2. The dispersion relations of the energy bands for the delta chain with t= 1.\nThe lowest band is completely flat at zero energy In this case, the value of D0equals\nthe number of the unit cells.\nnumber of sites |Λ|, it suggests that the lowest band is flat at zero energy. Figure 1\nshows an example of a lattice system in which the lowest band is flat, called a delta\nchain. The realization of this lattice system with an optical lattice is also discussed [59].\nThe dispersion relations of this lattice system are shown in Fig. 2.\nNow we are ready to state our theorem.\nTheorem 1. Consider the SU( N) Hubbard model (3)withT≥0andNc=D0. The\nmodel exhibits SU( N) ferromagnetism if and only if the |Λ0|×|Λ0|matrix\u0010\n(P0)x,y\u0011\nx,y∈Λ0\nis irreducible.\n2.3. Proof of Theorem 1\nIn this subsection, we prove Theorem 1. We first prove the following lemma.\nLemma 1. One can take a subset I⊂Λwith|I|=D0and a basis {µz}z∈Iofh0in\nsuch a way that for each z∈I, the basis vector µz= (µz(x))x∈Λsatisfies µz(z)̸= 0and\nµz(z′) = 0 for any z′∈I\\{z}.\nProof. Our proof is essentially the same as the proof of Lemma 11.16 in Ref. [35]. We\nsee that the rank of Tis|Λ| −D0since dim h0=D0. Then there exists a subset Λ′⊂Λ\nwith|Λ′|=|Λ| −D0such that the determinant of the submatrix ( tx,y)x,y∈Λ′is nonzero,Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 8\nFigure 3. In the delta chain, the vector µzsatisfying the condition of Lemma 1 can\nbe defined to be localized at the black site. This vector has a component of 1 at the\nblack site zand−1/√\n2 at the two white sites adjacent to z.\nand any ( |Λ| −D0+ 1)×(|Λ| −D0+ 1) submatrix of Thas determinant zero [60].\nLetI= Λ\\Λ′. We find that, for arbitrary z∈I, the submatrix ( tx,y)x,y∈Λ′∪{z}has\ndeterminant zero, which implies that this matrix has a zero eigenvalue. We denote the\ncorresponding eigenvector by ˜µz= (˜µz(x))x∈Λ′∪{z}. We can see that ˜ µz(z)̸= 0. This\nis because if ˜ µz(z) = 0, then (˜ µz(x))x∈Λ′is an eigenvector of ( tx,y)x,y∈Λ′with eigenvalue\nzero. This contradicts that the matrix ( tx,y)x,y∈Λ′has nonzero determinant. Thus, we\nhave µz(z)̸= 0. We then define a |Λ|-dimensional vector µz= (µz(x))x∈Λas\nµz(x) =(\n˜µz(x) ifx∈Λ′∪ {z},\n0 otherwise ,(16)\nforz∈I. We note that µz(z)̸= 0 for z∈Iandµz(z′) = 0 for z′∈I\\{z}. UsingP\ny∈Λ′∪{z}tx,y˜µz(y) = 0, we can see that\n⟨µz,Tµz⟩= 0. (17)\nBecause of the positive semidefiniteness of T, it holds that Tµz= 0. Since µz(z)̸= 0\nandµz(z′) = 0, the set {µz}z∈Iis linearly independent, and hence it is a basis of h0\nsatisfying the conditions of Lemma 1.\nIn the example shown in Fig. 1, the subset Ican be taken as the entire set of the\nblack sites. In this case, the vector µzis localized at a black site. This vector has\nnonzero components only at the black site and the two white sites adjacent to it. See\nFig. 3\nWith the basis {µz}z∈I, we can characterize Λ 0as\nΛ0={x∈Λ|µz(x)̸= 0 for some z∈I}. (18)\nThis can be seen as follows. Let {ψi}i=1,...,D 0be an orthonormal basis of h0. The\nprojection matrix P0is written as ( P0)x,y=PD0\ni=1ψi(x)ψi(y)∗. Suppose that µz(x) = 0\nfor all z∈Ifor some x∈Λ. Then we see that ψi(x) = 0 for all i= 1, . . . , D 0since\nthe vector ψican be written as a linear combination of {µz}z∈I. Therefore, we have\n(P0)x,x=PD0\ni=1ψi(x)ψi(x)∗= 0, which means that x /∈Λ0ifµz(x) = 0 for all z∈I.Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 9\nConversely, suppose that for some x∈Λ, there exists z∈Isuch that µz(x)̸= 0. Let Pz\nbe the projection matrix onto the one-dimensional subspace spanned by µz. Then we\nsee that ( P0)x,x≥(Pz)x,x. Since µz(x)̸= 0, ( Pz)x,x>0. Therefore, we have ( P0)x,x̸= 0,\nand hence, x∈Λ0. Thus, Λ 0={x∈Λ|µz(x)̸= 0 for some z∈I}. For example, in\nthe delta chain, the set of the eigenvectors with eigenvalue zero {µz}z∈Ican cover the\nentire lattice system, thus Λ 0= Λ.\nWe write µz∼µz′if there is a site x∈Λ such that µz(x)µz′(x)̸= 0. We say that\nthe basis {µz}z∈Iis connected if there is a sequence {zi}i=0,...,nwith zi∈Isuch that\nz0=z,zn=z′, andµzi−1∼µzifori= 1, . . . , n . We also write µz≁µz′if there is no\nsitexsuch that µz(x)µz′(x)̸= 0.\nThen we can prove the following lemma.\nLemma 2. Consider the SU( N) Hubbard model with Nc=D0. The model exhibits the\nSU(N) ferromagnetism if and only if the basis {µz}z∈Iis connected.\nProof. Since the hopping matrix Tis positive semidefinite, ˆHhopis also positive\nsemidefinite. This can be seen as follows. Since dim h=|Λ|and dim h0=D0, there are\n|Λ|−D0linearly independent eigenvectors of Twith positive eigenvalues. We denote the\neigenvectors by ϕi(i= 1, . . . ,|Λ| −D0), which satisfy Tϕi=λiϕiwith λi>0. Since\nthey can always be taken to be orthogonal to each other, we assume that ⟨ϕi,ϕj⟩=δi,j.\nWith the vectors µzandϕi, we define a new set of operators\nˆa†\nz,α=X\nx∈Λµz(x)ˆc†\nx,α, (19)\nˆb†\ni,α=X\nx∈Λϕi(x)ˆc†\nx,α. (20)\nThey satisfy\n{ˆaz,α,ˆaw,β}={ˆbi,α,ˆbj,β}={ˆaz,α,ˆbi,β}= 0, (21)\n{az,α, a†\nw,β}=δα,β⟨µz,µw⟩, (22)\n{ˆbi,α,ˆb†\nj,β}=δα,βδi,j, (23)\n{ˆaz,α,ˆb†\ni,β}= 0, (24)\nwhere the last line follows since ⟨µz,ϕi⟩= 0. Because the hopping matrix can be written\nastx,y=P|Λ|−D0\ni=1 λiϕi(x)ϕi(y)∗, we can represent the hopping Hamiltonian as\nˆHhop=NX\nα=1|Λ|−D0X\ni=1λiˆb†\ni,αˆbi,α. (25)\nSince ˆb†\ni,αˆbi,α≥0 and λi>0,ˆHhopis positive semidefinite. The interaction term ˆHint\nis also positive semidefinite because ˆ nx,αˆnx,β= (ˆcx,αˆcx,β)†ˆcx,αˆcx,β≥0. Hence, the entire\nHamiltonian ˆHHubis positive semidefinite.Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 10\nWe define the fully polarized states as follows. First, the fully polarized state with\ncolor αis given by\n|Φallα⟩= Y\nz∈Iˆa†\nz,α!\n|Φvac⟩, (26)\nwhere |Φvac⟩is the normalized vacuum state for ˆ cx,α. We can easily see that the |Φallα⟩\nis an eigenstate of ˆHHubwith zero energy. Since ˆHHub≥0, the state |Φallα⟩is the\nground state of ˆHHubinHD0(Λ). Due to the SU( N) symmetry, one can obtain other\nground states of the following form\n|ΦM1,M2,...,M N⟩=\u0010\nˆFN,1\u0011MN···\u0010\nˆF2,1\u0011M2|Φall 1⟩, (27)\nwhere M1=D0−PN\nα=2Mα. We also refer to the states of the form (27) as fully\npolarized states. It is easily seen that\nˆC2|Φallα⟩=D0(N−1)\n2\u0012D0\nN+ 1\u0013\n|Φallα⟩, (28)\nwhich means that the state |Φallα⟩has the maximum eigenvalue of ˆC2inHD0(Λ).\nBecause [ ˆC2,ˆFα,β] = 0, the fully polarized states of the form (27) also have the same\neigenvalue for ˆC2. Thus, all the fully polarized states are ground states of ˆHHubwith\nthe maximum eigenvalue of ˆC2inHD0(Λ).\nIn the following, we prove that there are no other ground states if and only if\n{µz}z∈Iis connected. Let |ΦGS⟩be an arbitrary ground state of ˆHHubinHD0(Λ). In\ngeneral, we can express the state as\n|ΦGS⟩=X\nI1,...,IN⊂I\n˜I1,...,˜IN⊂˜If\u0010\n{Iα},{˜Iα}\u0011\n×\n Y\nz1∈I1ˆa†\nz1,1!\n··· Y\nzN∈INˆa†\nzN,N!\nY\ni1∈˜I1ˆb†\ni1,1\n···\nY\niN∈˜INˆb†\niN,N\n|Φvac⟩, (29)\nwhere f({Iα},{˜Iα}) is a coefficient, and Iαand˜Iαare subsets of Iand˜I={1,2, . . . ,|Λ|−\nD0}, respectively, such thatPN\nα=1\u0010\n|Iα|+|˜Iα|\u0011\n=D0. The ground state satisfies\nˆHHub|ΦGS⟩= 0, and the inequalities ˆHhop≥0 and ˆHint≥0 imply that ˆHhop|ΦGS⟩= 0\nand ˆHint|ΦGS⟩= 0. Since the hopping Hamiltonian can be expressed as in Eq. (25) and\nˆb†\ni,αˆbi,α≥0 for all i= 1, . . . ,|Λ| −D0andα, the condition ˆHhop|ΦGS⟩= 0 leads to\nˆbi,α|ΦGS⟩= 0 for all i= 1, . . . ,|Λ| −D0andα= 1, . . . , N. (30)\nSimilarly, the equation ˆHint|ΦGS⟩= 0 reduces to\nˆcx,αˆcx,β|ΦGS⟩= 0 for any x∈Λ and α̸=β. (31)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 11\nHere we consider the condition (30). Noting the anticommutation relations (24), we\nfind that the ground state |ΦGS⟩consists only of ˆ a†\nz,αoperators, i.e., the state |ΦGS⟩is\nwritten as\n|ΦGS⟩=X\nI1,...,IN⊂Ig({Iα}) Y\nz1∈I1ˆa†\nz1,1!\n··· Y\nzN∈INˆa†\nzN,N!\n|Φvac⟩, (32)\nwhere g({Iα}) is a coefficient, andPN\nα=1|Iα|=D0.\nThen we examine Eq. (31). We first consider the case where x=z∈I. In general,\nit holds that\n{ˆcx,α,ˆa†\nz,β}=δα,βµz(x), (33)\nfor any x∈Λ and z∈I. In particular, when x=z∈I, we have the following\nanticommutation relations\n{ˆcz,α,ˆa†\nz′,β}=δα,βδz,z′µz(z) (34)\nfor all z, z′∈I. Using the anticommutation relations (34), we find that g({Iα}) = 0 if\nthere is a pair of colors αandβsuch that Iα∩Iβ̸=∅. SincePN\nα=1|Iα|=D0, we have\n∪N\nα=1Iα=Iwhen Iα∩Iβ=∅. Therefore, the ground state takes the form\n|ΦGS⟩=X\nαC(α) Y\nz∈Iˆa†\nz,αz!\n|Φvac⟩, (35)\nwhere α= (αz)z∈Irepresents a color configuration over I, and the sum is taken over all\npossible color configurations.\nWe next consider Eq. (31) for x∈Λ\\I. With the use of Eq. (33), the condition (31)\nyields\nX\nz1<z2X\nγ\nγz1=β,γz2=αsgn(z1, z2;I)µz1(x)µz2(x)\n×(C(γ)−C(γz1↔z2))\nY\nz∈I\\{z1,z2}ˆa†\nz,γz\n|Φvac⟩= 0, (36)\nwhere sgn( z1, z2;I) is a sign factor arising from exchanges of the fermion operators. The\nconfiguration γz1↔z2is obtained from γby swapping γz1andγz2. Since all the states in\nthe sum are linearly independent, we find that\nµz1(x)µz2(x) (C(γ)−C(γz1↔z2)) = 0 . (37)\nfor all z1, z2∈Isuch that z1̸=z2andγ. When z1andz2satisfy µz1∼µz2, then\nby definition, there is a site x∈Λ\\Isuch that µz1(x)µz2(x)̸= 0. Consequently, we\nhave C(γ) =C(γz1↔z2) ifµz1∼µz2. When the basis {µz}z∈Iis connected, for anyFlat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 12\nz, z′∈I, we can take a sequence z1=z, z2, . . . , z n=z′such that µzi∼µzi+1. Thus,\nwe see C(γ) =C(γz↔z′) for any z, z′∈I. By noting that any permutation of αcan be\nobtained by repeatedly swapping two colors, we have\nC(α) =C(β), (38)\nwhere βis any color configuration obtained as a permutation of α. As a result, when Mα\nin Eq. (10) is fixed so thatPN\nα=1Mα=D0, the ground state is unique in HM1,...,M N(Λ).\nWe can verify that the states satisfying Eq. (38) are actually the fully polarized\nstates defined by Eqs. (26) and (27). To see this, we use a concept of a word. A\nword w= (w1, . . . , w D0) is a sequence in which wi∈ {1, . . . , N }for all i= 1, . . . , D 0.\nWe denote the number of occurrences of αinwby|wα|. The set of words for which\n|wα|=Mαis defined by W(M1,···MN) ={w||wα|=Mαfor all α}. For example,\nW(2,0,1) consists of (1 ,1,3), (1 ,3,1), and (3 ,1,1). With the notation, the ground\nstate in HM1,...,M N(Λ) satisfying Eq. (38) is written as\n\f\f\f˜ΦM1,...,M NE\n=X\nw∈W(M1,...,M N)ˆa†\nz1,w1ˆa†\nz2,w2···ˆa†\nzD0,wD0|Φvac⟩, (39)\nwhere we have denoted the set IbyI={z1, z2, . . . , z D0}. Noting the commutation\nrelations [ ˆFα,β,ˆa†\nz,γ] =δα,γˆa†\nz,β, we see that\n(ˆF2,1)M2|Φall 1⟩=M2!X\nw∈W(D0−M2,M2,0,...,0)ˆa†\nz1,w1ˆa†\nz2,w2···ˆa†\nzD0,wD0|Φvac⟩.(40)\nBy repeating the same procedure, we obtain\f\f\f˜ΦM1,...,M NE\nis the same as |ΦM1,...,M N⟩up\nto a normalization. Therefore, the unique ground state\f\f\f˜ΦM1,...,M NE\nis the fully polarized\nstate.\nIf the basis {µ}z∈Iis not connected, we can construct a different ground state\nother than fully polarized states. Hence, the SU( N) Hubbard model exhibits the SU( N)\nferromagnetism if and only if the basis {µz}z∈Iis connected.\nThe degeneracy of the ground states does not depend on how we take Iand{µz}z∈I.\nHence, the connectivity of the basis characterized by Lemma 1 is independent of the\nchoice of Iand{µz}z∈I.\nFinally, we show that the irreducibility of\u0010\n(P0)x,y\u0011\nx,y∈Λ0is equivalent to the\nconnectivity of {µz}z∈I.\nLemma 3. The matrix\u0010\n(P0)x,y\u0011\nx,y∈Λ0is irreducible if and only if {µz}z∈Iintroduced\nin Lemma 1 is connected.\nProof. We first show that non-connectivity of {µz}z∈Ileads to reducibility of\u0010\n(P0)x,y\u0011\nx,y∈Λ0. Let Iand{µz}z∈Ibe a subset and a basis characterized by Lemma 1.\nWe assume that Ican be decomposed as I=I1∪I2with I1∩I2=∅,I1̸=∅andFlat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 13\nI2̸=∅, in which µz≁µz′for any z∈I1andz′∈I2. When µz≁µz′, it holds that\n⟨µz,µz′⟩= 0 since µz(x)µz′(x) = 0 for any x∈Λ. We then define subsets Λ 1and Λ 2as\nΛj={x∈Λ|µz(x)̸= 0 for some z∈Ij}forj= 1,2. (41)\nBecause µz≁µz′for any z∈I1andz′∈I2, we see that Λ 1∩Λ2=∅, and obviously\nΛ0= Λ 1∪Λ2. By taking a linear combination of {µz}z∈I1, we obtain an orthonormal set\n{ψ(1)\ni}i=1,...,|I1|which spans the same space spanned by {µz}z∈I1. Similarly, we can take\nan orthonormal set {ψ(2)\nj}j=1,...,|I2|which spans the same space spanned by {µz′}z′∈I2.\nThe subspace h0is spanned by the orthonormal basis {ψ(1)\ni}i=1,...,|I1|∪ {ψ(2)\nj}j=1,...,|I2|,\nand the projection matrix P0can be written as\n(P0)x,y=|I1|X\ni=1ψ(1)\ni(x)\u0010\nψ(1)\ni(y)\u0011∗\n+|I2|X\nj=1ψ(2)\nj(x)\u0010\nψ(2)\nj(y)\u0011∗\n. (42)\nSince the states {ψ(1)\ni}i=1,...,|I1|are obtained by taking a linear combination of {µz}z∈I1,\nψ(1)\ni(x) is zero when x∈Λ2, and similarly ψ(2)\nj(x) = 0 when x∈Λ1. Therefore, we have\n(P0)x,y= 0 when x∈Λ1andy∈Λ2. This, together with the Hermiticity of P0, implies\nthat the matrix\u0010\n(P0)x,y\u0011\nx,y∈Λ0is reducible.\nWe then show that reducibility of\u0010\n(P0)x,y\u0011\nx,y∈Λ0implies that {µz}z∈Iis not\nconnected. Assume that Λ 0can be decomposed as Λ 0= Λ 1∪Λ2with Λ 1∩Λ2=∅,\nΛ1̸=∅and Λ 2̸=∅, in such a way that ( P0)x,y= 0 for any x∈Λ1andy∈Λ2. Let P1\nandP2be|Λ| × |Λ|matrices defined as\n(Pj) =(\n(P0)x,yifx, y∈Λj,\n0 otherwise ,(43)\nforj= 1,2. Both P1andP2are also projection matrices, and we see that P0=P1+P2\nandP1P2=P2P1= 0. Let Ibe the |Λ|×|Λ|identity matrix. Because I−Pj≥0, we can\nrepeat the same argument in Lemma 1 for I−Pjforj= 1,2. Then, for j= 1 and 2, we\ncan take a subset Ij⊂Λ with |Ij|= dim ker( I−Pj) and a basis {µ(j)\nz}z∈Ijof ker( I−Pj)\nin such a way that µ(j)\nz(z)̸= 0 and µ(j)\nz(z′) = 0 for any z′∈Ij\\{z}. Since µ(j)\nzis an\nelement of ker( I−Pj), it holds that\nPjµ(j)\nz=µ(j)\nz. (44)\nWe can also see that ⟨µ(1)\nz,µ(2)\nz′⟩= 0 because\n⟨µ(1)\nz,µ(2)\nz′⟩=⟨P1µ(1)\nz,P2µ(2)\nz′⟩\n=⟨µ(1)\nz,P1P2µ(2)\nz′⟩\n= 0, (45)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 14\nwhere we have used Eq. (44), P†\n1=P1andP1P2= 0. From the relation P1P2= 0 and\nEq. (44), we find\nP0µ(j)\nz=P0Pjµ(j)\nz=P2\njµ(j)\nz=µ(j)\nz, (46)\nwhich means that µ(j)\nz∈h0. We note that Tr P0= dim h0=D0, and similarly,\nTrP1=|I1|, and Tr P2=|I2|. Since Tr P0= TrP1+ TrP2, we have\nD0=|I1|+|I2|. (47)\nTherefore, the set of vectors {µ(1)\nz}z∈I1∪ {µ(2)\nz′}z′∈I2is a basis of h0characterized by\nLemma 1. From Eq. (44), we see that µ(j)\nz(x) can be nonzero only if x∈Λjforj= 1\nand 2, which implies that\nµ(1)\nz≁µ(2)\nz′for any z∈I1andz′∈I2. (48)\nHence, the basis {µ(1)}z∈I1∪ {µ(2)\nz′}z′∈I2is not connected.\nBy combining Lemmas 2 and 3, we obtain the desired result, Theorem 1.\n3. Rigorous results for the ferromagnetic SU( N) Kondo lattice model\nAs an application of Theorem 1, we shall establish a theorem about SU( N)\nferromagnetism of the SU( N) Kondo lattice model with a flat band.\n3.1. The SU( N) spin operators\nWe first introduce the standard Schwinger fermion representation of the SU( N) group.\nLet the N×Nmatrices GA(A= 1, . . . , N2−1) be the generators of the SU( N) group\nsuch that\nTr\u0000\nGAGB\u0001\n=δA,B, (49)\n[GA, GB] =N2−1X\nC=1ifA,B,CGC, (50)\nN2−1X\nA=1\u0000\nGA\u0001\nα,β\u0000\nGA\u0001\nµ,ν=δα,νδβ,µ−1\nNδα,βδµ,ν, (51)\nwhere fA,B,Care structure constants of the su(N) Lie algebra. Let ˆ c†\nαand ˆcαbe\nfermion creation and annihilation operators, where α= 1, . . . , N . They satisfy\n{ˆcα,ˆcβ}={ˆc†\nα,ˆc†\nβ}= 0 and {ˆcα,ˆc†\nβ}=δα,β. With these operators, one can obtain\nthe standard Schwinger fermion representation as\nˆSA=NX\nα,β=1ˆc†\nα\u0000\nGA\u0001\nα,βˆcβ. (52)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 15\nWe see that\n[ˆSA,ˆSB] =N2−1X\nC=1ifA,B,C ˆSC, (53)\nwhich means that the operators ˆSAform a representation of the su(N) Lie algebra. In\nthe following, we call the operators of the form Eq. (52) the SU( N) spin operators.\n3.2. The Hamiltonian of the SU( N) Kondo lattice model\nHere, we define the Hamiltonian of the SU( N) Kondo lattice model. Let Λ be a finite\nlattice. The operators ˆ c†\nx,αand ˆcx,αare creation and annihilation operators for itinerant\nfermions, which were introduced in Section 2. The operators ˆf†\nx,αand ˆfx,αare creation\nand annihilation operators for localized fermions at site x∈Λ with color α. They satisfy\n{ˆfx,α,ˆfy,β}={ˆf†\nx,α,ˆf†\ny,β}= 0, (54)\n{ˆfx,α,ˆf†\ny,β}=δα,βδx,y, (55)\n{ˆcx,α,ˆfy,β}={ˆc†\nx,α,ˆf†\ny,β}={ˆcx,α,ˆf†\ny,β}={ˆc†\nx,α,ˆfy,β}= 0. (56)\nThe number operator of localized fermion at site xwith color αis denoted by\nˆn(f)\nx,α=ˆf†\nx,αˆfx,α. In the following, the number operator of the itinerant fermions ˆ nx,α\nis denoted by ˆ n(c)\nx,α= ˆc†\nx,αˆcx,α. We also write ˆ n(c)\nx=PN\nα=1ˆn(c)\nx,α. We denote the Fock\nspace of the operators ˆ cx,αandˆfx,αbyH′(Λ). At each x∈Λ, we define the SU( N) spin\noperators for itinerant and localized fermions by using the representation of Eq. (52) as\nˆsA\nx=NX\nα,β=1ˆc†\nx,α\u0000\nGA\u0001\nα,βˆcx,β, (57)\nˆSA\nx=NX\nα,β=1ˆf†\nx,α\u0000\nGA\u0001\nα,βˆfx,β, (58)\nrespectively. The total SU( N) spin operator is given by ˆSA\ntot=P\nx∈ΛˆSA\ntot,x, where\nˆSA\ntot,x= ˆsA\nx+ˆSA\nx. The Hamiltonian of the SU( N) Kondo lattice model is defined by\nˆHKLM=NX\nα=1X\nx,y∈Λtx,yˆc†\nx,αˆcy,α+JKX\nx∈ΛN2−1X\nA=1ˆsA\nxˆSA\nx. (59)\nIn the following, we study the case where the SU( N) spins of itinerant and localized\nfermions are ferromagnetically coupled. That is, we assume that JK<0. We also\nassume that there is exactly one fermion described by ˆf†\nx,α(α= 1, . . . , N ) localized\nat each site x∈Λ, i.e., at each site, there is the SU( N) spin in the fundamental\nrepresentation corresponding to a single box . In what follows, we only consider the\nsubspace W(Λ) = {|Ψ⟩ ∈ H′(Λ)|ˆn(f)\nx|Ψ⟩=|Ψ⟩for all x∈Λ}, where ˆ n(f)\nx=PN\nα=1ˆn(f)\nx,α.Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 16\nThe SU( N) Kondo lattice model has the SU( N) symmetry, which can be seen from\nthe following relations\n[ˆHKLM,ˆSA\ntot] = 0 for all A= 1, . . . , N2−1. (60)\nThe SU( N) symmetry can also be described by using the color raising and lowering\noperators. The color raising and lowering operators are given as\nˆFα,β\ntot=X\nx∈Λ\u0010\nˆc†\nx,αˆcx,β+ˆf†\nx,αˆfx,β\u0011\nforα̸=β, (61)\nand the total number of itinerant and localized fermions with color αis given as\nˆFα,α\ntot=X\nx∈Λ\u0000\nˆn(c)\nx,α+ ˆn(f)\nx,α\u0001\n. (62)\nThe SU( N) symmetry of the model implies that\n[ˆHKLM,ˆFα,β\ntot] = 0. (63)\nThe total number of itinerant and localized fermions is given by ˆNtot=PN\nα=1ˆFα,α, and\nthe Hamiltonian (59) also commutes with ˆNtot. We note that the Hamiltonian (59) also\nsatisfies\n[ˆHKLM,ˆNc] = 0, (64)\nwhich means the model conserves the number of itinerant fermions described by ˆ c†\nx,α.\nTherefore, the numbers of itinerant fermions and the number of localized fermions are\nindependently conserved. In the following, we denote the eigenvalue of ˆNtot,ˆNc, and\nˆFα,α\ntotbyNtot, Nc, and Lα, respectively. Since it holds that ˆ n(f)\nx|Ψ⟩=|Ψ⟩in the subspace\nW(Λ), we haveP\nx∈Λˆn(f)\nx|Ψ⟩=|Λ||Ψ⟩. When the number of itinerant fermions is fixed\ntoNc, then Ntot=Nc+|Λ|. As in Section 2, we define the subspaces H′\nNtot(Λ) and\nH′\nL1,...,L N(Λ) by\nH′\nNtot(Λ) = {|Ψ⟩ ∈ H′(Λ)|ˆNtot|Ψ⟩=Ntot|Ψ⟩}, (65)\nH′\nL1,...,L N(Λ) = {|Ψ⟩ ∈ H′(Λ)|ˆFα,α\ntot|Ψ⟩=Lα|Ψ⟩for all α= 1, . . . , N }. (66)\nTo define the SU( N) ferromagnetism, we again define the Casimir operator ˆCtot,2\nof the SU( N) group as\nˆCtot,2=1\n2 NX\nα,β=1ˆFα,β\ntotˆFβ,α\ntot−ˆN2\ntot\nN!\n. (67)\nHere we define the SU( N) ferromagnetism of the SU( N) Kondo lattice model as follows.\nDefinition 2. Consider the Hamiltonian (59) with a fixed Ntot. We say that the model\nexhibits SU( N) ferromagnetism if any ground state |ΨGS⟩has the maximum eigenvalue\nofˆCtot,2inH′\nNtot(Λ),i.e.,\nˆCtot,2|ΨGS⟩=Ntot(N−1)\n2\u0012Ntot\nN+ 1\u0013\n|ΨGS⟩. (68)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 17\n3.3. Theorem for the SU( N) Kondo lattice model with a flat band\nIn the following, we again assume that T≥0 and h0= ker Tis not empty. The\ndimension of h0is denoted by D0>0. With the basis introduced in Lemma 1, we can\ndefine the set of operators ˆ a†\nz,αas in Eq. (19). We then define the fully polarized states\nwith the same color αfor itinerant and localized fermions as\n|Ψallα⟩= Y\nz∈Iˆa†\nz,α! Y\nx∈Λˆf†\nx,α!\n|Ψvac⟩, (69)\nwhere |Ψvac⟩is the normalized vacuum state for ˆ cx,αand ˆfx,αoperators. We also define\nthe state of the form,\n|ΨL1,...,L N⟩=\u0010\nˆFN,1\ntot\u0011LN···\u0010\nˆF2,1\ntot\u0011L2|Ψall 1⟩, (70)\nwhere L1=D0+|Λ|−PN\nα=2Lα. We also call the states of the form (70) fully polarized\nstates. In the same manner as Eq. (28), it can be checked that\nˆCtot,2|Ψallα⟩=Ntot(N−1)\n2\u0012Ntot\nN+ 1\u0013\n|Ψallα⟩, (71)\nwhere Ntot=D0+|Λ|. We can see that the fully polarized states of the form (70)\nalso have the same eigenvalue of ˆCtot,2, and thus all the fully polarized states have the\nmaximum eigenvalue of ˆCtot,2.\nUsing Theorem 1, we can prove the following theorem:\nTheorem 2. Consider the Hamiltonian (59) with Ntot=D0+|Λ|andJK<0. The\nferromagnetic SU( N) Kondo lattice model exhibits SU( N) ferromagnetism when the\nsubset Λ0introduced in Theorem 1 satisfies Λ0= Λ, and the matrix\u0010\n(P0)x,y\u0011\nx,y∈Λ0is\nirreducible.\nBefore we proceed with the proof of Theorem 2, let us remark that the ferromagnetic\nSU(N) Kondo lattice model on the delta chain shown in Fig. 1 exhibits SU( N)\nferromagnetism. As mentioned earlier, in the delta chain, Λ 0= Λ holds true. Since\nthe irreducibility of P0is also proven, according to Theorem 2, this is an example of the\nground state being SU( N) ferromagnetic.\n3.4. Proof of Theorem 2\nWe decompose the Hamiltonian (59) as\nˆHKLM=ˆHHub+JK\u0012\n1−1\nN\u0013\nˆNc+X\nx∈ΛˆVx, (72)\nwhere ˆHHubis the Hamiltonian of the SU( N) Hubbard model,\nˆHHub=NX\nα=1X\nx,y∈Λtx,yˆc†\nx,αˆcy,α+UX\nα<βX\nx∈Λˆn(c)\nx,αˆn(c)\nx,β, (73)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 18\nwith U >0. The local interaction ˆVxis defined by\nˆVx=−UX\nα<βˆn(c)\nx,αˆn(c)\nx,β−JK\u0012\n1−1\nN\u0013\nˆn(c)\nx+JKN2−1X\nA=1ˆsA\nxˆSA\nx. (74)\nIn the proof, we use the following lemma.\nLemma 4. The local interaction ˆVxis positive semidefinite when |JK|/U > N/ 2.\nProof. To prove this, we study the eigenvalues of ˆVx, which is denoted by Vxin the\nfollowing. We can express ˆVxas\nˆVx=−U\n2ˆn(c)\nx(ˆn(c)\nx−1)−JK\u0012\n1−1\nN\u0013\nˆn(c)\nx\n+JK \n1\n2N2−1X\nA=1ˆSA\ntot,xˆSA\ntot,x−1\n2N2−1X\nA=1ˆsA\nxˆsA\nx−1\n2N2−1X\nA=1ˆSA\nxˆSA\nx!\n. (75)\nSince we consider the subspace W(Λ), the operator1\n2PN2−1\nA=1ˆSA\nxˆSA\nxis the quadratic\nCasimir operator for the fundamental representation, and its eigenvalue is\nC2\u0010\u0011\n=1\n2N(N2−1). (76)\nBecause ˆ n(c)\nxcommutes with ˆVx, we can fix the number of itinerant fermions, which\nis denoted by n(c)(n(c)= 0, . . . , N ). In this sector, the operator1\n2PN2−1\nA=1ˆsA\nxˆsA\nxis the\nquadratic Casimir operator for the representation\n...n(c) , (77)\nand its eigenvalue is given by\nC2\n...n(c)\n=N+ 1\n2Nn(c)(N−n(c)). (78)\nTherefore, when n(c)is fixed, the local interaction ˆVxacts as\nˆVx=−U\n2n(c)(n(c)−1)−JK\u0012\n1−1\nN\u0013\nn(c)\n+JK \n1\n2N2−1X\nA=1ˆSA\ntot,xˆSA\ntot,x−N+ 1\n2Nn(c)(N−n(c))−1\n2N(N2−1)!\n. (79)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 19\nWhen the number of itinerant fermions is n(c), there are two possible total SU( N) spins\nin the representations\n...n(c)+ 1 , ...n(c) , (80)\nand the eigenvalues of the quadratic Casimir operator for these representations are\nC2\n...n(c)+ 1\n=N+ 1\n2N(n(c)+ 1)( N−n(c)−1), (81)\nand\nC2\n...n(c)\n=n(c)+ 1 +N+ 1\n2N(n(c)+ 1)( N−n(c)−1). (82)\nWhen n(c)= 0, there is only one possible total SU( N) spin described by . When\nn(c)=N, the itinerant fermions form the SU( N) singlet, in which the SU( N) spin\ntransforms in the trivial representation corresponding to the Young diagram with no\nbox. Thus, in this case, there is also one possible total SU( N) spin in the representation\n. The eigenvalue of the quadratic Casimir operator for the representation is\ngiven by Eq. (76). In this way, we find\nVx= 0 for n(c)= 0, (83)\nVx=−U\n2n(c)(n(c)−1)−2JKn(c)for ...n(c)+ 1 (1≤n(c)≤N−1), (84)\nVx=−U\n2n(c)(n(c)−1)−JK(n(c)−1) for ...n(c) (1≤nc≤N−1), (85)\nVx=−U\n2N(N−1)−JK(N−1) for n(c)=N. (86)\nWhen n(c)= 0, we see that Vx= 0. For n(c)= 1, . . . , N −1, when JK<0, the eigenvalue\nof Eq. (84) is strictly larger than that of Eq. (85). One can easily see the eigenvalues (85)\ncannot be negative when |JK|/U > N/ 2. For n(c)=N, the eigenvalue (86) is positive\nwhen|JK|/U > N/ 2. Thus, all the eigenvalues of ˆVxare greater than or equal to zero,\nwhich implies that ˆVx≥0.\nWe note that in the decomposition Eq. (72), one can take the parameter U > 0\narbitrarily. For a given JK, there always exists Usuch that |JK|/U > N/ 2. In the\nfollowing proof, Uis assumed to be such that |JK|/U > N/ 2, and hence ˆVx≥0.Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 20\nHere we show that the fully polarized states (69) and (70) are ground states. Since\nˆHHub≥0 and ˆVx≥0, we get a lower bound\nˆHKLM≥ −D0|JK|\u0012\n1−1\nN\u0013\n, (87)\nwhere we have used Nc=D0because we consider the Hilbert space H′\nD0+|Λ|(Λ). We\ncan easily check that the fully polarized state |Ψall 1⟩satisfies\nˆHKLM|Ψall 1⟩=−D0|JK|\u0012\n1−1\nN\u0013\n|Ψall 1⟩, (88)\nand hence the state |Ψall 1⟩is a ground state of ˆHKLM. Due to the SU( N) symmetry,\nall the fully polarized states are ground states of ˆHKLM.\nIn the rest of the proof, we prove that there are no other ground states. Let |ΨGS⟩\nbe an arbitrary ground state, which satisfies\nˆHKLM|ΨGS⟩=−D0|JK|\u0012\n1−1\nN\u0013\n|ΨGS⟩. (89)\nWith the decomposition (72), noting that ˆHHub≥0 and ˆVx≥0, we find\nˆHHub|ΨGS⟩= 0, (90)\nand\nˆVx|ΨGS⟩= 0 for all x∈Λ. (91)\nWe first consider the condition (90). According to Theorem 1, the ground state |ΨGS⟩\ncan be written as\n|ΨGS⟩=X\nαX\nβC(α,β) Y\nz∈Iˆa†\nz,αz! Y\nx∈Λˆf†\nx,βx!\n|Ψvac⟩, (92)\nwhere α= (αz)z∈Iis a color configuration of itinerant fermions over the subset I,\nandβ= (βx)x∈Λis a color configuration of localized fermions over the lattice Λ. The\ncoefficients C(α,β) must be symmetric under permutations of α.\nThen we consider the condition (91). Using the local constraint ˆ n(f)\nx|ΨGS⟩=|ΨGS⟩\nand Eq. (51), we have\nˆVx|ΨGS⟩= \n−U\n2ˆn(c)\nx(ˆn(c)\nx−1) +|JK|X\nα<β\u0010\nˆf†\nx,βˆc†\nx,α−ˆf†\nx,αˆc†\nx,β\u0011\u0010\nˆcx,αˆfx,β−ˆcx,βˆfx,α\u0011!\n|ΨGS⟩.\n(93)\nSince ˆ n(c)\nx(ˆn(c)\nx−1)|ΨGS⟩= 0, the ground state satisfies\n|JK|X\nα<β\u0010\nˆf†\nx,βˆc†\nx,α−ˆf†\nx,αˆc†\nx,β\u0011\u0010\nˆcx,αˆfx,β−ˆcx,βˆfx,α\u0011\n|ΨGS⟩= 0, (94)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 21\nwhich leads to\n\u0010\nˆcx,αˆfx,β−ˆcx,βˆfx,α\u0011\n|ΨGS⟩= 0, (95)\nfor any x∈Λ and α̸=β. Using the anticommutation relations (33) and (56), from\nEq. (95), we obtain\n(−1)D0X\nz∈IX\nα\nαz=αX\nβ\nβx=βsgn(z;I)µz(x) (C(α,β)−C(α|αz=β,β|βx=α))\n×\nY\nz′∈I\\{z}ˆa†\nz′,αz′\n\nY\nx′∈Λ\\{x}ˆf†\nx′,βx′\n|Ψvac⟩= 0, (96)\nwhere α|αz=βis the color configuration obtained from αby replacing αzwith β, and\nsimilarly, β|βx=αis the color configuration obtained from βby replacing βxwith α. The\nfunction sgn( z;I) is a sign factor arising from exchanges of the fermion operators.\nSince all the states in the sum are linearly independent and αandβare arbitrary,\nwe get\nµz(x) (C(α,β)−C(α|αz=βx,β|βx=αz)) = 0 , (97)\nfor any x∈Λ,z∈I,α, andβ. By assumption, it holds that Λ 0= Λ, and hence, for\nanyx∈Λ, there exists z0∈Isuch that µz0(x)̸= 0. For such z0∈I, we obtain\nC(α,β) =C(α|αz0=βx,β|βx=αz0) for any αandβ. (98)\nNoting that C(α,β) is symmetric under the permutations of αandx∈Λ is arbitrary,\nwe see that\nC(α,β) =C(α|αz=βx,β|βx=αz), (99)\nfor all x∈Λ and z∈I. Using Eq. (99), we also find that C(α,β) =C(α,βx↔y),\nwhere βx↔yis obtained from βby swapping βxandβy. Since any permutation of can\nbe obtained by repeatedly swapping two colors, we have\nC(α,β) =C(α′,β′), (100)\nwhere ( α′,β′) is a permutation of ( α,β) with ( α,β) being a color configuration over I\nand Λ. Consequently, when we fix Lαfor all α= 1, . . . , N , the coefficient C(α,β) is a\nconstant, and hence the ground state is unique in H′\nL1,...,L N(Λ).\nFinally, we can also check that the unique ground state in H′\nL1,...,L N(Λ) is the fully\npolarized state |ΨL1,...,L N⟩in a similar way as in Section 2. Here we introduce a word\nw′= (w′\n1, . . . , w′\nD0, w′\nD0+1, . . . , w′\nD0+|Λ|) whose length is D0+|Λ|and denote the set of\nwords for which |w′\nα|=LαbyW′(L1, . . . , L N) ={w′||w′\nα|=Lαfor all α}. The ground\nstate in H′\nL1,...,L N(Λ) satisfying Eq. (100) is written as\n\f\f\f˜ΨL1,...,L NE\n=X\nw′∈W′(L1,...,L N)ˆa†\nz1,w′\n1···ˆa†\nzD0,w′\nD0ˆf†\n1,w′\nD0+1···ˆf†\n|Λ|,w′\nD0+|Λ||Ψvac⟩, (101)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 22\nwhere we have labeled each site x∈Λ by integers as x= 1, . . . ,|Λ|. With the\ncommutation relations [ ˆFα,β\ntot,ˆa†\nz,γ] =δα,γˆa†\nz,β, and [ ˆFα,β\ntot,ˆf†\nx,γ] =δα,γˆf†\nx,β, we find that\n\u0010\nˆF2,1\ntot\u0011L2|Ψall 1⟩=L2!X\nw′∈W′(D0+|Λ|−L2,L2,0,...,0)ˆa†\nz1,w′\n1···ˆa†\nzD0,w′\nD0ˆf†\n1,w′\nD0+1···ˆf†\n|Λ|,w′\nD0+|Λ||Ψvac⟩.\n(102)\nBy repeating the same calculations, we see that the states\f\f\f˜ΨL1,...,L NE\nand|ΨL1,...,L N⟩\nare the same up to a normalization. Therefore, the unique ground state\f\f\f˜ΨL1,...,L NE\nin\nH′\nD0+|Λ|(Λ) is indeed the fully polarized state.\n3.5. Remark\nHere we would like to comment on the stability of the flat-band ferromagnetism for\nthe SU( N) Kondo lattice model. In previous studies, rigorous results regarding the\nstability of the flat-band ferromagnetism in the SU(2) Hubbard model have been\nobtained [35, 61–65]. The extension of these results to the SU( N) case has also been\nmade for a particular class of systems [48, 49]. By combining these results with the\ntechnique for the proof of Theorem 2, we can also discuss the stability of flat-band\nferromagnetism for the ferromagnetic SU( N) Kondo lattice model in a mathematically\nrigorous way. To illustrate this, let us consider a sufficiently large interaction strength U\nsuch that the SU( N) Hubbard model exhibits SU( N) ferromagnetism and assume that\nthe other parameters are also in the range where SU( N) ferromagnetism occurs. For\nthe Kondo coupling JKsuch that |JK|/U > N/ 2, we can repeat the same argument as in\nthe proof of Theorem 2. In this way, one can establish rigorous results on the stability\nof ferromagnetism in the ferromagnetic SU( N) Kondo lattice model with a nearly flat\nband.\n4. Conclusion and remark\nIn this paper, we have established rigorous results on flat-band ferromagnetism for\nthe SU( N) Hubbard model and the ferromagnetic SU( N) Kondo lattice model. For\nthe former, we found the necessary and sufficient condition for the ground state to\nexhibit SU( N) ferromagnetism when the number of particles is equal to the degeneracy\nof the lowest-energy single-particle states. The condition says that the irreducibility\nof the projection matrix onto the space of the lowest-energy single-particle states is\nequivalent to the presence of SU( N) ferromagnetism. We also showed that this general\ntheory could be applied to the ferromagnetic SU( N) Kondo lattice model with the\nhopping term of itinerant fermions that has a flat band at the bottom. Specifically,\nwe considered the case in which the number of itinerant fermions is the same as the\ndegeneracy of the flat band, and each site is occupied by one localized fermion. We then\nproved that the model with a nonzero ferromagnetic Kondo coupling exhibits SU( N)Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 23\nferromagnetism when certain conditions with respect to the hopping matrix are satisfied.\nIn our setup, we have considered situations where the number of fermions is fixed to\nfully occupy the lowest band, resulting in an insulating system. However, it will be\nintriguing to investigate ferromagnetism with different fillings, at which the system is\nexpected to be metallic. Nevertheless, even in the conventional SU(2) case, this remains\ninherently difficult and still challenging. Therefore, addressing metallic ferromagnetism\nmay require new mathematical methods and physical perspectives.\nIn addition, we discussed the ferromagnetic SU( N) Kondo lattice model in\nTheorem 2. While the presence of ferromagnetic interaction is physically natural in the\ncontext of ultracold atomic systems, considering antiferromagnetic interaction is also of\ninterest. However, the approach employed in our proof is not readily applicable to the\nantiferromagnetic case because the proof for the Lemma 4 does not work. Therefore,\nanother method will be required to obtain rigorous results for the antiferromagnetic\nSU(N) Kondo lattice model. For example, for the antiferromagnetic SU(2) Kondo\nlattice model, a rigorous proof has been given that the model with one conduction\nelectron exhibits an incomplete ferromagnetic order [53]. It might be possible to extend\nthe result to the general SU(N) cases; however, this particular issue lies beyond the\nscope of our current study and is left for future investigation.\nFinally, we discuss a potential extension of Theorem 2. As we discussed, the SU( N)\nKondo lattice model is expected to be experimentally realizable with ultracold atomic\ngases. While the Kondo lattice model neglects the on-site interaction among itinerant\nfermions, in principle, such interaction can be present. Thus it is worth investigating\nflat-band ferromagnetism for the models with both the on-site interaction U > 0 and\nthe ferromagnetic Kondo coupling JK<0. Since it has been rigorously proved that\nthe SU( N) Hubbard and ferromagnetic Kondo lattice model with a flat band exhibit\nSU(N) ferromagnetism when certain conditions are met, it would be possible to establish\nrigorous results on flat-band ferromagnetism in the presence of both interactions.\nAcknowledgments\nK.T. was supported by JSPS KAKENHI Grant No. 21J11575. H.K. was supported in\npart by JSPS Grant-in-Aid for Scientific Research on Innovative Areas No. JP23H01086,\nJSPS KAKENHI Grant No. JP18K03445, Grant-in-Aid for Transformative Research\nAreas A “Extreme Universe” No. JP21H05191, and the Inamori Foundation.\nReferences\n[1] Bloch I, Dalibard J and Zwerger W 2008 Rev. Mod. Phys. 80885\n[2] Bloch I, Dalibard J and Nascimbene S 2012 Nat. Phys. 8267–276\n[3] Lewenstein M, Sanpera A and Ahufinger V 2012 Ultracold Atoms in Optical Lattices: Simulating\nquantum many-body systems (Oxford University Press)\n[4] Gross C and Bloch I 2017 Science 357995–1001\n[5] Sch¨ afer F, Fukuhara T, Sugawa S, Takasu Y and Takahashi Y 2020 Nat. Rev. Phys. 2411–425Flat-band ferromagnetism in the SU( N) Hubbard and Kondo lattice models 24\n[6] Gorshkov A V, Hermele M, Gurarie V, Xu C, Julienne P S, Ye J, Zoller P, Demler E, Lukin M D\nand Rey A 2010 Nat. Phys. 6289–295\n[7] Kanamori J 1963 Prog. Theor. Phys. 30275–289\n[8] Gutzwiller M C 1963 Phys. Rev. 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Stat. Phys. 170399–420"    },    {        "title": "2304.11536v2.Spin_triplet_superconductivity_from_quantum_geometry_induced_ferromagnetic_fluctuation.pdf",        "content": "Spin-triplet superconductivity from quantum-geometry-induced ferromagnetic fluctuation\nTaisei Kitamura,1,∗Akito Daido,1and Youichi Yanase1\n1Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan\n(Dated: November 28, 2023)\nWe show that quantum geometry induces ferromagnetic fluctuation resulting in spin-triplet superconductivity.\nThe criterion for ferromagnetic fluctuation is clarified by analyzing contributions from the effective mass and\nquantum geometry. When the non-Kramers band degeneracy is present near the Fermi surface, the Fubini-Study\nquantum metric strongly favors ferromagnetic fluctuation. Solving the linearized gap equation with the effective\ninteraction obtained by the random phase approximation, we show that the spin-triplet superconductivity is\nmediated by quantum-geometry-induced ferromagnetic fluctuation.\nIntroduction.— Unconventional superconductivity beyond\nthe canonical Bardeen-Cooper-Schrieffer theory shows rich\nphysical phenomena including high-temperature supercon-\nductivity and topological superconductivity. Various fluc-\ntuations arising from many-body interactions play the main\nrole in the Cooper pairing for unconventional superconductiv-\nity, and low-dimensional fluctuations are particularly favor-\nable. For example, it is argued that high-temperature super-\nconductivity in cuprates is mediated by two-dimensional an-\ntiferromagnetic fluctuation [1–3]. Also, in iron-based high-\ntemperature superconductors the extended s-wave pairing is\nmediated by orbital [4–6] or antiferromagnetic [7, 8] fluctua-\ntion [9–11].\nHowever, searching for topological superconductivity [12–\n15] with Majorana fermion [16–18] is an unresolved prob-\nlem of modern condensed matter physics, which is attributed\nto the fact that the platform for topological superconductivity\nis rare in nature. Spin-triplet superconductors are canonical\ncandidates, and it is expected that ferromagnetic fluctuation\nmediates the spin-triplet Cooper pairing. However, candidate\nmaterials are restricted to a few heavy-fermion systems with\nthree-dimensional multiple bands [19–26].\nIn the two-dimensional isotropic continuum models, ferro-\nmagnetic fluctuation is not favored because of the constant\ndensity of states (DOS), which may imply the absence of\ntwo-dimensional spin-triplet superconductivity. Even for the\nanisotropic lattice systems, most quasi-two-dimensional su-\nperconductors do not show ferromagnetic fluctuation and an-\ntiferromagnetic fluctuations are rather ubiquitous, as we men-\ntioned above for cuprates and iron-based compounds. Thus,\nspin-triplet superconductivity from ferromagnetic fluctuation\nis expected to require peculiar band structures, and the search\nfor such systems is challenging for both materials and theoret-\nical models. In this Letter, nevertheless, we propose a guid-\ning principle for realizing ferromagnetic fluctuation in two-\ndimensional systems by referring to the quantum geometry of\nBloch electrons, which is recently attracting much attention in\nvarious fields [27–46].\nThe importance of quantum geometry in superconductors\nhas recently been recognized as it gives correction to the su-\nperfluid weight [34–37] . In the flat-band systems [36, 37, 47–\n51] the superfluid weight from Fermi-liquid theory vanishes,\nand the quantum geometric contribution determines the su-\nlim𝒒→#𝜕$!𝜕$\"𝜒#(𝒒)>0lim𝒒→#𝜕$!𝜕$\"𝜒#(𝒒)<0(a)(b)Ferromagnetic fluctuationAntiferromagnetic fluctuationFIG. 1. Schematic figures for (a) ferromagnetic and (b) antiferro-\nmagnetic fluctuation. We illustrate the q-dependence of χ0(q).\nperfluid weight. The quantum geometry also plays essential\nroles in the monolayer FeSe [52] and some finite-momentum\nCooper pairing states [53–57]. However, how quantum geom-\netry affects the pairing mechanism of superconductivity has\nnot been revealed. This work elucidates a way to create a\npairing glue of unconventional superconductivity via quantum\ngeometry.\nTo show that the quantum geometry enables strong ferro-\nmagnetic fluctuation in two-dimensional systems, resulting\nin spin-triplet superconductivity, we elucidate the criterion\nfor ferromagnetic fluctuation in the multi-band system with\nSU(2) symmetry. We find that the criterion is given by the\ngeneralized electric susceptibility (GES) which is defined as a\nnatural extension of the electric susceptibility to metals. The\nGES contains the terms obtained by the effective mass and the\nquantum geometry.\nThe key physics of quantum-geometry-induced ferromag-\nnetic fluctuation, which is shown below, is nontrivial quantum\ngeometry, especially Fubini-Study quantum metric [28, 58],\nfrom non-Kramers band degeneracy. As shown in this Let-\nter, the dispersive Lieb lattice model with non-Kramers band\ndegeneracy shows strong ferromagnetic fluctuation by this\nmechanism. Solving the linearized gap equation with the ef-\nfective interaction calculated by the random phase approxima-\ntion (RPA), spin-triplet superconductivity is demonstrated.\nCriterion for ferromagnetic fluctuation in multi-band Hub-\nbard models.— We consider the multi-band Hubbard model\nwith SU(2) symmetry, which contains multiple degrees of\nfreedom such as orbitals and sublattices [59]. The SU(2) sym-\nmetry means that the spin-orbit coupling and the magnetic\nfield are absent. For the interacting Hamiltonian, we con-\nsider the onsite Coulomb interaction Ustrong enough for the\nsuperconducting transition, by assuming strongly correlated\nmaterials. We then focus on the momentum dependence ofarXiv:2304.11536v2  [cond-mat.supr-con]  26 Nov 20232\nthe fluctuation, which mainly determines the superconducting\nsymmetry [3]. While we consider two-dimensional systems,\nthe following discussions apply to three-dimensional systems.\nThroughout this paper, Uis treated in the RPA\nscheme. When the system has only one band, the spin\n(charge) susceptibility χs(c)(q, iΩn)can be obtained as\nχs(c)(q, iΩn) =χ0\ns(c)(q, iΩn)/(1∓U\n2χ0\ns(c)(q, iΩn))by us-\ning the bare spin (charge) susceptibility of noninteracting sys-\ntems, χ0\ns(c)(q, iΩn). The interaction does not change the po-\nsition of peaks in the momentum qspace. Therefore, also for\nmost multi-band systems, it is expected that the momentum\ndependence of fluctuations arises from the bare susceptibil-\nity. Because the low-frequency spin (charge) fluctuation plays\nthe dominant role in mediating superconductivity, hereafter\nwe focus on the static fluctuations at Ωn= 0.\nIn multi-band systems with SU(2) symmetry, the bare\nspin/charge susceptibilities hold the relationship χ0\ns(q) =\nχ0\nc(q) = 2 χ0(q)with the bare susceptibility χ0(q). Thus, our\nmain concern is the presence/absence of the peak of χ0(q)at\nq= 0, corresponding to the presence/absence of ferromag-\nnetic fluctuation. The structure of susceptibility χ0(q)around\nq= 0 is determined by the curvature limq→0∂qµ∂qνχ0(q)\nwithµ, ν=x, y [60]. As a result, the criterion for the fer-\nromagnetic fluctuation is given by the sign of the curvature\n(see Fig. 1). Ferromagnetic fluctuation may be present when\nlimq→0∂qµ∂qνχ0(q)is negative. Otherwise, ferromagnetic\nfluctuation is prohibited.\nThe curvature limq→0∂qµ∂qνχ0(q)itself has a physical\nmeaning. For the discussion, it is useful to consider the\ncharge susceptibility in insulators at zero temperature, in-\nstead of the spin susceptibility. Based on the Kubo formula,\nthe curvature expresses the correction to the charge density,\nδ⟨ˆn(r)⟩, by the external electric field Eν(r)as,δ⟨ˆn(r)⟩=\n−P\nµν∂rν(limq→01\n2∂qµ∂qνχ0\nc(q)Eν(r))[59]. This means\nthat the curvature limq=0∂qµ∂qνχ0(q)is the electric suscep-\ntibility. Thus, by generalizing the concept of the electric sus-\nceptibility to metals, we define the generalized electric sus-\nceptibility (GES) as χ0:µν\ne≡limq=0∂qµ∂qνχ0(q)[59].\nFormula of GES.— Here, we derive the formula of\nGES [59], χ0:µν\ne=χ0:µν\ne:geom +χ0:µν\ne:mass ,\nχ0:µν\ne:geom =\n2X\nnZdk\n(2π)2\u0012f′(ϵn(k))\n2gµν\nn(k) +f(ϵn(k))Xµν\nn(k)\u0013\n,\n(1)\nχ0:µν\ne:mass =−2X\nnZdk\n(2π)2f(2)(ϵn(k))\n12[mµν\nn(k)]−1,(2)\nwhere ϵn(k)is the energy of the noninteracting Hamiltonian\nσ0⊗H0(k), which follows H0(k)|un(k)⟩=ϵn(k)|un(k)⟩\nwith the Bloch wave function |un(k)⟩. Note that σ0is the unit\nmatrix of spin space and nis the band index. Thus, GES is\ngiven by the two terms, χ0:µν\ne:geom andχ0:µν\ne:mass .\nThe first term χ0:µν\ne:geom named quantum geomet-\nric term is determined by the geometric quan-tities, namely, the Fubini-Study quantum metric\ngµν\nn(k) =P\nm(̸=n)Aµ\nnm(k)Aν\nmn(k) + c .c.and the po-\nsitional shift Xµν\nn(k) =P\nm(̸=n)(Aµ\nnm(k)Aν\nmn(k) +\nc.c.)/(ϵm(k)−ϵn(k)) with the Berry connection\nAµ\nnm(k) = i⟨∂kµun(k)|um(k)⟩. This term arises from\npurely interband effects and is absent in single-band sys-\ntems. In this term, the contributions from the quantum\nmetric and the positional shift are competitive. First, the\nquantum metric [28, 58], which is the counterpart of the\nBerry curvature [61], represents the distance between two\nadjacent states and is a positive definite tensor. Therefore,\ncombined with negative f′(ϵn(k)), the contribution from the\nquantum metric is always negative, favoring ferromagnetic\nfluctuation. Second, the positional shift [29] means the shift\nof electrons by the external electric field. In insulators at\nzero temperature, the contribution from the positional shift\ncorresponds to the well-known formula of electric suscepti-\nbility [62]. This term can be rewritten as it is proportional\ntoFnm(k)(Aµ\nnm(k)Aν\nmn(k) + c .c.)with the integrand\nof the Lindhard function, Fnm(k,q) = ( f(ϵm(k))−\nf(ϵn(k+q)))/(ϵn(k+q)−ϵm(k))q→0− − − → Fnm(k).\nTherefore, this contribution is always positive, which favors\nantiferromagnetic fluctuation.\nImportantly, both quantum metric and positional shift di-\nverge at the non-Kramers band-degenerate point. Therefore,\nquantum geometry plays an essential role when non-Kramers\nband degeneracy exists. However, the total geometric term\ndoes not diverge because of the cancellation of two contribu-\ntions [59].\nThe effective-mass term χ0:µν\ne:mass of GES is the purely intra-\nband effect and is determined by the band dispersion through\nthe effective mass [mµν\nn(k)]−1=∂kµ∂kνϵn(k). In single-\nband systems, only this term is finite. This term can be pos-\nitive and negative. For the hyperbolic dispersion ϵn(k) =\nk2/2m, the effective-mass term is zero because the DOS and\neffective mass are constants, which means the absence of fer-\nromagnetic fluctuation [59].\nGES with non-Kramers band degeneracy.— Because the\nnon-Kramers band degeneracy enhances the quantum geom-\netry, we focus on the Lieb lattice, which has been realized in\nultracold atoms allowing us to tune the strength of U[63, 64],\nwith the experimental test in mind. The Lieb lattice hosts\nthe flat band with three-fold band degeneracy, and the ground\nstate shows the flat-band ferromagnetism [65]. To distin-\nguish the quantum-geometry-induced ferromagnetic fluctua-\ntion from the flat-band ferromagnetism, we study the disper-\nsive Lieb lattice model in which the second and third-nearest-\nneighbor hoppings are finite. Unlike the usual Lieb lattice\nwith only the nearest-neighbor hopping, the flat band becomes\ndispersive and the three-fold band degeneracy at the Mpoint\n[k= (π, π)] is partially lifted, while the two-fold degeneracy\nremains protected by the C4rotation symmetry [59].\nThe dispersive Lieb lattice model is illustrated in Fig. 2(a).\nThe Fermi surfaces for the chemical potential µc= 0.5,0.7,\nand0.9are shown in Fig. 2(b), and the band dispersion is in3\n𝑘!𝑘\"−𝜋−𝜋𝜋𝜋(a)(b)(c)t=1.00.15t0.4t0.15t0.2t(d)\nXΓMΓ-4-3-2-1 0 1 2\n 64 128 192Ek-4-3-2-1 0 1 2\n 64 128 192Ek 0 0.5 1 1.5 2 2.5\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1DOSµc𝜇=0.7𝜇=0.5𝜇=0.9\n 0 0.5 1 1.5 2 2.5\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1DOSµc\nFIG. 2. (a) The dispersive Lieb lattice model, unit-cell (gray box),\nand hopping integrals (blue arrows). The first-nearest-neighbor hop-\nping is taken as the unit of energy, t= 1. (b) The Fermi surface for\nµc= 0.5(green line), 0.7(blue line), and 0.9(red line). (c) The\nband dispersion. (d) The DOS.\nFig. 2(c). The band-degenerate point lies on the Fermi surface,\nwhen µc= 0.7. As shown in Fig. 2(d), the maximum of DOS\ncorresponds to µc= 0.7.\nIn Fig. 3(a), we show the chemical-potential dependence\nof GES χ0:xx\ne. In some regions near the Lifshitz transitions\n(µc≃ −0.1and0.9), the GES shows the dip structure. This\nstructure is induced by the effective-mass term. The effective-\nmass contribution from each band is proportional to an odd\nfunction f(2)(ϵn(k)), and therefore, the effective-mass term\ntends to cancel out between the states below and above the\nFermi energy. However, the cancellation is incomplete for µc\nnear the Lifshitz transition point, and thus, the effective-mass\nterm gives a negative GES. This is an understanding of why\nferromagnetic fluctuation appears at finite temperatures when\nthe Fermi surface is small, from the viewpoint of the GES.\nIn contrast, accompanying the band degeneracy on the\nFermi surface, we obtain the maximally negative value of GES\nχ0:xx\ne atµc= 0.7, which is dominated by the quantum geo-\nmetric contribution. As expected from the band degeneracy at\nthe M point, the quantum geometric term of the GES mainly\ncomes from the region near the M point. This is verified\nby the k-resolved quantum geometric contribution shown in\nFig. 3(b). We find a large negative contribution to the GES\nfrom the vicinity of the Mpoint, which in turn induces ferro-\nmagnetic fluctuation.\nAs we have mentioned, the quantum metric gives a nega-\ntive contribution to the GES, while the positional shift posi-\ntively contributes. Our results imply that the quantum met-\nric overcomes the positional shift when the band-degenerate\npoint lies on the Fermi surface. This can be intuitively un-\nderstood from the formula of the quantum geometric term.\nThe quantum metric contributes to the GES with f′(ϵn(k)),\nwhich is divergent on the Fermi surface at low temperatures,\n[f′(0)∝1/T]. On the other hand, Fnm(k)in the positional\nshift contribution is a regular function. Therefore, the quan-\n-7-6-5-4-3-2-1 0 1 2 3\n-0.2 0 0.2 0.4 0.6 0.8 1χeµcχmassχ geomχe\n-1-0.5 0 0.5 1 1.5\n 0 0.02 0.04 0.06 0.08 0.1χeTχmassχ geomχe\n-14-12-10-8-6-4-2 0 2 4\n 0 0.02 0.04 0.06 0.08 0.1χeTχmassχ geomχe\n(c)(d)(a)\n𝑘!−𝜋𝜋𝑘\"−𝜋𝜋(b)\n-7-6-5-4-3-2-1 0 1 2 3\n-0.2 0 0.2 0.4 0.6 0.8 1χeµcχmassχ geomχe\n-14-12-10-8-6-4-2 0 2 4\n 0 0.02 0.04 0.06 0.08 0.1χeTχmassχ geomχe-14-12-10-8-6-4-2 0 2 4\n 0 0.02 0.04 0.06 0.08 0.1χeTχmassχ geomχeFIG. 3. GES χ0:xx\ne of the dispersive Lieb lattice model. In (a), (c),\nand (d), the triangles, circles, and squares show χ0:xx\ne,χ0:xx\ne:geom , and\nχ0:xx\ne:mass , respectively. (a) The µcdependence for T= 0.01. (b) The\nquantum geometric contribution to the GES from each kpoint for\n(µc, T) = (0 .7,0.02). The inset shows the contribution near the\nMpoint with band degeneracy. (c) and (d) show the temperature\ndependence for µc= 0.7andµc= 0.65, respectively. The purple\nline in (c) is a fitting curve χ0:xx\ne:geom ≃ −0.0631779 /T+ 0.462022 .\ntum metric becomes significant in the presence of band degen-\neracy at low energies. Consistent with the intuitive explana-\ntion, the geometric term is negatively enhanced at low temper-\natures owing to the contribution of quantum metric, as shown\nin Fig. 3(c). The geometric term is well fitted by the scaling\nχ0:µν\ne:geom =a/T+bwith constants a, b. Thus, we conclude\nthat the quantum metric on the Fermi surface induces ferro-\nmagnetic fluctuation when the non-Kramers band degeneracy\nlies on the Fermi surface.\nHowever, when the band-degenerate point is slightly off\nthe Fermi surface and temperature decreases so that T≪\n|µc−0.7|, the negative geometric term is suppressed as shown\nin Fig. 3(d). This is consistent with the fact that the quan-\ntum metric contribution is a Fermi-surface term. As the band-\ndegenerate point moves away from the Fermi surface by much\nmore than T,f′(ϵn(k))near the Mpoint decays, and the\nquantum metric contribution is suppressed. At low tempera-\ntures, the positional shift contribution overcomes the quantum\nmetric contribution, and the quantum geometric term is pos-\nitive. Thus, in this case, the ferromagnetic-antiferromagnetic\ncrossover of fluctuation occurs as the temperature decreases.\nQuantum-geometry-induced ferromagnetic fluctuation.—\nThen, to justify the above discussion, we show\nthe bare spin susceptibility defined by χ0\ns(q) =\n2P\nnmRdk\n(2π)2Fnm(k,q)(1−Dnm(k,q)), where the\nquantum distance Dnm(k,q)≡1− |⟨un(k+q)|um(k)⟩|2\nis closely related to the quantum geometry. Quantum geome-\ntry suppresses χ0\ns(q)atq̸= 0 via nonzero quantum distance4\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n 2.2 2.4 2.6 2.8 3 3.2 3.4\n𝑞!−𝜋𝜋𝑞!−𝜋𝜋𝑞\"−𝜋𝜋\n𝑞\"−𝜋𝜋(a)(b)(c)(d)-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n 2.5 3 3.5 4 4.5 5 5.5\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n 2.5 3 3.5 4 4.5 5 5.5\nFIG. 4. The bare spin susceptibility in the dispersive Lieb lattice\nmodel. (a) χ0\ns(q)and (b) χ0\ns:band (q)for(µc, T) = (0 .7,0.01)\nwith the same color bar. (c) and (d) show χ0\ns(q)for(µc, T) =\n(0.65,0.05)and(0.65,0.01), respectively.\nDnn(k,q), which is expanded as ∼P\nµνgµν\nn(k)qµqν+···\nwith the quantum metric. However, χ0\ns(0)is not sup-\npressed, and ferromagnetic fluctuation is relatively enhanced.\nThe van Vleck susceptibility arising from Dnm(k,q)\nforn̸=mcorresponds to the positional-shift con-\ntribution to the GES. For comparison, we also define\nthe bare spin susceptibility without quantum geometry,\nχ0\ns:band (q) = 2P\nnRdk\n(2π)2Fnn(k,q), in which magnetic\nfluctuation is determined by only the effective-mass term. By\ncomparing these two quantities, we can elucidate the effects\nof quantum geometry.\nIn Figs. 4(a) and 4(b), we show χ0\ns(q)andχ0\ns:band (q)in\nthe dispersive Lieb lattice model for µc= 0.7. As ex-\npected by Fig. 3(a) showing the negative GES, χ0:µν\ne =\nlimq=0∂qµ∂qνχ0(q), the bare spin susceptibility shows fer-\nromagnetic fluctuation (Fig. 4(a)). However, antiferromag-\nnetic fluctuation is obtained when we neglect the quantum\ngeometry (Fig. 4(b)). Thus, we conclude that the quantum\ngeometry induces the ferromagnetic fluctuation. It is empha-\nsized that the maximum of DOS at µc= 0.7is not suffi-\ncient for the ferromagnetic fluctuation; the relative enhance-\nment of χ0\ns(0)compared to χ0\ns(q̸= 0) by the quantum dis-\ntance/quantum geometry is essential. Note that the momen-\ntum dependence of spin susceptibility plays an essential role\nin unconventional superconductivity [1, 3]. We also show\nχ0\ns(q)for(µc, T) = (0 .65,0.05)and(µc, T) = (0 .65,0.01)\nin Figs. 4(c) and 4(d), respectively. Consistent with Fig. 3(d),\nwe confirm the crossover from ferromagnetic to antiferromag-\nnetic fluctuation as the temperature decreases.\nSpin-triplet superconductivity.— Finally, we show that\nquantum-geometry-induced ferromagnetic fluctuation medi-\nates spin-triplet superconductivity. To see this, we set the\nonsite interaction as U= 0.86and solve the linearized gap\nequation, λt(s)∆ll′(k) =−1\nNβP\nk′ωnP\n{li}Vt(s)\nll1,l2l′(k−\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n 0 50 100 150 200 250 300 350 400 450\n 0 0.2 0.4 0.6 0.8 1 1.2\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1λµcsinglettriplet\n𝑞!−𝜋𝜋𝑞\"−𝜋𝜋(a)(b)\n 0 0.2 0.4 0.6 0.8 1 1.2\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1λµcsinglettripletFIG. 5. (a) The spin susceptibility obtained by RPA for (µc, T) =\n(0.7,0.01). (b) The eigenvalues of the linearized gap equation at\nT= 0.01. The blue and orange lines show the maximum eigen-\nvalue for spin-triplet and spin-singlet superconductivity, respectively.\nEigenvalues for all the irreducible representations are shown in Sup-\nplemental Materials [59].\nk′)Gl1l3(k′, iωn)∆l3l4(k′)Gl2l4(−k′,−iωn), using the effec-\ntive interaction obtained by RPA Vt(s)(q), which is ≃\n−1(3)\n4Uχs(q)Uin single-band systems [1, 3, 66–68] but here\nextended to multi-band systems [59]. Here, G(k, iωn)is the\nGreen function with the Matsubara frequency, iωn. The insta-\nbility of spin-triplet (singlet) superconductivity with the form\nfactor ∆(k)is determined by the maximum eigenvalue λt(s).\nWhile the mean-field formalism overestimates the transition\ntemperature, the dynamical effect of effective interaction is\nexpected not to alter the superconducting symmetry, as in the\ncases of3He [66] and cuprates [69].\nFigure 5(a) shows the spin susceptibility at µc= 0.7\nobtained by RPA. Ferromagnetic fluctuation is enhanced by\nthe Coulomb interaction, as we see from the comparison\nto Fig. 4(a). Eigenvalues of the linearized gap equation\nare shown in Fig. 5(b) for spin-singlet extended- s-wave (or-\nange line) and spin-triplet p-wave (blue line) superconductiv-\nity [59]. It is revealed that the spin-triplet superconductivity\nis stabilized around µc≃0.7and0.9corresponding to the\nnegative peak of GES in Fig. 3(a).\nEspecially, we obtain the largest eigenvalue at µc=\n0.7where quantum geometry induces ferromagnetic fluc-\ntuation. Combined with the large DOS, the strong ferro-\nmagnetic fluctuation enhanced by interaction gives a large\neigenvalue for spin-triplet superconductivity. Thus, we con-\nclude spin-triplet superconductivity from quantum-geometry-\ninduced ferromagnetic fluctuation.\nDiscussion.— In this Letter, we show that quantum ge-\nometry induces ferromagnetic fluctuation and results in spin-\ntriplet superconductivity. The Fubini-Study quantum metric\non the Fermi surface is an essential quantity for this mech-\nanism of magnetism and superconductivity. Using the dis-\npersive Lieb lattice model, we demonstrated that the non-\nKramers band degeneracy on the Fermi surface plays the cen-\ntral role in enhancing the quantum-geometry-induced phe-\nnomena. In the diverse studies on unconventional super-\nconductivity, the quantum geometry of electrons coupled to\nmany-body effects has not been focused on. Stimulated by re-\ncent developments in the topology and geometry of quantum5\nmaterials, we shed light on a route to spin-triplet supercon-\nductivity and, thereby, topological superconductivity.\nA question of interest is whether our theory can be applied\nto other systems as well. To answer this, we have calculated\nthe GES of Raghu’s model [70] for iron-based superconduc-\ntors [59]. This model has the non-Kramers band degener-\nacy at the Γpoint. Also in this model, the quantum geome-\ntry induces ferromagnetic fluctuation due to the non-Kramers\nband degeneracy. In addition, we confirmed the quantum-\ngeometry-induced ferromagnetic fluctuation in other models\nwith the flat band and various band touching including the\nusual Lieb lattice model [71]. Thus, a wide range of mate-\nrials with non-Kramers band degeneracy [72, 73] are candi-\ndates for quantum-geometry-induced ferromagnetism and su-\nperconductivity. We expect that future material-specific stud-\nies will be stimulated by our work. The exploration of two-\ndimensional materials with high tunability, e.g., by band en-\ngineering through heterostructures, gate voltage, strain, and\ntwist angle is also expected.\nWe are grateful to R. Hakuno, K. Nogaki, Y . Takahashi,\nT. Nomoto, and R. Arita, for fruitful discussions. This work\nwas supported by JSPS KAKENHI (Grant Nos. JP18H01178,\nJP18H05227, JP20H05159, JP21K13880, JP21K18145,\nJP22H01181, JP22H04476, JP22H04933, JP22J22520).\n∗kitamura.taisei.67m@st.kyoto-u.ac.jp\n[1] T. Moriya and K. Ueda, Adv. Phys. 49, 555 (2000).\n[2] C. C. 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B Condens.\nMatter 97, 134423 (2018).7\nSupplemental Materials:\nSpin-triplet superconductivity from quantum-geometry-induced ferromagnetic fluctuation\nS1. SPIN, CHARGE, AND GENERALIZED ELECTRIC SUSCEPTIBILITY\nIn this section, we show the detailed calculation of the spin, charge, and generalized electric susceptibility. While we focus on\ntwo-dimensional systems in the main text, the following discussion is written for systems in any dimension, including two and\nthree-dimensional systems.\nA. Multi-band Hubbard model with SU(2) symmetry\nWe consider the multi-band Hubbard model with SU(2) symmetry,\nˆH=ˆH0+ˆHint, (S1)\nˆH0=X\nkX\nσˆc†\nσ(k)H0(k)ˆcσ(k), (S2)\nˆHint=UX\nRX\nlˆnl↑(R)ˆnl↓(R), (S3)\nwhere ˆc†\nσ(k) = ( ˆc†\n1σ(k). . .ˆc†\nlσ(k). . .ˆc†\nfσ(k))is the creation operator of electrons with the wave vector k, spin σ=↑↓,\nand the internal degrees of freedom lsuch as orbitals and sublattices. The dimension of the internal degrees of freedom is\nrepresented by f.ˆnlσ(R) = ˆc†\nlσ(R)ˆclσ(R)is the particle density operator for land spin σat position R. The Fourier transform\nis defined by ˆc†\nlσ(R) =1√\nNP\nke−ik·(R+rl)ˆc†\nlσ(k).H0(k)is the matrix representation of the Fourier transform of hopping\nintegrals with the internal coordinate rl.Uis the onsite Coulomb interaction, and Nis the volume of the system. The SU(2)\nsymmetry is preserved in this model, since the spin-orbit coupling (SOC) and the magnetic field are absent. We ignore two-body\ninteractions other than the onsite Coulomb interaction, such as an inter-orbital interaction, for simplicity.\nB. Spin and charge susceptibility\nThe particle density operator for each spin is defined by,\nˆnσ(q, τ) =X\nkˆc†\nσ(k, τ)ˆcσ(k+q, τ), (S4)\nwhere ˆcσ(k, τ) =eτˆHˆcσ(k)e−τˆHis the imaginary-time representation with the imaginary time τ. For later calculation, we also\ndefine the matrix elements of particle density operators for each spin as,\nˆnσ:ll′(q, τ) =X\nkˆc†\nσl(k, τ)ˆcσl′(k+q, τ). (S5)\nUsing this, the spin susceptibility and the charge susceptibility are defined by,\nχs(q, iΩn) = χ↑↑(q, iΩn)−χ↑↓(q, iΩn)−χ↓↑(q, iΩn) +χ↓↓(q, iΩn),\n= 2χ↑↑(q, iΩn)−2χ↑↓(q, iΩn), (S6)\nχc(q, iΩn) = χ↑↑(q, iΩn) +χ↑↓(q, iΩn) +χ↓↑(q, iΩn) +χ↓↓(q, iΩn),\n= 2χ↑↑(q, iΩn) + 2χ↑↓(q, iΩn), (S7)\nχσσ′(q, iΩn) =1\nNZβ\n0dτeiΩnτ⟨Tτ[ˆnσ(q, τ)ˆnσ′(−q)]⟩, (S8)8\nwith the bosonic Matsubara frequency iΩnand the inverse temperature β.Tτrepresents the time-ordering product for τ. Here,\nwe used the relations ensured by the SU(2) symmetry χ↑↑(q, iΩn) =χ↓↓(q, iΩn)andχ↑↓(q, iΩn) =χ↓↑(q, iΩn). We also\ndefine the matrix representation ¯χσσ′(q, iΩn)ofχσσ′(q, iΩn)using the matrix element written by,\n[¯χσσ′(q, iΩn)]l1l′\n1,l2l′\n2=1\nNZβ\n0dτeiΩnτ⟨Tτ\u0002\nˆnσ:l′\n1l1(q, τ)ˆnσ′:l2l′\n2(−q)\u0003\n⟩. (S9)\nNoninteracting system\nThe spin (charge) susceptibility of a noninteracting system, namely the bare spin (charge) susceptibility, can be written as,\nχ0\ns(q, iΩn) = χ0\nc(q, iΩn) = 2 χ0(q, iΩn), (S10)\nχ0(q, iΩn) = χ0\nσσ(q, iΩn) =1\nNZβ\n0dτeiΩnτ⟨Tτ[ˆnσ(q, τ)ˆnσ(−q)]⟩,\n=−1\nNβX\nkωnTr [G(k+q, iωn+iΩn)G(k, iωn)]. (S11)\nWe used the property of noninteracting systems, χ0\n↑↓(q, iΩn) =χ0\n↓↑(q, iΩn) = 0 , which is satisfied by the SU(2) symmetry\nof Hamiltonian. Here, we define the Green function of noninteracting systems G(k, iωn) = [iωn−H0(k)]−1with fermionic\nMatsubara frequency ωn. Tr represents the trace for all degrees of freedom except for the spin. The matrix elements are obtained\nas,\n1\n2[¯χ0\ns(c)(q, iΩn)]l1l′\n1,l2l′\n2= [¯χ0(q, iΩn)]l1l′\n1,l2l′\n2\n=−1\nNβX\nkωn\u0002\nGl1l2(k+q, iωn+iΩn)Gl′\n2l′\n1(k, iωn)\u0003\n. (S12)\nAfter taking the sum of Matsubara frequency, we get,\nχ0(q, iΩn) =X\nnmZdk\n(2π)df(ϵn(k+q))−f(ϵm(k))\nϵm(k)−ϵn(k+q) +iΩn|⟨un(k+q)|um(k)⟩|2, (S13)\nwith the dimension of the system d. We can calculate the band dispersion ϵn(k)and Bloch wave function |un(k)⟩by the\neigenvalue equation, H0(k)|un(k)⟩=ϵn(k)|un(k)⟩.\nRandom phase approximation\nWe define the irreducible vertex,\n\u0002\nΓ0\u0003\nl1l′\n1,l2l2′=Uδl1l′\n1δl′\n1l2δl2l′\n2. (S14)\nIn the random phase approximation (RPA), the spin (charge) susceptibility and its matrix representation are given by,\nχs(c)(q, iΩn) =X\nll′\u0002\n¯χs(c)(q, iΩn)\u0003\nll,l′l′\n¯χs(c)(q, iΩn) =\u0002\n1∓Γ0¯χ0(q, iΩn)\u0003−1¯χ0\ns(c)(q, iΩn). (S15)\nC. Generalized electric susceptibility\nIn this subsection, we derive the generalized electric susceptibility using the Kubo formula and local thermodynamics.9\nElectric and charge susceptibility via Kubo Formula\nFirst, we show an alternative way to introduce charge susceptibility by using linear response theory. Based on the Kubo\nformula, the charge susceptibility of real-space and real-time representation χc(r, t)with a position rand real time tis defined\nby,\n⟨ˆn(r, t)⟩=ρ0−Z\ndr′Zt\n−∞dt′χc(r−r′, t−t′)ϕ(r′, t′). (S16)\nHere,⟨ˆn(r, t)⟩is the expectation value of the particle density operator, ˆn(r, t) =P\nl,σˆc†\nl,σ(r, t)ˆcl,σ(r, t), namely the charge\ndensity, ϕ(r, t)is an external scalar potential, and ρ0is the charge density in the absence of the external field. Also, ˆc†\nl,σ(r, t) =\neiˆHt/ℏˆc†\nl,σ(r)e−iˆHt/ℏis the Heisenberg representation of creation operator with the Dirac constant ℏ= 1 for the natural unit.\nAfter the Fourier transform with respect to real time, we get the frequency representation of the charge susceptibility,\n⟨ˆn(r, t)⟩=ρ0−Z\ndr′Z∞\n−∞dω\n2πe−iωt+δtχc(r−r′, ω)ϕ(r′, ω), (S17)\nχc(r, ω) =Zt\n−∞dt′eiω(t−t′)−δ(t−t′)χc(r, t−t′), (S18)\nϕ(r, t) =Z∞\n−∞dω\n2πe−iωt+δtϕ(r, ω). (S19)\nHere, the infinitesimal δensures that the external field vanishes at t→ −∞ . We can also define the frequency representation of\nthe charge density as,\n⟨ˆn(r)⟩(ω) = δ(ω)ρ0−Z\ndr′χc(r−r′, ω)ϕ(r′, ω), (S20)\n⟨ˆn(r, t)⟩=Z∞\n−∞dω\n2πe−iωt+δt⟨ˆn(r)⟩(ω). (S21)\nSince the correlation function should decay away from the external field at r, the integrand of Eq. (S20) contributes only when\nr−r′is sufficiently small. In contrast, we assume that the scalar potential spatially modulates on a length scale larger than that\nof the correlation function. Therefore, we can expand the scalar potential by r−r′as,\n⟨ˆn(r)⟩(ω)−δ(ω)ρ0=−Z\ndr′χc(r−r′, ω)ϕ(r−(r−r′), ω),\n=−∞X\nn=0Z\ndr′χc(r−r′, ω)\u0002\n(−rµ+r′\nµ)∂rµ\u0003n\nn!ϕ(r, ω),\n=−1\nNX\nq∞X\nn=0Z\ndr′eiq·(r−r′)χc(q, ω)\u0002\n(−rµ+r′\nµ)∂rµ\u0003n\nn!ϕ(r, ω),\n=−∞X\nn=0Zdq\n(2π)dZ\ndr′eiq·(r−r′)\u0002\n−i∂qµ∂rµ\u0003nχc(q, ω)ϕ(r, ω)\nn!,\n=−∞X\nn=0lim\nq→0\u0002\n−i∂qµ∂rµ\u0003nχc(q, ω)ϕ(r, ω)\nn!,\n(S22)\nwhere and hereafter, we take the sum of repeated indices, such as µ=x, y, z . For example, Eq. (S22) up to n= 2is explicitly\nwritten as,\n⟨ˆn(r)⟩(ω)−δ(ω)ρ0=−χc(0, ω)ϕ(r, ω) +ilim\nq→0X\nµ1∂qµχc(q, ω)∂rµϕ(r, ω)\n+1\n2lim\nq→0X\nµν∂qµ∂qνχc(q, ω)∂rµ∂rνϕ(r, ω). (S23)10\nThrough the analytic continuation ω+iδ→iΩn,χc(q, ω)corresponds to Eq. (S7).\nHere, we focus on χc(q, ω= 0) for which the system does not depend on time. In other words, we focus on the particle\ndensity in equilibrium. When the system is metal and/or at a finite temperature, an equilibrium charge with an external electric\nfield cannot be defined, since the electric current follows in metals or at finite temperatures. Therefore, we consider an insulator\nat zero temperature. Considering the Lehmann representation [S1], we see that the charge susceptibility satisfies the relationship\nχc(q, ω) =χ∗\nc(−q,−ω)which means χc(q) =χc(−q)(=χc(q,0)). Therefore, odd-order derivatives of χc(q,0)with respect\ntoqvanish. As a result, up to the second order of ∂rµ, the equilibrium charge density is obtained as,\n⟨ˆn(r)⟩ −ρ0=−χc(0)ϕ(r) +1\n2lim\nq→0∂qµ∂qνχc(q)∂rµ∂rνϕ(r),\n=−χc(0)ϕ(r)−∂rµ\u00121\n2lim\nq→0∂qµ∂qνχc(q)Eν(r)\u0013\n, (S24)\nwhere Eν(r) =∂rνϕ(r)is the external electric field. In the insulator at zero temperature, the first term vanishes and this directly\nmeans that1\n2limq→0∂qµ∂qνχc(q)is the electric susceptibility which gives the correction to the charge density, δ⟨ˆn(r)⟩=\n⟨ˆn(r)⟩ −ρ0.\nGeneralized electric susceptibility via local thermodynamics\nNext, to clarify the physical meaning of the quantity1\n2limq→0∂qµ∂qνχc(q)in metals and/or at a finite temperature, we use\nthe local thermodynamics [S2–S6]. The following discussion is based on Ref. S5. We consider the scalar potential arising from\nan inhomogeneous distribution of disorders, structural asymmetry, contact with a substrate, and so on, rather than an applied\nelectric field. The electric field is assumed to be in a small region near rcompared to the volume N. In this setup, while the local\nparticle number depends on rdue to the spatial variation of ϕ(r), the total particle number is assumed to be constant. Thus, the\nsystem is static and the charge current does not flow. The setup is discussed in more detail in Ref. S5.\nThe length scale of ϕ(r)is sufficiently longer than the decay length of the Green function. Therefore, ϕ(r)varies slowly in\nspace and the system around ris well approximated by the uniform Hamiltonian in which chemical potential µcis replaced by\nµc−ϕ(r). In local thermodynamics, starting from the above assumption, the Hamiltonian is expanded by r′−rthrough ϕ(r′)\naround r′=r. As a result, free energy depends on rthrough ϕ(r)and is expanded as ,\nF(r) =F0(µc−ϕ(r)) +Qµν(µc−ϕ(r))∂rµ∂rνϕ(r)\n−1\n2∂µcQµν(µc−ϕ(r))∂rµϕ(r)∂rνϕ(r) +O(ql)3, (S25)\nwhere F0(µc−ϕ(r))is the free energy of uniform Hamiltonian with chemical potential µc−ϕ(r). In Eq. (S25), Qµνis the\nthermodynamic electric quadrupole moment defined in Ref. S5.\nThe charge density is defined by,\n⟨ˆn(r)⟩=−∂µcF(r)\n=ρ0(µc−ϕ(r)) +∂rµ∂rνQµν(µc−ϕ(r))−1\n2∂2\nµcQµν(µc−ϕ(r))∂rµϕ(r)∂rνϕ(r),\n(S26)\nwhere ρ0(µc−ϕ(r)) =−∂µcF0(µc−ϕ(r))is the charge density with the chemical potential µc−ϕ(r). Therefore, in the linear\nresponse theory, this can be written as,\n⟨ˆn(r)⟩ −ρ0=−χc(0)ϕ(r)−∂rµ(∂µcQµνEν(r)). (S27)\nThe first term is the charge susceptibility since it is also defined by χc(0) =−limϕ(r)→0δρ(µ−ϕ(r))\nδϕ(r). Comparing Eqs. (S24)\nand (S27), we get the relationship,\n1\n2lim\nq→0∂qµ∂qνχc(q) =∂µcQµν. (S28)\nTherefore,1\n2limq→0∂qµ∂qνχc(q)is a thermodynamic quantity even in metals and at a finite temperature, and we call it gener-\nalized electric susceptibility as a naive generalization of the electric susceptibility to metals.11\nDerivation of generalized electric susceptibility in noninteracting systems\nWe derive the formula of the generalized electric susceptibility in noninteracting systems. Starting from Eq. (S11) at iΩn= 0,\nthe generalized electric susceptibility is written by,\nlim\nq→0∂qµ∂qνχ0(q) =1\nNβX\nkωnTr\u0002\n∂kµG(k, iωn)∂kνG(k, iωn)\u0003\n,\n=1\nNβX\nkωnX\nnm⟨un(k)|∂kµH0(k)|um(k)⟩\n(iωn−ϵn(k))2⟨um(k)|∂kνH0(k)|un(k)⟩\n(iωn−ϵm(k))2,\n=1\nNX\nkX\nnf(3)(ϵn(k))\n6⟨un(k)|∂kνH0(k)|un(k)⟩⟨un(k)|∂kνH0(k)|un(k)⟩\n+1\nNX\nkX\nn̸=m\u0012\nf′(ϵn(k)) +f′(ϵm(k)) + 2f(ϵm(k))−f(ϵn(k))\nϵn(k)−ϵm(k)\u0013\n×⟨un(k)|∂kµH0(k)|um(k)⟩\n(ϵn(k)−ϵm(k))⟨um(k)|∂kνH0(k)|un(k)⟩\n(ϵn(k)−ϵm(k)). (S29)\nThe first term is the effective-mass term χ0:µν\ne:mass while the second term is the quantum geometric term χ0:µν\ne:geom . Note that, when\ntwo bands are degenerate, contribution to the quantum geometric term from the two degenerated bands ˜n,˜matkis\nχ0:µν\ne:geom:˜ n˜m=f(3)(ϵ˜n(k))\n6⟨u˜n(k)|∂kνH0(k)|u˜m(k)⟩⟨u˜m(k)|∂kνH0(k)|u˜n(k)⟩+c.c. (S30)\nThus, the geometric term does not diverge even in the presence of band touching.\nThen, by using the Hellmann-Feynman theorem,\n⟨un(k)|∂kµH0(k)|um(k)⟩=δnm∂kµϵn(k) + (ϵn(k)−ϵm(k))⟨∂kµun(k)|um(k)⟩, (S31)\nthe effective-mass and quantum geometric terms are rewritten as\nχ0:µν\ne:mass =1\nNX\nkX\nnf(3)(ϵn(k))\n6∂kµϵn(k)∂kνϵn(k),\n=−X\nnZdk\n(2π)df(2)(ϵn(k))\n6∂kν∂kµϵn(k), (S32)\nχ0:µν\ne:geom =1\nNX\nkX\nn̸=m\u0012\nf′(ϵn(k)) +f′(ϵm(k)) + 2f(ϵm(k))−f(ϵn(k))\nϵn(k)−ϵm(k)\u0013\n×⟨∂kµun(k)|um(k)⟩⟨um(k)|∂kνun(k)⟩,\n=X\nnZdk\n(2π)d\nf′(ϵn(k))X\nm(̸=n)⟨∂kµun(k)|um(k)⟩⟨um(k)|∂kνun(k)⟩+c.c.\n+2f(ϵn(k))X\nm(̸=n)⟨∂kµun(k)|um(k)⟩⟨um(k)|∂kνun(k)⟩+c.c.\nϵm(k)−ϵn(k)\n. (S33)\nHere, ∂kν∂kµϵn(k)is the effective mass, whileP\nm(̸=n)⟨∂kµun(k)|um(k)⟩⟨um(k)|∂kνun(k)⟩+c.c andP\nm(̸=n)(⟨∂kµun(k)|um(k)⟩⟨um(k)|∂kνun(k)⟩+c.c.)/(ϵm(k)−ϵn(k))are the quantum metric and the positional\nshift, respectively. Note that this formula is equivalent to the formula derived by ∂µcQµνof noninteracting systems.\nAs for the contribution from the quantum metric, f′(ϵn(k))is negative and the quantum metric has a positive value. Therefore,\nthe quantum metric always gives a negative contribution to the generalized electric susceptibility. In contrast, the contribution\nfrom the positional shift can be rewritten as,\nχ0:µν\nshift=X\nkX\nn̸=mZdk\n(2π)df(ϵm(k))−f(ϵn(k))\nϵn(k)−ϵm(k)\u0000\n⟨∂kµun(k)|um(k)⟩⟨um(k)|∂kνun(k)⟩+c.c\u0001\n.\n(S34)12\nSince the band resolved quantum metric, ⟨∂kµun(k)|um(k)⟩⟨um(k)|∂kνun(k)⟩+c.c, and the Lindhard function, (f(ϵm(k))−\nf(ϵn(k)))/(ϵn(k)−ϵm(k)), are positive, this contribution is always positive.\nGeneralized electric susceptibility of isotropic continuum model in two dimension\nWe consider the two-dimensional isotropic continuum model whose Hamiltonian is given by\nH(k) =ϵ(k)−µc=X\nµ=x,yk2\nµ\n2m−µc. (S35)\nIn this model, the generalized electric susceptibility can be written as,\nχ0:µµ\ne =−Zdk\n(2π)2f(2)(ϵ(k)−µc)\n61\nm,\n=−Z∞\n−µcdϵf(2)(ϵ)\n61\nmm\n2π,\n=−\u0014f′(ϵ)\n12π\u0015∞\n−µc=f′(µc)\n12π, (S36)\nsince the density of states is constant, m/2π. Note that the geometric term is absent in the single-band model. Therefore,\nthe generalized electric susceptibility χ0:µµ\ne vanishes at low temperatures which satisfy µc≫T; ferromagnetic fluctuation is\nprohibited in two-dimensional isotropic continuum models.\nS2. DISPERSIVE LIEB LATTICE MODEL\nA. Hamiltonian\nWe introduce the Hamiltonian of the dispersive Lieb lattice model. For comparison, we also show the usual Lieb lattice\nmodel. In Figs. S1(a) and S1(b), the hopping integrals of the dispersive and usual Lieb lattice models are schematically shown.\nIn contrast to the usual Lieb lattice model, the dispersive Lieb lattice model includes second- and third-nearest-neighbor hopping\nintegrals. Thus, the noninteracting Hamiltonian for the dispersive Lieb lattice model and the usual Lieb lattice model are written\nas\nH0:d(k) =\nϵA(k)−µc ϵx(k) ϵy(k)\nϵx(k) ϵB(k)−µcϵxy(k)\nϵy(k) ϵxy(k)ϵC(k)−µc\n, (S37)\nH0:l(k) =\n−µcϵx(k)ϵy(k)\nϵx(k)−µc0\nϵy(k) 0 −µc\n, (S38)\nrespectively. Here, we define\nϵA(k) =−2t3(coskx+ cos ky), (S39)\nϵB(k) =−2t3coskx−2t′\n3cosky, (S40)\nϵC(k) =−2t3cosky−2t′\n3coskx, (S41)\nϵx(k) =−2tcoskx/2, (S42)\nϵy(k) =−2tcosky/2, (S43)\nϵxy(k) =−4t2coskx/2 cosky/2, (S44)13\n(a)(b)\n(c)t=1.00.15t0.4t0.15t0.2t(d)\nXΓMΓ-4-3-2-1 0 1 2 3\n 64 128 192Ek-4-3-2-1 0 1 2 3\n 64 128 192EkXΓMΓt=1.0BCA\nFIG. S1. (a) The dispersive Lieb lattice and (b) the usual Lieb lattice. The unit cell (gray box) and three sublattices A, B, and C are shown.\nHopping integrals (blue arrows) are illustrated with a unit of the nearest-neighbor hopping. (c) and (d) show the band dispersion of the\ndispersive and usual Lieb lattice models, respectively.\nwith(t, t2, t3, t′\n3) = (1 .0,0.4,0.15,0.2). The energy dispersion for the dispersive and usual Lieb lattice models is shown in\nFigs. S1(c) and S1(d), respectively. Owing to the long-range hopping, the flat band in the original Lieb lattice gets dispersion.\nIn the dispersive Lieb lattice model, the third-nearest-neighbor hopping of the A sublattice (blue circles in Fig. S1(a)) is not\nequivalent to that of the B(C) sublattice (red circles in Fig. S1(a)). Therefore, three-fold band degeneracy at Mpoint is partially\nlifted and reduced to two-fold band degeneracy (see Fig. S1(c)).\nB. Non-Kramers band degeneracy at Mpoint\nNext, we discuss the non-Kramers band degeneracy in the (dispersive) Lieb lattice at the Mpoint based on the D4hpoint group\nsymmetry.\nPeriodic basis of Hamiltonian\nSince we adopt the Fourier transform with the internal position of sublattices, the Hamiltonian does not satisfy the Brillouin-\nzone periodicity, i.e., H0:d(l)(k)̸=H0:d(l)(k+G), with a reciprocal lattice vector G. Because this basis is not convenient for the14\nsymmetry analysis, we introduce the Hamiltonian with a periodic basis where the Fourier transform does not include the inter-\nnal position of sublattices, ˜H0:d(l)(k) =V(k,aB,aC)H0:d(l)(k)V†(k,aB,aC)withV(k,aB,aC) = diag(1 , eik·aB, eik·aC).\nHere,aBandaCare the internal positions of the sublattices, B and C, respectively. We set the internal position of the sub-\nlattice A as the origin, i.e. aA= (0,0,0). In this basis, the vector representation of the annihilation operator is written as,\nV(k,aB,aC)ˆc(k).\nSymmetry operation\nWe consider the symmorphic point group where the point-group element ˆg={pg}does not include any translation operation\nand its operation on real-space coordinates is given by ˆgr=pgr. The operation of ˆgon the Hilbert space is defined by the\nfollowing relation,\nˆgˆcl(R)ˆg−1= ˆcgl(R′), (S45)\npgR+pgrl=R′+rgl, (S46)\nwhere glis the transformed sublattice index by the symmetry operation. Here and hereafter, we omit the spin index for simplicity.\nThe wave-vector representation is transformed by the symmetry operation as,\nˆgeik·rlˆcl(k)ˆg−1=1√\nNeik·rlX\nRe−ik·(R+rl)ˆcgl(R′)\n=1√\nNX\nRe−ipgk·pgRˆcgl(R′)\n=1√\nNepgk·pgrlX\nR′e−ipgk·(R′+rgl)ˆcgl(R′)\n=epgk·pgrlˆcgl(pgk). (S47)\nTherefore, its vector representation for the dispersive Lieb lattice model can be written by,\nˆgV(k,aB,aC)ˆc(k)ˆg−1=V(pgk, pgaB, pgaC)Dgˆc(pgk)\n=V(pgk, pgaB, pgaC)DgV†(pgk,aB,aC)V(pgk,aB,aC)ˆc(pgk),\n(S48)\nwhere [Dg]ll′=δl,gl′is the representation matrix of the symmetry operation ˆgwith respect to the sublattice degree of freedom.\nFrom this, we obtain the representation matrix for the symmetry operation,\nUg(k) =V(pgk, pgaB, pgaC)DgV†(pgk,aB,aC). (S49)\nSymmetry analysis\nIn the following, we decompose the representation matrix of the symmetry operation at the Mpoint into the irreducible\nrepresentations of the point group D4h. If there is a two-dimensional representation in the decomposition, the eigenspectrum at\ntheMpoint has at least one set of doubly degenerate eigenstates.\nFor our purpose, it is sufficient to consider C4v, a subgroup of D4h, since there should be a two-dimensional irreducible\nrepresentation of D4hwhen we have that of C4v. The representation matrices at kM= (π, π,0)are given by\nUC4(kM) =\n1 0 0\n0 0−1\n0 1 0\n, Uσv(kM) =\n1 0 0\n0 1 0\n0 0−1\n, Uσd(kM) =\n1 0 0\n0 0 −1\n0−1 0\n, (S50)\nand so on. Here, g=Cnis the nfold rotational symmetry, and σvandσdare the mirror reflection symmetry whose mirror\nplanes are rotated from each other by π/4.15\nBased on Eq. (S50), the character table for the C4vpoint group is summarized in Table S1. Characters of the representation\nmatrix for a symmetry operation ˆg,ξUg(kM)(g)≡tr[Ug(kM)], are written by ξUg(kM)(g) =ξA(g)+ξE(g)with characters of the\nirreducible representations ξA(g)andξE(g). As a result, we conclude that the eigenvalues at the Mpoint are generally given by a\npair of doubly degenerate eigenstates and a non-degenerate eigenstate. In the usual Lieb lattice, the condition ϵA(k) =ϵB(C)(k)\nleads to the accidental three-fold band degeneracy. However, in the dispersive Lieb lattice, the additional third-nearest neighbor\nhopping lifts the accidental degeneracy while preserving the two-fold band degeneracy protected by the C4symmetry.\nTABLE S1. The character table of the C4vpoint group with the representation matrix at the Mpoint.\n12C4C22σv2σd\nUg(kM)3 1 −1 1 1\nA1 1 1 1 1 1\nA2 1 1 1 −1−1\nB1−1 1 1 1 −1\nB2 1−1 1 −1 1\nE 2 0 −2 0 0\nS3. LINEARIZED GAP EQUATION WITH RPA\nTo study the superconductivity, we solve the following two equations self-consistently:\nλ∆ll′(k) =−1\nNβX\nk′ωnX\nl1l2Vt(s)\nll1,l2l′(k−k′)Fl1l2(k′, iωn), (S51)\nFll′(k, iωn) =X\nl1l2Gll1(k, iωn)∆l1l2(k)Gl′l2(−k,−iωn). (S52)\nHere, λis the eigenvalue of the linearized gap equation and ∆ll′(k)is the gap function. The matrix representation of the effective\ninteraction Vll1,l2l′(k)for spin-triplet and spin-singlet pairings are obtained by RPA as\nVt(k) = Γ0\u0014\n−1\n4¯χs(k)−1\n4¯χc(k)\u0015\nΓ0, (S53)\nVs(k) = Γ0\u00143\n4¯χs(k)−1\n4¯χc(k)\u0015\nΓ0+ Γ0, (S54)\nrespectively. We ignore the Ωn-dependence of the effective interaction Vt(s)\nll1,l2l′(k), corresponding to the mean-field approxima-\ntion.\nS4. SUPERCONDUCTIVITY IN THE DISPERSIVE LIEB LATTICE\nIn this section, we show the detailed results of superconductivity in the dispersive Lieb lattice model described by Eq. (S37).\nThe dispersive Lieb lattice belongs to the D4hpoint group. In the presence of D4hpoint group symmetry, superconducting\nstates are classified into ten irreducible representations. However, since we consider a purely two-dimensional system with\nSU(2) symmetry, the C2rotation is equivalent to the space inversion. Therefore, A1g, A2g, B1g, B2g, andEurepresentations are\nallowed while A1u, A2u, B1u, B2u, and Egrepresentations are prohibited. Here, we set the temperature T= 0.01and show the\nchemical potential dependence of λfor all the allowed irreducible representations in Fig. S2. Either the spin-triplet Eupairing\n(blue line) or the spin-singlet A1gpairing (orange line) is dominant.\nHere, we discuss the dominant A1gandEusuperconducting states. We show the k-dependence of the A1g-gap functions at\nµc= 0.76in Fig. S3. Figures S3(a) and S3(b) show the gap functions for the intra-sublattice pairing on the B sublattice and the16\n 0 0.2 0.4 0.6 0.8 1 1.2\n 0  0.2  0.4  0.6  0.8  1λ\nµcA1g\nA2g\nB1g\nB2g\nEu\nFIG. S2. The eigenvalues λof the linearized gap equation for the dispersive Lieb lattice model. We set T= 0.01andU= 0.86. The orange,\ngreen, purple, red, and blue lines correspond to the A1g, A2g, B1g, B2g, andEurepresentations.\n𝑘!−𝜋𝜋𝑘!−𝜋𝜋𝑘\"−𝜋𝜋(a)(b)𝑘\"−𝜋𝜋B sublatticeC sublattice\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6\nFIG. S3. The k-dependence of the A1g-gap functions at µc= 0.76for intra-sublattice pairings on (a) the B sublattice and (b) the C sublattice.\nAs shown in Fig. S1(a) by red circles, the B (C) sublattice lies on the right (top) of the A sublattice. These two sublattices are related to each\nother by C4rotation.\nC sublattice illustrated in Fig. S1(a). The other components of the gap function are less dominant than these components. The\nsymmetry of superconductivity corresponds to the extended- s-wave superconductivity.\nFigure S4 shows the k-dependence of the Eu-gap functions at µc= 0.7. In the two-dimensional Eurepresentation, two\nindependent bases i.e. pxandpyare present, corresponding to the degenerate pairing states. Therefore, we show Fig. S4(a)-\n(c) for the pxpairing state while Fig. S4(d)-(f) for the pystate. Panels (a) and (d) [(b) and (e)] are the intra-sublattice pairing\ncomponent on the B [C] sublattice, while (c) and (f) show the inter-sublattice pairing component between the B and C sublattices.\nThe other components of the gap function are less dominant. In the Eurepresentation, any linear combination of the two\nindependent bases is allowed. Considering that full-gap superconducting states are thermodynamically stable to maximize the\ncondensation energy, the px+ipy-pairing state, namely, the time-reversal-symmetry-broken chiral p-wave pairing state may be\nfavored. Some nodes in Fig. S4, such as on the kx= 0 andkx= 0 lines are gapped. However, the chiral p-wave pairing state\nis degenerate with other p-wave pairing states such as pxˆx+pyˆydue to the spin degree of freedom of spin-triplet Cooper pairs.\nSince the degeneracy is protected by the SU(2) symmetry, it is lifted by the SOC.17\n𝑘!−𝜋𝜋𝑘!−𝜋𝜋𝑘\"−𝜋𝜋(a)(b)𝑘\"−𝜋𝜋\n𝑘!−𝜋𝜋(c)𝑘\"−𝜋𝜋\n𝑘!−𝜋𝜋𝑘!−𝜋𝜋𝑘\"−𝜋𝜋(d)(e)𝑘\"−𝜋𝜋\n𝑘!−𝜋𝜋(f)𝑘\"−𝜋𝜋B sublatticeC sublatticeB-C sublattice\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\nFIG. S4. The k-dependence of the Eu-gap functions at µc= 0.7. (a)-(c) and (d)-(f) show the pxandpybasis of Eurepresentation,\nrespectively. (a) and (d) [(b) and (e)] are the intra-sublattice pairing component on the B [C] sublattice while (c) and (e) are the inter-sublattice\npairing component between the B and C sublattices.\nS5. ANOTHER EXAMPLE OF QUANTUM-GEOMETRY-INDUCED FERROMAGNETIC FLUCTUATION : RAGHU’S\nMODEL\nTo show another example of quantum-geometry-induced ferromagnetic fluctuation, we consider Raghu’s model [S7] for iron-\nbased superconductors. Using the Pauli matrix and the unit matrix for the orbital space ρµandρ0, the Hamiltonian of the\nRaghu’s model is given by,\nH0:r(k) = (h0(k)−µc)ρ0+hxy(k)ρx+hz(k)ρz. (S55)\nHere, we define,\nh0(k) =−(t1+t2)(cos kx+ cos ky)−4t3coskxcosky, (S56)\nhz(k) =−(t1−t2)(cos kx−cosky), (S57)\nhxy(k) =−4t4sinkxsinky, (S58)\nwith(t1, t2, t3, t4) = (−1.0,1.3,−0.85,−0.85).\nThe band dispersion of the Raghu’s model is shown in Fig. S5(a). We see the band degeneracy at ΓandMpoints. The\ngeneralized electric susceptibility in this model is shown Figs. S5(c) and S5(d), where the total susceptibility χ0:xx\ne, the quan-\ntum geometric term χ0:xx\ne:geom , and the effective-mass term χ0:xx\ne:mass are plotted. In Fig. S5(c) showing the chemical potential\ndependence, we find the negative peak of χ0:xx\ne:geom nearµc= 2.8, where the band-degenerate point lies on the Fermi surface.\nFurthermore, the geometric term χ0:xx\ne:geom is negatively enhanced as the temperature decreases (see Fig. S5(d) for the tempera-\nture dependence). These behaviors are similar to the dispersive Lieb lattice model discussed in the main text and indicate the\nquantum-geometry-induced ferromagnetic fluctuation. The bare spin susceptibility χ0\ns(q)with the peak at q= 0 is shown in\nFig. S5(b), and the ferromagnetic fluctuation is confirmed. Thus, we conclude that the quantum geometry induces ferromagnetic\nfluctuation due to non-Kramers band degeneracy also in the Raghu’s model.18\n-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2\n 1 1.5 2 2.5 3χeµcχmassχ geomχ-3-2-1 0 1 2 3-3-2-1 0 1 2 3\n 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\n-8-6-4-2 0 2 4\n 64 128 192Ek𝑞!−𝜋𝜋(a)(b)𝑞\"−𝜋𝜋\nXΓMΓ(c)(d)\n-12-10-8-6-4-2 0 2 4 6\n 0 0.02 0.04 0.06 0.08 0.1χeTχmassχ geomχ\nFIG. S5. Results in the Raghu’s model. (a) The band dispersion, (b) the bare spin susceptibility χ0\ns(q)for(µc, T) = (2 .8,0.002) , (c) the\nchemical potential dependence of the generalized electric susceptibility χ0:xx\ne atT= 0.01, and (d) the temperature dependence of χ0:xx\ne at\nµc= 2.8. The geometric term and effective-mass term are also shown in (c) and (d).\n∗kitamura.taisei.67m@st.kyoto-u.ac.jp\n[S1] The Lehmann representation of charge susceptibility is written by\nχc(q, ω) =−1\nNZX\nab\u0010\ne−βEa−e−βEb\u0011⟨a|ˆn(q)|b⟩⟨b|ˆn(−q)|a⟩\nℏω+Ea−Eb+iδ. (S59)\nHere,Zis the partition function. |a⟩andEaare the eigenstate and the eigenvalue of the many-body Hamiltonian ˆH.\n[S2] A. Shitade, H. Watanabe, and Y . Yanase, Phys. Rev. B Condens. Matter 98, 020407 (2018).\n[S3] A. Shitade, A. Daido, and Y . Yanase, Phys. Rev. B Condens. Matter 99, 024404 (2019).\n[S4] Y . Gao, D. Vanderbilt, and D. Xiao, Phys. Rev. B Condens. Matter 97, 134423 (2018).\n[S5] A. Daido, A. Shitade, and Y . Yanase, Phys. Rev. B Condens. Matter 102, 235149 (2020).\n[S6] T. Kitamura, J. Ishizuka, A. Daido, and Y . Yanase, Phys. Rev. B Condens. Matter 103, 245114 (2021).\n[S7] S. Raghu, X.-L. Qi, C.-X. Liu, D. J. Scalapino, and S.-C. Zhang, Phys. Rev. B Condens. Matter 77, 220503 (2008)."    },    {        "title": "1301.4294v1.Field_Free_Synthetic_Ferromagnet_Spin_Torque_Oscillator.pdf",        "content": "arXiv:1301.4294v1  [cond-mat.mes-hall]  18 Jan 2013Field-Free Synthetic-Ferromagnet Spin Torque Oscillator\nYan Zhou1, Jiang Xiao (萧江)2,3,∗Gerrit E. W. Bauer4,5, and F. C. Zhang1,6\n1Department of Physics, The University of Hong Kong, Hong Kon g, China\n2Department of Physics and State Key Laboratory of Surface Ph ysics, Fudan University, Shanghai, China\n3Center for Spintronic Devices and Applications, Fudan Univ ersity, Shanghai, China\n4Institute for Materials Research, Tohoku University, Send ai, Japan\n5Kavli Institute of NanoScience, Delft University of Techno logy, Delft, The Netherlands\n6Center of Theoretical and Computational Physics, Univ. of H ong Kong, Hong Kong, China\n(Dated: March 21, 2022)\nWe study the magnetization dynamics of spin valve structure s with a free composite synthetic\nferromagnet (SyF) that consists of two ferromagnetic layer s coupled through a normal metal spacer.\nA ferromagnetically coupled SyF can be excited into dynamic al precessional states by an applied\ncurrent without external magnetic ﬁelds. We analytically d etermine the stability of these states\nin the space spanned by the current density and SyF interlaye r exchange coupling. Numerical\nsimulations conﬁrm our analytical results.\nThe transfer of angular momentum between the mag-\nnetic layers of current-driven spin valves (spin-transfer\ntorque) has not been so long ago predicted1,2and ex-\nperimentally conﬁrmed.3,4The implied eﬃcient electrical\ncontrol of magnetizations motivated the pursue of new\nresearch directions. When the current density exceeds\na critical value, the spin-transfer torque can switch the\nmagnetization to a diﬀerent static conﬁguration without\nthe necessity of applied magnetic ﬁelds, which makes it\nattractive for next generation Magnetoresistive Random\nAccess Memory (MRAM) application.3,5–7Under an ex-\nternal magnetic ﬁeld, the spin-transfer torque can also\ndrive the magnetization into sustainable coherent oscil-\nlations spanning a wide frequency range from a few MHz\nto several hundred GHz.3,5–11High frequency magnetic\noscillations generate a coherent microwave voltage signal\nthrough the Giant Magnetoresistance (GMR) in metallic\nspin valves or through the Tunneling Magnetoresistance\n(TMR) in magnetic tunnel junctions (MTJs). This eﬀect\ncan be used in so-called spin-torque oscillators (STO),\nwhich has many advantages including wide tunability,12\nveryhigh modulation rates,13,14compact devicesize, and\nhigh compatibility with standard CMOS processes.15,16\nThus STO is appealing for high frequency microwave ap-\nplications including microwave emitters, modulators and\ndetectors.17However, the necessity of an applied mag-\nnetic ﬁeld up to ∼1 Tesla has greatly limited the poten-\ntial of these STOs for microwave generation and wireless\ncommunication applications. Recently, various solutions\nhave been proposed to enable zero-ﬁeld operation, viz.\nSTO with a perpendicularly magnetized ﬁxed18or spin\nvalves with out-of-plane magnetized free layer,19,20mag-\nnetic vortex oscillators,21–26wavy-torque STO by judi-\ncially choosing free and ﬁxed layer materials with diﬀer-\nent spin diﬀusion lengths,27and a tilted magnetization\nof the ﬁxed layer with respect to the ﬁlm plane.28–32\nRecently, synthetic ferromagnets (SyFs) composed of\ntwo ferromagnetic layers separated by a very thin non-\nmagnetic spacer have been used to replace the free layerofa spin valve orMTJ.33–39SyF based spintronic devices\nhave the advantage of higher thermal stability, smaller\nstray magnetic ﬁelds, faster switching speed and reduced\nthreshold switching current as compared to single fer-\nromagnetic free layers.33–39Kleinet al.39predicted that\nan anti-ferromagneticallycoupled SyF layer with uncom-\npensated magnetization can generate microwave oscilla-\ntions at zero applied magnetic ﬁeld.\nHere we predict that a ferromagnetically coupled SyF\ncan also be driven into dynamical precessional states,\nwhich, however, are surrounded in parameter space by\nstatic canted states with non-collinear magnetizations.\nWe use an analytical approach to determine the stability\nregimes of the SyF system and conﬁrm results by numer-\nical simulations.\nWe study a spin torque nanodevice with synthetic fer-\nromagnetic free layers as shown in Fig. 1. The left\nferromagnetic ﬁlm forms the ﬁxed polarizer with mag-\nnetization m0/bardblˆz, and the SyF consists of two ferro-\nmagnetic layers FM 1and FM 2of thickness d1,2with a\nparamagnetic spacer. The unit vectors describing the\nmagnetization orientation are m1for FM 1andm2for\nFM2. For simplicity, we assume that the SyF layers\nFIG. 1. A spin valve structure with an SyF free layer, where\nFM0is the ﬁxed layer and FM 1,2layers are (anti-) ferromag-\nnetically coupled.2\nare made of the same materials with identical satura-\ntion magnetization Ms. The exchange coupling strength\nreadsEC=−JSm1·m2, where JandSare the cou-\npling energy per unit area and the cross section area of\nthe sample, respectively. This corresponds to an eﬀec-\ntive coupling ﬁeld Hc\ni=Jm¯i/(µ0Msdi), where i= 1,2\nand¯i= 3−i,µ0is the vacuum magnetic susceptibil-\nity.m1andm2can be parallel or anti-parallel at zero\napplied ﬁeld, corresponding to the non-local Ruderman-\nKittel-Kasuya-Yoshida (RKKY) exchange ferromagnetic\n(J >0) or antiferromagnetic ( J <0) coupling, respec-\ntively. The spacer between FM 0and FM 1is presumed\nthick enough that the RKKY coupling with the ﬁxed\nlayer is negligibly small. Although the dynamic dipo-\nlar coupling may be responsible for the apparent reduc-\ntion ofstatic magnetization40orlinewidth ofthe current-\ninduced spin wave mode41, it is estimated to be much\nsmaller for our case compared to the shape anisotropy\nﬁeld and the other ﬁelds due to current-induced spin\ntorque and interlayer exchange coupling and therefore\ndisregarded39.\nLetP0,1be the spin current polarization by m0,1such\nthat the spin current density in the two spacers are P0j\nandP1jwithjthe electric current density. The corre-\nsponding spin-transfer torques on m1andm2are given\nby the projections:\nNST1=γ/planckover2pi1j\n2eµ0Msd1m1×(P0m0−P1m2)×m1,(1a)\nNST2=γ/planckover2pi1j\n2eµ0Msd2P1m2×m1×m2, (1b)\nwithγthe gyromagnetic ratio and P0(P1) are in general\nfunctions of the angle θ=∠(m0,m1) (∠(m1,m2)).1,42\nSpin pumping causesenhanceddampingin aferromag-\nnetic layer by emitting spin current into the adjacent\nnon-magnetic layers.43This emitted spin pumping cur-\nrent can exert a torque on the second layer. Disregarding\nthe backﬂow and diﬀusion in the spacer layer, the torque\ndensity acting on midue to spin pumping from m¯ican\nbe written as\nNSPi=βm¯i×˙m¯i−[(βm¯i×˙m¯i)·mi]mi(2)\nwhereβis the eﬀective enhanced damping due to spin\npumping. It has been shown that Eq. (2) gives rise to a\ndynamic exchange interaction that can induce synchro-\nnization of the magnetization dynamics in two neighbor-\ning ferromagnetic layers even for wide spacers.44In the\nresults below we fully include the spin pumping. How-\never, in contrast to multilayers excited by microwaves,44\nwe observehere only small correctionsdemonstrating the\ndominance of charge current-induced torques.\nThe dynamics is described by the coupled Landau-\nLifshitz-Gilbert-Slonczewski (LLGS) equations,45,46\n˙mi=−γmi×Hi+αmi×˙mi−NSPi−NSTi,(3)whereαis the sum of the intrinsic Gilbert and the\nspin pumping induced damping.44The eﬀective mag-\nnetic ﬁelds Hiconsist of shape anisotropy and RKKY\nexchange coupling and can be written as,\nHi=2Ku\nµ0Ms[mi·ez]ez+Jm¯i\nµ0Msdi. (4)\nFor simplicity, we consider d1=d2=d(equal magne-\ntization) for the rest of the paper unless otherwise speci-\nﬁed. We linearize Eq. (3) in the vicinity of four collinear\nequilibrium states, i.e.↑↑,↑↓,↓↑,↓↓, and assume mi=\nλiˆz+uiwithλi=±anduidenoting the small trans-\nverse magnetization component. After the linearization\nand the Fourier transform ui(t) =/integraltext˜ui(ω)e−iωtdω/2π,\nEq. (3) becomes\n/parenleftBig\nˆAω+ˆV/parenrightBig/parenleftbigg\n˜u1\n˜u2/parenrightbigg\n= 0 (5)\nwith\nˆA=/parenleftbigg\n1−iαλ1iβλ2\niβλ11−iαλ2/parenrightbigg\n, (6a)\nˆV=ω0/parenleftbigg\nλ10\n0λ2/parenrightbigg\n+ωJ/parenleftbigg\nλ2−λ1\n−λ2λ1/parenrightbigg\n+iωj/bracketleftbigg\nP0/parenleftbigg\n−λ10\n0 0/parenrightbigg\n+P1/parenleftbigg\nλ1λ2−1\n1−λ1λ2/parenrightbigg/bracketrightbigg\n,(6b)\nwithω0= 2γKu/µ0Ms,ωJ=γJ/µ0Msd,ωj=\n(/planckover2pi1/2e)(γj/µ0Msd). The frequency of the normal modes\nare given by the eigenvalues of ˆW=−ˆA−1ˆV: Ω1and\nΩ2. When any of the Im Ω 1,2>0, the system is unsta-\nble, implying that an inﬁnitesimal perturbation will lead\nto magnetizationdynamics with amplitudes that initially\nincrease exponentially in time.\nThe above results allow us to calculate the stability re-\ngions for the ↑↑,↑↓,↓↑,↓↓phases in the space of typical\nexperimental parameters: angle-independent P1=P2=\nP= 0.5,d= 3 nm, Ku= 8×104J/m3,j∼108A/cm2\nandJ∼1 mJ/m2.39,47To analytically construct the sta-\nbility diagram as shown in the top-left panel of Fig. 2(a),\nwe ﬁrst calculate the eigenvalues for each given set of\n[j,J] as given by Eq. (5). Then we determine whether\nany of the four collinear static states (diﬀerent combi-\nnations of [ λ1,λ2]) is stable or not. For example, both\nthe imaginary part of the eigenvalues of ↑↑conﬁguration\n[λ1= +1,λ2= +1] are negative when j/lessorsimilar0. Therefore\n↑↑is stable in the blue region. In this way, we quickly\nmap the parameter space for any given set of [ j,J] and\nconstruct the entire stability diagram consisting of four\ncollinear magnetization conﬁgurations. The spin torque\ndrives the SyF to the parallel ↑↑conﬁguration for nega-\ntive currents j. For positive currents, the ↓↑conﬁgura-\ntion is preferred. These results can be understood from\nEq. (1). In a small region the antiparallel ↑↓state exists\nfor negative Jand small j(i.e.in the vicinity of the neg-\native vertical axis but not visible in the ﬁgure due to the\nscale). Although it seems that the ↓↓state also occupies3\n0 5 10 1\n  \n0 5 10 1\n0 20 40 1\n  \n0 5 10 1\n  \n0 20 40 1\n  \n0 20 40 1\n  \n/ / \nt (ns)  t (ns)  (2) (1) \n(6) (4) (3) \n(5) \nt (ns)  1 2 4 3 5 \n6 \nj (10 7 A/cm 2) J (10 -2  mJ/m 2) \n-5              0                5 0 5 \nj (10 7 A/cm 2) (a) (b) \n(c) \nz \nx y \nm2 m1 -5               0                5 \n 1 \n2 \n \nFIG. 2. (Color online) Dynamical phase diagram in the pa-\nrameter space of currents and RKKY coupling strengths. (a)\nPhase diagram calculated analytically by Eq. (5); none of th e\nfour states ↑↑,↑↓,↓↑,↓↓is stable in the white region. (b)\nPhase diagram calculated by numerically solving the LLGS\nEq. (3). The purple are the STO phase, and the white one\nthe canted state. (c) The time evolution of the polar angles\nθ1,2=∠(m1,2,ˆz) at the six diﬀerent points indicated in the\nphase diagram. In the third subﬁgure of (c), the solid and\ndashed lines correspond to diﬀerent sets of initial conditi ons.\nthe fourth quadrant ( j >0,J <0), this triangular region\nis hysteretic, i.e.↓↓and↓↑may both appear depending\non the history.\nMost importantly, there is a white/purple region in\nwhich none of the four static collinear states is stable,\ntherefore it must be either in a dynamical STO or static\ncanted state. To leading order of α, we ﬁnd from Eq. (5)\nan approximate boundary for the white region:\nupper:ωJ=ωj, (7a)\nlower:ωJ=/radicalBig\n4ω2\n0+ω2\nj−2ω0+αω2\n0\nωj,(7b)\nwhich is plotted as the black dashed lines in Fig. 2(a,b),\nmatchingthenumericallyobtainedboundariesalmostex-\nactly. Eq. (7) is calculated from the eigenvalue anal-\nysis based on Eq. (5) with perturbation from the four\nstatic collinear states. This method is equivalent to that\nused by Bazaliy et al.48A fully analyticalsolution for the\nboundary between STO and static canted phase turned\nout to be intractable. due to the complexity of Eq. (5)\nfor non-collinear states.\nWe now present numerical solutions of the LLGS\nEq. (3) including damping, spin torque and RKKY cou-\n15 \n   5 \n0 \nj (10 8 A/cm 2) f (GHz) \n10 -3 10 -2 10 -1 \nj (10 8 A/cm 2) STO canted STO canted a.u. \n10 \n       1        2        3       4       5        1        2        3       4       5 \nFIG. 3. (Color online) Power spectrum for m0·m1(left) and\nm1·m2(right)as afunctionofcurrentdensity jandfrequency\nfatJ= 0.25 mJ/m2, corresponding to the black line in the\ntop right panel of Fig. 2.\npling. We summarize the dynamics of the coupled m1\nandm2in Fig. 2(b), in which we conﬁrm the phase\nboundaries in the analytical analysis in Fig. 2(a). In\naddition, we can now map the STO phase by the pur-\nple color. The rest of the white region consists of static\ncanted states. In Fig. 2(c) we show the six diﬀerent SyF\nconﬁgurations that may exist depending on the current\nand RKKY coupling strength. Point 5 corresponds to an\nSTO state, in which both m1,2are undergoing large an-\ngle precessions, which result in a large magnetoresistance\noscillations attractive for applications.\nFor the STO phase, we study the power spectrum\nof the magnetoresistance due to the magnetization os-\ncillation of m1,2, which is approximated by R(t) =\nR0+∆R1m0·m1+∆R2m1·m2. Fig. 3 shows the Fourier\ntransform of m0·m1(left) and m1·m2(right) as a func-\ntion of currentdensity jatJ= 0.25 mJ/m2(correspond-\ning to the black line in Fig. 2(b)). The clear higher order\nharmonic modes are evidence of the non-linearities in the\nSTO dynamics. Fig. 3 also demonstrate that the oscil-\nlation frequency of the device can be continuously tuned\nby the current at zero applied magnetic ﬁeld and thus\npotentially be utilized for nano-scale microwave applica-\ntions. It should be noted that the frequency rangecan be\nfurther tuned by tens of GHz by adopting a larger Kuor\ntaking into account the easy-plane anisotropy ﬁeld (de-\nmagnetization ﬁeld).\nThe STO phase studied in this work diﬀers from that\nstudied by Klein et al.[39]. The STO phase found\nby Klein et al.arises only in an anti-ferromagnetically\n(J <0) coupled uncompensated SyF (M1=Msd1S <\nMsd2S=M2), in which the total magnetization for the\nSyF is opposite to that of m0. However, the STO phase\nfound in our study appears in the ferromagnetically cou-\npled SyF with J >0 and does not require M1/negationslash=M2. Fur-\nthermore, we were not able to reproduced the STO phase\nfound by Klein et al. for an uncompensated and antipar-\nallelSyF. Wecheckedtheeﬀect ofanangulardependence\nof the prefactor Pithat take into account the eﬀects of a\nspin accumulation42. The boundaries of the white region\nwill shift noticeably, but we ﬁnd no qualitative changes.\nThediﬀerenceswithRef. 39mightbeduetootherdetails4\nin handling spin transport.\nFinally, we note that our approach can be readily ex-\ntended from bi-layer to multilayer systems in which each\nlayer is exchange-coupled with its neighbouring layers\n(unpublished). This may provide a novel route to ef-\nfectively synchronize a large network of spin torque os-\ncillators.\nIn conclusion, we predict that the ferromagnetically\ncoupled SyF can be driven into STO states without the\nneed of applying magnetic ﬁelds. The resulting STOstates display large angle precession, therefore generat-\ning a large power output. In addition to dynamical STO\nstates, static canted states are also possible in the same\nstructure at slightly diﬀerent applied current densities.\nOur ﬁndings may guide the experimental eﬀort towards\nthe ﬁeld-free STO for real applications.\nWe acknowledge support from University Research\nCommittee (Project No. 106053) of HKU, the Univer-\nsity Grant Council (AoE/P-04/08) of the government\nof HKSAR, the National Natural Science Foundation of\nChina (No. 11004036, No. 91121002), the FOM founda-\ntion, DFG Priority Program SpinCat, and EG-STREP\nMACALO.\n∗Corresponding author: xiaojiang@fudan.edu.cn\n1J. C. Slonczewski, J. Magn. Magn. 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China  \n*Corresponding author: qianfan@buaa.edu.cn .  \n \n \n \n \n \n \n \n \n \n \n  2 / 30 \n Abstract  \nLow-dimensional semiconducting ferromagnets have attracted considerable attention \ndue to their promising applications as nano -size spintronics. However, realizing robust \nferromagnetic couplings that can survive at high temperature is restrained by  two \ndecisive factors: super -exchange coupling s and anisotropy. Despite widely explored  \nlow-dimensional anisotropy, strengthening super -exchange coupling s has rarely been  \ninvestigated . Here, we found  that ligands with lower electronegativity can strengthen \nferromagnetic super -exchange couplings and further propose d the ligand modulation \nstrategy to enhance the Curie temperature of  low-dimensional ferro magnets. Based on \nthe metallic CrX 2 (X = S, Se, Te ) family, substituting ligand atoms by halides can form \nstable semiconducting phase as CrSe Cl, CrSe Br and CrTeBr. It is  interesting to  \ndiscover  that, the nearest ferromagnetic super -exchange coupling s can be  strengthened  \nwhen substituting ligands from S to  Se and Te . Such evolution originates from the \nenhanced  electron hopping integral  and reduced energy intervals  between d and p orbits . \nWhile the second nearest anti -ferromagnetic coupling s are also  benefitted  due to \ndelocalized p-p interaction s. Finally, ligand modulation strategy is applied in other \nferro magnetic monolayers, further verifying our theory  and providing  a fundamental  \nunderstanding on controlling super -exchange coupling s in low-dimension.  \n \n \n \n \n \n \n  3 / 30 \n I. INTRODUCTION  \nTwo-dimensional (2D) materials have attracted tremendous interest in recent years \nand there are increasing 2D layered materials with unique physical and chemical \nproperties that have been theoretically predicted and fabricated in experiments since the \ndiscovery of graphene.[1] Although great success has been achieved in 2D materials, \nrealizing low dimensional ferromagnetism remains as a critical topic. Appealing 2D \nferromagnets have promising application s in nano -size spintronic devices, which \nenables both the low energy consumption and high storage density. Besides,  when \nforming heterojunctions with topological insulators,  ferromagnetic monolayers can \nintroduce magnetic proximity effect and break the time reversal symmetr y, further \nopening up a gap in the surface states and realizing quantum anomalous Hall effect.[2][3] \nNevertheless, long -range ferromagnetic order in low dimension has only been observed \nin few mate rials, and there are three famous monolayer phases , as CrI 3, Cr 2Ge2Te6, and \nFe3GeTe 2, have been received considerable attention.[4] Extensive investigated CrI 3 \nmonolayer is semiconducting Ising ferromagnet, with out -of-plane easy axis. Although \nsuch anisotropy is remarkable, ferromagnetic couplings via super -exchange inter action \namong the Cr -I-Cr path is rather week, resulting in the Curie temperature as low as 45 \nK.[5][6] Cr2Ge2Te6 is a Heisenberg magnet and the long -range ferromagnetic order can \nbe established by external magnetic field, but is also limited by the weak ferromagnetic \nsuper -exchange coupling s.[7] On the other aspect, Fe 3GeTe 2 monolayer exhibits \nmetallic properties with iterant ferromagnetism, whose T C is around 68 K , but can be \ntuned to the room temperature through an ionic gate.[8][9] The iterant  ferromagnetic \nexchange via carriers in Fe3GeTe 2 is much stronger than super -exchange, but \nsemiconducting ferromagnets would be more desirable due to their moderate band gaps \nand promising applications as transistors and spintronics. Therefore, it is necessary to  4 / 30 \n further explore semiconducting magnets w ith robust ferromagnetic couplings that can \nsurvive at high temperature.  \nHowever, the realization of strong low -dimensional ferromagnets depend s on two \ndecisive factors : anisotropy and super -exchange coupling s. For the first anisotropy \nrequirement, Mermin -Wagner theorem regulates that long -range magnetic order cannot \nexist in the isotropic two -dimensional system.[10] Nevertheless, only a small anisotrop y \nis enough to  open up a sizable gap in the magnon spectra, thus stabilizing magnetic \norders against finite temperature.[11] Experiments on Cr 2Ge2Te6 indicate that external \nmagnetic field can also introduce anisotropy, and such strategy is promising to be \napplied in other systems only if they are not strong XY magnets. On the other aspect, \nsuper -exchange theory has long been established by Goodenoug h, Kanamori and \nAnderson (GKA), which mainly focuses on ionic compounds with oxygen as ligand .[12-\n15] While in low dimension, large ligands as S/Se/Te/Br/I can rather stabilize the \nmonolayer structure.  However, ligands other than oxygen will lead to the en hanced \ncovalency and further deviate from the ionic picture, and the modification of super -\nexchange coupling s via controlling the degree of d-p hopping process can thus be \nrealized. Furthermore, substituting ligands has already been achieved  in the MoSSe \nsystem ,[16][17] which suggests  the feasibility of this ligands modulation strategy . \nHowever, the e xploration on the role of ligands and their effect on magnetic properties \nhave rarely been studied, but is rather essential for the modulation on the strength o f \nexchange couplings , especially in low -dimension . \nIn the present work, we have applied ligand m odulation strategy on the metallic \n1-T CrX 2 (X = S, Se, Te) family, since Cr based compounds are known to exbibit \nferromagnetism in low -dimension. Through multiple screening rules, CrSeBr, CrSeCl \nand CrTeBr were selected as the semiconducting Janus monolayer with  the robust  5 / 30 \n ferromagnetic order, while CrTeCl adopts antiferromagnetic coupling s. To understand \ntheir magnetic properties, we concentrate on exchange integrals, and the nearest \nexchange couplings can be determined by three kinds of super -exch ange process. It is \ndiscovered that, varying ligands from S  to Se and Te  can enhance the electron hopping \nintegral  and reduce energy intervals between d and p orbits  at the same time , thus \nstrengthening ferromagnetic couplings. Further calculation reveals that the second \nnearest exchange integral can also be significantly affected by ligands, where p-p \nhopping process in telluride compounds is much more effective due to the delocaliz ed \nfeature of p electrons. Finally, we examined our theory on a series of r eported low -\ndimensional magnets, further demonstrating our discoveries. Our work not only reveals \na new family of low -dimensional magnets with mixed ligands, but also provides a \nfundamental understanding on the modulation of  super -exchange coupling s. \n \nII. COMPUTATIONAL METHODS  \nFirst-principles calculation has been performed in the framework of density \nfunctional theory using the Vienna Ab initio Simulation Package (VASP).[18][19] \nGeneralized gradient approximation (GGA) exchange -correlation was described by the \nPerder -Burke -Ernzerhof (PBE) formulatio n.[20] The projector augmented wave (PAW) \npseudopotentials[21][22] was adopted to describe the interaction between electrons and \nnuclei. An energy cu toff 500eV was employed for the plane wave basis. The criteria of \nthe total energy convergence and the atomic force tolerance was set to 10-5 eV and 0.01 \neV/ Å respectively. For describing the Fermi -Dirac distribution function, a Gaussian \nsmearing of 0.02eV was used. The 13 ×13×1 Gamma -centered Monkhorst -Pack grids[23] \nwas employed to sample the Brillouin zone for the relaxation of all structures. While \nfor electronic structure calculation, denser grids were set to 24 × 24× 3 . Considering the  6 / 30 \n localized nature of 3d electrons for transition metals, the DFT+U meth od was \nadopted.[24] The effective U -J value was tested ranging from 1 to 5 eV, and for magnetic \nand electronic calculations, the U -J value was set to 2.8eV in accordance with the \nprevious work.[25][26] To describe the electronic structure more precisely, the HSE06 \n(Heyd -Scuseria -Ernzerhof) hybrid functional[27][28] was further applied . To analyze \nbonding and anti -bonding states in detail, Crystal Occupation Hamilton Population \n(COHP) was applied through using LOBSTER.[29-32]  \nIn order to validate that the Janus monolayer structure for CrSeX, CrTeX (X= Br, \nCl) are with the lowest energy, the structure prediction was performed using the ab-\ninitio  random structure searching method (CALYPSO) based on particle swarm \noptimization (PS O) algorithm.[33][34] The Janus structure was reproduced among more \nthan 10,000 generated structures and the dynamical stability was verified based on the \nphonon spectrum simulated by PHONONPY. In order to further test the stability at \nroom temperature, the ab-initio  molecular dynamics (AIMD)[35][36] simulation was \napplied using the Nose heat bath scheme. The canonical ensemble at 300K was adopted \nto simulate the thermal stability for the 4 ×4 supercell of CrXY (X = Se, Te; Y= Cl, Br).  \n \nIII. RESULTS AND DISCUSSIONS  \nA. Structure and stability  \nBased on the metallic 1 -T phase of CrX 2 (X = S, Se, Te), we have replaced one \nlayer of S/Se/Te by Cl/Br/I and further screen those candidates through comparing the \ntotal energies of a series of common 2D structures with the same atomic ratio (Figure \nS1). CrSeCl, CrSeBr, CrTeCl and CrTeBr  are selected out, which retain the 1 -T phase \nas the most stable atomic configuration (Table S1). Structure prediction algorithm is \nadditionally performed, further verifying that their gro und states as the 1 -T Janus  7 / 30 \n monolayer. Figure 1a shows the atomic configuration of CrSeBr while others are \npresented in Figure S2. Four candidates possess hexagonal symmetry belonging to the \nspace group 156. And the inversion symmetry is broken due to the replacement of one \nlayer of original ligand atoms by halides. Among this series, the bond angles for Cr -\nSe/Te/Cl/Br -Cr are all around 90° , suggesting a favored ferromagnetic super -exchange \ncouplings based on the GKA rule.[37][38] Lattice dynamics (phonon dispersio n relations) \nwere further calculated shown in Figures 1b and S2. The lack of imaginary modes  \ndemonstrates their excellent dynamical stability. Corresponding phonon density of \nstates is presented in the right panel of Figure 1b, indicating that three higher  optical \nmodes are mainly contributed by the correlative vibration of Cr and Se atoms while \nother three lower lying optical bands correspond to Se and Br atoms. Ab-initio  \nmolecular dynamics simulation was additionally performed, which shows that the \nhoneyc omb network remains intact after simulating 20 ps at 300 K, validating their \nthermodynamic stabilities at room temperature (Figure S3).  \n \n \nFIG. 1 . a. Top view and side views of atomic configuration for CrSeBr; b. Phonon \ndispersion relation and phonon density of states (P h-DOS) for CrSeBr.  \n \n 8 / 30 \n B. Magnetic and electronic properties  \nMagnetic properties are further calculated, and magnetic ground states are firstly \nderived by comparing total energies of different magnetic configurations (Figure S4). \nIt is found that CrSeCl, CrSeBr and CrTeBr all favor ferromagnetic couplings, and the \nenergy of FM order is ~ 40 meV per magnetic cation lower than that of the stripy AFM  \norder. CrTeCl, in contrast, prefers the stripy AFM order by ~ 3 meV per magnetic cation.  \nFor three ferromagnetic candidates, the overall magnetic moment of a single unit cell \nis 3 μ B. While local magnetic moments of Cr atoms are calculated to be ~ 3.3 μ B, based \non both PBE+U and HSE06 scheme (Table 1), suggesting the rationality of our adopted \nU value. It is worth to note that there is a sizable induced magnetic moment on ligands, \nespecially for Se and Te with the value around -0.3 μ B, being similar the sp in-polarized \niodine in the CrI 3 system. Magnetic anisotropy energy (MAE) induced by spin -orbit \ncoupling (SOC) is further calculated, where MAE is defined as E out-of-plane – Ein-plane per \nunit cell. As shown in Table 1, the existence of easy -plane indicates  that CrSeCl, \nCrSeBr are weak XY magnets while CrTeBr is relatively strong. To restore the long -\nrange  magnetic order in finite temperature, we have briefly explored two strategies to \nintroduce anisotropy. The first strategy is forming heterojunction with a nisotropic CrI 3 \nmonolayer , and such ferromagnetic substrate resembles the applied  the external \nmagnetic field. And, for weak XY magnets as CrSeCl and CrSeBr, the easy -axis can be  \nrestored in the whole heterojunction system, and the strengthen of it is simi lar with the \nremarkable value for CrI 3. The second strategy  is about uniaxial strain engineering , and \nwe have applied a series of in -plane uniaxial strain and further calculated the value of \nMAE (Figure S5 and Table S2). Results indicate that uniaxial stra in can break  the in-\nplane isotropy, and s uch geometric anisotropy will correlate with spin directions via \nSOC, whose magnitude is small but can also open the gap in magnon spectra  and further  9 / 30 \n repress quantum fluctuations. Finally,  for the CrTeCl with the s tripy AFM order, it \nexhibits  the strong in-plane anisotropy  (~ 1656  μeV) with the easy axis along the [100] \ndirection .  \nNext,  since the magnetic anisotropy is small, we quantitatively describe the \nmagnetic exchange interactions based on the Heisenberg spin Hamiltonian:  \nH = -J1 ∑Si⃗⃗⃗ <𝑖𝑗>·Sj⃗⃗  – J2 ∑Si⃗⃗⃗ <<ij>>·Sj⃗⃗    \nWhere <ij> and << ij>> represent the first and second nearest couplings , which  is \ngenerally sufficient to describe  the exchange  coupling s in magnetic monolayers .[39][40] \nThe calculated exchange parameter s as J1 and  J2 are listed in Table 1. Generally,  this \nseries exhibit ferromagnetic J 1 ~ 30 me V, and it is almost ten times of the value for \nCrI 3.[41] While the second nearest exchange interaction J 2 favors AFM order, which is \nsignificantly large r for telluride compound s. To further identify whether there exists a \ngeneral evolution trend of J1 and J 2 in these compounds, we further explored other \nCrXY (X = S/Se/Te, Y = Cl/Br/I) candidates, although they are metastable phases. As \npresented in Table S3 -S5, it is interesting to discover that J 1 gradually increases when \ninvolving from S to Se and Te, while J 2 decreases and will reach a n especially  large \nvalue for telluride compound s.  \n \n \n \n \n \n \n \n  10 / 30 \n TABLE I.  Magnetic moment of Cr based on PBE+U and HSE 06 scheme, exchange \nintegrals (J 1 and J 2), magnetic anisotropy energy (MAE) for monolayer phase s and \nheterojunction s. \nProperties  CrSeCl   CrSeBr  CrTeBr  CrTeCl  \nMagnetic moment (PBE+U)/  μB 3.32 3.34 3.39 3.49 \nMagnetic moment (HSE 06)/ μB 3.22 3.24 3.48 3.38 \nJ1/meV  28.23  29.48  34.94  33.49  \nJ2/meV  -5.51 -4.53 -14.83  -17.83  \nMAE (Monolayer)/μeV  114 56 1298  1162  \nMAE (Heterojunction)/ μeV  1375  878 -628  \n \n \nFIG. 2. Projected band structure and corresponding COHP for a. spin -up and b. spin -\ndown channel  in CrSeBr, where r ed, green and blue dots represent projected electronic \nstates in Cr, Se and Br atoms. c. Differential charge density for CrSeBr (left panel), \n 11 / 30 \n where ye llow and green region denotes charge accumulation and depletion with iso -\nsurface being set to 0.01 e/ Å. Spin density for CrSeBr (right panel), where red and green \nregion represent two kinds of spin components, with iso -surface as 0.01 e/ Å3. d. \nSchematic re presentation of the evolution of electronic states based on ligand field \ntheory, where red, green and blue energy levels denote those electronic states that are \nmainly occupied by Cr, Se and Br atoms respectively.  \n \nTo further  explore magnetic couplings, we firstly presented the electronic structure \nof CrSeBr in Figure 2a and 2b based on DFT+U scheme, as a typical representative. \nMore precise HSE06 hybrid functional scheme is also performed, and band structures \nare presented in  Figure S6. For both simulation method s, the band structure exhibits the \nsimilar semiconducting nature with indirect  moderate  band gap  as 1.46 eV  (DFT+U) \nand 2.43 eV (HSE06) . Band structures for other monolayers  are further shown in Figure \nS6, sharing the similar features . Atomic differential charge density along with the spin \ndensity is then plotted in Figure 2c . And it shows that,  charge deplet es around Cr atoms \nand accumulates  in the region of Cr-Br/Se bonds , while  Cr atoms are still responsible \nfor the large spin polarization . In combination with the projected band structures \n(Figure 2a and 2b)  and ligand field theory, the splitting and rearrangement of electronic \nstates  can be  clarified and illustrated as Figure 2b. Generally, in CrXY (X = S/Se/Te, Y \n= Cl/Br/I) compounds , the valance state for Cr is +3, indicating its d2sp3 hybridization \nfeature with the formation of 6 bonding and anti -bonding pairs, which is a typical case \nfor the octahedral ligand field. Therefore, the six hybridized bonding states, \ncorresponding to a1g, twofold eg and threefold t1u orbits with different symmetries, are \nfilled by 12 electrons: 9 from the p orbits of Se/Br atom and 3 from s/dx2-y2/dxy orbits of \nCr atom. The remained non -bonding t2g orbits, composed of dxz/dyz/dz2, are half filled by  12 / 30 \n three left electrons with spin -up component according to Hund’s rule, representing a \nstable electronic configuration. Such bonding states can be demonstrat ed by the Crystal \nOccupation Hamilton Population (COH P) analysis. As shown in the right panel of \nFigure 2a and 2b, the six lower lying bands, composed of a1g, twofold eg and threefold \nt1u, correspond to the COHP bonding peaks for both Cr -Se and Cr -Br bonds . While eg* \norbits above Fermi level exhibit strong anti -bonding feature , being consistent with the \nformer analysis. It is worth to note that, for threefold nonbonding t2g states right below \nFermi level, moderate anti -bonding peaks will appear. We can identify such anti -\nbonding nature in t2g states as the consequence of super -exchange process in the \nfollowing text. Next, after taking the exchange field into account, the spin -up and spin -\ndown states will split. As shown in Figure 2b, t2g↓ states are shifted far above Fermi \nlevel, while occupied t2g↑ states sit right below empty  eg*↑ states , further being followed \nby t1u states mainly composed of p orbits . \n \nC. Super -exchange mechanism and evolution of exchange integrals  \nBased on electronic structures, magnetic exchange mechanism  can be elaborated . \nFor semiconducting ferromagnetic CrSeBr series, there are two major kinds of \nexchange couplings : 1) direct exchange between magnetic cations; 2) super -exchange \n(SE) via p orbits of ligands. Firstly, direct exchange is originated from the overlapping \namon g two cations’ non -orthogonal states, thus being AFM and sensitive to  \ndistance s.[42] Since magnetic cations are separated by ligands, their interaction is rather \nweek for 3d elements. Only heavy transition metals need to further consider their \nmetallic interactions.[43] Next, for the second super -exchange process, it is responsible \nfor the strong ferromagnetic coupling s. In these Janus monolayers, geometry allows \nCr1-Se-Cr2 and Cr 1-Br-Cr2 bind with each other at a right angle, and the re are three  13 / 30 \n kinds of super -exchange scenarios based on different involved orbital symmetries , as \nshown in Figure 3 a. The first mechanism (SE1) can be expressed as t2g-px/py-t2g, which \nmeans that, the t2g states in Cr 1 and Cr 2 will form π bonds with px and py states in the \nsame ligand. Such π bond is described as the partial covalent bond by Goodenough,[13] \nwhich will allow spin -down electrons in the ligand to hop into t2g orbits. While spin -up \nelectrons left in the px/py state can ferromagnetically exchange with each other based on \nHund’s rule for such onsite orthogonal orbits. Second super -exchange process (SE2) \ncan be presented as  eg*-px/py-eg*, where electron hopping happens in eg*-px/y via partial \ncovalent σ bond, and ferromagnetic order is further maintained by the onsite orthogonal \npx and py exchange. Two former super -exchange process can be quantitatively expressed \nas:[38]  \n𝐽𝑡2𝑔𝑒𝑔⁄−𝑡2𝑔𝑒𝑔⁄𝑆𝐸12⁄ ~−𝑡𝑝𝑑𝑚2𝑡𝑝𝑑𝑚′2𝐽𝐻𝑝\n∆2(2∆+𝑈𝑝𝑝)2 \nWhere 𝑡𝑝𝑑𝑚 is the π type hopping integral  (𝑡𝑝𝑑𝜋) and σ type hopping integral ( 𝑡𝑝𝑑𝜎) in \nSE1 and SE2, respectively. 𝐽𝐻𝑝 is the Hund’s couplings in ligands, while ∆ and 𝑈𝑝𝑝 are \nthe energy  interval  between involved d and p orbits  and the onsite Coulomb interaction \nin p orbits. Finally, f or the third mechanism that happens  among t2g and empty eg* via a \nsingle p orbit, it can be described as: [38]   \n𝐽𝑡2𝑔−𝑒𝑔𝑆𝐸 ~−𝑡𝑝𝑑𝜋2𝑡𝑝𝑑𝜎2\n∆2(𝐽𝐻𝑇𝑀\n(2𝛥+𝑈𝑝𝑝)2+𝐽𝐻𝑇𝑀\n𝑈𝑑𝑑2) \nWhere 𝐽𝐻𝑇𝑀and 𝑈𝑑𝑑 are Hund’s couplings and onsite Coulomb interaction in magnetic \ncations. Since  𝐽𝐻𝑇𝑀 is much larger than 𝐽𝐻𝑝, the third ferromagnetic interaction plays the \ndominant  role. Furthermo re, such scenario can be described as t2g-p-eg*, where spin -up \nelectrons can form σ bond with eg* and spin -down electrons will hop into t2g orbits via \nπ bond simultaneously. And the whole interaction process can only be allow ed for the  14 / 30 \n ferromagnetic spin configuration. To conclude, for super -exchange in CrXY \ncompounds, there are  two basic hopping process denoted as t2g-p and eg*-p, and they \nare further  connected by either two onsite orthogonal p orbits (SE1 and SE2) or one \nsingle p orbit  (SE3), further forming  the complete super -exchange scenario.  \n \n \nFIG. 3. a. Sch ematic represent of three kinds of super -exchange (SE) mechanisms as \nSE1, SE2 and SE3. b. Partial density of states (PDOS) for CrSeBr and the e volution of \nintegrated eg*↑ states. c. The evolution of -ICOHP, magnetic moment on ligands and  the \nenergy interval ∆ between involved d and p orbits (for spin -up channel)  with regard to \nligands .  \n \n 15 / 30 \n After identifying three basic super -exchange scenarios , the evolution of the \nnearest exchange integral s with regard to ligand s can thus be clarified . As reflected \nfrom the above quantitative expressions, there  are two major factors that determine the \nstrength of super -exchange  couplings : d-p hopping integral (𝑡𝑝𝑑𝜋 for t2g-p hopping and \n𝑡𝑝𝑑𝜎 for eg*-p hopping) and the energy  interval  ∆ between involved d and p orbits . \nFirstly, f or the d-p hopping integral,  it can be reflected and described by three aspects: \nintegrated area for unocc upied  eg*↑ and t2g↓eg*↓ in the partial density of states (PDOS), \nintegrated COHP (ICOHP) for the  d-p interaction, and the spin polarization on ligands. \nWe firstly presented p artial density of states for CrSeBr in the upper panel of Figure 3 b, \nwhere occupied t2g↑ and empty eg*↑ and t2g↓eg*↓ are denoted as the  shaded area. Varying \nligands from S to Se and Te, it is found that integrated area s for unoccupied  eg*↑ and \nt2g↓eg*↓ states are decreasing (Figure 3b and Figure S7 ). And this trend suggest s that, for \nheavier ligands, electrons can hop  from p orbits  into unoccupied  d states more \neffectively and further lead to  the partially occupied eg*↑ and t2g↓eg*↓ states, thus \ndemonstrating the benefitted hopping integral . Furthermore, the hopping process will \nfurther introduce anti -bonding characteristics which originally belong to the eg*↑ and \nt2g↓eg*↓ states, also illustrating the observed anti -bonding peaks right below Fermi level \nas in the previous COHP analysis. Secondly, with the enhanced electron hopping, it can \nbe understood that electrons can be  more shared by  transition metal and ligand with the \nhigher covalency , rather than electrostatic interaction between magnetic cation and \nligand anion as oxygen in conventional ionic compounds.  Therefore, we quantitively \nevaluated the strength of covalency of d-p bonds by the integrated COHP (ICOHP) \nvalue. As shown in Figure 3 c, bond strength gradually decreases from ionic to covalent \nfeature, further being accompanied by the increasing J 1 at the same time.  Thus, when \nevolving from S to Se and Te, the decreased electronegativity of ligand can result in  16 / 30 \n such enhanced covalency. As for the third aspect, due to the more effective σ type \nhopping  between eg*↑ and p↑ orbits than the π type hopping  between t2g↓ and p↓, the \nspin-up electrons left on p orbits  would be  lesser than the spin-down electrons, thus \ninducing spin polarization on ligands with the negative magnetic moment. As presented \nin the upper panel of Figure 3c, varying ligands from S to Se and Te, the magnetic \nmoment on ligands is magnified , further reflecting the promoted electron hopping  \nprocess . On the other aspect, the energy  interval  between involved d and p orbits  are \nalso gradually  reduc ed as shown in Figure 3c  (for spin up) and Figure S8 (for spin \ndown) , which originate from the weakened splitting field of ligands with lower \nelectronegativity. Therefore, bonding and anti -bonding states as eg and eg* are with a \ncloser energy interval, and such reduced energy  gaps are expected to benefit  the super -\nexchange process. Huang et al.  regulated this point from a theoretical view and further \nachieved the reduced gaps via doping heavy transition metals rather t han controlling \nligands.[39]  \nOn the other hand, the large J 2 in telluride  compounds  can also be clarified by \nmodeling the interacting scenario as a “cation -anion -anion -cation” system,[38] which \ncan form right angle with each other because of the Janus monolayer geometry. There \nexist two kinds of interaction scenarios, which both involves the interaction between  \ntwo adjacent ligands and can be  denoted as super -super -exchange (SSE). As illustrated \nin Figure 4a  and 4b , we further classify them into SSE1 and SSE2 , which corresponds \nto t2g-p1-p2-t2g and eg*-p1-p2-eg* interaction. Similar to the former nearest exchange \ncouplings,  t2g/eg*-p hopping always exists, and is  further connected by  the formation of \nbonds between off-site p orbits , rather than the Hund’s couplings in  onsite  p orbits . Due \nto the orbital symmetry matching, SSE1 and SSE2 in Figure 4a and 4b can result in the \nσ and π bond s among adjacent p orbits respectively . Such p-p interaction  favors the   17 / 30 \n opposite spin -polarization direction on ligands and further produces  the AFM coupling \nbetween t2g/eg* orbits in the second nearest Cr pairs. Therefore, both mechanisms for J 2 \nall produce AFM couplings and generally exist in the Janus monolayer, and its strength \nis largely  determined by the offsite p-p bonds . In our CrXY system, it is found that, \nvarying from S to Se  and Te , J2 slightly decreases and reaches an anomalous large \nnegative value in telluride system. It can be understood that , the metallic property of \ntelluride ligand possesses  the delocalized and longer interaction ability of p electrons, \nand thus the exchange among two anions can be  significantly amplified. To demonstrate \nsuch mechanism, we have presented spatial charge distribution for the two -fold eg* \nstates at Γ point (Figure 4c), where the significant p orbit components can be observed \ndue to the d-p hybridization . Compared to selenium, telluride compounds are with a \nmore delocalized feature and can be more benefi cial for  the p-p interaction. \nFurthermore, integrated COHP for X -X and Y -Y bonds (X = Se/Te, and Y = Cl/Br) are  \nalso presented in Figure 4d. And such bond ing strength among adjacent ligands can be \nused as a quantitative  index to describe the offsite p-p bonds. Results  show that, for \nligands with tight electronic shell, the interaction among adjacent halides as Cl and Br \nis negligible. Therefore, Se -Se and T e-Te dominate the p-p couplings in SSE, and \ncompared to selenium compounds, Te-Te possess es a larger -ICOHP value and  can \nstrengthe n the  anti-ferromagnetic J 2 more effectively .  18 / 30 \n  \nFIG. 4. a. Schematic representation of two kinds of super -super -exchange (SSE) \nmechanisms as a. SSE1 and b. SSE2. c. Spatial charge distribution for double \ndegenerate eg* orbits at Γ point, for CrSeBr and CrTeCl respectively, where iso -surface \nis set to 0.01 e/ Å3. d. Evolution of -ICOHP for X -X and Y -Y bonds (X = Se/Te, Y = \nCl/Br).  \n \nNext, we would like to give some further suggestions on achieving robust \nferromagnetism in two -dimensional monolayer. Firstly, under the premise of t2g3eg*0 \nelectronic configuration, three kinds of super -exchange process will lead to \nferromagnetic order, which is the general feature of this electronic structure. \nFurthermore, experimental achieved ferromagnetic monolayer as CrI 3 and CrGeTe 3 are \n 19 / 30 \n both with the t2g3eg*0 electronic configuration. And we can further modulate super -\nexchange couplings in t2g3eg*0 via substituting ligands  with lower electronegativity , to \nstrengthen the d-p covalency  and thus benefit the electron hopping process . However, \nheavier ligands , especially for Te,  will also bring a side effect as the delocalized p \nelectrons and further  enhance  the second -nearest interaction. To verify the credence of \nour theory, we examine other t wo widely explored semiconducting ferro magnetic \nsystems, as CrY 3 (Y = Cl/Br/I) and Cr2Ge2X6 (X = S/Se/Te), which are both with the \nt2g3eg*0 electronic configuration.[5][6][7] The evolution of J 1 against ligands  is plotted in \nFigure S 9 and S10 , which demonstrates that, the nearest ferromagnetic exchange \nintegrals can both be  strengthened by ligands with the lower electronegativity , being \nthe same as the CrXY family. However, d ue to the different geometries, the second -\nnearest exchange integral will differ (ferromagnetic J2 for CrY 3 and anti -ferromagnetic \nJ2 for Cr 2Ge2X6, being consistent with the previous literature[41][44]), but their \nmagnitudes can both be magnified with the ligand evolution towards heavier element s \n(Figure S 9 and S10 ), again verifying our theory.  \n \nIV. CONCLUSION  \nIn conclusion, we have theoretically explored the role of ligands played in \nmodulating super -exchange process in semiconducting ferromagnets. Based on the \nmetallic phase  of CrX 2 (X = S/Se/Te), a layer of original ligands is replaced by halides, \nwhich results in a series of Janus  monolayer as CrSeBr, CrSeCl and CrTeBr, with robust \nferromagnetic couplings. Three kinds of super -exchange paths are revealed , being \nferromagnetic due to the special t2g3eg*0 electronic configuration. Detailed analysis on  \nexchange integral s shows that, the nearest ferromagnetic J1 gradually increases with the \nligands with the lower electronegativity . And we demonstrated that , heavier ligands can  20 / 30 \n strengthen  the electron hopping integral and reduce the energy interval  between d and \np orbits  at the same time , further benefiting the super -exchange process.  As for the \nsecond nearest J 2, its magnitude can also be enlarged due to the more delocalized p \nelectrons.  Therefore, we propose a fundamental understanding on the modulat ion of  \nferromagnetic couplings in low -dimensional semiconductors via ligands, serving as a \ntheoretical guidance on the further engineering of magnetic materials.  \n \nACKNOWLEDGEMENTS  \nJ. X. thank s Feng Zhi Ning and Ning Kang for generous support  and helpful discussions . \nQ.F.Z. was supported by National Key Research and Development Program of China \n(No. 2017YFB0702100) and National Natural Science Foundation of China \n(11404017).  D.L. acknowledges support by the European Regional Development Fund \nin the IT4Innovations national supercomputing center -Path to Exascale project, No.                                                                                                                                                                               \nCZ.02.1.01/0.0/0.0/16_013/0001791 within the Operational Programme Research,                                                                                                                  \nDevelopment and Education and by the Ministry of Education by Czech Science \nFoundation project No. 17 -27790S, and grant No. 8J18DE004 of Ministry of Education, \nYouth, and Sport of the Czech Republic.         \n \nREFERENCE  \n[1] H. C. Neto, F. Guinea,  N. M. R.  Peres, K. S.  Novoselov, and A. K.  Geim,  Rev. \nMod. Phys.  81, 109  (2009) . \n[2] M. Li, W. Cui, J. Yu, Z. Dai, Z. Wang,  F. Katmis, W. Guo, and J.  Moodera,  Phys. \nRev. B  91, 014427  (2015) . \n[3] M. 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R. China.  \n2. IT4Innovations & Nanotechnology Centre, VSB -Technical University of Ostrava, \n17.listopadu 2172/15, Ostrava CZ -70800, Czech Republic.  \n3. Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, P. \nR. China  \n*Corresponding authors: qianfan@buaa.edu.cn .  \n \n \n \n \n \n \n \n \n \n \n  25 / 30 \n Section SI. Structure and Stability  \nStarted from the 1 -T CrS 2, CrSe 2 and CrTe 2 monolayer phase, we have replaced \none layer of S/Se/Te by Cl/Br/I and further compare the energies of other common 2D \nstructures, to select out components with the 1 -T phase as the most stable structure. \nHere, three kinds of 2D structures are co nsidered, which are 1 -T phase, 2 -H phase and \nthe monolayer structure of the bulk phase CrOCl, as shown in Figure S1. And the most \nstable structure for the Cr based system with different ligand atoms is summarized in \nTable S1.  \n \n \nFigure S1. The geometric st ructure of a. 1 -T phase, b. 2 -H phase and c. CrOCl  phase (the monolayer \nunit of bulk phase CrOCl), respectively. Here, M denotes metallic atoms while X and Y represent \nligand atoms.  \n \nTable S1. The most stable structure for CrXY composites, where X = S/Se/T e and Y = Cl/Br/I.  \nCrXY  Phase  CrXY  Phase  CrXY  Phase  \nCrSCl  CrOCl  CrSeCl  1-T  CrTeCl  1-T  \nCrSBr  CrOCl  CrSeBr  1-T  CrTeBr  1-T \nCrSI  CrOCl  CrSeI  CrOCl  CrTeI  CrOCl  \n \n 26 / 30 \n  \nFigure S2. The geometric structure and the corresponding phonon spectra of 1 -T a. CrSeCl, b. \nCrTeBr, c. CrTeCl respectively, where side view, top view and phonon spectra are presented from \ntop to bottom.  \n \n \nFigure S3. Atomic configurations for a. CrSeCl, b. CrSeBr, c. CrTeBr and d. CrTeCl after 20ps \nAIMD simulation at 300K . \n \n \n \n \n \n \n \n \n 27 / 30 \n Section SII. Magnetic Properties  \n \nFigure S4. Three kinds of magnetic configurations for 1 -T phase CrXY  (where X and Y refer to \ndifferent ligands): a. ferromagnetic order; b. c. anti -ferromagnetic orders.  \n \nFigure S5. Schematic representation of the applied uniaxial strain.   \n \nTable S2.  Magnetic anisotropic energy (MAE) for CrSeCl, CrSeBr and CrTeBr, when ±  1.5% \nuniaxial strain is applied in the x direction (shown in Figure S5), and MAE is defined as E x – Ey. \n MAE CrSeCl /μeV  MAE CrSeBr /μeV  MAE CrTeBr /μeV  \n1.5%  24.04  19.84  156.29  \n-1.5%  -13.58  -25.43  -206.96  \n \nTable S3. Exchange integrals for CrX Cl ( X = S/Se/Te)  \n CrSCl  CrSeCl  CrTeCl  \nJ1 / meV  25.14  28.23  33.49  \nJ2 / meV  -1.88 -5.51 -17.83  \n \n 28 / 30 \n Table S 4. Exchange integrals for CrXBr ( X = S/Se/Te)  \n CrSBr  CrSeBr  CrTeBr  \nJ1 / meV  25.61  29.48  34.93  \nJ2 / meV  -1.54 -4.53 -14.62  \n \nTable S 5. Exchange integrals for CrXI ( X = S/Se/Te)  \n CrSI  CrSeI  CrTeI  \nJ1 / meV  23.42  29.41  36.27  \nJ2 / meV  -1.32 -3.40 -10.40  \n \nFigure S6. Calculated band structure based on HSE06 (upper panel) and PBE+U (lower panel) \nexchange -correlation scheme for a. CrSeCl, b. CrSeBr and c. CrTeBr, respectively  \n \n 29 / 30 \n  \nFigure S7 . The evolution of integ rated t2g↓eg*↓ states with regard to ligands.  \n \nFigure  S8. The evolution of the energy gap ∆ between involved d and p orbits.  \n \nFigure S 9. The e volution of the nearest and the second nearest exchange integral for CrY 3 (Y = \nCl/Br/I) . \n 30 / 30 \n  \nFigure S9.  The evolution of the nearest and the second nearest exchange integral for  Cr2Ge2X6 (X \n= S/Se/Te) . \n \n"    },    {        "title": "2104.11878v2.Theory_of_magnetic_inertial_dynamics_in_two_sublattice_ferromagnets.pdf",        "content": "Signatures of magnetic inertial dynamics in\ntwo-sublattice ferromagnets\nRitwik Mondal1;2\n1Department of Spintronics and Nanoelectronics, Institute of Physics ASCR, v.v.i.,\nCukrovarnická 10, Prague 6, 162 53, Czech Republic\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120\nUppsala, Sweden\nE-mail: mondal@fzu.cz\nAbstract. The magnetic inertial dynamics have been investigated for one sublattice\nferromagnets. Here, we develop the magnetization dynamics in two-sublattice\nferromagnets including the intra- and inter-sublattice inertial dynamics. First, we\nderive the magnetic susceptibility of such a ferromagnet. Next, by ﬁnding the poles of\nthe susceptibility, we calculate the precession and nutation resonance frequencies. Our\nresults suggest that while the resonance frequencies show decreasing behavior with\nthe increasing intra-sublattice relaxation time, the eﬀect of inter-sublattice inertial\ndynamics is contrasting.\n1. Introduction\nUltrafast manipulation of electrons’ spin remains at the heart of future generation spin-\nbased memory technology [1–3]. It has been observed that a fs laser pulse is capable of\ndemagnetizing a ferromagnetic material [4–6]. On the other hand, using these ultrashort\npulses, magnetic switching has been reported in ferrimagnetic [7–9] and ferromagnetic\nmaterials [10, 11]. These observations have been explained through the spin dynamics\nwithin Landau-Lifshitz-Gilbert (LLG) equation of motion [12–15].\nThe phenomenological LLG spin dynamics consists of spin precession and a\ntransverse damping [16–18]. Such an equation of motion has been derived from a\nrelativistic Dirac theory, where the transverse damping is found to originate from spin-\norbit coupling [19–22]. However, at ultrashort timescales, the traditional LLG equation\nneeds to be supplemented by several other spin torque terms [23]. Especially, at the\nultrafast timescales, the magnetic inertia becomes particularly relevant [24]. The eﬀect\nof magnetic inertia has been incorporated within extended LLG dynamics as a torque\ndue to the second-order time derivative of the magnetization M(r;t). The inertial LLG\n(ILLG) equation of motion reads [25–27]\n@M\n@t=M\u0002\u0014\n\u0000\rH+\u000b\nM0@M\n@t+\u0011\nM0@2M\n@t2\u0015\n; (1)arXiv:2104.11878v2  [cond-mat.other]  1 Sep 2021Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 2\nwhereM0andHdeﬁne the ground state magnetization and an eﬀective ﬁeld,\nrespectively. The ﬁrst and second terms in Eq. (1) represent the traditional LLG\nequation [18]. The inertial spin dynamics in the last term of Eq. (1) gives rise to\nthe spin nutation [28, 29]. The ILLG equation signiﬁes the fact that the dynamics of\na magnetic moment shows precession with nutation at ultrafast timescales, followed by\ntransverse damping [24]. The ILLG equation has schematically been depicted in Fig. 1.\nA simple dimension analysis shows that the transverse damping is characterized by\na dimensionless parameter \u000b, and the inertial dynamics are strengthened by inertial\nrelaxation time \u0011. The ILLG dynamics have been derived within the relativistic Dirac\nM(r, t)H\nPrecession\nNutationDamping\nFigure 1: Schematic depiction of ILLG equation of motion.\nframework as well, where it shows that the Gilbert damping \u000band inertial relaxation\ntime\u0011are tensors [30]. In particular, the relativistic theory derives that the Gilbert\ndamping dynamics is associated with the imaginary part of the susceptibility, while the\ninertial dynamics is given by the real part [31]. Such ﬁndings are found to be consistent\nwith a linear response theory of ferromagnet [32]. The inertial dynamics have also been\nderived within classical mechanics of a current loop [33]. Eq. (1) has been applied to\na single sublattice ferromagnet beyond ferromagnetic resonance (FMR), observing an\nadditional peak due to nutation resonance [34–36]. While the FMR peak appears at the\nGHz regime, the nutation resonance peak appears at the THz regime [37]. The ILLG\nequation has also been applied to antiferromagnets and ferrimagnets, and it has been\npredicted that the spin nutation should be better detected in antiferromagnets as it is\nexchange enhanced [38].\nRecently, the spin nutation resonance has been observed for ferromagnets in the\nexperiment [39]. Indeed, the nutation resonance peak has been seen at around 0.5\nTHz. Note that the experiment was performed in two-sublattice ferromagnets namely\nCoFeB and NiFe. For two-sublattice ferromagnet, the inter-sublattice exchange energies\nbecome important. Here, we describe the inertial eﬀects in a two-sublattice ferromagnet\ncoupled by the Heisenberg exchange interaction. We follow the similar procedureSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 3\nof Ref. [38] and derive the magnetic susceptibility. We not only consider the intra-\nsublattice inertial dynamics, but alsothe inter-sublattice dynamics. Our results suggest\nthat there are two precession resonance peaks: one at GHz regime and another at THz\nregime. Similarly, two nutation peaks can also be observed, both are at the THz regime.\nBy calculating the precession and nutation resonance frequencies, we observe that the\nresonance frequencies decrease with increasing intra-sublattice relaxation time, however,\nthe scenario is diﬀerent for inter-sublattice inertial dynamics.\n2. Theory of intra- and inter-sublattice inertial dynamics in two-sublattice\nferromagnets\nThe inertial dynamics for antiferromagnets have been introduced in Ref. [38]. For\ntwo-sublattice magnetic systems having magnetization MAandMB, forAandB\nrepresenting the two-sublattice, the ILLG equations of motion can be recast as\n@MA\n@t=\u0000\rA(MA\u0002HA) +\u000bAA\nMA0\u0012\nMA\u0002@MA\n@t\u0013\n+\u000bAB\nMB0\u0012\nMA\u0002@MB\n@t\u0013\n+\u0011AA\nMA0\u0012\nMA\u0002@2MA\n@t2\u0013\n+\u0011AB\nMB0\u0012\nMA\u0002@2MB\n@t2\u0013\n(2)\n@MB\n@t=\u0000\rB(MB\u0002HB) +\u000bBB\nMB0\u0012\nMB\u0002@MB\n@t\u0013\n+\u000bBA\nMA0\u0012\nMB\u0002@MA\n@t\u0013\n+\u0011BB\nMB0\u0012\nMB\u0002@2MB\n@t2\u0013\n+\u0011BA\nMA0\u0012\nMB\u0002@2MA\n@t2\u0013\n(3)\nIn each ILLG dynamics, the ﬁrst term represents the spin precession around an eﬀective\nﬁeldHA=B. The intra- and inter-sublattice Gilbert damping dynamics have been\ndenoted by the second and third terms, respectively. Similarly, the last two terms deﬁne\ninertial dynamics. While the intra-sublattice Gilbert and inertial dynamics have been\nweighed by \u000bAA=BBand\u0011AA=BB, the same for inter-sublattice dynamics are denoted by\n\u000bAB=BAand\u0011AB=BA. From a simple dimension analysis, it is clear to show that the\nGilbert damping parameters \u000bare dimensionless, in contrast, the inertial relaxation\ntimes\u0011have a dimension of time [26, 30]. It is worth mentioning that the Gilbert\ndamping\u000bhas been calculated for several materials within ab initio frameworks [32, 40–\n52], while there are also proposals to calculate the inertial relaxation time within\nextended breathing Fermi surface model [53–55]. These ILLG equations have been\ncontemplatedtoforecastthesignaturesofinertialdynamicsincollinearantiferromagnets\nand ferrimagnets [38].\nWe consider that the two-sublattice ferromagnet is aligned collinear at the ground\nstate such that MA=MA0^zandMB=MB0^z. The ferromagnetic system is under\nthe application of an external Zeeman ﬁeld H0=H0^z. Then, the free energy of the\nconsidered two-sublattice system can be considered as the sum of Zeeman, anisotropy,Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 4\nand exchange energies as\nF(MA;MB) =\u0000H0(MAz+MBz)\u0000KA\nM2\nA0M2\nAz\u0000KB\nM2\nB0M2\nBz\u0000J\nMA0MB0MA\u0001MB;\n(4)\nwhereKAandKBare anisotropy energies and Jis the isotropic Heisenberg exchange\nwithJ > 0for ferromagnetic coupling. To calculate the linear response properties of\nthe system, we consider that the small deviations of magnetization mA(t)andmB(t)\nwith respect to the ground state are induced by the transverse external ﬁeld hA(t)and\nhB(t). We calculate the eﬀective ﬁeld in the ILLG equation as the derivative of free\nenergy in Eq. (4) to the corresponding magnetization\nHA=\u0000@F(MA;MB)\n@MA=\u0012\nH0+2KA\nM2\nA0MAz\u0013\n^z+J\nMA0MB0MB\n=1\nMA0(H0MA0+ 2KA+J)^z+J\nMA0MB0mB;(5)\nHB=\u0000@F(MA;MB)\n@MB=\u0012\nH0+2KB\nM2\nB0MBz\u0013\n^z+J\nMA0MB0MA\n=1\nMB0(H0MB0+ 2KB+J)^z+J\nMA0MB0mA:(6)\nWe then expand the magnetization around the ground state in small deviations,\nMA=MA0^z+mA(t)andMB=MB0^z+mB(t). Essentially, with the eﬀective ﬁelds\nin Eqs. (5) and (6) along with the magnetization, the linear response for sublattice A\nprovides\n@mA\n@t=\u0000\rA\nMA0(H0MA0+ 2KA+J) [mAy^x\u0000mAx^y]\u0000\rAJ\nMB0[mBx^y\u0000mBy^x]\n\u0000\rAMA0[hAx^y\u0000hAy^x] +\u000bAA\u0014@mAx\n@t^y\u0000@mAy\n@t^x\u0015\n+\u000bABMA0\nMB0\u0014@mBx\n@t^y\u0000@mBy\n@t^x\u0015\n+\u0011AA\u0014@2mAx\n@t2^y\u0000@2mAy\n@t2^x\u0015\n+\u0011ABMA0\nMB0\u0014@2mBx\n@t2^y\u0000@2mBy\n@t2^x\u0015\n; (7)\nobtaining the dynamics for two components xandyas\n\rAMA0hAx=\rA\nMA0(H0MA0+ 2KA+J)mAx\u0000\rAJ\nMB0mBx+\u000bAA@mAx\n@t+\u000bABMA0\nMB0@mBx\n@t\n\u0000@mAy\n@t+\u0011AA@2mAx\n@t2+\u0011ABMA0\nMB0@2mBx\n@t2; (8)\n\rAMA0hAy=\rA\nMA0(H0MA0+ 2KA+J)mAy\u0000\rAJ\nMB0mBy+\u000bAA@mAy\n@t+\u000bABMA0\nMB0@mBy\n@t\n+@mAx\n@t+\u0011AA@2mAy\n@t2+\u0011ABMA0\nMB0@2mBy\n@t2: (9)\nIn the circular basis deﬁned by mA\u0006=mAx\u0006imAyandhA\u0006=hAx\u0006ihAy, the equationsSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 5\ncan be put together\n\rAMA0hA\u0006=\rA\nMA0(H0MA0+ 2KA+J)mA\u0006\u0000\rAJ\nMB0mB\u0006+\u000bAA@mA\u0006\n@t+\u000bABMA0\nMB0@mB\u0006\n@t\n\u0006i@mA\u0007\n@t+\u0011AA@2mA\u0006\n@t2+\u0011ABMA0\nMB0@2mB\u0006\n@t2: (10)\nSimilarly, one can calculate the linear response of the sublattice B in the circular basis\ndeﬁned bymB\u0006=mBx\u0006imByandhB\u0006=hBx\u0006ihByas\n\rBMB0hB\u0006=\rB\nMB0(H0MB0+ 2KB+J)mB\u0006\u0000\rBJ\nMA0mA\u0006+\u000bBB@mB\u0006\n@t+\u000bBAMB0\nMA0@mA\u0006\n@t\n\u0006i@mB\u0007\n@t+\u0011BB@2mB\u0006\n@t2+\u0011BAMB0\nMA0@2mA\u0006\n@t2: (11)\nWe deﬁne the response functions mA\u0006;mB\u0006;hA\u0006;hB\u0006/e\u0006i!tand \nA=\n\rA\nMA0(H0MA0+ 2KA+J)and \nB=\rB\nMB0(H0MB0+ 2KB+J). To simplify the\nexpressions, we introduce the following: \u0000AA=\rAMA0,\u0000BB=\rBMB0,\u0000AB=\rAMB0\nand\u0000BA=\rBMA0such that \u0000AA\u0000BB= \u0000AB\u0000BA. The linear response Eqs. (10) and\n(11) can be written in a matrix formalism\n\u0010hA\u0006hB\u0006\u0011\n=0\nBB@1\n\u0000AA(\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!)\u00001\n\u0000AB\u0012\rAJ\nMA0\u0007i!\u000bAB+!2\u0011AB\u0013\n\u00001\n\u0000BA\u0012\rBJ\nMB0\u0007i!\u000bBA+!2\u0011BA\u00131\n\u0000BB(\nB\u0006i!\u000bBB\u0000!2\u0011BB\u0000!)1\nCCA\u0010mA\u0006mB\u0006\u0011\n:\n(12)\nFor ﬁnding the susceptibility, we recall m\u0006=\u001f\u0006\u0001h\u0006such that the susceptibility matrix\nderives as\n\u001fAB\n\u0006=1\nD\u00060\nBB@1\n\u0000BB(\nB\u0006i!\u000bBB\u0000!2\u0011BB\u0000!)1\n\u0000BA\u0012\rBJ\nMB0\u0007i!\u000bBA+!2\u0011BA\u0013\n1\n\u0000AB\u0012\rAJ\nMA0\u0007i!\u000bAB+!2\u0011AB\u00131\n\u0000AA(\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!)1\nCCA;\n(13)\nwhere the determinant is expressed as\nD\u0006=1\n\u0000AA\u0000BB\u0000\n\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!\u0001\u0000\n\nB\u0006i!\u000bBB\u0000!2\u0011BB\u0000!\u0001\n\u00001\n\u0000AB\u0000BA\u0012\rAJ\nMA0\u0007i!\u000bAB+!2\u0011AB\u0013\u0012\rBJ\nMB0\u0007i!\u000bBA+!2\u0011BA\u0013\n:(14)\nNote that the intra-sublattice dynamical parameters enter in the diagonal elements of\nthe susceptibility matrix, however, the inter-sublattice dynamics are reﬂected in the oﬀ-\ndiagonal elements. Such a susceptibility matrix has been obtained with intra- and inter-\nsublattice Gilbert damping dynamics for antiferromagnets [56]. To ﬁnd the resonanceSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 6\nfrequencies, one has to solve the equation setting D\u0006= 0. Therefore, a fourth-order\nequation in frequency is obtained\nA\u0006!4+B\u0006!3+C\u0006!2+D\u0006!+E\u0006= 0; (15)\nwith the following coeﬃcients\nA\u0006=\u0011AA\u0011BB\u0000\u0011AB\u0011BA; (16)\nB\u0006= (\u0011AA+\u0011BB)\u0007i (\u000bAA\u0011BB+\u000bBB\u0011AA)\u0006i (\u000bAB\u0011BA+\u000bBA\u0011AB); (17)\nC\u0006= 1\u0007i (\u000bAA+\u000bBB)\u0000(\nA\u0011BB+ \nB\u0011AA)\u0000\u000bAA\u000bBB\n\u0000\u0012\rA\nMA0\u0011BA+\rB\nMB0\u0011AB\u0013\nJ\u0000\u000bAB\u000bBA; (18)\nD\u0006=\u0000(\nA+ \nB)\u0006i (\nA\u000bBB+ \nB\u000bAA)\u0006i\u0012\rA\nMA0\u000bBA+\rB\nMB0\u000bAB\u0013\nJ;(19)\nE\u0006= \nA\nB\u0000\rA\rB\nMA0MB0J2: (20)\nTheanalyticalsolutionoftheabove-mentionedequationisverycumbersome. Therefore,\nwe adopt the numerical techniques for solving Eq. (15). The solution of the above\nequation results in four frequencies, two of them correspond to the precession resonance\n(!p)ofeachsublatticeandtheothertwobelongtothenutationresonance( !n). Thereal\nand imaginary parts of the resonance frequency are denoted by ReandIm, respectively.\nFor example, the precession resonance frequencies are !p=Re(!p) + i Im(!p), while\nthe nutation resonance frequencies are !n=Re(!n) + iIm(!n). Comparing Eq. (15), a\nsimilar equation has been obtained for antiferromagnets and ferrimagnets [38], however,\nwithout the inter-sublattice inertial dynamics. We mention that the inter-sublattice\nGilbert damping dynamics have extensively been discussed [56, 57]. Therefore, we will\nnot consider in the following discussions. In particular, we allow \u000bAB=\u000bBA= 0, and\ncalculate the inertial eﬀects on precession and nutation resonances.\n3. Numerical results\nTo calculate the resonace frequencies, we numerically solve the Eq. (15) for two-\nsublattice ferromagnets having same magnetic moments in each sublattice i.e., MA0=\nMB0. We use the following parameters: \rA=\rB= 1:76\u00021011T\u00001-s\u00001,J= 10\u000021J,\nKA=KB= 10\u000023J,\u000bAA=\u000bBB=\u000b= 0:05,\u000bAB=\u000bBA= 0. Theconsideredexchange\nand anisotropy energies have similar order of magnitude as typical ferromagnets e.g., Fe\n[58]. The chosen Gilbert damping \u000b= 0:05is within the ab initio reported values [51].\nFor inertial relaxation times, even though, the ab initio calculation suggests about fs\ntimescales for transition metals [55], the recent experiment predicts it to be a higher\nvalue up to several hundreds of fs [39]. Therefore, in what follows, we have considered\nthe inertial relaxation times ranging from fs to ps.Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 7\n10−1100101102103\nη(fs)10−1100101102103ωFM\n±/2π(THz)\n1/η(a)\nRe/parenleftbig\nωu\np+/parenrightbig\nRe/parenleftbig\nωl\np+/parenrightbig\nRe/parenleftbig\nωu\nn−/parenrightbig\nRe/parenleftbig\nωl\nn−/parenrightbig\n10−1100101102103\nη(fs)0.000.010.020.030.040.05Im(ω±)/Re(ω±)|FM\n(b)\nIm/parenleftbig\nωu\np+/parenrightbig\n/Re/parenleftbig\nωu\np+/parenrightbig\nIm/parenleftbig\nωl\np+/parenrightbig\n/Re/parenleftbig\nωl\np+/parenrightbig\nIm/parenleftbig\nωu\nn−/parenrightbig\n/Re/parenleftbig\nωu\nn−/parenrightbig\nIm/parenleftbig\nωl\nn−/parenrightbig\n/Re/parenleftbig\nωl\nn−/parenrightbig\nFigure 2: The calculated resonance frequencies as a function of intra-sublattice inertial\nrelaxation time for two-sublattice ferromagnets using MA0=MB0= 2\u0016B. (a) The\nprecession and nutation resonance frequencies and (b) the eﬀective Gilbert damping\nhave been plotted.\n3.1. Intra-sublattice inertial dynamics\nTo focus on the intra-sublattice inertial dynamics, we set the inter-sublattice relaxation\ntime to zero i.e., \u0011AB=\u0011BA= 0, keeping the same inertial relaxation time in two-\nsublattice\u0011AA=\u0011BB=\u0011. Withthissetofspeciﬁcations, thecalculatedfrequencieshave\nbeen shown in Fig. 2. One can see that there exist two precession resonance frequencies\n(positive) and the corresponding two nutation resonance frequencies (negative). We\ndenote these two positive precession frequencies as !u\np+and!l\np+, while the two\nnegative nutation frequencies are !u\nn\u0000and!l\nn\u0000. The superscripts “u” and “l” denote\nthe upper and lower frequencies, respectively. These results are in contrast with the\nobservation in antiferromagnets or ferrimagnets, where one positive and one negative\nprecession (and nutation) frequencies are expected [38]. Nevertheless, the quantitative\ncomparison of the calculated frequencies agrees with those of the ferrimagnets, where\nthe upper (THz), and lower (GHz) frequency precession resonances are called an\nexchange and ferromagnetic modes, respectively [38, 59]. Similar to antiferromagnets\nand ferrimagnets [38], the resonance frequencies decrease with the intra-sublattice\ninertial relaxation time in the case of two-sublattice ferromagnets. Especially, the lower\nnutation resonance frequency scales with 1=\u0011, while the upper one shows deviation\nfrom 1=\u0011at higher relaxation times. This deviation from 1=\u0011has been noticed in\ntwo nutation modes for antiferromagnets and ferrimagnets [38]. An interesting feature\nis that the precession and nutation frequencies cross each other at certain inertial\nrelaxation times in ferromagnets. Such crossing was not observed in antiferromagnets\nand ferrimagnets [38]. The crossing happens especially with the upper precession mode\nwith lower nutation mode as seen in Fig. 2(a). However, we note that crossing of theseSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 8\n12345MA0/MB00.51.01.52.02.53.0Re°!up+¢/2º(THz)(c)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps\n12345MA0/MB00.010.020.030.040.05Im°!up+¢/Re°!up+¢(d)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps12345MA0/MB00.030.040.050.06Re°!lp+¢/2º(THz)(a)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps\n12345MA0/MB00.030.040.05Im°!lp+¢/Re°!lp+¢(b)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps\nFigure 3: The calculated precession resonance frequencies as a function of MA0=MB0\nfor two-sublattice ferromagnets, at several intra-sublattice inertial relaxation times.\n(a) The real part of the lower precession resonance frequencies, (b) the eﬀective\ndamping of lower resonance mode, (c) the real part of the upper precession resonance\nfrequencies, (d) the eﬀective damping of upper resonance mode, has been plotted.\ntwo modes have positive and negative frequencies, meaning that the upper precession\nmode (!u\np+) has a positive rotational sense, however, the lower nutation mode ( !l\nn\u0000) has\nthe opposite rotational sense in circular basis.\nThe inertial dynamics aﬀect the eﬀective Gilbert damping in a system. This has\nbeen demonstrated in Fig. 2(b) for two-sublattice ferromagnet by the ratio of imaginary\nand real parts of the calculated frequencies. We have used the same Gilbert damping\nfor both the sublattices \u000b\u00180:05and therefore, the eﬀective damping remains the same\nat smaller inertial relaxation times. However, the eﬀective damping decreases with\nincreased relaxation times, a fact that is consistent with the results of antiferromagnets\n[38]. It is observed that the decrease in eﬀective damping is exactly the same for\nprecession and corresponding nutation modes. Moreover, the upper precession mode\nis inﬂuenced strongly, which has already been observed for ferrimagnets [38].Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 9\nNext, we calculate the inﬂuence of diﬀerent sublattice magnetic moment ( MA06=\nMB0) on inertial dynamics. In particular, we compute the precession resonance\nfrequencies as a function of the ratio of magnetic moments ( MA0=MB0), at several\ninertial relaxation times in Fig. 3. We observe that the resonance frequencies decrease\nwith increasing diﬀerence in the magnetic moments. Such reduction is less visible in\ncase of lower precession frequencies e.g., Fig. 3(a), however, more prominent in upper\nprecession frequencies in Fig. 3(c). However, the diﬀerence of frequencies calculated\nat several relaxation times are similar for MA0=MB0andMA06=MB0. The latter\nsuggests that the inertial dynamics do not get quantitatively inﬂuenced by the same or\ndiﬀerent sublattice magnetic moments. A similar conclusion can also be made from the\ncomputation of eﬀective damping in Figs. 3(b) and 3(d). The eﬀective damping for the\nupper and lower precession modes remains almost constant (with a very small positive\nslope) for a higher ratio of MA0=MB0.\n3.2. Inter-sublattice inertial dynamics\n10−1100101102\nη/prime(fs)10−1100101102103ωFM\n±/2π(THz)\n(a)\nRe/parenleftbig\nωu\np+/parenrightbig\nRe/parenleftbig\nωl\np+/parenrightbig\nRe/parenleftbig\nωu\nn−/parenrightbig\nRe/parenleftbig\nωl\nn−/parenrightbig\n10−1100101102\nη/prime(fs)0.000.010.020.030.040.05Im(ω±)/Re(ω±)|FM\n(b)\nIm/parenleftbig\nωu\np+/parenrightbig\n/Re/parenleftbig\nωu\np+/parenrightbig\nIm/parenleftbig\nωl\np+/parenrightbig\n/Re/parenleftbig\nωl\np+/parenrightbig\nIm/parenleftbig\nωu\nn−/parenrightbig\n/Re/parenleftbig\nωu\nn−/parenrightbig\nIm/parenleftbig\nωl\nn−/parenrightbig\n/Re/parenleftbig\nωl\nn−/parenrightbig\nFigure 4: The calculated resonance frequencies as a function of inter-sublattice inertial\nrelaxation time for two-sublattice ferromagnets using MA0=MB0= 2\u0016B. The\nintra-sublattice inertial relaxation time was kept constant \u0011= 100fs. (a) The\nprecession and nutation resonance frequencies and (b) the eﬀective Gilbert damping\nhave been plotted.\nTo investigate the inter-sublattice inertial dynamics, we set the intra-sublattice\nrelaxation time as \u0011AA=\u0011BB=\u0011= 100fs. Such a relaxation time is lower than the\nexperimental ﬁndings in two-sublattice ferromagnets [39]. In fact, the direct comparison\nof Eq. (2) with the Eq. (2) of Ref. [39] provides \u0011\u0018\u000b\u001c. With the experimental\nﬁndings for CoFeB, \u000b= 0:0044and\u001c= 72ps (see Table 1 in Ref. [39]), we calculate\n\u0011= 316fs. We compute the eﬀect of inter-sublattice inertial dynamics as a function of\n\u0011AB=\u0011BA=\u00110in Fig. 4 considering \u00110<\u0011. As we mentioned earlier, the overlapping ofSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 10\nprecession ( !u\np+) and nutation ( !l\nn\u0000) frequencies at the intra-sublattice relaxation time\n\u0011= 100fs can be seen. We observe that the upper precession resonance frequency ( !u\np+)\nincreases, while the lower one ( !l\np+) decreases very small with inter-sublattice relaxation\ntimes. A similar conclusion can be made for nutation frequencies. This is in contrast to\nthe observation of intra-sublattice inertial dynamics as discussed above. A divergence\nin the upper nutation frequency can be noticed at the limit \u00110!\u0011. Such divergence can\nbe explained through the coeﬃcient Ain Eq. (15). At the limit \u00110!\u0011, the coeﬃcient of\nfourth power in frequency becomes A=\u0011AA\u0011BB\u0000\u0011AB\u0011BA=\u00112\u0000\u001102!0, which brings\nthe fourth-order equation into an eﬀective third-order equation in frequency.\nA similar observation can also be concluded from the calculation of eﬀective\ndamping in Fig. 4(b). Similar to the intra-sublattice inertial dynamics, the eﬀective\ndamping of the precession and corresponding nutation mode behaves exactly the same\nfor the inter-sublattice inertial dynamics. We observe that the damping of upper\nprecession and nutation modes increases with inter-sublattice inertial relaxation time,\nhowever, it is the opposite for lower precession and nutation modes. Therefore, we\nconclude that the eﬀect of intra- and inter-sublattice inertial dynamics are contrasting.\n4. Conclusions\nTo conclude, we have incorporated the intra- and inter-sublattice inertial dynamics\nwithintheLLGequationofmotionandcalculatedtheFMRresonancefortwo-sublattice\nferromagnets. To this end, we ﬁrst derive the magnetic susceptibility that is a tensor.\nTo calculate the resonance frequencies, we ﬁnd the poles of the susceptibility. Without\nthe inertial dynamics, there exist two precession modes in a typical two-sublattice\nferromagnet. The introduction of inertial dynamics shows two nutation resonance\nfrequencies corresponding to the precession modes. We note that these precession\nand nutation resonances can be excited by right and left circularly polarised pulses,\nrespectively, and vice-versa within a circular basis. The precession and nutation\nfrequenciesdecreasewiththeintra-sublatticerelaxationtimeasalsohasbeenseeninthe\ncase of antiferromagnets in previous work [38]. However, at certain relaxation times, the\nprecessionandnutationfrequenciesoverlapwitheachother. Notethattheseoverlapping\nprecessionandnutationfrequencieshaveoppositerotationalsenseincircularbasis, thus,\nthey can be neatly realised in the experiments. The inter-sublattice inertial dynamics\nincrease the resonance frequencies and eﬀective damping for upper precession mode,\nhowever, have opposite eﬀect on lower precession mode in two-sublattice ferromagnets.\n5. 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B 64(17)\n174402 URL https://link.aps.org/doi/10.1103/PhysRevB.64.174402\n[59] Schlickeiser F, Atxitia U, Wienholdt S, Hinzke D, Chubykalo-Fesenko O and Nowak\nU 2012 Phys. Rev. B 86(21) 214416 URL https://link.aps.org/doi/10.1103/\nPhysRevB.86.214416"    },    {        "title": "1203.3826v1.Universal_low_temperature_tricritical_point_in_metallic_ferromagnets_and_ferrimagnets.pdf",        "content": "Universal low-temperature tricritical point in metallic ferromagnets and ferrimagnets\nT. R. Kirkpatrick1, D. Belitz2;3\n1Institute for Physical Science and Technology,\nand Department of Physics, University of Maryland,\nCollege Park, MD 20742, USA\n2Department of Physics and Institute of Theoretical Science,\nUniversity of Oregon, Eugene, OR 97403, USA\n3Materials Science Institute,\nUniversity of Oregon, Eugene, OR 97403, USA\n(Dated: November 30, 2018)\nAn earlier theory of the quantum phase transition in metallic ferromagnets is revisited and gener-\nalized in three ways. It is shown that the mechanism that leads to a \ructuation-induced \frst-order\ntransition in metallic ferromagnets with a low Curie temperature is valid, (1) irrespective of whether\nthe magnetic moments are supplied by the conduction electrons or by electrons in another band, (2)\nfor ferromagnets in the XY and Ising universality classes as well as for Heisenberg ferromagnets, and\n(3) for ferrimagnets as well as for ferromagnets. This vastly expands the class of materials for which\na \frst-order transition at low temperatures is expected, and it explains why strongly anisotropic\nferromagnets, such as UGe 2, display a \frst-order transition as well as Heisenberg magnets.\nPACS numbers: 64.70.Tg; 05.30.Rt; 75.50.Cc; 75.50.Gg\nI. INTRODUCTION, AND RESULTS\nQuantum phase transitions are a subject of great\ninterest.1,2In contrast to classical or thermal phase tran-\nsitions, which occur at a nonzero temperature Tc>0 and\nare driven by thermal \ructuations, quantum phase tran-\nsitions occur at zero temperature, T= 0, as a function\nof some non-thermal control parameter and are driven\nby quantum \ructuations. In this paper we will focus\non quantum phase transitions in metallic systems. For\nreasons discussed below, these transitions are especially\ninteresting.\nA prototypical quantum phase transition is the one\nfrom a paramagnetic metal to a ferromagnetic metal.\nIndeed, the earliest theory of a quantum phase transi-\ntion was the Stoner theory of ferromagnetism.3Stoner\nassumed that the conduction electrons are responsible\nfor the ferromagnetism, and developed a mean-\feld the-\nory that describes both the classical and the quantum\nferromagnetic transition. In an important paper, Hertz\nlater derived a Landau-Ginzburg-Wilson (LGW) func-\ntional for this transition by considering a simple model\nof itinerant electrons that interact only via a contact po-\ntential in the particle-hole spin-triplet channel.1Hertz\nanalyzed this (dynamical) LGW functional by means of\nrenormalization-group (RG) methods. He concluded that\nthe critical behavior in the physical dimensions d= 2 and\nd= 3 is mean-\feld-like. That is, as far as the static crit-\nical exponents of the transition at T= 0 are concerned,\nhe concluded that Stoner theory is exact in d= 2 and\nd= 3.\nIn the mid 1990s it was realized that the above con-\nclusion is not correct. The problem is that in metals\natT= 0 there are gapless particle-hole excitations that\ncouple to the magnetic order-parameter \ructuations andin\ruence the quantum critical behavior for all dimensions\nd\u00143. In Hertz's theory this coupling is taken into ac-\ncount only in an approximation that does not su\u000ece for\nyielding the leading critical behavior. Technically, Hertz\ntheory treats the fermionic soft modes in a tree approx-\nimation, whereas describing their in\ruence on the criti-\ncal behavior requires taking into account fermionic loops.\nPhysically, a correct description of any phase transition\nmust treat the order parameter \ructuations and all soft\nmodes that couple to them on equal footing.\nA theory that takes into account these e\u000bects was de-\nveloped by the present authors and T. Vojta. In Ref. 4\nit was shown that the quantum phase transition from a\nmetallic paramagnet to an itinerant ferromagnet in the\nabsence of quenched disorder in d= 2 andd= 3 is\ngenerically discontinuous, or of \frst order, in contrast to\nthe second-order transition with mean-\feld critical be-\nhavior predicted by Hertz theory.5The mechanism be-\nhind this phenomenon is analogous to what is known\nas a \ructuation-induced \frst-order transition in super-\nconductors and liquid crystals.6There, soft \ructuations\nof the electromagnetic vector potential (in superconduc-\ntors) or the nematic order parameter (in liquid crystals)\ncouple to the order parameter and e\u000bectively change the\nsign of the cubic term in the equation of state, leading to\na \frst-order transition. In the quantum magnetic case,\nthe role of the additional soft modes is played by the\nfermionic particle-hole excitations mentioned above that\nare massless at T= 0. Since these modes acquire a mass\natT > 0, the tendency towards a \frst-order transition\ndiminishes with increasing temperature. This leads to a\ntricritical point at a temperature Ttc>0 that separates\na line of continuous transitions at T > T tcfrom a line\nof \frst-order transitions at T < T tc. In a later paper\nwith Rollb uhler, the e\u000bects of a magnetic \feld Hwere\ninvestigated.7It was found that in the space spanned byarXiv:1203.3826v1  [cond-mat.str-el]  16 Mar 20122\nFIG. 1: Generic phase diagram of a metallic magnet in the\nspace spanned by temperature ( T), magnetic \feld ( H), and\nthe control parameter ( t). Shown are the long-range or-\ndered magnetic (LRO) and paramagnetic (PM) phases, lines\nof second-order transitions, surfaces of \frst-order transitions\n(\\tricritical wings\"), the tricritical point (TCP), and the two\nquantum critical points (QCP). The long-range order can be\nof ferromagnetic or ferrimagnetic type, and the electrons caus-\ning the long-range order can be in the same band as the con-\nduction electrons, or in a di\u000berent band. See the text for\nfurther explanation.\nT,H, and the control parameter, tricritical wings, or\nsurfaces of \frst-order transitions, emanate from the tri-\ncritical point and terminate in a pair of quantum critical\npoints in the T= 0 plane. The wing boundaries at T >0\nare given by lines of critical points that are reminiscent\nof a conventional liquid-gas critical point and connect\nthe tricritical point with the quantum critical points at\nT= 0. The resulting generic phase diagram is shown in\nFig. 1. This general picture is in good agreement with\nexperimental results for low-Curie-temperature metallic\nferromagnets, including ZrZn 2,8UGe 2,9URhGe,10and\nMnSi.11,12\nIn this paper we generalize our previous theory in three\nimportant ways. First, we show that our previous results,\nwhich had been derived under the same assumption made\nby Stoner and by Hertz, namely, that the magnetism is\ncaused only by itinerant electrons, remain valid in metal-\nlic systems where the magnetism is caused by electrons\nin a di\u000berent band than the conduction electrons.\nSecond, we show that the results are notrestricted\nto Heisenberg ferromagnets, contrary to what was im-\nplied in Refs. 4 and 13. Rather, they apply equally\nwell to metallic XY or Ising magnets, since the mag-\nnetic moments couple to conduction electrons whose\nspins have three degrees of freedom. This is an im-\nportant point, since some of the relevant materials are\nstrongly anisotropic magnets, including UGe 2(easy axis)\nand URhGe (easy plane).\nThird, we show that the phase diagram shown in Fig.1 also applies to generic metallic ferrimagnets. Ferrimag-\nnets are materials that spontaneously develop both a ho-\nmogeneous and a staggered magnetization at the same\ncritical value of either the temperature (for a classical\ntransition) or a non-thermal control parameter (for a\nquantum transition). Physically, this can happen when\nmagnetic moments of unequal magnitude on a bipartite\nlattice align in opposite directions.14\nThe unifying principle behind these generalizations is\nthe realization that coupling a homogeneous magneti-\nzation to conduction electrons will produce the same\nresults irrespective of the microscopic origin of the\nmagetization.15As a result, the phase diagram depicted\nschematically in Fig. 1 is valid for generic metallic ferro-\nmagnets in addition to itinerant ones, for ferromagnets of\nXY or Ising type in addition to Heisenberg magnets, and\nfor ferrimagnets as well as for ferromagnets. In all cases\nwe also consider the e\u000bects of nonmagnetic quenched dis-\norder. In Ref. 4 it was shown that this type of disor-\nder leads to an interesting phase diagram with a num-\nber of multi-critical points, and that su\u000eciently strong\nquenched disorder causes the \frst-order paramagnetic-\nto-ferromagnetic transition in metals to become second\norder. We will see that the same result holds for metallic\nferrimagnets. Experimentally, the e\u000bects of disorder on\neither one of these transitions have not yet been studied\nsystematically.\nII. THEORY\nWe now derive the results listed in Sec. I. To this end\nwe are interested in a theory that describes the magne-\ntization or order-parameter (OP) \feld M, the fermionic\ndegrees of freedom described by Grassmann-valued \felds\n\u0016 and , and the coupling between them. Accordingly,\nthe action will have three parts:\nA[M;\u0016 ; ] =AOP[M] +~AF[\u0016 ; ] +~Ac[M;\u0016 ; ];\n(2.1a)\nand the partition function is given by\nZ=Z\nD[M]D[\u0016 ; ]e\u0000A[M;\u0016 ; ]: (2.1b)\nWe are, however, not interested in a complete descrip-\ntion of the fermionic degrees of freedom; rather, we want\nto restrict ourselves to the fermionic soft modes and in-\ntegrate out the massive modes in the simplest approxi-\nmation that respects the symmetries of the problem to\narrive at an e\u000bective Landau-Ginzburg-Wilson (LGW)\ntheory in terms of soft modes only. If we denote the soft\nfermionic degrees of freedom collectively by q, and the\nmassive ones by P, we formally have\nZ=Z\nD[M;q]e\u0000ALGW[M;q]; (2.2a)3\nwhere\nALGW[M;q] =AOP[M]\u0000lnZ\nD[P]e\u0000~AF[q;P]\n\u0002e\u0000~Ac[M;q;P]\n\u0011 A OP[M] +AF[q] +Ac[M;q]:(2.2b)\nAs we will see later, the qare matrices formed by bilinear\nproducts of the fermion \felds, qnm(x;y) =\u0016 n(x) m(y)\nwith (n+ 1=2)(m+ 1=2)<0, and the Pare given by\nthe same products with ( n+ 1=2)(m+ 1=2)>0. Here\n n(x)\u0011 (x;!n) is the temporal Fourier transform of\nthe Grassmann \feld  (x), wherex\u0011(x;\u001c) comprises\nthe real-space position xand the imaginary-time variable\n\u001cin a Matsubara formalism, and !n= 2\u0019T(n+ 1=2)\nis a fermionic Matsubara frequency. \u0016 n(x) is de\fned\nanalogously.\nThis separation of soft and massive fermionic modes\nqandP, respectively, integrating out Pin a suitable\napproximation, and determining the consequences of the\ncoupling between qandM, is the central objective of\nthis paper. For the separation we will make use of the\ngeneral theory developed in Ref. 16.\nA. Order parameter, and coupling to fermions\nWe are interested in magnetic order, and hence the\nappropriate order-parameter \feld is the magnetization\nM(x). We write the magnetization as a part m(x) whose\naverage is the homogeneous magnetization, and a part\nn(x) whose average is a staggered magnetization,\nM(x) =m(x) +n(x)NX\nj=1cos(kj\u0001x): (2.3)\nHere thekjareNwave vectors that characterize the\nstaggered magnetic order, and both m(x) andn(x) are\nslowly varying in space and time. In particular, their\nFourier expansions contain only wave numbers that are\nsmall compared to the norms of the kj.\nIn a paramagnetic state the expectation values of m\nandnare both zero. At a transition to a ferromagnetic\nstate the expectation value of mbecomes nonzero while\nthat ofnremains zero; at a transition to an antifer-\nromagnetic state the converse is true. A ferrimagnetictransition is characterized by both mandnacquiring a\nnonzero expectation value at the same point in parame-\nter space. In this sense there is only one order parameter\n\feld for a ferrimagnetic transition; this fact will be impor-\ntan later. For the purposes of the present paper, a crucial\nquestion is the coupling of the order-parameter \ructua-\ntions to the soft fermionic degrees of freedom. Since the\nsoft parts of the latter are soft at zero wave number,\nthe leading coupling is to m. The fermions also couple\nton, but this leads to subleading e\u000bects since the stag-\ngered magnetization is soft at a nonzero wave number.\nWe will neglect this coupling in what follows. Physically,\nthe near-homogeneous magnetization \rutuations act as\na magnetic \feld proportional to mthat couples to the\nelectronic spin density\nns(x) =X\na;b\u0016 a(x)\u001bab b(x): (2.4a)\nHere\u001b= (\u001bx;\u001by;\u001bz)\u0011(\u001b1;\u001b2;\u001b3) denotes the Pauli\nmatrices, and a;b= (\";#)\u0011(+1;\u00001) are spin indices.\nThe coupling takes the form of a Zeeman term\n~Ac[M;\u0016 ; ] =cZ\ndxm(x)\u0001ns(x); (2.4b)\nwithca coupling constant. As we will see, the spin den-\nsity contains both massive and massless modes, so only\npart of Eq. (2.4b) contributes to Ac[M;q] in Eq. (2.2b).\nWe will discuss this separation next.\nB. Fermionic soft modes\nIn this subsection we separate the massless fermionic\nmodes from the massive ones by means of the technical\napparatus developed in Ref. 16. Here we will quote only\nas much of this formalism as is necessary for the further\ndevelopment, see Ref. 16 for additional details.\nThe soft fermion excitations are all two-particle ex-\ncitations; the related correlation functions are those of\nbilinear products of fermion \felds. The latter commute\nwith each other, and with individual fermion \felds, and\nhence are isomorphic to classical \felds. Denoting these\nclassical \felds by Q, we de\fne a classical matrix \feld\nQnm(x;y)\u0018=i\n20\nBB@\u0000 n\"(x)\u0016 m\"(y)\u0000 n\"(x)\u0016 m#(y)\u0000 n\"(x) m#(y) n\"(x) m\"(y)\n\u0000 n#(x)\u0016 m\"(y)\u0000 n#(x)\u0016 m#(y)\u0000 n#(x) m#(y) n#(x) m\"(y)\n\u0016 n#(x)\u0016 m\"(y)\u0016 n#(x)\u0016 m#(y)\u0016 n#(x) m#(y)\u0000\u0016 n#(x) m\"(y)\n\u0000\u0016 n\"(x)\u0016 m\"(y)\u0000\u0016 n\"(x)\u0016 m#(y)\u0000\u0016 n\"(x) m#(y)\u0016 n\"(x) m\"(y)1\nCCA: (2.5)\nHere \\\u0018=\" means \\isomorphic to\"; technically, the isomor- phism is implemented by means of a Lagrange multiplier4\n\feld, see below. We also de\fne the Fourier transform of\nQ,\nQnm(k;p) =1\nVZ\ndxdye\u0000ik\u0001x+ip\u0001yQnm(x;y):(2.6a)\nIt is further useful to de\fne\nQnm(k;q) =Qnm(k+q=2;k\u0000q=2) (2.6b)\nand\nQnm(x) =Qnm(x;x) =1\nVX\nqeiq\u0001xX\nkQnm(k;q):\n(2.6c)\nThe 4\u00024 matrixQnmcan be expanded in a spin-\nquaternion basis\nQnm(x;y) =3X\nr;i=0(\u001cr\nsi)i\nrQnm(x;y); (2.7)\nwhere\u001c0=s0=112is the unit 2\u00022 matrix, and\n\u001c1;2;3=\u0000s1;2;3=\u0000i\u001b1;2;3. An explicit inspection of\nthe 16 matrix elements shows that r= 0;3 represents\nthe particle-hole channel, i.e., products of the form \u0016  ,\nwhereasr= 1;2 represents the particle-particle channel,\ni.e., products of the form \u0016 \u0016 or  . For our purposes\nwe will need only the particle-hole degrees of freedom.\nIt was shown in Ref. 16 (see also Ref. 17) that a crucial\ncriterion for separating the fermionic degrees of freedom\ninto soft and massive modes is given by the relative signs\nof the frequency arguments of the matrix elements Qnm.\nAccordingly, we write\ni\nrQnm(x) =i\nrqnm(x) \u0002(\u0000!n!m) +i\nrPnm(x) \u0002(!n!m)\n(i= 1;2;3) (2.8)\nHere \u0002 is the step function, and we use the fact that in\nthe spin-triplet channel ( i= 1;2;3) the expectation value\nof theQ-matrix vanishes (this is since the fermionic de-\ngrees of freedom described by Qdo not by themselves\nhave long-ranged magnetic order; see the discussion at\nthe end of the current subsection), so that qandPrep-\nresent \ructuations. In what follows we will absorb the\nstep functions into the matrix \felds qandP, i.e., writ-\ningqnmimpliesn\u00150 andm< 0 andPnmimplies either\nn\u00150 andm\u00150 orn <0 andm < 0. Thei\nrqare the\nspin-quaternion elements of a matrix\nqnm(x) =X\nr;i(\u001cr\nsi)i\nrqnm(x): (2.9a)\nIt is also useful to de\fne an adjoint matrix\nq+\nnm(x) =X\ni;r(\u001c+\nr\ns+\ni)i\nrqmn(x); (2.9b)\nwhere\u001c+\nrands+\niare the hermitian conjugates of \u001crand\nsi, respectively. In addition, the theory contains a \feldq =nm(x) that has the same properties as qnm(x) except\nfor di\u000berent propagators, see below. The origin of q =is\nthe Lagrange multiplier \feld \u0015that constrains the bilin-\near products of fermion \felds to the q. In various places\nin the theory q\u0000\u0015\u0011q =appears, and the \u0015-propagator\nequals minus the q-propagator for noninteracting elec-\ntrons, whereas cross-correlations between qand\u0015vanish.\nThe net e\u000bect of \u0015is therefore to subtract the noninter-\nacting part of the q-propagator wherever the combination\nq\u0000\u0015occurs.\nTheqcorrelation functions are the basic soft modes in\nthe theory, see below. However, due to nonlinear cou-\nplings thePcouple to the qand thus have a soft compo-\nnent. This e\u000bect can be expressed by expanding Pin a\npower series in q. To quadratic order in qand to lowest\norder in the fermion interaction one \fnds\nP12(k)\u0019 \u0000 2iX\n3X\np'(3)\n132(p;k\u0000p)'\u00001\n13(p)'\u00001\n32(k\u0000p)\n\u0002\u0002\nq =13(p)q =+\n32(k\u0000p) +q =+\n13(p)q =32(k\u0000p)\u0003\n:(2.10)\nHere and it what follows we use a simpli\fed notation for\nfrequency indices, 1 \u0011n1, etc. We have dropped con-\ntributions to Pof higher order in q, and a contribution\nthat is linear in the interaction and linear in q, see Ref.\n16; neither will be needed for our purposes. We also have\nomitted a term quadratic in qand quadratic in the inter-\naction, which leads to less singular contributions to the\nfree energy than the one we keep. Note the frequency re-\nstrictions inherent in Eq. (2.10): sgn ( !n1) = sgn (!n2) =\n\u0000sgn (!n3). Here\n'12(k) =1\nVX\npG1(p)G2(p\u0000k) (2.11)\nwith!n1!n2<0 implied, and\n'(3)\n132(k1;k2) =1\nVX\npG1(p)G3(p\u0000k1)G2(p\u0000k1\u0000k2)\n(2.12)\nwhereG1(p)\u0011G(p;i!n1) is the single-particle Green\nfunction.'12has a scaling form\n'12(k) =NF2\u0019G\nk'd(Gi\n1\u00002=k)\n\u0011'(k;\n1\u00002): (2.13)\nwhereGis a coupling constant whose bare value is the\ninverse Fermi velocity, G= 1=vF,NFis the density of\nstates per spin at the Fermi level, and \n 1\u00002=!n1\u0000!n2.\nInd= 2;3, and for free electrons, we \fnd explicitly\n'd=2(z) = sgn (Im z)=p\n1\u0000z2; (2.14a)\n'd=3(z) =\u0000i\n2ln\u00121\u0000z\n\u00001\u0000z\u0013\n; (2.14b)\nwhich we recognize as the hydrodynamic part of the Lind-\nhard function. Equations (2.13) and (2.14) re\rect the5\nsoft particle-hole excitations with a linear momentum-\nfrequency relation in a metallic electron system. In par-\nticular,'(k;\nn= 0)/1=jkj, and'(k= 0;\nn)/\n1=\nn.18For later reference we also note the following\nidentities that hold for a special form of '(3):\n'(3)\n121(k;\u0000k) =\u0000'(3)\n212(k;\u0000k) =\u0000@\n@i!n1'12(k)\n\u0011'(3)(k;\n1\u00002): (2.15)\nThe fermionic action can be expressed in terms of q\nandP, and by using Eq. (2.10) and its generalizations\nto higher order one obtains a fermionic soft-mode action\nentire in terms of q. For our purposes we need only theGaussian part of this action, which reads\nAF[q] =\u00008X\nkX\n1;2\n3;4X\nr=0;33X\ni=0i\nrq12(k) \u0000i\n12;34(k)i\nrq34(\u0000k):\n(2.16a)\nHere 1\u0011n1etc., and the Gaussian vertex is given by\n\u0000i\n12;34(k) ='\u00001\n12(k) +\u000e1\u00002;3\u000042T\ri(2.16b)\nwith\ri=0=\u0000\rsand\ri=1;2;3=\rt;i, where\rs>0 and\n\rt;i>0 are the spin-singlet and spin-triplet interaction\namplitudes. The fermionic Gaussian propagator is given\nby the inverse of the vertex. One \fnds\nhi\nrq12(k)j\nsq34(\u0000k)i=1\n16\u000ers\u000eij\"\n\u000e13\u000e24'12(k)\u00002\riT\u000e1\u00002;3\u00004'12(k)'34(k)\n1\u00002\ri\u001f(0)\n1\u00002(k)#\n; (2.17a)\nwhere\n\u001f(0)\n1\u00002(k)\u0011\u001f(0)(k;\n1\u00002) =\u0000TX\n34\u000e1\u00002;3\u00004'34(k): (2.17b)\nWe see that the q-propagator is given in terms of ', and hence is soft. The \felds q =that enterP, Eq. (2.10), are\ncharacterized by Gaussian propagators\nhi\nrq =12(k)j\nsq34(\u0000k)i=hi\nrq12(k)j\nsq =34(\u0000k)i=hi\nrq12(k)j\nsq34(\u0000k)i (2.17c)\nand\nhi\nrq =12(k)j\nsq =34(\u0000k)i=\u00001\n8\riT\u000e1\u00002;3\u00004'12(k)'34(k)\n1\u00002\ri\u001f(0)\n1\u00002(k): (2.17d)\nThe last expression is just the interacting part of the\nq-propagator, Eq. (2.17a), as was mentioned after Eq.\n(2.9b).\nThe interaction amplitudes in the Gaussian fermionic\nvertex, Eq. (2.16b), warrant some comments. First, we\nnote that the three spin-triplet amplitudes \r1;2;3\nt are in\ngeneral not identical in a cyrstalline solid, and they do\nnot need to be for what follows. Second, we comment\non the two cases that result from the magnetism being\ncaused by the conduction electrons, or by electrons in\na band di\u000berent from the conduction band, respectively.\nLet us \frst assume the latter case, which is the concep-\ntually more straightforward one. Then AF[q], which de-\nscribes the conduction electrons, is independent of the\nmagnetism and contains interactions in both the spin-\nsinglet and spin-triplet channels. The only restriction is\nthat the latter are weak enough to not lead to magnetism\nby themselves. The conduction electrons are a\u000bected by\nthe magnetization, which acts as an e\u000bective magnetic\n\feld, and this is described by the Zeeman coupling term,Eq. (2.4b). The other possibility, which is conceptually\nmore complex, is that the magnetism is caused by the\nconduction electrons themselves. In this case the mag-\nnetic order parameter and the soft modes qdescribe de-\ngrees of freedom for electrons in the same band. The\nmagnetic order parameter then should be thought of as\nderiving from the spin-triplet interaction between the\nconduction electrons, e.g., via a Hubbard-Stratonovich\ndecoupling of the latter. This leaves the bare action AF\nwith a spin-singlet interaction only. However, as long as\nthe latter is present, a spin-triplet interaction will always\nbe generated under renormalization. The action AFwill\ntherefore again contain a spin-triplet interaction ampli-\ntude, albeit one that is much weaker than the one in the\nunderlying action that describes the system before the\nseparation of magnetic and fermionic degrees of freedom.\nThis is the case that was discussed, for ferromagnetism,\nin Ref. 13, which used phenomenological and symmetry\narguments to construct the fermionic part of the action.\nFinally, we mention that we assume the conduction elec-6\ntrons, in the absence of a nonzero magnetization (i.e.,\nwith the coupling constant cin Eq. (2.4b) put equal to\nzero), to indeed have three soft spin-triplet excitations at\nT= 0, which are given by Eqs. (2.17) with i= 1;2;3.\nThis is not necessarily the case. For instance, an external\nmagnetic \feld gives two of these three channels (the ones\ntransverse to the \feld) a mass, and a small concentration\nof magnetic impurities will make all three channels mas-\nsive without having signi\fcant other e\u000bects. However,\nin general the energy scales associated with these e\u000bects\nwill be small, and they will lead to a small reduction, but\nnot a complete suppression, of the tricritical temperature\nin Fig. 1. We will discuss this point in more detail in Sec.\nIII.\nC. Coupling between the order parameter and the\nfermionic soft modes\nWe are now in a position to separate the Zeeman term,\nEq. (2.4b), into parts where the order parameter couples\nto soft and massive fermionic modes, respectively. If we\nde\fne a temporal Fourier transform of the magnetization\n\feldmby\nmn(x) =p\nTZ1=T\n0d\u001c ei\nn\u001cm(x;\u001c); (2.18)\nwith \nn= 2\u0019Tn a bosonic Matsubara frequency, then\nwe can write Eq. (2.4b) in the form\n~Ac[M;Q] = 2cp\nTZ\ndxX\nn3X\ni=1mi\nn(x)\n\u0002X\nr=0;3(\u00001)r=2X\nmtr [(\u001cr\nsi)Qm;m+n(x)]:(2.19)\nBy expressing Qin terms of qandPby means of Eq.\n(2.8), andPin terms of q =by means of Eq. (2.10), we\nobtain the desired coupling Ac[M;q] between the order-\nparameter \ructuations and the fermionic soft modes q.D. Generalized Mean-Field Theory\nAn e\u000bective action, Ae\u000b[M] in terms of the order pa-\nrameter alone can be obtained by integrating out the\n\feldsq,\nAe\u000b[M] = lnZ\nD[q]eALGW[M;q]: (2.20)\nIn general the evaluation of this expression is very di\u000e-\ncult. However, it can be evaluated exactly within a gener-\nalized mean-\feld approximation that was \frst employed\nin the context of liquid crystals and superconductors6\nand is de\fned as follows. First, we ignore temporal and\nspatial variations of the order parameter, i.e. we treat\nthe \feldsm(x) andn(x) in Eq. (2.3) as numbers. If we\nassume ordering in the 3-direction, we have\nMi(x)\u0019\u000ei32\n4m+nNX\nj=1cos(kj\u0001x)3\n5; (2.21a)\nwhich implies\nmi\nn(x)\u0019\u000ei3\u000en0m=p\nT : (2.21b)\nThis mean-\feld approximation for the order parameter\nmeans that only the part of Qthat is diagonal in fre-\nquency space, i.e., Pmm, contributes to Eq. (2.19). This\nin turn means that the contribution to Pthat is linear\ninq, which we had dropped from Eq. (2.10), does not\ncontribute. Second, we restrict ourselves to quadratic\norder inq. That is, we treat the fermionic soft modes in\na Gaussian approximation with a \fxed magnetic order\nparameter. The validity of these approximations will be\ndiscussed in Sec. III B.\nWith these approximations the action Acthat couples\nqand the order parameter is quadratic in qand can be\nwritten\nAc[m;q] = 8X\nr;s=0;3X\ni;ji\nrq12(k)ij\nrs\u0000c\n12;34(k)j\nsq34(\u0000k): (2.22a)\nHere\nij\nrs\u0000c\n12;34(k) =\u000e13\u000e244cm\u0012\n0 1\n\u00001 0\u0013\nrs0\nB@0 0 0 0\n0 0 1 0\n0\u00001 0 0\n0 0 0 01\nCA\nij'(3)\n121(k;\u0000k)'\u00002\n12(k); (2.22b)\nand we have used Eq. (2.15). The matrices give the values ofij\nrs\u0000cfor the 4 possible values of ( r;s) and the 16 possible\nvalues of (i;j).\nThe integral over qin Eq. (2.20) can now easily be carried out. For the free-energy density f=\u0000TAe\u000b=Vwe\nobtain\nf=f0(m;n) + \u0001f(m): (2.23a)7\nHeref0=\u0000TAOP=Vis the mean-\feld free energy in the absence of a coupling to the fermionic soft modes. For\n\u0001f(m), which is the contribution to the free energy due to this coupling, one \fnds\n\u0001f(m) =2\nVX\nk0\nTX\nnlnN(k;\nn;m); (2.23b)\nwhereP0\nkdenotes a wave vector sum such that jkj<\u0003 with \u0003 an ultraviolet cuto\u000b, and\nN(k;\nn;m) =\u000016c2\rt;1\rt;2m2\n2\nn\u0010\n'(3)(k;\nn)\u00112\n'\u00004(k;\nn) +'\u00004(k;\nn)Y\ni=1;2h\n1\u00002\rt;i\u001f(0)(k;\nn)i\n:(2.23c)\nThe equation of state is obtained by minimizing the\nfree energy density. In the absence of a coupling be-\ntween the order parameter and the fermionic soft modes\nthis amounts to minimizing f0, which yields the ordinary\nmean-\feld equation of state. For a ferromagnet, the lat-\nter has the usual Landau form. For a ferrimagnet, the\nequation of state depends on details of the magnetic or-\nder. It can be complicated and describe several di\u000berent\nphases, see, e.g., Ref. 19. However, generically the \frst\nphase encountered as one approaches from the paramag-\nnetic state is entered via a second-order transition. After\nminimizing f0and expressing nin terms of mone thus\nhas again an ordinary mean-\feld equation of state given\nby\nh=rm+um3+O(m5); (2.24)\nwherehis an external magnetic \feld in the 3-direction,\nu>0, and the transition occurs at r= 0.20In Appendix\nA we recall a very simple model that leads to this result.\nThe second term on the right-hand side of Eq. (2.23a)\ngives an additional contribution to the equation of state,\nwhich then reads\nh=rm+um3\u000064mc2\rt;1\rt;2\n\u00021\nVX\nk0\nT1X\nn=1\n2\nn\u0000\n'(3)(k;\nn)\u00012'\u00004(k;\nn)\nN(k;\nn;m):\n(2.25)\nThis is the desired generalized mean-\feld equation of\nstate which takes into account the coupling of the order\nparameter to the fermionic soft modes.\nE. Discussion of the Generalized Mean-Field\nEquation of State\nWith some e\u000bort the integrals in Eqs. (2.23b) and\n(2.25) can be explicitly performed. However, the salient\npoints can be seen by simple scaling considerations and\ndimensional analysis. Equations (2.11) and (2.13) im-\nply that the frequency \n nscales as the wavenumber\nk, \nn\u0018k, and that '(k;\nn)\u00181=k\u00181=\nn, which\nalso can be seen explicitly from Eqs. (2.14). Equation(2.15) implies that '(3)(k;\nn)\u00181=k2\u00181=\n2\nn. Equa-\ntion (2.23c) then shows that there is a length scale Lm,\nor a corresponding frequency scale !m, that scales as\nLm\u00181=!m\u00181=m. If one attempts to expand \u0001 f(m),\nEq. (2.23b), in powers of matT= 0, then nonanalytici-\nties will occur at next-to-leading order for all d\u00143.\nAn alternative way to describe this mechanism is to\nsay that of the three soft fermionic spin-triplet excita-\ntions, Eq. (2.17a) with r=s= 0;3 andi=j= 1;2;3,\ntwo (namely, the ones transverse to the order parameter\ndirection) acquire a mass due to the coupling between\nthe fermions and the order parameter m, as can be seen\nexplicitly from Eq. (2.22b). This acquisition of a mass\nby a generic soft mode due the spontaneous breaking of\na continuous symmetry is an example of the Anderson-\nHiggs mechanism,22{24even though the broken symmetry\nin this case is not a gauge symmetry, see the discussion\nin Sec. III A. It implies in turn that the free energy is a\nnonanalytic function of m.\nAt nonzero temperatures the singularities are cut o\u000b\nbyTaccording to m\u0018T. That is, a crossover occurs\nfromm-scaling to T-scaling when the Zeeman splitting\nis comparable to the temperature, or the thermal length\nscaleLT/1=Tis comparable to the magnetic length\nscaleLmmentioned above. Taking into account the sign\nofN, Eq. (2.23c), one \fnds schematically, for 1 <d< 3,\n\u0001f(m) =\u0000vm2(m2+T2)(d\u00001)=2; (2.26a)\nand ford= 3\n\u0001f(m) =v\n8m4ln(m2+T2); (2.26b)\nwherev>0 is a positive constant.\nThe most important aspects of this result, as far as\nthe order of the transition is concerned, are the sign of\nvand the power of matT= 0. For all d\u00143 there\nis a negative term in the free energy that dominates the\nm4in the Landau free energy and hence necessarily leads\nto a \frst-order transition. Another way to see this is by\nexpanding \u0001 f(m), Eq. (2.26a), in powers of mforT >0.\nThe leading term is proportional to \u0000m4=T3\u0000d. That is,\nthere is a negativem4term whose prefactor diverges as\nT!0 for alld\u00143, which implies that there will be a\ntricritical point at some temperature. The free energy for8\nFIG. 2: Schematic sketch of the free energy for three values\nof the parameter r. The \frst-order transition occurs at r=\nr1>0. It pre-empts the second-order transition of Landau\ntheory which would occur at r= 0.\nthree di\u000berent values of ris plotted schematically in Fig.\n2. For this schematic free energy, the equation of state\nin the case d= 3, for which many experimental results\nexist, takes the form\nh=rm+v\n2m3ln(m2+T2)\n+m3\u0012\nu+v\n4m2\nm2+T2\u0013\n:(d= 3) (2.27)\nAlso of interest is the other physical dimensionality, d=\n2, where the equation of state reads\nh=rm\u00002vm(m2+T2)1=2\n+m3\u0012\nu\u0000v\n(m2+T2)1=2\u0013\n:(d= 2) (2.28)\nHere the analyticity is stronger than in the 3- dcase, with\na negative m2-term in the equation of state at T= 0.\nThis is particularly interesting in the case of Ising mag-\nnets, which display long-range order in d= 2 even at\nT > 0. The case of Heisenberg and XY magnets, which\ndo not show true long-range order in d= 2 except at\nT= 0, is more complicated.\nThese are the same results that were obtained using\na more phenomenological theory of the fermionic soft\nmodes in Ref. 13. They were discussed extensively in that\nreference, as well as in Refs. 4 and 7. There is no need to\nrepeat this discussion here, and the salient features are\nsummarized by the schematic phase diagram shown in\nFig. 1. The important conclusion of the current paper is\nthat the validity of these results, in addition to itinerant\nHeisenberg ferromagnets, extends to metallic ferromag-\nnets where the magnetism is not due to the conduction\nelectrons, to metallic ferromagnets in the XY or Ising\nuniversality class, and also to metallic ferrimagnets. The\nonly condition is that the conduction electrons are not\nsubject to strong spin-symmetry breaking e\u000bects such asmagnetic impurities. We note in passing that an inter-\nesting system is provided by the easy-plane ferromagnet\nURhGe, where an in-plane magnetic \feld transverse to\nthe magnetization has been used to tune the transition,\naccess the tricritical point, and map out the tricritical\nwings.10This situation requires a re\fnement of the the-\nory presented above, which will be reported elsewhere.21\nIII. DISCUSSION, AND CONCLUSION\nWe now discuss our results, before concluding with a\nsummary.\nA. The mechanism behind the \frst-order transition\nThe mechanism that leads to the \frst-order tran-\nsition discussed in Sec. II E is precisely analogous to\nthe \ructuation-induced \frst-order transition discussed in\nRef. 6 for the BCS-superconductor transition and the\nnematic-to-smectic-A transition in liquid crystals. An\nimportant physical ingredient is an underlying \\generic\"\nsoft mode, i.e., one that is not related to the phase tran-\nsition in question, but couples to the order parameter.\nIn the case of liquid crystals this soft mode is the ne-\nmatic Goldstone mode, in the case of superconductors,\nthe vector potential, in the present case, the spin-triplet\nparticle-hole excitation. At the transition of interest, this\nsoft mode acquires a mass that is given in terms of the\nnonzero expectation value of the order parameter. This\ngeneral mass-generating mechanism was \frst pointed out\nby Anderson, and is now known as the Anderson-Higgs\nmechanism.22{24This coupling of the order parameter to\nunderlying soft modes leads to a non-analytic term in the\nLandau free energy that is dominant over the usual quar-\ntic term and has a negative sign, leading to a \frst-order\ntransition. It should be stressed that this is only one way\nto realize a \ructuation-induced \frst-order transition; an-\nother one, for instance, is realized by a \u001e4-theory with a\ncubic anisotropy.25The current realization is analogous\nto the case of scalar electrodynamics studied by Cole-\nman and Weinberg in a particle-physics context.26It is\nalso worthwhile noting that the analogy between super-\nconductors on one hand, and liquid crystals and quantum\nmagnets on the other, breaks down in the ordered phase.\nIn the former case, the Goldstone mode gets absorbed\ninto the longitudinal component of the vector potential,\nwhich is massive, and there is no soft mode in the ordered\nphase. In the latter, there are Goldstone modes in the\nordered phases, namely, a \\smecton\" with an anisotropic\ndispersion relation in the smectic-A phase (Ref. 27, see\nalso Ref. 28) and magnons in the magnetic phase.9\nB. Universality of the \frst-order transition, and\nthe validity of the generalized mean-\feld theory\nExperimentally, all examples of clean low-T cferromag-\nnets (for disordered systems, see below; ferrimagnets so\nfar have not been systematically studied from this point\nof view) show a \frst-order transition if the Curie temper-\nature is suppressed far enough. There is not a single ex-\nample of a quantum critical point in zero magnetic \feld.\nWhile this is consistent with the generalized mean-\feld\ntheory theory presented in Sec. II, it is somewhat surpris-\ning when compared with the case of liquid crystals, where\nan analogous theory also predicts a \frst-order transition.\nIn this case, in stark contrast to that of quantum mag-\nnets, the observed transition is usually of second order,\nand only recently have examples of a (weakly) \frst-order\ntransition been found.29These observations beg the ques-\ntion whether in the case of quantum magnets the gener-\nalized mean-\feld approximation is more generally valid\nthan in classical systems.\nTo discuss this point, we \frst observe that we have\nmade three approximations to treat the action given by\nEq. (2.1a). First, we have integrated out the fermionic\nmassive modes in a saddle-point approximation that re-\nspects the Ward identity that governs the soft-mode\nstructure of the system.16,30Second, we have kept the\nsoft fermionic degrees of freedom only to Gaussian or-\nder in the soft modes q. Third, we have treated the\norder parameter in a mean-\feld approximation. These\napproximations are not independent of one another, and\nthe \frst two simpli\fcations do not constitute any addi-\ntional approximation over and above the last one. This\ncan be seen as follows.\nThe mean-\feld approximation for the order parame-\nter means that the fermionic degrees of freedom describe\nan interacting electron system that is spin-polarized by\nthe coupling to the homogeneous magnetization, which\nacts as an e\u000bective external magnetic \feld. The state\nof the fermionic subsystem is thus described by a stable\nFermi-liquid \fxed point. Corrections to the fermionic\nsoft-mode action due to massive degrees of freedom are\nirrelevant with respect to this \fxed point by at least one-\nhalf power of frequency or wavenumber in all dimensions,\nand thus cannot change the properties of system.17Sim-\nilarly, only the terms quadratic in qcontribute to the\n\fxed-point action; all higher-order terms are irrelevant\nby power counting. Keeping terms of higher order in q\nwill therefore renormalize the parameters of the theory,\nbut it cannot change its structure. In particular, it can-\nnot change the sign of the term in the equation of state,\nEqs. (2.27, 2.28), that is due to the soft fermionic \ructu-\nations and leads to the \frst-order transition.\nThis leaves the mean-\feld approximation for the order\nparameter to be discussed. If the \frst-order transition\natr=r1occurs far from the second-order transition at\nr= 0 that is pre-empted by it (see Fig. 2), then order-\nparameter \ructuations are negligible and the results of\nthe generalized mean-\feld theory are qualitatively cor-rect. If, however, the \frst-order transition occurs close\nto the putative second-order one, i.e., if the minimum\nin the free energy in Fig. 2 is very shallow, then it is\nless clear whether order-parameter \ructuations can be\nneglected. One key di\u000berence between classical liquid\ncrystals and quantum magnets is that in the former case,\nthe system is below the upper critical dimension d+\nc= 4\nfor the (unrealized) phase transition that would occur\nin the absence of any coupling between the smectic or-\nder parameter and the nematic soft modes. In contrast,\nthe quantum magnetic systems are above the correspond-\ning upper critical dimension d+\nc= 1 that follows from\nHertz theory, and even with that coupling taken into ac-\ncount, ordinary mean-\feld theory becomes exact, as far\nas the description of the phase transition is concerned,\nford>3.31This strongly suggests that order-parameter\n\ructuations are of much less importance in the case of\nquantum magnets, and it provides a possible explanation\nof the fact that the observed transition is universally of\n\frst order.\nIrrespective of these observations, the role of order-\nparameter \ructuations in quantum magnets is a topic\nthat warrants additional work. For the case where the\nmagnetism is not produced by the conduction electrons,\nthis will require an action that properly describes lo-\ncalized magnetic moments and their \ructuations, e.g.,\nthe one given in Ref. 32. For itinerant magnets, i.e., if\nthe magnetism is due to the conduction electrons them-\nselves, the theory developed in Sec. II will apply, but\nthe order-parameter \ructuations and the fermionic ex-\ncitations both need to be kept, along the lines of the\nphenomenological theory of Ref. 13. The latter reference\ngave a scenario that can lead to a second-order transi-\ntion in the magnetic case. It would also be interesting to\nexperimentally study quantum ferromagnets or ferrimag-\nnets ind= 2, where order-parameter \ructuations will be\nstronger than in d= 3.\nC. The e\u000bects of quenched disorder\nSo far we have discussed the case of clean or pure mag-\nnets. Impurities, modeled by quenched disorder, have\nimportant e\u000bects that are both needed to understand ex-\nperimental observations in certain systems, and to pre-\ndict e\u000bects that can serve to ascertain that the \frst-order\ntransition in pure samples is indeed due to the posited\nmechanism.\nQuenched disorder changes the soft-mode spectrum of\nthe fermions. It gives the ballistic soft modes that are\nrepresented by Eqs. (2.17) as mass, and leads to new soft\nmodes that are di\u000busive. In the context of the current\ntheory, this change has two principal e\u000bects. First, it\ncuts o\u000b the nonanalyticity in the clean equation of state,\nEqs. (2.27, 2.28). Second, it leads to a new nonanalytic\nterm in the equation of state that has the opposite sign\nand whose prefactor vanishes in the clean limit.31The\nresulting schematic generalized Landau theory has been10\ndiscussed in Ref. 4. A more detailed model discussion\nthat allows for semi-quantitative predictions of the e\u000bects\nof disorder will be presented elsewhere;21here we just\npresent the most pertinent aspects of such a model cal-\nculation. A good representation of the mean-\feld equa-\ntion of state for realistic values of the magnetization, the\ntemperature, and the disorder, is\nh=rm+v1=4\n4(kF`)3=2m3\nm3=2+ (bT)3=2\n+v\n2m3ln\u0002\ncm2+ (1=kF`+bT)2\u0003\n+um3;(3.1)\nwhich generalizes Eq. (2.27) in the presence of quenched\ndisorder. Here the magnetic \feld hand the temperature\nTare measured in units of the Fermi energy \u000fFand the\nFermi temperature TF, respectively, and the magnetiza-\ntionmis measured in units of the conduction electron\ndensity (we put \u0016B= 1). The dimensionless coupling\nconstantvis proportional to the fourth power of the ef-\nfective spin-triplet interaction amplitude of the conduc-\ntion electrons. It is a measure of how strongly corre-\nlated the conduction electrons are, and it is bounded\nabove by a stability criterion that requires v.0:5.\nkFis the Fermi wave number of the conduction elec-\ntrons, and `is the elastic mean-free path. Within a\nDrude model, and for good metals, one has approxi-\nmatelykF`\u00191;000=(\u001a0=\u0016\ncm), with\u001a0the residual\nelectrical resistivity. candbare dimensionless constants\nthat are equal to c= 1=45 andb= 3\u0019in a model\ncalculation.21The second factor in the second term on\nthe right-hand side is a reasonable representation, for re-\nalistic parameter values, of a more complicated scaling\nfunction\nm3=2g(kF`m;bT=m )\u0019m3\nm3=2+ (bT)3=2(3.2)\nthat depends on the disorder in addition to the temper-\nature, and we have dropped the last term in Eq. (2.27)\nfrom Eq. (3.1) since one generically expects v\u001cu.\nAtT= 0, and in a clean system, Eq. (3.1) yields a\n\frst-order transition at r1=vm2\n1=4, where the magneti-\nzation discontinuously jumps from m= 0 tom=m1=\ne\u0000(1+2u=v)=2. Withu\u00190:14 andv\u00190:02 this yields\nm1\u00194\u000210\u00003, which is reasonable for a weak ferromag-\nnet. Similarly, there is a tricritical temperature given\nbyTtc=TF= (1=b) exp(\u0000u=v); with the same parame-\nter values this yields Ttc=TF\u001910\u00004, orTtc\u001910 K for\nTF= 100;000 K, which is also reasonable. This tricritical\npoint gets destroyed by quenched disorder on the order of\nkF`\u0019bTtc=TF\u00191;000, or a residual resisitivity on the\norder of\u001a0\u00191\u0016\ncm. At this point the second term on\nthe right-hand side of Eq. (3.1) is still very small, and the\ncritical behavior at the resulting quantum critical point\nis given by ordinary mean-\feld exponents except extrely\nclose to the transition, where it crosses over to the crit-\nical behavior derived in Ref. 17. For instance, in this\nasymptotic region the critical exponents \fand\u000e, de\fnedbym(h= 0)/jrj\fandm(r= 0)/h1=\u000e, respectively,\nare given by \f= 1=2 and\u000e= 3=2, as opposed to the\nmean-\feld values \f= 1=2 and\u000e= 3. Only for sub-\nstantially larger values of the disorder, \u001a0\u0019100\u0016\ncm\nwith the above parameters, does the asymptotic critical\nbehavior extend over a sizeable range of rvalues (up to\njrj\u00190:01). This observation explains why an experi-\nment on Ni xPd1\u0000x, which shows a ferromagnetic transi-\ntion at a very small value of x(x\u00190:025) corresponding\nto weak disorder, found mean-\feld exponents consistent\nwith Hertz theory,33whereas Bauer et al.34found non-\nmean-\feld exponents, at least some of which were con-\nsistent with Ref. 17, in URu 2\u0000xRexSi2, where the ferro-\nmagnetic transition occurs at x\u00190:15 with the residual\nresistivity on the order of \u001a0\u0019100\u0016\ncm.35\nD. Conclusion\nIn conclusion, we have extended a previous theory\nof quantum ferromagnets in several important ways.\nWe have shown that the mechanism that leads to the\nparamagnet-to-ferromagnet transition at low tempera-\nture ind= 3 andd= 2 to be generically of \frst order,\nwhich was \frst reported in Ref. 4, is valid in anisotropic\nferromagnets, in ferrimagnets, and in metallic ferromag-\nnets where the conduction electrons are not the source of\nthe magnetization, in addition to the case of isotropic\nitinerant ferromagnets originally considered. This ex-\nplains why the low-temperature transition is observed\nto be of \frst order in highly anisotropic ferromagnets,\nand it much expands the class of materials for which this\nphenomenon is predicted. For clean magnets, an e\u000bec-\ntive theory of soft fermionic modes recently developed\nin Ref. 16 has provided a technical basis that improves\non the phenomenological theory of Ref. 13. In the pres-\nence of quenched disorder, the theory allows for a semi-\nquantitative description of the suppression and ultimate\ndestruction of the tricritical point. A sizeable range of\ndisorder exists where the observable critical behavior is\npredicted to be mean-\feld like, whereas for very large\ndisorder the asymptotic critical region, which is char-\nacterized by non-mean-\feld Gaussian critical exponents,\nexpands and eventually eliminates the mean-\feld region.\nAppendix A: A simple mean-\feld model of a\nferrimagnet\nHere we recall a very simple mean-\feld model of the\ntransition from a paramagnet to long-range ferrimagnetic\norder.14Consider a one-dimensional chain of alternating\nmagnetic moments \u0016a,\u0016bthat are antiferromagnetically\ncoupled. Weiss theory assumes that the a-moments and\nb-moments are subject to e\u000bective magnetic \felds\nBa=\u0000\u0015Mb (A1a)\nBb=\u0000\u0015Ma; (A1b)11\nrespectively, where \u0015>0. The magnetizations Ma;bare\ngiven by the Brillouin expressions\nMa=\u0017\u0016atanh(\u0016aH=T +\u0016aBa=T);(A2a)\nMb=\u0017\u0016btanh(\u0016bH=T +\u0016bBb=T):(A2b)\nHereHis an external magnetic \feld, Tis the tem-\nperature, and \u0017is the number of magnetic moments\nof each species. If one de\fnes reduced magnetic \felds\nha;b=H=\u0017\u0016a;b\u0015, a reduced temperature t=T=\u0017\u0016a\u0016b\u0015,\nand reduced moments ma;b=Ma;b=\u0017\u0016a;b, then one sees\nthat the Weiss mean-\feld equations (A1, A2) have a so-\nlutionma=\u0000mb= ~m, where ~mis the solution of the\nusual mean-\feld equation of state\nh=r~m+ ~m3=3 +O( ~m5); (A3)wherer=t\u00001. This simple model thus describes a\ntransition at t= 1 to ferrimagnetic order where the\nhomogeneous magnetization is given by m=Ma+\nMb=\u0017(\u0016a\u0000\u0016b) ~mand the staggered magnetization\nn=Ma\u0000Mb=\u0017(\u0016a+\u0016b) ~mis proportional to m.\nAcknowledgments\nWe gratefully acknowledge discussions and correspon-\ndence with Greg Stewart, Je\u000b Lynn, and Nick Butch.\nThis work was supported by the National Science Foun-\ndation under Grant Nos. DMR-09-29966, and DMR-09-\n01907.\n1J. Hertz, Phys. Rev. B 14, 1165 (1976).\n2S. Sachdev, Quantum Phase Transitions (Cambridge Uni-\nversity Press, Cambridge, 1999).\n3E. C. Stoner, Proc. Roy. Soc. London A 165, 372 (1938).\n4D. Belitz, T. R. Kirkpatrick, and T. Vojta, Phys. Rev.\nLett. 82, 4707 (1999).\n5In these and related statements we denote by dthe spatial\ndimensionality of the system; spin space is always consid-\nered to be three-dimensional.\n6B. I. Halperin, T. C. Lubensky, and S.-K. Ma, Phys. Rev.\nLett. 32, 292 (1974).\n7D. Belitz, T. R. Kirkpatrick, and J. Rollb uhler, Phys. Rev.\nLett. 94, 247205 (2005).\n8M. Uhlarz, C. P\reiderer, and C. Hayden, Phys. Rev. Lett.\n93, 256404 (2004).\n9V. Taufour, D. Aoko, G. Knebel, and J. Flouquet, Phys.\nRev. Lett. 105, 217201 (2010).\n10E. A. Yelland, J. M. Barraclough, W. Wang, K. V.\nKamenev, and A. D. Huxley, Nature Physics 7, 890 (2011).\n11C. P\reiderer, S. R. Julian, and G. G. Lonzarich, Nature\n(London) 414, 427 (2001).\n12MnSi is actually a helimagnet, see Ref. 36, but the pitch\nwavelength of the helix is large compared to the atomic\nlength scale and for the purposes of the present discussion\nthe magnetic order can be approximated as ferromagnetic.\nSee also Ref. 37.\n13T. R. Kirkpatrick and D. Belitz, Phys. Rev. B 67, 024419\n(2003).\n14C. Kittel, Introduction to Solid State Physics (Wiley, New\nYork, 1996).\n15This is true within the framework of the generalized mean-\n\feld theory whose validity is discussed in Sec. III B. If\norder-parameter \ructuations are important it is possible\nthat the behavior is less universal.\n16D. Belitz and T. R. Kirkpatrick, Phys. Rev. B xx, xxxxxx\n(2012), (arXiv:1112.5916).\n17D. Belitz and T. R. Kirkpatrick, Phys. Rev. B 56, 6513\n(1997).\n18In Ref. 13 the propagator was modeled as '(k;\nn)/\n1=(jkj+\u0019\nn=2vF) independent of the dimensionality.\nWhile this has the correct scaling behavior, it can lead,in explicit calculations of certain observables in certain di-\nmensions, to nonzero prefactors of nonanalyticities when\nthe exact prefactor is zero, see Ref. 16.\n19M. L. Plumer, A. Caill\u0013 e, and K. Hood, Phys. Rev. B 40,\n4958 (1989).\n20In any given material it is of course possible that the pa-\nrameteruin the Landau theory is negative, leading to a\n\frst-order transition even within Landau theory. However,\nthis will not be the case generically, whereas the general-\nized mean-\feld theory predicts a generic \frst-order transi-\ntion.\n21Yan Sang, D. Belitz, and T.R. Kirkpatrick, unpublished\nresults.\n22P. W. Anderson, Phys. Rev. 130, 439 (1963).\n23P. W. Higgs, Phys. Lett. 12, 132 (1964).\n24P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964).\n25D. J. Wallace, J. Phys. C 6, 1390 (1973).\n26S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).\n27P. G. DeGennes and J. Prost, The Physics of Liquid Crys-\ntals(Clarendon, Oxford, 1993).\n28T. R. Kirkpatrick and D. Belitz, Phys. Rev. B 80, 075121\n(2009).\n29A. Yethiraj, R. Mukhopadhyay, and J. Bechhoefer, Phys.\nRev. E 65, 021702 (2002).\n30T. R. Kirkpatrick and D. Belitz, Phys. Rev. Lett. 108,\n086404 (2012).\n31D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod.\nPhys. 77, 579 (2005).\n32N. Read and S. Sachev, Phys. Rev. Lett. 75, 3509 (1995).\n33M. Nicklas, M. Brando, G. Knebel, F. Mayr, W. Trinkl,\nand A. Loidl, Phys. Rev. Lett. 82, 4268 (1999).\n34E. D. Bauer, V. S. Zapf, P.-C. Ho, N. Butch, E. J. Freeman,\nC. Sirvent, and M. B. Maple, Phys. Rev. Lett. 94, 046401\n(2005).\n35N. P. Butch and M. B. Maple, J. Phys. Cond. Matt. 22,\n1642204 (2010).\n36Y. Ishikawa, K. Tajima, D. Bloch, and M. Roth, Solid\nState Commun. 19, 525 (1976).\n37C. P\reiderer, G. J. McMullan, S. R. Julian, and G. G.\nLonzarich, Phys. Rev. B 55, 8330 (1997)."    },    {        "title": "1006.3856v1.Intrinsic_Ferromagnetism_in_Eu_doped_ZnO.pdf",        "content": "1 \n Intrinsic Ferromagnetis m in Eu doped ZnO \n \nM.H.N. Assadia, Y.B. Zhanga, 1, M. Ionescub, P. Photongkama, and S. Lia \n \na School of Materials Science and Engineering, The University of New South Wales, NSW \n2052, Australia \nb Australian Nuclear Science and Technol ogy Organization, Sydney, NSW 2234, Australia  \n \nWe report room temperature ferromagnetism in as-implanted Eu doped ZnO (ZnO:Eu).  To address the origin of ferromagnetism ab initio  calculations of \nZnO:Eu system are performed.  Results show that the ferromagnetism is induced by ZnO point defects as Eu ions in perfect ZnO tend to align \nantiferromagnetically.  \n1. Introduction \nDiluted magnetic semiconductors (DMSs) have attracted enormous in terests because of \ntheir potential for innovative spintronics app lication [1].  In order to achieve Curie \ntemperatures ( T\nC) above room temperature, DMSs are normally fabricated by incorporating \ntransition metal or rare earth ions into a nonmagnetic semiconduc tor host lattice [2].  Among \nmany candidates, ZnO-based DMSs with unique technological applications have recently \nbeen in the focus of an intense and ongoing re search [3].  The net magnetization in DMS \nmaterials should not arise from ferromagnetic  inclusions (secondary phases), but from \nlocalized magnetic moments of separated ions, being distributed uniformly in the host and \nferromagnetically aligned via an indirect of ma gnetic coupling.  Due to the highest magnetic \nmoment that can be born on a single ion, Eu is  considered as magnetic dopants in ZnO to \nachieve a high magnetization DMS.  However, the solubility of Eu in ZnO is strongly limited \nby the lattice distortion associated with the much larger ionic radius of the rare earth atoms.  \nThe magnetic interactions between  the localized impurity atoms as well as lattice distortions \ncaused by these atoms play an important role in determining atomic arrangements in such \nmaterials.  In this paper, we report on ferroma gnetic properties at r oom temperature of Eu-\ndoped ZnO epitaxial thin films using ion beam  techniques.  Then by using theoretical ab initio  \ntechniques, we reveal the arra ngement of Eu dopants in the host lattice and relate the \nferromagnetic interactio n among Eu ions to ZnO’s native defects.  \n \n2. Sample preparation \nEpitaxial ZnO (0001) thin films for ion im plantation were supplied by Nanovation Inc. \nFrance.  The films were grown on 10 × 10× 0.5mm c-Al 2O3 substrates by Pulse laser \ndeposition (PLD).  Each film has approximate ly 100 nm thickness.  The ion implantation was \nperformed at room temperature by using a me tal vapour vacuum arc (MEVVA) ion source \nunder vacuum, 2 × 10-6 mbar.  The Europium (Eu) ions we re implanted at 45 kV with ion \nbeam current 30 mA.  Two Eu-implanted ZnO samples have been prepared at different \nimplantation time.  The amount of Eu in thes e two samples was characterized by Rutherford \nbackscattering experiment was 1.253 × 1016 and 1.670 × 1016 atom/cm2.  These values are \nequivalent to 3% and 4% by mole respectively.  The magnetic properties of Eu-doped were \nstudied by Quantum Design MP MS SQUID magnetometer. \n   \n                                                \n \n1 Author to whom correspondence should be addressed; electronic mail: y.zhang@unsw.edu.au 2 \n 3. Results \nFig. 1 shows isothermal magnetizat ion curves of as-implanted Zn 0.97Eu0.03O and \nZn0.96Eu0.04O films as a function of an external magnetic field applied parallel to the film \nsurface measured at room temperature.  It demonstrates that both of films possess ferromagnetism at room temperature with co ercive fields of 50 Oe and 1.81 and 2.23 µ\nB/Eu for \nZn0.97Eu0.03O and Zn 0.96Eu0.04O respectively.  This result indicates slight change in \nmagnetization per Eu ion as Eu concentratio n increases.  Ferromagnetism found in these \nsamples can not be caused by Eu metal clus ters or oxide forms such as. EuO and Eu 2O3 since \nXRD results show no secondary phases. \n  \n \nFig. 1. M-H curve of Zn 0.97Eu0.03O and Zn 0.96Eu0.04O at 300K measured by SQUI DS magnetometer, after the \nsubtraction of diamagnetic contribution of Al2O3 substrat e.  The external applied field was applied parallel to \nthe film surface \n \n4. Theoretical investigation  \nIn our theoretical study, total energy (tE) calculations were performed using plane-\nwave pseudopotential approach of density functi onal theory as implemented in CASTEP code \n[4] within the framework of generalized-gradi ent approximation.  Ultrasoft pseudopotentials \nwere represented in reciprocal space and Eu’s 4 f were treated as valence electrons.  An energy \ncut-off of 800 eV for plane- wave basis set and a 3×3×1 Γ-centered k-point grid for integration \nover reciprocal space were  used.  To model the doped ZnO, a 32-atomic 2 a×2a×2c ZnO \nsupercell was adopted for calculations with tw o Eu ions substituting Zn sites.  This was \nnecessary for calculations of the relative energies of ferromagnetic (FM) and \nantiferromagnetic (AFM ) spin alignments.  The difference be tween these two energies per Eu \nion is defined as E∆ = (t\nAFME  − t\nFME) / 2, which indicates the fe rromagnetic phase stability. \nIn order to investigate the aggregation tendency among Eu ions, two spatial \narrangements were studied, Configuration 1 in  which Eu ions were separated by only one \noxygen ion  and Configuration 2 in which Eu ions were  separated by a chain of -O-Zn-O- ions \nas shown in Fig. 2(a) and Fig. 2(c) respectively.   \nFor both of configurations geometry re laxation was performed allowing internal \ncoordinates to relax until the Ca rtesian components of atomic fo rces acting on all ions in the \nsupercell was smaller than 0.05 eVÅ-1 and simultaneously the energy converged to 10-5 eV \nper step, per atom.  The relaxed structures for both c onfigurations are repr esented in Fig. 2(b) \nand Fig.2(d) respectively.  Et, ∆E and the Eu-Eu distance ( DEu-Eu) are presented for both \nconfigurations in Table 1. \nEt for Configuration 2 is lower by 44 meV than that for Configuration 1, indicating that \nEu ions in ZnO do not tend do aggregate via an oxygen ion.  This lowers the chance for nono-\nscale aggregation or s econdary phase formation. 3 \n \n \nFig.2. Schematic representation of the ZnO wurtzite structure, with Eu ions incorporated into the host lattice in \nConfiguration 1, unrelaxed (a) and relaxed (b) and Configuration 2, unrelaxed (c) and relaxed (d).  Note the \nmagnitude of the relaxation, particularly of Eu-O bond length.  The magnitude of the relaxation in Eu-O bond \nhas decreased significantly as Eu-Eu separation has increased. \n \nDue to Eu’s larger radius, it is expected that  incorporation of Eu ions in ZnO introduces \nlocal lattice distortion.  Such lattice distor tion is examined by analysing the difference \nbetween the Eu-O bond length in unrelaxed and relaxed structures of both configurations.  \nAccording to Fig. 2(a) the Eu-O bond length in unr elaxed structure which is identical to Zn-O \nbond length in un-doped ZnO, is 1.991 Å along c direction and 1.975 Å within ab plane. In \nConfiguration 1, after relaxation of ionic coordinates the Eu-O bond length expanded \ndramatically to 2.300 Å and 2.312 Å (~ 14%) along c direction and 2.248 Å and 2.299 Å (~ \n14%) within ab plane. \nAs the separation of Eu ions increases in configuration 2, a slight decrease in the \nmagnitude of the Eu-O bond length expansion is observed.  Its Eu-O bond length is expended \nto 2.363 Å (12%) along c directio n and 2.210 (11%).  This indi cates the expansion strain \ncaused by substitutional Eu ions reduces as their separation increase.  \nTable 1. E\nt, ∆E and DEu-Eu are presented for the ZnO:Eu system.  Et and DEu-Eu are shown for \nthe relaxed AFM structure .  Et is presented with respect to an  arbitrary origin which puts the \nhighest calculated energy at zero. \n \nConfiguration Et(meV) ∆E(meV) DEu-Eu(Å) \n1 (near) 0 -20 3.753 \n2 (far) -44 -40 4.460 \n \nMagnetically ∆E is -20 meV for Configuration 1 and -40 meV for Configuration 2.  \nNegative values for ∆E in both configurations suggest an antiferromagnetic interaction \nbetween Eu ions separated either by a single oxygen ion or the chain of -O-Zn-O- ions at 0 K.  \nSince the energy associated with thermal electroni c fluctuations in a given temperature is in \nthe range of ~ kBT, approximately 25.8 meV for T = 300 K, these values of  ∆E do not \nguarantee any effective coupling at room temper ature. Therefore, paramagnetic behaviour is \npredicted at room the ZnO:Eu system. \nIn the next step, simulation was repeated on defective ZnO:Eu systems.  Two different \ndefects were considered: (a) oxygen vacancy (V O) and (b) interstitial Zn (Zn I).  They were \nintroduced into Configuration 2.  ∆E for the system with V O was -5 meV which excludes the \npossibility of V O mediating the observed ferromagnetic inte raction in the ZnO:Eu system.  In \nthe defective system with Zn I, ∆E rises to +152.2 meV, which proves that the ferromagnetic 4 \n state is much more stable than antife rromagnetic state in the presence of Zn I.  In this system \nthe magnitude of ∆E is almost 5 times larger than kBT at room temperature which guarantees \nTC above 300 K for this system. \n \n5 Conclusion \nRoom temperature magnetism was observed in for Zn 0.97Eu0.03O and Zn 0.96Eu0.04O \nsystems.  In search for the origin of observe d magnetism, theoretical investigation revealed: \n(1) Eu ions do not tend to aggregate via oxygen, wh ich reduces the chance of the formation of \nnano-scale magnetic secondary phase. (2) Ferrom agnetic interaction is mediated by defects \nsuch as Zn I.  However V O does not stabilizes the ferromagnetic phase. \n \nAcknowledgments \n This work was supported by Australian Research Council (Grant Nos. DP0770424 and \nDP0988687) and Australian Institute  of Nuclear Science and E ngineering (AINSE award, \nAINGRA09118). \n \nReferences \n                                                 \n[1] Ohno H 1998 Science  281 951 \n[2] Sato K and Katayama-Yoshida H 2002 Semicond. Sci. Tech.  17 367  \n[3] Pan F, Song C, Liu X J, Yang Y C and Zeng F 2008 Mat. Sci. Eng. R.  62 1 \n[4] Segall M D, Lindan P J D, Probert M J, Pick ard C J, Hasnip P J, Clark S J, and Payne M \n C 2002 J. Phys.-Condens. Mat.  14 2717  \n "    },    {        "title": "1201.3680v1.Spin_selective_Kondo_insulator__Cooperation_of_ferromagnetism_and_Kondo_effect.pdf",        "content": "arXiv:1201.3680v1  [cond-mat.str-el]  18 Jan 2012Spin-selective Kondo insulator: Cooperation of ferromagn etism and Kondo eﬀect\nRobert Peters∗and Norio Kawakami\nDepartment of Physics, Kyoto University, Kyoto 606-8502, J apan\nThomas Pruschke\nDepartment of Physics, University of G¨ ottingen, 37077 G¨ o ttingen, Germany\n(Dated: July 27, 2018)\nWepropose thenotion ofspin-selectiveKondoinsulator, wh ichprovidesafundamentalmechanism\nto describe the ferromagnetic phase of the Kondo lattice mod el with antiferromagnetic coupling.\nThis unveils a remarkable feature of the ferromagnetic meta llic phase: the majority-spin conduction\nelectrons show metallic- while the minority-spin electron s show insulating-behavior. The resulting\nKondogap in the minority spin sector, which is due tothe coop eration of ferromagnetism andpartial\nKondo screening, evidences a dynamically-induced commens urability for a combination of minority-\nspin electrons and parts of localized spins. Furthermore, t his mechanism predicts a nontrivial\nrelation between the macroscopic quantities such as electr on magnetization, spin polarization and\nelectron ﬁlling.\nPACS numbers: 71.10.Fd 71.27.+a 71.30.+h 75.20.Hr\nEven 30 years after their discovery heavy-fermion sys-\ntems attract much attention due to their fascinating\nproperties. Apart from being Fermi liquids with ef-\nfective mass thousand times as large as the free elec-\ntron one, they show all kinds of competing or coexisting\nphases, and at the boundaries between these phases one\nfrequently observes quantum phase transitions, accom-\npanied by barely understood non-Fermi liquid behavior\n[1–3]. Heavy fermion compounds usually include lan-\nthanides or actinides with open 4 f- or 5f-shells, which in\nthe simplest theoretical modeling can be viewed as a reg-\nular lattice of local moments coupled to the conduction\nelectrons. This coupling typically leads to twocompeting\nmechanisms: the long-ranged RKKY interaction and the\nlocal Kondo screening. While the RKKY interaction fa-\nvorsa magnetically ordered state, the Kondo screening is\nusuallyconsideredtoformaparamagneticheavy-fermion\nstate. The competition of these two mechanisms can be\neasily understood in terms of the Doniach phase diagram\n[4].\nWhile in most heavy-fermioncompounds the magnetic\norder is antiferromagnetic, there are a certain class of\ncompounds showing ferromagnetic order. For example,\nthe recently discovered YbNi 4P2is a ferromagnetically\norderedheavy-fermioncompound which seems to be very\nclose to a quantum critical point [5]. Taking such fer-\nromagnetic heavy fermion compounds as motivation we\nanalyze in detail the mechanism stabilizing the ferromag-\nnetic state. An interesting question in this context is, if\nand how the Kondo eﬀect accounts for the ferromagnetic\nstate [6–9].\nIn this letter, we propose a spin-selective Kondo in-\nsulator, where the Kondo screening plays an essential\nrole in stabilizing the ferromagneticmetallic state at zero\ntemperature, which elucidates a previously unrecognized\nfeature of the ferromagnetic phase: the majority-spin(minority-spin)conductionelectronsareinametallic(in-\nsulating) state. We claim that this notion is not spe-\nciﬁc to certain choices of system parameters but is fun-\ndamental and ubiquitous for the ferromagnetic phase in\nthe Kondo lattice model. Due to partial Kondo screen-\ning, parts of the local moments are bound to the elec-\ntrons, resulting in a dynamically-induced commensura-\nbility which is essential for producing the gap in the mi-\nnorityspinelectrons. Weﬁndthatthiscommensurability\ncondition leads to a nontrivial relation between electron\nmagnetization, spin polarization and electron ﬁlling.\nThe competition or cooperation between the magnetic\nphase mediated by the RKKY interactionand the Kondo\nscreening can be modeled via a Kondo lattice model with\nantiferromagnetic coupling between the local moments\nand the conduction electrons. The Kondo lattice model\nreads [4, 10, 11],\nH=t/summationdisplay\n<i,j>σc†\niσcjσ+J/summationdisplay\ni/vectorSi/vector si\n/vector si=c†\niσm/vector ρσmσnciσn,\nwherec†\niσcreates an electron on site iwith spin-direction\nσ,/vector ρrepresentsthe vectorofPauli-matrices,and /vectorSirepre-\nsentsthelocalspinswhicharecoupledtotheelectronsvia\nan antiferromagnetic spin-spin interaction with strength\nJ >0.\nTo solve the Kondo lattice model we use the dynam-\nical mean ﬁeld theory (DMFT) [12–14]. DMFT maps\nthe lattice model onto a quantum impurity model with\na fermionic bath being determined self-consistently. Al-\nthough being an approximation to real systems, DMFT\nhas provided many insights into the physical properties\nand can even captures subtle diﬀerences in the lattice\ngeometry. For solving the impurity model, we use the\nnumerical renormalization group (NRG) [15, 16], which2\n-0.5 00.5\nω/W00.511.5ρ(ω)W\n-0.5 00.5\nω/W-0.0500.05\nω/W\nincreasing n↓ increasing n↑n↓n↑\n0.011\n0.028\n0.077\n0.091\n0.1340.054\n0.108\n0.181\n0.197\n0.241n↓n↑\nFigure 1: (Color online) Spin-resolved spectral functions for\nthe ferromagnetic state in the Kondo lattice model J/W=\n0.25. The inset shows a magniﬁcation around the Fermi en-\nergy for the spin-down component illustrating the gap in the\nspectral function.\nis able to reliably calculate spectral functions at very low\ntemperatures [17, 18].\nFirst, we brieﬂy summarize the known DMFT results\nfor the Kondo lattice model [10, 11, 19, 20] (A discussion\non the RKKY interaction within DMFT can be found in\n[19].) At half ﬁlling there is a pronounced antiferromag-\nnetic N´ eel state for weak coupling, which vanishes with\nincreasingcouplingstrength Jviaacontinuoustransition\nto a paramagnetic insulating state, the Kondo insula-\ntor. Doping slightly away from half ﬁlling this transition\nchanges into a transition between an antiferromagnetic\nstate (possibly spin-density-wave) and a paramagnetic\nmetallic state. Especially the paramagnetic state around\nhalf ﬁlling is dominated by the Kondo eﬀect, where the\nKondoscreeningoflocalizedspinsresultsin alargeFermi\nsurface accompanied by a narrow band and a gap close\nto the Fermi energy. Away from half ﬁlling the eﬀects\nof Kondo screening become less important as there is an\nimbalance between local moments and available conduc-\ntion electrons. Such a tendency might be even stronger\nwhen the system enters a ferromagnetic state realized\nat low ﬁllings, because an additional imbalance between\nspin-up and spin-down electrons arises. Contrary to this\nnaive expectation, however, we demonstrate here that\nthe Kondo screening plays an essential role even in the\nferromagnetic phase. In particular, we reveal that the\ncooperation of ferromagnetism and Kondo screening can\nrealize a novel kind of Kondo insulating state in the fer-\nromagnetic metallic phase.\nFigure1shows the local spin-resolved spectral-\nfunctions calculated in the ferromagnetic phase for a\nBethe lattice with antiferromagnetic Kondo coupling\nJ/W= 0.25 (bandwidth W= 4t). For this coupling\nstrength the ferromagnetic phase extends from a nearly1 2 3 4 56\nW/J0.0010.010.11gap size ∆/Wn↓=0.1\nn↓=0.2\n0.1 0.2n↓0.10.20.3gap size ∆/WJ/W=0.3\nJ/W=0.4\nJ/W=0.5\nFigure 2: (Color online) Gap width ∆ /Win the minority-\nspin spectral function depending on the spin-coupling Jand\nthe occupation nc\n↓. The temperature of the system is T/W=\n3·10−4. The lines in the left panel are ﬁts as ∼exp(−a/J).\nempty system to approximately nc=nc\n↑+nc\n↓= 0.5.\nOne ﬁnds a striking diﬀerence in the spectral functions,\nwhich has not been recognized previously, for the major-\nity spin ( nc\n↑) and the minority spin ( nc\n↓). While in the\nmajority-spin spectral function a peak at the Fermi en-\nergyω= 0 and a dip for ω >0 can be found, there is\na gap at the Fermi energy in the minority-spin spectral\nfunction. It is important to note that such a gap is not\npresent in the ferromagnetic phase for a Kondo lattice\nmodel with ferromagnetically coupled spins. We propose\nthat this gap in the spectral function is due to a partial\nKondo screening of the localized spins, which results in\nan intriguing state: although the ferromagnetic state is\nmetallic, only the majority-spin electrons contribute to\nthe low-temperature properties, in particular transport.\nTheminorityspins,eventhoughnotcompletelydepleted,\nform an insulator, which we name spin-selective Kondo\ninsulator .\nIncreasing the occupation number, the dip in the\nmajority-spinspectral function movescloser to the Fermi\nenergy and becomes more pronounced. Eventually, the\nferromagnetic state is replaced by a paramagnetic state,\nforwhichthespectralfunctionsforbothspin-components\nsuﬀer from the typical suppression of the DOS for ω >0\ndue to Kondo screening. Increasing the occupation to-\nwards half ﬁlling this dip becomes deeper and ﬁnally\nmoves to the Fermi energy, forming the Kondo insula-\ntor.\nClear evidence showing that the above insulating gap\nis indeed caused by the Kondo screening can be found\nin the dependence of the gap width ∆ on the occupation\nand coupling strength, shown in Fig. 2. The left panel\nin Fig.2displays the dependence of the gap width on\nthe coupling strength. For this purpose the minority-\nspin occupancy was kept constant (also resulting in a3\n0 0.2 0.4 0.60.8\nFilling n=n↑+n↓00.51Magnetization\n0.2 0.4 0.60.8\nFilling n=n↑+n↓<m>=<n↑>-<n↓>\n-<Sz>\ncommensurabilityJ/W=0.3 J/W=0.5\nc-electrons\nf-electrons\nsinglet\nnc↓+nf↓= 1\nFigure 3: (Color online) Upper panel: Magnetization and\n“commensurability”, ( nc\n↓+nf\n↓), for two diﬀerent coupling\nstrengths and diﬀerent occupation numbers calculated for a\nBethe lattice at T/W= 3·10−4. The electron magnetization\nis shown as /angbracketleftm/angbracketright=nc\n↑−nc\n↓, while the spin expectation value\n/angbracketleftSz/angbracketrightis shown mirrored as −/angbracketleftSz/angbracketright. The commensurability-\ncondition is explained in the text. Lower panel: sketch of th e\nlocal conﬁguration (see text).\nnearly constant majority-spin occupancy). The depen-\ndency on Jperfectly obeys a Kondo temperature-like\nform ∆ ∼exp(−a/J) with a ﬁtting constant a, sug-\ngesting that the Kondo physics is essential for the gap-\nformation. As a function of increasing ﬁlling the gap\nwidth decreases monotonically, as shown in the right\npanel of Fig. 2. As soon as the ferromagnetic phase van-\nishes, the gap at the Fermi energy closes, too, and the\nminority and majority spectral functions look similar to\nthe right panel in Fig. 1.\nLet us now elucidate the basic physics behind this fer-\nromagnetic state. In Fig. 3the magnetization of the\nconduction electrons /angbracketleftm/angbracketright=nc\n↑−nc\n↓and the polariza-\ntion of the localized spins −/angbracketleftSz/angbracketrightis shown. Note that\nthe local spin-polarization always has the sign opposite\nto the conduction electron magnetization due to the an-\ntiferromagnetic coupling, thus −/angbracketleftSz/angbracketrighthas the same sign.\nIncreasing the number of conduction electrons, the mag-\nnetization of the electrons ﬁrst increases due to increas-ing ﬁlling, and eventually decreases again due to the\nsuppression of the ferromagnetic state. On the other\nhand, the spins are almost fully polarized for a nearly\nempty lattice, with monotonically decreasing polariza-\ntion for increasing conduction electron number. In the\nspirit of a pseudo-fermion representation, let us assume\nthat the localized spins are actually formed by a local\nhalf-ﬁlled and strongly-interacting energy level so that\n/angbracketleftSz/angbracketright= (nf\n↑−nf\n↓)/2 andnf\n↑+nf\n↓= 1 (deﬁning nf\nσas the\nspin-dependent occupation of this level). Remarkably,\nwe ﬁnd that the following nontrivial commensurability\ncondition holds within the ferromagnetic state:\nnc\n↓+nf\n↓= 1, (1)\nas can be seen in Fig. 3. Note that Eq. ( 1) is equiva-\nlent tonf\n↑=nc\n↓. It should be noticed that this condition\nis nota priori given but is generated dynamically due\nto many-body eﬀects. To clarify the origin of the above\ncommensurability we propose that a partial local Kondo-\nsinglet is formed in which /angbracketleftnc\n↓/angbracketrightmajority- and minority-\nelectrons participate, thus combining all spin-down con-\nduction electrons together with a part of the f-electrons\nand the spin-up conduction electrons to a Kondo spin-\nsinglet. Here, we have assumed that spin-down is the\nminority-spin direction. The remaining majority-spin\nconduction electrons and spin-down f-electrons form a\nferromagnetic state. (see a sketch in the lower panel of\nFig.3). That the number of spin-down electrons in-\ncludingf- and conduction-electrons sums up to unity\ngives a commensurable situation, which results in a gap\nattheFermi energy. Ontheotherhand, forthe majority-\nspins there is not such a commensurabilitycondition, but\nnc\n↑+nf\n↑=nc=nc\n↑+nc\n↓holds. Therefore, this par-\ntial Kondo screening results in an insulating state for\nthe minority-spin, while the majority-spin electrons re-\nmain metallic. For this reason we have called this state\na “spin-selective Kondo insulator”. The commensurabil-\nity condition ( 1) smoothly connects to the Kondo insu-\nlator at half ﬁlling, suggesting that this ferromagnetic\nstate should exist up to half ﬁlling. However, our results\nclearlyshowthat there is atransition fromthis ferromag-\nnetic phase to a paramagnetic state at electron ﬁllings\nfornc> nferro. This is only possible, if the expectation\nvalue/angbracketleftSz/angbracketrightjumps, leading to a discontinuous phase tran-\nsition at nferro. Our ﬁnding of the discontinuous transi-\ntion completely agrees with the recent analytical results\nshowing that non-analytic terms prevent the continuous\ntransition from a ferromagnet to a paramagnet [21].\nA further important consequence deduced directly\nfrom the commensurability condition ( 1) is a nontrivial\nrelation between electron magnetization, spin polariza-\ntion and occupation number:\n2/angbracketleftSz/angbracketright+/angbracketleftm/angbracketright=/angbracketleftnc/angbracketright−1. (2)\nThis formula connects these three quantities which are\notherwise independent from each other. By arranging it-4\nFigure 4: (Color online) Momentum-resolved spectral func-\ntions for the square lattice and J/W= 0.3,n=nc\n↑+nc\n↓=\n0.25,m=nc\n↑−nc\n↓= 0.1. The right side always shows a\nmagniﬁcation of the left side around the Fermi energy ω= 0\nrepresented by the green line. From top to bottom the ﬁg-\nures show the majority-spin and the minority-spin spectral\nfunction, respectively.\nFigure 5: (Color online) Momentum-resolved occupation\nnumber n(k) (same parameters as in Fig. 4). Left (right)\npanel shows the majority- (minority-) spin component. The\ndotted line represents the Fermi surface for non-interacti ng\nelectrons (for the majority-spin this lines coincides with the\nshown surface). Note that for improving the contrast, the\noccupation for the minority-spin electrons is displayed in the\nintervaln(k)∈[0,0.3].\nself in this way the system can gain an additional energy\noriginating from the partial Kondo screening. Note that\nit should be possible to verify such a relation experimen-\ntally.\nThe formation of the gap in the spectral function\ndoes not depend on the lattice geometry. For exam-\nple, it can also be found in DMFT calculations for\na two-dimensional square lattice. Figure 4shows the\nmomentum-resolved spectral functions for J/W= 0.3\nandnc\n↑+nc\n↓= 0.25 for a square lattice. The right\npanels are magniﬁcations around the Fermi energy. In\nthe minority-spin spectral function (bottom right) the\ngap can be clearly seen. While the majority-spin elec-\ntrons are renormalized for ω >0 with a ﬁnite life-time,\nthe minority-spin electrons are renormalized for ω <0.\nThis behavior is also visualized in Fig. 5, in which themomentum-resolved occupation number is shown. While\nfor the majority-spin electrons the occupation number\ndistribution looks like in the non-interacting case, for the\nminority-spin electrons this function is actually smeared\nout as compared to a reference non-interacting system,\ni.e. we indeed observe a large Fermi volume here.\nUsing DMRG, we have conﬁrmed that the spin-\nselectiveKondoinsulatortogetherwith the commensura-\nbility condition can be found for the ferromagnetic phase\nofthe one-dimensional(1D) Kondolattice model, too. In\nfact, previous calculations for the ferromagnetic ground\nstateobservedamagnetization Stot= 1/2(L−Nc)[22](L\nsystem length, Ncelectron number), which supports the\ncommensurability condition, and two separated bands in\nthe spectral functions [23]. From these precise analyses\nof the 1D model, we conclude that our ﬁnding is ubiqui-\ntous for the Kondo lattice model and not an artifact of\nDMFT.\nIn conclusion we have clariﬁed the physics behind the\nferromagnetic metallic phase realized in the Kondo lat-\ntice model. We have demonstrated that the cooperation\nof ferromagnetism and partial Kondo screening results\nin an intriguing phase, here named spin-selective Kondo\ninsulator, where an insulating state is stabilized for the\nminority-spin electrons while the majority-spin electrons\nare still metallic. We believe that the mechanism pro-\nposed here, the dynamically generated commensurabil-\nity, should be generic for the ferromagnetic phase in the\nKondo lattice models. It alternatively provides the non-\ntrivial relation between the electron magnetization, spin\npolarization and occupation number, for which the sys-\ntem can gain a maximum of additional energy. The pro-\nposed relation between the macroscopic quantities might\nbe conﬁrmed in experiments. Good candidates in this\ncontext are ferromagnetic heavy fermion compounds, es-\npecially compounds having a large Kondo temperature.\nFor such compounds a veriﬁcation of the above stated\nrelation might be possible. Furthermore, spin-resolved\ntransport measurements should show metallic majority-\nspin but insulating minority-spin electrons as well as a\nlarge Fermi surface for the minority-spin component.\nWe acknowledge fruitful discussions with A. Koga.\nRP thanks the Japan Society for the Promotion of\nScience (JSPS) and the Alexander von Humboldt-\nFoundation. TP also gratefully acknowledges support\nby JSPS through the Bridge program. NK is supported\nby KAKENHI (Nos. 21540359, 20102008) and JSPS\nthrough its FIRST Program.\n∗peters@scphys.kyoto-u.ac.jp\n[1]P. Coleman, Handbook of Magnetism and Advanced Mag-\nnetic Materials (John Wiley and Sons, 2007), p. 95.\n[2]P. Coleman and A. Schoﬁeld, Nature 443, 226 (2005).\n[3]P. Gegenwart and Q. Si, Nature Physics 4, 186 (2008).5\n[4]S. Doniach, Physica B 91, 231 (1977).\n[5]C. Krellner, S. Lausberg, A. Steppke, M. Brando, L. Pe-\ndrero, H. Pfau, S. Tenc´ e, H. Rosner, F. Steglich, and\nC. Geibel, New Journal of Physics 13, 103014 (2011).\n[6]W. Lee, H. Ku, and R. Shelton, Phys. Rev. B 38, 11562\n(1988).\n[7]N. Perkins, J. Iglesias, M. Nunez-Regueiro, and B. Co-\nqblin, Europhysics Letters 79, 57006 (2007).\n[8]S. Yamamoto and Q. Si, Proc. Natl. Acad. Sci. USA 107,\n15704 (2010).\n[9]G. Li, G. Zhang, and L. Yu, Phys. Rev. B 81, 094420\n(2010).\n[10]C. Lacroix and M. Cyrot, Phys. Rev. B 20, 1969 (1979).\n[11]P. Fazekas and E. Muller-Hartmann, Z. Phys. B: Con-\ndens. Matter 85, 285 (1991).\n[12]W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324\n(1989).\n[13]A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg,\nRev. Mod. Phys. 68, 13 (1996).\n[14]T. Pruschke, M. Jarrell, and J. Freericks, Adv. Phys. 44,187 (1995).\n[15]K. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n[16]R. Bulla, T. Costi, and T. Pruschke, Rev. Mod. Phys.\n80, 395 (2008).\n[17]R. Peters, T. Pruschke, and F. Anders, Phys. Rev. B 74,\n245114 (2006).\n[18]A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99,\n076402 (2007).\n[19]R. Peters and T. Pruschke, Phys. Rev. B 76, 245101\n(2007).\n[20]J. Otsuki, H. Kusunose, and Y. Kuramoto, J. Phys. Soc.\nJpn.78, 034719 (2009).\n[21]D. V. Efremov, J. J. Betouras, and A. Chubukov,\nPhys. Rev. B 77, 220401(R) (2008).\n[22]H.Tsunetsugu, M. Sigrist, andK.Ueda, Rev.Mod.Phys.\n69, 809 (1997).\n[23]S. Smerat, U. Schollw¨ ock, I. P. McCulloch, and\nH. Schoeller, Phys. Rev. B 79, 235107 (2009)."    },    {        "title": "0803.3922v1.Current_induced_persistent_magnetization_in_a_relaxorlike_manganite.pdf",        "content": "arXiv:0803.3922v1  [cond-mat.str-el]  27 Mar 2008Current-induced persistent magnetization in a relaxorlik e manganite\nH. Sakai1and Y. Tokura1,2,3\n1Department of Applied Physics, University of Tokyo, Tokyo 1 13-8656, Japan\n2Cross-Correlation Materials Research Program (CMRG), RIK EN, Wako 351-0198, Japan\n3Multiferroics Project, ERATO, Japan Science and Technolog y Agency (JST), Tokyo 113-8656, Japan\nA single crystal of 7% Fe-doped (La 0.7Pr0.3)0.65Ca0.35MnO3shows up as a typical relaxor fer-\nromagnet , where ferromagnetic metallic and charge-orbital-ordere d insulating clusters coexist with\ncontrollable volume fraction by external stimuli. There, t he persistent ferromagnetic metallic state\ncan be produced by an electric-current excitation as the ﬁla mentary region, the magnetization in\nwhich is increased by ∼0.4µBper Mn. A clear distinction from the current heating eﬀect in a\nmagnetic ﬁeld, which conversely leads to a decrease in ferro magnetic fraction, enables us to bi-\ndirectionally switch both the magnetization and resistanc e by applying the voltages with diﬀerent\nmagnitudes.\nCorrelated electron systems with competing ordered\nstates often show a dramatic phase change in response\nto even a minute external stimulus. One such example\nis a hole-doped perovskite manganite, in which a keen\ncompetition between a charge-orbital-ordered insulating\n(CO) phase and a ferromagnetic metallic (FM) one is a\nmajorsourceformagnetic-ﬁeld-inducedphasetransitions\nor colossalmagnetoresistance(CMR) phenomena.[1] The\ndiﬀerence in free energy between the two states is so\nsmall near the bicritical phase boundary that an exter-\nnal perturbation other than a magnetic ﬁeld can also\ndrive the insulator-metal transition.[2, 3, 4, 5] Among\nthem, the electric-ﬁeld-induced collapse of the CO into\nthe conducting, perhaps FM, state has been intensively\ninvestigated since its ﬁrst discovery in Pr 0.7Ca0.3MnO3\ncrystal.[2] For such a current excitation, there has always\nbeen some concern about the Joule heating since the in-\nduced low-resistance state is occasionally similar in a re-\nsistance value to the high-temperature state due to the\ninherent semiconducting nature. In CMR manganites,\nthe observation of persistent changes in magnetization as\nwell as in resistance may be the direct evidence for the\ngenuine electric action on the phase change, although\nsuch studies have been rare[6, 7, 8]. In particular, the\ncurrent production of the persistent ferromagnetic state\nis also an intriguing issue in the light of electromagnet\nfunction in nanometric materials.\nIn this Letter, we have investigated electric-current-\ninduced eﬀects on a CMR manganite, especially on Fe-\ndoped (La 0.7Pr0.3)0.65Ca0.35MnO3. The Fe doping sup-\npresses the FM correlation while stabilizing the CO one,\nleading to the phase-separatedground state with the FM\nand CO clusters coexisting.[9] The feature is analogous\nto the case of Cr-doped CO manganites[10, 11], although\nthecharge-orbitalorderremainsshort-ranged( ≤2nm)[9]\nand forms the macroscopically near-isotropic lattice and\nelectronic structure in the present case. Such a system\nhas been known as a relaxor ferromagnet ,[11] showing\nglassy magnetic and magnetotransport properties, such\nas magnetic-ﬁeld annealing eﬀects and long-time relax-\nation phenomena. Furthermore, the controlled volumefraction of the FM and CO states can be durably ﬁxed\nat low temperatures due to the multistability character-\nistic of the relaxor system. This will give an ideal arena\nfor investigating the intrinsic eﬀect of external perturba-\ntions accompanying temporary heating, since the persis-\ntent change can be detected free from the heating prob-\nlem,aswasdoneinthephotoirradiationcase.[12]Wehere\npresent a direct evidence for the current-induced persis-\ntent FM state in distinction from the heating eﬀect in\nthe relaxor ferromagnet.\nAsinglecrystalof(La 0.7Pr0.3)0.65Ca0.35Mn0.93Fe0.07O3\nwas grown by the ﬂoating zone method, as detailed\nelsewhere[9]. Electrodes, made of heat-treatment-type\nsilver paint, were formed on the both end surfaces of the\ncrystal specimen. Current-voltage ( I-V) characteristics\nwere measured for the crystal connected in series with\nthe load resistor ( RL=1 MΩ) [see the inset to Fig. 1(b)].\nThe sample resistance Rwas measured at a constant\nvoltage Vmeaswith a two-probe conﬁguration; the\ncontact resistance was ∼10 Ω and can be safely ignored\nin the low-temperature high-resistance state. For the\nsimultaneous measurement of the magnetization Mand\nR, the sample mounted in a superconducting quantum\ninterference device magnetometer was connected to the\nI-Vmeasurement system through a co-axial cable.\nFigure 1(a) shows the typical I-Vcharacteristics at 4\nK for a crystal of (La 0.7Pr0.3)0.65Ca0.35Mn0.93Fe0.07O3,\nwhich is insulating down to the lowest temperature [see\nFig. 1(b)]. For this measurement, the sample was ﬁrst\ncooled to 4 K at 0 T and then a bias voltage was swept\nas 0 V→Vmax→0 V. This sweep was repeated with\nsuccessively increasing Vmaxfrom 250 V to 900 V. For\nVmax≤450 V, the system remains at the initial high-\nresistance state, exhibiting the nonlinear I-Vcurve but\nwith no (or minimal) hysteresis. For Vmax=500 V, how-\never, the I-Vcurve displays a marked hysteresis, indicat-\ningthatthe systemisswitchedtothe low-resistancestate\nwith the Rdrop by∼5 orders of magnitude. Similar hys-\nteretic curveswith smallerwidths arealsoobservedwhen\nVmaxis further increased up to 700 V, accompanying a\ngradualdecreasein Rbyanotheroneorderofmagnitude.2\nFIG. 1: (Color online) (a) I-Vcharacteristics for a single\ncrystal of (La 0.7Pr0.3)0.65Ca0.35Mn0.93Fe0.07O3at 4 K in zero\nmagnetic ﬁeld. Inset: change in resistance Ras a func-\ntion of the maximum voltage Vmax(resistance-sensing volt-\nageVmeas= 250 V). (b) Temperature proﬁles of resistance\n(Vmeas=50 V) in warming runs at 0 T after applying Vmax\n(0, 450 and 900 V) at 4 K. Data measured after magnetic-\nﬁeld cooling (FC) at 1 T are also shown. Inset: a schematic\ndiagram of the circuit; the electrode distance d∼70µm, the\nelectrode area A∼700×800µm2, and the load resistance\nRL=1 MΩ.\nForVmax≥800 V, a sudden increase of current occurs at\n∼750 V probably due to the current heating eﬀect. In\nfact, this lower-resistance state is not persistent and eas-\nilyvanishesfor V≤750V. Inthehigh-voltageregime, the\ncurrentheatingwouldtransientlyraisethelocaltempera-\nture of the crystal above the glass transition temperature\n(vide infra ), leading to no further persistent decrease in\nR. The inset to Fig. 1(a) summarizes the change in R\nversusVmax, measured after each voltage sweep.\nWe show in Fig. 1(b) the temperature proﬁles of R\nin warming runs at 0 T after applying the bias voltage\n(0, 450, and 900 V) at 4 K at 0 T. As a comparison,\nwe also display the corresponding data measured after\nmagnetic-ﬁeld cooling (FC) at 1 T. Regardless of a cur-\nrent or a magnetic ﬁeld, the induced low-resistance state\nsurvives only at low temperatures (below ∼50 K), where\nthe system exhibits a spin-glass phase, and it returns to\nthe pristine high-resistance state by warming tempera-\nture. Therefore, the eﬀect of a current excitation should\nbe attributed not to the dielectric breakdown caused by\nsome permanent lattice-structural damage, but to the\nmodiﬁcation of the local electronic/lattice state, i.e., the\npersistent increase in the metallic volume fraction.\nTo directly detect the possible ferromagnetism in the\ncurrent-induced metallic state, we have concurrently\nmeasured MandRwith applying bias voltages. Fig-\nure 2(a) displays their variation at 3 K after FC at 0.4\nT. Dotted vertical arrows and voltage values indicate the\ntiming for the voltage sweep and its Vmax, respectively,\nwhereVmaxwas successively increased. The correspond-\nFIG. 2: (Color online) (a) Variation of magnetization (∆ M)\nand resistance ( Vmeas= 150 V) for a single crystal of\n(La0.7Pr0.3)0.65Ca0.35Mn0.93Fe0.07O3(d∼50µm,A∼300×400\nµm2) at 3 K at 0.4 T after FC. Dotted vertical arrows indicate\nthe timing for applying bias voltages. (b) I-Vcharacteristics\ncorresponding to the bias voltage sweeps ( RL=1 MΩ). Inset:\nresistance R(Vmeas=50 V) versus magnetization Mat 3 K\nat 0.4 T after FC with various magnitudes of magnetic ﬁelds.\ningI-Vcurves are shown in Fig. 2(b). Mis barely\nsaturated at 0.4 T while the system shows still insulat-\ning down to 3 K. Note that temporal drift of Mreﬂects\nthe slow dynamics in the glassy system. When applying\nVmax≤250 V, no change was discerned in MwhileR\nexhibits a small drop. By further increasing Vmaxup to\n300 and 350 V, we observed a marked increase in Mas\naccompanied by a large drop in R. This ensures that\nthe FM state is induced persistently by a current excita-\ntion, which we hereafter call the “intrinsic” eﬀect. After\napplying Vmax=400 V, on the other hand, we found a\ndecrease in Mas well as an increase in R. Such a de-\ncrease in FM fraction may stem from the rapidtempera-\nture cycle, consisting of the abrupt temperature rise due\nto the Joule heating and the subsequent rapid cooling\nback to 3 K. This was already observed for Vmax=400\nV in Fig. 2(b) as a sharp increase in current at ∼350 V\nin a voltage-increasing run and a decrease at ∼230 V in\na voltage-decreasing run. The slowly ﬁeld-cooled state\nshould be the least aﬀected in principle by the slowsim-\nilar thermal cycles. However, the rapid cooling may lead3\nto a smaller volume fraction of the FM state, since its\nevolution induced by a magnetic ﬁeld is subject to the\nlong-time relaxation eﬀect.[11] Note that the Joule heat-\ning similarly observed at zero ﬁeld (for Vmax≥800 V)\ngives no inﬂuence on the FM fraction, as shown in Fig.\n1(a). Thus, the heating phenomenon in a magnetic ﬁeld\nis clearly separated as the extrinsic electric-current eﬀect\nfrom the intrinsic one; they have the opposite impacts on\nthe FM volume fraction.\nThe total increase in Minduced by the intrinsic cur-\nrent eﬀect amounts to 2 .0×10−5emu at 3 K at 0.4 T. In\ncase of the FM state produced homogeneously over the\ncrystal, the net current-induced Mwould be as small\nas 3.3×10−3µBper Mn. This is, however, completely\ncontradictory to the large Rdrop by∼5 orders of mag-\nnitude. It is likely that the FM state is induced only\nlocally, forming the ﬁlamentary path between the elec-\ntrodes. When the FM clusters are evenly distributed\nover this crystal, the values of MandRat 3 K at 0.4 T\nshould have the relation shown in the inset to Fig. 2(b),\nwhich was obtained by changing the FM fraction with\nvarying the magnitude of a magnetic ﬁeld in FC. Pro-\nvided that only the ﬁlamentary region formed by current\nﬂowing changes the FM fraction obeying the above rela-\ntion, we can roughly estimate its cross-sectional size so\nas to make the experimental results self-consistent; this\nleads to the conclusion that the FM state with M∼1.4\nµBper unit cell and R∼1.4×105Ω [indicated by the\nthick horizontal arrow in the inset to Fig.2(b)] should be\nformed in ∼1/120 of the area of the crystal ( ∼400×300\nµm2). Consequently, ∆ Minduced in such a ﬁlamentary\nregion is ∼0.40µBper Mn, being comparable to the\nphotoinduced case.[12] The estimated sub-mm size of the\nﬁlament also appears to be consistent with that observed\nby the microscopy for Pr 0.7Ca0.3MnO3crystal.[13]\nMaking use of the opposite actions of the intrinsic and\nextrinsic (heating) current-excitation eﬀects, we demon-\nstrate the cyclic electric control of both MandR. Fig-\nure 3 shows the typical switching characteristics with\nchanging the magnitude of the applied voltage, measured\nat 20 K at 0.4 T. The detailed operation is as follows.\nAfter FC to 20 K, we ﬁrst checked that the applica-\ntion ofVmax≤300 V exhibited the intrinsic eﬀect while\nVmax≥320 V was enough to cause the heating eﬀect.\nWe then set the state by applying 320 V as the initial\none, where the FM fraction was reduced by the afore-\nmentioned heating and subsequent quenching processes.\nBy applying 300 V to this state, a fraction of the FM\nstatewasrecoveredduetotheintrinsiccurrent-excitation\neﬀect, exhibited as an increase in M(by∼6×10−5\nemu) and a decrease in R(by∼1 order of magnitude).\nThe subsequent application of320V gaveagaintransient\nheating and decreased the FM component, returning M\nandRback almost to the starting values. Repeating\nthese two processes (“intrinsic” and “heating”) leads to\nthe cyclic switching as shown in Fig. 3.\n\u0001 \u0002 \u0000 \u0003 \u0004 \u0005 \u0006 \u0007 \b\n\t \n \u000b \f \r\n\u000e \u000f \u0010 \u0011 \u0012\n\u0013 \u0014 \u0015 \u0016 \u0017\n\u0018 \u0019 \u001a \u001b \u001c\u001d \u001e\n\u001f\n !\n\"\n# $\n%& ' ( ) * + ,\n-./0123456789:;<\n=>\n?@ABCDEFGHIJKLMNO\nP Q R ST U V W X Y\nZ [ \\ ]^ _ ` a b c d e f g hi j k lm n o p q r s t u\nv w x yz { | }\n~      \n    \n   \n   \n    \n   \n¡ ¢£\n¤¥\n¦ § ¨ © ª« ¬ \nFIG. 3: (Color online) Switching operation of magne-\ntization (ﬁlled circle) and resistance (open circle) for a\n(La0.7Pr0.3)0.65Ca0.35Mn0.93Fe0.07O3single crystal ( d∼50\nµm,A∼300×400µm2) at 20 K at 0.4 T after FC. The\nmagnetization and resistance ( Vmeas=150 V) were measured\nat regular intervals. Vertical arrows indicate the timing f or\napplying the switching voltages.\nThis switching operation can be performed by using\nthe voltage pulses (a few hundreds milliseconds), which\nis reminiscent of the colossal electroresistance memory\n(CERM) eﬀect, as has been reported for the hole-doped\nCO manganites.[14, 15] The CERM phenomenon is the\nreversible resistance switching with bipolar (sometimes\nnonpolar) electric pulses, observed at room temperature\nfor various transition-metal oxide ﬁlms.[16, 17] Its mech-\nanism is still under controversy but one of the plausible\nscenarios is that a ﬁlamentary conducting path gener-\nated by the electric stress, such as the soft breakdown,\nis aﬀected by the redox reaction due to the current heat-\ning eﬀect.[18] Our result, on the other hand, presents\nthe similarresistanceswitchingbycontrollingthe volume\nfraction of the (ﬁlamentary) metallic state, which can be\nregarded as a purely electronic CERM without suﬀering\nfrom the redox reaction (e.g. oxygen ion drift), although\nit works only at low temperatures at the moment.\nInconclusion, wehaveobservedapersistentincreasein\naferromagnetic-metallic(FM) volumefractionin asingle\ncrystalof (La 0.7Pr0.3)0.65Ca0.35Mn0.93Fe0.07O3by apply-\ning voltages (300-500 V) between the electrodes (50-70\nµm) . The current-induced FM state is anticipated to\nform a ﬁlamentary pathway, where the net increase in\nmagnetization is estimated as ∼0.4µBper Mn. Utiliz-\ning the intrinsic current-excitation eﬀect and the Joule\nheating one in a magnetic ﬁeld, the latter of which con-\nversely decreases the FM fraction, we have demonstrated\nreproducible switching of both the magnetization and re-\nsistance by changing the magnitude of the pulse voltage.\nSuch an electric control of the conducting and magnetic\nstates for a relaxorlike manganite may ﬁnd a route to\noxide electronic devices in the future.\nWe thank R. Kumai, Y. Onose, and S. Iguchi for4\nfruitful discussions. This work was partly supported by\nMEXT TOKUTEI (16076205) and JSPS Fellows\n[1] Y. Tokura, Rep. Prog. Phys. 69797 (2006).\n[2] A. Asamitsu, Y. Tomioka, and Y. Tokura, Nature (Lon-\ndon)38850 (1997).\n[3] K.Miyano, T.Tanaka, Y.Tomioka, andY.Tokura, Phys.\nRev. Lett. 784257 (1997).\n[4] V. Kiryukhin, D. Casa, J. P. Hill, B. Keimer, A.\nVigliante, Y. Tomioka, and Y. Tokura, Nature (London)\n386813 (1997).\n[5] Y. Moritomo, H. Kuwahara, Y. Tomioka, and Y. Tokura,\nPhys. Rev. B 557549 (1997).\n[6] J. Stankiewicz, J. Ses´ e, J. Garc ´ia, J. Blasco, and C. Rillo,\nPhys. Rev. B 6111236 (2000).\n[7] A. Guha, N. Khare, A. K. Raychaudhuri, C. N. R. Rao,\nPhys. Rev. B 62R11941 (2000).\n[8] G. Garbarino, C. Acha, P. Levy, T. Y. Koo, S-W.\nCheong, Phys. Rev. B 74100401 (2006).\n[9] H. Sakai, K. Ito, R. Kumai, and Y. Tokura, Phys. Rev.B76155112 (2007).\n[10] B. Raveau, A. Maignan, and C. Martin, J. Solid State\nChem.130162 (1997).\n[11] T. Kimura, Y. Tomioka, R. Kumai, Y. Okimoto, and Y.\nTokura, Phys. Rev. Lett. 833940 (1999).\n[12] Y. Okimoto, Y. Ogimoto, M. Matsubara, Y. Tomioka,\nT. Kageyama, T. Hasegawa, H. Koinuma, M. Kawasaki,\nand Y. Tokura, Appl. Phys. Lett. 801031 (2002).\n[13] M. Fiebig, K. Miyano, Y. Tomioka, and Y. Tokura, Sci-\nence2801925 (1998).\n[14] S. Q. Liu, N. J. Wu, and A. Ignatev, Appl. Phys. Lett.\n762749 (2000).\n[15] A. Sawa, T. Fujii, M. Kawasaki, and Y. Tokura, Appl.\nPhys. Lett. 854073 (2004).\n[16] A. Beck, J. G. Bednorz, Ch. Gerber, C. Rossel, and D.\nWidmer, Appl. Phys. Lett. 77139 (2000).\n[17] S. Seo, M. J. Lee, D. H. Seo, E. J. Jeoung, D.-S. Suh, Y.\nS. Joung, I. K. Yoo, I. R. Hwang, S. H. Kim, I. S. Byun,\nJ.-S. Kim, J. S. Choi, and B. H. Park, Appl. Phys. Lett.\n855655 (2004).\n[18] Y. Ogimoto, Y. Tamai, M. Kawasaki, and Y. Tokura,\nAppl. Phys. Lett. 90143515 (20"    },    {        "title": "2204.12094v1.Bulk_domain_Meissner_state_in_the_ferromagnetic_superconductor_EuFe___2___As___0_8__P___0_2______2____Consequence_of_compromise_between_ferromagnetism_and_superconductivity.pdf",        "content": "Bulk domain Meissner state in the ferromagnetic superconductor EuFe 2(As 0:8P0:2)2: Consequence\nof compromise between ferromagnetism and superconductivity\nWentao Jin,1,\u0003Sebastian Mühlbauer,2,yPhilipp Bender,3Yi Liu,4Sultan Demirdis,5Zhendong\nFu,6Yinguo Xiao,7Shibabrata Nandi,8Guang-Han Cao,9Yixi Su,5,zand Thomas Brückel8, 5\n1School of Physics, Beihang University, Beijing 100191, China\n2Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universität München, D-85748 Garching, Germany\n3Physics and Materials Science Research Unit, University of Luxembourg,\n162A Avenue de la Faiencerie, L-1511 Luxembourg, Grand Duchy of Luxembourg\n4College of Science, Zhejiang University of Technology, Hangzhou 310023, China\n5Jülich Centre for Neutron Science JCNS at Heinz Maier-Leibnitz Zentrum (MLZ),\nForschungszentrum Jülich GmbH, Lichtenbergstrasse 1, D-85747 Garching, Germany\n6Neutron Platform, Songshan Lake Materials Laboratory, Dongguan 523808, China\n7School of Advanced Materials, Peking University Shenzhen Graduate School, Shenzhen 518055, China\n8Jülich Centre for Neutron Science JCNS and Peter Grünberg Institut PGI,\nJARA-FIT, Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany\n9Department of Physics, Zhejiang University, Hangzhou 310027, China\nSmall-angle neutron scattering (SANS) measurements are performed on the ferromagnetic superconductor\nEuFe 2(As0:8P0:2)2(Tsc= 22:5K) to probe the delicate interplay between ferromagnetism and superconduc-\ntivity. A clear signature of large ferromagnetic domains is found below the ferromagnetic ordering temperature\nTC= 18.5 K. In a small temperature interval of \u00181.5 K below TC, additional SANS signal is observed, of\nwhich the indirect Fourier transform reveals characteristic length scales in between \u001880 nm to \u0018160 nm.\nThese nanometer-scaled domain structures are identiﬁed to result from an intermediate inhomogeneous Meiss-\nner effect denoted domain Meissner state, which was recently observed on the surface of EuFe 2(As0:79P0:21)2\ncrystals by means of magnetic force microscopy [V . S. Stolyarov et al:, Sci. Adv. 4, 1061 (2018)], ascribing\nto the competition between ferromagnetism and superconductivity. Our measurements clearly render the do-\nmain Meissner state as a bulk phenomenon and provide a key solution to the mystery regarding the intriguing\ncoexistence of strong ferromagnetism and bulk superconductivity in these compounds.\nThe antagonistic nature of ferromagnetism (FM) and super-\nconductivity (SC) make the coexistence of these two states of\nmatter quite rare, as the strong exchange ﬁelds from a ferro-\nmagnet generally destroy the singlet Cooper pairing via the\nparamagnetic effect1. Theoretically, there are several scenar-\nios in which SC and FM can reach a compromise. Firstly, by\nforming multidomains or a so-called “cryptoferromagnetic”\nstructure such as spiral alignment of spins, the ferromagnetic\nstate can lower its detrimental effect on the SC2,3. Secondly,\nthe superconducting Cooper pairs can be \"polarized\" by the\nexchange ﬁelds and show non-zero momentum, leading to\ninhomogeneous SC in space called the Fulde-Ferrell-Larkin-\nOvchinnikov (FFLO) state4,5. Thirdly, the internal magnetic\nﬁeld from FM may penetrate the superconductor in the form\nof vortices, leading to a spontaneous vortex state6.\nThe coexistence of SC and FM was previously revealed\nin uranium-based heavy-fermion compounds UGe 2, URhGe,\nUCoGe and intercalated iron-selenide (Li,Fe)OHFeSe7–10.\nHowever, what coexists with the SC in most of these com-\npounds is weak FM with a small ordered moment. Recently,\nthe observation of an intriguing coexistence of bulk SC and\nstrong FM in EuFe 2As2-family iron pnictides (Eu122) with\nthe superconducting critical temperature Tsc\u001822 K and fer-\nromagnetic ordering temperature TC\u001817 K has attracted\nmuch attention11–14. Neutron diffraction experiments have\nconﬁrmed the ferromagnetic ordering of localized Eu2+mo-\nments with a huge moment of \u00187\u0016Bper Eu atom in the su-\nperconducting ground state, induced by either chemical dop-\ning into the parent compound or application of hydrostaticpressure15–19. The unprecedentedly large saturated moment,\nrelatively high TCandTsc, and broad temperature range in\nwhich SC and FM coexists, make the Eu122 ferromagnetic su-\nperconductors (FMSCs) quite unique and promising for pos-\nsible applications in superconducting spintronics.\nUnfortunately, only very limited studies have been carried\nout to understand the coexistence mechanism in the Eu122\nFMSCs. Through in-depth magnetometry measurements, Jiao\net al:proposed a spontaneous vortex state as the solution\nfor the coexistence of FM and SC in Eu(Fe 0:91Rh0:09)2As220.\nUsing magnetic force microscopy (MFM), Stolyarov et al:\nhas systematically investigated how FM and SC ﬁght with\neach other at nanoscale as a function of temperature in\nEuFe 2(As0:79P0:21)221. Importantly, evolved from the homo-\ngeneous Meissner state (HMS) for TC<T<Tsc, an inho-\nmogeneous domain Meissner state (DMS) characterized by\nstriped domains with the width of \u0018100 to\u0018200 nm was ob-\nserved in a very narrow temperature range ( \u00181 K) just below\nTC. Upon further cooling, based on the intermediate DMS, a\ndomain vortex-antivortex state (DVS) characterized by coex-\nisting Abrikosov vortices and ferromagnetic domains of larger\nsize ( l\u0018350 nm) is ﬁnally established well below TC, well\nconsistent with expectations from the theory of magnetic do-\nmain phases in FMSCs with purely electromagnetic interac-\ntion22. However, as MFM is a surface-sensitive techniqe, con-\nﬁrming the existence of intermediate DMS using bulk probes\nis crucial for understanding the delicate interplay between SC\nand FM in the Eu122 FMSCs.\nIn this Letter, using small-angle neutron scattering (SANS)arXiv:2204.12094v1  [cond-mat.supr-con]  26 Apr 20222\n(D)\n mid Q, T=5K, H=0.25T\n0 -0.01 -0.02 -0.03 0.01 0.02 0.03-1\nq [Å]xH\n mid Q, T=18K, H=0T\n00.010.020.03\n-0.01\n-0.02\n-0.03 -1\nq [Å]y\n0 -0.01 -0.02 -0.03 0.01 0.02 0.03-1\nq [Å]xmid Q, T=5K, H=0T\n0 -0.01 -0.02 -0.03 0.01 0.02 0.03-1\nq [Å]x\nLog Ia b(A)\nmid Q, T=5K, H=0.25T\n0 -0.01 -0.02 -0.03 0.01 0.02 0.03-1\nq [Å]x\nLog I Hmid Q, T=5K, H=0.5T\n0 -0.01 -0.02 -0.03 0.01 0.02 0.03-1\nq [Å]xH(B)\n(E) (F)A BSec. A, T=18K\nSec. B, T=18Ksector analysis, H=0T\n Rocking angle [°]-5  0     5010203040Sec. A, T=5K\nSec. B, T=5K Sector Intensity / Std. mon. [a.u.](C)\n00.010.020.03\n-0.01\n-0.02\n-0.03 -1\nq [Å]y\nFigure 1: Two-dimensional (2D) SANS patterns measured at different temperatures and magnetic ﬁelds. All 2D data shown represent the\ncenter of rocking scan of \u00065\u000e. A high-temperature background obtained at T= 25 K is subtracted, and the strong signal around the direct\nbeam is masked. (A) and (B) show the zero-ﬁeld SANS patterns at T= 18 K and T= 5 K, respectively, with the crystallographic orientations\nin tetragonal notation marked by the white arrows in (B). (D) and (E) show the SANS patterns measured at T= 5 K in a longitudinal magnetic\nﬁeld of Hl= 0.25 T and 0.5 T parallel to the incident neutron beam along c, respectively. (F) shows the SANS pattern measured at T= 5 K\nin a transverse magnetic ﬁeld of Ht= 0.25 T along the [1, -1, 0] direction perpendicular to the incident neutron beam. The white azimuthal\nsectors in (A), (D) and (F) illustrate the regions in which the intensities are integrated radially into I(q)as shown in Fig. 3, and the horizontal\nsectors in (A) marked A and B illustrate the regions in which the intensities are integrated and shown in (C) as a function of the rocking angle.\nas a bulk probe, we ﬁnd a clear signature of DMS in single-\ncrystal samples of the EuFe 2(As0:8P0:2)2FMSC. In a small\ntemperature interval of \u00181.5 K below TC, additional SANS\nsignal revealing characteristic length scales in between \u001880\nnm to\u0018160 nm is observed, in addition to large and smooth\nferromagnetic domains that exhibit characteristic scattering\nintensity ofI(q)/q\u0000423. These additional length scales van-\nish with decreasing tempature and increasing magnetic ﬁeld,\nsuggesting the important role of the intermediate DMS in the\ncompetition between SC and FM in the Eu122 FMSCs.\nSingle crystals of EuFe 2(As0:8P0:2)2were grown using a\nself-ﬂux method24, and used for measurements on the diffuse\nscattering cold-neutron spectrometer DNS and small-angle\nneutron scattering instrument SANS-1 at MLZ25,26. Sup-\nporting DC magnetization measurements were performed on\na Quantum Design magnetic property measurement system\n(MPMS) using one crystal from the same batch. Details of\nneutron scattering and magnetization measurements are pre-\nsented in the Supplemental Material27(see, also, Ref. 28–30\ntherein for similar magnetization data in other Eu122 FM-\nSCs). As presented below, superconducting and ferromag-\nnetic transitions occur below Tsc= 22.5 K and TC= 18.5\nK, respectively.Figure 1 summarizes the SANS results obtained using the\nmediumQ-range setting. As the background recorded at T\n= 25 K in the paramagnetic state is subtracted, the displayed\nSANS scattering intensity around q= 0 is of purely magnetic\norigin, arising from the scattering from large-scale ferromag-\nnetic domains. Figs. 1(A) and 1(B) show the data at T= 18 K\nand 5 K, respectively, measured in zero ﬁeld ( H= 0 T) after\na zero-ﬁeld-cooling (ZFC) process. As shown in Fig. 1(C),\na comparison between the rocking scans at these two temper-\natures integrated using the horizontal sector A or B clearly\nindicates an increasing intensity with decreasing temperature,\ndue to a signiﬁcant increase of the Eu2+moment at 5 K. In\naddition, the magnetic SANS intensity in zero ﬁeld displays\na star-shaped anisotropy with a four-fold symmetry, as shown\nin Figs. 1(A), 1(B), and S1B27. Such an anisotropy is believed\nto be due to the pinning of ferromagnetic domains below TC\nfollowing the high-symmetry crystallographic directions, re-\nsulting in a wider coverage in real-space length scales along\na/baxes and a smaller coverage along the reciprocal a\u0003/b\u0003di-\nrections31.\nApplying a longitudinal magnetic ﬁeld parallel to the in-\ncident neutron beam along the caxis effectively suppresses\nthe magnetic SANS intensity, as shown in Figs. 1(D) and3\n1(E) at T= 5 K, due to the suppression of in-plane ferro-\nmagnetic ﬂuctuations around q= 0 by the longitudinal ﬁeld.\nIn the meantime, the application of a longitudinal ﬁeld rotates\nthe ferromagnetic domains towards the ﬁeld direction along\nc, leading to a homogenous azimuthal distribution of the in-\nplane length scale and a symmetric SANS pattern. In con-\ntrast, for a transverse ﬁeld applied parallel to the abplane and\nperpendicular to the neutron beam, as shown in Fig. 1(F), a\ntwo-leaf anisotropy is observed as a result of the rotation of\nferromagnetic domains towards the ﬁeld direction along [1,\n-1, 0].\nFig. 2(A) shows the DC magnetization data of one\nEuFe 2(As0:8P0:2)2crystal, which suggests the appearances of\nsuperconducting and ferromagnetic transitions below Tsc=\n22.5 K and TC= 18.5 K, respectively27. The temperature\nand ﬁeld dependences of the integrated magnetic SANS in-\ntensity are presented in Figs. 2(B-D). The TCvalue deter-\nmined from the temperature dependence of magnetic SANS\nsignal in zero ﬁeld, as shown in Fig. 2(B), is perfectly consis-\ntent with those determined from the magnetization data and\nthe zero-ﬁeld neutron diffraction measurement, respectively.\nWhile the total integrated intensity of (1 1 0) Bragg reﬂection\ncorresponding to the ferromagnetic ordering with a magnetic\npropagation vector of k= 0 shows a typical Landau’s order-\nparameter behavior, the magnetic SANS intensity is character-\nized by an additional hump around \u001817.5 K, associated with\nthe appearance of the intermediate DMS, as will be discussed\nbelow, due to the interplay between FM and SC.\nFigs. 2(C) and 2(D) summarize the effects of longitudi-\nnal and transverse ﬁeld, respectively, on the temperature de-\nFigure 2: DC magnetization of an EuFe 2(As0:8P0:2)2crystal mea-\nsured in a magnetic ﬁeld of 10 Oe parallel to the caxis in ZFC and\nFC processes (A), the temperature dependences of the total integrated\nintensity of the (1 1 0) ferromagnetic reﬂection measured at the DNS\nspectrometer and the magnetic SANS signal (B) in zero ﬁeld, as well\nas the effects of a longitudinal (C) and transverse (D) ﬁeld on the\nmagnetic SANS intensity. In (B), the two datasets are scaled to each\nother with respect to those at 5 K. In (C) and (D), the temperature\ndependences are measured in FC processes under different ﬁelds.pendences of integrated magnetic SANS intensity. The ap-\nplication of either a longitudinal or transverse ﬁeld tends to\nsuppress the intrinsic ferromagnetic ﬂuctuations and accord-\ningly the onset temperature of magnetic SANS scattering. The\nSANS signal completely vanishes in a longitudinal ﬁeld of Hl\n= 1 T and a tranverse ﬁeld of Ht= 0.5 T. Furthermore, the\nhump exhibited close to \u001817.5 K in zero ﬁeld almost disap-\npears for Hl= 0.5 T and Ht= 0.125 T, respectively, suggesting\nthe destructive role of the external ﬁeld against the DMS.\nTheq-dependences of the magnetic SANS intensity of\nEuFe 2(As0:8P0:2)2at different temperatures, integrated using\nthe white sectors as schematically sketched in Figs. 1(A),\n1(D) and 1(F) for the zero-, longitudinal- and transverse-ﬁeld\ncases, respectively, were summarized in Fig. 3. As shown\nin Fig. 3(A), in zero ﬁeld, all the I(q)curves for T<TC\nshow an overall sloping background with the power-law be-\nhavior ofI(q)/Sq\u00004, known as the Porod’s law expected\nfor scattering from smooth ferromagnetic domains with ﬂat\ninterfaces independent of q(Sis the domain wall area in our\ncase)32. The increasing SANS intensity and vertical intercept\nofI(q)curves with decreasing temperature indicates the in-\ncrease of the ferromagnetic moment and the growth of ferro-\nmagnetic domains upon cooling. Interestingly, for the tem-\nperatures closely below TCin an interval of\u00181.5 K, theI(q)\ncurves clearly exhibit a double-hump structure superimposed\non a sloping background.\nUsing the indirect Fourier transform (IFF) method intro-\nduced in Ref. 33, the real-space pair distance distribution\nfunctionP(r)in EuFe 2(As0:8P0:2)2forT= 17:5K,18K and\n18:5K atH= 0T can be extracted according to the relation\nI(q) = 4\u0019Z1\n0P(r)sin(qr)\nqrdr + background\nand plotted in Fig. 3(F), where characteristic length scales in\nbetween\u001880 to\u0018160 nm are identiﬁed to show up at these\nintermediate temperatures. These additional length scales re-\nvealed here are completely consistent with the previous obser-\nvation of an intermediate inhomogeneous DMS using MFM,\ncharacterized by striped domains with the width in between\n100 to 200 nm, in a very narrow temperature range ( \u00181\nK) just below TCin EuFe 2(As0:79P0:21)221. As shown by\nthe solid curves in Fig. 3(A), the superimposition of these\nnanometer-scaled structures with larger smooth ferromagnetic\ndomains ﬁts the I(q)curves perfectly. Upon further cooling\n(T\u001cTC), the double humps smear out when the FM is fully\ndeveloped (see Fig. S1A for a comparison between 18 K and\n5 K)27, indicating that these additional length scales exist only\nin an intermediate state residing in a very narrow temperature\nrange below TCand reﬂect the competition between FM and\nSC. Therefore, we regard them as convincing evidences of the\nappearance of DMS in EuFe 2(As0:8P0:2)2, but more impor-\ntantly, in a bulk form, as SANS is a bulk probe of large-scale\nstructure and magnetism23.\nFuthermore, the q-dependences of the magnetic SANS in-\ntensity at different temperatures in longitudinal and transverse\nﬁelds were plotted in Figs. 3(B-C) and 3(D-E), respectively.\nIn a longitudinal ﬁeld of Hl= 0.25 T (Fig. 3(B)) and Hl= 0.5\nT (Fig. 3(C)), the double-hump feature can still be indentiﬁed4\n-1\n q [Å]10-210-1\n-1\n q [Å]10-210-1-1\n q [Å]10-210-1\n10-610-410-2100 Int. Intensity / Std. mon.10-610-410-2100 Int. Intensity / Std. mon.H=0T H=0.25T H=0.5T\nH=0.25T H=0.125Tlongitudinal H longitudinal H\ntransverse H transverse H(A) (B)\n(E) (D)(C)\n0 1 2 302468-5\nx10\n3\nr [10  Å]P(r) [a.u.]17.5K, 0T\n18K, 0T\n18.5K, 0T\n17K, 0.125T , \n12.5K, 0.25T ,transverse H\ntransverse H(F) ﬁt 17.5K\nﬁt 18K\nﬁt 18.5K\nﬁt 17K5K\n15K\n17.5K\n18K\n18.5K5K\n15K\n17K\n18K5K\n15K\n16K\n5K\n15K\n16K\n17K6K\n10K\n12.5K\nﬁt 12.5K\nFigure 3: q-dependences of the radially integrated magnetic SANS intensity at different temperatures in zero magnetic ﬁeld (A), a longitudinal\nﬁeld of Hl= 0.25 T (B) and Hl= 0.5 T (C), a transverse ﬁeld of Ht= 0.125 T (D) and Ht= 0.25 T (E), respectively. The I(q)curves with\na double-hump feature collected at the intermediate temperatures just below TC, for the zero-ﬁeld and transverse-ﬁeld case, were ﬁtted using\nthe IFF method, as the solid lines in (A), (D) and (E) represent. (F) shows the extracted real-space pair distance distribution function P(r),\nrevealing the existence of the DMS at intermediate temperatures closely below TC.\nat intermediate temperatures, but less pronounced. However,\ntheI(q)behavior observed in a transverse ﬁeld is qualitatively\ndifferent, in which the double humps are basically replaced by\na single hump. As shown in Fig. 3(F), ﬁtting the I(q)curves\nforHt= 0.125 T at T= 17 K and Ht= 0.25 T at T= 12.5\nK using the same IFF method yields a smaller length scale in\nbetween\u001850 to\u001870 nm, suggesting a more effective role\nof transverse ﬁeld in killing the DMS into domains with even\nsmaller sizes. The general tendency of I(q)evolving with de-\ncreasing temperature in external ﬁelds is similar to the zero-\nﬁeld case, with the total magnetic SANS intensity increasing\nand the broad feature associated with nanometer-scaled DMS\nsmearing out.\nIn stark contrast to the previous SANS results on\nisostructural 122-family iron-based superconductors, such as\nKFe 2As2and doped BaFe 2As234–37, where either a disordered\nor ordered vortex lattice was observed in the superconducting\nstate, no evidence of Abrikosov vortices can be identiﬁed in\nEuFe 2(As0:8P0:2)2belowTsc. Even for the temperatures in\nbetweenTCandTsc, where a homogeneous Meissner state is\nexpected, the large background of strongly ﬂuctuating FM of\nEu together with the imperfection of coaligned crystals makes\nthe detection of the superconducting vortex lattice almost im-\npossible despite considerable counting efforts. However, a\nclear signature of the formation of smooth and large ferro-\nmagnetic domains outside the resolution range of the SANSinstrument (\u0018300 nm) is found below TC= 18:5K, featured\nby characteristic scattering with I(q)/q\u00004. This is well con-\nsistent with the observation of the domain vortex-antivortex\nstate (DVS) characterized by ferromagnetic domains of large\nsize ( l\u0018350 nm) well below TCby MFM technique21.\nSummarizing all results obtained in this study, the\ntemperature-ﬁeld phase diagrams for EuFe 2(As0:8P0:2)2un-\nder applied longitudinal and transverse ﬁelds are constructed\nand plotted in Fig. 4. Most importantly, in a small tempera-\nture interval of\u00181.5 K below TC(TDMS <T<TC, where\nTDMS denotes the temperature below which the double-hump\nfeature associated with the DMS is no longer pronouncedly\nvisible), when the ferromagnetic correlations are just estab-\nlished, additional SANS signals in form of double humps as-\nsociated with previously proposed nanometer-scaled Meissner\ndomains (DMS) is observed, suggesting that the decoration of\nSC on the FM in forms of additional large-scale domain struc-\ntures is in fact of a bulk nature. When driven away from the\nnarrow phase region of the DMS by lowering the tempera-\nture, these Meissner domains with the size of \u0018100-200 nm\nare quickly buried into larger ferromagnetic domains when the\nFM gets enhanced. Further SANS measurements in magnetic\nﬁelds in more details will be crucial for better understanding\nthe effects of external ﬁelds on the evolution of length scales\nof the intermediate DMS and manipulating the interplay be-\ntween the FM and SC.5\nFigure 4: Temperature-ﬁeld phase diagrams for EuFe 2(As0:8P0:2)2\nunder an applied longitudinal (a) and transverse (b) magnetic ﬁeld,\nrespectively. TCdenotes the temperature below which the ferro-\nmagnetic correlations get established. Tscdenotes the superconduct-\ning transition temperature, which is supposed to be suppressed very\nslowly under 2 T for EuFe 2(As1\u0000xPx)213. NS, HMS, DMS and DVS\ndenote the normal state above Tsc, the homogeneous Meissner state\nin between TscandTC, the domain Meissner state in between TCand\nTDMS, and the domain vortex-antivortex state below TDMS, respec-\ntively. TDMS is determined according to the temperature at which\nthe magnetic SANS intensity in Fig. 2(C,D) displays an additional\nfeature and the I(q)proﬁles shows a clearly identiﬁed double-bump\nstructure.In conclusion, by systematically investigating the magnetic\nSANS intensity of the FMSC EuFe 2(As0:8P0:2)2as a func-\ntion of temperature and magnetic ﬁeld, we conﬁrm that the\nDMS observed in the previous surface-sensitive MFM mea-\nsurement is factually of a bulk nature and exists only in a\nnarrow temperature and magnetic ﬁeld range as an interme-\ndiate state. Ascribing to the subtle interplay between FM and\nSC in the Eu122 FMSCs, these nanometer-scaled Meissner\ndomains are ﬁnally buried into bulk ferromagnetic domains\nwith much larger size upon cooling, corresponding to the DVS\nin the ground state. The temperature-ﬁeld phase diagrams of\nEuFe 2(As0:8P0:2)2under applied longitudinal and transverse\nmagnetic ﬁelds are constructed to describe the competition be-\ntween SC and FM. Our results provide a key solution to the\nmystery regarding the intriguing coexistence of strong ferro-\nmagnetism and bulk superconductivity in the Eu122 FMSCs.\nAcknowledgments\nWe would like to acknowledge Susanne Mayr for the assis-\ntance with the co-alignment of the crystals, and Wenhe Jiao\nfor stimulating discussions. This work is based on experi-\nments performed at the SANS-1 instrument operated by Tech-\nnische Universität München and DNS instrument operated by\nJülich Centre for Neutron Science (JCNS) at the Heinz Maier-\nLeibnitz Zentrum (MLZ), Garching, Germany. This work\nwas supported by the National Natural Science Foundation of\nChina (Grant No. 12074023) and National Key Research and\nDevelopment Program of China (2016YFA0300202), and the\nFundamental Research Funds for the Central Universities in\nChina.\n\u0003Electronic address: wtjin@buaa.edu.cn\nyElectronic address: Sebastian.Muehlbauer@frm2.tum.de\nzElectronic address: y.su@fz-juelich.de\n1L. N. 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The formula can involve correctly a ferromagnetism h as been induced by eﬀective mass\ndiﬀerence between up- and down-spin electrons. Then, this e ﬀective mass mismatch ferromagnet\nand standard Stoner ferromagnet have been employed in this p aper. As an application of the\nformulation, we have studied the tunneling eﬀect for juncti ons including spin-triplet p-wave super-\nconductor, where we choose a normal insulator for the insula ting region whereas our formula can\ntreat a ferromagnetic insulator. Then, we have been able to d evote our attention to features of\nferromagnetic metal. The conductace spectra show a clear di ﬀerence between two ferromagnets\ndepending upon the way of normalization of the conductance. Especially, a essential diﬀerence is\nseen in zero-bias conductance peaks reﬂecting characteris tics of each ferromagnets. From obtained\nresults, it will be suggested that the measurements of the tu nneling conductance in the junction\nprovide us a useful information about the mechanism of itine rant ferromagnetism in metals.\nPACS numbers: 72.25.-b, 74.25.F-, 74.20.Rp\nI. INTRODUCTION\nAndreev reﬂection(AR), which occurs at the inter-\nface of the junctions involving superconductors, is one\nof the most important elemental processes in the trans-\nport through the superconducting junctions [1]. A the-\nory of transport taking into account the AR was for-\nmulated by Blonder, Tinkham and Klapwijk referred\nto as BTK theory[2]. The BTK theory enables us to\nprobe the pairing state of superconductors. For exam-\nple, for a junction consists of a normal metal and an\nunconventional superconductor, the quantum interfer-\nence eﬀect between the injected and the Andreev re-\nﬂected particles, which feel mutually the diﬀerent sign\nof superconducting pair potential through the scattering\nevent at the interface of the junction, forms the so-called\nzero-energy Andreev bound states(ZABS)[3, 4]. Indeed,\nthe ZABS originated from the d-wave symmetry of su-\nperconducting pair potential have been observed as the\nzero-biasconductance peaks(ZBCPs) in tunneling exper-\nimentofhigh TCcupratesuperconductors[5–7]according\nto theoretical prediction of Tanaka and Kashiwaya(TK)\nformula[4]. The ZBCPs reﬂecting the ZABS are the es-\nsential feature of the electrical conductions in normal\nmetal/insulator/unconventionalsuperconductorjunction\nand provide us the information of the pairing symmetry\nof superconductor[8–10]. On the other hand, in ferro-\nmagnet/insulator/conventionalsuperconductor junction,\nthe measurements for low energy transport via AR oﬀer\nthe opportunity to probe the magnetic property such as\n∗Electronic address: yoshida.nobukatsu@nihon-u.ac.jp\n†Electronic address: yamashiro.masashi@nihon-u.ac.jpthe polarization of ferromagnetic materials[11–13]. The\nAR in this junction is suppressed by the exchange ﬁeld\nin the ferromagnet layer. As a result, the conductance\nat low energy of the junction is suppressed respond-\ning to the polarization of ferromagnet. The behavior\nof ZBCPs in ferromagent/insulator/unconventional su-\nperconductor junction have been studied to understand\nthe characteristic properties of unconventional supercon-\nductors and to utilize the properties as applications of\nspintronics[14–16]. In these junctions, the ferromagnet\nhas been described within the Stoner model based on a\npicture of free electrons. However, in some materials, the\nother descriptions of ferromagnetism are required. The\nferromagnetism kinetically driven by a spin-dependent\nbandwidth asymmetry, or, equivalently, by an eﬀective\nmass splitting between ↑- and↓-spin particles [17–22] is\nan interesting model giving itinerant ferromagnet.\nRecently, Annunziata, et al. analyzed\nthe charge and spin transport in ferromag-\nnet/insulator/superconductor(F/I/S) junctions with\ntaking account above mentioned the spin-dependent\nbandwidth asymmetry ferromagnet (SBAF) making the\neﬀective masses have diﬀerent values in ferromagnet[23–\n25]. They clariﬁed that from the knowledge of the\ncritical transmission angle the measurement of the\neﬀective mass diﬀerence among the particles could be\npossible[23]. In there, it has been also shown that the\nF/I/S junction is an eﬀective probe to investigate the\nmechanism of ferromagnetism and the pairing symmetry\nof unconventional superconductor. Furthermore, it is\nsuggested that the F/I/S junction can be useful as\nswitching device using the spin current in the case of\nthe symmetry of superconductor is conventional s-wave\ncase[24] and be possible as a spin-ﬁltering device[25].\nThey have studied on F/I/S junctions with several types2\nof superconducting symmetries as the conventional s-\nwave, unconventional dx2−y2-wave, and the time reversal\nsymmetry broken dx2−y2+isordx2−y2+dxystates.\nMoreover, although zero bias anomaly of diﬀerential\nresistance has been shown experimentally in Sr 2RuO4-Pt\npoint contact experiment[26], more recently, Kashiwaya\net. al.[27] has shown the ZBCP which is direct evidence\nof the ZABS by tunneling spectroscopic experiment of\nSr2RuO4-Au junction. There is much interest and im-\nportance to investigate problems on spin triplet p-wave\nsymmetry nature because the p-wave pairing, especially,\na chiralpx±ipystate breaking time-reversal symmetry\nis one of the best candidates for bulk superconducting\nstate of Sr 2RuO4[27–34].\nIn this paper, a formulation of the tunnel-\ning conductance for charge and spin currents in\nferromagnet/ferromagnetic-insulator/superconductor\n(F/FI/S) junctions will be presented by taking the ef-\nfective mass diﬀerence leading the spin-band asymmetry\nbetween ↑- and↓-spin particles in ferromagnet [23–25]\ninto our previous theory[14]. Although the formulation\ncan be used for singlet and Sz= 0 triplet superconduc-\ntors, we will study a chiral p-wave state. Our formula\nhas general form being able to include the ferromagnetic\ninsulator, however, a normal insulator surrounded by the\nferromagnet and the superconductor is considered for\nthe insulating layer to get pure characteristic features of\na ferromagnetism in a ferromagnet. It is found that the\nnormalized conductance spectra shows a clear diﬀerence\nbetween Stoner and spin-band asymmetry ferromagnets\n(STF and SBAF for abbreviation). The present results\nmay be helpful in investigations of the mechanism of\nferromagnet.\nThis paper is organized as follows. In Sec.II we ex-\nplain a theoretical model and derive a formulation fol-\nlowing our previous method based on the BTK theory.\nThe results for ferromagnet/insulator/chiral p-wave su-\nperconductor junctions are presented in Sec.III. Finally\nthe results are summarized in Sec.IV.\nII. MODEL AND FORMULATION\nFor the model of formulation, we consider a two-\ndimensional ballistic F/FI/S junction with semi-inﬁnite\nelectrodes shown in ﬁgure 1. A ﬂat interface is assumed\nto be located at x=0, and the ferromagnetic insulator\nfor up(down) spin is described by a potential V↑(↓)(x)\n= (V0+ (−)Vex)δ(x), where δ(x),V0andVexare theδ\nfunction, a nonmagnetic barrier amplitude and a mag-\nnetic barrier amplitude, respectively.\nFor the ferromagnetism in the F electrode, we adopt\ntwo kinds models of mechanisms shown in Fig2. One\nof these is the standard Stoner model in which the fer-\nromagnetism is induced by the exchange potential lead-\ning to the rigid energy shift between ↑-spin and ↓-spin\nbands. The other is a spin bandwidth asymmetry model\nproposed by Hirsch[18], in which the bandwidth is tuned\nFIG. 1: Schematic illustration of scattering processes of a n\ninjected electron with ↑-spin at the F/FI/S ballistic junction.\nHere,θ↑,θAR, andθSare injection, Andreev reﬂection, and\ntransmission angles, respectively. It is assumed that the n or-\nmal reﬂection at the interface is totally specular, then nor mal\nreﬂection angle is also given by θ↑. For the case of ↓-spin\nelectron, it can be depicted by ﬂipping ↑by↓in the ﬁgure.\nrelatively by the ratio of the eﬀective masses between ↑-\nand↓-spin particles. Although in the following the free\nparticle-like spectra of parabolic type is assumed as nor-\nmal electronic dispersion relation, we suppose to deﬁne\nthe concept of bandwidth for the above description. It\nimplies that there can be some relations between this\ndescription and some eﬀective one-band tight binding\nmodel permitting the eﬀective masses of carriers being\nproportional to the inverse of the width of the bands\nwhere the carriers get itinerancy. Hence, only giving dif-\nferent values of the masses for ↑- and↓-spin electrons\nyields a bandwidth asymmetry.\nThe spatial dependence of the pair potential is taken\nas ∆(r) = ∆Θ( x) for simplicity. In addition, we\nconsider the Sz= 0 pairing states, where the ele-\nments of pairpotential are given by ∆ ↑,↑= ∆↓,↓=\n0 and ∆ ↑,↓=−∆↓,↑for the singlet pairing state or\n∆↑,↓= ∆↓,↑for triplet pairing state. Thus, the eﬀec-\ntive Hamiltonian(Bogoliubov-de Gnnes (BdG) equation)\nof the system can be reduced the decoupled equation for\nthe eigen states ( uF(S)↑(↓)(r),vF(S)↓(↑)(r))Tand is given\nby\n/parenleftbigg\nHσ\n0(r) ∆(r)\n∆∗(r)−H¯σ\n0(r)/parenrightbigg/parenleftbigg\nuF(S)σ(r)\nvF(S)¯σ(r)/parenrightbigg\n=E/parenleftbigg\nuF(S)σ(r)\nvF(S)¯σ(r)/parenrightbigg\n.(1)\nHereEis the energy of the quasipaticle and Hσ\n0(r) is\nthe single particle Hamiltonian for σ-spin where ¯ σ=−σ.\nIn the Ferromagnet side, the single particle Hamilto-\nnian is given by Hσ\n0(r) =−/planckover2pi12∇2/2mσ−ρUex−EFM\nwhereσ=↑,↓,mσis the eﬀective mass for σ-band par-\nticle,ρ= +1(−1) for↑(↓)-spin,Uexis the exchange\npotential and EFMis the Fermi energy. The Hσ\n0in3\nFIG. 2: A sketch of dispersion relation between energy and\nwave number, and Fermi surface for STF (left side) and for\nSBAF (right side). It is assumed that free electron model\nwith rigidly energy shift in STF and diﬀerent eﬀective masse s\nfor each spin in SBAF.\nthe superconductor side is given by H0(r) =Hσ\n0(r) =\n−/planckover2pi12∇2/2mS−EFSwhere the msandEFSare the ef-\nfective mass of the quasiparticle and the Fermi energy,\nrespectively. To describe the Fermi surface diﬀerence, we\nassumeEFM=EFS=EF.In the following, we apply the quasiclassical approxi-\nmation where EF≫( E, ∆(k) ) and the k-dependence\nof ∆(k) is replaced by the angle θSbetween the direc-\ntion of the trajectory of quasiparticles in the supercon-\nductor and the interface normal. In the quasiclassical\napproximation, the wave vectors of k↑(↓)andkELQ(HLQ)\nare given by k↑(↓)=/radicalbig(2m↑(↓)//planckover2pi12)(EF+(−)Uex) and\nkELQ(HLQ)=kS=/radicalbig\n2mSEF//planckover2pi12, respectively, where\nELQ (HLQ) indicates electronlike (holelike) quasiparti-\ncles. For example, we assume the injection of ↑-spin elec-\ntronsfrom the ferromagnetatan angle θ↑tothe interface\nnormalasshowninFig.1. Therearefourpossiblescatter-\ning trajectories exist; Andreev reﬂection with angle θAR\nasholesbelonging to ↓-spin band, normalreﬂection(NR),\ntransmission to superconductor as ELQ, and transmis-\nsion as HLQ. These four processes are described in same\nway for↓-spin electrons with changing the scattering an-\ngleθ↑toθ↓. Sincethetranslationalsymmetryholdsalong\nthey−axis, the parallel momentum components of all\ntrajectoriesareconserved kσsinθσ=k¯σsinθ¯σ=kSsinθS.\nThe angles θσ,θ¯σandθSdiﬀer from each other except\nwhenUex= 0 and m↑=m↓=mS, which means retrore-\nﬂectiverly of AR broken by the exchange ﬁeld and the\neﬀective masses diﬀerence. The BdG equations are re-\nduced to the eﬀective one-dimentional equation due to\nthe translational invariance along y-axis of the Hamilto-\nnian. Thus, the solutionsofthe BdGequationsfor σ-spin\nelectron injections are described as\n/parenleftbigg\nuFσ(x <0)\nvF¯σ(x <0)/parenrightbigg\n=/parenleftbigg\n1\n0/parenrightbigg\neikσxcosθσ+a¯σ/parenleftbigg\n0\n1/parenrightbigg\neik¯σxcosθ¯σ+bσ/parenleftbigg\n1\n0/parenrightbigg\ne−ikσxcosθσ(2)\n/parenleftbigg\nuSσ(x >0)\nvS¯σ(x >0)/parenrightbigg\n=cσ/parenleftbiggu+\nv+e−iφ+/parenrightbigg\neikSxcosθS+dσ/parenleftbigg\nv−eiφ−\nu−/parenrightbigg\ne−ikSxcosθS(3)\nwith\nu±=/radicalBigg\n1\n2/parenleftbigg\n1+Ω±\nE/parenrightbigg\n, v±=/radicalBigg\n1\n2/parenleftbigg\n1−Ω±\nE/parenrightbigg\nΩ±=/radicalBig\nE2−|∆±|2, (4)\neiφ±=∆±\n|∆±|,∆+= ∆(θS),∆−= ∆(π−θS) (5)\nwhere the probability coeﬃcients a¯σ,bσ,cσ, anddσare for AR, NR, transmission ELQ and HLQ. These coeﬃcients\nare calculated from the boundary conditions at x= 0,\nu(v)Fσ(¯σ)(x= 0) =u(v)Sσ(¯σ)(x= 0), (6)\n/planckover2pi12\n2mSduSσ\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0−/planckover2pi12\n2mσduFσ\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0=VσuSσ(x= 0) (7)\n/planckover2pi12\n2mSdvS¯σ\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0−/planckover2pi12\n2m¯σdvF¯σ\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0=V¯σvSσ(¯σ)(x= 0) (8)4\nAs explained in our previous paper, the reﬂection pro-\ncess depends upon the size relation of the Fermi surfaces\nbetween FM and SC. In the following, we will consider\na situation where k↓< kS< k↑, andm↑/mS=mS/m↓\nwithm↑> mS> m↓. Following the BTK theory with\ntaking care of the probability conservation of quasiparti-\ncle ﬂow,\n|bσ|2+vf,¯σ\nvf,σ|a¯σ|2+vs,+\nvf,σ|cσ|2+vs,−\nvf,σ|dσ|2= 1 (9)\nthe conductance GC(S)\nS,σfor theσ-spin charge(spin) cur-\nrent through the system can be calculated by\nGC(S)\nS,σ= 1+(−)vf,¯σ\nvf,σ|a¯σ|2− |bσ|2(10)\nwherevf,σ=/planckover2pi1kσ/mσis the groupvelocities ofthe σ-spinparticles in ferromagnet and vs,+(−)=/planckover2pi1kS/mSis that\nof the ELQ (HLQ) in superconductor. It is much worth\nto note that our conductance formula GC(S)\nS,σis diﬀerent\nfrom former works[23–25]. On our way of formulating\nthe conductance as an extension of previous formulation\n[8, 14] to the present situation, it has to be needed for\ncorrecttreatment of mass mismatch in a same metal that\nthe coﬃcient of AR a¯σshould be given by the ratio of\ngroup velocities rather than that of wavenumbers as a\nconsequence of conservation law of particle ﬂow. Using\nthe obtained AR ( a¯σ) and NR ( bσ) coeﬃcients in the\nsame way as the previous paper[14] based on the TK\nformula[4], the charge(spin) conductance for each spin ↑\nand↓can be formulated by\nGC(S)\nS,↑=GN,↑1− |Γ+Γ−|2(1−GN,↓)+(−)GN,↓|Γ+|2\n|1−Γ+Γ−/radicalbig\n1−GN,↓/radicalbig\n1−GN,↑exp[i(ϕ↓−ϕ↑)]|2Θ(|θS| −θC)\n+[1−Θ(|θS| −θC)]GN,↑1− |Γ+Γ−|2\n|1−Γ+Γ−/radicalbig\n1−GN,↑exp[i(ϕ↓−ϕ↑)]|2(11)\nGC(S)\nS,↓=GN,↓1− |Γ+Γ−|2(1−GN,↑)+GN,↑|Γ+|2\n|1−Γ+Γ−/radicalbig\n1−GN,↓/radicalbig\n1−GN,↑exp[i(ϕ↑−ϕ↓)]|2Θ(|θS| −θC) (12)\nwith\nGN,↑(↓)=4λ↑(↓)\n(1+λ↑(↓))2+Z2\n↑(↓),exp(iϕ↑(↓)) =1−λ↑(↓)+iZ↑(↓)\n1+λ↑(↓)+iZ↑(↓),\nZ↑(↓)=Z0,↑(↓)\ncosθS, Z 0,↑(↓)=2ms(V0−(+)Vex)\n/planckover2pi12kS,Γ±=v±\nu±,\nλ↑(↓)=/radicalBigg\nγ−1(1)+γ−1/2(1/2)\ncos2θS/parenleftbig\n1−γ−1/2(1/2)+(−)χ/parenrightbig\n, (13)\nwhereθC≡cos−1/radicalbig\nγ−1/2(χ−1+γ1/2) ( or\nsin−1/radicalbig\nγ−1/2(1−χ) ) is the critical angle of the\nAR measured in the superconductor side. Here,\nχ=Uex/EF(0≤χ≤1) andγ=m↑/m↓≥1. In the\nabove,GN,σcorresponds to the conductance when the\nsuperconductor is in the normal state. We calculate the\nnormalized conductance deﬁned by\nGC(S)\nT(eV) =/integraltextπ/2\nπ/2dθScosθS(P↑GC(S)\nS,↑+P↓GC(S)\nS,↓)\n/integraltextπ/2\nπ/2dθScosθS(P↑GN,↑+P↓GN,↓)\n(14)where the polarization Pσforσ-spin is expressed as\nP↑=γ(1+χ)\nγ(1+χ)+1−χ, P ↓=1−χ\nγ(1+χ)+1−χ.\nIt is noted in general that the normalized conductance\nwill be deﬁned alternatively corresponding to the actual\nexperiments.\nAbove formulas (2.11), (2.12), and (2.14) can re-\nproduce former formulas of tunneling conductance for\njunctions including triplet superconductor (TS). For\nm↑=m↓, these eqations coincide to that of STF/I/TS5\nTABLE I: Numerical values of the magnetization M, the nor-\nmalized exchange interaction χ=Uex/EF, and themass ratio\nγ=m↑/m↓. For the value M= 0.25, there are two cases,\none is pure STF, χ= 0.25 andγ= 1, and the other is pure\nSBAF,χ= 0 and γ= 5/3. All other values of Mare in the\nsame way except the case of M= 0.\nmagnetization Mexchange Int. χmass mismatch γ\n0 0 1\n0.25 0.25 1\n0.25 0 5/3\n0.5 0.5 1\n0.5 0 3\n0.75 0.75 1\n0.75 0 7\n0.99 0.99 1\n0.99 0 200\njunction[14], and for m↑=m↓andUex= 0, the conduc-\ntance formula for N/I/TS junction[8] is reproduced.\nIII. RESULTS\nAt ﬁrst, we notice about the growth of the magne-\ntization Mfor STF or SBAF. Using the polarization\nPσ,Mis given by M=P↑−P↓. For pure STF case\n(γ= 1), the magnetization is equal to the magnitude of\nexchange splitting M=χ(=Uex/EF). For pure SBAF\ncase(χ= 0), the Misgivenby M= (γ−1)/(γ+1). Thus,\nthe half metal state in SBAF case is unphysical situation\nbecause γ=∞. Figure 3 shows the Min SBAF case\nas a function of γ. It can be seen that the growth rate\nofMbecomes very gradual over γ≈50. From this, one\ncan expect the clear diﬀerences of transport properties\ndepending on Mbetween STF and SBA near the half\nmetallic limit. Hereafter, we call “strong ferromagnetic\nregime”asaregionunderandnearthehalfmetalliclimit.\nInthefollowingsubsections,weapplyourconductance\nformula to ferromagnet/insulator/tripletsuperconductor\n(F/I/TS) junction (F referredto as STF or SBAF) where\nVex= 0. Asthepairingpotential, atriplet p-wavestateis\nemployedbychoosing∆ ↑↓(θS) =∆↓↑(θS) =∆0exp(iθS),\n∆↑↑(θS) = ∆↓↓(θS) = 0 for opposite spin pairing. And in\naddition, we choose some sets of parameters ( χ,γ) giving\nthe same M={0,0.25,0.5,0.75,0.99}shown in Table I\nso as to get clear characteristics of each ferromagnets.\nA. Distinction between STF and SBAF\nTo investigate a consequence of the diﬀerent mecha-\nnism of the magnetization, avoiding any eﬀects of the\nnormal barrier we consider the highly transparent junc-\ntion in the metallic limit ( Z0= 0). In this case, the\nnormalized total conductances GC\nT(eV) show same trend\nthat conductance values inside the energy gap eV <∆0\nare reduced when the value of magnetization Mis in-0 101Magnetization M M\nχ\nγSTFSBAF\n0 101\n100 20000.51Magnetization M M\nχ\nγSTFSBAF\nFIG. 3: The magnetization Mas a function of γin the pure\nSBAF, and inserted panel is Mas a function of χin the pure\nSTF.\n0 1 2012\n0 0.5 1012Normalized Conductance\nNormalized ZBC\nNormalized Energy Magnetization eV/ ∆0 MGTC(      )eV\nGTC(    )0(a) (b)\nSBAF\nSTF0.0 M Magnetization\n0.25\n0.5\n0.75\n0.99\nSolid: SBAF\nDotted: STF\nFIG. 4: Normalized conductance spectra for the charge cur-\nrentGC\nT(eV) in the metallic limit, Z= 0 in (a), and in (b),\nheightofconductanceatzeroenergy isplottedas afunction of\nthe magnetization Mfor STF (red line) and for SBAF (black\nline).\ncreased for both STF and SBAF (Fig.4(a)). It indicates\nthat the retro-reﬂectivity of the AR is broken due to the\ninduced M. However, the Mdependence of reduction\nforGC\nT(eV <∆0) is diﬀerent for each of them. The dif-\nference can be seen more clearly in the M-dependence\nof conductance values at eV= 0,GC\nT(0), in Fig4.(b). It\nis found that the suppression of GC\nT(0) for SBAF case\nis weaker rather than that for STF case without weak\nmagnetizationregime,0 .0≤M≃0.2andathalfmetallic\nlimit,M= 1.0. To clearthe reasonofdiﬀerent Mdepen-\ndence of conductances for SBAF case and STF case, we\nshow the critical angle of AR as a function of Min Fig.5.\nTheθCfor both SBAF and STF cases decreases with in-\ncreasingM. Itisfoundthatthediﬀerencebetweenangles\nis getting larger from M∼0.2 to∼0.9, and convergesto\nzeroatM= 1.0. Fornearlyhalfmetallic limit M= 0.99,\ntheθCis almost suppressed in the STF case, while there\nstill remains in the SBAF case. The critical angles for\nSTF and SBAF are θC= cos−1√\nM(= sin−1√\n1−M)\nandθC= cos−1/radicalBig\n1−(1−M\n1+M)1/2(= sin−1/radicalBig\n(1−M\n1+M)1/2),\nrespectively. Then, it is clear that θCin SBAF case is6\n0 0.5 100.250.5\nMagnetization MθC Normalized critical angle/ π\nSBAF\nSTF\nFIG. 5: Critical angles of AR as functions of Mfor STF\n(black line) and for SBAF (red line). As described in main\ntext, these angles are determined by the conservation con-\ndition for momenta parallel to the interface, k↓sinθ↓=\nkSsinθS. Such as in the present model, i.e., the ferromag-\nnetism is given by a mismatch in kinetic energy the AR crit-\nical angle is determined by θC=θSfor the case of θ↓=π/2\nbecause of satisfying the condition k↓< kS.\nlarger than that in STF case for same Mexcept non-\nmagnetic state, M= 0 and half metal state, M= 1.\nConsequently, as shown in Fig.4, the GC\nT(eV <∆) in\nSBAF case is larger than that in STF case.\nB. Ferromagnetic feature on ZBCP\nIt has been shown theoretically that the ZBCP in\nF/I/S junction would be useful for measuring the mag-\nnetization of ferromagnet[14, 15]. In here, we study the\nvalidity of the ZBCP for the distinction of ferromag-\nnets. Figure 6 shows the conductance GC\nT(eV) for the\njunction in the tunneling limit Z= 5. The ZBCPs\nseen in both STF and SBAF cases are attributed to the\nanisotropy of the pair potential of p-wave superconduc-\ntor. For STF case, the previous results[14] have been\nreproduced(Fig.6(a)). In contrast, there are some diﬀer-\nences for SBAF case. Especially, it is found that the con-\nductance near eV= 0 increases slightly with increasing\nM(Fig.6(b)). This opposite behavior can be seen more\nclearly in the Mdependence of ZBCPiFig.6(c). With in-\ncreasing M, in contrast to the monotonically decreasing\nbehavior of STF case, the ZBCP in SBAF case increases\nup to a certainvalue of Min strongferromagneticregime\nand then, suddenly decreases toward the half metallic\nlimit where the ZBCPs in both cases are suppressed per-\nfectly. Cause of this opposite behavior could be reduced\nto the deﬁnition of normalization way since the magni-\ntude of ZBCP being a constant value in non-normalized\ncase depends on the conductance as the superconductor\nis in normal state. There is other deﬁnition of normal-\nization by using the AR critical angle measured in the\nferromagnet side[14, 15]. However, in that case, the AR\ncritical angle itself depends on and is controlled by the0123\n0 1 20123\n0 0.5 10123\nMagnetization MNormalized Energy eV/ ∆0\nSBAF\nSTFNormalized ZBCP Normalized Conductance GTC(      )eV GTC(    )0(a)\n(b)\n(c)Magnetization Magnetization M\n0.0\n0.25\n0.5\n0.75\n0.99\nFIG. 6: Normalized conductance spectra of the charge cur-\nrentGC\nT(eV) in the tunneling limit, Z= 5 for STF/I/TS\njunction (a) and for SBAF/I/TS junction (b). And in (c),\nheight of ZBCPs is plotted as a function of the magnetization\nMfor STF (red line) and for SBAF (black line).\nmagnitude of M, as a results, even the normalization\ndepends on M. Accordingly, in order to avoid the inﬂu-\nence ofM, we alternatively calculate an angle averaged\nconductance deﬁned as in the following,\nQS=QS,↑+QS,↓\nQS,σ=/integraltextπ/2\nπ/2dθScosθSPσGC\nS,σ\n/integraltextπ/2\nπ/2dθScosθS.\nWe show the calculated results of the angle averaged\nconductance QSin Fig.7 which, in both STF and SBAF\ncases, show same tendency to decrease as increasing\nM(Fig.7(a)). Similarly, the ZBCP is decreasing function\nofM(Fig.7(b)). It is also shown that the reduction ratio7\n0 1 200.10.2\n0 0.5 100.10.2\nNormalized Energy eV/ ∆0Averaged Conductance\nAveraged ZBCP\nMagnetization M(a) (b)\nSBAF\nSTFSolid: SBAF\nDotted: SBAFMagnetization M\n0.0\n0.25\n0.5\n0.75\n0.99QS(      )eV\nQS(    ) 0\nFIG. 7: Angle averaged conductance spectra QSas a func-\ntion of magnetization (a) and ZBCPs vs. magnetization\nstrength (b) in the tunneling limit Z= 5. Here, the con-\nductance for SBAF case is indicated as solid line and for STF\ncase is as dotted line.\n0 100.51\n0 100.1Averaged Conductance QN\nM(a)\nQN\nQNQN= QN  +  QN\nSolid: SBAF\nDotted: STF\nMagnetization\nAveraged Conductance QNQN= QN  +  QN\nQN\nQN\nM Magnetization(b)\nFIG. 8: Magnetization dependence of angle averaged normal\nconductance QN,QN,↑, andQN,↓for SBAF case (solid line)\nand STF case (dotted line) in the tunneling limit for the case\nof the superconductor is in normal state, (a) for Z= 0, (b)\nforZ= 5.\ndiﬀers in each of both cases as same as that in metallic\nlimit. Thus, the opposite behavior seen in normalized\nconductance would reduce to the conductance in normal\nstate. Therefore, it is noticed that the conductance of\nthe junction forthe superconductorbeing innormalstate\nplay an important role on our attention for two diﬀerent\nferromagnetisms.\nIn order to clarify the diﬀerence between STF case\nand SBAF case more, we calculate the conductance in\nferromagent/normal metal (F/I/N) junction for both\nin metallic and in tunneling limits. The angle aver-\naged conductance in F/I/N junction QN=/summationtext\nσQN,σ\nis deﬁned in similar way to that in F/I/S junction\nreplacing GC\nS,σbyGC\nN,σ. The calculated results of\nQN,σfor both Z0= 0 and Z0= 5 are shown in\nFig.8. The angle resolved conductance GC\nN,σforσ-spin\nis rewritten by GC\nN,σ= 4cosθS˜λσ/((cosθS+˜λσ)2+\nZ2\n0,σ) where ˜λσ=/radicalbig\ncos2θS+ρχin STF/I/N and\n˜λσ=γ−ρ/2/radicalbig\ncos2θS+(γρ/2−1) in SBAF/I/N junc-\ntions. Here, we mention properties of M-dependence of\nGC\nN,σthrough χorγin advance of descriptions aboutQN. In STF/I/N junction, the GC\nN,↑increases following\ngrowth of the magnetization, i.e., with increasing χsince\nthe gain of Fermi energy due to the band shift is larger\nthan the Fermi surface eﬀect[14] acting as an eﬀective\nbarrier between STF and normal metal, under the con-\nservation of the momentum along y−direction. On the\nother hand, because there is no Fermi energy gain from\nspread of the band width due to the eﬀective mass mis-\nmatch in SBAF and the inﬂuence of the eﬀective barrier\narising from the Fermi surface eﬀect becomes stronger\nwith the increase of γ, theGC\nN,↑in SBAF/I/N junction\ndecreases with increasing γand become zero in the limit\nofγ→ ∞.GC\nN,↓for both STF and SBAF cases decreases\nwith increasing the magnetization caused by χorγ.\nIn the metallic limit Z0= 0 (Fig.8(a)), it is found that\ntheQN,↑in STF/I/N junction increases with increasing\nMin contrast to QN,↓decreasing toward zero in half\nmetal state. In this case, Mis given directly as M=χ.\nThus, the total conductance QN=QN,↑+QN,↓is re-\nduced slightly by the Fermi surface eﬀect with increasing\nMup to∼0.7. In SBAF/I/N junction, we can see simi-\nlar behavior in QN,↑(↓). The increase of QN,↑is owing to\nP↑which is an increasing function of γ. However, near\nthehalfmetalliclimit, QN,↑reducesrapidlyreﬂectingthe\nbehavior of GN,↑which is a decreasing function of Mto-\nward zero at M= 1(γ=∞) as mentioned above. Thus,\nas shown in Fig.4, the GS\nT(eV) in SBAF/I/S junction\ndecreases slowly with increasing Mwith comparing to\nthat in STF/I/S junction. The diﬀerence between STF\nand SBAF becomes more clearly in the tunneling limit\nZ= 5(Fig.8(b)). With increasing M,QN,σin STF case\nvariesin rapidly ratherthan that in SBAF case. This is a\ndiﬀerence of a barrier eﬀect felt by particles with σ-spin\nin each cases. The barrier potential simply becomes rel-\natively lower for particles with ↑-spin and higher for par-\nticles with ↓-spin in the STF case due to the rigid Fermi\nenergyshift. However, the particlesin SBAF directly feel\nthebarrierpotentialbecausethereisnoshift oftheFermi\nenergy. Thus, in SBAF/I/N junction, the increase of the\nmagnitude of QN,↑due toP↑is suppressed by the Fermi\nsurface eﬀect and barrier potential and then, QN,↑is get-\nting lowerwith increasing Min contrastto the STF case.\nTherefore, the QNin SBAF/I/N junction shows the op-\nposite behavior ofthat in STF/I/Njunction. As a result,\nthe normalized conductance GS\nT(eV) in SBAF/I/S junc-\ntion increases due to the reduction of the QNdepending\nonM(Fig.6(b)-(c)). Indeed, as shown in Fig.7, the angle\naveraged conductance QS(eV)s for both STF and SBAF\ncase show same trend on varying M. Thus, it can be\nconclude that the measurement of QNwill be also useful\nto identify the STF and SBAF. However, we emphasize\nthat the measurement of ZBCP originated from ZABS\nis more powerful probe to investigate ferromagnet than\nthat ofQN. Because, two QNs seemingly show drasti-\ncally diﬀerent behavior depending on Mfor enough large\nZ0(Fig.8(b)), by carefully looking of the ﬁgure, diﬀer-\nences of each values of QNs are not so large for same M\nexcept strong ferromagnetic regime. Therefore, it seems8\nthat an experimental distinction will become more diﬃ-\ncult on measurement of QN. The ZBCP is getting more\nclear for larger Z0, then which can be expected to play a\nrole of good manifestation of the diﬀerence of STF and\nSBAF.\nIV. SUMMARY\nIn summary, we have derived a formula of the\ntunneling conductance in ferromagnet/ferromagnetic-\ninsulator/superconductor with antiparallel spin pair-\ning junction by extending our previous theory for\nstandard Stoner ferromagnet (STF) so as to in-\nclude spin-band asymmetry ferromagnet (SBAF) orig-\ninated from eﬀective mass mismatch between parti-\ncles with opposite spins. Applying the formulation to\nferromaget/insulator/ p-wave superconductor junctions,\ndiﬀerences between pure STF and pure SBAF have been\ninvestigated intensively. We found that, with growing\nthe magnetization, the diﬀerence becomes clear in tun-\nneling conductance. The clarity of diﬀerence between\nSTF and SBAF depends on the way of normalization of\nconductance and comes out more clearly in ZBCP nearhalf-metallic limit. The obtained results suggest that the\nmeasurement of ZBCP may be useful for discriminating\nmechanism of ferromagnetism.\nAlthough our formulation includes the ferromagnetic\ninsulator, we have studied only the normal insulating\nbarrier case in this paper. The spin-ﬁltering eﬀect have\nbeen expected in the ferromagnetic insulator[14] or in\nferromagnet given by the eﬀective mass mismatch[25].\nThen, as an interesting future problem we will study ex-\ntensively the spin-ﬁltering eﬀect in junctions of including\nboth ferromagnetic insulator and mass mismatch ferro-\nmagnet connected to superconductors of s-,d-wave and\nbroken time reversal symmetry pairing states. Moreover,\nit will be an important issue that the proximity eﬀect is\ntakeninto accountto the presentformulationby carrying\nout the self-consistent calculation of the pairing poten-\ntial in order to analyze the actual experiments. 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Yamasaki, M. Koy-\nanagi, N. Yoshida, and Y. Tanaka, Physica C 339, 287\n(2000)"    },    {        "title": "2009.01977v1.Detection_of_Ferromagnetic_Resonance_from_1_nm_thick_Co.pdf",        "content": "1  Detection of Ferromagnetic Resonance from 1 nm-thick Co  Shugo Yoshii, Ryo Ohshima, Yuichiro Ando, Teruya Shinjo and Masashi Shiraishi †  Department of Electronic Science and Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan  †Corresponding author (shiraishi.masashi.4w@kyoto-u.ac.jp)  Abstract To explore the further possibilities of nanometer-thick ferromagnetic films (ultrathin ferromagnetic films), we investigated the ferromagnetic resonance (FMR) of 1 nm-thick Co film. Whilst an FMR signal was not observed for the Co film grown on a SiO2 substrate, the insertion of a 3 nm-thick amorphous Ta buffer layer beneath the Co enabled the detection of a salient FMR signal, which was attributed to the smooth surface of the amorphous Ta. This result implies the excitation of FMR in an ultrathin ferromagnetic film, which can pave the way to controlling magnons in ultrathin ferromagnetic films.   2  Introduction Nanometer-thick films, so-called ultrathin films, have been collecting broad attention because of their abundant spintronics nature [1-3]. Chiba and co-workers carried out a pioneering study, in which 1 nm-thick ferromagnetic Co film exhibited sizable modulation of its magnetization under an applied gate voltage [1]. Indeed, the Curie temperature of the ultrathin Co was lowered to ca. 320 K and was modulated by gating, which gave a great impact to the spintronics field. A subsequent study using 0.4 nm-thick ultrathin Co revealed greater potential of magnetization control by combining ionic gating [2]. More recently, studies on the spintronic nature of ultrathin films have been expanded to include nonmagnetic metals, and the discovery of a gate-tunable inverse spin Hall effect in ultrathin Pt (2 nm thickness) has opened the field of tunable spin-orbit interaction (SOI) [3]. A common physics among the magnetization control of ultrathin ferromagnetic films and the tunable SOI in ultrathin nonmagnetic films is a substantial shift of the Fermi level under a strong gate electric field [4,5].  Investigation of spin physics using ferromagnetic resonance (FMR) is pivotal in modern spintronics. It has been used in a wide range of studies from both fundamental and applied physics, including those related to the generation of spin current [6] and a spin-torque diode effect [7]. FMR takes place under the simultaneous application of a static external magnetic field and microwave, which enables uniform rotation of magnetization. The uniform magnetization rotation generates a spin wave, a quantized state of which is magnon. Whilst magnons have attracted tremendous attention in spintronics because of their spin current propagation ability [8], they have also attracted great attention in the field of hybrid quantum systems because of the potential of creating strong coupling states with another magnon [9], a photon [10] and a superconducting qubit [11].  Despite the strong potential of magnons in spintronics and other condensed-matter fields, generation of FMR, of which excitation is indispensable to generate magnons in ferromagnets, in ultrathin ferromagnetic metals is not easy. In fact, the thickness of ferromagnetic metal films often used in FMR studies in spintronics is typically greater than 5 nm, because the surface roughness of the substrate beneath the ferromagnetic metal hampers the observation of a clear FMR signal. However, given the success of the substantial magnetization control in ultrathin Co by gating [1,2], efficient and tunable magnon creation in ultrathin ferromagnetic metals under FMR can provide a path to electric-field control of magnons, because the number of magnons is proportional to the square root of the total 3  magnetization and the coupling strength between magnons and photons is collectively enhanced by square root of number of magnons [12]. For the achievement, the first milestone is excitation of FMR with a sufficiently small resonance field and the half-width at half-maximum in ultrathin ferromagnetic metals, resulting in magnon excitation, which has not been sufficiently achieved. In this study, we report the realization of FMR in a 1 nm-thick Co film, which has not been previously demonstrated, by depositing a Ta buffer layer beneath the Co layer. The key to achieving FMR is utilization of smooth surface of amorphous Ta layer.   Results  Figures 1(a) and (b) show sample structures and measuring setup, respectively. We prepared two different types of samples: One is SiO2/Co (type-A), and the other is SiO2/Co/Ta (type-B) (see Fig. 1(a)). In type-A samples, a Co thin film of 1,2,3 or 5 nm thickness was deposited onto a SiO2 (300 nm)/Si substrate using radio-frequency magnetron sputtering. In type-B samples, a Ta buffer layer of 3 nm thickness was deposited beneath the 1 nm-thick Co. FMR measurements were performed using a TE011 (transverse electric mode) cavity of an electron spin resonance system. Figure 2(a) shows the FMR spectra of the four type-A samples, where the thickness of the Co film was varied. The resonance field and the linewidth of the spectra changed dramatically with the Co thickness. More importantly, an FMR signal was barely observed when the thickness of the Co was less than 2 nm. To circumvent the problem of the missing FMR in thin Co films, we introduced Ta as a buffer layer (type-B sample). Amorphous Ta is widely recognized to possess a quite smooth surface [1,2,13], although direct observation of the smooth surface of Ta was difficult without exposing the surface to air in our experimental setup and structural analyses such as x-ray diffraction cannot be implemented due to the thin Ta layer (3 nm).  Hence, we expected that the insertion of Ta beneath ultrathin Co enables formation of a flat and continuous Co film with nm-thick, yielding a sharp FMR spectrum from the Co. In fact, Chiba and co-workers observed the anomalous Hall effect from 0.4 nm-thick Co grown on 3 nm-thick Ta, which is compelling evidence for the formation of a continuous film [1,2]. Figure 2(b) shows a comparison of the FMR spectra of the Co (1 nm) and Co (1 nm)/Ta (3 nm) samples. The difference in the FMR spectra is clearly discernable, and the FMR spectrum of the Co (1 nm)/Ta (3 nm) sample is, in fact, quite obvious. Given that amorphous Ta possesses a smooth surface, the clear FMR spectrum of the Co is 4  attributed to the insertion of the Ta buffer layer with a smooth surface beneath the 1 nm-thick Co. Figure 2(c) shows the resonance fields, the half-width at half-maximum (∆µ0H) of the FMR spectra, and the Gilbert damping constant a as functions of the Co layer thickness of the type-A and -B samples. The resonance field and the ∆µ0H were obtained by deconvolution of the integral form of the FMR spectra (for more detail, see Methods and Supplementary Information). The Gilbert damping constant a was calculated using the following equation, 𝛼=∆𝜇!𝐻\t∙𝛾2𝜋𝑓\"#$,\t\t\t\t\t\t\t\t\t\t\t\t\t(1) where 𝛾 is the gyromagnetic ratio of Co and fres is the applied microwave frequency [14]. Whilst the resonance field, the half-width at half-maximum of the FMR spectra, and the Gilbert damping constant monotonically increased with decreasing the Co thickness, the 1 nm-thick Co layer overlying a Ta buffer layer exhibited noticeable suppression of them. Thus, it is corroborated that the insertion of an amorphous Ta layer beneath an ultrathin Co layer is an efficient approach to excite FMR in the 1 nm-thick Co.  Since the thickness of the amorphous Ta layer in the previous studies was fixed at 3 nm [1,2], the thickness of the Ta buffer layer was varied from 2 to 5 nm in increments of 1 nm and the FMR of the 1 nm-thick Co on Ta buffer layers of various thickness was measured. The FMR spectra of the Co layers with Ta buffer layers of various thickness and the Gilbert damping constant of each sample are shown in Figs. 3(a) and 3(b), respectively. Neither the FMR spectra nor the magnitude of the Gilbert damping constant is dependent on the Ta thickness, which suggests that the insertion of a 3 nm-thick Ta buffer layer is sufficient to induce formation of a flat Co layer. Notably, an FMR signal was not observed from the Co (1 nm)/ Ta (1 nm) sample (see Supplementary Information), which directly indicates that an excessively thin Ta buffer layer does not allow exciting the FMR in a 1 nm-thick Co layer.   To better understanding the aforementioned phenomena, we prepared a Co (1 nm)/\t𝕆 /Ta (3 nm) sample, where the surface of the Ta was intentionally oxidized (𝕆 denotes that the sample was exposed to air at this sample fabrication step). Figure 4(a) shows a comparison of the FMR spectra of the Co (1 nm)/Ta (3 nm) and Co (1 nm)/\t𝕆 /Ta (3 nm) samples. A substantial difference in the FMR spectra is observed; an FMR signal was not observed from the Co (1 nm)/\t𝕆 /Ta (3 nm) sample albeit a 3 nm-thick Ta buffer layer was introduced. The lack of an FMR signal 5  from the Co (1 nm)/\t𝕆 /Ta (3 nm) sample is thus attributed to the oxidized surface of the Ta. As aforementioned, the insertion of an amorphous Ta layer beneath the 1 nm Co facilitates the formation of a flat and continuous Co film because the surface of the amorphous Ta is smooth. Meanwhile, the surface roughness of the oxidized 3 nm-thick Ta was measured to be almost the same as that of the SiO2 substrate; a roughness of ca. 1 nm was observed for both samples by atomic force microscopy (AFM), whilst the grain sizes of these two samples slightly differed (see Figs. 4(b)). Here, to note is that surface of amorphous Ta cannot be measured by AFM without exposing the sample surface to air in our measuring setup. Hence, we deduce that the surface of the oxidized Ta loses sufficient smoothness, which can hinder the formation of a flat and continuous 1 nm-thick Co film. These results unequivocally rationalize that inserting a Ta buffer layer and maintaining its smooth surface play crucial roles in the growth of an ultrathin Co layer that can generate salient FMR.   Discussion The upshift of the resonance field as a function of the Co thickness in type-A samples (see Figs. 2 (a) and (c)) is attributed to a decrease of the total magnetization. Chiba and co-workers observed strong suppression of magnetization in 1 nm-thick Co, resulting in a substantial decrease of the Curie temperature [1]. Hence, the upshift of the resonance field is due to weaker magnetization of the ultrathin Co, consistent with the previous study [1]. Meanwhile, the missing FMR spectra is attributed to roughness of the Co film. We used SiO2/Si substrates, of which surface is not sufficiently smooth. Indeed, it was previously found that the FMR spectra of Ni80Fe20 (Py) were strongly dependent on the substrates and that the FMR linewidth of Py grown on a SiO2 substrate was greater than the linewidths of Py grown on yttrium-iron-garnet and non-doped diamond substrates [15]. The broader FMR spectrum of Py on a SiO2 substrate is ascribed to the roughness of the a SiO2 substrate, which hampers isotropic magnetization and uniform magnetization precession under FMR. The results obtained in the present study are consistent with those reported in the literature [15].  In chronicle of FMR studies of thin ferromagnetic films, a couple of studies were implemented about detection of FMR spectra from ultrathin Co [16,17] almost two decades ago. The thickness of the Co films in those 6  previous studies is comparable to that in our present study. Meanwhile, the resonance field and the FMR linewidth in those studies were roughly 500 mT and 60 mT (note that the linewidth of 60 mT is equivalent to the half-width at half-maximum of roughly 70 mT) [16], respectively (the similarly large resonance field was also reported in the literature [17], whereas the linewidth was not discussed in the study). Such the large resonance field and the broad FMR linewidth are attributed to a fact that the ferromagnetism was quite weak and the magnetization precession under FMR was not uniform, i.e., the quality of the Co was poor. Furthermore, the FMR spectrum exhibited the single branch only when the thickness of the Co was greater than 2 nm in that study [16]. Given that the resonance field and the half-width at half-maximum of the FMR spectra in our study are smaller and roughly 80 mT and 15 mT, respectively, and that the single FMR spectrum can be seen from the 1 nm-thick Co, the ultrathin Co used in our study simultaneously possesses sufficiently strong ferromagnetism and uniform FMR unlike in the previous studies, i.e. we experimentally demonstrated that the quality of the ultrathin Co with a Ta buffer layer is sufficiently good and the ultrathin Co is quite available for future magnon spintronics and quantum hybrid systems.  In summary, we achieved FMR excitation from an ultrathin Co film with a thickness of 1 nm by inserting an amorphous Ta buffer layer. The smoothness of the amorphous Ta played a crucial role in a formation of a flat and ultrathin Co layer. These findings provide a new pathway to the electric-field control of magnons using ultrathin ferromagnetic metals.  Methods In type-A samples, a Co thin film of 1,2,3 or 5 nm thickness was deposited using radio-frequency magnetron sputtering. The base pressure of the sputtering system was kept to be lower than 2.5×10-5 Pa, and the flow rate and partial pressure of the Ar gas were set to be 5 sccm and 0.5 Pa, respectively; and the sputtering temperature was room temperature (RT). The deposition rate of the Co was 0.8 nm/min. A SiO2 layer (10 nm) was deposited onto the Co film to prevent from oxidation of the Co layer. In type-B samples, a Ta buffer layer of 3 nm thickness was deposited, where the deposition rate of the Ta was 3.83 nm/min. The substrate was the same as that used for the type-A samples. During the deposition of Co and Ta, the sample holder was rotated at 20 rpm.  7  FMR measurements were performed using a TE011 (transverse electric mode) cavity of an electron spin resonance system (JEOL JES-FA 200); an external magnetic field under microwave irradiation was applied parallel to the sample plain, and the microwave frequency and power were set to be 9.12 GHz and 10 mW, respectively (see Fig. 1(b)). All of the FMR measurements were carried out at RT.  Deconvolution of the FMR spectra was carried out by using the integral form of the FMR spectra. In addition to the Lorentzian (symmetric) component, an asymmetric component was taken into account. The fitting function used is 𝐹(𝜇!𝐻)=𝐴$%&(∆)!*)\"()!*,)!*#$%)\"-(∆)!*)\"−2𝐴.$%&/()!*,)!*#$%)()!*,)!*#$%)\"-(∆)!*)\"+𝑎(𝜇!𝐻)+𝑏, where Asym and Aasym are symmetric and asymmetric components, respectively, and, a and b are constants.    8  References 1. Chiba, D., Fukami, S., Shimamura, K., Ishiwata, N., Kobayashi K., & Ono, T. 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Tsukahara, A., Ando, Y., Kitamura, Y., Emoto, H., Shikoh, E., Delmo, M.P., Shinjo T. & Shiriashi, M. Self-induced inverse spin Hall effect in permalloy at room temperature. Phys. Rev. B 89, 235317 (2014).  16. Purcell, S.T., van Kestere, H.W., Cosman, E.C., Zeper W.B. & Hoving, W. Magnetic properties of ultrathin epitaxial Co films on a Pd (111) single crystal. J. Appl. Phys. 69, 5640-5642 (1991).  17. Gieniusz, R., Stupakiewicz, A., Liedke, O., Maziewski, A., Gogol P. & Beauvillain, P. FMR study of ultrathin Co magnetic films on vicinal Si (111) substrates. J. Magn. Magn. Mater. 272-276, e911-e912 (2004).     10  Acknowledgement This work is supported in part by a Grant-in-Aid for Scientific Research (S), “Semiconductor spincurrentronics” (No. 16H06330) and Izumi Science and Technology Foundation. The authors thank Prof. Daichi Chiba of Osaka University, Japan, for his fruitful discussion and suggestions about the growth of an amorphous Ta layer. Author contributions M. S., Y. A. and R. O. conceived the experiments. S.Y. fabricated samples, collected data and analyzed results. R.O. helped in the experiments. S.Y., R.O. and M.S. wrote the manuscript. All authors discussed the results.  Competing interest The authors declare no competing interests.  Additional information  Supplementary information is available for this paper at XXX. Correspondence and requests for materials should be addressed to M.S.     11  Figure captions Figure 1. (a) Schematics of the sample structures of the type-A and the type-B samples. The thickness of the Co (tCo) in the type-A samples was 1, 2, 3, or 5 nm. The samples were capped with 10 nm-thick SiO2 to prevent oxidization of the Co. (b) Schematic of the setup used for FMR measurements.  Figure 2. (a) FMR spectra of sample-A Co of 1, 2, 3, and 5 nm in thickness. (b) Comparison of the MR spectra of the Co (1 nm) and Co (1 nm)/Ta (3 nm) samples. (c) The Co thickness dependence of the resonance field µ0Hres (upper panel), the half-width at half-maximum of the FMR spectra DH (middle panel), and the Gilbert damping constant a (lower panel).  Figure 3. (a) FMR spectra from the Co (1 nm)/Ta (tTa nm) samples. The thickness of the Ta (tTa) was changed from 2 to 5 nm. (b) The Ta thickness dependence of the Gilbert damping constant a.  Figure 4. (a) Comparison of the FMR spectra of the Co (1 nm)/Ta (3 nm) and Co (1 nm)/\t𝕆 /Ta (3 nm) samples. (b) Atomic force microscopic views of the surfaces of  the SiO2 substrate and the oxidized Ta.    12  Figures    \n Fig. 1(a) Yoshii et al  \n Fig. 1(b) Yoshii et al   \n13    Fig. 2(a)(b)(c) Yoshii et al.    \n14    \n  Fig. 3(a)(b) Yoshii et al.   \n15   Fig 4(a). Yoshii et al.   \n Fig 4(b) Yoshii et al  \n"    },    {        "title": "1007.5432v3.Diamagnetic_susceptibility_of_spin_triplet_ferromagnetic_superconductors.pdf",        "content": "arXiv:1007.5432v3  [cond-mat.supr-con]  6 Oct 2010Diamagnetic susceptibility of spin-triplet\nferromagnetic superconductors\nH. Belich1,2, Octavio D. Rodriguez Salmon1, Diana V. Shopova3,\nand Dimo I. Uzunov1,3†\n1International Institute of Physics, Universidade Federal de Rio Grande do Norte,\nav. Odilon Gomes de Lima, 1722, 59078–400, Natal (RN), Brazi l.\n2Universidade Federal do Esp´ ırito Santo (UFES), Departame nto de F´ ısica e\nQu´ ımica, Av. Fernando Ferrari 514, Vit´ oria, ES, CEP 29075 -910, Brazil.\n3Collective Phenomena Laboratory, G. Nadjakov Institute of Solid State Physics,\nBulgarian Academy of Sciences, BG-1784 Soﬁa, Bulgaria.\n†Corresponding author: d.i.uzunov@gmail.com\nKey words : Ginzburg-Landau theory, thermodynamic property, supercon ductiv-\nity, ferromagnetism, magnetization, phase diagram.\nPACS: 74.20.De, 74.20.Rp\nAbstract\nWe calculate the diamagnetic susceptibility in zero extern al magnetic ﬁeld\nabove the phase transition from ferromagnetic phase to phas e of coexistence\nof ferromagnetic order and unconventional superconductiv ity. For this aim we\nuse generalized Ginzburg-Landau free energy of unconventi onal ferromagnetic\nsuperconductor with spin-triplet electron pairing. A poss ible application of\nthe result to some intermetallic compounds is brieﬂy discus sed.\nIn certain ferromagnetic unconventional superconductors the phase transition to su-\nperconductivity states occurs in the domain of stability of ferroma gnetic phase (an\nexample is the itinerant ferromagnet UGe 2[1, 2, 3]). This seems to be a general\nfeature of ferromagnetic superconductors with spin-triplet elec tron pairing [4, 5, 6]\n(see also reviews [7, 8]). In such situation the thermodynamic prope rties near the\nphase transitionlinemaydiﬀer fromthoseknown forthe supercond ucting-to-normal\n1PT\nPcFMN\nFSTFS(P)TF(P)\n1\n2\nC\nFigure 1: An illustration of the T−Pphase diagram of UGe 2(details are omitted): N – normal\nphase, FM - ferromagnetic phase, FS - phase of coexistence of fe rromagnetic order and supercon-\nductivity, TF(P) andTFS(P) are the respective phase transition lines (solid line corresponds to\nsecond order phase transition, dashed lines correspond to ﬁrst o rder phase transitions; 1 and 2 are\ntricritical points; Pc∼1.6 GPa is the critical pressure; TF(0)∼53 K;TFS<1.22 K; the loop C\nindicates a small domain ( T <0.3 K,P∼16 GPa) where the shape of the phase diagram is not\nwell established by available experimental data.\nmetal transition. We show this by using the example of diamagnetic su sceptibility\nabove the phase transition line of superconducting transition in spin -triplet ferro-\nmagnetic superconductors. This is the line in the temperature-pre ssure (T−P)\nphase diagram (Fig. 1), which separates the pure ferromagnetic p hase (FM) and the\nphase (FS) of coexistence of ferromagnetic order and supercon ductivity. Here we\npresent the result for diamagnetic susceptibility which follows from t he Ginzburg–\nLandau theory for such type of superconductors [4, 5, 6]. We ou tline the main steps\nof calculation of diamagnetic susceptibility in the Gaussian approximat ion. At the\nend we brieﬂy discuss the possible application of our results to real s ystems.\nFollowing notations and results in Refs. [4, 7, 8], we present the GL fr ee energy\n(ﬂuctuation Hamiltonian) of spin-triplet ferromagnetic supercond uctors, which is\nessential in the present consideration, namely\nH=/integraldisplay\nd3x/braceleftBig\nˆH0[ψ(x)]+ˆHM[ψ(x)]/bracerightBig\n(1)\nby the energy densities\nˆH0=/planckover2pi12\n4m3/summationdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n∇−2ie\n/planckover2pi1cA/parenrightbigg\nψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+as|ψ|2(2)\nand\n2ˆHM=iγ0M·(ψ×ψ∗)+ρM2·ψ2(3)\nIn Eqs. (2)–(3), ψ(x) ={ψj(x);j= 1,2,3}is three dimensional vector ﬁeld with\ncomplex components ψj, which represents thesuperconducting order, Misthespon-\ntaneous magnetization, the vector potential Ais related to the magnetic induction\nbyB=H+4πMand obeys the Coulomb gauge ( ∇·A= 0);as=αs(T−Ts),γ0\nandρare positive material parameters, and 2 eand 2mare the charge and the eﬀec-\ntive mass of the electron Cooper pairs, respectively. We neglect th e possible spatial\nanisotropy, which is usually represented in the gradient terms of th e Hamiltonian H\n(see, e.g., Ref. [4, 7, 8]).\nOur task is to calculate the equilibrium free energy\nF=−β−1ln/integraldisplay/productdisplay\nx∈VDψ(x)exp(−βˆH), (4)\nin the volume V=LxLyLzof the superconductor and the diamagnetic susceptibility\nper unit volume in zero external magnetic ﬁeld, given by χ= [−∂2F/V∂2H]H=0;\nβ−1=kBT. In Eq. (4), the functional integral is taken over both real [ ℜψ(x)] and\nimaginary [ ℑψ(x)] parts of the complex ﬁeld ψ(x), i.e.,Dψ(x)≡dℜψ(x)dℑψ(x).\nNote that for temperatures near TFS(P) we can always set β≈βFS= 1/kBTFS\n(see, e.g., [9]).\nAs far as the behaviour in FM phase in a close vicinity of curve TFS(P) is of interest\nto our consideration, (see Fig. 1), the magnetization Mhas a magnitude |M| ≡M,\ngiven byM(T,P) = [αf(T−TF)/bf]1/2, i.e., the result from the standard Landau\ntheory of ferromagnetic transitions with parameters af=αf(T−TF) andbf[4]\nFm=afM2+bf\n2M4, (5)\nwhereaf=αf(T−TF), andbf>0. Therefore, in our consideration M(T,P) is a\nknown thermodynamic quantity, which is established by the exhaust ive thermody-\nnamic analysis of the phases in the unconventional superconducto r in [4].\nWe choose the magnetization M= (0,0,M) and the external magnetic ﬁeld H=\n(0,0,H) to lie along the ˆ z-axis. Then the ﬁrst term in Eq. (3) takes the simple\nformM(ψ×ψ∗)z=M(ψ1ψ∗\n2−c.c.). Under the supposition of uniform external\nmagnetic ﬁeld H, we take the gauge of the vector potential AasA= (−By,0,0),\nand following classic papers [10, 11, 12], we can represent the ﬁelds ψj(x) by the\nseries\n3ψj(x) =1\nLxLz/summationdisplay\nqcj(q)ϕj(q,x) (6)\nin terms of the eigenfunctions\nϕj(q,x) =1\n(LxLz)1/2ei(kx+kz)un(y) (7)\nof the operator [ i/planckover2pi1∇+(2e/c)A]2/4m, corresponding to the eigenvalues\nE(q) =/parenleftbigg\nn+1\n2/parenrightbigg\n/planckover2pi1ωc+/planckover2pi12\n4mk2\nz, (8)\nspeciﬁed by the quantum number n= 0,1,...,∞, the wave vector components kx\nandkz, and the cyclotron frequency ωc= (eB/mc). In Eq. (6), the function un(y)\nis related to the Hermite polynomials Hn(y) by\nun(y) =Ane−(y−y0)2\n2a2\nHHn/parenleftbiggy−y0\naH/parenrightbigg\n, (9)\nwhereA−1\nn= (aB2nn!√π)1/2[13],y0=a2\nBkx, andaB= (/planckover2pi1c/2|e|B)1/2;B=|B|.\nNow the ﬂuctuation Hamiltonian becomes H=/summationtext\nqˆH(q) with\nˆH(q) =/summationdisplay\nj˜E(q)cj(q)c∗\nj(q)\n+iγ0M[c1(q)c∗\n2(q)−c.c.], (10)\nwhere\n˜E(q) =E(q)+as+ρM2. (11)\nApplying the unitary transformation,\nc1(q) =i√\n2[−φ+(q)+φ−(q)] (12a)\nc2(q) =1√\n2[φ+(q)+φ−(q)] (12b)\nrenders the ﬂuctuation Hamiltonian as a sum of squares of ﬁeld comp onentsc3(q),\nandφ±(q), and the free energy (4) can be calculated as usual Gaussian inte grals\nover the same ﬁelds.\n4Following approximations, justiﬁed in Ref. [12], we obtain the result\nF\nV=µB2/parenleftBigg\n1\na1/2\n−+1\na1/2\n0+1\na1/2\n+/parenrightBigg\n, (13)\nwhere\na±(γ0) =as+ρM2±γ0M, (14)\na0≡a±(0), andµ=e2kBT/24π/planckover2pi1c2m1/2. Having in mind that ∂/∂H=∂/∂B, the\nﬂuctuation diamagnetic susceptibility in Gaussian approximation take s the form\nχ(T) =−2µ/parenleftBigg\n1\na1/2\n−+1\na1/2\n0+1\na1/2\n+/parenrightBigg\n, (15)\nIn contrast to usual superconductors [12], where the contribut ion to the free energy\nfrom the diamagnetic currents is represented by a single term, her e we have three\nterms with labels 0, and ±which exactly correspond to the contributions of the ﬁeld\ncomponents c3, andφ±, respectively.\nNow one should use known results [4, 5, 6, 7, 8] to analyze the singula rities of free\nenergy in a close vicinity (0 <T−TFS≪TFS) to the phase transition curve TFS(P)\nin the FM phase ( TF> T > T FS), whereM(T,P) = [αf(TF−T)/bf]1/2and, for\nsome real intermetallic compounds, for example, UGe 2, the condition ( TF−TFS)≫\n(T−TFS) is satisﬁed. We shall brieﬂy discuss the behaviour of the free ener gy (13)\nnear the left-hand part of the curve TFS(P), where the phase transition FM-FS is of\nsecond order. For this case the critical temperature TFS(P) is given in Refs. [5, 6].\nIn the present notations TFS(P) is deﬁned by the equation\nTFS=Ts−ρ\nαs∆+γ0\nαs∆1/2, (16)\nwhere ∆ ≡[M(TFS)]2=αf(TF−TFS)/bf>0. Expanding a0(T), anda±(γ0,T) to\nﬁrst order in ( T−TFS), one may easily check that a−(TFS) = 0 and\na−(T)≈˜a−(T−TFS), (17)\nwhere\n˜a−=αs−ραf\nbf+γ0α1/2\nf\n2[bf(TF−TFS)]1/2, (18)\n5whilea0anda+remainpositiveat TFS:a0(TFS) =γ0∆1/2, anda+(TFS) = 2a0(TFS).\nTherefore, only one of all three ﬂuctuation diamagnetic contribut ions in Eqs. (13)\nand (15) will generate singularity of the free energy and the typica l divergence of\nsusceptibility. Keeping only the singular term in Eq. (13), we obtain th at in a close\nvicinity of line TFS(P), wherea−≪min(a0,a+),\nχ(T) =χ0\n(T−TFS)1/2, (19a)\n(T >T FS), where the scaling amplitude χ0is given by\nχ0=−2µ\n˜a1/2\n−. (19b)\nNote thata±(γ0)>0 is a condition for the stability of the FM phase and, therefore,\nthe quantity ˜ a−is always positive for TFS(P)<T <T F(P).\nThe formulae (19a) and (19b) are our main result. This scaling relatio n [9] is of\ntypical Gaussian type with an inverse root dependence on ( T−TFS) whereas the\nscaling amplitude χ0contains an essentially new information. Compared to known\nresultforusualsuperconductors[12],theﬂuctuationdiamagnet icsusceptibility(19a)\ncontains an extra factor (˜ a−)−1/2, which depends on the material parameters of the\nunconventional ferromagnetic superconductor. The value of th e new susceptibility\namplitude factor (˜ a−)−1/2in Eq. (19b) should be taken at TFS(P) for any pressure\nPof interest. Thus in evaluating the parameter ˜ a−we may use the Eq. (16) for\nTFS(P).\nIn some real systems the Eq. (18) can be simpliﬁed. For example, in U Ge2,Ts∼0\nK [5, 6],TF≫TFS[1] and, therefore, one may use ˜ a−≈(αs−ραf/2bf). This\nresult is obtained with the help of Eq. (16). In itinerant ferromagne ts with uniaxial\nanisotropyas, forexample, UGe 2, bothphasesFMandFSmayoccurintwo domains\nwith opposite magnetizations |M|=±M. Here we have considered FM and FS\nwithM >0. In the domains of FM, where M <0, the singular parts of the free\nenergy and the susceptibility will be given by the terms, containing th e quantity\na+. Because of the invariance of the Eqs. (13) and (15) with respect to the change\na±→a∓, the results presented by Eqs. (13), (15), and (19a)–(19b) ar e valid in both\ndomains of the FM and ψ-ﬂuctuations corresponding to any domain ( M≶0) of\nFS [4].\nWe have used the Gaussian approximation, which is not valid in the critic al re-\ngion [9] of anomalous ﬂuctuations. However, the critical region of r eal ferromag-\nnetic superconductors with spin-triplet electron pairing is often ve ry narrow and,\n6hence, virtually of no interest. Therefore, the present results c an be reliably used\nin interpretation of experimental data for real itinerant ferroma gnets, which exhibit\nlow-temperaturespin-triplet superconductivity triggeredbythe ferromagneticorder.\nReferences\n[1] S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Hase lwimmer,\nM.J. Steiner, E. Pugh, I. R. Walker, S.R.Julian, P. Monthoux, G.G.L onzarich,\nA. Huxley. I. Sheikin, D. Braithwaite, and J. Flouquet, Nature (Lon don) 406\n(2000) 587.\n[2] A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois, D. Braithw aite, R. Calem-\nczuk, and J. Flouquet, Phys. Rev. B 63 (2001) 144519.\n[3] N. Tateiwa, T. C. Kobayashi, K. Hanazono, A. Amaya. Y. Haga. R . Settai, and\nY. Onuki, J. Phys. Condensed Matter 13 (2001) L17.\n[4] D. V. Shopova, and D. I. Uzunov, Phys. Lett. A 313 (2003) 139 ; Phys. Rev. B\n72 (2005) 024531.\n[5] M. G. Cottam, D. V. Shopova, and D. I. Uzunov, Phys. Lett. A 3 73 (2008) 152.\n[6] D. V. Shopova and D. I. Uzunov, Phys. Rev. B 79 (2009) 064501 .\n[7] D. V. Shopova, and D. I. Uzunov, Bulg. J. Phys. 32 (2005) 81.\n[8] D. V. Shopova and D. I. Uzunov, in: Progress in Ferromagnetism Research, ed.\nby V. N. Murray (Nova Science Publishers, New York, 2006) pp. 223 . ISBN:\n1-59454-469-7.\n[9] D. I. Uzunov, Introduction to the theory of critical phenomen a (World Scien-\ntiﬁc, Singapore, 1993); 2nd Edition: World Scientiﬁc, New Jersey, 2 010.\n[10] H. Schmidt, Z. Physik 216 (1968) 336.\n[11] A. Schmid, Phys. Rev. 180 (1968) 527.\n[12] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part II ( Pergamon,\nLondon, 1980).\n[13] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathemat ical Functions\n(Dover, New York, 1965).\n7"    },    {        "title": "0708.3231v1.Current_induced_torques_due_to_compensated_antiferromagnets.pdf",        "content": "arXiv:0708.3231v1  [cond-mat.mes-hall]  23 Aug 2007Current-induced torques due to compensated antiferromagn ets\nPaul M. Haney and A. H. MacDonald\nDepartment of Physics, The University of Texas at Austin, Au stin, Texas, 78712-0264, U.S.A.\nWe analyse the inﬂuence of current induced torques on the mag netization conﬁguration of a fer-\nromagnet in a circuit containing a compensated antiferroma gnet. We argue that these torques are\ngenerically non-zero and support this conclusion with a mic roscopic NEGF calculation for a circuit\ncontaining antiferromagnetic NiMn and ferromagnetic Co la yers. Because of symmetry dictated\ndiﬀerences in the form of the current-induced torque, the ph ase diagram which expresses the depen-\ndence of ferromagnet conﬁguration on current and external m agnetic ﬁeld diﬀers qualitatively from\nits ferromagnet-only counterpart.\nPACS numbers: 85.35.-p, 72.25.-b,\nIntroduction — Current-induced torques in noncollinear\nferromagnetic metal circuits were predicted over 10 years\nago[1, 2], and have since been the subject of an extensive\nandquite successfulbodyofexperimentalandtheoretical\nresearch. Almost all studies of current-induced torques\nconsider either their role in ferromagnetic (F) spin valve\ncircuits [3, 4, 5, 6, 7] or their inﬂuence on magnetic do-\nmain wall motion [8, 9, 10, 11, 12]. In both cases, the\ncurrent-induced torques can be understood as following\nfrom the transfer of conserved spin angular momentum\nfrom current-carryingquasiparticlestothe magneticcon-\ndensate, hence the term spin-transfer torque. It has re-\ncently been predicted that current-induced torques are\ngenerically present whenever non-equilibrium quasiparti-\ncles interact with non-collinear magnetic order parame-\nters,even[13,14]incircuitscontainingonlyantiferromag-\nnetic (AF) elements. Experiments[15] have established\na dependence of unidirectional exchange bias ﬁelds on\ncurrent, providing indirect evidence that current-induced\ntorques are present in AFs. In this Letter we analyse the\ninﬂuence of current-induced torques on a F thin ﬁlm in\na circuit containing a compensated AF. Because of a key\ndiﬀerence in symmetry compared to purely F spin-valve\ncircuits, we ﬁnd that the phase diagram which expresses\nthe dependence of the magnetic conﬁgurationof the F on\ncurrent and external magnetic ﬁeld diﬀers qualitatively\nfrom the familiar F only spin-valve phase diagram[16].\nIn particular, we ﬁnd that transport currents can drive\nthe F to a stable steady state with magnetization per-\npendicular to the AF layer moments. In the following\nparagraphs we argue on symmetry grounds for the form\nof the current-induced torque, and explore its robustness\nby performing a fully microscopiccurrent-induced torque\ncalculation for a circuit containing Co and NiMn layers.\nWe then turn our attention to the construction of the\nF state phase diagram implied by equations of motion\nwhich include the current-induced torque term, and con-\nclude with a discussion of experimental implications.\nCurrent-induced Torques due to Compensated\nAntiferromagnets — The total current-induced torque\nacting on a F nanoparticle can always[1] be expressed in\nterms of the diﬀerence between incoming and outgoing\nFIG. 1: Current-induced torques due to a compensated an-\ntiferromagnet. The arrows above the structure indicate the\nelectron ﬂux spin direction. The white arrows indicate the\nensuing current-induced torques on the FM.\nspin currents. The presence of a ferromagnet will in\ngeneral induce a nonzero spin current at the AF-F\ninterface. When spin-polarized electron ﬂux from an AF\nwith orientation ˆ nAFenters a F with orientation ˆ nF,\nthe spin current entering F will have some component in\nthe ˆnAFdirection. It follows that, just as in the familiar\ncase where both materials are F, a current-induced\ntorque will act in the plane deﬁned by ˆ nAFand ˆnF, as\nillustrated in Fig. (1). (Out of plane torques are also\nnon-zero but tend to be much smaller.) Spin-invariance\nof the overall circuit implies that the in-plane torque\nmust be an odd function of the angle θbetween ˆ nFand\nˆnAF, and that it can therefore be expanded in terms of\na sin-only Fourier series, vanishing for both parallel and\nantiparallel collinear conﬁgurations. Most AF materials\nused in magnetoelectronics are fully compensated, i.e.\nthe spin-density sums to zero (or nearly so) in every\nlattice plane perpendicular to the current direction.\nIn this case, reversal of the AF moment direction is\nequivalent to a lateral translation which cannot inﬂuence\nthe current-induced torque. It follows that in the\ncompensated AF case the torque is invariant under\nθ→θ+π, restricting its Fourier expansion to terms\nproportional to sin(2 nθ). The torque therefore vanishes\nwhen ˆnFis perpendicular to ˆ nAF, and undergoes a\nsign change for θ→π−θ, as illustrated in Fig. (1).\nThe property that the torque acting on a F due to a\ncompensated AF vanishes not only for collinear but also2\nfor perpendicular orientations is primarily responsible\nfor the novel current-induced torque phase diagram that\nwe discuss below.\nCurrent-Induced Torques for Co/NiMn — We employ a\nnon-equilibrium Green’s function (NEGF) approach[17]\nfor microscopic calculations of magneto-transport prop-\nerties and current-induced torques. Quasiparticle Hamil-\ntonians are constructed using density functional theory\nwithin the local spin density approximation (extended to\nallow noncollinear spin conﬁgurations), norm-conserving\npseudopotentials, and an s,p,dsingle-zeta atomic or-\nbital basis set. The induced torque per-current can be\ncalculated atom by atom[17]:\n/vector˙S\nI=µB\ne/integraltext\ndk/bardbl/summationtext\nα,β(/vector∆α,β×/vector mtr\nβ,α)/integraltext\ndk/bardblT(ǫF). (1)\nThe right-hand-side of Eq. (1) expresses the misalign-\nment between the non-equilibrium spin-density, /vector mtr\nα,β,\nand the spin-dependent part of the exchange-correlation\npotential, /vector∆α,β. HereT(ǫF) is the transmissionprobabil-\nity,k/bardbllabels transverse channels, α,βare orbital labels,\nandαis summed only over orbitals centered on the atom\nof interest. When the many-body Hamiltonian is spin ro-\ntationally invariant, the current-induced torque on each\natom is equal to the net spin ﬂux out of the atom.\nWe apply this approach to a system with a single in-\nterface between antiferromagnetic NiMn and ferromag-\nnetic Co. The crystal structure of NiMn is face centered\ntetragonal,withNiandMnlayersalternatinginthe(001)\ndirection[18]. The Ni atoms are approximately nonmag-\nnetic, while the Mn atoms form a compensated antifer-\nromagnetic 2-dimensional lattice within each plane (See\nFig. (2)). In our calculation, we use a= 3.697˚A, with\nac/aratio of 0 .9573 for NiMn and, following Ref. [19],\na lattice-matched tetragonal structure for Co with a c/a\nratio chosen to conserve its experimental atomic volume.\nThe results shown here are for current in the (001) di-\nrection, perpendicular to the interface between Co and\nNi terminated NiMn. The current through the interface\nhas a polarization P= (T↑−T↓)/(T↑+T↓) = 6.4% when\nthe F and AF moments are collinear, with the largercon-\nductance forthe ferromagnetmajorityspins. To evaluate\nthe current-induced torques present in the system, we ro-\ntate the Co layer magnetization orientation by an angle\nθwith respect to the NiMn moment direction and use\nEq. (1).\nFig. (3) shows the total torques acting on the AF\nand F order parameters as a function of θ. The torque\nacting on the F closely follows the form anticipated on\nsymmetry grounds above. We associate the small torque\natθ=π/2 with weak ferromagnetism which is induced\nin the top (Ni) layer of NiMn. In NiMn only the diﬀer-\nence between the torques on the two sublattice of the AF\ndrives the order parameter. The current-induced torque\ntends to drive the orientation of downstream material\nFIG. 2: Illustration of the NiMn-Co interface model.\n(AF or F) parallel with that of the upstream, and to\ndrive the upstream material orientation perpendicular to\nthe downstream (so that for electron ﬂow from AF to F,\nthe F tends to align to AF, and the AF tends to become\nperpendicular to F, within their common plane).\nWe have also considered Mn terminated NiMn adja-\ncent to Co. In this case the last Mn layer acquires a net\nmagnetic moment in the direction of Co. The current-\ninduced torquesdonot showasclean ofasin2 θbehavior,\nbut a combination of sinθandsin2θ. We conclude that\nthe absence of odd nsinnθtorques is closely tied to the\ndegree of compensation at the AF interface.\n04590135180−0.5−0.2500.250.5\nRelative orientationSTT/I (µB/e)STT/I acting on AFM and FM\nAFM\nFM\nFIG. 3: Current-induced torques per current acting on the or -\nder parameters of the AF and F layers vs relative orientation .\nIn the AF the order parameter is driven by the diﬀerences\nbetween torques on opposite sublattices. Units are µB/e.\nPhase Diagram for a pinned antiferromagnet — We now\nconsider the implications of this new form of current-\ninducedtorqueforsystemswiththeusualthinﬁlmgeom-\netry, assumming for the sake of deﬁniteness that the AF\nmomentdirection ˆ nAFispinnedandliesintheplane,and\nthat the external magnetic ﬁeld His applied in the same\ndirection. We use a spherical coordinate system for the F\nmoment direction, taking ˆ nAFas the polar direction and\nthe ˆxdirection as the ﬁlm normal (the demagnetizing\nﬁeld is denoted by Hd). We assume that a non-magnetic\nspacer layer is placed between the F and AF layers so3\nthat exchange interactions negligible. Just as in the pure\nF case, a spacer layer is not expected to have a large im-\npact on current-induced torques. We also omit easy-axis\nanisotropy; its inclusion wouldn’t substantially change\nthe picture described below. With these ingredients the\npolar and azimuthal torques acting on the ferromagnet\nare:\nΓθ=−1\n2sin(θ)sin(2φ)Hd+sin(2θ)HCI;\nΓφ= sin(θ)H+1\n2sin(2θ)cos2(φ)Hd. (2)\nHere we have parameterized the current-induced torque\nbyHCIand chosen a sign convention in which HCI<0\nwhen it favors perpendicular alignment. Steady-state so-\nlutions satisfy Γ φ= Γθ= 0 and are stable for small devi-\nations when Gilbert damping is included. We havedeter-\nmined the stability regions of the steady state solutions\ndiscussed below by following the procedure described in\nRef. 20. We present all of our results in terms of the\ndimensionless ﬁelds h=H/HdandhCI=HCI/Hd.\nInthe absenceofthecurrentinduced torques,themag-\nnetization simply lines up with the magnetic ﬁeld applied\nin the easy plane. The inﬂuence of the current-induced\ntorque on F is particularly dramatic for HCI<0. Be-\ncause the torques then tend to push the magnetization\nperpendicular to ˆ nAF, the ﬁeld-aligned solution is stable\nonly for:\nhCI≥ −α\n2/parenleftbigg\n|h|+1\n2/parenrightbigg\n, (3)\nwhereαis the Gilbert damping parameter. For suf-\nﬁciently strong currents and weak external ﬁelds, a\nperpendicular-to-plane steady state becomes stable:\nθ=π\n2+h;\nφ=−2hCIh+nπ . (4)\nThese equations have been derived assuming that hand\nhCIare small. In the above nis an even integer for\nsolutions which point approximately in the +ˆ xdirection,\nand an odd integer for the −ˆxdirection. The region of\nstability for this solution is:\nhCI≤ −α\n2/parenleftbigghsinh−2cos2h\nhsinh−cos2h/parenrightbigg\n, (5)\nwhere the the fraction on the r.h.s. of the above in-\nequality must be negative, implying that |h|<0.608, or\nequivalently |H|< µ0Ms(0.608).\nThe stability of this counter-intuitive stable steady\nstate is explained in Fig. (4). This ﬁgure illustrates\nthe situation when the excursions from the easy plane\nare small. For simplicity we ﬁrst consider no external\nﬁeld. In the absence of the current-induced torque asmall ﬂuctuation out ofthe easy plane would initiate pre-\ncession about the hard axis which damps back into the\neasy plane. The presence of the sin2 θtorque, however,\ndrives the magnetization ˆ mperpendicular to ˆ nAFwithin\ntheir common plane. As ˆ mprecesses around the hard-\naxis, this torque vector has a component which points\nout of the easy plane. If the angle between ˆ nAFand the\nin-plane component of ˆ misβ, the magnitude varies as\nΓx= 2HCImxsin2β, as shown in the ﬁgure. The crucial\npoint is that this torque is always positive throughout\nthe precession. When this torque exceeds the damping,\nthe out-of-plane conﬁguration is stabilized. The eventual\nout-of-plane orientation can be +ˆ xor−ˆxdepending on\nthe direction of the initial ﬂuctuation out of plane. The\npresence of an applied ﬁeld changes the trajectory of the\nmagnetization upon excursions from the easy-plane. For\na suﬃciently large applied ﬁeld, the torque is unable to\nstabilize the out-of-plane conﬁguration, and no steady\nstate is reached.\n−101\n−1.5−1−0.500.510  Out of plane\nTorque      Hard−axis\nβ\nz y m\nnAFx\nFIG. 4: Saddle shape illustrates the out-of-plane torque vs β\nfor small excursions of the magnetization orientation ˆ mfrom\nthe easy plane. The out-of-plane torque is always positive.\nInteresting new steady states can in principle also be\ninduced by the current-induced torque for HCI>0. For\n|H| ≤Hd, the steady state stability analysis identiﬁed\nconﬁgurations in which the magnetization is approxi-\nmately anti-aligned with the applied ﬁeld:\nθ= cos−1(−h)\nφ=−2hCIh. (6)\nwhichisstablefortherangeofappliedﬁeldsandcurrents:\nhCI≥α\n2/parenleftbigg2−h2\n3h2−1/parenrightbigg\n(7)\nForHCI>0 and|H| ≥Hd, the equilibrium solutions\naremz=±1. In this case the stability condition for the\nmagnetization anti-aligned with the ﬁeld is:\nhCI≥α\n2/parenleftbigg\n|h|−1\n2/parenrightbigg\n. (8)\nThese anti-aligned states occur only if the magnetization\nis initially nearly anti-aligned with the applied ﬁeld. The4\nreason for their stability is that this form of the current-\ninduced torque does not distinguish between +ˆ zand−ˆz\n- it merely tends to make to direct the F to the near-\nest available ˆ z-axis, even if it’s opposite to the applied\nﬁeld. The region for such a solution is shown in Fig. (5),\nlabelled ±z. This misaligned steady state may not be\nexperimentally relevant however because it occurs only\nwhen the magnetization is initially nearly anti-aligned to\nan applied ﬁeld of ﬁnite magnitude |H|> Hd/radicalbig\n1/3.\nFig. (5) shows the xandzcomponents of the magne-\ntization as a function of applied ﬁeld and current, deter-\nmined numerically. We have taken the damping α=.01.\nAlso shown is the magnitude of the power spectrum peak\nofz(t) (labelled “P Z”) - a nonzero value indicates a pre-\ncessing solution. Also shown is the stability boundaries\ndeﬁned by Eqs. (3, 5, 7, 8). The numerics verify the sta-\nbility of the unusual out-of-plane and ﬁeld-anti-aligned\nsolutions. The conversion of the dimensionless hCIinto\na real current density for a material with demagnetiza-\ntion ﬁeld of 1 T is J= (hCIt)×3.8·109A/cm2, wheret\nis the thickness of the F layer in nm.\nFIG. 5: Magnetic conﬁguration (M x,Mz) and peak of power\nspectrum P z(arbitrary units) versus applied ﬁeld and cur-\nrent. Also shown is stability boundaries found analyticall y\n(the labels ±x,±z refer also to solutions which point approx-\nimately in these directions). The stability boundary plot a lso\nshows the reduced out-of-plane solution space for negative to\npositive ﬁeld sweep with a dashed line.\nThedataforeach( h, hCI) pointofFig. (5) isobtained\nbeginning from an initial condition close to the solution\ngiven by Eq. (4). These equilibrium solutions are not\nuniversal attractors, and are attained for a subset of ini-\ntial conditions. To see the eﬀect of initial conditions, we\nhave also swept the applied ﬁeld from negative to posi-\ntive for each applied current, using the slightly perturbed\nﬁnal coordinates of a trajectory as the initial conditionfor the next value of applied ﬁeld. The out-of-plane so-\nlution space is reduced, shown by the dotted line in Fig.\n(5) in the stability boundaries plot.\nWe now comment on the experimental possibilities of\nseeing these eﬀects. In the preceding analysis, we assume\nthattheAFisﬁxed. Thiscanbeaccomplishedbyplacing\nalargeF adjacentto the AF, sothat the AF ispinned via\ntheexchangebiaseﬀect(theoverallstackstructurewould\nbe pinning F - AF - spacer - free F). The presence of this\npinning F may inﬂuence the dynamics of the free F, but\nits signature should be very distinct from the inﬂuence of\nthe AF layer on the free F. The orientation of the free F\nshould be observablefrom magnetoresistanceeﬀects with\nthe pinning ferromagnet.\nA virtue of the out-of-plane F conﬁguration is that the\nsurface of the AF need not be single domain for its ob-\nservation. As long as the magnetization of the AF is\ncompensated and points in the plane (which is the pre-\nferred direction for NiMn [21, 22]), diﬀerent orientations\nof domains at the AF surface should cooperatively push\nthe F out of the plane. The encouraging aspect of this\nproposal is that the signature of the AF current-induced\ntorque is so unique, helping to provide a distinguished\ncharacteristic for its observation.\nAcknowledgments — We would like to acknowledge\nvery helpful conversations with Maxim Tsoi and Olle\nHeinonen. This work was supported in part by Seagate\nCorporationand by the National Science Foundation un-\nder grant DMR-0606489, and the computational work\nwas supported by Texas Advanced Computational Cen-\nter (TACC).\n[1] Slonczewski, J. Magn. Magn. Mat. 62, 123, (1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] M. Tsoi et al., Phys. Rev. Lett. 81, 493(E) (1998).\n[4] M. Tsoi et al., Nature 406, 46 (2000).\n[5] J. A. Katine et al., Phys. Rev. Lett. 84, 3149 (2000).\n[6] F. J. Albert et al., Phys. Rev. Lett., 89, 226802 (2002).\n[7] M. R. Pufall, W. H. Rippard, and T. J. Silva, App. Phys.\nLett.83, 323 (2003).\n[8] G. S. Beach et al., Phys. Rev. Lett. 97, 057203 (2006)\n[9] J. Grollier et al., Appl. Phys. Lett. 83, 509 (2003).\n[10] A. Yamaguchi et al., Phys. Rev. Lett. 92, 077205 (2004).\n[11] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno,\nNature428, 539 (2004).\n[12] M. Hayashi et al., Phys. Rev. Lett. 96, 197207 (2006).\n[13] A. S. N´ u˜ nez, R. A. Duine, Paul Haney, and A. H. Mac-\nDonald, Phys. Rev. B 73, 214426 (2006).\n[14] P. M. Haney et al., Phys. Rev. B 75, 174428 (2007).\n[15] Z. Wei et al., Phys. Rev. Lett. 98, 116603 (2007).\n[16] S. I. Kiselev et al., Nature 425, 380 (2003).\n[17] P. M. Haney et al., Phys. Rev. B 76, 024404 (2007).\n[18] L. P´ al et al., J. App. Phys. 39, 538 (1968).\n[19] T. C. Schulthess and W. H. Butler, J. App. Phys. 83,\n7225 (1998).\n[20] Ya. B. Bazaliy, B. A. Jones, and Shou-Cheng Zhang,5\nPhys. Rev. B 69, 094421, (2004).\n[21] A. Sakuma, J. Mag. Magn. Mat. 187, 105 (1998).\n[22] J.S. Kasper, J.S. Kouvel, J. Phys. Chem. Solids 11, 231(1959)."    },    {        "title": "1301.1649v2.Integrable_Heisenberg_Ferromagnet_Equations_with_self_consistent_potentials.pdf",        "content": "arXiv:1301.1649v2  [physics.gen-ph]  12 Mar 2013Integrable Heisenberg Ferromagnet Equations with self-co nsistent\npotentials\nZh.Kh. Zhunussova, K.R. Yesmakhanova, D.I. Tungushbaeva,\nG.K. Mamyrbekova, G.N. Nugmanova and R. Myrzakulov∗\nEurasian International Center for Theoretical Physics and Department of General\n&Theoretical Physics, Eurasian National University, Astan a 010008, Kazakhstan\nAbstract\nIn this paper, we consider some integrable Heisenberg Ferro magnet Equations with self-\nconsistent potentials. We study their Lax representations . In particular we give their equiva-\nlent counterparts which are nonlinear Schr¨ odinger type eq uations. We present the integrable\nreductions of the Heisenberg Ferromagnet Equations with se lf-consistent potentials. These in-\ntegrable Heisenberg Ferromagnet Equations with self-cons istent potentials describe nonlinear\nwaves in ferromagnets with some additional physical ﬁelds.\nContents\n1 Introduction 1\n2 Preliminaries 2\n3 The (1+1)-dimensional M-XCIX equation 3\n3.1 Lax representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n3.2 Shcr¨ odinger-type equivalent counterpart . . . . . . . . . . . . . . . . . . . . . . . . 4\n3.3 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n3.3.1 Principal chiral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n3.3.2 Heisenberg ferromagnetic equation . . . . . . . . . . . . . . . . . . . . . . . 5\n4 The (1+1)-dimensional M-LXIV equation 5\n5 The (1+1)-dimensional M-XCIV equation 6\n5.1 Lax representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n5.2 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n5.2.1 The M-XCIX equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n5.2.2 The M-LXIV equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n5.3 Equivalent counterpart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n6 Conclusion 8\n1 Introduction\nNonlinear eﬀects are fundamental part of many phenomena in diﬀer ent branches of sciences. Such\nnonlinear eﬀects are modelled by nonlinear diﬀerential equations (ND E). One of important parts\nof NDE is integrable NDE, which sometimes also called as soliton equation s. Integrable spin\nsystems (SS) are one of main sectors of integrable NDE and play inte resting role in mathematics\n∗The corresponding author. Email: rmyrzakulov@gmail.com\n1in particular in the geometry of curves and surfaces. On the other hand, integrable SS play cruical\nrole in the description of nonlinear phenomena in magnets.\nIn this paper, we study some integrable Myrzakulov equations with s elf-consistent potentials.\nWe present their Lax representations as well as their reductions. Finally we give their equivalent\ncounterparts which have the nonlinear Schr¨ odinger equation typ e form.\nThe paper is organized as follows. In Sec. II, we give the basic facts from the theory of\nthe Heisenberg ferromagnet equation. In Sec. III, we investigat e the (1+1)-dimensional M-XCIX\nequation. Next, we study the (1+1)-dimensional M-LXIV equation in Sec. IV. In Sec. V we\nconsider the (1+1)-dimensional M-XCIV equation. Finally, we give co nclusions in Sec. VI.\n2 Preliminaries\nFirst example of integrable SS is the so-called Heisenberg ferromagn etic model (HFM) which reads\nas [1]-[2]\nSt=S∧Sxx, (2.1)\nwhere∧denotes a vector product and\nS= (S1,S2,S3),S2= 1. (2.2)\nThe matrix form of the HFM looks like\niSt=1\n2[S,Sxx], (2.3)\nwhere\nS=Siσi=/parenleftbigg\nS3S+\nS−−S3/parenrightbigg\n. (2.4)\nHereS2=I, S±=S1±iS2,[A,B] =AB−BAandσiare Pauli matrices\nσ1=/parenleftbigg\n0 1\n1 0/parenrightbigg\n, σ2=/parenleftbigg\n0i\n−i0/parenrightbigg\n, σ3=/parenleftbigg\n1 0\n0−1/parenrightbigg\n. (2.5)\nNote that the HFM (2.1) is Lakshmanan equivalent [1] to the nonlinear Schr¨ odinger equation\n(NSE)\niϕt+ϕxx+2|ϕ|2ϕ= 0. (2.6)\nAlso we recall that between the HFE (2.1) and NSE (2.6) takes place t he gauge equivalence [2].\nIn literature diﬀerent types integrable and nonintegrable SS have b een proposed (see e.g. [3]). As\nexamples of such extensions we here present the following two integ rable equations:\ni) the Myrzakulov-XXXIV (M-XXXIV) equation [3]\nSt−S∧Sxx−uSx= 0, (2.7)\nut+ux+α(S2\nx)x= 0. (2.8)\nii) the Myrzakulov-I (M-I) equation [3]\nSt−(S∧Sy+uS)x= 0, (2.9)\nux+S·(Sx∧Sy) = 0. (2.10)\nSome properties of these and other integrable and nonintegrable S S were studied in [5]-[36]. Also\nnote that the M-I equation (2.9)-(2.10) we write sometimes as [3]\nSt−S∧Sxy−uSx= 0, (2.11)\nux+S·(Sx∧Sy) = 0. (2.12)\nOf course that both forms of the M-I equation that is Eq.(2.9)-(2.1 0) and Eq.(2.11)-(2.12) are\nequivalent each to others. In this paper we study some integrable g eneralizations of the HFM\n(2.1).\n23 The (1+1)-dimensional M-XCIX equation\nThe (1+1)-dimensional Myrzakulov-XCIX equation (or shortly M-X CIX equation) reads as [3]-[4]\nSt+0.5ǫ1S∧Sxx+2\nωS∧W= 0, (3.1)\nWx+2ωS∧W= 0, (3.2)\nwhere∧denotes a vector product and\nS= (S1,S2,S3),W= (W1,W2,W3), (3.3)\nHereαis a real function, S2=S2\n1+S2\n2+S2\n3= 1,SiandWiare some real functions, ωandǫiare\nreal constants. In terms of components the system (3.1)-(3.2) takes the form\nS1t+0.5ǫ1(S2S3xx−S3S2xx)+2\nω(S2W3−S3W2) = 0, (3.4)\nS2t+0.5ǫ1(S3S1xx−S1S3xx)+2\nω(S3W1−S1W3) = 0, (3.5)\nS3t+0.5ǫ1(S1S2xx−S2S1xx)+2\nω(S1W2−S2W1) = 0, (3.6)\nW1x+2ω(S2W3−S3W2) = 0, (3.7)\nW2x+2ω(S3W1−S1W3) = 0, (3.8)\nW3x+2ω(S1W2−S2W1) = 0. (3.9)\nOn the other hand, the system (3.1)-(3.2) can be rewritten as\niSt+0.25ǫ1[S,Sxx]+1\nω[S,W] = 0, (3.10)\niWx+ω[S,W] = 0, (3.11)\nwhere\nS=Siσi=/parenleftbiggS3S−\nS+−S3/parenrightbigg\n, W=Wiσi=/parenleftbiggW3W+\nW−−W3/parenrightbigg\n. (3.12)\nHereS±=S1±iS2, W±=W1±iW2,[A,B] =AB−BA, σ iare Pauli matrices.\n3.1 Lax representation\nLet us consider the system of the linear equations\nΦx=UΦ, (3.13)\nΦt=VΦ. (3.14)\nLet the Lax pair U−Vhas the form [3]-[4]\nU=−iλS, (3.15)\nV=λ2V2+λV1+i\nλ+ωV−1−i\nωV0, (3.16)\nwhere\nV2=−iǫ1S, (3.17)\nV1= 0.25ǫ1[S,Sx], (3.18)\nV−1=V0=/parenleftbigg\nW3W+\nW−−W3/parenrightbigg\n. (3.19)\nWith such U,Vmatrices, the equation\nUt−Vx+[U,V] = 0 (3.20)\nis equivalent to the M-XCIX equation (3.1)-(3.2). It means that the M-XCIX equation (3.1)-(3.2)\nis integrable by the Inverse Tranform Method (ITM).\n33.2 Shcr¨ odinger-type equivalent counterpart\nOur aim in this section is to ﬁnd the Shcr¨ odinger-type equivalent cou nterpart of the M-XCIX\nequation. To do is, let us we introduce the 3 new functions ϕ,pandηas\nϕ=αeiβ, (3.21)\np=−/bracketleftbigg\n2S−W3−(S3+1)W−+S−2W+\nS3+1/bracketrightbigg\neiς, (3.22)\nη= 2S3W3+S−W++S+W−, (3.23)\nwhere\nα= 0.5(S2\n1x+S2\n2x+S2\n3x)0.5, (3.24)\nβ=−i∂−1\nx/bracketleftbiggtr(SxSSxx)\ntr(S2x)/bracketrightbigg\n, (3.25)\nς= exp/bracketleftbigg\niθ−1\n2∂−1\nx/parenleftbiggS+S−\nx−S+\nxS−\n1+S3/parenrightbigg/bracketrightbigg\n(3.26)\nandθ=const. It is not diﬃcult to verify that these 3 new functions satisfy the fo llowing equations\niϕt+ǫ1(0.5ϕxx+|ϕ|2ϕ)−2ip= 0, (3.27)\npx−2iωp−2ηϕ= 0, (3.28)\nηx+ϕ∗p+ϕp∗= 0, (3.29)\nIt is nothing but the nonlinear Schr¨ odinger-Maxwell-Bloch equation (NSMBE). It is well-known\nthat the SMBE is integrable by IST. Its Lax representation reads a s [29]-[28]\nΨx=AΨ, (3.30)\nΨt=BΨ, (3.31)\nwhere\nA=−iλσ3+A0, (3.32)\nB=λ2B2+λB1+B0+i\nλ+ωB−1. (3.33)\nHere\nA0=/parenleftbigg\n0ϕ\n−ϕ∗0/parenrightbigg\n, (3.34)\nB2=−iǫ1σ3, (3.35)\nB1=ǫ1A0, (3.36)\nB0= 0.5iǫ1α2σ3+0.5iǫ1σ3A0x, (3.37)\nB−1=/parenleftbiggη−p\n−p∗−η/parenrightbigg\n. (3.38)\n3.3 Reductions\n3.3.1 Principal chiral equation\nLet us we set ǫ1= 0. Then the M-XCIX equation reduces to the equation\niSt+1\nω[S,W] = 0, (3.39)\niWx+ω[S,W] = 0. (3.40)\nIt is nothing but the principal chiral equation. As is well-known that it is integrable by ITM. The\ncorresponding Lax pair is given by\nU=−iλS, (3.41)\nV=−iλ\nω(λ+ω)W. (3.42)\n43.3.2 Heisenberg ferromagnetic equation\nNow let us we assume that W= 0. Then the M-XCIX equation reduces to the equation\niSt+0.25ǫ1[S,Sxx] = 0. (3.43)\nIt is the HFM (2.1) within to the simplest scale transformations.\n4 The (1+1)-dimensional M-LXIV equation\nThe (1+1)-dimensional M-LXIV equation (or shortly M-LXIV equat ion) reads as [3]:\niSt+ǫ2i[Sxxx+6(βS)x]+1\nω[S,W] = 0, (4.1)\niWx+ω[S,W] = 0. (4.2)\nThe corresponding Lax pair is given by\nU=−iλS, (4.3)\nV=λ3V3+λ2V2+λV1+i\nλ+ωV−1−i\nωV−1, (4.4)\nwhere [3]\nV3=−4iǫ2S, (4.5)\nV2= 2ǫ2SSx, (4.6)\nV1=ǫ2i(Sxx+6βS), (4.7)\nV−1=W=/parenleftbiggW3W+\nW−−W3/parenrightbigg\n(4.8)\nwithβ=rq= 0.125tr[(Sx)2]. The formulas (3.21)-(3.23) gives us the Schrodinger equivalent of\nthe (1+1)-dimensional M-XCIV equation. It has the form (see e.g. [35]-[36])\niqt+iǫ2(qxxx+6rqqx)−2ip= 0, (4.9)\nirt+iǫ2(rxxx+6rqrx)−2ik= 0, (4.10)\npx−2iωp−2ηq= 0, (4.11)\nkx+2iωk−2ηr= 0, (4.12)\nηx+rp+kq= 0. (4.13)\nThis system is nothing but the Hirota-Maxwell-Bloch equation. Its La x representation reads as\nΨx=AΨ, (4.14)\nΨt= [−4iǫ2λ3σ3+B]Ψ, (4.15)\nwhere\nA=−iλσ3+A0, (4.16)\nB=λ2B2+λB1+B0+i\nλ+ωB−1. (4.17)\nHere\nB2= 4ǫ2A0, (4.18)\nB1= 2iǫ2rqσ3+2iǫ2σ3A0x, (4.19)\nA0=/parenleftbigg\n0q\n−r0/parenrightbigg\n, (4.20)\nB0=ǫ2(rxq−rqx)σ3+B01, (4.21)\nB01=/parenleftbigg0 −ǫ2qxx−2ǫ2rq2\nǫ2rxx+2ǫ2qr20/parenrightbigg\n, (4.22)\nB−1=/parenleftbigg\nη−p\n−k−η/parenrightbigg\n. (4.23)\n5This system we can reduce to the form\niqt+iǫ2(qxxx+6δ|q|2qx)−2ip= 0, (4.24)\npx−2iωp−2ηq= 0, (4.25)\nηx+δ(q∗p+p∗q) = 0. (4.26)\n5 The (1+1)-dimensional M-XCIV equation\nThe Myrzakulov-XCIV equation or shortly M-XCIV equation reads a s [3]:\niSt+0.5ǫ1[S,Sxx]+ǫ2i[Sxxx+6(βS)x]+1\nω[S,W] = 0, (5.1)\niWx+ω[S,W] = 0. (5.2)\n5.1 Lax representation\nThe Lax pair of the M-XCIV equation (5.1)-(5.2) is given by\nU=−iλS, (5.3)\nV=λ3V3+λ2V2+λV1+i\nλ+ωV−1−i\nωV−1, (5.4)\nwhere [3]\nV3=−4iǫ2S, (5.5)\nV2=−2iǫ1S+2ǫ2SSx, (5.6)\nV1=ǫ1SSx+ǫ2i(Sxx+6βS), (5.7)\nV−1=W=/parenleftbigg\nW3W+\nW−−W3/parenrightbigg\n(5.8)\nwithβ=rq= 0.125tr[(Sx)2].\n5.2 Reductions\nThe M-XCIV equation admits some integrable reductions. For examp le, it has the following\nintegrable reductions.\n5.2.1 The M-XCIX equation\nLetǫ2= 0. Then the M-XCIV equation takes the form\niSt+0.5ǫ1[S,Sxx]+1\nω[S,W] = 0, (5.9)\niWx+ω[S,W] = 0. (5.10)\nIt has the Lax pair of the form\nU=−iλS, (5.11)\nV=λ3V3+λ2V2+λV1+i\nλ+ωW−i\nωW, (5.12)\nwhere [3]\nV2=−2iǫ1S, (5.13)\nV1=ǫ1SSx, (5.14)\nW=/parenleftbiggW3W+\nW−−W3/parenrightbigg\n. (5.15)\n65.2.2 The M-LXIV equation\nNow let us consider the case ǫ1= 0. In this case the M-XCIV equation transforms to the equation\niSt+ǫ2i[Sxxx+6(βS)x]+1\nω[S,W] = 0, (5.16)\niWx+ω[S,W] = 0. (5.17)\nThe corresponding Lax pair reads as\nU=−iλS, (5.18)\nV=λ3V3+λ2V2+λV1+i\nλ+ωV−1−i\nωV−1, (5.19)\nwhere [3]\nV3=−4iǫ2S, (5.20)\nV2= 2ǫ2SSx, (5.21)\nV1=ǫ2i(Sxx+6βS), (5.22)\nV−1=W=/parenleftbiggW3W+\nW−−W3/parenrightbigg\n(5.23)\nwithβ=rq= 0.125tr[(Sx)2].\n5.3 Equivalent counterpart\nTo ﬁnd the Schrodinger equivalent, we again us the formulas (3.21)- (3.23). Finally the Schrodinger\nequivalent of the (1+1)-dimensional M-XCIV equation has the form (see e.g. [35]-[36])\niqt+ǫ1(qxx+2rq2)+iǫ2(qxxx+6rqqx)−2ip= 0, (5.24)\nirt−ǫ1(rxx+2r2q)+iǫ2(rxxx+6rqrx)−2ik= 0, (5.25)\npx−2iωp−2ηq= 0, (5.26)\nkx+2iωk−2ηr= 0, (5.27)\nηx+rp+kq= 0. (5.28)\nThis system is nothing but the Hirota-Maxwell-Bloch equation. Its La x representation reads as\nΨx=AΨ, (5.29)\nΨt= [−4iǫ2λ3σ3+B]Ψ, (5.30)\nwhere\nA=−iλσ3+A0, (5.31)\nB=λ2B2+λB1+B0+i\nλ+ωB−1. (5.32)\nHere\nB2=−2iǫ1σ3+4ǫ2A0, (5.33)\nB1= 2iǫ2rqσ3+2iǫ2σ3A0x+2ǫ1A0, (5.34)\nA0=/parenleftbigg\n0q\n−r0/parenrightbigg\n, (5.35)\nB0= (iǫ1rq+ǫ2(rxq−rqx))σ3+B01, (5.36)\nB01=/parenleftbigg0 iǫ1qx−ǫ2qxx−2ǫ2rq2\niǫ1rx+ǫ2rxx+2ǫ2qr20/parenrightbigg\n, (5.37)\nB−1=/parenleftbigg\nη−p\n−k−η/parenrightbigg\n. (5.38)\n7Ifp=δk∗,r=δq∗, this system we can reduce to the form\niqt+ǫ1(qxx+2δ|q|2q)+iǫ2(qxxx+6δ|q|2qx)−2ip= 0, (5.39)\npx−2iωp−2ηq= 0, (5.40)\nηx+δ(q∗p+p∗q) = 0. (5.41)\nNote that the (1+1)-dimensional HMBE (5.55)-(5.56) admits the fo llowing integrable reduc-\ntions.\ni) The NSLE as ǫ1−1 =ǫ2=p=η= 0:\niqt+qxx+2δ|q|2q= 0. (5.42)\nii) The (1+1)-dimensional complex mKdV eqation as ǫ1=ǫ2−1 =p=η= 0:\nqt+qxxx+6δ|q|2qx= 0. (5.43)\niii) The (1+1)-dimensional Schrodinger-Maxwell-Bloch equation as ǫ1−1 =ǫ2= 0:\niqt+qxx+2δ|q|2q−2ip= 0, (5.44)\npx−2iωp−2ηq= 0, (5.45)\nηx+δ(q∗p+p∗q) = 0. (5.46)\niv) The (1+1)-dimensional complex mKdV-Maxwell-Bloch equation as ǫ1=ǫ2−1 = 0:\nqt+qxxx+6δ|q|2qx−2p= 0, (5.47)\npx−2iωp−2ηq= 0, (5.48)\nηx+δ(q∗p+p∗q) = 0. (5.49)\nv) The following (1+1)-dimensional equation as ǫ1=ǫ2= 0:\nqt−2p= 0, (5.50)\npx−2iωp−2ηq= 0, (5.51)\nηx+δ(q∗p+p∗q) = 0. (5.52)\nor\nqxt−2iωqt−4ηq= 0, (5.53)\n2ηx+δ(|q|2)t= 0. (5.54)\nvi) The following (1+1)-dimensional equation as δ= 0:\niqt+ǫ1qxx+iǫ2qxxx−2ip= 0, (5.55)\npx−2iωp−2η0q= 0, (5.56)\nwhereη0= 0. AgainwenotethatallthesereductionsareintegrablebyIST.T hecorrespondingLax\nrepresentations we get from the Lax representation (5.29)-(5.3 0) as the corresponding reductions.\n6 Conclusion\nHeisenberg ferromagnet models play an important role in modern the ory of magnets. These are\nnonlinear partial diﬀerential equations. Some of these models are in tegrable by the Inverse Scat-\ntaring Method that is they are soliton equations. In this paper, we h ave studied some Heisenberg\nferromagnet equations (models) with self-consistent potentials. We have presented their Lax rep-\nresentations. Also we have found their Schr¨ odinger type equivale nt counterparts.\n8References\n[1] M. Lakshmanan. Phys. Lett. A, 340,199 (2012).\n[2] L. Takhtajian Astrophys. Space Sci. 340,199 (2012).\n[3] R. Myrzakulov. On some integrable and nonintegrable soliton equa tions of magnets I-IV\n(HEPI, Alma-Ata, 1987).\n[4] R. Myrzakulov. arXiv:1008.4486; arXiv:1204.1093; arXiv:1301.164 9;\n[5] MyrzakulovR., MartinaL,MyrzakulKur., MyrzakulovR, Soliani G. JournalofMathematical\nPhysics, V.42, 3, P.1397-1417 (2001). 1.291\n[6] Myrzakulov R, Danlybaeva A.K, Nugmanova G.N. Theoretical and M athematical Physics,\nV.118, 3, P. 441-451 (1999). 0.65\n[7] Myrzakulov R., Lakshmanan M., Vijayalakshmi S., Danlybaeva A. Jo urnal of Mathematical\nPhysics, V.39, 7, P. 3765-3771 (1998). 1.941\n[8] Myrzakulov R., Vijayalakshmi S., Syzdykova R., Lakshmanan M. Jo urnal of Mathematical\nPhysics, V.39, 4, P. 2122-2139 (1998). 1.941\n[9] Myrzakulov R., Nugmanova G., Syzdykova R. Journal of Physics A : Mathematical & Theo-\nretical, V.31, 47, P.9535-9545 (1998). 1.564\n[10] Myrzakulov R., Vijayalakshmi S., Nugmanova G., Lakshmanan M. P hysics Letters A, V.233,\n4-6, P. 391-396 (1997). 1.632\n[11] Myrzakulov R., Daniel M., Amuda R. Physica A., V.234, 3-4, P.715-7 24 (1997). 1.373\n[12] Myrzakulov R., Makhankov V.G., Pashaev O.. Letters in Mathemat ical Physics, V.16, N1,\nP.83-92 (1989) 1.819\n[13] MyrzakulovR., MakhankovV.G., MakhankovA. PhysicaScripta, V .35, N3, P. 233-237(1987)\n1.204\n[14] Myrzakulov R., Pashaev O.., Kholmurodov Kh. Physica Scripta, V.3 3, N4, P. 378-384 (1986)\n1.204\n[15] Anco S.C., Myrzakulov R. Journal of Geometry and Physics, v.60 , 1576-1603 (2010)\n[16] A.J. Lopez-Revelles, R. Myrzakulov, D. Saez-Gomez, Physical Review D, 85, N10, 103521\n(2012).\n[17] K. Bamba, R. Myrzakulov, S. Nojiri, S. D. Odintsov, Physical Re view D, 85, N10, 104036\n(2012).\n[18] M. Duncan, R. Myrzakulov, D. Singleton. Phys. Lett. B, 703, N4, 516-518 (2011).\n[19] R. Myrzakulov, D. Saez-Gomez, A. Tureanu. General Relativit y and Gravitation, 43, N6,\n1671-1684 (2011)\n[20] V. Dzhunushaliev, V. Folomeev, R. Myrzakulov, D. Singleton. Ph ysical Review D, 82, 045032\n(2010)\n[21] I. Brevik, R. Myrzakulov, S. Nojiri, S. D. Odintsov. Physical Re view D,86, N6, 063007(2012)\n[22] E. Elizalde, R. Myrzakulov, V.V. Obukhov, D. Saez-Gomez. Class ical and Quantum Gravity,\n27, N8, 085001-12 (2010)\n[23] Myrzakulov R., Rahimov F.K., Myrzakul K., Serikbaev N.S. On the ge ometry of stationary\nHeisenberg ferromagnets : ”Non-linear waves: Classical and Quan tum Aspects”, Kluwer\nAcademic Publishers, Dordrecht, Netherlands, P. 543-549 (2004 )\n9[24] Myrzakulov R., Serikbaev N.S., Myrzakul Kur., Rahimov F.K. On con tinuous limits of some\ngeneralized compressible Heisenberg spin chains Journal of NATO Sc ience Series II. Mathe-\nmatics, Physics and Chemistry, V 153, P. 535-542 (2004)\n[25] Myrzakulov R., Martina L., Kozhamkulov T.A., Myrzakul Kur. Inte grable Heisenberg ferro-\nmagnets and soliton geometry of curves and surfaces In book: ”N onlinear Physics: Theory\nand Experiment. II”. World Scientiﬁc, London, P. 248-253 (2003)\n[26] Myrzakulov R. Integrability of the Gauss-Codazzi-Mainardi eq uation in 2+1 dimensions. In\n”Mathematical Problems of Nonlinear Dynamics”, Proc. of the Int. Conf. ”Progress in Non-\nlinear sciences”, Nizhny Novgorod, Russia, July 2-6, 2001, V.1, P.31 4-319 (2001)\n[27] Martina L, Myrzakul Kur., Myrzakulov R, Soliani G. Journal of M athematical Physics, V.42,\n3, P.1397-1417 (2001).\n[28] K. Porsezian, K. Nakkeeran. Phys. Rev. Lett., 74, 2941 (1995).\n[29] S.P. Burtsev, I.R. Gabitov. Phys. Rev. A, v.49, 2065 (1994).\n[30] Chen Chi, Zhou Zi-Xiang. Darboux Tranformation and Exact Solutions of the Myrzakulo v-I\nEquations , Chin. Phys. Lett., 26, N8, 080504 (2009)\n[31] Zhao-Wen Yan, Min-Ru Chen, Ke Wu, Wei-Zhong Zhao.J. Phys. S oc. Jpn., 81, 094006(2012)\n[32] Yan Zhao-Wen, Chen Min-Ru, Wu Ke, Zhao Wei-Zhong. Commun. T heor. Phys., 58, 463-468\n(2012)\n[33] K.R. Esmakhanova, G.N. Nugmanova, Wei-Zhong Zhao, Ke Wu. Integrable inhomogeneous\nLakshmanan-Myrzakulov equation , [nlin/0604034]\n[34] Zhen-Huan Zhang, Ming Deng, Wei-Zhong Zhao, Ke Wu. On the integrable inhomogeneous\nMyrzakulov-I equation , [nlin/0603069]\n[35] Chuanzhong Li, Jingsong He, K. Porsezian. Rogue waves of the Hirota and the Maxwell-Bloch\nequations , [arXiv:1205.1191]\n[36] Chuanzhong Li, Jingsong He. Darboux transformation and positons of the inhomogeneous\nHirota and the Maxwell-Bloch equation , [arXiv:1210.2501]\n10"    },    {        "title": "1701.02832v1.Emergent_incommensurate_correlations_in_the_frustrated_ferromagnetic_spin_1_chains.pdf",        "content": "Emergent incommensurate correlations in the frustrated ferromagnetic spin-1 chains\nHyeong Jun Lee and MooYoung Choi\nDepartment of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 08826, Korea\nGun Sang Jeon\u0003\nDepartment of Physics, Ewha Womans University, Seoul 03760, Korea\n(Dated: June 30, 2018)\nWe study the frustrated ferromagnetic spin-1 chains, where the ferromagnetic nearest-neighbor\ncoupling competes with the antiferromagnetic next-nearest-neighbor coupling. We use the density\nmatrix renormalization group to obtain the ground states. Through the analysis of spin-spin cor-\nrelations we identify the double Haldane phase as well as the ferromagnetic phase. It is shown\nthat the ferromagnetic coupling leads to incommensurate correlations in the double Haldane phase.\nSuch short-range correlations transform continuously into the ferromagnetic instability at the tran-\nsition to the ferromagnetic phase. We also compare the results with the spin-1/2 and classical spin\nsystems, and discuss the string orders in the system.\nPACS numbers: 75.10.Jm, 75.50.Ee, 75.40.Mg\nI. INTRODUCTION\nOne-dimensional quantum spin systems have attracted\nmuch interest for the past decade. One of the reasons is\nthat quantum \ructuations play a more dominant role in\ntheir ground states than for higher-dimensional systems.1\nGenerally quantum \ructuations suppress long-range or-\nder of a many-body system in low dimensions. In the\npresence of antiferromagnetic nearest-neighbor (NN) in-\nteractions the integer/half-integer Heisenberg spin chains\nwere suggested to have ground states with/without exci-\ntation gaps;2,3this was con\frmed by extensive numeri-\ncal studies in the spin-1/24,5and in the spin-14,6{11sys-\ntems. The resulting ground state of the spin-1 system\nis known to have a \fnite gap9and a \fnite correlation\nlength.10,12It was also veri\fed8that the ground state is\nconnected to the A\u000feck-Kennedy-Lieb-Tasaki (AKLT)\nstate,13,14which was originally obtained in the presence\nof bilinear and biquadratic interactions. In addition to\ntheoretical studies, there have been extensive experimen-\ntal studies:15{22Some of the materials considered can\nbe understood as one-dimensional spin-1 systems, giv-\ning supports for the theoretical results. The data on\nNi(C 2H8N2)2NO2ClO 4provide evidence for the Haldane\ngap15,16and additional spin-1/2 degrees of freedom17,18\nat the singlet-bond-broken sites in the presence of impu-\nrities. Gapped excitations were also observed in CsNiCl 3\nwith a correlation length smaller than the prediction\nfrom the theory,20{22which is now explained by weak\ncouplings between spin-1 chains. The zigzag spin chain\nin NaV(WO 4)2manifests a spin gap in the magnetic\nsusceptibility.23Measurements of magnetic properties of\nANi 2V2O8(A=Pb,Sr) have also shown that they are de-\nscribed as a Haldane gapped phase without long-range\norder.24{26\nFrustration, which is caused by the competition of\ntwo or more exchange interactions or the system geom-\netry, turns out to generate enormous quantum emer-\ngent phenomena in exotic phases. Extensive numer-ical studies on frustrated spin chains have been per-\nformed.5,27{30One good example is the spin-1 chain with\ntwo types of antiferromagnetic interactions, one between\nNN spins and the other between next-nearest-neighbor\n(NNN) spins.28{30The chain exhibits a discontinuous\nphase transition between two kinds of the AKLT states;\nthey are distinguished by the patterns of singlet bonds\nin the chain. The two phases are also characterized by\nthe existence of the hidden order due to breaking of the\nZ2\u0002Z2symmetry.\nOn the other hand, we have another kind of frustrated\nsystem, so-called frustrated ferromagnetic one, where the\nNN coupling is ferromagnetic. Numerous studies have\nbeen performed on such frustrated ferromagnetic sys-\ntems, especially for the spin-1/2 case.5,31{38It has been\nrevealed that the spectrum data in cuprate materials such\nas LiCu 2O2and LiCuVO 4can be explained as the prop-\nerties of the frustrated ferromagnetic system with good\nagreement.37It is also found that interesting phase tran-\nsitions exist at zero temperature in this system. This\nmotivates us to examine a frustrated ferromagnetic spin-\n1 system, of which the full understanding still lacks.\nIn this paper, we study frustrated ferromagnetic quan-\ntum spin-1 systems in one dimension. We obtain the\nground state through the use of the density-matrix-\nrenormalization-group (DMRG) method, and analyze the\nresulting spin-spin correlation functions. It is found that\nincommensurate short-range correlations are induced in\nthis system. Such correlations turn out to be smoothly\nconnected to the ferromagnetic phase at the transition\nwith divergent correlation length. We also discuss the\nstring order parameters in the double Haldane phase.\nThis paper is organized as follows. In Sec. II, we de-\nscribe the model and the method. Section III is devoted\nto the presentation and the discussion of numerical re-\nsults. Finally, a summary is given in Sec. IV.arXiv:1701.02832v1  [cond-mat.str-el]  11 Jan 20172\nII. MODEL AND METHOD\nWe consider a one-dimensional spin-1 system with NN\nand NNN couplings. The model is described by the\nHamiltonian\nH=J1X\ni^Si\u0001^Si+1+J2X\ni^Si\u0001^Si+2 (1)\nwhere ^Si\u0011(^Sx\ni;^Sy\ni;^Sz\ni) is a vector spin-1 operator at the\nith site. The NN and the NNN exchange couplings are\ndenoted by J1andJ2, respectively. We investigate the\nHamiltonian with ferromagnetic NN couplings ( J1<0)\nand antiferromagnetic couplings ( J2>0) at zero tem-\nperature in this work.\nIn the absence of the NN coupling ( J1= 0) the sys-\ntem reduces to two decoupled subchains; each consists of\nNNN pairs interacting with the antiferromagnetic NNN\ninteraction. The individual subchain lies in the Haldane\nphase, which is characterized by an excitation gap, expo-\nnentially decaying spin-spin correlations, and long-range\nstring order.10Such a phase is called a double Haldane\nphase, in which singlet bonds are formed between NNN\npairs.\nIn the presence of the antiferromagnetic NN coupling\n(J1>0) it was revealed28,29that the double Haldane\nphase undergoes a discontinuous transition to the Hal-\ndane phase at J2=J1'0:744. At the transition the sys-\ntem exhibits a jump in the string order parameter with\nthe bulk gap and the correlation length remaining \fnite.\nThe transition is attributed to the breaking of a hidden\nZ2\u0002Z2symmetry in the Haldane phase.38\nWhen the ferromagnetic NN coupling is dominant\n(jJ1j\u001dJ2), the ground state is expected to be a fer-\nromagnetically ordered state with total spin Stot=L\nwith gapless spin-wave excitations.39It is also known\nthat the ferromagnetic phase is robust against additional\ncompeting interactions. In the intermediate region the\nthermodynamic behavior for the S=1 chains is less clear\nin contrast with spin-1/2 chain systems.5,31{33,35,36\nIn this work, we use the DMRG method9,10,40with\nthe in\fnite algorithm to obtain the ground state of the\nsystem. We adopt slightly modi\fed open boundary con-\nditions, which are sketched in Fig. 1(a): the last two NN\nspins interact with J2. In the standard open boundary\nconditions in Fig. 1(b) the system tends to have two free\n1/2-spins at each end, one on each subchain. The anti-\nferromagnetic interaction J2in the modi\fed open bound-\nary conditions helps the two free spins in standard open\nboundary conditions to form a singlet bond, leading to\nan e\u000bective \fnite-size calculation in the double Haldane\nphase. We also performed comparative computation in\nthe standard open boundary conditions and found no sig-\nni\fcant qualitative di\u000berence in the bulk state.\nWe have performed calculations in chains up to L=\n200. The spin-spin correlations in di\u000berent sizes turn out\nto collapse to a single curve in a wide range of coupling\nparameters as will be displayed later. We estimate the\nuncertainty in the data by the maximum deviation from\nL(b)(a)FIG. 1. (Color online) Schematic representation of a spin-1\nchain in (a) modi\fed and (b) standard open boundary con-\nditions. The NN coupling J1and the NNN coupling J2are\nrepresented by solid and dashed lines, respectively. Shaded\nregions denote the singlet bonds which are expected to be\nformed between two half-spins of two sites in the double Hal-\ndane phase with J1<0 andJ2>0.\nthe average value for the systems with size L\u001550 and\nmark the error bars when they are larger than the size of\nsymbols. We have also checked out the convergence of the\ndata with respect to m, wheremis the number of states\nper block after truncation, by increasing mconsecutively.\nThe truncation errors in our calculations with m= 200\nare of the order of 10\u00005to 10\u00007for the double Haldane\nphase, in which the main interest of this work lies. They\nare larger in the central region of the double Haldane\nphase due to the increase of frustration. In the region\nof the ferromagnetic phase the truncation errors reduce\nsigni\fcantly below 10\u000015. We have performed the calcu-\nlation with mup to 250, and found that most physical re-\nsults do not depend signi\fcantly on m. Somem-sensitive\nphysical quantities such as the correlation length have\nbeen presented together with the extrapolation value to\nm=1, and relevant error bars have been marked when\nthey are larger than the size of the symbols. Henceforth\nwe will denote the energy and the length in units of J2\nand of the lattice constant, respectively, throughout this\npaper.\nIII. RESULTS AND DISCUSSION\nTo examine how the system evolves as the couplings\nvary, we \frst calculate the spin-spin correlation function\nCS(l) de\fned by\nCS(l)\u0011h^Si\u0001^Si+li: (2)\nIn order to minimize \fnite-size e\u000bects, we take site isuch\nthat both sites iandi+lare as far from the boundaries\nas possible. Figure 2(a) shows the spin-spin correlation\nfunction for J1= 0. The ground state for J1= 0 can be3\n-1 0 1(a) \n-1 0 1(b) CS(l)\n-1 0 1(c) \n-1 0 1\n0 25 50 75 100(d) \nl\nFIG. 2. (Color online) Spin-spin correlation functions as\na function of the separation lbetween the spins at J1=\n0;\u00000:5;\u00001, and\u00002 from top to bottom. The period of the\noscillation is 4 (i.e. k=\u0019=2) atJ1= 0 and becomes incom-\nmensurate as J1decreases.\nsimply understood in terms of the two completely decou-\npled chains, resulting in \fnite spin-spin correlations only\nfor even-integer separations lwith vanishing correlations\nfor odd-integer separations. Within each subchain the\nsystem lies in the Haldane phase, which exhibits short-\nrange antiferromagnetic correlations. Such spin-spin cor-\nrelations are in good agreement with those plotted in\nFig. 2(a). The correlations decay exponentially with the\nseparation l, superposed by the oscillating correlations\nwith the period of four.\nWe also plot the spin-spin correlations for various fer-\nromagnetic NN couplings J1in Fig. 2. When J1is \fnite,\nthe two subchains are no longer decoupled and \fnite cor-\nrelations show up for odd-integer separations. Interest-\ningly, the correlations still display oscillating behavior\napparently with a single period, which is di\u000berent from\n4, the value for J1= 0; it increases monotonously with\nthe increase ofjJ1j. It is also of interest to note that the\noscillating period does not always appear commensurate\nwith the lattice period. Another conspicuous feature is\nthat the decay of the correlations becomes slower as the\nferromagnetic NN coupling becomes stronger. While the\ncorrelations become negligible around l\u001930 forJ1= 0,\nwe can observe clear oscillations for l&60 atJ1=\u00002.\n 0 2 4 6 8\n-1 -0.5  0  0.5  1S (k)\nk / πFIG. 3. (Color online) Spin structure factor S(k) for var-\nious NN couplings J1. The data for J1= 0,\u00000:5,\u00001, and\n\u00002 with various lengths L= 80 to 200 are marked by (red)\nsquares, (green) circles, (blue) triangles, and (purple) inverted\ntriangles. The structure factor has two peaks in the double\nHaldane phase, which are not divergent in the thermodynamic\nlimit. AsJ1is reduced, the peaks move closer to k= 0 and the\nsystem exhibits incommensurate short-ranged correlations.\nUntilJ1reaches the value Jc\u0011\u00004 the system does not\nshow any abrupt change in the correlations, which implies\nthat in the region of Jc< J 1<0 the system remains\nin the double Haldane phase. For J1< Jc, the spin-\nspin correlation function takes just the constant value of\nunity, independent of l, and the total spin of the ground\nstate turns out to be Stot=L, signifying that the sys-\ntem is in the ferromagnetic phase. The critical value into\nthe ferromagnetic phase for S= 1 is the same as that\nin the case of classical spin systems5andS= 1=2 spin\nsystems,31,33{36as pointed out in Ref. 41.\nFor a quantitative analysis of the evolution of the states\nwithJ1varied, we examine the spin structure factor\nS(k)\u0011X\nleiklCS(l);\nwhich is the Fourier transform of CS(l). We have used\nfast Fourier transform algorithms for CS(l) to obtain\nS(k) and plot the resulting spin structure factor in Fig. 3\nfor various values of J1. ForJ1= 0 we have broad peaks\natk=\u0006\u0019=2, which re\rects oscillations with period 4.\nThe peak broadens due to the exponentially decaying\ncorrelations. AsjJ1jincreases, the positions of two peaks\nmove towards k= 0. It is also clear that the peak grad-\nually becomes narrower with the increase of J1. This\nresult forS(k) gives quantitative support to all the obser-\nvations on the spin-spin correlation function CS(l) men-\ntioned above. The structure factor remains \fnite even\nwhenLis increased inde\fnitely, and the system does not\nexhibit long-range spin order in this region.\nWe can determine the pitch angle k\u0003of the spin correla-\ntions by the position of the maximum in S(k) on the side\nof positivek. ForJ1= 0, the peak is located at k\u0003=\u0019=2,4\n 0 0.5 1\n-5 -4 -3 -2 -1  0  1  2  3ferromagneticdouble Haldane\nHaldanek*/ π\nJ110-1100\n10-1100101\nFIG. 4. (Color online) The pitch angle k\u0003as a function\nof the NN couplings J1. The pitch angles, which are deter-\nmined by the maximum position in the structure factor, are\nmarked by (green) solid circles. The pure NNN-AKLT state\n(J1= 0) hask\u0003=\u0019=2 re\recting the short-ranged antiferro-\nmagnetic correlation in each subchain. The antiferromagnetic\ncorrelation becomes incommensurate at \fnite J1in the dou-\nble Haldane phase. We also plot the results of the classical\nspin system, arccos( \u0000J1=4) (dot-dashed line), and those of\nthe spin-1/2 system from Ref. 35 (triangles) and Ref. 38 (di-\namonds) for comparison. The data marked by a solid line as\nwell as the transition point to the Haldane phase are quoted\nfrom Ref. 29. The inset shows the pitch angle in a log-log plot\nforJ1\u0000Jc>0 withJc=\u00004 and the dashed line represents\nthe best power-law \ft to the data. The data points without\nerror bars have errors not larger than the size of the symbols.\nwhich is consistent with the NNN AKLT state, the pro-\ntotype state in the double Haldane phase. In Fig. 4 we\nplotk\u0003as a function of J1. AsjJ1jincreases,k\u0003decreases\nmonotonously from k\u0003=\u0019=2 forJ1= 0 tok\u0003= 0 for\nJ1=Jc, and the ground state connects smoothly with\nthe ferromagnetic state at J1=Jc. The decreasing curve\ncorresponds to a convex-up function.\nSuch behavior is reminiscent of the spiral state which\nshows up in the classical spin system. In the presence\nof NN ferromagnetic couplings and NNN antiferromag-\nnetic couplings, the classical spins exhibit a spiral state\nfor\u0000Jc< J 1< Jcwith the wave number given by\nq= arccos(\u0000J1=4).5Similar behaviors of the pitch angle\nwere also reported in previous numerical studies of the\nspin-1/2 chains via the transfer-matrix DMRG method35\nand the in\fnite time evolving block decimation algorithm\nmethod37,38. For comparison, we have also plotted the\ndata for classical and spin 1/2 chains with a dot-dashed\nline and empty symbols, respectively, in Fig.4. In all\nthe three systems the pitch angle k\u0003reduces with the\nincrease ofjJ1jstarting from k\u0003=\u0019=2 atJ1= 0, and\napproaches k\u0003= 0 continuously at the transition into\nthe ferromagnetic phase. In the case of the spin-1/2\nchains, the plateau-like region persists near J1= 0 up\ntoJ1\u0019\u00002. On the other hand, the classical spin systemdisplays rather gradual decrease even near J1= 0. The\ncurve ofk\u0003for the spin-1 chain locates between the classi-\ncal and the spin-1/2 systems, which may be attributed to\nthe reduction of quantum \ructuations in spin-1 systems\nin comparison with spin-1/2 systems. It would be in-\nteresting to study variations of the pitch angle for higher\nspins, which should re\rect the e\u000bects of both the changes\nin quantum \ructuations and the alternating behavior of\ninteger and half-integer spins.\nIn the inset of Fig. 4, we plot k\u0003as a function of J1\u0000Jc\nin the log-log scale. Near the critical point the pitch angle\ndisplays the power-law behavior\nk\u0003\u0018(J1\u0000Jc)\u000b: (3)\nThe best \ft in the range \u00003:9\u0014J1\u0014\u00003:0 gives the\nexponent\u000b= 0:47(3). Although the best-\ft value of \u000b\nis a bit smaller than \u000bcl= 0:5 for the classical spiral\nstate, the two values are consistent within numerical er-\nrors. Numerical errors in the exponent are mainly due to\nthe large relative errors near Jc. It is also notable that\nthe curve of k\u0003versusJ1\u0000Jcis slightly convex up in the\nlog-log plot, which tends to give an additional underes-\ntimate of\u000bin the power-law \ft over a \fnite window of\nJ1.\nIn Fig. 4 we also plot the pitch angle in the case of\nJ1>0: the solid line represents the data from an earlier\nstudy29forS= 1 and the data points denoted by solid\ncircles are obtained from our calculation. The double\nHaldane phase in this region gives pitch angles in the\nrange\u0019=2<k\u0003<\u0019.28,29The pitch angle increases with\nJ1and becomes commensurate with k\u0003=\u0019aroundJ1\u0019\n2:67. Remarkably, the incommensurate spin correlations\npersist even in the region of the Haldane phase, which\nsets in atJ1\u00191:34. This is in contrast to the fact that\nthe pitch angle reduces to k\u0003= 0 exactly at the transition\nto the ferromagnetic phase for J1<0.\nIn order to examine the correlation length, we divide\nthe spin correlation function CS(l) by the oscillating fac-\ntor cos(k\u0003l) and estimate the correlation length \u0018by \ft-\nting it to the exponential decay \u0018exp(\u0000l=\u0018). In Figs. 5\nand 6 one can see that CS(l)=cos(k\u0003l) forJ=\u00001 and\u00002\ndecays exponentially in a wide range of l. Figure 5 also\ndemonstrates that the quantities in di\u000berent sizes exhibit\nalmost the same exponentially-decaying behavior except\nfor near the edges. The decay tends to become slower as\nthe number mof the kept states is increased. In Fig. 7\nwe plot the correlation length \u0018estimated from the best\n\ft as a function of J1for various values of mas well as in\nthe limitm!1 . We have also reproduced the data for\nJ1>0 from an earlier work.28ForJ1>0 our numerical\nresults are consistent with the peak associated with the\ntransition between the Haldane and the double Haldane\nphases. For J1<0 the correlation length is enhanced\nasJ1approaches Jc. We also note that there exists a\nsmall bump around J1= 0, which signi\fes stronger spin\n\ructuations near the decoupled subchains.\nWe also examine string order and double-string order\nforJ1<0, which can be probed by the use of the follow-5\n10-810-4100\n 0  50  100  150(a) CS(l) / cos( k*l)\nl 0  50  100  150  200(b) \nl\nFIG. 5. (Color online) Semi-log plot of the spin-spin corre-\nlation function CS(l) divided by cos( k\u0003l) for several values of\nLand (a)J1=\u00001; (b)J1=\u00002. The pitch angle k\u0003is de-\ntermined by the maximum position of the structure factor for\neachJ1. The data denoted by (red) squares, (green) circles,\n(blue) triangles, and (pink) inverted triangles correspond to\nL= 50;100;150, and 200, respectively.\n10-810-4100\n 0  50  100  150(a) CS(l) / cos( k*l)\nl 0  50  100  150  200(b) \nl\nFIG. 6. (Color online) Semi-log plot of the spin-spin cor-\nrelation function CS(l) divided by cos( k\u0003l) for several values\nofmand (a)J1=\u00001; (b)J1=\u00002. The pitch angle k\u0003is\ndetermined in the same way as in Fig. 5. The data denoted\nby (green) circles, (blue) triangles, and (pink) inverted trian-\ngles correspond to m= 50;100, and 150, respectively, and the\nsolid lines are the best linear \fts.\ning nonlocal correlators38,42,43\nO1(l;l0)\u0011\u0000*\nSz\nl2\n4exp0\n@l0\u00001X\nj=l+1i\u0019Sz\nj1\nA3\n5Sz\nl0+\nand\nO2(l;l0)\u0011*\nSz\nlSz\nl+12\n4exp0\n@l0\u00002X\nj=l+2i\u0019Sz\nj1\nA3\n5Sz\nl0\u00001Sz\nl0+\n:\n 0 10 20 30 40\n-5 -4 -3 -2 -1  0  1  2  3ξ\nJ1ferromagneticdouble Haldane HaldaneFIG. 7. (Color online) Correlation length as a function of\nthe NN coupling J1form=100 [(green) squares], 150 [(blue)\ntriangles], 200 [(pink) inverted triangles], and its extrapola-\ntion tom=1[(red) circles]. As we approach the phase\nboundary to the ferromagnetic phase the correlation length\nbecomes very large and is expected to diverge at J=\u00004.\nSolid lines denote the data quoted from Ref. 28. The data\npoints without error bars have errors not larger than the size\nof the symbols.\n0.000.050.100.15\n-5 -4 -3 -2 -1  0O\nJ1O1O2\n10-610-410-2100\n-4 -3 -2 -1  0\nFIG. 8. (Color online) String order parameters as a function\nof the NN coupling J1. We represent O1andO2by (red)\nsolid circles and (green) empty circles, respectively. The inset\ndisplays the semi-log plot of O2. The data points without\nerror bars have errors not larger than the size of the symbols.\nThe string order parameters O1andO2can then be de-\n\fned by\nO1\u0011 lim\njl\u0000l0j!1O1(l;l0); (4)\nO2\u0011 lim\njl\u0000l0j!1O2(l;l0) (5)\nin the limit of in\fnite separations. Figure 8 shows string\norder parameters O1andO2versusJ1. ForJ1= 0, we\nhave nonzero O2with vanishing O1, which is a typical\ncharacteristic of the double Haldane phase. On the other6\n10-310-210-1100\n 0  20  40  60  80  100|Cκ(l)|\nl10-310-210-1100\n 0  20  40  60  80  100\nFIG. 9. (Color online) Semi-log plot of the chirality correla-\ntion function C\u0014(l) forJ1=\u00003. The data denoted by (red)\nsquares, (green) circles, (blue) triangles, and (pink) inverted\ntriangles correspond to m= 50;100;150, and 200, respec-\ntively.\nhand, in the ferromagnetic phase ( J1<Jc) bothO1and\nO2vanish. As J1decreases from zero, O2also reduces.\nIt is remarkable that O2is signi\fcantly reduced for\nJ1.\u00002 while the transition into the ferromagnetic\nphase occurs at J1=\u00004. Such an abrupt reduction\nmay suggest the possible existence of emergent phase be-\ntween the double Haldane phase and the ferromagnetic\nphase. However, the detailed analysis has revealed that\nthe string nonlocal correlator O2(l;l0) shows quite dis-\ntinct behavior for \u00004<J 1.\u00002 from that of the ferro-\nmagnetic phase where the string order is absent. In the\nferromagnetic phase O2(l;l0) shows a clear exponential\ndecay withjl\u0000l0j, implying that the corresponding string\norder parameter O2vanishes in the thermodynamic limit.\nFor\u00004< J 1.\u00002, in contrast, O2(l;l0) remains \fnite\nalthough its values are very small. The semi-log plot of\nO2, which is presented in the inset of Fig. 8, also sup-\nports a direct transition from the double Haldane phase\nto a ferromagnetic one. It demonstrates that the de-\ncreasing behavior is rather consistent with the transition\natJ1=\u00004 although a rapid decrease of O2starts around\nJ1\u0019\u00002.\nIt is also worth while to examine whether a chiral phase\nis present between the double Haldane phase and the fer-\nromagnetic phase. In the spin-1/2 case the chiral phase\nhas been reported in a close vicinity of the region,37,38\nand accordingly it is a strong candidate for a new phase\nif it emerges also in the spin-1 case. We have thus com-\nputed the chirality correlation function de\fned by\nC\u0014(l)\u0011h^\u0014z\ni^\u0014z\ni+li (6)\nwith\n^\u0014z\ni\u0011^z\u0001(^Si\u0002^Si+1) =^Sx\ni^Sy\ni+1\u0000^Sy\ni^Sx\ni+1: (7)\nFigure 9 presents C\u0014(l) forJ1=\u00003, whereO2is sig-\nni\fcantly reduced. The chirality correlation function ex-hibits an apparent exponential decrease, which indicates\nthe absence of a chiral phase. This also implies that the\nemergence of a new phase is less probable. We presume\nthat the great reduction in O2for\u00004< J 1<\u00002 is\ncaused by the large ferromagnetic \ructuations which are\nenhanced markedly in that region as can be seen in Fig. 7.\nIV. SUMMARY\nWe have investigated the one-dimensional spin-1 sys-\ntem frustrated by the combination of the spin exchange\ninteractions: the ferromagnetic one between NN spins\nand the antiferromagnetic one between NNN spins. Via\nthe DMRG calculations we have con\frmed explicitly that\nthe ferromagnetic phase transition for S= 1 occurs at\nJ1=\u00004(\u0011Jc),41,44below which the spin-spin correla-\ntion function becomes constant. The robustness of Jc\nsuggests that quantum \ructuations due to the quantum\nnature of the spin do not a\u000bect signi\fcantly the ferromag-\nnetic phase transition; this is in sharp contrast with the\nfact that other phase boundaries of the frustrated spin\nchains exhibit strong dependence on Sand a variety of\ndistinct phases.27Such interesting results may be tested\nthrough experiments on ultracold atoms.45{47\nIn the double Haldane phase, on the other hand, the\nsystem shows short-ranged antiferromagnetic spin-spin\ncorrelations. Such behavior remains robust in a wide\nrange of the NN ferromagnetic interaction strength J1up\ntoJc. The pitch angle of the incommensurate spin-spin\ncorrelations decreases from \u0019=2 asJ1approaches Jc, van-\nishing atJ1=Jc. Similar behaviors were also reported\nin earlier works on the classical and spin-1/2 systems. It\nhas been revealed that the results of the spin-1 system are\ncloser to the classical results than those of the spin-1/2\none. We have also discussed the behavior of string order\nparameters in the double Haldane phase in the presence\nof ferromagnetic NN couplings. It has been found that\nthe string order parameter O2undergoes a substantial\nreduction far above Jc. However, detailed analysis has\nsuggested that a new phase is less likely to emerge in\nthat region. It is presumed that the enhancement in the\nferromagnetic \ructuations gives rise to such substantial\nreduction in O2.\nVarious extensions of our study are expected to pro-\nduce fruitful results in future study. The introduction\nof the anisotropic interaction of the XXZ-type27in our\nmodel will reduce the rotational symmetry and possi-\nbly give rise to another symmetry-broken state such as\nthe chiral or the dimer-like ones. 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Jpn. 71, 319 (2002)."    },    {        "title": "1811.12090v2.Ferromagnetic_Neutron_Stars_in_Scalar_Tensor_Theories_of_Gravity.pdf",        "content": "arXiv:1811.12090v2  [astro-ph.HE]  27 Feb 2019Ferromagnetic Neutron Stars in Scalar-Tensor Theories of G ravity\nZeinab Rezaei1,2∗andHabib Youseﬁ Dezdarani1\n1Department of Physics, Shiraz University, Shiraz 71454, Ir an.\n2Biruni Observatory, Shiraz University, Shiraz 71454, Iran .\nAbstract\nFerromagnetic spin ordering can take place in neutron stars . This phase transition alters the\nneutron star equation of state. Here, applying the scalar-t ensor theories of gravity, we investigate\nthe structure of neutron stars which are in the ferromagneti c phase. Considering the equation of\nstate of ferromagnetic neutronmatter with Skyrme-typeint eractions at zero temperatureandusing\nthe scalar-tensor theories of gravity with suﬃciently nega tive coupling constant, we explore the\nspontaneousscalarization inferromagneticneutronstars . Inthisregard, themassversusthecentral\ndensity, the proﬁles of scalar ﬁeld and mass and density, the central scalar ﬁeld for diﬀerent central\ndensities, and the mass-radius relation of ferromagnetic n eutron stars are presented. Moreover, we\ninvestigate the inﬂuences of the coupling constant on the sc alarization of ferromagnetic neutron\nstars. The eﬀects of the coupling constant on the critical den sities of scalarization and the scalar\ncharge of ferromagnetic neutron stars are also calculated. In addition, we study the maximum\nvalue of the coupling constant at which the spontaneous scal arization takes place presenting the\ninﬂuence of the neutron matter equation of state on this crit ical value.\n∗Corresponding author. E-mail: zrezaei@shirazu.ac.ir\n1I. INTRODUCTION\nFerromagnetic spin state in neutron stars is one of the most intere sting subjects for both\nnuclear and particle astrophysicists. Diﬀerent investigations verif y that this ordering takes\nplace in neutron stars [1–14]. The conditions for a ferromagnetic tr ansition in neutron stars\nin the framework of exact general-relativistic equilibrium hydrodyna mics and applying the\nrelativistic equation of state (EOS) for interacting hard-core nuc leons have been presented\n[1]. They showed that a ferromagnetic transition in these stars is no t inconsistent with\nstellar equilibrium in possible superdense stars. The ferromagnetic t ransition has been\nexplored in a dense neutron matter system employing the Landau’s F ermi liquid function for\na hard-sphere gas of neutrons [2]. Applying the relativistic Hartree -Fock approach to a pure\nneutronsystem withthedensities appropriateforneutronstars shows thattheferromagnetic\ntransition is speciﬁed by the Fock terms [3]. Within a relativistic Hartr ee-Fock approach\nand using diﬀerent combinations of mesons and couplings, it has been clariﬁed that a π-\nmeson pseudoscalar coupling prefers the ferromagnetic phase in t he pure neutron system\n[4]. Studying the neutron matter with Skyrme forces veriﬁes that w ith the parameters\nwhich provide a very good parameterization of realistic neutron mat ter calculations, the\nferromagnetic spin ordering occurs in the neutron matter [5]. The Landau Fermi liquid\ntheory conﬁrms that the spontaneous magnetization takes place in dense degenerate neutron\nsystem and this phase transition determines the neutron star den sity and magnetic ﬁeld [6].\nStudying the highly dense nuclear matter with the relativistic mean-ﬁ eld approach conﬁrms\nthat when the axial-vector interaction is negative enough, the sys tem holds ferromagnetism\n[7]. Employing the Hartree-Fock theory for a system of nucleons int eracting through a\ncentral spin-isospin schematic force and using the technique of th e anomalous propagator,\nthe critical values of the interaction that leads to a transition to a f erromagnetic phase have\nbeen obtained [8]. The axial anomaly acting on the parallel layers of ne utral pion domain\nwalls which spontaneously formed at high density in chiral nonlinear sig ma model results in\na ferromagnetic phase at high density for degenerate neutron ma tter [9]. Noting the chiral\neﬀective model incorporating magnetic ﬁelds and the chiral anomaly , it has been shown\nthat with an axial vector meson condensation, there will be a possib le realization of a QCD\nferromagnetic phase and ferromagnetic magnetars [10]. The calcu lation in the framework of\na relativistic σ+ω+π+ρHartree-Fock approach veriﬁes that the interaction of protons with\n2neutrons leads to the lower densities of spontaneous magnetizatio n compared to the pure\nneutron matter [11]. Considering the presence of the protons with neutrons in a Hartree-\nFock methed with Skyrme forces also leads to the reduction of the d ensity of ferromagnetism\n[12]. For asymmetric nuclear matter in the framework of a Fermi liquid theory with the\nSkyrme eﬀective interaction, the system experiences a phase tra nsition to the spin polarized\nstate with the oppositely directed spins of neutrons and protons, i.e. antiferromagnetic spin\nstate [12]. However, considering the phase transition of symmetric nuclear matter in Fermi\nliquid theory with the Skyrme eﬀective interaction conﬁrms that the ferromagnetic spin\nstate is preferable over the antiferromagnetic one [13]. The critica l density of ferromagnetic\ntransition in neutron matter with Skyrme-type interactions decre ases with temperature [14].\nScalar-tensor theories of gravity (STTs) which are among the ext ensions of general rela-\ntivity(GR)havebeenstudied anddeveloped byFierz[15], Jordan[16], BransandDicke [17],\nBergman [18], Nordtvedt [19], Wagoner [20], Damour and Esposito-Fa rese [21]. These the-\nories which include one or more scalar ﬁelds coupled to matter, predic t scalar gravitational\nwave that can be detected by the laser interferometer [22, 23]. Th ese theories of gravity\nhave been extensively applied to investigate the relativistic compact objects [24–38]. It has\nbeen conﬁrmed that a wide class of STTs passes the weak-ﬁeld grav itational tests indicating\nnonperturbative strong-ﬁeld deviations away from GR in neutron s tars [24]. The instability\nin spherically symmetric stars in STTs is induced by the scalar ﬁeld for s ome ranges of the\nvalue of the ﬁrst derivative of the coupling function [25]. In the grav itational collapse of\nneutron star toward a black hole, the amplitude of the gravitationa l wave increases with\nthe value of the parameter of the coupling function [26]. Using a turn ing point method, the\nsecular stability of neutron stars in STTs against spherically symmet ric perturbations has\nbeen studied [27]. Considering spherical neutron star models in STTs , it has been shown\nthat with some conditions on the second derivative of the coupling fu nction and on star’s\nmass, there exist two strong-scalar-ﬁeld solutions as well as the u sual weak-ﬁeld one [28].\nSolving the ﬁeld equations describing the equilibrium of rapidly rotating neutron stars in\nscalar-tensor theories of gravity shows that the deviations of th e rapidly rotating scalar-\ntensor neutron stars from the GR solutions is signiﬁcantly larger th an in the static case [29].\nStudying the eﬀect of a mass term in the spontaneous scalarization of neutron stars conﬁrms\nthat this addition is a natural extension to the model that avoids th e observational bounds\non the STTs [30]. Investigation of slowly rotating neutron stars in sc alar-tensor theories\n3with a massive gravitational scalar ﬁels veriﬁes that that mass, rad ius and moment of in-\nertia for neutron stars in massive scalar-tensor theories can diﬀe r drastically from the pure\nGR solutions [31]. For lower sound velocity in the core of neutron star s, the maximum mass\nlimit of the scalarized neutron stars is larger than that of in GR, while f or the stiﬀ EOSs\nwith high sound velocity, the maximum mass limit in GR is larger than the on e in STT\n[32]. Applying diﬀerent realistic equations of state, including pure nuc lear matter, nuclear\nmatter with hyperons, hybrid nuclear and quark matter, and pure quark matter to study\nthe scalarization of static and slowly rotating neutron stars shows that the magnitude of\nthe scalarization is correlated with the value of the gravitational po tential at the center of\nthe star [33]. The coupling to additional degrees of freedom in massle ss STTs leads to new\nfamilies of modes inthe spectrum ofpulsating neutron stars, withno counterpart inGR [34].\nRelativistic stars in degenerate higher-order STTs which evade the constraint on the speed\nof gravitational waves imposed by GW170817 have been investigate d [35]. Their results\nindicate that for high density stars, the mass-radius relation chan ges from the one in GR.\nFor the coupling constant conﬁned to values provided by the astro nomical observations, the\nmaximum compactness of neutron stars in GR is higher than the one in STT [36]. Apply-\ning scalar-tensor theory with massive ﬁeld with self-interaction ter m in the potential to the\nneutron star models demonestrates that the self-interaction te rm suppresses the scalar ﬁeld\nand large deviations from pure GR is observed [37]. Using a class of sca lar-tensor theories of\ngravity indistinguishable from GR in the weak ﬁeld regime but with signiﬁc ant deviations in\nstrong ﬁelds, it has been shown that the maximum mass of a diﬀerent ially rotating neutron\nstar increases signiﬁcantly for scalarized solutions and such stars can reach larger angular\nmomenta [38]. Regarding the above discussions, the EOS of ferroma gnetic neutron matter\ncan have signiﬁcant eﬀects on the scalarized neutron stars. The m ain goal of this work is\ninvestigating the eﬀects of the ferromagnetic neutron matter on the structure of neutron\nstars in STTs. Considering the ferromagnetic neutron matter with Skyrme-type interac-\ntions within the STT, we investigate the structure as well as the spo ntaneous scalarization\nof ferromagnetic neutron stars.\n4II. FORMALISM\nWe note a homogeneous system of Nneutrons with the spin-up number density n↑, spin-\ndown number density n↓, and total number density n=n↑+n↓. The Skyrme-like eﬀective\ninteractions have the standard form [14, 39],\nV(r1,r2) =t0(1+x0Pσ)δ(r)\n+1\n2t1(1+x1Pσ)(K′2δ(r)+δ(r)K2)+t2(1+x2Pσ)K′.δ(r)K\n+1\n6t3(1+x3Pσ)[ρ(R)]γδ(r)+iW0(σ1+σ2).[K′×δ(r)K], (1)\nwithr=r1−r2,R= (r1+r2)/2,K= (∇1− ∇2)/2i, andPσ= (1 +σ1.σ2)/2. Besides,\nK, the relative momentum acts on the right and K′is its conjugate and acts on the left.\nMoreover,t0,t1,t2,t3,x0,x1,x2,x3,γ, andW0are the Skyrme-force parameters. The\ntotal energy of neutron matter per particle is calculated by E/N=/angb∇acketleftψ|H|ψ/angb∇acket∇ight/Nthrough\nthe Hartree-Fock method in which ψandHdenote the wave function and the Hamiltonian\nof system, respectively. For the non-ferromagnetic neutron ma tter (NFM), the energy per\nparticle is as follows [39],\nENFM/N=3\n5/planckover2pi12\n2m(3π2n)2/3+1\n4t0(1−x0)n+1\n24t3(1−x3)nγ+1+3\n40(3π2)2/3Θn5/3,(2)\nwith Θ =t1(1−x1)+3t2(1+x2). We also consider a system of polarized neutron matter\nwith the spin-up number density n↑, spin-down number density n↓, and spin polarization\nparameter ∆ = ( n↑−n↓)/n. The energy per particle for ferromagnetic (FM) neutron matter\nis given by [14],\nEFM/N=/planckover2pi12\n2mτ↑+τ↓\nn+1\n4n[2t2(1+x2)][τ↑n↑+τ↓n↓]\n+1\n4n[t1(1−x1)+t2(1+x2)][τ↑n↓+τ↓n↑]\n+1\nn[t0(1−x0)+1\n6t3(1−x3)nγ]n↑n↓, (3)\nwith\nτσ=3\n10(3π2n)2/3n(1±∆)5/3, (4)\nin which + and −denote the up and down spin projections. It is possible to calculate t he\nequilibrium value of the spin polarization parameter, i.e. ∆ min, at each number density\n5nfrom Eq. (3). By the equilibrium value, we mean the value at which the e nergy is\nminimum. Besides, the energy at each ∆ mincan be calculated using Eq. (3). Here, we\nemploytheSLy230amodeltodescribetheneutronmatterEOS[39]. Atlowerdensities, ∆ min\nis equal to zero and neutron matter is fully unpolarized. At a density about 5.40ρ0in which\nρ0= 1.66×1017kg/m3,thisparameterspontaneouslyincreasesanditreachesto1atad ensity\nabout 10.41ρ0. Neutron matter with the densities between 5 .40ρ0<ρ<10.41ρ0is partially\npolarized. Besides, with the densities higher than 10 .41ρ0, the neutron matter becomes fully\npolarized and it is in ferromagnetic state. Applying the ﬁrst law of the rmodynamics, i.e.\nP=n2(∂(E/N)\n∂n), the EOSs for the NFM and FM neutron matter are calculated. Fig. 1\nshows the EOSs of non-ferromagnetic and ferromagnetic neutro n matter in Sly230a model.\nFor both NFM and FM neutron matter, the pressure grows by incre asing the density. At\nρ/ρ0P /P0\n0246810121400.511.522.5 NFM\nFM\nFIG. 1: Pressure, P, of the non-ferromagnetic (NFM) and ferromagnetic (FM) neu tron matter in\nSLy230a model versus the density, ρ, withP0= 5×1034kg/ms2andρ0= 1.66×1017kg/m3.\nmost values of ρ, the rate at which the pressure increases is higher for NFM neutro n matter.\nTherefore, the EOS of FM neutron matter is softer than NFM one. As wee see in the\nfollowing, the softening of the EOS aﬀects the properties of the sc alarized neutron stars, i.e.\nmass, radius, and spontaneous scalarization. The main goal of this work is investigating the\neﬀects of neutron matter ferromagnetism on the properties of n eutron stars in STTs.\nTo describe the neutron stars in STTs, we start with the spacetime line element in the\n6Einstein frame for a spherical symmetry static neutron star as fo llows [40],\nds2=−N(r)2dt2+A(r)2dr2+r2(dθ2+sin2θdφ2), (5)\nin whichN(r) andA(r) are the metric functions. The function A(r) is related to the mass\nbyA(r) := [1−2m(r)/r]−1/2. We also consider the neutron star as a perfect ﬂuid with\nstress-energy tensor as follows,\n˜Tµν= ˜ǫ˜uµ˜uν+ ˜p(˜gµν+ ˜uµ˜uν), (6)\nwith the ﬂuid’s total energy density ˜ ǫ, pressure ˜p, and 4-velocity ˜ u. It should be noted that\ntilde presents the quantities in Jordan frame. Besides, we denote t he physical metric or\nJordan metric by ˜ gµν:=a(φ)2gµνin whicha(φ) andφare the coupling function and scalar\nﬁeld, respectively. For the coupling function a(φ), we consider two models as follows [24, 40],\nModel 1 (M1) :a(φ) = [cosh/parenleftBig√\n3β(φ−φ0)/parenrightBig\n]1\n3β,\nα(φ) =1√\n3tanh(√\n3β(φ−φ0)), (7)\nModel 2 (M2):a(φ) =e1\n2β(φ−φ0)2,\nα(φ) =β(φ−φ0), (8)\nin whichβis the coupling constant and we assume φ0= 0. In addition, α(φ) :=dlna(φ)\ndφ. It\nhasbeen veriﬁed that anonperturbative ampliﬁcationmechanism of the coupling strength of\nthe scalar ﬁeld exists when the logarithm of the coupling function has a suﬃciently negative\ncurvature around φ0[24]. They showed the existence of strong-ﬁeld deviations from GR in\nSTTs with β/lessorsimilar−4. Besides, the negative values of βresult in the scalar ﬁeld nonlinearities\nand these reinforce the naturally attractive character of scalar interactions [43]. We know\nthat the binary pulsar experiments set constraints on the value of the coupling constant i.e.\nβ >−4.8 [44]. Moreover, the pulsar-white dwarf binary PSR J0348+0432, p ut a bound\non the coupling constant β≥ −4.5 [44]. These constraints are in the massless scalar ﬁeld\ncase. However, adding a mass term to the scalar ﬁeld potential res ults in the extension\nto the model that avoids these observational bounds [30]. In fact , the coupling constant\ncan be much smaller than −4.5 for massive STTs [31]. We apply diﬀerent values for the\ncoupling constant which are lower and larger than the lower limit set by the binary pulsar\nexperiments. Consideration of the values smaller than the lower limit f rom the observations\n7is justiﬁed because in the present study which is the ﬁrst investigat ion of the scalarized FM\nneutron stars, we are interested on the eﬀects of the coupling co nstant onthe scalarization of\nthese stars. In addition, the calculations with β <−4.5 in the massless case give the upper\nlimit for the deviations from GR in the massive case [38]. We also calculate the maximum\nvalue of the coupling constant at which the spontaneous scalarizat ion takes place in NFM\nand FM neutron stars. This determines the inﬂuence of the neutro n matter EOS on this\ncritical value. This value has previously calculated using a polytropic e quation of state to\nbe−4.35 in the nonrotating star [27] and −3.9 in rapidly rotating stars [29].\nByapplying some calculations, theﬁeld equations inSTTs leadto thefo llowing equations\nwhich describe the structure of neutron stars in STTs [40],\ndm\ndr= 4πr2a4˜ǫ+r\n2(r−2m)/parenleftBigdφ\ndr/parenrightBig2\n, (9)\ndlnN\ndr=4πr2a4˜p\nr−2m+r\n2/parenleftBigdφ\ndr/parenrightBig2\n+m\nr(r−2m), (10)\nd2φ\ndr2=4πra4\nr−2m/bracketleftbigg\nα(˜ǫ−3˜p)+r(˜ǫ−˜p)dφ\ndr/bracketrightbigg\n−2(r−m)\nr(r−2m)dφ\ndr, (11)\nd˜p\ndr=−(˜ǫ+ ˜p)/bracketleftbigg4πr2a4˜p\nr−2m+r\n2/parenleftBigdφ\ndr/parenrightBig2\n+m\nr(r−2m)+αdφ\ndr/bracketrightbigg\n, (12)\ndmb\ndr=4πr2a3˜ρ/radicalBig\n1−2m\nr, (13)\nin whichmbdenotes the baryonic mass. These are the generalized Tolman-Opp enheimer-\nVolkoﬀ equations in STTs. The boundary conditions to solve these eq uations together with\nthe neutron matter EOS are as follows,\nm(0) =mb(0) = 0,lim\nr→∞N(r) = 1, φ(0) =φc,lim\nr→∞φ(r) = 0,\ndφ\ndr(0) = 0,˜p(0) =pc,˜p(Rs) = 0, (14)\nwhereRsis the radius of the star. Using a fourth-order Runge-Kutta algor ithm [41], we\nintegrate the above equations. The integration is done with the bou ndary conditions at\nr= 0. In addition, with a guess for the scalar ﬁeld at the center, i.e. φ(0) =φc, we do the\niteration on φcsuch that the following condition satisﬁes [24, 40],\nφs+2ψs/radicalbig\n˙ν2s+4ψ2sarctanh/bracketleftBigg/radicalbig\n˙ν2s+4ψ2s\n˙νs+2/Rs/bracketrightBigg\n= 0. (15)\n8In the above equation, the subscript s denotes the quantities on t he surface of star. Besides,\nψs:= (dφ/dr)sand ˙νs:= 2(dlnN/dr)|s=Rsψ2\ns+2ms/[Rs(Rs−2ms)]. Moreover, the ADM\nmass,MADM, and scalar charge, ω, are given by [24, 40],\nMADM=R2\ns˙νs\n2/parenleftbigg\n1−2ms\nRs/parenrightbigg1\n2\nexp/bracketleftBigg\n−˙νs/radicalbig\n˙ν2s+4ψ2sarctanh/parenleftBigg/radicalbig\n˙ν2s+4ψ2s\n˙νs+2/Rs/parenrightBigg/bracketrightBigg\n,(16)\nω=−2MADMψs/˙νs. (17)\nIt should be mentioned that the scalar charge is introduced throug hthe asymptotic behavior\nof the scalar ﬁeld with r→ ∞as follows [21],\nφ(r) =ω/r+O(1/r2). (18)\nIII. RESULTS AND DISCUSSION\nA. Mass versus the Central Density\nFig. 2 demonstrates the NFM and FM neutron star mass as a functio n of the central\ndensity for the STT and GR in two models (M1 and M2) of coupling funct ion for diﬀerent\nvalues of the coupling constant. In M1 for diﬀerent values of the co upling constant and M2\nfor higher values of the coupling constant, the neutron star mass increases by increasing the\ndensity to a special value of the central density. For the densities higher than that special\nvalue, the neutron star mass is constant. This special value of the central density is greater\nfor NFM neutron stars. At lower densities, the mass of FM neutron stars is greater than\nthat of NFM ones. But at higher densities, the mass of NFM neutron stars is higher than\nthat of FM stars. This is due to this fact that the EOS of NFM neutro n matter is stiﬀer\nthan the EOS of FM neutron matter. In addition, in M1 for diﬀerent v alues ofβand in M2\nfor higher values of β, for both NFM and FM neutron stars, the results of GR and STT are\nnearly equal. This is while the results of STT and GR are diﬀerent for low er values of β.\nFor the cases that the results of GR and STT are diﬀerent, the neu tron stars are scalarized.\nIn M2 with lower values of β, the mass of the stars with lower central densities in STT is\nsmaller than GR. However, for some stars with higher densities, the mass in STT is greater\nthan GR. As we see in the following, this fact that the result of STT ho w is diﬀerent from\nGR is related to the variation of the central scalar ﬁeld respect to t he central density. For\n9ρc/ρ0M /Msun\n5 10 150.511.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)M1\nβ=-6.0\nρc/ρ0M /Msun\n5 10 150.511.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)M1\nβ=-5.3\nρc/ρ0M /Msun\n5 10 150.511.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)M1\nβ=-4.5\nρc/ρ0M /Msun\n5 10 150.511.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)M2\nβ=-6.0\nρc/ρ0M /Msun\n5 10 150.511.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)M2\nβ=-5.3\nρc/ρ0M /Msun\n5 10 150.511.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)M2\nβ=-4.5\nFIG. 2: Neutron star mass, M,versus the central density, ρc,for NFM and FM neutron stars in\nSTT and GR applying M1 and M2 with diﬀerent values of the coupli ng constant, β.\nNFM neutron stars with high densities, the results of STT and GR are the same. Therefore,\nthere is no scalarized NFM neutron stars with high central density. However, for the FM\nneutron stars with densities greater than a special value, the res ults of GR and STT are not\nthe same. This diﬀerence continues to highest values of the density considered in this work,\ni.e.ρc= 14ρ0. Therefore, even the FM neutron stars with high densities are sca larized\nunlike NFM stars. In fact, this phenomenon (scalarization of FM neu tron stars with high\ndensities) is the main distinction of NFM and FM neutron stars. For th e lower values of β,\nthe deviation of STT from GR is more considerable. Besides, the diﬀer ence of the results\nin STT and GR is more signiﬁcant in M2 compared to M1. In Table I, we pre sent the\nmaximum mass for NFM and FM stars. According to these results, fo r NFM neutron stars\nin two models, the maximum mass decreases by increasing the coupling constant. The\ndiﬀerence of maximum mass in two models decreases by increasing β. In FM stars unlike\nthe NFM ones, the maximum mass in two models grows with the increase ofβ. So we can\nconclude that the eﬀects of βon the maximum mass depend on the EOS of neutron matter.\n10Maximum Mass ( M/circledottext)\nModel β STT (NFM) STT (FM)\n1−6.0 2.15 1.32\n−5.3 2.09 1.33\n−4.5 1.93 1.36\n2−6.0 2.48 1.16\n−5.3 2.20 1.27\n−4.5 1.93 1.36\nTABLE I: Maximum mass of NFM andFM neutronstarsin STTfor diﬀe rent values of thecoupling\nconstant in two models. Besides, the maximum mass for NFM and FM neutron stars in GR is\n2.15M/circledottextand 1.36M/circledottext, respectively.\nMaximum Compactness\nModel β STT (NFM) STT (FM)\n1−6.0 0.30 0.16\n−5.3 0.28 0.16\n−4.5 0.25 0.17\n2−6.0 0.29 0.13\n−5.3 0.28 0.15\n−4.5 0.25 0.16\nTABLE II: Maximum compactness of NFM and FM neutron stars in S TT for diﬀerent values of the\ncoupling constant in two models. Besides, the maximum compa ctness for NFM and FM neutron\nstars in GR is 0 .33 and 0 .17, respectively.\nWith stiﬀer EOS (NFM neutron matter), the maximum mass reduces a sβgrows. However,\nwith softer EOS (FM neutron matter), the maximum mass increases by increasing β. In\naddition, for NFM neutron stars, the maximum mass in M2 is greater t han M1. However,\nthis diﬀerence of two models is opposite for the FM stars. It should w e noted that the\nmaximum mass of FM stars in STT is always smaller than or equal to one in GR. In both\n11gravities, the maximum mass of FM stars is lower than the NFM ones. T his is due to the\nfact that the EOS of FM neutron matter is softer than the NFM one . Table II also gives\nthe maximum compactness of NFM and FM neutron stars. By increas ingβ, the maximum\ncompactness of stars in STT, reduces for NFM stars while it grows f or FM ones. Moreover,\nfor almost all NFM and FM neutron stars, the maximum compactness in STT is lower than\nthe one in GR. This eﬀect is in agreement with the one reported in Ref. [36].\nB. Proﬁles of Scalar Filed, Mass, and Density\nFig. 3 shows the proﬁle of scalar ﬁeld for NFM and FM neutron stars in STT with two\nmodels. The value of scalar ﬁeld is nonzero in each point of the stars. Moreover, for diﬀerent\nvalues ofβ, at each distance to the center of star, M2 predicts higher values for the scalar\nﬁeld compared to M1. This is because in M2, the coupling of scalar ﬁeld t o metric is more\nr(km)φ(r)\n0 5 10 1500.050.10.150.20.250.30.35NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)β=-6.0\nr(km)φ(r)\n0 5 10 1500.050.10.150.20.250.30.35NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)β=-5.3\nr(km)φ(r)\n0 5 10 1500.050.10.150.20.250.30.35NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)β=-4.5\nFIG. 3: Proﬁle of scalar ﬁeld in NFM and FM neutron stars for tw o models with diﬀerent values\nof the coupling constant, β. The values of the central density are ρc= 7ρ0andρc= 12ρ0for the\nNFM and FM neutron stars, respectively.\nsigniﬁcant compared to M1. The rate at which the scalar ﬁeld reduce s is higher in M2.\nOur calculations verify that by increasing β, the scalar ﬁeld in neutron star decreases. This\nis due to the fact that for lower values of β, the coupling of scalar ﬁeld to metric is more\nsigniﬁcant. In addition, the scalar ﬁeld for FM neutron star is smaller than the NFM one.\nTherefore, with the softer EOS, the magnitude of the scalar ﬁeld is smaller than the one\nwith the stiﬀer EOS. The diﬀerence between the proﬁle of scalar ﬁeld in two models is more\nconsiderable for NFM neutron stars. Moreover, with diﬀerent valu es ofβ, for both NFM\n12and FM neutron stars, the diﬀerence of M1 and M2 is more important in the center of star\ncompare to its surface.\nFig. 4 presents the proﬁle of mass for NFM and FM neutron stars in S TT with two\nmodels. According to Fig. 4, the mass proﬁles in two models are diﬀere nt for both NFM\nr(km)m(r) /Msun\n0 5 10 1500.511.52NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-6.0\nr(km)m(r) /Msun\n0 5 10 1500.511.52NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-5.3\nr(km)m(r) /Msun\n0 5 10 1500.511.52NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-4.5\nFIG. 4: Same as Fig. 3 but for the proﬁle of mass.\nand FM neutron stars. This diﬀerence is more signiﬁcant for NFM sta rs. With higher values\nofβ, the proﬁles approach to each other. The mass proﬁle of FM stars in two models is\nlower than the NFM one. This is due to the fact that the FM EOS is soft er than NFM one.\nFig. 5 shows the proﬁle of density for NFM and FM neutron stars in ST T with two models.\nFor the proﬁle of density, the diﬀerence between two models is more considerable at lower\ncoupling constants. This diﬀerence is nearly negligible for FM stars. F ig. 5 conﬁrms that\nr(km)ρ(r) /ρ0\n0 5 10 1524681012NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-4.5\nr(km)ρ(r) /ρ0\n0 5 10 1524681012NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-5.3\nr(km)ρ(r) /ρ0\n0 5 10 1524681012NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-6.0\nFIG. 5: Same as Fig. 3 but for the proﬁle of density.\nby increasing β, the density becomes equal to zero at smaller distances to the cen ter of star\n(i.e. stars with smaller radii). Moreover, at each coupling constant, M2 predicts larger radii\n13for NFM stars. It should be noted that for FM neutron stars, the eﬀect of coupling constant\non the radius is not considerable.\nC. Central Scalar Field versus the Central Density\nFig. 6 shows the central scalar ﬁeld versus the central density fo r NFM and FM neutron\nstars. For both NFM and FM stars considering two models with all valu es of the coupling\nconstant, the central scalar ﬁeld is zero at lower densities. For st ars with zero central\nscalar ﬁeld, the solutions of GR and STT are equal. Considering both N FM and FM stars,\nthe scalar ﬁeld increases by increasing the density from a certain va lue and therefore the\nspontaneous scalarization takes place. However, the densities at which the scalarization\ntakes place are diﬀerent for NFM and FM neutron stars. In this wor k, we denote the ﬁrst\ncritical density of scalarization by ρcr1. The GR and STT solutions are diﬀerent at nonzero\nscalar ﬁelds. In both models for all values of β, the scalar ﬁeld of NFM stars becomes zero at\na value of density (second critical density of scalarization, ρcr2). Moreover, for these stars,\nthe scalar ﬁeld remains zero up to high densities. Consequently, the high density NFM\nneutron stars are not scalarized. However, for FM neutron star s, the scalar ﬁeld increases\nmonotonically by increasing the density and it does not become zero e ven at high densities.\nTherefore, the high density FM neutron stars are also scalarized. This is the main diﬀerence\nof NFM and FM stars. This phenomenon is related to the one explained in Fig. 2. In fact,\nthe diﬀerence of the GR and STT solutions up to high densities for the mass verses the\ndensity in FM stars is a result of the nonzero scalar ﬁeld in these star s. Therefore, the EOS\nof star aﬀects its scalarization. It can be seen from Fig. 6 that with lower values of β, the\nﬁrst critical densities of scalarization for NFM and FM stars are mor e close to each other.\nBesides, for lower values of β, the ranges of density at which the NFM and FM stars are\nscalarized have more overlap with each other. For NFM stars in two m odels, the value of\nρcr1increases as the coupling constant grows, in agreement with the re sult of Ref. [36]. In\naddition, for these stars, ρcr2decreases by increasing β. It can be concluded that the range\nat which the NFM stars are scalarized is larger for lower values of β. With lower values of\nthe coupling constant, ρcr1in FM stars is smaller than NFM ones. Therefore, with lower\nvalues ofβ, lower density FM stars can be also scalarized. We can found that th e range of\nscalarization in FM neutron stars is greater than NFM one. In fact, softening of EOS leads\n14ρc/ρ0φc\n5 10 1500.10.20.30.4NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-6.0\nρc/ρ0φc\n5 10 1500.10.20.30.4NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-5.3\nρc/ρ0φc\n5 10 1500.10.20.30.4NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-4.5\nFIG. 6: Central scalar ﬁeld, φc, versus the central density, ρc, for NFM and FM neutron stars\napplying M1 and M2 with diﬀerent values of the coupling consta nt,β. Note that in GR we have\nφc= 0.\nto a larger range of scalarization. This eﬀect agrees with the result reported in Ref. [42].\nAnother result of Fig. 6 is that by increasing the coupling constant, the maximum value of\nthe central scalar ﬁeld decreases. For that reason, neutron st ars are more scalarized with\nlower values of β. This result has also reported in Refs. [24, 26, 36]. Fig. 6 also veriﬁes that\nM2 predicts higher values of the scalar ﬁeld compared to M1. Theref ore, the stars are more\nscalarized in M2.\nD. Mass-Radius Relation\nFig. 7 presents the mass-radius relation for NFM and FM neutron st ars in STT and\nGR considering two models with diﬀerent values of β. The mass-radius relation of neutron\nstars which are not scalarized is equal in STT and GR. With diﬀerent va lues ofβin M1\nand with high values of βin M2, for NFM and FM neutron stars, the smaller stars have\nlarger masses. Besides, for that conditions, the bigger stars hav e lower masses. In fact, they\nare gravitationally bound stars. Fig. 7 shows that in two models for n eutron stars, the\ndeviation of STT from GR is more signiﬁcant with lower values of β. According to Fig. 7,\nfor massive NFM neutron stars, the results of STT and GR are equa l. However, for massive\nFM neutron stars, these results are not the same. This is due to th e fact that high density\nFM stars unlike the NFM ones are scalarized (see Fig. 6). In addition, this deviation in\nM2 is more considerable than M1. Because M2 predicts more scalariza tion for neutron stars\n15R (km)M (Msun)\n101214161811.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)\nM1\nβ=-6.0\nR (km)M (Msun)\n101214161811.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)\nM1\nβ=-5.3\nR (km)M (Msun)\n101214161811.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)\nM1\nβ=-4.5\nR (km)M (Msun)\n101214161811.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)\nM2\nβ=-6.0\nR (km)M (Msun)\n101214161811.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)\nM2\nβ=-5.3\nR (km)M (Msun)\n101214161811.522.5\nNFM (STT)\nFM (STT)\nNFM (GR)\nFM (GR)\nM2\nβ=-4.5\nFIG. 7: Mass versus the radius for NFM and FM neutron stars in S TT and GR applying M1 and\nM2 with diﬀerent values of the coupling constant, β. The lines which show the allowed region for\nneutron stars are also presented.\ncompared to M1. For lower values of β, the deviation of STT from GR in the mass-radius\nrelation takes place in a greater region. This is a result of the fact th at with lower values of\nβ, the range of scalarization is bigger (see Fig. 6). For scalarized NFM stars (specially with\nβ=−6.0 in M2), the massive stars are bigger while the lower mass ones are sm aller. This\nmeans that the slop of mass-radius relation for these stars is diﬀer ent from the one in GR.\nIn fact, in these cases, the stars are self bound. However, the s calarized FM neutron stars\nare still gravitationally bound. It is clear from Fig. 7 that the deviatio n of STT from GR is\nmore signiﬁcant for NFM stars compared to FM ones. Therefore, it can be concluded that\nthe EOS of neutron matter aﬀects the amount of deviation of STT f rom GR. The scalarized\nNFM neutron stars are larger in size compared to the GR solutions. I n addition, for FM\nneutron stars, the mass of scalarized stars is lower than the star s in GR. This is similar to\nthe result of Ref. [40].\n16βρcr1/ρ0\n-6-5.5 -5-4.5510152025\nNFM\nFM\nβρcr2/ρ0\n-6-5.5 -5-4.56.577.588.599.510\nNFM\nFIG. 8: First, ρcr1, and second, ρcr2, critical density of scalarization in NFM and FM neutron sta rs\nin M1 versus the coupling constant, β.\nE. Critical Density of Scalarization\nIn this part, we are going to investigate the eﬀects of the coupling c onstant on the critical\ndensity of scalarization for NFM and FM stars. Fig. 8 gives the ﬁrst a nd second critical\ndensities for diﬀerent stars in M1 versus the coupling constant. Fo r both NFM and FM\nstars, the ﬁrst critical density, ρcr1, increases as the coupling constant grows. This means\nthat for higher values of β, the stars become scalarized at higher densities. It should be\nnoted that our results conﬁrm that the critical density of scalariz ation does not depend on\nthe coupling function model. It is clear from Fig. 8 that for the most v alues ofβ, the ﬁrst\ncritical density inFM stars is higher thanthe one in NFMstars. Moreo ver, the rate at which\nρcr1grows with βis greater for FM stars. The eﬀects of the spin polarization of neut ron\nmatter onρcr1is more signiﬁcant when the coupling constant is higher. Fig. 8 also sho ws the\nsecond critical density of scalarization for NFM stars versus β. It should be mentioned that\nsince the FM neutron stars are scalarized up to high densities, the s econd critical density of\nscalarization is not deﬁned for the FM stars. In NFM stars, the sec ond critical density, ρcr2,\ndecreases as βgrows. Regarding the increase of ρcr1withβ, it is possible to conclude that\nfor NFM neutron stars, the range of density at which the scalariza tion takes place decreases\nas the coupling constant grows. Therefore, the solution of STT ap proaches to GR when β\n17βρcr1/ρ0\n-4.5-4.45-4.4-4.35-4.3510152025\nNFM\nFM\nβρcr2/ρ0\n-4.5-4.45-4.4-4.35-4.36.577.58\nNFM\nFIG. 9: Same as Fig. 8 but for a smaller range of the coupling co nstant.\nincreases.\nTo explore how the neutron matter EOS aﬀects the critical value of β, i.e. the maximum\nvalue of the coupling constant at which the scalarization takes place , we have presented the\ncritical densities of scalarization in a smaller range of βin Fig. 9. It is clear that in the case\nof FM neutron stars, the maximum value of the coupling constant is h igher compared to\nthe NFM neutron stars. Therefore, in FM neutron stars, the ran ge of the coupling constant\nat which the stars are scalarized is more extended. Our results con ﬁrm that the critical\nvalues ofβare−4.35 and−4.29 for NFM and FM stars, respectively. According to our\nresults, for NFM neutron stars with β=−6.0 andβ=−4.35, the ﬁrst critical densities are\nρcr1= 3.87ρ0andρcr1= 6.21ρ0, respectively. Besides, with β=−6.0 andβ=−4.35, the\nsecond critical densities are ρcr2= 9.72ρ0andρcr2= 6.48ρ0, respectively. Moreover, for FM\nstars withβ=−6.0 andβ=−4.29, the values of the ﬁrst critical density are ρcr1= 3.67ρ0\nandρcr1= 23.09ρ0, respectively.\nF. Scalar Charge in Ferromagnetic and Non-Ferromagnetic Neutron Stars\nFig. 10 presents the scalar charge versus the mass for NFM and FM neutron stars. For\nboth NFM and FM low mass stars with all values of βin two models, the scalar charge\nis zero. For all NFM neutron stars with the masses higher than a spe cial value, the scalar\n18charge grows when the mass increases. For the NFM neutron star s, the mass at which the\nscalar charge becomes nonzero increases with the increase of β. This is in agreement with\nthe result of Ref. [43]. For NFM neutron stars with high masses, the scalar charge again\nbecomes zero. The range of mass with nonzero scalar charge is pre cisely the range at which\nthe central scalar ﬁeld is nonzero (see Fig. 6). In fact, the more s calarized the neutron\nstar, the more amount of scalar charge for star. Fig. 10 shows th at the mass at which the\nscalar charge again becomes zero reduces as βincreases. In addition, the maximum value\nof scalar charge decreases as the coupling constant grows. The s tars with nonzero scalar\ncharge are the scalarized neutron stars. The range of mass with n onzero scalar charge is the\nsame in two models. However, M2 predicts more scalar charge compa red to M1. Because\nthe neutron stars are more scalarized in M2. For FM stars in M1 with lo wer values of β,\nM (Msun)ω\n0.511.5 22.500.20.40.60.811.21.4 NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-6.0\nM (Msun)ω\n0.511.522.500.20.40.60.811.21.4 NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-5.3\nM (Msun)ω\n0.511.522.500.20.40.60.811.21.4 NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-4.5\nFIG. 10: Scalar charge, ω, versus the mass, M, for NFM and FM neutron stars applying M1 and\nM2 with diﬀerent values of the coupling constant, β.\ni.e.−6.0 and−5.3, the scalar charge is an increasing function of mass. This is due to t he\nfact that in these conditions, the scalar ﬁeld is an increasing functio n of both density and\nmass (see Figs. 2 and 6). However, for FM stars in M1 with β=−4.5 and also in M2 with\ndiﬀerent values of β, the scalar charge decreases when the mass grows. It is due to th e fact\nthat according to Figs. 2 and 6, by increasing the density the scalar ﬁeld increases while the\nmass decreases. Therefore, for FM stars, the slope of scalar ch arge versus the mass depends\non the model of coupling function as well as the coupling constant. B y increasing β, the\nmaximum value of scalar charge in FM neutron stars like the NFM ones d ecreases. The\nmaximum value of scalar charge in NFM stars in two models with diﬀerent values ofβis\ngreater than the one in FM stars. The curve related to the scalar c harge of FM neutron\n19stars is not a closed curve unlike the one related to the NFM stars. T his is a result of this\nfact that the FM neutron stars remain scalarized up to high densitie s.\nFig. 11presentsthescalarchargeofNFMandFMneutronstarsve rsustheircompactness.\nIn two models with all values of βfor NFM and FM neutron stars with low compactness,\nthe scalar charge is zero. In NFM neutron stars with the compactn ess higher than a special\nM /Rω\n0.050.10.150.20.250.300.20.40.60.811.21.4 NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-6.0\nM /Rω\n0.050.10.150.20.250.300.20.40.60.811.21.4 NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-5.3\nM /Rω\n0.050.10.150.20.250.300.20.40.60.811.21.4 NFM (M1)\nNFM (M2)\nFM (M1)\nFM (M2)\nβ=-4.5\nFIG. 11: Same as Fig. 10 but for the scalar charge versus the co mpactness of star, M/R.\nvalue, the scalar charge increases with the compactness. Besides , the scalar charge again\nbecomes zero for NFM stars with high compactness. However, for FM stars, the scalar\ncharge increases monotonically with the increase of the compactne ss. Fig. 11 conﬁrms that\nthe compactness at which the scalar charge becomes nonzero is hig her for NFM neutron\nstars compared to FM ones. This means that the FM stars with lower compactness can\nalso have scalar charge. For all stars in two models, the value of com pactness at which the\nscalar charge becomes nonzero increases when the coupling const ant grows. It is due to the\nfact that with lower values of |β|and approaching to GR, the more compactness is needed\nto have the scalar charge. In addition, for both NFM and FM neutro n stars in two models,\nthe range of compactness at which the scalar charge is nonzero de creases by increasing β. It\nmeans that with the lower values of |β|and approaching to GR, the chance for ﬁnding the\nstars with scalar charge is lower.\nIV. SUMMARY AND CONCLUDING REMARKS\nThe structure of ferromagnetic (FM) neutron stars in scalar-te nsor theories of gravity has\nbee studied. To describe the neutron star, we employ the equation of state of FM neutron\n20matter with Skyrme-type interactions. We found that the soft EO S of FM neutron matter\nleads tothe lower values oftheneutron star mass comparedto the non-ferromagnetic(NFM)\none. Our results conﬁrm that with the lower values of the coupling co nstant, the results of\nSTT deviate signiﬁcantly from the ones in GR. In the cases which the r esults of STT are\ndiﬀerent from GR, the neutron stars are scalarized. For high dens ity NFM neutron stars,\nthe results of STT and GR are the same and there is no scalarized NFM neutron stars\nwith high central density. However, for the FM neutron stars up t o high central density\nconsidered in this work, the results of GR and STT are not the same a nd these stars even\nwith high densities are scalarized. We found that the densities at whic h the scalarization\ntakes place are not equal for NFM and FM neutron stars. Our calcu lations show that for\nboth NFM and FM neutron stars, the ﬁrst critical density of scalar ization increases as the\ncoupling constant grows. In addition, in NFM stars, the second crit ical density decreases as\nthe coupling constant increases. The range of scalarization in FM ne utron stars is greater\nthan NFM ones. For both NFM and FM stars, the maximum value of the central scalar ﬁeld\nreduces as the coupling constant increases. We showed that the d eviation of STT from GR\nin the mass-radius relation is more signiﬁcant with lower values of the c oupling constant.\nBesides, this deviation in the mass-radius relation as well as the scala rization are seen in a\ngreater region when the coupling constant takes lower values. We f ound that the deviation\nof STT from GR is more signiﬁcant for NFM stars compared to FM ones . It means that the\nEOS of neutron matter aﬀects the amount of deviation of STT from GR. For NFM neutron\nstars, the scalarized ones are larger in size compared to the GR solu tions. Moreover, for\nFM neutron stars, the mass of scalarized stars is lower than the on es in GR. In FM neutron\nstars, the maximum value of the coupling constant at which the star s are scalarized is higher\ncompared to the NFM neutron stars. Our results verify that the c ompactness at which the\nscalar charge becomes nonzero is greater for NFM neutron stars compared to FM ones. Our\nwork determines the magnetic eﬀects of neutron stars on the pro perties of these stars in the\nSTTs, i.e. the proﬁle of scalar ﬁeld, the scalarization and its critical d ensities, the scalar\ncharge, and the deviation of STT from GR. In fact, we conclude tha t when one considers\nthe neutron stars in ferromagnetic phase within the STTs, it is nece ssary to note that the\nneutron star EOS has signiﬁcant eﬀects on the behaviour of these stars in STTs. 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Mendes and N. Ortiz, A Mathematica [Wolfram Rese arch, Inc., Mathematica\n10.2, Champaign, IL (2015)] notebook containing routines t o construct equilibrium solu-\n24tions in STTs and complementary plots to this paper is availa ble at the public repository,\nhttps://bitbucket.org/nestor ortiz/ highly compact neut ron starsinstts.git.\n[42] M. Salgado, D. Sudarsky, and U. Nucamendi, Spontaneous scalarization , Phys. Rev. D 58\n(1998) 124003 [arXiv:gr-qc/9806070].\n[43] T. Damour and G. Esposito-Farese, Tensor-scalar gravity and binary-pulsar experiments ,\nPhys. Rev. D. 54(1996) 1474 [arXiv:gr-qc/9602056].\n[44] P. C. C. Freire, N. Wex, G. Esposito-Farese, J. P. W. Verb iest, M. Bailes, B. A. Jacoby, M.\nKramer, I. H. Stairs, J. Antoniadis, and G. H. Janssen, The relativistic pulsarwhite dwarf\nbinary PSR J1738+0333 II. The most stringent test of scalartens or gravity , Mon. Not. Roy.\nAstron. Soc. 423(2012) 3328 [arXiv:1205.1450].\n25"    },    {        "title": "2105.04740v1.Pressure_enhanced_ferromagnetism_in_layered_CrSiTe3_flakes.pdf",        "content": "Pressure-enhanced ferromagnetism in layered CrSiTe3 flakes Cheng Zhang+1,2, Yue Gu+3, Le Wang+2, Lianglong Huang2, Ying Fu2, Cai Liu2, Shanmin Wang2, Jia-Wei Mei*2,5, Xiaolong Zou*3, Jun-Feng Dai*2,4 1. School of Physics, Harbin Institute of Technology, Harbin, 150001, People's Republic of China 2. Shenzhen Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China 3. Shenzhen Geim Graphene Center, Tsinghua- Berkeley Shenzhen Institute & Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, P. R. China 4. Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen 518055, China 5. Shenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices, Southern University of Science and Technology, Shenzhen 518055, China  + The authors contribute to this work equally * Corresponding authors:  daijf@sustech.edu.cn; xlzou@sz.tsinghua.edu.cn; meijw@sustech.edu.cn  Abstract: The research on van der Waals (vdW) layered ferromagnets have promoted the development of nanoscale spintronics and applications. However, low-temperature ferromagnetic properties of these materials greatly hinder their applications. Here, we report pressure-enhanced ferromagnetic behaviours in layered CrSiTe3 flakes revealed by high-pressure magnetic circular dichroism (MCD) measurement. At ambient pressure, CrSiTe3 undergoes a paramagnetic-to-ferromagnetic phase transition at 32.8 K, with a negligible hysteresis loop, indicating a soft ferromagnetic behaviour. Under 4.6 GPa pressure, the soft ferromagnet changes into hard one, signalled by a rectangular hysteretic loop with remnant magnetization at zero field. Interestingly, with further increasing pressure, the coercive field (𝑯𝒄) dramatically increases from 0.02 T at 4.6 GPa to 0.17 T at 7.8 GPa, and the Curie temperature (𝑻𝒄𝒉: the temperature for closing the hysteresis loop) also increases from ~36 K at 4.6 GPa to ~138 K at 7.8 GPa. The influences of pressure on exchange interactions are further investigated by density functional theory calculations, which reveal that the in-plane nearest-neighbor exchange interaction and magneto-crystalline anisotropy increase simultaneously as pressure increases, leading to increased 𝑯𝒄 and 𝑻𝒄𝒉 in experiments. The effective interaction between magnetic couplings and external pressure offers new opportunities for both searching room-temperature layered ferromagnets and designing pressure-sensitive magnetic functional devices.  Recently, layered ferromagnets coupled by van der Waals (vdW) force such as CrGeTe3,1 CrI3,2 and Fe3GeTe2,3, 4 have gained significant interest, because they not only provide an ideal platform to study the fundamental physics of magnetic interaction in two-dimensional (2D) limit, but also offer great potential in the nanoscale spintronics applications. So far, the intrinsic Curie temperatures (Tc) of known 2D ferromagnets are much lower than room temperature (CrI3, 45K; GrGeTe3, 30 K; Fe3GeTe2, 68 K),1-3 hence, considerable efforts have been devoted to increasing their Tc and making the long-range ferromagnetism more robust. Naturally, enhancing the ferromagnetic exchange interaction can contribute to the increase of Tc.5 For 2D ferromagnets, magneto-crystalline anisotropy breaks the spin-rotational symmetry and can also stabilize 2D Ising-type ferromagnetism. Therefore, searching for new routes to improve these two magnetic interactions is of great significance to the applications of 2D ferromagnets. Compared with bulk cases, the magnetic states of 2D ferromagnets show greater controllability by external stimuli,3, 6-11 which may improve their magneto-crystalline anisotropy and exchange interaction, thereby expanding their applications and deepening our understanding of the underlying mechanism of magnetism in 2D systems. Hydrostatic pressure is one of the practical and easy-to-implement routes for controlling 2D magnetic states12-16. It can effectively tune the distance between adjacent atomic layers, and bond lengths and bond angles of intralayer atoms in 2D magnets, leading to modulation of interlayer and intralayer exchange interactions as well as magneto-crystalline anisotropy. Experimentally, for bilayer CrI3,7, 17 hydrostatic pressure can switch the magnetic orders between antiferromagnetic and ferromagnetic states. Under pressure, Fe3GeTe2 shows the reduction of the local moment on Fe ions as well as pressured-induced increase of the electronic itinerancy.18, 19 Moreover, a spin reorientation is reported in layered ferromagnetic insulator CrGeTe3 under pressure,20 highlighting the effective control of magneto-crystalline anisotropy by pressure. With similar crystal structure as CrGeTe3 (Figure 1a),  CrSiTe3 also exhibits soft ferromagnetic behavior without any remnant magnetization under zero field.21  After the replacement of Ge by smaller Si, the application of hydrostatic pressure should further shorten the distance between 3d atoms. Therefore, the exchange interaction and the magnetic behaviors could be greatly influenced given the small bandgap of CrSiTe3.22 Meanwhile, it has been reported that ferromagnetic CrSiTe3 undergoes a transition to superconducting phase under a pressure of ~7.5 GPa,23 which makes a detailed understanding of  the evolution of magnetic behaviors very important.  In this work, a CrSiTe3 flake was capsulated in a diamond anvil cell (DAC) with pressure up to 10 GPa and the ferromagnetic behaviours under pressure were investigated via in-situ high-pressure magnetic circular dichroism (MCD) microscopy. We found that CrSiTe3 experiences a transition from soft to hard ferromagnetic states with a remanent magnetization at zero field appearing beyond 4.6 GPa. Importantly, a further increase of pressure up to 7.8 GPa dramatically increases the coercive field (𝐻#) and Curie temperature (𝑇#$), indicating the enhancement of magneto-crystalline anisotropy and exchange interaction. Our density functional theory (DFT) calculations also reveal enhanced magneto-crystalline anisotropy energy and in-plane nearest-neighbour exchange interaction (J1) as a function of pressure, corroborating experiment observations.  High pressure experiments were performed in a DAC with pressure up to 10 GPa. In order to load sample into DAC, we first mechanically exfoliated CrSiTe3 flakes from signal crystal with a size larger than 50×50\t𝜇𝑚. However, the sample thickness was in the scale of nanometer (~100 nm). Then they were transferred onto the culet of diamond using PC films. The entire sample was then covered by BN flakes to prevent sample degradation. After filling the pressure-transmitting medium and ruby balls, the entire DAC was mounted onto the cold-finger of low-temperature cryostat with temperature in the range between 16 K and 300 K. The out-of-plane ferromagnetic states were probed using in situ high-pressure MCD spectroscopy. Figure 1b shows a schematic diagram of experimental setup for MCD measurement in a reflection geometry under high-pressure environment. The details about this technique can be found in method section. High quality CrSiTe3 single crystals were grown by the flux method (see method for details). The representative sample size was around 5×7\t𝑚𝑚% as shown in the inset of Figure 1c. CrSiTe3 has a 2D honeycomb layered structure coupled by vdW force (Figure 1a). The XRD results in Figure 1c and Figure S1 indicate the high quality of our sample. And the magnetic susceptibility (c) measurement in Figure 1d reveals a ferromagnetic order (FM) with Curie temperature Tc of 32.8 K. The value of cc is more than four times cab at 2 K, which suggests that the crystallographic c-axis is the easy axis, consistent with the previous report.24 The MCD measurement also shows a saturated magnetization as the applied field increase above 0.05 T at 16 K (Figure 1e) due to the ferromagnetic properties. However, a hysteresis loop is absent in MCD measurement when we sweep the field back and forth. It indicates a soft ferromagnetic behavior in bulk CrSiTe3, which has a similar magnetic behavior as that of CrGeTe39. Overall, at ambient pressure, CrSiTe3 exhibits a quasi-2D Ising ferromagnetic orders below a Curie temperature of ~32.8 K with a negligible coercivity.   The pressure-induced MCD changes will reveal more important information about interlayer or intralayer exchange interaction due to the shrink of distance between magnetic atoms. To study the evolution of magnetic properties of CrSiTe3, we measure the MCD signals in a CrSiTe3 flake at a temperature of 16 K under five representative hydrostatic pressures of 0.7, 2.5, 4.6, 6.2, and 7.8 GPa, respectively. As shown in Figure 2a, at 0.7 and 2.5 GPa, the magnetic behaviour of CrSiTe3 is characterized by a linear relation between the magnetization and the applied magnetic field below ±0.15 T and a saturated magnetization above that value. However, no measurable hysteresis loop, namely, remanent magnetization at 𝜇&𝐻=0\t𝑇, can be found under two pressures. Although the saturated magnetization indicate ferromagnetic behaviour due to spin-spin exchange interaction, no hysteresis loop suggests that the barrier between two spin polarized states is very low. So this is considered as the soft ferromagnetic state. Surprisingly, at 4.6 GPa, a typical rectangular hysteresis loop appears as applied field scans in the same range. The average coercive field is evaluated to be 0.02 T at 16 K. Because of the appearance of remnant magnetic order at zero field, we consider the corresponding magnetic state of CrSiTe3 as the hard ferromagnetic state. As pressure increases further, the area of hysteresis loop increases and the corresponding coercive field also increases from 0.02 T at 4.6 GPa to 0.17 T at 7.8 GPa. Above 7.8 GPa, we cannot detect any MCD signal due to a structural transition at ~ 7.5 GPa accompanied by a ferromagnetic-paramagnetic phase transition11 (data are not shown). As summarized in Figure 2b, the coercive field dramatically increases as pressure increases, with a threshold pressure around 4 GPa. Since the coercive field 𝐻# is proportional to the easy-axis single-ion magnetic anisotropy, we infer that the uniaxial anisotropy energy in CrSiTe3 dramatically increases above threshold pressure, which is significantly different from the magnetic behaviors of Fe3GeTe2 under pressure7. Figure 3 shows the temperature-dependence of MCD measurement as a function of magnetic field under different pressures. At 0.7 and 2.5 GPa, the MCD signals gradually change from a linear relation with saturation to a pure linear relation within ±0.3 T as the temperature increases. This indicates a transition from ferromagnetic to paramagnetic states, consistent with susceptibility measurement at normal pressure. From the temperature dependence of MCD signals, we can roughly extract that the transition temperature at two pressures is ~ 32 K at 0.7 GPa (Figure 3a) and ~ 34 K at 2.5 GPa (Figure 3b), respectively. Compared with the Curie temperature at normal pressure, the negligible difference of transition temperature indicates the weak influence of pressure on magnetic behaviour. According to mean field theory for ferromagnetic Ising model, the critical transition temperature is proportional to spin-spin exchange interaction (J).  Obviously, J is not dramatically changed in this pressure region. For the hard ferromagnetic state at 4.6 GPa (Figure 3c), the hysteretic loop can remain until 40 K. Here, we define the temperature for the closure of hysteresis loop as the Curie temperature for hard ferromagnetic state (𝑇#$) and mainly discuss the influence of pressure on it. As pressure increases further, the hysteresis loop can survive up to higher temperature, which is evaluated to be 100 K at 6.2 GPa (Figure 3d) and 138 K at 7.8 GPa (Figure 3e), respectively. Since the proportional relation between Curie temperature and J in Ising model, the dramatically increases of 𝑇#$ indicates the inter- or intra-layer exchange interaction is enhanced by passing through a threshold pressure (around 4 GPa). Experimentally, the increased pressure mainly decreases the interlayer atomic distance and intralayer bond length and angle between atoms, which correspond to tune the interlayer and intralayer exchange interaction. However, we cannot separate them simply based on our experimental measurements. To understand the origin of increasing Tc and 𝐻# under pressure, DFT calculations were employed to investigate the response of structural, electronic, and magnetic properties of CrSiTe3 to external pressure. CrSiTe3 crystallizes in the R3__ space group, with the monolayers stacked in ABCABC sequence, as shown in Figure 1a. To describe possible structural transition, we also consider two other stacking orders, i.e., AA and inclined AA sequences illustrated in the insets of Figure S2, both of which correspond to C2/m space group. The energies of these three structures as a function of pressure is shown in Figure S2. It is noted that even when the pressure increases to 12 GPa, the energies of these two C2/m phases are still higher than that of R3__ phase. Therefore, we focus on the R3__ phase hereafter.  To explore the magnetic properties of CrSiTe3 under pressure, we have considered various magnetic orders, including ferromagnetic (FM), interlayer antiferromagnetic (AFM) with intralayer FM coupling, and three intralayer AFM with interlayer FM coupling. The four AFM states are named as interlayer-AFM (iAFM), Neel-AFM (nAFM), Stripy-AFM (sAFM), and Zigzag-AFM (zAFM). The last three states are shown in Figure 4a, according to their different intralayer arrangement for local magnetic moments. The calculated pressure-dependent energies for these states by DFT+U method are displayed in Figure 4b, which clearly shows that FM is always the ground state. More importantly, the energy difference between FM and AFM orders generally increases as pressure increases, which suggests that the stability of ferromagnetism is enhanced by pressure.  The evolution of band structures under pressure is shown in Figure S3, along with the orbital projection to Cr and Te atoms. The top valence bands and bottom conduction bands are contributed by Te atoms and hybrid Cr and Te orbitals, respectively. Under pressure, top valence bands shift up significantly, whereas bottom conduction bands move down slightly. Accordingly, the metallicity mainly originates from Te orbitals, and the Mott and magnetic behaviors remain largely unchanged. To give a quantitative estimation of Tc of CrSiTe3, we adopted mean-field theory (MFT). Based on the Heisenberg Hamiltonian 𝐻=-𝐽'(𝑺'·𝑺(',(, where Si is the spin operator of Cr on site i (~ 3/2 in our case), and Jij is the exchange coupling constant between spins on site i and j, Tc can be evaluated as 𝑇#=−%*(*,-)/0!2-1∑𝐽'('2(4 with n the number of neighboring sites.3 Here, the intralayer first, second and third nearest-neighbor exchange interactions J1, J2, and J3, as well as interlayer first nearest-neighbor exchange interaction J4 were considered, as illustrated in the insets of Figure 4c. As shown in Figure 4c, J2 having the smallest absolute value varies in a narrow range between -0.5 and 0.5 meV. In contrast, J1, J3 and J4 are always negative, which tend to stablize FM order. Under pressure, J3 shows a slow and monotonic decline, while J1 and J4 decrease rapidly after 4 GPa, with a slight increase for J4 after 8 GPa. In particular, the absolute values of J1 and J4 are much larger than those of J2 and J3, with J1 always stronger than J4. Given the number of neighboring sites for J1 (n = 3) is larger than that for J4 (n = 1), the contribution of J1 in the Hamiltonian is much greater than J4. Therefore, the interlayer first nearest-neighbor exchange interaction J1 plays a dominant role in the enhancement of ferromagnetism. Except for the intial slight decrease by about 19.4 K below 3 GPa, Tc increases rapidly, reaching 325K under 8 GPa, which is about 132 K higher than that under 0 GPa. Although the MFT method generally overestimates Tc,25 the obtained variation of Tc is consistent with our experimental results from 0 to 7.8 Gpa. The increase of FM interaction under pressure can be qualitatively understood by the change in the gap between spin-up unoccupied eg and occupied t2g orbitals (see schematic in Figure 4d for virtual hopping), which shows a monotonic decrease versus pressure (Figure S4), giving rise to enhanced FM exchange coupling.26 Besides DFT+U method, we also adopted the HSE06 functional to estimate Tc under different pressures by considering intralayer J1, J2, J3 only to reduce the computational cost, and the results shows similar trend for Tc, as shown in Figure S5. The experimentally observed change in 𝐻# can be rationalized by analysing magneto-crystalline anisotropy energy (MAE), which is defined as the energy difference between states with magnetic moments along in-plane and out-of-plane directions, respectively. The positive MAE indicates easy-axis magnetization, while negative one suggests easy-plane magnetization.  Our calculations shown that MAE is highly sensitive to the Hubbard U parameter as shown in Table S1. Thus, we adopted the HSE06 method instead to calculated MAE, as shown in Figure 4e. It can be clearly seen that as the pressure increases, MAE keeps positive and increases monotonically from 0.39 meV/f.u. at 0 GPa to 0.46 meV/f.u. at 6 GPa, followed by a slight decrease. These results are consistent with the enhanced coercive field and harder ferromagnetic behaviors under pressure in our experiment. In conclusion, we have demonstrated a behaviour of soft ferromagnetic to hard ferromagnetic transition in CrSiTe3 flakes by means of hydrostatic pressure. The rectangular hysteretic loop, signed by a remnant magnetization at zero field, appears above 4.6 GPa, with increased coercive field (𝐻#) and Curie temperature (𝑇#$) as pressure increases further. Our DFT calculations justified that the in-plane nearest-neighbor exchange interaction and magneto-crystalline anisotropy contribute to increased 𝐻# and 𝑇#$ in experiments. Our results will be helpful in understanding the underlying mechanism of magnetic exchange interaction and open up an novel route to effectively tune magnetic properties in 2D materials. Acknowledgement We would like to thank Prof. Haizhou Lu from SUSTech for helpful discussions. J.F. acknowledges the support from the National Natural Science Foundation of China (11974159) and the Guangdong Natural Science Foundation (2021A1515012316). J.W.M was partially supported by the program for Guangdong Introducing Innovative and Entrepreneurial Teams (No. 2017ZT07C062), and Shenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices (No. ZDSYS20190902092905285). J.W.M and L.W. was supported by Guangdong Basic and Applied Basic Research Foundation (No. 2020B1515120100). L.W. was supported by China Postdoctoral Science Foundation (2020M682780). Theoretical part was supported by the National Natural Science Foundation of China (11974197, 51920105002), Guangdong Innovative and Entrepreneurial Research Team Program (No. 2017ZT07C341). Conflict of Interest  The authors declare no conflict of interest. Keywords  2D ferromagnetic materials, exchange interaction, high-pressure magnetic circular dichroism (MCD) spectroscopy Methods Sample growth CrSiTe3 crystals were grown by using the Si-Te eutectic as flux. High-purity elements Cr grains (99.996%), Si pieces (99.9999%), Te blocks (99.9999%) were weighed in the molar ratio Cr:Si:Te = 1:2:6, and placed in an alumina crucible, then sealed in a fully evacuated quartz tube. The crucible was heated to 1373 K and dwell for 10 hrs, then cooled slowly to 973 K in 150 hrs, where the flux was spun off by a centrifuge. The single crystal XRD and powder (obtained by grounding the single crystals) XRD were performed on a Rigaku Smartlab-9kW diffractometer with Cu Kα radiation (lKa1 = 1.54056 Å) at room temperature. The magnetic susceptibility (c) was performed on a Quantum Design MPMS3 SQUID magnetometer. Magnetic circular dichroism measurements under high pressure The ferromagnetic behaviours of CrSiTe3 flake under high pressure were measured in in situ magnetic circular dichroism (MCD) system (Figure 1b). Here, a 632.8 nm HeNe laser was selected as an excited light, and focused onto sample within DAV by a long working distance objective with spot size of around 2 μm. The polarized state of excited beam was modulated by a photoelastic modulator (PEM), so that it changed between left-handed and right-handed circularly polarization with a frequency of 50 KHz. The reflected MCD signal, namely the intensity difference between two circularly polarized lights, is collected by a Si detector and recorded by an lock-in amplifier. During measurements, the out-of-plane magnetic field, generated from a supper-conducting loop, was scanned within 0.3 T. Hence, we can get the MCD intensity as a function of applied magnetic field. Density functional theory (DFT) first-principles calculations  Our density functional theory (DFT) calculations were done adopting the Vienna ab initio Simulation Package (VASP),27, 28 with Perdew–Burke–Ernzerhof parameterization of the generalized gradient approximation (GGA)29 and projector augmented wave (PAW) method.30 describing the exchange-correlation and ion-electron interactions, respectively. A cutoff energy of 500 eV was chosen for plane waves basis set. The structural optimization was performed until the force on each atom and energy were converged to 0.01 eV/Å and 1 ×10-6 eV, respectively. Van der Waals (vdW) interaction was included using the optB86b-vdW functional.31 The DFT+U method was used to treat the onsite coulomb interaction of Cr d orbitals properly.32 Similar as CrGeTe3,1 the choice of Hubbard U is crucial to determine the electronic and magnetic properties of CrSiTe3. We tested different values of U (Table S1), and finally chose U = 0.5 to ensure the appropriate ground state of CrSiTe3, which shows the correct ground magnetic structure, i.e., interlayer ferromagnetic coupling and the easy-axis magnetization along the out-of-plane direction. With 6 × 6 × 2 k-point sampling, the optimized bulk lattice constants are a = b = 6.81 Å, and c = 20.48 Å. In addition, Hybrid functional (HSE06) with 3 × 3 × 1 k-point sampling was used for the calculation of magneto-crystalline anisotropy energy.33     FIGURES AND CAPTIONS \n Figure 1: (a) Side view of lattice structure of CrSiTe3. The red arrows indicate the magnetization direction. (b) Schematic of in situ high-pressure magnetic circular dichroism (MCD) experimental setup with the lowest temperature of 16 K and the highest pressure of 12 GPa, respectively. (c) The single crystal X-ray diffraction pattern of CrSiTe3. Inset is the optical image of single crystal CrSiTe3. (d) Magnetic susceptibility of CrSiTe3 measured on a single crystal with field cooling (FC) and zero-field cooling (ZFC) modes under a magnetic field of 2000 Oe applied along the c-axis and ab-plane, respectively. (e) Magnetization of a CrSiTe3 flake as a function of magnetic field (H) along c axis, measured at 16 K at normal pressure.   \n Figure 2: (a) MCD signals as a function of applied magnetic field at 16 K as pressure changes from 0.7 GPa to 7.8 GPa. (b) Extracted average critical field for spin-flip transition as a function of applied pressures.  \n Figure 3: Temperature dependence of MCD signals in the CrSiTe3 flake under five pressures, namely (a) 0.7, (b) 2.5, (c) 4.6, (d) 6.2 and (e) 7.8 GPa, respectively. The red arrows indicate the temperature for closure of hysteresis loop at 4.6, 6.2 and 7.8 GPa, respectively.    \n Figure 4: (a) Ferromagnetic and three antiferromagnetic orders: Neel-AFM, Stripy-AFM, and Zigzag-AFM. (b) The calculated energy difference between different magnetic orders and ground FM state as a function of pressure.  (c) The calculated intralayer first, second and third nearest-neighbor exchange interactions J1, J2, J3, interlayer nearest-neighbor exchange interactions J4 (illustrated in the insets), and Curie temperature Tc as functions of pressure. (d) Schematics for local coordination and virtual hopping between Cr-d orbitals intermediated by Te-p orbitals. 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"    },    {        "title": "1602.01976v1.Quantum_transport_in_ferromagnetic_graphene__Role_of_Berry_curvature.pdf",        "content": "Quantum Transport in Ferromagnetic Graphene : Role Of Berry Curvature\nDebashree Chowdhury\u0003, Banasri Basuy1\n1Physics and Applied Mathematics Unit, Indian Statistical Institute,\n203 Barrackpore Trunk Road, Kolkata 700 108, India\nThe magnetic e\u000bects in ferromagnetic graphene basically depend on the principle of exchange inter-\naction when ferromagntism is induced by depositing an insulator layer on graphene. Here we deal\nwith the consequences of non-uniformity in the exchange coupling strength of the ferromagnetic\ngraphene. We discuss how the in- homogeneity in the coordinate and momentum of the exchange\nvector \feld can provide interesting results in the conductivity analysis of the ferromagnetic graphene.\nOur analysis is based on the Kubo formalism of quantum transport.\nGraphene [1] is a two dimensional material consisting of a single layer of carbon atoms arranged in a honeycomb or\nchicken wire structure. The familiar pencil-lead, which is known as graphite, consists of layers of carbon atoms tightly\nbonded in the plane. This graphite layers are graphene and it is the thinnest as well as strongest material known\ntill now. Graphene can conduct electricity as e\u000eciently as copper and outperforms all other materials as a conductor\nof heat. Graphene is almost completely transparent, yet so dense that even the smallest atom helium cannot pass\nthrough it. Unlike in ordinary semiconductors, the \fgure of the dispersion relation is cone like which meet at a point,\nthe Dirac point. The energy-momentum plot of quasiparticles behaves as if they were massless electrons, so-called\nDirac fermions, that travel at a constant speed with a small but noteworthy fraction of the speed of light. Undoped\ngraphene has a Fermi energy coinciding with the energy of the conical points. This have completely \flled valence\nband and empty conduction band and there exists no bandgap in between and as such graphene is an example of\ngapless semiconductor and the Hamiltonian near KandK0points can be written as HK=\u0000i~v~ \u001br;HK0=HT\nK;\nwhere~ \u001bare Pauli matrices. This form of the Hamiltonian is a two dimensional analogue of the Dirac Hamiltonian\nof massless fermions but instead of c, we have Fermi velocity vF(\u0019c=300). The ultra \rat geometry, high electron\nmobility and excellent intrinsic transport properties make graphene a unique material in condensed matter physics.\nBesides, the long spin \rip length of graphene makes it a promising candidate for spintronic applications.\nThe importance of ferromagnetism in industry and modern technology is well known. The ferromagnetism is the\nbasis for many electrical and electro-mechanical devices, like electromagnets, electric motors, generators, transformers\nand also the magnetic storag devices as tape recorders and hard disks. After the observation of graphene in isolation it\nwas very natural for the scientists to search ferromagnetism in graphene. There are variety of ways for the experimental\nrealization of magnetized graphene or more precisely graphene with spin imbalance. There may exists some intrinsic\nferromagnetic correlations in graphene. Use of an insulating ferromagnetic substrate or adding a magnetic material or\nmagnetic dopants or defect on top of the graphene sheet may be other options to achieve ferro-magnetism in graphene.\nIn particular, by depositing ferromagnetic insulating layer (FI) on graphene, magnetization is induced through the\nexchange proximity interaction (EPI). Induction of large exchange splitting has been demonstrated by depositing\nferromagnetic insulator EuO on graphene [2].\nOn the otherhand, study of the gauge \felds in spintronics [3, 4], a study of the quantum mechanical spin property\nof carriers and its application to technology, is also a topic of recent attraction. Although there are many paths and\ntechniques to study the transport of spin through any solid, advantage in working with gauge \felds help us to extend\nthe usual electric and magnetic \felds analysis in the richer realm of gauge \felds. The Berry phase, a fundamental\nresult in quantum mechanics, has important consequences in spintronics. The Bery phase results from cyclic, adiabatic\ntransport of quantum states with respect to any parameter (eg. real space coordinate vector ~ r, or the momentum\nspace vector ~k). The discovery of intrinsic spin Hall e\u000bect triggered the importance of Berry phase theory in the\ncontext of spin orbit coupling. Attention was also paid to \fnd the e\u000bects of ~kspace gauge \felds in graphene, optics\nand exciton systems.\nThe induced magnetic e\u000bects in graphene which basically rely on the principle of exchange interaction is a topic of\nrecent interest. In this paper, we focus on the non-uniform exchange coupling and have studied the physics of Hall\nconductivity of a spin-orbit coupled (SOC)system via the Kubo formula approach. The exchange vector may be a\nfunction of coordinate, momentum or time. There is an experimental demonstration which shows that the exchange\ncoupling in case of the deposition of FM insulators on graphene is momentum dependent. Moreover, space dependent\nexchange \feld is used to study the spin lens con\fguration [5]. The motivation of this paper is to present the role of\n\u0003Electronic address:debashreephys@gmail.com\nyElectronic address:sribbasu@gmail.comarXiv:1602.01976v1  [cond-mat.mes-hall]  5 Feb 20162\nBerry curvature in the analysis of the Hall conductivity for the inhomogeneous exchange coupling and momentum\ndependent exchange coupling.\nThe organization of the paper is as follows: Hamiltonian of the model for the ferromagnetic graphene is developed\nin section I. Section II deals with the discussion of the spin conductivity analysis for a momentum dependent exchange\nvector showing the importance of the Berry curvature in this context. The next section contains the derivation of the\nBerry curvature for space dependent exchange coupling. Finally we summarize in the last section.\nI. THE HAMILTONIAN\nWe consider a thin insulating ferromagnetic material deposited on the top of a graphene sheet with substrate\ninduced SOC so that the semiclassical theory of spin Hall e\u000bect in an undoped ferromagnetic graphene can be\ndeveloped through the Hamiltonian [2]\nH=vF~ \u000b:~k+EF+~ \u001b:~h+V(r) +\u0015G[~ \u001b\u0002~rrV(r)]:~k; (1)\nwhere\u000bis equal to the unit matrix in spin space. vFis the Fermi velocity, ~ \u001b= (\u001bx:\u001by;\u001bz) is the Pauli matrix and\nthe Fermi energy is given by EF. The second term indicates the exchange Hamiltonian ( Hex) due to the interaction\nbetween the local magnetization of the ferromagnet and the surface Dirac fermions and ~his the exchange energy\nvector.V(~ r) is the total potential present in the system which includes the potential due to external electric \feld and\ncrystal lattice potential Vcrys. The last term denotes the spin-orbit coupling term.\nIn the absence of an external magnetic \feld, in undoped ferromagnetic graphene ( EF= 0), with a constant exchange\nenergy a speci\fc type of charge Hall e\u000bect has been predicted [2]. In this case, the charge Hall e\u000bect is generated by\nspin Hall mechanism. Within the semiclassical theory of spin-orbital dynamics of carriers, a longitudinal electric \feld\nproduces a pure charge transverse current with no polarization of spin.\nOur motivation here is to study the spin-orbital dynamics of the carriers for the momentum dependent exchange\ncoupling within the semi-classical framework. We can write the Hamiltonian of the system with momentum dependent\nexchange \feld as\nH=vF~ \u000b:~k+EF+~ \u001b:~h(~k) +V(r) +\u0015G[~ \u001b\u0002~rrV(r)]:~k (2)\nCollecting only the dynamical terms, we can rewrite the Hamiltonian as\nH(~k) =EG(~k) +EF+~ \u001b:(~h+~ m)(~k) =\u000f(~k) +~ \u001b:~M(~k); (3)\nwhere\u000f(~k) =EG(~k) +EF; ~ m(~k) =\u0015G~rrV(r)\u0002~kand~M(~k) = (~h+~ m)(~k):We do our analysis for exchange energy\n~ \u001b:~h(~k)<Fermi energy EF;such that\u000bis equal to the unit matrix in the spin space and the carriers are electron like\nwith up and down spin.\nII. CONDUCTIVITY ANALYSIS\nThe generalised spin orbit Hamiltonian (3) with momentum dependent exchange can now be used to derive the\nHall conductivity using the Kubo method and it can be shown that the conductivity is intimately connected to the\n~k\u0000space Berry curvature.\nWe here use the Matsubara Greens function technique such that the conductivity can be obtained from the Kubo\nformula [6] as\n\u001bxy= lim\n!!0i\n!Qij(!+i\u000e); (4)\nwhere\nQij=1\n\n\fX\nk;ntrfJi(~k)G[k;i(!n+\u0017m)]Jj(~k)G(~k;i!n)g: (5)\nG(~k;i!n) is the single particle Greens function, \n is the system area, Ji(~k)s are the current operator. !nand\u0017mare\nthe fermionic and bosonic Matsubara frequencies and can be expressed as !n= (2n+ 1)=\fand\u0017m= 2m\u0019=\f . The3\nsingle particle Greens function is given by\nG(~k;i!n) = (i!n\u0000H(~k))\u00001\n=1\n2[1 +Mi(~k)\u001bi]=M\ni!n\u0000\u000f(~k)\u0000M+1\n2[1\u0000Mi(~k)\u001bi]=M\ni!n\u0000\u000f(~k) +M\n=R+\ni!n\u0000E+(~k)+R\u0000\ni!n\u0000E\u0000(~k)(6)\nwhereM=q\nMi(~k)Mi(~k); R\u0006=1\n2[1 +Mi(~k)\u001bi=M] andE\u0006(~k) =\u000f(~k)\u0006M:\nUsing the Hamiltonian (4), the current operator Ji(~k) can be derived as\nJi(~k) =@H(~k)\n@ki=@\u000f(~k)\n@ki+@Mj(~k)\n@ki\u001bj; (7)\nwherei;jare the space indices.\nQij(i\u0017m) is then given by\nQij(i\u0017m) =1\n\nX\ns;t=\u0006X\nktr[Ji(~k)Rs(~k)Jj(~k)Rt(~k)]\ni\u0017m\u0000Es(~k) +Et(~k)(nt\u0000ns)(~k); (8)\nwhere (nt\u0000ns)(~k) = (nF(Et(~k))\u0000nF(Es(~k));andnF(\u000ft;s(~k)) denotes the Fermi distribution function. With the help\nof Matsubara sum, the expression of \u001bijcan now be recast as\n\u001bij= lim\n!!0i\n!Qij(!+i\u000e): (9)\nInserting the value of Qxyin the expression we can write\n\u001bij= lim\n!!0i\n!1\n\nX\ns;t=\u0006X\nktr[Ji(~k)Rs(~k)Jj(~k)Rt(~k)]\n!+i\u000e\u0000Es(~k) +Et(~k)(nt\u0000ns)\n=\u0000i\n\nX\nktr[Ji(~k)R\u0000(~k)Jj(~k)R+(~k)\u0000Ji(~k)R+(~k)Jj(~k)R\u0000(~k)]\n(E+(~k)\u0000E\u0000(~k))2(n+\u0000n\u0000)\n=\u0000i\n\nX\nktr[Ji(~k)R\u0000(~k)Jj(~k)R+(~k)\u0000Ji(~k)R+(~k)Jj(~k)R\u0000(~k)]\n4M2(n+\u0000n\u0000)(~k)\n=\u0000i\n\nX\nkD(~k)\n4M2(n+\u0000n\u0000)(~k); (10)\nwhereD(~k) =tr[Ji(~k)R\u0000(~k)Jj(~k)R+(~k)\u0000Ji(~k)R+(~k)Jj(~k)R\u0000(~k)]:Explicitly in terms of the exchange vector, we\ncan write\nD(~k) =1\n2tr[(@\u000f(~k)\n@kx+@M\u000b(~k)\n@kx\u001b\u000b)@M\f\n@kyM\r\nM(\u001b\f\u001b\r\u0000\u001b\r\u001b\f)]\n=2i\u000f\u000b\f\r\nM@M\u000b\n@kx@M\f\n@kyM\r; (11)\nwhere\u000b;\f; \r are the space indices. Here we have used the fact that tr(\u001b\u000b\u001b\f\u001b\r\u0000\u001b\u000b\u001b\r\u001b\f) = 4i\u000f\u000b\f\r:Thus the Hall\nconductivity in the x-y plane can be written as\n\u001bxy=1\n2M3\nX\nk\u000f\u000b\f\r@M\u000b\n@kx@M\f\n@kyM\r(n+\u0000n\u0000)(~k): (12)\nThis expression shows the dependence of the Hall conductance on the exchange \feld ~h(~k) and the electric \feld due\nto the potential V(~ r) as~M(~k) = (~h+~ m)(~k):In case of constant ~h=jhj~ n;the amplitude of the spin Hall mechanism\ninduced charge Hall conductance is linearly proportional to exchange splitting h.4\nBy converting the summation into integration over the \frst Brillouin zone one can write the Hall conductivity in\nx-y plane as [6]\n\u001bxy=1\n2\nZ\nFBZcxcy\n4\u00192d2~k \n~M:@~M\n@kx\u0002@~M\n@ky!\n; (13)\nwherecxcy= \n:From the well known de\fnition of the Berry curvature in momentum space, eqn.(13) indicates that\nthe conductivity can be written as as\n\u001bxy=1\n8\u00192Z\nFBZd2k\nz(~k); (14)\nwhere \nz(~k) is the Berry curvature in momentum space.\nThe Berry curvature for spin Hall systems carries a spin dependence and is equal but opposite for spin up and down\nelectrons i.e \n\"\nz(~k) =\u0000\n#\nz(~k). Thus eqn (14) indicates that we have the spin Hall conductivity as\n\u001bsH=\u001b\"\nxy\u0000\u001b#\nxy; (15)\nwhereas the charge conductivity is given by\n\u001bc\nxy=\u001b\"\nxy+\u001b#\nxy; (16)\nwhich is e\u000bectively zero. Thus for a undoped ferromagnetic graphene system the inhomogeneity in the exchange\ncoupling can produce a pure transverse spin current with equal number of spin up and spin down electrons in our\nsystem.\nLet us now consider the situation when the external electric potential is absent. We then have V(~ r) =Vcrystal;\nonly. In that case, the Hall conductivity is given as\n\u001bxy=1\n8\u00192Z\nFBZd2~k \n~h:@~h\n@kx\u0002@~h\n@ky!\n: (17)\nwhich shows the existence of \fnite Hall conductivity due to the momentum dependence of the exchange vector.\nIt is amazing to note that in undoped ferromagnetic graphene, momentum dependence of the exchange vector can\ninduce Hall conductance even in the absence of an electric \feld. Since the momentum dependence of exchange coupling\nfor ferromagnetic graphene has already been demonstrated in an experiment [7], speci\fc choice for the momentum\ndependence may produce some impressive result.\nIII. ANALYSIS WITH SPACE DEPENDENT EXCHANGE COUPLING\nIt is known that the space dependent exchange \feld can be used to study the spin lens con\fguration. For an\ninhomogeneous exchange vector the Hamiltonian for ferromagnetic graphene can be written as\nH=vF~ \u000b:~k+~ \u001b:~h(~ r) (18)\nIn this case, we have not considered any external \feld. For the sake of simplicity the spin orbit coupling (SOC) term\nis also not considered. Actually, our motivation is to see the role of inhomogeneous exchange vector. Variation of the\nexchange \feld can induce spin Berry gauge as well. We can write this spin gauge as\nA\"#\na=\u0000i\u001c\n\"#;~h(r)j@\n@raj\"#;~h(r)\u001d\n=@ha(~ r)\n@raA\"#\na(h) (19)\nThis can also be written as\nA\"#\na(h) =\u001c\n\"#;~hj@\n@haj\"#;~h\u001d\n(20)\nwherea=i;j;k are the space indices. A\"#\na(h) is the exchange \feld dependent Berry gauge \feld appearing due to the\ninhomogeneity of the exchange \feld vector.5\nA physical \feld can be generated due to the presence of the gauge A\"#\na(h) It is quite obvious that the zcomponent\nof the curvature only exists and is given by\n\nc(~ r) =@A\"#\ny\n@x\u0000@A\"#\nx\n@y=@ha\n@x@hb\n@y(@A\"#\nb\n@ha\u0000@A\"#\na\n@hb) (21)\nFor further analysis, in a standard notation of unit vector we write the exchange vector as\n~h=h(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012 ) (22)\nwhere\u0012is the polar angle and \u001eis the azimuthal angle.\nIn the adiabatic approximation, the carriers remain in the same spin eignestates for h(r1)andh(r2);and the \ripping\nbetween states is forbidden. Thus we can write the adiabatic gauge as\nA\"#\nad(h) =\u00061\n2(1\u0000cos\u0012)rh\u001e (23)\nwhere\u0006denotes\"eigenstate parallel ( #anti- parallel) to h(~ r). Thus the ~ rspace Berry curvature is given by,\n\nc(~ r) =@ha\n@x@hb\n@y\u000fabc(\u0006hc\nh3): (24)\nThis demonstrates that in a ferromagnetic graphene, for the space dependent exchange vector, in the adiabatic\napproximation a physical \feld is generated which is the well known Berry curvature. This expression of Berry\ncurvature shows that it is the source of monopole, if they exist. Further, it can be shown that in ferromagnetic\ngraphene, if the spin orbit interaction is taken into account along with the inhomogeneous exchange vector, the Berry\ncurvature is the origin of the spin orbit torque [8].\nIV. SUMMARY\nNow we summarize our results. In ferromagntic graphene, conductivity depends on the inhomogeneity of the\nexchange vector in momentum space. In this system momentum dependence of the exchange vector can induce Hall\nconductance even in the absence of an electric \feld. We have also shown that this time reversal (TR) symmetric\nsystem is a potential candidate for pure spin current source. This study is not only mathematically interesting, it\nhas some physical consequences as momentum dependence of exchange coupling for ferromagnetic graphene is also\nexperimentally veri\fed [7]. Our analysis has shown the importance of ~k\u0000and~ r\u0000space Berry curvature in the spin\ntransport of ferromagnetic graphene.\n[1] A. H. Castro Neto, F. Guinea, N. M. R. Peres ,K. S. Novoselov and A. K. Geim, REVIEWS OF MODERN PHYSICS, 81,\n(2009).\n[2] Babak Zare Rameshti and Malek Zareyan, Appl. Phys. Lett. 103, 132409 (2013).\n[3] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004).\n[4] T Fujita, M B A Jalil and S G Tan, New Journal of Physics 12, 013016 (2010).\n[5] A. G. Moghaddam and M. Zareyan, Phys. Rev. Lett. 105, 146803 (2010).\n[6] R. Kubo, J. Phys. Soc. Jpn. 12, 570, (1957).\n[7] H. Miyazaki, T. Ito, H. J. Im, S. Yagi, M. Kato, K. Soda, and S. Kimura, Phys. Rev. Lett. 102, 227203 (2009).\n[8] Debashree Chowdhury and B. Basu, Annals of Physics 355, 338,(2015)."    },    {        "title": "2309.12497v1.Ferromagnetic_quantum_critical_point_in_a_locally_noncentrosymmetric_and_nonsymmorphic_Kondo_metal.pdf",        "content": "Ferromagnetic quantum critical point  in a locally noncentrosymmetric  and nonsymmorphic Kondo \nmetal  \n \nSoohyeon Shin1,*,†, Aline Ramires2,*,§, Vladimir Pomjakushin3, Igor Plokhikh1, and Ekaterina \nPomjakushina1 \n \n1 Laboratory for Multiscale Materials Experiments, Paul Scherrer Institut, Villigen 5232, \nSwitzerland  \n2 Condensed Matter Theory Group, Paul Scherrer Institut, Villigen 5232, Switzerland  \n3 Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, Villigen 5232 PSI, \nSwitzerland  \n \nCorresponding authors: † soohyeon.shin@psi.ch , § aline.ramires@psi.ch  \n* These authors contributed equally to this work  \n \nAbstract  \n    Quantum critical points (QCPs), zero -temperature phase transitions, are windows to \nfundamental quantum -mechanical phenomena associated with universal behaviour and can \nprovide parallels to the physics of black holes. Magnetic QCPs have been extensively investigated \nin the vicinity of antiferromagnetic order. H owever, QCPs are rare in metallic ferromagnets due \nto the coupling of the order parameter to electronic soft modes  1,2. Recently, antisymmetric spin -\norbit coupling in noncentrosymmetric  systems was suggested to protect ferromagnetic QCPs 3. \nNonetheless, multiple centrosymmetric materials host FM QCPs, suggesting a more general \nmechanism behind their protection. In this context, CeSi 2-δ, a dense Kondo lattice crystallising in \na centrosymm etric structure, exhibits ferromagnetic order when Si is replaced with Ag. We report \nthat the Ag -substitution controls the strength of the Kondo coupling, leading to a transition \nbetween paramagnetic and ferromagnetic Kondo phases. Remarkably, a ferromagne tic QCP accompanied by concurrent strange -metal behaviour emerges. Herein, we suggest that, despite the \ncentrosymmetric structure, spin -orbit coupling arising from the local noncentrosymmetric \nstructure, in combination with nonsymmorphic symmetry, can prot ect ferromagnetic QCPs. Our \nfindings present a unique example of Kondo coupling -driven ferromagnetic QCP through chemical \ndoping and offer a general guideline for discovering new ferromagnetic QCPs.  \n \nIntroduction  \n    Quantum critical point s (QCP), zero -Kelvin phase transition s that are a consequence of the \nHeisenberg uncertainty principle, exhibit exotic quantum phenomena associated with order \nparameter fluctuations controlled by non -thermal tuning parameters  4. Non-Fermi liquid (NFL) or \nstrange metallic  behaviour , a departure from the standard Fermi liquid (FL) behaviour associated \nwith normal metals, generally  appears near QCPs due to the strong scattering  between conduction \nelectrons  and quantum fluctuations  5. Strange -metal behaviour has been investigated in various \nantiferromagnetic (AFM) QCPs  6. In contrast, QCPs  in ferromagnetic (FM) compounds  are rare , \nas they are often  cut out by a first-order transition 7, or smeared out by  inhomogene ity 8 or short -\nrange order  9,10. Theory developed by Be litz, Kirkpatrick, and Votja (BKV) showed that while in \nclean systems the FM quantum phase transition (QPT) is generally first order due to the coupling \nof the magnetization to soft electronic modes, in disordered systems the QPT can be associated \nwith a s econd order phase transition, in other words, with a legitimate QCP 11,12. Indeed, it was \nobserved that the Curie temperature of the itinerant F Ms UCo 1-xFexGe 13 and Mn 1-xCrxSi2 14 is \ndriven to zero by doping, and the observed critical exponents corroborate the predictions of the \nBKV theory. Although rare, local -moment FM QCPs with concurrent NFL behaviour have been \nreported in clean systems such as CeRh 6Ge4 and YbNi 4(P1-xAsx)2 5,15,16.     More precisely, t he avoided FM QCP s have been ascribed to soft modes or massless two -particle \nexcitations of a generic Fermi liquid when spin -orbit coupling  (SOC)  is negligible 1,2. The  \ndiverging spin susceptibility of the Fermi liquid at zero temperature and magnetic field contributes \nto the fr ee energy with non -analytic terms in the magnetization, making the phase transition first \norder at low temperatures  1,2,17. However, it has been suggested that strong spin -orbit coupling  \nintroduced by the breaking of spatial inversion symmetry in noncentro symmetric materials \nintroduces a cut -off to this divergence and can restore the second -order nature of the QPT 3. Indeed, \na FM QCP has been reported for CeRh 6Ge4, a heavy fermion material that crystallizes in a \nnoncentrosymmetric hexagonal structure ( P6_\nm2, no. 187) 5. However, a FM QCP  has also been \nreported in YbNi 4(P1-xAsx)2, which adopts the centrosymmetric tetragonal structure ( P42/mnm , no. \n136) 15. A better understanding of the key ingredients for the existence of FM QCPs is highly \ndesirable.  \n    CeSi 2-δ is the first dense Kondo FM in which ordered Ce moments are reduced due to Kondo \nscreening. Figure 1a shows that Ce and Si atoms are locate d at 4a(0,3/4,1/8) and 8e(0, 1/4, \n0.2910(2) ) Wyckoff position, respectively, in the  centrosymmetric  ThSi 2-type tetragonal structure \n(I41/amd , no. 141) 18. Stoichiometric CeSi 2 exhibits a paramagnetic ground state , with the Ce \nmoments Kondo -screened by the conduction electrons,  as schematically  depicted on the left of Fig. \n1b 19. When Si -site deficiency  (δ) is larger than 0.15,  the Ce moments order ferromagnetically \nbelow T = 11 ∼ 14 K , with the Curie temperature increasing with δ 20. In addition, early pressure \nexperiments on CeSi 1.81 suggested the existence of a FM QCP near P = 13.1 kbar 21. However, the \nconcurrent NFL behaviour associated with the δ-driven putative FM QCP has not been \ninvestigated because of a structural transition  20. Nonetheless, it has been reported  that replacing \nSi with Ag also induces FM ordering  without any structural transition  18. Here, we report a  potential  FM QCP and concurrent NFL  behaviour  by controlling Ag -doping in the dense Kondo lattice \nCeSi 2-δ and suggest the importance of local noncen trosymmetricity and nonsymmorphicity for the \ngeneral stabilization of FM QCPs.  \nResults  \n    Figure 1 c is a magnetic phase diagram of Ce(Si 1−xAgx)2-δ showing  various magnetic ground \nstates, i.e., a paramagneti c (PM) state  at 0  x < 0.1, a FM state  at 0.1  x < 0.2 2, and an AFM state  \nat 0.2 3 < x  0.30. The transition temperatures were determined by kinks or peaks of a derivative \nof magnetic susceptibility , electrical resistivity  , specific heat capacity C, and imaginary part \nof AC susceptibility  AC (for details, see Extended Data Fig. E1 ). As shown in Fig. 1 b, neutron \npowder diffraction experiments reveal that the Ce moments lie within a tetragonal plane in the FM \nstate.  On the other hand,  ordered moments  are perpendicular to the tetragonal plane in the AFM \nstate (for details, see  Extended Data Fig. E2, Table E1, and the Supplementary Information SI ). \nNeutron diffraction experiments allow for an estimation of the magnitude of the ordered moments. \nFor x = 0.1, the moments are estimated to be 0.71(2) B, smaller than the expected value  of ~1.25 \nB 22 expected for  Ce ions in this crystalline environment. The reduced ordered moments  are fully \nrecovered in the antiferromagnetic state with 1.29(1) B at x = 0.35, indicating that the Kondo \nhybridization (blue spheres in Fig. 1 b) becomes weaker with increasing x. The FM order is \nsuppressed to zero -temperature  at a critical concentration  xc ~ 0.0 7, characterising a putative FM \nQCP  5,8,15. \n    Figure 1d shows the temperature -dependent magnetic susceptibility  (T) of three different \nmagnetic ground states. FM and AFM  transition temperatures are indicated by TC and TN, \ndetermined by a peak of /T and kink of (T), respectively.  In the PM state,  field-dependent magnetization  M(0H=7T) of x = 0.1 is 0.42 B, indica ting that the ordered Ce moments in the FM \nstate are reduced  to approximately 1/3 of Ce3+ moment . As shown in Fig. 1e, the fully local ised \nCe moments in the AFM state ( x  0.27) are  continuously  suppressed with decreasing x towards \nthe putative FM QCP, indicating that the Kondo hybridization is systematically controlled by Ag -\nsubstitution . The continuous suppression of Ce moments and FM transition temperature with \ndecreasing x suggests  that the  FM order  is controlled  by the Kondo couplin g strength . The size of \nthe ordered moment (empty squares) deduced from the neutron experiments is in line with the \ntendency of M(0H=7 T) (filled squares).  \n    Figure 2a shows that the peak in C/T due to the FM transition  decreases with decreasing x, as \nindicated by arrows, reaching T = 0 K  near the critical concentration  xc = 0.0 7. Bulk property \nmeasurements , C/T and (T), do not show any feature of magnetic ordering  down to 0.36 K  at x  \nxc, indicating the FM QCP at xc ~ 0.07 . As shown in Fig. 2b, large entropy accumulation is \nobserved near xc, with C/T ~ 0.6 J/mol/K2 at T = 0.36 K, comparable  to other Ce -based heavy -\nfermion compounds 5,23,24. C/T, obtained by subtracting T3 contribution from C/T, follows a \n−logT dependence, the hallmark of NFL behaviour, for x = 0.06 and 0.05, down to the lowest \nmeasured temperature. The NFL behaviour near  xc is also supported by electrical transport data. \nTemperature -dependent electrical re sistivity (T) of x = 0 and 0.07 are plotted in Fig. 2c. For x = \n0,   T2 follows the standard FL theory, while for x = 0.07,   T, another hallmark of NFL \nbehaviour in the temperature range of 0.45 K < T < 5 K . Furthermore, the transverse \nmagnetoresistance (MR) for x = 0 at T = 1 K exhibits an H2 dependence, associated with normal -\nmetal behaviour, whereas the MR for x = 0.07 shows a n H-linear dependence up to 0H = 14 T. \nSimilar behaviour has been reported for FeSe 1-xSx and La 2-xCexCuO 4 due to quantum critical fluctuations 25,26. The divergence of  C/T, as well as  T- and H-linear  at x ~ xc, indicates  that the \nFM QCP appears in the dense Kondo ferromagnet  CeSi 1.97 by Ag -substitution . \n    Figure 3a summarises the quantum critical behaviour around the FM QCP in Ce(Si 1−xAgx)1.97. \nThe upper panel of Fig. 3a shows that C/T at T = 0.34 K exhibits a peak at x ~ xc. In addition, as \nplotted on the right ordinate in the upper panel of Fig. 3a, the exponent n defining the temperature \ndependence of the resistivity, (T) = 0 + ATn, is near unity, indicating a linear -in-temperature \nbehaviour around xc. The lower panel di splays a colour plot of n, calculated from  ∂ln(ρ−ρ0)/∂lnT, \nshowing the expected funnel shape of NFL behaviour due to the magnetic quantum fluctuations, \nas reported in other FM and AFM QCPs 5,27. The FL line in the phase diagram was obtained from \nthe electrical resistivity measurements, below which (T) follows T2 dependence (for details, see \nExtended Data Fig. E3).  \nDiscussion  \n    Non-Fermi liquid behaviour can be observed in the absence of QCPs when metallic FMs are \ntuned by chemical doping 28-33 due to Griffiths phase effects 34. Chemical doping induces \ninhomogeneity to the Kondo coupling strength, giving rise to Kondo -cluster -glass behaviour \nobserved in CePd 1-xRhx 32,33 and CeCu 1-xNix 31. The scaling law of the quantum Griffiths phase is \ngiven by ’AC  C/T  T − 1, M  H, with 0 <   < 1 29. For instance, CePd 1-xRhx exhibits the NFL \nbehaviour without a diverging Grüneisen ratio, and experimental results are well explained by the \nabove scaling laws 33. However, Ce(Si 1−xAgx)1.97 near xc exhibits a different scaling behaviour, \nwith ’AC  T−1.7, C/T  -logT, and M  H0.5 (for details, see Extended Data Fig. E4 ). The absence \nof Griffiths phase is further confirmed by the absence of hysteresis between zero -field-cooled and field-cooled (T) in the temperature range where the (T) exhibits a T-linear dependence ( T < 5 \nK). \n    Recent studies of Si -substitution in the FM quantum critical matter CeRh 6Ge4 suggested that \nimpurities might play an important role in the phenomenology of the FM QCP 35. In contrast to \nthe NFL behaviour near the pressure -induced FM QCP, the electrical r esistivity near the critical \nSi concentration doesn’t exhibit the T-linear behaviour although the Curie temperature seems to \nbe suppressed continuously. At x > xc, CeRh 6(Ge 1-xSix)4 exhibits single -ion Kondo physics due to \nthe inhomogeneous Kondo screening,  which was not observed in Ce(Si 1-xAgx)1.97. The T-linear \n(T) and the absence of inhomogeneous Kondo screening near x ~ xc suggest that the effects of  the \ndisorder are negligible in Ce(Si 1-xAgx)1.97.  \n    Based on the experimental evidence supporting the  FM QCP in Ce(Si 1-xAgx)1.97, we now  address \nthe origin of its stability.  Recent work by Kirkpatrick and Belitz  3 showed  how inversion symmetry \nbreaking in (globally) noncentrosymmetric  materials can provide a loophole to avoid the general \nmechanism that transforms a FM QCP in to a first order phase transition. In noncentrosymmetric \nmaterials, antisymmetric spin -orbit coupling (SOC) splits the electronic bands in such a way that \nthe two -particle excitations that couple to the magnetic order parameter acquire a mass and \ntherefore do not contribute with nonanalytic terms to the free energy. This proposal lays out a \nnatural explanation for the observation of signatures of FM quantum criticali ty in the \nnoncentrosymetric  heavy fermions CeRh 6Ge4 5,16 and UIr  36. Nevertheless, previous work by the \nsame authors suggests that this loophole is valid only for materials that have global inversion \nsymmetry breaking  37-39. If the material is only locally  noncentrosymmetric, an effective chiral \nsymmetry  guarantees the double  degeneracy of the electronic structure and therefore the presence \nof soft modes that can couple to the magneti zation, leading to a first order phase transition in the same fashion as i n Fermi liquids without SOC. One notable exception to this conclusion is the case \nof systems for which extra lattice symmetries guarantee that the coupling between the distinct \nchirality sectors is zero 37, which  makes the material  behave effectively as tw o independent \nglobally noncentrosymmetric systems. In the presence of terms that couple the two chiral sectors, \nthe nonanalyticity is restored, but its strength is reduced by a factor of |(k)|2/|(k)|2 38,39 ((k) \ncorresponds to the coupling between the two chirality sectors and (k) to the antisymmetric  SOC). \nThe calculations of Belitz and Kirkpatrick were done within the lowest order loop expansion 39. \nMost recently, Miserev and collaborators 40 showed that second -order perturbation  theory in the \nelectron -electron interaction restores the nonanaliticities  in the spin s usceptibility, and therefore \nthe first -order nature of the FM transition in noncentrosymmetric  systems. Even though this result \nseems robust, the phenomenology presented here suggests that the prefactors accompanying such \nhigher -order nonanaliticities are much smaller than for Fermi liquids in the absence of SOC, \nresulting in a weaker effect, allow ing for the experimental observation of NFL behaviour around \nputative FM QCP in these systems.  \n    Motivated by th ese observation s, we construct ed Table I. Remarkably, most of the heavy \nfermion FMs that display a second order  phase transition and signature s of quantum criticality are \neither globally noncentrosymmetric, or locally noncentrosymmetric and nonsymmorphic. We , \ntherefore,  infer that nonsymmorphicity is a key ingredient in stabili zing these FM QCPs, as in \nnonsymmorphic materials the ratio |(k)|2/|(k)|2 can be made arbitrarily small if the Fermi surfaces \nare located close to the BZ boundaries. A rough trend of the ratio |(k)|2/|(k)|2 can be captured by \nthe atomic distances between inequivalent sublattices sites. The larger the inter sublattice distances, \nthe smaller the hopping amplitudes between the sublattices, what corresponds to a weaker coupling \nbetween sectors with opposite chir ality. For the particular case of CeSi 2, Ce and Ce’ sublattices are not centers of inversion, despite the fact that the lattice is globally centrosymmetric. The center \nof inversion lies at the midpoints between Ce and Ce’. Furthermore, the two sublattices are related \nby a nonsymmorphic symmetry, as depicted in Fig. 3c (a more detailed discussion is given in the \nSI). 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Ce ground state in CePd 1-xNix. Physica \nB: Condensed Matter  230-232, 162 -164 (1997). \nhttps://doi.org:https://doi.org/10.1016/S0921 -4526(96)00578 -9 \n49 Sidorov, V. A.  et al.  Quenching of ferromagnetism in β -UB 2C and UNiSi 2 at high pressure. \nJournal of Physics: Conference Series  273, 012014 (2 011). https://doi.org:10.1088/1742 -\n6596/273/1/012014  50 Hidaka, H., Takahashi, S., Shimizu, Y., Yanagisawa, T. & Amitsuka, H. Pressure -Induced \nQuantum Critical Point in Ferromagnet U 4Ru7Ge6. Journal of the Physical Society of Japan  \n80, SA102 (2011). https://doi.org:10.1143/JPSJS.80SA.SA102  \n  Tables  \nTable I: Summary of known heavy -fermion FM displaying either a first or second order \nphase transition (QCP).  We highlight the reported type of QCP, in addition to the space group \nof each material, drawing s pecial attention to the presence of absence of global (Glob.) and local \n(Loc.) inversion symmetry and nonsymmorphicity. In addition, the last column (D sub) gives the \ndistances between sublattices of lanthanide or actinide sites, which trend corroborates th e \ndiscussion in the main text. Exceptions of the proposed picture are highlighted by (*) and (**).  \nYbNi 4(P1-xAsx)2 and U 4Ru7Ge6, with the second order phase transition ascribed to the one -\ndimensional nature of the crystalline structure and a particularly large distance between \ninequivalent U sites relieving the need for nonsymmorphicity, respectively. Among the materials \nthat disp lay a first order phase transition, exceptions to this rule include U(Co, Rh)Ge and U(Co, \nRh)Al, which can be understood in terms of the small distances between sublattices, what enhances \nthe ISH. Another exception is U 3P4, which is globally noncentrosymme tric and nonsymmorphic, \nbut displays a first order phase transition. Despite the simple chemical formula, this system has a \nunit cell with exceptionally many atoms, which might make this simple picture not directly \napplicable.  \nMaterial  Order  Type  Space \ngroup  Inversion \nGlob./Loc.  Symm orphic  Dsub \nYb(Cu,Ir) 2Si2 41,42 1st Local  I4/mmm  Y/Y Y 3.924  \nUGe 2 43 1st Itinerant  Cmmm  Y/N Y 3.854  \nU(Co,Rh)Ge * 44 1st Itinerant  Pnma  Y/N N 3.480  \nU(Co,Rh)Al*  45,46 1st Itinerant  P6_\n2m N Y 3.394  \nU3P4 * 47 1st Itinerant  I4_\n3d N N 3.841  \n       \nCe(Pd,Pt) 1-xNix 48 2nd Local  Cmcm  Y/N N 3.868  \nCeSi 1.81 21 2nd Local I41/amd Y/N N 4.055  \nUNiSi 2 49 2nd Local  Cmcm  Y/N N 4.060  \nCeRh 6Ge4 5,16 2nd Local P6_\nm2 N Y 3.855  \nUIr 36 2nd Itinerant  P21 N N 3.637  \nYbNi 4(P1-xAsx)2** 15 2nd Local P42/mmm  Y/Y N 3.857  \nU4Ru7Ge6 ** 50 2nd Itinerant Im3_\nm Y/N Y 4.144  \n \n  Figures and captions  \nFigure 1  \n \nFig. 1 Magnetic phase diagram and Kondo coupling of Ce(Si 1-xAgx)2-. a, Crystal structure of \nCe(Si 1-xAgx)2-. Yellow, blue, and grey spheres represent Ce, Si, and Ag atoms, respectively. Ce \nand Ce’ indicate two different Ce sublattices distinguished by different local atomic environments. \nb, Schematic drawings display magnetic structures of three different magne tic ground states, and \nthe strength of Kondo screening (translucid blue spheres). ord indicates the size of the ordered \nmagnetic moments determined by neutron scattering experiments in units of Bohr magneton B. c, \nMagnetic phase transition temperatures, determined by measuring magnetic susceptibility , \nspecific heat capacity C, and electrical resistivity , are plotted as a function of Ag -doping x. PM, \nFM, and AFM represent the paramagnetic, ferromagnetic, and antiferromagnetic states, \nrespectively. d, Magnetic susceptibility for PM, FM, and AFM states is plotted as a function of \ntemperature. TC and TN indicate transition temperatures of FM and AFM . e, The ordered magnetic \nmoment (open squres) and magnetisation value (closed squares) at 0H = 7 T and T = 1.85 K are \nplotted as a function of x. The dashed line indicates the expected value for Ce3+ of Ce(Si 1-xAgx)1.9. \n  \nFigure 2 \n \nFig. 2 Non -Fermi -liquid behaviour near the FM QCP of Ce(Si 1-xAgx)1.97. a, Specific heat \ncapacity divided by temperature C/T of Ce(Si 1-xAgx)1.97 is plotted as a function of temperature. TC \nindicates the ferromagnetic transition temperature. b, C/T, obtained from subtracting T3 term \nfrom C, is plotted as a function of temperature for x near the critical concentration. The magenta \nline represents C/T  –logT at low temperatures. c, Electrical resistivity  after subtracting residual \nresistivity 0 for x = 0 and 0.07 is plotted as a function of temperature. The red lines are the least -\nsquares fitting using  – 0  Tn. d, Transverse magnetoresistance MR for x = 0 and 0.07 were \nmeasured up to 0H = 14 T at T = 1 K. The red lines are the least -squares fitting using MR  H. \n  \nFigure 3  \n \nFig. 3 Quantum criticality near the FM QCP protected by nonsymmorphic symmetry . a, In \nthe upper panel, C/T at T = 0.34 K  and resistivity temperature exponent n, estimated by the least -\nsquares fitting using ρ(T) = ρ0 + ATn, are plotted as a function of x on the left and right ordinate, \nrespectively. In the lower panel, TC and TFL are plotted as a function of x overlaid on the colour \nplot of n on the T – x plane. b, Crystal structure highlighting the two inequivalent Ce subl attices \nindicated by Ce and Ce’, and the nonsymmorphic symmetry operation that takes one sublattice \ninto another, a combination of mirror and translation symmetries. c, Ratio of inter -sublattice \nhopping ISH and spin -orbit coupling SOC plotted on the Brillo uin zone (BZ) edges. Note that this \nratio goes to zero along some lines (black colour) due to nonsymmorphic symmetries. The dark \ngray colour around these lines indicates regions within which the ISH dominates over SOC.  \n  \nMethods  \nMaterials synthesis  and characterisation  \nPolycrystalline samples of Ce(Si 1-xAgx)2- were synthesized by the arc -melting technique that \nwas shown elsewhere 1. Ce (rod, 99.9%, ChemPUR), Si (lump, 99.9999%, Alfa Aesar), and Ag \n(granule, 99.99%, ChemPUR) were prepared i n a stoichiometric molar ratio. The weighted \nelements in stoichiometric composition were melted several times after flipping over to improve \nthe sample homogeneity. To improve the crystal quality, the arc -melted buttons covered by Ta -\nfoil were sealed in an  evacuated silica tube for a thermal annealing at 850 C for 10 days. Phase \npurity was investigated by powder x -ray diffraction measurements (PXRD) using a Bruker D8 \nAdvance with Cu -cathode. All cases do not show any impurity phases. Crystal structure and \nstoichiometry were investigated by single -crystal diffraction using STOE STADIVARI \ndiffractometer with Mo K  radiation (0.71073 Å). For details of doping -dependent lattice \nparameters and chemical compositions, see Extended Data Table E2. All single -crystal  x-ray \ndiffraction results are displayed in SI. Note that the nominal x was used for 0  x  0.20 because \nthe Si -site deficiency is negligible and lattice parameters are systematically controlled with \nnominal x, while actual x was used for nominal value of  0.25  x  0.35.  \nNeutron powder diffraction and refinement  \nThe crystal and magnetic structures were studied by neutron powder diffraction using the high -\nresolution powder diffractometer for thermal neutrons (HRPT) 2 at the Swiss Spallation Neutron \nSource SINQ at Paul Scherrer Institut (PSI), Switzerland. About 2 g  of each powder was loaded \nin 6 mm vanadium containers. Diffraction patterns were collected at temperatures of  1.8 and 15 K using neutrons with wavelengths of 1.494 and 2.45 Å. All diffraction data were analysed using \nthe programs of the FullProf software  suite  3. The symmetry analysis of the magnetic structures \nwas done using the Bilbao crystallographic server 4 and the ISODISTORT tool based on \nISOTROPY software  5,6. \nBulk property measurements  \nElectrical resistivity measurements were performed using the st andard four -probe (25 μm Pt \nwires) technique applying a current of 1 mA on the polished surface of bar -shaped specimens. \nElectrical resistance and heat capacity were measured by physical property measurement system \n(PPMS, Quantum Design) with He -3 insert. Magnetization measurements were performed on a \nsuperconducting quantum interference device (SQUID) installed in the magnetic property \nmeasurement system (MPMS, Quantum Design), in the temperature and magnetic field ranges \nfrom T = 1.8 to 300 K and  0H = 0 to 70 kOe, respectively.  \n1 Cordruwisch, E., Kaczorowski, D., Rogl, P., Saccone, A. & Ferro, R. Constitution, structural \nchemistry and magnetism in the ternary system Ce –Ag–Si. Journal of Alloys and Compounds  320, \n308-319 (2001). https://doi.org:https://doi.org/10.1016/S0925 -8388(00)01474 -2 \n2 Fischer, P.  et al.  High -resolution powder diffractometer HRPT for thermal neutrons  at SINQ. \nPhysica B: Condensed Matter  276-278, 146 -147 (2000). \nhttps://doi.org:https://doi.org/10.1016/S0921 -4526(99)01399 -X \n3 Rodríguez -Carvajal, J. Recent advances in magnetic structure determination by neutron powder \ndiffraction. Physica B: Condensed Matter  192, 55 -69 (1993). \nhttps://doi.org:https://doi. org/10.1016/0921 -4526(93)90108 -I \n4 Aroyo, M. I., Perez -Mato, J.M., Orobengoa, D., Tasci, E., De La Flor, G., Krov, A. \nCrystallography online: Bilbao crystallographic server. Bulgarian Chemical Communications  43, \n15 (2011) . \n5 Campbell, B. J., Stokes, H. T., Tanner, D. E. & Hatch, D. M. ISODISPLACE: a web -based tool \nfor exploring structural distortions. Journal of Applied Crystallography  39, 607 -614 (2006). \nhttps://doi. org:doi:10.1107/S0021889806014075 . \n6 Stokes, H. T. & Hatch, D. M. Isotropy Subgroups of the 230 Crystallographic Space Groups .  \n(1988).   \nAcknowledgments  \nS.S. and E .P. would like to thank Eric D. Bauer  and Hanoh Lee  for the fruitful discussion. Neutron \nexperiments were performed at SINQ in Paul Scherrer Institut, Switzerland. Electrical resistance, \nmagnetization, and heat capacity measurements were performed in Laborato ry for Multiscale \nMaterials and Experiments at Paul Scherrer Institut. This project has been supported by  the Swiss \nNational Science Foundation  SNSF Project No. 200021 _188706.  AR acknowledges support from \nthe Swiss National Science Foundation through Ambiz ione Grant No. 186043.  \nAuthor contribution  \nS.S., A.R.,  and E .P. initiated and lead the project. S.S. synthesised the crystals and performed the \nbulk property measurements. S.S. and V.P. performed the neutron experiments and analysed the \nresults. I.P. perfo rmed and analysed the single -crystal x -ray diffraction experiments. S.S., A.R., \nand E.P. wrote the manuscript with input from all authors.  \nCompeting interest declaration  \nThe authors declare no competing interests.  \nData availability  \nThe data supporting this study are available via the Zenodo repository \n(DOI: 10.5281/zenodo.8363352 ). \n  Extended  data  figures and tables  \n \nExtended Data Fig. E1. Bulk  property anomalies at the ferromagnetic phase transition.  a, \nTemperature -dependent imaginary part of AC magnetic susceptibility  ”(T) for x = 0.09 exhibits \na peak at the ferromagnetic phase transition TC. b, Temperature -dependent normalised electrical \nresistivity (T) changes a curvature below TC. The red line indi cates the linear behaviour above T \n= TC. c, Temperature -dependent specific heat capacity divided by temperature C(T)/T exhibits the \nFM transition below which C(T)/T deviates from the steep increase, and this temperature is \nconsistent with other FM anomalies. d, The FM transition peak in  ”(T) of various Ce(Si 1-xAgx)1.97 \nshifts to lower temperatures with decreasing x. e, (T) of x = 0.09 and 0.075 shows the suppressi on \nof TC with decreasing x. (T) of x = 0.07 doesn't change curvature down to T = 0.45 K, showing \nthe linear behaviour from the lowest measured temperature to T = 6 K.  \n  \n \nExtended Data Fig. E2. Neutron powder diffraction results and obtained magnetic structures . \na (b) , In the upper panel, the black and red colour symbols are neutron powder diffraction patterns \nof x = 0.15 ( x = 0.30) collected at T = 15 and 1.8 K, respectively. All Bragg peaks are  assigned to \na structure space group I41/amd. In the lower panel, the black open symbols represent the \ndifference pattern of x = 0.15 ( x = 0.30) obtained by subtracting the pattern collected at T = 15 K \nfrom the one at T = 1.8 K. The red line in the lower panel is a fitting result using the Shubnikov \nmagnetic space group Imm'a'  (I41'/a'm'd ) for x = 0.15 ( x = 0.30). c (d), Schematic for the magnetic \nstructure was obtained from Rietveld refinement using Imm'a'  (I41'/a'm'd ) for FM phase 0.15  x \n< 0.22 (for AFM phase 0.23  x  0.30). Greenish, blue, and grey colour spheres represent Ce, Si, \nand Ag atoms, respectively. Red c olour arrows represent the ordered magnetic moments in the FM \n(AFM) phase.  \n  \nExtended Data Table  E1. Refinement results of neutron powder diffraction data for x = \n0.15, 0.30, and 0.35 of  Ce(Si 1-xAgx)1.9. \nx SG a MSG b a (Å) c (Å) m  2 c \n0.15 I41/amd Imm'a'  4.1962(1)  14.1808(3)  0.69(2)  1.03/1.03  \n0.30 I41/amd I41'/a'm'd  4.2061(1)  14.4714(3)  1.11(1)  1.01/1.01  \n0.35 I41/amd I41'/a'm'd  4.2098(2)  14.5782(8)  1.29(1)  1.11/1.11  \na Structure space group used for fitting the pattern collected at T = 15 K.  \nb Magnetic  space group used for fitting the pattern collected at T = 1.8 K. \nc The goodness of fit for difference pattern using the Le Bail/Rietveld refinement . \n \n   \n \nExtended Data Fig. E3. Normalised electrical resistivity and Fermi liquid behaviour.  a, \nElectrical resistivity  normalised by  (T=300K)  is plotted as a function of temperature. b, Fermi -\nliquid temperature TFL below which (T) = FL(T) = 0 + AT2 was determined by the temperature \nwhere (T) – FL(T) = 0, indicated by downward arrows.  \n  \n \nExtended Data Fig. E4. The least -squares fittings of ’AC(T) and M(H), and (T) for x = \n0.07. a, The real part of the AC magnetic susceptibility  ’AC(T) for x = 0.07 was fitted using the \nequation of ’AC(T) = a + bTc, and the temperature exponent of c = -1.76(4) was obtained. b, \nField -dependent magnetisation M(H) for x = 0.07 was fitted using M(H) = d + eHg, and the field \nexponent of g = 0.489(6) was obtained. c, The temperature -dependent magnetic susceptibility \n(T) for x = 0.07 was measured with a zero -field-cooled (ZFC) and field -cooled (FC) process, \nshowing the absence of difference between ZFC and FC data.  \n \n  \n \nExtended Data Table E2. Single -crystal x -ray diffraction (XRD) results obtained at T = 120 \nK of Ce(Si 1-xAgx)2-. \nNominal x Refined x Refined    a (Å) c (Å) \n0 0 0.034(3)  4.1660(7)  13.967(3)  \n0.02 0.020(4)  0.008(8)  4.1733(9)  13.997(5)  \n0.05 0.041(3)  0.018(6)  4.1792(8)  14.065(4)  \n0.07 0.055(1)  0.045(2)  4.1904(3)  14.051(1)  \n0.10 0.091(2)  0.018(4)  4.2001(5)  14.116(1)  \n0.15 0.135(3)  0.030(6)  4.2015(4)  14.192(1)  \n0.20 0.185(2)  0.030(4)  4.2092(11)  14.351(6)  \n0.25 0.227(1)  0.046(2)  4.2139(6)  14.450(3)  \n0.30 0.252(1)  0.096(2)  4.2214(4)  14.471(3)  \n0.35 0.300(2)  0.100(4)  4.2244(7)  14.659(3)  \n \n Supplementary Information\nFerromagnetic quantum critical point in a locally non-centrosymmetric and\nnon-symmorphic Kondo metal\nSoohyeon Shin,1,\u0003Aline Ramires,2,\u0003Vladimir Pomjakushin,3\nIgor Plokhikh,1Marisa Medarde,1and Ekaterina Pomjakushina1\n1Laboratory for Multiscale Materials Experiments,\nPaul Scherrer Institut, Villigen 5232, Switzerland\n2Condensed Matter Theory Group, Paul Scherrer Institut, Villigen 5232, Switzerland\n3Laboratory for Neutron Scattering and Imaging,\nPaul Scherrer Institut, Villigen 5232 PSI, Switzerland\n(Dated: September 20, 2023)\nI. THEORETICAL DISCUSSION\nCe(Si 1\u0000xAgx)1:9crystallizes in the ThSi 2-type tetragonal structure with space group I4 1/amd\n(#141), which is globally centrosymmetric. Note, thought, that none of the atomic sites are centers\nof inversion, characterizing this system as locally noncentrosymmetric. The metallic state at low\ntemperatures displays a large speci\fc heat coe\u000ecient of the order of 0 :1J=molK2, indicating that\nthis is a heavy Fermi liquid, with e\u000bective quasiparticles that carry spectral weight from f-electrons\nassociated with the Ce atoms. This observation motivates us to model the low-temperature metallic\nstate based on e\u000bective orbitals at the two inequivalent Ce sublattices.\nThe minimal Hamiltonian for a locally noncentrosymmetric system can be written in terms of\nPauli matrices \u001ciand\u001bi(i= 1;2;3), and the corresponding two-dimensional identity matrices\n\u001c0and\u001b0, encoding the sublattice (1 ;2) = (Ce;Ce0) and spin (\";#) degree of freedom (DOF),\nrespectively:\nH=X\nk\ty\nk[\u0018k\u001c0\n\u001b0+ \u0001 1k\u001c1\n\u001b0+ \u0001 2k\u001c2\n\u001b0+gk\u0001(\u001c3\n\u001b)]\tk; (1)\nwith the basis \ty\nk= (cy\nk1\";cy\nk1#;cy\nk2\";cy\nk2#). Herecy\nk\u000b\fis a creation operator for an electron with\nmomentum kin sublattice \u000bwith spin\f.\u0018kcorresponds to the intra-sublattice hopping, \u0001 ik\ncorrespond to inter-sublattice hopping (ISH), and gkencodes spin-orbit coupling (SOC). As in-\nversion symmetry exchanges sublattices ( P=\u001c1\n\u001b0), the Hamiltonian respects global inversion\n\u0003These two authors contributed equally2\nsymmetry if \u0018kand \u0001 1kare even and \u0001 2kandgkare odd in momentum. Note that the sublattice\nDOF introduced here is in direct correspondence to the chirality DOF discussed by Kirkpatrick\nand Belitz [1{3].\nBelow we give the details of the construction of the lowest order terms based on a tight-\nbinding model for the Ce sites for materials in the family of CeSi 2. The crystal structure is\nbody centered tetragonal, with lattice vectors t1= (\u0000a=2;a=2;c=2),t2= (a=2;\u0000a=2;c=2), and\nt3= (a=2;a=2;\u0000c=2), whereaandcare the in-plane and out-of-plane lattice constants, respec-\ntively. The nearest neighbours Ce sites in the same sublattice are within the xy-plane, separated\nby\u00111= (a;0;0),\u00112= (\u0000a;0;0),\u00113= (0;a;0), and\u00114= (0;\u0000a;0). The lowest order terms in the\ntight-binding picture take the form (same form for HCe0\u0000Ce0):\nHCe\u0000Ce=tX\nhi;ji;\u001bcy\ni1\u001bcj1\u001b+h:c: (2)\n=tX\nhi;ji;\u001bX\nkcy\nk1\u001be\u0000ik\u0001riX\nk0ck01\u001beik0\u0001rj+h:c:\n=tX\nk;k0;\u001bcy\nk1\u001bck01\u001bX\nie\u0000i(k\u0000k0)\u0001ri4X\nn=1eik0\u0001\u0011n+h:c:\n=tX\nk;\u001bcy\nk1\u001bck1\u001bh\neikxa+e\u0000ikxa+eikya+e\u0000ikyai\n+h:c:\n=X\nk;\u001bcy\nk1\u001bck1\u001b2t[cos(kxa) + cos(kya)] +h:c:;\nThe last form allows us identify \u0018k= 2t11[cos(kxa) + cos(kya)]\u0000\u0016, wheretis the corresponding\nhopping amplitude and \u0016the chemical potential.\nAssuming a Ce site is located at the origin, the nearest Ce0sites are located at \u000e1= (a=2;0;c=4),\n\u000e2= (0;a=2;\u0000c=4),\u000e3= (0;\u0000a=2;\u0000c=4), and\u000e4= (\u0000a=2;0;c=4). Based on this information, we\ncan write down the ISH from Ce to Ce0atoms:\nHCe\u0000Ce0=t0X\nhi;ji;\u001bcy\ni1\u001bcj2\u001b+h:c: (3)\n=t0X\nhi;ji;\u001bX\nkcy\nk1\u001be\u0000ik\u0001riX\nk0ck02\u001beik0\u0001rj+h:c:\n=t0X\nk;k0;\u001bcy\nk1\u001bck02\u001bX\nie\u0000i(k\u0000k0)\u0001ri4X\nn=1eik0\u0001\u000en+h:c:\n=t0X\nk;\u001bcy\nk1\u001bck2\u001bh\nei(kxa=2+kzc=4)+ei(kya=2\u0000kzc=4)+ei(\u0000kya=2\u0000kzc=4)+ei(\u0000kxa=2+kzc=4)i\n+h:c:\n=X\nk;\u001bcy\nk1\u001bck2\u001b2t0h\neikzc=4cos(kxa=2) +e\u0000ikzc=4cos(kya=2)i\n+h:c:;3\nwheret0is the corresponding hopping amplitude. The last form allows us to identify \u0001 1k=\n2t0cos(kzc=4)[cos(kxa=2) + cos(kya=2)] and \u0001 2k= 2t0sin(kzc=4)[cos(kxa=2)\u0000cos(kya=2)].\nThe x- and y-components of the SOC can be derived considering an e\u000bective staggered electric\n\feld generated by the noncentrosymemtric paths between nearest Ce atoms in the same sublattice.\nThe staggered electric \feld is generated by the Si atoms and has a component in the z-direction.\nThe electric \feld changes sign from one Ce sublattice to another, and can be written as:\nHSOCx = [\u000e1;m\u0000\u000e2;m]X\nhi;ji;\u001b;\u001b0\u000bcy\nim\u001b[\u001bx]\u001b\u001b0cjm\u001b0+h:c: (4)\n= [\u000e1;m\u0000\u000e2;m]X\nk;\u001b;\u001b0\u000bcy\nkm\u001b[\u001bx]\u001b\u001b0ckm\u001b0[eikya\u0000e\u0000ikya] +h:c:\n= [\u000e1;m\u0000\u000e2;m]X\nk;\u001b;\u001b0cy\nkm\u001b[\u001bx]\u001b\u001b0ckm\u001b02i\u000bsin(kya) +h:c:;\nand\nHSOCy = [\u000e1;m\u0000\u000e2;m]X\nhi;ji;\u001b;\u001b0\u000bcy\nim\u001b[\u001by]\u001b\u001b0cjm\u001b0+h:c: (5)\n= [\u000e1;m\u0000\u000e2;m]X\nk;\u001b;\u001b0\u000bcy\nkm\u001b[\u001by]\u001b\u001b0ckm\u001b0[e\u0000ikxa\u0000e\u0000ikxa] +h:c:\n= [\u000e1;m\u0000\u000e2;m]X\nk;\u001b;\u001b0cy\nkm\u001b[\u001by]\u001b\u001b0ckm\u001b0(\u00002i)\u000bsin(kxa) +h:c:;\nfrom what we can identify gx(k) = 2\u000bsin(kya) andgy(k) =\u00002\u000bsin(kxa).\nThe z-component of the SOC can be derived considering an e\u000bective tight binding model with a\nstaggered electric \feld connecting bonds linking next-next-nearest neighbours in the plane. In this\ncase the e\u000bective electric \feld can be thought of as generated by the complementary Ce sublattice\nand lies on the xy-plane:\nHSOCz = [\u000e1;m\u0000\u000e2;m]X\nhi;ji0;\u001b;\u001b0\u000bcy\nim\u001b[\u001by]\u001b\u001b0cjm\u001b0+h:c: (6)\n= [\u000e1;m\u0000\u000e2;m]X\nk;\u001b;\u001b0\fcy\nkm\u001b[\u001bz]\u001b\u001b0ckm\u001b0[ei(2kx+ky)a\u0000ei(kx+2ky)a+ei(\u0000kx+2ky)a\u0000ei(\u00002kx+ky)a\n+ei(\u00002kx\u0000ky)a\u0000ei(\u0000kx\u00002ky)a+ei(kx\u00002ky)a\u0000ei(2kx\u0000ky)a]\n= [\u000e1;m\u0000\u000e2;m]X\nk;\u001b;\u001b0cy\nkm\u001b[\u001bz]\u001b\u001b0ckm\u001b04\f[sin(kxa) sin(2kya)\u0000sin(kya) sin(2kxa)];\nfrom what we can identify gz(k) = 4\f[sin(kxa) sin(2kya)\u0000sin(kya) sin(2kxa)].4\nNote that the ISH \u0001 1kand \u0001 2kare both zero at some high symmetry points at the Brillouin\nzone (BZ) edge:\n•At lines on the kz= 2\u0019=cplane with kx=\u0006ky. Note that this includes the high symmetry\npointZ= (0;0;2\u0019=c);\n•At lines with ( kx;ky) = (\u0006\u0019=a;\u0006\u0019=a) for anykz. Note that this includes the high symmetry\npointsX= (\u0019=a;\u0019=a; 0) andP= (\u0019=a;\u0019=a;\u0019=c );\n•Forkz= 0 along a line passing through the Xpoint, characterized by the condition\ncos(kxa=2) =\u0000cos(kya=2). Note that this line is an extension of the diagonal lines on\nthekz= 2\u0019=cplane if one considers the stacking of the BZ for the body centered tetragonal\nsystem.\nIn summary, the ISH term is strictly zero at the high symmetry points Z,X, andP. It is\nalso zero along the horizontal high symmetry lines connecting the ZandXpoints and along the\nvertical high symmetry lines passing through the XandPpoints.\nNote that the SOC components also have zeros, but in di\u000berent regions of the BZ:\n•The x-SOC term, proportional to sin( kya), is zero for ky=f0;\u0006\u0019=ag,8kxandkz.\n•The x-SOC term, proportional to sin( kxa), is zero for kx=f0;\u0006\u0019=ag,8kyandkz.\n•The z-SOC term, proportional to sin( kxa) sin(2kya)\u0000sin(kya) sin(2kxa), is zero along the\nplanes with kx= 0,ky= 0, andkx=\u0006ky,8kz. Is it also zero at the planes with kx=\u0006\u0019=a\nandky=\u0006\u0019=a, where the x- and y- components of SOC are zero. Furthermore, the z-SOC\nterm is zero for kz= 0 along a line passing through the Xpoint, and along the vertical line\npassing through the X and P points, where HCe\u0000Ce0also vanishes.\nFigure S1 summarizes these results, showing in dark color the regions at the BZ boundary where\nthe SOC dominates over ISH.5\nFIG. S1. Brillouin zone (BZ) of CeSi 2, a body-centered tetragonal system, and the respective form factors\nassociated with inter-sublattice hopping (ISH) processes, \u0001(k) =f\u00011(k);\u00012(k)g, and spin-orbit coupling\n(SOC),\r(k) = f\rx(k);\ry(k);\rz(k)g. The colour scheme is such that white means maximum (up to\nnormalization for each term) and black means zero. The rightmost panels give the form factor of the ratio\nj\u0001(k)j=j\r(k)jfor two values of r=t0=\u000band\f= 0, indicating that ISH can be parametrically smaller\nthan SOC in certain regions at the BZ surfaces. For small values of r=t0=\u000b, these regions become more\nextended. The red points in the top left BZ indicate the high symmetry points.6\nII. SUPPLEMENTS FOR EXPERIMENTAL RESULTS\nA. single-crystal x-ray di\u000braction results\nCrystals of Ce(Si 1\u0000xAgx)2\u0000\u000ewere mounted on the MiTeGen MicroMounts loop and used for\nx-ray structure determination. Measurements were performed at T= 120 K using the STOE\nSTADIVARI di\u000bractometer equipped with a Dectris EIGER 1M 2R CdTe detector and with an\nAnton Paar Primux 50 Ag/Mo dual-source using Mo K \u000bradiation (\u0015= 0.71073 \u0017A) from a micro-\nfocus x-ray source and coupled with an Oxford Instruments Cryostream 800 jet.\nData reduction was performed with X\u0000Area package (Version 2.1, STOE & Cie GmbH,\nDarmstadt, Germany, 2022). The intensities were corrected for Lorentz and polarization e\u000bects,\nand frame scaling along with an empirical absorption correction using spherical harmonics was\napplied using X\u0000Area package. Crystal structures were solved using charge \ripping algorithm\nimplemented in Super\rip [4] supplied with JANA2020 [5].7\nTable S1. Details of single-crystal diffraction experiments for Ce(Si1-xAgx)2-d. Measurements done at T = 120 K, measured with MoKα radiation (0.71073Å), space group I41/amd (tetragonal, No. 141, origin choice 1).  x = 0 x = 0.02 x = 0.05 x = 0.07 x = 0.10 a in Å 4.1660(7) 4.1733(9) 4.1792(8) 4.1904(3) 4.2001(5) c in Å 13.967(3) 13.997(5) 14.065(4) 14.0517(13) 14.1163(17) V in Å3 242.41(7) 243.77(11) 245.66(10) 246.74(3) 249.02(5) Rint 0.0113 0.0268 0.022 0.0189 0.0219 Δmax/Δmin 0.47e/ -0.41e 1.18e/ -1.25e 1.26e/ -1.71e 1.26e/ -1.27e 2.36e/ -2.35e h, k, l range -7 < h < 7 -7 < h < 7 -6 < h < 7 -7 < h < 6 -6 < h < 7 -5 < k < 7 -6 < k < 4 -6 < k < 7 -6 < k < 7 -7 < k < 6 -23 < l < 23 -23 < l < 23 -22 < l < 23 -24 < l < 23 -23 < l < 23 θ range 5.11 - 37.41 5.1 - 37.18 5.09 - 37.18 5.08 - 37.72 5.06 - 37.26 N reflections all/ merged/ > 3σ 3920/ 192/ 158 5520/ 192/ 158 5544/ 193/ 159 7105/ 201/ 165 6414/ 195/ 163 D in g ⸱cm3 / μ in mm-1 5.3532/ 19.323 5.4355/ 19.523 5.4864/ 19.685 5.5225/ 19.801 5.6294/ 20.148 RF( >3σ)/ wRF(> 3σ)/ RF(all)/ wRF(all) 0.0092/ 0.0214/ 0.0181/ 0.0227 0.0133/ 0.0319/ 0.0209/ 0.0334 0.0116/ 0.0258/ 0.0211/ 0.0269 0.0045/ 0.0084/ 0.0134/ 0.0090 0.0065/ 0.0152/ 0.0172/ 0.0175 χ2(>3σ)/χ2(all) 1.0/1.04 1.0/1.06 1.05/1.11 1.01/1.05 1.07/1.02   x = 0.15 x = 0.20 x = 0.25 x = 0.30 x = 0.35 a in Å 4.2015(4) 4.2092(11) 4.2139(6) 4.2214(4) 4.2244(7) c in Å 14.1920(19) 14.351(6) 14.450(3) 14.471(2) 14.659(3) V in Å3 250.53(5) 254.26(14) 256.58(7) 257.88(5) 261.60(9) Rint 0.0272 0.0148 0.0274 0.0257 0.0344 Δmax/Δmin 1.19e/ -1.15e 0.56e/ -0.68e 0.67e/ -0.65e 0.83e/ -0.81e 1.24e/ -0.69e h, k, l range -6 < h < 7 -7 < h < 5 -7 < h < 7 -7 < h < 7 -6 < h < 6 -7 < k < 5 -7 < k < 7 -7 < k < 7 -4 < k < 7 -6 < k < 6 -24 < l < 24 -24 < l < 23 -17 < l < 23 -24 < l < 24 -19 < l < 23 θ range  5.05 - 37.35 5.04 - 37.15 5.03 - 37.48 5.02 - 34.77 N reflections all/merged/above 3σ 5751/ 196/ 162 3957/ 204/ 166 5626/ 200/ 163 3891/ 206/ 169 4243/ 178/ 142 D in g ⸱cm-3 / μ in mm-1 5.7754/ 20.631 5.9027/ 21.04 6.0192/ 21.421 6.0936/ 21.665 6.1993/ 22.003 RF( >3σ)/ wRF(> 3σ)/ RF(all)/ wRF(all) 0.0126/ 0.0345/ 0.0204/ 0.0371 0.0084/ 0.0136/ 0.0137/ 0.0136 0.0068/ 0.0106/ 0.0145/ 0.0107 0.0086 0.0189 0.0155 0.0193 0.0135/ 0.0301/ 0.0203/ 0.0307 χ2(>3σ)/χ2(all) 1.01/1.04 0.99/1.1 1.03/1.13 1.0/1.09 0.99/1.1   8\nTable S2. Atomic coordinates and displacement parameters. Ce is in 4e (0,0,0) position, Si/Ag mixed site 8e (0, 0, z). Atom Occupancy x/a y/b z/c Uiso x = 0 Ce1 1 0 0 0 0.00406(5) Si1 0.983(6) 0 0 0.58399(6) 0.0069(2) x = 2 Ce1 1 0 0 0 0.00876(7) Si1 0.980(4) 0 0 0.58397(7) 0.0109(3) Ag1 0.020(4) x = 5 Ce1 1 0 0 0 0.00691(6) Si1 0.959(3) 0 0 0.58393(6) 0.0088(2) Ag1 0.041(3) x = 0.07 Ce1 1 0 0 0 0.00443(2) Si1 0.9447(8) 0 0 0.583927(19) 0.00526(9) Ag1 0.0553(8) x =0.10 Ce1 1 0 0 0 0.00462(4) Si1 0.9083(16) 0 0 0.58389(3) 0.00505(13) Ag1 0.0917(16) x = 0.15 Ce1 1 0 0 0 0.00707(8) Si1 0.865(3) 0 0 0.58367(5) 0.0081(2) Ag1 0.135(3) x = 0.20 Ce1 1 0 0 0 0.00443(4) Si1 0.8141(11) 0 0 0.58341(3) 0.00484(14) Ag1 0.1859(11) x =0.25 Ce1 1 0 0 0 0.00526(4) Si1 0.7730(8) 0 0 0.583155(19) 0.00613(11) Ag1 0.2270(8) x = 0.30 Ce1 1 0 0 0 0.00442(5) Si1 0.7476(13) 0 0 0.58302(3) 0.00564(13) Ag1 0.2524(13) x = 0.35 Ce1 1 0 0 0 0.00705(9) Si1 0.700(2) 0 0 0.58310(4) 0.0083(2) Ag1 0.300(2)   9\nTable S3. Components of anisotropic tensor (U12 = U23 = U13 = 0). Atom U11 U22 U33 x = 0 Ce1 0.00344(8) 0.00344(8) 0.00531(10) Si1 0.0056(4) 0.0072(4) 0.0078(4) x = 0.02 Ce1 0.00734(12) 0.00734(12) 0.01159(14) Si1/Ag1 0.0094(6) 0.0095(6) 0.0137(5) x = 0.05 Ce1 0.00584(10) 0.00584(10) 0.00905(12) Si1/Ag1 0.0083(4) 0.0075(4) 0.0107(4) x = 0.07 Ce1 0.00419(4) 0.00419(4) 0.00490(4) Si1/Ag1 0.00591(16) 0.00485(16) 0.00502(13) x = 0.10 Ce1 0.00422(6) 0.00422(6) 0.00542(7) Si1/Ag1 0.0062(2) 0.0044(2) 0.0046(2) x = 0.15 Ce1 0.00611(13) 0.00611(13) 0.00900(15) Si1/Ag1 0.0092(4) 0.0067(4) 0.0084(4) x = 0.20 Ce1 0.00381(7) 0.00381(7) 0.00568(9) Si1/Ag1 0.0035(3) 0.0070(3) 0.00401(19) x = 0.25 Ce1 0.00471(6) 0.00471(6) 0.00634(9) Si1/Ag1 0.0086(2) 0.00457(19) 0.00516(18) x = 0.30 Ce1 0.00458(8) 0.00458(8) 0.00410(9) Si1/Ag1 0.0081(2) 0.0050(2) 0.00382(19) x = 0.35 Ce1 0.00628(13) 0.00628(13) 0.00861(18) Si1/Ag1 0.0107(4) 0.0061(4) 0.0080(3)  10\nB. Neutron powder di\u000braction analysis\nHere, we address the details of Rietveld re\fnements for neutron di\u000braction data. Neutron\npowder di\u000braction (NPD) patterns, obtained at T= 15 K above ordering temperatures, were\nanalysed by Rietveld re\fnement using the FullProf suite [6], and all cases were \ft by the room\ntemperature structure I41=amd with decreased lattice parameters. As shown in Extended Data\nFigs. E2a and b, NPD patterns exhibit additional intensities due to the magnetic scattering below\nordering temperatures. Di\u000berence patterns, obtained by subtracting the pattern collected at T=\n15 K from the pattern at T= 1.8 K, were \ft by Le Bail model in order to check ordering wave\nvectors k's. For both ferromagnetic (FM) and antiferromagnetic (AFM) cases, k= 0 shows the\nbest result. Using crystallographic information obtained by Rietveld re\fnement, possible magnetic\nspace groups were investigated using Bilbao crystallographic server [7] and ISODISTORT tool\nbased on ISOTROPY software [8, 9]. Figure S2 shows all possible magnetic subgroups that give\nnon-zero magnetic moments for magnetic Ce ions located on Wckyo\u000b position 4 a(0, 3/4, 1/8)\nof spacegroup I41=amd (no. 141) when k= 0. As shown in Fig. S2b, six maximal symmetry\nsubgroups were adopted for \ftting di\u000berence patterns, and the orthorhombic Imm0a0(no. 74.559)\nand the tetragonal I40\n1=a0m0d(no. 141.556) gives the same \ftting quality with Le Bail \fttings for\nferromagnetic (FM) and antiferromagnetic (AFM) structures, respectively. The obtained magnetic\nstructures are depicted in Extended Data Figs. E2c and d. The unit cell transformations from\nparent paramagnetic space groups are given by the following relations (A, B, C) = 1/4 + (-b, a,\nc) inImm0a0and (A, B, C) = (a, b, c) in I40\n1=a0m0d, where the capital letter and lower case are\nthe basis vectors of the magnetic and the parent paramagnetic space group, respectively.11\nFIG. S2. Magnetic subgroup graph for CeSi 2\u0000d. a, The graph displays all magnetic subgroups that\ngive non-zero magnetic moments for Ce sites 4 a(0, 3/4, 1/8) in a space group I41=amd (no. 141) when\nthe magnetic ordering wave vector kis 0.b.Only the maximal magnetic subgroups are displayed and red\ncircles indicate subgroups Imm0a0(no. 74.559) and I40\n1=a0m0d(no. 141.556) giving the best re\fnement\nresults for ferromagnetic (FM) and antiferromagnetic (AFM) structures, respectively.12\n[1] T. R. Kirkpatrick and D. Belitz, \\Quantum ferromagnetic transition in clean dirac metals,\" EPL (Eu-\nrophysics Letters) 127, 57003 (2019).\n[2] T. R. Kirkpatrick and D. Belitz, \\Soft modes and nonanalyticities in a clean dirac metal,\" Phys. Rev.\nB99, 085109 (2019).\n[3] D. Belitz and T. R. Kirkpatrick, \\Magnetic quantum phase transitions in a clean dirac metal,\" Phys.\nRev. B 100, 174433 (2019).\n[4] Lukas Palatinus and Gervais Chapuis, \\Super\rip{a computer program for the solution of crystal struc-\ntures by charge \ripping in arbitrary dimensions,\" Journal of Applied Crystallography 40, 786{790 (2007).\n[5] V\u0013 aclav Pet\u0014 r\u0013 \u0010\u0014 cek, Michal Du\u0014 sek, and Luk\u0013 a\u0014 s Palatinus, \\Crystallographic computing system jana2006:\ngeneral features,\" Zeitschrift f ur Kristallographie-Crystalline Materials 229, 345{352 (2014).\n[6] Juan Rodr\u0013 \u0010guez-Carvajal, \\Recent advances in magnetic structure determination by neutron powder\ndi\u000braction,\" Physica B: Condensed Matter 192, 55{69 (1993).\n[7] Mois I Aroyo, Juan Manuel Perez-Mato, Danel Orobengoa, EMRE Tasci, Gemma de la Flor, and Asel\nKirov, \\Crystallography online: Bilbao crystallographic server,\" Bulg. Chem. Commun 43, 183{197\n(2011).\n[8] Harold T Stokes, Dorian M Hatch, Branton J Campbell, and David E Tanner, \\Isodisplace: a web-based\ntool for exploring structural distortions,\" Journal of Applied Crystallography 39, 607{614 (2006).\n[9] Harold T Stokes, S van Orden, and Branton J Campbell, \\Isosubgroup: an internet tool for generating\nisotropy subgroups of crystallographic space groups,\" Journal of Applied Crystallography 49, 1849{1853\n(2016)."    },    {        "title": "1809.05466v1.Non_collinearity_and_spin_frustration_in_the_itinerant_kagome_ferromagnet_Fe3Sn2.pdf",        "content": "arXiv:1809.05466v1  [cond-mat.str-el]  14 Sep 2018LETTER TO THE EDITOR\nNon-collinearity and spin frustration in the itinerant\nkagome ferromagnet Fe 3Sn2\nL.A. Fenner1, A.A. Dee1, A.S. Wills1,2,∗\n1Chemistry Department, UCL, 20 Gordon Street, London WC1H 0A J, UK\n2London Centre for Nanotechnology, 17-19 Gordon Street, Lon don WC1H 0AH,\nUK\nE-mail:a.s.wills@ucl.ac.uk\nAbstract.\nFrustrated itinerant ferromagnets, with non-collinear st atic spin structures,\nare an exciting class of material as their spin chirality can introduce a Berry phase\nin the electronic scattering and lead to exotic electronic p henomena such as the\nanomalous Hall eﬀect (AHE).\nThis study presents a reexamination of the magnetic propert ies of Fe 3Sn2,\na metallic ferromagnet, based on the 2-dimensional kagome b ilayer structure.\nPreviously thought of as a conventional ferromagnet, we sho w using a combination\nof SQUID measurements, symmetry analysis and powder neutro n diﬀraction, that\nFe3Sn2is a frustrated ferromagnet with a temperature-dependent n on-collinear\nspin structure. The complexity of the magnetic interaction s is further evidenced\nby a re-entrant spin glass transition ( Tf≃80K) at temperatures far below the\nmain ferromagnetic transition ( TC= 640K).\nFe3Sn2therefore providesarareexample ofafrustrateditinerant ferromagnet.\nFurther, as well as being of great fundamental interest our s tudies highlight the\npotential of Fe 3Sn2for practical application in spintronics technology, as th e AHE\narising from the ferromagnetism in this material is expecte d to be enhanced by the\ncoupling between the conduction electrons and the non-triv ial magnetic structure\nover an exceptionally wide temperature range.\n1. Introduction\nThe discovery of unconventional magnetic and electronic phenome na in conductors is\nimportant for the development of spintronics: information techno logy based on the\napplication and control of electronic spin. The range of mechanisms being enlisted to\nengineer exotic electronic properties is steadily growing, and include s eﬀects such as\nthe complex quasi-two-dimensional multiband Fermi surface of the Fe-based pnictide\nsuperconductors [1], centrosymmetry breaking by magnetic orde r (e.g.TbMnO 3\n[2]), double exchange (manganites)[3], and competing interactions b etween diﬀerent\nmoment types, e.g.d- andf- moments ( e.g.RECrSb 3series [4]). One particularly\nintriguing avenue for research are conductors with non-collinear s tatic spin structures,\nas their chirality can introduce a Berry phase in the electronic scatt ering and lead\nto spin-dependent eﬀects, such as extraordinarily large values of the anomalous Hall\neﬀect (AHE). This mechanism for the AHE was ﬁrst developed by Mat let al.[5]\nand Yeet al.[6] to account for the unusual behaviour in La 1−xCaxMnO3. The\nBerry phase mechanism has also been conﬁrmed to explain the AHE be haviourLetter to the Editor 2\nwell in a variety of systems including the spinel CrCu 2Se4[7] and thin ﬁlms of\nMn5Ge3[8], and has been proposed to account for the AHE that occurs belo w 100K\nin the semiconducting pyrochlore Nd 2Mo2O7, which features a canted spin-ice-like\nferromagnetic spin structure below T C= 89K [9, 10]. There is, however, some\ncontroversy surrounding the actual mechanism for AHE in Nd 2Mo2O7: Yasuiet al.\n[11] and Sato [12] analysed the magnetic ﬁeld-dependence of the sp in structure, from\nwhich they calculated the spin chirality and predicted the Hall resistiv ity, and found\nthat neither the spin chirality mechanism, nor any of the other curr ently known AHE\nmechanisms, can account for the behaviour of this material. This ob servation reopens\nfundamental questions over the origin of the AHE in frustrated ma gnets.\nFrustrated magnets, where conventional magnetic order is ‘frus trated’ by a\ncompetition between the diﬀerent magnetic exchange interactions and a large ground\nstate degeneracy, have proven to be one of the simplest domains in which to engineer\nextraordinary electronic eﬀects. The list of experimentally observ ed exotic ground\nstates is ever increasing and includes the spin glass states of (H 3O)Fe3(SO4)2(OH)6\n[13, 14, 15], SrCr 9xCa12−9xO19[16] and Y 2Mo2O7[17]; the quantum spin liquid\nstates ofHerbertsmithite[18] and Kapellasite (ZnCu 3(OH)6Cl2)[19]; the spin ice states\nof bulk Dy 2Ti2O7and Ho 2Ti2O7[20] and the nano-engineered realisations of the\nspin ices[21, 22]. Magnetic frustration can also lead to very rich magn etic phase\ndiagrams, e.g.forgadoliniumgalliumgarnet(GGG)[23], Gd 2Ti2O7[24]andtheseries\nLixMn2O4[25]. Further, much eﬀort is currently focussed on the degenerat e manifold\nitself as a medium able to support new phenomena, such as order-by -disorder[26],\nKasteleyn transitions [27], the formation of eﬀective magnetic mono poles [28], and\ntopological spin glass behaviour [29, 13, 14]. All of these studies ar e, however, on\ninsulators and progressin the ﬁeld of frustrated itinerant magnet s has been very much\nhindered by the lack of model systems with which to explore and test the developing\ntheories.\nIn this article we introduce Fe 3Sn2as a new non-collinear and frustrated itinerant\nferromagnet based on a kagome bilayer structure. While the mater ial has been\nknown for many years [30] there is much confusion over its magnetic properties, with\nthe analysis of early M¨ ossbauer [31, 32] and powder neutron diﬀra ction data [33]\nbeing hindered by diﬃculties and inconsistencies. The authors of the se early papers\nconcluded that the spins in Fe 3Sn2lie approximately along the c-axis above 250K,\nand undergo a gradual rotation into the abplane below 250K, remaining collinear\nthroughout the rotation. Our reexamination of the magnetic prop erties of Fe 3Sn2\nfollowed from the hope that the spins on the Fe-sublattice are actu ally frustrated,\nwhich would lead to characteristic ﬂuctuations and exotic spin-depe ndent conduction\nproperties. Here, we show using a combination of theoretical and e xperimental\ntechniques, that spin frustration is both allowed and present in Fe 3Sn2. Firstly,\nthe presence of spin frustration is indicated by temperature-dep endent magnetisation\nmeasurements, which reveal the presence of competing magnetic interactions and\nevidence a re-entrant spin glass component below ≃80K. Symmetry analysis is\nthen applied to demonstrate that ferromagnetism in Fe 3Sn2is not restricted to being\ncollinear, thereby hinting at the rich physics that is possible in this mat erial. Further,\nthe analysis of powder neutron diﬀraction data in terms of both collin ear and non-\ncollinear magnetic models is presented.\nThese ﬁndings indicate that Fe 3Sn2is a particularly notable candidate for\nspintronics applications as the high Curie temperature ( TC>600K), and the possible\nfrustration enhancement to the AHE expected for a ferromagne t, would allow accessLetter to the Editor 3\nat room temperature to the eﬀective control of spin polarised cur rents [34], as well as\nproviding new routes for the conversion of magnetic data into an ele ctrical signal in\ndevices such as sensors and nonvolatile magnetic memory[35].\nThe crystal structure of Fe 3Sn2is shown in ﬁgure 1. Originally believed to\nbe monoclinic [30], the crystal structure was later corrected by sin gle crystal X-ray\ndiﬀraction and found to be best described by the space group R¯3m[36]. The Fe ions\noccupythe18 hcrystallographicsite(0.4953,0.5047,0.1131),andformbilayersof oﬀset\nkagome networks. These kagome layers are in turn made up of 2 size s of equilateral\ntriangles, with Fe–Fe distances of 2.732 ˚A and 2.582 ˚A; this is shown by the diﬀerently\ncolouredtrianglesin the ﬁgure. The Fe–Fe distance forming the bilay eris 2.584 ˚A. The\nSn ions occupy two distinct crystallographic sites, Sn1 (0.0000, 0.00 00, 0.1041) and\nSn2 (0.0000, 0.0000, 0.3303); the ﬁrst of these lie within the kagome layers, and the\nsecond lie between the kagome bilayers. The reﬁned values of the lat tice parameters\n(a=b= 5.3147,c= 19.7025˚A with respect to the tripled hexagonal unit cell) of the\nsample used in these studies are in good agreement with those of pre vious studies [36].\nFigure 1. The crystal structure of Fe 3Sn2reﬁned using neutron diﬀraction data\ncollected on the D20 diﬀractometer with neutrons of 1.3 ˚A. The Fe ions form\nbilayers of oﬀset kagome networks, and the Sn ions lie in the c entre of the kagome\nhexagons and between the bilayers. The blue and red triangle s (colour online)\nindicate the larger and smaller equilateral triangles resp ectively.\n5g of bulk Fe 3Sn2powder were prepared by grinding stoichiometric amounts of\nFe and Sn powders (purities) [37] in a glovebox. The mixed powder was pelletised and\nsealed into a silica ampoule that had been put under vacuum (10−5mbar) and ﬂushed\nout with argon three times, and then ﬁnally backﬁlled with argon to 3.5 mbar in order\nto reduce Sn evaporation. The pelletized sample was heated to 1073 K in a muﬄe\nfurnace at 1K/min. At ﬁrst the progress of the reaction was chec ked by x-ray powder\ndiﬀraction (Bruker D4 Endeavor with Cu K- αradiation, equipped with a graphite\nsecondary monochromator to eliminate the Fe ﬂuorescence) ever y few days, and the\npellets were reground, repelletized and sealed into an ampoule each t ime. However,\nit was found that the reaction is complete after 1 week and that no r egrinding step\nis required. Each time an ampoule was removed from the furnace it wa s quenched by\nsubmersion into cold water, as Fe 3Sn2is only stable between 873K and 1088K [38].\nX-ray powder diﬀraction (D4 Endeavor) showed that the sample wa s∼95% Fe 3Sn2\nphase; the remainder consisted of FeSn 2and FeSn phases.\nMagnetic measurements were performed on Fe 3Sn2powder using a Quantum\nDesign MPMS-7 dc-SQUID magnetometer, and the oven insert was u sed for\nmeasurementsabove300K.Thedc-susceptibilitywasmeasuredbe tween5Kand700KLetter to the Editor 4\nin ﬁeldsof100Oeand1000Oe, andthe ﬁelddependence wasmeasure dup to10,000Oe\nat temperatures from 2K to 300K. The sample was held within a piece o f aluminium\nfoil which was attached to the end of the sample rod with copper wire for all of the\nstudies [39]. A straw sleeve was used at low temperature to prevent sample movement\nin the cryogenic gas ﬂow.\nThe magnetic susceptibility ( χdc) of Fe 3Sn2between 5K and 700K, in ﬁelds of\n100 and 1 000Oe, is shown in ﬁgure 2(a). The transition into the ferr omagnetic state\ndetermined from the maximum in dχ/dTvs.TisTC≃640K, in fair agreement with\nthe approximate values of 612K [38] and 657K [31] derived from M¨ o ssbauer data by\nprevious workers. This transition is believed to be to a state in which t he spins lie\nalong the c-axis. On cooling, the 1 000Oe data shows that this ferromagnetic r esponse\nsaturatesuntil at ∼520Kanothercomponentcausesthesusceptibilitytoincrease. We\nsuggest that it is at this temperature that the spins begin their rot ation towardsthe ab\nplane. This transition is continuous until at ∼60K the susceptibility decreases. The\nsuppression of this drop by ﬁeld cooling is characteristic of a spin glas s component\nat temperatures far below the main ferromagnetic transition, and provides further\nevidence of underlying magnetic frustration in this itinerant magnet . The separation\nof the zero-ﬁeld fooled and ﬁeld cooled data allow the spin glass freez ing temperature\nto be estimated as Tf≈80K. It is possible that this transition is actually the onset\nof a second ferromagnetic component, however this seems rathe r unlikely as there is\nonly one crystallographic magnetic iron site in the Fe 3Sn2crystal structure. Field-\ndependent studies, shown for 150K in ﬁgure 2(b), indicate that th ere is very little\ncoercivity and that the magnetisation saturates in ﬁelds close to 10 000Oe, reaching\na maximum value of ∼1.9µBFe−1. This value changes little in the range 5 −300K,\nand is signiﬁcantly less than that expected for localised Fe moments, indicating that\nthe Fe valence electrons are shared between localised and itinerant environments.\n0 100 200 300 400 500 600 700 0510 15 20 25 30 35 40 45   χ (emu mol -1 )\nTemperature (K) -10000 -5000 0 5000 10000 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 \n  B Magnetisation (μ  Fe -1 )\nField (Oe) (a) (b) \n zero field cooled \n field cooled 100 Oe  \n field cooled 1000 Oe \nFigure 2. (a) Magnetic susceptibility of Fe 3Sn2in applied ﬁelds of 100Oe and\n1000Oe, between 5K and 700K. The zero ﬁeld cooled curve was me asured in\na ﬁeld of 100Oe. (b) A hysteresis loop showing the ﬁeld-depen dence of the\nmagnetisation of Fe 3Sn2in ﬁelds up to 10 000Oe at 150K.\nIn order to investigate whether the low temperature transition inv olves a secondLetter to the Editor 5\nferromagnetic component or to a re-entrant spin glass phase, we measured the\nthermoremanent magnetisation (TRM) of Fe 3Sn2as a function of time, t. The sample\nwas cooled from 300K to 25K (below Tf) in zero ﬁeld, then after a wait time, tw(2400\nto 16900s), a ﬁeld of 50Oe was applied, and the relaxation of the TRM was measured\nas a function of t. The recorded curves, shown in ﬁgure 3 (a), are all well ﬁtted by\nthe usual function used to describe the relaxation of spin glasses: a superposition of a\nstretched exponential and a constant term, MTRM=M1+M0exp[−(t/τ)1−n] [40],\nwhereM1is the constant term, M0is the initial TRM, τis the characteristic time\nconstant and nis the exponent. The relaxation of the TRM in ferromagnets, on the\nother hand, is usually best ﬁtted by a power law of the form MTRM=M1+M0t−γ\n[40], where M1is a constant, M0is the initial TRM and γis the power law exponent.\nThis suggests that the low temperature transition is of a spin glass n ature.\nThe relaxation of the TRM shows a clear dependence on tw, which is typical in a\nnon-equilibrium, spin-glass phase: the longer the tw, the slower the relaxation of the\nTRM [15, 41, 42]. Conversely, the relaxation in a ferromagnetic phas e is expected to\nshow negligible dependence on tw[40]. If a stationary (equilibrium) part is subtracted\nfrom our TRM curves and they are plotted against t/twan almost full aging scaling\nis observed (ﬁgure 3 (b)). This further indicates that the low temp erature phase\ntransition involves a spin glass component, rather than a ferromag netic one.\n100 1000 10000 100000 2.45 2.46 2.47 2.48 2.49 \n8256 \n11472 16900 2400  MTRM  (emu g -1 )\nt (s) 0.001 0.01 0.1 1 10 0.748 0.750 0.752 0.754 0.756 0.758 \n  \nA = 0.1 \n  α = 0.005 \nt / t w 2400 \n 8256 \n 11472  16900  \nMFC  - A(τ 0/ t) α\nTRM / M (a) (b) \ntw (s) tw (s) \nFigure 3. (a) Relaxation of the thermoremanent magnetization at 25K, for\ndiﬀerent waiting times, tw. The solid lines are the best ﬁts of the data to the\nequation, MTRM=M1+M0exp[−(t/τ)1−n]. (The relative positions of the\ncurves along the yaxis should not be taken as meaningful as their separations\nare within the error of the ﬁeld produced by the SQUID.) (b) Th e aging part of\nthe TRM, normalised to the ﬁeld cooled value of the magnetisa tion as a function\noft/tw. The stationary part, A(τ0/t)α, whereτ0is a microscopic time, has been\nsubtracted from the data. The scaling constants Aandαagree well with those\nfor the AgMn spin glass [41].\nIn order to determine how the competing energy scales within Fe 3Sn2are\nmanifested in the ordering of the atomic moments, powder diﬀractio n data were\ncollected with neutrons of wavelength 1.3 ˚A using the high ﬂux diﬀractometer D20\nat the ILL. Approximately 2g of sample was held in a 10mm diameter van adium canLetter to the Editor 6\n 8 18 28 38  48  58 68  -50  0  55  100  150  200  250 \n 2-theta (deg)  Counts × 10 3 (arb. units) \nFigure 4. Fit to the powder neutron diﬀraction pattern of Fe 3Sn2. The upper\ntick marks indicate the positions of peaks predicted from th e nuclear phase, the\nlower indicate those in the magnetic phase. The circles corr espond to the observed\nscattering, the line shows the calculated diﬀraction patte rn and the diﬀerence is\ngiven below. Signiﬁcant contamination from the cryomagnet leads to the increase\nin background between 30 and 38◦; regions where scattering from the sample\nenvironment are distinct from that of Fe 3Sn2were excluded from the reﬁnement.\nThe data were collected at 300K using the D20 diﬀractometer w ith neutrons\nof wavelength 1.3 ˚A. The ﬁnal goodness of ﬁt parameters were χ2= 113.9 and\nRwp= 9.87 with 51 reﬁned parameters.\nwith temperature being controlled using a cryomagnet. Data were t aken at 300, 150,\nand 6K (all below the Curie temperature, T C= 640K) in zero magnetic ﬁeld. The\nbasis vectors that describe the diﬀerent symmetry types of magn etic structure were\ncalculated using the technique of representational analysis embod ied in the program\nSARAh[43]. Analysisofthe crystaland magnetic structurewascarriedou t using data\noverthe angularrange8◦≤2θ≤75◦, usingFullprof[44] togetherwith SARA h-Reﬁne.\nRepresentational analysis indicates that the magnetic represent ation for the Fe\ncrystallographic site (18 h) is decomposed into the irreducible representations (IRs)\nof the little group of the propagation vector Gk=R¯3maccording to Γ Mag=\n1Γ(1)\n1+2Γ(1)\n2+2Γ(1)\n3+1Γ(1)\n4+3Γ(2)\n5+3Γ(2)\n6, where the subscript numbering follows\nthat given in the works of Kovalev [45] and the superscript indicate s the order of\nthe IRs. Inspection of their associated basis vectors (BVs) reve als that there are two\nferromagnetic IRs with uncompensated components along the c-axis and in the ab\nplane, respectively: Γ 3and Γ5. Further, Γ 3corresponds to an umbrella structure in\nwhich an ordered antiferromagnetic component is also allowed in the abplane such\nthat the moments are restricted to the local acmirror planes of the individual kagome\ntriangles perpendicular to the kagome plane, a structure similar to t hat found in the\nFe-jarosites [46]. Γ 5spans 6 basis vectors (BVs) and as such corresponds to a complex\nmagneticstructuretype madeup ofcomponentsthat arebothfe rromagnetic(in the abLetter to the Editor 7\nTable 1. Reﬁned values of the moments, their angles from the principl e\ncrystallographic axes and the goodness-of-ﬁt parameters f or models of collinear\nand non-collinear ordering in Fe 3Sn2as a function of temperature. All deviations\ncorrespond to standard errors, determined from the errors i n the reﬁned\nparameters, except for those of the angles in the non-collin ear model, which\ncorrespond to the spread of angles of the individual moments .\nCollinear Non-collinear (Γ 3⊕Γ5)\nT(K)θ(◦)φ(◦)µB(Fe)Rwpθ(◦)φ(◦)µB(Fe)Rwp\n300 0±0 20.6±3.6 2.27±0.13 9.85 0 ±39.9 21.6±8.7 2.19±0.15 9.87\n150 0±0 17.5±8.1 1.82±0.18 9.71 0 ±83.1 32.7±10.9 1.63±0.23 9.51\n6 0±0 65.9±7.9 1.90±0.17 9.79 0 ±31.6 68.6±6.7 1.95±0.70 9.63\nplane) and antiferromagnetic (in the abplane and ||c). Γ5also has the notable quality\nthat it allows the moments on the diﬀerent Fe-sites to be of unequal sizes. These\ncalculations indicate a possible richness in the orderings of the Fe-mo ments that can\noccur in Fe 3Sn2at the atomic level: a transition from a state with all the moments\nalongthe c-axis to one with the moments in the abplane does not requirethe moments\nto be either collinear or equal in magnitude. As there is no symmetry r equirement for\nthe moments to be collinear and equal, it follows that the key experime ntal challenge\nis to determine the degree of non-collinearity and the variation in the moment sizes,\nas both of these, and the ﬂuctuations associated with them, could lead to anomalous\nelectron transport eﬀects.\nReﬁnement ofthe powderneutrondiﬀractiondatawascarriedout using2models:\na simple collinear model in which the angle away from the c-axis of a set of identical\nmoments was reﬁned, and a non-collinear model in which the weighting coeﬃcients\nof the diﬀerent BVs calculated by representational theory were r eﬁned. Both models\nindicate that at 300K (ﬁgure 4) the moments lie largely along the c-axis (1). There\nis essentially no diﬀerence between the quality of the ﬁts for the collin ear and non-\ncollinear (Γ 3⊕Γ5) models, and in both cases the deviation of the average moment\ndirection from the c-axis is approximately φ∼ ±21◦. On cooling, both models show\nthe moments to be ﬂopping into the abplane, with only a small discrepancy appearing\nin their ability to ﬁt the experimental diﬀraction data: the magnetic s cattering at\n≃25.1◦and≃28.3◦is better ﬁtted by the non-collinear structures of Γ 3⊕Γ5at both\n6 and 150K (ﬁgure 5). The observation of ordered Fe-moments of ∼2µBis in good\nagreement with prior M¨ ossbauer [31] and powder diﬀraction stud ies [33]. No changes\nin the average reﬁned magnitude of the moments were observable in this experiment,\nindicating that this is not the main drive for the spin reorientationtra nsition. Further,\nthe similarities between the magnetic diﬀraction patterns and reﬁne d models at 6 and\n150K indicate that the spin structures in the intermediate and spin g lass phases are\nclosely related. Unfortunately, the large background from the sa mple environment\nprevents any comment from being made about the strength of the diﬀuse scattering\nassociatedwith the disorderedspin glasscomponent, and howit cha ngesupon cooling.\nThe equivalence in the quality of the ﬁts from the collinear and non-co llinear\nmodels indicate that unpolarised powder neutron diﬀraction does no t have the\nsensitivity required to unambiguously pin down the degree of canting together with\nthe variation in the sizes of the magnetic moments. Previous attemp ts to improveLetter to the Editor 8\n150K collinear \n 22  24  26  28  30  32 \n 2-theta (deg)  Counts × 10 3 (arb. units) \n -5  7  19  43  55 \n 31 300K collinear \n 22  24  26  28  30  32 \n 2-theta (deg)  Counts × 10 3 (arb. units) \n -5  7  19  43  55 \n 31 \n300K non-collinear \n 22  24  26  28  30  32 \n 2-theta (deg)  Counts × 10 3 (arb. units) \n -5  7  19  43  55 \n 31 150K non-collinear \n 22  24  26  28  30  32 \n 2-theta (deg)  Counts × 10 3 (arb. units) \n -5  7  19  43  55 \n 31 6K non-collinear \n 22  24  26  28  30  32 \n 2-theta (deg)  Counts × 10 3 (arb. units) \n -5  7  19  43  55 \n 31 6K collinear \n 22  24  26  28  30  32 \n 2-theta (deg)  Counts × 10 3 (arb. units) \n -5  7  19  43  55 \n 31 \n300K \n 150K \n 6K \nac\nb\nFigure 5. The top panel shows the goodness of ﬁt of the collinear magnet ic\nstructure model to the data in the region 21◦≤2θ≤32◦, at 6K, 150K and\n300K (left to right). The middle panel shows the goodness of ﬁ t of the non-\ncollinear model in the same region at the three temperatures . The bottom panel\nshows the reﬁned non-collinear magnetic structures which c orrespond to the plots\nin the middle panel.\nthe quality of the ﬁt of the collinear model through modiﬁcation of th e Fe-form factor\nare not well justiﬁed [33] and lead to a unsatifying model. Rather, we argue that\nthe non-collinear model is to be preferred as it allows resolution of diﬃ culties in the\ninterpretationofearly57Fe and119Sn M¨ ossbauerdata[31,32], andaconsistentpicture\nof the temperature-dependent spin transition of Fe 3Sn2to be constructed.\nIn the early studies the authors concluded that the magnetic stru cture features 2\ncomponents. The ﬁrst has population αand is a collinear ferromagnetic component\nwhere the moments lie almost parallel with the c-axis above 250K, gradually rotate\ntowardsthe abplaneoncoolingbelow250K,andlieinthe abplaneatlowtemperature,\nremaining collinear throughout the rotation. A second contribution was required\nto model the rotation of the moments on warming; it involves moment s in the ab\nplane and has a population (1 −α) that decreases slowly on warming. Our model\nof non-collinear ferromagnetism allows an alternative interpretatio n of these data:\nthe spins continuously rotate from the abplane to the cdirection up to ∼520K,\nand feature a non-collinear component that is temperature depen dent. The slow\nrotation then indicates that the near balance of the energy scales responsible is\ntemperature insensitive. The spin glass transition at low temperatu re (Tf≃80K)\nis then to a phase with the moments largely within the abplane, a situationLetter to the Editor 9\nreminiscent of the anisotropy-induced spin glass state of the kago me antiferromagnet\n(H3O)Fe3(SO4)2(OH)6[13, 14].\nIn conclusion, we show that Fe 3Sn2is a rare example of a frustrated itinerant\nmagnet. Three transitions are observed upon cooling: the ﬁrst at TC= 640K is from\nthe paramagnetic phase to a collinear ferromagnetic phase with the moments collinear\nwith the c-axis. On cooling from ∼520K to ∼75K the moments rotate from the\nc-axis into the abkagome plane. During this transition symmetry restrictions that\nrequire the moments to be collinear and of equal size are relaxed, allo wing a non-\ntrivial ferromagnetic structure to develop. The energy scales re sponsible for this spin\nstructure are at present unclear, though the Dzyaloshinsky-Mo riya interaction [47],\nwhich is allowed on the kagome lattice, is an obvious candidate. [48].\nOn further cooling below ∼75K the competition between magnetic interactions\nleads to a transition to a re-entrant spin glass phase. The origins of this spin glass\nphase are unclear as such behaviour is more commonly observed in hig hly disordered\nferromagnets, e.g.Fe0.7Al0.3[49], whereas Fe 3Sn2is not a disordered system. Further\nwork is also required to understand the role that the high degree of frustration and\nthe 2-dimensional ﬂuctuations expected from the underlying kago me lattice play in\nthe magnetism of this material.\nOurstudiesonFe 3Sn2alsoindicateitspotentialforuseinspintronicsforbothspin\ninjection and applications based on the AHE. The latter may be encha nced above the\nvalues expected for a conventional ferromagnet by a coupling bet ween the conduction\nelectrons and the non-trivial ferromagnetic spin structure. Suc h a coupling is likely as\na largeHall resistivity has recently been observed at room tempera ture in the granular\nalloy ﬁlms with the composition Fe 68Sn32of around 60 times greater than the Hall\nresistivity of pure Fe, though details of the underlying magnetic str ucture are not\ncurrently known [50].\nReferences\n[1] Kamihara Y, Hiramatsu H, Hirano M, Kawamura R, Yanagi H, K amiya T and Hosono H 2006\nJ. Am. Chem. Soc. 12810012–3; Wang F, Zhai H, Ran Y, Vishwanath A and Lee D-H 2009\nPhys. Rev. Lett. 102047005\n[2] Kimura T, Goto T, Shintani H, Ishizaka K, Arima T and Tokur a Y 2003 Nature42655–8\n[3] Anderson PW and Hasegawa H 1955 Phys. Rev. 100 675\n[4] Inamdar M, Thamizhavel A and Ramakrishnan S 2009 J. Alloy. Compd. 48028; Inamdar M,\nThamizhavel A and Ramakrishnan S 2008 J. 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Phys. 931851–3"    },    {        "title": "2306.06659v1.Ferromagnetic_Superconductivity_in_Two_dimensional_Niobium_Diselenide.pdf",        "content": "1 \n Ferromagnetic  Superconductivity in Two -dimensional Niobium Diselenide  \nTingyu Qu1,2, Shangjian Jin2,3, Fuchen Hou4, Deyi Fu3, Junye Huang5, Darryl Foo Chuan Wei3, Xiao Chang5, \nKenji Watanabe6, Takashi Taniguchi6, Junhao Lin4, Shaffique Adam2,3,5,7, Barbaros Özyilmaz1,2,3, 5,8,* \n1 NUS Graduate School, Integrative Sciences and Engineering Programme  (ISEP) , National University of \nSingapore, Singapore, Singapore.  \n2 Department of Physics, Nationa l University of Singapore, Singapore, Singapore.  \n3 Centre for Advanced 2D Materials, National University of Singapore, Singapore, Singapore.  \n4 Department of Physics, Southern University of Science and Technology, Shenzhen, China.  \n5 Department of Materials Science and Engineering, National University of Sin gapore, Singapore, Singapore.  \n6 National Institute for Materials Science, 1 -1 Namiki, Tsukuba 305 -0044, Japan.  \n7 Yale -NUS College, 16 College Ave West, Singapore , Singapore.  \n8 Institute for Functional Intelligent Materials (I -FIM) , National University of Singapore, Singapore, Singapore.  \n* e-mail: barbaros@nus.edu.sg  \n \nThe co -existence of ferromagnetism and superconductivity becomes possible through  \nunconventional pairing in the superconducting state[1]. Such  materials  are exceedingly rare in \nsolid -state sys tems but are promising platforms  to explore topological phases[2],[3], such as \nMajorana bound states[4]–[6]. Theoretical investigations date back to the late 1950s[7],[8], but only \na few system s have  so far  been expe rimentally identified  as potential hosts[9]–[13]. Here, we \nshow that atomically -thin niobium diselenide (NbSe 2) intercalated with  dilute  cobalt  atoms \nspontaneously displays  ferromagnetism  below the superconducting transition temperature ( 𝑇C). \nWe elucidate the origin of this phase by constructing a magnetic tunnel junction that consists \nof cobalt and cobalt -doped niobium diselenide (Co -NbSe 2) as the two ferromagnetic electrodes , \nwith an ultra -thin boron nitride as the tunnelling barrier. At a temperature well below  𝑇C, the \ntunnelling magnetoresistance shows a bistable state, suggesting a ferromagnetic order  in Co -\nNbSe 2. We propose a Ruderman –Kittel –Kasuya –Yosida  exchange coupling mechanism based \non the spin-triplet  superconducting order parameter  to mediate such ferromagnetism . We \nfurther perform n on-local  lateral spin valv e measurement s to confirm  the origin of the \nferromagneti sm. The observation of  Hanle precession  signals  show  spin diffusion  length up to \nmicrometre s below 𝑇C, demonstrating an intrinsic spin-triplet nature  in superconducting NbSe 2. \nOur discovery of superconductivity -mediated ferromagnetism opens the door to an alternative \ndesign of  ferromagnetic superconductors . 2 \n Introduction  \nFerromagnetic  superconductor s display intrinsic co-existence of  ferromagnetism and \nsuperconductivity[1]. Though the nature  of Cooper pairs in such materials is still under \ndebate[14]–[16], a spin-triplet pairing state is generally accepted  as a necessary  ingredient[1],[11]. \nSo far , only a few system s have been  shown to meet this requirement . For example, in uranium -\nbased compounds and stacked graphene , Pauli limit violation is used  as the hallmark  of spin-\ntriplet Cooper pairs[11]–[13]. However, a n intriguing question remains as whether a \nsuperconducting condensate could spontaneously display ferromagnetism.  \nRecently, transition metal dichalcogenide (TMD) superconductors have  emerg ed as an \nalternative  candidate to realize  such unconventional superconductivity.  Few-layer TMDs \nnaturally host anisotropic pairing channels, which support additional symmetry breaking and \nunusual superconducting properties such as Ising protection[17]–[19], spin-triplet  pairing \nstates[18]–[21] and topological superconductivity[22]–[25]. In the family of TMD superconductors, \nniobium diselenide (NbSe 2) stands out because of its high  superconducting transition \ntemperature ( 𝑇C)[17] and the expected  sizeable singlet -triplet mixing[19],[20]. In addition, as \npredicted,  when intercalated with magnetic atoms , a spin-triplet superconducting  host could \ngive rise to  Ruderman –Kittel –Kasuya –Yosida (RKKY) -mediated  ferromagnetism[26]. In this \npaper, we use spin -dependent transport measurement s to show  that below 𝑇C, NbSe 2 becomes \nferromagnetic when intercalated  with dilute cobalt (Co) atoms.  \nWe develop two distinct device configurations. First, we construct a vertical magnetic \ntunnel junction (MTJ) with evaporated Co and Co -doped NbSe 2 (Co-NbSe 2) as the two \nferromagnetic electrode s and BN as the tunnelling  layer in -between.  We study  \nsuperconductiv ity-mediated ferromagnetism via spin -dependent tunnelling measurements \ninside the gap of NbSe 2. Second, we utilize the ferromagnetic Co -NbSe 2 electrodes as a lateral \nspin injector/detector  for pristine  NbSe 2 channel s and study  magnetic  switching  and Hanle \nprecession  via non-local spin valve (NLSV) measurement s. These results directly confirm that \nNbSe 2 hosts spin-triplet pairing states and that it is the former  that gives rise to  ferromagnetism \nin Co -NbSe 2 below 𝑇C. Our system is promising for studying  the co-existence of \nferromagnetism and superconductivity and  building emerging quantum devices in two-\ndimensional (2D)  limit . \n \n  3 \n Device Design and Magnetic Intercalation  \nWe select ultra -thin (2 ~ 3 atomic layers) hexagonal boron nitride (BN) to encapsulate NbSe 2. \nPatterned Co lines are deposited onto the BN/NbSe 2 heterostructure via e -beam evaporation \nand form the ferromagnetic contacts. Here, the ultra -thin BN plays two additional roles.  At \ndevice level, it acts as a spin dependent tunnelling barrier. More crucially , through our \nfabrication process, it also allows for interstitial doping of NbSe 2 (See Methods) (Fig. 1a, left \npanel) . As evident from the mass c ontrast scanning transmission electron microscopy (STEM) \nimage, no obvious Co clusters are observed (Fig. 1b, top panel). Based on the electron energy \nloss spectroscopy (EELS), the Co doping concentration ( 𝑛Co) decays exponential ly with a n \naverage  value of around  3% and a penetration depth of around a -few nm (Fig. 1b, bottom panel). \nWe conclude that few -layer BN likely serves as a  stable buffer laye r assist ing the intercalation \nof Co of van der Waals (VdW) structure  (Extended Data 1) . \nThe magnetic properties  are characterized by  three different transport measurements , \nnamely, a four -probe charge transport measurement to characterize  the Co-doped NbSe 2 \nchannel resistance, a two -probe magnetic  transport  measurement of the  tunnelling \nmagnetoresistance (TMR) in a magnetic tunnel junction (MTJ) consisting of Co/BN/Co -NbSe 2, \nand a non -local lateral spin valve measurement  with Co -NbSe 2 as an injector /detector and \nundoped NbSe 2 as the channel (Fig. 1a, right panel and Extended Data 2 ). We start our \ndiscussion with the four -probe measurement to  understand the impact of Co intercalation on \nthe superconductivity of multi -layer NbSe 2. Encouragingly,  the superconducting state is \npreserved  suggesting  that the Co doping in NbSe 2 is sufficiently  dilute (See Supplementary \nInformation). Further, voltage –current ( 𝑉–𝐼) measurements in the vicinity of superconducting \ntransition (around 𝑇C) follows a power -law ( 𝑉~𝐼𝛼) and its non-linearity becomes more \nprominent with decreasing temperature  (Fig. 1c). This can be well fitted by the theoretical \nmodel of Berezinskii –Kosterlitz –Thouless (BKT) transition[27],[28], implying that our Co -doped \nsuperconducting layers are in the 2D regime. Such an  observation is surprising because the \nthicknesses of our samples are all around 10 ~ 12 nm  (Extended Data 3). It suggests that the \nCo intercalation decouples the bulk NbSe 2 withou t breaking  the superconductivity. In fact, the \ndecoupling of the superconducting layers becomes  more prominent  with increasing Co \nconcentration . For such samples, despite a reduction of the zero -field 𝑇C, the in -plane upper \ncritical field ( 𝐵C2||) is enhanced and can exceed  the Pauli limit ( 𝐵Pauli ) (Fig. 1d and \nSupplementary Information). The  𝐵C2||𝐵Pauli⁄  vs. 𝑇C relation  of our  Co-NbSe 2 behaves 4 \n similarly to that of few -layer NbSe 2 flakes[17], which  also agrees with the recent discovery of \nreduced dimensionality in an intercalated bulk NbSe 2 achieved by a different mean s[29]. We \nalso studied devices where the BN buffer layer  was transparent  or missing altogether . In neither \ndevice s, we observed a supercondu cting transition . \nSuperconductivity -mediated  Ferromagnetism  \nNext, we study the phase diagram of our Co -NbSe 2 (Device A) with temperature and magnetic \nfield as two independent variables. In pristine NbSe 2, the field -free superconductivity persists \nat all temperatures below 𝑇C. In contrast, at 𝐵=0, although our Co -NbSe 2 exhibits a \nsuperconducting state below 𝑇C0 (~ 6.6 K), a resistive phase re-emerges below a second critical \ntemperature 𝑇K (𝑇K<𝑇C0) such that at base temperature (1.5 K), the resistance increases to \nnearly 40% of that just above 𝑇C. Surprisingly, this unusual phase can be completely screened \nout under a finite magnetic  field ( 𝐵K), where our system re -enters the non -resistive state (re -\nentrant superconductivity) (Fig. 2a). We also notice that 𝑇K is anisotropic under the in -plane \nand out -of-plane field directions (Extended Data 4 ). The resistance anomaly triggered by the \nsuperconducting transition resembles  a failed superconductor and hints to a fermionic phase \ninduced in a bosonic host[30],[31]. However, our system differs in one critical aspect, namely, the \nexistence of a non -resistive state at intermediate temperature and magnetic fi eld range s (𝑇K < \n𝑇 < 𝑇C and 𝐵K < 𝐵 < 𝐵C2||) for a finite  range of 𝑛Co (See Extended Data 5 and Supplementary \nInformation) . We will later argue that 𝑛Co  corresponds to a length  scale comparable to the \nsuperconducting coherence length for ferromagnetism to emerge inside the gap.  \nHaving confirmed that the resistance anomaly results from the Co intercalation, we \ninvestigate the magnetic properties of this state. Here, Co and Co -NbSe 2 form the  two \nferromagnets in our MTJ s and we study their voltage -bias dependent tunnelling spectrum at \nfixed temperatures and fixed fields. At low temperature and  small field, we observe a \nprominent zero -bias peak (ZBP) in the differential conductance ( 𝑑𝐼/𝑑𝑉). Notably, both the \ntemperature  range  and the field range are consistent with the resistive state in the phase diagram \noccurring at 𝑇<𝑇K and 𝐵<𝐵K. At temperatures above 𝑇K and magnetic fields above 𝐵K, the \nZBP vanishes, which likewise matches the condition s for the non -resistive phase in Fig. 2a. To \nfurther understand the origin of ZBP, we fit the tunnelling spectrum using the p-wave pairing \nmode l[32]. It captures well both the magnitude of the ZBP and the observed critical temperature \nat 𝑇K. These results hint towards mediated magnetism inside the superconducting  gap[33] (See \nMethods and Supplementary Information ). To rule out  any spurious origins  that could arise 5 \n from  the Co contact[34],[35], we turn our attention to  tunnelling  magnetoresistance  (TMR) \nmeasurements of  the Co/BN/Co -NbSe 2 MTJ (Fig. 3a).  \nWe first measur e TMR as a function of 𝐵|| along the easy  axis of the Co electrode. We \nensure a constant current (≤ 0.1 µA) through the junction such that the potential in Co -NbSe 2 \nfalls inside gap. Below 𝑇K, we observe hysteretic  switching of the TMR with a magnitude ( ∆𝑅) \nthat gradually increases with decreasing temperature such that it saturates at 500 Ω ( ∆𝑅𝑅⁄ ~ \n10%)  at base temperature  (𝑇=1.5 K) (See top panel in Fig. 3b, Fig. 3c, and top panel in Fig. \n3d). This demonstrates the existence of two ferromagnets of distinct coercive fields[36]. Using \nanisotropic magnetoresistance (AMR) measurements on the Co contact  alone , we identify that \nthe higher switching field originates from  the Co electrode (Extended Data 6). Therefore, the  \nsecond swit ch at lower field must result from the switching of  Co-NbSe 2 (Fig. 3b, bottom \npanel) . Critically, at temperature s above 𝑇K but still below 𝑇C, no bistable states in the TMR \nare observed  (Extended Data 7). Note that  the observation of an anti -parallel configuration also \nrules out  proximity -induced ferromagnetism in NbSe 2 as the proximity effect would \nspontaneously favour  a parallel configuration . Next, we fix the temperature below 𝑇K and study \nthe bias -dependence of the TMR in more detail (Fig. 3e). The bias current determines the \npotential difference across the tunnel junction (Fig. 3 d, bottom panel) , thus it can be used to \ntune the potential relative to  the superconducting gap. Only  for bias currents that confine the \npotential deep inside the gap, a clear hysteretic switching in TMR  is observed . \nWe now present a phenomenological model based on our observation that the \nferromagnetism only occurs below 𝑇K and when the bias is  inside the superconducting gap. \nSince  the ferromagnetic order only exist s with dilute Co doping , we can rule out  direct \nexchange  couplin g. We also note that the effect only appears above  a certain  threshold  of Co \ndoping , namely,  a minimum  separation distance between the Co atoms is needed. This suggest s \nindirect exchange coupling, namely, RKKY interaction . Above 𝑇C, RKKY should be  \nsuppressed for two reasons  in the metallic state . First, the Fermi velocity ( 𝜈F) is high, leading \nto an oscillation of the  RKKY coupling strength ( 𝐽RKKY )[37],[38]. Second, the strong intrinsic spin \norbit coupling lead s to rapid dephasing[39]. Also , if the Fermi level is outside the \nsuperconducting gap, RKKY is dominated by filled electrons and the subsequent oscillating  \n𝐽RKKY  suppresses the long -range ferro magnetic coupling. But below 𝑇C and inside the gap , the \nelectrons are screened out  and the RKKY can now be mediated by superconducting carriers . \nRecently, Parkin S. and co -workers have indeed reported the coupling of two individual 6 \n magnetic moments  (within 5 -nm range)  in superconducting niobium by scanning tunnelling \nmicroscopy measurement[40]. Theoretically, ferromagnetic RKKY in superconducting \ncondensate prefers  a spin -triplet ground state (Fig. 3f)[26]. In addition,  for RKKY  to emerge  \nbelow 𝑇C, the magnetic moment separation should be comparable to the superconducting \ncoherence length ( 𝜉GL)[41]. Indeed, this condition  is met when both temperature and magnetic \nfield are  below critical thresholds , such that 𝑇<𝑇K<𝑇C and 𝐵<𝐵K<𝐵C2||, and with the \nbias deep inside the gap . Only then, 𝜉GL becomes small er than the magnetic coupling length  \nand the RKKY interaction  can be mediated by sufficient superconducting carriers  \n(Supplementary Information).  \nSpin -triplet Pairing State  \nOur model critically depends on the existence of a triplet state . While N bSe2 has been reported \nto host such state , a direct proof is missing . Proximity -induced spin supercurrents[42],[43] cannot \nprobe intrin sic triplet pairs in the superconductor . Our approach is to use Co -NbSe 2 instead as \na superconducting ferromagnet to enable direct spin injection  into the gap of NbSe 2. To do so, \nwe build a lateral non -local spin valve with a few-layer Nb Se2 as the channel material (Device \nB) with three different channel lengths  (Fig. 4a).  \nWe first note that  devices with either a direct Co contact  or devices  with a much thicker \nBN barrier  (≥ 5 layers) , limiting the Co intercalation , did not show any non-local signal  (See \nSupplementary information) . Only f or device  where  𝑛Co is around 3%, we observe spin signals  \nin the superconducting state . Fig. 4 b shows a 2D mapping of the non -local resistance (𝑅NL) as \na function of 𝐼Inj and 𝐵|| along the easy axis of Co -NbSe2 (set at −40 mT and swept from 0 to \n+40 mT) , where an anti -parallel can be  clearly resolved where the largest switching magnitude \nof 𝑅NL (∆𝑅NL) is more than 150% of the background (Extended Data 8).  Individual  field \nsweeps are shown in Fig. 4c, where we plot 𝑅NL by scanning 𝐵|| with different injection \ncurrent s (𝐼Inj). We observe nearly symmetric h ysteretic switching between two distinct  \nresistance states  at several  representative 𝐼Inj provided the injection bias  is inside the gap . \nTo unambiguously probe the spin-triplet state in NbSe 2, channel length -dependent  Hanle \nmeasurement s are performed, where the injector and detector are polarized by 𝐵|| before an \nout-of-plane field ( 𝐵⊥) is swept for in-plane spin precession. First, w e see nearly symmetric \nHanle precession curves under 𝐼Inj=0.1 μA and 𝑇=100  mK (Extended Data 9a). Second, \nthe precession  occurs only for temperature s below 𝑇C and when  𝐼Inj is inside the gap  (Extended 7 \n Data 9b). Third, we observe a clear length dependence of the precession signals.  Fig. 4d shows \n𝑅NL vs. 𝐵⊥ as a function of channel lengths  (𝐿CH) equal to  0.6 m, 0.8 m and 3 .0 m. For \n𝐿CH= 0.8 m, the signal changes sign  at 𝐵⊥=±0.125 T, and then saturates at ±0.5 T, \nshowing  a clear signature of spin precession.  For 𝐿CH = 3.0 m, the Hanle curve shows even \ntwo additional  peaks at  ±0.4 T, implying  a 180 phase rotation of the spins. A second spin -\nphase rotation at longer  channel length  suggests the spin  information  is well protected  in the \nsuperconducting channel. Using Bloch equation s in the diffusive regime, we estimate  the spin \ndiffusion length  (𝜆S) in the order  of micro metre s (See Methods and Supplementary \nInformation) . Such long-range spin transport through the superconducting channel  rules out \nany proximity effect, and  strongly suggest  an intrinsic triplet -pairing state  in NbSe 2. \nSummary and Outlook  \nIn summary,  we discovered  superconductivity -mediated  ferromagnetism via magnetic Co \nintercalation in atomically -thin NbSe 2. The ferromagneti sm in the superconducting condensate  \noccurs due to  the presence  of a spin-triplet pairing state , which is demonstrated by our spin -\ndependent transport measurement s. 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The charge transport can be characterized by the channel voltage ( across \ncontacts #2 and #3 ) in a four -probe measurement. b, Scanning transmission electron microscopy \n(STEM) image of the Co/BN/Co -NbSe 2 heterojunctions (top panel) and the corresponding depth \nprofiles of 𝑛Co on the NbSe2 side (bottom panel). c, 𝑉–𝐼 curves for the Co -intercalated NbSe 2 devices \nwith temperatures ranging from below 𝑇C to just above 𝑇C. Inset: Temperature dependence of the \nexponent 𝛼 deduced from the power -law, 𝑉~𝐼𝛼. As indicated by the black dashed line, 𝛼 approaches 3 \nat 𝑇 = 6.3 K. d, 𝐵C2||𝐵Pauli⁄  vs. 𝑇C for Co -intercalated NbSe 2 devices with different doping levels.  \nThe devices with low 𝑛Co, high 𝑛Co and without  (w/o)  𝑛Co correspond to Devices A, C and E , \nrespectively.  \n  \n11 \n  \nFig. 2 | Ferromagnetic  phase below 𝑻𝐂 and the YSR state s. a, The ( 𝐵||, 𝑇) phase diagram. Our system \nundergoes from a normal state (N) to a superconducting state (SC) at 𝑇<𝑇C. Under the condition of \n𝑇<𝑇K and 𝐵<𝐵K, a resistance anomaly phase occurs, corresponding to a ferromagnetic  phase  \ncoexisting with superconductivity (FM+SC). b, 𝑅−𝑇 curves by line cuts with discrete 𝐵|| in (a). A \nconstant AC current (1 µA) is applied through the channel . c-d, Two -probe 𝑑𝐼/𝑑𝑉 curves with \ntemperature ( c) and field ( d) sequences. The ZBP occurs under the condition 𝑇<𝑇K and 𝐵<𝐵K. \n  \n12 \n  \nFig. 3 | Direct s pin-dependent  evidence for s uperconductivity -mediated  ferromagnetism. a , MTJ \nmeasurement schematics. Co and Co -NbSe 2 act as two ferromagnets that are separated by the thin BN \ntunnelling  barrier. The TMR is measured as a function of 𝐵|| along the easy axis of the Co contact. b, \nA comparison between the TMR of Co/BN/Co -NbSe 2 and the AMR of Co contact at 𝑇 = 2.0 K. The \nTMR and AMR are measured with two separate AC current s at 0.1 A and 10 A, respectively.  The \nswitching in TMR is up to 500 Ω while the most prominent AMR effect is around 0.15 Ω.  c-e, \nTemperature ( c) and bias-current ( e) dependences of the TMR signals through Co -NbSe2 MTJ. Zero -\nfield 𝑅−𝑇 curve (top panel) and 𝑑𝐼/𝑑𝑉 curve inside the superconducting gap (bottom panel) at 𝑇=\n2.0 K are shown in (d). The labelled coloured dots in the top (bottom) panel in ( d) correspond to the \nfive different temperatures (biases) in ( c) and ( e), respectively.  Signature  of ferromagnetism (the \nswitching in TMR) only exists below 𝑇K and with the energy inside ∆SC. f, Schematics of a \nferromagnetic RKKY (FM RKKY) interaction with spin-triplet pairs as the exchange carriers. A \nferromagnetism  is always preserved because of  the exchange by spin-triplet  pairs . \n  \n13 \n  \nFig. 4 | Non -local  spin supercurrent and Hanle precession. a , Micrograph of the device  for non-local \nmeasurement . The three different channels are labelled by #1  (~0.6 µm) , #2 (~0.8 µm)  and #3  (~3.0 \nµm), respectively. b, 2D mapping of 𝑅NL as a function of 𝐵|| and 𝐼Inj at 𝑇 = 53 mK . A sign change of \n𝑅NL occurs at finite 𝐵|| near the coercive field of the Co detector and at finite 𝐼Inj for the potential inside \nthe gap. A DC bias is swept from –5 µA to +5 µA with 0.0 5 µA in each step as the injection source and \nan AC bias equal to 0.0 5 µA is applied to resolve the non-local signal  with a Lock -in amplifier . c, \nExamples of non -local switches with different 𝐼Inj at 𝑇 = 53 mK . The injection current tunes the \nmagnitude of the switching but does not obviously affect the switching fields. d, 𝑅NL by sweeping 𝐵⊥ \nat 𝑇 = 46 mK , with 𝐿CH = 0.6, 0.8 and 3.0 µm, respectively. The red lines are fitted by  our modified  \nHanle precession model.  \n  \n14 \n Acknowledgements  \nB. Ö. acknowledges the support by the National Research Foundation, Prime Minister’s Office, \nSingapore, under its NRF Investigatorship ( Grant No.  NRF -NRFI2018 -8) and Medium -Sized \nCentre Programme.  D-F-C. W. and S. A. would like to acknowledge the Singapore Na tional \nScience Foundation Investigator Award (Grant No. NRF -NRFI06 -2020 -0003).  F.H. and J.L. \nwould like to acknowledge the support from National Natural Science Foundation of China \n(Grant No.11974156), the Science, Technology and Innovation Commission of S henzhen \nMunicipality (No. ZDSYS20190902092905285 and KQTD20190929173815000), and also the \nassistance of SUSTech Core Research Facilities, especially technical support from Cryo -EM \nCenter and Pico -Centre that receives support from Presidential fund and Deve lopment and \nReform Commission of Shenzhen Municipality. K.W. and T.T. acknowledge support from the \nElemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), \nJST. We thank Alexander Hamilton , Feixiang Xiang , Manfred Sigrist and Zijin Lei for the \nfruitful discussions.  \n \nAuthor Contributions  \nB. Ö. initiated, coordinated, and supervised the work. T. Q. , D. F. and J. H.  fabricated the \ndevices  and performed transport measurement s. T. Q., B. Ö. and S . J. performed data analysis. \nX. C. assisted the data analysis. F. H. and J. L. performed the DF -TEM, HR -TEM and STEM \ncharacterizations. S. J., D-F-C. W. and S. A.  provided theor y work. K.W. and T.T. grew the \nBN crystals. T. Q., B. Ö., and S. J. co-wrote the manuscript.  \n \nCompeting Interests  \nThe authors declare no competing interests.  \n \n \n  15 \n Methods  \nDevice design and  fabrication.  The NbSe 2 flakes are mechanically exfoliated from bulk crystals \n(grown by HQ Graphene) by Scotch tapes onto a Si/SiO 2 substrate. The typical geometry of the NbSe 2 \nis about 10 -µm long, and a 1~2 -µm wide. The high -quality h -BN (grown by K. Watanabe and T. \nTaniguchi f rom National Institute for Materials Science, Japan) is exfoliated onto a photoresist -coated \n(PMMA 495 -A4) Si wafer for the ease of hunting ultra -thin layers (2~4 layers) via optical contrast. The \nultra-thin h -BN is then stacked onto the NbSe 2 by a semi -dry transfer method. Here, the ultra-thin BN \nlayer plays three roles. First, it serves as an encapsulation layer that protects NbSe 2 from oxidation. \nSecond, it is the diffusion barrier where atomic defeats are induced and enables the implantation of \ndilute f erromagnetism into NbSe 2. Third, at least one layer of BN remains relatively intact after the \ndoping process, which plays an important role as the tunnelling barrier in the magnetic tunnel junction. \nThe BN/NbSe 2 stack then undergoes a high -vacuum (< 10–6 Torr) thermal annealing (210 ºC for 6 hrs) \nto remove the bubbles formed during the transfer. An e -beam lithography with acceleration voltage of \n30 kV is employed to pattern the contacts, followed by an e -beam evaporation of Co/Ti films (35 nm/7.5 \nnm, deposi tion rate: 0.5 Å/s) under an ultra -high vacuum condition (5×10–8 Torr). We leverage the  e-\nbeam radiation during the pattern writing to induce short -range disorders in the ultra -thin BN and creates \natomic defects in NbSe 2. And the evaporated Co atoms carry sufficient kinetic energy to diffuse through \nthe disorders in BN and intercalate into NbSe 2. Au contacts are deposited separately via a thermal \nevaporator. The heterostructure goes through another high -vacuum thermal annealing to improve the \nquality of all  contacts.  \n \nMaterials characterization.  The cross -sectional STEM specimens were prepared using a Cryo -\nfocused Ion Beam in ultra -high vacuum (< 10–6 mbar) in a liquid -nitrogen temperature environment. \nSTEM imaging, EDS and EELS analysis of vertical heterojunctions of Co/BN/NbSe 2 were performed \non an FEI Titan Themis with an X -FEG electron gun and a DCOR aberration corrector operating at 300 \nkV, at roo m temperature. The inner and outer collection angles for the STEM images ( 𝛽1 and 𝛽2) were \n48 and 200 mrad, respectively, with a convergence semi -angle of 25 mrad. The beam current was 100 \npA for imaging and spectrum collection. In the elemental analysis,  the Co concentration ( 𝑛Co in Fig. \n1b) is defined as the ratio of detected Co atoms to the total atoms of all the detected elements. This ratio \nand its error bars were calculated using a quantitative analysis model of EDS and EELS spectrum with \ncommercial  codes that are standard and routinely used in this type of elemental assessment.  \n \nLow -temperature measurement. The electrical transport measurements were performed in a variable \ntemperature insert (with a magnetic field up to 10 T and temperature down to 1.5 K for Device A) and \na dilution refrigerator (with an in -plane magnetic field up to 7 T and out -of-plane magn etic field up to \n2 T and temperature down to 40 mK for Devices B, C, D  and E). A lock -in amplifier (with a frequency \nof 13 Hz) and a source measure unit (Keithley) were applied for the output and acquisition of the AC \nand DC signals, respectively. A four -probe measurement is used to analyse the 𝑉–𝐼 and 𝑅–𝑇 \ncharacteristics, a two -probe measurement is used to analyse the differential conductance and TMR \nsignals, and a non -local measurement is used to characterize the spin transport. The in -plane magnetic \nfield is applied along the easy -axis of the Co contacts. For the characterization of the AMR of Co \ncontacts, a four -probe measurement is performed, where the magnetic field is parallel/antiparallel to the \nbias current through the easy axis of the Co contac t. \n \nSuperconducting gap symmetries and YSR states.  To analyse the temperature and magnetic field \ndependence s of superconducting gap and YSR states , we fit our differential conductance data with a \nmodified Bardeen –Cooper –Schrieffer (BCS) theory[32],[44], 16 \n  𝑑𝐼\n𝑑𝑉(𝑉)=𝜎𝑁\n2𝜋∫ 𝑑𝐸(𝑑𝑓(𝜖)\n𝑑𝜖|𝜖=𝐸−𝑒𝑉)∫ 𝑑𝜃 𝑅𝑒[𝐸−𝑖𝛤\n√(𝐸−𝑖𝛤)2−𝛥0cos2(𝑙𝜃)]2𝜋\n0∞\n−∞ \n             +𝜎𝑍𝐵𝑃𝑒𝑥𝑝 (−𝑉2\n2𝜎2), (1) \n   \nwhere 𝜎𝑁 is the background normal state differential conductance, 𝛤 is the quasiparticle broadening and \nΔ0cos (𝑙𝜃) is the superconducting gap size with the angular -dependent factor 𝑙 (𝑙=0,1,2,3 for s-, p-, \nd-, and f-wave superconducting order parameters, r espectively). We only consider 𝑙=0 and 1 for \nsinglet and triplet fittings. The second term in Equation (1) is added to contain the YSR states. \n \nSuperconducting coherence length.  To analyse the suppression of 𝑇C by magnetic field, we introduce \nthe pair -breaking equation[45] under an external magnetic field. Since the thickness is much thinner than \nthe parallel penetration depth 𝜆||≈69 nm[46], we apply the pair -breaking equation in thin film,  \n ln𝑇C\n𝑇C0=𝛹(1\n2)−𝛹(1\n2+𝛼\n2𝜋𝑘B𝑇C),    (2) \nwhere 𝛼=𝐷𝑒2𝐵∥2𝑑2/6ℏ for parallel field and 𝛼=𝐷𝑒𝐵⊥ for perpendicular field. The superconducting \ncoherence length 𝜉GL0=𝜉GL(𝑇=0,𝐵=0) can be extracted by t he electronic diffusion constant 𝐷=\n16𝑘𝐵𝑇𝑐0𝜉GL0/ℎ[47]. The fitting results by Equation (2) are shown in Fig. 1d, and the extracted values of \n𝜉GL0 are shown in Extended Data 10 . \n \nHanle pr ecession and non -local resistance.  For the Hanle precession measurement, the injector and \ndetector are first polarized in parallel or antiparallel  directions  to each other by an in -plane magnetic \nfield. Then the in -plane magnetic field is cancelled,  and a n out -of-plane field to superconduct ing flake  \nis performed. For 𝐿CH = 0.6 and 0.8 µm, we follow the standard Hanle model[48] below,  \n 𝑅NL=𝑅0∫𝑑𝑡1\n√4𝜋𝐷𝑡𝑒−𝐿CH2\n4𝐷𝑡𝑒−𝑡\n𝜏cos(𝜔𝐿𝑡)∞\n0,    (3) \nwhere 𝑅0 is a scale factor for the  resistance, 𝐷 is the spin diffusion constant, 𝐿CH is the centre -to-centre \ndistance between the injector and detector , 𝜏 is the spin relaxation time, 𝜔𝐿=𝑔𝜇𝐵𝐵⊥/ℏ is the Larmor \nfrequency with Landé 𝑔-factor ( 𝑔=2), 𝜇𝐵 is the Bohr magneton and ℏ is the Plank constant . The spin \nrelaxation lengths are comput ed by 𝜆s=√𝐷𝜏 ~ 0.62 and 1.06  µm for the two channels, respectively.  \nHowever, for 𝐿CH = 3.0 µm, the standard Hanle model cannot describe the  second peak (without a sign \nchange) at 𝐵⊥ ≈ ±0.4 T , which suggests that  Meissner effect should be taken into account  (given NbSe 2 \nis a Type -II superconductor) . Therefore, we introduce a term incorporating  the vort ices in the \nsuperconducting channel , and the modified Hanle model is in the form  below,  \n 𝑅NL=𝑅0∫𝑑𝑡1\n√4𝜋𝐷𝑡𝑒−𝐿CH2\n4𝐷𝑡𝑒−𝑡\n𝜏[1−𝑎𝑣+𝑎𝑣cos(𝜔𝐿𝑡)]∞\n0,    (4) \nwhere  𝑎𝑣=1−exp (−𝐵/𝐵𝑣) is a parameter accounting for the proportion of the transverse area with \nvortices  responsible for the  screening effect of magnetic field , and  𝐵𝑣 is a measure of the magnetic field \nresponsible  for the  area density of vortices.   17 \n  \nExtended Data 1  | TEM characterizations of Co intercalation.  a, Cross -section STEM image of Co -\nintercalated NbSe 2 heterojunction. The BN region is determined by electron energy loss spectra (EELS), \nwhere a crystalline monolayer or bilayer BN always remains after the deposition of Co. b, Energy \ndispersive spectrum (EDS) maps for the elements Co, N, Nb, Se. Scale bar: 5nm. c, Line cuts of the \nelemental abundance perpendicular to the Co/BN/Co -NbSe 2 interface, extracted from EELS. d, \nEvolution of the VdW interlayer gap (red) and intralayer gap (blue), extracted from STEM cross -\nsections (inset), as a function of distance to the BN/NbSe 2 interface (in number of NbSe 2 monolayers). \nThe horizontal red and blue dashed lines respectively refer to the interlayer and intralayer gaps in the \npristine bulk NbSe 2. Scale bar: 2 nm. Inset: Schematic definition about the intralayer and i nterlayer.  \n  \n18 \n  \nExtended Data 2 | Device morphology.  a, Optical micrograph of Device A. The Co and Au contacts \nare deposited onto the BN -encapsulated channel (NbSe 2) in an alternate pattern, where both the Co -\ndoped and -undoped SC regions can be characterized. The easy axes of the Co contacts are aligned in \ncross with the channel for the ease of spin injection into the channel. b, The AFM image of ( a) in phase \ncontrast. c, Thickness  profile of the BN flake ( the probed region is denoted by the yellow dashed box \nin b). d, Optical micrograph of Device B.  d, Optical micrograph of Device B. An optical filter \n(wavelength ~ 560 nm) is applied to increase  the outline contrast  for BN and NbSe 2. For Hanle \nprecession measurements, the three characterized channels (from the bottom p anel to the top panel in \nFig. 4d) are between contacts #2 and #3, #3 and #4, and #4 and #5, respectively. e, The AFM image of \n(d). f, Thickness profile of the NbSe 2 flake (the probed region is denoted by the red dashed line in e). \n \n19 \n  \nExtended Data 3 | BKT transition in Co -intercalated NbSe 2 and the fitting models. a, b, 𝑉–𝐼 \ncurves for non -Co-intercalated ( a) (Device D) and Co -intercalated ( b) NbSe 2 (Device A) with \ntemperatures ranging from base  to just above 𝑇C. c, Temperature dependence of the exponent α deduced \nfrom the power -law, 𝑉~𝐼𝛼. As indicated by the red dashed line, α approaches 3 at 𝑇 = 6.3 K for the \nCo-intercalated sample. For the non -Co-intercalated sample, α shows an abrupt jump from around 1 to \naround 6 from 𝑇C to just below 𝑇C. d, Temperature dependence of the resistance at zero field. With the \ncondition 𝑇BKT <𝑇<𝑇C, 𝑅(𝑇) follows the relation in the form of 𝑅=𝑅0∙𝑒−𝑎𝑇̂−12⁄, where 𝑅0 and a \nare parameters reflecting the properties of the specific superconductor and 𝑇̂=𝑇𝑇𝐵𝐾𝑇⁄ −1 is the \nreduced temperature. The red line is a fit to BKT transition, yielding 𝑇BKT  ≈ 6.35 K . \n  \n20 \n  \nExtended Data 4 | Re-entrant superconductivity  under out -of-plane field . a, 𝑅–𝑇 curve at zero \nmagnetic field for Device A. Inset: measurement set -up. b, 𝑅–𝐵⊥ curve  at 𝑇= 1.5 K for Device A. A \nre-entrant superconductivity occurs at around 2 T and the system enters the normal state at around 5 T. \nc, 𝑅–𝑇 curve at zero magnetic field for Device B. Inset: measurement set -up. d, 𝑅–𝐵⊥ curve at 𝑇= 80 \nmK for Device A. A re -entrant superconductivity occurs at around 1.6 T. \n  \n21 \n  \nExtended Data 5 | Doping control of the RRR , 𝑻𝐊 and 𝑻𝐂. a, 𝑅–𝑇 curves  of several  devices with \ndifferent Co doping levels  from 273 K  to base temperature . The RRR is defined as 𝑅𝑅𝑅 =𝑅273 𝐾𝑅𝑇C+ ⁄ , \nwhere 𝑅𝑇C+ refers to the resistance just above 𝑇C. b, 𝑅–𝑇 curves  of the corresponding  devices in (a) \nfrom 𝑇C+ to base temperature . The red dashed lines are references for zero resistance.  For all the four \ndevices, a constant AC current (1 µA) is applied through the channel  for the measurement . We note that \nfor th e ferromagnetic phase to emerge, a critical Co inte rcalation is needed. For the two devices (A and \nB) showing the resistance anomaly and spin switching have similar Co doping levels estimated by the \nRRR (with the value around 7.6 ~ 8.2, which corresponds to average Co concentration around 3~5% by \nSTEM). For the devices without sufficient Co doping, we do not observe any signature of \nsuperconductivity -mediated  ferromagnetism, and the RRR is no less than 9 . \n  \n22 \n  \nExtended Data 6 | Comparison between the AMR of Co contact and the TMR of in Co/BN/Co -\nNbSe 2 MTJ. a,  Measurement set -up for Co AMR. A constant current is passed along the easy axis of \nthe Co stripe where the magnetic field is swept. b, Bias-current dependence of the AMR signals of the \nCo contact. The AMR is only prominent if the bias current is large ( > 2 µA) and its magnitude is in \nmΩ scale. c, Measurement set -up for Co/BN/Co -NbSe 2 TMR. A constant current is through the \nCo/BN/Co -NbSe 2 and a non -ferromagnet (Au) is used as ground. d, Bias-current dependence of the \nTMR signals of the MTJ. The TMR is only p rominent if the bias  inside ΔSC and the magnitude is in \n100-Ω scale.  \n  \n23 \n  \nExtended Data 7 | Temperature dependence of the two -probe MTJ resistance at zero field ( 𝑹𝟎) \nand the switching magnitude in TMR. The 𝑅0 increases with cooling down below 𝑇C due to the \nenhanced ΔSC. The TMR (%) only emerges below 𝑇K and saturates at around 1.5% below 2.5 K. 𝐼AC = \n0.02 µA.  \n  \n24 \n  \nExtended Data 8 | Hanle precession signals in the superconducting spin valve.  a, 2D mapping of \n𝑅NL as a function of 𝐵|| and 𝐼Inj. b, 𝑅NL as a function of 𝐼Inj under two discrete magnetic fields \nhighlighted by the blue dashed line (parallel regime) and red dashed line (anti -parallel regime) in ( a). \n  \n25 \n  \nExtended Data 9 | Hanle precession measurements  in the superconducting spin valve.  a, 𝑅NL as a \nfunction of the out-of-plane  magnetic field  with 𝐼Inj = 0.1 µA . The opposite signs of the two peaks  in \nthe precession curves correspond to the parallel and antiparallel configurations between the injector and \ndetector . b, Injection bias current ( 𝐼Inj) dependence of the Hanle precession signals at 46 mK. The grey \ndashed line  is taken at 𝐼Inj = 0.05 µA and 𝑇 = 4.0 K  (>𝑇C0). \n  \n26 \n  \nExtended Data  10 | Basic parameters about our devices . \n"    },    {        "title": "0803.2175v1.Current_induced_noise_and_damping_in_non_uniform_ferromagnets.pdf",        "content": "arXiv:0803.2175v1  [cond-mat.mes-hall]  14 Mar 2008Current-induced noise and damping in non-uniform ferromag nets\nJørn Foros,1Arne Brataas,1Yaroslav Tserkovnyak,2and Gerrit E. W. Bauer3\n1Department of Physics, Norwegian University of Science and Technology, 7491 Trondheim, Norway\n2Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n3Kavli Institute of NanoScience, Delft University of Techno logy, 2628 CJ Delft, The Netherlands\n(Dated: November 1, 2018)\nIn the presence of spatial variation of the magnetization di rection, electric current noise causes a\nﬂuctuatingspin-transfer torque that increases the ﬂuctua tions of the ferromagnetic order parameter.\nBy the ﬂuctuation-dissipation theorem, the equilibrium ﬂu ctuations are related to the magnetiza-\ntion damping, which in non-uniform ferromagnets acquires a nonlocal tensor structure. In biased\nferromagnets, shot noise can become the dominant contribut ion to the magnetization noise at low\ntemperatures. Considering spin spirals as a simple example , we show that the current-induced noise\nand damping is signiﬁcant.\nPACS numbers: 72.70.+m, 72.25.Mk, 75.75.+a\nElectric currents induce magnetization dynamics in\nferromagnets. Three decades ago, Berger1,2showed that\nan electric current passing through a ferromagnetic do-\nmain wall exerts a torque on the wall. The cause of this\nspin-transfer torque is the reorientation of spin angular\nmomentum experienced by the electrons as they adapt to\nthe continually changingmagnetization. Subsequently, it\nwas realized that the same eﬀect may also be present in\nmagnetic multilayers3. In the latter case, the torque may\ncause reversal of one of the layers, while in the former,\nit may cause domain wall motion. The early ideas have\nbeen conﬁrmed both theoretically and experimentally4.\nRecently, the importance of noise in current-induced\nmagnetization dynamics has drawn attention. Although\noften noise is undesired, it may in some cases be quite\nuseful. Wetzels et al.5showed that current-induced mag-\nnetization reversal of spin valves is substantially sped up\nby an increased level of current noise. The noisy cur-\nrent exerts a ﬂuctuating torque on the magnetization6.\nRavelosona et al.7reported observation of thermally as-\nsisted depinning of a narrow domain wall under a cur-\nrent. Thermally-assistedcurrent-driven domain wall mo-\ntion has also been studied theoretically8,9.\nThe present paper addresses current-induced magneti-\nzation noise in non-uniformly magnetized ferromagnets.\nThe spatial variationof the magnetization direction gives\nrise to increased magnetization noise; by a ﬂuctuating\nspin-transfer torque, electric current noise causes ﬂuc-\ntuations of the magnetic order parameter. We take\ninto account both thermal current noise and shot noise,\nand show that the resulting magnetization noise is well\nrepresented by introducing ﬁctitious stochastic magnetic\nﬁelds. By the ﬂuctuation-dissipation theorem (FDT),\nthe thermal stochastic ﬁeld is related to the dissipation\nof energy, or damping, of the magnetization. The FDT\nhence constitutes a simple and eﬃcient way to evaluate\nthe damping, providing also a physical explanation in\ntermsofcurrentnoiseand spin-transfertorque. Since the\ncorrelator of the stochastic ﬁeld in general is inhomoge-\nnous and anisotropic, the damping is a nonlocal tensor.\nAs a simple and illuminating example we consider ferro-magnetic spin spirals, for which the ﬁeld correlator and\ndamping become spatially independent. It is shown that\nfor spirals with relatively short wavelength ( ∼20nm),\nthe current-induced noise and damping is substantial.\nSince half a wavelength of a spin spiral can be consid-\nered as a simple model for a domain wall, this suggests\nthat current-induced magnetization noise and damping\nshould be an issue for narrow domain walls.\nIt is instructive to start with an introduction to the\nFDT for uniform (single-domain) ferromagneticsystems,\ncharacterized by a time-dependent unit magnetization\nvectorm(t) and magnetization magnitude Ms(the sat-\nuration magnetization). The spontaneous equilibrium\nnoise of such macrospins is convenientlydescribed by the\ncorrelator Sij(t−t′) =/angbracketleftδmi(t)δmj(t′)/angbracketright, whereδmi(t) =\nmi(t)−/angbracketleftmi(t)/angbracketrightis the random deviation of the magne-\ntization from the mean value at time t. The brackets\ndenote statistical averaging at equilibrium, while iandj\ndenote Cartesian components. The magnetization ﬂuc-\ntuations are assumed weak, so that they to ﬁrst order are\npurely transverse to the equilibrium (average) direction\nof magnetization. Applying a weak external magnetic\nﬁeldh(ext)(t), the magnetization can be excited from the\nequilibrium state. Assuming linear response, the result-\ning transverse change in magnetization is\n∆mi(t) =/summationdisplay\nj/integraldisplay\ndt′χij(t−t′)h(ext)\nj(t′),(1)\ndeﬁning the transverse magnetic susceptibility χij(t−t′)\nas the causal response function. The FDT relates this\nsusceptibility to the equilibrium noise correlator10:\nSij(t−t′) =kBT\nMsV/integraldisplay\ndωe−iω(t−t′)χij(ω)−χ∗\nji(ω)\ni2πω,(2)\nwhereTis the temperature and Vis the volume of the\nferromagnet.\nThe spontaneous equilibrium ﬂuctuations δm(t) may\nbe regardedto be caused by a ﬁctitious random magnetic\nﬁeldh(t) with zero mean. We can derive an alternative\nform of the FDT in terms of the correlator /angbracketlefthi(t)hj(t′)/angbracketright.2\nTo do so, simply note that Eq. (1) implies that δmi(ω) =/summationtext\njχij(ω)hj(ω) in Fourier space. Inverting this relation,\nit follows from Eq. (2) that\n/angbracketlefthi(t)hj(t′)/angbracketright=kBT\nMsV/integraldisplay\ndωe−iω(t−t′)[χ−1\nji(ω)]∗−χ−1\nij(ω)\ni2πω,\n(3)\nwhereχ−1\nij(ω) is theij-component of the Fourier trans-\nformed inverse susceptibility tensor.\nThe magnetic susceptibility can be found from the\nLandau-Lifshitz-Gilbert (LLG) equation of motion. The\nstochasticLLGequationdescribesmagnetizationdynam-\nics and noise in both uniform as well as non-uniform fer-\nromagnets, and reads\ndm\ndt=−γm×[Heﬀ+h+h(ext)]+α0m×dm\ndt.(4)\nHereγisthegyromagneticratio, Heﬀisaneﬀectivestatic\nmagnetic ﬁeld determining the equilibrium state, h(t) is\nthe aboverandomnoise-ﬁeld, h(ext)(t) is the weakexcita-\ntionintroducedin Eq. (1), and α0isthe Gilbert damping\nconstant. Linearizing this equation in the magnetic re-\nsponse to h(ext)(t), we ﬁnd the inverse susceptibility\nχ−1=1\nγ/bracketleftbigg\nγ|Heﬀ|−iωα0iω\n−iω γ|Heﬀ|−iωα0/bracketrightbigg\n(5)\nwritteninmatrix(tensor)formintheplanenormaltothe\nequilibriummagnetizationdirection. Notethat the static\nﬁeld has here been assumed local and magnetization in-\ndependent. While not valid in most realistic situations,\nthis simple form for the eﬀective ﬁeld captures the key\nphysics of interest here, since only the dissipative part\nof the susceptibility (the Gilbert damping term) aﬀects\nthe noise. Inserting Eq. (5) into Eq. (3), we get the\nwell-known result11\n/angbracketlefthi(t)hj(t′)/angbracketright=2kBTα0\nγMsVδijδ(t−t′),(6)\nwhereiandjdenote components orthogonalto the equi-\nlibrium magnetization direction. This expression relates\nthe equilibrium noise, in terms of h, to the damping or\ndissipationofenergyin the ferromagnet. It maybe noted\nthat in thin ferromagneticﬁlms in good electrical contact\nwith a metal, the equilibrium noise and corresponding\nGilbert damping has been shown to be substantially en-\nhanced. This is due to the transfer of transverse spin\ncurrent ﬂuctuations in the neighbouring metal to the\nmagnetization6,12.\nWe now turn our attention to a more complex sys-\ntem, i.e., a metallic ferromagnet whose direction of mag-\nnetization mis varying along some direction in space,\nsay, the y-axis. It is assumed that the spatial variation\nis adiabatic, i.e., slow on the scale of the ferromagnetic\ncoherence length. The ferromagnet is furthermore as-\nsumed to be translationally invariant in the x- andz-\ndirections, and its magnetization magnitude is taken to\nbe constant and equal to the saturation magnetizationMs. In general, the dynamics and ﬂuctuations of such\na magnetization texture depend on position. Due to the\nspatial variation of the magnetization, longitudinal (i.e.,\npolarized parallel with the magnetization) spin current\nﬂuctuations transfer spin angular momentum to the fer-\nromagnet. The resulting enhancement of the magnetiza-\ntion noise is described by introducing a random magnetic\nﬁeld, whose correlator is inhomogenous and anisotropic,\nunlike Eq. (6). By the FDT, the correlator is related\nto the magnetization damping, that acquires a nonlo-\ncal tensor structure. In the following we make use of\nthe fact that the time scale of electronic motion is much\nshorter than the typical precession period of magneti-\nzation dynamics, as implicitly done already in Eq. (6).\nWe shall disregard spin-ﬂip processes and the associated\nnoise. Spin-ﬂipcorrectionsinFe, Ni, andCoareexpected\nto be small because the spin-ﬂip lengths are long com-\npared to the length scale of spatial variation (domain\nwall width) we consider. Spin-ﬂip is important in Py.\nHowever, domain walls in Py are so wide that the eﬀects\ndiscussed here are not important anyway. We therefore\ndo not discuss spin-ﬂip scattering.\nIt is convenient to transform the magnetization tex-\nture to a rotated reference frame, deﬁned in terms of the\nequilibrium (average) magnetization direction m0(y) =\n/angbracketleftm(y,t)/angbracketrightof the texture. The three orthonormal unit\nvectors spanning this position-dependent frame is ˆ v1=\nˆ v2×ˆ v3,ˆ v2= (dm0/dy)/|dm0/dy|andˆ v3=m0. The\nlocal gauge\nU(y) =/bracketleftbigˆ v1(y)ˆ v2(y)ˆ v3(y)/bracketrightbigT, (7)\ntransforms the magnetization, and hence the relevant\nequations involving the magnetization, to this frame.\nThat is, Um0(y)≡˜m0=ˆ z, where the tilde indicates\na vector in the transformed frame. We note also that\nUˆ v1=ˆ xandUˆ v2=ˆ y, and that Uis orthogonal, i.e.,\nU−1=UT= [ˆ v1ˆ v2ˆ v3].\nWe consider a charge current Iﬂowing through the\nferromagnet along the y-axis. Assuming that the equi-\nlibrium magnetization direction m0(y) changes adiabat-\nically, the electrons spins align with the changing mag-\nnetization direction when propagating through the tex-\nture. Thespincurrentisthenanywherelongitudinal,and\nhence given by Is(y) =Ism0(y). The alignment of the\nelectrons spins causes a torque τ(y) =dIs(y)/dyon the\nferromagnet. Since dIs(y)/dyis perpendicular to m0(y),\nthe torque can be written τ(y) =−m0(y)×[m0(y)×\ndIs(y)/dy], or˜τ(y) =Uτ(y) =−˜m0×[˜m0×UdIs(y)/dy]\nin the transformed representation. When I= 0, which\nwe will take in the following, Is= 0 and ˜τ= 0, on aver-\nage. However, at T/negationslash= 0 thermal ﬂuctuations of the spin\ncurrent result in a ﬂuctuating spin-transfer torque\n∆˜τ(y,t) =−∆Is(t)˜m0×[˜m0×Udm0(y)\ndy],(8)\nwhere ∆Is(t) are the time-dependent spin current ﬂuctu-\nationswith zeromean, propagatingalongthe y-direction.3\nThe action of the ﬂuctuating torque on the magnetiza-\ntion is described by the LLG equation if we, by conserva-\ntion of angular momentum, add the term γ∆τ/(MsA)\non the right hand side. Here Ais the cross section\n(in thexz-plane) of the ferromagnet. Linearizing and\ntransforming the LLG equation to the rotated reference\nframe, it is seen that the ﬂuctuating torque (8) can\nbe represented by a random magnetic ﬁeld ˜h′(y,t) =\n∆Is(t)/MsA)[˜m0×Udm0(y)/dy], analogous to h(t) dis-\ncussed above. Using Eq. (7)\n˜h′(y,t) =−∆Is(t)\nMsA/vextendsingle/vextendsingle/vextendsingle/vextendsingledm0(y)\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆx, (9)\ni.e., the (transformed) current-induced random ﬁeld\npoints in the x-direction.\nThe longitudinal spin current ﬂuctuations ∆ Is(t) can\nbe found by Landauer-B¨ uttiker scattering theory6,13.\nDisregarding spin-ﬂip processes, the spin-up and spin-\ndown electrons ﬂow in diﬀerent and independent chan-\nnels. In the low-frequency regime, in which charge is in-\nstantly conserved, longitudinal spin current ﬂuctuations\nare perfectly correlated throughout the entire ferromag-\nnet. Hence, the thermal spin current ﬂuctuations are\ngiven by6,13\n/angbracketleft∆Is(t)∆Is(t′)/angbracketright=¯h2\n(2e)22kBT(G↑+G↓)δ(t−t′),(10)\nwhereG↑(↓)istheconductanceforelectronswiththespin\naligned (anti)parallel with the magnetization. This ex-\npression is simply the Johnson-Nyquist noise generalized\nto spin currents6. We ﬁnd from Eqs. (9) and (10)\n/angbracketleft˜h′\nx(y,t)˜h′\nx(y′,t′)/angbracketright=2kBTξxx(y,y′)\nγMsVδ(t−t′) (11)\nfor the correlator of the current induced random ﬁeld.\nHere we have deﬁned\nξxx(y,y′) =γ¯h2σ\n4e2Ms/vextendsingle/vextendsingle/vextendsingle/vextendsingledm0(y)\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledm0(y′)\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingle(12)\nwithσ= (G↑+G↓)L/Athe total conductivity. Recall\nthat˜h′\ny(t) =˜h′\nz(t) = 0. Eq. (11) describes the nonlo-\ncal anisotropic magnetization noise due to thermal cur-\nrent ﬂuctuations in adiabatic non-uniform ferromagnets.\nThis excess noise vanishes with the spatial variation of\nthe magnetization. As a consequence of Eq. (10), the\nrandom ﬁeld correlator depends nonlocally on the mag-\nnetization gradient.\nAccording to the FDT, the thermal noise is related to\nthe magnetization damping. Since the noise correlator\n(11) is inhomogeneous and anisotropic, the correspond-\ning damping must in general be a nonlocal tensor. To\nevaluate the damping, we hence need the spatially re-xyz\n/c108/c47/c50/c113\nFIG. 1: An example of a non-uniform ferromagnet. The mag-\nnetization rotates with wavelength λin theyz-plane, forming\na spin spiral.\nsolved version of the FDT, which reads\n/angbracketleftδ˜mi(y,t)δ˜mj(y′,t′)/angbracketright=kBT\nMsA/integraldisplay\ndωe−iω(t−t′)\n×χij(y,y′,ω)−χ∗\nji(y′,y,ω)\ni2πω,\n(13)\nin the transformed representation. Here δ˜m(y,t) =\nUδm(y,t) =δmx(y,t)ˆ x+δmy(y,t)ˆ yare the spatially\ndependent transformed magnetization ﬂuctuations. The\nsusceptibility is deﬁned by\n∆˜mi(y,t) =/summationdisplay\nj/integraldisplay /integraldisplay\ndy′dt′χij(y,y′,t−t′)˜h(ext)\nj(y′,t′),\n(14)\nanalogous to Eq. (1), but with the external ﬁeld\nand magnetic excitations transformed: ˜h(ext)\nj(y,t) =\nUh(ext)\nj(y,t) and ∆ ˜m(y,t) =U∆m(y,t). The suscep-\ntibility in the local gauge frame diﬀers from Eq. (5) and\nhas to be determined. It is straightforward to gener-\nalize Eqs. (13) and (14) to the case of general three-\ndimensional dynamics.\nWe may substitute ˜h(ext)\nj(y′,t′) by˜h′\nj(y′,t′) in Eq. (14)\ntoﬁndthe ﬂuctuations δ˜m(y,t)ofthemagnetizationvec-\ntor caused by the spin-transfer torque. Combining this\nexpression with Eqs. (13) and (11), we arrive at an inte-\ngral equation for the unknown susceptibility, from which\nthe nonlocal tensor damping follows. Instead of ﬁnding\na numerical solution for an arbitrary texture, we con-\nsider here a ferromagnetic spin spiral as shown in Fig. 1,\nfor which the description of magnetization noise can be\nmappedontothemacrospinproblem. Asimpleanalytical\nresult can then be found, allowing for a comparison with\nEq. (6), and hence an estimate of the relative strength\nand importance of the current-induced noise and damp-\ning.\nSpin spirals can be found in some rare earth metals14\nand in the γ-phase of iron15, and are described by\nm0(y) = [0,sinθ(y),cosθ(y)], where θ(y) = 2πy/λ=qy,\nwithλthe wavelength of the spiral. Then dm0(y)/dy=\nq[0,cosθ(y),−sinθ(y)] so that|dm0(y)/dy|=q. As em-\nphasized earlier, our theory is applicable when the wave-\nlengthismuchlargerthanthemagneticcoherencelength.4\nFor transition metal ferromagnets, the coherence length\nis of the order of a few ˚ angstr¨ om. From Eq. (12) we\nﬁndξxx=γ¯h2σq2/(4e2Ms). The current-induced noise\ncorrelator (11) for spin spirals is hence homogeneous,\n/angbracketleft˜h′\nx(t)˜h′\nx(t′)/angbracketright=2kBTξxx\nγMsVδ(t−t′),(15)\nsimilar to Eq. (6), but anisotropic. The problem of relat-\ning noise to damping in terms of the FDT can therefore\nbe mapped exactly onto the macrospin problem: The\ntransformation (7) can be used to show that equations\nanalogous to Eqs. (1)-(6) are valid for the spin spiral,\nwhen analyzed in the local gauge frame. It is then seen\nthat the damping term corresponding to Eq. (15) is\n˜m×← →ξd˜m\ndt(16)\nin the transformed representation. Here\n← →ξ=/parenleftbigg\nξxx0\n0 0/parenrightbigg\n(17)\nis the 2×2 tensor Gilbert damping in the xy-plane.\nHence,ξxxis the enhancement of the Gilbert damping\ncaused by the spatial variation of the magnetization and\nthe spin-transfer torque. Due to its anisotropic nature,← →ξis inside the cross product in Eq. (16), ensuring that\nthe LLG equation preserves the length of the unit mag-\nnetization vector ˜m.\nIn order to get a feeling for the signiﬁcance of the\ncurrent-induced noise and damping, we evaluate← →ξnu-\nmerically for a spin spiral with wavelength 20 nm, and\ncompare with α0. Taking parameter values for α0,Ms,\nandσfrom Refs.16,17,18,19, we ﬁnd ξxx≈5α0for Fe (with\nα0= 0.002), and ξxx≈4α0for Co (with α0= 0.005).\nHence, current-induced noise and damping in spin spi-\nrals can be substantial. Considering half a wavelength\nof the spin spiral as a simple domain wall proﬁle, these\nresults furthermore suggest that a signiﬁcant current-\ninduced magnetization noise and damping should be ex-\npected in narrow (width ∼10 nm) domain walls in typ-\nical transition metal ferromagnets. The increased noise\nlevel should assist both ﬁeld- and current-induced do-\nmain walldepinning7,9,20. The increaseddamping shouldbe important for the velocity of current-driven walls,\nwhich recent theoretical and experimental advances sug-\ngest is inversely proportional to the damping4. The in-\ncreased noise and the tensor nature of the Gilbert damp-\ning should be taken into account in micromagnetic sim-\nulations.\nSo far we have only considered thermal current noise;\nlet us ﬁnally turn to shot noise. With the voltage U\nacross the ferromagnet turned on, a nonzero current I\nﬂows in the y-direction. Disregarding spin-ﬂip processes,\nthe resulting spin current shot noise is6,13\n/angbracketleft∆I(sh)\ns(t)∆I(sh)\ns(t′)/angbracketright=¯h2\n(2e)2eUFGδ(t−t′) (18)\nat zero temperature. Here the superscript (sh) empha-\nsizesthatwearenowlookingatshotnoise. TheFanofac-\ntorFis between 0 and 1 for non-interacting electrons21.\nWhen the length of the metal exceeds the electron-\nphonon scattering length λep, shot noise vanishes13,21.\nλepis strongly temperature dependent, and can at low\ntemperatures exceed one micron in metals. To ﬁnd the\ncontribution from shot noise to the magnetization noise,\nsimply replace Eq. (10) with Eq. (18) in the above cal-\nculation of the random-ﬁeld correlator. In experiments\non current-induced domain wall motion, typical applied\ncurrent densities are about j= 108A/cm24. At low\ntemperatures, the ratio of shot noise to thermal current\nnoise,eUF/2kBT, can then exceed unity for long (but\nnot longer than λep) ferromagnetic (e.g. Fe) wires. Shot\nnoise can hence be expected to be the dominant contri-\nbution to the magnetization noise at low temperatures.\nIn summary, we have calculated current-induced mag-\nnetization noise and damping in non-uniform ferromag-\nnets. Taking into account both thermal and shot noise,\nwe evaluated the ﬂuctuating spin-transfer torque on the\nmagnetization. The resulting magnetization noise was\ncalculated in terms of a random magnetic ﬁeld. Em-\nploying the FDT, the corresponding enhancement of the\nGilbert damping was identiﬁed for spin spirals.\nThis work was supported in part by the Research\nCouncil of Norway, NANOMAT Grants No. 158518/143\nand 158547/431, and EC Contract IST-033749 “Dyna-\nMax”.\n1L. Berger, J. Appl. Phys. 49, 2156 (1978).\n2L. Berger, J. Appl. Phys. 55, 1954 (1984).\n3J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n4C. H. Marrows, Adv. Phys. 54, 585 (2005).\n5W. Wetzels, G. E. W. Bauer, and O. N. Jouravlev, Phys.\nRev. Lett. 96, 127203 (2006).\n6J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 95, 016601 (2005).\n7D. Ravelosona, D. Lacour, J. A. Katine, B. D. Terris, and\nC. Chappert, Phys. Rev. Lett. 95, 117203 (2005).8G. Tatara, N. Vernier, and J. Ferr´ e, Appl. Phys. Lett. 86,\n252509 (2005).\n9R. A. Duine, A. S. N´ u˜ nez, and A. H. MacDonald, Phys.\nRev. Lett. 98, 056605 (2007).\n10L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Statis-\ntical Physics, Part 1 (Pergamon Press, 1980), 3rd ed.\n11W. F. Brown, Phys. Rev. 130, 1677 (1963).\n12Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n13Y. M. Blanter and M. B¨ uttiker, Phys. Rep. 336, 1 (2000).5\n14J. Jensen and A. K. Mackintosh, Rare Earth Magnetism\n(Oxford University Press, 1991).\n15M. Marsman and J. Hafner, Phys. Rev. B 66, 224409\n(2002).\n16S. M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974).\n17K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n18CRC Handbook of Chemistry and Physics (CRC Press,1985), 66th ed.\n19American Institute of Physics Handbook (McGraw-Hill,\n1963), 2nd ed.\n20J. P. Attan´ e, D. Ravelosona, A. Marty, Y. Samson, and\nC. Chappert, Phys. Rev. Lett. 96, 147204 (2006).\n21A. H. Steinbach, J. M. Martinis, and M. H. Devoret, Phys.\nRev. Lett. 76, 3806 (1996)."    },    {        "title": "1903.06896v1._S___1_2_ferromagnetic_Heisenberg_chain_in_a_verdazyl_based_complex.pdf",        "content": "APS/123-QED\nSecond institution and/or address This line break forced\nS=1/2 ferromagnetic Heisenberg chain in a verdazyl-based complex\nN. Uemoto1, Y. Kono2, S. Kittaka2, T. Sakakibara2, T. Yajima2,\nY. Iwasaki1, S. Miyamoto1, Y. Hosokoshi1, and H. Yamaguchi1\n1Department of Physical Science,\nOsaka Prefecture University, Osaka 599-8531, Japan\n2Institute for Solid State Physics,\nthe University of Tokyo, Chiba 277-8581, Japan\n(Dated: March 19, 2019)\nWe present a model compound for the S=1/2 ferromagnetic Heisenberg chain composed of the\nverdazyl-based complex [Zn(hfac) 2][4-Cl-o-Py-V-(4-F) 2].Ab initio MO calculations indicate a pre-\ndominant ferromagnetic interaction forming an S=1/2 ferromagnetic chain. The magnetic suscep-\ntibility and speci\fc heat indicate a phase transition to an AF order owing to the \fnite interchain\ncouplings. We explain the magnetic susceptibility and magnetization curve above the phase tran-\nsition temperature based on the S=1/2 ferromagnetic Heisenberg chain. The magnetization curve\nin the ordered phase is described by a conventional AF two-sublattice model. Furthermore, the\nobtained magnetic speci\fc heat reproduces the almost temperature-independent behavior of the\nS=1/2 ferromagnetic Heisenberg chain. In the low-temperature region, the magnetic speci\fc heat\nexhibitsp\nTdependence, which is attributed to the energy dispersion in the ferromagnetic chain.\nPACS numbers: 75.10.Jm,\nI. INTRODUCTION\nOne-dimensional (1D) spin systems have been inten-\nsively investigated both experimentally and theoretically,\nover the last several decades. In particular, the S= 1/2\nantiferromagnetic (AF) Heisenberg chain is of fundamen-\ntal importance in strongly correlated quantum many-\nbody systems, because it is one of the few systems for\nwhich a nontrivial ground state is precisely known. The\nground state of the S= 1/2 AF Heisenberg chain is\ndescribed by a Tomonaga-Luttinger liquid, which is a\nquantum critical state with a k-linear energy disper-\nsion [1]. Such linear dispersion results in the well-known\nT-linear behavior of the low-temperature speci\fc heat.\nAlthough many examples of the S= 1/2 AF Heisenberg\nchain have been reported so far, the number of exam-\nples ofS= 1/2 ferromagnetic Heisenberg chain is much\nsmaller. Nevertheless, theoretical investigation of the S\n= 1/2 ferromagnetic Heisenberg chain have been per-\nformed over many decades. The ground state is a fer-\nromagnetic order, but there is no ordered state at any\n\fnite temperature. Bethe demonstrated that the exact\nmany-body wave function of such a system can be ex-\npressed analytically [2], and Bonner and Fisher studied\nthe thermodynamic properties [3]. Takahashi explained\nthe low-temperature thermodynamic properties by using\nthe modi\fed spin-wave approximation [4]. It is con\frmed\nthat the magnetic susceptibility diverges as T\u00002with de-\ncreasing temperature and that the low-temperature spe-\nci\fc heat exhibits ap\nTdependence associated with the\nk2dependence of the energy dispersion [5{8]. In the case\nof an actual quasi-1D ferromagnetic system, theoretical\nstudy suggests that predominant 1D ferromagnetic inter-actions realize a dimensional crossover of Bose-Einstein\ncondensation (BEC) [9].\nFrom the experimental point of view, several Cu com-\nplexes have been actively investigated as the best real-\nizations of the S= 1/2 ferromagnetic Heisenberg chain\nmainly in the 1980s [10{12]. In (C 6H11NH3)CuX 3(X=Cl\nand Br), the almost temperature-independent speci\fc\nheats are well explained by using ferromagnetic chain\nmodels, while low-temperaturep\nTdependence is not\nobserved due to the appearance of large peaks asso-\nciated with the phase transition to long-range-order\n(LRO) [12]. Neutron scattering studies on these Cu\ncomplexes demonstrated the k2dependence of the spin-\nwave dispersion in the chain direction [13, 14]. Although\nthese Cu complexes have strong ferromagnetic chain\ninteractions, their magnetic properties exhibit slightly\nanisotropic behavior. Thus, organic radical compounds\nwith isotropic nature owing to orbital quenching have\nattracted interest as a model system of ferromagnetic\nHeisenberg chain. From the 1990s, several organic rad-\nical compounds, e.g. \f-p-NPNN,\r-p-NPNN, BImNN,\nF4BImNN, and \f-BBDTA-GaBr 4, have been reported\nto form the S= 1/2 ferromagnetic Heisenberg chain [15{\n22]. Their magnetic properties in the ordered phase and\nvicinity of the phase transition temperature are examined\nin connection with the pursuit for organic magnet.\nIn order to design Heisenberg spin systems, we have\nalso focused on organic radical compounds. One of the\nsuitable candidates is verdazyl radical, which can ex-\nhibit a delocalized \u0019-electron spin density in nonplanar\nmolecular structures. The \rexibility of the molecular or-\nbitals in the verdazyl radical o\u000bers tunability of the in-\ntermolecular magnetic interactions by molecular design.\nRecently, we demonstrated that the verdazyl radical canarXiv:1903.06896v1  [cond-mat.str-el]  16 Mar 20192\nform a variety of unconventional S= 1/2 Heisenberg spin\nsystems, such as the quantum pentagon, random honey-\ncomb, and fully-frustrated square lattice, which have not\nbeen realized in conventional inorganic materials [23{25].\nFurthermore, in contrast to conventional organic radical\nsystems, the verdazyl radical facilitates the formation of\nferromagnetic interactions, and thus we have realized a\nvariety ofS= 1/2 Heisenberg spin systems with 1D fer-\nromagnetic interactions [26{28]. For instance, antiferro-\nmagnetically coupled ferromagnetic chains form a two-\ndimensional honeycomb lattice in 2-Cl-6-F-V [27], and\nthe frustrated square lattice in ( o-MePy-V)PF 6contains\nalternating ferromagnetic chains [28]. In 3-I-V, which\nforms a spin ladder with a predominant ferromagnetic-\nleg interaction, quasi-1D BEC was actually observed near\nthe saturation \feld [29].\nIn this paper, we present a new verdazyl-based\ncomplex. We successfully synthesized single crystals\nof [Zn(hfac) 2][4-Cl-o-Py-V-(4-F) 2] [hfac = 1,1,1,5,5,5-\nhexa\ruoroacetylacetonate, 4-Cl- o-Py-V-(4-F) 2= 3-(4-\nCl-2-pyridyl)-1,5-bis(4-\ruorophenyl)-diphenylverdazyl].\nAb initio molecular orbital (MO) calculations indicate\nthat a predominant ferromagnetic interaction forms an\nS=1/2 ferromagnetic chain. The magnetic susceptibility\nand speci\fc heat show a phase transition to an AF order\nowing to the \fnite AF interchain couplings. We explain\nthe magnetic susceptibility, magnetization curve, and\nmagnetic speci\fc heat above the phase transition tem-\nperature based on the S=1/2 ferromagnetic Heisenberg\nchain. Furthermore, we observep\nTdependence of the\nmagnetic speci\fc heat in the low-temperature region.\nII. EXPERIMENTAL AND NUMERICAL\nMETHOD\nWe synthesized 4-Cl- o-Py-V-(4-F) 2through a conven-\ntional procedure for verdazyl radical [30]. A solution of\n[Zn(hfac) 2]\u00012H2O (510 mg, 0.99 mmol) in 15 ml of hep-\ntane was re\ruxed at 60\u000eC. A solution of 4-Cl- o-Py-V-(4-\nF)2(380 mg, 0.99 mmol) in 2 ml of CH 2Cl2was slowly\nadded, and stirring was continued for 1 h. After the\nmixed solution cooled to room temperature, a dark-green\ncrystalline solid of [Zn(hfac) 2][4-Cl-o-Py-V-(4-F) 2] was\nseparated by \fltration and washed with heptane. The\ndark-green residue was recrystallized using acetonitrile\nat 10\u000eC.\nSingle crystal X-ray di\u000braction (XRD) experiment\nwas performed by using a di\u000bractometer with an imag-\ning plate (R-AXIS RAPID, Rigaku) with graphite-\nmonochromated Mo-K \u000bradiation at room temperature.\nThe single crystal XRD data are re\fned by using the\nSHELX software [31]. The structural re\fnement was car-\nried out using anisotropic and isotropic thermal parame-\nters for the nonhydrogen atoms and the hydrogen atoms,\nrespectively. All the hydrogen atoms were placed at the\ncalculated ideal positions\nThe magnetic susceptibility and magnetization curveswere measured using a commercial SQUID magnetome-\nter (MPMS-XL, Quantum Design) above 1.8 K and a\ncapacitive Faraday magnetometer with a dilution refrig-\nerator down to 80 mK. The experimental results were cor-\nrected for the diamagnetic contribution of \u00001:65\u000210\u00004\nemu mol\u00001, which is determined based on the QMC anal-\nysis to be described and close to that calculated by Pas-\ncalfs method. The speci\fc heat was measured with a\ncommercial calorimeter (PPMS, Quantum Design) us-\ning a thermal relaxation method above 1.9 K and a\nhandmade apparatus by a standard adiabatic heat-pulse\nmethod with a dilution refrigerator down to about \u001870\nmK. There is no signi\fcant di\u000berence in magnetic prop-\nerties between the \feld directions owing to the isotropic\ngvalue of\u00182.00 in verdazyl radical systems. Therefore,\nall experiments were performed using small randomly ori-\nented single crystals.\nAb initio MO calculations were performed using the\nUB3LYP method with the basis set 6-31G in the Gaus-\nsian 09 program package. The convergence criterion was\nset at 10\u00008hartree. For the estimation of intermolecular\nmagnetic interaction, we applied our evaluation scheme\nthat have been studied previously [32].\nThe QMC code is based on the directed loop algorithm\nin the stochastic series expansion representation [33].\nThe calculations for the S= 1/2 ferromagnetic Heisen-\nberg chain was performed for N= 1024 under the pe-\nriodic boundary condition, where Ndenotes the system\nsize. It was con\frmed that there is no signi\fcant size-\ndependent e\u000bect. All calculations were carried out using\nthe ALPS application [34, 35].\nIII. RESULTS AND DISCUSSION\nA. Crystal structure and magnetic model\nThe crystallographic data for the synthesized\n[Zn(hfac) 2][4-Cl-o-Py-V-(4-F) 2] are summarized in Table\nI, and the molecular structure is shown in Fig. 1(a). The\nverdazyl ring (which includes four N atoms), the upper\ntwo phenyl rings, and the bottom pyridine ring are\nlabeled R 1, R2, R3, and R 4, respectively. The dihedral\nangles of R 1-R2, R 1-R3, R 1-R4are approximately\n9\u000e, 51\u000e, and 9\u000e, respectively. Each molecule has a\ndelocalized S=1/2. The result of the MO calculation\nfor [Zn(hfac) 2][4-Cl-o-Py-V-(4-F) 2] molecule indicate\nthat approximately 61 % of the total spin density is\npresent on R 1. Further, while R 2and R 3each account\nfor approximately 14 % and 18 % of the relatively large\ntotal spin density, R 4accounts for less than 7 % of\nthe total spin density. Therefore, the intermolecular\ninteractions are caused by the short contacts related to\nthe R 1, R2, and R 3rings. Since Zn(hfac) 2has a low spin\ndensity less than 1 % of the total spin density, it works\nas a spacer between verdazyl radicals, resulting in the\nlow dimensionality of the spin model. We focus on the\nstructural features related to the 4-Cl- o-Py-V-(4-F) 2to3\nTABLE I: Crystallographic data for [Zn(hfac) 2][4-Cl-o-Py-V-\n(4-F) 2].\nFormula C 29H15ClF14N5O4Zn\nCrystal system Orthorhombic\nSpace group Pbca\na=\u0017A 10.7183(6)\nb=\u0017A 19.0465(10)\nc=\u0017A 32.8749(16)\nV/\u0017A36711.3(6)\nZ 8\nDcalc/g cm\u000031.711\nTemperature RT\nRadiation Mo K \u000b(\u0015= 0.71075 \u0017A)\nTotal re\rections 7625\nRe\rection used 3283\nParameters re\fned 487\nR[I >2\u001b(I)] 0.0673\nRw[I >2\u001b(I)] 0.1697\nGoodness of \ft 1.002\nCCDC 1889996\nconsider intermolecular interactions.\nWe evaluated the intermolecular exchange interactions\nof all molecular pairs within 4.0 \u0017A through the ab initio\nMO calculations and found one predominant ferromag-\nnetic interaction. Its value was evaluated as J=kB=\n\u000010:7 K, which is de\fned in the Heisenberg spin Hamilto-\nnian given byH=JP\n<i;j>Si\u0001Sj, whereP\n<i;j>denotes\nthe sum over the neighboring spin pairs. The molecular\npair associated with this interaction is related by an a-\nglide re\rection symmetry and has N-C short contact of\n3.51 \u0017A, as shown in Figs. 1(b) and (c). Accordingly, a\nuniformS= 1/2 ferromagnetic chain is formed by the\nexpected interaction Jalong thea-axis, as shown in Fig.\n1(d). The Zn(hfac) 2acts as a spacer between the 1D\nchains, as shown in Fig. 1(e). The other intermolecu-\nlar interactions associated with interchain couplings are\nevaluated to be less than approximately 1/100 of Jin\nabsolute values, which enhances the 1D character of the\npresent spin model. Considering strong dependence on\nthe calculation method and basis set in the MO calcula-\ntion, such small interchain couplings do not have enough\nreliability [36], and thus we evaluate the e\u000bective inter-\nchain couplings through a mean-\feld analysis of the ex-\nperimental results, as discussed later.\nB. Magnetic susceptibility\n@ Figure 2 shows the temperature dependence of the\nmagnetic susceptibility ( \u001f=M=H ) at 0.05 T. The con-\ntribution of JFappears in the temperature dependence of\n\u001fT, which increases with decreasing temperatures down\nto approximately 0.7 K, as shown in the upper inset of\nFig. 2. At temperature above 2 K, the Curie-Weiss lawis followed, \u001f=C=(T\u0000\u0012W). The estimated Curie con-\nstant is about C= 0.373 emu\u0001K/mol, which is close to\nthe expected value for noninteracting S= 1/2 spins, and\nthe Weiss temperature is estimated to be \u0012W= 3.2 K.\nBelow 0.7 K, \u001fTdecreases with decreasing temperature,\nindicating the existence of additional weak AF interac-\ntions. Furthermore, we observe a discontinuous change\nin\u001fat 0.31 K, which indicates a phase transition to a\nthree-dimensional LRO. In the ordered phase, \u001fbecomes\nalmost temperature independent, which suggests that a\nspin-\rop transition \feld is lower than 0.05 T as will be\ndiscussed later. As shown in the lower inset of Fig. 2,\nthe discontinuous change shifts to a lower temperature at\na higher \feld of 0.1 T, which indicates that the ordered\nstate is an AF ordering of the ferromagnetic chains.\nThe MO calculations show the formation of the S=\n1/2 ferromagnetic chain, and the one-dimensionality is\nenhanced owing to the almost nonmagnetic Zn(hfac) 2\nbetween the chain structures. Accordingly, we calcu-\nlated the magnetic susceptibility \u001fQMC for theS=\n1/2 ferromagnetic Heisenberg chain by using the QMC\nmethod. Given the radical-based compound, we assume\nthe isotropic g-value of 2.00. The experimental result\nexhibits the contribution of weak AF interchain interac-\ntions as represented by the broad peak of \u001fT, and thus\nwe consider an AF mean \feld to reproduce the observed\nbehavior. The mean-\feld approximation is represented\nas:\n\u001f=\u001fQMC\n1 + (zJ0=Ng2\u00162\nB)\u001fQMC(1)\nwherezis the number of nearest-neighbor spins, J0is\nthe interchain exchange interaction, Nis the number\nof spins, and \u0016Bis the Bohr magneton. We obtained\ngood agreement between the experimental and calculated\nresluts above the phase transition temperature, including\nthe broad peak of \u001fT, by using the parameters J=kB=\n\u00008.8 K andzJ0=kB= 0.32 K, as shown in Fig. 2 and its\nupper inset. A clear AF contribution can be con\frmed\nby comparing the calculated results of the isolated fer-\nromagnetic chain and the ferromagnetic chain with the\nAF mean \feld, as shown in Fig. 2 and its upper inset.\nThe Weiss temperature for the 1D chain is given by \u0012W\n=\u00002JS(S+ 1)=3kBfrom the mean-\feld approximation,\nand we obtaine \u0012W= 4.4 K, which is close to the evalu-\nation from the Curie-Weiss \ftting.\nC. Magnetization curve\nFigure 3(a) shows the magnetization curve at 1.8 K\nabove the phase transition temperature. The saturation\nvalue of 0.96 \u0016B/f.u. indicates that the purity of the radi-\ncals is approximately 96 %. The dominant ferromagnetic\ncontribution can be con\frmed by comparing the observed\nresult and the Brillouin function for free S= 1/2 spins,\nand the experimental result is well explained by the S=\n1/2 ferromagnetic Heisenberg chain with J=kB=\u00008.8 K,4\nFIG. 1: (color online) (a) Molecular structure of [Zn(hfac) 2][4-Cl-o-Py-V-(4-F) 2]. (b) Molecular pair associated with the\ndominant ferromagnetic interaction J. Hydrogen atoms are omitted for clarity. The broken line indicates N-C short contact.\n(c) Crystal structure forming a 1D chain along the a-axis, and (d) the corresponding S= 1/2 ferromagnetic chain. (e) Crystal\nstructure viewed parallel to the chain direction. The broken line encloses molecules comprising each chain structure.\nas shown in Fig. 3(a). The slight deviation is considered\nto originate from the contribution of weak AF interchain\ninteractions as in the case of the magnetic susceptibility.\nFigure 3(b) shows the magnetization curve at 0.08 K.\nThe samples used are the same as those for the magnetic\nsusceptibility, and the experimental result corresponds to\nthe magnetization in the AF ordered phase. We observe\nan almost linear increase with increasing \felds up to ap-\nproximately 0.15 T. The observed linear increase at 0.08\nK indicates that the magnetic behavior in the ordered\nphase can be described by a classical AF two-sublattice\nmodel, in which the spins on each ferromagnetic chain\nform one sublattice moment. The colinear two-sublattice\nis aligned along the easy-axis under zero-\feld conditions.\nFor the external \feld parallel to the the easy-axis, the dis-\ncontinuous spin-\rop phase transition occurs at a certain\n\feld. Above the spin-\rop transition \feld, two sublattices\nare tilted to the \feld direction with equivalent angles. For\nthe external \feld perpendicular to the easy-axis, two sub-\nlattices are tilted from the easy-axis with equivalent. In\nthe present case, we have not observed anomalous behav-\nior associated with the spin-\rop transition in the magne-\ntization curve of randomly oriented single crystals. Thus,\nthe transition \feld can be evaluated to be less than 0.01\nT, which is consistent with the small magnetic anisotropy\nin the organic radical systems. The magnetization curvebecomes almost identical for any \feld direction above the\nspin-\rop phase transition, yielding the almost tempera-\nture independent behavior of the magnetic susceptibility\nat 0.05 T in the ordered phase. Accordingly, we can con-\nsider an isotropic spin system for the analysis even in\nthe ordered phase. Two sublattices are coupled by the\nAF interchain interactions, and a mean-\feld approxima-\ntion gives the magnetization curve at T= 0, expressed as:\nMmean =g2\u0016BH=2zJ0, whereHis the external magnetic\n\feld. We obtained good agreement between the experi-\nmental and calculated results by using zJ0=kB= 0:21 K\n(as shown in Fig. 3(b)), which is close to the value ob-\ntained from the analysis of the magnetic susceptibility.\nD. Speci\fc heat\nThe experimental results for the speci\fc heat Cpat\nzero-\feld clearly exhibit a \u0015-type sharp peak, which is\nassociated with the phase transition to the AF LRO, as\nshown in Fig. 4(a). Because we used di\u000berent samples\nfrom those used for the magnetization measurements, the\nphase transition slightly shifts to a lower temperature.\nWe ascribe this disparity to a high sensitivity of the in-\nterchain couplings to the purity of the radicals and/or\nslight impurities. We evaluated the magnetic speci\fc5\nFIG. 2: (color online) Temperature dependence of magnetic\nsusceptibility ( \u001f=M=H ) of [Zn(hfac) 2][4-Cl-o-Py-V-(4-F) 2]\nat 0.05 T. The upper inset shows the temperature dependence\nof\u001fT. The solid lines with open circles and open squares\nrepresent the calculated results for the isolated ferromagnetic\nchain and the ferromagnetic chain with the AF mean \feld,\nrespectively. The lower inset shows \u001fin the vicinity of the\nphase transition temperature at 0.05 and 0.1 T.\nheatCmby subtracting the lattice contribution Cland\nassumedClin the low-temperature region approximated\nas:Cl=a1T3+a2T5+a3T7. The constants a1\u0000a3\nare determined to reproduce the following magnetic spe-\nci\fc heat calculated by the QMC method. As a result,\nClwith the constants a1= 0:12,a2=\u00003:6\u000210\u00003, and\na3= 5:7\u000210\u00005was evaluated. The value of a1cor-\nresponds to a Debye temperature of 25 K, which is not\nvery di\u000berent from those for other verdazyl radical com-\npounds, but is a slightly smaller value [26, 27]. This\nsmall Debye temperature reduces the applicable temper-\nature region of the Debye's T3law, and thus we had to\nalso consider T5andT7terms to evaluate the magnetic\nspeci\fc heat up to approximately 5 K. The smaller value\nof Debye temperature is consistent with the molecular\narrangement of the present compound, in which there\nis no large overlap of molecular orbitals causing strong\nAF coupling. We calculated the magnetic speci\fc heat\nfor theS= 1/2 ferromagnetic Heisenberg chain with\nJ=kB=\u00008.8 K by using the QMC method and obtained\ngood agreement between the experiment and calculation\nabove the phase transition temperature, as shown in Fig.\n4(a). Here, we con\frm that the calculated speci\fc heat\nis consistent with those reported in previous theoretical\nworks [3, 6{8].\nThe phase transition temperature decreases with in-\ncreasing \feld, as shown in Fig. 4(b). Such \feld depen-\nFIG. 3: (color online) Magnetization curve of [Zn(hfac) 2][4-\nCl-o-Py-V-(4-F) 2] at (a) 1.8 K and (b) 0.08 K. The broken\nline represents the Brillouin function for S= 1/2 at 1.8 K.\nThe solid lines represent the calculated results for (a) the S\n=1/2 ferromagnetic chain at 1.8 K by using QMC method\nand for (b) the AF two-sublattice model at zero temperature\nby using the mean-\feld approximation.\ndence of the phase transition temperature is consistent\nwith that for \u001fand also indicates the AF LRO. At 0.03\nand 0.1 T, the phase transition temperature becomes\nlower than the experimental temperature, but the magne-\ntization curve indicates that the spins are not fully polar-\nized yet. Thus, we expect that the low-temperature spe-\nci\fc heat above 0.03 T arises from the gapless 1D ferro-\nmagnetic dispersion. As shown in the inset of Fig.4(a), in\nthe low-temperature region, the magnetic speci\fc heats\nat 0.03 T and 0.1 T actually showp\nTbehavior, which\nclearly demonstrates the contribution of k2dispersion in\n1D ferromagnetic chain [3, 6{8].\nIV. SUMMARY\nWe successfully synthesized single crystals of the\nverdazyl-based complex [Zn(hfac) 2][4-Cl-o-Py-V-(4-F) 2].\nAb initio MO calculations indicated the formation of\nanS=1/2 ferromagnetic chain. The magnetic suscep-\ntibility and speci\fc heat indicated a phase transition to\nan AF order owing to weak interchain interactions, and\nthe low-temperature magnetization curve exhibited a lin-6\nFIG. 4: (color online) Speci\fc heat of [Zn(hfac) 2][4-Cl-o-Py-\nV-(4-F) 2]. (a) Total speci\fc heat Cpand its magnetic con-\ntributionCmat 0 T. The solid line with squares represents\nthe calculated result for the S=1/2 ferromagnetic Heisenberg\nchain. (b) Low-temperature region of Cmat 0, 0.01, 0.03,\nand 0.1 T. The inset shows thep\nTdependence of the low-\ntemperature Cmat 0.03 and 0.1 T.ear increase. We explained the magnetic susceptibility\nabove the phase transition temperature on the basis of\ntheS=1/2 ferromagnetic Heisenberg chain with an AF\nmean \feld by combining the QMC method and mean-\n\feld approximation. The magnetization curve in the or-\ndered phase was described by the conventional AF two-\nsublattice model considering the AF interchain couplings.\nFurthermore, we reproduced the almost temperature-\nindependent magnetic speci\fc heat for the S=1/2 fer-\nromagnetic Heisenberg chain. In the low-temperature\nregion, we observedp\nTdependence of the magnetic spe-\nci\fc heat, which indicates the contribution of k2disper-\nsion in the 1D ferromagnetic chain.\nAcknowledgments\nThis research was partly supported by Grant for\nBasic Science Research Projects from KAKENHI (No.\n17H04850 and No. 18H01164) and the Matsuda Foun-\ndation. A part of this work was performed as the joint-\nresearch program of ISSP, the University of Tokyo.\n[1] T. 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Yamaguchi, Polyhedron 20, 1235 (2001)."    },    {        "title": "0803.0085v1.Phase_separation_and_the_effect_of_quenched_disorder_in__Pr__0_5_Sr__0_5_MnO_3_.pdf",        "content": "arXiv:0803.0085v1  [cond-mat.str-el]  3 Mar 2008Phase separation and the eﬀect of quenched\ndisorder in Pr0.5Sr0.5MnO3\nA K Pramanik and A Banerjee\nUGC-DAE Consortium for Scientiﬁc Research, University Campus, K handwa Road\nIndore-452017, M.P, India.\nE-mail:alok@csr.ernet.in\nAbstract. The nature of phase separation in Pr0.5Sr0.5MnO3has been probed\nby linear as well as nonlinear magnetic susceptibilities and resistivity me asurements\nacross the 2nd order paramagnetic to ferromagnetic transition ( TC) and 1st order\nferromagnetic to antiferromagnetic transition ( TN). We found that the ferromagnetic\n(metallic) clusters, which form with the onset of long-range order in the system at\nTC, continuously decrease their size with the decrease in temperatur e and coexist\nwith non-ferromagnetic (insulating) clusters. These non-ferrom agnetic clusters are\nidentiﬁed to be antiferromagnetic. Signiﬁcantly, it is shown that the y do not arise\nbecause of the superheating eﬀect of the lower temperature 1st order transition. Thus\nreveals unique phase coexistence in a manganite around half-doping encompassing\ntwo long-range order transitions. Both the ferromagnetic and an tiferromagnetic\nclusters form at TCand persist much below TN. Substitution of quenched disorder\n(Ga) at Mn-site promotes antiferromagnetism at the cost of ferr omagnetism without\nadding any magnetic interaction or introducing any signiﬁcant lattice distortion.\nMoreover, increase in disorder decreases the ferromagnetic clus ter size and with 7.5%\nGa substitution clusters size reduces to the single domain limit. Yet, a ll the samples\nshow signiﬁcant short-range ferromagnetic interaction much abo veTC. Resistivity\nmeasurements also reveal the novel phase coexistence identiﬁed from the magnetic\nmeasurements. It is signiﬁcant that, increase in disorder up to 7.5% increases the\nresistivity of the low temperature antiferromagnetic phase by abo ut four orders.\nPACS numbers: 75.47.Lx, 75.30.Kz, 75.40.GbPhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 2\n1. Introduction\nPhysical properties and their variation with external stimuli or inte rnal disorder in\nhalf doped perovskite manganites with generic formula R0.5A0.5MnO3(where R and\nA respectively stand for trivalent rare earth and divalent alkaline ea rth elements) are\ntopics of signiﬁcant current interest [1, 2]. Coexistence of contr asting phases, namely,\nferromagnetic (FM) - metallic (M) and antiferromagnetic (AF) - insu lating (I) is a\ncommon occurrence in manganites around half doping [1]. This phase coexistence or\nphase separation (PS) is considered to be responsible factor for o bserved functional\nproperties as well as argued to be main impetus for bi-critical phase competition [2].\nHowever, the origin of PS and the nature of coexisting phases rema in a matter of\ndebate [3, 4]. It is widely considered that FM-M and AF-I phases have similar energies\nin manganites around half-doping as a result of which small perturba tions, compared to\nthermal energy, can cause colossal changes [1, 2, 5]. This proximit y in energies of two\ncontrasting phases remains despite considerable change in the nat ure of spin, charge\nand orbital ordering of the ground state with the variation in ionic ra dii of R/A atoms,\nwhich aﬀects the Mn-O-Mn bonds and the one electron bandwidth [1 , 2]. The decrease\nin average radii of R/A atoms brings about decrease in bandwidth an d the ground state\nof the system changes from so called A-type to CE-type AF struct ure. This indicates\nthat the nature of PS and the eﬀect of quenched disorder may not remain identical as\nspin, charge and orbital ordering changes across the bandwidth.\nIt has been shown that ‘ﬁnite-size’ clusters with large uncompensa ted spins are\ncreated when quenched disorder is introduced in the Mn-site of nar row bandwidth\nPr0.5Ca0.5MnO3having CE-type AF and charge ordered-insulating ground state [6].\nElectronic phase separation gives rise to interesting eﬀects within t hese clusters having\nsome similarities with other transition metal oxides like cuprates and n ickelates\n[7, 8]. Quenched disorder in the form of magnetic ions (a few % of Cr an d Co)\ninNd0.5Ca0.5MnO3andPr0.5Ca0.5MnO3respectively, gives rise to coexisting AF-I\nand FM-M phases with many unusual physical properties at low temp erature [9, 10].\nInterestingly, recent studies have shown that the ground state ofPr0.5Ca0.5MnO3\nchanges to FM-M even with minimal substitution of non-magnetic diso rder (2.5% Al) at\nMn-site, without introducing any signiﬁcant structural distortion [4, 11]. Signiﬁcantly,\nthe conductivity of Pr0.5Ca0.5MnO3increases when such disorder is introduced in\nthe Mn-O-Mn network [12]. Apart from chemical substitution, inter nal disorder in\nvarious forms also lead to remarkable eﬀects. For example, anisotr opic stress in\nNd0.5Sr0.5MnO3, having larger bandwidth than Pr0.5Ca0.5MnO3, has signiﬁcantly\nchanged the orbital ordering (OO) and other physical properties [13]. In addition,\nPr0.5Sr0.5MnO3, which is an intermediate bandwidth system with A-type AF-I\nground state, shows strain induced dimensionality crossover trigg ering a localization-\ndelocalization transition [14]. In the classic work of Imry and Ma [15], it was argued\nthat random quenched disorder arising from lattice defects, disloc ations or chemical\nsubstitution can destabilize the long range order system favoring f ormation of ‘ﬁnite-Phase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 3\nsize’ clusters. Appearance of coexisting clusters and their size re gulation by disorder in\nmanganites is also shown from simulation work by Moreo et al.[16]. In another study,\nBurgyet al.proposed chemical disorder driven inhomogeneous states across the ﬁrst\norder transition [17]. Recently, the fragility of both A-type and CE- type AF-I ground\nstate against quenched disorder is shown from computer simulation [18]. Following\nthese experimental and theoretical developments, it becomes imp erative to investigate\nseriously the nature of PS and the eﬀect of disorder in diﬀerent half doped manganites.\nIn this context, Pr0.5Sr0.5MnO3(PSMO) is very attracting system considering the\nA-type AF ground state where FM layers are coupled antiferromag netically exhibiting\nquasi two dimensional (2D) metallic behavior [19, 20]. Moreover, th is composition is at\nthe phase boundary of FM and AF ground states [21] thus, making it quite vulnerable\ntowards tendency for PS, which has been observed at low tempera ture from the55Mn\nNMR study [22]. Further, PSMO has a second order paramagnetic (P M)-I to FM-M\ntransition around 270 K and a ﬁrst order FM-M to AF-I transition ar ound 140 K [23].\nImryandWortis[24]hadpredictedtheroundingofﬁrstordertran sitionduetoquenched\nrandom disorder and beyond certain percentage the nature of tr ansition is expected\nto change from ﬁrst to second order. Recently, it is shown that th e transformation\nkinetics of the ﬁrst order transition in PSMO is hindered, resulting in t unable coexisting\nfraction of kinetically arrested high-T (FM) phase with equilibrium AF- I phase at low\ntemperature [4]. Hence, substitution of quenched disorder in this s ystem will reveal\nthe eﬀect on both the phase transitions as well as on the nature of PS in the same\nsystem. However, introducing quenched disorder in manganite in th e form of chemical\nsubstitutions remains a challenging task, as introduction of dissimilar ion(s) at R/A or\nMn-site can lead to change in structure, which modiﬁes the original s ystem completely.\nApart from this, introduction of magnetic elements will also modify th e basic system\nby introducing additional magnetic interactions.\nWe report here a detail study of PS in PSMO and the eﬀect of substit utional\ndisorder (Ga) at the Mn-site. Substitution of Ga neither add any ma gnetic interaction\nnor introduce any signiﬁcant lattice distortion since Ga3+beingd10element has zero\norbitalorspinmoment andhasmatching ionicradiiwiththeexisting Mn3+[12]. Earlier\nstudy on the same substitution in PSMO has mainly focused on magnet o-transport\nproperties and shown that high-T FM phase is suppressed and low-T AF phase is\nenhanced duetothequenched disorder [25]. Thus disorder promot estheAFinteractions\nat the cost of FM in PSMO, which is contrary to the observation in Pr0.5Ca0.5MnO3\nfor the similar kind of substitution [11, 4]. The signiﬁcant ﬁndings of pr esent study are\nsummarized below. We show drastic decrease in FM transition temper ature (TC) and\nincrease in AF transition ( TN) with Ga substitution (upto 7.5%) without any signiﬁcant\nchange in structure. The change in TCwith Ga follows simple mean-ﬁeld variation\nindicating that there is no unwanted magnetic interaction is added to the system by\nthe quenched disorder (Sec. 3.1). The coexisting FM and non-FM clu sters arise at\nhigh temperature just below TCand persists much below TNfor all the samples. Size\nof the FM clusters reduces with the increase in quenched disorder ( Sec. 3.2). ThePhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 4\nTable 1. Structural and ﬁtting parameters determined from the Rietveld p roﬁle\nreﬁnementofthe powderXRD patternsforthe seriesPr 0.5Sr0.5Mn1−xGaxO3. HereO1\nrefers to the apical oxygen of the perovskites and O2 is the equat orial oxygen which\nlies in the plane of the perovskite layer. The average Mn valence stat e of Mn was\ndetermined by iodometric titration. The extreme right column shows the percentage\nchange (∆%) of the parameters between the end members of the s eries (i.e. between\nx=0 and 7.5%) .\nGa (x) 0% 2 .5% 5 .0% 7 .5% ∆%\na (˚A) 5.4027 5.4068 5.4044 5.4046 +0.07\nc (˚A) 7.7814 7.7820 7.7817 7.7791 -0.03\nV (˚A3) 227.13 227.49 227.24 227.23 +0.04\nMn-O1 ( ˚A) 1.9454 1.9455 1.9451 1.9448 -0.03\nMn-O1-Mn 180o180o180o180o0.0\nMn-O2 ( ˚A) 1.9255 1.9274 1.9274 1.9304 +0.25\nMn-O2-Mn 165.5o165.32o164.89o163.65o-1.1\nMn3+% 51.74 50.39 47.51 45.04 -\nMn4+% 48.22 47.1 47.48 47.45 -\nMn Valance (Av) 3.4819 3.4827 3.4994 3.5126 +0.8\nFM phase fraction changes continuously as the temperature is red uced from TCand\nis tracked using second-order susceptibility which directly probes t he variation in the\nspontaneous magnetization (Sec. 3.3). An attempt is made to ident ify the nature of\nthe coexisting non-FM phase in the FM regime and is found to be AF in na ture.This\nsigniﬁcant observation implies that both the FM and AF phase s form at TC(Sec. 3.4).\nIt is interesting to note that though the measured magnetic momen t at 2K matches\nwell with the fully spin aligned value, the moment found from the ﬁtting to Curie-\nWeiss Law above TCgives much higher value indicating the existence of FM short-range\norder above TC(Sec. 3.5). The intriguing nature of the coexisting phases in PSMO is\nsubstantiated through resistivity measurements which also shows drastic eﬀect of the\nquenched disorder (Sec. 3.6). Signiﬁcance of this study in the cont ext of PS and the\neﬀect of quenched disorder in the intermediate bandwidth system a round half-doping is\nprovided in concluding section (Sec. 4).\n2. Experimental Details\nPolycrystalline samples of Pr0.5Sr0.5Mn1−xGaxO3series with x = 0.0, 0.025, 0.05 and\n0.075 have been prepared by the standard solid state ceramic rout e usingPr6O11,\nSrCO 3,MnO2andGa2O3withpuritymorethan99 .99%andﬁnalsinteringtemperature\nwas used 1500oCfor 36 hours. The x-ray diﬀraction (XRD) measurements were don e\nwith Rigaku Dmax 300 diﬀractometer with CuKαradiation at room temperature. All\nthe samples were found to be in the single phase and XRD pattern wer e analyzed\nby the Rietveld proﬁle reﬁnement programme by Young et al.[26]. To estimate the\nMn3+/Mn4+ratio, Iodometric redox titration has been done using sodium thiosu lphatePhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 5\nTable 2. Transition temperatures have been calculated from the diﬀerentia l change\nof the ac susceptibility with the temperature for the series Pr0.5Sr0.5Mn1−xGaxO3.\nLast row shows measured demagnetization factor (N) at tempera ture≈0.99TC.\nGa (x) 0% 2 .5% 5.0% 7.5%\nTC(K) 269.2 236.41 201.17 174.26\nTN(K) 124.72 133.59 158.85 -\nN0.308 0.533 0.657 -\n(Na2S2O3,5H2O) and potassium iodide (KI). The average atomic concentration in\nsystem has been found out through energy dispersive analysis of x -ray (EDAX) attached\nto transmission electron microscope (TECNAI G2-20FEI). Low ﬁeld AC susceptibility\nmeasurements were performed with the home-made AC-suscepto meter [27]. DC\nmagnetizations were measured with a home-made vibrating sample ma gnetometer\n(VSM) [28] and Quantum Design 14 Tesla VSM (PPMS). DC resistivity me asurements\nwere done with the standard four-probe method.\nRoom temperature XRD pattern has been ﬁtted and found to be in t etragonal\nI4/mcm space group. All the Rietveld reﬁnement parameters and the titra tion results\nhave been given in Table 1. The goodness of the ﬁt , which is deﬁned as the ratio\nofRwp/Rexphas been found to be around 1.4 for the whole series [26]. It has bee n\nfound that there is no major structural distortion due to doping, as seen in Table 1.\nTitration results show average Mn valence enhances with the doping . This indicates\nthat substituted Ga has preferentially replaced Mn3+and it is also expected from ionic\nsize matching ( Mn3+= 0.65˚A andGa3+= 0.62˚A). The average concentration of\nchemical constituents found through EDAX matches with the nomin al concentration\nwithin experimental accuracy which is ±0.5% for the lowest Ga doped compound.\n3. Results and discussions\n3.1. Variation of T Cand T Nwith quenched disorder\nThe real and imaginary part of the ﬁrst order ac susceptibility ( χR\n1andχI\n1respectively)\nmeasured for this series in 0.5 Oe ac ﬁeld and frequency of 131 Hz is sh own in the Fig.\n1a and 1b respectively. Parent compound (x = 0) shows two magnet ic phase transitions\nwith lowering temperature (PM to FM and FM to AF). For the doped sa mples, PM to\nFM transition temperature ( TC) decreases and FM to AF transition temperature ( TN)\nincreases systematically. Transition temperatures are calculated from the maximum\nchange in dχ/dTwith temperature and given in the Table 2. For x = 0 compound,\nTCmatches well with the reported single crystal value [23]. Imaginary p art of the ac\nsusceptibility which signiﬁes the magnetic loss also shows sharp chang es around TCand\nTN.\nThe large systematic change in TC(≈100 K) without considerable structural\ndistortion in present series is intriguing as the substituted (nonmag netic) disorder doesPhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 6\nnot add any magnetic interaction in the system. We have calculated t he tolerance\nfactor (τ) for the series and found very small change in τ(≈0.03 %) between the end\ncompositions (0% and 7.5%). To get similar change in TCby substitution at Ln-site\nin half doped compound Ln0.5−xSr0.5MnO3, large change in τ(around 0.4 - 0.9 %)\nis required [29]. Thus, the reduction of TCinPr0.5Sr0.5Mn1−xGaxO3arises from site\ndilution of the magnetic lattice and we attempted to explain this by mea n ﬁeld theory\n(MFT), considering only nearest neighbor interactions. According to MFT, TCcan be\nexpressed as [30]:\nTC=2S(S+1)zJ\n3kB(1)\nWhere S is the average spin per magnetic ion, kBis the Boltzmann constant, z is\nthe number of nearest neighbor magnetic atoms and J is the exchan ge integral. For\nmanganite, Eq. 1 has to be modiﬁed considering possible magnetic inte ractions viz. i)\nSuperExchange(SE) Mn3+−O−2−Mn3+,Mn4+−O−2−Mn4+andii)DoubleExchange\n(DE)Mn3+−O−2−Mn4+. Considering these three interactions, variationof TCwith Al\nsubstitution at the Mn-site in LaMnO 3+δwas explained [31]. However, for the present\ncase we consider only an average magnetic interaction and if the Ga s ubstitution (x)\nis random in the magnetic lattice, then it scales with the number of nea rest neighbour\n(z) of Eq. 1. Thus a simple site dilution by Ga will result in a linear variation ofTC\nwith x. Interestingly, Fig. 2 shows this linear variation of the experim entally found TC\nwith Ga substitution (x) for the present series. This linear scaling of TCindicates that\ni) no signiﬁcant complication is introduced due to site dilution or the ave rage exchange\nintegral is not modiﬁed and ii) substitution is random. It is interesting that this simple\nmean ﬁeld approach has led to the qualitative understanding of the e ﬀect of disorder on\nmodiﬁcation of TCfor such a complicated system. Similar analysis for large bandwidth\nmanganite, La0.7Sr0.3Mn1−xM′\nxO3(M′= Al, Ti) was used to explain linear variation of\nTCwiththeamount ofAlaswell asTi substitutions [32]. Further, itwas shownfromthe\nextrapolation of the linear variation, that TCvanishes only when Mn3+−O−2−Mn4+\nDE interaction vanishes by the selective substitution of either Mn4+orMn3+by Ti\nor Al respectively. This was attributed to the dominance of DE inter action for this\nlarge bandwidth system. However, for the present Pr 0.5Sr0.5Mn1−xGaxO3series, such\nextrapolation (from Fig. 2) leads to vanishing of TCat x≈0.21, when there are\nsubstantial amount of both Mn4+andMn3+are present. This indicates the reduction\nin the strength of DE interaction with the decrease in bandwidth.\nMore drastic eﬀect of quenched disorder is there on the TN, which appears to be\ncounterintuitive, and certainly opposite to general antiferromag netic systems or even\nthe 2-dimentional antiferromagnetic manganite [33]. It is clear fro m Table-2 that TN\nincreases monotonically with substitution. However, the variation o fTNdoes not follow\nlinear scaling with quenched disorder (x). Earlier study has shown th at substitution of\ntrivalent (In, Ga) and tetravalent (Sn, Ti) nonmagnetic elements a t Mn site in PSMO\nincreases and decreases TNrespectively [25]. Hence, respective change in Mn ionic\nconcentration and minute structural modiﬁcation arising from diso rder can modify TNPhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 7\nin opposite way. As seen in Table 1, Mn-O bond length in basal (O2) and apical (O1)\nplane has very small positive and negative changes respectively with disorder in this\nseries. This can make FM state unstable and assist this A-type AF st ate (where FM\ninteraction is realized along the basal plane and these planes are ant iferromagnetically\ncoupled along the apical direction) to set in at higher temperature, hence increasing\nTNwith disorder. As mentioned earlier, substitution of Ga in the Mn-site of PSMO\nis one of the simplest route to introduce quenched disorder in the ma gnetic lattice of\nthis system. However, the present experimental observations a ppear to contradict the\ntheoretical proposal of fragility of A-type AF state against quen ched disorder [18].\n3.2. Phase separation and the eﬀect of quenched disorder on i t\nWe have probed the phase separation from the thermal hysteres is (TH) in ac-\nsusceptibility. Fig. 3a shows χR\n1measured in 0.2 Oe ﬁeld and frequency of 131 Hz\nfor the parent compound (x = 0) in both heating and cooling cycles. I t is evident that\nχR\n1shows ﬁnite (TH) which starts immediately below TCand persists much below TN.\nThe amount of TH [ χR\n1(cooling) - χR\n1(heating)] as measured in 9 Oe ac-ﬁeld and 131\nHz frequency is given as a function of temperature in Fig. 3b for the present series.\nIn general, TH in ac- χhas been shown to be the generic feature of ﬁrst-order phase\ntransition (FM-AF) [34]. But in addition to a peak around TN, Fig. 3b shows that\nthe TH starts immediately below the respective TCs and persists through the FM phase\nfor all the samples. This indicates the inhomogeneous nature of the FM state where\nﬁnite size FM clusters coexist with non-FM one. Higher value of χR\n1in the cooling\nrun indicates that cooling run contains more FM fraction than the he ating run. It is\nsigniﬁcant that the TH appears just below TCor with the onset of long range ordering in\nthe system and persists below TN. Moreover, similar TH is also observed in variation of\nspontaneous magnetization measured throughhigher order (eve n order) susceptibility as\nwell as in resistivity and will be discussed in later sections. This new type of PS across\ntwo long-range order magnetic transitions in the same sampl e is rather uncommon in\nliterature.\nTo understand the nature of the inhomogeneous FM state we have studied the\nchange in magnetic anisotropy as a function of both temperature a nd composition. Ac-\nsusceptibility shows sharp peak in low ﬁelds immediately below TC, as shown in Fig.\n1a. Enhancement of this peak height with decrease in ﬁeld can be obs erved for x=0\nsample in Fig. 3a for 0.2 Oe compared to that shown in Fig. 1a for 0.5 Oe. Such peak is\ncommonly known as Hopkinson’s peak and mainly originates due to the rapid increase in\nanisotropy immediately below TC, in particular when it exceeds the applied ac ﬁeld [35].\nWhen both shape as well as magnetocrystalline anisotropies are pre sent, the intrinsic\nmagnetic susceptibility ( χint) is related to the measured low ﬁeld susceptibility ( χmes)\nin the following way [36]:\nχ−1\nint(T) =χ−1\nmes(T)−4πN(T) (2)\nWhere N(T) = Nd+NK(T),Ndis called demagnetization factor which dependsPhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 8\non the sample shape anisotropy and gives demagnetization ﬁeld Hd= 4πNdM[37].\nThe magnetocrystalline anisotropy is taken care by NKand for polycrystalline FM\nsample,Ndis the dominant factor to anisotropy. For a FM, χintdiverges at TCin\nthe absence of external magnetic ﬁeld and Nddepends only on the shape or dimension\nof the sample. Hence, the measured susceptibility in the ferromagn etic region of a\nhomogeneous FM is expected to remain almost constant as a functio n of temperature\n(as shown in Ref. [38]). However, signiﬁcant temperature depende nce in susceptibility\nwithin FM phase (Fig. 1a) indicates variation of magnetic anisotropy a rising from\nthe temperature dependence in the Ndfor the inhomogeneous FM state. Following\nthe protocol given in Ref. [36], we have calculated Ndat diﬀerent temperatures in\nFM regime from low ﬁeld dc magnetization data. We have also calculated coercive\nﬁeld (HC) from the hysteresis loops at various temperatures. Fig. 4 shows thatNd\nincreases with lowering temperature for x = 0 compound and exceed s the value (0.495)\nwhich was estimated from the dimension of the bulk sample [39]. This clea rly indicates\nthat the Ndfor an inhomogeneous FM is not given by the sample dimension but by\nthe dimension of the ‘ﬁnite-size’ FM clusters within it. Moreover, its v ariation with\ntemperature follows the variation in the magnetic anisotropy result ing in concomitant\nincrease in HCas shown in Fig. 4. Thus we infer from Fig. 4, that the FM clusters\nin this system spontaneously change their dimensions even well within the FM state\nright from their formation, and corroborate the conclusion drawn from the observed\nthermal hysteresis in Fig. 3. It may be argued that, as we decreas e the temperature\nand approach TN, FM cluster size decreases resulting in increase in surface anisotro py\ncontributing substantially to the total magnetic anisotropy of the system.\nTo understand the eﬀect of quenched disorder on the inhomogene ous FM state, we\nhave measured the variation of NdandHCwith temperature for other members of the\nseries and found qualitatively similar behaviour (not shown) as shown in Fig. 4. Table\n2 shows the value of Ndat≈0.99TCs of respective samples, with x = 0.0, 0.025 and\n0.05. It is clear that Ndincreases with increase in quenched disorder (x) though the\nbulk dimension of all the samples are roughly same. It also implies, from the argument\ngiven above, that increase in quenched disorder reduces the size o f the FM clusters.\nSuch variation in ‘ﬁnite-size’ cluster with disorder qualitatively agree s with the earlier\nstudies on cuprate [40]. In the present system, FM clusters form im mediately below TC\nand coexist with non-FM clusters. Size of these FM clusters reduce with decrease in\ntemperature or increase in quenched disorder.\nWegivefurtherevidenceofthedecreaseinFMclustersizewithincre aseinquenched\ndisorder fromtheﬁelddependence ofac- χ. Fig. 5ashows χR\n1forparent compoundinthe\nmeasuring ac-ﬁeldrangeof0.5to8Oe. Itisclearthatthe Hopkinson’s peak immediately\nbelowTC, though observed in 0.5 Oe, reduces and χR\n1increase with increase in ﬁeld.\nAs mentioned earlier, Hopkinson’s peak which is an outcome of the rapid increase in\nanisotropy compared to the measurement ﬁeld, is expected to dec rease with increase\nin ﬁeld. Higher ﬁeld will help in overcoming higher pinning potentials and th e domain\nwalls will move further within the multi-domain FM clusters, giving rise to concomitantPhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 9\nincrease in χR\n1. On the contrary, such ﬁeld dependence is absent for the x= 0.075\nsample (Fig. 5b) indicating that the cluster size is so small that it cann ot support\nmultiple domains. We plot the diﬀerence between the χR\n1measured in 8 Oe and 0.5\nOe at 0.99 TCof the respective samples as a function of disorder (x) in Fig. 5c. Th e\nobserved decrease in ﬁeld dependence indicates that the decreas e in FMcluster size with\nincrease in disorder (x) actually reduces the number of domains with in a cluster and\napproaches the single-domain limit for x=0.075, where no ﬁeld depend ence is expected.\nThus the decrease in cluster size will reduce the extent of the doma in wall motion as the\nﬁeld is increased. Consequently, the diﬀerences between the magn etic losses for 8 Oe\nand 0.5 Oe i.e. the diﬀerence between the corresponding χI\n1will also reduce, as shown\nin the Fig. 5c.\n3.3. Temperature variation of FM cluster size probed throug h non-linear susceptibility\nWe have utilized non-linear ac-susceptibility to probe the phase-sep arated state in\nthis series; through the eﬀective study of variation of spontaneo us magnetization\nas a function of temperature as well as quenched disorder. Non-lin ear magnetic\nsusceptibilities are important experimental tools which are used to u nravel intricacies\nabout the magnetic states in diﬀerent systems [6, 41, 42]. However , their magnitude\nbeingcoupleoforderssmallerthanthelinearpart, seriouseﬀortsa reinvolvedtomeasure\nthem. In general, magnetization (m) can be expanded in terms of ma gnetic ﬁeld (h) as\nm=m0+χ1h+χ2h2+χ3h3+χ4h4+... (3)\nWherem0is the spontaneous magnetization, χ1(≈∂m/∂h) is linear and χ2,χ3,χ4,\netc. are non-linear susceptibilities. The even order susceptibilities ( χ2,χ4, ...) arise\nfor the systems which hold no inversion symmetry for m with respect to applied ﬁeld\ni.e m(h) /negationslash= -m(-h). In another way, presence of symmetry-breaking ﬁeld is required\nfor the experimental observation of χ2in ac susceptibility measurement. The origin of\nsuch symmetry breaking ﬁeld can be either a superimposed externa l dc ﬁeld or internal\nﬁeld in a ferromagnet arising from spontaneous magnetization. For this reason, the\nsecond order susceptibility ( χ2≈∂2m/∂h2) has been probed to study the presence of\nspontaneous magnetization in diﬀerent compounds [43, 44]. For fer romagnet, as the\ntemperature is decreased through TC, spontaneous magnetization appears giving rise to\nthe symmetry breaking ﬁeld and resulting asymmetry in magnetizatio n can be deﬁned\nas ∆m = m(h) - (-m(-h)). Across the TC, sharp increase in spontaneous magnetization\nwill result in a sharp increase in ∆m. The rate of this increase will slow do wn as it is\ncooled further below TCand may approach saturation far below TC. Consequently, χ2\nwhich is proportional to ∂∆m/∂hwill show a sharp negative peak around TC, similar to\na peak expected for internal χ1(≈∂m/∂h). However, this peak in χ2is expected to be\nnegative since the internal ﬁeld crated by spontaneous magnetiza tion is opposite to the\ndirection to the external ac-ﬁeld. Moreover, χ2shows sharp feature whenever there is\nsharp change in internal ﬁeld and ﬁnite value as long as there is variat ion in the internal\nﬁeld or spontaneous magnetization.Phase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 10\nWe have measured χ2in the absence of external dc magnetic ﬁeld while heating as\nwell as cooling the samples. Fig. 6 shows χ2for the series measured in 9 Oe ac ﬁeld\nand fundamental frequency of 131 Hz. While cooling from the higher temperature, χ2\nfor the samples x = 0, 0.025 and 0.05 appears with the onset of FM and shows sharp\nnegative peak around TC. Below TCfor all the samples we found a ﬁnite value of χ2\nin FM state whose magnitude decreases with decrease in temperatu re. On further\ncooling, around TN,χ2again shows comparatively small negative peak. Both the\npeaks around TCandTNindicate sudden change in internal ﬁeld. In addition, the\nsmall peak around TNshows considerable thermal hysteresis and shifts in peak position\n(insets of Fig. 6). This conﬁrms a broad ﬁrst order FM-AF transitio n in the form\nof supercooling and superheating of FM/AF phases. More signiﬁcan tly, there is a\nconsiderable diﬀerence in χ2between the heating and cooling runs in FM regime and\naroundTC. This clearly indicates diﬀerence in FM phase fraction in heating and co oling\nrun. Though such thermal hysteresis in χ2can be justiﬁed for a broad ﬁrst order FM\nto AF transition but the same around the second order PM to FM tra nsition and\nalso within the FM state is noteworthy. Thus the coexisting phase fr actions varies\nimmediately with the onset of long range ordering. The presence of χ2for x = 0.075\ncompoundinwidetemperaturerangeshowsconclusivelytheexisten ceofferromagnetism\nin this compound. However, this FM phase is rather inhomogeneous in dicated by the\nbroadness of the peak over a wide temperature range. This ﬁnding is quite remarkable\nas susceptibility/magnetization behavior of this compound (see Figs . 5b) resembles spin\nglasslikewhere χ2cannotexist. Moreover, magnetizationofthiscompoundlookssimila r\nto that of x = 0.06 compound of the same series in Ref. [25], where it is c oncluded that\nFM vanishes within this amount of substitution. But we show in this stu dy that, long\nrange FM in PSMO exists even at higher concentration of nonmagnet ic substitution.\nThus, observed thermal hysteresis in χ2conclusively shows that coexisting FM phase\nfraction changes as a function of temperature right from the formation around TCbut\ncontinues to exist much below TN.\n3.4. Phase inhomogeneity and identiﬁcation of high-T non-F M phase\nNow we attempt to identify the nature of the coexisting non-FM pha se found\nimmediately below TC. Probable nature of non-FM phase could be the high-T PM\nor low-T AF phase. Intuitively this phase may be considered to be AF s ince in this\nsystem both FM and AF phases are considered to have similar energie s. It is interesting\nto note that our recent study on the parent compound has clearly shown that ﬁrst-order\nﬁeld-temperature induced FM-AF transition process is kinetically ar rested resulting in\npersistence of high-T FM phase fraction down to the lowest temper ature (5K), much\nbelow the closure of the hysteresis of the ﬁrst-order process [4]. We have measured\ndc-magnetization of the parent compound in 517 Oe following diﬀeren t protocol. Fig.\n7a shows considerable amount of TH in FM regime between ﬁeld cooled c ooling (FCC)\nand ﬁeld cooled warming (FCW) magnetization, indicating that the coe xisting phasePhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 11\nfractions change during cooling and heating cycles. This measureme nt ﬁeld (517 Oe)\nbeing much above the coercive ﬁeld discards the possibility that TH ar ise due to the\neﬀects of local anisotropy. In addition, the same measurement ha s been repeated on\nthe powder sample (after crushing the pellet) and we got TH similar to the bulk (not\nshown) which rule out any possibility arising form other factors like st rain. Again, we\nhave measured magnetization in 517 Oe while cooling in zero ﬁeld, followin g a rigorous\nprotocol similar to Ref. [45]. In this protocol, we have started coo ling the sample in zero\nﬁeld fromabove roomtemperature. At each measurement temper ature, we isothermally\napply 517 Oe ﬁeld and measure the magnetization. After that, we iso thermally reduce\nthe ﬁeld to zero, cool the sample to the next lower measurement te mperature and repeat\nthe above procedure for the successive lower temperatures. Ma gnetization measured in\nthis protocol is shown as M(517, 0) in Fig. 7a. If the coexisting phas e is PM (having\nthe same structural symmetry with FM) then with an application of ﬁ eld, FM clusters\nwill grow freely irrespective of the way the ﬁeld is applied and M(517, 0 ) will match\nwith FCC data. But the coexisting AF phase, structurally diﬀerent f rom FM phase,\nwill not grow freely as ﬁeld is applied after cooling in zero ﬁeld due to the energy barrier\nbetween the AF and FM phases. Consequently M(517, 0) will be less t han FCC data. It\nmay be noted that in PSMO, PM and FM phase are having same structu re but there is\nstructuralchangearoundFM-AFtransition[46]. Fig. 7aclearlysho wsM(517,0)departs\nfrom FCC below TCand gives lesser value than the M(FCC) throughout the measuring\ntemperature. This may be considered as strong evidence that the coexisting non-FM\nphase is AF and not PM. To check whether application of ﬁeld at each m easurement\ntemperature for M(517, 0) modiﬁes the corresponding phase fra ction irreversibly, we\nhave measured M(virgin) by the following protocol. Each time we coole d the sample\nfrom the PM-state to target temperature in zero ﬁeld, isotherma lly applied 517 Oe\nto measure magnetization M(virgin) for that temperature and rep eat the complete\nprocedure for all other temperatures . As evident in Fig. 7a, ther e is no diﬀerence\nbetween M(virgin) and M(517,0).\nThe above experiment strongly suggests that the AF phase also fo rm around TC\nand coexist with the FM phase in the ferromagnetic region. In this re gion, an energy\nbarrier separates the FM and AF phase and the later has higher ene rgy than the former.\nThe higher energy AFphase will bein metastable statewhich has been conﬁrmed by the\ntime dependent magnetization. Magnetization has been measured a s a function of time\n(t) after cooling the sample in zero ﬁeld from PM phase to the target temperature and\nsubsequently applying a measuring ﬁeld. Normalized magnetization (M (t)/M(0)) as a\nfunction of t is shown in Fig. 7b. A continuous increase in magnetizatio n is observed\nafter application of 100 Oe at 225 K and 150 K (within FM regime) and th e same has\nalso been observed after application of 5 kOe at 225 K. This suggest s that the coexisting\nmetastable phase (AF) tries to overcome the energy barrier to ac hieve the low energy\nFM state. The presence of AF phase again checked by collecting FCC and FCW data\nat 517 Oe but cooling was done down to only 170 K which is well within the F M phase\nand much above TN. Fig. 7c shows a TH indicating presence of AF phase. This is aPhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 12\nTable 3. Measured µeffdetermined from ﬁtting to Curie-Weiss law, calculated µeff\nconsidering spin only value, measured and calculated saturation mom ent (µ) values\ndetermined from the magnetization data for the series Pr 0.5Sr0.5Mn1−xGaxO3.\nGa (x) 0% 2 .5% 5.0% 7.5%\nMeasured µeff(µB/f.u.) 5.532 5.479 5.549 5.449\nExpected µeff(µB/f.u.) 4.396 4.286 4.161 4.038\nExpected µ(µB/f.u.) 3.516 3.428 3.325 3.225\nMeasured µ(µB/f.u.) 3.508 3.281 3.261 3.176\nsigniﬁcant observation since it suggests that presence of A F phase at high temperature\nis not due to superheating phenomenon because the sample has not approached the TN\nwhile cooling. This also clearly indicates that both the FM a nd AF phases form with the\nonset of long-range order at TC.Neutron diﬀraction or other microscopic experimental\ntools could be used to directly conﬁrm this signiﬁcant observation. H owever, it may not\nbetrivial toidentify such small changes inthecoexisting phase frac tionsunambiguously.\n3.5. Evidence of short range FM interaction above TC\nMeasured magnetization in 1 Tesla ﬁeld shows Curie-Weiss behaviour, M/H = χ=\nC/(T−θP) only from temperatures which are much higher than the respectiv eTCs\n(Fig. 8). Eﬀective Bohr magneton ( µeff) per formula unit (f.u.) calculated from the\nﬁtting to Curie-Weiss law are signiﬁcantly larger than the expected s pin only value\n(µeff=g/radicalBig\nS(S+1)) and are given in Table 3. This indicates that the short-range\nFM interactions exist much above TC. Existence of short range FM interaction much\naboveTCwas proposed for La0.67Ca0.33MnO3compound where magnetic clusters were\ndetected from small angle neutron scattering by De Teresa et al.[47]. Such kind of\nmagnetic clusters in PM region have been shown for single crystal PS MO [48] and\nother manganites in both single crystal [49] as well as in polycrystallin e sample [50]\nand thought to be an intrinsic property of manganites [1]. It may be n oted here that\nthe ﬁeld induced FM state at 2K gives almost fully aligned spin moment at 14 Tesla.\nThe measured magnetic moment ( µ) per f.u. at 2K in 14 Tesla as well as the expected\nmoment per f.u. calculated for spin only value from the respective Mn3+andMn4+\ncontent is given in Table 3. The measured and calculated moment for s pin only value\nmatch reasonably well for all the samples.\n3.6. Phase coexistence and the resistivity measurement\nResistivity measurement also substantiates the remarkable phase coexistence shown\nfrom the magnetic measurements. Fig. 9a shows resistivity for x = 0 compound\nmeasured while heating and cooling. A metal to insulator transition (M IT) has been\nobserved around 270 K in agreement with the single crystal measur ement [23]. As\nthe FM-AF transition is approached while cooling, resistivity shows st eep increasePhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 13\naccompanied by a huge thermal hysteresis, which is characteristic s of ﬁrst order phase\ntransition. Minor hysteresis loops (MHLs) are shown to be a useful method to conﬁrm\nas well as study the disorder broadened ﬁrst order phase transit ion [34, 51]. Following\nsimilar protocol we have recorded MHLs by measuring resistivity while cooling, to\ndiﬀerent temperatures (113, 104, 95 and 90 K) and then heating f rom those points\nto all the way above TN(Fig. 9b). MHLs are also recorded in heating cycle (not shown)\nconﬁrming disorder broadened ﬁrst order transition and related v ariation of coexisting\nmetallic and insulating phase fractions. However, it is noteworthy th at TH in resistivity\nremainsmuchabove TN(itremainsdistinctupto225KwellwithinFMregion)asshown\nin the inset of Fig. 9a. It is rather signiﬁcant that resistivity being a percolativ e process,\nsuch TH well within the FM region conﬁrms that fraction of FM ( metallic) clusters\nchanges right from their formation at higher temperature as also concluded from the\nmagnetic measurements.\nFig. 10 shows resistivity for all the samples. No clear MIT was observ ed in the Ga\nsubstituted samples and resistivity increases with the substitution at all temperatures.\nHowever, the signature of FM to AF transition is clear from the kink in resistivity\nappearing around the TNof the respective samples with an exception for x = 0.075. The\neﬀect of quenched disorder is drastic and shows an increase by abo ut 0.5×104times\nin resistivity at 92 K, well within AF phase, with only 7.5% substitutional disorder.\nThough increase in resistivity with quenched disorder is observed in a nother half doped\nbilayer manganite [33] but the observed colossal increase in the pre sent series is rather\nintriguing since here also the resistivity is governed by the electrical conduction in the\nFMlayersoftheA-typeAFstructure. Itmaybementionedhere, t hatinanearlierstudy\ndecrease in resistivity has been observed with disorder for the CE t ype AF state [12].\nIn view of these and also in the context of recent theoretical deve lopments [18], detailed\ninvestigation of the resistivity behaviour of the present series nee ds to be undertaken.\n4. Conclusion\nIn summary, we have studied the structural, magnetic and transp ort properties of\nhalf doped Pr0.5Sr0.5MnO3and the eﬀect of quenched disorder (Ga substitution)\nin the magnetic lattice without introducing any additional magnetic int eractions or\nsigniﬁcant structural distortion. Substitution of Ga in the Mn-site has drastic but\nopposite eﬀects on the FM and AF transitions. Increase in substitu tional disorder\ndecreases TCbut increases TNand points toward reduction in the strength of double-\nexchange interaction in this intermediate bandwidth system. The FM state is found\nto be inhomogeneous and the FM cluster size decreases with decrea se in temperature\nor increase in quenched disorder. The electronic phase separation gives rise to thermal\nhysteresis in the size of this clusters right from their formation at TCwhich is evident\nfrom the TH in the susceptibility or spontaneous magnetization. This system shows\na novel phase coexistence, both FM and AF clusters form with the onset of long-range\norder in the system at TCand persist down to the lowest temperature. Moreover, thePhase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 14\nshort range FM interaction exists much above TC. Resistivity also shows the signature\nof the variation of coexisting phases and substantiates the conclu sion drawn from the\nmagnetic measurements. At lower temperature (in AF state), res istivity shows orders of\nmagnitudeincrease withquenched disorder. The observed novel p hasecoexistence needs\nfurther attention and required to be probed through microscopic tools and supporting\ntheoretical work.\n5. Acknowledgment\nWe express our sincere thanks to Dr. P. Chaddah for many helpful discussions\nregarding phase coexistence. We acknowledge Dr. N. P. Lalla for XR D and EDAX\nmeasurements. We thank Mr. Kranti Kumar and Mr. K. Mukherjee for the help during\nthe measurements. 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B73104401\n[50] Lu W J, Sun Y P, Song W H and Du J J 2006 Solid State Commun. 138200\n[51] Singh K J, Chaudhary S, Chattopadhyay M K, Manekar M A, Roy S B and Chaddah P 2002\nPhys. Rev. B65094419Phase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 16\n/s48/s50/s52/s54/s56\n/s40/s97/s41/s99\n/s49/s82\n/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41/s32/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s55/s46/s53/s32/s37/s53/s32/s37/s50/s46/s53/s32/s37/s48/s32/s37\n/s32/s72\n/s97/s99/s32/s61/s32/s48/s46/s53/s32/s79/s101\n/s70/s114/s101/s113/s32/s61/s32/s49/s51/s49/s32/s72/s122\n/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/s48/s46/s52\n/s55/s46/s53/s32/s37/s53/s32/s37/s50/s46/s53/s32/s37/s48/s32/s37/s40/s98/s41/s99\n/s49/s73\n/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41/s32\n/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50\n/s32\n/s84/s32/s40/s75/s41\nFigure 1. (a) Real and (b) Imaginary parts of ﬁrst order ac susceptibility ( χR\n1and\nχI\n1)) measured in an 0.5 Oe ac ﬁeld and 131 Hz frequency have been plott ed as a\nfunction of temperature for the series Pr0.5Sr0.5Mn1−xGaxO3.\n/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56/s49/s54/s48/s50/s48/s48/s50/s52/s48/s50/s56/s48/s84\n/s67/s32/s40/s75/s41\n/s120\nFigure 2. Variation of TChas been plotted as a function of x for the\nPr0.5Sr0.5Mn1−xGaxO3series. This ﬁgure shows linear decrease in TCwith increase\nin disorder.Phase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 17\n02468\nx = 0.0(a)χ1R (emu/mole) \n Heating\n Cooling\n100 150 200 250 300012\nTH (emu/mole)(b)TH (emu/mole)\nT (K) 0.0 %\n 2.5 %\n 5.0 %\n0.000.010.02\n  7.5 %\nFigure 3. (a) Real part of ﬁrst order ac susceptibility ( χR\n1) measured in 0.2 Oe ac\nﬁeld and 131 Hz during heating and cooling for x = 0 compound. (b) Tem perature\nvariation of amount of thermal hysteresis (deﬁned in the text) is s hown for the ﬁrst\norder susceptibility which has been measured in 9 Oe and 131 Hz for th e series\nPr0.5Sr0.5Mn1−xGaxO3.\n0.20.40.60.81.0\n CalculatedExperimental\nT (K)80 120 160 200 240 280060120180240\nTCNdHC (Oe)\nFigure 4. Left axis shows temperature variation of coercive force HCand right axis\nshows experimentally measured demagnetization factor (N d) in the FM regime as a\nfunction of temperature for x = 0 compound. The ﬁlled symbol is the calculated N d\nfor the same compound.Phase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 18\n/s48/s50/s52/s54/s56/s49/s48\n/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s48/s32/s37/s40/s97/s41/s32/s56/s32/s79/s101\n/s32/s53/s32/s79/s101\n/s32/s51/s32/s79/s101\n/s32/s49/s32/s79/s101\n/s32/s48/s46/s53/s32/s79/s101\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54/s32/s56/s32/s79/s101\n/s32/s53/s32/s79/s101\n/s32/s51/s32/s79/s101\n/s32/s49/s32/s79/s101\n/s32/s48/s46/s53/s32/s79/s101\n/s55/s46/s53/s32/s37/s40/s98/s41/s99\n/s49/s82\n/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s84/s32/s40/s75/s41\n/s40/s99/s41/s99\n/s56/s32/s79/s101/s32/s45/s32 /s99\n/s48/s46/s53/s32/s79/s101/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s120/s32/s82/s101/s97/s108\n/s32/s73/s109/s97/s103/s105/s110/s97/s114/s121\nFigure 5. (a) Temperature variation of real part of ac-susceptibility ( χR\n1) measured\nin diﬀerent ac ﬁeld and 131 Hz frequency for x = 0 compound. (b) The same ﬁeld\ndependence has been given for x = 0.075 compound. (c) Diﬀerence o fχbetween 8\nOe and 0.5 Oe ﬁelds as measured at 0.99T Cfor real and imaginary parts have been\nplotted as a function of composition (x).Phase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 19\n/s45/s54/s45/s52/s45/s50/s48\n/s48/s37/s99\n/s50/s114\n/s32/s40/s126/s49/s48/s45/s51\n/s32/s101/s109/s117/s47/s109/s111/s108/s101/s46/s79/s101/s41/s32\n/s45/s51/s45/s50/s45/s49/s48\n/s50/s46/s53/s37\n/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48\n/s53/s37\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s48/s46/s48/s49/s50/s45/s48/s46/s48/s48/s56/s45/s48/s46/s48/s48/s52/s48/s46/s48/s48/s48\n/s55/s46/s53/s37\n/s84/s32/s40/s75/s41\nFigure 6. Temperature variation of the real part of second order suscept ibility (χR\n2)\nmeasuredduringheating andcoolingin an acﬁeld of9Oe and frequenc y131Hz for the\nseriesPr0.5Sr0.5Mn1−xGaxO3. The arrow shows the direction of temperature cycle.\nInsets show the magniﬁed view of the same plots around the broad ﬁ rst order FM-AF\ntransition which clearly depict the thermal hysteresis associated w ith the ﬁrst order\ntransition due to supercooling and superheating of FM/AF phases.Phase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 20\n/s49/s50/s48 /s49/s56/s48 /s50/s52/s48 /s51/s48/s48/s48/s54/s48/s48/s49/s50/s48/s48/s49/s56/s48/s48/s50/s52/s48/s48/s51/s48/s48/s48\n/s40/s97/s41\n/s72/s32/s61/s32/s53/s49/s55/s32/s79/s101/s77/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s84/s32/s40/s75/s41/s32/s77/s32/s40/s70/s67/s67/s41\n/s32/s77/s32/s40/s70/s67/s87/s41\n/s32/s77/s32/s40/s53/s49/s55/s44/s48/s41\n/s32/s77/s32/s40/s86/s105/s114/s103/s105/s110/s41/s32\n/s49/s54/s48 /s50/s48/s48 /s50/s52/s48 /s50/s56/s48/s50/s52/s48/s48/s50/s54/s48/s48/s50/s56/s48/s48/s51/s48/s48/s48\n/s40/s99/s41\n/s72/s32/s61/s32/s53/s49/s55/s32/s79/s101/s77/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s84/s32/s40/s75/s41/s48 /s50/s48/s48/s48 /s52/s48/s48/s48/s49/s46/s48/s48/s48/s49/s46/s48/s48/s53/s49/s46/s48/s49/s48/s49/s46/s48/s49/s53\n/s40/s98/s41\n/s50/s50/s53/s32/s75/s44/s32/s53/s32/s107/s79/s101/s50/s50/s53/s32/s75/s44/s32/s49/s48/s48/s32/s79/s101/s49/s53/s48/s32/s75/s44/s32/s49/s48/s48/s32/s79/s101/s77/s32/s40/s116/s41/s47/s77/s40/s48/s41\n/s116/s32/s40/s115/s41\nFigure 7. (a) DC Magnetization as a function of temperature has been measu red\nin 517 Oe ﬁeld for x = 0 compound. Measurements were done in FCC, FC W,\nM(517,0)andM(virgin) mode(deﬁned in text). (b) Time (t) depende nce ofnormalized\nmagnetization has been plotted at diﬀerent temperature and applie d ﬁeld. (c) This\nplot shows FCC and FCW magnetization measured in the same ﬁeld but c ooling was\ndone only down to 170 K (much above TN), which show reasonable TH in the FM\nstate.Phase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 21\n/s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48 /s52/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s53/s32/s37/s55/s46/s53/s32/s37\n/s50/s46/s53/s32/s37/s48/s32/s37/s99\n/s100/s99/s45/s49\n/s32/s40/s79/s101/s46/s109/s111/s108/s101/s46/s101/s109/s117/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s72/s32/s61/s32/s49/s32/s84\nFigure 8. Inverse dc susceptibility measured in 1 T magnetic ﬁeld has been plott ed\nas a function of temperature for the series Pr0.5Sr0.5Mn1−xGaxO3. Straight lines\nshow the Curie-Weiss law ﬁtting of susceptibility data above TC. Data for x = 0.025\ncompound has been shifted to higher temperature by 10 K for clarit y.\n/s48 /s54/s48 /s49/s50/s48 /s49/s56/s48 /s50/s52/s48 /s51/s48/s48/s48/s46/s48/s49/s48/s46/s49\n/s84\n/s67/s120/s32/s61/s32/s48/s40/s97/s41\n/s32/s67/s111/s111/s108/s105/s110/s103\n/s32/s72/s101/s97/s116/s105/s110/s103/s49/s53/s48 /s49/s55/s53 /s50/s48/s48 /s50/s50/s53/s51/s46/s53/s51/s46/s54/s76/s111/s103/s40 /s114 /s41/s32/s40/s79/s104/s109/s45/s99/s109/s41\n/s84/s32/s40/s75/s41/s114 /s32/s40/s49/s48/s45/s51\n/s79/s104/s109/s45/s99/s109/s41\n/s84/s32/s40/s75/s41\n/s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s48/s46/s48/s48/s52/s48/s46/s48/s48/s56/s48/s46/s48/s49/s50\n/s40/s98/s41/s114 /s32/s40/s79/s104/s109/s45/s99/s109/s41\n/s84/s32/s40/s75/s41\nFigure 9. (a) The semi-log plot of resistivity as a function of temperature for x = 0\ncompound measured during heating and cooling. The arrow indicates the direction of\ntemperature cycle. Inset shows the magniﬁed view of thermal hys teresis in resistivity.\n(b) Minor hysteresis loop (MHL) around FM-AF phase transition has been plotted as\na function of temperature. This plot shows disorder broadened ﬁr st order transition.\nThe lines inside the envelop curve represent the data recored in hea ting cycle.Phase separation and the eﬀect of quenched disorder in Pr0.5Sr0.5MnO3 22\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48/s49/s48/s48\n/s55/s46/s53/s37\n/s53/s37\n/s50/s46/s53/s37\n/s48/s37/s76/s111/s103/s40 /s114 /s41/s32/s40 /s87 /s45/s99/s109/s41\n/s84/s32/s40/s75/s41\nFigure 10. Resistivity data has been presented for the Pr0.5Sr0.5Mn1−xGaxO3series\nas a function of temperature in a semi-log plot."    },    {        "title": "1307.6481v1.Phase_transitions_in_2D__J__1__J__2___model_with_arbitrary_signs_of_exchange_interactions.pdf",        "content": "arXiv:1307.6481v1  [cond-mat.str-el]  24 Jul 2013Phase transitions in 2D J1−J2model with arbitrary signs of exchange interactions\nA.V. Miheyenkov+∗, A.V. Shvartsberg∗, A.F. Barabanov+\n+Institute for High Pressure Physics RAS, 142190 Moscow (Tro itsk), Russia\n∗Moscow Institute of Physics and Technology, 141700 Dolgopr udny, Russia\n(Dated: January 23, 2018)\nThe ground state of the S= 1/2J1−J1Heisenberg model on the 2D square lattice with arbi-\ntrary signs of exchange constants is considered. States wit h diﬀerent spin long-range order types\n(antiferromagnetic checkerboard, stripe, collinear ferr omagnetic) as well as disordered spin-liquid\nstates are described in the frames of one and the same analyti cal approach. It is shown inter alia,\nthat the phase transition between ferromagnetic spin liqui d and long-range order ferromagnet is a\nsecond-order one. On the ordered side of the transition the f erromagnetic state with rapidly varying\ncondensate function is detected.\nPACS numbers:\nInvestigationofthe two-dimensionalfrustratedHeisen-\nberg model is of current importance for understanding\nmagnetic properties of various layered compounds. Spin\nsubsystem of CuO2planes in cuprate high-temperature\nsuperconductors (HTSC) can be described by J1−J2\nHeisenbergmodelon the squarelattice with spin S= 1/2\nand antiferromagnetic signs of both exchange constants.\nIntensively studied layered vanadium oxides can be de-\nscribed in the frames of same model, but not only with\nantiferromagnetic exchanges.\nIn the classical limit S≫1 at zero temperature three\ntypes of long-range order (LRO) are realized: ferromag-\nnetic (FM), Neel antiferromagnetic(AFM) andcolumnar\n(stripe). At the points J2/|J1|= 0.5 there are ﬁrst or-\nder phase transitions from checkerboard AFM order to\nstripe for J1>0 and from stripe to ferromagnetic order\nforJ1<0, at point J1= 0,J2=−1 there is a transition\nfrom AFM to FM order. The positions of the better-\nstudied vanadates on the classical J1–J2model phase di-\nagram are shown in Fig. 1 (the data from Refs. 1,2).\nAtT/negationslash= 0 long-rangeorder due to Mermin-Wagnerthe-\nJ1J2\nJ2/J1 → −∞Stripe\nNeelFMBaCdVO(PO4)2SrZnVO(PO4)2BaZnVO(PO4)2Li2VOSiO4\nLi2VOGeO4\nPbVO3\nVOMoO4\nFIG. 1: Phase diagram of the J1–J2Heisenberg model on the\n2D square lattice in the classical limit. Dots represent the\nrelations between J1andJ2for the better-studied vanadates\n(data from1,2).oremis impossible for any spin, at T= 0 for large SLRO\nexists throughout the ” J1−J2-circle”. Nevertheless, it is\ngenerally accepted that for S= 1/2 even for T= 0 spin\nﬂuctuations near phase transition points lead the sys-\ntem to one of the singlet states without LRO and with\nnonzero spin gap. The structure of disordered phases\nremains debatable. Usually the following states are con-\nsidered: spin liquid, conserving translational and SU(2)\nsymmetry of the Hamiltonian; plaquette lattice cover-\ning, which breaks translational symmetry, but conserves\nSU(2) symmetry; and states that break both transla-\ntional and SU(2) symmetry.\nIn the present work the ground state of 2D J1−J2\nHeisenberg model is investigated in the frames of spheri-\ncally symmetric self-consistent approach (SSSA) for two-\ntime retardedGreen’sfunctions (Refs. 3,4, seealsorecent\nreview in Ref. 5). This approach automatically conserves\nSU(2) symmetry of the Hamiltonian, translational sym-\nmetry and spin constraint on the site. Unlike previous\ntreatments of S= 1/2 model, we investigate the entire\nphasediagramforarbitraryvaluesof J1andJ2, including\ncases ofJ1<0,J2>0 andJ1<0,J2<0.\nIn the quantum limit S= 1/2, the ﬁrst quadrantofthe\ndiagram 0 ≤ϕ≤π/2, tanϕ=J2/J1,J1, J2>0, has\nbeen studied up to now most extensively. In this case a\ndisordered state (spin liquid) appears between AFM and\nstripe LRO phases. The transitions AFM — spin liquid\n— stripe phase are continuous in the frames of SSSA.\nTheregion J1<0,J2>0forS= 1/2ofthe phasedia-\ngramhasbeeninvestigatedinframesofSSSAin6,7, where\nthe the ﬁrst order transition between FM LRO state and\nspin liquid has been stated. As it will be seen hereafter,\nunlike6,7, our consideration leads a continuous second-\norder transition between the mentioned states, the prop-\nerties of FM state being signiﬁcantly modiﬁed near the\ntransition.\nBeforediscussingthe phasediagraminthe wholerange\nof the angle ϕ, let us write down the Hamiltonian Hand\nthe form of spin-spin Green’s function Gzz(ω,q), which\ncan be obtained in the frames of SSSA3–7(for SSSA\nGzz=Gxx=Gyy;/angbracketleftSα\ni/angbracketright= 0, α=x,y,z).\nH=J1\n2/summationdisplay\ni,g/hatwideSi/hatwideSi+g+J2\n2/summationdisplay\ni,d/hatwideSi/hatwideSi+d (1)2\nGzz(ω,q) =/angbracketleftSz\nq|Sz\n−q/angbracketrightω=Fq\nω2−ω2q(2)\nFq=−8[J1(1−γg)cg+J2(1−γd)cd] (3)\nγg(q) =1\nzg/summationdisplay\ngeiqg=1\n2(cos(qx)+cos(qy)) (4)\nγd(q) =1\nzd/summationdisplay\ndeiqd= cos(qx)cos(qy) (5)\nwhereg,dare vectorsof nearest and next-nearest neigh-\nbors,cR=/angbracketleftSz\nnSz\nn+R/angbracketright— spin-spin correlation functions\non the corresponding coordination spheres, zg=zd= 4\n— number of sites on the ﬁrst and the second coordina-\ntion spheres. Hereafter all the energetical parametersare\nset in the units of J=/radicalbig\nJ2\n1+J2\n2.\nFor further analysis, it is convenient to represent the\nspinexcitationspectrum ω2\nq(4,5)inthreefollowingforms:\nω2\nq= 2A(q)(1−γg)(1−γg+δFM(q)) =\n=−2A(q)(1−γg)(1+γg+δAFM(q)) =\n=−2A(q)(1−γg)(1+γd+δStripe(q)) (6)\nExpressions for Aandδfrom (6) are rather unwieldy,\nand we do not present them completely. We will just\npresent the form of AδAFMas an example:\nAδFM= 8J1J2(/tildewidecdg−/tildewidecg)+J2\n1(1−20/tildewidecg+8/tildewidecd+4/tildewidec2g)+\n+1−γd\n1−γg[8J2J1(/tildewidecdg−/tildewidecg)+J2\n2(1−20/tildewidecd+8/tildewidec2g+4/tildewidec2d)] (7)\nIn (7) the correlators /tildewidecr=αcrare written in one\nvertexαapproximation (5,6). Five correlators cr(r=\ng, d,2g, rgd=|g+d|,2d) and vertex correction α\nare obtained self-consistently through the Green’s func-\ntionGzz. The additional condition is the exact sum rule\nfulﬁllment /angbracketleft/hatwideS2\ni/angbracketright= 3/4.\nThe introduced parameters δFM(q),δAFM(q), and\nδStripe(q) have a clear physical meaning and deﬁne the\nspin excitation spectrum basic properties. For all phases\n— three ordered (AFM, stripe, and FM), and spin liquid\n— the spin gap is closed at the zero point Γ= (0,0). In\nthe FM phase the spectrum around Γis quadratic in q,\nfor other phases it is linear. Near the transitions to FM\nfrom the neighboring phases the spectrum around Γhas\nthe form ωq∼q/radicalBig\nδFM+q2\n4. SoδFMdictates the conver-\nsion from FM spectrum ωq∼q2toωq∼q. In the AFM\nphase the spin gap is closed not only in Γ, but also in\nAFM point Q= (π,π). When approaching to the AFM\nphase from the neighboring phases the spectrum around\nQisωq∼√δAFM+κ2,κ=|Q−q|, i.e.δAFMdirectly\ndeﬁnes the gap ∆ AFMin the spectrum. For the stripe\nphase and it’s neighborhood the situation is similar to\nthat for AFM phase (with corresponding substitutions,\nthe role of control point goes from Qto to stripe points\nX= (0,π),(π,0)).Thus, vanishing of any of the three parameters δFM,\nδAFM,δStripedeﬁnes transition to the corresponding or-\ndered phase and simultaneous alteration of the spectrum\nnearthe correspondingcontrolpoint (transition from lin-\near to quadratic for FM and vanishing of the spin gap in\nthe Dirac spectrum for two others). For the spin liquid\nthe spectrum gap is opened in the whole Brillouin zone\nexceptΓ.\nLet us depict in more detail the description of the spin\nLRO. The structure factor has the form\ncq=/angbracketleftSz\nqSz\n−q/angbracketright=−1\nπ/integraldisplay\ndωn(ωq)ImGzz(ω,q) =\n=Fq\n2ωq(2n(ωq)+1) (8)\nheren(ωq) is Bose function. Correlation functions are\nexpressed through the structure factor as\ncR=/angbracketleftSz\nnSz\nn+R/angbracketright=/summationdisplay\nqcqeiqR=\n=ccond/summationdisplay\nq0eiq0R+1\n4π2/integraldisplay\ndqeiqRFq\n2ωq(9)\nwhere the condensate part is\nccond= lim\nT→01\n4π2/integraldisplay\ndqn(ωq)Fq\nωq(10)\nAtT→0δ-like peaks in the structure factor can ap-\npear at some points q0of the Brillouin zone (where ωq\ntendstozero),this peaksbeinginducedbytheBosefunc-\ntionn(ωq). Then the condensate term ccondappears in\nthe correlation functions cR. This corresponds to the\nLRO existence ( cconddeﬁnes spin-spin correlator at the\ninﬁnity). The term without n(ωq) on the right hand side\nof (9) goes to zero as R→ ∞.\nFor AFM and stripe long-range orders the condensate\nterm appears as the spectrum ωqvanishes correspond-\ningly at the points QandX. As mentioned above, the\nspectrum near this points is (in the correspondingphase)\nωq∼κ, where κ=|q−Q|or|q−X|. The Green’s\nfunction numerator Fqdoes not vanish at this points.\nThe spectrum linearity and nonzero Fqvalue constitute\nthe condition for condensate to appear3.\nIn the presence of FM LRO spin condensate appears\nat the point Γ. Near this point the Green’s function\nnumerator Fq∼q2, so the spectrum near Γis to be also\nquadratic ωq∼q2for the condensate to appear.\nNote that if the third exchange interaction J3is added\nto the model, the helical LRO can also be realized. In\ntheJ1−J2−J3model the condensate peak point in the\nstructure factor can be located not only at Γ,Q, orX,\nbut also at arbitrary incommensurate point on the side\nor diagonal of the Brillouin zone8.\nFig. 2 shows the phase diagram at T→0, the con-\ndensates and correlators corresponding to ﬁrst three co-\nordination spheres being depicted. Fig. 3 represents\nspin gaps in the symmetrical points. In the interval\n0≤ϕ≤ϕ1= 0.051 AFM LRO is realized: spin gap\nat AFM point Qis zero, ∆ Q= 0, the spectrum near Q\nis linear in |q−Q|, there is a nonzero AFM condensate\ncAFM\ncond.3\n−0.1−0.0500.05Correlatorsϕ1ϕ2\nϕ3 ϕ4 ϕ6\n0 π/2 π 3π/2 ϕcg\ncdc2g\nStripeFM AFM\nFIG. 2: The condensate ccond(absolute value of spin-spin\ncorrelator at inﬁnity) and spin-spin correlators on the ﬁrs t\nthree coordination spheres dependence on ϕ(J1= cosϕ, J2=\nsinϕ). Black bold line shows ccond, blue solid line — cg,\nred dotted — cd, green dash dotted — c2g. The points of\nphase transitions are marked on the x-axis: ϕ1– AFM →\nSL1transition, ϕ2— SL1→Stripe,ϕ3— Stripe →SL2,ϕ4\n— SL2→FM1(for the rescaled vicinity of ϕ4see Fig.4), ϕ6\n— FM2→AFM transition. See text for details.\nAtϕ=ϕ1condensate cAFM\ncondvanishes, AFM gap ∆ Q\nopens, and the spectrum becomes ungapped in the whole\nBrillouine zone, except trivial zero point Γ, where it\nremains linear. The system turns to spin liquid state\n(let’s denote it by SL1), which is realized in the interval\nϕ1≤ϕ≤ϕ2= 1.111. In this phase LRO is absent, and\nshort-range order transforms with growing ϕfrom the\nAFM-like ( cg<0,|cg|> cd> c2g>0) to the one typical\nfor stripe phase ( cd<0, c2g>0,|cd|> c2g>|cg|). At\nthe sametime spin gap at the point Qpassesthroughthe\nmaximum, and the gap at stripe points Xmonotonically\ndecreases (Fig. 3).\nAtϕ=ϕ2the stripe gap ∆ Xvanishes, the spectrum\nat stripe points Xbecomes linear, and condensate cStripe\ncond\nbecomes nonzero, the system turns to the LRO stripe\nphase, which is realized in the interval ϕ2≤ϕ≤ϕ3=\n2.141. Note the very interesting point ϕ=π/2, where\n−0.04−0.0200.020.040.060.080.1Gaps/50\nϕ1 ϕ2 ϕ3 ϕ4 ϕ6∆Q/50∆X/50\nStripeFM AFM\n0 π/2 π 3π/2 ϕ\nFIG. 3: Spin gaps ∆ Qand ∆ X(∆Q/50 and ∆ X/50 are\nshown) at the points Q= (π,π) (green dashed line) and\nX= (0,π),(π,0) (blue dash-dotted) of the Brillouin zone as\nfunctions of ϕ(J1= cosϕ, J2= sinϕ). Black solid line —\ncondensate ccond. All the points ϕ1–ϕ6are the same as in\nFig.2.−0.1 −0.050    0.05 −0.1 CorrelatorsCg\nCdC2g\nϕ4ϕ4a\nϕ5 ϕ−0.27−0.26−0.25\nE\nϕ4 ϕ5\nFIG. 4: Condensate ccondand correlators cg,cd,c2gevolu-\ntion from spin liquid SL2to ferromagnetic state FM2. As in\nFig.2, black bold line — ccond, blue solid — cg, red dotted —\ncd, green dash dotted — c2g.ϕ4corresponds to the transi-\ntion from spin liquid SL2to ferromagnetic state FM1, in the\nnarrow region ϕ4–ϕ4ashort-range FM order is absent, while\nlong-range FM order is present (see text), ϕ5— the border\nbetween ferromagnetic regions FM1and FM2(see text).\nInset: black solid line — energy per site εof the present\nwork, green dashed lines — energy extrapolation for the solu -\ntions SL2and FM2from Ref. 6. The intersection corresponds\nto ﬁrst-order transition between spin liquid and ferromagn et,\nstated in6.\nJ1= 0,J2= 1. The lattice with this exchange couplings\nsplits into two noninteracting AFM sublattices. Then\nit is obvious that cd(π/2) =cg(0),c2g(π/2) =cd(0)\n(see Fig. 2). Note that at ϕ=π/2, as in AFM phase,\n∆Q= 0 (see Fig. 3), however, this does not lead to AFM\nLRO, because F(Q,ϕ=π/2) also vanishes, and as a re-\nsultcAFM\ncond(ϕ=π/2) = 0.\nAt the point ϕ=ϕ3stripe gap ∆ Xopens, and the sys-\ntem again turns to the spin liquid state (SL2), realized in\nthe interval ϕ3< ϕ < ϕ 4= 2.712 (but the structure of\nthe short-range order diﬀers from that at ϕ1≤ϕ≤ϕ2).\nItisworthnoting, thatthenext-nearestneighborcorrela-\ntorcdremainsnegativethroughoutthe SL2existence, i.e.\nthe short-range order is not rearranged to the FM-like,\nwherecg,cd,c2g>0. The absolute value of cdalmost ev-\nerywhere, except tiny region near ϕ4, is larger than the\nnearest neighbor correlator cg.\nLet us emphasize ones more, that for all the mentioned\nphases the spectrum near Γ= (0,0) is linear in q.\nAtϕ=ϕ4there again appears a phase with LRO\n(ferromagnetic) and nonzero corresponding condensate\ncFM\ncond. Spectrum near the point Γbecomes quadratic\ninq(and the gap ∆ Γ(ϕ4) = 0). Fig. 2 and Fig. 3\ndemonstrate, that two regions are distinguishable in this\nphase — FM1and FM2. FM1covers in the tiny inter-\nvalϕ4< ϕ < ϕ 5= 2.733. Here condensate cFM\ncondgrows\nrapidly with the increase of ϕfromcFM\ncond= 0 to the\nmaximal value cFM\ncond= 1/12. Note, that near ϕ4FM\nLRO without FM short-range order is realized, cd<0\n(the corresponding interval is ϕ4< ϕ < ϕ 4a= 2.713).\nForϕ≥ϕ5(FM2region) all the correlators and the\ncondensate are equal to 1 /12 and the vertex correction\nα= 3/23,6).\nFM1region was not detected (Fig. 4) in6. The inset\nof Fig. 4 shows the energy at the transitions SL2→FM14\n00.20.40.60.811.21.4\nϕγAFM\nSpin liquidStripe\nπ/8 π/4 3π/4\nFIG. 5: Dependence of spin liquid SL1borders position on\nthe damping parameter γ(see text for details).\n→FM2. Dashed line is the extrapolatin of SL2energy\nto the intersection with the FM2energy (from6). Based\non this extrapolation, it was concluded in6, that a ﬁrst\norder transition occurs near the intersection point. Our\nconsideration leads to a conclusion (see Fig. 4), that the\nenergy derivative is continuous between SL2and FM1.\nNote that standard FM2solution3,6exists also for an-\nglesϕ < ϕ 5, down to ϕ=π−arctan(1 /2), but in this\nregion it happens to be metastable relative to FM1and\nSL2.\nAt the angles ϕ≥ϕ5the FM2solution is realized up\ntoϕ6= 3π/2. This point is a very special one. At ϕ6the\nlattice is splitted into two noninteracting sublattices. At\nϕ→ϕ6−0there is no frustration with respect to the FM\norder, at ϕ→ϕ6+0 — no frustration with respect to the\nAFM order. Therefore it is physicallyobvious that in the\nquantum limit atransition between these twophases isof\nthe ﬁrst order, as do our calculationsconﬁrm. Let us alsonote that, as it can be seen from Fig. 3, at ϕ→ϕ6+0\n(J1= +0,J2=−1), AFM condensate, i.e. absolute\nvalue of the spin-spin correlation function at inﬁnity, is\nmuch largerthan in the ”standard”AFM ( ϕ= 0,J1= 1,\nJ2= 0), and is equal to FM condensate at ϕ→ϕ6−0.\nIt means that FM next-nearest neighbor exchange with\nzero nearest exchange leads to stronger AFM order, than\nnearest AFM exchange with zero next-nearest one.\nIn conclusion let us note, that the signiﬁcant spin ex-\ncitations damping can be expected near the transitions\nspin liquid →LRO phase. Accounting for the damping\ncan shift the boundary of the corresponding transition.\nThis is demonstrated in Fig. 5, where the dependence\nof SL1phase boundaries on the damping parameter γis\nrepresented. We used the simple semiphenomenological\napproximation for the Green’s function Gzz\nγ, conserving\ncorrect analytical properties (see5for details).\nGzz\nγ(ω,q) =Fq\nω2−ω2q+iωγ(11)\nIt can be seen in Fig. 5, that the SL phase boundaries\nare sensitive to the value of damping. Nethertheless, our\nestimates show, that there are no topological modiﬁca-\ntions of the phase diagram for any reasonable values of\ndamping.\nTo summarize, in the present work the entire phase\ndiagram of the 2D J1−J2S= 1/2 Heisenberg model is\nconsidered in the frames of one and the same approach.\nIt is shown, that the transitions between all ordered and\ndisordered phases are continuous, except the transition\nFM→AFM at J1= 0,J2=−1.\nThis work is supported by Russian Foundation for Ba-\nsic Research, grant 13-02-00909a.\n1R.Nath, A.A.Tsirlin, H.Rosner, andC.Geibel, Phys.Rev.\nB78, 064422 (2008).\n2A.A. Tsirlin and H. Rosner, Phys. Rev. B79, 214417\n(2009).\n3H. Shimahara and S. Takada, J. Phys. Soc. Jpn. 60, 2394\n(1991).\n4A.F. Barabanov and V.M. Berezovsky, JETP 79, 627\n(1994); J. Phys. Soc. Jpn. 63, 3974 (1994).\n5A.F. Barabanov, A.V. Mikheenkov, and A.V. Shvartsberg,Theor. Math. Phys. 168, 1192 (2011).\n6M. Hartel, J. Richter, D. Ihle, and S.-L. Drechsler,\nPhys. Rev. B84, 104411 (2011).\n7M.Hartel, J.Richter, O.Gotze, D.Ihle, andS.-L.Drechsler ,\nPhys. Rev. B87, 054412 (2013).\n8A.V. Mikheyenkov, A.V. Shvartsberg, N.A. Kozlov, and\nA.F. Barabanov, JETP Lett. 93, 377 (2011)."    },    {        "title": "2003.13327v2.Impact_of_Next_Nearest_Neighbor_hopping_on_Ferromagnetism_in_Diluted_Magnetic_Semiconductors.pdf",        "content": "arXiv:2003.13327v2  [cond-mat.str-el]  15 Dec 2020Impact of Next-Nearest-Neighbor hopping on Ferromagnetis m\nin Diluted Magnetic Semiconductors\nSourav Chakraborty1, Subrat K Das2,∗, and Kalpataru Pradhan1,†\n1CMP Division, Saha Institute of Nuclear Physics, HBNI, Kolk ata 700064, India\n2SKCG Autonomous College, Paralakhemundi, Odisha 761200, I ndia\nBeing awide bandgapsystem GaMnNattracted considerable in terest after thediscoveryofhighest\nreported ferromagnetic transition temperature TC∼940 K among all diluted magnetic semiconduc-\ntors. Later, it became a debate due to the observation of eith er very low TC∼8 K or sometimes\nabsence of ferromagnetism. We address these issues by calcu lating the ferromagnetic window, TCvs\np, within the t−t′Kondo lattice model using a spin-fermion Monte-Carlo metho d on a simple cubic\nlattice. The next-nearest-neighbor hopping t′is exploited to tune the degree of delocalization of the\nfree carriers to show that the carrier localization (deloca lization) signiﬁcantly widens (shrinks) the\nferromagnetic window with a reduction (enhancement) of the optimum TCvalue. We correlate our\nresults with the experimental ﬁndings and explain the ambig uities in ferromagnetism in GaMnN.\nI. INTRODUCTION\nSearchfor high TCferromagnetismin diluted magnetic\nsemiconductors (DMSs) has been a topic of core impor-\ntance over the last two decades in view of potential tech-\nnological applications1–4. A DMS, with complementary\nproperties of semiconductors and ferromagnets, typically\nconsists of a non-magnetic semiconductor (e.g., GaAs or\nGaN) doped with a few percent of transition metal ions\n(e.g., Mn) onto their cation sites. The coupling between\nelectron states of the impurity ions and host semiconduc-\ntors drives the long-range ferromagnetism. The ultimate\ngoal is to demonstrate the dual semiconducting and fer-\nromagnetic properties of DMS at room temperature.\nMn doped GaAs (GaMnAs)5–10is one of the most\nextensively investigated DMS for which the highest re-\nportedTCis limited to 200 K [11,12]. Meanwhile, wide\nbandgap based DMSs have attracted substantial atten-\ntion after the discovery of room temperature TCin Mn\ndoped GaN (GaMnN)13–15. Wide bandgap materials are\npreferredovernarrowbandgapsemiconductorslikeGaM-\nnAs for two useful reasons: (i) possibility of room tem-\nperature ferromagnetism and (ii) suitability of its band\nstructure for spin injection16. But, the ferromagnetic\nstate in GaMnN is still a debated topic17,18. In the\nsearch of high TC, non-magnetic ions (like K, Mg, and\nCa) are also considered as potential dopants in nitride-\nbased wide bandgap semiconductors such as GaN and\nAlN19–22. Calculations show that the induced magnetic\nmoment for Ca substitution of Ga (single donor) in GaN\nis 1.00µB21, while it increases to 2.00 µBfor K substitu-\ntion22(K substitution of Ga is a double donor). Interest-\ningly Ga vacancy induces even a largermagnetic moment\n(∼3µB)21–23in GaN.\nIn order to avoid the complication arising due to metal\nions, cation-vacancy-induced intrinsic magnetism are ac-\n∗skdiitk@gmail.com\n†kalpataru.pradhan@saha.ac.intively investigated in wide bandgap nitride-based mate-\nrials23–25. The strong localization of defect states favors\nspontaneous spin polarization that leads to the forma-\ntion of local moments23. Usually the high formation en-\nergyof such cation vacancies due to unpaired electrons of\nthe anions around the vacancy sites prohibits us to have\nenough vacancy concentration that is required for col-\nlective magnetism26. Theoretical studies show that the\nformation energy can be reduced by applying an external\nstrain27. Overall, there is still no consensus regarding a\npathway to engineer high TCnitride-based DMS.\nAfter a considerable amount of eﬀorts has been given\nto the transition metal doped GaN based DMS, there is\nstill a lack of fundamental understanding of the origin of\nmagnetism. In the present work, we focus on certain as-\npects of the Mn doped GaAs and GaN like systems using\na model Hamiltonian study to address this fundamen-\ntal issue. The nature of ferromagnetism in GaMnAs is\nreasonably well understood3,4, and so is regarded as the\nmodel system to understand other similar DMSs. Here,\na few percentage of Mn2+ion (S = 5/2) replaces Ga3+,\ntherebycontributingaholetothehostvalenceband(VB)\nwhich mediate the magnetic interaction between the Mn\nspins. But the hole density (holes per Mn ions) is smaller\nthan1duetoAsantisites28(AsGa)andMninterstitials29\n(MnI) which act as double donors. It is well known that\nco-doping and post-growth annealing are some eﬀective\ntechniques to alter the hole density30,31. These holes re-\nside in the shallow acceptor level introduced by Mn ions\nin the host band gap ∼0.1 eV above the VB32–35re-\nﬂecting the long-range nature of magnetic interactions\nbetween the Mn ions. If these levels form a distinctspin-\npolarized impurity band (IB) for a ﬁnite impurity con-\ncentrationxthen the location of the Fermi energy EF\nwill play a crucial role in determining the TC. Qualita-\ntively, in this IB picture maximum TCis expected when\nimpurity band is half ﬁlled and supposed to decrease if\nEFis near the top or bottom of the band. In fact, the\nnon-monotonic ferromagnetic window is reported in ex-\nperiments for a wide range of hole density [36,37]. This\nis in agreement (disagreement) with the predictions of2\nRef. 10 in the high (low) compensation regime, reveling\nthe decisive role of sample structures along with compen-\nsation on the TCin DMS.\nMn interstitial is the crucial source of compensation as\nit’s removal improves both, the hole concentration and\nthe magneticallyactiveMn ions. Yu et al[38] haveshown\nthat theMnIconcentration reduces drastically by dif-\nfusing from the thin GaMnAs ﬁlm to the growth surface.\nThey also showed that all MnIcan be removed in case\nof thickness d <15 nm. Due to the eﬀective removal\nofMnIand interfacial eﬀects the TCis reported to be\n173 K ford= 50 nm [10], 185 K for d∼25 nm [39,40],\nand 191 K for d= 10 nm [12]. In comparison to thin\nﬁlms removing MnIfrom the bulk systems ( d≥60 nm)\nis diﬃcult, thereby limiting the TCto 120 K [36,37,41].\nOverall, the hole density, aﬀected by disorder, is very\nmuch dependent upon the growth process, the thickness\nand the structure of the DMS. In addition, structural\ndefects formed during the growth process can aﬀect the\nelectronic structure and hence the TCof DMS. In spite\nof a prolonged and intensive scientiﬁc eﬀorts GaMnAs is\nstill far from the room temperature applications.\nGaMnN seems to be a potential candidate to over-\ncome the above issue with TCover 300 K [13–15]. How-\never, achieving a ferromagnetic state in GaMnN is often\nchallenging17,18and the physical origin of the ferromag-\nnetism in this material still remains controversial due to\nthe contradicting experimental reports42–44. In contrast\nto GaMnAs, Mn is a deep acceptor in GaMnN form-\ning a distinct narrow IB that is ∼1.5 eV above the VB\nmaximum. Consequently, the hole mediated interactions\nbetween the Mn ions are short range in nature16,45–49.\nWherep-type co-doping (Mg in the case of GaMnN) has\nshown to enhance the saturation magnetization50, the\ntheoretical investigations found that extrinsic doping of\np-typegeneratingdefectssuchasGavacanciesreducethe\nstability of the ferromagnetic state51. In addition, the\ncoexistence of Mn2+(majority) and Mn3+(minority)52\nand the characteristics of defect states51–53have made\nthe nature of ferromagnetism in GaMnN more compli-\ncated compared to GaMnAs. So the theoretical studies\nto understand the ferromagnetism in GaMnN remains\nelusive to date.\nAim of this paper is to shed light on the unresolved\naspects of high TCferromagnetism in GaMnN. We con-\nsider thet−t′Kondo lattice model and calculate the\nmagnetic and the transport properties using a traveling\ncluster approximation based spin-fermion Monte-Carlo\nmethod54on a simple cubic lattice. Degree of delocal-\nization of the free carriers and hence the magnetic prop-\nerties are exploited by tuning the next-nearest-neighbor\n(NNN) hopping t′. We start with a brief introduction\nto the model Hamiltonian and the methodology of our\napproach. Next, the organization of this paper is three-\nfold: First we establish appropriate set of parameters for\nGaMnAs and GaMnN like systems. The electronic and\nmagnetic properties of GaMnAs are investigated in the\nsecond part. And, ﬁnally we calculate and connect ourresults with GaMnN.\nII. MODEL HAMILTONIAN AND\nMETHODOLOGY\nWe consider the diluted Kondo lattice Hamilto-\nnian55–58\nH=−t/summationdisplay\n/angbracketleftij/angbracketrightσc†\niσcjσ−t′/summationdisplay\n/angbracketleft/angbracketleftij/angbracketright/angbracketrightσc†\niσcjσ+JH/summationdisplay\nmSm./vector σm−µ/summationdisplay\nini,\nwherec†\niσ(ciσ) is the fermion creation (annihilation) op-\nerator at site iwith spinσ.tandt′are the nearest-\nneighbor( /an}bracketle{tij/an}bracketri}ht) and the NNN hopping parameters( /an}bracketle{t/an}bracketle{ti,j/an}bracketri}ht/an}bracketri}ht,\nrespectively. The third term is the Hund’s coupling JH\n(>0) between the impurity spin Smand the itinerant\nelectrons/vector σm(represented by Pauli spin matrices) at ran-\ndomly chosen site m. We consider the spin Smto be\nclassical and absorb it’s magnitude 5/2 into JHwithout\nloss of generality. Direct exchange interaction between\nthe localized spins due to magnetic moment clustering is\nneglected by avoiding the nearest neighbor Mn pairing.\nThe overall carrier density pis controlled through the\nchemical potential ( µ) given in the last term. µis cho-\nsen self consistently during the thermalization process\nto get the desired pat each temperature. For impurity\nconcentration xwe have 103xnumber of spins and pis\ndeﬁned as the holes per Mn impurity site. We consider\nx= 0.15-0.25 in a simple cubic lattice, where as GaAs\nis face centered cubic with four atoms per unit cell. So,\nthe impurity concentration we have taken for qualitative\nanalysis in simple cubic lattice is four times to that of fcc\nlattice. Therefore x= 25% for the impurity concentra-\ntion corresponds to roughly 6.25% Mn in the fcc systems\nlike GaMnAs59. We choose t= 0.5 eV by comparing the\nbare bandwidth (= 12 t) of our model to that of the real-\nistic bandwidth 6 eV for the host III-V semiconductors.\nOther parameters such as JH,t′, and temperature Tare\nscaled with t.\nThe model Hamiltonian incorporating spatial ﬂuctua-\ntions due to randomly distributed magnetic impurities,\nas in DMSs, must be carried out for a reasonably large\nsystem size for better results of the physical quantities\nsuch asTC[55,58]. We use the exact diagonalization\nbased classical Monte-Carlo method to anneal the sys-\ntem towardsthe ground state at ﬁxed carrierdensity and\ntemperature. First the classical spin Smis updated at a\nsite and in this background of new spin conﬁguration\nthe internal energy is calculated by exact diagonalization\nof the carriers. Then the proposed update is accepted\nor rejected by using the Metropolis algorithm. A single\nsystem sweep composed of the above processes repeated\nover each classical spin once. Note that the exact di-\nagonalization grows as O(N4) per system sweep and is\nnumerically too expensive for a system size of N= 103,\nwhere at each temperature we require at least over 1000\nsystem sweeps to anneal the system properly. We avoid3\n-8-6-4 -2 0\nω (in t)00.050.10.150.20.25\nN(ω)JH = 4.0\nJH = 6.0\nJH = 8.0\nJH = 10.0\n468 10 12\nJH0246810Ew and Eb (in t)Ew\nEb\n468 10 12\nJH100200300400500\nPR\n468 10 12\nJH0.010.0150.020.0250.03\nσdc(a) (b)\n(c) (d)\nFIG. 1: Shows the (a) density of states N(ω) with the for-\nmation of IB for diﬀerent values of the Hund’s coupling JH.\nFermi energy is set at zero; (b) change in the binding energy\nEband theEwwithJHshowing the localization-induced nar-\nrowing of the IB (the double arrow shows the width of the IB\nforJH= 6); (c) variation of the participation ratio with JH\ndistinguishing the extended states from the localized stat es,\nand (d) decrease in the dcconductivity (in units of πe2//planckover2pi1a)\nwithJHindicating carrier localization as in (c). All calcu-\nlations are made at ﬁxed impurity concentration x= 0.25,\ncarrier concentration p= 0.2, and temperature T= 0.05.\nthe sizelimitation byemployingaMonte-Carlotechnique\nbased on travelling cluster approximation54,60in which\nthe computational cost drops to O(N×Nc3) for each\nsystem sweep. Here Ncis the size of the moving clus-\nter reconstructed around the to-be-updated site and the\ncorresponding Hamiltonian is diagonalized rather than\nthat of the full lattice. This allows us to handle a sys-\ntem size of N= 103using a moving cluster of size Nc=\n63. All physical quantities are averagedoverten diﬀerent\nrandom conﬁgurations of magnetic impurities.\nIII. FORMATION OF THE IMPURITY BAND\nThe nature of the IB plays the key role in determining\nthe ferromagnetic state which solely depends on the ex-\nchange interaction JHand the amount of the magnetic\nimpuritiesxin the system. Ultra-fast transient reﬂectiv-\nity spectra61and magnetic circular dichroism measure-\nments36show the existence of a preformed IB inside the\nbandgap of in GaMnAs. We start our calculation for x\n= 0.25, where a separated IB starts to form for JH= 4\neven at relatively high temperature T= 0.05 as shown\nin the density of states (DOS) N(ω) =/an}bracketle{t1\nN/summationtext\nαδ(ω−ǫα)/an}bracketri}ht\nin Fig.1(a). Here the binding energy Eb= (bottom ofthe IB - top of the VB) ∼0.2t, where the small ﬁnite\ndensity of states between the VB and the IB is due to\nthe broadening used to calculate the DOS. We deﬁne the\nquantityEw(= top of the IB - top of the VB), which\nmust be smaller than the bandgap of the host semicon-\nductor. So Ew-Ebis the width of the impurity band.\nWith increase in the local Hund’s coupling the carriers\nget localized at the impurity sites, consequently the IB\nbecomes narrower and also moves away from the VB as\nevident fromFig. 1(b). Fromthese resultsnext we ﬁx the\nJHvalues to mimic GaMnAs and GaMnN like systems.\nGaMnAs is a low bandgap ( ∼1.5 eV) system with\nlong-ranged ferromagnetic interaction where the Ebis\nonly about ∼0.1 eV. Hence we choose JH= 4 for GaM-\nnAs forwhich Eb∼0.1eV(0.2t) and Ewis∼1.5eV(3t).\nDirect measurements yield JH= 1.2 eV - 3.3 eV [6–8]\nfor GaMnAs. Note that we absorbed the impurity spin\nmagnitude 5/2into JHwhich scaleswith t(= 0.5 eV). So\nourJHvalue is in the range as reported in experiments.\nIn contrast, the bandgap of GaMnN is ∼3.4 eV. And,\nthe IB is distinctly separated from the VB located at an\nenergy∼1.5 eV (Eb) above the VB implying the short-\nrange character of the ferromagnetic interactions. So in\nthis casewe choose JH= 10, where Eb∼2.75eV and Ew\nis∼3.5 eV (7t). Later, we will see that the NNN hop-\npingt′hardly alter the Ewvalue but signiﬁcantly aﬀects\nthe ferromagnetic state.\nThe degree of structural or magnetic disorder is in-\nversely proportional to the participation ratio PR =\n1//summationtext\ni(ψi\nl)4, where{ψl}are the quasiparticle wave func-\ntions. PR together with the DOS provide an extensive\npicture of both spectral and spatial features of quasipar-\nticle states. The participation ratio provides a measure\nof the number of lattice sites over which the state is ex-\ntended. For normalized wave functions the PR can range\nbetweenNfor a fully extended state and 1 for a site-\nlocalized state. In Fig. 1(c) we plot the PR of the state\nat the Fermi energy ( EF) withJHat ﬁxedp= 0.2 and T\n= 0.05. For the chosen Hund’s couplings JH= 4 and 10\nthe states are extended over ∼400 sites and only over ∼\n150 sites, respectively. It reﬂects the fact that the long-\nand the short-range nature of the exchange interactions\nin GaMnAs and GaMnN like systems are automatically\naccounted in the calculations.\nThen we calculate the dcconductivity by using the\nKubo-Greenwood formula62,63\nσ(ω) =A\nN/summationdisplay\nα,β(nα−nβ)|fαβ|2\nǫβ−ǫαδ(ω−(ǫβ−ǫα)) (1)\nwithA=πe2//planckover2pi1a, whereais the lattice spacing.\nnα(Fermifactors ) =f(µ−ǫα) andǫα,ǫβare the\ncorresponding eigen energies. And, fαβ=/an}bracketle{tψα|jx|ψβ/an}bracketri}ht\nare the matrix elements of the current operator jx=\nit/summationtext\ni,σ(c†\ni+x,σci,σ−h.c.). Finally, the dcconductivity is\nobtained by averaging the conductivity over a small low4\n0 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8\np00.0050.010.0150.020.0250.03\nTcJH=4\nJH=1010 20 30 40 50\nNconf00.0050.010.0150.020.0250.03\nTc\nJH=4\nJH=10\n0 0.010.02 0.030.04 0.050.060.07 0.08\nT00.20.40.60.81S(0)JH=4, N=123\nJH=4, N=103\nJH=10, N=123\nJH=10, N=103\n0 0.02 0.04 0.060.08\nT00.020.040.060.08\nσdcJH=4 \nJH=10\n0 0.02 0.04 0.06 0.08\nT5.975.985.99\nµ(a) (b)\n(c) (d)p=0.2\np=0.2p=0.2\nJH=4\nFIG. 2: Displays various physical quantitiesfor JH=4and 10\nat ﬁxed x=0.25. In case of ﬁxed carrier density calculations,\np=0.2. It demonstrates the (a) ferromagnetic structure fact or\nS(0) for two system sizes 103and 123, which are almost in-\ndistinguishable. The arrows point the TCvalues; (b) TCwith\nerror vs the number of conﬁgurations Nconfclarifying that\nNconf= 10 is reasonably good for our qualitative investiga-\ntions; (c) ferromagnetic windows TCvspshowing the local-\nization induced widening of the FM window in case of JH=10,\nand (d)dcconductivity vs temperature illustrating the more\nmetallicity ofthelong-range interactingsystems ( JH=4)com-\npared to the short-range interacting systems ( JH=10). The\ninset shows the variation of chemical potential µwith tem-\nperature for JH=4, required to set the desired p= 0.2.\nfrequency interval ∆ ωdeﬁned as\nσav(∆ω) =1\n∆ω/integraldisplay∆ω\n0σ(ω)dω.\n∆ωis chosen three to four times larger than the mean\nﬁnite-size gap of the system (determined by the ratio\nof the bare bandwidth and the total number of eigenval-\nues). Thisprocedurehasbeenbenchmarkedinaprevious\nwork63. The conductivity for ﬁxed p=0.2 atT= 0.05\nis shown in Fig. 1(d). The decrease in conductivity with\nJHsubstantiates the fact that the carriers get localized\nwith Hund’s coupling as seen in Fig. 1(a)-(c).\nIV. FERROMAGNETIC WINDOWS FOR t′= 0\nIn order to see the eﬀects of localization on ferromag-\nnetism we estimate the TCfrom the ferromagnetic struc-\nture factor S(0), whereS(q) =1\nN/summationtext\nijSi·Sjeiq·(ri−rj)\n(qare the wave vectors). The average structure factors\nforJH=4 and 10 are shown in Fig. 2(a) forp=0.2 us-\ning system sizes 103and 123. As the data of these twosystem sizes are barely distinguishable from each other,\nso we useN= 103for all calculations in this work. We\nestimateTCfrom each structure factor and then aver-\nage it over ten diﬀerent conﬁgurations, which is suﬃcient\nenough as shown in Fig. 2(b).TCvalue remains more or\nless same with the number of conﬁgurations Nconf. The\nerror forJH=4 andp=0.2 is found to decrease with the\nnumber of conﬁgurations and is in the acceptable range\nforNconf=10 for our qualitative investigations. And, for\nJH=10 andp=0.2 the error is insigniﬁcant i.e. the error\nbars are smaller than point sizes for all diﬀerent Nconf\nwe considered.\nNext we plot the ferromagneticwindows for JH=4 and\n10 in Fig. 2(c). The range of the FM window for JH=4\nis fromp=0 to 0.3. In the higher hole density regime the\ncarriers hopping gets restricted due to large delocaliza-\ntion length, and as a result kinetic energy is minimized\nand hence the TCis suppressed. On the other hand car-\nriers are less extended for JH=10 [see Fig. 1]. Conse-\nquently, the carrier hopping is stimulated to gain kinetic\nenergy resulting in wider FM window. In addition, in\nFig.2(d) we plot the dcconductivity in a wide range\nof temperature to corroborate the fact that the carriers\nare relatively more localized for JH=10 as compared to\nJH=4. All calculations with temperature are carried out\nfor ﬁxed carrierdensity p. The standard procedure to set\nthe desired pat all temperatures is by varying the chem-\nical potential µaccordingly with temperature as shown\nforJH=4 andp=0.2 in the inset.\nThe nature of the carriers that mediate the ferromag-\nnetism and in turn controls the TCdepends upon the\nlocation of the IB relative to the VB. Where, for JH=\n4 (GaMnAs-like) the gap is very small, that for JH= 10\n(GaMnN-like) is large, clearly displaying a separated IB\n(see Fig. 1). Keeping asidethe GaMnNcase, in literature\ntherearetwoconﬂictingtheoreticalviewpointsonthena-\nture of the carriers in GaMnAs. In one of those extreme\nlimits the IB is very much boardened and indistinguish-\nable from valence band, knownas the VB picture. In this\napproach within the mean-ﬁeld Zener model, the mag-\nnetic impurities induce itinerant carriersin the VB of the\nhost materials, which mediate the long-range magnetic\ninteractions9,10. It has been generally accepted because\nof it’s ability to explain a variety of features of GaM-\nnAs3,9,10,18,64–69. The key prediction of this approach is\nthatTCincreases monotonically with both the eﬀective\nMn concentration and the carrier density10. However,\nthis modelis contradictedby electronicstructurecalcula-\ntions45,70,71and argued that ferromagnetism in GaMnAs\nis determined by impurity-derived states that are local-\nized. This is commonly known as the IB picture. Several\nexperiments on the optical72–75and transport76,77prop-\nerties have reported that EFexists in the IB within the\nbandgap of GaMnAs. Results from resonant tunneling\nalso suggests that the VB remains nearly non-magnetic\nin ferromagnetic GaMnAs and does not merge with the\nIB78. This picture successfully explains the nonmono-\ntonic variation of TCwithpobserved in Refs. [36,37].5\n-8-6-4 -2 0 2\nω00.050.10.150.20.25\nN(ω)JH=2\nJH=4\n0 0.1 0.2 0.3 0.4 0.5 0.6\np00.0050.010.0150.020.0250.03\nTc\nJH=2\nJH=40 0.02 0.04 0.060.08\nT01020304050\nρ(a) (b)\nJH=2T=0.05JH=4p=0.2\np=0.2\nFIG. 3: Shows the (a) density of states N(ω) forJH= 2 and\n4 at ﬁxed p=0.2 and T=0.05. Fermi energy is set at zero.\nThere is no signature of IB for JH= 2, and (b) ferromagnetic\nwindows TCvspforJH= 2 and 4. Inset plots the resistivity\n(in units of /planckover2pi1a/πe2) Vs temperature, at p=0.2, indicating\nmore metallicity in case of JH= 2. We ﬁxed x= 0.25 for\nthese calculations.\nThis is in clear disagreement with the prediction of the\nvalence band picture9,10. However, recently it was also\nsuggested that both the mechanisms can be active simul-\ntaneously in GaMnAs17. In spite of all eﬀorts the issue\nof IB picture versus VB picture is still inconclusive.\nInthisbattleofbands79wheredoourassumptionofIB\npicture for JH= 4 in Fig. 1stands? As we have consid-\neredx= 25% in a N = 103system, in the ideal situation\nthe IB picture can be assigned when the participation ra-\ntio is within 250 i.e. the carriers are located only at the\nmagnetic sites. DOS along with PR in Fig. 1reveal that\nfor higher values of JH(=6 or more) the carriers are re-\nstricted to the magnetic sites [see Fig. 1(c)] and so can be\ncategorized in the IB model. But, in case of JH= 4 the\nIB is verycloseto the VB andso thereis signiﬁcantprob-\nability of hopping of the holes from the magnetic to host\nsites. In fact, due to this hopping process, the participa-\ntion ratio for JH= 4 [see Fig. 1(c)] is∼400. This shows\nthat there is signiﬁcant mixing between the VB and IB.\nInterestingly, even in the mixed nature of the carriers in\ncase ofJH= 4 theTCvaries non-monotonically unlike in\nthe valenceband picture10. Sothereisanaturalcuriosity\nto check the TCtrend in the pure VB picture in our cal-\nculation. For this we consider the lower Hund’s coupling\nJH= 2. The DOS plotted in Fig. 3(a) showsthat there is\nno signature of IB at all. Also, the calculated PR of the\nFermi state for p=0.2 is∼800. Clearly, this comes in the\ncategory of VB picture with more metallicity compared\ntomoderatelyandstronglycoupledsystems(seetheinset\nof Fig.3(b)). Most interestingly, the TCshows an opti-\nmization behavior with respect to pas in the case of JH\n= 4 [see Fig. 3(b)]. So we found that the non-monotonic\nbehavior of TCis independent of the VB and the IB pic-\ntures. Similar results alsofound by other techniques such\nas spin wave and earlier MC calculations55,59,80,81.-8-6-4 -2 0\nω (in t)00.050.10.150.20.25\nN(ω)JH = 10.0 , V=0.0\nJH = 4.0 , V=6.0\n4 6 8 10\nJH + V024681012Ew and Eb (in t)Ew (V=0)\nEb (V=0)\nEw (JH=4.0)\nEb (JH=4.0)\n4 6 8 10\nJH + V100200300400500\nPR\nV=0\nJH=4.0468 10\nJH + V0.010.0150.020.0250.03\nσdcV=0\nJH=4.04 6 8 10JH + V123Ew-Eb (in t)V=0\nJH=4\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7p00.0050.010.0150.020.0250.03\nTc\nJH = 10.0 , V=0.0\nJH = 4.0 , V=6.0(a)\n(b)\n(c) (d)\nFIG. 4: Presents the comparison of various physical quanti-\nties between the t−JHand the t−JH−Vmodels at ﬁxed\nx= 0.25. It compares the (a) density of states N(ω) for two\nsets of parameters ( JH,V)=(4,6) and (10,0), where features\nof the IBs are shown to match completely. Fermi energy is set\nat zero; (b) variation of the binding energy Eband theEwfor\ndiﬀerent sets of ( JH,V) values. In the x-axis JH+Vis var-\nied in two diﬀerent ways: (i) by varying Vwith ﬁxed JH=4\nand (ii) by varying JHwith ﬁxed V=0. Second one is the\nt−JHmodel for which the physical quantities are replotted\nfrom Fig 1. This shows that although EbandEwdiﬀer from\none representation to other but the width of the IBs match\nwell in the whole parameter range [see the inset]; (c) variat ion\nof the participation ratio and the dcconductivity (in units of\nπe2//planckover2pi1a) [in the inset] with ( JH+V) values as in (b). There is\none-to-one correspondence between them, and (d) ferromag-\nnetic windows TCvspshowing a good match for the two sets\nof parameters as in (a). In (a)-(c) the calculations are carr ied\nout at ﬁxed p= 0.2 and T= 0.05.\nV. CORRESPONDENCE BETWEEN t−JHAND\nt−JH−VMODELS\nThe properties of the IB and hence the ferromagnetic\nwindowcanbetunedbyvaryingthebindingenergyofthe\ncarriers. Hence it is worthful to highlight the t−JH−V\nmodel at this point before proceedingwith the NNN hop-\nping term in the Hamiltonian. Here the potential term is\nrepresented by/summationtext\nmVmnmwithVm=V at impurity sites\nand 0 otherwise. Apart from the magnetic nature of the\nHund’s term both JHandVact as the trapping centers\nfor the carriers at the impurity sites. So it would be in-\nteresting to check whether the t−JH−Vmodel can be\nqualitatively replaced by a only t−JHmodel or not, in\nthe parameter regime we consider. We benchmark our\nresults by comparing these two models. Fig. 4(a) shows\nthe DOSs for ( JH,V) = (10, 0) and (4, 6), where the\nIB is seen to be unaﬀected. Fig. 4(b) presents the bind-6\ning energy Eband theEwfor diﬀerent sets of ( JH,V)\nvalues. In the x-axis JH+Vis deﬁned in two diﬀerent\nways; (i) by varying V with ﬁxed JH= 4 representing\nthet−V−JHmodel and (ii) by varying JHwith ﬁxed\nV = 0 representing the t−JHmodel. Although the Ew\nand theEbdiﬀer from one representation to other for the\nwhole parameter range the widths of the IBs match well\n[see the inset]. Consequently, the PR and the conductiv-\nity results (see Fig. 4(c) and it’s inset) for t−JH−V\nmodel is more or less same to t−JHmodel. The com-\nparison of the ferromagnetic windows for both the set of\nparameters ( JH,V)= (10, 0) and (4, 6) indicate that the\nt−JH−Vmodel can be qualitatively replaced with a\nsuitable choice of t−JHonly, shown in Fig. 4(d). There-\nforeforsimplicity wespeciﬁcallyexplorethe t−JHmodel\nfor our further investigations.\nVI. EFFECTS OF NEXT NEAREST NEIGHBOR\nHOPPING FOR JH=4\nIn the recent past Dobrowolskaet al.[36] demonstrated\nthe existence of a preformed IB in GaMnAs and the TC\nis decided by the location of the Fermi energy within the\nimpurity band. In this picture the states at the center\nof the impurity band are extended resulting in maximum\nTC. And, the TCgets reduced towards both the top and\nthe bottom ends of the band due to localized states. In\nthe process insulator-metal-insulator (I-M-I) transition\nis observed with carrier density. Most importantly, they\nobserved the ferromagnetic state in a wide range of hole\ndensityp∼0.1-0.9. In Fig. 2(c) our FM window ranges\nonlyfromp=0to 0.3for JH= 4. Sonow wearegoingto\ninvestigate this mismatch by taking the impact of NNN\nhopping on the carrier mobility and magnetic properties\ninto account.\nWe start with the comparisonof the spin-resolved den-\nsity of states for t′= 0 and 0.2 at ﬁxed p= 0.2 and T\n= 0.004 [see Fig. 5(a)] for which ground states are fer-\nromagnetic. In both the cases the impurity band is spin\npolarized, while the VB remains more or less unpolar-\nized. In our holepicture positive t′acts as a localizing\nagent which can be visualized from the DOS, where the\nIB becomes narrow and shifts away from the VB. This\nis also apparently clear from the PR shown in Fig. 5(b),\nwhere the quasiparticle states in case of t′=0.2 are lo-\ncalized compared to t′=0 in the whole range of p. It is\nalso important to note that t′doesn’t alter the value of\nEw(∼3t) which is well within the bandgap of the host.\nAlternatively, higher JHcan also localize the carriers (as\nshown in Fig. 1(a)) and ultimately boarden the FM win-\ndow (see Fig 2(c)), butEwbecomes larger [see Fig. 1(b)]\nthan the bandgap which is not physically acceptable for\nnarrow bandgap host like GaAs.\nWe present the ferromagnetic window, TCvsp, for\nGaMnAs in Fig. 5(c). TheTCoptimizes around p=0.15\nand the ferromagnetism is restricted to a small window\nofp= 0-0.3 for t′=0. At higher hole concentration the-6 -5 -4 -3 -2 -1 0 1\nω-0.15-0.1-0.0500.050.10.15\nN(ω)\nt'=0.0\nt'=0.2\n0 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9p00.0050.010.0150.020.0250.03\nTct'=0.0\nt'=0.20 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9p100200300400500\nPRt'=0.0\nt'=0.2\n0 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9p00.010.020.030.040.050.060.07\nσdc\nt'=0.0\nt'=0.20 0.02 0.04T050100150200\nρ(a) (b)\n(c)\n(d)T=0.05\nT=0.004\nT=0.004p=0.4p=0.1\np=0.9t'=0.2\nFIG. 5: The eﬀects ofthenext-nearest-neighborhopping( t′=\n+0.2)andit’scomparison to t′=0are shownforvarious phys-\nical quantities at ﬁxed JH= 4 and x= 0.25. It presents the\n(a) spin-resolved density of states at ﬁxed T= 0.004, where\nthe IB shrinks and moves away from the VB due to carrier\nlocalization. The Fermi energy is set at zero; (b) change in\nthe participation ratio (PR) with pat ﬁxedT= 0.05 showing\nthe higher degree of localization for t′= +0.2; (c) displays\nthet′-induced broadening of the ferromagnetic window TCvs\np, and (d) dcconductivities (in units of πe2//planckover2pi1a) withpat\nﬁxedT= 0.004. I-M-I is conﬁrmed from the resistivity (in\nunits of/planckover2pi1a/πe2) Vs temperature plot for diﬀerent carrier den-\nsities, in the inset. The localization driven I-M-I transit ion is\nconsistent with the results presented in (c).\ncarrier mobility is suppressed due to larger delocaliza-\ntion length in GaMnAs, see Fig. 5(b). One can remo-\nbilize the carriers by reducing their overlap with a mild\nlocalization of the carriers which is stimulated by the\nNNN hopping parameter t′= 0.2 as shown in Fig. 5(b).\nConsequently, the ferromagnetism is activated and the\nwindow [see Fig. 5(c)] becomes wider ( p= 0-0.8) as ob-\nserved in the experiments ( p∼0.1-0.9) [30,36]. In order\nto correlate the magnetic and transport properties we\nplot the low temperature ( T= 0.004) dcconductivity\nin Fig.5(d). In a carrier-mediated magnetic system a\nminimum amount of carrier is necessary to initiate the\nmagnetic interactions, and at higher pthe magnetism is\nsuppressed due to the decrease in carrier mobility. Hence\nin these regimes the system is insulating and in interme-\ndiatepthe system is metallic resulting in higher TC. For\nbotht′= 0 and 0.2 conductivity goes through IMI tran-\nsition with optimization around the same value of pas in\ncase ofTCvspwindow, which supports the above carrier\nlocalization picture and also qualitatively matches with\nthe experiment36. Resistivity vs Temperature plot in the\ninset of Fig. 5(d) explicitly shows the IMI transition as\nwe increase the hole density. Experimental data together7\n-10 -8 -6-4 -2 0 2\nω-0.1-0.0500.050.10.150.20.25\nN(ω)t'=0.0\nt'=0.2\n0 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9\np00.010.020.03\nTct'=0.0\nt'=0.2\n-10 -8 -6-4 -2 0 2\nω-0.1-0.0500.050.10.150.20.25\nN(ω)t'=0.0\nt'=-0.2\nt'=-0.5\n0 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9\np00.010.020.030.040.050.06\nTct'=0.0\nt'=-0.2\nt'=-0.5(a) (b)\n(c) (d)\nFIG. 6: The eﬀects of the t′= +0.2 and it’s comparison to t′\n= 0 are presented for (a) the spin-resolved density of states\nat ﬁxed T= 0.004, where the IB moves away from the VB\ndue to carrier localization and (b) the ferromagnetic windo w\nTCvspshowing the localization-induced broadening. The\neﬀects of the t′=-0.2 and -0.5 with it’s comparison to t′=\n0 are presented for (c) the spin-resolved density of states a t\nﬁxedT= 0.004, where the IB extended towards the VB due\nto carrier delocalization, and (d) the ferromagnetic windo ws\nshowing the delocalization-induced shrinkening. The Ferm i\nenergy is set at zero. We ﬁxed JH= 10 and x= 0.25 for all\ncalculations.\nwith our ﬁndings hint the presence of NNN-hopping in\nGaMnAs like systems. But further probe and investiga-\ntions are necessary to establish this scenario.\nVII. EFFECTS OF NEXT NEAREST\nNEIGHBOR HOPPING FOR JH=10\nNow we study the GaMnN with the chosen Hund’s\ncouplingJH= 10. The spin-resolved DOS and the FM\nwindows are evaluated with the same set of parameter\nvalues as in Fig. 5. Fig.6(a) and (b) show that the eﬀect\nof NNN hopping on the IB and FM window are qualita-\ntively similar as in the case of JH= 4. Quantitatively,\ntheeﬀectoflocalizationdueto t′ismuchmoreprominent\nforJH= 10 and as a result the deduction of TCis sig-\nniﬁcant. But, the electronic structure calculations reveal\nthat the Ga defects in GaMnN introduce states between\nthe VB and the IB which depopulate the IB and in turn\ndestroy the ferromagnetism in GaMnN51. We mimic the\nsituation by introducing negative NNN hopping which\ndelocalize the carriers and consequently broaden the IB\ntowards the VB. This can be seen from the DOS plotted\nin Fig.6(c) fort′=-0.2and-0.5alongwith t′=0atﬁxed\np= 0.2 andT= 0.004. Note that the binding energy Eb0 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9p00.010.020.030.040.05\nTct'=0.0\nt'=-0.2\nt'=-0.5\n0 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9p00.010.020.030.04\nTct'=0.0\nt'=-0.2\nt'=-0.5-10 -8 -6-4 -2 0 2\nω-0.1-0.0500.050.10.15\nN(ω)t'=0\nt'=-0.2\nt'=-0.5\n-0.5-0.4 -0.3 -0.2 -0.1 0\nt'0.050.060.070.080.090.1\npMn\n-0.5 -0.4 -0.3 -0.2 -0.1 0\nt'00.0050.01\nσdc(a) (b)\n(c) (d)x=0.2\nx=0.15x=0.2 x=0.2\np=0.1\nFIG. 7: The eﬀects of the t′= -0.2 and -0.5 with it’s compar-\nison tot′= 0 at ﬁxed x= 0.2 are shown for (a) the spin-\nresolved density of states at ﬁxed T= 0.004 and (b) the\nferromagnetic windows TCvsp. For higher degree of delo-\ncalization the FM window becomes signiﬁcantly narrow. (c)\nplots the average carrier density per magnetic impurity sit e\n(pMn) vst′at ﬁxedp=0.1 and T= 0.05, which reveals the out\nﬂow of carriers to non-magnetic sites with delocalization. In\nconsequence the overall conductivity of the system increas es\nas shown in the inset. We ﬁnd similar narrowing of FM win-\ndow forx= 0.15 and presented in (d). We considered JH=\n10 for all calculations.\ndecreases but Ewremains more or less unaﬀected (i.e.\nEwis within the bandgap of host GaMnN). As positive\nand negative t′play opposite roles in the system, so the\nferromagnetic window shrinks and the optimum TCin-\ncreases with carrier delocalization, as shown in Fig. 6(d).\nThe solubilityofMn in GaAs andGaNis low, sowe es-\ntablish ourﬁndings by calculatingthe ferromagneticwin-\ndowsforlowerimpurityconcentrations. Firstweconsider\nx= 0.2 and the results for the spin-resolvedDOS and the\nFM windows are presented in Fig. 7(a) and (b) respec-\ntively. The IBs show qualitatively similar evolution with\nt′as in case of x= 0.25. Apart from the disappearance\nof ferromagnetism in the higher pregime, interestingly,\nthe magnetism also vanishes for verylow carrierdensities\nfort′= -0.5 making the FM window furthermore narrow.\nNote that we have considered the relative carrier density\ni.e. number of carrier per Mn impurity site as in experi-\nments. So, in the low xand lowerpregime the magnetic\nsites accumulate a tiny amount of holes resulting in a\nweaker magnetic interactions. Here, if we increase the\ncarrier mobility the holes get deplete from the magnetic\nto the non-magnetic sites which further suppresses the\nspin-spin couplings. The out ﬂow of carriers is displayed\nin Fig.7(c) where we plot the average carrier density at\nthe magnetic sites pMnvst′at ﬁxedp=0.1. Eventually,8\nthe ferromagnetism vanishes at lower pas in case of t′\n= -0.5. On the other hand, the overall conductivity of\nthe system increases with the degree of delocalization as\nshown in the inset. We ﬁnd similar results for x= 0.15\n[Fig.7(d)]. The vanishing ferromagnetism in both lower\nand higher pregimes for t′= -0.5 makes the ferromag-\nnetic window signiﬁcantly narrow, which suppresses the\nprobability of getting a FM state. In experiments the\npresence of defects makes the sample preparation very\ncrucial and our results indicate that unless the system\nhas a favourable combination of xandpin a narrow win-\ndow then there is a higher chance to observe either low\nTCor absence of ferromagnetism. In addition, the sharp\nincrease in the optimum TCin a thin window of pclar-\niﬁes the room temperature ferromagnetism occasionally\nobserved in experiments.\nVIII. CONCLUSIONS\nIn conclusion, we investigated the magnetic and the\ntransport properties of III-V DMSs using a classical\nMonte-Carlomethodwithinthe t−t′Kondolatticemodel\non a simple cubic lattice. We have shown that the car-rier mobility induced by the NNN hopping t′plays a vi-\ntal role in determining the ferromagnetic states in both\nGaMnAs and GaMnN like systems. In case of GaMnAs\na small positive t′(that helps to localize the carriers) is\nshown to be necessary to reproduce the robustness of the\nferromagnetic states in a wide range of carrier concen-\ntration as observed in experiments. On the other hand,\nif we delocalize the carriers by activating negative t’ the\nferromagnetic window signiﬁcantly shrinks with an en-\nhancement of the optimum value of TCin GaMnN. We\ncorrelate our ﬁndings with the experimental results and\nsuggest that Ga like vacancy in GaMnN that depopulate\nthe IB triggers high TCin low hole density. 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Blackman3 \n1 Department of magnetic nanostructures, Institute f or physics of microstructures RAS,  \nNizhniy Novgorod, GSP -105, 603950,  Russia  \n2 Department of physics and nanoelectronics, Lobachevsky State University of Nizhniy Novgorod,  \nNizhniy Novgorod, Gagarin Avenue, 23, 603950, Russia  \n3 Department of Physics, Univ ersity of Leicester, Leicester, LE1 7RH, UK  \n(Submitted on 26 December 2013, Revised 4 February 2014 ) \nWe present a theoretical investigation of  magnetostatic interaction  effects  in geometrically  frustrated  arrays  of anisotropic  \nmultilayer  ferromagnetic nanoparticles  arranged in  different spatial ly configured  systems  with triangular  symmetry . We show \nthat the interlayer magnetostatic interaction  significantly expands the  opportunities to create  magnetically frustrated systems . \nThe e ffects of  the magnetostatic  interaction in magnetization reversal processes  and the  possibilit y to control  the ferromagnetic \nresonance  spectrum in such systems  are d iscussed . \n1. Introduction  \nThe effects  of magnetostatic interaction  in the regular arrays  of anisotropic  single -domain  ferromagnetic  nanoparticles \narranged in  two-dimensional lattice s with different spatial symmetry (in the literature , such systems are called  “artificial  \nspin ice” ) are the subject of the intensive investigations in the last decade [1-6]. The increasing interest in these  objects  is \nassociated primarily with the possibility to investigate the fundamental properties of geometrically frustrated magnetic \nsystems using  a relatively  simple model.   \nThe recent progress in the e -beam  nanolithography  techniques enables the fabrication  of super dense arrays of  \nnanoparticles , which  demonstrate  an unusual collective behavior connected with the competition between configuration \nentropy effects, dipolar interaction  and a rtificial anisotropy  [7-9]. It was shown that artificial  spin ice systems demonstrate  \nthe considerable changes in coercivity, switching fields and scenario of magnetization reversal. In particular, the dynamic \nswitching  in an external magnetic field is accompanied by the effects of the effective magnetic charges ordering [ 9] and the \nappearance of the exotic states called “magnetic monopoles” [ 10,11].  \nThe basic idea of our work is the investigation of the influence of magnetostatic interaction on the magnetic states and \nhigh frequency properties of the frustrated magnetic systems based on multilayer stacks of binary  nanoparticles . In this case \none can expect a considerable expansion of the spectrum of magnetic states and a significant increase in the averaged \nmagnetostatic energy  at a lattice site  due to the strong  dipolar coupling between particles  in the neighborin g layers.  \nFurthermore,  in multilayer systems the effects  of geometrical  frustration  can be significantly expanded due to the variation \nof the  magnetic  moments for the particles  located in different  layers by varying the materials and layer thicknesses .  \nIn the current letter we concentrate our attention on two aspects of the magnetostatic interaction in multilayer stack \narrays. First is the realization of exotic magnetic configurations in the magnetization reversal process, which remain stable \nafter removal  of the external magnetic field. The other interesting problem is the influence of intralayer  and interlayer  \nmagnetostatic  interaction s on the spectrum of ferromagnetic resonance and magnetostatic spin waves  in the ordered arrays \nof multilayer stacks  as th e prototype s of artificial 3-D magnonic crystals  [12-15]. In particular,  we show that  the spectrum \nof collective modes of ferromagnetic resonance in such systems strongly depends  on the spatial  configuration of the \nmagnetic  moments of particles and can be significantly  changed  by switching of magnetic states in an  external magnetic \nfield. From a practical point  of view such systems  are promising for  the development  of tunable microwave  devices  [16-18] \nfor civil and military applications .  \n2. The methods  of calculations  \nWe investigate d the effects of  magnetostatic  interaction and  ferromagnetic resonance in  arrays of the elliptical \nnanodisks ( a × b × h) arranged in a triangular grating (Fig. 1 ). To simplify the calculations we used the theoretical model of  \nanisotropic  dipoles common ly used  for the description of such systems [4, 19, 20]. We assumed that the magnetic field of \nthe nanoparticles corresponds to the field of  a uniformly magnetized sphere with built -in anisotropy corresponding to the \nshape anisotropy  of an elliptical disc. In this approximation the energy of  a system  can be presented as   \n 2 2 2\n3 5( ) 3( )( ) 1 1\n2 2i j i ij j ij\nxi xi yi yi zi zi i ex\ni j i i ij ijM M M R M RW N M N M N M M HR R            \n          , (1) \n                                                \nCorresponding author : mironov@ipmras.ru   2 where iM\n is magnetic moment of i-th particle; xiN, yiN and ziNare demagnetizing factors along the main axes of \nelliptical disc; ijRis the separation between disc’s centers; exH\n is an external magnetic field .  \nTo find the eigenfrequencies  of the ferromagnetic resonance  in the particle array s we solved  the system  of Landau -\nLifshitz equations without  external magnetic field. The oscillations  of the magnetic  moment of i-th particle  in the array  are \ndescribed by the following equation:  \ni iiH MtM \n ,      (2) \nwhere  is gyromagnetic ratio , iH\n is the effective  magnetic field, which  is generally  defined as  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1. The schematic drawing of the elliptical particles array on tri angular lattice . \ni j di\nj iH H H\n   .      (3) \nHere jH\n is the stray field from j-th particle at the location of i-th particle  and diH\n is demagnetizing field, which is defined \nas \ndi iH NM  \n, \nwhere N\n is the tensor of demagnetizing factors:  \n\n\n\n\n\nzyx\nNNN\nN\n0 00 00 0\n. \nThe FMR eigenfrequency of the single elliptical disc is defined by effective field s of shape anisotropy  \n0 ( )( )x y z y SN N N N M     ,      (4) \nwhere SM is the saturation magnetization.  \nTo find the spectrum  of eigenfrequencies  for the interacting particles  array  we solved  the linearized equation  (2) with \n0exH\n. The magnetic moment  of each particle and  acting  field were re presented as : \n,st\ni i iM M m t    \njst\nj j h H H\n , \nwhere  st\niM\n is an average  static  magnetic  moment of the particle ; im t is an  alternating  magnetic moment  \n st\ni im t M ; st\njH\n is a static field;  jh\n is a high frequency magnetic field  acting on the  i-th particle from the  j-th \nparticle , (jh<<st\njH). Then the  linearized system  of Landau -Lifshitz equations can be written as follows : r jM\n ijR \ni j \niM\n a  b  h z \nx y  3 st st st i\ni j i i j i\nj i j imM h Nm m H NMt \n                                    .   (5) \nFurther, we  always assume  that the effective field  associated with  the particle shape  anisotropy  more than the fields  of the \nother particles (( ) ; ( )st\nx y i z y i jN N M N N M H  \n), i.e. precession of the  magnetic mo ments occurs  mainly around  the \nanisotropy axes  of each particle. However t he spectrum of eigenfrequencies  in such systems  is determined by the  \nmagnetostatic  interaction and strongly depends on the spatial configuration  of the magnetic moments.  \nThe remagnet ization  effects in interacting arrays under an external magnetic field were investigated by Monte Carlo  \nsimulations based on the model of anisotropic spheres . In the calculations we  have neglected any deviation  of the vectors \niM\n out of the plane of  the elliptical  disc and used the simple model of uniaxial anisotropy. In this case the magnetic \nenergy of particles array can be represented as  \n2 *\n3 5( ) 3( )( ) 1sin ( )2i j i ij j ij\ni i i ex\ni j i i ij ijM M M R M RW KV M HR R \n          \n          ,  (6) \nwhere K is anisotropy constant; V is the volume of particle; i is the angle characterizing the direction of magnetic \nmoment of i-th particle; *\ni is the angle characterizing the direction of anisotropy axis of i-th particle . The normalized \nenergy can be r epresented in the following form:  \n2 * *sin ( )i i i ex\ni iW m H       .    (7) \nHere  is the normalized magnetostatic  energy  \n 0 0 0 0\n3 53 1\n2i ij j ij i j\ni j ij ijm r m r m m\nr r                \n ,     (8) \nwhere 0im is a unit vector of the magnetic mo ment of i-th particle, ijr is normalized separation between particle centers. \nThe p arameter  and normalized external magnetic field exH\n are expressed as follows : \n2\n3sM V\nR K ,      (9) \nanex\nexHHH*,       (10) \nwhere anH is effective field of anisotropy (an s H K M ); sM is a saturation magnetization of particle material. The \nparameter  is the ratio of the magnetostatic energy to the energy of anisotropy.  \nThe magnetization reversal processes and stationary distribution of the magnetic moments in the part icle’s arrays \ncorresponding to the minimum energy (7) were simulated by  the Monte Carlo method.  \n3. Magnetostatic interaction in one -layer s ystems  \nOne of the  simplest systems  of interacting  binary  particles is a  system  with basic  a building block of  three particles  \nlocated  in the plane  at an angle of  120°  (Fig. 2).  \n \n \n \n \n \n \n \n \n \n \n \nFig. 2. The p ossible configurations  of magnetic moments  in an array  of three particles  with triangular \nsymmetry. Parameters : v is the number of ways to  implement the  state; q is an effective  magnetic \ncharge  on the lattice site ; m is an effective  magnetic mom ent on the lattice site ;  is normalized \ndensity of magnetostatic energy . (A) (B) \nv         2                 6 \nq         3                 1 \nm        0                 2  \nε       1.01         - 0.34  4 As it is known this system has two  states  differ ing in  the spatial configuration of  the magnetic  moments [6]. These states \ndiffer in the  probability  of their realization at  random in itial conditions ( this quantity is proportional to the number of  ways \nto implement  this state  v), the effective  magnetic charge  on the lattice site  q, the effective  magnetic moment  on the lattice \nsite m, and the normalized density of magnetostatic energy ε. As is seen , the state ( A) corresponding to the highest  effective  \nmagnetic charge  and zero mean  magnetic moment  (“magnetic monopole ”) has the highest energy . The quasi -homogeneous \nstate ( B) with effective  charge q = 1 and  nonzero average  magnetic moment  has a lower energy.  \n3.1. Magnetization reversal  in the one -layer  three particles array  \nThe form of the magnetization reversal  of the three particle array  in an uniform external  magnetic field  depends on the \nratio, , of the anisot ropy energy to the magnetostatic  interaction . The hysteresis  curves  and schemes of transitions \nbetween  different states for  the array  of three interacting particles  are shown in Fig. 3.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3. The h ysteresis  curve s and the scheme s of transitions  between different states of three  particles array : (a) 0, \n(b) 0.1 , (c) 0.5 , (d) 1. The  configuration s of the magnetic moments  of the particles  corresponds \nto the points on  the hysteresis curve s. The d irection of the external  magnetic field is  shown in the diagram s by \narrows  located above the  array  of particles.  \n(1) (2) (3) (4) \n(1) (2) (3) (4) (b) 0.1  \n(1) (2) (1) (2) \n(c) 0.5  \n(1) \n(1) (3) \n(2) \n(2) (3) (a) 0 \n(1) (3) \n(2) \n(d) 1 \n(1) (2) (3)  5 If the anisotropy is dominant as in Fig. 3a (here, the magnet ostatic interaction is switched off completely, =0), the high \nenergy state is realized ( “magnetic monopole ” state (2) in Fig. 3a) at intermediate reversing field. In this case  we can \nprepare two stable sta tes (A) and (B), in the notation of Fig. 2, by an external magnetic field.  As the interaction energy is \nincreased in comparison with anisotropy energy, the high energy state (B) becomes unstable. For exam ple, the hysteresis \ncurve for 0.1  (in Fig. 3b ) demonstrates o nly a narrow step  corresponding to the state (B), while for =0.5 (Fig 3c) \nthere is a continuous transition between state (A) and its complete reversal. As the interaction ene rgy is further increased \n(negligible anisotropy),  1 , as in Fig. 3d, the hysteresis loop collapses. The ground state  of the array  at zero external  \nmagnetic field is the  vortex state ( state (2) in the transition scheme  of Fig. 3d). \n3.2. Ferromagnetic resonance  in single  layer systems  \nWe calculated  the dependence of the the FMR frequencies  of the three particle array on parameter r (the distan ce \nbetween center of particle and center of array). The increasing magnetostatic interaction  between particles leads  to a \nsplitting of  the FMR  spectrum. The depend ences of normalized FMR  frequencies  on the nondimensional parameter 3/V r  \nfor the region of parameter 0.1  are shown in Fig . 4.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4. The dependences of normalized FMR  frequencies  on the distance  between center of the particles  \nand array center for two possible magnetic moment configurations.  The FMR fre quencies are \nnormalized on the resonance  frequency of single particle.  \nAs is seen  from Fig.  4a the interaction between particles  in a high -energy  configuration  (A) leads to the splitting of  the \nFMR  spectrum  into two branches and shifting to lower frequencies. On the other hand, for the low-energy configuration  \n(B) the splitting into three branches  is observed  (Fig 4b). Two branches  are shifted to the higher frequencies  and one to \nlower frequencies . Since t he high -energy  configuration  (A) has a higher symmetry than configuration  (B) the lower branch \nof spectrum in Fig. 4a is doubly  degenerate .  \nThe three  elements  group  considered  above  can be combined into  the structures  with different symmetry , leading to \nadditional  changes  in the spectrum of  FMR. Here we  consider three  possible  spatial configurations  of association in  \nsymmetrical  two-dimensional arrays (Fig.  5) and discuss the  changes in the FMR spectra  from parameter r* (distance \nbetween centers  of particle groups ). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 5. Three spatial configurations with quasi -homogeneous state for association of three particle group . \nThe parameter r* is the distance between  canters  of particle groups. (I) \n(II) \n(III) r* 0,00 0,02 0,04 0,06 0,080,9951,0001,0051,0101,015 \n 0,00 0,02 0,04 0,06 0,080,9800,9850,9900,9951,000\n  \nV / r3 v / v0 \nV / r3 v / v0 \n(a) (b)  6  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \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 \nFig. 6. The dependences of normalized FMR  frequencies  on the distance between centers  \nof particle groups . \nIn the configuration  (I) arrays  interact mainly only by the two outermost particles.  This leads to an  additional  monotonic \nsplitting of  the high frequency  modes in the  FMR  spectrum  (see Fig.  6a and compare with Fig.  5b). A different situation  is \nobserved  for the configurations  (II) and (III).  The interaction  energy of  the arrays  in these configurations  is less  than in the \nconfiguration  (I), and as a result  lower frequency splitting  is observed  (see Fig. 6b and Fig. 6c). However  these \nconfigurations  implemented  a nonmonotonic changing FMR spectrum that is the effect of competition between interaction \nof three elements inside group and intergroup  interaction .  0,0 0,2 0,4 0,6 0,8 1,00,920,940,960,981,001,020,0 0,2 0,4 0,6 0,8 1,00,991,001,011,021,031,041,05 0,0 0,1 0,2 0,3 0,41,001,051,101,151,201,25\n  \n(a) v / v0 \nr / r* \n(b) \nr / r* v / v0 \n(c) v / v0 \nr / r*  7 4. The two-layer systems  \nThe magnetostatic interaction energy strongly depends on the distance between the particles and can be estimated as  \n2 2\nint 3M VWd . \nIn this connection the stack arrangement of e lliptical particles (one above the other , see Fig. 7a ) leads to an essential \nincrease in the interaction energy. In addition, it can be easy realize d technologically.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 7. The schematic image of two-layer stack  (a) and array of three two -layer particles  \nwith triangular symmetry  (b). \nWe investigated the  peculiarities of the magnetostatic  interaction  in an array  of two -layer  particles with  triangular  \nsymmetry ( Fig. 7b).  The dependences of magnetostatic interaction energies for different con figurations of magnetic \nmoments on the interlayer separation are presented in Fig. 8 (a). When the distance between the laye rs is large (parameter \nr / d is close to zero) the interaction between the layers is small and in this system three energy -degenerat e states (I), (II) \nand (III) are implemented. The degrees of degeneracy are (I) – 2; (II) – 2; (III) – 4 respectively (see Fig.  8b). The close \napproach of layers removes the  degeneracy and the energy spectrum is split into eight components due to the inter layer \ninteraction.  The energies of one group states are shifted to higher energies (u) and the energies of other state s are shifted to \nlower energies (d). In the region of parameters r / d 0.6 ÷ 1 we observe the crossing of energ y levels from completely \ndifferent configurations  (I, II and III)  as the interaction of particles in the layer becomes comparable with the interaction of \nthe particles located in different layers.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 8.  (a) is the  dependences of normalized magnetostatic energ ies for different configurations of \nmagnetic moments  on interlayer separation. (b), (c) and (d) a re possible configurations of \nmagnetic moments in the array of three two -layer particles.  The moment configurations \nfor the upper and lower layers are shown si de by side in different colors.  (b) d r d\n2 1 \n(a) \n(I) (II) \n(III) (u) (d) (u) (d) \n(u1) (u2) (d1) (d2) (c) (b) \n(d) (I) \n(III) (II) \nr / d  \n(a) (u) \n(d)  8 0,0 0,1 0,2 0,3 0,4 0,50,800,850,900,951,00 \n (a) \nV / d 3 v / v0 \n0,0 0,1 0,2 0,3 0,4 0,51,01,11,21,3\n  \nV / d 3 (b) v / v0 4.1. The ferromagnetic resonance in two -layer stack  \nWe calculated FMR  spectra  for different configurations of  a two -layer  stack. The  dependencies of  the FMR  frequency  \non the distance  between the layers  for the stack  in a st ate with  ferromagnetic and  antiferromagnetic order  are presented i n \nFig. 9. Qualitatively,  these curves  are similar to  the spectra of  two-dimensional array  of three particles (see Fig. 4), but the \ndegree of frequency splitting  for the stack  is much  more th an for two -dimensional  array.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 9.  The dependences of normalized FMR  frequencies  on the separation  between layers in the \ntwo-layer stack for different spatial configurations of moments. (a) is the case of \nferromagnetic ordering, (b) is for antiferromagnetic ordering.  \n4.2. Magnetization reversal  in two-layer  particle array  \nWe have calculated the hysteresis  curves  and the transition schemes between different states for two -layer  particle arrays  \nfor different values of the r / d ratio ( Fig. 10(a -c)). We show specifically results for the =0.5 case, for which a single layer \nthree island cluster displayed a continuous transition between type (A) moment configurations (see Fig. 3c). The effect of \nincreasing the ratio of the int erlayer to intralayer coupling can be seen in the sequence Fig. 3c (equivalent to r / d = 0), \nFig. 10a ( r / d = 0.45), Fig. 10b ( r / d = 0.58) Fig. 10c ( r / d = 0.83). At low interlayer interaction ( r / d = 0.45) we have the \nsituation similar to the one -layer system (Fig. 3c). In this case there is only one stable state with quasi -uniform orientation \nof magnetic moments in layers and ferromagnetic interlayer ordering. The in termediate case (Fig. 10b ) has  two stable states , \nwhich correspond to ferromagnetic and antiferromagnetic interlayer ordering. Increasing the interlayer coupling further \n(Fig. 10 c) leads to a closing up of the hysteresis loop, and the formation of an antiferromagnetic ordered st ate with zero \naveraged magnetic moment in zero field . Thus th e variation of distance between layers enables the effective controlling the \nmagnetic states in two -layer system .  9 \n(1) (2) (1) (2) \n(b) r / d = 0.58 \n     = 0.5 \n(1) (2) (1) (2) r / d = 0.83 \n     = 0.5 \n(c) \n(1) \n(a) (2) r / d = 0.45 \n     = 0.5 \n(1) (2)  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 10. The h ysteresis  curves  and the schemes of transitions  between different s tates of two -layer  particle \narray s for the case  α = 0.5 and two interlayer spacings:  (a) r / d =  0.45; (b) r / d =  0.58; ( c) r / d \n=  0.83. The moment configurations for the upper and lower layers are shown side by side in \ndifferent colors.   10 4.3. The f erromagnetic resonance in  two-layer systems  \nWe have  calculated the spectra for two stable states of two-layer  particles  array ordered on the lattice  with triangular \nsymmetry (Fig 11(a,b)). The calculations show ed that the two-layer arrays  have a more compl icated splitting  of the FMR  \nspectrum in comparison  with one -layer system . In particular the dependencies  of normalized FMR frequencies  of the array \nin the antiferromagnetic interlayer ordering have the  local extrem es at some distances  between the layers  (Fig.  11(b)). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 11. The depende nces of normalized FMR  frequencies  on the separation  between layers in the two -\nlayer three particle groups for different spatial configurations of moments.  (a) is for ferromagnetic \ninterlayer ordering.  (b) is for antiferromagnetic interlayer ordering . \n5. The three-layer systems  \nThe energy of interaction  in multilayer systems can be controlled  also by variation of the  magnetic  moment magnitude \nfor the particles  located  in different layers  [21]. Fig. 12 shows  a schematic drawing of a system of  three particles  arranged  \none above the other, and the  state diagram  of this system  depending on the  interlayer distance  and the ratio  of the magnetic \nmoments  of the particles. We assumed that  the moments  of the outer  particles  are the same. In such system  there are three  \nstable configurations with  different  magnetostatic  energy.  As can be seen  from Fig.  12b, at the critical point  with the \nparameters  12 13/d d = 0.5 and m2 / m1 = 0.2 there is  a frustrated  state,  in which the energ ies of all magnetic moment s \nconfigurations are equal.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 2. The three -layer stack (a) and the diagram of possible states in dependence of magnetic moments  \nrelation and separations between layers  (b). \nd12/d13 \nm2 / m1 (m1 = m3) (b) 12d\n23d13d\n3 2 1 (a) 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,40,900,951,001,05\n  \n(a) v / v0 \nr / d 0,0 0,2 0,4 0,6 0,8 1,01,0001,0251,050\n  v / v0 \nr / d (b)  11 0,00 0,02 0,04 0,06 0,081,001,021,041,06 \n (b) \nV / d 3 v / v0 \n0,00 0,02 0,04 0,06 0,080,960,981,00\n  \n(a) \nV / d 3 v / v0 \n0,0 0,1 0,2 0,3 0,4 0,50,80,91,01,11,21,3\n  \nV / d 3 (c) v / v0 \n0,0 0,4 0,8 1,2 1,60,500,751,001,251,501,752,00\n  \nV / d 3 v / v0 \n(d) 5.1. The ferromagnetic resonance in  three -layer stack  \nWe calculated the FMR  spectra  for different configurations of  a three -layer  stack.  Fig. 13 shows the  dependenc es of the \nFMR  frequency  on the distance  between the layers  for the stack  in a state with  ferromagnetic , antiferromagnetic  and mixed \nferromagnetic - antiferromagnetic order ing. Qualitatively,  these curves  are similar to  the spectra of  the two-dimensional \narray  of three particles ( Fig. 4), but the degree of frequency splitting  for the stack  is much  more than for two -dimensional  \narray.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 3. The dependences of normalized FMR  frequencies  on the separation  between layers in the three -layer \nstack for different spatial configurations of moments. (a) is the case of ferromagnetic ordering, (b) is \nfor antiferromagnetic  ordering , (c) is for mixed ferromagnetic - antiferromagnetic ordering and (d) is \nfor the stack with the reduced magnetic moment of the middle layer  (m1= m3, m2 / m1 = 0.5) . \nAlso we have  considered the stack with differe nt mag netic moments in the layers ( Fig. 13d). The r elation of magnetic \nmoment of central layer m2 to moments of outer laye rs (m1 = m3) is equal 0.5. In this case we have initial splitting due to \nthe difference of magnetic moments and t he dependences of normalized FMR  frequenc ies have a local extremum.  Thus \nallowing difference s between the magnetic moments of layers provides an  additional way of control ling the FMR spectrum.  \n5.2. Magnetization reversal  in three -layer particle array  \nThe results of hysteresis calculations for three -layer stacks of three i dentical particle arrays on a triangular lattice are \npresented in Fig. 14. We again consider the =0.5 case and display the hysteresis plots for different interlayer spacing: (a) \nr / d = 0.45;  (b) r / d = 0.58;  (c) r / d = 0.83. When the interlayer couplin g is not strong ( r / d = 0.45), the system implements \na simple magnetization reversal process through the state with antiferromagnetic ordering of the mag netic moments in \nneighboring layers (Fig. 14a). Increasing the interlayer interaction leads to additio nal configuration states encountered \nduring the hysteresis cycle, as shown in Fig. 14b for r / d = 0.58. In this case the state (1) becomes unstable. The transition \nfrom state (1) to state (2) involves just a flip of the moment of one particle in the centr al layer (indicated in red in scheme). \nThe other two particles in the central layer then experience a flip in the transition to state (3). The low average moment \nstate (3) is essentially the same as state (2) in Fig. 14a. The small average moments in the t wo plots are slightly different \nbecause of slight differences in deviations from easy axis alignment. The transition from state (3) to state (4) involves \nmoment reversals in all three layers, so we have the same state  as (3)  but with the opposite directed average magnetic \nmoment. Fig.  14c shows the behavior as the interlayer coupling is increased still further to r / d = 0.83. The transition from \nstate (1) to state (2) is similar to that in Fig. 14b, but in this case state (2) becomes unstable. Here we have  only one \nantiferromagnetic ordered stable state (3) similar to the state (3) in Fig.  14b. Again the transition from state (3) to state (4) \ninvolves moment reversals in all three layers, so we have practically the same state with the opposite directed aver age \nmagnetic moment.   12 As the number of layers is increased, the number of states traversed in a hysteresis cycle also incr eases, reflecting the \nmore complex sequence of moment reorientation processes that is taking place.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 4. The h ysteresis  curves  and the schemes of transitions  between different states of three -layer particles \narray  for  = 0.5 case:  (a) r / d = 0.45; (b) r / d = 0.58; (c) r / d = 0.83 . The c onfiguration  of the \nmagnetic moments  of the particles  corresponds to the points on  the hysteresis curve . The one -particle \nflip in transition schemes in the pictures (b) and (c) is indicated in red.  \n(1) \n(3) \n(2) \n(3) (2) (b) r / d = 0.58 \n     = 0.5 (1) \n(2) \n(1) (2) (a) r / d = 0.45 \n     = 0.5 \n(1) \n(3) \n(2) \n(2) (3) (c) r / d = 0.83 \n     = 0.5 (4) \n(4)  13  \n5.3. The ferromagnetic resonance in  three -layer system  \nThe Fig. 15 shows the  dependencies  of the FMR  frequenc ies on the distance  between the layers  for three -layer  particle \narray  in the state with  ferromagnetic (Fig. 1 5a), antiferromagnetic interlayer order  (Fig. 1 5b), in the state with partial \nferromagnetic and antiferromagnetic ordering (Fig. 1 5c) and for the system with different magnetic moments in \nantiferromagnetic configuration  (Fig. 1 5d). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 5. The dependences of normalized FMR  frequencies  on the separation  between layers in array of the \nthree -layer  stacks for different spatial configurations of moments. (a) is for the ferromagnetic ordering. \n(b) is for antiferromagnetic ordering.  (c) is for the state with partial ferromagnetic  and antiferromagnetic \nordering.  (d) is for the stack s with the reduced ma gnetic mo ment of the middle layer (m1= m3, m2 / m1 = \n0.5) in the state with  antiferromagnetic interlayer ordering . The middle layer with the reduced magnetic \nmoment is highlighted in red.  \nAnalogous to the two -layer system the dependencies of FMR spectra  on the distance between the layers for three -layer \nsystem are nonmonotonic as a consequence of competition between the interlayer  and interlayer interactions. Note that the \nuse of the stacks with d ifferent layer thickness or different layer materials substan tially extends the possibilities in \ncontrolling of ferromagnetic resonance in such systems.  \n6. Discussion   \nThus, the calculations showed that the magnetostatic interaction between the particles in the layer and the interlayer \ninteraction strongly affect t he magnetization reversal dynamics in an external magnetic field and lead to the complex \nsplitting of FMR spectra.  \nIn the calculations, we used a simple model of anisotropic point dipoles. However, the real particle s are distributed \nsystems with a complex  character of the magnetic interaction s and this leads to inhomogeneous magnetization reversal 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,40,9250,9500,9751,0001,025 \n r / d (a) v / v0 \n0,0 0,2 0,4 0,6 0,8 1,0 1,20,9751,0001,0251,0501,075 \n r / d (b) v / v0 \n0,0 0,2 0,4 0,6 0,8 1,0 1,20,9500,9751,0001,0251,050\n  \nr / d (c) v / v0 \n0,0 0,4 0,8 1,2 1,6 2,00,500,751,001,25\n  \nr / d v / v0 \n(d)  14 modes and complication of the FMR spectrum. Along with the uniform coherent precession modes in such  systems , there  \nexist nonhomogeneous modes [22 -24] associated  with the internal spin resonances and edge effects. However for the \nsystems consisting of small magnetic nanoparticles the FMR spectrum of ground modes of uniform prece ssion and the \nbasic mechanisms of the splitting of resonance frequencies are described adequately by the simple model  we have used . \nIt was shown, that the systems consisting of one -layer elements with only in -plane interaction s display  the complicated \nbut weak splitting of FMR spectra (Figs. 4,6). The arrangement of elliptical discs in multi layer stacks systems leads to the \nconsiderable increase of interparticle magnetostatic interaction and as a consequence to the appeara nce of unusual states \nduring magnetization reversal and significant changes in the FMR spectra. It allows us to construct the different systems \nwith fine tuning of ferromagnetic resonance. For example, we have demonstrated that in two -layer system (consisting of \nthree two -layer stacks, located at 120°) there are two stable states (Fig. 10). The first state has a quasi -uniform  moment’s \ndistribution (with ferromagnetic interlayer ordering)  and can be realized in a system with low interlayer interact ion \n(Fig.  10a). The second state has zero average magnetic moment (with antiferromagnetic interlayer ordering)  and can be \nrealized i n the system with high interlayer interaction (Fig. 10c). In the system with intermediate interlayer  interaction  there \nare both these states, which can be switched by magnetization reversal in an external magnetic field (Fig. 10b). The FMR \nspectra correspo nding to these two states are presented in Fig. 11 and have the fundamental differences. The quasi -uniform \nstate has the spectrum shifted to lower frequencies, while the state with zero average moment has th e spectrum with four \nbranches shifted to higher f requencies. Therefore such two -layer particles arrays with simply recognized magnetic states \ncan be used in the magnetic labeling systems.  \nWe have shown that adding a third layer in the stack leads to an extra frustration of magnetic state s and complicated  \nsplitting in FMR spectra. Depending on the interlayer coupling in such systems the states with ferro magnetic, \nantiferromagnetic and mixed ferromagnetic - antiferromagnetic interlayer ordering are realized (Fig.  14). This significantly \nexpands the possibil ities to control of the FMR spectrum in such systems (Fig. 15 (a-c)). Moreover the FMR spectrum \nsplitting in three -layer systems is considerably larger than in the case of two -layer systems.  \nThe great perspectives for the engineering of magnetic states and  FMR spectra are associated with the possibility of \nvarying the ratio of magnetic moments for the particles belonged to different layers and interlayer spacings. In particular, \nwe have considered the three -layer system with a reduced magnetic moment of the  particles in the middle layer. Such \nsystems can achieve an increase the splitting of the spectra due to the initial frequency splitting for the differed particles \nforming the array (Fig. 15d).  \n \n7. Conclusion  \n \nThus we have demonstrated that multilayer  ferromagnetic nanoparticles  significantly expand  the opportunities to create  \nthe magnetically frustrated systems . The different spatial arrangement, variation of magnetic moments and interlayer \nspacings enable the fine tuning of possible stable magnetic states  and effective control of FMR  spectrum in such systems  \nthat is very important from the standpoint of practical applications .  \nAcknowledgement s \nAuthors thanks K.D.  Bessmertni y for assistance and A.A.  Fraerman for the useful  discussion s. \nThis work was suppor ted by grants of Russian Foundation for Basic Research , the p rogram s of Russian Academy of \nSciences  and The Ministry of Education  and Science of Russian Federation . \nReferences   \n1. R.F. Wang, C. Nisoli, R.S. Freitas, J. Li, W. McConville, B.J. Cooley, M.S. Lun d, N. Samart h, C. Leighton, V.H. \nCrespi, P.  Schiffer - Artificial “spin ice” in a geometrically frustrated lattice of nanoscale ferromagnetic islands , Nature, \n439, 303 -306 (2006).  \n2. C. Nisoli, R. Wang, J. Li, W.F. McConville, P.E. Lammert, P. Schiffer, V.H. Crespi - Ground state lost but degeneracy \nfound: The effective thermodynamics of artificial spin ice , Phys. Rev. Lett., 98, 217203 (2007).  \n3. A. Remhof, A. Schumann, A. Westphalen, H. Zabel, N. Mikuszeit, E.Y. Vedmedenko, T. Last, U. Kunze - \nMagnetostatic int eractions on a square lattice // Phys. Rev. B, 77, 134409 1 -6 (2008).  \n4. A. Westphalen, A. Schumann, A. Remhof, H. 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Adeyeye - Coupled oscillations in noncollinear microscale rectangular magnets , Phys . Rev. \nB, 82, 214422 (2010).  \n24. V.V. Kruglyak, P.S. Keatley, A. Neudert, R.J. Hicken, J.R. Childress, J.A. Katine - Imaging collective magnonic modes \nin 2D arrays of magnetic nanoelements, Phys. Rev. Lett., 104, 027201 (2010).  "    },    {        "title": "1412.1281v1.Paramagnetic_spin_pumping.pdf",        "content": "arXiv:1412.1281v1  [cond-mat.mes-hall]  3 Dec 2014Paramagnetic spin pumping\nY. Shiomi1and E. Saitoh1,2,3,4\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2WPI Advanced Institute for Materials Research,\nTohoku University, Sendai 980-8577, Japan\n3CREST, Japan Science and Technology Agency, Tokyo 102-0076 , Japan and\n4Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai 319-1195, Japan\n(Dated: July 16, 2018)\nAbstract\nWe have demonstrated spin pumpingfrom a paramagnetic state of an insulator La 2NiMnO 6into\na Pt ﬁlm. Single-crystalline ﬁlms of La 2NiMnO 6which exhibit a ferromagnetic order at TC≈270\nK were grown by pulsed laser deposition. The inverse spin Hal l voltage induced by spin-current\ninjection has been observed in the Pt layer not only in the fer romagnetic phase of La 2NiMnO 6but\nalso in a wide temperature range above TC. The eﬃcient spin pumping in the paramagnetic phase\nis ascribable to ferromagnetic correlation, not to ferroma gnetic order.\nPACS numbers: 72.25.Pn, 72.25.Mk, 75.47.Lx, 75.76.+j\n1A spin current in nonmagnetic metals consists of electrons with oppo site spins moving\nin opposite directions. This spin ﬂow which does not accompany a net c harge ﬂow, called a\npure spin current, is a key ingredient in spintronics devices [1]. A pure spin current can be\ndetected electrically by the inverse spin Hall eﬀect (ISHE) using met als with strong spin-\norbit interaction such as Pt. In the ISHE, electrons with opposite s pins are scattered into\ndiﬀerent directions by the spin-orbit interaction, as shown in Fig. 1( a). Then, the inverse\nspin Hall voltage, V, arises along the direction parallel to js×σ, wherejsandσare the\nspatial direction and the spin-polarization direction of the spin curr ent, respectively [Fig.\n1(a)].\nSpin pumping is a powerful tool to generate pure spin currents; wh en a magnetization\nprecession motion is induced in a magnet e.g.by ferromagnetic resonance (FMR), a spin\ncurrent is injected into an adjacent nonmagnetic metal. During the precessional motion of\nmagnetization, the spin angular momentum in the magnet is transfer red across the interface\nto an adjacent metal via the spin exchange at the interface, as sh own in Fig. 1(b). The spin\npumping has been studied using FMR so far in ferri- or ferro-magnet |paramagnetic-metal\nhybrid structures [2–14]. Pure spin currents produced by spin pum ping were ﬁrst observed\nby use of the ISHE in Ni-Fe |Pt double layers [2–7]. Recently, the spin pumping from a ferri-\nmagnetic insulator Y 3Fe5O12toa neighboring Pt ﬁlmwas discovered [9]. Spin pumping from\nmagnetic insulators, which does not accompany spin-polarized elect ric currents across the\ninterface [10], opened new interests in the spintronics ﬁeld from view points of applications\nand also of science of spin currents [11–16].\nIn the present study, we have developed a new insulator for spin pu mping: an oxide\ninsulator La 2NiMnO 6, whose Curie temperature, TC, is close to room temperature. Using\nLa2NiMnO 6|Pt devices, we have investigated spin pumping in the temperature ra nge from\n200 K to 350 K. Below TC, the spin current generated from La 2NiMnO 6by FMR spin\npumping is observed in an attached Pt layer via the ISHE. Furthermo re, clearly, the spin-\ncurrent signal appears also at paramagnetic resonance in La 2NiMnO 6even far above room\ntemperature, although La 2NiMnO 6is in the paramagnetic state and it has no long-range\nspontaneousmagnetization; theISHEsignalisobservedinthePtla yerinawidetemperature\nrange up to 350 K. This result is the evidence of spin pumping without f erromagnetic order.\nLa2NiMnO 6ﬁlms were grown on (001) SrTiO 3(STO) substrates by the pulsed laser de-\nposition (PLD) technique. Polycrystalline targets were ablated with a KrF excimer laser\n2light (λ= 248 nm) with a repetition rate of 2 Hz. During the deposition, STO su bstrates\nwere kept at 800◦C and pure oxygen gas of 0 .3 torr was continuously supplied into the\ngrowth chamber. After the thin-ﬁlm growth, the ﬁlms were anneale d at 800◦C in 400 torr\noxygen gas for 1 hour, and then cooled down to room temperature at the rate of 15◦C/min.\nThe crystal structures of the ﬁlms were evaluated by reﬂection h igh energy electron diﬀrac-\ntion (RHEED) and X-ray diﬀraction (XRD) using Cu Kαradiation. Clear streaks from\nLa2NiMnO 6ﬁlms were observed in RHEED patterns (see [17]). Resistivity and mag netiza-\ntion for the ﬁlms were measured in a superconducting magnet with an electrometer and a\nvibrating sample magnetometer, respectively. For the ISHE measu rements, a 10-nm-thick\nPt layer was sputtered in an Ar atmosphere on the top of the La 2NiMnO 6ﬁlms (4×2\nmm2). The simultaneous measurement of microwave resonance and dc v oltage was per-\nformed using coplanar-type waveguides [18]. The ISHE measuremen t at room temperature\nwas conducted by use of an electromagnet, while that at low and high temperatures was\ncarried out in a superconducting magnet with the incident microwave power of 1 W due\nto relatively large transmission loss of incident microwave energy [17], the small resonance\nabsorption of La 2NiMnO 6(∼10µW), and larger voltage noises at higher temperatures. We\napplied microwave bymeans ofa signal generator oravector netwo rk analyzer andmeasured\nresonance absorption with a microwave power meter. DC voltage ge neration between the\nends of the Pt layer at spin resonance was detected with a nanovolt meter.\nWe show X-ray diﬀraction patterns for our samples in Fig. 2(b). Sha rp (002) and (004)\ndiﬀraction peaks of the La 2NiMnO 6ﬁlm were clearly observed, indicating that La 2NiMnO 6\nﬁlmswereepitaxiallygrownonthe(001)STOsubstrates. Theout-o f-planelatticeparameter\nis 3.85˚A, which is almost the same as that of highly-ordered ﬁllms [19, 20]. As shown in\nFig. 2(b), clear Laue fringes appear around (002) and (004) reﬂe ctions, indicating high\ncrystalinity, ﬂat surface, and homogeneity of the growth ﬁlm [21]. F rom the period of the\nfringe oscillation, thickness ( t) of the La 2NiMnO 6ﬁlm was estimated to be 80 nm.\nThe La 2NiMnO 6ﬁlm was conﬁrmed to be highly insulating; sheet resistance Rsis higher\nthan 20 MΩ below 350 K. The high resistance at room temperature en ables us to study the\nspin pumping free from electric conduction in La 2NiMnO 6. Temperature ( T) and magnetic\nﬁeld (H) dependence of magnetization ( M) is shown in Figs. 2(c) and 2(d), respectively. A\nclear ferromagnetic-paramagnetic transition is observed at the C urie temperature TC∼270\nK [Fig. 2(c)], which is almost the same as the bulk value [22]. As shown in Fig . 2(d), the\n3saturation magnetization at 10 K is almost 5 µB/f.u. in accordance with the expected value\nfor the ferromagnetic ordering of Ni2+(S= 1) and Mn4+(S= 3/2) ions. Magnetic-ﬁeld\ndependence of Mexhibits a ferromagnetic hysteresis behavior below TC(with a coercive\nﬁeld of∼50 mT at 10 K). On the other hand, Mat 300 K shows linear H-dependence,\ncharacteristic of the paramagnetic phase [Fig. 2(d)].\nThe ISHE measurement was performed in both ferromagnetic and p aramagnetic states of\nLa2NiMnO 6, as illustrated in Fig. 1(a), where spin dynamics in the La 2NiMnO 6was excited\nby applying a microwave. Here, spin dynamics in a ferromagnetic stat e is characterized by\nferromagnetic resonance (FMR), while that in a paramagnetic stat e by electron paramag-\nnetic resonance (EPR). FMR corresponds to a precessional motio n of the order parameter,\ni.e.magnetization, of a ferromagnet. On the other hand, EPR is microw ave absorption\nby individual spin excitation in an external magnetic ﬁeld [Fig. 1(c)]; wh en microwave fre-\nquency is equal to the magnitude of spin-energy splitting induced by the external magnetic\nﬁeld, transition across the splitting is excited resonantly.\nIn Fig. 3, we show results of the spin-pumping measurement in a La 2NiMnO 6|Pt device\nat room temperature, where La 2NiMnO 6is in the paramagnetic state. Figure 3(a) shows\na power spectrum of the microwave reﬂected from the La 2NiMnO 6|Pt,P, as a function of\nmagnetic ﬁeld ( H) at 300 K. Here, incident microwave frequency ( f) and power ( Pin) are\nf= 5 GHz and Pin= 100 mW, respectively. The power spectrum shows dips around ±0.2\nT, which correspond to microwave absorption by EPR in La 2NiMnO 6. Voltage signal in\nthe Pt layer, V, measured simultaneously with the Pmeasurement is shown as a function\nofHin Fig. 3(b). At this temperature, the La 2NiMnO 6is in its paramagnetic phase and\nthere is no long-range spontaneous magnetization. Nevertheless , at the ﬁeld where EPR\noccurs, clear electromotive force peaks appear [see Fig. 3(b)]. Th e La2NiMnO 6ﬁlm is a\ngood insulator whose resistance is larger than 50 MΩ at room temepr ature, meaning that\nthe voltage signal is generated in the Pt layer. The sign of the voltag e peak is reversed by\nreversing the Hdirection and the magnitude of the peak increases linearly with microw ave\npower [17]. As shown inFig. 3(c), we also conﬁrmed that the voltage signal disappears when\nHis applied along a perpendicular direction to the ﬁlm plane ( H||σ||js). These results are\nconsistent with the symmetry of ISHE [ V||(js×σ)].\nTo further conﬁrm that the voltage signal appears in the paramag netic state, we show,\nin Fig. 3(d), the fdependence of the resonance magnetic-ﬁeld, Hr, determined from the\n4peak positions in the ISHE measurements at 300 K. In FMR, the relat ion between Hrand\nthe resonance frequency, fr, is given by the Kittel formula: 2 πfr=γ/radicalBig\nµ0Hr(µ0Hr+Meff),\nwhereMeffistheeﬀectivespontaneousmagnetizationwhichalsoincludessurfa ceanisotropy.\nAsshowninFig. 3(d), frdependence of Hrislinearat300K,whichindicatesthat Meff= 0,\nconﬁrming the EPR condition (2 πfr=γµ0Hr). From the slope value, we estimated γto be\nγ= 1.88×1011T−1s−1, which is almost the same as the free electron’s gyromagnetic ratio\n(γe= 1.76×1011T−1s−1). This result demonstrates again the spin-current injection from\nthe paramagnetic state.\nThe observed peak voltage ( V0) at EPR is not due to heating eﬀects induced by mi-\ncrowave resonance absorption. To examine possible contamination of thermoelectric voltage\nin the Pt layer, we show, in Fig. 3(b), the Nernst-eﬀect signal induc ed by a temperature\ngradient externally applied perpendicularly to the ﬁlm plane. Here, te mperature diﬀerence\nwas applied with the use of a 1kΩ-resistance heater attached on th e STO substrate and the\napplied power = 0 .4 W was set to almost the same as the incident microwave power in the\nISHE measurement ( Pin= 0.1-1 W). As shown in Fig. 3(b), the observed thermoelectric\nsignal is proportional to Hand its magnitude is very small; this voltage corresponds to\nthe ordinary Nernst eﬀect (see also [17]). Because the magnitude o f the observed electro-\nmotive force signal at EPR is much greater than the thermoelectric voltage despite very\nsmall resonance-absorption power ∼10µW, we can safely rule out possible heating-induced\neﬀects in the ISHE measurements. The spin-pumping voltage norma lized by the resonance\nabsorption, which indicates the eﬃciency of paramagnetic spin pump ing, is∼30 nV/10\nµW = 0.003 V/W at 5 GHz [Figs. 3(a) and 3(b)], which is comparable to that for the FM R\nspin pumping in Y 3Fe5O12|Pt bilayers [12].\nWe show temperature ( T) dependence of the voltage signals between 200 K and 350 K,\nin Fig. 4(a). Voltage peaks which correspond to the ISHE induced by spin pumping are\nobserved at each T; it is also noted that small spin Seebeck voltage signals [23] are obser ved\nat 200 K and 210 K as a broad background, which is asymmetric with re spect toH, because\nof the heating in the waveguide due to microwave energy loss (see als o [17]). Below TC≈270\nK, the resonance ﬁeld, or the voltage-peak position Hrsigniﬁcantly changes with T, while\nHris almost constant above TC. In Fig. 4(b), Meffvalues are estimated from the Hrvalues\nusing the Kittel formula at each T. AsTincreases from 200 K, the Meffvalue rapidly\ndecreases and becomes zero above TC. The temperature dependence of Meffis similar to\n5that of the observed magnetization ( M) at 0.3 T shown in Fig. 2 (c). The result conﬁrms\nagain that the La 2NiMnO 6is in its paramagnetic ( Meff= 0) phase at 300 K where the\nISHE signal was clearly observed.\nA short-range ferromagnetic correlation above TC, which was observed in La 2NiMnO 6\n[24–27], can be responsible for the persistent spin-pumping signals in such a wide T-range\naboveTCin La2NiMnO 6|Pt; the short-range correlation was observed above TCup to 350-\n400 K,e.g.by Raman spectroscopy [24], and also manifests itself as small devia tion of the\ninverse susceptibility fromtheCurie-Weiss behavior above TC[Fig. 2(c)]. The spin-pumping\namplitude is proportional to the two-point correlation function for local magnetization [28],\nwhile the static magnetization is proportional to local magnetization itself. Thanks to\nthe short-range correlation, the spin pumping signal comparable t o the ferromagnetic spin\npumping is observed even in the paramagnetic state of La 2NiMnO 6. Though such a short-\nrange correlation has been observed also in other ferromagnets a boveTC[29, 30], the high\nTCand strong short-range correlation of La 2NiMnO 6enable the eﬃcient paramagnetic spin\npumping at room temperature.\nWeﬁttheISHEvoltagesignalsaroundmicrowaveresonance(EPRor FMR)usingLorentz\nfunctions and plot the obtained half-maximum full-width (HMFW, ∆ H) and peak magni-\ntude|V0|as a function of Tin Figs. 4(c) and 4(d), respectively. As Tincreases from 200\nK, ∆Hshows a minimum at 250 K, and then increases gradually above TC[Fig. 4(c)]. On\nthe other hand, the peak voltage |V0|increases with increasing Tfrom 200 K and shows a\nmaximum at 250 K [Fig. 4(d)]. Above 250 K, |V0|decreases with increasing T, but it is\nstill prominent even far above TCup to 350 K [highlighted in red in Fig. 4(a)]. As shown\nin Figs. 4(c) and 4(d), the overall Tdependence of |V0|is similar to that of the inverse of\nthe linewidth, 1 /∆H; the observed enhancement of |V0|aroundTCcan be attributed to the\nsuppression of ∆ H.\nThe linewidth ∆ H, which reﬂects the damping for spin precession, exhibits Arrhenius -\ntypeTdependence above TC, as shown in Fig. 4(c): ∆ H∝exp[−∆E/kBT]. Here, the\nactivation energy ∆ Efor ∆His 0.17 eV. We found that this ∆ Evalue is the same as\nthat for the electrical conductivity of the La 2NiMnO 6ﬁlm,σ[= 1/(Rst)], as shown in Fig.\n4(c). The conductivity-like T-dependence of ∆ Himplies that spin relaxation occurs via\nthermally excited carriers above TC. The exponential Tdependence of |V0|(∝1/∆H) above\nTC[Fig. 4(d)] shows that the paramagnetic spin pumping is irrelevant to spin waves possibly\n6subsisting above TC,e.g.paramagnons, which should be related with the Tdependence of\nﬁeld-induced magnetization.\nAs for the lower- Tbehavior of ∆ H(∼1/V0), we note that such a minimal resonance\nlinewidth and a maximal resonance absorption near TCas observed in Figs. 4(c) and 4(d)\nhave been reported in doped perovskite manganese oxides A1−xA′\nxMnO3(A=La, Pr, ...;\nA′=Ca, Sr, Ba,...) [31]. For La 0.67Ca0.33MnO3[32] and La 0.67Sr0.33MnO3[33], asTap-\nproached TCfrom above, the resonance linewidth was reported to go through a minimum\nand then to increase abruptly below TC; this is similar to the present case [Fig. 4(c)]. In the\nprevious literature [31, 34], the increase in the linewidth below TChas been attributed to\nintrinsic magnetic inhomogeneity [31] or a nonuniform demagnetizat ion tensor [34]. Similar\neﬀects may cause the increase in ∆ HbelowTCfor the present La 2NiMnO 6samples.\nIn summary, we have demonstrated spin pumping above the Curie te mperature in\na double-perovskite oxide insulator La 2NiMnO 6. The paramagnetic spin pumping in\nLa2NiMnO 6|Pt which we found operates at room temperature and, in spite of th e absence of\nlong-range magnetic order, its eﬃciency can be comparable to that of typical spin-pumping\ndevices including ferromagnets. Since spin mixing conductance which has been believed to\nbe the language of spin pumping cannot be deﬁned for the paramagn etic state, the param-\nagnetic spin pumping will require expansion of the concept of spin pum ping.\nWe thank M. Kawasaki, A. Sawa, and Y. Kajiwara for experimental h elp and H. Adachi,\nD. Hou, R. Iguchi, K. Uchida, and Y. Fujikawa for fruitful discussio ns. This work was\nsupported by CREST-JST “Creation of Nanosystems with Novel Fu nctions through Process\nIntegration”, Grant-in-Aid for Scienctiﬁc Research (A) (242440 51) from MEXT, the Murata\nScience Foundation, and the Nippon Sheet Glass Foundation for Mat erials Science and\nEngineering.\n[1] S. Maekawa, S.O. Valenzuela, E. Saitoh, and T. Kimura, Sp in Current, (Oxford University\nPress, 2012).\n[2] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys . Lett.88, 182509 (2006).\n[3] M.V. Costache, M. Sladkov, S.M. Watts, C.H. van der Wal, a nd B.J. van Wees, Phys. Rev.\nLett.97, 216603 (2006).\n7[4] A. Azevedo, L.H. Vilela Le˜ ao, R.L. Rodr´ ıguez-Su´ arez , A.B. Oliveira, and S.M. Rezende, J.\nAppl. Phys. 97, 10C715 (2005).\n[5] O. Mosendz, et al.Phys. Rev. Lett. 104, 046601 (2010).\n[6] K. Ando, et al.J. Appl. Phys. 109, 103913 (2011).\n[7] A.Azevedo, L.H.Vilela-Le˜ ao, R.L.Rodr´ ıguez-Su´ are z, A.F.LacerdaSantos, andS.M.Rezende,\nPhys Rev. B 83, 144402 (2011).\n[8] R.H. 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Rev. B 53, 6521-6525 (1996).\n[33] M.T. Causa, et al.Phys. Rev. B 58, 3233-3239 (1998).\n[34] C.A. Ramos, M.T. Causa, M. Tovar, X. Obradors, and S. Pi˜ nol, J. Magn. Magn. Mater.\n177-181 , 867-868 (1998).\n9FIG. 1: (a) A schematic illustration of the inverse spin Hall eﬀect induced by spin pumping in\nLa2NiMnO 6|Pt structures. (b) A schematic illustration of spin pumping . Spin precession induces\na pure spin current in the adjacent layer. (c) A schematic ill ustration of electron paramagnetic\nresonance (EPR). In the presence of H, energy levels of parallel-spin and anti-parallel-spin st ates\nare separated: the Zeeman eﬀect. When the microwave frequenc y (f) is equal to the Zeeman\nenergy splitting (2 πf=γH), electron spins precess resonantly.\n10FIG. 2: (a) Crystalstructureof orderedLa 2NiMnO 6. Below thecritical temperature( TC≈270 K),\nLa2NiMnO 6exhibits ferromagnetism due to superexchange interaction between Ni2+and Mn4+.\n(b) 2θ-θX-ray diﬀraction patterns for the 80-nm-thick La 2NiMnO 6(LNMO) ﬁlm on the SrTiO 3\n(STO) substrate. (c) Temperature ( T) dependence of magnetization ( M) atµ0H= 0.3 T for the\nLa2NiMnO 6ﬁlm.Tdependence of 1 /Mis also shown in (c). TCis estimated to be ∼270 K from\nthe Curie-Weiss law, wherethe dotted line is guides to the ey es. (d) Magnetic-ﬁeld ( H) dependence\nof magnetization ( M) at several temperatures.\n11FIG.3: (a),(b) Magneticﬁeld( H)dependenceof(a) theEPRspectrum( P)and(b)theinversespin\nHallvoltage ( V)fortheLa 2NiMnO 6|Pt(10nm)ﬁlm. Themicrowave frequencyiskeptat5GHzand\nthe incident power is 100 mW. The measurement temperature is 300 K. A thermoelectric (Nernst)\nsignal measured under a perpendicular temperature gradien t (see text) is shown for comparison in\n(b); only a tiny normal Nernst signal (with a positive slope) is observed. (c) Comparison of voltage\nsignal (V) between the cases of in-plane H(H⊥js) and perpendicular-to-plane H(H||js). A\ndotted line is a guide for the eyes. (d) A contour plot of Vas functions of frequency ( f) andHat\n300 K. Incident microwave power ( Pin) is 0.8 W. Dotted lines (green) are guides to the eyes.\n12FIG. 4: (a) Magnetic ﬁeld ( H) dependenceof inverse spinHall voltage ( V) at various temperatures.\nThe microwave frequency ( f) is 5 GHz. (b) Temperature ( T) dependence of the resonance ﬁeld\n(Hr) and eﬀective spontaneous magnetization ( Meff). TheMeffvalues are estimated using the\nKittel formula. (c),(d) Temperature ( T) dependence of (c) the full-width at half-maximum (∆ H)\nand (d) the magnitude ( |V0|) of the peaks in the inverse spin Hall voltage signals. The so lid\ncurves in (c) and (d) are ﬁts to Arrhenius equations. Tempera ture (T) dependence of electrical\nconductivity ( σ) for the La 2NiMnO 6ﬁlm is shown for comparison in (c). The dotted vertical lines\nin (b)-(d) indicate TC(≈270 K).\n13Supplemental Material for “Paramagnetic spin pumping”\nY. Shiomi and E. Saitoh\nS1. RHEED pattern for the La 2NiMnO 6thin ﬁlm\nReﬂection high energy electron diﬀraction (RHEED) is a useful meth od to conﬁrm the\nordering of Bsite ions in the double-perovskite ( A2B′B′′O6) structure [M. Hashisaka, et\nal.Appl. Phys. Lett. 89, 032504 (2006)]. In Fig. 5, we show a RHEED pattern for the\nLa2NiMnO 6ﬁlm with a [110] beam incidence. In the ordered double-perovskite st ructure,\nB′andB′′cations are alternately arranged at the Bsites [Figs. 2(a) and 5(a)]. As shown\nin Fig. 5(b), clear streaks from a twofold superstructure are obs erved for the La 2NiMnO 6\nﬁlm, which shows the ordering of Mn4+and Ni2+ions in the Bsites.\nFIG. 5: (Fig. S1)( a) A schematic illustration of the crystal structure of order ed La2NiMnO 6. (b)\nReﬂection high energy electron diﬀraction (RHEED) pattern f or the La 2NiMnO 6thin ﬁlm with a\n[110] beam incidence. Clear twofold streaks are observed.\n14S2. Spin Seebeck eﬀect for the La 2NiMnO 6|Pt ﬁlms\nInferromagnet |paramagnetic-metal bilayers, ISHE voltageisgeneratedinaparam agnetic\nmetal as a result of a perpendicular temperature gradient (the lon gitudinal Seebeck eﬀect)\n[K. Uchida, et al.Appl. Phys. Lett. 97, 172505 (2010).]. We have investigated the\nlongitudinal spin Seebeck eﬀect for the La 2NiMnO 6|Pt bilayer ﬁlms, as shown in Fig. 6.\nHere, a 1kΩ-resistance heater is attached on the Pt layer to gene rate a temperature gradient\nalong the perpendicular direction. As shown in Fig. 6(b), a negative s pin Seebeck signal\nis observed at 200 K and it becomes smaller in magnitude monotonically w ith increasing\nsystem temperature. The magnitude ofvoltage signal ( |V|) at 100mT isshown asa function\nof temperature ( T) in Fig. 6(c). The temperature dependence of |V|is almost proportional\nto that of magnetization ( M) measured at 100 mT (black curve). We note that estimation\nof “paramagnetic” spin Seebeck contribution is diﬃcult at room temp erature, because the\nmagnitude of voltage signal is very small at room temperature [Figs. 3(b) and 6(b)] and\nbecause paramagneticspin Seebeck voltage should belinear with the magnetic ﬁeld similarly\nto the ordinary Nernst signal in Pt.\nAs shown in Fig. 4(a), in the spin-pumping measurements at 200 K and 210 K, small\nspin-Seebeck voltage is observed as a background in addition to the spin-pumping signals.\nThe sign of the spin Seebeck voltage is negative in a positive magnetic ﬁ eld, which indicates\nthat the Pt side is warmed in the spin-pumping measurements. Hence , the origin of the\nspin Seebeck signal is not a temperature increase in La 2NiMnO 6caused by the microwave\nresonance absorption, but is ascribable to that in the waveguide or Pt due to microwave\nenergy loss. It is noted that possible temperature elevation in the L a2NiMnO 6|Pt ﬁlm dur-\ning the spin-pumping measurements does not aﬀect our conclusion, because ferromagnetic\ncorrelation becomes weaker at higher temperatures.\n15FIG. 6: (Fig. S2)( a) A schematic illustration of the experimental setup for the spin Seebeck eﬀect.\nTemperature diﬀerence in the perpendicular direction is app lied using a chip resistance (1 kΩ)\nheater attached on the Pt layer. ( b) Magnetic-ﬁeld ( H) dependence of spin Seebeck voltage ( V) at\nsome temperatures in the La 2NiMnO 6|Pt ﬁlm. The applied power to the heater ( P) is 25 mW. ( c)\nTemperature ( T) dependence of spin Seebeck voltage ( V) at 100 mT. The temperature diﬀerence\n(∆T) is roughly estimated to be 10 K from the change in resistance of the Pt layer. Magnetization\n(M) at 100 mT is shown for comparison (black curve).\n16S3.S11parameters for the coplanar waveguides\nIn Fig. 7, we compare S11parameters for the reﬂection-type (short-end) coplanar wave g-\nuides used in the present study [the same design as that in Z. Qiu, et al., Appl. Phys.\nLett.103, 182404 (2013)] between room-temperature (RT) measuremen ts and low- and\nhigh-temperature measurements in PPMS. Since the microwave cab le is longer in PPMS\nmeasurements than in RT ones, the microwave loss is larger.\nFIG. 7: (Fig. S3) S11parameters for the coplanar waveguides in room-temperatur e (RT) measure-\nments and low- and high-temperature measurements in PPMS.\n17S4. Power dependence of the inverse spin Hall voltage induced by spin pumping\nat room temperature\nIn Fig. 8(a), we show magnetic-ﬁeld ( H) dependence of the inverse spin Hall voltage ( V)\ninduced by spin pumping at diﬀerent microwave power levels ( Pin= 100 mW to 1 W). Here,\nthe measurement was done at room temperature and microwave fr equency is 5 GHz. With\nincreasing Pin, the magnitude of the peak voltage increases monotonically. The va lues of\nvoltage peaks are plotted as a function of Pinin Fig. 8(b). The power dependence is almost\nlinear, which is the same as the cases in ferromagnetic spin pumping.\nFIG. 8: (Fig. S4) (a) Magnetic-ﬁeld ( H) dependence of the inverse spin Hall voltage ( V) induced\nby spin pumping at room temperature at diﬀerent microwave pow er levels ( Pin). The data are\nvertically shifted for clarity. (b) Power ( Pin) dependence of the inverse spin Hall voltage induced\nby spin pumping at room temperature.\n18S5. Analysis on resonance linewidth ( ∆H) at room temperature\nInFig. 9, we show theresonance linewidth forLa 2NiMnO 6andLa 2NiMnO 6|Pt which was\nmeasured on the colanar waveguides and using a cylindrical 9 .44-GHz TE 011cavity. Figure\n9(b) shows EPR spectra for La 2NiMnO 6and La 2NiMnO 6|Pt measured in the cavity. Clear\nresonance spectra are observed around 320 mT. In measuremen ts for several sets of samples,\nthe magnitude of the full-width at half-maximum (∆ H=√\n3∆Hpp) is slightly scattered\naround 20 mT for both La 2NiMnO 6and La 2NiMnO 6|Pt as shown in Fig. 9(a), and precise\nestimation of the possible linewidth enhancement by Pt coating was diﬃ cult.\nFigure 9(a) shows frequency dependence of the full-width at half- maximum (∆ H) for\nLa2NiMnO 6|Pt measured on the waveguides at room temperature; the green t riangles indi-\ncate the ∆ Hvalues obtained from Lorentzian ﬁts to the ISHE voltage shown in Fig . 3(d).\n∆Hincreases almost linearly with frequency. At 5 GHz, almost half of ∆ Horiginates in\nextrinsic origins independent of microwave frequency. From the va lue of the slope, we obtain\nα=γµ0∆H/(2πf)≈0.06 for La 2NiMnO 6|Pt.\nOn the temperature dependence of ∆ H, it is notable that according to a former study\non EPR for polycrystalline La 2NiMnO 6[S. Zhou, et al.Appl. Phys. Lett. 91, 172505\n(2007)], temperature dependence of ∆ Hfor La 2NiMnO 6aboveTCis similar to that of ∆ H\nobtained from ISHE voltage in La 2NiMnO 6|Pt [∆H∝exp[−∆E/kBT] in Fig. 4(c)]. This\nresult suggests that the possible enhancement of the linewidth by P t coating changes with\nFIG. 9: (Fig. S5) (a) Frequency dependenceof the linewidth ( ∆H) at room temperature. Thedata\nrepresented as green triangles were obtained from Lorentzi an ﬁts to the ISHE voltage in Fig. 3(d).\n(b) Resonance spectra for La 2NiMnO 6and La 2NiMnO 6|Pt. The measurement was performed in a\ncylindrical cavity and microwave power was set to be 10 mW.\n19temperature in proportion to exp[ −∆E/kBT], if the linewidth enhancement is present also\nfor paramagnetic spin pumping.\n20"    },    {        "title": "1312.6628v1.Ferromagnetism_in_Mn_doped_Sb2Te.pdf",        "content": "1 \n Ferromagnetism in Mn-doped Sb 2Te \n \nHuixia Luo, Quinn Gibson, Jason Krizan and R. J. Cava \n \nDepartment of Chemistry, Princeton University, Princeton, New Jersey \n08544, USA \n \nAbstract  \n       We report that Sb 2Te, a natural superlattice phase  consisting of two elemental Sb 2 \nlayers interleaved with single Sb 2Te3 layers, becomes fe rromagnetic at low \ntemperatures on doping with small percentages of Mn. Ferromagnetism appears for Mn concentrations as low as Sb\n1.98Mn 0.02Te, where a ferromagnetic T c of ~ 8.6 K is \nobserved. T c decreases with increasing Mn content in the stoichiometric materials but \nincreases with increasing Te excess in materials of the type Sb 1.93-yMn 0.07Te1+y, \nstarting at ~ 3 K at y = 0 and reaching a T c of ~ 8.9 K at y = 0.06.   \n \n \n \nKeywords : Mn-doped Sb 2Te; diluted ferromagn etic semiconductor \n \n  \n \n \n \n \n 2 \n 1. Introduction \n      Dilute magnetic semiconductors (D MSs), which are prepared by doping transition \nmetals into nonmagnetic semiconductor hosts, have attracted considerable interest for \nboth their scientific value and their potential spintronics applications. Ferromagnetism \nin semiconductors at low temperatures has been reported for various II-VI [1,2], IV-\nVI [3,4], and III-V [5,6] compounds doped with  transition metals such as Mn, Cr, V, \nand Fe. Recently, it has been experimentally shown that topological insulators based \non the small band gap semiconductors Bi 2Te3 [7,8] and Sb 2Te3 [9,10] become \nferromagnetic at low temperatures when doped with Mn, as does Cr-doped \n(Bi,Sb) 2(Te,Se) 3 [11,12]. Bi 2-xMn xTe3, for example, attains a T c = 12 K for x = 0.09, \nand the Curie temperature of Sb 1.985Mn 0.015Te3 is reported to reach as high as 17 K. \nSb2Te is a small band gap semiconductor in the Sb-Te system that is also expected to \ndisplay topological surface states [13]; it is  a natural superlattice phase made of layers \nof the elemental semimetal Sb and the normal valence semiconductor Sb 2Te3. \nBecause the interaction of topological surface states with ferromagnetism is expected \nto yield interesting effects, it is therefor e of interest to determine whether Mn doping \nof Sb 2Te, i.e. in Sb 2-xMn xTe, induces ferromagnetism at low temperatures. \n      First principles electronic structure calculations predict that the parent Sb 2Te \nmaterial is a small band gap semiconductor and a topological insulator [13,14]. It is one member of the (Sb\n2)n(Sb 2Te3)m infinitely adaptive series, whose crystal structure \n(see inset of Figure 1a) can be regarded as a natural superlattice in the c-axis direction. \nThe structure of Sb 2Te consists of single quintuple layers of Sb 2Te3 stacked with two \nbi-layer sandwiches of Sb, i.e. in the layer stacking sequence -{[Te-Sb-Te-Sb-Te]-[Sb-Sb]–[Sb-Sb]}-. The Sb\n2 layers and their stacking are similar to those found in \nelemental Sb [15-17]. This compound crystallizes in the trigonal space group P3m1 \n(#164) with a = 4.272(1) Å, c = 17.633(1) Å [18].  \n      Here we report the synthesis and characterization of Mn-doped Sb 2Te in \npolycrystalline form. Curie temperatures of around 8.6 K are observed at very low doping levels, i.e. for Sb\n1.98Mn 0.02Te. For higher Mn contents, we find that the \nferromagnetic T c depends on the Sb:Te ratio; Sb 1.89Mn 0.07Te1.04 and  Sb1.87Mn 0.07Te1.06  \ndisplay T cs near 7 and 8.9 K, respectively.  \n \n2. Experimental     3 \n       Polycrystalline Sb 2-xMn xTe (x = 0, 0.005. 0.01, 0.02, 0.04, 0.06, 0.07) and Sb 1.93-\nyMn 0.07Te1+y (y = 0, 0.02, 0.04, 0.06)  samples were prepared by a solid state reaction \nmethod. Stoichiometric quantities of high-purity elemental Sb (99.9999 %), Mn \n(99.99 %) and Te (99.999 %) were sealed in clean evacuated quartz ampoules. The \nMn was purified of oxygen before use. All the ampoules were heated up to 1000 oC, \nwhere the samples were melted and held overnight, after which they were quenched in water. Silver-colored polycrystalline boules were obtained. The samples were \nconfirmed to be single phase by powder X-ray diffraction (XRD) using a Bruker D8 \ndiffractometer with Cu K α radiation and a graphite diffracted beam monochromator. \nDC magnetization and Hall measurements were performed in a Quantum Design \nPhysical Property Measurement System (PPMS).   \n3. Results and Discussion     \n      Figures 1b and c show the powder XRD patterns of the polycrystalline Sb\n2-\nxMn xTe (x = 0, 0.005. 0.01, 0.02, 0.04, 0.06, 0.07) and Sb 1.93-yMn 0.07Te1+y (y = 0, 0.02, \n0.04, 0.06) samples, respectively. All the Mn-doped Sb 2Te samples studied show \nsingle phase Sb 2Te in the diffraction patterns - no impurity MnTe or MnTe 2 peaks \nwere observed. For slow cooled rather than quenched samples, Mn-Te phases were \npresent in the X-ray patterns, indicating a lower Mn solubility limit in the Sb 2Te \nphase at lower temperatures.  \n      Figure 2a shows the zero-field cooled temperature dependence of the magnetic \nsusceptibility for the polycrystalline Sb 2-xMn xTe (x = 0.005. 0.01, 0.02, 0.04, 0.06, \n0.07) series, under an applied magnetic fiel d, H = 10 kOe. The inset of Figure 2a \nemphasizes the susceptibilities of Sb 2-xMn xTe at low temperature. The susceptibilities \nfor Sb 2-xMn xTe (x = 0.005, 0.01, 0.02) are negative at high temperature, due to core \ndiamagnetism. All the Mn-doped crystals have a paramagnetic susceptibility at low temperature, which is ascribed to the increasing paramagnetic contribution from the \ndoped Mn. A transition to a ferromagnetic state is seen in the Sb\n2-xMn xTe samples at \nlow temperatures, as the observed magnetization becomes positive and large below T c. \nAbove the transition temperature, the magnetic susceptibility ( χ) can be fit to the \nCurie-Weiss law, χ-χ0 = C/(T- θ), where C is the Curie constant, θ is the paramagnetic \nCurie temperature, and χ0 is the temperature independent contribution to the \nsusceptibility. The fits were performed in the temperature range of 40 – 100 K; the M 4 \n vs. H curves at 40 K and higher are linear to beyond applied fields of 10 kOe (Figure \n3a), showing the validity of employing  χ = M/H at H = 10 KOe. The effective \nmagnetic moment (P eff) per Mn ion can be obtained by using P eff = (8C)1/2. Figure 2b \nshows the low temperature inverse susceptibility plots, 1/( χ-χ0) vs. T. The inverse \nsusceptibilities for x = 0.005 and 0.01 show straight line behavior, which is due to the \nweak local moment from the small amount of doped-Mn. As the Mn doping increases, \nthe effective moment per mol-Mn decreases. The fitted parameters are summarized in Table 1.    \n      Figure 3b shows the field-dependent magnetization curves at 2 K for Sb\n2-xMn xTe \n(x = 0.02, 0.04, 0.06, 0.07) samples, with an increasing field from 0 to 90 kOe.  The \nmagnetization values reach ~ 0.05, 0.08, 0.088 and 0.09 μB f.u.-1 at 2 K at 90 kOe, \nrespectively.  A comparison of narrow hysteresis loops on varying the applied field \nfrom -3 to 3 kOe for Sb 2-xMn xTe (x = 0.02, 0.04, 0.06, 0.07) samples is shown in \nFigure 4. Small coercivities (H cs) of ∼ 300 Oe, 380 Oe, 420 Oe and 420 Oe were \nobserved for the Sb 1.98Mn 0.02Te, Sb 1.96Mn 0.04Te, Sb 1.94Mn 0.06Te, and Sb 1.93Mn 0.07Te \nsamples, respectively, which indicate that the samples are soft ferromagnets. The H cs \nfor our Sb 2-xMn xTe (x = 0.02, 0.04, 0.06, 0.07) samples are similar to the previously \nreported single crystal Bi 1.91Mn 0.09Te3 (Hc ∼ 350 Oe, 2 K) [19] samples, although they \nare much lower than has been reported for V-doped Sb 2Te3 (H c ∼ 12 kOe, 2 K) [20]. \nThe Sb 2Te materials with higher Mn content show greater remnant magnetization, M r. \nFor x = 0.07, a remnant magnetization M r = 0.0085 μB f.u.-1is obtained. For x = 0.06, \nx = 0.04 and x = 0.02, M r = 0.0074, M r = 0.0060 and M r = 0.0037 μB f.u.-1, \nrespectively.  \n      To better determine the temperature of the ferromagnetic transitions, we apply the Arrott plot criterion for T\nc determination in disordered systems [21]. Figure 5 presents \nthe Arrott plots for polycrystalline Sb 2-xMn xTe (x = 0.02, 0.04, 0.06, 0.07) samples. In \nsuch plots, H/M versus M should behave as H/M = A’(T-T c)+B’M2+C’M4 at values of \nH above those of the magnetic hysteresis. At T c, the intercept changes sign. For \nsecond order transitions, B’ is positive. On the other hand, theory predicts that the \n(high-field) isotherms of M2 vs H/M are parallel lines for ferromagnets and the \nisotherm at T c passes through zero. As shown in Figure 5, the ferromagnetic transition \nin Sb 1.98Mn 0.02Te is second order as expected and the T c is close to 8.6 K. The 5 \n ferromagnetic transition temperatures for the higher Mn-content materials \nSb1.94Mn 0.06Te and in Sb 1.93Mn 0.07Te (see Figures 5c, d) are lower, ~ 3 K.  \n      We hypothesized that the T c for Sb 1.98Mn 0.02Te is greater than that for \nSb1.93Mn 0.07Te in spite of its lower Mn content due to the fact that for the higher Mn-\ncontent samples the p-type carrier concentration is t oo high. Therefore we designed a \nsecond series of materials, Sb 1.93-yMn 0.07Te1+y (y = 0, 0.02, 0.04, 0.06), for comparison. \nIn this case, up to a doping level where Fermi energy pinning results in no further \nchange, the p-type carrier concentration is expected to decrease as Te is substituted in \nthe Sb site, donating electrons. \n      Figure 6a shows the zero-field cooled temperature dependence of the magnetic \nsusceptibilities for the polycrystalline Sb 1.93-yMn 0.07Te1+y (y = 0, 0.02, 0.04, 0.06) \nsamples with the applied magnetic field, H = 10 kOe. The inset of Figure 6a \nemphasizes the susceptibilities of polycrystalline Sb 1.93-yMn 0.07Te1+y (y = 0, 0.02, 0.04, \n0.06) at low temperature. All the Sb 1.93-yMn 0.07Te1+y samples have a paramagnetic \nsusceptibility at low temperature, which is consistent with the polycrystalline Sb 2-\nxMn xTe samples, and is ascribed to the paramagnetic contribution from the Mn. The \ncomparison of the low temperature inverse susceptibility plots, 1/( χ-χ0) vs. T is \nshown in Figure 6b. The fitted parameters are summarized in Table 2. Again, the fits \nwere performed in the temperature range of 40 – 100 K; the M vs. H curves at 40 K \nand higher are linear to beyond applied fields of 10 kOe (Figure 7a), showing the \nvalidity of the use of  χ = M/H at H = 10 KOe.       \n      The field-dependent magnetization curves at 2 K for polycrystalline Sb 1.93-\nyMn 0.07Te1+y (y = 0, 0.02, 0.04, 0.06) on increasing field from 0 to 90 kOe are shown \nin Figure 7b.  Their magnetizations values reach ~ 0.09, 0.09, 0.08 and 0.1 μB f.u.-1 at \n2 K at 90 kOe for Sb 1.93-yMn 0.07Te1+y (y = 0, 0.02, 0.04, 0.06), respectively.  A \ncomparison of the narrow hysteresis loops on varying the applied field from -3 to 3 \nkOe for polycrystalline Sb 1.93-yMn 0.07Te1+y (y = 0, 0.02, 0.04, 0.06) samples is shown \nin Figure 8. Comparing the four samples, the coercivities (H cs) are the same, but the \nremnant magnetization M r changes; the higher the Te content, the larger the remnant \nmagnetization (M r) values. \n      Figure 9 presents the Arrott plots for the samples of Sb 1.93-yMn 0.07Te1+y (y = 0, \n0.02, 0.04, 0.06). The data show a systematic increase in T c with increasing Te \ncontent, beginning at ~ 3 K for Sb 1.93Mn 0.07Te and reaching a maximum of ~ 8.9 K for 6 \n Sb1.87Mn 0.07Te1.06. This is consistent with our expectation that excess Te acts as an n-\ntype dopant and therefore that the p-type carrier concentration for Sb 1.93Mn 0.07Te is \ntoo high to support the maximum T c possible at that Mn doping content in this phase. \nThis conclusion is confirmed by our measurements of the carrier concentrations of the \nTe-excess samples. All samples are p-type. The carrier concentrations for the Sb 1.93-\nyMn 0.07Te1+y  materials are found to be 3.4 x 1021 cm-3,  2.0  x 1021 cm-3, 0.6 x 1021 cm-\n3, and 1.2 x 1021 cm-3 for y = 0, 0.02, 0.04 and 0.06 respectively. Thus T c increases as \nthe p-type carrier concentration decreases. The Te is not an efficient dopant, only a \nsmall number of electrons ar e produced per number of Te dopants, and increased Te  \nconcentration beyond a few percent excess does not decrease the net carrier \nconcentration to low values or to n-type behavior. This is consistent with the well-\nknown difficulty in attaining low p-type concentrations or  n-type behavior in the \nsemiconductor Sb 2Te3 [16,22]. \n 4. Conclusions \n       A series of substitution of Mn-doped Sb\n2Te has been investigated. A Curie \ntemperature of ~ 8.6 K is observed in Sb 1.98Mn 0.02Te; this is a relatively low magnetic \nion content (1 % of the metal sites) for inducing ferromagnetism in a semiconductor. \nTc decreases with increasing Mn content in materials of the type Sb 2-xMn xTe up to the \nsolubility limit of x = 0.07, but higher T c’s are again obtained for x = 0.07 when \nexcess Te is employed to adjust the hole carrier concentration. This ferromagnetic \ntopological insulator system provides an avenue for studying the interactions of \nferromagnetism with topological surface states . Further study would be of interest.   \n \nAcknowledgements \n   This work was supported by the DARPA MESO program grant N66001-11-1-4110. \n   \n \n \n \n 7 \n References \n1. A. Haury, A. Wasiela, A. Arnoult, J. Cibert, S. Tatarenko, T. Dietl, and Y. Merle d'Aubigné, \nPhys. Rev. Lett. 79, 511 (1997). \n2. H. Saito, V. Zayets, S. Yamagata, and K. Ando, Phys. Rev. Lett.  90, 207202 (2003). \n3. T. Story, R. R. Ga ła\u001fzka, R. B. Frankel, and P. A. Wolff, Phys. Rev. Lett. 56, 777 (1986). \n4. Y. D. Park, A. T. Hanbicki, S. C. Erwin, C. S. Hellberg, J. M. Sullivan, J. E. Mattson, T. F. \nAmbrose, A. Wilson, G. Spanos, and B. T. Jonker, Science 295, 651 (2002).  \n5. D. Kitchen, A. Richardella, J. M. Tang, M. E. Flatte, and A. Yazdani , Nature 442, 436 (2006). \n6. P. Mahadevan, and A. Zunger, Phys. Rev. B  68, 075202 (2003). \n7. Y. S. Hor, P. Roushan, H. Beidenkopf, J. Seo, D.  Qu, J. G. Checkelsky, L. A. Wray, D. Hsieh, Y. \nXia, S. Y. Xu, D. Qian, M. Z. Hasan, N. P. Ong, A. Yazdani, and R. J. Cava, Phys. Rev. B   81, \n195203 (2010).  \n8. L. Cheng, Z. G. Chen, S. Ma, Z. d. Zhang, Y. Wang, H. Y. Xu, L. Yang, G. Han, K. Jack, G. Lu, \nand J. Zou, J. Am. Chem. Soc. 134, 18920 (2012). \n9. J. S. Dyck, P. Svanda, P Lostak, J Horak, W Chen, and C Uher, J. Appl. Phys. 94, 7631 (2003). \n10. J. Choi, S. Choi, J. Choi, Y. Park, H. M. Park, H.  W. Lee, B. C. Woo, and S. Cho, Phys. Status \nSolidi (b) 241, 1541 (2004). \n11. J. Zhang, C. Z. Chang, P. Tang, Z. Zhang, X. Feng, K. Li, L. L. Wang, X. Chen, C. Liu, W. \nDuan, K. He, Q. K. Xue, X. Ma, and Y. Wang, Science, 339, 1582  (2013). \n12. C. Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L. L. Wang, Z. \nQ. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S. C. Zhang, K. He, Y. Wang, L. Lu, X. C. \nMa, and Q. K. Xue, Science 340, 167 (2013). \n13. Q. Gibson et al. to be published. \n14. Y. Takagaki, A. Giussani, J. Tominaga, U. Jahn, and R. Calarco, J Phys.: Condens. Matter 25, \n345801 (2013). \n15. C. S. Barrett, P. Cucka, and K. Haefner, Acta Cryst. 16, 451 (1963). \n16. R. J. Cava, H. Ji, M. K. Fuccillo, Q. D. Gibson, and Y. S. Hor, J. Mater. Chem. C 1, 3176 (2013). \n17. V. Agafonov, N. Rodier, R. Céolin, R. Bellissent, C. Bergman and J. P. Gaspard,  Acta \nCryst.  C47, 1141-1143 (1991). \n18. V. Agafonov, N. Rodier, R. Céolin, R. Bellissent, C. Bergman, and J. P. Gaspard, Acta \nCrystallogr. Sect. C  47, 1141  (1991). \n19. Y. S. Hor, P. Roushan, H. Beidenkopf, J. Seo, D.  Qu, J. G. Checkelsky, L. A. Wray, D. Hsieh, Y. \nXia, S. Y. Xu, D. Qian, M. Z. Hasan, N. P. Ong, A. Yazdani, and R. J. Cava, Phys. Rev. B  81, \n195203 (2010). \n20. J. S. Dyck, P. Hájek, P. Lošt ’ák, and C. Uher, Phys. Rev. B 65, 115212 (2002). \n21. I. Yeung, R. M. Roshko, and G. Williams, Phys. Rev. B  34, 3456 (1986). \n22. Z. Starý, J. Horák, M. Stordeur, a nd M. Stölzer, J. Phys. Chem. Solids 49, 29 (1988). \n \n \n 8 \n Figures Captions \nFigure 1.  (Color online) (a) The crystal structure of Sb 2Te. The unit cell is shown as solid lines. The Sb \nand Te atoms are plotted as blue and red spheres, respectively. The Sb 2 bi-layer and Sb 2Te3 layer \nbuilding blocks are described in ball-and-stick and polyhedral forms, respectively; (b) The powder \nXRD patterns for Sb 2-xMn xTe (x = 0, 0.005. 0.01, 0.02,  0.04, 0.06, 0.07) and (c) Sb 1.93-yMn 0.07Te1+y (y = \n0, 0.02, 0.04, 0.06). \n \nFigure 2.  (Color online) (a) Zero-field cooled temperature-dependent dc magnetic susceptibility \nmeasured at 10 kOe applied magnetic field for the Sb 2-xMn xTe (x = 0.005. 0.01, 0.02, 0.04, 0.06, 0.07) \nsamples. (b)  Temperature dependence of the inverse susceptibility for the Sb 2-xMn xTe (x = 0, 0.005. \n0.01, 0.02, 0.04, 0.06, 0.07) samples. \n \nFigure 3.  (Color online) (a) The magnetic-field-dependent magnetization of the Sb 2-xMn xTe (x = 0, \n0.005. 0.01, 0.02, 0.04, 0.06, 0. 07) at 40 K. (b) The magnetic-field-dependent magnetization of Sb 2-\nxMn xTe (x = 0.02, 0.04, 0. 06, 0.07) at 2 K.  \n \nFigure 4.  (Color online) The hysteresis M vs. H loops of Sb 2-xMn xTe (x = 0.02, 0.04, 0.06, 0.07) \nsamples at 2 K with applied field varied from -3 to 3 kOe.  \n \nFigure 5.  (Color online) Arrott plots at various temperatures near T c for the polycrystalline Sb 2-xMn xTe, \n(a) x = 0.02, (b) x = 0.04, (c) x = 0.06, (d) x = 0.07 samples. (e) Inverse susceptibility (a’) inferred from \nthe intercept of Arrott plots (H/M = a’+ b’M2) of Figure 5a, b, c, d. (f) Detail of the regime near T c of \nFigure 5e. \n \nFigure 6.  (Color online) (a) Zero-field cooled temperature-dependent dc magnetic susceptibility \nmeasured at 10 kOe applied magnetic field for the polycrystalline Sb 1.93-yMn 0.07Te1+y (y = 0, 0.02, 0.04, \n0.06) samples. (b)  Temperature dependence of the inverse susceptibility for the polycrystalline Sb 1.93-\nyMn 0.07Te1+y (y = 0, 0.02, 0.04, 0.06) samples. \n \nFigure 7.  (Color online) (a) The magnetic-field-dependent magnetization of the polycrystalline Sb 1.93-\nyMn 0.07Te1+y (y = 0, 0.02, 0.04, 0.06) at 40 K. (b) The magnetic-field-dependent magnetization of the \npolycrystalline Sb 1.93-yMn 0.07Te1+y (y = 0, 0.02, 0.04, 0.06) samples at 2 K.  \n \nFigure 8.  (Color online) The hysteresis MH loops of the polycrystalline Sb 1.93-yMn 0.07Te1 +y (y = 0, 0.02, \n0.04, 0.06) samples at 2 K with applied field from -3 to 3 kOe.  \n \nFigure 9.  (Color online) Arrott plots at various temperatures near T c for the polycrystalline Sb 1.93-\nyMn 0.07Te1+y (a) y = 0, (b) y = 0.02, (c) y = 0.04, (d) y = 0.06 samples. (e) Inverse susceptibility (a’) \ninferred from the intercept of Arrott plots (H/M = a’+ b’M2) of Figure 9a, b, c, d. (f) Detail of the \nregime near T c of Figure 9e. 9 \n  \nTable 1 . Parameters from fitting the magnetic susceptibility of Sb 2-xMn xTe (x = 0, \n0.005. 0.01, 0.02, 0.04, 0.06, 0.07) to the Curie-Weiss law and Arrott plots.               \nx C \n(emu mol-1 \n Oe-1 f.u.-1) χ0  \n(emu mol-1 \n Oe-1 f.u.-1) θ \n(K) Peff \n(μB/mol-Mn) Tc \n(K) M at 90 kOe \nat 2 K \n(μB/f.u.) \n0.005 0.01963 -0.00042 -5.05 5.60 - - \n0.01 0.03770 -0.00040 -0.85 5.49 - - \n0.02 0.06639 -0.00035 5.57 5.15 8.6 0.05 \n0.04 0.10278 -0.000013 3.04 4.53 4.1 0.08 \n0.06 0.10834 -0.000215 5.29 3.8 3.6 0.088 \n0.07 0.11587 0.000015 2.89 3.63 3 0.09 \n                                                                                                                                                  \nTable 2 . Parameters from fitting the magnetic susceptibility of Sb 1.93-yMn 0.07Te1+y (y = \n0, 0.02, 0.04, 0.06) to the Curie-Weiss law and Arrott plots.           \ny C \n(emu mol-1 \n Oe-1 f.u.-1) χ0  \n(emu mol-1 \n Oe-1 f.u.-1) θ \n(K) Peff \n(μB/mol-Mn) Tc \n(K) M at 90 kOe  \nat 2 K \n(μB/f.u.) \n0 0.11587 0.000015 2.89 3.63 3 0.09 \n0.02 0.10384 0.000059 8.19 3.44 5 0.09 \n0.04 0.10042 -0.00024 8.07 3.42 7 0.08 \n0.06 0.09985 0.000151 15.76 3.38 8.9 0.1 \n \n    \n \n    \n \n  10 \n  \nFigure 1 \n \n     20 30 40 50 60Intensity (arb. units) Intensity (arb. units) \n(c)Sb1.93-yMn0.07Te1+y\n  \nx = 0.06\nx = 0.04\nx = 0.02\nx = 0.01\nx = 0.005Sb2-xMnxTe\ny = 0y = 0.02y = 0.04y = 0.06(b)\n20 30 40 50 60\n 2θ  (degree)\n  \n \n             \n      \n \n   \nSb  Te \nSb  (a) 11 \n Figure 2 \n0 70 140 210 2800.0000.0050.0100.0150.0200.0250.030\nSb2-xMnxTe\nH = 10 kOeSb2-xMnxTe\nH = 10 kOe(b) 1/(χ−χ0) (mol f.u. Oe-1 emu-1)  \n  χ (emu f.u.-1 Oe-1 mol-1) x=0.005\n x=0.01\n x=0.02\n x=0.04\n x=0.06\n x=0.07(a)\n0 1 02 03 04 00.000.010.02\n  \n0 2 04 06 08 0 1 0 00400800120016002000\nx=0.06x=0.04\nx=0.07x=0.01\nx=0.02\nT (K)\n   \n T (K)x=0.005\n \n \nFigure 3   \n0 7 14 21 280.0000.0020.0040.0060.0080.0100.012 x = 0.06(b)\nx = 0.005x = 0.01x = 0.02x = 0.04x = 0.06x = 0.07 Sb2-xMnxTe\n  T = 40 K\nH (kOe)Sb2-xMnxTe\n   T = 2 K  \nx = 0.07\nx = 0.04M (μΒ f.u.-1)\nH (kOe)x = 0.02M (μΒ f.u.-1)(a)\n0 2 04 06 08 0 1 0 00.000.010.020.030.040.050.060.070.080.09 \n  \n \n \n \n \n  12 \n  \n  \n \nFigure 4 \n-3 -2 -1 0 1 2 3-0.028-0.021-0.014-0.0070.0000.0070.0140.0210.028\nx = 0.06x = 0.04\nx = 0.02\n  M (μB f.u.-1)\nH (kOe)x = 0.07\nSb2-xMnxTe\n   T = 2 K\n 13 \n Figure 5\n0.000 0.001 0.002 0.003 0.004 0.00505001000150020002500\n0.000 0.002 0.004 0.006 0.008 0.0100200400600800100012001400\n0.000 0.005 0.010 0.015 0.02002004006008001000H/M (kOe emu-1)\nH/M (kOe emu-1)2 K10 K\n5 K15 K20 K\nH/M (kOe emu-1)(b)  \n H/M (kOe emu-1)(a) 25 K\n2 K5 K10 K15 K20 K25 K\n0.00 0.01 0.02 0.03 0.04 0.050100200300400500600700\n    Tc ∼ 3.6 K  \nTc ∼ 4.1 K    Tc ∼ 3 K  \n Tc ∼ 8.6 K  M2 (emu2)    Tc ∼ 8.6 K\nSb1.98Mn0.02Te\nM2 (emu2)\n   \n      Tc ∼ 4.1 K\nSb1.96Mn0.04Te\n0 4 8 12 16 20 24-1500-1000-500050010001500\n T (K)\na' (kOe mol-1)Sb2-xMnxTe\n  x = 0.02\n  x = 0.04\n  x = 0.06\n  x = 0.07\n  a' (kOe mol-1)\n2468-450-300-1500150300\nT (K)  x = 0.02\n  x = 0.04\n  x = 0.06\n  x = 0.07\n  2 K2 K(d)\n5 K7K10 K15 K20 K\n   \nM2 (emu2)25 K\n    Tc ∼  3.6 K\nSb1.94Mn0.06Te(c)\n(e) (f) Sb2-xMnxTe5 K10 K15 K20 K\n   \nM2 (emu2)25 K\n    Tc ∼ 3 K\nSb1.93Mn0.07Te\n  \n \n 14 \n Figure 6 \n0 50 100 150 200 250 3000.0000.0060.0120.0180.0240.0300.036(b) y = 0\n y = 0.02\n y = 0.04\n y = 0.06\n  \n  \nT (K) y = 0\n y = 0.02\n y = 0.04\n y = 0.06(a)χ (emu f.u. mol-1)\n0 1 53 04 50.000.010.020.03\n  \n0 2 04 06 08 0 1 0 00150300450600750900\nSb1.93-yMn0.07Te1+y\n   μ0H = 10 kOeSb1.93-yMn0.07Te1+y\n   μ0H = 10 kOe\n1/(χ−χ0) (mol f.u.-1 emu-1)\nT (K)\n  \n \nFigure 7  \n0 7 14 21 280.0000.0030.0060.0090.0120.015(b)\nSb1.93-yMn0.07Te1+y\n      T = 40 Ky = 0.04\ny = 0.02y = 0   M (μB f.u.-1 )\nH (kOe)Sb1.93-yMn0.07Te1+y\n      T = 2 Ky = 0.06M (μB f.u.-1 )\nH (kOe)(a)\n0 2 04 06 08 0 1 0 00.000.020.040.060.080.10 \ny = 0\ny = 0.04y = 0.02y = 0.06\n \n  \n \n   15 \n Figure 8 \n-3 -2 -1 0 1 2 3-0.04-0.03-0.02-0.010.000.010.020.030.04\ny = 0.02y = 0.04y = 0.06\n  M(μB f.u.-1 )\nH (kOe)Sb1.93-yMn0.07Te1+y\n       T = 2 Ky = 0\n \n \n       \n \n \n      16 \n Figure 9 \n0.000 0.005 0.010 0.015 0.02002004006008001000\n      Tc ∼ 8.9 K\nSb1.87Mn0.07Te1.06      Tc ∼7 K\nSb1.89Mn0.07Te1.04     Tc ∼ 5 K\nSb1.91Mn0.07Te1.02(b)\n2 K5 K10 K15 K20 K\n   \nM2 (emu2)25 K\n    Tc ∼ 3 K\nSb1.93Mn0.07Te(a)25 K\n20 K\n15 K\n10 K\n5 K\n2 K\n(c) (d)\n(e) (f)0.000 0.003 0.006 0.009 0.0120200400600800100012001400H/M (kOe emu-1)\nH/M (kOe emu-1) H/M (kOe emu-1)H/M (kOe emu-1)\n  \n  \nM2 (emu2)\n0.0000 0.0015 0.0030 0.0045 0.00600500100015002000\n7 K\n  \n  \nM2 (emu2)2 K5 K10 K15 K20 K25 K 25 K\n20 K\n15 K\n10 K7 K5 K\n2 K\n0.000 0.003 0.006 0.009 0.012020040060080010001200\n   \n M2 (emu2)\n0 5 10 15 20 25-40004008001200\nT (K) y = 0\n y = 0.02\n y = 0.04\n y = 0.06 y = 0\n y = 0.02\n y = 0.04\n y = 0.06\nT (K)\n  \n02468 1 0 1 2-800-600-400-2000200400600a' (kOe emu-1)a' (kOe emu-1)\nTc= 8.9 K Tc= 7 K Tc= 5 K Sb1.93-yMn0.07Te1+y\n  \nSb1.93-yMn0.07Te1+y\nTc= 3 K \n "    },    {        "title": "1105.1916v1.Triplet_superconductivity_in_a_ferromagnetic_vortex.pdf",        "content": "arXiv:1105.1916v1  [cond-mat.supr-con]  10 May 2011Triplet superconductivity in a ferromagnetic vortex\nMikhail S. Kalenkov,1Andrei D. Zaikin,2,1and Victor T. Petrashov3\n1I.E. Tamm Department of Theoretical Physics, P.N. Lebedev P hysical Institute, 119991 Moscow, Russia\n2Institute of Nanotechnology, Karlsruhe Institute of Techn ology (KIT), 76021 Karlsruhe, Germany\n3Department of Physics, Royal Holloway, University of Londo n, Egham, Surrey TW20 0EX, United Kingdom\nWe argue that odd-frequency triplet superconductivity can be conveniently realized in hybrid\nsuperconductor-ferromagnet (SF) structures with a ferrom agnetic vortex. We demonstrate that\ndue to proximity-induced long-range triplet pairing such S FS junctions can sustain appreciable\nsupercurrent which can be directly measured in experiments .\nA normal metal (N) sandwiched between two super-\nconductors (S) can become superconducting as a result\nof penetration of Cooper pairs from the superconduct-\ning electrodes. The range of penetration is set by the\nso-called thermal length ξTwhich can easily reach sev-\neral micrometers at suﬃciently low temperatures [1–3].\nThe situation changes drastically if the normal metal is\nreplaced by a ferromagnet (F). The quantum mechanical\nexchange interaction on the F-side then destroys conven-\ntional spin-singlet Cooper pairs within a few nanometers\n(the so-called paramagnetic eﬀect) [4]. Experiments to\ndetermine actual superconducting penetration depths in\nferromagnets intensiﬁed more than a decade ago, when\ntechniques were developed to fabricate hybrid nanoscale\nSF structures with well controlled geometries. Several\ngroups [5–9] reported an unexpectedly strong inﬂuence\nof superconductors that stimulated new theoretical ef-\nforts in a search for a sustainable superconductivity that\nis compatible with the exchange interaction. During\nthe last decade several theoretical mechanisms were sug-\ngested [10–16], some of which were successfully realized\nexperimentally [17–22]. A recent comprehensive review\nof the status of the ﬁeld was given in Ref. [23].\nCommon to all mechanisms of long range proximity\neﬀect in ferromagnets is the generation of triplet super-\nconductivity within highly inhomogeneous ferromagnetic\nregions adjacent to superconductors. The systems stud-\nied up to date include intrinsically inhomogeneous ferro-\nmagnets [17], half-metallic ferromagnets with spin-active\nFS interfaces [18, 19], and engineered multilayersconsist-\ning of magnetic and non-magnetic materials [20, 21].\nIn this letter we address a diﬀerent situation of\nproximity-induced long rangetriplet pairing in ferromag-\nnets with magnetic vortex structure. Magnetic vortices\nare stable in systems intermediate between very small,\n10 nm scale magnets, which behave as single giant spins,\nand macroscopicmagnets with dimensions exceeding ∼1\nµm. The magnetic structure in such mesoscopic mag-\nnets is the result of a competition between exchange,\nanisotropy, and dipolar energies and depends strongly on\ntheir shape. The latter property allows magnetic nano-\nengineering using modern nanolithography [24] opening\npossibility of investigating SFS structures with diﬀer-\ning magnetic structures. Recently mesoscopic magneticFIG. 1: (Color online) SFS junction formed by two supercon-\nducting electrodes connected via ferromagnetic vortex.\nstructures have attracted a lot of attention due to their\nremarkable transport properties [25–27], as well as the\nprospect of technological applications for magnetic stor-\nage of information of unprecedented density [28]. Below\nwewilldemonstratethatmesoscopicferromagneticstruc-\ntures can turn superconducting if attached to supercon-\nducting electrodes.\nThe model and quasiclassical formalism. We will con-\nsider a ferromagnetic ﬁlm of thickness dlocated in the\nxyplane with magnetization forming a vortex. This ﬁlm\nis partially covered by two superconducting electrodes\nthus forming an SFS contact as it is shown in Fig. 1.\nOur main goal is to analyze superconducting correlations\nthat penetrate into a ferromagnetic vortex from the elec-\ntrodes. In order to accomplish this goal we will employ\nthe quasiclassical Usadel equations [2, 29] for energy-\nintegrated matrix Matsubara-Green functions ˇG. E.g.\nin the ferromagnet with diﬀusion constant Dthese equa-\ntions read\niD∇(ˇG∇ˇG) = [ˇΩ,ˇG],ˇG2= 1, (1)\nwhere\nˇG=/parenleftbiggˆGˆF\nˆF+ˆG+/parenrightbigg\n,ˇΩ =/parenleftbiggiωnˆ1−ˆσh ∆ˆ1\n−∆∗ˆ1−iωnˆ1+ˆσh/parenrightbigg\n(2)2\nare 4×4 matrices in Nambu and spin spaces. Their\ncommutator in Eq. (1) and below is denoted by square\nbrackets. Accordingly, ˆG,ˆF,ˆF+andˆG+are 2×2 matri-\nces in the spin space, ωn=πT(2n+1) is the Matsubara\nfrequency, his the exchange ﬁeld in the ferromagnet and\nˆσ= (ˆσ1,ˆσ2,ˆσ3) represents the Pauli matrices in the spin\nspace. The same equations (1), (2) hold also for super-\nconducting electrodes, one should only replace Dby the\ndiﬀusion constant in the corresponding electrode. The\nsuperconducting order parameter ∆ equals to zero in the\nferromagnet, while in two superconducting terminals it is\nrespectively ∆ = ∆ 1exp(iχ/2) and ∆ = ∆ 2exp(−iχ/2)\nwith real ∆ 1,2andχbeing the superconducting phase\ndiﬀerence across our SFS junction.\nEquations (1) should be supplemented by appropri-\nate boundary conditions at each of the two SF-interfaces\nwhich account for electron transfer across these inter-\nfaces. In what follows we will assume that there exist\ntunnelbarriersatbothSFinterfaceswiththecorrespond-\ning tunneling resistances r1,2. In the tunneling limit it\nsuﬃces to employ Kuprianov-Lukichev boundary condi-\ntions [30] at each SF-interface. E.g., at the interface be-\ntween the ﬁrst superconducting electrode ( z >0) and the\nferromagnet ( z <0) these boundary conditions read\n2r1σˇGF∂zˇGF= [ˇGF,ˇGS1], (3)\nwhereˆGFandˆGS1are respectively the Green functions\nat the F- and S-sides of the ﬁrst interface and σis the\nDrude conductivity of a ferromagnet. Analogous bound-\nary conditions hold for the second SF-interface.\nLong-range triplet pairing in a ferromagnetic vortex.\nThe presence of tunnel barriers at both SF-interfaces ef-\nfectively implies weak electron tunneling regime in which\ncase the proximity eﬀect remains small and it suﬃces to\nlinearize Usadel equations in the ferromagnet as\nD∇2ˆF−2ωnˆF−i{ˆF,h(r)ˆσ}= 0.(4)\nIn Eq. (4) we restrict Matsubara frequencies to be pos-\nitiveωn>0 and denoted the anticommutator by curly\nbrackets. A similar equation holds for the function ˆF+.\nIn general magnetization patterns in thin ferromag-\nnetic ﬁlms depend on the ﬁlm geometry and are inﬂu-\nenced by the following trade-oﬀ. On one hand, magneto-\nstatic energy minimum is reached provided the ﬁlm mag-\nnetization remains in-plane. On the other hand, in some\nregions, such as, e.g., vortex cores, local magnetization\ncangoout-of-planein orderto minimize the exchangeen-\nergy. Asthe magneticcoreradiustypicallyremainssmall\nas compared to the superconducting coherence length,\nin the following we will assume that magnetization lies\nin-plane everywhere in the ferromagnet, see Fig. 1. In\nsuﬃciently thin ﬁlms the exchange ﬁeld hdepends only\non in-plane coordinates ( x,y) and can be represented as\nh= (hcosθ,hsinθ) whereθ=θ(x,y). In this case the\nspin structure of the anomalous Green function ˆFinsidethe ferromagnet can be chosen in the following form\nˆF=F0+ˆσmFh+ˆσ[ez,m]Ft, (5)\nwhereF0describes the singlet pairing component, while\nFhandFtcorrespondtotwodiﬀerenttripletcomponents.\nIn Eq. (5) we also introduced in-plane and normal to the\nplane unity vectors m=h/handez. Combining Eqs.\n(5) and (4) we arrive at the following equations for the\nabove components:\nD∇2F0−2ωnF0= 2ihFh, (6)\nDFh=DFt∇2\nρθ+2D(∇ρFt,∇ρθ)+2ihF0,(7)\nDFt=−DFh∇2\nρθ−2D(∇ρFh,∇ρθ),(8)\nwhere we deﬁned the diﬀerential operator\nD=D∇2−D(∇ρθ)2−2ωn (9)\nand distinguished ∇and∇ρas respectively 3d and 2d\n(in-plane) gradient operators.\nNote that Eqs. (6) and (7) contain the exchange ﬁeld\nhthus providing the characteristic length scale both for\nF0andFhof order ξh∼/radicalbig\nD/h. At the same time,\nEq. (8) does not contain the h-term and, hence, typi-\ncal variations of Ftoccur on a much longer length scale\nξT∼/radicalbig\nD/T≫ξh. This observation illustrates the dif-\nference between the two triplet components FhandFt\nand constitutes the essence of the long range proximity\neﬀect in SFS structures: while the components F0and\nFhdecay already in the vicinity of an SF-interface, the\ntriplet component Ftsurvives deep inside the ferromag-\nnet provided the temperature remains suﬃciently low.\nBefore turning to the solution of Eqs. (6)-(8) let us\nperform some further simpliﬁcations. Firstly, we will ne-\nglect both magnetic anisotropy and stray ﬁeld eﬀects. In\nthis case outside the magnetic vortex core the function θ\nobeys the equation\n∇2\nρθ= 0, (10)\nwhich allows to drop the ﬁrst terms in the right-hand\nside of Eqs. (7) and (8). Secondly, we will assume the\nferromagneticﬁlm to be suﬃciently thin d<∼ξT, in which\ncase the dependence of the long-range triplet component\nFton the coordinate zcan be neglected. Then, integrat-\ning Eq. (8) over zwe obtain\nDρFt=−2D(∇ρFh,∇ρθ),Fh=1\nd/integraldisplay0\n−dFhdz,(11)\nwhereFhis the average value of Fhcomponent over the\nferromagneticﬁlm thickness and Dρis deﬁned by Eq. (9)\nwith∇2→ ∇2\nρ.\nEq. (11) accountsfordiﬀusion ofthe long-rangetriplet\ncomponent Ftacross the ferromagnet with nonuniform\nin-plane magnetization. It demonstrates that non-zero3\nFtis generated in the parts of the ferromagnet where\nboth∇ρθand∇ρFhdiﬀer from zero. The condition\n∇ρθ∝ne}ationslash= 0 obviously holds everywhere in the ferromag-\nnetic plane since the magnetization remains non-uniform\nthere. As for the averaged component Fh, it vanishes\ntogether with its gradient at distances exceeding ∼ξh\nfrom SF-interfaces. In the immediate vicinity of such\ninterfaces Fhis non-zero, but its in-plane gradient re-\nmains small because in the main approximation it only\ndepends on the absolute value of the exchange ﬁeld h,\ncf. Eq. (7). The gradient ∇ρFhbecomes appreciable\nonly in the region of the ferromagnet just below the edge\nof the superconducting ﬁlm where Fhchanges abruptly.\nWith this in mind we arrive at the following result for\nthe long-range triplet component\nFt(ρ) =iD2\nhσd/summationdisplay\nk=1,2FSk\nrk/integraldisplay\nlkPρ,ρ′\nωn(∇′\nρθ(ρ′),nlk(ρ′))dlk,\n(12)\nwhich holds inside the ferromagnetic ﬁlm. Here FSkis\nanomalous Green function in the bulk of the k-th super-\nconductor and nlkis the outer unity vector normal to\nthe superconducting plane Sk(see Fig. 1). Integration\ncontours lkin Eq. (12) are lines in the xyplane corre-\nsponding to the edge of the superconductor Skand in the\nferromagnet kernel Pρ,ρ′\nωnobeys the equation\nDρPρ,ρ′\nωn=δ(ρ−ρ′), (13)\nwith boundary conditions ∂Pρ,ρ′\nωn/∂n= 0. We also note\nthat Eq. (12) can easily be generalized to the case of\narbitrary ∇2\nρθnot obeying Eq. (10).\nTriplet pairing and Josephson eﬀect. As triplet pairing\namplitude can survive deep in the ferromagnet, at suﬃ-\nciently low temperatures our SFS junction can sustain\nappreciable supercurrent which is converted from singlet\nto triplet and back in the vicinity of SF-interfaces. In\norder to evaluate this supercurrent we will employ the\nstandard expression for the current density\nj=πσT\n2eIm/summationdisplay\nωn>0Sp[ˆF∇ˆF+−ˆF+∇ˆF],(14)\nwhere the trace is taken over the spin degree of freedom.\nCombining Eqs. (5), (12) with (14) we recover the sinu-\nsoidal current-phase relation I(χ) =Icsinχwith\nIc=2πTD3\neh2σdr1r2/summationdisplay\nωn>0∆1∆2/radicalbig\n(ω2n+∆2\n1)(ω2n+∆2\n2)(15)\n×/integraldisplay\nl1,l2Pρ1,ρ2\nωn(∇ρθ(ρ1),nl1(ρ1))(∇ρθ(ρ2),nl2(ρ2))dl1dl2\nNote that in the course of our derivation we always as-\nsumed the proximity eﬀect to be suﬃciently weak. This\nassumption is satisﬁed under the condition\n1\nr1,2σ/radicalbigg\nD\nh≪/braceleftBigg\n1, d >∼ξh,\nd/radicalbig\nh/D, d <∼ξh.(16)FIG. 2: (Color online)Typical spatial distribution ofthel ong-\nrange superconducting triplet component Ftinduced in the\nferromagnetic disk with vortex-like magnetization by one s u-\nperconducting electrode ( x > R/2,z >0) with real ∆.\nEq. (15) together with its validity condition (16) repre-\nsents the central result of our analysis which fully deter-\nmines the Josephson critical current of an SFS junction\nwith a ferromagnetic vortex. Actually this result applies\nnot only to vortex conﬁgurations but also to a broader\nclass of non-uniform magnetization patterns.\nLet us now assume that our ferromagnetic ﬁlm has the\nform of a disk with radius Rand vortex-like magnetiza-\ntion pattern with vortex core located in the disc center.\nThen the function θequals to ϕ+π/2 for clockwise or\nϕ−π/2 for counterclockwise magnetization, where ϕis\nthe azimuthal angle (see Fig. 1). Eq. (10) is fulﬁlled\nin this case. Remarkably, Eq. (15) yields exactly the\nsame result for quite diﬀerent magnetization patterns:\nvortex-like ( θ=ϕ±π/2), antivortex-like ( θ=−ϕor\nθ=−ϕ+π) and hedgehog-like ( θ=ϕorθ=ϕ+π)\nstates. This property holds since the function ∇ρθre-\nmains the same (up to a sign) for all these magnetization\npatterns. Note, however, that for the last two patterns\nstray magnetic ﬁeld is not conﬁned to the disc center and\nmay inﬂuence superconductivity in the electrodes.\nForillustration, typicalspatialproﬁleofthe long-range\nsuperconductingtripletcomponent Ftinducedby onesu-\nperconducting electrode in the ferromagnetwith a vortex\nis schematically depicted in Fig. 2. As it was expected,\nFtis most eﬃciently generated close to the edge of a su-\nperconductor where the scalar product |(∇ρθ(ρ),nl(ρ))|\nreaches its maximum values. Provided the proximity ef-\nfect remains weak, the total value of Ftis given by a\nsuperposition of independent contributions from two su-\nperconducting electrodes, cf. also Eq. (12).\nAs one can observe in Fig. 2, the long-range triplet\ncomponent Ftpenetrating into the ferromagnet can take\nboth positive and negative values. Thus, depending on\nthe magnetization pattern inside the ferromagnetic ﬁlm\nit is possible to realize both zero- and π-junction states\nin our structure. The latter regime can be reached, e.g.,\nby implementing certain asymmetry in SF contacts.\nWe further consider a symmetric situation, set ∆ 1,2=\n|∆|and assume that the relevant Thouless energy εTh∼\nD/(2R)2remains smaller than the superconducting gap4\nFIG. 3: (Color online) Icversus temperature (normalized by\nthe critical temperature Tc) in SFS junctions containing a\nferromagnetic vortex at diﬀerent values of R. The edges of\nsuperconducting electrodes (contours l1,2) are chosen to coin-\ncide with straight lines y=±R/2.\n|∆|. Then in the limit T≪εThfrom Eq. (15) we ﬁnd\nIc∼D2εTh/(edh2r1r2σ), (17)\nwhile at intermediate temperatures εTh≪T≪ |∆|the\nJosephson current follows the standard exponential de-\npendence on temperature\nIc∼TD2\nedh2r1r2σexp/parenleftBig\n−L/radicalbig\n2πT/D/parenrightBig\n(18)\nwhereLis an eﬀective distance between the two SF\ncontacts which depends on geometry details (obviously\nL= 2Rfor small area contacts). For illustration the\nJosephson critical current Icis also plotted in Fig. 3 as\na function of temperature for diﬀerent values of R.\nOur result for Icin SFS systems turns out to be by\nthe factor ∼ε2\nTh/h2smaller than that for conventional\ndiﬀusive SNS junctions with identical geometry, cf., e.g.,\n[3, 30]. The critical current of our SFS structure can\nfurther be increased by a proper choice of the system\nparameters. For a simple estimate of possible maximum\nvalues of Iclet us employ Eq. (17) at the border of its\napplicability range (16). Then for T≪εThandd>∼ξh\nwe obtain\nIc∼D2σ/(eR2dh)∼(ξh/d)2εTh/(eRN),(19)\nwhereRNis the normal state resistance of the ferromag-\nnetic ﬁlm between two superconducting electrodes. This\nestimate is also supported by our independent calcula-\ntion (not presented here) which yields contributions to\nIc∝1/hin higher orders in barrier transmissions. Eq.\n(19) demonstratesthatfor d>∼ξhonecanexpecttoreach\nvalues of Iconly by the factor ∼ξ2\nh/d2smaller that the\nabsolutemaximum Ic∼εTh/eRNachievedfor SNS junc-\ntions [31]. Actually, the latter maximum value can also\nbe reached, but only for extremely thin ﬁlms d<∼ξh(cf.\nEqs. (17), (16)) with large values of RN.In summary, we demonstrated that long-range triplet\nsuperconductivity can coexist with a ferromagnetic vor-\ntex and evaluated the supercurrent across SFS junctions\ncontaining such vortex. For properly chosen system pa-\nrameters the eﬀect is well in the measurable range and\ncan be directly tested in future experiments. This work\nwas supported in part by DFG, by RFBR under grant\n09-02-00886and by British EPSRC grant EP/F01689/1.\n[1] C.J. Lambert and R. Raimondi, J. Phys.: Condens. Mat-\nter10, 901 (1998).\n[2] W. Belzig et al., Superlatt. Microstruct. 25, 1251 (1999).\n[3] A.A. Golubov, M.Yu. Kupriyanov, and E. Il’ichev, Rev.\nMod. Phys. 76, 411 (2004).\n[4] A.I. Buzdin, L.N. Bulaevskii, and S.V. Panyukov, JETP\nLett.35, 178 (1982).\n[5] V.T. Petrashov et al., JETP Lett. 59, 523 (1994).\n[6] M. Giroud et al., Phys. Rev. B 58, R11872 (1998).\n[7] V.T. Petrashov et al., Phys. Rev. Lett. 83, 3281 (1999).\n[8] J. Aumentado and V. Chandrasekhar, Phys. Rev. B 64,\n054505 (2001).\n[9] P. Nugent, I.A. Sosnin, and V.T. Petrashov, J. Phys.:\nCondens. Matter 16, L509 (2004).\n[10] F.S. Bergeret, A.F. Volkov, and K.B. Efetov, Phys. Rev.\nLett.86, 4096 (2001); Rev. Mod. Phys. 77, 1321 (2005).\n[11] A. Kadigrobov, R.I. Shekhter, and M. Jonson, Europhys.\nLett.54, 394 (2001).\n[12] M. Eschrig et al., Phys. Rev. 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B 81, 094436 (2010).\n[25] P. Bruno et al., Phys. Rev. Lett. 93, 096806 (2004).\n[26] A.A. Fraerman and O.G. Udalov, Phys. Rev. B 77,\n094401 (2008).\n[27] A. Neubauer et al., Phys. Rev. Lett. 102, 186602 (2009).\n[28] R.P. Cowburn et al., Phys. Rev. Lett. 83, 1042 (1999).\n[29] K.D. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n[30] M.Yu. Kuprianov and V.F. Lukichev, Sov. Phys. JETP\n67, 1163 (1988).\n[31] A.D. Zaikin and G.F. Zharkov, Fiz. Nizk. Temp. 7, 375\n(1981) [Sov. J. Low Temp. Phys. 7, 184 (1981)]; P. Dubos\net al., Phys. Rev. B 63, 064502 (2001)."    },    {        "title": "0810.1094v1.Thermodynamic_properties_of_the_itinerant_boson_ferromagnet.pdf",        "content": "arXiv:0810.1094v1  [cond-mat.stat-mech]  7 Oct 2008Thermodynamic properties of the itinerant-boson ferromag net\nChengjun Tao, Peilin Wang, Jihong Qin, and Qiang Gu\nDepartment of Physics, University of Science and Technolog y Beijing, Beijing 100083, P.R. China\n(Dated: November 4, 2018)\nThermodynamics of a spin-1 Bose gas with ferromagnetic inte ractions are investigated via the\nmean-ﬁeld theory. It is apparently shown in the speciﬁc heat curve that the system undergoes two\nphase transitions, the ferromagnetic transition and the Bo se-Einstein condensation, with the Curie\npoint above the condensation temperature. Above the Curie p oint, the susceptibility ﬁts the Curie-\nWeiss law perfectly. At a ﬁxed temperature, the reciprocal s usceptibility is also in a good linear\nrelationship with the ferromagnetic interaction.\nPACS numbers: 75.40.Cx, 75.10.Lp, 75.30.Kz, 03.75.Mn\nI. INTRODUCTION\nThe realization of spinor Bose-Einstein conden-\nsation in optical traps1,2has stimulated enormous\ninterest in magnetic properties of quantum Bose\ngases3,4,5,6,7,8,9,10,11,12,13,14,15. In optical traps, the hy-\nperﬁne degree of freedom of conﬁned atoms, such as\n87Rb, isreleasedand thereforethe atomcanexhibit mag-\nnetism. More intriguingly, an exchange-like spin-spin in-\nteraction can be present between atoms. In the F= 1\n87Rb atoms, the interaction is ferromagnetic3, so the\n87Rb gas appears to be a prototype of itinerant-boson\nferromagnet4,5,6,7,8.\nFerromagnetism is one of the central research themes\nin condensed matter physics16,17. Two types of ferro-\nmagnetism have already been intensively studied: local-\nmomentferromagnetismanditinerant-electronferromag-\nnetism. Although particles in these two systems obey\ndiﬀerent statistics, they both share some common fea-\ntures. For example, both ferromagnets have a Curie\npoint, above which the susceptibility conforms to Curie-\nWeiss law. Nonetheless, from the theoretical point of\nview, the origin of Curie-Weiss law is quite diﬀerent\nfor these two systems. In insulators it is due to local\nthermal spin ﬂuctuations and can be easily explained in\nthe mean-ﬁeld approximation. On the other hand, in\nitinerant-electron ferromagnets the Curie-Weiss law may\nbe caused by the mode-mode coupling between spin ﬂuc-\ntuationsandthetheoreticaltreatmentismuchmorecom-\nplicated17. An appropriate theory is the self-consistent\nrenormalization (SCR) theory18which goes beyond the\nHartree-Fock approximation and the random-phase ap-\nproximation. TheSCRtheorysucceedsinexplainingvar-\nious magnetic properties of itinerant-electron ferromag-\nnets and is also extended to treat the speciﬁc heat19.\nThe87Rb gas provides opportunity to study the third\ntype of ferromagnetism. Ho4, Ohmi and Machida5have\nstudied its ground state properties and the spin-wave\nspectrum. The long wavelength spectrum is linear in k,\nthe wave vector, as in the two former cases. In our previ-\nous papers, we have investigated the ﬁnite-temperature\nproperties, especially the Curie point8. We suggest that\nthe phase diagram in itinerant bosons should be more\ncomplicated than the other two ferromagnets, becausethe Bose system has an intrinsic phase transition, other\nthan the ferromagnetictransition. An interesting conclu-\nsion we arrived is that its Curie point, TF, is never be-\nlow the Bose-Einstein condensation temperature, TC, re-\ngardlessofthe magnitude ofthe ferromagneticcoupling8.\nKis-Szabo et algot the same point later9. However, ther-\nmodynamics of the itinerant-boson ferromagnet has not\nyet been investigated systematically so far.\nThe purpose of this paper is to calculate the thermo-\ndynamic quantities of ferromagnetic bosons. As in the\nfermion case, the speciﬁc heat and magnetic susceptibil-\nity are of the most interest. In Section 2, we introduce\nthe mean-ﬁeld approximation to deal with ferromagnetic\ninteraction, taking the spin-1 Bose gas as an example.\nIn Section 3, phase transitions are discussed by calcu-\nlating the free energy and speciﬁc heat. In Section 4,\nthe susceptibility above the Curie point is calculated. A\nsummary is given in the last section.\nII. THE MEAN-FIELD APPROXIMATION\nThe spin-1 Bose gas with ferromagnetic couplings is\ndescribed by the following Hamiltonian,\nˆH=/summationdisplay\nσ/integraldisplay\ndrˆψ†\nσ(r)/parenleftbigg1\n2m∇2−σhe/parenrightbigg\nˆψσ(r)\n−1\n2Is/integraldisplay\ndrˆS(r)·ˆS(r), (1)\nwhereˆψσ(r) is the quantum ﬁeld operator for annihi-\nlating an atom in spin state |σ/an}b∇acket∇i}htat siter. For a spin-1\ngas,σ= +1,0,−1. The parameter hedenotes the ex-\nternal magnetic ﬁeld. The last term represents the ferro-\nmagnetic exchange between two diﬀerent bosons meet-\ning at site randIs(>0) is the exchange constant.\nˆS={ˆSx,ˆSy,ˆSz}are the spin operators, which can be\nexpressed via the 3 ×3 Pauli matrices, for example,\nˆSz=/parenleftbigˆψ†\n+1ˆψ†\n0ˆψ†\n−1/parenrightbig\n1 0 0\n0 0 0\n0 0−1\n\nˆψ+1\nˆψ0\nˆψ−1\n.(2)\nWithin the mean-ﬁeld approximation, we treat the spin-\ndependent interactions as a molecular ﬁeld except of a2\nparticle with itself,\n−1\n2ˆS·ˆS≈ −/an}b∇acketle{tˆS/an}b∇acket∇i}ht·ˆS+1\n2/an}b∇acketle{tˆS/an}b∇acket∇i}ht·/an}b∇acketle{tˆS/an}b∇acket∇i}ht=−MˆSz+1\n2M2,(3)\nwhereM=/an}b∇acketle{tˆSz/an}b∇acket∇i}htis the ferromagnetic order parameter.\nThen the eﬀective Hamiltonian for the grand canonical\nensemble reads,\nˆH−ˆNµ=/summationdisplay\nkσ[ǫk−µ−σ(hm+he)]ˆnkσ+1\n2M2IsN,(4)\nwhereǫkis the kinetic energy for free particles, hm=\nIsMis called the molecular ﬁeld, similar to the Stoner\ntheory for fermion gases16;µis the chemical potential; N\nis the total particle number. The grand thermodynamic\npotential can be worked out in a standard way,\nΩ =−kBTlnTrexp[−ˆH−ˆNµ\nkBT]\n=−(kBT)5\n2Vm3\n2\n(2π¯h2)3\n2/summationdisplay\nσf5\n2/parenleftbiggµ+σh\nkBT/parenrightbigg\n+1\n2M2IsN,(5)\nwhereh=hm+he,mis the mass of particle, and fis\nthe polylogarithm function deﬁned by\nfn(x)≡∞/summationdisplay\nk=1(ex)k\nkn, (6)\nwherex≤0. The mean-ﬁeld self-consistent equations\nare derived from the grand thermodynamic potential,\nn=−1\nV/parenleftbigg∂Ω\n∂µ/parenrightbigg\nT,V+n0\n=/parenleftbiggkBTm\n2π¯h2/parenrightbigg3\n2/summationdisplay\nσf3\n2/parenleftbiggµ+σh\nkBT/parenrightbigg\n+n0; (7a)\nM=−1\nV/parenleftbigg∂Ω\n∂he/parenrightbigg\nT,V+n0\n=/parenleftbiggkBTm\n2π¯h2/parenrightbigg3\n2/bracketleftbigg\nf3\n2/parenleftbiggµ+h\nkBT/parenrightbigg\n−f3\n2/parenleftbiggµ−h\nkBT/parenrightbigg/bracketrightbigg\n+n0; (7b)\nwherenis the density of particles, n0is the density of\ncondensed one and M≡NM\nVis the magnetization. n0is\nzero unless the temperature is below the BEC point Tc.\nIII. THE FREE ENERGY AND SPECIFIC HEAT\nIn our previous investigations, we showed that the sys-\ntem exhibits two phase transitions, the Bose-Einstein\ncondensation (BEC) and the ferromagnetic transition8.\nThe condensation temperature TCand the Curie tem-\nperatureTFare calculated by solving the self-consistent/s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s45/s49/s46/s52/s45/s49/s46/s50/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s32/s70/s47/s86\n/s32/s116/s32/s80/s77\n/s32/s70/s77 /s32\nFIG. 1: Free energies of the FM and PM states with the\nferromagnetic coupling I= 1. The two curves cross at the\ntemperature tf≈0.80, which is just the FM transition point.\nequations. We ﬁnd that TFis never below TCfor all\nsystems with a ﬁnite ferromagnetic exchange ( Is/ne}ationslash= 0).\nHowever, one can get another solution to the Eqs. (7),\nwithM= 0 at all temperatures. It means the system\ndoes not undergo a ferromagnetic transition at all, but\nremainsinparamagnetic(PM)stateatlowtemperatures.\nActually, whether there exists a Curie point in the fer-\nromagnetic Bose gases is still a controversial question.\nSome researchers suppose that the Bose gas can not be\nmagnetized spontaneously at low temperatures even if\nthe ferromagnetic exchange is present20.\nIn order to single out the physically correct solution,\none has to compare the free energy of the the ferromag-\nnetic (FM) state and the PM state. The relationbetween\nthe free energy and the grand thermodynamic potential\nhas the form:\nF= Ω+Nµ . (8)\nFor computational convenience, the temperature Tand\nexchange interaction Isare re-scaled, as did in Ref. [8],\nby the following formula: t= [3ζ(3\n2)]−2\n3T/T0andI=\n[3ζ(3\n2)]−2\n3Is/(kBT0), where\nT0=1\nkB/parenleftbiggn\n3ζ(3\n2)/parenrightbigg2\n3/parenleftbigg2π¯h2\nm/parenrightbigg\nis the condensation temperature of ideal spin-1 Bose gas.\nHereinafter, all the numerical results are obtained by set-\ntingn=kB=2π¯h2\nm= 1. Figure 1 shows the free energy\nof unit volume for the gas with I= 1.0. It shows clearly\nthat the free energy of FM state is lower than that of the\nPM state at the low temperature region, which demon-\nstrates that the FM state should be more stable than\nPM state. Therefore, the low temperature state has a\nspontaneous magnetization. In experiments, the total\nspin of the ferromagnetic spinor condensate is observed\ntobeconserved,whichiscalledthe spinconservationrule\nin some literatures13,14. However, the spin conservation\nrule holds only globally, not locally. In the theoretical3\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s51/s54/s57/s67\n/s116/s32/s73/s32/s61/s32/s48/s46/s52\n/s32/s73/s32/s61/s32/s49/s46/s48\nFIG. 2: Speciﬁc heats of spinor Bose gases with the coupling\nI= 1.0 and 0.4. The dotted vertical lines serve to guide the\neye to see the transition points.\ntreatment of Ref. [20], the spin conservation rule is im-\nposed by introducing a lagrangian multiplier. It is over-\nconstrained in some sense, so that the spontaneous mag-\nnetization can not be established. Recent experiments\nand theories indicate some domain structures should be\nformed and each domain is magnetized11,12,15, where the\nconservation law for the total spin can be restored natu-\nrally.\nThe FM transition is induced by the FM coupling and\nthe transition temperature is about tf≈0.8 for the Bose\ngas withI= 1.0. When the temperature goes down fur-\nther, the BEC then occurs, which is the intrinsic phase\ntransition of Bose gases. To demonstrate diﬀerent fea-\ntures of the two transitions, we now calculate the speciﬁc\nheat of unit volume,\nC=1\nV/parenleftbigg∂U\n∂T/parenrightbigg\nB,V, (9)\nwhereUis the internal energy\nU=F−TS= Ω−T(∂Ω\n∂T)+Nµ (10)\n=3V(kBT)5\n2m3\n2\n2(2π¯h2)3\n2/summationdisplay\nσf3\n2/parenleftbiggµ+σh\nkBT/parenrightbigg\n.\nAs shown in Fig. 2, for the system with I= 1.0, the\nspeciﬁc heat exhibits a jump discontinuity at tf≈0.8,\nfrom the PM state to the FM state. This is a charac-\nteristic feature of the Landau-type of second-order phase\ntransition. And similar behaviors have been observed in\nthe speciﬁc heat of ferromagnetic insulators or itinerant-\nfermion ferromagnets16,17. The BEC occurs at tc≈0.5,\nwhere the speciﬁc exhibits a bend. But speciﬁc heat is\ncontinuous at the BEC point, similar to that of a free\nBose gas. The results indicate that the critical behaviors\nare diﬀerent at the two transition points on the mean-\nﬁeld level.IV. THE CURIE-WEISS LAW\nFor a ferromagnet, the susceptibility above the Curie\npoint is of special interest. As already studied, the sus-\nceptibility is well described by Curie-Weiss law both in\nthe insulating ferromagnet and the itinerant-electron fer-\nromagnet. In this section we calculate the susceptibility\nfor the itinerant-boson ferromagnet.\nThe susceptibility can be derived from Eqs. (7). Dif-\nferentiatingboth sidesofthe twoequationsand removing\nthe termdµ, the following equation are deduced,\ndM=fd/parenleftbiggIsM+he\nkBT/parenrightbigg\n, (11)\nwhere\nf=/parenleftbiggkBTm\n2π¯h2/parenrightbigg3\n2/bracketleftbigg\nf1\n2/parenleftbiggµ+h\nkBT/parenrightbigg\n+f1\n2/parenleftbiggµ−h\nkBT/parenrightbigg/bracketrightbigg\n−/parenleftbiggkBTm\n2π¯h2/parenrightbigg3\n2/bracketleftBig\nf1\n2/parenleftBig\nµ+h\nkBT/parenrightBig\n−f1\n2/parenleftBig\nµ−h\nkBT/parenrightBig/bracketrightBig2\n/summationtext\nσf1\n2/parenleftBig\nµ+hσ\nkBT/parenrightBig.(12)\nAbove the Curie point, the magnetization M(thenh=\nIsM+he) diminishes correspondingly when the external\nﬁeldhetends to zero. So the second term in the above\nequation is omitted and then fhas a simple form:\nf≈2/parenleftbiggkBTm\n2π¯h2/parenrightbigg3\n2\nf1\n2/parenleftbiggµ\nkBT/parenrightbigg\n. (13)\nThus the zero-ﬁeld susceptibility of unit volume is given\nby\nχ=/parenleftbigg∂M\n∂he/parenrightbigg\nT,V=1\nkBTf−1−n−1Is.(14)\nThe susceptibility χis a function of the coupling Isand\ntemperature T. Figure3 shows1 /χandχversusIat dif-\nferent given temperatures. As shown in the inset of Fig.\n3, the susceptibility becomes larger as the coupling Iin-\ncreasing. It is physically reasonable since the the system\nwith larger Ican be magnetized more easily. At a given\ntemperature, χdiverges as Iapproaches a critical value.\nIt is worth noting that the inverse of the susceptibility is\nin a good linear relationship with the coupling.\nThe susceptibility versus temperature is shown in Fig.\n4. One can immediately ﬁnd that the susceptibility meet\nquite well with the Curie-Weiss law in a very large tem-\nperature region. Seeing that the Curie-Weiss law is very\ndiﬃcult to be derived for the itinerant- fermionferromag-\nnet, it is really surprising that we get it for the itinerant-\nbosonferromagnet just based on the mean-ﬁeld approx-\nimation.\nIn order to discuss the Curie-Weiss law in a more ex-\nplicit way, we proceed to carry out a semi-analytical cal-\nculation to deduce the linear dependence of 1 /χon the4\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s52/s56/s49/s50/s49/s54\n/s32/s32\n/s73/s32\n/s73\n/s32/s32/s49/s47/s32/s116/s32/s61/s32/s48/s46/s52\n/s32/s116/s32/s61/s32/s48/s46/s54\n/s32/s116/s32/s61/s32/s48/s46/s56\nFIG. 3: Magnetic susceptibilities versus ferromagnetic co u-\nplings of spinor Bose gases at temperature t= 0.4,0.6 and\n0.8.\n/s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s32\n/s116\n/s116/s32/s73 /s32/s61/s32/s48/s46/s50\n/s32/s73 /s32/s61/s32/s48/s46/s52\n/s32/s73 /s32/s61/s32/s49/s46/s48/s49/s47\n/s32/s32\nFIG. 4: Magnetic susceptibilities versus temperatures of\nspinor Bose gases on coupling I= 0.2,0.4 and 1.0.\ntemperature. The ﬁrst step is to analyze the tempera-\nturedependenceof f. Itisquitecomplicated,becausethe\nchemical potential µis an implicit function of the tem-\nperature. We consider a limit case that the parameter Is\nis quite small, when Tfis close toTc. Soµis close to zero\nin the vicinity of Tf. Accordingto the asymptotic behav-\nior of the polylogarithm function: f3\n2(x)≈ζ(3\n2)−2√πx\nandf1\n2(x)≈/radicalbig\nπ/xasx→0−, we get the following\nequations from Eqs. (7a) and (13) respectively,\nn≈3/parenleftbiggkBTm\n2π¯h2/parenrightbigg3\n2/bracketleftbigg\nf3\n2(0)−2/radicalbigg\n−πµ\nkBT/bracketrightbigg\n.(15)\nand\nf≈2/parenleftbiggkBTm\n2π¯h2/parenrightbigg3\n2/radicalBigg\n−kBTπ\nµ. (16)\nSubstitute Eqs. (15) and (16) into Eq. (14), we get\nχ−1=nk−2\nB\n12π/parenleftbiggm\n2π¯h2/parenrightbigg−3\nT−1\n2/parenleftBig\nT−3\n2\n0−T−3\n2/parenrightBig\n−n−1Is.\n(17)In the vicinity of TFwhich is only slightly larger than\nT0, Eq. (17) could be further simpliﬁed to\nχ−1≈nk−2\nB\n8π/parenleftbiggm\n2π¯h2/parenrightbigg−3\nT−3\n0(T−T0)−n−1Is\n=9ζ2(3\n2)\n8πn−1kB/bracketleftbigg\nT−/parenleftbigg\nT0+8π\n9ζ2(3\n2)kBIs/parenrightbigg/bracketrightbigg\n.(18)\nThus the eﬀective FM transition temperature is deﬁned\nas\nTf=T0+8π\n9ζ2(3\n2)kBIs.\nSo far the Curie-Weiss law is derived. We note that the\nderivation is only valid in small Iscases.\nInthe high temperaturelimit, onecanalsoeasilyprove\nthatχ−1is linearly dependant on T. In this case, −µ\nkBT\nhas a quite large value, so that\nf1\n2/parenleftbiggµ\nkBT/parenrightbigg\n≈f3\n2/parenleftbiggµ\nkBT/parenrightbigg\n≈eµ\nkBT\naccording to Eq. (6). Combining Eqs. (7a), (13) and\n(14), it yields\nχ−1=n−1(kBT−Is). (19)\nWe estimate this equation holds in the range of t>∼10.\nV. SUMMARY\nIn summary, we calculate thermodynamic quantities\nof the spinor Bose gas with ferromagnetic interactions.\nSuch kind of investigations has already been performed\nintensively for the ferromagnetic fermions, while few as\nyet for bosons. Based on a mean-ﬁeld approximation,\nwe show that the system undergos a ferromagnetic phase\ntransitionﬁrst, thenthe Bose-Einsteincondensationwith\nthe temperature decreasing. The speciﬁc heat shows a\njump discontinuity at the Curie point and a bend at the\nBose-Einstein condensation temperature, indicating that\ncritical behaviors are diﬀerent near the two transition.\nThe more surprising result is that the mean-ﬁeld theory\nyield the magnetic susceptibility which satisﬁes perfectly\nthe Curie-Weiss law over a wide range of temperature.\nThis work is supported by the National Natural Sci-\nence Foundation of China (Grant No. 10504002), the\nFok Yin-Tong Education Foundation, China (Grant No.\n101008), and the Ministry of Education of China (Grant\nNo. NCET-05-0098).5\n1D.M. Stamper-Kurn et al., Phys. Rev. Lett. 80, 2027\n(1998); J. Stenger et al., Nature (London) 396, 345 (1998).\n2M.D. Barrett, J.A. Sauer, and M.S. Chapman, Phys. Rev.\nLett.87, 010404 (2001).\n3N.N. Klausen, J.L. Bohn, and C.H. Greene, Phys. Rev. A\n64, 053602 (2001).\n4T.L. Ho, Phys. Rev. Lett. 81, 742 (1998);\n5T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822\n(1998).\n6T. Isoshima, K. Machida, and T. Ohmi, Phys. Rev. A 60,\n4857 (1999).\n7M. Ueda, Phys. Rev. A 63, 013601 (2000).\n8Q. Gu and R.A. Klemm, Phys. Rev. A 68, 031604 (2003);\nQ. Gu, K. Bongs, and K. Sengstock, Phys. Rev. A 70,\n063609 (2004).\n9K. Kis-Szabo, P. Szepfalusy, and G. Szirmai, Phys. Rev.\nA72, 023617 (2005); G. Szirmai, K. Kis-Szabo, and P.\nSzepfalusy, Eur. Phys. J. D 36, 281 (2005).\n10W. Zhang et al., Phys. Rev. Lett. 95, 180403 (2005).\n11J. Mur-Petit et al., Phys. Rev. A 73, 013629 (2006).\n12Q. Gu and H. Qiu, Phys. Rev. Lett 98, 200401 (2007).13H. Schmaljohann et al., Phys. Rev. Lett. 92, 040402\n(2004); J. Kronj¨ ager et al., Phys. Rev. A 72, 063619\n(2005).\n14M.-S. Chang et al., Phys. Rev. Lett. 92, 140403 (2004);\nM.-S. Chang et al., Nature Physics 1, 111 (2005).\n15L.E. Sadler et al., Nature 443, 312(2006).\n16P. Mohn, Magnetism in the Solid state: An Introduction\n(Springer-Verlag, Berlin, 2003).\n17T. Moriya, Spin ﬂuctuations in itinerant electron mag-\nnetism(Springer-Verlag, Berlin, 1985).\n18K.K. Murata and S. Doniach, Phys. Rev. Lett. 29, 285\n(1972); T. Moriya and A. Kawabata, J. Phys. Soc. Jpn.\n34, 639 (1973).\n19Y. Takahashi, J. Phys.: Condens. Matter 11, 6439 (1999);\nY. Takahashi and H. Nakano, J. Phys.: Condens. Matter\n16, 4505 (2004).\n20T. Isoshima, T. Ohmi, and K. Machida, J. Phys. Soc. Jpn.\n69, 3864 (2000); W. Zhang, S. Yi, and L. You, Phys. Rev.\nA70, 043611 (2004)."    },    {        "title": "1801.01623v1.Magnetoresistance_originated_from_charge_spin_conversion_in_ferromagnet.pdf",        "content": "arXiv:1801.01623v1  [cond-mat.mes-hall]  5 Jan 2018Magnetoresistance originated from charge-spin conversio n in\nferromagnet\nTomohiro Taniguchi∗\nNational Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba 305-8568, Japan\n(Dated: October 1, 2018)\nAbstract\nTransverse magnetoresistance in a ferromagnetic/nonmagn etic/ferromagnetic trilayer originated\nfrom charge-spin conversion by anomalous Hall eﬀect is inves tigated theoretically. Solving the\nspin diﬀusion equation in bulk and using the spin-dependent L andauer formula at the ferromag-\nnetic/nonmagnetic interface, an analytical formula of the transverse resistivity is obtained. The\ncharge-spin conversion by the anomalous Hall eﬀect contribu tes to the magnetoresistance in a man-\nner proportional to the square of the spin anomalous Hall ang le. The angular dependence of the\nmagnetoresistance is basically identical to that of planar Hall eﬀect, but has an additional term\nwhich depends on the relative angle of the magnetizations in two ferromagnets.\n∗Electronic address: tomohiro-taniguchi@aist.go.jp\n1I. INTRODUCTION\nSpin Hall eﬀect originated from spin-orbit interaction in nonmagnetic heavy metal has\nattracted much attention from both fundamental and applied phy sics [1–3]. The spin Hall\neﬀect generates electric (charge) currents from spin currents andvice versa [4–8], which is\ncalled charge-spin conversion. The charge-spin conversion in a non magnet has been studied\nexperimentally in ferromagnetic/nonmagnetic multilayers by severa l methods, such as non-\nlocal spin current diﬀusion [9, 10], the combination of the inverse spin Hall eﬀect and spin\npumping [11, 12], spin-torque ferromagnetic resonance [13, 14], ﬁr st and second harmonic\nHall voltage [15, 16], and spin Hall magnetoresistance [17–19].\nThe spin-orbit interaction in a ferromagnet gives anomalous Hall eﬀe ct [20, 21], where\nan electric voltage is generated in the direction perpendicular to bot h the magnetization\nand an external electric ﬁeld. Note that the current generated b y the anomalous Hall eﬀect\nis spin-polarized. Recently, a theory of spin current generation an d spin-torque excitation\nby the anomalous Hall eﬀect was proposed [22]. Magnetoresistance eﬀects originated from\nthe charge-spin conversion by the anomalous Hall eﬀect were also p redicted in a single\nferromagnet [23] and ferromagnetic/nonmagnetic bilayer [24]. Ver y recently, on the other\nhand, the observations of the charge-spin conversion by the ano malous Hall eﬀect, or related\nphenomena, have been reported in a multilayer including two ferroma gnets [25–30].\nIn this paper, we investigate the magnetoresistance eﬀect in a fer romag-\nnetic/nonmagnetic/ferromagnetic trilayer originated from the ch arge-spin conversion by the\nanomalous Hall eﬀect. By solving the diﬀusion equation of spin accumu lation and the spin-\ndependent Landauer formula, an analytical formula of the transv erse resistivity is derived.\nIn addition to the magnetoresistance found in previous works, whe re the angular depen-\ndence is the same with the planar Hall eﬀect, a contribution to the ma gnetoresistance which\ndepends on the relative angle of the magnetizations in two ferromag nets is revealed.\nThe paper is organized as follow. In Sec. II, we describe the system stud-\nied in this paper. In Sec. III, the transverse magnetoresistance in the ferromag-\nnetic/nonmagnetic/ferromagnetic trilayer is calculated. The conc lusion is summarized in\nSec. IV.\n2II. SYSTEM DESCRIPTION\nThe system we consider is a ferromagnetic(F 1)/nonmagnetic(N)/ferromagnetic(F 2) tri-\nlayer structure shown in Fig. 1(a). We denote the unit vector point ing in the magnetization\ndirection in the F k(k= 1,2) layer as mkand the thickness of the layer as dk. An external\nelectric ﬁeld is applied along xdirection. We assume that the F 2layer shows the anoma-\nlous Hall eﬀect, and injects spin currents into the F 1layer placed along zdirection. The\nanomalous Hall eﬀect also provides electric currents ﬂowing in ydirection given by [24]\nJcy=σAH/bracketleftbigg\nm2z−/parenleftbiggσAH\nσF/parenrightbigg\nm2xm2y/bracketrightbigg\nEx\n+(β−ζ)σAH\nem2x∂zδµF2.(1)\nHere,σFandσAHare the longitudinal and transverse (anomalous Hall) conductivities , re-\nspectively. The spin polarization of these conductivities in respectiv e areβandζ[22].\nThe spin accumulation deﬁned from an electrochemical potential ¯ µsof the spin- s(s=↑,↓)\nelectrons is denoted as δµ= (¯µ↑−¯µ↓)/2.\nThe ﬁrst and second terms on the right hand side of Eq. (1) are the electric currents\ngenerated from the anomalous Hall eﬀect. On the other hand, the last term represents a\ncontribution from the charge-spin conversion by the anomalous Ha ll eﬀect [23, 24]. A factor\nϑ≡(β−ζ)σAH\nσF, (2)\ncharacterizestheeﬃciencyofthecharge-spinconversionbythe anomalousHalleﬀect. There-\nfore, let us deﬁne Eq. (2) as the spin anomalous Hall angle. The phys ical meaning of the\nspin anomalous Hall angle is as follows. A factor σAH/σFis the anomalous Hall angle [21],\ncharacterizing the ratio of the transverse voltage to the extern al voltage. The spin polar-\nization of the transverse current is given by the product of the an omalous Hall angle and\nthe spin polarization, ζ, i.e.,ζ(σAH/σF). Note that the electrons scattered to the transverse\ndirection creates the charge accumulation. The electrons then mo ve along the direction of\nan internal electric ﬁeld generated by the charge accumulation. Th e electric current due\nto this internal ﬁeld is also spin polarized, where the spin polarization is given by β. The\nnet spin polarization decreases because of this motion of the electr ons. As a result, the\ntotal spin polarization of the spin current generated by the anoma lous Hall eﬀect is given\nby|(β−ζ)σAH/σF|, which is the spin anomalous Hall angle deﬁned in Eq. (2).\n3In experiments related to spin Hall eﬀect such as, for example the m easurements of the\nharmonic Hall voltage [16, 30], the spin current was measured from the planar Hall eﬀect\nin the transverse direction. The angular dependence of the planar Hall eﬀect is described\nbymxmy. As can be seen in Eq. (1) and the results shown in our previous work [24], the\nhigher order term of the anomalous Hall eﬀect and the contribution from the charge-spin\nconversion, corresponding to the second and third terms of Eq. ( 1), respectively, have the\nsame angular dependence as mxmy. We should emphasize here that our previous work [24]\nfocuses on a ferromagnetic/nonmagnetic bilayer. On the other ha nd, as will be shown in the\nnext section, the present trilayer has another contribution to th e magnetoresistance whose\nangular dependence is described by not only by mxmybut also by m1·m2.\nIII. MAGNETORESISTANCE IN FERROMAGNETIC/NONMAGNETIC TRI-\nLAYER\nThe spin accumulation δµFin the F 2layer should be evaluated to calculate the magne-\ntoresistance from Eq. (1). The spin accumulations obey the diﬀusio n equation [5, 31],\n∂2δµF\n∂z2=δµF\nℓ2, (3)\nwhereℓis the spin diﬀusion length. The spin accumulation in the F 2layer is related to the\nspin current density in the F 2layer via [22]\nJsi=−/planckover2pi1σF\n2e2∂iδµF2−/planckover2pi1σAH\n2e2ǫiαkmα∂kδµF2\n−/planckover2pi1βσF\n2e2∂i¯µF2−/planckover2pi1ζσAH\n2e2ǫiαkmα∂k¯µF2,(4)\nwhere ¯µ= (¯µ↑+ ¯µ↓)/2, andǫijkis the Levi-Civita asymmetric tensor. The spin current\ndensity in the F 1layer is obtained in a similar way by neglecting terms related to σAH. The\nboundary conditions of Eq. (3) are given by the spin currents at th e boundaries. For the\nF2layer, the spin current is zero at the outer boundary, z= 0. On the other hand, we\ndenote the spin current at the F 2/N interface as JF2→N\ns. For simplicity, we assume that the\npenetration depth of the transverse spin current in the F 2layer is suﬃciently short [32–37],\nand therefore, only the component of JF2→N\nsparallel to m2survives inside the F 2layer.\n4Consequently, the solution of δµFin the F 2layer is given by\nδµF2=4πe2ℓ\n(1−β2)hσFsinh(d2/ℓ)/braceleftbigg/planckover2pi1(β−ζ)σAH\n2em2yEx/bracketleftbigg\ncosh/parenleftbiggz−d2\nℓ/parenrightbigg\n−cosh/parenleftBigz\nℓ/parenrightBig/bracketrightbigg\n−m2·JF2→N\nscosh/parenleftBigz\nℓ/parenrightBig/bracerightBig\n.(5)\nThe spin accumulation in the F 1layer is obtained in a similar way. The spin current at the\nF/N interface is given by the spin-dependent Landauer formula [38 ],\nJF→N\ns=1\n2πS/bracketleftbigg(1−γ2)g\n2m·(δµF−δµN)m−grm×(δµN×m)/bracketrightbigg\n, (6)\nwheregandγare the dimensionless interface conductance and its spin polarizatio n, respec-\ntively. The conductance gis related to the interface resistance rviar= (h/e2)S/g, whereS\nis the cross-section area. The real part of the mixing conductanc e is denoted as gr, whereas\nthat of the imaginary part is assumed to be zero, for simplicity [39]. Th e spin accumulation\nin the ferromagnet is δµF=δµFm, whereas δµNis the spin accumulation in the nonmagnet.\nSubstituting Eq. (5) into Eq. (6), the spin current at the F 2/N interface is rewritten as\nJF2→N\ns=−/planckover2pi1g∗(β−ζ)σAH\n2egFm2ytanh/parenleftbiggd2\n2ℓ/parenrightbigg\nExm2\n−1\n2πS[g∗(m2·δµN)m2+grm2×(δµN×m2)],(7)\nwheregFandg∗are deﬁned as\ngF\nS=h(1−β2)σF\n2e2ℓ, (8)\n1\ng∗=2\n(1−γ2)g+1\ngFtanh(d2/ℓ). (9)\nThe spin current at the F 1/N interface is obtained in a similar way.\nWe assume that the thickness of the nonmagnet is suﬃciently thinne r than its spin\ndiﬀusion length, as usually adopted in experiments [29]. Thus, the sp in current in the\nnonmagnet is conserved, i.e., JF1→N\ns+JF2→N\ns=0. Then, the spin accumulation in the\nnonmagnet is given by [22]\nδµN=−2π/planckover2pi1[1+λ(m1·m2)2]g∗(β−ζ)σAH\n2egF(gr+g∗)[1−λ2(m1·m2)2]tanh/parenleftbiggd2\n2ℓF/parenrightbigg\nm2yExSm2\n−2π/planckover2pi1λ(m1·m2)g∗(β−ζ)σAH\n2egF(gr+g∗)[1−λ2(m1·m2)2]tanh/parenleftbiggd2\n2ℓF/parenrightbigg\nm2yExSm2×(m1×m2).(10)\nHere, we deﬁne λ= (gr−g∗)/(gr+g∗). We note from Eq. (7) that\nm2·JF2→N\ns=−/planckover2pi1g∗(β−ζ)σAH\n2egFtanh/parenleftbiggd\n2ℓ/parenrightbigg\nEx−g∗\n2πm2·δµN. (11)\n5Substituting Eq. (11) together with Eq. (10) into Eq. (5), the solu tion ofδµF2is obtained.\nFrom Eq. (1), we deﬁne the averaged current density in the ydirection as\nJcy≡1\nd2/integraldisplayd2\n0dzJcy\n=σAH/bracketleftbigg\nm2z−/parenleftbiggσAH\nσF/parenrightbigg\nm2xm2y/bracketrightbigg\n+(β−ζ)σAH\ned2m2x[δµF2(z=d2)−δµF2(z= 0)].(12)\nWe also deﬁne the transverse resistivity as ρT=−(Jcy/Ex)/σ2\nF[40]. Using Eqs. (5), (10),\nand (11), we ﬁnd that the transverse resistivity is given by\nρT=−ρF/parenleftbiggσAH\nσF/parenrightbigg\nm2z+ρF/parenleftbiggσAH\nσF/parenrightbigg2\nm2xm2y+[∆ρ1+∆ρ2(m1,m2)]m2xm2y,(13)\nwhereρF= 1/σFis the resistivity. The ﬁrst two terms in Eq. (13) are the convention al\ntransverse resistivities resulting from the anomalous Hall eﬀect. O n the other hand, the last\ntwo terms originate from the charge-spin conversion by the anoma lous Hall eﬀect. The term\n∆ρ1is the resistivity found in our previous work [23, 24] given by\n∆ρ1\nρF=ℓ\n(1−β2)d2/bracketleftbigg(β−ζ)σAH\nσF/bracketrightbigg2/bracketleftbigg\n2−g∗\ngFtanh/parenleftbiggd2\n2ℓ/parenrightbigg/bracketrightbigg\ntanh/parenleftbiggd2\n2ℓ/parenrightbigg\n. (14)\nOn the other hand, ∆ ρ2in Eq. (13) is a new term found in this study. This resistivity\ndepends on the relative angle of the magnetizations in the F 1and F 2layers through the\ntermm1·m2, and is given by\n∆ρ2\nρF=−ℓ\n(1−β2)d2/bracketleftbigg(β−ζ)σAH\nσF/bracketrightbigg2g∗\ngF[1+λ(m1·m2)2]g∗\n(gr+g∗)[1−λ2(m1·m2)2]tanh2/parenleftbiggd2\n2ℓ/parenrightbigg\nm2xm2y.\n(15)\nThe resistivities ∆ ρ1and ∆ρ2are proportional to the square of the spin anomalous Hall\nangledeﬁnedbyEq. (2). Theterm∆ ρ1m2xm2yinEq. (13)hasthesameangulardependence\nas the planar Hall eﬀect. On the other hand, the term ∆ ρ2(m1,m2)m2xm2ydepends not\nonly onm2xm2ybut also on ( m1·m2)2. Figures 1(b) and 1(c) show examples of ∆ ρ2/ρF\nas a function of the rotation angle ϕof the magnetization m2= (cosϕ,sinϕ,0) in the F 2\nlayer, where the magnetization in the other (F 1) layer points to the direction (b) m1=ex\nand (c)m1= (ex+ey)/√\n2. The angular dependence of the planar Hall eﬀect ( ∝m2xm2y=\nsinϕcosϕ) is also shown by the dotted line as for guide. We referred the values of the\nparameters from typical experiments and calculations; ℓ= 1.0 nm,ρF= 300 Ωnm, β= 0.90,\nr= 0.25 kΩnm2,γ= 0.50,σAH/σF= 0.015,ζ= 1.50, andd1=d2= 2.0 nm [19, 39, 41].\n6We assume that the parameters are the same between two ferrom agnets, except σAHwhich\nis zero in the F 1layer, for simplicity. It is clearly shown that the resistivity ∆ ρ2m2xm2y\nhas a slightly diﬀerent angular dependence from the planar Hall eﬀec t, and depends on the\ndirection of the magnetization m1in the F 1layer. Therefore, the resistivity ∆ ρ2can be\ndistinguished from the planar Hall eﬀect when a material having a larg e spin anomalous\nHall angle is found. We also note that the spin Nernst eﬀect [42–44] in nonmagnets provides\nphysical phenomena similar to that originating from the spin Hall eﬀec t. Therefore, we\nexpect that the magnetoresistance calculated in this paper will be d iscovered not only by\nthe anomalous Hall eﬀect but also by the anomalous Nernst eﬀect, w hich was recently shown\nto contribute to the charge-spin conversion in ferromagnets [45 ].\nIV. CONCLUSION\nIn conclusion, a theoretical framework on the transverse magne toresistance in a ferro-\nmagnetic/nonmagnetic/ferromagnetic trilayer originating from ch arge-spin conversion by\nanomalous Hall eﬀect was developed. 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Lau, S. Takahashi, S. Mitan i, and M. Hayashi, Sci. Adv. 3,\ne1701503 (2017).\n[44] A. Bose, S. Bhuktare, H. Singh, V. G. Achanta, and A. Tula purkar, ”Observation of Spin\nNernst eﬀect in Platinum”, arXiv:1704.06788.\n[45] T. Taniguchi, J. Phys. Soc. Jpn. 85, 074705 (2016).\n9F1\nF2Nd1\nd2\nxyzm1\nm2V\nExAHE\nΔρ2/ρF (10 -4 )0.2\n0.1\n0\n-0.1\n-0.2\nΔρ2/ρF (10 -4 )0.2\n0.1\n0\n-0.1\n-0.2\n-0.3(b) (a) (c)\nrotation angle of m2, φ (deg)0 90 180 270 360\nrotation angle of m2, φ (deg)0 90 180 270 360m1=ex m1=( ex+ey)/√2\n0\nFIG. 1: (a) Schematic view of the system under consideration . An external electric ﬁeld Exis\napplied along the xdirection. The anomalous Hall eﬀect (AHE) in the F 2layer injects spin current\ninto the F 1layer. Transverse magnetoresistance is derived from the vo ltage generated along the\nydirection. (b), (c) Dependences of Eq. (15) on the rotation a ngleϕof the magnetization\nm2= (cosϕ,sinϕ,0) in the F 2layer for (b) m1=exand (c)m1= (ex+ey)/√\n2. The function\n−sinϕcosϕis also shown by the dotted line.\n10"    },    {        "title": "2306.05040v1.Highly_spin_polarized_carriers_and_strong_ferromagnetism_in_doped_perovskite_antiferromagnetic_semiconductors.pdf",        "content": "Highly spin-polarized carriers and strong ferromagnetism\nin doped perovskite antiferromagnetic semiconductors\nHong Jian Zhao,1, 2, 3, 4Longju Yu,1Yanchao Wang,1, 3,∗Laurent Bellaiche,5and Yanming Ma1, 4, 3, †\n1Key Laboratory of Material Simulation Methods and Software of Ministry of Education,\nCollege of Physics, Jilin University, Changchun 130012, China\n2Key Laboratory of Physics and Technology for Advanced Batteries (Ministry of Education),\nCollege of Physics, Jilin University, Changchun 130012, China\n3State Key Laboratory of Superhard Materials, College of Physics, Jilin University, Changchun 130012, China\n4International Center of Future Science, Jilin University, Changchun 130012, China\n5Physics Department and Institute for Nanoscience and Engineering,\nUniversity of Arkansas, Fayetteville, Arkansas 72701, USA\n(Dated: June 9, 2023)\nIn semiconductor spintronics, the generation of highly spin-polarized carriers and the efficient\nprobe of spin order (due to strong ferromagnetism) – at or above room temperature – are cru-\ncial because it allows for the design of spin-based semiconductor devices. Usually, such goals were\nfulfilled in room-temperature ferromagnetic semiconductors, being rare materials in nature. While\nroom-temperature antiferromagnetic semiconductors are plentiful, the possibility for creating highly\nspin-polarized carriers and strong ferromagnetism in these materials remain to be unraveled. Here,\nwe explore such a possibility by first-principles simulations, working with CaTcO 3and NaOsO 3per-\novskites – being room-temperature antiferromagnetic semiconductors. We find that doping them\nby electrons or holes results in these materials to be highly spin-polarized, carrying enormous ferro-\nmagnetic moments. Doping electrons with moderate carrier density can yield strong ferromagnetism\nin them, with the ferromagnetic moments being comparable to that in typical ferromagnetic semi-\nconductors. Our work thus indicates the merit of perovskite antiferromagnetic semiconductors in\nspintronics – for a possible replacement of ferromagnetic semiconductors.\nIntroduction.– Ferromagnetic semiconductors are of vital\nsignificance in semiconductor spintronics [1–4]. Mate-\nrials of this kind present giant Zeeman-type spin split-\ntings and strong ferromagnetism, where the former al-\nlow for the generation of highly spin-polarized carriers\nand the latter permits the efficient probe of spin or-\nder [1, 2, 5]. One shortcoming of ferromagnetic semi-\nconductors is that the Curie temperatures of these mate-\nrials are usually far lower than room temperature [1, 6],\nmaking the ferromagnetic semiconductors lose their prac-\ntical applications. In this regard, ferrimagnetic semi-\nconductors and related materials – such as asymmetric\nand bipolar antiferromagnetic semiconductors [7], being\nstill rare at room temperature – may come to the res-\ncue, serving as the “weak version” of ferromagnets (see,\ne.g., Refs. [4, 6, 8–10]). On the other hand, there are a\nvariety of perovskite semiconductors being antiferromag-\nnetic with N´ eel temperatures being above room temper-\nature [11, 12]. Some of these perovskites ( e.g., YFeO 3\nand CaTcO 3) can host Zeeman-type spin splittings and\nweak ferromagnetism [13, 14]. Typically, the Zeeman-\ntype spin splitting can reach ∼78 meV in CaTcO 3per-\novskite [14], being sizable – albeit a tiny value compared\nwith∼2 eV in YTiO 3[15]; The magnitudes of mag-\nnetic moments in antiferromagnets are much smaller than\nthose in ferromagnets ( e.g.,∼0.06µB/f.u. in antiferro-\nmagnetic YFeO 3[13]versus ∼0.8µB/f.u. in ferromag-\nnetic YTiO 3[16, 17]). At all events, this suggests a route\nto utilize perovskite antiferromagnetic semiconductors as\nalternatives for ferromagnetic semiconductors in spin-tronics, if the following questions are positively answered:\n(i) Is it possible to generate highly spin-polarized carri-\ners in antiferromagnetic semiconductors? (ii) Is there an\navenue to realize strong ferromagnetism in antiferromag-\nnetic semiconductors?\nIn this Letter, we address the aforementioned ques-\ntions by first-principles simulations. We focus on per-\novskite CaTcO 3and NaOsO 3, whose N´ eel temperatures\nare∼800 K [18] and ∼410 K [19–22], respectively. Be-\nlow N´ eel temperatures, these materials are all antiferro-\nmagnetic semiconductors [19–24]. We show that doping\nCaTcO 3and NaOsO 3by electrons or holes leads them\nto be highly spin-polarized, due to the sizable Zeeman-\ntype spin splittings of energy levels around their valence\nband maximum (VBM) and conduction band minimum\n(CBM). The doped carriers, carrying enormous ferromag-\nnetic moments, significantly enhance the ferromagnetism\nin antiferromagnetic CaTcO 3and NaOsO 3. In partic-\nular, doping electrons with the carrier density of 0.125\ne/f.u. (ebeing the charge of electron) gives rise to ferro-\nmagnetic moments of ∼0.3µB/f.u. in CaTcO 3and∼0.2\nµB/f.u. in NaOsO 3, an order of magnitude being com-\nparable to those in perovskite ferromagnetic semiconduc-\ntors ( e.g.,∼0.8µB/f.u. in YTiO 3[16, 17]).\nSpin splittings versus spin-polarized carriers.– To be-\ngin with, we revisit the possibilities for creating spin-\npolarized carriers in semiconductors by doping. In cen-\ntrosymmetric non-magnetic semiconductors, the elec-\ntron’s spin energy levels associated with + Sαand−Sα\n(α=x, y, z ), the spin magnetization of the electron,arXiv:2306.05040v1  [cond-mat.mtrl-sci]  8 Jun 20232\nare degenerate everywhere in kspace ( kbeing the wave\nvector). Doping such semiconductors is not expected\nto yield spin-polarized carriers because the carriers oc-\ncupy + Sαand−Sαstates in a symmetrical manner [see\nFig. 1(a)]. As for the non-centrosymmetric non-magnetic\nsemiconductors, a k-dependent effective magnetic field\nBeff(k) is emerged because of the spin-orbit coupling\n(SOC) [25, 26]. Such a field couples with electron’s spin\nviaBeff(k)·σ[25] and yields, e.g., Rashba-/Dresselhaus-\ntype spin splittings [27, 28], where σ≡(σx, σy, σz) is\nthe vector formed by Pauli matrix σα(α=x, y, z ).\nIn other words, the + Sαand−Sαlevels become non-\ndegenerate at some non-zero k[see Fig. 1(b)]. Nonethe-\nless, the lack of magnetism indicates the odd-function\nfeature of Beff(k), namely, Beff(−k) =−Beff(k) [25];\nAs a consequence, each ±Sαstate at any wave vector\nkcan find its degenerate partner ( i.e.,∓Sαstate) at\nthe corresponding −k. In such sense, the doped carri-\ners are not spin-polarized either in non-centrosymmetric\nnon-magnetic semiconductors [see Fig. 1(b)].\nFIG. 1. Schematizations of the band structures ( i.e.,\nband energies Ekversus the wave vector k) in electron-doped\nsemiconductors with null spin splitting [panel (a)], Rashba-\n/Dresselhaus-type spin splitting [panel (b)], tiny Zeeman-type\nspin splitting [panel (c)], and sizable Zeeman-type spin split-\nting [panel (d)]. The split energy levels with positive and\nnegative spin magnetization values are sketched in red and\nblue curves (and marked by red and blue arrows), respec-\ntively. The horizontal dash line in each panel denotes the\nFermi energy level Ef.\nIn ferromagnetic or ferrimagnetic materials, there ex-\nists an effective magnetic field Beff\nαbeing independent of\nk. The interaction between Beff\nαfield and electron’s spin,\nindicated by Beff\nασα, causes Zeeman-type spin splittings\nin these materials [2, 29]. If so, the degeneracy between\n+Sαand−Sαstate ( αbeing a particular direction deter-\nmined by Beff\nα) can be broken by the Beff\nαeffective mag-\nnetic field [see Fig. 1(d)]. Recent studies indicate that\nsome antiferromagnetic semiconductors showcase weak\nferromagnetism together with Zeeman-type spin split-\ntings (see, e.g., Ref. [14, 30, 31]). In such sense, thedoped carriers in antiferromagnetic semiconductors oc-\ncupy the + Sαand−Sαstates asymmetrically and thus\ncan be spin-polarized [32, 33]. Antiferromagnetic semi-\nconductors with tiny Zeeman-type spin splittings around\nthe CBM (VBM) [34] gain doped electrons (holes) that\nare slightly spin-polarized [Fig. 1(c)], while semiconduc-\ntors with sizable Zeeman-type splitting splittings yield\ncarriers being significantly (even 100%) spin-polarized\n[Fig. 1(d)]. Provided that the Zeeman-type spin split-\ntings are large enough around the VBM or CBM, the\ndoped carriers may carry sizable ferromagnetic moment\n– reinforcing the weak ferromagnetism in antiferromag-\nnetic semiconductors.\nFIG. 2. Schematizations of the ( Gx, Fz) and ( Gz, Fx) mag-\nnetic configurations in Pbnm ABX 3perovskite. In panels (a)\nand (b), the predominant G-ype and the weak F-type vectors\n(carried by Bions – represented by purple spheres) are de-\nnoted by cyan and red arrows, respectively. For displaying\nclarify, the AandXions are not shown in the schematiza-\ntions.\nDoped perovskite antiferromagnets: the spin-polarized\ncarriers.– In the following, we will demonstrate that our\naforementioned scenario can be realized in antiferromag-\nnetic ABX 3perovskites. We recall that various ABX 3\nperovskites ( e.g., rare-earth orthoferrites [11, 35], rare-\nearth orthochromates [11, 36], SrTcO 3[37], CaTcO 3[18]\nand NaOsO 3[19, 20]) adopt the Pbnm crystallographic\nspace group with the predominant G-type antiferromag-\nnetic vectors carried by Bions. Symmetry analyses and\nexperiments [11, 35, 36, 38, 39] indicate that two mag-\nnetic configurations – termed as ( Gx, Fz) and ( Gz, Fx)\nmagnetic configurations and sketched in Fig. 2 – can\npresent weak ferromagnetism (due to spin canting) in\nPbnm perovskite with G-type antiferromagnetism. To be\nspecific, aligning the predominant G-type vectors along\nxdirection (denoted by Gx) causes weak ferromagnetic\nvectors along z(i.e., being F-type and denoted by Fz).\nSimilarly, the predominant Gzantiferromagnetic vectors\nyield the weak ferromagnetic Fxvectors. Theories sug-\ngest that ( Gx, Fz) and ( Gz, Fx) magnetic configurations3\naccommodate Zeeman-type spin splittings that are mi-\ncroscopically rooted in SOC, where the magnitudes of\nZeeman-type spin splittings depend on the strength of\nSOC [14]. The split spin energy levels are associated\nwith spin magnetization ±Szfor (Gx, Fz) configuration\nand±Sxfor (Gz, Fx) configuration.\nFIG. 3. The band structures of CaTcO 3with the ( Gx, Fz)\nmagnetic configuration. The zero energy is set as the Fermi\nlevel Ef. Panels (a) and (b) correspond to electron ( nfree=\n+0.125e/f.u. ) and hole ( nfree =−0.125e/f.u. ) dopings,\nrespectively. The color bar denotes the spin magnetization\n±Sz(positive value for + Szand negative value for −Sz). In\nthe horizontal axis, the k-points in the high-symmetry path\nare represented by the reduced coordinates, with respect to\nthe reciprocal lattice vectors.\nNext, we select NaOsO 3, CaTcO 3and YFeO 3antifer-\nromagnetic perovskites ( G-type) as our platforms, where\nNaOsO 3exhibits strong SOC, CaTcO 3moderate SOC,\nand YFeO 3weak SOC. We calculate the band struc-\ntures of these materials with respect to ( Gx, Fz) and\n(Gz, Fx) magnetic configurations, upon carrier doping\n(carrier density nfree being ±0.125e/f.u. ) – the pos-\nitive and negative nfree values corresponding to elec-\ntron and hole dopings, respectively. The band struc-\ntures of CaTcO 3with ( Gx, Fz) magnetic configuration\nare shown in Fig. 3. Sizable Zeeman-type spin split-\ntings occur around the CBM (respectively, VBM) of\nCaTcO 3doped with electrons (respectively, holes). Upon\nelectron doping, the extra electrons predominantly oc-\ncupy the −Szstates with tiny concentration occupy-\ning + Szstates [Fig. 3(a)]. As a consequence, the −Sz\nspin-polarized electrons are injected into the material.\nThe hole doping in CaTcO 3[with ( Gx, Fz) configura-\ntion], similarly, creates holes that mostly occupy the −Sz\nstates, yielding −Szspin-polarized holes. Moving to the\n(Gz, Fx) configuration, the situations regarding Zeeman-type spin splittings are similar to those in ( Gx, Fz) config-\nuration, as shown in Fig. S1 of the Supplementary Mate-\nrial (SM) [40] (Note that our SM contains Refs. [41–60]).\nIn the ( Gz, Fx) configuration of CaTcO 3, the doped elec-\ntrons or holes prefer occupying the + Sxstates rather\nthan the −Sxstates.\nOur above discussion is basically valid for NaOsO 3\nwith ( Gx, Fz) and ( Gz, Fx) magnetic configurations (see\nFig. S2 of the SM). By saying “ basically ” we mean that (i)\nthe carriers in NaOsO 3predominantly occupy the ±Sα\nlevel [ e.g.,α=zin (Gx, Fz) configuration or α=x\nin (Gz, Fx)] – resembling the cases in CaTcO 3, and (ii) a\nsmall amount of carriers may also occupy ∓Sαlevel – the\npartner of the ±Sαlevel. In such sense, the carriers in\nNaOsO 3are partially spin-polarized. As for YFeO 3, the\nSOC is negligible and the Zeeman-type spin splittings are\ntiny. Figure S3 in the SM indicates that the spin mag-\nnetization |Sα|in the vicinity of the doping level [ α=z\nfor (Gx, Fz) and α=xfor (Gz, Fx)] is almost vanishing.\nFurthermore, the doped carriers occupy + Sαand−Sα\nstates in a nearly symmetrical manner (see Fig. S4 of the\nSM). Hence, the spin polarization of the doped carriers\nin YFeO 3is rather insignificant.\nDoped perovskite antiferromagnets: the enhanced\nferromagnetism.– In the undoped Pbnm NaOsO 3,\nCaTcO 3and YFeO 3, the predominant G-type antiferro-\nmagnetic vectors tend to align along the xaxis (com-\npared with yandzaxes), as predicted by our first-\nprinciples calculations. This yields the ( Gx, Fz) mag-\nnetic configuration in these materials and coincides with\nthe experiments (see e.g., Refs. [13, 18, 20] and Sec-\ntion I of the SM). Furthermore, the calculated magnetic\nmoments ( Mz, Mx) for NaOsO 3, CaTcO 3and YFeO 3\nare (0 .016,−0.003), (0 .057,−0.059) and (0 .066,−0.063)\nµB/f.u. , respectively, with (i) Mzbeing associated with\nFzin the ( Gx, Fz) configuration, and (ii) Mxbeing asso-\nciated with Fxin the ( Gz, Fx) configuration [61].\nWe now explore the effect of carrier doping on ferro-\nmagnetism in these perovskite antiferromagnets. First\nof all, our first-principles calculations, at the level of\ncollinear magnetism, show that the G-type magnetic\nphase in NaOsO 3, CaTcO 3and YFeO 3are more stable\nupon doping with nfree=±0.125e/f.u. , compared with\ntheA-,C-, and F-type magnetic phases (see Section III\nof the SM). That NaOsO 3becomes a G-type antiferro-\nmagnetic metal upon electron doping is basically consis-\ntent with the results shown in Refs. [62, 63]. In the fol-\nlowing, our discussion will be limited within the ( Gx, Fz)\nand ( Gz, Fx) magnetic configurations.\nFigure 4(a) shows the Mz– in the ( Gx, Fz) magnetic\nconfiguration – of CaTcO 3, NaOsO 3and YFeO 3as a\nfunction of nfree. In CaTcO 3, doping electrons (holes)\nreinforces its ferromagnetic component along the −zdi-\nrection (+ zdirection). This can be interpreted in the\nfollowing way. As shown in Fig. 3(a), the doped elec-\ntrons are −Szpolarized (see above) which carries the4\n/uni00000010/uni00000013/uni00000011/uni00000017/uni00000010/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Mz/uni00000003/uni0000000bB/f.u./uni0000000c\n/uni0000000b/uni00000044/uni0000000c\n/uni00000031/uni00000044/uni00000032/uni00000056/uni000000323/uni00000026/uni00000044/uni00000037/uni00000046/uni000000323/uni0000003c/uni00000029/uni00000048/uni000000323\n/uni00000010/uni00000013/uni00000011/uni00000014/uni00000013 /uni00000010/uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013\nnfree/uni00000003/uni00000003/uni0000000b/uni00000048/uni00000012/uni00000049/uni00000011/uni00000058/uni00000011/uni0000000c/uni00000010/uni00000013/uni00000011/uni00000017/uni00000010/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Mx/uni00000003/uni0000000bB/f.u./uni0000000c\n/uni0000000b/uni00000045/uni0000000c\nFIG. 4. The magnetic moments of CaTcO 3, NaOsO 3and\nYFeO 3as a function of carrier density nfree. Panels (a) and\n(b) correspond to the ( Gx, Fz) and ( Gz, Fx) magnetic config-\nurations, respectively.\nferromagnetic moments along −zdirection. As for hole\ndoping, the holes are −Szpolarized as well; Yet, the hole\ndoping is equivalent to removing electrons from the −Sx\nstates of CaTcO 3, creating the ferromagnetic moments\nalong + zdirection. Regarding the ( Gz, Fx) configura-\ntion of CaTcO 3, the doping effect on ferromagnetism is\nreadily understood by examining Fig. 4(b) and Fig. S1\nof the SM. Furthermore, the trends for the magnetic mo-\nments of NaOsO 3(versus nfree) follow those of CaTcO 3.\nThe physical interpretation is akin to the case of CaTcO 3\n(see Fig. S2 of the SM). Unlike CaTcO 3and NaOsO 3, the\nmagnetic moments of YFeO 3remain inert upon carrier\ndoping for both ( Gx, Fz) and ( Gz, Fx) magnetic config-\nurations (see Fig. 4). This results from the insignificant\nspin polarization of the doped carriers in YFeO 3, and\ncoincides with our discussion in the Section “ Doped per-\novskite antiferromagnets: the spin-polarized carriers ”.\nTo finish this section, we emphasize that carrier dop-\ning can significantly enhance the ferromagnetism in anti-\nferromagnetic CaTcO 3and NaOsO 3. Strikingly, doping\nCaTcO 3with electrons ( nfree = +0 .125e/f.u. ) yields\nferromagnetic moment of the order of ∼0.3µB/f.u. in\nboth ( Gx, Fz) and ( Gz, Fx) magnetic configurations. As\nfor NaOsO 3, doping electrons with nfree= +0 .125e/f.u.results in Mxof∼0.2µB/f.u. in the ( Gz, Fx) config-\nuration. Such ferromagnetic moments are comparable\nto those in the typical ferromagnetic perovskites ( e.g.,\n∼0.8µB/f.u. in YTiO 3[16, 17]). For CaTcO 3and\nNaOsO 3, the electron doping level nfree= 0.125e/f.u.\ncorresponds to the volume concentrations of ∼2×1021\ncm3, a concentration that could be accessible by electro-\nstatic doping (see, e.g., Refs. [64, 65]). Furthermore, we\nare aware of an avenue to realizing the electron doping\nin NaOsO 3(i.e., via the replacement of Na with Mg), as\nproposed by Ref. [66] – focusing on the insulator-to-metal\ntransition in NaOsO 3driven by doping.\nSummary and perspective.– By first-principles simula-\ntions, we have shown that CaTcO 3and NaOsO 3anti-\nferromagnetic perovskites enable the creation of highly\nspin-polarized carriers by either electron doping or hole\ndoping, thanks to the sizable Zeeman-type spin splittings\nof the electrons’ energy levels in the vicinity of CBM or\nVBM. The doped electrons or holes carry enormous fer-\nromagnetic moments and significantly enhance the ferro-\nmagnetism in antiferromagnetic CaTcO 3and NaOsO 3,\ncompared to their undoped neutral cases. Promisingly,\nthe N´ eel temperatures of CaTcO 3and NaOsO 3are all\nabove the room temperature. This implies a route to (i)\ninject highly spin-polarized carriers (and generate spin\ncurrent), and (ii) create sizable ferromagnetic moments\nin doped perovskite antiferromagnetic semiconductors, at\nroom temperature and in the absence of magnetic field.\nAs a perspective, it is possible to foreseen the importance\nof room-temperature perovskite antiferromagnetic semi-\nconductors – exhibiting sizable Zeeman-type spin split-\ntings around their VBM and CBM – in spintronics. We\nhope that our work can motivate the utilization of such\nantiferromagnetic semiconductors in the design and fab-\nrication of room-temperature spintronic devices.\nAcknowledgements.– We acknowledge the support from\nthe National Natural Science Foundation of China\n(Grants Nos. 12274174, T2225013, 52288102 and\n12034009). L.B. thanks the Office of Naval Research\n(ONR) under Grant No. N00014-17-1-2818 and the\nVannevar Bush Faculty Fellowship (VBFF) Grant No.\nN00014-21-1-2086 from the Department of Defense.\nL.J.Y. acknowledges the support from the International\nCenter of Future Science, Jilin University. The calcula-\ntion was performed in the high-performance computing\ncenter of Jilin University. The authors thank Prof. P.\nLiu at Chinese Academy of Sciences (Institute of Metal\nResearch) for the valuable discussion regarding the\nHubbard Ucorrection in NaOsO 3.\n∗wyc@calypso.cn\n†mym@jlu.edu.cn5\n[1] S. Wolf, D. Awschalom, R. Buhrman, J. Daughton,\nv. S. von Moln´ ar, M. 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The result was obtained by derivation of the sp in waves spectrum by means of kinetic\nequation.\nPACS numbers: 03.75.Ss, 05.30.Fk, 71.10.Ay, 71.10.Ca\nIntroduction.\nFerromagnetism in a system of itinerant fermions is\ntraditionally explained in terms of physical representa-\ntions ﬁrst introduced by Stoner [1]. It arises due to the\nshort range repulsion between the particles with the op-\nposite spins favoringthe ﬁnite magnetic polarizationhin-\ndered by the kinetic energy grown caused by the Pauli\nexclusion principle. This physical idea has found more\nexact formulation in frame of the Fermi gas model [2].\nAfter appearance of the Landau Fermi liquid theory [3]\nthere was developed the corresponding description of fer-\nromagnetic Fermi liquid [4, 5]. The fast development of\ncold gases physics has animated the interest to the prob-\nlem of itinerant ferromagnetism [6]. The Stoner model\nhas started to be the subject of more careful investiga-\ntions.\nStill, theoretically there is the question: can a gas of\nspin-upandspin-downfermionswithpurerepulsiveshort\nrange interaction be ferromagnetic? On the one hand\nthere is rigorousstatement [7] based on the Tan relations\nthat in both one and three dimensions, a Stoner instabil-\nityto asaturatedferromagnetforrepulsivefermionswith\nzero rangeinteractions is ruled out at any ﬁnite coupling.\nHowever, it rules out only saturated ferromagnetism for\nthehomogeneous gas but not partially polarized states.\nThe condition of the zero range interaction seems to be\nalso quite restrictive.\nOn the other hand the problem of transition to the fer-\nromagnetic state in a two-component repulsive Fermi gas\nhasbeen addressedin manystudies (see[2, 8–16]. In par-\nticular, there were established that in the ﬁrst order per-\nturbation theory [2] the ferromagnetic phase transition\nis of the second order and occurs at kFa=π/2 whereas\nthe second order perturbation theory [11] predicts the\nﬁrst order phase transition at kFa= 1.054. Finally non-\nperturbativestudy[16] returnsusbacktothe phasetran-\nsition of the second order at kFa= 0.858. These treat-\nments were also performed under the assumption that\nonly phases with homogeneous magnetizationcan be pro-\nduced. So, the properties of the Stoner-Hubbard model\nstill are under active theoretical investigation.\nLeaving apart the problem of a Fermi liquid phase\ntransition to the ferromagnetic state here we discuss the\nstability of this state in respect to the transversal inho-mogenious magnetization perturbations. The transverse\nspin wave dispersion in a ferromagnetic Fermi liquid at\nT= 0 in the assumption of absence of the quasiparticle\nexcitations attenuation has been obtained by Abrikosov\nand Dzyaloshinskii [4]. There was calculated the spin\nwaves spectrum ω=D′′k2and the reactive part of dif-\nfusion constant D′′was found proportional to the spon-\ntaneous magnetization. The theory has been criticized\nby C.Herring [17] who has remarked, that in a ferromag-\nnetic Fermi liquid the quasiparticle states with, say, spin\nup will no longer be closed to the Fermi surface of quasi-\nparticles with the opposite spins. So, they ”will have a\nﬁnite, rather than an inﬁnitesimal, decay rate”. In its\nturn, the spin wave dispersion has to require an imagi-\nnary part: D′′→ −i(D′+iD′′). Later the reactive part\nof spin wave spectrum has been calculated by Moriya\n[18] in frame of RPA approach applied to the ferromag-\nnetic Fermi gas with Hubbard type interaction. Again\nthe spin wave attenuation was not found. The problem\nwas addressed in the paper by the author[19].\nKinetic equation approach. The dispersion law\nω=ωL+(D′′−iD′)k2(1)\nfortransversalspinwaveswasfoundforFermiliquidboth\nin paramagnetic and ferromagnetic states. Here ωL=\nγH0is the Larmor frequency, H0is the external ﬁeld,\nγis the gyromagnetic ratio. To escape to look after γ\nand|γ|we treat γas a positive constant. The dispersion\nlaw is derived making use the Silin kinetic equation [20]\nunder conditions that the dispersive part is much smaller\nthan the scattering rate, the latter in its turn is much\nsmaller than the liquid polarization γHassumed small\nin comparison with the Fermi energy [19]\n|D|k2<<1\nτ<< γH. (2)\nThe polarization in the paramagnetic state can be cre-\nated either by the external ﬁeld or by the pumping. In\nthe former case it is determined by the external ﬁeld and\nthe Landau molecular ﬁeld parameter Fa\n0>−1\nγH=γH0\n1+Fa\n0, (3)2\nwhereas in the ferromagnetic state, when Fa\n0<−1, it\nhas the ﬁnite value\nγH= 4√\n6εF/radicalBigg\n1+Fa\n0\nFa\n0(4)\neven in the external ﬁeld or the pumping absence. The\ncorresponding magnetic moment density is given by\nM=γ2N0\n4H. (5)\nHereN0is the quasiparticles density of states at the\nFermi surface. We put /planckover2pi1= 1 throughout the paper.\nThe quasiparticles scattering rate in the polarized\nFermi liquid is [21]\n1\nτ=A((2πT)2+(γH)2), (6)\nHere,A=const∝m∗3W,m∗is the eﬀective mass and\nWis square modulus of the matrix element of the oppo-\nsite spin quasiparticles short range potential of interac-\ntion. The scattering rate has a ﬁnite value1\nτ∝(γH)2\neven atT= 0. This statement for spin polarized para-\nmagnetic Fermi liquid has been checked and conﬁrmed\nexperimentally [22].\nUnder fulﬁllment conditions (2) the reactive D′′and\nthedissipative D′partsofthediﬀusionconstantaregiven\nby the following expressions [19] obtained taking into ac-\ncount the amplitude Fa\n1of the ﬁrst angular harmonic of\nthe quasiparticles exchange interaction.\nTABLE\nPolarized paramagnetic Ferromagnetic at H0= 0\nD′′=v2\nF(1+Fa\n0)/parenleftBig\n1+Fa\n1\n3/parenrightBig\n3/parenleftBig\nFa\n0−Fa\n1\n3/parenrightBig\nγHD′′=v2\nFFa\n0/parenleftBig\n1+Fa\n1\n3/parenrightBig\nγH\n3(4εF)2/parenleftBig\nFa\n0−Fa\n1\n3/parenrightBig>0\n|D′′| ∝1\nHD′′∝H\nD′=v2\nF(1+Fa\no)/parenleftBig\n1+Fa\n1\n3/parenrightBig\n3/parenleftBig\nFa\n0−Fa\n1\n3/parenrightBig2\n(γH)2τ>0D′=−v2\nF(Fa\n0)2/parenleftBig\n1+Fa\n1\n3/parenrightBig\n3(4εF)2/parenleftBig\nFa\n0−Fa\n1\n3/parenrightBig2\nτ<0\nD′∝const |D′| ∝H2\nThe reactive part of diﬀusion coeﬃcient D′′in ferro-\nmagnetic Fermi liquid derived from kinetic equation [19]\nliterally coincides with result of Moriya [18] obtained in\nframe of RPA at Fa\n1= 0 applied to the Hubbard model.\nOne can provealso the exact correspondenceof these two\napproaches for the D′′in polarized paramagnetic Fermi\nliquid.\nSign of coeﬃcient D′′in polarized paramagnetic state\ncan be positive or negative depending on sign of the dif-\nferenceFa\n0−Fa\n1\n3. In ferromagnetic state where Fa\n0<−1,\nandFa\n0−Fa\n1\n3<0 the sign of coeﬃcient D′′is always\npositive.The dissipative parts D′of the diﬀusion coeﬃcients in\nthe polarized paramagneticFermi liquid and in the ferro-\nmagnetic Fermi liquid have the opposite signs. As result\nthe amplitude of transversal inhomogeneous deviations\nof magnetization\nδM∝exp(ikx−iωt) = exp(ikx−iD′′k2t−D′k2t) (7)\nattenuates in a polarized paramagnetic Fermi liquid but\nit grows up in a ferromagnetic Fermi liquid demonstrat-\ning instability of the Stoner state in respect of such de-\nviations.\nField theory approach. The calculation of reactive\npart in ferromagnetic Fermi liquid was performed by\nDzyaloshinskiiandKondratenko[5]. Theyhavenotethat\nthe amplitude of forward scattering [23] for the quasipar-\nticles with the opposite spins is proportional to the static\ntransverse susceptibility which is divergent in a isotropic\nferromagnet. In application to the spin polarized param-\nagneticFermiliquidonemustusethedivergenceoftrans-\nverse susceptibility at frequency equal to the Larmor fre-\nquency. Following this schema and taking into account\nthe imaginary self-energy of quasiparticles [19] one can\nobtain the dispersion laws for ferromagnetic Fermi liquid\nω≈/bracketleftbig\nγH−ic(γH)2/bracketrightbigk2\np2\nF, (8)\nand for paramagnetic Fermi liquid\nω−ωL≈/bracketleftbig\n−(γH)−1+ic/bracketrightbig\nv2\nFk2. (9)\nHere,cis positive constant. For dilute Fermi systems\nc∝m∗σ(in dimensional units c∝m∗σ//planckover2pi12), where σ\nis the cross-section for the s-wave scattering of particles\nwith opposite spins.\nWesee: bothinparamagneticandferromagneticFermi\nliquid cases the polarization dependence and the sign of\nthe reactive part of spin diﬀusion coeﬃcient D′′derived\nby the ﬁeld theoretical method are in correspondence\nwith kinetic equation results. As for the dissipative part\nD′a correspondence is absent. In contrast to the kinetic\nequation approach here we have : D′>0 for ferromag-\nnetic Fermi liquid and D′<0 for the polarized param-\nagnetic one. It is diﬃcult to consider the latter result as\nsatisfactory because phenomenological kinetic theory are\nconﬁrmed by the microscopic derivation of kinetic equa-\ntion for the paramagnetic Fermi gas with repulsion by\nmeans of the Keldysh technique [24]. In our opinion it\nmeans that the method developed in the paper [5] is ap-\npropriateforthecalculationofreactivepartofthansverse\nspin wave dispersion but it is inapplicable to calculation\nof spin wave dissipation.3\nConclusion\nThe problem of existence of itinerant ferromagnetism\nin system of Fermi particles with repulsive interaction\nwas recently addressed in the experiments with ultra-\ncold two component Fermi gas (6Li atoms in the lowest\ntwo hyperﬁne states [6]). By the increasing of magnetic\nﬁeld toward the Feshbach resonance [25] one can reach\nthe pairing instability that is formation of stable gaseous\nphase consisting of two opposite spin atom molecules.\nHowever, the molecule production is slow, as it requires\nthree-body process. So, one can expect that fast enough\nﬁeld magniﬁcation prepares the system state character-\nized by large ( comparable to inter-particle distance) op-\nposite spin atoms scattering length a >0 formally corre-\nsponding to the strong short range repulsion that leads\nto the formation offerromagneticstate. The latter is cer-\ntainly metastable but one can hope that at fast nonadia-\nbatic increase of scattering amplitude the ferromagnetic\nstate will be formed faster than more stable gas of Li 2\nmolecules.\nThe observable experimental signatures of cloud of\nfermions in the harmonic optical trap as a function of\ninteraction has been calculated [26]. They are charac-\nterized by the maximum of cloud size and minimum of\nkinetic energy at phase transition to the ferromagnet\nstate. The corresponding nonmonotoneous dependences\nhas been measured [6] but no magnetization grown has\nbeen resolved. Later there was demonstrated, that the\npairing instability is always stronger than the ferromag-\nnetic one [27]. Moreover, there was found theoretically\n[28]andthenconﬁrmedexperimentally[29]thatthemax-\nimum atom loss at a magnetic ﬁeld below Feshbach res-\nonance is the result of processes of atomic dimers forma-\ntion and relaxation. The authors of [29] conclude that\nunlike to the previous observations [6] ” ...the sample\nremains in the paramagnetic phase for a wide range in\nthe strength of interactions and wait time.”\nSo, the cold atomic gases do not undergo a ferromag-\nnetic phase transition. Being disappointed by their ﬁnd-\ningsthe authorsformulateevenmorestrongerstatement:\n”...the Nature does not realize a strongly repulsive Fermi\ngas with short range interaction, and the widely used\nStoner model is unphysical.”\nIndeed, one cannot point out a real ferromagnetic sys-\ntem described by the Stoner model. As an apparent ex-\nception could be the Quantum Hall Ferromagnetic state\nformed according to terminology accepted in the exper-\nimental publications [30, 31] as result ”a magnetic-ﬁeld-\ninduced Stoner transition”. In fact the mechanism form-\ning this state has nothing common with the short range\nStoner-Hubbard repulsion between the particles with an-\ntiparallel spin. On the contrary, it originates from the\nlong range attractive exchange interaction between the\nelectrons with parallel spins (for review see S.M.Girvin\n[32]) ﬁrst introduced in application to the itinerant elec-\ntron systems by Felix Bloch [33].Here we have demonstrated the intrinsic instability in\nrespect to the transversal inhomogeneous magnetization\nperturbation of an isotropic itinerant ferromagneticstate\ndescribed in terms ofthe Fermi liquid theoryvalid for the\nsystems with short range interparticle repulsion. The re-\nsult is obtained by means of derivation of dispersion of\nthe transversal spin waves making use the Silin kinetic\nequation. Due to the instability of an isotropic itiner-\nant ferromagnetic state one can conclude even stronger\nthan that was done in the paper [29]: if the Nature does\nrealize a strongly repulsive Fermi gas with short range in-\nteraction it does not lead to the formation of ferromagnet\nstate.\nAs for ferromagnetic metals there are a lot properties\ndiﬀering real itinerant ferromagnets from the isotropic\nmodel describing Fermi gas with short range repulsion\nbetween the particles. The simplest distinction is that\nin ferromagnetic metals the two-moment approximation\ndoes not work. The quasiparticle exchange interaction\nshouldbe expandednot by the Legendrepolynomialsbut\nby the eigen functions of the irreducible representation\nof the point crystal symmetry group taking into account\nspin-orbital interaction. The ferromagnetism in metals\nshould be treated taking into account semi-itinerant na-\nture of quasiparticles and the anisotropy of the quasipar-\nticles interactionoriginatingfrom the spin-orbitaleﬀects.\n[1] E. Stoner, Philos. Mag. 15, 1018 (1933).\n[2] K. Huang, Statistical mechanics , Wiley, New York, 1987.\n[3] L.D.Landau, ZhETF 30, 1058 (1956) [SovietPhys. JETP\n3, 920 (1956)]; ZhETF 32, 59 (1957) [Soviet Phys. JETP\n5, 101 (1957)]\n[4] A. A. Abrikosov and I. E. Dzyaloshinskii, Zh.\nEksp.Teor.Fiz 35, 771 (1958) [Sov.Phys.JETP 8, 535\n(1958)].\n[5] I. E. Dzyaloshinskii, P. S. Kondratenko, Zh.\nEksp.Teor.Fiz. 70, 1987 (1976) [Sov.Phys.JETP\n43, 1036 (1976)].\n[6] G.-B. Jo, Y.-R. Lee, J.-H. Choi, C. A. Christensen, T. H.\nKim, J. H. Thywissen, D. E. Pritchard, and W. Ketterle,\nScience325, 1521 (2009).\n[7] M. Barth, W. Zwerger, arXiv: 1101.5594.\n[8] S. Kanno, Progr. Theor. Phys. 44, 813 (1970).\n[9] D. Belitz, T. R. Kirkpatrick, T. Vojta, Phys. Rev. Lett.\n82, 4707 (1999).\n[10] H. Heiselberg,Phys. Rev. A 63, 043606 (2001).\n[11] R. A. Duine and A. H. MacDonald, Phys. Rev. Lett. 95,\n230403 (2005).\n[12] G .J. Conduit, B. D. Simons, Phys. Rev. A 79, 053606\n(2009); G .J. Conduit et al, Phys. Rev. Lett. 103, 207201\n(2009).\n[13] S. Pilati, G. Bertaina, S. Giorgini, and M. Troyer, Phys .\nRev. Lett. 105, 030405 (2010).\n[14] S.-Y. Chang et al, Proc. Nat. Acad. Sci. 108, 51 (2011).\n[15] H. Heiselberg,Phys. Rev. A 83, 053635 (2011).\n[16] L. He and X.-G. Huang, arXiv:1106.1345\n[17] C. Herring ”Exchange Interactions among Itinerant Ele c-4\ntrons” Chapter XIV, pp.345-385, in ”Magnetism” v.IV,\nedited by G. T. Rado and H. Suhl, Academic Press, NY\nand London, 1966.\n[18] T. Moriya Spin ﬂuctuations in itinerant electron mag-\nnetism(Springer-Verlag, Berlin, 1985).\n[19] V. P. Mineev, Phys. Rev. 72, 144418 (2005).\n[20] V. P. Silin, Zh. Eksp.Teor.Fiz. 33, 1227 (1957) [Sov.\nPhys.JETP 6, 945 (1958)].\n[21] V. P. Mineev, Phys. Rev. B 69, 144429 (2004).\n[22] S. Perisanu and G. Vermeulen, Phys. Rev B 73214519\n(2006).\n[23] L. D. Landau , Zh. Eksp.Teor.Fiz. 35, 97 (1958) [Sov.\nPhys. JETP 8, 70 (1959)].\n[24] D. I. Golosov and A. E. Ruckenstein, Phys.Rev.Lett. 74,\n1613 (1995); Journ. of Low Temp.Phys. 112, 265 (1998).\n[25] C. Chin, R. Grimm, P. Julien and E. Tiesinga, Rev.Mod.\nPhys.82, 1225 (2010).[26] L. J. LeBlanc, J. H. Thywissen, A. A. Burkov, A.\nParamekanti, Phys. Rev. A 80013607 (2009).\n[27] D. Pekker, M. Babadi, R. Sensarma, N. Zinner, L. Pollet,\nM.W.Zwierlein, and E. Demler, Phys. Rev. Lett. 106,\n050402 (2011).\n[28] S. Zhang and T.-L. Ho, New J. Phys. bf 13, 055003\n(2011).\n[29] C. Sanner, E. J. Su, W. Huang, A. Keshet,J. Gillen, and\nW. Ketterle, arXiv: 1108.2017.\n[30] B. A. Piot, D. K. Maude, M. Henini, Z. R. Wasilevski,\nK. J. Friedland, R. Hey, K. H. Ploog, A. I. Toropov, R.\nAirey and G. Hill, Phys. Rev. B 72, 245325 (2005).\n[31] A. S. Zhuravlev, A. B. Van’kov, L. V. Kulik, I. V.\nKukushkin, V. E. Kirpichev, S.H.Smet, K. v. Klitzing, V.\nUmansky, and W. Wegsheider, Phys. Rev. B 77, 155404\n(2008).\n[32] S. M. Girvin, ArXiv: cond-mat / 9907002.\n[33] F. Bloch, Z. Phys. 57, 545 (1929)."    },    {        "title": "2008.06875v2.Magnon_dispersion_in_bilayers_of_two_dimensional_ferromagnets.pdf",        "content": "Magnon dispersion in bilayers of two-dimensional ferromagnets\nLara C. Ortmanns1;2;3, Gerrit E. W. Bauer1;4, and Yaroslav M. Blanter1\n1Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Institute of the Theory of Statistical Physics, RWTH Aachen, 52056 Aachen, Germany\n3Department of Microtechnology and Nanoscience (MC2),\nChalmers University of Technology, SE-41298 G¨ oteborg, Sweden and\n4WPI-AIMR and Institute for Materials Research and CSRN, Tohoku University, Sendai 980-8577, Japan\nWe determine magnon spectra of an atomic bilayer magnet with ferromagnetic intra- and both ferro- and anti-\nferromagnetic interlayer coupling. Analytic expressions for the full magnon band of the latter case reveal that\nboth exchange interactions govern the fundamental magnon gap. The inter and intralayer magnetic ordering\nare not independent: a stronger ferromagnetic intralayer coupling e \u000bectively strengthens the antiferromagnetic\ninterlayer coupling as we see from comparison of two bilayer systems. The trivial topology of these exchange-\nanisotropy spin models without spin-orbit interaction excludes a magnon thermal Hall e \u000bect.\nI. INTRODUCTION\nTwo-dimensional van der Waals magnets (2DvdWM) [1]\nare a unique platform to study magnetism in 2 +\"dimensions\n[2–4]. Two-dimensional order is associated with strong in-\ntrinsic thermal ﬂuctuations [3, 5] and characteristic quantum\nphases [3], o \u000bering a new test bed for competing interactions,\nsuch as Heisenberg and anisotropic exchange [6] with dif-\nferent range, Dzyaloshinskii-Moriya (DM) [7, 8] and other\nspin-orbit couplings [9, 10], and magnetodipolar interactions\n[11, 12] in a rich variety of elements and crystal structures.\nThe parameters of many properties are highly tunable by elec-\ntric gating [13, 14] or by strain [15, 16]. Of particular inter-\nest is the control of the magnetic anisotropy that modulates\nthe spin ﬂuctuations and allows to study cross-overs between\ndi\u000berent types of spin Hamiltonians [3]. 2DvdWM can be\nstacked with themselves or other materials into multilayers\n[17–19, 22] or structured into nanodevices and directly ac-\ncessed by scanning probe microscopy or other surface sensi-\ntive experimental techniques [21].\nIn this young ﬁeld, many basic questions are still open.\nOnly recently the magnon energy dispersion has been calcu-\nlated, which is essential for understanding the spin dynamics\nand transport [24]. For compounds with a hexagonal lattice\nsuch as CrI 3and CrBr 3[25] as considered here we may expect\na magnon dispersion relation similar to that of the \u0019-electron\nbands of graphene — a minimum at k=0 and two degener-\nate Dirac points per unit cell at an intermediate energy. This\nwas conﬁrmed by an analytic expression for a 2DvdWM with\nferromagnetic (FM) exchange interactions [26–28]. However,\nbilayers with FM intra- and inter-layer exchange interaction\nshow characteristic di \u000berences with bilayer graphene in terms\nof the degeneracy and dispersion close to the Dirac points\n[24]. To date, the magnon dispersion for bilayers with antifer-\nromagnetic (AFM) coupling has to the best of our knowledge\nbeen computed only numerically [23, 24].\nHere we extend previous theories by including a more gen-\neral form of the perpendicular plane magnetic anisotropy. For\nthe bilayer with FM intra- and AFM inter-layer exchange, we\nreport analytical results for the full spectrum by a method in-\ntroduced by Colpa [29]. We analyze the interplay of FM intra-\nand AFM interlayer couplings as reﬂected in the fundamentalgap and total energy. The analytic solutions facilitate access\nto non-trivial topological properties such as the magnon Hall\ne\u000bect. For the class of perpendicular-plane anisotropy mod-\nels without magnetization texture or spin-orbit interaction the\ntopology is trivial, however.\nThe manuscript is organized as follows: In Sec.II, we deﬁne\nthe most general spin Hamiltonian of 2DvdWM. In Sec.III,\nwe review results on magnon spectra of an FM monolayer\nwith di \u000berent types of anisotropy and a bilayer with FM intra-\nand interlayer coupling. In Sec.IV we present our main re-\nsults, i.e., an analytic derivation of the dispersion for a bilayer\nwith FM intra-and AFM interlayer exchange coupling. We\nconsider ﬁrst isotropic exchange coupling for di \u000berent spin\nconﬁgurations and subsequently include perpendicular spin\nanisotropy. We analyze the e \u000bect of the magnetic order on\nthe fundamental gap as well as total energy. Finally we com-\npute the magnon Chern numbers of the energy bands. Sec.V\nsummarizes our conclusions and gives an outlook.\nII. THE MODEL\nOur starting point is the Heisenberg Hamiltonian with\nanisotropic terms that for a magnetic monolayer has the form\n[4]\nHsl=\u0000X\nhi;ji;\u000b(Ji j~Si\u0001~Sj+ \u0003\u000bS(\u000b)\niS(\u000b)\nj)\u0000X\niA\u0010\nS(z)\ni\u00112:(1)\nHere Ji jis the exchange interaction between spins that favors\nferromagnetic ( Ji j>0) or antiferromagnetic ( Ji j<0) order of\nthe classical ground state, respectively. Because the exchange\ninteraction is short-ranged, that between nearest neighbors\nhi jidominates, while more distant ones can be disregarded.\nAis the single-ion anisotropy perpendicular to the plane, and\n\u0003\u000bparameterizes an anisotropy in the exchange interaction in\na direction \u000b. These parameters depend on the material and\ncan be tuned externally such as by an applied magnetic ﬁeld\nor a gate voltage. In this paper we disregard the single-ion\nanisotropy ( A=0) but retain the anisotropic exchange assum-\ning out-of-plane anisotropy, \u0003z= \u0003,\u0003x= \u0003 y=0, noting that\nto leading order Aand\u0003are equivalent. We disregard any\nspin-orbit interactions at this stage.arXiv:2008.06875v2  [cond-mat.mtrl-sci]  6 May 20212\nIII. REVIEW OF FM MONO- AND FM BILAYERS\nWe ﬁrst review the Holstein-Primako \u000btransformation, the\nmethod of choice to treat the low frequency spectrum of\nspin Hamiltonians, as applied to FM monolayers [6, 24]\nwith isotropic exchange interaction (III A). Afterwards, we\nreview di \u000berent types of anisotropy in the FM coupling\nof the monolayer[6] (III B). Finally we consider a FM bi-\nlayer for isotropic exchange coupling as well as out-of-plane\nanisotropy and review the dispersion (III C) [24]. This section\nserves essentially for ﬁxing the geometry and the notation.\nA. General method\nThe Holstein-Primako \u000b(HP) transformation of the Hamil-\ntonian (1) replaces the local spin operators Sjin favor of Bo-\nson operators aj[30, 31]:\nS+\nj=p\n2s0BBBBBB@1\u0000ay\njaj\n2s1CCCCCCA1=2\naj;\nS\u0000\nj=p\n2say\nj0BBBBBB@1\u0000ay\njaj\n2s1CCCCCCA1=2\n;\nS(z)\nj=s\u0000ay\njaj: (2)\nAt low temperatures or weak excitation we may disregard all\nbut the zeroth order in a=p\n2sin the series expansion of the\nsquare root. A single boson excitation ha+ai=1 changes\nthe spin projection \u0001Sz=~parallel to the quantization axis\nzand perpendicular to the plane. After subtracting the con-\nstant ground state energy, the Hamiltonian with FM exchange\ninteraction and zero anisotropy ( \u0003 = A=0) reads\nH=\u00002JsX\nhi;jiay\njai+2JsZ n:nX\niay\niai: (3)\nZn:n=3 is the number of nearest neighbors of magnetic\ncations on a hexagonal lattice. The lattice can be spanned by a\ntriangular Bravais lattice with a two-atomic basis (see Fig. 1).\nTransformation to momentum space leads to non-interacting\nmagnons\nH=X\nk;r=\u0006~!r;kay\nr;kar;k; (4)\nwith energies[6]\nE\u0006(k)=~!\u0006;k=2Js(3\u0006jckj): (5)\nHere ck=1+e\u0000i~k~a1+e\u0000i~k~a2is the structure factor of the\nlattice with unit cell vectors ~a1;~a2;as depicted in ﬁgure 1.\nThis dispersion is isomorphic with the \u0019-electrons in mono-\nlayer graphene, as shown in Fig. 2 for the ﬁrst BZ. It has a\nminimum and maximum at the \u0000-point ( k=0) and two non-\nequivalent Dirac cones at the KandK0corners at energy 6 Js.\nwith conical dispersion.\nFIG. 1: Direct triangular Bravais lattice with a two-atomic basis A;B\n(crosses).Basis vectors ~a1,~a2span the primitive unit cell as indicated\nby dashed lines. Blue circles indicate lattice point and ais the lattice\nconstant.\nFIG. 2: Energy dispersion of an FM monolayer with isotropic ex-\nchange coupling along high symmetry directions in the ﬁrst BZ. K,\nK0are the inequivalent Dirac points.\nB. Anisotropies\nIn CrI 3[18] the magnetic anisotropy has an easy axis along\nˆz, i.e. perpendicular to the plane of the material. The Hamil-\ntonian (1) becomes\nˆH=\u0000JX\nhi;ji\u0010\nSx\niSx\nj+Sy\niSy\nj\u0011\n\u0000(J+ \u0003)X\nhi;jiSz\niSz\nj(6)\nwith\u0003;J>0. The dispersion [6]\nE\u0006(k)=(6(J+ \u0003)s\u00062Jsjckj); (7)\nis shifted by 6 \u0003scompared to the isotropic case. This shift\nreﬂects the suppression of the Goldstone mode of rotationally\nsymmetric systems by opening a spin wave gap at the \u0000-point.\nIn the expansion of the HP-transformation, we restricted to\nleading order, thereby neglecting magnon-magnon interac-\ntions that become relevant at ﬁnite temperature. A mean-ﬁeld\ntreatment of higher order bosonic operators renormalizes the\nexchange coupling constants, and thereby also the spin wave\ngap[6].\nWe model an easy-plane anisotropic FM with J>0 and\n\u0003<0 in the Hamiltonian (1). We eliminate the non-bilinear\nterms of the bosonic operators akby a Bogoliubov transfor-\nmation [32], which leads to quadratic forms of Bose operators3\nassigned to at most to two sublattices with spectrum:\nE\u0006=Jsp\nR\u0006S;\nR=36+4(1+\u0003\nJ)jckj2; (8)\nS=24(1+\u0003\n2J)jckj:\n\u0003 =\u0000Jrecovers the XY-model with dispersion [27],[6]\nE\u0006=6Jsr\n1\u0006jckj\n3(9)\nplotted in Fig. 3. The general monolayer Hamiltonian (8) have\nrecently studied in Ref. [28]. Note that E\u0006is proportional\nto the square root of the energy in the isotropic case. The\neasy-plane anisotropy was observed in a monolayer of CrCl 3\n[2, 33–35], which should therefore be a good system to study\nphase transitions in 2D.\nFIG. 3: Energy dispersion for an FM monolayer with easy-plane\nanisotropy \u0003 =\u0000J. For further explanation see the text.\nC. FM bilayer\nFor a bilayer with FM intra- and interlayer coupling\nJk;J?>0 and without anisotropies we arrive at the Hamil-\ntonian\nˆH=\u00002JkX\nhi;ji~Si\u0001~Sj\u00002J?X\nhi;ji~Si\u0001~Sj; (10)\nwhere the ﬁrst and second terms describe intra- and interlayer\ncoupling, respectively. We adopt the ratio of J?=0:26Jk\nas predicted for CrI 3by ﬁrst-principles calculations [18].\nWe consider here ABtype stacking of 2D hexagonal lattices\nwith a lateral shift by [2 =3;1=3] unit vectors (see Fig. 4)\n[18], which corresponds to the FM low-temperature crystal-\nlographic phase of bulk CrI 3[17, 19, 22]. We chose a unit cell\nfor a bilayer with four atoms, A-atoms A1 in the bottom-layer\n(1) and A2 in the top-layer (2) as well as B-atoms B1 and B2\n(see Fig. 4). Each A-(B)-atom has three nearest neighbors in\nthe same layer belonging to the B-(A)-sublattice. The atoms\nA2 on top of B1 form another pair of nearest-neighbours per\nunit cell. The magnon band structure consists now of four\nrather than two energy bands [24]\nE[1]\n\u0006=12Jks\u00064Jksjckj (11)\nE[2]\n\u0006=12Jks+4J?s\u00064sq\nJ2\n?+J2\nkjckj2; (12)which reﬂects the more complex unit cell. The lowest band\nFIG. 4: A bilayer with AB-stacking as for example in bulk BiI 3crys-\ntals. The primitive unit cell (dashed blue lines) contains four atoms,\nA1 (green-rimmed black dot) of bottom layer (label 1), B2 (red-\ngreen) of top layer 2 and the stacked pair of atoms B1-A2 (black-\ngreen cross) with A2 on top of B1. The basis vectors ~a1,~a2of the\nbilayer-lattice are the same as for the monolayer and are shown as\nblue arrows.\nE[1]\n\u0000is gapless at the origin because in the absence of any\nanisotropy the system is invariant with respect to a global spin\nrotation. At the Dirac points K,K0, the structure factor van-\nishes and E[2]\n+=(12sJk+8sJ?),E1=12sJk, where E1is\nthreefold degenerate. This spectrum di \u000bers from that of the\n\u0019-electrons in bilayer graphene, which are two-fold degener-\nate at the KandK0points with parabolic dispersion [36]. The\nwave functions at the Dirac points read\n\t[1]\nK(0)=1p\nNX\njeiK(0)~Rjay\nj;A1j0i; (13)\n\t[2]\nK(0)=1p\nNX\njeiK(0)~Rjay\nj;B2j0i; (14)\n\t[3]\nK(0)=1p\n2NX\njeiK(0)~Rj\u0010\nay\nj;B1+ay\nj;A2\u0011\nj0i; (15)\n\t[4]\nK(0)=1p\n2NX\njeiK(0)~Rj\u0010\nay\nj;B1\u0000ay\nj;A2\u0011\nj0i; (16)\nwherejj\u000bidenotes the position of the site on sublattice \u000b.\nThe eigenstate \t[1]\nK(0)(\t[2]\nK(0)) is localized to sublattice A1(B2)\nin layer 1(2) :The sublattices A2 and B1 are coupled by\nJ?, which generates an in phase or acoustic mode \t[3]with\nlower energy E1or out-of-phase \u0019-shifted optical mode \t[4]\nat higher energy E[2]\n+.\n\t[1];\t[2];\t[3]correspond to excitations in which the spins\non the same sublattice and layer, separated by aalong (1;0) or\n(\u00001\n2;p\n3\n2);precess with a relative phase shift2\u0019\n3. The spin pre-\ncession therefore reﬂects the structure of the hexagonal lattice\nbonds at the Dirac-points. In the Appendix we demonstrate\nthat these modes also solve the Landau-Lifshitz equation for\ncoupled classical spins.\nA perpendicular anisotropy can be modeled by the coupling\nconstants J(zz)\nk,J(zz)\n?and blue shifts the frequencies,4\nE[1]\n\u0006=12sJzz\nk+2s(Jzz\n?\u0000J?)\u00062sq\n(Jzz\n?\u0000J?)2+4J2\nkjckj2 (17)\nE[2]\n\u0006=12sJzz\nk+2s(Jzz\n?+J?)\u00062sq\n(Jzz\n?+J?)2+4J2\nkjckj2 (18)\nMoreover, the triple degeneracy at the Dirac points is reduced\nto a double one.\nIV . BILAYER WITH FM INTRA- AND AFM INTERLAYER\nEXCHANGE COUPLING\nWe ﬁrst derive new results for the dispersion for a bilayer\nwith FM intralayer and AFM interlayer isotropic exchange in-\nteractions (IV A). Subsequently, we include a perpendicular-\nplane anisotropy and focus on the analysis of the fundamental\ngap at \u0000(IV B). The topology in terms of the Berry curvature\nis subject of Section IV C.\nA. Isotropic exchange interaction\nSeveral papers discuss the impact of stacking [2, 18, 19, 22,\n37] on interlayer magnetic coupling of a CrI 3bilayer. Depend-\ning on the type of involved interlayer orbital hybridizations,\nthe corresponding coupling of the modeling spin Hamiltonian\nis FM or AFM type. For AB stacking, it has been shown by\ndensity functional theory calculations [19] that both NN and\nNNN interactions determine the order of the bilayer ground\nstate: There are one NN neighbor and 16 NNN within a unit\ncell, the NN contributing with AFM coupling whereas the\nNNN contributing with FM coupling, so that in total, inter-\nlayer magnetism in AB stacking is strongly FM. As magnetic\ninterlayer order can be tuned by application of an electro-\nstatic gate [14] or a magnetic ﬁeld [21], however, we ﬁnd it\ninstructive to discuss both types of interlayer coupling (FMand AFM) for the same type of stacking. Here we choose\nAB-stacking for simplicity and an AFM interlayer magnetism\nof the bilayer, as is induced by the NN couplings of the AB-\nstacking.\nWe ﬁrst calculate the energy dispersion of a bilayer with\nisotropic exchange coupling for di \u000berent spin directions and\nintra/interlayer coupling strengths Jk=J?. The Hamiltonian\nreads with Jk;J?>0\nˆH=\u00002JkX\nhi;ji2fintrag~Si\u0001~Sj+2J?X\nhi;ji2finterg~Si\u0001~Sj: (19)\nAgain, the ﬁrst sum includes the three in-plane nearest neigh-\nbors of a local moment on site i, while the second sum runs\nover closely spaced dimers A2,B1 between the layers. When\nSz=sfor the spins in the top layer (2), Sz=\u0000sin the bottom\nlayer (1) minimizes the classical ground state energy E0. The\nmagnons a+\ni;aiare the excitations. We apply the HP- transfor-\nmation and expand Eq. (19) to leading order in the magnon\noperators, thereby disregarding magnon-magnon interactions,\nwhich is valid at low temperatures. In a mean-ﬁeld approxi-\nmation, higher terms only renormalize the exchange constants\n[20], as conﬁrmed by experiments work on bilayer CrI 3[21],\nat a temperature T=0:033J. Therefore\nS+(\u0000)\ni;\u000b2=p\n2sa(+)\ni;\u000b2;Sz\ni;\u000b2=s\u0000a+\ni;\u000b2ai;\u000b2; (20)\nS\u0000(+)\ni;\u000b1=p\n2sa(+)\ni;\u000b1;Sz\ni;\u000b1=\u0000s+a+\ni;\u000b1ai;\u000b1: (21)\nThe subscripts refer to atom \u000b2fA;Bgof lattice cell iin layer\n\u00172f1;2g. The magnon Hamiltonian then reads\nˆH\u0000E0=\u00002sJkP\nh(i;\u000b\u0017);(j;\u000b0\u0017)i;\u000b,\u000b0(a+\nj;\u000b0\u0017ai;\u000b\u0017+h:c:)+6sJkP\ni;\u000b\u0017a+\ni;\u000b\u0017ai;\u000b\u0017\n+2sJ?P\ni(a+\ni;A2a+\ni;B1+h:c:)+2sJ?P\ni(a+\ni;A2ai;A2+a+\ni;B1ai;B1): (22)\nAs common for antiferromagnetic order, the classical ground\nstate is not an eigenstate of the Hamiltonian since\na+\ni;A2a+\ni;B1j#i 1j\"i 2/j\"i 1j#i 2,0: (23)\nWe can accommodate this issue by writing the Hamiltonian in\nreciprocal space as [29]\nˆH\u0000E0=Ec+X\n~k(~a+\n~k;~a\u0000~k)D(~a~k;~a+\n\u0000~k)T(24)where~a~k=(a~k;A1;a~k;B1;a~k;A2;a~k;B2),Eca constant to be dis-\ncussed later, and Dis the 8\u00028-matrix\nD= \nA B\nB A!\n(25)5\nin which\nA=0BBBBBBBBBBBB@3Jks\u0000Jksc\u0003\nk\n\u0000Jks ck3Jks+J?s0\n03Jks+J?s\u0000Jksc\u0003\nk\n\u0000Jksck 3Jks1CCCCCCCCCCCCA;(26)\nB=0BBBBBBBBBBBB@0\n00J?s\nJ?s00\n01CCCCCCCCCCCCA; (27)\nandckis again the structure factor of the hexagonal lattice.\nKowalska’s framework [32] is not applicable for four sub-\nlattices. Instead, we diagonalize the Hamiltonian by a para-\nunitary transformation Tof operators ( ~a~k;~a+\n\u0000~k)Tto the bosonic\noperators~\r~k[29]:\n(~\r~k;~\r+\n\u0000~k)T=T(~a~k;~a+\n\u0000~k)T(28)\nsuch that\nˆH\u0000E0\u0000Ec=X\n~k(~a+\n~k;~a\u0000~k)Ty(Ty)\u00001DT\u00001T(~a+\n~k;~a\u0000~k)y\n=~X\n~k(~\r+\n~k;~\r\u0000~k)diag(!1;::;! 4;!1;::;! 4)(~\r+\n~k;~\r\u0000~k)y\n=24X\n~k;r=1~!r \n\r+\nr;~k\rr;~k+1\n2!\n; (29)provided that Dis positive-deﬁnite. Ecis a further constant\nthat will be speciﬁed below. Tis para-unitary in the sense that\nT\u0011Ty=\u0011; (30)\nwith\u0011=diag( I4;\u0000I4);where Inis the unit matrix with dimen-\nsionn, which ensures that the \r(+)\nr;~kobey bosonic commutation\nrelations. (\u00151;:::;\u0015 8) :=(!1;::;! 4;\u0000!1;::;\u0000!4) are the para-\nvalues of D\n(D\u0000\u0015i\u0011)~vi=0 (31)\nwith para-vectors ~vi. Eq. (31) can be written as an eigenvalue\nproblem by multiplying by \u0011from the left\n(\u0011D\u0000\u0015iI)~vi=0: (32)\nDiagonalizing the non-Hermitian matrix \u0011Dleads to a set of\nfour positive and four negative eigenvalues \u0006\u0015icorresponding\nto the two twofold degenerate energy bands\nE\u0006=ss\n3JkJ?+9J2\nk \n1+jckj2\n9!\n\u0006p\n3Jkq\n3J2\n?+(12J2\nk+4JkJ?)jckj2: (33)\nThe di \u000berence in energy bands for bilayers with AFM and\nFM order can be traced to the matrix \u0011. In physical terms, two\nAFM-coupled sublattices ( A2-B1) generate two mode fami-\nlies that are exchanged by a \u0019-rotation of the bilayer and hence\nare degenerate. The additional symmetry is also responsible\nfor the degenerate ground state of the AFM bilayer. Breaking\nthe interlayer symmetry by perpendicular electric and mag-\nnetic ﬁelds removes the degeneracy [24].\nThe dispersion (33) is plotted in Fig. 5. We ﬁnd a dif-\nference4Ebetween the the zero-point energy of the magnon\nsystem and the classical ground state energy E0=\u000012NJks2\u0000\n2J?Ns2\n4E=Ec+X\n~k4X\nr=1~!r=\u0000Ns(12Jk+2J?)+X\n~k4X\nr=1~!r;(34)\nsee also Eq. (29). The ﬁrst term on the right hand side Ec=\nE0=s, arises from quantum ﬂuctuations of the z-component,\nwhile the second term reﬂects transverse ﬂuctuations cause.\nIn the following we disregard these zero-point ﬂuctuations,but recommend their study in a future project.\nAround the Dirac-points K;K0, the dispersion can be ex-\npanded up to second order in kas\nE+(k)=p\n3Jkss\n3+2J?\nJk+3\n8a2Jks3+6Jk\nJ?q\n9+6J?\nJkk2;\nE\u0000(k)=3Jks\u00001\n8a2Jks \n1+6Jk\nJ?!\nk2: (35)\nThe AFM coupling J?therefore opens a gap of the order\nsJ?atK,K0, leading to a quadratic rather than the linear\ndispersion found for the FM monolayer, but di \u000berent e \u000bec-\ntive masses. This gap implies a possible non-trivial topology.\nHowever, the Chern numbers are found to be zero for each\nbranch, which we indicate in Section IV C.6\nFIG. 5: Dispersion of a bilayer with AFM inter-layer and FM intra-\nlayer coupling, isotropic exchange coupling constants and a ratio of\ninter- vs. intralayer coupling J?=0:26Jk. We observe a gap of order\nJ?sat the Dirac points with quadratic instead of the linear dispersion\nof the FM monolayer in Fig. 2.B. Anisotropy\nNext, we introduce an out-of-plane anisotropy with Jzz\nk>\nJk,Jzz\n?>J?. The matrix Athen reads\nA=0BBBBBBBBBBBBBB@3Jzz\nks\u0000Jksc\u0003\nk\n\u0000Jksck3Jzz\nks+Jzz\n?s0\n03Jzz\nks+Jzz\n?s\u0000Jksc\u0003\nk\n\u0000Jksck 3Jzz\nks1CCCCCCCCCCCCCCA;\nwhile Bis not a \u000bected. We can still derive an analytic expres-\nsion for the energy dispersion\nE\u0006=sp\n2r\n18Jzz2\nk+6Jzz\nkJzz\n?+Jzz2\n?\u0000J2\n?+2J2\nkjckj2\u0006q\n(6Jzz\nkJzz\n?+Jzz2\n?\u0000J2\n?)2+[(12Jzz\nk+2Jzz\n?)2\u00004J2\n?]J2\nkjckj2: (36)\nand plot them in Figure 6 for coupling constants Jzz\nk=1:3Jk;\nJzz\n?=0:56Jk;J?=0:26Jk. Here we adopt again a ratio of\n0:26 between inter- and intra layer coupling. We assume that\nFM and AFM ordered layers are both AB stacked and that\nthe ratio between inter and intra-layer coupling (0 :26 for FM\nCrI 3[18]) only changes sign. Actually, AFM ordered CrI 3has\nboth a di \u000berent ( AB0) stacking and the interlayer exchange is\nsmaller with an inter /intra layer ratio of \u00000:018. Other con-\nstants are known for monolayer CrI 3[6, 38] and can be tuned,\nfor example, by an electrostatic gate.[14]. Here we chose\nthem to enhance the visibility of the e \u000bects in the ﬁgures. The\nFIG. 6: Magnon dispersion of a bilayer with AFM inter-layer and FM\nintra-layer coupling with anisotropic exchange coupling Jzz,Jxx=\nJyy=J. Here the inter-layer couplings J?=0:26Jk,Jzz\n?=0:56Jk\nand intra-layer coupling Jzz\nk=1:3Jk.\nanisotropy blue-shifts the lower band edge \u0018Jksrelative to the\nzero-point energy E0\u0000Ns(12Jzz\nk+2Jzz\n?)+PN\n~k=1P4\nr=1~!rand\nincreases the gap at the Dirac points ( \u0018J?sfor the isotropicAFM-bilayer) to\u0018Jzz\n?s.\nWe now analyze the fundamental gap ~!\u0000(~k=0) (see Eq.\n(36)) plotted in Figure 7(a) as a function of the FM coupling\nstrength JkforJ?=1:0J0,Jzz\nk\u0000Jk=1:0J0andJzz\n?\u0000J?=\n0:3J0. In a simple FM the gap\n\u0001FM/s(Jzz\nk\u0000Jk) (37)\ndepends on Jkonly via anisotropy. The anisotropy gap in a\npure AFM, on the other hand,\n\u0001AFM/sq\n(Jzz\n?\u0000J?)(Jzz\n?\u0000J?+2J?) (38)\ndepends not only on the anisotropy Jzz\n?\u0000J?, but also on the\nAFM coupling strength J?[39]. The increase of the intra-\nlayer FM coupling increases the gap E\u0000(~k=0) according to\nEq.(36), which by the reduced number of thermal magnons is\nequivalent to an enhanced AFM coupling.\nWe analyze this e \u000bect by computing the gap of a hypothet-\nical structure in which the contributions from Eq. (37) of the\nFM and Eq. (38) of the AFM coupling at ~k=0 are clearly\nseparated. The stacking of two ferromagnetic monolayers in\nthis “bilayer (II)” is slightly shifted such that there are two\nAFM-coupled dimer pairs A2\u0000B1 and A1\u0000B2 with coordi-\nnation number ZAFM=0:5 (see Figure 7b (right)) compared\nto the original coordination number ZAFM=1 for the single\ndimer-pair in bilayer (I) (see Figure 7b (left)). The gap of this\nmodiﬁed system\nE\u0000(k=0)=sq\n((Jzz\nk\u0000Jk)ZFM+(Jzz\n?\u0000J?)ZAFM) ((Jzz\nk\u0000Jk)ZFM+(Jzz\n?\u0000J?)ZAFM +2J?ZAFM) (39)7\nFIG. 7: (a) Magnon gaps for a realistic (black line) and a hypothetical\nbilayer (blue line) as a function of FM coupling strength Jk. The\nanisotropy is constant with Jzz\nk\u0000Jk=1:0J0,Jzz\n?\u0000J?=0:3J0and\nJ?=1:0J0. (b)(left) Realistic bilayer schematic with coordination\nnumbers ZAFM=1 and ZFM=3. (b)(right) Hypothetical bilayer\nschematic with ZAFM=0:5 and ZFM=3.\ndoes not depend explicitly on Jk, but on Jzz\nk\u0000Jk, see Figure\n7(a) (blue line) and Eq.(39). For Jk=0, the gap 3 J0s=s(Jzz\nk\u0000\nJk)ZFMof bilayer (I) is governed by the anisotropy of the FM\nintralayer exchange only, while the AFM coupling does not\ncontribute to the gap. The gaps converge to \u00183:61J0sonly\nwhen the FM coupling in bilayer (I) Jk'5J?. This result\nsuggests that a strong FM intra-layer coupling in the realistic\nstructure (I) increases the AFM inter-layer coupling, while in\nthe limit of weak FM coupling, the AFM order of the classical\nGS is less stable than in bilayer (II) (see green arrows in Figure\n7(b)).\nThis statement is corroborated by the ﬁnite-wave vector\nmagnon dispersion \u0001Ek;0=E\u0000(~k)\u0000E\u0000(0) as a function of\nthe FM coupling. The zero- k-magnon is that of an interlayer\nAFM in its classical GS. As \u0001Ek;0measures the energy cost\nof exciting a ﬁnite- k-magnon, it thereby measures the AFM\ncoupling strength. The right panel of ﬁgure 8 shows an \u0001Ek;0,\nwhich indeed increases with Jkfor both bilayers (I) and (II).\nThe left panel of ﬁgure 8 shows the di \u000berence \u0001Eh\nk;0\u0000\u0001Er\nk;0\nof a hypothetical and a real bilayer for di \u000berent points along\nthe\u0000\u0000Kdirection in the ﬁrst BZ, which decreases with in-\ncreasing Jk, conﬁrming that the real bilayer approaches the\ne\u000bective AFM coupling strength of the hypothetical bilayer\nfor large Jk:This shows that in the limit of strong intralayer\ncoupling, magnetic order does no longer depend on the choice\nof stacking in our speciﬁc case.\nFIG. 8: (Left) Di \u000berence \u0001Eh\nk;0\u0000\u0001Er\nk;0between the hypothetical and\na realistic bilayer structure as a function of FM coupling as a func-\ntion of ( kx;0)[\u0019\na] along the \u0000\u0000Kdirection in the ﬁrst BZ. (Right) En-\nergy di \u000berence \u0001E\u0000\nk;0between a magnon with wavevector (1 :2;0)[\u0019\na]\nand zero wavevector in the lower band as a function of FM coupling\nstrength Jkfor bilayers I (green) and II (violet).\nC. Topology\nThe topology of the magnon spectrum is reﬂected by the\nBerry curvature \nnk=rk\u0002hunkjirkjunkiof the n=\u0006bands\n(36), where unkis the periodic (Bloch) part of the wave func-\ntion [40]. For a Dirac-like spectrum, the Berry curvature is8\nFIG. 9: The Berry curvature \nxy;n(k) for the bands E+(left) and E\u0000\n(right) of a bilayer with AFM interlayer and FM intralayer coupling.\nThe exchange coupling constants are chosen as in Fig. 6\nlarge in the vicinity of the Dirac points, which dominate the\ntopological properties [23] as illustrated by Fig. 9. Their signs\nare opposite at Dirac points K,K0, which means that the Chern\nnumber vanishes for each band. The topology for the bilayers\nin the anisotropic exchange model without spin-orbit interac-\ntion is therefore trivial, without protected edge states inside\nthe gap. The thermal Hall conductivity, which is often used\nto probe topological properties of systems with a Dirac-like\nspectrum, is proportional to the product of the Bose distri-\nbution function times the Berry curvature \nxy;n(k) integrated\nover the ﬁrst BZ [41] and vanishes as well.\nThis corresponds to the general fact that a non-vanishing\nthermal Hall conductivity has so far been predicted for CrI 3\nmonolayer systems with the anisotropy contributions to the\nspin Hamiltonian of the Kitaev model [28] or the DMI\n[27, 42]. More generally, Costa et al [43] described magnons\nin monolayer CrI 3by an itinerant fermion model based on\nﬁrst-principles calculations, thereby circumventing model as-sumptions for the anisotropy. They showed that the spin-orbit\ncoupling of iodine is essential for a non-trivial topology.\nV . CONCLUSIONS\nWe report analytical expressions for the magnon band struc-\nture of bilayers of two-dimensional ferromagnets with (anti-)\nferromagnetic interlayer exchange coupling and perpendicular\nanisotropy, complementing previous numerical analysis [24].\nAn analytic expression for the fundamental gap reveals AFM\nand FM contributions that can be modeled by an e \u000bective co-\nordination number. As the comparison of the spectral proper-\nties between our real bilayer system and the hypothetical toy\nmodel have shown, an increasing FM coupling in the real bi-\nlayer leads e \u000bectively to a stronger AFM interlayer coupling.\nThe spectral properties refer to the analysis of the spectral gap\nas well as the energy cost associated with adding an additional\nmagnon to the system. 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APPENDIX: CLASSICAL CONSIDERATION OF\nFM-BILAYER EIGENMODES\nHere we show that the magnon modes at the Dirac points\ncan be derived from a purely classical torque cancellation ar-gument.\nCentral to the Landau-Lifshitz equation is the torque ~\u001cex-\nperienced by a spin by a magnetic ﬁeld ~H:\n~\u001c=d~Si\ndt=\r\u00160~Si\u0002~H; (40)\nwhere\r=\u0000g\u0016B<0 is the gyromagnetic ratio for the elec-\ntron and\u00160the permeability of free space. The coupling to\nneighboring spins can be taken into account by an e \u000bective\nﬁeld~He\u000b[31]\n~He\u000b=\u00002\ng\u00160\u0016BX\nj2<i>Ji j~Sj; (41)\nwhere\u0016Bis the Bohr magneton and gthe Land ´e-factor. Then\nd~Si\ndt=\r\u00160~Si\u0002~He\u000b; (42)\nWhen a spin belongs to a classical ground state that does not\nprecess, the torques cancel\n0=Jk~Si\u0002(~S1+~S2+~S3) (43)\n=Jksˆez\u0002~Stot (44)\nor\n0=X\nj2<i>Sx\nj=X\nj2<i>Sy\nj:\nIn modes (13)-(16) the excitation is equally distributed over\nthe lattice, so that the in-plane components Sk\n1=Sk\n2=Sk\n3.\nThe only solution is then given by a relative phase shift of\n2\u0019\n3which agrees with the eigenmodes at Dirac points K,K0\nobtained by diagonalizing the magnon Hamiltonian."    },    {        "title": "1308.1961v1.Ferromagnetism_of_the_Repulsive_Atomic_Fermi_Gas__three_body_recombination_and_domain_formation.pdf",        "content": "Ferromagnetism of the Repulsive Atomic Fermi Gas: three-body recombination and\ndomain formation\nIlia Zintchenko1, Lei Wang1;2and Matthias Troyer1\n1Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland and\n2Beijing National Lab for Condensed Matter Physics and Institute of Physics,\nChinese Academy of Sciences, Beijing 100190, China\nThe simplest model for itinerant ferromagnetism, the Stoner model, has so far eluded experimental\nobservation in repulsive ultracold fermions due to rapid three-body recombination at large scattering\nlengths. Here we show that a ferromagnetic phase can be stabilised by imposing a moderate optical\nlattice. The reduced kinetic energy drop upon formation of a polarized phase in an optical lattice\nextends the ferromagnetic phase to smaller scattering lengths where three-body recombination is\nsmall enough to permit experimental detection of the phase. We also show, using time depend-\nent density functional theory, that in such a setup ferromagnetic domains emerge rapidly from a\nparamagnetic initial state.\nPACS numbers: 67.85.Lm, 75.10.Lp, 03.75.Ss\nThe ground-state of a repulsive gas of fermions with\ncontact interaction was \frst predicted by Stoner [1] in\n1933 to be ferromagnetic and a precise value for the crit-\nical interaction strength in a homogeneous system was\nrecently obtained with di\u000busion Monte Carlo simulations\n[2{4]. While ultracold fermionic gases should provide a\ncontrolled environment to study this phenomenon, the\ninstability of a strongly repulsive gas towards molecule\nformation has so far prevented experimental realization\nof this phase. Repulsive fermions on the positive side of\nthe Feshbach resonance live on the meta-stable repuls-\nive branch [5]. When three atoms, one with the oppos-\nite spin of the other two, come close to each other two\natoms with opposite spin will form bosonic molecules and\nthe other one carries the binding energy away. The rate\nof such process increases rapidly with scattering length.\nThe lifetime of the gas is therefore largely governed by\nthe interaction strength and the spatial overlap between\nthe two spin species.\nA recent experiment [6] presented evidence for a pos-\nsible ferromagnetic state formed after rapid increase of\nthe scattering length. The nature of this phase has, how-\never, been questioned [7{9] as the peaks in kinetic energy,\ncloud size and loss rate observed in [6] are only indirect\nevidence for ferromagnetic domains [7], and it has been\nshown that molecule formation is dominant at large inter-\naction strengths [8, 9]. Several papers have proposed to\nreduce the recombination rate by using a polar molecu-\nlar gas with dipole interactions and positive scattering\nrange [10, 11], narrower Feshbach resonances [8, 9, 12, 13]\nand fermions with unequal mass [14, 15]. Although these\napproaches might prove promising in future experiments,\nthey all change the microscopic physics of the Stoner\nmodel.\nHere we suggest new strategies which achieve the\nsame goal of stabilising the ferromagnetic phase, yet pre-\nserve the microscopic physics and thus pave the way to-\nwards experimental realization of Stoner ferromagnetism.\nFigure 1: Phase diagram of the homogeneous repulsive Fermi gas\nas a function of temperature T=TFand interaction strength kFas.\nThe white dashed line indicates the paramagnet-ferromagnetic\nphase transition and the colour scale the polarization\nP= (n\"\u0000n#)=(n\"+n#).\nFirstly, the lifetime of the system can be increased by re-\nducing the overlap volume between polarized domains\nwith di\u000berent spins. This can be achieved using elong-\nated traps with larger aspect ratios, but turns out to be\ninsu\u000ecient to stabilise the phase by itself. A more ef-\nfective way is to reduce the critical scattering length, at\nwhich ferromagnetism sets in, by imposing a shallow op-\ntical lattice. The ferromagnetic transition is determined\nby a competition between the loss of kinetic energy and\ngain of interaction energy. Since the optical lattice re-\nduces the kinetic energy scale, the ferromagnetic state is\nstabilised. A similar e\u000bect is found in \rat band ferro-\nmagnetism [16{18]\nTo obtain quantitative results for the phase diagrams\nwe use Kohn-Sham density functional theory [19, 20]\nwhere the exchange-correlation energy is treated withinarXiv:1308.1961v1  [cond-mat.quant-gas]  8 Aug 20132\na local spin density approximation (LSDA) which has\nbeen widely used for materials simulations and more re-\ncently for ultracold atomic gases [21{24]. The LSDA\nexchange-correlation functional is obtained by solving\na uniform system at zero temperature with di\u000busion\nMonte-Carlo [3, 4], where interactions between fermionic\natoms with opposite spin are modelled by a hard-sphere\npotential with scattering length as.\nBefore discussing our proposal to stabilise the ferro-\nmagnetic phase we investigate whether thermal \ructu-\nations, which can signi\fcantly a\u000bect of the stability of\nthe ferromagnetic phase [25], might be responsible for\nthe absence of a stable ferromagnetic phase in exper-\niments. To quantify the e\u000bect of non-zero temperat-\nure we employ \fnite-temperature density functional the-\nory with a zero temperature exchange-correlation cor-\nrection [26{29]. The resulting phase diagram, presen-\nted in Fig. 1, is in general agreement with previous res-\nults [30, 31]. We observe almost full polarization in the\ncurrently experimentally accessible temperature regime\nT\u00180:25TF[8, 13] and thermal \ructuations are thus not\nthe dominant mechanism destabilising the ferromagnetic\nphase.\nAn equally important question is that of the time scale\nover which ferromagnetic domains form from an initially\nparamagnetic state, which should be within the capab-\nilities of current experimental setups for the observation\nof ferromagnetic domains to remain plausible. We ad-\ndress this issue within the time-dependent Kohn-Sham\nformalism [32] ignoring the e\u000bect of three-body recom-\nbination. The system is evolved in pancake shaped har-\nmonic con\fnement in the presence of thermal noise after\na quench of the interaction strength k0\nFasto 1:2. Already\naftert\u0018250tFferromagnetic domains with a feature size\n\u001810k\u00001\nFform (Fig. 2), which is within the resolution cur-\nrently achievable in the lab. The ferromagnetic phase is\nalso signi\fcantly more stable with a total recombination\nrate reduced by a factor \u00183 relative to the initial para-\nmagnetic gas. In the experiment [8], more than half of\nthe particles remain in the meta-stable repulsive branch\nafter 250tF. However, ferromagnetic domains were not\nobserved indicating that three-body recombination has\nmore signi\fcant e\u000bects beyond reducing the fraction of\nmeta-stable fermions in the system.\nWe thus focus on three-body recombination and de-\ntermine its e\u000bect on the stability of the ferromagnetic\nphase. The loss rate \u0000 = \u0000@tN=N (Nis the total num-\nber of particles in the system) is inversely proportional\nto the lifetime of the system and can be computed as [33]\n\u0000\u0018a3\nsX\n\u001bZ\ndrZ\njr0\u0000rj<asdr0\"F(r)g\u0016\u001b\u001b\u001b(r;r;r0) (1)\nwhere\"F(r) = ~2(3\u00192(n\"+n#))2=3=2mis the\nlocal Fermi energy, n\u001b(r) is the local density of\neach spin, mthe atom mass and g\u0016\u001b\u001b\u001b(r;r;r0) =\nFigure 2: Total magnetisation hjmjiin pancake shaped harmonic\ncon\fnement for N= 440. Starting from a paramagnetic phase the\nsystem evolves in the presence of thermal noise for t2[0; 250]tF.\nTopology of ferromagnetic domains at t=f50;150;250gtFis\nshown in the insets. As the gas remains unpolarised away from\nthe center, the maximum magnetisation is \u00180:8.\nh^ y\n\u0016\u001b(r)^ y\n\u001b(r)^ y\n\u001b(r0)^ \u001b(r0)^ \u001b(r)^ \u0016\u001b(r)iis the three-body\ncorrelation function which gives the probability of \fnd-\ning three particles with spin \u0016 \u001b\u001b\u001b [34] at locations rrr0\nrespectively. Here \u0016 \u001brefers to the opposite spin of \u001b. In\nthis way the total loss rate is in units of energy and the\nmicroscopic contribution, represented by the pre-factor\na3\nstogether with the integration over r0, is decoupled\nfrom the many-body background given by the correl-\nation function. The three-particle correlation function\nis commonly [30, 35, 36] treated phenomenologically as\ng0\n\u0016\u001b\u001b\u001b(r;r;r0) =n\u0016\u001b(r)n\u001b(r)n\u001b(r0), where the density is\nassumed to be constant within the integral over r0. This\nform disregards quantum mechanical corrections due to\nexchange e\u000bects and violates the Pauli principle. Within\nthe Kohn-Sham formalism a more accurate representa-\ntion is\ng\u0016\u001b\u001b\u001b(r;r;r0) =g0\n\u0016\u001b\u001b\u001b(r;r;r0)\u0000n\u0016\u001b(r)j\u001a\u001b(r;r0)j2(2)\nwhere\u001a\u001b(r;r0) is the one-body density matrix. The Pauli\nprinciple is restored in this formulation as g\u0016\u001b\u001b\u001b(r;r;r)\u0011\n0, thus leading to a better estimate of three-body losses.\nAs molecule formation in the gas is a three-body pro-\ncess and requires both species of fermions to be close to\neach other, the recombination rate (1) is determined by\nthe total volume where the two spin species are mixed. In\na polarised system this amounts to the overlap between\nferromagnetic domains at their boundaries, whose struc-\nture strongly depends on the external con\fning poten-\ntial. To further investigate this e\u000bect, we impose a ci-\ngar shaped harmonic potential, which has been used\nin a number of recent experiments [8, 13]. With cyl-\nindrical symmetry around the z-axis the potential is in\nthe formVHO(r;z) = 1=2(!2\nrr2+!zz2) with aspect ra-\ntio\u0015=!r=!z. We will consider the spin-balanced case3\n(a) Total density\n(b) Magnetisation\n(c) Recombination rate\nFigure 3: Total density n\"+n#, magnetisation n\"\u0000n#and loss\nrate \u0000 in harmonic con\fnement for the aspect ratio \u0015= 5,\nk0\nFas\u00181:4 andN\"=N#= 969. The zandraxes are along the\nhorizontal and vertical directions respectively.\nN\"=N#=N=2 and normalise the loss rate \u0000 by the\ncritical recombination rate \u0000 c, above which the system is\nunstable. This value can be estimated from the experi-\nmental observation of a maximum tolerable kFas\u00180:4\nfor6Li atoms in an optical dipole trap [13].\nWith equal number of spin-up and down particles, the\nferromagnetic ground state exhibits a domain wall across\nthe middle of the trap, as shown in Fig. 3b. Due to repul-\nsion between the two polarized domains there is a drop\nin total density along this region with a width \u00180:5k\u00001\nF\n(see Fig. 3a), which is, however, below the resolution of\ncurrent experiments [8]. As shown in Fig. 3c), particle\nloss occurs predominantly at the domain wall in the cen-\nter of the trap and in the low-density halo on the surface\nof the clouds where the gas remains unpolarised due to\nlow densities.\nOne can thus expect that larger aspect ratios may re-\nduce losses due to a smaller area of the domain wall. To\ninvestigate this quantitatively we present the results of\nour calculations for the recombination rate in Fig. 4 as\na function of interaction strength kFaa. Starting from\na non-interacting gas in the paramagnetic ground state,\nthe recombination rate increases until phase separation\ndue to ferromagnetism sets in and a maximum is reached\natkFas\u00181:1, after which total losses are again reduced.\nA similar dependence of the loss rate on scattering length\nwas observed in the experiment [6] and theoretical stud-\nies [36, 37]. As the trap is compressed around the z-axis\nat constant volume \u0016 != (!2\nr!z)1=3and system size N\n(see inset of Fig. 4), the aspect ratio increases and the\nrecombination rate is indeed signi\fcantly reduced. How-\never, with a six times lower recombination rate at \u0015= 14\nFigure 4: Normalised recombination rate \u0000 =\u0000cin harmonic\ncon\fnement for aspect ratios \u0015=f2;5;10;14gand\nN\"=N#= 560 at di\u000berent interaction strengths. Inset:\ndependence of \u0000 =\u0000con the aspect ratio for k0\nFas\u00181:6. From\nsimulations of di\u000berent system sizes we checked that the results\nare not sensitive to the number of atoms in the system.\ncompared to \u0015= 2, the loss rate is still above the critical\nvalue \u0000c.\nWe now turn to the most promising way of stabil-\nising the ferromagnetic phase, by reducing the critical\nscattering length of the ferromagnetic phase transition.\nThis can be achieved by imposing a shallow optical lat-\ntice [24], which reduces the kinetic energy loss at the\nphase transition and thus favors the ferromagnetic state.\nFigure 5 shows the phase diagram at quarter \flling\n(\u0016n= 0:5) in a shallow 3D cubic optical lattice VOL(r) =\nV0P3\ni=1sin2(\u0019ri=d), whereV0is in units of recoil en-\nergyER=~2\u00192\n2md2anddis the lattice constant. At quarter\n\flling \u0016n= 0:5, the maximum tolerable kFas\u00180:4 corres-\nponds toas= 0:16d, which is nearly half the critical scat-\ntering length predicted by di\u000busion Monte Carlo [3, 4].\nWe see that a ferromagnetic phase exists at this scat-\ntering length for a lattice depth of about V0\u00182:5ER.\nHowever, the loss rate is likely signi\fcantly a\u000bected by\nthe presence of a periodic potential. To address this issue\nwe calculate the recombination rate of the paramagnetic\nstate in an optical lattice by integrating over the unit cell\nEq.(1).\nThe loss rate initially grows as the lattice is ramped up\nsince the density increases at the center of the unit cell\n(inset of Fig. 5). In very deep lattices the loss rate is re-\nduced when one approaches the regime where the physics\nis well described by the single band Hubbard model and\ntriple occupation of a lattice site is reduced. In Fig. 5 we\nshow contour lines of constant loss rate \u0000 and indicate\nby grey shading the regime where we expect three body\nlosses to be above the critical experimental value. We\n\fnd that despite the increased three-body losses in a shal-\nlow lattice, the ferromagnetic phase is stable in a wide\nregion of the phase diagram. Our calculations have not4\nFigure 5: Contours of constant normalised loss rate \u0000 =\u0000cat\nquarter \flling \u0016 n= 0:5 in an optical lattice. In the shaded region\n\u0000>\u0000c. Red (yellow) region denotes the fully (partially) polarized\nferromagnetic phase. Inset: Normalised loss rates \u0000 =\u0000cfor\nas=f0:12;0:14;0:16gat di\u000berent lattice depths. The dotted line\nis \u0000c. From simulations at di\u000berent \fllings bellow \u0016 n= 1:0, we\nchecked that the physics remains qualitatively the same.\ntaken into account the quantum Zeno e\u000bect [38] which\nwill further suppress three body recombination.\nIn summary, we have shown that although larger as-\npect ratios in harmonic con\fnement can signi\fcantly re-\nduce the total recombination rate, the gas remains un-\nstable. However, in an optical lattice the ferromagnetic\nphase extends down to small enough scattering lengths\nwhere three-body recombination is below the critical\nvalue. Experimental veri\fcation of our results will be\nsolid con\frmation of the Stoner model of itinerant ferro-\nmagnetism.\nWe thank S. Pilati, W. Ketterle, A. Volosniev and\nL. Tarruell for discussions. Simulations were performed\non the Swiss Center for Scienti\fc Computing (CSCS)\ncluster Monte Rosa in Lugano, Switzerland and the Bru-\ntus cluster at ETH Zurich. 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We investigate the magnetic properties of spin-1 /2 charged Fermi gases with ferromagnetic cou-\npling via mean-ﬁeld theory, and ﬁnd the interplay among the p aramagnetism, diamagnetism and ferromag-\nnetism. Paramagnetism and diamagnetism compete with each o ther. When increasing the ferromagnetic\ncoupling the spontaneous magnetization occurs in a weak mag netic ﬁeld. The critical ferromagnetic cou-\npling constant of the paramagnetic phase to ferromagnetic p hase transition increases linearly with the\ntemperature. Both the paramagnetism and diamagnetism incr ease when the magnetic ﬁeld increases. It\nreveals the magnetization density ¯Mincreases ﬁrstly as the temperature increases, and then rea ches a\nmaximum. Finally the magnetization density ¯Mdecreases smoothly in the high temperature region. The\ndomed shape of the magnetization density ¯Mvariation is diﬀerent from the behavior of Bose gas with\nferromagnetic coupling. We also ﬁnd the curve of susceptibi lity follows the Curie-Weiss law, and for a\ngiven temperature the susceptibility is directly proporti onal to the Land´ e factor.\nPACS.XX.XX.XX No PACS code given\n1 Introduction\nMagnetism of electron gases has been one of the central\nissues in condensed matter physics. At low temperature,\nFermi particles ﬁll the Fermi level from the lowest energy\nvalues, subject to the Pauli exclusion principle, which is\ndiﬀerent from the Bose gas. Since the observation of Bose-\nEinstein condensation, the trapped ultracold Fermi gases\nhave attracted great interest [1,2,3,4]. The measurement\nof magnetic susceptibility for ultracold Fermi gases gives\nagreement with the Pauli paramagnetism [5]. In the mag-\nnetism of magnetized pair-fermion gases, it is shown that\nthe intrinsic spin play an important role in relativistic\nparamagnetism or diamagnetism [6].\nBesides the ideal gases, interaction need to be consid-\nered for further understanding the magnetism of quan-\ntum gases. While the Heisenberg model can usually be\nused to explain the magnetic properties of quantum gases.\nWithin the two diﬀerent large- Nformulations, the low-\ntemperaturepropertiesofquantumHeisenbergmodelshave\nbeen investigated [7], both in ferromagnetic (FM) and\nantiferromagnetic (AFM) situation. A Heisenberg’s FM\nmodel has also been used to deal with the high tempera-\nture susceptibility [8].\nNot only localized electrons, but also the ferromag-\nnetismofitinerantelectronshasreceivedagreatdealofat-\ntention. The development of itinerant electron magnetism\najhqin@sas.ustb.edu.cnhas been outlined with emphasis on spin ﬂuctuations [9].\nResearch on itinerant ferromagnetism of a trapped two-\ndimensional atomic gas has shown that the eﬀective in-\nteraction strength is unaﬀected by the particle number\ndensity, although the FM phase is enhanced [10]. The\nFM phase transition of a two-dimensional itinerant elec-\ntrons Stoner Hamiltonian has been studied with quantum\nMonte Carlo calculations. It is shown that a ﬁrst-order\nFM transition occurred for short screening lengths with\na screened Coulomb interaction [11]. In spite of an in-\nﬁnitesimal value of the coupling can induce a FM phase\ntransition for spinor Bose gases[12], the Stoner coupling of\nFermi gases cannot lead to a FM phase transition unless it\nis larger than a threshold. The mechanisms of the Curie-\nWeisslawfortheitinerantelectronFMmaterialhavebeen\ninvestigated through the 1 /dexpansion theory [13].\nIn this paper, by using the mean-ﬁeld theory, the mag-\nnetic properties of charged spin-1 /2 Fermi gases with FM\ninteractions are investigated. Our results uncover a com-\npetition among paramagnetism, diamagnetism and ferro-\nmagnetism. We also present a comparison with the results\nof charged spin-1 Bose gas with FM interactions which\nhavebeen obtainedpreviously[14].As theincreaseoftem-\nperature,thereisnotapseudo-criticaltemperatureforthe\ncharged spin-1 /2 Fermi gases with FM interaction, which\nis diﬀerent from the case of Bose gas. The relationship of\nsusceptibility and temperature obey the Curie-Weiss law.\nIn section 2, a model consisting of Landau diamagnetism,\nPauli paramagnetism and the FM eﬀect is constructed.2 Baobao Wang et al.: Magnetic properties of spin-1 /2 Fermi gases with ferromagnetic interaction\nThen the magnetization density and susceptibility are cal-\nculated respectively. Section 3 presents a detailed discus-\nsion of obtained results. In section 4, we give a brief sum-\nmary.\n2 The Model\nWe consider a spin-1 /2 Fermi gas with FM couplings of N\nparticles, with the eﬀective Hamiltonian written as\n¯H−µN=DL/summationdisplay\nj,kz,σ/parenleftbig\nǫl\njkz+ǫze\nσ+ǫm\nσ−µ/parenrightbig\nnjkzσ,(1)\nwhereµis the chemicalpotential ofthe system. The quan-\ntized Landau levels of the charged fermions are\nǫl\njkz= (j+1\n2)/planckover2pi1ω+/planckover2pi12k2\nz\n2m∗, (2)\nwherej= 0,1,2,...labels diﬀerent Landau levels, and\nω=qB/(m∗c) is the gyromagnetic frequency, with charge\nq, eﬀective mass m∗and the magnetic induction intensity\nB. The degeneracy of the Landau levels is\nDL=qBS\n2π/planckover2pi1c, (3)\nwhereSis the section area of x-y plane of the system.\nThe Zeeman energy levels associated with the spin de-\ngree of freedom,\nǫze\nσ=−g/planckover2pi1(qB/m∗c)σ=−gσ/planckover2pi1ω, (4)\nwheregistheLand´ efactor,and σdenotesthespin-zindex\nof Zeeman state |F= 1/2,mF=σ/an}bracketri}ht(σ= 1/2,−1/2).\nThe contribution to the eﬀective Hamiltonian from the\nFM couplings [15] is\nǫm\nσ=−4Iσ(m+2σnσ), (5)\nwhereIdenotes FM coupling and spin polarization m=\nn1\n2−n−1\n2.\nThen we obtain the grand thermodynamic potential\nΩT/negationslash=0=−1\nβlnTre−β(¯H−µN)\n=−1\nβDL/summationdisplay\nj,kz,σln/bracketleftBig\n1+e−β(ǫl\njkz+ǫze\nσ+ǫm\nσ−µ)/bracketrightBig\n,(6)\nwhereβ= 1/(kBT). Through the Taylor expansions and\nintegral over kz, we have\nΩT/negationslash=0=−ωV\n/planckover2pi12(m∗\n2πβ)3\n2∞/summationdisplay\nl=1\n/summationdisplay\nσ(−1)l+1l−3\n2e−lβ[1\n2/planckover2pi1ω−gσ/planckover2pi1ω−4Iσ(m+2σnσ)−µ]\n1−e−lβ/planckover2pi1ω,(7)\nwhereVis the volume of the system. A similar treatment\nhas been used to deal with the diamagnetism of scalarBose gases [16,17,18]. We have extended it to further\nstudy the competition between diamagnetism and param-\nagnetism of charged spin quantum gases [19,20]. As far as\nFM interaction is concerned, it is shown that the mean-\nﬁeld theory is still eﬀective in understanding the main\nphysics of magnetism [12,14,15].\nSome compact notations for the class of sums are in-\ntroduced for simplicity,\nFσ\nτ[α,δ] =∞/summationdisplay\nl=1(−1)l+1lα/2e−lβ/planckover2pi1ω[1\n2−gσ−4Iσ(m+2σnσ)\n/planckover2pi1ω−µ\n/planckover2pi1ω+δ]\n(1−e−lβ/planckover2pi1ω)τ .\n(8)\nThen we may rewrite equation (7) as\nΩT/negationslash=0=−ωV\n/planckover2pi12(m∗\n2πβ)3\n2/summationdisplay\nσFσ\n1[−D,0],(9)\nwhereD= 3 is the space dimensionality.\nThen the particle number density n=N/Vcan be\nobtained through the grand thermodynamic potential,\nn=−1\nV/parenleftbigg∂ΩT/negationslash=0\n∂µ/parenrightbigg\nT,V\n=x(m∗\n2πβ/planckover2pi12)3\n2/summationdisplay\nσFσ\n1[−1,0],(10)\nwherex=β/planckover2pi1ω.\nTaking the grand thermodynamic potential derivative\nwith respect to the magnetic induction intensity B, the\ntotal magnetization density can be obtained\nMT/negationslash=0=−1\nV/parenleftbigg∂ΩT/negationslash=0\n∂B/parenrightbigg\nT,V\n=q/planckover2pi1\nm∗c(m∗\n2πβ/planckover2pi12)3\n2\n/summationdisplay\nσ/braceleftbigg\nFσ\n1[−3,0]+x/bracketleftbigg/parenleftbigg\ngσ−1\n2/parenrightbigg\nFσ\n1[−1,0]−Fσ\n2[−1,1]/bracketrightbigg/bracerightbigg\n.\n(11)\nIn external magnetic ﬁeld H, we have\nB=H+4πM. (12)\nIt is convenient to introduce some dimensionless pa-\nrameters, such as t=T/T∗,¯M=m∗cM/(n/planckover2pi1q),¯ω=\n/planckover2pi1ω/(kBT∗),¯I=In/(kBT∗),¯µ=µ/(kBT∗),¯m=m/n,¯nσ=\nnσ/n,h=q/planckover2pi1H/(m∗ckBT∗), andx= ¯ω/t, the character-\nistic temperature of the system T∗is given by kBT∗=\n2π/planckover2pi12n2\n3/m∗.Accordingly,wecanre-expressequations(10),\n(11) and (12),\n1 = ¯ωt1\n2/summationdisplay\nσ¯Fσ\n1[−1,0], (13a)Baobao Wang et al.: Magnetic properties of spin-1 /2 Fermi gases with ferromagnetic interaction 3\n¯MT/negationslash=0=t3\n2\n/summationdisplay\nσ/braceleftbigg\n¯Fσ\n1[−3,0]+¯ω\nt/bracketleftbigg/parenleftbigg\ngσ−1\n2/parenrightbigg\n¯Fσ\n1[−1,0]−¯Fσ\n2[−1,1]/bracketrightbigg/bracerightbigg\n,\n(13b)\n¯ω=h+4πγ¯M, (13c)\nwhereγ= (q2n1/3)/(2πm∗c2), and\n¯Fσ\nτ[α,δ] =∞/summationdisplay\nl=1(−1)l+1lα/2e−l¯ω\nt/bracketleftBig\n1\n2−gσ−4¯Iσ( ¯m+2σ¯nσ)\n¯ω−¯µ\n¯ω+δ/bracketrightBig\n/parenleftbig\n1−e−l¯ω\nt/parenrightbigτ ,\n(14)\nwhere ¯µis the dimensionless parameter of the chemical\npotential, which can be determined from the mean-ﬁeld\nself-consistent calculations.\nAt last, we calculated the susceptibility of charged\nFermi gases with FM couplings. From the formula χM=\n(∂M\n∂H)T,V,the derivationofequation(11)forthe magnetic\nﬁeldH, the expression of susceptibility can be obtained\nas follows,\nχM=γb\n¯ω−4πγb, (15)\nwith\nb= 2¯ωt1/2/summationdisplay\nσ{(gσ−1\n2)¯Fσ\n1[−1,0]−¯Fσ\n2[−1,1]}−¯ω2t−1/2\n/summationdisplay\nσ{2(gσ−1)¯Fσ\n2[1,1]−(gσ−1\n2)2¯Fσ\n1[1,0]−2¯Fσ\n3[1,2]}.\n(16)\n3 Results and discussions\nIn the following discussions we will focus on the competi-\ntion of magnetism, and explore the factors that determine\nthe magnetic competition. Meanwhile a comparison with\nthe results of Bose gases with FM coupling will also be\npresented.\nFirstly,thedimensionlessmagnetizationdensity ¯Mand\n¯m= ¯n1/2−¯n−1/2versus¯Iis shown in ﬁgure 1. ¯Icis used\nto describe the critical value of reduced FM coupling con-\nstant of the paramagnetic phase to ferromagnetic phase\ntransition. We can ﬁnd the crossover of ¯Mfrom ﬁgure\n1(a). When the reduced FM coupling constant ¯Iis smaller\nthan¯Ic,¯Mis always equal to zero. And then ¯Mbegins to\nincrease with increasing ¯Ifrom¯Ic. It suggests that there\nexists a spontaneous magnetization with the increase of\n¯Iin the weak magnetic ﬁeld. Figure 1(b) indicates that\nalthough the value of land´ e factor gchanges, the evolve-\nment of ¯mversus¯Ialmost overlaps. And the value of ¯Ic\nis identical for the given reduced temperature, in despite\nof the values of land´ e factor are diﬀerent. While the criti-\ncal value of FM coupling constant ¯Icapproximates to 0.5,/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s98/s41\n/s32/s103/s61/s48/s46/s51\n/s32/s103/s61/s48/s46/s52\n/s32/s103/s61/s48/s46/s53\n/s32/s103/s61/s48/s46/s54\n/s32/s103/s61/s48/s46/s56\n/s32/s103/s61/s49/s46/s48\n/s32/s109\n/s73/s40/s97/s41/s32\n/s32/s77\nFig. 1. (a) The reduced magnetization density ¯M, (b) ¯m=\n¯n1\n2−¯n−1\n2as a function of reduced FM coupling constant ¯Iat\nreduced temperature t= 1.5 and the reduced magnetic ﬁeld\nh= 0.005. The Land´ e factor gis chosen as 0.3 (solid line),\n0.4 (dashed line), 0.5 (dotted line), 0.6 (dash dotted line) , 0.8\n(dash dot dotted line), and 1.0 (short dashed line).\nwhich can be evaluated from ﬁgure 1. The internal ﬁeld\ncomes from the spontaneous magnetization results in the\ndiamagnetism [14]. Figure 1 shows that ferromagnetism\nexceeds the diamagnetism with increasing ¯I.\nSince¯Icis an important parameter in the transfor-\nmation between FM phase and paramagnetic phase. The\nthreshold of ¯Icvs temperature is plotted in ﬁgure 2. The\nFM phasesituates above ¯Ic, while the paramagneticphase\nlocates under the ¯Ic. We can ﬁnd that the value of ¯Ic\nincreases linearly with the increase of temperature. The\nresults is similar to the Bose gas with FM coupling [14],\nalthough they submit to diﬀerent statistical rules, respec-\ntively. It indicates that the crossover from paramagnetic\nphase to FM phase is more diﬃcult with increasing the\ntemperature.\nAfter studying the spontaneous magnetization in the\nweak magnetic ﬁeld, ﬁgure 3 is plotted in order to un-\nderstand the inﬂuence of Land´ e factor g. In ﬁgure 3, the\nmagnetic ﬁeld is chosen as h= 0.8 and the reduced tem-\nperature t= 1.5. The horizontal line ¯M= 0 in ﬁgure\n3(a) is used to distinguish the region of paramagnetism\nand diamagnetism. It is shown that ¯Mis a negative value4 Baobao Wang et al.: Magnetic properties of spin-1 /2 Fermi gases with ferromagnetic interaction\n/s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s32/s73\n/s67\n/s32\n/s116\nFig. 2. ¯Icversus reduced temperature t phase diagram of\ncharged spin-1 /2 Fermi gases at magnetic ﬁeld h= 0.005.\nwhengis small, which mainly derives from the diamag-\nnetic contribution. From ﬁgure 3(b) we can ﬁnd that ¯ m\nincrease with increasing of Land´ e factor guntil approxi-\nmates saturation at higher ¯I. It indicates that the larger\n¯Ithe larger ¯ mfor the identical value of Land´ e factor g,\nwhere the Land´ e factor denotes the intensity of the para-\nmagnetism [19,20]. From ﬁgure 3(a) and 3(b), we can ﬁnd\nthattheenhanceofferromagnetismstimulatestheincreas-\ning of paramagnetism. That is ferromagnetism cooperates\nwith paramagnetism to confront diamagnetism in a ﬁnite\nmagnetic ﬁeld. This is diﬀerent from the result of the Bose\ngas with FM coupling [14]. Where it is faintly aﬀected by\nthe FM coupling in the evolvement of magnetization den-\nsity with g.\nFigure 4 is plotted to get more insight of the depen-\ndence of the magnetization density on magnetic ﬁeld. The\nsystem presents diamagnetism when gis small. Further-\nmore, there exists a threshold gcwhen 0.3< g <0.6,\nwhich making ¯M <0 whileg < gc. This is independent\nof the magnetic ﬁeld. From the inset of ﬁgure 4(a), we\ncan ﬁnd that the magnetization density approximates to\na small positive value at ﬁrst when g= 0.6. While g≥0.6,\nthe magnetization density increases ﬁrstly with increasing\nthe magnetic ﬁeld, and then declines up to reach a satura-\ntion value. It is shown that the paramagnetism competes\nwith diamagnetism in this deﬁnite FM coupling situation./s48 /s49 /s50/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s40/s98/s41\n/s32/s109\n/s103/s32/s73 /s61/s48/s46/s48\n/s32/s73 /s61/s48/s46/s49\n/s32/s73 /s61/s48/s46/s51\n/s32/s73 /s61/s48/s46/s52\n/s32/s73 /s61/s48/s46/s53/s32\n/s32/s77/s40/s97/s41\nFig. 3. (a) The reduced magnetization density M, (b) ¯m=\n¯n1/2−¯n−1/2versus Land´ e factor gat reduced temperature\nt= 1.5 andh= 0.8, where ¯I=0.0 (solid line), 0.1 (dashed\nline), 0.3 (dotted line), 0.4 (dash dotted line), and 0.5 (da sh\ndot doted line).\nThe increasing of Land´ e factor promotes paramagnetism,\nwhile the magnetic ﬁeld facilitates diamagnetism.\nFor a more detailed understanding of the paramag-\nnetism and diamagnetism respectively, now we turn to\nexamine the paramagnetization density ¯Mp=g¯mand\ndiamagnetization density ¯Md=¯M−¯Mpin ﬁgure 5. It is\nshown that both the paramagnetization density and dia-\nmagnetization density increases with enhancing the mag-\nnetic ﬁeld at ﬁrst. And then the paramagnetization den-\nsity¯Mptends to saturate when the magnetic ﬁeld inten-\nsity is strong. While the absolute value of diamagnetiza-\ntion density |¯Md|increases with the increase of magnetic\nﬁeld intensity. When the magnetic ﬁeld continues to in-\ncrease, the diamagnetism is close to a saturated value.\nTherefore for the higher magnetic ﬁeld the total magne-\ntization density ¯Mreaches to saturated values. This is\nin conformity with the result of ﬁgure 4 in qualitatively.\nWith the increase ofmagnetic ﬁeld, the contribution come\nfrom ferromagnetism, paramagnetism and diamagnetism\nreach to maximum, so the peak of total magnetization\ndensity appears when Land´ e factor gis 0.8 and 1.0 in\nﬁgure 4. Whereas the diamagnetism increases faster than\nparamagnetismasthemagneticﬁeldcontinuestoincrease.Baobao Wang et al.: Magnetic properties of spin-1 /2 Fermi gases with ferromagnetic interaction 5\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53\n/s48 /s53 /s49/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s40/s98/s41\n/s32/s103/s61/s48/s46/s49\n/s32/s103/s61/s48/s46/s51\n/s32/s103/s61/s48/s46/s54\n/s32/s103/s61/s48/s46/s56\n/s32/s103/s61/s49/s46/s48\n/s32/s109\n/s104/s104 /s32\n/s32/s77/s40/s97/s41/s32\n/s32/s77\n/s104\nFig. 4. (a) The reduced magnetization density M, (b) ¯m=\n¯n1/2−¯n−1/2as afunction of reducedmagnetic ﬁeld hatt= 1.5\nandI= 0.1, where Land´ e factor g=0.1(solid line), 0.3 (dashed\nline), 0.6 (dottedline), 0.8 (dashdottedline), and1.0 (da shdot\ndotted line). The inset of ﬁgure 4(a) depicts the relationsh ip of\nreduced magnetization density ¯Mversus magnetic ﬁeld hwith\nthe magnetic ﬁeld region lies between 0 and 10.\nIt accounts for the total magnetization density decreases\nwith the increase of magnetic ﬁeld until nearly saturated,\nwhich can be found in ﬁgure 4.\nThe characteristic parameter γhas been set to 10−6in\nﬁgures 1∼5, where the particle number density is 8 /nm3\nand the charge and mass are evaluated from a thin elec-\ntron gas. To further investigate the FM phase transition\nof the charged spin-1 /2 Fermi gases in a broad tempera-\nture region including low temperature, γ= 10 is assumed.\nFigure 6 shows ¯Mand ¯mas a function of the tempera-\nturetwheng= 0.8 in magnetic ﬁeld h= 0.005. It de-\nnotes that the maximal ¯Moccurs at a deﬁnite tempera-\nture, and then decreases in both lower temperature and\nhighertemperature regimes.Moreover, ¯Mdecreasesfaster\nin the low temperature region than the case of the Bose\ngas with FM coupling [14]. It reﬂects the diamagnetism\nstrengthens greatly in this region for Fermi gas. With in-\ncreasing the temperature, a ﬂat decline appears when ¯M\nis close to zero, which can be seen clearly from the inset\nof ﬁgure 6(a). This is obviously diﬀerent from the Bose\ngas. In our previous study on Bose gas with FM coupling/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s45/s49/s46/s48/s48/s45/s48/s46/s55/s53/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s40/s98/s41\n/s32/s103/s61/s48/s46/s49\n/s32/s103/s61/s48/s46/s51\n/s32/s103/s61/s48/s46/s54\n/s32/s103/s61/s48/s46/s56\n/s32/s103/s61/s49/s46/s48\n/s32/s77\n/s100\n/s104/s32\n/s32/s77\n/s112/s40/s97/s41\nFig. 5. (a) The reduced paramagnetization density Mp, (b)\nthe reduced diamagnetization density ¯Mdas a function of re-\nduced magnetic ﬁeld hatt= 1.5 andI= 0.1, where Land´ e\nfactorg=0.1 (solid line), 0.3 (dashed line), 0.6 (dotted line),\n0.8 (dash dotted line), and 1.0 (dash dot dotted line).\n[14], a sharp decline emerges when ¯Mapproaches to zero,\nwhich suggeststhat there exists a pseudo-condensatetem-\nperature in the transition from ferromagnetism to para-\nmagnetism. However, at high temperature region for the\ncharged Fermi gases with FM interaction, ¯Mand ¯mare\nasymptotic with respect to the zero point for diﬀerent\nvalues of ¯I. This demonstrates that there does not ex-\nist the pseudo-critical temperature for Fermi gases. The\ndiﬀerence between Fermi gases and Bose gases may be\nattributed to the diﬀerent statistical distribution.\nFigure 7 plots the evolution of reciprocal of suscepti-\nbility with the reduced temperature t, where the charac-\nteristic parameter γis reset as 10−6. It is shown that the\ncurves of 1 /χis proportional to the reduced temperature\nt. This suggests that the susceptibility of Fermi gaseswith\nFM coupling conforms to the Curie-Weiss law. When the\ntemperature is ﬁxed, the susceptibility increases with the\nincrease of the Land´ e factor g. Since the Land´ e factor g\ndenotes the strength of paramagnetism, it suggests that\nthe increase of susceptibility is mainly attributed to the\ncontribution of paramagnetism at the ﬁxed temperature\nand magnetic ﬁeld.6 Baobao Wang et al.: Magnetic properties of spin-1 /2 Fermi gases with ferromagnetic interaction\n/s48 /s53 /s49/s48 /s49/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53\n/s49/s48/s46/s48 /s49/s50/s46/s53 /s49/s53/s46/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56\n/s40/s98/s41\n/s32/s109\n/s116/s32/s73 /s61/s48/s46/s49\n/s32/s73 /s61/s48/s46/s53\n/s32/s73 /s61/s48/s46/s54\n/s32/s73 /s61/s48/s46/s56\n/s32/s73 /s61/s49/s46/s48/s32\n/s32/s77/s40/s97/s41\n/s32\n/s32/s77\n/s116\nFig. 6. (a) The reduced magnetization density ¯M, (b) ¯m=\n¯n1/2−¯n−1/2versus reduced temperature tat reduced magnetic\nﬁeldh= 0.005 and Land´ e factor g= 0.8, where I=0.1 (solid\nline), 0.5 (dashed line), 0.6 (dotted line), 0.8 (dash dotte dline),\nand 1.0 (dash dot dotted line). The inset of ﬁgure 6(a) depict s\nthe smooth decline of reduced magnetization density ¯Mwith\nthe reduced temperature lies between 10 and 15.\n4 Summary\nThis paper has revealed the interplay among paramag-\nnetism, diamagnetism and ferromagnetism of the charged\nspin-1/2 Fermi gases with FM interaction. The paramag-\nnetic eﬀect is described by the Land´ e factor g. Our re-\nsults show that the ferromagnetism overcomes the dia-\nmagnetism when ¯I >¯Icin weak magnetic ﬁeld. ¯Icin-\ncreaseslinearly with the increaseof temperature. With in-\ncreasing g, the gas presents paramagnetism from diamag-\nnetism. The reduced magnetization density ¯Mdeclines in\nthe high magnetic ﬁeld region, which indicates that the\ncontribution comes from diamagnetism enhances in the\nregion of strong magnetic ﬁeld. As the increase of temper-\nature, the magnetization density asymptotically approxi-\nmates to zero, which is diﬀerent from the case of Bose gas\nwith FM coupling. It indicates that there is not a pseudo-\ncritical temperature for charged Fermi gases with FM in-\nteraction. The diagram of susceptibility is in accordance\nwith the Curie-Weiss law./s49 /s51 /s53 /s55 /s57 /s49/s49/s48/s46/s48/s54/s46/s48/s120/s49/s48/s55/s49/s46/s50/s120/s49/s48/s56/s49/s46/s56/s120/s49/s48/s56/s50/s46/s52/s120/s49/s48/s56/s51/s46/s48/s120/s49/s48/s56\n/s32/s103/s61/s48/s46/s55\n/s32/s103/s61/s48/s46/s56\n/s32/s103/s61/s48/s46/s57\n/s32/s103/s61/s49/s46/s48\n/s32/s103/s61/s49/s46/s53/s32\n/s116/s32/s45/s49\nFig. 7. The reciprocal of susceptibility 1 /χversus reduced\ntemperature tat reduced magnetic ﬁeld h= 0.005 and reduced\nFM coupling constant ¯I=0.5. The Land´ e factors: g=0.7 (solid\nline), 0.8 (dashedline), 0.9 (dottedline), 1.0 (dash dotte d line),\nand 1.5 (dash dot dotted line).\nWe would like to thank Professor Qiang Gu for the helpful\ndiscussions. BW and JQ are supported by the Fundamental\nResearch Funds for the Central Universities under Grant No.\nFRF-TP-14-074A2, the Beijing Higher Education Young Elite\nTeacher Project under Grant No. 0389, and the National Nat-\nural Science Foundation of China under Grant No. 11004006,\nand HG is supported by the National Natural Science Founda-\ntion of China under Grant No. 11274032.\nAll authors contributed equally to this paper.\nReferences\n1. J. Noronha, D.J. Toms, Phys. Lett. A 267, 276 (2000)\n2. S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phy s.\n80, 1215 (2008)\n3. I. Bloch, Nature Physics 1, 23 (2005)\n4. D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, T. Esslinge r,\nScience340, 1307 (2013)\n5. Y.R. Lee, T.T. Wang, T.M. Rvachov, J.H. Choi, W. Ket-\nterle, M.S. Heo, Phys. Rev. A 87, 043629 (2013)\n6. J. Daicic, N.E. Frankel, R.M. Gailis, V. Kowalenko, Phys.\nRep.237, 63 (1994)\n7. D.P. Arovas, A. Auerbach, Phys. Rev. B 38, 316 (1988)Baobao Wang et al.: Magnetic properties of spin-1 /2 Fermi gases with ferromagnetic interaction 7\n8. G.S. Rushbrooke, P.J. Wood, Proceedings of the Physical\nSociety. Section A 68, 1161 (1955)\n9. T. Moriya, Journal of magnetism and magnetic materials\n100, 261 (1991)\n10. G.J. Conduit, Phys. Rev. A 82, 043604 (2010)\n11. G.J. Conduit, Phys. Rev. B 87, 184414 (2013)\n12. Q. Gu, R. A. Klemm, Phys. Rev. A 68, 031604 (2003)\n13. E. Miyai, F.J. Ohkawa, Phys. Rev. B 61, 1357 (2000)\n14. J.H. Qin, X.L. Jian, Q. Gu, J. Phys.: Condensed Matter\n24, 366007 (2012)\n15. C.J. Tao, P.L. Wang, J.H. Qin, Q. Gu, Phys. Rev. B 78,\n134403 (2008)\n16. D. J. Toms, Phys. Rev. B 50, 3120 (1994)\n17. D. J. Toms, Phys. Rev. D 51, 1886 (1995)\n18. G. B. Standen, D.J. Toms, Phys. Rev. E 60, 5275 (1999)\n19. X.L. Jian, J.H. Qin, Q. Gu, Phys. Lett. A 374, 2580 (2010)\n20. X.L. Jian, J.H. Qin, Q. Gu, J. Phys.: Condensed Matter\n23, 026003 (2011)"    },    {        "title": "1005.3426v1.Non_local_spin_sensitive_electron_transport_in_diffusive_proximity_heterostructures.pdf",        "content": "arXiv:1005.3426v1  [cond-mat.supr-con]  19 May 2010Non-local spin-sensitive electron transport in diﬀusive p roximity heterostructures\nMikhail S. Kalenkov\nI.E. Tamm Department of Theoretical Physics, P.N. Lebedev P hysics Institute, 119991 Moscow, Russia\nAndrei D. Zaikin\nInstitute for Nanotechnology, Karlsruhe Institute of Tech nology (KIT), 76021 Karlsruhe, Germany and\nI.E. Tamm Department of Theoretical Physics, P.N. Lebedev P hysics Institute, 119991 Moscow, Russia\nWe formulate a quantitative theory of non-local electron tr ansport in three-terminal disordered\nferromagnet-superconductor-ferromagnet structures. We demonstrate that magnetic eﬀects have\ndiﬀerent implications: While strong exchange ﬁeld suppres ses disorder-induced electron interference\nin ferromagnetic electrodes, spin-sensitive electron sca ttering at superconductor-ferromagnet inter-\nfaces can drive the total non-local conductance g12negative at suﬃciently low energies. At higher\nenergies magnetic eﬀects become less important and the non- local resistance behaves similarly to\nthe non-magnetic case. Our predictions can be directly test ed in future experiments on non-local\nelectron transport in hybrid FSFstructures.\nPACS numbers: 74.45.+c, 72.25.Ba, 73.23.-b, 74.78.Na\nI. INTRODUCTION\nThe phenomenon of Andreev reﬂection (AR)1is well\nknownto be responsiblefortransportofsubgapelectrons\nacrossan interface between a normal metal ( N) and a su-\nperconductor ( S). While this phenomenon is essentially\nlocalin hybrid proximitystructureswith only one NSin-\nterface,thesituationinmultiterminaldeviceswithtwoor\nmoreNSinterfaces (such as, e.g., NSNstructures) can\nbemorecomplicatedbecauseinadditiontolocalARelec-\ntrons can suﬀer non-local or crossed Andreev reﬂection\n(CAR)2. ThisphenomenonofCARenablesdirectexperi-\nmental demonstration of entanglement between electrons\nin spatially separated N-electrodes and can strongly in-\nﬂuence non-local transport of electrons in hybrid NSN\nsystems3,4.\nNon-local electron transport in the presence of CAR\nwas recently investigated both experimentally5–10and\ntheoretically3,4,11–19demonstratingarichvarietyofphys-\nical processes involved in the problem. For instance, the\neﬀect of CAR on the subgap non-local conductance of\nNSNstructuresis exactlycompensated byelastic cotun-\nneling (EC) provided only the lowest order terms in NS\ninterface transmissions are accounted for3. Taking into\naccount higher order processes in barrier transmissions\neliminatesthisfeatureandyieldsnon-zerovaluesofcross-\nconductance4. One can also expect that interactions11or\nexternal ac bias12can lift the cancellation between EC\nand CAR contributions already in the lowest order in\nbarrier transmissions.\nAnother non-trivial issue is the eﬀect of disorder. The-\noretical analysis of CAR in diﬀerent disordered NSN\nstructures was carried out in Refs. 13–17. In particular,\nit was demonstrated17that an interplay between CAR,\nquantum interference of electrons and non-local charge\nimbalance dominates the behavior of diﬀusive NSNsys-\ntems being essential for quantitative interpretation of a\nnumber of experimental observations7–9.Yet another important property of both local and non-\nlocal Andreev reﬂection processes is that they essentially\ndepend on spins of scattered electrons. Hence, CAR\nshould be sensitive to magneticproperties ofnormalelec-\ntrodes. This sensitivitywasindeed demonstratedalready\nin the ﬁrst experiments on ferromagnet-superconductor-\nferromagnet ( FSF) structures5where the dependence\nof non-local conductance on the polarization of ferro-\nmagnetic terminals was found. Theoretical analysis of\nspin-resolved CAR was carried out in Ref. 3 in the low-\nest order order in tunneling and in Refs. 18, 19 to all\norders in the interface transmissions. This analysis re-\nvealed a number of non-trivial features of non-local spin-\ndependent electron transport which can be tested in fu-\nture experiments.\nNote that previous work3,18,19merely concentrated\non ballistic electrodes whereas in realistic experiments\none usually deals with diﬀusive hybrid FSFstructures.\nTherefore it is highly desirable to formulate a theory\nwhich would adequately describe an interplay between\ndisorder and spin-resolved CAR. This is the main goal\nof the present paper. The structure of our paper is as\nfollows. In Sec. 2 we will formulate our model and\noutline our basic formalism of quasiclassical Green func-\ntions. This formalism will be employed in Sec. 3 where\nwe present the solution of Usadel equations and derive\ngeneralexpressions for the non-localspin-dependent con-\nductance and resistance for diﬀusive three-terminal FSF\nstructures at diﬀerent directions of interface magnetiza-\ntions. Concluding remarks are presented in Sec. 4 of our\npaper.\nII. MODEL AND BASIC FORMALISM\nLet us consider a three-terminal diﬀusive FSFstruc-\nture schematically shown in Fig. 1. Two ferromagnetic\nterminals F1andF2with resistances rN1andrN2and\nelectricpotentials V1andV2areconnected to asupercon-2\nFIG. 1: FSF structure under consideration.\nducting electrode of length Lwith normal state (Drude)\nresistance rLand electric potential V= 0 via tunnel bar-\nriers. The magnitude of the exchange ﬁeld h1,2=|h1,2|\nin both ferromagnets F1andF2is assumed to be much\nbigger than the superconducting order parameter ∆ of\ntheS-terminal and, on the other hand, much smaller\nthat the Fermi energy, i.e. ∆ ≪h1,2≪ǫF.\nThe latter condition allows to perform the analysis of\nourFSFsystem within the quasiclassical formalism of\nUsadel equations for the Green-Keldyshmatrix functions\nG. In each of our metallic terminals these equations can\nbe written in the form20\niD∇(ˇG∇ˇG) = [ˇΩ+eV,ˇG],ˇG2= 1,(1)\nwhereDis the diﬀusion constant, Vis the electric poten-\ntial,ˇGandˇΩ are 8×8 matrices in Keldysh-Nambu-spin\nspace (denoted by check symbol)\nˇG=/parenleftbigg˘GR˘GK\n0˘GA/parenrightbigg\n,ˇΩ =/parenleftbigg˘ΩR0\n0˘ΩA/parenrightbigg\n,(2)\n˘ΩR=˘ΩA=/parenleftbigg\nε−ˆσh∆\n−∆∗−ε+ˆσh/parenrightbigg\n, (3)\nεisthe quasiparticleenergy,∆( T)isthesuperconducting\norder parameter which will be considered real in a super-\nconductor and zero in both ferromagnets, h≡h1(2)in\nthe ﬁrst (second) ferromagnetic terminal, h≡0 outside\nthese terminals and ˆσ= (ˆσ1,ˆσ2,ˆσ3) are Pauli matrices\nin spin space.\nRetarded and advanced Green functions ˘GRand˘GA\nhave the following matrix structure\n˘GR,A=/parenleftbiggˆGR,AˆFR,A\n−ˆFR,A−ˆGR,A/parenrightbigg\n. (4)\nHere and below 2 ×2 matrices in spin space are denoted\nby hat symbol.\nHavingobtainedtheexpressionsfortheGreen-Keldysh\nfunctions ˇGone can easily evaluate the current density j\nin our system with the aid of the standard relation\nj=−σ\n16e/integraldisplay\nSp[τ3(ˇG∇ˇG)K]dε, (5)\nwhereσis the Drude conductivity of the corresponding\nmetal and τ3is the Pauli matrix in Nambu space.In what follows it will be convenient for us to employ\nthe so-calledLarkin-Ovchinnikovparameterizationofthe\nKeldysh Green function\n˘GK=˘GR˘f−˘f˘GA,˘f=ˆfL+τ3ˆfT,(6)\nwhere the distribution functions ˆfLandˆfTare 2×2\nmatrices in the spin space.\nFor the sake of simplicity we will assume that magne-\ntizations of both ferromagnets and the interfaces (see be-\nlow) are collinear. Within this approximation the Green\nfunctions and the matrix ˇΩ are diagonalin the spin space\nand the diﬀusion-like equations for the distribution func-\ntion matrices ˆfLandˆfTtake the form\n−D∇/parenleftBig\nˆDT(r,ε)∇ˆfT(r,ε)/parenrightBig\n+2ˆΣ(r,ε)ˆfT(r,ε) = 0,(7)\n−D∇/parenleftBig\nˆDL(r,ε)∇ˆfL(r,ε)/parenrightBig\n= 0, (8)\nwhere\nˆΣ(r,ε) =−i∆ImˆFR, (9)\nˆDT=/parenleftBig\nReˆGR/parenrightBig2\n+/parenleftBig\nImˆFR/parenrightBig2\n, (10)\nˆDL=/parenleftBig\nReˆGR/parenrightBig2\n−/parenleftBig\nReˆFR/parenrightBig2\n. (11)\nThe function ˆΣ(r,ε) diﬀers from zero only inside the su-\nperconductor. It accounts both for energy relaxation of\nquasiparticles and for their conversion to Cooper pairs\ndue to Andreev reﬂection. The functions ˆDTandˆDLac-\nquire space and energy dependencies due to the presence\nofthesuperconductingwireandrenormalizethediﬀusion\ncoeﬃcient D.\nThe solution of Eqs. (7)-(8) can be expressed in terms\nof the diﬀuson-like functions ˆDTandˆDLwhich obey the\nfollowing equations\n−D∇/bracketleftBig\nˆDT(r,ε)∇ˆDT(r,r′,ε)/bracketrightBig\n+2ˆΣ(r,ε)ˆDT(r,r′,ε) =δ(r−r′),(12)\n−D∇/bracketleftBig\nˆDL(r,ε)∇ˆDL(r,r′,ε)/bracketrightBig\n=δ(r−r′).(13)\nThesolutionsofUsadelequation(1)ineachofthemet-\nals should be matched at SF-interfaces by means of ap-\npropriateboundaryconditionswhichaccountforelectron\ntunneling between these terminals. The form of these\nboundary conditions essentially depends on the adopted\nmodel describing electron scattering at SF-interfaces.\nHere we stick to the model of the so-called spin-active\ninterfaces21which takes into account possibly diﬀerent\nbarrier transmissions for spin-up and spin-down elec-\ntrons. This model was already extensively used for the-\noretical description of diﬀerent physical phenomena, in-\ncluding spin-resolvedCAR in ballistic structures18,19and\nJosephson eﬀect with triplet pairing22,23. Here we em-\nploy this model in the case of diﬀusive electrodes and\nalso restrict our analysis to the case of tunnel barriers3\nwith channel transmissions much smaller than one. In\nthis case the corresponding boundary conditions read24\nAσ+ˇG+∂xˇG+=GT\n2[ˇG−,ˇG+]\n+Gm\n4[{ˆσmτ3,ˇG−},ˇG+]+iGϕ\n2[ˆσmτ3,ˇG+],(14)\n−Aσ−ˇG−∂xˇG−=GT\n2[ˇG+,ˇG−]\n+Gm\n4[{ˆσmτ3,ˇG+},ˇG−]+iGϕ\n2[ˆσmτ3,ˇG−],(15)\nwhereˇG−andˇG+are the Green-Keldysh functions from\nthe left ( x <0) and from the right ( x >0) side of the in-\nterface,Ais the eﬀective contact area, mis the unit vec-\ntor in the direction ofthe interfacemagnetization, σ±are\nDrude conductivities of the left and right terminals and\nGTis the spin-independent part of the interface conduc-\ntance. Along with GTthere also exists the spin-sensitive\ncontribution to the interface conductance which is ac-\ncounted for by the Gm-term, whereas the Gϕ-term arises\ndue to diﬀerent phase shifts acquired by scattered quasi-\nparticles with opposite spin directions.\nEmploying the above boundary conditions we can es-\ntablish the following linear relations between the distri-\nbution functions at both sides of the interface\nAσ+ˆDT\n+∂xˆf+T=Aσ−ˆDT\n−∂xˆf−T\n= ˆgT(ˆf+T−ˆf−T)+ ˆgm(ˆf+L−ˆf−L),(16)\nAσ+ˆDL\n+∂xˆf+L=Aσ−ˆDL\n−∂xˆf−L\n= ˆgL(ˆf+L−ˆf−L)+ ˆgm(ˆf+T−ˆf−T),(17)\nwhere ˆgT, ˆgL, and ˆgmare matrix interface conductances\nwhich depend on the retarded and advanced Green func-\ntions at the interface\nˆgT=GT/bracketleftBig/parenleftBig\nReˆGR\n+/parenrightBig/parenleftBig\nReˆGR\n−/parenrightBig\n+/parenleftBig\nImˆFR\n+/parenrightBig/parenleftBig\nImˆFR\n−/parenrightBig/bracketrightBig\n,\n(18)\nˆgL=GT/bracketleftBig/parenleftBig\nReˆGR\n+/parenrightBig/parenleftBig\nReˆGR\n−/parenrightBig\n−/parenleftBig\nReˆFR\n+/parenrightBig/parenleftBig\nReˆFR\n−/parenrightBig/bracketrightBig\n,\n(19)\nˆgm=Gmˆσm/parenleftBig\nReˆGR\n+/parenrightBig/parenleftBig\nReˆGR\n−/parenrightBig\n.(20)\nThe current density (5) can then be expressed in terms\nof the distribution function ˆfTas\nj=−σ\n4e/integraldisplay\nSp[ˆDT∇ˆfT]dε. (21)\nIII. SPECTRAL CONDUCTANCES\nLet us now employ the above formalism in order to\nevaluate electric currents in our FSFdevice depicted in\nFig. 1. The current across the ﬁrst ( SF1) interface canbe written as\nI1=1\ne/integraldisplay\ng11(ε)[f0(ε+eV1)−f0(ε)]dε\n−1\ne/integraldisplay\ng12(ε)[f0(ε+eV2)−f0(ε)]dε,(22)\nwheref0(ε) = tanh( ε/2T),g11andg12arelocal and non-\nlocal spectral electric conductances. Expression for the\ncurrent across the second interface can be obtained from\nthe above equation by interchanging the indices 1 ↔2.\nSolving Eqs. (7)-(8) with boundary conditions (16)-(17)\nwe express both local and nonlocal conductances ˆ gij(ε)\nin terms of the interface conductances and the function\nˆD. The corresponding results read\nˆg11(ε) =/parenleftbigˆRT\n2ˆML+ˆRT\n2ˆRL\n2ˆR1m−ˆRL\n1ˆR2\n2m\n+ˆRT\n12ˆRL\n12ˆR2m−ˆR1mˆR2\n2m/parenrightbigˆK,(23)\nˆg12(ε) = ˆg21(ε) =/parenleftbigˆRT\n12ˆML+ˆRT\n2ˆRL\n12ˆR1m\n+ˆRL\n12ˆR1mˆR2m+ˆRT\n12ˆRL\n1ˆR2m/parenrightbigˆK,(24)\nwhere we deﬁned\nˆMT,L=ˆRT,L\n1ˆRT,L\n2−(ˆRT,L\n12)2, (25)\nˆK−1=ˆMTˆML+ˆR2\n1mˆR2\n2m−ˆRT\n2ˆRL\n2ˆR2\n1m\n−2ˆRT\n12ˆRL\n12ˆR1mˆR2m−ˆRT\n1ˆRL\n1ˆR2\n2m(26)\nand introduced the auxiliary resistance matrix\nˆRT\n1= ˆg1T(ε)[ˆg1T(ε)ˆg1L(ε)−ˆg2\n1m(ε)]−1\n+D1ˆDT\n1(r1,r1,ε)\nσ1+DSˆDT\nS(r1,r1,ε)\nσS,(27)\nThe resistance matrices ˆRT\n2,ˆRL\n1andˆRL\n2can be obtained\nby interchanging the indices 1 ↔2 andT↔Lin Eq.\n(27). The remaining resistance matrices ˆRT,L\n12andˆRjm\nare deﬁned as\nˆRT,L\n12=ˆRT,L\n21=DSˆDT,L\nS(r1,r2,ε)\nσS,(28)\nˆRjm= ˆgjm(ε)[ˆgjT(ε)ˆgjL(ε)−ˆg2\njm(ε)]−1,(29)\nwherej= 1,2. The spectral conductance gijcan be\nrecoveredfromthe matrix ˆ gijsimplybysummingup over\nthe spin states\ngij(ε) =1\n2Sp[ˆgij(ε)]. (30)\nIt is worth pointing out that Eqs. (23), (24) deﬁn-\ning respectively local and nonlocal spectral conductances\nare presented with excess accuracy. This is because the\nboundary conditions (14)-(15) employed here remain ap-\nplicable only in the tunneling limit and for weak spin\ndependent scattering |Gm|,|Gϕ| ≪GT. Hence, strictly\nspeaking only the lowest order terms in Gm1,2andGϕ1,2\nneed to be kept in our ﬁnal results.4\nIn ordertoproceedit is necessarytoevaluatethe inter-\nface conductances as well as the matrix functions ˆDT,L\n1,2,S.\nRestricting ourselves to the second order in the interface\ntransmissions we obtain\nˆg1T(ε) =GT1ˆνS(r1,ε)+G2\nT1∆2θ(∆2−ε2)\n∆2−ε2ˆU1(ε),(31)\nˆg1L(ε) =GT1ˆνS(r1,ε)−G2\nT1∆2θ(ε2−∆2)\nε2−∆2ˆU1(ε),(32)\nˆg1m(ε) =Gm1ˆνS(r1,ε)ˆσm1, (33)\nand analogous expressions for the interface conductances\nof the second interface. The matrix function\nˆU1(ε) =D1\n2σ1/braceleftBig\nRe/bracketleftbig\nC1(r1,r1,2h+\n1)+C1(r1,r1,2h−\n1)/bracketrightbig\n−ˆσm1Re/bracketleftbig\nC1(r1,r1,2h+\n1)−C1(r1,r1,2h−\n1)/bracketrightbig/bracerightBig\n(34)\nwithh±\n1=h1±εdeﬁnes the correction due to the prox-\nimity eﬀect in the normal metal.\nTakingintoaccounttheﬁrstordercorrectionsinthein-\nterface transmissions one can derive the density of states\ninside the superconductor in the following form\nˆνS(r,ε) =|ε|θ(ε2−∆2)/radicalbig\n|ε2−∆2|\n+DS\nσS∆2\n∆2−ε2/summationdisplay\ni=1,2/bracketleftBigg\nGTiReCS(r,ri,2ωR)\n−ˆσmiGϕiImCS(r,ri,2ωR)/bracketrightBigg\n,(35)\nwhere\nωR=\n\n√\nε2−∆2, ε > ∆,\ni√\n∆2−ε2,|ε|<∆,\n−√\nε2−∆2, ε < ∆,(36)\nand the Cooperon Cj(r,r′,ε) represents the solution of\nthe equation\n/parenleftbig\n−D∇2−iε/parenrightbig\nC(r,r′,ε) =δ(r−r′) (37)\nin the normal metal leads ( j= 1,2) and the supercon-\nductor (j=S). In the quasi-one-dimensional geometry\nthe corresponding solutions take the form\nCj(xj,xj,ε) =tanh(kjLj)\nSjDjkj, j= 1,2,(38)\nCS(x,x′,ε) =sinh[kS(L−x′)]sinhkSx\nkSSSDSsinh(kSL), x′> x,(39)\nwhereSS,1,2are the wire cross sections and k1,2,S=/radicalbig\n−iε/D1,2,S.\nSubstituting Eq. (35) into Eqs. (31) and (32) and\ncomparing the terms ∝G2\nT1we observe that the tunnel-\ning correction to the density of states dominates over theterms proportional to ˆU1which contain an extra small\nfactor/radicalbig\n∆/h≪1. Hence, the latter terms in Eqs. (31)\nand (32) can be safely neglected. In addition, in Eq. (35)\nwe also neglect small tunneling corrections to the super-\nconducting density of states at energies exceeding the\nsuperconducting gap ∆. Within this approximation the\ndensityofstatesinsidethesuperconductingwirebecomes\nspin-independent ˆ νS(r,ε) = ˆσ0νS(r,ε). It can then be\nwritten as\nνS(r,ε) =|ε|/radicalbig\n|ε2−∆2|θ(ε2−∆2)\n+DS\nσS∆2θ(∆2−ε2)\n∆2−ε2/summationdisplay\ni=1,2GTiReCS(r,ri,2ωR).(40)\nAccordingly, the interface conductances take the form\nˆg1T(ε) = ˆg1L(ε) =GT1νS(r1,ε), (41)\nˆg1m(ε) =Gm1νS(r1,ε)ˆσm1. (42)\nIn the limit of strong exchange ﬁelds h1,2≫∆ and\nsmall interface transmissions considered here the prox-\nimity eﬀect in the ferromagnets remains weak and can\nbe neglected. Hence, the functions ˆDT,L\n1(r1,r1,ε) and\nˆDT,L\n2(r2,r2,ε) can be approximated by their normal\nstate values\nˆDT,L\n1(r1,r1,ε) =σ1rN1ˆ1/D1, (43)\nˆDT,L\n2(r2,r2,ε) =σ2rN2ˆ1/D2, (44)\nrNj=Lj/(σjSj), j= 1,2, (45)\nwhererN1andrN2are the normal state resistances of\nferromagnetic terminals. In the the superconducting\nregion an eﬀective expansion parameter is GT1,2rξS(ε),\nwhererξS(ε) =ξS(ε)/(σSSS) is the Drude resistance\nof the superconducting wire segment of length ξS(ε) =/radicalbig\nDS/2|ωR|. In the limit\nGT1,2rξS(ε)≪1, (46)\nwhich is typically well satisﬁed for realistic system pa-\nrameters, it suﬃces to evaluate the function ˆDT\nS(x,x′,ε)\nfor impenetrable interfaces. In this case we ﬁnd\nˆDT\nS(x,x′,ε) =\n\n∆2−ε2\n∆2CS(x,x′,2ωR),|ε|<∆,\nε2−∆2\nε2CS(x,x′,0),|ε|>∆.(47)\nWe note that special care should be taken while calculat-\ningDL\nS(x,x′,ε) at subgap energies, since the coeﬃcient\nDLin Eq. (8) tends to zero deep inside the supercon-\nductor. Accordingly, the function DL\nS(x,x′,ε) becomes\nsingular in this case. Nevertheless, the combinations\nˆRL\nj(ML)−1andˆRL\n12(ML)−1remain ﬁnite also in this\nlimit. At subgap energies we obtain\nˆRL\n1(ˆML)−1=ˆRL\n2(ˆML)−1=ˆRL\n12(ˆML)−1\n=1\nrN1+rN2+2(∆2−ε2)ed/ξS(ε)\n∆2rξS(ε)GT1GT2,(48)5\nwhered=|x2−x1|is the distance between two SF\ncontacts. Substituting the above relations into Eq. (24)we arrive at the ﬁnal result for the non-local spectral\nconductance of our device at subgap energies\ng12(ε) =g21(ε) =∆2−ε2\n∆2rξS(ε)exp[−d/ξS(ε)]\n2[rN1+1/gT1(ε)][rN2+1/gT2(ε)]\n×\n1+m1m2Gm1\ngT1(ε)Gm2\ngT2(ε)∆2\n∆2−ε21\n1−ε2\n∆2+rN1+rN2\n2rξS(ε)GT1GT2exp[−d/ξS(ε)]\n,|ε|<∆.(49)\nEq. (49) represents the central result of our paper.\nIt consists of two diﬀerent contributions. The ﬁrst of\nthem is independent of the interface polarizations m1,2.\nThis term represents direct generalization of the result17\nin two diﬀerent aspects. Firstly, the analysis17was car-\nried out under the assumption rN1,2gT1,2(ε)≪1 which\nis abandoned here. Secondly (and more importantly),\nsuﬃciently large exchange ﬁelds h1,2≫∆ of ferromag-\nnetic electrodes suppress disorder-induced electron inter-\nference in these electrodes and, hence, eliminate the cor-\nresponding zero-biasanomalyboth in local25–27and non-\nlocal17spectral conductances. In this case with suﬃcient\naccuracyone can set gTi(ε) =GTiνS(xi,ε) implying that\nat subgap energies gTi(ε) is entirely determined by the\nsecond term in Eq. (40) which yields in the case of quasi-\none-dimensional electrodes\ngT1(ε) =∆2GT1rξS(ε)\n2(∆2−ε2)/bracketleftBig\nGT1+GT2e−d/ξS(ε)/bracketrightBig\n,(50)\ngT2(ε) =∆2GT2rξS(ε)\n2(∆2−ε2)/bracketleftBig\nGT2+GT1e−d/ξS(ε)/bracketrightBig\n.(51)\nNote, that if the exchange ﬁeld h1,2in both normal\nelectrodes is reduced well below ∆ and eventually is\nset equal to zero, the term containing ˆU1(ε) in Eqs.\n(31), (32) becomes important and should be taken into\naccount. In this case we again recover the zero-bias\nanomaly25–27gTi(ε)∝1/√εand from the ﬁrst term in\nEq. (49) we reproduce the results17derived in the limit\nh1,2→0.\nThe second term in Eq. (49) is proportional to the\nproductm1m2Gm1Gm2and describes non-local magne-\ntoconductance eﬀect in our system emerging due to spin-\nsensitiveelectronscatteringat SFinterfaces. Itisimpor-\ntant that – despite the strong inequality |Gmi| ≪GTi–\nboth terms in Eq. (49) can be of the same order, i.e. the\nsecond (magnetic) contribution can signiﬁcantly modify\nthe non-local conductance of our device.\nIn the limit of large interface resistances\nrN1,2gT1,2(ε)≪1 the formula (49) reduces to aFIG. 2: (Color online) Local (long-dashed line) and non-\nlocal (short-dashedandsolid lines) spectral conductance snor-\nmalized to its normal state values. Here we choose rN1=\nrN2= 5rξS(0),x1=L−x2= 5ξS(0),x2−x1=ξS(0),\nGT1=GT2= 4Gm1= 4Gm2= 0.2/rξS(0). Energy de-\npendence of non-local conductance is displayed for paralle l\n(P)m1m2= 1 and antiparallel (AP) m1m2=−1 interface\nmagnetizations. Inset: The same in the limit of low energies .\nmuch simpler one\ng12(ε) =g21(ε) =rξS(ε)\n2exp[−d/ξS(ε)]\n×/bracketleftbigg∆2−ε2\n∆2gT1(ε)gT2(ε)+m1m2Gm1Gm2∆2\n∆2−ε2/bracketrightbigg\n.\n(52)\nInterestingly, Eq. (52) remains applicable for arbitrary\nvalues of the angle between interface polarizations m1\nandm2and strongly resembles the analogous result for\nthe non-local conductance in ballistic FSFsystems (cf.,\ne.g., Eq. (77) in Ref. 18). The ﬁrst term in the square\nbrackets in Eq. (52) describes the fourth order contribu-6\nFIG. 3: (Color online) Non-local resistance (normalized to\nits normal state value) versus temperature (normalized to t he\nsuperconducting critical temperature TC) for parallel (P) and\nantiparallel (AP) interface magnetizations. The paramete rs\nare the same as in Fig. 2.\ntion in the interfacetransmissionswhich remainsnonzero\nalso in the limit of the nonferromagnetic leads17. In con-\ntrast, the second term is proportional to the product of\ntransmissions of both interfaces, i.e. only to the second\norder in barrier transmissions3,18. This term vanishes\nidentically provided at least one of the interfaces is spin-\nisotropic.\nContrary to the non-local conductance at subgap en-\nergies, both local conductance (at all energies) and non-\nlocal spectral conductance at energies above the super-\nconducting gap are only weakly aﬀected by magnetic ef-\nfects. Neglecting small corrections due to Gmterm in\nthe boundary conditions we obtain\nˆg11(ε) =ˆRT\n1(ˆMT)−1,ˆg22(ε) =ˆRT\n2(ˆMT)−1,(53)\nˆg12(ε) =g21(ε) =ˆRT\n12(ˆMT)−1,|ε|>∆.(54)\nEqs. (53) and (54) together with the aboveexpressions\nfor the non-local subgap conductance enable one to re-\ncover both local and non-local spectral conductances of\nour system at all energies. Typical energy dependencies\nfor both g11(ε) andg12(ε) are displayed in Fig. 2. For in-\nstance, we observe that at subgap energies the non-local\nconductance g12changes its sign being positive for par-\nallel and negative for antiparallel interface polarizations.\nHaving established the spectral conductance matrix\ngij(ε) one can easily recover the complete I−Vcurves\nfor our hybrid FSFstructure. In the limit of low bias\nvoltages these I−Vcharacteristics become linear, i.e.\nI1=G11(T)V1+G12(T)V2, (55)\nI2=G21(T)V1+G22(T)V2, (56)FIG. 4: (Color online) The same as in Fig. 3 for the following\nparameter values: rN1=rN2= 5rξS(0),x1=L−x2=\n5ξS(0),x2−x1=ξS(0),GT1=GT2= 50Gm1= 50Gm2=\n0.025/rξS(0).\nwhereGij(T) represent the linear conductance matrix\ndeﬁned as\nGij(T) =1\n4T/integraldisplay\ngij(ε)dε\ncosh2ε\n2T. (57)\nIt mayalsobe convenientto invertthe relations(55)-(56)\nthus expressing induced voltages V1,2in terms of injected\ncurrents I1,2:\nV1=R11(T)I1−R12(T)I2, (58)\nV2=−R21(T)I1+R22(T)I2, (59)\nwhere the coeﬃcients Rij(T) deﬁne local ( i=j) and\nnonlocal ( i∝ne}ationslash=j) resistances\nR11(T) =G22(T)\nG11(T)G22(T)−G2\n12(T),(60)\nR12(T) =R21(T) =G12(T)\nG11(T)G22(T)−G2\n12(T)(61)\nand similarly for R22(T). In non-ferromagnetic NSN\nstructures the low temperature non-local resistance\nR12(T→0) turns out to be independent of both the\ninterface conductances and the parameters of the normal\nleads17. However, this universality of R12does not hold\nanymore provided non-magnetic normal metal leads are\nsubstituted by ferromagnets. Non-local linear resistance\nR12of ourFSFstructure is displayed in Figs. 3, 4 as\na function of temperature for parallel ( m1m2= 1) and\nantiparallel ( m1m2=−1) interface magnetizations. In\nFig. 3 we show typical temperature behavior of the non-\nlocalresistanceforsuﬃcientlytransparentinterfaces. For7\nboth mutual interface magnetizations R12ﬁrst decreases\nwithtemperaturebelow TCsimilarlytothenon-magnetic\ncase. However, at lower Timportant diﬀerences occur:\nWhile in the case of parallel magnetizations R12always\nremains positive and even shows a noticeable upturn at\nsuﬃciently low T, the non-local resistance for antiparal-\nlelmagnetizationskeepsmonotonouslydecreasingwith T\nand may become negative in the low temperature limit.\nIn the limit of very low interface transmissions the tem-\nperature dependence of the non-local resistance exhibits\na well pronounced charge imbalance peak (see Fig. 4)\nwhich physics is similar to that analyzed in the case of\nnon-ferromagnetic NSNstructures4,16,23. Let us point\nout that the above behavior of the non-local resistance\nis qualitatively consistent with available experimental\nobservations5.\nIV. CONCLUDING REMARKS\nIn this paper we developed a quantitative theory\nof non-local electron transport in three-terminal hy-\nbrid ferromagnet-superconductor-ferromagnetstructures\nin the presence of disorder in the electrodes. Within\nour model transfer of electrons across SFinterfaces is\ndescribed in the tunneling limit and magnetic proper-\nties of the system are accounted for by introducing ( i)\nexchange ﬁelds h1,2in both normal metal electrodes\nand (ii) magnetizations m1,2of bothSFinterfaces (the\nmodel of spin-active interfaces). The two ingredients(i) and (ii) of our model are in general independent\nfrom each other and have diﬀerent physical implications.\nWhile the role of (comparatively large) exchange ﬁelds\nh1,2≫∆ is merely to suppress disorder-induced inter-\nference ofelectrons25–27penetrating from a superconduc-\ntor into ferromagnetic electrodes, spin-sensitive electron\nscattering at SFinterfaces yields an extra contribution\nto the non-local conductance which essentially depends\non relative orientations of the interface magnetizations.\nFor anti-parallel magnetizations the total non-local con-\nductance g12and resistance R12can turn negative at suf-\nﬁciently low energies/temperatures. At higher tempera-\ntures the diﬀerence between the values of R12evaluated\nfor parallel and anti-parallelmagnetizations becomes less\nimportant. Atsuchtemperaturesthenon-localresistance\nbehaves similarly to the non-magnetic case demonstrat-\ning, e.g., a well-pronounced charge imbalance peak23in\nthe limit of low interface transmissions.\nWebelievethat ourpredictionscanbe directlyusedfor\nquantitative analysis of future experiments on non-local\nelectron transport in hybrid FSFstructures.\nAcknowledgments\nThis work was supported in part by DFG and by\nRFBRgrant09-02-00886. M.S.K.alsoacknowledgessup-\nport from the Council for grantsof the Russian President\n(GrantNo. 89.2009.2)andfromtheDynastyFoundation.\n1A.F. Andreev, Sov. Phys. JETP 19, 1228 (1964).\n2J.M. Byers and M.E. Flatte, Phys. Rev. Lett. 74, 306\n(1995); G. Deutscher and D. Feinberg, Appl. Phys. Lett.\n76, 487 (2000).\n3G. Falci, D. Feinberg, and F.W.J. Hekking, Europhys.\nLett.54, 255 (2001).\n4M.S. Kalenkov and A.D. Zaikin, Phys. Rev. B 75, 172503\n(2007); JETP Lett. 87, 140 (2008).\n5D. Beckmann, H.B. Weber, and H. v. L¨ ohneysen, Phys.\nRev. 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B 74,\n214510 (2006).\n15R. Melin, Phys. Rev. B 73, 174512 (2006).\n16D.S. Golubev and A.D. Zaikin, Phys. Rev. B 76, 184510\n(2007).\n17D.S. Golubev, M.S. Kalenkov, and A.D. Zaikin, Phys. Rev.\nLett.103, 067006 (2009).\n18M.S. Kalenkov and A.D. Zaikin, Phys. Rev. B 76, 224506\n(2007).\n19M.S. Kalenkov and A.D. Zaikin, Physica E 40, 147 (2007).\n20See, e.g., W. Belzig, F. Wilhelm, C. Bruder, G. Sch¨ on, and\nA.D. Zaikin, Superlatt. Microstruct. 25, 1251 (1999).\n21M. Eschrig, Phys. Rev. B 80, 134511 (2009).\n22M. Eschrig, J. Kopu, J. C. Cuevas, and G. Sch¨ on, Phys.\nRev. Lett. 90, 137003 (2003); M. Eschrig and T. Lofwan-\nder, Nat. Phys. 4, 138 (2008).\n23A.V. Galaktionov, M.S. Kalenkov, and A.D. Zaikin, Phys.\nRev. B77, 094520 (2008); M.S. Kalenkov, A.V. Galak-\ntionov, and A.D. Zaikin, Phys. Rev. B 79, 014521 (2009).\n24D. Huertas-Hernando, Yu.V. Nazarov, and W. Belzig,\nPhys. Rev. Lett. 88, 047003 (2002); A. Cottet, D. Huertas-\nHernando, W. Belzig, and Yu.V. Nazarov, Phys. Rev. B\n80, 184511 (2009).\n25A.F. Volkov, A.V. Zaitsev, and T.M. Klapwijk, Physica C\n210, 21 (1993).8\n26F.W.J. Hekking and Yu.V. Nazarov, Phys. Rev. Lett. 71,\n1625 (1993).27A.D. Zaikin, Physica B 203, 255 (1994)."    },    {        "title": "0705.2944v1.Evidence_for_ferromagnetic_spin_pairing_superconductivity_in_UGe__2___A____73__Ge_NQR_study_under_pressure.pdf",        "content": "arXiv:0705.2944v1  [cond-mat.supr-con]  21 May 2007Evidence for ferromagnetic spin-pairing superconductivi ty in UGe 2:\nA73Ge-NQR study under pressure\nA. Harada,1,∗S. Kawasaki,1H. Mukuda,1Y. Kitaoka,1Y. Haga,2\nE. Yamamoto,2Y.¯Onuki,2,3K. M. Itoh,4E. E. Haller,5and H. Harima6\n1Department of Materials Engineering Science, Osaka Univer sity, Osaka 560-8531, Japan\n2Advanced Science Research Center, Japan Atomic Energy Rese arch Institute, Tokai, Ibaraki 319-1195, Japan\n3Department of Physics, Osaka University, Osaka 560-0043, J apan\n4Department of Applied Physics and Physico-Informatics, Ke io University, Yokohama 223-8522, Japan\n5University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA\n6Department of Physics, Faculty of Science, Kobe University , Nada, Kobe 657-8501, Japan\n(Dated: November 3, 2018)\nWe report that a novel type of superconducting order paramet er has been realized in the fer-\nromagnetic states in UGe 2via73Ge nuclear-quadrupole-resonance (NQR) experiments perfo rmed\nunderpressure ( P). Measurements of the nuclear spin-lattice relaxation rat e (1/T1) have revealed an\nunconventional nature of superconductivity such that the u p-spin band is gapped with line nodes,\nbut the down-spin band remains gapless at the Fermi level. Th is result is consistent with that\nof a ferromagnetic spin-pairing model in which Cooper pairs are formed among ferromagnetically\npolarized electrons. The present experiment has shed new li ght on a possible origin of ferromag-\nnetic superconductivity, which is mediated by ferromagnet ic spin-density ﬂuctuations relevant to\nthe ﬁrst-order transition inside the ferromagnetic states .\nThe coexistence of magnetism and superconductiv-\nity (SC) has recently become an important topic in\ncondensed-matter physics. The recent discovery of SC\nin ferromagnets UGe 2[1, 2] and URhGe [3] has been a\ngreat surprise because the Cooper pairs are inﬂuenced by\na non-vanishing internal ﬁeld due to the onset of ferro-\nmagnetism (FM), which is believed to prevent the onset\nof SC. In the ferromagnet UGe 2with a Curie temper-\natureTCurie= 52K at ambient pressure ( P= 0), the\nemergence of P-induced SC has observed in the Prange\nof 1.0-1.6GPa [1, 2]. It is noteworthy that the SC in\nUGe2disappears above Pc∼1.6GPa, beyond which FM\nis suppressed. The SC and FM in this compound have\nbeen shown to be cooperative phenomena [4]. The su-\nperconducting transition temperature ( Tsc) is the high-\nest atPx∼1.2GPa, where a ﬁrst-order transition occurs\nfrom FM2 to FM1 as Pincreases. Here, it should be\nnoted that ferromagnetic moments are increased in the\nﬁrst-order transition from FM1 to FM2 as functions of\ntemperature and pressure, as shown in Fig. 1(a) [5, 6, 7].\nTheP-induced SC in UGe 2coexists with FM1 and FM2\nexhibiting the large magnetization of an order 1 µB/U\neven for the case of TCurie∼30K [6]. Therefore, the\nonset of SC is proposed to be suitable for the formation\nof a spin-triplet pairing state rather than a spin-singlet\npairing state [2]. However, there are few reports that\naddress the type of order parameter is realized in FM1\nand FM2. In a previous study, an unconventional nature\nof the SC has been suggested from the measurement of\nthe73Ge-NQR nuclear spin-lattice relaxation rate 1 /T1\n[6]. However, it has not been well understood whether\nthe presence of the residual density of states (RDOS) atFIG. 1: (color online). (a) Pressure versus temperature phase\ndiagram of UGe 2near the superconducting phase [6]. The Tsc\nvalues of FM1 (open squares) and FM2 (open triangles), and Px\nvalue determined in this study are plotted. (b) Crystal stru cture of\nUGe2with the ferromagnetic moment at the U site below TCurie.\nthe Fermi level in the SC state is intrinsic or not, sug-\ngestingthe occurrenceofa possible extrinsic eﬀect due to\nthe presence of any impurity and/or imperfection in the\nsample [6]. In particular, it is unclear why SC emerges\nwith the highest Tscwhen the ﬁrst-order transition oc-\ncurs from FM1 to FM2 at Px∼1.2GPa. In order to gain\ninsight into this issue, further experiments are required\nforunderstandinga P-inducedevolutionintheFMstates\nand a novel order-parameter symmetry emerging in the\nFM states in UGe 2.\nIn this letter, by performing the73Ge-NQR measure-\nments under pressure at zero ﬁeld ( H= 0) on a newly\npreparedsample, wereportthattheSCinthiscompound\niscausedbythe formationofup-spinCooperpairs, where\nthe gap opens only at the up-spin band in FM1 and FM22\nbut not at the down-spin band [8, 9, 10]. The ferro-\nmagnetic spin-pairing SC is considered to be mediated\nbyferromagneticspin-densityﬂuctuations relevanttothe\nﬁrst-order transition inside the ferromagnetic states.\npresent  \nprevious FM1 (b) (a) Paramagnetic\n8.0 8.4 8.8 9.2Echo-Intensity (arb. units)\nFrequency (MHz)8.0 8.4 8.8 9.2state\nFIG. 2: (color online). Comparison of the73Ge-NQR spectra\nof the present and previous samples of UGe 2in (a) paramagnetic\nstate and (b) FM1. The spectra at (a) P= 1.9GPa and (b) P=\n1.41 and 1.3GPa are shown by solid and open circles, respecti vely,\ndemonstrating that the present sample has better quality th an the\nprevious samples [6].\nApolycrystallinesample enrichedby73Ge wascrushed\ninto coarse powder for the NQR measurement and an-\nnealed to maintain its quality. The NQR experiments\nwere performed by the conventional spin-echo method at\nH= 0 in the frequency ( f) range of 5-11MHz at P=\n1.17, 1.2, 1.24, and 1.41GPa. Hydrostatic pressure was\napplied by utilizing a NiCrAl-BeCu piston-cylinder cell\nﬁlled with Daphne oil (7373) as a pressure-transmitting\nmedium. The value of Pat low temperatures was deter-\nminedfromthe TscofSnmeasuredbytheresistivitymea-\nsurement. Thepossibledistributionofthepressureinside\nthe sample was less than 3% in the present experimental\nsetup. A3He-4He dilution refrigerator was used to ob-\ntain the lowest temperature of 50mK. Figures 2(a) and\n2(b) show the NQR spectra in paramagnetic (PM) phase\nand FM1 phase, respectively. The linewidths in these\nNQR spectra are narrower for the present sample than\nfor the previous sample, demonstrating that the quality\nof the present sample is signiﬁcantly higher than that\nof the previous sample. Moreover, the NQR- T1measure-\nments reveal that the present sample exhibits the highest\nvalue ofTsc= 0.75K obtained thus far at P= 1.24GPa,\nensuring higher quality than before.\nFigure 3(a) shows the NQR spectra for the PM phase\nat 4.2K and P= 1.9GPa where FM1 is completely sup-\npressed. They reveal a structure consisting of separated\npeaks associated with three inequivalent Ge sites in one\nunit cell in the crystal structure illustrated in Fig. 1(b)\n[4, 6]. The number of Ge1 sites is twice that of Ge2 and\nGe3 sites in one unit cell. The Ge1 site is closely located\nalong the uranium (U)-zigzag chain, while the other two\nsites Ge2 and Ge3 are located outside this zigzag chain.\nFromtheanalysisoftheNQRspectraforFM1in thepre-\nvious experiment [6], it was demonstrated that the onset\nof FM1 induces an internal ﬁeld Hint= 0.9T at the Ge\nsites that additionally causes about the Zeeman split-1.24  GPa\n1.17  GPa1.2  GPa(c)(b)(a)\n(d)Ge1\nGe2Ge3 1.9  GPaEcho-Intensity (arb. units)\n5 6 7 8 9 10 11\nFrequency (MHz)(e)1.41  GPa\nFIG. 3: (color online). (a)73Ge-NQR spectra at 4.2K in the\nP-induced paramagnetic phase. The NQR spectra in (b), (c), (d ),\nand (e) represent in the ferromagnetic phases at 1.4K and P=\n1.41, 1.24, 1.2 and 1.17GPa, respectively. The dashed lines in the\nﬁgures indicate the simulated results (see text).\nting in each Ge-NQR spectrum. Furthermore, the an-\ngle between the principal axis for the nuclear quadrupole\nHamiltonian and a direction of Hintwas determined as\nθ∼π/3. When the ﬁrst-order transition occurs from\nFM1 to FM2, the spectral shape changes signiﬁcantly\nfrom the spectra at P= 1.24 and 1.41GPa to those\natP= 1.17 and 1.2GPa, as shown in Figs. 3(b)-(e).\nFrom the analysis of the spectra, it is estimated that\nHint= 0.9T for FM1 increases to Hint= 1.8T for FM2\n(seeFig.5(b)). Thesuddenincreasein Hintinthenarrow\nPrange should be relevant to the ﬁrst-ordertransition at\nPx. In such a case, the spectra near Pxare expected to\nreveala mixture of both domains of FM1 and FM2 in the\nnarrow range of Pclose toPxdue to an inevitable dis-\ntribution of P. In fact, the respective spectra at P= 1.2\nand 1.24GPa are composed of spectra arising from FM1\n(P= 1.41GPa) and FM2 ( P= 1.17GPa) with the ra-\ntios of 1:9 and 7:3, respectively, as shown by the dashed\nlines in Figs. 3(b) and 3(c). By considering an inevitable\nPdistribution (∆ P= 0.04GPa) as a Gaussian func-\ntiongivenbyexp[ −([P−P0]/[∆P/(2√\nln2)])2], weobtain\nPx= 1.23GPa. The present experimental results reveal\nthat the ﬁrst-order transition occurs at Px= 1.23GPa.\nFigure 4(a) shows the Tdependences of 1 /T1for FM1\nand FM2 at pressures that are slightly lower and higher\nthanPx= 1.23GPa, respectively. Clearly, 1 /T1for FM1\nand FM2 decreases without any indication of coherence\npeak just below Tsc, which provides evidence for the uni-\nform coexistence of the unconventionalSC and ferromag-\nnetism. In the previous study [6], the line-node gap\nmodel with RDOS Nresat the Fermi level was applied\nto interpret a systematic evolution in the superconduct-3\ning energy gap ∆ and a fraction of RDOS Nres/N0. Here\nN0is the density of state (DOS) at the Fermi level in\nthe FM phases. It should be noted that all the data of\n1/T1are uniquely determined in the present sample, but\nnot in the previous sample [6]. Therefore, we could not\nexclude the fact that the RDOS is present in the previ-\nous sample due to some impurity eﬀect. Similarly for the\npresent sample with higher quality than that of the pre-\nvious sample, the application of the line-node gap model\nwith the RDOS allows us to estimate Nres/N0= 0.50,\n0.48, 0.29, and 0.30and Tsc= 0.45, 0.55, 0.75, and 0.25K\n(±0.05K) at P= 1.17, 1.2, 1.24, and 1.41GPa, respec-\ntively. It should be noted that Tscdecreases from 0.45K\nto 0.25K although Nres/N0does decrease from 0.50 to\n0.30 atP= 1.2GPa in FM2 and at P= 1.41GPa in\nFM1. These results demonstrate that the presence of\nthe RDOS in the superconducting state is not due to the\nimpurity eﬀect but intrinsicin origin. Althoughsomeim-\npurity and/or imperfection based eﬀects, if any, are not\ncompletely ruled out, we state that the observationofthe\nhighestTsc= 0.75K ensures that the present sample is\none of the best quality samples reported thus far.\nFirst, we address whether or not the RDOS is asso-\nciated with a self-induced vortex state in the SC + FM\nuniformly coexisting state. By assuming the Abrikosov\ntriangular vortex lattice, a coherence length ξ∼130˚A\n[7], and an internal magnetic ﬁeld H= 0.125T [11], we\nconsider that only 3% of Nres/N0arises from the normal\nstate inside the self-induced vortex core in the SC+FM\nstate, which does not agree with the experimental re-\n1.17  GPa 1.2  GPa 1.41  GPa 1.24  GPa \nN \nTemperature (K)FM2 FM2 FM1 FM1( 1/T1T )1/2\nN N N \n0.1 1 0.1 1 0.1 10.20.40.60.8\n0.1 1TscTscTsc\nTsc1.17  GPa 1.2 GPa 1.41  GPa 1.24 GPa \nFM2 FM2 FM1 FM1\nTscTscTsc\nTsc\n(b)(a)\n0 0 0 0 (sec-1/2)  K0.1 1 0.1 110-310-210-11001011 / T1 (sec-1)\n0.1 1 0.1 1~T ~T~T ~T-1/2\nFIG. 4: (color online). (a) Temperature dependences of 1 /T1for\nFM2 at P= 1.17 and 1.2GPa measured at f= 7.68MHz and for\nFM1 at P= 1.24 and 1.41GPa measured at 7.09 and 7.07MHz,\nrespectively. (b) Temperature dependences of (1 /T1T)1/2related\nto the DOS in either the SC state or the normal state at each\nP. The solid curves represent the results calculated based on the\nferromagnetic spin-pairing model (see the text).sult. Alternatively, in another promising scenario that\nexplainsthe RDOS, weconsidera nonunitaryspin-triplet\npairing model [8]. In this model, the superconducting\nenergy gap opens only in the up-spin band parallel to\nthe magnetization of FM phases, but not in the down-\nspin band that remains gapless. We begin by assigning\npossible nuclear relaxation processes of Ge-NQR T1in\nthe ferromagnetic states. One of them is caused by the\ntransversal component of ﬂuctuations of internal mag-\nnetic ﬁeldsat the Gesites originatingfrom intrabandand\ninterband transitions across the Fermi level at each up-\nspin band and down-spin band. Another one is causedby\nonly the interband spin-ﬂip transition across the Fermi\nlevel between the up-spin band and down-spin band. By\nconsidering these relaxation processes, 1 /T1Tin the FM\nstate is expressed as\n1\nT1T∝2t2(T)cos2θ+[t1(T)+2t2(T)+t3(T)]sin2θ,\nt1(T) =1\nkBT/integraldisplay∞\n0dEN2\n↓(E)f(E)[1−f(E)],\nt2(T) =1\nkBT/integraldisplay∞\n0dEN↑(E)N↓(E)f(E)[1−f(E)],\nt3(T) =1\nkBT/integraldisplay∞\n0dEN2\n↑(E)f(E)[1−f(E)],\nwheret1(T),t2(T), andt3(T) indicate the former con-\ntributions and t2(T) represents the latter contribution\nwhich is only possible for θ= 0. When the energy de-\npendence of the DOS is neglected near the Fermi level,\nall contributions of t1=N2\n0↓,t2=N0↑N0↓, andt3=N2\n0↑\nare independent of temperature. Here, N0↑andN0↓are\nthe DOS at the up-spin and the down-spin bands at the\nFermi level in the normal FM state, respectively. θis an\nangle between the quantization axis of the73Ge-nuclear-\nquadrupole Hamiltonian and that of Hintat the Ge site\nin the FM state, which is estimated as θ∼π/3 from the\nanalysis of the NQR spectra in the FM state, and f(E)\nis a Fermi distribution function. In the ferromagnetic\nspin-pairing model, the line-node gap of ∆( φ) = ∆0cosφ\nis assumed only for the density of states Ns↑(E) at the\nup-spin band, but not for Ns↓(E) at the down-spin band.\nIt should be noted that if θ= 0,t2(T) should behave as\n1/T1∝T2well below Tsc. In the present case, because\nofθ∼π/3, the gapless term t1gives rise to the RDOS\nat the Fermi level in the superconducting state, as shown\nin Fig. 4(b). In fact the experimental results are actually\nin good agreement with this theoretical model, as indi-\ncated by the solid lines in Figs. 4(a) and 4(b). There-\nfore, the SC energy gap ∆ and N0↑/N0are estimated as\n2∆/kBTsc∼3.7, 3.8, 4.0, and 3.7 with N0↑/N0= 0.57,4\n0.57, 0.82, and 0.80 at P= 1.17, 1.2, 1.24, and 1.41GPa,\nrespectively. Here, N0(P) =N0↑(P)+N0↓(P).\nIn order to gain further insight into the novel SC state,\nFig. 4(b) shows the Tdependence of (1 /T1T)1/2related\nto the DOS at the Fermi level in either the SC or the\nnormal FM state. As shown in Fig. 5(c), the most in-\nteresting ﬁnding is that N0↑(P) dramatically increases\nasPincreases slightly from P= 1.2 to 1.24GPa across\nPx= 1.23GPa, accompanying the sudden reduction of\nHintshown in Fig. 5(b). By contrast, N0↓(P) remains\nalmost constant and the T-linear coeﬃcient of the spe-\nciﬁc heat γgradually increases with PacrossPx[13].\nThese results reveal that the Fermi level in FM2 is lo-\ncated just above a sharp peak in the majority up-spin\nband, and as Pincreases across Px, it shifts down to-\nward the peak when it enters the FM1 phase. In the\nferromagnetic spin-pairing SC state, the large DOS in\nthe up-spin band in FM1 enhances Tsc, leading to the\nhighest value of Tsc= 0.75K, whereas its reduction in\nFM2 decreases Tsc, asshownin Figs. 5(a)and 5(c). How-\never, it should be noted that as Pincreases further up\ntoP= 1.41GPa, even though N0↑(P) for FM1 remains\nrather larger than that for FM2, Tsc= 0.25±0.05K for\nFM1 atP= 1.41GPa becomes lower than Tsc= 0.45K\nfor FM2 at P= 1.17GPa . In this context, the large\nincrease in N0↑(P) is not always a main factor that in-\ncreasesTsc. Rather, the ﬁrst-order transition from FM2\nto FM1 at Pxis responsible for the mediation of the up-\nspin Cooper pairing. The longitudinal FM spin ﬂuctua-\ntions along the a-axis [14] softens in energy at the critical\nend point for the ﬁrst-order transition at Px. Therefore,\nwe suggest that this longitudinal FM ﬂuctuations along\nthea-axiswould be a mediator of the ferromagneticspin-\npairing SC where the majority up-spin band in the FM\nphases is gapped, while the minority down-spin band is\nnot.\nIn another context, it is predicted that Txcould be\nidentiﬁed with the formation of a simultaneous charge-\nand spin-density wave (CSDW) induced by the imper-\nfect nesting of the Fermi surface for the up-spin band\nand hence the superconducting pairing is mediated by\nCSDW ﬂuctuations around Px[15]. Although the NQR\nspectrum does not directly evidence an onset of static\nCSDW states, the remarkable increase in N0↑(P) across\nPxis relevant to the nesting at the Fermi surface for the\nup-spin band below Tx.\nIn conclusion, the73Ge-NQR measurements under\npressure on well characterized UGe 2have revealed that\nthe superconducting energygap opens only with the line-\nnode at the Fermi level in the majorityup-spin band, but\nthe down-band remains gapless. It is therefore concluded\nthat ferromagnetic spin-pairing SC occurs in UGe 2. We\nhave also shown that the ﬁrst-order transition from FM1\nto FM2 at Px= 1.23GPa occurs because the Fermi\nlevel is located just on the peak in the DOS of the up-\nspin band. The ferromagnetic spin-density ﬂuctuationsFIG. 5: (color online). Pressure dependence of (a) Tsc; (b) inter-\nnal magnetic ﬁeld, Hint, at the Ge1 site; and (c) relative Pdepen-\ndence of N0↑(solid circles) and N0↓(solid squares) estimated from\nthe ferromagnetic spin-pairing model on a scale of (1 /T1T)1/2and\ntheT-linear coeﬃcient of the speciﬁc heat γ[13] (open squares)\nacrossPx(see text). It should be noted that N0↑dramatically in-\ncreases when the ﬁrst-order transition from FM2 at P= 1.2GPa\nto FM1 at 1.24GPa occurs across Px= 1.23GPa.\nemerging in the vicinity of the critical end point for this\nﬁrst-order transition are considered to be the mediator\nof the onset of novel SC realized in ferromagnet UGe 2.\nWe thank H. Kotegawa, N. Tateiwa, S. Watanabe,\nand K. Miyake for fruitful discussions and comments.\nThis study was supported by Grant-in-Aid for Creative\nScientiﬁc Researchi15GS0213); the Ministry of Educa-\ntion, Culture, Sports, Science and Technology (MEXT);\nand the 21st Century COE Program (G18) supported by\nthe Japan Society for the Promotion of Science (JSPS).\nA.H. wasﬁnanciallysupportedbyaGrant-in-AidforEx-\nploratory Research of MEXT (No. 17654066).\n∗aharada@nmr.mp.es.osaka-u.ac.jp\n[1] S.S.Saxena, P.Agarwal, K.Ahilan, F.M.Grosche, R.K.\nW. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker,\nS. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley,\nI. Sheikin, D. Braithwaite, and J. Flouquet, Nature 406,\n587 (2000).\n[2] A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois,\nD. Braithwaite, R. Calemczuk, and J. Flouquet, Phys.\nRev. B63, 144519 (2001).\n[3] D. Aoki, A. D. Huxley, E. Ressouche, D. Braithwaite,\nJ. Flouquet, J. P. Brison, E. Lhotel, and C. Paulsen,\nNature413, 613 (2001).\n[4] A. Harada, S. Kawasaki, H. Kotegawa, Y. Kitaoka,\nY. Haga, E. Yamamoto, Y. ¯Onuki, K. M. Itoh,\nE. E. Haller, and H. Harima, J. Phys. Soc. Jpn. 74, 2675\n(2005).\n[5] C. Pﬂeiderer and A. D. Huxley, Phys. Rev. Lett. 89,\n147005 (2002).\n[6] H. Kotegawa, A. Harada, S. Kawasaki, Y. Kawasaki,\nY. Kitaoka, Y. Haga, E. Yamamoto, Y. ¯Onuki,5\nK. M. Itoh, E. E. Haller, and H. Harima, J. Phys. Soc.\nJpn.74, 705 (2005).\n[7] N. Tateiwa, T. C. Kobayashi, K. Hanazono, K. Amaya,\nY. Haga, R. Settai, and Y. ¯Onuki, J. Phys.: Condens.\nMatter13, L17 (2001).\n[8] T. Ohmi and K. Machida, Phys. Rev. Lett. 71, 625\n(1993).\n[9] K. Machida and T. Ohmi, Phys. Rev. Lett. 86, 850\n(2001).\n[10] I. A. Fomin, JETP Letters 74, 111 (2001).\n[11] The internal magnetic ﬁeld His derived from H= (1−\nn)M, where nis the demagnetization factor and Mis\nthe magnetization per volume. Here, the ferromagnetic\nmomentsis givenby1 µB/U[12]andthedemagnetizationfactor forthissamplegeometry(approximatelyspherical)\nisn= 1/3.\n[12] N. Tateiwa, K. Hanazono, T. C. Kobayashi, K. Amaya,\nT. Inoue, K. Kindo, Y. Koike, N. Metoki, Y. Haga,\nR. Settai, and Y. ¯Onuki, J. Phys. Soc. Jpn. 70, 2876\n(2001).\n[13] N.Tateiwa, T.C. Kobayashi, K.Amaya, Y.Haga, R.Set-\ntai, and Y. ¯Onuki, Physica B 312-313 , 109-111 (2002).\n[14] A. D. Huxley, S. Raymond, and E. Ressouche, Phys. Rev.\nLett.91, 207201 (2003).\n[15] S. Watanabe and K. Miyake, J. Phys. Soc. Jpn. 71, 2489\n(2002)."    },    {        "title": "0811.0384v1.Critical_thickness_for_itinerant_ferromagnetism_in_ultrathin_films_of_SrRuO__3_.pdf",        "content": "arXiv:0811.0384v1  [cond-mat.str-el]  3 Nov 2008Critical thickness for itinerant ferromagnetism in ultrat hin ﬁlms of SrRuO 3\nJing Xia,1,2W. Siemons,1,3G. Koster,1,3M.R. Beasley,1,4and A. Kapitulnik1,2,4\n1Geballe Laboratory for Advanced Materials, Stanford Unive rsity, Stanford, California, 94305\n2Department of Physics, Stanford University, Stanford, CA 9 4305\n3Faculty of Science and Technology and MESA+ Institute for Na notechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE, Enschede, The Ne therlands\n4Department of Applied Physics, Stanford University, Stanf ord, CA 94305\nUltrathin ﬁlms of the itinerant ferromagnet SrRuO 3were studied using transport and magnto-\noptic polar Kerr eﬀect. We ﬁnd that below 4 monolayers the ﬁlm s become insulating and their\nmagnetic character changes as they loose their simple ferro magnetic behavior. We observe a strong\nreduction in the magnetic moment which for 3 monolayers and b elow lies in the plane of the ﬁlm.\nExchange-bias behavior is observed below the critical thic kness, and may point to induced antifer-\nromagnetism in contact with ferromagnetic regions.\nPACS numbers: 75.70.i, 75.60.d, 71.45.Gm\nSrRuO 3is an itinerant ferromagnet with an or-\nthorhombically distorted cubic perovskite structure, ex-\nhibiting a transition to a ferromagnetic state at Tc∼160\nK that was shown to be dominated by transverse ﬂuc-\ntuations of robust local moments of size ∼1.6µB[1],\nthe largest of any 4-d ferromagnet. While at high-\ntemperatures, in the paramagnetic phase, it exhibits a\n“bad metal” behavior in the limit of kFℓ∼1 [2] sug-\ngesting that Fermi liquid theory may not be valid, the\nobservationofquantum oscillationsin the electrical resis-\ntivity of high-quality thin ﬁlms of SrRuO 3demonstrated\nthat the ground state of this system is a Fermi liquid\n[3]. At the same time, the degree of electron correlation\nin SrRuO 3has been found to be a strong function of\nruthenium deﬁciency [4]. To understand the contrast in\nthe behavior of SrRuO 3between high and low temper-\natures, appropriate perturbations such as disorder and\nreduced dimensionality, may be used that directly dis-\nturb the magnetic and transport properties of the sys-\ntem. Indeed, recent studies of the thickness dependence\nof the transport and electronic structure of SrRuO 3ﬁlms\n[5, 6] concluded that a metal-insulator transition (MIT)\noccurs in these ﬁlms at a critical ﬁlm thickness of 4 or 5\nmonolayers (ML), depending on disorder. However, the\nreported island-like microstructure showing coalescence\nof three-dimensional patches, and the inability to study\nthe nature of the magnetism, hinder any possible un-\nderstanding of the observed transition. Since this may\nbe the ﬁrst example of the interplay between itineracy,\nferromagnetism disorder and dimensionality, better ﬁlms\ngrowth and a more direct probe of magnetism are needed\nto establish the important ingredients of the physics in-\nvolved.\nIn this paper we present new results on the MIT in ul-\ntrathinSrRuO 3ﬁlmsandtheirassociatedmagneticprop-\nerties. We show that in homogeneous ﬁlms of SrRuO 3\na metal-insulator transition (MIT) occurs at a critical\nthickness below 4 ML. While Tcdrops rapidly below ∼10\nML, the size of the moment remains unchanged from its1.6µBin thick ﬁlms [1], and the easy axis which has been\ncloser to normal for thick ﬁlms, becomes even more nor-\nmal. However, below the critical thickness the easy axis\nof the moment plummets to the plane of the ﬁlm, and an\nexchange-biasbehavioremerges, suggestingthe existence\nofantiferromagnetic(AFM)regionsinthediﬀerentlayers\nthat interact with the remaining ferromagnetic regions.\nExamination of the transport properties of the measured\nﬁlms shows an increase in the resistance with decreasing\nthickness. At 4 ML the extrapolated low-temperature\nsheet-resistance is of order ∼7 kΩ, jumping up 8 orders\nof magnitude in 3 ML ﬁlms.\nSrRuO 3samples used in our experiment were grown\nby Pulsed Laser Deposition (PLD). The samples were\ngrown in a vacuum chamber with a background pressure\nof 10−7Torr. A 248 nm wavelength KrF excimer laser\nwas employed with typical pulse lengths of 20-30ns. The\nenergy density on the target is kept at approximately\n2.1 J/cm2. All ﬁlms were grown on TiO 2terminated\nSrTiO 3substrates[10], at 700C, with a a laserrepetition\nrate of 4 Hertz. We have calibrated the deposition rate\nmultiple times throughout the process by performing x-\nray reﬂectivity on thicker samples. The thickness of the\nﬁlms range from 2 to 25 ML, each with uncertainty of\nonly few laser pulses, which is equivalent to a very small\nfraction of a 1 ML (approx. 20 pulses per 1 ML).\nAtomic force microscope(AFM) images(Fig. 1), taken\nimmediately following deposition, indicate that between\n2and7ML,SrRuO 3ﬁlms showhomogeneouscoverageof\nthe substrate, with two-dimensional stripe shaped steps\nfollowing the ( ∼0.2◦) miscut of the substrate. These\ntwo-dimensional steps are 1 ML in height and are typ-\nically 100 nm in width. Moreover, unlike previous re-\nports [6, 12], no three-dimensional island growth was ob-\nserved, indicating an atomically smooth surface and sin-\ngle domain structure in these ﬁlms. The observed step-\nlike growth seems to fade at thicknesses above 9 ML as\nthe steps mostly coalesce, suggesting a transition from a\ngrowth mode of two-dimensional layer-by-layer to a step2\nﬂow mode, in agreement with earlier reports [11].\nFIG. 1: AFM images of: (a) SrTiO 3substrate before deposi-\ntion, (b) 2 ML, (c) 5 ML and (d) 9 ML (see text).\nFig. 2 shows the resistivity of the ﬁlms through the\ntransition, measured from room temperature down to\n4.2K. The ferromagnetic transition was noticeable in all\nﬁlms of 4 ML and above, however the transition becomes\nbroad and diﬃcult to determine for the very thin ﬁlms.\nWe note that while the extrapolated low-temperature\nsheet-resistance of the 4 ML ﬁlm is of order ∼7 kΩ, the\nlow-temperature resistance of the 3 ML ﬁlm increases\nmore than 8 orders of magnitude, much higher than the\nquantum of resistance for two-dimensions of h/e2∼26\nkΩ. Thus it is clear that a metal-insulator transition has\noccurred in between these two thicknesses.\nFIG. 2: Resistivity data of SrRuO 3ﬁlms. Arrows point to\nthe location of the ferromagnetic transition, determined f rom\nthe derivative of the resistivity (see e.g. top panels for 3 M L,\n4 ML and thick ﬁlm derivatives).\nThe magnetic properties of the ﬁlms were determined\nfrom Polar Kerr eﬀect (PKE) measurements, which isonly sensitive to the out-of-plane component of the mag-\nnetization[13]. Whileingeneralforthin-ﬁlmsmagnetism\nthe Kerrsignalislarge[13], forultrathin ﬁlms(approach-\ning 1 ML) of weak ferromagnets, especially deposited\non strongly birefringent substrates, these measurements\nmay become diﬃcult because of a small signal that may\nbe masked by the rotation of polarization due to the sub-\nstrate. In our case, SrRuO 3ﬁlms are deposited on mis-\ncut substrates of SrTiO 3which are very strongly linearly\nbirefringent. To overcome the above diﬃculties we have\nused a zero-area-loop Sagnac interferometer [16], which\ndirectly detects the circular birefringence in the magne-\ntized sample. This unique design is based on a Sagnac\nloop in which two counter-propagating beams with op-\nposite circular polarization reﬂect from the sample while\ncompletinga Sagnacloop. This designwhich wasﬁrstin-\ntroduced by Xia et al.[16] is capable of measuring time-\nreversal-symmetrybreaking eﬀects with a shot-noise lim-\nited sensitivity of 100 nanorad/√\nHzat a power of 10\nµWatt, while being completely immune to any reciprocal\neﬀects in the sample, such as linear birefringence [17].\nFor the results reported in this paper we used a normal-\nincidence conﬁguration, measuring at a wavelength of\n1550 nm with a beam focused on the sample to a spot\nsize of 3 µm and in the temperature range of 0.3 K to\nroom temperature. Since the optical penetration depth\nat the used frequency is of order 200 nm, while the thick-\nest sample used was only 8.8 nm, the signal measured, to\na very good approximation, was simply proportional to\nthe area density of the magnetic moment[13].\nFig. 3 shows the evolution of the PKE measured on\nthe samples from 4 ML to 22 ML thick samples. Hys-\nteresis loops were obtained by recording the Kerr signal\natthelowesttemperature(typically0.4K)whileramping\nan out-of-plane magnetic ﬁeld, and then subtracting the\nlinear paramagnetic response from the SrTiO 3substrate\nand diamagnetic response from the optical components\nin the fringing magnetic ﬁeld. After the magnetic ﬁeld\nwas turned oﬀ, the PKE was measured as a function of\ntemperature while the ﬁlms were warmed to room tem-\nperature. This allowed the determination of the Curie\ntemperature Tcand the angle of the easy axis.\nThe temperature dependent remanent-Kerr-eﬀect of\nthe 2 and 3 ML ﬁlms did not show any ferromagnetic\ntransition down to 0.4 K, to a resolution of ±0.2µrad.\nHowever,we suggestthat the magnetictransitionmay be\ndeduced from the resistivity data that shows sharp up-\nturnin the resistivityofthe3 MLﬁlm below ∼25K(note\nthe logarithmic axis!). The resistivity of the 2ML ﬁlm\ncouldnotbe measuredbelow100Kdueto thelargeresis-\ntance. Of course, this assertion still needs to be checked.\nMagnetization curves of 2 and 3 ML ﬁlms at 0.4 K are\nshown in Fig. 4a. These were obtained by ﬁrst cooling in\na ﬁeld, then performing a hysteresis loop, followed by a\nsubtraction of the (diamagnetic) contribution of the ﬁber\nstrand in the magnet. The S-shape and ﬁnite opening of3\nFIG. 3: Panels a-d are PKE Hysteresis loop for SrRuO 3ﬁlms\nof diﬀerent thickness takenat 4K, with magnetic ﬁeld applie d\nperpendicular to the plane of the ﬁlm. panels e-h show the\ntemperature dependence remanent PKE signal measured at\nzero magnetic ﬁeld duringwarmup, after a positive saturati on\nmagnetic ﬁeld was turned oﬀ at the lowest temperature.\nboth curves indicate that the low-temperature phase of\nthese ﬁlms have ferromagnetic component with moments\nthat lie entirely in the plane of the ﬁlm. The open loops\nare non-symmetric with respect to zero-ﬁeld, reminiscent\nof exchange-bias behavior [18]. Exchange-bias (EB) phe-\nnomena originate at the interface of FM and AF regions,\nwhere uncompensated AF moments result in a bias mag-\nnetic ﬁeld, causing the hysteresis loop of the FM to be\nshifted away from the origin [18]. Indeed, the hypothe-\nses of both, remanent in-plane ferromagnetism and EB\nare further supported by magnetoresistance (MR) mea-\nsurements at 0.3 K shown in Fig. 4b. We note that the\nmaximum MR observed is ∆ R/R∼0.005, a very small\neﬀect when compared to spin-scattering dominated MIT.\nThe ﬁrst sharp hysteresis loop is obtained for the 4\nML sample (Fig. 3a), pointing to ferromagnetism with\nan almost perpendicular moment [19]. Turning oﬀ the\nmagnetic ﬁeld, a remanent signal is observed ( θR\nK(T)),\nthat disappears at Tc(Fig. 3e). Similar data for other\nsamples is given in Fig. 5 where we show the thickness\ndependence of the saturation Kerr signal ( θS\nK(T), which\nisdeterminedasthehighestpointofthehysteresisloopin\nFig. 3), the Tcof the layers,and the variation ofthe easy-\naxis for all ferromagnetic ﬁlms. Fig. 5a shows that the\nFIG. 4: a) Hysteresis loop for the 2 ML and 3 ML samples.\nInsets show the region near the origin where the exchange-\nbias nature of the loop is clear. 3 ML is biased to the left\nand the 2 ML is biased to the right. Thick-dots mark the\ncrossing of the ﬁeld axis. b) Magnetoresistance for the 3 ML\nsample. Loop starts at S, continues to A, then B, and ends\nat A. Subsequent loops trace the A-B-A loop.\nsaturatedKerrsignalisproportionaltothe ﬁlm thickness\nfrom the thick (22 ML) down to the thinnest samples (4\nML), extrapolating to zero thickness. Since we argued\nthat all these ﬁlms are in the very thin limit compared\nto the penetration depth of the light, this result clearly\nshows that the thick ﬁlm moment ( ∼1.6µB) does not\nchange, and that that all layersare ferromagnetic. Below\n4 ML the saturation Kerr signal plummets, indicating a\nmuch smaller moment of ∼0.2µB.\nFig. 5b shows the thickness dependence of Tc. To de-\ntermine the temperature above which no ferromagnetism\nis observed we magniﬁed the region near the transition\nas show in the inset to that ﬁgure. While in general the\nmagnetization vanishes at Tcwith an exponent smaller\nthan unity, domain structure and reorientation in ﬁlms\nmay result in apparent lower transition temperature ( Te)\n[20]. We therefore deﬁne two critical temperatures as\nshow in the inset, and plot both in Fig. 5b. We note\nthat it is Tc, the temperature at which the Kerr signal\nvanishes, which smoothly extrapolates to the thick ﬁlms\nlimit and therefore to previously published data on 3-\ndimensional SrRuO 3ﬁlms [2]. Both, TeandTccannot\nbe measured below 4 ML. We note that the anomaly in\nthe resistivity, measured by taking the derivative of the\nresistivity curves, agrees with Tcfor the very thin ﬁlms\nand continues towards the thick ﬁlms limit as expected4\n[2].\nFIG. 5: Thickness dependencies of (a) saturated Kerr signal\n(/squaresolid) at the lowest temperature; (b) Curie temperature Tc(/trianglesolid),\nextrapolated Curie temperature Te(•), Resistivity anomaly\n(/square); and (c) the angle Φ ( /triangledownsld) between ﬁlm normal and mag-\nnetic easy axis at the lowest temperature. Dashed line in (a)\nis the linear ﬁt of the data point between 4 and 22 ML. Insert\nin (b) shows how TcandTeare determined.\nTo determine the magnetic anisotropy angle, we cal-\nculate Φ = cos−1(θR\nK/θS\nK). This is plotted in Fig. 5c. It\nwas previously found that for high-quality epitaxial thick\nﬁlms and at low temperatures Φ ∼30◦[2]. In Fig. 5c we\nshowthat Φ decreasesfrom 22◦in the case of22ML ﬁlm,\nto 14◦in 4 ML ﬁlm. Thus, the fact that below 4 ML the\nmoment is almost entirely in the plane ( Fig. 4), hence\ngoingin the complete opposite wayto the trend we found\nabove, is just another conﬁrmation that a phase transi-\ntion occurred between 3 and 4 monolayers.\nThe above observations point to a unique phase tran-\nsition that occurs as a function of thickness. Thick ﬁlms\nare itinerant and ferromagnetic with the moment point-\ning almost entirely in the direction perpendicular to the\nplane of the ﬁlms. However, when made thin, the mo-\nments seem to plummet into the plane of the ﬁlms, with\nAFM layers appearing in contact with FM layers thus\ninducing exchange-bias behavior. The origin of AFMregions may be a consequence of both, increased dis-\norder and the emerging two-dimensional physics, and\nthe observed thickness-driven phase-transition could ei-\nther be a natural consequence of enhanced correlations.\nThe strong reduction in the moment, from ∼1.6µBto\n∼0.2µBmay further indicate a proximity to a quantum\ncritical point in which coupling to ﬂuctuations causes the\nreduction in the moment.\nDiscussions with L. Klein are greatly acknowledged.\nFabricationofthe Sagnacsystem wassupported by Stan-\nford’s Center for Probing the Nanoscale, NSF NSEC\nGrant 0425897. Work at Stanford was supported by the\nDepartment of Energy Grant DE-AC02-76SF00515.\n[1] J. S. Dodge et al., Phys. Rev. B 60, R6987 (1999).\n[2] L. Klein, et al., Phys. Rev. Lett. 77, 2774 (1996).\n[3] A. P. Mackenzie et al., Phys. Rev. B 58, R13318 (1998).\n[4] W. Siemons, G. Koster, A. Vailionis, H. Yamamoto,\nD.H.A. Blank, and M.R. Beasley, Phys. Rev. B 76,\n075126 (2007).\n[5] G. Herranz, B. Martnez, J. Fontcuberta, F. S´ anchez, C.\nFerrater, M.V.Garca-Cuenca, andM.Varela, Phys.Rev.\nB 67, 174423 (2003).\n[6] D. Toyota et al., Appl. Phys. Lett. 87, 162508 (2005).\n[7] L.Antognazza, K.Char, T.H.Geballe, L.L.H.King, and\nA.W.Sleight, Appl. Phys. Lett. 63, 1005 (1993).\n[8] Michael Feigenson, James W. Reiner, and Lior Klein,\nPhys. Rev. Lett. 98, 247204 (2007).\n[9] Z. Q.QiuandS.D.Bader, Rev.Sci. Inst.71, 1243(2000).\n[10] G. Koster, B.L. Kropman, G. J. H. M. Rijnders, D. H.\nA. Blank, and H. Rogalla, Appl. Phys. Lett. 73, 2920\n(1998).\n[11] J. Choi, C. B. Eom, G. Rijnders, H. Rogalla, and D. H.\nA. Blank, Appl. Phys. Lett. 79, 1447 (2001).\n[12] G. Herranz, B. Martinez, J. Fontcuberta, F. S´ anchez,\nM. V. Garcia-Cuenca, C. Ferrater, and M. Varela, Appl.\nPhys. Lett. 82, 85 (2003).\n[13] See e.g. S. D. Bader, J. Magn. Magn. Mater. 100, 440\n(1991).\n[14] L. Klein, J. S. Dodge, T. H. Geballe, A. Kapitulnik, A.\nF. Marshall, L. Antognazza, and K. Char, Appl. Phys.\nLett. 66, 2427 (1995).\n[15] G. Herranz et al., J. Appl. Phys. 97, 10M321 (2005).\n[16] Jing Xia, P. T. Beyersdorf, M. M. Fejer, and A. Kapitul-\nnik, Appl. Phys. Lett., 89, 062508 (2006).\n[17] Jing Xia, Y. Maeno, P. T. Beyersdorf, M. M.Fejer, and\nA. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006).\n[18] For a recent review see e.g. J. Nogu´ es and I.K. Schuller ,\nJ. Magn. Magn. Mater. 192, 203 (1999).\n[19] We note that below 9 ML we occasionally observed step\nstructure which is usually attributed to competition be-\ntween domain wall and coherent reversal in ultrathin\nmagnetic ﬁlms: see e.g. R. A. Hyman A. Zangwill, and\nM. D. Stiles, Phys. Rev. B 58, 9276 (1998).\n[20] O. Riss, A. Tsukernik, M. Karpovsky and A. Gerber, J.\nMagn. Magn. Mat 298, 73 (2006)."    },    {        "title": "1107.5954v2.Coexistence_of_Ferromagnetism_and_Superconductivity_in_Noncentrosymmetric_Materials_with_Cubic_Symmetry.pdf",        "content": "Coexistence of Ferromagnetism and Superconductivity\nin Noncentrosymmetric Materials with Cubic Symmetry\nTitus Neupert\nCondensed Matter Theory Group, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland\nManfred Sigrist\nInstitute for Theoretical Physics, ETH Z urich, 8093 Z urich, Switzerland\n(Dated: November 4, 2018)\nThis is a model study for the emergence of superconductivity in ferromagnetically ordered phases\nof cubic materials whose crystal structure lacks inversion symmetry. A Ginzburg-Landau-type\ntheory is used to \fnd the ferromagnetic state and to determine the coupling of magnetic order to\nsuperconductivity. It is found that noncentrosymmetricity evokes a helical magnetic phase. If the\nwavelength of the magnetic order is long enough, it gives rise to modulations of the order parameter\nof superconductivity, both in modulus and complex phase. At magnetic domain walls the nucleation\nof superconductivity is found to be suppressed as compared to the interior of ferromagnetic domains.\nI. INTRODUCTION\nIf the unit cell of a three-dimensional crystal is non-\ncentrosymmetric, i.e., not invariant under the parity\noperation, the spatial inversion symmetry is broken.\nNoncentrosymmetricity allows for the Dzyaloshinskii-\nMoriya interaction that entails the breaking of spin ro-\ntation symmetry due to antisymmetric spin-orbit cou-\npling (ASOC). Many noncentrosymmetric material fea-\nture unusual properties, when ordering phenomena such\nas superconductivity or magnetic order, that break other\nsymmetries in addition to inversion and spin rotation, are\npresent. For example, ASOC and magnetic order lead to\na helical magnetic structure in MnSi1and Ba 2CuGe 2O72,\nwhere they give rise to magnetic \feld-induced ferroelec-\ntricity3.\nSuperconductivity with a noncentrosymmetric crystal\nstructure generically appears in a mixed parity state4.\nIn addition, many of the known noncentrosymmetric su-\nperconductors, such as CePt 3Si5,6, CeRhSi 37, and UIr8,\nshow states of magnetic order and in some regions of\nthe phase diagram superconductivity coexists with the\nmagnetic order. A simple consideration of the combined\naction of ASOC and time-reversal symmetry breaking\nmagnetic \felds or magnetization on the energy bands re-\nveals the possibility of spatial modulations of the order\nparameter of superconductivity (\fnite- q-pairing)9{12.\nThe aim of this paper is to study noncentrosymmet-\nric systems which show coexistence of magnetic order\nand superconductivity. To be concrete, we will focus\non a cubic crystal without inversion center described by\nthe point group Oand restrict our study to the case\nof ferromagnetic order, where the wavelength of mod-\nulations of the magnetization is much longer than the\nlattice spacing. We will use the generalized Ginzburg-\nLandau-approach. In sec. II we determine the nature of\nthe helical ferromagnetic phase in presence of ASOC. In\nsec. III we subsequently study, how the superconducting\nstate is altered on the background of this magnetic order.\nWe \fnd that noncentrosymmetricity causes the magneticmoment to follow a helical modulation and introduces a\nnew length scale for the superconducting order parame-\nter. Depending on the ratio of the magnetic wavelength\nand this length scale, the order parameter of supercon-\nductivity either remains homogeneous or exhibits a mod-\nulation both in complex phase and absolute value in this\nmagnetic phase. Finally, in sec. IV we consider a limit\nwhere the magnetic state resembles a \flamentary domain\nstructure. We show that superconductivity will nucleate\nin the interior of the domains rather than at the domain\nwalls.\nII. MAGNETIC STATE\nIn order to determine the magnetic state, the free en-\nergy density is expanded in the magnetization m(r) as a\nthree-dimensional order parameter. The expression must\nbe invariant under time reversal symmetry and under the\nsymmetry transformations of the cubic point group O.\nSpin-orbit coupling ties rotations of spin and orbital de-\ngrees of freedom together, such that the free energy has\nto be invariant under a simultaneous rotation in both\nspin and orbital space. The magnetization m(r) belongs\nto the irreducible (vector) representation \u0000 4of the point\ngroupO. The second-order terms of the free energy read\nF(2)\nM:=Z\nd3rn\n\u000bm2+\u001c0\u0000\u001c1\n2(r\u0002m)2+\u001c2(r\u0001m)2\n+\u001c3X\ni;j\u0002\n3\u000eij(@imi)2\u0000(@imi)(@jmj)\u0003\n+\u001c0+\u001c1\n4X\ni6=j(@imj+@jmi)2+#m\u0001(r\u0002m)o\n;(1)\nwhere#and\u001ci(i= 0:::3) are temperature-independent\nphenomenological parameters and \u000bhas the usual tem-\nperature dependence \u000b=\u000b0(T\u0000TM), with\u000b0>0. Here,\nTMis the transition temperature for a state of homoge-\nneous magnetization. The e\u000bect of noncentrosymmetric-\nity manifests itself in the presence of a Lifschitz-invariantarXiv:1107.5954v2  [cond-mat.supr-con]  19 Oct 20112\nterm proportional to #12,13, which would not be allowed\nhad we considered the centrosymmetric cubic point group\nOh. We demand that F(2)\nMis bound from below towards\narbitrarily strong modulations of m(r). Amongst others\nthis leads to the condition \u001c0>0.\nThe vector \feld m(r) that minimizes F(2)\nMis given by a\nhelix with a wavevector k0. Depending on the parameters\n\u001ci(i= 0:::3),k0is either aligned parallel to a coordinate\naxis (6-fold degenerate) or parallel to a body diagonal (8-\nfold degenerate). Let us assume k0=k0ezfrom here on.\nIn that case, the magnetization takes the form\nm(r) =m0[sin(\u0006k0z)ex+ cos (k0z)ey]: (2a)\nThe two signs correspond to two energetically degenerate\nchiralities and k0=j#j=(2\u001c0). The magnitude m0of\nthe magnetization would be determined by terms of the\norderm4. For the time being we ignore these terms and\nconsiderm0as a parameter. The transition temperature\nto the magnetic state is given by\nT\u0003\nM=TM+#2\n4\u000b0\u001c0(2b)\nand is always larger than TM, owing to \u001c0>0. In the\nlimit of a centrosymmetric system ( #!0) we recover\na state of homogeneous magnetization with the corre-\nsponding transition temperature, TM.\nIII. SUPERCONDUCTIVITY IN THE\nMAGNETIC PHASE\nIn this section, we study the in\ruence of homogeneous\nand helical magnetic order as given by eq. (2a) on the\nsuperconducting order. We assume the temperature to\nbe su\u000eciently below the transition temperature to the\nmagnetic phase, such that an emergent superconducting\norder parameter does not change the magnetization con-\nsiderably. For simplicity, a complex scalar superconduct-\ning order parameter \u0011(r) belonging to the \u0000 1representa-\ntion of the point group Ois considered. The expansion\nof the free energy up to the order \u00112reads\nFSC:=Z\nd3rn\naj\u0011j2+bjD\u0011j2+um2j\u0011j2+\niv(\u0011\u0003m\u0001D\u0011\u0000\u0011m\u0001D\u0003\u0011\u0003)o\n;(3)\nwhere D=r\u0000 2ieA=(~c) with Abeing the vector po-\ntential. Here, b,uandvare temperature-independent\nphenomenological parameters and ahas linear temper-\nature dependence a=a0(T\u0000TSC) witha0>0.TSC\ndenotes the critical temperature of superconductivity for\nm= 0. Demanding that FSCis bound form below if \u0011\nhas strong spatial \ructuations necessitates b >0. The\nterm proportional to urepresents the paramagnetic de-\npairing e\u000bects of the magnetization on the Cooper-pair\nformation, thus we assume u>0. Noncentrosymmetric-\nity is again re\rected by the presence of a Lifschitz-termproportional to v, which is forbidden in case with inver-\nsion symmetry. Note that the Lifschitz-term introduces\na new length-scale j\u0018Ljfor the superconducting order,\nwhere\u0018L:=b=(vm0) and is not singular at the transi-\ntion to the superconducting phase. In the following we\nshall assume the limit of strong type-II superconductiv-\nity in the sense that the length scales k\u00001\n0,\u0018L, and the\ncoherence lengthp\nb=aare assumed to be much smaller\nthan the magnetic penetration depth \u0015.12In this limit,\nwe can neglect the e\u000bect of the vector potential Aand\nreplace D!r in eq. (3).\nA. Homogeneous Magnetization\nBefore addressing the helical magnetic order, we \frst\nconsider the e\u000bect of a homogeneous magnetization m=\nm0eyon the onset of superconductivity, i.e., we take the\nlimitk\u00001\n0!1 . Minimizing the functional (3) with re-\nspect to\u0011straightforwardly yields a modulation of the\norder parameter as\n\u0011(x) =\u00110eiy=\u0018L: (4a)\nThe transition temperature is then given by\nT(1)\nSC=Thom\nSC+b\na0\u00182\nL(4b)\nand shows the advantage of the phase modulation of the\norder parameter as compared to the transition temper-\nature to a homogeneous superconducting state Thom\nSC =\nTSC\u0000um2\n0=a0. The appearance of these phase modula-\ntions of\u0011in noncentrosymmetric superconductors in a\nhomogeneous magnetic \feld was already pointed out in\nvarious other studies9{12,14.\nB. Helical Magnetization\nWe now turn to the more subtle e\u000bect of the helical\nmagnetization given by eq. (2a) on the superconducting\nstate. The variation of FSCwith respect to \u0011yields\n0 =\u0002\n@2\n\u0010+A(B) + 2Bcos(2\u0010)\u0003\n\u0011k;'(\u0010); (5a)\nwhere a Fourier transformation of the superconducting\norder parameter in the coordinates xandywas per-\nformed and ( kx;ky) = (kcos';ksin').\nWe introduce here the parameters\nA(B) :=4a0(Thom\nSC\u0000T)\nbk2\n0\u0000\u0012B\u0018Lk0\n2\u00132\n; (5b)\nB:=4k\n\u0018Lk2\n0; (5c)\nand thezcoordinate is substituted by \u0010as\u0010:=\n(k0z\u0000'\u0000\u0019=2)=2.\nIn eq. (5a) we identify Mathieu's di\u000berential equa-\ntion with the variable \u0010. Mathieu's equation cannot be3\nAB\n0 –0.5 –1 –1.51\n–1\nFIG. 1. (Color online) Portion of the stability diagram of\nMathieu's equation. Equation (5b) de\fnes a parabola AT(B).\nAs the temperature decreases, the parabola migrates from\nleft to right. The transition temperature is reached when the\nparabola touches the region of stability (dots on dashed curve)\nat some critical value Bc, which in turn determines k. If the\ncurvature of the parabola at k= 0 is larger than that of the\nstability region (dotted curve), there will be no contact with\nthe region of stability until the transition to the homogeneous\nstateBc= 0.\nsolved analytically in a closed form. However, one can\nidentify a region in the the A-B-parameter space where\nthe solutions of eq. (5a) are bounded for all \u00102R.\nThis stability region is displayed in \fg. 1 as a shaded\narea. For parameter values ( A;B) inside this stability\nregion, the system would be in a stable superconduct-\ning state. Equation (5b) de\fnes a parabola AT(B) of\nenergetically degenerate parameter values in the A-B-\nplane. These parabolas can be labeled by temperature\nTvia the temperature dependence of A. For su\u000eciently\nhigh temperatures these parabolas do not intersect with\nthe region of stability. As the temperature is lowered,\nthe superconducting instability occurs when the parabola\nAT(B) touches \frst the boundary of the (shaded) sta-\nbility region in \fg.1. Di\u000berent types of solutions de-\npend on the curvature of the parabola around B= 0\nwhich should be compared with that of the boundary to\nthe stability region, which can expanded at B= 0 to\nA(B)\u0019\u0000B2=2 + 7B4=12815. If the curvature of AT(B)\nis larger than that of the stability region, corresponding\nto the condition\nj\u0018Ljk0>p\n2; (6)\nthen the touching point is at Bc= 0 (see dotted line\nin Fig. 1) such that with k= 0 the superconducting\norder parameter is homogeneous. For the condition op-\nposite to eq. (6) the touching point is at \fnite values of\nBc\u0019\u0006f 16(2\u0000\u00182\nLk2\n0)=7g1=2(\u001c1) yielding a modulated\norder parameter with a \fnite k, as shown by the dashed\nline in \fg. 1. Equation (6) can be seen as an analogue to\nthe condition on the Ginzburg-Landau parameter \u0014that\nFIG. 2. (Color online) Schematic picture of the superconduc-\ntivity order parameter \u0011'(r) given by eq. (7) that nucleates in\nthe presence of the sketched helical magnetization. It exhibits\nan amplitude modulation in the z-direction, and in addition\nto that features a phase modulation perpendicular to the z-\ndirection (depicted by the colour gradient).\nappears in the discussion of the vortex phase of supercon-\nductors. As in our case, we compare a superconducting\nlength scale ( \u0018L) with a magnetic length scale ( k\u00001\n0) and\nobtain an inhomogeneous superconducting state if the\nsuperconducting length scale is shorter.\nFor the inhomogeneous case, the solution features a\ncontinuous circular degeneracy in the kx-ky-plane, pa-\nrameterized by '. The elementary solution is given in\nterms of the lowest order even Mathieu function ce 015by\n\u0011'(r) =\u00110exp\u0002\nik2\n0\u0018LBc(xcos'+ysin')\u0003\n\u0002ce0(k0z\u0000'\u0000\u0019=2;Bc):(7)\nThis form of the order parameter is sketched in \fg. 2\ntogether with the helical magnetic state. The general\nsolution is a superposition \u0011n:=Pn\nl=1cl\u0011'lwith the\ncomplex-valued coe\u000ecients cl. To \fnd the set of coe\u000e-\ncientscland phases 'lthat minimizes the free energy,\nwe use Abrikosov's parameter \fde\fned by\n\f:=\nj\u0011j4\u000b\nhj\u0011j2i2; (8)\nwhereh:::iis the spatial average. The solution that min-\nimizes\fis realized. A minimum requires that the par-\ntial derivatives @'l\fand@cl\fvanish for all l. For the\nphases, this yields the condition 'i\u0000'j=\u0019Nij=2;8i;j\nwith someNij2Z. An explicit evaluation of \ffor the\nremaining cases n= 1:::4 reveals the optimal solution\nto be\n\u0011opt;\u0006\n' =\u0011'+i\u0011'\u0006\u0019=2: (9)\nFrom this result we see that the order parameter ac-\nquires a long-wavelength phase and amplitude modula-\ntion perpendicular to the wavevector of the helical mag-\nnetic order k0and an amplitude modulation with the\nsame wavevector as the magnetic order. To lowest order\ninB2\nc/(2\u0000k2\n0\u00182\nL), the transition temperature of this\ninhomogeneous superconducting state is given by\nT(2)\nSC=Thom\nSC+bk2\n0\n14a0\u0000\n2\u0000k2\n0\u00182\nL\u00012; (10a)4\nand the superconducting order parameter has the ap-\nproximate form\n\u0011opt;\u0006\n'=0(r)\u0019\u00110p\n2n\neip\n2Bck0x\u0002\n1 +Bc\n2cos(2k0z)\u0003\n+ie\u0006ip\n2Bck0y\u0002\n1\u0000Bc\n2cos(2k0z)\u0003o\n;\n(10b)\ntaking 0<Bc\u001c1 and\u0018Lk0\u0019p\n2.\nIV. MAGNETIC SOLITONS AND\nSUPERCONDUCTIVITY\nThe analysis of the free energy F(2)\nMshowed that ASOC\nfavors an inhomogeneous magnetic state. So far we ig-\nnored in the discussion the explicit form of the fourth\norder terms in the free energy expansion. Besides \fx-\ning the magnitude of the magnetic moment, these terms\nmay also introduce additional features such as crystal\nanisotropy of the magnetic moments. The fourth order\nterms allowed within the point group O, ignoring any\ngradient terms, read\nF(4)\nM:=Z\nd3r \n3\n2\f0m4+ 2\f1X\nim4\ni!\n: (11)\nHere,\f0and\f1are temperature-independent phe-\nnomenological parameters. The expression shows that\nnot all directions of the magnetization are degenerate in\nenergy. If we simply insert the helical solution (2a) we\nobtain\nF(4)\nM=m4\n0Z\nd3r\u0014\nconst. +\f1\n2cos(4k0z)\u0015\n; (12)\nwhich does not minimize this part of the free energy. It\nis instructive to solve for the magnetization m(r), that\nminimizes the total magnetic free energy F(2)\nM+F(4)\nM,\nin order to understand the e\u000bect of the anisotropy term\n\f1qualitatively. As before, we assume that the mag-\nnetization has no component in ez-direction and will\nbe a function of zonly. With the complex notation\nM(z) =mx(z) +imy(z), the variational equation reads\n0 =\u0000\n\u0000\u001c0@2\nz+i#@z+\u000b\u0001\nM(z)\n+ 3(\f0+\f1)jM(z)j2M(z) +\f1\u0016M3(z):(13a)\nTo linear order in \f1, the solution is given by\nM(z) =m0eik0z\u0014\n1 +\f1\n2\f0#2\u00004\u000b\u001c0\n9#2\u00004\u000b\u001c0e\u00004ik0z\u0015\n(13b)\nwhere now m2\n0=\u000b0(T\u0003\nM\u0000T)=[3(\f0+\f1)]. Thus, the\nanisotropy term pins the magnetization m(r) parallel to\nexorey(parallel to ex\u0006ey) for\f1<0 (for\f1>0), be-\nsides a modulation of the amplitude jm(r)j=jM(z)j. In\nthe limit of a strong anisotropy, the magnetic state could\nbe viewed as parallel planes of magnetic solitons, where\neach soliton twists the magnetization-vector by 90\u000e.We will now address the question, how superconduc-\ntivity nucleates in the presence of such a \flamentary\nmagnetic structure. From our analysis in the previous\nsection, we know that in the limit j\u0018Ljk0>p\n2 a ho-\nmogeneous superconducting order nucleates despite the\nmodulated magnetic background. In the opposite case,\nj\u0018Ljk0<p\n2, for which a spatially modulated supercon-\nducting order is found, \frst of all the solitons in the\nmagnetization lift the continuous degeneracy of the solu-\ntion (9) parameterized by '.\nThen, the question arises whether the superconduct-\ning order parameter nucleates at the soliton (domain\nwall) position or rather in the interior of the domains.\nTo address this, let us consider the nucleation of bound\nstates at an isolated domain wall at which the mag-\nnetization is tilted from m(r)kexforz! \u00001 to\nm(r)keyforz!1 . We shall for the moment assume\nthat the amplitude jm(r)j\u0011jM(z)j=m0is constant\nacross the domain wall. The magnetization can thus be\nparametrized by a single function \u0012(z) =\u0000\u0012(\u0000z) with\n\u0012(z!\u00061 ) =\u0006\u0019=4 as\nM(z) =m0exp [i\u0012(z) +i\u0019=4]: (14a)\nInsertion of the magnetization (14a) in the free energy\nfor the superconducting order parameter, Eq. (3), yields\nupon Fourier transformation in the xandycoordinates\nthe variational equation\n\u0014\n\u0000@2\nz+2k\n\u0018LV(z)\u0015\n\u0011k(z) =\u0000\u0014a\nb+um2\n0\nb+ 2k2\u0015\n\u0011k(z);\n(14b)\nwhere\nV(z) :=\u0000sin[\u0012(z) +\u0019=4]\u0000cos[\u0012(z) +\u0019=4]:(14c)\nHere, we have chosen kx=ky\u0011kin accordance with\nthe symmetry of the problem and to obtain a binding po-\ntential. Eq. (14b) is reminiscent of the one-dimensional\nSchr odinger equation with the potential V(z) and an en-\nergy eigenvalue given by the square bracket on the rhs.\nThis analogy immediately delivers the inequality\n\u00002k\n\u0018Lp\n2>a\nb+um2\n0\nb+ 2k2; (15a)\nsince the energy of the lowest bound state is always larger\nthan the potential minimum V(0) =\u0000p\n2. Via the tem-\nperature dependence of a, the energy eigenvalue of the\nlowest bound state determines the temperature at which\nthe superconductivity nucleates at the domain wall. The\ntransition temperature Tdw\nSCof the bound state therefore\nsatis\fes\nTdw\nSC<T SC\u0000um2\n0\na0+b\na0\u0018L=T(1)\nSC; (15b)\nwhereT(1)\nSCis the bulk transition temperature de\fned in\neq. (4b). This inequality holds independent of the con-\ncrete form of the function \u0012(z). We conclude that super-\nconductivity will nucleate in the interior of the domains\nrather than at the magnetic solitons (domain walls).5\nThis result relies on the assumption that jm(r)jis spa-\ntially constant. If we relax this assumption and consider\nthe case in which the magnetization is suppressed near\nthe domain wall to a value m0\n0<m 0, the superconduct-\ning order near the domain wall would be less a\u000bected by\nthe paramagnetic depairing e\u000bect, represented by the pa-\nrameteru. The upper bound for Tdw\nSCin eq. (15b) then\nexceedsT(1)\nSC, thereby opening the way for an reversion of\nthe inequality Tdw\nSC<T(1)\nSC.\nV. CONCLUSIONS\nIn our study on the coexistence of ferromagnetism and\nsuperconductivity in noncentrosymmetric materials we\nfound that ASOC, represented by Lifschitz-terms in the\nfree energy expansions, gives rise to unusual modulations\nof the order parameters. Where a centrosymmetric ma-\nterial would have a homogeneously magnetized ferromag-\nnetic ground state, an arbitrarily small parity violation\ngenerates a state of helical magnetization.\nThe possible modulations in the superconducting or-\nder parameter in presence of magnetic order were found\nto be governed by the ratio of two length scales, j\u0018Ljand\nthe characteristic length of magnetic modulations k\u00001\n0.\nWhen the magnetic length scale is larger, the supercon-\nducting order is not homogeneous. This extends to the\nlimit of homogeneous magnetization k\u00001\n0!1 , where a\ncomplex phase modulation was found. In presence of thehelical magnetic phase, a state of simultaneous complex\nphase and amplitude modulation develops, if the mag-\nnetic wavelength is large enough.\nWe closed our discussion with the consideration of a\nmagnetic state of \flamentary solitons, which is obtained\nfor a strong anisotropy parameter in the magnetic free en-\nergy. We found that this will force the order parameter\nof superconductivity to nucleate with the same \flamen-\ntary structure, creating a stripe-like state with maxima\nin between two solitons.\nThe complex phase winding of the order parameter of\nnon-centrosymmetric superconductors that are exposed\nto a magnetic \feld was noticed theoretically a long time\nago14, but an experimental veri\fcation of this state is still\nlacking. The inhomogeneous superconducting states that\nwe \fnd in this work show both complex phase and ampli-\ntude modulations. The latter would entail an anisotropic\nresistivity drop at the criticality for current directions\nperpendicular and parallel to the wavevector of the mod-\nulation and are therefore more accessible to an experi-\nmental observation. In fact, such anisotropic supercon-\nducting transitions were reported for the antiferromag-\nnetically ordered noncentrosymmetric CeRhSi 3in ref. 7,\nwhere superconductivity and magnetism show and inter-\nesting interplay.\nWe would like to thank D.F. Agterberg, N. Hayashi,\nV. Mineev, Y. Yanase and S. Guerrero for helpful discus-\nsions. This work was \fnancially supported by the Swiss\nNational Science Foundation and the NCCR MaNEP.\n1P. Bak and M. Jensen, J. Phys. C, 13, L881 (1980).\n2A. Zheludev, G. Shirane, Y. Sasago, N. Kiode, and K.\nUchinokura, Phys. Rev. B, 54, 15163-15170 (1996).\n3H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, and\nY. Tokura, Phys. Rev. Lett. 105, 137202 (2010).\n4M. Sigrist, D. Agterberg, P. Frigeri, N. Hayashi, R. Kaur,\nA. Koga, I. Milat, K. Wakabayashi, and Y. Yanase, J.\nMagn. Magn. Mater., 310, 536-540 (2007).\n5E. Bauer, G. Hilscher, H. Michar, Ch. Paul, E. Scheidt, A.\nGribanov, Yu. Seropegin, H. Noel, M. Sigrist, and P. Rogl,\nPhys. Rev. Lett., 92, 027003 (2004).\n6K. Samokhin, E. Zijlstra, and K. Bose, Phys. Rev. B, 69,\n094514 (2004).\n7N. Kimura, Y. Muro, and H. Aoki, J. Phys. Soc. Jpn., 76,\n051010 (2007).\n8T. Akazawa, H. Hidaka, H. Kotegawa, T. Kobayashi,S.Fukushima, E. Yamamoto, Y. Haga, R. Settai, and Y.\nOnuki, Physica B, 378-380 , 355-358 (2006).\n9O. V. Dimitrova and M. V. Feigelman, Pisma Zh. Eksp.\nTeor. Fiz. 78, 1132 (2003) [JETP Lett. 78, 637 (2003)].\n10R. P. Kaur, D. F. Agterberg, and M. Sigrist, Phys. Rev.\nLett. 94, 137002 (2005).\n11D. Agterberg and R. Kaur, Phys. Rev. B, 75, 064511\n(2007).\n12V. Mineev and M. Sigrist, arXiv:0904.2962 (to be pub-\nlished).\n13K. Samokhin, Phys. Rev. B, 70, 104521 (2004).\n14V. M. Edelstein, Zh. Eksp. Teor. Fiz., 95, 2151 (1989) [Sov.\nPhys. JETP 68, 1244 (1989)].\n15M. Abramowitz and I. Stegun, Handbook of Mathematical\nFunctions (United States Department of Commerce, Na-\ntional Bureau of Standards, 1964)."    },    {        "title": "1008.2239v1.Neutral_skyrmion_configurations_in_the_low_energy_effective_theory_of_spinor_condensate_ferromagnets.pdf",        "content": "arXiv:1008.2239v1  [cond-mat.quant-gas]  13 Aug 2010Neutral skyrmion conﬁgurations in the low-energy eﬀective theory of spinor\ncondensate ferromagnets\nR. W. Cherng1and E. Demler1\n1Physics Department, Harvard University, Cambridge, MA 021 38\n(Dated: October 30, 2018)\nWe study the low-energy eﬀective theory of spinor condensat e ferromagnets for the superﬂuid\nvelocity and magnetization degrees of freedom. This eﬀecti ve theory describes the competition\nbetween spin stiﬀness and a long-ranged interaction betwee n skyrmions, topological objects familiar\nfrom the theory of ordinary ferromagnets. We ﬁnd exact solut ions to the non-linear equations\nof motion describing neutral conﬁgurations of skyrmions an d anti-skyrmions. These analytical\nsolutions provide a simple physical picture for the origin o f crystalline magnetic order in spinor\ncondensate ferromagnets with dipolar interactions. We als o point out the connections to eﬀective\ntheories for quantum Hall ferromagnets.\nI. INTRODUCTION\nFor systems with broken symmetries, low-energy ef-\nfective theories provide a simple and powerful tool for\ndescribing the relevant physics. The focus is on the long-\nwavelength Goldstone modes which emerge from the mi-\ncroscopic degrees of freedom. This often reveals the con-\nnections between seemingly unrelated systems that share\nthe same pattern of symmetry breaking. The most well-\nknown example is that of the complex scalar ﬁeld used\nin the low-energy theories of bosonic superﬂuids [1, 2],\nfermionic superconductors [3], and XY spin systems [4].\nEﬀective theories also bring to light subtle topological\neﬀectsthatmaybecomeimportantatlowenergies. Gold-\nstone modes often carry a non-trivial topology which can\ngive rise to the appearance of topological defects. For ex-\nample, in two spatial dimensions Kosterlitz and Thouless\npointed out the crucial role of vortices for the complex\nscalar ﬁeld [4, 5]. Vortices are point-like topological de-\nfects around which the phase of the complex scalar ﬁeld\nwinds by an integer multiple of 2 π.\nUltracold atomic systems provide an ideal testing\ngroundforlow-energyeﬀective theories. The microscopic\ndegreesoffreedomarewell-isolatedfromtheenvironment\nexperimentally and well-understood theoretically. The\nchallenge is in describing how these microscopic degrees\nof freedom organize at low energies in the presence of\nnon-trivial interactions. When only one internal hyper-\nﬁne level is important, the phenomenon of scalar Bose-\nEinstein condensation is given by the low-energy theory\nofthecomplexscalarﬁeld[6–8]. Aseriesofgroundbreak-\ning experiments have observed vortex lattices in rotating\ncondensates [9, 10] as well as evidence for the role of vor-\ntices in the equilibrium Kosterlitz-Thoulesstransition for\ntwo-dimensional condensates [11].\nForultracoldatomswithacomplexinternallevelstruc-\nture, there are various patterns of symmetry breaking.\nThis lead to rich possibilities and challenges for low-\nenergy eﬀective descriptions. Spinor condensate ferro-\nmagnets are one such system which has seen a number of\nimportant experimental advancements (see Ref. [12]).\nThis includes the development of optical dipole trapsused for preparation [13] and phase-contrast imaging\nused for detection [14] in S= 187Rb. In addition to the\nphase degree of freedom familiar from single-component\ncondensates, the magnetization naturally arises as a de-\nscription of the low-energy spin degrees of freedom. A\nvector quantity sensitive to both the population and co-\nherences between the three hyperﬁne levels, the magne-\ntization can be directly imaged in experiments [14].\nOne of the most striking observations in spinor con-\ndensate ferromagnets is the spontaneous formation of\ncrystalline magnetic order [15, 16]. From an initial\nquasi-two-dimensional condensate prepared with a uni-\nform magnetization, a crystalline lattice of spin domains\nemerges spontaneously at suﬃciently long times. The\npresence of a condensate with magnetization sponta-\nneously breaks global gauge invariance and spin rota-\ntional invariance. Additionally, crystalline order for the\nmagnetization breaks real space translational and rota-\ntional symmetry. Previous works have pointed out the\ncrucial role of dipolar interactions in driving dynami-\ncal instabilities within the uniform condensate towards\nstates with crystalline order [17–19]. Numerical analysis\nof the full multi-component Gross-Pitaevskii equations\nsuggest dipolar interactions can give rise to states with\ncrystalline order [18, 20].\nIn this and a companion paper [21], we take a com-\nplementary approach and focus on the low-energy eﬀec-\ntive theory of two-dimensional spinor condensate ferro-\nmagnets. This eﬀective theory describes the interaction\nbetween the superﬂuid velocity and the magnetization\ndegrees of freedom. Previous work has derived the equa-\ntions of motion for this eﬀective theory [22, 23]. We\nextend this result to demonstrate how the Lagrangian\nfor the eﬀective theory can be written as a non-linear\nsigma model in terms of the magnetization alone. The\neﬀect of the superﬂuid velocity is to induce a long-ranged\ninteraction term between skyrmions, topological defects\nfamiliar from the theory of ordinary ferromagnets [24].\nIn contrast to the point-like topological defects of vor-\ntices, skyrmions describe extended magnetization tex-\ntures which carry a quantized topological charge.\nIn the companion paper [21], we study how symmetry2\ngroups containing combined real space translational, real\nspace rotational, and spin space rotational symmetry op-\nerations can be used to classify possible crystalline mag-\nnetic orders. Within each symmetry class, we ﬁnd min-\nimal energy conﬁgurations describing non-trivial crys-\ntalline conﬁgurations.\nThe main purpose of this paper is to give a simple\nphysical physical picture for the origin of crystalline or-\nder in terms of neutral conﬁgurations of skyrmions and\nanti-skyrmions. Long-rangedskyrmion interactions force\nmagnetization conﬁgurations to have net neutral collec-\ntions of topological defects. We show this explicitly\nby ﬁnding exact analytical solutions for the non-linear\nequations of motion describing both localized collections\nof skyrmions and anti-skyrmions as well as extended\nskyrmion and anti-skyrmion stripes.\nSkyrmions have non-trivial spin conﬁgurations that\nspontaneously break translational and rotational invari-\nance in real space. Proposed originally in high energy\nphysics as a model for mesons and baryons [25, 26],\nskyrmionshavefound applicationsin a number of diverse\nﬁelds including quantum hall ferrogmagnets [27, 28], and\nmagnetically ordered crystals [29–32].\nA neutral collection of such topological objects is\nable to take advantage of the dipolar interaction en-\nergy without a large penalty in the skyrmion interac-\ntion energy. Since a neutral skyrmion conﬁguration has\nthe same topological number as the uniform magnet, its\nstability is not ensured by topology alone. However,\nthe scale invariance of the skyrmion interaction energy\nrules out the most straightforward instability of bring-\ning skyrmionsand anti-skyrmionsclosertogether. Essen-\ntially, the skyrmion interaction energy forces the charge\ndensities of a skyrmion and anti-skyrmion pair to shrink\nas the distance between them shrinks so that the overall\nenergy remains the same.\nThe analytical solutions we ﬁnd without dipolar inter-\nactions closely resemble the minimal energy conﬁgura-\ntions found numerically in the presence of dipolar inter-\nactions. The role of dipolar interactions can then be seen\nas stabilizing these non-trivial solutions of the eﬀective\ntheory. We point out that magnetic dipolar interactions\nare small. Thus it is a good starting point to ﬁnd states\nwhich are static nontrivial solutions of the system with-\nout dipolar interactions.\nThe eﬀective theory of spinor condensate ferromag-\nnets is essentially identical to that of the quantum Hall\nferromagnets [27, 33]. Although the microscopic de-\ngrees of freedom are fermionic electrons, the magneti-\nzation order parameter is bosonic. The Coloumb inter-\naction between electrons then gives a contribution to the\nskyrmion interaction for the eﬀective theory. The result-\ning skyrmion interaction is qualitatively the same as the\none for spinor condensate ferromagnets. This suggests\nthestudyofspinorcondensateferromagnetsmayhavein-\nteresting connections to quantum hall ferromagnets and\nvice versa.\nThe plan of this paper is as follows. In Sec. II wereview the theory of ordinary ferromagnets and how\nskyrmion solutions arise from the non-linear Landau-\nLifshitz equations of motion. We then proceed to re-\nview how these skyrmion solutions are used in the low-\nenergy eﬀective theory of quantum Hall ferromagnets in\nSec. III. Next we derive the low-energy eﬀective the-\nory for spinor condensate ferromagnets in Sec. IV. In\nparticular, we demonstrate how a long-ranged skyrmion\ninteraction term (which also appears for quantum hall\nferromagnets) arises from coupling of the magnetization\nto the superﬂuid velocity.\nIn Sec. V, we discuss how to interpret the mathemati-\ncalstructureofskyrmionsolutionsforordinaryferromag-\nnets in terms of a separation of variables. This approach\nallowsus to use ﬁnd new exact solutions for the oridinary\nferromagnet. More importantly, it also allows us to gen-\neralize the skyrmion solutions of the ordinary ferromag-\nnet to ﬁnd analytical solutions for the spinor condensate\nferromagnet. Theselattersolutionsdescribebothneutral\ncollections of localized skyrmions and anti-skyrmions as\nwell as extended stripe conﬁgurations. Finally, we dis-\ncuss how the analytical solutions we ﬁnd oﬀer insight\ninto quantum hall ferromagnets and spinor condensate\nferromagnets with dipolar interactions in Sec. VI\nII. SKYRMIONS IN FERROMAGNETS\nWe begin by reviewing the theory of ordinary two-\ndimensional ferromagnets described by the following La-\ngrangian, Hamiltonian, and Landau-Lifshitz equations of\nmotion [34]\nL=−S/integraldisplay\ndtd2xA(ˆn)·∂tˆn−/integraldisplay\ndtH\nH=S\n4/integraldisplay\nd2x∇(ˆn)2\n∂tˆn=1\n2ˆn×∇2ˆn (1)\nwhere ˆnis a three component real unit vector and A(ˆn)\nis the unit monopole vector potential. The order param-\neter ˆndescribes the magnetization and is a unit vector\nliving on the sphere. Calculating the variation of the La-\ngrangian to derive the Landau-Lifshitz equations [35, 36]\ncan be done by using δˆn=δw×ˆn. This is consistent\nwith the constraint |ˆn|= 1 sinceδˆn·ˆn= 0 by construc-\ntion. The variation of the term/integraltext\ndtA(ˆn)·∂tˆnis clearest\nwhen using its geometric interpretation as the area on\nthe sphere swept by ˆ n. While we include time depen-\ndent terms for completeness, in this paper we will only\nconsider static solutions\nIn addition to the trivial uniform solution, there are\nnon-trivial soliton solutions to the Landau-Lifshitz equa-\ntions called skyrmions. By parameterizing\nˆn=/bracketleftbigsin(α)cos(β) sin(α)sin(β) cos(α)/bracketrightbigT(2)3\nFIG. 1: A single localized skyrmion carrying 4 πnet skyrmion\ncharge in the ordinary ferromagnet (left). Neutral conﬁgur a-\ntion consisting of a localized skyrmion carrying +4 πskyrmion\ncharge in a negative background carrying −4πskyrmion\ncharge in the spinor condensate ferromagnet (right). Notic e\nthe +ˆz(−ˆz) meron carrying +2 π(+2π) net skyrmion charge\nat the origin (at inﬁnity) for the ordinary ferromagnet are\nmapped to a skyrmion (anti-skyrmion) carrying +4 π(-4π)\nnet skyrmion charge in the spinor condensate ferromagnet.\nRed (blue) background indicates positive (negative) skyrm ion\ndensityq, black 2D arrows the superﬂuid velocity vfor the\nspinor condensate ferromagnet, and shaded 3D arrows the\nmagnetization ˆ n.\nwe seeαcontrols ˆnz, the ˆzcomponent of the magnetiza-\ntion whileβcontrols the canonically conjugate variable\ngiving the orientation of ˆ nx, ˆny, the ˆx, ˆycomponents of\nthe magnetization.\nThe minimal energy solutions within each topological\nsector are called skyrmions and can be written in the\nform [24]\ntan(α/2)eiβ= exp[f(x+iy)] = exp[u(x,y)+iv(x,y)]\n(3)\nwheref(z) is a holomorphic function of zwith real part\nu(x,y) and imaginary part v(x,y). The function f(z)∼\nnlog(z−z0) canhavelogarithmicsingularities[52]. Since\nβhas to be 2πperiodic, the residues ofthese singularities\nmust be integers. This implies exp[ f(x+iy)] can only\nhave zeros or poles. The resulting spin conﬁguration has\nˆnz= +1(ˆnz=−1)atzeros(poles) ofexp[ f(z)] while ˆnx,\nˆnywind anti-clockwise (clockwise) along a path circling\nthe zeros (poles) in a anti-clockwise direction.\nThesesolutionsdescribetopologicaldefectsofordinary\nferromagnets. For ﬁxed boundary conditions, smooth,\nﬁnite energy conﬁgurations are separated into distinct\nclasses characterizedby a quantized topological invariant/integraltext\nd2xq(x) = 4πNwhereNis the number of times ˆ n\ncovers the sphere. Here\nq=ǫµνˆn·∇µˆn×∇νˆn (4)\nis the skyrmion density which is positive for f(z) holo-\nmorphic. From here on, lower Greek indices (upper Ro-\nman indices) refer to real space (order parameter) com-\nponents.\nConsider the single skyrmion solution shown in the left\nof Fig. 1. It corresponds to f(z) = log(z), carries net\nFIG. 2: Unit cell for a lattice of localized skyrmions carryi ng\n8πnet skyrmion charge per unit cell in the ordinary ferromag-\nnet (left). Unit cell for a neutral conﬁguration carrying ze ro\nnet skyrmion charge per unit cell in the spinor condensate\nferromagnet (right). Notice +ˆ z(−ˆz) merons are mapped to\nskyrmions (anti-skyrmions). Red (blue) background indica tes\npositive (negative) skyrmion density q, black 2D arrows the\nsuperﬂuid velocity v, and shaded 3D arrows the magnetiza-\ntion ˆn.\nskyrmion charge 4 π, and can be decomposed into a +ˆ z\nmeron at the origin and a −ˆzmeron at inﬁnity. Essen-\ntially, a meron can be thought of as half of a skyrmion\nand characterized by two signed quantities: the direction\nof the magnetization at the core and the orientation of\nthewindingawayfromthecore. Thesignoftheskyrmion\ndensity is the product of the sign of these two quantities.\nFor example, the +ˆ zmeron at the origin to the left of\nFig. 1 carries net skymrion charge 2 π.\nFor spinor condensate ferromagnets, scale invariance\nof the skyrmion interaction term guarantees stability\nagainst trivial rescaling. However, this may change if\nwe introudce a short distance cutoﬀ or quantum ﬂuctua-\ntions.\nNotice the Lagrangian in Eq. 1 has translational, ro-\ntational, and scale invariance in real space as well as ro-\ntational invariance in spin space. The single skyrmion\nsolution spontaneously breaks all of these symmetries.\nHowever, the action is invariant for f(z) = log[f0(z−z0)]\nwith complex constants z0andf0. Taking solutions with\nz0∝negationslash= 0 corresponds to spatially translating the solution\nwithz0= 0, solutions with Re[ f0]∝negationslash= 1 correponds to\nspatially rescaling the the solution with f0= 1, and so-\nlutions with Im[ f0]∝negationslash= 0 corresponds to rotation of the\nf0= 1 solution. In addition, the action is the same for\nOˆnwhereOis a rotation matrix describing solutions re-\nlated by spin space transformations.\nCollections of multiple skyrmions are described by\nf(z) having multiple singularities. A square lattice of\nskyrmions with net skyrmion charge 8 πper unit cell is\nshown in the left of Fig. 2. It can also be decomposed\ninto a collection of two +ˆ zmerons and two −ˆzmerons\narranged antiferromagnetically. In addition, there are\nsolutions tan( α/2)eiβ= exp[f(x−iy)] withf(¯z) anti-\nholomorphic characterized by ˆ x, ˆycomponents winding\nin the opposite direction compared to holomorphic solu-\ntions andqnegative.4\nIII. QUANTUM HALL FERROMAGNETS\nAfter describing the structure of skyrmion solutions\nfor ordinary ferromagnets, we now brieﬂy review how\nthey are used in the study of quantum Hall ferromagnets\n[27, 33]. For the low-energy eﬀective theory of quantum\nHall ferromagnets, spin 1 /2 electrons in a magnetic ﬁeld\nare described by a two component Chern-Simons com-\nposite boson theory. The bosons couple to the phys-\nical gauge ﬁeld due to the magnetic ﬁeld and a ﬁcti-\ntious Chern-Simons gauge ﬁeld which attaches appro-\npriate ﬂux quanta to change from bosonic to fermionic\nstatistics. At appropriate ﬁlling fractions, the physical\nand Chern-Simons ﬂuxes cancel and the resulting quan-\ntum Hall plateau is described as a condensate of compos-\nite bosons.\nThe Lagrangian and Hamiltonian for the resulting ef-\nfective theory is given by\nL=−1\n2/integraldisplay\ndtd2xA(ˆn)·∂tˆn−/integraldisplay\ndtH\nH=/integraldisplay\nd2x/bracketleftbigg1\n8(∇ˆn)2+1\n2/vectorB·ˆn/bracketrightbigg\n+1\n8/integraldisplay\nd2xd2y[q(x)−¯q]G(x−y)[q(y)−¯q] (5)\nwhere/vectorBis the magnetic ﬁeld giving rise to a linear Zee-\nmanshift,q(x) istheskyrmiondensityand ¯ q=/integraltext\nd2xq(x)\nis the net skyrmion charge.\nThe coupling of the bosons to the Chern-Simons ﬁeld\ngives rise to the skyrmion interaction term. At long dis-\ntances, the Chern-Simons ﬁeld ties the skyrmion density\nto the deviation of the physical electron density from its\nbackground value. In fact skyrmions are lowest energy\nquasiparticles at a ﬁlling factor of one [27, 37]. Thus, the\nskyrmiondensityinheritstheColoumbinteractionforthe\nelectron density. Away from quantum Hall plateaus, the\nbackground value ¯ qis non-zero and the singular |x−y|−1\nbehavior in G(x−y) forces ﬁnite energy conﬁgurations\nto have non-zero net skyrmion charge.\nThus the focus is on static conﬁgurations carrying net\nskyrmion charge. The skyrmion solutions of the ordinary\nferromagnet have non-zero ¯ qand provide a good qualita-\ntive description of quantum Hall ferromagnetsawayfrom\nquantum Hall plateaus. For example, a sqaure lattice of\nskyrmions as shown in Fig. 2 carries a net charge of\n8πper unit cell and can be used as a starting point for\nstudying conﬁgurations carrying ﬁnite ¯ q. Although such\nsolutionstakeintoaccountthespinstiﬀnessandthelong-\nranged divergence of the skyrmion interaction, quantita-\ntive results require additional analysis arising from the\ndetailed form of skyrmion interaction and the linear Zee-\nman shift. This usually involves numerical minimization\n[37–40] of the Hamiltonian in Eq. 5.IV. SPINOR CONDENSATE FERROMAGNETS\nHaving reviewed known results on ordinary and quan-\ntum Hall ferromagnets, we now consider spin Sspinor\ncondensate ferromagnets described by the microscopic\ncondensate wavefunction Ψ, a 2S+ 1 complex vector\n[41, 42]. The microscopic Gross-Pitaevskii Lagrangian\nis given by\nL=/integraldisplay\ndtiΨ†∂tΨ−/integraldisplay\ndtH−/integraldisplay\ndtHS\nH=/integraldisplay\nd2x/bracketleftbigg1\n2m|∇Ψ|2+g0(Ψ†Ψ)2+gs(Ψ†/vectorFΨ)2/bracketrightbigg\n(6)\nwhere/vectorFare spin matrices and g0>0 gives the spin-\nindependent contact interaction strength while gs<0\ngives the spin-dependent contact interaction strength\nwhich favors ﬁnite magnetization. Here, HSdenotes\nadditional spin dependent interactions such as the\nquadratic Zeeman shift and dipolar interactions.\nAs was the case for quantum Hall ferromagnets, we\nexpect a simpler description to emerge at low energies.\nThe resulting low-energy eﬀective theory should only in-\nvolve the condensate phase φand local magnetization\nˆnwhich describe the order parameters of the system.\nPrevious work has shown this can be done at the level\nof the equations of motion [22]. In this section, we ex-\ntend this result to derive the Lagrangian and Hamilto-\nnian for the eﬀective theory solely in terms of the magne-\ntization. However, the skyrmion density acts as a source\nof voriticity for the superﬂuid velocity. Thus, the ef-\nfect of the superﬂuid phase is to induce a logarithmic\nvortex-vortex interaction between skyrmions. The re-\nsulting non-linear sigma model is essentially identical to\nthat of the quantum Hall ferromagnetbut with skyrmion\ninteraction G(x−y)∼log(x−y) having logarithmic be-\nhavior instead of |x−y|−1behavior.\nWe begin by considering energies below the scale of\nspin-independent g0andferromagneticspin-dependent gs\ncontact interactions. The condensate has ﬁxed density\nΨ†Ψ=ρand fully polarized magnetization Ψ†/vectorFΨ=\nSρˆnwhere/vectorFare spin matrices. The states that satisfy\nthese constraints are parameterized solely in terms of the\nlow-energy degrees of freedom\nΨ=√ρeiφψˆn, ˆn·/vectorFψˆn=Sρψˆn(7)\nwhereφdescrribes the phase of the condensate and ψˆnis\na fully polarized unit spinor with ˆ ndescribing the orien-\ntation of the magnetization. Although φis not directly\nobservable, the superﬂuid velocity is a physical quantity\nvµ=∇µφ−iψ†\nˆn∇µψˆn (8)\nwhich has contributions from both φandψˆn.\nIn this paper, we are primarily interested in the com-\npetition between the spin stiﬀness and superﬂuid kinetic5\nenergy. In the companion paper [21], we address the ef-\nfect of magnetic dipolar interactions. From here on, we\nconsider the case HS= 0. From the Gross-Pitaevskii\nLagrangian in Eq. 6, the Berry’s phase term becomes\niΨ†∂tΨ=−ρ∂tφ−SρA(ˆn)·∂tˆnwhile the kinetic energy\nterm is givenby |∇Ψ|2=Sρ/2(∇ˆn)2+ρv2and the inter-\naction terms give constants. This gives the Lagrangian\nand Hamiltonian as\nL=−S/integraldisplay\ndtd2xA(ˆn)·∂tˆn−/integraldisplay\ndtH\nH=/integraldisplay\nd2x/bracketleftbiggS\n4(∇ˆn)2+1\n2v2/bracketrightbigg\n(9)\nwhere we take ρ=m= 1 for simplicity. Notice for ﬁxed\nρ, the∂tφterm is a total derivative which we exclude.\nCompared to the Lagrangian describing ordinary ferro-\nmagnetsinEq. 1, thereisanadditionalsuperﬂuidkinetic\nenergy term vµvµ.\nThe global phase φenters the Lagrangian quadrati-\ncally and only through v. The equation of motion for\nφgives∇µvµ= 0 implying the superﬂuid velocity is di-\nvergenceless. This follows from vdescribing transport of\nthe density ρ, a conserved quantity which is locally ﬁxed\nat low energies due to the spin-independent contact in-\nteraction. This implies that vµvµonly depends on the\ndivergenceless part of v. In momentum space, this is\nvµ(+k)[δµν−kµkν/k2]vν(−k) which can also be written\nasFµν(+k)Fµν(−k)/2k2.\nHere we have introduce the analog of the ﬁeld strength\ntensorFµν=∇µvν−∇νvµ,alocalquantitythatdepends\nonly on the divergenceless part of v. In two dimensions,\nthere is only one non-zero component to Fµν. From Eq.\n8, this is given by the skyrmion density Fxy=−Fyx=\nSq. Here we assume that the condensate phase φdoes\nnot contribute to Fxythrough vortex-like singularities.\nThis is valid because the vortex core energy is large.\nThe above results give vsolely in terms of ˆ nas\n∇µvµ= 0, ǫ µν∇µvν=Sq (10)\nwithqthe skyrmion density. Gradients in the order pa-\nrameter ˆnarise in part from phase gradients in the con-\ndensate wavefunction Ψ. Topologically non-trivial mag-\nnetization conﬁgurations can thus give rise to vorticity\ndescribed by a non-zero curl ǫµν∇µvν∝negationslash= 0.\nBy introducing the two-dimensional logarithmic\nGreen’s function −∇2G(x) =δ(x) we can write the\nsuperﬂuid kinetic energy Fµν(+k)Fµν(−k)/2k2in real\nspace and obtain\nL=−S/integraldisplay\ndtd2xA(ˆn)·∂tˆn−/integraldisplay\ndtH\nH=S\n4/integraldisplay\nd2x(∇ˆn)2+S2\n2/integraldisplay\nd2xd2yq(x)G(x−y)q(y)\n(11)\nwith the corresponding equations of motion given by\n(∂t+vµ∇µ)ˆn=1\n2ˆn×∇2ˆn (12)along with Eq. 10 for the superﬂuid velocity solved by\nvµ=Sǫµν∇νΦ, −∇2Φ =q(13)\nwhereΦ(x) =/integraltextd2xG(x−y)q(y) has the interpretationof\nthe two-dimensional Coloumb potential associated with\nq.\nCompared to the Landau-Lifshitz equations describing\nordinary ferromagnets in Eq. 1, the replacement ∂t→\n∂t+vµ∇µdescribes the advection of the magnetization\nby the superﬂuid velocity [22]. This advective term arise\nfromvariationofthesuperﬂuidkineticenergyterminEq.\n9 or equivalently from the skyrmion interaction term in\nEq. 11. Recall that we include the time dependence for\ncompleteness and focus only on static solutions.\nThe skyrmion density qgives the vorticity for the\nsuperﬂuid velocity v. Thus, the second term in the\nHamiltonian above gives the pairwise logarithmic inter-\naction energy between vortices. In the thermodynamic\nlimit, the logarithmic divergence of G(x−y) at large dis-\ntances forces ﬁnite energy conﬁgurationsto have zero net\nskyrmion density/integraltext\nd2xq(x) = 0.\nNotice Eq. 11 for spinor condensate and Eq. 5 for\nquantum hall ferromagnets have the same form. Al-\nthoughG(x−y) behaves as log |x−y|for the former and\n|x−y|−1forthelatter, bothgivesingularcontributionsat\nlong wavelengths. However, the important absence of a\nﬁnite backgroud value ¯ qin the skyrmion interaction im-\nplies conﬁgurations for spinor condensate ferromagnets\nmust have zero net skyrmion charge.\nV. EXACT SOLUTIONS WITH NEUTRAL\nSKYRMION CHARGE\nIn quantum hall systems density deviations from the\nincompressible state cause skyrmions. Density ﬂuctua-\ntions with zero net average (such as impurities) cause\nspin textures with zero net skyrmion number. States\nwith non-zero skyrmion number occur away from quan-\ntum hall plateaus. Thus the analytical skyrmion solu-\ntions carrying net charge for the ordinary ferromagnet\noﬀered insight into more complicated case of quantum\nHall ferromagnets away from the quantum Hall plateau.\nFor spinor condensate ferromangets, we will show how\nnet neutral solutions with skyrmions and anti-skyrmions\nwithout dipolar interactions oﬀer insight into the more\ncomplicated case with dipolar interactions.\nIn this section, we ﬁnd exact analytical solutions\nfor spinor condensate ferromagnets with logarithmic\nskyrmion interactions in the absence of dipolar inter-\nactions. We study the eﬀect of including dipolar in-\nteractions numerically after a symmetry analysis in the\ncompanion paper [21]. The exact solutions we ﬁnd here\ngreatly resemble the numerical solutions in the compan-\nion paper. As we discuss in Sec. VI, the interpretation\nof the exact solutions in terms of neutral collections of\nskyrmions and anti-skyrmions oﬀers physical insight into6\nthe morecomplicated numerical solutionsof the compan-\nion paper.\nTo ﬁnd exact solutions with zero net skyrmion charge,\nit is vital to include to eﬀect of the skyrmion interac-\ntion term. Recall it is the long wavelength divergence\nof this term that forces conﬁgurations to have zero net\nskyrmion charge. Although this cannot be done exactly\nfor|x−y|−1interactions as in quantum hall ferromag-\nnets, itispossibleforlogarithmicinteractionsasinspinor\ncondensate ferromagnets. Physically, this is because the\nlogarithmic interaction arises solely from the superﬂuid\nkinetic energy which is scale invariant just like the spin\nstiﬀness term.\nWe begin with the parameterization of ˆ nin Eqs. 2, 3,\nused in the skyrmion solutions of the ordinary ferromag-\nnet. Notice α,βprovide a set of orthogonal coordinates\nfor the sphere describing the order parameter space of\nˆn. Forf(x+iy) =u(x,y)+iv(x,y) holomorphic, u(x,y)\nandv(x,y)provideasetoforthogonalcoordinatesforthe\nplane describing real space. So for ordinary ferromag-\nnets, skyrmion solutions are given by α= 2tan−1(eu)\nandβ=v(see also Eq. 3), which can be understood as\na separation of variables.\nNoticeα(u) is a function of uonly while β(v) is a\nfunction of vonly. Each orthogonal coordinate of the\norder parameter space α,βis a function of only one or-\nthogonal coordinate of real space u,v, respectively. The\nreason why using uandvas coordinates is tractable\nis because they satisfy the Cauchy-Riemann equations\n∂xu= +∂yv,∂yu=−∂xv. In particular, this implies\n∇u· ∇v= 0 meaning countour lines of constant uare\nperpendicular to countour lines of constant vas required\nfor orthogonal coordinates. In addition, both ∇2u= 0\nand∇2v= 0 satisfy Laplace’s equation. The above two\nidentities simplify expressions involving ∇2which arise\nin the equations of motion. In particular, when changing\nvariables from ( x,y) to (u,v), the Laplacian retains its\nform∂2\nx+∂2\nx∝∂2\nu+∂2\nv.\nAn alternative interpretation of the above separation\nof variables is as follows. Given an arbitrary conﬁgura-\ntion for ˆn, consider the contourlines ofconstant ˆ nz, the ˆz\ncomponent of the magnetization. For contour lines with\nˆnz∝negationslash=±1 that form closed curves, consider the winding\nnumber of ˆnx+iˆny= sin(α)eiβ. For smooth conﬁgura-\ntions, this is a quantized integer that cannot change be-\ntween neighboring contours which do not cross ˆ nz∝negationslash=±1.\nThis implies the winding number is constant in regions\nbetween contourswith ˆ nz∝negationslash=±1. Label diﬀerent contours\nof ˆnzbyuand the label coordinate along each contour\nbyv.\nIn order to have a non-zero winding number, βmust\nhavesomedependence on vandtheminimaloneis β∝v.\nIn principal, βcan also depend on uand have some non-\nmonotonic dependence on v, but linear dependence is the\nsmoothest one compatible with non-zero winding num-\nber. We see that skyrmion solutions can be interpreted\nin the above manner along with the additional condi-\ntion thatuandvare mutually othogonal and satisfy\nFIG. 3: For an arbitrary smooth magnetization conﬁgura-\ntion, contour lines of ˆ nzand the phase of ˆ nx+iˆnyprovide a\nnatural coordinate system. For skyrmion solutions of the or -\ndinary ferromanget, this gives an orthogonal coordinate sy s-\ntem with contour lines intersecting at right angles. Using\nthe same ansatz for spinor condensate ferromagnets where\ncontour lines intersect at right angles allows us to solve th e\nnon-linear and non-local equations of motion. The magneti-\nzation and contour lines are shown for the single skyrmion\n(left) for the ordinary ferromagnet and neutral conﬁgurati on\n(right) for the spinor condensate ferromagnet. Hue indicat es\norientation of ˆ nx, ˆnycomponents of the magnetization and\nbrightness gives the ˆ nzcomponent with white (black) indi-\ncating ˆnz= +1 (ˆnz=−1).\nLaplace’s equation. Physically, these additional condi-\ntions onuandvcan be understood as a consequence of\nminimizing the spin stiﬀness. The relationship between\ncontour lines, winding number, and the exact solutions\nwe discussed in this section is shown in Fig. 3.\nWith this viewpoint, we can now generalize the\nskyrmion solutions for ordinary ferromagnets and also\nﬁnd new solutions for spinor condensate ferromagnets.\nForf(x+iy) =u(x,y)+iv(x,y) holomorphic, we take\nα=α(u), β =kv (14)\nfor the parameterization of Eq. 2. Compared to the\nskyrmion solutions of Eq. 3 with tan( α/2) = exp[u] for\nordinary ferromagnets, we allow for general dependence\nα(u) for spinor ferromagnets. Whereas for ordinary fer-\nromagnets we only need to solve ∇2ˆn= 0, for spinor\ncondensate ferromagnets we need to solve Eqs. 12, 13\nwith∂t= 0.\nIn addition, we include a constant of proportionality\nβ=kvinstead ofβ=v. Recall for ordinary ferromag-\nnets,β= +vandf(x+iy) holomorphic give skyrmion\nsolutions with positive skyrmion density qwhileβ=−v\nandf(x−iy) antiholomorphic give anti-skyrmion solu-\ntions withq. We can treat both types of solutions with\njustf(x+iy) holomorphic by allowing β=kvwithk\npositive or negative.\nThere are two cases to consider for f(z). The ﬁrst is\nwhenf(z)isapolynomialin zwith nosingularities. This\nwill turn out to describe skyrmion and anti-skyrmion\nstripe and domain wall conﬁgurations for both ordinary\nand spinor condensate ferromagnets. The second case is\nwhenf(z) has singularities. Since ˆ nshould be single-7\nvalued,βand thuskvcan only have constant 2 πNdis-\ncontinuities with Ninteger. This implies f(z) can only\nhave logarithmic singularities. These solutions will turn\nouttosimplybethelocalizedskyrmionconﬁgurationsfor\nordinaryferromagnetsandneutralcollectionsoflocalized\nskyrmions and anti-skyrmions for spinor condensate fer-\nromagnets.\nNext we consider Eq. 13 for the superﬂuid velocity v.\nForthe parameterizationin Eqs. 2, 14, we seefrom Eq. 4\nthat the skyrmion density qonly depends on u. We thus\ntakeΦ(u)toonlydependon uwhichreducestheequation\n−∇2Φ =qto−Φ′′(u) =q(u). From here on primes\ndenote derivatives with respect to u. By solving for Φ( u)\nwe can then obtain vby diﬀerentiating. Explicitly, we\nobtain\nq=−kcos(α)′|∂zf|2,vz=iSk[C+cos(α)]∂zf(15)\nwherevz=vx−ivy,∂zf=∂xf−i∂yfandCis a con-\nstantofintegrationphysicallydescribinga uindependent\nconstant contribution to the superﬂuid velocity.\nWe now proceed to analyze Eq. 12 for spinor conden-\nsate ferromagnets. By substituting the results of Eq. 15\naboveand the parameterizationin Eqs. 2, 14, we ﬁnd the\nˆzcomponent of Eq. 12 is automatically satisﬁed. In ad-\ndition, the ˆxand ˆycomponents are proportional to each\nother and reduce to a second ordinary diﬀerential equa-\ntion forα(u). For completeness, we can use the same\napproach to analyze Eq. 1 for ordinary ferromagnets us-\ning the same parameterization in Eqs. 2, 3.\nFor ordinary and spinor condensate ferromagnets, the\nequations of motion in Eq. 1 and Eq. 12 reduce to\n2α′′= +k2sin(2α)\n2α′′=−4SCk2sin(α)−(2S−1)k2sin(2α) (16)\nrespectively. Notice the equations of motion for ordinary\nferromagnets are formally given by the S= 0 limit for\nspinor condensate ferromagnets. Recall the spin stiﬀness\nterm scales linearly with Swhereas the superﬂuid kinetic\nenergyscalesquadraticallywith S. Forspinorcondensate\nferromagnets in the limit S→0, the superﬂuid kinetic\nenergy is negligible compared to the spin stiﬀness and\nthe ordinary ferromagnet is recovered. From here on, we\nconsider the more general equation of motion for spinor\ncondensate ferromagnets.\nInterpreting uas time, this equation is that of a clas-\nsical particle with coordinate αand momentum α′. The\ntotal energy E=K+Uis a constant of motion where\nK= (α′)2/2 is the kinetic energy while the periodic po-\ntentialUand equation of motion are\nU(α) =−2SCk2cos(α)−2S−1\n4k2cos(2α)\nα′=/radicalbig\nE−U(α) (17)\nwithS= 0 for ordinary ferromagnets. For this type of\nsolution, the total energy calculated from the Hamilto-nian in Eq. 11 is given by\nH=N/integraldisplay\ndudvS\n2/bracketleftbigg2S+1+4C2S\n4k2−E+α′2/bracketrightbigg\n(18)\nand similarly, S= 0 for the term in brackets for ordinary\nferromagnets.\nThe integer Ncomes from changing variables ( x,y)\nto (u,v) and taking into account each ( u,v) may occur\nfor multiple ( x,y). Mathematically, it is given by the\ndegree offviewed as a map from the complex plane to\nitselfC→C. For localized skyrmion solutions for the\nordinary ferromagnet, it physically corresponds to the\nskyrmion number. As an example, f(z) = logzfor the\nsolutionsshowninFig. 1(seeEq. 21forageneralization)\nand each value of ( u,v) occurs exactly once for ( x,y)\nranging over the plane and thus N= 1. Forf(z) =\nlog[(ϑ(z−λ,i)ϑ(z−λ∗,i))/(ϑ(z+λ,i)ϑ(z+λ∗,i))] with\nϑ(z,τ)the elliptictheta functionand λ= (1+i)/2forthe\nsolutionsshowninFig. 2(seeEq. 22forageneralization)\nand each value of ( u,v) occurs exactly twice for ( x,y)\nranging over the unit cell and thus N= 2.\nHere we comment on the signiﬁcance of the parame-\ntersCandE. Diﬀerent values of CandEcorrespond to\nsolutions with diﬀerent boundary conditions . Physically,\nCcontrols a constant backgroundcontribution to the su-\nperﬂuid velocity. The parameter Econtrols the relative\nscaling of the two components of the coordinate system.\nFor example, for doubly periodic stripe solutions which\nwe will discuss later, Econtrols the aspect ratio of the\nunit cell.\nFormally,EandCare constants of integration for the\nequations of motion. Since Eq. 16 is a second order\ndiﬀerential equation, we have to specify both α(u) and\nα′(u), with the latter given indirectly by the constant of\nmotionE. In addition, Centers through integration of\nEq. 15 relating the skyrmion density to the superﬂuid\nvelocity.\nThese considerationsmean that the total energyin Eq.\n18 cannot be directly compared for diﬀerent CandE. In\nparticular, one should not consider minimizing the total\nenergyHwith respect to EandC. Speciﬁc values of E,\nandCwill be selected by terms beyond the non-linear\nsigma model considered in this paper.\nA. Localized skyrmions and anti-skyrmions\nWe begin by considering the case of localized\nskyrmions for ordinary ferromagnets and neutral collec-\ntionsoflocalizedskyrmionsandanti-skyrmionsforspinor\ncondensate ferromagnets. This corresponds to f(z) hav-\ning logarithmic singularities. The singularities should be\nof integer magnitude and kshould also be an integer.\nRequiring a well-behaved, ﬁnite energy solution gives\nrise to several constraints. Consider Eq. 15 for the\nskyrmion density qand supeﬂuid velocity v. Since\n∂zfdiverges at the logarithmic singularities, we require\ncos(α)′andC+ cos(α) to vanish. In addition, a ﬁnite8\nregion in (x,y) near logarithmic singularities is mapped\nto an inﬁnite region in ( u,v). The constant term in Eq.\n18 for the total energy is then integrated over an inﬁnite\ninterval. Thus, we also require it to vanish.\nThese constraints uniquely specify E,Cand the\nasymptotic value α−∞. For spinor condensate ferromag-\nnets,E=k2(1 +6S)/4,C=±1, cos(α−∞) =∓1 with\nS= 0 for ordinary ferromagnets. For C=±1, we ﬁnd\nfor spinor condensate ferromagnets the solution of Eq.\n16 and the total energy of Eq. 18 given by\nα(u) = cos−1(∓1)−2cot−1(√\n2Ssinh(ku))\nH= 4πNSk2/parenleftbigg\n1+2Stan−1(√\n2S−1)√\n2S−1/parenrightbigg\n(19)\nwhich describes either a 0 →2πor−π→+πkink solu-\ntion. ForC=±1 we ﬁnd for ordinary ferromagnets\nα(u) = cos−1(∓1)+2tan−1(eku)\nH= 2πNSk2(20)\nwhich describes in contrast either a 0 →πor−π→0\nkink solution.\nConsider the classical mechanics problem describing\nthe evolution of α(u) in Eq. 17. For these solutions,\nElies at the maximum giving rise to kink solutions.\nFor ordinary ferromagnets, the kinks connect 0 →πor\n−π→0 and carry net positive or negative skyrmion\ncharge,respectively. Forspinorcondensateferromagnets,\nthe kinks connect −π→+πor 0→2πand carry net\nneutralskyrmioncharge. Theneutralconﬁgurationscon-\nsist of regions of oppositely charged skyrmion and anti-\nskyrmions. These regions are separated by lines where\nthe skyrmion density qvanishes, the magnetization ˆ nis\nalong ˆz, and the superﬂuid velocity vis large.\nForf(z) having a ﬁnite number of logarithmic singu-\nlarities, we can write\nf(z) = log/bracketleftBigg\nf0/producttextNa\nn=1(z−an)\n/producttextNb\nm=1(z−bm)/bracketrightBigg\n(21)\nwith the degree Nof the function f(z) given by N=\nmax(Na,Nb). Here,an(bn) give the locations of +ˆ z\n(−ˆz) merons each carrying net skyrmion charge 2 πfor\nthe ordinary ferromagnet. In contrast, an(bn) give the\nlocationsofskyrmions(anti-skyrmions)eachcarryingnet\nskyrmion charge +4 π(−4π) for the spinor condensate\nferromagnet. We show the corresponding plots of ˆ n,q,\nandvin Fig. 1 for f(z) = log(z). The single skyrmion\nsolutionfortheordinaryferromagnetwith S= 0isshown\non the left and a neutral conﬁguration of one skyrmion\nand one anti-skyrmion for the spinor condensate ferro-\nmagnet with S= 1 is shown on the right.\nFor a periodic lattice of logarithmic singularities,\nf(z) = log/bracketleftBigg\nf0/producttextNa\nn=1ϑ(z−an,τ)\n/producttextNb\nm=1ϑ(z−bm,τ)/bracketrightBigg\n(22)\nFIG. 4: Neutral stripe conﬁguration satisfying non-trivia l\ncorner boundary conditions. Red (blue) background indicat es\npositive (negative) skyrmion density q, black 2D arrows the\nsuperﬂuid velocity v, and shaded 3D arrows the magnetiza-\ntion ˆn.\nwheref0is a constant, and 1, τgive the basis vectors\ngenerating the latttice in complex form, and ϑ(z,τ) is\nthe elliptic theta function. The elliptic theta function\nϑ(z,τ) is essentially uniquely speciﬁed by the quasiperi-\nodic condition\nϑ(z+n+mτ,τ) = exp(−πim2τ−2πimz)ϑ(z,τ) (23)\nand holomorphicity. Just as Eq. 21 is built up from the\nlinear polynomials ( z−z0) which are holomorphic and\nvanish at one point in the complex plane, Eq. 22 is built\nup fromϑ(z,τ) which are holomorphic and vanish at one\npoint in the unit cell. For a discussion of theta functions\nin the quantum Hall eﬀect, see Ref. [43].\nIn the lattice case, Na=Nband/summationtextan=/summationtextbnin\norder to have f(z) periodic. This restriction comes from\nrequiringf(z+n+mτ) =f(z) and using Eq. 23. The\ndegreeNof the function f(z) per unit cell is given by\nN=Na=Nb. Again,an,bngivethe locationsofmerons\n(skyrmions or anti-skyrmions) for the ordinary (spinor\ncondensate) ferromagnet. Fig. 2 shows plots for f(z)\nhaving a lattice of logarithmic singularities with N=\nNa=Nb= 2,f0= 1,τ=i,a1=−a2= (1+i)/2,b1=\n−b2= (1−i)/2. Again the ordinary (spinor condensate)\nferromagnet is on the left (right).\nB. Stripe conﬁgurations\nNow we turn to case of stripe conﬁgurations described\nbyf(z) polynomial in z. The behavior of α(u) solutions\ncontrolledbythepotential inEq. 17changesas Ecrosses\ncritical points dU/dα= 0 of the potential.\nFor completeness, we ﬁrst brieﬂy consider f(z) given\nby higher order polynomials. The corresponding stripe9\nsolutions are not doubly periodic, but satisfy non-trivial\nboundary conditions. For example, we show the mag-\nnetization ˆn, skyrmion density q, and superﬂuid veloc-\nityvforf(z) =iz2in Fig. 4. This solution satis-\nﬁes corner boundary conditions with zero normal compo-\nnent to both the superﬂuid velocity vand spin current\nJi\nµ= ˆnivµ−ǫijkˆnj∇µˆnk/2.\nFrom here on, we focus on f(z) =izwith the corre-\nsponding solutions doubly periodic and describing stripe\nconﬁgurations. The diﬀerent types of behavior for α(u)\nareillustratedschematicallyalongwiththe resultingcon-\nﬁgurations for ˆ nin Fig. 5.Eabove the global maximum\ncorresponds to α(u) monotonic in uwhich from here on\nwe denote as M. This solution describes a periodic stripe\nsolution with ˆ nzvarying over the entire range ±1.Eat\na local maximum corresponds to a kink solution for α(u)\nconnecting α1toα2denoted as Kα2α1. This solution de-\nscribes a single domain wall conﬁguration in ˆ nzand is\nthe analog of the localized solution described earlier. E\nbelow a local maximum corresponds to α(u) oscillating\nnear a ﬁxed value α0denoted as Oα0. Finally,Ebelow\nthe global minimum is forbidden, denoted as F.\nNotice that for S= 0,S= 1/2,S >1/2, the cos(2 α)\nterm in the potential U(α) of Eq. 17 is negative, zero,\nandpositive. Forthemontonicsolutions M,thisdoesnot\naﬀect the qualitative behavior of the resulting periodic\nstripe conﬁgurations. For kink solutions Kconnecting\n0→2πor−π→+π(0→πor−π→0), the result-\ning single domain wall carrys zero net skyrmion charge\n(positive or negative skyrmion charge) for spinor con-\ndensate (ordinary) ferromagnets. Oscillatory solutions\nOalso have diﬀerent behavior with oscillations centered\nabout ˆnz≈ ±1 (ˆnz≈0) for spinor condensate (ordinary)\nferromagnets.\nFor ordinary ferromagnet with S= 0, we parametrize\nE=k2(1+2δ)/4 and ﬁnd the solution of Eq. 16 and the\ntotal energy of Eq. 18 given by\nα(u) =π/2±am/parenleftBig\nku√\n1+δ,(1+δ)−1/parenrightBig\n¯H=Sk2\n2/bracketleftbigg\n−δ\n2+θ(1+δ)/bracketrightbigg\n(24)\nwhere¯His the total energy density given by the aver-\nagingHover the unit cell. Also, am( x,m) is the Jacobi\namplitude function and we deﬁne\nθ(m) =mRe[E(m−1)]\nRe[K(m−1)](25)\nwithK(E) the complete elliptic integral of the ﬁrst (sec-\nond) kind. For δ <−1 the solution is forbidden F. For\n−1≤δ <0 there are two oscillatory solutions at the\n±π/2 minimaO±π/2. Forδ= 0 there are two kink solu-\ntionsK+π\n0andK0\n−π. Forδ>0 the solution is monotonic\nM. We show the classiﬁcationofsolutions alongwith the\ntotalenergydensityforordinaryferromagnetsinthe bot-\ntom left of Fig. 6. Notice the solutions and total energy\ndensity do not depend on C. This is because Centersthrough the superﬂuid velocity which is absent from the\nLagrangian for ordinary ferromagnets.\nFor spinor condensate ferromagnets with S= 1/2, we\nparameterize the constant of motion E=|C|k2(1 +2δ)\nand ﬁnd the solution of Eq. 16 and the total energy\ndensity from Eq. 18 given by\nα(u) = cos−1(C/|C|)+2am/parenleftBig\nku/radicalbig\n|C|(1+δ),(1+δ)−1/parenrightBig\n¯H=k2/bracketleftbigg(|C|−1)2\n8−δ|C|\n2+|C|θ(1+δ)/bracketrightbigg\n(26)\ndependsonlyon |C|. Forδ<−1thesolutionisforbidden\nF. For−1≤δ <0 there is one oscillatory solution Oπ\n(O0) forC <0 (C >0). Forδ= 0 there is one kink\nsolutionK2π\n0(K+π\n−π) forC <0 (C >0). Forδ >0 the\nsolution is monotonic M. We show the classiﬁcation of\nsolutions along with the total energy density for S= 1/2\nspinor condensate ferromagnets in the bottom right of\nFig. 6.\nFor spinor condensate ferromagnets with S >1/2 we\nparameterize the constant of motion E= (2S−1)k2(1+\n2δ)/4 and the constant C= (2S−1)γ/2S. Withγ=τ\nandδ=−1+2στwhereσ=±1 we ﬁnd one solution for\nEq. 16 while the total energy density from Eq. 18 given\nby\nα(u) = Arg/bracketleftBigg\n−τ−σ+2/radicalbig\nτ(τ−σ)s(u)+τs(u)2\n1+στ−στs(u)2/bracketrightBigg\n¯H=Sk2\n2[h0+(2S−1)j0] (27)\nwhere we deﬁne the auxiliary function\ns(u) = sin(ku/radicalbig\n(1−στ)(2S−1)) (28)\nin the solution for α(u) and the functions\nh0=((2S−1)τ−σS)2\n4S\nj0=/braceleftBigg\nστ√τ−σ0≤στ≤1\n0 otherwise(29)\nfor the total energy density. With γ=rsinh(τ) and\nδ=−1+2rsinh(τ) we ﬁnd two solutions for Eq. 16 and\nthe total energy density from Eq. 18 given by\nα±(u) =Arg/bracketleftbigg\n−cosh(τ/4)e±iw(u)∓sinh(τ/4)\nsinh(τ/4)e±iw(u)∓cosh(τ/4)/bracketrightbigg\n¯H=Sk2\n2[h+(2S−1)j∓(2S−1)k] (30)\nwhere we deﬁne the auxiliary function\nw(u) =am/bracketleftbigg\nku/radicalbig\n2r(2S−1),1+r−rcosh(t)\n2r/bracketrightbigg\n(31)10\nFIG. 5: Stripe conﬁgurations for diﬀerent boundary conditi ons inS≥1/2 spinor condensate (bottom row) and S= 0 ordinary\nferromagnets (top row). Red (blue) background indicates po sitive (negative) skyrmion density q, black 2D arrows the superﬂuid\nvelocity v, and shaded 3D arrows the magnetization ˆ n. Notice ˆ nx, ˆny(ˆnz) wind horizontally (oscillate vertically). The ﬁrst\ncolumn shows schematic plots of the classical periodic pote ntialU(α) controlling the evolution of the angular variable αfor\nˆnz= cos(α). Labels for the corresonding type of solution are below the dashed lines indicating the corresponding constant of\nmotionE.\nForEabove the maximum of U,α(u) is montonic in the coordinate ugiving rise to periodic stripe conﬁgurations labeled M\nwith ˆnzcovering the entire range ±1. ForEat a potential maximum, kink solutions connecting the maxim aα1,α2give rise\nto single domain wall conﬁgurations in ˆ nzlabeledKα2α1. For a given value of C, kink solutions occur at one speciﬁc value of\nE. ForEbelow the maximum, αoscillates about a minimum located at α0giving rise to periodic stripe conﬁguration with ˆ nz\noscillating about cos( α0).\nNotice that monotonic solutions Mare qualitatively similar for both S≥1/2 spinor condensate and S= 0 ordinary ferromag-\nnets. In contrast, spinor condensate (ordinary) ferromagn ets have a single (two distinct) 2 π(π) kink solutions Kcentered about\nα= 0 (α=±π/2). In addition, there is a single (two distinct) oscillator y solutions Oalso centered about α= 0 (α=±π/2)\nfor spinor condensate (ordinary) ferromagnets.\nin the solutions for α(u) and the functions\nh=1−(2S−1)δ\n2+(2S−1)2β2\n4S\nj=Re[2√rE(ω)]\nRe[K(ω)/√r]+\nRe[2√r[cosh(t/2)2Π(−sinh(t/2)2,ω)−K(ω)]]\nRe[K(ω)/√r]\nk=/braceleftBiggRe[πrsinh(t)]\nRe[K(ω)√\n2/r], Oπ,O0phase\n0 otherwise\nω=1+r−rcosh(τ)\n2r(32)\nforthetotalenergydensitywhereΠ( m,n)isthecomplete\nelliptic integral of the third kind. We show the classiﬁca-\ntion of solutions along with the total energy density for\nS >1/2 spinor condensate ferromagnets in the top row\nof Fig. 6. The boundaries between solutions of diﬀerent\ntypes are given by δ=−1+ 2β,δ=−1−2β,δ=β2.For increasing γ, notice kink solutions evolve from just\noneK2π\n0through a region with two K2π−α\nαandK+α\n−α,\nto just one K+π\n−πforγ <−1,−1< γ <+1, +1< γ,\nrespectively. The kink solutions separate the monotonic\nsolutionsMfrom the oscillatory solutions. For increas-\ningγ, the oscillatory solutions also evolve from just one\nOπto a region with two OπandO0, to just one O0for\nδ <−1−2γ,−1 + 2γ≥δ≥ −1−2γ,δ <−1 + 2γ,\nrespectively.\nVI. DISCUSSION\nHaving presented a uniﬁed description of both local-\nized and extended stripe solutions in ordinary and spinor\ncondensate ferromagnets, we now turn to how these solu-\ntions oﬀer insight into diﬀerent physical phenomena. We\nﬁrst considerquantum Hall systems. As discussed in Sec.\nIII, conﬁgurations for quantum Hall ferromagnets away\nfrom quantum Hall plateaus carry net skymrion charge11\n/Minus101Γ/Minus101∆S/GreΑter1/Slash12\nOΠ O0OΠ,O0K02ΠK/MinusΠ/PlusΠKΑ2Π/MinusΑ,K/MinusΑ/PlusΑM\nF\n/Minus101C/Minus101∆S/EquΑl1/Slash12\nOΠ O0K02ΠK/MinusΠ/PlusΠM\nF\n/Minus101C/Minus101∆S/EquΑl0\nO/MinusΠ/Slash12,O/PlusΠ/Slash12K/MinusΠ0,K0ΠM\nF\nFIG. 6: Classiﬁcation of solutions (2D plots) and total ener gy density (3D plots) for stripe conﬁgurations given diﬀere nt\nboundary conditions in S >1/2 (top row) and S= 1/2 (bottom left) spinor condensate as well as S= 0 (bottom right)\nordinary ferromagnets. Fig. 5 illustrates the correspondi ng conﬁgurations. γcontrols a constant contribution to the superﬂuid\nvelocity. δcontrols the energy of an associated classical mechanics pr oblem givingthe evolution of ˆ nz= cos(α), the ˆzcomponent\nof the magnetization. Monotonic solutions Mhave ˆnzcovering the entire range ±1. Kink solutions Kα2α1describe single domain\nwall conﬁgurations with ˆ nzconnecting cos( α1) to cos( α2). Oscillatory solutions Oα0have ˆnzoscillating about cos( α0). Notice\nkink solutions Kalways separate monotonic Mfrom oscillatory Osolutions. For S >1/2 spinor condensate ferromagnets,\nnotice the two distinct oscillatory and kink solutions for e achγ,δin the region near the origin which are absent for S= 1/2.\nForS= 0 ordinary ferromagnets, there is no dependence on C.\n[27, 33, 37, 39,40]. Thus, solutionsforthe ordinaryferro-\nmagnet describing collections of localized skyrmions car-\nrying net charge as shown in the left of Figs. 1 and 2\nhave been used extensively in this regime.\nHowever, we showed in Sec. V that these solutions of\nlocalized topological objects can be derived in a uniﬁed\nframework along with extended stripe solutions. There\nhave been a number of studies on the possibility of quan-\ntum Hall states with stripe order. At high Landau levels\nand with frozen spin degrees of freedom, Coloumb inter-\naction may directly favor charge density waves as pre-\ndicted theoretically [44, 45] and veriﬁed experimentally\n[46, 47]. Such states are not directly comparable to the\nstripe solutions we describe which have ﬁxed total den-\nsityand stripe order in the relative density . However,\nstripe order has also been proposed [48–50] and experi-\nmental evidence observed [51] in the context of quantum\nHall bilayers. Here, even though the total density be-\ntween layers is ﬁxed, both interlayer coherence and rela-\ntive density imbalancecandevelop. The isospin degreeof\nfreedom that arises can be used to deﬁne an appropriate\nmagnetization vector ˆ n. Here, the phase of the interlayer\ncoherence gives the orientation of ˆ nx, ˆny, while the rel-\native density imbalance gives ˆ nz. States with skyrmionstripe order and winding ˆ nx, ˆnyhave been proposed that\nare direct analogs of the conﬁgurations shown in the top\nrow of Fig. 5.\nFor spinor condensate ferromagnets, experiments at\nBerkeley suggest the possibility of a condensate with\ncrystalline magnetic order [15, 16]. This crystalline or-\nder arises from an eﬀective dipolar interactions modiﬁed\nby rapid Larmor precession and reduced dimensional-\nity. It can drive dynamical instabilities of the uniform\nstate which occur in a characteristic pattern [17, 18].\nModes controlling the component ˆ nparallel (perpendic-\nular) to the magnetic ﬁeld in spin space are unstable\nalong wavevectors perpendicular (parallel) to the mag-\nnetic ﬁeld in real space. Instabilities of this type can give\nrise to the spin textures shown in the stripe solutions of\nthe bottom row in Fig. 5. Here, ˆ nzis modulated along\ntheydirection while ˆ nx, ˆnywind along the xdirection.\nIn the companion paper [21], we have performed a\nsystematic numerical study of minimal energy conﬁgu-\nrations for spinor condensate ferromagnets with dipolar\ninteractions. This is made possible by the use of sym-\nmetry operations combining real space and spin space\noperations to distinguish diﬀerent symmetry classes of\nsolutions. For applied magnetic ﬁeld in the plane ˆB= ˆx12\nFIG. 7: Numerically optimized conﬁguration for two-\ndimensional spinor condensate ferromagnets with an eﬀecti ve\ndipolar interaction modiﬁed by rapid Larmor precession. Th e\nmagnetic ﬁeld ˆB= ˆxinducing Larmor precession lies along\nthe horizontal axis in the plane. Lattice constants are a/bardbl= 90\nµm anda⊥= 42µm. Red (blue) background indicates pos-\nitive (negative) skyrmion density q, black 2D arrows the su-\nperﬂuid velocity v, and shaded 3D arrows the magnetization\nˆn.\ncorresponding to current experiments, we show the low-\nest energy conﬁguration in Fig. 7.\nNotice ˆnzis modulated between ±1 just as in the\nmonotonic Msolutions shown in Fig. 5. In addition,\nthe ˆnx, ˆnycomponents wind along the horizontal axis.\nHowever, notice the winding in the ˆ nx, ˆnycomponents is\nnot uniform as in the solutions we ﬁnd in this paper. In\naddition, the winding changes from clockwise to counter-\nclockwise halfway along the horizontal axis. For the so-\nlutions we ﬁnd in this paper, the skyrmion density forms\nstripes of opposite charge parallel to the horizontal axis.\nFor the lowest energy conﬁguration in Fig. 7, the non-\nuniform winding leads to concentration of the skyrmion\ndensity in smaller regions and modulation in the sign of\nthe skyrmion density along the xaxis. We ﬁnd minimal\nenergy conﬁgurations in other symmetry classes are gen-\nerally of this type with ˆ nzoscillating between ±1 alongy\nand ˆnx, ˆnywinding along x. However, the detailed formof the winding along xvaries for diﬀerent classes.\nThus we see that the exact solutions for spinor conden-\nsate ferromagnets without dipolar interactions provides\na more transparent physical picture for the numerical so-\nlutions with dipolar interactions. This can be seen as fol-\nlows. The solutions we ﬁnd in this paper describe overall\nneutral collections of skyrmion and anti-skyrmion topo-\nlogical objects. The neutrality constraint comes from the\nlong-ranged divergence of the skyrmion interaction and\nremains even when considering additional spin interac-\ntionssuch asthe dipolar interaction. Morever, skyrmions\nand anti-skyrmions themselves have a non-trivial spin\ntexturewhichisevidentinFig. 5showingthesolutionsof\nthis paper. When dipolar interactions are included, such\nspintexturescantakeadvantageofthe gainin dipolarin-\nteraction energy without chaning their qualitative struc-\nture. However, quantitative details for minimal energy\nconﬁgurations such as the one shown in Fig. 7 require\ndetailed analysis of the competition between dipolar in-\nteractions, skyrmion interactions, and spin stiﬀness.\nIn conclusion, we have presented the low-energy eﬀec-\ntive theory of spinor condensate ferromagnets. This ef-\nfective theory describes the superﬂuid velocity and mag-\nnetization degrees of freedom and can be written as a\nnon-linearsigmamodel with long-rangedinteractionsbe-\ntween skyrmions, the topological objects of the theory.\nQuantum Hall ferromagnets share a similar eﬀective the-\nory with long-ranged skyrmion interactions. 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Lett. 99, 126804 (2007).\n[52] The convention in literature on skyrmions in ordinary\nferromagnets is to take tan( α/2)eiβ=g(x+iy) with\ng(z) allowed to have isolated zeros or poles. Taking\nf(z) = logg(z) and allowing f(z) to have logarithmic\nsingularities is equivalent and can be more easily gener-\nalized to spinor condensate ferromagnts."    },    {        "title": "1911.04121v5.Domain_wall_motion_in_a_diffusive_weak_ferromagnet.pdf",        "content": "Domain wall motion in a di\u000busive weak ferromagnet\nFaluke Aikebaier1,\u0003and Tero T. Heikkil a1,y\n1Department of Physics and Nanoscience Center, University of Jyv askyl a,\nP.O. Box 35 (YFL), FI-40014 University of Jyv askyl a, Finland\n(Dated: November 3, 2021)\nWe study the domain wall motion in a disordered weak ferromagnet, induced by injecting a spin\ncurrent from a strong ferromagnet. Starting from the spin di\u000busion equation describing the spin\naccumulation in the weak ferromagnet, we calculate the force and torque acting on the domain wall.\nWe also study the ensuing domain wall dynamics, and suggest a possible measurement method for\ndetecting the domain wall motion via measuring the additional resistance.\nI. INTRODUCTION\nCurrent-driven domain wall motion has been an active\n\feld of research due to its applications in memory-storage\ndevices1. Following a series of phenomenological theoret-\nical works2{5and experimental con\frmations,6{10a mi-\ncroscopic theory of domain wall motion was presented\nmore than a decade ago.11The essential mechanism of\nsuch e\u000bects is the transfer of momentum and spin to the\nlocal magnetization due to a force and a (spin) torque,\nrespectively, exerted by a spin polarized current pass-\ning through the domain wall.12However, spin-polarized\ncurrents may reduce the spin torque e\u000eciency with an\nincreasing temperature due to Joule heating.13,14\nOne suggestion to reduce the Joule heating is to replace\nthe spin-polarized charge current with the pure spin cur-\nrent to induce the domain wall motion. Such pure spin\ncurrents have been realized in a lateral spin valve geome-\ntry,15{17see for example Fig. 1. The scenario in this case\nis as follows. A spin polarized current is injected from a\nferromagnet to a nonmagnetic material, transported and\nabsorbed by the second ferromagnet containing a domain\nwall. The absorbed pure spin current then induces a do-\nmain wall motion. It was shown that the domain wall\nmotion in this case is also very e\u000ecient, in terms of the\nchange of the magnetization at the interface of the fer-\nromagnet where the spin current is absorbed. The force\nand torque in this structure have also been calculated\nfor a case of weak impurity scattering,18but the ensuing\ndomain wall dynamics have not yet been studied theo-\nretically.\nOne important feature of the pure spin current com-\npared to the spin-polarized current is that it decays\nwithin a length scale called spin-relaxation length, due\nto the spin-relaxation processes. In fact, spin relaxation\nsigni\fcantly a\u000bects the current-driven domain wall mo-\ntion.12For example, the spin relaxation of conduction\nelectrons is one of the most relevant mechanisms for\nthe damping of the domain wall motion. Moreover, it\nenhances the nonadiabaticity parameter of the domain\nwalls close to the adiabatic limit.19,20In disordered fer-\nromagnets, it has also been shown that the domain wall\nmotion is very e\u000ecient even in the case of weak ferromag-\nnetism with low spin polarization.21Therefore, studying\nthe domain wall dynamics in the presence of pure spincurrent without the accompanied charge current may give\nrise to interesting new features.\nHere we consider a similar structure with the one in\nRef. 18, except that the nonmagnetic metal is replaced\nby a weak ferromagnet containing a domain wall, and\na spin polarized current is injected from a strong ferro-\nmagnetic electrode. We de\fne the concepts of the \"weak\"\nand \"strong\" ferromagnets based on the size of the spin\npolarization and the possibility of using the spin di\u000bu-\nsion equation to describe the two systems. In particu-\nlar, in the strong ferromagnet we assume a spin-polarized\nFermi surface, described by spin-dependent densities of\nstatesN\u001b, di\u000busion constants D\u001band conductivities\n\u001b\u001b=e2N\u001bD\u001b.22In this case, we can study the spin\npolarized current in a homogeneous ferromagnet by writ-\ning di\u000busion equations separately for the two spin bands.\nOn the other hand, the weak ferromagnet has a weakly\nspin-split Fermi surface (small exchange \feld) for which\n\u001b\"=\u001b#. In this case we can include the Hanle precession\nterm into the kinetic equations and therefore rigorously\ndescribe spin accumulation in the case of an inhomoge-\nneous magnetization.\nThe spin polarized current injected from the strong fer-\nromagnetic electrode creates a spin accumulation in the\nweak ferromagnet which decays exponentially due to the\nspin relaxation processes. This spin accumulation can\nbe described by a spin di\u000busion equation with spin in-\ndependent parameters, and it describes a spin current in\na disordered wire. The solutions for the position depen-\ndent spin accumulation around the domain wall allows us\nto compute the force fand torque \u001czon the domain wall\nresiding at a distance Xfrom the injector. We show that\nthey are characterized by three length scales: domain\nwall size\u0015, spin relaxation length `s, and the magnetic\nlength`h. These length scales can in principle show up in\nany order, and we \fnd how the force and torque depend\non the order of those scales. In particular, due to the\nspin relaxation both the force and torque are exponen-\ntially decaying as functions of the distance of the domain\nwall from the injector, similar to the case in Ref. 18. We\nalso study the resulting domain wall dynamics, and show\nthat the domain wall motion with decaying force and\ntorque has its characteristic features. In particular, the\ndynamics can cross between di\u000berent dynamic regimes\ndepending on the position of the domain wall, and de-arXiv:1911.04121v5  [cond-mat.mes-hall]  26 Feb 20202\npending on the hierarchy of the length scales a\u000becting\nthe relative size of force and torque: In the case of a\nlarge torque and weak force, the domain wall motion can\ncross over from the unpinned motion for \u001cz(X)> k?\u000b0\nto the limit of intrinsic pinning with \u001cz(X)<k?\u0015, even-\ntually stopping the domain wall. Here k?is a quantity\ncharacterizing the hard-axis anisotropy. On the other\nhand, if the force dominates and is large enough close to\nthe injector, there is a crossover between oscillatory dy-\nnamics forf(X)>\u000b0k?and linearly (in time) decaying\ndynamics for f(X)<\u000b0k?. Here\u000b0describes damping.\nWe also suggest a possible measurement of the domain\nwall motion via the changes in the injection resistance,\nlinked to the dependence of the injection resistance on\nthe local spin accumulation at the position of the contact.\nSince the latter depends on the position of the domain\nwall, so does the injection resistance.\nThe outline of the paper is as follows. We \frst in-\ntroduce the model, a weak ferromagnet containing a do-\nmain wall in contact with a spin-polarized ferromagnetic\ninjector, in Sec. II. We also solve the spin di\u000busion equa-\ntion with proper boundary conditions which describes the\nspin accumulation in this model. The force and torque\ndue to the spin current are calculated in Sec. III. We\nstudy the domain wall dynamics in Sec. IV, and the\npossible measurement method accessing this dynamics\nin Sec. V before the conclusions in Sec. VI.\nII. MODEL AND METHOD\nWe study the domain wall motion in the structure in\nFig. 1. A spin polarized current is injected from a strong\nferromagnet to a di\u000busive weak ferromagnet containing a\ndomain wall. The injected current circulates on the left\nside of the injector, and a spin accumulation is induced in\nthe weak ferromagnet. The decaying spin accumulation\nresults in a spin current in both directions, capable of\ninducing a force and a torque on the domain wall.\nOn the right side of the injector, the weak ferromagnet\ncontains a domain wall, and the magnetization is inho-\nmogeneous. The inhomogeneity is shown in the exchange\n\feld as\nh=h(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012); (1)\nwherehis the strength of the exchange splitting. Here\n\u0012and\u001eare the in-plane and out-of-plane components of\nthe magnetization angle. For domain wall motion, \u001eis\nonly a function of time,11and the rotation is described by\nthe angle\u0012. A N\u0013 eel domain wall is energetically favoured\nin thin \flms, namely, the rotation of the magnetization\nhappens in the plane of the domain wall ( \u001e= 0). Then\nzXλWeak ferromagnetic metal0VII=0⃗PICharge  CurrentSpin  CurrentFIG. 1. Schematic view of the structure considered in this\npaper. A spin polarized current is injected from a strong ferro-\nmagnet to a di\u000busive weak ferromagnet containing a domain\nwall.\n\u0012can be expressed by a variational ansatz23\n\u0012(z) =\u0019\u0002\u0012\nz\u0000X\u0000\u0015\n2\u0013\n+\u0019\n\u0015\u0012\nz\u0000X+\u0015\n2\u0013\n\u0002\u0012\nz\u0000X+\u0015\n2\u0013\n\u0002\u0012\nX+\u0015\n2\u0000z\u0013\n;\n(2)\nwhere \u0002(z) is the Heaviside step function, Xis the posi-\ntion of the domain wall center, and \u0015is the domain wall\nsize. The variational ansatz to the rotation angle, instead\nof the typically used hyperbolic functions12with slightly\nlower domain wall energy, brings certain conveniences to\nthe analytical treatment of the problem while capturing\nthe essential physics of the domain wall. Since the deriva-\ntive of\u0012(x) is a constant inside the domain wall, the spin\ndi\u000busion equation, which describes the nonequilibrium\nspin accumulation, can be simpli\fed [see Eq. (4)]. The\nnonanalyticity of the derivative of \u0012(x) at the domain wall\nboundary can be transformed into boundary conditions\nof the spin di\u000busion equation [see Eq. (5) to Eq. (7)].\nThe spin accumulation in the weak ferromagnet is de-\nscribed by a spin di\u000busion equation in Eq. (A3). With\nthe domain wall structure in Eq. (2), it can be written as\n~D@2\nzs=~\n\u001css\u00002h\u0002s;\nwhereDis the di\u000busion constant, \u001csis the spin-\rip re-\nlaxation time, and s= (s1;s2;s3) is a spin accumu-\nlation vector. The spin-relaxation length is de\fned as\n`s=pD\u001cs.\nWe can use an SU(2) gauge transformation to treat\nthe exchange \feld as homogeneous. We de\fne a rotation\nmatrix as\n^R=ei\u001b2\u0012=2ei\u001b3\u001e=2;\nso that we can write the spin accumulation as\ns\u0001\u001b=^Rys0\u0001\u001b^R; (3)\nwhere\u001b= (\u001b1;\u001b2;\u001b3) is a vector of Pauli spin matri-\nces. Here the rotated spin accumulation s0= (s0\n1;s0\n2;s0\n3)3\nsatis\fes the following spin di\u000busion equation\n~D^@2\nzs0=~\n\u001css0\u00002h^z\u0002s0; (4)\nwhere ^z= (0;0;1),^@zY=@zY+@z\u0012(z)(^y\u0002Y) from\nthe fact that ^@zY\u0001\u001b=@zY\u0001\u001b\u0000[A;Y\u0001\u001b] =@zY\u0001\u001b+\n@z\u0012(x)(^y\u0002Y)\u0001\u001b, in whichA=i\u001b2@z\u0012(z)=2 is an SU(2)\ntype vector potential, Y=s0or@xs0, and ^y= (0;1;0).\nThe derivative of \u0012(z) divides the weak ferromagnet into\nthree regions. In the domain wall region it is a constant,\nand to the left and the right sides of the domain wall\nregion,\u00120(z) = 0. However, \u00120(z) is discontinuous at\nthe boundary of the domain wall. Therefore, we need a\nboundary condition to describe a continuous spin accu-\nmulation.\nWe can integrate Eq. (4) at the boundary of the do-\nmain wall, and obtain the boundary conditions\n@zs0\n1jz\u0006\nb\u0000@zs0\n1jz\u0007\nb=\u0000\u0019\n\u0015s0\n3jz\u0006\nb; (5)\n@zs0\n2jz\u0006\nb\u0000@zs0\n2jz\u0007\nb= 0; (6)\n@zs0\n3jz\u0006\nb\u0000@zs0\n3jz\u0007\nb=\u0019\n\u0015s0\n1jz\u0006\nb: (7)\nAt the domain wall edges z=z\u0006\nb=\u0006(X\u0006\u0015=2), and\n\u0006refers to the right and left sides of the domain wall\nboundary.\nThe second group of boundary conditions represent the\ninjection of the spin polarized current. As we show in\nAppendix C, the spin injection from a contact with a\nstrong ferromagnet with magnetization oriented in the z\ndirection and biased with potential Vcan be described\nwith the spin currents at the injection point,\n~D@zs0\n1= 0 (8)\n~D@zs0\n2= 0 (9)\n~D@zs0\n3=kI~D(s0\n3\u0000PI\rVN 0); (10)\nwherekIis an injector transparency, PIis an injector po-\nlarization (see Appendix C for precise de\fnitions of these\nquantities in terms of the properties of a ferromagnetic\ninjector wire), Vis the voltage at the injector, and N0is\nthe density of states at the Fermi level. The voltage is\nrescaled by a factor \r[de\fned in Eq. (C7)], due to fact\nthat the spin accumulation in the weak ferromagnet is\na\u000bected by the spin accumulation in the injector, see the\ndetails in Appendix C.\nMaking the equations dimensionless, we \fnd that the\ndomain wall physics is here described by three length\nscales: (i) domain wall size \u0015, (ii) spin relaxation length\n`s, and (iii) the magnetic length lh=p\n~D=h. The latterindicates the length within which a non-collinear compo-\nnent of the spin accumulation rotates a full period around\nthe local magnetization direction. This is an important\nscale since both the force and the torque depend on such\nnon-collinear components, as shown in Eqs. (16,17).\nThe \"phase diagram\" of di\u000berent dynamical regimes\ndepends on two dimensionless parameters corresponding\nto the ratios of these scales. In addition, the injector spin\npolarization PIdescribes the e\u000eciency of spin injection\n(the size of spin current for a given amount of charge\ncurrent), whereas the interface transparency parameter\nkIdetermines how strongly the resistance of the injector\ndepends on the domain wall position.\nIn many strong ferromagnetic metals like iron and\ncobalt, the exchange splitting his of the order of 1 eV.24\nThis then leads to a very small lh, of the order of the\natomic lattice spacing. For a weak ferromagnet, for ex-\nample CuNi, it is around 0 :05 eV.25This leads to a\nmagnetic length lhbetween 10 to 25 nm.25,26On the\nother hand, depending on the exact materials or sam-\nple properties (e.g. thickness and concentration of Ni),\nthe domain wall size \u0015and the spin-relaxation lengths\n`sof CuNi range from 15 to 25 nm27(estimated from\nmeasured anisotropy energy and exchange sti\u000bness con-\nstants) and from 7 to 25 nm28, respectively. This yields\n\u0015=lh\u00180:5:::1:5 andlh=`s\u00180:4:::3:6. As there are also\nother materials with weak ferromagnetism, we also can-\nnot exclude the other possibilities. In order to under-\nstand various properties of the domain wall motion in-\nduced from a spin current, we also consider these ratios\noutside of these ranges in the following discussions.\nWith the boundary conditions in Eq. (5) to Eq. (7)\nand in Eq. (8) to Eq. (10), we can solve the rotated spin\ndi\u000busion equation in Eq. (4). They can be solved analyt-\nically (see Appendix B), but the solutions are in general\nquite lengthy. Rather, we plot the components of the\nspin accumulation for an example set of parameters as a\nfunction of position in Fig. 2(a,b). We can see that s0\n1is\na monotonously increasing (decreasing) function of posi-\ntion in region to the left (right) side of the domain wall,\nand reaches a minimum in the domain wall center. The\nsecond component of spin accumulation s0\n2smoothly goes\nto zero away from the domain wall center. Compared to\nthe spin accumulation in the case without the domain\nwall,s0\n3changes sign in the domain wall region and ex-\nponentially decreases in region to the right of the domain\nwall.\nThe unrotated spin accumulation is given by Eq. (3).\nMore speci\fcally, we can write\ns1= cos\u001e(s0\n1cos\u0012+s0\n3sin\u0012)\u0000s0\n2sin\u001e (11)\ns2=s0\n2cos\u001e+ sin\u001e(s0\n1cos\u0012+s0\n3sin\u0012) (12)\ns3=s0\n3cos\u0012\u0000s0\n1sin\u0012: (13)\nThe unrotated components of the spin accumulation are\nplotted for\u001e= 0 in Fig. 2(c,d). Compared to the rotated4\n-6-3036-0.08-0.06-0.04-0.0200.02\n-6-303600.10.20.3-6-3036-0.15-0.1-0.0500.05\n-6-3036-0.100.10.20.3(a)(c)(d)(b)\nXX\nXX\nFIG. 2. Solutions of the spin di\u000busion equation. The solu-\ntions in the rotated space are shown in (a) and (b), and in the\nunrotated space are shown in (c) and (d). We also compare\ns0\n3ands3with the spin accumulation in the case of homoge-\nneous magnetization (no domain wall). Here the results are\nplotted for `s= 3:2lh,PI= 0:5,kIlh= 0:5, and\u0015=lh. The\ninjector is placed at x= 0, whereas the domain wall center is\natX= 0:5\u0015marked in the \fgure.\nsolution,s2remains the same but s1changes sign on the\ntwo sides of the domain wall center, and s3also makes a\ndi\u000berence compared to the case without the domain wall.\nIn the next section, we use these spin accumulations to\ncalculate the force and torque.\nIII. FORCE AND TORQUE\nThe force and torque acting on the domain wall are\ngiven by11,12\nF=\u0000Z\nd3zrh\u0001s (14)\nTz=\u0000Z\nd3z(h\u0002s)z; (15)\nwhere exchange \feld his given in Eq. (1), and the compo-\nnents of the spin accumulation s= (s1;s2;s3) are shown\nin Eq. (11) to Eq. (13). Substituting these to the force\nand torque in Eq. (14) and Eq. (15), we obtain\nF=\u0000h\u0019W\n\u0015Z\ndzs0\n1 (16)\nTz=\u0000hWZ\ndzs0\n2sin\u0012; (17)\nwhereWis the cross sectional area of the weak ferro-\nmagnet.The force and torque as a function of the domain wall\npositionXare plotted in Fig. 3 for a few sets of param-\neters. The common feature of all the cases are that both\n0123400.0020.0040.0060.0080.010.012\n0123400.010.020.030.040123400.0050.010.0150.020.025\n0123400.010.020.030.04(b)(a)(c)(d)\nFIG. 3. Force and torque for smaller domain walls in (a,b),\nand for larger domain walls in (c,d), as a function of domain\nwall center position X\u0000X0, whereX0=\u0015=2 is the shortest\ndistance of the domain wall center to the right of the injector.\nHere the results are plotted for `s= 3:2lh,PI= 0:5, and\nkIlh= 0:5.\ndecay exponentially as a function of X. This is due to\nthe fact that the spin accumulation and the resulting spin\ncurrent, which induces the domain wall motion, decays\nexponentially within the spin-relaxation length `s. These\nfeatures are also very similar to the ones in Ref. 18. From\nFig. 3(a,c), we can see that the force is independent of the\ndomain wall size for small domain walls, and it is smaller\nfor larger domain walls. On the other hand, the torque\nhas a nonmonotonic dependence on the domain wall size\n\u0015, as shown in Fig. 3(b,d). It \frst increases as \u0015increases\nup to of the order of lh, and then becomes smaller for\nlarger domain walls. This is not the same with the case\nof current driven domain wall motion, where the torque\nis much larger than the force for larger domain walls.12\nThis is due to the fact that when a spin relaxation length\n`sis smaller than the domain wall size \u0015(`s< \u0015), due\nto the decaying spin current, less spins are transferred to\nthe domain wall. This results in the smaller torque for\nlarger domain wall sizes in Fig. 3(d).\nThe dependence of the force and the torque on the spin\nrelaxation length are shown in Fig. 4. We can see that the\ntorque is a monotonously decreasing function of the in-\nverse relaxation length, i.e., decreasing spin relaxation in-\ncreases the torque, as expected from the fact that torque\nresults from spin transfer. On the other hand, the force\nis a non-monotonic function of lh=`s. It also decays if the\nspin relaxation becomes strong (i.e., lh\u001d`s). However,\nit also becomes small for a small magnetic length lh\u001cls.\nThis is due to the fact that contrary to the torque, which5\n012340246810-3\n00.511.5200.0050.010.0150123400.020.040.060.080.1\n00.511.5200.020.040.06(a)(c)(b)(d)\nFIG. 4. Force and torque for smaller domain walls in (a,b),\nand for larger domain walls in (c,d), as a function of inverse\nspin relaxation length `s. Here the force and the torque are\nplotted for the domain wall position X0=\u0015=2. The parame-\nters used in the calculations are PI= 0:5 andkIlh= 0:5.\nwithin our model only comes from the domain wall region\n(that is where \u00126= 0 in Eq. (17)), the force depends on the\nspin accumulation component s0\n1also around the domain\nwall. However, for small lh, this component oscillates\nrapidly, and thus the average force becomes small. Anal-\nogously, both the force and the torque become smaller\nfor larger\u0015=lh. This is due to the oscillations of the spin\naccumulation inside the domain wall region.\nIn order to get a further insight on the relative mag-\nnitudes of the force and torque, we examine the adia-\nbaticity parameter \fs=\u0015F=Tzas a function of lh=`sfor\ndi\u000berent\u0015in Fig. 5. Since FandTzboth decay in the\nsame manner, \fsis independent of the distance Xfrom\nthe injector. Comparing the values of \fsin Fig. 5(a)\nand (b), we can see that \fsis indeed smaller for larger\ndomain walls, but the spin relaxation also plays an im-\nportant role. We can see that \fs\u001d1 for strong spin\nrelaxation, i.e., force is much larger than the torque. On\nthe other hand, the torque is much larger than the force\nfor large domain walls \u0015&lh, provided the spin relax-\nation length is also longer than lh[Fig. 5(b)]. For small\ndomain walls \u0015\u001clh,\fsis proportional to \u0015\u00001. We can\nestimate\fsin this limit for lh<`sby\n\fs=8\n\u0019lh\n\u0015l2\nh\n`2s: (18)\nThis is plotted in Fig. 5(a) as the black dashed curve.\nThis behavior can be compared to the case of strong\nferromagnets in the ballistic limit11. There the only\nnon-adiabaticity (non-vanishing \fs) comes from the \f-\nnite\u0015F=\u0015. The spin di\u000busion equation employed here\nassumes that the Fermi wavelength \u0015Fis much smaller\nthan any other length scale. However, we see that inthis case other length scales, such as lhand`sgovern the\nbehavior of the adiabaticity parameter.\n00.20.40.60.810204060\n00.20.40.60.810246(a)(b)\nFIG. 5. Adiabaticity parameter \fsas a function of the in-\nverse spin-relaxation length `sfor di\u000berent domain wall sizes.\nThe results are plotted for PI= 0:5 andkIlh= 0:5. The\nanlaytical estimate for \fsin Eq. (18) is shown as the black\ndashed curve in (a).\nIV. DOMAIN WALL DYNAMICS\nIn the absence of an external pinning and a negligible\ndomain wall mass,29the dynamic equations of domain\nwall motion are11,12\n_\u001e+\u000b0_X\n\u0015=\u0015\n~NSF (19)\n_X\u0000\u000b0\u0015_\u001e=K?\u0015\n2~Ssin(2\u001e) +\u0015\n~NSTz; (20)\nwhere\u001eis the out-of-plane angle in Eq. (1), \u000b0is the\nGilbert damping parameter of the local magnetization,\nK?is the perpendicular anisotropy energy, and Sis the\nsize of the localized spin. Also, N= 2\u0015W=a3\n0is the\nnumber of spins in the domain wall, and a0is the lattice\nconstant. The force and torque are given in Eq. (14) and\nin Eq. (15), respectively.\nThe unit of FandTz=\u0015ish\rVN 0W. In order to make\nthe dynamic equations dimensionless, we multiply\nt0=~NS\n\u0015h\rVN 0W=2~S\na3\n0N0h\rV\nto both sides of Eq. (19) and Eq. (20), and after reorga-\nnizing the terms, write\n_X\n\u0015=1\n1 +\u000b2\n0h\n\u000b0f+\u001cz\n\u0015+k?sin(2\u001e)i\n(21)\n_\u001e=1\n1 +\u000b2\n0h\nf\u0000\u000b0\u001cz\n\u0015\u0000\u000b0k?sin(2\u001e)i\n: (22)\nHere we de\fned\nf=\u0000\u0019\n\u0015\rVN 0Z\ndxs0\n16\n\u001cz=\u00001\n\rVN 0Z\ndxs0\n2sin\u0012\nk?=K?S2\na3\n0N0h\rV:\nWe \frst discuss the case where the force is much larger\nthan the torque ( \fs&1). We can see from Fig. 5 that this\nis the case for small domain walls and large domain walls\nwith strong spin relaxation lh\u001d`s. For convenience we\nconsider a small domain wall \u0015\u001clh. The full numerical\nsolutions of the dynamic equations of domain wall motion\nin Eq. (21) and Eq. (22) are shown in Fig. 6.\nIf the force is a constant f=f0in the absence of\nthe torque, Eq. (22) yields _\u001e= 0 forf0< \u000b 0k?. Then\nthe domain wall moves with a constant velocity and a\nconstant out-of-plane angle\n_X=\u0015f0\n\u000b0; (23)\n\u001e=1\n2arcsin\u0012f0\n\u000b0k?\u0013\n: (24)\nIn the spin current induced domain wall motion, the force\ndecays as a function of the domain wall position X. If we\nwrite the force as f=f0e\u0000X=` s, then _\u001e!0 fort!1 ,\nand this yields\n_X=\u0015f0\n\u000b0e\u0000X=` s: (25)\nThis equation can be solved as\nX=X(0) +`slog\u0014\n1 +f0\u0015t\n`s\u000b0\u0015\n; (26)\nand\n_X=f0`s\u0015\n`s\u000b0+f0\u0015t;\nwhereX(0) is the domain wall position where _\u001e!0.\nThis is exempli\fed by the curves in Fig. 6(a,b). There,\nthe blue curve shows the behavior in the case where the\nforce is everywhere below \u000b0k?, and where _\u001e!0 at\naroundt\u0019200t0. From Eq. (22), we can also determine\n\u001e=1\n2arcsin\u0014f0`s\nk?e\u0000X(0)=`s\n`s\u000b0+f0\u0015t\u0015\n: (27)\nIff0> \u000b 0k?, the constant force leads to an oscilla-\ntory domain wall motion. This is known as the Walker\nbreakdown.30The red dash-dotted curves in Fig. 6(a,b)\nshows the situation where the force is initially above this\nthreshold, and only as the domain wall has moved fur-\nther from the injector fgets below this threshold (around\nt&1500t0). After that the domain wall motion follows\nEq. (25).\n050010001500200000.20.40.6\n030006000900000.050.10.150.2\n0300060009000-2.5-2-1.5-1-0.5010-40500100015002000-0.0100.010.020.030.04(c)(d)(b)(a)FIG. 6. Full numerical solutions of the dynamic equations of\ndomain wall motion in Eq. (21) and Eq. (22). The case where\nthe force is much larger than the torque is shown in (a,b),\nand the one where the torque is much larger than the force\nis shown in (c,d). In (a,b) we use \u0015= 0:01lhand`s= 3:2lh.\nIn (c,d)\u0015= 20lhand`s= 100lh. The other parameters used\nin the calculations are PI= 0:5,kIlh= 0:5,X0=\u0015=2, and\n\u000b0= 0:2. In the inset of (c,d), the results are shown for a\nsmaller time scale.\nFrom Fig. 5, we can see that the torque is much larger\nthan the force for large domain walls and weak spin re-\nlaxation. In the case of a constant torque in the absence\nof the force, the domain wall does not move if \u001c0\nz<k?\u0015.\nThe reason is that the perpendicular anisotropy energy\ndescribed by the coe\u000ecient k?absorbs the torque com-\npletely. This is known as intrinsic pinning.11Other-\nwise, if\u001c0\nz> k?\u0015, the domain wall moves with a \f-\nnite velocity. Similar to the force, we can write the\ntorque as\u001cz=\u001c0\nze\u0000X=` s. When the torque decays until\n\u001cz(X(t))< k?\u0015so that _\u001e!0, the domain wall stops\nmoving. It takes a longer time for a smaller k?to absorb\nthe torque completely. The domain wall position and _\u001e\nas a function of time for a decaying torque are plotted in\nFig. 6(c,d).\nWe next examine the domain wall motion in the pres-\nence of both force and torque ( \fs\u00191). In the case\nof constant force and torque, a small force is enough to\ndestroy the intrinsic pinning. The domain wall moves\nwith a constant velocity, see Eq. (23). This is also the\ncase with decaying force and torque, and the domain wall\nmotion follows Eq. (25). We can use Eq. (22) to obtain\n\u001efor_\u001e!0 as\n\u001e=1\n2arcsin\u00141\nk?\u0012f0\n\u000b0\u0000\u001c0\nz\n\u0015\u0013\u000b0`s\n`s\u000b0+f0\u0015te\u0000X(0)=`s\u0015\n:\n(28)\nThe numerical solutions of the dynamic equations of the\ndomain wall motion in Eq. (21) and Eq. (22) in the\npresence of comparable force and torque are plotted in7\n00.511.5210401234\n00.511.5210400.050.10.1503000600090000246\n010002000300040005000-101234510-3(c)(a)(b)(d)\nFIG. 7. Domain wall dynamics in the presence of compa-\nrable force and torque for di\u000berent k?in (a,b). Forward and\nbackward moving domain walls for di\u000berent domain wall sizes\nin (c,d) for a \fxed k?= 1:2jf0=\u000b0\u0000\u001c0\nz=\u0015j. In (a,b) we use\n\u0015=lhand`s= 3:2lh. In (c)`s= 3:2lh, and in (d) `s= 100lh.\nThe other parameters used in the calculations are PI= 0:5,\nkIlh= 0:5,X0=\u0015=2, and\u000b0= 0:2. In (c,d) we use a small\narrow to denote when the voltage changes sign.\nFig. 7(a,b)\nIn the above discussions, the voltage at the injector is\nconsidered to be positive V > 0. If the voltage changes\nsign at some instant of time, then the sign of the force\nand torque also changes, and they start pulling the do-\nmain wall instead of pushing it. This leads to the re-\nversed motion of the domain wall. The reversed domain\nwall motion for small domain walls \u0015<lhare shown in\nFig. 7(c). In this case, the domain wall reverses back to\nits original position X= 0 at an equal amount of time as\nthe one needed to push it further. The reversed domain\nwall motion for large domain walls with weak spin relax-\nation is shown in Fig. 7(d). In this case the domain wall\nhad stopped before the sign change of the injected spin\ncurrent.\nThe above analysis is based on the dynamics described\nby Eqs. (19) and (20), with force and torque obtained\nfrom the solutions of the spin di\u000busion equations. Those\nequations were derived12by assuming a clean ferromag-\nnet and an instant electronic response to the domain wall\nmotion. It was shown29,31{37that taking into account the\ndelayed electron dynamics, extra \"inertial\" terms propor-\ntional to \u001eand Xcan also appear, leading for example\nto a hysteretic dynamics of the domain wall. The prefac-\ntor of those terms, an e\u000bective mass of the domain wall,\nis proportional to the time it takes for the electrons to\ntraverse the domain wall width \u0015. If\u0015is large compared\nto the elastic mean free path, as assumed in the present\nmanuscript, this e\u000bective mass is also likely to change\nfrom the ballistic limit considered in Ref. 29. This iswhy we did not yet consider its possible e\u000bect on the\ndynamics in the present manuscript.\nV. DOMAIN WALL RESISTANCE\nThe current induced from the injector electrode is\ngiven by [see Appendix C, Eq. (C9)]\nI=G[\u0000\u0000\rV+PIs3(0)=N0];\nwhereGis the conductance of the injector, and \u0000 is de-\n\fned in Eq. (C8). We can see from Appendix B that the\nspin accumulation is linear in the injection voltage V.\nTaking that into account allows us to include an extra\nresistance that depends on the relaxation of s3along the\nwire. In particular, we may study this extra resistance\nin the presence of the domain wall at position X, and\nwithout it (formally X!1 ). This domain wall resis-\ntance provides a direct method to detect the domain wall\nmotion.\nIf we denote the spin accumulation at the position\nof the injector as s3(0) =\u0016zPI\rVN 0, where\u0016z=\n\u0016z(X;lh;`s;kI;\u0015) is a dimensionless quantity, then the\ncurrent through the contact can be written as I=\nG(\u0000\u0000 +\u0016zP2\nI)\rV. The spin accumulation thus adds a\n\"spin resistance\"\nRs=1\nG\u0016zP2\nI\r: (29)\nThe contribution of the domain wall to the spin resistance\nin Eq. (29) can be found by taking the di\u000berence of Rs\nwith the resistance in the absence of the domain wall R0\ns\nasRdw=Rs\u0000R0\ns. Here\nR0\ns=1\nG\u0016z(X!1 )P2\nI\r=2 +kI`s\nGP2\nIkI`s\r;\nwhere\u0016z(X!1 ) is determined from Eq. (B1), and kIis\nthe injector transparency. Again the analytic formula for\nRdwis long, but we show its behavior for some selected\nparameters in Fig. 8.\nWe can see that the domain wall contribution to the\nresistanceRdwreduces exponentially as the domain wall\nmoves away from the injector, as is natural due to the fact\nthatRdwdepends on the size of the spin accumulation\naround the domain wall. Close to the injector X=\u0015=2\n[Fig. 8(c)], the domain wall contribution is maximal for\nlh\u001c`sand for\u0015\u0019lh.\nVI. CONCLUSION\nIn conclusion, we have studied the domain wall motion\nin weak ferromagnets in a non-local spin-injection setup.\nWe have used a spin-di\u000busion equation to calculate the\nspin accumulation and evaluated the force and torque\nacting on the domain wall. Both decay exponentially as8\n00.511.5200.10.20.30.40.5\n00.511.5200.20.40.6(a)(c)(b)\nFIG. 8. The additional resistance Rdwintroduced from\nthe domain wall plotted as a function of the domain wall\nposition in (a,b), and the dependence of the maximum Rdw\natX0plotted as a function of \u0015and`\u00001\nsin (c). In (a) Rdwis\nplotted for di\u000berent domain wall sizes and `s= 3:2lh, and in\n(b) for di\u000berent spin relaxation lengths for \u0015= 2lh. The other\nparameters used in the calculations are kIlh= 0:5,PI= 0:5,\nandX0=\u0015=2.\na function of domain wall position. We have studied the\ndomain wall dynamics and have showed that the domain\nwall motion exhibits interesting features due to the de-\ncaying force and torque. For example, if the force close\nto the injector is larger than the torque and a threshold\nfor Walker breakdown, the domain wall exhibits \frst an\noscillatory dynamics, but further from the injector spin\nrelaxation necessarily takes the force below that thresh-\nold value, resulting into an algebraically decaying domain\nwall speed. On the other hand, for a large torque close\nto the injector, compared to both the force and an intrin-\nsic pinning value due to anisotropy, the relatively steady\ninitial motion ceases when the torque becomes smaller\nthan the intrinsic pinning value, and the domain wall\nessentially stops. Since the sign of both the force and\nthe torque depend on the sign of the injection current,\nthe domain wall motion can be reversed by reversing the\nsign of the current. This is why the pure spin current\ncan also be used to pull the domain wall back towards\nthe injector. Besides the analysis of the force and torque\nand their result on the dynamics, we have also described\na means to detect the domain wall position via monitor-\ning the injection resistance that depends on the domain\nwall position.Our model is an alternative description of domain wall\nmotion compared to majority of the models12dealing\nwith essentially ballistic electron systems. In those cases\nthe only relevant length scales are the domain wall size\nand the Fermi wavelength. We show how in disordered\nsystems and weak ferromagnets there may be also other\nessential length scales governing the domain wall dynam-\nics, especially the magnetic length lhand the spin relax-\nation length `s. Our approach is made possible by the\nuse of the spin di\u000busion equation also in the presence of\ninhomogeneous magnetism, which would not be straight-\nforward when the spin polarization in the ferromagnet is\nlarge. To be able to use this equation, we hence need\nto assume weak ferromagnetism, which limits the appli-\ncability range of our approach. On the other hand, it\nprovides hints on the types of e\u000bects expected also in the\ncase of strong ferromagnets for which, to our knowledge,\nan analogous theory does not exist.\nMoreover, our model can be used as a reference in\nthe study of the domain wall motion in superconduc-\ntor/ferromagnetic insulator (S/FI) hybrid structures. In\nsuch structures, the spin accumulation in the supercon-\nductor is described by a set of kinetic equations, which\nreduce to the spin di\u000busion equation in the normal state.\nThe equilibrium properties of such a structure are stud-\nied in Ref. 23. A thermally induced torque is studied in a\nsuperconductor/ferromagnet bilayer in the clean limit38,\nbut to our knowledge this has not been extended to the\nexperimentally more relevant disordered limit.\nACKNOWLEDGMENTS\nWe thank M. Silaev for discussions. This work was\nsupported by the Academy of Finland project number\n317118.\nAppendix A: Spin di\u000busion equation\nAs a useful tool in describing the electronic transport\nproperties of magnetic materials, we start by the spin\ndependent Boltzmann equation in the di\u000busive limit22,39\n(@t\u0000Dr2)fz(r;\u000f;t) =\u00001\n\u001csfz(r;\u000f;t); (A1)\nwhereDis the di\u000busion constant (in a weak ferromagnet\nassumed independent of the spin index), fz=f\"\u0000f#\nandf\u001bis the distribution function of electrons with spin\n\u001b=\"=#, and\u001csis the spin-\rip relaxation time. This\nequation has been widely used in spintronics, for example\nin the description of the spin accumulation at an interface\nbetween a ferromagnet and a nonmagnetic metal.40\nHere we assume a quasistatic description, where the\ndomain wall moves slowly compared to the electrons, so\nthat the spin di\u000busion equation can be solved in the static\ncase. We can estimate the validity range of this assump-\ntion by comparing the time scale \u001c=`s\u000b0=(f0\u0015) of do-9\nmain wall motion through a length `s(given in Eq. (26))\nto the characteristic electron di\u000busion time D=`2\nsthrough\na similar length scale. The quasistatic approximation is\nvalid when \u001c\u001dD=`2\ns, or\neV\u001c~D\n\u0015`s2S\u000b0\nN0a3\n0h\r; (A2)\nwhere we assumed f0of the order of unity. In this limit\nwe can hence disregard the time derivative in Eq. (A1).\nIn the case of an inhomogeneous exchange \feld, other\nspin components should be taken into account, and we\ncan replace fsbyf\u0001\u001b= (fx;fy;fz)\u0001\u001b, where\u001b=\n(\u001b1;\u001b2;\u001b3) is a vector of Pauli spin matrices. Consider-\ning the Heisenberg equation of motion for f\u0001\u001b, and sub-\nstituting back to the Boltzmann equation (see a similar\nderivation in Refs. 41 and 42, except that those articles\nwrite an opposite sign of the Zeeman energy term) we\nobtain for a steady state,\nDr2f\u0001\u001b=1\n\u001csf\u0001\u001b+i\n~hg\u0016B\n2B\u0001\u001b;f\u0001\u001bi\n;\nwhere the other component of the commutator is the Zee-\nman energy. There, g= 2 is theg-factor,\u0016Bis the Bohr\nmagneton, and Bis the magnetic \feld. By denoting\nh=g\u0016BB=2 and reorganizing the terms, we obtain\n~Dr2f=~\n\u001csf\u00002h\u0002f;\nwhere we used the relation ( a\u0001\u001b)(b\u0001\u001b) = 2i(a\u0002b)\u0001\u001b.\nThis equation is an extension of Eq. (A1) to the case with\ninhomogeneous magnetization, as it reduces to Eq. (A1)for the case of homogeneous magnetization in the steady\nstate.\nIntegrating over energy on the two sides, we \fnally\nobtain the spin-di\u000busion equation\n~Dr2s=~\n\u001css\u00002h\u0002s; (A3)\nwhere\ns(r) =N0Z\nd\u000ff(r;\u000f)\nis the spin accumulation at position r. The spin di\u000busion\nequation was used to describe the spin Hanle e\u000bect in\nferromagnet-normal metal-ferromagnet systems.41,43,44\nHere we use it to calculate the spin accumulation in a\nweak ferromagnet, including the Hanle e\u000bect from the\ninhomogeneous exchange \feld.\nThe spin current is given by the derivative of the spin\naccumulation\nj(r) =~Drs(r):\nThe spin current is a tensor, as it depends on position\nfor all three spin components. This spin current plays an\nimportant role in the domain wall motion.\nAppendix B: Spin accumulation with inhomogenous\nmagnetization\nSince the rotation angle in Eq. (2) is a step function,\nthe spin di\u000busion equation in Eq. (4) is separated into\nthree regions. On the left and right side of the domain\nwall, the general solution of Eq. (4) is given by\ns0\n1=1\n\u0011fcosh(z\u0011\u0017) [\u0011C1icos(z\u0011\u0016) + (\u0016C2i+\u0017C4i) sin(z\u0011\u0016)] + [(\u0017C2i\u0000\u0016C4i) cos(z\u0011\u0016) +\u0011C3isin(z\u0011\u0016)] sinh(z\u0011\u0017)g\ns0\n2=1\n\u0011fcosh(z\u0011\u0017) [\u0011C3icos(z\u0011\u0016) + (\u0000\u0017C2i+\u0016C4i) sin(z\u0011\u0016)] + [(\u0016C2i+\u0017C4i) cos(z\u0011\u0016)\u0000\u0011C1isin(z\u0011\u0016)] sinh(z\u0011\u0017)g\ns0\n3=ez=`sC5i+e\u0000z=`sC6i;\nwherei= 1;3 refers to the left and right side of the domain wall, and Cniare constants which are determined from\nthe boundary conditions. Here we also de\fned\n\u0011=\u00124\nl4\nh+1\n`4s\u00131=4\n\u0016= sin\u00141\n2arctan\u00122`2\ns\nl2\nh\u0013\u0015\n\u0017= cos\u00141\n2arctan\u00122`2\ns\nl2\nh\u0013\u0015\n:10\nIn the domain wall region the solutions are given by\ns0\n1=\u0000C12ezk1(\u000b2+`\u00002\ns\u0000k2\n1)\n2\u000bpN1k1+C22e\u0000zk1(\u000b2+`\u00002\ns\u0000k2\n1)\n2\u000bpN1k1\u0000C32ezk2(\u000b2+`\u00002\ns\u0000k2\n2)\n2\u000bpN2k2\n+C42e\u0000zk2(\u000b2+`\u00002\ns\u0000k2\n2)\n2\u000bpN2k2\u0000C52ezk\u0003\n2(\u000b2+`\u00002\ns\u0000k\u00032\n2)\n2\u000bpN2k\u0003\n2+C62e\u0000zk\u0003\n2(\u000b2+`\u00002\ns\u0000k\u00032\n2)\n2\u000bpN2k\u0003\n2\ns0\n2=C12ezk1al2\nh\n36\u000b\f2pN1k1\u0000C22e\u0000zk1al2\nh\n36\u000b\f2pN1k1+C32ezk2bl2\nh\n72\u000b\f2pN2k2\u0000C42e\u0000zk2bl2\nh\n72\u000b\f2pN2k2+C52ezk\u0003\n2b\u0003l2\nh\n72\u000b\f2pN2k\u0003\n2\u0000C62e\u0000zk\u0003\n2b\u0003l2\nh\n72\u000b\f2pN2k\u0003\n2\ns0\n3=C12ezk1\npN1+C22e\u0000zk1\npN1+C32ezk2\npN2+C42e\u0000zk2\npN2+C52ezk\u0003\n2\npN2+C62e\u0000zk\u0003\n2\npN2;\nwhere\n\u000b=\u0019\n\u0015\n\f=2\n4\u000b6+90\u000b2\nl4\nh+36\u000b4\n`2s+1\n2s\n\u00004\u0012\n\u000b4\u000012\nl4\nh\u000012\u000b2\n`2s\u00133\n+ 4\u0012\n\u000b6+90\u000b2\nl4\nh+36\u000b4\n`2s\u001323\n51=3\nand\nk1=s\n1\n3\u0012\n\u00002\u000b2+\f+3\n`2s+\u000b4\u000012=l4\nh\u000012\u000b2=`2s\n\f\u0013\nk2=vuut1\n12\"\n\u00008\u000b2+ 2i(i+p\n3)\f+12\n`2s+2(1 +ip\n3)(\u0000\u000b4+ 12=l4\nh+ 12\u000b2=`2s)\n\f#\nare the solutions of the following characteristic equation\n\u000b4\u0012\nk2\u00001\n`2s\u0013\n+\"\n4\nl4\nh+\u0012\nk2\u00001\n`2s\u00132#\u0012\nk2\u00001\n`2s\u0013\n\u00002\u000b2\u00122\nl4\nh\u0000k4+1\n`4s\u0013\n= 0:\nThe other coe\u000ecients are\na=\u0012\n\u000b4+\u000b2\f+\f2\u000012\nl4\nh\u00132\n+12\u000b2\n`2s\u0012\n\u00002\u000b4\u00002\u000b2\f+\f2+24\nl4\nh\u0013\n+144\u000b4\n`4s\nb=\u0000(1\u0000ip\n3)\u000b8\u0000(1 +ip\n3)\f4\u000048\f2\nl4\nh\u0000144(1\u0000ip\n3)\nl8\nh+\u000b6\u0014\n\u00002\u0010\n1 +ip\n3\u0011\n\f+24\n`2s\u0010\n1\u0000ip\n3\u0011\u0015\n+ 2\u000b2\"\n\u0000(1\u0000ip\n3)\f3+12(1 +ip\n3)\f\nl4\nh+12\f2\n`2s\u0000144(1\u0000ip\n3)\nl4\nh`2s#\n+ 6\u000b4\"\n\f2+4(1 +ip\n3)\f\n`2s+ 4(1\u0000ip\n3)\u00121\nl4\nh\u00006\n`4s\u0013#\nN1=(1 +l2\nhk2\n1)\b\na2l4\nh+ 324\f4\u0002\n\u000b4\u0000(k2\n1\u0000`\u00002\ns)2+ 2\u000b2(k2\n1+`\u00002\ns)\u0003\t\n1296\u000b2\f4k2\n111\n0102030-25-20-15-10-505\n036912-0.01-0.00500.005\n036912-0.01-0.0050036912-0.0100.010.020.03\n03691200.020.040.06\n0102030-20020406080036912-0.02-0.0100.010.02\n036912-0.03-0.02-0.010\n0369120.3060.3080.310.312(a)(b)(c)(d)(e)(f)(g)(h)(i)\nFIG. 9. Coe\u000ecients in the general solutions of the spin di\u000busion equation. Here the results are plotted for `s= 3:2lh,PI= 0:5,\nkIlh= 0:5, andX0=\u0015=2.\nN2=(1 +l2\nhjk2j2)\b\njbj2l4\nh+ 1296\f4\u0002\n\u000b4+jk2\n2j2+`\u00004\ns\u00002`\u00002\nsRe(k2\n2) +\u000b2(2`\u00002\ns\u00002Re(k2\n2) + 4jk2j2)\u0003\t\n5184\u000b2\f4jk2j2:\nThe unknown coe\u000ecients are determined from the boundary conditions. However, these coe\u000ecients are too long\nto be printed here and rather have to be shown numerically.45We plot some of the coe\u000ecients for di\u000berent values of\nthe domain wall size \u0015in Fig. 9.\nThe solutions also yield C52=C\u0003\n32andC62=C\u0003\n42, which also imply real-valued spin accumulation. Moreover, the\ncoe\u000ecients in region i= 3 are very similar with those in region i= 1, but with opposite signs ( C13,C33andC63).\nForX\u001d`s, we also \fnd\nC61=\u0000C63=kI`sPI\rVN 0\n2 +kI`s; (B1)\nbut in general the expression is more complicated.\nThe solutions in the domain wall region can hence be written as\ns0\n1=\u0000\u000b2+`\u00002\ns\u0000k2\n1\n2\u000bpN1k1\u0000\nC12ezk1\u0000C22e\u0000zk1\u0001\n\u0000Re\u0014\u000b2+`\u00002\ns\u0000k2\n2\n\u000bpN2k2\u0000\nC32ezk2\u0000C42e\u0000zk2\u0001\u0015\n(B2)\ns0\n2=al2\nh\n36\u000b\f2pN1k1\u0000\nC12ezk1\u0000C22e\u0000zk1\u0001\n+ Re\u0014bl2\nh\n36\u000b\f2pN2k2\u0000\nC32ezk2\u0000C42e\u0000zk2\u0001\u0015\n(B3)\ns0\n3=1pN1\u0000\nC12ezk1+C22e\u0000zk1\u0001\n+2pN2Re\u0000\nC32ezk2+C42e\u0000zk2\u0001\n: (B4)\nThe unrotated spin accumulation is given by the transformation in Eq. (3). We use these results to calculate the\nforce and torque in the equations of domain wall motion.12\nAppendix C: Description of the injector\nWe consider the case where the spin polarized current\nis injected to the weak ferromagnet from a strong ferro-\nmagnet attached to it a position z= 0. Since we assume\nthe injector magnetization to be homogeneous, we can\nwrite the spin di\u000busion equation separately in the two\nspin directions collinear with the magnetization of the\nstrong ferromagnet. If we assume that the current is in-\njected into the weak ferromagnet from a wire placed in\ntheydirection, the spin-di\u000busion equation in the injector\nbecomes42\n@2\nysI\n\u001b=sI\n\u001b\u0000sI\n\u0016\u001b\n2l2\u001b;\nwhere \u0016\u001bis the opposite spin to \u001b=\"=#andl\u001b=pD\u001b\u001c\u001b\nis the spin-dependent spin relaxation length. The general\nsolution of this equation can be written as\nsI\n\"=#=l2\n#(C1+C2y) +l2\n\"(C3+C4y)\nl2\ntot\n\u0006l2\n#=\"\nl2\ntoth\n(C3\u0000C1) cosh\u0010y\nl\u0011\n+l(C4\u0000C2) sinh\u0010y\nl\u0011i\n;\nwherel2\ntot=l2\n\"+l2\n#andl=p\n2l\"l#=ltot. The unknown\ncoe\u000ecients can be determined from the boundary condi-tion39\n\u001bI\n\"=#AT@ysI\n\"=#(0) =1\nRIh\nsw\n\"=#(0)\u0000sI\n\"=#(0)i\n; (C1)\nwhere\u001bI\n\u001b=e2N\u001bD\u001bis the spin dependent conductiv-\nity in the injector, N\u001bis the density of states of spin \u001b\nat the Fermi level, ATis the cross-sectional area of the\ntunnelling junction, RIis the resistance of the contact\nbetween the injector and the wire, and sw\n\u001bis the spin\ndensity for spin \u001bcreated at the weak ferromagnet. If\nthe voltage is applied at a distance Laway from the con-\ntact, then we have two more boundary conditions\nsI\n\"(\u0000L) +sI\n#(\u0000L) =VN0\nsI\n\"(\u0000L)\u0000sI\n#(\u0000L) = 0;\nwhere the upper equation states that the average poten-\ntial of the electrons at the distance LisV(e= 1), and\nthe lower indicates the vanishing of the spin accumula-\ntion in the electrode where the voltage is applied.\nWith the determined coe\u000ecients, we can write the chemical potential and the spin accumulation at the position of\ninjection as\n\u0016I(0)N0=sI\n\"(0) +sI\n#(0)\n=l2\ntotVN0\u001b\"\u001b#+ 2aILh\nl2\n#\u001b\"sw\n#(0) +l2\n\"\u001b#sw\n\"(0)i\nl2\ntot\u001b\"\u001b#+aIL(l2\n\"\u001b#+l2\n#\u001b\") +aIl(aILl2\ntot+l2\n#\u001b#+l2\n\"\u001b\") tanh (L=l)\n+n\naILl2\ntot\u0016w(0)N0+l2\ntotVN0\u001bF+ (l2\n\"\u0000l2\n#)h\n\u001b\"sw\n#(0)\u0000\u001b#sw\n\"(0)io\naIltanh(L=l)\nl2\ntot\u001b\"\u001b#+aIL(l2\n\"\u001b#+l2\n#\u001b\") +aIl(aILl2\ntot+l2\n#\u001b#+l2\n\"\u001b\") tanh (L=l)(C2)\nsI\n3(0) =sI\n\"(0)\u0000sI\n#(0) =aIll2\ntoth\naILsw\n3(0) +VN0(\u001b\"\u0000\u001b#)=2\u0000\u001b\"sw\n#(0) +\u001b#sw\n\"(0)i\ntanh(L=l)\nl2\ntot\u001b\"\u001b#+aIL(l2\n\"\u001b#+l2\n#\u001b\") +aIl(aILl2\ntot+l2\n#\u001b#+l2\n\"\u001b\") tanh (L=l); (C3)\nwhereaI= 1=(RIAT) and\u001bI\nF= (\u001bI\n\"+\u001bI\n#)=2. Here we also de\fned the chemical potential and the spin accumulation\nin the weak ferromagnet as \u0016w(0)N0=sw\n\"(0) +sw\n#(0) andsw\n3(0) =sw\n\"(0)\u0000sw\n#(0).\nWe assume for simplicity that the injector and the weak ferromagnetic wire cross sections are equal. De\fning the\ninjector transparency \u0014I= 1=(\u001bwRIAT), we can write the boundary condition analogous to Eq. (C1) for the weak\nferromagnet wire as\n@zsw\n\"=#(0) =\u0014Ih\nsw\n\"=#(0)\u0000sI\n\"=#(0)i\n;\nwhere\u001bwis the conductivity in the weak ferromagnet. We then write this boundary condition in terms of \u0016w(0) and\nsw\n3(0), and choose the zero point of potential so that \u0016w(0) = 0. By substituting sI\n\"=#(0) in Eq. (C2) and Eq. (C3),\nwe obtain for l\"=l#\n\u0012\n@z\u0016w(0)N0\n@zsw\n3(0)\u0013\n=\u0014I\naIltanh(L=l)(aIL+\u001bI\nF) +aIL\u001bI\nF+\u001bI\n\"\u001bI\n#\u0012aIltanh(L=l)\u001bI\nF+\u001bI\n\"\u001bI\n#aIL(\u001bI\n\"\u0000\u001bI\n#)=2\naIltanh(L=l)(\u001bI\n\"\u0000\u001bI\n#)=2aIL\u001bI\nF+\u001bI\n\"\u001bI\n#\u0013\u0012\n\u0000VN0\nsw\n3(0)\u0013\n:13\nThis equation leads to an Onsager relation for the cur-\nrent through the contact\n\u0012\n@z\u0016w(0)N0\n@zsw\n3(0)\u0013\n=\u0012\n\u0000kIPIkI\nPIkIkI\u0013\u0012\n\u0000\rVN 0\nsw\n3(0)\u0013\n;(C4)\nwhere the injector polarization and transparency are de-\n\fned as\nPI=aIL(\u001bI\n\"\u0000\u001bI\n#)=2\naIL\u001bI\nF+\u001bI\n\"\u001bI\n#=L(\u001bI\n\"\u0000\u001bI\n#)\n2(L\u001bI\nF+\u001bI\n\"\u001bI\n#RIAT)(C5)\nkI=\u0014I(aIL\u001bI\nF+\u001bI\n\"\u001bI\n#)\naIltanh(L=l)(aIL+\u001bI\nF) +aIL\u001bI\nF+\u001bI\n\"\u001bI\n#\n=(L\u001bI\nF+\u001bI\n\"\u001bI\n#RIAT)=\u001bw\nltanh(L=l)(L+RIAT\u001bI\nF) +L\u001bI\nFRIAT+\u001bI\n\"\u001bI\n#R2\nIA2\nT;\n(C6)\n\r=l\nLtanh\u0012L\nl\u0013\n(C7)\nand\n\u0000 =L\nlaIltanh(L=l)\u001bI\nF+\u001bI\n\"\u001bI\n#\ntanh(L=l)(aIL\u001bI\nF+\u001bI\n\"\u001bI\n#)\n=L\nlltanh(L=l)\u001bI\nF+\u001bI\n\"\u001bI\n#RIAT\ntanh(L=l)(L\u001bI\nF+\u001bI\n\"\u001bI\n#RIAT):(C8)The second row of Eq. 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B 67, 085319 (2003).\n45The Mathematica codes used to solve for the coe\u000ecients\nwill be provided together with the published paper."    },    {        "title": "1211.3033v1.About_the_two_spin_channel_model_for_ferromagnetic_excitations_and_spin_dependent_heat_transfer_equations.pdf",        "content": "About the two spin-channel model for ferromagnetic excitations\nand spin-dependent heat transfer equations\nJ.-E. Wegrowe\u0003\nEcole Polytechnique, LSI, CNRS and CEA/DSM/IRAMIS, Palaiseau F-91128, France\n(Dated: April 18, 2022)\nAbstract\nThe two spin-channel model is generalized to the case of transport of ferromagnetic excitations\nin electric conductors and insulators. The two channels are de\fned by reducing the ferromagnetic\ndegrees of freedom to a bivaluated variable, i.e. to an e\u000bective spin one-half. The reduction is\nperformed after de\fning the local magnetic con\fguration space by a sphere \u0006 x, and integrating\nthe relevant physical quantities over the two hemispheres \u0006\"\nxand \u0006#\nx. The con\fguration space\nis then extended to the xdirection for non-uniform magnetization excitations. The transport\nequations for both magnetic moments and magnetic energy are deduced, including the relaxation\nfrom one channel to the other. The heat transport equations for ferromagnets is deduced.\nPACS numbers: 75.47.-m, 72.25.-b, 85.80.Lp\n\u0003Electronic address: jean-eric.wegrowe@polytechnique.fr\n1arXiv:1211.3033v1  [cond-mat.mes-hall]  13 Nov 2012∑\n∑FIG. 1: Illustration of the \u0006 xcon\fguration space of ferromagnetic degrees of freedom at point\nx. The separation in two hemispheres \u0006 x= \u0006\"\nx[\u0006#\nxallows the two e\u000bective spin-channels to be\nde\fned. The arrow in the center represents a mean value of the magnetization in the con\fguration\nspace.\nIn the context of spintronics, the e\u000bect of spin-injection and spin-accumulation is easy to\ndescribe on the basis of the two spin-channel model for electric carriers, if the magnetization\nis locally de\fned by the microscopic spin one-half s=\u0006~=2 [1{5]. In that case, the electric\ncarriers de\fne without ambiguity the two channels at any points xof the material: one\nchannel for the up spin ~=2 and the other channel for the down spin \u0000~=2, with a \fxed\nquanti\fcation axis. However, in the case of ferromagnetic interactions between electric car-\nriers (e.g. for spin-waves excitations), this de\fnition is a-priori not valid since the magnetic\ncarriers are not longer de\fned locally by a spin one -half. Beyond, in the case of electric\ninsulators, the de\fnition of two spin-channel seems to be problematic because it cannot be\nbased on an assembly of delocalized quasi-particles.\nWe show in this paper that the two spin-channel model can nevertheless be generalized to\nany kind of macroscopic ferromagnetic excitations (quantal or classical). This generalization\nis based on a reduction method that allows the continuous magnetic degrees of freedom to\nbe reduced to a bivaluated variable at each point xof the usual space, i.e. to a local e\u000bective\n\\spin\" one-half. The reduction is performed after de\fning the local magnetic con\fguration\nspace over the sphere \u0006 x(section I below), and integrating the relevant physical quantities\n2over each hemisphere (see Fig. 1) for each point xof the usual space (section II). The con\fg-\nuration space is then extended to the xdirection for non-uniform magnetization excitations,\nand the transport equations are deduced (section III).\nThis generalization of the two spin-channel model shows that the e\u000bects generated by\nspin-polarized heat currents are similar to that generated by electric spin-polarized currents.\nTypically, the spin-accumulation e\u000bect occurring in spintronics devices can be generalized to\nthermal spin-accumulation [7]. The similarity of the transport equations of heat - together\nwith the usual Seebeck and Nernst e\u000bects - could explain the so-called spin-Seebeck and\nspin-Nernst e\u000bects observed recently on various materials [8{14].\nA. De\fnition of the localized ferromagnetic con\fguration space \u0006x\nIn this section, we focus on the localized uniform ferromagnetic moment ~M(x) =Ms~ er(x)\nde\fned with radial unit vector ~ er(x) and magnetization at saturation Msat pointx. The\napproach used is the mesoscopic non-equilibrium thermodynamic theory (MNET) [15, 16]\napplied to rotational brownian motion [17, 18].\nIn order to treat statistically the ferromagnetic degrees of freedom, a statistical ensemble\nof a large number of ferromagnetic moments is de\fned on the con\fguration space \u0006 x(Fig.\n1). Each magnetic moment is described by its position f\u0012;'gon the sphere \u0006 xof radius\nMs. The angle \u0012is associated to the radial unit vector ~ e\u0012and the angle 'is associated to\nthe azimuth unit vector ~ e'.\nThe statistical distribution of the magnetic moments on the sphere \u0006 xis then de\fned by\nthe density \u001aF(\u0012;';x) per units of solid angle d\n =sin\u0012d\u0012d' [19] and per unit length \u000ex.\nThe function \u001aF(\u0012;') is a solution of the rotational Fokker-Planck equation derived at the\nend of this subsection. At equilibrium, the density is given by the Boltzmann distribution\nde\fned by the ferromagnetic potential VF(\u0012;';x), and such that the e\u000bective magnetic\n\feld vanishes ~Heff=\u0000~rVF= 0 (~Heffcontains all deterministic contributions: external\nmagnetic \feld, dipolar \feld, anisotropy \feld, exchange \feld, etc). For out-of-equilibrum\nstates, \ructuations and di\u000busion plays a fundamental role. The distribution is given by the\nferromagnetic chemical potential that takes the form \u0016F=kTln (\u001aF) +VF[15, 19, 20].\nThe generalized force ~Heff+~h\u0011\u0000~r\u0006\u0016Fproduces a current of ferromagnetic moments\n~JF(\u0012;';x) =\u001aFd~ ur=dt, that is \rowing on the surface of the sphere. This generalized force\n3contains a deterministic part ~Heff=\u0000~rVFand a di\u000busive part ~h= (kT=\u001aF)~r\u001aF[17, 20].\nIf the ferromagnetic system is closed, the magnetic moments are conserved on \u0006 xso that:\nd\u001aF\ndt=\u0000div\u0006~JF: (1)\nOn the other hand, the ferromagnetic energy uFis also described by a the continuity\nequation. However, the system under interest is not adiabatic and the conservation equation\nfor the ferromagnetic energy takes the form:\nduF\ndt=\u0000div\u0006~Ju+r (2)\nwhererdescribes the dissipation in the environment and ~Juis the \rux of ferromagnetic\nenergy in the con\fguration space \u0006 x. If the temperature T(x) is uniform over the con\fg-\nuration space \u0006 x(i.e. at each point x), the \rux of heat is zero, and the power dissipated\nby the ferromagnetic system is reduced to the e\u000bect of the \rux of magnetic moments: the\nferromagnetic power dissipated or stored at each point xof the ferromagnet reads\nPF=\u0000Z\n\u0006~JF:~r\u0006\u0016Fd\n (3)\nThe application of the second law of thermodynamics allows the transport equation to be\ndeduced by writing the relation that links the generalized \rux to the generalized force.\nBoth quantities, \rux and forces, are then related by the Onsager matrix of the transport\ncoe\u000ecients \u0016L[19, 21, 22]:\n~JF=\u0000\u0016L~r\u0006\u0016F(4)\nwhere the \row ~JFis a two component vector de\fned with the unit vectors f~ e';~ e\u0012gof\n\u0006x. Accordingly, the Onsager matrix is a 2x2 matrix de\fned by four transport coe\u000ecients\nfL\u0012\u0012;L\u0012';L'\u0012;L''g. The Onsager reciprocity relations impose that L\u0012'=\u0000L'\u0012. Further-\nmore, assuming that the dissipation is isotropic, we have L\u0012\u0012=L''. Introducing \u000bas the\nratio of the o\u000b-diagonal to the diagonal coe\u000ecients; \u000b=L\u0012'=L\u0012\u0012, the ferromagnetic kinetic\nequation is de\fned by two ferromagnetic transport coe\u000ecients LF=L\u0012'=\u001aFand\u000b:\n\u0016L=\u001aFLF0\n@\u000b1\n\u00001\u000b1\nA (5)\nRewriting Eq. (4) in the reference frame f~ er;~ e\u0012;~ e'g, and recalling that the current is the\ndensity multiplied by the velocity we obtain the well known LL equation:\n4d~ er\ndt=\u0000LF\u001a\n~ er\u0002\u0012\n~Heff+kT\n\u001a~r\u001a\u0013\n+\u000b~ er\u0002\u0012\n~ er\u0002\u0012\n~Heff+kT\n\u001a~r\u001a\u0013\u0013\u001b\n: (6)\nThe equivalence between the LL equation and the phenomenological Gilbert equation [23]\ngives the relation between the coe\u000ecients \u000bandLFon the one hand, and the Gilbert\ncoe\u000ecients \u0011and the gyromagnetic ratio \ron the other hand:\n\u000b=\u0011\rMs\nLF=\r\nMs(1+\u000b2)(7)\nInserting Eq. (5) and Eq. Eq. (4) into the continuity equation Eq. (1) leads to the\nFokker-Planck equation in the con\fguration space \u0006 x:\nd\u001aF\ndt=~r\u0006:n\n\u001aF~ er\u0002\u0010\nLF~Heff+D~r\u0006\u001aF\u0011\n+\u000b\u001aF~ er\u0002\u0010\n~ er\u0002\u0010\nLF~Heff+D~r\u0006\u001aF\u0011\u0011o\n:(8)\nwhereD=LFkT=\u001aFis the di\u000busion coe\u000ecient. This result is well known [17, 18]. The goal\nis to generalize the description to non-uniform ferromagnets (i.e. performing the extension\nof the con\fguration space to the neighbors \u0006 x\u0006\u000ex, in order to describe transfer of magnetic\nmoments (beyond Eq. (1)) and transfer of energy. The objective of the next section is to\nsimplify the problem by de\fning \fst the two channel model for magnetic excitations.\nB. The two spin-channel model: reduction of the ferromagnetic degrees of freedom\nto a bivaluted variable.\nThe concept of \\spin-channels\" is inspired from spin-dependent transport studies (or\n\\spintronics\"), for which the electronic spin is a bivaluated degree of freedom with s=\u0006~=2\ncorresponding to the two states \"and#for \fxed quantization axis.\nWe will show in the following the a ferromagnetic bivaluated variable can be de\fned by\nreducing the continuous degrees of freedom of the magnetization to a bivaluated variable,\ni.e. to an e\u000bective spin . This reduction is performed by the integration of the relevant\nquantities over the two hemispheres \u0006\"\nxand \u0006#\nx, such that \u0006 x= \u0006\"\nx[\u0006#\nx(as shown in Fig.\n1). We can then de\fne the chemical potentials of the two hemispheres by:\n\u0016F\nl(x) =Z\n\u0006l\nx\u0016(\u0012;';x)d\n; (9)\n5The total chemical potential is given by the sum over the two hemisphere \u0016F=\u0016F\n\"+\u0016F\n#,\nand the di\u000berence de\fnes a \\ pumping force \" [22] \u0001\u0016=\u0016F\n\"\u0000\u0016F\n#. The number of magnetic\nmomentsnF\nlper unit length \u000exfor each hemisphere reads:\nnF\nl(x) =Z\n\u0006l\nx\u001aF(';\u0012;x)d\n; (10)\nso that the density of magnetic moments per unit length is nF\n0=nF\n\"+nF\n#and the di\u000berence\nbetween the hemispheres is \u0001 nF=nF\n\"\u0000nF\n#.\nLet us de\fne the scalar JF\nlby the expression:\nJF\nl(x) =Z\n\u0006l\nxdiv\u0006~JFd\n; (11)\nand the di\u000berence\n\u0001JF(x) =Z\n\u0006\"\nxdiv\u0006~JFd\n\u0000Z\n\u0006#\nxdiv\u0006~JFd\n: (12)\nFrom Eq. (1), Eq. (10) and Eq. (11) we have:\ndnF\n\"\ndt=JF\n\"=\u0001JF\n2\ndnF\n#\ndt=JF\n#=\u0000\u0001JF\n2(13)\nwhere the last equality in the left hand side is obtained by noting that the total numbers\nof magnetic moments on the sphere n0=n\"+n#is constant, so that dnF\n0=dt=JF\n\"+JF\n#= 0.\nNote that \u0001JFis the \rux of the magnetic moments \rowing from the hemisphere \u0006\"\nxto\nthe other hemisphere \u0006#\nx, and the \\ pumping force \" \u0001\u0016is thermodynamically conjugate to\nthe \\\rux\"\u0001JF\n2in the sense that the product P\"#= \u0001JF\u0001\u0016=2 is the ferromagnetic power\nexchanged between the two hemispheres.\nIn the same way as above, the ferromagnetic energy uF(\u0012;';x) is integrated over the two\nhemispheres. The reduced variable reads:\nUF\nl(x) =Z\n\u0006l\nxuF(';\u0012;x)d\n; (14)\nand from Eq. (2) we have:\ndUF\n\"\ndt=JU\n\"+R=2\ndUF\n#\ndt=JU\n#+R=2(15)\n6whereR=2 =R\n\u0006\"rd\n =R\n\u0006#rd\n, and the energy currents JU\nlare de\fned on each point\nxby the relation:\nJU\nl(x) =Z\n\u0006l\nxdiv\u0006~JUd\n: (16)\nIf the system is adiabatic total energy U0=U\"+U#on the sphere \u0006 xwould constant and\nwe would have, as for the current of magnetic moments:\nJU\n\"=\u0001JU\n2\nJU\n#=\u0000\u0001JU\n2;(17)\nwhere \u0001JU=JU\n\"\u0000JU\n#. In conclusion, even for an isolated and uniform ferromagnetic\nparticle (i.e. adiabatic ferromagnetic system), a current of energy is \rowing from one hemi-\nsphere to the other at any point x. In other terms, an e\u000bective \\spin-\rip relaxation\" has\nbeen de\fned for ferromagnetic excitations.\nC. Transport equations for heat current along the xdirection.\nA ferromagnetic wire can be modeled by a succession of segments of thickness \u000exand\nsection unity [6]. At each position x, the magnetization of volume \u0006\u000exis described in the\ncon\fguration space \u0006 xwith the corresponding chemical potential. The con\fguration space\nof the total ferromagnetic wire of length lis then the continuous limit ( \u000ex!0,N!1 and\nN\u000ex =l) of a chain of Nferromagnetic con\fguration spaces :::[\u0006x\u0000\u000ex[\u0006x[\u0006x+\u000ex[:::\n(see Fig. 2).\nLet us \frst consider an open system that is able to exchange magnetic moments with its\nenvironment [6]. In this case, a mechanism of transport of magnetic moments would take\nplace in the xdirection, and the conservation equations for magnetic moments through the\nspacex(Eq. (13)) would take the same form as in the case of spin-dependent transport in\nthe two channel model for electric conductors:\ndn\"\ndt=\u0000@JF\n\"\n@x\u0000_ F\ndn#\ndt=\u0000@JF\n#\n@x+_ F(18)\nwhere _ F\u0011\u0001JF=(2\u000ex) is the e\u000bective spin-\rip relaxation, in the sense that the \rux _ Fis\nthe velocity of the transformation of \\spin up\" \"into a \\spin down\" #under the action of\nthe chemical a\u000enity \u0001 \u0016F[4{6, 19, 21, 22, 24].\n7x\nx x + δx x - δx\nFIG. 2: Illustration of the con\fguration space \u0006 for a non-uniform ferromagnet of length l=N\u000ex\nand section unity, de\fned as a chain of Nspheres: \u0006 = :::[\u0006x\u0000\u000ex[\u0006x[\u0006x+\u000ex[:::at the\ncontinuous limite ( N!1;\u000ex!0).\nHowever, in a ferromagnet, magnetic moments are localized so that a spin-injection mech-\nanism carried by the magnetic excitations is not expected. The magnetic moments cannot be\ntransmitted from the con\fguration space \u0006l\nxto its neighbors \u0006l\nx\u00061andthe local con\fguration\nspace is closed from the point of view of the ferromagnetic degrees of freedom.\nIn contrast, since the system under interest is not adiabatic energy is transmitted from\nthe con\fguration space \u0006l\nxto its neighbors \u0006l\nx\u00061. Typically, this situation corresponds to\nferromagnetic resonance or heating performed at an extremity of the ferromagnetic device\n[8{14]. Space-dependent and spin-dependent heat currents JU\nl(x) are then produced and\n\row throughout the sample and through the interfaces. Furthermor, as shown in Eq. (15),\nthe energy \rux \u0001JUis produced inside the con\fguration space \u0006 x, \rowing from the subspace\n\u0006\"\nxto the subspace \u0006#\nx. Generalizing Eq. (15) with the contribution of the neighbors \u0006 x\u00061,\nwe have:\ndU\"\ndt=\u0000@JU\n\"\n@x\u0000_ th+R=2\ndU#\ndt=\u0000@JU\n#\n@x+_ th+R=2;(19)\nwhere _ th\u0011\u0001JU\n2\u000exis due to the relaxation between the two channels. The heat current\ngenerated by the energy carriers is given by the relation:\nJq\nl=JU\nl\u0000\u0016lJF\nl (20)\nWe are here interested in the case JF\nl= 0. The di\u000busion equations for each channel can\nthen be deduced [7]:\nJq\nl=\u0000Ll@\u0016F\nl\n@x+\u0015l@T\n@x(21)\n8D. Conclusion\nA non-equilibrium thermodynamic approach has been used in order to establish a two\nchannel model applied to macroscopic ferromagnetic excitations. The reduction of the ferro-\nmagnetic degrees of freedom from two continuous coordinates f\u0012;'gto a bivaluated scalar\nvariablel(and a quanti\fcation axis) is explicitly performed, and applied to the relevant\nphysical observables, i.e. the density of magnetic moments, the energy density, and the\ncorresponding currents. The complexity of the transport properties of magnetic moments\nand magnetic energy (including di\u000busion) can be reduced to the kinetic equations of two\nstates\"and#at each position xin space. This method allows the spintronics concept\nofspin-injection to be generalized to any kind of magnetic excitations. In particular, it is\nshown that thermal spin-injection can be performed without magnetic carriers, but with\nspin-dependent heat currents only. The validity of the usual technics of spintronics has been\nextended to the case of ferromagnetic excitations (quantal or classical) occurring in electric\nconductors or electric insulators.\n[1] I. A. Campbell, A. Fert, A. R. Pomeroy, Evidence for Two Current Conduction in Iron Phil\nMag. 15, 977 (1967).\n[2] A.G. Aronov, Spin injection in metals and polarization of nuclei , Zh. Eksp. Teor. Fiz. Pisma\nRed. 24, 37 (1976) [Sov. Phys. JETP Lett. 24, 32 (1976)].\n[3] M. Johnson, R. H. Silsbee, Interfacial charge-spin coupling: Injection and detection of spin\nmagnetization in metals Phys. Rev. Lett. 55, 1790 (1985)\n[4] P. C. van Son, H. van Kempen, P. Wyder, Boundary Resistance of the Ferromagnetic-\nNonferromagnetic Metal Interface Phys. Rev. Lett. 58, 2271 (1998)\n[5] T. Valet, A. Fert, Theory of the perpendicular magnetoresistance in magnetic multilayers Phys.\nRev. B 48, 7099 (1993) .\n[6] J. -E. Wegrowe, Thermodynamic approach of generalized Landau-Lifshitz-Gilbert equation with\nspin-polarized current , Phys. Rev. B 62, 1067 (2000).\n[7] J.-E. Wegrowe, D. Lacour, and H-J. Drouhin, Thermal Spin-Accumulation in Electric Con-\nductors and Insulators , arXiv:1207.3281 (2012).\n9[8] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh,\nObservation of the spin Seebeck e\u000bect , Nature 455, 778 (2008),\n[9] C. M. Jaworsky, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers,\nObservation of the spin-Seebeck e\u000bect in a ferromagnetic semiconductor , Nature Mater. 9,\n898 (2010).\n[10] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, Observation of longitudinal spin-\nSeebeck e\u000bect in magnetic insulators Appl. Phys. Lett. 97, 172505 (2010).\n[11] M. Erekhinsky, F. Casanova, I. K. Schuller, and A. Sharoni, Spin-dependent Seebeck e\u000bect in\nnon-local spin valce devices Appl. Phys. Lett. 100, 212401 (2012).\n[12] G. L. da Silva, L. H. Viela-Leao, S. M. Rezende, and A. Azevedo, Spin current injection by\nSeebeck and spin pumping e\u000bects in yttrium iron garnet/Pt structures J. Appl. Phys. 111,\n07C513 (2012).\n[13] S. Y. Huang, W. G. Wang, S. F. Lee, J. Kwo, and C. L. Chien, Intrisic Spin-Dependent\nThermal Transport Phys. Rev. Lett. 107, 216604 (2011).\n[14] A. V. Chumak, A. A. Serga, M. B. Jung\reisch, R. Neb, D. A. Bozhko, V. S. Tiberkevich,\nand B. Hillebrand, Direct detection of magnon spin transport by inverse spin Hall e\u000bect Appl.\nPhys. Lett. 100, 082405 (2012).\n[15] P. Mazur, Fluctuations and non-equilibrium thermodynamics , Physica A, 261, 451 (1998)\n[16] D. Reguera, J. M. G. Vilar and J. M. Rubi, The Mesoscopic Dynamics of Thermodynamic\nSystem J. Phys. Chem. B, 109, 21502 (2005).\n[17] W. F. Brown Jr., Thermal Fluctuations of Spingle-Domain Particle Phys. Rev. 130, 1677\n(1963).\n[18] W. T. Co\u000bey, Yu. P. Kalmykov, The Langevin equation , World Scienti\fc Series in contempo-\nrary Chemical Physics Vol. 27 (third Edition), 2012\n[19] J.-E. Wegrowe, S. M. Santos, M.-C. Ciornei, H.-J. Drouhin and M. Rubi, Magnetization\nreversal driven by spin-injection: A di\u000busive spin-transfer e\u000bect , Phys. Rev. B, 174408 77\n(2008).\n[20] Y. L. Raikher and V. I. Stepanov, Nonlinear dynamic susceptibilities and \feld-induced bire-\nfringence in magnetic particle assembllies , Adv. Chem. Phys. 129, 419 (2004).\n[21] J.-E. Wegrowe, Spin transfer from the point of view of the ferromagnetic degrees of freedom ,\nSolid State Com. 150519 (2010).\n10[22] J.-E. Wegrowe, H.-J. Drouhin, Spin-Currents and Spin-Pumping Forces for Spintronics , En-\ntropy 13, 316 (2011).\n[23] J.-E. Wegrowe and C. Ciornei Magnetization dynamics, gyromagnetic ratio, and inertial e\u000bects\nAm. J. Phys. 80, 607 (2012).\n[24] M.-C. Ciornei, J. M. Rubi and J.-E. Wegrowe, Magnetization dynamics in the inertial regime:\nNutation predicted at short time scales Phys. Rev. B 83, 020410(R) (2011).\n11"    },    {        "title": "1111.1778v1.Unusual_ferromagnetism_in_nanoparticles_of_doped_oxides_and_manganites.pdf",        "content": "Unusual ferromagnetism in nanoparticles of\ndoped oxides and manganites\nVatsal Dwivedi\nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign, IL 61801, USA\nA. Taraphder\nDepartment of Physics and Center for Theoretical Studies\nIndian Institute of Technology, Kharagpur 721302, India\nAbstract\nThe observation of unusually large ferromagnetism in the nanoparticles of doped\noxides and enhanced ferromagnetic tendencies in manganite nanoparticles have been\nin focus recently. For the transition metal-doped oxide nanoparticles a phenomeno-\nlogical `charge transfer ferromagnetism' model is recently proposed by Coey et al.\nFrom a microscopic calculation with charge transfer between the defect band and\nmixed valent dopants, acting as reservoir, we show how the unusually high ferromag-\nnetic response develops. The puzzle of nanosize-induced ferromagnetic tendencies in\nmanganites is also addressed within the same framework where lattice imperfections\nand uncompensated charges at the surface of the nanoparticle are shown to reorganize\nthe surface electronic structures with enhanced double exchange.\n1arXiv:1111.1778v1  [cond-mat.str-el]  8 Nov 20111 Introduction\nIn recent years, there has been a \rood of reports on ferromagnetic (FM) tendencies associ-\nated with nanoparticles and thin \flms. There are two major class of systems that show this\nbehaviour, the transition metal-doped (transparent) oxides and the colossal magnetoresis-\ntive (CMR) manganites. The bulk samples of many of the former systems are nonmagnetic,\ne.g. CuO, TiO 2while the manganites are antiferromagnetic (AFM) with charge and orbital\norder.\nOne aspect of this FM tendency is that it is almost certainly linked to the inhomo-\ngeneities, for example, the dopant cations for the oxides and the surface states and/or\nthe defects in the nanosize materials for manganites. The FM tendency is not present in\nwell-crystallized bulk samples. Secondly, the FM order is present in only a fraction of the\nsample volume. In doped oxides, the magnetization, however, is much higher than would\ncome from the dopants alone. In view of this strange observation, it has been argued that\nthe e\u000bect is not just an `impurity e\u000bect', rather the dopant impurities must in some way\ndramatically modify the electronic organization of the entire system (or a considerable part\nthereof). If a fraction of the lattice sites are used to explain the magnetic e\u000bects, it is not\nnecessary that the fraction of electrons in those sites only are responsible for the observed\nmagnetic properties: there can be transfer of electrons from other sites to the sites in\nquestion, namely, the surface, interfaces or the defects.\nSuch a phenomenological idea has been proposed [1, 2] by Coey et al. to account for\nthe high temperature FM in transition metal doped oxides (e.g., Fe:TiO 2, Fe:CuO and\nothers). The key idea here is that the magnetism is coming from the regions of defects\n(like phase segregated regions, stripes, twin boundaries) coupled to the dopants. Electrons\nin the defect band are coupled with the mixed valent dopant cations transferring electrons\nbetween the two subsystems. The chemical potential of the correlated defect band is tuned\nby the dopant electrons (which form a ` reservoir ' level). It is possible that depending on\nthe position of the chemical potential with respect to the van Hove singularity of the defect\nband, a Stoner type instability can be tuned leading to a high temperature magnetic long\nrange order (LRO). In particular, if the chemical potential is close to the band singularity,\nthe Stoner criterion U\u001a(\"F) = 1 is easily satis\fed, where Uis the local Coulomb repulsion\nin the defect band.\nIn order to make these ideas more concrete, we take a locally correlated itinerant electron\nsystem coupled to a reservoir, which can transfer electrons or holes to the band. For the\ndoped oxides, the itinerant band is formed by delocalization of electrons in the defect\n2bands [1, 2] as discussed above. In the manganite nanoparticles, the surface electrons\nform the analogue of the `defect' band while the imperfections, broken/dangling bonds and\nexcess charges present at the surface [3, 4, 5] act as the reservoir. The itinerant system\nis derived from the bulk, hence it can be expected to have characteristics of the bulk. Its\nchemical potential is dictated by the \flling. The charge reservoir, owing to its ability to\ntransfer electrons (holes), can alter the \flling of the band. For instance, for low \fllings,\nthe reservoir can transfer electrons leading to an increase in \flling and for higher \flling\nthe opposite could happen, depending on the location of the reservoir. This could pin the\nchemical potential of the itinerant band close to the peak in the density of states (DOS)\nleading to an enhancement of ferromagnetism.\nWe work out such models for two cases. Firstly, the phenomenological calculations of\nCoey et al. oncharge transfer ferromagnetism is studied more carefully starting from a\nHamiltonian and the results compared with their phenomenological arguments. This puts\nthe model on a microscopic basis. Second, for the manganites, the usual Zener double\nexchange model is treated in the presence of a reservoir as argued above, using Monte\nCarlo simulations in tandem with exact diagonalization of the Fermionic part [3]. The rest\nof this paper is organized as follows: section 2 describes the microscopic model for charge\ntransfer ferromagnetism, and section 3 describes the model and results for manganites.\n2 Doped oxide nanoparticles\n2.1 Introduction\nThe concept of a charge reservoir in the context of inducing ferromagnetism was intro-\nduced by Coey et al. (charge transfer ferromagnetism model [1, 2]) to explain the ferro-\nmagnetism observed in nanostructures (thin \flms, nanoparticles) of insulating oxides like\nrutile (TiO 2) [1] orCuO [2] doped with 1-5% of iron. In this case, the charge reservoir is\ndue to the multiple oxidation states of iron, which can donate (or accept) an electron via\nthe ionization process Fe2+ !Fe3++e\u0000. This reservoir can transfer electrons to the\nband formed by defects, twin boundaries, stripes, phase segregated regions, referred to, in\ngeneral, as the `defect band' . Experimentally, M ossbauer spectroscopy shows iron in both\nFe2+andFe3+oxidation states in these oxides.\nFor the defect band, at zero temperature, the condition of ferromagnetism is given by\nthe Stoner criterion U\u001a(\"F) = 1, where Uis the Stoner integral . Hence, the origin of\nmagnetization proposed by Coey et al. is the transfer of electrons from reservoir to the\n3conduction band, leading to Fermi level moving closer to the band singularity and Stoner\nsplitting of the two spin channels of the defect band. Using this and a model DOS, they\nshow that charge transfer to or from a reservoir into a narrow, defect-related band can give\nrise to the inhomogeneous Stoner-type wandering axis ferromagnetism that qualitatively\nreproduces the unusual magnetic properties of these systems - high Curie temperature,\nanhysteretic temperature-independent magnetization curves, a metallic or insulating FM\nground state and a moment that may exceed that of the dopant cations.\n2.2 Model and calculation\nBased on the ideas discussed above, the model can be represented by a Hamiltonian of\nitinerant electrons on a square lattice with on-site Hubbard interaction, coupled to a narrow\nreservoir (width of which is taken zero here). Such a Hamiltonian can be written as:\nH=X\n<i;j>\u001b(\u0000t\u0000\u000eij\u0016)cy\ni\u001bcj\u001b+\"DX\ni\u001bdy\ni\u001bdi\u001b\n+VX\ni\u001b(cy\ni\u001bdi\u001b+dy\ni\u001bci\u001b) +UX\nini\"ni#; (1)\nwhereci\u001b=cy\ni\u001bare the annihilation/creation operators for the itinerant electrons, di\u001b=dy\ni\u001bare\nthe same for the reservoir, Uis the on-site repulsion (acting only in the defect band), V\nis the coupling to the reservoir and \"Dis the position of the reservoir with respect to the\npeak of the defect band DOS. The parameters of the Hamiltonian are varied to obtain the\nphase diagrams. Typical values, for example, are given in several reviews [1]. In order to\nproceed, we use the mean-\feld approximation for the Hubbard term\nni\"ni#=hn\"ini#+hn#ini\"\u0000hn\"ihn#i\nThe self-consistent solutions can be found out easily from\nH=X\n<k>\u001b\u0010\n\"kcy\nk\u001bck\u001b+\"Ddy\nk\u001bdk\u001b+V(cy\nk\u001bdk\u001b+dy\nk\u001bck\u001b)\u0011\n+UX\nk(hn\"ink#+hn#ink\") (2)\nwhere, for a square lattice, we employ a tight binding dispersion \"k=\u00002t(coskx+cosky)\u0000\n~\u0016;~\u0016=\u0016+U\n2whose DOS has a weak logarithmic singularity at zero energy. The self-\n4Figure 1: The variation of magnetization on the U\u0000~\u0016plane with (left to right, top and bottom\ncolumn) no reservoir, reservoir at the peak of DOS, reservoir to the left and right of the band peak.\nThe color code represents magnetization. Insets in each of the \fgures show the position of the\nreservoir (broadened for a nonzero V) with respect to the defect band. There is an enhancement\nof FM (red/yellow regions) when the Fermi level is close to the peak of the defect band (shown in\ninsets).\nconsistency equations for hn\u001biwere solved over a momentum grid in the \frst Brillouin zone\ntill convergence within 0.1% is reached.\n2.3 Results\nThe magnetization M=hn\"i\u0000hn#iwas calculated as the Fermi level is moved and for\nvarious \fxed positions of the reservoir with respect to the defect band. The result, shown\nin Fig. 1, clearly shows the e\u000bect of adding a reservoir to the system - the magnetization\nis enhanced when the Fermi level is close to the reservoir. It shows how ferromagnetic\nregions in the phase space arise. The location of the FM region in the phase space strongly\ndepends on the relative positions of the Fermi energy \u0016and\"D. As\u0016passes through the\nreservoir, there is a strong mixing of the band and reservoir electrons. The Fermi level gets\npinned at the resonant level and Stoner criterion is easily met. There is, of course, always\na strong FM enhancement when the Fermi level is close to the peak of the defect band. In\nthe absence of reservoir, the system exhibits a electron-hole symmetry at half-\flling, as the\nband is symmetric about the peak. The defect band is expected to be spin-split when the\n5Fermi level is close to the peak of the DOS (Stoner splitting), where the characteristics of\nthis splitting is symmetric about the peak of the DOS due to the symmetry of the band.\nAs discussed above, the e\u000bect of reservoir is most pronounced when the reservoir is placed\no\u000b the peak of the DOS, thereby breaking the symmetry in the splitting. A higher splitting\nis seen when the Fermi level is close to the reservoir.\nNote that this mechanism is independent of the nature of the underlying lattice. In fact,\nthe special nesting in a square lattice generally favors AFM ground state at half-\flling. But\nthe mechanism discussed here can appear at any \flling and the special topology of the Fermi\nsurface is easily destroyed in a real system by beyond nearest-neighbor hopping.\nThe high magnetization obtained when the Fermi level is close to the reservoir is qual-\nitatively similar to that obtained by Coey et al. from phenomenological calculations using\na Lorentzian defect band [1] and a Lorentzian band for the reservoir [6].\n3 Manganites\n3.1 Introduction\nManganites came back to focus about seventeen years back owing to the discovery [7] of\ncolossal magnetoresistance in these compounds. These are a class of manganese compounds\nof composition A xB1\u0000xMnO 3(A,B = La, Ca, Ba, Sr, Pb, Nd, Pr), which crystallize in the\ncubic structure of the perovskite mineral CaTiO 3[8]. The basic unit of all the manganites\nis the MnO 6octahedron with corner-shared oxygen and the central Mn3+=4+ion.\nFor manganites, the active electronic levels are the 5-fold degenerate d-levels of the\nMn3+=4+. In the octahedral environment of MnO 6thed5is split into three-fold degenerate\nt2glower level and two-fold degenerate egupper level. The t2glevels are electronically inert\nand can be treated as localized spins with magnitude S= 3=2. These localized spins are\ncoupled to the itinerant egelectrons via Hund's coupling. The itinerant electron system\nforms a band, the \flling of which is controlled by the divalent cation doping.\nThe understanding of the magnetic e\u000bects in manganites is governed by the double ex-\nchange model by Zener [12], which gives a mechanism for hopping in the eglevels. The\nhopping is explained by the degeneracy of the Mn3+\u0000O2\u0000\u0000Mn4+and Mn4+\u0000O2\u0000\u0000Mn3+con-\n\fgurations. It involves a simultaneous transfer of an electron from Mn3+to O2\u0000and from\nO2\u0000to Mn4+. But as the electrons in the itinerant band are coupled to the localized, t2g\nelectrons via a Hund's coupling, the localized electron at a site will favour an egelectron\nof parallel spin on the site.\n6It is well known that manganites have a large Hund's exchange. In the limit this is\nin\fnite, each of the 5 d-orbitals will be spin split, the `wrong spin' orbitals are never\npopulated (as there are only 3 or 4 electrons in 3d orbitals of Mn ion). In this limit, the\ndouble occupancy at each orbital is also irrelevant. It su\u000eces to work with three degenerate\nt2gorbitals and one egorbital.\nThe nanostructured manganites show unusual magnetic behavior, di\u000berent from the\nbulk. It has been observed in several manganites that the charge ordered, AFM manganites,\nwhen reduced to nanosize, develop ferromagnetic tendencies [9], presumably with the charge\norder also destabilized. Two possible scenarios have been put forward to `explain' this: (i)\nthe nanosize e\u000bectively increases the surface pressure, P\u0018S=R, whereSis the surface\ntension and Ris the radius of the nanograin, assumed spherical. This excess pressure is\nsupposed to destroy the charge order [10]. Pressure induced melting of charge and AF\norder has indeed been seen in bulk manganites. (ii) The enhancement of FM state comes\nfrom intrinsic causes [3, 4, 5]. The reconstruction at the surface of a nanograin reorganizes\nelectronic states and favours double exchange. This would favour the FM tendencies over\nthe superexchange between Mn ions and, at su\u000eciently small sizes, completely destabilize\nAFM order. The second view is emboldened by the observation [5] that the excess pressure\non a typical nanograin is about 2-3 GPa, too low to melt the charge and AFM order.\nBesides, recent neutron scattering experiments [11] show no observable strain e\u000bects in\nbulk LaCaMnO 3up to about 30 GPa pressure.\n3.2 Model and calculation\nFollowing the discussions above, we use a single band (coming from the lone egorbital)\nHamiltonian coupled to a reservoir at each site. The basic manganite Hamiltonian is an\nitinerantegelectron system coupled to a localized t2gelectrons via Hund's coupling and a\nsuperexchange interaction between the localized electrons (leading to antiferromagnetism).\nThe localized t2gelectrons are treated as classical spins of magnitude S= 3=2 pointing at\nan angle\u0012with the spin quantization axis (taken z-axis here). A schematic of this model\nis depicted in Fig. 2. The Hund's coupling is taken as JH!1 . In this limit, the hopping\nintegral is modi\fed by the projection of spin at site ionto its nearest neighbour j[13]. The\n7(a)\n (b)\nFigure 2: (a) The various levels at each site : The bottom t2glevel with a localised classical\nspin, the middle itinerant eglevel coupled to t2glevel via Hund's coupling and the top reservoir\nlevel coupled to the middle level. (b)The various states in Monte Carlo simulations, with the\nlocalised spin azimuthal angle \u0012icolour-coded in [0;\u0019]. The \fgures indicate, from left to right, a\nferromagnetic, phase segregated and an antiferromagnetic state, obtained as Jis increased for a\ngiven\u0016.\noverall Hamiltonian is\nH=X\n<i;j>\u0012\n\u0000tcos\u0012\u0012i\u0000\u0012j\n2\u0013\n\u0000\u000eij\u0016\u0013\ncy\nicj+\nVX\ni(cy\nidi+dy\nici) +\"DX\nidy\nidi+\n~JX\n<i;j>cos (\u0012i\u0000\u0012j) (3)\nwhereci\u001b=cy\ni\u001bare the annihilation/creation operators for electrons in the band, di\u001b=dy\ni\u001bare\nthe operators for the reservoir, Vis the coupling to the reservoir, \"Dis the position of the\nreservoir with respect to the Fermi level and ~J=9\n4Jis the superexchange parameter. In\ncomputation, all parameters are normalized by the hopping parameter t.\nFor this o\u000b-diagonal disordered Hamiltonian we use a hybrid Monte Carlo simulation [3]\nwhere the Fermionic part was solved by exact diagonalization and the annealing over clas-\nsical variables were performed by Metropolis algorithm. The simulations were carried out\non a 12\u000212 square grid with periodic boundary condition. A vector \u0002 = [ \u00121;\u00122;:::\u0012N]\nuniformly distributed in [0 ;\u0019] is chosen to start with, where \u0012iis the azimuthal angle of the\nlocalized classical spin at site i. At each step, two \u0012i's were modi\fed by a random amount\nin\u0002\n\u0000\u0019\n16;\u0019\n16\u0003\nand the Hamiltonian was diagonalized for this new \u0002 vector. The choice be-\ntween the new and the old states was done using the Metropolis-Hastings algorithm. The\nsystem was annealed in this fashion from \f= 1 to\f= 25 in 100,000 iterations. The\n8spin-spin correlation and free energy was averaged out over a further 50,000 iterations.\n3.3 Results\nWe obtain the results for both with and without the reservoir. In order to obtain the phase\ndiagram, simulations were carried out for various parameter values. Typical magnetic\ncon\fgurations that appear in the ground states of the MC simulation are shown in Fig.\n2(b). Ferromagnetic, phase segregated and antiferromagnetic states are shown. It is not\nalways possible to delineate di\u000berent phases (particularly close to a phase transition). In\nthe theormodynamic limit, there will be a small splitting between the spin up and spin\ndown bands. However, for the small number of sites we work in (12 \u000212 = 144), a better\nquantitative method is to \fnd the nearest neighbor spin spin correlations, which is -1 and\n+1 for saturated AFM and FM, respectively. The phase boundary is given by hSiSji= 0.\nThe resulting phase diagram is plotted in Fig. 3(a) and Fig. 3(b). The value of \u0016is varied\nfrom\u00004 to 0, corresponding to zero to half \flling, i.e., the value of xvarying from 0 to 1\nin, say, La 1\u0000xCaxMnO 3.\nClearly the presence of a huge number of states leads to a situation where there could\nwell be degenerate (or nearly degenerate) solutions for the ground state. The competing\ninteractions of superexchange and double exchange favouring the AFM and FM correlations\nrespectively, lead to \frst order transitions and consequent phase segregation [3]. Indeed,\nsimilar situation obtains here too. As observed in the simulations, the phase transition\noccurs via a phase segregated state. It is worthwhile to mention the possibility of a canted\nspin state [14]. From a mean \feld calculation, it was shown that the AFM-FM transition\nin the double exchange Hamiltonian should proceed through two continuous transitions\nvia a spin canted state. However, the canted state has never been found to be the ground\nstate in any simulation [3]. The transition is \frst order with a phase-segregated (two-phase\ncoexistence) region as we also con\frm.\nThe addition of reservoir introduces two new parameters, the coupling Vand the posi-\ntion of the reservoir, \"Dwith respect to the Fermi level. In all the calculations, the Fermi\nlevel and the position of the reservoir are held constant, and the coupling is turned on from\nV= 0 toV= 0:3. The spin-spin correlation is computed, for the two cases of zero coupling\n(which is equivalent to no reservoir) and with a \fnite coupling. These calculations were\nrepeated for various positions of reservoir with respect to the Fermi level. The variation in\nspin-spin correlation is calculated for \u0016= 0 (\flling = 0 :5) and\u0016=\u00001 (\flling = 0 :2), as\nplotted in Fig. 4(a) and 4(b), respectively.\n9(a)\n (b)\nFigure 3: Phase diagram of the manganite system in absence of reservoir on (a) J\u0000\u0016and\n(b)J\u0000Nplanes. The colour codes are same as in previous \fgure.\n(a)\n (b)\nFigure 4: Variation in the spin-spin correlation due to reservoir at (a) \u0016= 0and (b)\u0016=\u00001\n10On addition of the reservoir (with \"D=\u00001:0; V= 0:3) to the system below the Fermi\nlevel at half-\flling, the \flling of the band decreased from 0.5 to 0.47; electrons have been\ndrained to the reservoir. At half \flling, each site has one electron and delocalization is not\npossible. However, when holes are introduced in this system, the holes will delocalize all\nover the lattice, leading to signi\fcant kinetic energy gain (enhanced double exchange) and\nferromagnetism is enhanced in the ground state. As seen in Fig. 4(a), a comparison between\nthe dashed ( V= 0:3) and contunous lines ( V= 0) indicate enhanced FM correlation in\nthe former, indeed higher spin spin correlation imply more ferromagnetic tendency. The\ncorresponding location and broadening of the reservoir is also shown in the inset. The\npresence of the reservoir moves the peak of the band towards higher energy by V2=t,\nthereby leading to decrease of the \flling of band for \u0016= 0. Hence, the reservoir induces\nferromagnetism by introducing holes in the band. For the second case of \u0016=\u00001; \"D=\u00061\n(Fig. 4(b)), the reservoir acts as a donor and drives the system towards ferromagnetism\nagain. Here, as the original band-\flling is low (0 :2 in this case), the delocalization energy\ncan be increased only by addition of electrons. The reservoir now acts as a donor and\naccomplishes the same objective as above.\n4 Conclusions\nIt is shown that the charge transfer ferromagnetism model introduced by Coey et al. is\nable to predict the hugely enhanced ferromagetic order seen in the transition metal doped\noxide nanoparticles qualtitatively. More importantly, the observed enhancement of FM\ntendencies in nanoparticles of manganites also \fnds a resolution within the same framework.\nSurface magnetic probes like \u0016SR may be a possible tool that can establish the proposed\nscenario in manganites.\nReferences\n[1] J. M. D. Coey, P. Stamenov, R. D. Gunning, M. Venkatesan and K. Paul, New J. Phys. 12 (2010)\n053025.\n[2] J. M. D. Coey, Kwanruthai Wongsaprom, J. Alaria and M. Venkatesan, J. Phys. D: Appl. Phys. 41\n(2008) 134012.\n[3] E. Dagotto, T. Hotta and A. Moreo, Phys. Rep. 344 (2001) 1.\n[4] H. Zenia, G. A. Gehring, G. Banach, and W. M. Temmerman, Phys. Rev. B 71 (2005) 024416.\n11[5] S. Kundu, T. K. Nath, A. K. Nigam, T. Maitra and A. Taraphder, arXiv:1006.2943 (2010): Jour.\nNanoscience and Nanotech. (2011) to be published .\n[6] Vatsal Dwivedi, P. Stamenov, C. Porter and J. M. D. Coey, Unpublished results.\n[7] S. Jin, T. H. Tiefel, M. McCormack, R. A. Fastnacht, R. Ramesh and L. H. Chen, Science 264 (1994)\n413.\n[8] Colossal Magnetoresistive Oxides edited by Y.Tokura, Gordon and Breach Science, Singapore, 2000.\n[9] S. S. Rao, K. N. Anuradha, S. Sarangi, and S. V. Bhat, Appl. Phys. Lett. 87, 182503 (2005).\n[10] Tapati Sarkar, A. K. Raychaudhuri, and Tapan Chatterji, Appl. Phys. Lett. 92, 123104 (2008).\n[11] Z. Jir\u0013 ak, E. Hadov\u0013 a, O. Kaman, K. Kn\u0013 \u0010\u0014 zek, M. Mary\u0014 sko, E. Pollert, M. Dlouh\u0013 a and S. Vratislav,\nPhys. Rev. B 81 (2010) 024403.\n[12] C. Zener and R. R. Heikes, Rev. Mod. Phys. 25 (1953) 191.\n[13] P. W. Anderson and H. Hasegawa, Phys. Rev. 100 (1955) 675.\n[14] P. G. de Gennes, Phys. Rev. 118 (1960) 141.\n12"    },    {        "title": "0903.2245v2.Magnetic_screening_properties_of_superconductor_ferromagnet_bilayers.pdf",        "content": "arXiv:0903.2245v2  [cond-mat.supr-con]  28 Sep 2009Magnetic screening properties of superconductor-ferroma gnet bilayers\nManuel Houzet1and Julia S. Meyer2,3\n1CEA, INAC, SPSMS, F-38054 Grenoble, France\n2Department of Physics, The Ohio State University, Columbus , Ohio 43210, USA\n3Universit´ e Joseph Fourier, F-38041 Grenoble, France\n(Dated: May 25, 2018)\nWe study theoretically the magnetic screening properties o f thin, diﬀusive superconduc-\ntor/ferromagnet bilayers subject to a perpendicular magne tic ﬁeld. We ﬁnd that the eﬀective pene-\ntration depth characterizing the magnetic response oscill ates with the thickness of the ferromagnetic\nlayer on the scale of the ferromagnetic coherence length.\nPACS numbers: 74.25.Nf,74.45.+c,74.78.-w\nWhile superconductor-normal metal (SN) struc-\ntures have been intensively studied for decades,\nsuperconductor-ferromagnet (SF) structures have only\nbecome accessible recently because of the much reduced\nlength scales in ferromagnets. Due to their incompatible\nspin properties, the proximity eﬀect between a singlet su-\nperconductor and a ferromagnet leads to a variety of un-\nusual phenomena1,2. Through the exchange ﬁeld acting\nontheelectronspinsinthe ferromagnet,Cooperpairsac-\nquire a ﬁnite momentum δkwhich leads to an oscillatory\nbehavior of the anomalous Green function3. Observable\nconsequences are, e.g., a non-monotonic dependence of\nthe transition temperature4,5,6and the density of states\nat the Fermi level7,8on the thickness of the F layer in\nSF bilayers, and the possibility of π-Josephson junctions\nat certain thicknesses of the F layer in superconductor-\nferromagnet-superconductor (SFS) trilayers9,10,11.\nWhile most experiments on hybrid systems use resis-\ntive measurements, screening of an external magnetic\nﬁeld oﬀers an alternative tool to study the proximity\neﬀect. These measurements probe deeply into the su-\nperconducting state because they provide both the mag-\nnitude and temperature dependence of the eﬀective su-\nperﬂuid density. Various conﬁgurations for the magnetic\nresponse can be considered. The magnetization of SN\nhybrids with a magnetic ﬁeld applied parallel to their in-\nterface has been addressed theoretically in Ref.12, with\nstill debated experimental results in the case of SN cylin-\nders13,14. Alternatively, the screening properties of thin\nﬁlms can be probed by measuring the mutual inductance\nof two coils positioned on opposite sides of the sam-\nple15,16. The mutual inductance can be related to the\ncomplex conductivity of the ﬁlm which in turn can be\nrelated to the screening length λor the superﬂuid den-\nsityρS. To be precise, in SF bilayers, one measures the\nsuperﬂuid density ρS∝λ−2integrated over the width of\nthe bilayer or an eﬀective screening length\nλ−2\neﬀ=d−1\nS/integraldisplaydF\n−dSdx λ−2(x), (1)\nwheredSanddFare the thicknesses of the supercon-\nducting and ferromagnetic layer, respectively, and xis\nthe coordinate normal to the interface. First experimen-tal results on Nb/Ni bilayers have been reported using\nthis setup in Ref.17, where a non-monotonic dependence\nof the eﬀective screening length λ−2\neﬀon the thickness of\nthe Ni layer has been observed.\nIn this paper, we study the screening length λeﬀof\na SF bilayer subject to a weak perpendicular magnetic\nﬁeld. The main assumptions are that (i)the exchange\nﬁeldhin the ferromagnet is much larger than the super-\nconducting order parameter ∆, (ii)the system is in the\ndirty limit and, thus, the Usadel equation18can be used,\n(iii)the screening length λeﬀis much larger than the\nthickness d=dS+dFof the bilayer, and (iv)the width\ndSof the superconducting (S) layer is smaller than the\nsuperconducting coherence length ξS=/radicalbig\nDS/(2πTc0),\nwhereDSis the diﬀusion constant and Tc0is the transi-\ntion temperature of the bare S layer. Our main result is\nthat the screening length displays an oscillatory behavior\nwith the thickness of the ferromagnet.\nDuetothenormalizationcondition ˆ g2= 1ofthequasi-\nclassical Usadel Green function ˆ g, it can be parametrized\nby an angle θsuch that the normal Green function G=\ncosθwhereas the anomalous Green function F= sinθ.\nThe system is then described by four coupled equations\nin terms ofthe angles θSon the S side ofthe SF interface,\nθ0on the ferromagnetic (F) side of the SF interface, and\nθFat the ferromagnet-vacuum interface.\nThe Usadel equation of the F layer, −DF∇2θ+\n2ihsinθ= 0, can be integrated to yield\n2√\niy=/integraldisplayθ0\nθFdθ1√cosθF−cosθ, (2)\nwherey=dF/ξFandξF=/radicalbig\nDF/his the ferromag-\nnetic coherence length, with the diﬀusion constant DFof\nthe F layer. The boundary condition imposing current\nconservation at the SF interface19can be expressed as\nsin(θS−θ0) = 2√\niβ/radicalbig\ncosθF−cosθ0,(3)\nwhereβ=RbσF/ξF. HereRbis the interface resistance\nper square, and σFis the conductivity of the F layer. In\nthe limit dS≪ξS, the Usadel equation of the S layer,\n−DS∇2θ+ 2ωsinθ= 2∆cos θ, whereωis a fermionic\nMatsubara frequency, can be simpliﬁed by an expansion2\nin smallspatialvariationsoftheangle θacrossthe Slayer\ncombined with the boundary condition (3). One obtains\nωsinθS+2√\niα/radicalbig\ncosθF−cosθ0= ∆cos θS,(4)\nwhereα=DSσF/(2σSdSξF) andσSis the conductivity\nof the S layer. Finally, the self-consistency equation for\nthe order parameter ∆ reads\n∆ =πTλBCSℜ/bracketleftBigg/summationdisplay\nωsinθS/bracketrightBigg\n, (5)\nwhereλBCSis the BCS coupling constant.\nIn diﬀusive superconductors, the screening length\nλdescribes the local (London) current response20to\na vector-potential, j=−1/(µ0λ2)A, where λ−2=\n(2πTµ0σS//planckover2pi1)/summationtext\nωF2is proportional to the superﬂuid\ndensity. Here µ0isthe vacuumpermeability. In SFbilay-\ners, the eﬀective screening length is related to the angles\nθthrough the equation\n1\nλ2\neﬀ=2πTµ0σS\n/planckover2pi1ℜ\n/summationdisplay\nω/parenleftbig\nsin2θS+γy/integraldisplay\n0dxsin2θ(x)/parenrightbig\n,(6)\nwhereγ=σFξF/(σSdS). Using the Usadel equation\nof the F layer, the integral over xcan be traded for an\nintegral over θ, namely dx=1\n2√\ni(cosθF−cosθ)−1/2dθ,\nranging from θFtoθ0.\nIn general the set of equations (2)-(5) can be solved\nnumerically only, but in some limiting casesan analytical\nsolution is possible. At T= 0, a simpliﬁcation occurs\nbecause the sums over ωcan be replaced by integrals,\nand subsequently the integration over ωcan be traded\nfor an integration over θSusing the Usadel equation21.\nIt is then suﬃcient to solve the Usadel equation at ω=\n0 forθS(0). In particular, using Eqs. (2) and (3), the\nUsadel equation of the S layer (4) can be brought into\nthe form ( ω+F(θS))sinθS= ∆cosθS, yielding dω=\n−(∆/sin2θS+F′(θS))dθS. Using this trick, the zero\ntemperature gap ∆ is given as\nln∆\n∆0=ℜ\nlntanθS(0)\n2+θS(0)/integraldisplay\n0dθSF′(θS)sinθS\n,(7)\nwhere ∆ 0is the zero-temperaturegap of the bare S layer,\na result which can then be used to compute λ−2\neﬀ.\nIn the following, we provide analytical results for the\neﬀective screening length in two limits, namely (i)for\na system without barrier β= 0 and (ii)for a system\nwith a strong barrier β≫1. In both cases, solutions are\npresented for small parameters α. For convenience, we\nintroduce the notation ˜ x= (1+i)xforx=α,β,y.\nIn the absence of a barrier β= 0, the boundary condi-\ntion (3) imposes that the angles on both sides of the SF\ninterface, θ0andθS, are the same.IfdF≫ξF, the angle θFis small, and Eq. (2) yields\nθF= 8tanθS\n4exp[−˜y]. Thus, we can simplify Eq. (4) to\nyield\nωsinθS+2˜αsinθS\n2/parenleftBigg\n1−8e−2˜ytan2θS\n4\nsin2θS\n2/parenrightBigg\n= ∆cosθS.(8)\nTreating α≪∆0perturbatively, one ﬁnds θS(0) =π\n2+\nδθS, where δθS=−√\n2(˜α/∆0)/parenleftbig\n1−16e−2˜y(3−2√\n2)/parenrightbig\n,\nand\nδ∆ =−2α/parenleftBig√\n2−ln(1+√\n2)/parenrightBig\n(9)\n−16\n3ℜ/bracketleftbig\n˜αe−2˜y/bracketrightbig/parenleftBig\n3ln(1+√\n2)+4−5√\n2/parenrightBig\n.\nThis solution describes gapless superconductivity with\na ﬁnite density of states ν(0) at the Fermi level in the\nsuperconductor that oscillates with the thickness of the\nferromagnet: ν(0) =−ν0ℜ[δθS], where ν0is the density\nof states in normal state. The equation for the screening\nlength takes the form\nλ2\n0\nλ2\neﬀ(0)= 1−/parenleftbig\na1−a2(cos2y+sin2y)e−2y/parenrightbigα\n∆0(10)\n+2√\n2\n3πγ/parenleftbigg\n1+a3ye−2ycos2y−a4α\n∆0/parenrightbigg\n,\nwhereλ−2\n0=πµ0σS∆0//planckover2pi1is the inverse screening length\nof the bare S layer at zero temperature, and the coeﬃ-\ncientsaiare positive22. Both, the contributions to the\neﬀective screening length from the S layer and from the\nF layer ( ∝γ), oscillate on the length scale of the ferro-\nmagnetic coherence length.\nIf on the other hand dF≪ξF, the variation of the\nangleθis small acrossthe F layerand, thus, θS−θF≪1.\nUsing Eq. (2), one obtains θS=θFcosh˜y. Inserting this\nrelation into the Usadel equation of the S layer results in\n/parenleftbigg\nω+2iαy+4\n3αy3cosθS/parenrightbigg\nsinθS= ∆cos θS.(11)\nWe ﬁnd δθS=−2iαy/∆0andδ∆ =−π\n3αy3. Note that\nbecauseθS(0) =π\n2+iφ, whereφreal,thedensityofstates\npossesses a gap in this regime. Using Eq. (11) to convert\nthe integral over ωto an integral over θS, the screening\nlength is given by\nλ2\n0\nλ2\neﬀ(0)= 1−π\n3(1+16\n3π2)α\n∆0y3+γy.(12)\nEq. (12) predicts an increase in λ−2\neﬀas long as dF<\nξ2\nF/ξSbefore it starts to decrease. The regime dF∼\nξFconnecting the results Eq. (10) and (12) is treated\nnumerically (see below).\nIn the opposite limit of a strong barrier, β≫1, both\nθ0andθFare small, if the F layer is not too thin, y≫\nβ−1. Eq. (2) then yields θ0=θFcosh˜y, and, using the3\nboundary condition, the Usadel equation of the S layer\ncan be rewritten as\n/parenleftbigg\nω+α\nβ−α√\n2iβ2tanh˜ycosθS/parenrightbigg\nsinθS= ∆cosθS.(13)\nEq. (13) yields δθS=−α/(β∆0), and thus no gap in thedensity of states, ν(0) =ν0α/(β∆0), while\nδ∆ =−α\nβ+πα\n8β2ℜ[(1−i)coth˜y]. (14)\nThe screening length is given as\nλ2\n0\nλ2\neﬀ(0)= 1−(1+2\nπ)α\nβ∆0+(π\n8+2\n3π)α\nβ2∆0ℜ[(1−i)coth˜y]−γ\n16β2ℜ/bracketleftbigg4iy+(1+i)sinh2˜y\nsinh2˜y/bracketrightbigg\n.(15)\nAgainthe eﬀectivescreeninglengthoscillatesonthescale\nofξF. However, these oscillations are suppressed due to\nthe large barrier. For y≫1, the oscillatory function in\nthe contribution of the S layer has the same form as the\none in (10) whereas the oscillating part of the contribu-\ntion from the F layer is proportional to ye−2ysin(2y).\nIn the case of a very thin F layer, y≪β−1, the varia-\ntion of the angle θis small across the F layer, see above.\nThe boundary condition at the SF interface then simpli-\nﬁes, and the Usadel equation of the S layer yields\n/parenleftbig\nω+2iαy+4αβy2cosθS/parenrightbig\nsinθS= ∆cos θS.(16)\nWe ﬁnd δθS=−2iαy/∆0andδ∆ =−παβy2. As for\nthe case without a barrier, the density of states in the\nthin ﬁlm regime is gapped. Using Eq. (16) to convert\nthe integral over ωto an integral over θS, the screening\nlength is given by\nλ2\n0\nλ2\neﬀ(0)= 1−π(1+16\n3π2)αβ\n∆0y2+γy.(17)\nThe inverse screening length increases in the very narrow\nregimedF< ξ3\nF/(βξ2\nS).\nThus, we ﬁnd oscillations of the screening length both\nin the absence of a barrier and in the presence of a strong\nbarrier. The amplitude of oscillations in the latter case is\nsuppressed, however. In both cases, analytic results can\nbe found for small thicknesses y < y∗and large thick-\nnessesy > y∗, wherey∗∼min{1,β−1}denotes the po-\nsition of the ﬁrst strong minimum. The vicinity of this\nminimum is not accessible to analytic solution.\nTo ﬁnd a solution in this regime, we note that Eq. (2)\nyields a general relation between θ0andθF, namely\nsinθ0\n2=1/radicalBig\n1+cot2θF\n2cn2/parenleftbig\n˜y,cos2θF\n2/parenrightbig,(18)\nwhere cn is the Jacobi elliptic function. Using Eq. (18),\nthe boundary condition (3) yields θSas a function of θF.\nInserting this solution into the other equations, the set\nof coupled Eqs. (4) and (5) can then be solved numeri-\ncally. The thickness dependence of the low-temperature0.0 0.5 1.0 1.5 2.0 2.5 3.0dF\nΞF0.00.20.40.60.81.0Λ02\nΛeff2/LParen1dF/RParen1\n2.02.53.03.54.00.1400.1450.1500.1550.160\nFIG. 1: Oscillation of the inverse screening length 1 /λ2\neﬀas\na function of dFat temperature T= 0.1Tc0. Here we use the\nparameters α= 1.2,β= 1, and γ= 0.6. The inset magniﬁes\nthe weak maximum at dF≈2.5ξF.\nscreening length is displayed in Fig. 1. The minimum at\ndF∼ξFis clearly visible whereas further oscillations at\nlargerdFare very small.\nFurthermore, the numeric solution allows one to de-\nscribe the temperature dependence of the screening\nlength. Fig. 2 shows the ﬁnite temperature curves for\ndiﬀerent values of dF/ξF. The oscillations of the zero-\ntemperature screening length mirror the oscillations of\nthe critical temperature aswell as the slope ofthe screen-\ning length close to Tc.\nIn the vicinity of the critical temperature Tcan an-\nalytic solution is possible for all parameter values. For\nthe SF bilayer, the critical temperature is given by the\nsolution of the equation1\nlnTc\nTc0= Ψ/parenleftbigg1\n2/parenrightbigg\n−ℜ/bracketleftbigg\nΨ/parenleftbigg1\n2+1\n2πTcτs/parenrightbigg/bracketrightbigg\n,(19)\nwhere the (complex) relaxation time τsreads\nτ−1\ns=√\n2iαtanh(√\n2iy)\n1+√\n2iβtanh(√\n2iy). (20)\nEqs. (19,20) are obtained by linearizing (2)-(5) in small4\nθclose to the transition. For τ−1\nssmall,Tc=Tc0−\nπ\n4ℜ/bracketleftbig\nτ−1\ns/bracketrightbig\n. For|τ−1\ns|= ∆0/2, the transition temperature\nvanishes according to Eq. (19). Note, however, that for\nlargeτ−1\nsthe transition typically becomes ﬁrst order23.Expansion of equations (2)-(5) up to cubic order\nthen yields the temperature dependence of the screen-\ning length λeﬀclose toTc. Namely,\n1\nλ2\neﬀ(T)=µ0σS\n/planckover2pi1πTc∆2(T)ℜ/bracketleftbigg/parenleftbigg\n1+γ\n4√\n2i2˜y+sinh(2˜y)\n(cosh˜y+˜βsinh˜y)2/parenrightbigg\nΨ(1)/parenleftbigg1\n2+1\n2πTcτs/parenrightbigg/bracketrightbigg\n. (21)\nWe see that the contribution of the F layer to λ−2\neﬀdis-\nplays an oscillatory dependence on its thickness. Fur-\nthermore, both Tcand ∆(T) oscillate. In particular,\n∆2(T)\nTc−T=4πTc/parenleftbig\n1−ℜ/bracketleftbig\n(2πTcτs)−1Ψ(1)(z)/bracketrightbig/parenrightbig\n−ℜ/bracketleftbig\nΨ(2)(z)−f(τs,β,y)Ψ(3)(z)/bracketrightbig,(22)\natT < T c, wherez=1\n2+(2πTcτs)−1and\nf(τs,β,y) =1\n48(2πTcτs)−11\n(1+˜βtanh˜y)3(23)\n×/parenleftbigg\n4(1+2˜βtanh˜y)2−2˜y+sinh(2˜y)\nsinh˜ycosh3˜y/parenrightbigg\n.\nNote that the simple relation λ−2∝∆2does not hold in\nthe presence of the F layer. The slope of λ−2\neﬀclose toTc\nhas its own dependence on the thickness of the F layer\nand the relaxation rate τ−1\ns.\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n0.000.050.100.150.200.250.300.35T\nTc00.000.050.100.150.20Λ02\nΛeff2/LParen1T/RParen1\n/SolidUpTriangledF/EquaΛ3.ΞF/MedSolidDiamonddF/EquaΛ1.3ΞF/SolidSquaredF/EquaΛ0.9ΞF/SolidCircledF/EquaΛ0.5ΞFFIG. 2: Temperature dependence of λ−2\neﬀ. The parameters\nused are the same as in Fig. 2, and four diﬀerent thicknesses\nare shown: dF/ξF= 0.5, 0.9, 1.3, and 3. The corresponding\ncritical temperatures are Tc(0.5) = 0.36Tc0,Tc(0.9) = 0.17Tc0\n(close to the minimum), Tc(1.3) = 0.23Tc0, andTc(0.5) =\n0.31Tc0(close to the asymptotic value for dF/ξF≫1).\nIn conclusion, we have shown that the screening length\nof SF bilayers displays a oscillatory dependence on the\nthickness of the F layer. Analytic solutions have been\nfound in various regimes and a general solution has been\ndetermined numerically. The obtained non-monotonic\ndependence of the screening length has been observed\nexperimentally17. Our method can be easily extended to\nmore complicated situations such as multilayers where\nunusual features of the proximity eﬀect have been pre-\ndicted1,2.\nAcknowledgments\nWe would like to acknowledge A. Buzdin and T.\nLemberger for many discussions. JSM thanks the\nCEA/INAC/SPSMS for hospitality.\n1A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n2F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys.77, 1321 (2005).\n3E. A. Demler, G. B. Arnold, and M. R. Beasley, Phys. Rev.\nB55, 15174 (1997).\n4A. I. Buzdin and M. Yu. Kupriyanov, JETP Lett. 52, 487\n(1990).5Z. Radovic, M. Ledvij, L. Dobrosavljevic-Grujic, A. I.\nBuzdin, and J. R. Clem, Phys. Rev. B 44, 759 (1991).\n6J. S. Jiang, D. Davidovic, D. H. Reich, and C. L. Chien,\nPhys. Rev. Lett. 74, 314 (1995).\n7A. I. Buzdin, Phys. Rev. B 62, 11377 (2000).\n8T. Kontos, M. Aprili, J. Lesueur J, and X. Grison, Phys.\nRev. Lett. 86, 304 (2001).5\n9A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov, JETP\nLett.35, 178 (1982).\n10A. I. Buzdin and M. Yu. Kupriyanov, JETP Lett. 53, 321\n(1991).\n11V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V.\nVeretennikov, A. A. Golubov, and J. Aarts, Phys. Rev.\nLett.86, 2427 (2001).\n12A. D. Zaikin, Solid State Comm. 41, 533 (1982).\n13P. Visani, A. C. Mota, and A. Pollini, Phys. Rev. Lett 65,\n1514 (1990).\n14F. Bernd M¨ uller-Alinger and A. C. Mota, Phys. Rev. Lett\n84, 3161 (2000).\n15S. J. Turneaure, E. R. Ulm, and T. R. Lemberger, J. Appl.\nPhys.79, 4221 (1996).\n16S.J.Turneaure, A.Pesetski, andT. R.Lemberger, J. Appl.Phys.83, 4334 (1998).\n17T. R. Lemberger, I. Hetel, A. J. Hauser, and F. Y. Yang,\nJ. Appl. Phys. 103, 07C701 (2008).\n18K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n19M. Y. Kupriyanov and V. F. Lukichev, Sov. Phys. JETP\n67, 1163 (1988).\n20See e.g. P. G. De Gennes, Superconductivity Of Metals And\nAlloys(Addison Wesley, 1989).\n21I. Baladi´ e and A. I. Buzdin, Phys. Rev. B 67, 014523\n(2003).\n22The numerical values of the coeﬃcients are a1= 1.77,a2=\n3.82,a3= 6.44, anda4= 3.19.\n23S. Tollis, Phys. Rev. B 69, 104532 (2004)."    },    {        "title": "1104.2279v1.Properties_of_Ferromagnetic_Superconductors.pdf",        "content": "arXiv:1104.2279v1  [cond-mat.str-el]  12 Apr 2011Properties ofFerromagnetic Superconductors\nDai Aokia,Fr´ ed´ ericHardyb, AtsushiMiyakec,a, ValentinTaufoura, TatsumaD.Matsudad,a,\nJacquesFlouqueta\naSPSMS, UMR-ECEA /UJF-Grenoble 1,INAC, 38054 Grenoble, France\nbKarlsruher Institut f¨ ur Technologie, Institut f¨ ur Festk ¨ orperphysik, 76021 Karlsruhe, Germany\ncKYOKUGEN,Osaka University, Toyonaka, Osaka 560-8531, Jap an\ndAdvanced Science Research Center, Japan Atomic EnergyAgen cy, Tokai, Ibaraki 319-1195, Japan\nAbstract\nThanks to the discovery in the last decade of three uranium fe rromagnetic superconductors,\nUGe2, URhGe and UCoGe, the fascinating aspects of the interplay b etween the triplet state\nof Cooper pairing and ferromagnetism have emerged. Further more, as the ferromagnetic prop-\nerties in the normal state are quite di ﬀerent with respect to the proximity of the ferromagnetic–\nparamagnetic instabilities, the feedback with the coexist ence of superconductivity gives rise to\nquite diﬀerent boundaries in pressure and magnetic ﬁeld. Special att ention is given on the lo-\ncation of the materials with respect to the tricriticality a nd on the reinforcement of SC in a\ntransverse ﬁeld response with respect to the direction of th e FM sublattice magnetization. The\notherfactsoftheinterplaybetweenFMandSCis brieﬂymenti oned.\nKeywords:\nheavyfermion,unconventionalsuperconductivity,ferrom agnetism,UGe 2,UCoGe,URhGe\n1. Introduction\nThe discovery of superconductivity(SC) in the ferromagnet UGe2has opened a new chapter\nintheexoticdomainofunconventionalsuperconductivity[ 1]. Thetrendisthattheferromagnetic\n(FM) interaction between highly renormalizedquasipartic lesis the source of SC pairing. In the\nthreeIsingferromagneticsuperconductorsUGe 2,URhGe[2] andUCoGe [3],it appearsthatthe\nCooperpairscondensein the equalspin pairingstate (ESP) w ith↑↑and↓↓spin carriers. Inthis\nreview, we give a schematic view of the phenomenon. This arti cle is quite complementary to\nthe paper recently published in J. Phys. Soc. Jpn. for the 100 years of superconductivity [4].\nWe focuson temperature( T), pressure ( P) and magnetic ﬁeld ( H) phase diagrams, in particular\non the precise location of the FM and FM +SC phases, and the PM (paramagnetic)and PM +SC\nboundaries. In these compounds, the occurrence of SC is stro ngly related to the e ﬀective mass\nenhancementassociatedwiththeferromagneticinstabilit ywhichoccursinUCoGeatthecritical\npoint (Pc,T=0) where FM is collapsed, while in UGe 2two distinct ferromagnetic phases FM1\nand FM2 are separated by Px. The new feature of these Ising ferromagnets is that the ﬁeld\nresponseoftheFM–PMinstabilityisquiteanisotropicbetw eenH/bardblM0andH⊥M0,whereM0\nis the sublattice magnetization. Furthermore, when the sys tem moves towards Pc, the FM-PM\ntransition line, TCurie(P), becomes ﬁrst order and the occurrence of tricriticality a t (TTCP,PTCP)\nleadstotheexistenceofaﬁeldinducedFMphasewhichextend sbeyondPctillaquantumcritical\nendpoint(QCEP) at( PQCEP,HQCEP).\nPreprint submitted to Comptes Rendus Physique October 27, 2018Inthisarticle,ﬁrstwebrieﬂydescribethefeaturesofitin erantferromagnetismandSCpairing\nmediated by ferromagnetic ﬂuctuations and then we summariz e the normal-state properties of\nUGe2,URhGeandUCoGeandcommentontheirinﬂuenceontheappeara nceofSCatzeroﬁeld.\nWediscusstheoccurrenceoftricriticalityinUGe 2andtheHreinforced/reentrantSCfor H⊥M0\ninURhGeandUCoGeinthecontextoftheirlongitudinalandtr ansverseﬁeldresponses[5,6,7].\nInconclusion,we givea shortlist ofotheraspectsoftheint erplaybetweenFM andSC.\n2. Ferromagnetismin itinerantelectronicsystem\nA majorbreakthroughin the understandingof FM in itinerant systemsappearedin 1985with\nMoriya’sself-consistentrenormalization(SCR)theoryof spinﬂuctuationsinaHubbardscheme.\nIn this model, the Fermi liquid regime, which is characteriz ed by a speciﬁc-heat linear term\nγand aT2dependent resistivity below a temperature TI, collapses when the transition from a\nlong range FM order to a PM ground state occurs at a characteri stic pressure Pc. On the other\nhand, the non-Fermi liquid (NFL) regime between TIandTIIexpands before recovering a high\ntemperature domain TIII(see Fig. 1(a)) [8, 9]. Pressure often tunes the system from F M to PM\nsince it increases the electronic bandwidth Wand thus decreases the density of states N(εF).\nBelowPc,UN(εF) is larger than 1 while above Pc,UN(εF) is smaller than 1, where Uis the\nonsite Coulombrepulsionand N(εF) is the electronicdensityof state at theFermi level. Table 1\nsummarizesthe expected Pdependenceof TI,TIIandTCurietogetherwith the variationof TCurie\nwithM0.\nTable 1: Pressure dependence of TI,TII,TCurieand the relation between the sublattice magnetization M0andTCuriefor\n3D FM systems.\nTI TII TII/TI TCurie TCurie(M0)\nFM (P−Pc)3/2(P−Pc)3/4(P−Pc)−3/4(P−Pc)3/4M03/2\nTable2: Pressuredependence of γ,χQ=0andAnearPcfor 3DFM systems.\nγ χ Q=0 A\nFM log( P−Pc) (P−Pc)−1(P−Pc)−1\nTable3: Temperature variation of C/T,chiQ=0and resistivity for 3D and 2DFM systems.\nC/T1/χQ=0ρ∼Tn\nFM 3D−logT T4/3T5/3\nFM 2D T−1/3−TlogT T4/3\nBelowTI, the eﬀective mass m∗taken fromγ∼m∗kFdiverges as log( P−Pc), while the\nuniform susceptibility χ(0) and the resistivity inelastic term Adiverge as1\nP−Pc. In addition, the\n2temperature variation of C/Tandχ(0) and the exponent nof the resistivity term depend on the\nmagneticdimensionalityas shownin Table 3. However,there is a majordiﬀerencebetweenFM\nand antiferromagnetic(AFM) quantumcritical points. For t he latter no divergenceof m∗occurs\natPcandγvaries asγ0(P−Pc)1/2. The (P−Pc)1/2singularity leads to a divergence of the\nGr¨ uneisen parameter Ωe=−∂logγ/∂logVatPc. For a second-orderphase transition, both the\nentropy(S)andthethermalexpansion( ∂S/∂P)collapseat Pc(Fig.1(a)).\nForFM systems, it iswell establishedbothexperimentally[ 10] andtheoretically[11] thatthe\ndivergence of m∗atPcis inhibited by the occurrence of a ﬁrst-order transition at Pc(Fig. 1(b))\nwhich is characterized by discontinuities ∆M0,∆V0inM0and volume V0, respectively. As the\nentropyreacheszeroat P=PcaccordingtotheClausius-Clapeyronrelation( dP/dT=∆S/∆V),\nthe initial ( T,P) line at very low temperature must be vertical. If ∆M0is small, the quantum\nphase transition at Pcwill only be weakly ﬁrst-order and strong ﬂuctuations will p ersist, being\nalmost like the second orderphase transition. Thusfor a str ong ﬁrst ordertransition(large ∆M0\nandlarge∆V) thereisa largediﬀerencebetween Pcand˜Pc.\nThe SCR theory was developed for 3D itinerant magnets and ext ended to the case of heavy\nfermion systems (HFS) with the simple idea that the bandwidt hWis renormalized to a Kondo\nenergykBTKcharacteristic of the strong local nature of the magnetism a nd its ﬂuctuations[12].\nThisleadstoastronglyrenormalizedbandmass mBandafurtherenhancementof m∗∗duetothe\nFM quasiparticle interactions [13]. Very often mBandm∗∗have comparable amplitudes. Thus,\ntheimageofinterferingquasiparticlesisthatofinterfer ingwaveswithalargedi ﬀractionpattern\ngivenbythestronglocal characterofthemagnetism.\nIn the case of cerium HFS, the e ﬀect of pressure is to switch the system from a magnetically\nordered state to a PM ground state. This is due to the strong Pincrease of the Kondo energy\nkBTKincomparisonwith theindirectintersitecoupling,givenb ytheRuderman,Kittel, Kasuya,\nYosida(RKKY)interaction. PressuredrivestheCesystemsf romatrivalentconﬁguration(witha\n4f-shell occupationnumber nf∼1)toa tetravalentconﬁgurationwith nf∼0. Accordingtothe\n4felectron-holesymmetry, nfcanvaryfrom nf=14(Yb2+)tonf=13(Yb3+)inytterbiumHFS\nand magneticgroundstates appearunder pressure [9]. For ur aniumcompounds,it is di ﬃcult to\npredict the pressure dependence of TCuriebecause the ﬂuctuations now occur between the two\nmagneticconﬁgurationsU3+andU4+.\n3. Cooperpairingandferromagnetism\nSoon after the elaboration of the BCS theory of s-wave superconductivity, [14] the problem\nof coexistence of SC and FM was discussed by V. Ginzburg. He no ticed that ﬁnding SC in\nferromagnetsisasprobableasﬁndingnon-ferromagneticSC inlargemagneticﬁelds[15]. How-\never, the relevance of FM spin ﬂuctuations for SC was pointed out in 1966 [16]. The exis-\ntence of an anisotropic BCS state was illustrated by the p-wave superﬂuidity observedin liquid\n3He [17, 18, 19]. p-wave SC transitions for paramagnon mediated SC in nearly FM systems\nwere ﬁrst calculated by Layzer and Fay in 1971 [20]. However, it is only in 1980 that Fay and\nAppel publishedthe ﬁrst paperconcerningthe variationof TscthroughPcin the limited context\nofthesocalledequalspinpairing(ESP)statewith ↑↑and↓↓quasiparticles(Fig.1(b))[21]. The\nESP interactionwith ↑↑and↓↓componentsof the triplet channel with an angularmomentum q\nisrelatedtothe noninteractingLindhardresponseofthesp inχ↑\n0andχ↓\n0bytherelation:\nV↑↑=∆2χ↓\n0\n1−U2χ↓\n0(q)χ↑\n0(q)(1)\n3TomediateSCwitha ↑↑minority-spincomponent,amajority-spincomponent ↓↓isrequired. As\nshown in Fig. 1(b), the ↑↑minority-spin carriers ﬁrst condense in the FM state and SC c orre-\nsponds to a two-band model. In the PM region, both components condense at the same critical\ntemperature. However,ifFMabruptlydisappearsthroughaﬁ rst-ordertransitionat Pcinsteadof\n˜Pc, itis thenclearthat thesingularityat Pccouldbesuppressed(Fig.1(b)).\nInthetheoryofFayandAppelperformedforthesecondordert ransition,thesuperconducting\ncritical temperature Tscisdescribedby\nTsc=ωcexp/parenleftBigg\n−1+λz\nλ∆/parenrightBigg\n, (2)\nwhereλzis the renormalized-mass parameter and λ∆is the interaction parameter. ωc, which is\nbasicallyproportionalto TI,vanishesat Pc. Closeto Pc,thisformuladiﬀersfromthewell-known\nMcMillan-likeformula,\nTsc∼T0exp(−1/λ), (3)\nwith\nλ=λ∆\n1+λz, (4)\nand where T0is a characteristic cuto ﬀenergy. Outside around Pc, they are basically the same.\nForURhGe,a simplerexpressionwaschosen,[13]\nTsc∼T0exp/parenleftBigg\n−m∗\nm∗∗/parenrightBigg\n(5)\nm∗=mB+m∗∗, (6)\nwhere the quasiparticle e ﬀective mass m∗is the sum of the band mass mBand the correlation\nmassm∗∗. Here 1+λz=m∗/m0andλ∆=m∗∗/m0, wherem0is the free electron mass. Further\ncalculations of Tsc(P) show that Tschas only weak minima at Pc[22]. Additional discussions\nconcerningthecoexistenceofFMandSCcanbefoundinRefs.[ 23,24,25,26,27]. Calculations\ninthePMsideof Pc,forAFandFMinteractions,wereperformedusingtheEliash bergformalism\nforthequasi-2Dand3Dcases[28]withspeciﬁcapplications tocubicandtetragonalsymmetries\nas a functionof the electronic or magneticanisotropy [29]. In general, a spin singlet is favored.\nHowever,foratripletstate,pairingisonlycausedbylongi tudinalﬂuctuationswhereastransverse\nﬂuctuations are pair-breaking and impurity scattering is s trongly enhanced at Pc[30]. Thus, it\nis notsurprisingthat triplet SC in ferromagnetshasonly be endiscoveredin Ising ferromagnets.\nFinally, the possible SC order parameters in ferromagnetic materials have been classiﬁed using\ngeneralsymmetryargumentsforcubicandorthorhombicstru ctures[31, 32,33].\n4. Threeferromagneticsuperconductors: UGe 2, URhGeandUCoGe\nSC was discovered in the three uranium ferromagnets UGe 2, URhGe and UCoGe where a\nstrong 5felectronic component exists at the Fermi energy in the densi ty of states. Thus, the\n5felectronsarestronglydelocalized. Figure2summarizesth eirmaincharacteristicparameters.\nFor UGe 2, FM appearsat a ratherhightemperature, TCurie∼52K,whichis quitecomparableto\ntherenormalizedbandwidth W. TheexistenceoftheFermisurfacewillbefeltbelow T∼W/10.\nAt ambient pressure the speciﬁc heat exhibits a clear jump at TCurie, as shown in Fig.3(a) [34].\nAtTx∼25K, a crossover occurs between two interfering FM phases; F M2 with a sublattice\n4magnetization M0≈1.5µBandFM1 with M0≈1µB[35]. SC appearsonlyunderpressurewith\na maximum Tmax\nsc∼0.7K, at the pressure Px∼1.2GPa, where the ground state switches from\nFM2toFM1[36]. Aclearspeciﬁc-heatanomalywasdetectedat Tx(P)forP∼Px[37]. Finally,\nFM disappearsat Pc∼1.49GPa.\nURhGe with TCurie=9.5K is of great experimental interest because it becomes supe rcon-\nducting with Tsc=0.27K at ambient pressure (Fig. 3(b)). Thus, it o ﬀers the possibility of\napplying a larger variety of experimental methods for under standing its superconducting prop-\nerties.TCurieappears well below the characteristic temperature related to the bandwidth ( W).\nThelow-temperatureSommerfeldcoe ﬃcientisequalto 160mJ /K2mol[7, 38]. With increasing\npressure, TCurieincreases while TSCdecreases [39]. Thus, URhGe can be considered as a good\nexampleoftheinterplaybetweenSC andFM whichisfarfromth ecritical regimearound Pc.\nAgain,TCurieismuchlowerthan WinUCoGe. However,thespeciﬁcheatanomalyat TCurie∼\n3K, shown in Fig. 3, is very broad and highly dependent on the s ample purity. In fact, there is\nevidencefrom NMR measurementsthat the transitionis indee dﬁrst order[40] and the speciﬁc-\nheat anomalyresults from the discontinuouschange in entro pyatTCurieand strong ﬂuctuations,\nwhichindicatethatthemagneticcoherencelengthremainsc onstantoveranextended Twindow.\nUCoGe is an unique example of ferromagnetic superconductiv ity at ambient pressure with a\nrather small ordered moment M0∼0.05µB. With increasing pressure, TCuriedecreases and\nvanishesat PcwhileTscinitiallyraisesandexhibitsabroadmaximumat Pc[41,42,43]. Another\nunique feature of UCoGe is its low carrier concentration whi ch implies that the contribution of\nthe Co3dstatesto thedensityofstates isnotnegligibleat theFermi energy[44, 45].\nAsthesecompoundsallhaveanorthorhombicstructure,itis interestingtostudytheirthermal-\nexpansioncoeﬃcientsalongthethree principalaxes a,bandc, [34, 38, 46, 7] whicharerelated\nto the uniaxial pressure derivative of TCurievia the Ehrenfest and Clausius-Clapeyron relations\nfor second-order and ﬁrst-order transitions, respectivel y. As illustrated in Fig. 4, the thermal-\nexpansioncoeﬃcientsalongthethreeaxesdonotshowthesamevariationwit huniaxialpressure\natTCurie. Forthesethreecompounds,alargenegativedropof αbisobservedat TCurie. Itmustbe\nnotedthatforUGe 2,thecrossoverregimeat Tx∼25KfromFM1toFM2ismarkedbyextrema\ninαaαbandαcwhich do not coincide in position. Above the critical pressu rePc, where the\nsystemswitchesfromFM1toFM2througharealﬁrstordertran sition,thejumpsmeasuredalong\nthethreeaxishavetooccuratthesametemperature[47]. InU CoGe,thethermalexpansionwas\nalsomeasuredbelow Tsc. Thevolumechangesat TCurieandTscareoppositeinsign,asobserved\ninotherhighlyanisotropicmaterialslikeURu 2Si2wherethevolumechangesatthehiddenorder\ntransitionand Tscareoppositein sign,aswell.\nTheratioofthevolumethermal-expansioncoe ﬃcienttothespeciﬁcheatgivestheopportunity\nto calculate the electronic Gr¨ uneisen parameter Ωe(T). Above TCurie, the three compoundshave\na positive Gr¨ uneisen coe ﬃcient: the pressure derivative of the entropy, dS/dPis negative. For\nURhGe,thissignremainsthesameoncoolingthrough TCuriesinceTCurieincreaseswithpressure.\nHowever,a sharp sign changeoccursforUCoGe and UGe 2(dS/dPbecomespositive)in excel-\nlent agreement with the observation that TCuriecollapses at 1GPa and 1 .5GPa in UCoGe and\nUGe2, respectively. In UGe 2, it is interesting to remark that TCurieis comparable to Wand that\nthe electronicGr¨ uneisenparameterin thePM phaseis quite close to zero. ForUCoGe, Ωe(T)is\nalready large and temperature independent above TCuriewith a value quite similar to that of the\nintermediatevalenceCe compounds.\n55. (T,P)phase diagram: interplayofSC, PMand FM\nFigure 6 showsschematic ( T,P)phase diagramsof UGe 2, URhGe and UCoGe. In UGe 2, SC\nis squeezed between the two ﬁrst-order transitions at PxandPc[1, 36, 48]. The robust ﬁrst-\nordernatureofthesetransitionsmakesit di ﬃculttoestablishwhetherSCexistshomogeneously\nin the FM2 and PM phases and a deﬁnite conclusion is still unde r debate. Furthermore, the\nFermi surfacechangesbetween theFM2 and the PM states [49, 5 0]. Two diﬀerentmodelswere\nproposed to explain the maximum of TscatPx. In the ﬁrst one, SC is mediated by the charge\ndensity wave or spin density wave (CDW /SDW) ﬂuctuations at Px[51] while the second one\ninvokes a twin-peak structure in the electronic density of s tates [52]. No extra superstructures\nwereobservedat Px. ThetransitionfromFM2 toFM1 seemsrestrictedto aswitch b etweentwo\nFMstateswithconsequenceson λzandλ∆reproducingratherwellthepressurevariationof Tsc.\nIn URhGe, the situation correspondsto the behavior predict edin the FM domainfor P≪Pc\nwiththeparticularitythat TCurieincreaseswith PwhileTscdecreasesanddisappearsabove4GPa\n(Fig.6(b))[53,54].\nInUCoGe, TCurieandTsctendtomergewithincreasingpressure[41,43]andtheFMano maly\nis no longer detected in resistivity and susceptibility mea surements when Tsc≈TCurie. Thus,\nTCurieseems to collapse suddenly under pressure leaving a wide max imum in the pressure de-\npendence of Tsc. At least, the observationof the SC anomalyin the PM side ind icates that bulk\nsuperconductivityexistsinthePM domain[42].\n6. Longitudinaland transversemagneticﬁeld response\nIntheseIsingferromagnets,themagneticﬁeldleadstoapar ticularresponsewhenitisapplied\neither parallel or perpendicular to the initial sublattice magnetization M0(oriented along the a\naxisforUGe 2andalongthe caxisforURhGeandUCoGe).\nInUGe 2,theFMtransitionat TCurieatambientpressureisofsecondorderandtheapplication\nof a magnetic ﬁeld parallel to M0weakens the FM correlations. Thus, the FM speciﬁc-heat\nanomaly is rapidly reduced and shifts to higher temperature with increasing H;TCurieseems to\nincrease with Hbutγreaches the band-mass value γBwhen the ﬁeld strength is comparable to\nthe molecularﬁeld. InUGe 2, thismolecularﬁeld isverylarge ∼200T.\nHowever, the nature of the transition at TCuriechanges under pressure from second to ﬁrst\norder at the tricritical point ( TTCP,PTCP) [55, 56]. When the ﬁeld is applied along the sublattice\nmagnetization M0, the occurrence of this tricriticality gives rise to in-ﬁel d FM wings that open\nat the TCP and terminate at quantum critical end-points loca ted at (PQCEP,HQCEP) forT=0\n(see Fig. 7). The pressure di ﬀerencePQCEP−Pcis related to the pressure di ﬀerence˜Pc−Pc\nwhich correlateswith the jump ∆M0observedat Pc. UGe2representsan ideal case forstudying\nFM tricriticality since ∆M0∼0.9µBis large and the TCP ( TTCP=24K,PTCP=1.42GPa) and\nthe QCEP ( PQCEP≈3.5GPa,HQCEP≈18T) are accessible with present laboratory equipments.\nFigure7 showsthe phasediagramof UGe 2forH/bardblM0. The (Tx,Px) lineterminatesat a critical\npoint in the H=0 plane. UGe 2switches fromFM2 to FM1 at H=Hxand fromPM to FM1 at\nHc. Both ﬁelds HcandHxwill end at a QCEP. High magnetic-ﬁeld measurements, H>20T,\nare necessary to clarify the QCEP for Hc. The transition from FM1 to FM2 that occurs at Hx\nhas a strong feedback on SC as illustrated by the unusual temp erature dependence of Hc2(T)\n(see Fig. 8) [57]. At H =0, changes of the e ﬀective mass enhancement were observedat Pxand\nPc, respectively. Thus, similar changes should also occur at HcandHxforPc<P<PQCEP;\non approaching PQCEP,HQCEPwell deﬁned maximum of m∗(H) must appear. An interesting\n6point is the possible concomitant occurrence of Lifshitz tr ansition which is associated with the\ntopological change of Fermi surfaces [58]. This may add anot her feedback for the treatment of\nthe metamagnetictransition.\nForH/bardblM0, a weakening of the FM correlationssimilar to that of UGe 2at ambient pressure\nis observed in URhGe and UCoGe. In URhGe, the FM transition is second order at P=0 and\nmoves away from tricriticality as Pis increased since∂TCurie/∂P>0. For UCoGe, as already\nmentioned, the PM-FM transition may be ﬁrst order [59]. Howe ver,M0is already weak at\nambientpressureanditdecreaseswith P. Itisthussuspectedthat PQCEPwill beverycloseto Pc\nandthatHQCEPwill beratherlow.\nHowever,spectaculare ﬀectsarisefor H⊥M0. Thetransverseresponseleadstoadecreaseof\nTCurie, whichcan bedescribedusingthe Landaufreeenergy[60]. Fi gure9 showsschematically\ntheﬁeld variationof TCurie(H)andγ(H)forH/bardblM0andH⊥M0inURhGe. Ifγgoesthrougha\nmaximum, a ﬁeld enhancementof m∗∗accompaniedby an enhancementof Tscoccurswhen the\ninduced transverse magnetic component along the hard axis, e.g.χbHbalongb-axis, becomes\ncomparableto M0(whereχbistheinitialslopeofmagnetizationalong baxis). Table4givesthe\nestimatedcharacteristicﬁeldsalongthethreeaxesforthe threeuraniumSC ferromagnets.\nTable 4: Susceptibilities and characteristic ﬁelds of UGe 2,URhGeand UCoGe.\nχaχbχcHaHbHc\n(µB/T) (T)\nUGe20.006 0.0055 0.011 230 250 122\nURhGe 0.006 0.03 0.01 66 13 40\nUCoGe 0.0024 0.006 0.029 29 12 2.5\n7. ReinforcementofSC in thetransverseresponse\nIn URhGe, the susceptibility along the hard magnetization a xisb,χb=∂Mb/∂H, is large\nin comparison with the easy axis c, (χb/χc∼3). At a ﬁeld HR/bardblb, a reorientation of the\nmagnetic moment occurs and the easy axis changes from the cto thebaxis. In a restricted\nﬁeldrangecenteredaround HR,reentrantSCappears. Figure10showsschematicmagnetiza tion\ncurves and the temperature dependence of Mat diﬀerent ﬁelds H/bardblb-axis.TCurieis marked by\na maximum ofχb. The coeﬃcient of the magnetization T2term is linked to the Sommerfeld\ncoeﬃcient, according to the thermodynamic Maxwell relation ∂γ/∂H=∂2M/∂T2. As shown\nhere,TCuriedecreaseswithincreasingﬁeldandissuppressedat HRatlowtemperatures. Theﬁeld\ndependenceofthe e ﬀectivemass m∗(H)obtainedfromthe Maxwellrelationand directspeciﬁc-\nheatmeasurementsareshowninFig.11. Theenhancementofth eeﬀectivemasswithincreasing\nﬁeldH/bardblbis at the origin of the reentrant SC (RSC) illustrated in Fig. 12. Using the simple\nformula,Tsc∼T0exp(−m∗/m∗∗),Hc2can be calculatedwithin the orbitallimit: Hc2∼(m∗Tsc)2.\nExcellent agreement is obtained for the magnetic ﬁeld range where RSC is observed (Fig. 13).\nMoreover,knowingthe Pdependenceof m∗(H),RSCispredictedtocollapseat PRSC∼2GPaas\nobservedexperimentally[54]. At0K,wenoticethatalinear extrapolationofM(H),for H/bardblb,\nfromH>HRtoH=0 exhibits a non-zero intercept, suggesting that the reorie ntation process\ndoesnotcorrespondtoatransitiontothePMregime. Thepres ervationoftheFMphasesuggests\n7that the FM Fermi surface is rather robust during the reorien tation process, in good agreement\nwith theweaksingularitiesofthe thermoelectricpowerdet ectedatHR[61].\nIn UCoGe, for the same ﬁeld strength, no reorientation is exp ected sinceχcis larger thanχb\nandχa. However,the transverse response, when HreachesHb, leads to an unusual dependence\nofHc2(T) (as shown in Fig. 13). It is related to a ﬁeld enhancement of m∗as reﬂected by the\nenhancement of A(H) forH/bardblbwhenHapproaches Hb. ForH/bardblca strong decrease of Ais\ndetected (Fig. 14). The calculated Fermi surface of UCoGe in the FM phase is quite di ﬀerent\nfromthat in the PM phase, [44] andthe system is close to a FM–P M instability. Hence, it could\nbe possible that the transversemagneticﬁeld drivesthe sys tem throughthe FM–PM singularity.\nEvidencecouldbegivenbytherecentobservationoflargeva riationsofthethermoelectricpower\natHb[61]. In recent Shubnikov-de Haas experiments that measure the Fermi surface and the\ncyclotroneﬀectivemass,a quitelarge Hresponseisdetected[62].\n8. Conclusion andremarks\nWe havepresentedthe( T,P,H)phasediagramsofthethreeuraniumferromagneticsuperco n-\nductors,UGe 2,URhGeandUCoGe. We havefocusedontheenhancementofe ﬀectivemassand\nits relation to SC. We emphasize that the magnetic singulari ties atPc,Px,Hx, andHcare often\nassociated with a Fermi surface reconstruction related to t he ﬁrst-order nature of the magnetic\ntransition in UGe 2and UCoGe. The full determination of the Fermi surface, as a f unction of P\nandH, is expectedsoon thanksto progressesin the crystal purity . The case of URhGe which is\nlocated far from FM–PM instability and far from tricritical ity seems to be the ideal example of\nFM superconductivity.\nAn interesting aspect of FM superconductivityconcernsthe inﬂuence of FM on macroscopic\nphenomena such as the Meissner e ﬀect [63]. Other topics are SC in FM domain walls and the\nphenomenaassociated with the relative orientationof the S C order parameter to the magnetiza-\ntion, the eﬀect of SC on FM domain structure [64]. Of course as the interna l ﬁeld is large with\nrespect to the lower critical ﬁeld Hc1(∼10−3T), spontaneousvortexformationmay alreadyoc-\ncuratH=0. ItisonlyrecentlythatcarefulDCmagnetizationmeasure mentwererealizedinthe\ncase of UCoGe (4πM∼0.01T), no full Meissner e ﬀect, i.e. no indication of Hc1was detected\nat least for H/bardblc-axis.[65, 66].\nAdvancesin theﬁeldhavebeenmainlyachievedthroughthedi scoveryofnewsystems. Even\nnowthemaingoalistodiscoveraverycleansystemlikeCe-11 5heavyfermionsuperconductors\nwhere large and pure single crystals are easily available. U nfortunatelyup to now, high quality\nsingle crystal growth of UCoGe and URhGe is a di ﬃcult task and for UGe 2SC appears only\nunder pressure squeezed between two ﬁrst order transitions . It is worthwhile to remark that SC\ninFMmaterialshasonlybeendetectedinuraniumintermetal liccompoundswithIsingtypeFM,\nconﬁrming the key role of longitudinal ﬂuctuation and for ma terials with quite moderate heavy\nfermion character ( γ≤160mJ/K2mol) by comparison to the large value of γ(>1J/K2mol)\nreported for prototype d-wave superconductors (CeCu 2Si2, Ce-115) close to their AF–PM in-\nstability. Maybe due to the low value of Tscprovided by FM longitudinal ﬂuctuations and the\nsensitivity to disorderat bothFM andSC onsets, a moderater enormalizedbandwidthis a quite\nfavorableconditionforthe coexistenceofFM andSC. Upto no w,allattemptsto discoverSC in\notherFMmaterialshavefailedwithCe ferromagneticheavyf ermioncompounds.\n8Acknowledgments\nWe thank J. P. Brison, A. Buzdin, S. Fujimoto, H. Harima, K. Ha sselbach, E. Hassinger, L.\nHowald, K. Ishida, W. Knafo, G. Knebel, H. Kotegawa, L. Malon e, C. Meingast, V. Michal, V.\nMineev, K. Miyake, C. Paulsen, S. Raymond, R. Settai, I. 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Jpn. 80(2011) 013705.\n[63] D. V.Shopova and D.I. Uzunov: Phys.Rev. B 72(2005) 024531.\n[64] A. I.Buzdin and A.S.Mel’nikov: Phys.Rev. B 67(2003) 020503.\n[65] K. Deguchi, E.Osaki, S. Ban, N. Tamura, Y. Simura, T.Sak akibara, I. Satoh and N.K. Sato: J. Phys. Soc. Jpn. 79\n(2010) 083708.\n[66] C. Paulsen et al.to bepublisehd.\n10TTCurie\nPPcTITIIIII\nPc\nT TCurie\nP Pc\nTTCurie\nTsc\nP Pc(b)TTord\nTTCPPTITIIIII\nPc(a)\n(c)\n(d)~\nFigure1: (a)MagneticphasediagramfromtheSCRtheory. Bel owTI,theFermiliquidpropertiesareobserved. Foraﬁrst\norder transition, Pccan reach ˜Pcif∆M0is close to zero (b). (c)Superconducting phase diagram near the ferromagnetic\ninstability, from Fay and Appel. (c)Phase diagram of UCoGe.\n110T\nTxTCurieW~400K\nUGe2scmax(0.7K, P=1.2GPa)\n20K 52K\n0T\nTCurie W~400K\nURhGescmax(0.27K, P=0)\n9.5K\n0T\nTCurie W~400K\nUCoGesc(0.6K)\n3KM0 // a-axis\n(~1.5 µB)\nM0 // c-axis\n(~0.4 µB)\nM0 // c-axis\n(~0.05 µB)\nFigure 2: Characteristic energy scales of the three ferroma gnetic superconductors, UGe 2,URhGe and UCoGe.\n12500\n400\n3000020406080100 12050100150200250\n200\n100\n0C/T (mJ/K2mol)\n15 10 5 0\nT (K)T (K)\nURhGeTCurie\nTSCUGe2\n(b)\n(c) 80\n60\n40\n20\n0C/T (mJ/K2mol)\n543210\nT (K)TCurieTSC\nUCoGe70\n60\n50\n40C/T (mJ/K2mol)\n105 0H (T)H // c-axis\n0.45 K\n (a)(C–Cph)/T (mJ/K2mol)\nTxTCurie\nFigure 3: Speciﬁc heat of UGe 2,URhGeand UCoGe. Thephonon contribution is subtracted for UGe2[34,38, 7].\n13T/TCuriea\nb\nb\nbc\nacac\nUGe2\nURhGe\nUCoGe-40-30-20-1001020α (10-6 K-1) α (10-6 K-1) α (10-6 K-1)15\n10\n5\n0\n-5\n-10\n-15\n3\n2\n1\n0\n-1\n-2\n-3\n-41 2 0\nT/TCurie1 2 0\nT/TCurie1 2 0\nFigure 4: Temperature dependence of the thermal-expansion coeﬃcients along a,bandc-axis in UGe 2, URhGe and\nUCoGe [34, 38,46].\n14Ωe (T)20\n0\n0 0.5 1.0 1.5 2.0-20\n-40\n-60\n-80\n-10060\n40\nT/TCurieUGe2URhGeUCoGe\nFigure 5: Temperature dependence of the Gr¨ uneisen paramet er in UGe 2,URhGe and UCoGe [38].\nT\nTCurie\nTx\nPxPcTTCP\nP(a)   UGe 2 (b)   URhGe (c)   UCoGe\n~ W\nFM2FM1T\nTCurieWT\nTCurie\nP P\n~ 4 GPaW\nsc sc sc\nFigure 6: Schematic ( T,P)phasediagrams of UGe 2,URhGe and UCoGe.\n15PxPx\nPT HT = 0 K\nFM2FM1\nPM\nPUGe2\nH\nPcPc\nTTCP, PTCPTCurie\nTx\nHQCEP, PQCEPHQCEP\nPQCEP\nFigure7: Schematic( T,P,H)phasediagramofUGe 2. Theinsetshowsthe( H,P)planeat T=0. Veryrecentexperiments\nallow usto locate the position of the QCEPfor Hc. TheQCEP for Hxrequires anew set of high ﬁeld measurements.\nH\nTHxFM2\nSC FM1UGe2\nFigure 8: Schematic temperature dependence of the upper cri tical ﬁeld Hc2of UGe 2forH/bardblM0(a-axis).\n16T\nTCurie\nH // c // M0\nH // bURhGe\nγγ\nΗ\nΗHR\nHR\nFigure 9: Schematic ( T,H)phase diagram for H/bardblb(hard-axis) and H/bardblc(easy-axis). Theﬁeld dependence of γis also\ndepicted for both cases.\n17M\nHH // M 0\n(c-axis)\n(b-axis)H _ M 0\nFigure 10: Schematic magnetization curves at low temperatu re and the temperature dependence of magnetization at\nconstant ﬁelds H/bardblb-axis in URhGe [38].\n180.18\n0.17\n0.16\n0.15\n0.14\n0.13\n0.12γ (J/K2mol)\n15 10 5 0\nH (T)H // b-axis\nc-axis\nURhGe\nFigure 11: Field dependence of the Sommerfeld coe ﬃcientγobtained from the Maxwell relation ( b-axis) and direct\nspeciﬁc-heat measurements at 0 .4K(c-axis) in URhGe [38,7].\n190.5\n0.4\n0.3\n0.2\n0.1\n0TscH→0 (K)URhGe\n40\n30\n20\n10\n0Hc2 (T)\n15 10 5 0\nH (T)(a)\n(b)\nFigure 12: Calculated TscandHc2based on the ﬁeld dependence of m∗in URhGe[13].\n2020\n15\n10\n5\n0H (T)\n10 5 0\nT (K)URhGe\nH // b-axisRSC\nSCHR\nFM TCurie\n3 2 1 0\nT (K)UCoGe\nH // b-axis\nTCurie\nSC FM(a) (b)\nFigure 13: ( H,T) phase diagrams of URhGeand UCoGefor H/bardblb-axis [7, 6].\n2125\n20\n15\n10\n5\n0Hc2 (T)\n1.0 0.5 0\nT / TscUCoGe\nH // b-axis\na-axis\nc-axis\n1.5\n1.0\n0.5\n0√A / A 0\n20 15 10 5 0\nH (T)H // b-axis\na-axis\nc-axisUCoGe(a)\n(b)\nFigure 14: (a) Temperature dependence of Hc2along the orthorhombic directions of UCoGe (b) Field depend ence of\neﬀective mass (the normalized√\nA) [6].\n22"    },    {        "title": "1011.3658v1.Ferromagnetism_in_Dilute_Magnetic_Semiconductors.pdf",        "content": "Ferromagnetism in Dilute Magnetic Semiconductors \np\nR.pdapSilvapNeves a,pA.pFerreirapdapSilva apandpR.pKishore b,* p\np\napInstitutopdepFísica,pUniversidadepFederalpdapBahia ,pCampuspOndinap\n40210p340pSalvador,pBahia,pBrazilp\np\nbpInstitutopNacionalpdepPesquisaspEspaciaisp–pINPE/L ASp\n12210p970pSãopJosépdospCampos,pSãopPaulo,pBrazil.p\np\nAbstract \nWepstudypthepferromagnetismpofpGa 1LxMn xAsppbypusingpapmodelpHamiltonian,pbasedponpanp \nimpuritypbandpandpthepantiLferromagneticpexchangepi nteractionpbetweenpthepspinspofpMnpatomspandp \nthepchargepcarrierspinpthepimpuritypband.pBasedponp thepmeanpfieldpapproachpwepcalculatepthep \nspontaneouspmagnetizationpaspapfunctionpofptemperat urepandpthepferromagneticptransitionp \ntemperaturepaspapfunctionpofpthepMnpconcentration.p TheprandompdistributionpofpMnpimpuritiespisp \ntakenpintopaccountpbypMatsubarapandpToyozawaptheory pofpimpuritiespinpsemiconductors.pWep \ncomparepourpresultspwithpexperimentspandpotherptheo reticalpfindings.p\np\nPACS:p75.25.Lj,p75.30.Hx,p75.50.Cc,p75.50.Ppp\np\n \n1.  Introduction \n \nDilutepmagneticpsemiconductorsp(DMS)pareppromisingp\nforptechnologicalpapplicationspaspwellpaspinteresti ngpfromp \nthepbasicpphysicsppointpofpview.pPossiblepapplicati onspexistp \ninpspinpelectronicsp(spintronics)p[1].pwhichpemploy pthepspinp \ndegreepofpfreedompofpelectronspinpadditionptoptheir pcharge.p \nThispmaypallowpthepincorporationpofpferromagneticpe lementsp \nintopsemiconductorpdevices,pandpthuspthepintegratio npofpdatap \nprocessingpandpmagneticpstorageponpapsinglepchip.pS incepthep \nelectronicpspinpispapquantumpmechanicalpdegreepofpf reedom,p \nquantumpinterferencepeffectspcouldpbepexploitedpinp devices,p \neventuallypleadingptopthepdesignpofpquantumpcompute rsp[2].p\nAp lotp ofp workp hasp beenp donep onp disorderp effectsp inp\nnonmagneticpsemiconductorspandpmetalsp[3].pOnly,pdu ringp \nthep lastp fewp yearsp disorderp effectsp inp DMSp havep bee np \nconsidered.pTheypcanpbepstrongpdueptoptheppresencep ofpap \nhighp concentrationp ofp chargedp impurities;p thep typic alp \ndistancepbetweenpthesepdefectspisproughlypofpthepsa meporderp \naspthepFermipwavelength.p\nApkeypcomponentpofpspintronicspispthepdevelopmentpo fp \nnewp ferromagneticp semiconductors.p Followingp thep \nsuccessfulpdevelopmentpofpGa 1LxMn xAsp[4]ppandpIn 1LxMn xAs pp\n[5] pasp ferromagneticp semiconductorsp (withp xp ~p 1Lp 10%)p \nusingp carefulp lowp temperaturep molecularp beamp epitax yp \n(MBE)ptechnique,pintensivepworldwidepactivityphaspl edptop \nclaimspofpferromagnetismp(somepatproomptemperaturep andp \nabove)pinpseveralpDMSplikepGaMnPp[6],pGaMnNp[7],pGe Mnp \n[8],pGaMnSbp[9]petc.pItpispatppresentpunclear,pwhet herpallp \nthesepreportspofpferromagnetismp(particularlypatpro ompp\n*pCorrespondingpauthor.p\nTel:55L12L32086714;pELmailp:prkishore.br@gmail.comptemperaturesp orp above)p arep indeedp intrinsicp magneti cp \nbehaviorp orp arep arisingp fromp clusteringp andp segrega tionp \neffectspassociatedpwithpvariouspMnpcomplexesp(which phavep 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thep ferromagnet icp \ntransitionptemperaturepT Cpandpthepmagneticpmomentponpholep\ndensitiespn hpandpthepMnpionpnumberpdensitypn ip.pp\nThep existencep ofp DMSp ferromagnetismp seemsp top bep \nindependentpofpthepsystempbeingpmetallicporpinsulat ing.pForp \nGaMnAs,p bothp metallicp andp insulatingp samplesp arep \nferromagneticp withp thep transitionp temperaturep being p \ntypicallyphigherpforpmetallicpsystemspalthoughpthis pmaypnotp \nnecessarilyp bep ap genericp behavior.p Manyp otherp syste ms,p\nexhibitingp DMSp ferromagnetismp [6L9],p arep howeverp \nstronglypinsulating.pAdditionally,pinpGa 1LxMn xAspsamples,p\nexhibitingp reentrantp metalLinsulatorp transitions,p t hep \nferromagneticp behaviorp appearsp top bep completelyp \ncontinuousp asp ap functionp ofp xp withp thep onlyp observa blep \neffectp beingp ap variationp inp T C.p Thep realp DMSp carriersp 2p\np\nmediatingpthepferromagneticpinteractionpbetweenpthe plocalp \nmomentsparepgenerallyplikelyptopbepfarpfrompfreepho lespinpthep \nvalencepbandpofpthephostpsemiconductorpmaterial.pTh eypare,p \ninpallplikelihood,pextendedporpboundpcarrierspinpth epimpurityp\nband,pwhichpformspinptheppresencepofpthepMnpdopants .pTherep \nisp ap greatp dealp ofp directp experimentalp supportp forp thep \nrelevancep ofp thisp impurityp bandp picturep forp DMSp \nferromagnetism,p bothp firstp principlep bandp structure p \ncalculationsp[10] ppandpMontepCarlopsimulationp[11] ppconfirmp \nthep impurityp bandp naturep ofp thep carriersp activep inp DMSp \nferromagnetism.pp\nInpthepwakepofpMatsubarapandpToyozawapmethodp(MT)p \nforp ap disorderedp systemp [12] p,p wep studyp thep magneticp \npropertiespofpGaMnAs,pdescribedpbypstatespofpholesp formingp \nanpimpuritypband.pFollowingpthepmeanpfieldpapproxim ation,p\nwepcomparepourpresultspwithpthatpofpBerciupandpBhat tp(BB) p\n13 .pTherepareptwopmainpdifferencespbetweenpourpapproa chp \nandp thatp ofp BBp [13]:p 1.p wep performp ourp calculations p \nconsideringpanpinfiniteplatticepandp2.pwepmodelpanp effectivep\nantiferromagniticp (AFM)p exchangep interactionp coupli ngp\nbetweenpthepimpuritypandpholespspinspconsideringpth epMnp \nsuperlatticepaspanppaveragepofppSC,pBCCpandpFCCplat tices,p\nwhichpispapsimplepestimatepofptheprandompdistributi onpofpMnp \nimpuritiespinpGaAs.pp\nInpSectionp2pwepdescribepthepmodelpHamiltonian,pthe p \nmeanp fieldp approximation,p andp thep MatsubaraLToyozaw ap \napproachp top considerp thep randomp distributionp ofp the p \nmagneticp impurities.p p Inp sectionp 3p wep presentp thep \ncalculationsp andp discussionsp ofp ourp results.p Finall yp inp \nSectionp4pwepgivepapbriefpsummarypofppourpwork.p\n2.  Formalism \np\nFollowingp(BB)p[13]p,pwepusepthepelectronpformalism ,p \nalthoughpthepchargepcarrierspinpGa 1LxMn xAsparepholes.pItpisp \nanpequivalentpsystempdopedpwithphypotheticalpdonors ,pwithp \nimpurityp levelsp belowp ap conductionp band.p Sincep inp t hisp \nwork,pourpmainpinterestpisptopconsiderpthepeffectpo fpdisorder,p \nthispwillpnotpchangepthepessentialpaspectspofpthepp roblem,pasp \ndiscussedpbypBBp[13].pWepconsiderpexactlypthepsamep modelp \nHamiltonianpaspthatpofpBBp[13].pInppresencepofpthep magneticp \nfieldp H,pthispHamiltonianpispgivenpasp\np\nΗp=o ∑\nσijtij oc iσ†cjσo+o ∑\nijJij oSio. s joooooooooooo\nooooooooooooooooooooo∑ogµ BH∑\nisizo∑og´oµ BoH ∑\niSizppppppppppppppppppp(1)p\np\nwherepc iσ †p andpc iσ parepthepcreationpandpannihilationpoperatorsp \nforp holesp ofp spinp σp atp thep sitep I,p andp J ij p isp thep antiL\nferromagneticp(AFM)pexchangepinteractionpbetweenpsp insp \nofpholespandpMnpions.pThepfirstptermpdescribespthep chargep \ncarriersp hoppingp betweenp Mnp sitesp withp ap hoppingp tij ,p \ndependentponpthepdistancep jiRR−.pWeprestrictpourselvesp top thep uncompensatedp samplesp inp whichp tii p canp bep \nconsideredpapconstantpindependentpofpsitep i.pWepshallpdefinep \nthepenergypscalepsuchpthatpp tii o=o0 .pTheptransferpintegralp tij p\nwillpbepcalculatedpfromp1sphydrogenplikepwavepfunct ions.p\nThepsecondptermpdescribespthepAFMpinteractionpbetwe enp \nSMn pandp shpspins,pandptheplastptwoptermspdescribepthepenergyp\ninteractionpwithpthepexternalpmagneticpfield.p\nWep treatp thep secondp termp inp thep molecularp fieldp \napproximationpasp\np\n                  S io. s joo=ooS Mn os jzooo+oos hoS izo–oS Mn os hpppppppppppppppppppp(2)p\np\nwherep SMn o=o<oS izo> pandp shp=p <os jzo> parepassumedptopbep \nconfigurationallypaveragedpvaluespofpMnpandpholepsp inspandp \narepthuspindependentpofpthepsitepindex.pWithinpthis pmeanp \nfieldpapproximation,pthepHamiltonianp(1)pbecomesp\n( )[ ]∑ −′− =\niz\niBz\ni B heff HgHgsJ s S H µ µp\nMneffj\nijiij i\niiMneff SJcctccSJ − + + ∑ ∑ σ\nσσ σ\nσσσ† †\n2pppppppppppppppppp(3)p\np\nwherep Jeff o=o ∑\niJij o=oJo ∑\niexpo (pLp2p jiRR− p/pBa)p.pp\nandp Bapp isp thep Bohrp radius.p Forp thisp molecularp fieldp \nHamiltonian,pwepgetp\np\np\nooooooooooooooooooooooooooS Mn o=oBSo[β(g´ µBHpLpshpJ eff )],pppppppppppppppppppp(4)p\np\nwhere,ppBS(x) o=o (Sp+1/2p) coth [(pSp+p1/2p)xp] o–o(1/2) cotho (x/2)p\nispthepBrillouinpfunctionpandpβp=p1/(k BT).pFollowingpBBp13p\nwephavepassumedpthatpforpMnpatomspspinpSp=p5/2.pp\nNowptopobtainp shpwepconsiderpthepGreen´spfunctionp[14]p\np\nppppppppppppppppppppG ijσp(t)p=pLiθp(t)p<p[c iσ(t) p,ppcjσ†]+p>pppppppppppppppppppp(5)p\np\nwhere,p[A p,ppB]σpp=pABp+σpBAp;ppσp=p+porpL.pBypconsideringp \nthepFourierptransformp\np\npppppppppppppppppG ijσp(t)p=p(1/2π)∫∞\n∞−Gijσp(ω)pe Lipωpt pdtpppppppppppppppppp(6)p\np\nwep getp thep equationp ofp motionp ofp thep Green´sp functi onppppppp \nGij (ωp)paspp\nppppppppppppppp(ωp–pt σ)pG ijσ p(ω)p=pδ ij p+p ∑\nltiloGljσ (ω)ppppppppppppppp(7)p\nwherepp\np\nooooooooooooooooooooot σo=o(σo/2)(S Mn oJ eff pLogµBH) .pppppppppppppppppppppppppppp(8)p\np3p\np\npppppNow,pconsideringpthepcompletelyprandompdistrib utionpofp \nthepmagneticpimpurities,pwepusepthepapproachpofpMat subarap \nandp Toyozawap top obtainp thep configurationallyp averag edp \ndiagonalpGreen`spfunctionpasp[12,14]p\np\nppppppppppppppppp<pG iiσ(ω)p>pp=p(1/(ωp–pt σp ))pζp(ωp–pt σ)ppppppppppppppppp(9)p\np\nwherep\nppppppppppppppppppppppppppppζp(ωp)p=p1/(1p–pηp(ωp)) ppppppppppppppppppppppppppp(10)p\np\nandpp\noooooooooooo \n)()(1)(\n8)()(2\n23\nkk\ntntdkn\nii\nωωζ ωπωζωη\n−=∫ ooo (11)p\nwhereptp( k) ispthepFourierptransformpofpp  \np\npppppppppp \n) exp() 1 (21 1\n0 j iB j iB ij a att RR RR −−−+−=− −(12)pppppppp\np\ninpthepreciprocalpspace.pHerep t0pispthepionizationpenergypofp1sp \nstate.ppp\np\nppppTheppaveragedpGreenpfunctionppp<pG iiσ(ω)p>ppcanpbepusedp \ntopobtainppthepdensitypofpstatespperpimpuritypforps pinpσpasp\np\nooooooooD σ(ωo)o=o Lp(p1/πp) olim εo→o0oIm o<oG iiσo(ωo+oioεo)o> pppppp(13)p\np\nWepusepthisppdensitypofpstatesptopobtainpp shpaspp\np\npppppppp shoo=o o(1/2) o∑σoo oσ∫odωoD σ(ωp) o(pe βp(ωp–pµ)p p+p1p) L1ppppppp(14)p\np\nwherepthepFermipenergypµpcanpbepobtainedpfrompthepn umberp \nofpholespperpimpurityp po=on ho/onipasp\np\npppppppp=p o∑σoo o∫odωoD σ(ωp)p(pe βp(ωp–pµ)p p+p1p) L1ppppppp oooooooooooooo(15)p\np\n3.  Results and Discussions \n \nppppForp ourp calculations,p wep takep thep samep paramete rsp asp \nusedpbypBBp13 .pForpexamplepweptake,pthepionizationpenergyp \nt0p=p112.4pmeV=1Ry,pthepBohrpradiusp Ba=7.8pÅpandpJp=p15p \nmeV.p Thep numberp densityp n ip ofp thep Mnp impuritiesp isp \ncalculatedpfrompthepexpressionp nio=o4xo/oa 3p,pwherep a=p5.65p \nÅ,pisptheplatticepconstantpofpGaASplattice.pNow,pfir stpwepfixp \nthepvaluepofp xptopcalculatep nipandpassumepthepinitialpvaluepofp \nSMn o=o5/2 .pThispenablepusptopcalculatepD σ(ω)pandp µpfromp \nEqs.p(13)pandp(15)prespectivelypandpthenp shpfrompEq.p(14)p \nforpapfixedpvaluepofptemperaturepT.pThepvaluepofp shpobtainedp \nispusedptopcalculatepS Mn pfrompEq.p(4).pThispprocedurepisp\nrepeatedptillpselfLconsistencypispachieved.pThispal lowedpusptop calculatep SMn pandp shpaspapfunctionpofp kBT/J pforpvariouspvaluep \nofp ppandp xpandpcomparepitpwithpthepresultspofppBBp[13].p\nWep havep alsop donep ourp selfp consistentp calculations, p \nconsideringpthatpMnpimpuritiesparepplacedpinpanpord eredp \nlattice.pWepassumepthatpthisporderedplatticepispanp averagepofp \nSC,p BCCp andp FCCp latticesp inp contrastp top BBp [13]p who p \nconsideredp thatp thep Mnp impuritiesp arep distributedp i np SCp \nstructure.pForporderedpsystem,pEqs.p(14p)pandp(15)p reduceptop\np\npppppppppppppppppppppp ∑∫−=σσσπ )( )16(,13\nk k s tfdhppppppp(16)pp\nppppppppppppppppppppppp ∫∑−=σσ π )( )8 (,13\nkktfd p pppppppppppp(17)p\np\nwherep σ σttt+=k kandp 1)() 1 ()(−−+=µβ\nσσk\nkte tf p isp \nthepFermipdistributionpfunction.pThepexpressionpfor pS Mnp\nispgivenpbypEq.p(4).p\np\nWephavepshownpourpresultspinpFiguresp1L4.pInpFig.p1 ,pwep \nhavepplottedpthepdensitypofpstatesp(DOS)pforpapcomp letelyp \nrandompdistributionpofppMnpimpuritiespcorresponding ptopxp=p \n0.05pandppp=p0.1pandpcomparedpitpwithpthatpofppBBp[ 13].ppInp \nFigp2pourpresultspshowpthatpthepdisorderedpsystemph aspap \nsignificantp differencep inp thep behaviorp ofp magnetiza tionp \ncurvespcomparedptoptheporderedpcase.pAccordingptopB Bp[13] pp\nandp C.p Timmp1,p inp disorderedp systemsp thep averagep \nmagnetizationpofpMnpspinspaspapfunctionpofptemperat urephasp\nmorep pronouncedp decreasep inp thep lowp temperaturesp an dp \ndecreasedpmorepslowlypinpthepregionpofpintermediate pandp \nhighp temperatures,p leadingp top higherp ferromagneticp\ntransitionptemperaturep TCp[15]pthanpinpthepcasepofporderedp \nsystems.  Althoughp TCppresentphigherpvaluespinpdisorderedp \nsystems,p thep magnetizationp duep top Mnp spinsp fallsp \ndramaticallypfrompthepsaturationpvalue,punlikeporde redppcasep \nwherep therep isp ap lessp markedp fallp inp thep regionp ofp lowp \ntemperaturesp withp ap considerablep magnetizationp forp\nintermediateptemperatures.pp\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\nFIG.p1pImpuritypbandpdensitypofpstatespforpthepspin psplittingpatp To=o \n0Ko withp xo=o0.05o andpppo=o0.1 p.pTheppcalculationspfrompBBp 13 ppforp -300-250-200-150-100 -50 0 50 100 012345\n  DOS (a. u. )\nEnergy (meV) Disordered \n  D/barb3up/barb3up /barb3up/barb3up\n  D/barb3down/barb3down /barb3down/barb3down \nBB[13] \n  D/barb3up/barb3up /barb3up/barb3up \n  D/barb3down/barb3down /barb3down/barb3down \n 4p\np\nthep completelyp randomp distributionp ofp impuritiesp ar e oalso o\npresentedp.p\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\nFIG.p2pThepaveragepMnpandpcarrierpspinsp SMn pandp sh /p pforp \nbothporderedpandpdisorderedpsystemspaspapfunctionpo fp kBT/J .pp\nForpthepsakepofpcomparisonpthepresultspofpBBp[13]pf orpbothp \norderedp andp disorderedp systemp withp completelyp rando mp \ndistributionp(crd)pofpimpuritiesparepalsopshown.p\np\nFig.3pshowspthepeffectpofpmagneticpfieldponp SMn pandp sh/p p.pItp \nshowsp thatp thep applicationp ofp magneticp fieldp influe ncesp \nmorepstronglypthepspinspofpMnpimpuritiespthanpholep spins.pItp \nshouldpbepnotedpthat,pexceptpatpveryplowptemperatur espandp \nverypnearptoptransitionptemperature,pourpvaluespfor p SMn  ( \nsh/p )parepalwaysplowerp(phigherp)pthanpthatpofpBBp[13]p .p\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\nFIG.p3pThepaveragepMnpandpcarrierpspinsp SMn pandp sh /p paspap \nfunctionp ofp kBT/J p withp andp withoutp applicationp ofp anp \nexternalpmagneticpfield.pForpcomparison,pthepresult spofpBBp \n[13]parepalsopshown.pp\np\nInp Fig.p 4p wep havep shownp calculationp ofp thep transiti onp \ntemperature oTcp asp ap functionp ofp xp forp bothp orderedp andp \ndisorderedp samples.p Resultsp arep comparedp withp thep experimentalp resultsp ofp oMatsukurap etp al. p p[16]p andp \nBeschotenppetpal.p[17].pAlthoughpT Cphasptopbephigherpforpthep \ndisorderedp system,p thep spontaneousp magnetizationp ha sp ap \nresidualp valuep onp ap largep extensionp ofp thep temperat urep \nvalues.pThispcanpbepexplainedpbypthepformationpofpc lustersp \nofpimpurities.pEachpclusterpcanphavepapmagnetizatio npinpap \ngivenp direction,p providingp severalp differentp region sp withp \nmagnetizationpinprandompdirections,pwhichpcanpbepex plainedp \nbyp thep longp tailp foundp inp thep curvesp ofp spontaneous p \nmagnetizationpforpdisorderedpsystems.p p\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\np\nFIG.p 4p Comparisonp betweenp theoreticalp (disorderedp a ndp \nordered)pandpthepexperimentalpvaluespofp TCo. \np\n4. Conclusions \n \nBrieflypwephavepstudiedpthepdisorderpeffectspinpGaM nAsp \ndilutepmagneticpsemiconductorspdescribedpbypanpimpu rityp \nbandp usingp Matsubarap andp Toyozawap approachp forp \ndisorderedpsemiconductors.pWephavepcomparedpourpres ultsp \nwithpthatpofpBBp[13]ppandpthepexperimentalpresultsp ofpT c.p\nSincepthispmodelpispapplicablepforpthepinfinitepsiz epofpthep \nsample,pitpcanpbepusefulptopclarifypsomeppointsptha tparepnotp \nyetpwellpunderstoodpbecausepofpthepfinitepsizepcalc ulations.pp\np\nAcknowledgements \n \nThispworkpwasppartiallypsupportedpbypthepBrazilianp pp\nagenciespFAPESBpandpCNPQ.p\np\nReferences \n \n[1] pS.pA.pWolfpetpal.,pSciencep294,p1488p(2001);pT.pDie tlpetp \nal.,Sciencep 287,p 1019(2000);p C.p Timm,p J.p Phys.p Cond .p \nMatterp15,pR1865p(2003);pI.pZuticpetpal,pRev.pMod.p Phys.p \n76,p323p(2004).p0.0 0.2 0.4 0.6 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 \n  sh/p , S Mn \nkBT/J        p = 0.10 \n        x = 0.00926 \n Disorded \n Ordered \n BB [13](crd) \n BB [13](ordered) \n0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 \n  sh/p , S Mn \nkBT/J   H = 0T \n  BB [13], H = 0T            \n  H = 5T \n  BB [13], H = 5T            \n  H = 10T \n  BB [13], H = 10T 0.00 0.02 0.04 0.06 040 80 120 \n  TC(K) \nMn composition (x)  Disodered (p=0.1) \n Ordered (p=0.1) \nExperimental data \nRef.[16] \nRef.[17] 5p\np\n[2] pC.pH.pBennettpandpD.pP.pDiVincenzo,pNature,p404,p24 7p \n(2000).p\n[3] pP.pA.pLeepandpT.pV.pRamakrishnan,pRev.pMod.pPhys.p5 7,p \n287p (1985);p B.p Kramerp andp A.p MacKinnon,p Rep.p Prog.p\nPhys.p56,p1469p(1993).p\n[4] pH.pOhno,pSciencep281,951p(1998)p\n[5]pH.pOhnopet.pal,pPhys.pRev.pLett.p68,p2664p(1992 ).p\n[6] pN.pTheodoropouloupet.palpPhys.pRev.pLett.p89,p10720 3p \n(2002).p\n[7] pM.pL.pSeedpet.pal,pAppl.pPhys.pLett.p79,p3473p(2001 ).p\n[8] pY.pD.pParkpet.pal,pSciencep295,p652p(2002).p\n[9] pX.pChenpet.pal,pAppl.pPhys.pLett.p81,p51p(2002).p\n[10]pB.pL.pSheu,petpal.,pPhys.,pRev,pLett.,p99 , 227205p(2007).p\n[11]p pG.pA.pFiete,pG.pZaránd,pandpK.pDamle,pPhys.pRev.,p \nLett.,p91,p097202p(2003);pG.pA.pFiete,petpal.,pPhys .pRev.pBp \n72,p045212p(2005).p\n[12] pT.pMatsubarapandpY.pToyozawa,pProg.pTeor.pPhys.,p26 ,p \n739p(1961).pp\n[13]pM.pBerciupandpR.pN.pBhatt,pPhys.pRev.pLett.p87 ,p \n107203p(2001),pPhyspRev.pBp69,p045202p(2004),pPhys. pRev.p \nLett.p90,p029702p(2003),pC.pTimm,pF.pSchäfer,pandpF .pvonp \nOppen,pPhys.pRev.pLett.p90,p029701p(2003).p\n[14]pA.pFerreirapdapSilva,pR.pKishore,pI.pC.pdapCun hapLima,p \nPhys.pRev.pBp23,p4035p(1981),pR.pKishore,pI.pC.pdap Cunhap \nLima,pM.pFabbripandpA.pFerreirapdapSilva,pPhys.pRev .pBp26,p \n1038p(1982).p\n[15] pJ.pL.pXu,pM.pvanpSchilfgaarde,pG.pD.pSamolyuk,pPhys .p\nRev.pLett.p94,p097201p(2005).p\n[16] pF.pMatsukura,pH.pOhno,pandpY.pSugawara,pPhys.pRev.p\nBp57,pR2037p(1998).p\n[17] pB.pBeschoten,petpal.,pPhys.pRev.pLett.p83,p3073p(19 99).p\np\np"    },    {        "title": "0708.1681v1.Spin_valve_effect_by_ballistic_transport_in_ferromagnetic_metal__MnAs____semiconductor__GaAs__hybrid_heterostructures.pdf",        "content": "Spin valve effect by ballistic transport in ferr omagnetic metal (MnAs) / \nsemiconductor  (GaAs) hybrid heter ostructur es \n \nPham  Nam  Hai1, Yusuke Sakata1, Masafum i Yokoyam a1, \n Shinobu Ohya1,2, and Masaaki Tanaka1,2 \n \n1Department of Electr onic Engineering, The University of Tokyo, \n7-3-1 Hongo, Bunkyo-ku, Tokyo 1 13-8656, Japan  \n \n2Japan Science and Technology Agency , \n4-1-8 Honcho, Kawaguchi-shi, Saitama 332-0012, Japan \n \nAbstract \nWe dem onstrate the spin valve ef fect by  ballis tic transpo rt in f ully epitaxial \nMnAs ferrom agnetic metal / GaA s sem iconductor / GaAs:MnAs granular hybrid \nheterostructures. The GaAs:MnAs m aterial contains ferromagnetic MnAs nanoparticles \nin a GaAs m atrix, and acts as a spin inject or and a spin detector . Although the barrier \nheight of the GaAs/MnAs interface was f ound to be very sm all, relatively lar ge \nmagnetoresistance was observed. This result sh ows that by using ball istic transpo rt, we \ncan realize a lar ge spin valv e effect without inserting a high tunnel barrier at the \nferrom agnetic m etal / sem iconductor interface. \nPACS num bers: 72.25.Dc, 72.25.Hg, 75.47.Jn, 85.75.-d\n 1 In spin tronics app lications, m etal-based p assive dev ices such as  giant \nmagnetoresistance (GMR) head sen sors a nd m agnetic rando m access m emory (MRAM)  \nhave achieved considerable success. Using the spin degre es of freedom  in three -terminal \nactive sem iconductor devices is then a natu ral extension of spintronics research. \nRecently , novel devices such as spin field-ef fect transis tor (spin FET) [1] and spi n \nmetal-oxide-sem iconductor field-ef fect tran sistor (spin MOSFET) [ 2] have been \nproposed as new building blocks for future el ectronics. Those devi ces are expected to \nhave the advantages of non- volatility and low power cons umption of magnetic devices, \nas well as high-speed operation of sem iconduc tor devices [3]. The m ost basic operations  \nof these sem iconductor -based spintronic de vices require electrical injection of \nspin-polarized carriers into a sem iconducto r channel and detection of them  by FM \nelectrodes, i.e. the spin valv e effect in FM / SC / FM hybr id heterostruct ure. Unlike the  \nmetal-based  spin va lves, however , this hyb rid spin valve stru cture has a serious  \nproblem ; due to the lar ge conductivity m ismatch between ferrom agnetic m etals and \nsemiconductors in the dif fusive transport regi me, the im balance of injected m ajority and \nminority spins in the sem iconductor chan nel becom es extrem ely sm all. This \n“conductivity m ismatch” problem  has been confirm ed in recent theoretical and \nexperim ental studies [4-12]. It h as been well e stablished tha t to overcom e this problem a \n 2tunnel or Schottky barrier has to be inserted  at the interface of FM  and SC [4-5,13-14]. \nHowever , such a high  resis tance interf ace is  not preferred, becau se it d rastically  \ndecreases the current driving capability when used in active transport devices. \n In this letter , we show that by usi ng ballistic transport of spin-polarized \nelectrons in a SC chann el, we can o btain a large spin v alve effect with out ins erting a \nhigh res istance interface. In the case of ba llistic transpo rt, the h igh resis tivity of \nsemiconductors is no longer relevant, thus  the conductivity m ismatch problem  may not  \noccur . Consequently , a ballistic sem iconduc tor spintronic device can  utilize both the \nnon-volatility of m agnetic de vices and the high-speed operation of sem iconductor  \ndevices [15,16]. Indeed, the spin FET and the spin MOSFE T are assumed to work under \nthe ballistic transpo rt regime. Nevertheless, r ealization of ballistic tr ansport in FM / SC / \nFM spin valve structures has been very  challenging. Firstly , the length of the \nsemiconductor channel must be shorter than th e mean free path of electrons, that is, it  \nmust be a s cale of several ten s of nan ometers or shorter . Seco ndly, the interface between \nFM and SC must be very sm ooth and free of disorders to av oid loss of spin selectivity \n[15,17], thus any surface treatm ent techniques such as etching  or sputterin g should be  \nexcluded. T hirdly, because the bias voltage Vhalf, at which the spin valve ratio is redu ced \nby half is typically severa l hundreds of mV , the Schottk y barrier at the FM / SC \n 3interface m ust be low  enough to  allow balli stic tunneling of el ectrons from  the \nelectrodes to the conduction band of the sem iconductor spacer with a sm all bias voltage. \nTo satisfy these conditions , we have perform ed epitaxial growth of FM / SC / \nFM spin valve stru ctures using m olecular beam  epitaxy (MB E). Figure 1 shows our spin \nvalve device structure grown on a p+ GaAs(001)  substrate. The structure consists of a \nnon-doped GaAs se miconductor laye r with the thickness of tGaAs = 10 - 30 nm \nsandwiched by two ferrom agnetic MnAs electrode s. The bottom  electrod e is a granu lar \nthin film , in which ferrom agnetic MnAs nanopar ticles with size of 5 nm  in diam eter are \nembedded in a thin f ilm GaAs m atrix (ref erred to as GaAs:MnAs). The top electrode is  \na 20 nm –thick type-A  MnAs thin film  [18].  The growth procedure was as follows. \nFirstly , we grew a 20 nm-thick Be-doped GaAs buf fer layer on a p+GaAs(001) substrate  \nat 580°C. After cooling the substrate tem perature to 300° C, we grew  a 5 nm –thick \nGa0.957Mn 0.043As thin film. Then, the structure wa s annealed at 580°C for 20 m inutes in \nthe MBE growth chamber , during which pha se separation occurred in the GaMnAs \nlayer and MnAs nanoparticles were for med in  the GaAs m atrix [19-21]. After that, the \nsubstrate temperature was c ooled to 300°C again and a GaAs spacer layer with \nthickness of 10 – 30 nm was grown. Finally , a 20 nm -thick type-A  MnAs thin film  was \ngrown at 260°C as a top electrode. After co mpleting the growth, post growth annealing  \n 4was carried out in the growth  cham ber at 320°C for 10 m inutes to im prove the structural \nand m agnetic properties of the top MnAs f ilm. By growin g the GaAs  space r lay er at \n300°C, we can suppress dif fusion of residua l Mn atom s from the GaAs:MnAs electrode  \nto the GaAs spacer while  maintain ing high crystal quality of GaAs for ballistic transpo rt. \nThe GaAs layers grown  at 300°C are sem i-insulating and sh ow an electron m obility of \n1000 cm2/Vs at room  temperature, corresponding to the m ean free path of 10 nm . The  \nballistic transport of electr ons through the 300°C-grown GaAs layer at low te mperature  \nwas confirmed by observing the resonant tunnel ing ef fect in AlAs / GaAs / AlAs double \nbarrier diodes with a 300°C-grown GaAs quantum  well. \nIt is worth noting that the over growth of high-quality sem iconductor layer on \ntop of a FM layer is very dif ficult. The uni que GaAs:MnAs electrode in our spin valve, \nhowever , allows the over growth of a high- quality and atom ically  controlled GaAs \nsemiconductor spacer layer [ 19-21]. Furtherm ore, the MnAs nanopa rticles have been \nshown to work well as a spin inje ctor and a spin detecto r [21] . Although the crystal \nstructure of MnAs (NiAs type hexagonal) is dif ferent from  that of GaAs (zinc-blende), \nthe in terfaces between  the elect rodes and the GaAs spacer are very  smooth, as  revealed  \nby the streaky patterns of the reflection high ener gy electron  diffraction patterns \nobserved at each layer . The barrier height betw een MnAs and GaAs in our spin valve \n 5structure was found to be very sm all, as discussed below . \nFigures 2 shows the tGaAs dependence of the resistances R of four spin valves \nwith tGaAs = 10, 15, 20, 30 nm  plotted in two ways; (a) log( R)- tGaAs with a bias vo ltage \nV of 1 mV , and (b) R- (tGaAs)2 with V = 50 mV . Assum ing direct  tunneling of electrons \nthrough the GaAs rectangular -type tunnel barrie r as shown in the inset of Fig. 2(a), we \ndeduce that the barrier height φ < 1 m eV from  the gradient of the log( R) - tGaAs line by \nusing the WKB approxim ation. This resu lt is s urprising bu t seem s con sisten t with the \nfact the Ferm i level of MnAs lies above  the m inimum of the X valley of AlAs  \nsemiconductor in MnAs/GaAs/AlAs/GaAs:Mn As m agnetic tunnel junctions (MTJs) \n[21]. The conductance across the MnAs/GaAs in terface in o ur sam ples is com parable or \neven better than that of the Co/GaAs interface with φ ~ 10 meV  reported recen tly [22]. \nWith this very sm all barrier he ight, ele ctrons can eas ily transpo rt from the MnAs \nelectrodes to the conduction band of GaAs by the Fowler -Nordheim  (FN) tunneling \nwhen the bias is higher ( V = 50 mV), as shown in the inset of Fig. 2(b). The FN \ntunneling current is given by \n32\nGaAs GaAs 3/2 MnA s\n2\nGaAs GaAs82exp83πφπφ\n=− tm em VSImh t heV,   (1) \nwhere V is the bias voltage,  S is the ef fective area thr ough which the current flows,  h is \nthe Plank constant, e is the electronic char ge,  and  are the ef fective MnA s mGaAsm\n 6electron m ass of MnAs and GaAs, respectively . When  eV >>φ, the tGaAs dependence of \nthe exponential part of e quation (1) is weak, thus  /=RVIis proportional to  (tGaAs)2. \nFigure 2(b) shows that R measured at 50 mV  is proportional to (tGaAs)2, revealing the FN \ntunneling nature of electron transport. In Fig. 2(c), we show the conductance ( G)–bias \nvoltage ( V) characteristics of the fo ur spin  valv e devices. The G–V characteristics are \nclearly dif ferent from  that of a typical Si mmons-type tunnel juncti on with a rectangular \nbarrier (Ref. [23]), but can be explained by the FN-tunneling based equation (1). In the \nlow bias areas (area I) where the exponential part of equati on (1) is sm aller than unity , \nG non-linearly increases w ith the increasing bias V. In the interm ediate bias are a (area \nII), G linearly incr eases with th e increasing  V since  the exp onentia l part of equation (1) \nhas reached unity . Finally , in the hig h bias area (area III), G increases non-linearly again \nwith the increasing bias. The non-linear increase of G in area III can  be attribu ted to the \nparallel conduction through th e GaAs m atrix where MnAs nanoparticles do not exist. \nThis parallel conduction results in  the decrease of spin valve ra tios at lar ge bias voltages, \nas discussed later . \nFigures 3(a)-(d) show the m agnetic-field dependence of th e res istance of the \nspin valve d evices  measured at 7  K with a bias of 50 mV . The black and red curves are \nmajor and m inor loops, respectiv ely. The major loops are superpo sition of the \n 7magnetoresistance (MR) com ponents of the ferrom agnetic electrodes (gradual change) \nand the spin  valve ef fect (abr upt jumps of resistance). The hysteresis o bserved in the \nminor loops indicates that th e resistance jum ps correspond to  the m agnetization reversal \nof the GaAs:MnAs nanoparticles.  If we assum e purely d iffusive transpo rt of electro ns in \nthe GaAs spacer , we can estim ate the spin valve ratio by [6]: \n (Spin Valve ratio) = 2sf\n2 MnAs GaAs\nGaAs GaAs8\n\nrlPrt,    (2) \nwhere  is the spin polariz ation of  MnAs, and  are the products of the \nresistivity by the sp in dif fusion lengt h for MnAs and GaAs, respectively , and is the  \nspin-dif fusion length of GaAs. Using  = 0.5 (R ef. [24]),  = 10PMnAsrGaAsr\nsf\nGaAsl\nl PMnAs GaAs/r r-6, = \n10 µm and = 10 nm , we get a spin valve ratio of 10sf\nGaAs\nGaAst-6. This spin valve ra tio \nexpected for purely dif fusive transport is 4 or ders of magnitude s maller than that of our \nexperim ental results. Consequently , the appear ance of the spin valve ef fect of several %  \nin our structures, even when there is no high tunnel or Schottky barriers at the interfaces, \nindicates that the electron transport is ballistic. \nFigure 4(a) shows the bias depen dence of the spin valv e ratio [defined as \n(Rmax-RH=0)/RH=0] measured at 7 K, where Rmax is the m aximum  resistance and RH=0 is \nthe res istance at zero m agnetic field. The sam ple with tGaAs = 10 nm  has the lar gest spin \nvalve ratio with a m aximum  of 8.2 % when th e bias voltage V < 120 mV . Howe ver, \n 8when V > 120 m V, its spin valve ratio decrease s rapidly and  becom es smaller than that \nof the sam ple with tGaAs = 30 nm . This is because when V > 120 mV , the parallel \nconduction becom es largest for sample with tGaAs = 10 nm  but sm all for the sam ple with  \ntGaAs = 30 nm. Despite the parallel  conduction, the spin valve effect was observed up to \n300 mV  for all sam ples in our experim ents. Figure 4(b) shows the tem perature \ndependence of the spin valve ratios m easured with a b ias voltage of  50 mV. The spin \nvalve ra tios of all sam ples decr eased with increasing tem perature, and disappeared at \nabout 90 K. In contrast, the MR component (t he gradual change of  resistance with the \nmagnetic field sweep) w as observed up to room  temperature. The absence of the s pin \nvalve ef fect at T > 90 K is not due to the tem perature dependence of the m agnetization \nof MnAs,  since the tunneling m agnetoresist ance ef fect of MnAs  / GaAs / AlAs /  \nGaAs:MnAs (with φ = 5 nm  MnAs nanoparticles) MTJs was observed up to roo m \ntemperature [25]. The decrease of the spin va lve ef fect with increasing temperatu re and  \nits disappearance at T > 90 K are thus a natural m anifestation of the electron transport \nmechanism  in the GaA s channel; when th e electron tr ansport chang es from ballistic  \nregim e to purely dif fusive regim e with  increasing tem perature, the conductivity \nmismatch occurs and destroys the spin valve ef fect. \nIn conclu sion, we have observed relative ly lar ge spin va lve ef fect by ballis tic \n 9transport in MnAs / GaAs / GaAs:MnAs hybrid  heterostructure, even when the barrier \nheight of M nAs/GaAs interface is v ery sm all. Our experim ental results have shown that \nby choosing a proper combination of FM / SC ma terials and using bal listic transport, we  \ncan obtain a lar ge spin valve ef fect without using a high resistance in terface. Noting that \nthe silicon technology node has reached 45 nm , it is expected that the ballistic transport \nof electrons in sem iconductors can be utiliz ed to m ake active sem iconductor -based spin \ndevices for future electronics. \nThis work was supported by JST -SORST/ PREST O, Grant-in-Aids for  \nScientific Research from MEXT , the Speci al Coordination Program s for Prom oting \nScience and T echnology , and Kurata Mem orial Hitachi Sci. & T ech. Foundation. One of \nthe authors (Pham  Nam Hai) thanks th e JS PS Research Fellowships for Y oung \nScientists.\n 10Refer ences\n \n[1]  S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). \n[2]  S. Sugahara and M. Tanaka, Appl. Phys. Lett. 84, 2307 (2004). \n[3]  Em erging research  devices,  International technology roadmap for semiconductor  \n2005 Edition . (http://www .itrs.net/Links/2005ITRS/Hom e2005.htm ) \n[4]  G. Schm idt, D. Ferrand, L. W. Molenkam p, A. T. Filip and B. J. Van Wees, Phys. \nRev. B 62, R4790 (2000). \n[5]  E. I. Rashba, Phys. Rev . B 62, R16267 (2000). \n[6]  A. Fert and H. Jaf fres, Phys. Rev . B 64, 184420 (2001). \n[7]  Y. Q. Jia, R. C. Shi, and S. Y. Choii, IEEE Trans. Magn. 32, 4707 (1996). \n[8]  P. R. Hammar , B. R. Bennett, M. J.  Yang, and M. Johnson, Phys. Rev . Lett. 83, \n203 (1999). \n[9]  F. G. Monzon, H. X. Tang and M. L. Roukes, Phys. Rev . Lett. 84, 5022 (2000). \n[10] S. Gardelis, C. G. Smith, C. H. W. Barnes, E. H. Linfield and D. A. Ritchie, Phys. \nRev. B 60, 7764 (1999). \n[11] A. T . Filip, B. H. Hoving, F . J. Jedem a, B. J.  Van Wees, B. Dutta and S. Bor ghs, \nPhys. Rev . B 62, 9996 (2000). \n[12] C.-M. Hu1, J. Nitta, A. Jensen, J. B.  Hansen and H. T akayanagi, Phys. Rev . B 63, \n125333 (2001). \n[13] A. T. Hanbicki, B. T . Jonker , G. Itskos, G . Kioseoglou and A . Petrou, Appl. Phys.  \nLett. 80, 1240 (2001). \n[14] X. Jiang, R. W ang, R.  M. Shelby , R. M. M acfarlane, S. R. Bank, J. S. Harris, S. S. P . \nParkin, Phys. Rev . Lett. 94, 056601 (2005). \n[15] G . Kirczenow , Phys. Rev . B 63, 054422 (2001). \n 11 \n[16] D. Grundler , Phys. Rev . B 63, 161307 (2001). \n[17] G . Schm idt and L. W. Molenkam p, Sem icond. Sci. Technol. 17, 310 (2002). \n[18] M. Tanaka, J. P . Harbison, M. C. Park, Y . S. Park, T. Shin and G. M. Rothber g, Appl. \nPhys. Lett. 65, 1964 (1994).  \n[19] J. De Boeck, R. Oesterholt, A. V an Esch, H. Bender , C. Bruynseraede , C. V an \nHoof and G . Borghs, Appl. Phys. Lett. 68, 2744 (1996). \n[20] H. Shim izu, M. Miyam ura and M. Tanaka, Appl. Phys. Lett. 7 8, 1523 (2001). \n[21] P . N. Hai, M. Y okoyam a, S. Oh ya and M. T anaka, Appl. Phys. Lett. 89, 242106 \n(2006). \n[22] T. Trypiniotis, D. H. Y. Tse, S. J. S teinm uller, W. S. Cho and J. A. C. Bland, IEEE \nTrans. Magn. 43, 2872 (2007). \n[23] J. G . Simmons. J. Appl. Phys. 34, 1793 (1963). \n[24] R. P . Panguluri, G . Tsoi, B. Nadgorny , S. H. Chun, N. Sa marth and I. I. Mazin, Phys. \nRev. B 68, 201307 (2003). \n[25] P . N. Hai, M. Yokoyam a, S. Ohya and M. Tanaka, Physica E 32, 416 (2006) \n 12Figur e legends \n \nFig. 1.  Schem atic structure of our spin valve devices  which c onsist of  MnAs thin f ilm (20 nm ) \n/ GaAs (10-30 nm ) / GaAs:MnAs  (5 nm ) grown on a p+GaAs (001) substrate. \n \nFig. 2.  Transport cha racteristics of  spin valve struc tures.  (a) GaAs-thickness  (tGaAs) \ndependence of the resistance R of four spin valv es with tGaAs = 10, 15, 20, 30 nm  plotted as \nlog(R) - tGaAs. The inset shows the band diagram  of MnAs / GaAs / Ga As:MnAs for the case \nof direct tunneling of electr ons through the rectangular -type GaAs tunnel barrier . The  \nresistances were m easured at 7 K with a bias vo ltage of  1 mV. The black line shows the f itted \nlog(R) - tGaAs by W KB approxim ation. The gr adient of  the fitted line  revea ls an ef fectiv e \nbarrier heig ht sm aller th an 1 m eV.  (b) tGaAs dependence of the resistance R of the above four \nspin valves plotted as R- (tGaAs)2. The inset shows the band di agram  of MnAs  / GaAs / \nGaAs:MnAs for the case of FN-tunneling of el ectrons through the tria ngular GaA s barrier . \nThe resistan ces were m easured at 7 K with a bi as voltage of 50 mV . The dashed line is a guide  \nto the eyes.  R is proportional to  (tGaAs)2, revealing the FN-tunnelin g nature of electron \ntranspo rt across the M nAs / GaAs interfaces.   (c)  conductance ( G) – bias voltage ( V) \ncharacteristics of the four spin v alves. The G – V characteristics are clea rly dif ferent from  that \nof a tunnel junction, but can be  explained by equation (1). The G – V characteristics can be \n 13divided into three areas: area I where the exponentia l part of equation (1) is s maller than unity \nthus G non-linearly increases with the increasing V, area II where the exponential part of \nequation (1) approaches unity  thus G is proportional to V, and area III where th e parallel \nconduction through the GaAs matrix l eads to non-linear  increasing of G with V. The thin solid  \nline shows the borders of those areas. \n \nFig. 3.  Magnetic-field dependence  of the r esistance of  the spin va lve s tructure  with  (a) tGaAs = \n30 nm . (b) tGaAs = 20 nm. (c) tGaAs = 15 nm . (d) tGaAs = 10 nm. The resis tances were m easured \nat 7 K with a bias of 50 mV . The m agnetic field was applied in plane along  the easy \nmagnetization axis [ 2011 ] of the MnAs thin f ilm, which is para llel to the GaAs[ 110] \nazim uth. The black and red curv es are m ajor and m inor loops, respectively . The m ajor loops \nare superposition of the m agnetoresistance (MR)  com ponents of the fe rrom agnetic electrodes \n(gradual ch ange) and the spin valve ef fect (abrupt jum ps of resi stance). The hysteresis  \nobserved in the m inor loops indicates that  the resistance jum ps correspond to the \nmagnetization reversal of the GaAs:MnAs nanoclusters. \n \nFig. 4. (a) Bias dependen ce of th e spin valv e ratios . The spin va lve ratios (de fined as \n(Rmax-RH=0)/RH=0) were m easured at 7 K. The sam ple with tGaAs = 10 n m has the la rgest sp in \nvalve ratio with a m aximum  of 8.2 % when the bias voltage V < 120 mV . When V > 120 mV , \n 14its sp in va lve ratio de creases rapidly  and becom es smaller th an that of  the sam ple with tGaAs = \n30 nm  due to the parallel conduction . (b) T emperatu re dependence of the spin valv e ratio s \nmeasured at a bias vo ltage of 50 mV . The sp in valve ratios decreas ed with increasin g \ntemperature, and disappeared at 90 K. \n 15 \nFig. 1. Hai et al.  \n 16 \nFig. 2. Hai et al.  \n 17 \nFig. 3. Hai et al.  \n 18 \nFig. 4. Hai et al.  \n 19"    },    {        "title": "0906.0246v1.Current_voltage_characteristics_of_tunable_ferromagnet_silicon_ferromagnet_channels_in_the_spin_blockade_regime.pdf",        "content": "arXiv:0906.0246v1  [cond-mat.mes-hall]  1 Jun 2009Current-voltage characteristics of tunable\nferromagnet-silicon-ferromagnet channels in the\nspin blockade regime\nD V Khomitsky\nDepartment of Physics, University of Nizhny Novgorod, Gagarin Av enue 23, 603950\nNizhny Novgorod, Russian Federation\nE-mail:khomitsky@phys.unn.ru\nAbstract. The steady current-voltage characteristics of ferromagnet-s ilicon-\nferromagnet channels with tunable emitter and collector polarizatio ns are investigated\nin the presence of spin blockade generalizing the model developed by Pershin Yu V\nand Di Ventra M (2008 Phys. Rev. B77073301). The dependence of the critical\ncurrent on both collector and emitter polarizations is obtained analy tically. It is found\nthat the current amplitude in the channel can be eﬀectively tuned b y varying the dif-\nference between the collector and emitter ferromagnet polarizat ions which allows to\nperform the magnetic manipulation of the electrical current in wide c lass of both n-\nand p-doped, low- and high-Ohmic semiconductor channels coupled t o ferromagnetic\nleads.\nPACS numbers: 72.25.Dc, 72.25.Mk, 73.23.Hk\nSubmitted to: J. Phys.: Condens. MatterCurrent-voltage characteristics of tunable F-Si-F channe ls 2\n1. Introduction\nThe progress of spintronics and physics of heterostructures wh ich can be observed\nduring the last years [1, 2] is focused on various physical phenomen a, and one of\nthem which attracts a considerable attention is the spin-dependen t transport through\nsemiconductor/spin-polarized junctions [3, 4, 5, 6, 7, 8, 9, 10, 11 , 12, 13, 14, 15]. The\nphysics of carrier polarization and its inﬂuence on transport in comp osite structures\nsuch as semiconductor/ferromagnet has been studied both theo retically [4, 6, 7, 8, 9,\n12, 13, 14, 15] and experimentally [3, 5, 10, 11]. One of the models de scribing the\nspin-resolved carrier concentrations and currents at the junct ion is the two-component\ndrift-diﬀusion model [6, 13, 14] which predicted highly nonlinear and s aturating current-\nvoltage dependence at a single semiconductor/feromagnet junct ion due to the eﬀect of\nspin blockade [13, 14]. In this model the detailed structure of the ch arge and current\ndistribution at the junction area [4, 6] as well as the Schottky barr iers [7, 8], the charge\nredistribution eﬀects [9], and the bound states [12] are not taken in to consideration.\nStill, the qualitative and distinguishable behaviour of current satura tion due to the\neﬀect of spin blockade is reliably predicted under various system par ameters such as\nthe junction/semiconductor resistance ratio. The spin blockade r egime arises from the\nspatial distribution of the spin-minority carriers which cannot ente r the ferromagnet\nregion and form a cloud near the junction which growth prevents th e further increase of\nspin-majority carrier transport if the current exceeds a thresh old value called the critical\ncurrent. Further studies have shown the importance and promisin g applications of this\neﬀect also for non-stationary phenomena such as spin memory eﬀe cts [15]. The models\ndescribed above were applied mainly to GaAs-based semiconductor c hannels, but is is\nknown that the silicon-based structures are also of big interest fo r spintronics due to\nthe dominating place of silicon in currently available electronic technolo gies. More, the\ntechnologies of fabricating the silicon/ferromagnet structures s uch as Si/Si:Mn formed\non a basis of diluted magnetic semiconductors have been intensively d eveloped during\nthe last few years [16, 17] which makes their future applications in sp intronics promising\nand creates certain questions about the phenomena described ab ove. Is there a spin\nblockade regime in a silicon/ferromagnet junction at speciﬁc values o f applied voltage,\ncarrier mobility and concentration ? If so, what is the critical curre nt density and how\nit depends on the silicon and ferromagnet parameters such as the c arrier polarization\nin ferromagnets and the conductivity of the semiconductor chann el? How deep can\nwe modulate the current in the channel by manipulating the polarizat ion of emitter\nor collector ferromagnets relative to each other? In the present manuscript we study\nthese problems in the framework of a simple but eﬀective model of tr ansport in the spin\nblockaderegime[13,14]whichwegeneralizeforthecaseofarbitrar ycarrierpolarizations\nin the emitting and collecting ferromagnetic regions of the channel a s well as for wide\nrange of low- and high-Ohmic n-doped and p-doped silicon samples. It is found that the\ncurrent can be deeply modulated by changing the spin alignment in the emitter and/or\ncollector ferromagnet since the critical current density is very se nsitive to it. We ﬁnd theCurrent-voltage characteristics of tunable F-Si-F channe ls 3\nanalytical expression for the critical current density and calculat e the current-voltage\ndependencies for various combinations of the channel/contact re sistance ratios, as well\nas for n- and p-type of doping with both high and low concentrations . The manuscript\nis organized as follows: in Section 2 we derive a model generalizing the d escription of\nthe spin blockade regime for the two-ferromagnet channel with ar bitrarypolarizations in\nthe emitter and collector ferromagnets and discuss the propertie s of the critical current\ndensity, in Section 3 we plot and discuss the current-voltage chara cteristics for various\ncombinations of system parameters, and the conclusions are given in Section 4.\n2. The model\nThe schematic view of the ferromagnet-silicon-ferromagnet chan nel is shown in Figure1.\nThe collector ferromagnet with the junction resistance r1is separated from the emitter\nferromagnet by a bulk silicon channel with length Lwhich we consider as exceeding the\nspin diﬀusion length lsgiven by [6, 13, 14]\nls=2D\nµE+/radicalig\nµ2E2+4D/τs(1)\nwhereDandµarethecarrier diﬀusion coeﬃcient anddrift mobility, respectively, E\nis the electric ﬁeld inside the channel, and τ∼10 ns is the typical spin relaxation time\n[2]. One can see from (1) that lsis maximal at zero electric ﬁeld when ls=√Dτs\nand when the diﬀusion coeﬃcient and spin relaxation time are big, or, s inceDis\ncoupled to the mobility via the Einstein relation D=µkBT/e[18], it is clear that\nthe spin diﬀusion length grows with the mobility. Since the mobility in the b ulk silicon\nis typically lower than the one for GaAs [19], for the given parameter s of the silicon the\ncondition L >> l sisalreadyfulﬁlledif L≥10mkmwhich isareasonablechannel length\nof bulk semiconductor structures. Hence, from the point view whe re the spin-resolved\nconcentration decay length is considered, the distance between e mitter and collector\nferromagnets here can be taken as inﬁnite which simpliﬁes the bound ary conditions.\nThe key parameters of our model are the variable polarization degr eesαandβin the\ncollector and emitter ferromagnets describing the state of their n on-ideality as well as\nthe chosen direction of polarization for the majority of carriers. F or example, the pair\nα= 1,β= 1/2 corresponds to the previously investigated case [14] where the carriers\narefullypolarizedinthecollector andfully unpolarizedintheemitter. C orrespondingly,\nthe choice α= 0.8,β= 0.7 describes the situation when the collector and emitter are\nhighly polarized in the same direction while the values α= 0.8,β= 0.3 describe the\nopposite polarization at the emitter. We adopt the labeling j1(2)for the spin-resolved\ncurrent with spins in ferromagnets aligned up (down). If jis the total current density,\nthan the spin-resolved current at the collector j1=αjwhich means that α= 1(0)\ncorresponds to an ideal ferromagnet where magnetic moments ar e all aligned up (down),\nandα= 1/2 describes the unpolarized current. The same labeling is adopted fo r the\ncarrier polarizationgeneratedby theemitter ferromagnet withpo larization β. The mainCurrent-voltage characteristics of tunable F-Si-F channe ls 4\nFigure 1. Schematic view of a ferromagnetic (F) - silicon (Si) - ferromagnetic\n(F) channel with variable polarization degrees αandβin collector and emitter\nferromagnets. The channel and contact zero-bias resistances arer0andr1, and the\ndirection of the carrier motion Iis shown (for the electrons the actual current is −I).\nThe spin blockade may occur in the contact region at the collector (le ft F) if the\npolarizations αandβsigniﬁcantly diﬀer from each other, which enhances the contact\nresistance from the zero-bias value r1as long as the current builds up.\npurpose of the present paper is to ﬁnd out how deep is the inﬂuence of the various α\nandβdiﬀerences reﬂecting the polarization switch on the total electrica l current in the\nchannel. Weinclude inourmodel thechannel andcontact zero-bias resistances r0andr1\n(see Figure 1) with variable ratio, and the direction of the carrier mo tion is labeled as I\n(for the electrons the actual current is −I). The spin blockade may occur in the contact\nregion at the collector (left F in Figure 1) if the polarizations αandβsigniﬁcantly diﬀer\nfrom each other, which enhances the contact resistance from th e zero-bias value r1as\nlong as the current builds up.\nThe model for the spin-resolved current densities j1,2and spin-resolved\nconcentrations n1+n2=NwhereNis the total carrier concentration is well-known\n[6, 13, 14] and consists of the continuity equation\ne∂n1,2\n∂t= div/vectorj1,2+e\n2τs(n2,1−n1,2) (2)\nand the equation for the spin-resolved current\n/vectorj1,2=en1,2µE+eD∇n1,2. (3)\nThese equations should beaccompanied by theboundary conditions atthecollector\ncontactx= 0 and at the other boundary of the channel xmax=L >> l swhich is in our\nproblem can be considered as x=∞. In this case the steady-state solution of (2) exists\n[13, 14] which has the form of two exponents decaying to their conc entrations deﬁned\nby the boundary conditions at the emitter. In our model the emitte r ferromagnet is\ndescribed by thearbitrarypolarization βwhich generalizes theunpolarized case β= 1/2\nconsidered previously [13, 14], so\nn1(∞) =βN (4)\nn2(∞) = (1−β)N. (5)\nThe steady-statesolution of(2) satisfying theboundaryconditio n(5) attheemitter\nand the normalizing condition n1+n2=Nhas the formCurrent-voltage characteristics of tunable F-Si-F channe ls 5\nn1(x) =βN−Ae−λx(6)\nn2(x) = (1−β)N+Ae−λx, (7)\nwhereλ= 1/lsis the inverse decay length deﬁned in (1). The parameter Ashould\nbe determined from the boundary condition at the collector when x= 0. If the collector\nis a non-ideal ferromagnet with the polarization α, then the spin-resolved current at the\ncollector\nj1(0) =αj, (8)\nand the other boundary condition at the emitter j1(∞) =βjis satisﬁed\nautomatically. The total current density jis related to the electric ﬁeld in the channel\nvia usual relation j=eµNEwhich allows to construct a closed equation for the current\ndensity and applied voltage. From (8) it follows that the conductivity of the collector\njunction is proportional to the concentration of the majority car riers, and at the absence\nof the current the junction resistance has a predetermined value r1while the silicon\nchannelwithlength Landcross-section Sisdescribedbytheresistancedeﬁnedinausual\nway by the carrier mobility, concentration and geometric dimensions asr0=L/(eµNS).\nAfter a simple algebra we obtain the Ohmic law I(r0+r1(I)) =Vwhich reads in our\ncasesimilar totheoneobtainedpreviously [14] foranon-polarizedem itter with β= 1/2,\nI\nr0+r11\n1−2/radicalig\n1+8[jcS\nI]2−1\n=V, (9)\nbut in our generalized model the parameter jcdeﬁned as the critical current density\n[14] depends on both the collector and emitter ferromagnet polariz ationsαandβ:\njc=j0\nc/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt2\n/parenleftigα−β/2\nβ/parenrightig2−1\n4(10)\nwhere\nj0\nc=eN/radicaligg\nD\n2τs(11)\nis the critical current density for the case of fully polarized collecto r ferromagnet\nwithα= 1 and fully unpolarized emitter with β= 1/2 [14]. Indeed, for α= 1 and\nβ= 1/2 the value of jcfrom (10) is equal to j(0)\nc.\nThe main property of equation (9) determining the current-voltag e characteristic\nis the current saturation occurring at current densities compara ble tojc. Thus, it is of\ninterest to investigate the critical current density (10) inmore de tails with respect to the\ntunable polarization parameters αandβof the collector and emitter ferromagnets. In\nFigure 2 the three-dimensional plot jc=jc(α,β) is shown for the critical current densityCurrent-voltage characteristics of tunable F-Si-F channe ls 6\nFigure 2. (Colour online) Dependence of critical current density jcmeasured in units\nofj(0)\ncon the polarizations αandβin collector and emitter ferromagnets. The case of\nfully polarizedcarriersin the collectorwith α= 1and fully unpolarizedelectronsin the\nemitterβ= 1/2 considered in [14] is shown by a reference point A when jc=j(0)\nc. The\nmagnitude of jcdiverges along the line α=βwhich means that in the case of a perfect\ncoincidence of the spin alignment in the emitter and collector the critic al current can\nnever be reached and there is no spin blockade in this limit. In the oppo site limit of\nmaximum diﬀerence between αandβthe value of jcis considerably smaller than j(0)\nc,\nso the spin blockade here occurs at lower current densities.\n(10) measured in units of j(0)\nc. The case of fully polarized carriers in the collector with\nα= 1 and fully unpolarized electrons in the emitter β= 1/2 considered in [14] is shown\nby a reference point A when jc=j(0)\nc. It is clear from (10) that jcdiverges along the\nlineα=βwhich means that in the case of a perfect coincidence of the spin align ment\nin the emitter and collector the critical current can never be reach ed and there is no\nspin blockade in this limit. In the opposite limit of maximum diﬀerence betw eenαand\nβthe value of jcis considerably smaller than j(0)\nc, so the spin blockade here occurs at\nlower current densities. The observed large variations of the critic al current density are\nimportant for the desired high degree of control onthe electrical current ﬂowing through\nthe ferromagnet-silicon-ferromagnet channel which can be achie ved by variations of atCurrent-voltage characteristics of tunable F-Si-F channe ls 7\nFigure 3. Current-voltage characteristics for a F-Si-F channel with L= 10 mkm\nandS= 5 mkm2shown in Figure 1 with (a) n-doped and (b) p-doped silicon with\nhigh carrier concentration N= 1019cm−3. On each plot the three families of curves\ncorrespond to three positions of the ”spin valve” emitter while the c arriers at the\ncollector are always highly polarized with α= 0.8. Each family has two curves labeled\n1and 2 (bold) which correspondsto the lowand highratiosofthe jun ction and channel\nresistances r1/r0= 1/5 andr1/r0= 5/1. The solid curves N1, N2 describe the case\nof non-polarized carriers at the emitter with β= 0.5 which was considered in [14], the\ndashed curves P1, P2 are for the emitter polarization β= 0.7 which is close to the\ncollector polarization α= 0.8, and the dash-dotted curves A1,A2 are for the emitter\npolarization β= 0.3 which diﬀers signiﬁcantly from the one of the collector. For both\ntype of carriers and for both low and high ratios of the contact/ju nction resistance\nthe current saturation is observed being the indication of the spin b lockade regime.\nThe switch between diﬀerent emitter polarizations leads to strong m odulations of the\ncurrent saturation amplitude, creating the possibility of the magne tization control on\nthe current.\nleast one ferromagnet polarization, say the emitter polarization β. Below we shall see\nthat these expectations are conﬁrmed by the current-voltage d ependencies in both n-\ndoped and p-doped channels with both low- and high-Ohmic resistanc e.\n3. Current-voltage characteristics\nThe current-voltage characteristics for n-doped and p-doped s ilicon channels with high\nand low carrier concentration are shown in Figures 3 and 4, respect ively.Current-voltage characteristics of tunable F-Si-F channe ls 8\nThe data for the room temperature drift mobility of electrons and h oles in the bulk\nsilicon doped with various concentrations was taken from the stand ard reference plots\n[19], allowing to estimate both channel resistance for a bulk silicon sam ple with L= 10\nmkm and S= 5 mkm2and the reference critical current density j(0)\ncfrom (11). In\nFigure 3 the results are shown for the silicon channel with high carrie r concentration\nN= 1019cm−3where the reference critical current density j(0)\ncis within the range of\n160...180 mkA /mkm2, and the channel resistance r0= 25...30 Ohm. On each plot\nthe three families of curves correspond to three positions of the ” spin valve” emitter\nwhile the carriers at the collector are always highly polarized with α= 0.8. Each\nfamily has two curves labeled 1 and 2 (bold) which corresponds to the low and high\nratios of the junction and channel resistances r1/r0= 1/5 andr1/r0= 5/1. The solid\ncurves N1, N2 describe the case of non-polarized carriers at the e mitter with β= 0.5\nwhich was considered in [14], the dashed curves P1, P2 are for the em itter polarization\nβ= 0.7 which is close to the collector polarization α= 0.8, and the dash-dotted curves\nA1,A2 are for the emitter polarization β= 0.3 which diﬀers signiﬁcantly from the one\nof the collector. For both type of carriers and for both low and high ratios of the\ncontact/junction resistance ratio the current saturation is obs erved being the indication\nof the spin blockade regime. The switch between diﬀerent emitter po larizations leads to\nstrong modulations of the current saturation amplitude, creating the possibility of the\nmagnetization control on the current.\nTo compare the results for low-Ohmic samples with the high-Ohmic one s, we\npresent in Figure 4 the current-voltage characteristics for the s ilicon channel doped\nwith low carrier concentration N= 1014cm−3where the reference critical current\ndensity ismuchlower andiswithintherangeof0 .002...0.003mkA /mkm2. Thechannel\nresistance is correspondingly much higher, and we consider r0= 200...500 kOhm for\nn-doped and p-doped samples, respectively. The labeling of all curv es is the same as\nin Figure 3, and the maximum achievable currents are much lower due t o signiﬁcantly\nsmaller carrier concentration which decreases the conductivity. O ne can draw the same\nconclusionsaboutthespinblockademanifestationandpolarizationc ontrolofthecurrent\namplitudes mentioned above which obviously can be applied also for the case of low\ncarrier concentrations. We can conclude that the main goal of the present model which\nis the achievement of a deep current modulation by the magnetizatio n switch at the\nemitter can be reached in both low- and highly-doped silicon samples, w ith both n-\ntype and p-type doping and with both low and high ratios of the junct ion/channel\nresistances. Hence, the proposed model of the tunable current -voltage characteristics in\na ferromagnet-silicon-ferromagnet channel seems to be applicab le to a rather wide range\nof ferromagnet-semiconductor structures.\n4. Conclusions\nWe have studied the steady-state current-voltage characteris tics of ferromagnet-silicon-\nferromagnet channels with long bulk silicon sample having the length ex ceeding theCurrent-voltage characteristics of tunable F-Si-F channe ls 9\nFigure 4. SameasinFigure3butforthe (a)n-dopedand(b) p-dopedsiliconch annels\nwith low carrierconcentration N= 1014cm−3. The labeling of curves is the same as in\nFigure 3, and the maximum achievable currents are lower due to signiﬁ cantly smaller\ncarrier concentration which decreases the conductivity. The sam e conclusions about\nthe spin blockade manifestation and polarization control of the cur rent amplitudes\nmentioned in the caption for Figure 3 can be applied here for the case of low carrier\nconcentrations.\nspin diﬀusion length. The current behaviour was investigated in the p resence of spin\nblockade regime at the collector junction, and the dependence of t he critical current on\nboth collector and emitter polarizations has been obtained analytica lly. It was found\nthatthecurrent amplitudecanbeeﬀectively tunedbyvaryingthed iﬀerence between the\ncollector and emitter ferromagnet polarizations which allows to perf orm the magnetic\nmanipulation on the electrical current in wide class of both n- and p-d oped, low- and\nhigh-Ohmic semiconductor channels coupled to ferromagnetic leads .\nAcknowledgments\nThe author is grateful to E.S. Demidov for many helpful discussions . The work\nwas supported by the Ministry of Education and Science RF under th e Program\n”Development of the Higher School Research Potential” (Grant No . 2.1.1/2833).Current-voltage characteristics of tunable F-Si-F channe ls 10\n[1] Awschalom D D, Loss D and Samarth N (eds) 2002 Semiconductor Spintronics and Quantum\nComputation (Nanoscience and Technology) (Berlin: Springer)\n[2] Zˇ uti´ c I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. 76323\n[3] Kawakami R K, Kato Y, Hanson M, Malajovich I, Stephens J M, Joh nston-Halperin E, Salis G,\nGossard A C and Awschalom D D 2001 Science294131\n[4] Zˇ uti´ c I, Fabian J and Das Sarma S 2002 Phys. Rev. Lett. 88066603\n[5] Epstein R J, Malajovich I, Kawakami R K, Chye Y, Hanson M, Petro ﬀ P M, Gossard A C and\nAwschalom D D 2002 Phys. Rev. B65121202(R)\n[6] Yu Z G and Flatt´ e 2002 Phys. Rev. B66201202(R)\n[7] Albrecht J D and Smith D L 2002 Phys. Rev. B 66113303\n[8] Albrecht J D and Smith D L 2003 Phys. Rev. B 68035340\n[9] Yu Yue, Li Jinbin and Chui S T 2003 Phys. Rev. B 67193201\n[10] Stephens J, Berezovsky J, McGuire J P, Sham L J, Gossard A C a nd Awschalom D D 2004 Phys.\nRev. Lett. 93097602\n[11] Crooker S A, Furis M, Lou X, Adelmann C, Smith D L, Palmstrom C J a nd Crowell P A 2005\nScience3092191\n[12] Dery H and Sham L J 2007 Phys. Rev. Lett. 98046602\n[13] Pershin Yu V and Di Ventra M 2007 Phys. Rev. B75193301\n[14] Pershin Yu V and Di Ventra M 2008 Phys. Rev. B77073301\n[15] Pershin Yu V and Di Ventra M 2008 Phys. Rev. B78113309\n[16] Demidov E S et al 2006 Pis’ma Zh. Eksp. Theor. Fiz. 83664 (2006 Journal of Experimental and\nTheoretical Physics Letters 83568)\n[17] Demidov E S et al 2009 Journal of Magnetism and Magnetic Materials 321690\n[18] Smith R A 1978 Semiconductors (Cambridge: Cambridge University Press)\n[19] Sze S M 1981 Physics of Semiconductor Devices (New York: John Wiley and Sons)"    },    {        "title": "1906.07302v2.Ferromagnetism_and_superconductivity_in_twisted_double_bilayer_graphene.pdf",        "content": "Ferromagnetism and superconductivity in twisted double bilayer graphene\nFengcheng Wu1and Sankar Das Sarma1\n1Condensed Matter Theory Center and Joint Quantum Institute,\nDepartment of Physics, University of Maryland, College Park, Maryland 20742, USA\n(Dated: April 2, 2020)\nWe present a theory of competing ferromagnetic and superconducting orders in twisted double\nbilayer graphene (TDBG). In our theory, ferromagnetism is induced by Coulomb repulsion, while\nsuperconductivity with intervalley equal-spin pairing can be mediated by electron-acoustic phonon\ninteractions. We calculate the transition temperatures for ferromagnetism and superconductivity as\na function of moir\u0013 e band \flling factor, and \fnd that superconducting domes can appear on both the\nelectron and hole sides of the ferromagnetic insulator at half \flling. We show that the ferromagnetic\ninsulating gap has a dome shape dependence on the layer potential di\u000berence, which provides an\nexplanation to the experimental observation that the ferromagnetic insulator only develops over a\n\fnite range of external displacement \feld. We also verify the stability of the half-\flled ferromagnetic\ninsulator against two types of collective excitations, i.e., spin magnons and valley magnons.\nI. INTRODUCTION\nMoir\u0013 e superlattices form in van der Waals bilayers with\na small orientation misalignment and/or lattice constant\nmismatch. Recently moir\u0013 e bilayers have emerged as a\nplatform to study fundamental physics of strongly in-\nteracting systems, in view of the discovery of correlated\ninsulating and superconducting states in twisted bilayer\ngraphene1,2. Moir\u0013 e superlattices often generate spatial\ncon\fnement for low-energy electrons, suppress electron\nkinetic energy, and therefore e\u000bectively enhance interac-\ntion e\u000bects. Evidences of correlated insulating and su-\nperconducting states have so far been reported in three\ngraphene-based moir\u0013 e systems, including twisted bilayer\ngraphene1{11, twisted double bilayer graphene12{16, and\nABC trialyer graphene on hexagonal boron nitride17{19.\nTwisted bilayer graphene (TBG) is a subject under\nintense theoretical study20{62, but the exact nature of\nthe correlated insulating (CI) and superconducting (SC)\nstates in TBG remains unsettled. The half-\flled corre-\nlated insulator in TBG crosses over to a metallic state\nby a strong perpendicular or parallel magnetic \feld1,2,\nwhich possibly rules out spin-polarized ferromagnetic\nstates, but leaves a large number of possible non-FM\nstates as candidates, e.g., valley polarized state, and\ncharge/spin/valley density wave states to name a few.\nBy contrast, there appears to be good experimental\nevidence that the half-\flled CI in twisted double bilayer\ngraphene (TDBG) with a twist angle \u0012around 1:3\u000eis fer-\nromagnetic, because the correlation driven insulating gap\nhas been found to be enhanced by an in-plane magnetic\n\feld12{16. Possible signature of SC domes in adjacent to\nthe CIs has also been reported in TDBG12,13. These ex-\nperimental discoveries identify TDBG as another impor-\ntant moir\u0013 e system with strong interaction e\u000bects. More-\nover, TDBG represents a simpler as well as a more tun-\nable system compared to TBG, because moir\u0013 e bands of\nTDBG can be controlled by an out-of-plane electric dis-\nplacement \feld and its \frst moir\u0013 e conduction band can be\nenergetically isolated from neighboring bands, whereasthe \frst moir\u0013 e valence and conduction bands in TBG are\ntypically connected via Dirac points.\nIn this paper, we theoretically study TDBG FM and\nSC orders in its \frst moir\u0013 e conduction band. In our the-\nory, ferromagnetism is driven by Coulomb repulsion as\nin Stoner model, but superconductivity is mediated by\nelectron-phonon interactions. FMCI can occur at half\n\flling when spin majority and minority bands are sep-\narated in energy by Coulomb exchange interaction. We\n\fnd that the FM insulating gap is tunable by a layer po-\ntential di\u000berence Uthat is generated by an external out-\nof-plane displacement \feld. This tunability originates\nfrom the strong dependence of the moir\u0013 e bands on U,\nand agrees with experimental observations12{16. We also\ncalculate spin and valley magnon spectrum and verify the\nstability of the FMCI.\nAway from half \flling, the state is generally metallic,\nwhich can be susceptible to superconducting instability\nat low temperature. Because electron-acoustic phonon\ninteractions in graphene mediate both spin singlet and\nspin triplet intervalley Cooper pairing60, superconduc-\ntivity can take place even in the presence of ferromag-\nnetism, as long as the spinless time-reversal symmetry is\npreserved. We estimate the transition temperatures of\nferromagnetism and superconductivity as a function of\n\flling factor, and \fnd that superconducting domes can\nappear on both sides of the half-\flled ferromagnetic in-\nsulator.\nOur paper is organized as follows. We describe the\nmoir\u0013 e Hamiltonian and band structure of TDBG in Sec-\ntion II. We \fnd that the moir\u0013 e bands are tunable by\nU, and the van Hove singularity in the non-interacting\ndensity of states can be tuned from below to above half\n\flling of the \frst moir\u0013 e conduction bands. This feature\nallows the controlling of many-body physics using the\nout-of-plane displacement \feld. We present theory for\nferromagnetism and superconductivity, respectively, in\nSections III and IV, and make a brief conclusion in Sec-\ntion V. Some technical details of the theory are given in\nAppendices A and B.arXiv:1906.07302v2  [cond-mat.str-el]  1 Apr 20202\n(a) (b)\n𝐴1 𝐵1\n𝐴2 𝐵2\n𝐴1 𝐵1\n𝐴2 𝐵2𝑈/2\n𝑈/6\n−𝑈/6\n−𝑈/2\nFIG. 1. (a) Top and (b) side view of twisted double bilayer\ngraphene. The top and bottom bilayers are marked by red\nand blue colors.\nII. MOIR \u0013E BANDS\nWe study TDBG with a small twist angle \u0012relative\nto the AB-AB stacking con\fguration, and calculate the\nmoir\u0013 e band structure using a continuum Hamiltonian\ngeneralized from TBG63to TDBG64{69. Within the\ncontinuum approximation, \u0006Kvalleys are treated sep-\narately. For each AB bilayer graphene, we use the fol-\nlowingk:pHamiltonian in + Kvalley\nH0(k) =0\nB@\u00010~v1k\u0000~v2k+\r1\n~v1k+ 0 ~v3k\u0000~v2k+\n~v2k\u0000~v3k+ 0 ~v1k\u0000\n\r1~v2k\u0000~v1k+\u000101\nCA;(1)\nwhich is in the basis of A1,B1,A2andB2sites [Fig. 1(a)]\nfrom one AB bilayer graphene. k\u0006stands forkx\u0006iky.\nParameter values are taken as\n(v1;v2;v3) = (0:844;\u00000:045;\u00000:091)\u0002106m/s;\n\r1= 361 meV ;\u00010= 15 meV;(2)\nwhich are extracted from ab initio results of Ref. 70. The\nmoir\u0013 e Hamiltonian in + Kvalley is given by\nH+K=\u0012\nhb(k)~T(r)\n~Ty(r)ht(k)\u0013\n; (3)\nwherehb(k) andht(k) arek:pHamiltonians for bot-\ntom and top bilayer graphene, and are equal to\nH0[^R(+\u0012=2)(k\u0000\u0014+)] andH0[^R(\u0000\u0012=2)(k\u0000\u0014\u0000)], respec-\ntively. Here ^R(\u0006\u0012=2) are rotation matrices and \u0014\u0006=\n[4\u0019=(3aM)](\u0000p\n3=2;\u00071=2). The moir\u0013 e period aMis ap-\nproximately a0=\u0012, wherea0is the monolayer graphene\nlattice constant. ~T(r) is the tunneling between bottom\nand top bilayer graphene, which varies spatially with the\nmoir\u0013 e period as speci\fed by\n~T(r) =\u0012\n0T(r)\n0 0\u0013\n;\nT(r) =w0T0+w1(e\u0000ib+\u0001rT+1+e\u0000ib\u0000\u0001rT\u00001);\nTj=\u001b0+ cos(2\u0019j=3)\u001bx+ sin(2\u0019j=3)\u001by(4)\n(a) (b)\n402002040𝜀(meV)\nത𝐾′ തΓ ത𝐾ഥ𝑀𝜃=1.24∘\n𝑈=45meV\n+Kvalley𝒞+𝐾=+2\n𝒞+𝐾=−2\n0.0 0.2 0.4 0.6 0.8 1.001234\nnnMDOS eV1nm2\n𝑛/𝑛𝑠\n1\n2\n3\n4\n5𝑈(meV)\n35\n40\n41.6\n45\n50DOS(eV−1nm−2)FIG. 2. (a) Moir\u0013 e bands in + Kvalley along high-symmetry\nlines. The blue (red) band is the \frst conduction (valence)\nband above (below) charge neutrality point, and has a Chern\nnumber of +2 ( \u00002) in +Kvalley. (b) Non-interacting density\nof states (DOS) per spin and per valley as a function of \flling\nfactor. The DOS are tunable by the layer-dependent potential\nU, and peaks at half \flling when U\u001941:6 meV.\u0012is 1:24\u000e\nfor numerical studies presented in this paper.\nwhere we only keep tunneling terms between adjacent\nlayers, andb\u0006are moir\u0013 e reciprocal lattice vectors given\nby [4\u0019=(p\n3aM)](\u00061=2;p\n3=2).w0andw1are two tunnel-\ning parameters, which in general have di\u000berent numerical\nvalues due to layer corrugation in the moir\u0013 e pattern22,71.\nWe takew0= 88 meV and and w1= 100 meV. An\nout-of-plane electric displacement \feld generates a layer\ndependent potential, which can be parametrized using a\nsingle parameter Uas illustrated in Fig.1(b). The point\ngroup symmetry of TDBG is D3in the absence of the\ndisplacement \feld ( U= 0), and is broken down to C3\nwhenUis \fnite.\nA representative moir\u0013 e band structure is shown in\nFig. 2 for\u0012= 1:24\u000eandU= 45 meV. The \frst con-\nduction band in Fig. 2(a) is isolated in energy from other\nbands, narrow in bandwidth( \u001813 meV), and topologi-\ncally nontrivial with a Chern number of +2 in + Kval-\nley. Because of time-reversal symmetry, the correspond-\ning moir\u0013 e band in \u0000Kvalley has the opposite Chern\nnumber.\nWe \fnd that the band dispersion can be drastically\ncontrolled by the potential U, as demonstrated by the\nenergy contour plots of the \frst moir\u0013 e conduction band\nshown in Fig. 3. In particular, the van Hove saddle\npoints can be e\u000bectively moved in the momentum space\nby tuningU. At a critical U\u001944 meV, three van Hove\nsaddle points merge to the corner of the moir\u0013 e Brillouin\nzone (MBZ), forming a high-order saddle point72. Cor-\nrespondingly, the density of states (DOS) for the non-\ninteracting band has a strong dependence on U, and the\nvan Hove singularity in the DOS can be tuned from below\nto above half \flling by varying U[Fig. 2(b)]. This strong\ndependence of the moir\u0013 e bands on Uhas implications on\ninteraction physics, as we explain in the following.3\n(a) 𝑈=35meV (b) 𝑈=44meV (c) 𝑈=50meV\n𝜀(meV) 12.0 25.5\n 𝜀(meV) 13.9 27.1\n 𝜀(meV) 15.2 28.3\nFIG. 3. Energy contours for the \frst moir\u0013 e conduction band in + Kvalley for di\u000berent values of U. Yellows lines mark Fermi\nsurface at the van Hove energy. Red points mark inequivalent van Hove saddle points, which merge to one higher-order saddle\npoint in (b).\nIII. FERROMAGNETISM\nA. Ferromagnetic Ground State\nMany-body interactions are e\u000bectively enhanced for\nelectrons in the nearly \rat moir\u0013 e bands. Here we study\n\ratband ferromagnetism driven by Coulomb repulsion\nusing a momentum-space formalism19,64,67. We only re-\ntain the \frst conduction band for simplicity, and the\nsingle-particle Hamiltonian projected onto this band is\nH0=X\nk;\u001c;s\"k;\u001ccy\nk;\u001c;sck;\u001c;s; (5)\nwherekis momentum measured relative to the center of\nthe MBZ,\u001c=\u0006is the valley index, srepresents spin\n(\",#) andcy\nk;\u001c;sis the fermion creation operator. \"k;\u001cis\nthe spin independent moir\u0013 e band energy; its valley de-\npendence is determined by time reversal symmetry, and\n\"k;\u001c=\"\u0000k;\u0000\u001c.\nWe project Coulomb interaction onto the \frst conduc-\ntion band, and the interacting Hamiltonian has the form\nH1=1\n2AX\nV(\u001c\u001c0)\nk1k2k3k4cy\nk1;\u001c;scy\nk2;\u001c0;s0ck3;\u001c0;s0ck4;\u001c;s;\nV(\u001c\u001c0)\nk1k2k3k4=X\nqV(q)O(\u001c)\nk1k4(q)O(\u001c0)\nk2k3(\u0000q);\nO(\u001c)\nk1k4(q) =X\n\u001b;`Z\ndreiq\u0001r\b\u0003\n\u001c;k1;\u001b;`(r)\b\u001c;k4;\u001b;`(r);(6)\nwhereAis the system area and \b \u001c;k(r) is the Bloch wave\nfunction for the \frst conduction band in valley \u001cKand\nat momentum k. The indices \u001band`respectively la-\nbel sublattices and layers. By time-reversal symmetry,\n\b\u001c;k(r) = \b\u0003\n\u0000\u001c;\u0000k(r). In the plane wave matrix elementO(\u001c)\nk1k4(q), the momentum qcan di\u000ber from k1\u0000k4by\nmoir\u0013 e reciprocal lattice vectors. Hamiltonian H1rep-\nresents density-density interaction, and preserves spin\nSU(2) and valley U(1) symmetry. In fact, H1has an\nenlarged SU(2)\u0002SU(2) symmetry, which stands for an\nindependent spin rotational symmetry within each val-\nley. Short-range interactions (e.g., atomic scale on-site\nHubbard repulsion), which we do not study explicitly,\nbreaks the SU(2) \u0002SU(2) symmetry down to spin SU(2)\nsymmetry.\nThe Coulomb interaction V(q) can be screened by di-\nelectric environment and nearby metallic gates. We as-\nsume that TDBG is encapsulated by an insulator (typi-\ncally boron nitride), and is in the middle to two metallic\ngates, which generate an in\fnite series of equally spaced\nimage charges with alternating signs. Under this im-\nage charge approximation, the screened Coulomb poten-\ntial in momentum space is V(q) = 2\u0019e2tanh(qd)=(\u000fq),\nwhere\u000fis the dielectric constant of the encapsulating\ninsulator, and dis the vertical distance between the\ntop (bottom) metallic gate and TDBG. We take dto\nbe 50 nm for all calculations presented in the follow-\ning. The Coulomb interaction energy scale is set by\nEC=e2=(\u000faM). At\u0012= 1:24\u000e,aM\u001911:4nm, and\nEC\u001912 meV for \u000f= 10. Since the typical Coulomb in-\nteraction energy scale is comparable to the bandwidth ( \u0018\n10 meV), there is a strong tendency towards symmetry-\nbreaking phases driven by interactions. The system is\ncharacterized by almost-\rat narrow noninteracting bands\nwith large Coulomb energy, a classic situation for the\nmanifestation of strong correlation physics.\nWe use Hartree-Fock (HF) approximation and assume\nthat both moir\u0013 e periodicity and valley U(1) symmetry are\npreserved, but allow spin polarization, motivated by the\nexperimental evidence of ferromagnetism12{14in TDBG.4\n30 35 40 45 500.00.51.01.52.02.53.03.5\nUmeVGapmeV𝜖=9\n𝜖=10\n𝜖=11\n𝜖=11.9ΔFM(meV)\n𝑈(meV)\nFIG. 4. The ferromagnetic insulating gap \u0001 FMat half \flling\nas a function of the layer potential di\u000berence Ufor di\u000berent\nvalues of dielectric constant \u000f. \u0001FMhas a dome shape, which\nexplains the experimental observation12{16that the FMCI ap-\npear only over a \fnite range of displacement \feld.\nThis leads to the following mean-\feld HF Hamiltonian\nHMF=X\nk;\u001c;sEk;\u001c;scy\nk;\u001c;sck;\u001c;s;\nEk;\u001c;s=\"k;\u001c+ \u0006k;\u001c;s;\n\u0006k;\u001c;s=1\nAX\nk0;\u001c0;s0V(\u001c\u001c0)\nkk0k0knF(Ek0;\u001c0;s0)\n\u00001\nAX\nk0V(\u001c\u001c)\nkk0kk0nF(Ek0;\u001c;s)(7)\nwhere the quasiparticle energy Ek;\u001c;sincludes moir\u0013 e band\nenergy\"k;\u001cand Hartree-Fock self energy \u0006 k;\u001c;s, andnF\nis the Fermi-Dirac occupation number. By projecting\nthe interaction onto the \frst conduction band, we ne-\nglected self-energy induced by interaction with all other\noccupied bands. This is expected to be a reasonable ap-\nproximation for TDBG because of the energetic separa-\ntion of neighboring bands from the \frst conduction band.\nThe neighboring band e\u000bects could, in principle, be in-\ncluded in the theory, if necessary, either by developing\na brute-force multiband mean \feld theory or perturba-\ntively, with the higher bands contributing paramterically\nsmaller terms because of the large energy denominators\nassociated with band separations.\nWe denote the electron \flling factor as \u0017=n=ns,\nwherenis the electron density, and nsthe density for\n4 electrons per moir\u0013 e cell. At half \flling \u0017= 1=2 that\ncorresponds to 2 electrons per moir\u0013 e cell, the mean-\n\feld theory in Eq. (7) leads to two distinct symmetry-\nbroken states, namely, valley-polarized state and valley-\nunpolarized state, which are degenerate at this particular\n\flling. Short-range interactions can break this symme-\ntry, as explained in Appendix A. Because the moir\u0013 e con-\nduction bands carry a valley-contrast Chern number, the\nvalley polarized state supports quantum anomalous Halle\u000bect (QAHE). To our knowledge, QAHE has not yet\nbeen observed in TDBG. Therefore, we leave the valley\npolarized state at \u0017= 1=2 in TDBG to future study, and\nfocus on valley unpolarized states in the following.\nBecause the Hamiltonian H=H0+H1has the\nenlarged SU(2)\u0002SU(2) symmetry, a valley unpolarized\nstate can have independent spin polarization in the two\nvalleys. For example, the two valleys can have ei-\nther parallel or antiparallel spin polarization. How-\never, atomic-scale on-site Hubbard interaction explic-\nitly breaks SU(2) \u0002SU(2) down to SU(2) symmetry, and\nselects the ferromagnetic state in which spins of the\ntwo valleys are polarized along the same direction (see\nAppendix A). In the following, we only consider val-\nley unpolarized state with identical spin polarization in\nthe two valleys. Therefore, we make the ansatz that\n\u0006k;\u001c;s= \u0006\u0000k;\u0000\u001c;s. This mean-\feld ansatz preserves spin-\nless time-reversal symmetry.\nWith the above ansatz, we solve the mean-\feld theory\nin Eq. (7) self consistently. One characteristic quantity\nis the zero-temperature ( T= 0) ferromagnetic gap \u0001 FM\nthat separates the occupied spin majority states from\nthe unoccupied spin minority states at \u0017= 1=2. We plot\n\u0001FMas a function of the layer dependent potential Ufor\ndi\u000berent dielectric constant \u000fin Fig. 4, which shows that\n\u0001FMhas a dome shape and is positive only over a \fnite\nrange ofUfor large\u000f. The dome shape in \u0001 FMcorrelates\nwith the non-interacting DOS [Fig. 2(b)], which peaks\nat\u0017= 1=2 atU\u001941meV. A larger non-interacting\nDOS at\u0017= 1=2 implies a stronger instability towards\nsymmetry breaking, and therefore, a larger interaction\ndriven energy gap. However, we note that this argument\nis only qualitative, as \u0001 FMdoes not exactly follow the\nnon-interacting DOS. An important conclusion we can\ndraw from Fig. 4 is that the FM insulating gap is tunable\nby an external displacement \feld, which agrees with the\nexperimental observation that the FMCI at \u0017= 1=2 only\ndevelops over a \fnite range of displacement \feld12{16.\nB. Magnon Spectrum\nA positive FM insulating gap \u0001 FMindicates that the\nFM state is a good ansatz at the mean-\feld level. To ex-\namine whether the FM state is stable beyond mean-\feld\ntheory, we calculate the energy spectrum for one-magnon\ncollective excitations. There are two types of magnons for\nthe FM insulator at half \flling, namely, spin magnons\nand valley magnons. The spin magnons involve collec-\ntive particle-hole transitions from the occupied spin ma-\njority band to the unoccupied spin minority band within\nthe same valley, as illustrated in Fig. 5(a); the valley\nmagnons are collective particle-hole transitions that \rip\nthe valley index, as shown in Fig. 5(b). We calculate the\nspin and valley magnon spectrum separately by solving\ntheir corresponding Bethe-Salpeter equations, following\nthe theory developed in Ref. 73. Details of the theory\ncan also be found in Appendix B.5\nK' K0246810Spin Wave meV\n10(a)\n−𝐾 +𝐾\n↓\n↑ ↑↓\nK' K0246810Valley Wave meV\n10(b)\n−𝐾 +𝐾\n↓\n↑ ↑↓ℰ𝑆(meV)\nℰ𝑉(meV)\n𝜖=10 𝜖=10\nFIG. 5. (a) Excitation spectrum for spin magnons (illustrated\nin the upper panel). The blue line marks the gapless spin\nwave mode. (b) Excitation spectrum for valley magnons (il-\nlustrated in the upper panel). The valley magnon spectrum\nis fully gapped. Uis 45 meV for the calculations.\nRepresentative spectra for spin and valley magnons\nare shown in Figs. 5(a) and 5(b), respectively. The spin\nmagnon spectrum has gapless spin wave modes, consis-\ntent with the Goldstone's theorem, as the continuous\nSU(2)\u0002SU(2) symmetry is spontaneously broken in the\nferromagnetic state. In fact, the SU(2) symmetry asso-\nciated with each valley is broken, so there are two spin\nwave modes, one for each valley. The overall spin exci-\ntation spectrum is nonnegative in Fig. 5(a), showing the\nstability of the FM insulator against spin-magnon exci-\ntations.\nBy contrast, the valley magnon spectrum shown in\nFig. 5(b) is gapped. This is consistent with the fact that\nthere is only U(1) symmetry in the valley space, and\nthe FM insulator does not break this valley U(1) sym-\nmetry. The positive valley magnon spectrum indicates\nthe stability of the FM insulator against valley-magnon\nexcitations, and also implies that the FM insulator is en-\nergetically more favorable than inter-valley density wave\nstate.\nBased on the spectrum shown in Fig. 5, we conclude\nthat the half-\flled FMCI in TDBG can be stable against\none-magnon collective excitations.\nC. Mean-Field Transition Temperature\nWe now turn to \fnite temperature physics and cal-\nculate the mean-\feld transition temperature TFMfor\nthe FM phase. To determine TFM, we de\fne \u0006(\u0006)\nk;\u001c=\n(\u0006k;\u001c;\"\u0006\u0006k;\u001c;#)=2. AtTFM, \u0006(\u0000)\nk;\u001cis in\fnitesimally small,\nand the self-consistent equation (7) can be linearized as\n0 0.5 1010203040506070𝑛/𝑛𝑠𝑇FM\n𝑇SC 𝑇SC𝑇(K)\n−𝐾 +𝐾Pairing\n↓\n↑↓\n↑FIG. 6. Mean-\feld transition temperatures for ferromag-\nnetism (blue lines) and superconductivity (red lines) as a\nfunction of \flling factor n=n s.Uis 45 meV and \u000fis 11 for\nthe calculation. The inset is a schematic illustration for the\ninter-valley equal-spin pairing in the ferromagnetic state.\nfollows\n\u0006(\u0000)\nk;\u001c=X\nk0M(\u001c)\nkk0\u0006(\u0000)\nk0;\u001c;\nM(\u001c)\nkk0=\u00001\nAV(\u001c\u001c)\nkk0kk0@nF(E)\n@EjE=\"k0+\u0006(+)\nk0;\u001c;(8)\nfrom which TFMcan be obtained by requiring the largest\neigenvalue of the matrix M(\u001c)to be 1.\nA representative plot of TFMas a function of \flling\nfactor is shown in Fig. 6, where ferromagnetism develops\nover a large range of \flling factors with TFMup to few\ntens of kelvin. We note that our mean-\feld theory over-\nestimates the tendency towards ordering, as \ructuations\nlike spin waves are neglected in the estimation of TFM.\nThe ferromagnetic state at half \flling can be an insulator\nat zero temperature, when the spin majority bands are\nfully \flled and separated from the empty spin minority\nbands by an energy gap \u0001 FM. Away from half \flling,\nthe ferromagnetic state is generically metallic with spin\ndependent Fermi surfaces.\nIV. SUPERCONDUCTIVITY\nThe metallic state away from half \flling can be sus-\nceptible to superconducting instability due to enhanced\nelectron-phonon interaction in moir\u0013 e \ratband systems.\nHere we study superconductivity mediated by electron-\nacoustic phonon interactions. The in-plane acoustic lon-\ngitudinal phonon modes mediate e\u000bective electron attrac-\ntion as follows\nHatt=\u0000g0X\n\u001b;\u001b0;`;sZ\ndr^ y\n+\u001b`s^ y\n\u0000\u001b0`s0^ \u0000\u001b0`s0^ +\u001b`s;(9)6\nwhere ^ \u001c\u001b`s(r) is the electron \feld operator at the coarse-\ngrained position rassociated with valley \u001cK, sublattice\n\u001b=A;B, layer`= 1;2;3;4 and spins=\";#. In Eq. (9),\nwe only retain attractive interactions that pair electrons\nfrom opposite valleys. The coupling constant g0is given\nbyD2=(\u001amv2\ns), whereDis the deformation potential, \u001am\nis the mass density of monolayer graphene, and vsis the\nvelocity of acoustic longitudinal phonon. Using D= 30\neV,\u001am= 7:6\u000210\u00008g/cm2,vs= 2\u0002106cm/s, we\nestimateg0to be 474 meV nm2. Here we neglect retar-\ndation e\u000bects in the phonon mediated electron attraction\nfor simplicity.\nAs we showed previously in Ref. 60, the attraction\nin Eq.(9) can be decomposed into four di\u000berent pair-\ning channels that are distinguished by their orbital and\nspin characters: (1) intrasublattice spin-singlet s-wave\npairing, i.e., ( isy)ss0^ y\n+\u001b`s^ y\n\u0000\u001b`s0; (2) intersublattice spin-\ntripletp-wave pairing, e.g., Fss0^ y\n+A`s^ y\n\u0000B`s0, whereF\ncan be any one of the three symmetric tensors ( s0\u0006sz)=2\nandsx; (3) intersublattice spin-singlet d-wave pairing,\ne.g., (isy)ss0^ y\n+A`s^ y\n\u0000B`s0; and (4) intrasublattice spin-\ntripletf-wave pairing, i.e., Fss0^ y\n+\u001b`s^ y\n\u0000\u001b`s0. Thes-wave\nandf-wave pairings are only distinguished by their spin\ncharacters, and the same is true for panddpairings.\nThe angular momenta of intersublattice Cooper pairs\narise from the valley-contrast sublattice chirality under\nC3rotation59,60. InABbilayer graphene, one of the\nsublattices in each layer [ A1andB2sites in Fig. 1(b)] is\npushed to higher energy by interlayer tunneling. There-\nfore, intersublattice pairing is energetically less favorable\ncompared to intrasublattice pairing in TDBG. In the fol-\nlowing, we only consider interactions that pair electrons\non the same sublattice, and project such interactions onto\nthe \frst moir\u0013 e conduction band. The projected pairing\nHamiltonian is\nHp=\u00001\nAX\ngkk0cy\nk;+;scy\n\u0000k;\u0000;s0c\u0000k0;\u0000;s0ck0;+;s;\ngkk0=g0AX\n\u001b;`Z\ndrj\b+;k;\u001b;`(r)j2j\b+;k0;\u001b;`(r)j2;(10)\nwhere we only keep interactions that pair electrons with\nopposite momenta, i.e., momentum kin +Kvalley and\nmomentum\u0000kin\u0000Kvalley. The pairing Hamiltonian\nHpalso has the SU(2) \u0002SU(2) symmetry, and supports\nboth spin singlet s-wave and spin triplet f-wave pairings.\nBecause of the ferromagnetism induced by Coulomb re-\npulsion, equal spin pairing is more favored compared to\nspin singlet pairing. Therefore, we consider intervalley\npairing between electrons with the same spin, which leads\nto the following Bardeen-Cooper-Schrie\u000ber (BCS) mean-\n\feld Hamiltonian\nHBCS=\u0000X\nk;s(\u0001k;scy\nk;+;scy\n\u0000k;\u0000;s+ H.c.);\n\u0001k;s=1\nAX\nk0gkk0hc\u0000k0;\u0000;sck0;+;si:(11)By combining the BCS Hamiltonian HBCSand the ef-\nfective single-particle Hamiltonian HFM[Eq. (7)] that is\nrenormalized by the Coulomb interaction, we obtain the\nsuperconducting linearized gap equation\n\u0001k;s=X\nk0\u001f(s)\nkk0\u0001k0;s;\n\u001f(s)\nkk0=gkk0\nA1\u00002nF(Ek0;+;s)\n2(Ek0;+;s\u0000\u0016);(12)\nwhere\u0016is the chemical potential, and Ek;\u001c;s=\"k;\u001c+\n\u0006k;\u001c;sis the e\u000bective band energy including the self en-\nergy. We have used the spinless time-reversal symmetry,\nwhich implies Ek;\u001c;s=E\u0000k;\u0000\u001c;s, to simplify the super-\nconducting susceptibility \u001f(s). Because of this symmetry,\nferromagnetism does not lead to depairing e\u000bect for su-\nperconductivity with intervalley equal-spin pairing. In\nEq. (12), spin up and down channels have independent\ngap equations. The superconducting transition temper-\natureTSCis reached when the largest eigenvalue of \u001f(s)\nis 1. Fig. 6 plots TSCas a function of \flling factor,\nand shows two superconducting domes respectively on\nthe two sides of the half-\flled ferromagnetic state. In\nFig. 6, we take the value of g0(the attractive interaction\nstrength) to be three times of 474 meV nm2(the value\nobtained from the above electron-acoustic phonon cou-\npling parameters) in order to get a value of TSCon the\norder of 1 K. We note that TSCis exponentially sensitive\ntog0as well as the moir\u0013 e band \ratness. A quantitative\nstudy ofTSCis beyond the scope of this paper. In any\ncase, the experimental parameters are not known with\nsu\u000ecient accuracy for a quantitative estimate of TSCat\nthis stage of development of the \feld. The main purpose\nof this section is to point out the possibility of phonon-\nmediated spin triplet pairing in a ferromagnetic system.\nWe discuss the e\u000bect of an in-plane magnetic \feld Bk\nonTSCin thef-wave channel. If the parent state for\nsuperconductivity is spin unpolarized, then TSCcan be\nslightly enhanced by Bkin the low-\feld regime, because\nZeeman energy leads to an e\u000bective spin dependent chem-\nical potential shift62,67,74. On the other hand, if the par-\nent state already has maximum spin polarization allowed\nby a given \flling factor, then an externally applied Bk\n\feld can no longer change the amount of spin polariza-\ntion, andTSCis reduced by Bkdue to orbital e\u000bect62,67.\nIn Ref.13,TSCis found to be slightly enhanced by weak\nBk\feld, indicating that the superconducting state has\nspin triplet pairing but with no spin polarization. Our\nmean-\feld phase diagram in Fig. 6 likely overestimates\nthe \flling range for ferromagnetism. We emphasize that\nthere is always a superconducting instability in a par-\ntially \flled band regardless of the presence or absence\nof ferromagnetism in our theory, where the superconduc-\ntivity is mediated by electron-phonon interactions and\nferromagnetism is driven by Coulomb repulsion. An ad-\nditional signature of electron-acoustic phonon interaction\nin TDBG is that phonon scattering can lead to large\nlinear-in-Tresistivity in transport above some crossover\ntemperatures75.7\nFinally, We note that experimental signatures of SC in\nTDBG are not yet conclusive, as discussed in detail in\nRef. 16.\nV. CONCLUSION\nIn conclusion, we have presented a theory of ferromag-\nnetism induced by Coulomb repulsion and superconduc-\ntivity mediated by electron-acoustic phonon interactions\nin moir\u0013 e bands of TDBG. In our theoretical phase dia-\ngram, there can be a ferromagnetic correlated insulator\nat half \flling, and superconducting domes on both the\nelectron and hole sides of the half-\flled insulator. Fer-\nromagnetism and superconductivity are two prototypical\norders that can occur in moir\u0013 e \rat bands, while there\nare many other possible competing and/or intertwined\norders, such as nematicity that breaks rotational symme-\ntry and density wave state that breaks moir\u0013 e translation\nsymmetry76. In TDBG, there is experimental evidence\nthat states with both spin and valley polarization are\npossibly stabilized at 1/4 and 3/4 \fllings by a \fnite in-\nplane magnetic \feld13,14,16. CIs at these factors could\nalso be spin and/or valley polarized states. Because of\nthe valley contrast Chern numbers in the non-interacting\nmoir\u0013 e bands, valley polarized CIs can also display quan-\ntum anomalous Hall e\u000bects. Our work should be viewed\nas a step towards a full quantitative theory of the po-\ntentially very rich TDBG phase diagram. A note-worthy\nqualitative feature of the current work is the possibility,\nalready apparent at the mean \feld level, that SC and\nFMCI phases, although they arise from di\u000berent interac-\ntions (electron-phonon for SC and electron-electron for\nFMCI), could compete with each other in TDBG moir\u0013 e\n\ratband with the FM phase centered around half-\flling\nand the SC domes manifesting on both electron- and\nhole-doped sides of half-\flling. The fact that this could\nbe the experimental TDBG situation may indicate that\nour theory captures some essential qualitative aspect of\nmoir\u0013 e interaction physics although our use of mean \feld\ntheory (and many other approximations, e.g., neglect\nof higher bands) exaggerates the quantitative stability\nof the symmetry-broken phases compared with experi-\nments.\nTDBG and other related moir\u0013 e systems represent a\nhighly tunable platform, where moir\u0013 e band structure can\nbe e\u000bectively controlled by the out-of-plane displacement\n\feld, as revealed by our theoretical study. This feature\nallows in situ control of band structure, and provides un-\nprecedented opportunities to study many-body physics.\nVI. ACKNOWLEDGMENT\nF. W. thanks Y.-T. Hsu, X. Li, and R.-X. Zhang for\ndiscussions. This work is supported by Laboratory for\nPhysical Sciences.Appendix A: Short-Range Interactions\nWe show that short-range interactions, in particular,\nthe atomic scale on-site Hubbard repulsion, explicitly\nbreak the SU(2)\u0002SU(2) symmetry down to spin SU(2)\nsymmetry, and favors ferromagnetic states in which spins\nin the two valleys are polarized along the same direction.\nThe on-site Hubbard repulsion on monolayer graphene\nhoneycomb lattice is given by\nH2=U0X\nR;\u001bby\nR\u001b\"by\nR\u001b#bR\u001b#bR\u001b\"\n=U0\nN0X\nkiX\n\u001bby\nk1\u001b\"by\nk2\u001b#bk3\u001b#bk4\u001b\";(A1)\nwhereU0is the on-site Hubbard interaction, Ris the\nlattice vector, \u001bis the sublattice index ( A,B),Nis the\nnumber of unit cells in the monolayer, and the prime on\nthe summation of the second line is the momentum con-\nservation constraint, i.e., k1+k2is equivalent to k3+k4\nmodulo reciprocal lattice vectors. To obtain a continuum\nmodel, we only keep states near \u0006Kpoints:\nH2\u0019U0\nN0X\nki0X\n\u001ciX\n\u001bby\nk1\u001c1\u001b\"by\nk2\u001c2\u001b#bk3\u001c3\u001b#bk4\u001c4\u001b\";(A2)\nwhere\u001c=\u00061 is the valley index, and the prime on the\nsummation of \u001ciimplies the valley conservation \u001c1+\u001c2=\n\u001c3+\u001c4due to momentum conservation. In the operator\nby\nk\u001c\u001bs, the momentum kis measured relative to \u001cK. We\nmake a Fourier transformation to introduce the coarse-\ngrained real-space position r:\nby\nk\u001c\u001bs=1p\nAZ\ndreik\u0001r^ y\n\u001c\u001bs(r); (A3)\nwhereAis the system area. The on-site repulsion can\nthen be transformed to a continuum Hamiltonian with\nlocal interaction:\nH2\u0019u00X\n\u001ciX\n\u001bZ\ndr^ y\n\u001c1\u001b\"^ y\n\u001c2\u001b#^ \u001c3\u001b#^ \u001c4\u001b\";(A4)\nwhereu0=U0A0, andA0=p\n3a2\n0=2 is the area per unit\ncell in the monolayer. The local repulsion in Eq. (A4)\ncan swap the valley indices of a pair of electrons, and\ntherefore, break the SU(2) \u0002SU(2) symmetry down spin\nSU(2) symmetry.\nWe project H2in Eq. (A4) to the \frst moir\u0013 e conduc-\ntion band, perform Hartree-Fock decomposition using the\nansatz given in the main text, and obtain the following\nmean-\feld Hamiltonian:\n~H2=X\n\u001c;k\b\n[1\nAX\n\u001c0k0u(\u001c\u001c0)\nkk0hcy\nk0;\u001c0;#ck0;\u001c0;#i]cy\nk;\u001c;\"ck;\u001c;\"\n+[1\nAX\n\u001c0k0u(\u001c\u001c0)\nkk0hcy\nk0;\u001c0;\"ck0;\u001c0;\"i]cy\nk;\u001c;#ck;\u001c;#\t\n;\n(A5)8\nu(\u001c\u001c0)\nkk0=u0AX\n\u001b`Z\ndrj\b\u001c;k;\u001b;`(r)j2j\b\u001c0;k0;\u001b;`(r)j2:(A6)\nIt is clear from Eq. (A5) that the local repulsion fa-\nvors a valley unpolarized but spin polarized ferromag-\nnetic state with spins associated with the two valley po-\nlarized to the same direction.\nThe energy scale for the short-range repulsion is EU=\nU0a2\n0=a2\nM, which is about 2 meV using U0= 5 eV, and\nis an order-of-magnitude weaker compared to the long-\nrange Coulomb interaction. The atomic-scale on-site\nHubbard interaction acts a weak anisotropy that breaks\nSU(2)\u0002SU(2) down to SU(2) symmetry, and selects a\nparticular set of states out of an SU(2) \u0002SU(2) multiplet.\nFor examples, the short-range Hubbard interaction aligns\nspins associated with the two valleys in the ferromagnetic\nphase, and suppresses s-wave but not f-wave pairing in\nthe superconducting phase.\nAppendix B: Theory for magnon excitations\nIn this Appendix, we present the theory for magnon\nexcitations, which has been discussed in moir\u0013 e systems\nin Refs. 73 and 77. The spin magnon states can be\nparametrized as follows\njQiS=X\nkzk;Qcy\nk+Q;+;#ck;+;\"jFMi (B1)\nwherejFMiis the half-\flled FM insulating state in which\nspin#bands in both valleys are empty, zk;Qare vari-\national parameters, and Qis the momentum of themagnon. In the magnon state jQiS, we make a single\nspin \rip from the occupied spin \"band to unoccupied\nspin#band within the same + Kvalley. Variation of the\nmagnon energy with respect to zk;Qleads to the following\nBethe-Salpeter equation\nES(Q)zk;Q=X\nk0H(Q)\nkk0zk0;Q;\nH(Q)\nkk0= (Ek+Q;+;#\u0000Ek;+;\")\u000ek;k0\u00001\nAV(++)\nk0(k+Q)(k0+Q)k;\n(B2)\nwhere the \frst part in H(Q)\nkk0is the quasiparticle energy\ncost of the particle-hole transition, and the second part\nrepresents the electron-hole attraction. The eigenvalue\nES(Q) represents the energy of spin magnons. We note\nthat there is another spin wave mode in \u0000Kvalley, which\ncan be formulated in a similar way as Eq. 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Box 35 (YFL), FI-40014 University of Jyv¨ askyl¨ a, Finl and\n2Institute of Physics, University of Tartu, Tartu, EE-50411 , Estonia\n3Moscow Institute of Physics and Technology, Dolgoprudny, 1 41700 Russia\n(Dated: December 24, 2019)\nSuperconductor-ferromagnet-superconductor Josephson j unctions are known to exist in the 0 and\nπstates with the transitions between them controlled by the t emperature and ferromagnetic in-\nterlayer thickness. We demonstrate that these transitions can be controlled also by the external\nmagnetic ﬁeld directed perpendicular to the layers. By vary ing the ratio of diﬀusion coeﬃcients in\nsuperconducting and ferromagnetic layers, these ﬁeld-con trolled transitions can be made detectable\nfor arbitrary large value of the exchange energy in the ferro magnet. We also show that the 0- πtran-\nsitions in the perpendicular ﬁeld can be observed as the spec iﬁc features of the ﬂux-ﬂowconductivity\ndependencies on the ferromagnetic thickness in accordance with recent experimental results.\nSuperconductor-ferromagnet-superconductor (SFS)\njunctions such as shown schematically in Fig. 1 are\nknown to have either 0 or πground state Josephson\nphase diﬀerence1,2. Switching between these states\ncontrolled by the parameters such as temperature Tand\nferromagnetic interlayer thickness dFare governed by\nthe oscillations of the Cooper pair wave functions as a\nresult of the energy splitting between the spin-up and\nspin-down states introduced by the exchange ﬁeld h3,4.\nTransitions between the 0 and π-states have been\nobserved experimentally as the strong oscillations of\nthe critical current of a junction5–8. Theπstate can\nbe also revealed in the closed electric loop with inte-\ngrated SFS junction by the appearance of spontaneous\nsupercurrents9. Nowadays, the π-junction state of SFS\nattracts much attention due to applications in the ﬂux-\nquantum logic based memory cells10–13and supercon-\nducting qubit implementations14,15.\nAlthough the 0- πtransition can be observed by chang-\ningthe temperatureforweakferromagnetswith small h5,\nthis can be more challenging in systems with h≫Tc0\nwhere we denote Tc0to be the bulk critical temperature\nof the superconducting layer. This can be illustrated us-\ning the temperature-thickness phase diagram in Fig. 2A\ncalculated at B= 0 as described below for the structure\nshown schematically in upper left panel in Fig. 1. For\nlarge exchange ﬁelds, h= 20Tc0, the boundary between\n0 andπstates is almost vertical, so that one should con-\ntroldFwith very high precision to spot the region of\ntemperature-controlled transition. The origin of this be-\nhaviour can be understood by considering the complex\nlengthξ−1\nF=/radicalbig\n(T+ih)/DFwhich determines the be-\nhaviour of superconducting correlations in the F layer\ncharacterized by diﬀusion coeﬃcient DF. Forh≫Tc0\nthe scale is temperature-independent for the considered\nregimeT < T c0and hence the 0- πswitching occurs at\nthe same dFfor all temperatures. Thus the only way to\nswitch SFS regularly from 0 to πstate in this case is to\nscan over dFwhich requires fabrication and measuringSFS\ndS2dFdS\nπ00 −π transition under magnetic field\n00B\nB\nvL\nAV00\nEjtrCrossover of the flux -flow resistivity\nB\nvL\nAVπ0\nEjtr\nFIG. 1: Upper panels: SFS structure where the\nincreasing perpendicular magnetic ﬁeld Bresults in the\ntransition from πto 0-state. Lower panel: Crossover of\nthe ﬂux-ﬂow resistivity ρffof the SFS structure upon\nthe transition from 0 to πstate with the change of F\nlayer thickness. vLis the velocity of Abrikosov vortex\n(AV) shown by the yellow color, jtris the transport\ncurrent, E=ρffjtris the electric ﬁeld.\nmany samples.\nHere we show that this situation can be improved by\nintroducingthe additionalcontrolparameterwhichis the\nmagnetic ﬁeld Bperpendicular to the layers. Unlike\ntemperature-driven 0- πtransition which requires weak\nF and ﬁne tuning of the F thickness, we show that the\ninterval of F thicknesses suitable for ﬁeld-driven transi-\ntion can be made arbitrarily wide for any exchange ﬁeld\nby reducing the diﬀusion constant of S with respect to\nthe one in F.\nIn the case ofapplied perpendicular magnetic ﬁeld, the\nscale of oscillations in the F layer is determined at small2\n0.00.10.20.30.40.5\ndF/dS0.00.20.40.60.8T/Tc020\n0-state0-state\nπ-statenormal stateh/Tc0=4\nAPhase diagram ( B=0)\n0.00.10.20.30.40.5\ndF/dS0.00.20.40.6Tc/Tc0B/H(0)\nc2=00-brπ-br\nB/H(0)\nc2=0.3\nDS=DF\nDS=0.01 DFB0,π branches of Tc(B)\n0.050.100.150.200.25\nd0π\nF/dS0.00.10.20.30.4Hc2/H(0)\nc2 D\nS=\nD\nFDS=0.01DFCF thickness of 0 −π transition vs field\nFIG. 2: (Color online) (A) Zero-ﬁeld B= 0 phase diagram for the ground states of SFS junction with red an d blue\ncurves corresponding to h/Tc0= 20 and h/Tc0= 4, respectively. Boundaries restrict the 0 state either from nor mal\none at higher temperatures (solid) or π-state at diﬀerent dF(dotted). The transition from πto normal state is\nshown by dashed curve. For panel A we have considered DS=DF. (B) Field-dependent critical temperatures of the\ntransitions to 0 and πstates shown by solid and dashed curves, respectively, and their ﬁ rst crossing points (dots).\nRed:B/H(0)\nc2= 0; blue, black: 0 .3 forDS/DF= 1 and 0 .01, respectively, and H(0)\nc2is the upper critical ﬁeld in the\nbulk. Vertical dotted line is 0- πtransition for B= 0. (C) Field dependencies of the 0- πtransition thickness d0π\nF\ndetermined by the ﬁrst crossing of the 0 and πbranches of Tc(B) for diﬀerent DS/DFratios. Note that each point\non the curves corresponds to the temperature Tc(B=Hc2,d0π\nF). Dots indicate the case Tc(B=Hc2,d0π\nF)→0. For\ncomparison the value of d0π\nF(B= 0,T= 0) is shown by dashed line. The maximal interval of the ﬁeld-contro lled 0-π\ntransition occuring at T→0 is shown by the red arrow. In (B, C) h/Tc0= 20, and in all panels dS= 3.3ξ, where\n2πTc0ξ2=DF.\ntemperatures T≪qbyξ−1\nF(B) =/radicalbig\n(q+ih)/DFwhere\nq=eBDF. Then, even for arbitrary large hthe orbital\neﬀect can introduce signiﬁcant shift of ξFifq∼h. This\nconditions can be achieved if the orbital depairing can\nbe made suﬃciently strong. The largest values of qcan\nbe obtained near the upper critical ﬁeld B=Hc2. By\ntaking into account estimation eHc2DS∼1, the regime\nq∼hrequires the diﬀusion coeﬃcient in the supercon-\nductorDSmuch smaller than that in the ferromagnet\nDS≪DF. This condition can be always achieved by\nintentionally adding impurities and decreasing electron\nscattering time in dirty S. The eﬀect is demonstrated in\nFigs. 2B,D as signiﬁcant shift of the F thickness segre-\ngating0 and πstates from its zero-ﬁeldvalue by applying\nmagnetic ﬁeld.\nIn the intermediate region of perpendicular magnetic\nﬁelds 0< B < H c2the SFS junction is in the mixed\nstate, which means that it is pierced by the Abrikosov\nvortex lines. The natural question is how the 0 and πsu-\nperconducting states manifest themselves in the vortex\nbehaviour. In principle, the discrepancy between distri-\nbutions of the gap order parameter in these states results\nin the diﬀerent response to the applied magnetic ﬁeld.\nThis was revealed recently, for instance, by superﬂuid-\ndensity measurements16.\nThe characteristic feature of the mixed state is a non-\nzero resistivity which occurs due to the dissipative mo-\ntion of mobile vortex lines in the superconducting envi-\nronment. Below we show that distinct gap proﬁles of the\n0 andπ-state lead also to the diﬀerence in ﬂux-ﬂow re-\nsistivity of SFS. We demonstrate that one can detect 0- π\ntransitions measuring the qualitative change in the de-\npendence of resisitivity on dFin the perpendicular mag-\nnetic ﬁeld.Below we present theoretical description consistent\nwith available ﬂux-ﬂow resistivity data for SFS17and\ndiscuss low-ﬁeld ﬂux-ﬂow resistivity experiment, where\nresisitivity of SFS is proportional to the vortex density\nin agreement with Bardeen-Stephen theory. We argue\nthat in the π-state the relevant numeric proportional-\nitycoeﬃcientexhibits universal h-independentbehaviour\nproviding a way to distinguish between the 0 and πstate\nof SFS by single ﬂux-ﬂow resisitivity measurement.\nModel. We start with the formalisim of quasiclassi-\ncal Green’s function (GF)18generalized to describe non-\nequilibrium spin states in diﬀusive superconductors19,\nˇg(t1,t2,r) =/parenleftbigg\nˆgRˆgK\n0 ˆgA/parenrightbigg\n, where ˆ gR/A/Kare the re-\ntarded/advanced/Keldysh components which are deter-\nmined by the Keldysh-Usadel equation\n{ˆτ3∂t,ˇg}t=ˆ∂rˆD(ˇg◦ˆ∂rˇg)−i[ˆτ3ˆH,ˇg]t,(1)\nwhereˆDis the diﬀusivity tensorwhich can be anisotropic\nand space-dependent, ˆH=ˆσh−ˆτ1ˆ∆, and ˆτiand ˆσi\n(i= 0,1,2,3) are Pauli matrices in Nambu and spin\nspace,his the exchange ﬁeld. The gap function ˆ∆ =\n|∆|e−iˆτ3ϕ, whereϕis the gap phase, is determined by\nthe self-consistency condition\n∆ =πλTrˆgK\n12(t,t)/4, (2)\nwhereλis the coupling constant ﬁnite in S layers. In\nEq. (1), the commutator operator is deﬁned as [ X,g]t=\nX(t1)g(t1,t2)−g(t1,t2)X(t2), similarly for anticommu-\ntator{,}t. The symbolic product operator is given by\n(A◦B)(t1,t2) =/integraltext\ndtA(t1,t)B(t,t2) and covariant diﬀer-\nentialsuperoperatorreadsas ˆ∂r=∂r−ieA[ˆτ3,·], wheree3\nis the elementary charge. The diﬀusion coeﬃcient is dif-\nferent in S and F regions. For simplicity we assume that\nF is isotropic Dz=Dx,y=DFwhile S is anisotropic\nwithDz=DFandDx,y=DS. The anisotropy assump-\ntion does not aﬀect results qualitatively since they rely\non the diﬀerence of diﬀusion coeﬃcients in the direction\nperpendicular to the magnetic ﬁeld, that is along the lay-\ners.\nFirst, we start with the equilibrium problem of the\nmagnetic-ﬁeld driven 0- πtransitions. Our goal is to ﬁnd\nthe range of parameters where the system undergoes this\ntransition with changing magnetic ﬁeld from 0 to Hc2in\nthe direction perpendicular to SF interface, B=Bz. To\ndetermine such parameters it is enough to compare the\nstates at the end points of this interval, namely at B= 0\nand atB=Hc2. We assume h=hzin F layer, put\nˆg(t1,t2) =/integraltext∞\n−∞ˆg(ε,t)e−iε(t1−t2)dε\n2π, wheret= (t1+t2)/2,\nandusegradientexpansionfortime-convolutionproducts\nto obtain equations for the temperature GF by replacing\n−iεwith Matsubara frequency ωn.\nIn the absence of magnetic ﬁeld B= 0, we deter-\nmined the lowest-energy state of the SFS system on the\nT,dFplane by evaluating the free energy20using self-\nconsistent distribution of ∆( z) and corresponding GF21.\nBy comparing numerical values of the 0 and πbranches\nof free energy, we obtained the ﬁrst-order 0- πtransition\nlines shown in Fig. 2A. Previosly, thermodynamic 0- π\ntransition was discussed only within Ginzburg-Landau\ntheory22.\nOne can see that for parameters h≫Tc0typical for\nstrong ferromagnets such as Co the 0- πtransition curve\nis almost vertical, that is the threshold thickness d0π\nFde-\npends on the temperature very weakly. Practically, this\nmeans that it quite diﬃcult to choose dFin the range\nwhere SFS system has temperature-controlled 0- πtran-\nsition.\nTo ﬁnd how magnetic ﬁeld changes critical tempera-\ntures of the 0 and π-states we generalize the multi-mode\napproach used previously for SF bilayers23,24. Using the\nsymmetryofsolutionswereducethe SFSproblemtothat\nof the SF bilayer with diﬀerent boundary conditions at\nfree F interface corresponding to 0 and πstates. We con-\nsider gauge A=yBxand apply the Abrikosov ansatz\n∆ =/summationdisplay\nmCmeimpy˜∆(x−mx0,z), (3)\nˆg12n=/summationdisplay\nm,σCmeimpyfσn(x−mx0,z)ˆσσ.(4)\nHere anomalous Matsubara GF (4) is extended into spin\nspace by introducing 2ˆ σσ= ˆσ0+σˆσ3, where σ=±.\nOther notations are conventional for lattice solution,\nnamely, |Cm|= 1,pis deﬁned by lattice symmetry,\nx0=pL2\nHandL−2\nH= 2eB. Next we separate vari-\nablesfσn(x,z) = Ψ0(x)ασn(z) and˜∆(x,z) = Ψ0(x)β(z),\nwhereΨ 0=e−L−2\nHx2/2iszeroLandauleveleigenfunction,\nto obtain Usadel Eq. in the form\nDz∂2\nzασn−2[(ωn+q+iσh)ασn+iβ] = 0,(5)together with self-consistency condition β=\nλπiT/summationtext\nσ,n≥0ασn. Here q=eBDxis the orbital\nenergy. We solve21Eq. (5) together with boundary\nconditions and self-consistency equation by means\nof multi-mode approach23,24yielding the the upper\ncritical ﬁeld Hc2=Hc2(T) or ﬁeld-dependent critical\ntemperature Tc=Tc(B).\nTheresultingdependencies Tc=Tc(dF) fordiﬀerent B\nare shown in Fig. 2B. For small magnetic ﬁelds, we have\nintersecting 0 and πbranches resulting in the oscillatory\nbehaviour of Tc(dF)23,25,26. For larger B, there appear\nintervals of dFwith only one stable state, either 0 or π\nas shown by blue curves in Fig. 2B. The 0- πtransitions\noccur at the values ofthickness d0π\nFdetermined by the in-\ntersection of 0 and πbranches of Tc(dF) (shown by dots\nin Fig. 2B). Fig. 2C demonstrates the magnetic ﬁeld\ndependence of d0π\nFconﬁrming our qualitative arguments\nabout its high sensitivity to the ratio of diﬀusion coeﬃ-\ncients in F and S layers. For DF/DS> h/T c0(black line\nin Fig.2C ) there is a strong variation of threshold thick-\nness with ﬁeld as compared with almost no dependence\nofd0π\nFin the opposite case (blue line in Fig.2C).\nTo understand the SFS behaviour under applied mag-\nnetic ﬁeld it is enough to compare the endpoints which\nare the states at B= 0 and at B=Hc2shown in Figs.\n2A,C, respectively. In Figs. 2C the 0 states at B=Hc2\nare on the left of the corresponding solid curve while π\nstates at B= 0 and T= 0 are on the right of the dashed\nline. For larger Tthe shift of dashed line is negligible as\ncan be inferred from Fig.2A. From comparison of dashed\nand solid black lines in Figs. 2C one can see that for\nDS= 0.01DFthere is a wide interval of dFwhere the\n0-πtransition with necessity occurs when varying mag-\nnetic ﬁeld from 0 to Hc2at ﬁxedTanddF. This inter-\nval bounded by d0π\nF(Hc2) curve and d0π\nF(B= 0) value is\nshown by the red arrow in Fig. 2C.\n−10010\nx/ξ−3−2−10123z/ξ\nSFSvortex axisA0-state\n0.000.160.320.480.640.80\n−10010\nx/ξ−3−2−10123z/ξ\nSFSvortex axisBπ-state\n0.000.120.240.360.480.600.72Gap amplitude | |/ 0\nFIG. 3: (Color online) Gap proﬁle |∆|/∆0normalized\nto the bulk gap ∆ 0in the vortex cell of SFS for the 0\n(A) and π-state (B). Calculations have been done for\ndF/dS= 0.17,h/Tc0= 6 and DS=DF. White\nhorizontal lines correspond to SF interfaces.\nFlux-ﬂow resistivity . At intermediate values of\nmagnetic ﬁeld 0 < B < H c2SFS system is in the mixed\nstate consisting of Abrikosov vortex (AV) lines shown\nschematically in lower panels of Fig. 1. Vortex struc-\nture transforms due to the proximity eﬀect27,28. Such a\ntransformation is diﬀerent in the 0 and πstates of the\nSFS system which aﬀects their dynamical properties as4\nshown below.\nTo calculate the structure of individual vortices\nat ﬁnite magnetic ﬁelds we use the circular cell\napproximation29–32, where the unit cell of the hexago-\nnal vortex lattice hosting a single vortex is replaced by a\ncircular cell with the centre at the point of superconduct-\ning phase singularity. Inside circular cell, the gap and\nmagnetic ﬁeld distributions are taken radially symmetric\nwith respect to the cell centre. At that, the circular-\ncell radius is uniquely deﬁned by magnetic induction,\nrc=/radicalbig\nφ0/(πB) so that there is exactly one ﬂux quan-\ntumφ0=π/epassing through the unit vortex cell21.\nCalculated gap proﬁle inside the cell is shown in Fig. 3.\nThe controlled motion of the vortices can be pro-\nduced by applying transport current jtrwhich exerts\nthe Lorentz force FL=φ0jtr×zon each vortex due\nto interaction with its local magnetic ﬁeld. Vortex mo-\ntionwithvelocity vLproducesperpendicularelectricﬁeld\nE=B×vLas shown in Fig. 1. This ﬁeld causes energy\ndissipation due to the ohmic losses inside the normal vor-\ntex core which can be expressed as the viscous friction\nF=−ηvL, whereηis vortex viscosity. In the steady-\nstate regime, F+FL= 0, we obtain E=ρffjtr, where\nρff=φ0B/ηis ﬂux-ﬂow resistivity.\nTo calculate viscosity η, we consider microscopic\nexpression33,34for the force Facting due to non-\nequilibrium environment21. The latter is determined by\nthe vortex-motion induced deviations of electron distri-\nbution function from the Fermi-Dirac one which obey\nkinetic equations derived21from the Keldysh part of the\nKeldysh-Usadel Eq. (1). The coeﬃcients in kinetic equa-\ntions are determined by the vortex structure that we ﬁnd\nfrom equilibrium problem as explained above. We con-\nsider low-temperature regime where the nonequlibrium\nstates have subgap energies and therefore their contri-\nbutions relax at the distances of the order of coherence\nlength. Thisisdiﬀerentfromthe vicinityof Tcwherevor-\ntex motion in multilayered systems is determined by the\nrenormalization of long-range charge imbalance mode35.\nIn Fig. 4B we show the calculated ρfffor stable\nparts of the 0 and πbranches. The intersection of these\nbranches points to the ﬁrst-order 0- πtransition whose\nposition scales with ξF∼1/√\nh. The crossover be-\nhaviour of ρffat 0-πtransition can be understood quali-\ntatively using the Bardeen-Stephen expression ρff/ρn=\nβ−1B/Hc2, where β∼1 is determined by the partic-ular microscopic model and ρnis normal-state resistiv-\nity. The inverse upper critical ﬁeld of SFS trilayer21,24,36\nH−1\nc2(dF) is shown in Fig. 4B. The variations of H−1\nc2(dF)\nfollow closely the behaviour of ρff. Their oscillation in\nthe vicinity of the 0- πtransition is caused by the su-\nperconductivity suppression with dFin the 0 and its\nenhancement in the πstate. In the considered low-\ntemperature regime β≈0.77 for usual single-band dirty\nsuperconductors37. Fig. 4D demonstrates dependencies\nβ(dF) for SFS sandwich. We see that bulk value is ap-\nproached in the limit dF→0, that is in the absence of\nF layer. For ﬁnite dF, the function β(dF) passes in the\n0-state through the maximum whose height exceeds bulk\nvalue 0.77 for not very weak F. At that, in the π-state\nβ <0.77 approaches universal h-independent asymptotic\nweakly varying with dF. These signatures of βcan be\nused for distinguishing the state of SFS with the help of\nthe single ﬂux-ﬂow resistivity experiment without fabri-\ncating and measuring many samples.\nResults shown in Fig. 4A are in qualitative agreement\nwith measurements demonstrating the increase followed\nby the saturation of ﬂux-ﬂow resistivity in SFS trilayer\nwiththegrowthof dF17. Althoughoscillationsof ρff(dF)\nand 0-πtransition point were not directly detected in\nthis experiment, even in such a case ﬂux-ﬂow resisitivity\nmeasurements allow to distinguish between samples in\nthe 0 and πstates by means of the βvalue as discussed\nabove.\nTo conclude, we have demonstrated the possibility of\nthe 0-πtransitions in the SFS structure driven by the\nperpendicular magnetic ﬁeld. These transitions can be\nachieved in the wide interval of the F layer thicknesses\nprovidedtheSlayerhasmuchsmallerdiﬀusioncoeﬃcient\nthan F layer. In contrastto the temperature-drivenones,\nthe magnetic ﬁeld-driven 0- πtransitions can be realized\nin principle for arbitrary large exchange ﬁeld h≫Tc.\nBesides that we have found indications of 0- πtransitions\nin the ﬂux-ﬂow conductivity of SFS structure. This be-\nhaviourisin thequalitativeagreementwith experimental\nobservations.\nThis work was supported by the Academy of Fin-\nland (Project No. 297439), Russian Science Foundation\n(Grant No. 19-19-00594) and the European Regional\nDevelopment Fund (Mobilitas Pluss grant MOBTP152).\nIt is our pleasure to acknowledge discussions with M.\nYu. Kupriyanov, M. M. Khapaev, N. Pompeo and A.S.\nMel’nikov.\n∗mikesilaev@gmail.com\n1A. I. Buzdin, L. Bulaevskii, and S. Panyukov, JETP. Lett.\n35, 147 (1982).\n2A. I. Buzdin and M. Y. Kupriyanov, JETP. Lett. 53, 321\n(1991).\n3A. I. Buzdin, Rev. 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Buzdin2\n1Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia\n2Institut Universitaire de France and University Bordeaux,\nLOMA UMR-CNRS 5798, F-33405 Talence Cedex, France\n(Dated: August 10, 2018)\nWe studypeculiarities of proximity eﬀect in clean supercon ductor – ferromagnet structures caused\nby either spatial or momentum dependence of the exchange ﬁel d. Even a small modulation of the\nexchange ﬁeld along the quasiparticle trajectories is show n to provide a long range contribution\nto the supercurrent due to the speciﬁc interference of parti cle- and hole- like wave functions. The\nmomentum dependence of the exchange ﬁeld caused by the spin – orbit interaction results in the\nlong – range superconducting correlations even in the absen ce of ferromagnetic domain structure\nand can explain the recent experiments on ferromagnetic nan owires.\nThe exchange ﬁeld hin ferromagnetic (F) metals is\nwell known to destroy Cooper pairs resulting, thus, in a\nstrong decay of superconducting (S) correlations in the\nF material and suppression of Josephson current in SFS\njunctions (see Refs. 1, 2 for review). Considering the\nquantum mechanics of quasiparticle excitations this de-\nstructive eﬀect of the exchange ﬁeld can be viewed as a\nconsequence of a phase diﬀerence γ∼L/ξh= 2Lh//planckover2pi1VF\ngained between the electron- and hole- like parts of the\ntotal wave function at the path of the length L. Both\nin the clean and dirty limits the measurable quantities\nshould be calculated as superpositions of fast oscillating\ncontributions eiγfrom diﬀerent trajectories and, thus,\nrapidly vanish with the increasing distance from the SF\nboundary.\nThis textbook physical picture appears to be in sharp\ncontrastwith anumber ofrecentexperiments [3–8] which\npoint to an anomalously large length of decay of super-\nconducting correlations inside the F metal. As we can\njudge from the observation [8] of a noticeable supercur-\nrent through a Co nanowire, this decay length can be of\nthe order of half a micrometer which well exceeds typi-\ncal coherence lengths in ferromagnets both in the clean\nand dirty limits. In the dirty limit such strong proxim-\nity eﬀect can hardly be explained even taking account\nof long–range triplet correlations [2] induced by the ex-\nchange ﬁeld inhomogeneity.\nNaturally, the inhomogeneity of the ﬁeld hcaused\nby the ferromagnetic domain structure can improve the\nconditions of Cooper pair survival in the clean limit as\nwell. To suppress the destructive trajectory interference\nmentioned above the domain structure should cancel the\nphase gain γfor a certain group of quasiparticle trajec-\ntories. A simple example of such phase gain compen-\nsation can be realized in a clean junction consisting of\ntwo F layers with opposite orientations of magnetic mo-\nment [9, 10]. On the other hand in the diﬀusive limit\nthis compensation eﬀect vanishes [11]. Note, that the ex-\nchange ﬁeld inhomogeneity along the quasiclassical tra-jectory experiencing multiple reﬂections from the ferro-\nmagnet surface can appear even in the absence of the\nspatial domain structure. Indeed, the exchange ﬁeld act-\ning on band electrons in a solid with a ﬁnite spin – orbit\ninteraction should obviously depend on the quasiparticle\nmomentum [12]: h=h(k). The normal quasiparticle\nreﬂection is accompanied, of course, by the change in the\nmomentum direction, and, thus, by the change in the\nexchange ﬁeld. The momentum dependent hﬁeld can\nstrongly aﬀect the phase gain γalong the trajectories\neven in the F sample prepared in a single domain state\n(as it has been done in experiments with Co nanowires\n[8]).\nThe goal of this paper is to show that in the clean\nlimit there exists a possibility to cancel the particle –\nhole phase diﬀerence for a large group of quasiclassical\ntrajectories due to either spatial or momentum depen-\ndence of the exchange ﬁeld. Such set of trajectories pro-\nvides a long–range contribution to the Josephson cur-\nrent through a ferromagnetic system which decays at the\nlength scale characteristic for a nonmagnetic metal. We\nconsider two generic examples which illustrate the above\nscenario of a long–range proximity eﬀect: (i) Josephson\ntransport through a pair of ferromagnetic layers with a\nstepwiseexchangeﬁeld distribution; (ii) Josephsontrans-\nport through a nanowire with a specular electron re-\nﬂection at the surface and exchange ﬁeld varying with\nthe changing quasiparticle momentum. See Supplemen-\ntal Material at [URL will be inserted by publisher] for\ndetails of calculations.\nJosephson transport through a ferromagnetic bilayer. Let\nus start from the simplest model illustrating the origin\nof the quasiparticle interference suppression: Josephson\njunction containingtwo ferromagneticlayersofthe thick-\nnessesd1andd2, respectively (see Fig.1). Here we con-\nsider the limit of short junction d1+d2≪ξs, whereξs\nis the superconducting coherence length. The exchange\nﬁeldsh1andh2in the layers are rotated at the angle\nα. Following the quasiclassical procedure considered in2\n/c113xSL SRd1d2\n/c97 sh1h2\nFigure 1: (Color online) Josephson junction containing two\nferromagnetic layers. Linear quasiparticle trajectory is shown\nby the red dashed line.\nRef. 13 we ﬁnd the current – phase relation:\nI=/summationdisplay\nnIn=/summationdisplay\nnansinnϕ∝angbracketleft(n,nF)cosnγ∝angbracketright\n∝angbracketleft(n,nF)∝angbracketright,(1)\nwherenis the unit vector normal to the junction plane,\nnFis the unit vector along the trajectory, and anare\nthe coeﬃcients of the Fourier expansion for the current\n– phase relation ISNS(ϕ) for zero exchange ﬁeld, i.e.,\nfor superconductor – normal metal junction of the same\ngeometry. The angular brackets denote the averaging\nover diﬀerent quasiclassical trajectories. The ﬁrst two\ncoeﬃcients in this expansion take the form:\nan=4eT\n/planckover2pi1N(−1)n−1∞/summationdisplay\nm=0/parenleftBig\nµm−/radicalbig\nµ2m−1/parenrightBign\n, n= 1,2,\n(2)\nwhereµm= 2π2T2(2m+ 1)2/∆2\n0+ 1, ∆ 0is the\ntemperature dependent superconducting gap, N=\ns−1\n0/integraltext\nds/integraltext\ndnF(nF,n),s−1\n0=kF/2π(s−1\n0= (kF/2π)2)\nfor 2D (3D) junctions, and the integral/integraltext\n...dsis taken\nover the junction cross–section. The factor Nis deter-\nminedbythenumberoftransversemodesinthejunction:\nN∼S/s0, whereSis the junction cross–section area.\nThe phase γcan be found from the singlet part of the\nanomalous quasiclassical Green function:\nfs(s=sR) = cosγ\ntaken at the right superconducting electrode. Here we\nuse a standard parametrization [14] f=fs+ftˆσ, where\nˆσisaPaulimatrixvectorinthespinspace. Thefunctions\nfs,ftsatisfy the linearized Eilenberger equations written\nfor zero Matsubara frequencies\n−i/planckover2pi1VF∂sfs+2hft= 0,−i/planckover2pi1VF∂sft+2fsh= 0,(3)\nand the conditions fs(s=sL) = 1,ft(s=sL) = 0 at\nthe left superconducting electrode. Solving the above\nequations for the particular bilayer geometry we ﬁnd:\ncosγ= cos2α\n2cos/parenleftbiggd1+d2\nξhcosθ/parenrightbigg\n+sin2α\n2cos/parenleftbiggd1−d2\nξhcosθ/parenrightbigg\n,\n(4)where cos θ= (n,nF). This expression allows us to write\nthe ﬁrst harmonic in the current – phase relation in the\nform:\nI1=/bracketleftbigg\ncos2α\n2Ic1/parenleftbiggd1+d2\nξh/parenrightbigg\n+sin2α\n2Ic1/parenleftbiggd1−d2\nξh/parenrightbigg/bracketrightbigg\nsinϕ,\n(5)\nwhereIc1(d/ξh) is the critical current of the ﬁrst har-\nmonic in a SFS junction with a homogeneous exchange\nﬁeldh. The interference eﬀects discussed in introduction\nresult in the power decay of the critical current Ic1vs\nthe F layer thickness d:Ic1∝d−1/2for a 2D junction\n[15] and Ic1∝d−1for a 3D junction [16]. Taking sym-\nmetric case d1=d2we immediately get a long–range\ncontribution to the Josephson current\nδIc1= sin2α\n2Ic1(0)sinϕ , (6)\nwhich does not decay with the increasing distance be-\ntween the S electrodes. It is important to note that this\ncontribution does not vanish for an arbitrary nonzero an-\nglebetween the magnetic moments in the F layers.\nLong–rangebehaviorcan be observedfor a second har-\nmonic in the current – phase relation as well. Indeed,\ncalculating the average ∝angbracketleft(n,nF)cos2γ∝angbracketrightwe ﬁnd a non-\nvanishing long–range supercurrent contribution even for\nd1∝negationslash=d2:\nδIc2=a2sin2α\n2sin2ϕ . (7)\nNote, that the emergence of long–range proximity eﬀect\nfor high harmonics in Josephson relation is in a good\nagreementwith recenttheoreticalﬁndings in Refs.17, 18.\nJosephson current through a ferromagnetic wire. We now\nproceed with the consideration of a more complicated\nexample of the interference phase suppression in a fer-\nromagnetic wire where the quasiclassical trajectories of\nelectrons and holes experience multiple specular reﬂec-\ntions from the wire surface (see Fig. 2a). The particular\ngeometry shown in Fig. 2a can be considered as a rough\nmodel for experiments on Co nanowires [8]. For simplic-\nity we restrict ourselves to the case of a 2D junction.\nTaking account of the spin-orbit interaction inside the\nferromagnet we obtaine the exchange part of the eﬀec-\ntive Hamiltonian for the band electrons depending on\nthe quasi-momentum ( k) orientation:\nˆHex=/summationdisplay\nijβij(k)h0iσj=h(k)ˆσ,\nwhereh0is a pseudo vector determined by the ferro-\nmagnetic moment. Assuming the absence of the system\nanisotropy described by a polar vector we ﬁnd the sim-\nplest form of the resulting exchange ﬁeld: h=h0+\nβsok−2\nF(h0,k)k, whereβsois a constant determined by\nthe spin – orbit interaction, and kFis the Fermi momen-\ntum.3\nDL\n/c113 Fy\nxSL SR\na)x0\nD\nL/c113y\nxSL SR\nFy0\nb)\nFigure 2: (Color online) Josephson transport through a\nnanowire in the overlap (a) and edge (b) geometries. The\nquasiparticle trajectories are shown by the red dashed line s.\nThe exchange ﬁeld along the quasiparticle trajectory\nexperiencing the reﬂection at the wire surface should\nchange its direction. Thus, we obtain the problem de-\nscribed by Eqs. (3) with a periodic exchange ﬁeld along\nthe trajectory characterized by a given angle θand a cer-\ntain starting point at the superconductor surface. The\nsame equations for each trajectory can be of course de-\nrived for a periodic domain structure. Let us consider\nﬁrst the problem of calculating the band spectrum ǫ(k)\nin the ﬁeld hvarying with the period 2 D/sinθ:\n−i/planckover2pi1VF∂sfs+2hft=ǫ(k)fs, (8)\n−i/planckover2pi1VF∂sft+2fsh=ǫ(k)ft. (9)\nThe solution can be written in the Bloch form:\n/parenleftbiggfs\nft/parenrightbigg\n=eiks/parenleftbiggfsk\nftk/parenrightbigg\n,\nwherefsk(s+2D/sinθ) =fsk(s) andftk(s+2D/sinθ) =\nftk(s). One can see that provided this solution corre-\nsponds to the energy branch ǫσ(k) there exist another so-\nlution (f∗\ns,−f∗\nt) corresponding to the energy −ǫσ(k). On\nthe other hand the latter solution corresponds also to the\nenergyǫ˜σ(−k) and, thus, we obtain the following sym-\nmetry property of the band spectrum: ǫ˜σ(−k) =−ǫσ(k),\nwhere the indices σand ˜σdenote diﬀerent branch num-\nbers. The full set of energy branches can be split in\nsuch pairs provided the number of branches is even. For\nan odd number of branches there is always one branch\nwhich does not have a partner. For this branch we ob-\ntainǫσ(−k) =−ǫσ(k) and, thus, this spectrum branch\ncrosses the zero energy level at k= 0:ǫσ(0) = 0. The\ncorresponding phase gain γappears to vanish for trajec-\ntories containing an integer number of periods shown in\nFig. 2a and, therefore, the solution with k= 0 and ǫ= 0\nprovides a long–range contribution to the supercurrent.\nFor the sake ofdeﬁniteness we choosethe ﬁeld h0to be\ndirected along the wire axis xand obtain the exchangeﬁeld in the form: h=x0hx+y0hy(s), where hx(θ)≃h0\nis constant along the trajectory and hy(s) is a periodic\nfunction with zero average. In the interval −D/sinθ <\ns < D/sinθthehyﬁeld component is deﬁned by the\nexpression hy=βsoh0sinθcosθsigns. Introducing the\nFourier expansions\nhy=/summationdisplay\nqHqeiqs, Hq=−i˜h2sinθ\nDq,\nfs,tx,ty= eiks/summationdisplay\nqFs,x,y(k+q)eiqs,\nwe rewrite the Eqs. (8) and (9) in the form:\n(/planckover2pi1VF(k+q)−ǫ)Fs(k+q)+2hxFx(k+q)\n+2/summationdisplay\n˜qHq−˜qFy(k+ ˜q) = 0, (10)\n(/planckover2pi1VF(k+q)−ǫ)Fx(k+q)+2hxFs(k+q) = 0,(11)\n(/planckover2pi1VF(k+q)−ǫ)Fy(k+q)\n+2/summationdisplay\n˜qHq−˜qFs(k+ ˜q) = 0. (12)\nHereq,˜q=qm=π(2m+1)sinθ/D,mis an integer, and\n˜h=βsoh0sinθcosθ.\nTo get the solution for a small periodic ﬁeld hywe\nuse a perturbative approach similar to the nearly free\nelectron approximation in the band theory of solids and\nrestrict the number of interacting Fourier harmonics in\nthe expansions. For this purpose it is instructive to con-\nsider the limit of zero periodic potential hyand separate\nthree types of solutions: (i) the solution ( Fs,Fx,Fy) =\n(0,0,1)δq−pcorresponding to the energy ǫ0=/planckover2pi1VF(k+p)\n(ii) the solutions ( Fs,Fx,Fy) = (1,±1,0)δq−p±corre-\nsponding to the energies ǫ±=/planckover2pi1VF(k+p±)±2hx. Here\npandp±are arbitrary reciprocal lattice vectors. The\nabove modes should strongly interact provided the res-\nonant condition ǫ0=ǫ+=ǫ−is fulﬁlled. Such reso-\nnance is possible for the case when the value 2 hx//planckover2pi1VF\nequals to a certain reciprocal lattice vector qm. Close\nto such Bragg – type resonance we see that the dom-\ninant harmonics correspond to the following choice of\nreciprocal lattice vectors: p= 0,p±=∓qm. Writing\nthe solution as a superposition of these three harmon-\nics we ﬁnd renormalized spectral branches ǫ0=/planckover2pi1VFk,\nǫ±=/planckover2pi1VFk±/radicalbig\n(/planckover2pi1VFqm−2hx)2+8|Hqm|2and corre-\nsponding eigenfunctions. Applying now the boundary\nconditions at s= 0 for the superposition of the above\neigenfunctions we ﬁnd the amplitude of the singlet com-\nponent corresponding to the energy branch ǫ0andk= 0:\nfsm=8|Hqm|2cos(qms)\n(/planckover2pi1VFqm−2hx)2+8|Hqm|2.\nAt the surface of a right superconducting electrode we\nshould take the coordinate sto be equal to the integer\nnumber of periods. We also need to sum up the above4\nresonant expressions over all Fourier harmonics of the\nperiodic potential:\nfs(s=sR) =∞/summationdisplay\nm=08|Hqm|2\n(/planckover2pi1VFqm−2hx)2+8|Hqm|2.\nThe precision of such resonant – type expression has\nbeen also conﬁrmed by the numerical solution of the\nEqs. (8) and (9) carried out using the transfer matrix\nmethod. Note, that we omit here the contribution from\nthe solutions corresponding to the branches ǫ±: these\nfunctions correspond to a nonzero quasimomentum and,\nthus, should gain a ﬁnite phase factor along the trajec-\ntory length. During averaging over diﬀerent trajectories\nthis phase factor causes the suppression of the resulting\nsupercurrentcontributionwith the increasingwirelength\nL.\nThe starting point of the trajectory varies in the in-\nterval ∆x= 2D/tanθand, as a consequence, the long –\nrange ﬁrst harmonic in current – phase relation takes the\nform:\nI1=a1sinϕπ/2/integraldisplay\n0dθcosθfs(sR).\nAssuming the resonances to be rather narrow we approx-\nimate them by the delta – functions and obtain:\nI1=a1sinϕ/summationdisplay\nm√\n2π/planckover2pi1VF˜h(θm)\nh2xDsin2θm.\nwhere sin θm= 2hxD/π/planckover2pi1VF(2m+1). In the limit D≫\n/planckover2pi1VF/2hxone can replace the sum over mby the integral:\nI1≃a1√\n2π/2/integraldisplay\n0dθ˜h(θ)\nhx(θ)cosθsinϕ≃a1√\n2\n3βs0sinϕ .\nCertainly, the above long–range eﬀect in the ﬁrst har-\nmonic is rather sensitive to the system geometry: taking,\ne.g., the system sketched in Fig. 2b we will not obtain the\nfull cancellation of the phase γbecause the trajectories\nin this case do not contain integer number of exchange\nﬁeld modulation periods. However, similarly to the case\nof bilayer the long–range eﬀect is still possible for higher\nharmonics. We apply the above perturbative procedure\nfor the calculation ofthe full fsfunction for the geometry\nshown in Fig. 2b. The second harmonic in the current–\nphase relation reads\nI2=a2sin2ϕπ/2/integraldisplay\n0dθcosθ/parenleftbig\n2∝angbracketleftf2\ns(sR)∝angbracketrighty0−1/parenrightbig\n,(13)\nwhere∝angbracketleft...∝angbracketrighty0= (1/D)/integraltextD\n0...dy0denotes averaging over\nthe starting point of the trajectory y0(see Fig. 2b).Keeping only the terms linear in the small |Hqm|ampli-\ntude we get the following expression for the long–range\npart of the second harmonic I2:\nI2=a2sin2ϕ/summationdisplay\nm√\n2π/planckover2pi1VF˜h(θm)\nh2xDsin2θm≃a2√\n2\n3βs0sin2ϕ.\nWe emphasize that the second harmonic of Josephson\ncurrent in both above examples is negative because of\nthe condition a2<0.\nNotethattheabsenceofthedecayofthesingle-channel\ncritical current was pointed out in Ref. 19 as a pos-\nsible source of the long-ranged proximity eﬀect in Co\nnanowires. However the averaging of the phase gain for\ndiﬀerent modes strongly decreases the critical current.\nIn contrast the results presented in this Letter demon-\nstrate that in the ballistic regime the spin-orbit inter-\naction generates the non-collinear exchange ﬁeld which\nproduces the long – range Josephson current. This con-\nclusion is always true for the second harmonic in the cur-\nrent – phase relation and for some geometries it may be\nalso valid for the ﬁrst harmonic. Therefore our ﬁndings\nprovide a natural explanation of the recent experiments\nwith Co nanowire [8]. To discriminate between two pro-\nposed mechanisms of the long ranged eﬀect, the studies\nof higher harmonics in Josephson current-phase relations\ncould be of major importance. Also it should be interest-\ning to verify on experiment the predicted simple angular\ndependence (6) of the critical current in S/F/S junctions\nwith composite interlayer.\nThe authorsthank R. Shekhterfor valuable comments.\nThis work was supported, in part, by European IRSES\nprogram SIMTECH (contract n.246937), the Russian\nFoundation for Basic Research, FTP Scientiﬁc and ed-\nucational personnel of innovative Russia in 2009-2013,\nandthe programofLEA”PhysiqueTheoriqueet Matiere\nCondensee”.\n[1] A. I. Buzdin, Rev. Mod. Phys., 77, 935 (2005).\n[2] F. S.Bergeret, A.F. Volkov, andK.B. Efetov, Rev.Mod.\nPhys.,77, 1321 (2005).\n[3] J. W. A. Robinson, J. D. S. Witt, and M. G. Blamire,\nScience 329. 59 (2010).\n[4] T. S. Khaire, M. A. Khasawneh, W. P. Pratt, Jr., and N.\nO. Birge, Phys. Rev. Lett. 104, 137002 (2010).\n[5] I. Sosnin, H. Cho, and V. T. Petrashov, A. F. Volkov,\nPhys. Rev. Lett. 96, 157002 (2006).\n[6] R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G.\nMiao, G. Xiao, and A. Gupta, Nature, 439, 825 (2006).\n[7] M. Giroud, H. Courtois, K. Hasselbach, D. Mailly, and\nB. Pannetier, Phys. Rev. B 58, R11872 (1998).\n[8] Jian Wang, Meenakshi Singh, Mingliang Tian, Nitesh\nKumar, Bangzhi Liu, Chuntai Shi, J. K. Jain, Nitin\nSamarth, T. E. Mallouk & M. H. W. Chan, Nature\nPhysics, 6, 389 (2010).5\n[9] Ya. M. Blanter and F. W. J. Hekking, Phys. Rev. B 69,\n024525 (2004).\n[10] Z. Pajovic, M. Bozovic, Z. Radovic, J. Cayssol and A.\nBuzdin, Phys. Rev. B 74, 184509 (2006).\n[11] B. Crouzy, S. Tollis, and D. A. Ivanov, Phys. Rev. B 75,\n054503 (2007).\n[12] A. Kadigrobov, Z. Ivanov, T. Claeson, R. I. Shekhter,\nand M. Jonson, Europhys. Lett. 67, 948 (2004).\n[13] A. I. Buzdin, A. S. Melnikov, and N. G. Pugach, Phys.\nRev. B83, 144515 (2011).\n[14] T. Champel, T. L¨ ofwander, and M. Eschrig, Phys. Rev.\nLett.100, 077003 (2008).\nThierry Champel1,2, Tomas Lo”fwander1,3, andMatthias Eschrig\n[15] F. Konschelle, J. Cayssol, and A. I. Buzdin, Phys. Rev.\nB78, 134505 (2008).\n[16] A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov,\nJETP Lett. 35, 178 (1982) [Pis’ma Zh. Eksp. Teor. Fiz.\n35, 147 (1982)].\n[17] L. Trifunovic, Phys. Rev. Lett. 107, 047001 (2011).\n[18] L. Trifunovic, Z. Popovic, and Z. Radovic, Pys. Rev. B\n84, 064511 (2011).\n[19] F. Konschelle, J. Cayssol, and A. Buzdin, Phys. Rev. B\n82, 180509 (2010)."    },    {        "title": "1110.1278v1.Ferromagnetism_in_YbCu2Si2_at_high_pressure.pdf",        "content": "arXiv:1110.1278v1  [cond-mat.str-el]  6 Oct 2011Ferromagnetism in YbCu 2Si2at high pressure\nA. Fernandez-Pañella, D. Braithwaite, B. Salce, G. Laperto t and J. Flouquet\nSPSMS, UMR-E CEA / UJF-Grenoble 1, INAC, 38054 Grenoble, Fra nce∗\nWe demonstrate from detailed ac susceptibility and calorim etry studies under hydrostatic pressure\nthat YbCu 2Si2probably orders ferromagnetically at high pressure. The ( p,H,T ) phase diagram\nshows that the transition temperature increases with press ure but also with an applied magnetic\nﬁeld. We suggest that many ytterbium systems may show a trend towards ferromagnetism and\nwe discuss the possible reasons for this. We also examine the implications, including the potential\nof YbCu 2Si2and other Yb compounds for further studies of the rich physic al properties that may\noccur near a ferromagnetic critical point.\nPACS numbers: 71.27.+a, 62.50.-p\nKeywords: magnetism, quantum criticality, intermediate v alence state\nI. INTRODUCTION\nA key point for the understanding of quantum phe-\nnomena at a magnetic point is the comaprison be-\ntween cerium and ytterbium systems. Both Yb and\nCe can assume a magnetic (Ce3+,4f1/Yb3+,4f13) conﬁg-\nuration or ﬂuctuations between this and a nonmagnetic\n(Ce3+,4f1/Yb3+,4f13) state. The cleanest way to tune a\nsystem to its critical point is with high pressure. In both\nfamilies pressure tends to decrease the occupancy of the\n4f shell, leading to opposite eﬀects on the magnetism. In\ncerium systems pressure tends to drive the system from\nmagnetic order to a paramagnetic ground state whereas\nin ytterbium systems magnetic order tends to be induced\nby pressure. The free Yb3+ion (J=7/2) with 13 electrons\nin the 4f shell can be considered as the hole analogue of\nCe3+. In a real lattice, however, there are important dif-\nferences, as summarized in a recent review1: the deeper\nlocalization of the 4f electrons, and the stronger spin-\norbit coupling in Yb can lead to a diﬀerent hierarchy of\nthe signiﬁcant energy scales (Kondo temperature, Crys-\ntal ﬁeld). Moreover, in Yb the valence state is able to\nchange far more between 2 and 3, whereas in cerium sys-\ntems the range is generally much more restricted close to\n3. Yb systems are therefore rather easily in an interme-\ndiate valence state with ﬂuctuations between the integer\nvalues, and this can be the case even when long range\nmagnetic order is present.1\nHowever a more basic element in order to compare the\nphysics at a critical point of any ytterbium system with\nits cerium counterpart is the nature of the order that oc-\ncurs. In the cerium systems where quantum criticality is\ninduced, the magnetically ordered phase is established at\nambient pressure. It can therefore be fully characterized\nby neutron scattering and magnetization measurements\nand in most known cases is antiferromagnetic (AFM).\nOn the other hand in ytterbium where a critical point is\ninduced with pressure, the magnetic order will appear\nunder pressure, often too high for neutron scattering,\nand no direct determination of the magnetic structure\nis possible. There is some evidence that in several ytter-\nbium compounds ferromagnetic (FM) correlations mightdominate. YbInNi 4orders ferromagnetically2or at least\nwith a strong FM component.3In YbInCu 4magnetic\norder is induced at rather low pressure (3 GPa) which\nhas allowed magnetization measurements showing a FM\nstate.4In other systems evidence is more indirect: FM\nﬂuctuations have been shown in YbRh 2Si25from trans-\nport measurements it was also suggested that the order\nthat appears at high pressure (8 GPa) in YbRh 2Si2and\nYbIr 2Si2might be FM.6Magnetoresistance at high pres-\nsures also point to possible FM correlations and order in\nYbNi 2Ge2.7YbCu 2Si2is one of the clearest cases where\nmagnetic order can be induced with pressure, this having\nbeen conﬁrmed by resistivity,8calorimetry,9and Möss-\nbauer studies.10However none of these techniques can\ngive direct information on the type of order, nor distin-\nguish between AFM and FM order. We show here ac\nsusceptibility measurements under high pressure which\nshow direct evidence for FM order in this system.\nII. EXPERIMENTAL DETAILS\nHigh-quality single crystals were grown by an indium\nﬂux method (using MgO crucibles) as described in detail\nelsewhere.9The crystals were characterized by resistivity\nand the residual resistivity ratio (RRR 0.7−300K= 220) at-\ntests to the excellent quality with a corresponding resid-\nual resistivity of 0.3 µΩ. Measurements under pressure\nwere carried out using diamond-anvil cells. As pressure\ntransmitting medium we used argon which has excellent\nhydrostatic conditions up to 10 GPa. Measurements were\nperformed between 1.5 K to 20 K and from nearly am-\nbient pressure to 13 GPa in a He cryostat with the in\nsitupressure-tuning device.11Pressure was determined\nfrom the ruby ﬂuorescence by placing some ruby chips in\nthe pressure chambers. The ac susceptibility technique\nwe used is an adaptation of the technique developed in\nCambridge.12We directly placed the sample inside the\npickup coil (350 µm of external diameter, 10 turns with\na 12 µm insulated Cu wire). An ac ﬁeld of 0.1 mT was\ngenerated by the primary coil outside the pressure cham-\nber at a frequency of 721 Hz. The magnetic ﬁeld was\napplied parallel to the easy caxis of magnetization.13,142\nSpeciﬁc heat was measured by an ac calorimetry tech-\nnique detailed elsewhere.15The sample was heated by a\nlaser modulated by a mechanical chopper at about 722 Hz\nwhich was found, from the signal-frequency dependence,\nhigh enough to thermally decouple the sample from its\nenvironment. The temperature oscillations which are in-\nversely proportional to the speciﬁc heat were obtained\nvia the voltage measured from the Au/AuFe thermocou-\nple soldered directly on the sample and its thermoelec-\ntric power. A small constant magnetic ﬁeld up to 0.45 T\ncould be superposed using a superconducting magnet.\nIII. RESULTS AND DISCCUSSION\nA. Ac susceptibility and ac calorimetry under\npressure\nFigure 1(a) shows the low-temperature dependence of\nthe ac susceptibility at selected pressures. For pressures\nabove 6.5 GPa we ﬁnd the onset of an increase of the\nac susceptibility ( χ′) for T < 4 K, which becomes a clear\npeak for p > 8 GPa corresponding to the magnetic transi-\ntion which we followed up to nearly 13 GPa. The broad-\nening of the peak and the decreasing intensity of the sig-\nnal with increasing pressures could be caused by homo-\ngeneity eﬀects inside the pressure chamber. Unlike with a\ndc magnetization measurement, distinguishing between a\nFM and an AFM transition with ac susceptibility is not\nas trivial due to irreversibility eﬀects in the FM state\nχ′can decrease with decreasing temperature below the\nCurie temperature (T Curie ) and frequently shows a peak\nas found here. Indeed a similar eﬀect was seen in χ′in\nYbInCu 4.16In order to determine the nature of the mag-\nnetic order we have therefore made an estimate of the\nabsolute value of the susceptibility. This was done using\ntwo methods. First we calculated the value using estima-\n12\n10\n8\n6\n4\n2\n0/s99' (arb. units)\n12108642\nT (K)12.9 GPa\n9.2 \n8.6\n6.57.58.08.3a\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0/s99' (10-6m3/mol)\n12840\nT (K) 8.9 GPa b\n χ'experiment\n \nCurie Law:  \n µeff= 1.25µB\n µeff= 4.54µB\nCurie-Weiss Law:\n µeff= 3.32µB \n        and Tc= 2.9 K\nFIG. 1. (Color online) (a) ac susceptibility versus tempera -\nture for diﬀerent pressures below and above the critical pre s-\nsure. (b) Susceptibility at 8.9 GPa compared with expected\nCurie dependences for: µeff= 4.54 µB(dashed line) and µeff\n= 1.25 µB(dotted line).tions of the sample and pick-up coil geometries and ﬁlling\nfactor.12Second we compared the signal from the sample\nwith the superconducting transition of a Pb sample with\nthe same dimensions in the same set-up. The two meth-\nods gave similar results with less than 20 %diﬀerence.\nIn Figure 1(b) the susceptibility at 8.9 GPa is compared\nwith two Curie laws (red lines) and a Curie-Weiss law\n(green line). The experimental χ′contains an unknown\nbut roughly constant background, so the zero has been\ntaken at the highest temperature measured (15 K). First\nwe compare the experimental curve with the Curie law\ncorresponding to the eﬀective moment of µeff= 1.25\nµBwhich was determined experimentally by Mössbauer\nspectroscopy at a similar pressure.10This value is simi-\nlar to that found in other Yb systems with the tetragonal\n(ThCr 2Si2) structure showing magnetic order and com-\npatible with the moment expected for Yb3+taking into\naccount the eﬀects of the crystalline electric ﬁeld (CEF)\nand Kondo screening.17–19The experimental data show\nclearly a much larger magnetic signal. We also show the\nCurie law for an eﬀective magnetic moment correspond-\ning to the theoretical value for the Yb3+free ion, µeff=\n4.54µB. Even in this case the experimental data still di-\nverges much more quickly than the theoretical curve be-\nlow 5 K. The agreement of the experimental data above\n5 K is probably coincidental and can also be compatible\nwith a smaller moment and a positive Curie-Weiss tem-\nperature. Finally, the green line shows the best ﬁt for\nthis data with a Curie-Weiss law (with T CW= 2.9 K\nandµeff= 3.52 µB). This value for µeffis surprisingly\nlarge. This could be due to a combination of pressure\nand sample inhomogeneity, and the very strong slope\nof dT M/dP, causing a spread of ordering temperatures\nwithin the sample. All these points strongly lead toward\nferromagnetism. Of course from these results we cannot\n5\n4\n3\n2\n1\n0T (K)\n14121086420\np (GPa)4\n2χ ' (a.u.)\n4321\nT (K)1.5\n1.4\n1.3C/T (a.u)\nTM\n RRR ~ 200\n             200 \n             100\n               18\nFIG. 2. (Color online) ( p,T) phase diagram: Diamonds and\ntriangles represent T Mobtained from ac susceptibility and ac\ncalorimetry respectively in this study. Circles are the res ults\nfrom previous measurements (full circles Ref.8 and open cir -\ncles Ref.9). Lines are guides for the eyes. The inset shows th e\ncriterion used to determine T M. The improvement in sample\nquality (higher RRR) seems to lead to systematically higher\nordering temperatures3\nexclude a more complex magnetic structure like canted\nantiferromagnetism as recently suggested in YbInNi43\nbut we need at least a signiﬁcant FM component. The\npressure dependence of the magnetic temperature T M\nobtained by ac-susceptibility is summarized in Figure 2\nand agrees well with the data found from calorimetry\nmeasurements on the same batch. Interestingly the or-\ndering temperatures are found to be much higher than\nreports on previous batches of crystals, possibly reﬂect-\ning the improved crystal quality.\nB. Magnetic ﬁeld eﬀect on the ac susceptibility\nand ac calorimetry under pressure\nFurther proof of FM order is found in the behavior\nwhen a small constant magnetic ﬁeld is applied along\nthec-axis. Susceptibility and speciﬁc heat curves for se-\nlected pressure around 8.5 GPa are shown in Figure 3\n(a) and (b) respectively. In both cases, the amplitude of\nthe transition decreases with the application of low mag-\nnetic ﬁelds but at the same time, T Mis clearly shifted\ntowards higher values for increasing ﬁelds. This trend\ncan be better observed in the phase diagram (T M, H) of\nFigure 3 (d) were we have plotted the transition temper-\natures versus the applied magnetic ﬁelds. This response,\n4\n3\n2\n1C/T (arb.units)\n54321\nT (K) 8.5 GPa\n0.1\n0.2\n0.45µ0H= 0 Ta 1.7\n1.6\n1.5χ' (arb. units)\n1086420\nT (K) 8.6 GPa\n 0 T\n0.1\n0.2\n0.4b\n5\n4\n3χ' (arb. units)\n8642\nT (K)0 T7.5 GPa\n0.1 \n0.2\n0.3\n0.4c\n4\n3\n2T (K)\n0.4 0.0\nµ0H (T)d\n8.5 GPa\n7.5\nFIG. 3. (Color Online) Eﬀect of magnetic ﬁeld: (a) ac\ncalorimetry curves at 8.5 GPa and magnetic ﬁelds up to 0.45\nT. (b) Ac susceptibility vs temperature at 8.6 GPa and se-\nlected magnetic ﬁelds. (c) Susceptibility curves at 7.5 GPa\nunder magnetic ﬁeld. The transition is induced for ﬁelds\ngreater than 0.2 T. (d) Magnetic dependence of T Mnear 8.5\nGPa (blue curves) and 7.5 GPa (red curve) obtained from\nspeciﬁc heat (open circles) and susceptibility (full circl es and\ntriangles) measurements.\nwhere T Curie increases with ﬁeld, is consistent with the\nusual behaviour of a ferromagnetic system under mag-\nnetic ﬁeld3, and is contrary to the usual behavior of an\nantiferromagnet. Surprisingly, as shown in Figure 3(c),\nat 7.5 GPa where no transition is visible down to 1.5 Kat zero ﬁeld we found we could induce a metamagnetic\ntransition by the application of dc ﬁelds above 0.2 T. The\nrapid decrease of the intensity in χ′is similar to the eﬀect\nseen in YbInCu4,16and probably mainly reﬂects the loss\nof any ac response in the FM state once a single domain\nhas been created with the dc ﬁeld, as well as the fact that\nunder dc ﬁeld the transition becomes a crossover as the\nmagnetic ﬁeld also breaks time-reversal symmetry from\na partially to fully polarized state.\nA complete phase diagram of the ( p, H, T ) phase dia-\ngram of T Mis shown in Figure 4. We can observe that\nthe trend of the increase of T Munder magnetic ﬁeld at\n8.5 GPa is valid for all pressures we measured. Interest-\ningly we ﬁnd that T Mdoes not signiﬁcantly change for\nany pressure at very low magnetic ﬁelds (up to 0.1 T).\nThe reason for this is not clear, but it could be that this\nis the ﬁeld necessary to obtain a single domain and that\nbelow this value the internal ﬁeld depends weakly on the\napplied ﬁeld.\nFIG. 4. (Color online) Complete Phase diagram ( p, H, T M) at\nlow-magnetic ﬁelds and high pressures. Red points and green\npoints correspond to the values of T Mat zero ﬁeld (from\nFig.2) and under ﬁeld above the critical pressure respectiv ely.\nOrange dots are the values of T Mat 7.5 GPa (from Fig.3c),\nclose to the critical pressure, and the black points indicat e\nwhere ac-susceptibility curves detected no transition dow n to\n1.5 K.\nNow we discuss why ytterbium systems might favor\nFM rather than AFM order in contrast to most cerium\ncompounds. One simple idea stems from the recent theo-\nretical works showing how magnetic ﬁeld can control the\nvalence transition.20The application of ﬁeld can induce\nthe ﬁrst order valence transition (FOVT) accompanied\nwith strong metamagnetic and magnetostriction eﬀects.\nConversely if the volume change and valence transition\nare induced by pressure, the spontaneous magnetization\ncan easily appear. More precisely close to a valence tran-\nsition FM would be favored by the presence of large non-\nlocal dynamical deformation of the lattice.1,20This could\nexplain why ferromagnetism is more common in ytter-\nbium where the valence changes with pressure are much4\nlarger than in cerium systems. We turn now to the conse-\nquences of FM rather than AFM order. Ferromagnetism\nin YbCu 2Si2leads us to inquire about the nature of the or-\nder of the transition at low temperatures. From the phase\ndiagram it appears already that magnetic order appears\nrather suddenly at T > 1 K inspiring speculation that the\ntransition could be ﬁrst-order.9This is now less surpris-\ning as it seems a rather general property of ferromagnets,\nand so far no example has been found of a FM quantum\ncritical point (QCP), with a second-order transition down\nto T = 0. The experimental studies of several itinerant\nferromagnets like ZrZn 2,21Co(S (1−x)Sex)2,22MnSi23and\nUGe 224all show a tricritical point where the paramagnet-\nferromagnet transition changes from a second-order to a\nﬁrst-order phase transition when it is driven towards a\nQCP by applying either external or chemical pressure.\nThese results might suggest that in YbCu 2Si2the tran-\nsition also becomes ﬁrst-order near the critical pressure.\nAnother consequence is that FM ﬂuctuations are\nless favorable than AFM ones for magnetically induced\nsuperconductivity.25This might be one of the reasons\nthat superconductivity is far more elusive in ytterbium\nsystems than in cerium compounds. The ﬁrst-order tran-\nsition also changes the picture. In UGe 2which is so far\nthe only system where superconductivity occurs at a FM\ncritical point with a large ordered moment, supercon-\nductivity appears only within the FM region. Supercon-\nductivity, if it exists in YbCu 2Si2, might occur at very\nlow temperature as the 4f levels in Yb-based compounds\nare narrower than in Ce-based ones which leads to lower\ncharacteristic temperatures, extreme sensitivity to sam-\nple quality and an appearance not necessarily where it is\nexpected in the phase diagram.\nIV. CONCLUSIONS\nAc susceptibility and ac calorimetry measurements in\nhigh-quality single crystals of YbCu 2Si2under high pres-\nsure and small magnetic ﬁelds strongly suggest that the\nmagnetic order induced for p > 8GPa is FM. Pres-\nsure measurements were performed in diamond-anvil cellswith argon as pressure-transmitting media and the re-\nsults conﬁrmed the ( p,T) phase diagram of YbCu 2Si2\npreviously reported but surprisingly, the magnetic tran-\nsition temperatures found in this study are signiﬁcantly\nhigher. We might attribute the diﬀerence to the improve-\nment of the crystal quality. We have compared the low\ntemperature susceptibility curves with diﬀerent ﬁts; two\nCurie laws with µeff= 1.25 µBandµeff= 4.54 µBre-\nspectively largely underestimate the experimental values\nat low temperature. A Curie-Weiss law with T CW= 2.9\nK and µeff= 3.52 µBgives the best agreement with\nthe experimental data. Even though this value of the\neﬀective magnetic moment is larger than the value ex-\npected from Mössbauer spectroscopy, these results point\nstrongly to ferromagnetism. Further proof for FM in\nthis system was obtained by superimposing a small mag-\nnetic ﬁeld. The ( p,H,T ) phase diagram clearly shows that\nthe transition temperature increases with magnetic ﬁelds\nlarger than 0.1 T for all pressures we measured and simul-\ntaneously the transition is rapidly suppressed under ﬁeld.\nThese trends are experimentally consistent with the be-\nhavior of a FM system under magnetic ﬁeld. Of course\nwe can not neglect a more complex magnetic structure\nbut a signiﬁcant FM component is at least observed. FM\nseems to appear abruptly at about 8 GPa, suggesting a\nﬁrst-order transition. This is what it has been observed\nin all ferromagnets up to now: near the vicinity of a QCP\nthe FM transition becomes ﬁrst order. Due to the avail-\nability now of extremely high quality crystals, YbCu 2Si2\nis therefore a good candidate to explore in great accu-\nracy the phase diagram around the critical pressure at\nvery low temperature and under magnetic ﬁeld. It is an\nexperimental challenge due to the rather high pressure\ninvolved but could be very rewarding.\nACKNOWLEDGMENTS\nWe Thank J.-P. Sanchez and J.-L. Tholence for use-\nful discussions. This work was supported by the French\nNational Research Agency (ANR) through the contracts\nECCE and DELICE.\n∗Corresponding author: daniel.braithwaite@cea.fr\n1J. Flouquet and H. Harima, Arxiv preprint\narXiv:0910.3110 (2009).\n2J. L. 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B, 63, 054529\n(2001)."    },    {        "title": "2206.05770v3.Interference_phenomena_in_Josephson_junctions_with_ferromagnetic_bilayers__Spin_triplet_correlations_and_resonances.pdf",        "content": "Interference phenomena in Josephson junctions with ferromagnetic bilayers:\nSpin-triplet correlations and resonances\nDanilo Nikoli\u0013 c,1, 2Mihajlo Vanevi\u0013 c,1Alexander I. Buzdin,3, 4and Zoran Radovi\u0013 c1, 5\n1Department of Physics, University of Belgrade, Studentski trg 12, 11158 Belgrade, Serbia\n2Fachbereich Physik, Universit at Konstanz, D-78457 Konstanz, Germany\n3University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence Cedex, France\n4World-Class Research Center \\Digital Biodesign and Personalized Healthcare\",\nSechenov First Moscow State Medical University, Moscow 119991, Russia\n5Serbian Academy of Sciences and Arts, Kneza Mihaila 35, 11000, Belgrade, Serbia\n(Dated: August 31, 2022)\nWe study the Josephson e\u000bect in planar SF 1F2S junctions that consist of conventional s-wave\nsuperconductors (S) connected by two metallic monodomain ferromagnets (F 1and F 2) with arbi-\ntrary transparency of interfaces. We solve the scattering problem in the clean limit based on the\nBogoliubov - de Gennes equation for both spin-singlet and odd in frequency spin-triplet pairing\ncorrelations. We calculate numerically the Josephson current-phase relation I(\u001e). While the \frst\nharmonic of I(\u001e) is completely generated by spin-singlet and short-range spin-triplet superconduct-\ning correlations, for noncollinear magnetizations of ferromagnetic layers the second harmonic has\nan additional long-range spin-triplet component. Therefore, for a strong ferromagnetic in\ruence,\nthe long-range spin-triplet contribution to the second harmonic dominates. We \fnd an exception\ndue to the geometric resonance for equal ferromagnetic layers when the \frst harmonic is strongly\nenhanced. Both \frst and second harmonic amplitudes oscillate with ferromagnetic layer thicknesses\ndue to 0\u0000\u0019transitions. We study the in\ruence of interface transparencies and \fnd additional\nresonances for \fnite transparency of the interface between ferromagnetic layers.\nI. INTRODUCTION\nThe interplay between superconductivity and mag-\nnetism in proximity heterostructures [1{4] has been at-\ntracting considerable interest for decades, see for exam-\nple Refs. [5{16]. Remarkably, odd in frequency spin-\ntriplet pairing correlations may occur in SFS Joseph-\nson structures comprised of superconductors with spin-\nsinglet pairing and a metallic ferromagnet [17, 18]. In\nthe case of a homogeneous ferromagnet, the triplet pair\namplitude has zero total spin projection on the magne-\ntization axis. This amplitude, as well as the spin-singlet\namplitude, decay over the short length scale determined\nby the exchange energy hin the ferromagnet. The char-\nacteristic coherence length in ferromagnet is given by\n\u0018F=~vF=hand\u0018F=p\n~D=h (D=vF`=3 is the dif-\nfusion coe\u000ecient with `being the electronic mean free\npath) in the clean and di\u000busive limit, respectively. The\nsituation is quite di\u000berent for an inhomogeneous ferro-\nmagnet where spin-triplet pair amplitudes with \u00061 total\nspin projection on the magnetization axis emerge [17].\nThese amplitudes decay on substantially larger length\nscales determined by temperature, \u0018F=~vF=(kBT)\nin the clean and \u0018F=p\n~D=(kBT) in the di\u000busive\nlimit [12].\nA simple realization of a Josephson junction with an in-\nhomogeneous ferromagnet is the SF 1F2S heterostructure\nwith two monodomain ferromagnets having noncollinear\nin-plane magnetizations [19{30]. However, in such prox-\nimity structures the long-range spin-triplet component\nof the supercurrent consists only of even harmonic am-\nplitudes [22{25]. In the case of strong ferromagnets the\nshort-range components are suppressed and the secondharmonic is dominant in the current-phase relation. Odd\nharmonic amplitudes can be long-ranged only in hetero-\njunctions with three or more ferromagnetic layers [31{43].\nNote that the anharmonic current-phase relation can\nbe expanded as I(\u001e) =I1sin\u001e+I2sin 2\u001e+:::, where\nthenth harmonic amplitude Incorresponds to the phase-\ncoherent transport of nCooper pairs [23]. Junctions\nwith a pure second harmonic exhibit degenerated ground\nstates for\u001e= 0 and\u0019at the so-called 0 \u0000\u0019transi-\ntion [4, 44]. A small contribution of other harmonics lifts\ndegeneracy and leads to the coexistence of stable and\nmetastable 0 and \u0019states [45{48].\nA long-ranged supercurrent has been observed in Nb\nJosephson junctions with Ni- and Co-based ferromag-\nnetic multilayers [49{53]. An enhanced second harmonic\nin the long-ranged supercurrent has been observed in\nmesa-heterostructures of cuprate superconductors and\nferromagnetic bilayers of manganite and ruthenate [54],\nwhile a pure second harmonic has been observed in\nNbN=GdN=NbN junctions [55].\nA dominant second harmonic, I2\u001dI1, can be realized\nin the regime of highly asymmetric SF 1F2S junctions [22{\n25]. The physical picture behind this e\u000bect is the fol-\nlowing: At the SF interface the exchange \feld generates\nspin-triplet correlations with 0 spin projection. Penetrat-\ning into the next ferromagnetic layer with misoriented\nmagnetization they mix forming long-range spin-triplet\ncorrelations with \u00061 spin projection. Therefore, for a\nfully developed spin-triplet proximity e\u000bect one of two\nferromagnetic layers should be su\u000eciently thin to pro-\nvide a large short-range spin-triplet amplitude with zero\nspin projection at the interface between ferromagnetic\nlayers in order to generate large long-range spin-tripletarXiv:2206.05770v3  [cond-mat.supr-con]  30 Aug 20222\namplitudes with \u00061 spin projection. The other ferro-\nmagnetic layer should be su\u000eciently thick to \flter out\nthe short-range correlations [56].\nIn this paper we study the Josephson e\u000bect in clean\nplanar (three-dimensional) SF 1F2S junctions that consist\nof conventional s-wave superconductors and two metal-\nlic monodomain ferromagnets (equal strength and di\u000ber-\nent thicknesses) with arbitrary transparency of the inter-\nfaces. We calculate numerically the Josephson current-\nphase relation I(\u001e) by using the Bogoliubov-de Gennes\nformalism. In particular, we calculate the \frst and sec-\nond harmonic amplitudes, I1andI2. The long-range\nsecond harmonic is well-pronounced for an overall strong\nferromagnetic in\ruence due to the contribution of the\nodd-frequency spin-triplet correlations with \u00061 spin pro-\njection. However, for equal thicknesses of ferromagnetic\nlayers the spin-singlet contribution to the \frst harmonic\nis dominant due to geometric resonances even for a strong\nferromagnetic in\ruence. In a previous paper the results\nfor linear (one-dimensional) SF 1F2S structures were illus-\ntrated only for equal ferromagnetic layers and the in\ru-\nence of the long-range spin-triplet correlations was com-\npletely hidden [20]. In subsequent papers using the same\napproach [29, 30], the interplay between the geometric\nresonances and spin-triplet correlations was not studied\nexplicitly. Here, we focus on this subject.\nBoth the \frst and the second harmonic oscillate with\nferromagnetic layer thicknesses due to the 0 \u0000\u0019tran-\nsitions. For \fnite transparency of interfaces the super-\ncurrent is suppressed, with higher harmonics being more\na\u000bected. A lower transparency of the interface between\nferromagnetic layers, where the long-range spin-triplet\ncorrelations are generated, has a nontrivial impact on\nthe interference phenomena: For certain thicknesses of\nthe ferromagnetic layers we \fnd additional geometric res-\nonances.\nThe paper is organized as follows. In Sec. II we present\nthe model and the solution. In Sec. III we present and\ndiscuss the numerical results for the Josephson current\nand harmonic amplitudes. Finally, the concluding re-\nmarks are given in Sec. IV.\nII. MODEL\nA. The Bogoliubov-de Gennes equations for SF1F2S\nheterojunctions\nWe consider a clean planar (three-dimensional)\nSI1F1I2F2I3S heterojunction that consists of two super-\nconductors (S), two uniform monodomain ferromagnetic\nlayers (F 1and F 2), and three nonmagnetic interfacial po-\ntential barriers between metallic layers ( I1\u0000I3), depicted\nin Fig. 1. Superconductors are described in the frame-\nwork of BCS formalism, while for ferromagnets we use\nthe Stoner model with a spatially-dependent energy shift\n2h(r) between the spin subbands. The model and meth-\nods are the same as in the previous papers [20, 29, 30].\nSS𝑥𝑦𝑧𝛼1𝑥𝑦𝑧𝛼2F1F2\n𝑑!0𝑑!+𝑑\"FIG. 1. Schematic representation of an SF 1F2S junction.\nTwo ferromagnetic layers F 1and F 2of thicknesses d1andd2,\nrespectively, are coupled to two superconducting electrodes\n(S). The magnetization vectors lie in the yzplane at angles\n\u000b1and\u000b2with respect to the zaxis. The insulating inter-\nfaces between the superconducting and ferromagnetic layers\nare denoted asI1\u0000I3.\nElectronlike and holelike quasiparticles with energy\nEand spin projection \u001b=\";#are described by u\u001b(r)\nandv\u001b(r), respectively, where ris the spatial coordi-\nnate. Using the four-component wave function \t( r) =\n[u\"(r);u#(r);v\"(r);v#(r)]T, the Bogoliubov-de Gennes\nequation has the following form:\n\u0014H\t(r) =E\t(r); (1)\nwith \u0014Hbeing a 2\u00022 matrix in particle-hole space\n\u0014H=\u0012^H(r)^\u0001\n^\u0001\u0003\u0000^H(r)\u0013\n; (2)\nwhere each block itself is a 2 \u00022 matrix in spin space, such\nthat, ^H(r) =H0(r)^1\u0000h(r) sin[\u000b(r)]^\u001c2\u0000h(r) cos[\u000b(r)]^\u001c3\nand ^\u0001(r) = \u0001( r)^\u001c1. Here, ^\u001ciare Pauli matrices, ^1 is\nthe unity matrix, and H0(r) =\u0000~2r2=2m+W(r) +\nU(r)\u0000\u0016. The chemical potential is denoted by \u0016,W(r) =P\niWi\u000e(x\u0000xi) is the potential of the barriers at the\ninterfaces, and U(r) is the electrostatic potential. The x\naxis is chosen to be perpendicular to the layers, whereas\nx1= 0,x2=d1, andx3=d1+d2are coordinates of the\ninterfaces. At zero temperature the di\u000berence \u0016\u0000U(r)\nis equal to the Fermi energy EF. The in-plane ( y\u0000z)\nmagnetizations of the two F layers are not collinear in\ngeneral, and the magnetization orientation is de\fned by\nthe angle\u000b(r) with respect to the zaxis. We choose\n\u000b(r) =\u000b1for 0< x < d 1in F 1, and\u000b(r) =\u000b2for\nd1<x<d 1+d2in F 2(see Fig. 1).\nFor simplicity, we assume equal magnitudes of the ex-\nchange \feld in the ferromagnets, h(r) =h\u0002(x)\u0002(d1+\nd2\u0000x) where \u0002( x) stands for the Heaviside step func-\ntion. We also assume that the e\u000bective electron mass\nmis constant throughout the layers and that the mean\nFermi energy of the ferromagnets, EF= (E\"\nF+E#\nF)=2,\nis equal to the Fermi energy of superconductors. How-\never, the in\ruence of the mismatch of e\u000bective electron3\nmasses and Fermi energies is similar to the in\ruence of\n\fnite interfacial transparency: an increase of the normal\nrefection [46, 57, 58].\nWe take the pair potential \u0001( r) in the form\n\u0001(r) = \u0001[ei\u001e1\u0002(\u0000x) +ei\u001e2\u0002(x\u0000d1\u0000d2)];(3)\nwhere \u0001 is the bulk superconducting gap. The macro-\nscopic phase di\u000berence across the junction is \u001e=\u001e1\u0000\u001e2.\nThe temperature dependence of \u0001 is given by \u0001( T) =\n\u0001(0) tanh(1 :74p\nTc=T\u00001) with \u0001(0) being the pair po-\ntential at zero temperature and Tcthe critical tempera-\nture [59]. In general, \u0001( r) should be determined self-\nconsistently [29, 42]. However, for simplicity we use\nthe stepwise pair potential given in Eq. (3), since self-\nconsistency will not alter our results qualitatively [26].\nB. Solution of the scattering problem\nThe parallel component of the wave vector, kk, is con-\nserved due to the translational invariance of the junc-\ntion in the direction perpendicular to the xaxis. Conse-\nquently, the wave function can be written in the form\n\t(r) = (x)eikk\u0001r; (4)\nwhere (x) = [u\"(x);u#(x);v\"(x);v#(x)]Tsatis\fes the\nfollowing boundary conditions:\n (x)jxi+0= (x)jxi\u00000= (xi); (5)\nd (x)\ndx\f\f\f\f\nxi+0\u0000d (x)\ndx\f\f\f\f\nxi\u00000=ZikF (x): (6)\nHere,Zi= 2mWi=~2kF(i= 1;2;3) are parameters [60]\nthat measure the transparency of the interfaces (i.e.,\nbarrier strengths) located at xi= 0;d1;d1+d2, and\nkF=p\n2mEF=~2is the Fermi wave vector.\nThe four independent solutions of the scattering prob-\nlem for Eq. (1) correspond to the four types of quasi-\nparticle injection processes: an electronlike or a holelike\nquasiparticle injected from either the left or the right su-\nperconducting electrode. When an electronlike quasipar-\nticle is injected from the left, the solutions of Eq. (1) for\nthe left superconductor ( x <0) written in the compact\nform are\n\u0012\nu\u001b\nv\u0016\u001b\u0013\n=\u0012\n\u0016uei\u001e1=2\n\u0016ve\u0000i\u001e1=2\u0013\neik+x+b1\u001b\u0012\n\u0016uei\u001e1=2\n\u0016ve\u0000i\u001e1=2\u0013\ne\u0000ik+x\n+a1\u001b\u0012\n\u0016vei\u001e1=2\n\u0016ue\u0000i\u001e1=2\u0013\neik\u0000x; (7)\nand for the right superconductor ( x>d 1+d2)\n\u0012\nu\u001b\nv\u0016\u001b\u0013\n=c1\u001b\u0012\n\u0016uei\u001e2=2\n\u0016ve\u0000i\u001e2=2\u0013\neik+x+d1\u001b\u0012\n\u0016vei\u001e2=2\n\u0016ue\u0000i\u001e2=2\u0013\ne\u0000ik\u0000x;\n(8)where \u0016\u001bis opposite to \u001b=\"#, \u0016u=p\n(1 + \n=E)=2,\n\u0016v=p\n(1\u0000\n=E)=2, and \n =p\nE2\u0000\u00012. Constants\na1\u001b;b1\u001b;c1\u001b, andd1\u001bcorrespond to Andreev and normal\nre\rections, direct transmission, and nonlocal Andreev re-\n\rection, respectively. For the left ferromagnetic layer\nF1(0<x<d 1) solutions of Eq. (1) are\n\u0012\nu\"\nu#\u0013\n=c1\u0012\nicos\u000b1\n2\n\u0000sin\u000b1\n2\u0013\neiq+\n\"x+c2\u0012\nicos\u000b1\n2\n\u0000sin\u000b1\n2\u0013\ne\u0000iq+\n\"x\n+c3\u0012\nisin\u000b1\n2\ncos\u000b1\n2\u0013\neiq+\n#x+c4\u0012\nisin\u000b1\n2\ncos\u000b1\n2\u0013\ne\u0000iq+\n#x;\n(9)\n\u0012\nv\"\nv#\u0013\n=c5\u0012\nicos\u000b1\n2\n\u0000sin\u000b1\n2\u0013\neiq\u0000\n\"x+c6\u0012\nicos\u000b1\n2\n\u0000sin\u000b1\n2\u0013\ne\u0000iq\u0000\n\"x\n+c7\u0012\nisin\u000b1\n2\ncos\u000b1\n2\u0013\neiq\u0000\n#x+c8\u0012\nisin\u000b1\n2\ncos\u000b1\n2\u0013\ne\u0000iq\u0000\n#x:\n(10)\nSolutions for the right ferromagnetic layer F 2(d1<x<\nd1+d2) can be obtained by substituting \u000b1!\u000b2, with\na new set of constants c0\n1;:::;c0\n8.\nWhen a holelike quasiparticle is injected from the left,\nthe solutions of Eq. (1) for the left superconductor ( x<\n0) are given by\n\u0012\nu\u001b\nv\u0016\u001b\u0013\n=\u0012\n\u0016vei\u001e1=2\n\u0016ue\u0000i\u001e1=2\u0013\ne\u0000ik\u0000x+b2\u001b\u0012\n\u0016vei\u001e1=2\n\u0016ue\u0000i\u001e1=2\u0013\neik\u0000x\n+a2\u001b\u0012\n\u0016uei\u001e1=2\n\u0016ve\u0000i\u001e1=2\u0013\ne\u0000ik+x; (11)\nwhile for the right superconductor ( x>d 1+d2)\n\u0012\nu\u001b\nv\u0016\u001b\u0013\n=c2\u001b\u0012\n\u0016vei\u001e2=2\n\u0016ue\u0000i\u001e2=2\u0013\ne\u0000ik\u0000x+d2\u001b\u0012\n\u0016uei\u001e2=2\n\u0016ve\u0000i\u001e2=2\u0013\neik+x:\n(12)\nConstantsa2\u001b;b2\u001b;c2\u001b, andd2\u001bdescribe analogous pro-\ncesses as in the case of an injected electronlike quasipar-\nticle given earlier.\nSolutions for ferromagnetic layers in the case a hole-\nlike quasiparticle injected from the left can be obtained\nby substituting ci!Ciandc0\ni!C0\niin solutions for\nthe case of an injected electronlike quasiparticle. The\nlongitudinal x- components of the wave vectors in the\nsuperconductors are given by\nk\u0006=r\n2m\n~2(EF\u0006\n)\u0000k2\nk; (13)\nwhile their counterparts in the ferromagnetic layers read\nq\u0006\n\u001b=r\n2m\n~2(EF\u0006E+\u001a\u001bh)\u0000k2\nk: (14)\nThe sign\u0006in the superscript corresponds to the sign of\nthe quasiparticle energy, whereas \u001a\u001b= +1 (\u00001) is related\nto the spin projection \u001b=\"(#).4\nSolutions for quasiparticles with opposite spin orien-\ntations are nontrivially coupled: in the superconductors,\nEqs. (7) and (8) and Eqs. (11) and (12), as well as in the\nferromagnets, Eqs. (9) and (10). In that manner, both\nthe usual and spin-\rip Andreev re\rections are taken into\naccount.\nAll the unknown 48 coe\u000ecients in the above solutions,\nin both electronlike and holelike scattering problems, are\ndetermined from the boundary conditions, Eqs. (5) and\n(6), at the three interfaces.\nC. The Josephson current\nThe Josephson current can be calculated from the lin-\near superposition of amplitudes the normal and anoma-\nlous Andreev re\rections [61], a1\u001banda2\u001b,\nI(\u001e) =e\u0001\n2~X\n\u001b;kk;!nkBT\n2\nn(k+\nn+k\u0000\nn)\u0014a1\u001bn(\u001e)\nk+n\u0000a2\u001bn(\u001e)\nk\u0000n\u0015\n:\n(15)\nHere,\u001e=\u001e1\u0000\u001e2is the superconducting phase di\u000berence,\nanda1\u001bn;a2\u001bn,k\u0006\nn, and \nn=p\n!2n+ \u00012are obtained\nfrom the corresponding quantities shown in the previous\nsection by performing the analytic continuation, E!\ni!n. The Matsubara frequencies are !n= (2n+1)\u0019kBT,\nwithn= 0;\u00061;\u00062;:::and the temperature T.\nFor nonmagnetic (SNS and SIS) Josephson junctions\na1\u001banda2\u001bare\u001bindependent and related by particle-\nhole symmetry, a1(\u001e) =a2(\u0000\u001e). However, for SFS\njunctions (with homogeneous/inhomogeneous magneti-\nzation), when the odd-frequency spin-triplet correlations\n(short/long range) are generated, the amplitudes a1\u001band\na2\u001bare\u001bdependent. In that way, the spin-mixing pro-\ncesses are included.\nPerforming a summation over kkby employingP\nkk!\nA(2\u0019)\u00002R\nd2k, we obtain\nI(\u001e) =\u0001\u0019\nRNekBTZ\u0019=2\n0d\u0012sin\u0012cos\u0012\u0002\n\u0002X\n\u001b!nk+\nn+k\u0000\nn\n4\nn\u0012a1\u001bn(\u001e)\nk+n\u0000a2\u001bn(\u001e)\nk\u0000n\u0013\n:(16)\nHere,RN= 2\u00192~=Ae2k2\nFwithAbeing the cross section\nof the junction and we assume kk=kFsin\u0012, since we deal\nwith standard BCS superconductors, where \u0001 =EF\u0018\n10\u00003\u000010\u00004.\nIn general, the current-phase relation is an anharmonic\n2\u0019-periodic function and can be expanded as\nI(\u001e) =I1sin\u001e+I2sin 2\u001e+:::; (17)\nwhere thenth harmonic amplitude Incorresponds to the\nphase-coherent transport of nCooper pairs.\n0.0 0.2 0.4 0.6 0.8 1.0-0.050.000.050.00.10.2\nφ/πeRNI/π∆\n0ππ\nπ/4π/4\nπ/2π/2\n3π/43π/4\n0SNS\nkFd1= 500\nkFd2= 500\nkFd1= 10\nkFd2= 990(a)\n(b)FIG. 2. The Josephson current in SF 1F2S junctions as a\nfunction of the superconducting phase di\u000berence \u001efor the\nferromagnetic layer thicknesses (a) d1=d2= 500k\u00001\nFand (b)\nd1= 10k\u00001\nF,d2= 990k\u00001\nF, forh=EF= 0:1,T=Tc= 0:1, and\ndi\u000berent relative angles between the magnetizations: \u000br= 0,\n\u0019=4,\u0019=2, 3\u0019=4,\u0019. The Josephson current for SNS junction\n(h= 0) with the thickness d1+d2= 1000k\u00001\nFis shown in the\npanel (a) for comparison (dotted line).\nIII. RESULTS AND DISCUSSION\nWe illustrate our results on SF 1F2S planar junc-\ntions with relatively weak ferromagnets h=EF= 0:1\nat low temperature T=Tc= 0:1. The ferromagnetic\ncoherence length is \u0018F=~vF=h= 20k\u00001\nF. Super-\nconductors are characterized by the bulk pair poten-\ntial at zero temperature \u0001(0) =EF= 10\u00003which corre-\nsponds to the superconducting coherence length \u0018S(0) =\n~vF=\u0019\u0001(0) = 636 k\u00001\nF. The Josephson current is nor-\nmalized to \u0019\u0001=eRNas usual [see Eq. (16)]. The total\nthickness of the ferromagnetic bilayer is kept constant,\nd1+d2= 1000k\u00001\nF= 50\u0018F= 1:57\u0018S(0):\nA. Fully transparent interfaces\nThe current-phase relation in a junction with fully\ntransparent interfaces, Z1=Z2=Z3= 0, for vari-\nous values of the relative angle between magnetizations,\n\u000br=\u000b1\u0000\u000b2, and equal thicknesses of the ferromag-\nnetic layers is shown in Fig. 2(a). It can be seen that the5\n20 40 60 80-0.050.000.05\n4004204404604805000.000.050.10\n(a) (b)\n(c) (d)\nkFd1 kFd1eRNIc/π∆ eRNI1,2/π∆\nI1 I1I2 I2\nFIG. 3. The critical current in SF 1F2S junctions with mutu-\nally orthogonal magnetizations \u000br=\u0019=2 and fully transpar-\nent interfaces, Z1=Z2=Z3= 0, shown as a function of the\nF1layer thickness d1: (a) thin and (b) thick F 1layer. The\ntotal thickness is d1+d2= 1000k\u00001\nF. The amplitudes of the\n\frst (solid line) and the second harmonic (dashed line) of the\nJosephson current are shown in (c) and (d).\ncurrent is completely suppressed for the parallel mag-\nnetizations, \u000br= 0, and increases almost monotonously\nwith a misorientation of magnetizations up to I(\u001e) of the\ncorresponding SNS junction ( h= 0) for the antiparallel\nmagnetizations, \u000br=\u0019, which has been observed experi-\nmentally [62]. The current-phase relation is a practically\nuniversal function of the ferromagnetic in\ruence, which\nis measured by the product of thickness and the exchange\n\feld strength, d\u0001h[46]. This explains the cancellation of\nferromagnetic in\ruence in the case of equal thicknesses\nand equal strengths of the ferromagnets. However, in\nthat case no signi\fcant in\ruence of the triplet correla-\ntions was found even for noncollinear magnetizations [20].\nThis we explain now by a dominant \frst harmonic due\nto the geometric resonance e\u000bect [see Fig. 3(d)].\nA dominant second harmonic can be seen in Fig. 2(b)\nfor highly unequal thicknesses of the ferromagnetic layers,\nkFd1= 10 andkFd2= 990, and noncollinear magneti-\nzations. In contrast to the case of equal ferromagnetic\nlayers, the critical current is not a monotonous function\nof the misorientation angle \u000br. It almost vanishes for\n\u000br= 0;\u0019and reaches the maximum for \u000br=\u0019=2. This\nis a manifestation of the long-range spin-triplet proximity\ne\u000bect in ferromagnetic bilayers where the \frst harmonic\nis suppressed and the phase-coherent transport of two\nCooper pairs becomes dominant [22{26].\nTo illustrate the role of ferromagnetic bilayer asymme-\ntry, we calculate the critical current Icand the ampli-\ntudes of the \frst and the second harmonic, I1andI2,\nas functions of the F 1layer thickness, d1, keeping the\n0.0 0.2 0.4 0.6 0.8-0.0050.0000.005\n0.0 0.2 0.4 0.6 0.8 1.0-0.0050.0000.0050.000.050.10\n-0.04-0.020.000.020.04\nφ/π φ/πeRNI/π∆\n0 0π ππ\n0(a) (b)\n(c) (d)SN 1N2S SF1F2S\nSF1F2S SF1F2Sπ/2\nπ/2\nπ/2(0,1,0)(0,1,0)\n(1,0,1)\n(1,0,1)(1,1,1)\n(1,1,1)FIG. 4. The Josephson current in (a) SN 1N2S and (b){\n(d) SF 1F2S junctions with layer thicknesses d1= 10k\u00001\nF,\nd2= 990k\u00001\nF, and di\u000berent barrier strengths ( Z1;Z2;Z3) at\nthe interfaces. Relative angles between the magnetizations\n\u000br= 0,\u0019=2,\u0019in SF 1F2S junctions are indicated in the plots.\ntotal thickness constant, kF(d1+d2) = 1000. The rela-\ntive angle between the magnetizations is \u000br=\u0019=2 (the\nstrongest e\u000bect of spin-triplet correlations) and the inter-\nfaces are fully transparent, Z1=Z2=Z3= 0. Results\nare shown in Fig. 3. When d1approaches d2we can see\nthe rise of the I1amplitude due to the geometric reso-\nnance. Because of that, the \frst harmonic is dominant\nfor equal ferromagnetic layers and the spin singlet and\nspin triplet with zero spin projection correlations prac-\ntically generate the supercurrent [20]. In ferromagnetic\nbilayers only even harmonics (the second is the largest)\ncan be generated by long-range spin-triplet correlations\nwith\u00061 spin projections [23].\nThe characteristic oscillations of I1(d1);I2(d1), and\nIc(d1) are due to 0\u0000\u0019transitions with the period\npractically equal to the ferromagnetic coherence length\n\u0018F= 20k\u00001\nF. Note that in the clean limit the critical\ncurrentIcis minimum but not zero at the 0 \u0000\u0019transi-\ntion [8, 9, 46].\nB. Finite interfacial transparencies\nThe role of \fnite interfacial transparencies is illus-\ntrated in Figs. 4-6. For comparison, the current-phase re-\nlation for a clean SN 1N2S (h= 0) junction with kFd1=\n10,kFd2= 990 is shown for various interfacial barrier\nstrengths, see Fig. 4(a). With decreasing transparency\nthe supercurrent is suppressed in comparison to the fully\ntransparent case [see the dotted curve in Fig. 2(a)]. In\nthis case the \frst harmonic is dominant. The supercur-\nrent of SF 1F2S junctions with \u000br= 0;\u0019=2;\u0019and di\u000ber-6\nFIG. 5. The \frst harmonic amplitude (solid line) and the\nsecond harmonic amplitude (dashed line) of the Josephson\ncurrent-phase relation in SF 1F2S junctions with orthogonal\nmagnetizations \u000br=\u0019=2 as a function of the F 1layer thick-\nnessd1, for total thickness d1+d2= 1000k\u00001\nF, and for dif-\nferent barrier strengths at the interfaces Z= (Z1;Z2;Z3):\n(a)Z= (0;0;0), (b)Z= (0;1;0), (c)Z= (0;3;0), and (d)\nZ= (1;0;1). Additional geometric resonances are pointed to\nby arrows: (b) and (c).\nent interfacial transparencies is shown in Figs. 4(b)-(d).\nNote that for collinear magnetizations, \u000br= 0;\u0019, the\ncurrent is short ranged and for orthogonal magnetiza-\ntions,\u000br=\u0019=2, the dominant second harmonic is due\nto the long-range spin-triplet correlations. It can be seen\nthat a lower transparency of the interface between ferro-\nmagnets is less destructive than lower transparencies of\nthe interfaces between superconductors and neighboring\nferromagnetic layers. The depairing e\u000bect of normal re-\n\rection at the SF interfaces is stronger due to the direct\nsuppression of the Andreev process.\nThe in\ruence of \fnite interfacial transparencies on the\n\frst and the second harmonics is quite di\u000berent. A \frst\nharmonic is generated by the phase-coherent transport\nof one Cooper pair, while the second harmonic is de-\ntermined by the phase-coherent transport of two Cooper\npairs. In Fig. 5 the \frst harmonic amplitude (solid curve)\nand the second harmonic amplitude (dashed curve) are\nshown as functions of d1forkF(d1+d2) = 1000,\n\u000br=\u0019=2, and di\u000berent transparencies of the interfaces,\n0.0 0.2 0.4 0.6 0.8 1.00.000.050.10\nφ/πeRNI/π∆SF1F2S\nZ= (0,1,0)\nkFd1= 500\nkFd2= 500\n0π/2π\nSN1N2SFIG. 6. The Josephson current in SF 1F2S junctions with\nequal layer thicknesses d1=d2= 500k\u00001\nFand interfacial\nbarrier strengths Z2= 1,Z1=Z3= 0, shown for di\u000ber-\nent relative angles between magnetizations \u000br= 0,\u0019=2,\u0019.\nThe Josephson current in SN 1N2S junction with the same\nlayer thicknesses and barrier strengths is shown for compari-\nson (dotted line).\nZ= (Z1;Z2;Z3). It can be seen that both I1andI2am-\nplitudes are suppressed by decreasing the transparency of\nthe interfaces, the \frst harmonic amplitude being much\nless a\u000bected.\nNew geometric resonances and ampli\fcations of I1\nemerge for a \fnite transparency of the interface between\nferromagnetic layers [see Figs. 5(b) and 5(c)]. Besides\nthe resonant ampli\fcation of I1ford1=d2, we \fnd reso-\nnant ampli\fcations at d1=d2=3;d2=5;:::. This e\u000bect is\nrelated to the multiple re\rections that lead to the emer-\ngence of electron and hole quasiclassical trajectories with\na canceled phase accumulation.\nThe current-phase relations for equal ferromagnetic\nlayers and \fnite interfacial transparency between them\nare shown in Fig. 6. The critical currents are approxi-\nmately two times smaller than in the fully transparent\ncase [see Fig. 2(a)]. We can see a peculiar ampli\fca-\ntion of the Josephson current for antiparallel magnetiza-\ntions,\u000br=\u0019, in comparison with nonmagnetic layers.\nThis e\u000bect was previously reported for SFIFS Joseph-\nson junctions with antiparallel orientations of magneti-\nzations [63], and for junctions between superconductors\nwith ferromagnetic exchange \felds [64, 65].\nIV. CONCLUSIONS\nWe have studied the Josephson e\u000bect in clean planar\nSF1F1S junctions with arbitrary transparencies of the\ninterfaces between the layers. By solving the scatter-\ning problem for the Bogoliubov-de Gennes equation, we\nhave calculated numerically the current-phase relation,7\nthe critical current, and \frst and second harmonic ampli-\ntudes. For relatively a weak exchange \feld, h=EF= 0:1,\nmutually orthogonal magnetizations, \u000br=\u0019=2, and very\nunequal thicknesses of the ferromagnetic layers, d1\u001cd2,\na well-pronounced second harmonic is obtained as a sig-\nnature of the long-range spin-triplet correlations. On\nthe other hand, for equally thick ferromagnetic layers,\nd1=d2, the spin-singlet contribution to the \frst har-\nmonic is enhanced due to the geometric resonance, and\ndominates even for thick layer junctions (strong ferro-\nmagnetic in\ruence) with orthogonal magnetizations.\nBoth resonant and spin-triplet e\u000bects qualitatively\npersist in the presence of impurities or moderate dis-\norder (see, for example, the quasiclassical analysis in\nRefs. [22, 25, 26]). In experiments the resonances can\nbe recognized as more sinusoidal I(\u001e), while the long-\nrange spin-triplet correlations in the Josephson junctions\nwith ferromagnetic bilayers lead to the more anharmonic\ncurrent-phase relation due to the dominant second har-\nmonic.\nBoth the \frst and the second harmonic amplitude show\ncharacteristic oscillations with varying thicknesses of the\nferromagnetic layers. The critical current oscillates in the\nsame manner. This is due to the 0 \u0000\u0019transitions and\nthe period of oscillations is the ferromagnetic coherence\nlength\u0018F.For a \fnite transparency of interfaces the supercur-\nrent is suppressed, with higher harmonics being more\na\u000bected. A low transparency of the F 1=F2interface,\nwhere the long-range spin-triplet correlations are gener-\nated, has a nontrivial impact on the interference phe-\nnomena and consequently to the current-phase relation.\nFor certain thicknesses of the ferromagnetic layers in ad-\ndition tod1=d2new geometric resonances occur at\nd1=d2=3;d2=5;:::, making the \frst harmonic dominant\neven in asymmetric junctions.\nACKNOWLEDGMENTS\nThe work was supported by Serbian Ministry of Edu-\ncation, Science and Technological Development, Project\nNo. 171027. Z.R. also acknowledges the support of the\nSerbian Academy of Sciences and Arts, Grant No. F87.\nA.I.B. acknowledges support by the Ministry of Science\nand Higher Education of the Russian Federation within\nthe framework of state funding for the creation and devel-\nopment of World-Class Research Center \\Digital Biode-\nsign and Personalized Healthcare\", Grant No. 075-15-\n2022-304. 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Maryland NanoCenter, University of Maryland, College Pa rk, MD 20742, USA\nIt is shown that the current-induced torques between a ferro magnetic layer and an antiferromag-\nnetic layer with a compensated interface vanish when the fer romagnet is aligned with an axis of\nspin-rotation symmetry of the antiferromagnet. For proper ly chosen geometries this implies that\nthe current induced torque can stabilize the out-of-plane ( or hard axis) orientation of the ferromag-\nnetic layer. This current-induced torque relies on phase co herent transport, and we calculate the\nrobustness of this torque to phase breaking scattering. Fro m this it is shown that the torque is not\nlinearly dependent on applied current, but has an absolute m aximum.\nI. INTRODUCTION\nCurrent-induced torques result from the interaction\nbetween conductionelectronspins andthe magnetization\nof a sample when current ﬂows through it. This torque\nis generally present when the magnetization is spatially\nnonuniform, and has been extensively studied in the\ncontext of magnetic domain walls and spin valve struc-\ntures. Since its theoretical prediction1,2, extensive stud-\nieshaveled toatheoreticalframeworkofcurrent-induced\ntorque in ferromagnets that describes experimental re-\nsults with quantitative success.3It has been proposed\nthat current-induced torques also exist in antiferro-\nmagnetic systems.4,5Previous theoretical studies con-\nsidered systems composed entirely of antiferromagnetic\nlayers4,6,7as well as experimental8and theoretical9–11\nsystems with both ferromagnetic and antiferromagnetic\nlayers. Theoretical work has also focused on antiferro-\nmagnet textures.12–15Experiments have demonstrated\ncurrent-induced torque in materials with other types\nof complex magnetic ordering, such as skyrmion lat-\ntices. Recent theory16andexperiment17haveshownthat\nantiferromagnets exhibit anisotropic magnetoresistance,\ndemonstrating a coupling between magnetic order and\ncharge transport these materials.\nAntiferromagnets exhibit an array of magnetic order-\ning, such as spin density waves that are commensurate\nor incommensurate with the lattice, and conﬁgurations\nwith multiple spin density waves. As shown in Ref. 9,\nthe symmetry properties of the antiferromagnetic layer\ncan lead to torques in multilayers with qualitatively dif-\nferent properties than conventional spin valves. In par-\nticular, a collinear compensated antiferromagnetic layer\ninterface (with each spin in the ±ˆzdirection, which we\ncall a 1Q spin structure) leads to a torque which vanishes\nwhen the ferromagnet is perpendicular to the ˆ zdirec-\ntion. This torque can stabilize the hard-axis orientation\nof the ferromagnet in systems where the antiferromag-\nnet is pinned. Here we treat similar systems (see Fig.\n1a), and compute the current-induced torque on the fer-\nromagnetic layer. (Previous works have investigated thecurrent-induced torque on the antiferromagnetic layer in\nsuch systems.10,11)\nIn this work we consider a system where the antifer-\nromagnetic layer has a 3Q spin structure (see Fig. 1b).\nThis is qualitatively diﬀerent than the previously studied\n1Q antiferromagnet because the 3Q structure has only a\nsingleaxis of spin rotational symmetry (3-fold in this\ncase), whereas for the 1Q antiferromagnet alldirections\nperpendicular to the ˆ zdirection are axes of 2-fold spin\nrotational symmetry. We show that an important con-\nsequence of the reduced symmetry of the 3Q antiferro-\nmagnet is that the current-induced torque stabilizes the\nout-of-plane magnetic orientation only when the magne-\ntization is initialized nearby this orientation (in contrast,\nthe 1Qantiferromagnetdrives anyinitial orientationout-\nof-plane). In this work we additionally determine the ef-\nfects of phase breaking scattering: The current-induced\ntorque relies on phase coherence, and quantifying the ro-\nbustness with respect to scattering is important to gauge\nthe feasibility of observing these eﬀects in real systems.\nOur results are easily generalized to multilayers com-\nposed of a free ferromagnet layer, and a ﬁxed magnetic\nlayer whose spin conﬁguration has an axis of n-fold ro-\ntational symmetry. The key property of the torque is\nthat: if the ferromagnetic layer is aligned with an axis\nof spin-rotational symmetry of the ﬁxed layer, then the\ncurrent-induced torque (in fact, all torques) must van-\nish. This is seen by recognizing that, by assumption,\nthe system is invariant with respect to spin rotations\nabout the ferromagnet orientation by some angle φn,\nand any nonzero torque (which must be perpendicular\nto the ferromagnet orientation) does not respect this\nsymmetry. For conventional spin valves, this statement\nimplies the well known fact that the current-induced\ntorque vanish when the ferromagnet layers are aligned or\nanti-aligned. Identifying the points where the current-\ninduced torque vanishes is important because the torque\nmay drive the magnetization to these ﬁxed points. For\nproperlydesignedantiferromagnet-ferromagnetmultilay-\ners this property of the torque can stabilize the out-of-\nplane magnetic orientation.9This is because this orien-2\ntation, being a maximum of the magnetic free energy,\nrepresents a ﬁxed point for the conventional micromag-\nnetic torques. In the absence of current-induced torques,\nthis out-of-plane conﬁguration is an unstable ﬁxed point;\nhowever if the current-induced torque drives the fer-\nromagnet to this orientation and exceeds the damping\ntorque, it can stabilize this conﬁguration, as shown by\nmicromagnetic simulations in Ref. 9.\nII. METHOD\nTo calculate the current-induced torques, we use the\nnonequilibrium Green’s function technique within a tight\nbinding representation. This is a well established ap-\nproach to calculating the transport properties of mag-\nnetic thin ﬁlms. We highlight the most important details\nhere. The system is taken to consist of two semi-inﬁnite\nelectrodes, with a scatteringregionplaced between them.\nThereis a diﬀerence Vappin the electrochemicalpotential\nof the two electrodes. The central quantity is the density\nmatrixρ:\nρ=i\n2π/integraldisplayEF−Vapp/2\n−∞[Gr(E)−Ga(E)]dE+\n/integraldisplayEF+Vapp/2\nEF−Vapp/2Gr(E)ΓL(E)Ga(E)dE.(1)\nwhereGr,a(E) = (E−HC−Σr,a\nL(E)−Σr,a\nR(E))−1.\nHCis the scattering region Hamiltonian, and Σr\nLis the\nself energy which describes the electronic coupling be-\ntweenthescatteringregionandthesemi-inﬁnite left lead;\nit is given by Σr\nL=V†\nC,Lgr\n0,L(E)VC,L, whereVC,Lis the\ncoupling matrix element between the left lead and cen-\ntral region, and g0,Lis the surface Green’s function of\nthe isolated semi-inﬁnite left lead. The same form of self\nenergy holds for the right lead.\nAs noted in previous works,7phase coherence plays a\ncentral role in a number of the antiferromagneticsystems\nstudied so far. To explore the robustness of the torques\nin this system, we include an additional self energy Σ Sin\nGreen’s function which describes elastic, phase breaking\nscattering. Its form is:\nΣS(E) =iD(Gr(E)−Ga(E)) (2)\nwhereDparameterizes the scattering. A discussion of\nthe parameter Din terms of real material properties and\ntemperature is given in Sec. (III).\nWe assumethe spin-orbitcouplingis negligible, sothat\nthe current-induced torque on the ferromagnet layer is\nsimply given by the transverse component of incoming\nspin current ﬂux. For our geometry, the net spin cur-\nrent has real space velocity in the ˆ ydirection. The spin\ncurrent operator /vectorJ(y) is then:ˆ/vectorJ(y) =/summationdisplay\nj∈R(y)\nk∈L(y)\ns,s′i/bracketleftBig\nc†\nj,s/vector σs,s′ck,s′tj,k−h.c./bracketrightBig\n,(3)\nwhereR(y) are the set of sites with coordinate y′greater\nthany, andL(y) are the set of sites with coordinate y′\nless than y./vector σsigma is the vector of Pauli matrices,\nand we take the hopping tj,kbetween all sites jandk\nto be spin independent. We present results in terms of\ntorque per current (units of µB/e), which represents the\nspin torqueeﬃciency. The absolutevalueofthis quantity\ndetermines the critical current needed to drive magnetic\ndynamics.\nAs discussed in Refs. 18 and 19, it is sometimes nec-\nessary to compute the entire energy integral (both terms\nin Eq. 1) in order to ﬁnd the current-induced torques.\nThis is particularly the case when the torques in question\nare present in equilibrium (which is itself dependent on\nthe symmetries of the system, as discussed in Ref. 18).\nWe checked explicitly that the current-induced torque in\nquestionforthese systems aredominated by the nonequi-\nlibrium contribution to the density matrix (the second\nterm of Eq. 1), and present only this contribution in the\nresults (the remaining “energy integral” contribution is\nseveral orders of magnitude smaller in all the cases we\nchecked). We take the Fermi energy to be EF= 3.75t,\nand use a dense k-point mesh to converge the transport\nintegrals, up to 10002k-points for a unit cell having 4\natoms per layer.\nA schematic of the overall system is shown in Fig.\n1a. It consists of semi-inﬁnite ferromagnetic and anti-\nferromagnetic layers, separated by a nonmagnetic spacer\nwhich is 3 atomic layers thick. The layers are fcc, with\ninterfaces in the [111] direction. We use two diﬀerent\nspin structures for the antiferromagnet. One is a 3Q\nspin structure, depicted in Fig. 1b. The spin structure\nin the (111) planes is shown in Fig. 1c, which shows the\n3-fold symmetry of the spin in the x−zplane. Each spin\nalso has a component along the yaxis (into or out of the\npage). Atomswithnoarrowintheﬁgurehaveaspinfully\naligned in the +ˆ ydirection, while other atoms’ spins are\npartially canted in the −ˆydirection, so that the net bulk\nspinvanishes. Commonantiferromagneticmaterialssuch\nas FeMn are predicted to have a 3Q ground state,21,23,29\nconsistent with measurements24,25, although there is not\ncomplete consensus between all the experimental data.\nTo further explore the consequences of the antiferromag-\nnet symmetry, we also consider a system where the y\ncomponent of the spins are set to 0. This artiﬁcial sys-\ntem retains the 3-fold symmetry in the x−zplane, but\nis also symmetric under sy↔ −sy. We refer to this as\nthe “no-canting” antiferromagnet. We emphasize that\nour primary results generalize to any antiferromagnet for\nwhich there is an axis of n-fold spin rotational symmetry,\nas explained in the introduction.\nWe present the angular variation of the torque on the3\nFIG. 1: (a) Overall system geometry (b) The crystal and\nspin structure for the 3Q state. The spins at the corners of\nthe interior box all point inward. (c) The spin on the [111]\ninterface of the lattice from (b). The small black, medium\nred, and large gray dots represent atoms in diﬀerent layers\n(i.ewith diﬀerent y-values). The spin of dots without an ar-\nrow is completely in the ˆ ydirection, while other spins are\npartially canted in the ˆ ydirection. (d) Spherical coordinate\nsystem used to describe the torques on the ferromagnet. The\nblue (dark) spins in the x−zplane represent the 3-fold sym-\nmetric spins of the antiferromagnetic layer, and the skinni er\nred arrow represents the orientation of the ferromagnet lay er.\nferromagnet layer in terms of spherical coordinates, as\nshown in Fig. 1d. The ˆ ydirection is the hard axis of\nthe F, which is taken to coincide with the axis of 3-fold\nsymmetry of the antiferromagnet. As explained in the\nintroduction, this alignment of hard axis and the anti-\nferromagnet axis of spin rotational symmetry is crucial\nfor the out-of-plane orientation to be stabilized by the\ncurrent-induced torque. The ˆ zdirection is along one of\nthe spins of the antiferromagnetic layer. We utilize sim-\nilar schematics as Fig. 1d in the next section to show\nthe relative orientation of the ferromagnet layer with the\nspins of the antiferromagnet.\nIII. RESULTS\nThe current-induced torque on the ferromagnetic layer\nfor a no-canting antiferromagnetic system is shown in\nFig. 2. Unlike the current-induced torque in a conven-\ntional spin valve, whose magnitude has a simple sin( θ)\ndependence, we ﬁnd a more complex angular dependence\nfor the torque. We ﬁrst ﬁx φ= 0◦and vary the ferromag-\nnet orientation from θ= 0 to 360◦. These orientations\nare in the easy plane. The torques conform to the 3-fold\nsymmetry, varying approximately as sin(3 θ), as shown\nin Fig. 2a. For ﬁxed φ= 90◦, sweeping the polar angle\nFIG. 2: The angular dependence of the current-induced\ntorque (CIT) on the ferromagnet for the system with no an-\ntiferromagnetic canting in the y-direction (the “no canting”\nsystem). The black dashed line is the torque in the ˆφdi-\nrection, and the gray line with markers is the torque in the\nˆθdirection. (a) shows the torque as when the ferromagnet\nis coplanar with the antiferromagnet spins, which shows a\nsin(3θ) dependence. (b) shows an intermediate angle, and (c)\nshows the torque as the ferromagnet orientation is normal to\ntheplaneoftheantiferromagnet spins. Inthiscase, thetor que\nvaries as sin(2 θ). The diagrams to the right of the plots show\nthe direction of antiferromagnet spins in the x-z plane, and\nwith a circle representing the angles of the ferromagnet lay er\nin the plot.\nθtakes the magnetization out of the easy plane, through\nthe hard axis direction. The torques in this case are\nshown in Fig. 2c. The torques vary as sin(2 θ), again\nas required by symmetry. For ﬁxed φ= 45◦, varying\nθtakes the ferromagnet on an “oﬀ-axis” orbit, and the\ntorque exhibits more complex angular dependence.\nFigure 3 showssimilar results for the 3Q spin structure\nfor the same set of magnetic orientations. The reduction\nin symmetry due to the inequivalence of syand−syleads\nto more complex behavior of the torque. For φ= 0◦, we\nnotetheinvarianceofthetorqueunder θ→θ+120◦. Key\ndata points are shown in Fig. 3c by the black arrows. As\nargued in the introduction, when the ferromagnet layer\nis aligned to the axis of 3-fold symmetry, the current-\ninduced torque vanishes.\nTogainafullerviewofthecurrent-inducedtorquenear\nthe out-of-plane ﬁxed point, we show the torque in the\nvicinity of these points in Fig. 4. For the no-canting\nsystem, the +ˆ yand−ˆyﬁxed point are equivalent. For\nelectrons ﬂowing from the antiferromagnet to the ferro-\nmagnet, these are stable ﬁxed points. For the 3Q an-\ntiferromagnet, on the other hand, the +ˆ yand−ˆyﬁxed\npoints are inequivalent. In this case, we ﬁnd the +ˆ yis a\nstable attractor, while the −ˆyis an elliptic ﬁxed point.4\nFIG. 3: The angular dependence of the torque on the ferro-\nmagnet for the system with no 3Q spin ordering of the an-\ntiferromagnet. The black dashed line is the torque in the ˆφ\ndirection, and the gray line with markers is the torque in the\nˆθdirection. (a) shows that the torque again varies as sin(3 θ)\nwhen the ferromagnet layer is conﬁned to the x−zplane\n(easy plane). (b) shows complex angular dependence for the\nferromagnet layer oriented along an axis of low symmetry.\n(c) shows that the torque vanishes when the ferromagnet is\naligned to the axis of 3-fold symmetry of the antiferromagne t\n(arrows indicate these points).\nThe nature of the ﬁxed point (stable, unstable, elliptic,\netc.) is parameter dependent, making it diﬃcult to make\ngeneralstatements about the prevalence of diﬀerent ﬁxed\npoints.\nFor antiferromagnetic systems it is also important to\ndistinguish between stable ﬁxed points to which any ini-\ntial magnetization vector is driven (global attractors),\nand those ﬁxed points for which only an initial magne-\ntization vector nearby is driven (local attractors). In-\nspection of Fig. 2a shows that, if the magnetization is\nin thex−zplane, the torque driving it to the out-of-\nplane direction is quite weak (in this case, the relevant\ntorque is in the ˆφdirection). On the other hand, if the\nmagnetization is near the z−yplane (Fig. 2c), the\ntorque driving it to the out-of-plane orientation (Γ θ) is\nmuch stronger. Rather than characterizing the ﬂow of\nthe current-induced torque ﬁeld for any particular sys-\ntem in detail (which is highly parameter dependent), we\nsimply emphasize that an experiment is more likely to\nobserve these torques if the magnetization is initially in\nthe out-of-plane before the current is applied. Applica-\ntion of a current can stabilize this conﬁguration, so that\nsubsequent removal of the applied ﬁeld does not result in\nthe magnetization returning to the easy plane.\nIn contrast to the current-induced torque in non-\ncollinear ferromagnets, the current-induced torque in\nFIG. 4: A zoom-in view of the torques on the ferromagnet\nlayer near the ﬁxed point of the current-induced torque. (a)\nshows the result for the “no canting” system, where the ±y\nﬁxed points are equivalent. The red dot on the sphere on\nthe right represents the magnetic orientation shown in the\nleft panel. The dark blue arrows represent the orientation\nof the antiferromagnet spins. (b) shows the result for the\n3Q system. The torques indicate that the +ˆ yorientation is\na stable ﬁxed point. (c) shows that the −ˆyorientation is\nan elliptic ﬁxed point. (The three blue (dark) arrows of (a)\nhave no ˆ ycomponent, while for (b) and (c), the three similar\nblue (dark) arrows are canted, acquiring a small positive ˆ y\ncomponent.)\nmany antiferromagnet systems rely on phase coherence.7\nThis is because the eigenstates of the bulk antiferro-\nmagnet are degenerate Kramer’s doublets with opposite\nspins. A distribution of these eigenstates carries no net\nspin current. However, spin-dependent reﬂection at the\nferromagnet interface leads to a superposition of these\ndegenerate states, which results in a nonzero spin polar-\nization of the current in the antiferromagnet. The com-\nponent of this spin current perpendicular to the ferro-\nmagnet is responsible for the torque on the F, and van-\nishes as the coherence between the states is destroyed.\nThe requirement of ballistic (or quasi-ballistic) transport\nimposes more stringent requirements on the existence\nof current-induced torques in antiferromagnets than in\nferromagnets. Materials should be nearly single crystal,\nand scattering (from e.g. phonons) should be minimized.\nIn order to estimate the acceptable limits of electron-\nphonon scattering, we add an elastic scattering channel5\nFIG. 5: (a) The magnitude of current-induced torque near\nthe ﬁxed point of the “no-canting” system as a function of\nthe elastic scattering parameter D/t2. The parameters used\nin the curve ﬁt are: Γ 0= 0.0478 (µB/e), A= 670t−2. (b)\nThe same plot for the 3Q system. The ﬁt applies only to (a).\ntotheGreen’sfunctionself-energyasdescribedinSec. II.\nFig. 5 shows how increased scattering decreases current-\ninduced torque near the out-of-plane ﬁxed point of the\nno-canting system. Here the scattering parameter Dis\nnormalized by the square of the hopping matrix element\nt.\nTo place the result of Fig. 5 in context, we write Din\nterms of material properties. For simplicity, we focus on\njust one phase breaking process: elastic acoustic phonon\nscattering. Ouraimistoexplicitlyshowthatthecurrent-\ninducedtorque, asafunctionoftheappliedcurrent, hasa\nmaximum absolute value. Depending on materials prop-\nerties and temperature, other scattering processes may\nbe more important. In any event, for acoustic phonon\nscattering, Dtakes the form:26\nD=E2\nakBT\nρv2a3≡D0T (4)\nwhereEaistheelasticdeformationpotential, ρisthema-\nterialdensity, visthespeedofsound, aisthelatticespac-\ning, and Tis the temperature. The linear Tdependence\nreﬂectsthe increasedthermalpopulationofphononswith\nincreasing temperature. Other scattering process ( e.g.\nelectron-electron scattering, inelastic phonon scattering)\ndepend on Tdiﬀerently; generally D∝Tpwherepvaries\nfrom 0.5 to 3 (see Ref. 27 and references within).\nJoule heating may increase the importance of thermal\neﬀects: for current density Jﬂowing through a material\nwith resistivity Ω, thermal conductivity κ, and length L\nalong the current direction (in this case, the ˆ y-direction),\nthe spatially averaged temperature increases by a fac-\ntor on the order of J2L2Ω/κ. To stabilize the out-of-\nplane magnetic orientation requires a current density of\nαγMstF/2g28, wheregis the current-induced torque per\ncurrent, αis the damping, γis the gyromagnetic ratio,\nMsis the saturation magnetization of the ferromagnet\nlayer, and tFis the thickness of the ferromagnet layer.\nFor the no-canting system, the current-induced torque\nper current is g= 0.05µB/e. According to this esti-\nmate and typical material parameters, this leads to acritical current density on the order of 1012A/m2. Tak-\ningρ= 10−7Ω·m, κ= 50 W /(m·K),L= 50 nm\nleads to only a modest increase in temperature, less than\n10 K. The other parameters of Eq. 4 for metals are typ-\nicallyEa= 10 eV, ρ= 104kg/m3, v= 5000 m /s, a=\n0.35 nm. In total, we ﬁnd a Dparameter on the order of\n10−5eV2to10−4eV2. In lightofFig. 5, this impliesthat\nelastic phonon scattering does not immediately destroy\nthecurrent-inducedtorquefortheno-cantingsystem. On\nthe other hand, the much weaker current-induced torque\nper current of the 3Q system ( g= 4×10−4µB/e) re-\nquires a 100-fold increase in the current to stabilize the\nout-of-plane orientation, a current density which exceeds\nthe maximum these systems can accommodate.\nWe’veobservedthatthe current-inducedtorquedecays\nas 1/Dfor the no-canting system. This is not universal\nbehavior. Indeed, the current-induced torque in the 3Q\nsystem is nonmonotonic with scattering parameter D.30\nDespite its non-universality, we ﬁnd it instructive to as-\nsume such a dependence in order to derive closed form\nexpressions for the maximum current-induced torque as\na function of applied current density. Recalling that Dis\nproportional to T, we ﬁnd the absolute current-induced\ntorque Γ abs(units of torque) varies with current as:\nΓabs(J) =Γ0J\n1+AD0(T0+BJ2), (5)\nwhere Γ 0is the current-induced torque in the absence\nof scattering (recall Γ 0has units of torque per current),\nT0is the sample temperature in the absence of cur-\nrent,B=L2ρ/κdescribes the system’s susceptibility\nto current-induced heating, and D0is deﬁned in Eq. 4.29\nThe absolute current-induced torque has a maximum -\nfor current densities that are too large, the magnitude\nof the current-induced torque decreases due to increased\nscattering from Joule heating. The maximum absolute\ncurrent-induced torque is given by:\nΓmax\nabs=Γ0\n3L/radicalBigg\n2κ\nρD0A(1+D0T0A), (6)\nThe parameters Γ 0andAare entirely system speciﬁc,\nand related to the spin dependent transport properties\nof a system, and their robustness with respect to scatter-\ning. If the above maximum torque exceeds the damping\ntorqueαγMstF/2g, then the out-of-plane conﬁguration\ncan be stabilized by the current-induced torque. Intu-\nitively, it’s advantageous to use a low Msmaterial in\norder to reduce the critical current, and a thin multi-\nlayer to reduce heating. For scattering processes with\ndiﬀerent functional dependence on T, a similar line of\nreasoning applies, although the speciﬁc form of the max-\nimum absolute current-induced torque will diﬀer. It’s\nstraightforward to show that a Tpdependence of Dre-\nsults in a maximum current-induced torque expression\nsimilar to Eq. 6, where the expression inside the square\nroot is taken to the power 1 /2p.6\nIV. CONCLUSION\nThis work demonstrates the role of symmetry and\nphase coherence eﬀects in the current-induced torque\npresent between ferromagnet and antiferromagnetic lay-\ners with a compensated interface. Basic symmetry ar-\nguments identify the ﬁxed points of the current-induced\ntorque. We demonstrate that for an antiferromagnetic\nlayer with a 3Q spin structure, the current-induced\ntorque has a complex angular dependence, and the ﬁxed\npoints for the current-induced torque are generally only\nlocal attractors. This is important because experiments\ndesigned to drive the ferromagnet to these ﬁxed points\nmust initialize the ferromagnet suﬃciently nearby. We\nalsoshowviaexplicitcalculationstheprimaryroleplayed\nby phase coherence for these torques, and show an in-\nverse relationship between the magnitude of the current-\ninduced torque and the phase breaking scattering pa-rameter. In the antiferromagnetic system with planar\nspins(the no-cantedsystem), weﬁndthecurrent-induced\ntorque to be suﬃciently robust to scattering to stabilize\nthe out-of-plane magnetic orientation, while for the 3Q\nordered antiferromagnet, the current-induced torque is\ntoo weak to stabilize this orientation. We expect that\nthe robustness of this torque to scattering should be sys-\ntem speciﬁc, determined by which scattering processes\nare dominant and the system electronic structure.\nV. ACKNOWLEDGEMENTS\nA.P. acknowledges support under the Cooperative Re-\nsearch Agreement between the University of Maryland\nand the National Institute of Standards and Technol-\nogyCenter forNanoscaleScience and Technology, Award\n70NANB10H193, through the University of Maryland.\n1L. Berger, Phys. Rev. B 54, 9353 (1996).\n2J. Slonczewki, J. Magn. Magn. Mater. 159, L1 (1996).\n3D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n4A.S. N´ u˜ nez, R.A. Duine, Paul Haney, A.H. 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We expect that ﬂuctuations or deviations from ordered\nconﬁgurations will decrease the eﬀectiveness of symmetry-\nbased torques.\n23G. Malcolm Stocks, W. A. Shelton, Thomas C. Schulthess,\nBalazs jfalussy, W. H. Butler, and A. Canning, J. Appl.\nPhys.91, 7355 (2002).\n24S. Kawarazaki, Y. Sasaki, K. Yasuda, T. Mizusaki and A.\nHirai, J. Phys.: Condens. Matter 2, 5747 (1990).\n25S. J. Kennedy and T. J. Hicksm, J. Phys. F: Met. Phys.\n17, 1599 (1987).\n26M. Lundstrom, Fundamentals of Carrier Transport 2nd ed.\nCambridge University Press (2000).\n27P. Mohanty, E. M. Q. Jariwala, and R. A. Webb, Phys.\nRev. Lett. 78, 3366 (1997).\n28This expression for thecurrent density requiredto stabili ze\nthe out-of-plane orientation assumes that there is no ap-\nplied magnetic ﬁeld, or other sources magnetic anisotropy.\nIn this case, the damping torque from the hard-axis\nanisotropy for a magnetization with small tilt angle βaway\nfrom the hard-axis is γαMsβ, while the current-induced\ntorque is 2 gJβ/t F. Equating these two leads to the form\nof current density given in the text.\n29We assume that the material paramters in Eq. 4are weakly\ntemperature dependent in this analysis.\n30For the 3Q system, the current-induced torque decays\nmonotonically with scattering parameter Dfor each state\n(i.eeachk-point) individually. However the sign of the\ncurrent-induced torque varies by state, so that there is\npartial cancellation when summing over all states. The in-\ncrease of the total current-induced torque at small Dis\nthe result of less cancellation as the states’ torque, as eac h\nstate’s contribution changes slightly."    },    {        "title": "0902.1630v1.Nucleation_of_superconductivity_and_vortex_matter_in_superconductor___ferromagnet_hybrids.pdf",        "content": "arXiv:0902.1630v1  [cond-mat.supr-con]  10 Feb 2009TOPICAL REVIEW\nNucleation of superconductivity and vortex matter in\nsuperconductor – ferromagnet hybrids\nA Yu Aladyshkin1,2, A V Silhanek1, W Gillijns1, and V V Moshchalkov1,\n(1)INPAC – Institute for Nanoscale Physics and Chemistry, Nano scale Superconductivity and Magnetism and\nPulsed Fields Group, K.U.Leuven, Celestijnenlaan 200D, B– 3001 Leuven, Belgium\n(2)Institute for Physics of Microstructures, Russian Academy of Sciences, 603950, Nizhny Novgorod,\nGSP-105, Russia\nE-mail:aladyshkin@ipm.sci-nnov.ru, alejandro.silhanek@fys.k uleuven.be\nAbstract. The theoretical and experimental results concerning the th ermodynamical and low-frequency trans-\nport properties of hybrid structures, consisting of spatia lly-separated conventional low-temperature supercon-\nductor (S) and ferromagnet (F), is reviewed. Since the super conducting and ferromagnetic parts are assumed\nto be electrically insulated, no proximity eﬀect is present and thus the interaction between both subsystems is\nthrough their respective magnetic stray ﬁelds. Depending o n the temperature range and the value of the exter-\nnal ﬁeldHext, diﬀerent behavior of such S/F hybrids is anticipated. Rath er close to the superconducting phase\ntransition line, when the superconducting state is only wea kly developed, the magnetization of the ferromagnet\nis solely determined by the magnetic history of the system an d it is not inﬂuenced by the ﬁeld generated by the\nsupercurrents. In contrast to that, the nonuniform magneti c ﬁeld pattern, induced by the ferromagnet, strongly\naﬀect the nucleation of superconductivity leading to an exo tic dependence of the criticaltemperature TconHext.\nDeeper in the superconducting state the eﬀect of the screeni ng currents cannot be neglected anymore. In this\nregion of the phase diagram various aspects of the interacti on between vortices and magnetic inhomogeneities\nare discussed. In the last section we brieﬂy summarize the ph ysics of S/F hybrids when the magnetization of\nthe ferromagnet is no longer ﬁxed but can change under the inﬂ uence of the superconducting currents. As a\nconsequence, the superconductor and ferromagnet become tr uly coupled and the equilibrium conﬁguration of\nthis “soft” S/F hybrids requires rearrangements of both, su perconducting and ferromagnetic characteristics, as\ncompared with “hard” S/F structures.\nSome ﬁgures in this paper are in color only in the electronic v ersion.CONTENTS 2\nContents\n1 Introduction 5\n2 Nucleation of superconductivity in S/F hybrids (high-tem perature limit) 8\n2.1 Ginzburg-Landau description of a magnetically coupled S/F hybrid system . . . . . . . . . . . . 8\n2.2 Magnetic conﬁnement of the OP wave function in an inhomogeneou s magnetic ﬁeld: general\nconsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n2.3 Planar S/F hybrids with ferromagnetic bubble domains: theory . . . . . . . . . . . . . . . . . . . 13\n2.4 Planar S/F hybrids with ferromagnetic bubble domains: experimen ts . . . . . . . . . . . . . . . . 15\n2.5 S/F hybrids with 2D periodic magnetic ﬁeld: theory and experiment s . . . . . . . . . . . . . . . 18\n2.6 Mesoscopic S/F hybrids: theory and experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n3 Vortex matter in non-uniform magnetic ﬁelds at low tempera tures 27\n3.1 London description of a magnetically coupled S/F hybrid system . . . . . . . . . . . . . . . . . . 27\n3.2 Interaction of a point magnetic dipole with a superconductor . . . . . . . . . . . . . . . . . . . . 27\n3.3 Magnetic dots in the vicinity of a plain superconducting ﬁlm . . . . . . . . . . . . . . . . . . . . 30\n3.4 Planar S/F bilayer hybrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40\n3.5 Stray ﬁeld-induced Josephson junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42\n4 Hybrid structures: superconductor – soft magnets 45\n5 Conclusion 50CONTENTS 3\nList of main notations\nAcronyms\nGL Ginzburg-Landau,\nDWS domain-wall superconductivity,\nF ferromagnet or ferromagnetic,\nOP order parameter,\nRDS reverse-domain superconductivity,\nS superconductor or superconducting\n1D one-dimensional\n2D two-dimensional\nLatin letters\nAvector potential, corresponding to the total magnetic ﬁeld: B= rotA,\navector potential, describing to the nonuniform component of the m agnetic ﬁeld, b= rota,\nBtotal magnetic ﬁeld: B=Hext+b,\nbnonuniform component of the magnetic ﬁeld induced by ferromagne t,\ncspeed of light,\nDsthickness of the superconducting ﬁlm,\nDfthickness of the ferromagnetic ﬁlm (or single crystal),\nfabsolute value of the normalized OP wave function: f=/radicalbig\n(Reψ)2+(Imψ)2,\njextthe density of the external current: rot Hext= (4π/c)jext,\njsthe density of superconducting currents,\nGsffree (Gibbs) energy of the S/F hybrid,\nGmterm in the free energy functional accounting for the spatial var iation of the magnetization,\nHextexternal magnetic ﬁeld,\nHexexchange ﬁeld,\nHc1lower critical ﬁeld: Hc1= Φ0ln(λ/ξ)/(4πλ2),\nHc2upper critical ﬁeld: Hc2= Φ0/(2πξ2) =H(0)\nc2(1−T/Tc0),\nH(0)\nc2upper critical ﬁeld at T= 0:H(0)\nc2= Φ0/(2πξ2\n0),\nhseparation between superconducting and ferromagnetic ﬁlms,\nLangular momentum of Cooper pairs (vorticity): ψ=f(r)eiLϕ,\nℓHmagnetic length: ℓH=/radicalbig\nΦ0/(2π|Hext|),\nℓ∗\nbeﬀective magnetic length determined by a local magnetic ﬁeld b∗\nz:ℓ∗\nb=/radicalbig\nΦ0/(2π|b∗z|),\nℓψtypical width of the localized OP wave function,\nMmagnetization of the ferromagnet,\nMsmagnetization of the ferromagnet in saturation,\nm0dipolar moment of a point-like magnetic particle,\nRsradius of the superconducting disk,\nRfradius of the ferromagnetic disk-shaped dots,\nRdposition of a point-magnetic dipole: Rd={Xd,Yd,Zd},\nTc0superconducting critical temperature at B= 0,\nwperiod of the one-dimensional domain structures in ferromagnet\nGreek letters\nα,βconstants of the standard expansion of the density of the free e nergy with respect to |Ψ|2,\nǫ(0)\nvself-energy of the vortex line per unit length: ǫ(0)\nv= (Φ0/4πλ)2lnλ/ξ,\nΘ the OP phase: Θ = arctan(Im ψ/Reψ),\nλtemperature-dependent magnetic ﬁeld (London) penetration len gth:λ=λ0//radicalbig\n1−T/Tc0,CONTENTS 4\nλ0magnetic ﬁeld penetration length at T= 0,\nξtemperature-dependent superconducting coherence length: ξ=ξ0//radicalbig\n1−T/Tc0,\nξ0Ginzburg-Landau coherence length at T= 0,\nπ3.141592653...,\nρelectrical resistivity,\nΦ0magnetic ﬂux quantum: Φ 0=π¯hc/e≃2.07 Oe·cm2,\nΨ superconducting order parameter (OP) wave function,\nΨ0OP saturated value, Ψ 0=/radicalbig\n−α/β,\nψnormalized OP wave function, ψ= Ψ/Ψ0\nCoordinate systems\nThroughout this paper we use both cartesian reference system ( x,y,z) and cylindrical reference system\n(r,ϕ,z), wherez−axis is always taken perpendicular to the superconducting ﬁlm/disk.CONTENTS 5\n1. Introduction\nAccording to the classical Bardeen–Cooper–Schrieﬀer theory of superconductivity, the ground state of the\nsuperconducting condensate consists of electron pairs with oppo site spins (the so-called spin-singlet state)\nbounded via phonon interactions [1, 2]. As early as 1956, Ginzburg [3] pointed out that this fragile state of\nmattercouldbedestroyedbytheformationofahomogeneousfer romagneticorderingofspinsifitscorresponding\nmagnetic ﬁeld exceeds the thermodynamical critical ﬁeld of the sup erconductor. Later on, Matthias et al.\n[4, 5, 6] demonstrated that besides the orbital eﬀect (i.e. a pure e lectromagnetic interaction between the\nferromagnetic and superconducting subsystems), there is also a strong suppression of superconductivity arising\nfrom the exchange interaction which tends to align the spins of the e lectrons in detriment of Cooper-pair\nformation. AndersonandSuhl [7]predictedthatacompromisebet weentheseantagonisticstatescanbeachieved\nif the ferromagnetic phase is allowed to break into domains of size muc h smaller than the superconducting\ncoherence length ξin such a way that from the superconductivity point of view, the net magnetic moment\naverages to zero. Alternatively, Larkin and Ovchinnikov [8] and Fu lde and Ferrel [9], theoretically predicted\nthat superconductivity can survive in a uniform ferromagnetic sta te if the superconducting order parameter is\nspatially modulated.\nIn general terms, the eﬀective polarization of the conduction elec trons, either due to the external ﬁeld\nHext(orbital eﬀect) or the exchange ﬁeld Hex(paramagnetic eﬀect), leads to a modiﬁcation (suppression and\nmodulation) of the superconducting order parameter. Typically, in ferromagnetic metals the exchange ﬁeld is\nconsiderably higher than the internal magnetic ﬁeld and it dominates the properties of the system. However,\nin some cases, where both ﬁelds can have opposite directions, an eﬀ ective compensation of the conduction\nelectronspolarizationcanoccurandconsequentlysuperconduct ivitycanberecoveredathighﬁelds Hext≃ −Hex\n(Jaccarinoand Peter [10]). Bulaevskii et al.[11] gavean excellent overviewof both experimental and theoretic al\naspects of coexistence of superconductivity and ferromagnetis m where both orbital and exchange eﬀects are\ntaken into account.\nThe progressive development of material deposition techniques an d the advent of reﬁned lithographic\nmethods have made it possible to fabricate superconductor-ferr omagnet structures (S/F) at nanometer scales.\nUnlike the investigationsdealing with the coexistence ofsupercondu ctivity and ferromagnetismin ferromagnetic\nsuperconductors(forreviewseeFlouquetandBuzdin [12]), the fe rromagneticandsupercondictingsubsystemsin\nthe artiﬁcial heterostructures can be physically separated. As a consequence, the strong exchange interaction is\nlimited to acertaindistance aroundthe S/Finterfacewhereasthe w eakerelectromagneticinteractioncanpersist\nto longer distances into each subsystem. In recent reviews, Izyu movet al.[13], Buzdin [14] and Bergeret et al.\n[15] discussedin detail the roleofproximityeﬀects in S/F heterostr ucturesdominated byexchangeinteractions †.\nIn order to unveil the eﬀect of electromagnetic coupling it is imperat ive to suppress proximity eﬀects by\nintroducing an insulating buﬀer material between the S and F ﬁlms. In an earlier report, Lyuksyutov and\n†In particular, trilayered S/F/S structures with transpare nt S/F interfaces allow to realize Josephson junctions with an arbitrary\nphase diﬀerence between the superconducting electrodes, w hich depend on the thickness of the ferromagnetic layer (see , e.g., papers\nof Proki´ c et al.[16], Ryazanov et al.[17], Kontos et al.[18], Buzdin and Baladie [19], Oboznov et al.[20] and references therein).\nThe antipode F/S/F heterostructures attract a considerabl e attention in connection with the investigation of unusual properties of\nsuch layered hybrid structures governed by the mutual orien tation of the vectors of the magnetization in the “top” and “b ottom”\nferromagnetic layers (see, e.g., papers of Deutscher and Me unier [21], Ledvij et al.[22], Buzdin et al.[23], Tagirov [24], Baladi´ e et\nal.[25], Gu et al.[26, 27], Pe˜ na et al.[28], Moraru et al.[29], Rusanov et al.[30], Steiner and Ziemann [31], Singh et al.[32]).\nArray□of□magnetic□dots\ncovered□by□superconducting□film\nSuperconducting□film□with□array\nof□magnetic□dots□on□topPlanar□hybrid□containing□domain\nstructure□in□ferromagnic□film\nMesoscopic□and□individual\nS/F□hybridsS\nSS\nSS\nM\nFigure 1. (color online) Typical examples of considered S/F hybrid sy stems with dominant orbital interaction.CONTENTS 6\n1990 1995 2000 2005 20100510152025303540\nYearNumber of publicationsFlux−coupled S/F hybrids: Dynamics of research activity\nFigure 2. (color online) The histogram shows an increase of the number of publications dealing with the\ninvestigations of the S/F hybrids where the conventional lo w−Tcsuperconductors interact with magnetic\ntextures mainly via stray magnetic ﬁelds: blue bars corresp ond to the experimental papers, while white bars\nrefer to pure theoretical contributions.\nPokrovsky [33] addressed the physical implications of both electro magnetic coupling and exchange interaction\nin the S/F systems deep into the superconducting state.\nIn the present review we are aiming to discuss the thermodynamic an d low-frequency transport phenomena\nin the S/F hybrid structures dominated by electromagnetic interac tions. We focus only on the S/F hybrids\nconsisting of conventional low −Tcsuperconductors without weak links ‡. The S/F heterostructures with pure\nelectromagnetic coupling can be described phenomenologically using G inzburg-Landau and London formalisms\nratherthan sophisticated microscopicalmodels. Some typical exa mples ofsuch structuresfound in the literature\nare shown schematicallyin Fig. 1. As an illustration of the continuous g rowthof interest in S/F heterostructures\nwith suppressed proximity eﬀect we refer to Fig. 2, which shows the number of the publications during the last\ntwo decades.\nThe review is organized as follows. Section 2 is devoted to the nucleat ion of the superconducting order\nparameterunder inhomogeneousmagnetic ﬁelds, induced by single d omain walls and periodic domain structures\nin plain ferromagnetic ﬁlms or by magnetic dots. A similar problem for ind ividual symmetric microstructures\nwas reviewed by Chibotaru et al.[40]. Section 3 is devoted to the static and dynamic properties of S/F systems\nat low temperatures when the superconducting OP becomes fully de veloped and the screening eﬀects cannot\nbe disregarded any longer. The vortex pinning properties of the S/ F hybrids have been recently analyzed by\nV´ elezet al.[41] and the fabrication of ordered magnetic nanostructures has been earlier considered by Mart´ ın\net al.[42]. In the last section 4 we brieﬂy introduce the problem of “soft” m agnets in combination with\nsuperconducting materials, where now the superconducting curr ents and the magnetic stray ﬁeld emanating\nfrom the ferromagnetic material mutually inﬂuence each other. In the conclusion, we formulate a number of\nrelevant issues that, to our understanding, remain unsettled and deserve further investigations. The appendix\nsummarizes the experimental and theoretical research activities on the considered S/F heterostructures, where\nwe present a classiﬁcation based on the choice of materials for the e xperimental research and on the used model\nfor the theoretical treatment.\nImportantly, we would like to note already in the Introduction that t he literature and references used\nby the authors in this review by no mean can be considered as a comple te set. Due to dynamic and rather\ncomplex character of the subject and also to the limited space in this review, inevitably quite a lot of important\n‡Ferromagnetic dots are shown to induce an additional phase d iﬀerence in Josephson junctions, leading to a signiﬁcant mo diﬁcation\nof the dependence of the Josephson critical current Icon the external magnetic ﬁeld Hext(so-called Fraunhofer diﬀraction pattern,\nsee, e.g., textbook of Barone and Paterno [34]), which becom es sensitive to the magnetization of ferromagnetic particl e (Aladyshkin\net al.[35], Vdovichev et al.[36], Fraerman et al.[37], Held et al.[38], Samokhvalov [39]).CONTENTS 7\nand interesting contributions could have been missed and, therefo re, in a way, the used references reﬂect the\n“working list” of publications the authors of this review are dealing wit h.CONTENTS 8\n2. Nucleation of superconductivity in S/F hybrids (high-te mperature limit)\n2.1. Ginzburg-Landau description of a magnetically couple d S/F hybrid system\nDerivation of the Ginzburg-Landau equations\nInordertodescribehybridstructures,consistingofatype-IIs uperconductorandaferromagnet,forthecasethat\nno diﬀusion of Cooper pairs from superconductor to ferromagnet takes place, the phenomenological Ginzburg-\nLandau (GL) theory can be used. As a starting point we consider th e properties of S/F hybrids for external\nmagnetic ﬁelds Hextbelow the coercive ﬁeld of the ferromagnet which is assumed to be re latively large. In this\ncase the magnetization of the ferromagnet Mis determined by the magnetic history only and it does neither\ndepend on Hextnor on the distribution of the screening currents inside the superc onductor. Such a “hard-\nmagnet approximation” is frequently used for a theoretical treat ment and it appears to be approximately valid\nfor most part of the experimental studies presented in this sectio n. The review of the properties of hybrid S/F\nsystems consisting of superconductors and soft magnets will be p resented later on in section 4.\nFollowing Landau’s idea of phase transitions of the second kind, the e quilibrium properties of a system close\nto the phase transition line can be obtained by minimization of the free energy functional (see, e.g, textbooks\nof Abrikosov [43], Schmidt [44], Tinkham [45]):\nGsf=Gs0+Gm+/integraldisplay\nV/braceleftBigg\nα|Ψ|2+β\n2|Ψ|4+1\n4m/vextendsingle/vextendsingle/vextendsingle/vextendsingle−i¯h∇Ψ−2e\ncAΨ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+B2\n8π−B·M−B·Hext\n4π/bracerightbigg\ndV, (1)\nwhere the integration should be performed over the entire space †. HereGs0is a ﬁeld– and temperature–\nindependent part ofthe free energy, α=α0(Tc0−T),α0andβare positive temperature–independent constants,\nΨ is an eﬀective wave function of the Cooper pairs, B(r) = rotA(r) is the magnetic ﬁeld and the corresponding\nvector potential, Tc0is the critical temperature at B= 0,eandmare charge and mass of carriers (e.g.,\nelectrons), and cis the speed of light. The term Gm, which will be explicitly introduced in the last section 4,\naccounts for the self-energy of the ferromagnet which depends on the particular distribution of magnetization.\nThistermseemstobeconstantforhardferromagnetswithaﬁxed distributionofmagnetization, thereforeitdoes\nnot inﬂuence the OP pattern and the superconducting current dis tribution in hard S/F hybrids. Introducing a\ndimensionless wave function ψ= Ψ/Ψ0, normalized by the OP value Ψ 0=/radicalbig\na0(Tc0−T)/bin saturation, one\ncan rewrite Eq. (1) in the following form\nGsf=Gs0+Gm+/integraldisplay\nV/braceleftBigg\nΦ2\n0\n32π3λ2/parenleftBigg\n−1\nξ2|ψ|2+1\n2ξ2|ψ|4+/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇ψ+i2π\nΦ0Aψ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg\n+B2\n8π−B·M−B·Hext\n4π/bracerightbigg\ndV, (2)\nexpressed via the temperature-dependent coherence length ξ2= ¯h/[4ma0(Tc0−T)], the London penetration\ndepthλ2=mc2b/[8πe2a0(Tc0−T)], and the magnetic ﬂux quantum Φ 0=π¯hc/|e|.\nAlthough the Ginzburg-Landau model was proven to be consistent only at temperatures close to the\nsuperconducting critical temperature (Gorkov [46]), the applicab ility of this model seems to be much broader,\nat least from a qualitative point ofview. After minimization ofthe free energy functional Eq. (2) with respect to\nthe order parameter(OP) wavefunction ψandArespectively, one can derive the two coupled Ginzburg-Landau\nequations [43, 44, 45]:\n−ξ2/parenleftbigg\n∇+i2π\nΦ0A/parenrightbigg2\nψ−ψ+|ψ|2ψ= 0, (3)\nrot rotA=4π\ncjs+4πrotM+4π\ncjext, (4)\n†Hereafter we used the Gauss (centimeter-gram-second) syst em of units, therefore all vectors B,M, andHexthave the same\ndimensionality: [ B]=Gauss (G), [ M]=Oersted (Oe), [ Hext]=Oersted (Oe).CONTENTS 9\nwhere\njs=c\n4π|ψ|2\nλ2/parenleftbiggΦ0\n2π∇Θ−A/parenrightbigg\nrepresents the density of superconducting currents, while jext= (c/4π)rotHextis the density of the currents\ncorresponding to external sources, and Θ is the OP phase, ψ(r) =f(r)eiΘ(r).\nLinearized GL equation\nIt is quite natural to expect that at the initial stage of the format ion of superconductivity [i.e. close to the\nphase transition line Tc(Hext), which separates the normal and superconducting state in the T−Hextplane], the\ndensity of the superconducting condensate will be much smaller tha n the fully developed OP value: |ψ|2≪1.\nThis allowsone to neglect: (i) the nonlinearterm |ψ|2ψin Eq. (3) and, (ii) the correctionsto the vectorpotential\nAcaused by the screening currents in Eq. (4), since the supercurr entsjsare also proportional to |ψ|2. Thus,\nthe nucleation of superconductivity can be analyzed in the framewo rk of the linearized GL equation [43, 44, 45]:\n−/parenleftbigg\n∇+i2π\nΦ0A/parenrightbigg2\nψ=1\nξ2ψ, (5)\nin a given magnetic ﬁeld described by the vector potential distributio n\nA=1\nc/integraldisplayjext(r′)\n|r−r′|d3r′+/integraldisplayrotM(r′)\n|r−r′|d3r′, (6)\nThe solution of Eq. (5) consists of a set of eigenvalues (1 /ξ2)n, corresponding to the appearance of certain OP\npatternψn, for every value of the applied magnetic ﬁeld Hext. The critical temperature of the superconducting\ntransitionTcis determined by the lowest eigenvalue of the problem: Tc=Tc0/braceleftbig\n1−ξ2\n0(1/ξ2)min/bracerightbig\n.\nThe phase boundary for plain superconducting ﬁlms\nFirst, wewouldliketopresentthewell-knownsolutionofthelinearized GLequationEq. (5)correspondingtothe\nOPnucleation in a plain superconducting ﬁlm, inﬁnite in the lateraldirec tionand placed in atransverse uniform\nmagnetic ﬁeld Hext=Hextz0[43, 44, 45]. Taking the gauge Ay=xHext, one can see that Eq. (5) depends\nexplicitly on the x−coordinate only, therefore its general solution can be written in th e form:ψ=f(x)eiky+iqz,\nwhere the wave vectors kandqshould adjust themselves to provide the maximization of the Tcvalue. Using\nthis representation in Eq. (5), it is easy to see that the spectrum o f eigenvalues (1 /ξ2)nis similar to the energy\nspectrum of the harmonic oscillator but shifted: (1 /ξ2)n= 2π(2n+1)Hext/Φ0+q2and the (1/ξ2) minimum\n(the maximum of Tc) corresponds to n= 0 andq= 0 for any Hextvalue, (1/ξ2)min= 2πHext/Φ0. The critical\ntemperature of the superconducting transition †as a function of a uniform transverse magnetic ﬁeld is given by\nTc=Tc0/bracketleftbig\n1−2πξ2\n0Hext/Φ0/bracketrightbig\n, or\n1−Tc\nTc0=|Hext|\nH(0)\nc2, (7)\nwhereH(0)\nc2= Φ0/(2πξ2\n0) is the upper critical ﬁeld at T= 0. The inversely proportional dependence of the\nshift of the critical temperature 1 −Tc/Tc0on the square of the OP width ℓ2\nHcan be interpreted in terms\nof the quantum-size eﬀect for Cooper pairs in a uniform magnetic ﬁe ld. It should be mentioned that the\neﬀect of sample’s topology on the eigenenergy spectrum (1 /ξ2)nbecomes extremely important for mesoscopic\nsuperconducting systems, whose lateral dimensions are compara ble with the coherence length ξ. Indeed,\nthis additional conﬁnement of the OP wave function signiﬁcantly mod iﬁes the OP nucleation in mesoscopic\nsuperconductors and the corresponding phase boundaries Tc(Hext) diﬀer considerably from that typical for\nbulk samples and ﬁlms inﬁnite in the lateral directions (see Chibotaru et al.[40], Moshchalkov et al.[47, 48],\nBerger and Rubinstein [49]).\n†It is well known that superconductivity nucleates in the for m of a Gaussian-like OP wave function ψ(x,y) =e−(x−x0)2/2ℓ2\nHeiky,\nlocalized in the lateral direction at distances of the order of the so-called magnetic length ℓ2\nH= Φ0/(2π|Hext|) and uniform over\nthe ﬁlm thickness. The oscillatory factor eikydescribes the displacement of the OP maximum positioned at x0=kΦ0/(2πHext)\nwithout a change of the (1 /ξ2)minvalue. It is interesting to note that the conﬁnement of the OP wave function is determined by\nthe magnetic length ℓH, i.e. the OP width is a function of the external ﬁeld Hext. On the other hand, the temperature-dependent\ncoherence length ξis a natural length scale describing the spatial OP variatio ns. The equality ℓ2\nH=ξ2deﬁnes the same phase\nboundary in the T−Hextplane as that given by Eq. (7).CONTENTS 10\n2.2. Magnetic conﬁnement of the OP wave function in an inhomo geneous magnetic ﬁeld: general\nconsiderations\nThe main focus in this section is to describe the nucleation of the supe rconducting order parameter in a static\nnon-uniform magnetic ﬁeld Hext+b(r) based on a simple approach †. This method makes it possible to see\ndirectly a correspondence between the position of the maximum of t he localized wave function ψand the\ncritical temperature Tcin the presence of spatially-modulated magnetic ﬁeld b(r), generated by ferromagnet.\nFor simplicity, we assume that the thin superconducting ﬁlm is inﬁnite in the (x,y)−plane, i.e. perpendicular\nto the direction of the external ﬁeld Hext=Hextz0. This allows us to neglect the possible appearance of\nsuperconductivity in the sample perimeter (surface superconduc tivity [43, 44, 45]) and focus only on the eﬀect\narising from the nonuniform magnetic ﬁeld.\nImportance of out-of-plane component of the ﬁeld\nIt should be emphasized that the formation (or destruction) of su perconductivity in thin superconducting ﬁlms\nis sensitive to the spatial variation of the out-of-plane component of the total magnetic ﬁeld. Indeed, the upper\ncritical ﬁelds H⊥\nc2andH/bardbl\nc2for the out-of-plane and in-plane orientation for a uniform applied m agnetic ﬁeld can\nbe estimated as follows [43, 44, 45]:\nH⊥\nc2∼Φ0\nξ2, H/bardbl\nc2∼Φ0\nξDs, (8)\nwhereDsis the thickness of the superconducting sample. For rather thin su perconducting ﬁlms and/or close\nto the superconducting critical temperature Ds≪ξ=ξ0(1−T/Tc0)−1/2, therefore H⊥\nc2≪H/bardbl\nc2. In other\nwords, superconductivity will generally be destroyed by the out-o f-plane component of the magnetic ﬁeld\nrather than by the in-plane component, and thus, to a large exten t, the spatial distribution of the out-of-\nplane component determines the OP nucleation in thin-ﬁlm structure s. Since a uniform magnetic ﬁeld is known\nto suppress the critical temperature, one can expect that the h ighestTcvalue should correspond to the OP\nwave function localized near regions with the lowest values of the per pendicular magnetic ﬁeld |Bz(r)|provided\nthatBz=Hext+bz(r) varies slowly in space.\nIf the ﬁeld Hextexceeds the amplitude of the internal ﬁeld modulation (i.e. Hext<−maxbzand\nHext>−minbz), the total magnetic ﬁeld is non-zero in the whole sample volume, and the favorable positions\nfor the OP nucleation are at the locations of minima of |Bz(r)|=|Hext+bz(r)|. If the characteristic width ℓψ\nof the OP wavefunction, which will be deﬁned later, is much less than t he typical length scale ℓbof the magnetic\nﬁeld variation, then locally the magnetic ﬁeld can be considered as unif orm at distances of the order of ℓψand\nit approximately equals to min |Hext+bz(r)|. Then, using the standard expression for the upper critical ﬁeld\nEq. (7) and substituting the eﬀective magnetic ﬁeld instead of the a pplied ﬁeld, one can obtain the following\nestimate for the phase boundary\n1−Tc\nTc0≃min|Hext+bz(r)|\nH(0)\nc2,|Hext| ≫max|bz|. (9)\nAccording to this expression, the dependence Tc(Hext) is still linear asymptotically even in the presence of a\nnonuniform magnetic ﬁeld. However, the critical ﬁeld will be shifted u pwards (for Hext>0) and downward (for\nHext<0) by an amount close to the amplitude of the ﬁeld modulation. Genera lly speaking, such a “magnetic\nbias” can be asymmetric with respect to the Hext= 0 provided that max bz(x)∝ne}ationslash=|minbz(x)|.\nFor relatively low Hextvalues, when the absolute value of the external ﬁeld is less than the amplitude of\nthe ﬁeld modulation ( −maxbz<Hext<−minbz), thez−component of the total magnetic ﬁeld Hext+bz(x)\nbecomes zero locally somewhere inside the superconducting ﬁlm. As a result, superconductivity is expected to\nappear ﬁrst near the positions where Hext+bz(r0) = 0 [see panel (a) in Fig. 3]. It is natural to expect that\nthe details of the OP nucleation depend strongly on the exact topolo gy of the stray ﬁeld as well as the ﬁeld\ngradient near the lines of zero ﬁeld. As an example we will analyze the f ormation of superconductivity in a thin\nsuperconducting ﬁlm placed in an inhomogeneous magnetic ﬁeld modula ted along certain direction.\nOP nucleation in a magnetic ﬁeld modulated in one direction\n†We introduce the following notations: b= rotacharacterizes the non-uniform component of the magnetic ﬁe ld only, while\nB=Hext+b= rotAis the total magnetic ﬁeld distribution, the external ﬁeld Hextis assumed to be uniform.CONTENTS 11\nxBz(x) =bz(x) +HextBz(x) = (dbz/dx)x0(x−x0)\n|ψ(x)|(a)\nℓψ\nℓbx0\n−4−202402468ε0\nQmin ε0=0.904(b)\nFigure 3. (color online) (a) Schematic representation of the OP wave f unctionψ(x) localized near the point\nx0, where the z−component of the total magnetic ﬁeld Bz=Hext+bzvanishes. Provided that the OP width is\nmuch smaller than the typical length scales of the magnetic ﬁ eld (ℓψ≪ℓb), the actual ﬁeld distribution Bz(x)\ncan be approximated by a linear dependence Bz(x)≃(dbz/dx)x0(x−x0).\n(b). Energy spectrum ε0vs.Qof the model problem Eq. (11), after Aladyshkin et al.[51].\nFollowing Aladyshkin et al.[50], we estimated the dependence of Tc(Hext) for a thin superconducting ﬁlm\nin the presence of a non-uniform magnetic ﬁeld modulated along the x−direction, where ( x,y,z) is the\nCartesian reference system. Let the external ﬁeld be oriented p erpendicular to the plane of the superconducting\nﬁlm,Hext=Hextz0, while the z−component of the total magnetic ﬁeld vanishes at the point x0, i.e.\nHext+bz(x0) = 0. The vector potential, corresponding to the ﬁeld distribution Bz(x) =Hext+bz(x), can\nbe chosen in the form Ay(x) =xHext+ay(x), wherebz=day/dx. Since there is no explicit dependence on\nthey−coordinate in the linearized GL equation Eq. (5), the solution uniform over the sample thickness can be\ngenerally found as ψ(x,y) =fk(x)eikyand the absolute value of the OP satisﬁes the following equation:\n−d2fk\ndx2+/parenleftbigg2π\nΦ0xHext+2π\nΦ0ay(x)−k/parenrightbigg2\nfk=1\nξ2fk. (10)\nNow the parameter kcannot be excluded by a shift of the origin of the reference system , therefore one should\ndetermine the particular kvalue in order to minimize (1 /ξ2\n0)(1−Tc/Tc0) and thus, to maximize the Tcvalue.\nFor an unidirectional modulation of the ﬁeld, the curves of zero ﬁeld , where we expect the preferable\nOP nucleation, are straight lines parallel to the y−axis, and their positions depend on the external ﬁeld,\nx0=x0(Hext). Expanding the vector potential inside the superconducting ﬁlm in a power series around the\npointx0, one can get\nAy(x)≃x0Hext+ay(x0)+1\n2b′\nz(x0)(x−x0)2+...\nThis localapproximationis valid aslongas |b′′\nz(x0)ℓψ/b′\nz(x0)| ≪1.Introducinganew coordinate τ= (x−x0)/ℓψ\nand the following auxiliary parameters ℓψandQk\nℓψ=3/radicalBigg\nΦ0\nπ|b′z(x0)|, Qk=−3/radicalBigg\nΦ0\nπb′z(x0)/parenleftbigg2π\nΦ0x0Hext+2π\nΦ0ay(x0)−k/parenrightbigg\n,\nwe can reduce Eq. (10) to the bi-quadratic dimensionless equation\n−d2f\ndτ2+(τ2−Qk)2f=εf ,whereε=ℓ2\nψ\nξ2\n0/parenleftbigg\n1−T\nTc0/parenrightbigg\n. (11)\nThus, the problem of the calculation of the highest Tcvalue in the presence of an arbitrary slowly-varying\nmagnetic ﬁeld as a function of both the external ﬁeld and the param eters of the “internal” ﬁelds is reduced\nto the determination of the lowest eigenvalue ε0=ε0(Q) of the model equation Eq. (11). As was shown in\nRef. [51], the function ǫ0(Q) is characterized by the following asymptotical behavior: ε0(Q)≃Q2+√−Qfor\nQ≪ −1 andǫ0(Q)≃2√QforQ≫1 and it has the minimum value εmin= 0.904 [the panel (b) in Fig. 3].CONTENTS 12\nExtracting Tcfromε0= (1−Tc/Tc0)·ℓ2\nψ/ξ2\n0, the approximate expression of the phase boundary takes the\nfollowing form:\n1−Tc\nTc0≃ξ2\n0\nℓ2\nψmin\nkε0(Qk)≃ξ2\n0/parenleftbigg\nπ|b′\nz(x0)|\nΦ0/parenrightbigg2/3\n,−maxbz<Hext<−minbz. (12)\nIf there are several points x0,iwhere the external ﬁeld compensates the ﬁeld generated by the f erromagnetic\nstructure, then the right-hand part of Eq. (12) should be minimize d with respect to x0,i. The application of\nEq.(12) for the model cases bz= 4Msarctan(Df/x) (single domain wall) and bz=B0sin(2πx/w) (periodic\ndomain structure) were given in Ref. [50].\nOP nucleation in axially-symmetrical magnetic ﬁeld\nSimilar to the discussion above, one can expect that in the presence of an axially-symmetrical magnetic ﬁeld\nsuperconductivity will nucleate in the form of ring-shaped channels of radiusr=r0, whereHext+bz(r0) = 0.\nThe independence of the linearized GL equation Eq. (5) on the angula rϕ−coordinate results in a conservation\nof the angular momentum (vorticity) Lof the superconducting wave function. Thus, a nonuniform magne tic\nﬁeld makes it possible to have an appearance of giant (multiquanta) v ortex states, which are energetically\nunfavorable in plain (non-perforated) large-area superconduct ing ﬁlms, but have been observed in mesoscopic\nsuperconductors(Moshchalkov et al.[47, 48], Bergerand Rubinstein [49]) andnanostructuredﬁlms with a ntidot\nlattices (Baert et al.[52], Moshchalkov et al.[53]). Expanding the vector potential in the vicinity of r0and\nrepeating similar transformations as above, one can get the followin g approximate expression †for the phase\ntransition line (Aladyshkin et al.[51, 54])\n1−Tc\nTc0≃ξ2\n0\nℓ2\nψ/parenleftBigg\nmin\nLε0(QL)−ℓ2\nψ\n4r2\n0/parenrightBigg\n,−maxbz<Hext<−minbz,\nwhere the parameters ℓψ=3/radicalBig\nΦ0/(π|dbz/dr|r0) andQL=−/bracketleftbig\n2πr0Aϕ(r0)/Φ0−L/bracketrightbig\nℓψ/r0depend on the external\nﬁeld. Thus, this model predicts ﬁeld-induced transitions between g iant vortex states with diﬀerent vorticities.\nSince the vorticity Lis a discrete parameter, the changes of the favorable Lvalue while sweeping the external\nﬁeld leads to abrupt changes in dTc/dHext. Similar periodic oscillations of Tcwere originally observed by Little\nand Parks [55, 56] for a superconducting cylinder in a parallel magne tic ﬁeld and later for any mesoscopic\nsuperconductor in a perpendicular magnetic ﬁeld (for a review see C hibotaru et al.[40]).\nEﬀect of nonuniform magnetic ﬁeld on two-dimensional elect ron gas\nItisinterestingtonotethatthereisaformalsimilaritybetweenthe linearizedGLequation(5)andthestationary\nSchr¨ odinger equation (see, e.g., Ref. [57]) for a charged free spin less particle in a magnetic ﬁeld\n−¯h2\n2m/parenleftbigg\n∇−ie\n¯hcA/parenrightbigg2\nψ=Eψ, (13)\nwhereψis the single particle wave function, eis the charge and mis the mass of this particle. Based on this\nanalogy, one can map the results, obtained for a normal electronic gas in the ground state, on the properties of\nthe superconducting condensate near the “superconductor–n ormal metal” transition on the T−Hextdiagram.\nIn particular, the eﬀect of a unidirectional magnetic ﬁeld modulation on the energy spectrum of a two-\ndimensional electronic gas (2DEG) was analyzed by M¨ uller [58], Xue an d Xiao [59], Peeters and Vasilopoulos\n[60], Peeters and Matulis [61], Wu and Ulloa [62], Matulis et al.[63], Ibrahim and Peeters [64], Peeters et al.\n[65], Gumbs and Zhang [66], Reijniers and Peeters [67, 68], Nogaret et al.[69, 70]. Remarkably, in a magnetic\nﬁeld varying linearly along certain direction, quasi-classical electron ic trajectories propagating perpendicularly\nto the ﬁeld gradient ∇Bzare conﬁned to a narrow one-dimensional channel localized around the region where\nBz= 0 [58]. A more detailed numerical treatment revealed a lifting of the w ell-known degeneracy of the Landau\nstates on the centers of the Larmor’s orbit, inherent to electron s in a uniform magnetic ﬁeld. Periodic magnetic\nﬁeld patterns with zero and nonzero average value was shown to tr ansform the standard Landau spectrum\nE= ¯hωc(n+ 1/2) into a periodic E(ky) dependence, describing the broadening of the discrete Landau le vels\n†Note that the case of a point magnetic dipole approximation f ails forL= 0 since the maximum of the OP wave function is\nlocated atr= 0 and this theory cannot correctly describe neither the OP n ucleation nor the phase boundary Tc(Hext) at negative\nHextvalues close to the compensation ﬁeld B0=−maxbz(r).CONTENTS 13\ninto minibands as in the case of one-dimensional potential (here ωc=|e|Hext/(mc) is the cyclotron frequency).\nAn oscillatory change of the width of energy bands as His swept was shown to give rise to oscillations in the\nmagnetoresistance of 2DEG at low Hextvalues, which reﬂect the commensurability between the diameter of\ncyclotronorbitattheFermilevelandtheperiodofthemagneticﬁe ldmodulation[59]. Thementionedoscillatory\nmagnetoresistance due to commensurability eﬀects was later on co rroborated experimentally by Carmona et\nal.[71]. The inﬂuence of two-dimensional magnetic modulations on the sin gle-particle energy spectrum was\ntheoretically considered by Hofstadter [72]. A modiﬁcation of the sc attering of two-dimension electrons due to\nthe presence of either ferromagnetic dots or superconducting v ortices was shown to lead to a nontrivial change\nof the conductivity in various hybrid systems: 2DEG/superconduc tor (Geim et al.[73], Brey and Fertig [74],\nNielsen and Hedegard [75], Reijniers et al.[76]) and 2DEG/ferromagnet (Khveshchenko and Meshkov [77], Ye\net al.[78], Solimany and Kramer [79], Ibrahim et al.[80], Sim et al.[81], Dubonos et al.[82], Reijniers et\nal.[83, 84]). The eﬀect of inhomogeneous magnetic ﬁeld on the weak-loc alization corrections to the classical\nconductivity of disordered 2DEG was considered by Rammer and She lankov [85], Bending [86], Bending et al.\n[87, 88, 89], Mancoﬀ et al.[90], Shelankov [91], Wang [92].\n2.3. Planar S/F hybrids with ferromagnetic bubble domains: theory\nThe aforementioned Ginzburg-Landau formalism can be applied in the case of a non-uniform magnetic ﬁeld\ngenerated by the domain structure of a plain magnetic ﬁlm. The prob lem of the OP nucleation in planar\nS/F hybrid structures was theoretically analyzed for hard ferrom agnets characterized by an out-of-plane\nmagnetization M=Mz(x)z0by Aladyshkin et al.[50], Buzdin and Melnikov [93], Samokhin and Shirokoﬀ [94],\nAladyshkin et al.[95], Gillijns et al.[96]. It is also worth mentioning the pioneer paper of Pannetier et al.[97],\nwhere the OP nucleation in a periodic sinusoidal magnetic ﬁeld generat ed by a meander-like lithographically-\nprepared metallic coil was considered.\nThe distribution of the vector potential a(r) can be obtained, either by integration of the last term in the\nr.h.s. ofEq.(6)orbyadirectconsiderationofthemagnetostaticp roblem. Providedthatthewidthofthedomain\nwallsδis much smaller than other relevant length scales, the ﬁeld distributio ns can be calculated analyticallyfor\nsome simple conﬁgurations (Aladyshkin et al.[95], Sonin [98, 99]). Choosing the gauge Ay=xHext+ay(x,z),\none can easily see that the linearized GL equation Eq. (5) does not de pend on the y−coordinate, hence we\ncan generally ﬁnd the solution in the form ψ(x,y,z) =fk(x,z) exp(iky), where the function fk(x,z) should be\ndetermined from the following 2D equation:\n−∂2fk\n∂x2−∂2fk\n∂z2+/bracketleftbigg2π\nΦ0ay(x,z)+2π\nΦ0xHext−k/bracketrightbigg2\nfk=1\nξ2fk, (14)\nIf the superconducting ﬁlm has insulating interfaces at the top and bottom surfaces and A·n= 0 at the surface\n∂Vsof the superconductor, one should apply the standard boundary conditions: ∂fk/∂n/vextendsingle/vextendsingle/vextendsingle\n∂Vs= 0 (here nis the\nnormal vector).\nThe spatial distribution of the magnetic ﬁeld, induced by the periodic 1D domain structure, strongly\ndepends onthe relationshipbetweenthe width ofthe magneticdoma inswandthe thicknessofthe ferromagnetic\nﬁlmDfas well as on the distance hbetween superconductor and ferromagnet (Fig. 4). If the supe rconducting\nﬁlm thickness Dsis much smaller than the typical length scales of the nonuniform magn etic ﬁeld (wandDf),\nin a ﬁrst approximation one can neglect the OP variations in the z−direction and omit the term ∂2fk/∂z2in\nEq. (14). As a result, the OP nucleation in a thin superconducting ﬁlm is determined by the spatial proﬁle of\nthe perpendicular magnetic ﬁeld only, while the eﬀect of the parallel ﬁ eld can be ignored.\nCriterium for the development of domain-wall superconduct ivity\nIn this section we will discuss the possibility of localizing the OP wave fun ction near a domain wall at zero\nexternal ﬁeld. Obviously, the regime of domain-wall superconduct ivity (DWS) can be achieved only if the\ntypicalb∗\nzvalue inside the magnetic domains is rather large in order to provide an exponential decay of the\nOP, described by the eﬀective magnetic length ℓ∗\nb=/radicalbig\nΦ0/2π|b∗z|, within a half-width of the domain: ℓ∗\nb<w/2.\nFor thick ferromagnetic ﬁlms ( w/Df≪1) the magnetic ﬁeld inside domains is almost uniform and it can be\nestimated as b∗\nz≃2πMs, giving us the rough criterion of the realization of the DWS regime and the criticalCONTENTS 14\n−1 0 1−2−1012Bz(x)/(4Ms)\nx/w148w/Df=2.5\nFigure 4. (Color online) Transverse z−component of the magnetic ﬁeld, induced by one-dimensional periodic\ndistribution of magnetization with the amplitude Msand the period w, calculated at the distance h≪Df\nabove the ferromagnetic ﬁlm of a thickness Df, adapted from Aladyshkin and Moshchalkov [95].\ntemperature T(0)\ncatHext= 0:π2Msw2/Φ0>1 and (Tc0−T(0)\nc)/Tc0≃2πMs/H(0)\nc2. Of course, 2 πMs/H(0)\nc2\nshould be less than unity otherwise superconductivity will be totally s uppressed.\nBy applying an external ﬁeld on the order of the compensation ﬁeld, Hext≃2πMs, one can get local\ncompensation of the ﬁeld above the domains with opposite polarity an d a doubling of the ﬁeld above the\ndomains of the same polarity. Since superconductivity is expected t o form at regions with zero ﬁeld (which\narewwide), the maximal critical temperature can be estimated as follows : (Tc0−Tmax\nc)/Tc0≃ξ2\n0/w2(a\nconsequence of the quantum-size eﬀect for Cooper pairs in nonun iform magnetic ﬁeld). Therefore, Tmax\ncwill\n−2 −1 0 1 2−2−1.5−1−0.50\nHext/(4Ms)(Df/ξ0)2 (1−Tc/Tc0)(a)w/Df=14\nw/Df=8\nw/Df=2.5\n8π Ms D2\nf/Φ0=1.5\n0 10 20 30 40 5000.511.52\nw/Df8π Ms Df2/Φ0\nI − small periodII − intermediate\nperiodIII − large periodI − No reentrant superconductivity\nII − Reentrant superconductivity\nIII − Reentrant superconductivity\ndTc/d|Hext| > 0 at Hext=0(b)\nFigure 5. (Color online) (a) Examples of the phase transition lines Tc(Hext) for a planar S/F structure\ncontaining a periodic 1D domain structure ( Msis the saturated magnetization, Dfis the ferromagnetic ﬁlm\nthickness,wis the period of the domain structure, the thickness of the su perconducting ﬁlm Ds≪(Df,w)\nand the separation between superconducting and ferromagne tic ﬁlmsh/Df≪1), adapted from Aladyshkin and\nMoshchalkov [95]. Black dashed line correspond to the Tc(Hext) dependence in the absence of the non-uniform\nﬁeld.\n(b) Diﬀerent regimes of localized superconductivity in the presence of a 1D domain structure in the Ms−w\nplane, obtained numerically for Ds≪(Df,w) andh/Df≪1, adapted from Aladyshkin and Moshchalkov [95].\nIn regions II and III the phase boundary Tc(Hext) exhibits reentrant superconductivity. The slope dTc/d|Hext|\natHext= 0 can be positive (III), zero (II) or negative (II, near the s eparating line I–II). Region I corresponds\nto the monotonic Tc(Hext) dependence.CONTENTS 15\nexceedT(0)\nc, pointing out to the non-monotonous Tc(Hext) dependence for the same Msandwparameters,\nwhich are necessary to have the DWS regime at Hext= 0. The typical phase boundary Tc(Hext), corresponding\ntothe DWS regimeat Hext= 0andshowninpanel(a)in Fig.5, ischaracterizedbythepresence ofapronounced\nreentrant behavior and the parabolic dependence of TconHextat low ﬁelds (curve labelled w/Df= 8). This\ntype of phase boundary was predicted by Pannetier et al.[97] for a superconducing ﬁlm in a ﬁeld of parallel\nmetallic wires carrying a dc current, and by Buzdin and Melnikov [93] an d Aladyshkin et al.[50, 95] for planar\nS/F hybrids.\nLocalized superconductivity in S/F hybrids for w/Df≪1andw/Df≫1\nFor S/F hybrids with smaller periods of the ﬁeld modulation ( π2Msw2/Φ0≪1) the OP distribution cannot\nfollow the rapid ﬁeld variations and, as a consequence, at Hext≃0 there is a broad OP wave function, spreading\nover several domains and resulting in an eﬀective averaging of the n onuniform magnetic ﬁeld. In this case the\ncritical temperature was shown to decrease monotonically with incr easing|Hext|[curve labelled w/Df= 2.5 in\nFig. 5(a)], similar to the case of superconducting ﬁlms in a uniform mag netic ﬁeld. By applying an external\nﬁeld, one can shrink the width of the OP wave function and localize it wit hin one half-period above the\ndomains with opposite magnetization. The interplay between both, t he external ﬁeld and the periodic magnetic\nﬁeld, which determines the resulting OP width, leads to a sign change o f the second derivative d2Tc/dH2\next.\nAt highHextvalues the width of the OP wave function, positioned at the center o f the magnetic domain,\nis determined by the local ﬁeld Bloc≃ |Hext| −2πMs, therefore we come to a biased linear dependence\n1−Tc/Tc0≃/vextendsingle/vextendsingle|Hext| −2πMs/vextendsingle/vextendsingle/H(0)\nc2. These qualitative arguments were supported by numerical solutio ns of\nthe linearized GL equation [95].\nThe casew/Df≫1 should be treated separately since the z−component of the ﬁeld inside the magnetic\ndomains is very inhomogeneous: the absolute value |bz(x)|reaches a minimum b∗\nz= 8πMsDf/wat the domain\ncenter, while the maximal value is still equal to 2 πMsat the domain walls. It was shown that the |bz(x)|\nminima are favorable for the OP nucleation at Hext= 0. In this regime, the OP localization in the center of the\ndomains at Hext= 0 is possible as long as 2 π2MswDf/Φ0>1. At the same time the nucleation near domain\nwalls is suppressed by the mentioned ﬁeld enhancement near the dom ain walls. The sudden displacement of the\nlocalized OP wave function between the centers of the domains of po sitive and negative magnetization, when\ninverting the Hextpolarity, results in a new type of phase boundary Tc(Hext) with a singularity at Hext= 0\n[95]. It is important to note that, for w/Df≫1 andHext= 0 the critical temperature increases linearly with\nalmost the same slope dTc/d|Hext|=Tc0/H(0)\nc2as theTcvalue decreases in an applied uniform magnetic ﬁeld\n[curve labelled w/Df= 14 in Fig. 5 (a)].\n2.4. Planar S/F hybrids with ferromagnetic bubble domains: experiments\nOP nucleation in perpendicular magnetic ﬁeld\nTo the best of our knowledge, the ﬁrst observation of reentrant superconductivity †in planar S/F hybrids was\nreported by Yang et al.[101] who measured the electrical resistance of a superconduct ing Nb ﬁlm grown on top\nof a ferromagnetic BaFe 12O19substrate characterized by an out-of-plane magnetization. Lat er on, the same\nsystem Nb/BaFe 12O19was examined by Yang et al.in Ref. [102]. From the parameters typical for the domain\nstructureinBaFe 12O19singlecrystalsandNb ﬁlms ( Ms≃102Oe,w≃2µm,Df≃90µm,H(0)\nc2≃30kOe), the\nfollowing estimates can be obtained: w/Df≃0.02,π2Msw2/Φ0>102and 2πMs/H(0)\nc2≃0.02. Therefore, such\na ferromagnet is suitable for the realization of the DWS regime at Hext= 0. The appearance of these localized\nsuperconducting paths guided by domain walls was shown to result in a broadening of the superconducting\nresistive transition at low magnetic ﬁelds. As the ﬁeld Hextis ramped up, the superconducting areas shift away\nfromthedomainwallstowardsthewiderregionsabovethedomainswit hanoppositepolarity(so-calledreversed-\ndomain superconductivity, RDS) where the absolute value of the to tal magnetic ﬁeld is minimal because of the\n†The experimental observation of the inﬂuence of a periodic m agnetic ﬁeld, generated by an array of parallel wires with cu rrent\nIﬂowing alternatively in opposite directions, on the proper ties of an Al superconducting bridge was reported by Panneti eret al.\n[97]. Since the max |bz| ∝I, reentrant superconductivity can be realized for rather hi ghIvalues, as it was shown experimentally. It\nshould be noted that already in sixties Artley et al.[100] experimentally studied the eﬀect of the domain walls i n a thin permalloy\nﬁlm on the superconducting transition of a thin indium ﬁlm.CONTENTS 16\n−3−2−1 0123−3−2−10123\nHext/Hc(a)Hret\nMrem(Hret)T=300K\nT=5KM (105 A/m)\n−2 −1 0−3−2−10123(c)\ntimeHext\nHs=104 Oe\nHret\nHret/HcHext=0, T=5 K Mrem (105 A/m)(b)\nFigure 6. (color online) Preparation of the magnetic state in a ferrom agnetic Co/Pt ﬁlm with a desirable\nremanent magnetization Mrem.\n(a) Magnetization loops M(Hext) at 300 K (triangles) and 5 K (squares), the magnetic ﬁeld axi s is normalized\nby the corresponding coercive ﬁelds Hc, adapted from Gillijns et al.[96];\n(b) Remanent magnetization Mrem, measured at 5 K and Hext= 0 after saturation in positive ﬁelds (up to 104\nOe) and subsequent application of a returning ﬁeld Hret[this procedure is shown schematically in panel (c)],\nadapted from Gillijns et al.[96].\ncompensation eﬀect. As a consequence, the superconducting cr itical temperature Tcincreased with increasing\n|Hext|up to 5 kOe. Once the external ﬁeld exceeds the saturation ﬁeld Hsof the ferromagnet ( Hs≃5.5 kOe\nat low temperatures), the domain structure in the ferromagnet d isappears and the phase boundary abruptly\nreturns back to the standard linear dependence (1 −Tc/Tco)≃ |Hext|/H(0)\nc2. Since the width and the shape\nof the magnetic domains continuously depend on the external ﬁeld, the theory developed in section 2.2 is not\ndirectly applicable for the description of the experiment, although it qualitatively explains the main features of\nthe OP nucleation in such S/F systems.\nSubstituting Nb by a superconductor with a smaller H(0)\nc2value (e.g., Pb with H(0)\nc2≃1.8 kOe) allows one\nto study the eﬀect of the superconducting coherence length ξ0=/radicalBig\nΦ0/2πH(0)\nc2on the localization of the OP. It\nwas shown by Yang et al.[103] that the increase of the Ms/H(0)\nc2ratio suppresses the critical temperature of the\nformationofdomain-wallsuperconductivityatzeroexternalﬁeld ‡, thereforesuperconductivityinPb/BaFe 12O19\nhybrids appeared only near the compensation ﬁelds above the reve rsed domains.\nDirect visualization of localized superconductivity in Nb/PbFe 12O19structures was performed by Fritzsche\net al.[104]. The basic idea of this technique is the following: if the sample temp erature becomes close to a\nlocal critical temperature at a certain position ( x,y), then a laser pulse, focused on that point, can induce the\nlocal destruction of superconductivity due to heating. The obser ved increase of the global resistance Rof the\nsuperconducting bridge can be associated with the derivative dR(x,y)/dT. By varying the temperature and\nscanning the laser beam over the Nb bridge under investigation, it is p ossible to image the areas with diﬀerent\ncritical temperatures. For example, it allows one to attribute the f ormation of well-deﬁned regions with rather\nhigh local critical temperatures above magnetic domains at the com pensation ﬁeld with the appearance of\nreversed-domain superconductivity.\nThe eﬀect of the amplitude of the ﬁeld modulation on the OP nucleation was considered by Gillijns et al.\n[105, 106] on thin-ﬁlm trilayered hybrid F/S/F structures. In cont rast to the BaFe 12O19single crystal discussed\nabove, the multilayered Co/Pd ﬁlms are characterized by a high resid ual out-of-plane magnetization, Ms∼102\nOe, almost independent on the external ﬁeld at |Hext|< Hcoer, whereHcoer≃103Oe is the typical coercive\nﬁeld at low temperatures. The use of two ferromagnetic ﬁlms with slig htly diﬀerent coercive ﬁelds allowed\nthem to prepare diﬀerent magnetic conﬁgurations and thus to con trol the amplitude of the nonuniform ﬁeld\n‡In section 2.3 we argued that the critical temperature, T(0)\nc, at zero external ﬁeld is proportional to 1 −2πMs/H(0)\nc2.CONTENTS 17\n1 mm 1 mm 1 mm 1 mm(a) (b) (c) (d)\nNegatively□magnetized□domainsPositively□magnetized□domains\n−400−300−200−100 01002003004000.50.70.91.11.31.5\nHext (Oe)Tc (K)Hret=0\n−3.93 kOe\n−4.15 kOe\n−4.42 kOe\n−4.55 kOe−4.61 kOe−5.00 kOe(e)\nFigure 7. (color online) (a)–(d) MFM images obtained at T= 300 K for Hretvalues equal to -1.75 kOe (a),\n-2.00 kOe (b), -2.50 kOe (c), -3.00 kOe (d), the coercive ﬁeld H300K\nc= 1.91 kOe, adapted from Gillijns et al.\n[96]. The dark (bright) color represents domains with posit ive (negative) magnetization.\n(e) A set of experimental phase boundaries Tc(Hext) obtained for the same bilayered S/F sample (a\nsuperconducting Al ﬁlm on top of a Co/Pt multilayer) in vario us magnetic states measured after the procedure\nof an incomplete demagnetization: Hext= 0⇒Hext= 10 kOe ⇒Hext=Hret⇒Hext= 0 for various\nreturning ﬁelds Hretindicated on the diagram, the coercive ﬁeld H5K\nc= 3.97 kOe, adapted from Gillijns et al.\n[96].\ninside the superconductor due to the superposition of the partial stray ﬁelds via an appropriate demagnetizing\nprocedure. Theeﬀectivedoublingoftheamplitudeoftheinternalﬁ eldforaconﬁgurationwithtwodemagnetized\nferromagnetic ﬁlms (containing bubble domains) leads to a broadenin g of the temperature interval, where the\nTc(Hext) line demonstrates the non-monotonous behavior. In other word s, the critical temperature of domain-\nwall superconductivity at Hext= 0 expectedly decreases as the eﬀective magnetization increases . In addition,\nthe enhancement of the internal ﬁeld results in a shift of the Tcmaxima to higher Hextvalues, what corresponds\nto reversed-domain superconductivity.\nThespatialextensionwheretheﬁeldcompensationtakesplace,isa crucialparameterdeﬁningthenucleation\nof superconductivity: an OP trapped in a broader region results in a higherTcvalue and vice versa (Gillijns et\nal.[96], Aladyshkin et al.[107]). The magnetic state of the ferromagnet can be reversibly ch anged after the\nfollowing procedure of an incomplete demagnetization : Hext= 0⇒Hext=Hs⇒Hext=Hret⇒Hext= 0,\nwhereHsis the saturation ﬁeld (see Fig. 6). As a result, one can obtain any de sirable remanent magnetization\n−Ms<Mrem<Ms(as well as any average width of the magnetic domains) by varying th eHretvalue. At\nHret<0 the formation of the negative domains decreases the average wid th of the positive domains and\nit causes a drastic lowering of the height of the Tcpeak, positioned at negative ﬁelds and attributed to the\nappearance of superconductivity above large positive domains [cur vesHret=−3.93 kOe and Hret=−4.15 kOe\nin Fig. 7]. This observation is a direct consequence of the increase of the ground energy of the “particle-in-a-\nbox” with decreasing width of the box. When Mremis close to zero, thus indicating the presence of an equal\ndistribution of positive and negativedomains, a nearly symmetric pha se boundary with two maxima of the same\namplitude is recovered [curve Hret=−4.55 kOe in Fig. 7]. For higher Hretvalues, the ﬁrst peak, located at\nnegative ﬁelds, disappears, whereas the peak at positive ﬁelds shif ts up in temperature and is displaced to lower\nmagnetic ﬁeld values [curves Hret=−4.61 kOe and Hret=−5.00 kOe in Fig. 7]. This second peak eventually\nevolves into a linear phase boundary when the ferromagnetic ﬁlm is fu lly magnetized in the negative direction.\nOP nucleation in parallel magnetic ﬁeld\nWe would like to mention a few related papers devoted to the nucleatio n of superconductivity in various planar\nS/F structures, where superconducting and ferromagnetic laye rs were not electrical insulated and thus an eﬀect\nof exchange interaction can not be excluded.\nAn appearance of domain walls in permalloy ﬁlm leading to dips in a ﬁeld depe ndence of the resistivityCONTENTS 18\nof a superconducting Nb ﬁlm was observed by Rusanov et al.[108]. The position of these resistivity minima\nwere found to be dependent on the sweep direction of the in-plane o riented external ﬁeld. An opposite eﬀect\n(maxima of resistivity at temperatures below Tc0) was observed by Bell et al.[109] for thin-ﬁlm amorphous S/F\nstructures consisting of superconducting MoGe ﬁlms and ferroma gnetic GdNi layers. It was interpreted as ﬂow\nof weakly pinned vortices induced by the stray ﬁeld of magnetic doma ins in the ferromagnetic layers. Zhu et al.\n[110] demonstrated that the domain structure in multilayered CoPt ﬁlms can be modiﬁed by applying in-plane\nexternal ﬁeld during the deposition process. This deposition ﬁeld do es not change the overall perpendicular\nmagnetic anisotropy of the Co/Pt ﬁlms but it induces a weak an in-plan e magnetic anisotropy and eventually\nalters the domain patterns. Indeed, after demagnetizing with an in -plane ac magnetic ﬁeld oriented along the\ndeposition ﬁeld direction, one can prepare the domain structure in a form of largely parallel stripe domains. In\ncontrast to that, the same multilayered structure fabricated at zero ﬁeld or demagnetized with out-of-plane ac\nﬁeld exhibits a nearly random labyrinth-type domain pattern. Sweep ing the external ﬁeld H/bardbl\next, one can control\nthe arrangement of domain walls, and drive the S/F hybrid from norm al to superconducting state for the same\ntemperature and magnetic ﬁeld.\n2.5. S/F hybrids with 2D periodic magnetic ﬁeld: theory and e xperiments\nIn this section we continue the discussion concerning the propertie s of large-area S/F hybrids in which the\nnonuniform magnetic ﬁeld is created by regular arrays of ferromag netic dots. The fabrication of such magnetic\nstructures makes it possible to achieve full control of the spatial characteristics of the nonuniform magnetic\nﬁeld (both the topology and the period), which can be eventually des igned practically at will. One can expect\nthat due to the ﬁeld-compensation eﬀect the inhomogeneous magn etic ﬁeld modulated in both directions will\naﬀect the OP nucleation in the same way as for the planar S/F struct ures. However, the 2D periodicity of the\nmagneticﬁeldnaturallyleadstotheappearanceofwell-deﬁnedcomm ensurabilityeﬀectsforsuchhybridsystems.\nIn other words, a resonantchangein the thermodynamicaland tr ansportpropertiesofthe superconducting ﬁlms\nappears as a function of the external magnetic ﬁeld Hext, similar to that observed for superconductors with\nperiodic spatial modulation of their properties (e.g., perforated su perconducting ﬁlms [52, 53]). These matching\nphenomena takeplace at particular Hext,nvalues that can be used asindicatorsallowingus to ﬁnd a relationship\nbetween the most probable microscopic arrangement of the vortic es in the periodic potential and the global\ncharacteristics of the considered S/F hybrids measured in the exp eriments.\nCommensurate solutions of the GL equations.\nThe periodic solutions of the GL equations in the presence of a nonun iform 2D periodic magnetic ﬁeld can\nbe constructed by considering one or more magnetic unit cells (of to tal areaS) and applying the following\nboundary conditions:\nA(r+bk) =A(r)+∇ηk(r), (15)\nΨ(r+bk) = Ψ(r) exp(2πiηk(r)/Φ0),\nwherebk,k={x,y}are the lattice vectors, and ηkis the gauge potential (Doria et al.[111]). The gauge\ntransformation Eq. (15) is possible provided that the ﬂux induced b y the external magnetic ﬁeld HextSthrough\nthe chosen area Sis equal to an integer number of ﬂux quanta, nΦ0, which gives us the matching ﬁelds\nHext\nn=±nΦ0/S.\nThe formation of diﬀerent vortex patterns in S/F hybrids, contain ing square arrays of the magnetic dots\nwith perpendicular magnetization, at Hext= 0 were studied by Priour and Fertig [112, 113] and Miloˇ sevi´ c\nand Peeters [114, 115]. Since the total ﬂux through an underlying s uperconducting ﬁlm, inﬁnite in the lateral\ndirection, equals to zero, vortices cannot nucleate without corre spondingantivortices, keeping the total vorticity\nzero. It was shown that the number of the vortex-antivortex pa irs depends on the dipolar moment of the\nmagnetic dot. The equilibrium vortex phase, corresponding to the m inimum of the free energy functional, can\nexhibit a lower symmetry than the symmetry of the nonuniform magn etic ﬁeld. In order to get such vortex\nstates of reduced symmetry, the GL equations with periodic bound ary conditions should be considered in a\nlarge supercell (2 ×2, 4×4 etc.). In most cases vortices are conﬁned to the dot regions, wh ile the antivortices,\ndepending on the dot’s magnetization, can form a rich variety of reg ular lattice states with broken orientational\nand mirror symmetries (Fig. 8). The creation of vortex-antivorte x pairs in case of ferromagnetic dots withCONTENTS 19\n(a)□□□□N =2,□N =2v av (b)□□□□N =2.5,□N =2.5v av (c)□□□□N =3,□N =3v av\nFigure 8. (coloronline)Contour-plots ofthe Cooper pairdensity |ψ|2ofstable vortex phases insuperconducting\nthin ﬁlms in the presence of a magnetic dot array with dipolar moments oriented perpendicular to the ﬁlm plane,\nadapted from Priour and Fertig [112]. The images in (a), (b), and (c) correspond to magnetic dot strengths of\n3.61, 4.33, and 4.95 fundamental ﬂux units, respectively (t he dot’s dipole moment is speciﬁed by the positive\nﬂux passing through each unit cell). The Cooper pair density is highest in red and lowest in blue regions.\nAntivortices appear as small blue spots between the magneti c dots depicted by black squares. The number of\nvorticesNvand antivortices Navper unit cell are indicated on the plots. The calculations we re performed for\na large 4 ×4 supercell with periodic boundary conditions and for the fo llowing parameters: κ=λ/ξ= 4, the\nperiod of the magnetic dot array is 6.25 ξ, the lateral size of the magnetic dot is 2 ξ×2ξ.\nin-plane magnetization was considered by Miloˇ sevi´ c and Peeters [ 116]. As expected, vortex-antivortex pairs\nappear under the poles of each magnet according to their speciﬁc in homogeneous magnetic ﬁeld keeping the\ntotal ﬂux through the superconducting ﬁlm equals to zero.\nThe inﬂuence of the external ﬁeld on the formation of symmetrical and asymmetrical commensurate vortex\nconﬁgurations in thin superconducting ﬁlms in the presence of a squ are array of out-of-plane magnetized dots\nwas investigated by Miloˇ sevi´ c and Peeters [114, 115, 117, 118]. T he simulations were carried out only for\nsome discrete values Hext,nof the external ﬁeld, corresponding to the magnetic ﬂux quantiza tion per magnetic\nsupercell of area S. Vortices were shown to be attracted by the magnetic dots in the p arallel case (at Hext,n>0\nforMz>0) and repelled in the antiparallel case (at Hext,n<0 forMz>0). In the parallel case the vortex\nconﬁgurations for the integer matching ﬁelds are similar to that for the vortex pinning by regular arrays of\nantidots.\nThe temperature dependence of the magnetization threshold for the creation of vortex-antivortexpairs was\nconsidered in Ref. [118]. It was noted that the system will not neces sarily relax to the ground state, if there are\nmetastable states, corresponding to local minima in the free energ y. As long as the given vortex state is still\nstable and it is separated from other stable vortex conﬁgurations by a ﬁnite energy barrier (analogous to the\nBean-Livingston barrierfor the vortex entry into superconduct ing samples), then the vorticity remains the same\neven when changing temperature. However, an increase in temper ature resulting in a decrease of the height\nof the energy barriers and strengthening of the thermal ﬂuctua tions, can eventually cause a phase transition\nbetween the vortex states with diﬀerent number of vortices. The modiﬁcation of the ground state (at Hext= 0)\nby the creation of extra vortex-antivortex pairs, when changing temperature and/or increasing the Msvalue\nmanifests itself as cusps in the phase boundary separating superc onductor from the normal metal phase in the\nMs−Tdiagram, similar to the Little-Parks oscillations in the Tc(Hext) dependence [55, 56].\nOscillatory nature of the phase transition line (in-plane d ot’s magnetization)\nThe inﬂuence of two-dimensional square arrays of micron-sized, in -plane magnetized particles (SmCo, GdCo,\nFeNi) on the electrical resistance of a superconducting Nb ﬁlm, usu ally interpreted as ﬁeld-induced variations of\nthe critical temperature, were experimentally studied by Pannetie ret al.[97], Otani et al.[119], Geoﬀroy et al.\n[120]. The oscillatory dependence of the resistivity ρon the perpendicularly oriented external ﬁeld Hextwith a\nperiod ∆Hextclose to Φ 0/a2was observed at T <Tc0only when the dots had been magnetized before cooling\n(hereais the period of the magnetic dot array). The appearance of minima in theρ(Hext) dependence, which\nare reminiscent of the Little-Parks oscillations for multiply-connect ed superconductors and superconducting\nnetworks (see the monograph of Berger and Rubinstein [49] and re ferences therein), were attributed to theCONTENTS 20\n−20 −10 0 10 207.0457.0477.0497.051\nHext (Oe)Tc (K)Array of Gd−Co dots on top of Nb film\nFigure 9. (color online) Phase transition line of a superconducting N b ﬁlm with a ferromagnetic Gd 33Co67\nparticle array (the period 4 µm), adapted from Otani et al.[119].\nvariation of the critical temperature, Tc=Tc(Hext) due to the ﬂuxoid quantization in each magnetic unit cell.\nTunable ﬁeld-induced superconductivity for arrays of out- of-plane magnetized dots\nThe fabrication of periodic arrays of multilayered Co/Pd and Co/Pt m agnetic dots with out-of-plane magnetic\nmoments allows one to observe both the matching phenomena and th e modiﬁed OP nucleation due to the\nﬁeld-compensation eﬀect. The stray ﬁeld induced by positively magn etized dots has a positive z−component\nof the magnetic ﬁeld under the dots and a negative one in the area in b etween the dots. Thus, the magnetic\nﬁeld in the region in between the dots will be eﬀectively compensated a tHext>0, stimulating the appearance\nof superconductivity at nonzero Hextvalues (magnetic ﬁeld-induced superconductivity). Lange et al.[121]\ndemonstrated that the Tcmaximum is located at Hext= 0 for the demagnetized magnetic dots and it is shifted\ntowards a certain Hext,nwhich depends on the dot’s magnetization [see the panel (a) in Fig. 10 ]. This quantized\nshift of the Tcwas attributed to the ﬁeld compensation in the interdot areas acco mpanied by an annihilation\nof the interstitial antivortices under the action of the external ﬁ eld, since (i) the number of antivortices is\ndetermined by the magnetic moment of the dots; (ii) the interstitial antivortices can be fully cancelled only at\nthe matching ﬁelds Hext,n=nΦ0/a2(Miloˇ sevi´ c and Peeters [122]). Thus, the appearance of periodic kinks in\ntheTc(Hext) phase boundary with a period coinciding with the ﬁrst matching ﬁeld H1can be associated with\nthe ﬂuxoid quantization, conﬁrming that superconductivity indeed nucleates in multiply connected regions of\nthe ﬁlm.\nThe results of further investigations on similar hybrid systems (an a rray of micron-sized Co/Pt dots on top\nof an Al ﬁlm) was presented by Gillijns et al.[123, 124, 125]. As a consequence of the rather large diameter\nof the dots, the demagnetized dot’s state microscopically corresp onds to a magnetic multidomain state with\nvery weak stray ﬁeld. As it was demonstrated in Ref. [123], the rem anent magnetization of the dots, which\nwere initially demagnetized, depends monotonously on the maximal ap plied ﬁeld (excursion ﬁeld) Hret. Thus\nthe total remanent magnetic moment of the dot becomes variable a nd tunable, hereby changing the inﬂuence\nof the ferromagnet on the superconductor in a continuous way. I t was found that a gradual increase of the\ndot’s magnetization from zero to a certain saturated value results (i) in a quantized displacement of the main\nTcmaximum towards nH1(nis integer) due to the quantized character of the ﬁeld-induced sup erconductivity\n[the panel (b) in Fig. 10]; (ii) in an enhancement of the local Tcmaxima, attributed to the formation of\na commensurate vortex phase at discrete matching ﬁelds, which be comes more pronounced as compared to\nRef. [121].\nThe eﬀect of changing the average remanent magnetization Mremand the radius Rfof the magnetic dots\non the superconducting properties of an Al ﬁlm deposited on top of a periodic array of such dots was studied\nby Gillijns et al.[124, 125]. Indeed, once the dot’s magnetization becomes saturat ed, the only way to furtherCONTENTS 21\n−40−30−20−100102030407.187.27.227.24−3H1−2H1−H1H12H13H1\nHext (Oe)Array of Co/Pd dots on top of Pb filmTc (K)(a)\n−20−10 0102030401.261.281.31.321.34\n−3H1−2H1−H1H12H13H14H15H16H17H1\nHext (Oe)Tc (K)Array of Co/Pt dots covered by Al film\n(b)\nM0=0 M0=180 Oe\nM0=260 Oe\nM0=320 Oe\nM0=360 Oe\nFigure 10. (color online) Field-induced superconductivity in a super conducting ﬁlm with an array of magnetic\ndots:\n(a) TheTc(Hext) dependences obtained for a Pb ﬁlm after demagnetization of the ferromagnetic dot array (the\nbrown central curve marked by circles), saturation of the do ts in a large positive Hext(the red right curve\nmarked by diamonds), and saturation in a large negative Hext(the blue left curve marked by squares), adapted\nfrom Lange et al.[121]. The period of the lattice was 1.5 µm. The arrows depict the corresponding matching\nﬁelds.\n(b) Superconducting transition Tc(H) of an Al ﬁlm for diﬀerent magnetic states of the square array of the\nferromagnetic dots of the period 2 µm, adapted from Gillijns et al.[123]. By increasing the magnetization a\nclear shift of Tc(Hext) and a decrease of Tmax\ncis observed.\nincrease the magnetic ﬂux from each magnet can be achieved by incr easing the lateral size of the dots. It was\nexperimentally found that the larger the Rfvalue, the smaller the necessary ∆ Mneeded to shift the main\nTc(Hext) maximum by one matching ﬁeld.\nBoth the ﬁeld compensation and matching eﬀects in plain Al ﬁlms with a s quare array of ferromagnetic\nCo/Pt disks were investigated by Gillijns et al.[96] and Aladyshkin et al.[107]. Due to the presence of the\nout-of-plane magnetized dots, there are threediﬀerent areas, where the OP can potentially nucleate: above the\npositive or negative domains, inside the magnetic dot, and in between the dots. In the demagnetized state\nthe interdot ﬁeld is close to zero, therefore superconductivity st arts to nucleate at this position at relatively\nlow magnetic ﬁelds, resulting in an almost linear phase boundary cente red atHext= 0. By magnetizing the\ndots positively (i) the amplitude of the ﬁeld between the dots increas es negatively and (ii) the typical width\nof the positive domains becomes larger than that for negative doma ins. Therefore the peak, associated with\nthe OP localization between the dots, shifts towards positive ﬁelds. In addition, a second local Tcmaximum,\ncorresponding to the appearance of superconductivity above th e broader positive domains, appears, while\nthe OP nucleation above narrower negative domains is still suppress ed. For negatively magnetized dots the\nreversed eﬀect occurs. It is important to note, that the amplitud e of the main Tcpeak, corresponding to the\nOP nucleation between the dots, remains almost constant, since th e mentioned area of the OP localization is\nalmost independent of the dot’s magnetic state (Fig. 11).\nIndividual ferromagnetic dots above/inside superconduct ing ﬁlms\nMarmorkos et al.[126]studiedapossibilitytocreategiantvorticesbyaferromagnet icdiskwithout-of-plane\nmagnetization embedded in thin superconducting ﬁlm within full nonline ar, self-consistent Ginzburg-Landau\nequations. Later using the same model Miloˇ sevi´ c and Peeters [12 7, 128, 129] considered the formation of\nvortex-antivortex structures in plain superconducting ﬁlms, inﬁn ite in the lateral direction, in the ﬁeld of an\nisolated ferromagnetic disk with out-of-plane magnetization within t he full nonlinear Ginzburg-Landau theory.\nAntivortices were shown to be stabilized in shells around a central co re of vortices (or a giant vortex) with\nmagnetization-controlled ”magic numbers” (Fig. 12). The transitio n between the diﬀerent vortex phases while\nvarying the parameters of the ferromagnetic dot (namely, the ra dius and the magnetization) occurs through theCONTENTS 22\n−100 −50 0 50 1001.11.151.21.251.31.35\nHext (Oe)Tc (K)Array of CoPt dots covered by Al film\nDemagnetized\nMagnetized\nHret=0−2.25 kOeHret=−3.50 kOe\nFigure 11. (color online) The phase boundaries Tc(Hext) for an S/F hybrid, consisting of an Al ﬁlm and\nan array of magnetic dots, in the demagnetized state, in the c ompletely magnetized state in positive direction\nas well as in several intermediate magnetic states, adapted from Gillijns et al.[96] and Aladyshkin et al.\n[107]. The period ∆ Hextof theTcoscillations, which are distinctly seen in the curve corres ponding to the\nmagnetized states, is equal to 5.1 Oe and it exactly coincide s with the matching ﬁeld, i.e., ∆ Hext= Φ0/S,\nwhere Φ 0= 2·10−7Oe·cm2is the ﬂux quantum and S= 4µm2is the area of the unit cell. Note that ﬁeld and\ntemperature intervals shown here are much broader than thos e presented in Fig. 10 (b).\ncreation of a vortex-antivortex pair under the magnetic disk edge .\n2.6. Mesoscopic S/F hybrids: theory and experiments\nIn all the description of nucleation of superconductivity so far, we have ignored the eﬀects of the sample’s\nborders. It is well known that the OP patterns in mesoscopic super conducting samples with lateral dimensions\ncomparabletothe superconductingcoherencelength andmagnet icpenetrationdepth, is substantiallyinﬂuenced\nby the geometry of the superconductor (see review of Chibotaru et al.[40] and references therein). As a\nresult, the presence of the sample’s boundaries allows the appeara nce of exotic states (giant vortex states,\nvortex clusters, shell conﬁgurations etc.), otherwise forbidden for bulk superconductors and non-patterned plain\n(a)   Nv=4, Nav=4 (b)   Nv=5, Nav=5 (c)   Nv=6, Nav=6 (d)   Nv=7, Nav=7\nFigure 12. (color online) Contour-plots of the Cooper pair density |ψ|2, illustrating the appearance of vortex-\nantivortex shell structures in large-area superconductin g ﬁlm in the ﬁeld of perpendicularly magnetized disk\nfor diﬀerent magnetic moments mof the ferromagnetic particle: m/m0= 25 (a), 29 (b), 35 (c) and 38 (d),\nwherem0=Hc2ξ3, by courtesy of M.V. Miloˇ sevi´ c. The highest |ψ|values are shown in lighter shades and\nthe lowest densities in darker shades. It is important to not e that only the central part of the superconducting\nﬁlm (45ξ×45ξ) is shown here, black solid line schematically depicts the e dge of the magnetic dot. Red circles\ncorrespond to the vortex cores, while blue squares mark the p osition of antivortices. The number of vortices Nv\nand antivortices Navare indicated on the plots. The simulations were performed f or the following parameters:\nκ=λ/ξ= 1.2, the lateral size of superconducting sample is 256 ξ×256ξ, the radius of ferromagnetic disk is\n4.53ξ, and the thicknesses of superconductor and ferromagnet are equal to 0.1 ξ.CONTENTS 23\nsuperconducting ﬁlms. Since, as we have pointed out above, a nonu niform magnetic ﬁeld is an alternative way\nto conﬁne the superconducting OP in a certain HextandTrange, mesoscopic S/F hybrids seem to be of interest\nfor studying the interplay between diﬀerent mechanisms of conﬁne ment of the superconducting condensate.\nIt is important to note, that the screening eﬀects can still be omitt ed provided that the lateral size of the\nthin superconducting sample is smaller than the eﬀective penetratio n depthλ2D=λ2/Ds. In this case the\nself-interaction of the superconducting condensate can be take n into account solving the nonlinear decoupled\nGL equation\n−ξ2/bracketleftbigg\n∇+2πi\nΦ0A(r)/bracketrightbigg2\nψ−ψ+|ψ|2ψ= 0, (16)\nwhere the vector potential distribution is given by the external so urces and the ferromagnet only [see Eq. (6)].\nInterplay between diﬀerent regimes of the OP nucleation\nAs we anticipated above, a very interesting phenomenon in mesosco pic S/F hybrids is the interplay between\ncompeting regimes of the OP nucleation, which can be clearly seen in th e case of a small-sized magnetic\ndot of radius Rfplaced above a mesoscopic superconducting disk, Rf≪Rs. Indeed, the |ψ|maximum can be\ngenerallylocated either at the centralpart of the superconduct ing disk, close to the magnetic dot (magnetic-dot-\nassisted superconductivity) or at the outer perimeter of the sup erconducting disk (surface superconductivity).\nFor a positively magnetized dot the regime of the magnetic-dot-ass isted superconductivity, associated with\nthe appearance of superconductivity in the region with compensat ed magnetic ﬁeld, can be realized only for\nHext<0. At the compensation ﬁeld ( |Hext| ≃B0) and provided that/radicalbig\nΦ0/(2πB0)≪Rsthe enhancement of\nthez−component of the ﬁeld near the disk edge acts as a magnetic barrier for the superconducting condensate\nand it prevents the edge nucleation of superconductivity even in sm all-sized superconductors ( B0being the\nmaximum of the self-ﬁeld of the magnetized dot). The OP nucleation n ear the magnetic dot becomes possible,\nif the critical temperature T(0)\ncof the formation of localized superconductivity with the OP maximum a t the\nsuperconducting disk center\n1−T(0)\nc\nTc0≃2πξ2\n0\nΦ0/vextendsingle/vextendsingleHext+B0/vextendsingle/vextendsingle.\nexceeds the the critical temperature for the edge nucleation reg ime\n1−Tc3\nTc0≃0.592πξ2\n0\nΦ0/vextendsingle/vextendsingleHext/vextendsingle/vextendsingle,\ncorresponding to the critical ﬁeld of surface superconductivity Hc3= 1.69Hc2[43, 44, 45]. Due to the diﬀerent\nslopesdT(0)\nc/dHextanddTc3/dHextand the oﬀsets, one can conclude that the edge OP nucleation regim e\napparently dominates both for positive and large negative Hextvalues. Only in the intermediate ﬁeld range the\nhighest critical temperature corresponds to the formation of su perconductivity near the magnetic particle.\nLittle-Parks oscillations in mesoscopic samples\nThe nucleation of superconductivity in axially-symmetrical mesosco pic S/F structures (e.g., superconducting\ndisks or rings in the ﬁeld of a perpendicularly magnetized ferromagne tic circular dot) were studied theoretically\nby Aladyshkin et al.[54], Cheng and Fertig [130], Miloˇ sevi´ c et al.[131, 132], and experimentally by Golubovi´ c\net al.[133, 134, 135, 136] and Schildermans et al.[137]. Due to the cylindrical symmetry of the problem,\nsuperconductivity was found to appear only in the form of giant vor ticesψ(r,θ) =fL(r)exp(iLθ), whereLis\nthe angular momentum Lof the Cooper-pairs (vorticity). The appearance of vortex-ant ivortex conﬁgurations\nin superconducting disks of ﬁnite radius at temperatures close to Tcis possible, although these states were\npredicted to be metastable states.\nThe observed periodic cusp-like behavior of the Tc(Hext) dependence was attributed to the ﬁeld-induced\ntransition between states with diﬀerent vorticity similar to that of m esoscopic superconductors in a uniform\nmagnetic ﬁeld. However, the stray ﬁeld, induced by the magnetized dot, was shown to be responsible for a\npeculiar asymmetry of the oscillatory Tc(Hext) phase boundary and a shift of the main Tcmaximum towards\nnonzeroHextvalues [54, 133, 134, 135]. The mentioned abrupt modiﬁcation of the preferable nucleation regime\nwhen sweeping Hextcan lead to a double change in the slope of the Tc(Hext) envelope from Tc0/H(0)\nc3to\nTc0/H(0)\nc2[54, 137] (see Fig. 13). The restoration of the slope close to Tc0/H(0)\nc2can be interpreted as an eﬀectiveCONTENTS 24\n−200 −100 0 100 2000.50.70.91.11.3\nHext (Oe)Tc (K)Co/Pt disk on top of mesoscopic Al disk\nRerefence sample\nMagnetized,\nHm=3.4 kOe\nMagnetized,\nHm=2.0 kOe Demagnetized\nFigure 13. (color online) The phase transition lines Tc(Hext), obtained experimentally for the mesoscopic\nS/F hybrid system with a 0 .1Rncriterion for three diﬀerent magnetic states (completely m agnetized, partly\nmagnetized and demagnetized states), RnandHmbeing the normal-state resistance and the magnetizing ﬁeld ,\nadapted from Schildermans et al.[137]. The considered S/F system consists of superconducti ng Al disk covered\nby ferromagnetic Co/Pt multilayered ﬁlm of the same radius Rs=Rf= 0.825µm. Black solid line represents\ntheTc(Hext) dependence for the reference Al disk of the same lateral siz e.\nelimination of the boundary eﬀects in mesoscopic S/F samples at the c ompensation ﬁeld (near the main Tc\nmaximum). Interestingly, the nonuniform magnetic ﬁeld can be used to control the shift in the ﬁeld dependence\nof the maximal critical current Ic(Hext) for a bias current ﬂowing through the superconducting loop, whic h\nallows one to tune the internal phase shift in superconducting netw orks (Golubovic et al.[136]).\nItisimportanttonotethattheperiodicity∆ HextoftheLittle-Parksoscillationsinthe Tc(Hext)dependence\nis explicitly given by the area where the superconducting OP is conﬁne d. For edge nucleation only the area\nenclosed by the superconductor determines the period of the osc illations, which can be roughly estimated\nas ∆Hext≃Φ0/R2\ns. However in the case of the magnetic-dot-assisted nucleation (in t he vicinity of the\ncompensationﬁeld) the areaofthe OPlocalizationis determined byth e spatialcharacteristicsofthe nonuniform\nmagnetic ﬁeld (either the dot’ radius Rfor the vertical separation between the dot and the superconduc tor\nZd), therefore ∆ Hext≃Φ0/max{R2\nf,Z2\nd}. As a consequence, the change of the nucleation regimes manifest s\nitself as an abrupt modiﬁcation of the oscillatory Tc(Hext) dependence. In particular, both the amplitude and\nthe period of the Little-Parks oscillations become much larger, prov ided that (Zd,Rf)≪Rsand the OP wave\nfunction localizes far from the sample’s edges (Aladyshkin et al.[54], Carballeira et al.[138]).\nSymmetry-induced vortex-antivortex patterns\nMesoscopic S/F hybrids of a reduced symmetry (e.g., structures c onsisting of a superconductor/ferromagnet\ndisks and/or regular polygons) represent nice model systems for studying symmetry-induced phenomena. The\nformation of diﬀerent vortex-antivortex conﬁgurations was stu died theoretically by Carballeira et al.[138]\nand Chen et al.[139] for mesoscopic superconducting squares with a circular ferr omagnetic dot magnetized\nperpendicularly, and experimentally by Golubovic et al.[140] for a superconducting Al disk with a magnetic\ntriangle of Co/Pt on top. It was shown that the symmetry-consist ent solutions of the Ginzburg-Landau\nequations †, earlier predicted for mesoscopic superconducting polygons by Ch ibotaruet al.[40], are preserved\nfor regular superconducting polygons in the stray ﬁeld of ferroma gnetic disk. However, since spontaneously\nformed vortices and antivortices interact with the magnetic dot in a diﬀerent way, it leads to a modiﬁcation\nof the symmetry-induced vortex patterns (see Fig. 14). In part icular, the dot can be used to enlarge these\nvortex-antivortexpatterns, thus facilitating their experimenta l observationwith local vortex-imagingtechniques\n†By symmetry-consistent solutions we mean those vortex conﬁ gurations reﬂecting the symmetry of the problem. For instan ce, in\na mesoscopic superconducting square with vorticity L= 3, the state consisting of four vortices and a central antiv ortex may have\nlower energy than the conﬁguration of three equidistant vor tices, which breaks the square symmetry (Chibotaru et al.[141]).CONTENTS 25\nR =0.4a,□□□□□□□□-5.25<H a / <-3.5f ext2/c70/c48(b)□□No□magnetic□dot\n (a)□□Magnetic□dot□on□top\n5<H a / <6.3ext2/c70/c48\nFigure 14. (color online) A comparison between the vortex-antivortex patterns that can be observed for L= 3\n(a) with and (b) without the magnetic dot on top of the superco nducting square, ais the lateral size of the\nsample, adapted from Carballeira et al. [138]. As can be seen, the vortex-antivortex pattern rotat es 45◦and\nexpands dramatically in the presence of the magnetic dot.\n(“magnetic lensing”).\nEmbedded magnetic particles\nDoria [142, 143] and Doria et al.[144, 145] studied theoretically the formation of vortex patterns induced\nby magnetic inclusions embedded in a superconducting material but e lectrically insulated from the multiply-\nconnected superconductor. Since in the absence of an external ﬁeld, ﬂux lines should be closed, vortex lines\nare expected to start and end at the magnetic inclusions. The calcu lations, performed in the framework of GL\ntheory for a mesoscopic superconducting sphere with a single magn etic point like particle in its center, reveal\nthat the conﬁned vortex loops arise in triplets from the normal cor e when the magnetic moment reaches the\nscale deﬁned by m0= Φ0ξ/2π. Therefore a vortex pattern is made of conﬁned vortices, loops a nd also broken\nloops that spring to the surface in the form of pairs. This vortex st ate provides a spontaneous vortex phase\nscenario for bulk superconductors with magnetic inclusions, where the growth of the vortex loops interconnects\nneighboring magnetic inclusions.\nEﬀect of the ﬁnite thickness of the superconducting ﬁlm\nThe eﬀect of the ﬁnite thickness of the superconducting ﬁlm on the Tc(Hext) dependence was studied\ntheoretically by Aladyshkin et al.[54], Aladyshkin and Moshchalkov [95], and Schildermans et al.[137].\nIt follows from Maxwell’s equations that the faster the spatial varia tion of the magnetic ﬁeld in the lateral\n(x,y)−directions, the faster the decay of the ﬁeld in the transverse z−direction. However the OP variation\nalong the sample’s thickness can be eﬀectively suppressed by a requ irement of vanishing the normal derivative\n∂ψ/∂nof the superconducting OP at the top and bottom surfaces of the superconductor. Indeed, the\nFigure 15. (color online) Conﬁned vortex loops arise in sets of threes i nside a mesoscopic superconducting\nsphere (radius 15 ξ), as shown here for two consecutive values of the point-like magnetic moment that occupies\nits center: (a) m/m0= 15; (b)m/m0= 20, by the courtesy of M. M. Doria.CONTENTS 26\nexternal ﬁeld applied along the z−direction makes the OP variations over the superconducting ﬁlm thic kness\nmore energetically unfavorable than in the lateral direction, espec ially forHextvalues of the order of the\ncompensation ﬁeld or higher. The ﬁeld dependence Tc(Hext) for rather thick superconducting ﬁlms in the\nhigh-ﬁeld limit is similar to the phase transition line described by Eq. (7). This behavior results from the\neﬀective averagingof the inhomogeneous magnetic ﬁeld by the quas i–uniform OP wave function over the sample\nthickness, which substantially weakens the eﬀect arising from the ﬁ eld modulation in the lateral direction.\nIn other words, only the lateral inhomogeneity leads to the anomalo usTc(Hext) dependence and reentrant\nsuperconductivity in particular, while the transverse ﬁeld non-unif ormity masks this eﬀect. Interestingly, the\nlocation where superconductivity starts to nucleate shifts towar ds the point with zero total magnetic ﬁeld (not\nonlyz−component of the magnetic ﬁeld) as the thickness of superconduc tor increases (Aladyshkin et al.[54]).CONTENTS 27\n3. Vortex matter in non-uniform magnetic ﬁelds at low temper atures\n3.1. London description of a magnetically coupled S/F hybri d system\nIn section 2 we have reviewed the theoretical modelling within the GL f ormalism and the experimental data\nconcerning the superconducting properties of S/F hybrids rathe r close to the phase transition line Tc(Hext). In\nthis section we consider the properties of S/F hybrids for a fully dev eloped superconducting OP wave function,\ni.e. forT≪Tc0. In this limit, the superconducting properties of the S/F hybrids ca n be correctly described by\nthe London theory, omitting any spatial variations of the OP. As be fore, we assume that the coercive ﬁeld of the\nferromagnet is much higher than the upper critical ﬁeld of the supe rconductor. This guarantees that, during\nthe investigation of the superconducting properties, no changes in the ferromagnetic element(s) will occur.\nAt rather low temperatures one should take into account that the magnetic ﬁeld, induced by screening\n(Meissner) currents or by vortices, can no longer be neglected an d will strongly interact with the ferromagnet.\nThe free energy functional of the S/F hybrid can be written in the f ollowing form [44, 45]\nGsf=Gs0+Gm+/integraldisplay\nV/braceleftbiggλ2\n8π/parenleftbig\nrotB/parenrightbig2+B2\n8π−B·M−B·Hext\n4π/bracerightbigg\ndV. (17)\nAs before, Gs0is the self-energy of superconductor, while the term Gmdepends on the magnetization of\nferromagnet only. Assuming that Gmis constant for a given distribution of magnetization and minimizing this\nfunctionalGsfwith respect to the vector potential A, one can obtain the London-Maxwell equation:\nB+λ2rot rotB= Φ0/summationdisplay\niδ(r−Rv,i)z0+4πλ2rotrotM, (18)\nwhere summation should be done over all vortices, positioned at poin ts{Rv,i}andδ(r) is the Dirac delta-\nfunction†. In Eq. (18) we assumed that each vortex line, parallel to the z−axis and carrying one ﬂux quantum,\ngenerates a phase distribution Θ = ϕwith rot( ∇Θ) = rot ([ z0×r0]/r) =δ(r)z0[here (r,ϕ,z) are cylindrical\ncoordinates with the origin chosen at the vortex]. It should be notic ed that the solution of Eq. (18) gives the\nmagnetic ﬁeld distribution for a given vortex conﬁguration, and in or der to ﬁnd the ﬁeld pattern one should\nﬁnd the minimum of the total free energy Gsfwith respect to the vortex positions {Rv,i}.\n3.2. Interaction of a point magnetic dipole with a supercond uctor\nNext we consider the generic problem of the interaction of a superc onductor with a point magnetic dipole\npositioned at a height Zdabove the superconducting sample. Introducing the OP phase, wh ich determines the\ndensity of supercurrents js=c/(4πλ2)[Φ0/(2π)∇Θ−A], the free energy functional Eq. (17) at Hext= 0 can\nbe rewritten as Gsf=Gs0+Gm+Gs, where\nGs=/integraldisplay/braceleftbiggΦ2\n0\n32π3λ2(∇Θ·∇Θ)−Φ0\n16π2λ2(∇Θ·A)−1\n2B·M/bracerightbigg\ndV (19)\nsimilar to that obtained by Erdin et al.[146]. Due to the linearity of Eq. (18), we can introduce the ﬁeld\nBv= rotAv, generated by a single vortex line positioned at r=Rv, and the ﬁeld Bm= rotAm, corresponding\nto the screening (Meissner) current distribution induced by the ma gnetic dipole ‡. As a result, for the particular\ncase of a point magnetic dipole, M=m0δ(r−Rd), the energy of the system Gsdepending on the supercurrents\ncan be represented as a sum of the self-energy of the vortex ǫ(0)\nv= (Φ0/4πλ)2lnλ/ξ, which is independent on\nthe dipolar moment, and a term Gint, responsible for the interaction between the dipole and supercond uctor\n(e.g., Wei et al.[147], Carneiro [148]):\nGint=−1\n2m0·Bm(Rd)−m0·Bv(Rd). (20)\n†We follow the standard deﬁnition of the δ−function:δ(x,y) = 0 for any x∝negationslash= 0,y∝negationslash= 0, and−∞/integraltext\n−∞−∞/integraltext\n−∞δ(x,y)dxdy= 1.\n‡These ﬁelds BvandBmare the solutions of the following diﬀerential equations\nBv+λ2rot rotBv= Φ0δ(r−Rv)z0,\nBm+λ2rot rotBm= 4πλ2rotrotM.CONTENTS 28\nThe ﬁrst term in Eq. (20) corresponds to the interaction between the magnetic dipole and the local ﬁeld at the\ndipole’s position Bm(Rd) due to the screening currents. The second term in Eq. (20) gives the energy of the\ninteraction between the vortex and the magnetic dipole, which cons ists of two parts: (i) the “hydrodynamical”\ninteractionbetweencirculatingsupercurrentsduetothevortex ononehandandthedipoleontheotherhand; (ii)\nthe interactionbetween the strayﬁeld ofthe vortexwith the magn etic moment. Interestingly both contributions\nturn out to be equal to ( −1/2)m0·Bv(Rd).\nInteraction between a point magnetic dipole and the Meissne r currents\nIn the case of a superconductor without vortices [the so-called Me issner state, rot( ∇Θ) = 0], the interaction\nbetween the dipole and the superconductor is given by Um= (−1/2)m0·Bm(Rd). It can easily be seen that\nthis energy contribution is always positive, since the screening curr ents induced by a dipole generate a magnetic\nﬁeld which is opposite to the orientation of the dipole. This is in principle j ust a reformulation of the fact that\na magnetic dipole will be repelled by a superconducting ﬁlm with a force f=−∂Um/∂Rd. The related problem\nof levitating ferromagnetic particle over various superconducting structures was considered theoretically by Wei\net al.[147], Xu et al.[149], Coﬀey [150, 151] and Haley [152, 153]. In particular, in the limiting caseZd/λ≫1\nthe magnetic levitation force was found to vary linearly with λ, regardless of the shape of the magnet. The\nfact thatλ(as well as the vertical component of the force) shows an expone ntial temperature dependence for\ns−wave superconductors, linear −Tford−wave superconductors, and quadratic −T2dependence in a wide low-\ntemperature range for materials with s+idsymmetry of the gap, can assist in distinguishing the type of pairing\nin real samples based on magnetic force microscopy (MFM) measure ments [149].\nCreation of vortices by the stray ﬁeld of a magnetic dipole\nIn reality, magnetic dipoleswhich haveahigh dipolarmoment orareloca lizedrathercloseto the superconductor\ncan suppressthe Meissnerstate due to the dipole’s strayﬁeld. The possible appearanceofa vortexwould change\nthe total energy of the S/F hybrid system by Gs=ǫ(0)\nv+ǫmv, whereǫmv∝ −m0accounts for the dipole-vortex\ninteraction. Clearly, one can make Gs<0 by increasing m0and therefore it will become energetically favorable\nto generate vortices in the system. Such a scenario for the forma tion of the mixed state was considered, e.g.,\nby Weiet al.[147, 154] for a vertically magnetized dipole placed above a thin super conducting ﬁlm. As a\nconsequence of Eq. (20), the appearance of a vortex line in the su perconductor drastically changes the resulting\nforce acting on the magnetic dipole (Xu et al.[149], Wei et al.[147, 154]).\nIt is important to notice that in the case of superconducting ﬁlms inﬁ nite in the lateral direction and\ncooled in zero magnetic ﬁeld, a spontaneously induced vortex should be accompanied by an antivortex in order\nto provide ﬂux conservation imposed by Maxwell’s equations †. The destruction of the Meissner state in a\nzero-ﬁeld-cooled superconducting ﬁlm was theoretically studied by Melnikov et al.[155] and Aladyshkin et\nal.[156]. In this case the formation of the mixed state occurs via the pe netration of vortex semi-loops which\nsplit into vortex-antivortex pairs. The local suppression of the Be an-Livingston energy barrier [157], which\ncontrols the process of vortex penetration, takes place when th e screening current density will be of the order\nof the depairing current density. The threshold distance Z∗\nd=Z∗\nd(T), corresponding to the suppression of\nthe energy barrier can be experimentally detected by measuring a n onzero remanent magnetization as long\nas pinning is relevant. Interestingly, such non-contact technique allows one to estimate the depairing current\ndensity and its temperature dependence in thin superconducting Y Ba2Cu3O7ﬁlms [155, 156, 158]. Since the\nsurface energy barrier in superconductors with a ﬂat surface is k nown to be suppressed by applying an external\nﬁeld on the order of the critical thermodynamical ﬁeld Hcm= Φ0/(2√\n2πλξ) [43], one can get a rough criterion\nfor the persistence of the vortex-free state in the presence of a magnetic dipole: m0/Z3\nd≤Hcm. A diﬀerent\ncriterion should be obtained in case the dipole is already present close to the superconductor and the whole\nsystem is cooled down below the superconducting temperature. Un der this ﬁeld-cooled condition a lowercritical\nmagnetization to induce a vortex-antivortex pair is expected.\nTheproblemoftheformationofvortex-antivortexpairsinsuperc onductingﬁlmsinthepresenceofvertically\nand horizontally magnetized dipole at Hext= 0 was considered theoretically by Miloˇ sevi´ c et al.[159, 160],\nCarneiro [148]. It was shown that an equilibrium vortex pattern could consist of spatially separated vortices\nand antivortices (for rather thin superconducting ﬁlms) or, when the superconducting ﬁlm thickness increases\n†In mesoscopic superconducting systems, the returning ﬂux l ines generated by an out-of-plane magnetized dipole can byp ass the\nborder of the sample and the existence of an antivortex is not required.CONTENTS 29\nand becomes comparable to the London penetration depth, curve d vortex lines, which start and terminate at\nthe surface of the superconductor.\nMagnetic pinning\nNow we will discuss the properties of S/F hybrids where the magnetiz ation of the ferromagnet and the distance\nbetween ferromagnet and superconductor are assumed to be ﬁx ed resulting in a constancy of the interaction\nenergy with the screening currents induced by the ferromagnet. In this case, any variation of the free energy\nof the S/F hybrid in the presence of vortices (either induced by the ferromagnetic element or by external\nsources) can be attributed to the rearrangements of the vorte x pattern. The part of the interaction energy Gint,\nproportional to the magnetization of the ferromagnet and sensit ive to the vortex’ positions, is usually called the\nmagnetic pinning energy Up.\nIn order to illustrate the angular dependence of the interaction be tween a point magnetic dipole of ﬁxed\nmagnetization and a vortex, we refer to the following expression (C arneiro [148], Miloˇ sevi´ c et al.[159, 160])\nUp(r⊥,0) =Φ2\n0\n2πλ/bracketleftbigg\n−1\n2/parenleftbiggm0,z\nΦ0λ/parenrightbiggDs\n(r2\n⊥+Z2\nd)1/2+Ds\n2r⊥/parenleftbiggm0,z\nΦ0λ/parenrightbigg/parenleftbigg\n1−Zd\n(r2\n⊥+Z2\nd)1/2/parenrightbigg\ncosφ/bracketrightbigg\n,(21)\nobtained for a thin superconducting ﬁlm Ds≪λand for vortex-dipole separations smaller than the eﬀective\npenetration depth λ2D=λ2/Ds. Herem0,xandm0,zare the in-plane and out-of-plane components of the\ndipolar moment, r⊥=/radicalbig\nx2+y2,Rd= (0,0,Zd) is the position of the dipole, φis the angle between the x-axis\nand the vector position of the vortex in the plane of the supercond uctorr⊥. The resulting pinning potentials\nfor an in-plane and out-of-plane magnetized dipole, derived from Eq . (21), are shown in Fig. 16. The pinning of\nvortices (and antivortices) in the superconducting ﬁlms of a ﬁnite t hickness on the magnetic dipole at Hext= 0\nwas analyzed by Miloˇ sevi´ c et al.[159, 160] and Carneiro [148, 161]. In addition, Carneiro considere d the\nmagnetic pinning for the case of a dipole able to rotate freely in the pr esence of the external magnetic ﬁeld\n[162, 163, 164] or external current [162].\nIn Ref. [148] it was shown that this magnetic pinning potential has a d epth ofm0/(4πλ), penetrates a\ndistanceλinto the ﬁlm whereas its range parallel to the ﬁlm surface is a few times λ. This ﬁnding points out\nthe relevance of the penetration depth λto characterize the purely magnetic pinning potential. Since typical\nλvalues are similar for a broad spectrum of superconducting materia ls, this suggests that magnetic pinning\nrepresents a promising way of increasing the critical current not o nly in conventional superconductors but also\nin high-Tcsuperconductors. Notice that the fact that the magnetic pinning range is determined by λsets a limit\nfor the minimum distance between magnetic particles, beyond which t he vortex lines cannot resolve the ﬁeld\nmodulation and therefore the pinning eﬃciency decreases. Having in mind that many practical applications\n−4−2024\n−4−2024−4−20\nx/λ y/λUp/U0, U0=Φ02/(πDs)(a)  M=m0 z0δ(x,y,z−Zd)\n−4−2024\n−4−2024−101\nx/λ y/λUp/U0, U0=Φ02/(πDs)(b)  M=m0 x0δ(x,y,z−Zd)\nFigure 16. (color online) The spatial dependence of the energy of a sing le vortex in a ﬁeld of point magnetic\ndipole,M=m0δ(x,y,z−Zd), calculated according to Eq. (21) for the height Zd/λ= 1 and the dipole strength\nm0/(Φ0λ) = 10:\n(a) vertically magnetized dipole, m0=m0z0,\n(b) horizontally magnetized dipole, m0=m0x0.CONTENTS 30\ntypically involve high- Tcsuperconductors with ξ≪λ, it becomes clear that this maximum density of pinning\nsites to trap individual vortices is much lower than that limited due to c ore pinning.\nIt is important to mention that real nanoengineered ferromagnet ic elements are quite far from point dipoles\nbut rather consist of extended volumes of a magnetic material, suc h as magnetic dots and stripes. In this case,\ndue to the principle of superposition, the magnetic pinning energy ne eds to be integrated over the volume of\nthe ferromagnet Vfto determine the interaction between a vortex line and a ﬁnite size pe rmanent magnet\nUp(r) =−/integraldisplay\nVfM(r′)·Bv(r−r′)d3r′. (22)\nThe magnetic pinning in an inﬁnite superconducting ﬁlm produced by ind ividual ferromagnetic objects\nof ﬁnite size was considered for the following cases: a ferromagnet ic sphere magnetized out-of-plane (Tokman\n[165]), magnetic disks, rings, rectangles, and triangles magnetized out-of-plane (Miloˇ sevi´ c et al.[160, 166]),\nmagnetic bars and rectangular dots magnetized in-plane (Miloˇ sevi ´ cet al.[167]), circular dots magnetized either\nin-plane and out-of-plane (Erdin et al.[146], Erdin [168]), circular and elliptic dots and rings magnetized\nout-of-plane (Kayali [169, 170], Helseth [171]), columnar ferromag netic rod (Kayali [172]). The analysis of\nthe pinning properties of a vortex in a semi-inﬁnite superconducting ﬁlm due to magnetic dots was done by\nErdin [173]. Similar to the case of a point magnetic dipole, the extended magnetic objects are able to generate\nvortex-antivortex pairs provided that the size and the magnetiza tion are large enough.\nWe would like to note that the attraction of a superconducting vort ex to a source of inhomogeneous\nmagnetic ﬁeld (e.g., coil on a superconducting quantum interferenc e device or magnetized tip ofa magnetic force\nmicroscope) makes it possible to precisely manipulate the position of a n individual vortex. Such experiments\nwere performed by Moser et al.[174], Gardner et al.[175], Auslaender et al.[176] for high −Tcsuperconducting\nthinﬁlmsandsinglecrystalsinintermediatetemperatureswhenintrin sicpinningbecomeweaker. Thistechnique\napparently opens unprecedent opportunities, e.g., for a direct me asuring of the interaction of a moving vortex\nwith the localdisorderpotential and forpreparingexoticvortexs tates likeentangledvortices(Reichhardt [177]).\nEquation (22) shows that the pinning potential does not only depen d on the size and the shape of the\nferromagnetic elements but also on the particular distribution of th e magnetization (i.e. on their exact magnetic\nstate). It is expected that the average pinning energy is less eﬃcie nt for magnetic dots in a multidomain state\nwhereas it should reach a maximum for single domain structures. In o ther words, if the size of the magnetic\ndomains is small in comparison with λor the separation distance Zd, then a vortex line would feel the average\nﬁeld emanating from the domains and the magnetic pinning should be st rongly suppressed. This ﬂexibility of\nmagnetic pinning centers makes it possible to tune the eﬀective pinnin g strength, as we shall discuss below.\nThe question now arises whether the pinning potential produced by a magnetic dipole will remain eﬃcient\nwhen a vortex-antivortexpair is induced by the magnetic dipole. In o rder to answer this question it is necessary\nto minimize the mutual interaction between the induced vortex-ant ivortex pair, the magnetic element and the\ntest vortex generated by the external ﬁeld. The magnetic momen t – test vortex interaction is a linear function\nofm0always favoring the test vortex to sit on the positive pole of the mag net, whereas the induced currents –\ntest vortexinteraction will follow the Little-Parks oscillations (due t o the creation of vorticesby the dipole). For\nthe case of an in-plane dipole, Miloˇ sevi´ c et al.[167] demonstrated that all these terms produce a subtle balance\nof forces which lead to a switching of the optimum pinning site from the positive to the negative magnetic pole,\nasm0is increased. For the case of out-of-plane magnetic dipole, once a v ortex-antivortex pair is induced, the\ntest vortex will annihilate the antivortexleaving a single vortex on to p of the magnetic dot (Gillijns et al. [123]).\n3.3. Magnetic dots in the vicinity of a plain superconductin g ﬁlm\nThe rich variety of possibilities considered theoretically in the previou s section for an individual ferromagnetic\nelement in close proximity to a superconducting ﬁlm represents an ex perimental challenge in part due to the\ndiﬃculties associated with recording small induced signals. A very suc cessful way to overcome this limitation\nconsists of studying the average eﬀect of a periodic array of dots at the expense of blurring sharp transitions,\nsuch as the vortex-antivortex generation, or inducing new collect ive phenomena associated with the periodicity\nof the magnetic templates. An important experimental condition th at should be satisﬁed in order to justify\nthe analogies drawn between individual dipoles and arrays of dipoles, is the lack of magnetostatic interactionsCONTENTS 31\n−50 −25 0 25 500246810\n−4H1−3H1−2H1−H1H12H13H14H1\nMagnetized\nDemagnetized\nReferenceArray of Co dots covered by Pb film\nHext (Oe)M (10−4 emu)\nFigure 17. (color online) Upper half of the magnetization loop Mvs.HextatT/Tc0= 0.97, for a\nsuperconducting Pb ﬁlm (50 nm thickness) on top of a triangul ar lattice of Au/Co/Au dots (period 1.5 µm\ncorresponding to a ﬁrst matching ﬁeld of 9.6 Oe) before and af ter magnetizing the dots (ﬁlled and open symbols,\nrespectively), and for a reference Pb 50 ﬁlm (solid line), ad apted from van Bael et al.[196]. The curve for the\nmagnetized dots is slightly shifted upwards for clarity.\nbetween neighboring dipoles. In other words, it is necessary to ens ure that the ﬁeld generated by a dipole at the\nposition of its nearest neighbors lies below the coercive ﬁeld of the ch osen ferromagnetic materials (Cowburn et\nal.[178, 179], Novosad et al.[180, 181]).\nCommensurability eﬀects in S/F hybrids with periodic array s of magnetic dots\nIn the early 1970’s, Autler [182, 183] proposed that a periodic arra y of ferromagnetic particles should give rise\nto an enhancement of the critical current of the superconductin g material. Recent developments on lithographic\ntechniques have made it possible to prepare superconducting stru ctures containing a regular array of magnetic\ndots (Co, Ni, Fe) at submicrometer scale of desirable symmetry in a c ontrolled way (Otani et al.[119], Geoﬀroy\net al.[120], Mart´ ın et al.[184, 185, 186, 187], Morgan and Ketterson [188, 189], Hoﬀmann et al.[190], Jaccard\net al.[191], Villegas et al.[192, 193, 194], Stoll et al.[195], van Bael et al.[196, 197, 198, 199, 200, 201],\nvan Look et al.[202]). The S/F hybrids containing periodic arrays of magnetic elemen ts with out-of-plane\nmagnetization (multilayered Co/Pt and Co/Pd structures) were fa bricated and investigated by van Bael et al.,\n[198, 199, 200, 201, 203, 204] and Lange et al.[205, 206]. In all these works, it was found that the presence\nof the lattice of magnetic dots leads either to (i) a resonant change in the magnetoresistance ρ(Hext) and the\nappearance of pronounced equidistant minima of resistivity with the periodH1= Φ0/S0determined by the\nsize of the magnetic unit cell S0[119, 120, 184, 185, 186, 187, 189, 190, 191, 193, 192, 194, 195 ]; (ii) or to the\npresence of distinct features as peaks or plateaus in the ﬁeld depe ndence of the critical current Ic(Hext) or in\nthe magnetization curve M(Hext) [188, 189, 190, 196, 198, 199, 200, 201, 202, 203, 204, 205, 20 6].\nSuch matching eﬀects are also very well known for spatially-modulat ed superconducting systems with\nantidot lattices without ferromagnetic constituents (e.g., Baert et al.[52], Moshchalkov et al.[53], Hebard et al.\n[207], Rosseel et al.[208], Harada et al.[209], Metlushko et al.[210]). These periodic anomalies are commonly\nexplained in terms of commensurability eﬀects between the vortex la ttice governed by the external ﬁeld and the\nartiﬁcially-introduced pinning potential at an integer number of vor tices per unit cell at Hext=±nH1(withn\ninteger). Typically these lithographically deﬁned arrays are limited to a minimal period of the unit cell of the\norder of a few hundred nanometers, giving rise to a maximum matchin g ﬁeldH1≃10−102Oe. Alternative\nmethods for introducing more closely packed particles and thus larg erH1values can be achieved by Bitter\ndecoration (Fasano et al.[211, 212], Fasano and Menghini [213]) or a diversity of self-assemb led techniques\n(Goyalet al.[214], Villegas et al.[215], Welp et al.[216, 217], Vinckx et al.[218], Vanacken et al.[219]).\nThe ﬁeld dependence of the critical current Ic(Hext) and magnetization M(Hext), can be symmetrical or\nasymmetrical with respect to Hext= 0 depending on the dot’s net magnetization (compare Figs. 17 and 1 8).CONTENTS 32\n−80 −40 0 40 80−6−4−20246\n−3H1−2H1−H1H12H13H1\nPositively magnetized\nHext (Oe)M (10−4 emu)Array of Co/Pt dots covered by Pb film\n(a)\n−80 −40 0 40 80−6−4−20246\n−3H1−2H1−H1H12H13H1\nNegatively magnetized\nHext (Oe)M (10−4 emu)Array of Co/Pt dots covered by Pb film\n(b)\nFigure 18. (color online) Magnetization curves Mvs.HextatT= 7 K (Tc0= 7.17 K,T/Tc0= 0.976) for a\nsuperconducting Pb ﬁlm (50 nm thickness) on top of a Co/Pt dot array (the period is 1.0 µm, the ﬁrst matching\nﬁeld is 20.68 Oe) with all dots aligned in the positive (a) and negative (b) direction, adapted from van Bael et\nal.[204].\nThe latter takes place for arrays of magnetic dots with out-of-pla ne magnetization [198, 199, 200, 201, 203,\n204, 205, 206]. Indeed, vertically magnetized dots with average ma gnetic moment ∝an}b∇acketle{tm∝an}b∇acket∇i}htz>0, similar to point\nmagnetic dipoles, produce a stronger pinning potential for vortice s (atHext>0 when∝an}b∇acketle{tm∝an}b∇acket∇i}ht ∝ba∇dblHext) than for\nantivortices ( Hext<0). In contrast to that, in-plane magnetized dots are able to pin vo rtices and antivortices\nat the magnetic poles equally well [see Eq. (21) and Fig. 16]. This explain s the experimentally observed ﬁeld\npolarity-dependent (asymmetric) pinning for arrays of out-of-p lane magnetized particles (Fig. 18 and Fig. 19).\nThe interaction between vortices and a periodic array of hard magn etic dots on top or underneath a plain\nsuperconductingﬁlmwithintheLondonapproximation †wastheoreticallyanalyzedbyHelseth[225], Lyuksyutov\n†This issue seems to be part of a more general problem of the int eraction of vortex matter with a periodic potential regardl ess the\nnature of the pinning in the superconducting system (see, e. g., Reichhardt et al.[220, 221, 222, 223, 224] and references therein).\nIn this review we discuss only the results obtained for the S/ F hybrids, keeping in mind that similar eﬀects can be observe d for\n−4−3−2−10123400.20.40.60.81\n−H1H12H13H1\nHext/H1jc/jGLMs=250 Oe\nMs=510 Oe\nMs=1400 Oe\nFigure 19. (color online) Field dependence of the critical depinning c urrentjc, calculated for a periodic array\nof out-of-plane magnetized dots. The values of the dot’s mag netization are indicated in the plot. jGLis the\ndensity of the depairing (Ginzburg-Landau) current, T/Tc0= 0.9, adapted from Miloˇ sevi´ c and Peeters [122].\nIt is worth noting that (i) jc(Hext) is asymmetric similar to that shown in Fig. 18, and (ii) the q uantized\ndisplacement of the jcmaximum toward nonzero Hextvalue is sensitive to the magnetization of the dots.CONTENTS 33\n−1000 −500 0 500 100010−21001021042Rs=100 nm × 1000\n180 nm × 100\n270 nm × 10\n340 nm(a)Array of Ni dots covered by Nb film\nHext (Oe)ρ (µΩ cm)\n−1000 −500 0 500 100010−410−21001021042Rs=100 nm × 1000\n180 nm × 100\n230 nm × 10\n300 nm(b)Array of Ag dots covered by Nb film\nHext (Oe)ρ (µΩ cm)\nFigure 20. (color online) (a) Field dependence of the electrical resis tanceρfor a superconducting Nb ﬁlms\ncovering periodic arrays of the magnetic Ni dots with diﬀere nt dot’s diameter (indicated in the plot), but for\nthe same lattice constant 400 nm, adapted from Hoﬀmann et al.[190]. The curves are shifted by factors of 10\nfrom each other for clarity.\n(b) Dependences ρ(Hext) for samples with diﬀerent diameters of non-magnetic Ag dot s, adapted from Hoﬀmann\net al.[190].\nand Pokrovsky [226], ˇS´ aˇ sik and Hwa [227], Erdin [228], Wei [229, 230], Chen et al.[231]. These calculations\nshow that at Hext= 0 for out-of-plane magnetized dots, vortex-antivortex pairs c an be created in thin-ﬁlm\nsuperconductors with the vortices always sitting on top of the mag netic dot and the antivortices located in\nbetween the dots. For in-plane magnetized dots (or magnetic bars ), the vortex and antivortex will be located\nat opposite sides of the magnetic dots as described above for individ ual magnetic dipoles. Unlike the case of an\nisolated dipole, the threshold magnetization value needed to create a vortex-antivortex pair is also a function\nof the period of the lattice (Miloˇ sevi´ c and Peeters [114]). Direct v isualization of vortex-antivortex pairs via\nscanning Hall probe microscopy was achieved for a square array of in-plane dots by van Bael et al.[197] and\nfor out-of-plane dots by van Bael et al.[204] and Neal et al.[232].\nItisknownthatthepreferredvortexconﬁgurationinahomogene ousdefect-freesuperconductingﬁlmshould\nbe close to a triangular lattice because of the repulsive vortex-vor tex interaction [43, 44, 45]. The artiﬁcially-\nintroduced pinning appears to be the most eﬀective provided that e ach vortex is trapped by a pinning center,\ni.e. when the symmetry of the pinned vortex lattice coincides with tha t imposed by the topology of the internal\nmagnetic ﬁeld. The transition between square and distorted triang ular vortex lattice, induced by variation\nof the strength of the periodic pinning potential and the characte ristic length scale of this interaction, was\nconsidered by Pogosov et al.[233] for superconductors with a square array of pinning centers . Experimentally\nthe ﬁeld-induced reconﬁguration of the vortex lattice (from rect angular to square) for superconducting Nb ﬁlms\nand rectangular arrays of circular magnetic Ni and Co dots was rep orted by Mart´ ın et al.[185] and Stoll\net al.[195] as an abrupt increase of the period of the oscillation in the ρ(Hext) dependence (resulting from\nthe shrinkage of the period of the vortex lattice) and decrease of the amplitude of such oscillations (due to a\nweakening of the eﬀective pinning) while increasing |Hext|.\nThe dependence of the magnetic pinning in superconducting Nb ﬁlms o n the diameter of the Ni dots was\nstudied by Hoﬀmann et al.[190]. They found that more minima appear in the magnetoresistance (or maxima\nin the critical current) as the lateral dot’s size increases, indicatin g thus an enhanced pinning (the panel (a)\nin Fig. 20). This eﬀect can be caused by the two parameters which inc rease with the dot size: the total\nmagnetic moment ∝an}b∇acketle{tm∝an}b∇acket∇i}ht(proportional to the dot’s volume Vf=πR2\nfDf) and the area on the order of πR2\nfwhere\nsuperconductivity might be locally suppressed due to the high stray ﬁeld or proximity eﬀect. In addition, larger\nmagnetic dots can stabilize giant vortices carrying more than one ﬂu x quantum.\nnon-magnetic patterned superconductors as well.CONTENTS 34\n−60 −30 0 30 60−3−2−101\n−3H1−2H1−H1H12H13H1\nPositively magnetized\nNegatively magnetizedArray of Co/Pt antidots covered by Pb film\nHext (Oe)M (10−4 emu)\nFigure 21. (color online) Magnetization curves M(Hext) atT= 7.05 K (Tc0= 7.20 K,T/Tc0= 0.972) of a\nsuperconducting Pb ﬁlm on top of a magnetic Co/Pt antidot lat tice (the period is 1.0 µm, the ﬁrst matching\nﬁeld is 20.68 Oe) after saturation in a positive ﬁeld ( Mz>0, open circles) and after saturation in a negative\nﬁeld (Mz<0, ﬁlled circles), adapted from Lange et al.[234, 235].\nPeriodic arrays of magnetic antidots\nThe antipode of arrays of magnetic dots is a perforated ferromag netic ﬁlm (so-called magnetic antidots), which\nalso produces a periodic magnetic ﬁeld. This system can be regarded as the limiting case of big magnetic dots\nwith a diameter larger than the period of the periodic lattice.\nMagnetic antidots in multilayered Co/Pt ﬁlms, characterized by an ou t-of-plane remanent magnetization,\nand their inﬂuence on the superconducting properties of Pb ﬁlms we re studied by Lange et al.[205, 234, 235,\n236]. From magnetostatic considerations, such submicron holes in a ferromagnetic thin ﬁlm generates a very\nsimilarﬁeldpatternasanarrayofmagneticdotsofthesamegeomet ry,butwithoppositesign. Asaconsequence,\nthe enhanced magnetic pinning and the pronounced commensurabilit y peaks in the M(Hext) dependence are\nobserved for the opposite polarity of the external ﬁeld (i.e. at Hext<0 for positively magnetized ﬁlm and vice\nversa), see Fig. 21. However the matching eﬀects are considerab ly weakened in the demagnetized state of the\nCo/Pt multilayer with holes as compared with the demagnetized array of magnetic dots, thus indicating that\nan irregular domain structure eﬀectively destroys a long-range pe riodicity [205].\nVan Bael et al.[237] and Raedts et al.[238] explored perforated Co ﬁlm with in-plane anisotropy.\nIn this case the magnetic ﬁeld distribution becomes non-trivial since such magnetic antidots eﬀectively pin\nmagnetic domain walls which generate a rather strong magnetic ﬁeld. As a result, neither matching eﬀects nor\npronounced asymmetry were observed in the magnetization curve s of the superconducting layer, but only an\noverall enhancement of the critical current after the sample was magnetized along the in-plane easy axis, in\ncomparison with the demagnetized state.\nAnisotropy of the transport characteristics and guidance o f vortices\nIn any periodic array of pinning centra, transport properties suc h as magnetoresistance ρ(H) and the critical\ncurrentJc(H), exhibit a dependence not only on the absolute value of Hext, but also on the direction of the\napplied transport currentwith respect to the principal translatio n vectorsof the periodic pinning array(Villegas\net al.[193, 194], Soroka and Huth [239], V´ elez et al.[240], Silhanek et al.[241], W¨ ordenweber et al.[242]).\nInterestingly, the direction of the Lorentz force fL=c−1[j×B] and the drift velocity of the vortex lattice do not\ngenerally coincide. It was demonstrated that for rectangular arr ays of magnetic dots the minimum of resistivity\ncorresponds to a motion of the vortex lattice along the long side of t he array cell. Such behavior was predicted\nby Reichhardt et al.[223] by numerical simulations indicating that a rectangular array of pinning centersCONTENTS 35\n0 45 90 135 18000.20.40.60.81\n m0\n FLβJc/Jd\nβ, degree\nFigure 22. (color online) Critical current, Jc, for a vortex pinned by a dipole array as a function of the angl e\nβbetween the magnetic moments and the driving force, adapted from Carneiro [243]. Here Jdis the depairing\ncurrent.\nFigure 23. (a) Magneto-optical image of a non-patterned superconduct ing Pb disk at T= 2 K andHext= 50\nOe, demonstrating an isotropic ﬂux penetration, after Gheo rgheet al.[246]. White corresponds to a high local\nmagnetic ﬁeld and black to zero local ﬁeld. (b) Magneto-opti cal image of a circular Pb sample decorated by\nfully magnetized Co/ Pt dots, obtained at T= 2 K andHext= 72 Oe, after Gheorghe et al.[246]. The external\nﬁeld is applied parallel to the dot’s magnetic moment.\n(c) Magneto-optical image of ﬂux entry in a superconducting MoGe ﬁlm at H||\next= 16.5 Oe,T= 4.5 K\nfollowing the preparation of stripe domain structures in a p ermalloy ﬁlm by turning on and oﬀ an in-plane ﬁeld\nofH/bardbl\next= 1 kOe at an angle of 45◦with respect to the sample edge at T > Tc0, after Vlasko-Vlasov et al.\n[247]. The brightness of the magneto-optical contrast corr esponds to the vortex density. The large yellow arrow\nshows the preferential ﬂux entry direction coinciding with the direction of the stripe domains in the Py ﬁlm.\nThe thin solid line with arrow marks the sample edge. (d) Same as in panel (c) after application and switching\noﬀ ofH/bardbl\next= 1 kOe along the sample edge at T >Tc0, after Vlasko-Vlasov et al.[247].\ninduces an easy direction of motion for the vortex lattice (and large r dissipation as well) along the short side of\nthe array cell.\nSimilar anisotropic transport properties was studied by Carneiro [2 43] for the case of a periodic array\nof in-plane magnetic dipoles. In order to illustrate the angular depen dence of the critical depinning current\non the angle βbetween the direction of the injected current and the magnetic mo ment of in-plane oriented\ndipoles we refer to Fig. 22. Interestingly, Verellen et al.[244] showed that this resulting guided vortex motion\nin square arrays of magnetic rings can be rerouted by 90◦simply by changing the dipole orientation or can\neven be suppressed by inducing a ﬂux-closure magnetic vortex sta te with very low stray ﬁelds in the rings.\nSimilar anisotropic vortex motion was recently observed in Nb ﬁlms with a periodic array of one-dimensional\nNi lines underneath by Jaque et al.[245]. The mentioned channelling of vortices lead to an anisotropic vor tex\npenetration that has been directly visualized by means of magnetoo ptics experiments [Gheorghe et al.[246],\nsee Fig. 23(b)].\nMechanisms of pinning in S/F hybrids\nItshouldbenoticedthatthemagneticpinningoriginatingfromthesp atialmodulationofthe“internal”magneticCONTENTS 36\nﬁeld generally competes with so-called core pinning resulting from str uctural inhomogeneities in real samples\n(either regular or random defects). In addition to random intrinsic pinning, the fabrication of an array of\nmagnetic particles naturally leads to an alteration of the local prope rties of the superconducting ﬁlm (e.g., due\nto proximity eﬀects, corrugation of the superconducting layer or local suppression of the critical temperature).\nAsaconsequence, both magneticandstructuralmodulationshar ethesameperiodicity, andaclearidentiﬁcation\nof the actual pinning type becomes diﬃcult.\nA direct comparison of the pinning properties of arrays of magnetic vs. nonmagnetic dots have been\naddressed by Hoﬀmann et al.[190] and Jaccard et al.[191]. These reports show that even though both\nsystems display commensurability features, the pinning produced b y magnetic arrays of Ni dots is substantially\nstronger than that produced by non-magnetic Ag particles (Fig. 2 0). In our opinion, the main issue whether the\nenhanced pinning for the sample with ferromagnetic Ni dots actually arises from purely magnetic interactions\nand not from an additional suppression of the local critical temper ature, e.g. due to the enhanced magnetic\nﬁeld near magnetic dots, remains unclear. In principle, the most str aightforward way to distinguish the two\ncompeting pinning mechanisms is the mentioned ﬁeld-polarity of the ma gnetic pinning for the S/F hybrids with\ndots magnetized perpendicularly, i.e. exploring the broken ﬁeld-pola rity symmetry.\nClear evidence of the ﬁeld-polarity dependent pinning properties ha s been reported by Gheorghe et al.\n[246] in Pb ﬁlms on top of a square array of [Co/Pt] 10dots with a well deﬁned out-of-plane magnetic moment.\nIn this work the authors show that the critical current of the hyb rid system can be increased by a factor of\n2 when the magnetic dots are switched from low stray ﬁeld in the dema gnetized state (disordered magnetic\nmoment) to high stray ﬁeld in the magnetized state (nearly single dom ain state) at temperatures as low as\nT≃0.3Tc0(see Fig. 24). Additional evidence of an increase of the critical cur rent at low temperatures (far\nfrom the superconducting/normal phase boundary) produced b y magnetic dots was reported by Terentiev et\nal.[248, 249, 250].\nTunable pinning centers\nAn apparent advantage of using magnetic pinning centra is their ﬂex ibility (tunability) in contrast to core\npinning on structural inhomogeneities. Indeed, according to Eq. ( 22) the magnetic pinning should be sensitive\nto the particular distribution of magnetization inside the ferromagn etic elements. Depending on the geometrical\ndetails of the dot and the magnetic anisotropy of the chosen mater ial a huge variety of magnetic states can\nbe found. For instance, domain formation is expected to be suppre ssed for structures with lateral dimensions\n123456703691215\n〈m〉z/Vf=1.00 Ms\n〈m〉z/Vf=0.63 Ms\n〈m〉z/Vf=0.25 Ms\n〈m〉z/Vf=0\n〈m〉z/Vf=−1.00 MsArray of Co/Pt dots covered by Pb film\nT (K)jc (105 A/cm2)\nFigure 24. (color online) Temperature dependence of the critical curr ent density jc, estimated from magneto-\noptical images, for a superconducting Pb ﬁlm with square arr ay of the ferromagnetic Co/Pt dots on top,\nin various magnetic states of the dots: demagnetized ( ◦), fully magnetized parallel conﬁguration ( ⋄), fully\nmagnetized antiparallel conﬁguration (square), partiall y magnetized parallel, ∝angb∇acketleftm∝angb∇acket∇ightz= 0.25MsVf(∆) and\n∝angb∇acketleftm∝angb∇acket∇ightz= 0.63MsVf(∇), adapted from Gheorghe et al.[246]. The dashed lines are guides to the eye.CONTENTS 37\n−2000 −1000 0 1000 2000−1−0.500.51\nVortex annihilation\nVortex nucleationVortex annihilationVortex nucleation\nH||\next (Oe)M||/MsArray of Fe dots covered by Al film\n(a)\nT=6.0 K\n−2000 −1000 0 1000 200010−1100\nH||\next (Oe)ρ/ρnArray of Fe dots covered by Al film\n(b)\nT=1.25 K\nFigure 25. (color online) (a) Normalized magnetization M/bardbl/Msvs. in-plane applied ﬁeld H/bardbl\next(Msis the\nsaturated magnetization) for an array of Fe dots with averag e diameter of 140 nm and average interdot distance\nof 180 nm measured at T= 6 K (above the critical temperature of superconducting Al ﬁ lm), adapted from\nVillegas et al.[215]. Brown diamonds correspond to a virgin state of the Fe d ots, the magnetic state depicted\nby red (blue) circles obtained after saturation in positive (descending branch) and negative (ascending branch)\nmagnetic ﬁelds, respectively. The ﬁlled circles schematic ally show the regions where the magnetic vortex is\nexpected to take place. Vertical dashed lines mark the coerc ive ﬁelds.\n(b) Normalized resistivity ρvs. in-plane applied ﬁeld H/bardbl\nextfor the same sample at T= 1.25 K (below the critical\ntemperature of superconducting Al ﬁlm), ρnis the normal-state resistance. Open (blue) and ﬁlled (red) circles\nmark the curves measured from negative and positive saturat ion respectively, while brown diamonds correspond\nto the virgin state, adapted from Villegas et al.[215].\nsmaller than tens of nm (Raabe et al.[251]), whereas for larger sizes the magnetic sample breaks into dom ains\nof diﬀerent orientation (Seynaeve et al.[252]). The exact transition from single domain to multi-domain\nstructures depends on the shape, dimensions, temperature and particular material, among other parameters.\nMore recently, Villegas et al.[215, 253] and Hoﬀmann et al.[254] experimentally investigated the switching\nof the ferromagetic dots from single domain to magnetic vortex sta te while sweeping the external ﬁeld and the\ninﬂuence of their stray ﬁelds on the resistivity of the S/F hybrid sam ple (Fig. 25). The interaction between a\nvortex in a superconducting ﬁlm and a magnetic nanodisk in the magne tic vortex state was studied theoretically\nby Carneiro [255]. For magnetic dots big enough to host a multi-domain state it is possible to tune the average\nmagnetic moment by partially magnetizing the sample in a ﬁeld lower than the saturation ﬁeld or even recover\nthe virgin state by performing a careful degaussing procedure sim ilar to that shown in Fig. 6 (Gillijns et al.\n[96, 123], Lange et al.[256]). Interestingly, recently Cowburn et al.[178] showed that small disks of radius\nabout 50 nm made of supermalloy (Ni-80%, Fe-14%, Mo-5%) lie in a single domain state with the magnetization\nparalleltothediskplaneandwiththepropertythattheirdirectionc anbereorientedbysmallappliedﬁelds. This\nsystems represents the closest experimental realization of in-pla ne free-rotating dipoles, which was theoretically\nanalyzed by Carneiro [162, 163, 164] within the London formalism.\nWhatever the mechanism of pinning produced by the magnetic dots, either core or electromagnetic, it\nis now a clearly established fact that changing the domain distribution in each dot has profound eﬀects on\nthe superconducting pinning properties as demonstrated, for ex ample, by van Bael et al.[196], van Look et\nal.[202]. This result points out the importance of performing a careful study of the magnetic properties of\nthe dots in order to identify the domain size, shape, distribution, an d stable states. Van Bael et al.[196]\npresented the ﬁrst report directly linking changes in the hysteres is loop of a superconducting Pb ﬁlm when\nthe underlying submicron Co islands are switched from 2 ×2 checkerboard magnetic domain pattern to single\ndomain structures.\nAs we pointed out above, unfortunately both, the multidomain stat e and the magnetic vortex state,\nstill involve a sizable component of the magnetic stray ﬁeld which even tually inﬂuences the response of the\nsuperconducting properties by locally suppressing the order para meter. In other words, it is actually notCONTENTS 38\nFigure 26. (color online) Diﬀerent magnetic states realized in square permalloy micro-loops depending on the\ndirection of the applied magnetic ﬁeld, by courtesy of Metlu shkoet al.\npossible to completely switch oﬀ the magnetic pinning using singly conne cted structures. It has been recently\nshown that a way to partially circumvent this limitation can be achieved by using multiply connected ring-like\nmagnetic structures. In this case, if a ﬂux-closure state is induce d in the magnetic ring, in principle, there is\nalmost no stray ﬁeld present, besides small ﬁelds due to domain walls a t the sharp corners of the ring. Indeed,\na two-dimensional magnetic material of ring like shape of group symm etryCncan be set in two ﬂux closure\nstates of opposite chirality and n(n−1) diﬀerent polarized states. In a square loop, for instance, 12 st ates\ncorresponding to six diﬀerent dipole directions with two opposite dipo le orientations are expected. If the net\ndipolar moment is parallel to the side of the square, the ﬁnal state is named horseshoe state whereas if the\ndipole is along the diagonal of the square, it is called onion state. Figur e 26 shows the diﬀerent topologically\nnon-equivalent magnetic states for a square ring of magnetic mate rial with in-plane magnetization. Clear\nexperimental evidence of ON/OFF magnetic pinning potentials induce d by these type of multiply-connected\nferromagnetic structures have been demonstrated by Silhanek et al.[257, 258, 259]. It is worth mentioning that\nthe S/F structures investigated in Ref. [258] exhibit two well disting uished phases corresponding to a disordered\nphase when the sample is in the as-grown state and an ordered phas e when the sample is magnetized with an\nin-plane ﬁeld. These order-disorder transitions manifest themselv es as an enhancement of submatching features\nin the ﬁeld dependence of the critical current which cannot be expla ined from a simple rescaling of the response\ncorresponding to the disordered phase.\nRandom (disordered) magnetic inclusions\nEarly studies of the inﬂuence of ferromagnet on the superconduc ting state were performed in the sixties\nby Strongin et al.[260], Alden and Livingston [261, 262] and Koch and Love [263] for a d ispersion of ﬁne\nferromagnetic particles (Fe, Gd, Y) in a superconducting matrix. T hese reports motivated further experimental\nand theoretical investigations of the inﬂuence of randomly distribu ted particles on/underneath/inside\nsuperconducting materials, which continue nowadays (Sikora and M akiej [264, 265], Wang et al.[266],\nLyuksyutov and Naugle [267, 268, 269], Santos et al.[270], Kuroda et al.[271], Togoulev et al.[272],\nKruchinin et al.[273], Palau et al.[274, 275], Haindl et al.[276, 277], Snezhko et al.[278], Rizzo et al.\n[279], Stamopoulos et al.[280, 281, 282, 283, 284], Suleimanov et al.[285], Xing et al.[286, 287, 288]). In\nmost of these investigations no precautions were taken to electric ally isolate the magnetic particles from the\nsuperconducting material which presumably results in a substantia l core pinning due to proximity eﬀects.\nXinget al.[287] reported on controlled switching between paramagnetic †and diamagnetic Meissner eﬀect\nin S/F nanocomposites consisting of Pb ﬁlms with embedded single-dom ain Co particles. These authors argue\nthat in this particular system the paramagnetic Meissner eﬀect att ributed to the superconducting part only,\noriginates from the spontaneous formation of vortices induced by the ferromagnetic inclusions. Therefore, the\ndiﬀerent contributions of the external ﬁeld and the spontaneous vortices to the resulting magnetization of the\nsample, make it possible to manipulate the sign of Meissner eﬀect by ch anging the orientation of the magnetic\n†The paramagnetic Meissner eﬀect in various superconductin g systems is discussed in the review of Li [289].CONTENTS 39\n−100 −50 0 50 100−4−3−2−10\nHext (Oe)M (10−4 emu)Plain Co/Pt film covered by Pb film\ns=0.10\ns=0.30\ns=0.50\ns=0.70\ns=0.85\nFigure 27. (color online) Bottom parts of the magnetization curves Mvs.Hextfor a superconducting Pb ﬁlm\ncovering a Co/Pt multilayer. The curves M(Hext) corresponding to the diﬀerent values of the parameters s,\nwhich is deﬁned as the fraction of magnetic moments that are p ointing up ( m>0) relative to the total number\nof magnetic moments: s= 0.1 (open circles), s= 0.3 (diamonds), s= 0.5 (crosses), s= 0.7 (triangles) and\ns= 0.85 (ﬁlled circles), adapted from Lange et al.[256].\nmoments embedded in the supercondicting matrix ‡.\nVortex dynamics in a periodic magnetic ﬁeld\nHere we wantto brieﬂy discuss the peculiaritiesof low-frequencyvo rtexdynamics in nonuniform magnetic ﬁelds.\nMagnetictemplatesplacedinthevicinityofasuperconductingﬁlmnot onlyinduceschangesinthestaticpinning\nproperties but also in the overall dynamic response of the system. Langeet al.[206] demonstrated that the\nvortex-antivortex pairs induced by an array of out-of-plane mag netized dots lead to a strong ﬁeld polarity\ndependent vortex creep as evidenced in the current-voltage cha racteristics. This result shows that in S/F\nhybrids with perpendicular magnetized dots vortices and antivortic es experience a diﬀerent pinning strength.\nA theoretical study of the dynamic evolution of these interleaved la ttices of vortices and antivortices in the case\nof in-plane point like dipoles has been recently addressed by Carneiro [291] and Lima and de Souza Silva [292].\nA more subtle eﬀect, namely magnetic-dipole-induced voltage rectiﬁ cation was predicted by Carneiro [243].\nUnlikeconventionalratchetsystems §, in the particularcaseofmagneticratchet, induced byin-planemag netized\ndots, the motion of vortices is in the opposite direction than the mot ion of antivortices, thus giving rise to a\nﬁeld-polarity independent rectiﬁcation (Silhanek et al.[258, 259], de Souza Silva et al.[303]). This magnetic\ndipole-induced ratchet motion depends on the mutual orientation a nd strength of the local magnetic moments\nthus allowing one to control the direction of the vortex drift. In so me cases, a nonzerorectiﬁed signal is observed\neven atHext= 0 resulting from the interaction between the induced vortex-ant ivortex pairs by the magnetic\ndipoles [303]. It is worth emphasizing that in the case of in-plane magne tic dipoles treated by Carneiro [243],\nthe inversion symmetry is broken by the stray ﬁeld of the dipoles, th us giving rise to diﬀerent depinning forces\nparallel and anti-parallel to the dipoles orientations, as shown in Fig. 22.\n‡Previously, Monton et al.[290] reported on an experimental observation of the parama gnetic Meissner eﬀect in Nb/Co\nsuperlattices in ﬁeld-cooled measurements, however the or igin of this eﬀect remains unclariﬁed.\n§Early theoretical studies showed that a vortex lattice subm itted to an oscillatory excitation in the presence of a non-c entro-\nsymmetric pinning potential gives rise to a net drift ∝angb∇acketleftv∝angb∇acket∇ightof the vortex lattice which in turn generates a dc voltage sig nal\nVdc=/integraltext\n[∝angb∇acketleftv∝angb∇acket∇ight×Hext]·dlalong the direction of bias current (Zapata et al.[293], Lee et al.[294], Wambaugh et al.[295]).\nThese predictions are in agreement with recent experimenta l results obtained for purely superconducting systems (Vil legaset al.\n[296], W¨ ordenweber et al.[297], van de Vondel et al.[298], Togawa et al.[299], de Souza Silva et al.[300], Wu et al.[301],\nAladyshkin et al.[302]).CONTENTS 40\n3.4. Planar S/F bilayer hybrids\nIn this section we shall discuss the properties of continuous planar S/F structures which have macroscopically\nlarge lateral dimensions. As before, the superconducting and fer romagnetic ﬁlms are assumed to be electrically\ninsulated from each other.\nAppearance of vortices in planar S/F structures\nThe interaction of the Meissner currents and the currents induce d by vortex lines with a one-dimensional\ndistribution of the magnetization (both single domain walls, periodic do main structures and magnetic bars) in\nthe London approximation was considered by Sonin [99], Genkin et al.[304], Bespyatykh and Wasilevski [305],\nBespyatykh et al.[306], Helseth et al.[307], Laiho et al.[308], Traito et al.[309], Erdin [310], Bulaevskii\nand Chudnovsky [311, 312], Kayali and Pokrovsky [313], Burmistrov and Chtchelkatchev [314], Ainbinder and\nMaksimov [315], Maksimova et al.[316, 317]. It was found that in order to create vortex-antivorte x pairs\nin the S/F bilayer with out-of-plane magnetization at Hext= 0 (and thus keeping the total ﬂux through the\nsuperconducting ﬁlm zero) the amplitude of the magnetization Msshould overcome the following threshold\nvalue [305, 308, 309]\nM⊥\nv−av=Hc1\n4αDs\nw∝Φ0Ds\nλ2wlnλ/ξ,\nwhereαis a numerical factor of the order of unity and the period wof the domain structure is assumed\nto be ﬁxed (the hard-magnet approximation). This estimate corre sponds to the case when the width of the\ndomain walls is much smaller than other relevant length scales. The crit ical magnetization M⊥\nv−avdecreases\nmonotonically with decreasing superconducting ﬁlm thickness Ds. The equilibrium vortex pattern appearing in\nthe superconductingﬁlm at Hext= 0andMs>M⊥\nv−avconsistsofstraightvortices, arrangedin one-dimensional\nchains, with alternating vorticities corresponding to the direction o f the magnetization in the ferromagnetic\ndomains [305, 308, 309]. The parameters of such vortex conﬁgura tion with one or two vortex chains per half-\nperiod was analyzed by Erdin [310]. It was shown that in equilibrium the vortices in the neighboring domains\nare halfway shifted, while they are next to each other in the same do main. Alternatively, as the thickness Ds\nincreases, the vortex conﬁguration, consisting of vortex semi-lo ops between the ferromagnetic domains with\nopposite directions of the magnetization, becomes energetically fa vorable [308, 309] provided that Ms>M⊥\nloops,\nwhere\nM⊥\nloops=Hc1\n8ln(w/πλ)∝Φ0\nλ2lnλ/ξ\nln(w/πλ).\nThe destruction of the Meissner state in the S/F bilayer with in-plane magnetization was considered by\nBurmistrov and Chtchelkatchev [314]. Since the out-of-plane comp onent of the ﬁeld, which is responsible for\nthe generation of the vortex, is maximal near the domain wall (unlike from the previous case) and goes to zero\nin the center of magnetic domains, one can consider only a single doma in wall. At Hext= 0 a creation of a\nsingle vortex near the Bloch-type domain wall of width δcorresponds to the condition\nM/bardbl\nv≃Hc1\n4πλ\nDf×/braceleftbigg2λ/δ,\n1−32λ/(π2δ),πδ/(4λ)≪1\nπδ/(4λ)≫1.\nMagnetic pinning and guidance of vortices in planar S/F stru ctures\nIrrespective of whether the domain structure in the ferromagne tic layer is spontaneously created or was present\nbeforehand, the spatial variation of the magnetization will lead to a n eﬀective vortex pinning (Bespyatykh et\nal.[306], Bulaevskii et al.[318]). However, there are discrepancies in the estimates concern ing the pinning\neﬀectiveness. Indeed, Bulaevskii et al.[318] argued that superconductor/ferromagnet multilayers of n anoscale\nperiod can exhibit strong pinning of vortices by the magnetic domain s tructure in magnetic ﬁelds below the\ncoercive ﬁeld when the ferromagnetic layers exhibit strong perpen dicular magnetic anisotropy. The estimated\nmaximum magnetic pinning energy for a single vortex in such a system is about 100 times larger than the\ncore pinning energy produced by columnar defects. In contrast t o that, Bespyatykh et al.[306] have shown\nthat the eﬀectiveness of magnetic pinning of vortices in a layered sy stem formed by an uniaxial ferromagnet,\ndoes not considerably exceed the energy of artiﬁcial pinning by a co lumn-type defect, regardless the saturation\nmagnetization of the ferromagnet. The limitation of the pinning ener gy is caused by the interaction of externalCONTENTS 41\nvortices with the spontaneous vortex lattice formed in the superc onducting ﬁlm when the magnetization of the\nferromagnetic ﬁlm exceeds the critical value (see Eq. 23).\nTherehavebeennumerousexperimentalinvestigationscorrobor atingtheenhancementofthecriticalcurrent\nin planar S/F hybrids. It was shown that the presence of a bubble do main structure in Co/Pt ferromagnetic\nﬁlms with out-of-plane magnetization modiﬁes the vortex pinning in su perconducting Pb ﬁlms (Lange et al.\n[205, 256, 319, 320]), leading to an increase of the width of the magn etization loop M(Hext) as compared with\na uniformly magnetized S/F sample (Fig. 27). The crossover betwee n an enhanced magnetic pinning on bubble\nmagnetic domains observed at low temperatures and a suppressed magnetic pinning at temperatures close to\nTcfor a demagnetized S/F bilayer can be possibly associated with an incr ease of an eﬀective penetration length\nλ2/Ds, characterizing the vortex size, and an eﬀective averaging on the small-scale variation of the nonuniform\nmagnetic ﬁeld provided that λ2/Dsconsiderably exceed the period of the magnetic ﬁeld (Lange et al.[320]).\nInterestingly, the parameters of the bubble domain structure (t he size and the density of domains of both signs\nof magnetization) can be controlled by demagnetization similar to tha t reported in Refs. [96, 123]. A three-fold\nenhancement of the critical depinning current in Nb ﬁlms fabricated on top of ferromagnetic Co/Pt multilayers\nwas observed by Cieplak et al.[321, 322] based on magnetization measurements and on the analys is of the\nmagnetic ﬁeld distribution obtained by using a 1D array of Hall sensor s. The mentioned enhancement of the\nmagnetic pinning takes place in the ﬁnal stages of the magnetization reversal process, and it can attributed to\nresidual un-inverted dendrite-shaped magnetic domains.\nHigh-resolution magneto-optical imaging performed by Goa et al.[323] in superconducting NbSe 2single\ncrystals and ferrite-garnet ﬁlms demonstrates that the stray ﬁ eld of Bloch domain walls can be used to\nmanipulate vortices. Indeed, depending on the thickness of the sa mple, the vortices are either swept away\nor merely bent by the Bloch wall.\nVlasko-Vlasov et al.[247, 324] and Belkin et al.[325, 326] studied the anisotropic transport properties of\nsuperconducting MoGe and Pb ﬁlms and NbSe 2single crystals which are in the vicinity of a ferromagnetic\npermalloy ﬁlm. In these works a quasi-one-dimensional distribution o f magnetization can be achieved by\napplying a strong enough in-plane ﬁeld Hext>300Oe, which aligns the domain walls in a desired direction.\nSuch a domain structure was maintained even after switching oﬀ the external magnetic ﬁeld. Magneto-optical\nmeasurements directly display the preferential direction of the vo rtex entry in the presence of the perpendicular\nmagnetic ﬁeld along the domain walls [the panel (c) and (d) in Fig. 23]. By reorienting the magnetic domains\nusing combinations of dc and ac ﬁelds, it is possible to rearrange curr ent patterns in the S/F bilayer and thus\nmanipulate its conductivity. The presence of this rotatable periodic stripe-like magnetic domain structure\nwith alternating out-of-plane component of magnetization results in a diﬀerence in the critical depinning\ncurrent density between cases when the magnetic domain stripes a re oriented parallel and perpendicular to\nthe superconducting current: J/bardbl\nc> J⊥\nc. For planar thin-ﬁlm Pb/Py structures Vlasko-Vlasov et al.[324]\nobserved a pronounced magnetoresistance eﬀect yielding four or ders of magnitude resistivity change in a few\nmillitesla in-plane ﬁeld. In addition, the S/F bilayer exhibits commensura bility features that are related to the\nmatching of the Abrikosov vortex lattice and the magnetic stripe do mains (Belkin et al.[326]). The matching\neﬀects are less apparent than for S/F hybrids with magnetic dots, although commensurability becomes more\npronounced as temperature is lowered. This result can be explained by the gradual decrease in the λvalue,\nwhich leads to stronger modulation of the magnetic ﬁeld in the superc onductor at lower temperatures and\nconsequently, to more prominent magnetic interaction with ferrom agnetic domain structure.\nIt is interesting to note that the eﬀect of magnetic domains on the p inning of vortices, was also observed\nin high−Tcsuperconductors such as YBa 2Cu3O7−δ(Garc´ ıa–Santiago et al.[327], Jan et al.[328], Zhang et\nal.[329], Laviano et al.[330]). At the same time the inﬂuence of the ferromagnet on the nuc leation in the\nhigh−Tcsuperconductors should be rather small due to extremely small co herence length (of the order of few\nnanometers).\nCurrent compensation eﬀect and ﬁeld-polarity dependent cr itical current\nA superconducting square with in-plane magnetized ferromagnet o n top was proposed by Miloˇ sevi´ c\net al.[331] as a potential ﬁeld and current compensator, allowing to impro ve the critical parameters of\nsuperconductors. Indeed, such a magnet generates stray ﬁeld s of the same amplitude but opposite signs at\nthe poles of the magnet, therefore the ﬁeld-compensation eﬀect leads to the enhancement of the upper criticalCONTENTS 42\n0 100 200 300 400 5000246\nI (mA) −6 0 6 V (V)0.3\n0.0\n−0.3Hext=500 Oe\nIc+Ic−Ic+\nIc−\nF\nS\nHextIxM\nHext (Oe)Ic (mA)Co bar on top of Nb stripe\nFigure 28. (color online) Diode eﬀect in the Nb bridge (2 µm width) with the in-plane magnetized Co stripe\non top: Experimental dependence of I+\nc(the critical current in the x−direction, see the geometry of the S/F\nsystem on the inset) and I−\nc(the critical current in the opposite direction) on the exte rnal magnetic ﬁeld Hext\napplied in the y−direction (T= 4.2 K,Tc0= 9.2 K), adapted from Vodolazov et al.[333]. In the inset the dc\nI−Vcharacteristic of our hybrid system Hext= 500 Oe is presented, showing a pronounced diode eﬀect.\nﬁeld equally for both polarities of the external ﬁeld. The supercond ucting state was shown to resist much higher\napplied magnetic ﬁelds for both perpendicular polarities. In addition, such ferromagnet induces two opposite\nscreening currents inside the superconducting ﬁlm plane (in the per pendicular direction to its magnetization),\nwhich eﬀectively compensates the bias current, and therefore su perconductivity should persist up to higher\napplied currents and ﬁelds. These eﬀects have been recently stud ied experimentally by Schildermans et al.\n[332] in an Al/Py hybrid disk of 1.7 µm diameter where a ﬁnite dipolar moment lying in the plane of the\nstructure was achieved by pinning magnetic domains with the contac t leads used for electrical measurements.\nVodolazov et al.[333] and Touitou et al.[334] considered an alternative experimental realization of the\ncurrent compensator, consisting of a superconducting bridge an d a ferromagnetic bar magnetized in-plane and\nperpendicularly to the direction of the bias current. Such geometr y allows one to weaken the self-ﬁeld of\nthe superconducting bridge near its edge and thus to enhance the total critical current corresponding to the\ndissipation-free current ﬂow. Since the self-ﬁeld compensation oc curs only for a certain direction of the current\n(for ﬁxed magnetization), the presence of magnetized coating lea ds to a diode eﬀect – the current-voltage\nI−Vdependence becomes asymmetrical (Fig. 28). Later the similar diﬀe rence in critical currents ﬂowing in\nopposite directions was studied experimentally by Morelle and Moshch alkov [335] for a system consisting of a\nsuperconducting Al strip placed close to a perpendicularly magnetiz ed Co/Pd rectangle and Vodolazov et al.\n[336] for Nb/Co bilayer in the presence of titled external magnetic ﬁ eld.\n3.5. Stray ﬁeld-induced Josephson junctions\nJosephson junctions consist of weak links between two supercond ucting reservoirs of paired electrons.\nCommonly, these junctions are predeﬁned static tunnel barriers that, once constructed, can no longer be\nmodiﬁed/tuned. In contrast to that, a new concept of the Josep hson junctions with a weak link generated\nby the local depletion of the superconducting condensate by a “ma gnetic barrier” from a micro/nano-patterned\nferromagnet can be realized (Sonin [98]). Interestingly, this type of devices oﬀer an unprecedented degree of\nﬂexibility as it can be readily switched ON/OFF by simply changing betwee n diﬀerent magnetic states using\nan in-plane ﬁeld. This switching process is fully reversible, and non-vo latile since does not require energy to\nkeep one of the magnetic states.\nA pioneer investigation of the properties of superconducting weak links achieved by local intense magnetic\nﬁelds was performed by Dolan and Lukens [337]. The sample layout use d by these authors and their typicalCONTENTS 43\nAl\nPb Pb(a)\nIHext\nPy(b)\nIUnquenched□□state\nPb,□Sng=0.3-0.4 m /c109\nPyIM Pb,□SnV Vw=3-14 m /c109\nw=1-5 m /c109M Areas□with□highest\nmagnetic□field\nWeak□□links\n(c)\nPyIM Pbw=5 m /c109 Hext\nFigure 29. (color online) (a) Sample layout investigated by Dolan and L ukens [337]: An uniform Al bridge\nwas covered with a superconducting Pb strips (dashed rectan gles) everywhere but in a small region near the\ncenter of the bridge. Due to the ﬂux expulsion from the Pb stri ps the local magnetic ﬁeld is primarily conﬁned\nto this gap.\n(b) Sample conﬁguration investigated by Clinton and Johnso n [338, 339]: a Pb (or Sn) transport bridge is\npartially covered with a ferromagnetic Py strip with in-pla ne magnetic moment M. WhenMis parallel to the\nbridge a strong stray ﬁeld depletes the superconducting ord er parameter in a small region near the border of\nthe Py bar (quenched state) thus inducing a weak link.\n(c) Schematic presentation of magnetoquenched supercondu cting quantum interferometer, consisting of two\nsuperconducting Pb bridges connected in parallel and perma lloy ﬁlm on top, after Eom and Johnson [342].\ndimensions is schematically shown in Fig. 29 (a) and it consists of an Al b ridge locally covered by a plain Pb\nﬁlm which has a thin gap of width gand spans the width of the Al strip at its center. By applying an exte rnal\nmagnetic ﬁeld, the Pb ﬁlm screens the magnetic ﬁeld due to the Meissn er eﬀect in the whole Al bridge but\nmagnify its intensity at the gap position. This eﬀect leads to a local re gion of suppressed superconductivity\nwhich givesrise to dc and ac Josephsoneﬀect asevidenced by a ﬁnite criticalcurrent and the presenceof Shapiro\nsteps in the current-voltagecharacteristicsat Vn=n¯hω/(2e) when the system was irradiated with rf-excitations\nwith frequency ω= 2πf,nis integer. Interestingly, the Josephson-like features appear fo r applied ﬁelds in the\nshield gap approximately equal to the upper critical ﬁeld of the Al ﬁlm .\nAn alternative method to obtain a ﬁeld-induced weak link has been mor e recently introduced by Clinton\nand Johnson [338, 339, 340, 341]. The basic device consist of a bilaye r of a thin superconducting strip and a\nferromagnetic layer with in-plane magnetic moment overlapping the w idth of the bridge [see panel (b) in Figure\n29]. When the magnetic moment is parallel to the superconducting br idge the dipolar fringe is strong enough to\nlocally suppress the superconducting order parameter across th e bridge (quenched state) and thus create a weak\nlink. This eﬀect can be turned oﬀ by simply magnetizing the ferromagn etic layer perpendicular to the transport\nbridge with an external in-plane dc ﬁeld or by a current pulse in a sepa rate transport line [341]. Clearly, the\nproposed switchable Josephson junction seems to be very attrac tive for potential technological applications,\nsince energy is required only to change the magnetic states, which a re thereafter maintained in thermodynamic\nequilibrium. Later on based on the same idea Eom and Johnson [342] pr oposed a switchable superconducting\nquantum interferometer consisting of a ferromagnetic Py ﬁlm part ly covering two parallel superconducting Pb\nbridges fabricated in a loop geometry. The dependence of the volta geV, induced on this superconducting loop\nat injection of stationary bias current, on the perpendicular magn etic ﬁeldHextis shown in Fig. 30(b) and it\nreminds the standard Fraunhofer diﬀraction pattern (Barone an d Paterno [34]).CONTENTS 44\n0 1 2 3 4036912\n∆V= ¯hω/(2e)Py strip on top of Pb bridgeV, µV\nI, µA(a)\nac Josephson effect\nrf off\n2 4 6 82468\n∆H= Φ0/SloopPy film on top of Pb loopV, µV\nHext, Oe(b)\ndc Josephson effect\nFigure 30. (color online) (a) The I−Vcurves obtained for a plain superconducting Pb bridge (2 µm wide)\nsubjected in the inhomogeneous magnetic ﬁeld, quenched sta te [see the panel (b) in Fig. 29] for diﬀerent\nintensities of rf-irradiation, adapted from Clinton and Jo hnson [339]. The experiment was carried out at T= 5\nK,T/Tc0≃0.76,Hext= 0, frequency fof the radio signal equal to 0.75 GHz.\n(b) TheI−Vdependence obtained for a superconducting Pb loop of the wid th 4.5µm with rectangular hole\n1.5×7.0µm2covered by permalloy ﬁlm [see the panel (c) in Fig. 29], adapt ed from Eom and Johnson [342]. This\ncurve demonstrates the short period oscillations with the p eriod determined by the area of the superconducting\nloopSloop.CONTENTS 45\n4. Hybrid structures: superconductor – soft magnets\nThus far, we have discussed the inﬂuence that a ferromagnet has on the superconducting properties of S/F\nhybrids, assuming that the magnetization of the ferromagnet Mremains practically unaltered. In this last\nsection, weconsiderthe possibilitythat the magnetization Mcanbe changedeitherdueto the externalmagnetic\nﬁeld or by the superconducting screening currents induced by the magnetic subsystem, which are particularly\nrelevant at low temperatures. This situation could, in principle, be ac hieved by using paramagnetic materials\nor soft ferromagnetic materials with a low coercive ﬁeld.\nThe equilibrium propertiesof“superconductor– softmagnet”hyb ridstructures(so-calledsoft S/Fhybrids)\ncan be obtained phenomenologically by the minimization of the Ginzburg -Landau energy functional Eq. (1) or\nthe London energy functional Eq. (17), in which the term Gmresponsible for the self-energy of the ferromagnet\nbecomes important\nGm=1\n2M2s/integraldisplay\nVf/parenleftBig\nJ/bardbl|∇Mx|2+J/bardbl|∇My|2+J⊥|∇Mz|2/parenrightBig\ndV−/integraldisplay\nVf2πQM2\nzdV, (23)\nwhereJ/bardblandJ⊥characterize the exchange interaction between spins in a uniaxial f erromagnet with respect to\nthe in-plane and out-of-plane direction, Qis a quality factor taking into account the internal anisotropy of th e\nferromagnet and determining the preferable orientation of the ma gnetization (either in-plane or out-of-plane).\nEquation (23) describes the energy cost for having a slowly varying spatial distribution of the magnetization †\nand, in particular, it describes the energy of a domain wall in a ferrom agnet. In some cases (for instance, for\nrapidMvariations typical for ferromagnets with domain walls of rather sma ll width), in order to simplify the\nproblem, the increase of the free energy given by Eq. (23) can be t aken into account phenomenologically by\nsubstituting Gmby a ﬁxed term Gdwrepresenting the energy of a domain wall.\nModiﬁcation of the domain structure in a ferromagnetic ﬁlm b y the superconducting screening currents\nThe inﬂuence of superconducting environment (both substrate o r coating) on the equilibrium width of magnetic\ndomains in ferromagnetic ﬁlms was considered theoretically by Sonin [9 9], Genkin et al.[304], Sadreev [344],\nBespyatykh et al.[345, 346], Stankiewicz et al.[347, 348], Bulaevskii and Chudnovsky [311, 312], Daumens\nand Ezzahri [349]. In particular, one can expect a prominent chang e in the equilibrium period of a one-\ndimensional domain structure at Hext= 0 for rather thick ferromagnetic ﬁlms ( Df≫w) with out-of-plane\nmagnetization. Indeed, the Meissner currents, induced by the fe rromagnet, will decrease the magnetic ﬁeld\ninside the superconductor (usual ﬂux expulsion eﬀects) and signiﬁ cantly increase the magnetic ﬁeld inside the\nferromagnet. As a consequence, the density of the free energy of the ferromagnet, given by B2/8π−B·Mor,\nequivalently, by H2/8π−2πM2\nz, raisesfor a given Mzdistribution. However, the total energy of the S/F system\ncan be lowered by a decrease of the period of the ferromagnetic do mains: the smaller the period, the faster the\ndecay of Haway from the surfaces of the ferromagnetic ﬁlm. Thus, it is expec ted that the equilibrium width of\nmagnetic domains in planar S/F bilayer becomes smaller below the critica l temperature of the superconducting\ntransition as compared with the state T >Tc0. In contrast to that, for thin ferromagnetic ﬁlms ( Df≪w) the\nopposite behavior is predicted: the domain width in the free ferroma gnetic ﬁlm should be smaller than that for\nthe same ﬁlm on top of a superconducting substrate [348]. This can b e understood by taking into account the\nchange of the far-zone demagnetizing ﬁeld characteristics. In ad dition, Stankiewicz et al.[348] argued that the\neﬀect of the superconducting substrate on the period of domain s tructure in ferromagnetic ﬁlms with in-plane\nmagnetization is rather small as compared with that for the out-of -plane magnetized ferromagnets. However,\nan increase of the magnetostatic energy of the S/F hybrids at T <Tc0due to the Meissner currents results also\nin a shrinkage of the equilibrium width of an isolated 180◦Bloch wall, separating two ferromagnetic domains\nwith in-plane magnetization, in the vicinity of the superconducting su bstrate, as was predicted by Helseth et\nal.[307].\nThe foreseen decrease of the period of the domain structure in a f erromagnetic garnet ﬁlm in contact with\na superconducting Pb ﬁlm was recently investigated by magneto-op tical imaging (Tamegai et al.[350]). It was\ndemonstrated that the shrinkage depends both on temperature and the thickness of superconducting coating\n†The theory of superconductor – soft ferromagnet systems nea r the “ferromagnet–paramagnet” transition has been consid ered by\nLiet al.[343] within Ginzburg-Landau formalism.CONTENTS 46\n5 5.5 6 6.500.20.40.60.81Pb film on top magnetic garnet film\nT (K)Shrinkage(a)\nFigure 31. (color online) (a) The temperature dependence of the shrink age ratio of the width of magnetic\ndomains in a hybrid structure consiting of a Pb ﬁlm ( Ds= 300 nm) on top of a magnetic garnet ﬁlm in\ncomparison with the same ferromagnetic ﬁlm without superco nducting coating, adapted from Tamegai et al.\n[350].\n(b) The magneto-optical image of four segments of the superc onducting Pb/garnet ﬁlm structure, diﬀering by\nthe thickness of the Pb ﬁlm ( Ds= 0 nm, 100 nm, 200 nm, 300 nm in a counter-clock-wise directio n), taken at\nT= 5.0 K, adapted from Tamegai et al.[350].\nlayer. The temperature dependence of the shrinkage factor sevaluated by comparing the average width ∝an}b∇acketle{tw∝an}b∇acket∇i}htof\nthe magnetic domain width in regions with and without the supercondu cting Pb ﬁlm is shown in Fig. 31(a). It\npoints out that the lower the temperature, the narrower the mag netic domains are ( s= 0.47 atT= 5.0 K).\nAlteration of magnetization of ferromagnetic dots by the su perconducting screening currents\nThe Meissner currents also inﬂuence the magnetic states and the p rocess of magnetization reversal in\nferromagnetic disks placed above a superconductor. It is well kno wn that a uniformly magnetized (single\ndomain) state is energetically favorable for radii Rfsmaller than some threshold value R∗\nf(for a given thickness\nof the dotDf), while the magnetic vortex state can be realized for Rf>R∗\nf. The typical M(Hext) dependence\nfor ferromagnetic disks for Rf> R∗\nfwas already shown in Fig. 25. The dependence R∗\nfvs.Df(the phase\ndiagram in the “diameter–height” plane) in the presence of a bulk sup erconductor, characterized by the London\npenetration depth λ, was investigated numerically by Fraerman et al.[351] and later analytically by Pokrovsky\net al.[352]. It was shown that the smaller λ, the smaller the critical diameter R∗\nfbecomes for a given dot’s\nthickness. The transitions between the two magnetic states can b e induced also by increasing the external\nmagnetic ﬁeld: the magnetic vortex, possessing an excess energy at zero ﬁeld, becomes energetically favorable\nfor ﬁnite external ﬁelds (the magnetic-vortex nucleation ﬁeld). A lthough the energy of the interaction between\nthe superconductor and ferromagnet is expected to be much sma ller than the self-energy of the ferromagnetic\nparticle (for realistic λvalues), it could lead to an experimentally observable decrease in the magnetic vortex\nnucleation ﬁeld H/bardbl\nnucland increase in the magnetic vortex annihilation ﬁeld H/bardbl\nann(Fig. 32).\nThe appearance of a spontaneous magnetization of individual S/F h ybrids, consisting of an Al bridge and\ndemagnetizedNidotsontop, uponcoolingthroughthesupercond uctingtransitiontemperatureat Hext= 0, was\nreported by Dubonos et al.[353]. Indeed, the reshuﬄing of magnetic domains in the submicron fe rromagnetic\ndisk, caused by temperature-dependent screeningofthe domain ’sstrayﬁelds by the superconductor, can explain\nthe observed appearance of nonzero magnetization of the ferro magnet at low temperatures. More recently, the\nmodiﬁcation of the magnetic state of Nb/Co and Nb/Py superlattice s induced by screening currents in the\nsuperconducting Nb ﬁlms was studied experimentally by Monton et al.[290, 354, 355] and Wu et al.[356].\nKruchinin et al.[273] demonstrated theoretically that a superconducting environ ment modiﬁes the\nmagnetostatic interaction between localized magnetic moments (em bedded small ferromagnetic particles),\nresulting either in parallel or antiparallel alignment of neighbor dipolar moments at Hext= 0. The crossoverCONTENTS 47\nbetween these regimes depends on the ratio of the interparticle sp acing and the London penetration depth,\nand thus preferable “magnetic” ordering (ferromagnetic vs. ant iferromagnetic arrangements) can be tuned by\nvarying temperature.\nMixed state of soft S/F hybrid structures\nThe magnetostatic interaction between a vortex-free supercon ducting ﬁlm and a uniformly magnetized\nferromagnetic ﬁlm at Hext= 0 may cause the spontaneous formation of vortices in the superc onductor\nand magnetic domains in the ferromagnet in the ground state of plan ar S/F bilayers with perpendicular\nmagnetization. Lyuksyutov and Pokrovsky [357], and Erdin et al.[358] argued that the ground state of\nthe S/F system could be unstable with respect to the formation of s uperconducting vortices. Indeed, for a\nuniformly magnetized S/F bilayer, characterized by a magnetization of the ferromagnetic ﬁlm per unit area\nm=MsDf, the magnetostatic interaction between the superconductor an d the ferromagnet changes the total\nenergy of an isolated vortex line to εv=ε(0)\nv−mΦ0[146] as compared with the self-energy of the vortex in the\nsuperconducting ﬁlm ε(0)\nvwithout a ferromagnetic layer. As a consequence, the formation o f vortices becomes\nenergetically favorable as soon as εv<0 (either for rather large mvalues or at temperatures close to Tc0where\nε(0)\nvvanishes). However, as the lateral size of the S/F system increas es, the averaged vortex density nvwould\ngenerate a constant magnetic ﬁeld Bz≃nvΦ0along thez−direction which can lead to an energy increase\nlarger than the gain in energy due to creation of vortices. Hence, in order for the vortex phase to survive,\nthe ferromagnetic ﬁlm should split in domains with alternating magnetiz ation in a ﬁnite temperature range at\nT < Tc0. As long as the magnetic domain width exceeds the eﬀective penetra tion depth, the energy of the\nstripe domain structure seems to be minimal (Fig. 33). Interesting ly, the interaction between a single vortex\nin a superconducting ﬁlm and the magnetization induced by this vorte x in the adjacent ferromagnetic ﬁlm can\ncross over from attractive to repulsive at short distances (Helse th [171]).\nCarneiro studied the interaction between superconducting vortic es and a superparamagnetic particle with\nconstant dipolar moment, which is assumed to be able to freely rotat e, in the London model [162, 163, 164].\nIt was found that, due to the rotational degree of freedom, the pinning potential for superconducting vortices\ndiﬀers signiﬁcantlyfromthat forapermanent dipole. In particular, the interactionbetween the superconducting\n010020030004008001200\nH||ann (Oe)\n2Rf(a)\nλ=∞λ=50 nm\n0100200300−400−2000200400 H||nucl (Oe)\n2Rf(b)\nλ=∞\nλ=50 nm\n−1200 −600 0 600 1200−1−0.500.51\nλ=∞\nλ=50 nm\nH||\next (Oe)single domain state\nmagnetic\nvortex state\nsingle domain state(c)M||/Ms\nFigure 32. (color online) (a) The dependence of the magnetic vortex ann ihilation ﬁeld H/bardbl\nann, corresponding to\nthe transition from the magnetic vortex state to a single-do main state, on the diameter of disk 2 Rf, calculated\nfor an isolated ferromagnetic disk of 20 nm thickness (i.e, w ithout superconductor, λ=∞, open circles) and\nfor the same disk placed above a bulk superconductor ( λ= 50 nm, ﬁlled circles), adapted from Fraerman et al.\n[351].\n(b) The dependence of the magnetic vortex nucleation ﬁeld H/bardbl\nnucl, corresponding to the transition from single-\ndomain state to the magnetic vortex state for the same proble m, adapted from Fraerman et al.[351].\n(c) The magnetization curve M/bardbl/Msvs. in-plane external ﬁeld H/bardbl\nextdemonstrating the process of the\nmagnetization reversal for the magnetic disk (20 nm thickne ss and 100 nm diameter) for λ=∞(open circles)\nandλ= 50 nm (ﬁlled circles), adapted from Fraerman et al.[351]. Thus, the screening eﬀect increases the\nwidth of the Hextinterval at the ascending and descending branches of the mag netization curve where magnetic\nvortex state is energetically favorable.CONTENTS 48\nvortexand the magnetic dipole can be tuned by applying an in-plane ex ternalﬁeld: the correspondingdepinning\ncritical current was shown to be anisotropic and its amplitude poten tially varies by as much as one order of\nmagnitude. Later on, this approach was generalized by Carneiro [2 55] for hybrid systems consisting of thin\nsuperconducting ﬁlm and a soft ferromagnetic disks in the magnetic vortex state (similar to that for Refs.\n[351, 352] but considering a vortex line inside the superconducting s ample).\nA new method of pinning vortices in S/F epitaxial composite hybrids co nsisting of randomly distributed\nGd particles incorporated in a Nb matrix was reported by Palau et al.[275, 274]. Since the size of Gd particles\nare much smaller than the coherence length and the interparticle dis tance is much shorter than the penetration\ndepth, this regime of collective magnetic pinning diﬀers both from con ventional core and magnetic pinning\nmechanisms. In this case, since a vortex “feels” a homogeneous su perconductor (for length scales on the order\nofλ), pinning eﬀects are expected to be small. However, due to the loca l ﬁeld of a vortex, the Gd particles\ncan be magnetized and a moving vortex would lead to hysteretic losse s in the magnetic particles, which in turn\nresults in an increased pinning (for decreasing magnetic ﬁelds).\nSuperconductor – paramagnet hybrid structures\nAn alternative way of modifying the superconducting properties of soft hybrid structures is by using\nparamagnetic constituents, characterized by zero or very low re manent magnetization. Such superconductor-\nparamagnet hybrids with a magnetization M= (µ−1)H/4πdepending on the external ﬁeld ( µis the magnetic\npermeability) in the presence of transport current were consider ed theoretically by Genenko [359], Genenko\nand Snezhko [360], Genenko et al.[361, 362, 363] for µ≫1. It was predicted that the paramagnetic material\nplaced near superconducting stripes and slabs can drastically modif y the current distribution in such hybrids,\nthus, suppressing the current enhancement near the supercon ducting sample’s edges inherent for any thin-ﬁlm\nsuperconductor in the ﬂux-free current-carrying state. As a c onsequence, the current redistribution leads to an\nincrease of the threshold value of the total bias current corresp onding to the destruction of the Meissner state.\nIn other words, the magnetically shielded superconductors even in the Meissner state are able to carry without\ndissipation rather high transport current comparable with the typ ical current values for a regime of strong\nﬂux pinning [359, 361, 362, 363]. A survival of the Meissner state fo r thin-ﬁlm superconducting rings carrying\na current and placed between two coaxial cylindrical soft magnets was studied by Genenko et al.[364, 365]\nand Yampolskii et al.[366]. The similar problem concerning with the distribution of magnetic ﬁ eld inside and\noutsideasuperconductingﬁlamentsheathedbyamagneticlayer,a swellasthe magnetizationofsuchastructure\nin the region of reversible magnetic behavior in the Meissner state wa s considered by Genenko et al.[367]. The\nformation of the mixed state in various superconductor/paramag net structures in the presence of transport\nFS(a)T <T<Tc0 Curie\nFS(b)T<T <Tc0 Curie js\nFigure 33. (color online) (a) Uniform state of the S/F bilayer above the superconducting critical temperature,\n(b) Spontaneously formed magnetic domain structure and cou pled chains of superconducting vortices with\nalternating vorticity at T <Tc0, adapted from Erdin et al.[358]. Solid arrows correspond to the magnetization\nof the ferromagnet, while black arrows schematically show t he circulating superconducting currents.CONTENTS 49\ncurrent, or an external magnetic ﬁeld Hextor the ﬁeld of hard magnetic dipoles, were analyzed by Genenko\net al. [368], Genenko and Rauh [369], Yampolskii and Genenko [370], and Yampolskii et al.[371, 372, 373]\nin the framework of the London model. The Bean-Livingston barrier against the vortex entry in shielded\nsuperconducting ﬁlaments was shown to strongly depend on the pa rameters of the paramagnetic coating and,\nas a result, the critical ﬁeld at which the ﬁrst vortex enters can be enhanced [368].\nSince for the superconductor/paramegnet hybrids there are no changes neither in the vortex structure\nin the Meissner state of superconductor or in the magnetic state o f paramagnetic elements, characteristics\nof superconductor–paramagnet hybrids are presumed to be rev ersible, and ac losses should be minimal for\nsuch structures. It stimulated the implementation of paramagnet ic and ferromagnetic coatings in high- Tc\nsuperconductors in order to improve the critical current and red uce the ac-losses (Majoros et al.[374], Glowacki\net al.[375], Horvat et al.[376, 377], Duckworth [378], Kovac et al.[379], Touitou et al.[334], Pan et al.[380],\nJoosset al.[381], Gu et al.[382], G¨ om¨ ory et al.[383, 384]).\nAlthough, strictly speaking, permalloy is a ferromagnet with rather low-coercive ﬁeld, it can behaves\nqualitatively similar to paramagnetic materials. Indeed, the magnetiz ation vector for thin-ﬁlm structures\ndeviates from in-plane orientationif a perpendicular externalﬁeld is applied. Such rotation ofthe magnetization\nof the dot toward the out-of-plane direction while sweeping the ext ernal ﬁeld gives rise to a z−component of\nmagnetization depending on the external ﬁeld. The eﬀect of the st ray ﬁeld generated by soft permalloy dots on\nthecriticalcurrent IcofthesuperconductingAlloopswasexperimentallystudiedbyGolub ovi´ candMoshchalkov\n[385]. The monotonous decrease in the period of the oscillation on the Ic(Hext) with increasing the Hextvalue\nwas interpreted as a ﬂux enhancement due to the increase of the o ut-of-plane component of the dot’s magnetic\nmoment. In this sense soft magnetic materials are promising candida tes for the design of a linear magnetic ﬂux\nampliﬁer for applications in superconducting quantum interference devices.CONTENTS 50\n5. Conclusion\nWe would like to conclude this work by formulating what, we believe, are the most exciting unsolved issues and\npossible interesting directions for further studies of S/F hybrid sy stems.\nSpontaneous formation of a vortex lattice and domain patter ns.As we discussed in Section 4, the\nmagnetostatic interaction between a vortex-free superconduc ting ﬁlm and a uniformly and perpendicularly\nmagnetized ferromagnetic ﬁlm at zero external ﬁeld, may lead to a s pontaneous formation of vortex-antivortex\npairs in the superconductor accompanied by alternating magnetic d omains in the ferromagnet in the ground\nstate (see Fig. 33 and Refs. [33, 358]). To the best of our knowledg e, thus far there are no experimental results\nconﬁrming this prediction. In part, this is likely due to the diﬃculties of combining a very low coercive ﬁeld\nferromagnetic materials, needed to guarantee the free accommo dation of magnetic domains, and a well deﬁned\nout-of-plane magnetic moment.\nMagnetic pinning. Based on London equations we have clearly deﬁned the magnetic pinn ing energy as\nm0·Bv(see section 3). Since London equations are valid as long as core con tributions are negligible, in\nprinciple this simple relationship holds for materials with κ=λ/ξ≫1 and low temperatures. Unfortunately\nin the vast majority of the experimental reports so far, it seems t hat these conditions are not satisﬁed. Any\nother contributions such as local suppression of Tc, proximity eﬀect, or local changes in the mean free path,\nwhich are not accounted for within London approximation, could lead to a deviation from the treatment in the\nframework of London model. The problem that remains unsolved so f ar is the identiﬁcation of the most relevant\nmechanisms of vortex pinning in S/F hybrid systems.\nThermodynamic properties of S/F heterostructures. Although the electric transport in superconduc-\ntor/ferromagnet hybrid systems has been intensively studied dur ing the past decades, very little is known\nabout their thermodynamic and thermal properties such as their e ntropy, speciﬁc heat, thermal conductivity,\netc. From an academic point of view it would be very relevant to invest igate the nature of the phase transitions,\nor present thermodynamic evidence of conﬁnement of the superc onducting order parameter. On the other hand,\nestimating the heat released when the system changes its state mig ht also provide useful information for devices\nbased on S/F heterostructures.\nElectromagnet-superconductor hybrids. Most of the research performed so far has been focussed on the\neﬀects of an inhomogeneous ﬁeld generated by a ferromagnetic lay er onto a superconducting ﬁlm. As was early\ndemonstrated by Pannetier et al.[97] in principle there is no diﬀerence whether this inhomogeneous ﬁeld is\nthe stray ﬁeld emanating from a permanent magnet or the magnetic ﬁeld generated by micro (nano) patterned\ncurrent carrying wire on top of the surface of the ﬁlm. The great a dvantage of the latter is the degree of\nﬂexibility and control in the design of the magnetic landscape and the external tuneability of its intensity. Such\nelectromagnet-superconductor hybrids represent a promising a lternative for further exploring the basic physics\nbehind S/F hybrid.\nTime resolved vortex creation and annihilation. Josephsonπ-junctions giverisetospontaneousformationof\nhalf-integer ﬂux quanta, so called semiﬂuxons (Hilgenkamp et al.[386]). It has been theoretically demonstrated\nthatforlongJosephsonjunctionswithzigzag π-discontinuitycornersthegroundstatecorrespondstoaﬂatph ase\nstate for short separation between corners whereas an array o f semiﬂuxons is expected for larger separations.\nInterestingly, by applying an external bias current it is possible to f orce the hopping of these semiﬂuxons\nbetween neighboring discontinuities (Goldobin et al.[387]. This hopping of semiﬂuxons could be identiﬁed\nthrough time resolve ac measurements with drive amplitudes above t he depinning current. There are clear\nsimilarities between these arrays of semiﬂuxons in zigzag Josephson systems and the vortex-antivortex arrays\nin S/F systems. Indeed, recently Lima and de Souza Silva [292] have s hown theoretically that the dynamics of\nthe vortex-antivortex matter is characterized by a series of cre ation and annihilation events which should be\nreﬂected in the time dependence of the electrical ﬁeld. Experiment al work corroborating these predictions are\nrelevant for understanding the dynamics of ﬂux annihilation and cre ation in S/F systems.\nAll in all, the strive to comprehend the ultimate mechanisms ruling the in teraction between ferromagnets\nand superconductors has made this particular topic an active theo retical and experimental line of research.\nWe believe that these vigorous eﬀorts will inspire further developme nts in this area of solid state physics andCONTENTS 51\nperhaps motivate new applications of technological relevance.CONTENTS 52\nSummary of experimental and theoretical researches\nCo/Pt BaFe12O19 Co, Fe, Ni Fe/Ni (Py) Other\nCo/Pd PbFe12O19 ferromagnets\nS/F hybrids consisting of large-area superconducting and p lain non-patterned\nferromagnetic ﬁlms (single crystals) with domain structur e\nPb ﬁlms [205, 256, 319, 320] [103] [324] [350]\n[388]\nNb ﬁlms [105, 106, 110, 321] [101, 102] [290, 354, 355, 389] [108, 356] [276, 390, 391]\n[322, 392] [104] [393, 394, 395, 396]\nAl ﬁlms [96, 107]\nOther low −Tc [397, 398] [271, 376, 377, 380] [100, 247] [109, 323]\nﬁlms [325, 326]\nHigh−Tcﬁlms [328, 334] [327] [374, 375, 379, 382] [399] [329, 330, 378]\n[383, 384, 399] [381, 400, 401]\n[402, 403]\nS/F hybrids consisting of large-area superconducting ﬁlm a nd ferromagnetic elements:\nsingle particles, periodic arrays of magnetic dots (antido ts) and stripes\nPb ﬁlms [121, 198, 199, 200] [196, 197, 198, 199]\n[201, 203, 204, 205] [200, 201, 202, 205]\n[206, 232, 234, 235] [237, 238, 257, 286]\n[236, 246, 404, 405] [287, 288, 303, 406]\nNb ﬁlms [280, 281, 282, 284] [184, 185, 186, 187] [97, 254] [119, 120, 263]\n[283] [188, 189, 190, 191] [407, 408] [274, 275, 276]\n[192, 193, 194, 195] [277, 409, 410]\n[240, 245, 248, 249]\n[250, 296, 407, 411]\n[412, 413, 414, 415]\n[416, 417]\nAl ﬁlms [96, 107, 123, 124] [215, 253, 303] [244, 258, 259] [278]\n[125, 418, 419]\nOther low −Tc [397] [211, 260, 261, 262]\nﬁlms [272, 279]\nHigh−Tcﬁlms [285, 420] [155, 156, 158]\n[174, 175, 176]\nLaterally conﬁned and mesoscopic S/F hybrids\nPb ﬁlms [340, 421] [339, 340, 341]\n[342]\nNb ﬁlms [333, 336] [341]\nAl ﬁlms [133, 134, 135, 136] [353] [332, 385, 422] [337]\n[137, 140, 335, 405]\n[423, 424]\nOther low −Tc [272, 367] [338]\nﬁlms\nTable 1. Summary of experimental research on vortex matter in the S/F hybrids with dominant orbital\ninteraction (suppressed proximity eﬀect).CONTENTS 53\nGinzburg–Landau London Newton–like\nformalism formalism description of\nvortex dynamics\nBilayered and multilayered [50, 93, 94, 95] [98, 99, 146, 173, 270, 304]\nlarge-area S/F structures [96, 105, 106, 343] [305, 306, 307, 308, 309, 310]\n(superconducting ferromagnets) [425] [311, 312, 313, 314, 318, 344]\n[345, 346, 347, 348, 349, 357]\n[358, 426, 427, 428, 429, 430]\n[431, 432, 433, 434, 435, 436]\n[437]\nIndividual F elements [51, 54, 63, 76] [146, 147, 148, 149, 150, 153]\nover large-area S ﬁlms [127, 128, 129] [154, 155, 156, 159, 160, 161]\n(inside bulk superconductors) [162, 163, 164, 165, 166, 167]\n[168, 169, 170, 171, 172, 173]\n[255, 273, 278, 291, 351, 352]\n[431]\nArrays of F dots elements [112, 113, 114, 115] [187, 225, 226, 227, 228, 229] [231, 243, 291]\nover large-area S ﬁlms [116, 117, 118, 122] [230, 243, 420, 404, 431] [292, 303, 404]\n(inside bulk superconductors) [124, 142, 143, 182] [416, 417, 438]\n[183, 419]\nLaterally conﬁned and [40, 54, 126, 130] [151, 152, 267, 268, 269, 315]\nmesoscopic S/F structures [131, 132, 134, 135] [316, 317, 333, 439]\n[137, 138, 139, 140]\n[144, 145, 331, 423]\nHybrid structures with [359, 360, 361, 362, 363, 364]\nparamagnetic elements [365, 366, 367, 368, 369, 370]\n[371, 372, 373]\nTable 2. Summary of theoretical research on vortex matter in the S/F h ybrids with dominant orbitalinteraction\n(suppressed proximity eﬀect)CONTENTS 54\nAcknowledgements\nThe authors are grateful to G. Carneiro, M.M. Doria, A.A. Fraerma n, A.S. Mel’nikov, M.V. Miloˇ sevi´ c, A.V.\nD.A. Ryzhov, Samokhvalov, M.A. Silaev, T. Tamegai, J.E. Villegas, V.K. V lasko-Vlasov, D.Yu. Vodolazov, J.\nvan de Vondel for the valuable comments and remarks which certain ly improved the quality of this review.\nThis work was supported by the K.U. Leuven Research Fund GOA/20 04/02 program, NES–ESF program,\nthe Belgian IAP, the Fund for Scientiﬁc Research – Flanders (F.W.O.–V laanderen), the Russian fund for Basic\nResearch, by Russian Academy of Sciences under the program “Qu antum physics of condensed matter” and\nthe Presidential grant MK-4880.2008.2 (A.Yu.A.). A.V.S. and W.G. are g rateful for the support from the\nF.W.O.–Vlaanderen.\n[1] J. Bardeen, L.N. Cooper, and J.R. Schrieﬀer. 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The rigorous method we use is based\non the transformation of original hamiltonian into positiv e semideﬁnite form. This technique is\nindependent of the spatial dimesion and does not require int egrability of the model. The obtained\nferromagnetism is connected to dispersionless bands but in a much broader sense than ﬂat band\nferromagnetism requires, where on every site a Hubbard term is present. In our case only a small\npercentage of, even randomly distributed, sites are only in teracting.\n1Recent observation of ferromagnetism in polythiophene compound s1has generated a\nwidespan interest and a heightened research eﬀort to develop plas tic ferromagnets and more\ngenerally to understand ferromagnetism in systems made entirely o f nonmanetic elements.\nAs a result several theories have emerged to describe ferromagn etism in these systems2,\nwith particular focus on ferromagnetism due to dispersionless band s3,4or periodic Anderson\nmodel5.\nThese theories however, are centered on two particle description s, on exact diagonaliza-\ntionsofsmall samples, ormean ﬁeldtypeofapproximations6, i.e., exploring weakinteracting\nlimits. This contrasts with the recent band structure calculations7, which revealed the fact\nthat the on-site Coulomb repulsion magnitude in these systems is act ually relatively high,\nand may even reach 10eV causing strong correlation eﬀects. Thus , exploring possibilities\nfor other techniques compatible with strong correlation eﬀects is n eeded for describing the\norigin of ferromagnetism in these systems.\nIn this Letter we present the result of an exactcalculation applied to a multiband Hub-\nbard model, by which it can be shown rigorously that ferromagnetic g round state appears\nwell below quarter ﬁlling. A similar method was used previously8in polythiophene type\nstructures in the high doping limit, i.e., well over the half ﬁlled case. In t he present case\nhowever, working well below half ﬁlling, our results are much broader than the ﬂat band\napproach3,4because we obtain ferromagnetism with sparse and even random pr esence of the\nlocal Coulomb repulsion, i.e., a Hubbard term is not required on every s ite.\nOur analytical approach proceeds in three steps: the transform ation of the Hamiltonian\ninto positive semideﬁnite form, the construction of the ground sta te and the proof of their\nuniqueness. The technique is independent of the spatial dimension a nd does not require\nintegrability of themodel. The methodwas previously applied toconst ruct the exact ground\nstates for the Hubbard chains with diﬀerent geometrical structu res9and even for two-10and\nthree-dimensional periodic Anderson model11. Details of the method are described in Ref.12.\nFollowing through with these steps, we start with i) transformation of a Hamiltonian\ninto positive semideﬁnite form . A positive semideﬁnite operator ˆPis deﬁned as an operator\nwhich has only non-negative expectation values with all components |χ/an}bracketri}ht ∈ Hof the Hilbert\nspaceH, i.e./an}bracketle{tχ|ˆP|χ/an}bracketri}ht ≥0. However, any Hamiltonian, ˆH, which describes a physical system\nis always bounded below and hence, can be written as ˆH=ˆP+C, where ˆPa positive\nsemideﬁnite and Ca scalar. It follows that the most general wave vector |Ψg/an}bracketri}htsatisfying\n2ˆP|Ψg/an}bracketri}ht= 0 is the ground state wave vector of ˆHwith ground state energy Eg=C.\nTo show how straightforward the method is, we exemplify it’s applicat ion with the two-\ndimenisonal Hubbard model, given by the well-known Hamiltonian:\nˆH=/summationdisplay\ni,σ(txˆc†\ni+x,σˆci,σ+tyˆc†\ni+y,σˆci,σ+H.c)+ˆHU,ˆHU=U/summationdisplay\niˆni,σˆni,−σ,(1)\non a square lattice with Bravais vectors x,yand periodic boundary conditions. Here we\nused standard notations, where ˆ c†\nj,σis the electron creation operator with spin projection σ,\nˆnj,σ= ˆc†\nj,σˆcj,σis the number operator, txandtyare the hopping matrix elements connecting\nnearest neighbour lattice sites in xandydirections, and U >0 represents the strength of\nthe on-site Coulomb repulsion.\nPer deﬁnition ˆHUis a positive semideﬁnite operator, however the total Hamiltonian, ˆH\nis not. Hence, we perform a transformation on ˆHto obtain one. For this, to each square,\nwith coordinates i,i+x,i+x+y,i+y, we attach two block operators\nˆAi,σ=a2ˆci+x,σ+a3ˆci+x+y,σ+a4ˆci+y,σ,ˆBi,σ=b1ˆci,σ+b2ˆci+x,σ+b4ˆci+y,σ,(2)\nso as the starting Hamiltonian, Eq.(1), transforms into\nˆHAB=/summationdisplay\ni,σ(ˆA†\ni,σˆAi,σ+ˆB†\ni,σˆBi,σ) =ˆH−ˆHU+qˆN, (3)\nˆP=ˆHAB+ˆHU,C=−qN, with the number of electrons, N, ﬁxed. The obtained ˆHAB\nis a positive semideﬁnite operator, and hence ˆP=ˆHAB+ˆHUalso. The only task left is to\ncalculate the coeﬃcients of the block operators, ˆAi,σandˆBi,σ, for which the transformation\ninto Eq.(3) gives:\ntx=b∗\n2b1+a∗\n3a4, ty=b∗\n4b1+a∗\n3a2, ty+x=ty−x=b∗\n4b2+a∗\n4a2= 0,\nq=|b1|2+|b2|2+|b4|2+|a2|2+|a3|2+|a4|2. (4)\nThis system of equations represents the matching conditions . Obtaining a solution for these\nmatching conditions implies a solution for the Hubbard Hamiltonian. This is usually not\nan easy task, as these equations are coupled, complex algebraic no n-linear equations, but it\ncan be done in some restricted ˆHparameter space, e.g., see, Refs.9–11and even in disordered\nsystems13.\nHaving a solution for the matching equations, we can easily go to the s econd step in our\napproach, namely ii)theconstructionofthegroundstate , i.e.,|Ψg/an}bracketri}ht. Thesolutionwilldepend\n3on the structure of ˆP, however the most common case is when ˆPoperator contains terms\nof the form/summationtext\ni,σˆA†\ni,σˆAi,σ,/summationtext\ni,σˆB†\ni,σˆBi,σ. In these cases the ground state is constructed with\nthe help of a block operator ˆC†\nj,σwhich anticommutes with ˆAi,σandˆBi,σ, i.e.,{ˆAi,σ,ˆC†\nj,σ′}=\n{ˆBi,σ,ˆC†\nj,σ′}= 0, for all possible values of all indices. Namely, if |χ/an}bracketri}ht=/producttext\ni,σˆC†\ni,σ|0/an}bracketri}htwhere|0/an}bracketri}ht\nis the bare vacuum, then ˆHAB|χ/an}bracketri}ht= 0. In order however, to obtain the ground state |Ψg/an}bracketri}htof\nˆHwe need|χ/an}bracketri}htto control all positive semideﬁnite operators in ˆP. In other words |χ/an}bracketri}hthas to\nbe inserted in the kernel14of all positive semideﬁnite operators existing in ˆP. This process\nis easily implemented by imposing some restrictions ( i,σ)∈ Mon the validity domain of\n|χ/an}bracketri}ht, to be determined separately on model basis, after which the true ground state becomes\n|Ψg/an}bracketri}ht=/producttext\n(i,σ)∈MˆC†\ni,σ|0/an}bracketri}ht.\nThe last step in our approach is iii) the proof of their uniqueness . For the most general\ncase, when the ground state |Ψg(m)/an}bracketri}htisMfold degenerate (i.e. m= 1,2,...,M), the proof of\nthe uniqueness is done in two steps. In the ﬁrst step, we prove tha t for all possible mvalues\n|Ψg(m)/an}bracketri}ht ∈Ker(ˆP) is true. In the second step, we verify that any arbitrary wave ve ctor\n|ν/an}bracketri}ht ∈Ker(ˆP) can be expressed as a linear combination of the |Ψg(m)/an}bracketri}htterms, see Ref.11,12,15.\nIn the non-degenerate case the steps are the same, but applied o nly to the m= 1 ground\nstate component.\nIn the following, we apply the above method to two cases of pentago n chains. First we\nanalysethepentagonchainwithoutexternallinks, seeFig. (1). Th issystemisaconductor, a\nconjugated polymer of great interest which has not been analyzed yet with rigurous methods\nonly way above the half ﬁlled concentration regime. Each pentagon c ell contains four sites\nper cell. The cell deﬁned at any site i, see, the ﬁrst cell of Fig. (1), has four adjacent sites\nati+rn, wheren= 1,2,3,4, and by convention r2= 0. For a ﬁxed n, the sites i+rnare\nbelonging to the n-th sublattice. The on-site potentials and hopping transfer we use d the\nnotations shown on the second and third cell of the pentagon chain of Fig. (1).\nWith the above notations, the non-interacting part of the Hamilton ian becomes ˆH0=\n/summationtext\nσ/summationtextNc\ni=1{[t(ˆc†\ni+r1,σˆci,σ+ˆc†\ni+a,σˆci+r1,σ)+tnˆc†\ni+r3,σˆci+r4,σ+t′(ˆc†\ni,σˆci+r3,σ+ˆc†\ni+r4,σˆci+a,σ)+H.c.]+\nǫ′\n0ˆni+r1,σ++ǫ′\n1(ˆni+r3,σ+ ˆni+r4,σ)+ǫ′\n2ˆni,σ)},whereNcrepresents the number of cells. There\nare 4Nclattice sites in the system and Nelectrons.\nWhile,theinteractingpartoftheHamiltonianis ˆHU=/summationtextNc\ni=1[U0ˆni+r1,↑ˆni+r1,↓+U1(ˆni+r3,↑ˆni+r3,↓+\nˆni+r4,↑ˆni+r4,↓)+U2ˆni,↑ˆni,↓],where, since in the positions i+r1, (i+r3,i+r4), andidiﬀerent\ntype of atoms are potentially present holding in order the on-site on e-particle potentials\n4r1\nr4\n3r\nε'0ε'1ε'1\nε'2t t\nt' t'\ntnA2\nA3 A1... ...\njj−a+r1\nj−a+r3j−a+r4i+r1\ni\nj+a−r1j+a−r4j+a−r3\ni+a\ni+r i+r43\nFIG. 1. The pentagon chain without external links. The ﬁrst p entagon at site ishows the site\ncoordinates; the on-site potentials and the hopping transf er mattrixes are shown on the second and\nthird pentagons, respectively. The fourth pentagon depict s the triangular regions on which the\nblock operators of Eq. (7) are deﬁned. Finally, the last two p entagons present the sites (connected\nwith dotted line as a guide to the eye) which contribute to the block operator ˆB†\nj,σdeﬁned in Eq.\n(12).\nǫ′\n0,ǫ′\n1andǫ′\n2, three diﬀerent U0,U1,U2>0 on-site Coulomb repulsion (Hubbard interaction)\nvalues are used. One has the Hubbard Unat the site where the on-site potential is ǫ′\nn.\nHence, the total Hamiltonian will be simply ˆH=ˆH0+ˆHUand using the technique\npreviously detailed, see, Eqs. (2) - (4), the block operators are d eﬁned as:\nˆA1,i,σ=a1,1ˆci+r1,σ+a1,2ˆci,σ+a1,3ˆci+r3,σ,\nˆA2,i,σ=a2,1ˆci+r1,σ+a2,3ˆci+r3,σ+a2,4ˆci+r4,σ,\nˆA3,i,σ=a3,1ˆci+r1,σ+a3,4ˆci+r4,σ+a3,5ˆci+a,σ. (5)\nThese operators span16a pentagon cell as depicted in the fourth cell of Fig.(1). Using\nperiodic boundary conditions ˆH0transfroms into:\nˆH0=/summationdisplay\nσNc/summationdisplay\ni=13/summationdisplay\nm=1ˆA†\nm,i,σˆAm,i,σ. (6)\nWe are interested to ﬁnd the ground state solution well below quart er ﬁlling, hence we work\nin the condition N≤Nc. For the solution of the matching conditions, for real hopping\n5matrix elements and conditions tn>0,ǫ′\n1−tn>0 we obtained:\na1,1=eiφ1|a1,1|, a1,2=eiφ1t\n|a1,1|, a1,3=eiφ1t′\nt|a1,1|,\na2,1=eiφ2|a2,1|, a2,3=−eiφ2t′\nt|a1,1|2\n|a2,1|, a2,4=−eiφ2ttn\nt′|a2,1|\n|a1,1|2,\na3,1=eiφ3|a3,1|, a3,4=eiφ3t′\nt|a3,1|, a3,5=eiφ3t\n|a3,1|, (7)\nwhereφm,m= 1,2,3arearbitraryphases. InEqs. (7)theHamiltonianparameters t,t′,tn,ǫ′\n1\ncan be arbitrary chosen, while ǫ′\n0,ǫ′\n2are given by the conditions ǫ′\n0= [t2/(t′2tn)](ǫ′2\n1−t2\nn),\nandǫ′\n2= 2t′2/(ǫ′\n1−tn). These last two conditions provide the lowest ﬂat band of the band\nstructure.\nSinceˆH0has the simple expression (6), we look for the ground state wave fu nction in the\nform\n|Ψg/an}bracketri}ht=N≤Nc/productdisplay\ni=1ˆB†\ni,σi|0/an}bracketri}ht, (8)\nwhere|0/an}bracketri}htis the bare vacuum, and ˆB†\ni,σisatisﬁes for all n= 1,2,3 the relation\n{ˆAn,i,σ,ˆB†\ni′,σ′\ni′}= 0, (9)\nwherei,i′,σ,σ′\ni′are arbitrary. Since only one type of canonical Fermi operator is d eﬁned on\neach site, Eq. (8) is true if the ˆB†\ni′,σ′\ni′operators do not overlap, or the neighbouring operators\noverlap at least on one site.\nThe ﬁrst case, when the ˆB†\ni′,σ′\ni′operators do not overlap, would mean a localized and para-\nmagnetic ground state of the general form ˆB†\ni,σ=x1ˆc†\ni+r1,σ+x3ˆc†\ni+r3,σ+x4ˆc†\ni+r4,σ. However,\nthere isn’t any value of x1,x2,x3, except x1=x3=x4= 0, which would satisfy Eq. (9),\nhence there is no solution in this case.\nTo search for a solution in the second case, i.e., when the ˆB†\ni′,σ′\ni′operators overlap, we\ndeﬁneˆB†\ni,σas shown on the last two cells of Fig. (1), namely:\nˆB†\ni,σ=x1ˆc†\ni+r1,σ+x2ˆc†\ni,σ+x3ˆc†\ni+r3,σ+x4ˆc†\ni+r4,σ\n+y1ˆc†\ni−a+r1,σ+y3ˆc†\ni−a+r3,σ+y4ˆc†\ni−a+r4,σ, (10)\nand the solution to (9) is:\nx4=−t\nt′x1, x3=tǫ′\n1\nt′tnx1, x2=−t\nt′2tn(ǫ′2\n1−t2\nn)x1,\ny1=x1, y3=x4=−t\nt′x1, y4=x3=tǫ′\n1\nt′tnx1. (11)\n6Consequently, the ˆB†\ni,σoperator becomes\nˆB†\ni,σ=x1[−t(ǫ′2\n1−t2\nn)\nt′2tnˆc†\ni,σ+(ˆc†\ni+r1,σ+ˆc†\ni−a+r1,σ)+tǫ′\n1\nt′tn(ˆc†\ni+r3,σ+ˆc†\ni−a+r3,σ)\n−t\nt′(ˆc†\ni+r4,σ+ˆc†\ni−a+r3,σ)]. (12)\nThe (unnormalized) ground state wave function at 1 /8 ﬁlling (e.g. N=Nc) becomes a\nsaturated ferromagnet\n|Ψg/an}bracketri}ht=Nc/productdisplay\ni=1ˆB†\ni,σ|0/an}bracketri}ht, (13)\nwhereσis ﬁxed. Below 1 /8 ﬁlling the block operator ˆB†\ni,σis still given in Eq. (12), and the\nground state will have the (8) form. But, a geometrical degenera cy occurs: only overlaping\nˆB†\ni,σoperators will have the same spin index. Hence, the ground state w ill be composed\nfrom ferromagnetic clusters which if don’t overlap, will have arbitra ry spin orientations. We\nshould also point out that Eq. (13) corresponds to the half ﬁlled lowe r ﬂat band. The\nobtained solution is true for arbitrary large U0,U1,U2>0 Hubbard terms.\nNext, we analyse the second model of a pentagon chain, namely the pentagon chain\nwith external links and antennas, see Fig. (2). This chain is also a con ductor, and we are\ngoing to show in the following that the obtained results are qualitative ly the same as in the\nprevious case. The new pentagon chain, with external links and ant ennas connected to the\npentagons, is shown in Fig. (2). The cell now contains six sites and co nsequently, there will\nbe six sublattices in the system. The cell deﬁned at any site i, see, the ﬁrst cell of Fig. (2)\nhas six adjacent sites at i+rn, where now n= 1,2,...,6, andr3= 0 by convention. For a\nﬁxedn, the sites i+rnare belonging to the n-th sublattice.\nWith the on-site potentials and hopping matrix elements deﬁned on th e second and third\ncellofFig. (2),thestartingHamiltonian ˆH=ˆH0+ˆHUbecomes: ˆH0=/summationtext\nσ/summationtextNc\ni=1{[tfˆc†\ni+r1,σˆci+r2,σ+\ntcˆc†\ni+r6,σˆci+a,σ+tnˆc†\ni+r4,σˆci+r5,σ+t(ˆc†\ni+r2,σˆci,σ+ˆc†\ni,σˆci+r4,σ+ˆc†\ni+r5,σˆci+r6,σ+ˆc†\ni+r6,σˆci+r2,σ)+H.c.]+\nǫ′\n0(ˆni+r1,σ+ ˆni+r2,σ)+ǫ′\n1(ˆni+r4,σ+ ˆni+r5,σ)+ǫ′\n2(ˆni,σ+ ˆni+r6,σ)},while the interacting part of\nthe Hamiltonian is now ˆHU=/summationtextNc\ni=1[U0(ˆni+r1,↑ˆni+r1,↓+ ˆni+r2,↑ˆni+r2,↓) +U1(ˆni+r4,↑ˆni+r4,↓+\nˆni+r5,↑ˆni+r5,↓)+U2(ˆni,↑ˆni,↓+ˆni+r6,↑ˆni+r6,↓)].In the interacting part of the Hamiltonian, since\nin positions ( i+r1,i+r2), (i+r4,i+r5), and (i,i+r6) diﬀerent type of atoms are present,\nthree are three diﬀerent U0,U1,U2>0 local Coulomb repulsion values. Also, the number of\nlattice sites on the chain is 6 Nc, and the number of electrons is N.\n7... A1A3A\n2A\nA54\nii+r1\ni+r2r1\nr4r2\nr6i+r6\ni+a 5rε ε1 1' '\nε00\n'ε'2ε2ε' 'tn\nt t\nt t\ntftc\ni+r4i+r5j\nx1x2x4x5\nx6y3\ny yy\n2y\n3\n4 5...\nFIG. 2. The pentagon chain with external links and antennas. The ﬁrst pentagon at site ishows\nthe site coordinates; the on-site potentials and the hoppin g transfermattrixes are shown on the\nsecond pentagon. The third pentagon depicts the ﬁve regions (three triangular and two bond\ndomains) on which the block operators from Eq. (18) are deﬁne d. Finally, the last two pentagons\npresent the sites (connected with dotted line as a guide to th e eye) which contribute to the block\noperator ˆBdagger\nj,σdeﬁned in Eq. (23). The coeﬃcients of each individual site co ntributions ( xi,yi)\nare also shown.\nUsing the same approach as for the previously analysed case, we ob tain for the block\noperator ˆB†\ni,σ:\nˆB†\ni,σ=x1[ˆc†\ni+r1,σ−ǫ′\n0\ntfˆc†\ni+r2,σ+ǫ′\n0\ntfˆc†\ni+r4,σ−ǫ′\n0ǫ′\n1\ntftnˆc†\ni+r5,σ+ǫ′\n0(ǫ′2\n1−t2\nn)\nttftnˆc†\ni+r6,σ\n−(ǫ′\n1−tn)sign(tc) ( ˆc†\ni+a+r1,σ−ǫ′\n0\ntfˆc†\ni+a+r2,σ+ǫ′\n0\ntfˆc†\ni+a+r5,σ\n−ǫ′\n0ǫ′\n1\ntftnˆc†\ni+a+r4,σ+ǫ′\n0(ǫ′2\n1−t2\nn)\nttftnˆc†\ni+a,σ) ]. (14)\nThe (unnormalized) ground state wave function at 1 /12 ﬁlling (e.g. N=Nc) becomes a\nsaturated ferromagnet\n|Ψg/an}bracketri}ht=Nc/productdisplay\ni=1ˆB†\ni,σ|0/an}bracketri}ht, (15)\nwhereσis ﬁxed. Below 1 /12ﬁlling the expression of ˆB†\ni,σremains asgiven inEq. (14), but in\n(15) a geometrical degeneracy occurs, only overlaping ˆB†\ni,σoperators will have the same spin\n8index, and the ground state will be constructed from ferromagne tic clusters which if not in\ncontact, willhavearbitraryspinorientation. Thegroundstategiv enbyEq. (15)corresponds\nto a half ﬁlled lowest ﬂat band. The obtained solution however, occur s for arbitrary large\nU0,U1,U2>0 Hubbard interaction terms. Similar situations for other compound s have been\nintensively analyzed in literature22–24.\nInsummary, byemploying arigorousanalyticalmethodwehavecons tructedexactground\nstates for multiorbital pentagon Hubbard chains. The ferromagn etism what we found well\nbelow half ﬁlling originate from the multi-orbital polygon chains which yie ld dispersionless\nband in the presence of site-dependent Coulomb intercation. Ndependent ground states\nwe have obtained for N≤Nc, and the system is conducting for N < N c. AtN=Nc, the\nferromagnetism emerges since in the ground state wave vector all contributing terms have\nthe same ﬁxed spin projection. The proof of the uniqueness of our results can be made along\nthe lines of Refs.12,15.\nOrganic ferromagnets have attracted much attention as a challen ging target. In particu-\nlar, organic magnets consisting entirely of non-magnetic elements is of fundamental as well\nas practical interest. Ordinary ferromagnets consist of magnet ic elements and even one-\ndimensionals models which exhibit ferromagnetism exploit electrons in dorforbitals. In\nthepresence of stronginteraction, forexample such intheKondo latticecase, the felectrons\nare responsible for ferromagnetism which, as it was shown in Refs.17using non-Abelian den-\nsity matrix renormalization group18, order due to scattering with the conduction electrons.\nSince hopping is energetically most favorable for conductions electr ons which preserver their\nspins, called coherent hopping, this tends to align the localized felectron spins17.\nBut, in the cases analysed in this Letter only non-magnetic elements are present in the\npentagon chain. We can rightfully ask the question how magnetism ca n arise in these\nsystems? The answer to this question is that the Coulomb intercatio ns are capable of\nturning itinerant system into a ferromagnetic phase in an extended parameter region. The\nmagnetism arises as an eﬀect of the electron-electron repulsion wh en the adjacent block\noperators which yield the ground state wave vector overlap and int uitively the spin has to\nalign to lower the repulsion energy due to Pauli’s principle.\nContinuing the above agurment, due to the overlaping adjacent blo ck operators, in our\nmodel we do not even need all sites to be interacting, it is enoguh to h ave merely one site\nto be intercating in each cell. To show this, let us consider ﬁrst the pe ntagon chain wihtout\n9external links. The sites contributing to the block operators of th e ground state wave vector\nare shown in Fig. (1) with dotted lines (last two pentagon of the ﬁgur e). Consequently, at\nN=Ncnumber of electrons, ferromagnetism will appear even if thereis on ly onesite ineach\npentagon with non zero local Coulomb repulsion, namely one of the sit es with coordinates\nr1,r3,r4. On these sites, even a random distribution of one local Coulomb rep ulsion on each\ncell yields ferromagnetism. We note that in this case 75% of sites are non-interacting (three\nsites from four in each cell), i.e. without Hubbard interaction.\nThe same is true for the case of pentagon chains with external links . If one site per\npentagon has a Hubbard U attached to it, in between sites with coor dinatesr1,r2,r4,r5,r6,\nsee, Fig. (2). In this case 83.3% of sites are non-interacting (ﬁve s ites out of six in each cell).\nThis shows that surprizingly, the complete absence of magnetic ato ms with sparse and even\nrandompresence ofthelocalCoulombrepulsion canleadtoferroma gnetism. Thisunderlines\nthat the conditions in which we obtained ferromagnetism are much br oader than those ﬁxed\nby ﬂat-band ferromagnetism, where on every site of the system U >0 is required25. Hence,\nthe obtained solutions point to a new route for the design of ferrom agnetic chain polymners.\nRegarding the experimental observation of ferromagnetism, we h ave to point out that\nthe required electron doping of the pentagon chains can be achieve d19by changing the\nFermi level by selecting appropriate side groups or by ﬁeld-eﬀect d oping in a double-layer\ntransisitor structure20. Indeed, depending on the applied doping levels pentagon polymers\ncan be turned21into ferromagnets, spin glasses or simple paramagnetic polymers.\nAcknowledgements\n(1) For M. Gul´ acsi this research was realized in the frames of TAMO P 4.2.4. A/2-11-1-\n2012-0001 ”National Excellence Program - Elaborating and operat ing an inland stu-\ndent and researcher personal support system”. The project w as subsidized by the\nEuropean Union and co-ﬁnanced by the European Social Fund.\n(2) Zs. Gul´ acsi kindly acknowledges ﬁnancial support provided by Alexander von Hum-\nboldt Foundation, OTKA-K-100288 (Hungarian Research Funds fo r Basic Research)\nandTAMOP4.2.2/A-11/1/KONV-2012-0036(co-ﬁnancedbyEUand EuropeanSocial\nFund).\n101A. A. Correa, et al., Synth. Met. 121 (2001) p. 1836; O. R. Nascimento, et al., Phys. Rev. B67\n(2003) 144422; F. R. de Paula, et al., 320 (2008) p. 193; S. Majumdar, et al., arXiv: 0905.2021.\n2Y. Suwa, et al., Phys. Rev. B68 (2003) p. 174419; R. Arita, et al., Phys. Rev. Lett. 88 (2002)\np. 127202; ibid., Phys. Rev. B68 (2003) p. 140403.\n3A. Mielke and H. Tasaki, Commun. Math. Phys. 158 (1993) p. 341 .\n4A. Mielke, Phys. Lett. A174 (1993) p. 443; J. Phys. A: Math. Ge n 32 (1999) p. 8411.\n5Y. Suwa, et al., Phys. Rev. B82 (2010) p. 235127.\n6See, for example M. Gulacsi and Z. Gulacsi, Phys. Rev. B33 (19 86) p. 6147; Z. Gulacsi and M.\nGulacsi, Phys. Rev. B36 (1987) p. 699; and Z. Gulacsi, M. Gula csi and I. Pop, Phys. Rev. B37\n(1988) p. 2247.\n7T. O. Wehling, et al., Phys. Rev. Lett. 106 (2011) p. 236805.\n8Z. Gulacsi, A. Kampf and D. Vollhardt, Phys. Rev. Lett. 105 (2 010) p. 266403.\n9Z. Gulacsi, A. Kampf and D. Vollhardt, Phys. Rev. Lett. 99 (20 07) p. 026404; Z. Gulacsi and\nI. Orlik, Jour. Phys. A34 (2001) L359.\n10I. Orlik and Z. Gulacsi, Phil. Mag. Lett. 78 (1998) p. 177; P. G urin and Z. Gulacsi, Phys. Rev.\nB64 (2001) p. 045118; Z. Gulacsi and M. Gulacsi, Phys. Rev. B7 3 (2006) p. 014514.\n11Z. Gulacsi and D. Vollhardt, Phys. Rev. Lett. 91 (2003) p. 186 401;ibid., Phys. Rev. B72 (2005)\np. 075130.\n12Z. Gulacsi, A. Kampf and D. Vollhardt, Prog. Theor. Phys. Sup pl. 176 (2008) p. 1.\n13Z. Gulacsi, Phys. Rev. B69 (2004) p. 054204; Phys. Rev. B66 (2 002) p. 165109; Eur. Phys. Jour.\nB30 (2002) p. 295.\n14The kernel Ker(ˆO) of an arbitrary operator ˆOis a Hilbert subspace containing all wave vectors\n|φ/an}bracketri}htwith the property ˆO|φ/an}bracketri}ht= 0.\n15Z. Gulacsi, Int. J. Mod. Phys. B27 (2013) p. 1330009.\n16M. Gulacsi, Phil. Mag. B76 (1997) p. 731; M. Gulacsi, H. van Be ijeren and A. C. Levi, Phys.\nRev. E47 (1993) p. 2473.\n17I. P. McCulloch, et al., J. Low Temp. Phys. 117 (1999) p. 323; I. P. McCulloch, et al., Phil.\nMag. Lett. 81 (2001) p. 869; I. P. McCulloch, et al., Phys. Rev. B65 (2002) p. 052410.\n18I. P. McCulloch and M. Gulacsi, Aust. J. Phys. 53 (2000) p. 597 ;ibid., Phil. Mag. Lett. 81\n11(2001) p. 447; ibid., Europhys. Lett 57 (2002) p. 852.\n19A. Opitz, et al., New J. Phys. 10 (2008) p. 065006.\n20K. Ueno, et al., Nat. Mater. 7 (2008) p. 855; H. Yuan, et al., Adv. Funct. Mater. 19 (2009)p.\n1046.\n21M. J. Panzer, et al., Appl. Phys. Lett. 86 (2005) p. 022104; M. J. Panzer and C. D. F risbie, J.\nAm. Chem. Soc. 127 (2005) p. 6960.\n22S. Capponi, O. Derzhko, A. Honecker, et al., Phys. Rev. B88 (2013) p. 144416.\n23O. Derzhko, J. Richter, O. Krupnitska, et al., Phys. Rev. B88 (2013) p. 094426.\n24M. Maksymenko, A. Honecker, R. Moessner, et al., Phys. Rev. Lett. 109 (2012) p.096404.\n25In order to present this statement in mathematical terms, we mention that for example in the\nﬁrst paper of Ref.[4] (Phys.Lett A174,443(1993)), if 75-80 % of the sites are non-interacting,\nEq.(18) on pg. 445 is no more valid, hence the proof started on pg.444 is no more correct.\n12"    },    {        "title": "2006.00348v1.Magnetization_dynamics_in_proximity_coupled_superconductor_ferromagnet_superconductor_multilayers.pdf",        "content": "Magnetization dynamics in proximity-coupled\nsuperconductor/ferromagnet/superconductor multilayers\nI. A. Golovchanskiy1;2;3, N. N. Abramov2, V. S. Stolyarov1;3, V. I. Chichkov2, M. Silayev1;4,\nI. V. Shchetinin2, A. A. Golubov1;5, ,V. V. Ryazanov2;6, A. V. Ustinov2;7;8, M. Yu. Kupriyanov1;9\n1Moscow Institute of Physics and Technology, State University,\n9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia;\n2National University of Science and Technology MISIS, 4 Leninsky prosp., Moscow, 119049, Russia;\n3Dukhov Research Institute of Automatics (VNIIA), 127055 Moscow, Russia;\n4Department of Physics and Nanoscience Center, University of Jyv askyl a,\nP.O. Box 35 (YFL), Jyv askyl a FI-40014, Finland;\n5Faculty of Science and Technology and MESA+ Institute for Nanotechnology,\nUniversity of Twente, 7500 AE Enschede, The Netherlands;\n6Institute of Solid State Physics (ISSP RAS), Chernogolovka, 142432, Moscow region, Russia;\n7Physikalisches Institut, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany;\n8Russian Quantum Center, Skolkovo, Moscow 143025, Russia;\n9Skobeltsyn Institute of Nuclear Physics, MSU, Moscow, 119991, Russia\nIn this work, magnetization dynamics is studied in superconductor/ferromagnet/superconductor\nthree-layered \flms in a wide frequency, \feld, and temperature ranges using the broad-band ferro-\nmagnetic resonance measurement technique. It is shown that in presence of both superconducting\nlayers and of superconducting proximity at both superconductor/ferromagnet interfaces a massive\nshift of the ferromagnetic resonance to higher frequencies emerges. The phenomenon is robust and\nessentially long-range: it has been observed for a set of samples with the thickness of ferromagnetic\nlayer in the range from tens up to hundreds of nanometers. The resonance frequency shift is char-\nacterized by proximity-induced magnetic anisotropies: by the positive in-plane uniaxial anisotropy\nand by the drop of magnetization. The shift and the corresponding uniaxial anisotropy grow with\nthe thickness of the ferromagnetic layer. For instance, the anisotropy reaches 0.27 T in experiment\nfor a sample with 350 nm thick ferromagnetic layer, and about 0.4 T in predictions, which makes\nit a ferromagnetic \flm structure with the highest anisotropy and the highest natural resonance fre-\nquency ever reported. Various scenarios for the superconductivity-induced magnetic anisotropy are\ndiscussed. As a result, the origin of the phenomenon remains unclear. Application of the proximity-\ninduced anisotropies in superconducting magnonics is proposed as a way for manipulations with a\nspin-wave spectrum.\nI. INTRODUCTION\nLast two decades can be associated with a remark-\nable progress in areas of spin condensed matter physics,\nnamely, in spintronics1,2and magnonics3,4. Develop-\nments in spin physics have also advanced research in\nsuperconducting systems: by hybridizing superconduct-\ning and ferromagnetic orders intriguing physics emerges\nand new device functionality can be achieved, which\nis inaccessible in conventional systems. Thus, su-\nperconducting spintronics5can be viewed as a way\nfor manipulation with spin states employing an inter-\nplay between ferromagnetic an superconducting spin\norders. A long list of examples includes supercon-\nductor/ferromagnet/superconductor (S/F/S) josephson\njunctions6that can be employed as phase pi-shifters7\nand memory elements8,9, F/S/F-based spin valves10, and\nmore complex long-range spin-triplet superconducting\nsystems11{14. Superconducting spintronics necessarily\ninvolves the superconducting proximity15between fer-\nromagnetic and superconducting subsystems. On the\nother hand, superconducting magnonics can be viewed\nas manipulation with eigen-states of collective spin ex-\ncitations via their interaction with a superconductingsubsystem16{18. In contrast to superconducting spintron-\nics, in superconducting magnonics the proximity e\u000bect\nappears to be undesirable due to a possible suppression\nof fundamental characteristics of superconducting sub-\nsystem and consequently, degradation of the magnonic\nspectrum19.\nRecently, a qualitatively new manifestation of super-\nconductor/ferromagnet hybridization has been reported,\nwhich in a way merges both areas the superconduct-\ning spintronics and the superconducting magnonics. In\nRefs.20,21a drastic increase of the ferromagnetic res-\nonance frequency has been observed in superconduc-\ntor/ferromagnet/superconductor three-layers in presence\nof superconducting proximity between superconducting\nand ferromagnetic layers. The origin of the phenomenon\nremains unclear. Possible explanations that has been\nproposed so far are attributed to incorporation of the\nspin-triplet superconducting pairing mechanism20or to\nan interplay of magnetization dynamics with the vor-\ntex/Meissner state of superconducting layers21. No con-\nvincing explanation has been provided so far.\nIn this paper, we report a detailed experimental study\nof the e\u000bect of superconducting proximity in S/F/S\nheterostructures on magnetization dynamics in the F-\nlayer. Experiments are performed using a broad-bandarXiv:2006.00348v1  [cond-mat.supr-con]  30 May 20202\nferromagnetic resonance (FMR) measurement technique\nin magnetic \feld, frequency and temperature domains.\nThis work is organized as follows. Section II gives exper-\nimental details. Section III provides experimental results:\nmicrowave ferromagnetic resonance absorption spectra\nat \feld-frequency domain at di\u000berent temperatures and\ntheir quantitative analysis. For a complete picture, we\nalso suggest to review previous research studies on sim-\nilar systems (see Refs.20{22). Section IV is devoted to\ndiscussion of experimental results where we state that\nthe e\u000bect of superconducting proximity in S/F/S sys-\ntems can not be explained employing concepts of the su-\nperconducting Meissner screening or of the vortex phase.\nWhile the origin of the phenomena remains unclear at\nthis stage, the authors suspect a contribution of spin-\ntriplet superconductivity. Section V demonstrates capa-\nbilities of the e\u000bect for manipulation of the spin-wave\nspectrum in S/F/S-based continuous \flms and magnonic\ncrystals.\nII. EXPERIMENTAL DETAILS\nFIG. 1. Schematic illustration of the investigated chip-\nsample. A series of S/F/S \flm rectangles is placed directly\non top of the central transmission line of the co-planar waveg-\nuide. Magnetic \feld His applied in-plane along the x-axis.\nMagnetization dynamics is studied by measuring\nthe ferromagnetic resonance absorption spectrum us-\ning the VNA-FMR approach23{25. A schematic il-\nlustration of the investigated chip-sample is shown in\nFig. 1. The chip consists of 150 nm thick supercon-\nducting niobium (Nb) co-planar waveguide with 50 Ohm\nimpedance and 82-150-82 \u0016m center-gap-center dimen-\nsions. The waveguide is fabricated on top of Si/SiO x\nsubstrate using magnetron sputtering of Nb, optical\nlithography and plasma-chemical etching techniques.\nA series of niobium/permalloy(Py=Fe 20Ni80)/niobium\n(Nb/Py/Nb) \flm structures with lateral dimensions X\u0002\nY= 50\u0002140\u0016m and spacing of 25 \u0016m along the x\u0000axis\nis placed directly on top of the central transmission line\nof the waveguide using optical lithography, magnetronsputtering and the lift-o\u000b technique. Importantly, depo-\nsition of Nb/Py/Nb three-layers is performed in a single\nvacuum cycle ensuring an electron-transparent metallic\nNb/Py interfaces. A 20-nm-thick Si spacing is deposited\nbetween Nb co-planar and Nb/Py/Nb threelayers in or-\nder to ensure electrical insulation of the studied samples\nfrom the waveguide. Five di\u000berent samples has been fab-\nricated and measured with di\u000berent thickness of super-\nconducting (S) and ferromagnetic (F) layers (see Tab. I).\nOne of samples was fabricated with an additional insu-\nlating (I) layer at one of S/F interfaces.\nSample ID S(Nb) F(Py) I(AlO x)S(Nb)\nS1 110 19 0 110\nS2 110 19 0 7\nS3 85 22 10 115\nS4 140 45 0 140\nS5 110 350 0 110\nTABLE I. Parameters of studied samples.\nThe experimental chip was installed in a copper sample\nholder and wire bonded to PCB with SMP RF connec-\ntors. A thermometer and a heater were attached directly\nto the holder for precise temperature control. The holder\nwas placed in a superconducting solenoid inside a closed-\ncycle cryostat (Oxford Instruments Triton, base temper-\nature 1.2 K). The response of experimental samples was\nstudied by analyzing the transmitted microwave signal\nS21(f;H) with the VNA Rohde & Schwarz ZVB20. For\nexclusion of parasitic box resonance modes from consid-\neration, all measured spectra S21(f;H) have been \frst\nnormalized with S21(f) at\u00160H= 0:3 T, and then dif-\nferentiated numerically in respect to H. The response of\nexperimental samples was studied in the \feld range from\n-0.22 T to 0.22 T, in the frequency range from 0 up to\n18 GHz, and in the temperature range from 1.7 to 11 K.\nIII. EXPERIMENTAL RESULTS:\nFERROMAGNETIC RESONANCE IN\nPROXIMITY-COUPLED S/F/S SYSTEMS\nFigure 2 illustrates the studied phenomenon using\nS(Nb)/F(Py)/S(Nb) sample with 110 nm thick Nb lay-\ners and 19 nm thick Py layer. This sample is referred\nto as S1. Thickness of Py layer is selected for direct\ncomparison of obtained results with previous research\nstudies20,21. Figures 2a,b show FMR absorption spec-\ntradS21(f;H)=dH atT= 2 K (a), which is far below\nthe superconducting critical temperature Tcof Nb, and\natT= 9 K (b), which corresponds to Tc. Both spec-\ntra contain a single \feld-dependent spectral line, i.e., the\nFMR absorption line. FMR absorption spectra at dif-\nferent temperatures have been \ftted with the Lorentz\ncurve and the dependencies of the resonance frequency\non magnetic \feld fr(H) have been extracted. Figure 2c3\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s40/s97/s41/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s45 /s50/s46 /s48/s120 /s49/s48/s45 /s52\n/s45 /s49/s46 /s54/s120 /s49/s48/s45 /s52\n/s45 /s49/s46 /s50/s120 /s49/s48/s45 /s52\n/s45 /s56/s46 /s48/s120 /s49/s48/s45 /s53\n/s45 /s52/s46 /s48/s120 /s49/s48/s45 /s53\n/s45 /s50/s46 /s55/s120 /s49/s48/s45 /s50/s48\n/s52/s46 /s48/s120 /s49/s48/s45 /s53\n/s56/s46 /s48/s120 /s49/s48/s45 /s53\n/s49/s46 /s50/s120 /s49/s48/s45 /s52\n/s49/s46 /s54/s120 /s49/s48/s45 /s52\n/s50/s46 /s48/s120 /s49/s48/s45 /s52/s84 /s32/s61/s32/s49/s46/s55/s32/s75\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s84 /s32/s61/s32/s57/s46/s48/s32/s75\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s45 /s51/s46 /s48/s48/s120 /s49/s48/s45 /s52\n/s45 /s50/s46 /s52/s48/s120 /s49/s48/s45 /s52\n/s45 /s49/s46 /s56/s48/s120 /s49/s48/s45 /s52\n/s45 /s49/s46 /s50/s48/s120 /s49/s48/s45 /s52\n/s45 /s54/s46 /s48/s48/s120 /s49/s48/s45 /s53\n/s48/s46 /s48/s48\n/s54/s46 /s48/s48/s120 /s49/s48/s45 /s53\n/s49/s46 /s50/s48/s120 /s49/s48/s45 /s52\n/s49/s46 /s56/s48/s120 /s49/s48/s45 /s52\n/s50/s46 /s52/s48/s120 /s49/s48/s45 /s52\n/s51/s46 /s48/s48/s120 /s49/s48/s45 /s52\n/s40/s98/s41\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s32/s49/s46/s55/s32/s75\n/s32/s53/s46/s57/s32/s75\n/s32/s55/s46/s49/s32/s75\n/s32/s55/s46/s57/s32/s75\n/s32/s56/s46/s53/s32/s75\n/s32/s57/s46/s48/s32/s75/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102\n/s114/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84\n/s40/s99/s41\nFIG. 2. a,b) FMR absorption spectra dS21(f;H)=dH for S1 sample measured at T= 2 K>T c(a) andT= 9 K.Tc(b). The\ngrayscale is coded in absolute units. c) Dependencies of the FMR frequency on magnetic \feld fr(H) at di\u000berent temperatures\nfor S1 sample.\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48\n/s32/s83/s49\n/s32/s83/s49/s32/s102/s105/s116\n/s32/s83/s50\n/s32/s83/s51/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s102/s105/s101/s108/s100/s32 /s72\n/s97/s44/s32/s109/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\n/s40/s97/s41\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s49/s46/s48/s50/s49/s46/s48/s52/s49/s46/s48/s54/s49/s46/s48/s56/s49/s46/s49/s48\n/s32/s83/s49\n/s32/s83/s50\n/s32/s83/s51/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s32/s109/s97/s103/s110/s101/s116/s105/s115/s97/s116/s105/s111/s110/s32 /s77\n/s101/s102/s102/s44/s32/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\n/s40/s98/s41\nFIG. 3. The dependence of the anisotropy \feld Ha(a) and e\u000bective magnetization Meff(b) on temperature. Black square\ndots correspond to S1 S/F/S sample, red circular dots correspond to S2 S/F/s' sample, and blue diamond dots correspond to\nS3 S/F/I/S sample. Green curve in (a) is the \ft of Ha(T) with Eq. 2, which yields the following parameters: \u00160Ha0= 77 mT,\nTc= 9:0 K,p= 3:7.\ncollects resonance curves fr(H) that are measured at dif-\nferent temperatures. Basically, Fig. 2 demonstrates the\nessence of the phenomenon: it shows that upon decreas-\ning the temperature below Tcthe resonance curve fr(H)\nshifts gradually to higher frequencies. For instance, upon\ndecreasing the temperature the frequency of the natu-\nral FMRfr(H= 0) increases from about 0.5 GHz at\nT\u00159 K to about 8.5 GHz at T= 1:7 K.\nFMR curves fr(H) in Fig. 2c follow the typical Kittel\ndependence for thin in-plane-magnetized ferromagnetic\n\flms at in-plane magnetic \feld:\n(2\u0019fr=\u00160\r)2= (H+Ha) (H+Ha+Meff) (1)\nwhere\u00160is the vacuum permeability, \r= 1:856\u0002\n1011Hz/T is the gyromagnetic ratio for permalloy, Ha\nis the uniaxial anisotropy \feld that is aligned with theexternal \feld, and Meff=Ms+Mais the e\u000bective\nsaturation magnetization, which includes the saturation\nmagnetization Msand the out-of-plane anisotropy \feld\nMa. The \ft of FMR curves in Fig. 2c with Eq. 1 yields\nthe dependence of superconducting proximity-induced\nanisotropy \felds HaandMeffon temperature given in\nFig. 3 with black squares.\nFigure 3 shows that at T > T cthe anisotropy \feld is\nnegligible\u00160Ha\u0018 \u00002\u000210\u00004T and the e\u000bective magne-\ntization is\u00160Meff\u00191:1 T. These parameters are typical\nfor permalloy thin \flms. Also, at T >T cno dependence\nofHaandMeffon temperature is observed. At T <T c\nupon cooling the anisotropy \feld Haincreases gradually\nand reaches \u00160Ha\u001978 mT atT= 2 K. This value is\nwell consistent with previous studies on samples with the\nsame thickness of Py layer20,21. The dependence Ha(T)\ncan be characterized by \ftting it with the following ex-4\npression\nHa=Ha0(1\u0000(T=Tc)p) (2)\nwhereHa0is the e\u000bective anisotropy \feld at zero temper-\nature,Tcis the critical temperature, and pis a free expo-\nnent parameter. The \ft of Ha(T) with Eq. 2 is shown in\nFig. 3a with blue curve and yields the zero-temperature\nanisotropy \u00160Ha0= 77 mT.\nImportantly, the e\u000bective magnetization also demon-\nstrates a temperature dependence: upon cooling \u00160Meff\ndrops by about 70 mT. Such e\u000bect was has not been ob-\ntained in previous studies20,21due to instrumental limi-\ntations. The drop \u0000\u0001Meffand the uniaxial anisotropy\n\feldHaat 2 K are roughly equal. Thus, we state that su-\nperconductivity in S/F/S structure a\u000bects the magneti-\nzation dynamics by inducing positive in-plane anisotropy\nand by the drop of e\u000bective magnetization.\nAs the next step, following Ref.20, we con\frm that\nboth superconducting layers are required for development\nof the e\u000bect of superconducting proximity on magnetiza-\ntion dynamics, and that electrical conductivity, i.e. the\nproximity, also is required to take place at both S/F in-\nterfaces. The following S(Nb)/F(Py)/s'(Nb') sample is\nstudied with 110 nm thick S(Nb) layer, 19 nm thick Py\nlayer, which are similar to S1 sample, and thin 7 nm\nthick s'(Nb') layer . This sample is referred to as S2 (see\nTab. I). The upper s'(Nb') layer of S2 sample is argued to\nbe non-superconducting due to its small thickness, below\nthe superconducting coherence length and the London\npenetration depth, and due to the action of the inverse\nproximity e\u000bect. Yet, the upper layer is expected to re-\nproduce the microstructure of the upper Nb/Py inter-\nface. Basically, S2 sample represent S1 S/F/S sample\nwith a removed superconducting layer. FMR absorption\nspectra of S2 sample show no noticeable temperature de-\npendence, which is consistent with previous studies20,\nand practically match with the spectrum of S1 sample\natT&Tc(Fig. 2b). Fitting procedures of FMR spec-\ntra and of resonance curves for S2 sample yield Ha(T)\nandMeff(T) dependencies that a shown in Fig. 3 with\nred circular dots. The anisotropy \feld Ha(T) in Fig. 3a\nis negligible, it varies in the range from \u00005\u000210\u00004T\nto\u00003\u000210\u00004T and shows no dependence on tempera-\nture. The e\u000bective magnetization curve Meff(T), being\nat\u00160Meff\u00191:072 T atT >T c, shows a minor increase\nby\u00160\u0001Meff\u00193 mT upon decreasing temperature and\ncrossingTc. Note that variation of Meffwith tempera-\nture for S2 sample is opposite to one for S1 sample.\nNext, the following S(Nb)/F(Py)/I(AlO x)/S(Nb) sam-\nple is studied with thicknesses of Nb and Py layers similar\nto S1 and S2 samples, and additional insulating layer at\none of S/F interfaces. The sample is refereed to S3 (see\nTab. I). Basically, S3 sample represent S1 S/F/S sample\nwith suppressed conductivity at one of S/F interfaces.\nFMR absorption spectra of S3 sample shows no notice-\nable temperature dependence, which is consistent with\nprevious studies20. Blue diamond dots in Fig. 3 show\nHa(T) andMeff(T) dependencies for S3 sample. Theanisotropy \feld Ha(T) in Fig. 3a is negligible, though is\nslightly higher than one for S1 and S2 samples. It varies\nin the range from 3 to 5 mT and shows insigni\fcant de-\npendence on temperature. The e\u000bective magnetization\ncurveMeff(T), varies in the range from 1.1 up to 1.2 T\nand shows a minor drop by \u00160\u0001Meff\u001910 mT in vicinity\ntoTc. Therefore, with S2 and S3 samples we con\frm that\nboth superconducting layers are required for development\nof the e\u000bect of superconducting proximity on magneti-\nzation dynamics and that superconducting proximity is\nrequired to take place at both S/F interfaces.\nAs a crucial step, the dependence of phenomenon\non the thickness of the F-layer is revealed. Fig-\nure 4 demonstrates this dependence with a di\u000berent\nS(Nb)/F(Py)/S(Nb) sample with 140 nm thick Nb layers\nand 45 nm thick Py layer. This sample is referred to as\nS4 (see Tab. I). Figure 4a collects resonance curves fr(H)\nthat are measured at di\u000berent temperatures. It shows\nthat upon decreasing the temperature below Tcthe reso-\nnance curve fr(H) shifts gradually to higher frequencies\nfollowing the same trend as for S1 sample. Comparison\nof Fig. 4a with Fig. 2a immediately indicates that the ef-\nfect of the superconducting proximity in S/F/S systems\non magnetization dynamics is substantially stronger for\nthe thicker S4 sample: upon decreasing the temperature\nthe frequency of the natural FMR increases from about\n1 GHz atT= 10 K up to about 14.5 GHz at T= 3 K.\nIn other terms, by increasing the thickness of the F layer\nby a factor of 2.3 the enhancement of the natural FMR\nfrequency of S/F/S sample in superconducting state at\nT\u001cTchas increased by a factor of 1.6.\nThe \ft of FMR curves in Fig. 4a with Eq. 1 yields\nthe dependence of superconducting proximity-induced\nanisotropy \felds HaandMeffon temperature that are\ngiven in Fig. 4b,c with black squares. Figure 4b shows\nthat atT > T cthe anisotropy \feld is negligible as in\ncase of S1, S2 and S3 samples. At T <T cupon cooling\nthe anisotropy \feld Haincreases gradually and reaches\n\u00160Ha\u0019200 mT at T= 2 K.\nThe temperature dependence of the e\u000bective magneti-\nzationMeff(T) given in Figure 4c is more complex and\nis qualitatively di\u000berent from one for S1 sample. Upon\ncooling\u00160Meff\frst drops from 1.2 T at T >T cto about\n0.6 T atT.Tcand than increases gradually up to\nabout 1.03 T at T= 2K. We argue that such tempera-\nture dependence can be explained by \feld dependence of\nproximity-induced parameters. Indeed, at \fxed T < T c\nat upper-right section of a resonance absorption spec-\ntrumS21(f;H) superconductivity is partially suppressed\nby external \feld and microwave radiation, and therefore\nHais expected to be reduced while Meffis expected\nto be increased as compared to lower-left section of the\nspectrum. This phenomenon can be illustrated by \ftting\nof FMR curves in Fig. 4a with Eq. 1 in the limited \feld\nrange. Red circular dots in Fig. 4b,c show temperature\ndependencies of HaandMeffobtained by \ftting only\npart of FMR curves at \u00160H < 90 mT. Figure 4c shows\nthat the drop of MeffatT.Tcis signi\fcantly reduced:5\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102\n/s114/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84/s32/s51/s46/s48/s32/s75\n/s32/s53/s46/s54/s32/s75\n/s32/s54/s46/s57/s32/s75\n/s32/s55/s46/s53/s32/s75\n/s32/s55/s46/s57/s32/s75\n/s32/s56/s46/s51/s32/s75\n/s32/s49/s48/s32/s75\n/s40/s97/s41\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48\n/s40/s98/s41/s32/s111/s98/s116/s97/s105/s110/s101/s100/s32/s117/s115/s105/s110/s103/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s101/s110/s116/s105/s114/s101/s32/s102/s105/s101/s108/s100/s32/s114/s97/s110/s103/s101\n/s32/s111/s98/s116/s97/s105/s110/s101/s100/s32/s117/s115/s105/s110/s103/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s99/s117/s116/s111/s102/s102/s32\n/s48/s72 /s60/s57/s48/s32/s109 /s84\n/s32/s102/s105/s116/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s102/s105/s101/s108/s100/s32\n/s48/s72\n/s97/s44/s32/s109/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s49/s46/s49/s49/s46/s50/s32/s111/s98/s116/s97/s105/s110/s101/s100/s32/s117/s115/s105/s110/s103/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s101/s110/s116/s105/s114/s101/s32/s102/s105/s101/s108/s100/s32/s114/s97/s110/s103/s101\n/s32/s32/s111/s98/s116/s97/s105/s110/s101/s100/s32/s117/s115/s105/s110/s103/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s99/s117/s116/s111/s102/s102/s32\n/s48/s72 /s60/s57/s48/s32/s109 /s84/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s32/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32\n/s48/s77\n/s101/s102/s102/s44/s32/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\n/s40/s99/s41\nFIG. 4. a) Dependencies of the FMR frequency on magnetic \feld fr(H) at di\u000berent temperatures for S4 sample. b,c) The\ndependence of the anisotropy \feld Ha(b) and e\u000bective magnetization (c) on temperature. The data in b,c that is shown with\nblack square dots was obtained by \ftting fr(H) in the entire \feld range from 0 up to 200 mT. The data in b,c that is shown with\nred circular dots was obtained by \ftting fr(H) in the cut-o\u000b \feld range from 0 up to 90 mT. Green curve in (b) shows the \ft\nofHa(T), which is obtained using the cut-o\u000b \feld range, with Eq. 2, which yields the following parameters: \u00160Ha0= 196 mT,\nTc= 9:0 K,p= 7:7.\nupon cooling \u00160Meff\frst drops from 1.2 T at T > T c\nto about 0.8 T at T.Tcand than increases gradually\nup to about 1.03 T at T= 2K. Green curve in Fig-\nure 4b shows the \ft of Ha(T), which is obtained using\nthe cut-o\u000b \feld range, with Eq. 2. The \ft yields the\nzero-temperature anisotropy \u00160Ha0= 196 mT. Overall,\nthe drop \u0000\u0001Meffand the induced Haat 2 K are are\nroughly equal as in case of S1 sample: the anisotropy\n\feld\u00160Ha0= 196 mT while the drop of the e\u000bective\nmagnetization \u00160\u0001Meff\u0019 \u0000170 mT.\nImportantly, FMR parameters of the S1 sample, Ha(T)\nandMeff(T) in Fig. 3, are mostly unchanged when ob-\ntained using the same limited range of magnetic \felds\n\u00160H < 90 mT. This fact can be explained by frequency\ndependence of proximity-induced anisotropy \felds. In-\ndeed, resonance frequencies for S1 sample are typically\nby a factor of 2 lower than for S4 sample. Therefore,\nthe superconducting state of S-layers in S1 sample is less\na\u000bected by microwave radiation than in S4 sample.\nFigure 5 demonstrates the e\u000bect of the superconduct-\ning proximity in S/F/S systems on magnetization dy-\nnamics for a di\u000berent S(Nb)/F(Py)/S(Nb) sample with\na radically thicker 350 nm thick Py layer. This sam-\nple is referred to as S5 (see Tab. I). Figure 5a collects\nresonance curves fr(H) that are measured at di\u000berent\ntemperatures; it shows that upon decreasing the temper-\nature below Tcthe resonance curve fr(H) shifts gradually\nto higher frequencies following the same trend as for S1\nand S4 samples. However, the enhancement of the FMR\nfrequency upon decreasing temperature at T < T cis so\nintense that the FMR curve approaches the instrumen-\ntal frequency band limit already at T\u00188 K (note the\ntemperature range in legend of Fig. 5a). Comparison of\nFig. 5a with Figs. 2a and 4a con\frms that the e\u000bect of\nthe superconducting proximity in S/F/S systems on mag-netization dynamics enhances with growing thickness of\nthe F-layer. Upon decreasing the temperature the fre-\nquency of the natural FMR of S5 sample increases from\nabout 1 GHz at T > T cup to about 17 GHz already\natT= 8 K. Proximity to the superconducting critical\ntemperature, insu\u000ecient signal-to-noise ratio, parasitic\nbox modes, did not allow to \ft resonance curves consid-\nering both HaandMeffin Eq. 1 as \ftting parameters.\nTherefore, the \ftting routine was modi\fed for S5 sam-\nple as follows. First, fr(H) curves have been \ftted at\nT > T cwith Eq. 1. The \ft yields \u00160Meff\u00191:076 T\nand\u00160Ha\u00181 mT. Next, fr(H) curves at T < T chave\nbeen \ftted with Eq. 1 considering magnetization \fxed at\n\u00160Meff= 1:076 T and considering Haas the only \ftting\nparameter. The dependence Ha(T) is given in Fig. 5b\nwith black squares. It shows that the e\u000bective anisotropy\n\feld reaches \u00160Ha\u00190:27 T at 8 K. Note that by \fxing\nMeffthe so-obtained anisotropy \feld Hais expected to\nbe underestimated since according to Meff(T) depen-\ndencies for S1 and S4 samples Meffshould actually drop\natT < T c. Green curve in Figure 5b shows the \ft of\nHa(T) with Eq. 2. The \ft yields the extrapolated zero-\ntemperature anisotropy \u00160Ha0= 375 mT, which is also\nexpected to be underestimated.\nSummarizing experiential \fndings, superconductivity\nin S/F/S three-layers shifts the FMR to higher frequen-\ncies. The shift can be quanti\fed by the proximity-\ninduced positive in-plane anisotropy Haand by a drop\nof e\u000bective magnetization Meff. BothHaand the drop\nofMeffare roughly equal and are \feld-, frequency- and\ntemperature-dependent. The phenomenon requires both\nsuperconducting layers of S/F/S and presence of super-\nconducting proximity at both S/F interfaces. The phe-\nnomenon shows a dependence on the thickness of the F-\nlayer: for thicker F-layer the shift of the FMR frequency6\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48\n/s32/s56/s46/s48/s32/s75\n/s32/s56/s46/s49/s32/s75\n/s32/s56/s46/s50/s32/s75\n/s32/s56/s46/s51/s32/s75\n/s32/s56/s46/s52/s32/s75\n/s32/s56/s46/s53/s32/s75\n/s32/s56/s46/s54/s32/s75\n/s32/s56/s46/s55/s32/s75\n/s32/s57/s46/s53/s32/s75/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102\n/s114/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84\n/s40/s97/s41\n/s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s40/s98/s41/s32/s72\n/s97/s32/s100/s97/s116/s97\n/s32/s102/s105/s116/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s102/s105/s101/s108/s100/s32\n/s48/s72\n/s97/s44/s32/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\nFIG. 5. a) Dependencies of the FMR frequency on magnetic \feld fr(H) at di\u000berent temperatures for S5 sample. b) The\ndependence of the anisotropy \feld Haon temperature. Green curve in (b) shows the \ft of Ha(T) with Eq. 2, which yields the\nfollowing parameters: \u00160Ha0= 375 mT, Tc= 8:74 K,p= 13:9\nis substantially stronger. In addition, it should be noted\nthat (i) no dependence of the FMR spectrum on the in-\nput power has been observed in the range of input power\nfrom -15 dB to 0 dB; (ii) all measured spectra for all\nsamples are \feld-reversible; and (iii) no dependence of\nthe FMR linewidth on experimental parameters could be\nnoted owing partially to insu\u000ecient signal-to-noise ratio.\nAs a \fnal remark it should be noted that, technically,\nsamples S4 and S5 demonstrate the highest natural FMR\nfrequencies and corresponding in-plane anisotropies for\nin-plane magnetized ferromagnetic \flm systems ever re-\nported (see, for instance, Ref.26for comparison).\nIV. DISCUSSIONS: POSSIBLE ORIGIN OF\nPROXIMITY-INDUCED ANISOTROPIES IN\nS/F/S SYSTEMS\nA natural initial guess for the origin of the e\u000bect of\nsuperconducting proximity in S/F/S systems on mag-\nnetization dynamics is the Meissner screening of exter-\nnal \feld, the so-called lensing e\u000bect18,27. For instance,\none could employ \ruxometric or magnetometric demag-\nnetizing factors28,29of the system for estimation of a\nhypothetical diamagnetic moment in Nb layers that in-\nduces magnetostatic \feld Ha. However, this estimation\nis not required since the following set of unful\flled con-\nditions points towards irrelevance of the lensing e\u000bect in\ndiscussed experiments: (i) In case of the lensing e\u000bect\nthe induced Hais not a constant but a \feld-dependent\nquantity18. (ii) In case of the lensing e\u000bect the induced\nHashould decrease with increasing thickness of the F-\nlayer. (iii) The lensing e\u000bect should hold for S/F/I/S\nstructure (S3 sample) and should be only halved for S/F\nstructure (S2 sample) . (iv) The \feld that is induced\nby the lensing e\u000bect can not exceed the \frst critical\n\feld, which in Nb is about 100 mT (see values of Ha\nin Figs. 4b and 5b). None of the above hypothetical ef-fects does take place. In addition, consideration of the\nlensing e\u000bect does not clarify possible origin of the drop\nof magnetization \u0001 MeffatT <T cin Figs. 3b and 4c.\nIn fact, S/F (S2) and S/F/I/S (S3) structures may\nevidence the e\u000bect of Meissner screening on precessing\nmagnetization in thin \flm geometry. Meissner screening\nis expected to show itself in the absence of the in-plane\nanisotropy and in the presence of small negative out-of-\nplane uniaxial anisotropy. The later might be indicated\nby a small variation of MeffatT\u0018Tcin Figs. 3c and 4c.\nThe next hypothetical candidate for impact on mag-\nnetostatic state of the F-layer is the vortex phase. The\nfollowing set of unful\flled conditions evidence that the\nvortex phase can not have any e\u000bect on magnetization\ndynamics: (i) The e\u000bect of the vortex phase should hold\nfor S/F/I/S structure (S3 sample). (ii) Presence of the\nvortex phase that is induced by the external magnetic\n\feld should, in the \frst place, lead to hysteresis in the\nabsorption spectrum due to pinning30. (ii) The density of\na vortex phase that is induced by the external \feld is ex-\npected to be \feld-dependent leading to \feld-dependence\nof hypothetical vortex-phase-induced anisotropies. (iii)\nPresence of the vortex phase in superconducting thin\n\flms induces only insigni\fcant total magnetic moments\nand corresponding stray \felds. In addition, low expected\ndensity and arbitrary nature of the out-of-plane vortex\nphase unfavor its possible contribution.\nMechanisms that are considered above are limited to\nmagnetostatic interactions between F- and S-subsystems.\nAlternative explanations imply electronic correlations be-\ntween superconducting and ferromagnetic subsystems.\nFor instance, in Refs.31{33the electromagnetic proxim-\nity e\u000bect and spin polarization in planar superconductor-\nferromagnet structures are discussed. The electromag-\nnetic proximity e\u000bect implies presence of the supercon-\nducting condensate in the ferromagnetic layer and induc-\ntion of screening currents in the S/F system as a response7\non magnetic moment34rather than on magnetic \feld.\nWhile in general the electromagnetic proximity e\u000bect is\ndiamagnetic and induces magnetic \feld that counteracts\nthe magnetization, at certain thicknesses of the F-layer\nthe so-called paramagnetic electromagnetic proximity ef-\nfect can take place, which induces magnetic \feld along\nthe magnetization31. However, large thickness of F-layers\nin our experiments of 20, 40 and 350 nm in comparison to\nthe typical electron correlation length of singlet pairs in\nferromagnets35{37\u0018F\u00181 nm, and predicted oscillating\nbehaviour of the sign of induced \feld with the thickness of\nthe F-layer rule-out contribution of the electromagnetic\nproximity e\u000bect on magnetization dynamics in considered\nS/F/S systems.\nAlso, one can rule-out possible contribution of the\nspin-inverse proximity e\u000bect or the so-called spin-\nscreening38,39. The spin-screening considers accu-\nmulation of spins with polarization opposite to F-\nmagnetization in a thin layer of the S-subsystem of the\norder of the coherence length in vicinity to the S/F in-\nterface. Such spin orientation could possibly produce\nstray \felds of a required direction along magnetization in\nthe F-layer. Yet, owing to thin \flm geometry and small\ndemagnetizing factors28,29of the system an implausibly\nlarge magnetization of the spin-polarized area is required\nfor induction of the observed Ha, which is far above su-\nperconducting critical \felds.\nAnother possible explanation for the e\u000bect of supercon-\nducting proximity in S/F/S systems on magnetization\ndynamics is provided in the very \frst report of the ef-\nfect. In Ref.20it is proposed that the e\u000bective anisotropy\n\feld is produced due to interaction of magnetization with\nspin-polarized spin-triplet superconducting electrons via\nthe spin-transfer torque mechanism40{43. This mecha-\nnism requires presence of spin-triplet superconducting\npairs as a necessary ingredient. In Ref.20it is proposed\nthat the spin-triplet superconductivity is induced by the\ndynamically precessing magnetization in accordance with\nthe Ref.44. However, such mechanism requires large fre-\nquency of magnetization precession that should be com-\nparable to the depairing frequency and is inconsistent\nwith the frequency range of reported results.\nThus, we state that at this stage even a qualitative\nexplanation of the e\u000bect of superconducting proximity\nin S/F/S systems on magnetization dynamics is unavail-\nable. Yet, long-range nature of the phenomenon and the\nmandatory S/F/S symmetry of the phenomenon are sig-\nnatures for a role of spin-triplet superconductivity35.\nV. PROSPECTS OF THE PROXIMITY EFFECT\nFOR APPLICATION IN MAGNONICS\nThe e\u000bect of the superconducting proximity in S/F/S\nsystems on magnetization dynamics can be e\u000bective in\nmagnonics for variation of the FMR frequency or for\nmodulation of the spin-wave velocity. In this section,\nmicromagnetic simulations are employed45for calcula-tion of spin-wave spectra for S/F/S-based continuous\n\flms and periodic structures in the magnetostatic sur-\nface wave (MSSW) geometry46,47, following Refs.48{50.\nThe following micromagnetic parameters of studied F-\nlayers are considered, which correspond to S1 sample:\nthickness of F-layer d= 20 nm, the saturation magne-\ntization\u00160Ms= 1 T, the anisotropy \feld Ha= 0, the\napplied \feld \u00160H= 0:02 T, the exchange sti\u000bness con-\nstantA= 1:3\u000210\u000011J/m, and the gyromagnetic ratio\n\u00160\r= 2:21\u0002105m/A/s. The excitation \feld pulse has\nthe maximum frequency fmax= 30 GHz, the gaussian\nspatial pro\fle with the width at half-maximum of 200\nnm, and the amplitude of 0.001Ms. In simulations, the\ndiamagnetic (Meissner) contribution of S-subsystem on\nmagnetization dynamics was accounted via the method of\nimages17,51in case of continuous S-layers and via the dia-\nmagnetic representation of superconductors18,19in case\nof a \fnite-size S-elements. The e\u000bect of the supercon-\nducting proximity in S/F/S is represented by a local uni-\naxial anisotropy \feld \u00160Hs= 0:07 T that corresponds to\nS1 sample (see Fig. 3a).\nFigure 6a collects simulation results for continuous\nthin \flms. Blue solid curves show a typical dispersion\ncurve for MSSW in the plain F-\flm that is obtained\nwith simulations in absence of any contribution from S-\nsubsystem. Simulation results are well con\frmed by the\nanalytical dispersion relation51, shown with blue dashed\ncurves. Red solid and dashed lines show dispersion curve\nof MSSW in the S/F bilayer in presence of magnetostatic\ninteraction between the S and the F subsystems. The\nmagnetostatic interaction is accounted using the method\nof images17,51. It shows that in presence of magneto-\nstatic interaction the dispersion is nonreciprocal: the fre-\nquency pass-band for positive wavenumbers is approxi-\nmately doubled as compared to the pass-band for nega-\ntive wavenumbers. The nonreciprocity is a known prop-\nerty of MSSWs, which emerges due to asymmetry of the\nferromagnetic \flm across its thickness or due to asymme-\ntry of its surrounding (see Ref.51for details). Purple solid\nand dashed lines show dispersion curve of MSSW in the\nS/F/S three-layer in presence of the proximity-induced\nuniaxial anisotropy Hsbut absence of magnetostatic in-\nteraction between the S and the F subsystems. It shows\nthat at zero wavenumber the di\u000berence in frequencies be-\ntween the plain \flm and the \flm with uniaxial anisotropy\nis maximum and corresponds to the di\u000berence in FMR\nfrequencies. Upon increasing the wavenumper the dif-\nference in frequencies reduces. Black solid and dashed\nlines show dispersion curve of MSSW in the S/F/S three-\nlayer in presence of both the proximity-induced uniaxial\nanisotropy Hsand of the magnetostatic interaction be-\ntween the S and the F subsystems. Comparison of these\ncurves with dispersions in plain F \flm, in S/F bilayer\nand in F \flm with proximity-induced uniaxial anisotropy\nHsindicates that both the proximity-induced anisotropy\nand the magnetostatic screening a\u000bect the kinetics of\nspin waves. The magnetostatic screening is the domi-\nnating e\u000bect on spin-wave velocity at the range of higher8\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s53/s49/s48/s49/s53/s50/s48/s50/s53/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s87/s97/s118/s101/s110/s117/s109 /s98/s101/s114/s32/s49/s47 /s44/s32 /s109 /s45 /s49/s32/s80/s108/s97 /s105/s110 /s32/s70 /s77/s32/s102/s105/s108/s109/s44/s32/s115/s105/s109/s117 /s108/s97 /s116/s105/s111/s110 /s115\n/s32/s70 /s77/s47/s83 /s67/s32/s98/s105/s108/s97 /s121 /s101 /s114/s44/s32/s115/s105/s109/s117 /s108/s97 /s116/s105/s111/s110 /s115\n/s32/s83 /s67/s47/s70 /s77/s47/s83 /s67/s32/s116/s104 /s114/s101 /s101 /s108/s97 /s121 /s101 /s114/s44/s32/s115/s105/s109/s117 /s108/s97 /s116/s105/s111/s110 /s115\n/s32/s70 /s77/s32/s102/s105/s108/s109/s32/s119 /s105/s116/s104 /s32/s83 /s67/s45/s105/s110 /s100/s117 /s99 /s101 /s100/s32/s97 /s110 /s105/s115/s111/s116/s114/s111/s112/s121 /s44/s32/s115/s105/s109/s117 /s108/s97 /s116/s105/s111/s110 /s115\n/s32/s80/s108/s97 /s105/s110 /s32/s70 /s77/s32/s102/s105/s108/s109/s44/s32/s116/s104 /s101 /s111/s114/s121 \n/s32/s70 /s77/s47/s83 /s67/s32/s98/s105/s108/s97 /s121 /s101 /s114/s44/s32/s116/s104 /s101 /s111/s114/s121 \n/s32/s83 /s67/s47/s70 /s77/s47/s83 /s67/s32/s116/s104 /s114/s101 /s101 /s108/s97 /s121 /s101 /s114/s44/s32/s116/s104 /s101 /s111/s114/s121 \n/s32/s70 /s77/s32/s102/s105/s108/s109/s32/s119 /s105/s116/s104 /s32/s83 /s67/s45/s105/s110 /s100/s117 /s99 /s101 /s100/s32/s97 /s110 /s105/s115/s111/s116/s114/s111/s112/s121 /s44/s32/s116/s104 /s101 /s111/s114/s121 \n/s40/s97/s41\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s48/s46/s48/s48/s48/s50/s48/s48/s46/s48\n/s87/s97/s118/s101/s110/s117/s109 /s98/s101/s114/s32/s49/s47 /s44/s32 /s109 /s45 /s49/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s109/s97/s120\n/s109/s105/s110\n/s40/s98/s41\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s87/s97/s118/s101/s110/s117/s109 /s98/s101/s114/s32/s49/s47 /s44/s32 /s109 /s45 /s49/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s48/s46/s48/s48/s48/s50/s48/s48/s46/s48\n/s109 /s105/s110/s109 /s97/s120\n/s40/s99/s41\nFIG. 6. a) Dispersion curves for MSSW that propagate in continuous F, S/F, and S/F/S \flms. Dispersion curves that are\nobtained numerically are shown with solid lines. Dispersion curves that are obtained using analytical expression are shown with\ndashed lines. b,c) Spin-wave spectra of S/F/S based magnonic crystals that are formed in MSSW geometry with the lattice\nparametera= 1\u0016m. Inserts in (b,c) show schematic illustrations of the considered magnonic crystals.\nwavenumbers, while the proximity-induced anisotropy is\ndominating in vicinity to 0 wavenumbers.\nFigures 6b,c show the spin-wave spectrum of S/F/S-\nbased magnonic crystals where periodicity of the dis-\npersion is reached by periodic location of S/F/S-three-\nlayered areas. Figure 6b shows the spectrum of the hy-\nbrid magnonic crystal that consists of alternating F and\nS/F/S sections (see the inset) with the lattice period\na= 1\u0016m, the width of F-section 0.5 \u0016m, and the thick-\nness of S-layers 120 nm. Calculating this spectrum both\nthe diamagnetic representation of S-stripes18,19and local\nS/F/S-induced anisotropy are considered. The spectrum\ncan be characterized as conventional one: it consist of\nallowed and forbidden bands, the forbidden bands are\nopened at Brillouin wavenumbers 1 =2a. The width of\nband gaps reduces at higher frequencies. For instance,\nthe \frst (lower-frequency) band gap is of width about\n1.8 GHz, and the second band gap is of width 1 GHz.\nFigure 6c shows the spectrum for an alternative re-\nalization of the hybrid magnonic crystal, which consists\nof alternating F/S and S/F/S sections (see the inset).\nLower S-subsystem forms a continuous layer, so the struc-\nture is spatially asymmetric in respect to the z-axis. For\nthis structure similar geometrical parameters are consid-\nered: the lattice period a= 1\u0016m, the width of F/S-\nsection 0.5 \u0016m, and the thickness of the upper S-layers\n120 nm. Calculating this spectrum the diamagnetic rep-\nresentation of S-stripes18,19, the image method17,51have\nbeen used for \fnite-size and continuous superconduct-\ning elements, respectively. The e\u000bect of the proxim-\nity in S/F/S sections is represented by the same local\nanisotropyHs. The spectrum for this spatially asymmet-\nric structure is di\u000berent. The spectrum consist of allowed\nand forbidden bands. The forbidden bands are of similar\nwidth as in Fig. 6b: the \frst (lower-frequency) band gap\nis of width about 1.7 GHz, and the second band gap isof width 0.9 GHz. However, spatial asymmetry induces\nnonreciprocity of the spectrum and indirect location of\nband gaps away from Brillouin wavenumbers.\nIt should be noted that in both cases the e\u000bect of the\nproximity in S/F/S sections is dominating for formation\nof band gaps: in absence of this e\u000bect forbidden bands\nare not obtained. This can be explained by a rather weak\ndiamagnetic response of S-subsystems on spin waves with\nconsidered wavelength. However, diamagnetic response\nof S-subsystems does a\u000bect frequency and wavenumber\nposition of allowed and forbidden bands .\nAs a \fnal remark we should note that for magnonic\ncrystals with thicker F-layers the bandwidth of the for-\nbidden bands is expected to increase correlating with\nthe zero-temperature anisotropy \feld. In particular, the\nbandwidth of the forbidden bands for a S/F/S-based\nmagnonic crystal with F-layer of thickness of a few hun-\ndreds of nm is expected to be comparable with values for\nbi-componental magnonic crystals52,53.\nVI. CONCLUSION\nSummarizing, magnetization dynamics is studied\nin superconductor/ferromagnet/superconductor multi-\nlayers in presence of superconducting proximity. It\nis shown that superconductivity in S/F/S three-layers\nshifts the FMR to higher frequencies. Presence of\nboth S-layers and proximity at both S/F interfaces are\nmandatory for the phenomenon. The frequency shift\nis quanti\fed by the proximity-induced positive in-plane\nanisotropy Haand by a drop of e\u000bective magnetization\nMeff. BothHaand the drop of Meffare comparable.\nThe phenomenon shows a dependence on the thickness of\nthe F-layer: for thicker F-layer the shift of the FMR fre-\nquency is substantially stronger. For two studied samples9\nwith thickness of the F-layer 45 and 350 nm the highest\nnatural FMR frequencies and corresponding anisotropies\nare reached among in-plane magnetized ferromagnetic\nsystems. At the current stage even a qualitative explana-\ntion of the e\u000bect of superconducting proximity in S/F/S\nsystems on magnetization dynamics is unavailable.\nApplication of the proximity-induced anisotropies for\nmanipulation with the spin-wave spectrum is demon-\nstrated for continuous \flms and periodic magnonic\ncrystals. 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We solve thecorresponding Fokker-Planckequ ationand showthat thenon-equilibrium\nnoise yields to a non-monotonous dependence of the precessi on spectrum linewidth on the current.\nMagnetization dynamics of a ferromagnet under inﬂu-\nence of a spin polarized current is a subject of intensive\ninvestigations (for recent reviews see Refs. [1, 2]). It\nwas realized [3, 4] that the spin current may transfer the\nangular momentum to the ferromagnet, resulting in a\ntorque acting on its magnetization direction. In the case\nof a small ferromagneticdomain the torque may lead to a\nrotation of the magnetization as a whole, rather than to\nan excitation of spin waves. This phenomenon, allowing\nfor an electronic manipulation of the magnetization, has\na promise for a number of potential applications.\nThe eﬀect has been recently observed [5, 6, 7, 8, 9] in a\nsetup, wherethe spin-torquemagnitudeanddirectionare\ntuned to compensate exactly the dissipation force acting\non the magnetization of the ”free” ferromagnetic layer.\nThis leads to an undamped precession which is detected\nthroughthe induced microwaveradiation. Boththe spec-\ntral width and the generated power exhibit a strong de-\npendence on the current ﬂowing through the interface of\nthe twoferromagnets. Itwasshownlater[10, 11] thatthe\nequilibriumthermalnoise, ﬁrstintroducedindynamicsof\nmicromagnets by F-L. Brown [12], may partially account\nfor the observed linewidth.\nOn the other hand, since the experiments are per-\nformed under non-equilibrium conditions (spin current\nstrong enough to balance the dissipation), one needs to\naddress non-thermal sources of noise as well. The most\nessentialofthemisthe spin shot noise associatedwiththe\ndiscreteness of spin passing through the interface. This\nphenomenon may be accounted for by adding a ﬂuctuat-\ning part to the spin current vectorin the Slonczewski’s\ntorque term of Landau-Lifshitz-Gilbert (LLG) equation\nIs→Is+δIs(t). The resulting stochastic LLG equation\nfor the unit vector m=M/Min the direction of the\nmagnetization Mtakes the form\ndm\ndt=−γ[m×Heﬀ]+α(θ)/bracketleftbigg\nm×dm\ndt/bracketrightbigg\n+γ\nMV/bracketleftbig\nm×[(Is+δIs)×m]/bracketrightbig\n. (1)\nHereγis gyromagnetic ratio, Heﬀ=−∂F/∂Mis the\neﬀective magnetic ﬁeld, which includes both an external\nﬁeld and magnetic anisotropy, and Vis a volume of thefree ferromagnet. Gilbert damping α(θ) is renormalized\nby the coupling to the ﬁxed ferromagnet [2, 13] and is\nthus dependent on a relative orientation angle θof the\nﬁxed and free ferromagnets.\nOne could expect that the ﬂuctuating part of the spin\ncurrent vector δIs(t) is preferentially directed along the\nspin polarization of the incoming electron ﬂux, i.e. along\nIs. This is notthe case, however, due to the quantum\nnature of the eﬀect. Indeed, each spin-ﬂip event transfers\nexactly one /planckover2pi1unit of the angular momentum to the free\nferromagnet. Due to the uncertainty principle, direction\nof an ensuing magnetization rotation is completely ran-\ndom. As a result, the ﬂuctuating part of the spin torque\nmust have an isotropic correlator\n∝angbracketleftδIs,iδIs,j∝angbracketright= 2D(θ)δijδ(t−t′), (2)\nwherethevariance D(θ) doesnotdependonthe cartesian\nindexesi,j=x,y,z, but may depend on the mutual ori-\nentationofthe twoferromagnets. Becauseofthe isotropy\nof the stochastic torque, it can be equally well repre-\nsented by a ﬂuctuating magnetic ﬁeld Heﬀ→Heﬀ+h(t),\ninstead of the ﬂuctuating spin ﬂux δIs(t). In the lat-\nter case the correlator of the stochastic ﬁelds reads as\n∝angbracketlefthi(t)hj(t′)∝angbracketright= 2D(MV)−2δijδ(t−t′). This type of non-\nequilibrium noise was considered in Ref. [19] in context\nof NFN structures.\nHereweconsideramodelofamagnetictunnel junction\n(MTJ) Ref. [14]. Magnetization dynamics of the free\nferromagnet is described using Holstein-Primakoﬀ (HP)\nparametrization [15] of its total spin operator by deriv-\ningsemiclassicalequationsofmotionsforHP bosons. We\nemploy Keldysh formalism to allow for non-equilibrium\nconditions, i.e. voltage bias between the two ferromag-\nnets [16, 17, 18]. After integrating out the fermionic\ndegrees of freedom in the second order in both tunnel-\ning and spin-ﬂip processes, we obtain an eﬀective ac-\ntion for HP bosons, which encapsulates deterministic\nforces (external magnetic ﬁeld and spin-torque) along\nwith the stochastic term. The latter containsaninforma-\ntion about both equilibrium and non-equilibrium noise\ncomponents.\nThe result of the program, outlined above, is the fol-2\nlowing noise correlator\nD(θ) =MV\nγα0T+/planckover2pi1\n2Isf(θ)coth/parenleftbiggeV\n2T/parenrightbigg\n,(3)\nwhereα0is a bare Gilbert damping of an isolated grain\nandVis a voltage bias between the two ferromagnets.\nThe non-equilibrium part of the noise is proportional to\nthe spin-ﬂipcurrentI sf(θ). Thelattercountsthe number\nofrealspin ﬂip events (as opposed to the spin current I s\nwhich is associated with virtual spin ﬂips). In the MTJ\nsetup we found for the spin-ﬂip conductance\ndIsf(θ)\ndV=/planckover2pi1\n4e/bracketleftbigg\nGPsin2/parenleftbiggθ\n2/parenrightbigg\n+GAPcos2/parenleftbiggθ\n2/parenrightbigg/bracketrightbigg\n; (4)\nGP=G+++G−−;GAP=G+−+G−+,\nwhere we adopted notations of Ref. [14] for the par-\ntial conductances Gσσ′between the spin-polarized bands\nof the two ferromagnets. Here GP/APstay for elec-\ntric conductances in parallel (P) and antiparallel (AP)\nconﬁgurations, with GP≥GAP. The electric conduc-\ntance of the MTJ in an arbitrary orientation is given by\ndIe(θ)/dV=GPcos2(θ/2)+GAPsin2(θ/2). Notice that\nthe spin shot noise is minimal for P orientation and max-\nimal for AP one – exactly opposite to the charge current\nand the chargeshot noise. In the same notations the spin\nconductance is [14]\ndIs\ndV=/planckover2pi1\n4e(G++−G−++G+−−G−−),(5)\nwhere the angular dependence is already explicitly taken\ninto account in Eq. (1). A non-polarized current, i.e.\nG+σ=G−σ, does not exert deterministic spin torque,\nnevertheless it still induces the shot noise torque on the\nferromagnet. This eﬀect was recently discussed in the\ncontext of NFN structures [19].\nThe same calculation also leads to a renormalization\nof Gilbert damping coeﬃcient in LLG equation (1)\nα(θ) =α0+/planckover2pi1γ\neMV/parenleftbiggdIsf(θ)\ndV/parenrightbigg\n, (6)\nwhere the spin-ﬂip conductance is given by Eq. (4).\nSuch an enhancement of the dissipation due to the spin\ntransport between the two ferromagnets is discussed in\nRefs. [2, 13]. In compliance with the ﬂuctuation–\ndissipation theorem, the equilibrium ( V→0) noise cor-\nrelator is given by D=TMVα(θ)/γ[12]. Due to the\nrenormalization, the damping is minimal in P orienta-\ntion. The angular dependence of the damping term se-\nlects a unique angle, where the spin-torque compensates\ndissipation. Upon increasing the spin current above the\ncritical one, such an angle gradually increases from zero\ntoπ.Below we restrict ourselves to a setup, where both the\nmagnetic ﬁeld and the spin current are directed along\nthez-axis speciﬁed by the magnetization direction of the\nﬁxed ferromagnet. Transforming Eq. (1) to the spher-\nical coordinates [20], one obtains two coupled Langevin\nequations for the relative angle θ(t) and the azimuthal\nangleφ(t)\n˙θ=−sinθTz(θ)+cotθ˜D(θ)+ξθ(t),(7)\n˙φ= Ω(θ)+cscθξφ(t); (8)\nTz(θ) =γ′(α(θ)Hz+Is/(MV)) ; Ω = γHz−αTz,\nwhereγ′=γ/(1+α2) and˜D(θ) =D(θ)γγ′/(MV)2. Here\nξθ(t) andξφ(t) are two uncorrelated random noises with\nthe correlators\n∝angbracketleftξθ(t)ξθ(t′)∝angbracketright=∝angbracketleftξφ(t)ξφ(t′)∝angbracketright= 2˜D(θ)δ(t−t′).(9)\nBoth Eq. (1) and Eqs. (7), (8) should be interpreted in\nthe sense of retarded regularization, or Ito calculus [20].\nLet us ﬁrst analyze deterministic dynamics described\nby Eqs. (7), (8) with ˜D →0. For a strong enough\n(and negative for positive Hz) spin current the condition\nTz(θ) = 0 may be satisﬁed for a certain angle ¯θ. Notice\nthat it is the angulardependence of the enhanced Gilbert\ndamping [2, 13], which is responsible for the angle selec-\ntivity. InsuchacaseEq.(8)describesastableundamped\nprecession with the frequency Ω( ¯θ) =γHz. The inten-\nsity of the induced microwave radiation [5, 6, 7, 8, 9] is\ngiven by the square of the oscillating magnetic moment,\ni.e. proportional to sin2¯θ.\nTo analyze eﬀects of the noise we shall assume that\nα(¯θ)≪1, allowing for time scales separation. The fast\nvariable is the azimuthal angle φ(t), while the angle θ(t)\nis the slow one. For a ﬁxed slow variable θEqs. (8), (9)\nlead to the Lorentzian shape of the emitted microwave\npower spectrum\nS(ω,θ)∝sin2θ2csc2θ˜D(θ)\n[ω−Ω(θ)]2+/bracketleftBig\ncsc2θ˜D(θ)/bracketrightBig2.(10)\nTo compare with the observed power spectrum this ex-\npression should be averaged over the stationary prob-\nability distribution P(θ) of the slow degree of freedom\nS(ω) =/integraltext\nsinθdθP(θ)S(ω,θ). The distribution function\nP(θ,t) obeys the Fokker-Planck equation which follows\nfrom Eqs. (7), (9) [12, 20]\n˙P=1\nsinθ∂θ/bracketleftBig\nsin2θTz(θ)P+sinθ∂θ/parenleftBig\n˜D(θ)P/parenrightBig/bracketrightBig\n.(11)\nThe stationary solution of Eq. (11) is given by\nP(θ) =1\nZ˜D(θ)exp/braceleftBigg\n−/integraldisplayθ\n0sinθ′dθ′Tz(θ′)\n˜D(θ′)/bracerightBigg\n,(12)3\nwhere constant Zis chosen to satisfy the normalization\ncondition/integraltextπ\n0sinθdθP(θ) = 1. For a weak noise the sta-\ntionary distribution function has a sharp maximum close\nto the angle ¯θ, where the deterministic spin-torque com-\npensates the dissipation.\nFigure 1 shows the calculated spectral linewidth at the\nhalf maximum as a function of the applied voltage. The\norigin of the non-monotonous dependence may be un-\nderstood by inspection of Eq. (10). The initial decline\nis due to the geometric factor csc2θin the width of the\nLorentzian. As the voltage increases, so does the angle\nwhere the distribution function exhibits the maximum.\nDue to csc θ, coming from the noise term on the r.h.s. of\nEq. (8), the amplitude of the phase noise decreases lead-\ning to a narrowingof the powerspectrum. The initial de-\ncrease of the spectral width was discussed in Ref. [10, 11]\nin terms of the equilibrium thermal noise [12].\nThe subsequent increase of the spectral width at yet\nlarger voltages was observed in a number of experiments\n[6, 7, 8] and, to the best of our knowledge, remained\nunexplained. It naturally comes about due to the non-\nequilibrium component of the noise. Indeed the noise\ncorrelator (3) grows with the applied voltage due to the\ngrowth of the spin-ﬂip current. The dependence of the\nspin-ﬂip current I sf(θ) on the bias is faster than the lin-\near because of the increase of the spin-ﬂip conductance,\nEq. (4), with the operation angle ¯θon top of the over-\nall proportionality of I sfto the bias V. As a result, for\neV/greaterorsimilar2Tthe noise variance Dgrows rapidly, leading to\nthe broadening of the power spectrum, cf. Eq. (10).\nAnother consequence of the non-equilibrium noise is\nthe saturation of the spectral linewidth at small tem-\nperatures [9]. Since the devices are always operated at\ncurrents larger than the critical one, the noise intensity\n(3) is ﬁnite D(θ) = (/planckover2pi1/2)Isf(θ) even at T→0. Thus de-\ncreasing the temperature one should observe saturation\nof the linewidth at T∼eV, provided the induced damp-\ning (the second term on the r.h.s. of Eq. (6)) is larger\nthan the bare one.\nIn the remainder of the paper we outline derivation of\nEqs. (1) – (6). The MTJ is modeled by the two itiner-\nant ferromagnets, whose majority ( σ= +) and minority\n(σ=−) bands aredescribed by the operators c†\nkσ,ckσfor\nthe ﬁxed ferromagnet and d†\nlσ,dlσfor the free layer. We\nfound convenient to work in the instantaneous reference\nframe, where the magnetization of the free layer points\nin thez-direction. The corresponding Hamiltonian takes\nthe following form\nH0=/summationdisplay\nk,σǫkσc†\nkσckσ+/summationdisplay\nlσ(ǫl−JSzσ)d†\nlσdlσ−γS·H\n−J(S+s−+S−s+)+\n/summationdisplay\nkl,σσ′Wσσ′\nklc†\nkσdlσ′+h.c.\n,\nhere the spin-dependent tunneling matrix elements are/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s44/s48/s49/s48/s48/s44/s48/s49/s53/s48/s44/s48/s50/s48/s48/s44/s48/s50/s53\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56\n/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48/s57/s48/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32 ∆ω\n/s86/s111/s108/s116/s97/s103/s101/s32/s86/s32/s78/s111/s114/s97/s109/s108/s105/s122/s101/s100/s32/s112/s111/s119/s101/s114/s32/s32/s112\n/s32/s86/s111/s108/s116/s97/s103/s101/s32/s86\n/s32/s65/s118/s101/s114/s97/s103/s101/s32/s97/s110/s103/s108/s101/s32/s60 θ>\nFIG. 1: (Color online) Calculated linewidth of the microwav e\nspectral power vs. applied voltage in units of temperature.\nInset: calculated microwave power (solid line) and the av-\nerage precession angle (dashed line) vs. voltage. Parame-\nters:MV/(/planckover2pi1γ) = 10, /planckover2pi1γHz= 3K,T= 1K,α0= 0.01.\nConductances in units e2/h:GP= 0.181,GAP= 0.019.\ndIs/dV= 0.01e.\nWσσ′\nkl=∝angbracketleftσ|σ′∝angbracketrightW, where the spin-transformation matrix\nis∝angbracketleftσ|σ∝angbracketright=e−iσφ/2cosθ/2 and∝angbracketleftσ|σ′∝angbracketright=eiσφ/2sinθ/2;\nands=1\n2/summationtext\nlσσ′d†\nlσ/vector σσσ′dlσ′is the spin of itinerant elec-\ntrons, while s±=sx±isy. We have explicitly accounted\nfor the interactions of the itinerant electrons in the free\nlayer with its totalspinS=MV/γ. To make the latter\na dynamical variable we use HP parametrization [15]\nSz=S−b†b;S−=b†/radicalbig\n2S−b†b;S+=/radicalbig\n2S−b†bb,\nwhereb†,bare usual bosonic operators.\nNext we write the corresponding action in terms of\ncomplex fermionic and bosonic ﬁelds ckσ(t),dlσ(t),b(t),\nwhere the time variable runs along the closed Keldysh\ncontour [16, 17, 18]. We then transform to two-\ncomponent vector notations in terms of symmetric (clas-\nsical ”cl”) and antisymmetric (quantum ” q”) combina-\ntions of the forward and backward propagating ﬁelds.\nOne should keep in mind that the distribution functions\nof thecanddfermions have a relative shift of the chem-\nical potentials by eV. The fermionic ﬁelds may be inte-\ngrated out exactly and the remaining bosonic eﬀective\naction expanded to the second order in the tunneling\namplitude Wand to the ﬁrst and second orders in the\nspin-ﬂip processes S±s∓. The corresponding processes\nare represented by the diagrams of Fig. 2. The approx-\nimations are justiﬁed by the weakness of tunneling and\nlargeness of S≫/planckover2pi1.\nThe resulting action for the complex bosonic ﬁelds\nbcl(t),bq(t) takes the form S=S0+S1+S2where the\nsubscript indicates the order in spin-ﬂips processes. Here4\nFIG. 2: Diagrams for spin-ﬂip processes: (a) the ﬁrst order,\ndescribing the spin-torque; (b) the second order, describi ng\nthe spin shot noise along with the enhanced damping. Solid\n(dashed)lines denote electronic propagators in thefree (ﬁ xed)\nlayers. Bold dashed lines are propagators of HP bosons (spin -\nﬂips). Tunneling vertices are denoted by circles with cross es.\nthe bare action is [21]\nS0=/integraldisplay\ndt¯bq(t)/parenleftBig\ni∂tbcl(t)+γ/radicalbig\nS/2H+/parenrightBig\n+c.c.(13)\nThe ﬁrst order correction in spin-ﬂip amplitude is repre-\nsented by diagram of Fig. 2a. This is a virtual transition\ninto an opposite spin band with a subsequent tunneling\nout of the free layer into the ”correct” spin band of the\nﬁxed magnet. The latter process is possible for θ∝negationslash= 0,π,\ndue to a ﬁnite Wσσ′\nkl. The net result is transferring an-\ngular momentum /planckover2pi1to the total spin of the free layer, i.e.\nthe deterministic spin-torque [3, 4]. The corresponding\ncontribution to the action is\nS1=i√\n2S/integraldisplay\ndt¯bq(t)Issinθe−iφ+c.c., (14)\nwhere I sis given by Eq. (5) with Gσσ′=4πe2\nh|W|2νσ\ncνσ′\nd\nandνσ\nc,dare densities of states of the two ferromagnets\nin theσband. The second order processes in spin-ﬂips\nare depicted by the diagram of Fig. 2b. These are real\n(i.e. Golden rule) processes, which matrix elements in-\nclude the spin-ﬂips. They lead to dissipation as well as\nﬂuctuations. The corresponding action is\nS2=/integraldisplay\ndt/bracketleftbigg\nα(θ)(¯bq∂tbcl−¯bcl∂tbq)+2i\nSD(θ)¯bqbq/bracketrightbigg\n,(15)\nwhereD(θ) andα(θ) are given by Eqs. (3), (4) and (6)\n(without internal dissipation α0).\nOne then decouples the last term on the r.h.s. of\nEq. (15) by means of the complex Hubbard-Stratonovich\nﬁeldδI+(t) = Is,x+iIs,y. The remaining action is linear\ninbq(t) and¯bq(t). It constitutes thus resolution of func-\ntionalδ-functions of the ﬁrst order Langevin equations\nonbcl(t) =/radicalbig\nMV/(2γ)m+(t) and its complex conjugate.\nThose are nothing but m±components of Eq. (1), with\nthe noise intensity given by Eqs. (2)–(4) [21].\nTo conclude: in the presence ofthe spin-torque, caused\nby a spin polarized current, the LLG equation should be\nmodiﬁed to include a stochastic Langevin term whichaccounts for the shot-noise, associated with the spin cur-\nrent. This term is diﬀerent from the previously dis-\ncussed thermal stochasticity in LLG equation [12], be-\ncause of its non-equilibrium origin. We have derived the\ncorresponding noise correlator in the MTJ setup. We\nhave argued that the non-equilibrium noise manifests it-\nself in a non-monotonous voltage dependence and low-\ntemperature saturation of the linewidth of the precession\npower spectrum.\nWe are grateful to P. Crowell, A. Levchenko, D.\nPfannkuche and V. Kagalovsky for useful discussions.\nA. C. and J. S. acknowledge ﬁnancial support from\nDFG through Sonderforschungsbereich 508 and Sonder-\nforschungsbereich 668. A.K. was supported by the NSF\ngrant DMR-0405212 and by the A. P. Sloan foundation.\n[1] D. C. Ralph and M. D. Stiles, arXiv:cond-mat/0711.4608\n(2007).\n[2] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375, (2005).\n[3] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1(1996).\n[4] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[5] S. I. Kiselev et al., Nature 425, 380 (2003).\n[6] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n[7] W. H. Rippard, M. R. Pufall, and S. E. Russek, Phys.\nRev. B 74, 224409 (2006).\n[8] Q. Mistral et al., Appl. Phys. Lett. 88, 192507 (2006).\n[9] J. C. Sankey, I. N. Krivorotov, S. I. Kiselev, P. M. Bra-\nganca, N. C. Emley, R. A. Buhrman, and D. C. Ralph,\nPhys. Rev. B 72, 224427 (2005).\n[10] V. Tiberkevich, J. Kim, and A. Slavin, ArXiv:cond-\nmat/0709.4553 (2007).\n[11] J. Kim, V. Tiberkevich, and A. N. Slavin, Phys. Rev.\nLett.100, 017207 (2008).\n[12] W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963).\n[13] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[14] J.C. Slonczewski and J.Z. Sun, J. Magn. Magn. Mater.\n310, 169 (2007).\n[15] T. Holstein and H. Primakoﬀ, Phys. Rev . 58, 1098\n(1940).\n[16] L. V. Keldysh, Zh. Eksp. Theor. Fiz. 47, 1515 (1964);\n[Sov. Phys. JETP 20, 1018 (1965)].\n[17] A. Kamenev, Many-body theory of non-equilibrium\nsystems, inNanophysics: Coherence and Transport ,\nH. Bouchiat, et al.(editors); pp. 177-246, Elsevier, Am-\nsterdam, 2005.\n[18] R.A. Duine, A.S. Nunez, J. Sinova and A.H. MacDonald\nPhys. Rev. B 75, 214420 (2007).\n[19] J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. Bauer,\nPhys. Rev. Lett. 95, 016601 (2005).\n[20] N.G. van Kampen, Stochastic Processes in Physics and\nChemistry , North-Holland, Amsterdam, 2001.\n[21] Intheinstanteneousreference frame mz= 1andm±= 0,\nwhile ˙mz= 0 and ˙ m±∝negationslash= 0. Therefore all the terms in\nthe action ∼bcl(t)∝m+(t) do not contribute to the\nequations of motion and are omitted. We thus keep only5\nterms∼∂tbclalong with those which contain only bq."    },    {        "title": "0710.5585v1.Devitrification_of_a_glass_like_arrested_ferromagnetic_phase_in_La0_5Ca0_5MnO3.pdf",        "content": "1 \n Devitrification of a glass-like arrested ferromagnetic phase in La 0.5Ca0.5MnO 3. \nA. Banerjee, Kranti Kumar and P. Chaddah. \nUGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, \nIndore-452017, INDIA. \n \nAbstract  \nMagnetization measurements in La 0.5Ca0.5MnO 3 manganite show that the high-\ntemperature long-range ferromagnetic-metallic phase transforms to antiferromagnetic-insulating phase, although a fraction of ferromagnetic-metallic phase undergoes glass-like kinetic arrest and coexists at low temperature with the equilibrium antiferromagnetic-insulating phase. We show here through resistivity measurements that the residual arrested ferromagnetic-metallic fraction can be converted to the equilibrium antiferromagnetic-insulating phase by successive annealing at higher temperatures, possibly through heterogeneous nucleation of equilibrium phase. Significantly, larger fractions of this glassy ferromagnetic-metallic phase can be obtained by cooling in higher fields and larger conversion to equilibrium antiferromagnetic-insulating phase results.  \n \nGlasses form below a temperature where dynamics is arrested, preserving the high-\ntemperature structure while avoiding the first-order liquid-solid transformation at the \nhigher temperature T\nC. Rapid cool-down is needed to arrest the transformation kinetics for \nmetallic alloys, while glass-formers like O-terphenyl vitrify easily because the kinetics is \narrested at a temperature T K that is closer to T C1-3. Arrest of kinetics can also inhibit a first-\norder process where both the phases on either side of the transition have long-range \nstructural, including magnetic, order. It has been shown recently that glass-like arrest of \nkinetics intervenes in first-order magnetic transformations and results in coexisting phases \nwith competing magnetic orders4-8. In La 0.5Ca0.5MnO 3 manganite, the high-temperature \nlong-range ferromagnetic-metallic phase transforms to antiferromagnetic-insulating phase, \nalthough a fraction of ferromagnetic-metallic phase coexists at low temperature9. \nMagnetization measurements show that this ferromagnetic-metallic phase has undergone \nglass-like kinetic arrest and coexists at low temperature with the equilibrium \nantiferromagnetic-insulating phase, similar to other half-doped manganites8,10. We show \nhere through resistivity measurements that the residual arrested ferromagnetic-metallic 2 \n fraction can be converted to the equilibrium antiferromagnetic-insulating phase by \nsuccessive annealing at higher temperatures1. The restoration of kinetics in the arrested \nphase presumably results in heterogeneous nucleation of equilibrium phase. We obtained \nlarger fractions of this glassy ferromagnetic-metallic phase by cooling in higher fields and \nfound better conversion to equilibrium antiferromagnetic-insulating phase, reminiscent of \ndevitrification of structural glasses into crystallites. The T C can be varied over a wide range \nwith magnetic field and the difference between T K and T C can be tuned in the same sample. \nThis advantage may help shed light on the physics of structural glasses. \n \nAlthough glasses have been used for many centuries and variety of materials \nincluding monatomic metallic liquids are capable of glass formation11, a quantitative \nunderstanding of the glass transition remains a major scientific challenge. Glasses form by \nfreezing the structure of a liquid; but all amorphous solids are not glass1,2. The general \nprinciple underlying diverse glasses is the arrest of the kinetics of the first-order transition, \nwhich connects two symmetrically incompatible orders, freezing the higher energy state \ninto a non-ergodic state at low temperature.  The concept of glass-like arrest of kinetics has \nrecently been invoked to successfully explain various anomalies in disorder broadened \nfirst-order magnetic transitions as a function of field and temperature for materials like colossal magnetoresistance manganites\n5-8,10.  These have long-range magnetic orders on \neither side of the transition. However, the high-temperature phase persists in the low-\ntemperature region where it is energetically unstable, and the lack of dynamics triumphs \nover thermodynamics. In pure systems, the first-order transition can occur along a sharply \ndefined (H C, T C) line in the 2-control variable (H, T) space. Due to disorders, different \nregions having length-scale of the order of the correlation-length can have different T C \nresulting in the transition line broadening into a band; as also do the lines corresponding to supercooling (H*, T*) and superheating (H**, T**) limits\n12,13. Similarly, if the glass-like \nkinetic arrest is to occur below a (H K, TK) line for a given cooling rate in the pure system, \ndisorder broadens this into a band consisting of quasi-continuum of (H K, T K) \nlines5,7,8,10,14,15.  By traversing the 2-control variable H-T space, tunable coexisting \nfractions of arrested and equilibrium phases have been observed because of this disorder \nbroadening. Devitrification is an evidence of glassy state1-3 which is recently demonstrated \nfor such glassy magnetic state of variety of systems7,8,10,14. The present study focuses on a 3 \n half-doped manganite, La 0.5Ca0.5MnO 3, similar to the sample  where Loudon et al. have \nproposed a ‘nucleation and growth’ process for transformation from high-temperature \nferromagnetic-metallic (FMM) to low-temperature antiferromagnetic-insulating (AFI) \nphase9. Coexistence of FMM with AFI regions at low temperature was shown, and here we \nexplore their origin and nature. \n \nPolycrystalline La 0.5Ca0.5MnO 3 sample has been prepared through a well-established \nchemical route known as ‘pyrophoric method’. High purity (>99.9%) La 2O3, CaCO 3 and \nC4H6MnO 4.4H 2O are taken in stoichiometric quantities as starting materials. These \nmaterials are dissolved in aqueous nitric acid and the resulting solutions are mixed together \nwith triethanolamine (TEA). The complex solution is heated to dehydrate and decompose \nleaving behind organic-based, black, fluffy precursor powder. This dried mass is then \ngrounded to fine powder, palletized and then calcined at 10000C for 3 hrs in oxygen \natmosphere. The powder x-Ray diffraction (XRD) was carried out using an 18 kW Rigaku \nRotaflex RTC 300 RC diffractometer with Cu-K α radiation. Rietveld profile refinement of \nXRD pattern confirms that the sample is in single phase without any detectable impurity \nand crystallizes in orthorhombic structure with ‘pnma’ space group. The resistivity and \nmagnetic measurements are performed using commercial set-ups (14Tesla-PPMS-VSM, \nM/s. Quantum Design, USA).      \n \nFigure-1(a) shows the magnetization as a function of temperature measured in 1 \nTesla field under various protocols. The hysteresis between field-cooled cooling (FCC) and \nfield-cooled warming (FCW) paths indicates a disorder-broadened first-order transition \nfrom FMM to AFI with reducing temperature. However, a substantial magnetization in the \nlow temperature antiferromagnetic phase, similar to Ref. 9, suggests the persistence of \nferromagnetic phase. Because of disorder-broadening of (H*, T*) and (H K, TK) lines into \nbands, the fraction of glass-like arrested state can be controlled by cooling in different \nfields5-8, 15.  Based on the measurements similar to those reported earlier7,8 in the glass-like \nkinetically arrested magnetic systems, we propose that the AFI state is in equilibrium at \nlow temperature and the FMM phase fraction exists as kinetically arrested glassy or non-\nergodic state. We can collect larger fractions of glass-like arrested FMM phase at 5K by \ncooling in 6T field and then reducing field to 1T. This glass-like arrested FMM state 4 \n devitrifies on heating as depicted in Fig.1 by the rapid fall in magnetization, of the 6T-\ncooled states, which approach the equilibrium AFI phase while warming. This half-doped \nmanganite should have the spin-aligned value of 3.5 µB/Mn accompanied by metallic \nconductivity9. The observed 1T-FCC magnetization of 0.61 µB/Mn at 5K can be attributed \nto a frozen FMM phase fraction of about 17%, which is close to the percolation threshold \nfor electrical conductivity. Hence, around this FMM phase fraction the drastic resistivity \nchanges are a more sensitive tool, compared to the magnetization, to probe small changes \nin the phase fractions. Figure 1(b) shows the heating and cooling cycles of the zero-field \nresistivity. Apart from the expected thermal hysteresis, the decrease in resistivity with the \ndecrease in temperature below ~ 70K reflects the presence of FMM phase fraction in AFI \nmatrix. Similar to magnetization, the zero-field resistivity also shows larger fraction of \nFMM phase when the sample is cooled in 6T (Fig. 1b). Different values of magnetization \nand resistivity in the same measurement temperature (5K) and field (1T for magnetization \nand zero for resistivity) indicate the presence of non-ergodic states. The concomitant sharp \ndecrease in magnetization and increase in resistivity, of the 6T cooled state, around 20 K as \nshown in Fig. 1 (a) and (b) results when this arrested non-ergodic FMM phase devitrifies to \nAFI phase on warming8.   \n \nThis glass-like arrested FMM state resembles the conventional structural glasses in \nvarious respects. We now show that it partially crystallizes to equilibrium AFI state after \nannealing, similar to the route followed to produce nano-crystals within the metallic glasses or to the formation of glass ceramic\n1.  Fig 2 (a) shows resistivity in zero-field while \nwarming after cooling from 320K in zero field and again after a temperature cycle from 5K \nto 140K to 5K without applying any field. The increase in resistivity after annealing to \n140K indicates that the system is ripening to its equilibrium state similar to crystallization, \nin this case to AFI phase at low temperature. For conventional systems such an effect can \ntake place from heterogeneous nucleation process. When contrasted with the initial cool-\ndown, we now have an abundance of AFI seeds to trigger this process when energy \nbecomes available to the system16. It is well known that larger fraction of a glass can be \ndevitrified from a partially crystalline state by successive annealing1. Similar effect of \nsuccessive annealing at progressively higher temperatures has been observed in resistivity \n(as also in magnetization, not shown here). Fig. 2(b) shows the resistivity in zero-field at 5 \n 5K as a function of progressively higher annealing temperature after cooling the sample to \n5K in zero field.  Each temperature cycle to progressively higher temperature produces \nlarger factions of AFI phase and the system approaches the equilibrium state overcoming a \nhierarchy of barriers. Back conversion to FMM phase occurs, however, as the annealing \ntemperature is raised above 150K. \n \nWe now investigate whether the devitrification and heterogeneous nucleation is \ninfluenced by the initial glass-like kinetically arrested FMM phase fraction. This fraction \ncan be tuned by cooling in different fields8; is about 17% when cooled in a field of 1T, but \nrises to about 90% when cooled in 6T [Figure 1(a)]. This fraction remains fixed when the \nfield is reduced to zero at 5K. We show in Figures 3(a) and 3(b) the results of annealing to \n140K on these two cases. The initial higher FMM phase fraction for higher cooling field is \nevidenced by lower resistivity at 5K in zero field. However, after annealing to 140K the \nbehaviour reverses, the lower resistivity of the 6T-cooled state becomes much higher \ncompared to 1T-cooled state and exceeds the measurement range of the instrument below \n38K. Similarly, the 1T-cooled state, initially having lower resistivity compared to zero \nfield cooled state, shows higher resistivity after annealing to 140K [Figures 2(a) and 3(a)]. \nThe effect of successive annealing on 6T- and 1T-cooled states is shown in Figure 3(c). Annealing to temperatures higher than 80K clearly brings out a conversion to larger \nequilibrium AFI state when the starting glass-like kinetically arrested FMM phase fraction \nis larger. The effect is drastic for annealing temperature of 120K and beyond. This result \nmay appear counterintuitive but may have some connection with the folklore that “hot \nwater can freeze faster than cold”\n17 or the everyday experience that bigger pieces of glass \nare more susceptible to shattering when dropped. \n \nThe ease with which field (and temperature) can be varied contrasts with the current \nexperimental efforts involved in varying pressure (and temperature) to explore the energy \nlandscape of glassy systems11,18. Since T C varies strongly with magnetic field, so does the \ndifference between T K and T C; cooling in different fields allows us to arrest varying \nfractions of glass-like FMM phase without changing the rate of cooling5-8.  Such glass-like \nkinetic arrest of first-order magnetic transitions, especially where the transition is \nbroadened by disorder, should enable studying the physics of structural glasses.  6 \n  \nReferences : \n1. Greer, A. L. Metallic glasses. Science 267, 1947 (1995). \n2. Brawer, Steven. Relaxation in viscous liquids and glasses (The American \nCeramic Society, Inc., Columbus, Ohio, 1985). \n3. Debenedetti, P. G. and Stillinger, F. H. Supercooled liquids and glass transition. \nNature  410, 259 (2001). \n4. Chattopadhyay, M. K., Roy, S. B. & Chaddah, P. Kinetic arrest of the first-order \nferromagnetic-to-antiferromagnetic transition in Ce(Fe 0.96Ru0.04)2: Formation of \na magnetic glass. Phys. Rev. B  72, 180401(R) (2005). \n5. Kumar, Kranti, Pramanik, A. K., Banerjee, A., Chaddah, P., Roy, S. B., Park, S., \nZhang, C. L. & Cheong, S. –W. Relating supercooling and glass-like arrest of \nkinetics for phase separated systems: Doped CeFe 2 and (La, Pr, Ca)MnO 3. Phys. \nRev. B  73, 184435 (2006).  \n6. Wu, Weida, Israel, Casey, Hur, Namjung, Soonyong, Park, Cheong, Sang-Wook \n& De Lozane, Alex. Magnetic imaging of a supercooling glass transition in a \nweakly disordered ferromagnet. Nature Materials  5, 881 (2006). \n7. Banerjee, A., Mukherjee, K., Kumar, Kranti & Chaddah, P. Ferromagnetic \nground state of the robust charge-ordered manganite Pr 0.5Ca0.5MnO 3 obtained by \nminimal Al substitution. Phys. Rev. B 74, 224445 (2006). \n8. Banerjee, A., Pramanik, A. K., Kumar, Kranti & Chaddah, P. Coexisting tunable \nfractions of glassy and equilibrium long-range-order phases in manganites. J. \nPhys.: Condens. Matter  18, L605 (2006). \n9. Loudon, James C., Mathur, Neil D. & Midgley, Paul A. Charge-ordered \nferromagnetic phase in La 0.5Ca0.5MnO 3. Nature  420, 797 (2002). \n10. Rawat, R., Mukherjee, K., Kumar, K., Banerjee, A. Chaddah, P. Anomalous \nfirst-order transition in Nd 0.5Sr0.5MnO 3: an interplay between kinetic arrest and \nthermodynamic transitions.  J. Phys.: Condens. Matter  19, 256211 (2007). 7 \n 11. Bhat, M. H. et al. Vitrification of a monatomic metallic liquid. Nature  488, 787 \n(2007). \n12. Soibel, Alex, Zeldov, Eli, Rappaport, Michael, Myasoedov, Yuri, Tamegai, \nTsuyoshi, Ooi, Shuuichi, Konczykowski, Marcin & Geshkenbein, Vadim B. \nImaging the vortex melting process in the presence of disorder. Nature  406, 282 \n(2000). \n13. Roy, S. B., Perkins, G. K., Chattopadhyay, M. K., Nigam, A. K., Sohhey, K. J. \nS., Chaddah, P., Caplin, A. D. & Cohen, L. F. First Order Magnetic Transition in \nDoped CeFe 2 Alloys: Phase Coexistence and Metastability. Phys. Rev. Lett.  92, \n147203 (2004). \n14. Roy, S. B. et al. Devitrification of the low temperature magnetic-glass state in \nGd5Ge4. Phys. Rev. B  75, 184410 (2007). \n15. Chaddah, P., Banerjee, A. and Roy, S. B. Correlating supercooling limit and \nglass-like arrest of kinetics for disorder-broadened 1st order transitions: \nrelevance to phase separation. Preprint at <http://www.arXiv.org/cond-\nmat/0601095> (2006). \n16. Cacciuto, A., Auer, S. and Frenkel, D. Onset of heterogeneous crystal nucleation \nin colloidal suspensions. Nature  428, 404 (2004). \n17. Jeng, Monwhea. Hot water can freeze faster than cold?!?. Preprint at \n<http://www.arXiv.org/cond-mat/0512262> (2005). \n18. Mishima, Osamu & Stanley, Eugene. The relationship between liquid, \nsupercooled and glassy water. Nature  396, 329 (1998). \n \nAcknowledgement:  DST, Government of India is acknowledged for funding the 14 \nTesla-PPMS-VSM.  \n 8 \n 0 100 200 30010-310-210-1100101\nMe a s u r i ng field zero Cooling\n Wa rming\n Wa rming \n        after 6T cooled  ρ (kΩ.cm )\nT (K)0 100 200 3000123\nbaM (µB / Mn ) FCC-1T\n FCW -1T\n 6T cooled \nMe a s u r i ng field 1 Te s la\n \nFigure 1:  Temperature (T) dependence of Magnetization (M) in 1 Tesla field and \nresistivity (R) in zero field of La 0.5Ca0.5MnO 3 measured under different protocols. a, M \nvs. T while cooling (FCC) in 1T field from 320K to 5K and again while warming (FCW) from 5K shows the thermal hysteresis accompanying the first-order ferromagnetic to antiferromagnetic transition. After cooling the sample in 6T field the field is reduced isothermally to 1T at 5K and M is measured while warming. The large value of M at 5K reflects a dominant arrested ferromagnetic phase, and its devitrification starts around 20K. b,  R vs. T while cooling in zero field from 320K to \n5K and again while warming from 5K shows the thermal hysteresis accompanying the first-order metallic to insulating transition. After cooling the sample in 6T field, the field is reduced isothermally to zero at 5K and R is measured while warming. The low value of R at 5K reflects a large fraction of arrested metallic phase, and its devitrification starts around 20K. Thus the high temperature ferromagnetic-metallic phase (FMM) undergoes a first-order transition to antiferromagnetic-insulating (AFI) phase at low temperature. However, multivalued M at 5K and 1T and also different values of R at 5K in zero field indicate the presence of non-ergodic states. Their devtrification while warming is indicated by sharp changes in M and R around 20K.  9 \n 0 50 100 1500300600900\nat 5K  & 0Tb  ρ (k Ω.cm )\nA nnealing T (K)  ρ (k Ω.cm )0 100 200 3000204060a Cooled and m easured\nin zero field\n Initial wa rming\n Wa rming after annealing \n           at 140K\nT (K)\n \nFigure 2:   Resistivity as a function of temperature in zero field.  a, Resistivity while \nwarming from 5K after cooling from 320K in zero field. Again the sample is cooled from 320K to 5K, then warmed to 140K and cooled back to 5K. Resistivity is measured while warming after this temperature cycle. The large difference in the resistivity values at low temperature, between the initially cooled state and that after annealing to 140K, arises from the devitrification through the heterogeneous nucleation process. During initial cool-down the entire sample was FMM and the AFI phase formed through homogeneous nucleation. After warming from 5K and annealing at 140K, seeds of AFI phase enabled heterogeneous nucleation, reducing the residual FMM fraction.   b, \nResistivity values at 5K as a function of annealing temperatures. After cooling the sample to 5K, resistivity is measured while cycling the temperature between 5K and the successive higher annealing temperatures. Progressively higher resistivity values at 5K after successive annealing indicate additional devitrification toward equilibrium AFI phase. This successive annealing produces higher resistivity at 5K than the single annealing shown in a. Annealing above 150K causes back conversion to FMM phase \nand resistivity drops. The cooling/heating rates and the temperature interval between the data points are maintained all through these measurements. 10 \n 0 50 100 150 2000246 c ρ (kΩ.cm ) ρ (~103 * kΩ.cm )\nA nnealing T(K) Cooled in 1T\n Cooled in 6T0 100 200 30010-2100102104 \nC ooled in 6 Te s la from  320KMe a s u red in Zero field Initial wa rming in 0T\n Wa rming after return \n          from  140K  in 0Tb Initial wa rming in 0T\n Wa rming after return \n          from  140K  in 0T\nT (K)0 100 200 30010-2100102a ρ (kΩ.cm )\nT (K)C ooled in 1 Te s la from  320KM easured in Zero field\n \nFigure 3:  Effect of larger initial fractions of the glass-like FMM phase produced after initial \ncooling in 1T and 6T on the single-shot annealing, and on successive annealing, is studied through \nzero-field resistivity measurements. a, We cool from 320K in 1T to 5K and then isothermally \nreduce the field to zero. Resistivity in zero field is measured while warming from 5K. The sample \nis again cooled from 320K to 5K in 1T and then after isothermally reducing the field to zero, \nwarmed to 140K and cooled back to 5K. Resistivity is measured while warming after this temperature cycle without changing the field condition (maintained as zero). b, Resistivity in zero \nfield while warming similar to the protocol of a but this time the cooling field 6T is used during \ncooling from 320K to 5K. The lower value of R at 5K after cooling from 320K in 6T and then \nisothermally reducing the field to zero, reflects a larger fraction of arrested metallic phase. The \nlarge difference in the resistivity values between the initially cooled state and that after annealing to 140K in both these cases, as in figure 2a and 2b, arises from the heterogeneous nucleation process. \nThe resistivity below 38K exceeds the measurement range of the instrument. c, Resistivity values at \n5K as a function of annealing temperatures. After cooling the sample to 5K once in 1T (and whole \nprocess is repeated with 6T), resistivity is measured while cycling the temperature between 5K and \nthe successively higher annealing temperatures. Progressively higher resistivity values at 5K after successive annealing indicate additional devitrification toward equilibrium AFI phase. This \nsuccessive annealing produces higher resistivity at 5K than the single annealing of the respective \ncases, and the 6T-cooled case gives the largest AFI fraction.  Its resistivity at 5K after annealing to \n130K, 140K, 150K and 160K exceeds the measurement range of the instrument. Annealing above \n150K causes back conversion to FMM phase and resistivity drops. "    },    {        "title": "1012.2397v1.Spin_orbit_driven_ferromagnetic_resonance__A_nanoscale_magnetic_characterisation_technique.pdf",        "content": "arXiv:1012.2397v1  [cond-mat.mes-hall]  10 Dec 2010Spin-orbit driven ferromagnetic resonance: A nanoscale ma gnetic\ncharacterisation technique\nD. Fang,1H. Kurebayashi,1J. Wunderlich,2,3K. V´ yborn´ y,3L. P. Zˆ arbo,3\nR. P. Campion,4A. Casiraghi,4B. L. Gallagher,4T. Jungwirth,3,4and A. J. Ferguson1,∗\n1Microelectronics Group, Cavendish Laboratory, Universit y of Cambridge,\nJJ Thomson Avenue, Cambridge CB3 0HE, UK\n2Hitachi Cambridge Laboratory, Cambridge CB3 0HE, UK\n3Institute of Physics ASCR, v.v.i., Cukrovarnick´ a 10, 162 5 3 Praha 6, Czech Republic\n4School of Physics and Astronomy,\nUniversity of Nottingham, Nottingham NG7 2RD, UK\nAbstract\nWe demonstrate a scalable new ferromagnetic resonance (FMR ) technique based on the spin-\norbitinteraction. AnalternatingcurrentdrivesFMRinuni formferromagneticstructurespatterned\nfrom the dilute magnetic semiconductors (Ga,Mn)As and (Ga, Mn)(As,P). This allows the direct\nmeasurement of magnetic anisotropy coeﬃcients and damping parameters for individual nano-\nbars. By analysing the ferromagnetic resonance lineshape, we perform vector magnetometry on\nthe current-induced driving ﬁeld, observing contribution s with symmetries of both the Dresselhaus\nand Rashba spin-orbit interactions.\n1Ferromagnetic resonance (FMR) is the most common technique for exploring spin-\ndynamics phenomena and for the magnetic characterisation of fer romagnets.1However,\npreviously developed FMR techniques, based on exciting the magnet ic system by an ex-\nternal alternating magnetic ﬁeld from a resonant cavity2–4or a micro-waveguide,5–8struggle\nto simultaneously achieve scalability of the technique to nano-size ob jects, uniformity of the\nexcitation ﬁeld, and the range of available excitation frequencies. W e introduce an FMR\ntechnique applicable to individual nanomagnets in which the FMR driving ﬁeld is generated\nin the probed magnet itself. The excitation is driven by the eﬀective ﬁ eld generated by\nan alternating electrical current passing through the ferromagn et, which results from the\ncombined eﬀect of the spin-orbit (SO) coupling and exchange intera ction.9–11Our SO-FMR\ncan be operated at tuneable frequencies and we demonstrate its s ensitivity and scalabil-\nity by measuring the variation of micromagnetic parameters of lithog raphically patterned\n(Ga,Mn)As and (Ga,Mn)(As,P) nano-bars.\nFMR induced by driving an alternating current directly through the p robed sample has\nbeen previously demonstrated for speciﬁc non-uniform magnetic n ano-devices such as spin-\nvalves.12,13The experiments utilised the spin-transfer torque in which spin-pola rised electri-\ncal current acts on spatially varying magnetisation14and can be viewed as a macroscopic\nangular momentum transfer eﬀect. Our SO-FMR (Figure 1a) does n ot require the speciﬁc\nsamples with a non-collinear magnetisation proﬁle. The method can be applied to a broad\nrange of systems including uniformly polarised nanomagnets. This is b ecause the eﬀective\nﬁeld utilised in the SO-FMR does not rely on the spatial variation of the magnetisation vec-\ntor but on a microscopic non-collinearity of individual electron spins d ue to their relativistic,\nSO-coupled band structure. Speciﬁcally, when an electrical curre nt traverses through the\nuniformly magnetised material, the resulting non-equilibrium distribut ion of occupied states\nin the SO-coupled carrier bands yields a non-equilibrium spin polarisatio n.15–17The polari-\nsation produces a transverse component of the internal exchan ge ﬁeld (can be viewed as an\neﬀective magneticﬁeld) andatorqueis appliedto themagnetisationv ector.9,18This current-\ninduced eﬀective ﬁeld is generic to ferromagnets with SO-coupling an d inversion asymmetry\nin their band structure. Previously it has been utilised for magnetisa tion switching in the\nferromagnetic semiconductor (Ga,Mn)As10and for domain nucleation in a Pt/Co/AlO x\nstack.11\nThemicro andnano-barsemployed inour SO-FMRstudy arepattern ed by electron beam\n2lithography on 25 nm-thick ﬁlms of (Ga 0.94,Mn0.06)As and (Ga 0.94,Mn0.06)(As0.9,P0.1), grown\nby low-temperature molecular beam epitaxy. The (III,Mn)V ferrom agnetic semiconductors\nused in our study are particularly favourable systems for observin g and exploring SO-FMR\nbecause of the compatibility of the material with advanced semicond uctor nanofabrication\ntechniques, because the carrier bands have strong SO-coupling, and the (III,Mn)V nanos-\ntructures have a rich phenomenology in their micromagnetic parame ters. In the following\ntext we demonstrate our scalable SO-FMR technique in lithographica lly patterned bars of\nwidth ranging from several µm’s to 80 nm (Figure 1b).\nIn order to drive SO-FMR we pass a microwave-frequency current through the nano-bar.\nThis is achieved by wire-bonding the sample between an open-circuit c oplanar transmis-\nsion line and a low-frequency connection which also provides a microwa ve ground (Fig-\nure 1c). Since the microwave excitation ﬁeld originates from the mat erial properties, only\na 2-terminal nano-bar (a resistor) is required in our experiment, e nabling simple and rapid\nsample fabrication. For detection of FMR we utilise a frequency mixing eﬀect based on\nthe anisotropic magnetoresistance (AMR).2–7When the magnetisation precession is driven,\na time-dependent change ∆ R(t) in the longitudinal resistance from the equilibrium value\nRoccurs (due to the AMR). The resistance oscillates with the same fr equency as the mi-\ncrowave current, therefore causing frequency mixing and a direc tly measurable dc voltage\nVdcis generated across the nano-bar. This voltage is our observable p roviding a direct probe\nof the amplitude and phase of the magnetisation precession with res pect to the microwave\ncurrent.\nWe ﬁrst show measurements on a 80 nm-wide nano-bar patterned in the [1¯10] direction\nfrom the (Ga,Mn)(As,P) epilayer. The magnetic ﬁeld dependence of Vdcis measured at\ndiﬀerent microwave frequencies and taken at a temperature of 6 K . The frequency of the\nincident current is ﬁxed while an external dc magnetic ﬁeld H0is swept and a well-deﬁned\nresonance peak appears (Figure 2a). The peak is well-ﬁtted by the solution of the Landau-\nLifshitz-Gilbert (LLG) equation, which describes the dynamics of pr ecessional motion of the\nmagnetisation:\nVdc=Vsym∆H2\n(H0−Hres)2+∆H2+Vasy∆H(H0−∆H)\n(H0−Hres)2+∆H2(1)\nHereHresis the ﬁeld at which resonance occurs and ∆ His the linewidth (half width at\nhalf maximum) of the FMR peak. The resonance lineshape is a combinat ion of symmetric\n3and anti-symmetric Lorentzian functions with amplitudes VsymandVasy, respectively. Their\nrelative contributions are determined by the phase of the driving ﬁe ld with respect to the\ncurrent, and the direction of the driving ﬁeld (see Equation 3 & 4).\nFigure 2b plots the frequency-dependence of the resonance ﬁeld Hres. It is described by\nthe equation for ferromagnetic resonance:19\n/parenleftbiggω\nγ/parenrightbigg2\n=µ2\n0(Hres+H′\nani)(Hres+H′′\nani) (2)\nwhereH′\naniandH′′\naniare terms containing the demagnetisation and anisotropy energies of\nthe ferromagnet (see Methods). A gyromagnetic constant γcharacteristic for Mn2+spins\nof 176 GHz/T (g-factor 2) is used for the ﬁtting. This, together w ith the good agreement\nbetween the observed peaks and the ﬁtted results from the LLG e quation, conﬁrms that we\nobserve the coherent precession of Mn spins.\nTheFMRlinewidth(∆ H= ∆Hinhomo+αω/γ)describesthedampingintheferromagnetic\nsystem. The broadband nature of our setup allows us to determine the inhomogeneous\n(2.5mT)andfrequency-dependent contributionstothedamping( Figure2c)thatcorrespond\ntoGilbert-dampingconstant α=0.023. UsingavectorﬁeldcryostatwealsoperformtheSO-\nFMR measurements for diﬀerent orientations of the external mag netic ﬁeld. In Figure 2d we\npresent the data from an in-plane scan of the magnetic ﬁeld showing that there is a strong\nuniaxial anisotropy perpendicular to the bar direction. By analysing the peak positions\n(Figure 2e) using Equation 2 we quantify the anisotropy ﬁelds and ﬁn dµ0H2/bardbl=−180 mT\n(uniaxial) and µ0H4/bardbl= 68 mT (biaxial).\nWenow demonstrate that SO-FMR can beapplied to comparative inve stigations of nano-\nbarswheretheanisotropiesdiﬀerfrombulkvalues. Theeﬀectofst rain-relaxation, duetothe\nlithographic patterning, on the magnetic anisotropy of (Ga,Mn)As n ano-bars has previously\nbeen studied by electrical transport20–22and optically-detected FMR.8We ﬁrst compare\nthe eﬀect of strain-relaxation between 500 nm bars under compre ssive ((Ga,Mn)As) and\ntensile ((Ga,Mn)(As,P)) growth strain. The in-plane anisotropies ar e studied; although\n(Ga,Mn)(As,P) is out-of-plane magnetised23, the applied ﬁeld H0brings the magnetisation\ninto plane. In (Ga,Mn)As we observe an additional uniaxial contribut ion to the anisotropy\n(µ0HU= 32 mT) along the bar (Figure 3a & c) with a similar magnitude to previou s\nreports.8,20,22By contrast in the (Ga,Mn)(As,P) nano-bar (Figure 3b & c) the sign o f the\nuniaxial anisotropy ( µ0HU=−50.1 mT) has reversed andthe easy axis is now perpendicular\n4to the bar. This can be understood in terms of the sign of the strain relaxation: these\nmaterials become magnetically easier in the direction of most compres sive (least tensile)\nstrain. So when the tensile strain of the (Ga,Mn)(As,P) nano-bar re laxes, it introduces an\neasy axis perpendicular to the bar (Figure 3d). Furthermore we me asure (Ga,Mn)(As,P)\nbars of diﬀerent widths and observe a decrease in the strain-relax ation induced anisotropy\nfrom the 80 nm bar ( µ0HU=−270 mT) to the 500 nm bar ( µ0HU=−50.1 mT), and almost\nno eﬀect of strain-relaxation in the 4 µm bar (µ0HU=−10.5 mT).\nAs well as being able to determine the patterning-induced change in a nisotropy, we also\ncompare the damping among the nano-bars of diﬀerent sizes. The f requency-dependent\nterm (related to damping) increases for decreasing bar width: α= 0.004 (4µm-wide), 0.006\n(500 nm) and 0.023 (80 nm). The signiﬁcantly higher value of Gilbert da mping at 80 nm\ncompared with the 500 nm and 4 µm bars may be due to damage during the etching process.\nThe frequency-independent term is relevant in the case of strain r elaxation as it indicates\nthe inhomogeneity of anisotropy ﬁelds within the bar itself. The inter mediate case of 500 nm\nshows greater inhomogeneity ∆ Hinhomo= 9.9 mT than the 4 µm bar ∆ Hinhomo= 5.4 mT,\nexplained by the increased variation in local anisotropy. By contras t, for 80 nm bar reduces\nto ∆Hinhomo= 2.5 mT, indicative of a high degree of strain-relaxation.\nTo characterise SO-FMR we must understand the direction and amp litude of the eﬀective\nﬁeldheﬀthat drives magnetisation precession. Similar to the experiments on STT-FMR in\nspin-valves12,13we are able to perform vector magnetometry on the driving ﬁeld fro m the\nangle dependence of the amplitude of the FMR peak. For a vector dr iving ﬁeld heﬀ(t) =\n(hx,hy,hz)eiωtin-phase with the microwave current I(t) = (Ix,0,0)eiωt, the amplitudes of\nthe two components of the FMR peak are:\nVsym(θ) =I∆R\n2Asymsin(2θ)hz (3)\nVasy(θ) =I∆R\n2Aasysin(2θ)(hxsinθ+hycosθ) (4)\nwhere ∆Ris the AMR coeﬃcient of the ferromagnetic sample, θis the angle between the\napplied ﬁeld H0and the current I, andAsym(asy)are constants determined by the magnetic\nanisotropies. Hence by decomposing the resonance lineshape into VsymandVasy, and by\nmeasurements of the AMR and magnetic anisotropies we are able to d educe the components\nofheﬀ.\nNo component of Vsymis seen to behave as sin(2 θ), indicating that the driving ﬁeld heﬀ\n5is predominantly in-plane. Figure 4a shows the angle-dependence of Vasyfor a 500 nm-\nwide (Ga,Mn)As bar patterned in the [1 ¯10] direction. We see that Vasy(θ) comprises a\n−sin(2θ)cos(θ) term, indicating that the driving ﬁeld is perpendicular to I. In a [110]\ndevice (Figure 3a) the amplitude of Vasyhas the opposite sign, indicating that the driving\nﬁeld has reversed. For nano-bars along [100] and [010] (Figure 3b) , theVasycurve is a\nsuperposition of sin(2 θ)sin(θ) and sin(2 θ)cos(θ) functions, showing that the driving ﬁeld\nconsists of components both parallel and perpendicular to I.\nThese data are most clearly seen by plotting the dependence of the magnitude and di-\nrection of the eﬀective ﬁeld on the current (nano-bar) orientatio n (Figure 3c). Two con-\ntributions to the driving ﬁeld are observed with diﬀerent symmetry, heﬀ=hR+hD. The\nﬁeldshRandhDhave angular dependence on Ireminiscent of the angular dependence of\nRashba and Dresselhaus SO ﬁelds in the momentum space, respectiv ely.24,25The ﬁeld with\nDresselhaus symmetry, as previously observed in magnetisation sw itching experiments,10\nis due to the diagonal elements in the strain tensor (due to the lattic e mismatch between\nGaAs substrate and (Ga,Mn)As). Therefore hDchanges sign between the (Ga,Mn)As and\n(Ga,Mn)(As,P) materials (comparing Figure 4c and 4d). The Rashba s ymmetry ﬁeld hR\ncan be modelled by oﬀ-diagonal elements in the strain tensor. This st rain is not physically\npresent in the crystal structure of (Ga,Mn)As epilayers. It has b een introduced, however,\nin previous studies to model the in-plane uniaxial anisotropy presen t in (Ga,Mn)As and the\nﬁtted values of this eﬀective oﬀ-diagonal strain are typically sever al times smaller than the\ndiagonal, growth-induced strain.26This is consistent with the observed smaller magnitude\nofhR= 6.5µT thanhD= 18µT (values given at j= 105Acm−2). BothhDandhRare\nmeasured to be linear in current density (Figure 4e & f). We observe a larger magnitude\nofhDat a given current density in the (Ga,Mn)(As,P) nano-bars. This is ex plained by\nthe larger magnitude of the growth strain and larger resistivity (lar gerEat given j) of\n(Ga,Mn)(As,P) as compared to the (Ga,Mn)As ﬁlm.23\nIn conclusion, we perform variable-frequency FMR experiments on individual micro and\nnano-bars of uniform ferromagnetic semiconductors (Ga,Mn)As a nd (Ga,Mn)(As,P). The\nFMR is driven by a torque at microwave frequencies whose origin lies in t he internal eﬀective\nﬁeld (due to the SO-coupling and exchange interaction) of the prob ed ferromagnet. We have\ndemonstrated the utility of our SO-FMR technique by determining th e rich characteristics of\nmagnetic anisotropy ﬁelds and damping coeﬃcients in the studied nan oscale ferromagnetic\n6semiconductor samples. Inaddition, we have performed vector ma gnetometry onthe driving\nﬁeld allowing us to measure a previously unobserved contribution to t he current-induced\nﬁeld in the studied ferromagnets with symmetry of the Rashba SO-in teraction. Our work\ndemonstrates a new scalable FMR technique which provides an unpre cedented method to\nperform magnetic characterisation of uniform ferromagnetic nan ostructures and to study\nthe nature of the current-induced eﬀective magnetic ﬁeld in SO-co upled ferromagnets.\nWe acknowledge fruitful discussions with Ion Garate, Allan H. MacDo nald and Leonid\nRokhinson and support from EU Grants FP7-214499 NAMASTE, FP7 -215368 SemiSpin-\nNet, ERC Advanced Grant, from Czech Republic Grants AV0Z10100 521, KAN400100652,\nLC510, KJB100100802 and Preamium Academiae, DF acknowledges s upport from Cam-\nbridgeOverseas Trusts andHitachi Cambridge Laboratory, A.J.F. acknowledges thesupport\nof a Hitachi research fellowship.\n∗Electronic address: ajf1006@cam.ac.uk\n1Vonsovski ˇi, S. V.Ferromagnetic Resonance (Pergamon, Oxford, 1966).\n2Goennenwein, S. T. B. et al.Electrically detected ferromagnetic resonance. Appl. Phys. Lett.\n90(2007).\n3Mecking, N., Gui, Y. S. & Hu, C.-M. Microwave photovoltage an d photoresistance eﬀects in\nferromagnetic microstrips. Phys. Rev. B 76(2007).\n4Hui, X.et al.Electric detection of ferromagnetic resonance in single cr ystal iron ﬁlm. Appl.\nPhys. Lett. 93(2008).\n5Costache, M. V., Watts, S. M., Sladkov, M., van der Wal, C. H. & van Wees, B. J. Large cone\nangle magnetization precession of an individual nanopatte rned ferromagnet with dc electrical\ndetection. Appl. Phys. Lett. 89(2006).\n6Costache, M. V., Sladkov, M., van der Wal, C. H. & van Wees, B. J . On-chip detection of\nferromagnetic resonance of a single submicron Permalloy st rip.Appl. Phys. Lett. 89(2006).\n7Yamaguchi, A. et al.Broadband ferromagnetic resonance of Ni 81Fe19wires using a rectifying\neﬀect.Phys. Rev. B 78(2008).\n8Hoﬀmann, F. et al.Mapping the magnetic anisotropy in (Ga,Mn)As nanostructur es.Phys.\nRev. B80(2009).\n79Manchon, A. & Zhang, S. Theory of spin torque due to spin-orbi t coupling. Phys. Rev. B 79\n(2009).\n10Chernyshov, A. et al.Evidence for reversible control of magnetization in a ferro magnetic ma-\nterial by means of spin-orbit magnetic ﬁeld. Nature Phys. 5, 656–659 (2009).\n11Miron, I. M. et al.Current-driven spin torque induced by the Rashba eﬀect in a fe rromagnetic\nmetal layer. Nature Mater. 9, 230–234 (2010).\n12Tulapurkar, A. A. et al.Spin-torque diode eﬀect in magnetic tunnel junctions. Nature438,\n339–342 (2005).\n13Sankey, J. C. et al.Spin-transfer-driven ferromagnetic resonance of individ ual nanomagnets.\nPhys. Rev. Lett. 96(2006).\n14Myers, E. B., Ralph, D. C., Katine, J. A., Louie, R. N. & Buhrma n, R. A. Current-induced\nswitching of domains in magnetic multilayer devices. Science285, 867–870 (1999).\n15Edelstein, V. Spin polarization of conduction electrons in duced by electric current in two-\ndimensional asymmetric electron systems. Solid State Commun. 73, 233–235 (1990).\n16Inoue, J., Bauer, G. E. W. & Molenkamp, L. W. Diﬀuse transport a nd spin accumulation in a\nRashba two-dimensional electron gas. Phys. Rev. B 67(2003).\n17Silov, A. Y. et al.Current-induced spin polarization at a single heterojunct ion.Appl. Phys.\nLett.85, 5929–5931 (2004).\n18Garate, I. & MacDonald, A. H. Inﬂuence of a transport current on magnetic anisotropy in\ngyrotropic ferromagnets. Phys. Rev. B 88(2009).\n19Liu, X. & Furdyna, J. K. Ferromagnetic resonance in Ga 1−xMnxAs dilute magnetic semicon-\nductors. J. Phys.: Cond. Matt. 18, R245– R279 (2006).\n20H¨ umpfner, S. et al.Lithographic engineering of anisotropies in (Ga,Mn)As. Appl. Phys. Lett.\n90(2007).\n21Wunderlich, J. et al.Local control of magnetocrystalline anisotropy in(Ga,Mn) As microdevices:\nDemonstration in current-induced switching. Phys. Rev. B 76(2007).\n22Wenisch, J. et al.Control of magnetic anisotropy in (Ga,Mn)As by lithography -induced strain\nrelaxation. Phys. Rev. Lett. 99(2007).\n23Rushforth, A. W. et al.Molecular beam epitaxy grown (Ga,Mn)(As,P) with perpendic ular to\nplane magnetic easy axis. J. Appl. Phys. 104(2008).\n24Dresselhaus, G. Spin-orbit coupling eﬀects in zinc blende st ructures. Phys. Rev. 100, 580–586\n8(1955).\n25Bychkov, Y. A. & Rashba, E. I. Oscillatory eﬀects and the magne tic susceptibility of carriers\nin inversion layers. J. Phys. C: Solid State Phys. 17, 6039–6045 (1984).\n26Zemen, J., Kuˇ cera, J., Olejn´ ık, K. & Jungwirth, T. Magneto crystalline anisotropies in\n(Ga,Mn)As: Systematic theoretical study and comparison wi th experiment. Phys. Rev. B 80\n(2009).\n9Figure 1, Principle of the experiment and its setup. a, Precession of the mag-\nnetisation vector Maround the total magnetic ﬁeld Htot.Mis subject to a damping torque\nταdue to energy dissipation, which causes the magnetic motion to relax towardsHtot. The\ndriving torque τSOdue to current-induced eﬀective ﬁeld counters the eﬀect of damp ing, and\nleads to steady-state motion ∂M/∂t=−γM×Htot.b,SEM image of a 80 nm-wide bar,\npatterned from the (Ga,Mn)(As,P) wafer. c,Schematic of the experimental setup.\n10Figure 2, Spin-orbit driven ferromagnetic resonance. a, Vdcmeasured at 8, 10\nand 12 GHz (circles) on the 80 nm-wide device. The resonance peaks are clearly observed\nand can be well-described by Equation 1 (solid lines are the ﬁtted resu lts). The diﬀerence\nin the signal level at diﬀerent ωis caused by the frequency-dependent attenuation of the\nmicrowave circuit. b,The resonance ﬁeld Hresas a function of the microwave frequency\n(black triangles). The red solid line is the ﬁtted results to Equation 2. c,Frequency-\ndependence of the FMR linewidth ∆ H(black squares). The data are ﬁtted to a straight line\nto extract information on ∆ Hinhomoandα.d,Vdcmeasured from in-plane rotational scans\nof the external ﬁeld H0. The colour scale represents the magnitude of the voltage. ϕis the\nanglebetween themagnetisationvector Mandthe[100]crystalline axis. e,Angle-plotofthe\nresonance ﬁeld Hres, which is extracted by ﬁtting to each FMR peak using Equation 1 (blac k\ncircles). The red line is a ﬁtting curve to Equation 2 to calculate the ma gnetic anisotropy.\n11Figure 3, SO-FMR on devices patterned from diﬀerent materia ls and with\nvarious sizes. a, Hres(ϕ) measured from an in-plane rotational scan on a 500 nm-wide\n(Ga,Mn 0.06)As bar (patterned along the [010] axis). The circles are measurem ent data, and\nthe solid line is the ﬁtted results to Equation 2. The black arrow marks the long axis of the\nnano-bar. b,Hres(ϕ) measured on a (Ga,Mn 0.06)(As,P 0.1) device with identical shape and\norientation. c,Comparison of the in-plane anisotropy ﬁelds Hibetween the two samples.\nd,Schematic of the strain relaxation in the compressively-strained (G a,Mn)As and and\ntensile-strained (Ga,Mn)(As,P) nanostructures. e,Comparison of the magnetic anisotropy\n(in terms of the proﬁles of Hres) among 80, 500 and 4000 nm-wide (Ga,Mn)(As,P) bars. f,\nThe linewidth ∆ Hof the FMR signals measured on the three devices.\n12Figure 4, Characterisation of the driving ﬁeld in both (Ga,M n)As and\n(Ga,Mn)(As,P) devices. a–b, Amplitudes of the anti-symmetric part of the FMR sig-\nnalVasy, measured on a group of 500 nm-wide (Ga,Mn)As bars (circles), pat terned along\ndiﬀerent crystalline directions. The solid lines are ﬁtted results to Eq uation 4. c,Plot\nof the magnitude and direction of the current-induced eﬀective ﬁe ldheﬀmeasured on the\n(Ga,Mn)As nano-bars, scaled for a current density j= 105A/cm2.d,Similar plot for heﬀ\nmeasured on the (Ga,Mn)(As,P) devices. e–f,Current density dependence of hDandhRin\nboth (Ga,Mn)As and (Ga,Mn)(As,P) nano-bars. A second horizonta l scale is included for\nthe electric ﬁeld, calculated from the device resistance (values give n in Methods).\n13"    },    {        "title": "1306.1044v1.Field_induced_ferromagnetism_in_one_dimensional_tight_binding_lattices.pdf",        "content": "epl draft\nField-induced ferromagnetism in one-dimensional tight-binding\nlattices\nGiuseppe Della Valle(a)andStefano Longhi\nDipartimento di Fisica, Politecnico di Milano and\nIstituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche\nPiazza L. da Vinci 32, I-20133 Milano, Italy\nPACS 71.10.Fd { Lattice fermion models (Hubbard model, etc.)\nPACS 75.10.-b { General theory and models of magnetic ordering\nPACS 75.78.Jp { Ultrafast magnetization dynamics and switching\nAbstract { We theoretically show the possibility to induce magnetic ordering in non-magnetic\none-dimensional systems of strongly interacting electrons hopping on a tight-binding lattice. Our\nanalysis is provided within the framework of the t1-t2Hubbard Model, assuming non-zero second\nneighbor hopping rate. It is shown that a high-frequency electric \feld can be exploited to induce\narti\fcial ferromagnetism and eventually control anti-ferromagnetic/ferromagnetic phase transi-\ntion. Our analysis is validated by numerical simulations in a low-density system of 2-particles on\na lattice with 11 sites.\nIntroduction. { The appearance of ferromagnetism\nin strongly-interacting quantum systems is the subject of\na long-lasting research in solid-state physics. Since the\nearly sixties, many e\u000borts have been devoted to prove\nthe existence of large-spin and eventually saturated-spin\nground state in Hubbard-type models [1,2]. Nagaoka fer-\nromagnetism has been demonstrated for one hole in an\nhalf-\flled band in the limit of in\fnite electron interaction\n[3]. Lieb ferromagnetism for half-\flled bipartite lattices\n[4] and Mielke-Tasaki ferromagnetism in \rat band sys-\ntems [5{7] are other important cases where the appearance\nof ferromagnetism has been rigorously established. An-\nother approach to the theoretical description of ferromag-\nnetism within the Hubbard model considers both nearest-\nneighbor hopping rate t1and next-nearest-neighbor hop-\nping ratet2(the so-called t1-t2Hubbard model). This led\nto the demonstration of the M uller-Hartmann ferromag-\nnetism in the limit of in\fnite interaction Uand low elec-\ntron density n[8]. A generalization of the latter scenario to\na \fnite interaction energy Uand for arbitrary particle den-\nsitynhas been reported by Pieri and coworkers in Ref. [9].\nIn that work, extended numerical simulations of the one-\ndimensional t1-t2Hubbard model indicate the existence of\nfully polarized ferromagnetic states in an extended region\n(a)giuseppe.dellavalle@polimi.itof theU\u0000nplane, provided that t1t2<0 and the ratio\n\u0000t2=t1is su\u000eciently large [9]. More recent applications\nof the one-dimensional t1-t2Hubbard model to ferromag-\nnetism have led to the demonstration that the order of\nferromagnetic transition is determined by the quantum-\nliquid phase of the system [10]. Ferromagnetic ordering\nhas been demonstrated in Wigner lattices, where long-\nrange Coulomb repulsion dominates over kinetic energy of\nelectrons [11]. Also, the onset of t1-t2Hubbard ferromag-\nnetism has been predicted and thoroughly investigated in\ntwo-dimensional lattices [12,13].\nThough the t1-t2route to ferromagnetism has been\nwidely explored, in all previous studies a static (i.e. time-\nindependent) Hubbard Hamiltonian has been considered\nand no attempts to investigate the e\u000bects of an external\ndriving \feld on the magnetic ordering within t1-t2Hub-\nbard model have been so far reported.\nOn the other hand, time-dependent Hubbard models,\ndescribing the e\u000bects of external driving \felds or param-\neter modulations, can provide a fertile ground to con-\ntrol the properties of correlated-particles systems. Exam-\nples include \feld-controlled super\ruid to Mott-insulator\nphase transition [14, 15], dynamic unbinding transitions\nin a periodically-driven fermionic Mott-insulator at half-\n\flling [16], switching of the interaction from repulsive to\nattractive [17], and control of correlated tunneling and su-\np-1arXiv:1306.1044v1  [cond-mat.str-el]  5 Jun 2013G. Della Valle et al.\nperexchange spin interactions by ac \felds [18].\nIn this paper we consider a t1-t2Hubbard model de-\nscribing the hopping motion of Ninteracting electrons on\na one-dimensional lattice driven by an external ac electric\n\feld. The main result of our analysis is that, in the high\nfrequency regime, the time-dependent Hubbard model re-\nsults in an e\u000bective static Hubbard model with renormal-\nizedt0\n1-t0\n2hopping rates. The capability to control the\nmagnitude and relative sign of t0\n1andt0\n2by the external\n\feld in a broad range of values shows the possibility to\ninduce magnetic ordering transitions of the ground state\nof the system. The predictions of the asymptotic analy-\nsis are con\frmed by direct numerical simulations of the\ntime-periodic Hubbard model in case of N= 2 interacting\nelectrons.\nThe driven t 1-t2Hubbard model. { In the pres-\nence of an external driving electric \feld E(t), thet1-t2\nHubbard Hamiltonian of a one-dimensional system of in-\nteracting electrons reads as follows:\n^H=^Hhop1+^Hhop2+^Hint+^Hdrive (1)\nwhere\n^Hhop1=\u0000\u0016ht1L\u00001X\nj=1X\n\u001b=\";#\u0010\n^ay\nj;\u001b^aj+1;\u001b+ ^ay\nj+1;\u001b^aj;\u001b\u0011\n(2)\n^Hhop2=\u0000\u0016ht2L\u00002X\nj=1X\n\u001b=\";#\u0010\n^ay\nj;\u001b^aj+2;\u001b+ ^ay\nj+2;\u001b^aj;\u001b\u0011\n(3)\n^Hint =ULX\nj=1Y\n\u001b=\";#^nj;\u001b (4)\n^Hdrive =edE(t)LX\nj=1X\n\u001b=\";#j^nj;\u001b: (5)\nIn previous equations, Lis the number of lattice sites, d\nis the lattice period, Uis the on-site Coulomb repulsion\nenergy, ^ay\nj;\u001bis the fermionic creation operator that creates\none electron at site jwith spin\u001b(j= 1;2;:::;L ;\u001b=\";#),\nand ^nj;\u001b= ^ay\nj;\u001b^aj;\u001bis the spin\u001bparticle number operator\nat lattice site j.\nThe state vector of the system j (t)iin Fock space rep-\nresentation can be written as:\nj (t)i=X\nn;mf(n;m;t)jn;mi; (6)\nwheref(n;m;t) is the complex amplitude for the jn;mi=\njn1;n2;:::nL;m1;m2;:::mLiFock basis element, represent-\ning a state with njelectrons occupying the jlattice site\nwith spin#andmjelectrons occupying the jlattice site\nwith spin\", withnj;mjtaking only the two values 0 and\n1, owing to the anti-commutation rules of the Fermi op-\nerators. Given above decomposition of the state vector,\nstandard projection technique provides the following evo-lution equation for the amplitudes f(n;m;t):\ni\u0016hdf(n;m;t)\ndt=X\ns;qhn;mj^Hjs;qif(s;q;t): (7)\nRenormalized t0\n1-t0\n2Hubbard model and \feld-\ninduced magnetic ordering transition. { The dy-\nnamics of the system under a high-frequency driving \feld\ncan be at best captured after the substitution:\nf(n;m;t) =g(n;m;t) exp2\n4\u0000i\b(t)LX\nj=1j(nj+mj)3\n5;\n(8)\nwhere\n\b(t) =ed\n\u0016hZt\n0dt0E(t0): (9)\nThis way, the amplitude probabilities g(n;m;t) satisfy the\nfollowing coupled equations:\nidg(n;m;t)\ndt=1\n\u0016hX\ns;qhn;mj^Hhop1+^Hhop2js;qi (10)\n\u0002exp [i\u001a(n;m;s;q)\b(t)]g(s;q;t)\n+1\n\u0016hX\ns;qhn;mj^Hintjs;qig(s;q;t):\nIn Eq. (10) we have set\n\u001a(n;m;s;q) =LX\nj=1j(nj\u0000sj+mj\u0000qj): (11)\nNote that since the e\u000bect of the tunneling Hamiltonians\n^Hhop1and ^Hhop2is to shift one electron from lattice site\njto lattice site j\u00061 orj\u00062 respectively, the nonvan-\nishing matrix elements entering in Eq. (10) correspond to\n\u001a(n;m;s;q) =\u00061 for ^Hhop1and to\u001a(n;m;s;q) =\u00062 for\n^Hhop2. If we now assume that the external driving \feld\nis periodic with period T= 2\u0019=!, in the high-frequency\nlimit!\u001dt1;t2;U, the rapidly oscillating exponential\nterms on the right-hand side of Eq. (10) can be replaced\nby their time average over one oscillation cycle of the ac\n\feld [14], resulting in an e\u000bective renormalization of the\nhopping amplitudes [19]. In particular, for a sinusoidal\ndriving \feld E(t) =E0sin(!t), the averaging procedure\nyields for the renormalized hopping rates t0\n1;t0\n2the follow-\ning expressions:\nt0\n1=t1hexp[\u0006i\b(t)]i=t1J0(\u0000) (12)\nt0\n2=t2hexp[\u0006i2\b(t)]i=t2J0(2\u0000) (13)\nwhereh:::i= 1=TRT\n0dt:::, \u0000 =edE 0=(\u0016h!), andJ0is the\nBessel function of \frst kind and zero order. Note that\nthe ratiot0\n2=t0\n1=rt2=t1withr=J0(2\u0000)=J0(\u0000), can be\nmade negative and arbitrarily large in modulus provided\nthatz0=2<\u0000<z0, beingz0'2:405 the \frst zero of J0,\n(Fig. 1). Hence the e\u000bect of the driving \feld is basically\np-2Field-induced ferromagnetism in 1D\nFig. 1: The renormalization ratio rbetween second and \frst\nneighbor tunneling rates as a function of \u0000. The red shaded\narea corresponds to z0=2<\u0000< z0. Red circle represents the\nvalue \u0000 = 2 :2 used in the simulations of Fig. 2.\nto renormalize the original t1andt2hopping rates, with\na renormalization ratio rthat can be tuned over a wide\nrange of values. Our analysis is valid for an arbitrary den-\nsity of electrons n=N=L, and thus the results reported in\nRef. [9] can be readily applied to the e\u000bective t0\n1-t0\n2Hub-\nbard Hamiltonian. In particular, it is known that for any\ngiven density of electrons and for any given interaction en-\nergyU, there exists a threshold value for the ratio \u0000t0\n2=t0\n1\nabove which the ground state of the system becomes fer-\nromagnetic [9]. By proper control of the ratio E0=!we\ncan thus actively turn the ground state of the system into\na ferromagnetic state, regardless the value of the hopping\nratest1;t2and of the interaction energy Uof the original\n(static) Hubbard Hamiltonian.\nNumerical results and discussion. { To validate\nthe previous analysis we consider the simplest case of\nN= 2 electrons with opposite spins, for which numerical\nsimulations can be easily performed for a reasonably large\nnumber of lattice sites. In this case, Fock space represen-\ntation of the state vector of the system reads as follows:\nj (t)i=X\nn;mcn;m(t)^ay\nn;#^ay\nm;\"j0i; (14)\nwherecn;m(t) is the complex amplitude for the basis state\ncorresponding to one electron with spin #at lattice site n\nand one electron with spin \"at sitem. By inserting the\nansatz of Eq. (14) into previous Eqs. (1)-(5), we end up\nwith the following evolution equations for the amplitude\nprobabilities cn;m:\nidcn;m\ndt=\u0000t1(cn\u00001;m+cn+1;m+cn;m\u00001+cn;m+1)\n\u0000t2(cn\u00002;m+cn+2;m+cn;m\u00002+cn;m+2)\n+\u0014U\n\u0016h\u000en;m+ \u0000!(n+m) sin(!t)\u0015\ncn;m:(15)\nThe behaviour of the two-electron system under high fre-\nFig. 2: (a) Quasi-energy spectrum of the driven t1-t2lattice as\na function of interaction energy. (b) Zoom-in of the \frst two\nquasi-energies. (c) Expectation value of the total spin operator\n^S2\ntotfor the \frst two quasi-energy (QE) states of the system.\nquency driving with a strong electric \feld can be under-\nstood by inspecting the quasi-energies and quasi-energy\nstates of the system described by Eq. (15), computed ac-\ncording to standard Floquet theory [20]. As an exam-\nple, in Fig. 2(a) we show the computed quasi-energy spec-\ntrumE0for parameter values t2= 0:05t1,!= 10t1and\n\u0000 = 2:2 (see red circle in Fig. 1), with Uranging be-\ntween 0 and t1. Within the t1-t2Hubbard model, an\nincrease of Uabove a critical value is known to be re-\nsponsible for the onset of ferromagnetism [9, 21]. The\nquasi-energy spectrum shown in Fig. 2(a) clearly indicates\na typical Mott-Hubbard band-gap formation at a certain\nvalue of the interaction energy, with a bounce of higher\nlevels emerging from a broader spectrum of lower lying\nlevels. These higher levels correspond to bound-particle\nstates (doublons). The onset of magnetic ordering in the\nsystem can be investigated by computing the expectation\nvalue of the total spin operator ^S2\ntot, whose eigenvalues\nare given by Stot(Stot+ 1), withStotthe total spin of the\np-3G. Della Valle et al.\nFig. 3: (a) Energy spectrum of the equivalent t0\n1-t0\n2static lattice\nas a function of interaction energy. (b) Zoom-in of the \frst two\nenergy levels. (c) Expectation value of the total spin operator\n^S2\ntotfor the \frst two eigenstates of the system. Energy values\nare normalized to gt0\n1=t1being g= 1=J0(\u0000), for a better\ncomparison with the results of Fig. 2.\nsystem, that ranges from 0 (anti-ferromagnetic state) to a\nmaximum value Smax(saturated ferromagnetic state) that\ndepends on the number of particles, namely Smax=N=2\norSmax=L\u0000N=2 respectively below or above particle\ndensityn= 1, that is half-\flling [21]. A zoom-in of the\nspectrum showing the \frst two quasi-energies is reported\nin Fig. 2(b), whereas the corresponding expectation value\nof^S2\ntotis reported in Fig. 2(c). The expectation value of\n^S2\ntoton the ground and \frst excited quasi-energy states\nis computed as a time-average over one oscillation cycle.\nHowever, in the high-frequency regime of our simulations,\nthe periodic part of the quasi-energy state (i.e. the Flo-\nquet mode) is composed of a dominating dc term plus\nan ac correction made of small harmonic terms (see e.g.\n[20] and references therein). Therefore, the total spin of\nthe system in the quasi-energy states is almost station-\nary. Note that as the interaction energy approaches acritical value UC'0:43t1, the \frst two quasi-energies at-\ntain a level crossing, and the ground quasi-energy state\nexperiences a transition from anti-ferromagnetic (AFM)\nordering, with Stot= 0, to a saturated ferromagnet (FM),\nwithStot= 1. The accuracy of the asymptotic analysis\ncan be checked by comparing the behaviour of the driven\nlattice with the equivalent static (undriven) lattice with\nrenormalized tunneling rates t0\n1andt0\n2given by Eqs. (12)\nand (13) with \u0000 = 2 :2. Numerically computed energy\nspectrum aside with ground-state and \frst excited state\nenergies and total spin expectation value for the equivalent\nstatic lattice are reported in Fig. 3. The excellent agree-\nment with the results of Fig. 2 con\frms the accuracy of our\ne\u000bective Hubbard model with renormalized hopping rates.\nThe above results suggest the possibility to manipulate\nmagnetic ordering in non magnetic systems of strongly-\ninteracting electrons hopping on a one-dimensional tight-\nbinding lattice. Interestingly, a weak low-frequency mod-\nulation of the electric \feld amplitude can be exploited to\nswitch the system between AFM and FM states, once the\nhigh-frequency \feld has driven the system close to the\nphase transition point. It is worth noting that since r(\u0000)\nis not bound from below, our approach o\u000bers enough de-\ngrees of freedom to attain AFM/FM transition in a broad\nregion ofU-t2parameters, by acting on the \feld ampli-\ntude. This is illustrated in Fig. 4 for the simple case of\nN= 2 electrons discussed above. The \fgure shows the\ncritical value of the normalized driving \feld amplitude \u0000\nin the plane ( t2=t1,U=t1) at which AFM/FM transition\noccurs.\nFig. 4: Critical value of \u0000, separating AFM and FM phases, as\na function of second neighbor tunneling rate t2and interaction\nenergy U, normalized to t1.\nConclusions. { In this work we analytically demon-\nstrated that under high-frequency strong electric \felds a\nnon-magnetic system of Ninteracting electrons hopping\non a one-dimensional tight-binding lattice with non-zero\nnext-nearest neighbors hopping rate can be driven into a\nferromagnetic quasi-energy state. A numerical validation\nprovided for the simple case of N= 2 electrons with oppo-\nsite spin on a \fnite lattice with 11 sites, con\frmed the ana-\nlytical prediction. Our results suggest that the system can\np-4Field-induced ferromagnetism in 1D\nbe driven to the anti-ferromagnetic/ferromagnetic transi-\ntion point in a controllable way and for a broad range of\nparameters. The possibility to actively induce ferromag-\nnetism in non magnetic media can disclose novel oppor-\ntunities and stimulate further developments in arti\fcial\nmagnetism, with potential applications to spintronics and\nquantum magnetic devices. Also, the capability of control-\nling the ratio between \frst and second neighbor tunneling\nrates in tight-binding lattices can be pro\ftably exploited\nin cold atom systems to access quantum simulation of t1-t2\nHubbard ferromagnetism [22].\nREFERENCES\n[1]Hubbard J. ,Proc. R. Soc. London Ser. A ,276(1963) 238.\n[2]Kanamori J. ,Prog. Theor. Phys. ,30(1963) 275.\n[3]Nagaoka Y. ,Phys. Rev. ,147(1966) 392.\n[4]Lieb E. H. ,Phys. Rev. Lett. ,62(1989) 1201.\n[5]Tasaki H. ,Phys. Rev. Lett. ,69(1992) 1608.\n[6]Mielke A. ,Phys. Lett. A ,174(1993) 443.\n[7]Mielke A. and Tasaki H. ,Commun. Math. Phys. ,158\n(1993) 341.\n[8]Muller-Hartmann E. ,J. Low. Temp. Phys. ,99(1995)\n349.\n[9]Pieri P., Daul S., Baeriswyl D., Dzierzawa M. and\nFazekas P. ,Phys. Rev. B ,54(1996) 9250.\n[10]Daul S. ,Eur. Phys. J. B ,14(2000) 649.\n[11]Daghofer M., Noack R. M. and Horsch P. ,Phys.\nRev. B ,78(2008) 205115.\n[12]Taniguchi H., Morita Y. and Hatsugai Y. ,Phys. Rev.\nB,71(2005) 134417.\n[13]Pandey S. and Singh A. ,Phys. Rev. B ,75(2007)\n064412.\n[14]Longhi S. ,Phys. Rev. B ,77(2008) 195326.\n[15]Zenesini A., Lignier H., Ciampini D., Morsch O. and\nArimondo E. ,Phys. Rev. Lett. ,102(2008) 100403.\n[16]Hassler F., R uegg A., Sigrist M. and Blatter G. ,\nPhys. Rev. Lett. ,104(2010) 220402.\n[17]Tsuji N., Oka T., Werner P. and Aoki H. ,Phys. Rev.\nLett.,106(2011) 236401.\n[18]Chen Y.-A., Nascimbene S., Aidelsburger M., Atala\nM., Trotzky S. and Bloch I. ,Phys. Rev. Lett. ,107\n(2011) 210405.\n[19]Eckardt A., Weiss C. and Holthaus M. ,Phys. Rev.\nLett.,95(2005) 260404.\n[20]Grifoni M. and H anggi P. ,Phys. Rep. ,304(1998) 229.\n[21]Tasaki H. ,J. Phys.: Condens. Matter ,10(1998) 4353.\n[22]Zaleski T. A. and Kope \u0013c T. K. ,J. Phys. B: At. Mol.\nOpt. Phys. ,43(2010) 085303.\np-5"    },    {        "title": "1408.3675v1.Effects_of_impurity_on_tunnel_magnetoresistance_in_a_ferromagnetic_electrode_carbon_nanotube_ferromagnetic_electrode_junctio.pdf",        "content": "arXiv:1408.3675v1  [cond-mat.mes-hall]  15 Aug 2014Eﬀects of impurity on tunnel magnetoresistance in a ferroma gnetic electrode/carbon\nnanotube/ferromagnetic electrode junction\nA. Ahmadi Fouladi, J. Vahedi and M. Soleymani\nDepartment of Physics, Sari Branch, Islamic Azad Universit y, Sari, Iran.\n(Dated: December 17, 2018)\nEﬀects of impurity on the spin-dependent transport in a sing le wall carbon nanotube spin-valve,\nas ferromagnetic electrode/carbon nanotube/ferromagnet ic electrode model junction is numerically\ninvestigated. Using a generalized Green’s function method and the Landauer-B¨ uttiker formalism,\nthe impurity conditions are determined by randomly substit ution of carbon atoms in the honeycomb\ncarbon nanotube lattice by nitrogen and boron atoms. We have found that transport characteristics,\nincluding the spin-dependent current and tunnel magnetore sistance are strongly inﬂuenced by the\nimpurity eﬀects. We think that the results of the present rep ort could be useful for designing the\nfuture spintronic devices.\nPACS numbers: 85.35.Ds, 85.65.+h, 81.07.Nb, 72.10.Di\nI. INTRODUCTION\nIn recent years nanotechnology has attracted atten-\ntion of numerous researchers due to its vast potential\napplications1–22. Spintronics is one of the most attrac-\ntive research areas of nanotechnology. The development\nof spintronic devices based on the tunnel magnetoresis-\ntance (TMR), giant magnetoresistance (GMR) and spin-\nvalve eﬀects has changed magnetic memory applications.\nIn the conventional spin-valve geometry, a non-magnetic\nspacer, which controls the total resistance of the device,\nthe spin polarization and transport, is sandwiched be-\ntween two magnetic contacts. Among many suggestions\nfor possible non-magnetic spacers, organic materials pro-\nvide desirable properties like ﬂexibility, inexpensive ma-\nterial and production methods have attracted investiga-\ntions in the organic electronics1–12. Besides, the spin-\norbit coupling and hyperﬁne interactions in the organic\nmaterials are very weak so that the spin memory can be\nas long as a few seconds. Therefore the spin-ﬂip pro-\ncess during transport can be neglected3,9. Such prop-\nerties make them ideal for spin-polarized transport ap-\nplications. Among many types of organic materials, the\ncarbon nanotube (CNT) is a promising candidate as a\ncomponent in spintronic devices. Spin-polarized trans-\nport through the CNT contacted by ferromagnetic leads\nhas also been recently investigated1,12–17.\nIn the other hand, understanding of the consequence\nof doping on the electronic eﬃciency of devices based on\nlow-dimensional carbon-structures is a key-point for the\nfuture development of carbon based nanoelectronics and\nto the ﬁnding of novel functionalities. Namely, since the\nelectronic properties of CNTs are strongly related to the\ndelocalised electron system, clearly any modiﬁcation of\nthe CNTs will inﬂuence these properties. Therefore, by\na suitable choice of the type of modiﬁcation the elec-\ntronic properties of CNTs can be intentionally tuned.\nThe tuning of electronic properties is also referred to as\nimpurity doping the CNTs. Also the substitution of car-\nbon atoms in the honeycomb lattice by atoms with a dif-\nferent number of valence electrons in general introducesadditional states in the density of states. Nitrogen (N)\nand boron (B) are the natural choice for doping since\nthey diﬀer only by one in its number of valence electrons\ncompared to carbon atoms. There have been several\ntheoretical23–25studies on the doping of CNTs by B and\nN atoms. Also several authors26–35have used diﬀerent\nexperimental methods to prepare doped CNT with nitro-\ngen and boron. In this paper, we have numerically inves-\ntigated the coherent spin-dependent transport through\n(5,0) zigzag CNT with 1.56 nm length attached to ferro-\nmagnetic 3-dimensional leads (FM/CNT/FM junction),\nas depicted in Figs. 1(a) and 1(b) for parallel and anti-\nparallel spin alignments, respectively. Using the Green’s\nfunction method and relying on the Landauer-B¨ uttiker\nformalism, we have studied the inﬂuence of impurities\non spin-dependent transport properties and TMR in a\nFM/CNT/FM model junction. In this workthe impurity\nconditions are determined by randomly replacing carbon\natoms in the honeycomb CNT lattice with nitrogen and\nboron atoms.\nII. COMPUTATIONAL SCHEME\nAgeneralizedHamiltonianforthe FM/CNT/FMjunc-\ntion can be expressed as;\nH=HCNT+HL+HR+HC. (1)\nThe ﬁrst term is the Hamiltonian for the CNT within the\nnon-interacting picture given by\nHCNT=/summationdisplay\nn,σεnd†\nn,σdn,σ−/summationdisplay\n<n,m>,σtC−C(d†\nn,σdm,σ+H.c.).(2)\nwherenruns over the π-orbitals of a carbon atom on\nthe CNT. The operator d†\nn,σ(dn,σ) creates (annihilates)\nan electron with spin σ≡↑or↓at siten.εnis the on-\nsite energy of the carbon atom and tC−Cis the nearest\nneighbour electron hopping integral. Within the non-\ninteracting picture, the single-band tight binding Hamil-\ntonian of the left (right) ferromagnetic electrode can be2\nFIG. 1: A schematic FM/CNT/FM junction model for (a)\nparallel and (b) anti-parallel spin alignments.\nwritten as;\nHβ=/summationdisplay\niβ,σ{(ε0−σ.Jβ)c†\niβ,σciβ,σ\n−t(c†\niβ,σciβ+1,σ+c†\niβ+1,σciβ,σ)}.(3)\nwherec†\niβ,σ(ciβ,σ) denotes the creation (annihilation) op-\nerator of an electron with spin σat siteiin the electrode\nβ(=LorR).tandε0are the nearest neighbour hopping\nintegral and the on-site energy, respectively. Here −σ.Jβ\nistheinternalexchangeenergyand Jβdenotesthemolec-\nular ﬁeld at site iβ. The last term of Eq. 1, HCdenotes\nthe coupling between CNT and FM electrodes\nHC=/summationdisplay\nn,σ/summationdisplay\ni,σtc(n,σ,i)(c†\ni,σdi,σ+H.c), (4)\nwhere the matrix elements tc(n,σ,i)represent the coupling\nstrength between CNT, and FM electrodes are taken to\nbetc. For a complete system, i.e., CNT with two FM\nelectrodes, the spin-dependent Green’s function can be\nwritten as;\nGσ(E) =/bracketleftbig\nE1−HCNT−ΣL,σ−ΣR,σ]−1,(5)\nΣL,σand ΣR,σare the self-energy matrix due to coupling\nof CNT to the left and right electrodes respectively and\ngiven by\nΣβ,σ(E) =τCNT,βgβ,στβ,CNT, (6)\nwhereτis the hopping matrix that couples CNT to the\nFM electrodes, and gβ,σis the surface Green’s function\nof FM electrodes which is given by36;\ngβ,σ(n,m;z) =/summationdisplay\nkψk(rn)ψ∗\nk(rm)\nz−ε0+σ.Jβ−E(k),(7)wherern≡(xn,yn,zn),k≡(lx,ly,kz) ,z=E+iη,\nψk(rn) =2√\n2/radicalbig\n(Nx+1)(Ny+1)Nzsin(lxxnπ\nNx+1)\n×sin(lyynπ\nNy+1)sin(kzzn), (8)\nand\nE(k) = 2t/bracketleftbig\ncos(lxπ\nNx+1)+cos(lyπ\nNy+1)+cos(kza)/bracketrightbig\n.(9)\nHere,lx,y(= 1,...,Nx,y) are integers, Nξwithξ=x,y,z\nis the number of lattice sites in the ξdirection and k∈\n[−π/a,π/a]. The number of atoms at the cross-section\nof FM electrodes are taken to be Nx=Ny= 5. We\nassume that only the central atom at the electrode cross-\nsection connected to the one carbon atom at the end of\nCNT lattice. The broadening matrix Γ L,σ(ΓR,σ) can be\ncalculated through the expression,\nΓβ,σ=−2Im(Σβ,σ). (10)\nNeglecting the spin-ﬂip scattering eﬀects, one may con-\nsider transport of spin-up and spin-down electrons sep-\narately. The spin-dependent transmission probability of\nan electron in the CNT can be calculated using the fol-\nlowing relation37:\nTσ(E) =Tr(ΓL,σGr\nσΓR,σGa\nσ). (11)\nWhereGr\nσandGa\nσaretheretardedandadvancedGreen’s\nfunction. Based on non-equilibrium Green’s function\nmethod, the spin-dependent current as a function of\nthe applied bias voltage in the low-bias limit can be\ncomputed in the framework of the Landauer-B¨ uttiker\nformula37:\nIσ(Va) =e\nh/integraldisplay+∞\n−∞Tσ(E)/bracketleftbig\nfL−fR/bracketrightbig\ndE, (12)\nwherefL(R)=f(E−µL(R)) is the Fermi distribution\nfunction at the left (right) electrodewith chemicalpoten-\ntialµL(R)=EF±eVa\n2and Fermi energy EF. Forthe sake\nofsimplicity, here we have assumed that the total voltage\nis dropped across CNT/electrode interface, and this as-\nsumption does not highly aﬀect the qualitative aspects of\ncurrent-voltage characteristics. In fact, the electric ﬁeld\ninsidetheCNT,especiallyforshortlengthCNT,seemsto\nhave an insigniﬁcant eﬀect on the I−Vcharacteristics.\nOn the contrary, for longer CNT and higher bias volt-\nages, the electric ﬁeld inside the CNT may play a more\nconsiderablerole depending on the structure of the CNT,\nbut yet the eﬀect is very small38. The tunnel magnetore-\nsistance (TMR) can be deﬁned as a relative change in\nthe current of the system when magnetization of two FM\nelectrodes switch between parallel ( P) and anti-parallel\n(AP) conﬁgurations, hence: TMR≡IP−IAP\nIP. In fact\nthe TMR is associated with asymmetry of the density of\nstates for two spin channels in the FM electrodes.3\nIII. RESULTS AND DISCUSSION\nBased on the formalism described in section 2, we\nrepresent the results of numerical calculations. Tight-\nbinding parameters for FM electrodes chosen to be ε0=\n3eV,t= 1eVand|Jβ|= 1.25eV. In our calculation,\nthe on-site energy of C, N and B atoms are assumed to\nbe 0, -2.50 and +2.33 eV, respectively39,40. The nearest\nneighbour electron hopping integrals are tC−C=−3eV,\ntC−B=−2.7eVandtC−N=−3.14eV. As a reference\nenergy, the Fermi energy of FM electrodes is set EF= 0\nand the coupling between CNT and two FM electrodes\nset astc= 0.3eV. Magnetizationdirectionin the left fer-\nromagnetic electrode is ﬁxed in the + ydirection, while\nthe right electrode is free to be ﬂipped into either the + y\nor−ydirection by an external magnetic ﬁeld. For paral-\nlel (P) alignment of magnetization in the FM electrodes,\nspin-up and spin-down electrons encounter a symmetric\nstructure, while for anti-parallel ( AP) alignment these\nelectronsencounter an asymmetricstructure. A lowtem-\nperature of T= 11Kis taken to avoid spin ﬂip in the\nelectron transport progress2.\nIn Fig.2, we have plotted surface density of states\n(SDOS) of the isolated FM electrodes and density of\nstates (DOS) of the isolated CNT for pure and randomly\ndoped B and N atoms with 8% concentrations in both\nparallel and anti-parallel conﬁgurations. The discrete-\nness of molecular levels is observed from DOS graphs.\nFurthermore, Fig.2 shows that dopant shifts HOMO and\nLUMO levels away from the Fermi energy. Therefore, in\nthe presence of dopants, symmetry of DOS about the\nFermi energy (for pure case) is broken. The shift of\nHOMO and LUMO levels depends on concentration of B\nand N atom doping. B doping (N doping) of CNT shifts\nHOMO and LUMO levels towards the higher (lower)\nenergies41. Consequently,HOMO-LUMOenergygapwill\nchange.\nWethencalculatedDOSandlogarithmicscaleoftrans-\nmission function for FM/CNT/FM junction for Pand\nAPconﬁgurations, which are presented in Fig. 3. As it\ncan be seen, when isolated CNT contacted to FM elec-\ntrodes, the contribution of the electronic levels of the\nelectrode surface increases DOS. However, we can see a\nremarkable diﬀerence between DOS of the PandAP\nconﬁgurations, which can eﬀect on transmission function\nas shown in Figs. 3(a) and 3(b).\nThe resonance peaks in transmission probability are\nassociated with eigenenergies of CNT. When electrons\ngo from the left to right FM electrode one through CNT,\nelectron waves propagating along diﬀerent branches of\nCNT honeycomb lattice may suﬀer a relative phase shift\nbetween themselves. Consequently, there might be con-\nstructiveor destructiveinterferencedue to the superposi-\ntion of the electronic wave functions along various path-\nways. Therefore, transmission probability changes. Also,\nwe observe some anti-resonant states appear in trans-\nmission probability in Figs.3(c) and 3(d). These anti-\nresonant states are related to the quantum interferenceeﬀect. In general, transmission probabilities are higher\nforPalignment than APalignment. This diﬀerence\nis associated with the asymmetry of SDOS of the FM\nelectrodes for spin-up and spin-down electrons as shown\nin Fig.2 and quantum tunnelling phenomenon through\na CNT.In the presence of N and B atoms, with shift-\ning HOMO and LUMO levels, the position of resonance\npeaks changes (inset of Fig.3 (c)).\nIn order to provide a deep understanding of spin-\ndependent transport, we have plotted current-voltage\ncharacteristicsofFM/CNT/FM system for(a) Pand (b)\nAPconﬁgurations in Fig.4. It is observed that the pres-\nence of impurities have a profound eﬀect on the current\namplitude. Threshold voltage changes in the presence of\nB and N atoms, owing to altering the energy gap. By de-\ncreasing energy gap, the nearest molecular levels to the\ngap is less separated from the Fermi level, and smaller\nvoltage is needed for turning current on. In Pconﬁg-\nuration the majority (minority) electrons with spin-up\n(spin-down) in the left FM electrode move into the ma-\njority (minority) states in the right FM electrode by tun-\nnelling through CNT. For APconﬁguration the majority\n(minority) electrons with spin-up (spin-down) in the left\nelectrode move into the minority (majority) states in the\nright FM electrode. Therefore, total current amplitude\ninPalignment is higher than APalignment.\nIn Fig.5 we have depicted TMR ratio as a function\nof applied bias voltage in the absence and presence of\nimpurities. As illustrated, TMR ratio shows maximum\nvalue (78%) for lower bias voltages. By increasing ap-\nplied voltage, TMR shows a sharp drop and after that\nit varies slowly. In the low bias voltages for maximum\nvalue of TMR, there is no molecular level available in\nthe gap region of CNT so that electrons can tunnel be-\ntweenthechemicalpotentialoftheleftandrightFMelec-\ntrodes. Then, in the low-voltage regime current is small\nand TMR ratio is large. When bias voltageincreases, the\nelectrochemicalpotentialsinelectrodesareshiftedgradu-\nally, andsomeoftheenergystatesarepushedupbetween\nthe chemical potential ofthe right and left electrodesand\ncurrent of tunnelling through junction increases remark-\nably. Our results are qualitatively in agreement with the\nexperimental measurements41. Moreover, because of the\ndiﬀerent energy gaps in the absence and presence of im-\npurity, we can see that the ﬁrst sharp decrease of TMR\nhappens for diﬀerent values of bias voltage. As a result\none may tune TMR ratio with applied voltage and also\nimpurities.\nIn Fig.6, we have plotted TMR as a function of ap-\nplied bias voltage for diﬀerent amount of B atoms as an\nimpurity concentration. Because of the diﬀerent HOMO-\nLUMO gap of CNT in the presence of diﬀerent amounts\nof impurities, we can see that the decrease of TMR oc-\ncurs for various values of bias voltage. Also for higher\namount of impurity (12%), maximum value of TMR ra-\ntio decreases.4\nIV. CONCLUSIONS\nTo summarize, basedon Landauer-B¨ uttikerformalism,\nwehavestudied the eﬀectsofimpurityonspin-dependent\ntransport through a single wall carbon nanotube (CNT)\nattached to ferromagnetic (FM) 3-dimensional leads as\nFM/CNT/FM model junction. We choose (5,0) zigzag\nCNT and study eﬀects of randomly replaced carbon\natoms in the honeycomb CNT lattice with nitrogen and\nboron atoms. Our results indicate that spin transport\ncharacteristics, including the spin-dependent current-\nvoltage characteristics, the transmission probability and\ntunnel magnetoresistance (TMR) are strongly inﬂuencedby type of impurity as well as its concentration. Re-\nsults show that the existence of impurities can change\nthe HOMO-LUMO energy gap. Consequently, current\namplitude and threshold voltage undergo changes. 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Phys. 794724 (1996)5\n/s32/s83/s112/s105/s110/s45/s117/s112\n/s32/s83/s112/s105/s110/s45/s100/s111/s119/s110\n/s45/s48/s46/s48/s51 /s45/s48/s46/s48/s50 /s45/s48/s46/s48/s49 /s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51/s45/s50/s45/s49/s48/s49/s50\n/s48/s46/s48/s48/s48/s49 /s48/s46/s48/s48/s48/s50 /s48/s46/s48/s48/s48/s51 /s48/s46/s48/s48/s48/s52 /s48/s46/s48/s48/s48/s49 /s48/s46/s48/s48/s48/s50 /s48/s46/s48/s48/s48/s51 /s48/s46/s48/s48/s48/s52 /s48/s46/s48/s48/s48/s49 /s48/s46/s48/s48/s48/s50 /s48/s46/s48/s48/s48/s51 /s48/s46/s48/s48/s48/s52 /s45/s48/s46/s48/s51 /s45/s48/s46/s48/s50 /s45/s48/s46/s48/s49 /s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51/s32\n/s32/s32/s76/s101/s102/s116/s45/s101/s108/s101/s99/s116/s114/s111/s100/s101/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s32/s32/s32\n/s32/s32/s32\n/s32\n/s68/s79/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\n/s32/s32/s32\n/s32/s32\n/s32/s32\n/s32/s65/s110/s116/s105/s112/s97/s114/s97/s108/s108/s101/s108\n/s32/s83/s112/s105/s110/s45/s117/s112\n/s32/s83/s112/s105/s110/s45/s100/s111/s119/s110/s32/s83/s112/s105/s110/s45/s117/s112\n/s32/s83/s112/s105/s110/s45/s100/s111/s119/s110/s78/s45/s100/s111/s112/s101/s100/s32/s67/s78/s84/s66/s45/s100/s111/s112/s101/s100/s32/s67/s78/s84/s80/s117/s114/s101/s32/s67/s78/s84/s82/s105/s103/s104/s116/s45/s101/s108/s101/s99/s116/s114/s111/s100/s101\nFIG. 2: Surface density of states of the isolated FM electrod es and density of states of the isolated CNT for the pure case\nand randomly doped B and N atoms with 8% concentrations for pa rallel (top panel) and anti-parallel (bottom panel) spin\nalignments.\n/s50/s52/s54/s56\n/s45/s50 /s45/s49 /s48 /s49 /s50/s48/s50/s52/s54/s45/s50/s48/s45/s49/s48/s48\n/s45/s50 /s45/s49 /s48 /s49 /s50/s45/s54/s48/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s45/s48/s46/s53/s48 /s45/s48/s46/s50/s53 /s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48/s45/s49/s48/s45/s53/s48/s32/s32\n/s32/s32/s32/s112/s117/s114/s101/s32/s99/s97/s115/s101\n/s32/s66/s45/s100/s111/s112/s101/s100\n/s32/s78/s45/s100/s111/s112/s101/s100\n/s32/s32\n/s32/s32/s68/s79/s83/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s41 /s40/s98/s41/s40/s97/s41\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s80/s97/s114/s97/s108/s108/s101/s108\n/s65/s110/s116/s105/s112/s97/s114/s97/s108/s108/s101/s108\n/s32/s32\n/s32/s32\n/s65/s110/s116/s105/s112/s97/s114/s97/s108/s108/s101/s108/s40/s100/s41/s32/s112/s117/s114/s101/s32/s99/s97/s115/s101\n/s32/s66/s45/s100/s111/s112/s101/s100\n/s32/s78/s45/s100/s111/s112/s101/s100/s40/s99/s41\n/s80/s97/s114/s97/s108/s108/s101/s108\n/s32/s32\n/s32/s32/s76/s111/s103/s32/s40/s84/s40/s69/s41/s41\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\nFIG. 3: Density of states ((a) and (b)) and logarithmic scale of transmission function ((c) and (d)) versus energy for the\nFM/CNT/FM junction in PandAPconﬁgurations for the pure case and randomly doped B and N ato ms with 8% concentra-\ntions.6\n/s49/s50/s51/s52/s53\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s49/s50\n/s32/s32\n/s32/s32/s32/s112/s117/s114/s101/s32/s99/s97/s115/s101\n/s32/s66/s32/s97/s116/s111/s109/s115\n/s32/s78/s32/s97/s116/s111/s109/s115\n/s32/s32\n/s32/s32/s65/s110/s116/s105/s112/s97/s114/s97/s108/s108/s101/s108/s80/s97/s114/s97/s108/s108/s101/s108\n/s40/s98/s41/s40/s97/s41/s67/s117/s114/s114/s101/s110/s116/s32/s40 /s65/s41\n/s86/s111/s108/s116/s97/s103/s101/s32/s40/s86/s41\nFIG. 4: Current-voltage ( I−V) characteristics for (a) Pand (b)APconﬁgurations for the pure case and randomly doped B\nand N atoms with 8% concentrations\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s54/s54/s54/s56/s55/s48/s55/s50/s55/s52/s55/s54/s55/s56/s56/s48\n/s32/s112/s117/s114/s101/s32/s99/s97/s115/s101\n/s32/s66/s32/s97/s116/s111/s109/s115\n/s32/s78/s32/s97/s116/s111/s109/s115/s84/s77/s82/s32/s40/s37/s41\n/s86/s111/s108/s116/s97/s103/s101/s32/s40/s86/s41\nFIG. 5: TMR ratio as afunction of an applied bias for the purec ase and randomly doped B and Natoms with 8% concentrations\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s54/s52/s54/s56/s55/s50/s55/s54/s56/s48\n/s32/s32/s32\n/s32/s84/s77/s82/s32/s40/s37/s41\n/s86/s111/s108/s116/s97/s103/s101/s32/s40/s86/s41/s32/s40/s97/s41\n/s32/s40/s98/s41\n/s32/s40/s99/s41\n/s32/s40/s100/s41\nFIG. 6: TMR ratio as a function of an applied bias for (a) pure c ase and randomly doped B atom with (b) 4%, (c) 8% and (d)\n12% concentrations."    },    {        "title": "1108.1286v1.Thermoelectric_detection_of_ferromagnetic_resonance_of_a_nanoscale_ferromagnet.pdf",        "content": "Thermoelectric detection of ferromagnetic resonance of a nanoscale ferromagnet\nF. L. Bakker,\u0003J. Flipse, A. Slachter, D. Wagenaar, and B. J. van Wees\nPhysics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, The Netherlands\n(Dated: August 10, 2021)\nWe present thermoelectric measurements of the heat dissipated due to ferromagnetic resonance of\na Permalloy strip. A microwave magnetic \feld, produced by an on-chip coplanar strip waveguide,\nis used to drive the magnetization precession. The generated heat is detected via Seebeck mea-\nsurements on a thermocouple connected to the ferromagnet. The observed resonance peak shape is\nin agreement with the Landau-Lifshitz-Gilbert equation and is compared with thermoelectric \fnite\nelement modeling. Unlike other methods, this technique is not restricted to electrically conductive\nmedia and is therefore also applicable to for instance ferromagnetic insulators.\nPACS numbers: 76.50.+g, 75.78.-n, 72.15.Jf, 85.80.Fi\nThermal e\u000bects in ferromagnetic materials are sub-\nject to extensive research since the discovery of the\nspin-Seebeck e\u000bect [1{3]. Recently, spin dynamics and\n(spin-) caloritronics, two popular branches of spintron-\nics, started to come together as spin pumping induced\nby spin dynamics has been proposed as the origin of the\nspin-Seebeck e\u000bect [4, 5]. Magnetization dynamics has\nbeen studied thoroughly in magnetic systems as it is an\nimportant mechanism for future spintronic applications,\ne.g. for microwave generators [6, 7] and spin sources via\nspin pumping [8, 9]. However, dissipation mechanisms\nthat accompany magnetization dynamics, and cause lo-\ncal heating, are still not fully understood [10, 11].\nHere, we focus on a new aspect, the coupling between\nmagnetization dynamics and the generation of heat. We\ndeduce from thermoelectric measurements on a Permal-\nloy (Py) island the heat dissipation during ferromagnetic\nresonance. This on-chip detection technique, based on\nthe Seebeck e\u000bect, o\u000bers a novel method for characteriz-\ning ferromagnetic resonance and hence is distinctly di\u000ber-\nent from other techniques, such as scanning thermal mi-\ncroscopy [12, 13]. Due to the thermal detection, electrical\ncontact to the ferromagnet is in principle not required.\nHence, this method allows for FMR measurements on\nnon-conductive materials like ferromagnetic insulators.\nWhen a ferromagnet is brought into resonance, energy\nis absorbed from the applied microwave \feld. This en-\nergy causes the magnetization ~Mto precess around an\ne\u000bective \feld ~Hand the motion is well described by\nthe Landau-Lifshitz-Gilbert (LLG) equation d~M=dt =\n\u0000\r~M\u0002~H+(\u000b=M s)~M\u0002d~M=dt with\r= 176 GHz/T the\ngyromagnetic ratio. The last term in the LLG equation\ndescribes the damping of the magnetization towards the\ndirection of the e\u000bective \feld ~H, using the phenomeno-\nlogical damping parameter \u000b. This process is purely\ndissipative and converts magnetostatic energy into heat.\nDuring ferromagnetic resonance (FMR) this continuous\ndissipation leads to heating of the ferromagnetic mate-\nrial.\nIn this experiment, we measured the temperature of a\nferromagnet while subject to a microwave magnetic \feld.A ferromagnetic strip is placed close to the shortened\nend of a coplanar strip waveguide (CSW) as shown in\nFig. 1. Microwave power is applied to the CSW, leading\nto an out of plane rf magnetic \feld. A static magnetic\n\feldh0is applied along the easy axis of the magnet. In\naddition, a thermocouple consisting of a NiCu and Pt\nwire is connected to the ferromagnet by a Au bridge. In\nthis way, the temperature can be measured by making\nuse of the Seebeck e\u000bect. The Seebeck e\u000bect describes\nthe generation of a voltage due to a temperature gradi-\nent,rV=\u0000SrT, withSthe material dependent See-\nbeck coe\u000ecient. The voltage that develops across the\nPt wire is di\u000berent than the voltage that develops across\nthe NiCu wire, leading to a nonzero voltage between the\ntwo wires. This thermovoltage scales with the Seebeck\ncoe\u000ecients ( SPt\u0000SNiCu) and the temperature di\u000berence\n(T1\u0000T0). Note that NiCu is chosen because of its rel-\natively high Seebeck coe\u000ecient ( SNiCu = -40\u0016V=K and\nSPt= -5\u0016V=K ). In the following, we calculate the tem-\nT1VPt\nNiCu\nT0\nCSW\nIrfhrf\nh0Pyx\ny\nFIG. 1. (color online) Concept of the thermoelectric detec-\ntion of ferromagnetic resonance. A coplanar strip waveguide\ngenerates a microwave magnetic \feld hrfin the ^zdirection\nacting on a small ferromagnetic strip. A static magnetic \feld\nh0is applied along the ^ xaxis. When the strip is brought into\nresonance, it absorbs energy from the \feld which is dissipated\nas heat. The dissipation is detected by a Pt-NiCu thermocou-\nple which probes the temperature T1of the ferromagnet with\nrespect to a reference temperature T0via the Seebeck e\u000bect.arXiv:1108.1286v1  [cond-mat.mes-hall]  5 Aug 20112\nperature rise during FMR from the dissipated power.\nThe energy of a ferromagnetic particle in a magnetic\n\feld, the Zeeman energy, is given by:\nE=\u0000Z\nV~M\u0001~BdV (1)\nwith~Mthe magnetization, ~B=\u00160\u0010\n~Hext+~HD=2\u0011\nthe\nsum of the externally applied magnetic \feld and the de-\nmagnetizing \feld and Vthe volume of the particle [14].\nHere,~HDis divided by two to compensate for counting\neach volume twice in the integral. For this experiment, a\nstatic magnetic \feld, h0, is applied in the ^ xdirection and\na driving rf \feld, hrfcos!t, in the ^zdirection, making\n~B= (h0\u0000Nxmx=2;\u0000Nymy=2;hrfcos!t\u0000Nzmz=2)\nwithNx,NyandNzthe demagnetization factors. The\ndissipation energy can now be calculated from the time\nderivative of E, assuming a uniform ~Mand~B:\ndE\ndt=\u0000V \nd~M\ndt\u0001~B+~M\u0001d~B\ndt!\n(2)\nwhere the \frst part of Eq. 2 expresses the dissipation due\nto the magnetization motion and the second part the en-\nergy absorbed from the microwave \feld. In equilibrium,\nthe absorption of energy from the microwave \feld equals\nthe dissipation,hdE=dti= 0, leading to heating of the\nferromagnet. In order to \fnd an expression for the dis-\nsipated power, we use a procedure similar to Ref. [15]\nwhere the magnetization dynamics is described by the\nlinearized LLG equation. We assume that for small angle\nprecessional motion dmx=dt= 0 such that mxis constant\nand the solution to the Landau-Lifshitz-Gilbert (LLG)\nequation can be written in terms of the sum of in-phase\nand out-of-phase susceptibilities. The components my\nandmzare now de\fned as my=\u001f0\ny!1cos!t+\u001f00\ny!1sin!t\nandmz=\u001f0\nz!1cos!t+\u001f00\nz!1sin!twith:\n\u001f0\ny=\u000b!2(!y+!z)\n(!2\u0000!y!z)2+ (\u000b!)2(!y+!z)2\n\u001f00\ny=!(!2\u0000!y!z)\n(!2\u0000!y!z)2+ (\u000b!)2(!y+!z)2(3)\nHere,!is the frequency of the driving rf \feld, !y=\n\r(h0\u0000(Nx\u0000Ny)ms),!z=\r(h0\u0000(Nx\u0000Nz)ms),\n!1=\rhrfandmsthe saturation magnetization. The\n\u001f0\nzand\u001f00\nzare related via \u001f0\nz=\u000b\u001f0\ny\u0000!y=!\u001f00\nyand\n\u001f00\nz=\u000b\u001f00\ny+!y=!\u001f0\ny, respectively. With these expres-\nsions formx,my,mzandBx,By,Bzone can easily \fnd\nits time derivatives and calculate the relevant dot prod-\nuct of Eq. 2. We are not interested in high frequency\nvariations in the dissipated power and hence, average out\nall contributions that vary with a frequency !or 2!and\n\fnd:\n\u001cdE\ndt\u001d\ndissipation=\u001f00\nz!2\n1!V\n2\r(4)\n1 /uni03BCmIrf\n1 2NiCu PtFMI+V+V-I-Irfa) b)FIG. 2. (color online) a) Scanning electron microscope (SEM)\nimage of the thermoelectric FMR device. b) Image of a device\nwith four contacts for dc AMR detection of FMR.\nFrom this expression we can deduce that the resonance\npeak shape of the dissipated power is determined by\n\u001f00\nzand scales with the applied microwave \feld and fre-\nquency. In order to convert this power into a tempera-\nture rise, we make use of 3D \fnite element thermoelectric\nmodeling. For details about the modeling we refer to ear-\nlier publications [16, 17].\nThe samples are fabricated using a three-step electron\nbeam lithography process on top of a thermally oxidized\nSi substrate. A scanning electron microscope (SEM) im-\nage of the investigated devices is shown in Fig. 2. The\ndevices consist of a 50nm thick Py strip (2 \u0016m\u0002400 nm)\nclose to a 100 nm thick Au coplanar strip waveguide. The\nCSW is made using an optical lithography process. In the\nthermoelectric device (Fig. 2a), there are two 40 nm thick\ncontacts (Pt and NiCu) forming a thermocouple. The\nPy island is connected to the thermocouple by a highly\nthermal conductive Au contact (thickness: 120 nm). The\nother side of the thermocouple is connected by 120 nm\nthick Au contacts to the bonding pads. In the case of the\nanisotropic magnetoresistance (AMR) device (Fig. 2b),\nfour 120 nm thick Au contacts directly connect to the Py\nstrip. The NiCu is deposited by DC sputtering to pre-\nserve the original alloy composition (45% Ni, 55% Cu).\nTo avoid lift-o\u000b problems a double-layer resist technique\nwith a large undercut (PMMA-MA and PMMA 950K) is\nused. The Au, Pt and Py are deposited using an e-beam\nevaporator (base pressure 1 \u000210\u00007mbar) and a single\nlayer resist (PMMA 950K). Prior to the Au deposition,\nthe NiCu, Pt and Py surfaces are cleaned with Ar ion\nmilling.\nFor the measurements, we have used a frequency mod-\nulation method to obtain a better signal to noise ratio\nand to remove background voltages due to heating of\nthe CSW short. The microwave \feld frequency is al-\nternated between two di\u000berent values with a separation\nof 5 GHz. A lock-in ampli\fer, tuned to the same fre-\nquency (17 Hz), measures the di\u000berence in dc voltage\nacross contacts 1 and 2 (Fig. 2a) between the two fre-\nquencies (V=Vf=high\u0000Vf=low ). Because of the large3\nseparation between Vf=low andVf=high , they can not both\nful\fll the resonance condition at a speci\fc magnetic \feld.\nWith this method, one e\u000bectively measures the di\u000berence\nin the Seebeck or AMR voltage when the ferromagnet is\nin- and o\u000b-resonance. All measurements were performed\nat room temperature.\nFig. 3a shows the measured Seebeck voltage as a func-\ntion of magnetic \feld for di\u000berent rf \feld frequencies (10\n- 20 GHz) for 12 dBm rf power. The position of the peaks\nand dips correspond to the resonance \feld for flowand\nfhigh, respectively. We have plotted the peak position\nas a function of the applied rf frequency in Fig. 3b and\nfound peak heights ranging from 46 nV at 10 GHz to\n105 nV at 17 GHz. For a uniform precessional mode, the\nresonance \feld is related to !by the Kittel equation [18]:\n!2=\r2(h0\u0000(Nx\u0000Ny)ms)(h0\u0000(Nx\u0000Nz)ms) (5)\nThe line corresponds to a \ft of Eq. 5 and the almost\nperfect \ft con\frms the uniform precessional motion. We\nobtained the following \ftting parameters: Nx= 0:01,\nNy= 0:09,Nz= 0:90 and\u00160ms= 1:11 T. These param-\neters have been used to calculate the dissipated power of\nEq. 4 for di\u000berent frequencies. However, in order to do\nthis accurately one \frst need to determine the magnitude\nof the rf magnetic \feld experimentally.\nTo obtain the correct experimental value for the mag-\nnitude of the rf magnetic \feld, we have measured the dc\nanisotropic magnetoresistace (AMR) in a dedicated de-\nvice with a four-terminal geometry (shown in Fig. 2b).\nThe AMR e\u000bect describes the dependence of the resis-\ntance on the angle \u0012between the current ~Iand the di-\nrection of magnetization ~MbyR=R0\u0000\u0001Rsin2\u0012, where\nR0is the resistance of the strip when ~Iand~Mare par-\nallel, and \u0001 Rthe di\u000berence in resistance between the\nparallel and perpendicular alignment of ~Iand~M. A mea-\nsurement of Ras a function of a perpendicular applied\nmagnetic \feld is plotted in Fig. 4c. From this measure-\nment, we determined the magnitude of the AMR e\u000bect\nand found \u0001 R=R 0= 1:5 %. For a steady resonant pre-\ncession, the average cone angle \u0012cof the precession can\nnow be extracted from the observed AMR voltage.\nFig. 4a displays the AMR voltage versus magnetic \feld\nfor di\u000berent microwave \feld frequencies. For this mea-\nsurement, we have used a dc current Idcof 300\u0016A.\nThe obtained voltage now corresponds to the dc cur-\nrent multiplied with the resistance change, being V=\nIdc\u0001Rsin2\u0012c, and the precession angle can be extracted.\nUsing Eq. 3 and an expression for the average cone angle:\n\n\u00122\nc\u000b\n=!2\n1(\u001f02\ny+\u001f002\ny+\u001f02\nz+\u001f002\nz)=2 (6)\none can deduce hrf=!1=\rfrom a \ft of the measured\npeak height, and the result is plotted in Fig. 4b for a\nmicrowave power of 12 dBm. The \feld strength is found\nto decrease twofold when the frequency is increased from\n10 to 20 GHz. We attribute this to frequency dependent\nSeebeck Voltage [ /uni03BCV ]\nMagnetic Field [ mT ]10 GHz20 GHz\n12 GHz\n11 GHz13 GHz14 GHz15 GHz16 GHz17 GHz18 GHz19 GHz\n0 50 100 150 200 250 300 3508101214161820\nMagnetic Field [mT]Frequency [GHz]a)\nb)\nSeebeck voltage [nV]\nMagnetic Field [mT]−400 −200 0 200 40000.20.40.60.811.21.41.6\n15 GHz17 GHz\n16 GHz18 GHz19 GHz20 GHz\n0 50 100 150 200 250 300 350050100150200\n010203040506070\nPower [nW]c)\n10 GHz\n20 GHz15 GHzFIG. 3. (color online) a) Series of Seebeck voltage versus\nmagnetic \feld measurements for 11 di\u000berent frequencies. The\ntraces are o\u000bset by 150 nV for clarity. Due to the modula-\ntion technique using two driving frequencies that are 5 GHz\napart, peaks and dips are observed at the resonance \felds for\nboth frequencies. b) Frequency versus the magnetic \feld at\nthe center of the resonance peak. The line corresponds to\na \ft of the Kittel equation. c) Generated power and corre-\nsponding Seebeck voltage calculated using Eq. 4 and thermo-\nelectric \fnite-element modeling for multiple frequencies. The\nmeasured peak heights of a) are indicated by the black dots.\nattenuation of the microwave signal, leading to smaller rf\n\felds at higher frequencies.\nNow we can calculate, using \fnite element modeling in\nComsol Multiphysics, the Seebeck voltage that is gener-\nated due to the heating of the ferromagnet. In this model\nwe impose the constant heat \rux, given by Eq. 4, through\nthe top layer of the ferromagnet and solve the thermo-\nelectric model [16, 17]. Both, the heat \rux and the calcu-\nlated Seebeck voltage are plotted in Fig. 3c (solid lines)\nfor multiple frequencies. Peak heights ranging from 98\ntill 197 nV are calculated. For comparison, the observed\npeak height of Fig. 3a is replotted in Fig. 3c as black\ndots.\nFor a \fxed rf \feld strength, the Seebeck voltage should\nincrease monotonically with frequency due to an increas-\ning dissipation (see Eq. 4). However, because of the ex-\nperimental variation in rf \feld strength for di\u000berent fre-\nquencies, a speci\fc relation between the calculated See-\nbeck voltage and the frequency is found (Fig. 3c). The\nexperimental data is in agreement with the calculations\nwithin a factor of two and follows partially the same4\n−400 −200 0 200 4000123456V [/uni03BCV ] \nMagnetic Field [mT]10 GHz20 GHz\n12 GHz\n11 GHz13 GHz14 GHz15 GHz16 GHz17 GHz18 GHz19 GHz\n10 12 14 16 18 20 220.511.5\nFrequency [GHz]\n /uni03BC0h1 [mT]\n−200 −100 0 100 20015.615.715.8\nR [/uni03A9] \nMagnetic Field [mT]h0     Ia) b)\nc)\nFIG. 4. (color online) a) AMR voltage vs. magnetic \feld.\nPeaks and dips are observed for the resonance \felds of the\ntwo driving frequencies (5 GHz apart). The di\u000berent traces\nare o\u000bset for clarity reasons. b) The magnitude of the\nrf \feld is extracted from the peak height of the resonance\n(V=Idc\u0001Rsin2\u0012c). c) Anisotropic magnetoresistance mea-\nsurement with ~Iand~Bperpendicularly aligned.\ntrend. This discrepancy is attributed to small sample\nto sample variations in the rf \feld strength. Since the\nAMR measurements are performed on a separate device,\nthe \felds can di\u000ber for the thermoelectric device. More-\nover, small shifts in the contact area of the thermocouple\ncan lead to changes in the heat transport and hence, dif-\nferent thermo voltages.\nFurthermore, circulating rf currents combined with an\noscillating magnetoresistance at the same frequency can\ncause dc voltages via a rectifying e\u000bect and mimic the\nobserved thermal behavior in our devices [15]. We have\nexcluded these e\u000bects by using a similar device with an\nAu-Au thermocouple such that the Seebeck e\u000bect van-\nishes. For this device we observed a \rat background\nvoltage without peaks and dips. We note that thermal\nvoltages can be of importance in other device geometries\nwhere ferromagnets are electrically connected to nonmag-\nnetic metals. For example, detection of interface voltages\nthat arise due to spin-pumping [19, 20] cannot be easily\ndistinguished from generated Seebeck voltages. Thor-\nough temperature or material dependent measurements\nmight o\u000ber a solution to discriminate between both mech-\nanisms.\nIn conclusion, we have demonstrated a new thermo-\nelectric detection technique for ferromagnetic resonance.\nThe observed resonance peaks are in good agreement\nwith the LLG equation and thermoelectric \fnite element\nmodeling. Additionally, this technique can be applied on\nthe nanoscale and is not limited to conductive ferromag-\nnetic media. Thermal detection o\u000bers a valid, alterna-\ntive method for studying the dissipation, i.e. the Gilbert\ndamping term, in nanoscale ferromagnetic islands. We\nhope that these results stimulate research for new physi-\ncal e\u000bects that arises from coupling between magnetiza-tion dynamics and caloritronics.\nWe would like to acknowledge B. Wolfs, M. de Roosz\nand J.G. Holstein for technical assistance. This work is\npart of the research program of the Foundation for Fun-\ndamental Research on Matter (FOM) and supported by\nNanoLab and the Zernike Institute for Advanced Mate-\nrials.\n\u0003f.l.bakker@rug.nl\n[1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n[2] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom,\nJ. P. Heremans, and R. C. Myers, Nature Materials 9,\n898 (2010).\n[3] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi,\nJ. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai,\nG. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature\nMaterials 9, 894 (2010).\n[4] J. Xiao, G. E. W. Bauer, K. C. Uchida, E. Saitoh, and\nS. Maekawa, Phys. Rev. B 81, 214418 (2010).\n[5] H. Adachi, J.-i. Ohe, S. Takahashi, and S. Maekawa,\nPhys. Rev. B 83, 094410 (2011).\n[6] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature 425, 380 (2003).\n[7] D. Houssameddine, U. Ebels, B. Dela et, B. Rodmacq,\nI. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-\nP. Michel, L. Prejbeanu-Buda, M.-C. Cyrille, O. Redon,\nand B. Dieny, Nature materials 6, 441 (2007).\n[8] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and\nB. I. Halperin, Physical Review B 66, 060404 (2002).\n[9] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B 66, 224403 (2002).\n[10] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[11] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[12] R. Meckenstock, Rev. Sci. Instrum. 79, 041101 (2008).\n[13] F. Sakran, A. Copty, M. Golosovsky, D. Davidov, and\nP. Monod, Applied Physics Letters 84, 4499 (2004).\n[14] J. Coey, Magnetism and magnetic materials (Cambridge\nUniversity Press, 2010).\n[15] M. V. Costache, S. M. Watts, M. Sladkov, C. H. van der\nWal, and B. J. van Wees, Applied Physics Letters 89,\n232115 (2006).\n[16] F. L. Bakker, A. Slachter, J.-P. Adam, and B. J. van\nWees, Physical Review Letters 105, 136601 (2010).\n[17] A. Slachter, F. L. Bakker, and B. J. van Wees, ArXiv\ne-prints (2011), arXiv:1107.3290 [cond-mat.mes-hall].\n[18] C. Kittel, Introduction to Solid State Physics -7th ed.\n(John Wiley & Sons, Inc., New York, 1995).\n[19] X. Wang, G. E. W. Bauer, B. J. van Wees, A. Brataas,\nand Y. Tserkovnyak, Phys. Rev. Lett. 97, 216602 (2006).\n[20] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der\nWal, and B. J. van Wees, Physical Review Letters 97,\n216603 (2006)."    },    {        "title": "1101.4782v1.Charge_and_spin_transport_through_a_ferromagnet_insulator_unconventional_superconductor_junction.pdf",        "content": "arXiv:1101.4782v1  [cond-mat.supr-con]  25 Jan 2011Charge and spin transport through a ferromagnet/insulator /unconventional\nsuperconductor junction\nGaetano Annunziata, Mario Cuoco, Paola Gentile, Alfonso Romano, C anio Noce\nCNR-SPIN, I-84084 Fisciano (Salerno), Italy\nDipartimento di Fisica “E. R. Caianiello” Universit` a di Sa lerno,\nI-84084 Fisciano (Salerno), Italy\n(Dated: August 22, 2018)\nWe analyze the charge and spin transport through a ballistic ferromag-\nnet/insulator/superconductor junction by means of the Bog oliubov-de Gennes equations.\nFor the ferromagnetic side we assume that ferromagnetism ma y be driven by an unequal mass\nrenormalization of oppositely polarized carriers, i.e. a s pin bandwidth asymmetry, and/or by a\nrigid splitting of up-and down-spin electron bands, as in a s tandard Stoner ferromagnet, whereas\nthe superconducting side is assumed to exhibit a d-wave symmetry of the order parameter, which\ncan be pure or accompanied by a minority component breaking t ime-reversal symmetry. Several\nremarkable features in the charge conductance arise in this kind of junction, providing useful\ninformation about the mechanism of ferromagnetism in the fe rromagnetic electrode, as well as\nof the order parameter symmetry in the superconducting one. In particular, we show that when\na time-reversal symmetry breaking superconductor is consi dered, the use of the two kinds of\nferromagnet mentioned above represents a valuable tool to d iscriminate between the diﬀerent\nsuperconducting mixed states. We also explain how this junc tion may mimic a switch able to turn\non and oﬀ a spin current, leaving the charge conductance unch anged, and we show that for a wide\nrange of insulating barrier strengths, a spin bandwidth asy mmetry ferromagnet may support a spin\ncurrent larger than a standard Stoner one.\nI. INTRODUCTION\nTransport in itinerant ferromagnet/insulator/super-\nconductor (F/I/S) junctions is a fundamental issue in\ncondensed matter physics for its deep implications in\nelectronics and spintronics,1and for the opportunity it\noﬀers to test the physical properties of ferromagnetic and\nsuperconducting materials via point contact2,3or scan-\nning tunneling4,5measurements. Moreover, this type of\nhybrid structure may serve as a playground for a wealth\nof interesting quantum mechanical eﬀects pertaining to\ntheinterplaybetweenspinandchargedegreesoffreedom.\nIn fact, beyond the possibility of a direct estimation of\nthe gapmagnitude in conventionalsuperconductors, tun-\nneling conductance measurements oﬀer the opportunity\nto probe also the superconducting order parameter sym-\nmetry in unconventional superconductors. This prop-\nerty has made this kind of measurements fundamental\nin ﬁnding clues about the symmetry of the new families\nof superconductors recently discovered, for which there\nis a general consensus that they cannot be considered\nas conventional. For example, one of the strongest evi-\ndences supporting d-wave symmetry for high- Tccuprates\nis the zero bias conductance peak (ZBCP) revealed in\nab-plane tunneling conductance from normal metals.6In\nsomecuprates, suchas forinstance YBa 2Cu3O7−δ,7,8the\nexistence of a subdominant component in the order pa-\nrameter possibly breaking time-reversal symmetry is still\na matter of debate and, in this respect, exploiting the\ninterplay between magnetism and superconductivity in\ntunneling experiments is one of the standard routes to\ninvestigate this issue. Generally, by using a ferromag-\nnetic electrode in tunneling experiments it is possible tochange the relative contributions of up and down elec-\ntrons to the total density of states or, in the half-metal\nlimit, to isolate a single spin channel.\nTointerpretthelargeamountoftunneling experiments\nperformed on F/I/S junctions involving an unconven-\ntionalsuperconductor,9–16fundamentaltheoriesoftrans-\nport, such as in particular the one by Blonder, Tinkham\nand Klapwijk (BTK),17have been suitably extended to\ntakeintoaccountallpossiblesymmetriesofthesupercon-\nducting order parameter. In this context, the ferromag-\nnetic electrode has been predominantly described within\nthe Stoner model, relying on the assumption that the\nbands associated with the two possible electron spin ori-\nentationshaveidenticaldispersion, but arerigidlyshifted\nin energy by the exchange interaction. However, Stoner\nmodel may prove to be insuﬃcient to describe real fer-\nromagnets because many terms deriving from Coulomb\nrepulsion are eliminated from the full Hamiltonian, al-\nthough in some situations their contribution can be im-\nportant.18,19\nThe complexityofferromagnetismin metals is testiﬁed\nby the wide range of manifestations it exhibits in nature.\nAs relevant examples of this variety, we mention the fer-\nromagnetic transition metals Fe, Co, and Ni and their\nalloys,20weak metallic ferromagnets such as ZrZn 221,22\nand Sc 3In,23,24colossal magnetoresistance manganites\nsuch as La 1−xSrxMnO3,25and rare earth hexaborides\nsuch as EuB 6.26,27Therefore, when theoretically mod-\nelling F/I/S junctions, it may be important to assume\nfor the magnetism in the F electrode microscopic scenar-\nios other than the Stoner one. Among them, of pecu-\nliar interest is a form of itinerant ferromagnetism driven\nby a gain in kinetic energy deriving from a spin depen-2\ndent bandwidth renormalization, or, equivalently, by an\neﬀective mass splitting between up- and down-spin carri-\ners.28–33The interplay of superconductivity with this ki-\nnetically driven ferromagnetism has been recently shown\nto originate diﬀerent features compared to the Stoner\ncase, concerning the phenomena of coexistence, proxim-\nity and transport. More precisely, we have studied the\noccurrence of the coexistence of ferromagnetism and s-\nwave singlet superconductivity within a model where the\nmagnetic moments are due to a kinetic exchange mecha-\nnism, andwehaveshownthat thedepairedelectronsplay\na crucial role in the energy balance, and that when their\ndynamicaleﬀect issuchthat toundresstheeﬀectivemass\nof the carriers which participate in the pairing, a coex-\nisting ferromagnet-superconducting phase can be stabi-\nlized.34Then, wehaveexactlysolvedanextended version\nof the reduced BCS model for particles that get paired\nin the presence of a polarization arising from spin de-\npendent bandwidths and we have calculated the ground-\nstate phase diagram in the full parameter space of the\npair coupling and the bandwidth asymmetry as a func-\ntion of ﬁlling for diﬀerent types of spectrum topologies.35\nWe have also investigated the proximity eﬀect within\na junction made of an unconventional superconductor\nand a ferromagnet in the clean limit with high barrier\ntransparency, and we have shown that the two above-\nmentioned mechanisms for ferromagnetism lead to dif-\nferent features as concerns the formation at the interface\nof dominant and sub-dominant superconducting compo-\nnents as well as their propagation in the ferromagnetic\nside.36Finally, a F/I/S ballistic junction with a conven-\ntionals-wave superconductor has been used to distin-\nguish whether itinerant ferromagnetism in the F elec-\ntrode is due to exchange splitting or to spin-dependent\nmass renormalization of up- and down-spin electrons.37\nWe have also shown that under appropriate conditions\nthe spin dependent conductance of minority carriers can\nbe larger than for majority carriers below the energy gap\n∆0, and lower above it, suggesting that the junction, in\na suitable range of microscopical parameters, may work\nas a spin-ﬁltering device.38\nIn this work we carry out the investigation of the in-\nterplay between diﬀerent types of ferromagnetism and\nsuperconductivity analyzing charge and spin transport\nthrough a ballistic F/I/S junction where various uncon-\nventional symmetries for the superconducting electrode\nare considered. The problem is handled by solving the\nBogoliubov-de Gennes (BdG) equations39within an ex-\ntended Blonder-Tinkham-Klapwijk approach, here for-\nmulated for a two-dimensional F/I/S junction. As it\nis well known, this method has been generalized in the\nlast years to take into account higher dimensionalities,\nanisotropic forms of the superconducting order param-\neter, diﬀerent Fermi energies for the two sides of the\njunction, and a spin–ﬂip interfacial scattering.40–52We\ninvestigate the behavior of the charge and the spin con-\nductance, revealing several noteworthy features arising\nfrom the interplay between unconventional superconduc-tivity and each of the two kinds of ferromagnetism spec-\niﬁed above. The diﬀerences emerging in the two cases\nare shown to provide relevant indications on the physical\nproperties of the materials constituting both the ferro-\nmagnetic and the superconducting electrode of the junc-\ntion. Moreover, we also show that the behavior of the\ncharge conductance in the case of pure d-wave materials\nis diﬀerent from that found when a minority component\nbreaking time-reversalsymmetry (BTRS) is also present.\nIn this case, the new features emerging around zero-bias\nvoltageexhibitadiﬀerentbehaviordependingonwhether\na Stoner or a mass mismatch ferromagnet is considered,\nthus providing a clear indication on the nature of the mi-\ncroscopic mechanism underlying ferromagnetism in the\nF layer. We would like to notice that our assumption of\na bulk character of these BTRS components is justiﬁed\nby the two-dimensional character of the junction that we\nanalyze. Indeed, with this two-dimensional geometry, a\nnodeless broken time reversal symmetry state may ap-\npear throughout the S side of the junction, and this is\nconsistent with the BTRS dx2−y2+isordx2−y2+idxy\ncombinations here assumed.\nWe also report on the eﬀect of the mass asymmetry\non spin conductance for conventionaland unconventional\nsuperconducting electrodes and we show that under spe-\nciﬁc conditions the mass mismatch greatly enhances spin\ncurrent, so that a spin bandwidth asymmetry ferromag-\nnet can lead to a spin current much larger than the that\nproduced by a Stoner ferromagnet, at the same polariza-\ntion. Hence, we explain how a F/I/S junction can work\nas a switch able to turn on and oﬀ a spin current, leaving\nthe charge current unchanged.\nThe paper is organized as follows. In Section II we\nformulate the microscopic model and the related method\nof solution. In Section III we present the results, dis-\ncussing them in three diﬀerent Subsections concerning:\n(A) the magnetization in the ferromagnetic side, (B) the\ncharge conductance through a junction with a supercon-\nductor having a pure dx2−y2-wave symmetry or a broken\ntime-reversal symmetry of dx2−y2+isordx2−y2+idxy\ntype, and (C) the spin conductance through a junction\nwith conventional and unconventional superconductors.\nFinally, Section IV is devoted to the conclusions.\nII. MODEL AND FORMALISM\nThe system under study is built up of two semi-inﬁnite\nlayers connected by an inﬁnitely thin insulating barrier\nresulting in an interfacial scattering potential of the form\nV(r) =Hδ(x). We choose an interface lying along the y\ndirection at x= 0 (see Fig. 1) so that the region x <0\n(from now on the F side) is occupied by an itinerant\nferromagnet (a Stoner or a spin bandwidth asymmetry\nferromagnet, or a combination of the two), while the re-\ngionx >0 (from now on the S side) is occupied by a\nsinglet superconductor (so there is no need to specify the\nspin quantization axis). We point out that though in the3\nfollowingwerefer tofree particle-likespectraofparabolic\ntype for which the concept of bandwidth is in principle\nill-deﬁned, we nonetheless imagine to link this descrip-\ntion to some eﬀective one-band tight-binding model, al-\nlowing to relate the inverse of the mass of the carriers to\nthe width of the eﬀective bands where itinerancy takes\nplace. In this way, a bandwidth asymmetry is generated\nby simply assuming diﬀerent values of the masses for up-\nand down-spin electrons.\nWe describe the excitations propagating through the\njunction by means of the single-particle Hamiltonian\nHσ\n0=/bracketleftbig\n−/planckover2pi12∇2/2mσ−ρσU−EF/bracketrightbig\nΘ(−x)\n+/bracketleftbig\n−/planckover2pi12∇2/2m′−E′\nF/bracketrightbig\nΘ(x)+V(r),(1)\nwhereσ=↑,↓,mσis the eﬀective mass for σ-polarized\nelectrons in the F side, ρ↑(↓)= +1(−1),Uis the ex-\nchange interaction, EFis the Fermi energy of the ferro-\nmagnet, Θ( x) is the unit step function, m′andE′\nFare\nthe quasiparticleseﬀective mass and the Fermi energyfor\nthe superconductor, respectively.\nThe BdG equations read as\n/parenleftbigg\nHσ\n0∆\n∆∗−H¯σ\n0/parenrightbigg/parenleftbigg\nuσ\nv¯σ/parenrightbigg\n=ε/parenleftbigg\nuσ\nv¯σ/parenrightbigg\n, σ=↑,↓,(2)\nwhere ¯σ=−σand (uσ,v¯σ)≡Ψσis the energy eigen-\nstate in the electron-holespaceassociatedwith the eigen-\nvalueε(excitation energies are measured from the Fermi\nlevel). Eqs. (2) admit an analytical solution in the ap-\nproximation of a rigid superconducting pair potential,\ni.e. ∆(r) = ∆(θ′)Θ(x), whereθ′is the angular vari-\nable for the S side (see Fig. 1). The Hamiltonian invari-\nance under y-directed translations permits to factorize\nthe part of the eigenstate parallel to the interface, i.e.\nΨσ(r) =eik/bardbl·rψσ(x), reducing the eﬀective dimensional-\nity of the problem.\nLooking at Fig. 1, we observe that at the inter-\nface four scattering processes are possible for an elec-\ntron injected from the F side with spin σand momen-\ntumk+\nσ(k+\nσ=/bracketleftbig/parenleftbig\n2mσ//planckover2pi12/parenrightbig\n(EF+ρσU+ε)/bracketrightbig1/2):a) An-\ndreev reﬂection (AR) resulting in a hole with momen-\ntumk−\n¯σ(k−\n¯σ=/bracketleftbig/parenleftbig\n2m¯σ//planckover2pi12/parenrightbig\n(EF+ρ¯σU−ε)/bracketrightbig1/2) belong-\ning to the opposite spin band and a Cooper pair trans-\nmitted in the superconductor; b) normal reﬂection; c)\ntransmission as electron–like quasiparticle with momen-\ntumk′+\nσ(k′+\nσ=/bracketleftBig/parenleftbig\n2m′//planckover2pi12/parenrightbig/parenleftBig\nE′\nF+/radicalbig\nε2−|∆σ+|2/parenrightBig/bracketrightBig1/2\n);\nd) transmission as hole–like quasiparticle with momen-\ntumk′−\nσ(k′−\nσ=/bracketleftBig/parenleftbig\n2m′//planckover2pi12/parenrightbig/parenleftBig\nE′\nF−/radicalbig\nε2−|∆σ−|2/parenrightBig/bracketrightBig1/2\n),\nwhere ∆ σ±=|∆σ±|eiφ±\nσis the pair potential felt by\nelectron-like (+) and hole-like ( −) quasiparticles. We\nnotice that the spin dependence of ∆ σ±comes out from\nthe diﬀerent trajectories followed by up- and down-spin\nquasiparticles. Which of these processes actually takes\nplace depends on the energy, momentum and spin orien-\ntation of the incoming electrons, as well as on the inter-\nfacial barrier strength, the polarization in the F side andF S\nΘΣ\nΘΣΘ'Σy\nxa\nb\nΒ\nb c\nd\na\nFIG. 1: (Color online) Scheme of the planar F/I/S junction\nanalyzed in the paper. Here, θσ,θ¯σ, andθ′\nσare injection,\nAndreev reﬂection, and transmission angles, respectively , for\nelectrons andquasiparticles with spin σ.βis the angle formed\nbythecrystallographic aaxisofa d-wavesuperconductorwith\nthexaxis.\nthe symmetry of the superconducting order parameter in\nthe S side.\nFor standard low-biased F/I/S junctions, one has\nEF,E′\nF≫(ε,|∆|), so that one can apply the Andreev\napproximation53and ﬁx the momenta on the Fermi sur-\nfaces. In this case the solutions of BdG equations for the\ntwo sides of the junction can be written as\nψF\nσ(x) =eikF\nσ,xx/parenleftbigg\n1\n0/parenrightbigg\n+aσeikF\n¯σ,xx/parenleftbigg\n0\n1/parenrightbigg\n+bσe−ikF\nσ,xx/parenleftbigg\n1\n0/parenrightbigg\n(3)\nψS\nσ(x) =cσeik′F\nσ,xx/parenleftbiggu+\ne−iφ+v+/parenrightbigg\n+dσe−ik′F\nσ,xx/parenleftbigg\neiφ−u−\nv−/parenrightbigg\n(4)\nwhere\nu±=/radicalBigg\nε±/radicalbig\nε2−|∆σ±|2\n2ε\nv±=/radicalBigg\nε∓/radicalbig\nε2−|∆σ±|2\n2ε,\nand the superscript F in the wave-vectors denotes that\nthey are taken on the Fermi surfaces.\nThe boundary conditions at the interface allow for the\ncalculation of the probability amplitude coeﬃcients aσ,\nbσ,cσ,dσfor the four scattering processes. We have\nψF\nσ(0) =ψS\nσ(0) (5a)\nmσ\nm′duS\nσ\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0−duF\nσ\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0=2H mσ\n/planckover2pi12uS\nσ(0) (5b)\nm¯σ\nm′dvS\n¯σ\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0−dvF\n¯σ\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0=2H m¯σ\n/planckover2pi12vS\n¯σ(0).(5c)\nEq. (5) show that the mass asymmetry explicitly renor-\nmalizes the interface barrier strength H, giving rise to4\na dependence of this quantity on the spin of the carri-\ners. This eﬀect, which under suitable conditions leads\nto a diﬀerent behavior of these carriers across the bar-\nrier, allows to infer that the presence of spin depen-\ndent electron masses in Eq. (5) may mimic a spin active\nbarrier, in the sense that electrons with opposite spin\nfeel diﬀerent values of the barrier height. A junction\nwith a mass mismatch ferromagnet can thus induce an\neﬀective spin-active interfacial eﬀect, which for speciﬁc\nchoices ofHhas been shown to produce a minority–spin\ncharge conductance component higher than the corre-\nsponding majority-spin one.38The dimensionless param-\neterZ= 2m′Hπ2/(/planckover2pi12k′\nF) everywhere in following will\nconveniently characterize the strength of the interfacial\nscattering.\nThe chargeand spin diﬀerential conductancesat T= 0\nand energy ε, i.e. at bias voltage V=ε/e,ebeing the\nelectron charge, are calculated from the ratio between\nthe charge and spin ﬂuxes across the junction and the\nincident ﬂuxes at that bias. They can be easily obtained\nfrom the probabilities associated with the four processes\nlisted above,41and for each spin channel they can be\nwritten as\nGσ(ε,θ) =Pσ/parenleftBigg\n1+kF\n¯σ,x\nkFσ,x|aσ(ε,θ)|2−|bσ(ε,θ)|2/parenrightBigg\n(6)\nΣσ(ε,θ) =Pσ/parenleftBigg\n1−kF\n¯σ,x\nkFσ,x|aσ(ε,θ)|2−|bσ(ε,θ)|2/parenrightBigg\n,(7)\nwhereθis the angle formed by the momentum of the\nelectrons propagatingfrom the F side with respect to the\nnormal to the interface (see Fig. 1), and the polarization\nPσ=nσ/(n↑+n↓) is the fraction of electrons occupying\ntheσ-spin band of the metallic ferromagnet.\nThe measured conductances take contributions from\na range of angles determined by the experimental con-\nditions. This range is limited from above due to the\nconservation of the momentum parallel component\nkF\nσsinθ=kF\n¯σsinθ¯σ=k′Fsinθ′\nσ, (8)\nwhereθ¯σandθ′\nσare AR and the transmission angles, re-\nspectively, for electrons and quasiparticles with spin σ.\nFrom this equation it is easy to verify the existence of\ncritical angles above which these processes are no more\npossible, resulting in virtual AR41and normal reﬂection,\nrespectively. The angularly averaged diﬀerential conduc-\ntances for given spin orientation are then deﬁned as41\n/angb∇acketleftGσ(ε)/angb∇acket∇ight=/integraldisplayθσ\nC\n−θσ\nCdθcosθ Gσ(ε,θ)//integraldisplayθσ\nC\n−θσ\nCdθcosθ(9)\n/angb∇acketleftΣσ(ε)/angb∇acket∇ight=/integraldisplayθσ\nC\n−θσ\nCdθcosθΣσ(ε,θ)//integraldisplayθσ\nC\n−θσ\nCdθcosθ,(10)\nwhereθσ\nCis the critical angle for the transmission of σ-\nspin electrons.Finally, the net averagedchargeandspin conductances\nare respectively deﬁned as\n/angb∇acketleftG(ε)/angb∇acket∇ight=/angb∇acketleftG↑(ε)/angb∇acket∇ight+/angb∇acketleftG↓(ε)/angb∇acket∇ight (11)\n/angb∇acketleftΣ(ε)/angb∇acket∇ight=/angb∇acketleftΣ↑(ε)/angb∇acket∇ight−/angb∇acketleftΣ↓(ε)/angb∇acket∇ight. (12)\nIII. RESULTS\nThe results here obtained for the F/I/S junction are\ngrouped in three following distinct Subsections concern-\ning (A) the magnetization in the F side, (B) the charge\nconductance for a dx2−y2-wave or broken time-reversal\nstates (BTRS) associated with dx2−y2+isanddx2−y2+\nidxypairing symmetry superconducting electrode, and\n(C) the spin conductance for the above choices of the or-\nder parameter and, for a comparison, for a conventional\ns-wave superconducting S-side. To appreciate the eﬀect\nof mass asymmetry, we neglect Fermi energies mismatch\neﬀects and ﬁx EF=E′\nF.\nA. MAGNETIZATION\nThe spin bandwidth asymmetry in the F side directly\naﬀects the density ofstates per spin orientation, and con-\nsequently the net polarization. Given the single-particle\nHamiltonian(1), weﬁndthatinthetwo-dimensionalcase\nthe ground-state magnetization M≡P↑−P↓is given by\nM2D=(X+1)Y\nX(Y−1)+Y+1−1−X\nX(Y−1)+Y+1,(13)\nwhereX=U/EFandY=m↑/m↓. Eq. (13) correctly\nreduces to known results for a pure Stoner ferromagnet\nwhenY→1.42On the other hand, when Y→0(∞)\nwe precisely reproduce the half-metal limit M→ −1(1).\nFor a ﬁxed value of the exchange splitting, the mass\nmismatch enhances the net polarization for m↑> m↓\n(Y >1) and hinders it the other way around ( Y <1).\nThe situation is illustrated in Fig. 2, where the den-\nsity plot of the magnetization at T= 0 in the ( m↑/m↓,\nU/EF)parameterspaceisshownforone-,two-andthree-\ndimensional ferromagnets, together with three isomagne-\ntizationcurvesplottedtoclarifythemagnetizationtrend.\nEach point corresponds to a diﬀerent realization of the\nferromagnetic order in the sense that the relative weights\nof the exchange splitting and the mass mismatch are de-\nterminedbythecoordinatesofthatpoint, whilethevalue\nofMisﬁxedalongtheisomagnetizationcurves(thesolid,\ndashed and dotted lines of Fig. 2). We see that though\nthequalitativebehaviorisindependent onthedimension-\nality, for the chosen band dispersion in the ferromagnet,\none alwaysﬁnds M3D>M2D>M1Dwhen evaluated for\nthe samem↑/m↓andU/EFvalues.5\nM<000.20.40.60.81\n012345678\nm↑/m↓00.20.40.60.81U/EF3DM<000.20.40.60.81\n3D\n00.20.40.60.81U/EF\nACE\nB D F2DM<000.20.40.60.81\n00.20.40.60.81U/EF1D\nFIG. 2: (Color online). Density plot of the ground state\nmagnetization as a function of the mass mismatch and the\nnormalized exchange interaction, for one-, two- and three-\ndimensional ferromagnetic electrodes. As shown in the leg-\nend on the right, lighter color regions are associated with\nhigher values of the magnetization. For clarity, only three\niso-magnetization curves are plotted in all panels, corres pond-\ning toM= 0.25 (solid line), M= 0.50 (dashed line), and\nM= 0.75 (dotted line). In the middle panel, referring to\nthe dimensionality considered in this paper, we depict six\nrepresentative points: A and B correspond to two diﬀer-\nent microscopic states with the same macroscopic magneti-\nzation M = 0.25, A reprsenting a standard Stoner ferromag-\nnet (m↑/m↓= 1), and B a purely spin bandwidth asymme-\ntry ferromagnet ( U/EF= 0). The same holds for the (C,D)\nand (E,F) couples of points, referring to higher values of th e\nmagnetization ( M= 0.50 andM= 0.75, respectively). The\nvalues assumed by the microscopic parameters in the above\nmentioned six states are summarized in Table I.\nB. CHARGE TRANSPORT\nWe have analyzed F/I/S conductance spectra in two\ndimensions in the entire parameter space, excluding the\nregions corresponding to M <0 (indicated in black inU/EFm↑/m↓M\nA 0.25 10.25B 0 5/3\nC 0.50 10.50D 0 3\nE 0.75 10.75F 0 7\nTABLE I: Values of the normalized exchange interaction\nU/EF, the mass mismatch m↑/m↓and the magnetization M\nfor the six illustrative points displayed in the middle pane l of\nFig. 2.\nFig.2) sincethey aremirrorimagesofthosewith positive\nM, assumingthat m↑/m′=m′/m↓forY >1. We notice\nthat with this choice critical angles for AR and transmis-\nsion exist only for majority electrons. In the following\nSubsections we discuss the results for the six represen-\ntative points highlighted in Fig. 2, which correspond to\na pure Stoner ferromagnet (STF), i.e. m↑/m↓= 1, and\na pure spin bandwidth asymmetry ferromagnet (SBAF),\ni.e.U/EF=0, for three diﬀerent values of the magneti-\nzationM= 0.25,0.50,0.75 (the corresponding values of\nm↑/m↓andU/EFare reported in Table I). F/I/S con-\nductance spectra will be shown for various symmetries of\nthe superconducting order parameter, emphasizing the\ndiﬀerences in transport between STF/I/S and SBAF/I/S\njunctions. We ﬁnally notice that spectra change continu-\nously as one moves along an isomagnetization curve from\na point corresponding to a STF to a point corresponding\nto a SBAF.\n1. F/I/ dx2−y2JUNCTION\nIt is well known54,55that in N/I/S junctions involving\na normal metal and a dx2−y2superconductor, a zero-\nbias conductance peak (ZBCP) develops in the tunnel-\ning limit, this peak becoming narrower and narrower as\nincreasing values of the interfacial barrier strength are\nconsidered. This ZBCP is the consequence of the pres-\nence of an Andreev bound state56(ABS) at the Fermi\nenergy, induced by the change in sign of the pair poten-\ntial across line nodes. It implies that electron-like and\nhole-like quasiparticles specularly reﬂected at the inter-\nface always ﬁnd the “right” sign of the pair potential\nto be Andreev reﬂected. In this case the ABS is at the\nsame energy for every quasiparticle trajectory, i.e. for\nevery angle θ. When the normal metal in the junction is\nreplaced by a STF, the ZBCP is lowered because of the\npresence of the ferromagnetic polarization which inhibits\nARs and can be splitted in two sub-peaks developing\nsymmetrically at ﬁnite energies,42,43depending on inter-\nfacial scattering strength. The splitting of the ZBCP is6\nclearly visible in the angle-resolved charge conductance,\nwhile in the angle-averaged one it is distinguishable only\nforhighmagnetization. However,when the interfacebar-\nrier strength Zis reduced, this structure becomes better\ndeﬁned since the two peaks get more separated, though\nless pronounced.\nNow, let us investigate how this picture is modiﬁed\nwhen a SBAF is taken into account. We remind that,\nwhen the superconducting electrode has d-wave symme-\ntry, the pair potential felt by electrons (+) and holes (–)\nis ∆σ,±= ∆0cos[2(θ′\nσ∓β)], whereβis the angle formed\nby the crystallographic aaxis of the superconductor with\nthexaxis (see Fig. 1). We here ﬁx β=π/4 to analyze a\ndx2−y2-wave superconductor with line nodes perpendicu-\nlar to the interface. In Fig. 3 we show the averaged dif-\nferential conductance spectra evaluated at the six points\nhighlighted in Fig. 2 and listed in Table I, in the limit\nof full transparency of the barrier (left panel) and for an\nintermediate value of Z(right panel). Comparing the\nbehavior of a SBAF/I/ dx2−y2and a STF/I/d x2−y2junc-\ntion, we ﬁnd qualitative deviations in the charge con-\nductance which become more and more signiﬁcant as\nincreasing values of the magnetization and of the bar-\nrier strength are considered (see Fig. 3). It is found\nthat with the increase of the magnetization the ZBCP is\nlowered rapidly and eventually smeared out in the STF\ncase, whereas in the SBAF case the ZBCP is more ro-\nbust against the polarization of the F-side. The drop of\nthe zero bias conductance may be attributed to the fact\nthat for a given injection angle, when Mincreases above\na threshold, the AR processes for the incident electron\nwith spin up is suppressed and only the AR of spin down\nelectrons contributes to the ZBCP. We point out that\nthis behavior can be rigorously proved considering that\nthe ABS amplitude decreases with increasing exchange\nﬁeld due to the sensitivity of Andreev reﬂections to spin\npolarization, represented in BTK-type models by a sup-\npression in the Andreev term coeﬃcient. This picture\nis slightly modiﬁed when a SBAF is considered, since in\nthis case the barrier, according to Eq. (6), may be spin\nselective, assisting the conductance of the two spin chan-\nnels in a diﬀerent way. This eﬀect results into a charge\nconductance always larger than the one obtained in the\ncorrespondingSTF casewith the samemagnetization M.\nFinally, we notice that with increasing Z, i. e. when\nwe move from the metallic limit towards the tunneling\none, the averaged charge conductance here obtained re-\nproduces the well-known behavior previously reported in\nthe literature.40,41\n2. F/I/S-BTRS JUNCTION\nIt is generally accepted that for many unconventional\nsuperconductors a subdominant component of the order\nparameter breaking time-reversal symmetry can be in-\nduced whenever translational symmetry is broken, e.g.\nnear surfaces, interfaces and vortices.60,61,64For some/Minus1.0/Minus0.5 0.0 0.50.40.60.81.01.21.41.6\n/CurlyEpsilon/Slash1/CapDelta0/LeΣΣG/Greater\n/Minus1.0/Minus0.5 0.0 0.5 1.0\n/CurlyEpsilon/Slash1/CapDelta0B\nD C\nE FAZ/Equal5 Z/Equal0\nFIG. 3: (Color online). Averaged diﬀerential conductance\nspectra for a junction with a dx2−y2-wave superconducting\nelectrode, evaluated in the states corresponding to the six\npoints indicated in Fig. 2 and listed in Table I, in the metall ic\nlimitZ= 0 (left panel) and for intermediate barrier trans-\nparency Z= 5 (right panel).\nmaterials, such as e.g. YBCO,8there is controversy\naboutthesymmetryofthesecondarycomponent, namely\nif the order parameter is of the dx2−y2+is- ordx2−y2+\nidxy-wave type. Furthermore, the splitting of the ZBCP,\nleading to the formation of symmetric peaks at ﬁnite\nbias, has been interpreted7,57,62as a signature of the ad-\nmixture of an imaginary pair potential component with\nthe dominant dx2−y2-wave one, corresponding to a time-\nreversal broken symmetry state.60,63The peak splitting\nreﬂects the fact that the zero-energystates are shifted by\na positive or negative amount due to the Doppler shift\nof a ﬁnite vector potential, and the good agreement be-\ntweentheoryandexperimentssuggeststhattheexistence\nof BTRS is a plausible explanation for the origin of the\npeak splitting of the charge conductance.\nThus, motivated by the fact that charge transport in\njunctions with a superconducting electrode could be a\nvaluable probe of the order parameter symmetry, we\ncompare here transport through F/I/S junctions having\ndx2−y2+isordx2−y2+idxyBTRS states in the S side\nand a SBAF or a STF in the F side. When the super-\nconducting electrode has dx2−y2+is- ordx2−y2+idxy-\nwave symmetry, the pair potential felt by electrons (+)\nand holes ( −) is ∆s\nσ,±= ∆1cos[2(θ′\nσ∓π/4)] +i∆2and\n∆d\nσ,±= ∆1cos[2(θ′\nσ∓π/4)] +i∆2sin[2(θ′\nσ∓π/4)], re-\nspectively. We have analyzed spectra for several values\nof ∆1and ∆ 2but for brevity we show here the results\nonly for ∆ 1≈0.968∆0and ∆ 2= 0.25∆0. We notice that\nfor this choice of ∆ 1and ∆ 2the gap amplitude is ∆ 0for\nθ′=π/4. In Fig. 4 the averaged charge conductance\nis plotted considering the two above-mentioned BTRS\nsuperconductors for a F/I/S junction with a STF (left\npanel) and a SBAF (right panel), for two representative\nvalues of the barrier strength Zand for a magnetization\nMequalto0.5. Aninspectionofthisﬁguresuggeststhat,7\nfor highZ, the junction exhibits for both kinds of ferro-\nmagnet a zero-bias charge response diﬀerent for the two\nBTRS states, implying that STF/I/S or SBAF/I/S junc-\ntions are equally useful to discriminate between BTRS\norder parameters involved in the S-side. For complete-\nness, it is worth stressing that the charge conductance\nin the SBAF case is always larger than the one obtained\nin the STF one, the diﬀerence being only quantitative.\nIn the low-barrier limit, the spectra for the two BTRS\nstates almost coincide for a STF, while for a SBAF they\nare clearly more distinguishable. Therefore, we can state\nthat in the high transparency limit a SBAF/I/S junction\nmay be seen as a more powerful tool than a STF/I/S one\nto discriminate between the two BTRS states. The ori-\ngin of the diﬀerent behavior of the conductance at zero\nbias for STF and SBAF electrodes lies in an ABS at zero\nenergy which is present in the case of dx2−y2+idxy-wave\nsymmetry (only for particular angles65), but not in the\ndx2−y2+isone. As explained in Section II, this eﬀect\nis clearly visible for a SBAF, because this kind of ferro-\nmagnetic electrode introduces an extra eﬀective barrier\nwhich aﬀects the charge transport of the hybrid struc-\nture, and pushing actually the junction toward the tun-\nneling regime where ABSs become the dominant channel\nfor transport.\n/Minus1.0/Minus0.5 0.0 0.50.60.70.80.91.01.11.21.3\n/CurlyEpsilon/Slash1/CapDelta0/LeΣΣG/Greater\n/Minus1.0/Minus0.5 0.0 0.5 1.0\n/CurlyEpsilon/Slash1/CapDelta0D\n1\n5Z isxyidC\n1\n5Z isxyid\nFIG. 4: (Color online). Averaged diﬀerential conductance\nspectra for a junction with a dx2−y2+is(solid lines) and a\ndx2−y2+idxy(dashed lines) superconducting electrode, eval-\nuated at the points C (STF, left panel) and D (SBAF, right\npanel) indicated in Fig. 2, in the intermediate ( Z=5) and high\ntransparency( Z=1) regime. Werecall that themagnetization\nisM=0.5 for both panels.\nC. SPIN TRANSPORT\nWe have analyzed the averaged spin conductance\n/angb∇acketleftΣ(ε)/angb∇acket∇ightdeﬁned in Eq.(12), for the same unconventional\npairing symmetries taken into account in the previous\nSubsection. For comparison, we have also considereda superconducting electrode characterized by a conven-\ntionals-wave pairing. Although several choices of the\nmagnetization Min the F electrode and of the barrier\nstrengthZhave been considered, in Fig. 5 we limit our-\nselves to the presentation of the spin conductance curves\nin the case M= 0.75 andZ= 5 (lower values of Zand\nMdo not qualitatively alter our results). In this ﬁgure\nsolid and dashed lines refer to the case of a junction with\nan STF and with a SBAF, respectively, the diﬀerent col-\norsbeing associatedwith diﬀerent superconducting order\nparametersymmetries. For dx2−y2-wavepairing, the spin\nconductance is non-vanishing at every ﬁnite bias and its\nproﬁle exhibits, at low biases, the well-known V-shaped\nbehavior typically produced by the gapless excitations\nassociated with nodes of the order parameter. On the\nother hand, for the two BTRS states considered here the\nspin conductance starts being non-zero at a ﬁnite bias,\ncorresponding to the energy of the minority component\nbreaking time reversal, and this activated behavior is re-\nlated to the nodeless properties of BTRS. Moreover, for\nthe three kinds of unconventional pairing symmetry con-\nsidered here, the spin conductance for biases lower than\nthe energy gap ∆ 0is always larger for a junction with\na STF than for a junction with a SBAF. Above ∆ 0this\ndiﬀerence in magnitude gets appreciably larger, and for\na given kind of ferromagnet /angb∇acketleftΣ(ε)/angb∇acket∇ightbecomes practically\nindipendent on the speciﬁc pairing symmetry.\nFig. 6 shows the relative gain in the spin conduc-\ntance of the SBAF contribution /angb∇acketleftΣ(F)/angb∇acket∇ightwith respect to\nthe STF one /angb∇acketleftΣ(E)/angb∇acket∇ight, deﬁned as ∆Σ( F|E) = (/angb∇acketleftΣ(F)/angb∇acket∇ight −\n/angb∇acketleftΣ(E)/angb∇acket∇ight)//angb∇acketleftΣ(E)/angb∇acket∇ight, as a function of the barrier height at\na ﬁxed bias ε/∆0=1.01 immediately above the energy\ngap ∆ 0. For comparison, we have calculated the same\nquantity also for the case of a junction with an s-wave\nsuperconductor (orange curve). We see that for a barrier\nheightZlowerthan approximately15 the gain is positive\nonly for an s-wave superconductor and it can be as high\nEF\nZ/Equal5dx2/Minusy2\ndx2/Minusy2/Plusi s\ndx2/Minusy2/Plusi dxy\n0.0 0.5 1.0 1.5 2.00.00.10.20.30.40.50.60.7\n/CurlyEpsilon/Slash1/CapDelta0/LeΣΣ/CapSigma/Greater\nFIG. 5: (Color online) Averaged diﬀerential spin conductan ce\nspectra evaluated at the points E and F reported in Fig. 2,\nfor unconventional superconducting electrodes.8\nas 100%. We have checked that this peculiar eﬀect is\nrelated to the presence of the superconducting electrode.\nIndeed, analyzing the spin conductance in STF/I/N and\nSBAF/I/N junctions, i.e. junctions where the supercon-\nductor is replaced by a normal metal, we have found that\nin the STF case the spin current is always greater than\nin the SBAF one. Looking separately at Andreev and\nnormal reﬂection probabilities, we have veriﬁed that this\nextra spin current can be ascribed to the fact that ma-\njority electrons coming from a SBAF have a zero proba-\nbility of being normally reﬂected at the gap edge, while\nelectrons coming from a STF have a ﬁnite residual prob-\nability to undergo the same process. For completeness,\nin the inset we have reported the averaged diﬀerential\nspin conductance for a junction with an s-wave super-\nconductor in the two cases of a SBAF (dotted line) and\na STF (solid line), obtained for the same choice of pa-\nrameters adopted in Fig. 5. We see that /angb∇acketleftΣ(ε)/angb∇acket∇ightis always\nzero below the energy gap; indeed in such situation the\nelectrons cannot enter the superconductor side as quasi-\nparticles because there are no quasiparticles states in the\ngap. Nevertheless, by Andreev reﬂection, they can cross\nthe interface and decay into the Cooper pair condensate,\nthus preventing a spin current ﬂow.\nFor spintronics applications, the ability to perform op-\nerations acting on spin currents but not on charge cur-\nrents is in general highly desirable. The results presented\nabove allow to individuate a particular situation where\nthis is possible using F/I/S junctions with a SBAF elec-\ntrode. For an s-wave superconductor in the case of a\n0.0 0.5 1.0 1.5 2.00.10.20.30.40.50.60.7\n/CurlyEpsilon/Slash1/CapDelta0/LeΣΣ/CapSigma/Greater\nse\ndx2/Minusy2\ndx2/Minusy2/Plusi s\ndx2/Minusy2/Plusi dxy/CurlyEpsilon/Slash1/CapDelta0/Equal1.01Z/Equal5\n0 5 10 15 20 25 30/Minus1.0/Minus0.50.00.51.0\nZ/CapDelta/CapSigma/LParen1F/VertBar1E/RParen1\nFIG. 6: (Color online) Relative gain in spin conductance\nof a SBAF with respect to a STF, ∆Σ( F|E) = (/angbracketleftΣ(F)/angbracketright −\n/angbracketleftΣ(E)/angbracketright)//angbracketleftΣ(E)/angbracketright, as a function of the barrier height Z, at\na bias value immediately above the energy gap ∆ 0, i. e.\nε= 1.01∆0. In the inset we have plotted the spin averaged\ncurrent for an s-wave electrode, for the same choice of the\nparameters adopted in Fig. 5.ﬁnite barrier strength, it has been recognized that the\nchargeconductanceispeakedaroundthegapedge.37,41,45\nOn the other hand, we have previously shown that the\nspin current is zero below the energy gap ∆ 0and rises\nabruptly just above it (inset of Fig. 6). If we then make\nthe voltage across the junction vary between two limit-\ning bias values ε1<∆0andε2>∆0such that charge\nconductance is the same, it is possible to turn from a sit-\nuation where a certain charge current is passing through\nthe junction while spin current is zero ( ε=ε1), to a\ncase where the spin current is diﬀerent from zero and the\ncharge conductance remains unaﬀected ( ε=ε2). Since\nthe upper bias ε2below which the switch state is “on”\nfalls only slightly above ∆ 0, we expect that the spin cur-\nrent through the device will be much greater if it is gen-\nerated by a SBAF rather than by a STF, given the ap-\npreciable diﬀerence between the two cases visible in the\ninset of Fig. 6.\nIV. CONCLUSIONS\nIn this paper we have studied the conductance spectra\nof ferromagnetic/insulator/superconductor hybrid struc-\ntures, developing an extension of the standard BTK ap-\nproach to the case of a ferromagnetic electrode exhibit-\ning either a standard Stoner exchange mechanism or a\nmass mismatch-driven ferromagnetism. We have investi-\ngated the eﬀects induced by these twodiﬀerent sourcesof\nmagnetization comparing the averaged charge and spin\nconductancesofSTF/I/Sjunctions(whereonlyexchange\nsplitting is present) and SBAF/I/S junctions (where only\nmass mismatch is present), for various symmetries of the\norder parameter in the superconducting electrode. Our\nanalysis has revealed several diﬀerences between the two\ncases. For the charge conductance, we have found a nar-\nrower and higher peak in the SBAF/I/ dx2−y2case com-\npared to the STF/I/ dx2−y2one, this ﬁnding being po-\ntentially useful for the experimental detection of a mass\nmismatch contribution to the magnetization.\nSince the Andreev reﬂection is phase sensitive, the on-\nset and amplitude of Andreev bound states, manifesting\nthemselves in the zero bias conductance peak, is a signa-\nture of the symmetry of the order parameter. For this\nreason,wehavealsoinvestigatedthetransportproperties\nof a junction with a superconductor exhibiting a broken\ntime-reversal symmetry of dx2−y2+isordx2−y2+idxy\ntype. Inthe high transparencylimit, wehavefound adif-\nferent behavior aroundzero bias of SBAF/I/ dx2−y2+idxy\nand STF/I/ dx2−y2+idxyjunctions, such that the use of\na SBAF allows to discriminate more eﬃciently between\nBTRS states with dx2−y2+isordx2−y2+idxypairing\nsymmetry than STF does. Indeed, as previously dis-\ncussed a SBAF ferromagnetic electrode introduces an ex-\ntra eﬀective barrier which aﬀects the charge transport of\nthe hybrid structure, driving the junction toward a tun-\nneling regime where ABSs is the dominant channel for\ntransport.9\nAs far as the spin transport is concerned, we have\nshown that the averaged spin conductance in a STF/I/S\njunction is greaterthan in a SBAF/I/S one for all the su-\nperconducting symmetries analyzed here, except for the\ncase of a conventional s-wave superconducting electrode.\nWe have also shown that a F/I/S junction with an s-\nwave superconductor can work as a switch able to turn\non and oﬀ a spin current, leaving the charge current un-\nchanged. In particular, our results show that for a wide\nrange of interfacial barrier strengths, the spin current\npassingthroughthe junction when the state ofthe switch\nis “on” is larger if the ferromagnetic electrode is a SBAF\nrather than a STF. This relative increase in spin current\ncan be very high, and for particular values of the barrier\nstrength a gain of up to 100% can be reached.Finally, we point out that the theoretical framework\nbehind the calculation presented in this paper is simple\nenough to allow analytic solutions in the whole relevant\nparameterspace. We leave for future work more complex\napproaches able to include the eﬀect of spin-ﬂip scat-\ntering, more realistic band structures, non-equilibrium\ntransport, as well as a self-consistent treatment of the\npair potential.\nV. ACKNOWLEDGMENTS\nWe wish to thank sincerely Prof. Jacob Linder for\nuseful discussions.\n1I.ˇZuti´ c, J. Fabian, and S.D. Sarma, Rev. Mod. Phys. 76,\n323 (2004).\n2R. J. Soulen Jr., J. M. Byers, M. S. Osofsky, B. Nadgorny,\nT. Ambrose, S. F. Cheng, P. R. Broussard, T.Tanaka, J.\nNowak, J. S. Moodera, A. Barry, and J. M. D. Coey, Sci-\nence282, 85 (1998); S. K. Upadhyay, A. Palanisami, R.\nN. Louie, and R. A. Buhrman, Phys. Rev. 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B 64, 224502 (2001)."    },    {        "title": "2005.13850v1.Spin_Pumping_Induced_Non_Linear_Electric_Current_on_the_Surface_of_a_Ferromagnetic_Topological_Insulator.pdf",        "content": "Spin-Pumping-Induced Non-Linear Electric Current on the Surface of a Ferromagnetic Topological\nInsulator\nYusuke Hama1,\u0003and Kentaro Nomura2, 3\n1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan\n2Institute of Material Research, Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan\n3Center for Spintronics Research Network, Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan\n(Dated: May 29, 2020)\nWe investigate the spin-pumping-induced electric current on the surface of a three-dimensional topological\ninsulator hybridized with a ferromagnet, namely, ferromagnetic topological insulator. In order to do this, we\nestablish the microscopic formalism and construct the perturbation theory using a Keldysh Green’s function\napproach. We analyze how this electric current is generated by an exchange interaction and an external ac mag-\nnetic ﬁeld, which is the driving force of ferromagnetic resonance as well as the spin pumping. The mechanism is\nas follows. First, the ferromagnetic resonance is driven and a zero-momentum magnon emerges. It is the ﬂuctu-\nation from the saturation magnetization pointing parallel to the precession axis of the ferromagnetic resonance.\nAfter then, the spin pumping is generated with the zero-momentum magnon being the carrier of spin. The zero-\nmomentum magnon and the topological insulator surface state couples through the exchange interaction and the\nspin carried by the magnon is transferred to it. Owing to the spin-momentum locking, the transferred spin is\nconverted into the momentum of topological insulator surface state leading to the generation of electric current\nﬂowing perpendicular to the precession axis of the ferromagnetic resonance. It is quadratic in the amplitude of\nexternal ac magnetic ﬁeld whereas it is linear to the strength of the exchange interaction. The associated electric\nvoltage is described by the spectrum of zero-momentum magnon. The non-linearity of spin-pumping-induced\nelectric current in the ac magnetic ﬁeld as well as the linearity in the exchange-interaction strength reﬂects that\nthe surface of ferromagnetic topological insulator has a high-performing functionality of generating the electric\ncharge current by magnetic controlling.\nI. INTRODUCTION\nQuantum technologies for hybridizing two or more sub\nquantum systems have been advancing rapidly with many\ntypes of elements ranging from solid-state systems to atomic-\nmolecular and optical systems having been used, for exam-\nple, electrons and nuclei in GaAs semiconductors, nitrogen-\nvacancy centers in diamonds, superconducting qubits, and\natoms and cavities composing cavity quantum electrodynamic\nsystems [1–13]. The functionalities of these hybrid quantum\nsystems are superior to or richer than those of any individual\nsub quantum systems and are characterized in the way they\nare composed of. By selecting sets of sub quantum systems\nwhich are the best choices to engineer the hybrid quantum\nsystem which has the functionality to perform the task you are\naiming, it becomes a powerful tool to execute quantum-state\ncontrolling, quantum information processing, and spintronics.\nThe key issue for spintronics is to perform a high-efﬁcient\nconversion of electric charge and spin degrees of freedom\nor the coherent controlling of electricity and magnetism with\nlowering sufﬁciently an energy consumption (Joule heating).\nIn order to accomplish these tasks, we have to search for ma-\nterials having potentials to create physical processes which\ncan be utilized for them and use these materials to engineer\nquantum devices. Examples include the non-magnetic heavy\nmetals with strong spin-orbit interaction which exhibits (in-\nverse) spin Hall effect like Pt and materials composed of metal\nand oxide possessing Rashba interfaces [14–24]. Recently,\n\u0003current address: Quemix Inc., 2-11-2 Nihombashi, Chuo-ku, Tokyo 103-\n0027, Japantopological insulator (TI) is considered to be a good candi-\ndate for a component of spintronics devices because TI ex-\nhibits bulk state with strong spin-orbit coupling as well as\nsurface state whose spin and momentum are strongly coupled\nwhich is called the spin-momentum locking (high-efﬁcient\nconvertibility of spin and electric charge current) [25–28].\nIn addition, the hybrid quantum system of magnetic materi-\nals and TI, namely, the magnetic TI, has been intensively in-\nvestigated from both theoretical and experimental points of\nview [29–59]. Including the quantum anomalous Hall ef-\nfect, the magnetic TI exhibits rich quantum phenomena ow-\ning to the composition of magnetism and spin-momentum\nlocking (multifunctionality). Because of the multifunction-\nality and the high-efﬁcient convertibility of spin and electric\ncharge current, the magnetic TI is considered to be one of\nthe promising candidate for spintronics devices and a large\nnumber of investigations have been made toward this goal\n[29, 38–45, 47, 51, 52, 54, 55, 57–59]. Although great ef-\nforts have been made for this, we still have not satisfactorily\nachieved the microscopic understanding of physics at the in-\nterface between the magnet and TI. For instance, we have not\nunderstand satisfactorily the way and how efﬁciently the spin\ntransferred from the magnet can be converted into the elec-\ntric current and/or voltage (spin pumping and the associated\nphenomena; inverse spin Hall effect and inverse Edelstein ef-\nfect) whereas the electric current of TI being converted into\nthe magnetization dynamics and/or a spin current (spin-orbit\ntorque, spin Hall effect, and Edelstein effect). Such complex-\nities are arising from the fact that the spin current is not a con-\nserved current in the macroscopic systems and the difﬁculties\nto distinguish whether the contribution to the electric charge\ncurrent under observation is coming from the surface state orarXiv:2005.13850v1  [cond-mat.mes-hall]  28 May 20202\nthe bulk state. It is important and an urgent issue to challenge\nanalyzing these problems in order to achieve a deeper under-\nstanding of the conversion between the electric current (orbital\ndegrees of freedom) and the magnetization dynamics (spin po-\nlarization as well as the spin current) in the magnetic TI, the\nphysics at the interface between magnets and TI surface state\nboth theoretically and experimentally, and further, to realize\nthe coherent controlling of TI surface state and magnetization\ntoward spintronics application.\nIn this paper, we will focus on the physics of TI surface\nstate and construct the microscopic theory for the quantum\ntransport phenomena at the interface between ferromagnet and\nTI. In order to do this, we use a Keldysh (non-equilibrium or\ncontour-time) Green’s function approach. We investigate the\nelectric current of TI surface state as well as the associated\nelectric voltage induced by the spin pumping originating in\nthe ferromagnetic resonance (FMR) driven by an external ac\nmagnetic ﬁeld. We analyze in detail how this electric current\nis created by the ac magnetic ﬁeld and the exchange interac-\ntion between the localized spin in the ferromagnet and the TI\nsurface state. We show that when the spin is carried from the\nzero-momentum magnon, which is created by the FMR, to\nthe TI surface state through the exchange interaction, due to\nthe spin-momentum locking this carried spin is converted into\nthe momentum. Then correspondingly, the electric current is\ninduced in the direction perpendicular to the precession axis\nof FMR, namely, spin-pumping-induced electric current. It\nis the quadratic response to the ac magnetic ﬁeld whereas it\nscales linearly to the strength of exchange coupling. On the\nother side, the associated electric voltage has a structure rep-\nresented by the spectrum of zero-momentum magnon which\nclearly reﬂects that the driving force of this electric voltage is\nthe spin pumping. Our result enables us to understand clearly\nnot only the mechanism of the spin-pumping-induced electric\ncurrent and its characteristic, but it also gives us a qualitative\nexplanation for the experimental results reported previously\n[42, 59].\nThis paper is organized as follows. In Sec. II, we present\nour microscopic model of the composite system of ferromag-\nnet and TI surface state. Then, we construct the formalism for\ndescribing the time evolution of this system using the Keldysh\nGreen’s function approach. Based on it, we present a mathe-\nmatical representation for the electric current of TI surface\nstate at the non-equilibrium steady state. Next, to calculate\nthis electric current we establish the perturbation theory for\nthe Keldysh Green’s function where the external ac magnetic\nﬁeld and the exchange interaction are regarded as perturbative\nterms. In Sec. III, which presents the main result of this pa-\nper, we discuss in detail the generation of electric current of TI\nsurface state induced by the spin pumping as well as the asso-\nciated electric voltage. By analyzing the structure of Feynman\ndiagram for the perturbative Green’s function, we discuss the\nmechanism of the spin-pumping-induced electric current as\nwell as its characteristics. Then, we make a comparison be-\ntween our result and the experimental results [42, 59] through\nthe characteristic of electric voltage. Sec. IV is devoted to the\nconclusion and outlook of this paper.\nFIG. 1. Schematic illustration of the ferromagnetic TI. FM is rep-\nresented by the Hamiltonian HFMwhile the surface state of TI is\nexpressed by HTI. The localized spin in FM couples with the spin\nof TI surface state through the exchange interaction Vexc. The total\nHamiltonian of this system is given by H=HFM+HTI+Vexc:\nII. MICROSCOPIC THEORY\nIn this section, we ﬁrst present our microscopic model for\nthe ferromagnetic TI. Based on it, we establish the formalism\nto describe the time evolution of this system generated by the\nspin pumping. Then we evaluate the electric current of the\nTI surface state at the non-equilibrium steady state using the\nKeldysh Green’s function approach. We do this by construct-\ning the perturbation theory for the Keldysh Green’s function\nso that the ac magnetic ﬁeld and the exchange interaction are\ntreated as perturbative terms.\nA. Modeling and Formalism\nThe ferromagnetic TI is the composite system of a ferro-\nmagnet (FM) and the three-dimensional TI. We take a spa-\ntial cartesian coordinate so that the xyplane is parallel to the\ninterface between the FM and TI whereas the zaxis perpen-\ndicular to it. The surface state of TI appears in the xyplane.\nThe TI surface state and a localized spin in the FM are cou-\npled through the exchange interaction. The illustration of fer-\nromagnetic TI is presented in Fig. 1. Experimentally, this\nsystem is created by doping the magnetic atoms (for instance,\nCr, V , and Mn) to the TI or implementing the heterostructure\nof ferromagnetic materials (e.g., a ferromagnetic insulator as\nwell as metal including EuS, EuO, YIG, and permalloy such\nasNi81Fe19and CoFeB) and the TI [33, 34, 36–40, 42, 45–\n54, 56–59]. The examples of three-dimensional TI include\ntetradymites Bi2Se3andBi2Te3[25–28, 60]. Hereinafter, let\nus focus on the interface between FM and TI and model the\ncomposite system of localized spin at this interface and the\nTI surface state (let us call it the surface of ferromagnetic\nTI). The spin pumping and the associated inverse spin Hall\neffect in the heterostructure systems composed of ferromag-\nnetic metal (or ferromagnetic insulator such as YIG) and Pt\nor NiPd alloy have been modeled in [19]. By referring to it,\nwe model the surface of ferromagnetic TI for describing the\nspin pumping process and the associated electric-current gen-\neration. It is described by the Hamiltonian H=H0+Vexc;3\nwhereH0=HFM+HTIwithHTI=\u0016HTI\n0+Himp. The\nHamiltonian \u0016HTI\n0is the unperturbed Hamiltonian of the TI\nsurface state consisting of the spin-momentum-locking term\nwith the dispersion relation being measured from the chem-\nical potential \u0016TI:\u0016HTI\n0= (HTI\n0\u0000\u0016TINTI). The operator\nNTIis the number operator of TI surface state. Hereinafter,\nlet us take the chemical potential \u0016TIto be equal to the Fermi\nenergy of TI and denote it as \u000fF.Himpis an impurity potential\nterm and assume to be spin independent (non magnetic). HFM\nis the unperturbed Hamiltonian of FM and take its chemical\npotential to be zero (\u0016FM= 0):Vexcis the exchange inter-\naction between the localized spin in FM and TI surface state.\nThe Hamiltonians \u0016HTI\n0andHimpare given by\n\u0016HTI\n0=Z\nd2x y\n\u000b0(x)\u0010\nH(0)\nTI(x)\u0000\u000fF1\u0011\n\u000b0\u000b \u000b(x);(1)\nHimp=Z\nd2x y\n\u000b0(x)Himp\n\u000b0\u000b(x) \u000b(x); (2)\nwhere\u0010\nH(0)\nTI(x)\u0011\n\u000b0\u000b=\u0000i~vF(\u001by@x\u0000\u001bx@y)\u000b0\u000b; (3)\nHimp\n\u000b0\u000b(x) =NimpX\niimp=1Vimp(x\u0000Ximp\niimp)\u00011\u000b0\u000b: (4)\nThe operators  \u000b(x)and y\n\u000b(x)are the annihilation and cre-\nation operators of the TI surface state at the two-dimensional\nspatial coordinate x= (x;y), respectively. The index \u000b=\";#\ndescribes the spin degrees of freedom of TI surface state. The\nsummation is taken for two repeated indices \u000band\u000b0in Eqs.\n(1) and (2).vF\u00185:0\u0002105m/s is the Fermi velocity while \u001bx\nand\u001byare the Pauli matrices. Vimp(x\u0000Ximp\niimp)in Eq. (4) is\nthe impurity potential and the vector Ximp\niimp= (Ximp\niimp;Yimp\niimp)\nis the coordinate of iimp-th impurity. Nimpis the total num-\nber of impurities. 1\u000b0\u000bis the two by two unit matrix. For\nthe details of TI-surface-state ﬁeld operators  \u000band y\n\u000b0, see\nsubSec. A 1 in Appendix A. The Hamiltonian HFMis given\nby\nHFM=\u0000JnxX\nhijiSi\u0001Sj+~\rX\niB0Sy\ni: (5)\nThe three-component vector Si= (Sx\ni;Sy\nj;Sz\ni)represents\nthe localized spin of the FM at the spatial coordinate ri=\n(rx\ni;ry\ni):The indices iandjruns from 1 to NlocwithNloc\ndenoting the total number of localized spin at the inter-\nface between FM and TI. Jnxis the strength of the nearest-\nneighboring exchange interaction. The summationP\nhijiis\ntaken for nearest-neighboring pairs. For any i, the localized\nspinSisatisﬁesS2\ni= (Sx\ni)2+ (Sy\ni)2+ (Sz\ni)2=S0(S0+ 1)\nwithS0its spin magnitude. \ris the gyromagnetic ratio of lo-\ncalized spin. The static magnetic ﬁeld B0is applied to the y\ndirection and the saturation magnetization is created along this\ndirection. Hereinafter we will not include the demagnetizing\ncoefﬁcient for simplicity. The exchange interaction Vexchas\nthe form\nVexc=\u0000JexcX\niX\na=x;y;zsa(ri)Sa\ni; (6)\nFIG. 2. Diagrammatic representation of the time evolution of fer-\nromagnetic TI surface. At far past ( t=\u00001), the ferromagnetic TI\nsurface is in the thermal equilibrium state described by the grand-\ncanonical ensemble \u001aGC(H;\f;\u000f F):After then, at t=t0the exter-\nnal magnetic ﬁeld Hext(t)is applied and the FMR as well as the spin\npumping are driven. The time evolution of this system is represented\nby the density matrix \u001aH(t). At sufﬁciently a long time ( t\u001dt0),\nthe surface of ferromagnetic TI is in the non-equilibrium steady state\nand the associated quantum transport of TI surface state is generated.\nwhereJexcis the strength of the exchange interaction.\nsa(ri) = y\n\u000b0(ri)(\u001ba\n\u000b0\u000b=2) \u000b(ri)is the spin density of TI\nsurface state at the coordinate ri.\nNext, let us discuss the time evolution of this system. At\ninitial time ( t=\u00001), the ferromagnetic TI is in the thermal\nequilibrium state with the temperature T. It is represented by\nthe grand-canonical ensemble with its density matrix\n\u001aGC(H;\f;\u000f F) =exp (\u0000\fH)\nTr (exp (\u0000\fH)); (7)\nwhere\f\u00001=kBTwithkBthe Boltzmann con-\nstant. Note that the TI-Fermi-energy dependence\nis included in the Hamiltonian H:Att=t0, we\napply an ac external magnetic ﬁeld Bext(t) =\nBext\u0000\nsin\u0000\nsgn(B0)\u0001!extt\u0001\n;0;cos\u0000\nsgn(B0)\u0001!extt\u0001\u0001\n,\nwhere sgn(B0) = +1 (\u00001)whenB0>0 (<0). Here we\nhave taken a circular polarized light. Bextand!extare its\namplitude and frequency, respectively. This triggers the ferro-\nmagnetic resonance (FMR). The system at t > t 0is going to\nbe described by the total Hamiltonian H(t) =H+Hext(t)\nwhereHext(t)is given by\nHext(t) =~\rX\na=x;zX\niBext\na(t)Sa\ni: (8)\nFor later convenience, we decompose the total Hamiltonian\nH(t)into the formH(t) =H0+H0(t)withH0(t) =\nVexc+Hext(t):The precession axis of the FMR is along the\nydirection owing to the static magnetic ﬁeld B0. Once the\nFMR is triggered, a spin transfer occurs from FM to the TI4\nsurface state mediated by the exchange interaction Vexc, i.e.,\nspin pumping. As a result, a spin polarization as well as an as-\nsociated non-equilibrium state is generated on the surface of\nTI. Such a physical process (the time evolution of the system\natt>t 0) is represented by the density matrix [62, 63]\n\u001aH(t) =U(t;t0)\u001aC(H;\f;\u000f F)Uy(t;t0); (9)\nwhere the time-evolution operator U(t;t0)is given by\nU(t;t0) =Texp\u0012\n\u0000i\n~Zt\nt0H(t0)dt0\u0013\n; (10)\nwith the symbol Tdenoting the time-ordering product of real\ntime. By using the density matrix in Eq. (9), the expectation\nof a physical operator Aatt>t 0is expressed by\nhA(t)i= Tr [AH(t)\u001aC(H;\f;\u000f F)]; (11)\nwhereAH(t) =Uy(t;t0)AU(t;t0).\nX\u000b\n=\nTr(X\u001aC(H;\f;\u000f F))represents the thermal average taken\nwith respect to the Hamiltonian H. At sufﬁciently a long\ntime (t\u001dt0), the non-equilibrium steady state is realized\nand the quantum transport phenomena of TI surface state is\ngenerated. To summarize the above description, in Fig. 2 we\npresent the diagrammatic structure of time evolution of the\nferromagnetic TI surface.\nSince the microscopic formalism for the time evolution of\nthe ferromagnetic TI surface as well as that for the expectation\nof the physical operators have been established, let us discuss\nthe quantum transport phenomena on the surface of ferromag-\nnetic TI at the non-equilibrium steady state. When the spin\npumping is driven by the FMR, the y-polarized spin is injected\nfrom FM to TI surface. We write the spin current associated\nwith this spin pumping process as Jspin\ny;z. The ﬁrst subscript y\ndenotes the direction of the spin polarization whereas the sec-\nond subscript zdescribes the ﬂowing direction of spin current.\nThrough the exchange interaction Vexc, the spin current ﬂows\ntoward the TI surface. Some portion of Jspin\ny;z is going to be\nconverted into the momentum (the electric current ﬂowing on\nthe surface of TI) due to the spin-momentum locking. Besides\nthat, it might be converted into other types of phenomena, for\ninstance, a dissipation process like a spin relaxation process or\nthe spin current which bounces back to FM. From such a con-\nsideration, the exact evaluation of the spin current Jspin\ny;z and\nhow efﬁciently it is converted into the electric current of TI\nsurface state are very difﬁcult tasks. This is because it is hard\nto mathematically deﬁne the spin current since the spin is not\nthe conserved quantity or the spin current is not the conserved\ncurrent in the macroscopic system like a mili-meter-scale sys-\ntem. On the other hand, what has been observed in the ex-\nperiment is the electric voltage induced by the spin pumping\n[42, 59]. By taking into account of this fact, although there\nare some theoretical approaches which treat mathematically\nthe spin current and calculate the spin-to-charge conversion\nefﬁciency using a concept such as spin-mixing conductance\n[15, 19, 22], we do not take such approaches. Instead, we con-\nsider that the y-polarized spin carried from FM to TI surface\nvia spin-pumping process is going to be mainly converted intothe electric charge current of TI surface state. Therefore, in-\nstead of calculating the spin current Jspin\ny;zdirectly and analyze\nhow efﬁciently it is converted into the electric charge current,\nwe calculate directly the electric charge current of the TI sur-\nface state and analyze how it is created by the ac magnetic\nﬁeld and the exchange interaction. Here we calculate the x-\ncomponent of electric charge current density jx(x). It is given\nbyjx(x) =\u0000evF\u0000\n y\n\u000b0(x)\u001by\n\u000b0\u000b \u000b(x)\u0001\n=\u00002evFsy(x);with\n\u0000e(<0)the electric charge and sy(x)is they-component\nspin density of TI surface state at the coordinate x. Such an\nequivalence of the x-component of electric charge current and\nthey-component spin originates in the spin-momentum lock-\ning. By denoting the annihilation and creation operators of\nTI surface state ﬁeld in the Heisenberg picture with respect to\nH(t)as H\u000b(x;t)and y\nH\u000b(x;t), respectively, from Eq. (11)\nthe expectation of the x-component electric current density at\ntimetis given by\nhjx(x;t)i=\u0000evF\u001by\n\u000b0\u000b\n y\nH\u000b0(x;t) H\u000b(x;t)\u000b\n: (12)\nB. Keldysh Green’s Function and Perturbation Theory\nOur next task is to rewrite the expectation value of electric\ncurrent density in Eq. (12) with the Keldysh Green’s function\nand evaluate it by constructing the perturbation theory where\nthe perturbative term is H0(t) =Hext(t) +Vexc. Then, what\nwe evaluate at the end is the spatial and temporal averaged\nelectric current density at the non-equilibrium steady state. It\nis deﬁned by\n\u0016jx=Zd2x\nVZt0+T\nt0dt\nThjx(x;t)i; (13)\nwhereVthe area of TI surface. The time Tis given by\nT= 2\u0019Ntime=!extwithNtime a positive integer. We as-\nsume it to be very large to describe that we are taking the\nlong-time average (Ntime\u001d1). By analyzing the structure\nof perturbative Keldysh Green’s function, we investigate how\nthe TI-surface-state electric current \u0016jxis generated by the spin\npumping in terms of the ac external magnetic ﬁeld and the ex-\nchange interaction.\nFirst, we rewrite the expectation value of electric current\ndensity in Eq. (12) by the ﬁeld operators in the interaction\npicture. We denote the creation and annihilation operators of\nTI surface state in the interaction picture as  y\nH0\u000band H0\u000b,\nrespectively. The expectation value of x-component electric\ncurrent density at the non-equilibrium steady state becomes\n[61]\n\njx(x;t)\u000b\n=ievF\u001by\n\u000b0\u000blim\nt0!t+\nx0!xG<\n\u000b\u000b0(xt;x0t0); (14)\nwhereG<\n\u000b\u000b0(xt;x0t0)is the lesser component of full real-time\nGreen’s function. t+is the time which is inﬁnitesimally later\nthant:t+=t+\u000f+\ntwith\u000f+\nta positive inﬁnitesimal. The\nlesser Green’s function G<\n\u000b\u000b0(xt;x0t0)is redescribed by the5\nFIG. 3. Schematic for the closed contour C. It consists of two\nsub contours C\u0000andC+:The sub contour C\u0000starts from\u001c=\u00001\nand ends at\u001c= +1whereasC+begins from\u001c= +1and reaches\n\u001c=\u00001. The variables tandt0are the real times which are obtained\nby performing the real-time projection on the contour times \u001cand\n\u001c0, respectively. As shown in the diagram in Fig. 2, the contour C\ndescribes the time evolution of the ferromagnetic TI surface such that\nat the far past ( t!\u00001 ) the thermal-equilibrium state represented\nby\u001aGC(H;\f;\u000f F)was realized, and due to the external ﬁeld Hext(t),\nat sufﬁciently a long time ( t!+1) the non-equilibrium state is\ngenerated.\nKeldysh (contour-time) Green’s function deﬁned by [62, 63]\niGC;\u000b\u000b0(x\u001c;x0\u001c0) =D\nTC\u0002\nUexc\nCUext\nC H0\u000b(x\u001c) y\nH0\u000b0(x0\u001c0)\u0003E\n0;\n(15)\nwhere\nX\u000b\n0= Tr(X\u001aGC(H0;\f;\u000f F))is the thermal average\ntaken with respect to the unperturbed Hamiltonian H0. The\ncontourCis the closed path as shown in Fig. 3 and is rep-\nresented by the time variable called the contour time. Let us\ndenote it as \u001c. The symbol TCrepresents the time-ordering\noperator for contour times belonging to C. For instance, if\n\u001c1< \u001c2we haveTC[A1(\u001c1)A2(\u001c2)] =\u0006A2(\u001c2)A1(\u001c1). We\nobtain the positive sign after we exchanged the order between\nA1(\u001c1)andA2(\u001c2)if this exchange was bosonic (exchanging\neven numbers of fermionic operators) while we get the nega-\ntive sign if the exchange was fermionic (exchanging odd num-\nbers of fermionic operators). The contour Cconsists of two\nsub contours C\u0000andC+:The sub contour C\u0000starts from\n\u001c=\u00001 and reaches \u001c= +1while the sub contour C+\nbegins from \u001c= +1and ends at \u001c=\u00001. Such a struc-\nture represents that the timescale of dynamics we are focusing\non is when the non-equilibrium steady state is realized. It is\nwhen sufﬁciently a long time has passed since we applied the\nexternal ﬁeld Hext(t)(at timet0). In order to describe such\na situation, the limit t0!\u00001 is going to be taken while\nfor timet, which is the time when the non-equilibrium state\nwe are focusing on is realized, we take t!1 . The reason\nwe have the two sub contours C\u0000andC+is because, as de-\nscribed in Eq. (11), the physical operators are sandwiched be-\ntween the two time evolution operators Uy(t;t0)andU(t;t0).\nNote that the temporal structure of contour Cis equivalent\nto the structure of time evolution presented in Fig. 2. Thecontour times \u001cand\u001c0in Eq. (15) belong to C\u0000andC+,\nrespectively. In such a temporal conﬁguration, the Keldysh\nfunctionGC;\u000b\u000b0(x\u001c;x0\u001c0)becomes the lesser Green’s func-\ntion via real-time projection. For more details on the real-time\nprojection of the Keldysh Greens’ function formalism see sub-\nSec. B 1 in appendix B.\nThe operatorsUexc\nCandUext\nCare the time-evolution opera-\ntors along the contour Cgenerated by VexcandHext, respec-\ntively. They are deﬁned by\nUext\nC= exp\u0012\n\u0000i\n~Z\nCd~\u001cHext\nH0(~\u001c)\u0013\n;\nUexc\nC= exp\u0012\n\u0000i\n~Z\nCd\u0014\u001cVexc\nH0(\u0014\u001c)\u0013\n: (16)\nHext\nH0(~\u001c)andVexc\nH0(\u0014\u001c)in the above equation are written by\nthe ﬁeld operators in the interaction picture at the contour\ntime ~\u001cor\u0014\u001c. In order to perform the perturbative calcu-\nlation, we rewrite the Hamiltonians Hext\nH0(~\u001c)andVexc\nH0(\u0014\u001c)\nin the momentum representation and reorganize the unper-\nturbed and perturbed terms. For doing this, let us intro-\nduce the Fourier transform of the spin density for TI sur-\nface state. It is given by sa(x) =V\u00001P\nkeik\u0001xsa(k)\nwherek= (kx;ky)is the two-dimensional wavevector of\nTI surface state and sa(k) =P\nk0 y\n\u000b0(k0)(\u001ba\n\u000b0\u000b=2) \u000b(k0+\nk)witha=x;y;z . The (inverse) Fourier trans-\nform of the ﬁeld operator of TI surface state is given\nby H0\u000b(xt) =\u0000\n1=p\nV\u0001P\nkeik\u0001x H0\u000b(kt); H0\u000b(kt) =\u0000\n1=p\nV\u0001R\nd2xe\u0000ik\u0001x H0\u000b(xt):Besides the TI-surface-state\nﬁeld operator in the momentum representation, we introduce\nthe magnon ﬁeld operators represented in the momentum\nspace by re-expressing the localized spin with them (Holstein-\nPrimakoff tranformation). They are given by\nSy\ni=\u0000sgn(B0)0\n@S0\u00001\nNlocX\npp0ay(p0)a(p)ei(p\u0000p0)\u0001ri1\nA;\nS\u0000\ni=Sz\ni\u0000iSx\ni=8\n<\n:q\n2S0\nNlocP\npe\u0000ip\u0001riay(p);(B0<0)q\n2S0\nNlocP\npeip\u0001ria(p);(B0>0);\nS+\ni=Sz\ni+iSx\ni=8\n<\n:q\n2S0\nNlocP\npeip\u0001ria(p);(B0<0)q\n2S0\nNlocP\npe\u0000ip\u0001riay(p);(B0>0);\n(17)\nwherea(p)(ay(p)) denotes the annihilation (creation) oper-\nator of magnon with momentum p= (px;py). The annihi-\nlation and creation operators of magnon satisfy the commuta-\ntion relation [a(p);ay(q)] =\u000e(p\u0000q)with all others being\nzero. By using the spin density sa(k)and the magnon ﬁeld\noperatorsa(p)anday(p0), the Hamiltonian of the surface of\nferromagnetic TI is re-expressed in the momentum space as6\n\u0016HTI\n0=~vFX\nk \u000b0(k)\u0000\n\u001by(kx+kx\n0)\u0000\u001bxky\u0000\u000fF1\u0001\n\u000b0\u000b \u000b(k); (18)\nHimp=1\nVX\nkq\u000bvimp(q)\u001aimp(q) y\n\u000b(k+q) \u000b(k); (19)\nHFM\n0=X\np\u000fFM\npay(p)a(p);\nVexc=8\n<\n:\u0000q\nS0\n2NlocV2P\nqp\u0010\nJexc\n(qp)s\u0000(q)a(p) +Jsd\u0003\n(qp)s+(\u0000q)ay(p)\u0011\n+Jexc\nVP\npp0ay(p0)a(p)sy(p0\u0000p);(B0<0)\n\u0000q\nS0\n2NlocV2P\nqp\u0010\nJexc\n(qp)s+(q)a(p) +Jsd\u0003\n(qp)s\u0000(\u0000q)ay(p)\u0011\n\u0000Jexc\nVP\npp0ay(p0)a(p)sy(p0\u0000p);(B0>0);\n(20)\nHext(t) =~\rBextr\nNlocS0\n2\u0010\nay(0)e\u0000i!extt+a(0)ei!extt\u0011\n; (21)\nwherevimp(q) =R\nd2xe\u0000iq\u0001xVimp(x)and\u001aimp(q) =PNimp\ni=1e\u0000iq\u0001Xi. We will take vimp(0) = 0 .\u000fFM\np =\nzJnxS0(1\u0000\rp)+~\rejB0jis the dispersion relation of magnon\nwith\rp=z\u00001P\n\u001ae\u0000ip\u0001\u001a: zis the number of nearest-\nneighboring sites for localized spins and \u001arepresents the\nnearest-neighboring-site vector. The quantity Jexc\n(qp)is deﬁned\nbyJexc\n(qp)=P\niJexcei(q+p)\u0001ri:By comparing the Hamilto-\nnian \u0016HTI\n0in Eq. (1) and that in Eq. (18), we see that be-\ncause of the exchange interaction the Dirac point of TI sur-\nface state (the point where the dispersion of TI surface state\nbecomes zero) is shifted to the momentum k0= (kx\n0;0)with\nkx\n0= sgn(B0)(JexcS0n2D\nloc)=(2~vF). Heren2D\nloc=Nloc=Vis\nthe two-dimensional number density of localized spins. For\nthe convenience, we perform the Fourier transformation on\nthe ﬁeld operators of TI surface state  \u000b(x)and y\n\u000b0(x)by\nthe shifted momentum ~k=k+k0:As a result, the formula\nof Hamiltonian HTI\n0in Eq. (18) described by the shifted mo-\nmentum ~kis going to be equivalent to that in Eq. (1) rep-\nresented by the original momentum k. Hereinafter we just\nsimply write the shifted momentum ~kask. Note that with-\nout the impurity effect, the TI surface state exhibits the linear\ndispersion relation \u000fTI\nk=~vFkwithk=p\n(kx)2+ (ky)2.\nConsequently, the surface of ferromagnetic TI is remodeled\nas the hybrid quantum system of magnon and TI surface state\nwith the Hamiltonians in Eqs. (18) - (21).\nNext, in order to construct the perturbation the-\nory for the Green’s function GC;\u000b\u000b0(x\u001c;x0\u001c0)in\nEq. (15) let us perform the Fourier transforma-\ntion with taking the limit x0!x. We have\nGC;\u000b\u000b0(x\u001c;x0\u001c0) =V\u00001P\nkk0ei(k\u0000k0)xGC;\u000b\u000b0(k\u001c;k0\u001c0).\nHereGC;\u000b\u000b0(k\u001c;k0\u001c0)is given by GC;\u000b\u000b0(k\u001c;k0\u001c0) =\n\u0000iD\nTC\u0002\nUexc\nCUext\nC H0\u000b(k\u001c) y\nH0\u000b0(k0\u001c0)\u0003E\n0. Then, we\nperform the perturbative expansion on GC;\u000b\u000b0(k\u001c;k0\u001c0)by\nexpanding the two operators Uext\nCandUexc\nCin Eq. (16) with\nrespect toHext\nH0(\u0014\u001c)andVexc\nH0(~\u001c), respectively. It is going tobe represented in the form\nGC;\u000b\u000b0(k\u001c;k0\u001c0) =1X\nn=01X\nn0=0G(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0): (22)\nWe have used the superscript ( n;n0)in the right-hand side of\nEq. (22) to describe that the perturbative Green’s function\nG(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is in then-th order of Hextwhile it is\nin then0-th order of Vexc. Note that the Green’s functionP1\nn0=0G(0;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is the full thermal-equilibrium\nGreen’s function since it does not contain the external-ﬁeld\nHamiltonian Hext. At the non-equilibrium steady state, what\nwe observed in the experiment is the deviation (ﬂuctuation)\nfrom the thermal-averaged value at thermal equilibrium.\nThus, we calculate and show the expectation value of\njx(x;t)\u000b\nin Eq. (14) as well as the spatial and temporal\naveraged electric current \u0016jxin Eq. (13) for n\u00151:\nAs a result, the perturbative Green’s function\nG(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is expressed by the unperturbed Keldysh\nGreen’s functions of TI surface state and magnon given by\niG0\nC;\u000b\u000b0(x\u001c;x0\u001c0) =D\nTC\u0002\n H0\u000b(x\u001c) y\nH0\u000b0(x0\u001c0)\u0003E\n0;\n(23)\niD0\nC(q\u001c;q0\u001c0) =D\nTC\u0002\naH0(q\u001c)ay\nH0(q0\u001c0)\u0003E\n0: (24)\nG0\nC;\u000b\u000b0(x\u001c;x0\u001c0)is the unperturbed Keldysh Green’s function\nof the TI surface state while D0\nC(q\u001c;q0\u001c0)is that of magnon.\nThe Fourier transform of G0\nC;\u000b\u000b0(x\u001c;x0\u001c0)is given as\nG0\nC;\u000b\u000b0(x\u001c;x0\u001c0) =V\u00001P\nkk0ei(k\u0000k0)xG0\nC;\u000b\u000b0(k\u001c;k0\u001c0).\nNote that bothG0\nC;\u000b\u000b0(k\u001c;k0\u001c0)andD0\nC(q\u001c;q0\u001c0)are diago-\nnal in momentum: G0\nC;\u000b\u000b0(k\u001c;k0\u001c0) =G0\nC;\u000b\u000b0(k;\u001c;\u001c0)\u000ekk0\nandD0\nC(q\u001c;q0\u001c0) =D0\nC(q;\u001c;\u001c0)\u000eqq0. To obtain the\nphysical observables like the electric current of TI sur-\nface state, we project the contour times onto the real-time\naxis. Then, the Keldysh Green’s functions G0\nC;\u000b\u000b0(x\u001c;x0\u001c0)\nandD0\nC(q\u001c;q0\u001c0)are rewritten by the unperturbed real-time7\nGreen’s functions: G0\nC;\u000b\u000b0(k;\u001c;\u001c0)!\u0016g\u0017\n\u000b\u000b0(k;t\u0000t0)and\nD0\nC(q; ~\u001c;~\u001c0)!\u0016D~\u0017(q;~t\u0000~t0). Here\u0017;~\u0017= t;<;>; ~tde-\nnoting the time-ordered, lesser, greater, and anti-time-ordered\ncomponents, respectively. t;t0;~t, and ~t0are real-time vari-\nables introduced by the real-time projection and correspond\nto\u001c;\u001c0~\u001c, and ~\u001c0, respectively. After the Keldysh Green’sfunctions are transformed into the real-time Green’s functions\nthey are represented by the differences of two real-time vari-\nables. As a result, the perturbative Keldysh Green’s function\nG(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is redescribed as products of unperturbed\nreal-time Green’s functions. Its formula can be organized with\nthe retarded and advanced components in the momentum-\nfrequency representation given by\n\u0016gr\n\u000b\u000b0(k;!) =(1+~H0)\u000b\u000b0\n2\u0010\n!+!F\u0000!TI\nk+i\n2\u001crel\nTI\u0011+(1\u0000~H0)\u000b\u000b0\n2\u0010\n!+!F+!TI\nk+i\n2\u001crel\nTI\u0011\n\u0016ga\n\u000b\u000b0(k;!) =(1+~H0)\u000b\u000b0\n2\u0010\n!+!F\u0000!TI\nk\u0000i\n2\u001crel\nTI\u0011+(1\u0000~H0)\u000b\u000b0\n2\u0010\n!+!F+!TI\nk\u0000i\n2\u001crel\nTI\u0011; (25)\n\u0016Dr(0;!) =1\n!\u0000!FM\n0+i\u000b!;\u0016Da(0;!) =1\n!\u0000!FM\n0\u0000i\u000b!; (26)\nwhere 1is the two by two unit matrix and ~H0is given by\n~H0= \n0\u0000i(kx\u0000iky)\nki(kx+iky)\nk0!\n: (27)\nThe Green’s functions \u0016gr(a)\n\u000b\u000b0(k;!)and \u0016Dr(a)(0;!)in Eqs.\n(25) and (26) are the retarded (advanced) components of TI-\nsurface-state and zero-momentum magnon Green’s functions,\nrespectively. The frequencies !TI\nk;!F, and!FM\n0are deﬁned\nby!TI\nk=~\u00001\u000fTI\nk,!F=~\u00001\u000fF, and!FM\n0=~\u00001\u000fFM\n0=\n\rjB0j, respectively. \u001crel\nTIis the relaxation time of the TI sur-\nface state due to the impurity effect Himpwhile the constant \u000b\nappearing in the magnon Green’s function is the Gilbert damp-\ning constant. We put bars on top of these Green’s functions\nto express that we have taken into account the impurity and\ndamping effects.\nThe perturbation theory for the Keldysh Green’s function in\nthe above way enables us to clearly explore how the electric\ncurrent on the surface of TI is induced by the spin pumping in\nterms of the external ac magnetic ﬁeld and the exchange inter-\naction. In Appendix A, we present the details for the real-time\nGreen’s functions of TI surface state as well as the derivation\nof retarded and advanced components of impurity-averaged\nGreen’s functions given in Eq. (25) using the imaginary-time\nGreen’s function formalism. Moreover, we describe the real-\ntime Green’s functions of magnon and then discuss the deriva-\ntion of retarded and advanced Green’s functions in Eq. (26)\nusing the Landau-Lifshitz-Gilbert equation. In Appendix B,\nwe present the detailed description for the Keldysh Green’s\nfunction formalism as well as the relation between Keldysh\nGreen’s function and real-time Green’s function. Further,\nwe show some formulas of Keldysh Green’s function formal-\nism and by applying them we demonstrate the derivation of\nimpurity-averaged Green’s functions of TI surface state for\nthe retarded, advanced, lesser, and greater components.III. SPIN-PUMPING-INDUCED NON-LINEAR ELECTRIC\nCURRENT\nSince we have established the perturbation theory, we now\nevaluate the electric current of the TI surface state induced by\nthe spin pumping. Let us ﬁrst present the diagrammatic rep-\nresentation of our perturbative Green’s function. Based on it,\nwe microscopically analyze how the electric current is gen-\nerated by the external magnetic ﬁeld and the exchange inter-\naction. Then, we show the structure of spin-pumping-induced\nelectric current as well as the associated electric voltage repre-\nsented by the static external magnetic ﬁeld, the amplitude and\nthe frequency of ac magnetic ﬁeld, the exchange-interaction\nstrength, the relaxation time originating in the non-magnetic\nimpurity, and the Gilbert-damping constant. Finally, we com-\npare our result of the electric voltage with the experimental\nresults [42, 59].\nLet us evaluate the right-hand side of Eq. (22). We denote\nthe expectation of spatial and temporal averaged x-component\nelectric current corresponding to the term G(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)\nas\u0016j(n;n0)\nx:First, we can show that the spatial and temporal\naveraged electric current \u0016j(1;1)\nx is zero. This implies that the\nsurface of ferromagnetic TI does not show the linear response\nin the ac external magnetic ﬁeld. Next, let us present the next-\nleading-order term \u0016j(2;1)\nx. In order to obtain this, we calculate\nthe perturbative Green’s function G(2;1)\nC;\u000b\u000b0(k\u001c;k0\u001c0)given in\nthe right-hand side of Eq. (22). First, we expand the time-\nevolution operators Uext\nCandUexc\nCwith respect to Hext\nH0(~\u001c)\nandVexc\nH0(\u0014\u001c), respectively. With using the Wick’s theorem\nthe perturbative Green’s function G(2;1)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is given\nin terms of the unperturbed Keldysh Green’s functions of TI\nsurface state and magnon as8\nG(2;1)\nC;\u000b\u000b0(k\u001c;k0\u001c0) =isgn(B0)\u0012\n\u0000i\n~\u00133\u0000\n~\rBext\u00012Jexcn2D\nlocS0\n4Z\nCd~\u001c1d~\u001c2d\u0014\u001c1e\u0000i!ext(~\u001c1\u0000~\u001c2)X\npp0k1\n\u0002\u0002\nD0\nC(0; ~\u001c1;~\u001c2)D0\nC(p; 0+)\u000ep;p0+D0\nC(0; ~\u001c2;\u0014\u001c1)D0\nC(0; \u0014\u001c1;~\u001c1)\u000ep;0\u000ep0;0\u0003\n\u0002h\nG0\nC;\u000b\u000b0\n1(k;\u001c;\u0014\u001c1)\u001by\n\u000b0\n1\u000b1G0\nC;\u000b1\u000b0(k0; \u0014\u001c1;\u001c0)\u000ek;k1\u000ek0;k1+p0\u0000p\n\u0000G0\nC;\u000b\u000b0(k; 0+)\u001by\n\u000b0\n1\u000b1G0\nC;\u000b1\u000b0\n1(k1; 0+)\u000ek;k0\u000ek1;k1+p0\u0000pi\n: (28)\nWe note that in the above equation the positive inﬁnitesimal\ntime difference 0+forG0\nC;\u000b\u000b0(k; 0+)is equal tot0\u0000t:Since\n\u001c(=t)2C\u0000while\u001c0(=t0)2C+, the Green’s function\nG0\nC;\u000b\u000b0(k; 0+)is the lesser Green’s function. On the other\nhand, 0+forG0\nC;\u000b1\u000b0\n1(k1; 0+)is equal to\u001c+\n1\u0000\u001c1:The contour\ntimes\u001c+\n1and\u001c1both belong to the same sub contour C\u0016(\u0016=\n\u0000;+):\nSecond, we perform the real-time projection on the con-tour times ~\u001c1;~\u001c2, and \u0014\u001c1and rewrite the right-hand side of\nEq. (28) by the real-time Green’s functions of TI surface state\nand magnon. Then, the perturbative Keldysh Green’s func-\ntionG(2;1)\nC;\u000b\u000b0(k\u001c;k0\u001c0)in the right-hand side of Eq. (28) be-\ncomes the lesser real-time Green’s function which we write\nasG<(2;1)\n\u000b\u000b0(kt;k0t0)witht0=t+(see also Eq. (14)). Let us\ndenote the real-time variables corresponding to ~\u001c1;~\u001c2, and \u0014\u001c1\nas~t1;~t2, and \u0014t1, respectively. Then, by using the ﬁrst formula\nin Eq. (B10), we obtain\nG<(2;1)\n\u000b\u000b0(kt;k0t0) =isgn(B0)\u0012\n\u0000i\n~\u00133\u0000\n~\rBext\u00012Jexcn2D\nlocS0\n4Z\nd~t1d~t2d\u0014t1e\u0000i!ext(~t1\u0000~t2)\u0016Da(0;~t2\u0000\u0014t1)\u0016Dr(0;\u0014t1\u0000~t1)\n\u0002\u0010\n\u0016gr\n\u000b\u000b0\n1(k;t\u0000\u0014t1)\u001by\n\u000b0\n1\u000b1\u0016g<\n\u000b1\u000b0(k;\u0014t1\u0000t) + \u0016g<\n\u000b\u000b0\n1(k;t\u0000\u0014t1)\u001by\n\u000b0\n1\u000b1\u0016ga\n\u000b1\u000b0(k;\u0014t1\u0000t)\u0011\n\u000ekk0; (29)\nwhere we have usedR\nCd~\u001c1d~\u001c2D0\nC(0; ~\u001c1;~\u001c2) =R\nd~t1d~t2\u0010\n\u0016Dt\u0000\u0016D<\u0000\u0016D>+\u0016D~t\u0011\n(0;~t1\u0000~t2) = 0:\nFurther, the termG0\nC;\u000b\u000b0(k; 0+)G0\nC;\u000b1\u000b0\n1(l; 0+)in Eq. (29)\nvanishes since it describes the disconnected diagram: We\ndenote the real time variables t1andt0\n1which are the real-\ntime projection of the contour times \u001c1and\u001c0\n1, respectively.\nThey satisfy \u001c1< \u001c0\n1because in the exchange-interaction\nHamiltonian Vsdthe operator  y\n\u000b0\n1(l;\u001c0\n1)comes to the left\nside of \u000b1(l;\u001c1). When\u001c1;\u001c0\n12C\u0000we havet1< t0\n1and\nG0\nC;\u000b1\u000b0\n1(l;\u001c1;\u001c0\n1) = \u0016g<\n\u000b1\u000b0\n1(l;t1\u0000t0\n1)whereas for\u001c1;\u001c0\n12C+\nwe obtaint1>t0\n1andG0\nC;\u000b1\u000b0\n1(l;\u001c1;\u001c0\n1) = \u0016g<\n\u000b1\u000b0\n1(l;t1\u0000t0\n1).\nHence, we haveR\nCd\u001c1lim\u001c0\n1!\u001c+\n1G0\nC;\u000b1\u000b0\n1(l;\u001c1;\u001c0\n1) =R\ndt1\u0002\n\u0016g<\n\u000b1\u000b0\n1(l;0\u0000)\u0000\u0016g<\n\u000b1\u000b0\n1(l;0+)\u0003\n= 0 . Here 0\u0000is the\nnegative inﬁnitesimal. For the detail treatments on real-time\nprojection as well as the real-time integration see subSec.\nB 1 in Appendix B. Third, what we do is we perform the\nFourier transforms on the above Green’s functions as, for\ninstance, \u0016Da(0;~t1\u0000\u0014t1) =Rd~!\n2\u0019e\u0000i~!(~t1\u0000\u0014t1)\u0016Da(0;~!)and\n\u0016gt\n\u000b\u000b0\n1(k;t\u0000\u0014t1) =Rd!\n2\u0019e\u0000i!(t\u0000\u0014t1)\u0016gt\n\u000b\u000b0\n1(k;!):Then, using Eq. (B28) the right-hand side of Eq. (29) is\nrewritten by the retarded and advanced Green’s functions in\nEqs. (25) and (26). By performing the temporal integralsR\nd~t1d~t2d\u0014t1and from Eq. (13), the x-component averaged\nelectric current density of TI surface state becomes\nFIG. 4. Feynman diagram for the spin-pumping-induced electric cur-\nrent described by Eq. (30). It consists of two TI-surface-state Green’s\nfunctions (solid lines), two magnon Green’s functions (wavy lines),\nand two vertices denoted by crosses. As described by the orange cir-\ncle, the exchange interaction between the zero-momentum magnon\nand the TI surface state occurs at the right vertex leading to the gen-\neration of electric current on the surface of TI.9\n\u0016j(2;1)\nx= (\u0000evFsgn(B0))\u0012\n\u0000i\n~\u00133\u0000\n~\rBext\u00012\u0012JexcS0n2D\nloc\n4\u0013\n\u0016Dr(0;!ext)\u0016Da(0;!ext)\n\u0002Zd2kd!\n(2\u0019)3f(~!)h\u0010\n\u0016ga\n\u000b\u000b0\n1(k;!)\u001by\n\u000b0\n1\u000b1\u0016ga\n\u000b1\u000b0(k;!)\u0000\u0016gr\n\u000b\u000b0\n1(k;!)\u001by\n\u000b0\n1\u000b1\u0016gr\n\u000b1\u000b0(k;!)\u0011\n\u001by\n\u000b0\u000bi\n; (30)\nwheref(~!) =\u0000\n1 +e\f~!\u0001\u00001and we have taken a contin-\nuum limitV\u00001P\nk!R\nd2k=(2\u0019)2. The right-hand side of\nEq. (30) represents the way the electric current of TI sur-\nface state is induced by the spin pumping due to the external\nmagnetic ﬁeld Hextand the exchange interaction Vexc. To\nsee this clearly, let us describe \u0016j(2;1)\nx diagrammatically and\npresent this in Fig. 4. The solid and the wavy lines represent\nthe Green’s function of TI surface state (fermion line) and that\nof magnon (boson line), respectively. The two vertices are de-\nscribed by crosses where the energy and momentum conserve.\nThe gray and orange circles denote the amplitude of ac mag-\nnetic ﬁeldBextand the exchange-interaction strength Jexc,\nrespectively. The Pauli matrix in the left side originates in the\ngenerator of x-component electric current while the right one\nis coming from the y-component exchange interaction. The\nretarded Green’s function \u0016Dr(0;!ext)appearing in this dia-\ngram describes the emission process of magnon with the zero\nmomentum and the energy ~!extgoing from orange to gray\ncircles whereas the advanced Green’s function \u0016Da(0;!ext)\nrepresents the absorption process going from gray to orange.\nIn the diagram in Fig. 4, the energy and momentum of TI\nsurface state remains unchanged. This is because, ﬁrst, the\nemission and absorption processes of magnon of the energy\n~!extoccur with each process occurring once. Second, the\nmagnon does not carry momentum since the ac magnetic ﬁeld\nHextis spatially homogeneous and so does the Fourier trans-\nform of exchange interaction Jexc\n(qp): for they-component it is\ndescribed by the constant Jexc(see Eq. (20)). Since we have\noverlooked at the structure of our diagram, let us now ana-\nlyze the mechanism of the spin-pumping-induced electric cur-\nrent. Initially, the TI surface state is in the thermal equilibrium\n\u001aGC(H;\f;\u000f F)and the origin of Fermi sphere of TI surface\nstate is atk0= (0;0). When the FMR is triggered at t=t0,\nthe localized spin Sistarts to show its dynamics described by\nthe Landau-Lifshitz-Gilbert equation (see Eq. (A31)) and the\nmagnon of zero momentum and frequency !extemerges. It is\nthe ﬂuctuation of the saturation magnetization in the ydirec-\ntion created by the external magnetic ﬁeld B0. After then, the\nspin pumping occurs associating with the spin current Jpump\ny;z\nﬂowing from FM to the surface of TI. The zero-momentum\nmagnon is going to be the carrier of it. In other words, the\nspin current Jpump\ny;z is the ﬂow of zero-momentum magnon.\nThe magnon couples with the TI surface state through the ex-\nchange interaction Vexc. Then owing to the spin-momentum\nlocking, the magnon acts like an additional momentum of TI\nsurface state. This means that effectively TI surface state ex-\nperiences the coupling between the magnon as an electric ﬁeld\nbeing applied and a non-equilibrium state of TI surface stateis driven. Such a situation can be described as the deviation of\nthe TI-surface-state Fermi circle from the origin (see also Fig.\n5 (b)). On the other side, the TI surface state is affected by\nthe impurity potential Himpgiven by Eq. (2). Then as time\ngoes by, the effect of effective electric ﬁeld of magnon and the\nimpurity effect Himpare going to get balanced. As a result,\nthe TI surface state and the magnon both relax to the non-\nequilibrium steady state and the static electric ﬁeld is created\non the surface of TI. Let us call it the spin-pumping-induced\nelectric ﬁeld ESPI\nx. At the non-equilibrium steady state, the\nTI surface state experiences the ESPI\nxand the spin-pumping-\ninduced electric current \u0016j(2;1)\nx ﬂows on the TI surface as the\nresponse to it. To make the relation between \u0016j(2;1)\nx andESPI\nx\nclear, let us rewrite \u0016j(2;1)\nx with using the electrical conductiv-\nity\u001bTI\nxxas\u0016j(2;1)\nx=\u001bTI\nxxESPI\nx. As in the case of Dirac electrons\nin graphene, the electrical conductivity of TI surface state \u001bTI\nxx\ncan be calculated by using the Boltzmann equation [66]. We\nobtain\u001bTI\nxx=\u000fF\u001crel\nTI\n2~\u0001e2\n2\u0019~. On the other side, the formula\nofESPI\nxis obtained by evaluating the right-hand side of Eq.\n(30). For doing this, ﬁrst we remark that the denominator in\nthe right-hand side of Eq. (30) is a function of the dispersion\nof TI surface state ( !TI\nk=~\u00001\u000fTI\nk), and thus, it is the func-\ntion of the absolute k=p\n(kx)2+ (ky)2. Hence, it means\nthat the denominator in the right-hand side of Eq. (30) is an\neven and symmetric function of kxandky. Due to this fact,\nfor the numerator in the right-hand side of Eq. (30) the terms\nproportional to kxkyas well as (kx)2\u0000(ky)2vanish. As a\nresult, the only terms which remain are the products of two\ndiagonal elements of TI-surface state Green’s function, i.e.,\u0000\n\u0016ga(r)\n\"\"(k;!)\u00012and/or\u0000\n\u0016ga(r)\n##(k;!)\u00012. In the following evalu-\nation, we only retain the ﬁrst term of \u0016gr(a)\n\u000b\u000b0(k;!)in Eq. (25)\nsince only the electronic state in the vicinity of Fermi-energy\nlevel contributes to the electric transport. Next, we perform\nkand!integrals with using three types of approximations.\nWe ﬁrst do from the !integral and rewrite the integrand with\nthe derivative term @f(~!)=@(~!). As the ﬁrst approxima-\ntion, we take the low-temperature limit ( \f!1 ) and we ob-\ntain@f(~!)=@(~!) =\u0000\u000e(~!):By performing the !integral,\nthe integrand becomes the function of the relaxation time \u001crel\nTI,\nthe TI-surface-state dispersion !TI\nk, and the Fermi energy !F\n(=~\u00001\u000fF)given as1=2\u001crel\nTI\u0000\n(!TI\nk\u0000!F)2+(1=2\u001crel\nTI)2\u0001. Next, we rewrite\nthekintegral in the following way:d2k\n(2\u0019)2=~N(\u0018k)d\u0018k,\nwhere ~N(\u0018k) =(\u0018k+\u000fF)\n2\u0019(~vF)2is the density of states per volume\nand\u0018kis the energy of the TI surface state measured with re-\nspect to the Fermi energy. It is deﬁned by \u0018k=\u000fTI\nk\u0000\u000fF. As a10\nresult, the integrand becomes ~N(\u0018k)\u0002~=2\u001crel\nTI\n\u00182\nk+(~=2\u001crel\nTI)2. Then as\na second approximation, we regard only the electronic state in\nthe vicinity of Fermi surface contributes to the electric current.\nIn other words, the TI surface state depends weakly on the\ndensity of states ~N(\u0018k). Therefore, we take ~N(\u0018k)\u0019~N(0):\nOn the other side, the lower limit of \u0018k-integral is\u0000\u000fF. We\nconsider that on the surface of area Va huge number of elec-\ntrons are contained. Hence, as a third approximation we takethe number density of TI surface state nTI\n2Dto be sufﬁciently\nlarge. Since the number density nTI\n2Dis related to the Fermi\nenergy\u000fFas\u000fF=~vFq\n4\u0019nTI\n2D;we take\u000fF!1 (see also\nthe argument below Eqs. (A25) and (B19)). By using these\nthree types of approximations and performe the \u0018k-integral,\nwe have\nESPI\nx(!ext;B0) =\u0000sgn(B0)\u0012JexcS0n2D\nloc\n4evF\u001crel\nTI\u0013(\rBext)2\n(!ext\u0000!FM\n0)2+ (\u000b!ext)2; (31)\nwhere!FM\n0=\rjB0j. Consequently, when the FMR occurs\nwith the frequency !ext, the electric current \u0016j(2;1)\nx as well as\nthe electric ﬁeld ESPI\nxare induced by the spin pumping on\nthe surface of TI. It ﬂows perpendicular to the precession axis\n(yaxis) of FMR owing to the spin-momentum locking. It is\nproportional to the square of the ac-magnetic-ﬁeld amplitude\nBextdescribing that it is the non-linear (quadratic) response\nto the external ac magnetic ﬁeld. In other words, it is pro-\nportional to the power of the applied electromagnetic wave\n(microwave). Like the FMR (magnon) spectrum, the electric\ncurrent \u0016j(2;1)\nx (or the electric ﬁeld ESPI\nx) is described by the\nquantities\rBext,!ext,!FM\n0, and the Gilbert-damping con-\nstant\u000b. Indeed, the spectral function of magnon can be ob-\ntained by multiplying the factor \u000b!extto the third factor of\nESPI\nxin Eq. (31): \u000b!ext\u00021\n(!ext\u0000!FM\n0)2+(\u000b!ext)2. In other\nwords, the retarded Green’s function of magnon is equivalent\nto the magnetic susceptibility (see Eq. (A32) and the argument\nbelow it). Physically, this represents the absorption energy\nof localized spin which we need to drive the FMR (see also\nthe argument after Eq. (8) in [15]). The electric ﬁeld ESPI\nx\ndepends on both magnetic quantities including \rBext;\u000b, the\nexchange interaction strength Jexc, the density of localized\nspinn2D\nloc, and those of TI such as Fermi velocity vF, and\nthe relaxation time \u001crel\nTI. This is natural and reasonable be-\ncause the spin-pumping-induced electric ﬁeld ESPI\nxis real-\nized at the non-equilibrium steady state owing to the com-\nmensuration of the effective electric ﬁeld of magnon and the\nimpurity effect Himpmediated by the exchange interaction.\nBased on the Feynman diagram in Fig. 4, we can under-\nstand why the spin-pumping-induced electric current \u0016j(2;1)\nx\nis the quadratic response to the ac magnetic ﬁeld in the fol-\nlowing way. First, the ac magnetic ﬁeld is used to drive the\nFMR and the associated zero-momentum magnon which cou-\nples with the TI surface state through the exchange interac-\ntion. Second, to generate the transport phenomena of TI sur-\nface state we need to drive the magnon with the ac magnetic\nﬁeld once more. As a result, the electric current of TI surface\nstate becomes the quadratic response to the ac magnetic ﬁeld\nsuch that both the emission and absorption processes of zero-\nmomentum magnon occur. Indeed, this naturally reﬂects that\nFIG. 5. (a) Schematic of the generation of the electric ﬁeld EISHE\nx .\nWhen the spin current Jspin\ny;z is injected to the non-magnetic metal,\nowing to the spin-orbit interaction the inverse spin Hall effect is gen-\nerated so that the both electrons of spins polarized in the positive\nand negative ydirections accumulate on the edge of the sample. As\na result, the electric ﬁeld EISHE\nx and the associated electric current\njISHE\nc emerge. (b) Schematic of the generation of the spin-pumping-\ninduced electric current \u0016j(2;1)\nx. When we drive the FMR and the asso-\nciated spin pumping, the zero-momentum magnon couples with the\nTI surface state through the exchange interaction. Due to the spin-\nmomentum locking, the TI surface state effectively experiences the\nzero-momentum magnon as the electric ﬁeld. As a result, the spin-\npumping-induced electric ﬁeld ESPI\nxas well as the spin-pumping-\ninduced electric current \u0016j(2;1)\nx are generated (inverse Edelstein ef-\nfect). It is described as the ﬂow of Fermi circle of TI surface state.\nthe spin-pumping-induced electric current is generated by the\nelectromagnetic wave whose power is quadratic to the ampli-\ntude of ac magnetic ﬁeld. The spin-pumping-induced elec-\ntric current \u0016j(2;1)\nx (or the spin-pumping-induced electric ﬁeld\nESPI\nx) gets larger by raising the ac-magnetic-ﬁeld amplitude\nBext(or the power of electromagnetic wave) and by choosing\nthe ferromagnetic material exhibiting a strong exchange cou-\npling strength. Such a feature is reﬂecting that the surface of\nferromagnetic TI has a high-performing functionality of gen-\nerating the electric charge current by magnetic controlling.\nTo make our understanding on the spin-pumping-induced\nelectric current \u0016j(2;1)\nx better, let us compare it with the elec-\ntric ﬁeld associated with the inverse spin Hall effect by us-\ning the illustrations presented in Fig. 5. The inverse spin11\nHall effect occurs in, for instance, the hybrid system com-\nprise of FM and non-magnetic heavy metal exhibiting strong\nspin-orbit interaction, for example, the heterojunction of NiFe\nand Pt [19, 20, 22, 24]. When we inject the y-polarized spin\ncurrent to the non-magnetic metal ﬂowing in the zdirection,\ndue to the spin-orbit coupling both the electrons whose spins\nare polarized in the positive and negative ydirections ﬂow\nparallel into the xdirection and accumulate to the edge of\nsample. As a result, the electric ﬁeld, namely, EISHE\nx emerges\nin thexdirection. Simultaneously, the associated electric cur-\nrentjISHE\nc ﬂows in the same direction (Fig. 5(a)). This is\nthe phenomenon in a three-dimensional bulk system. In con-\ntrast, our spin-pumping-induced electric current \u0016j(2;1)\nx is the\nphenomenon intrinsic in the two-dimensional surface system.\nThe mechanism of its generation is not due to the accumula-\ntion of electrons on the edge of sample but due to the effective\nelectric ﬁeld of magnon via the spin-momentum locking. As\nillustrated in Fig. 5(b), the spin-pumping-induced electric cur-\nrent can be described as the ﬂow of Fermi circle of TI surface\nstate. It is nothing but the inverse Edelstein effect which is\nalso realized in systems possessing Rashba interfaces [22, 24].\nFinally, let us make a qualitative comparison between\nour result and the experimental results [42, 59]. What has\nbeen measured in these experiments are the electric voltage\nemerged on the surface due to the spin pumping. Therefore,\nwe calculate the spin-pumping-induced electric voltage and\ncompare its characteristic with the experimental results. Be-\nfore we give a detailed argument, we note here that in [42] the\ndirection of static magnetic ﬁeld B0(the precession axis of\nFMR) is taken to be parallel to the yaxis while in [59] it is\ntaken to be in the xaxis. Since the essence of physics does\nnot change, as we did in Sec. II A we take the precession axis\nof FMR to be in the yaxis (thus, the electric current of TI sur-\nface state or electric voltage emerges in the xdirection). To\nmake our argument clear and simple, in the following we in-\ntroduce the effective electric voltage by using ESPI\nx. First, as\nwe see in Eq. (31) the electric ﬁeld ESPI\nxis spatially homoge-\nneous along the xdirection. Thus, by multiplying ESPI\nxwith\nthe length of TI surface in the xdirectionlTI\nx, we obtain the\nelectric voltage in the xdirection and call it as VSPI\nx. Next,\nwe divideVSPI\nxby the factor\u0010\n\u0000JexcS0n2D\nloclTI\nx\n4evF\u001crel\nTIB2\n0\u0011\nbecause essen-\ntially its characteristic is represented by the Gilbert-damping\nconstant\u000b, the external frequency !ext, and the frequency\n!FM\n0=\rjB0j=\r(sgn(B0)B0). In addition, in the experi-\nment the external frequency !extis ﬁxed whereas the mag-\nnetic ﬁeldB0varies from positive to negative values. By\ntaking account of this, we take !extto be the positive con-\nstant and introduce the “spin-pumping-induced electric volt-\nage\u0016VSPI\nx” deﬁned as the function of B0as\n\u0016VSPI\nx(~B0) =\u0000\u0002(~B0)eP\n\u0010\n~B0\u00001\u00112\n+\u000b2\n+ \u0002(\u0000~B0)eP\n\u0010\n~B0+ 1\u00112\n+\u000b2; (32)\nwhere ~B0= (\rB0)=!extis the dimensionless magnetic ﬁeld\nFIG. 6. Plots of spin-pumping-inducedelectric voltage \u0016VSPI\nx de-\nﬁned by Eq. (32). The vertical axis represents the dimensionless\nmagnetic ﬁeld ~B0whereas the horizontal axis represents the spin-\npumping-induced electric voltage \u0016VSPI\nx. The blue, orange, green,\nand red curves are for P= 0:0100;0:0075;0:0050 , and 0:0025 , re-\nspectively. For all four curves, we take the Gilbert damping constant\n\u000bto be equal to 0.15.\nandP= (Bext)2. It is the quantity describing the power\nof electromagnetic ﬁeld which we apply to derive the FMR.\n\u0002(\u0006~B0)is the Heaviside step function. We plot \u0016VSPI\nx(~B0)\nin Fig. 6 by taking the electromagnetic-wave power Pas\na parameter while we ﬁx the Gilbert-damping constant \u000b\nto 0.15. Here we plot \u0016VSPI\nx for four different conditions;\nP= 0:0100;0:0075;0:0050 , and 0:0025 . The full width of\nhalf maximum is equal to the Gilbert damping constant \u000b:\nThe most striking features of \u0016VSPI\nx are (i) the emergence of\ntwo side peaks and (ii) the linear scaling of two peak val-\nues with respect to the power P; the two side peaks locate\nat~B0=\u00061:The values of two peaks have the same abso-\nlute values ( =eP=\u000b2) but the signs are opposite. Let us now\nlook at the experimental data [42, 59]. First, in [42] the (bulk\ninsulating) TIs were chosen as Bi 1:5Sb0:5Te1:7Se1:3naming\nBSTS and Sn-doped Bi 2Te2Se. On the other hand, for the\nferromagnetic material they choose Ni 81Fe19. Let us focus\non Figs. 3(b) or 4(a) and 3(c). Fig. 3(b) is the experimental\ndata of electric voltage for sample BSTS/Ni 81Fe19with the\nsample size of BSTS is 4\u00023\u00020:1mm3for four different\nmicrowave-power conditions; 0.2mW, 0.15mW, 0.10mW, and\n0.05 mW. Fig 4(a) is the result of electric voltage for samples\nBSTS/Ni 81Fe19with three different sample sizes of BSTS;\n4\u00021\u00020:1mm3,4\u00023\u00020:1mm3, and 2\u00021:5\u00020:2\nmm3. They are plotted as functions of static magnetic ﬁeld\nwhich corresponds to our B0(or~B0). Both of them show\ntwo side peaks as discussed in the previous description. Two\npeak spots appear symmetrically with respect to the origin of\nthe magnetic-ﬁeld axis and the two peak values have (almost)\nthe same absolutes with opposite signs. The similar result\nis reported in [59]. In this work, the magnetic TI was engi-\nneered by creating the heterostructure of YIG and Cr-doped\nTI: YIG/Cr 0:08(Bi0:37Sb0:63)1:92Te3. We will focus on Fig.\n2(c) where the electric voltage is plotted as a function of mag-\nnetic ﬁeld. It shows the similar features as the results shown\nin Fig. 3(b) or Fig 4(a) in [42]: the emergence of two side12\npeaks having opposite signs. The difference between the elec-\ntric voltage in Fig. 3(b) or Fig 4(a) in [42] and that in Fig.\n2(c) in [59] is that the signs of two peaks; in Fig. 3(b) the\npeak value at positive magnetic ﬁeld is negative while it is\npositive in Fig. 2(c) it is positive. Such an opposite-sign\nbehavior, however, is not essential for our analysis and we\nwill not refer to its origin. We note that in [51] the measure-\nment of spin-pumping-induced voltage was performed using\nthe bilayer systems of Bi 2Se3(TI) and CoFeB (ferromagnet).\nIn this experiment, it is considered that the dominant contri-\nbution to the spin-pumping-induced voltage is coming from\nthe inverse spin Hall effect (bulk state) rather than the inverse\nEdelstein effect (surface state). Thus, although the measured\nspin-pumping-induced voltage shows similar characteristics\n(Figs. 2 and 3(a) and (b)) with \u0016VSPI\nx, we will not make a\ncomparison with these experimental results. Next, let us take\na look at Fig. 3(c) in [42]. It shows the microwave-power de-\npendence of peak values for the BSTS sample size 4\u00021\u00020:1\nmm3. Both the positive and negative peak values become\nlarger as microwave power increases. To summarize, from the\nabove analysis we see that the physical behavior of our result\n\u0016VSPI\nxrepresented by Eq. (32) and Fig. 6 match qualitatively\nwith these experimental results.\nIV . CONCLUSION\nIn this paper, we have investigated the electric current on\nthe surface of ferromagnetic TI induced by the spin pumping.\nFirst, we have presented the microscopic model of ferromag-\nnetic TI surface and represented its time evolution. We have\nmathematically formulated how the system evolves from the\nthermal equilibrium state realized in the far past to the non-\nequilibrium steady state driven by the spin pumping (FMR).\nThen we have used the Keldysh Green’s function approach\nto analyze the generation of a spin-pumping-induced elec-\ntric current. We have calculated it by regarding the ac exter-\nnal magnetic ﬁeld and the exchange interaction as perturba-\ntive terms. In this way, we could clearly understand the way\nspin-pumping-induced electric current is generated by these\ntwo interactions. The mechanism is as follows. The FMR is\ntriggered by the ac magnetic ﬁeld and the magnon with the\nzero momentum emerges. It is the ﬂuctuation from the satura-\ntion magnetization. After then, the spin pumping is induced,\nand during such a process, the spin current ﬂows from FM to\nTI carried by the zero-momentum magnon. Through the ex-\nchange interaction, the zero-momentum magnon couples with\nthe TI surface state and the spin is exchanged between them.\nThen owing to the spin-momentum locking, it is converted\ninto momentum and effectively the TI surface state experi-\nences this additional momentum as the applied electric ﬁeld.\nOn the other hand, the TI surface state is affected by the non-\nmagnetic impurity. As a result, at the non-equilibrium steady\nstate these two effects commensurate and the static electric\nﬁeld, i.e., the spin-pumping-induced electric ﬁeld is created\nleading to the generation of the spin-pumping-induced elec-\ntric current. It scales quadratically to the ac magnetic ﬁeld\n(linear to the power of electromagnetic ﬁeld) while it is lin-ear to the strength of the exchange interaction. The effective\nelectric voltage \u0016VSPI\nx in Eq. (32) is expressed by the spec-\ntrum of zero-momentum magnon which clearly reﬂects that it\nis created by the spin pumping (FMR). The effective electric\nvoltage \u0016VSPI\nxshows two side peaks. They emerge when the\nabsolute of external frequency of ac magnetic ﬁeld becomes\nequivalent to the Zeeman gap of magnon. The absolutes of\nthese two peak values are the same while they have the oppo-\nsite signs. Further, the absolutes of two peaks are the increas-\ning function of the microwave power. Such characteristics of\nour effective voltage \u0016VSPI\nxshow qualitatively the good match-\ning with the experimental results of electric voltage reported\nin [42, 59]. Consequently, the spin-pumping-induced elec-\ntric current is the quantum phenomena intrinsic in the hybrid\nquantum system of TI surface state and the zero-momentum\nmagnon. It is the non-linear response to the ac magnetic ﬁeld.\nOur microscopic theory based on the Keldysh Green’s func-\ntion approach makes not only the mechanism as well as the\nstructure of spin-pumping-induced electric current (voltage)\nclear. We believe that our theory can be extended in many\nother types of quantum phenomena occurring at the inter-\nface between the magnetic materials and TI. For instance, we\nwould like to apply our Keldysh Green’s function approach\nto analyze the heat current as well as the spin Seebeck effect\nand the spin-orbit torque in the future. In addition, we become\nable to extract more information on magnets and TI. For in-\nstance, by measuring the peak value of electric voltage we can\nestimate the exchange-coupling strength. Another important\nand interesting issue is the Fermi-energy dependence on spin-\npumping-induced electric voltage. It is important to analyze\nwhether the contribution to electric transport quantities (for\ninstance, the electric voltage) is coming from the surface state\nor the bulk state [51, 54, 59]. The transport properties of Dirac\nelectrons in solids are affected by many types of elements. For\ninstance, the Fermi-energy dependence of Dirac-electron con-\nductivity in graphene differs whether the impurity potential\nis short-range (delta-function) type or long-range (Coulomb)\ntype [66]. For the TI surface state the characteristics of its\nconductivity is not only generated by the impurity effect but\nalso by a scattering process due to a magnetic texture such as\nskyrmion [67]. As our future work, we would like to explore\nthe rich transport phenomena on the surface of a magnetic TI\ninduced by the impurity potentials and the magnetic textures\nwith many types and analyze carefully the characteristics of\nelectric voltage as well as the electrical conductivity as func-\ntions of the Fermi energy.\nTo discuss our result from the application point of view,\nthe non-linear response in the magnetic ﬁeld as well as the\nlinear scaling in the exchange-coupling strength of the spin-\npumping-induced electric current clearly indicates that the\nsurface of ferromagnetic TI possesses the high-performing\nfunctionality of creating the electric charge current or volt-\nage by the magnetic controlling. When we think of engineer-\ning spintronics devices, the merit of using the ferromagnetic\nTI comparing to the hybrid system of FM and metal like the\nNiFe/Pt is the lower energy consumption: The joule heating is\nsuppressed for the ferromagnetic TI because the bulk is insu-\nlating while it is unavoidable for the FM/metal hybrid system13\nsince the bulk is metallic. By designing carefully the larger\nhybrid quantum systems based on the magnet and TI, we will\nbecome able to perform the coherent controlling of magnon\ndynamics and the quantum transport of TI surface state at the\ninterface, and consequently, make a high-efﬁcient conversion\nof the spin and the electric charge current (coherent control-\nling of the magnetism and the electricity). Such investigations\nlead to an important progress on the realization of magnetic-\nTI-based spintronics devices.\nACKNOWLEDGMENTS\nY . H thanks Kanta Asakawa for the having the discussion on\nthe basics of FMR experiment, Yuki Shiomi for having fruitful\ndiscussion on Ref. [42], Minoru Kawamura for discussing the\nphysical interpretation on the non-linearity of spin-pumping-\ninduced electric current, Hiroyasu Yamahara for the discus-\nsion on Refs. [42, 51] as well as the basics of FMR experi-\nment. This work was supported in part by the MEXT Grant-\nin-Aid for Scientiﬁc Research on Innovative Areas KAK-\nENHI Grant Number JP15H05870 (Y . H), and JSPS KAK-\nENHI Grants Nos. JP15H0584 and JP17K05485, JST CREST\nGrant No. JPMJCR18T2, and JSPS KAKENHI Grant No.\nJP20H01830 (K. N).\nAppendix A: Field Quantization, Real-Time Green’s Function,\nand Imaginary-time Green’s function\nIn this section, ﬁrst we present the details of ﬁeld quantiza-\ntion for the TI surface state. Then, we introduced the unper-\nturbed real-time Green’s function. Next we demonstrate the\nderivation of impurity-averaged Green’s function of TI sur-\nface state by using the imaginary-timeGreen’s function for-\nmalism. Further, we show the real-time Green’s functions\nof magnon and the retarded and advanced components of\nmagnon Green’s function including the Gilbert-damping ef-\nfect.\n1. Field Quantization and Real-Time Green’s Functions of TI\nSurface State\nThe spin-momentum-locking Hamiltonian of TI surface\nstate in the momentum space is given by (see also Eq. (3))\nHTI\n0;\u000b0\u000b(k) =~vF(kx\u001by\u0000ky\u001bx)\u000b0\u000b; (A1)\nwhere\u000b;\u000b0=\";#. The eigenvalues of the above Hamil-\ntonian are\u0006\u000fTI\nk=\u0006~vFkwithk=p\n(kx)2+ (ky)2:\nWe denote the positive and negative-energy plane-wave solu-\ntions asu(+)\nk(xt) =u(+)(k)ei(k\u0001x\u0000!TI(k)t)andu(\u0000)\nk(xt) =\nu(\u0000)(k)ei(k\u0001x+!TI(k)t), respectively. The eigenfrequency\n!TI(k)is obtained from \u000fTI\nkas!TI(k) = ~\u00001\u000fTI(k).\nThe vectors u(+)(k) = (u\"(k);u#(k))tandu(\u0000)(k) =(u(\u0000)\n\"(k);u(\u0000)\n#(k))tare two-column vectors with “t\" denot-\ning the transpose. We take them as\nu(+)(k) =1p\n2\u00121\nik+\nk\u0013\n; u(\u0000)(k) =1p\n2\u0012ik\u0000\nk\n1\u0013\n;\n(A2)\nwherek\u0006=kx\u0006iky. The two eigenvectors u(+)(k)and\nu(\u0000)(k)satisfy the completeness relations\nX\n\u000b=\";#u(+)y\n\u000b(k)u(+)\n\u000b(k) =X\n\u000b=\";#u(\u0000)y\n\u000b(k)u(\u0000)\n\u000b(k) = 1;\nX\n\u000b=\";#u(+)y\n\u000b(k)u(\u0000)\n\u000b(k) =X\n\u000b=\";#u(\u0000)y\n\u000b(k)u(+)\n\u000b(k) = 0:\n(A3)\nWith using the plane-wave solutions in Eq. (A2) and the com-\npleteness relations in Eq. (A3), we construct the ﬁeld operator\nof TI surface state. By denoting the ﬁeld operator (annihila-\ntion operator) of TI surface state as  \u000b(x), it is given by\n \u000b(x) =1p\nVX\nk;\u0015=\u0006\u0010\nu(\u0015)\n\u000b(k)eik\u0001x\u0011\n\u0001c(\u0015)(k); (A4)\nwherec(+)(k)andc(\u0000)(k)are annihilation operators of TI\nsurface state whose energy and momentum are (\u000fTI(k);k)\nand(\u0000\u000fTI(k);k), respectively. Vis the area of TI surface.\nFor the ground state, we choose the Dirac sea represented by\nj0i=Y\nkc(\u0000)y(k)j~0i; (A5)\nwherej~0iis the Fock state which satisﬁes c(\u0006)(k)j~0i= 0for\nanyk:Correspondingly, we rewrite the ﬁeld operator in Eq.\n(A4) as\n \u000b(x) =1p\nVX\nk\u0000\nu\u000b(k)eik\u0001xa(k) +v\u000b(k)e\u0000ik\u0001xby(k)\u0001\n;\n(A6)\nwherea(k) =c(+)(k);by(\u0000k) =c(\u0000)(k),u\u000b(k) =\nu(+)\n\u000b(k), andv\u000b(k) =u(\u0000)\n\u000b(\u0000k). The operator a(k)is\nannihilation operator of particle (electron) with the energy\n+\u000fTI(k)and momentum kwhileby(k)is creation opera-\ntor of anti-particle (hole) with the energy +~!TI(k)and\nmomentumk. They satisfy the anti-commutation relations\nfa(k);ay(k0)g=fb(k);by(k0)g=\u000e(k\u0000k0), and all the\nothers are zero. From these anti-commutation relations and\nEq. (A3), we have f \u000b(x); y\n\u000b0(x0)g=\u000e(x\u0000x0)and\nf \u000b(x); \u000b0(x0)g=f y\n\u000b(x); y\n\u000b0(x0)g= 0:By using the\noperator \u000b(x)in Eq. (A6) and its Hermitian conjugate the\nfree Hamiltonian, momentum operator, and number operator\nare described as\nHTI\n0=X\nk\u000fTI(k)\u0000\nay(k)a(k) +by(k)b(k)\u0001\n;\nPi=X\nk~ki\u0000\nay(k)a(k) +by(k)b(k)\u0001\nNTI=X\nk\u0000\nay(k)a(k)\u0000by(k)b(k)\u0001\n; (A7)14\nwherei=x;y: We have neglected the constants in HTI\n0and\nNTIwhich are the contribution from the Dirac sea. Here-\ninafter, we just write NTIasN. Based on the previous argu-\nment, next we consider the Hamiltonian \u0016HTI\n0=HTI\n0\u0000\u000fFN\n(see also Eq. (1)) with \u000fF(>0) the Fermi energy of TI.\nWe have taken the chemical potential of TI surface state to\nbe equal to\u000fF:Correspondingly, we re-describe the ﬁeld op-erator \u000b(x)in Eq. (A6) by three operators a(k),b(+)(k),\nandb(\u0000)(k): The operator a(k)is the annihilation operator of\nmomentumkwith its energy higher than the Fermi energy \u000fF.\nOn the other hand, the operator b(+)(k)is the annihilation op-\nerator of momentum kwith a positive energy which is lower\nthan\u000fF:b(\u0000)(k)is the annihilation operator of momentum k\nwith a negative energy [68]. The representation of operator\n \u000b(x)in terms ofa(k),b(+)(k), andb(\u0000)(k)is given as\n \u000b(x) =1p\nVX\nkh\n\u0002(k\u0000kF)ua\n\u000b(k)eik\u0001xa(k) +\u0010\n\u0002(kF\u0000k)vb(+)\n\u000b(k)b(+)y(k) +vb(\u0000)\n\u000b(k)b(\u0000)y(k)\u0011\ne\u0000ik\u0001xi\n; (A8)\nwhere \u0002(k\u0000kF) (\u0002(kF\u0000k))is the step function and\nkF= (~vF)\u00001\u000fF. The operators a(k),b(+)(k), andb(\u0000)(k)\nsatisfy the anti-commutation relation fa(k);ay(k0)g=fb(+)(k);b(+)y(k0)g=fb(\u0000)(k);b(\u0000)y(k0)g=\u000e(k\u0000k0)\nand all the others are zero. The two-column vectors u\u000b(k);\nvb(+)\n\u000b(k);andvb(\u0000)\n\u000b(k)are given by\nua(k) =1p\n2\u00121\nik+\nk\u0013\n; vb(+)(k) =1p\n2\u0012ik\u0000\nk\n1\u0013\n; vb(\u0000)(k) =1p\n2\u0012\n\u0000ik\u0000\nk\n1\u0013\n: (A9)\nFrom Eqs. (A8) and (A9), the Hamiltonian \u0016HTI\n0is expressed by the operators a(k),b(+)(k), andb(\u0000)(k)as\n\u0016HTI\n0=X\nkh\n\u0002(k\u0000kF)\u0018TI\na(k)ay(k)a(k) + \u0002(kF\u0000k)\u0018TI\nb(+)(k)b(+)y(k)b(+)(k) +\u0018TI\nb(\u0000)(k)b(\u0000)y(k)b(\u0000)(k)i\n; (A10)\nwhere\u0018TI\na(k) =\u0000\u0018TI\nb(+)(k) =\u000fTI(k)\u0000\u000fF, and\n\u0018TI\nb(\u0000)(k) =\u000fTI(k) +\u000fF. For deriving Eq. (A10), we have\nused the anti-commutation relation fb(+)(k);b(+)y(k0)g=\nfb(\u0000)(k);b(\u0000)y(k0)g=\u000e(k\u0000k0):Next, we introduce theﬁeld operator of TI surface state in the interaction picture with\nrespect to the Hamiltonian Let us denote it as  H0\u000b(xt)which\nis deﬁned by  H0\u000b(xt) =ei\u0016HTI\n0t=~ \u000b(x)e\u0000i\u0016HTI\n0t=~. It is de-\nscribed bya(k),b(+)(k), andb(\u0000)(k)as\n H0\u000b(xt) =1p\nVX\nk\u0010\n\u0002(k\u0000kF)ua\n\u000b(k)ei(k\u0001x\u0000!\u0018;TI\na(k)t)a(k) + \u0002(kF\u0000k)vb(+)\n\u000b(k)e\u0000i\u0010\nk\u0001x\u0000!\u0018;TI\nb(+)(k)t\u0011\nb(+)y(k)\n+vb(\u0000)\n\u000b(k)e\u0000i\u0010\nk\u0001x\u0000!\u0018;TI\nb(\u0000)(k)t\u0011\nb(\u0000)y(k)\u0011\n; (A11)\nwhere!\u0018;TI\na(k) = ~\u00001\u0018TI\na(k);!\u0018;TI\nb(+)(k) = ~\u00001\u0018TI\nb(+)(k),\nand!\u0018;TI\nb(\u0000)(k) = ~\u00001\u0018TI\nb(\u0000)(k). By using the ﬁeld oper-\nator H0\u000b(xt)in Eq. (A8) and its Hermitian conjugate\n y\nH0\u000b(xt), we introduce two-point real time Green’s func-tion in the interaction picture. Let us denote the time-\nordered, anti-time-ordered, retarded, and advanced Green’s\nfunctions as gt(0)\n\u000b\u000b0(xt;x0t0),g~t(0)\n\u000b\u000b0(xt;x0t0); gr(0)\n\u000b\u000b0(xt;x0t0),\nandga(0)\n\u000b\u000b0(xt;x0t0);respectively. They are given by15\ngt(0)\n\u000b\u000b0(xt;x0t0) =\u0000iTh H0\u000b(xt) y\nH0\u000b0(x0t0)i\u00160=gt(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ngt(0)\n\u000b\u000b0(k!);\ngt(0)\n\u000b\u000b0(k!) =(1+~H0(k))\u000b\u000b0\n2\u00141\u0000nf(\u000fTI\nk)\n!+!F\u0000!TI\nk+i\u0011+nf(\u000fTI\nk)\n!+!F\u0000!TI\nk\u0000i\u0011\u0015\n+(1\u0000~H0(k))\u000b\u000b0\n2\u0014\u0016nf(\u000fTI\nk)\n!+!F+!TI\nk+i\u0011+1\u0000\u0016nf(\u000fTI\nk)\n!+!F+!TI\nk\u0000i\u0011\u0015\n;\ng~t(0)\n\u000b\u000b0(xt;x0t0) =\u0000i~Th H0\u000b(xt) y\nH0\u000b0(x0t0)i\u00160=g~t(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ng~t(0)\n\u000b\u000b0(k!);\ng~t(0)\n\u000b\u000b0(k!) =(1+~H0(k))\u000b\u000b0\n2\u0014\u0000nf(\u000fTI\nk)\n!+!F\u0000!TI\nk+i\u0011+nf(\u000fTI\nk)\u00001\n!+!F\u0000!TI\nk\u0000i\u0011\u0015\n+(1\u0000~H0(k))\u000b\u000b0\n2\u0014\u0016nf(\u000fTI\nk)\u00001\n!+!F+!TI\nk+i\u0011+\u0000\u0016nf(\u000fTI\nk)\n!+!F+!TI\nk\u0000i\u0011\u0015\n;\ngr(0)\n\u000b\u000b0(xt;x0t0) =\u0000i\u0012(t\u0000t0)\u0001\u001a(0)\n\u000b\u000b0(xt;x0t0) =gr(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ngr(0)\n\u000b\u000b0(k!);\nga(0)\n\u000b\u000b0(xt;x0t0) =i\u0012(t0\u0000t)\u0001\u001a(0)\n\u000b\u000b0(xt;x0t0) =ga(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\nga(0)\n\u000b\u000b0(k!);\ngr(0)\n\u000b\u000b0(k!) =Zd\u0014!\n2\u0019\u001a(0)\n\u000b\u000b0(k\u0014!)\n!\u0000\u0014!+i\u0011ga(0)\n\u000b\u000b0(k!) =Zd\u0014!\n2\u0019\u001a(0)\n\u000b\u000b0(k\u0014!)\n!\u0000\u0014!\u0000i\u0011; (A12)\nwhereh\u0001\u0001\u0001i \u00140denotes the thermal average taken by the den-\nsity matrix\u001aGC(\u0014H0;\f;\u000f F)with \u0014H0=\u0016HTI\n0: \u0011is a positive\ninﬁnitesimal. The symbols Tand~Tare the time-ordering and\nanti-time-ordering operators, respectively. For the step func-\ntion\u0012(t\u0000t0)we have used \u0012(t\u0000t0) =\u0000Rd!\n2\u0019ie\u0000i!(t\u0000t0)\n!+i\u0011.\nThe functions nf(\u000f)and \u0016nf(\u000f)are given by nf(\u000f) =\u0000\n1 +e\f(\u000f\u0000\u000fF)\u0001\u00001and\u0016nf(\u000f) =\u0000\n1 +e\f(\u000f+\u000fF)\u0001\u00001.nf(\u000f)rep-\nresents the Fermi-Dirac distribution function for the energy\n\u000fwith the chemical potential \u0016which we take to be equalto\u000fF:\u0016nf(\u000f)is the one with the chemical potential equal to\n\u0000\u000fF. They are given by the thermal average of TI-surface-\nstate ﬁeld operators as nf(\u000fTI(k)) =hay(k)a(k)i\u00160= 1\u0000\nhb(+)y(k)b(+)(k)i\u00160and \u0016nf(\u000fTI(k)) =hb(\u0000)y(k)b(\u0000)(k)i\u00160:\nThe matrix ~H0is given by Eq. (27) or\n~H0(k) = \n0\u0000i(kx\u0000iky)\nki(kx+iky)\nk0!\n:\nThe spectral functions \u001a(0)\n\u000b\u000b0(xt;x0t0)and\u001a(0)\n\u000b\u000b0(k!)are\n\u001a(0)\n\u000b\u000b0(xt;x0t0) =hf H0\u000b(xt); y\nH0\u000b0(x0t0)gi\u00160=\u001a(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\n\u001a(0)\n\u000b\u000b0(k!);\n\u001a(0)\n\u000b\u000b0(k!) =\u0019h\n(1+~H0(k))\u000b\u000b0\u000e(!+!F\u0000!TI\nk) + (1\u0000~H0(k))\u000b\u000b0\u000e(!+!F+!TI\nk)i\n; (A13)\nwherefgin the above ﬁrst equation denotes the anticommu-\ntator:fX;Yg=XY+YX.Besides time-ordered, anti-time-ordered, retarded, and ad-\nvanced components, there are lesser and greater components\ndeﬁned by16\ng<(0)\n\u000b\u000b0(xt;x0t0) =ih y\nH0\u000b0(x0t0) H0\u000b(xt)i\u00160=g<(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ng<(0)\n\u000b\u000b0(k!);\ng<(0)\n\u000b\u000b0(k!) =i\u0019f(~!)h\n\u000e(!\u0000!TI\nk+!F)(1+~H0(k))\u000b\u000b0+\u000e(!+!TI\nk+!F)(1\u0000~H0(k))\u000b\u000b0i\n;\ng>(0)\n\u000b\u000b0(xt;x0t0) =\u0000ih H0\u000b(xt) y\nH0\u000b0(x0t0)i\u00160=g>(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ng>(0)\n\u000b\u000b0(k!);\ng>(0)\n\u000b\u000b0(k!) =\u0000i\u0019\u0000\n1\u0000f(~!)\u0001h\n\u000e(!\u0000!TI\nk+!F)(1+~H0(k))\u000b\u000b0+\u000e(!+!TI\nk+!F)(1\u0000~H0(k))\u000b\u000b0i\n; (A14)\nwheref(~!) =\u0000\n1 +e\f~!\u0001\u00001. The lesser and greater components of Green’s functions are related to the time-\nordered, anti-time-ordered, retarded, and advanced compo-\nnents through the relations [62, 65]\ng<(0)\n\u000b\u000b0(xt;x0t0) =gt(0)\n\u000b\u000b0(xt;x0t0)\u0000gr(0)\n\u000b\u000b0(xt;x0t0) =g~t(0)\n\u000b\u000b0(xt;x0t0) +ga(0)\n\u000b\u000b0(xt;x0t0);\ng>(0)\n\u000b\u000b0(xt;x0t0) =gt(0)\n\u000b\u000b0(xt;x0t0)\u0000ga(0)\n\u000b\u000b0(xt;x0t0) =g~t(0)\n\u000b\u000b0(xt;x0t0) +gr(0)\n\u000b\u000b0(xt;x0t0): (A15)\nThe above relation also holds for the Green’s functions in\nthe momentum-frequency representation. Further, from Eq.\n(A14) and the formula1\nz\u0000z0\u0006i\u0011= P\u0010\n1\nz\u0000z0\u0011\n\u0007i\u0019\u000e(z\u0000z0),\nwe have\ng<(0)\n\u000b\u000b0(k!) =f(~!)(ga(0)\n\u000b\u000b0(k!)\u0000gr(0)\n\u000b\u000b0(k!));\ng>(0)\n\u000b\u000b0(k!) =\u0000(1\u0000f(~!))(ga(0)\n\u000b\u000b0(k!)\u0000gr(0)\n\u000b\u000b0(k!)):\n(A16)\n2. Impurity-Averaged Imaginary-Time Green’s Function\nWe now include the impurity-potential effect and derive the\nimpurity-averaged Green’s function for the TI surface state.Let us denote a function described by the coordinates of im-\npurities asF(X1;:::;XNimp):The impurity average is de-\nﬁned by\nhF(X1;:::;XNimp)iimp\nave=ZNimpY\ni=1dXi\nVF(X1;:::;XNimp):\n(A17)\nTo perform this on the Green’s functions of TI surface state,\nwe derive the Dyson’s equation for the imaginary-time (Mat-\nsubara) Green’s functions with regarding the impurity poten-\ntial as the perturbation which is given by Eq. (4). First, let\nus introduce the unperturbed imaginary-time Green’s function\ndeﬁned by [61, 64, 65]\nG(0)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =\u0000\nTM H0\u000b(x\u001cM) y\nH0\u000b0(x0\u001c0\nM)\u000b\n0;GC=G(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM);\n=1\n\f~VX\nkmei\u0000\nk\u0001(x\u0000x0)\u0000i!m(\u001cM\u0000\u001c0\nM)\u0001\nG(0)\nM;\u000b\u000b0(k;i!m);\nG(0)\nM;\u000b\u000b0(k;i!m) =Zd!0\n2\u0019\u001a(0)\n\u000b\u000b0(k!0)\ni!m\u0000!0: (A18)\nHere\u001cM;\u001c0\nMare the imaginary times and TMrepresents\nthe imaginary-time ordering. The fermionic imaginary-time\nGreen’s functionG(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM)in the above equa-\ntion is anti-periodic with respect to the the imaginary time:G(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM) =\u0000G(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM\u0006\f~).\nCorrespondingly, the Matsubara frequency !mis given by\n!m= (2m+ 1)\u0019=(\f~)and the Fourier transform of the\nG(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM)for the temporal component be-17\ncomesG(0)\nM;\u000b\u000b0(x\u0000x0;i!m) =R\f~\n0ei!m(\u001cM\u0000\u001c0\nM)G(0)\nM;\u000b\u000b0(x\u0000\nx0;\u001cM\u0000\u001c0\nM)d(\u001cM\u0000\u001c0\nM).\u001a(0)\n\u000b\u000b0(k!)is the spectral function\ndeﬁned in Eq. (A13). By performing the analytic continua-\ntioni!m!!+i\u0011onG(0)\nM;\u000b\u000b0(k!)in Eq. (A18), we obtain\nthe retarded Green’s function gr(0)\n\u000b\u000b0(k!)in Eq. (A12). On theother side, we have the advanced Green’s function ga(0)\n\u000b\u000b0(k!)\nin Eq. (A12) by i!m!!\u0000i\u0011.\nNext, the Dyson equation owing to the impurity potential is\ngiven by [64]\nGM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =G0\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM)\n+Z\f~\n0d\u001cM\n1Z\nd2x1G0\nM;\u000b\u000b0\n1(x\u001cM;x1\u001cM\n1)Himp\n\u000b0\n1\u000b1(x1)GM;\u000b\u000b0(x1\u001cM\n1;x0\u001c0\nM): (A19)\nLet us rewrite the right-hand side of Dyson Equation (A19)\nasGM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =P\nnG(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM)whereG(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM)is in then-th order of impurity potential\nHimp:Its form is represented as\nG(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =Z\f~\n0d\u001cM\n1\u0001\u0001\u0001d\u001cM\nnZ\nd2x1\u0001\u0001\u0001d2xnHimp\n\u000b0\n1\u000b1(x1)\u0001\u0001\u0001Himp\n\u000b0n\u000bn(xn)\n\u0002G0\nM;\u000b\u000b0\nn(x\u0000xn;\u001cM\u0000\u001cM\nn)G0\nM;\u000bn\u000b0\nn\u00001(xn\u0000xn\u00001;\u001cM\nn\u0000\u001cM\nn\u00001)\u0001\u0001\u0001G0\nM;\u000b1\u000b0(x1\u0000x0;\u001cM\n1\u0000\u001c0\nM);\n(A20)\nwhere we have used G0\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =G0\nM;\u000b\u000b0(x\u0000\nx0;\u001cM\u0000\u001c0\nM). Let us perform the Fourier transfor-\nmations on the Green’s function G(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM).\nSince the impurity potential is time independent, theGreen’s function G(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM)is described as\nG(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =G(n)\nM;\u000b\u000b0(x;x0;\u001cM\u0000\u001c0\nM):Then the\nFourier transformation is given as G(n)\nM;\u000b\u000b0(x;x0;\u001cM\u0000\u001c0\nM) =\n(\f~V)\u00001P\nmkk0G(n)\nM;\u000b\u000b0(k;k0;i!m)ei(k\u0001x\u0000k0\u0001x0\u0000!m(\u001cM\u0000\u001c0\nM)):\nThe formula ofG(n)\nM;\u000b\u000b0(k;k0;i!m)is represented as\nG(n)\nM;\u000b\u000b0(k;k0;i!m) =1\nVnX\nk1;:::;kn\u00001h\nG(0)\nM;\u000b\u000bn(k;i!m)vimp(k\u0000kn\u00001)ih\nG(0)\nM;\u000bn\u000bn\u00001(kn\u00001;i!m)vimp(kn\u00001\u0000kn\u00002)i\n\u0001\u0001\u0001h\nG(0)\nM;\u000b3\u000b2(k2;i!m)vimp(k2\u0000k1)ih\nG(0)\nM;\u000b2\u000b2(k1;i!m)vimp(k1\u0000k0)i\nG(0)\nM;\u000b1\u000b0(k0;i!m)\n\u0002\u001aimp(k\u0000kn\u00001)\u001aimp(kn\u00001\u0000kn\u00002)\u0001\u0001\u0001\u001aimp(k3\u0000k2)\u001aimp(k2\u0000k1)\u001aimp(k1\u0000k0); (A21)\nwhere we have used (\f~)\u00001R\f~\n0d\u001cexp\u0000\ni(!m\u0000!m0)\u001c\u0001\n=\n\u000em;m0. In the above equation, all the Matsubara frequencies of\nn+1unperturbed Green’s functions in the right-hand side are\nequivalent due to time independence of impurity potential. We\nnow perform the impurity average on Eq. (A21) by assuming\nthat the total number of impurities Nimpis sufﬁciently large.We use the formulas, for instance [65],\nh\u001aimp(k\u0000k0)iimp\nave=Nimp\u000ek;k0;\nh\u001aimp(k)\u001aimp(k0)iimp\nave=Nimp\u000ek+k0;0+N2\nimp\u000ek;0\u000ek0;0:\n(A22)\nForn= 0;1;2, we have18\nhG(0)\nM;\u000b\u000b0(k;k0;i!m)iimp\nave=\u000ek;k0G(0)\nM;\u000b\u000b0(k;i!m);\nhG(1)\nM;\u000b\u000b0(k;k0;i!m)iimp\nave= 0;\nhG(2)\nM;\u000b\u000b0(k;k0;i!m)iimp\nave=\u000ek;k0G(0)\nM;\u000b\u000b1(k;i!m)\u0012\nnimpZd2q\n(2\u0019)2jvimp(q\u0000k)j2G(0)\nM;\u000b1\u000b0(q;i!m)\u0013\nG(0)\nM;\u000b0\u000b0(k0;i!m)\n\u0011\u000ek;k0\u0016G(2)\nM;\u000b\u000b0(k;i!m); (A23)\nwherenimp =Nimp=Vis the number density of impu-\nrities. To derive Eq. (A23), we have taken the contin-\nuum limitV\u00001P\nq!R\nd2q=(2\u0019)2. Further, we used\nv\u0003\nimp(q) =vimp(\u0000q)andvimp(0) = 0:The impurity-\naveraged Green’s function hG(n)\nM;\u000b\u000b0(k;k0;i!m)iimp\navein Eq.\n(A23) is diagonal with respect to momentum. Similarly,\nthe impurity-averaged Green’s function is diagonal in mo-\nmentum for n\u00153[64, 65]. As a result, when the impurity\naverage is taken on Green’s function, it restores the trans-\nlational symmetry. Let us express hGM;\u000b\u000b0(k;k0;i!m)iimp\nave\nandhG(n)\nM;\u000b\u000b0(k;k0;i!m)iimp\nave as \u0016GM;\u000b\u000b0(k;i!m)\u000ek;k0and \u0016G(n)\nM;\u000b\u000b0(k;i!m)\u000ek;k0, respectively. Their Fourier\ntransforms are given by hGM;\u000b\u000b0(x;x0;i!m)iimp\nave\u0011\n\u0016GM;\u000b\u000b0(x\u0000x0;i!m) =V\u00001P\nkeik\u0001(x\u0000x0)\u0016GM;\u000b\u000b0(k;i!m)\nandhG(n)\nM;\u000b\u000b0(x;x0;i!m)iimp\nave\u0011\u0016G(n)\nM;\u000b\u000b0(x\u0000x0;i!m) =\nV\u00001P\nkeik\u0001(x\u0000x0)\u0016G(n)\nM;\u000b\u000b0(k;i!m).\nNext, we reorganize the perturbative expansionP\nn\u0016G(n)\nM;\u000b\u000b0(k;i!m)by representing it as the sum of all\nirreducible Feynman diagrams. Let us denote the associated\nself-energy as \u0006imp(k;i!m)\u000b\u000b0:Then, the Dyson equation\nfor the impurity-averaged Green’s function \u0016GM;\u000b\u000b0(k;i!m)\nis expressed as\n\u0016GM;\u000b\u000b0(k;i!m) =G(0)\nM;\u000b\u000b0(k;i!m) +G(0)\nM;\u000b\u000b2(k;i!m)\u0006imp(k;i!m)\u000b2\u000b1\u0016GM;\u000b1\u000b0(k;i!m): (A24)\nTo evaluate the self-energy \u0006imp(k;i!m), we take the impu-\nrity potential as Vimp(x\u0000Ximp\ni) =v0\u000e(x\u0000Ximp\ni)with\nv0a constant and use the ﬁrst-Born approximation. Then,\nwe havevimp(q) =v0:Since the Fermi energy \u000fFis posi-\ntive, the term in the unperturbed Green’s function which con-tributes dominantly to this evaluation is 1=(i!m+!F\u0000!TI\nk).\nTherefore, when we perform the momentum integral for eval-\nuating \u0006imp(k;i!m), we just retain 1=(i!m+!F\u0000!TI\nk).\nLet us write the self-energy in the ﬁrst-Born approximation as\n\u0006imp\n1BA(k;i!m)\u000b\u000b0. It is evaluated as\n\u0006imp\n1BA(k;i!m)\u000b\u000b0=nimp\n~Zd2q\n(2\u0019)2jvimp(q\u0000k)j2G(0)\nM;\u000b\u000b0(k;i!m) =1\n2nimpv2\n0Z1\n\u0000\u000fFd\u0018~N(\u0018)1\ni\u000fm\u0000\u0018\n\u0019\u00001\n2~N(0)nimpv2\n0Z1\n\u00001d\u0018\u0018+i\u000fm\n\u00182+\u000f2m=\u0000i\n2\u0019~N(0)nimpv2\n0sgn(!m)\u0011\u0000i\n2\u001crel\nTIsgn(!m)\u000e\u000b;\u000b0; (A25)\nwhere\u000fm=~!m. The quantity ~NTI(\u0018)is the density of\nstates per volume of the TI surface state measured from the\nFermi energy. It is given by ~NTI(\u0018) = (\u000fF+\u0018)=\u0000\n2\u0019(~vF)2\u0001\n:\nFor going from the ﬁrst to second line of Eq. (A25), we have\nmade an approximation such that the density of state ~NTI(\u0018)\nincluded in the integrand can be set with ~NTI(0). This is\nbecause we can consider that the energy state in the vicin-\nity of Fermi level dominantly contributes to the self energy\n\u0006imp\n1BA(k;i!m)\u000b\u000b0:jqj;jkj'kFand ~N(\u0018)'~N(0). Fur-ther, we have replaced the lower limit \u0000\u000fFwith\u00001 since\nwe consider that the number of electrons included in the sur-\nface with its area Vis large enough and the number density\nof TI surface nTI\n2Dcan be taken as large. The relation between\n\u000fFandnTI\n2Dis given by\u000fF=~vFq\n4\u0019nTI\n2D, and hence, we\ntake\u000fF!1:The time\u001crel\nTIis the relaxation time of TI sur-\nface state owing to the impurity effect. We now derive the\nimpurity-averaged green’s function \u0016GM;\u000b\u000b0(k;i!m). From\nEqs. (A24) and (A25) we obtain19\n\u0016GM;\u000b\u000b0(k;i!m) =\u0014\u0010\nG(0)\nM(k;i!m)\u0011\u00001\n+i\n2\u001crel\nTIsgn(!m)\u00011\u0015\u00001\n\u000b\u000b0; (A26)\nwhere\n\u0010\nG(0)\nM(k;i!m)\u0011\u00001\n\u000b\u000b0= (!m+!F)1\u000b\u000b0+!TI\nk~H0;\u000b\u000b0:\n(A27)\nBy performing the analytic continuation i!m!!+\nisgn(!m)\u0011, consequently, we obtain the retarded and ad-\nvanced components of impurity-averaged Green’s functions\n\u0016gr\n\u000b\u000b0(k;!)and\u0016ga\n\u000b\u000b0(k;!)in Eq. (25). The retarded compo-\nnent is obtained for sgn(!m)>0while we get the advancedcomponent for sgn(!m)<0.\n3. Magnon Green’s Functions\nWe present the real-time magnon Green’s functions. First\nlet us show them without the damping effect. Like given in\nEq. (A12), the time-ordered, anti-time-ordered, retarded, and\nadvanced components of magnon Green’s functions in the in-\nteraction picture are\nDt(pt;p0t0) =\u0000iThaH0(pt)ay\nH0(p0t0)i0=\u000ep;p0Dt(p;t\u0000t0) =Zd!\n2\u0019e\u0000i!(t\u0000t0)\u000ep;p0Dt(p!);\nD~t(pt;p0t0) =\u0000i~ThaH0(pt)ay\nH0(p0t0)i0=\u000ep;p0D~t(p;t\u0000t0) =Zd!\n2\u0019e\u0000i!(t\u0000t0)\u000ep;p0D~t(p!);\nDt(p!) =1 +nb(\u000fFM\np)\n!\u0000!FMp+i\u0011\u0000nb(\u000fFM\np)\n!\u0000!FMp\u0000i\u0011; D~t(p!) =nb(\u000fFM\np)\n!\u0000!FMp+i\u0011\u00001 +nb(\u000fFM\np)\n!\u0000!FMp\u0000i\u0011;\nDr(pt;p0t0) =\u0000i\u0012(t\u0000t0)\u0001\u001a(0)(pt;p0t0) =\u000ep;p0Dr(p;t\u0000t0) =\u000ep;p0Zd!\n2\u0019e\u0000i!(t\u0000t0)Dr(p!);\nDa(pt;p0t0) =i\u0012(t0\u0000t)\u0001\u001a(0)(pt;p0t0) =\u000ep;p0Da(p;t\u0000t0) =\u000ep;p0Zd!\n2\u0019e\u0000i!(t\u0000t0)Da(p!);\n\u001a(0)(pt;p0t0) =\u000ep;p0e\u0000i!FM\np(t\u0000t0); Dr(p!) =1\n!\u0000!FMp+i\u0011; Da(p!) =1\n!\u0000!FMp\u0000i\u0011; (A28)\nwhereaH0(pt) =eiH0t=~a(p)e\u0000iH0t=~anday\nH0(pt) =eiH0t=~ay(p)e\u0000iH0t=~.nb(\u000f) =\u0000\ne\f\u000f\u00001\u0001\u00001is the Bose-\nEinstein distribution function. The lesser and greater Green’s\nfunctions are given by\nD>(pt;p0t0) =\u0000ihaH0(pt)ay\nH0(p0t0)i0=\u000ep;p0D>(p;t\u0000t0) =Zd!\n2\u0019e\u0000i!(t\u0000t0)\u000ep;p0D>(p!);\nD<(pt;p0t0) =\u0000ihay\nH0(p0t0)aH0(pt)i0=\u000ep;p0D<(p;t\u0000t0) =Zd!\n2\u0019e\u0000i!(t\u0000t0)\u000ep;p0D<(p!);\nD>(p!) =\u00002\u0019i\u000e(!\u0000!FM\np)\u0000\n1 +nb(\u000fFM\np)\u0001\n; D<(p!) =\u00002\u0019i\u000e(!\u0000!FM\np)nb(\u000fFM\np): (A29)\nThe magnon Green’s functions presented above satisfy ex-\nactly the same relations given in Eq. (A15). Further, from\nEqs. (A28) and (A29), we can verify that with similar to Eq.(A16) these components satisfy the relations\nD<(p!) =\u0000nb(~!) (Da(p!)\u0000Dr(p!));\nD>(p!) =\u0000\u0000\n1 +nb(~!)\u0001\n(Da(p!)\u0000Dr(p!)):(A30)20\nNext, we show the magnon Green’s functions including the\ndamping effect. This is obtained by solving the Landau-\nLifshitz-Gilbert equation\ndSi\ndt=\r\u0000\nB0+Bext(t)\u0001\n\u0002Si\u0000\u000b\nS0\u0012\nSi\u0002dSi\ndt\u0013\n;\n(A31)\nwhereB0= (0;B0;0)withB0a constant. For\nBext(t), we take the same magnetic-ﬁeld con-\nﬁguration as we did in Sec. II: Bext(t) =\nBext\u0000\nsin\u0000\nsgn(B0)\u0001!extt\u0001\n;0;cos\u0000\nsgn(B0)\u0001!extt\u0001\u0001\n.\nThe solution is given in the form Sy\ni=\u0000S0sgn(B0)and\nS\u0000\ni=Sz\ni\u0000iSx\ni=\u0016S\u0000\nie\u0000isgn(B0)!exttwhere \u0016S\u0000\niis a complex\nconstant. The demagnetizing coefﬁcient is going to be\nexcluded. By introducing Bext;\u0000=Bext;z\u0000iBext;xand\nthe magnetic susceptibility as \u001fmag, for sgn(B0)>0we\nre-expressS\u0000\niasS\u0000\ni=\u001fmagBext;\u0000[15]. As a result, we\nobtain\n\u001fmag=\rS0\n!ext\u0000!FM\n0+i\u000b!ext; (A32)\nwhere!FM\n0=\rjB0j, which is the Zeeman gap of magnon.\nWith eliminating the factor \rS0in Eq. (A32), we identify the\nmagnetic susceptibility \u001fmagwith the retarded Green’s func-\ntion\u0016Dr(0;!)in Eq. (26). On the other hand, for sgn(B0)<0\nwe identify the retarded function with the response (magnetic\nsusceptibility) of S+\ni=Sz\ni+iSx\nitoB+\ni=Bz\ni+iBx\niand\nthis is equal to \u001fmagin Eq. (A32). The advanced component\nis given by the complex conjugate of retarded component.\nAppendix B: Keldysh Green’s Functions\nIn this section, ﬁrst we present the formalism for the\nKeldysh Green’s function. Next, we show how the Keldysh\nGreen’s function is related to the real-time Green’s func-\ntion via the real-time projection. Further, with presenting\nsome useful formulas obtained by the the real-time projection,\nwe demonstrate the derivation of impurity-averaged real-time\nGreen’s functions.\n1. Real-Time Projection\nAs discussed in subsec. II B, our starting point is the full\nlesser Green’s function of TI surface state in Eq. (14). We\nrewrite this with the Keldysh Green’s function given by Eq.\n(15) or\niGC;\u000b\u000b0(x\u001c;x0\u001c0) =D\nTC\u0002\nUexc\nCUext\nC y\nH0\u000b0(x0\u001c0) H0\u000b(x\u001c)\u0003E\n0:\nThe time-evolution operators Uexc\nCandUext\nCin the above\nequation are deﬁned by Eq. (16) or\nUexc\nC= exp\u0012\n\u0000i\n~Z\nCd\u0014\u001cVexc\nH0(\u0014\u001c)\u0013\n;\nUext\nC= exp\u0012\n\u0000i\n~Z\nCd~\u001cHext\nH0(~\u001c)\u0013\n:The perturbative calculation is performed by expanding Uexc\nC\nandUext\nCwith respect to Vexc\nH0(\u0014\u001c)andHext\nH0(~\u001c), respectively.\nThen by taking thermal average on them, these perturbative\nexpansions are described by the unperturbed Keldysh Green’s\nfunctions given by Eqs. (23) and (24) or\niG0\nC;\u000b\u000b0(x\u001c;x0\u001c0) =D\nTC\u0002\n H0\u000b(x\u001c) y\nH0\u000b0(x0\u001c0)\u0003E\n0;\niD0\nC(q\u001c;q0\u001c0) =D\nTC\u0002\naH0(q\u001c)ay\nH0(q0\u001c0)\u0003E\n0:\nWe perform the real-time projection to the above Keldysh\nGreen’s functions in order to calculate the physical observ-\nables. We do this by classifying whether the contour time \u001c\nbelongs to the path C\u0000orC+while\u001c0toC\u0000orC+(see Fig.\n3). We have four different conﬁgurations. To represent this\nsituation clearly, let us introduce a two-by-two-matrix Green’s\nfunction (Schwinger-Keldysh Green’s function) [62]\n^G\u000b\u000b0(x\u001c;x0\u001c0) =\u0012^G\u0000\u0000\n\u000b\u000b0(xt\u0000;x0t0\u0000)^G\u0000+\n\u000b\u000b0(xt\u0000;x0t0+)\n^G+\u0000\n\u000b\u000b0(xt+;x0t0\u0000)^G++\n\u000b\u000b0(xt+;x0t0+)\u0013\n;\n(B1)\nwheret\u0006andt0\u0006are both real times. The compo-\nnent ^G\u0016\u00160\n\u000b\u000b0(xt\u0016;x0t0\u00160) (\u0016;\u00160=\u0007)is representing that\nthe contour time \u001c=t\u0016belongs to the contour C\u0016\nwhile\u001c0=t0\u00160belongs to C\u00160. The components\n^G\u0000\u0000\n\u000b\u000b0(xt\u0000;x0t0\u0000);^G\u0000+\n\u000b\u000b0(xt\u0000;x0t0+);^G+\u0000\n\u000b\u000b0(xt+;x0t0\u0000);and\n^G++\n\u000b\u000b0(xt+;x0t0+)are equivalent to time-ordered, lesser,\ngreater, and anti-time-ordered components, respectively.\nSimilarly, we introduce the Schwinger-Keldysh Green’s\nfunction of magnon given by\n^D(q\u001c;q0\u001c0) =\u0012^D\u0000\u0000(qt\u0000;q0t0\u0000)^D\u0000+(qt\u0000;q0t0+)\n^D+\u0000(qt+;q0t0\u0000)^D++(qt+;q0t0+)\u0013\n;\n(B2)\nwhere the components ^D\u0000\u0000(qt\u0000;q0t0\u0000);^D\u0000+(qt\u0000;q0t0+),\n^D+\u0000(qt+;q0t0\u0000);and ^D++(qt+;q0t0+)are equivalent to\ntime-ordered, lesser, greater, and anti-time-ordered Green’s\nfunctions, respectively.\nIn the following, let us show some examples of calculation\nfor the real-time projection on the Keldysh Green’s functions.\nWe will just write the time arguments of functions and omit\nthe arguments of spatial coordinate or momentum since what\nwe want to demonstrate here is the calculation for real-time\nprojection and integrals of real-time variables. We perform\nthe integral along the contour Cby decomposing it into C\u0000\nandC+and rewrite them by the real-time variables.\nFor practice, ﬁrst let us show the simplest example of inte-\ngral along the contour Cgiven by a single contour-time vari-\nable\u001c1. It has a form\nf(\u001c;\u001c0) =Z\nCd\u001c1g(\u001c;\u001c1)h(\u001c1;\u001c0)\n=Z+1\n\u00001\u001c\u00161\u00161\nzdt\u00161\n1f(t;t\u00161\n1)g(t\u00161\n1;t0); (B3)21\nwhere\n\u001c\u00161\u00160\n1z =\u0012\n\u001c\u0000\u0000\nz\u001c\u0000+\nz\n\u001c+\u0000\nz\u001c++\nz\u0013\n=\u0012\n1 0\n0\u00001\u0013\n; (B4)\nandt\u00161\n1is the real-time variable. Via the real-time projec-\ntion, let us rewrite the function f(\u001c;\u001c0)asf\u0017\u0016\u00160(t;t0). Heretandt0are real-time variables corresponding to the real-time\nprojection of the contour times \u001cand\u001c0, respectively. The\nsuperscript\u0017\u0016\u00160= t;<;>; ~twith\u0016;\u00160=\u0006. It describes\nthe situation such that \u001c2C\u0016while\u001c02C\u00160. For instance,\nwhen\u001c2C\u0000while\u001c02C+the function f(\u001c;\u001c0)becomes\nf<(t;t0). In the following, we list the four cases of f(\u001c;\u001c0)\ngiven as\nft(t;t0) =Z+1\n\u00001dt1\u0000\ngt(t;t1)ht(t1;t0)\u0000g<(t;t1)h>(t1;t0)\u0001\n;\nf<(t;t0) =Z+1\n\u00001dt1\u0000\ngt(t;t1)h<(t1;t0)\u0000g<(t;t1)h~t(t1;t0)\u0001\n;\nf>(t;t0) =Z+1\n\u00001dt1\u0000\ng>(t;t1)ht(t1;t0)\u0000g~t(t;t1)h>(t1;t0)\u0001\n;\nf~t(t;t0) =Z+1\n\u00001dt1\u0000\ng>(t;t1)h<(t1;t0)\u0000g~t(t;t1)h~t(t1;t0)\u0001\n: (B5)\nBy using the relations in Eq. (A15), Eq. (B5) can be re- described by the lesser, greater, retarded, and advanced com-\nponents as\nf<(t;t0) =Z+1\n\u00001dt1\u0000\ng<(t;t1)ha(t1;t0) +gr(t;t1)h<(t1;t0)\u0001\n;\nf>(t;t0) =Z+1\n\u00001dt1\u0000\ng>(t;t1)ha(t1;t0) +gr(t;t1)h>(t1;t0)\u0001\n;\nfr(t;t0) =Z+1\n\u00001dt1gr(t;t1)hr(t1;t0); fa(t;t0) =Z+1\n\u00001dt1ga(t;t1)ha(t1;t0): (B6)\nNext, we present an example of temporal function represented\nby two contour-time variables \u001c1and\u001c2given by\nf(\u001c;\u001c0) =Z\nCd\u001c1d\u001c2g(\u001c;\u001c2)h(\u001c2;\u001c1)l(\u001c1;\u001c0): (B7)Like we did in Eq. (B5), we perform the real-time projection\non\u001c1and\u001c2and rewrite them as t1andt2, respectively. As a\nresult, we have\nf<(t;t0) =Z+1\n\u00001dt1dt2\u0000\ng<(t;t2)ha(t2;t1)la(t1;t0) +gr(t;t2)h<(t2;t1)la(t1;t0) +gr(t;t2)hr(t2;t1)l<(t1;t0)\u0001\n;\nf<(t;t0) =Z+1\n\u00001dt1dt2\u0000\ng>(t;t2)ha(t2;t1)la(t1;t0) +gr(t;t2)h>(t2;t1)la(t1;t0) +gr(t;t2)hr(t2;t1)l>(t1;t0)\u0001\n;\nfr(t;t0) =Z+1\n\u00001dt1dt2(gr(t;t2)hr(t2;t1)lr(t1;t0)); fa(t;t0) =Z+1\n\u00001dt1dt2(ga(t;t2)ha(t2;t1)la(t1;t0)):(B8)\nEqs. (B6) and (B8) are called Langreth rules [62, 63]. As a last example, we demonstrate a calculation represented22\nby three contour-time variables ~\u001c1;~\u001c2;and\u001c1. The integral\nwhich we calculate is\nf(\u001c;\u001c0) =Z\nCd~\u001c1d~\u001c2d\u001c1l(~\u001c1;\u001c1)m(\u001c1;~\u001c2)n(\u001c;\u001c1)o(\u001c1;\u001c0):\n(B9)With using the real-time variables ~t1,~t2, andt1corresponding\nto~\u001c1;~\u001c2;and\u001c1, respectively, the right-hand side of Eq. (B9)\nis rewritten as\nf<(t;t0) =Z+1\n\u00001d~t1d~t2dt1la(~t1;t1)mr(t1;~t2)\u0000\nn<(t;t1)oa(t1;t0) +nr(t;t1)o<(t1;t0)\u0001\n;\nf>(t;t0) =Z+1\n\u00001d~t1d~t2dt1la(~t1;t1)mr(t1;~t2)\u0000\nn>(t;t1)oa(t1;t0) +nr(t;t1)o>(t1;t0)\u0001\n;\nfr(t;t0) =Z+1\n\u00001d~t1d~t2dt1la(~t1;t1)mr(t1;~t2)nr(t;t1)or(t1;t0);\nfa(t;t0) =Z+1\n\u00001d~t1d~t2dt1la(~t1;t1)mr(t1;~t2)na(t;t1)oa(t1;t0): (B10)\n2. Impurity-Averaged Real-Time Green’s Function\nLet us apply the Keldysh Green’s function formalism to de-\nrive the retarded, advanced, lesser, and greater components ofimpurity-averaged real-time Green’s functions.\nThe Dyson equation for the Keldysh Green’s function of TI\nsurface state due to the non-magnetic impurity effect is given\nby [16, 62, 63]\nGC;\u000b\u000b0(x\u001c;x0\u001c0) =G0\nC;\u000b\u000b0(x\u001c;x0\u001c0) +Z\nCd\u001c1Z\nd2x1G0\nC;\u000b\u000b0\n1(x\u001c;x1\u001c1)Himp\n\u000b0\n1\u000b1(x1)GC;\u000b1\u000b0(x1\u001c1;x0\u001c0);\n=G0\nC;\u000b\u000b0(x\u001c;x0\u001c0) +Z\nCd\u001c1Z\nd2x1GC;\u000b\u000b0\n1(x\u001c;x1\u001c1)Himp\n\u000b0\n1\u000b1(x1)G0\nC;\u000b1\u000b0(x1\u001c1;x0\u001c0): (B11)\nWe use the formulas given in Eq. (B6) and perform the real-\ntime projection on the contour times \u001c;\u001c0and\u001c1in Eq. (B11).Then, we obtain the Dyson equations for retarded, advanced,\nlesser, and greater Green’s functions given by23\ngr\n\u000b\u000b0(xt;x0t0) =gr(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)gr(0)\n\u000b\u000b0\n1(xt;x1t1)gr\n\u000b1\u000b0(x1t1;x0t0)\n=gr(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)gr\n\u000b\u000b0\n1(xt;x1t1)gr(0)\n\u000b1\u000b0(x1t1;x0t0)\nga\n\u000b\u000b0(xt;x0t0) =ga(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)ga(0)\n\u000b\u000b0\n1(xt;x1t1)ga\n\u000b1\u000b0(x1t1;x0t0)\n=ga(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)ga\n\u000b\u000b0\n1(xt;x1t1)ga(0)\n\u000b1\u000b0(x1t1;x0t0)\ng<\n\u000b\u000b0(xt;x0t0) =g<(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)\u0010\ng<(0)\n\u000b\u000b0\n1(xt;x1t1)ga\n\u000b1\u000b0(x1t1;x0t0) +gr(0)\n\u000b\u000b0\n1(xt;x1t1)g<\n\u000b1\u000b0(x1t1;x0t0)\u0011\n=g<(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)\u0010\ng<\n\u000b\u000b0\n1(xt;x1t1)ga(0)\n\u000b1\u000b0(x1t1;x0t0) +gr\n\u000b\u000b0\n1(xt;x1t1)g<(0)\n\u000b1\u000b0(x1t1;x0t0)\u0011\ng>\n\u000b\u000b0(xt;x0t0) =g>(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)\u0010\ng>(0)\n\u000b\u000b0\n1(xt;x1t1)ga\n\u000b1\u000b0(x1t1;x0t0) +gr(0)\n\u000b\u000b0\n1(xt;x1t1)g>\n\u000b1\u000b0(x1t1;x0t0)\u0011\n=g>(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)\u0010\ng>\n\u000b\u000b0\n1(xt;x1t1)ga(0)\n\u000b1\u000b0(x1t1;x0t0) +gr\n\u000b\u000b0\n1(xt;x1t1)g>(0)\n\u000b1\u000b0(x1t1;x0t0)\u0011\n;\n(B12)\nwhere we have used the relations gt=gr+g<=ga+g>\nandg~t=\u0000gr+g>=\u0000ga+g<as given in Eq. (A15).\nBased on Eq. (B12), we derive the impurity-averaged Green’s\nfunctions for all of these four components. Basically, we can\nobtain them by adopting the same argument given in subSec.\nA 2. At ﬁrst, let us focus on the retarded component. Weexpressgr\n\u000b\u000b0(xt;x0t0)in the perturbative expansion form as\ngr\n\u000b\u000b0(xt;x0t0) =P\nngr(n)\n\u000b\u000b0(xt;x0t0)wheregr(n)\n\u000b\u000b0(xt;x0t0)is\nin then-th order ofHimp. Then we perform the Fourier trans-\nformations\ngr\n\u000b\u000b0(xt;x0t0) =1\nVX\nkk0Zd!d!0\n(2\u0019)2ei((kx\u0000k0x0)\u0000(!t\u0000!0t0))gr\n\u000b\u000b0(k!;k0!0);\ngr(n)\n\u000b\u000b0(xt;x0t0) =1\nVX\nkk0Zd!d!0\n(2\u0019)2ei((kx\u0000k0x0)\u0000(!t\u0000!0t0))gr(n)\n\u000b\u000b0(k!;k0!0): (B13)\nWe take the impurity average on gr\n\u000b\u000b0(k!;k0!0)as well as\ngr(n)\n\u000b\u000b0(k!;k0!0). Let us denote them as hgr\n\u000b\u000b0(k!;k0!0)iimp\nave\nandhgr(n)\n\u000b\u000b0(k!;k0!0)iimp\nave, respectively. As a result, both of\nthem become diagonal in momentum and frequency repre-sented as\nhgr\n\u000b\u000b0(k!;k0!0)iimp\nave= \u0016gr\n\u000b\u000b0(k!)\u000ek;k0\u0001(2\u0019\u000e(!\u0000!0));\nhgr(n)\n\u000b\u000b0(k!;k0!0)iimp\nave= \u0016gr(n)\n\u000b\u000b0(k!)\u000ek;k0\u0001(2\u0019\u000e(!\u0000!0)):\n(B14)\nConsequently, the impurity average of gr\n\u000b\u000b0(xt;x0t0)\nis represented in the translational invariant form:\nhgr\n\u000b\u000b0(xt;x0t0)iimp\nave= \u0016gr\n\u000b\u000b0(x\u0000x0;t\u0000t0). The Fourier trans-\nform of impurity-averaged Green’s function \u0016gr\n\u000b\u000b0(k!)is ob-\ntained from the two-point Green’s function \u0016gr\n\u000b\u000b0(x\u0000x0;t\u0000t0)\nas\n\u0016gr\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei(k(x\u0000x0)\u0000!(t\u0000t0))\u0016gr\n\u000b\u000b0(k!): (B15)24\nNext, we re-sum the perturbative expansion \u0016gr\n\u000b\u000b0(k!) =P\nn\u0016gr(n)\n\u000b\u000b0(k!)and express it in terms of the self-energy. Here\nwe take the ﬁrst-Born approximation to evaluate this as we did\nin subSec. A 2. As a result, we obtain\n\u0016gr\n\u000b\u000b0(k!) =gr(0)\n\u000b\u000b0(k!) +gr(0)\n\u000b\u000b2(k!)\u0006r(0)\n\u000b2\u000b1(k!)\u0016gr\n\u000b1\u000b0(k!)\n=gr(0)\n\u000b\u000b0(k!) + \u0016gr\n\u000b\u000b2(k!)\u0006r(0)\n\u000b2\u000b1(k!)gr(0)\n\u000b1\u000b0(k!);\n(B16)\nwhere the self-energy \u0006r(0)\n\u000b2\u000b1(k!)is given by\n\u0006r(0)\n\u000b2\u000b1(k!) =nimp\n~VX\nqjvimp(q\u0000k)j2gr(0)\n\u000b2\u000b1(q!):(B17)Formally, we can solve Eq. (B16) for \u0016gr\n\u000b\u000b0(k!)as\n\u0016gr\n\u000b\u000b0(k!) =\u0014\u0010\ngr(0)(k!)\u0011\u00001\n\u0000\u0006r(0)(k!)\u0015\u00001\n\u000b\u000b1\u0010\ngr(0)(k!)\u0011\u00001\n\u000b\u000b0= (!+!F+i\u0011)1\u000b\u000b0\u0000!TI\nk~H0;\u000b\u000b0:\n(B18)\nNext, let us evaluate the self-energy \u0006r(0)\n\u000b2\u000b1(k!)in Eq. (B17).\nIn order to do this, we only retain the term proportional to\n!+!F\u0000!TI\nq+i\u0011ingr(0)\n\u000b2\u000b1. As a result, we have\n\u0006r(0)\n\u000b2\u000b1(k!) =1\n2nimpv2\n0Z1\n0d\u000fN(\u000f)1\n\u000fF\u0000\u000f+i\u0011=1\n2nimpv2\n0Z1\n\u0000\u000fFd\u0018~N(\u0018)1\n\u0000\u0018+i\u0011\n\u0019\u00001\n2~N(0)nimpv2\n0Z1\n\u00001d\u0018\u0018+i\u0011\n\u00182+\u00112=\u00001\n2\u0019~N(0)nimpv2\n0=\u0000i\n2\u001crel\nTI\u000e\u000b2\u000b1; (B19)\nwhereN(\u000f) =\u000f=\u0000\n2\u0019(~vF)2\u0001\nis the density of states per\nvolume of TI surface state with \u000f=~!.\u0018=\u000f\u0000\u000fFand\n~N(\u0018) = (\u000fF+\u0018)=\u0000\n2\u0019(~vF)2\u0001\n:The time\u001crel\nTIin the above\nequation is the same quantity appearing in Eq. (A25). As we\nderived Eq. (A25), in the above analysis we considered that\nthe quantum tranport phenomena is induced by the electrons\nin the energy state in the vicinity of Fermi surface and the\nnumber density of TI surface to be sufﬁciently large. Thus,\nwe assumejqj;jkj'kF;\u000f\u001c\u000fFand set ~N(\u0018) = ~N(0)with\ntaking\u000fF!1 for the lower limit in the ﬁrst line of above\nequation. From Eqs. (B16) and (B19), we have the formula\nof\u0016gr\n\u000b\u000b0(k!)and it is the same as the one given in Eq. (25).\nBy applying the similar argument, we can derive the advanced\ncomponents of impurtiy-averaged Green’s function \u0016ga\n\u000b\u000b0(k!).\nLike Eq. (B16), we have\n\u0016ga\n\u000b\u000b0(k!) =ga(0)\n\u000b\u000b0(k!) +ga(0)\n\u000b\u000b2(k!)\u0006a(0)\n\u000b2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!)\n=ga(0)\n\u000b\u000b0(k!) + \u0016ga\n\u000b\u000b2(k!)\u0006a(0)\n\u000b2\u000b1(k!)ga(0)\n\u000b1\u000b0(k!);\n(B20)\nwhere\n\u0006a(0)\n\u000b2\u000b1(k!) =nimp\n~VX\nqjvimp(q\u0000k)j2ga(0)\n\u000b2\u000b1(q!):(B21)The self-energy \u0006a(0)\n\u000b2\u000b1(k!)in Eq. (B21) can be evaluated\nby the same analysis which we did exactly for deriving Eq.\n(B19). We have \u0006a(0)\n\u000b2\u000b1(k!) =\u0000\u0006r(0)\n\u000b2\u000b1(k!):We solve Eq.\n(B20) for \u0016ga\n\u000b\u000b0(k!)and obtain\n\u0016ga\n\u000b\u000b0(k!) =\u0014\u0010\nga(0)(k!)\u0011\u00001\n\u0000\u0006a(0)(k!)\u0015\u00001\n\u000b\u000b1\u0010\nga(0)(k!)\u0011\u00001\n\u000b\u000b0= (!+!F\u0000i\u0011)1\u000b\u000b0\u0000!TI\nk~H0;\u000b\u000b0;\n(B22)\nwhich is equal to the advanced Green’s function in Eq. (25).\nLastly, we derive the impurity-averaged lesser Green’s func-\ntion referring to the analysis in [16]. By the similar analysis\nused for deriving Eqs. (B16) or (B20), we obtain\n\u0016g<\n\u000b\u000b0(k!) =g<(0)\n\u000b\u000b0(k!) +g<(0)\n\u000b\u000b0\n2(k!)\u0006a(0)\n\u000b0\n2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!)\n+gr(0)\n\u000b\u000b0\n2(k!)h\n\u0006r(0)\n\u000b0\n2\u000b1(k!)\u0016g<\n\u000b1\u000b0(k!) + \u0006<(0)\n\u000b0\n2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!)i\n;\n(B23)25\nwhere\n\u0006<(0)\n\u000b2\u000b1(k!) =nimp\n~VX\nqjvimp(q\u0000k)j2g<(0)\n\u000b2\u000b1(q!):(B24)From Eqs. (B18), (B22), and (B24), Eq. (B23) is rewritten as\n\u0016g<\n\u000b\u000b0(k!) = \u0016gr\n\u000b\u000b2(k!)\u0006<(0)\n\u000b2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!)\n+ \u0016gr\n\u000b\u000b2(k!)\u0014\u0010\ngr(0)(k!)\u0011\u00001\n\u0001g<(0)(k!)\u0001\u0010\nga(0)(k!)\u0011\u00001\u0015\n\u000b2\u000b1\u0016ga\n\u000b1\u000b0(k!): (B25)\nFirst, let us evaluate the second term in Eq. (B25). From Eqs. (A16), (B18), and (B22) we have\n\u0014\u0010\ngr(0)(k!)\u0011\u00001\n\u0001g<(0)(k!)\u0001\u0010\nga(0)(k!)\u0011\u00001\u0015\n\u000b2\u000b1=f(~!)\u0014\u0010\ngr(0)(k!)\u0011\u00001\n\u0000\u0010\nga(0)(k!)\u0011\u00001\u0015\n\u000b2\u000b1\n= 2i\u0011f(~!)1\u000b2\u000b1: (B26)\nBy taking the limit \u0011!0+, we see that the second term in Eq. (B25) vanishes. Next, let us evaluate the ﬁrst term in Eq.\n(B25) using Eqs. (25) and (A16). This becomes\n\u0016gr\n\u000b\u000b2(k!)\u0006<(0)\n\u000b2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!) =f(~!) 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Ivanov,4,5,2,∗and Franco Nori2,6\n1Institute of Physics, 03028 Kiev, Ukraine\n2CEMS, RIKEN, Saitama, 351-0198, Japan\n3Vernadsky Taurida National University, Simferopol, 95007 Ukraine\n4Institute of Magnetism, 03142, Kiev, Ukraine\n5Taras Shevchenko National University of Kiev, 01601, Ukrai ne\n6Department of Physics, The University of Michigan, Ann Arbo r, MI 48109-1040, USA.\n(Dated: October 17, 2018)\nAbstract\nWe analyze theoretically the novel pathway of ultrafast spi n dynamics for ferromagnets with\nhigh enough single-ion anisotropy (non-Heisenberg ferrom agnets). This longitudinal spin dynamics\nincludesthecoupled oscillations of themodulusofthemagn etization together withthequadrupolar\nspin variables, which are expressed through quantum expect ation values of operators bilinear on\nthe spin components. Even for a simple single-element ferro magnet, such a dynamics can lead to\nan inertial magnetization reversal under the action of an ul trashort laser pulse.\nPACS numbers: 75.10.Jm, 75.10.Hk, 78.47.J-, 05.45.-a\n1I. INTRODUCTION\nWhich is the fastest way to reverse the magnetization of either a ma gnetic particle or a\nsmall region of a magnetic ﬁlm? This question has attracted signiﬁcan t interest, both funda-\nmental and practical, for magnetic information storage.1Intense laser pulses, with durations\nless than a hundred femtoseconds, are able to excite the ultrafas t evolution of the total spin\nof a magnetically-ordered system on a picosecond time scale, see e.g . the reviews.2–4The\nlimitations for the time of magnetization reversal come from the cha racteristic features of\nthe spin evolution for a magnet with a concrete type of magnetic ord er. The dynamical\ntime cannot be shorter than the characteristic period of spin oscilla tionsT,T= 2π/ω0,\nwhereω0is the magnetic resonance frequency. For ferromagnets, the fr equency of standard\nspin oscillations (precession) is ω0,FM=γHr, whereγis the gyromagnetic ratio, and Hris\nan eﬀective ﬁeld of relativistic origin, like the anisotropy ﬁeld, which is u sually less than a\nfew Tesla. Thus, the dynamical time for Heisenberg ferromagnets cannot be much shorter\nthan one nanosecond. For antiferromagnets, all the dynamical c haracteristics are exchange\nenhanced, and ω0,AFM=γ√HexHr, whereHexis the exchange ﬁeld, Hex=J/2µB,Jis the\nexchange integral, and µBis the Bohr magneton, see Ref. 5. The excitation of terahertz spin\noscillations has been experimentally demonstrated for transparen t antiferromagnets using\nthe inverse Faraday eﬀect or the inverse Cotton-Mouton eﬀect.6–11The non-linear regimes\nof such dynamics include the inertia-driven dynamical reorientation of spins on a picosecond\ntime scale, which was observed in orthoferrites.10,11\nThe exchange interaction is the strongest force in magnetism, and the exchange ﬁeld\nHexcan be as strong as 103Tesla. The modulus of the magnetization is determined by\nthe exchange interaction, and the direction of the magnetization is governed by relativistic\ninteractions. It would be very tempting to produce a magnetization reversal by changing\nthe modulus of the magnetization vector, i.e., via the longitudinal dynamics of M. For such\na process, dictated by the exchange interaction, the character istic times could be of the\norder of the exchange time τex= 1/γHex, which is shorter than one picosecond. However,\nwithin the standard approach such dynamics is impossible. The evolut ion of the modulus\nof the magnetization, M=|M|, within the closed Landau-Lifshitz equation for the magne-\ntization only (or the set of such equations for the sublattice magne tizations, Mα), is purely\ndissipative.12This feature could be explained as follows: two angular variables, θandϕ,\n2describing the direction of the vector M, within the Landau-Lifshitz equation determine\nthe pair of conjugated Hamilton variables (cos θandϕare the momentum and coordinate,\nrespectively). Also, the evolution of the single remaining variable M=|M|, governed by a\nﬁrst-order equation can be only dissipative; see a more detailed disc ussion below. Moreover,\nthe exchange interaction conserves the total spin of the system , and the relaxation of the\ntotal magnetic moment of any magnet can be present only when acc ounting for relativis-\ntic eﬀects. Thus the relaxation time for the total magnetic moment is relativistic but it\nis exchange-enhanced, as was demonstrated within the irreversib le thermodynamics of the\nmagnon gas.12Note here that the relaxation of the magnetization of a single sublattice for\nmulti-sublattice magnets can be of purely exchange origin.13Recently, magnetization rever-\nsal on a picosecond time scale has been experimentally demonstrate d for the ferrimagnetic\nalloy GdFeCo, see Refs. 13,14. These results can be explained within t he concept of ex-\nchange relaxation, developed by Baryakhtar,15accounting for the purely exchange evolution\nof the sublattice magnetization.16Such an exchange relaxation can be quite fast, but its\ncharacteristic time is again longer than the expected “exchange tim e”τex= 1/γHex.\nThus, the ultrafast mechanisms of magnetization reversal impleme nted so far are: the dy-\nnamical (inertial) switching possible for antiferromagnets,10,11andthe exchange longitudinal\nevolution for ferrimagnets.13,14,16These are both quite fast, with a characteristic time of the\norder of picoseconds; but their characteristic times are longer th an the “ideal estimate”: the\nexchange time τex.\nInthiswork, wepresent atheoretical study ofthepossibility ofth edynamical evolutionof\nthe modulus of the magnetization for non-Heisenberg ferromagne ts with high enough single-\nionanisotropythatcanbecalled longitudinal spindynamics . Forsuchadynamics, an inertial\nmagnetization reversal ispossibleevenforasimplesingle-element ferromagnet. Longitudina l\ndynamics does not exist in Heisenberg magnets, and this dynamics ca nnot be described in\ntermsoftheLandau-Lifshitzequation, orusingtheHeisenbergHa miltonian, whichisbilinear\nover the components of spin operators for diﬀerent spins, see mo re details below in Sec. II.\nThe key ingredient of our theory is the inclusion of higher-order spin quadrupole variables.\nIt is known that for magnets with atomic spin S >1/2, allowing the presence of single-ion\nanisotropy, the spin dynamics is not described by a closed equation f or spindipolarvariable\n/an}bracketle{tS/an}bracketri}htalone (or magnetization M=−2µB/an}bracketle{tS/an}bracketri}ht).17–24Here and below /an}bracketle{t.../an}bracketri}htmeans quantum and\n(atﬁnitetemperature)thermalaveraging. Tobespeciﬁc, wecho osethespin-oneferromagnet\n3with single-ion anisotropy, the simplest system allowing this eﬀect. Th e full description of\nthese magnets requires taking into account the dynamics of quadrupolar variables,Sik=\n(1/2)/an}bracketle{tSiSk+SkSi/an}bracketri}ht, that represent the quantum averages of the operators, bilinea r in the\nspin components. Our theory is based on the consistent semiclassic al description of a full\nset of spin quantum expectation values (dipolar and quadrupolar) f or the spin-one system,\nwhich was investigated by many authors from diﬀerent viewpoints.17–24As we will show, the\nlongitudinal dynamics of spin, including nonlinear regimes, can be excit ed by a femtosecond\nlaser pulse. With natural accounting for the dissipation, the longitu dinal spin dynamics\ncan lead to changing the sign of the total spin of the system (longitu dinal magnetization\nreversal).\nII. MODEL DESCRIPTION\nThe Landau-Lifshitz equation was proposed many years ago as a ph enomenological equa-\ntion, and it is widely used for the description of various properties of ferromagnets. Con-\ncerning its quantum and microscopic basis, it is worth noting that this equation naturally\narises using the so-called spin coherent states .25,26These states can be introduced for any\nspinSas the state with the maximum value of the spin projection on an arbit rary axis n.\nSuch states can be parameterized by a unit vector n; the direction of the latter coincides\nwith the quantum mean values for the spin operator /an}bracketle{tS/an}bracketri}ht=Sn(dipolar variables). This\nproperty is quite convenient for linking the quantum physics of spins to a phenomenological\nLandau-Lifshitz equation. The use of spin coherent states is most eﬃcient when the Hamil-\ntonian of the system is linear with respect to the operators of the s pin components. If an\ninitial state is described by a certain spin coherent state, its quant um evolution will reduce\nto a variation of the parameters of the state (namely, the directio n of the unit vector n),\nwhich are described by the classical Landau-Lifshitz equation. Thu s, spin coherent states\nare a convenient tool for the analysis of spin Hamiltonians containing only operators linear\non the spin components or their products on diﬀerent sites. An impo rtant example is the\nbilinear Heisenberg exchange interaction, described by the ﬁrst te rm in Eq. (1) below.\nIn contrast to the cases above, for the full description of spin- Sstates, one needs to\nintroduceSU(2S+1) generalized coherent states.21–24The analysis shows that spin coherent\nstates areless natural forthedescription ofmagnets whose Ham iltoniancontainsproducts of\n4thespincomponentoperatorsatasinglesite. Suchtermsarepres entformagnetswithsingle-\nion anisotropy or a biquadratic exchange interaction. Magnets with non-small interaction\nof this type are often called non-Heisenberg . For such magnets, some non-trivial features,\nabsent for Heisenberg magnets, are known. Among them we note t he possibility of so-\ncalled quantum spin reduction; namely the possibility to have the value of|/an}bracketle{tS/an}bracketri}ht|less than\nits nominal value, |/an}bracketle{tS/an}bracketri}ht|< S, even for pure states at zero temperature. This was ﬁrst\nmentioned by Moriya,27as early as 1960. As an extreme realization of the eﬀect of quantum\nspin reduction, we note the existence of the so-called spin nematic p hases with a zero mean\nvalueofthespininthegroundstateatzerotemperature. Inthela sttwodecades, theinterest\non such states has been considerable, motivated by studies of mult icomponent Bose-Einstein\ncondensates of atoms with non-zero spin.28–31\nA signiﬁcant manifestation of quantum spin reduction is the appeara nce of an additional\nbranch of the spin oscillations, which is characterized by the dynamic s (oscillations) of the\nlength of the mean value of spin without spin precession.17,19–24The characteristic frequency\nof this mode can be quite high (of the order of the exchange integra l). For this reason,\nfor a description of resonance properties or thermodynamic beha vior of magnets, this mode\nis usually neglected, and the common impression is that the dynamics o f magnetic mate-\nrials with constant single-ion anisotropy K <(0.2-0.3)Jis fully described by the standard\nphenomenological theory. However, for an ultrafast evolution of the spin system under a\nfemtosecond laser pulse, one can expect a lively demonstration of t his longitudinal high-\nfrequency mode. Thus, it is important to explore the possible manife stations of the eﬀects\nof quantum spin reduction in the dynamic properties of ferromagne ts.\nThe simplest model allowing spin dynamics with eﬀects of quantum spin r eduction is\ndescribed by the Hamiltonian\nH=−1\n2/summationdisplay\nn,ℓ¯JSnSn+ℓ+K\n2/summationdisplay\nn(Sn,x)2, (1)\nwhereSnis the spin-one operator at the site n;¯J >0 is the exchange constant for nearest-\nneighbors ℓ, andK >0 is the constant of the easy-plane anisotropy with the plane yzas\nthe easy plane. The quantization axis can be chosen parallel to the z-axis and /an}bracketle{tS/an}bracketri}ht=/an}bracketle{tSz/an}bracketri}htez.\nFor the full description of spin S= 1 states, let us introduce SU(3) coherent states21–24\n|u,v/an}bracketri}ht=/summationdisplay\nj=x,y,z(uj+ivj)|ψj/an}bracketri}ht, (2)\n5where the states |ψj/an}bracketri}htdetermine the Cartesian states for S= 1 and are expressed in terms of\nthe ordinary states {|±1/an}bracketri}ht,|0/an}bracketri}ht}with given projections ±1,0 of the operator Szby means\nof the relations |ψx/an}bracketri}ht= (|−1/an}bracketri}ht−|+1/an}bracketri}ht)/√\n2,|ψy/an}bracketri}ht=i(|−1/an}bracketri}ht+|+1/an}bracketri}ht)/√\n2,|ψz/an}bracketri}ht=|0/an}bracketri}ht, with the\nreal vectors uandvsubject to the constraints u2+v2= 1,u·v= 0. All irreducible spin\naverages, which include the dipolar variable /an}bracketle{tS/an}bracketri}ht(average value of the spin) and quadrupole\naveragesSik, bilinear over the spin components, can be written through uandvas follows\n/an}bracketle{tS/an}bracketri}ht= 2(u×v),\nSik=1\n2/an}bracketle{tSiSk+SkSi/an}bracketri}ht=δik−uiuk−vivk. (3)\nAt zero temperature and within the mean-ﬁeld approximation, the s pin dynamics is de-\nscribed by the Lagrangian24\nL=−2/planckover2pi1/summationdisplay\nnvn(∂un/∂t)−W(u,v), (4)\nwhereW(u,v) =/an}bracketle{tu,v|H|u,v/an}bracketri}htis the energy of the system.\nWe are interested in spin oscillations which are uniform in space, and he nce we assume\nthat the discrete variables uandvhave the same values for all spins and are only dependent\non time. The frequency spectrum of linear excitations, which consis ts of two branches, can\nbe easily obtained on the basis of the linearized version of the Lagran gian (4). In the general\ncase, the system of independent equations for uandv, taking into account the aforemen-\ntioned constrains u2+v2= 1,u·v= 0, consists of four nonlinear equations, describing\ntwo diﬀerent regimes of spin dynamics. One regime is similar to that for an ordinary spin\ndynamics treated on the basis of the Landau-Lifshitz equation; it c orresponds to oscillations\nof the spin direction. The second regime corresponds to oscillations of the modulus of the\nmagnetization /an}bracketle{tS/an}bracketri}ht=S(t)ez, with the vectors uandvrotating in the xy-planeperpendicular\nto/an}bracketle{tS/an}bracketri}ht. This mode of the spin oscillations corresponds to the longitudinal sp in dynamics. It\nis convenient to consider these two types of dynamics separately. Particular non-linear lon-\ngitudinal solutions, with /an}bracketle{tS/an}bracketri}ht=s(t)ezanduz= 0, vz= 0, were found in Refs. 32,33. Note\nhere that the longitudinal dynamics is much faster than the standa rd transversal one, and\nthe standard spin precession (described by the Landau-Lifshitz e quation) at a picosecond\ntime scale just cannot develop. Therefore, these two regimes, lon gitudinal and transverse,\ncan be treated independently, and we limit ourselves only to the longit udinal dynamics with\n/an}bracketle{tS/an}bracketri}ht=s(t)ezanduz= 0, vz= 0.\n6III. LONGITUDINAL SPIN DYNAMICS\nTo describe the longitudinal spin dynamics, it is convenient to introdu ce new variables:\nthe spin modulus s= 2|u||v|= 2uvand angular variable γ, with\nu=u(excosγ−eysinγ),v=v(exsinγ+eycosγ), (5)\nIn this representation /an}bracketle{tSz/an}bracketri}ht=s, and the non-trivial quadrupolar variables are\n/an}bracketle{tSxSy+SySx/an}bracketri}ht=√\n1−s2sin2γand/angbracketleftbig\nS2\ny−S2\nx/angbracketrightbig\n=√\n1−s2cos2γ, with all other quantum\naverages being either zero (as the transverse spin components /an}bracketle{tSx,y/an}bracketri}htorSxz,Syz) or trivial,\nindependent on sandγ, as/an}bracketle{tS2\nz/an}bracketri}ht= 1. The mean-ﬁeld energy, written per one spin through\nthe variables s, γ, takes the form\nW(s,γ) =−J\n2s2−K\n4√\n1−s2cos2γ, (6)\nwhereJ=¯JZ,Zis the number of nearest neighbors. The ground state at√\n1−s2>0\ncorresponds to cos2 γ= 1, with the mean value of the spin s=±¯s, ¯s=√\n1−κ2<1, that\nis a manifestation of quantum spin reduction at non-zero anisotrop y. Here we introduce the\ndimensionless parameter κ=K/4J. For these variables, the Lagrangian can be written as\nL=/planckover2pi1s∂γ\n∂t−W(s,γ), (7)\nand/planckover2pi1sandγplay the role of canonical momentum and coordinate, respectively, with the\nHamilton function W(s,γ). The physical meaning of the above formal deﬁnitions is quite\nclear: the angular variable γdescribes the transformation of quadrupolar variables under\nrotation around the z-axis, with /planckover2pi1sas the projection of the angular momentum on this axis.\nA. Small oscillations\nLet us now start with the description of the dynamics of small-amplitu de oscillations.\nAfter linearization around the ground state, the equation leads to a simple formula for the\nfrequency of longitudinal oscillations\n/planckover2pi1ωl= 2J¯s= 2J√\n1−κ2, (8)\nwhich are in fact coupled oscillations of the projection of the spin and quadrupolar variables,\nsee Fig. 1.\n7One can see that, for a wide range of values of the anisotropy cons tant, likeκ <0.2-\n0.8, this frequency ωlis of the order of (1.8-1.2) J//planckover2pi1, i.e.,ωlis comparable to the exchange\nfrequencyJ//planckover2pi1. Thus thelongitudinal spindynamics isexpected tobequitefast. In contrast,\nstandard transversal oscillations for a purely easy-plane model ( 1) are gapless (they acquire\na ﬁnite gap when accounting for a magnetic anisotropy in the easy pla ne, which is usually\nsmall). Thus the essential diﬀerence in the frequencies of these tw o dynamical regimes is\nclearly seen.\nAt a ﬁrst glance, there is a contradiction between the concept of lo ngitudinal dynamics\ncaused by single-ion anisotropy and the result present in equation ( 8): the value of ωlis\nstill ﬁnite for vanishing anisotropy constant K; and it is even growing to the value 2 J//planckover2pi1\nwhenκ→0. This can be explained as follows: for a given energy, the ratio of am plitudes\nfor the oscillations of the spin variable sand quadrupolar variable γvanish atκ→0 as\nκ. In fact, for extremely low anisotropy the spin oscillations are not p resent in this mode,\nwhich becomes just a free rotation of the quadrupole ellipsoid of the formγ= 2Jt//planckover2pi1, with\ns= ¯s= const. Ona phase plane with coordinates( s,γ) this dynamics isdepicted by vertical\nstraight lines parallel to the γ-axis, see Fig. 2(c) below. We will discuss this feature in more\ndetail with the analysis of non-linear oscillations.\nB. Nonlinear dynamics and phase plane analysis.\nBefore considering damped oscillations, it is instructive to discuss dis sipationless non-\nlinear longitudinal oscillations. It is convenient to present an image of the dynamics as\na “phase portrait” on the plane momentum-coordinate ( s, γ), which shows the behavior\nof the system for arbitrary initial conditions. The phase trajecto ries in the plane without\ndissipation can be found from the condition W(s,γ) = const.\nTheenergy (6)hasaninﬁnite set ofminima, with s=±¯sandγ=πn,withequal energies\n(green ellipses on the Fig.2), and an inﬁnite set of maxima at s= 0 andγ=π/2+πn(red\nellipses on the Fig.2), here nis an integer. Only the minima with s= ¯sands=−¯s\nare physically diﬀerent; equivalent extremes with diﬀerent values of ncorrespond to the\nequal values of the observables and are completely equivalent. The minima on the phase\nplane correspond to foci with two physically diﬀerent equilibrium stat es with antiparallel\norientation of spin s=±¯s, and/angbracketleftbig\nS2\ny−S2\nx/angbracketrightbig\n=√\n1−¯s2,/an}bracketle{tS2\nz/an}bracketri}ht= 1,/an}bracketle{tSxSy+SySx/an}bracketri}ht= 0. The\n8FIG. 1: (color online) Graphic presentation of the variable ssandγand their evolution. The thick\nred arrow represents the mean value of the spin. The quadrupo lar variables are shown by the blue\nthree-axial ellipsoid with the directions of the main axis ( chosen to have /an}bracketle{tS1S2/an}bracketri}ht= 0):e3=ezand\ne1,e2. The half-axes of the ellipsoid are equal to/angbracketleftbig\nS2\n1/angbracketrightbig\n,/angbracketleftbig\nS2\n2/angbracketrightbig\nand/angbracketleftbig\nS2\n3/angbracketrightbig\n=/angbracketleftbig\nS2\nz/angbracketrightbig\n= 1. (a) the ground\nstate, (b) the standard transverse dynamics, i.e. the spin p recession. The other frames (c)-(e)\npresent the transient values of the variables in longitudin al oscillations. (c) and (e) correspond to\nthe longest and shortest length of the spin, and at the moment depicted in (d) the spin length\nequals to its equilibrium value, but the quadrupolar ellips oid is turned on the angle γwith respect\nto thex-axis. On the frames (c)-(e), the shape of the unperturbed el lipsoid is shown by light grey.\nsaddle points are located at the values s= 0 andγ=πn. The lines with s=±1 are\nsingular; these correspond to degenerate motion with γlinear in time γ=±tJ//planckover2pi1; the points\nat these lines where dγ/dtchange sign, can be treated as some non-standard saddle points.\nThe shape of the phase trajectories, i.e., the characteristic feat ures of oscillations, varies\nwith the change of the anisotropy parameter κ. Note ﬁrst the general trend, the relative\namplitude of the changes of the spin and γdepends on κ: the bigger κis, the larger values\n9of the change of spin are observed. The topology of the phase tra jectories change at the\ncritical value of the anisotropy parameter κ. At smallκ<1/2, the trajectories with inﬁnite\ngrowingγare present, and the standard separatrix trajectories connec t together diﬀerent\nsaddle points, see Fig. 2 (c). As mentioned above; only such trajec tories are present at the\nlimitκ→0, but, in fact, even for the small value of κ= 0.2 used in Fig.2 (c), the main\npart of the plane is occupied by the trajectories that change the s pin. At the critical value\nκ= 1/2,the separatrix trajectories connect the saddle points at s= 0,γ= 0 and the\ndegenerated saddle points at the singular lines with s=±1. At larger anisotropy κ>1/2,\nas in Fig. 2(a), the only separatrix loops entering the same saddle po int are present.\nThe minima, saddle point and the separatrix are the key ingredients f or the switching\nbetween the ground states with s= ¯sands=−¯s. The case of large anisotropy κ >1/2\nlooks like the standard one for the switching phenomena, whereas f or small anisotropy the\nsituation is more complicated.\nC. Damped longitudinal motion\nDampingisacrucial ingredient forthedynamical switching between d iﬀerent, butequiva-\nlent inenergy, states. Thehigh-frequency modeoflongitudinal os cillationshavehigh-enough\nrelative damping; as was found from microscopic calculations,32the decrement of longitudi-\nnal mode Γ = λωl, whereλ∼0.2. To account for the damping in the dynamic equations\nforsandγ, it is useful to consider a diﬀerent parametrization of the longitudin al dynamics.\nLet us now introduce a unit vector, σ=σ1e1+σ2e2+σ3e3, with components σ3=s,\nσ1=/an}bracketle{tS2\ny−S2\nx/an}bracketri}ht,andσ2=/an}bracketle{tS1S2+S2S1/an}bracketri}ht. Being written through σ, the equation of motion\ntakes the form of the familiar Landau-Lifshitz equation\n/planckover2pi1∂σ\n∂t= [σ×heﬀ]+R,heﬀ=−∂W\n∂σ, (9)\nwhereheﬀcan be treated as an eﬀective ﬁeld for longitudinal dynamics, and th e relaxation\ntermRis added. The equation of motion with R= 0 is fully equivalent to the Hamilton\nform of the equation found from (7), but the form of the dissipatio n is more straightforward\nin unit-vector presentation. The choice of the damping term in a sta ndard equation for\nthe motion of the transverse spin is still under debate, see.15,34But here the damping term\n10/s45/s48/s46/s56 /s45/s48/s46/s52 /s48/s46/s48 /s48/s46/s52 /s48/s46/s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50 /s47\n/s115\n(a)κ=0.6/s45/s48/s46/s56 /s45/s48/s46/s52 /s48/s46/s48 /s48/s46/s52 /s48/s46/s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50 /s47\n/s115\n(b)κ=0.5\n/s45/s48/s46/s56 /s45/s48/s46/s52 /s48/s46/s48 /s48/s46/s52 /s48/s46/s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50 /s47\n/s115\n(c)κ=0.2\nFIG. 2: (color online) Phase plane representations for diss ipation-free non-linear longitudinal spin\noscillations for diﬀerent values of the parameter κ;κ=0.6, 0.5 and 0.2 for panels (a), (b), and (c),\nrespectively. Thegreen andred ellipses present theminima andmaxima, respectively; thestandard\nsaddle points are depicted by red rectangles, while the stan dard separatrix trajectories are drawn\nby red lines. The singular trajectories with s=±1 and the separatrix trajectories entering the\nnon-standard saddle points on these lines s=±1 are shown by blue lines on the frames (a) and\n(c). For the critical value κ= 0.5, all the separatrix trajectories and the singular traject ories with\ns=±1 organize a common net; and on the corresponding frame (b) al l of them are presented by\nred lines.\n11can be written in the simplest form, as in the original paper of Landau and Lifshitz, R=\nλ[heﬀ−σ(heﬀσ)]. The arguments are as follows: (i) this form gives the correct valu e of the\ndecrement of linear oscillations, Γ = λωl; (ii) it is convenient for analysis, because it keeps\nthe condition σ2= 1. Finally, the equations of motion with the dissipation term of the\naforementioned form are:\n/planckover2pi1ds\ndt=−∂W\n∂γ−λ(1−s2)∂W\n∂s,/planckover2pi1dγ\ndt=∂W\n∂s−λ\n(1−s2)∂W\n∂γ. (10)\nThese equations describe the damped counterpart of the non-line ar longitudinal oscillations\ndiscussed in the previous subsection and present as phase portra its on Fig. 2. The character\nof the motion at not-too-large λcan be qualitatively understood from energy arguments.\nThe trajectories of damped oscillations in any point of the phase plan e approximately follow\nthe non-damped (described by equation W(s,γ) = const) ones, but cross them passing from\nlarger to smaller values of W, see Figs. 3 and 4. It happens that for the case of interest,\nthe dynamics is caused by the time-dependent stimulus. An action of the stimulus on the\nsystemcanbedescribedbyaddingthecorrespondingtime-depend entinteractionenergy∆ W\nto the system Hamiltonian, W→W(s,γ) + ∆W(s,γ,t). Within this dynamical picture,\n∆Wproduces an “external force” driving the system far from equilibr ium.\nTheanalysis isessentially simpliﬁed forapulse-like stimulus ofashort du ration∆t(much\nshorter than the period of motion, ωl∆t≪1). In this case, the role of the pulse is reduced\nto the creation of some non-equilibrium state, which then evolves as some damped nonlinear\noscillations described by the “free” equations (10) with ∆ W= 0. The phase plane method,\nwhich shows the behavior of the system for arbitrary initial conditio ns, is the best tool for\nthe description of such an evolution.\nIt is worth noting that the asymptotic behavior of the separatrix t rajectories at γ,s→0,\nis important for this analysis, and can be easily found analytically as\n/parenleftBigγ\ns/parenrightBig\nsepar=Rsepar=\n=1\n8κ/bracketleftBig\nλ(1+3κ)+/radicalbig\nλ2(1+3κ)2+16κ(1−κ)/bracketrightBig\n.(11)\nFirst let us start with the analysis for high-enough anisotropy. The corresponding phase\nportrait is present in Fig. 3. The general property of the phase pla ne is that the phase\ntrajectories cannot cross each other; they can only merge at th e saddle points. Thus the\ntrajectories coming to diﬀerent minima are stretched between two separatrix lines entering\n12/s45/s48/s46/s56 /s45/s48/s46/s52 /s48/s46/s48 /s48/s46/s52 /s48/s46/s56\n/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/s48/s46/s52/s50 /s47\n/s115\nFIG. 3: (color online) Phase plane representation of damped longitudinal spin oscillations for\nκ=0.6. Here and in Fig. 4, the dashed lines (obtained analytic ally before) represent the phase\ntrajectories withoutdissipation, whilethe full lines are trajectories for dissipation constant λ= 0.2,\nfound numerically. The separatrix lines are drawn by red cur ves.\nthe same saddle point from diﬀerent directions, as shown in Fig. 3. Fr om this it follows that\nany initial state with arbitrary non-equilibrium values of spin s(+0), but without deviation\nofγfrom its equilibrium value, evolve to the state with the same sign of the spin as for\ns(+0), and no switching occurs. On the other hand, if the initial cond ition is above the\nseparatrix trajectory, entering the saddle point, the evolution w ill move the system to the\nequivalent minimum with the sign of the spin opposite to the initial one, s(+0), realizing\nthe switching.\nFigure 4 shows the phase plane for equations (10) for the more com plicated case of\nlow anisotropy, demonstrating possible scenarios of the switching o f the sign of the spin\nvalue during such dynamics. Here the separatrix trajectories for the damped motion can\nbe monitored from their maxima, and the full picture of the behavior can be understood\nonly when including a few equivalent foci with γ= 0,±π,±2π, ect., with diﬀerent, but\nequivalent in energy, values of the spin, s=±¯s, ¯s=√\n1−κ2, and diﬀerent saddle points,\nlocated atγ= 0,±π,±2π. As for small anisotropy, the trajectories coming to diﬀerent\n13minima are located between two branches of the separatrix lines, bu t now this “separatrix\ncorridor” is organized by separatrix lines entering diﬀerent saddle p oints. The switching\nphenomena is also possible, but the process involves a few full turns of the variable γ.\nThe general regulation for any anisotropy can be formulated as fo llows: for realizing\nspin switching, one needs to have the initial deviation (reduction) of the spin value, and,\nsimultaneously, a non-zero deviation of the quadrupolar variable γ. To switch the positive\nspin value to negative, one needs to start from the states just ab ove the separatrix line\nentering the saddle point from positive values of s. The smaller the initial value of the spin,\nthe smaller value of γ(0) would realize the switching. From the asymptotic equation (11),\nthe corresponding ratio Rsepar=γ(0)/m(0) is smaller for small values of λ; but even when\nλ→0,it exceeds the value Rsepar(λ= 0) = 0.5/radicalbig\n(1−κ)/κ. Thus, the switching could occur\nfor non-zero values of κ.\nIV. INTERACTION OF THE LIGHT PULSE ON THE SPIN SYSTEM: CRE-\nATION OF THE INITIAL STATE FOR SWITCHING.\nLet us now consider the longitudinal spin evolution caused by a speciﬁ c stimulus: a\nfemtosecond laser pulse. The reduction of the spin to small values w as observed in many\nexperiments, and the only non-trivial remaining question is: how can we create a deviation\nof the quadrupolar variable γfrom its equilibrium value γ= 0. To ﬁnd this, we now con-\nsider possible mechanisms of light interaction with quadrupolar variab les of non-Heisenberg\nmagnets.\nThe interaction of the spin system of magnetically-ordered media an d light is described\nby the Hamiltonian (as above, written per spin) ∆ W= ¯εijv0Ei(t)E∗\nj(t)/16π, wherev0\nis the volume per spin, Ei(t) is the time-dependent amplitude of the light in the pulse,\n¯εij=d(ωε(spin)\nij)/dω,ε(spin)\nijis the spin-dependent part of the dielectric permittivity tensor,\nandωis the frequency of light. For the longitudinal dynamics considered h ere, circularly-\npolarized light propagating along the z-axis acts on the z-component of the spin via the\nstandard inverse Faraday eﬀect, with the antisymmetric part of ¯ ε(a)\nijas, ¯ε(a)\nxy=−¯ε(a)\nxy=sαF,\n14/s45/s49/s46/s48 /s45/s48/s46/s56 /s45/s48/s46/s54\n/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50 /s47/s32\n/s115\n(a)\n (b)\n/s48/s46/s54 /s48/s46/s56 /s49/s46/s48\n/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s50 /s47\n/s32/s115\n(c)\nFIG. 4: (color online) Phase plane for the damped spin evolut ions for low anisotropy, κ= 0.2. The\ncentral frame shows the full diagram; left and right panels d emonstrate the details of the behavior\nnear the equilibrium values s=−¯sands= ¯s, respectively. On this frame, the regions colored\nby green and yellow correspond to diﬀerent basins of attracti on with initial values leading to the\nequilibrium states with s= ¯sands=−¯s, respectively.\n15giving an interaction of the form\n∆Wcircular=sαFv0\n16π|Ecirc|2σ, (12)\nwhereEcircis the (complex) amplitude, σdescribes the pulse helicity: σ=±1 for right-\nhanded and left-handed circularly polarized laser pulses. To describ e qualitatively the result\nofthe actionofthelight pulse, let us nowassume that thepulse dura tionτpulseistheshortest\ntime of the problem. If the pulse duration is shorter than the period of spin oscillations, the\nreal pulse shape can be replaced by the Dirac delta function, |Ecirc|2→E2\npτpulseδ(t),where\nE2\np=/integraltext\n|Ecirc|2dt/τpulsecharacterizes the pulse intensity. (Note that this approximation\nis still qualitatively valid even for any comparable values of τpulseand 2π/ω) Then, using\nequations (10) one can ﬁnd the eﬀect produced by the pulse. Within this approximation, the\naction of a pulse leads to an instantaneous deviation of the variable γfrom its equilibrium\nvalue, which then evolves following the non-perturbed equations of motion (10). Keeping in\nmind that before the pulse action the system is in equilibrium, s(−0) = ¯sandγ(−0) = 0, it\nis straightforward to ﬁnd the values of these variables ( sandγ) after the action of the pulse,\ns(+0) andγ(+0). For our purposes, the non-equilibrium value of the quadrupo lar variable\nγis important:\nγ(+0) =−αF\n16π/planckover2pi1E2\npv0τpulseσ. (13)\nThecumulative actionofthecircularly-polarizedpulse, including anes sential reductionof\nthe spin value (caused either by thermal or non-thermal mechanis ms) and the deviation of γ\ndescribed by (13) could lead to the evolution we are interested here , switching the spin of the\nsystem. Note here that for standard spin reduction the polarizat ion of the light pulse is not\nessential,35–37whereas the values of γ(+0) are opposite for right- and left-handed circularly\npolarized pulses. These features are characteristic of the eﬀect described here. Note the\nrecent experiment where the role of circular polarization in spin switc hing for GdFeCo alloy\nwas mentioned, but the authors have attributed it to magnetic circ ular dichroism.38\nV. CONCLUDING REMARKS.\nLet us now compare the approach developed in this article with previo us results on\nsubpicosecond spin evolution. The ﬁrst experimental observation of demagnetization for\nferromagnetic metals under femtosecond laser pulses shows that the magnetic moment can\n16be quenched very fast to small values, much faster than one picos econd.35–37These eﬀects\nare associated with a new domain of the physics of magnets, femtomagnetism ,39and its\nanalysis is based on the microscopic consideration of spins of atomic e lectrons,40,41or itin-\nerant electrons.42Not discussing this fairly promising and fruitful domain of magnetism,\nnote that, to the best of our knowledge, no eﬀects of magnetizat ion reversal during this\n“femtomagnetic stage” has been reported in the literature. For e xample, the subpicosecond\nquenching processes for the ferromagnetic alloy GdFeCo are resp onsible for the creation of\na far-from-equilibrium state, but the evolution of this state, giving the spin reversal, can be\ndescribed within the standard set of equations for the sublattice m agnetizations.16\nIn contrast, here we propose some pathway to switch the sign of t he magnetic moment\nduring extremely short times, of order of the exchange time. It is s hown here that the spin\ndynamics for magnets with non-small single-ion anisotropy can lead t o the switching of the\nsign of the magnetic moment via the longitudinal evolution of the spin m odulus together\nwith quadrupolar variables, i.e., quantum expectation values of oper ators bilinear over the\nspin components SxandSy. It is worth to stress here that the “restoring force” for this\ndynamics is the exchange interaction , and the characteristic time is the exchange time. On\nthe other hand, to realize this scenario, one needs to a have non-H eisenberg interaction, e.g.,\nsingle-ion anisotropy, which couples the spin dipole and quadrupole va riables.\nObviously, this eﬀect is beyond the standard picture of spin dynamic s based on any\nclosed set of equations for the spin dipolar variables (i.e., the quantu m expectation values\nlinear on the spin components) alone. Note that our approach base d on the full set of\nvariablesfortheatomicspinis“moremacroscopic” thanthe“femto magnetic” approach,40–42\ndealing with electronic states. To realize this type of switching, it is ne cessary to have a\nsigniﬁcant coupling between dipolar and quadrupolar spin variables, w hich is present in\nmagnets with strong single-ion anisotropy. Such anisotropy is know n for the numerous\nmagnets based on anisotropic ions of transition elements such as Ni2+, Cr2+, Fe2+. As\nthe classic example, note nickel ﬂuosilicate hexahydrate NiSiF 6·6H2O, with spin-one Ni2+\nions, coupled by isotropic ferromagnetic exchange interaction and subject to high single-\nion anisotropy. For this compound, the strong eﬀect of quantum s pin reduction is known,\nwith its strength dependent on the pressure: the value of K/Jis growing with the pressure\nPresulting in the value /an}bracketle{tS/an}bracketri}ht= 0.6 atP= 6 kbar and leading to the transition to the\nnon-magnetic state with /an}bracketle{tS/an}bracketri}ht= 0 atP∼10 kbar.43,44\n17A number of recent experiments were done with rare-earth trans ition-metals com-\npounds.13,14,38,45Noteherearichvarietyofnon-linearspindynamicsobservedforth inﬁlmsof\ntheFeTballoyunder theactionoffemtosecond laserpulses.45However, thetheorydeveloped\nhere for simple one-sublattice ferromagnet cannot be directly app lied for the description of\nsuch compounds. Ferromagnetic order with high easy-plane anisot ropy is present for many\nheavy rare-earth elements, such as Tb and Dy at low temperature s.46This feature is known\nboth for bulk monocrystals,46and in thin layers and superlattices, see Ref.47and references\nwherein. Strictly speaking, in our article only spin-one ions were cons idered. Rare-earth\nmetals have non-zero values of both spin and orbital momentum, fo rming the total angular\nmomentum of the ion, and for their description the theory needs so me modiﬁcations. How-\never, we believe that the eﬀects of spin switching caused by quadru polar spin dynamics will\nbe present as well for such magnets with high values of atomic angula r momentum.\nThe scenario proposed here includes inertial features, with the ev olution of an initial\ndeviation from one equilibrium state to the other, located far from t he initial one. The\ninitial deviations should include both deviation of the magnetization an d of the quadrupolar\nvariable,γ. Thus the eﬀect is based on standard magnetization reduction, bu t it is helicity\ndependent as well. The necessary initial deviations can be created b y a light pulse of circular\npolarization and the possibility of switching depends on the connectio n of the initial spin\ndirection and the pulse helicity. The possible materials should satisfy a number of general\nconditions: they should be very susceptible to magnetization quenc hing, which is typical for\nmany materials, as well as a sizeable Faraday eﬀect, and they should also have spin one and\na high enough easy-plane anisotropy.\nThis work is partly supported by the Presidium of the National Acade my of Sciences of\nUkraine via projects no.VTs/157 (EGG)and No.0113U001823(BAI) andby the grants from\nState Foundation of Fundamental Research of Ukraine No. F33.2/ 002 (VIB and YuAF) and\nNo. F53.2/045 (BAI). 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Mackintosh, Rare Earth Magnetism: Structures and Excitations (Clarendon\nPress, Oxford, 1991).\n47A. T. D. Gr¨ unwald, A. R. Wildes, W. Schmidt, E. V. Tartakovsk aya, J. Kwo, C. Majkrzak,\nR. C. C. Ward, and A. Schreyer, Phys. Rev. B 82, 014426 (2010).\n21"    },    {        "title": "1002.3510v1.Paramagnetic_limiting_in_ferromagnetic_superconductors_with_triplet_pairing.pdf",        "content": "arXiv:1002.3510v1  [cond-mat.supr-con]  18 Feb 2010Paramagnetic limiting in ferromagnetic superconductors w ith triplet pairing\nV. P. Mineev\nCommissariat ` a l’Energie Atomique, INAC/SPSMS, 38054 Gre noble, France\n(Dated: April 23, 2022)\nThe spin susceptibility in the uranium ferromagnet superco nductors is calculated. There is shown\nthat the absence of superconductivity paramagnetic limita tion for the ﬁeld directions perpendic-\nular to the direction of the spontaneous magnetization is ex plained by the itinerant ferromagnet\nband splitting rather than by a rotation of magnetization to ward the external ﬁeld direction. The\nqualitative description of the upper critical ﬁeld tempera ture dependence is given.\nPACS numbers: 74.20.De, 74.20.Rp, 74.25.Dw\nI. INTRODUCTION\nIt is commonly believed that in the ferromagnet super-\nconducting uranium compounds1–3we deal with triplet\nsuperconductivity. In particular, it is due to the fact\nthat the upper critical ﬁeld strongly exceeds the Pauli\nlimit. However, the paramagnetic limitation of triplet\nsuperconductivity is inessential only then the external\nﬁeld direction is parallel to the spin quantization axis.\nOn the contrary, it is quite essential for the ﬁeld orienta-\ntion perpendicular to the spin quantization direction.4,5\nIn actuality the situation is the opposite. In two of ura-\nnium superconducting compounds URhGe and UCoGe\nthe upper critical ﬁeld in the direction of spontaneous\nmagnetization ˆ cis about the paramagnetic limiting ﬁeld\ninthismaterials. Atthesametimetheuppercriticalﬁeld\nin the perpendicular to magnetization directions is much\nhigher than the paramagnetic limiting ﬁeld.6–8Moreover\nthis property persists also in the reentrant superconduct-\ning state of the URhGe9,10, where the superconductivity\nis reappearing under the magnetic ﬁeld 12 Tesla in b\ncrystallography direction causing alignment of magneti-\nzation parallel to baxis. The additional ﬁeld oriented in\nacrystallography direction does not destroy the super-\nconducting state till to 20 Tesla ! The similar behavior\nwas recently found in the UCoGe.11,12\nOne can think that magnetization direction always fol-\nlows the direction of the external ﬁeld that prevents the\nsuppression of superconducting state like it is in the su-\nperﬂuid3He−Aand should be in a superconductor with\ntriplet pairing in the absence of spin-orbital coupling ﬁx-\ning the mutual orientation of spins quantization axis and\nthe crystalline symmetry directions.4In uranium com-\npounds the magnetic anisotropy is quite strong.13As re-\nsult, in the superconducting URhGe the ﬁeld oriented\nparalleltobaxiscausesonlytinyrotationofthemagneti-\nzationdirection9tilltoHc2≈1.3Teslamorethantwicely\nexceedingtheparamagneticlimitingﬁeld.6Hence, thero-\ntation of the magnetization cannot be responsible for the\nabsence of the paramagnetic limitations.\nHere weinvestigatetheoreticallysucha remarkablebe-\nhaviourofthe ferromagneticsuperconductors. Therewill\nbe given the microscopic derivation of the paramagnetic\nsusceptibility of the ferromagnet superconductors for theﬁeld orientation perpendicular to the direction of the\nspontaneous magnetization. The absence of Pauli lim-\nitations of superconductivity is found related with the\nitinerant ferromagnet band splitting rather than with\nthe magnetization rotation. The latter is also impor-\ntant at higher ﬁelds near the metamagnetic or magneti-\nzation rotation phase transition. Hence, the critical ﬁeld\nin the itinerant ferromagnets can be calculated ignoring\nthe paramagnet limitations. In conclusion we discuss the\nupper critical ﬁeld temperature dependence in the ura-\nnium compounds in moderate ﬁeld region.\nII. SPIN SUSCEPTIBILITY\nURhGe and UCoGe are the orthorhombic ferromag-\nnets with spontaneous magnetization oriented along c\ncrystallography axis. At the temperatures below the\nCurie temperature and in the absence of magnetic ﬁeld\ntheccomponent of magnetization has a ﬁnite value. The\nmagnetic ﬁeld applied along baxis creates the magneti-\nzation along its direction but decreases the magnetiza-\ntion parallel to c. Phenomenologically it is described by\nmeanstheLandaufreeenergyofferromagnetinmagnetic\nﬁeld14\nF=αz0(T−Tc)Mz2+βzMz4\n+αyMy2+βyzMz2My2−MyH.(1)\nHere the y,zare directions of the spin axes pinned to\n(b,c) crystallographic directions correspondingly. The\nﬁeld induced magnetization along b-direction is\nMy=H\n2(αy+βxyM2z). (2)\nSubstituting this value back in the eqn. (1) we obtain at\nβxyM2\nz/αy<1, that is certainly true not so far from the\nCurie temperature,\nF=α0z/parenleftbigg\nT−Tc+βyzH2\n4αz0α2y/parenrightbigg\nMz2+βz.Mz4.(3)\nHence, the Curie temperature\nTCurie(H) =Tc−βyzH2\n4αz0α2y(4)2\nis suppressed by the magnetic ﬁeld oriented along b-axis.\nThis type of behavior was observed in UCoGe.11The\nmagnetization along z-direction is also decreased\nM2\nz=αz0(Tc−T)\n2βz−βyzH2\n8α2yβz(5)\nThe ﬁeld dependence of magnetization components in\nURHGe has been reported in the paper9. For supercon-\nducting state realizing in the low ﬁeld region of the phase\ndiagram the upper critical ﬁield for the ﬁeld orientation\nalong b-axis does not exceed 1.3 Tesla.6At this ﬁeld the\nmagnetization in b-direction is at least 10 times smaller\nthan the magnetization along c-direction which is practi-\ncally ﬁeld independent.9Hence, the magnetic ﬁeld acting\non the electron spins in ˆ z-direction can be taken equal to\nexchange ﬁeld\nh= 4πMz(H= 0)ˆz. (6)\nThe ﬁeld in ˆ y-direction is\nB= (H+4πMy(H))ˆy. (7)\nIn that follows we shall assume that both phenomena\nferromagnetismandsuperconductivityaredeterminedby\nthe spin-up and the spin-down electrons ﬁlling two sepa-\nrate bands split by the exchange ﬁeld h∼Tc/µB. Then\nthe magnetic moment of the itinerant electron subsystem\nis given by\nM=µBT/summationdisplay\nn/integraldisplayd3k\n(2π¯h)3TrσˆG. (8)\nHereσ= (σx,σy,σz) are Pauli matices.\nIn the normal state the Green function in linear in\nrespect to Bapproximation is\nˆG=ˆGn−µBBˆGnσyˆGn, (9)\nwhere\nˆGn=/parenleftbigg\nGn+0\n0Gn−/parenrightbigg\n, Gn±=1\niωn−ξk±µBh.(10)\nWe obtain\nM=µBT/summationdisplay\nn/integraldisplayd3k\n(2π¯h)3[ˆz(Gn+−Gn−)\n−2µBBˆyGn+Gn−].(11)\nFor a ﬁnite value of the exchange ﬁeld this is equal to\nM=µB(N↑−N↓)h+B\nh. (12)\nHereN↑,↓arethe numbers ofelectronsin the spin-up and\nspin-down band. The corresponding susceptibility is\nχyy=µB(N↑−N↓)/h (13)On the other hand in absence of the band splitting that\nis ath= 0 the magnetic moment is\nM= 2µ2\nBN0B, (14)\nwhereN0is the density of states per one electron spin\nprojection. The susceptibility is given by the Pauli for-\nmula\nχyy(h= 0) = 2 µ2\nBN0. (15)\nThe superconducting state in two band itinerant fer-\nromagnet is built of pairing states formed either by spin-\nup electrons from one band or by spin-down electrons\nfrom another band.15–18This state is characterized by\ntwo component order parameter\nˆ∆ =/parenleftbigg\n∆k↑0\n0 ∆k↓/parenrightbigg\n. (16)\nThen instead eqn. (11) we obtain\nM=µBT/summationdisplay\nn/integraldisplayd3k\n(2π¯h)3[ˆz(Gs+−Gs−)\n−µBBˆy(Gs+Gs−+Gs−Gs++F+F†\n−+F−F†\n+)],(17)\nwhere\nGs±=−iωn−ξk±\nω2n+ξ2\nk±+|∆↑,↓|2, F±=∆↑,↓\nω2n+ξ2\nk±+|∆↑,↓|2\n(18)\nare the superconducting state Green functions and\nξk±=ξk∓µBh.\nThe straightforward calculation shows, that at the\nband splitting exceeding the superconducting gaps\nµBh≫ |∆±|, even at T= 0,\nT/summationtext\nn/integraltextdξ[2Gn+Gn−−2Gs+Gs−−F+F†\n−−F−F†\n+]\nT/summationtext\nn/integraltext\ndξ Gn+Gn−\n∼/summationdisplay\nαβ=↑,↓∆kα∆∗\nkβ\n(µBh)2ln(µBh)2\n∆kα∆kβ≪1 (19)\nIt implies that the susceptibility in the superconducting\nstatepracticallykeepsitsnormalstatevalue.19Thepara-\nmagnetic limiting ﬁeld formally proves to be of the order\nof the exchange ﬁeld\nHp≈h\nln(µBh/|∆|). (20)\nHence, so long the band splitting is larger than the\ngap, the paramagnetic suppression of the superconduct-\ning state by the ﬁeld perpendicular to the spontaneous\nmagnetization is absent.\nOn the contrary at h= 0 the formal calculation from\nthe equation(17) yields the susceptibility\nχyy(h= 0,T) = 2µ2\nBN0/integraldisplaydΩ\n4πY(ˆk,T),(21)3\nwhere\nY(ˆk,T) =1\n4T/integraldisplay+∞\n−∞dξ\ncosh2(/radicalbig\nξ2+∆2\nk/2T)\nis generalized Yosida function. The susceptibility\nχyy(h= 0,T) tends to zero at T→0.4Thus, the mag-\nnetic ﬁeld directed perpendicular to the Cooper pairs\nspins in a nonferromagnet superconductor with triplet\npairing suppress superconductivity like it does in the\nusual superconductors with singlet pairing.\nAll the formulated conclusions are valid at moderate\nmagnetic ﬁelds when My(H)≪Mz(H). At external\nﬁelds of the order of exchange ﬁeld H∼h, the equi-\nlibrium magnetzation align itself parallel to the external\nﬁeld. In this conditions the paramagnetic limitation of\nsuperconductivity is absent as well.\nIII. CONCLUDING REMARKS\nIn general there are three mechanisms of the magnetic\nﬁeld inﬂuence on the superconducting state in the su-\nperconductors with triplet pairing20: (i) the orbital de-\npairing, (ii) paramagnetic limiting, and (iii) stimulation\nor suppression of nonunitary superconductivity due to\nmagnetic ﬁeld dependence of density of states21.\nWe have demonstrated here that the superconducting\nstateintheitinerantsuperconductorswithtripletpairing\nis not a subject of the paramagnetic limiting. For the\ncompleteness let us brieﬂy look at the two other source\nof the ﬁeld inﬂuence.\nMaking use eqn. (17) one can show that the supercon-\nducting spontaneous magnetization\nMs= ˆzµB/bracketleftbigg\nN↑−N↓+(N′\n0↑|∆↑|2−N′\n0↓|∆↓|2)lnεF\nTs/bracketrightbigg\n.\n(22)\nis slightly modiﬁed in comparison with its normal state\nvalue\nMn= ˆzµB[N↑−N↓]. (23)\nHere,N′\n0↑andN′\n0↓are the energy derivatives of the den-\nsity of states at the Fermi level for the spin up and spin\ndown band correspondingly. The spontaneous magneti-\nzation change causes the corresponding energy shift un-\nder magnetic ﬁeld Hzparallel to ˆ zdirection that leads inits turn to the criticall temperature shift. To avoid the\ncumbersome formula we write it for the case of presence\nonly the spin-up pairing\nδTs\nTs=µBHzN′\n0↑\nN0↑lnεF\nTs(24)\nOn the other hand the magnetic ﬁeld Hdirected per-\npendicular to the direction of spontaneous magnetiza-\ntion does not cause a linear in Hshift in the free en-\nergy of superconducting state. Hence, for this ﬁeld direc-\ntion the third mechanism of magnetic ﬁeld inﬂuence on\nthe superconducting state is also ineﬀective. The state-\nment is valid in the moderate ﬁelds when the inequality\nMy(H)≪Mz(H) takes place.\nThus the only orbital mechanism suppression of super-\nconductivity is essential. Experimentally , in UCoGe for\nthe ﬁeld directed perpendicular to the spontaneous mag-\nnetization there was observedthe pronounced upper crit-\nical ﬁeld upward curvature7,8,11apparently related with\nthe magnetic ﬁeld dependence of the eﬀective mass in\nthis material11. Indeed, for an orthorhombic supercon-\nductor under magnetic ﬁeld directed along bdirection22\nthe Ginzburg-Landau formula for the critical tempera-\nture ( for simplicity we limit ourself by the one band\ncase) is\nTs(H) =Ts0/parenleftbigg\n1−CH\nm∗a(H)m∗c(B)/parenrightbigg\n,(25)\nwhereCis a constant with dimensionality m2/H. We\nsee, that the decreasing of the eﬀective mass with in-\ncreasing of magnetic ﬁeld followed by saturation of its\nﬁeld dependence found at moderate ﬁelds in the paper11\ninevitablycausestheappearanceoftheupwardcurvature\nin magnetic ﬁeld dependence of the critical temperature\nas well of the upper critical ﬁeld.\nAcknowledgments\nThis work was partly supported by the grant SINUS\nof Agence Nationale de la Recherche.\nThe author is indebted to J.-P. Brison for the enlight-\nening discussion and the interest to the paper.\n1S. S. Saxena, P. Agarval, K. Ahilan, F. M. Grosche, R. K.\nW. Hasselwimmer, M. J. Steiner, E. Pugh, I. R. Walker,\nS. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, I.\nSheikin, D. Braithwaite, and J. Flouquet, Nature406, 587\n(2000).\n2D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flou-\nquet, J. P. Brison, E. Lhotel and C. Paulsen, Nature413,613 (2001).\n3N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P.\nKlaasse, T. Gortenmulder, A. de Visser, A. Hamann, T.\nGorlach, and H.v.Lohneysen, Phys. Rev. Lett. 99, 067006,\n(2007).\n4V. P. Mineev and K. V. Samokhin, Introduction to Uncon-\nventional Superconductivity (Gordon and Breach Science4\nPublishers, Amsterdam, 1999).\n5C. H. Choi, J. A. Sauls, Phys. Rev. B 48, 13684 (1993).\n6F. Hardy and A. D. Huxley, Phys. Rev. Lett. 94, 247006\n(2005).\n7N. T. Huy, D. E. de Nijs, Y. Huang, and A. de Visser,\nPhys.Rev.Lett. 100, 007002 (2008).\n8E. Slooten, T. Naka, A. Gasparini, Y. K. Huang, and A.\nde Visser, Phys. Rev. Lett. 103, 097003 (2009).\n9F. Levy, I. Sheikin, B. Grenier, A. D. Huxley, Science309,\n1343 (2005).\n10F. Levy, I. Sheikin, A. D. Huxley, Nature Physics 3, 460\n(2007).\n11D. Aoki, T. D. Matsuda, V. Taufour, E. Hassinger, G.\nKnebel, and J. Flouquet, Journ. Phys. Soc. Japan, 78,\n113709 (2009).\n12Anne de Visser: JPSJ Online - News and Comments\n[November 10, 2009].\n13A. B. Shick, Phys. Rev. B 65, 180509(R) (2002).\n14One should stress the importance of the ﬁeld exact orien-\ntation perpendicular to the spontaneous magnetization. A\nﬁeld misalignment thatis apresence of theﬁeldcomponent\nalong c-direction transforms the second order phase tran-\nsition to the crossover. See eg L. D. Landau, E. M. Lifshitz\n”Statistical Physics”, Pergamon Press, Oxford, 1980.\n15V. P. Mineev and T. Champel, Phys. Rev. B 69, 144521\n(2004).\n16V. P. Mineev, Int. J. Mod. Phys. 18, 2963 (2004).17V. P. Mineev ”Coexistence of triplet superconductivity\nand itinerant ferromagnetism”, pp.68-73 in ”Advances\nin theoretical physics”, Landau Memorial Conference,\nChernogolovka, Russia, 22-26 June 2008, AIP Conference\nProceedings, vol. 1134; eds. V. Lebedev, M. Feigel’man;\nMelville, New York, 2009. And also arXiv:0812.2171v2.\n18V. P. Mineev, Journ. Low Temp. Phys. 158, 615 (2010).\n19The susceptibility in noncentrosymmetric superconductor s\n(see K. V. Samokhin, Phys. Rev. B 76, 094516 (2007)) has\nthe similar property.However, the transverse susceptibil ity\nin ferromagnet, thatis response to B⊥h, does not contain\na term originating of electrons near the Fermi surface.\n20I. A. Lu’yanchuk and V. P. Mineev, Zh. Eksp. Theor. Fiz.\n93,2045 (1987) [Sov. Phys. -JETP 661168 (1987)].\n21V. Ambegaokar, N. D. Mermin, Phys. Rev. Lett. 30, 81\n(1973).\n22Looking for the orbital suppression of superconductivity\nby the magnetic ﬁeld directed along baxis one can neglect\nby the action of spontaneous magnetization directed along\ncaxis on the electron charges. The reason for this is that\nthe latter magnetic ﬁeld can be estimated as µU/Vecwhere\nµUis magnetic moment per Uranium atom and Vecis the\nelementary cell volume. For the URhGe this is of the or-\nder 100 Gauss and it is even less in the case of UCoGe.\nThe scale of the measured upper critical ﬁelds noticeably\nexceeds such a small value."    },    {        "title": "1602.00439v1.Hubbard_models_with_nearly_flat_bands__Ground_state_ferromagnetism_driven_by_kinetic_energy.pdf",        "content": "arXiv:1602.00439v1  [cond-mat.str-el]  1 Feb 2016Hubbard models with nearly ﬂat bands:\nGround-state ferromagnetism driven by kinetic energy\nPatrick Müller,1Johannes Richter,1and Oleg Derzhko2,3, 1, 4\n1Institut für theoretische Physik, Otto-von-Guericke-Uni versität Magdeburg, P.O. Box 4120, 39016 Magdeburg, German y\n2Institute for Condensed Matter Physics, National Academy o f Sciences of Ukraine, Svientsitskii Street 1, 79011 L’viv, Ukraine\n3Department for Theoretical Physics, Ivan Franko National U niversity of L’viv, Drahomanov Street 12, 79005 L’viv, Ukra ine\n4Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy\n(Dated: February 2, 2016)\nWe consider the standard repulsive Hubbard model with a ﬂat l owest-energy band for two one-\ndimensional lattices (diamond chain and ladder) as well as f or a two-dimensional lattice (bilayer)\nat half ﬁlling of the ﬂat band. The considered models do not fa ll in the class of Mielke-Tasaki ﬂat-\nband ferromagnets, since they do not obey the connectivity c onditions. However, the ground-state\nferromagnetism can emerge, if the ﬂat band becomes dispersi ve. To study this kinetic-energy-\ndriven ferromagnetism we use perturbation theory and exact diagonalization of ﬁnite lattices. We\nﬁnd as a typical scenario that small and moderate dispersion may lead to a ferromagnetic ground\nstate for suﬃciently large on-site Hubbard repulsion U > U c, where Ucincreases monotonically\nwith the acquired bandwidth. However, we also observe for so me speciﬁc parameter cases, that (i)\nferromagnetism appears at already very small Uc, (ii) ferromagnetism does not show up at all, (iii)\nthe critical on-site repulsion Ucis a nonmonotonic function of the bandwidth, or that (iv) a cr itical\nbandwidth is needed to open the window for ground-state ferr omagnetism.\nPACS numbers: 71.10.-w, 75.10.Lp, 75.10.Jm\nKeywords: Hubbard model, ﬂat band, ferromagnetism\nI. INTRODUCTORY REMARKS\nExplaining ferromagnetism from a simple model of\nitinerant electrons such as the standard Hubbard model is\na long-standing problem in the condensed matter theory.\nAmong many routes leading to ferromagnetism the so-\ncalled ﬂat-band ferromagnetism of Mielke and Tasaki1–3\nis of special interest. On the one hand, many results\nfor Mielke-Tasaki ﬂat-band ferromagnetism have been\nobtained rigorously. On the other hand, this mecha-\nnism is important for material design, since it opens\ninteresting possibilities to obtain ferromagnetic mate-\nrials in which magnetic atoms are completely missing.\nIn brief, the mechanism of this kind of ferromagnetism\nlooks as follows.1–3Flat-band ground states (i.e., the\none-particle states from completely dispersionless band\nwhich is the lowest-energy one) can be considered as\none-particle states which are localized within small trap-\nping cells on a lattice.4–6Therefore, exact many-electron\nground states at low electron densities can be constructed\nsimply by ﬁlling the traps. Importantly, in the case of\nconnected (overlapping) traps, electrons being in sym-\nmetric spin states avoid the on-site Hubbard repulsion,\nand, as a result, these states remain within the ground-\nstate manifold for U >0with aU-independent energy.\nThus, the (degenerate) ground state consists of a set\nof ferromagnetic clusters. If the electron density ex-\nceeds a threshold value, a macroscopic wrapping ferro-\nmagnetic cluster appears and ferromagnetism dominates\nthe ground-state properties of thermodynamically large\nsystems.1–3,5,7–9This ferromagnetism is robust against\nperturbation, i.e., the ferromagnetic state remains sta-\nble for slightly perturbed models which have a moderatechange in the hopping integrals leading to a slightly dis-\npersive one-electron band.10,11\nThe above description of the emergence of ground-\nstate ferromagnetism is based on the assumption, that\nthe trapping cells have common sites, i.e., the so-called\nconnectivity condition is satisﬁed for the localized one-\nelectron states. In other words, the localized states over-\nlap and this was essential for the proofs in Refs. 1,2.\nOn the other hand, there are lattices which have lowest-\nenergy ﬂat bands but the traps do not have common\nsites (nonoverlapping or isolated traps). Those ﬂat-band\nlattices cannot support the above described mechanism\nfor ferromagnetism, since the trapped electrons cannot\nbe in contact with each other, and, thus are unable to\ncorrelate. Hence, ﬂat-band Hubbard models with iso-\nlated traps do not exhibit ferromagnetism at zero tem-\nperature, rather there is a macroscopically degenerate\n(i.e., the degeneracy grows exponentially with the system\nsize) ground-state manifold, where paramagnetic states\ndominate.12–15However, the macroscopically degenerate\nground-state manifold is very sensitive to small pertur-\nbations which may lead to subtle eﬀects of violations of\nthe ﬂat-band conditions. This scenario has been inves-\ntigated in Ref. 16 for the speciﬁc example of the frus-\ntrated diamond chain. It was demonstrated that the\nmacroscopically degenerate ground-state manifold with\nall traps ﬁlled by electrons results in a non-magnetic\nzero-temperature phase,13,16but small deviations from\nthe ideal ﬂat-band geometry of hopping integrals (which\nmakes the ﬂat band slightly dispersive) lead to a fully\npolarized ferromagnetic many-electron ground state if\nU > U c. The value of Ucdepends on the strength of\nthe deviation from the ideal geometry. Note that another2\nroute to ground-state ferromagnetism without connectiv-\nity condition in the ﬂat band was discussed in Ref. 17.\nIn the present paper we broaden and generalize our\nprevious study on the dispersion-driven ferromagnetism\nin ﬂat-band Hubbard systems.16As already mentioned\nabove, those studies referred to one particular lattice,\nnamely to an azurite-like18diamond-Hubbard chain.\nMoreover, analytical calculations presented in Ref. 16\nwere restricted to the fourth-order perturbation the-\nory for a two-cell chain. In the present study we ex-\ntend the analytical calculations to higher-orders pertur-\nbation theory this way validating the previous results.\nMore importantly, we consider other lattices with iso-\nlated trapping cells, the one-dimensional ladder and the\ntwo-dimensional bilayer. These new lattices have more\ndegrees of freedom to constitute deviations from the ideal\nﬂat-band geometry. Thus, we will demonstrate that\nthe dispersion-driven ferromagnetism is a rather general\nmechanism to establish ferromagnetic ground states in\nHubbard models having isolated trapping cells in the ﬂat-\nband limit. In addition to the analytical perturbation\ntheory, we also perform extensive exact-diagonalization\nstudies. Our analysis will, on the one hand, conﬁrm the\nconclusions derived from the study of the Hubbard dia-\nmond chain.16On the other hand, we will discuss further\nconsequences of deviations from the ideal ﬂat-band ge-\nometry on ferromagnetism. In particular, we ﬁnd that\nin some cases the required threshold on-site repulsion Uc\nmay be quite small, whereas in other cases ferromagnetic\nground states do not appear at all. There are also cases\nwhen ferromagnetic ground states appear only, if the ac-\nquired bandwidth exceeds a threshold, and then Ucbe-\ncomes a nonmonotonic function of the bandwidth. Our\nﬁndings are compactly collected in phase diagrams, ob-\ntained both by analytical treatment and exact diagonal-\nization, which indicate the regions of dispersion-driven\nground-state ferromagnetism.\nThe paper is organized as follows. After a brief de-\nscription of the models to be considered (Sec. II) and the\nmethods to be used (Sec. III) we pass to a discussion\nof the obtained results for the diamond chain (Sec. IV),\nthe ladder (Sec. V), and the bilayer (Sec. VI). We brieﬂy\nsummarize our results in Sec. VII. Several appendices\npresent some lengthy formulas which are relevant for the\ndiscussion in the main text of the paper.\nII. MODELS\nWe consider the standard repulsive one-orbital Hub-\nbard model with the Hamiltonian\nH=/summationdisplay\nσ=↑,↓H0,σ+HU,\nH0,σ=/summationdisplay\n(ij)tij/parenleftBig\nc†\ni,σcj,σ+c†\nj,σci,σ/parenrightBig\n, tij>0,\nHU=U/summationdisplay\nini,↑ni,↓, U >0,(2.1)+1,1+1,2 ,2\n,1,323\n1 31\nmm m\nm\nmt\nt tt t\n,1 +1,1+1,2 ,2\n2\n1122\n12\n21\nm mm m\nt\ntt\ntt\ntt\nttt\n1122\n21122\nx         y x     y m      ,m m  ,m mm\n,1,2\n  −1,1  +1     ,1\nFIG. 1: (Color online) Lattices considered in the present pa -\nper: The frustrated diamond chain, the frustrated two-leg\nladder, and the frustrated bilayer (from top to bottom). The\nsites are enumerated by two indexes m,i: The ﬁrst one enu-\nmerates the cells, m= 1,...,N, and the second one enumer-\nates the sites within a cell, i= 1,2,3(diamond) and i= 1,2\n(ladder and bilayer). The hopping integral for the vertical\nbond ist2, whereas the hopping integral along the bond con-\nnecting the sites m,iandm+1,jis denoted by tij, see also the\nmain text. For ideal ﬂat-band geometry tij=tand2t < t2\n(diamond and ladder) or 4t < t2(bilayer).\nwhere generally accepted notations are used in Eq. (2.1).\nWe investigate the Hubbard model (2.1) on two one-\ndimensional and one two-dimensional N-site lattices\nwhich are shown in Fig. 1, namely the frustrated diamond\nchain, the frustrated two-leg ladder, and the frustrated\nbilayer. In case of ideal ﬂat-band geometry all hopping\nintegrals tij=tare equal, except the hopping integral on\nthe vertical bond t2. Then one of the one-electron bands\nis strictly ﬂat and it becomes the lowest one, if t2is suf-\nﬁciently large. The localized-electron states are then lo-\ncated (trapped) on the vertical t2-bonds. Obviously, the\ntrapping cells do not have common sites, the connectiv-\nity condition is violated, and the zero-temperature state\nin the subspaces with n≤ Nelectrons are nonmagnetic.\nFrom Fig. 1 it is obvious, that the number of trapping\ncellsNfor the diamond chain and the ladder/bilayer is\nN=N/3andN=N/2, respectively.\nWe consider deviations from the ideal ﬂat-band geom-\netry of the following form: For the diamond chain, fol-\nlowing Ref. 16, we set t13=t32=t1∝ne}ationslash=t23=t31=t3,3\nt1+t3= 2t < t2(azurite-like geometry;18for more gen-\neral deformations see Ref. 19). It is convenient to param-\neterize the azurite-like distortion as follows:\nt1=t(1+δ), t3=t(1−δ);\nt=t1+t3\n2, δ=t1−t3\nt1+t3. (2.2)\nFor the ladder/bilayer t11,t12,t21, andt22may be diﬀer-\nent, but we assume t11+t12+t21+t22= 4tand2t < t2\n(ladder) or 4t < t2(bilayer). Again it is convenient to\nintroduce the following parameterization:\nt11=tl(1+δl), t12=tf(1+δf),\nt21=tf(1−δf), t22=tl(1−δl);\ntl=t11+t22\n2, δl=t11−t22\nt11+t22,\ntf=t12+t21\n2, δf=t12−t21\nt12+t21(2.3)\nwithtl+tf= 2t.\nIn the distorted systems the lowest ﬂat band with en-\nergyε1acquires a dispersion, i.e., ε1→ε1(κ), resulting\nin a nonzero bandwidth W1>0. In Ref. 16, the acquired\ndispersion was characterized by a parameter W1/w2,\nwherew2denotes the bandwidth of the dispersive bands\nfor the ideal ﬂat-band geometry (note that for the dia-\nmond chain there are two dispersive bands with identi-\ncal bandwidth). Furthermore, for the diamond chain we\nhaveW1≈2(t3−t1)2/t2,w2≈2(t3+t1)2/t2and there-\nforeW1/w2≈Ω2, whereΩ≡ |(t3−t1)/(t3+t1)|used in\nRef. 16 equals to |δ|, cf. Eq. (2.2). However, since for the\nHubbard ladder/bilayer the acquired bandwidth is not\nthe only relevant parameter that controls the emergence\nof ferromagnetism, we prefer to use throughout this paper\nthe above introduced parameters tandδfor the diamond\nchain and tl,tf,δl, andδffor the ladder/bilayer.\nIII. METHODS\nIn our study we use an analytical perturbation-theory\napproach and numerical exact diagonalization. Let us\nbrieﬂy explain these methods. The starting point of the\nperturbation theory is the splitting of the Hamiltonian\nHof the problem at hand into the main part (unper-\nturbed Hamiltonian) H0and the perturbation V, i.e.,\nH=H0+V. Then we use the perturbation-theory for-\nmulas given in Ref. 20 (see also Appendix A) to deter-\nmine the inﬂuence of the perturbation Von the degen-\nerate ground-state manifold. Since t2>0is the largest\nhopping integral and U >0, the main part consists of\nthe hopping terms on the vertical bonds and all on-\nsite repulsion terms. The perturbation consists of all\nother hopping terms. Next we have to ﬁnd all eigen-\nstates and eigenvalues of the unperturbed Hamiltonian\nH0. ForNsites and nelectrons there are altogether\nCn\n2N= (2N)!/[n!(2N−n)!]eigenstates. For example, forn=N= 2,3,4,5ladder problems we have 28, 220,\n1820, 15504 eigenstates, respectively. In the considered\nregime, i.e., dominating positive t2,U >0is suﬃciently\nlarge, and n=N, the ground state is 2n-fold degenerate,\ni.e., 4-, 8-, 16-, 32-fold degenerate for n=N= 2,3,4,5.\nIt has the form:\n|GS∝an}bracketri}ht=l†\n1,σ1...l†\nn,σn|vac∝an}bracketri}ht,\nl†\nm,σm=1√\n2/parenleftBig\nc†\nm,1,σm−c†\nm,2,σm/parenrightBig\n. (3.1)\nThe choice of the concrete linear combinations of states\n(3.1) used as a starting point of perturbation theory is\nrelated to the model with perturbation. Supposing an\neﬀective magnetic Heisenberg model for the low-energy\ndegrees of freedom,16the choice of ground states of the\nunperturbed Hamiltonian H0which account the SU(2)\nsymmetry of the Hubbard Hamiltonian is straightfor-\nward, for more details see Appendix B. The resulting\nperturbation-theory formulas up to the sixth order are\ncollected in Appendix A (see also Appendices C, D, and\nE). It is in order to mention here, that in the small- U\nlimit, in addition to the states (3.1), also states with two\nelectrons in one cell, become relevant. As a result, the\nperturbation theory starting from the set of states (3.1)\nmay fail for U→0, see below.\nTo perform the fourth and sixth order perturbation\ntheory we use the symbolic computation software Math-\nematica . To implement the symbolic calculation we used\nthe SNEG package, see Ref. 21, for Mathematica. The\npackage handles the non-commutative multiplication of,\ne.g., fermionic creation and annihilation operators. This\nis required to perform the perturbation theory in higher\norder for larger Hubbard clusters. For a compact sketch\nof the procedure see Appendix F.\nFor the numerical exact diagonalization we use J. Schu-\nlenburg’s spinpack .22,23This code allows the calculation\nof the ground state for the Hubbard model with a half-\nﬁlled lowest band up to N= 20sites. Thus, by consid-\nering various system sizes the ﬁnite-eﬀects can be esti-\nmated. The comparison of the results obtained by two\ndiﬀerent approaches ﬁnally allows to get a consistent de-\nscription of the ground-state phases of the considered\nHubbard systems.\nIV. DIAMOND CHAIN\nThe Hubbard model Hamiltonian on the diamond\nchain is given in Eq. (2.1) with the following explicit form\nforH0,σ:\nH0,σ=/summationdisplay\nm/bracketleftBig\nt2c†\nm,1,σcm,2,σ\n+t1/parenleftBig\nc†\nm,1,σcm,3,σ+c†\nm,3,σcm+1,2,σ/parenrightBig\n+t3/parenleftBig\nc†\nm,2,σcm,3,σ+c†\nm,3,σcm+1,1,σ/parenrightBig\n+H.c./bracketrightBig\n,(4.1)4\nsee Fig. 1. Eq. (4.1) corresponds to an azurite-like\ndeformation.18Furthermore, we assume half ﬁlling of the\nlowest nearly ﬂat one-electron band, i.e., the number of\nelectrons equals the number of cells n=N.\nExtensive exact-diagonalization calculations for this\nmodel were reported in Ref. 16. However, the analyti-\ncal treatment by perturbation theory was restricted to\nfourth-order calculations for the two-cell diamond chain\nwith open boundary conditions consisting of N= 5\nsites. (Note, that for the special diamond-chain geome-\ntry the second-order perturbation theory is not suﬃcient\nto describe ground-state ferromagnetism.16) In this pa-\nper we present the sixth-order perturbation theory and\nconsider also a larger cluster consisting of three cells in\nfourth-order perturbation theory. That allows to vali-\ndate the previous lower-order approach and promises a\nbetter agreement with exact diagonalization for larger\ndeviations from the ideal ﬂat-band geometry.\nThe results for the triplet and singlet energies calcu-\nlated for the cluster of N= 5sites with n= 2electrons\nup to the sixth order,\nEt=−2t2+E(2)+E(4)\nt+E(6)\nt+...,\nEs(U) =−2t2+E(2)+E(4)\ns(U)+E(6)\ns(U)+...,(4.2)\nare given in Appendix C. From the obtained data one\ncan see that with increasing of the order of perturbation-\ntheory calculations the analytical results for the triplet\nand singlet energies monotonically approach the exact-\ndiagonalization data from above. The critical on-site re-\npulsionUcis determined from the equation Et=Es(Uc).\nIn fourth-order perturbation-theory we get a compact\nformula16\nU(4)\nc\nt2=√\n16+65δ2+9|δ|\n1−δ2|δ|. (4.3)\nEq. (4.3) implies that in fourth order Uc/t2depends\nonly on the deviation from the ideal ﬂat-band geome-\ntry controlled by δ, but not on tort2. Unfortunately, in\nsixth order U(6)\ncobtained as a solution of the equation\nE(4)\nt+E(6)\nt=E(4)\ns(U(6)\nc)+E(6)\ns(U(6)\nc)has to be calculated\nnumerically, and cannot be presented in a compact ana-\nlytical form. By contrast to U(4)\nc, the sixth-order result\nU(6)\nc/t2weakly depends on t2, which was also found in\nour exact-diagonalization results. The corresponding re-\nsults for U(4)\ncandU(6)\ncare shown in Fig. 2. It is evident,\nthat the diﬀerence between the values of U(4)\ncandU(6)\ncat\nleast for small δ, where the perturbation theory is valid,\nis small (the diﬀerence in Fig. 2 becomes only visible if δ\nexceeds 0.4). Thus, we conﬁrm that the simple equation\n(4.3) describes the phase boundary surprisingly well.\nAnother way to extend the previous perturbation-\ntheory calculations of Ref. 16 is to enlarge the cluster\nsizes used for the perturbation theory. For that we con-\nsidern= 3electrons on the three-cell diamond chain\nwith open boundary conditions which has N= 8sites.\nAlready in fourth order the perturbation theory becomesFIG. 2: (Color online) Phase diagram for the Hubbard dia-\nmond chain. Ferromagnetic ground states appear for U > U c.\nUcis shown as a function of δ,t= 1, see Eq. (2.2). The var-\nious critical lines Uc(δ)are obtained by perturbation theory\nand exact diagonalization.\nmore ambitious, since we have to take into account much\nmore states, see Appendix C. Remarkably, for the larger\ncluster we get the same value of U(4)\ncas given in Eq. (4.3).\nOur results are summarized in Fig. 2, where we\nalso show some exact-diagonalization results obtained\nearlier.16This ﬁgure provides evidence, that the sixth-\norder perturbation-theory calculations ( N= 5) almost\ndo not change the predictions for Uc(δ)according to\nEq. (4.3), although there is a weak dependence of Uc/t2\nont2in agreement with exact-diagonalization data (com-\npare the curves PT6 for t2= 3andt2= 6in Fig. 2). The\nfact that Eq. (4.3) has been obtained now from calcula-\ntions for both two-cell and three-cell diamond chains (i.e. ,\nforN= 5andN= 8), also explains the good agreement\nof Eq. (4.3) with exact-diagonalization results for longer\nchains (e.g., for N= 6cells, see Fig. 2). Finally, we em-\nphasize again that our new results demonstrate that the\nformula for Ucgiven in Eq. (4.3) provides a simple and\nsuﬃciently precise criteria for emergence of ground-state\nferromagnetism in the Hubbard diamond chain.\nV. LADDER\nNext we consider as a new example for a ﬂat-band\nmodel with isolated trapping cells the Hubbard model\non a frustrated ladder, see Fig. 1. We point out at the\nbeginning that, by contrast to the diamond chain, there\nis no intermediate site between two trapping cells. The\nexplicit form for H0,σin Eq. (2.1) is\nH0,σ=/summationdisplay\nm/parenleftBig\nt2c†\nm,1,σcm,2,σ\n+t11c†\nm,1,σcm+1,1,σ+t12c†\nm,1,σcm+1,2,σ\n+t21c†\nm,2,σcm+1,1,σ+t22c†\nm,2,σcm+1,2,σ+H.c./parenrightBig\n,(5.1)5\nsee Fig. 1.\nUsing the notations of Eq. (2.3), the one-electron dis-\npersion relations for this model can be written in a com-\npact manner as follows:\nε1,2(κ) = 2tlcosκ\n∓/radicalBig\n(t2+2tfcosκ)2+4t2\nlδ2\nlcos2κ+4t2\nfδ2\nfsin2κ.(5.2)\nFlat-band geometry occurs when t11=t12=t21=t22=\ntortl=tf=t,δl=δf= 0and2t < t2. Thenε1(κ) =\nε1=−t2andε2(κ) =t2+4tcosκ > ε1.\nWe consider a quite general deviation from the ideal\nﬂat-band geometry, and assume only that t11+t12+t21+\nt22= 4tortl+tf= 2tand2t < t2. Thus after ﬁxing\ntlandtfwith the restriction tl+tf= 2t < t2we are\nleft with two free parameters, δlandδf[see Eq. (2.3)],\nconstituting a two-dimensional parameter region. Except\nthe general case of deformations, we will also consider two\nspecial deformations, (i) a symmetric deformation with\nt11=t22,t12=t21andt11∝ne}ationslash=t12(tl∝ne}ationslash=tf,δl=δf= 0)\nand (ii) a semi-symmetric deformation with t11=t12,\nt21=t22andt11∝ne}ationslash=t21(tl=tf=t,δl=δf=δ∝ne}ationslash= 0)\nwhich is identical to t11=t21,t12=t22andt11∝ne}ationslash=t12\n(tl=tf=t,δl=−δf=δ∝ne}ationslash= 0), since all results depend\nonly onδ2\nlandδ2\nf, see, e.g., Eq. (5.2). For case (i) the\ndispersion relation Eq. (5.2) becomes\nε1,2(κ) =∓t2+2(tl∓tf)cosκ, (5.3)\nwhereas for case (ii) translates into\nε1,2(κ) = 2tcosκ∓/radicalBig\n(t2+2tcosκ)2+4t2δ2.(5.4)\nIt is worth noting that the acquired bandwidth of the for-\nmer ﬂat band due to the symmetric deformation may be\nlarger than due to the semi-symmetric one. On the other\nhand, while the symmetric deformation does not lead to\nferromagnetic ground states at all, see below, the semi-\nsymmetric one produces ferromagnetic ground states for\nvery small U > U c, see below. Obviously, the acquired\nbandwidth as the only relevant parameter is insuﬃcient\nto characterize the capability to obtain ground-state fer-\nromagnetism.\nIn what follows we ﬁrst discuss perturbation-theory re-\nsults in comparison with exact-diagonalization data for\nladders up to N= 4cells (N= 8sites) and then present\nall analytical ﬁndings along with exact diagonalization\nforN= 12,16,20(N= 6,8,10) in phase diagrams.\nA. Two electrons and two cells\nWe begin with the case of n= 2electrons on the ladder\nofN= 2cells with open boundary conditions imposed.\nPerturbation-theory calculations for the energies of the\ntriplet state and the singlet state can be easily obtained\nby symbolic computation up to the sixth order:\nEt=−2t2+E(2)\nt+E(4)\nt+E(6)\nt+...,\nEs(U) =−2t2+E(2)\ns(U)+E(4)\ns(U)+E(6)\ns(U)+....(5.5)Here the second-order corrections are as follows:\nE(2)\nt=−t2\nlδ2\nl+t2\nfδ2\nf\nt2,\nE(2)\ns(U) =−(tl−tf)2\nt2−2t2\nlδ2\nl+t2\nfδ2\nf\n2t2+U−8(tl−tf)2\nU.(5.6)\nThe explicit lengthy expressions for the higher-order cor-\nrections are given in Appendix D. Typical dependences\nof low-lying energies on Uare shown in Figs. 3(a), 3(b),\nand 3(c) for a particular general deformation, a symmet-\nric deformation, and a semi-symmetric deformation, re-\nspectively.\nThe conclusions obtained from the formulas and plots\n(Fig. 3) of the singlet and triplet energies are as follows:\nIn the small- Ulimit the perturbation theory may fail,\ncf. Figs. 3(a) and 3(b). The reason for this has been\nmentioned above already: In the small- Ulimit some rel-\nevant excited states approach the ground-state manifold.\nThe deviation from the ideal ﬂat-band geometry leads\nto more drastic eﬀects and also to a larger diversity in\nthe energy dependence on Uthan for the diamond chain\nconsidered in the previous section. The behavior of Et\nandEs(U)shown in Fig. 3(a) for the general case quali-\ntatively resembles that for the diamond chain (cf. Fig. 8\nin Appendix C). On the other hand, the symmetric and\nsemi-symmetric cases are totally unlike. Namely, as long\nas the perturbation theory converges, for the symmetric\ndeformation, case (i), the singlet energy (circles and blue\ncurves) is always lower than the triplet energy (triangles\nand red curves), Es< Et=−2t2, see Fig. 3(b). Note\nthat all exact-diagonalization data also yield Es< Et\nfor the symmetric case. For the semi-symmetric case the\ntriplet energy becomes the lowest one, Et< Es(U), if\nUexceeds a very small critical value Uc, see Fig. 3(c).\n[For the case shown in Fig. 3(c) exact diagonalization\ngivesUc≈0.015and the perturbation-theory result is\nU(6)\nc= 0.] That means, ferromagnetism does not ap-\npear at all for the symmetric deformation, whereas for\nthe semi-symmetric case only a very small Uis required\nto promote its appearance. Next important diﬀerence in\ncomparison to the diamond-chain case is related to the\nenergy scale (compare Figs. 3 and 8): The splitting of\ntriplet and singlet for the ladder occurs already in the\nsecond order (and only in the fourth order for the dia-\nmond chain). This can be traced back to the diﬀerence in\nlattice geometries. Thus, for the ladder the second-order\nperturbation theory already provides useful results.\nThe above described features of the energy depen-\ndences on Ucan be understood by a more detailed anal-\nysis of the perturbation-theory treatment, see Appen-\ndices A and B. For that we consider the action of the per-\nturbation Von the triplet and singlet states, i.e., V|t,±1∝an}bracketri}ht,\nV|t,0∝an}bracketri}ht, andV|s∝an}bracketri}ht. The results depend on the symme-\ntry of the imposed deformation. Thus, for the symmet-\nric case V|t∝an}bracketri}ht= 0, butV|s∝an}bracketri}ht ∝(l†\na,↑l†\na,↓+l†\nb,↑l†\nb,↓)|vac∝an}bracketri}ht.\nAs a consequence, the unperturbed triplet energy −2t2\nremains unchanged after switching on V, whereas the6\nFIG. 3: (Color online) Energies of low-lying states (triple t –\nred, singlet – blue) as a function of on-site repulsion U(per-\nturbation theory up to sixth order and exact-diagonalizati on\ndata) for n= 2electrons on the ladder of N= 2cells (open\nboundary conditions). (a) t2= 3,t11= 0.85,t12= 0.95,\nt21= 1,t22= 1.2(general deformation). (b) t2= 3,\nt11=t22= 1.1,t12=t21= 0.9(symmetric deformation). (c)\nt2= 3,t11=t21= 1.1,t12=t22= 0.9(semi-symmetric de-\nformation); exact diagonalization yields Uc≈0.015, whereas\nthe perturbation-theory prediction is U(6)\nc= 0.\nunperturbed singlet energy −2t2decreases after switch-\ning onVand ferromagnetism cannot arise. Moreover,\nthe state V|s∝an}bracketri}htoverlaps with “dangerous” excited states\nofH0(which contain l†\na,↑l†\na,↓,l†\nb,↑l†\nb,↓and have the en-\nergy−2t2+UforU→0) leading to the failure of\nthe perturbation theory in the small- Ulimit. On theother hand, for the semi-symmetric case V|t∝an}bracketri}htcontains\nc†\nm,1,σc†\nm,2,σ|vac∝an}bracketri}htor(c†\nm,1,↑c†\nm,2,↓+c†\nm,1,↓c†\nm,2,↑)|vac∝an}bracketri}ht,\nwhereas V|s∝an}bracketri}ht ∝(c†\nm,1,↑c†\nm,1,↓−c†\nm,2,↑c†\nm,2,↓)|vac∝an}bracketri}ht. Since\nthe state (c†\nm,1,↑c†\nm,1,↓−c†\nm,2,↑c†\nm,2,↓)|vac∝an}bracketri}htis orthogonal\nto the dangerous excited states of H0, the perturbation\ntheory does not fail in the small- Ulimit. Moreover, the\nstatesV|t∝an}bracketri}htandV|s∝an}bracketri}hthave the same overlap integral with\nthe excited states of H0with the energies 0 and U, re-\nspectively. Therefore, the decrease of the triplet energy\nexceeds the decrease of the singlet energy instantaneously\nasU >0, i.e., ferromagnetism appears for inﬁnitesimally\nsmall positive U.\nIn second order the perturbation theory yields a com-\npact formula for the critical value of on-site repulsion Uc.\nUsing Eq. (5.6) we get\nU(2)\nc\nt2=5|tl−tf|+/radicalBig\n9(tl−tf)2+16(t2\nlδ2\nl+t2\nfδ2\nf)\n−(tl−tf)2+t2\nlδ2\nl+t2\nfδ2\nf\n×|tl−tf|.(5.7)\nObviously, for symmetric deformations, when tl∝ne}ationslash=tfand\nδl=δf= 0, Eq. (5.7) gives for U(2)\nc=−8t2<0, that\nis consistent with the absence of ferromagnetism in this\ncase. It is also obvious, that formula (5.7) yields U(2)\nc= 0\nfortl=tf, i.e., for t11−t12−t21+t22= 0. That criterion,\ntl=tf, holds for semi-symmetric deformations, where in\naddition also δl=δfis valid. However, in higher-order\nperturbation theory as well as in exact diagonalization we\nﬁnd that the constraint tl=tfdoes not imply Uc= 0,\nratherUcmay become large for the general case δl∝ne}ationslash=δf,\nifδlorδfbecome of the order of unity, see Fig. 6.\nSupposing that the energies behave smoothly as chang-\ning deformations, we can expect that there is a ﬁnite pa-\nrameter region in the vicinity of the symmetric case with-\nout ground-state ferromagnetism. Indeed, for tl∝ne}ationslash=tfthe\nsecond-order formula (5.7) leads to an elliptic shape in\ntheδl–δfplane given by\n/parenleftbiggtl\ntl−tfδl/parenrightbigg2\n+/parenleftbiggtf\ntl−tfδf/parenrightbigg2\n= 1. (5.8)\nWe illustrate this behavior in Fig. 4, where we also show\na few points obtained by exact diagonalization which\nare in qualitative agreement with the predictions from\nEq. (5.8). It is worthwhile to remark that Eq. (5.8) re-\nmains unaltered if interchanging tl↔tfandδl↔δf\n(this symmetry is also evident in Fig. 4). However, exact-\ndiagonalization data shown by symbols in Fig. 4 do not\nshow this symmetry present in the second-order results,\ni.e., it is not generally present in the model, cf., e.g.,\nEq. (5.2).\nB. Three (four) electrons and three (four) cells\nLet us discuss brieﬂy the perturbation theory for larger\nclusters. In the case of three electrons on the ladder7\nFIG. 4: (Color online) There is no ferromagnetic ground\nstates for the Hubbard ladder in the region around the origin\nof the plane δl–δf[t11−t12−t21+t22= 2(tl−tf)/negationslash= 0]. Ana-\nlytical predictions based on the second-order perturbatio n-\ntheory calculations (5.7) (lines) are compared with exact-\ndiagonalization data for N= 16,t2= 3(symbols) for several\nvalues of tlandtf,tl+tf= 2.\nof three cells we face a 23-fold degenerate ground state,\nwhich consists of the quadruplet |q∝an}bracketri}ht(total spin is 3/2)\nand two doublets |d1∝an}bracketri}htand|d2∝an}bracketri}ht(total spin is 1/2). We\nare interested in the energies Eq,Ed1, andEd2. In Ap-\npendix D, we provide explicit expressions for these ener-\ngies\nEq=−3t2+E(2)\nq+E(4)\nq+...,\nEdi(U) =−3t2+E(2)\ndi(U)+E(4)\ndi(U)+...,\ni= 1,2.(5.9)\nIn the case of four electrons on the ladder of four cells\nwe face a 24-fold degenerate ground state, which consists\nof the quintuplet |Q∝an}bracketri}ht(total spin is 2), three triplets |t1∝an}bracketri}ht,\n|t2∝an}bracketri}ht,|t3∝an}bracketri}ht(total spin is 1), and two singlets |s1∝an}bracketri}ht,|s2∝an}bracketri}ht(total\nspin is 0). In Appendix D, we provide explicit expressions\nfor their energies\nEQ=−4t2+E(2)\nQ+E(4)\nQ+...,\nEti(U) =−4t2+E(2)\nti(U)+E(4)\nti(U)+...,\ni= 1,2,3,\nEsj(U) =−4t2+E(2)\nsj(U)+E(4)\nsj(U)+...,\nj= 1,2.(5.10)\nWe report corresponding results for the energies up to\nthe fourth order along with exact-diagonalization data\nfor the general, symmetric, and semi-symmetric defor-\nmations for n=N= 3andn=N= 4in Appendix D.\nThe main features of these results resemble strongly theones discussed in the previous subsection for n=N= 2.\nTherefore, the main conclusions obtained from those data\nfor the energies of larger cells are consistent with those\ndiscussed in Sec. VA for two cells. Most remarkably,\nwithin the second-order perturbation theory, the critical\nvalueU(2)\ncfor the three-cell and four-cell clusters coin-\ncide with U(2)\ncfor the two-cell cluster, i.e., it is given by\nEq. (5.7).\nLet us ﬁnally mention that within the perturbation\ntheory for N= 4cells the fully polarized ferromagnetic\nstate (it is a quintuplet for N= 4) is in competition with\ntriplet and singlet states. We ﬁnd, cf. Fig. 11, that either\na singlet or the ferromagnetic quintuplet is the ground\nstate. This ﬁnding, that the fully polarized ferromagnetic\nstate competes with a nonmagnetic singlet state (but not\nwith partially polarized states) is supported by exact-\ndiagonalization data obtained for systems with an even\nnumber of cells N>4.\nC. Phase diagram\nIn this subsection we collect analytical and numerical\nﬁndings to construct the ground-state phase diagrams of\nthe Hubbard ladder. According to Eq. (2.3), there are\nthree parameters which characterize the ladder, i.e., tl\nandtfwithtl+tf= 2t < t2,δl, andδf. We set t2= 3,\ntl+tf= 2. After ﬁxing tlandtfwe are left with two\nfree parameters δlandδf. We consider the ﬁrst quad-\nrant of positive δlandδfin theδl–δfplane. We move\nthrough the quadrant by straight lines in the horizon-\ntal direction ( δfis ﬁxed,δlvaries), in the vertical direc-\ntion (δlis ﬁxed,δfvaries), as well as along the diagonal\nδl=δf=δ. Certainly perturbation-theory results are\nreasonable only for small deviations from the ideal ﬂat-\nband geometry. However, there are no such restrictions\nfor exact-diagonalization data.\nWe begin with a quite general case assuming tl=\n1.025,tf= 0.975andδl= 0. The dependence of Uc\nonδfis reported in Fig. 5. The ground state is ferro-\nmagnetic above the curves Uc(δf); this region is denoted\nas FM. In this case, the dependence of Ucon the ac-\nquired bandwidth is a nonmonotonic function: For small\nδfferromagnetism does not appear at all [in agreement\nwith Eq. (5.8)]; increasing δfbeyond a threshold value\nδf1ferromagnetism sets in and Ucdecreases with growing\nδf. Second-order perturbation theory, Eq. (5.8), predicts\nδf1≈0.051, exact diagonalization for N= 16 yields\nδf1≈0.053. Beyond δf≈0.4the critical repulsion Uc\nstarts to increase with increasing of δf. This behavior\nis obtained from both the fourth-order perturbation the-\nory and exact diagonalization for diﬀerent system sizes\nwith open and/or periodic boundary conditions imposed.\nThe second-order perturbation theory gives qualitatively\ncorrect results only for small δf<0.4. From exact-\ndiagonalization data for N= 16it is obvious that there\nis again a threshold value δf2(forN= 16 we found\nδf2≈3.25) above which no ferromagnetism appears.8\nFIG. 5: (Color online) Phase diagram in the quarter plane δf\n–U/t2for the ladder with t2= 3,tl= 1.025,tf= 0.975, and\nδl= 0obtained by perturbation-theory calculations and by\nexact diagonalization for N= 6,8,12,16with open and/or\nperiodic boundary conditions.\nFig. 5 illustrates a quite subtle interplay of the hopping-\nintegral geometry and the on-site Hubbard repulsion re-\nquired for establishing of ground-state ferromagnetism.\nNext we pass to the case tl=tf= 1. The depen-\ndences of Uconδf, onδl=δf=δ, and on δlare re-\nported in panels (a), (b), and (c) in Fig. 6, respectively.\nThe ground state is ferromagnetic above the curves Uc(δ);\nthis region is denoted as FM. We recall that in the case\ntl=tffrom Eq. (5.7) we get U(2)\nc= 0; nonzero values\nofUccome only from higher-order (in fact, fourth-order)\ncalculations. Furthermore, for the semi-symmetric defor-\nmation, i.e., δl=δf=δ, the perturbation theory yields\nU(4)\nc= 0. Obviously, higher-order processes should lead\nto ﬁnite values for Uc, as it is indicated by the exact-\ndiagonalization data shown Fig. 6(b).\nAs can be seen in Figs. 6(a) and 6(c), analytical results\nwhich refer to the case of N= 3,4cells with open bound-\nary conditions and exact-diagonalization data which refer\nto the case of N= 6,8,10cells are in a reasonable agree-\nment. By contrast to the parameter situation shown in\nFig. 5, in all cases presented in Fig. 6 ground-state fer-\nromagnetism can be obtained also for small deviations\nfrom the ﬂat-band geometry (controlled by δfand/or\nδl). Comparing the exact-diagonalization data for diﬀer-\nent system sizes Nwe observe that the ﬁnite-size eﬀects\nremain small, thus the discussed phenomenon should be\npresent for thermodynamically large systems, too.\nIt is in order to mention a special ﬁnite-size eﬀect that\nmay appear for large values of δland/orδf. In this limit,\nthe dominating hopping parameters may correspond to\ngeometries which do not ﬁt to the initial ladder structure.\nThus, for t11= 1 +δl,t22= 1−δland small δf, in\nthe limit of δl→ ∞ the legs of the ladder form two\nalmost decoupled chains. Such a ﬁnite simple Hubbard\nchain at quarter ﬁlling with an odd number of electrons\n(i.e., a chain of 6 or 10 sites with 3 or 5 electrons) has aFIG. 6: (Color online) Phase diagram in the quarter plane δ\n–U/t2for the ladder with t2= 3,tl=tf= 1obtained by\nfourth-order perturbation theory and by exact diagonaliza tion\n(note that second-order perturbation theory yields U(2)\nc= 0).\n(a)δl= 0,0.05. (b)δl=δf=δ; fourth-order perturbation-\ntheory calculations yield zero value for Uc. (c)δf= 0,0.05.\nferromagnetic ground state. Therefore, the limit of large\ndeviations, shown for completeness in our ﬁgures, goes\nbeyond the primary focus of discussing the dispersion-\ndriven ferromagnetism in systems with ladder geometry.9\nVI. BILAYER\nAs mentioned already, the mechanism leading to the\nemergence of ferromagnetism driven by kinetic energy is\nnot restricted to dimension D= 1. To illustrate this,\nwe consider the two-dimensional counterpart of the Hub-\nbard ladder, namely the frustrated bilayer, see Fig. 1.\nFrom the technical point of view, the two-dimensional\nmodel is more challenging, since the smallest cluster ap-\npropriate for perturbation theory and imaging the basic\ngeometry of the bilayer is built by ﬁve cells (a central cell\nwith four neighboring cells). Furthermore, in contrast to\nthe ladder for the exact diagonalization we do not have\na sequence of ﬁnite lattices of N= 12,16,20sites in\nD= 2. The smallest ﬁnite bilayer lattice with periodic\nboundary conditions has N= 16sites. Hence, we cannot\nprovide a detailed discussion of the bilayer model, rather\nwe will demonstrate for a particular parameter set that\nthe mechanism of kinetic-energy-driven ferromagnetism\nalso holds in D= 2.\nIn analogy to the ladder, for the bilayer one of the two\none-electron bands is ﬂat if t11=t12=t21=t22=tor\ntl=tf=t,δl=δf= 0and it becomes the lowest one\nif4t < t2. Within fourth-order perturbation theory we\nare able to calculate the energies of the fully polarized\nsextuplet (total spin 5/2) and of the quadruplets (total\nspin 3/2),\nES=−5t2+E(2)\nS+E(4)\nS+...,\nEqi(U) =−5t2+E(2)\nqi(U)+E(4)\nqi(U)+...,\ni= 1,2,3,4,(6.1)\nsee Appendix E. Hence, our perturbation-theory treat-\nment remains incomplete, since we cannot compare with\nthe energies of the ﬁve doublets with total spin 1/2.\nOn the other hand, the comparison with the exact-\ndiagonalization data for the ﬁve-cell cluster, where the\ndoublet states are taken into account, yields an excellent\nagreement between both approaches. That is because for\nthis cluster the level crossing between the sextuplet and\nthe lowest quadruplet takes place at the same Uas for\nthe crossing of sextuplet and the lowest doublet.\nAs a ﬁrst (remarkable) outcome we ﬁnd, that the\nsecond-order result U(2)\ncagain is given by Eq. (5.7). We\nshow numerical data for the critical repulsion Ucfor the\nset of parameters t2= 5,tl= 1.025,tf= 0.975, and\nδl= 0in the ground-state phase diagram presented in\nFig. 7 (cf. the corresponding phase diagram for the lad-\nder shown Fig. 5).\nBasically the same features as for the corresponding\nladder are also found for the phase diagram of the bi-\nlayer. However, it is obvious that Ucfor the ﬁnite lattice\nofN= 16sites with periodic boundary conditions is no-\nticeably above perturbation-theory results and the exact-\ndiagonalization results for N= 10sites. We argue that\nthe ﬁnite system of N= 10 sites with open boundary\nconditions is only a very rough model of thermodynam-\nically large bilayer, since only one (among ﬁve) verticalFIG. 7: (Color online) Phase diagram in the quarter plane δf\n–U/t2for the bilayer with t2= 5,tl= 1.025,tf= 0.975, and\nδl= 0obtained by perturbation-theory calculations and by\nexact diagonalization for N= 10,16with open and periodic\nboundary conditions.\nbond has the same environment as in inﬁnite lattice. The\nﬁnite system of N= 16sites with periodic boundary con-\nditions is free of this shortcoming.\nVII. CONCLUSIONS\nWe have used perturbation theory as well as exact di-\nagonalization of ﬁnite systems to examine the kinetic-\nenergy-driven emergence of ferromagnetic ground states\nin Hubbard models with a half-ﬁlled lowest-energy ﬂat\nband for lattices which do not obey the connectivity con-\ndition (isolated trapping cells). Generally speaking, if ( i)\nthe ﬂat band acquires a small dispersion this way allow-\ning to the previously localized electrons to correlate and\n(ii) the on-site Hubbard repulsion Uis suﬃciently strong\nthe ground state becomes ferromagnetic. However, the\nrelation between the required Ucand the acquired band-\nwidth might be quite intricate. Thus, for some deforma-\ntion geometries ferromagnetism does not appear at all,\nfor others it appears already for small U; in some cases\nUcis an increasing function of the deformation strength,\nwhereas in others it becomes nonmonotonic. The mech-\nanism leading to kinetic-energy-driven emergence of fer-\nromagnetism is studied in detail for one-dimensional sys-\ntems with isolated trapping cells. However, as it is\ndemonstrated for a speciﬁc two-dimensional system this\nmechanism works in higher dimensions as well. Although\nour analysis refers to ﬁnite systems, the observed ﬁnite-\nsize behavior indicates convincingly that such a scenario\nshould survive in the thermodynamic limit, too. Thus\nour main conclusion is that the described phenomenon\nis a quite general way of establishing ground-state ferro-\nmagnetism in the repulsive Hubbard model at low elec-\ntron densities around the ﬂat-band limit.\nFurthermore, for special examples, the diamond chain,\nthe ladder as well as the bilayer, we have obtained sim-10\nple analytical formulas, cf. Eq. (4.3) and Eq. (5.7),\nwhich amazingly well estimate the region of ground-state\nferromagnetism. From the technical point of view, we\nhave elaborated computer-adapted scheme for analytical\nperturbation-theory calculations up to the sixth order.\nFinally, it is in order to notice that experimental\nsearches for Mielke-Tasaki ﬂat-band ferromagnetism re-\nmain an ambitious goal of numerous experimental stud-\nies, see, e.g., Refs. 9,24. 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Richter, A. Voigt, S. Krüger, and C. Gros, J. Phys.\nA29, 825 (1996).\nAppendix A: Perturbation-theory formulas for the ground-s tate energy up to the sixth order\nIn this appendix, we present the perturbation-theory formu las up to the sixth order, which are used in our study.\nAlthough these formulas can be found in Ref. 20, we show them h ere for the reader’s convenience and the self-\nconsistency of the paper.\nFirst we split the N-cell Hamiltonian of the model Hinto the main part H0and the perturbation V, i.e.,H=H0+V.\nWe consider the subspace of n=Nelectrons. All eigenstates |α∝an}bracketri}htand their energies Eαof the unperturbed Hamiltonian11\nH0are known. We consider the ground state |GS∝an}bracketri}htof the unperturbed Hamiltonian H0, which is 2n-fold degenerate\n(each cell can be occupied either by up- or down-spin electro n). We denote the ground-state energy by EGS. Moreover,\nwe have ∝an}bracketle{tGS|V|GS∝an}bracketri}ht= 0. Since the ground states are degenerate, the choice of the gr ound states requires some\nconsideration. From Ref. 16 we know that the eﬀective Hamilt onian to describe the low-energy degrees of freedom\nis a Heisenberg Hamiltonian. Hence, we choose the set of grou nd states as a corresponding set of eigenstates of the\nHeisenberg model that way also implying the required SU(2) s ymmetry as well as the spatial symmetry of the clusters\nused for the perturbation theory (for details see Appendix B ). The lowest-order perturbation-theory corrections to\nthe ground-state energy EGSare as follows:\nE(2)\nGS=/summationdisplay′\nα∝an}bracketle{tGS|V|α∝an}bracketri}ht∝an}bracketle{tα|V|GS∝an}bracketri}ht\nEGS−Eα,\nE(3)\nGS=/summationdisplay′\nα/summationdisplay′\nβ∝an}bracketle{tGS|V|α∝an}bracketri}ht∝an}bracketle{tα|V|β∝an}bracketri}ht∝an}bracketle{tβ|V|GS∝an}bracketri}ht\n(EGS−Eα)(EGS−Eβ),\nE(4)\nGS=/summationdisplay′\nα/summationdisplay′\nβ/summationdisplay′\nγ∝an}bracketle{tGS|V|α∝an}bracketri}ht∝an}bracketle{tα|V|β∝an}bracketri}ht∝an}bracketle{tβ|V|γ∝an}bracketri}ht∝an}bracketle{tγ|V|GS∝an}bracketri}ht\n(EGS−Eα)(EGS−Eβ)(EGS−Eγ)−/summationdisplay′\nα/summationdisplay′\nβ∝an}bracketle{tGS|V|α∝an}bracketri}ht∝an}bracketle{tα|V|GS∝an}bracketri}ht∝an}bracketle{tGS|V|β∝an}bracketri}ht∝an}bracketle{tβ|V|GS∝an}bracketri}ht\n(EGS−Eα)2(EGS−Eβ),\nE(5)\nGS= (1,1,1,1)+1\n2(2,1,0,1)+1\n2(1,2,0,1)+1\n2(1,1,0,2)+1\n2(2,0,1,1)+1\n2(1,0,2,1)+1\n2(1,0,1,2),\nE(6)\nGS= (1,1,1,1,1)\n+1\n2(2,1,1,0,1)+1\n2(1,2,1,0,1)+1\n2(1,1,2,0,1)+1\n2(1,1,1,0,2)\n+1\n2(2,1,0,1,1)+1\n2(1,2,0,1,1)+1\n2(1,1,0,2,1)+1\n2(1,1,0,1,2)\n+1\n2(2,0,1,1,1)+1\n2(1,0,2,1,1)+1\n2(1,0,1,2,1)+1\n2(1,0,1,1,2)\n+1\n2(3,0,1,0,1)+3\n8(2,0,2,0,1)+1\n4(2,0,1,0,2)+3\n8(1,0,2,0,2)+1\n2(1,0,1,0,3);(A1)\nhere the superscript ‘prime’ means that the sum extends over all states of the unperturbed Hamiltonian H0except\nthe ground states. Moreover, we have introduced shorthand n otations20\n(k1,k2,...,k n) =∝an}bracketle{tGS|VR(k1)VR(k2)V...VR(kn)V|GS∝an}bracketri}ht,\nR(k)=/braceleftBigg−|GS∝an}bracketri}ht∝an}bracketle{tGS|, k = 0,/parenleftBig/summationdisplay′\nα|α/angbracketright/angbracketleftα|\nEGS−Eα/parenrightBigk\n, k >0(A2)\n(again the superscript ‘prime’ means that the sum extends ov er all states of the unperturbed Hamiltonian H0except\nthe ground state) in the formulas for E(5)\nGSandE(6)\nGS.\nIn the present study we are able to calculate the sixth-order corrections for the N= 2-cell cases, but fourth-order\ncorrections for the cases of N= 3,N= 4, andN= 5cells.\nAppendix B: Ground states of the unperturbed Hamiltonian\nThe energy of the 2n-fold degenerate (see Appendix A) unperturbed ground state s isEGS=−nt2. Before applying\nperturbation-theory formulas of Appendix A we have to const ruct within 2n-fold degenerate ground states the “correct”\n2nlinear combinations being SU(2) symmetric eigenstates of t he corresponding Heisenberg model of the perturbation-\ntheory clusters. The energy of all components of a SU(2) mult iplet is the same (i.e., are not splitted by the perturbation\nV). However, the energies of diﬀerent multiplets may become d iﬀerent after switching on perturbation, where at least\nsecond-order theory is required, since ∝an}bracketle{tGS|V|GS∝an}bracketri}ht= 0. Thus, the number of diﬀerent energies obtained by perturba tion\ntheory cannot exceed 2,3,6,10for the case of N= 2,3,4,5cells, respectively.\nWe begin with the case of N= 2cells (m= 1andm+1 = 2 in Fig. 1) and n= 2electrons. “Correct” unperturbed\nground states are as follows:\n|t,1∝an}bracketri}ht=l†\n1,↑l†\n2,↑|0∝an}bracketri}ht,|t,0∝an}bracketri}ht=1√\n2/parenleftBig\nl†\n1,↑l†\n2,↓+l†\n1,↓l†\n2,↑/parenrightBig\n|0∝an}bracketri}ht,|t,−1∝an}bracketri}ht=l†\n1,↓l†\n2,↓|0∝an}bracketri}ht,\n|s∝an}bracketri}ht=1√\n2/parenleftBig\nl†\n1,↑l†\n2,↓−l†\n1,↓l†\n2,↑/parenrightBig\n|0∝an}bracketri}ht, (B1)12\ni.e., the three components of the triplet states |t∝an}bracketri}htand the singlet state |s∝an}bracketri}ht. It is convenient to use shorthanded\nnotations | ↑↑∝an}bracketri}ht=l†\n1,↑l†\n2,↑|0∝an}bracketri}ht,| ↑↓∝an}bracketri}ht=l†\n1,↑l†\n2,↓|0∝an}bracketri}htetc. so that Eq. (B1) becomes\n|t,1∝an}bracketri}ht=| ↑↑∝an}bracketri}ht,|t,0∝an}bracketri}ht=1√\n2(| ↑↓∝an}bracketri}ht+| ↓↑∝an}bracketri}ht),|t,−1∝an}bracketri}ht=| ↓↓∝an}bracketri}ht,\n|s∝an}bracketri}ht=1√\n2(| ↑↓∝an}bracketri}ht−| ↓↑∝an}bracketri}ht ). (B2)\nWe pass to the case of N= 3cells (open boundary conditions) and n= 3electrons. “Correct” unperturbed ground\nstates are\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleq,3\n2/angbracketrightbigg\n=| ↑↑↑∝an}bracketri}ht,/vextendsingle/vextendsingle/vextendsingle/vextendsingleq,1\n2/angbracketrightbigg\n=1√\n3(| ↑↑↓∝an}bracketri}ht+| ↑↓↑∝an}bracketri}ht+| ↓↑↑∝an}bracketri}ht),/vextendsingle/vextendsingle/vextendsingle/vextendsingleq,−1\n2/angbracketrightbigg\n=1√\n3(| ↑↓↓∝an}bracketri}ht+| ↓↑↓∝an}bracketri}ht+| ↓↓↑∝an}bracketri}ht),/vextendsingle/vextendsingle/vextendsingle/vextendsingleq,−3\n2/angbracketrightbigg\n=| ↓↓↓∝an}bracketri}ht,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled1,1\n2/angbracketrightbigg\n=1√\n2(| ↑↑↓∝an}bracketri}ht−| ↓↑↑∝an}bracketri}ht ),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled1,−1\n2/angbracketrightbigg\n=1√\n2(| ↑↓↓∝an}bracketri}ht−| ↓↓↑∝an}bracketri}ht ),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled2,1\n2/angbracketrightbigg\n=1√\n6(| ↑↑↓∝an}bracketri}ht−2| ↑↓↑∝an}bracketri}ht+| ↓↑↑∝an}bracketri}ht),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled2,−1\n2/angbracketrightbigg\n=1√\n6(| ↑↓↓∝an}bracketri}ht−2| ↓↑↓∝an}bracketri}ht+| ↓↓↑∝an}bracketri}ht),(B3)\ni.e., the quadruplet |q∝an}bracketri}htand the two doublets |d1∝an}bracketri}ht,|d2∝an}bracketri}ht. The total spin of |d1∝an}bracketri}htand|d2∝an}bracketri}htis 1/2 and the ‘local’ sz\nj-values\nfor the sites j= 1,2,3are as follows: 0,±1/2,0for|d1∝an}bracketri}htand±1/3,∓1/6,±1/3for|d2∝an}bracketri}ht. The states given in Eq. (B3)\nare the eigenstates of (s1+s2+s3)2, ofsz\n1+sz\n2+sz\n3, and of the Hamiltonian H=s1·s2+s2·s3(three-site Heisenberg\nmodel with open boundary conditions) with the energies 1/2(|q∝an}bracketri}ht),0(|d1∝an}bracketri}ht), and−1(|d2∝an}bracketri}ht).\nNext we consider N= 4cells along a chain with open boundary conditions and n= 4electrons. The unperturbed\nSU(2) symmetric ground states are\n|Q,2∝an}bracketri}ht=| ↑↑↑↑∝an}bracketri}ht,...,|Q,−2∝an}bracketri}ht=| ↓↓↓↓∝an}bracketri}ht,\n|t1,1∝an}bracketri}ht=1\n2/radicalbig\n2−√\n2/bracketleftBig\n−| ↑↑↑↓∝an}bracketri}ht +(1−√\n2)| ↑↑↓↑∝an}bracketri}ht− (1−√\n2)| ↑↓↑↑∝an}bracketri}ht+| ↓↑↑↑∝an}bracketri}ht/bracketrightBig\n,...,\n|t2,1∝an}bracketri}ht=1\n2(| ↑↑↑↓∝an}bracketri}ht−| ↑↑↓↑∝an}bracketri}ht )+1\n2(−| ↑↓↑↑∝an}bracketri}ht +| ↓↑↑↑∝an}bracketri}ht),...,\n|t3,1∝an}bracketri}ht=1\n2/radicalbig\n2+√\n2/bracketleftBig\n−| ↑↑↑↓∝an}bracketri}ht +(1+√\n2)| ↑↑↓↑∝an}bracketri}ht− (1+√\n2)| ↑↓↑↑∝an}bracketri}ht+| ↓↑↑↑∝an}bracketri}ht/bracketrightBig\n,...,\n|s1∝an}bracketri}ht=1√\n6/bracketleftbigg1\n2/parenleftBig\n−1−√\n3/parenrightBig\n| ↑↑↓↓∝an}bracketri}ht−1\n2/parenleftBig\n1−√\n3/parenrightBig\n| ↑↓↑↓∝an}bracketri}ht+| ↑↓↓↑∝an}bracketri}ht\n+| ↓↑↑↓∝an}bracketri}ht−1\n2/parenleftBig\n1−√\n3/parenrightBig\n| ↓↑↓↑∝an}bracketri}ht+1\n2/parenleftBig\n−1−√\n3/parenrightBig\n| ↓↓↑↑∝an}bracketri}ht/bracketrightbigg\n,\n|s2∝an}bracketri}ht=1√\n6/bracketleftbigg1\n2/parenleftBig\n−1+√\n3/parenrightBig\n| ↑↑↓↓∝an}bracketri}ht−1\n2/parenleftBig\n1+√\n3/parenrightBig\n| ↑↓↑↓∝an}bracketri}ht+| ↑↓↓↑∝an}bracketri}ht\n+| ↓↑↑↓∝an}bracketri}ht−1\n2/parenleftBig\n1+√\n3/parenrightBig\n| ↓↑↓↑∝an}bracketri}ht+1\n2/parenleftBig\n−1+√\n3/parenrightBig\n| ↓↓↑↑∝an}bracketri}ht/bracketrightbigg\n, (B4)\ni.e., one quintuplet |Q∝an}bracketri}ht, the three triplets |t1∝an}bracketri}ht,|t2∝an}bracketri}ht,|t3∝an}bracketri}ht, and the two singlets |s1∝an}bracketri}ht,|s2∝an}bracketri}ht. These states are eigenstates\nof the Heisenberg Hamiltonian H=/summationtext3\ni=1si·si+1with the energies 3/4(|Q∝an}bracketri}ht),(−1 + 2√\n3)/4(|t1∝an}bracketri}ht),−1/4(|t2∝an}bracketri}ht),\n(−1−2√\n3)/4(|t3∝an}bracketri}ht),(−3+2√\n3)/4(|s1∝an}bracketri}ht), and(−3−2√\n3)/4(|s2∝an}bracketri}ht).13\nIn the case of N= 5cells and n= 5electrons relevant for the bilayer problem we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleS,5\n2/angbracketrightbigg\n=| ↑↑↑↑↑∝an}bracketri}ht,...,/vextendsingle/vextendsingle/vextendsingle/vextendsingleS,−5\n2/angbracketrightbigg\n=| ↓↓↓↓↓∝an}bracketri}ht,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleq1,3\n2/angbracketrightbigg\n=1\n2(| ↑↑↑↑↓∝an}bracketri}ht−| ↑↑↑↓↑∝an}bracketri}ht +| ↑↓↑↑↑∝an}bracketri}ht−| ↓↑↑↑↑∝an}bracketri}ht ),...,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleq2,3\n2/angbracketrightbigg\n=1\n2(| ↑↑↑↑↓∝an}bracketri}ht−| ↑↑↑↓↑∝an}bracketri}ht−| ↑↓↑↑↑∝an}bracketri}ht +| ↓↑↑↑↑∝an}bracketri}ht),...,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleq3,3\n2/angbracketrightbigg\n=1\n2(| ↑↑↑↑↓∝an}bracketri}ht+| ↑↑↑↓↑∝an}bracketri}ht−| ↑↓↑↑↑∝an}bracketri}ht−| ↓↑↑↑↑∝an}bracketri}ht ),...,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleq4,3\n2/angbracketrightbigg\n=1\n2√\n5(| ↑↑↑↑↓∝an}bracketri}ht+| ↑↑↑↓↑∝an}bracketri}ht− 4| ↑↑↓↑↑∝an}bracketri}ht+| ↑↓↑↑↑∝an}bracketri}ht+| ↓↑↑↑↑∝an}bracketri}ht),...,\n..., (B5)\ni.e., one sextuplet |S∝an}bracketri}htand the four quadruplets |q1∝an}bracketri}ht,|q2∝an}bracketri}ht,|q3∝an}bracketri}ht,|q4∝an}bracketri}ht. Note that the ﬁve doublets are not given here,\nsince they are not used for perturbation theory, cf. the disc ussion in Sec. VI. The geometry of the cluster is that\nof a Heisenberg star25with central spin s3, i.e., the choice given in Eq. (B5) corresponds to the eigens tates of the\nHeisenberg Hamiltonian H=s1·s3+s2·s3+s3·s4+s3·s5with the energies 1(|S∝an}bracketri}ht),1/2(|q1∝an}bracketri}ht,|q2∝an}bracketri}ht, and|q3∝an}bracketri}ht), and\n−3/2(|q4∝an}bracketri}ht).\nIn the present study we use Eqs. (B2) and (B3) for the diamond c hain, Eqs. (B2), (B3), and (B4) for the ladder,\nand Eq. (B5) for the bilayer. Since for the N= 5bilayer we compare the energies ESandEq1,Eq2,Eq3,Eq4only,\nthe formulas given in Eq. (B5) are suﬃcient for this purpose.\nAppendix C: Perturbation-theory results for the diamond ch ain\nn= 2electrons on the diamond chain of N= 2cells\nWe consider the case of open boundary conditions, i.e., N= 5. Corrections to the ground-state energy E(0)=−2t2\nup to the sixth order are as follows:\nE(2)=−(t3−t1)2\nt2; (C1)\nE(4)\nt=−(t3+t1)2(t3−t1)2\n2t3\n2+(t3−t1)4\nt3\n2,\nE(4)\ns(U) =−(t3+t1)2(t3−t1)2\n4t3\n2+(t3−t1)4\nt3\n2−(8t2+U)(t3−t1)4\n4t3\n2U−(t3+t1)2(t3−t1)2\n2(2t2+U)t2\n2−2(t3−t1)4\n(2t2+U)t2\n2;(C2)\nE(6)\nt=−(t3−t1)2/parenleftbig\nt4\n3−14t3\n3t1+34t2\n3t2\n1−14t3t3\n1+t4\n1/parenrightbig\n2t5\n2,\nE(6)\ns(U) =−(t3−t1)2/bracketleftbig\n192t4\n2(t3−t1)4+48t3\n2U(t3−t1)2/parenleftbig\n5t2\n3−2t3t1+5t2\n1/parenrightbig/bracketrightbig\n12t5\n2U2(2t2+U)2\n−(t3−t1)2/bracketleftbig\n4t2\n2U2/parenleftbig\n22t4\n3+21t3\n3t1−38t2\n3t2\n1+21t3t3\n1+22t4\n1/parenrightbig/bracketrightbig\n12t5\n2U2(2t2+U)2\n−(t3−t1)2/bracketleftbig\n4t2U3/parenleftbig\n2t4\n3+27t3\n3t1−34t2\n3t2\n1+27t3t3\n1+2t4\n1/parenrightbig/bracketrightbig\n12t5\n2U2(2t2+U)2\n−(t3−t1)2/bracketleftbig\nU4/parenleftbig\nt4\n3+12t3\n3t1−14t2\n3t2\n1+12t3t3\n1+t4\n1/parenrightbig/bracketrightbig\n12t5\n2U2(2t2+U)2. (C3)\nThe results up to the fourth order were reported in Ref. 16. In Fig. 8 we show dependences of the triplet and singlet\nenergies on Uobtained within diﬀerent orders of the perturbation theory according to Eqs. (C1), (C2), (C3) along14\nFIG. 8: (Color online) Energies of low-lying states (triple t – red, singlet – blue) as a function of the on-site repulsion Ufor\nn= 2electrons on N= 2cells of the diamond chain with open boundary conditions wit ht2= 3,t1= 0.9,t3= 1.1. The results\nup to the second, fourth, sixth orders are denoted by short-d ashed, long-dashed, solid lines, respectively. The result s of exact\ndiagonalization are shown by symbols. Note that the energie s of the triplet and the singlet coincide within the second or der,\nsee Eq. (C1).\nwith exact-diagonalization data for a typical set of hoppin g integrals t2= 3,t1= 0.9,t3= 1.1[t= 1,|δ|= 0.1,\nδ= (t1−t3)/(t1+t3)]. Obviously, in the limit U→0the perturbation theory fails, since it yields a singlet ene rgy\ntending to −∞whereas the exact-diagonalization data is ﬁnite. The reaso n for that is clear: Within the exploited\nscheme the speciﬁc states with two electrons having diﬀeren t spins in one cell are treated as excited states, however,\nin the small- Ulimit their energy approaches the ground-state energy; bei ng treated as excited states they lead to\nlarge denominators in the terms of the perturbation-theory series, see Eqs. (A1), (A2).\nn= 3electrons on the diamond chain of N= 3cells\nWe consider the case of open boundary conditions, i.e., N= 8. Corrections to the ground-state energy E(0)=−3t2\nup to the fourth order are as follows:\nE(2)=−2(t1−t3)2\nt2; (C4)\nE(4)\nq=(t1−t3)2/parenleftbig\n7t2\n1−26t1t3+7t2\n3/parenrightbig\n4t3\n2,\nE(4)\nd1(U) =(t1−t3)2/bracketleftbig\n−24t2\n2(t1−t3)2+t2U/parenleftbig\nt2\n1−50t1t3+t2\n3/parenrightbig\n+2U2/parenleftbig\n7t2\n1−23t1t3+7t2\n3/parenrightbig/bracketrightbig\n8t3\n2U(2t2+U),\nE(4)\nd2(U) =(t1−t3)2/bracketleftbig\n−40t2\n2(t1−t3)2−t2U/parenleftbig\n17t2\n1+14t1t3+17t2\n3/parenrightbig\n+14U2/parenleftbig\nt2\n1−3t1t3+t2\n3/parenrightbig/bracketrightbig\n8t3\n2U(2t2+U). (C5)\nSplitting of various SU(2) multiplets begins in the fourth o rder of perturbation theory. In Fig. 9 we show dependences\nof the quadruplet and doublets energies on Uobtained within diﬀerent orders of the perturbation theory according to\nEqs. (C4), (C5) along with exact-diagonalization data for t he same set of hopping integrals as in Fig. 8, i.e., t2= 3,\nt1= 0.9,t3= 1.1. At a ﬁrst glance one may be worry about the agreement between perturbation theory and exact\ndiagonalization. However, comparing the fourth-order res ults and the exact-diagonalization data for N= 2cells\nshown in Fig. 8 one can see a similar diﬀerence which is obviou sly improved by the the sixth-order calculations.15\nFIG. 9: (Color online) Ground-state energy as a function of t he on-site repulsion Uforn= 3electrons on N= 3cells of the\ndiamond chain (open boundary conditions) with t2= 3,t1= 0.9,t3= 1.1. Quadruplet energy (salmon) versus doublets energy\n(skyblue and magenta). The results up to the second and fourt h orders are denoted by short-dashed and long-dashed lines,\nrespectively. The results of exact diagonalization are sho wn by symbols. Note that the energies of the doublet and quadr uplet\nstates coincide within the second order, see Eq. (C4).\nAppendix D: Perturbation-theory results for the ladder\nn= 2electrons on the ladder of N= 2cells\nFor the two-cell ( N= 4) ladder (open boundary conditions) we have the following co rrections to the unperturbed\nground-state energy E(0)=−2t2:\nE(2)\nt=−(t11−t22)2+(t12−t21)2\n4t2,\nE(2)\ns(U) =−(t11−t12−t21+t22)2\n4t2−(t11−t22)2+(t12−t21)2\n2(2t2+U)−2(t11−t12−t21+t22)2\nU; (D1)16\nE(4)\nt=1\n64t3\n2/bracketleftbig\nt4\n11−4t3\n11t22+2t2\n11/parenleftbig\n3t2\n22−t2\n21−6t21t12−t2\n12/parenrightbig\n−4t11t22/parenleftbig\nt2\n22+3t2\n21−14t21t12+3t2\n12/parenrightbig\n+t4\n22−2t2\n22/parenleftbig\nt2\n21+6t21t12+t2\n12/parenrightbig\n+(t21−t12)4/bracketrightBig\n,\nE(4)\ns(U) =1\n64t3\n2U3(2t2+U)3/braceleftBig\n4096t6\n2(t11−t12−t21+t22)4+7680t5\n2U(t11−t12−t21+t22)4\n+t11/bracketleftBig\n67t2\n22−132t22(t12+t21)+66(t12+t21)2/bracketrightBig\n+21t3\n22−66t2\n22(t12+t21)+66t22(t12+t21)2\n+256t4\n2U2(t11−t12−t21+t22)/bracketleftbig\n21t3\n11+t2\n11(67t22−66(t12+t21))−(t12+t21)/parenleftbig\n21t2\n21+46t12t21+21t2\n12/parenrightbig/bracketrightbig\n+t2\n11/bracketleftBig\n406t2\n22−784t22(t12+t21)+386(t12+t21)2/bracketrightBig\n+32t3\n2U3/bracketleftBig\n55t4\n22−248t3\n22(t12+t21)+386t2\n22(t12+t21)2−8t22(t12+t21)/parenleftbig\n31t2\n21+57t12t21+31t2\n12/parenrightbig/bracketrightBig\n+32t3\n2U3/parenleftbig\n55t4\n21+256t3\n21t12+406t2\n12t2\n21+256t21t3\n12+55t4\n12/parenrightbig\n+32t3\n2U34t11/bracketleftbig\n−2(t12+t21)/parenleftbig\n31t2\n21+67t12t21+31t2\n12/parenrightbig/bracketrightbig\n+32t3\n2U3/bracketleftbig\n55t4\n11+8t3\n11(32t22−31(t12+t21))\n+4t11/bracketleftbig\n64t3\n22−196t2\n22(t12+t21)+t22/parenleftbig\n193t2\n21+392t12t21+193t2\n12/parenrightbig/bracketrightbig\n+t2\n11/bracketleftbig\n156t2\n22−311t22(t12+t21)+4/parenleftbig\n38t2\n21+77t12t21+38t2\n12/parenrightbig/bracketrightbig\n+t11/bracketleftbig\n100t3\n22−311t2\n22(t12+t21)+4t22/parenleftbig\n77t2\n21+158t21t12+77t2\n12/parenrightbig\n−(t12+t21)/parenleftbig\n97t2\n21+214t12t21+97t2\n12/parenrightbig/bracketrightbig\n+20t4\n22−97t3\n22(t12+t21)+4t2\n22/parenleftbig\n38t2\n21+77t12t21+38t2\n12/parenrightbig\n−t22(t12+t21)/parenleftbig\n97t2\n21+214t12t21+97t2\n12/parenrightbig/bracketrightbig\n+16t2\n2U4/bracketleftbig\n20t4\n11+t3\n11(100t22−97(t12+t21))+4/parenleftbig\n5t4\n21+25t3\n21t12+39t2\n21t2\n12+25t21t3\n12+5t4\n12/parenrightbig/bracketrightbig\n+2t2U5(t11−t12−t21+t22)/bracketleftBig\n23t3\n22−69t2\n22(t12+t21)+69t22(t12+t21)2−(t12+t21)/parenleftbig\n23t2\n21+30t12t21+23t2\n12/parenrightbig/bracketrightBig\n+2t2U5(t11−t12−t21+t22)/bracketleftBig\n23t3\n11+t2\n11(53t22−69(t12+t21))+t11/parenleftBig\n53t2\n22−138t22(t12+t21)+69(t12+t21)2/parenrightBig/bracketrightBig\n+U6(t11−t12−t21+t22)2/bracketleftBig\nt2\n11+2t11(t22−3(t12+t21))+t2\n22−6t22(t12+t21)+(t12+t21)2/bracketrightBig/bracerightBig\n.(D2)\nThe formulas for the sixth-order corrections are too length y to be presented here, although we use these formulas\nto produce the results reported in Figs. 3(a), 3(b), and 3(c) . The formulas in (D1), (D2) become simpler in two\nparticular cases introduced in Sec. V. For the symmetric def ormation we have\nE(2)\nt= 0,\nE(2)\ns(U) =−(t11−t12)2(8t2+U)\nt2U; (D3)\nE(4)\nt= 0,\nE(4)\ns(U) =(t11−t12)2/bracketleftBig\n512t3\n2(t11−t12)2+192t2\n2U(t11−t12)2+32t2U2(t11−t12)2+U3/parenleftbig\nt2\n11−6t11t12+t2\n12/parenrightbig/bracketrightBig\n4t3\n2U3.(D4)\nFor the semi-symmetric deformation we have\nE(2)\nt=−(t11−t21)2\n2t2,\nE(2)\ns(U) =−(t11−t21)2\n2t2+U; (D5)\nE(4)\nt=−t11t21(t11−t21)2\n2t3\n2,\nE(4)\ns(U) =−(t11−t21)2/bracketleftBig\n8t2t11t21+U(t11+t21)2/bracketrightBig\n2t2(2t2+U)3. (D6)17\nFurthermore, the sixth-order corrections are as follows:\nE(6)\nt= 0,\nE(6)\ns(U) =−1\n8t5\n2U5(16t2\n2+U2)/bracketleftbig\n(t11−t12)2/parenleftbig\n524288t7\n2(t11−t12)4+327680 t6\n2U(t11−t12)4+131072 t5\n2U2(t11−t12)4\n+1024t4\n2U3(t11−t12)2/parenleftbig\n35t2\n11−76t11t12+35t2\n12/parenrightbig\n+256t3\n2U4(t11−t12)2/parenleftbig\n28t2\n11−65t11t12+28t2\n12/parenrightbig\n+8t2\n2U5/parenleftbig\n121t4\n11−556t3\n11t12+886t2\n11t2\n12−556t11t3\n12+121t4\n12/parenrightbig\n+16t2U6(t11−t12)2/parenleftbig\n4t2\n11−17t11t12+4t2\n12/parenrightbig\n+U7/parenleftbig\nt4\n11−14t3\n11t12+34t2\n11t2\n12−14t11t3\n12+t4\n12/parenrightbig/parenrightbig/bracketrightbig\n(D7)\n(symmetric deformation) and\nE(6)\nt=(t11−t21)2/parenleftbig\nt4\n21+4t3\n21t11−26t2\n21t2\n11+4t21t3\n11+t4\n11/parenrightbig\n32t5\n2,\nE(6)\ns(U) =(t11−t21)2/parenleftbig\nt4\n21+4t3\n21t11−26t2\n21t2\n11+4t21t3\n11+t4\n11/parenrightbig\n(2t2+U)5\n+(t11−t21)2/bracketleftBig\nt2U(t11+t21)2/parenleftbig\n3t2\n21−14t21t11+3t2\n11/parenrightbig\n−2t21t11U2(t11+t21)2/bracketrightBig\n2t2\n2(2t2+U)5(D8)\n(semi-symmetric deformation).\nn= 3electrons on the ladder of N= 3cells\nFor the three-cell ( N= 6) ladder (open boundary conditions) we have the following co rrections to the unperturbed\nground-state energy E(0)=−3t2:\nE(2)\nq=−(t11−t22)2+(t12−t21)2\n2t2,\nE(2)\nd1(U) =1\n4/parenleftBigg\n−2t2\n11+t11(2t22+t12+t21)−2t2\n22+t22(t12+t21)−2/parenleftbig\nt2\n12−t12t21+t2\n21/parenrightbig\nt2\n−(t11−t22)2+(t12−t21)2\n2t2+U−4(t11+t22−t12−t21)2\nU/parenrightbigg\n,\nE(2)\nd2(U) =−1\n4t2U(2t2+U)/bracketleftbig\n(8t2+U)/parenleftbig\n3t2(t11+t22−t12−t21)2\n+U/parenleftbig\n2t2\n11+t11(2t22−3(t12+t21))+2t2\n22−3t22(t12+t21)+2/parenleftbig\nt2\n12+t12t21+t2\n21/parenrightbig/parenrightbig/parenrightbig/bracketrightbig\n; (D9)\nE(4)\nq=1\n16t3\n2/bracketleftbig\nt4\n11−4t3\n11t22+t2\n11/parenleftbig\n6t2\n22−3t2\n12−10t12t21−3t2\n21/parenrightbig\n+2t11t22/parenleftbig\n−2t2\n22+t2\n12+14t12t21+t2\n21/parenrightbig\n+t4\n22−t2\n22/parenleftbig\n3t2\n12+10t12t21+3t2\n21/parenrightbig\n−4t12t21(t12−t21)2/bracketrightbig\n, (D10)\nand the formulas for E(4)\nd1(U)andE(4)\nd2(U)are too lengthy to be presented here. Formulas given in Eqs. ( D9), (D10)\nare illustrated in Fig. 10, where we show the dependence of en ergies the quadruplet and doublets on Ufor three\ntypical sets of parameters.18\nFIG. 10: (Color online) Ground-state energy (up to the fourt h order of perturbation theory and exact-diagonalization d ata) as\na function of the on-site repulsion Uforn= 3electrons on the open ladder of N= 3cells. (a) t2= 3,t11= 0.85,t12= 0.95,\nt21= 1,t22= 1.2(general deformation). (b) t2= 3,t11=t22= 1.1,t12=t21= 0.9(symmetric deformation). (c) t2= 3,\nt11=t21= 1.1,t12=t22= 0.9(semi-symmetric deformation).19\nn= 4electrons on the ladder of N= 4cells\nFor the four-cell ( N= 8) ladder (open boundary conditions) we have the following co rrections to the unperturbed\nground-state energy E(0)=−4t2:\nE(2)\nQ=−3/parenleftbig\n(t11−t22)2+(t12−t21)2/parenrightbig\n4t2,\nE(2)\nt1(U) =1\n4/parenleftbig√\n2−2/parenrightbig\nt2U(2t2+U)/bracketleftBig\n2t2U/parenleftBig/parenleftBig\n21−13√\n2/parenrightBig\nt2\n11+2t11/parenleftBig/parenleftBig\n9−7√\n2/parenrightBig\nt22+5/parenleftBig\n2√\n2−3/parenrightBig\n(t12+t21)/parenrightBig\n+/parenleftBig\n21−13√\n2/parenrightBig\nt2\n22+10/parenleftBig\n2√\n2−3/parenrightBig\nt22(t12+t21)+/parenleftBig\n21−13√\n2/parenrightBig\nt2\n12+2/parenleftBig\n9−7√\n2/parenrightBig\nt12t21+/parenleftBig\n21−13√\n2/parenrightBig\nt2\n21/parenrightBig/bracketrightBig\n−1\n4/parenleftbig√\n2−2/parenrightbig\nt2U(2t2+U)/bracketleftBig\nU2/parenleftBig\n3/parenleftBig√\n2−2/parenrightBig\nt2\n11+2t11/parenleftBig√\n2t22+/parenleftBig\n3−2√\n2/parenrightBig\n(t12+t21)/parenrightBig\n+3/parenleftBig√\n2−2/parenrightBig\nt2\n22\n+/parenleftBig\n6−4√\n2/parenrightBig\nt22(t12+t21)+3/parenleftBig√\n2−2/parenrightBig\nt2\n12+2√\n2t12t21+3/parenleftBig√\n2−2/parenrightBig\nt2\n21/parenrightBig/bracketrightBig\n−16/parenleftbig\n2√\n2−3/parenrightbig\nt2\n2(t11+t22−t12−t21)2\n4/parenleftbig√\n2−2/parenrightbig\nt2U(2t2+U),\nE(2)\nt2(U) =1\n4/parenleftbigg−3t2\n11+2t11(t22+t12+t21)−3t2\n22+2t22(t12+t21)−3t2\n12+2t12t21−3t2\n21\nt2\n−2/parenleftbig\n(t11−t22)2+(t12−t21)2/parenrightbig\n2t2+U−8(t11+t22−t12−t21)2\nU/parenrightBigg\n,\nE(2)\nt3(U) =−1\n4/parenleftbig\n2+√\n2/parenrightbig\nt2U(2t2+U)/bracketleftBig\n2t2U/parenleftBig/parenleftBig\n21+13√\n2/parenrightBig\nt2\n11+2t11/parenleftBig/parenleftBig\n9+7√\n2/parenrightBig\nt22−5/parenleftBig\n3+2√\n2/parenrightBig\n(t12+t21)/parenrightBig\n+/parenleftBig\n21+13√\n2/parenrightBig\nt2\n22−10/parenleftBig\n3+2√\n2/parenrightBig\nt22(t12+t21)+/parenleftBig\n21+13√\n2/parenrightBig\nt2\n12+2/parenleftBig\n9+7√\n2/parenrightBig\nt12t21+/parenleftBig\n21+13√\n2/parenrightBig\nt2\n21/parenrightBig/bracketrightBig\n−1\n4/parenleftbig\n2+√\n2/parenrightbig\nt2U(2t2+U)/bracketleftBig\nU2/parenleftBig\n3/parenleftBig\n2+√\n2/parenrightBig\nt2\n11+2t11/parenleftBig√\n2t22−/parenleftBig\n3+2√\n2/parenrightBig\n(t12+t21)/parenrightBig\n+3/parenleftBig\n2+√\n2/parenrightBig\nt2\n22\n−2/parenleftBig\n3+2√\n2/parenrightBig\nt22(t12+t21)+3/parenleftBig\n2+√\n2/parenrightBig\nt2\n12+2√\n2t12t21+3/parenleftBig\n2+√\n2/parenrightBig\nt2\n21/parenrightBig/bracketrightBig\n−16/parenleftbig\n3+2√\n2/parenrightbig\nt2\n2(t11+t22−t12−t21)2\n4/parenleftbig\n2+√\n2/parenrightbig\nt2U(2t2+U),\nE(2)\ns1(U) =1\n4/parenleftBigg\n−3t2\n11+t11/parenleftbig/parenleftbig√\n3−3/parenrightbig\n(t12+t21)−2√\n3t22/parenrightbig\n+3t2\n22+/parenleftbig√\n3−3/parenrightbig\nt22(t12+t21)+3t2\n12−2√\n3t12t21+3t2\n21\nt2\n+/parenleftbig√\n3−3/parenrightbig/parenleftbig\n(t11−t22)2+(t12−t21)2/parenrightbig\n2t2+U+4/parenleftbig√\n3−3/parenrightbig\n(t11+t22−t12−t21)2\nU/parenrightBigg\n,\nE(2)\ns2(U) =1\n4/parenleftBigg\n−3t2\n11+t11/parenleftbig/parenleftbig\n3+√\n3/parenrightbig\n(t12+t21)−2√\n3t22/parenrightbig\n−3t2\n22+/parenleftbig\n3+√\n3/parenrightbig\nt22(t12+t21)−3t2\n12−2√\n3t12t21−3t2\n21\nt2\n−/parenleftbig\n3+√\n3/parenrightbig/parenleftbig\n(t11−t22)2+(t12−t21)2/parenrightbig\n2t2+U−4/parenleftbig\n3+√\n3/parenrightbig\n(t11+t22−t12−t21)2\nU/parenrightBigg\n;(D11)\nE(4)\nQ=1\n64t3\n2/bracketleftbig\n7t4\n11−28t3\n11t22+t2\n11/parenleftbig\n42t2\n22−22t2\n12−68t12t21−22t2\n21/parenrightbig\n+28t11t22/parenleftbig\n−t2\n22+t2\n12+6t12t21+t2\n21/parenrightbig\n+7t4\n22−2t2\n22/parenleftbig\n11t2\n12+34t12t21+11t2\n21/parenrightbig\n−(t12−t21)2/parenleftbig\nt2\n12+30t12t21+t2\n21/parenrightbig/bracketrightbig\n,(D12)\nand the formulas for E(4)\nt1(U),E(4)\nt2(U),E(4)\nt3(U),E(4)\ns1(U), andE(4)\ns2(U)are too lengthy to be presented here. In Fig. 11\nwe illustrate the dependence of the quintuplet, triplets, a nd singlets energies on Ufor three typical sets of parameters.20\nFIG. 11: (Color online) Ground-state energy (up to the fourt h order of perturbation theory and exact-diagonalization d ata) as\na function of the on-site repulsion Uforn= 4electrons on the open ladder of N= 4cells. (a) t2= 3,t11= 0.85,t12= 0.95,\nt21= 1,t22= 1.2(general deformation). (b) t2= 3,t11=t22= 1.1,t12=t21= 0.9(symmetric deformation). (c) t2= 3,\nt11=t21= 1.1,t12=t22= 0.9(semi-symmetric deformation).21\nAppendix E: Perturbation-theory results for the bilayer\nn= 5electrons on the bilayer of N= 5cells\nFor the ﬁnite-size bilayer cluster (star geometry) we have o btained the following corrections to the unperturbed\nground-state energy E(0)=−5t2:\nE(2)\nS=−(t11−t22)2+(t12−t21)2\nt2,\nE(2)\nq1(U) =E(2)\nq2(U) =E(2)\nq3(U)\n=1\n16/parenleftBigg\n−16t2\n11+t11(14t22+9(t12+t21))−16t2\n22+9t22(t12+t21)−2/parenleftbig\n8t2\n12−7t12t21+8t2\n21/parenrightbig\nt2\n−9/parenleftbig\n(t11−t22)2+(t12−t21)2/parenrightbig\n2t2+U−36(t11+t22−t12−t21)2\nU/parenrightBigg\n,\nE(2)\nq4(U) =1\n16/parenleftBigg\n−16t2\n11+t11(22t22+5(t12+t21))−16t2\n22+5t22(t12+t21)−2/parenleftbig\n8t2\n12−11t12t21+8t2\n21/parenrightbig\nt2\n−5/parenleftbig\n(t11−t22)2+(t12−t21)2/parenrightbig\n2t2+U−20(t11+t22−t12−t21)2\nU/parenrightBigg\n;(E1)\nE(4)\nS=1\n4t3\n2/bracketleftbig\nt4\n11−4t3\n11t22+2t2\n11/parenleftbig\n3t2\n22−t2\n12−6t12t21−t2\n21/parenrightbig\n−4t11t22/parenleftbig\nt2\n22+3t2\n12−14t12t21+3t2\n21/parenrightbig\n+t4\n22−2t2\n22/parenleftbig\nt2\n12+6t12t21+t2\n21/parenrightbig\n+(t12−t21)4/bracketrightbig\n, (E2)\nand the formulas for E(4)\nq1(U),E(4)\nq2(U),E(4)\nq3(U), andE(4)\nq4(U)are too lengthy to be presented here.\nAppendix F: Mathematica tutorial. n= 3electrons on the diamond chain of N= 8sites\nAfter installing and calling the SNEG package in a Mathemati ca sheet one basically needs the commands outlined\nbelow. At ﬁrst all appearing annihilation operators have to be deﬁned with the command\nsnegfermionoperators (c1,c2,...,c8,la,lb,...,dc);\nThe occurring numbers correspond to the lattice sites. We al so deﬁne new operators lA= (cA,1−cA,2)/√\n2[see\nEq. (3.1)], as well as dA= (cA,1+cA,2)/√\n2, to describe the ground states of the unperturbed Hamiltoni an. This is\nimplemented with the deﬁnition of rules, which are explaini ng the connection to the new set of operators:\nrules=snegold2newrules/parenleftbigg\n{c1(),...,c8()},{la(),...,dc()},/braceleftbiggc1()−c2()√\n2,...,c7()+c8()√\n2/bracerightbigg/parenrightbigg\nThe unperturbed Hamiltonian, which corresponds to Eq. (4.1 ), is called with the command\nH0=t2(hop(c1(),c2())+hop(c4(),c5())+hop(c7(),c8()))\n+U(hubbard(c1())+hubbard(c2())+hubbard(c3())+hubbard(c4()))\n+U(hubbard(c5())+hubbard(c6())+hubbard(c7())+hubbard(c8()))\nThe next step is solving the unperturbed system H0. If one constructs the basis-set with, e.g.,\nbasis=qbasis({la(),...,dc(),c3(),c6()})\nthe Hamilton matrix is built with the command\nham=makematricesbzop (H/. rules,basis[[4;;4]]),22\nwhere basis [[4;;4]]chooses the ﬁlling Qof the considered states. Here the ﬁlling corresponds to Q=−5 =−sites+\nelectrons . To get the Eigenenergies and the Eigenfunctions of H0one needs to call\nhamSz0 =Select[ham,First[#1] ={−5}&][[1,2]];\nvalues=maskOp(Eigensystem ,hamSz0);\nAs discussed in the previous sections it is advisable to comb ine the ground-state manifold of the unperturbed\nHamiltonian to respect the SU(2) symmetry. Since the Eigenf unctions are stored in values , a new set of Eigenfunctions\nis provided by, e.g.,\nFor[i= 1,i≤Length[values[[1]]],i++,maskOp(NormedVecs (i) =values[[2,i]])];\n...\nNormedVecsSU2 (138) = NormedVecs (136)+ NormedVecs (138)−2NormedVecs (139);\nNormedVecsSU2 (139) = NormedVecs (135)−NormedVecs (137);\n...\nThe next step is to set up the perturbation part with\nV=t1(hop(c6(),c8())+hop(c4(),c6())+hop(c3(),c5())+hop(c1(),c3()))\n+t3(hop(c2(),c3())+hop(c3(),c4())+hop(c6(),c7())+hop(c5(),c6()));\nVsz=makematricesbzop (V/.rules,basis[[4;;4]]);\nvsz1=Select[Vsz,First[#1] ={−5}&][[1,2]];\nThe Elements Vi,jthen are given by\nFor[i= 1,i≤Length[values[[1]]],i++,\nFor[j= 1,j≤Length[values[[1]]],j++, ElementsV (i,j) = (vsz1.NormedVecsSU2 (j)).NormedVecsSU2 (i)]]\nFinally, the energy corrections can be computed easily from the formulas (A1). The energy correction of the ﬁrst\norder, e.g., is given by E(1)\nGS=ElementsV (GS,GS)."    },    {        "title": "0802.3646v1.Tunnel_barrier_enhanced_voltage_signals_generated_by_magnetization_precession_of_a_single_ferromagnetic_layer.pdf",        "content": " 1  Tunnel barrier enhanced voltage signals generate d by magnetization precession of a single \nferromagnetic layer \n \nT. Moriyama,1 R. Cao,1 X. Fan,1 G. Xuan,2 B. K. Nikoli ć,1 Y . Tserkovnyak,3 J. Kolodzey,2 and \nJohn Q. Xiao1  \n1Department of Physics and Astronomy, Univer sity of Delaware, Newark, DE 19716, USA \n2Department of Electrical and Computer Engineeri ng, University of Delaware, Newark, DE 19716, USA \n3Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA \n \n \n \nAbstract: \nWe report the electrical detection of magnetization dynamics in an Al/AlO x/Ni80Fe20/Cu tunnel \njunction, where a Ni 80Fe20 ferromagnetic layer is brought into precession under the \nferromagnetic resonance (FMR) conditions. The dc  voltage generated across the junction by the \nprecessing ferromagnet is enhanced about an  order of magnitude  compared to the voltage signal \nobserved when the contacts in this type of mu ltilayered structure are ohmic.  We discuss the \nrelation of this phenomenon to magnetic spin pumping and speculate on other possible \nunderlying mechanisms responsible for the enhanced electrical signal. \n \nPACS numbers:  76.50.+g, 72.25.Mk, 72.25.Hg \n  2  In recent years, basic and applied research  in metal-based spintronics has shifted \nincreasingly from the static to the dynamic magnetic properties in hybrid nanostructures \ncomposed of ferromagnetic and normal metal layers  [1-5]. A variety of experimentally observed \nphenomena involving nonlocal magnetization dynamics  in magnetic multilayers are due to two \ncomplementary effects: (i) the transfer of spin angular momentum accompanying charge \ncurrents driven by the applied bias voltage between  ferromagnetic layers results in torques that \n(for sufficiently high current densities) ge nerate spontaneous magnetization precession and \nswitching [1]; and (ii) the precessi ng magnetization of  a ferromagnet ( FM) pumps spins into \nadjacent normal metal layers ( NM) with no applied bias [2, 5, 6]. In particular, the spin pumping \neffect [5] is a promising candidate for realizing a spin battery device [7] as a source of elusive \npure spin currents (not accompanied by any net charge transport) emitted at the FM/NM  \ninterface, where steady magnetization precession of the FM layer is sustained by the absorption \nof external rf radiation under the FMR conditions. Another promising application of \nmagnetization dynamics is microwave-assist ed reduction of the switching field of FM, which \ncould play an important role in  advanced recording media [4]. \nThus far, however, the spin pumping effect has been demonstrated mostly indirectly  as \nan additional contribution to the FMR linewidth in FM/NM  multilayers (where the NM is Pt, Pd, \nCu, etc.) that can be described as the inte rface-induced enhancement of the Gilbert dumping \nconstant [8-11]. The vigorous pursuit of direct el ectrical detection of spin pumping has led to \ntheoretical proposals [6] to use a single precessing FM as both the source and detector  of \npumped spin accumulation in NM layers. This adroit scheme has b een realized in a very recent \nexperiment [2] measuring the diff erence in voltages of the orde r of several hu ndred nanovolts  3  between two FM/NM  interfaces of a NM1/FM/NM2  trilayer.  \nIn this Letter, we report measuremen ts of the dc voltage across Al/AlO x/Ni80Fe20/Cu \ntunnel junctions with precessing magnetization of Ni 80Fe20. The surprisingly large observed \nvoltage is about μV , which seems to qualitatively  agree with the spin pumping theory [5, 6] but \nrequires an unreasonably large  spin-mixing conductance of the FM/tunnel-barrier contact. We \nconclude that a new nonequilibrium phenome non, which dynamically couples the spin and \ncharge degrees of freedom, exists in tunneling stru ctures. The rest of this  Letter presents details \nof our experiment, and we conclude by speculati ng on possible theoretical scenarios responsible \nfor these surprising experimental results. \nThe experimental setup is illustrated in Fi g. 1. A tunnel junction was fabricated on the \nsignal conductor of a coplanar waveguide (C PW) transmission line. The tunnel junction \nstructure of  Cu (100nm)/Al (10nm)/AlO x (2.3nm)/Ni 80Fe20 (20nm)/Cu (70nm)/Au (25nm) \nwas fabricated on a Si substrate with a 1 μm thick thermal oxide layer, by using magnetron \nsputtering deposition and conventional microfab rication processing. The bottom-most 100 nm \nCu layer was patterned into the CPW designed to have 50 Ω characteristic impedance in the \nabsence of the tunnel structure. The aluminum  oxide tunnel barrier was formed by plasma \noxidation. The size of the tunnel junction pillar was 50×50 μm2, and the dc junction resistance \nwas measured to be 67 k Ω. A microwave signal from a vect or network analyzer (Agilent \n8753B) was introduced through the CPW and generated a microwave magnetic field rfHthat \nwas linearly polarized in the plane of the tunne l junction. The external dc magnetic field (-120 \nOe to 120 Oe) was swept along the axis of the CPW (the y-direction), so that the magnetization \nchanged its direction within the x-y plane. The precessing spin mainly rotated around the y-axis.  4  Two electrical probe tips were  used to measure the dc voltage across the junction. The \nmicrowave input signal was varied from 0.7 GHz to 4 GHz, with power up to 18 dBm \namplitude-modulated with a 400 Hz sinusoidal signal to allow for lock-in detection. It should be \nmentioned that there was always a few tens of microvolt background voltage at the detector due \nto microwave noise. We found th at the background voltage was dire ctly proportional to the input \nmicrowave power.  We can thus maintain the sa me power applied to the device at different \nfrequencies by slightly tuning the nominal input power (±1 dB m) to maintain the same \nbackground voltage within 20% error.  This ma kes it possible to compare the data between \ndifferent frequencies without concerning the frequency dependence of the CPW impedance \nwhich changes the power input to the device slightly.   \nFigure 2 shows the dc voltage as a functi on of the external magnetic field in the \nAl/AlO x/Ni80Fe20/Cu tunnel junction. At each microwav e frequency, the voltage peaks of \nmagnitude VΔappear symmetrically at positive and negative fields. The peak field as a \nfunction of the microwave frequency shown in Fi g. 3 agrees well with the values we obtained \nfrom the flip-chip CPW FMR measurements. Th e Kittel formula [12] fits the data with \nreasonable parameters, 4 9 kGsMπ= , 19 OekH= , and gyromagnetic ratio \n-1 -10.0176 s Oeγ= , confirming that the dc voltage peak appears at the uniform FMR mode of \nthe Ni 80Fe20 layer. The peak magnitude reaches about 1 μV at 2GHz which is much larger than \nthe maximum value of about 250 nV at 14.5 GHz reported in Re f. [2] for a Pt/Ni 80Fe20/Al \nstructure. Figure 4 shows microwav e power and frequency dependence of VΔ, which increases \nwith increasing microwave power [Fig. 4 (a)]. We also plot VΔ as a function of precession \ncone angle in Fig. 4(b). Th e precession cone angle of Ni 80Fe20 was determined by the change in  5  the tunnel resistance at the FMR field in IrMn/Fe 70Co30/AlO x/Ni80Fe20 magnetic tunnel junctions \n[13] with 20mV bias voltage so that the dc voltage effect of the microvolt order we are \ndiscussing here can be neglected.  A clear dip in an tiparallel states and a pe ak in parallel states \nare observed corresponding to FMR fields, and the precession angle θ can be determined from \n()1c o s RR θ Δ∝ − .  At 10 dBm power input, the precessi on cone angle was around 17° [13].  \nAs the applied frequency increased, VΔ increases almost linearly,  as shown in Fig. 4(c). \nBefore attempting to interpret our results, we have to examine carefully the \nrectification effects which could induce similar dc voltage response.  Possible rectification effect \nmay arise from both the time-dependent anis otropy magnetoresistance (AMR) effect and the \nanomalous Hall effect (AHE) discussed in Refs.[ 14]. The current due to these two effects is \ngiven by \n()2RMρσρΔ′=− ⋅ + ×jj MM jM,   (1) \nwhere j is the current, σ is the conductivity, ρ is the resistivity, ρΔ is the \nmagnetoresistive anisotropy, and R is the anomalous Hall constant. The magnetization \nprecessing around the y-axis is described by the vector \n()s i ns i n ,c o s ,c o ss i nMt M M tωθθω θ =M . The microwave-induced current across the \njunction along the microwave electric fi eld direction (z-axis) is given by \n()() αω+ = t jcos,0,0j , where αis the phase difference with respect to the phase of spin \nprecession . The z -component of Eq. (1) is of in terest to our experiments, \n() θωαωρρ2 2sin cos cos ' t t j jz + Δ−= . The time average of zj′ is zero, allowing us to \nconclude that there is no dc component in the z -direction in our sample. The result holds even if  6  the precession axis fluctuates in the x-y plane.  Thus, in our sample c onfiguration we expect no \ndc voltage generated due to the rectification effect. We purposely broke  the tunnel barrier to \ninvestigate Al/Ni 80Fe20/Cu contact and also made Cu/AlO x/Al/Ni 80Fe20/Cu junctions. In both \ncases, no VΔ was observed within our measurement se nsitivity of about 100 nV . This implies \nthat the large VΔ in Al/AlO x/Ni80Fe20/Cu was indeed developed due to the AlO x/Ni80Fe20 \ninterface. \nLet us now try to interpret our results w ithin the framework of the standard spin \npumping theory [5-7].  At the FMR, a steady precession of the magneti zation of the FM layer \npumps a spin current into the adjacent NM according to [5, 7] \nRe Im4pump\nsddggdt dt π↑↓ ↑↓⎛⎞=× +⎜⎟⎝⎠mmIm=,  (2) \nwhere m is the unit vector along th e instantaneous direction  of the precessing magnetization \nand Reg↑↓ (Img↑↓) is the real (imaginary) part of the dimensionless interfacial spin-mixing \nconductance (in units of 2eh ) which describes spin tr ansport perpendicular to mat the \nFM/NM  interface [5, 15, 16]. For transparent intermetallic FM/NM contacts Img↑↓ is \ntypically neglected because of  being much smaller than Reg↑↓[5-7], while for low transparent \ncontacts we find  Re 0.5gg↑↓= and Im 0.5gg↑↓\u0011  (using the simple Stoner model for \nFM and random binary alloy w ith a gap in the energy spectru m for the tunnel barrier as NM \nlayer; g is the total charge conductanc e of the junction).  The possi bility for non-negligible  \nImg↑↓ for tunneling interfaces is also highlighted by recent first principles  calculations [17]. \nThe injected spin current builds up a spin accumulation sμ in the NM layer (close to the \nFM/NM  interface) when the spin -flip relaxation rate in NM is smaller than the spin injection rate.  7  This, in turn, drives a backward flowing spin current back\nsIinto the precessing FM [5]. The \nbackward flowing spin current parallel to the magnetization can be absorbed by the FM, in the \npresence of spin-flip processes . Due to spin-dependent bulk and interface conductances, this \nabsorbed spin current is converted  into charge accumulation at the FM/NM  interface [18]. The \nmaximum value of the voltage drop dcVacross the FM/NM  interface, at fixed frequency ω \nand cone angle θof magnetization precession (assuming the frequency is much greater than the \ncharacteristic spin-flip rate in the normal me tal and the ferromagnet is thicker than its bulk \nspin-diffusion length), is obtained for NM layer thickness much sma ller than its spin-diffusion \nlength dN << λdsN  as [6]:  \n()2\ndc 22 2 2sin cos\n2 () s i n ( 1 ) c o sF\nFN FpgVe gp gg p g g gω\nωω ω ω ω ωθθ ω\nηθ θ↑↓=\n−+ + + −=,  (3) \nwhere ()tanhN\nNN N s dgg dω η λ↑↓=  with the thickness dN and the spin diffusionN\nsdλ of the \nNM layer, Fg and Ng parameterize the transport properties of the bulk FM and NM, gω↑↓ \nis the real part of the eff ective spin-mixing conductance, ()( ) p gg ggω ωω ωω↑↓↑ ↓=− +  is the \ninterfacial spin-polarization, gggωωω↑↓=+  is the sum of spin-up and spin-down effective \nconductances of the FM/NM  interface. The “ef fective” interfacial transport quantities are in \ngeneral frequency-dependent since they have to be  evaluated for the interface resistance in series \nwith a NM resistor of length NM LDω ω=  over which the oscillati ng transverse components \nof spin accumulation in NM (with diffusion constantNMD) are averaged to zero, although this is \nnot important in practice for high-impeda nce tunnel barriers.  The voltage drops 1,2\ndcVemerging \nat each of the two FM/NM contacts will differ from each other when conductances gω↑↓ and/or \nspin-flip diffusion lengthsN\nsλon two sides of the multilayer ar e substantially different, as  8  observed by measuring 12\ndc dc VV VΔ= − on lateral Pt/Ni 80Fe20/Al device in a recent experiment \n[2].  \nTo compare the spin pumping theory with our results, we assume that spin-mixing \nconductance is governed by the AlO x/Ni80Fe20 interface, but there is no spin flipping inside the \nbarrier and spin accumulation is induced in the Al layer. Since the interface conductance gω \nfor the low transparency tunnel barri er is much smaller than smallFg, the first term in the \ndenominator of Eq. (3) reduces to ()2sinFNgηθ+ , so that Eq. (3) can be parameterized \nwithggωω↑↓,Nη,pω, and θ. Using 0.3 pω= for the AlO x/Ni80Fe20 interface, Fig. 4(b) shows \nthe best fitting (solid line) of our results by Eq. (3), where we extract 3.4 ggωω↑↓≈ and \n0Nη≈ from the fit. We found that Nη has to be set to zero in order to fit our data, which \nrequires that Ng is roughly comparable or smaller than gω↑↓, while we expect the former to be \nseveral orders of magnitude larger than the latter, using the measured tunneling conductance. \nThis is the first discrepancy between our results and an attempt to explai n them using standard \ninterfacial spin pumping theory originally devel oped [5, 7] and experimentally confirmed [2] for \nintermetallic FM/NM  contacts. On the other hand, linear fitting of VΔvs. frequency in Fig. 4(c) \nyields the slope of 0.85 μV/GHz, and 6.1 ggωω↑↓≈ at precession cone angle of 17° by Eq. (3). \nThese values are larger than the typical value 1 ggωω↑↓≈ for transparent intermetallic contacts, \nwhich is highly unexpected  when compared to standard estimates [16] of ggωω↑↓ for trivial \n(non-magnetic) tunnel contacts. \nV oltage generation based on the spin pumping mechanism [6] is based on spin injection \ninto the normal metal, across the FM/NM  interface, with its subsequent  diffusion, relaxation, and \nbackflow into the ferromagnet, which is ultimately responsible for the build-up of the voltage  9  drop across the contact. A tunnel barrier exponen tially impedes electron flows (and thus spin \ncurrents) across the FM/NM  contact, and one, therefore, would not expect  a significant voltage \ngeneration by the spin-pumping mechanism. This is  the reason why we were  not able to reach a \nquantitative agreement with the theory. The tunne l barrier essentially cuts off the normal metal \nfrom the FM, while a voltage probe may now be thought  of as a nonintrusive probe of dynamic \nprocesses within the FM. If the magnetization dynamics can  generate nonequilibrium spin \naccumulation inside the ferromagnet, in analogy with the pumped spin generation in the normal \nmetal (presumably requiring spin-orbit or other spin-flip processes in the FM), the voltage \nmeasured by the FM may in fact be probing this spin accumulation rather than a nonlocal spin \npumping process. Exploring this possibility re quires further theoretical  analysis and other \nnonlocal probes of the magnetization dynamics. Fi nally, we note that our theoretical discussion \ncompletely disregarded many-body effects due to  electron-electron inte ractions, which may \nmodify substantially the predictions of the standa rd spin pumping theory, especially if we drive \nthe magnetization dynamics beyond the linearized regime. \nIn conclusion, we observed a large dc voltage , of the order of microvolts, across the \nAl/AlO x/Ni80Fe20/Cu tunnel junctions, due to a dynamic spin and charge coupling driven by the \nprecessing magnetization of a single Ni 80Fe20 ferromagnetic layer at ferromagnetic resonance. \nBy short circuiting the tunnel ba rrier, we demonstrated that the observed dc voltage mainly \narises from the Al/AlO x/Ni80Fe20 contact. The phenomenon appears qualitatively similar to the \npredictions of the spin pumping formalism, but  a quantitative analysis shows a number of \ndiscrepancies with the standard theory. This  suggests a new nonequilibrium mechanism for the \nspin and charge coupling, which is responsible fo r the voltage generation much larger than that  10  observed very recently for intermetallic interfaces  [2]. We speculate on the role of intrinsic \ndynamic processes in the ferromagnet and the effect s of the electron-electr on interactions, as \npossible culprits for our observations, but a more t horough theoretical analysis is desirable in the \nfuture. \nWe thank M. D. Stiles and S.-T. Chui fo r illuminating discussions. This work was \nsupported by NSF DMR Grant No. 0405136, and DOE DE-FG02-07ER46374. 11   \nReferences:  \n[1] S. I. Kiselev, et al., Nature 425, 380 (2003). \n[2] M. V . Costache, et al., Phys. Rev. Lett. 97, 216603 (2006). \n[3] J. Grollier, et al., Phys. Rev. B 73, 060409 (2006). \n[4] T. Moriyama, et al., Appl. Phys. Lett. 90, 152503 (2007). \n[5] Y . Tserkovnyak, et al., Rev. Modern Phys. 77, 1375 (2005). \n[6] X. H. Wang, et al., Phys. Rev. Lett. 97, 216602 (2006). \n[7] A. Brataas, et al., Phys. Rev. B 66, 060404 (2002). \n[8] S. Mizukami, et al., J. Magn. Magn. Mater. 226, 1640 (2001). \n[9] B. Heinrich, et al., Phys. Rev. Lett. 90, 187601 (2003). \n[10] B. Heinrich, et al., J. Supercond. Novel Magn. 20, 83 (2007). \n[11] T. Gerrits, et al., J. Appl. Phys. 99, 023901 (2006). \n[12] C. Kittel, Introduction to solid state physics  (Wiley, Hoboken, NJ, 2005), 8th ed., Chap. 13. \n[13] T. Moriyama, et al., unpublished. \n[14] W. G. Egan, et al., J. Appl. Phys. 34, 1477 (1963). \n[15] K. Xia, et al., Phys. Rev. B 65, 220401 (2002). \n[16] A. Brataas, et al., Phys. Rep. 427, 157 (2006). \n[17] I. Turek, et al., J. Phys.: Condens. Matter 19, 365203 (2007). \n[18] S. Takahashi, et al., Phys. Rev. B 67, 052409 (2003). \n \n  12  Figure captions:  \nFIG. 1. (Color online) Schema tic diagram of the sample stru cture (a) and the measurement \ngeometry (b). The arrow in the Ni 80Fe20 layer indicates the magnetiza tion direction. An external \ndc field exH is applied in the y-direc tion and the rf magnetic field rfH is applied along the \nx-direction. A coplanar microwave probe f eeds microwave signals through the coplanar \nwaveguide. DC voltage across the junction (TJ) is measured between top of the junction and \nsignal line of the CPW.  \n \nFIG. 2. (Color online) The dc voltage VΔ generated across the Al/AlO x/Ni80Fe20/Cu tunnel \njunction as a function of the ex ternally applied static magnetic  field. The frequency of the \napplied rf field ranges from 1.8 to 2.8 GH z. The background voltage is subtracted for \ncomparison purpose. \n \nFIG. 3. (Color online) The frequency dependence of  static magnetic field at which the dc voltage \npeak (circles) appears in Fig. 2. The crosses label the frequency depe ndence of the resonance \nfield obtained from FMR measurem ents on flip-chip structures in the CPW line. The curve is a \nfit to the Kittel formula [12]. \n \nFIG. 4. The amplitude of the dc voltage VΔ measured across Al/AlO x/Ni80Fe20/Cu device as \nfunction of: (a) microwave power;  (b) precession cone angle; and (c) microwave frequency at 10 \ndBm. Solid lines in (b) and (c) are the f it to Eq. (3) as described in the text. \n  13   \n \n \nFig.1 T. Moriyama et al. 14  \n \nFig. 2 T. Moriyama et al. 15  \n \nFig.3 T. Moriyama et al.  16  \n \nFig. 4 T. Moriyama et al. \n "    },    {        "title": "1108.2582v1.Ferromagnetism_in_semiconductors_and_oxides__prospects_from_a_ten_years__perspective.pdf",        "content": "Ferromagnetism in semiconductors and oxides: prospects from a ten years' perspective\nTomasz Dietl1, 2,\u0003\n1Institute of Physics, Polish Academy of Science,\nal. Lotnik\u0013 ow 32/46, PL-02-668 Warszawa, Poland\n2Institute of Theoretical Physics, University of Warsaw, PL-00-681 Warszawa, Poland\n(Dated: August 15, 2011)\nOver the last decade the search for compounds combining the resources of semiconductors and fer-\nromagnets has evolved into an important \feld of materials science. This endeavour has been fuelled\nby continual demonstrations of remarkable low-temperature functionalities found for ferromagnetic\nstructures of (Ga,Mn)As, p-(Cd,Mn)Te, and related compounds as well as by ample observations\nof ferromagnetic signatures at high temperatures in a number of non-metallic systems. In this pa-\nper, recent experimental and theoretical developments are reviewed emphasising that, from the one\nhand, they disentangle many controversies and puzzles accumulated over the last decade and, on\nthe other, o\u000ber new research prospects.\nPACS numbers: 75.50.Pp\nIntroduction\nAdvances in the epitaxy of semiconductor compounds\nhave made it possible to fabricate quantum structures in\nwhich con\fned electrons or photons exhibit outstanding\nproperties and functionalities. Similarly, the atomic pre-\ncision of metal and oxide \flm deposition has allowed to\nmaster a number of striking spin transport phenomena.\nThe discovery of ferromagnetism in p-type Mn-doped IV-\nVI (ref. 1), III-V (refs 2{4), and II-VI (refs 5,6) com-\npounds has opened a road towards the development of\nmultifunctional materials systems bridging the resources\nof semiconductor quantum structures and ferromagnetic\nmultilayers as well as has enabled the study of collective\nmagnetic phenomena as a function of the spin and carrier\ndensities.\nOver the last ten years or so the \feld of ferromag-\nnetism in dilute magnetic semiconductors (DMSs) and\ndilute magnetic oxides (DMOs) has evolved into an im-\nportant branch of materials science. The comprehensive\nresearch on these systems has been stimulated by con-\ntinual demonstrations of outstanding low-temperature\nfunctionalities in (Ga,Mn)As, p-(Cd,Mn)Te, and re-\nlated structures7,8, some examples being spin-injection9,\nelectric-\feld10,11and electric-current12control of mag-\nnetism, tunneling anisotropic magnetoresistance in pla-\nnar junctions13and in the Coulomb blockade regime14,\nas well as current-induced domain displacement without\nthe assistance of a magnetic \feld15. These \fndings have\nput into focus the interplay of magnetization texture and\ndynamics with carriers' population and currents, a broad\ntopic of today's physics of spintronic materials. At the\nsame time, since the \frst report on (Ti,Co)O 2(ref. 16),\nthe persistence of spontaneous magnetization to above\nthe room temperature has been found for a number of\nDMOs and DMSs, and even for materials nominally con-\ntaining no transition metal (TM) impurities.\nHowever, despite the massive investigations, the origin\nand control of ferromagnetism in DMSs and DMOs is, ar-\nguably, the most controversial research topic in today'smaterials science and the condensed matter physics. As\nemphasized here, the abundance of contradicting views\nhas resulted from intertwined theoretical and experimen-\ntal challenges, requiring the application of cutting edge\ncomputational and materials nanocharacterisation meth-\nods, often becoming available only now. In this way,\nDMSs and DMOs emerge as an outstanding playground\nto test our understanding of unanticipated relationships\nbetween growth conditions and a self-organized alloy\nstructure17as well as between quantum localization, car-\nrier correlation, and ferromagnetism, and constitute an\nactive research direction18{20.\nWe begin this review by recalling the foundations of\nthep\u0000dZener model, proposed a decade ago to de-\nscribe the origin and properties of ferromagnetism in p-\ntype Mn-doped semiconductors21. A crucial role of the\nAnderson-Mott localization in the physics of these sys-\ntems is then discussed in the context of other models put\nforward to explain outstanding \fndings which have been\naccumulated over the recent years for (Ga,Mn)As. It is\nshown that the p\u0000dZener model describes a number of\nthermodynamic and micromagnetic properties of III-V\nand II-VI DMSs as well as continues to constitute a good\nstarting point to address the question about prospects\nof research on hole-mediated ferromagnetic semiconduc-\ntors. The second part of the review is devoted to the\norigin and control of high temperature ferromagnetism\nin DMSs and DMOs. We argue, exploiting results of ex-\ntensive nanocharacterisation works, that puzzling prop-\nerties of these compounds re\rect a highly non-random\ndistribution of magnetic cations. While a number of ap-\npealing functionalities has already been demonstrated for\nthese ferrmagnetic/semiconductor nanocomposites, we\nare only at the beginning of the road to demonstrate\ndevice structures of these emerging materials systems.arXiv:1108.2582v1  [cond-mat.mtrl-sci]  12 Aug 20112\nTHEp\u0000dZENER MODEL\nIn view of the progress in materials fabrication by\nepitaxial methods2{6,22it was timely a decade ago to\nunderstand the ferromagnetism in DMSs as well as to\nask whether the Curie temperature TCcan be raised\nto above 300 K from the 110 K observed at that time\nin (Ga,Mn)As containing only 5% of Mn (ref. 22).\nPhotoemission23as well as optical studies in the single\nimpurity limit24demonstrated that Mn provides both lo-\ncalised spins and itinerant holes mutually coupled by a\np\u0000dexchange interaction. Zener25\frst proposed the\nmodel of ferromagnetism driven by the exchange inter-\naction between band carriers and localized spins. In the\ncase of semiconductors, the Zener model26,27is equivalent\nto the approach developed by Ruderman, Kittel, Kasuya,\nand Yosida (RKKY), in which the Friedel oscillations of\nthe spin density are taken into account26.\nIn the proposed implementation of the Zener model the\nstructure of the valence subbands was described by the\nKohn-Luttinger six bands' kphamiltonian, taking the\nspin-orbit interaction into account21. Thermodynamic\ncharacteristics were then evaluated in the mean-\feld ap-\nproximation. At the same time, arguments were pre-\nsented why the model is valid even on the insulator side of\nthe Anderson-Mott metal-to-insulator transition (MIT),\nprovided that the holes remain weakly localised.\nThis approach was found to constitute an appropri-\nate minimal theory, capable to describe adequately the\nmagnitude of TCand of magnetic anisotropy \felds in-\nduced by biaxial strains in (Ga,Mn)As and p-(Zn,Mn)Te\n(ref. 21). It was also pointed out that GaN and ZnO\ncontaining appropriately high concentrations of both Mn\nspins (x&5%) and delocalised or weakly localised holes\nin the valence band ( p&3:5\u00011020cm\u00003) might support\nthe ferromagnetic order to above the room temperature.\nIt was underlined, however, that prior to the veri\fcation\nof this prediction, important issues of solubility limits\nand self-compensation as well as of the transition to a\nstrong-coupling case with the decreasing lattice constant\nneed to be addressed experimentally21.\nA GUIDE THROUGH OTHER MODELS\nExtensive studies over last ten years have made clear\nthat ferromagnetic DMSs and DMOs form two distinct\nclasses. The \frst class comprises p-type Mn-based DMSs,\nin which the ferromagnetism is associated with the pres-\nence of holes. Here, step by step improvements in growth\nprotocols and in post-grown processing have made it pos-\nsible to increase the Mn and hole densities, particularly\nin (Ge,Mn)Te (ref. 28) and (Ga,Mn)As (ref. 29{31), in\nwhich the magnitudes of TC's approach now 190 K at a\nvalue of the e\u000bective Mn concentration xe\u000bbelow 10%,\nas implied by the magnitude of saturation magnetization\n(see Fig. 1). While this evolution of TCis consistent\nwith thep\u0000dZener model, its basic foundation, namelythat in the concentration range relevant to ferromag-\nnetism the holes reside in the valence band in (Ga,Mn)As\nand related systems has been objected by two schools of\nthoughts:\n•Following pioneering ab initio work carried out for\n(In,Mn)As (ref. 32), it has been argued based on the\noutcome of available \frst principles methods that\nthe holes reside in band-gap states derived from the\nTMdlevels, so that the relevant spin-spin coupling\nmechanism is the double-exchange33,34.\n•A series of \fndings from optical and transport stud-\nies, and hard to reconcile with expectations for the\nholes moving in a weakly perturbed valence band,\nhave been taken as an evidence for the location of\nthe Fermi energy within a Mn-acceptor impurity\nband detached from the valence band or retaining\nthedcharacter of Mn dopants even in the region,\nwhere they overlap with the valence band on the\nmetallic side of the MIT35,36.\nTo the second class of ferromagnetic systems belongs\na broad range of semiconductors, oxides, and carbon\nderivatives, showing ferromagnetic-like features persist-\ning to above room temperature without the presence of\nitinerant holes or, in some cases, even without intentional\ndoping by TM impurities37{39. A number of diverging\nmodels has been proposed to explain the origin of this\nintriguingly robust ferromagnetism:\nFirst, in a long series of ab initio works a ferromag-\nnetic ground state has been found for dilute TM spins\neven in the absence of band carriers40. Following pio-\nneering contributions32,41, the e\u000bect has been assigned to\nthe ferromagnetic superexchange41or, more frequently,\nto the double exchange32that turns up, within the em-\nployed computation methodologies, if the states derived\nfrom the TM dlevels are partly occupied for one spin\ndirection40.\nSecond, it is supposed that electrons either residing\nin the conduction band42or forming bound magnetic\npolarons43mediate ferromagnetic couplings between di-\nlute TM spins. Alternatively, the presence of these cou-\nplings is assigned to carriers residing on defects, such as\nvacancies44or on residual impurities, such as hydrogen45.\nThird, the ferromagnetic response is related to spins of\nelectrons residing on point or extended defects and cou-\npled by an exchange interaction. Within this so-called d0\nmodel of high temperature ferromagnetism, the presence\nof magnetic impurities is either unnecessary or serves\nmerely to bring the Fermi level to the relevant defect\nstates46.\nFinally, it has been persistently suggested39,47that the\nlimited solubility of TMs' impurities in particular hosts\nmay result in the formation of nanoscale regions contain-\ning a large density of magnetic cations and, thus, speci-\n\fed by a high spin ordering temperature.3\n0 50 100 150 200 25001020304050\nTemperature (K)\n  Magnetization (emu/cm3)\nGe1-xMnxTe\n   x = 0.08\n0 1 02 03 04 05 06 07 004080120160200Ga1-xMnxAs\nSaturation magnetization (emu/cm3)\n  Curie temperature (K)\nFIG. 1: Experimental data for p-type DMSs \flms showing\nthe Curie temperature TCapproaching 200 K at the e\u000bective\nMn concentration xe\u000bbelow 10%. Upper panel: temperature\ndependence of magnetization in (Ge,Mn)Te with high (circles)\nand low (triangles) hole concentrations (after ref. 28); Lower\npanel:TCas a function of saturation magnetization MSatfor\nannealed (Ga,Mn)As \flms grown in various molecular beam\nepitaxy (MBE) systems (after ref. 30).\nWHERE DO THE HOLES RESIDE IN (Ga,Mn)As?\nAccording to the double exchange scenario33,34, the\nanti-crossing picture36,48,49, and the impurity band\nmodels4,35, the hole states retain impurity band charac-\nteristics in the density regime relevant to the ferromag-\nnetism of (Ga,Mn)As. For a comprehensive presentation\nof arguments in favour of such a scenario we remit the\nreaders to ref. 35.\nAnother view, shared by the present author and ex-\nposed in detail elsewhere50,51, is that similarly to other\ndoped semiconductors the carrier localization in p-type\nDMSs results from a collective e\u000bect of randomly dis-\ntributed scattering centres upon the Fermi liquid of\nstrongly correlated band carriers51{54. Guided by pre-\nvious extensive studies of non-magnetic semiconductors,we may anticipate that rather than absolute values, only\ncertain scaling characteristics of d.c., a.c., and tunneling\nconductivity tensors can be presently interpreted theoret-\nically near the MIT, at least at temperatures below the\nmomentum relaxation rate. This is in contrast to ther-\nmodynamic properties, such as electronic speci\fc heat,\nwhich are virtually unperturbed by disorder and elec-\ntronic correlation at the localization boundary52{54.\nEmpirically, the Anderson-Mott MIT occurs for the\ncarrier concentration pcat which the magnitude of the\nkinetic energy per band carrier, Ekin\u0019(3=5)EF, calcu-\nlated with no disorder diminishes to about one third of\nthe single impurity binding energy EI(ref. 55). In Fig. 2\nthe experimental values of EIfor Mn acceptors in vari-\nous III-V semiconductors are shown56. As seen,EIand,\nthus,pcis enhanced rather dramatically comparing to\nnon-magnetic acceptors, particularly on going from an-\ntimonides to nitrides through arsenides and phosphides.\nThis shift is caused by the p\u0000dhybridisation, whose\nimportance grows with the decreasing cation-anion bond\nlength, ultimately resulting in a transition to the strong\ncoupling limit, where the hole binding is dominated by\nthep\u0000dinteraction57.\n0.45 0.50 0.55 0.60 0.65-3-2-10\nInSbAlSbGaSb\nInAsInPGaAs AlAsGaP AlPInNGaN\n  Energy (eV)\nLattice constant (nm)\nFIG. 2: Experimental energies of Mn levels in the gap of III-\nV compounds with respect to the valence-band edges (after\nref. 56).\nAccording to the scaling theory and relevant\nexperiments51{54, the Anderson-Mott MIT is continuous.\nHence, the carrier localization length \u0018decreases rather\ngradually from in\fnity at the MIT towards the impurity\nBohr radius in the strongly localised regime, so that at\na length scale smaller that \u0018, the wave function retains\nan extended character. Such band-like carriers, whose\nquantum di\u000busivity vanishes at the MIT, were originally\nthought to mediate the long-range interactions between\nthe TM spins in DMSs in the whole density regime rele-\nvant to ferromagnetism21,26.4\nIn agreement with this view temperature dependent\nquantum corrections to the conductance58and the char-\nacter of tunneling DOS20are consistent in (Ga,Mn)As\nwith the expectations for Anderson-Mott localization of\nholes in the GaAs valence band. In particular, scanning\ntunneling microscopy data20, though a\u000bected presum-\nably by the proximity to the surface, point rather directly\nto the crucial importance of bothdisorder and carrier cor-\nrelation in the relevant range of Mn concentrations. At\nthe same time, the direct visualisation20of spatial \ructu-\nations in local DOS provides a support to the view19that\nthe disappearance of ferromagnetism with carrier local-\nization proceeds viaan intermediate superparamagnetic-\nlike phase.\nWithin the Anderson-Mott localization model, the\ne\u000bects of disorder and carrier correlation appear as\nsome broadening and Landau's renormalisation of va-\nlence band thermodynamic DOS atEF,\u001aFwhich deter-\nminesTCwithin the p\u0000dZener model (refs 21,26,59).\nIn this context, thermoelectric power Sat high tem-\nperatures is relevant, as it is a good measure of \u001aF, so\nthat its magnitude may tell between the valence band\nand impurity band pictures. Recent measurements of\nSin compensated (Ga,Mn)As were interpreted in terms\nof the anti-crossing model treating broadening of the\nimpurity band as an adjustable parameter49. In this\nway, a rather small di\u000berence between the magnitudes\nofSfor (Ga,Mn)As and GaAs:Be at given hole densi-\nties was explained. We note, however, that the Mott\nformulaS= (\u00192=3e)k2\nBT@EFln\u001b(EF) with\u001aFof the\nGaAs valence band58describes quantitatively not only\nthe data49for GaAs:Be but also for (Ga,Mn)As assuming\nthat the energy dependence of the apparent hole mobility\n\u0016=\u001b=ep changes from the value expected for acoustic\nphonon scattering, \u0016\u0018E\u00001=2\nF, to the one speci\fc for\nionised impurity scattering, \u0016\u0018E3=2\nF, when the degree\nof compensation increases.\nFurthermore, within the impurity-band models, the\nmagnitude of TCis predicted to reach a maximum, when\nthe Fermi level is shifted across the peak in the den-\nsity of the impurity states. The accumulated data for\n(Ga,Mn)As show that the value of TCdecreases mono-\ntonically when diminishing the hole concentration by\ngating60or by increasing the concentration of compen-\nsating donors49,61,62. On the other hand, it was found\nin recent studies that the magnitude of TCactually in-\ncreases by co-doping with Si donors48and, moreover, it\ngoes through a maximum as a result of the modulation\ndoping by Be acceptors63. As underlined in these works,\nthe \fndings are consistent with the impurity band sce-\nnario. However, the presented data reveal that the in-\ncrease ofTCin Si-doped samples and its decrease in the\nBe case is associated with, respectively, an enlargement\nand a reduction of the saturation value of magnetization\nand, thus, of the Mn concentration xe\u000bthat determines\nthe magnitude of TC. The results48,63can, therefore, be\nexplained by the known anticorrelation64betweenxe\u000b\nand the density of holes during the epitaxy, here reducedby Si donors48or \"siphoned o\u000b\" from the Be-doped bar-\nrier into the grown quantum well63.\nAt the same time, low values of \u0016(below 10 cm2/Vs),\ntaken as an evidence for the large magnitude of an ef-\nfective hole mass35,36, can be explained by the prox-\nimity to the MIT, where charge di\u000busion is much re-\nduced by quantum localization e\u000bects. Furthermore, re-\nferring to a shift of an a.c. conductivity maximum with\nthe hole density35, which contradicts the expectations of\nthe Drude-Boltzmann theory, we emphasise that the fre-\nquency dependent conductance near the MIT, at least up\nto frequencies of the order of the momentum relaxation\nrate, is dominated by quantum localization e\u000bects52{54,\nwhose presence may account for the observed anomalies.\nCURIE TEMPERATURE FOR\nCARRIER-MEDIATED FERROMAGNETISM IN\nIII-V DMSs\nIn Fig. 3 the highest values of TCfound to date in p-\ntype Mn-based III-V DMSs are reported29{31,65{68, and\ncompared to the early predictions of the p\u0000dZener\nmodel21,69for \fxed values of the Mn and hole concen-\ntrations. We see that the theory reproduces the chemi-\ncal trends and describes semi-quantitatively the absolute\nvalues ofTC. The observed trend re\rects a decrease of\nthep\u0000dexchange energy for larger cation-anion dis-\ntances as well as an enhanced role of the competing\nspin-orbit interaction in materials with heavier anions.\nAt the same time, the dependence of TCon the e\u000bec-\ntive Mn concentration30,61and the density of itinerant\nholes, changed in (Ga,Mn)As by Mn concentration, donor\ncompensation or by gating60,61,70, is consistent with the\np\u0000dZener model. Furthermore, the model describes\nproperly the magnitude of the strain-induced magnetic\nanisotropy71.\nUnfortunately, a more detailed comparison between\ntheory and experiment is hampered by the lack of in-\nformation on short-range antiferromagnetic interactions,\nwhich become progressively more important when the\nMn concentration increases, as well as by enduring di\u000e-\nculties in the accurate determination of the Mn and hole\ndensities. According to 3D atom probe investigations72,\nMn ions are distributed randomly in (Ga,Mn)As within\na 1 nm resolution. Nevertheless, from previous channel-\ning studies73we know that because of self-compensation,\na considerable portion of Mn ions occupies interstitial\npositions, and hence they act as double donors70, com-\npensating partly holes as well as Mn spins, due to a sup-\nposedly antiferromagnetic coupling between interstitial\nand substitutional Mn pairs. Furthermore, a large frac-\ntion of Mn, particularly in annealed samples, reside in a\nnear-to-the-surface MnO layer.\nThe applicability of the p\u0000dZener model for\n(Ga,Mn)As and related systems has been con\frmed by\nab initio studies in which inaccuracies of the LSDA are\npartly compensated by the LSDA + U approach74or by5\n10 100   \n \nInSbInAs\nGaSbGaP\nGaAs\nCurie temperature (K)Computed TC  \nxMn = 0.05 \np = 0.35 nm-3\n10 100   \n \nInSbInAs\nGaSbGaP\nGaAs\nCurie temperature (K)Highest TC\nreported for\np-type (III,Mn)V\nFIG. 3: Predictions of the p\u0000dZener model compared to ex-\nperimental data for p-type (III,Mn)V DMSs. Upper panel:\ncomputed values of the Curie temperature TCfor various\np-type semiconductors containing 5% of Mn and 3 :5\u00021020\nholes per cm3(after ref. 21; the value for (In,Mn)Sb is taken\nfrom ref. 69). Lower panel: the highest reported values\nfor (Ga,Mn)P (ref. 65); (Ga,Mn)As (ref. 29,30); (In,Mn)As\n(ref. 66); (Ga,Mn)Sb (ref. 67); (In,Mn)Sb (ref. 68).\nself-interaction corrections75. Furthermore, a number of\nferromagnetism models, tailored to DMSs without holes\nin the valence band, have been put forward, as reviewed\nelsewhere40,70. It is still unclear, however, whether a long\nrange ferromagnetic order can settle, say, above 10 K, if\nholes are bound to individual Mn acceptors in DMSs (the\nstrongly localized regime), so that the exchange interac-\ntion decays exponentially with the distance between spin\npairs. So-far, a ferromagnetic coupling between isolated\nnearest neighbor Mn pairs was revealed be scanning tun-\nneling microscopy in GaAs:Mn, and analyzed successfully\nin terms of a tight binding model76.\nIt is instructive to compare (Ga,Mn)As, (Ga,Mn)P,\nand (Ga,Mn)N containing the same concentration of Mn,\nsay, 6%, as in these three material systems EIdi\u000bers\nCurie temperature\nTM concentration, DOSstrong coupling\nstrong coupling\nweak coupling\nweak couplingVCA\nVCA\nMIT MITFIG. 4: Schematic dependence of TCon the concentration of\nmagnetic impurities and density of hole states at the Fermi\nlevel for a weak and a strong coupling. Higher values of TC\nare predicted within the virtual crystal and molecular \feld\napproximation for the strong coupling. However, the region,\nwhere the holes are localized and do not mediate the spin-spin\ninteraction is wider in the strong coupling case (after ref. 57).\nsigni\fcantly, according to the data collected in Fig. 2.\nThe (Ga,Mn)As \flms containing 6% of Mn and typically\nabove 1020holes per cm3are on the metallic side of the\nMIT22. In the case of Ga 0:94Mn0:06P the magnitude of\nEIis large enough to result in hole localization. How-\never, the magnitude of the conductance activation energy\n(ref. 65), a factor of ten smaller than EIfor GaP:Mn, in-\ndicates that the holes are only weakly localised. Accord-\ningly, also in this case the p\u0000dZener model can serve\nto explain the origin of ferromagnetic correlations.\nIn contrast, no information on hole transport is avail-\nable for the MBE-grown Ga 0:94Mn0:06N \flm77, indicating\nthat the strongly localised regime is reached. In line with\nthe notion that itinerant holes are necessary to observe\na coupling between diluted spins, the observed TCis as\nlow as 8 K. In terms of a schematic drawing presented in\nFig. 4, (Ga,Mn)N represents the strong coupling case,\nwhere the holes remain localised over a wide rage of\nMn concentrations, in contrast to both Ga 1\u0000xMnxAs, for\nwhich the MIT appears already below x= 2% for weakly\ncompensated samples50, and (Ga,Mn)P representing an\nintermediate case.\nIt is worth noting that there are indications of a non-\nrandom Mn distribution in another (Ga,Mn)N \flm grown\nby MBE (ref. 78). This may suggest that TC= 8 K con-\nstitutes actually an upper limit for TCin Ga 0:94Mn0:06N.\nThese \fndings demonstrate, therefore, that the cur-\nrent ab initio theory40, predicting TC\u001960 K for\nGa0:94Mn0:06N, still overestimates the signi\fcance of fer-\nromagnetic couplings in the case of DMSs with no valence\nband holes.6\nCOMPETING INTERACTIONS IN p-TYPE II-VI\nDMSs\nIn II-VI compounds, where Mn is an isoelectronic im-\npurity, it is possible to control independently the spin and\nthe carrier density. However, at given Mn and hole con-\ncentrations, TCis much lower in II-VI DMSs, comparing\nto III-V compounds, owing to a destructive in\ruence of\nthe short-range antiferromagnetic superexchange. This\ne\u000bect is less relevant in III-V DMSs, where Mn2+cen-\ntres are negatively charged, so that the enhanced hole\ndensity at closely lying Mn pairs compensates, at least\npartly, short-range antiferromagnetic interactions21.\nFollowing theoretical prediction26, carrier-induced fer-\nromagnetism was revealed in modulation doped p-type\n(Cd,Mn)Te/(Cd,Zn,Mn)Te:Mg heterostructures employ-\ning photoluminescence spectroscopy5,59. The character\nof magnetic anisotropy as well as the magnitude of TC\nand its evolution with the hole density, controlled by the\nelectric \feld and illumination, were found to be consis-\ntent with the p\u0000dZener model, adapted for this low\ndimensionality system. Interestingly, however, no mag-\nnetic hystereses have been detected below TC. According\nto extensive Monte-Carlo simulations, the e\u000bect re\rects\nfast magnetization dynamics, generated by the antiferro-\nmagnetic interactions at the borders of the hole layer79.\nFerromagnetic signatures were also found in epi-\nlayers of p-type (Zn,Mn)Te:N (refs 6,80) and n-type\n(Zn,Mn)O:Al (ref. 81). As shown in Fig. 5, under the\nhigh density of carriers, the low-temperature resistance\nacquires a hysteretic behaviour. This points to the ap-\npearance of a ferromagnetic order as well as demonstrates\ndirectly the existence of a strong coupling between car-\nriers and Mn spins. The temperature dependence of the\ncoercive force, together with magnetic susceptibility mea-\nsurements above 2 K, point to a magnetic ordering tem-\nperatureTC= 1:45 K in the case of Zn 0:0981Mn0:019Te:N\ncontaining 1 :2\u00011020holes per cm3. This value is in agree-\nment with the predictions of the p\u0000dZener model, pro-\nvided that the aforementioned antiferromagnetic inter-\nactions and the spin-orbit interaction are taken carefully\ninto account80. A similar experimental procedure leads\ntoTC= 160 mK in the case of Zn 0:097Mn0:03O:Al con-\ntaining 1:4\u00011020electrons per cm3(ref. 81). Taking the\ndi\u000berences in relevant parameters and, in particular, a\nthree times larger amplitude of the p\u0000dexchange in-\ntegral comparing to the s\u0000dcase, the experimentally\nobserved di\u000berence in the TCvalues between p-type and\nn-type materials can be readily explained.\nPROSPECTS FOR HIGHER TCIN DMSs\nA number of authors has reported the observation of\nroom temperature ferromagnetism in various semicon-\nductors and oxides containing supposedly randomly dis-\ntributed localised spins. However, it is probably fair to\nsay that so-far none of these \fndings has been con\frmed\nnature materials  | VOL 9 | DECEMBER 2010 | www.nature.com/naturematerials  969the magnitude of TC and its evolution with the hole density, which \nare controlled by the electric field and the illumination, was found to \nbe consistent with the p–d Zener model adapted for this low-dimen -\nsionality system. Interestingly, however, no magnetic hystereses have \nbeen detected below TC. According to extensive Monte Carlo simu -\nlations, the effect reflects fast magnetization dynamics generated by \nthe antiferromagnetic interactions at the borders of the hole layer79.\nFerromagnetic signatures were also found in epilayers of p-type \n(Zn,Mn)Te:N (refs  6, 80) and n-type (Zn,Mn)O:Al (ref.  81). As shown \nin Fig.  5, for a high carrier density the low-temperature resistance \nbecomes hysteretic. This points to the appearance of ferromagnetic \norder and directly demonstrates the existence of a strong coupling \nbetween carriers and Mn spins. The temperature dependence of the \ncoercive force, together with magnetic susceptibility measurements \nabove 2  K, point to a magnetic ordering temperature of TC = 1.45 K \nin the case of Zn0.0981Mn0.019Te:N with 1.2  × 1020 holes  cm−3. This value \nis in agreement with the predictions of the p–d Zener model, pro -\nvided that the aforementioned antiferromagnetic interactions and \nthe spin–orbit interaction are taken carefully into account80. A simi -\nlar experimental procedure shows that TC = 160 mK in the case of \nZn0.097Mn0.03O:Al with 1.4  × 1020 electrons  cm−3 (ref.  81). Given the \ndifference between relevant parameters and, in particular, the three -\nfold-greater amplitude of the exchange integral in the p–d case than in \nthe s–d case, the experimentally observed difference in the TC values \nbetween p-type and n-type materials can be readily explained.\nCarrier-mediated ferromagnetism with higher TC\nA number of authors have reported the observation of room-\ntemperature ferromagnetism in various semiconductors and \noxides containing supposedly randomly distributed localized spins. \nHowever, it is probably fair to say that so far none of these findings has been confirmed by other groups and none has resulted in the \ndemonstration of a device structure working at room temperature. \nAs I will argue in the following sections, if this robust ferromagnet -\nism is not an experimental artefact it can be explained by assuming \na non-random distribution of the magnetic ions.\nIt seems that despite ten years of extensive investigations, the \nconditions under which room-temperature ferromagnetism was \npredicted for nitrides and oxides21 have not yet been experimentally \nmet. In particular, no (Ga,Mn)N, (Zn,Mn)O or related compound \ncontaining a few per cent of randomly distributed magnetic cations \nand a few 1020 delocalized or weakly localized holes per cubic centi -\nmetre has so far been synthesized. However, in annealed (Ga,Mn)As \nthe effective Mn concentration, as judged from the saturation mag -\nnetization, now approaches 10%, although TC remains below 200  K.\nCan, therefore, carrier-mediated, indirect spin–spin coupling \nproduce ferromagnetic ordering that is stable up to room tempera -\nture? I note that a variant of the p–d Zener model explains the ori -\ngin of ferromagnetism in double-perovskite compounds, such as \nSr2CrReO6, where TC can reach 625  K (ref.  82) despite the distance \nbetween localized spins being as large as 0.6–0.7  nm. This shows the \npotential of this mechanism to support high-temperature ferromag -\nnetism possibly also in DMSs, where, in GaN and ZnO, the distance \nbetween second-nearest-neighbour cations is less than 0.5  nm.\nHowever, the obvious difficulty is to synthesize a p-type, wide-\nbandgap DMS system in which the hole density is large enough \nto result in a MIT even in the strong-coupling case, as sketched \nin Fig.  4. It has been suggested57—but not yet verified experimen -\ntally—that once the MIT is reached, and the bound states therefore \nwashed out by many-body screening, high values of TC (expected \nin the virtual-crystal and molecular-field approximations21) should \nemerge. Various methods allowing us to enlarge the hole density −0.05 0.00 0.05\nMagnetic ﬁeld (T)−0.5 0.0 0.5\nMagnetic ﬁeld (T)05101520\nn–(Zn,Mn)O p–(Zn,Mn)T e:N\n50 mK 0.1 K\n60 mK\n0.2 K\n75 mK\n0.3 K 100 mK\n0.45 K125 mK\n0.6 K 150 mK\n0.8 K200 mK\n1 K100\n50\n0∆Rxx (Ω)ab\nFigure  5 | resistive  indications  of ferromagnetism  in p-Zn0.981mn0.019te:n and n-Zn0.97mn0.03o:al. a, The temperature dependence of the hysteresis \nwidths at low temperatures and the magnetic susceptibility measurements above 2  K indicate that TC = 1.45 ± 0.05  K in p-Zn0.981Mn0.019T e:N with a hole \nconcentration of 1.2  × 1020 cm−3. b, The temperature and field scales are an order of magnitude smaller in n-Zn0.97Mn0.03O:Al with an electron concentration \nof 1.4  × 1020 cm−3, where TC = 160 ± 20 mK. Solid lines show changes of longitudinal resistivity in the magnetic field, DRxx, as measured for decreasing (blue \narrows) and increasing (red arrows) the field. Curves obtained at different temperatures are vertically shifted for clarity. The width of the hysteresis loops \nis seen to increase on lowering the temperature. Figures reproduced with permission from: a, ref.  80, © 2001 APS; b, ref.  81, © 2001 Springer.review article NATure MAT erIAls Doi: 10.1038/nmat2898\nnmat_2898_DEC10.indd   969 10/11/10   16:09:57\n© 20 Macmillan Publishers Limited. All rights reserved 10FIG. 5: Resistive indications of ferromagnetism in p-\nZn0:981Mn0:019Te:N and n-Zn 0:97Mn0:03O:Al. The temper-\nature dependence of the hystereses width at low tempera-\ntures as well as magnetic susceptibility measurements above\n2 K point to TC= 1:45\u00060:05 K in p-Zn 0:981Mn0:019Te:N\nwith the hole concentration 1 :2\u00011020cm\u00003. The temper-\nature and \feld scales are an order of magnitude smaller\nin n-Zn 0:97Mn0:03O:Al with the electron concentration 1 :4\u0001\n1020cm\u00003, whereTC= 160 \u000620 mK (adapted from refs. 80\nand 81).\nby other groups as well as none has resulted in the demon-\nstration of a device structure working at room temper-\nature. As we will argue in the following sections, if not\nrepresenting an experimental artifact, this robust ferro-\nmagnetism can be explained assuming a non-random dis-\ntribution of the magnetic ions.\nIt appears that despite ten years of extensive investiga-\ntions the conditions under which the room temperature\nferromagnetism was predicted for nitrides and oxides21\nhave not yet been experimentally met. In particular, no\n(Ga,Mn)N, (Zn,Mn)O, or a related compound containing\na few percent of randomly distributed magnetic cations\nanda few 1020delocalised or weakly localised holes per\ncm3has so-far been synthesised. On the other hand, in\nannealed (Ga,Mn)As the e\u000bective Mn concentration, as\njudged from the magnitude of the saturation magnetiza-\ntion, approaches now 10% but TCremains below 200 K.\nCan, therefore, carrier mediated indirect spin-spin cou-\npling produce a ferromagnetic ordering stable up to the\nroom temperature? We note that a variant of the p\u0000d\nZener model explains the origin of ferromagnetism in\ndouble perovskite compounds, such as Sr 2CrReO 6, where\nthe magnitudes of TCattain 625 K (ref. 82), despite that\nthe distance between localized spins is as large as 0.6 {\n0.7 nm. This shows a potential of this mechanism to sup-\nport high-temperature ferromagnetism, possibly also in\nDMSs, where the distance between second nearest neigh-\nbour cations is smaller than 0.5 nm in GaN and ZnO.\nHowever, the obvious di\u000eculty is to synthesise a p-\ntype wide band gap DMS systems, in which the hole\ndensity is large enough to result in a MIT, even in the\nstrong coupling case, as sketched in Fig. 4. It has been7\nsuggested57|but not yet veri\fed experimentally|that\nonce the MIT is reached, which means that the bound\nstates are washed out by many body screening, high val-\nues ofTC, expected within the VCA and MFA21, should\nemerge. Various methods allowing to enlarge the hole\ndensity above 1020cm\u00003without increasing the degree\nof disorder, such as gating as well as doping in a mod-\nulated fashion or by exploiting interfacial electric \felds,\nmay constitute the appropriate road towards achieving\na semiconductor showing high and tunable TCvalues.\nWe note that the enduring progress in the gate oxide\ndeposition11,19allows one to achieve an interfacial charge\ndensity of the order of 3 \u00011013cm\u00002, that is up to about\n3\u00011020per cm3.\nORIGIN OF HIGH TEMPERATURE\nFERROMAGNETISM\nPerhaps the most surprising development of the last\ndecade in the science of magnetic materials is abundant\nobservations of spontaneous magnetization persisting up\nto above room temperature in semiconductors and ox-\nides, in which no ferromagnetism at any temperature has\nbeen expected, particularly within the p\u0000dZener model.\nThese \fndings have o\u000bered prospects for a spread of spin-\ntronic functionalities much wider than it could initially\nbe anticipated. At the same time, they have generated\na considerable theoretical e\u000bort resulting in proposals of\nseveral novel mechanisms of exchange interactions be-\ntween diluted spins, designed to interpret robust ferro-\nmagnetism in magnetically doped or even magnetically\nundoped systems. Nevertheless, it appears that there is\nno visible convergence between particular experimental\n\fndings and theoretical models.\nOver the last years we start to realize that open d\nshells of magnetic impurities in non-magnetic solids not\nonly provide localised spins but viahybridisation with\nband states contribute signi\fcantly to the cohesive en-\nergy, particularly if TM impurities occupy neighboring\nsites. The resulting attractive force between magnetic\ncations leads to their aggregation invalidating the main\nparadigm of the DMS and DMO physics concerning the\nrandom distribution of TM spins. It may be anticipated\nthat the magnetic nanocrystals formed in this way as-\nsume the crystallographic form imposed by the matrix.\nAccordingly, the properties of these condensed magnetic\nsemiconductors (CMSs) may be not yet included in ma-\nterials compendia, so that it is a priori unknown whether\nthey are metallic or insulating as well as whether they ex-\nhibit ferromagnetic, ferrimagnetic or antiferromagnetic\nspin order. However, due to a large concentration of\nthe magnetic constituent within the CMSs nanocrystals,\ntheir spin ordering temperature is expected to be rela-\ntively high, typically above the room temperature.\nAs seen today, the experimental detection of a non-\nrandom spin distribution and possible contamination has\nbeen highly challenging in DMSs research39. Only re-cently the actual spatial distribution of TM cations in\nsome DMSs has been established by some groups, ow-\ning to the application of state-of-the-art element-speci\fc\nnanocharacterisation tools. While in some cases one\ndeals with elemental ferromagnetic metal nanoparticles,\nthe case of Co in ZnO (ref. 83), usually TM compounds\nare involved. Taking (Ga,Fe)N as an example, we note\nthat according to standard laboratory high-resolution x-\nray di\u000braction (HRXRD), the incorporation of Fe sim-\nply leads to a broadening of the GaN-related di\u000brac-\ntion maxima without revealing any secondary phases84.\nIn contrast, a much brighter synchrotron source has al-\nlowed to identify the presence of precipitates in the same\nsamples, as shown in Fig. 6, a counterpart of MnAs\nnanocrystals in GaAs (ref. 85,86). The appearance of\ncrystallographic phase separation in (Ga,Fe)N is sup-\nported by near-edge x-ray absorption \fne-structure (EX-\nAFS) studies87. The dominant ferromagnetic precipitate\nwas identi\fed as Fe 3N but in some cases nanocrystals\nin the form of an elemental ferromagnetic metal, Fe in\nthis case, are also visible88. At the same time, transmis-\nsion electron microscopy (TEM) with appropriate mass\nand strain contrast as well as electron dispersive spec-\ntroscopy (EDS), not only corroborated the outcome of\nsynchrotron XDR, but revealed also the aggregation of\nmagnetic cations without distorting the host wurtzite\nstructure under certain growth conditions88. This chem-\nical phase separation is known in the DMS literature\nas spinodal decomposition, independently of the micro-\nscopic mechanism leading to the aggregation of the TM\ncations.\nThe application of TEM with EDS allowed to\nevidence the chemical phase separation in annealed\n(Ga,Mn)As (ref. 86,89), (Zn,Cr)Te (ref. 90), (Al,Cr)N,\nand (Ga,Cr)N (ref. 91), whereas according to the re-\nsults summarised in Fig. 6, hexagonal nanocrystals were\ndetected in (Ga,Mn)N. Finally, we mention the case of\n(Ge,Mn), where under suitable growth conditions quasi -\nperiodically arranged nanocolumns are observed92, as\nshown in Fig. 7. Actually, a tendency to nanocolumn\nformation was also reported for (Al,Cr)N (ref. 91) and\n(Zn,Cr)Te (ref. 93). This demonstrates that growth con-\nditions can assist in controlling the nanocrystals shape.\nInterestingly, these two kinds of nanocrystal forms were\nreproduced by Monte-Carlo simulations94. Furthermore,\na strict correlation between ferromagnetic features and\nthe presence of CMS nanocrystals has been demonstrated\nfor these systems. However, the identi\fcation of the dom-\ninant microscopic mechanisms leading to robust spin or-\ndering, that is to a large magnitude of TCand magnetic\nanisotropy, awaits for detailed experimental and theo-\nretical studies for particular combinations of CMSs and\nhosts.\nAs expected, the lowering of the growth temperature\nand/or the increase of the growth rate88hampers the\naggregation of magnetic cations. Moreover, it has been\nsuggested that it is possible to change the TM charge\nstate and, therefore, the aggregation energy by co-doping8\n0.000.050.10 \n  6789 1 0 1 1 HIGH [Mn]\nMn pile-upMn-K βMn-K αGa-K β\nGa escape peakGa-K α\nENERGY (keV)6789 1 0 1 1 FLUORESCENCELOW [Mn]\nMn pile-upMn-K βMn-K αGa-K β\nGa escape peakGa-K α\n \nENERGY (keV)\n0 1 02 03 04 05 00.80.91.0 \nX (μm)PROFILES  Ga-K α \n25μm \nMn \nGa  \nScattering  \nMn-Kα \n \nFIG. 6: Evidence for crystallographic and chemical phase\nseparations in DMSs. Upper panel: synchrotron XRD\nand TEM results for (Ga,Fe)N showing the precipitation of\nhexagonal\u000f-Fe3N nanocrystals (after ref. 88). Lower panel:\nelement-speci\fc synchrotron radiation micro-probe analysis of\n(Ga,Mn)N showing aggregation of Mn cations (after ref. 78).\nwith shallow donors or acceptors95,96. This way of af-\nfecting the TM valency stems from the presence of band\ngap states derived from the dorbitals. These states\ntrap carriers supplied by shallow impurities, altering the\ncharge state of the magnetic cations and, hence, modi-\nfying their mutual interactions. Accordingly, co-doping\nof DMSs and DMOs with shallow acceptors or donors,\nduring either growth or post-growth processing, modi\fes\nthe valence providing a powerful mean for the control\nof the magnetic cation aggregation. These predictions\nare corroborated by experimental \fnding for (Ga,Mn)N\n(ref. 39), (Zn,Cr)Te (ref. 90), and (Ga,Fe)N (ref. 88),\nwhere remarkable changes in the ferromagnetic charac-\nteristics upon co-doping with shallow impurities have\nbeen found and correlated with the TM distribution.\n8\nFIG. 6: Evidence for crystallographic and chemical phase\nseparations in DMSs. Upper panel: synchrotron XRD\nand TEM results for (Ga,Fe)N showing the precipitation of\nhexagonal ǫ-Fe3N nanocrystals (after ref. 88). Lower panel:\nelement-speciﬁc synchrotron radiation micro-probe analy sis of\n(Ga,Mn)N showing aggregation of Mn cations (after ref. 78).\nsuggested that it is possible to change the TM charge\nstate and, therefore, the aggregation energy by co-doping\nwith shallow donors or acceptors95,96. This way of af-\nfecting the TM valency stems from the presence of band\ngap states derived from the dorbitals. These states\ntrap carriers supplied by shallow impurities, altering the\ncharge state of the magnetic cations and, hence, modi-\nfying their mutual interactions. Accordingly, co-doping\nof DMSs and DMOs with shallow acceptors or donors,\nduring either growth or post-growth processing, modiﬁes\nthe valence providing a powerful mean for the control\nof the magnetic cation aggregation. These predictions\nFIG. 7: Formation of nanocolumns in DMS by aggregation of\nMn cations. Upper panel: Mn-rich nanocolumns in (Ge,Mn)\nevidenced by: (a) HRTEM plane view and (b) Mn chemical\nmaps (after ref. 92). Lower panel: Monte Carlo simulation of\nchemical phase separation in (Zn,Cr)Te with seeding to init i-\nate the growth of nanocolumns and control of their diameter\nby Cr ﬂux (after ref. 94).\nare corroborated by experimental ﬁnding for (Ga,Mn)N\n(ref. 39), (Zn,Cr)Te (ref. 90), and (Ga,Fe)N (ref. 88),\nwhere remarkable changes in the ferromagnetic charac-\nteristics upon co-doping with shallow impurities have\nbeen found and correlated with the TM distribution.\nFinally, we note that depending on the growth condi-\ntions CMS nanocrystals can be distributed randomly or\naccumulate either at the interface with the buﬀer or at\nthe ﬁlm surface. Furthermore, TM impurities may deco-\nrate or diﬀuse along extended defects such as dislocations\nor grain boundaries97. This appears to explain an inverse\ncorrelation between samples’ quality and the appearance\nof high TCferromagnetism, noted by some authors in the\ncase of oxides98.\nSOME MODELING\nThe above qualitative considerations are supported by\nab initio studies. For instance, the computed energy\nchange associated with bringing two Ga-substitutional\nMn atoms to the nearest neighbor cation position is\nEd=−120 meV in GaAs and −300 meV in GaN,\nand reaches −140 and −350 meV in the case of a\ncation-substitutional Cr nearest neighbor pair in ZnTe\nand GaN, respectively90,99. In contrast, there is virtu-\nally no energy change associated with bringing two Zn-\nsubstitutional Mn atoms to the nearest neighbor cation\nsites in (Zn,Mn)Te, where Ed= 21 meV (ref. 90). This\ncan be associated to the fact that the Mn dstates lit-\ntle perturb the sp3tetrahedral bonds as both the lowerFIG. 7: Formation of nanocolumns in DMS by aggregation of\nMn cations. Upper panel: Mn-rich nanocolumns in (Ge,Mn)\nevidenced by: (a) HRTEM plane view and (b) Mn chemical\nmaps (after ref. 92). Lower panel: Monte Carlo simulation of\nchemical phase separation in (Zn,Cr)Te with seeding to initi-\nate the growth of nanocolumns and control of their diameter\nby Cr \rux (after ref. 94).\nFinally, we note that depending on the growth condi-\ntions CMS nanocrystals can be distributed randomly or\naccumulate either at the interface with the bu\u000ber or at\nthe \flm surface. Furthermore, TM impurities may deco-\nrate or di\u000buse along extended defects such as dislocations\nor grain boundaries97. This appears to explain an inverse\ncorrelation between samples' quality and the appearance\nof highTCferromagnetism, noted by some authors in the\ncase of oxides98.\nSOME MODELING\nThe above qualitative considerations are supported by\nab initio studies. For instance, the computed energy\nchange associated with bringing two Ga-substitutional\nMn atoms to the nearest neighbor cation position is\nEd=\u0000120 meV in GaAs and \u0000300 meV in GaN,\nand reaches \u0000140 and \u0000350 meV in the case of a\ncation-substitutional Cr nearest neighbor pair in ZnTe\nand GaN, respectively90,99. In contrast, there is virtu-\nally no energy change associated with bringing two Zn-\nsubstitutional Mn atoms to the nearest neighbor cation\nsites in (Zn,Mn)Te, where Ed= 21 meV (ref. 90). This\ncan be associated to the fact that the Mn dstates lit-\ntle perturb the sp3tetrahedral bonds as both the lower\nd5(donor) and the upper d6(acceptor) Hubbard levels\nare respectively well below and above the band edges in\nII-VI compounds100, so that there is no considerable dif-\nference between the band hybridisation involving Zn or\nMn. This conclusion is consistent with a large solubility9\n0.0 0.5 1.0 1.5 2.0 2.5 3.0-200-150-100-50050100\n  \n (Zn,Cr)Te\n (Ga,Cr)AsPairing energy (meV)\nNumber of holes in d-shell\nFIG. 8: Computed energy change Edresulting from bringing\ntwo Cr impurities to the nearest neighbor cation positions in\nZnTe and GaAs depending on the number of holes in the Cr\nd5shell (adapted from ref. 102).\nof Mn in II-VI compounds and the apparently random\ndistribution of Mn in these systems101.\nTo model the e\u000bect of co-doping, we note that the en-\nergy of the screened Coulomb interaction between two el-\nementary charges residing on the nearest neighbor cation\nsites in the GaAs lattice is Ed=\u0000280 meV. This value\nindicates that a change in the charge state can a\u000bect the\naggregation signi\fcantly, as the gain of energy associated\nwith bringing two Mn atoms close by is Ed=\u0000120 meV,\nas quoted above99. Accordingly, a surplus of charge on\nTM ions, comparing to non-magnetic cations, brought\nby co-doping with shallow dopants can overweight the\ngain of energy stemming from p\u0000dhybridization and\nimpede the nanocrystal assembling95,96. This picture is\ncon\frmed by ab initio computations within the LSDA\nfor (Ti,Cr)O 2(ref. 96) and (Zn,Cr)Te (refs 90 and 102).\nAs shown in Fig. 8, the value of Edattains a minimum\nin ZnTe when the two Cr cations are in the 2+ charge\nstate102(d4con\fguration). However, the computation\nresults shown in the same plot indicate that also in GaAs,\nEdgoes through a minimum for the Cr2+case, rather\nthan in the case of Cr3+pairs, as might be expected for\nIII-V compounds.\nFROM DILUTE TO NANOCOMPOSITE\nSYSTEMS\nIn view of the above discussion, the incorporation of\nTM impurities to semiconductors not only bridges ferro-\nmagnetic and semiconductor capabilities but also o\u000bers\na way to develop a new kind of nanocomposite systems\nconsisting of ferromagnetic and metallic nanocrystals co-\nherently buried in a semiconductor host. The applica-\ntion of embedded metallic and semiconducting nanocrys-\ntals is known to be on the way to revolutionise the per-formance of various commercial devices, such as \rash\nmemories, low current semiconductor lasers, and single\nphoton emitters. Similarly far reaching can be the use\nof nanocomposite semiconductor/ferromagnetic systems\ndue to their unique capabilities and to the possibility of\ncontrolling the shape (nanodots vs.nanocolumns) and\nsize by growth parameters and co-doping during the epi-\ntaxial process.\nIt has already been demonstrated that these nonocom-\nposites show strong magnetotrasport and magnetoopti-\ncal e\u000bects89,90, that could possibly allow these systems\nto be exploited as magnetic \feld sensors as well as in\nmagnetooptical devices. In particular, a combination of\na strong magnetic circular dichroism speci\fc to ferromag-\nnetic metals and weak losses characterising the semicon-\nductor hosts suggest possible functionalities as optical\nisolators as well as three-dimensional (3D) tunable pho-\ntonic crystals and spatial light modulators for advanced\nphotonic applications. As an interesting recent develop-\nment one can quote a spin battery e\u000bect demonstrated\nfor the GaAs:MnAs system in a magnetic \feld103.\nAs shown in Fig. 7, the controlled growth of\nnanocolumns of a ferromagnetic metal can allow to fab-\nricate in-situ ,e.g., a dense array of magnetic tunnel\njunctions94or Coulomb blockade devices. Thus, the\nmedia in question can be employed for low-power high-\ndensity magnetic storage, including spin-torque magnetic\nrandom access memories and race-track 3D domain-wall\nbased memories. If su\u000eciently high tunneling magnetore-\nsistance (TMR) will be found, one can envisage the appli-\ncation for \feld programmable logic (TMR-based connect-\ning/disconnecting switches) and even for all-magnetic\nlogic, characterised by low power consumption and radi-\nation hardness. Furthermore, embedded metallic nanos-\ntructures may serve as building blocks for all-metallic\nnanoelectronics and for high quality nanocontacts in na-\nnoelectronics, optoelectronics, and plasmonics as well\nas constitute media for thermoelectric applications104.\nWorth mentioning is also the importance of hybrid semi-\nconductor/ferromagnetic systems in various proposals of\nscalable quantum processors.\nd0FERROMAGNETISM AND BEYOND\nOrganic ferromagnets and quantum Hall ferromagnets\nare a proof that ferromagnetism is possible in materials\nwithout magnetic ions, albeit the corresponding TC's are\nso-far rather low, typically below 20 K. It has also been\nsuggested that a robust ferromagnetism can appear in\ncertain zinc-blende metals like CaAs, and be driven by a\nStoner instability in the narrow heavy hole band105,106,\na prediction awaiting for an experimental con\frmation.\nIt has been known for a long time that a number of\ndefects or non-magnetic impurities form localised para-\nmagnetic centers in various hosts. Some of these states\nmight show a large intra-center correlation energy Uthat\ncould ensure an adequate stability of the spins, even if10\ntheir density increases or the material is co-doped with\nshallow impurities. It is, therefore, tempting to relate\nthe presence of unexpected high-temperature ferromag-\nnetism in various oxides and carbon derivates to mag-\nnetic moments residing rather on nonmagnetic defects or\nimpurities than on open dshells of TMs38,107.\nAs shown theoretically108, a sizable exchange interac-\ntion takes place between valence-band holes residing on\nnon-magnetic acceptors and electrons in the conduction\nband. This demonstrates that band carriers can mediate\na Zener-type coupling between spins localised on defect\ncenters. Furthermore, if the spin concentration increases,\nso that either the Hubbard-Mott or the Anderson-Mott\ntransition is reached, the double exchange or Stoner-like\nmechanism might appear46. However, in each of these\ncases a clear correlation between magnetic and trans-\nport properties should be visible, analogous to the one\nobserved routinely in manganites, (Ga,Mn)As, and p-\n(Zn,Mn)Te. In particular, the fabrication of a spintronic\nstructure { like a magnetic tunnel junction { working up\nto high temperatures, would constitute a strong con\fr-\nmation of the existence of spin transport in these chal-\nlenging systems. Alternatively, defects and impurities,\nsimilarly to TM dopants, could form high spin aggre-\ngates in certain hosts. In this case, the presence of\nferromagnetic-like features should correlate with the ex-\nistence of defect agglomerations that can be revealed by\nemploying state-of-the-art nanocharacterisation tools.\nTo date, suggestions concerning defect-related high-\ntemperature ferromagnetism come only from global mag-\nnetization measurements. Therefore, it appears more\nnatural to assume at this stage that a small number\nof magnetic nanoparticles { that escaped from the de-\ntection procedure { account for the high-temperature\nferromagnetic-like behaviour of nominally nonmagnetic\ninsulators and semiconductors. Such nanoparticles could\nbe introduced during the synthesis or post-growth pro-\ncessing, and can reside in the sample volume, at disloca-\ntions or grain boundaries but also at the surface, interface\nor in the substrate. An instructive example is provided\nby the case of porous silicon109.\nSUMMARY\nWhere are we then after these ten years? With no\ndoubt (Ga,Mn)As and related compounds have consid-\nerably strengthen their position as an outstanding play-ground to develop and test novel functionalities unique\nto a combination of ferromagnetic and semiconductor\nsystems. Many of the concepts8, like spin-injection,\nelectric-\feld control of the TCmagnitude and magneti-\nzation direction, tunneling anisotropic magnetoresistance\nin planar junctions and in the Coulomb blockade regime,\ncurrent-induced domain displacement without assistance\nof a magnetic \feld, are being now developed in devices\ninvolving ferromagnetic metals, which may function at\nambient temperatures. Obviously, however, a further in-\ncrease ofTC, over the current record value of 190 K,\ncontinues to be a major goal in the \feld of DMSs.\nAt the same time, investigations of magnetically doped\nsemiconductors and oxides have faced us with unexpect-\nedly challenging issues, intractable by conventional ma-\nterials characterisation, computational, and theoretical\ntools. In addition to an interplay between phenomena\nspeci\fc to strongly correlated and disordered systems,\nencountered in doped semiconductors and manganites,\nDMSs and DMOs show a fundamentally new ingredient\nbrought about by a not anticipated correlation in the\nmagnetic ion distribution. We are now learning how to\nvisualise and control the magnetic ion aggregation in or-\nder to develop methods allowing to obtain lateral and ver-\ntical distributions as well as shapes of magnetic nanocrys-\ntals on demand. The ferromagnetic metal/semiconductor\nnanocomposites fabricated in this way o\u000ber a spectrum\nof so-far unexplored possibilities in various \felds of ma-\nterials science and device physics.\nAre there only two classes of magnetically doped semi-\nconductors and oxides showing ferromagnetic features,\nsay, above 10 K? Are ferromagnetic correlation mediated\nby valence band holes and embedded magnetic nanocrys-\ntals the only sources of spontaneous magnetization in\nthese systems? We will see in the years to come.\nAcknowledgments\nThe author's works described here were supported by\nthe FunDMS Advanced Grant of the European Research\nCouncil within the \"Ideas\" 7th Framework Programme\nof the EC, and earlier by ERATO project of Japan Sci-\nence and Technology Agency and Humboldt Foundation.\nThe fruitful and enjoyable collaboration with the groups\nof Alberta Bonanni, Shinji Kuroda, Hideo Ohno, and\nMaciej Sawicki is gratefully acknowledged.\n\u0003Electronic address: dietl@ifpan.edu.pl\n1Story, T., Galazka, R. R., Frankel, R. B. & Wol\u000b, P. A.\nCarrier-concentration{induced ferromagnetism in PbSn-\nMnTe. Phys. Rev. Lett. 56, 777{779 (1986).\n2Ohno, H., Munekata, H., Penney, T., von Moln\u0013 ar, S.\n& Chang, L. L. Magnetotransport properties of p-type\n(In,Mn)As diluted magnetic III-V semiconductors. Phys.Rev. Lett. 68, 2664{2667 (1992).\n3Ohno, H. et al. (Ga,Mn)As: A new diluted magnetic\nsemiconductor based on GaAs. Appl. Phys. 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Needs\nTheory of Condensed Matter Group, Cavendish Laboratory,\nJ J Thomson Avenue, Cambridge CB3 0HE, United Kingdom\n(Dated: November 6, 2018)\nFirst-principles density-functional-theory calculatio ns show that compression of alkali metals sta-\nbilizesopenstructureswithlocalized interstitial elect ronswhichmayexhibitaStoner-typeinstability\ntowards ferromagnetism. We ﬁnd ferromagnetic phases of the lithium-IV-type, simple cubic, and\nsimple hexagonal structures in the heavier alkali metals, w hich may be described as s-band ferro-\nmagnets. We predict that the most stable phases of potassium at low temperatures and pressures\naround 20 GPa are ferromagnets.\nPACS numbers: 71.15.Nc,75.10.Lp,62.50.-p,61.66.-f\nAmong the elements, bulk ferromagnetism is found\nonly in the ﬁrst row of the transition metals and in the\nlanthanides, where partially ﬁlled 3 dand 4felectronic\nshells of strongly-localizedorbitalsare present. Magnetic\norder is normally reduced and eventually destroyed by\nthe application of pressure because it tends to delocalize\nelectronicstates. The alkalimetalsarearchetypalnearly-\nfree-electron materials in which each atom contributes a\nsingle valence electron to a gas whose eﬀective interac-\ntion with the ionic cores is weak. They might there-\nfore be thought of as the least likely elements in which\nto ﬁnd bulk ferromagnetism. The alkali metals, lithium\n(Li), sodium(Na),potassium(K),rubidium(Rb), cesium\n(Cs), and probably francium (Fr) (the most unstable\nnaturally occurring element) adopt body-centered-cubic\n(bcc) phases under ambient conditions which, on com-\npression, transform to face-centered-cubic (fcc) phases\n[1]. Diamond-anvil-cell experiments have shown, how-\never, that they adopt more open structures at higher\npressures [1]. Density-functional theory (DFT) calcula-\ntions have reproduced the stability of the experimentally\nobserved phases and have given insights into their elec-\ntronic structures [2–6]. The most surprising result is that\na large amount of valence charge in the open structures\nresides within the interstitial regions rather than in close\nproximity to the ions. This corresponds to the formation\nof “electrides” in which the interstitial electrons form the\nanions [5–8]. The open structures correspond to well-\npacked ionic solids when both the alkali metal ions and\nthe centers of the interstitial electronic charges are des-\nignated as ionic positions [9].\nThe band structures of alkali metals deviate substan-\ntially from nearly-free-electron behavior under applied\npressure [2–6]. The occupied valence bands become\nﬂatter than the corresponding nearly-free-electron ones,\nwhichnarrowsthe occupiedvalencebandwidth. Ashcroft\nand coworkers [2, 3, 8] have attributed this phenomenonto the interaction of the valence electrons with the rela-\ntively incompressible ionic cores that occupy an increas-\ningly large fraction of the total volume as pressure is\nincreased. Under ambient conditions the eﬀective in-\nteraction between the valence electrons and ionic cores\nis weak, but it becomes strongly repulsive under pres-\nsure and forces valence electrons to occupy interstitial\npositions [2, 3]. The interstitial regions in close-packed\nstructures are numerous, but small, and “cutting up” the\nvalence charge into small regionsincreases the kinetic en-\nergy. The kinetic energy can, however, be reduced by\nadopting more open structures which have less numerous\nbut larger interstitial regions in which to accommodate\nthe valenceelectrons. Thiseﬀect evidently overcomesthe\nconcomitant increase in core-core repulsion energy.\nTheFermi surface/Braggplanemechanismforstabilis-\ning structures [10] involves the action of the Fourier com-\nponentsofthelatticepotentialonthedegenerateelectron\norbitals on the Bragg planes in reciprocal space. The to-\ntal energy is lowered if a structure is adopted in which\nBragg planes graze the Fermi surface, because the occu-\npied orbitals just inside the Bragg plane are lowered in\nenergy while those of the unoccupied orbitals just out-\nside are raised. This mechanism can lead to structural\ninstabilities and it is likely to be involved in determining\nthe details of the high-pressure structures of many of the\nalkalis [11, 12]. The Fermi surface/Bragg plane mecha-\nnism normally operates when g(E) is large in the region\naround the Fermi energy EF, but an alternative insta-\nbility comes into play when g(E) is also strongly peaked\naroundEF. In this case the energy may be lowered by\ntransferring electrons from one spin channel to the other,\nresulting in ferromagnetism (FM). This is the Stoner in-\nstability [13]. The key quantities in Stoner’s theory are\ng(EF) and the eﬀective interaction Ibetween the up and\ndown spin densities in the unpolarized state, with the\ninstability to FM occurring when g(EF)I >1.2\nMotivated by the above analysis, we have investigated\nmagneticorderinginthealkalisunderhigh-pressures. We\nsearched for low-enthalpy structures using ab initio ran-\ndomstructuresearching(AIRSS)[14,15], whichhasbeen\nsuccessfully applied to systems as diverse as metals un-\nder high pressures [5, 9] and molecular solids [16, 17].\nWe used the castep plane-wave DFT code [18] and\nthe Perdew-Burke-Ernzerhof (PBE) Generalized Gradi-\nent Approximation (GGA) density functional [19], and\nwe also present results obtained with a Thomas-Fermi\nscreened exchange functional (SX) [20] and the local spin\ndensity approximation (LSDA). We used ultrasoft pseu-\ndopotentials [21], treating all three electrons explicitly\nfor Li, and nine electrons for the other alkalis. Brillouin\nzone integration grids of spacing 2 π×0.05˚A−1were used\nforthe searchesand the low-enthalpystructureswerefur-\nther relaxed using a ﬁner grid spacing of 2 π×0.03˚A−1\nwhich gave very accurate enthalpy diﬀerences between\nthe phases. Large plane wave basis set energy cutoﬀs\nwere used. The phonon calculations were performed us-\ning a ﬁnite-displacement method and 64-atom supercells.\nWe performed spin-polarized calculations, starting\nsomerelaxationsin ahigh spin state with anaveragespin\ndensity of one electron per atom and others with zero av-\nerage spin density. In each calculation the spin density\nwas allowed to evolve freely as the structure was relaxed.\nWe also performed calculations without spin polariza-\ntion. The use of a wide variety of starting structures and\nspin states was found to be important in allowing many\ndiﬀerent spin and atomic conﬁgurations to be accessed.\nThe main searches were performed at pressures around\nthose at which experiments show that the fcc structures\ntransform to more open phases. We performed searches\nwith unit cells containing up to eight atoms, relaxing a\ntotal of about 1500 structures. Calculations for other\nknown structures of the alkalis which have more than\neight atoms per cell were also performed, namely a pe-\nriodic structure of space group I4/mcmwith 56 atoms\nwhich is a good analogue of the K-III (Rb-IV) incom-\nmensurate host-guest structure, and the 84-atom C2221\nstructure of Cs-III and the related 52-atom structure of\nRb-III [1].\nWe ﬁnd good agreement with the experimentally ob-\nserved phase transitions. Our calculated coexistence\npressure for K-fcc and K- I4/mcmof 19.8 GPa is close\nto the experimental transition pressure of 23 GPa [1].\nWe note that Marqu´ es et al.[23] have reported ﬁnding\nthe K-I4/mcmphase in some high-pressure experiments\nand the K-hP4 phase in others, which is consistent with\nour ﬁnding that K-hP4 is only slightly less stable than\nK-I4/mcm. Our coexistence pressure for the Rb-fcc and\nRb-C2221phases of 14.2 GPa is in excellent agreement\nwith the experimental transition pressure of 14 GPa [1].\nWeobtainacoexistencepressureforCs-fccandCs- C2221\nof 4.8 GPa, compared with the experimental transition\npressure of 4.2 GPa [1]. We ﬁnd Cs-IV to be stable from4.8–10 GPa, in excellent agreement with experiment [1].\nWe ﬁnd no region of stability for the observed Cs-III\nstructure which in our calculations only becomes more\nstable than the fcc phase at 4.9 GPa, although this is a\nsmall discrepancy. In general we ﬁnd that our calculated\ncoexistencepressurescanbebroughtintoagreementwith\nthe experimental data by rigidly shifting the enthalpy\ncurves by less than 5 meV, which indicates the high level\nof accuracy of our calculations.\nWe found I¯43d, simple cubic (sc)and simplehexagonal\n(sh) phasesofK,Rb, Cs andFr with strongFM ordering,\nand a weakly FM Cs-fcc phase, see Fig. 1. The enthalpy\nreductions due to the formation of FM moments in the\nsc phases are similar to those in the corresponding sh\nphases. The sh phases are more stable than the sc phases\nat lower pressures in K and Rb, and more stable at all\npressuresin Cs. Wedid notﬁndanyspin-polarizedstates\nofLiorNainourfully-convergedcalculations,butwecan\nobtain them by reducing the number of k-points. This\nshows that Li and Na are close to a FM instability and\ndemonstrates the importance of carefully studying the\nconvergence with respect to the k-point sampling, as we\nhave done. We predict that the FM K- I¯43dphase to be\nthe most stable in the range 18.5–20 GPa and the FM\nK-scphase to be the most stable in the range20–22GPa.\nThe FM ordering leads to an energy gain of a few tens\nof meV per atom. When the centers of the interstitial\nelectronic charges are designated as ionic positions, sc\nbecomes the CsCl structure, sh the MgB 2structure, and\nI¯43dthe Th 3P4structure [22], with the Cs, Mg and P\nsites being those of the cations and the Cl, B and Th\nsites being those of the interstitial electrons.\nTo explore the sensitivity to the density functional we\nalso performed calculations using a Thomas-Fermi SX\nfunctional [20] and the LSDA. For the SX calculations we\nused a screening wave vector of ks= 0.764 a.u., which\ncorresponds to a Wigner-Seitz radius of rs= 4.2 a.u.\nThis is larger than the value of rs= 3.4 a.u. obtained\nfrom the average valence charge density of FM K-sc at\n20 GPa, but the results are rather insensitive to reason-\nable variations in ksas it is proportional to the one-sixth\npower of the average charge density. Using the SX func-\ntional instead of PBE-GGA stabilizes FM K-sc over K-\nfcc by about 15 meV per atom. We expect that the sc\nphases of each of the alkalis would be further stabilized\nwith respect to the fcc phases when calculated with the\nSX functional. We did not ﬁnd magnetic ordering in\nK when using the LSDA functional which predicts the\nK-sc phase to be stable above about 20 GPa, in disagree-\nment with experiment. Given that the pressures of the\nobserved transitions calculated within PBE-GGA are in\nvery good agreement with experiment, we are inclined to\nbelieve that it gives a better description of compressed\nalkalis than the LSDA or SX functionals.\nCalculations of the harmonic vibrational modes of the\nK-fcc, K-sc and FM K-sc phases showed them to be dy-3\nnamically stable. The phonon modes of the sc phases are\nsubstantially softer than those of K-fcc, and including\nthe zero-point enthalpy stabilizes the K-sc and FM K-sc\nphases over K-fcc by about 25 meV per atom at 20 GPa.\nThe stability of K-sc and FM K-sc over K-fcc is slightly\nincreasedbyincluding the vibrationalcontributionto the\nfree energy at a temperature of 300 K, but the magnetic\nmoment will be reduced by disordering of the spins.\nThe valence electron density of states of the paramag-\nnetic and FM states of K-sc at 20 GPa are shown in Fig.\n2. In the paramagnetic system, EFis close to the top of\na large peak in g(E) and the system is ripe for a Stoner\ninstability. In the FM K-sc phase EFfalls just above the\npeak for the majority-spin band and well below the peak\nfor the minority spin band. The FM K-sc state is about\n35 meV per atom more stable than the paramagnetic K-\nsc phase, which is suﬃcient to make FM K-sc the most\nstable phase in the pressure range 20–22 GPa. The value\nofg(EF) for non-spin-polarized K-sc at 20 GPa is 1.4 per\neV per atom, so that the Stoner criterion is satisﬁed for\nI >0.7 eV, which is similar to the values deduced for\nFM in transition metals [24].\nThe spin densities of FM K-sc and FM K-sh at 20\nGPa are shown in Fig. 3. A large blob of spin-polarized\ncharge density resides on the interstitial site at the cen-\nter of the cube of K-sc, while the spin polarization on the\natomic sites is small. The spin polarization on the atoms\nis also small in FM K-sh, and the interstitial spin den-\nsity is more diﬀuse. The FM K-sc structure achieves its\nmaximum spin moment of about 0.72 electrons per atom\nat 22 GPa, while the maximum spin moment in FM K-\nsh of about 0.62 electrons per atom occurs at 20 GPa,\nand the maximum spin moment of K- I¯43dis about 0.4\nelectrons per atom. The FM phases might be described\nby a Hubbard-like model [25] using tight-binding sor-\nbitals centered on the interstitial regions. The magnetic\nstate would then be described as s-band FM, which is\nvery diﬀerent from the dorfband FM observed in the\ntransition metal and lanthanide elements.\nWe are not aware of any experimental evidence for\nmagnetic ordering in bulk alkali metals, although there\nis evidence of it in low-dimensional systems. Theoretical\nwork by Overhauser [26] suggested that charge-density\nand spin-density-wave instabilities might occur in alkali\nmetals at low pressures, but these ideas have not been\nwidely accepted. A DFT study by Zabala et al. [27]\nfound spontaneous magnetization in “Na wires” mod-\nelled by jellium, and Bergara et al. [28] found instabilities\nto FM in atomically-thin Li and Na wires. FM ground\nstates are, however, forbidden in strictly one-dimensional\nsystems by the Lieb-Mattis theorem [29] and, in quasi-\none-dimensional systems, quantum ﬂuctuations tend to\nsuppressFM.ExperimentalobservationsofFMinKclus-\nters incorporated within a zeolite [30] and anti-FM in K\nclusters in a nanographite-based host [31] have also been\nreported. We have used the AIRSS method to search for18 19 20 21 22\nPressure (GPa)-0.04-0.020.000.020.040.06Enthalpy (eV/atom)fcc\nsc FM\nsc\nsh FM\nsh\nK-III\nhp4\nI43d FM\nI43d\n12 13 14 15 16\nPressure (GPa)-0.0200.020.040.060.08Enthalpy (eV/atom)fcc\nsc FM\nsc\nsh FM\nsh\nRb-III\nI43d FM\nI43d\n2 4 6 8 10\nPressure (GPa)-0.1-0.0500.050.10.15Enthalpy (ev/atom)fcc\nfcc FM\nsc FM\nsc\nsh FM\nsh\nCs-III\nCs-IV\nCs-V\nI43d FM\nI43d\nFIG. 1: (color online). Enthalpy-pressure curves for the FM\nand paramagnetic phases of K, Rb, and Cs. Diﬀerences in\nenthalpyfrom thefcc phase are plotted. FM phases are shown\nas solid lines and paramagnetic phases as dashed lines.4\n-4 -2 0 2 4\nEnergy (eV)-1-0.500.51e-DOS (states/eV/atom)\nFIG. 2: (color online). The electronic density of states, g(E),\nof K-sc at 20 GPa in states per eV per atom. Data for the\nparamagnetic phase is shown in brown and for FM K-sc in\ngrey. The Fermi energy is shown as a vertical dotted line.\nFIG. 3: (color online). The spin density of FM K-sc (top) and\nFM K-sh (bottom) at 20 GPa. The purple spheres represent\nthe atoms and the spin density is shown in green.structures of small unsupported K clusters, ﬁnding that\nsome weakly anti-FM states are energetically favorable\nand that quite strongly FM phases with moments up to\nabout 0.5 electrons per atom occur at higher energies.\nElectride formation in alkali metal clusters at ambient\npressureishighlyunlikelybecausethebondsaretoolong,\nand the magnetic ordering must produced by some other\nmechanism. It might be possible to further stabilize FM\nphases of bulk alkali metals by alloying them with other\nalkalis or other species. Low-temperature experiments\nlooking for magnetism in compressed alkalis are needed\nto test our predictions.\nIn summary, we predict mechanically-stable FM\nphases of the heavier alkali metals to have low enthalpies\nat pressures just above the stability range of the fcc\nphases. FM phases with the Li-IV ( I¯43d), sc, and sh\nstructures can be described as s-band electride ferromag-\nnets. The FM K- I¯43dand K-sc phases are predicted to\nbe the most stable at low temperatures and pressures\naround 20 GPa.\nThe authors were supported by the Engineering and\nPhysical Sciences Research Council (EPSRC) of the UK.\n[1] M. I. McMahon and R. J. Nelmes, Chem. Soc. Rev. 35,\n943 (2006), and references therein.\n[2] J. B. Neaton and N. W. Ashcroft, Nature 400, 141\n(1999).\n[3] J. B. Neaton and N. W. Ashcroft, Phys. Rev. Lett. 86,\n2830 (2001).\n[4] M. Hanﬂand, K. Syassen, N. E. Christensen, and D. L.\nNovikov, Nature 408, 174 (2000).\n[5] C. J. Pickard and R. J. Needs, Phys. Rev. Lett. 102,\n146401 (2009).\n[6] Y. Ma, M. Eremets, A. R. Oganov, Y. Xie, I. Trojan, S.\nMedvedev, A. O. Lyakhov, M. Valle, and V. Prakapenka,\nNature458, 182 (2009).\n[7] H. G. von Schnering and R. Nesper, Angew. Chem. Int.\nEd. Engl. 26, 1059 (1987).\n[8] B. Rousseau and N. W. Ashcroft, Phys. Rev. Lett. 101,\n046407 (2008).\n[9] C. J. Pickard and R. J. Needs, Nature Materials 9, 624\n(2010).\n[10] H. Jones, Proc. Roy. Soc. A 147, 396 (1934).\n[11] G. J. Ackland and I. R. Macleod, New J. Phys. 6, 138\n(2004).\n[12] V. F. Degtyareva and O. Degtyareva, New J. Phys. 11,\n063037 (2009).\n[13] E. Stoner, Proc. R.Soc.London, Ser.A 169, 0339(1939).\n[14] C. J. Pickard and R. J. Needs, Phys. Rev. Lett. 97,\n045504 (2006).\n[15] C. J. Pickard and R. J. Needs, J. Phys.: Condens. Matter\n23, 053201 (2011).\n[16] C. J. Pickard and R. J. Needs, Nature Physics 3, 473\n(2007).\n[17] C. J. Pickard and R. J. Needs, Nature Materials 10, 757\n(2008).\n[18] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M.5\nI. J. Probert, K. Refson, and M. C. Payne, Z. Kristallogr.\n220, 567 (2005).\n[19] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.77, 3865 (1996).\n[20] S. J. Clark and J. Robertson, Phys. Rev. B 82, 085208\n(2010).\n[21] D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).\n[22] K. Meisel, Z. anorg. u. allgem. Chem. 240, 300 (1939).\n[23] M. Marqu´ es, G. J. Ackland, L. F. Lundegaard, G. Stin-\nton, R. J. Nelmes, M. I. McMahon, and J. Contreras-\nGarc´ ıa, Phys. Rev. Lett. 103, 115501 (2009).\n[24] O. K. Andersen, J. Madsen, U. K. Poulsen, O. Jepsen,\nand J. Koll´ ar, Physica B and C 86–88, 249 (1977).[25] J. Hubbard, Proc. Roy. Soc. A 276, 238 (1963).\n[26] A. W. Overhauser, Phys. Rev. B 3, 3173 (1971).\n[27] N. Zabala, M. J. Puska, and R. M. Nieminen, Phys. Rev.\nLett.80, 3336 (1998).\n[28] A. Bergara, J. B. Neaton, and N. W. Ashcroft, Int. J.\nQuant. Chem. 91, 239 (2003).\n[29] E. Lieb and D. Mattis, Phys. Rev. 125, 164 (1962).\n[30] Y. Nozue, T. Kodaira, and T. Goto, Phys. Rev. Lett. 68,\n3789 (1992).\n[31] K. Takai, S. Eto, M. Inaguma, T. Enoki, H. Ogata, M.\nTokita, and J. Watanabe, Phys. Rev. Lett. 98, 017203\n(2007)."    },    {        "title": "1002.0090v1.Exchange_coupling_and_magnetoresistance_in_CoFe_NiCu_CoFe_spin_valves_near_the_Curie_point_of_the_spacer.pdf",        "content": "arXiv:1002.0090v1  [cond-mat.mtrl-sci]  30 Jan 2010Exchange coupling and magnetoresistance in CoFe/NiCu/CoF e spin-valves\nnear the Curie point of the spacer\nS. Andersson and V. Korenivski1\nNanostructure Physics, Royal Institute of Technology, SE- 106 91 Stockholm,\nSweden\n(Dated: 1 November 2018)\nThermal control of exchange coupling between two strongly ferr omagnetic layers through a weakly ferromag-\nnetic Ni-Cu spacer and the associated magnetoresistance is invest igated. The spacer, having a Curie point\nslightly above room temperature, can be cycled between its parama gnetic and ferromagnetic states by vary-\ning the temperature externally or using joule heating. It is shown th at the giant magnetoresistance vanishes\ndue to a strong reduction of the mean free path in the spacer at ab ove∼30% Ni concentration — before the\nonset of ferromagnetism. Finally, a device is proposedand demonst rated which combines thermally controlled\nexchange coupling and large magnetoresistance by separating the switching and the read out elements.\nI. INTRODUCTION\nThermal control in spintronic devices and MRAM ap-\nplications has in recent years been of great interest due\nto the associated increase of stability and decrease in\npowerconsumption1–3. Recently, thermally excitedoscil-\nlations in nanocontacts, reaching frequencies of the order\nof GHz, have been predicted4. In this model a ferromag-\nnetic (FM) ﬁlm is separated from a small FM grain by\na point contact having a diameter of a few nanometers.\nDue to the high current densities reached in such a small\narea,the FMregionwithin thepoint contactreachesvery\nhightemperatures. When thelocaltemperature ishigher\nthan the FM Curie point the exchange coupling through\nthe point contact becomes vanishingly small.\nThe modelisbasedon twopremises. First, athermally\ncontrolled exchange coupling between two FM regions.\nSecond, an increase in resistance when the FM regions\ndecouple. The ﬁrst criterion can be met in a metallic sys-\ntem with two strong ferromagnets separated by a weakly\nFM spacer. If this can be combined with a large change\nin resistance both criteria would be met. To date, the\nlargest changes of resistance obtained in an all metal-\nlic structure have been from giant magnetoresistance5,6\n(GMR).\nIn this work we investigate the possibility of decou-\npling two strong ferromagnets separated by a weakly fer-\nromagnetic Ni-Cu alloy. To verify if the Ni-Cu alloy, in\nits paramagnetic phase, can be used as a GMR spacer\nwe study the eﬀects of adding nickel to a copper spacer\nin a spin-valve structure on the interlayer exchange cou-\npling and GMR. Finally, we design and implement an\nimproved thermionic spin-valve structure, in which the\nswitching and the read out layers are separated.\nII. EXPERIMENTAL DETAILS\nAll ﬁlms were deposited on thermally oxidized Si\nsubstrates using magnetron sputtering at a base pres-\nsure better than 5 ·10−8Torr. The argon pressure\nduring sputtering was kept at 3 mTorr. To demon-strate thermally controlled exchange coupling sam-\nples with structure Cu(90)/Ni 80Fe20(8)/Co 90Fe10(2)/\nNixCu1−x(t)/Co 90Fe10(5)/Ta(10) (thickness in nanome-\nters) were deposited. Three diﬀerent thicknesses, t=\n10,20,30 nm, of the weakly ferromagnetic Ni-Cu alloy\nwere used for studying the interlayerexchangein the sys-\ntem. Variation in xwas obtained by cosputtering Ni and\nCu onto Si substrates that had been cut into 90 x 10 mm\nstrips. In this way a compositional gradient was created\nalong the Si strips ranging from x= 0.2 tox= 0.9. By\ncutting the strips into smaller pieces a series of samples\nwith diﬀerent Curie temperatures were obtained.\nInordertoperformmagneticcharacterizationthesam-\npleswereplaced in alooptracerequipped with athin-ﬁlm\nheater. The temperature was controlled through a feed\nback loop using a type-T thermocouple in close contact\nwith the samples. Further investigations of the switch-\ning behavior at room temperature were performed using\na vibrating sample magnetometer (VSM).\nTo measure the eﬀect of Ni-Cu alloying on GMR,\nsamples with structure Ni 80Fe20(4) / Co 90Fe10(1) /\nNixCu1−x(3.5) / Co 90Fe10(2) were deposited. The Ni-\nCu alloys were cosputtered from Ni and Cu targets such\nthat the stoichiometry, x, could be varied by controlling\nthe relative diﬀerence in sputtering rates.\nTo measurecurrentin plane (CIP) GMR, thin Al wires\nwere bonded to the top of the samples. Before electrical\nmeasurements the separate switching of the NiFe/CoFe\nbi-layer and CoFe top layer was conﬁrmed using a mag-\nnetometer.\nIII. RESULTS AND DISCUSSION\nA Ni-Cu alloy was chosen for the weakly FM spacer\nbecause of the well known dependence of Curie point\non the Ni concentration7–9. The Ni-Cu spacer is used\nto separate a magnetically softer NiFe/CoFe bi-layer\nfrom a magnetically harder CoFe layer. Here NiFe and\nCoFe stand for Ni 80Fe20and Co 90Fe10, respectively. By\ncosputtering Ni and Cu, a series ofsamples with diﬀerent\nCurie temperatures were obtained.2\nApplied Field (Oe)Normalized Magnetization(a)\n(b)T = 25 Co\nT = 110 CoA.B.ReversibleIrreversible}\n}\nNiFe/CoFeCoFe20 0 -20 40 -40\n20 0 -20 40 -40\nApplied Field (Oe)-101-101Normalized Magnetization\nxxxxxxxxxx\nxxxx\nFIG. 1. Normalized magnetization versus applied magnetic\nﬁeld for a sample structure NiFe/CoFe/Ni xCu1−x(30)/ CoFe,\nx≈0.7 at (a) room temperature and(b) 110◦C. Uponheating\ntheNi-Cu alloy goes throughaferromagnetic toparamagneti c\nphase transition and the NiFe/CoFe bi-layer decouples from\nthe CoFe layer.\nA. Thermally controlled interlayer exchange coupling\nFig. 1 (a) shows the magnetization loop for a sam-\nple with a 30 nm thick Ni-Cu layer having a Curie point\njust above room temperature, x≈0.7. The shape of the\ncurve indicates that the strongly FM layers are weakly\nexchange coupled through the Ni-Cu alloy. Two distinct\nregions can be seen — similar to the magnetic state of a\nspring-magnet10. At point Ain Fig. 1 (a) the magnetic\nmoment of the soft NiFe/CoFe layer starts to rotate in\nthe external magnetic ﬁeld. This is a reversible rotation\ndue to the exchange coupling through the Ni-Cu alloy\nto the harder CoFe layer. The reversible switching con-\ntinues until point Bis reached. By comparing the mag-\nnitude of the magnetization at 25 Oe (point B) in Fig.1 (a) with that in Fig. 1 (b) in which the NiFe/CoFe\nlayer is heated to 110◦C and thereby decoupled from the\nCoFe layer, it can be seen that at room temperature the\nNiFe/CoFe layer has not yet ﬁnished rotating 180◦when\nthe hard layer switches. At point Bthe torque on the\nCoFe layer is too strong for it to remain in position and\nan irreversible rotation of all layers follows. This behav-\nior was conﬁrmed by VSM measurements of the same\nsample at room temperature. Starting at a positive ﬁeld,\nhigh enough so that all the moments were aligned, the\nmagnetization was measured while the external ﬁeld was\nswept to -20 Oe and then back again. After the ﬁeld re-\nversal at -20 Oe the magnetization backtracks the values\nmeasured before the reversal. The same behavior was\nseen for ﬁeld reversals up to -25 Oe indicating that the\nrotation is reversible up to this point. For reversal ﬁelds\nany higher than this, the magnetization does not back-\ntrack the values measured before the ﬁeld reversal. This\nconﬁrms that the switching behavior is irreversible for\nﬁelds higher than ±25 Oe.\nFig. 1 (b) shows the same sample at 110◦C. The Curie\npoint of the spacer has been reached and the soft and\nhard FM layers are essentially exchange decoupled as ev-\nidenced by the two distinct magnetization transitions at\napproximately 15 and 45 Oe. As the temperature is re-\nduced to room temperature the two magnetization tran-\nsitions shift towards each other and the sharp magneti-\nzation loop becomes signiﬁcantly skewed. At room tem-\nperature the curve shape is back to the one shown in Fig.\n1 (a). This thermally controlled interlayer exchange cou-\npling is perfectly reversible on thermal cycling within the\ngiven temperature range.\nForsampleswith the Curie point aroundroomtemper-\nature the spacer had to be 20 or 30 nm thick in order to\ncompletely diminsh the exchange coupling through the\nspacer. An explanation for this could be that the alloy is\nnot homogenous after cosputtering at room temperature\nbut contains regions with diﬀerent Curie points. If the\nspacer is too thin, these regions could extend to the alloy\ninterfaces and couple the two CoFe ﬁlms. Another pos-\nsible explanation is that the alloying is homogenous but\nthe two strong ferromagnets are coupled by exchange in-\nteractionsthroughthespacerevenattemperaturesabove\nthe Curie point. It has been indicated that exchange can\npropagatethrough paramagnetic regions on length scales\nof several nanometers11, which is believed to be due to\nenhancement of magnetic order in thin layers caused by\nthe proximity eﬀect of a strong ferromagnet.\nB. CoFe/Ni-Cu/CoFe spin-valve\nTo understand the CIP GMR in the above\nNiFe/CoFe/Ni-Cu/CoFe system, we have to consider the\nmean free path in the Ni-Cu spacer. When the spacer\nthickness is much larger than the mean free path the\nCIP GMR signal vanishes as exp( −tNiCu/λ)12. Hereλ\nis the electron mean free path in the spacer material and3\nCalculated from data in [16]\nCalculated from data in [17]\nCalculated from data in [18]\nOur measurements10\n10\n102\n1\n0\n0 20 40 60 80 100\nNi Concentration (at. %)\nMean Free Path (nm)\nFIG. 2. Electron mean free pathin bulkNi-Cu for diﬀerentNi\nconcentrations. The data points have been calculated using\nthe Drude model from published data on resistivity measure-\nments at 300, 273 and 250K16–18.CIP GMR (%)\n0 10 20 30 40 50 600123456\n77 K\n300 K\nNi Concentration (at. %)\nM / M\ns\nH (Oe)(a)(b)\n(c)1\n-1\n-110\n0\n-100 100 0\n2\n2\n2FIG. 3. (a) Current in plane (CIP) giant magnetoresistance\n(GMR) versus Ni concentration in a NiFe(4)/CoFe(1)/Ni-\nCu(3.5)/CoFe(2) spin-valve. The inset shows the normalize d\nmagnetization versus applied ﬁeld for a spin-valve with (b) 29\nat.% Ni in the spacer and (c) 35 at. % Ni in the spacer.\ntNiCuis the spacer thickness. In order to obtain a high\nGMR signal the thickness of the spacer should be com-\nparable or smaller than λ. Fig. 2 shows λfor diﬀerent Ni\nconcentrations from our measurements as well as calcu-\nlated from published experimental data using the Drude\nmodel. The mean free path decreases quickly with in-\ncreasing Ni concentration. In the interesting range of\nNi concentrations between 40% and 70% where the al-\nloy is weakly ferromagnetic7,λis down to ∼1 nm. As\nwas detailed in the previous section, our spacers with the\nCurie point close to room temperature have a minimum\nthickness of 20 nm in order to completely diminish the\ninterlayer exchange coupling. Assuming the thickness of\nthe weakly ferromagnetic spacer can be reduced by fur-ther material optimization, we next examine how thin a\nspacerwouldstillprovideameasurableGMR signal. The\nthinnest possible spacer to avoid any signiﬁcant RKKY\ncoupling is ∼3 nm13. We therefore choose this thickness\nand evaluate the CIP GMR versus Ni concentration.\nWe have fabricated samples with structure\nNiFe(4)/CoFe(1)/Ni-Cu(3.5)/CoFe(2). CIP GMR\nmeasurements for samples with diﬀerent Ni concentra-\ntions are shown in Fig. 3 (a). With pure Cu in the\nspacera signal of 3.7 % is measured at room temperature\n(5% at 77 K). When Ni is added to the spacer the signal\ndrops caused by a decrease in λ. At 29 at. % Ni the\nGMR has decreased by more than a factor of three,\nwhich we attribute to a sharp decrease of the mean free\npath on alloying. Fig. 3 (b) shows the magnetization\nversus applied ﬁeld for this sample. It can be seen that\nthe switching is still separate at this concentration.\nAt 35 at. % Ni the magnetic layers couple and the\nswitching is no longer separate, which is shown in Fig. 3\n(c). It can thus be concluded that even for a very thin\nCu-Ni spacer the CIP GMR signal essentially vanishes\nfor 29 at. % Ni concentration, where the layers are still\ndecoupled magnetically. The ﬁnal decrease to zero GMR\nat 35 at. % Ni is due to coupling through the spacer and\nnot to a decrease in λ.\nItwillbeinformativetopointoutthatbyusingcurrent\nperpendicular to the plane GMR instead of CIP the lim-\niting length scale would be the spin diﬀusion length14,\nlsf, and not λ. However, published experimental data\nindicate that lsfdecrease with the same rate as λand\nhas been measured to be 7.5 nm at 5◦K in an alloy with\n22.7% Ni15. This is still below the minimum thickness\nin our case of 20 nm needed to achieve reliable interlayer\ndecoupling.\nTo overcome the above limitations and use thermally\ncontrolled interlayer exchange coupling together with\nlarge magnetoresistance we propose a new design in\nwhich the GMR readout layerand the weaklyFM spacer\nare separated. A schematic is shown in the inset to Fig.\n4. An antiferromagnet (AFM) is used to exchange bias\na FM ﬁlm which works as a reference layer in the spin-\nvalve. The spin-valve uses a metallic spacer for GMR or\nan insulator for tunneling magnetoresistance. The spin-\nvalve spacer is only used for read out and can be opti-\nmized to give the highest possible magnetoresistance sig-\nnal. The top layer of the spin-valve is exchange coupled\nthrough a weakly FM alloy to a top pinned FM. At high\ntemperatures the coupling through the weakly FM alloy\nis negligible and the top layer of the spin-valve is free to\nrotate. However, at low temperatures it will be exchange\ncoupled to the top pinned FM. This results in full ﬂexi-\nbility when choosing the composition and Curie point of\nthe weakly ferromagnetic alloy while at the same time\nmaking it possible to achieve high magnetoresistance.\nTo demonstrate this new struc-\nture we have deposited samples of\nNiFe(3)/MnIr(15)/CoFe(4)/ Cu(3.5)/CoFe(4)/NiFe(6)/\nCoFe(2)/NiCu(20)/CoFe(2)/NiFe(3)/MnIr(12)/Ta(5)4\n0 -20 -40 40 -6002.5\n2.0\n1.5\n1.0\n0.5\n-80 20\nCIP GMR (%)\nApplied Field (Oe)\nxx\nxxxx\nxxxxxx\nAFM\nxx\nAFM\nWeak FM Alloy\nMetal / Insulator\n601.3·106A/cm2\n1.0·106A/cm2\n7.5·105A/cm2\nFIG. 4. CIP GMR for three diﬀerent current den-\nsities versus applied ﬁeld measured on a sample\nstructure NiFe(3)/MnIr(15)/CoFe(4)/Cu(3.5)/CoFe(4)/\nNiFe(6)/CoFe(2)/NiCu(20)/CoFe(2)/NiFe(3)/ MnIr(12)/\nTa(5). The inset shows a schematic of a device in which\nthermally controlled exchange coupling is separated from\na spin-valve read out. Either giant magnetoresistance or\ntunnelling magnetoresistance can be used for read out.\nwith 70 at. % Ni in the spacer ( TC≈100◦C). The\nsamples were patterned using photolithography into\nstrips 50 µm wide and 1 mm long and then bonded\nat the edges with aluminum wire. The resulting CIP\nGMR signal for diﬀerent current densities is shown in\nFIG. 4. When the current density is increased and the\ntemperature of the device is correspondingly raised due\nto joule heating, the top layer of the spin-valve decouples\nfrom the top pinned CoFe/NiFe bi-layer producing a\nstrong GMR signal. For a current density of 1 .3·106\nA/cm2,T > T C≈100◦C, the exchange decoupling is\ncomplete and the full CIP GMR signal of 2.5% for this\nstructure is obtained.IV. CONCLUSION\nThermally controlled exchange coupling between two\nstrongly FM ﬁlms separated by a weakly FM Ni-Cu\nspaceris demonstrated. At temperatures higher than the\nCurie point of the spacer the FM ﬁlms are decoupled. At\nlower temperatures the switching behavior can be sepa-\nrated into two regions — reversible and irreversible.\nIn a CoFe/Ni-Cu/CoFe spin-valve the CIP GMR sig-\nnal vanishes due to a sharp reduction of the mean free\npath on alloying for Ni concentrations above ∼30 at. %.\nA new design is proposed and demonstrated, combining\nthermally controlled exchangecoupling and largemagne-\ntoresistance, which may prove useful for applications in\ncurrent controlled magneto-resistive oscillators.\nACKNOWLEDGMENTS\nThis work was supported by EU-FP7-FET-STELE.\n1I. L. Prejbeanu, M. Kerekes, R. C. Sousa, H. Sibuet, O. Redon,\nB.Dieny and J. P.Nozi´ eres, J.Phys. Condens. Matter 19, 165218\n(2007).\n2R. S. Beech, J. A. Andersson, A. V. Pohm and J. M. Daughton,\nJ. Appl. Phys. 87, 6403 (2000).\n3J. M. Daughton and A. V. Pohm, J. Appl. Phys. 93, 7304 (2003).\n4A. Kadigrobov, S. I. Kulinich, R. I. Shekhter, M. Jonson and V .\nKorenivski, Phys. Rev. B 74, 195307 (2006).\n5G.Binasch, P.Gr¨ unberg, F. Saurenbach and W. Zinn, Phys. Re v.\nB39, 4828 (1989).\n6M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau and F.\nPetroﬀ, Phys. Rev. Lett. 61, 2472 (1988).\n7T. J. Hicks, B. Rainford, J. S. Kouvel and G. G. Low, Phys. Rev.\nLett.22, 531 (1969).\n8S. K. Dutta Roy and A. V. Subrahmanyam, Phys. Rev. Lett.\n177, 1133 (1969).\n9J. B. Sousa, M. R. Chaves, M. F. Pinheiro and R. S. Pinto, J.\nLow. Temp. Phys. 18, 125 (1975).\n10J. E. Davies, O. Hellwig, E. E. Fullerton, J. S. Jiang, S. D. Ba der,\nG. T. Zim´ anyi and K. Liu, Appl. Phys. Lett. 86, 262503 (2005).\n11A. Hernando and T. Kulik, Phys. Rev. B 49, 7064 (1994).\n12A. Barth´ el´ emy and A. Fert, Phys. Rev. B 43, 13124 (1991).\n13S. S. P. Parkin, R. Bhadra and K. P. Roche, Phys. Rev. Lett. 66,\n2152 (1991).\n14J. Bass and W. P. Pratt Jr., J. Phys. Condens. Matter 19, 183201\n(2007).\n15S. Y. Hsu, P. Holody, R. Loloee, J. M. Rittner, W. P. Pratt Jr.\nand P. A. Schroeder, Phys. Rev. B 54, 9027 (1996).\n16R. W. Houghton, M. P. Sarachik and J. S. Kouvel, Phys. Rev.\nLett.25, 238 (1970).\n17H. M. Ahmad and D. Greig, J. Phys. (Paris). 35, C4-223 (1974).\n18P. A. Schroeder, R. Wolf and J. A. Woollam, Phys. Rev. 138,\nA105 (1965)."    },    {        "title": "2201.06060v2.Ferromagnetic_resonance_modulation_in__d__wave_superconductor_ferromagnetic_insulator_bilayer_systems.pdf",        "content": "Ferromagnetic resonance modulation in d-wave superconductor/ferromagnetic\ninsulator bilayer systems\nYuya Ominato,1Ai Yamakage,2Takeo Kato,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China.\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: May 6, 2022)\nWe investigate ferromagnetic resonance (FMR) modulation in d-wave superconductor\n(SC)/ferromagnetic insulator (FI) bilayer systems theoretically. The modulation of the Gilbert\ndamping in these systems re\rects the existence of nodes in the d-wave SC and shows power-law\ndecay characteristics within the low-temperature and low-frequency limit. Our results indicate the\ne\u000bectiveness of use of spin pumping as a probe technique to determine the symmetry of unconven-\ntional SCs with high sensitivity for nanoscale thin \flms.\nI. INTRODUCTION\nSpin pumping (SP)1,2is a versatile method that can\nbe used to generate spin currents at magnetic junctions.\nWhile SP has been used for spin accumulation in vari-\nous materials in the \feld of spintronics3,4, it has recently\nbeen recognized that SP can also be used to detect spin\nexcitation in nanostructured materials5, including mag-\nnetic thin \flms6, two-dimensional electron systems7{9,\nand magnetic impurities on metal surfaces10. Notably,\nspin excitation detection using SP is sensitive even for\nsuch nanoscale thin \flms for which detection by con-\nventional bulk measurement techniques such as nuclear\nmagnetic resonance and neutron scattering experiment is\ndi\u000ecult.\nRecently, spin injection into s-wave superconductors\n(SCs) has been a subject of intensive study both theoret-\nically11{20and experimentally21{34. While the research\ninto spin transport in s-wave SC/magnet junctions is ex-\npected to see rapid development, expansion of the devel-\nopment targets toward unconventional SCs represents a\nfascinating research direction. Nevertheless, SP into un-\nconventional SCs has only been considered in a few recent\nworks35,36. In particular, SP into a d-wave SC, which is\none of the simplest unconventional SCs that can be real-\nized in cuprate SCs37, has not been studied theoretically\nto the best of our knowledge, although experimental SP\nin ad-wave SC has been reported recently38.\nIn this work, we investigate SP theoretically in a bi-\nlayer magnetic junction composed of a d-wave SC and\na ferromagnetic insulator (FI), as shown in Fig. 1. We\napply a static magnetic \feld along the xdirection and\nconsider the ferromagnetic resonance (FMR) experiment\nof the FI induced by microwave irradiation. In this setup,\nthe FMR linewidth is determined by the sum of the in-\ntrinsic contribution made by the Gilbert damping of the\nbulk FI and the interface contribution, which originates\nfrom the spin transfer caused by exchange coupling be-\nMicrowavex yz\nSpin current\nFerromagnetic resonanceInteractionFIG. 1. Schematic of the d-wave SC/FI bilayer system. The\ntwo-dimensional d-wave SC is placed on the FI. Precessional\nmotion of the magnetization is induced by microwave irradia-\ntion. The spins are injected and the magnetization dynamics\nare modulated because of the interface magnetic interaction.\ntween thed-wave SC and the FI. We then calculate the\ninterface contribution to the FMR linewidth, which is\ncalled the modulation of the Gilbert damping hereafter,\nusing microscopic theory based on the second-order per-\nturbation39{41. We show that the temperature depen-\ndence of the modulation of the Gilbert damping exhibits\na coherent peak below the transition temperature that\nis weaker than that of s-wave SCs11,13{15. We also show\nthat because of the existence of nodes in the d-wave SCs,\nthe FMR linewidth enhancement due to SP remains even\nat zero temperature.\nThe paper is organized as follows. In Sec. II, we in-\ntroduce the model Hamiltonian of the SC/FI bilayer sys-\ntem. In Sec. III, we present the formalism to calculate\nthe modulation of the Gilbert damping. In Sec. IV, we\npresent the numerical results and explain the detailed\nbehavior of the modulation of the Gilbert damping. In\nSec. V, we brie\ry discuss the relation to other SC sym-\nmetries, the proximity e\u000bect, and the di\u000berence between\nd-wave SC/FI junctions and d-wave SC/ferromagnetic\nmetal junctions. We also discuss the e\u000bect of an e\u000bectivearXiv:2201.06060v2  [cond-mat.mes-hall]  5 May 20222\nZeeman \feld due to the exchange coupling. In Sec. VI,\nwe present our conclusion and future perspectives.\nII. MODEL\nThe model Hamiltonian of the SC/FI bilayer system\nHis given by\nH=HFI+HdSC+HT: (1)\nThe \frst term HFIis the ferromagnetic Heisenberg\nmodel, which is given by\nHFI=\u0000JX\nhi;jiSi\u0001Sj\u0000~\rhdcX\njSx\nj; (2)\nwhereJ>0 is the exchange coupling constant, hi;ji\nrepresents summation over all the nearest-neighbor sites,\nSjis the localized spin at site jin the FI,\ris the gy-\nromagnetic ratio, and hdcis the static magnetic \feld.\nThe localized spin Sjis described as shown using the\nbosonic operators bjandby\njof the Holstein-Primako\u000b\ntransformation42\nS+\nj=Sy\nj+iSz\nj=\u0010\n2S\u0000by\njbj\u00111=2\nbj; (3)\nS\u0000\nj=Sy\nj\u0000iSz\nj=by\nj\u0010\n2S\u0000by\njbj\u00111=2\n; (4)\nSx\nj=S\u0000by\njbj; (5)\nwhere we require [ bi;by\nj] =\u000ei;jto ensure that S+\nj,S\u0000\nj,\nandSx\njsatisfy the commutation relation of angular mo-\nmentum. The deviation of Sx\njfrom its maximum value S\nis quanti\fed using the boson particle number. It is conve-\nnient to represent the bosonic operators in the reciprocal\nspace as follows\nbk=1p\nNX\nje\u0000ik\u0001rjbj; by\nk=1p\nNX\njeik\u0001rjby\nj;(6)\nwhereNis the number of sites. The magnon opera-\ntors with wave vector k= (kx;ky;kz) satisfy [bk;by\nk0] =\n\u000ek;k0. Assuming that the deviation is small, i.e., that\nhby\njbji=S\u001c1, the ladder operators S\u0006\njcan be approx-\nimated asS+\nj\u0019(2S)1=2bjandS\u0000\nj\u0019(2S)1=2by\nj, which\nis called the spin-wave approximation. The Hamiltonian\nHFIis then written as\nHFI\u0019X\nk~!kby\nkbk; (7)\nwhere we assume a parabolic dispersion ~!k=Dk2+\n~\rhdcwith a spin sti\u000bness constant Dand the constant\nterms are omitted.\nThe second term HdSCis the mean-\feld Hamiltonian\nfor the two-dimensional d-wave SC, and is given by\nHdSC=X\nk(cy\nk\";c\u0000k#)\u0012\n\u0018k \u0001k\n\u0001k\u0000\u0018k\u0013\u0012ck\"\ncy\n\u0000k#\u0013\n;(8)wherecy\nk\u001bandck\u001bdenote the creation and annihilation\noperators, respectively, of the electrons with the wave\nvectork= (kx;ky) and thexcomponent of the spin\n\u001b=\";#, and\u0018k=~2k2=2m\u0000\u0016is the energy of conduc-\ntion electrons measured from the chemical potential \u0016.\nWe assume that the d-wave pair potential has the form\n\u0001k= \u0001 cos 2\u001ekwith the phenomenological temperature\ndependence\n\u0001 = 1:76kBTctanh \n1:74r\nTc\nT\u00001!\n; (9)\nwhere\u001ek= arctan(ky=kx) denotes the azimuth angle of\nk. Using the Bogoliubov transformation given by\n\u0012ck\"\ncy\n\u0000k#\u0013\n=\u0012\nuk\u0000vk\nvkuk\u0013\u0012\rk\"\n\ry\n\u0000k#\u0013\n; (10)\nwhere\ry\nk\u001band\rk\u001bdenote the creation and annihilation\noperators of the Bogoliubov quasiparticles, respectively,\nandukandvkare given by\nuk=r\nEk+\u0018k\n2Ek; vk=r\nEk\u0000\u0018k\n2Ek; (11)\nwith the quasiparticle energy Ek=p\n\u00182\nk+ \u00012\nk, the mean-\n\feld Hamiltonian can be diagonalized as\nHdSC=X\nk(\ry\nk\";\r\u0000k#)\u0012\nEk 0\n0\u0000Ek\u0013\u0012\rk\"\n\ry\n\u0000k#\u0013\n:(12)\nThe density of states of the d-wave SC is given by43\nD(E)=Dn= Re\u00142\n\u0019K\u0012\u00012\nE2\u0013\u0015\n; (13)\nwhereDn=Am= 2\u0019~2is the density of states per spin of\nthe normal state, Ais the system area, and K(x) is the\ncomplete elliptic integral of the \frst kind in terms of the\nparameterx, where\nK(x) =Z\u0019=2\n0d\u001ep\n1\u0000xcos2\u001e: (14)\nD(E) diverges at E=\u0001 = 1 and decreases linearly when\nE=\u0001\u001c1 because of the nodal structure of \u0001 k. The\ndensity of states for an s-wave SC, in contrast, has a\ngap forjEj<\u0001. This di\u000berence leads to distinct FMR\nmodulation behaviors, as shown below.\nThe third term HTdescribes the spin transfer between\nthe SC and the FI at the interface\nHT=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+J\u0003\nq;k\u001b\u0000\n\u0000qS+\n\u0000k\u0001\n; (15)\nwhereJq;kis the matrix element of the spin transfer pro-\ncesses, and \u001b\u0006\nq= (\u001by\nq\u0006i\u001bz\nq)=2 andS\u0006\nk=Sy\nk\u0006iSz\nkare3\n(a) Spin transfer process (b) Self-energy\nJq,kJ*q,k\np/uni2191p+q/uni2193\np/uni2191p+q/uni2193\n−k −k\n/uni03A3R\nk(/uni03C9)=\nFIG. 2. (a) Diagrams of the bare vertices of the spin transfer\nprocesses at the interface. (b) Self-energy within the second-\norder perturbation.\nthe Fourier components of the ladder operators and are\ngiven by\n\u001b+\nq=X\npcy\np\"cp+q#; \u001b\u0000\n\u0000q=X\npcy\np+q#cp\"; (16)\nS\u0000\n\u0000k\u0019(2S)1=2by\nk; S+\nk\u0019(2S)1=2bk: (17)\nUsing the expressions above, HTcan be written as\nHT\u0019p\n2SX\np;q;k\u0010\nJq;kcy\np\"cp+q#by\n\u0000k+J\u0003\nq;kcy\np+q#cp\"b\u0000k\u0011\n:\n(18)\nThe \frst (second) term describes a magnon emission\n(absorption) process accompanying an electron spin-\rip\nfrom down to up (from up to down). A diagrammatic\nrepresentation of the interface interactions is shown in\nFig. 2 (a).\nIn this work, we drop a diagonal exchange coupling at\nthe interface, whose Hamiltonian is given as\nHZ=X\nq;kJq;k\u001bx\nqSx\nk: (19)\nThis term does not change the number of magnons in\nthe FI and induces an e\u000bective Zeeman \feld on electrons\nin the two-dimensional d-wave SC. We expect that this\nterm does not a\u000bect our main result because the coupling\nstrength is expected to be much smaller than the super-\nconducting gap and the microwave photon energy. We\nwill discuss this e\u000bect in Sec. V brie\ry.\nIII. FORMULATION\nThe coupling between the localized spin and the mi-\ncrowave is given by\nV(t) =\u0000~\rhacX\ni(Sy\nicos!t\u0000Sz\nisin!t); (20)wherehacis the amplitude of the transverse oscillating\nmagnetic \feld with frequency !. The microwave irra-\ndiation induces the precessional motion of the localized\nspin. The Gilbert damping constant can be read from\nthe retarded magnon propagator de\fned by\nGR\nk(t) =1\ni~\u0012(t)h[S+\nk(t);S\u0000\n\u0000k(0)]i; (21)\nwhere\u0012(t) is a step function. Second-order perturbation\ncalculation of the magnon propagator with respect to the\ninterface interaction was performed and the expression of\nthe self-energy was derived in the study of SP39{41. Fol-\nlowing calculation of the second-order perturbation with\nrespect to Jq;k, the Fourier transform of the retarded\nmagnon propagator is given by\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!); (22)\nwhere\u000bis the intrinsic Gilbert damping constant that\nwas introduced phenomenologically44{46. The diagram\nof the self-energy \u0006R\nk(!) is shown in Fig. 2 (b). From the\nexpressions given above, the modulation of the Gilbert\ndamping constant is given by\n\u000e\u000b=\u00002SIm \u0006R\nk=0(!)\n~!: (23)\nWithin the second-order perturbation, the self-energy\nis given by\n\u0006R\nk(!) =\u0000X\nqjJq;kj2\u001fR\nq(!); (24)\nwhere\u001fR\nq(!) represents the dynamic spin susceptibility\nof thed-wave SC de\fned by\n\u001fR\nq(!) =\u00001\ni~Z\ndtei(!+i0)t\u0012(t)h[\u001b+\nq(t);\u001b\u0000\n\u0000q(0)]i:(25)\nSubstituting the ladder operators in terms of the Bogoli-\nubov quasiparticle operators into the above expression\nand performing a straightforward calculation, we then\nobtain434\n\u001fR\nq(!) =\u0000X\npX\n\u0015=\u00061X\n\u00150=\u00061\u0012(\u0018p+\u0015Ep)(\u0018p+q+\u00150Ep+q) + \u0001 p\u0001p+q\n4\u0015Ep\u00150Ep+q\u0013f(\u0015Ep)\u0000f(\u00150Ep+q)\n\u0015Ep\u0000\u00150Ep+q+~!+i0; (26)\nwheref(E) = 1=(eE=kBT+ 1) is the Fermi distribution\nfunction.\nIn this paper, we focus on a rough interface modeled in\nterms of the mean J1and variance J22of the distribution\nofJq;k(see Appendix A for detail). The con\fgurationally\naveraged coupling constant is given by\njJq;k=0j2=J12\u000eq;0+J22: (27)\nIn this case, \u000e\u000bis written as\n\u000e\u000b=2SJ12\n~!Im\u001fR\nq=0(!) +2SJ22\n~!X\nqIm\u001fR\nq(!):(28)\nThe \frst term represents the momentum-conserved spin-\ntransfer processes, which vanish as directly veri\fed from\nEq. (26). This vanishment always occurs in spin-singlet\nSCs, including sandd-wave SCs, since the spin is\nconserved43. Consequently, the enhanced Gilbert damp-\ning is contributed from spin-transfer processes induced\nby the roughness proportional to the variance J22\n\u000e\u000b=2SJ22\n~!X\nqIm\u001fR\nq(!): (29)\nThe wave number summation can be replaced as\nX\nq(\u0001\u0001\u0001)!Dn\n2\u0019Z1\n\u00001d\u0018Z2\u0019\n0d\u001e(\u0001\u0001\u0001): (30)\nChanging the integral variable from \u0018toEand substi-\ntuting Eq. (26) into Eq. (29), we \fnally obtain\n\u000e\u000b=2\u0019SJ 22D2\nn\n~!Z1\n\u00001dE[f(E)\u0000f(E+~!)]\n\u0002Re\u00142\n\u0019K\u0012\u00012\nE2\u0013\u0015\nRe\u00142\n\u0019K\u0012\u00012\n(E+~!)2\u0013\u0015\n:\n(31)\nNote that the coherence factor vanishes in the above ex-\npression by performing the angular integral. The en-\nhanced Gilbert damping in the normal state is given by\n\u000e\u000bn= 2\u0019SJ 22D2\nn; (32)\nfor the lowest order of !. This expression means that \u000e\u000b\nis proportional to the product of the spin-up and spin-\ndown densities of states at the Fermi level7.IV. GILBERT DAMPING MODULATION\nFigure 3 shows the enhanced Gilbert damping constant\n\u000e\u000bas a function of temperature for several FMR frequen-\ncies, where \u000e\u000bis normalized with respect to its value in\nthe normal state. We compare \u000e\u000bin thed-wave SC shown\nin Figs. 3 (a) and (c) to that in the s-wave SC shown in\nFigs. 3 (b) and (d). The enhanced Gilbert damping for\nthes-wave SC is given by13\n\u000e\u000b=2\u0019SJ 22D2\nn\n~!Z1\n\u00001dE[f(E)\u0000f(E+~!)]\n\u0002\u0012\n1 +\u00012\nE(E+~!)\u0013\n\u0002Re\u0014jEjp\nE2\u0000\u00012\u0015\nRe\"\njE+~!jp\n(E+~!)2\u0000\u00012#\n;\n(33)\nwhere the temperature dependence of \u0001 is the same as\nthat for the d-wave SC, given by Eq. (9). Note that\nthe BCS theory we are based on, which is valid when\nthe Fermi energy is much larger than \u0001, is described by\nonly some universal parameters, including Tc, and inde-\npendent of the detail of the system in the normal state.\nWhen ~!=k BTc= 0:1,\u000e\u000bshows a coherence peak just\nbelow the transition temperature Tc. However, the co-\nherence peak of the d-wave SC is smaller than that of\nthes-wave SC. Within the low temperature limit, \u000e\u000bin\nthed-wave SC shows power-law decay behavior described\nby\u000e\u000b/T2. This is in contrast to \u000e\u000bin thes-wave SC,\nwhich shows exponential decay. The di\u000berence in the low\ntemperature region originates from the densities of states\nin thed-wave ands-wave SCs, which have gapless and full\ngap structures, respectively. When the FMR frequency\nincreases, the coherence peak is suppressed, and \u000e\u000bde-\ncays monotonically with decreasing temperature. \u000e\u000bhas\na kink structure at ~!= 2\u0001, where the FMR frequency\ncorresponds to the superconducting gap.\nFigure 4 shows \u000e\u000batT= 0 as a function of !. In\nthed-wave SC,\u000e\u000bgrows from zero with increasing !as\n\u000e\u000b/!2. When the value of \u000e\u000bbecomes comparable to\nthe normal state value, the increase in \u000e\u000bis suppressed,\nand\u000e\u000bthen approaches the value in the normal state.\nIn contrast, \u000e\u000bin thes-wave SC vanishes as long as the\ncondition that ~! < 2\u0001 is satis\fed. When ~!exceeds\n2\u0001,\u000e\u000bthen increases with increasing !and approaches\nthe normal state value. This di\u000berence also originates\nfrom the distinct spectral functions of the d-wave and\ns-wave SCs. Under the low temperature condition that\nT= 0:1Tc, the frequency dependence of \u000e\u000bdoes not5\n0.1 5.0\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2\n0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n0.1\n0.5\n1.0\n1.5\n2.03.04.05.0/uni210F/uni03C9/kBTc\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0\n0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n(a)  d-wave (b)  s-wave\n(c)  d-wave (d)  s-wave\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2\n0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0\n0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n0.1\n0.5\n1.0\n1.5\n2.03.04.05.0\nFIG. 3. Enhanced Gilbert damping \u000e\u000bas a function of tem-\nperatureT. The left panels (a) and (c) show \u000e\u000bin thed-\nwave SC in the low and high frequency cases, respectively.\nThe right panels (b) and (d) show \u000e\u000bin thes-wave SC in the\nlow and high frequency cases, respectively. \u000e\u000bnis the normal\nstate value.\nchange for the s-wave SC, and it only changes in the\nlow-frequency region where ~!.kBTfor thed-wave SC\n(see the inset in Fig. 4).\nV. DISCUSSION\nWe discuss the modulation of the Gilbert damping\nin SCs with nodes other than the d-wave SC consid-\nered in this work. Other SCs with nodes are expected\nto exhibit the power-law decay behavior within the low-\ntemperature and low-frequency limit as the d-wave SCs.\nHowever, the exponent of the power can di\u000ber due to\nthe di\u000berence of the quasiparticle density of states. Fur-\nthermore, in the p-wave states, two signi\fcant di\u000berences\narise due to spin-triplet Cooper pairs. First, the uni-\nform spin susceptibility \u001fR\nq=0(!) can be \fnite in the spin-\ntriplet SCs because the spin is not conserved. Second,\nthe enhanced Gilbert damping exhibits anisotropy and\nthe value changes by changing the relative angle between\nthe Cooper pair spin and localized spin35.\nIn our work, proximity e\u000bect between FIs and SCs\nwas not taken into account because the FMR modula-\ntion was calculated by second-order perturbation based\non the tunnel Hamiltonian. Reduction of superconduct-\n/uni210F/uni03C9=2/uni0394(T =0)\ns-waved-wave\n00.2\n2 4 6 81.2\n0.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n\n/uni210F/uni03C9/kBTc0.4\n0.0\n100 10.05\n0.000.1T/Tc=0.0FIG. 4. Enhanced Gilbert damping \u000e\u000bas a function of fre-\nquency!. The vertical dotted line indicates the resonance\nfrequency ~!= 2\u0001(T= 0). The inset shows an enlarged\nview in the low-frequency region.\ning gap due to the proximity e\u000bect15and e\u000bect of the\nsubgap Andreev bound states that appear in the ab-axis\njunction47would also be an important problem left for\nfuture works.\nPhysics of the FMR modulation for d-wave\nSC/ferromagnetic metal junctions is rather di\u000ber-\nent from that for d-wave SC/FI junctions. For d-wave\nSC/ferromagnetic metal junctions, spin transport is\ndescribed by electron hopping across a junction and\nthe FMR modulation is determined by the product\nof the density of states of electrons for a d-wave SC\nand a ferromagnetic metal. (We note that the FMR\nmodulation is determined by a spin susceptibility of\nd-wave SC, which in general includes di\u000berent informa-\ntion from the density of states of electrons.) While the\nFMR modulation is expected to be reduced below a SC\ntransition temperature due to opening an energy gap, its\ntemperature dependence would be di\u000berent from results\nobtained in our work.\nFinally, let us discuss e\u000bect of the diagonal exchange\ncoupling given in Eq. (19) (see also the last part of\nSec. II). This term causes an exchange bias, i.e., an e\u000bec-\ntive Zeeman \feld on conduction electrons in the d-wave\nSC, which is derived as follows. First, the x-component\nof the localized spin is approximated as hSx\nji \u0019S,\nwhich gives Sx\nk\u0019Sp\nN\u000ek;0. Next, the matrix element\nJq;k=0is replaced by the con\fgurationally averaged value\nJq;k=0=J1\u000eq;0. Consequently, the e\u000bective Zeeman\n\feld term is given by\nHZ\u0019EZX\np(cy\np\"cp\"\u0000cy\np#cp#); (34)\nwhere we introduced a Zeeman energy as EZ=J1Sp\nN.\nThis term induces spin splitting of conduction electrons6\nin thed-wave SC and changes the spin susceptibility of\nthe SC. The spin-splitting e\u000bect causes a spin excitation\ngap and modi\fes the frequency dependence in Fig. 4, that\nwill provide additional information on the exchange cou-\npling at the interface. In actual experimental setup for\nthed-wave SC, however, the Zeeman energy, that is less\nthan the exchange bias between a magnetic insulator and\na metal, is estimated to be of the order of 0 :1 erg=cm2.\nThis leads to the exchange coupling that is much less\nthanJ\u00180:1 meV for YIG48. Therefore, we expect that\nthe interfacial exchange coupling is much smaller than\nthe superconducting gap and the microwave photon en-\nergy though it has not been measured so far. A detailed\nanalysis for this spin-splitting e\u000bect is left for a future\nproblem.\nVI. CONCLUSION\nIn this work, we have investigated Gilbert damping\nmodulation in the d-wave SC/FI bilayer system. The\nenhanced Gilbert damping constant in this case is pro-\nportional to the imaginary part of the dynamic spin sus-\nceptibility of the d-wave SC. We found that the Gilbert\ndamping modulation re\rects the gapless excitation that\nis inherent in d-wave SCs. The coherence peak is sup-\npressed in the d-wave SC when compared with that in\nthes-wave SC. In addition, the di\u000berences in the spec-\ntral functions for the d-wave ands-wave SCs with gap-\nless and full-gap structures lead to power-law and ex-\nponential decays within the low-temperature limit, re-\nspectively. Within the low-temperature limit, \u000e\u000bin the\nd-wave SC increases with increasing !, while\u000e\u000bin the\ns-wave SC remains almost zero as long as the excitation\nenergy ~!remains smaller than the superconducting gap\n2\u0001.\nOur results illustrate the usefulness of measurement of\nthe FMR modulation of unconventional SCs for determi-\nnation of their symmetry through spin excitation. We\nhope that this fascinating feature will be veri\fed exper-\nimentally in d-wave SC/FI junctions in the near future.\nTo date, one interesting result of FMR modulation in\nd-wave SC/ferromagnetic metal structures has been re-\nported38. This modulation can be dependent on metallic\nstates, which are outside the scope of the theory pre-\nsented here. The FMR modulation caused by ferromag-\nnetic metals is another subject that will have to be clar-\ni\fed theoretically in future work.\nFurthermore, our work provides the most fundamental\nbasis for application to analysis of junctions with vari-\nous anisotropic SCs. For example, some anisotropic SCs\nare topological and have an intrinsic gapless surface state.\nSP can be accessible and can control the spin excitation of\nthe surface states because of its interface sensitivity. The\nextension of SP to anisotropic and topological supercon-\nductivity represents one of the most attractive directions\nfor further development of superconducting spintronics.\nAcknowledgments.| This work is partially supportedby the Priority Program of Chinese Academy of Sciences,\nGrant No. XDB28000000. We acknowledge JSPS KAK-\nENHI for Grants (No. JP20H01863, No. JP20K03835,\nNo. JP20K03831, No. JP20H04635, and No.21H04565).\nAppendix A: Magnon self-energy induced by a\nrough interface\nThe roughness of the interface is taken into account\nas an uncorrelated (white noise) distribution of the ex-\nchange couplings35, as shown below. We start with an\nexchange model in the real space\nHex=X\njZ\nd2rJ(r;rj)\u001b(r)\u0001Sj\n=X\nq;kJq;k\u001bq\u0001Sk: (A1)\nThe spin density \u001b(r) in the SC and the spin Sjin the\nFI are represented in the momentum space as\n\u001b(r) =1\nAX\nqeiq\u0001r\u001bq; (A2)\nSj=1p\nNX\nkeik\u0001rjSk; (A3)\nwhereAdenotes the area of the system and Nis the\nnumber of sites. The exchange coupling constant is also\nobtained to be\nJq;k=1\nAp\nNX\njZ\nd2rei(q\u0001r+k\u0001rj)J(r;rj): (A4)\nThe exchange model Hexis decomposed into the spin\ntransfer term HTand the e\u000bective Zeeman \feld term HZ\nasHex=HT+HZ.\nNow we consider the roughness e\u000bect of the interface.\nUncorrelated roughness is expressed by the mean J1and\nvarianceJ22as\n1p\nNX\njJ(r;rj) =J1; (A5)\n1\nNX\njj0J(r;rj)J(r0;rj0)\u0000J12=J22A\u000e2(r\u0000r0);(A6)\nwhereOis the con\fgurational average of Oover the\nroughness. 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Re-\ncent experiments have obtained two-dimensional (2D) room temperature ferromagnetic metals, such\nas monolayer MnSe 2. In this paper, we proposed a way to obtain room temperature ferromagnetic\nsemiconductors through metal-semiconductor transition. By the density functional theory calcu-\nlations, a room temperature ferromagnetic semiconductor is obtained in monolayer MnSe 2with a\nfew percent tensile strains, where a metal-semiconductor transition occurs with 2.2% tensile stain.\nThe tensile stains raise the energy of d orbitals of Mn atoms and p orbitals of Se atoms near the\nFermi level, making the Fermi level sets in the energy gap of bonding and antibonding states of\nthese p and d orbitals, and opening a small band gap. The room temperature ferromagnetic semi-\nconductors are also obtained in the heterostructures MnSe 2/X (X = Al 2Se3, GaSe, SiH, and GaP),\nwhere metal-semiconductor transition happens due to the tensile strains by interface of heterostruc-\ntures. In addition, a large magneto-optical Kerr effect (MOKE) is obtained in monolayer MnSe 2\nwith tensile strain and MnSe 2-based heterostructures. Our theoretical results pave a way to obtain\nroom temperature magnetic semiconductors from experimentally obtained 2D room temperature\nferromagnetic metals through metal-semiconductor transitions.\nI. INTRODUCTION\nIn spintronics, it is still a challenge in experiments to\nrealize room temperature ferromagnetic semiconductors.\nIn 2017, the successful synthesis of two-dimensional (2D)\nvan der Waals ferromagnetic semiconductors CrI 3[1] and\nCr2Ge2Te6[2] in experiments has attracted extensive at-\ntention to 2D ferromagnetic semiconductors. According\nto Mermin-Wagner theorem [3], the magnetic anisotropy\nis essential to produce long-range magnetic order in 2D\nsystems. Recently, more 2D ferromagnetic semiconduc-\ntors have been obtained in experiments, such as Cr 2S3\n[4], CrCl 3[5], CrBr 3[6], CrSiTe 3[7], CrSBr [8], where\nthe Curie temperatures TCare far below room tempera-\nture. On the other hand, the 2D van der Waals ferromag-\nnetic metals with high TChave been obtained in recent\nexperiments. For example, TC= 140 K in CrTe [9], 300\nK in CrTe 2[10], 344 K in Cr 3Te6[11], 160 K in Cr 3Te4\n[12], 280 K in CrSe [13], 300 K in Fe 3GeTe 2[14, 15], 270\nK in Fe 4GeTe 2[16], 229 K in Fe 5GeTe 2[17], 380 K in\nFe3GaTe 2[18], 300 K in MnSe 2[19], etc.\nRecent studies have shown that the physical properties\nof 2D materials are sensitive to external regulations, such\nas electric filed [14, 20–23], doping [24–28], surface func-\ntionalization [29, 30], intercalation [31, 32], heterostruc-\nture [33–48], strain [34, 49–53], etc. Among them, strain\nengineering is an effective technique to change the lat-\ntice and electronic structures and thus to control various\n∗gsu@ucas.ac.cn\n†gubo@ucas.ac.cnproperties of 2D materials. In contrast to bulk materi-\nals, 2D materials have stronger deformation capacity and\nthus can withstand greater elastic strain without frac-\nture, which shows great advantages in strain engineer-\ning. For example, monolayer MoS 2can sustain strains as\nlarge as 11% [54], monolayer FeSe can sustain strains up\nto 6% [55, 56], and single-layer graphene can even with-\nstand 25% elastic strain [57]. It has been observed that in\nfew-layer black phosphorus that biaxial tensile strain can\nincreases the band gap, while biaxial compressive strain\ncan decrease the band gap [49]. Under uniaxial strain up\nto 1.7%, a band gap reduction of 0.3 eV was observed in\nmonolayer MoS 2[50].\nIn addition, heterostructures can have a significant\nimpact on the magnetism of magnetic materials [38–\n40, 45–47]. Heterostructures also have a great impact\non the band structure. Especially, due to the effect\nof interlayer interaction and strain, metal-semiconductor\ntransition happens in some semimetal/semiconductor\nheterostructures, and a band gap Egcan be opened,\nsuch as Eg= 0.44 eV in silicene/GaP [41], 0.22 eV in\nsilicene/Ga 2SeS [42], 0.67 eV in CSe/BP [58], 13 meV in\ngraphene/MoSi 2N4[43] and 17 meV in Ge/SMoSe [44].\nAs a room temperature ferromagnetic material, MnSe 2\nhas attracted a lot of attentions due to its interest-\ning properties [36, 37, 59–66]. The structure of MnSe 2\nis shown in Fig. 1(a) with space group of P 3m1\n(164). The band structure from density functional the-\nory (DFT) with Perdew-Burke-Ernzerhof (PBE) pseu-\ndopotential shows semimetal properties [62–65]. Because\nPBE pseudopotential always underestimated the band\ngap, Heyd-Scuseria-Ernzerhof (HSE) hybrid functional\napproach is believed to give a better description of thearXiv:2312.15259v1  [cond-mat.mtrl-sci]  23 Dec 20232\nband structures [67]. By the DFT with HSE, MnSe 2was\ncalculated to be a semimetal [45, 63] or semiconductor\nwith small band gap of 0.01 eV [65]. Experimental results\nsuggest that MnSe 2at Bi 2Se3substrate has a small band\ngap [60]. On the other hand, the electronic properties of\nmonolayer MnSe 2are still unclear. It was reported that\ntensile strain could enhance the ferromagnetic properties\nof monolayer MnSe 2and weaken the metallicity [63, 65].\nIt has been discussed that TCcan be enhanced by tensile\nstrain in some 2D ferromagnetic semiconductors [38, 68–\n70] The evolution of band structures of monolayer MnSe 2\nwas calculated by DFT with PBE, giving no band gap\nwith tensile strain up to 8% [65]. Is it possible to open a\nband gap in these room temperature ferromagnetic met-\nals by some methods, and obtain room temperature fer-\nromagnetic semiconductors?\nIn this paper, we propose a way to obtain room\ntemperature ferromagnetic semiconductor by metal-\nsemiconductor transition in monolayer MnSe 2and\nMnSe 2-based heterostructures. The DFT calculation\nwith HSE hybrid functional shows that monolayer MnSe 2\nis room temperature ferromagnetic semimetal. A 2.2%\ntensile strain could open a small band gap in monolayer\nMnSe 2. An in-plane to out-of-plane magnetic anisotropy\ntransition happens with 1.7% tensile strain, and TCin-\ncreases with tensile strain. In addition, the heterostruc-\ntures MnSe 2/X with X= GaP [71–73], GaSe [74], SiH\n[75], and Al 2Se3[40] were studied. The room tempera-\nture ferromagnetic semiconductors are also obtained in\nthese heterostructures, where metal-semiconductor tran-\nsition occurs due to tensile stain by interfaces. Large\nmagneto-optical Kerr effect is found in monolayer MnSe 2\nwith a percent tensile strain and MnSe 2-based het-\nerostructures. Our theoretical results propose a way to\nobtain room temperature ferromagnetic semiconductors\nby metal-semiconductor transition in monolayer MnSe 2\nby applying tensile strain or building heterostructures.\nII. METHOD\nAll calculations were based on the DFT as imple-\nmented in the Vienna ab initio simulation package\n(VASP) [76]. The exchange-correlation potential is de-\nscribed by the PBE form with the generalized gradient\napproximation (GGA) [77]. The electron-ion potential\nis described by the projector-augmented wave (PAW)\nmethod [78]. We carried out the calculation of PBE + U\nwith U = 4 eV for 3d electrons in Mn [62–65]. The band\nstructures were calculated in HSE06 hybrid functional\n[67]. The plane-wave cutoff energy is set to be 650 eV.\nThe 9×9×1 Γ center K-point was used for the Brillouin\nzone (BZ) sampling. To obtain accurate results of mag-\nnetic anisotropy energy (MAE), K-points were chosen as\nΓ-centered 25 ×25×1. The density of states (DOS) were\nobtained from HSE and K-points of Γ-centered 18 ×18×1.\nThe structures of all materials were fully relaxed, where\nthe convergence precision of energy and force was 10−6eV and 10−2eV/˚A, respectively. The van der Waals\neffect is included with DFT-D3 method [79]. The Wan-\nnier90 code was used to construct a tight-binding Hamil-\ntonian [80, 81]. The WannierTools code was used to ob-\ntain a tight-binding band structure from tight-binding\nHamiltonian [82]. The Heisenberg-type Monte Carlo sim-\nulation was performed on a 50 ×50×1 lattice with 2500\nmagnetic points for MnSe 2. 1×105steps were carried for\neach temperature, and the last one-thirds steps were used\nto calculate the temperature-dependent physical quanti-\nties.\nIII. RESULTS AND DISCUSSION\nA. Monolayer MnSe 2\nThe crystal structure of monolayer MnSe 2is shown in\nFig. 1(a), which shows a triangle lattice. The calculated\nin-plane lattice constant is a0= 3.658 ˚A, in agreement\nwith previous calculations [62–65]. The band structure of\nmonolayer MnSe 2with HSE hybrid functional is shown\nin Fig. 1(b). The lowest band above Fermi level and the\nhighest band below Fermi level slightly overlap, showing\nthe semimetal behavior. The bands near Fermi level con-\ntain only one component of spins, showing the half-metal\nbehavior. In addition, the Monte Carlo results of magne-\ntization and susceptibility as a function of temperature\nfor monolayer MnSe 2is shown in Fig. 1(c), giving a high\nTC= 354 K.\nFIG. 1. (a) Top and side views of crystal structure of mono-\nlayer MnSe 2. (b) Spin polarized band structure of MnSe 2,\nobtained by the DFT calculation with HSE hybrid functional.\n(c) Magnetization and susceptibility of MnSe 2as a function of\ntemperature, obtained by the Monte Carlo simulation based\non a 2D Heisenberg model. The Curie temperature of mono-\nlayer MnSe 2is calculated as TC= 354 K, being consistent\nwith the experiment [19].3\nB. Monolayer MnSe 2with strain\nFIG. 2. For monolayer MnSe 2, tensile strain dependence of\n(a) band gap, (b) Curie temperature TC, and (c) magnetic\nanisotropy energy (MAE), obtained by the DFT and Monte\nCarlo calculations. Numerical results of four MnSe 2-based\nheterostructures are also included, where strain comes from\nthe interface of heterostructures. Experimental TCand MAE\nof heterostructure MnSe 2/GaSe [19] are included for compar-\nison.\nTo study the effect of strain on electronic properties,\nwe applied biaxial strain. The strain ratio is defined as\nϵ= (a−a0)/a0, where aanda0are the in-plane lat-\ntice constants with and without strain, respectively. The\nvariation of band gap of monolayer MnSe 2with tensile\nstrain calculated by DFT with HSE is shown in Fig. 2\n(a). A metal-semiconductor transition happens with a\ntensile strain of 2.2%. Fig. 2 (b) shows TCof monolayer\nMnSe 2as a function with tensile strain. TCincrease with\ntensile strain, in agreement with previous report [38, 68–\n70]. The MAE is defined as ( E⊥−E∥)/NMn, where E ⊥\nand E ∥are energies of MnSe 2with out-of-plane and in-\nplane magnetic polarization, respectively. NMnis the\nnumber of Mn atoms in a unit cell. For monolayer MnSe 2\nwithout strain, as shown in Fig. 2(c), the in-plane MAE\nof 0.75 meV/Mn is obtained, in agreement with the pre-\nvious report [63]. By applying tensile strain, an in-plane\nto out-plane MAE transition is obtained at 1.7% tensile\nstrain. Thus, monolayer MnSe 2will become a room tem-perature ferromagnetic semiconductor with out-of-plane\nMAE by applying tensile strains above 2.2%.\nFIG. 3. Metal-semiconductor transition of monolayer MnSe 2\ndue to strain. DFT results of partial density of state (PDOS)\nof (a) d orbitals of Mn and (b) p orbitals of Se without stain.\nPDOS of (c) d orbitals of Mn and (d) p orbitals of Se with\n4% tensile stain.\nIn order to analyze the metal-semiconductor transi-\ntion of monolayer MnSe 2under strain, the partial den-\nsity of states (PDOS) were calculated, as shown in Fig.\n3. PDOS of d orbitals of Mn and p orbitals of Se with-\nout strain is shown in Figs. 3(a) and 3(b), respectively.\nPDOS of Mn and Se with 4% tensile strain is shown in\nFigs. 3(c) and 3(d), respectively, with a small gap. The\ntensile stains raise the energy of d orbitals of Mn atoms\nand p orbitals of Se atoms near the Fermi level, making\nthe Fermi level lie in the energy gap of bonding and an-\ntibonding states of these p and d orbitals, and opening a\nsmall band gap.\nC. Heterostructures MnSe 2/X (X = GaP, GaSe,\nSiH, and Al 2Se3)\nConsidering the mismatch from substrate is an efficient\nway to provide strain for 2D material in experiments [83],\nwe constructed MnSe 2-based heterostructures with 2D\nsemiconductors to provide strain. GaP [71, 72], GaSe\n[74], SiH [75], and Al 2Se3[40] are nonmagnetic semicon-\nductors. The calculated results with PBE at monolayer\nlimit give band gaps of 1.22, 1.80, 2.18 and 1.69 eV, re-\nspectively, and lattice constants of 3.916, 3.814, 3.888 and\n3.788 ˚A, respectively. According to the lattice mismatch\nδ= 2×(a1−a2)/(a1+a2)×100%, the lattice mismatch\nof GaP, GaSe, SiH, and Al 2Se3with MnSe 2are 6.8%,\n4.2%, 6.1%, and 3.5%, respectively. The detailed results\nare given in Supplemental Material [84].\nWe consider different stacking models of heterostruc-\ntures, and the detailed data are given in Supplemental\nMaterial [84]. As shown in Fig. 4, all heterostructures4\nFIG. 4. Structure and spin polarized band structure of MnSe 2-based heterostructures, (a, e) MnSe 2/GaP, (b, f) MnSe 2/GaSe,\n(c, g) MnSe 2/SiH, and (d, h) MnSe 2/Al 2Se3. The band structures are obtained by the DFT calculation with HSE hybrid\nfunctional.\nare ferromagnetic semiconductors. The optimised lat-\ntice constants of MnSe 2/X with X = GaP, GaSe, SiH,\nand Al 2Se3are about 3.83, 3.76, 3.81, and 3.75 ˚A, re-\nspectively, with an effective tensile strain of 4.7%, 2.8%,\n4.2%, and 2.5%, respectively. The binding energy Ebfor\nMnSe 2/X is defined as Eb= (EMnSe 2/X−EMnSe 2−\nEX)/Ntot, where EMnSe 2/X,EMnSe 2, and EXrepresent\nthe total energies of heterostructure MnSe 2/X, mono-\nlayer MnSe 2, and monolayer X, respectively, and Ntot\nis the total atom number in a unitcell. The calculated\nresults are -0.24, -0.17, -0.10, and -0.20 eV/atom for X\n= GaP, GaSe, SiH, and Al 2Se3, respectively, indicat-\ning their stability. TCof heterostructures were obtained\nthrough DFT calculations and Monte Carlo simulation.\nAs shown in Fig. 2, the calculation results predict that\nMnSe 2/X with X = GaP, GaSe, SiH, and Al 2Se3are\nroom temperature magnetic semiconductors with out-\nof-plane MAE. It is noted that for the heterostructure\nMnSe 2/GaSe, Tc above room temperature and out-of-\nplane magnetization were observed in the experiment\n[19]. Thus, our calculation results in Fig. 2 are consistent\nwith the experiment of MnSe 2/GaSe [19]. The detailed\ncalculation results are given in Supplemental Material\n[84].\nHeterostructures not only provide effective strain, but\nalso induce interlayer interaction. The calculated re-\nsults of the most stable stacks of heterostructures is\nshown in Fig. 2. The calculated results of heterostruc-\ntures are different with those of monolayer MnSe 2with\nthe same tensile strain. For example, heterostructure\nMnSe 2/GaSe has a band gap of 0.18 eV, while mono-\nlayer MnSe 2with same lattice constants has a band gapof 0.08 eV. Heterostructure MnSe 2/Al 2Se3give a MAE\nof -1.11 meV/Mn, while monolayer MnSe 2with same lat-\ntice constants give a MAE of -0.36 meV/Mn. Therefore,\nthe interlayer interaction may play an important role in\nproperties of heterostructures.\nD. MOKE in monolayer MnSe 2and\nheterostructures MnSe 2/X\nWe investigated the magneto-optical Kerr effect for\nMnSe 2based structures. The Kerr rotation angle is given\nby:\nθK(ω) =Reεxy\n(1−εxx)√εxx, (1)\nwhere εxxandεxyare the diagonal and off-diagonal\ncomponents of the dielectric tensor ε, and ωis the fre-\nquency of incident light. The dielectric tensor εcan\nbe obtained by the optical conductivity tensor σas\nε(ω) =4πi\nωσ(ω) +I, where I is the unit tensor. The\ncalculated Kerr rotation angle as a function of photon\nenergy for monolayer MnSe 2with strain is shown in Fig.\n5 (a), the result for heterostructures MnSe 2/X with X =\nGaP, GaSe, SiH and Al 2Se3is shown in Fig. 5 (b). Ac-\ncording to our calculation results, there exist large Kerr\nrotation angles with out-of-plane magnetization, such as\nmonolayer MnSe 2with 2% and 4% tensile strain. On the\ncontrary, Kerr rotation angles with in-plane magnetiza-\ntion are small, such as monolayer MnSe 2without strain.\nIn addition, heterostructures MnSe 2/X with out-of-plane\nmagnetization also have a large Kerr rotation angle. The5\nFIG. 5. DFT results of Kerr rotation angle for MnSe 2with\n0%, 2%, 4% tensile strain and heterostructures MnSe 2/X (X\n= SiH, Al 2Se3, GaSe, and GaP). Experimental and numerical\nresults of Fe [85] are also included for comparison.\nexperimental result for Fe [85] and our DFT result for Fe\nbulk are also included for comparison. The Kerr rotation\nangles for stretched MnSe 2are about 4 times bigger than\nthat of bcc Fe. Detailed results of Kerr rotation angles\nare given in Supplemental Material [84].IV. CONCLUSION\nBased on the DFT calculations, we studied the prop-\nerties of monolayer MnSe 2with strain and MnSe 2-\nbased heterostructures. For monolayer MnSe 2, a metal-\nsemiconductor transition happens at 2.2% tensile strain,\nand an in-plane to out-of-plane MAE transition happens\nat 1.7% tensile strain. In addition, TCof monolayer\nMnSe 2increases with tensile strain. The heterostruc-\ntures MnSe 2/X with X = GaP, GaSe, SiH, and Al 2Se3\nwere studied, and the DFT calculation results show that\nthey are room temperature ferromagnetic semiconduc-\ntors with out-of-plane MAE. Large magneto-optical Kerr\neffect was found in monolayer MnSe 2with a few percent\ntensile strain and MnSe 2-based heterostructures. Our\nresults propose a way to obtain room temperature fer-\nromagnetic semiconductors by metal-semicondutor tran-\nsition in monolayer MnSe 2by applying a few percent\ntensile stain or building heterostructures.\nV. ACKNOWLEDGEMENTS\nThis work is supported by National Key R&D Program\nof China (Grant No. 2022YFA1405100), National Natu-\nral Science Foundation of China (Grant No. 12074378),\nChinese Academy of Sciences (Grants No. YSBR-030,\nNo. JZHKYPT-2021-08, No. XDB33000000), Beijing\nMunicipal Science and Technology Commission (Grant\nNo. Z191100007219013).\n[1] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. 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Klemm\nDepartment of Physics, University of Central Florida, Orla ndo, Florida 32816, USA\n(Dated: December 5, 2018)\nIn superconducting ferromagnets for which the Curie temper atureTmexceeds the superconduct-\ning transition temperature Tc, it was suggested that ferromagnetic spin ﬂuctuations coul d lead to\nsuperconductivity with p-wave spin triplet Cooper pairing. Using the Stoner model of itinerant fer-\nromagnetism, we study the feedback eﬀect of the p-wave superconductivity on the ferromagnetism.\nBelowTc, the ferromagnetism is enhanced by the p-wave superconductivity. At zero temperature,\nthe critical Stoner value for itinerant ferromagnetism is r educed by the strength of the p-wave pair-\ning potential, and the magnetization increases correspond ingly. More important, our results suggest\nthat once Stoner ferromagnetism is established, Tmis unlikely to ever be below Tc. For strong and\nweak ferromagnetism, three and two peaks in the temperature dependence of the speciﬁc heat are\nrespectively predicted, the upper peak in the latter case co rresponding to a ﬁrst-order transition.\nPACS numbers: 71.10.-w, 71.27.+a, 75.10.LP\nDue to the strong interplay between conventional su-\nperconducting (SC) and ferromagnetic (FM) states, the\nexplorationoftheir possiblecoexistencein the samecrys-\ntal might have seemed fruitless, but has nevertheless at-\ntracted a great deal of interest recently. This possible\ncoexistence was ﬁrst proposed by Ginzburg more than 50\nyears ago1. Several years later, Larkin and Ovchinnikov2\nand Fulde and Ferrell3independently developed a mi-\ncroscopic theory of this coexistence in the presence of\na strong magnetic ﬁeld, based upon a spatially inho-\nmogeneous SC order parameter, presently referred to as\nthe FFLO state. Meanwhile, Berk and Schrieﬀer sug-\ngested that conventional s-wave superconductivity in the\nparamagnetic phase above the Curie temperature Tmis\nsuppressed by critical ferromagnetic ﬂuctuations near to\nTm4. However, more recent calculations showed that\nconventional s-wave superconductivity can form in the\nweakly FM regime close to a quantum phase transition5.\nIn addition, Fay and Appel predicted that p-wave su-\nperconductivity could arise in itinerant ferromagnets6.\nTheir pioneering work indicated that longitudinal fer-\nromagnetic spin ﬂuctuations could result in a p-wave\n“equal-spin-pairing” SC state within the FM phase.\nExperimentally, a major development occurred with\nthe observation by Saxena et al.that UGe 2, nominally\nan itinerant FM compound, undergoes an SC transition\nat lowTcvalues under high pressure7. An SC state was\nalso found in other itinerant ferromagnets such as ZrZn 2\nand URhGe8,9. In each case, the regime of the SC phase\nappearscompletelywithinthatoftheFMphase, suggest-\ning a cooperative eﬀect between the SC and FM states.\nThese experimental achievements have stimulated re-\nnewed theoretical interest in the subject. Recently, a\nlarge eﬀort has been devoted to the understanding of the\nunderlying physics of the coexisting SC and FM states,\nwith a focus upon the SC pairing mechanism and theorbital symmetry of the SC order parameter. Although\nearlier works by Suhl and Abrikosov suggested that an\ns-wave pairing interaction between conduction electrons\ncould be mediated by ferromagnetically-ordered local-\nized spins, such as by impurities10,11, recent studies of\nthese SC ferromagnets7,8,9have assumed that the itin-\nerant electrons involved in both the FM and SC states\nare within the same band12,13,14,15,16,17. Some of these\nstudies assumed conventional s-wave pairing. For exam-\nple, Karchev et al.studied an itinerant electron model\nin which the same electrons are responsible for both the\nFM and SC states12. In that study, the Cooper pairs\nwere assumed to be in a spin-singlet state, and the ferro-\nmagnetism was described within the Stoner model. How-\never, the resulting SC ferromagnetic state was shown to\nbe energetically unfavorable when compared to the con-\nventional, nonmagnetic SC state13. A possible exception\nto this incompatibility could occur if the magnetic in-\nstability were to arise from a dynamic spin exchange in-\nteraction, as discussed by Cuoco et al.14. On the other\nhand, a number of other workers avoided the likely in-\ncompatibility of the SC and FM states by assuming a\nspin-triplet SC order parameter with p-waveorbital sym-\nmetry, for simplicity15,16,17. Kirkpatrick et al.indicated\nthat ap-wave SC state meditated by ferromagnetic spin\nﬂuctuations is more likely to coexist within the Heisen-\nberg FM phase regime than within the paramagnetic\nphase regime15. Machida and Ohmi studied the prop-\nerties of a p-wave SC ferromagnet phenomenologically16.\nMore recently, a microscopic model of the coexistence of\na nonunitary spin-triplet SC state with a weakly itin-\nerant FM state was developed by Nevidomskyy17. The\npresent nature of the SC coexistent with the FM state\nin these ferromagnetic superconductors is still somewhat\ncontroversial, although increasingly, additional experi-\nments on the U-based materials have provided increasing2\nsupport for a spin-triplet state rather than a spin-singlet\none18,19,20,21.\nMost theoretical studies have focused primarily on the\neﬀect of the established ferromagnetism upon the na-\nture of the coexistent superconductivity, as summarized\nabove. However, to fully understand the interplay be-\ntween the SC and FM states when they coexist, one\nshould also study the feedback eﬀect of the superconduc-\ntivity upon the ferromagnetism itself, as has been done\nin only one study to date17.\nHere we study explicitly the eﬀects of the p-wave pair-\ning on the FM ordering, using the Stoner model of itin-\nerant ferromagnetism as the starting point. We cal-\nculate the critical Stoner parameter Uc, the magneti-\nzationm, and the two parallel-spin p-wave gap func-\ntion magnitudes, ∆ ±, respectively, as functions of the\npair-interaction strength V. We also discuss ﬁnite-\ntemperature properties, including the T-dependencies of\nthese order parameters and the speciﬁc heat C(T).\nWe take the Hamiltonian for the ferromagnetic super-\nconductor to have the form\nHFM+SC=/summationdisplay\nk,σ(ǫk−µ−σM)c†\nkσckσ\n+1\n2V/summationdisplay\nk,k′\nσ,σ′VSC(k,k′)c†\nk,σc†\n−k,σ′c−k′σ′ck′σ,(1)\nwhereσ=±represent the single-particle spin states,\nand the single-quasiparticle part of Hcomprises the\nStoner model for itinerant electrons, where ǫkis the non-\nmagnetic part of the quasiparticle dispersion, µis the\nchemical potential and M=U(/an}bracketle{tn+/an}bracketri}ht − /an}bracketle{tn−/an}bracketri}ht)/2 is the\nmagnetic molecular-ﬁeld with Uthe Stoner exchange in-\nteraction, and Vis the sample volume. The pairing po-\ntential is taken to have the p-wave form22,VSC(k,k′) =\n−Vˆk·ˆk′. In weak coupling theory, Vis non-zero and\nassumed to be constant only within the narrow energy\nregion|ǫ−ǫF| ≤ωcnear to the Fermi energy ǫF, where\nωcis the energy cut-oﬀ.\nBecause of the pair-breaking eﬀects of the strong\nexchange ﬁeld in ferromagnets, we assume that only\nparallel-spin Cooper pairs can survive. Thus we set the\np-wave antiparallel-spin gap function ∆ 0= 0 and retain\nthe two gap functions with parallel-spin states mS=±1,\n∆±1. The SC order parameter is assumed to have the\nfollowing p-wave symmetry22, ∆±1(k) = (ˆkx+iˆky)∆±.\nThe Hamiltonian is treated via the Green function\nmethod within the mean-ﬁeld theory framework. In ad-\ndition to the normal Green function Gσ(k,τ−τ′) =\n−/an}bracketle{tTτckσ(τ)c†\nkσ(τ′)/an}bracketri}ht, the anomalous Green function de-\nscribing the pairing of electrons should be introduced,\nFσ(k,τ−τ′) =/an}bracketle{tTτckσ(τ)c−kσ(τ′)/an}bracketri}ht. Using the standard\nequation of motion approach, the Green functions arederived to be\nG±(k,ipn) =−(ipn+ǫk∓M)\np2n+(ǫk∓M)2+|∆±1(k)|2,\nF±(k,ipn) =∆±1\np2n+(ǫk∓M)2+|∆±1(k)|2,(2)\nwhere the pnare the Matsubara frequencies, and the FM\nand SC order parameters are respectively deﬁned as\nM=U\n2V/summationdisplay\nk(/an}bracketle{tnk+/an}bracketri}ht−/an}bracketle{tnk−/an}bracketri}ht),\n∆±1(k) =−1\nV/summationdisplay\nk′VSC(k,k′)F±(k′,τ= 0).(3)\nAll of the order parameters can be calculated using the\nabove Green functions. They are found to satisfy\nM=U\n2V/summationdisplay\nk/braceleftBigg\nǫ↑\nk[1−2f(E−)]\n2E−(k)−ǫ↓\nk[1−2f(E+)]\n2E+(k)/bracerightBigg\n,\n(4)\n∆±1(k) =−1\nV/summationdisplay\nk′VSC(k,k′)1−2f[E±(k′)]\n2E±(k′)∆±1(k′),\n(5)\nwhereǫ↑,↓\nk=ǫk−µ±M,E±(k) =/radicalBig\n(ǫ↓,↑\nk)2+|∆±1(k)|2,\nandf(E) is the Fermi function. The chemical poten-\ntialµis determined from the equation for the number of\nelectrons per unit volume, or particle density,\nn=1\nV/summationdisplay\nk/braceleftBigg\n1−ǫ↑\nk[1−2f(E−)]\n2E−(k)−ǫ↓\nk[1−2f(E+)]\n2E+(k)/bracerightBigg\n,\n(6)\nwhich is equal to unity at half ﬁlling.\nEquations(4), (5)and(6)with n= 1comprisetheself-\nconsistent equations for the ferromagnetic superconduct-\ning system. We solve the equations for the simple case of\na spherical Fermi surface at half ﬁlling. It is convenient\nto solve these equations by converting the summations\noverk-space to continuum integrals over energy,\nM=U\n32π2/integraldisplay∞\n0dε/integraldisplayπ\n0dθsinθ√\nε\n×\n\nε↑tanh[E−\n2T]\nE−−ε↓tanh[E+\n2T]\nE+\n\n,(7)\n1 =V\n32π2/integraldisplayǫF±+ωc\nǫF±−ωcdε/integraldisplayπ\n0dθ\n×/braceleftbigg√\nε·sin3θ\nE±tanh[E±\n2T]/bracerightbigg\n,(8)\nn=1\n16π2/integraldisplay∞\n0dε/integraldisplayπ\n0dθsinθ√\nε\n×\n\n2−ε↑tanh[E−\n2T]\nE−−ε↓tanh[E+\n2T]\nE+\n\n,(9)3\n/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48 /s50/s53/s48/s48 /s51/s48/s48/s48 /s51/s53/s48/s48/s49/s48/s46/s53/s49/s49/s46/s48/s49/s49/s46/s53/s49/s50/s46/s48/s49/s50/s46/s53/s49/s51/s46/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48\n/s49/s50/s46/s55/s52/s49/s50/s46/s55/s53/s49/s50/s46/s55/s54\n/s32/s32/s85\n/s99\n/s86/s32/s85\n/s99\n/s86\n/s32/s32/s32/s85\n/s99/s32/s85\n/s99\nFIG. 1: The Stoner point Uc(V) as a function of the p-wave\ninteraction strength VatT= 0. Inset: Enlargement of the\nregion 0 ≤V≤500.\nwhereǫF±=µ±M,ε↓,↑=ε−ǫF±, andE±=/radicalBig\n(ε↓,↑)2+sin2θ|∆±|2. In the above equations, the unit\nof energy is rescaled by the factor/planckover2pi12n2/3\n2m∗. The di-\nmensionless interactions UandVare thus deﬁned by\nU=U(/planckover2pi12n2/3\n2m∗)−1andV=V(/planckover2pi12n2/3\n2m∗)−1, and the dimen-\nsionless energies ǫF±,ε,ωc,E±,∆±, andµare deﬁned\nanalogously. The dimensionless temperature is deﬁned\nbyT=kBT(/planckover2pi12n2/3\n2m∗)−1. We choose ωc= 0.01ǫF, where\nǫFis the dimensionless Fermi energy at M=T= 0.\nBy solving the equations self-consistently, we can in-\nvestigate the interplay between the magnetism and the\nsuperconductivity in the coexisting state. This issue was\ndiscussed previously based on a similar framework, with\nthe emphasis placed on the eﬀects on the SC pairing due\nto the critical spin ﬂuctuations in FM compounds17. The\npresent work focuses on the reciprocal action, i.e., the in-\nﬂuence of the SC on the FM.\nAccording to Stoner theory, a Fermi gas can exhibit\nferromagnetism only when the eﬀective FM exchange is\nlarger than the critical Stoner point. For a system de-\nscribed by Eq. (1), Urepresents the eﬀective exchange\ninteraction. In the absence of the p-wave SC interaction,\nV= 0, the dimensionless Stoner point Uc(0)≈12.76104.\nForV/ne}ationslash= 0, we calculate Uc(V). As shown in Fig. 1,\ntheT= 0 Stoner point Uc(V) decreases as Vincreases,\nwhich impliesthat the p-waveCooperpairingreducesthe\nbarriertotheonsetofthemagnetizationoftheFermigas.\nWe note that Vmight be very small in a real system, so\nthe enhancement eﬀect of the superconductivity on the\nferromagnetism may be very weak. The inset of Fig. 1\nshowsthedetailsof Uc(V)intheregionofsmall V, where\nthe decreasing tendency of Uc(V) with increasing Vstill\ncan be seen clearly.\nTo further demonstrate the inﬂuence of the SC on the\nFM, we discuss the magnetization m≡ /an}bracketle{tn+/an}bracketri}ht − /an}bracketle{tn−/an}bracketri}htas\na function of VatT= 0. Here we use m= 2M/U\ninstead of Mto eliminate the dependence of Ucupon\nV. As shown in Fig. 2, m(V) increases with Vfor each/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48\n/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48 /s50/s53/s48/s48 /s51/s48/s48/s48 /s51/s53/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32/s86\n/s32/s109\n/s32/s32/s109\n/s86\nFIG.2: Plots oftheelectronic magnetization m≡ /angbracketleftn+/angbracketright−/angbracketleftn−/angbracketright\nas a function of the p-wave interaction strength VatT= 0\nfor ﬁxed values of U. From larger to smaller mat ﬁxed V,\nU= 12.8(shortdotted), 12.77(dashed), 12.761(solid), 12.743\n(dotted), 12.7 (dash-dotted)and12.495 (shortdashed). In set:\nEnlargement of the region 0 ≤V≤500.\ngiven value of U. ForU >Uc(0),m(0) is ﬁnite, since the\nsystem is spontaneously magnetized, and m(V) increases\nmonotonically from m(0), eventually reaching unity at a\nﬁniteV≤2300. For U <Uc(0), however, m(V) = 0 for\nV <Vc(U), and then m(V)/ne}ationslash= 0increasessharplywith V\nforV≥Vc(U), eventually reaching unity at V >2300.\nThe critical value Vc(U) corresponds to the reduction in\nthe Stoner point Uc(V) at which the onset of the ferro-\nmagnetism is induced, as pictured in Fig. 1. This is a\nsecond way in which the p-wave superconductivity can\nenhance the ferromagnetism.\nA similar eﬀect was found in the ferromagnetic spin-1\nBose gas which exhibits two phase transitions, the FM\ntransition and Bose-Einstein condensation (BEC). The\nBEC temperature increases with FM couplings and, on\nthe other hand, the FM transition is signiﬁcantly en-\n/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48 /s50/s53/s48/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56\n/s86\n/s65\n/s32/s32\n/s86/s32\n/s32\nFIG. 3: Plots of ∆+(dashed) and ∆−(solid) as functions of\nVatU= 12.77 andT= 0.VAis the value of Vat which\n∆−has a maximum, and ∆−→0 atV→∼2300, the point\nat which m→1 in Fig. 2.4\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s40/s99/s41/s40/s98/s41/s32\n/s32/s45\n/s32\n/s43\n/s32/s77\n/s32/s77/s39\n/s32/s32\n/s40/s97/s41\n/s32/s32/s32\n/s32/s32\n/s84\nFIG. 4: Shown are plots of the order parameters M(dotted),\n∆+(dashed), and ∆−(solid) as functions of Tin the coex-\nistence state for V= 300.M′(dash-dotted) is the magnetic\norder parameter when V= 0. (a) U= 12.79>Uc(0) and\nT′\nm>Tc+. (b)U= 12.77>Uc(0) but 0 <T′\nm<Tc+. (c)\nU= 12.76<Uc(0) but U >Uc(V). The ferromagnetism\nis induced due to the p-wave pairing ( M/negationslash= 0) even though\nM′= 0.\nhanced due to the onset of the BEC23. Considering that\ntriplet Cooper pairs behave somewhat like spin-1 bosons,\na FM superconductor is analogous to a FM Bose gas.\nFigure 3 displays plots of the p-wave SC order param-\neters,∆±as functions of VatT= 0 and U= 12.77,\njust above the V= 0 Stoner point Uc(0). Although\nwith increasing V,∆+rises monotonically, ∆−initially\nrises, reaches a maximum at VA, and then decreases at\nan increasing rate until it vanishes discontinuously when\nm(V) = 1. For U= 12.77,m(V)>0 is shown by the\ndashed curve in Fig. 2, so that ∆+>∆−for allV. Since\nmalsogrowswith V, themeannumberofspin-downelec-\ntronsdecreaseswithincreasing V, vanishingwhen m→1\natV≈2300, at and beyond which ∆−→0.\nWe now discuss the ﬁnite temperature properties of\nthe system. We deﬁne M′to be the magnetic order pa-\nrameter when V= 0, for which ∆±= 0. The Tde-\npendencies of the order parameters ∆±,M, andM′are\nobtained numerically and shown for V= 300 and three\ndiﬀerent Ucases in Fig. 4. The order parameters be-\ncome non-vanishing below their respective dimensionless/s48/s46/s48/s48/s48 /s48/s46/s48/s48/s52 /s48/s46/s48/s48/s56 /s48/s46/s48/s49/s50 /s48/s46/s48/s49/s54/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52\n/s32\n/s32/s45\n/s32\n/s43/s32\n/s84/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54\n/s32/s77/s77\nFIG. 5: Plots of the order parameters M(dotted), ∆+\n(dashed), and ∆−(solid) as functions of Tin the coexistence\nstate for V= 20 and U= 12.761<Uc(0).\ntransition temperatures Tc±,Tm, andT′\nm. In each case,\nthe SC orderparameters ∆±increasemonotonicallywith\ndecreasing TbelowTc±, respectively. In the FM super-\nconductor, Tc−<Tc+and∆−(T)<∆+(T), as shown\nin Figs. 4(a), 4(b), and 4(c). In addition, M′(T) also\nincreases monotonically with decreasing Tfor the ferro-\nmagnet in the absence of any superconductivity, as de-\npicted in Figs. 4(a) and 4(b) for the respective cases\nU >Uc(0) andT′\nm>Tc+and 0<T′\nm<Tc+. How-\never, the T-dependence of Mis non-trivial when p-wave\nsuperconductivity is present. In the ﬁrst case pictured\nin Fig. 4(a), M(T) =M′(T) forT′\nm>Tc+, as in the\nabsence of superconductivity. However, M(T) exhibits\nan upward kink at Tc+below which ∆+/ne}ationslash= 0. Then, for\nTc−<T <Tc+,Mincreases sharply with decreasing\nT, and exhibits a downward kink at Tc−below which\n∆−/ne}ationslash= 0. Below Tc−,M(T) then decreases monotoni-\ncally with T. This case was discussed previously in a\nsimilar scenario24.\nThe case T′\nm<Tc±not previously discussed is more\ninteresting. Two examples of this case with V= 300\nare shown in Figs. 4(b) and 4(c). In Fig. 4(b), the\nmagnetization M′forV= 0 (and ∆±= 0) is so weak\nthat 0<T′\nm<Tc−, but a non-vanishing Venhances\nthe magnetization, M, causing the actual dimensionless\nCurie temperature Tmto equal Tc+, below which both\n∆+(T)andM(T)becomediscontinuouslynon-vanishing,\nsignalingaﬁrst-ordertransition. Theirbehaviorsfor T <\nTc+=Tmare then qualitatively similar to that shown\nin Fig. 4(a), with ∆−(T)/ne}ationslash= 0 for T <Tc−, causing\na downward kink in M(T) atTc−, below which M(T)\ndecreases monotonically with T. For the more extreme\ncase when U <Uc(0) and T′\nm= 0 but U >Uc(V)\ndepicted in Fig. 4(c), the behaviors of the three order\nparameters are very similar to that shown in Fig. 4(b).\nConsidering that Vis usually small in real systems, a\ncase with V= 20 is checked, as shown in Fig. 5 where\nUis taken to be 12 .761, slightly lower than Uc(0) but\nlarger than Uc(20)≈12.7608. Fig. 5 looks very similar5\nto Fig. 4(c).\nAlthough we did not investigate the limit V→0+,\nthe examples with V= 300 and V= 20 of the case\nT′\nm<Tc+pictured in Figs. 4(b), 4(c) and Fig. 5 suggest\nthat in FM superconductors, the actual Curie tempera-\ntureTmis unlikely to ever be lower than the upper SC\ntransition temperature Tc+, even if the FM order were\nextremely weak. In other words, these examples argue\nagainst the possibility of a FM Tregime inside the p-\nwave triplet SC regime, with an actual Tm<Tc+. Anal-\nogously, it was shown that the ferromagnetic transition\nneveroccursbelowthe Bose-Einsteincondensationin the\nFM spin-1 Bose gas23. Moreover, the present results are\nto some extent consistent with the observed phase dia-\ngrams of UGe 27and ZrZn 28, and with the theoretical\ndiscussion of Walker and Samokhin25, who argued that\nthe superconductivity only occurs within the FM region.\nIn addition, this scenario is consistent with de Haas van\nAlphen experiments under pressure on UGe 227.\nHowever, very recent experiments on UCoGe under\npressure were interpreted as potentially having such a\nFM regime inside the SC regime near to the FM quan-\ntum critical point28. However, the dcresistance and ac\nsusceptibility measurements of TmandTc+could not de-\ntermine if there were a FM region inside the SC one for\npressures just below their extrapolated quantum critical\npressure pc, allowing for a ﬁrst-order phase transition at\nthe point when Tm=Tc+, beyond which only a parallel-\nspin triplet state exists28. Further experiments are en-\ncouraged to determine if the FM and SC phase regimes\nwith 0< Tm< Tc+at ﬁxed pressure actually exist in\nUCoGe.\nAs suggested by the results for the temperature depen-\ndencies of the order parameters, the FM superconduct-\ning system shows multiple phase transitions, which can\nbe determined experimentally from measurements of the\nspeciﬁc heat. The speciﬁc heat at constant volume for\nour model can be calculated from\nC(T) =T∂S\n∂T,\nwhere the electronic contribution to the entropy Scan\nbe derived from\nS=−/summationdisplay\nk,σ=±{f(Eσ)lnf(Eσ)\n+[1−f(Eσ)]ln[1−f(Eσ)]}.\nThe speciﬁc heat was calculated previously based on\na model of s-wave superconductivity coexisting with\nferromagnetism26. Fors-wave superconductors, there is\nonly one SC transition temperature Tc, at which there is\na jump in the speciﬁc heat at the second ordertransition.\nHowever, the case of a p-wave superconductor coexisting\nwith ferromagnetism is more interesting. In Figs. 6(a)\nand 6(b), the results for the speciﬁc heat corresponding\nto the cases pictured in Figs. 4(a) and 4(b) for the order\nparameters are shown. For the case U >Ucpictured in/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s52/s53 /s48/s46/s53/s48 /s48/s46/s53/s53/s48/s46/s53/s49/s48/s48/s46/s53/s49/s50/s48/s46/s53/s49/s52/s48/s46/s53/s49/s54\n/s32/s32/s67/s47/s84\n/s40/s98/s41\n/s32/s32/s67/s47/s84\n/s84/s40/s97/s41/s32/s32/s67/s47/s84/s32\n/s84\nFIG.6: Plotsoftheelectronic speciﬁcheatatconstantvolu me\nas a function of Tfor (a) a case corresponding to Fig. 4(a).\nThe inset shows a transition from ferromagnetic to paramag-\nnetic phase occurs at the Curie point Tm≈0.5. The dotted\ncurve denotes the speciﬁc heat of the free electron gas; (b) a\ncase corresponding to Fig. 4(b). The Curie point Tm=Tc+,\nat which the transition is ﬁrst order.\nFigs. 4(a) and 6(a), there are three phase transitions at\nthe temperatures Tc−<Tc+<Tm. In Fig. 6(b), an ex-\nample of the case T′\nm<Tc+whenV= 0 pictured in Fig.\n4(b) is shown. In this case with V= 300, there is a ﬁrst-\norder phase transition at Tm=Tc+, and a second-order\nphase transition at Tc−.\nIn conclusion, it is shown that p-wave triplet Cooper\npairingcanenhancetheferromagnetisminsuperconduct-\ning ferromagnets. This enhancement is most prominent\nfor the magnetic exchange interaction Uvery near to the\nStoner point Uc(0), the critical value for the strength of\nthe exchange interaction required for the onset of ferro-\nmagnetism in the absence of the p-wave pairing interac-\ntionV. With ﬁnite V,Uc(V) is reduced and the fer-\nromagnetic order parameter increases in magnitude with\nincreasing V. The temperature dependencies of the mag-\nnetic and parallel-spinsuperconductingorderparameters\nand of the speciﬁc heat are calculated. The results show\nthat the Curie temperature is unlikely to ever be lower\nthan the upper SC transition temperature, in agreement\nwith pressure measurements on UGe 227. This feature\nalso may be relevant to recent experiments on UCoGe28.\nThe temperature dependence of the speciﬁc heat exhibits\ntwo peaks for weak ferromagnetism in the coexistence\nstate, with a ﬁrst-order transition at the combined fer-\nromagnetic and upper p-wave SC transition, and a lower\nsecond-order p-wave SC transition. For strong ferromag-6\nnetism, thespeciﬁcheatexhibitsthreesecond-ordertran-\nsitions. Our results support the possible coexistence of\np-wave superconductivity with a ferromagnetic state.\nThis work was supported by the Key Project of the\nChinese Ministry of Education (No. 109011), the FokYin-Tung Education Foundation, China (No. 101008),\nandtheNCETprogramofChineseMinistryofEducation\n(NCET-05-0098). X. J. would like to thank Jihong Qin\nfor helpful discussions.\n∗Electronic address: qgu@sas.ustb.edu.cn\n1V. L. Ginzburg, Zh. Eksp. Teor. Fiz. 31, 202 (1956)[Sov.\nPhys. JETP 4, 153 (1957)].\n2A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz.\n47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)].\n3P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964).\n4N. K. Berk and J. R. Schrieﬀer, Phys. Rev. Lett. 17, 433\n(1966).\n5K. B. Blagoev, J. R. Engelbrecht and K. S. Bedell, Phys.\nRev. Lett. 82, 133 (1999).\n6D. Fay and J. Appel, Phys. Rev. B 22, 3173 (1980).\n7S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R.\nK. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. 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Caianiello\", Universit\u0012 a degli Studi di Salerno, I-84084 Fisciano (SA), Italy\n3Universit\u0012 a degli Studi di Salerno, I-84084 Fisciano (SA), Italy\n4Intesa Sanpaolo - Direzione Rischi Finanziari e di Mercato, Piazza Paolo Ferrari 10, 20121 Milano\n(Dated: June 8, 2022)\nWe study transport phenomena through a ballistic ferromagnet-superconductor-ferromagnet\n(F/S/F) junction, comparing the case in which the ferromagnetic order in the two F layers is of the\nstandard Stoner type with the case where it is driven by a spin mass mismatch (SMM). It is shown\nthat the two mechanisms lead to a di\u000berent behavior in the charge and the spin conductances, espe-\ncially when compared to the corresponding non-superconducting ferromagnet-normal-ferromagnet\n(F/N/F) junctions. In particular, when the injected current is perpendicular to the barrier, for high\nbarrier transparency and large magnetization of the F layers, the large mass mismatch gives rise to\nan enhancement of both low-bias charge and spin conductances of the F/S/F junction, which is not\nobserved in the equal-mass case. When all the allowed injection directions are considered, the low\nbias enhancement of the charge conductance for SMM leads still holds for high barrier transparency\nand large magnetization of the F layers. However, in the case of non-transparent interfaces, spin\ntransport with SMM ferromagnets exhibits an opposite sign response with respect to the Stoner\ncase at high biases for all magnetization values, also manifesting a signi\fcant ampli\fcation induced\nby superconductivity at the gap edge. The above mentioned di\u000berences can be exploited to probe\nthe nature of the electronic mechanism underlying the establishment of the ferromagnetic order in\na given material.\nPACS numbers:\nI. INTRODUCTION\nHeterostructures made of ferromagnetic (F) and su-\nperconducting (S) alternating layers exhibit a variety of\npeculiar phenomena occurring at the nanoscale range of\nlayer thicknesses1{3. Thanks to the great progress in the\npreparation of high-quality hybrid F/S systems achieved\nin the last years, their properties have been deeply inves-\ntigated in view of the design of new devices susceptible of\nrelevant applications in the \feld of electronics and spin-\ntronics3,4. The interest in the above-mentioned systems\nis however not limited to this context. Under speci\fc\nconditions, their behavior may provide relevant informa-\ntion on the type of ferromagnetism characterizing the F\nlayer as well as on the symmetry of the order parameter\nin the superconductor, allowing to distinguish among the\nvarious possible unconventional pairing states5.\nMost of the relevant e\u000bects arising in F/S structures,\nsuch as the spatial oscillations of the electronic density\nof states or the nonmonotonic dependence of the critical\ntemperature on the ferromagnet layer thickness, are ulti-\nmately related to the damped oscillatory behavior char-\nacterizing the propagation of the Cooper pair wave func-\ntion from the superconductor to the ferromagnet6. This\nis in turn due to the formation of Cooper pairs with a \f-\nnite center-of-mass momentum originating from the pres-\nence of the exchange \feld7. As far as transport is con-\ncerned, it is well known that in a N/S junction, where\nN denotes a normal metal, for energies below the super-\nconducting gap \u0001, conduction is only possible via An-dreev re\rection (AR) processes by which two electrons\nwith opposite spins, one above and the other one below\nthe Fermi energy, incident from the non-superconducting\nlayer, are transferred in the superconductor as a Cooper\npair8. This leaves holes in the normal system which give\nrise to a parallel conduction channel, in this way lead-\ning to a doubling of the normal-state conductance for\neV < \u0001 (Vis the applied voltage)9. When the nor-\nmal metal is replaced by a conventional Stoner ferromag-\nnet, the relative shift of the density of states for spin-up\nand spin-down electrons caused by the exchange interac-\ntion (Fig. 1) can be large enough that an electron with,\nsay, spin-up incident on the interface \fnds no spin-down\npartner to form a Cooper pair able to move to the su-\nperconducting layer10. Andreev re\rections at the inter-\nface are then blocked so that only single-particle exci-\ntations contribute to the conductance. As a result, the\nhigher is the exchange interaction in the ferromagnet, the\nstronger is the conductance suppression in the subgap en-\nergy range11.\nIn F/S junctions, Andreev re\rections occur as local\nprocesses at the superconductor interface and produce\na Cooper pair in the superconductor. In multiterminal\nF/S hybrid structures where the thickness of the S layer\nis of the order of the BCS superconducting coherence\nlength of the material, they can also manifest themselves\nas a nonlocal process12, referred to as crossed Andreev\nre\rection (CAR). In such a case, the retrore\rection of\nthe hole from an AR process, resulting from an incident\nelectron at energies less than the superconducting gap at\none lead, occurs in the second ferromagnetic lead with\nTypeset by REVT EXarXiv:2206.03335v1  [cond-mat.supr-con]  7 Jun 20222\nthe same charge transfer as in a normal AR process of a\nCooper pair in the superconductor. For CAR to occur,\nelectrons of opposite spin must exist at each non super-\nconducting electrode (so as to form the pair in the su-\nperconductor). Therefore such processes are expected to\nbe strongly suppressed in F/S/F junctions with parallel\nalignment of the F polarizations, while they can survive\neven at strong polarization in the case of F layers having\nopposite magnetization alignment. The reverse process\nof the CAR produces spatially separated entangled states\nof electrons by splitting Cooper pairs from the super-\nconducting condensate into the two external leads13,14.\nIn trilayer structures, CARs usually compete with other\ntransport processes, such as the normal re\rections, the\nlocal Andreev re\rections, and the elastic cotunneling, i.e.\nthe quantum mechanical tunneling of electrons between\nthe external leads via an intermediate state in the super-\nconductor.\nGenerally speaking, F/S/F trilayer structures o\u000ber a\nrich playground to investigate the interplay between su-\nperconductivity and ferromagnetism. For instance, in\nsuch structures theory predicts15that for parallel align-\nment of the magnetizations in the two ferromagnetic lay-\ners, the superconducting critical temperature is lower\nthan in the case of antiparallel alignment, and can even\nbe zero. The system can thus behave as a spin valve\nwhere superconductivity can be switched on and o\u000b by\nreversing the \feld direction in one of the two magnetic\nlayers. However, it has been experimentally veri\fed16{19\nthat as soon as collinear, i.e. parallel or antiparallel,\ncon\fgurations are considered, the critical temperature\nshift is of the order of millikelvins, i.e. relatively small\ncompared to the theory predictions. The smallness of\nthis spin valve e\u000bect has been supposed to be due to a\nnon-optimal choice of the layer thickness and/or the se-\nlected layer material20. Actually, in the collinear case\nmost of the experiments have been performed on sys-\ntems not satisfying the condition \u0018S=dS\u00151,\u0018SanddS\nbeing the coherence length and the thickness of the su-\nperconducting layer, respectively, which has been theo-\nretically demonstrated15to be a prerequisite for a large\nspin valve e\u000bect. On the other hand, di\u000berences arise\nwhen non-collinear con\fgurations are considered. In this\ncase the dependence of Tcon the angle \u000bbetween the two\nmagnetization directions is non-monotonic with a mini-\nmum for\u000b=\u0019=26. A good agreement between theory\nand experiments is in this case obtained taking explic-\nitly into account the odd triplet correlations generated\nvia proximity e\u000bect by the non-collinearity of the mag-\nnetizations21. Interestingly, it has been recently demon-\nstrated that in F/S/F trilayers based on d-wave super-\nconductors, such as in particular YBa 2Cu3O7\u0000\u000e(YBCO)\nsandwiched between insulating layers of ferromagnetic\nPr0:8Ca0:2MnO 3, the critical temperature shift between\nparallel and antiparallel con\fgurations can approach the\nvery large value of 2 K, with oscillations driven by the\nYBCO thickness over a length scale that is two orders\nof magnitude larger than the superconducting coherencelength\u0018S22.\nFIG. 1: Density of states for spin-up and spin-down electrons\nin the Stoner (a) and in the spin mass mismatch (SMM) case\n(b).\nHowever, the F/S/F heterostructures so far consid-\nered have been theoretically investigated by assuming\nfor the ferromagnetic layers Stoner-like models where\nthe bands associated with the two possible electron spin\norientations have the same dispersion and are rigidly\nshifted in energy by the exchange interaction [Fig. 1(a)].\nGiven the complexity of the forms in which the phe-\nnomenon of ferromagnetism manifests itself in metals,\nit may be relevant to perform the analysis of the above\nsystems referring to scenarios di\u000berent from the Stoner\none. Among them, we will consider here a form of itin-\nerant ferromagnetism driven by a gain in kinetic energy\nstemming from a spin-dependent bandwidth renormal-\nization [Fig. 1(b)], or, equivalently, from an e\u000bective spin\nmass mismatch (SMM) between spin-up and spin-down\nelectrons23,24. Such kind of ferromagnetism can be theo-\nretically described through microscopic approaches based\non an extended Hubbard model, where the exchange and\nnearest-neighbor pair hopping terms are explicitly taken\ninto account. Indeed, when treated within mean-\feld\napproaches, these contributions, generally neglected in\nstudies based on the Hubbard model, lead to quasiparti-\ncle energies for the two spin species which are not sim-\nply split, as in the Stoner picture, but acquire di\u000berent\nbandwidths or, equivalently, di\u000berent e\u000bective masses25.\nFor suitably low temperatures and in speci\fc parameter\nregimes, this spin-dependent mass renormalization can\nlead to the establishment of a ferromagnetic order which\narises from a gain in kinetic energy rather than in poten-\ntial energy as in the usual Stoner scheme. This kind of\nferromagnetism has experimentally been shown to be at\nthe origin of the optical properties of the colossal mag-\nnetoresistance in manganites26, in some rare-earth hexa-\nborides27as well as in some magnetic semiconductors28.\nAs far as theory is concerned, this is predicted to substan-\ntially a\u000bect the coexistence of ferromagnetism and super-\nconductivity29as well as proximity30and transport31{33\nphenomena in F/S bilayers. It may also play a role in\nthe stabilization of the Fulde-Ferrell-Larkin-Ovchinnikov\nphase in heavy-fermion systems34. Compared to the\nStoner case, the interplay of this form of ferromagnetism\nwith superconductivity is expected to give rise to di\u000ber-3\nent features in the behavior of several physical quantities.\nIn particular, this issue has been investigated for F/S bi-\nlayer with anisotropic singlet superconductors, showing\nthat the di\u000berent response predicted for the two kinds\nof ferromagnets allows to discriminate among the possi-\nble time-reversal symmetry-breaking states established in\nthe superconducting layer32. In that context it was also\nshown that in a wide range of interface transparencies\na SMM ferromagnet may support spin currents signi\f-\ncantly larger than a standard Stoner one.\nIn this paper, the comparison between the role played\nby Stoner and SMM ferromagnets is performed referring\nto a clean F/S/F trilayer, in analogy with the study pre-\nsented in Refs.31,32for a bilayer structure. In particu-\nlar, by analyzing the behavior of the di\u000berential charge\nand spin conductances, we show that when the F/S/F\njunction is based on SMM ferromagnets, it behaves dif-\nferently from junctions with ordinary Stoner ferromag-\nnets, both in the transparent and in the tunnel limit.\nBy comparing the transport properties of the junction in\nthe superconducting regime (F/S/F) with those of the\nnon-superconducting case (F/N/F), we \fnd that for the\nSMM mechanism the interplay between superconductiv-\nity and ferromagnetism is not detrimental to charge and\nspin transport, as for the Stoner mechanism, but instead\nboth charge and spin conductances through the F/S/F\njunctions get enhanced with respect to the F/N/F case in\nthe bias region where superconductivity is most e\u000bective.\nMoreover, a distinctive feature of the SMM mechanism\nemerges in the spin transport at large bias, at interme-\ndiate and low barrier transparency: the sign of the spin\ncurrent is opposite to the sign of the lead magnetization,\nwhile in the Stoner case this sign di\u000berence is not seen.\nWe clarify the origin of this behavior, also showing that\nthe presence of superconductivity is able to amplify the\nmagnitude of the spin current at the gap edge both in\nthe Stoner and in the SMM case.\nSuch results are presented in the paper as follows. In\nSection II we formulate the microscopic model based on\nthe Bogoliubov-de Gennes (BdG) equations, written in\neach region of the junction. In this framework, we discuss\nhow to solve the scattering problem in order to derive\nthe probability coe\u000ecients associated with the relevant\nscattering processes. Then we explain how to use such\ncoe\u000ecients to calculate the charge and spin conductance\nthrough the junction. The obtained results are discussed\nin Section III, in connection to the behavior of the scat-\ntering coe\u000ecients, both for fully transparent interfaces\nand in the tunnel limit. Finally, Section IV is devoted to\nthe conclusions.\nAdditional details about the applied formal procedure\nare provided in the Appendices. In Appendix A we report\nthe expression of the wave functions for the injection pro-\ncesses which are not reported in the main text; Appendix\nB contains the derivation of the probability current con-\nservation; in Appendix C we derive the spin-dependent\ncharge conductance through the junction; Appendix D\nreports the behavior of the critical injection angles be-low which the di\u000berent scattering processes are allowed;\n\fnally, Appendix E shows the derivation of a symmetry\nproperty characterizing the scattering amplitudes in the\nSMM case for transparent barriers and perpendicular in-\njection.\nII. THE MODEL\nWe consider a planar symmetric trilayer junction in the\nclean limit, made up of a superconducting layer of thick-\nnessL, sandwiched between two identical semi-in\fnite\nitinerant ferromagnets, as schematically shown in Fig. 2.\nThe junction lies in the xz-plane; the superconducting\nlayer is connected to the two ferromagnetic electrodes by\nthin, insulating interfaces, located at the positions z= 0\nandz=L, respectively.\nFIG. 2: Schematic representation of the considered symmet-\nric planar ferromagnet/superconductor/ferromagnet (F/S/F)\njunction.Lis the thickness of the superconducting layer.\nThe two ferromagnetic subsystems are assumed identical and\nsemi-in\fnite.\nThe barrier potential at the two interfaces is modelled\nas\nV(r) =H\u000e(z) +H\u000e(z\u0000L); (1)\nwhereHdenotes the potential amplitude at each inter-\nface, and\u000e(z) is the Dirac delta function. We also as-\nsume a rigid pairing potential for the superconducting\nside, such that\n\u00010(r) = \u0001\u0002(z)\u0002(L\u0000z); (2)\nwhere \u0002(z) is the Heavyside step function and \u0001 0is the\nBCS bulk gap.\nMoreover, we assume that the e\u000bective mass of the\nsystem is\nm\u0003(r;\u001b) =m\u001b\u0002(\u0000z) +m\u001b\u0002(z\u0000L) +mS\u0002(z)\u0002(L\u0000z):\nHere,m\u001bis the spin-dependent mass of electrons in each\nferromagnetic layer, while mSis the carrier mass in the\nsuperconducting one. In order to analyze the e\u000bects on\nthe transport across the junction which derive from the\nasymmetric mass renormalization, in comparison with\nthose due to the conventional Stoner-induced ferromag-\nnetism, we assume an equal exchange \feld in the two\nferromagnetic sides of the junction:\nh(r) =U\u0012(\u0000z) +U\u0012(L\u0000z):4\nWe will illustrate in the following the solution of the\nquantum problem of the electron propagation from one\nside of the junction to the other, then showing how to\ndetermine the di\u000berential charge and spin conductances\nthrough the junction in the ballistic limit.\nA. Bogoliubov-de Gennes equations\nThe single-particle Hamiltonian for a given spin pro-\njection\u001b=\";#reads as\nH\u001b(r) =\u0000~2\n2m\u0003(r;\u001b)r2+V(r)\u0000\u0016(r)\u0000\u001a\u001bh(r) (3)\nwhere\u001a\"(#)= +1(\u00001) and we have de\fned the chemical\npotential\u0016(r) as\n\u0016(r) =EF\nF\u0012(\u0000z) +ES\nF\u0012(z)\u0012(L\u0000z) +EF\nF\u0012(z\u0000L);\n(4)\nwithEF\nF(ES\nF) being the Fermi energy in the ferromag-\nnetic (superconducting) side. We assume that there is no\nFermi energy mismatch between the three subsystems, so\nthat\nEF\u0011EF\nF=ES\nF: (5)\nIn the absence of spin-\rip scattering, the spin-dependent\nfour-component BdG equations for each subsystem can\nbe decoupled into two subsets of two-component equa-\ntions, one for the spin-up electron-like and spin-down\nhole-like quasiparticle wavefunctions ( u\";v#), and the\nother one for the corresponding quasi-particle wavefunc-\ntions having opposite spin projection ( u#;v\"). The BdG\nequations for each subset are then:\n\u0012\nH\u001b(r) \u0001 0(r)\n\u0001\u0003\n0(r)\u0000H\u0016\u001b(r)\u0013\n\t\u001b=\"\t\u001b; (6)\nwhere \u0016\u001b=\u0000\u001band \t\u001b\u0011(u\u001b;v\u0016\u001b) is the energy eigen-\nstate in the electron-hole space associated with the eigen-\nvalue\". The Hamiltonian invariance under translation\nalong thex-direction allows to factorize the part of the\neigenstates which corresponds to the electron motion\nin the direction parallel to the interfaces direction, i.e.\n\t\u001b(r) =eikk\u0001r \u001b(z), hence reducing the BdG problem\nto the solution of e\u000bective one-dimensional equations.\nThe solutions of the BdG equations for electrons ( e)\nand holes (h) propagating in each ferromagnetic side are\n e\n\u0006;\u001b(z) =\u0012\n1\n0\u0013\ne\u0006iqe\n\u001bzz(7)\n h\n\u0006;\u001b(z) =\u0012\n0\n1\u0013\ne\u0006iqh\n\u001bzz(8)\nwhereqe(h)\n\u001bzis the projection along the zdirection of theelectron (hole) momentum, whose total amplitude is\nqe\n\u001b=r\n2m\u001b\n~2(EF+\u001a\u001bU+\") (9)\nqh\n\u001b=r\n2m\u001b\n~2(EF+\u001a\u001bU\u0000\") (10)\nwith\u001a\"(#)= +1 (\u00001). The plus sign in Eq.(7) refers\nto electrons propagating from the left to the right side,\nwhile the minus sign indicates electrons moving in the\nopposite direction. Since holes have opposite group ve-\nlocity direction with respect to electrons, in Eq.(8) the\nplus sign refers to hole motion from right to left while the\nminus one refers to hole propagation from left to right.\nIn the superconducting side, solutions for electron-like\nand hole-like quasiparticles are given by\n e\n\u0006;S(z) =\u0012\nu0\nv0\u0013\ne\u0006ike\nzz(11)\n h\n\u0006;S(z) =\u0012\nv0\nu0\u0013\ne\u0006ikh\nzz; (12)\nwhereke(h)\nzare thezcomponents of the electron-like\n(hole-like) quasiparticle momenta having amplitudes\nke=r\n2mS\n~2\u0010\nEF+p\n\"2\u0000\u00012\u0011\n(13)\nkh=r\n2mS\n~2\u0010\nEF\u0000p\n\"2\u0000\u00012\u0011\n; (14)\nandu0andv0are the coherence factors expressed as\nu0=vuut1\n2 \n1 +p\n\"2\u0000\u00012\n\"!\n(15)\nv0=vuut1\n2 \n1\u0000p\n\"2\u0000\u00012\n\"!\n: (16)\nWe apply the quasiclassical Andreev approximation, as-\nsuming that the processes of interest in our analysis in-\nvolve quasiparticles which are close to the Fermi energy,\nsuch thatEF\u001d(\";\u0001). As a consequence, qe\n\u001b=qh\n\u001b\u0011\nq\u001b=q\n2m\u001b\n~2(EF+\u001a\u001bU). We keep the energy depen-\ndence in the superconducting momenta occurring in the\nexponents of the superconducting wave functions given\nby Eqs. (11) and (12), in such a way to catch the inter-\nference e\u000bects in the S region. Using the Fermi energy\ncondition (5) and the relation ES\nF=~2(kS\nF)2\n2mS, we can de-\n\fne the renormalized momentum amplitudes in the fer-\nromagnetic and in the superconducting layers as\n~q\u001b\u0011q\u001b\nkS\nF=srm\u001b\nm\u0016\u001b(1 +\u001a\u001bX) (17)\n~ke\u0011ke\nkS\nF=q\n1 +p\n\u00182\u0000\u000e2 (18)\n~kh\u0011kh\nkS\nF=q\n1\u0000p\n\u00182\u0000\u000e2; (19)5\nrespectively, having introduced the adimensional quanti-\ntiesX=U=EF,\u0018=\"=EF,\u000e= \u0001=EF, with the condi-\ntionm\"=mS=mS=m#.\nB. The scattering problem\nThe total wave function of the F/S/F trilayer is ob-\ntained as a linear combination of the solutions of the BdG\nequations for each individual region. Here we extend\nFIG. 3: Schematic representation of the scattering processes\nof a carrier ( p=e;hfor electrons and holes, respectively)\ninjected from the left ferromagnetic side with spin projection\n\u001bat an angle \u0012\u001bwith respect to the direction perpendicular\nto the interfaces. The processes are described in detail in the\ntext.\nto the F/S/F trilayer the Blonder-Tinkham-Klapwjik\n(BTK) scattering theory35originally formulated for a\nN/S bilayer junction and then extended to a F/S one,\nin the case of a Stoner ferromagnet6or a SMM one36.\nThis kind of approach has also been used to investigate\na F/S/F junction, but only in the case of ferromagnetic\norder of the conventional Stoner type37{39. Rather, here\nthe approach will also be applied to a trilayer with ferro-\nmagnetic layers of the SMM type.\nWhen a carrier with spin \u001band momentum amplitude\nq\u001bis injected from the left ferromagnetic side with an\ninjection angle \u0012\u001b, its propagation gives rise to eight pos-\nsible scattering processes at the interfaces, as shown in\nFig. 3. Within the left F layer they include ( i) Andreev\nre\rection, converting the incident electron (hole) with\nspin\u001binto a hole (electron) with opposite spin \u0016 \u001b, which\npropagates with a momentum amplitude q\u0016\u001balong a direc-\ntion forming an angle \u0012A\n\u0016\u001bwith the normal to the interface\n(ap\u0016\u001bin Fig. 3), and ( ii) normal re\rection as an electron\n(hole) having spin \u001b, momentum amplitude q\u001band mov-\ning along a direction forming an angle \u0000\u0012\u001bwith the nor-\nmal direction to the interface ( bp\u001bin Fig. 3). Then, in\nthe S layer, close to the \frst interface, there occur ( i) the\ntransmission of an electron in the superconducting side,\npropagating with momentum amplitude keat an angle\n\u0012S\ne\u001bwith respect to the normal to the interface ( \u000bp\u001bin\nFig. 3), and ( ii) the transmission of a hole, propagating\nwith momentum amplitude khand negative momentum\ncomponent along the z-direction, at an angle \u0012S\nh\u001bwith\nrespect to the normal to the interface ( \fp\u0016\u001bin Fig. 3). In\nproximity of the second barrier, the relevant processes are(i) the re\rection of an electron with momentum ampli-\ntudekeforming an angle \u0012S\ne\u001bwith the normal direction\nto the interface ( \rp\u001bin Fig. 3), and ( ii) the re\rection\nof a hole having momentum amplitude khand angle\u0012S\nh\u001b\nmeasured from the normal to the interface ( \u0011p\u0016\u001bin Fig. 3).\nFinally, in the right F lead there occur ( i) the transmis-\nsion of an electron with spin \u001b, momentum amplitude q\u001b\nalong a direction forming an angle \u0012T\ne\u001bwith the normal\nto the interface ( cp\u001bin Fig. 3), and ( ii) the transmission\nof a hole with spin \u0016 \u001b, momentum amplitude q\u0016\u001balong a\ndirection forming an angle \u0012T\nh\u0016\u001bwith the normal to the\ninterface ( dp\u0016\u001bin Fig. 3). Due to the symmetry of the\njunction, if quasiparticles are injected from the right F\nside, identical processes will occur, with reversed sign for\nall propagation velocities.\nThe scattering angles associated with the above listed\nprocesses depend on the injection angle \u0012\u001bof the incident\nparticles. They can be calculated by using the conserva-\ntion of the momentum components which are parallel to\nthe interfaces:\nq\u001bk\u0011q\u001bsin\u0012\u001b=q\u0016\u001bsin\u0012A\n\u0016\u001b=ke(h)sin\u0012S\ne(h)\u001b\n=q\u001bsin\u0012T\n\u001b=q\u0016\u001bsin\u0012T\n\u0016\u001b: (20)\nThe scattering processes happening due to the injection\nof a quasiparticle p=e;hfrom one of the two ferromag-\nnetic sides in general occur with di\u000berent probabilities,\ndepending on the excitation energy \", the superconduct-\ning energy gap \u0001, the polarization of the ferromagnetic\nleads, and the barrier strength H. Consequently, the\nscattering processes enter the total wave function expres-\nsion with some unknown amplitudes to be determined by\nimposing the matching conditions for the wave functions\nat the interfaces.\nTaking into consideration the above listed scattering\nprocesses, the wave function for an electron which is in-\njected from the left F side at energy \", angle\u0012\u001band with\nspin\u001bcan be written as\n F\ne\u001bL(z) =\u0012\n1\n0\u0013\neiq\u001bz cos\u0012\u001b+ae\u0016\u001b\u0012\n0\n1\u0013\neiq\u0016\u001bz cos\u0012A\n\u0016\u001b\n+be\u001b\u0012\n1\n0\u0013\ne\u0000iq\u001bz cos\u0012\u001b(21)\nin the left F side ( z<0), as\n S\ne\u001b(z) =\u000be\u001b\u0012\nu0\nv0\u0013\neikez cos\u0012S\n\u001be\n+\fe\u0016\u001b\u0012\nv0\nu0\u0013\ne\u0000ikhz cos\u0012S\n\u001bh\n+\re\u001b\u0012\nu0\nv0\u0013\ne\u0000ikez cos\u0012S\n\u001be\n+\u0011e\u0016\u001b\u0012\nv0\nu0\u0013\neikhz cos\u0012S\n\u001bh (22)6\nin the S side (0 <z<L ), and as\n F\ne\u001bR(z) = ce\u001b\u0012\n1\n0\u0013\neiq\u001bz cos\u0012T\n\u001b+\nde\u0016\u001b\u0012\n0\n1\u0013\ne\u0000iq\u0016\u001bz cos\u0012T\n\u0016\u001b (23)\nin the right F side ( z > L ). The corresponding wave\nfunctions for the injection of a hole from the left F lead\nare reported in Appendix A.\nThe coe\u000ecients corresponding to the scattering pro-\ncesses occurring at the two interfaces are determined by\nimposing the following boundary conditions: the continu-\nity conditions of the wave functions at the two interfaces,\n F\np\u001bL(0) = S\np\u001b(0)\n F\np\u001bR(L) = S\np\u001b(L); (24)\nand the discontinuity of the wave function \frst derivative\nwith respect to the zspatial coordinate at the interface\nlocations due to the local barrier potentials. Such con-\nditions are derived by integrating the BdG equations in\nEqs. (6) over the zvariable in a very narrow range around\neach barrier, and read:\nd\ndzuS\n\u001b\f\f\f\nz=0\u0000mS\nm\u001bd\ndzuF\n\u001bL\f\f\f\nz=0=ZuS\n\u001b(0)\nd\ndzvS\n\u0016\u001b\f\f\f\nz=0\u0000mS\nm\u0016\u001bd\ndzvF\n\u0016\u001bL\f\f\f\nz=0=ZvS\n\u0016\u001b(0)\nmS\nm\u001bd\ndzuF\n\u001bR\f\f\f\nz=L\u0000d\ndzuS\n\u001b\f\f\f\nz=L=ZuS\n\u001b(L)\nmS\nm\u0016\u001bd\ndzvF\n\u0016\u001bR\f\f\f\nz=L\u0000d\ndzvS\n\u0016\u001b\f\f\f\nz=L=ZvS\n\u0016\u001b(L):(25)\nHere (uS\n\u001b;vS\n\u0016\u001b) are the components of the superconducting\nwave function  S\ne\u001b(z), whose explicit expression is given\nin (22), whereas ( uF\n\u001b\u000b;vF\n\u0016\u001b\u000b) with\u000b=L;Rare the com-\nponents of the ferromagnetic wave functions  F\ne\u001bL(R)(z)\nin the left ( L) and right ( R) side, respectively. Their ex-\nplicit expressions have been provided in Eqs. (21),(23).\nFinally, we have de\fned Z=2mSH\n~2kS\nF.\nThe probability amplitudes associated with each scat-\ntering process are obtained from the solution of the sys-\ntem in Eqs. (25). This is done by using the conservation\nof the current probability, in the form derived in Ap-\npendix B. For a carrier p=e;hinjected from left with\nenergy\"=~\u0018\u0001 along a direction forming an angle \u0012\u001b\nwith the direction normal to the interface, we \fnd that\nthe probabilities Ap\u001bfor local Andreev re\rections, Bp\u001b\nfor the specular re\rection, Cp\u001bfor the transmission with\nthe same charge in the right F lead and Dp\u001bfor the trans-\nmission with opposite charge in the right F lead are givenby:\nAp\u001b=m\u001b\nm\u0016\u001bq\u0016\u001bcos\u0012A\n\u0016\u001b\nq\u001bcos\u0012\u001bjap\u0016\u001b\u0010\n~\u0018;\u0012\u001b\u0011\nj2; (26)\nBp\u001b=jbp\u001b\u0010\n~\u0018;\u0012\u001b\u0011\nj2; (27)\nCp\u001b=cos\u0012T\n\u001b\ncos\u0012\u001bjcp\u001b\u0010\n~\u0018;\u0012\u001b\u0011\nj2; (28)\nDp\u001b=m\u001b\nm\u0016\u001bq\u0016\u001bcos\u0012T\n\u0016\u001b\nq\u001bcos\u0012\u001bjdp\u0016\u001b\u0010\n~\u0018;\u0012\u001b\u0011\nj2: (29)\nIn the case of injection from the right side, the expres-\nsions are exactly the same as Eqs. (26)-(29), due to the\nmirror symmetry of the junction.\nC. Charge and spin conductances\nThe knowledge of the coe\u000ecients (26)-(29) allows to\nobtain the expression of the charge and the spin con-\nductances, again referring to the extension of the BTK\napproach to the case of F/S/F trilayer. As for the case of\nthe single F/S junction36, the conductance can be more\nconveniently calculated into the left ferromagnetic side\nof the junction, where the current \row does not include\nsupercurrents. In the presence of an applied bias Vbe-\ntween the left and right side of the junction, four injection\nprocesses can occur, as graphically shown in Fig. 4: the\ninjection of an electron with spin \u001bfrom the left side of\nthe junction [Fig. 4(a)], the injection of a hole with spin \u001b\nfrom the left side [Fig. 4(b)], the injection of an electron\nwith spin\u001bfrom the right side [Fig. 4(c)], and the injec-\ntion of a hole with spin \u001bfrom the right side [Fig. 4(d)].\nAs derived in detail in Appendix C, by properly taking\ninto account all these processes38, the charge and spin\ncurrents,JcandJsrespectively, \rowing normally to the\ninterfaces due to the application of a voltage bias V, can\nbe written as\nJc(s)=J\"\u0006J# (30)\nwhere the spin-dependent current J\u001bis\nJ\u001b=e~X\n\u000f;\u00122F1q\u001b\nm\u001bcos\u0012\u0014\nf\u0012\n\u000f\u0000eV\n2\u0013\n\u0000f\u0012\n\u000f+eV\n2\u0013\u0015\n\u0002(Ae\u001b+Ce\u001b+Ah\u001b+Ch\u001b): (31)\nStarting from this expression, one can show (see Ap-\npendix C) that the spin-dependent angle-averaged di\u000ber-\nential conductance for an applied bias Vcan be written\nas\nG\u001b(E) =dJ\u001b\ndV=Z\u0012c\n\u001b\n0d\u0012G\u001b(E;\u0012) (32)\nwith E=eVand\nG\u001b(E;\u0012) =G0~q\u001bcos\u0012(Ae\u001b+Ce\u001b+Ah\u001b+Ch\u001b)\f\f\fE\n2;\u0012:\n(33)7\nFIG. 4: Processes involved in the calculation of the charge\nand spin conductance through the junction: (a) injection of\nan electron with spin \u001bfrom the left side of the junction; (b)\ninjection of a hole with spin \u001bfrom the left side; (c) injection\nof an electron with spin \u001bfrom the right side, (d) injection of\na hole with spin \u001bfrom the right side.\nHereG0=e2qF\n\u0019~is the conductance of the junction when\nthe three layers are all in the normal state, qFis the\nFermi momentum in the normal state, and ~ q\u001b=q\u001b=qF.\nThe de\fnition (32) of G\u001btakes into account that the ex-\nperimentally measured conductance takes contributions\nfrom a limited range of injection angles, depending on the\nexperimental conditions. This is speci\fed by the value of\n\u0012c\n\u001b, which is the critical incidence angle for electrons with\nspin\u001binjected from the left ferromagnet, above which\ntransmission processes to the right ferromagnet do not\noccur. An explicit evaluation of the critical angles char-\nacterizing the scattering processes taking place within\nthe junction is presented in Appendix D.\nIn terms of G\u001bthe charge and the spin conductance\nare de\fned as\nGc(E) =G\"(E) +G#(E) (34)\nGs(E) =G\"(E)\u0000G#(E): (35)III. RESULTS AND DISCUSSION\nWe assume a superconducting layer having thickness\nL= 5000=kF, which is of the order of the superconduct-\ning coherence lenght \u0018S= 2000=kF. In order to compare\nthe e\u000bects due to the two microscopic mechanisms re-\nsponsible for the ferromagnetism, we \fx the magnetiza-\ntion amplitude in the F leads, and then we choose pairs\nof states where the same value of magnetization is ob-\ntained either via a pure Stoner-like mechanism or via the\nmass splitting one only. Referring to Fig. 5, showing the\nmagnetization as a function of Xand of the mass ratio\nY=m\"=m#, a given pair of such states is represented by\ntwo points lying on the same isomagnetic curves (small\ndashed, dotted, dotted-dashed and large dashed lines for\nM= 0:25;0:5;0:75;0:95, respectively). Points such as A,\nC, E, and G correspond to m\"=m#and thus are rep-\nresentative of ferromagnetic states of pure Stoner origin,\nwhile points such as B, D, F along the horizontal axis\ncorrespond to pure SMM ferromagnetic states. The dif-\nferent cases analyzed in the following correspond to the\nvalues ofX,YandMreported in the table of Fig. 6.\nBy using Eqs. (30)-(33), we have calculated the charge\nand the spin conductance through the double junc-\ntion at di\u000berent values of the applied bias, of the po-\nlarization of the ferromagnetic leads, and of the bar-\nrier transparencies. The charge and spin conductances\nof the F/S/F junction have both been normalized to\nthe charge conductance of the corresponding ferromag-\nnetic/normal/ferromagnetic (F/N/F) junction, in order\nto better visualize the e\u000bects induced by the supercon-\nducting state.\n1. Charge conductance\nThe charge conductance of the F/S/F junction for par-\nticles injected perpendicularly to the barriers is shown\nin Fig. 7 in the Stoner case [Figs. 7(a)-7(c)] and in the\nSMM case [Figs. 7(d)-7(f)], for three di\u000berent values of\nthe barrier transparency. The most appreciable di\u000ber-\nences between the junction behavior in the presence of\nthe two di\u000berent mechanisms for ferromagnetism appear\nfor fully transparent interfaces ( Z= 0) and high values\nof the magnetization. Indeed, in this regime, while in the\nStoner case the charge conductance is signi\fcantly sup-\npressed at low bias with respect to the F/N/F case, on\nthe contrary in the SMM case it is enhanced when the\nvalue of the mass mismatch is increased.\nThis e\u000bect can be understood by taking into ac-\ncount the expression of G\u001b(E;\u0012), which explicitly de-\npends on the carrier linear momentum [Eq.(33)]. Since\na large magnetization directly a\u000bects the linear momen-\ntum of carriers involved in the transmission processes [see\nEq.(17)] in a more sizable way in the SMM case than in\nthe Stoner one, a large mass mismatch, such as the one\noccurring for the parameter choice corresponding to the\npoint H listed in the table of Fig. 6, is expected to in-8\nFIG. 5: Density plot of the magnetization for a two-\ndimensional ferromagnetic system where both Stoner and\nSMM mechanisms are responsible for the ferromagnetic state.\nIsomagnetic curves are shown: continuous, small dashed,\ndotted, dotted-dashed and large dashed lines correspond to\nM= 0;0:25;0:5;0:75;0:95, respectively. The parameter val-\nues associated with each marked point are given in Table 6.\nPoint H lies far away on the horizontal axis and cannot be\nproperly shown in the \fgure.\nFIG. 6: Chosen values of the magnetization and correspond-\ning microscopic parameters used to investigate separately the\npure Stoner case and the SMM one.\nduce a strong contribution to the charge conductance.\nCorrespondingly, at lower values of the magnetization,\nwhere the mass mismatch in SMM ferromagnets is re-\nduced, carrier momenta for spin-up and spin-down elec-\ntrons assumes more comparable values, which makes the\nSMM charge conductance more similar to the Stoner one.\nAside from the magnetization driven enhancement of\nthe particle linear momentum, another e\u000bect generally\ncontributes to determine the observed di\u000berences be-\ntween the SMM and the Stoner charge conductance, and\nit is linked to the Andreev re\rections which occur when\nsuperconductivity is switched on. They directly con-\ntribute to the conductance according to Eq.(33), and,\nFIG. 7: Voltage bias ( E=eV) dependence of the charge\nconductance at normal incident angle ( \u0012= 0) in the Stoner\ncase [(a)-(c)] and in the SMM case [(d)-(f)] at di\u000berent values\nof the barrier transparency Z: (a) and (d) refer to Z= 0, (b)\nand (e) to Z= 2, (c) and (f) to Z= 4. The chosen values of\nthe magnetization are M= 0;0:25;0:5;0:75;0:95 (red, green,\ncyan, violet and magenta lines, respectively).\nin particular, they play the major role in the scattering\nprocess for applied bias below the energy gap at large\nbarrier transparencies [see Figs.8 (a), 8 (c), and Fig.9 (a)].\nThe occurrence of Andreev scattering due to the super-\nconducting state counterbalances the detrimental e\u000bect\nthat ferromagnetism has on the charge conduction. In-\ndeed, both mechanisms responsible for ferromagnetism\ngenerally disadvantage the charge transport through the\njunction. The Stoner mechanism, via the energy shift be-\ntween the opposite spin electrons [Fig.1(a)], reduces the\navailable states for minority carriers; the SMM mecha-\nnism induces an unbalance between the velocities of car-\nriers with opposite spin, such that at positive magnetiza-\ntion values, spin-up electrons become slower than spin-\ndown ones, thus providing a reduced contribution to the\nconductance. Therefore, in the presence of superconduc-\ntivity, the lack of the energy shift between opposite spin\ndensity of states in the SMM case allows strong Andreev\nre\rections, which are instead suppressed by increasing\nthe magnetization in the Stoner case, due to a reduction\nof accessible states for spin-down holes. Therefore, the\nmore robust Andreev re\rections, together with a sizable\nlinear momentum ampli\fcation at large magnetization\nin the SMM case, can explain the very di\u000berent behav-\nior of the charge conductance of the F/S/F junctions in\nthe Stoner and in the SMM case for transparent barriers\n[Fig. 7(a) and 7(d)].\nDi\u000berently from what happens in the regime of highly9\nFIG. 8: Energy dependence of the probability coe\u000ecients for\nAndreev re\rections ( A) and transmission in the right ferro-\nmagnet as electrons ( C), for spin-up and spin-down injected\ncarriers [(a),(b) and (c), (d), respectively]. Here we have\nconsidered the case of Stoner ferromagnetic layers and high-\ntransparent barriers ( Z= 0). The values of Mand the related\ncolor lines are the same as those used in Fig. 7.\nFIG. 9: Energy dependence of the probability coe\u000ecients for\n(a) Andreev re\rections, (b) transmission into the right ferro-\nmagnet as electrons for spin-up injected carriers, in the case\nof SMM ferromagnetic layers and for perfectly transparent\nbarriers ( Z= 0). The same coe\u000ecients for spin-down elec-\ntrons are not shown because, as proved in Appendix E, they\ncoincide with those for spin-up ones. The values of Mand\nthe related color lines are the same as those used in Fig. 7.\ntransparent barriers, for \fnite transparency di\u000berences\nbetween the Stoner and the SMM case tend to become\nless and less appreciable as Zis increased, regardless of\nthe magnetization value in the F layers. This can be\nexplained in terms of the behavior of the Andreev re\rec-\ntions, which at low bias become strongly suppressed by\nincreasing Z[see for instance Figs. 10 and 11 for the case\nZ= 2]. The small contribution provided by the Andreev\nre\rections makes less and less e\u000bective the role played by\nthe large momentum values associated with a high value\nof the mass mismatch, thus leading to a similar behavior\nof the Stoner and the SMM charge conductance at any\nvalue ofM.\nSimilar results are obtained in the case of the charge\nconductance integrated over all possible injection angles,\nas shown by Fig. 12. Again we see a much larger low-\nbias weight in the SMM case than in the Stoner one for\nZ= 0 and large magnetization, this di\u000berence tending\nFIG. 10: Energy dependence of the probability coe\u000ecients in\nthe Stoner case for Andreev re\rections ( A) and transmission\nto the right ferromagnet as electrons ( C), for spin-up injected\nelectrons [respectively (a) and (b)] and spin-down injected\nelectrons [respectively (c) and (d)], in the low-transparency\nlimit ( Z= 2). The values of Mand the related color lines are\nthe same as those used in Fig. 7.\nFIG. 11: Energy dependence of the probability coe\u000ecients in\nthe SMM case for Andreev re\rections (A) and transmission\nto the right ferromagnet as electrons (C), for spin-up injected\nelectrons [respectively (a) and (b)] and spin-down injected\nelectrons [respectively (c) and (d)], in the low-transparency\nlimit ( Z= 2). The values of Mand the related color lines are\nthe same as those used in Fig. 7.\nto disappear at any Mwhen a lower and lower barrier\ntransparency is considered.\n2. Spin conductance\nThe behavior of the spin conductance is shown in\nFig. 13 in the case of electron incidence normal to the\ninterfaces. The results obtained for full transparency are\nshown in Fig. 13 (a) for the Stoner case and in Fig. 13 (d)\nfor the SMM one. In particular, the spin conductance\nincreases at all energies with increasing magnetization,10\nFIG. 12: Voltage bias dependence of the charge conductance\nintegrated over all the allowed injection angles in the Stoner\n(a)-(c) and in the SMM case (d)-(f) at di\u000berent values of\nbarrier transparency: (a) and (d) for Z= 0, (b) and (e) for\nZ= 2,(c) and (f) for Z= 4. The values of Mand the related\ncolor lines are the same as those used in Fig. 7.\nthis e\u000bect being at low bias much more pronounced in\nthe SMM case than in the Stoner one. This can be ex-\nplained noting that while with Stoner ferromagnets this\ntrend comes from the asymmetrization of the probability\ncoe\u000ecients and Fermi momenta of particles with oppo-\nsite spin, in the SMM case the probability coe\u000ecients at\n\u0012= 0 and Z= 0 are equal for spin-up and spin-down\nelectrons, so that the amplitude of the spin conductance\nis positive and determined by the di\u000berence between the\nFermi momenta of opposite spin electrons. As already\npointed out, such a momentum di\u000berence becomes very\nsigni\fcant in the SMM case for high magnetization val-\nues, due to the strong mass renormalization driving the\nferromagnetic order. Furthermore, below the energy gap,\nthe Andreev re\rections are very strong and almost insen-\nsitive to polarization, thus allowing a magnitude of the\nspin conductance much larger in the SMM case than in\nthe Stoner one, in particular at low bias.\nIn the presence of non transparent barriers and in par-\nticular in the tunnel limit, while the spin conductance\nof the F/S/F junction in the Stoner case systematically\nincreases as a function of the ferromagnetic polarization\n[Figs. 13 (b) and 13(c)], in the SMM case there are ranges\nof the applied voltage bias where the spin conductance\nbecomes negative, with an absolute value which slightly\nincreases with the magnetization [Figs. 13 (e) and 13(f)].\nSuch feature comes from the asymmetry in the e\u000bective\nmass of opposite spin particles which leads to a larger\nvelocity, and thus to a better transmission, of spin-down\nelectrons compared to the spin-up ones [Figs. 11(b) and\n11(d)], thus reversing the role of majority spin electrons\nFIG. 13: Voltage bias dependence of the spin conductance at\nnormal incident angle ( \u0012= 0) in the Stoner (a)-(c) and in the\nSMM case (d)-(f) at di\u000berent values of barrier transparency:\n(a) and (d) for Z= 0, (b) and (e) for Z= 2,(c) and (f) for\nZ= 4. The values of Mand the related color lines are the\nsame as those used in Fig. 7.\nin the transmission process with respect to the Stoner\ncase [Figs. 10(b) and 10(d)]. It emerges especially at bias\nvalues above the superconducting gap and at large barrier\nvalues, since in such regime this e\u000bect does not compete\nanymore with the occurrence of Andreev re\rections, and\ntransport through the junction is fully dominated by the\ntransmission of normal, non-superconducting quasiparti-\ncles.\nMore pronounced di\u000berences between the Stoner and\nthe SMM case are found when one considers the spin con-\nductance integrated over all possible incidence directions\n[see Fig. 14]. In the Stoner case the spin conductance is\nalways positive, for all values of the applied bias, the bar-\nrier transparency and the F layer magnetization. At low\nbias it almost vanishes, then becoming \fnite above the\nenergy gap, with a magnitude which increases with the\nspin polarization [Figs. 14(a)- 14(c)]. Below the gap, the\nspin conductance is negligible also in the case of SMM fer-\nromagnetic layers, but di\u000berently from the Stoner case,\natZ= 0 and larger bias values it is positive at small M\n[Fig. 14 (d)], then becoming more and more negative as\nincreasing values of Mare considered [Figs. 14 (e) and\n14(f)]. For \fnite values of Zwe \fnd a similar behavior,\nthe only di\u000berence being that at large bias the spin con-\nductance in the SMM case is always negative, regardless\nof the value of M.\nIn order to understand this behavior, it is important to\ntake into account that, according to the critical angle de-\npendence on lead magnetization (see Appendix D), while\nthe injection cone of spin-up electrons is strongly sup-11\nFIG. 14: Voltage bias dependence of the spin conductance\nintegrated over all the allowed injection angles in the Stoner\n(a)-(c), and in the SMM case (d)-(f) at di\u000berent values of\nbarrier transparency: (a) and (d) for Z= 0, (b) and (e) for\nZ= 2,(c) and (f) for Z= 4. The values of Mand the related\ncolor lines are the same as those used in Fig. 7.\npressed by the magnetization, spin-down electrons can\nenter the junction at any injection angle. Consequently,\nin the integrated spin current, there is a strong compe-\ntition between the contribution due to injected spin-up\nelectrons, which is dominant at \u0012= 0 but restricted to a\nvery limited angle range at increasing magnetization, and\nthe contribution due to the injection of spin-down elec-\ntrons, which is \fnite and sizable at all injection angles, in\nparticular in the SMM case. Such competition gives also\nrise to very small values of the total spin conductance at\nbias lower than 2\u0001.\nWe also notice that above 2\u0001, the most signi\fcant\ncontribution to the conductance comes from the exci-\ntation of normal quasiparticles, so that the sign of the\nspin conductance is dictated by the unbalance between\nthe normal transmission of opposite spin particles. In\nthe Stoner case one always gets positive values, increas-\ning with the polarization, since at any magnetization,\nspin-up electrons have in the right ferromagnetic layer\nmore accessible energy states than spin-down ones, due\nto the positive sign of the lead magnetization. On the\ncontrary, the negative values found in the SMM case [see\nFigs. 14 (d)-14(f)] come from the mass unbalance between\nspin-up and spin-down electrons, which favors the trans-\nmission of the faster down-spin electrons. On the other\nhand, in the bias regime below 2\u0001, Andreev re\rections\ncounterbalance this tendency, since they have an approxi-\nmately equal weight for spin-up and spin-down electrons,\nthus giving rise to a negligible spin current. Summariz-\ning, while at biases below the gap the superconducting\ne\u000bects dominate via the Andreev scattering, above thatvalue the dominant role is played by the ferromagnetic\norder which in the SMM case allows the spin-down cur-\nrent to dominate.\nNevertheless, the presence of superconductivity in-\nduces an enhanced DOS at the gap edge, thus ampli-\nfying the spin current for bias values of the order of 2\u0001:\naround that value, the total charge and spin conductance\nare characterized by a peak, which is tunable through\nthe polarization of the ferromagnetic leads, both in the\nStoner and in the SMM case [see Figs. 14 (c) and 14(f)].\nThe results we got show the emergence of peculiar ef-\nfects within the context of superconductor-based mag-\nnetic double junctions. In this framework, many studies\nhave been performed addressing spin and charge conduc-\ntance in F/N/F structures, mainly in the context of spin-\nvalve e\u000bects40{44, as well as in F/S/F double junctions,\nwhere the e\u000bects of the relative orientation of the two\nferromagnetic leads have been investigated. However,\nto our knowledge, no systematic study has been done\nin symmetric junctions on the spin response neither for\nF/N/F systems nor for F/S/F junctions as a function of\nthe amplitude of the magnetization of the ferromagnetic\nleads, exploring the role of di\u000berent metallic ferromag-\nnets. In particular, we are not aware of measurements\ndemonstrating a sign reversal for the spin conductance\nas a function of the ferromagnet polarization for normal\nincidence, or negative values of the integrated spin con-\nductance, in symmetric F/N/F junctions. We argue that\nthis may be ascribed to the fact that in realistic materi-\nals the mechanism responsible for ferromagnetism could\nbe a combination of both magnetic exchange (Stoner)\nand kinetic (mass mismatch) modi\fcations of the spin\ndependent electronic structure. Since the two mecha-\nnisms give rise to opposite spin currents, their simulta-\nneous presence in a given material can be responsible\nin the corresponding F/N/F junction for a spin conduc-\ntance that, on the average, is vanishing or di\u000ecult to\ndetect. In Fig.15, we show the results of the integrated\nspin conductance at Z= 2 for two representative cases of\nF/N/F junctions, together with the corresponding F/S/F\nones, where both ferromagnetic mechanisms are active\nwith di\u000berent polarizations [Fig.15(a)] or di\u000berent rela-\ntive weights [Fig.15(b)]. We have denoted by pSMM the\nweight of the spin mismatch mechanism with respect to\nthe Stoner one, and calculated the total spin conductance\nas\nGS= (1\u0000pSMM)GStoner\nS +pSMMGSMM\nS:\nWe observe that in the F/N/F case, the spin conductance\nis substantially featureless with incoherent oscillations in\nenergy. On the other hand, the use of the supercon-\nductor allows to exploit its characteristic energy scale,\nassociated with the superconducting gap \u0001, and focus\non a speci\fc energy window in the analysis of the trans-\nport properties. In our case the latter corresponds to the\nyellow-shaded region in Fig.15, where the energy depen-\ndence of the spin conductance is marked by distinctive\nfeatures around E= 2\u0001. In particular, we \fnd that12\nwhen the SMM mechanism is predominant, the spin con-\nductance exhibits a dip-peak structure developing from\nnegative to positive values that tends to evolve towards\na positive single peak as the weight of the Stoner mecha-\nnisms gradually increases. Based on that, we expect that\na measurement of the spin conductance of a F/S/F junc-\ntion may provide relevant hints on the extent to which\nthe two mechanisms compete between each other.\nFIG. 15: Integrated spin conductance at Z= 2 for two repre-\nsentative cases of F/N/F junctions, together with the cor-\nresponding F/S/F ones, where both ferromagnetic mecha-\nnisms occur with di\u000berent polarizations (a) or di\u000berent rel-\native weights (b). Here pSMMis the weight of the SMM mech-\nanism with respect to the Stoner one. The yellow-shaded\nregion marks the energy window where distinctive features\nemerge in the F/S/F junction.\nIV. CONCLUSIONS\nWe have presented a study of the transport phenomena\nin a clean F/S/F junction with parallel magnetization in\nthe two F layers, making a comparison between the case\nof Stoner-type ferromagnetic layers with the one where\nferromagnetism is driven by an asymmetric mass renor-\nmalization of carriers with opposite spin. We have shown\nthat charge and spin transport in this junction exhibits\ndi\u000berent features depending on the mechanism which is\nresponsible for the ferromagnetism, in the case of per-pendicular injection as well as when considering the inte-\ngrated behavior over all the allowed injection directions.\nIn particular, for transparent barriers, Andreev re\rec-\ntions are more robust in the SMM case than in the Stoner\none as the magnetization in the F layers is increased. As a\nconsequence, with SMM ferromagnets transport through\nthe junction is characterized by a signi\fcant ampli\fca-\ntion of the charge conductance with respect to the F/N/F\ncase, which is not observed in the Stoner case. Then, the\nspin conductance of the SMM junction monotonously in-\ncreases with the ferromagnetic exchange for particles in-\njected perpendicularly to the barriers, in opposition to\nthe nonmonotonous behavior versus magnetization found\nin the Stoner case. Finally, in the tunnel limit, while the\ncharge conductance assumes values which weakly depend\non the ferromagnetic mechanism, the spin conductance\nexhibits opposite signs in the SMM and in the Stoner case\nat large applied bias. In both cases, the superconducting\npairing enhances the amplitude of the spin current close\nto the gap edge.\nSo far, several magnetic materials have been found\nto exhibit properties that cannot be framed exclusively\nwithin a Stoner scenario26,27,45. This often happens in\nthe cases of half-metal ferromagnets, where the almost\nfull degree of spin polarization develops in regimes where\na mass mismatch is clearly distinguishable and high po-\nlarization values cannot be explained in terms of the\nStoner mechanism only46. For these systems, a theo-\nretical analysis where the role played by spin-dependent\nelectron masses is explicitly taken into account is likely to\nbe required. Within this context, the di\u000berent behavior\npredicted in the Stoner and in the SMM cases may pro-\nvide useful indications on the nature of the mechanism\noriginating the ferromagnetic order in a given ferromag-\nnetic material. Given the usual limitations in the exper-\nimental realization of heterostructures, the possibility of\nselecting the magnetization mechanism, in this way con-\ntrolling the spin of the carrier responsible for transport,\nmay turn out to be useful in the design of electronics\nand spintronics devices. In this context, it may also be\ninteresting to investigate to what extent the use of SMM\nferromagnets, instead of Stoner-like ones, a\u000bects the be-\nhavior of a F/S/F junction when treated as a spin valve.\nWe \fnally point out that in the present analysis the\nthree subsystems are all considered in the clean limit.\nConcerning the dependence of the results on the sam-\nple purity, we expect that in the regime of weak dis-\norder and in the absence of spin-\rip scattering in the\nsuperconductor, disorder is mainly a\u000becting the coher-\nence length, which becomes \u0018D\u0018p\n~D=\u0001 withDbeing\nthe di\u000busion constant and \u0001 the superconducting gap.\nThis renormalization implies that, with respect to the\ninvestigated clean con\fguration, similar results are ex-\npected by scaling the thickness Lof the superconductor.\nHowever, when the disorder introduces strong energy re-\nlaxation processes and spin \rip scattering, our results\nare no longer valid and a di\u000berent approach is required\nto deal with the spin di\u000busion in the superconductor.13\nFor instance, the analysis performed in Ref.47, involv-\ning Stoner type ferromagnets and spin \rip scattering in\nthe superconductor, showed that the spin current is sup-\npressed at bias below the superconducting energy gap,\nand a massive spin \rip occurs at energies close to the\ngap. These processes can of course modify the obtained\nresults near the gap edge. A complementary approach in-\ncluding an inelastic transport regime has been proposed\nin Refs.44,48, where it is shown that in a F/S/F struc-\nture the superconductor becomes a low-carrier system for\nspin transport, due to the opening of the gap, and thus\nthe accumulation spin signal is greatly enhanced with re-\nspect to a non-superconducting layer. On the basis of\nthis result, we argue that inelastic processes can help to\ndistinguish the spin signal when going from the normal to\nthe superconducting phase, and that might be applicable\nfor both Stoner and SMM ferromagnets. Along this line,\nwe point out that to the best of our knowledge, all the\nperformed studies in the presence of disorder deal with\nStoner ferromagnetic leads, while an investigation of the\ne\u000bects of disorder in the case of SMM-based ferromagnets\nis still lacking. This problem goes beyond the scope of\nthis work and will be considered in future investigations.\nAppendix A\nIn this Appendix we complement Eqs. (21)-(23) re-\nporting the expression of the wave functions in the three\nregions of the junction for injections other than the one\nof electrons with spin \u001bfrom the left F side.\nFor the injection of a hole with energy \"and spin\u001b\nfrom the left F side, the wave functions in the three re-\ngions of the junction are:\n F\nh\u001bL(z) =\u0012\n0\n1\u0013\ne\u0000iq\u001bz cos\u0012\u001b+ah\u0016\u001b\u0012\n1\n0\u0013\ne\u0000iq\u0016\u001bz cos\u0012A\n\u0016\u001b\n+bh\u001b\u0012\n0\n1\u0013\neiq\u001bz cos\u0012\u001b(36)\nforz<0;\n S\nh\u001b(z) =\u000bh\u001b\u0012\nu0\nv0\u0013\neikez cos\u0012S\n\u001be\n+\fh\u0016\u001b\u0012\nv0\nu0\u0013\ne\u0000ikhz cos\u0012S\n\u001bh\n+\rh\u001b\u0012\nu0\nv0\u0013\ne\u0000ikez cos\u0012S\n\u001be\n+\u0011h\u0016\u001b\u0012\nv0\nu0\u0013\neikhz cos\u0012S\n\u001bh (37)\nfor 0<z<L ;\n F\nh\u001bR(z) = ch\u0016\u001b\u0012\n0\n1\u0013\ne\u0000iq\u0016\u001bz cos\u0012T\n\u001b\n+dh\u001b\u0012\n1\n0\u0013\neiq\u0016\u001bz cos\u0012T\n\u0016\u001b (38)forz>L .\nThe expressions of the wave functions corresponding\nto the injection of an electron with energy \"and spin\u001b\nfrom the right F side can be obtained from Eqs. (21)-\n(23) by reversing the sign of all wavevectors. The same\nsubstitution can be applied to Eqs. (36)-(38) to get the\nwave functions corresponding to the injection of a hole\nwith energy \"and spin\u001bfrom the right ferromagnet.\nDue to the symmetry of the problem with respect to the\nsuperconducting layer, the probability amplitudes of the\nscattering processes corresponding to particle injection\nfrom the right side are equal to those of the corresponding\nprocesses for the same particle injection from the left side.\nAppendix B\nIn this Appendix we derive the probability current con-\nservation through the junction, showing where the mass\nasymmetry condition enters, and how it a\u000bects the ex-\npression of the probabilities associated with the scatter-\ning processes occurring in the junction.\nDenoting by P\u001b(r;t) =j\t\u001b(r;t)j2the probability den-\nsity to \fnd a particle at a given time tin the volume\nelementdraround the position r, we have\nd\ndtZ\ndrP\u001b(r;t) = 0; (39)\nwith the integrals extended to the whole space. Using\nthe Schr odinger equations for the spinors \t \u001b(r;t) and\n\t\u0003\n\u001b(r;t)\n{~@\n@t\t\u001b(r;t) =HBdG\n\u001b\t\u001b(r;t) (40)\n\u0000{~@\n@t\t\u0003\n\u001b(r;t) =HBdG\n\u001b\t\u0003\n\u001b(r;t) (41)\nand taking into account that, due to the independence of\nthe Hamiltonian of the time coordinate, the time depen-\ndence of the wave function can be factorized, we \fnally\nget:\n@P\u001b(r;t)\n@t=2\n~Imfu\u0003\n\u001b(r)^H\u001b(r)u\u001b(r)\u0000v\u0003\n\u0016\u001b(r)^H\u0003\n\u0016\u001b(r)v\u0016\u001b(r)g\n=~Im(u\u0003\n\u001b(r)^p2\nm\u001bu\u001b(r)\u0000v\u0003\n\u0016\u001b(r)^p2\nm\u0016\u001bv\u0016\u001b(r)):(42)\nIn the absence of magnetic \feld, the momentum opera-\ntor is ^p=\u0000{~rso that one \fnally gets the continuity\nequation\n@\n@tP\u001b(r;t) +r\u0001J\u001b(r) = 0 (43)\nwith the probability density current J\u001b(r) given by\nJ\u001b(r) = Im\u0014~\nm\u001bu\u0003\n\u001b(r)ru\u001b(r)\u0000~\nm\u0016\u001bv\u0003\n\u0016\u001b(r)rv\u0016\u001b(r)\u0015\n:\n(44)14\nThe expression of J\u001bclearly shows that an asymmetry\nin the e\u000bective mass of electrons with opposite spin also\nenters the formal expression of the probability density\ncurrent. Such expression can be used to derive the proba-\nbility coe\u000ecients associated with the scattering processes\ntaking place in the junction. In particular, from the prob-\nability density current conservation it follows that the to-\ntal current \rowing through a surface enclosing the whole\nsystem is zero. By considering the probability density\ncurrent \rowing through the interfaces, and thus along\nthez-direction, there are the following contributions: the\nprobability density current JI\n\u001bfor the incident particles\nwith spin\u001b, and the associated re\rected and transmitted\ncurrentsJR\n\u001bandJT\n\u001b, respectively. They are related by\nthe following equation:\nJI\n\u001b+JR\n\u001b=JT\n\u001b: (45)\nBy using the wave functions de\fned in each region of the\njunction in the representative case of injected electrons\nwith spin\u001b, we have that the projections of the currents\nin the direction perpendicular to the interfaces are:\nJI\n\u001b=~\nm\u001bq\u001bcos\u0012\u001b (46)\nJR\n\u001b=\u0000~\nm\u001bjbe\u001bj2q\u001bcos\u0012\u001b\u0000~\nm\u0016\u001bjae\u0016\u001bj2q\u0016\u001bcos\u0012A\n\u0016\u001b(47)\nJT\n\u001b=~\nm\u001bjce\u001bj2q\u001bcos\u0012T\n\u001b+~\nm\u0016\u001bjde\u0016\u001bj2q\u0016\u001bcos\u0012T\n\u0016\u001b:(48)\nBy applying Eq.(45) and dividing all the terms by the\ninjected current, we get the relation\n1 =A\u001b+B\u001b+C\u001b+D\u001b (49)\nwhich allows to de\fne the probability coe\u000ecients for\ngeneric injected particles p:Ap\u001b,Bp\u001b,Cp\u001b,Dp\u001b(being\np=e;hfor electrons and holes, respectively) as reported\nin Eqs. (26)-(29).\nAppendix C\nIn this Appendix we present the detailed derivation of\nthe junction conductance, following an extension of the\noriginal BTK approach35to the case of a F/S/F double\njunction with Stoner ferromagnets38.\nAs for the case of the single junction, we calculate the\nconductance in the left ferromagnetic side, where the cur-\nrent \row does not include supercurrents and the calcu-\nlation is therefore more convenient. Taking into account\nthe results presented in Appendix B, the charge current\n\rowing in the presence of an applied bias Vfrom the left\nto the right side of the junction can be calculated as:\nJ\u001b=~ImX\nl;\u001b;\u000fql\u0014f\u000f\nml\u001bu\u0003\nl\u001b@\n@zul\u001b+\n+(1\u0000f\u000f)\nml\u0016\u001bv\u0003\nl\u001b@\n@zvl\u001b\u0015\n: (50)Heref\u000fis the Fermi distribution function and the sub-\nscriptl= 1;4 refers to the four possible injection pro-\ncesses described in the main text and graphically shown\nin Fig. 4. We thus have q1=q3=eandq2=q4=\u0000e,\ne(<0) being the electron charge.\nWhen a bias potential Vis applied between the two\nF leads, by taking into account that the summation on\nthe energies involves the energy levels of the side where\nparticles are injected, we can write the following expres-\nsions for the normal components of the spin-dependent\ncurrents in the left F side:\nJ\u001b=e~X\n\u0012;\u000f2F1\u0014q\u001b\nm\u001bcos\u0012(1\u0000jbe\u001bj2)f\u0012\n\u000f\u0000eV\n2\u0013\n\u0000q\u0016\u001b\nm\u0016\u001bcos\u0012A\n\u0016\u001bjae\u0016\u001bj2\u0014\n1\u0000f\u0012\n\u0000\u000f\u0000eV\n2\u0013\u0015\u001b\n\u0000e~X\n\u0012;\u000f2F1\u001a\n\u0000q\u0016\u001b\nm\u0016\u001bcos\u0012A\n\u0016\u001bjah\u0016\u001bj2f\u0012\n\u000f\u0000eV\n2\u0013\n+q\u001b\nm\u001bcos\u0012(1\u0000jbh\u001bj2)\u0014\n1\u0000f\u0012\n\u0000\u000f\u0000eV\n2\u0013\u0015\u001b\n+e~X\n\u0012;\u000f2F2\u001a\n\u0000q\u001b\nm\u001bcos\u0012T\n\u001bj~ce\u001bj2f\u0012\n\u000f+eV\n2\u0013\n\u0000q\u0016\u001b\nm\u0016\u001bcos\u0012T\n\u0016\u001bj~de\u0016\u001bj2\u0014\n1\u0000f\u0012\n\u0000\u000f+eV\n2\u0013\u0015\u001b\n\u0000e~X\n\u0012;\u000f2F2\u001aq\u001b\nm\u001bcos\u0012T\n\u001bj~ch\u001bj2\u0014\n1\u0000f\u0012\n\u0000\u000f\u0000eV\n2\u0013\u0015\n+q\u0016\u001b\nm\u0016\u001bcos\u0012T\n\u0016\u001bj~dh\u0016\u001bj2f\u0012\n\u000f\u0000eV\n2\u0013\u001b\n: (51)\nTaking into account that 1 \u0000f(\u000f) =f(\u0000\u000f) and assum-\ning that the two ferromagnets are identical, so that the\nprobability amplitudes of scattering processes due to par-\nticles injected from the right side are equal to those for\nthe same particle injection from the left side (in particu-\nlar~cp\u001b=cp\u001b, and ~dp\u0016\u001b=dp\u0016\u001b), it is possible to write the\ncurrent as\nJ\u001b=Je\u001b+Jh\u001b\nwith\nJe\u001b=e~X\n\u0012;\u000f2F1q\u001b\nm\u001bcos\u0012\u0014\nf\u0012\n\u000f\u0000eV\n2\u0013\u0000\n1\u0000jbe\u001bj2\n\u0000q\u0016\u001b\nq\u001bm\u001b\nm\u0016\u001bcos\u0012T\n\u0016\u001b\ncos\u0012jde\u0016\u001bj2\u0013\n\u0000f\u0012\n\u000f+eV\n2\u0013\n(52)\n\u0012\njae\u0016\u001bj2q\u0016\u001b\nq\u001bm\u001b\nm\u0016\u001bcos\u0012A\n\u0016\u001b\ncos\u0012+jce\u001bj2\u0013\u0015\n=e~X\n\u0012;\u000f2F1q\u001b\nm\u001bcos\u0012\u0014\nf(\u000f\u0000eV\n2) (1\u0000Be\u001b\u0000De\u001b)\n\u0000f\u0012\n\u000f+eV\n2\u0013\n(Ae\u001b+Ce\u001b)\u0015\n(53)15\nand\nJh\u001b=\u0000e~X\n\u0012;\u000f2F1q\u001b\nm\u001bcos\u0012\u0014\nf\u0012\n\u000f+eV\n2\u0013\u0000\n1\u0000jbh\u001bj2\u0000\nq\u0016\u001b\nq\u001bm\u001b\nm\u0016\u001bcos\u0012T\n\u0016\u001b\ncos\u0012j~dh\u001bj2\u0013\n\u0000f\u0012\n\u000f\u0000eV\n2\u0013\u0012\njah\u0016\u001bj2q\u0016\u001b\nq\u001bm\u001b\nm\u0016\u001bcos\u0012A\n\u0016\u001b\ncos\u0012+j~ch\u001bj2\u0013\u0015\n=\u0000e~X\n\u0012;\u000f2F1q\u001b\nm\u001bcos\u0012\u0014\nf\u0012\n\u000f+eV\n2\u0013\n(1\u0000Bh\u001b\u0000Dh\u001b)\n\u0000f\u0012\n\u000f\u0000eV\n2\u0013\n(Ah\u001b+Ch\u001b)\u0015\n(54)\nFrom the conservation of the probability current, we \f-\nnally get:\nJ\u001b=e~X\n\u0012;\u000f2F1q\u001b\nm\u001bcos\u0012\u0014\nf(\u000f\u0000eV\n2)\u0000f(\u000f+eV\n2)\u0015\n\u0002(Ae\u001b+Ce\u001b+Ah\u001b+Ch\u001b): (55)\nBy considering that the sum over the allowed energies\nand injection angles can be written as\nX\n\u0012;\u000f2F1=Z\u0019=2\n0d\u0012Z\nd\u000fNF\n\u001b(\u000f)\nand that the spin dependent density of states NF\n\u001b(\u000f)\nin the two equivalent ferromagnets in the 2D case is\nconstant and equal to NF\n\u001b(\u000f) =m\u001b=(2\u0019~2), the spin-\ndependent charge conductance can be written as:\nG\u001b(E) =dJ\u001b\ndV=Z\u0019=2\n0d\u0012G\u001b(\u0012;E) (56)\nwhere E=eVand\nG\u001b(\u0012;E) =G0~q\u001bcos\u0012(Ae\u001b+Ce\u001b+Ah\u001b+Ch\u001b)\f\f\fE\n2;\u0012:\nHere ~q\u001b=q\u001b=qFandG0=e2qF\n\u0019~is the conductance of\nthe junction when the three layers are all in the normal\nstate.\nAppendix D\nIt is known that the measured conductance takes con-\ntributions from a range of incidence angles which de-\npends on the conditions under which experiments are\nperformed. For electrons and holes injected from the left\nside, two limiting angles have to be considered: (a) the\nincident angle above which local Andreev re\rections can\nnot occur, and (b) the limiting angle of incidence for\nthe transmission into the superconductor. Such limiting\nangles can be derived from the application of the conser-\nvation law given by Eq. (20).\nFIG. 16: Density plot of the critical angle for the local An-\ndreev re\rections of spin-up electrons (and holes) injected from\nthe left side (or equivalently from the right side since we\nhave considered the two ferromagnetic leads fully identical),\nas a function of the microscopic parameters X=U=EFand\nY=m\"=m#, which are assumed to be the same in the two\nferromagnetic leads. Isomagnetization lines for di\u000berent val-\nued of the magnetization are reported. Each of these lines\nalso corresponds to a \fxed critical angle value. No limitation\nto the injection direction of spin-down electrons (holes) holds\nwhen the magnetization is assumed positive.\nFIG. 17: Density plot of the critical angle for trasmission to\nSof spin-up electrons (and holes) injected from the left side\n(or equivalently from the right side, since we have considered\nthe two ferromagnetic leads identical), as a function of the\nmicroscopic parameters X=U=EFandY=m\"=m#. Iso-\nmagnetization lines for di\u000berent valus of the magnetization\nare also reported. No limitation to the injection direction of\ndown-spin electrons (holes) holds when the magnetization is\nassumed positive.\nIn the case where the two ferromagnets are identical,16\nFIG. 18: Critical injection angles for the Andreev re\rections\nof spin-up electrons (and holes) entering from the left side\n(or equivalently from the right side, since we have considered\nidentical ferromagnetic leads), as functions of the microscopic\nparameters X=U=EFandY=m\"=m#, in the Stoner (a)\nand in the SMM case (b). The dotted line in the two panels\nindicates the magnetization value M= 0:75.\nand assuming that the magnetization amplitude is posi-\ntive, we \fnd that in the case of spin-down injected par-\nticles, both local Andreev re\rections and particle trans-\nmissions into the S layer can occur independently of the\ninjection direction. Indeed, according to Eq. (20), the\nscattering angles de\fning the direction of the Andreev\nre\rections and the transmission in S are\n\u0012A\n\"= arcsin\u0014q#\nq\"sin\u0012#\u0015\n(57)\n\u0012T\n#= arcsin\u0014q#\nkS\nFsin\u0012#\u0015\n(58)\nwhere\u0012\u001bis the angle of the particles which are injected\nwith spin\u001b. When the lead magnetization is assumed\npositive,q#is less than q\", and is also less than kS\nF. Con-\nsequently, in the right side of Eqs. (57)-(58), the argu-\nment of the arcsine functions is always less than 1, and\nthis implies that both the scattering angles \u0012A\n\"and\u0012T\n#\nare well de\fned at all the injection angles \u0012#.On the other side, when injecting spin-up particles,\nAndreev re\rection and transmission in the superconduct-\ning layer can occur provided that the injection angle is\nless than the following critical values, respectively:\n\u0012cA\n#= arcsin\"s\n1\u0000X\nY(1 +X)#\n(59)\n\u0012cT\n\"= arcsin\"s\n1p\nY(1 +X)#\n: (60)\nIn the considered two-dimensional limit, the magnetiza-\ntion of each ferromagnetic lead, de\fned as M= (n\"\u0000\nn#)=(n\"+n#), withn\"(#)being the number of spin-up\n(spin-down) electrons in the ferromagnetic layer, can be\nexpressed as a function of the microscopic parameters X\nandYde\fned in the main text as31:\nM=(X+ 1)Y\u0000(1\u0000X)\n(1 +Y) +X(Y\u00001): (61)\nUsing this expression, it is possible to write down the\nlimiting angle for the Andreev processes as\n\u0012cA\n#= arcsin\"r\n1\u0000M\n1 +M#\n: (62)\nThis implies that the limiting angle for the Andreev re-\n\rections of spin-up electrons (and holes) does not depend\non the mechanism responsible for the ferromagnetic or-\nder, but only depends on the value assumed by the mag-\nnetization, as we plot in Fig. 16. On the other hand,\nthe critical angle for the transmission in S crucially de-\npends on the value of XandY, as shown in Fig. 17. For\nany \fxed value of the magnetization, it takes a smaller\nvalue in the case of the pure SMM mechanism compared\nto the Stoner one. However, the comparison between the\ncritical angle for transmission into S with that for the An-\ndreev re\rection shows that the actual limit to the injec-\ntion cone of particles from one ferromagnetic lead to the\nother comes from the Andreev re\rections, since the cor-\nresponding limiting angle is systematically smaller than\nthat for the transmission into the S side, both for the\nStoner [Fig. 18(a)] and for the SMM mechanism [Fig. 18\n(b)]. In principle, virtual Andreev re\rections character-\nized by imaginary momenta could also be allowed for in-\njection angles above \u0012cA\n#, but in this case we have found\nno solution to our system of equations.\nAppendix E\nHere we show that, di\u000berently from the Stoner case, for\nSMM ferromagnetic layers the probabilities correspond-\ning to the di\u000berent scattering processes at transparent\ninterfaces ( Z= 0) and perpendicular injection direction\n(\u0012\u001b= 0) are independent of the spin orientation of the\ninjected particles. This feature comes from a symmetry17\nbetween spin-up and spin-down carriers exhibited by the\nsystem of coupled linear equations (25), which only holds\nin the SMM case at Z= 0 and\u0012\u001b= 0. In the following we\ndemonstrate it in the case where carriers are electrons,\nbut it also holds in the hole case. To this purpose, we\nnote that system (25) can be written in a compact form\nas\n^Me\u001bXe\u001b=Ye\u001b; (63)where Xe\u001bis the vector of the unknown variables Xe\u001b=\n(ae\u0016\u001b;be\u001b;ce\u001b;de\u0016\u001b;\u000be\u001b;\fe\u0016\u001b;\re\u001b;\u0011e\u0016\u001b):In the case of in-\njected electrons with spin \u001b, the matrix ^Me\u001band the\nvector Ye\u001bare\n^Me\u001b=0\nBBBBBBBBBB@0\u00001 0 0 u0v0u0v0\n\u00001 0 0 0 v0u0v0u0\n0 0 \u0000ei~q\u001bl0 u0\u0006+\nev0\u0006\u0000\nhu0\u0006\u0000\nev0\u0006+\nh\n0 0 0 \u0000e\u0000i~q\u0016\u001blv0\u0006+\neu0\u0006\u0000\nhv0\u0006\u0000\neu0\u0006+\nh\n0iZ+ ~q\u001b=p\nY 0 0 u0\u0000v0\u0000u0v0\n\u0000~q\u0016\u001bp\nY+iZ 0 0 0 v0\u0000u0\u0000v0u0\n0 0 ei~q\u001bl(~q\u001b=p\nY+iZ) 0 \u0000u0\u0006+\nev0\u0006\u0000\nhu0\u0006\u0000\ne\u0000v0\u0006+\nh\n0 0 0 \u0000e\u0000i~q\u0016\u001bl(~q\u0016\u001bp\nY\u0000iZ)\u0000v0\u0006+\neu0\u0006\u0000\nhv0\u0006\u0000\ne\u0000u0\u0006+\nh1\nCCCCCCCCCCA\nand\nYe\u001b=\u0010\n1;0;0;0;~q\u001b=p\nY\u0000iZ;0;0;0\u0011\n;\nhaving de\fned \u0006\u0006\ne;h=e\u0006i~ke;hLandl=L kF.\nForZ= 0 and\u0012\u001b= 0, the above matrices for the\ntwo spin species are related to each other through the\nfollowing relations:\nS^Me\"=^Me#U (64)\nSYe\"=Ye#: (65)\nHere Sis a o\u000b-diagonal block matrix S=\u0011x\r0,\r0being\none of the 4x4 Dirac matrices, and \u0011xis a 8x8 matrix\nde\fned as\n\u0011x=\u0012\n0I4\nI40\u0013\n; (66)\nI4is the 4x4 unit matrix and Uis a diagonal matrix de-\n\fned as\nU= diag\u0000\ns2;\u00001;e{l\u0000;\u0000s2e{l\u0000;s;\u0000s;\u0000s;s\u0001\n;(67)\nwith \u0000 =\u00001+s2\ns, ands=Y1=4\n1. From Eqs. (64)-(65)\nit follows that the unknown vector for injected spin-\ndown electrons is linked to that for the injected spin-\nup ones through the relation: Xe#=UXe\". More-\nover, in the SMM case the probability vector de\fned asP\u001b= (Ae\u001b;Be\u001b;Ce\u001b;De\u001b), can be expressed in terms of\nthe unknown coe\u000ecients vector ~X\u001b= (ae\u0016\u001b;be\u001b;ce\u001b;de\u0016\u001b)\nas reported in Eqs. (26)-(29). In matrix form, we have\nP\u001b=~Xy\n\u001bR\u001b~X\u001b (68)\nwith\nR\"=0\nB@s20 0 0\n0 1 0 0\n0 0 1 0\n0 0 0s21\nCA (69)\nandR#= (R\")\u00001.\nUsing Eqs. (63)-(65), we can write\nP#=~Xy\n#R#~X# (70)\n=\u0010\nU~X\"\u0011y\nR#\u0010\nU~X\"\u0011\n(71)\n=~Xy\n\"\u0010\n~UyR#~U\u0011\n~X\" (72)\n=~Xy\n\"R\"~X\"=P\" (73)\nwith ~Uy~U= (R\")2. This demonstrates the equality of the\nprobability amplitudes for the injection of spin-up and\nspin-down electrons in the SMM case. 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H´ etet1\n11Laboratoire De Physique de l’ ´Ecole Normale Sup´ erieure,\n´Ecole Normale Sup´ erieure, PSL Research University,\nCNRS, Sorbonne Universit´ e, Universit´ e Paris Cit´ e ,\n24 rue Lhomond, 75231 Paris Cedex 05, France.\n(Dated: December 23, 2022)\nWe present a study on the trapping of hard ferromagnetic particles using alternating magnetic\nﬁelds, with a focus on planar trap geometries. First, we realize and characterize a magnetic Paul trap\ndesign for millimeter-size magnets based on a rotating magnetic potential. Employing a physically\nrotating platform with two pairs of permanent magnets with opposite poles, we show stable trapping\nof hard ferromagnets a centimeter above the trap and demonstrate that the particle shape plays a\ncritical role in the trapping. Finally, we propose a chip trap design that will open a path to studies\nof gyromagnetic eﬀects with ferromagnetic micro-particles.\nIn the 50’s, Wolfgang Paul proposed to employ alter-\nnating electric ﬁelds to solve the conundrum imposed\nby Maxwell’s equations [1] which prohibits levitation of\ncharged particles using static electric ﬁelds. Although\nthe electric ﬁeld is zero on average, the slight particle\ndisplacement during one period of the electric ﬁeld al-\nternation can lead to an eﬃcient dynamical stabilization\nof the particle. Since then, Paul traps found countless\napplications such as in mass spectrometry and RF spec-\ntroscopy [2], ground state cooling of atoms [3], quantum\ncomputing [4] or more recently to quantum engineering\nof levitating particles [5, 6]. Trapping magnets is also an\nimportant endeavor that gained interest lately. A par-\nticularly intriguing direction is the search for atomic-like\neﬀects on magnet motion, stemming from the spin de-\ngree of freedom [7–9]. Observing such eﬀects is under\nreach using trapped nano-ferromagnets or particles con-\ntaining a large number of polarized spins and could lead\nto several applications in gyroscopy, magnetometry [10],\nspin-mechanics [11, 12], or in fundamental tests of quan-\ntum mechanics [13].\nThere exists several magnetic levitation protocols for\nferromagnets. For instance, adding angular momentum\nto a magnet on top of toroidal magnetic ﬁeld can provide\nstable trapping. The mechanisms behind this so-called\nLevitron were attributed to the combined action of the\ngyroscopic stability and magnet’s precession [14], both\nof which act to continuously align the magnet precession\naxis to the local magnetic ﬁeld direction. Another widely\nused method is electromagnetic suspension (EMS) [15],\nwhich uses servo-loops to counteract deviations of the\nparticle motion away from a desired working point. A\nlast example that shows eﬃcient magnet levitation em-\nploys the diamagnetism of superconductors [12, 16–18],\nwith large reported mechanical quality factors [12, 19]\nand foreseeably high control in the engineering of the\nmagnet quantum motion.\nFollowing the same logic as for charged particles, a\ntrap that uses alternating magnetic ﬁelds may also oﬀer\nstrong harmonic conﬁnement of magnets together with\nroom temperature operation. A major diﬀerence between\nmagnetic Paul traps (MPT) and electric Paul traps is\nΩa)\nNS\nb)\n-0.02\n20-0.01\n200\n100.01\n100.02\n0 0-10 -10-20-20\n-0.015-0.01-0.00500.0050.010.015Ω\nx(mm) y(mm)Bz(x,y)TeslaFIG. 1. a) Sketch of the magnetic Paul trap. b) Numerical\nsimulations showing the projection of the magnetic ﬁeld vec-\ntor on thezaxis as a function of xandy, at a distance z= 1\ncm from the trap.\nthat the magnetic energy depends on the ferromagnet ori-\nentation and thus the angular dynamics must be also be\ncontrolled for stable levitation. A strong enough homoge-\nneous magnetic ﬁeld can nevertheless ﬁx the direction of\nthe magnet dipole moment without exerting any force on\nthe magnet. The method was in fact already realized [20]\nshortly after having been employed for trapping neutralarXiv:2212.11622v1  [quant-ph]  22 Dec 20222\natoms in the early days of Bose-Einstein condensation\n[21]. The authors of Ref. [20] used a combination of per-\nmanent magnets and time varying currents in Helmholtz\ncoils to enable ponderomotive conﬁnement. Similar traps\nfor large particles were then soon implemented [22–24].\nIn the present article, we demonstrate and characterize\na magnetic Paul trap using alternating magnetic ﬁelds\ncoming from physically rotating permanent magnets in a\nplanar geometry. We then propose a planar on-chip de-\nsign that will enable trapping of micro-metric particles,\nas well as enabling straightforward optical and inductive\ndetection of their motion.\nWe start by showing the trapping of a ferromagnet for\nall 6 degrees of freedom, namely the 3 center of mass\nmodes and the 3 librational modes, in a room tempera-\nture table-top set-up. Instead of the stationary-wave ge-\nometry typically employed in electric Paul traps, here we\nuse the running-wave version, where the ﬁeld curvatures\nin the (x,y) plane rotate over time. We rotate magnetic\nﬁelds using a physically rotating platform holding rigidly\nﬁxed permanent magnets to produce a ponderomotive\npotential in the ( x,y) plane.\nFigure 1-a) is a depiction of the platform that we em-\nploy. Opposite pairs of 5 mm square magnets are oriented\nso that their magnetic moments point to the same direc-\ntion. The magnets are glued on the periphery of a copper\nbaseplate. The distance between the center of the oppo-\nsite magnet pairs is 20 mm. The assembly is attached to\nthe spinning arm of the motor from a mechanical chop-\nper. The motor can make this system rotate at a maxi-\nmum frequency of 100 Hz, thus generating fast rotating\nmagnetic ﬁelds along the x,ydirections. The zdirection\ncan also be conﬁned when the particle shape is taken into\naccount, as will be explained later.\nConsider a particle of mass mwhich can move in a\nplane subjected to a saddle potential rotating at the fre-\nquency Ω/2π. We designate by ( x,y) the spatial co-\nordinate of the particle in the laboratory-ﬁxed frame\nand (X,Y ) the coordinate in the rotating saddle frame.\nAdding an extra homogeneous magnetic ﬁeld along z\nﬁxes the particle angle along the zdirection. As shown\nin the Supplemental Material (SM), because of the typ-\nically large moment of inertia of the particles we trap,\nonly a moderate homogeneous B ﬁeld (in the mT range)\nis enough to ensure that the particle magnetic moment\ndirection is not being deﬁned by the B ﬁelds from the\nrotating magnets.\nFigure 1-b) is the result of numerical simulations of\nthe static magnetic ﬁeld component along the zdirection\nas a function of xandyat a distance z= 1 cm from\nthe four-magnets trap. The magnetic ﬁeld reaches about\n0.1 T at maximum, and features a hyperbolic paraboloid\nshape close to ( x,y) = (0,0). In the rotating frame, the\nmagnetic ﬁeld projection along zaround the saddle point\nreads :\nBz(X,Y ) =1\n2B/prime/prime\nz(z)/parenleftbig\nX2−Y2/parenrightbig\n, (1)\nwhereB/prime/prime\nz(z) is the curvature of the magnetic ﬁeld com-ponent along z. In this frame, two forces must be added:\nthe centrifugal force Fcen=mΩ2(XeX+YeY) and the\nCoriolis force FCor=−2m(ΩeZ)×(˙XeX+˙YeY). The\nconﬁnement along xandycomes from the competition\nbetween FcenandFCor. In the laboratory-ﬁxed frame,\nthe magnetic ﬁeld is time-dependent and reads:\nBz(r,t) =B/prime/prime\nz(z)\n2/parenleftbig\n(x2−y2) cos (2Ωt)−2xysin (2Ωt)/parenrightbig.\n(2)\nThe coupled set of equations of motions for the center\nof mass coordinates subjected to Bz(r,t) can be solved\nstraightforwardly by averaging over one period of the\ntrap oscillation (see SM, section 1). This procedure leads\nto ponderomotive conﬁnements both in the xandydi-\nrections at trapping frequencies that are given by\nωx,y\n2π=Bsat|B/prime/prime\nz(z)|\nµ0πρmΩ. (3)\nHere,ρmis the magnet density and Bsat≈1 T is\nthe magnetization at saturation of the magnet. Using\nΩ/2π≈100 Hz, we obtainωx,y\n2πin the Hz range when\nthe magnet lies a centimeter above the trap. Notably,\nin this geometry, the magnetic ﬁeld is zero at the center\n(x,y) = (0,0) at any position along z. If the particle\nwas a point-like magnet, there would be no conﬁnement\nin thezdirection. A homogeneously magnetized particle\nwhose form is invariant by π/2 rotations about an axis\nes, will also not experience any force along zif the sym-\nmetry axisescoincides with ez. Indeed, the total force\nexerted by the four magnets on the whole body cancels\nout at any point in time. However, any deviation of the\nparticle symmetry axis from the zdirection imposed by\nthe external homogeneous magnetic ﬁeld may give rise\nto a net force along zonto the particle. When aver-\naged over a cycle of the trap rotation, the particle would\nfeel a ponderomotive force that pushes it away from the\nz= 0 point. Another situation which could give rise to\nsuch a force along zis when the particle is asymmetrical.\nTaking a parallelepiped with magnetic moment along z,\nwith a length lalongxand a square cross-section with a\nside length h<l , one ﬁnds (See SM section II) that the\ntime-dependent potential along the zdirection reads :\nEmag(r,t)≈1\n2mω2\nzcos (2ωt)z2, (4)\nwhere\nωz=/radicalBigg\nBsata2\n12µ0ρm(l2−h2). (5)\nNote that the radial potential is not impacted by this\nshape eﬀect to ﬁrst order. As expected the force is pro-\nportional to a2=∂2B/prime/prime\nz(z)/∂2zand the decrease of the\nmagnetic curvature B/prime/prime\nz(z) alongzgives rise to an out-\nward force when averaged over one oscillation period.\nThere will, in turn, be a local energy minimum thanks3\niii) i)ii)\nFIG. 2. Photos showing three magnets levitating above the rotating magnetic saddle: i) a 3 mm cube, ii) a 10 mm long cylinder\nwith a 5 mm diameter and iii) a parallelepiped with short side-length of 1 mm and a 5 mm ×5 mm, ﬂat square magnet. In\nthese experiments, the magnets do not rotate over time in any direction (see movie in the Supplementary materials).\nto gravity, thus conﬁning the particle at a distance d≈1\ncm from the trap (see SM section 1).\nExperimentally, we ﬁnd that at frequencies Ω /2πin the\n50 to 100 Hz range, millimeter scale magnets can stably\nlevitate for extended periods of time (more than hours)\nat a distance d≈15 mm from the rotating baseplate\nwhere magnets are glued. Fig 2 i) to iii) show three pho-\ntos of levitating magnets of diﬀerent shapes and sizes. All\nthree are neodymium iron boron magnets with a nickel\ncoating. The three magnets are perfectly stable for all 6\ndegrees of freedom and over extended periods of time. A\nvideo of a levitating magnet can also be found in the sup-\nplemental materials. We found that placing magnets in a\nsmall water container above the trap and then removing\nthe water using a syringe once they are stably trapped\nensures eﬃcient loading. The ease with which we can\ntrap under water most likely stems from the increased\ncapture volume as the damping is increased. After op-\ntimizing external homogeneous magnetic ﬁeld directions\nand amplitudes using an extra permanent magnet, all of\nthese magnets can be made stable angularly. The homo-\ngeneous magnetic ﬁeld can indeed change the orientation\nof the magnet without perturbing the particle center of\nmass, as long as the magnetic ﬁeld gradients at the parti-\ncle location are negligible compared to the eﬀective time-\naveraged magnetic ﬁeld gradients from the saddle. We\nalso found that the stator of the motor itself generates a\nstrong enough static magnetic ﬁeld that can also orient\nthe magnet.\nNumerical simulations (See SM, section 1-B) show\ngood agreement with the experimental observations. In\nparticular, the observed trapping height agrees with a\nmodel taking into account the particle shape. We also\nfound further evidence that conﬁnement along the z\ndirection arises because of the ﬁnite particle size by\nperforming experiments with a millimeter-size spherical\nmagnet. Conﬁnement along zwas not observed for such\nparticles, in agreement with Eq. 5, which predicts no con-\nﬁnement ifl≈hin the trap center. Note that this is true\nonly for small enough particles (See SM section I-B fora description of the limits of the model). In order to\nmeasure the zdependent force when this particle is dis-\nplaced from the trap center, we enclose the particle in a\nsmall glass tube and read-out its zmotion as the tube\nis displaced transversally. Fig. 3-b) shows a picture of\nthe set-up and Fig. 3-a) is a schematic showing the em-\nployed parametrization. Fig. 3-c) are data where the\ncenter of mass position zis monitored as a function of x.\nThe particle experiences no outward force when the tube\nis positioned such that |x|<2.5 mm and remains at the\nbottom of the tube. As the tube is displaced, the particle\nis lifted up, with a maximum height at |x|≈19 mm, and\nthen falls down again. This behavior is very well captured\nby numerical simulations where the outward force comes\nfrom the competition between ponderomotive force and\ngravity. Note also that, although extensive tests have\nbeen made, no trapping could be obtained after rotating\nthe whole apparatus upside-down, which shows that the\nzconﬁnement is not solely due to the rotating magnetic\nﬁeld and that gravity plays a role. One last check is the\ndependency of the particle position with Ω /2π. Fig. 3-\nd) shows the height zof a trapped 2.5 mm side cubic\nmagnet (Fig. 2), as a function of Ω /2π.zis seen to\nincrease as the trap rotation frequency is increased, as\nexpected from the ponderomotive nature of the outward\nforce competing with gravity.\nThe oﬀ-centered position of the particle along zis very\nconvenient for the large optical access required for optical\nread-out of the motion. It will also facilitate the particle\nlevitation in a vacuum chamber without having to inte-\ngrate the rotating platform. Such planar traps are thus\nvery attractive. As is the case with magnetic traps for\nneutral atoms or with ion traps, the most versatile way\nto trap particles while reaching high trapping frequencies\ntogether with open access is to employ chip-based designs\nconsisting of current-carrying wires. Here, we propose\nthe planar trap design depicted in Fig. 4-a) where an al-\nternating current runs through a micro-wire in an exter-\nnally applied homogeneous ﬁeld B0=B0ez. The total\napplied magnetic ﬁeld Btot(r,t) on the ferromagnet at4\nc)\nΩ\n76 80 84 88 92 9614.014.414.815.215.6 \nRotation frequency (Hz)Particle height z (mm)d)zxB0\n1416182022\n0x (mm)z (mm)-20 -10 10 20a) b)MagnetGlass tube\nFIG. 3. a) Schematic showing the parametrization of the particle coordinates. b) Picture of the set-up showing the glass tube\nwhere the particle is enclosed. c) Measured position of the center of mass of a spherical 1 mm diameter magnet as a function\nof the distance from the platform rotation axis. The central blue area indicates the range of xvalues where the particle sits\nat the bottom of the glass tube. d) Position of the center of mass of a square magnet along zas a function of the rotation\nfrequency Ω /2π.\nthe center of this ring is the sum of a homogeneous ﬁeld\nB0and an oscillating parabolic magnetic ﬁeld B1(r,t)\nwhich reads\nB1(r,t) =B/prime/prime\n1\n2/bracketleftBig\n(z2−1\n2(x2+y2))ez\n−(xzex+yzey)/bracketrightBig\ncos (Ωt), (6)\nwhere ( ex,ey,ez) is the laboratory-ﬁxed axis, ( x,y,z ) are\nspatial coordinates, B/prime/prime\n1is the curvature of B1(r,t) and\nΩ/2πis the frequency of the magnetic ﬁeld oscillation.\nWe consider a hard ferromagnetic neodymium micro-\nsphere with a radius a= 1µm. We use the zyz conven-\ntion for the three Euler angles ( α,β,γ) that parametrize\nthe orientation of the magnet as well as the body-ﬁxed\nmagnetic moment orientation µ(see SM, section II). The\nnorm of the magnetic moment is µ=BsatV/µ 0, whereV\nthe volume of the ferromagnet and Bsat≈1.0 T is the\nmagnetization at saturation. In the ( x,y,z ) basis, the\nmagnetic moment is expressed as\nµ=−µ(−cαs˜βcγ−sαsγ)ex\n−µ(cαsγ−sαs˜βcγ)ey−µc˜βcγez, (7)\nwhereciandsidesignate the cosine and sine function\nwith arguments iand˜βis the shifted nutation angle de-\nﬁned as ˜β=β−π/2. By performing a second order Tay-\nlor expansion of the microsphere magnetic energy Emag=\n−µ·Btot(r,t) around the position ( ˜β,γ,x,y,z ) = 0, weobtain:\nEmag=µB0/parenleftBigg\nγ2\n2+˜β2\n2/parenrightBigg\n−µB/prime/prime\n1\n2/parenleftbigg\nz2−1\n2(x2+y2)/parenrightbigg\ncos (Ωt). (8)\nWithin the stability conditions (See SM, section 1-B),\nthe alternating magnetic potential leads to a pseudo-\nharmonic conﬁnement of the center of mass modes. After\naveraging the potential over one period of the magnetic\nﬁeld oscillation, we obtain the secular frequencies:\nωγ=ω˜β=/radicalBigg\n5\n2B0Bsat\nµ0ρma2(9)\nand\n˜ωz= 2˜ωx= 2˜ωy=Ω\n2|qz|√\n2, (10)\nwhereqz=−2qx=−2qy= 2B/prime/prime\n1Bsat/(µ0ρmΩ2).Tak-\ningqz= 0.2, where the secular harmonic approxima-\ntion is valid and the following experimentally accessi-\nble values : B0= 10 mT, B/prime/prime\n1= 105T.m−2and\nΩ = (2π)2.0 kHz, we obtain ωγ=ω˜β= (2π)300 kHz\nand ˜ωz= 2˜ωx= 2˜ωy= (2π)100 Hz. These trapping fre-\nquencies are on the order of the trapping frequencies that\ncan be achieved using superconducting traps for micro-\nmagnets [12, 19], albeit here using a room temperature5\na)\nb)\nFIG. 4. a) Depiction of the current carrying trap design.\nb) Simulations showing the center of mass motion in the x\ndirection (blue curve) and the zdirection (red curve) for a\ntrapping frequency Ω /2π= 1.7 kHz, an initial position x0=\n0,z0= 0 and an initial velocity ˙ x0/negationslash= 0, ˙z0/negationslash= 0.\nset-up. Fig. 4-b) shows the result of numerical simula-\ntions where the center of mass motion for the x(blue\ncurve) and z(red curve) modes evolve as a function of\ntime for a Paul trap frequency Ω /2π= 1.7 kHz. We see\nthat as the magnet is displaced away from the position\nx,z= 0, a small high frequency oscillation appears after\nwhich the particle is brought back to x,z= 0. This ob-\nservation is in accordance with the expected principle of\nthe ponderomotive conﬁnement and in good agreement\nwith the above analytical calculations.\nThe shift of the center of mass position induced by\ngravity isz0=−g/˜ω2\nz≈ − 20µm. Such a shift will\ngenerate micromotion in the ezdirection as well as a\nsmall coupling between the center of mass and the an-\ngular modes. We found that the gravity can however\nbe compensated for by an external electric ﬁeld if the\nparticle is electrically charged or using a magnetic ﬁeld\ngradient such that B2=B/prime\n2(zez−x/2ex−y/2ey) withB/prime\n2=mg/µ = 8.0×10−2T.m−1. The coupling between\nthe center of mass and the angular modes induced by this\nextra ﬁeld can safely be neglected here (See SM, section\n2).\nWhile it is straightforward to experimentally gener-\nate the magnetic ﬁeld B0, the generation of an oscillat-\ning magnetic ﬁeld with a curvature B/prime/prime\n1= 105T.m−2\nrequires a carefully engineered trap. We propose to em-\nploy a high-purity gold double-loop micro-trap with radii\nr1= 100µm andr2= 200µm made by lithography, on a\nsilicon chip with alternating currents ik(t) =ikcos (ωt),\nk={1,2}with−2i1=i2. The condition i1/i2=−r1/r2\nfurther suppresses the magnetic ﬁeld at r= 0 while keep-\ning its harmonic dependency. The magnetic ﬁeld curva-\nture reads B/prime/prime\n1=−9\n16µ0i1/r3\n1. This type of microchip\ncan carry current of 105A/cm2without experiencing too\nmuch heating [25, 26]. With a 50 µm large and 2 µm\nthin gold layer, the current i1equals 0.1 A which leads\nto the desired value B/prime/prime\n1≈105T.m−2. The induced cur-\nrents generated by one loop onto the other are found to\nbe negligible (See SM, section 3). The Eddy currents cre-\nated inside the levitating ferromagnet are also negligible\ncompared to the gas dissipation process even under high\nvacuum (see SM, section 3). We thus expect no heating\nof the particle in this parameter regime.\nThe ability to control translational and rotational de-\ngrees of freedom of trapped particles with high precision\nhas led to new opportunities for fundamental and applied\nresearch [6]. The high sensitivities of levitated objects\nto forces and torques in the motional ground state mo-\ntivates research on tests of spin-mechanical coupling in\nthe mesoscopic regime where gyromagnetic eﬀects play a\nrole. It is arguably beneﬁcial to enrich the set of tools\navailable to release some constraints that may be present\nin some trapping systems. As an example, optical levi-\ntation is typically prone to particle heating [27–29] and\ntrapped charged particles to not lend themselves easily\nto approaching objects such as micro-fabricated optical\ncomponents or other two-level systems [11] at distances\nbelow the micron. The presented magnetic Paul trap\nseems not to suﬀer from these problems and will thus\nenrich this already multidisciplinary research ﬁeld.\nACKNOWLEDGMENTS\nWe would like to thank fruitful discussions with R. Fol-\nman, D. Budker, J. Wei, R. Blatt, O. Romero-Isart and\nP. Huillery. GH acknowledges funding from the Ile-de-\nFrance region in the framework of the DIM SIRTEQ. This\nproject also received funding from the European Union’s\nHorizon 2020 Research and Innovation Programme under\nGrant Agreement no. 731473 and 101017733.\n[1] W. Paul, Electromagnetic traps for charged and neutral\nparticles, Rev. Mod. Phys. 62, 531 (1990).[2] M. Welling, H. Schuessler, R. Thompson, and6\nH. Walther, Ion/molecule reactions, mass spectrometry\nand optical spectroscopy in a linear ion trap, Interna-\ntional Journal of Mass Spectrometry and Ion Processes\n172, 95 (1998).\n[3] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jeﬀerts,\nW. M. Itano, D. J. Wineland, and P. Gould, Resolved-\nsideband raman cooling of a bound atom to the 3d zero-\npoint energy, Phys. Rev. Lett. 75, 4011 (1995).\n[4] J. I. Cirac and P. Zoller, Quantum computations with\ncold trapped ions, Phys. Rev. Lett. 74, 4091 (1995).\n[5] M. Perdriat, C. Pellet-Mary, P. Huillery, L. Rondin,\nand G. H´ etet, Spin-mechanics with nitrogen-vacancy\ncenters and trapped particles, Micromachines 12,\n10.3390/mi12060651 (2021).\n[6] C. Gonzalez-Ballestero, M. Aspelmeyer, L. Novotny,\nR. Quidant, and O. Romero-Isart, Levitodynam-\nics: Levitation and control of microscopic ob-\njects in vacuum, Science 374, eabg3027 (2021),\nhttps://www.science.org/doi/pdf/10.1126/science.abg3027.\n[7] D. F. Jackson Kimball, A. O. Sushkov, and D. Budker,\nPrecessing ferromagnetic needle magnetometer, Phys.\nRev. Lett. 116, 190801 (2016).\n[8] C. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac,\nand O. Romero-Isart, Quantum spin stabilized magnetic\nlevitation, Phys. Rev. Lett. 119, 167202 (2017).\n[9] K. 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Schmiedmayer, Controlling cold atoms\nusing nanofabricated surfaces: Atom chips, Phys. Rev.\nLett.84, 4749 (2000).\n[26] L. Feenstra, L. M. Andersson, and J. Schmiedmayer,\nMicrotraps and atom chips: Toolboxes for cold atom\nphysics, General Relativity and Gravitation 36, 2317\n(2004).\n[27] A. T. M. A. Rahman, A. C. Frangeskou, M. S. Kim,\nS. Bose, G. W. Morley, and P. F. Barker, Burning and\ngraphitization of optically levitated nanodiamonds in\nvacuum, Scientiﬁc Reports 6, 21633 EP (2016).\n[28] L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and\nA. Nick Vamivakas, Observation of nitrogen vacancy pho-\ntoluminescence from an optically levitated nanodiamond,\nOptics Letters 38, 2976 (2013).\n[29] T. M. Hoang, J. Ahn, J. Bang, and T. Li, Electron spin\ncontrol of optically levitated nanodiamonds in vacuum,\nNature Communications 7, 12250 EP (2016).Supplemental Material\nPlanar Magnetic Paul Traps for Ferromagnetic Particles\nM. Perdriat, C. Pellet-Mary, T. Copie, G. H´ etet1\n1Laboratoire De Physique de l’ ´Ecole Normale Sup´ erieure,\n´Ecole Normale Sup´ erieure, PSL Research University,\nCNRS, Sorbonne Universit´ e, Universit´ e de Paris Cit´ e,\n24 rue Lhomond, 75231 Paris Cedex 05, France\nCONTENTS\nI. Rotating magnetic saddle 2\nA. The rotating saddle 2\nB. The rotating magnetic saddle 3\nII. Magnetic Paul trap with current carrying loops 5\nA. System and parametrization 6\nB. Hamiltonian of the system and equation of motion 7\nC. Derivation of the equation in the small motion limit 9\nD. Time-averaged equations 9\nE. Numerical value and elimination of the angular-CoM coupling terms 10\nIII. Calculation of the diﬀerent inductive current in the set-up 11\nA. Induction current in the micro-loops 11\nB. Induction current in the levitated ferromagnet 11\nReferences 12arXiv:2212.11622v1  [quant-ph]  22 Dec 20222\nI. ROTATING MAGNETIC SADDLE\nIn this section, we explain the origin of the conﬁnement of a particle subjected to a\nrotating saddle potential. We derive the stability condition and the macromotion equation\nas well as the resulting secular frequency. Last, we include particle size eﬀects in order to\nexplain the z-conﬁnement.\nA. The rotating saddle\nWe ﬁrst consider a particle of mass mwhich can move in a plane subjected to a saddle\npotential (non-magnetic at ﬁrst) rotating at the angular frequency ω. We designate by ( x,y)\nthe spatial coordinate of the particle in the laboratory-ﬁxed frame and ( X,Y ) the coordinate\nin the rotating saddle frame. The rotating saddle potential reads:\nU(X,Y ) =1\n2mω2\nr/parenleftBig\nX2−Y2/parenrightBig\n. (1)\nWriting the motion equation of the particle in the rotating frame, one adds the centrifugal\nforceFcen=mΩ2(XeX+YeY) and the Coriolis force FCor=−2m(ΩeZ)×(˙XeX+˙YeY).\nThe fundamental principle of dynamics gives the coupled set of equations:\n¨X−2Ω˙Y+/parenleftBig\nω2\nr−Ω2/parenrightBig\nX= 0, (2)\n¨Y+ 2Ω ˙X−/parenleftBig\nω2\nr+ Ω2/parenrightBig\nY= 0. (3)\nThis set of equations can be written as ˙U=AUwithU=t(X,Y, ˙X,˙Y) and\nA=\n0 0 1 0\n0 0 0 1\nΩ2−ω2\nr 0 0 2Ω\n0ω2\nr+ Ω2−2Ω 0\n. (4)\nThe particle is stable if and only if the eigenvalues of the matrix Ahave a negative real\npart. The eigenvalues verify the equality:\nλ4+ 2Ω2λ2+ Ω4−ω4\nr= 0, (5)\nand therefore:\nλ2=±ω2\nr−Ω2. (6)3\nThe eigenvalues have no positive real part if and only if Ω −ωr>0. Thus, we obtain the\nstability condition:\nΩ>ωr. (7)\nTo estimate the secular frequency, we will employ the approach proposed in [1]. The motional\nequation of the particle in the laboratory-ﬁxed frame reads:\n¨V+ω2\nrS(Ωt)V= 0, (8)\nwith\nV=t(x,y), (9)\nS(Ωt) =\ncos (2Ωt) sin (2Ωt)\nsin (2Ωt)−cos (2Ωt)\n. (10)\nUsing the transformation of the guiding-center [1]:\nW=V−1\n4/parenleftbiggωr\nΩ/parenrightbigg2\nS(ωt)/parenleftbigg\nV−1\nΩJ˙V/parenrightbigg\n,J=\n0−1\n1 0\n, (11)\nwe obtain a diﬀerential equation for W:\n¨W−1\n4ωr/parenleftbiggωr\nΩ/parenrightbigg3\nJ˙W+1\n4ω2\nr/parenleftbiggωr\nΩ/parenrightbigg2\nW=/parenleftbiggωr\nΩ/parenrightbigg4\nf/parenleftbigg\nω2\nrW,ωr˙W,ωr\nΩ/parenrightbigg\n, (12)\nwherefa linear function in ω2\nrW,ωr˙Wand analytic in ωr/Ω in a ﬁxed neighborhood of\nωr/Ω = 0. In the limit ωr/Ω→0, this equation results in a radial motion given by the\ncharacteristic conﬁning secular frequency ˜ ω= (2π)1\n2ω2\nr\nΩand by a precessional motion at the\ncharacteristic secular frequency ˜ ωprec=1\n4ω4\nr\nΩ3.\nB. The rotating magnetic saddle\nWe now consider the rotating magnetic saddle described in the main text. We model the\nlevitating magnet as a parallelepiped of length l, square cross-section with a side length h\nand volume V=l×h2. We suppose that the magnetic dipole of the magnet is oriented along\nz. The vector R= (X,Y,Z ) designates the spatial coordinate in the saddle rotating frame4\nand by r= (x,y,z ) the spatial coordinate in the laboratory-ﬁxed frame. The component\nalong theZdirection of the magnetic saddle in the rotating frame reads :\nBZ(R) =B/prime/prime\nZ(Z)\n2/parenleftBig\nX2−Y2/parenrightBig\n,. (13)\nto second order in X,Y . HereB/prime/prime\nZis an even function of Z. In the laboratory-ﬁxed frame,\nthis potential is time-dependent and reads:\nBz(r,t) =B/prime/prime\nz(z)\n2/parenleftBig\n(x2−y2) cos (2Ωt)−2xysin (2Ωt)/parenrightBig\n. (14)\nwhere Ω/2πis the rotation frequency of the magnetic saddle.\nWe suppose that an external homogeneous magnetic ﬁeld aligns the orientation of the\nmagnet in the zdirection and that the angle along the zaxis is also conﬁned. Experimen-\ntally, the conﬁnement is realized using an external permanent magnet. We can neglect the\ninﬂuence of the magnetic ﬁeld components in the xandydirections since it is perpendicular\nto the dipole of the magnet. The magnetic energy of an inﬁnitesimal element of volume dV\nthen equals to d Emag(r,t) =−M·B(r,t)dV =−MzBz(r,t)dV. We designate by ( x,y,z )\nthe spatial coordinate of the center of mass of the magnet. Integrating the magnetic energy\nover all the magnet volume, we obtain the total energy:\nEmag(r,t) =/integraldisplayh\n2+x\n−h\n2+x/integraldisplayl\n2+y\n−l\n2+y/integraldisplayh\n2+z\n−h\n2+z−M·BdV, (15)\nwhich yields\nEmag(r,t) =−MV/parenleftBigg/integraldisplayh\n2+z\n−h\n2+zB/prime/prime\nz(z/prime)\n2hdz/prime/parenrightBigg/parenleftBigg\ncos (2Ωt)/parenleftBigg1\n3/parenleftBiggh2\n4−l2\n4/parenrightBigg\n+x2−y2/parenrightBigg\n−2 sin (2Ωt)xy/parenrightBigg\n.\n(16)\nWe deﬁne the function F(z) =/integraltexth\n2+z\n−h\n2+zB/prime/prime\nz(z/prime)\n2hdz/primewhich is a even function of zbecauseB/prime/prime\nzis\na even function of z. We assume that the Taylor expansion of Fat second order in zis\nveriﬁed in the parameter values of zexplored such that F(z)≈a0+a2z2\n2. Keeping only the\nsecond order terms in the spatial coordinates, we obtain:\nEmag(r,t)≈−MVa 0/parenleftBig\ncos (2Ωt)/parenleftBig\nx2−y2/parenrightBig\n−2 sin (2Ωt)xy/parenrightBig\n−MVa2\n24/parenleftBig\nh2−l2/parenrightBig\ncos (2Ωt)z2.\n(17)\nWe obtain the energy:\nEmag(r,t)≈1\n2mω2\nr/parenleftBig\ncos (2Ωt)/parenleftBig\ny2−x2/parenrightBig\n+ 2 sin (2Ωt)xy/parenrightBig\n+1\n2mω2\nzcos (2Ωt)z2, (18)5\nwhere we introduced the relevant frequencies:\nωr/2π=/radicalBigg\n2Bsata0\nµ0ρm, (19)\nωz/2π=/radicalBigg\nBsata2\n12µ0ρm(l2−h2). (20)\nIn our experiment, the magnetic ﬁelds generated by the trap are strong enough so the\nmagnets are trapped outside the harmonic region along z. The position of the magnet\nin thezdirection is the result of a balance between the outward force from the rotating\nplatform and gravity. The pseudo- magnetic potential Ψ zalongz, namely the magnetic\nenergy averaged over one cycle of the trap rotation reads [2]:\nΨz=|∇z(µBz(z))|2\n4mΩ2, (21)\nwhereµ=BsatV/µ 0is the magnetic moment of the magnet. Bz(z) is the time independent\nprefactor in the energy. Here, using 15, we ﬁnd\nBz(z) =−MVF (z)1\n12(h2−l2). (22)\nFig 2, trace (i), shows the result of a numerical simulation of Ψ z, normalized to the particle\nvolumeVas a function of the distance from the trap center. ∇zF(z) was estimated by ﬁrst\nﬁtting each of the numerically-obtained static magnetic potentials along yat each position\nz. Fig.1b) in the main text, shows one such numerical simulation at a position z= 10 mm.\nThe modulus of the magnetic ﬁeld curvature is the same along x. This procedure enables\nto extractB/prime/prime\nz(z/prime). The calculation is followed by a numerical integration of B/prime/prime\nz(z/prime) alongz/prime,\nfrom−h\n2−zto−h\n2+z, and a subsequent derivation with respect to z. The parameters\nused in ﬁg 2 are l= 10 mm, h= 4 mm, Ω /2π= 80 Hz and Bsat= 1 T. Trace iii) is\nthe gravitational potential ρgz. Trace (iii) shows the sum of trace (i) and (ii), showing a\npotential minimum at d≈11.5 mm.\nII. MAGNETIC PAUL TRAP WITH CURRENT CARRYING LOOPS\nIn this section, we provide more details about the proposed current carrying loop design.\nWe give the equations of motion, in the limit of small displacement, taking into account\na magnetic ﬁeld gradient to compensate the eﬀect of gravity. We verify that the coupling\nbetween the angular and the center of mass (CoM) modes induced by the magnetic ﬁeld\ngradient term is negligible.6\n0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022020040060080010001200140016001800\nDistance from trap center (m)Potential energy (in J/m3)\n(i)(ii)(iii)\nFIG. 1. Numerical simulations showing the pseudo-potential energy along zas a function of the\ndistance from the trap center (trace i), the gravitational potential (trace (ii)) and the total potential\n(trace (iii)).\nγ,β\nxyz α\nFIG. 2. Schematics showing the parametrization of both the center of mass and angular degrees\nof freedom.\nA. System and parametrization\nWe designate by Oe1e2e3the body-ﬁxed reference frame of the magnet and by Oexeyez\nthe laboratory frame. We use the Euler angle u=t(α,β,γ ) in thezyz convention to7\nparametrize the angular motion of the magnet such thatt(e1,e2,e3) =tR(u)t(ex,ey,ez)\nwith:\nR(u) =Rz(α)Ry(β)Rz(γ) =\ncα−sα0\nsαcα0\n0 0 1\n\ncβ0sβ\n0 1 0\n−sβ0cβ\n\ncγ−sγ0\nsγcγ0\n0 0 1\n. (23)\nwherecν= cos (ν) andsν= sin (ν). Deriving the product, we get:\nR(u) =\ncαcβcγ−sαsγ−sαcγ−cαcβsγcαsβ\ncαsγ+sαcβcγcαcγ−sαcβsγsαsβ\n−sβcγ sβsγcβ\n. (24)\nThe magnetic momentum of the ferromagnet is supposed to be attached to the particle.\nWe consider that the magnetic momentum is oriented along the −e1axis such that µ=−µe1\nwithµ>0. In the laboratory-ﬁxed coordinate system, we have:\nµ=−µ((cαcβcγ−sαsγ)ex+ (cαsγ+sαcβcγ)ey−sβcγez) (25)\nWe perform the angular change of variable β=˜β+π/2 and obtain\nµ=−µ/parenleftBig\n(−cαs˜βcγ−sαsγ)ex+ (cαsγ−sαs˜βcγ)ey−c˜βcγez/parenrightBig\n(26)\nWe consider a total magnetic ﬁeld Btot(r,t) composed of three diﬀerent magnetic ﬁelds, a\nconstant ﬁeld B0, an oscillating harmonic magnetic ﬁeld B1(r,t) and a constant magnetic\nﬁeld gradient B2which reads:\nB0=B0ez, (27)\nB1(r,t) =B/prime/prime\n1\n2cos (Ωt)/parenleftBigg\nz2−x2+y2\n2/parenrightBigg\nez\n−B/prime/prime\n1\n2cos (Ωt) (xzex+yzey),(28)\nB2=B/prime\n2(zez−x/2ex−y/2ey). (29)\nB. Hamiltonian of the system and equation of motion\nThe total Hamiltonian reads:\nH=p2\n2m+L2\n2I−µ·Btot(r,t) +mgz, (30)8\nwhere p=pxex+pyey+pzezis the CoM momentum, L=Lxex+Lyey+Lzezis the angular\nmomentum. We have\nL2\n2I=(pα+pγsin˜β)2\n2Icos˜β2+p2\n˜β\n2I+p2\nγ\n2I. (31)\nThe angular momenta pα,pβandpγare linked to the angles by the relations:\np˜β=I˙˜β, (32)\npα=I( ˙α−˙γsin˜β), (33)\npγ=I( ˙γ−˙αsin˜β). (34)\nFinally, we get the equations of motion:\ndpx\ndt=µ·∂B\n∂x, (35)\ndpy\ndt=µ·∂B\n∂y, (36)\ndpz\ndt=−mg+µ·∂B\n∂z, (37)\ndpα\ndt=∂µ\n∂α·B, (38)\ndp˜β\ndt=1\nIcos˜β3(pαsin˜β+pγ)(pα+pγsin˜β) +∂µ\n∂˜β·B, (39)\ndpγ\ndt=∂µ\n∂γ·B. (40)\nWe used these equations to simulate the motion of the magnet.9\nC. Derivation of the equation in the small motion limit\nWe can calculate the forces and torques to ﬁrst order in the motion variables x,y,z, ˜β,γ\nand we obtain:\nFx=−µB/prime/prime\n1\n2cos (Ωt)x−µB/prime\n2\n2(cα˜β+sαγ), (41)\nFy=−µB/prime/prime\n1\n2cos (Ωt)y−µB/prime\n2\n2(sα˜β−cαγ), (42)\nFz=−mg+µB/prime\n2+µB/prime/prime\n1cos (ωt)z, (43)\nΓα= 0, (44)\nΓ˜β=−µB0˜β−µB/prime\n2\n2(cαx+sαy), (45)\nΓγ=−µB0γ−µB/prime\n2\n2(sαx−cαy). (46)\nIn order to compensate the displacement along the axis ezdue to the gravity, we use a\nﬁeld gradient which is deﬁned by the equality µB/prime\n2=mg. This condition does not depend\non the size of the particle since both the magnetic momentum and the mass depends on the\nvolume of the magnet. We obtain:\nB/prime\n2=µ0ρmg\nBsat. (47)\nWe takeµ0= 4π×10−7T.m.A−1,ρm= 7.0×103kg.m−3,g= 9.8 m.s−2andBsat= 1.0 T\nwhich gives the condition:\nB/prime\n2= 8.6×10−2T.m−1. (48)\nD. Time-averaged equations\nLet us now derive the secular motion. We introduce the qfactors:\nqz=−2qx=−2qy=2\nΩ2B/prime/prime\n1Bsat\nµ0ρm. (49)\nUnder the condition q≤0.4, we can average out the time-depending terms using the\nsecular approximation and we obtain:10\nFx/m=−˜ω2\nxx−/radicalBigg\nI\nmω2\nc(cα˜β+sαγ), (50)\nFy/m=−˜ω2\nyy−/radicalBigg\nI\nmω2\nc(sα˜β−cαγ), (51)\nFz/m=−˜ω2\nzz, (52)\nΓα/I= 0, (53)\nΓ˜β/I=−ω˜β2˜β−/radicalbiggm\nIω2\nc(cαx+sαy), (54)\nΓγ/I=−ωγ2γ−/radicalbiggm\nIω2\nc(sαx−cαy). (55)\nwith the characteristic frequencies:\nω˜β=ωγ=/radicalBigg\n5\n2B0Bsat\nµ0ρma2, (56)\n˜ωz/2 = ˜ωx= ˜ωy= Ω|qx|√\n2, (57)\nωc=/radicaltp/radicalvertex/radicalvertex/radicalbt/radicalBigg\n5\n2B/prime\n2Bsat\nµ0ρma. (58)\nE. Numerical value and elimination of the angular-CoM coupling terms\nExperimentally, we propose to use a magnetic ﬁeld B0= 10 mT and a curvature B/prime/prime\n1=\n105T.m−2. We ﬁxeqx= 0.1 which leads to the Paul trap frequency Ω = (2 π) 5.0×102Hz.\nUnder these values, we obtain the typical frequencies:\n˜ωx= ˜ωy= (2π) 3.5×101Hz, (59)\nω˜β=ωγ= (2π) 2.7×105Hz, (60)\nωc= (2π) 6.3×102Hz. (61)\nWe have ˜ωxω˜β<10ω2\ncso we can safely neglect the coupling between the angular and\nthe CoM modes.11\nIII. CALCULATION OF THE DIFFERENT INDUCTIVE CURRENT IN THE\nSET-UP\nIn this section, we examine the inﬂuence of the oscillating magnetic ﬁeld on both the\ntrapping mechanism and on the levitating particle. We conclude that for the protocol\nproposed, there are both negligible.\nA. Induction current in the micro-loops\nWe calculate the Eddy current generated by the loop 1 onto the loop 2. The magnetic\nﬂux inside the loop 1 equals:\nΦB,1=πr2\n1B2(t) =πr2\n1µ0i2\n2r2cos (Ωt). (62)\nThe circulation of the electric ﬁeld reads:\n/contintegraldisplay\nE1.dl= 2πr1E (63)\nThe Faraday’s law gives:\nE1=µ0i2Ω\n4r1\nr2sin (Ωt) (64)\nFinally, we obtain the Eddy current value normalized by the initial current in the loop:\ni2→1(t) =µ0σSΩ\n4r1\nr2i2sin (Ωt), (65)\nwhereσis the electrical conductivity of gold and S= 100µm2is the area of a slice of the\ngold lithography. Using i1/i2=−r1/r2, we get:\ni2→1(t)\ni1=−µ0σSΩ\n4sin (Ωt) (66)\nUsing the numerical values Ω = (2 π)2.0×103Hz,σ= 4.4×107S.m−1, we obtain that this\nratio is of the order of 10−5so we can safely neglect the Eddy current generated by a loop\nonto the other one.\nB. Induction current in the levitated ferromagnet\nInduction current in the levitated ferromagnetic sphere could induce some internal heating\nwhich could be problematic at low pressure. Technically, the levitated sphere does not feel12\nany oscillating magnetic ﬁeld at the equilibrium position (0 ,0,0). However, one has to take\ninto account the sphere volume and the magnetic ﬁeld value inside the sphere is of the order\nofBind/similarequalB/prime/prime\n1a2. The current density inside the sphere equals:\nj/similarequalΩσa3B/prime/prime\n1 (67)\nThe power dissipated by the Joule eﬀect then equals:\nP= Ω2σa9B/prime/prime2\n1 (68)\nThe dependance at the power of nine of the magnet size makes the induction current heating\nof the order of 10−28W. This is not suﬃcient to heat the internal temperature of the magnet\neven at ultra high vacuum.\n[1] O. Kirillov and M. Levi, A coriolis force in an inertial frame, 30, 1109 (2017).\n[2] H. G. Dehmelt, D. R. Bates, and I. Estermann, Radiofrequency spectroscopy of stored ions i:\nStorage**part ii: Spectroscopy is now scheduled to appear in volume v of this series. (Academic\nPress, 1968) pp. 53–72."    },    {        "title": "1608.01136v1.Evolution_of_magnetic_fluctuations_through_the_Fe_induced_paramagnetic_to_ferromagnetic_transition_in_Cr__2_B.pdf",        "content": "arXiv:1608.01136v1  [cond-mat.str-el]  3 Aug 2016Evolution of magnetic ﬂuctuations through the Fe-induced p aramagnetic to\nferromagnetic transition in Cr 2B\nD. Arˇ con,1,2,∗L. M. Schoop,3,4R. J. Cava,3and C. Felser4\n1Joˇ zef Stefan Institute, Jamova c. 39, 1000 Ljubljana, Slov enia\n2Faculty of Mathematics and Physics, University of Ljubljan a, Jadranska c. 19, 1000 Ljubljana, Slovenia\n3Department of Chemistry, Princeton University, Princeton , New Jersey 08544, USA\n4Max-Planck-Institut f¨ ur Chemische Physik fester Stoﬀe, 01 187 Dresden, Germany\n(Dated: October 15, 2018)\nIn itinerant ferromagnets, the quenched disorder is predic ted to dramatically aﬀect the ferro-\nmagnetic to paramagnetic quantum phase transition driven by external control parameters at zero\ntemperature . Here we report a study on Fe-doped Cr 2B, which, starting from the paramagnetic\nparent, orders ferromagnetically for Fe-doping concentra tionsxlarger than xc= 2.5%. In parent\nCr2B,11B nuclear magnetic resonance data reveal the presence of bot h ferromagnetic and antifer-\nromagnetic ﬂuctuations. The latter are suppressed with Fe- doping, before the ferromagnetic ones\nﬁnally prevail for x > x c. Indications for non-Fermi liquid behavior, usually assoc iated with the\nproximity of a quantum critical point, were found for all sam ples, including undoped Cr 2B. The\nsharpness of the ferromagnetic-like transition changes on movingaway from xc, indicating signiﬁcant\nchanges in the nature of the magnetic transitions in the vici nity of the quantum critical point. Our\ndata provide constraints for understanding quantum phase t ransitions in itinerant ferromagnets in\nthe limit of weak quenched disorder.\nPACS numbers: 76.60.-k, 75.50.Cc, 73.43.Nq, 76.50.+g\nI. INTRODUCTION\nItinerant ferromagnets are a class of materials for\nwhich the transition from the paramagnetic to ferromag-\nnetic state is considered as a canonical example of a sec-\nond order phase transition. When an external control\nparameter such as pressure suppresses the ferromagnetic\nstate, then these materials approach a quantum critical\npoint1(QCP) that separates the itinerant ferromagnetic\nstate from its paramagnetic counterpart.2In such cases,\nthe transition between two nearly degenerate magnetic\nground states is driven by nonthermal ﬂuctuations and\nthe system undergoes a quantum phase transition at zero\ntemperature. In an external magnetic ﬁeld, however, the\ntransition, before being completely suppressed, changes\ntoﬁrstorderbelowatricriticaltemperature.3,4Moreover,\nwhile itinerant ferromagnets are usually well described\nwithin the framework of standard Fermi liquid theory, in\nthe proximity of a QCP a non-Fermi liquid state can of-\nten be inferred5,6from resistivity measurements, i.e. the\nresistivity displays a power-law temperature dependence\nρ∝Tnwith an unconventional exponent of n <2.7\nThe existence of a QCP, the quantum critical region at\nﬁnite temperatures and non-Fermi liquid behavior has\nbeen experimentally veriﬁed on a number of clean itiner-\nant ferromagnets, such as 3 d-based MnSi,7,8ZrZn2(Ref.\n9) or NbFe 2(Ref. 10) and 5 f-based UGe 2, URhGe or\nUCoGe.11–14\nIn contrasttothe establishedcaseofQCPsin the clean\nitinerant ferromagnets mentioned above, the behavior in\nsystems with quenched disorder is less clear. Quenched\ndisorder is predicted to suppress the tricritical tempera-\nture until it vanishes at a critical value of disorder and\nthe transition changes back to second order with non-mean-ﬁeld exponents.2However, examples of itinerant\nferromagnets where the ferromagnetic transition is sup-\npressed by quenched disorderat the QCP are remarkably\nscarce. Therefore, there is a need for new model systems\nwhere the nature of the phase transition and possible\ndeviations from the Fermi-liquid state can be systemat-\nically investigated close to the QCP in the presence of\ndisorder.\nRecently, a paramagnetic to ferromagnetic phase tran-\nsition has been reported in lightly Fe-doped Cr 2B.15The\nparent Cr 2B is an intermetallic compound that crystal-\nlizes in an orthorhombic structure16,17(Fig. 1) with\nmany bands crossing the Fermi energy.18Due to the\nglide planes in the nonsymmorphic space group Fddd,\nthe presence of three-dimensional Dirac points in the\nelectronic structure is symmetry dictated and may ac-\ncount for the n−type carriers with high mobility mea-\nsured in Hall eﬀect experiments.18According to ﬁrst\nprinciplecalculations, thegroundstateofCr 2Bshouldbe\nantiferromagnetic.19Experimentally, antiferromagnetic\ncorrelations have indeed been inferred from the magne-\ntization data, although no transition to a long-range or-\ndered phase in Cr 2B has been found.15Upon Fe-doping,\ndetailed Arrott analysis of the temperature and the ﬁeld\ndependences of magnetization data revealed that a fer-\nromagnetic phase emerges at a critical Fe-concentration\nnearx= 0.02.15Moreover,the observedlogarithmic con-\ntribution to the heat capacity was taken as a hallmark of\nquantum criticality, and the power-law exponent n∼1\nin the temperature dependence of resistivity was taken\nto indicate a non-Fermi liquid state in doped samples.\nHowever, whether Fe-doped Cr 2B does indeed represent\na new family of intermetallics where quenched disorder\ndrives a ferromagnetic quantum phase transition calls for2\nadditional independent experimental conﬁrmation.\nMagnetic resonance techniques, such as nuclear mag-\nnetic resonance (NMR) and electron spin resonance\n(ESR), have proven to be extremely powerful probes of\nlocal spin susceptibilities and spin ﬂuctuations close to a\nQCP.20–26Here we report detailed11B NMR and ESR\nmeasurements on polycrystalline Fe-doped Cr 2B at dif-\nferent doping levels acrossthe paramagneticto ferromag-\nnetic transition. Surprisingly, we ﬁnd a strong tempera-\nture dependent shift of the11B NMR spectra that gen-\nerally follows a Curie-Weiss-like dependence, revealing\na crossover from predominantly antiferromagnetic corre-\nlations to predominantly ferromagnetic correlations at a\ncriticalFe-dopinglevelof x∼2.5%. Oncooling,weﬁnda\ncrossoverto a low-temperaturestate where the11B NMR\nshift displays a non-Fermi-liquid T−1/2-dependence, thus\ncomplyingwithwhatisexpectedinthevicinityofaQCP.\nSimultaneously, the appearance of a strong electron spin\nresonance signal and a substantial broadening of the11B\nNMR spectra provide evidence for ferromagnetic-like or-\ndering at low-temperatures that is completely absent at\nlower Fe-doping levels. The data presented here reveal\nhighly unusual behavior for Fe-doped Cr 2B, which can-\nnotbe simplyrationalizedwithin astandardFermi-liquid\nmodel. Moreover, the systematic evolution of the sharp-\nness of the ferromagnetic-like transition with Fe-doping\nlevelimpliessigniﬁcantchangesinthenatureofthephase\ntransition as the quenched disorder varies across the crit-\nical Fe-doping value for the quantum phase transition\n(QPT).\nFIG.1. (color online). (a)Theorthorhombiccrystal struct ure\nof Cr2B. (b) The local coordination of each boron site (green\npolyhedron) has eight nearest Cr atoms (large blue spheres)\nand two more B atoms (small green spheres) at apical posi-\ntions.II. EXPERIMENTAL METHODS\nFor this study, the polycrystalline Fe-doped Cr 2B sam-\nples used in Ref. 15 to characterize the magnetic and\ntransport properties of the Cr 2−xFexB system were em-\nployed. The Fe-doping range is between 0 and 5%. Ac-\ncording to powder x-ray diﬀraction, only very small frac-\ntions of non-magnetic Cr metal were identiﬁed in the\ndiﬀraction proﬁles in addition to the doped Cr 2B phase.\nThe samples were also characterized in detail by high\nresolution transmission electron microscopy and high an-\ngle annular dark-ﬁeld scattering transmission microscopy\nto exclude the possibility of Fe clustering. The mag-\nnetic susceptibility data excluded the presence of mag-\nnetic impurities.15\nFor the temperature-dependent ESR experiments, the\nsamples were sealed under helium in a standard 4 mm-\ndiameter silica tube (Wilmad Lab Glass) whereas for the\n11B NMR experiments, samples with a mass of around\n90 mg were directly inserted into the NMR coil. A con-\nventional continuous wave (cw) electron paramagnetic\nresonance spectrometer operating at a Larmor frequency\nESRνL= 9.6 GHz was employed to detect the electron\nspin resonance. The spectrometer is equipped with a\nstandard Varian E-101 microwave bridge, a Varian rect-\nangular TE102 resonance cavity, and an Oxford Cryo-\ngenicscontinuous-ﬂowhelium cryostat. The temperature\nstability was better than ±0.1 K over the entire temper-\nature range of measurements (4 −300 K).\nThe11B (nuclear spin I= 3/2) NMR spectra and\nthe spin-lattice relaxation rates were measured between\n5 and 300 K in a magnetic ﬁeld of 4 .7 T. The11B NMR\nshifts are determined relative to the Larmor frequency\n11νL= 64.167MHz, deﬁnedbyaBF 3Et2Ostandard. For\n11B NMR line shape measurements, asolid-echopulse se-\nquence,π/2−τ−π/2−τ−echo, was employed, with\na pulse length tw(π/2) = 1.9µs and an interpulse delay\nτ= 40µs. The complete polycrystalline NMR spec-\ntrum was obtained by summing the real part of spec-\ntra measured step-by-step at resonance frequencies sep-\narated by ∆ ν= 50 kHz. Since it was impossible to com-\npletely invert the11B nuclear magnetization, we used the\nsaturation-recoverypulse sequence for the spin-lattice re-\nlaxation rate measurements. Typically, the saturation\ntrain consisted of a sequence of 20 π/2 pulses separated\nby 50µs.\nBecause11B is a quadrupole nucleus, we model the\n11B NMR spectra with the general spin Hamiltonian\nH=HZ+HQ+HScomprisingthe nuclearZeeman ( HZ),\nnuclear quadrupole ( HQ) and11B shift ( HS) terms, re-\nspectively. The later gives rise to the11B NMR lineshift\nof the central −1/2↔1/2 transition, K, and includes\nthe temperature independent chemical shift, σ, and the\nKnightshift, Ks,whichoriginatesfromthehyperﬁnecou-\npling toitinerant electrons. The mostimportant term for\nour study is Ks, as it is directly proportional to the local\nelectronic susceptibility χ, e.g., its isotropic part is given\nbyKiso=as\nNAµ0χ, whereasis a hyperﬁne coupling con-3\n/s40/s98/s41\n/s53/s46/s48/s37/s32/s70/s101/s50/s46/s53/s37/s32/s70/s101/s50/s46/s48/s37/s32/s70/s101/s48/s46/s53/s37/s32/s70/s101\n/s32/s32/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41/s67/s114\n/s50/s66/s84 /s32/s61/s32/s51/s48/s48/s32/s75/s40/s97/s41\n/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s50/s48/s52/s48/s54/s48/s56/s48/s32/s49/s47/s50/s32/s40/s107/s72/s122/s41\n/s70/s101/s45/s100/s111/s112/s105/s110/s103/s32/s40/s37/s41/s52/s48/s48/s52/s50/s53/s52/s53/s48/s52/s55/s53/s53/s48/s48\n/s81/s32/s40/s107/s72/s122/s41\nFIG. 2. (color online). (a) Comparison of the room-\ntemperature11B NMR spectra (gray shaded area) measured\non polycrystalline Fe-doped Cr 2B samples at various Fe-\ndoping levels. Solid red lines are the powder lineshape ﬁts\nto a model that includes anisotropic shift and quadrupole ef -\nfects up to the second order. A single11B NMR component\nwas suﬃcient in all cases. (b) Fe-doping dependence of the\n11B NMRlinewidth broadeningparameter, δ1/2, (solid circles,\nleft scale) and of the quadrupole frequency νQ, (open circles,\nright scale). The solid and dashedblack lines are guides tot he\neye for the Fe-doping dependence of δ1/2andνQ, respectively.\nstant,NAis Avogadro’s number and µ0is the magnetic\npermeability of vacuum. When analyzing the11B NMR\nspectra of Cr 2B we include the11B anisotropic shift and\nthe quadrupole eﬀects up to the second order. Fitting of\nthe spectra thus yields, in addition to K, the quadrupole\nsplitting frequency νQ=3eVzzQ\nh2I(2I−1)and the asymmetry\nparameter η= (Vxx−Vyy)/Vzz, both deﬁned by the com-\nponents of the electric ﬁeld gradient (EFG) Vijat the\n11B site. Here Qandhare the11B quadrupole moment\nand the Planck constant, respectively. The powder NMR\nspectrum is computed as a histogram of resonance fre-\nquenciesobtainedbysummingoveruniformlydistributed\npolar and azimuthal angles of the magnetic ﬁeld orien-\ntation with respect to the quadrupole tensor principal\naxes.27,28We included the homogeneous broadening of\nthe line through the convolution of the computed spectra\nwith a Lorentzian line with a full-width-half-maximum\nδ1/2. The eﬀect of quadrupole splitting frequency distri-\nbution on11B NMR spectra was alsotested by using nor-\nmally distributed values of quadrupole frequencies with\nthe center at νQand width of ∆ νQ.\nIII. RESULTS AND DISCUSSION\nA. Homogeneity of the Cr 2B samples after Fe\ndoping\nThe room temperature11B NMR spectrum of Cr 2B\nsample [Fig. 2(a)] displays a characteristic quadrupolepowder lineshape with a clearly pronounced satellite\ntransition ( ±3/2↔ ±1/2) singularities ﬂanking the cen-\ntral peak that corresponds to a −1/2↔1/2 transi-\ntion. The spectrum is considerably shifted by K=\n−228(3) ppm relative to the11B NMR reference fre-\nquency. The central peak has a nearly Lorentzian\nlineshape thus implying that the broadening due to\nthe anisotropic shift interactions and the second-order\nquadrupole corrections is negligible. A lineshape ﬁt of\nthe spectrum yields a quadrupole splitting frequency\nofνQ= 471(8) kHz and an asymmetry parameter of\nη= 0.02(1). Surprisingly, we ﬁnd η≈0, although\nthe11B site symmetry – B is at the low-symmetry 16 g\n(0.125, 0.125, 0.4993) site [Fig. 1(b)]16,17– does not re-\nquire such a restriction. The homogeneous broadening is\nδ1/2= 24(1) kHz.\nLight doping of Cr 2B with 0.5% Fe induces almost no\nchange to the11B NMR lineshape [Fig. 2(a)] and the\nunconstrained ﬁt returns, compared to parent undoped\nCr2B, nearly the same νQ= 483(9) kHz, η= 0.02(1) and\nδ1/2= 24(1) kHz. As the level of Fe-doping increases,\nhowever, the11B NMR spectra drastically broaden, and\nfor 5% Fe doping the central peak also becomes slightly\nanisotropic. On extracting the parameters from the11B\nNMR lineshapeﬁts, weﬁrstnoticethat νQmarginallyre-\nduces with increasingFe concentration[Fig. 2(b)], i.e. to\n467(8) kHz, 465(8) kHz and 461(10) kHz in the 2%, 2.5%\nand 5% doped samples, respectively. This insensitivity\nofνQto the Fe-doping implies that the lattice around\nthe B-sites contributing to the measured spectra is not\nmarkedly perturbed for Fe-doping levels up to 5%. (We\nnote that it is possible that the B atoms sitting next to\nFe-dopant atoms experience signiﬁcantly diﬀerent EFG\nand hyperﬁne ﬁelds that shift and broaden the spectra\nbeyond the sensitivity of the present NMR experiments.)\nNevertheless, the local hyperﬁne ﬁelds at “weakly per-\nturbed”11B sites do change, judging from the Fe-doping\nvariationinthelinewidthparameter δ1/2, whichincreases\nto 31(1) kHz, 34(1) kHz and 64(2) kHz for 2%, 2.5% and\n5% Fe doping, respectively [Fig. 2(b)]. Simultaneously,\nsatellite transition singularities become signiﬁcantly less\npronounced, which explains the larger η= 0.08(1) for\nthe largest Fe-doping level. (We note that for the 5%\ndoped sample a11B NMR lineshape ﬁt that includes a\nsmall distribution of νQ, i.e. ∆νQ/νQ= 6%, describes\nthe experimental spectrum equally well.) Therefore we\nﬁnd that Fe-doping indeed introduces some local-site dis-\norder, which is rather sensitively picked up by the NMR\nparameters. However, because all11B NMR spectra can\nstill be simulated with a single well-deﬁned value of νQ,\nwe conclude that the Fe-doping must be rather homoge-\nneous for all samples, consistent with previous chemical\nand structural characterization.154\nB. The paramagnetic state of parent Cr 2B\nFig. 3a shows the temperature dependence of the cen-\ntral transition peak in parent Cr 2B. This peak retains\nits Lorentzian lineshape at all temperatures and shows\nalmost no broadening between room temperature and\nT= 4 K. The complete absence of broadening of the\npowder11B NMR spectra is a ﬁrm evidence that no\nstatic magnetic order is established. However, the spec-\ntra monotonically shift towards higher frequencies with\ndecreasing temperature. Between room temperature and\n50 K the11B NMR shift Kshows [Fig. 3(b)] a Curie-\nWeiss-like dependence\nK(T) =σ+B/(T−Tcw). (1)\nHereweidentifythe constant σ=−264(3)ppm asatem-\nperature independent chemical shift. On the other hand,\nthe second temperature-dependent Knight-shift contri-\nbution originates from the hyperﬁne interactions with\nthe unpaired electronic moments. The extracted nega-\ntive Curie-Weiss temperature Tcw=−12(3) K implies\nthat the involved itinerant electronic states are antiferro-\nmagnetically correlated and thus corroborates the mag-\nnetization data15and the ﬁrst principle calculations.19\nAtT≈35 K we notice a small discontinuity in K,\nbut for lower temperatures Kstill continues to increase\nwith decreasing temperature. Plotting the temperature\ndependence of the Knight shift Ks(T) =K(T)−σon\na log-log plot [inset to Fig. 3(b)], we notice a gradual\nchange of slope at around 35 K. Whereas at higher tem-\nperaturestheslopeof Ksindeedﬁtstoa T−1dependence,\nas established above, we ﬁnd that for lower temperatures\nKsdevelops approximately a T−1/2dependence before\nleveling oﬀ at the lowest temperatures. A very similar\nsequence of power-laws in the temperature dependence\nof the Knight shift has been reported for YbRh 2Si2,20\nand attributed to non-Fermi-liquid behavior in the vicin-\nity of a QCP.\nNext, the spin dynamics in the paramagnetic state of\nparent Cr 2B were probed through the11B spin-lattice re-\nlaxation rate 1 /T1. In striking contrast to the strongly\ntemperature dependent K, the spin-lattice relaxation\nrate divided by temperature, 1 /T1T, shows a much\nweaker temperature dependence [Fig. 3(c)]. Between\nroom temperature and 35 K, 1 /T1Tgradually decreases\nby∼30%, but then on further cooling it starts to in-\ncrease again.\nForcorrelatedmetals whereelectron-electronexchange\nenhancement eﬀects are important, the Korringarelation\nis29\n11T1TK2\ns=¯h\n4πkBγ2\ne\nγ2\n11β. (2)\nHereγeandγ11are the electronic and11B gyromag-\nnetic ratios, respectively. The Korringa factor βis intro-\nduced to account for the electron-electron exchange.29–31\nInserting the room-temperature value of 1 /T1T= 2.9·10−3s−1K−1andKs= 25 ppm into Eq. (2) we calcu-\nlateβ= 3.3 thus complying with the enhancement of\ndynamic spin susceptibility. In general, β >1 implies\nferromagnetic ﬂuctuations,30,31seemingly contradicting\nthe analysis of the Knight-shift data, which disclosed the\npresence of antiferromagnetic correlations. We addition-\nally note that, although the precise value of the Korringa\nfactorβdepends on the choice of σ, we still ﬁnd that the\nincrease of βwith decreasing temperature further cor-\nroborates the presence of ferromagnetic correlations. In\norder to explain this apparent contradiction, we suggest\nthat both ferromagnetic and antiferromagnetic spin ﬂuc-\ntuations are present, but the later are ﬁltered out at the\n11B site.Namely, the local coordination of11B site has\neight nearest neighboring Cr atoms and two more11B\nsites at the apical positions [Fig. 1(b)]. We next point\nout that near the Fermi level, the calculated density of\nstates is dominated by Cr dbands.19Therefore, the ap-\npropriate electron-nuclear Hamiltonian consists of a sum\nof transferred hyperﬁne coupling of11B at site kto the\neight nearestneighborCr electronspins at k. As aresult,\nthe contribution of antiferromagneticspin ﬂuctuations to\nthe spin-lattice relaxation may be suppressed.29,32The\nimportant conclusion from this part of our study is thus\nthatalthoughthemulti-bandCr 2Bparentcompounddis-\nplays both antiferromagnetic and ferromagnetic correla-\ntions, they are not suﬃciently strong to establish a long-\nrange magnetically ordered state. The observed anoma-\nlies at∼35 K in both K(T) and 1/T1Tdata imply the\npresence of subtle electronic changes that may preclude\nmagnetic ordering at low temperatures, but additional\nmore detailed experiments in this temperature range are\nrequired to unveil the nature of the undoped Cr 2B sys-\ntem.\nC. The ferromagnetic state in Fe-doped Cr 2B\nCr2B becomes ferromagnetic when 2% or more of the\nCr atoms are replaced by Fe.15In order to investigate\nthe ferromagnetic state we ﬁrst employ the electron spin\nresonance technique. For 5% Fe-doped Cr 2B in the high\ntemperature metallic paramagneticphase, no conducting\nelectron spin resonance(CESR) signal could be detected.\nHowever, as the temperature is decreased below ∼70 K,\nanasymmetricDyson-likeresonance33appearsat a g≈2\nresonance ﬁeld [Fig. 4(a)]. The Dyson lineshape of the\nresonance is a hallmark of a metallic state, thus ruling\nout previously undetected insulating iron-oxide impuri-\ntiesasapossiblesourceoftheobservedESRsignal. With\ndecreasing temperature, the signal gradually grows in in-\ntensity and shifts to lower resonance ﬁelds, signaling the\nonset and growth of internal magnetic ﬁelds in the ma-\nterial. The observed behavior is consistent with what\nis seen for ferromagnetic resonance in the metallic state,\nand thus agrees with the proposed ferromagnetic ground\nstate for 5% Fe-doped Cr 2B.\nAs the Fe-dopinglevel decreases,the appearanceofthe5\n/s45/s54/s48/s48 /s45/s52/s48/s48 /s45/s50/s48/s48 /s48 /s50/s48/s48 /s52/s48/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s67/s114\n/s50/s66\n/s52/s32/s75/s49/s54/s32/s75/s50/s55/s32/s75/s53/s48/s32/s75/s56/s48/s32/s75/s49/s49/s48/s32/s75/s49/s51/s48/s32/s75/s49/s54/s48/s32/s75/s50/s48/s48/s32/s75\n/s32/s32/s78/s77/s82/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s112/s112/s109/s41/s51/s48/s48/s32/s75/s40/s97/s41 /s67/s114\n/s50/s66\n/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48\n/s40/s99/s41/s40/s98/s41\n/s32/s32/s75 /s32/s40/s112/s112/s109/s41\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s52\n/s67/s114\n/s50/s66\n/s32/s32/s49/s47 /s84\n/s49/s84 /s32/s40/s49/s48/s45/s51\n/s32/s115/s45/s49\n/s75/s45/s49\n/s41/s32\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s49/s48 /s49/s48/s48/s49/s48/s49/s48/s48/s84/s45/s49/s47/s50/s75/s45 /s32/s40/s112/s112/s109/s41\n/s84/s32/s40/s75/s41/s84/s45/s49\nFIG. 3. (color online). Temperature dependences of (a) the\n11B central transition ( −1/2↔1/2) peak, (b) the shift of\nthe central transition peak, K, and the spin-lattice relaxation\nrate 1/T1divided by Tmeasured in parent undoped Cr 2B.\nThe vertical and horizontal dashed lines in (a) and (c) mark\nroom temperature values. The solid line in (b) is a ﬁt with a\nCurie-Weiss dependence to a negative Curie-Weiss tempera-\nture ofTcw=−12(3) K. The inset to (b) shows the temper-\nature dependence of the Knight shift, Ks, on a log-log scale,\nobtained after subtracting the chemical shift σfromK. Solid\nlines indicate the high-temperature slope of Ks∝T−1and\nthe low-temperature slope of Ks∝T−1/2.\nferromagnetic-like resonance is systematically shifted to\nlower temperatures. For instance, whereas at 40 K in 5%\nFe-doped sample the resonance signal is very strong, no\nsuch line was detected at the same temperature for the\n4% and 3.5% Fe-doping levels [Fig. 4(b)]. It is, however,\ndetected at lower temperatures for those compositions.\nIn order to quantitatively follow the ferromagnetic-like\nresonance, we next ﬁt the spectra to a Dyson lineshape33\n[Fig. 4(a)]. The extracted temperature dependences of\nthe intensity ofthe resonancepeak, which is proportional\nto the sample’s magnetization, aresummarized for diﬀer-\nent Fe-doping levels in Fig. 4(c). The magnetic order-\ning onset temperature Tcsystematically decreases from\n∼70 K at 5% doping to ∼30 K and then to ∼18 K\nfor 4% and 3.5% Fe-doped samples, respectively. In ad-\ndition, whereas the transition in the 5% doped sample\nis smeared over a large temperature interval, it is much\nsharper at lower doping-levels, which are closer to the\nQCP. This experimental observation may imply signiﬁ-\ncant changes in the nature of magnetic transition in the\nvicinity of the QCP.\nFurther insight into the development of local mag-\nnetic ﬁelds is provided by the11B NMR spectra mea-\nsured for 5% Fe-doped Cr 2B [Fig. 5(a)]. Whereas the\nspectra retain a characteristic quadrupole I= 3/2 pow-\nder lineshape with small anisotropic Knight shift inter-\naction [e.g., similar as at 300 K (Fig. 2)] on cooling\ndown to ∼140 K, broadening due to the anisotropic\nKnight shift interaction gradually begins to dominate\nthe spectra at lower temperatures. Below∼50 K the/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48/s45/s49/s48/s49/s50\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s51/s54/s57\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54\n/s53/s50/s32/s75\n/s52/s48/s32/s75\n/s51/s50/s32/s75\n/s50/s52/s32/s75\n/s49/s50/s32/s75/s69/s80/s82/s32/s115/s105/s103/s110/s97/s108/s32/s32/s40/s97/s114/s98/s46/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s32/s40/s109/s84/s41/s53/s32/s75/s40/s97/s41\n/s53/s37/s32/s70/s101/s215/s53\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s32/s40/s109/s84/s41/s83/s105/s103/s110/s97/s108/s32/s32/s40/s97/s114/s98/s46/s117/s46/s41 /s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s32/s40/s97/s114/s98/s46/s117/s46/s41/s40/s98/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s32/s40/s75/s41/s84 /s32/s61/s32/s52/s48/s32/s75\n/s32/s53/s37/s32/s70/s101\n/s32/s52/s37/s32/s70/s101\n/s32/s51/s46/s53/s37/s32/s70/s101/s40/s99/s41\nFIG. 4. (color online). (a) Temperature dependence of fer-\nromagnetic resonance in 5% Fe doped Cr 2B (black circles)\nmeasured at X-band (ESRνL= 9.6 GHz) frequencies. Solid\nred lines are ﬁts of the spectra to the Dyson lineshape. The\ndotted vertical line at 330 mTmarks theresonance ﬁeld of the\ng= 2 electron paramagnetic resonance signal. (b) Compari-\nson of ferromagnetic resonance spectra measured at 40 K for\n5% (black circles), 4% (red triangles) and 3.5% (blue square s)\nFe-doped Cr 2B, respectively. (c) Temperature dependence of\nthe intensities of the ferromagnetic resonance spectra for 5%\n(black circles), 4% (red triangles) and 3.5% (blue squares)\nFe-doped Cr 2B, respectively.\nspectra already display a lineshape that is reminiscent\nof anisotropic Knight shift interactions. Alternatively,\nthe lineshape broadening originating from the distribu-\ntion of Knight shifts is less probable because it is not\naccompanied also by the large quadrupole frequency dis-\ntribution. Therefore, for the ﬁts to the data, we assumed\ntemperature independent νQ= 461 kHz and η= 0.08,\nboth extracted from the room temperature spectra, and\nthat the isotropic Kand anisotropic Kanisoparts of the\nKnight-shift tensor are temperature dependent parame-\nters. The temperature dependences of KandKanisothus\nobtained are summarized in Fig. 5(b). Both parameters\nexhibit a very strong temperature dependence that can\nagain be described as a Curie-Weiss-like dependence be-\ntween room temperature and 70 K. Fitting the isotropic\nshiftK(T) to Eq. (1) yields σ=−270(8) ppm, which\nis nearly identical to the corresponding chemical shift of\nthe parent Cr 2B. On the other hand, we now ﬁnd a pos-\nitive Curie-Weiss temperature, TCW= 21(4) K, which\nis thus fully consistent with the dominant ferromagnetic\ncorrelations in heavily Fe-doped Cr 2B. The temperature\ndependence of Kanisoalso supports this ﬁnding.\nLarge broadening of the11B NMR spectra at low tem-\nperatures, reﬂected in the enhanced Kaniso, clearly ev-\nidences the development of static local magnetic ﬁelds\nand thus of spin-freezing. The ferromagnetic TCWsug-\ngests that the magnetic moments induced by Fe-doping\nfreeze into a spin state where ferromagnetic correlations\nprevail. However, what may be surprising is that the\nshift ofK(T) remains relatively small, i.e. on the order\nof∼300 ppm, which is thus only a fraction of Kaniso. We6\n/s53/s37/s32/s70/s101\n/s49/s56/s48/s32/s75\n/s32/s32\n/s50/s49/s32/s75/s53/s48/s32/s75/s55/s53/s32/s75/s57/s53/s32/s75/s49/s52/s48/s32/s75/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41/s40/s97/s41\n/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s32/s75 /s32/s40/s112/s112/s109/s41/s40/s98/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s50/s52\n/s32/s32/s49/s47 /s84\n/s49/s84 /s32/s40/s49/s48/s45/s51\n/s32/s115/s45/s49\n/s75/s45/s49\n/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40/s99/s41/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s75\n/s97/s110/s105/s115/s111/s32/s40/s37/s41\nFIG. 5. (color online). (a) The temperature evolution of\nthe11B NMR spectra for 5% Fe-doped Cr 2B. The solid red\nlines are ﬁts to a model with quadrupole and anisotropic\nKnight-shift interactions. In the model, we assumed tem-\nperature independent νQ= 461 kHz and η= 0.08. (b)\nThe temperature dependences of the isotropic (red circles,\nleft scale) and anisotropic (blue squares, right scale) par ts of\nthe Knight shift. A ﬁt of the11B NMR shift, K(T), to Eq.\n1 (solid red line) yields the chemical shift σ=−270(8) ppm,\nB= 9.7(9)·103ppm K and the ferromagnetic Curie-Weiss\ntemperature TCW= 21(4) K. The dashed blue line is a guide\nto the eye. (c) Temperature dependence of the11B NMR\nspin-lattice relaxation rate divided by temperature, 1 /T1T.\nthus conclude that the contact hyperﬁne (or the isotropic\npart of the transferred hyperﬁne) interaction with itiner-\nant electrons is nearly the same in Fe-doped Cr 2B com-\npared to parent undoped Cr 2B. The reason for this is\ncurrently unknown, but one of the possibilities is that\nwhen Fe is introduced into the lattice it creates localized\nstates that interact with11B mostly via long-range, e.g.\ndipolar, interactions. This may also explain the temper-\nature dependence of the spin-lattice relaxation rate [Fig.\n5(c)]. Compared to parent Cr 2B, the slightly shorter T1\nat room temperature yields 1 /T1T= 1.3·10−3s−1K−1,\nand therefore a ferromagnetically enhanced β≈8. On\ncooling below ∼70 K, 1/T1Tindeed starts to increase as\nexpected when close to the magnetic ordering. However,\nthe absence of a divergence in 1 /T1T, normally found at\nthe magnetic ordering temperature, and the broadening\nof the ferromagnetic-like transition observed in the ESR\ndata (Fig. 4) may be signatures of a distribution of fer-\nromagnetic freezing temperatures, and thus of a smeared\ntransitiontoaferromagnetic-likestatewith ahighdegree\nof disorder.\nD. Behavior close to the critical Fe doping\ncomposition\nFinally, we focus on Fe-doping levels lower than 2.5%,\ni.e. the samples with suppressed ferromagnetic order-ing temperature being thus close to the QCP.15Inspect-\ning the low-temperature11B NMR spectra of these sam-\nples we ﬁnd that they retain a characteristic powder\nquadrupole lineshape at all temperatures (insets to Fig.\n6). This proves the absence of the strong internal ﬁelds\nthatwouldbroadenthe11BNMRspectra,aswasthecase\nfor ferromagnetic 5% Fe-doped Cr 2B at higher temper-\natures. The absence of the ferromagnetic-like resonance\nsignal in these samples down to 4 K provides additional\nevidence for the absence of any magnetic order. On the\nother hand, the temperature dependence of K(T) is a\nstrong indication of a non-Fermi liquid state. Namely,\nbetween room temperature and ∼70 K the shift follows\na Curie-Weiss-like dependence (Fig. 6), which seems to\nbe a general characteristic of all the Cr 2B samples stud-\nied, both doped and undoped.\nFittingK(T) in the temperature interval between 300\nand 70 K reveals a signiﬁcant Fe-doping dependence of\nthe NMR-determined Curie-Weiss temperature. While\nwe ﬁnd a small but positive TCW= 8(3) K for 2.5%\ndoped samples and above, it is negative (antiferromag-\nnetic),TCW=−20(3) K, for the 2% doped material and\nremains negative to lower Fe-dopeing levels. Similar to\nthe case for parent Cr 2B, at lower temperatures Ks(T)\nincreaseswith a smallerpower-lawexponent, i.e. roughly\nasT−1/4, before leveling oﬀ at lowest temperatures.\nCompared to parent Cr 2B, the anomaly in the tem-\nperature dependence of K(T) shifts from 35 K to 24 K\nin the 2% Fe-doped sample (Fig. 6). Surprisingly, this\nanomaly is not clearly observed in the spin-lattice re-\nlaxation rates, which are temperature independent, i.e.\n1/T1T= 2.9·10−3s−1K−1, between 300 and 5 K. In\ncontrast to the 5% Fe-doped sample, there is no en-\nhancement in 1 /T1Tthat would suggest the develop-\nment of a ferromagnetic-like state at low temperatures.\nThe11B NMR shift and the spin-lattice relaxation rate\ndata thus unambiguously show that Fe-doped Cr 2B ma-\nterials at concentrations lower than the critical 2.5%\nvalue lack magnetic order, while systematically showing\nelectronic characteristics that deviate from conventional\nFermi-liquid behavior.\nIV. DISCUSSION AND CONCLUSIONS\nThe insensitivity at room-temperature of the\nquadrapole frequencies νQ[Fig. 2(b)] and11B NMR\nshifts [Fig. 2(a)] clearly demonstrates that Fe doping of\nCr2B at levels up to 5% does not signiﬁcantly perturb\nthe local structural and electronic environment at the\nB-sites. Yet, the emergence of the ferromagnetic-like\nresonance (Fig. 4) and the large broadening of the\n11B NMR spectra (Fig. 5) observed in 5% Fe-doped\nCr2B are consistent with the presence of magnetic\nordering at low temperatures when the Fe concentration\nexceeds the critical value of xc≈2.5%. Therefore, the\nquantum paramagnetic to ferromagnetic transition at\nzero temperature is indeed triggered by Fe doping at7\n/s32/s32\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41/s50/s37/s32/s70/s101\n/s53/s32/s75/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s50/s46/s53/s37/s32/s70/s101\n/s49/s53/s32/s75\n/s32/s32\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41\n/s52 /s49/s48 /s49/s48/s48 /s51/s53/s48/s49/s48/s49/s48/s48/s49/s48/s48/s48\n/s50/s37/s32/s70/s101\n/s84/s45/s49/s47/s52\n/s84/s45/s49\n/s32/s75/s32 /s45/s32 /s40/s112/s112/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\nFIG. 6. (color online). The temperature dependence of the\n11B NMR Knight shift, Ks=K−σ, for 2% Fe-doped Cr 2B\n(open circles). The Knight shifts are extracted directly fr om\nthe shifts of the11B NMR spectra, K, obtained by subtrac-\ntion of the chemical shift σ=−253 ppm. The solid red lines\nindicate that Ks∝T−1at high temperatures, and then grad-\nually changes to Ks∝T−1/4at low temperatures. Insets:\nLow-temperature11B NMR spectra (gray shaded area) mea-\nsured for 2% Fe-doped Cr 2B (T= 5 K) and 2.5% Fe-doped\nCr2B (T= 15 K). The solid red lines are lineshape ﬁts with\nK= 54 ppm, νQ= 496 kHz and δ1/2= 155 kHz (Cr 2B sam-\nple with 2% Fe doping) and K= 167 ppm, νQ= 450 kHz and\nδ1/2= 243 kHz (Cr 2B sample with 2.5% Fe doping).\nxc.So, what really changes after Fe-doping that drives\nsuch transition? Our11B NMR and ESR data highlight\ntwo primary factors that constrain the discussion of\nthe paramagnetic to ferromagnetic transition in this\nmaterial: the non-Fermi-liquid behavior and the simul-\ntaneous presence of antiferromagnetic and ferromagnetic\ncorrelations.\nThe non-Fermi-liquid behavior is revealed through the\nstrong temperature dependence of the11B Knight-shift\nfound across the entire phase diagram (Fig. 7). All sam-\nples surprisingly show a Curie-Weiss-like dependence of\nKsat high temperatures. One possible explanation for\nsuch a dependence is the presence of localized states in\nthe samples, originating either from defects (i.e. a slight\nnon-stoichiometry in case of the parent Cr 2B) or from\nan orbitally selective Mott transition34,35in this multi-\nband system. However, a crossover to a low-temperature\nstate where KsfollowsT−nwith a power-law exponent\nn≤1/2 contradicts both these possibilities. Rather we\nconclude that the Fe-doped Cr 2B materials family is in-\ndeed close to a QCP as originally suggested.15A very\nsimilar temperature dependence of the Knight-shift has\nbeen found in other archetypal materials close to a QCP,\nwhere such dependence was attributed to the existence\nof strong spin correlations in the metallic state.20\nThe second important ﬁnding relates to the origin of\n/s48 /s49 /s50 /s120\n/s67/s51 /s52 /s53 /s54/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s70/s101/s45/s100/s111/s112/s101/s100/s32/s67/s114\n/s50/s66\n/s70/s77\n/s32/s84\n/s67/s87 \n/s32/s84\n/s99\n/s32/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s70/s101/s45/s100/s111/s112/s105/s110/s103/s32/s40/s37/s41/s78/s70/s76\nFIG. 7. (color online). Dependence of the NMR-determined\nCurie-Weiss temperature TCW(open black circles) and mag-\nnetic ordering onset temperature Tc(solid blue circles) on the\nof Fe-doping concentration of Cr 2B. The dotted black and the\ndashed blue lines are a guides to the eye. A transition is seen\nfrom a paramagnetic non-Fermi liquid metal (NFL, yellow\nshading) with predominant antiferromagnetic correlation s to\na ferromagnetic metal (FM, blue shading) at the critical con -\ncentration xc= 2.5%.\nspinﬂuctuationsandtheirevolutionwithFe-doping. The\nanalysis of the11B spin-lattice relaxation data is consis-\ntent with ferromagnetic ﬂuctuations. These ferromag-\nnetic ﬂuctuations coexist with antiferromagnetic corre-\nlations deduced from the negative Curie-Weiss temper-\nature. Fe-doping does not signiﬁcantly aﬀect the fer-\nromagnetic ﬂuctuations. On the other hand, the Fe-\ndependence of TCW(Fig. 7) suggests that the antifer-\nromagnetic correlations gradually vanish as the doping\nlevel approachesthe critical Fe concentration. For higher\nFe-doping levels, ferromagnetic correlations prevail and\nas a result the magnetic ordering temperature monoton-\nically and rapidly increases with x.What remains to be\nanswered in future work is why Fe-doping aﬀects the fer-\nromagnetic correlations to a much lesser degree than the\nantiferromagnetic correlations.\nIn conclusion, we have systematically investigated the\neﬀect of Fe-doping on the magnetism in the intermetallic\ncompound Cr 2B.11B NMR and ESR data suggest that\nthese materials may indeed be close to a quantum crit-\nical point at the critical Fe-doping level of xc≈2.5%.\nThe data also reveal that antiferromagnetic and ferro-\nmagnetic correlations coexist in these materials, but are\ndiﬀerently aﬀected by Fe-doping. At xc, ferromagnetic\ncorrelations prevail and magnetic ordering is observed\nat higher doping levels. Our characterization thus indi-\ncates that intermetallic Cr 2B as an interesting material\nsystem where the eﬀect of weak quenched disorder on a\nferromagnetic quantum phase transition can be system-\natically studied.8\nACKNOWLEDGMENTS\nD.A. and C.F. acknowledge the ﬁnancial support by\nthe European Union FP7-NMP-2011-EU-Japan project\nLEMSUPER under contract no. 283214. D.A. also ac-\nknowledges discussions with Peter Jegliˇ c about the11BNMR spectra. D.A. and C.F. also acknowledge discus-\nsions with Peter Adler regarding the magnetic ground\nstate. The research at Princeton University was sup-\nported by the US Depaertment of energy, grant DE-\nFG02-98ER45706.\n∗e-mail: denis.arcon@ijs.si\n1M. Lavagna, Philos. Mag. B 81, 1469 (2001), URL\nhttp://www.tandfonline.com/doi/abs/10.1080/13642810 108208565 .\n2Y. Sang, D. Belitz, and T. R. Kirkpatrick,\nPhys. Rev. Lett. 113, 207201 (2014), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.113.207 201.\n3D. Belitz, T. R. 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Fisica dell’Universit` a di Pisa and INFN,\nLargo Pontecorvo 2, I-56127 Pisa, Italy and\n2Dip. Fisica dell’Universit` a di Roma “La Sapienza”\nand INFN, P.le Moro 2, I-00185 Roma, Italy\n(Dated: July 13, 2011)\nAbstract\nWe investigate the ferromagnetic-glassy transitions whic h separate the low-temperature ferro-\nmagnetic and spin-glass phases in the temperature-disorde r phase diagram of three-dimensional\nIsing spin-glass models. For this purpose, we consider the c ubic-lattice ±J(Edwards-Anderson)\nIsing model with bond distribution P(J) =pδ(J−1) +(1−p)δ(J+1), and present a numerical\nMonte Carlo study of the critical behavior along the line tha t marks the onset of ferromagnetism.\nThe ﬁnite-size scaling analysis of the Monte Carlo data show s that the ferromagnetic-glassy\ntransition line is slightly reentrant. As a consequence, fo r an interval of the disorder parameter p,\naroundp≈0.77, the system presents a low-temperature glassy phase, an i ntermediate ferromag-\nneticphase,andahigh-temperatureparamagneticphase. Al ongtheferromagnetic-glassy transition\nline magnetic correlations show a universal critical behav ior with critical exponents ν= 0.96(2)\nandη=−0.39(2). The hyperscaling relation β/ν= (1+η)/2 is satisﬁed at the transitions, so that\nβ/ν= 0.305(10). This magnetic critical behavior represents a new u niversality class for ferromag-\nnetic transitions in Ising-like disordered systems. Overl ap correlations are apparently not critical\nand show a smooth behavior across the transition.\nPACS numbers: 75.50.Lk,05.70.Fh,64.60.F-,05.10.Ln\n1I. INTRODUCTION\nSpin glass models are simpliﬁed, although still quite complex, models ret aining the main\nfeatures of physical systems which show glassy behavior in some re gion of their phase di-\nagram. They may be considered as theoretical laboratories where the combined eﬀects of\ndisorder and frustration can be investigated. Their phase diagram and critical behavior\ncan be used to interpret the experimental results for complex mat erials. Ising-like spin\nglasses, such as the ±JIsing model,1model disordered uniaxial magnetic materials char-\nacterized by random ferromagnetic and antiferromagnetic short -ranged interactions, such\nas Fe1−xMnxTiO3and Eu 1−xBaxMnO3; see, e.g., Refs. 2–4. The random nature of the\nshort-ranged interactions is mimicked by nearest-neighbor rando m bonds.\nThree-dimensional (3D) Ising spin glasses have been widely investiga ted. At low temper-\natures they present ferromagnetic and glassy phases, dependin g on the amount of frustra-\ntion. The critical behaviors along the ﬁnite-temperature paramag netic-ferromagnetic and\nparamagnetic-glassy (PG) transition lines have been accurately st udied.5–12On the other\nhand, the low-temperature behavior, in particular the nature of t he glassy phase and of the\nboundary between the ferromagnetic and glassy phases, is still de bated.\nIn this paper we focus on the low-temperature transition line which s eparates the ferro-\nmagnetic phase, characterized by a nonzero magnetization, and t he spin-glass (glassy) phase\nin which the magnetization vanishes while the overlap expectation valu e remains nonzero.\nWe consider the 3D ±JIsing model, deﬁned by the Hamiltonian1\nH=−/summationdisplay\n/angbracketleftxy/angbracketrightJxyσxσy, (1)\nwhereσx=±1, thesumisover thenearest-neighborsitesofacubiclattice, and theexchange\ninteractions Jxyare uncorrelated quenched random variables with probability distrib ution\nP(Jxy) =pδ(Jxy−1)+(1−p)δ(Jxy+1). (2)\nThe usual bimodal Ising spin glass model, for which [ Jxy] = 0 (brackets indicate the average\nover thedisorder distribution), corresponds to p= 1/2. Forp/negationslash= 1/2we have [ Jxy] = 2p−1/negationslash=\n0, and ferromagnetic (or antiferromagnetic) conﬁgurations are energetically favored.\nThe phase diagram of the cubic-lattice ±JIsing model is sketched in Fig. 1. We only\nconsider p≥1/2 because of the symmetry p→1−p. While the high-temperature phase is\n2/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1M\n1/2Is\nDBT\n1−pFP\nG\nFIG. 1: (Color online) Temperature-disorder phase diagram of the 3D ±JIsing model. The phase\ndiagram is symmetric under p→1−p(but for small values of pthe system is antiferromagnetic).\nalways paramagnetic (P), at low temperatures there is a ferromag netic (F) phase for small\nfrustration, i.e., small values of 1 −p, and a glassy (G) phase with vanishing magnetization\nforsuﬃciently largefrustration. InFig.1wedo notreportanylow- temperature mixedphase\nwith simultaneous glassy and ferromagnetic behavior as found in mea n-ﬁeld models13, for\nwhich, at present, there is no evidence.14,15The diﬀerent phases are separated by transition\nlines belonging to diﬀerent universality classes. They meet at a magne tic-glassy multicritical\npoint M located along the so-called Nishimori line16,172/T= ln[p/(1−p)], where the\nmagnetic and the overlap two-point correlation functions are equa l. Scaling arguments18,19\nshow that the transition lines must be all parallel to the Taxis at the multicritical point M.\nThe paramagnetic-ferromagnetic (PF) transition line starts at th e Ising transition of\nthe pure system at p= 1, at20TIs= 4.5115232(16), with a correlation-length exponent\nνIs≈0.6301(ν= 0.63012(16)fromRef. 21and ν= 0.63002(10)fromRef. 20). AlongthePF\nlinethemagneticcriticalbehaviorisuniversal,5andbelongstotherandomly-diluteIsinguni-\nversality class,22,23characterized by the correlation-length critical exponent νPF= 0.683(2).\nIt extends up to the multicritical point M, located at19pM= 0.76820(4), TM= 1.6692(3),\nwhose multicritical behavior is characterized by two even relevant r enormalization-group\n(RG) perturbations with RG dimensions y1= 1.02(5) and y2= 0.61(2). The paramagnetic-\nglassy (PG) transition line runs from M to the ﬁnite-temperature tr ansition at p= 1/2,\nat6TB= 1.11(1). The glassy critical behavior is universal along the PG line;6the overlap\n3correlation-length exponent is quite large,6–11νPG= 2.45(15). Finally, (at least) another\ntransition line is expected to separate the ferromagnetic and glass y phases. This is the\nferromagnetic-glassy (FG) transition line that marks the onset of ferromagnetism and which\nruns from M down to the point D at T= 0. The nature and the general features of this tran-\nsition line in Ising spin glasses are not known. Beside a few numerical wo rks atT= 0,14,15\nthis issue has never been investigated at ﬁnite temperature.\nAn interesting issue concerning the FG transition line is whether it is re entrant, which\nwould imply the existence of a range of values of pfor which the glassy phase is separated\nfromtheparamagneticphase byanintermediate ferromagneticph ase. Asproved in Refs. 16,\n17, ferromagnetism can only exist in the region p > pM, which implies that pD≥pM. We\nalso mention that, using entropicarguments applied to frustration, the FG phase boundary\nwas argued to run parallel to the Taxis,17,24i.e.,pD=pMfor anyT < T M, with the\ncritical behavior controlled by a T= 0percolation ﬁxed point.12The FG transition was\nnumerically investigated at T= 0 in Ref. 14, obtaining the estimate pD= 0.778(5) for the\ncritical disorder, which is slightly larger than pM= 0.76820(4). Thus, it suggests a slightly\nreentrant FG transition line, although its apparent precision is not s uﬃcient to exclude\npD=pM.\nIn this paper we study the nature of the FG transition. In particula r, we investigate\nwhether the magnetic variables show a continuous and universal cr itical behavior from M to\nD, and whether hyperscaling is violated as it occurs in some systems w hose critical behavior\nis controlled by a zero-temperature ﬁxed point, like the 3D random- ﬁeld Ising model.25\nNote that we focus on the low-temperature ferromagnetic trans ition line, which marks\nthe onset of ferromagnetism moving from the glassy phase with zer o magnetization. There\nis also the possibility that a second low-temperature transition line ex ists for larger values\nofp. In this case there would be a mixed low-temperature phase, in which ferromagnetism\nand glassy order coexist. This occurs in mean-ﬁeld models13such as the inﬁnite-range\nSherrington-Kirkpatrick model.26However, numerical T= 0 ground-state calculations in\nthe 3D±JIsing model on a cubic lattice14and in related models15do not seem to show\nevidence of a mixed phase and are consistent with a unique transition .\nIn this paper we present a Monte Carlo (MC) study of the critical be havior along the\nFG transition line. We perform simulations of ﬁnite systems deﬁned on cubic lattices of size\nL≤20. A ﬁnite-size scaling (FSS) analysis of numerical data at T= 0.5 andT= 1 as\n4a function of pshows that magnetic correlations undergo a continuous transition along the\nFG line. The critical behavior is universal, i.e., independent of Talong the line. For the\nmagneticcriticalexponentsweobtain ν= 0.96(2)and η=−0.39(2). Moreover, hyperscaling\nis veriﬁed. The FG transition line turns out to be slightly reentrant. I ndeed, we ﬁnd\npc= 0.7729(2) at T= 0.5 andpc= 0.7705(2) at T= 1, which are deﬁnitely larger than\nthe disorder parameter pM= 0.76820(4) at the multicritical point. Therefore, for a small\ninterval of the disorder parameter, around p≈0.77, the phase diagram presents three\ndiﬀerent phases: a low-temperature glassy phase, an intermediat e ferromagnetic phase, and\na high-temperature paramagnetic phase.\nNote that the critical behavior of the magnetic correlations along t he FG transition line\nshows a new universality class of ferromagnetic transitions in Ising- like disordered systems,\nwhichdiﬀersfromtherandomly-diluteIsinguniversalityclassdescrib ing thecriticalbehavior\nalong the PF transition line, and from the random-ﬁeld Ising universa lity class characterized\nby hyperscaling violation.\nThe general features of the phase diagram presented in Fig. 1 sho uld also characterize the\ntemperature-disorder phase diagram of other 3D Ising spin glass m odels with tunable disor-\nder parameters. For example, one may consider models with Gaussia n bond distributions,\nsuch as\nP(Jxy)∼exp/bracketleftbigg\n−(Jxy−J0)2\n2σ/bracketrightbigg\n, (3)\nwhere theparameters J0andσcontrol theamount ofdisorder (thepureferromagneticmodel\ncorresponds to J0>0andσ= 0). This distributionis alsocharacterized bythepresence ofa\nNishimori line T=σ/J0, where themagneticandtheoverlap two-point correlationfunctio ns\nare equal. We also mention that an analogous temperature-disorde r phase diagram, with\nthree transition lines meeting at a multicritical point like Fig. 1, is also fo und in 3D XY\ngauge glass models.27A similar phase diagram is also expected for other continuous spin\nglasses, like XY and Heisenberg spin glasses with bond distributions (2 ) or (3).\nThe paper is organized as follows. In Sec. II we describe the MC simula tions, and provide\nthe deﬁnitions of the quantities we consider. Sec. III presents th e FSS analysis of the MC\ndata, reporting the main results of the paper. Finally, in Sec. IV we d raw our conclusions.\nIn the appendix we report some details of the FSS analyses.\n5II. MONTE CARLO SIMULATIONS AND OBSERVABLES\nIn order to study the FG transition line, which connects points M and D in Fig. 1, we\nperform MC simulations of the ±JIsing model on cubic lattices of size Lwith periodic\nboundary conditions. We use the Metropolis algorithm, the random- exchange method, and\nmultispin coding. Implementation details can be found in Ref. 6. In the random-exchange\nsimulations we consider NTsystems at the same value of pand at diﬀerent temperatures in\nthe range Tmax≥Ti≥Tmin, withTmax/greaterorsimilar2 andTmin= 0.5. The value Tmaxis chosen so\nthat the thermalization at Tmaxis suﬃciently fast—typically we take Tmax/greaterorsimilarTM≈1.67—\nwhile the intermediate values Tiare chosen such that the acceptance probability for the\ntemperature exchange is at least 10%. We require one of the Tito be along the Nishimori\nline.16The results for this temperature value can be compared with the kn own exact results\nand thus provide a check of the MC code and the thermalization. Fina lly, one of the\ntemperatures always corresponds to T= 1. The parameter NTincreases with Land varies\nfromNT= 5 forL= 4 toNT= 19 for L= 20. Thermalization is checked by verifying\nthat disorder averages are stable when increasing the number of M C steps for each disorder\nrealization. We average over a large number Nsof disorder samples: Ns≈2×106samples\nforL= 4,6,8,Ns≈3×105forL= 10,Ns≈105forL= 12,Ns≈5×104forL= 16, and\nNs≈5×103forL= 20.\nThe simulations are quite costly, because of the very slow dynamics f or low temperatures.\nThis makes the computational eﬀort increase with a large power of t he lattice size. In our\nrange of values of L, the number of iterations which must be discarded for thermalizatio n\napparently increases as L8for our largest lattices (with an increasing trend with increasing\nL). Hence, taking into account the volume factor, theCPU time for e ach disorder realization\napparently increases as L11(but we should warn that its large- Lasymptotic behavior may\nbe even worse). In total, simulations took approximately 40 years o f CPU time on a single\ncore of a recent standard commercial processor.\nWe consider the magnetization and the magnetic correlation functio n deﬁned as\nm=1\nV[/angbracketleft|/summationdisplay\nxσx|/angbracketright], (4)\nG(x)≡[/angbracketleftσ0σx/angbracketright],\nwhere the angular and the square brackets indicate the thermal a nd the quenched average\n6over disorder, respectively. We deﬁne the magnetic susceptibility a nd the second-moment\ncorrelation length, respectively as\nχ≡/summationdisplay\nxG(x), (5)\nξ2≡1\n4sin2(qmin/2)/tildewideG(0)−/tildewideG(q)\n/tildewideG(q),\nwhereq= (qmin,0,0),qmin≡2π/L, and/tildewideG(q) is the Fourier transform of G(x). Moreover,\nwe consider the cumulants\nU4≡[µ4]\n[µ2]2, (6)\nU22≡[µ2\n2]−[µ2]2\n[µ2]2,\nwhere\nµk≡ /angbracketleft(/summationdisplay\nxσx)k/angbracketright. (7)\nAt the critical point Rξ≡ξ/L,U4, andU22(in the following we call them phenomenological\ncouplings and denote them by R) are expected to approach universal values in the large- L\nlimit (within cubic L3systems with periodic boundary conditions). In the ferromagnetic\nphase we have U4→1,U22→0, andRξ→ ∞, while in the glassy phase we expect Rξ→0.\nWe also deﬁne analogous quantities using the overlap variables qx≡σ(1)\nxσ(2)\nx, whereσ(1)\nx\nandσ(2)\nxaretwo independent replicas corresponding tothesamecouplings Jxy. Inparticular,\nwe consider ξoandUo\n4deﬁned by replacing the magnetic variables with the overlap variables\nin Eqs. (5) and (6).\nIII. FINITE-SIZE SCALING ANALYSIS\nIn this section we present a ﬁnite-size scaling (FSS) analysis of the M C data close to the\nFG transition line. We consider two values of the temperature, T= 0.5 andT= 1, below\nthe temperature TM= 1.6692(3) of the multicritical point M, and perform a FSS analysis\nas a function of p.\n70.225 0.226 0.227 0.228 0.229 0.230 0.231 0.232\n1-p0.70.80.9\nRξ\nL=4  \nL=6\nL=8\nL=10\nL=12\nL=16\nL=20T=0.5\nFIG. 2: (Color online) Estimates of RξatT= 0.5. The vertical lines show the location of the\nmulticritical point M: 1 −pM= 0.23180(4).\nA. Phenomenological couplings and universality\nTo begin with, we analyze the data at T= 0.5. In Fig. 2 we show the MC estimates\nofRξas a function of 1 −p. Analogous plots are obtained for U4andU22. The data\nfor diﬀerent lattice sizes clearly show crossing points, providing evid ence for a continuous\ntransition. They cluster at values of pwhich are deﬁnitely larger than pM, ruling out a\nvertical transition line from M to the T= 0 axis.\nIn the critical limit, the phenomenological couplings Rscale as\nR=fR[(p−pc)L1/ν], (8)\nwhere we have neglected analytic and nonanalytic scaling correction s. Equivalently, one can\ntest FSS by considering two diﬀerent couplings R1andR2. In the FSS limit R1=F12(R2),\nwhere the function F12(R2) is universal, i.e., identical in any model that belongs to a given\nuniversality class. Clear evidence of FSS is provided in Fig. 3, where th e phenomenological\ncouplings U4andU22are reported versus Rξ≡ξ/L. The data appear to rapidly approach\na nontrivial limit with increasing the lattice size. Scaling corrections ar e only visible in the\ncase ofU22, but they decrease with increasing L.\nIn order to determine the critical parameter pcand the exponent ν, we ﬁtU4,U22, and\n80.6 0.7 0.8 0.9Rξ0.20.30.4\nU22L=4  \nL=6\nL=8\nL=10\nL=12\nL=16\nL=20T=0.5\n0.6 0.7 0.8 0.9Rξ1.31.41.5\nU4L=4  \nL=6\nL=8\nL=10\nL=12\nL=16\nL=20T=0.5\nFIG. 3: (Color online) U4(bottom) and U22(top) vs RξatT= 0.5.\nRξ≡ξ/Lto Eq. (8). Details are reported in App. A1. We obtain\npc(T= 0.5) = 0.7729(2), ν= 0.96(2), (9)\nR∗\nξ= 0.764(6), U∗\n4= 1.331(5), U∗\n22= 0.305(2), (10)\nwhereR∗=fR(0) is the value of the phenomenological coupling Rat the critical point.\nScaling corrections turn out to be small.\nAn analogous FSS analysis can be performed at T= 1, with the purpose of checking\nuniversality, i.e., of determining whether all transitions along the FG lin e belong to the same\nuniversality class. For this purpose, we use the fact that, given an y pair of RG invariant\nquantities R1andR2, the FSS function R1=F12(R2) is universal. In Fig. 4 we plot U4\nandU22versusRξfor both T= 0.5 andT= 1. The plot of U4provides good evidence of\nuniversality: all data fall onto a single curve with remarkable precisio n. The results for U22\n9show instead signiﬁcant scatter, but they are also consistent with universality if one takes\ninto account scaling corrections: indeed, as Lincreases the data for T= 1 approach the\nT= 0.5 results.\nFor a more quantitative check, we must explicitly take into account s caling corrections at\nT= 1, since they aresigniﬁcantly larger than those observed at T= 0.5. For instance, ﬁts of\nthe phenomenological couplings at T= 1 to Eq. (8) show a somewhat large χ2/DOF (DOF\nis the number of degrees of freedom of the ﬁt). Moreover, the es timates show systematic\ntrends as the lattices with smaller values of Lare discarded in the ﬁt, see App. A1 for\ndetails. To include scaling corrections, we ﬁt the data to\nR=fR[(p−pc)L1/ν]+L−ωgR[(p−pc)L1/ν]. (11)\nThe smallest χ2/DOF is obtained for 0 .8/lessorsimilarω/lessorsimilar0.9. Correspondingly ν= 0.91(3), in\nsubstantial agreement with the estimate (9). Also the estimates o fR∗\nξ,U∗\n4, andU∗\n22, see\nApp. A1, are in agreement with the estimates (10) at T= 0.5. Therefore, all results\nstrongly support the universality of the critical behavior along the FG line. It is diﬃcult\nto estimate reliably the exponent ωfrom the data. It we assume universality and ﬁt the\nresults at T= 1 ﬁxing ν= 0.96(2), we obtain ω= 0.95(10). Note that the ﬁts of the data\natT= 0.5 give much larger values for ω, i.e.,ω/greaterorsimilar2, see App. A1. This is probably due\nto the fact that corrections with ω≈1 have very small amplitudes at T= 0.5, so that we\nare simply measuring an eﬀective exponent that mimicks the behavior of several correction\nterms.\nThe FSS ﬁts also provide estimates of pcatT= 1. We obtain\npc(T= 1) = 0.7705(2). (12)\nNote that pc(T= 1)> pM≈0.7682, conferming the reentrant nature of the FG transition\nline.\nB. Magnetic susceptibility\nAs discussed at length in Ref. 6, in the critical limit the magnetic susce ptibility scales as\nχ(p,L) =uh(p)2L2−ηfχ[(p−pc)L1/ν], (13)\n100.6 0.7 0.8 0.9Rξ0.20.30.4\nU22L=4     T=1\nL=6\nL=8\nL=10\nL=12\nL=16\nL=20\nL=12  T=0.5\nL=16\nL=20\n0.60 0.65 0.70 0.75 0.80 0.85 0.90Rξ1.21.31.41.5\nU4L=4     T=1\nL=6\nL=8\nL=10\nL=12\nL=16\nL=20\nL=12  T=0.5\nL=16\nL=20\nFIG. 4: (Color online) U4(bottom) and U22(top) vs RξatT= 1 and at T= 0.5 (only data with\nL≥12).\n0.6 0.7 0.8 0.9 1.0\nξ/L1.52.02.5(χ/uh2)ξ−2.39L=4   T=0.5\nL=8\nL=12\nL=16\nL=20\nL=4   T=1\nL=8\nL=12\nL=16\nL=20\nFIG. 5: (Color online) /tildewideχ≡χu−2\nhξ−2.39versusξ/LforT= 1 and T= 0.5.\n11whereuh(p) is related to the magnetic scaling ﬁeld and is an analytic function of p(and also\nof the temperature). Fits of χatT= 1 and T= 0.5 are good ( χ2/DOF of order 1) if we\ninclude all data such that L≥6, provided that uh(p) is taken into account (see App. A2\nfor details). We end up with the ﬁnal estimate\nη=−0.39(2). (14)\nSinceξ/Lis a function of ( p−pc)L1/νin the FSS limit, see Eq. (8), we can rewrite Eq. (13)\nas\nχ(p,L) =uh(p)2ξ2−ηFχ(ξ/L). (15)\nThe function Fχ(x) is universal apart from a multiplicative constant, which takes into a c-\ncount the freedom in the normalization of the function uh(p). In Fig. 5 we show the quantity\n/tildewideχ=χu−2\nhξ−2.39forT= 1 and T= 0.5. For each temperature the function uh(p) is deter-\nmined by ﬁtting the susceptibility data to Eq. (15), ﬁxing η=−0.39. Moreover, the scaling\nﬁelds are normalized so that /tildewideχ(T= 1,L= 16)≈/tildewideχ(T= 0.5,L= 16) for ξ/L≈0.8. If we\ndiscard the data with L= 4 and 8 at T= 0.5, all points fall on top of each other, conﬁrming\nuniversality.\nC. Evidence of hyperscaling\nSince the FG transition line extends up to T= 0, hence the critical behavior may be\ncontrolled by a zero-temperature ﬁxed point, hyperscaling might b e violated, as it happens\nin the 3D random-ﬁeld Ising model.25In order to check whether hyperscaling holds along\nthe FG line, we consider the magnetization, which is expected to beha ve asm∼L−β/νat\nthe critical point, and the magnetic susceptibility, which scales as χ∼L2−η. If hyperscaling\nholds,βandηare related by\nβ\nν=d−2+η\n2, (16)\n(in the present case d= 3), which guarantees that χ/m2scales as Ld. In order to verify\nwhether Eq. (16) holds, we consider H≡χ/(m2L3) and assume that it behaves as\nH≡χ\nm2L3∼LζfH[(p−pc)L1/ν]. (17)\n120.0 0.5 1.0 1.5\nLβ/νmt0.00.51.01.5L-β/νP(mt)L=6\nL=8\nL=12\nL=16\nL=20\nβ/ν=0.305T=0.5,  p=0.7729\nFIG. 6: (Color online) Scaling behavior of the distribution of the thermal averages of the magne-\ntization, at T= 0.5 andp=pc= 0.7729. We set β/ν= 0.305.\nIf hyperscaling holds, ζvanishes. A FSS analysis of the data at T= 0.5 andT= 1 gives\nthe rather stringent bound (details in App. A3)\n|ζ|<0.01, (18)\nwhich allows us to conclude, quite conﬁdently, that hyperscaling hold s. If this the case,\nusing estimates (14) and (9) of ηandν, we obtain\nβ/ν= (1+η)/2 = 0.305(10), β= 0.29(1). (19)\nAs a further check, we consider the sample distribution P(mt) of the thermal averages of\nthe magnetization\nmt≡1\nV/angbracketleft|/summationdisplay\nxσx|/angbracketright (20)\nat the critical point p=pc= 0.7729,T= 0.5, which is expected to behave asymptotically\nas\nP(mt)≈Lβ/νP(Lβ/νmt). (21)\nIn Fig. 6 we plot P(Lβ/νmt) usingβ/ν= 0.305. The data clearly show the expected\nscaling behavior. In conclusion, the numerical results do not show e vidence of hyperscaling\nviolations in the critical behavior of magnetic correlations.\nOur data for H(p,L) can also be used to provide further evidence of universality. Inde ed,\nif we use the fact that ξ/Lis a function of ( p−pc)L1/ν, see Eq. (8), we can rewrite Eq. (17)\n130.6 0.7 0.8 0.9 1.0\nξ/L1.101.151.20 HL=4   T=0.5\nL=8\nL=12\nL=16\nL=20\nL=4   T=1\nL=8\nL=12\nL=16\nL=20\nFIG. 7: (Color online) H≡χ/(m2L3) versusξ/LforT= 1 and T= 0.5.\nforζ= 0 as\nH(p,L) =FH(ξ/L)+O(L−ω), (22)\nwhereFH(x) should be the same at T= 0.5 and at T= 1 if all transitions along the FG\ntransition line belong to the same universality class. The plot of the da ta, see Fig. 7, clearly\nconﬁrms universality: all points fall onto a single curve.\nD. Overlap correlations\nIn our numerical study we also consider quantities involving the over lap variables, such\nasξo/LandUo\n4, deﬁned at the end of Sec. II. In Fig. 8 we show MC data up to L= 12\n(since their computation turned out to be signiﬁcantly more demand ing, we restricted the\nmeasurements for the lattices L= 16,20 to the magnetic correlations). Unlike the magnetic\nquantities, the overlap data do not show crossings in the interval o fpwe have investigated.\nApparently Uo\n4decreases continuously, while ξo/Lincreases as L→ ∞. This may reﬂect\nthe fact that the FG transition line separates two orderedphases with respect to the overlap\nvariables. Note that the diﬀerences between data at the same pandTand at diﬀerent values\nofLdecrease as 1 −pincreases. Hence, if there is a line in the ( p,T) plane where the overlap\nvariables show crossings, it must be such that 1 −p >0.234, i.e., it must lie in the region\np < pM, where no ferromagnetism is possible.\n140.222 0.224 0.226 0.228 0.230 0.232 0.234\n1-p1.01.11.2\nU4o\nL=4   T=1L=6L=8L=10L=12L=4   T=1/2L=6L=8L=10L=12\n0.222 0.224 0.226 0.228 0.230 0.232 0.234\n1-p0.51.01.52.0\nξo/L\nL=4   T=1L=6L=8L=10L=12L=4   T=1/2L=6L=8L=10L=12\nFIG. 8: (Color online) Estimates of ξo/L(bottom) and Uo\n4(top), deﬁned in terms of the overlap\nvariables, at T= 0.5 andT= 1.\nIV. CONCLUSIONS\nWe investigate the critical behavior along the ferromagnetic-glass y transition line of the\nT-pphase diagram of the cubic-lattice ±J(Edwards-Anderson) Ising model, cf. Eq. (1),\nwhich marks the low-temperature boundary between the ferroma gnetic phase and the glassy\nphase where the magnetization vanishes, i.e., the transition line that runs from M down to\nthe point D at T= 0 in Fig. 1.\nWe present a numerical study based on MC simulations of systems of size up to L= 20,\nobtaining MC estimates of several quantities at T= 0.5 andT= 1 (which are well below\nthe temperature TM= 1.6692(3) of the multicritical point M) as a function of the disorder\nparameter p. TheresultsoftheFSSanalysesareconsistentwiththetwocontin uousmagnetic\n15transitions belonging to the same universality class. The correspon ding critical exponents\nareν= 0.96(2) and η=−0.39(2). Since the critical line extends up to T= 0, the\ncritical behavior may be controlled by a zero-temperature ﬁxed po int. Correspondingly, it\nis possible to have hyperscaling violations, as it occurs in the 3D rando m-ﬁeld Ising model.\nOur MC results show that the hyperscaling relation β/ν= (1 +η)/2 is satisﬁed, so that\nβ/ν= 0.305(10) and β= 0.29(1). The FSS results provide a robust evidence of a universal\nmagnetic critical behavior along the FG transition line. A reasonable h ypothesis is that also\nthe zero-temperature transition belongs to the same universality class. This is supported\nby the available numerical data at T= 0. The numerical study of Ref. 14 for the ±J\nIsing model at T= 0, using lattice sizes up to L= 14, provided evidence of a magnetic\ntransition at pD= 0.778(5), with critical exponents ν= 1.3(3) and β= 0.2(1). Numerical\nanalyses15for other Ising spin-glass models at T= 0 give consistent values of the critical\nexponents, ν= 0.9(2) and β= 0.3(1) using data up to L= 12. These estimates are\nsubstantially consistent with our results along the FG transition line, supporting a universal\ncritical behavior along the FG transition from the multicritical point M down to the T= 0\naxis.\nWealso investigate the behavior of overlap correlations. They do no t appear to be critical\nand show an apparently smooth behavior across the FG transition. Our numerical results\ndo not show evidence of other transitions close to the transition line where ferromagnetism\ndisappears. Thus, they do not hint at the existence of a mixed ferr omagnetic-glassy phase,\nas found in mean-ﬁeld models,13in agreement with earlier T= 0 numerical studies.14,15\nTheFGtransitionlineisslightlyreentrant. Indeed, weﬁndthat pc= 0.7729(2)at T= 0.5\nandpc= 0.7705(2) at T= 1, which are deﬁnitely larger than pM= 0.76820(4), although\nthey are quite close. This implies that there exists a small interval of the disorder parameter,\naroundp≈0.77, showing three diﬀerent phases when varying T: with increasing the tem-\nperature, the system goes from the low-temperature glassy pha se with zero magnetization,\nto an intermediate ferromagnetic phase, and ﬁnally to the high-tem perature paramagnetic\nphase. Correspondingly, itﬁrst undergoes aglassy-ferromagne tic transitionwith ν= 0.96(2)\nand then a ferromagnetic-paramagnetic transition with ν= 0.683(2). We mention that a\nslightly reentrant low-temperature transition line, where ferroma gnetism disappears, also\noccurs in the phase diagram of the 2D ±JIsing model.28,29\nThe main features of the FG transition line are not expected to depe nd on the particular\n160.0 0.1 0.2 0.3 0.4 0.5\n1-p01234\nTIs\nPF\nMCP\nPG\nFG\nT=0\nN line\nFP\nG\nFIG. 9: (Color online) Numerical results for the phase bound aries of the cubic-lattice ±JIsing\nmodel (1) in the T-pplane. The dashed lines are interpolations of the data30.\ndiscretebonddistributionofthe ±JIsing model, cf. Eq. (2). Theyshouldalsoapplytomore\ngeneral distributions with tunable disorder parameters, such as t he Gaussian distribution\nreported in Eq. (3), and also to experimental spin glass systems wit h tunable disorder.\nWe conclude showing Fig. 9 which reports all available numerical result s for the phase\nboundaries of the cubic-lattice ±JIsing model (1) in the T-pplane, taken from Ref. 5 for\nthe PF transition line, from Ref. 19 for the multicritical point along th e Nishimori (N) line\nT= 2/ln[p/(1−p)], fromRef. 6 for the data along the PG line, from this paper along th e FG\nline, and from Ref. 14 for the T= 0 transition point. The dashed lines are interpolations\nof the data along the transition lines which satisfy the expected sca ling behavior at the\nmulticritical point where they meet, controlled by the crossover ex ponentφ= 1.67(10), see\nRefs. 19,28 for details.30\nAcknowledgments\nThe MC simulations were performed at the INFN Pisa GRID DATA cente r, using also\nthe cluster CSN4.\n17TABLE I: Results of combined ﬁts of U4,U22, andRξto Eq. (A1) without scaling corrections. χ2\nis the sum of the residuals in the ﬁt and DOF is the number of deg rees of freedom. Each column\ncorresponds to results in which only data satisfying L≥Lminare included. R∗≡fR(0) is the\nvalue of the phenomenological coupling at the critical poin t.\nT= 0.5\nLmin 4 6 8 10\nχ2/DOF 5594/289 567/229 203/169 79/109\nν 0.971(4) 0.964(5) 0.954(8) 1.00(2)\npc0.77230(1) 0.77275(2) 0.77284(3) 0.77281(5)\nR∗\nξ0.7453(2) 0.7564(3) 0.7592(6) 0.759(2)\nU∗\n41.3450(2) 1.3364(4) 1.3343(6) 1.334(2)\nU∗\n220.3046(2) 0.3045(3) 0.3057(6) 0.310(2)\nT= 1\nLmin 4 6 8 10\nχ2/DOF 10593/289 1842/229 365/169 98/109\nν 1.054(5) 0.995(5) 0.963(8) 0.982(20)\npc0.76819(2) 0.76920(2) 0.76975(3) 0.76994(5)\nR∗\nξ0.6826(2) 0.7019(3) 0.7147(6) 0.7220(17)\nU∗\n41.3973(3) 1.3779(4) 1.3662(6) 1.3613(19)\nU∗\n220.3094(3) 0.3020(4) 0.2977(5) 0.3013(17)\nAppendix A: Analysis details\n1. Phenomenological couplings\nIn order to determine the exponent νand the critical parameter pc, we analyze the\nphenomenological couplings U4,U22, andRξ≡ξ/L. In the critical limit each quantity R\nbehaves as\nR(p,L)≈fR[up(p)L1/ν]+uω(p)L−ωgR[up(p)L1/ν], (A1)\nwhere the nonlinear scaling ﬁelds up(p) anduω(p) are analytic functions of p. We have\nup(pc) = 0 while, in general, we expect uω(pc)/negationslash= 0. For both temperatures our data belong\n18to a small interval of values of p, so that we expect the approximations up(p)≈p−pc\nanduω(p)≈uω(pc) =aωto work well. To check it, we also performed ﬁts assuming\nup(p) =p−pc+k(p−pc)2. We did not ﬁnd any signiﬁcant diﬀerence.\nWe ﬁrst analyze the results at T= 0.5. We perform combined ﬁts of the three quantities\nto Eq. (A1) without scaling corrections (we set aω= 0). If the scaling functions fRare\napproximated by fourth-order polynomials, we obtain the results r eported in Table I. We\nreport estimates for diﬀerent Lmin: in each ﬁt we only include data satisfying L≥Lmin.\nCorrections are quite small and indeed the results corresponding t oLmin= 8 andLmin= 10\nmostly agree within errors. We also perform ﬁts that take into acco unt scaling corrections.\nWe ﬁxω, approximate gR(x) by a second-order polynomial, and repeat the ﬁt for several\nvalues of ωbetween 1 and 5. If we perform a combined ﬁt of U4andRξ(we include all\nresults with L≥4), the smallest χ2/DOF (DOF is the number of degrees of freedom of\nthe ﬁt) is obtained for 3 /lessorsimilarω/lessorsimilar4 and one would estimate ν= 0.96(1) and pc= 0.7729(1).\nIf instead we use U4,Rξ, and also U22we obtain ω≈2,ν= 0.95(1), and pc= 0.7731(1).\nThese results indicate that scaling corrections are quite small, and q uite probably cannot\nbe parametrized be a single correction term. Our best estimates of ωare simply eﬀective\nexponents that parametrize the contributions of several diﬀere nt correction terms, which are\nall relevant for our small lattice sizes.\nIf we compare all results, we end up with the estimates pc= 0.7729(2) and ν= 0.96(2),\nreported inEq. (9). Forthe phenomenological couplings at criticalit y,R∗≡fR(0), we obtain\nthe estimates reported in Eq. (10), i.e., R∗\nξ= 0.764(6),U∗\n4= 1.331(5) and U∗\n22= 0.305(2).\nThe ﬁnal estimates and their errors take into account the results of the ﬁts with and without\nscaling corrections.\nThe same analyses can be performed at T= 1. Combined ﬁts to Eq. (A1) without scaling\ncorrections give the results reported in Table I. It is quite clear tha t scaling corrections at\nT= 1 are larger then those at T= 0.5. The goodness of the ﬁt is worse and the ﬁt results\nshow systematic trends. It is however reassuring that they appa rently converge towards the\nestimates (9) and (10), in agreement with universality.\nIt is interesting to check whether scaling corrections can explain th e diﬀerences which\noccur among the results for T= 1 reported in Table I and the results obtained at T= 0.5.\nSince the results for U∗\n22atT= 1 are nonmonotonic as a function of Lmin, at least two\ncorrection terms must be included to explain the observed trend of the data. Therefore,\n19TABLE II: Estimates of the exponent ηobtained by ﬁts to Eq. (A3), where ˆfχis approximated by\na fourth-order polynomial and ˆ u(p) by a second-order polynomial. In each ﬁt we only include the\ndata which satisfy L≥Lmin. We ﬁx ν= 0.96(2) and the value of pc:pc= 0.7729(2) at T= 0.5\nandpc= 0.7705(2) at T= 1.\nT= 0.5 T= 1\nLminχ2/DOF η χ2/DOF η\n4 516/94 −0.414(6) 62/94 −0.393(6)\n6 39/74 −0.400(8) 18/74 −0.389(9)\n8 22/54 −0.397(12) 16/54 −0.389(12)\n10 11/34 −0.398(16) 6/34 −0.390(16)\nthe ﬁt of the U22data with a single scaling correction makes no sense. In any case, th e\nestimate obtained for Lmin= 10 diﬀers from the one reported in Eq. (10) by one combined\nerror bar, and therefore is in agreement with universality. We then perform combined ﬁts\nofU4andξ/Lto Eq. (A1), approximating gR(x) by a second-order polynomial and ﬁxing\nωto several values between 0.5 and 1.5. The smallest χ2/DOF is obtained for 0 .8/lessorsimilarω/lessorsimilar\n0.9. Correspondingly, we obtain pc= 0.7705(1), R∗\nξ= 0.765(10), and U∗\n4= 1.32(1). The\nestimates of the phenomenological couplings at criticality are now in v ery good agreement\nwith the estimates at T= 0.5. As for νwe obtain ν= 0.91(3), which is sligthly smaller\nthan, but still consistent with the estimate at T= 0.5. If we ﬁx ν= 0.96(2) as obtained at\nT= 0.5, we ﬁnd ω= 0.95(10),pc= 0.7704(1), R∗\nξ= 0.757(7),U∗\n4= 1.326(6).\nThese ﬁts provide an estimate of pcatT= 1. We quote the estimate pc= 0.7705(2)\nalready reported in Eq. (12), which satisﬁes the inequality pc/greaterorsimilar0.7700, which one would\nobtain from the results reported in Table I. It is unclear how reliable o ur estimates of ωare.\nIn any case, they suggest a value close to 1.\n2. Magnetic susceptibility\nWe analyze the magnetic susceptibility which should scale as\nχ(p,L) =uh(p)2L2−ηfχ[up(p)L1/ν], (A2)\n20TABLE III: Estimates of the exponent ζ. In each ﬁt we only include the data which satisfy\nL≥Lmin. We ﬁx ν= 0.96(2) and the value of pc:pc= 0.7729(2) at T= 0.5 andpc= 0.7705(2)\natT= 1.\nT= 0.5 T= 1\nLminχ2/DOF ζ χ2/DOF ζ\n4 88/98 −0.007(2) 161/98 −0.015(1)\n6 9/78 −0.003(2) 15/78 −0.009(2)\n8 8/58 −0.002(3) 3/58 −0.006(3)\n10 4/38 −0.005(5) 2/58 −0.005(5)\nwhereuh(p) is related to the magnetic scaling ﬁeld and is an analytic function of p; scaling\ncorrections have been neglected. In order to determine η, we perform ﬁts to\nlnχ= (2−η)lnL+ˆfχ[(p−pc)L1/ν]+ ˆu(p), (A3)\nwhereˆfχis approximated by a fourth-order polynomial and uh(p) is normalized so that\nˆu(p=pc) = 0. In this expression we have replaced up(p) withp−pc. Inclusion of the\nsecond-order term does not change the quality of the ﬁt and the r esults. Instead, even if\nthe interval in pis small, the function uh(p) cannot be approximated by a constant, hence\nˆu(p) cannot be set to zero. Indeed, the ﬁts in which ˆ u(p) is approximated by a second-\norder polynomial have a χ2/DOF which is signiﬁcantly smaller than those in which we\nset ˆu(p) = 0. For instance, for T= 0.5 andLmin= 6 (we ﬁx pcandν, see caption of\nTable II), we have χ2/DOF = 265 /76 and 39/74 for the ﬁt with ˆ u(p) = 0 and the ﬁt with\na second-order polynomial, respectively. The results of the ﬁts in w hich we ﬁx νandpcare\nreported in Table II. The results are very stable with Lminand are completely consistent\nwith universality. Note that, at variance with what is observed for t he phenomenological\ncouplings, corrections for T= 1 are apparently smaller than for T= 0.5. This may indicate\nthe presence of several corrections which cancel out for our va lues ofL. A conservative ﬁnal\nestimate is η=−0.39(2), already reported in Eq. (14).\n213. Hyperscaling\nIn order to study hyperscaling we consider the ratio\nH≡χ\nm2L3. (A4)\nIf hyperscaling holds, it should behave as\nH(p,L) =fh[up(p)L1/ν]≈fH[(p−pc)L1/ν], (A5)\nwhere we have neglected scaling corrections. In order to allow for a possible hyperscaling\nviolation we introduce a new exponent ζand assume that\nH(p,L) =LζfH[(p−pc)L1/ν]. (A6)\nTo determine ζwe perform ﬁts to\nlnH(p,L) =ζlnL+ˆfH[(p−pc)L1/ν], (A7)\nwhereˆfH(x) is approximated by a second-order polynomial. Fit results are repo rted in\nTable III. Here we ﬁx νandpcto the values determined above. The quality of the ﬁts is\nvery good and scaling corrections are apparently small for both va lues of the temperature.\nThe exponent ζis clearly compatible with zero, proving that hyperscaling is satisﬁed. More\nprecisely, we obtain the bound |ζ|<0.01, already reported in Eq. (18).\n1S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975).\n2A. Ito, H. Aruga, E. Torikai, M. Kikuchi, Y. Syono, H. Takei, P hys. Rev. Lett. 57, 483 (1986).\n3K. Gunnarsson, P. Svedlindh, P. Nordblad, L. Lundgren, H. 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For T≤1\nwe report the straight lines connecting the data at T= 1,0.5 andT= 0.5,0. Analogously, we\nproceed for the PG line, reporting the curve p−pM=cu2(p,T)φforp≥0.7 and a straight line\nfor 0.5≤p≤0.7.\n24"    },    {        "title": "2401.17554v1.Ferromagnetic_Semiconductors_and_Spintronic_Devices.pdf",        "content": "1 \nECE423_25 \n \nFerromagnetic Semiconductors and Spintronic \nDevices \n \nNazmul Hasan, Graduate Student Member, IEEE  \n \n \nAbstract— Ferromagnetic semiconductors play a crucial role \nin spintronic devices, enabling effective control of electron spin \nover charge. This study explores their unique properties, ongoing \nadvancements in spin control, and potential integration into next-\ngeneration semiconductor technologies. \n \nIndex Terms —Ferromagnetic semiconductors, electron spin, \nspintronics, device physics. \n \nI. INTRODUCTION \nISCOVERY of giant magnetoresistance (GMR) and \ntunneling magnetoresistance (TMR) during 1980s-\n2000s opened a new avenue in solid state physics to \nunderstand and control both charge transport and spin \nbehavior of carriers in semiconductors [1]. However, \nsemiconductors’ physical transport behavior is extraordinarily \nsensitive to defects, impurities, gate biases, and so on whereas \nmagnetism comes from an ordered state of collective \nelectronic formations which have a significant impact on \nmaterial’s optoelectronic properties. Introducing local \nmoments into typical semiconductors in 1970s led to the \nrealization that it is possible to tune materials’ physics by \ncombining semiconducting and magnetic behavior through \ndiluted magnetic semiconductors [2].  \nHowever, it has established that temperature dependent \nferromagnetic transition is possible into semiconductor \nmaterials when III-V materials are heavily doped with Mn \nutilizing valence band (VB) carriers. Strong interaction of VB \ncharge carriers of s shell with localized magnetic moments of \npartially filled d/f shell significantly affect magnetic order and \nthus carrier motion [3]. Besides, applying external magnetic \nfield also can alter semiconductors’ optoelectronic behavior \nthrough creating or manipulating the flow of spin-polarized \ncarriers [4]. Hence, manipulating materials’ properties \nutilizing their both semiconducting and magnetic behavior \nthrough controlling carrier concentrations and magnetic order \nby applying external magnetic field is become intriguing in the \nsolid-state physics for next generation electronics devices. \nSpintronic devices utilize this spin-current transport behavior \nfollowing various device engineering techniques for high-\nspeed information processing. \n \nThis work was accomplished as a term final paper for the Semiconductor \nDevices (ECE423) course under Professor Roman Sobolewski at the \nElectrical and Computer Engineering Department of the University of \nRochester. (Corresponding Author: Nazmul Hasan)  \nNazmul Hasan is with Electrical and Computer Engineering Department of \nthe University of Rochester, Rochester, NY 14620, USA (e-mail: \nnhasan5@ur.rochester.edu). \n To understand the functional ferromagnetic semiconductor \nspintronics, initiation physics of ferromagnetism in \nsemiconductors described with spintronics device principles in \nthe following section 2 and 3. Moreover, recent device \nengineering techniques from materials to scaling are also \nshortly drew up in section 4, and at last section practical \napplications perspectives are discussed.  \nII. ORIGIN OF FERROMAGNETISM IN SEMICONDUCTORS  \nIn semiconductor, universally, interplay between electronic \nspin degree of freedom, repulsive coulomb interactions \nbetween electrons, and the fermionic quantum statistics of \nelectrons arises ferromagnetism in materials where Pauli \nexclusion principle play a vital role to correlate the spin and \norbital behavior to determine particle exchange symmetry in \nelectronic wave functions [1], [5]. The repulsive coulomb \ninteractions between electrons strongly influence magnetic \norder of the material and aligning with Pauli’s principle \nelectrons’ groups sharing same spin state results to having \nquantum ground state of the system. In ferromagnetic case, a \nnonzero local spin density of states is associated with the \nmany-electron wave function of the system in the same \ndirection of space at ground state [3], [6]. This dependence of \nthe system energy on the orientation order of magnetic ion’s \nlocal moments is called exchange interaction which can be \ncontrolled by doping in the host semiconductor material. \nHence, fermi statistics explain the origin of ferromagnetic \nbehavior in semiconductors as illustrated in Fig. 1 below. \nMagnetic semiconductors usually have narrower carrier \nenergy bands where s orbital electrons strongly interact with \npartially filled d/f shell’s magnetic moments. \nConsidering Mn doped GaAs semiconductor, where the \nsystem energy is dependent on the exchange interaction of Mn \nbased on its local moments’ orientation. Intrinsically, itinerant \nexchange of electron spins can result in spontaneous spin \npolarization of the entire system as result of having same spin \nstate. Which will lead to double occupancy in each eigen \nstates with a large density of states at fermi level that in turn \nmakes possible to move electrons from one spin band to \nanother having system’s kinetic energy lower. Hence, Stoner \ninstability happens in the ferromagnetic system, and thus \nStoner mechanism of localized key spin failed in \nferromagnetism [4]. However, Heisenberg’s direct exchange \nin the local nature of two spin moments in effect of strong \nlocal coulomb interactions difference of singlet spin wave and \ntriplet spin wave suppress valence charges fluctuations. That \nleads to transferring nonmagnetic atom’s electron to the \nfavorable empty shell of magnetic atoms resulting in polarized D2 \nECE423_25 \n \nnonmagnetic atom by electrons local moment that is coupled \nwith neighboring magnetic atoms as shown in Fig. 1(b). This \nexchange coupled magnetic moments form parallel spin \nalignment ferromagnetically increasing hopping probability by \ndecreasing kinetic energy. Hence, conduction electrons tend to \nmaintain ferromagnetically favored system as the electron \nenergy is minimal though the ordering is dependent on the \ncurie temperature Tc of the system [7]. \n \n  Fig. 1. (a) & (b) Magnetic behavior mechanism for \nFerromagnetic materials, (c) Changes in band structure based \non spin orientation [8]. \nIII. SEMICONDUCTOR SPINTRONIC DEVICE PRINCIPLES  \nAdding spin degree of freedom in electrons has ushered in a \nnew era of versatility, allowing for precise control over the \nmagnetic spin behavior of electrons, particularly in the realm of \nnovel nanoelectronic devices. Semiconductor spintronics has \nintroduced crucial concepts in spin transport, spin injection, \nSilsbee-Johnson spin-charge coupling, spin-dependent tunneling, \nspin relaxation, and spin dynamics. This intricate interplay \nbetween varying symmetries and structures results in diverse \nforms of spin-orbit coupling within the semiconductor-based \nheterostructures system, yielding a spectrum of effective spin-\norbit Hamiltonians that provides a versatile and adaptable \nframework for manipulating and controlling spin-related \nphenomena in semiconductor materials [6]. \nAlike semiconductor doping concentration led quantum \ntunneling mechanism in ferromagnetic semiconductors also holds \npivotal significance for the field of spintronics particularly in \ncomprehending the transport of spin-polarized electrons through \nmaterials. Quantum tunneling is observed in ferromagnetic \nsemiconductors devices, particularly in magnetic tunnel junctions \n(MTJs) where two ferromagnetic layers are separated by a thin insulating barrier that allows electrons to traverse potential energy \nbarriers possessing their specific spin orientations resulting in a \ncontrolled flow of spin-polarized electrons [9]. According to \nquantum mechanics, there exists a finite probability that these \nelectrons can tunnel through the barrier despite its energy being \nhigher than the electrons' energy. The probability of tunneling is \ninfluenced by the spin alignment of the electrons and the \nmagnetic configuration of the ferromagnetic layers. When the \nmagnetizations of the layers are parallel, electrons with aligned \nspins face a lower tunneling barrier, leading to a higher \nprobability of tunneling which results in a spin-polarized current. \nUtilizing tunneling magnetoresistance (TMR) in devices \nleveraging this phenomenon for efficient manipulation and \ndetection of spin-polarized electrons through controlled resistance \nbased on the relative alignment of magnetization in ferromagnetic \nlayers [6]. \n \n  Fig.  2. (a) (i) Hanle effect results to manipulation of the \nspin-polarization in semiconductor, (ii) Energy band profile of \nFM/I/SC junction [10]; (b) Spin valve based GMR effect: \nparallel orientation facilitating easy flow for majority electrons \n(spin-up), antiparallel direction leads in a scattered flow of \nelectrons projected with DOS [9]; (c) Spin-orientation based \nelectron tunneling in FM heterostructure [11]. \n \nThe quantum tunneling mechanism is involved in spin-transfer \ntorque, a phenomenon where a spin-polarized current exerts \ntorque on a magnetic layer enables the precise control of \nmagnetization direction in spintronics devices like magnetic \nrandom-access memory (MRAM) [12]. Maintaining quantum \ncoherence during tunneling spin polarized electron transport \n(a) \n(b) \n(c) \n(a) \n(i) \n(b) \n  \n  \n \n(ii) \nc) 3 \nECE423_25 \n \nensures the integrity of spin information. The ability to selectively \ntransmit spin-polarized electrons through barriers allows for the \nefficient operation of spintronic components [13]. Quantum \ntunneling enables the creation of devices with reduced power \nconsumption and enhanced performance. As spintronics explores \navenues for quantum computing, the quantum tunneling \nmechanism in ferromagnetic semiconductors becomes a critical \nelement for the creation of qubits and the manipulation of \nquantum states, offering potential advantages in quantum \ninformation processing [13], [14]. \nIn ferromagnetic semiconductor-based spintronics devices, the \nimpact of spin-orbit coupling (SOC) is deeply rooted in the \nfundamental interplay between the intrinsic spin of electrons and \ntheir orbital motion. This interaction can be described by the \nRashba and Dresselhaus effects in which structural inversion \nasymmetry, such as in asymmetric semiconductor \nheterostructures results to the Rashba effect, while the \nDresselhaus effect arises from bulk inversion asymmetry, \ncommon in materials lacking structural inversion symmetry [15]. \nSpin-orbit coupling modifies the energy levels of electrons \ndepending on their spin orientation, effectively mixing different \nspin states. This coupling is a relativistic effect that becomes \nparticularly significant in ferromagnetic semiconductor devices, \ninstrumental in spin manipulation, notably through the spin-orbit \ntorque phenomenon. \n \n  Fig. 3. (a) Electron and Spin current behavior with \npolarizations [16] (b) Spin relaxation time ss vs. implanted \ndensity, spin relaxation time behavior on magnetic field [17], \n(c) charge-spin current conversion and (d) switching \nmechanism in FM devices, (e) Magnetization behavior with \napplied current pulse for different switching [9].  IV. FUNCTIONAL MATERIALS AND DEVICES FABRICATION  \n   Semiconductor properties can be manipulated through \nprocesses such as doping (by density and type), exposure to \nlight, and modulation by electrostatic gates which enable \namplification and transistor-like functionalities. Advancement \nin materials and device engineering, allowing for tailored \nfunctionalities and improved performance in ferromagnetic \nspintronic devices. The design and fabrication of such devices \ninvolve integrating different materials to create layered \nconfigurations that exploit the unique properties of each \ncomponent. For instance, combining ferromagnetic metals \nwith semiconductors or insulators enables the creation of \nspintronic devices with enhanced spin transport and \nmanipulation capabilities. Leveraging bandgap engineering, \ndiverse heterostructures can be constructed to tune \nferromagnetic semiconductors utilizing the density of states in \nspin-polarized energy bands. Recent studies also attempted to \nutilize the van der walls materials to make heterostructures in \nview of implementing next generation computing electronics \nenhancing spin electron transport, resistance, switching \nbehavior, spin injection modulation and so on.  \n \n \n  Fig. 4. Device configurations for (a) Heterostructure (b) \nlayered structure combined with ME/PE; (c)(i)Device \nSchematics of the FGT/MoS 2/FGT spin valve with h-BN \ncapping layer, (ii) Junction resistance and MR vs magnetic \nfield,  (iii) I-V curve for the device, (iv) spin transport w.r.t. \nspin valve is in antiparallel (left) and parallel (right) \nconfigurations [18]; (d) electrostatic strain methods; (e) MAE \nper atom and T C for strain for monolayer FM material [19]. \n \n  Employing an interacting device structure utilizing \ninteraction of charge carriers and spin waves can result in spin \nwave amplification by charge carriers drifting mobility [7]. In \ncase of spin wave modulation, mutual orientation between \nmagnetic and electric field is important to maintain in the (b) \n  \n (a) \n(d) \n(e) (c) (b) \n(c) \n \n (a) \n(i) (ii) (e) \n (d) (i) \n(ii) (iii) \n(iv) 4 \nECE423_25 \n \ndevice, where device and materials engineering like use of \nferromagnetic thin film with piezoelectric layer can modify \nthe device performance. Another most popular technique is \nfollowed by device researchers is spin valve structure, that a \ndevice based on ferromagnetic semiconductor thin film \nheterostructures that utilizes the Giant Magnetoresistive effect \nby tunneling spin-polarized charge carriers [11]. Moreover, \nstrain engineering is becoming an intriguing, applied approach \nto tune the ferromagnetic behavior in semiconductor \nheterostructures in a more controllable and precise way to \nadvance spintronics applications particularly using 2d van der \nwalls materials [14], [20].  \nIV. APPLICATIONS  \nFerromagnetic semiconductors, with their unique electronic \nand magnetic properties, find compelling applications in the \nrealm of spintronic device physics. In memory and logic \napplications, the integration of ferromagnetic semiconductors \nenables the creation of spintronic devices that capitalize on the \nspin degree of freedom of electrons. Non-volatile magnetic \nmemory, exemplified by Magnetoresistive Random Access \nMemory (MRAM), leverages the ferromagnetic properties for \nstable data storage with advantages such as low power \nconsumption and high-speed operation [8]. The utilization of \nferromagnetic semiconductors in Quantum dots enhances \nspintronic functionalities, contributing to the development of \ninnovative spin-based computing and data storage \ntechnologies [3]. The quantum nature of these materials allows \nfor the manipulation and storage of quantum information. In \nsensing applications, ferromagnetic semiconductors play a \ncrucial role, particularly in magnetoresistive sensors, where \nchanges in resistance due to magnetic fields enable sensitive \nand precise detection. Spintronic detectors, utilizing the \nproperties of ferromagnetic semiconductors, demonstrate \nefficacy in sensing magnetic fields and currents, presenting \nopportunities for applications across various industries, \nincluding automotive, healthcare, and electronics, where \nreliable and high-performance sensing and detection are \nimperative for technological advancement [6]. The versatile \nphysics of ferromagnetic semiconductors thus underpin their \nindispensable role in the ever-evolving landscape of \nspintronics. \nV. CONCLUSION  \nTaking advantage of having the ability to tune both electron \nand spin together in the ferromagnetic semiconductor, \nresearchers are emphasizing their efforts to come up with the \nmost efficient computing devices. Tunability by core \nsemiconductor variable to spin transport paved the way to \nscaling the devices for next generation spintronics utilizing the \nspin degree of freedom effect in device performances. Thus, \nunderstanding the device mechanism with controlled spin \npolarized electrons’ flow requires more efforts through \nmaterials and device engineering. Adhering to Moore's Law, \nachieving efficiency in nano-scaled devices requires mastery \nover both electron and spin behavior in ferromagnetic \nsemiconductor spintronic devices. ACKNOWLEDGMENT  \nThe author appreciates Professor Roman Sobolewski for \nguiding and reviewing the term paper summary, enhancing the \nmanuscript.  \nREFERENCES  \n[1] M. Tanaka, “Recent progress in ferromagnetic semiconductors and \nspintronics devices,” Jpn J Appl Phys , vol. 60, no. 1, p. 010101, Jan. \n2021, doi: 10.35848/1347-4065/abcadc.  \n[2] T. 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Çakır, “Strain–\nSpintronics: Modulating Electronic and Magnetic Properties of Hf 2 \nMnC2O2 MXene by Uniaxial Strain,” The Journal of Physical \nChemistry C , vol. 123, no. 19, pp. 12451–12459, May 2019, doi: \n10.1021/acs.jpcc.9b00594. "    },    {        "title": "1112.6036v1.Current_conservation_and_ratio_rules_in_magnetic_metals_with_Coulomb_repulsion.pdf",        "content": "arXiv:1112.6036v1  [cond-mat.supr-con]  28 Dec 2011EPJ manuscript No.\n(will be inserted by the editor)\nCurrent conservation and ratio rules in magnetic metals wit h\nCoulomb repulsion\nKosuke Odagiri\nElectronics and Photonics Research Institute, National In stitute of Advanced Industrial Science and Technology, Tsu kuba\nCentral 2, 1–1–1 Umezono, Tsukuba, Ibaraki 305–8568, Japan\nDecember 2011\nAbstract. From general considerations of spin-symmetry breaking ass ociated with (anti-)ferromagnetism\nin metallic systems with Coulomb repulsion, we obtain inter esting and simple all-order rules involving\nthe ratios of the densities of states. These are exact for fer romagnetism under reasonable conditions, and\nnearly exact for anti-ferromagnetism. In the case of ferrom agnetism, the comparison with the available\nexperimental and theoretical numbers yields favourable re sults.\nPACS.11.40.-q Currents and their properties – 75.10.-b General t heory and models of magnetic ordering\nContents\n1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1\n2 The ferromagnetic ratio rule . . . . . . . . . . . . . . 2\n3 Analysis of spin current conservation . . . . . . . . . 6\n4 Calculation of the parameters . . . . . . . . . . . . . 10\n5 Conclusions and Outlook . . . . . . . . . . . . . . . 13\n1 Introduction\n1.1 Theoretical background\nAs is obvious and well known [1], magnetic order breaks\nSU(2) spin symmetry (hereafter called SU(2) spin). This\ngives rise to gapless excitations in the form of Nambu–\nGoldstonemodeswhicharemagnonsand,accordingtothe\nGoldstone theorem, excitations with energy gap, which\nmay be called the Higgs bosons.\nThese excitations behave like elementary ﬁelds, and\ntheir interaction is central to spin-current conservation\nbut, at the same time, they comprise of electrons with\nwhich they interact: i.e., they are composite. This imposes\nsevere constraints on the properties of these ﬁelds, which\nwe aim to discuss and exploit in this paper.\nThesamesituation,ofGoldstoneﬁelds(i.e.,bothGold-\nstone and Higgs ﬁelds) that are themselves composite ob-\njects,arisesnotablyintwoproblemsinthecontextofhigh-\nenergy physics. The ﬁrst problem is that of axial symme-\ntrybreakingatlowenergyscalesdueto the SU(3) Cstrong\ninteraction.TheGoldstonebosonsherearepions.Thesec-\nond problem is that of electro-weak symmetry breaking.\nAlthough in the Standard Model, the Higgs ﬁeld, or the\ndoublet order-parameter ﬁeld for SU(2) L⊗U(1)Y, is an\nelementary scalar ﬁeld, it is an attractive possibility thatthis ﬁeld is composite and, for example, is composed of\nquarks which bind strongly at high scales due to some\ninteraction (N.B. not by the SU(3) Cstrong interaction\nwhich is weakly interacting at those scales). The low-\nenergy phenomenology would then not be dissimilar to\nthat of the Standard Model, but with some constraints on\nquantities such as the Higgs-boson mass.\nA new approach to these problems, due to Gribov,\nhave appeared in refs. [2,3,4,5]. These involve the idea\nof super-criticality and a self-consistent treatment of the\nfermion and Goldstone ﬁelds in the presence of the super-\ncritical interaction. We shall make use of, and extend, the\nmethods presented therein, to the case of magnetism. As\nfor the other approaches to these old problems, see, for\nexample, ref. [6] for an old approach to the ﬁrst problem,\nand ref. [7] for an overview of the various methods and\ntechniques developed to handle the second problem.\n1.2 Outline of the paper\nOur work concerns systems of electrons (or holes) which\ninteract under a generalized Coulomb exchange (i.e., ex-\nchange of a generic gapless photon). We consider the sys-\ntem in the spin-symmetry-broken phase that arise in fer-\nromagnetism and anti-ferromagnetism.\nOur aim is to obtain exact relations between quanti-\nties that characterizethe spin-symmetry-brokenphase us-\ning the Dyson–Schwinger equations. This is possible be-\ncause of the presence of the Ward–Takahashi identities\nwhich arise because of the conservation of spin symmetry.\nIt turns out that the formof the Coulomb interactiondoes\nnot aﬀect these relations. The Coulomb interaction does\naﬀect, for instance, the electronic self-energy, but these\nare incorporated in the relations in a general way.2 Kosuke Odagiri: Current conservation and ratio rules in ma gnetic metals with Coulomb repulsion\nBefore presenting the full analytical framework, we\nstart with the simple case of the discussion of ground-\nstate stability in ferromagnetism, in sec. 2. This gives rise\nto an exact rule for ferromagnetism which involves the\nelectronic densities of states. We discuss this case with\nillustrations and a physical interpretation.\nThefullframework,whichemploycurrent-conservation\ntechniques, is developed in secs. 3 and 4. In sec. 3, the\ninteraction is worked out and presented in the form of\nFeynman rules. In sec. 4, the parameters of the interac-\ntion is worked out. This gives rise to an exact rule for\nanti-ferromagnetism which involves the electronic densi-\nties of states, but which involves the bare spin exchange\nenergy.\nThe conclusions are stated at the end.\n2 The ferromagnetic ratio rule\nBefore we introduce the full framework, let us discuss the\nstability of the ferromagnetic ground state as an illustra-\ntive example. We do so because this is a relatively simple\nproblem, which does not require the full formalism, and\nwhich nevertheless leads to a strikingly simple and useful\nratio rule. We shall expand the methods introduced here\nto build the formalism later.\n2.1 Description of the system\nLet us write the electrons in the SU(2) spindoublet form:\nψa≡/parenleftbigg\nψ↑\nψ↓/parenrightbigg\n. (1)\nThe Lagrangian has the form:\nL=δabψ∗\na/parenleftbigg\ni∂\n∂t−ǫ(−i∇)−eΓµAµ/parenrightbigg\nψb+(photon K.E.)\n(2)\nAµis the electro-magnetic ﬁeld, of which we shall retain\nonly the electrostatic term A0later, as the contribution of\nthe 3-vector potential is suppressed by the speed of light.\nThe photon kinetic energy term may then be taken to be\n−(∇A0)2/2, which leads to an electrostatic 1 /rinterac-\ntion.ǫrefers to the dispersion relation of the electron. e\nis the electro-magnetic charge, which is deﬁned to be neg-\native for electrons. Γµis the vertex function, whose time\ncomponent is 1 for a Lagrangian of this form.\nIn the absence of magnetic order, the system is in-\nvariant under both the electromagnetic U(1) EMand the\nSU(2)spinrotations of ψ, where the latter is represented\nby\nU(φi)≡exp/parenleftbig\niσiφi/2/parenrightbig\n. (3)\nσi(i= 1,2,3) are the Pauli matrices and φiare the ro-\ntation angles. Note that we are referring to global rota-\ntions here. Invariance under local rotations requires the\npresence of gauge ﬁelds such as the electromagnetic ﬁeld,\nincluding the vectorpotential partand written in a gauge-\ninvariant fashion.There are two conserved currents, which are orthogo-\nnal. The ﬁrst is the U(1) EMcurrent:\nJµ\nEM=ψ∗\naδabΓµψb. (4)\nµrefers to the time-space four-vector index (=0,1,2,3).\nThe second is the SU(2) spincurrent. This is written as\nJµ,i\nspin=ψ∗\naσi\nabΓµψb. (5)\nLetusindicatethecurrentdiagrammaticallybyacross.\nWe see that current–current mixing, which we denote as\nΠµν\nmixingand whose lowest order term is given by\nµ\nσi\nabν\nδab\n(6)\nvanishes by symmetry for all ito all perturbative orders,\nand therefore the two currents are orthogonal.\nWhen there is magnetic order, there arises, locally, a\npreferred orientation of spin, let us say along ↓, and this\nbreaks the SU(2) spinsymmetry, viz:\nSU(2)spin−→U(1)z. (7)\nAs a result, there arises two Goldstone modes whose cou-\npling is proportional to linear combinations of σ1,2, and\na Higgs mode whose coupling is proportional to σ3. The\nresidual symmetry U(1) zrefers to the symmetry under\nrotation by the generator σ3:\nU(φ3)≡exp/parenleftbig\niσ3φ3/2/parenrightbig\n≡diag/parenleftBig\neiφ3/2,e−iφ3/2/parenrightBig\n.(8)\nThe form of the eﬀective theory will be discussed later.\n2.2 U(1) current mixing\nA result, which is almost trivial but possibly not pre-\nviously discussed explicitly, is that after this symmetry\nbreaking, the currents are no longer orthogonal. Equation\n(6) is easily calculated. Of particular interest is the 0 −0\ncomponent of eqn. (6) for i= 3 (i.e., the U(1) zcurrent:\ni= 1,2vanish) at zeroexternal energyand momenta. The\nvertexfunction Γ0being equal to 1 when the electrons are\nFermi-liquid-like for each spin orientation, we obtain\nΠ00\nmix= lim\nq→0/integraldisplaydd+1k\n(2π)d+1iG↑(k)G↑(q+k)−G↓(k)G↓(q+k).\n(9)\nHence\nΠ00\nmix=g↑(ǫF)−g↓(ǫF). (10)\nNote thatΠis deﬁned with a negative sign, that is Π=\n−A, whereAis the two-point amplitude. Gare the elec-\ntron Green’s functions. g(ǫF) are the densities of states at\nFermi energy.\nAs is well known, the Dyson–Schwinger all-order cor-\nrectionstoaﬁshdiagramsuchasthatindicatedineqn.(6)\nis incorporatedby replacingthe Green’sfunctions by theirKosuke Odagiri: Current conservation and ratio rules in mag netic metals with Coulomb repulsion 3\nall-order counterparts and one or the other of the ver-\ntices (and not both) by their all-order counterpart. Equa-\ntion (10) is an all-order expression in the sense that the\nGreen’s functions are arbitrary. However, the vertex cor-\nrection needs care. Let us consider the all-order correc-\ntion to the photon vertex. At zero external energy and\nmomenta, the time component of the all-order vertex is\ngiven by the energy derivative of G−1, by virtue of the\nWard–Takahashi identity. It follows that, if the all-order\nGreen’s function is given by\nG(E,k) =Z\nE−ǫ(k)+i0sgn(ǫ(k)−µ),(11)\nas is the case for the Fermi liquid, then the vertex is cor-\nrected byZ−1. On the other hand, there is Z2coming\nfromG2, and so the net result is proportional to Z.Z\nbeing the correct renormalizing factor for the density of\nstates, eqn. (10) is exact. It is not diﬃcult to see that a\nmore general form of Galso admits this property:\nG(E,k) =Z(k)\nf(E−ǫ(k)+i0sgn(ǫ(k)−µ)),(12)\nwherefis any function, so long as the density of states is\ndeﬁnable as the integral of G. Thus it is not a necessary\ncondition that the system is a Fermi liquid.\nReturning to eqn. (10), in general, g↑andg↓are not\nequal atthe Fermi surface,and thereforethe currentsmix.\nIt is worth noting here that the current mixing is zero\nin the case of anti-ferromagnetism, because the two sub-\nlattice contributions are equal and opposite.\nBefore proceeding, let us calculate the other ﬁsh dia-\ngrams. Both for the EM current and for the spin U(1) z\ncurrent, we obtain:\nΠ00\nEM=Π00\nz=g↑(ǫF)+g↓(ǫF). (13)\nAgain, this is an exact result provided that the Green’s\nfunctions are of the form eqn. (12) and the densities of\nstatescanbe deﬁned astheirintegrals.Notethat although\nwehaveretainedthesubscript‘EM’torefertoelectromag-\nnetism, in reality, we are analyzing electrostatics.\n2.3 Derivation of the ratio rule\nLet us now consider the stability of the ferromagnetic\nground state. To do so, a primary condition is the van-\nishing of the tadpole:\nσ3\n(14)\nasisrequiredbytheconditionthattherearenotermsthat\nare linear in the Higgs ﬁeld in the eﬀective Lagrangian. In\notherwords,theﬁrstderivativeoffreeenergyasafunction\nof the magnetic order parameter must vanish when the\nground state is stable. We will also need to check that the\nsecondderivativeispositive.This meansthe term whichisbilinear in the Higgs ﬁeld, or the self-energy of the Higgs\nboson, is positive. The Higgs self-energy is the same as\nΠ00\nzcalculated earlier on, up to the square of a coupling\nconstant. This is necessarily positive.\nEquation (14) is calculated easily, and we obtain\nAtadpole\nz=/integraldisplaydd+1k\n(2π)d+1iG↑(k)−G↓(k) =ρ↑−ρ↓.(15)\nρrefers to the total density of states of electrons. This\nis an exact expression since higher-order corrections to\nthe tadpole are taken into account by making Gall-order\nand vertex to be bare. This is always negative if ↓is the\npreferred orientation of spin.\nIt follows that eqn. (14) by itself is non-zero. How-\never, the currents mix, and we must incorporate the con-\ntribution of the photon tadpole, multiplied by the Higgs–\nphoton two-point function which has the same form as\nΠ00\nmixingcalculated in eqn. (10):\nσ3\nabδab\n(16)\nNow, to make this equation all-order, we must include the\nscreeningeﬀect in the photon propagator,and this hasthe\nsame form as Π00\nEMcalculated in eqn. (13). Altogether, we\nobtain\nAtadpole\nphoton part= (ρ↑+ρ↓)e×(g↑(ǫF)−g↓(ǫF))e\n−(g↑(ǫF)+g↓(ǫF))e2.(17)\nThe two contributions must vanish when added to-\ngether. Although the charge eappears here, whether one\ntakes the charge carriers to be electrons or holes is a mat-\nter of choice, so ecan be taken as constant. Hence,\ng↓(ǫF)−g↑(ǫF)\ng↓(ǫF)+g↑(ǫF)=ρ↓−ρ↑\nρ↓+ρ↑(18)\nor,\ng↑(ǫF)\ng↓(ǫF)=ρ↑\nρ↓. (19)\nThis is our ferromagnetic ratio rule.\nIn eqn. (18), the left-hand side is often called the spin\npolarization P,forexampleinthecontextoftunnelmagneto-\nresistance. The right-hand side is the magnetic moment\nnB=m/µBdivided by the number of carriers n, i.e.,\nP=nB\nn. (20)\nNote that the deﬁnition of nis ambiguous. However, it\nis a measure of the number of electrons or holes that are\nactively involved in the formation of ferromagnetic order.\nAs such, one would expect that its order is estimated by\nthe number of carriers in the conduction band. If so, we\nobtain a simple rule of the thumb:\n/braceleftbigg\ng↓>g↑(electrons),\ng↓<g↑(holes).(21)4 Kosuke Odagiri: Current conservation and ratio rules in ma gnetic metals with Coulomb repulsion\nThat is, the density of states of the majority-spin charge\ncarriers is always greater. According to this rule, one ex-\npects the density of states to be rising with energy for\nelectrons and decreasing with energy for holes. This can\nbe a useful rule of the thumb to establish whether a sys-\ntem with certain givendensity of states is likely to become\na ferromagnet.\n2.4 A diagrammatic derivation\nThe preceding derivation is formal and, we believe, com-\nplete, but it may appear baﬄing at the start that the sum\nof apparently only two contributions is suﬃcient to give\nall-order statements about the system.\nAnother way to derive the same result, in a more di-\nagrammatic fashion, is to consider the mixing between\nscreened photon and the Higgs boson. Let us denote the\nmixed states by /tildewideγand/tildewideh. Then the sum of the tadpoles\nσ3\nab/tildewideγ,/tildewideh\n(22)\nneeds to be zero, whereas the sum of the tadpoles\nδab/tildewideγ,/tildewideh\n(23)\nis non-zero, and gives a constant contribution to the en-\nergy levels of the states.\nThe sum over the former set of tadpoles can be ex-\npanded diagrammatically in terms of γand h as:\n+ + +···\n.(24)\nIt will be seen that if the sum of the ﬁrst two terms is\nzero, then the remaining contributions, which are propor-\ntional to the sum of ﬁrst two terms, vanish automatically.\nTherefore it suﬃces to calculate the sum of the ﬁrst two\nterms.\n2.5 Examples\nOne way to understand eqn. (20) is as a deﬁnition of n.\nThis number can then be compared with the other esti-\nmates of the number of carriers such as by the Hall eﬀect\nand with the nominal number of electrons or holes.\nAs aﬁrstexample,in thecaseofhalfmetals,andinthe\nidealcase,the spin polarizationwouldbe perfect, i.e., P=\n1. The magnetization will also be perfect, and eqn. (20)\nwill be satisﬁed so long as we take nto be equal to nB.\nNext, in the case of a Coulomb system whose renor-\nmalized dispersion relation is given exactly by ǫ(k) =\n(/planckover2pi1k)2/2me, the density of states is given by:\ng(ǫ) =1\nπ2/radicalbig\nm3eǫ/2, ρ=1\n3π2/radicalBig\nm3eǫ3\nF/2.(25)We have deﬁned ρas the integral over whole of the oc-\ncupied states. Equation (19) then admits the following\nsolutions only:\ng↑=g↓,or,g↑= 0. (26)\nThat is, either the system is a half metal or there is no fer-\nromagnetic order. This is consistent with the observation\nthatgroup1elementsarenotferromagnetic,andalsowith\nthe observation that the Fermi gas system with quadratic\n(bare) dispersion relation is only weakly ferromagnetic[8].\nLet us now consider the case of transition metals. In\neqn. (20),nBis the only quantity which is measured un-\nambiguouslyandaccurately. Pismeasurable,butdiﬀerent\nmethods yield diﬀerent results [9,10]. Furthermore, these\nexperimental numbers do not match with the results of\ntheoretical calculation [11].\nIn principle, theoretical numbers for gshould be com-\npared against theoretical numbers for ρ, and experimental\nnumbers for gshould be compared against the experimen-\ntal numbers for ρ. Let us ﬁrst look at the experimental\nnumbers.\nelement carriers PTPCnB\nFe electrons 0.40 0 .42∼0.46 2.22\nCo holes 0.35 0.42 1.72\nNi holes 0.23 0 .43∼0.465 0.606\nelementn=nB/P n nominalnH\nFe 4.8∼5.6 8 3.00\nCo 3.7∼4.9 3 0.520\nNi 1.3∼2.6 2 1.12\nTable 1. Experimental numbers for spin asymmetry P, mag-\nnetic moment nBper site,ncalculated as the ratio of these\ntwo, the nominal number of charge carriers, and the number of\ncharge carriers as measured using the Hall eﬀect. Two values\nforPare taken from refs. [9] and [10], respectively. nBis from\nref. [12].nHis calculated from RHlisted in ref. [13].\nIn tab. 1, we summarize the experimental numbers for\nP,nBandn. The calculated values of nare compared\nagainst the nominal number of charge carriers, i.e., the\nnumber of 4s and 3d electrons or holes, and with the num-\nberofcarrierscalculatedfromtheHallratio RH=−1/ne.\nWe see that the the values of ndo not seem to be in con-\ntradiction of the nominal number of chargecarriers,in the\nsensethatn<nnominalfor Fe andn≈nnominalforNi and,\narguably, Co. However, the variation in the experimental\nnumbers is too great to make a concrete statement.\nLet us now turn to the theoretical numbers.\nIn tab. 2, we show the numbers for g↓/g↑as estimated\nfrom ref. [11]. In the case of Fe, ρcould be estimated\nroughly by the eye as the ratio of the areas underneath\nthe density-of-state curves. For Co and Ni, this was not\npossible because ofthe longtails in these curves. However,\nthe relative size of gwas consistent with the nature of the\ncarriers. That is, the density of states was found to be\ngreater for the majority spin. All three cases are thus not\ninconsistent with eqn. (19).Kosuke Odagiri: Current conservation and ratio rules in mag netic metals with Coulomb repulsion 5\nelement g↓/g↑ρ↓/ρ↑P n =nB/P\nFe 2.1±0.2 2∼2.5 0.35±0.04 6.3±0.7\nCo 7.0±1.5>1 0.75±0.04 2.3±0.1\nNi 10 ±1.5>1 0.82±0.03 0.74±0.03\nTable 2. Theoretical numbers for g↓/g↑andρ↓/ρ↑. The num-\nbers forgwere measured using a ruler applied to the density-\nof-state plots of ref. [11]. The numbers for ρwere estimated\nby the eye and a ruler. The Fermi level is at the tail of the\ndensity of states for the case of Co and Ni, and rendering the\ndeﬁnition of ρdiﬃcult or arbitrary. We also show the values of\nPcorresponding to these values of g↓/g↑, and an estimate of\nnbased on the actual values of nBwhich are shown in tab. 1.\nWe also show the values of Pcalculated from g↓/g↑.\nThese are quite diﬀerent from the experimental numbers\nintroduced earlier. As a result, the numbers for ndiﬀer\nfrom before, if we use the same values of nB.\nTo summarize, it is diﬃcult to check the ratio rules\nquantitatively, at the present level of accuracy.\n2.6 Physical interpretation\nLet us discuss tadpole cancellation in a more intuitive\nfashion.\n(a)\nseahiggs (b)\nseaphoton\n(c)\nsea(d)\nsea\nFig. 1. The interaction of a conduction electron with the\nFermi sea of electrons.\nFigure 1 represents the interaction experienced by a\nconduction electron or holes due to the surrounding Fermi\nsea of (conduction) electrons or holes. Let us say that the\nchargecarriersare electrons. Note that ﬁg. 1a corresponds\nto eqn. (14), and ﬁg. 1d corresponds to eqn. (16).\nThe interaction shown in Fig. 1a involvesthe exchange\nof the density ﬂuctuation of spin, i.e., the Higgs boson.\nThis boson being a scalar, its exchange is always repulsive\nbetween like particles, i.e., between the same spin. Since,\nby deﬁnition, there are more majority-spin electrons thanminority spin electrons, this interaction makes majority-\nspin electrons more energetically unfavourable. That is,\nFig. 1a, or the density ﬂuctuation of spin, tends to sup-\npress magnetic order.\nThe exchange of the Higgs boson is not the only inter-\naction between the conduction electrons and the sea, and\nin Fig. 1b, we show the Coulomb exchange. This is always\nrepulsive, and is of the same magnitude for both type of\nelectrons, and so this diagram does not contribute to the\nformation or suppression of magnetic order.\nFigure 1b by itself is inﬁnite since the photon propaga-\ntor diverges at zero momentum transfer. This is, as usual,\nremedied by the screening eﬀect which is shown in ﬁg. 1c.\nThe screening eﬀect, such as that shown in ﬁg. 1c,\nusually suppresses the charge. This is because a negative\ncharge attracts positive charge, and this positive charge\ntends to cancel the negative charge.\nHowever,thecontributionofﬁg.1drequiresmorethought.\nThe sea electrons, which have negative charge, attracts\npositive charge. When this positive charge has the same\nspin as the conduction electron, i.e., when the positive\nchargesuppresses the electronic spin which is aligned with\nthe spin of the conduction electron, the positive chargeat-\ntracts this conduction electron. On the other hand, when\nthe positive charge has opposite spin to that of the con-\nduction electron, then the conduction electron is repelled.\nWhether the interaction of ﬁg. 1d tends to create mag-\nnetic order or suppress it depends on which type of spin is\nmore likely to be excited, i.e., on the density of states at\nthe Fermi surface. This is the meaning of the tadpole can-\ncellation. In other words, the ferromagnetic ground state\nis stable when the interaction due to the ﬂuctuation of\nspin density, which is mediated by the Higgs boson and\nwhich always suppresses the polarization of spin, is equal\nand opposite to the contribution due to the electrostatic\npolarization of ﬁg. 1d which, depending on circumstances,\ncan counteract it.\n2.7 Comparison with the Hubbard model\nWhen theCoulombinteractionisscreened,the interaction\nbecomes point-like in the limit of large screening, i.e.,\n1\ng↓(ǫF)+g↑(ǫF)−→ˆU, (27)\nwhereˆUis a constant which can be interpreted as the\non-site Coulomb repulsion Uup to a normalization. Let\nus now see what would happen if we were to start from\na theory which treats the on-site Coulomb repulsion Uas\nthe starting point, such as the Hubbard model.\nIn this case, eqn. (14) is unchanged, but eqn. (16) is\nmodiﬁed to take the following form:\nσ3\nabˆU\n. (28)\nHere, as is usual in the Hubbard model, the spin going\ninto the ﬁsh part must be opposite to the spin going into6 Kosuke Odagiri: Current conservation and ratio rules in ma gnetic metals with Coulomb repulsion\nthe tadpole. That is,\nAtadpole\nHubbard=ˆU[g↑(ǫF)ρ↓−g↓(ǫF)ρ↑],(29)\nin the place of eqn. (17). This would lead to diﬀerent con-\nsequences.\nThe origin of this discrepancy is clear. Expressed in\nterms of the screened Coulomb propagator, there are two\ncontributions that go into eqn. (28). One is the genuine\ntadpole-like contribution of the form eqn. (16). The con-\ntribution of this term is given by\nˆU[g↑(ǫF)−g↓(ǫF)](ρ↓+ρ↑), (30)\nso that this term has the same form as in eqn. (28). On\nthe other hand, there is a second contribution, which is\nthe self-energy correction:\nσ3\nab\n. (31)\nThe contribution due to this term is given by\nˆU[g↓(ǫF)ρ↓−g↑(ǫF)ρ↑]. (32)\nAdding together these two contributions yields eqn. (29).\nThe discrepancy comes because in our approach, the\nself-energy correction is absorbed in the all-order Green’s\nfunction, whereas in the Hubbard-model approach, this\nis not possible. In the Hubbard model, either both con-\ntributions are treated as a tadpole, or both contributions\nare treated as a self-energy correction. If the latter, one\nwill have the condition that the simple tadpole, with the\nself-energy corrections, by itself vanishes. This condition\nrequiresρ↑=ρ↓, and therefore we will never have a stable\nferromagnetic solution out of the Hubbard model.\nThis, in our opinion, is a limitation of the Hubbard\nmodel.Thelimitationisduetotheinabilitytotreatcurrent–\ncurrent mixing, which is the basis of our discussion in this\nsection.\nOne may still argue that on-site Coulomb repulsion is\npresent, physically. In other words, the eﬀective screened\nCoulomb propagator, which ordinarily gives a divergent\ncontribution at the origin up to a UV cut-oﬀ, is not really\ndivergent but only large and ﬁnite at the origin.\nIf so, this may be thought of as a variant of the UV\ncut-oﬀ of the Coulomb propagator Dphoton, which may be\nparametrized, for example, as\nDphoton(k) =1\n−k2−ak4, (33)\nwhereais a parameter (positive or negative). Even if this\nis not permissible as a ﬁeld theory, it is permissible as\nan UV (Pauli–Villars) regularization procedure. It will be\nseen that such a cut-oﬀ does not aﬀect our argument at\nall, since our discussion involves zero momentum photons.\nThe electronself-energywill be aﬀected bythe UVcut-oﬀ,\nbut this does not aﬀect our results explicitly.3 Analysis of spin current conservation\nLet us now move on to the formalism, which is required if\nwe are to go beyond the tadpole-level analysis of the pre-\ncedingsection.We adaptGribov’sanalysisofaxialcurrent\nconservation [2,3] to the context of spin current conserva-\ntion in systems with partial magnetic order.\nTo sum up in one phrase, our goal is to start from\nthe Coulombic system, which is deﬁned by eqn. (2), and\nsolve it as exactly as possible using the Dyson–Schwinger\nequations, under a number of assumptions.\nThe major assumption is that of spontaneous sym-\nmetry breaking. If the spin symmetry is broken sponta-\nneously, then the Goldstone theorem guarantees the pres-\nence of Goldstone and Higgs modes. These modes are,\nin terms of the initial Lagrangian, electronic excitations.\nHowever, in terms of the eﬀective theory that appears at\nthe end, they are elementary, and participate in the con-\nservation of the spin current. This is the main property\nthat allows us to solve the Dyson–Schwinger equations.\nThe other assumptions, such as the linear or quadratic\nform of the magnon dispersion relation and the constancy\nof exchange energy which are sometimes required, are ap-\nproximations, which we believe are viable, that can be\nlifted if one has the computational resources.\nThe resulting eﬀective Lagrangianis found to have the\nfollowing interaction term:\nLI\neﬀ∝ψ†Φ·σψ. (34)\nHereΦiis essentially the order-parameterﬁeld, but with a\ncertain formal diﬀerence which we shall discuss. We would\nlike to emphasize at this point that this equation is not\nour starting point. It is rather the end product of solving\nthe Coulombic system by means of the Dyson–Schwinger\nequations, with the aid of the Goldstone theorem and the\nWard–Takahashi identities.\nLet us start by discussing current conservation.As dis-\ncussed in the previous section, current is absolutely con-\nserved when the spin symmetry is conserved.\n∂\n∂xµJµ,i\nspin= 0. (35)\nCurrentconservationisreﬂectedinthefollowingWard–\nTakahashi identity:\nΓµ(q1−q2)µ=G−1\nλ1(q1)−G−1\nλ2(q2).(36)\nΓµis the vertex in the momentum space. λ1,2refer to\nthe spin states, but these are dummy indices here in the\nsense that G−1\nλ(q) is independent of λ. Thus eqn. (36)\nholds for any combination of spin, and therefore current\nis conserved. qared+1-vectors with components ( q0,q).\nq0is the energy and qis the spatial momentum, with\n/planckover2pi1= 1.\nTheWard–Takahashiidentityisviolatedinthesymmetry-\nbroken phase, since there is now an energy diﬀerence ∆E,\nwhich is the exchange energy, between the diﬀerent spin\nstates:\n∆E=G−1\n↓(q)−G−1\n↑(q). (37)Kosuke Odagiri: Current conservation and ratio rules in mag netic metals with Coulomb repulsion 7\nThis is the deﬁnition for the ferromagneticcase, when ∆E\nis positive if we take ↓to be the majorityspin state. In the\nanti-ferromagnetic case, ∆Eis positive in one sub-lattice\nand negative in the other.\nIfG−1\n↓andG−1\n↑are both linear in energy, ∆Eis given\nbyǫ↑−ǫ↓and is constant up to a possible dependence\non the spatial momentum q. In principle, ∆Edepends\nonq0andq. In particular, at the threshold, G−1would,\nin general, have a singular structure corresponding to the\nemissionandabsorptionofthe Goldstoneboson φthrough\nthe process e∗→eφ. The results of the previous section\nare stable against such corrections, as we have discussed.\nHowever,the resultsofthis sectionaremore easilyderived\nfor constant ∆E, which corresponds to the case of the\nFermi liquid whose exchange energy is constant.\nAfter the symmetry violation, the currents Jµ\n1,2are no\nlonger conserved, and the Ward–Takahashi identity is vi-\nolated by\nΓµ(q1−q2)µ∝G−1\nλ1(q1)−G−1\nλ2(q2)±∆E(λ1∝negationslash=λ2).(38)\nThis∆Econtribution is of the same form as the cou-\npling of the Goldstone boson φ, and current conservation\nis restored by including the contribution of the Goldstone\nboson. This is the case even when ∆Eis not constant.\nSpeciﬁcally, spin current conservation is restored for the\nvertex/tildewideΓwhich is modiﬁed by the inclusion of the Gold-\nstone boson,\n/tildewideΓµ=µ\n+µ\n.(39)\nAs before, the crosses indicate the spin-current vertices,\nand the dashed line indicates the Goldstone boson. There\nisnothing strangein thisresult, sincethe Goldstoneboson\narose in the ﬁrst place as the longitudinal component of\nthe spin current. After taking away the longitudinal com-\nponent, the remainingpart istransverseand thereforesat-\nisﬁes the Ward–Takahashi identity. The current–magnon\ntwo-point function which appears in the second term con-\nsists of fermionic and bosonic loop. The latter contains\nmagnons and the Higgs boson.\nThe Goldstone bosons φ1andφ2correspond to the\nSU(2) rotation perpendicular to the local orientation of\nspin (which is along z),\nU(φ1,φ2) = exp/bracketleftBigg\nif−12/summationdisplay\ni=1φiσi/bracketrightBigg\n, (40)\nand they correspond physically to the magnons. fis the\nGoldstone boson form factor which, by virtue of eqn. (39),\nis calculated as the strength of the current–Goldstone-\nboson two-point amplitude.\n3.1 The two-point function and the coupling with\nfermions\nWe have noted in eqn. (7) that there is a residual symme-\ntry associated with U(1) z, which is conserved. The statescan be classiﬁed according to the charges under this rota-\ntion group. First, we deﬁne the charge of ψ↑to be +1/2.\nThe remaining charges follow automatically, and we ob-\ntain the values listed in tab. 3. These are necessarily con-\nserved. Note that the U(1) EMcharges are efor the elec-\ntron/holeﬁelds and 0 for all others. These chargesare also\nconserved.\nﬁeld U(1) zcharge U(1) EMcharge\nψ+≡ψ↑ +1/2 e\nψ−≡ψ↓ −1/2 e\nφ+≡ −φ1+iφ2 +1 0\nφ−≡φ1+iφ2 −1 0\nh0≡h 0 0\nTable 3. The U(1) zand U(1) EMcharges of the ﬁelds. eis\npositive for holes and negative for electrons.\nIn order that the Ward–Takahashi identity is satisﬁed\nby the vertex of eqn. (39), the following identity needs to\nbe satisﬁed:\nµ×qµ=−fD−1\nφ(q).\n(41)\nHere,fis a constant of proportionality, and is the same\nquantity as that which appears in eqn. (40). qµis the\nmomentum ﬂowing into the two-point function from the\ncurrent (i.e., left to right). This present deﬁnition of fis\nmorerigorous.Notethat thisalsoﬁxesthesignof Dφ.Our\npresent deﬁnition corresponds to taking the couplings to\nbe real and taking the sign of Dφto be opposite to that\nfor scalar particles.\nGiven this deﬁnition of f, we can determine the cou-\npling constants with the fermions by the condition that\neqn. (39) satisﬁes the Ward–Takahashi identity. We then\nobtain the Feynman rules that are given in ﬁgs. 2a and b.\n(a)\n+1/2−1/2−1\n+f−1∆E(b)\n−1/2 +1 /2+1\n−f−1∆E\n(c)\n+1/2 +1 /20\n+f−1∆E(c)\n−1/2−1/20\n−f−1∆E\nFig. 2. The Feynman rules for the coupling of the magnons\nand the Higgs boson with the fermions.8 Kosuke Odagiri: Current conservation and ratio rules in ma gnetic metals with Coulomb repulsion\nFor example, for the conﬁguration of ﬁg. 2a, eqn. (39)\nyields the following Ward–Takahashi identity:\nqµΓµ+i2(−fD−1\nφ(q))Dφ(q)(f−1∆E) =G−1\n−−G−1\n+.(42)\nNote that the same Feynman rules can be obtained, less\nrigorously,by considering the rotation associated with the\nGoldstone bosons, eqn. (40), and considering its coupling\nwith the fermions in eqn. (2).\nThe vertices that involve the Higgs boson, which are\nshown in ﬁg. 2c and d, cannot be ﬁxed by this particular\ntype of Ward–Takahashi identity. However, they can be\nﬁxed by considering the current insertion in the three-\npoint amplitude, for example, as shown in ﬁg. 3.\n(a)\n/tildewideΓ\n+1/2−1/2\n−1/20(b)\n/tildewideΓ+1/2\n+1/2−1/20\n(c)\n/tildewideΓ+1/2\n−1/2+1 0\nFig. 3.The threediagrams whose summustsatisfy theWard–\nTakahashi identity.The crosses correspond tothemodiﬁed c ur-\nrent vertex deﬁned by eqn. (39).\nThe Ward–Takahashi identity applied to ﬁg. 3 also al-\nlows us to determine the magnon–magnon–Higgs vertex.\nHowever, for doing so, we need to know the form of Dφ(q)\nandDh(q). Let us therefore calculate the current–magnon\ntwo-point function of eqn. (41).\n(a)\nµ\nq f−1∆E+1/2\n−1/2(b)\nµ\nq+1\n0\nFig. 4. The fermionic (a) and bosonic (b) contributions to\nthe current–magnon two-point function.\nFor now, we calculate the fermionic loop, which is\nshown in ﬁg. 4a. It should be noted that the end result\nof this calculation is independent of whether we consider\nφ+orφ−. Using the Feynman rule of ﬁg. 2a, we obtain\niAµ\ntwo−point(q) =\ni4(−1)/integraldisplaydd+1k\n(2π)d+1ΓµG+(k)G−(k−q)(f−1∆E).(43)In particular, for the case q→0, we obtain the exact\nexpression:\nAµ\ntwo−point(q→0) =/integraldisplaydd+1k\n(2π)d+1if−1Γµ[G−(k)−G+(k)],\n(44)\nwhere we made use of eqn. (37). For Fermi liquids, Γ0= 1\nand, by symmetry, the spatial components of this ampli-\ntude usually vanishes at q= 0. We thus obtain\nAµ\ntwo−point(q→0) =f−1(ρ−−ρ+,0,0,0).(45)\nNotethatthisvanishesforthe caseofanti-ferromagnetism\nwhere there is no global spin asymmetry.\nWhen we compare eqn. (45) with eqn. (41), we see\nimmediately that\nf2=ρ−−ρ+(ferromagnetism), (46)\nand\n/braceleftbigg−D−1\nφ(q) =q0−q2/2mφ(ferromagnetism),\n−D−1\nφ(q) =q2\n0−u2q2(anti-ferromagnetism),\n(47)\nforsmallenergyandmomenta.Theinclusionofthebosonic\nloop does not alter this conclusion. The unusual negative\nsign ofDreﬂects the fact that the magnons are pseudo-\nscalar. That is, the ﬁelds are iφrather than φin our con-\nvention.\nWe need to calculate f2by other means, such as cal-\nculating the two-point amplitude for ﬁnite q, for anti-\nferromagnetism. uandmφare parameters which are in\nprinciple calculable by, for example, evaluating the ﬁnite\nqcase. However, the form of eqn. (43) implies mφ∼me\nandu∼vF.\nThere is actually a smarter method than to calculate\nthe ﬁnite-qcase (which is cumbersome), but the full cal-\nculation, in the case of anti-ferromagnetism, requires our\nknowledge of the bosonic three-point functions. Let us\ntherefore postpone the calculation of anti-ferromagnetic\nf2and other parameters for now.\nLet us summarize the results of this section up to here.\nFirstly, we summarize the propagators and the two-\npoint functions in ﬁg. 5. The Higgs-boson Green’s func-\ntions are given in ﬁg. 5b, with a constant energy gap ∆h.\n∆hisdeﬁnedastheenergygapforferromagnetismandthe\nenergy gap squared for anti-ferromagnetism. This deﬁni-\ntion is convenient when we discuss the bosonic three-point\nfunctions. Note that it is an approximation to say that\n∆his independent of momenta and energy. However, it\nbecomes easier to implement current conservation in this\nmanner. For completeness’s sake, we also list the screened\nphoton Green’s function (with the approximationthat the\nscreening,ΠEM, is constant) and the Higgs–photon mix-\ning. These are as given in the previous section.\nSecondly, the fermionic vertices are as given before in\nﬁg. 2. We did not list the photonic vertex, but this is\ngiven bye. Note that the couplings given in ﬁg. 2 can be\nsummarizedin the followingcompact form (c.f. eqn. (34)):\nLI\neﬀ= (f−1∆E)ψ†Φ·σψ, (48)Kosuke Odagiri: Current conservation and ratio rules in mag netic metals with Coulomb repulsion 9\n(a)\nq\nφ±\nF:−1\nq0−q2/2mφ+i0\nAF:−1\nq2\n0−u2q2+i0(b)\nq\nh\nF:+1\nq0−q2/2mφ−∆h+i0\nAF:+1\nq2\n0−u2q2−∆h+i0\n(c)\nq\nφ±\nF: (1,q/mφ)f\nAF: (q0,u2q)f\n(d)\nq\nγ\n+1\n−q2−e2ΠEM+i0(e)\nq\nγ h\nF:−e(f−1∆E)Πmixing\nFig. 5. The propagators and the two-point functions. ΠEM\nis given by the 0 −0 component of eqn. (13), and Πmixingis\ngiven by the 0 −0 component of eqn. (10).\nwhereΦis deﬁned by\nΦ= (φ1,φ2,−v+h). (49)\nφ1andφ2are as shown in tab. 3, and are given by\nφ1=1\n2(−φ++φ−), φ 2=i\n2(φ++φ−).(50)\nvis a parameter, which has the interpretation as the vac-\nuum expectation value of the Φﬁeld. In order that the\nenergy diﬀerence between the two states that is given by\neqn. (48) should agree with the actual energy diﬀerence\n∆E,vneeds to satisfy\nv=f/2. (51)\nΦis essentially a magnetic order-parameterﬁeld. This dif-\nfers from the more conventional form such as\nU(φ1,φ2)(0,0,v+h)T, (52)\nbut they match in the limit of small ﬁelds, up to some\ndiﬀerences in convention.\n3.2 Bosonic vertices\nLet us consider the Ward–Takahashi identity correspond-\ning to the amplitude described by ﬁg. 3.\nWe denote theinitial statemomentum tobe q1andthe\nﬁnal state momenta to be q2andq3.q2is for theh0bosonandq3is for the −1/2 fermion. We denote the momentum\nwhich goes into the vertex by q, so thatq+q1=q2+q3.\nIt is not necessary that the fermions and the bosons are\non shell, i.e., G−1\n+(q1) etc. need not be zero.\nUpon contraction with q, the ﬁrst two diagrams yield\nqµAµ\na=−i2f−1∆E/parenleftbig\n1−G−1\n+(q1)G−(q+q1)/parenrightbig\n(53)\nand\nqµAµ\nb=i2f−1∆E/parenleftbig\nG−1\n−(q3)G+(q3−q)−1/parenrightbig\n.(54)\nThe amplitude as a whole satisﬁes the Ward identity if\nthe third amplitude satisﬁes\nqµAµ\nc= 2i2f−1∆E/parenleftbig\n1+D−1\nh(q2)Dφ(q2−q)/parenrightbig\n.(55)\nThis requires vertices of the form shown in ﬁg. 6.\n(a)\n±1 ±10\n2f−1∆h(b)\n±1 0q1q2Γ\nF:−2(1,q1+q2\n2m)\nAF:−2(q1+q2)\nFig. 6. The bosonic three-point functions. In (b), q1+q2is a\nshort-hand notation for (( q1+q2)0,u2(q1+q2)).\n(a)\n/tildewideΓ\n0 +1\n+10(b)\n/tildewideΓ0\n+1−1 0\n(c)\n+1\n0\n+10(d)\n0 0±1 ±1\n−4f−2∆h\nFig. 7. The three diagrams (a–c) whose sum must satisfy the\nWard–Takahashi identity. The bosonic four-point function (d)\nis ﬁxed as a result.\nFinally, we require the Ward–Takahashi identity for\nthe sum of the three diagrams which are shown in ﬁg. 7a–\nc. We choose the momenta to be h(q1)→h(q2)+φ+(q3),\nwithq=q2+q3−q1being the four-momentum ﬂowing\ninto the current. We obtain\nqµAµ\na=−4i2f−1∆h(D−1\nh(q1)Dφ(q1+q)+1),(56)\nand\nqµAµ\nb=−4i2f−1∆h(D−1\nh(q2)Dφ(q2−q)+1).(57)10 Kosuke Odagiri: Current conservation and ratio rules in m agnetic metals with Coulomb repulsion\nThus we require\nqµAµ\nc= 8i2f−1∆h (58)\nin order that the Ward–Takahashi identity is satisﬁed.\nHence we obtain the Feynman rule shown in ﬁg. 7d.\n4 Calculation of the parameters\nIn the preceding section, we worked out the form of the\ntheory. Let us now work out the parameters.\nIn sec. 2, we used the condition of tadpole cancellation\nto work out a certain rule involving the ratios of density\nof states, that need to be satisﬁed in ferromagnetism.\nIn sec. 3, we presented the Feynman rules and were\nable to relate the current–magnon two-point function to\nthe form of f2and the bosonic propagators.\nWe now generalize these results, and work out the four\nparameters, which are (1) f2, (2)∆E, (3)∆hand (4)u\normφ.\nCorresponding to these four unknowns, we have four\nequations, which involve: (1) tadpole cancellation, (2) the\ntime component of the current–boson two-point function,\n(3) the space component of the same two-point function\nand (4) the Higgs-boson self-energy.\n4.1 Tadpole cancellation\nLet us start with the condition of tadpole cancellation.\nWe treated the ferromagnetic case in sec. 2. The anti-\nferromagneticcaseiscalculatedanalogously,butthemech-\nanism of cancellation is diﬀerent. This time, we have the\ncontribution of the magnon loop:\niAtadpole\nmagnon=i2(2f−1∆h)/integraldisplaydd+1k\n(2π)d+1Dφ(k),(59)\nwhich must be equal and opposite to the fermionic loop:\niAtadpole\nfermion= (−1)i2(f−1∆E)/integraldisplaydd+1k\n(2π)d+1(G+(k)−G−(k)).\n(60)\nNote that the sum over positive and negative ∆Eis im-\nplicit.\nThis gives us the following condition:\n−2∆h/integraldisplaydd+1k\n(2π)d+1iDφ(k) =/summationdisplay\nsublattices[(ρ+−ρ−)∆E].\n(61)\nNotethat∆Eispositivewhen ρ−>ρ+.Thatis,theright-\nhand side is negative. Let us introduce a more compact\nnotation:\n2∆h/integraldisplay\nDφ=ρM|∆E|. (62)\nThe convention is that ρM=/summationtext|ρ−+ρ+|is positive. The\nintegral ofDφis evaluated easily:\n/integraldisplay\nDφ=/integraldisplaydd+1k\n(2π)d+1i−1\nk2\n0−u2k2+i0=/integraldisplayddk\n2u(2π)d|k|.\n(63)Thisisdivergentatlargemomenta,andneedstobecut-oﬀ\nat|k|=K, whereK∼π/a. We then obtain\nρM|∆E|\n2∆h=/braceleftbigg\nK/4πu(d= 2),\nK2/8π2u(d= 3).(64)\nNote that the propagators are all-order, and this requires\nthat∆Eisbare,andthatthevertexcorrectionsarenotin-\ncluded in eqn. (59). Whether ∆Eis stable against higher-\nordercorrectionsdependsonthesizeofthecoupling f−1∆E\nand the relative size of ucompared with the electron ve-\nlocityv≈vF. The form of eqn. (59) corresponds to the\nall-ordervertex,and thereforethis equation, and eqn. (64)\nwhich follows from it, suﬀer from double counting. How-\never, this ambiguity, that is due to double counting, can-\ncels when we discuss the Higgs-boson self-energy later on.\n4.2 Current–magnon two-point function\nIn sec. 3, we calculated the fermionic contribution to the\ncurrent–magnon two-point function, which is shown in\nﬁg. 4a. We obtained\nAµ\nfermionic=−/integraldisplaydd+1k\n(2π)d+1i(f−1∆E)ΓµG+(k)G−(k−q).\n(65)\nFor the simple case of ǫ(k) = (/planckover2pi1k)2/2me, the vertex is\ngiven byΓµ= (1,(k−q/2)/me). The bosonic loop, which\ncorresponds to ﬁg. 4b, is written as\nAµ\nbosonic=\n/integraldisplaydd+1k\n(2π)d+1i(2f−1∆h)(−2(2k−q))Dh(k)Dφ(k−q) (66)\nThis is for the anti-ferromagnetic case. Note that the case\nof ferromagnetism requires a further twist, as we have not\nyet included the higgs–screened-photon mixing.\nLet us consider the limit of small external momentum,\nq→0. It is easy to see that the fermionic amplitude van-\nishes for anti-ferromagnetism. As for the bosonic ampli-\ntude, this vanishes for ferromagnetism because of the ab-\nsence of negative energy states. The amplitude vanishes\nfor anti-ferromagnetism also, but for a diﬀerent reason,\nnamely symmetry.\nLet us, instead of trying to evaluate these integrals for\narbitrary values of q, make use of the Ward–Takahashi\nidentities to replace the current–magnon two-point func-\ntions with the corresponding current–current two-point\nfunctions (c.f., ref. [2]).\nTo do so, we ﬁrst write down the current–current two-\npoint functions as\nΠµν\nfermionic=−/integraldisplaydd+1k\n(2π)d+1iΓµΓνG+(k)G−(k−q),(67)\nand\nΠµν\nbosonic=/integraldisplaydd+1k\n(2π)d+1i\n(−2(2k−q)µ)(−2(q−2k)ν)Dh(k)Dφ(k−q).(68)Kosuke Odagiri: Current conservation and ratio rules in mag netic metals with Coulomb repulsion 11\nBy virtue of the Ward–Takahashi identities, we obtain\nqνΠµν−fAµ=Cµ\nfermionic+Cµ\nbosonic,(69)\nwhereCµare given by\nCµ\nfermionic=/integraldisplaydd+1k\n(2π)d+1iΓµ(k,q−k)(G+(k)−G−(k−q))\n(70)\nand\nCµ\nbosonic=/integraldisplaydd+1k\n(2π)d+1i(−2(2k−q)µ)(Dh(k)+Dφ(k−q)).\n(71)\nWe now equate Awith the Feynman rules of ﬁg. 5c,\nand take the derivative with respect to qλin the limit of\nsmallq. For the ferromagnetic case, we obtain\nΠµλ\nfermionic/vextendsingle/vextendsingle/vextendsingle\nq→0+f2m−1\nφdiag(0,−I) =∂\n∂qλCµ\nfermionic/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0.\n(72)\nIstands for the spatial identity matrix. The 0 −0 com-\nponent of this equation is zero on the right-hand side and\nin the second term of the left-hand side, whereas the ﬁrst\nterm on the left-hand side is non-zero:\nΠ00\nfermionic/vextendsingle/vextendsingle\nq→0=−/integraldisplay\nG+G−=ρ−−ρ+\n∆E.(73)\nThis happens because of the approximation D−1\nφ(q) =\nq0−q2/2mφ. There is, in principle, a q2\n0term as well,\nthe omission of which is inconsistent with the 0 −0 com-\nponent of this equation. As for the spatial components, we\nobtain\nf2m−1\nφd=/integraldisplaydd+1k\n(2π)d+1i/bracketleftbigg\n(G+(k)+G−(k))d\n2me+G+(k)G−(k)v2\ne/bracketrightbigg\n.(74)\nHere,d/merefers to the second derivative of ǫ(k), whereas\nverefers to the ﬁrst derivative.\nLet us introduce the following shorthand notation:\nf2m−1\nφ=/angbracketleftbiggρ−+ρ+\n2me−(ρ−−ρ+)v2\ne\n∆Ed/angbracketrightbigg\n.(75)\nThere is an obvious generalization to the case of spatial\nasymmetry.Weexpect this ﬁnalresulttobe stableagainst\nhigher-order corrections, because the renormalization fac-\ntorsdue tothe vertexcorrectionandthe Green’sfunctions\ncancel.\nAs discussed in sec. 2, ( ρ−+ρ+) is not well-deﬁned.\nHowever, the ratio of ( ρ−+ρ+) against (ρ−−ρ+) is well\ndeﬁned because of eqn. (18). Furthermore, f2is given by\nρ−−ρ+.\nAsanorderestimation,wecansaythat vecanbetaken\nto be almost constant near the Fermi surfaces. It would\nbe a bad approximation to say that meis also constant,but we can introduce a quantity meto be the inverse of\nthe average inverse fermion mass. We then obtain\n2me\nmφ≈1\nP−2mev2\nF\n∆Ed. (76)\nPis the spin asymmetry. mφis necessarily positive, but\nmeneeds not be positive although we generally expect it\nto be. Ifmeis positive, then the inequality reads\nP/lessorsimilar∆Ed\n2mev2\nF. (77)\nLet us now turn to the anti-ferromagnetic case. Here\nwe need both the fermionic and the bosonic loops. Corre-\nsponding to eqn. (72), we now have\nΠµλ/vextendsingle/vextendsingle\nq→0+f2diag(1,−u2I) =∂\n∂qλCµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0.(78)\nThefermioniccontributionsareasgivenabove.Thebosonic\ncontributions are given by\nΠµλ\nbosonic/vextendsingle/vextendsingle/vextendsingle\nq→0=−16/integraldisplay\nkµkλDhDφ,(79)\nand\n∂\n∂qλCµ\nbosonic= 2diag(1,−u2I)/integraldisplay\nDh−Dφ,(80)\nusing the same notation as in eqn. (62). The integral over\nDφis given by eqn. (63), and is a positive quantity. The\nintegral over Dhis given by\n/integraldisplay\nDh=−/integraldisplayddk\n2(2π)d√u2k2+∆h,(81)\nand this is a negative quantity.\nAltogether, we obtain\nf2+2/integraldisplay\n(Dh−Dφ)−16/integraldisplay\nk2\n0DhDφ=−ρM\n|∆E|,(82)\nand\nf2+2/integraldisplay\n(Dh−Dφ)−16\nd/integraldisplay\nk2DhDφ\n=1\nu2/angbracketleftbiggρ−+ρ+\n2me−ρMv2\ne\n|∆E|d/angbracketrightbigg\n. (83)\n4.3 The Higgs-boson self-energy\nWe now come to the ﬁnal condition, namely that the\nHiggs-boson excitation energy ∆his given by the self-\nenergy diagrams which are shown in ﬁg. 8.\nThe fermionic contributionis similarto Π00\nzwhich was\ncalculated in sec. 2, and is given by\n−iΠh\na=−i4/integraldisplaydd+1k\n(2π)d+1(f−1∆E)2\n(G+(k)G+(k−q)+G−(k)G−(k−q)).(84)12 Kosuke Odagiri: Current conservation and ratio rules in m agnetic metals with Coulomb repulsion\n(a) (b)\n(c)\nFig. 8. The diagrams for the self-energy of the Higgs boson.\nHence\nΠh\na= (f−1∆E)2(g−(ǫF)+g+(ǫF)).(85)\nNote that∆Erefers to the bare quantity. This is because,\nﬁrstly, (∆E)2in eqn. (84) needs to be the product of the\nbare∆Eand the renormalized ∆E. However, the renor-\nmalization of ∆Egives rise to the renormalization factor\nZ−1which is opposite to the renormalization factor Zfor\neachpropagator.Itfollows,therefore,that ∆Eineqn.(85)\nactually refers to the bare quantity.\nAtq= 0 (and at zero temperature), the contributions\nof ﬁg. 8b and c are zero for ferromagnetism. Hence, for\nthe case of ferromagnetism, we obtain\n∆h=g−(ǫF)+g+(ǫF)\nρ−−ρ+(∆E)2. (86)\nIf the density of states gis a linear function, then ∆h=\n2∆Esince (ρ−−ρ+) is given by the area of a trapezium\nwhose two parallel sides are g−andg+, and whose height\nis∆E.Ifnot,and gisaconvexfunctioninbetween g−and\ng+as is the case for iron [11], ∆hwill be less than 2 ∆E.\nThis gives a useful estimate of the Higgs-boson excitation\nenergy, which can be tested experimentally.\nWe should remember that the Higgs–screened-photon\nmixing cannotbe neglected when the spin asymmetry Pis\nlarge. The actual value of ∆hwhere the resonance occurs\nwill be sensitive to the behaviour of the photonic modes\n(screened photon and plasmon).\nLet us now turn to anti-ferromagnetism. The contri-\nbution of ﬁg. 8b is given by\n−iΠh\nb=i4/integraldisplaydd+1k\n(2π)d+1(2f−1∆h)2Dφ(k)Dφ(k−q).(87)\nThis is divergent,but is imaginaryat zero temperature for\nq2\n0−u2q2>0 (which is where the Higgs mode needs to\nexist). We therefore omit this contribution for now. The\ncontribution of ﬁg. 8c is given by\n−iΠh\nc=i2/integraldisplaydd+1k\n(2π)d+1(−4f−2∆h)Dφ(k).(88)\nNow using eqn. (62), this reduces to\nΠh\nc=−4f−2∆h/integraldisplay\nDφ(k) =−2f−2ρm|∆E|.(89)Hence,\n∆h=Πh\na+Πh\nc\n= 2f−2|∆E|/bracketleftbigg1\n2(g−(ǫF)+g+(ǫ))|∆E|−ρm/bracketrightbigg\n.(90)\nWe thus obtain the anti-ferromagnetic ratio rule:\ng−(ǫF)+g+(ǫF)\n2ρM/|∆E|>1. (91)\nThis is satisﬁed if the density of states is a concave func-\ntion in between the two Fermi energies.\nLet us denote the concavity by δc, deﬁned as:\nδc=g−(ǫF)+g+(ǫF)\n2ρM/|∆E|−1. (92)\nThis then leads to\n2ρM|∆E|\n∆h=δ−1\ncf2. (93)\nBy eqn. (64), we then obtain\nf2=/braceleftbigg\nδcK/πu (d= 2),\nδcK2/2π2u(d= 3).(94)\nSmallδctherefore leads to strong coupling f−1∆E. We\nexpect physically that strong coupling tends to suppress\nmagnetism, because the oscillations between the two spin\nstates will become more frequent. Our results are not af-\nfected so long as the densities of states can be deﬁned.\nHowever,∆Ewill receive a large correction through the\nelectron self-energy.\n4.4 Summary of results at zero temperature\nLet us summarize our results.\nFor the case of ferromagnetism, the parameters ∆E,\nf2,mφand∆hare ﬁxed by the following constraints:\nρ+/ρ−=g+(ǫF)/g−(ǫF), (95)\nf2=ρ−−ρ+, (96)\nf2m−1\nφ=/angbracketleftbiggρ−+ρ+\n2me−(ρ−−ρ+)v2\ne\n∆Ed/angbracketrightbigg\n,(97)\nf2∆h= (g−(ǫF)+g+(ǫF))(∆E)2. (98)\nOutoftheseequations,whichareallnon-perturbative,the\nﬁrst three are relations that only involve all-order quanti-\nties. In the last equation, ∆Erefers to the bare quantity.\nIn eqn. (97), ∆Eis the all-order quantity, but is assumed\nto be more or less independent of energy and momenta\n(though generalization is possible).\nFor the case of anti-ferromagnetism, |∆E|,f2,uand\n∆hare ﬁxed by eqns. (64), (82), (83) and (93). Equations\n(64) and (93) involve |∆E|as a bare quantity and ∆h\nin eqn. (93) is ambiguous. Equations (82) and (83) only\ninvolve all-order quantities, but are dependent on the UV\ncut-oﬀ, as is the case in eqn. (64).\nWe obtained the rule δc>0, whereδcis a measure of\nconcavity and is deﬁned by eqn. (92).Kosuke Odagiri: Current conservation and ratio rules in mag netic metals with Coulomb repulsion 13\n4.5 Finite temperature analysis\nSince our results involve diagrams that are evaluated for\nq= 0, it is, in principle, straightforward to generalize\nthem to ﬁnite temperatures. However, the bosonic dia-\ngrams, which were zero in the case of ferromagnetism, be-\ncome non-zero at ﬁnite temperatures, and therefore the\nresulting expressions are messy.\nA full calculation is beyond the scope of this present\nanalysis, but let us present two representative results.\nFirst, for the case of anti-ferromagnetism, we have\nfound that the bosonic loop of ﬁg. 8b is real and diverges\nfor ﬁniteT:\nΠh\nb(T) =−(2f−1∆h)2\n32T3/integraldisplayddk\n(2π)dd\ndx/parenleftbigg\n−coth(x)\nx/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=u|k|/2T.\n(99)\nThis makes ∆hnegative, and so magnetic order is forbid-\nden. Inouropinion,this implies that inanti-ferromagnetic\nmetals, a genuine long-range order is not permitted, at\nleast at ﬁnite temperatures.\nSecond,letusconsiderhowthe ratioruleofeqn.(18)is\nmodiﬁed at ﬁnite temperatures. We now have the bosonic\ncontribution which reads\nAtadpole\nboson(T) =−(2f−1∆h)/integraldisplayddk\n(2π)d1\nexp((k2/2mφ)/T)−1.\n(100)\nThis is evaluated easily using standard methods. For the\ncase of three spatial dimensions, we obtain\nAtadpole\nboson(T) =−2f−1∆hζ(3/2)/parenleftbiggmφT\n2π/parenrightbigg3/2\n.(101)\nHere,ζ(3/2) = 2.612···. This contribution should be\nequal and opposite to the fermionic contributions, which\nare given by\nAtadpole\nfermion(T) = (f−1∆E)[−(ρ↓−ρ↑)T+(ρ↓+ρ↑)TP(T)].\n(102)\nHereρandPcorrespondto their ﬁnite-temperature coun-\nterparts:\nρT=/integraldisplay\nf((ǫ−µ)/T)g(ǫ)dǫ, (103)\ngT=−/integraldisplay\nf′((ǫ−µ)/T)g(ǫ)dǫ, (104)\nwherefis the Fermi distribution function. Hence\n−(ρ↓−ρ↑)T+(ρ↓+ρ↑)TP(T) = 2ζ(3/2)/parenleftbiggmφT\n2π/parenrightbigg3/2∆h\n∆E.\n(105)\n∆h,mφand∆Eare also functions of temperature.\nFor smallT, we can assume that only the ﬁrst term\non the left-hand side depends signiﬁcantly on T, and that\nthe parameters on the right-hand side can be taken as\nconstants. This then implies that the magnetization goes\ndown asT3/2, which is a well-known result. All of the\nparameters on the right-hand side are, in principle, mea-\nsurable. This can then be tested experimentally.5 Conclusions and Outlook\nWe presented a nonperturbative framework for treating\nmagneticorderinmetals,causedbyaCoulombinteraction\n(or generalized Coulomb interaction).\nWe obtained interesting ‘ratio rules’ involvingthe den-\nsitiesofstatesforbothferromagneticandanti-ferromagnetic\ncases. These involve all-order quantities (with the excep-\ntion of|∆E|) and can therefore be compared directly with\nthe experimental numbers, if they become available at\ngreater precision.\nWe have seen that the shape of the density-of-states\ncurve play an essential role in determining the possibil-\nity of magnetic ordering. The density of states must rise\nwith energy, when the charge carriers are electrons, for\nferromagnetism.\nForanti-ferromagnetism,thedensity-of-statescurvemust\nbe concave. However, we have seen that the radiative cor-\nrections,duetothemagnons,atﬁnitetemperaturesbreaks\nlong-range orders. In our understanding, this means that\ngenuine long-range anti-ferromagnetic order is not possi-\nble, at least at ﬁnite temperatures. More work is required\nto elucidate the nature of the ground state.\nThe case of magnetic insulators is not covered by this\nwork, in which the exchange energy ∆Eis considered to\nbe more or less independent of k.\nTwo cases require special attention, which we have not\nbeen able to discuss in much detail. The ﬁrst is the case\nof strong coupling, which occurs when f, or the vacuum-\nexpectation value vof the magnetic order-parameterﬁeld,\nis small. Here, we expect that the radiative corrections\nsuppressthemagneticorderandthatthesystemwillfavour\nthe paramagnetic state. The second is the case of large\nmagnon velocity u, in comparison with the electron ve-\nlocityvF, in the case of anti-ferromagnetism. Here, the\nresponse of the magnetic background becomes instanta-\nneous towards the movement of the electron. We hope to\nbe able to discuss this case in a separate publication [14]\nThe results of this work can be used to calculate arbi-\ntrary amplitudes, such as scattering amplitudes.\nThe methods presented in this work, being an adap-\ntation of Gribov’s analysis of axial-current conservation,\nis of a general nature. However, we are presently unaware\nof other possible applications of the methods presented\nherein.\nWe thank I. Hase, S. Sharma, K. Yamaji and T. Yanagisawa\nfor extensive and informative comments and discussions.\nWe have been informed by Dr. I. Hase that a phenomeno-\nlogical study of the correlation between densities of state s of\na material and its magnetic properties has previously been r e-\nported. However, we have not been able to locate this study.\nReferences\n1. See,forexample:H.NeubergerandT.Ziman,Phys.Rev.\nB 39(1989) 2608; H.Leutwyler, Phys.Rev. D 49(1994)\n3033.14 Kosuke Odagiri: Current conservation and ratio rules in m agnetic metals with Coulomb repulsion\n2. V. N. Gribov, Eur. Phys. J. C 10(1999) 71 [arXiv:hep-\nph/9807224]; ibid. 10(1999) 91 [arXiv:hep-ph/9902279].\n3. V. N. Gribov, Phys. Lett. B 336(1994) 243 [arXiv:hep-\nph/9407269].\n4. V.N. Gribov, Orsay lectures on conﬁnement , arXiv:hep-\nph/9403218, arXiv:hep-ph/9407269, arXiv:hep-\nph/9905285.\n5. For a review, see: Yu.L. Dokshitzer and D.E. Kharzeev,\nhep-ph/0404216.\n6. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122(1961)\n345; Phys. Rev. 124(1961) 246.\n7. G. Cvetiˇ c, Rev. Mod. Phys. 71(1999) 513.\n8. See, for example: D. P. Young, et al., Nature 397(1999)\n412, and references therein.\n9. P. M. Tedrow and R. Meservey, Phys. Rept. 238(1994)\n173.\n10. R. J. Soulen Jr., et al., Science 282(1998) 85.\n11. R. Maglic, Phys. Rev. Lett. 31(1973) 546; K. C. Wong,\nE.P.WohlfarthandD.M.Hum,Phys.Lett. A 29(1969)\n452; J. Callaway and C. S. Wang, Phys. Rev. B 7(1973)\n1096.\n12. C. Kittel, Introduction to solid state physics , 5th edition,\nJohn Wiley & Sons, Inc., New York, London, Sydney,\nToronto, 1976.\n13. D.E. Gray (contributingeditor), AIP Handbook , 3rdedi-\ntion, AIP, New York, 1972.\n14. See sec. III of K. Odagiri and T. Yanagisawa,\narXiv:1104.1247v1. This publication is currently under\nmajor revision."    },    {        "title": "2402.07171v1.Quantum_geometric_bound_for_saturated_ferromagnetism.pdf",        "content": "Quantum geometric bound for saturated ferromagnetism\nJunha Kang,1, 2, 3Taekoo Oh,4Junhyun Lee,5and Bohm-Jung Yang1, 2, 3, ∗\n1Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea\n2Center for Theoretical Physics (CTP), Seoul National University, Seoul 08826, Korea\n3Institute of Applied Physics, Seoul National University, Seoul 08826, Korea\n4RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n5Department of Physics and Astronomy, Center for Materials Theory,\nRutgers University, Piscataway, NJ 08854, United States of America\nDespite its abundance in nature, predicting the occurrence of ferromagnetism in the ground state\nis possible only under very limited conditions such as in a flat band system with repulsive interaction\nor in a band with a single hole under infinitely large Coulomb repulsion, etc. Here, we propose a\ngeneral condition to achieve saturated ferromagnetism based on the quantum geometry of electronic\nwave functions in itinerant electron systems. By analyzing multi-band repulsive Hubbard models\nwith an integer band filling, relevant to either ferromagnetic insulators or semimetals, we propose a\nrigorous quantum geometric upper bound on the spin stiffness. By employing this geometric bound,\nwe establish that saturated ferromagnetism is prohibited in the absence of interband coupling, even\nwhen the local Hubbard repulsion is infinitely large. As a corollary, this shows that saturated\nferromagnetism is forbidden in any half-filled Hubbard model. We also derive the condition that\nthe upper bound of the spin stiffness can be completely characterized by the Abelian quantum\nmetric. We believe that our findings reveal a profound connection between quantum geometry\nand ferromagnetism, which can be extended to various symmetry-broken ground states in itinerant\nelectronic systems.\nIntroduction.— Ferromagnetism is the simplest form of\nsymmetry-broken ground states whose fundamental ori-\ngin has been discussed by divergent perspectives, from\nHeisenberg’s localized electron picture [1] to Bloch’s itin-\nerant electron approach [2]. More recently, the modern\ntheory of ferromagnetism formulated based on the Hub-\nbard model [3] has led to various theorems about itiner-\nant ferromagnetism, including the following two distinct\nmechanisms. One is Nagaoka’s ferromagnetism [4–6],\nwhere a single hole in the valence band induces the spin\nalignment when the electron repulsion is infinite. The\nother is flat band ferromagnetism with repulsive inter-\naction, extensively studied by Mielke [7] and Tasaki [8].\nDespite the valuable insights delivered by them, these\ntwo representative theorems rely on the highly restric-\ntive form of the kinetic Hamiltonian without the consid-\neration of electronic wave functions inherent in itinerant\nelectronic systems.\nRecently, the quantum geometry of electronic wave\nfunctions has garnered increasing attention due to its rel-\nevance to various physical phenomena, such as anomalous\nLandau levels of flat bands [9–11], flat band superfluidity\n[12–14], and nonlinear Hall effect [15, 16], etc. Moreover,\nit has been shown that quantum geometry plays a promi-\nnent role in describing the interacting ground states with\nspontaneous symmetry breaking. For instance, in super-\nconductors, quantum geometry was shown to be deeply\nrelated to the superfluid weight [13, 17], the length scale\nof Cooper pairs [18, 19], and the dynamics of the Higgs\nmode [20]. Also, in excitonic ground states, quantum ge-\nometry was shown to induce anomalous Lamb shifts [21],\n∗bjyang@snu.ac.krcontribute to the exciton drift velocity [22], and stabilize\nexciton condensates [23]. In contrast, the role of quan-\ntum geometry in magnetic ground states remains largely\nunexplored [24, 25].\nIn this Letter, we investigate the effects of quantum\ngeometry on a class of ferromagnetic ground states at\ninteger filling, namely, saturated ferromagnetism, which\ncan appear in itinerant electronic systems where the total\nspin is maximally polarized. By examining the spin ex-\ncitations of the Hubbard model, we study the spin stiff-\nness, the inverse effective mass of the gapless magnon\ngenerated by breaking continuous spin rotational sym-\nmetry. We show that the spin stiffness has an upper\nbound characterized by the quantum geometry of the\nnon-interacting electronic bands, which shows the promi-\nnent role of interband coupling in stabilizing saturated\nferromagnetism. Using the relation between this geo-\nmetric upper bound and the positive-definiteness of the\nspin stiffness, we derive a no-go theorem for saturated\nferromagnetism. Explicitly, we show that saturated fer-\nromagnetism is strictly prohibited in the absence of the\ninterband coupling. This immediately implies that satu-\nrated ferromagnetism is forbidden in any half-filled Hub-\nbard model. We also derive the condition that the geo-\nmetric upper bound can be completely characterized by\nthe Abelian quantum metric, directly demonstrating its\nrelation with quantum geometry.\nSaturated ferromagnetism.— Saturated ferromag-\nnetism represents the most robust manifestation of fer-\nromagnetic behavior, characterized by the ground state\npossessing maximum total spin. The highly restricted\nform of this configuration allows us to represent the\nground state using only the information of the kinetic\nHamiltonian, considerably simplifying the problem [3].arXiv:2402.07171v1  [cond-mat.str-el]  11 Feb 20242\nFor its description, let us consider the following repulsive\nHubbard model\nH=NcX\nkNorbX\nα,β↑,↓X\nσh(k)αβc†\nkασckβσ+UNcX\niniα↑niα↓,(1)\nwhere h(k) and U > 0 indicate the kinetic Hamilto-\nnian and the local Hubbard repulsion, respectively. ckβσ\ndenotes the annihilation operator of an electron with\nmomentum k, orbital index α= 1, . . . , N orb, and spin\nσ=↑,↓.niασ≡c†\niασciασis the corresponding number op-\nerator. i= 1, . . . , N clabels the unit cells. In addition, we\ndefine a momentum-independent filling v≡Nocc/2Norb,\nassuming that Nocc∈Zelectrons exist per unit cell,\nwhich results in a total of Ntot=NcNoccelectrons. More\ndetailed information of the model is given in the Supple-\nmental Materials (SM).\nFollowing Ref. [3], we denote the local and total spin\noperators by Siα=1\n2P\nσ1,σ2c†\niασ1σσ1σ2ciασ2andStot=P\niαSiα, respectively ( σis the Pauli matrix, and we set\nℏ= 1). The system is said to exhibit saturated ferro-\nmagnetism if and only if\n(Stot)2|GS⟩=Smax(Smax+ 1)|GS⟩, (2)\nwhere Smax≡Ntot\n2, for any ground state |GS⟩. This\nimposes the constraints that Nocc∈ {1, . . . , N orb}, and\nv≤1\n2. A noteworthy feature of saturated ferromagnets\nis that we can entirely ascertain the eigenspace of the\nground state using the noninteracting Hamiltonian [3,\n26].\nTo clarify, let EGS=min(PNtot\njεj) be the ground\nstate energy such that εj∈S\nispec{h(ki)}, with a de-\ngeneracy d0. Then, the ground state manifold is a\nd0×(2Smax+1)-fold degenerate eigenspace characterized\nbyEGS, where the additional (2 Smax+1)-fold degeneracy\narises from the SU(2) symmetry of the Hubbard model.\nMore specifically, let |GSj⟩(j= 1, . . . , d 0) be fully ↑-\nspin polarized states with energy EGS. If we define the\nspin-lowering operator as S−\ntot≡Sx\ntot−iSy\ntot, the states\n(S−\ntot)m|GSj⟩(m∈ {0, . . . , 2Smax},j∈ {1, . . . , d 0}) form\na basis of the ground state manifold. Thus, we can per-\nform a SU(2) rotation to express the ground state to\nbe aligned with ↑-spins. Throughout this letter, we as-\nsume such ground state, and denote it by |Ω⟩. Moreover,\nd0= 1 when the kinetic Hamiltonian has no partially oc-\ncupied band. Then, the ground state can be represented\nby a single Slater determinant [3], and mean-field theory\nbecomes exact for any U. Below, we use the term “ferro-\nmagnetism” to mean this specific context unless specified\notherwise.\nSpin excitations.— Let us describe the spin excitation\nspectrum of saturated ferromagnets, generally composed\nof the Stoner continuum and the low-energy magnon\nmodes. Assuming a ground state |Ω⟩in which Noccoccu-\npied bands are fully ↑-spin polarized, the spin excitationoperator with momentum Qcan be described by\nΨ†\nQ=NcX\nkNorbX\nαNoccX\nnzkαn(Q)c†\nk+Qα↓ϕnk, (3)\nwhere zkαn(Q) is an arbitrary number and ϕ†\nnk=PNorb\nαc†\nkα↑|un(k)⟩αis the creation operator of the n-\nth occupied band in which |un(k)⟩is the eigenvector\nofh(k) with eigenvalue En(k). Then, we have |Ω⟩=QNocc\nnQNc\nkϕ†\nnk|0⟩, where |0⟩indicates the vacuum state.\nDefining |Q⟩ ≡Ψ†\nQ|Ω⟩, the spin excitation energy is\ngiven by\nE(Q) =⟨Q|H|Q⟩\n⟨Q|Q⟩− ⟨Ω|H|Ω⟩. (4)\nWe decouple the many-body terms in Eq. (4) using mean-\nfield theory and apply the variational principle to mini-\nmizeE(Q) with respect to z∗\nkαn(Q). This gives the fol-\nlowing eigenvalue equation\nX\nk′βn′HSE\nkαn,k′βn′(Q)zk′βn′(Q) =E(Q)zkαn(Q),(5)\nwhere the spin excitation Hamiltonian HSE(Q) is an\nNcNorbNocc×NcNorbNoccHermitian matrix given by\nHSE\nkαn,k′βn′(Q) = [h(k+Q)αβ−En(k)δαβ]δkk′δnn′\n+U\nNc[NcX\nqNoccX\nlα⟨ul(q)|ul(q)⟩αδkk′δnn′\n−α⟨un′(k′)|un(k)⟩α]δαβ. (6)\n(See SM for the details.)\nFor illustration, let us describe the spin excitation\nspectrum of a saturated ferromagnet on the kagome lat-\ntice shown in Fig. 1 (a). The tight-binding Hamiltonian\ncontaining only the nearest-neighbor hopping amplitude\nt >0 is given by\nh(k) = 2 t\n0 cosk·a1\n2cosk·a2\n2\ncosk·a1\n20 cosk·(a1−a2)\n2\ncosk·a2\n2cosk·(a1−a2)\n20\n−µI,\n(7)\nwhere a1= (1,0) and a2= (1\n2,√\n3\n2),µis the chemical\npotential, and Iis an identity matrix. This model has\na spin-degenerate flat band at the bottom. At 0 K, this\nmodel exhibits saturated ferromagnetism at any U > 0\nwhen the bottom flat band is half filled, i.e., Nocc= 1 and\nv= 1/6 [7, 27]. The corresponding spin-split mean-field\nband structure is shown in Fig. 1 (b). The relevant spin\nexcitation spectrum in Fig. 1(c), composed of the Stoner\ncontinuum and spin wave excitations, is obtained by di-\nagonalizing Eq. (6) at each Q. In general, there are Norb\nmagnons, which consists of 1 gapless Goldstone mode and3\nb\nMean-field band structure\nΓ Γ K M\nΓ Γ M Γ Γ K K Ma\nKagome lattice\nA\nB Ca1a2\n0246c d6\n4\n2\n0\nSpin excitation spectrum Magnon dispersion \n00.2 0.4 0.6 M\nΓ K\nb1b2/g1831(/g2193)\nℰ(/g2173) ℰ(/g2173)\nFIG. 1. Spin excitation spectrum of kagome ferromag-\nnet. (a) The kagome lattice. (b) The mean-field band struc-\nture for Nocc= 1, t= 1, U= 3, and µ=−2, where the\nlowest flat band in red is fully occupied. Red and blue lines\nrepresent the ↑and↓spin bands, respectively. (c) The spin\nexcitation spectrum. Red and grey lines correspond to the\nmagnon bands and Stoner continuum, respectively. (d) The\nmagnon band structure.\nNorb−1 gapped modes. When the ground state is ferro-\nmagnetic, the gapless magnon exhibits a quadratic dis-\npersion with an energy minimum at Q=0. Conversely,\na negative (or zero) magnon energy indicates that the\nferromagnetic ground state is unstable [28, 29].\nUpper bound for spin stiffness.— Now let us assume a\nferromagnetic ground state, and use perturbation theory\nto derive an upper bound of the spin stiffness:\nDµν≡∂µ∂νEm(Q)|Q=0. (8)\nHere, Em(Q) is the energy of the gapless magnon with\nmomentum Q. We reorganize Eq. (6) as HSE(Q) =\nHSE(0)+V(Q) and treat V(Q)≡ HSE(Q)−HSE(0) as\ntheUindependent perturbation.\nEq. (5) is exactly solved at Q= 0 with zk′βn′(0) =\n|un′(k′)⟩βandE(0) = 0. The assumption of ferromag-\nnetism prohibits E(0)<0, and thus Em(0) = 0 becomes\nthe ground state energy of the unperturbed Hamiltonian\nHSE(0). The magnon energy up to first order perturba-\ntion is therefore:\nE(1)\nm(Q) =⟨zGS(0)|V(Q)|zGS(0)⟩\n⟨zGS(0)|zGS(0)⟩, (9)\nwhere |zGS(0)⟩is the ground state of HSE(0).\nSince first order perturbation uses the ground state\nof the unperturbed Hamiltonian to calculate the energy\nof the perturbed Hamiltonian, it overestimates the true\nground state energy in general (see SM for details). To-\ngether with the fact that Em(Q) increases with U, we\nobtain the following inequality,\nEm(Q)|U<∞≤ Em(Q)|U→∞≤ E(1)\nm(Q). (10)\n1st  order perturbation \nU=10 5\nU=1 a\n00.5 11.5 \nM Γ KDxx Dyy ,\nDxy Dxx Dyy ,(1) (1) \nDxy (1) b\nℰ(/g2173) Dμν \n1 10 5 50100 500 10000812 24 /g94110 -2 \nUFIG. 2. Upper bound for the spin stiffness of the\nkagome saturated ferromagnet. (a) The gapless magnon\ndispersion. Red and blue lines correspond to U= 105and\nU= 1 cases, respectively. The black curve indicates the up-\nper bound E(1)\nm(Q). (b) Spin stiffness as a function of U. Due\ntoC3zsymmetry, Dxx=DyyandDxy= 0. The red and\nblue curves correspond to Dxx(Dyy) and Dxy, respectively.\nThe purple dashed lines are the upper bounds D(1)\nµµ, which\nare identical to the spin stiffness at U→ ∞ .\nUtilizing that both inequalities saturate at Q=0, we\ncan convert Eq. (10) to the inequality on spin stiffness as\nDµµ(U <∞)≤Dµµ(U→ ∞ )≤D(1)\nµµ, (11)\nwhere D(1)\nµν≡∂µ∂νE(1)\nm(Q)|Q=0is the first order spin\nstiffness (see SM for details).\nIn Fig. 2, we demonstrate the validity of Eq. (11) in\nthe kagome lattice. Interestingly, the second inequality\nof Eq. (11) saturates in Fig. 2 (b), which is also true\nfor a number of other lattice models. In such cases, the\nstrongly correlated limit is governed solely by the energy\nand quantum geometry of the single particle Hamilto-\nnian, as we shortly show.\nTo describe the gapless magnon dispersion near the Γ\npoint, we expand E(1)\nm(Q) in powers of Qµup to quadratic\norder to obtain\nE(1)\nm(Q) =QµQν\n2NcNoccNcX\nk[NoccX\nn∂µ∂νEn(k)+\n2NorbX\nm>N occNoccX\nn(Em(k)−En(k))χmn\nµν(k)],(12)\nwhere a summation over the µandνindices is assumed\n(see SM for details). Here, χmn\nµν(k) is the fidelity tensor\ndefined as χmn\nµν(k)≡ ⟨∂µum(k)|un(k)⟩⟨un(k)|∂νum(k)⟩,\nwhich characterizes the transition probability between\nthem-th and n-th band [10, 30]. Eq. (12) explicitly shows\nthatE(1)\nm(Q) is composed of two separate terms. One is\nthe sum of the electronic band curvature and the other\nis the sum of the transition probabilities between the un-\noccupied and occupied bands weighted by their energy\ndifference. Note that by unoccupied bands, we refer to\nthe (Norb−Nocc) unoccupied ↑-spin bands of h(k).\nNo-go theorem.— Using Eqs. (10), (11), and (12), we\nderive a no-go theorem that forbids saturated ferromag-\nnetism. Explicitly, we show that for systems described4\nby a repulsive Hubbard model, if the spin-polarized oc-\ncupied bands are either isolated or flat, saturated fer-\nromagnetism is forbidden when the fidelity between the\noccupied and unoccupied bands is zero. Also, as a corol-\nlary, we prove the absence of saturated ferromagnetism\ninanyhalf-filled Hubbard model. Since Dµνis a positive\ndefinite tensor for a ferromagnetic ground state, one can\nderive the no-go theorem by examining the condition for\nD(1)\nµν= 0.\nLet us consider the band structure of ↑-spin electrons\ndescribed by h(k). If the occupied bands are separated\nfrom the unoccupied bands by a direct gap, the first term\nof Eq. (12) is zero, since En(k) are smooth and periodic\nin the Brillouin zone. Likewise, if the occupied bands\nare flat, the same term vanishes as well, even when band\ncrossings with the unoccupied bands exist. In such cases,\nif we further assume no interband coupling between the\noccupied and unoccupied bands, i.e., χmn\nµν(k) = 0, the\nsecond term also vanishes, which completes the proof of\nthe no-go theorem.\nSpecifically, in the case of the half-filled Hubbard\nmodel with Nocc=Norb, saturated ferromagnetism in-\ndicates that all ↑-spin bands are fully occupied. Without\nany unoccupied bands, χmn\nµν(k) should be zero by defini-\ntion. Since the first term of Eq. (12) also vanishes due\nto the periodicity and smoothness of the energy bands,\nwe find that saturated ferromagnetism is strictly forbid-\nden in any half-filled repulsive Hubbard model in arbi-\ntrary dimension. A particularly important consequence\nis that any repulsive Hubbard model with a single or-\nbital cannot exhibit saturated ferromagnetism at integer\nfilling. We demonstrate the no-go theorem by examin-\ning the magnon dispersion of several half-filled Hubbard\nmodels in the SM.\nRelation to quantum geometry.— Let us discuss the\ngeometric meaning of the upper bound in Eq. (12). In\ngeneral, the geometry of the quantum state |un(k)⟩is\ncharacterzied by the quantum geometric tensor (QGT)\nGn\nij(k) whose real and imaginary parts correspond to the\nquantum metric and Berry curvature, respectively. Ex-\nplicitly,\nGn\nµν(k) =⟨∂µun(k)|[I−Pn(k)]|∂νun(k)⟩\n=X\nm̸=n⟨∂µun(k)|um(k)⟩⟨um(k)|∂νun(k)⟩\n=X\nm̸=nχnm\nµν(k), (13)\nwhere the projector Pn(k) is defined as Pn(k) =\n|un(k)⟩⟨un(k)|. This shows that the QGT Gn\nµν(k) of the\nn-th band is given by the summation of the fidelity tensor\nχnm\nµν(k) over all m̸=n. In multiband cases, the Abelian\nQGT can be generalized as [10]\nGµν(k)≡NoccX\nn=1⟨∂µun(k)|[I−Pocc(k)]|∂νun(k)⟩=NorbX\nm>N occNoccX\nnχnm\nµν(k), (14)\nwhere Pocc(k) =PNocc\nn=1|un(k)⟩⟨un(k)|. The similarity\nbetween Eq. (12) and Eq. (14) indicates the geometric\ncharacter of the spin stiffness.\nIn particular, when the occupied and unoccupied\nbands are fully degenerate among themselves, ∆( k)≡\nEm(k)−En(k) is independent of mandn, which leads\nto\nE(1)\nm(Q) =QµQν\n2NcNoccNcX\nk[NoccX\nn∂µ∂νEn(k) + 2∆( k)gµν(k)],\n(15)\nwhere gµν(k) = ( Gµν(k) +Gνµ(k))/2 is the Abelian\nquantum metric. This directly follows from Eq. (14),\nχmn\nµν(k) =χnm\nνµ(k), and the fact that Dµνis a symmet-\nric tensor. We note that Eq. (15) always holds for two-\norbital models with 1 occupied and 1 unoccupied band,\nwhich indicates that in such systems, a non-vanishing\nquantum metric is essential for the emergence of satu-\nrated ferromagnetism.\nTo demonstrate the effect of the quantum metric on\nspin stiffness, we compare the gapless magnon spectra of\ntwo models with identical electronic band structure but\ndifferent quantum metrics. Explicitly, let us consider a\nsystem in the wallpaper group p3 as in Fig. 3(a), and\nplace three orbitals whose C3zeigenvalues are 1, ω, and\nω2(ω=e2πi\n3), respectively, at the 1 aWyckoff position.\nHere, C3zindicates a 3-fold rotation about the z-axis.\nWe consider one occupied flat band at the Fermi level\n(E= 0) and two degenerate unoccupied bands with en-\nergyEunocc(k) = 3+cos k·a1+cosk·a2+cosk·(a1−a2).\nThe relevant Hamiltonian is given by\nh(k) =Eunocc(k)(I−P(k)), (16)\nwhere P(k) =|u1(k)⟩⟨u1(k)|is the occupied band pro-\njector. The corresponding electronic energy spectrum is\nshown in Fig. 3(b).\nIn this model, the Abelian quantum metric of the oc-\ncupied band is expressed as [31]\ngµν(k) =1\n2Tr∂µP(k)∂νP(k), (17)\nwhere the trace is performed on the orbital indices. Thus,\nwe can tune gµν(k) by changing P(k) or|u1(k)⟩.The\ntrivial model with zero quantum metric is constructed\nby using |u1(k)⟩= (1 ,0,0)T. Following Ref. [31], we\nconstruct the nontrivial model with\n|u1(k)⟩=1\n3\n1 +e−ik·a1+e−ik·a2\n1 +ωe−ik·a1+ω2e−ik·a2\n1 +ω2e−ik·a1+ωe−ik·a2\n, (18)\nwhose quantum metric is gµν(k) =δµν/6 in orthogonal\ncoordinates. From Eq. (15), we obtain D(1)\nµν= 0 and5\nc\nM Γ Kd\nℰ(/g2173)\n1 10 5 50100 500 1000Dxx \nUNontrivial metric (pert.) \nNontrivial metric \nTrivial metric \nU=10 5\n0123\nTrivial metric Nontrivial metric Nontrivial metric (pert.) \n00.2 0.4 0.6 0.8 1b a\na1a2\n1a \n-4 \n4-2 02\nkxky4\n2\n0/g1831(/g2193)\n0\nFIG. 3. Influence of the quantum metric on the spin\nstiffness. (a) Lattice structure belonging to wallpaper group\np3.a1= (1,0) and a2= (1\n2,√\n3\n2) are the primitive lattice vec-\ntors. (b) The band structure of the Hamiltonian in Eq. (16).\nThe system consists of a single occupied flat band at E= 0,\nwhich is separated from two degenerate dispersive bands. (c)\nGapless magnon spectrum for U= 105. (d) Dxxplotted as a\nfunction of U. In this model, Dxx=DyyandDxy= 0 due\ntoC3zsymmetry. We note that Dµν(U→ ∞ ) =D(1)\nµν. In\n(c) and (d), the red and black curves are numerical data and\nthe upper bound, respectively, for the model with nontrivial\nquantum metric. The green curves are calculated for the case\nwith zero quantum metric.\nδµνfor the trivial and nontrivial models, respectively.\nFig. 3(c) and (d) show the calculation of the gapless\nmagnon spectrum and spin stiffness. The two models\nshow drastically different behavior despite the same en-\nergy dispersion, which demonstrates the significance of\nquantum metric on the computed quantities. Moreover,\nthe quantum metric solely controls the model’s capability\nto host saturated ferromagnetism.\nLastly, when ∆( k) = ∆ is independent of kin an\ninsulator, Eq. (15) reduces to a simpler relation to the\nAbelian quantum metric:\nD(1)\nµν=2∆\nNcNoccNcX\nkgµν(k). (19)\nIn such cases, the lower bounds of the quantum met-\nric proposed in previous studies can be used to impose\ntopological constraint on D(1)\nµν. For instance, the well-\nknown inequality between the trace of the quantum met-\nric and the Berry curvature [13, 17] gives a lower bound\nof the spin stiffness imposed by the Chern number of\noccupied bands, indicating that the band topology may\nstabilize saturated ferromagnetism. Similarly, the fragile\nor obstructed atomic band topology can also give another\nlower bound that can be obtained from real space invari-\nants developed recently [31, 32] (See SM).\nDiscussion.— We have shown that quantum geometryplays an essential role in stabilizing saturated ferromag-\nnetism, relevant to ferromagnetic insulators or semimet-\nals [33–40]. This implies that ferromagnetic insulators\nor semimetals can be realized in materials which intrin-\nsically have strong interband couplings. This may be\nthe physical reason why most ferromagnets found in na-\nture are metals; for an insulator to be a ferromagnet,\na nontrivial quantum geometry is required. We empha-\nsize that the validity of these results is mathematically\nrigorous beyond mean-field theory (See SM). Although\nour theory is limited to saturated ferromagnetism, con-\nsidering that rigorous statements about the stability of\ngeneral ferromagnetism are either very rare or relying on\nthe restricted forms of the kinetic Hamiltonian [3], we\nbelieve that our work provides a significant insight for\nunderstanding the fundamental origin of ferromagnetism\nand its relation to the quantum geometry. Extending\nour theory to general broken symmetry ground states in\nitinerant electronic systems would be one important di-\nrection for future study.\nACKNOWLEDGMENTS\nWe thank Seung-Hun Lee for fruitful discussions re-\ngarding this work. 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Chen, “Magnetic\nweyl semimetal phase in a kagom´ e crystal,” Science 365,\n1282–1285 (2019).\n[41] Jen˝ o S´ olyom, Fundamentals of the Physics of Solids\n(Springer Berlin Heidelberg, 2010).8\nSupplemental Material for “ Quantum geometric bound for saturated ferromagnetism\n”\nCONTENTS\nS1. Tight-binding conventions 8\nS2. Mean-field Hamiltonian of saturated ferromagnets 9\nS3. Derivation of the spin excitation Hamiltonian 9\nS4. Spin excitations and the gapless magnon mode 12\nS5. Upper bound of the gapless magnon mode 14\nS6. Numerical calculations on the no-go theorem 16\nS7. Lower bounds of the quantum metric 17\nS8. Validity of the results 17\nS1. TIGHT-BINDING CONVENTIONS\nIn this Appendix, we establish the tight-binding conventions used throughout this letter. We consider a d-\ndimensional periodic lattice spanned by the primitive lattice vectors {a1, . . . ,ad}. The tight-binding Hilbert space\nis spanned by L¨ owdin orbitals c†\niασlabeled by the unit cell Ri(i= 1, . . . , N c), orbital ( α= 1, . . . , N orb), and spin\n(σ=↑,↓) indices. The momentum space basis is given by\nc†\nkασ=1√NcX\nieik·(Ri+τα)c†\niασ. (S1)\nThe tight-binding Hamiltonian can then be expressed as\nH=Hkin+Hint\n=X\nijαβσtαβ\nijc†\niασcjβσ+UX\niαc†\niα↑c†\niα↓ciα↓ciα↑\n=X\nkαβσh(k)αβc†\nkασckβσ+U\nNcX\nkk′qαc†\nk+qα↑c†\nk′−qα↓ck′α↓ckα↑, (S2)\nwhere we neglect Umklapp scattering of the repulsive Hubbard interaction ( U >0). We denote the eigenstates of the\nkinetic (noninteracting) Hamiltonian h(k)αβ=P\nie−ik·(Ri+τα−τβ)tαβ\ni0by\nh(k)|un(k)⟩=En(k)|un(k)⟩ (S3)\n(n= 1, . . . , N orb), where |un(k)⟩is an Norb×1 vector. In addition, we define the filling v≡Nocc/2Norb, where\nNocc∈Zelectrons exist per unit cell, resulting in a total of Ntot=NcNoccelectrons. Note that we restrict the\nproblem to cases where the band filling is an integer, which corresponds to an insulator or a semimetal where the\nband inversion occurs at the Fermi level.9\nS2. MEAN-FIELD HAMILTONIAN OF SATURATED FERROMAGNETS\nIn this Appendix, we review the mean-field properties of a ferromagnet. Following the main text, we assume that\nevery occupied state has spin ↑. The mean-field Hamiltonian obtained by decoupling the many-body terms of the\nsecond line of Eq. (S2) is given by\nHmf=X\nkαβσh(k)αβc†\nkασckβσ+UX\nkασ⟨nα−σ⟩c†\nkασckασ−UNcX\nαnα↑nα↓, (S4)\nwhere ⟨nασ⟩=⟨c†\niασciασ⟩is the unit cell-independent orbital filling. In the ( ↑,↓)T⊗(α1, . . . , α Norb)Tbasis, Hmf(k)\nis a (2 Norb×2Norb) Hermitian matrix given by\nHmf(k) =\u0012\nH↑↑(k)H↑↓(k)\nH↓↑(k)H↓↓(k)\u0013\n=\u0012\nh(k) 0\n0h(k)\u0013\n+U\u0012\ndiag(n↓) 0\n0 diag(n↑)\u0013\n, (S5)\nwhere nσ= (n1σ, . . . , n Norbσ). Since the ↓bands are empty ( n↓=0), the Hamiltonian of occupied states (which\nhave↑) is identical to h(k). This observation underscores that the ground state of a saturated ferromagnet is fully\ndetermined by the noninteracting Hamiltonian. As an example, we present the mean-field band structure of a kagome\nferromagnet in Fig. 1(b) of the main text.\nS3. DERIVATION OF THE SPIN EXCITATION HAMILTONIAN\nIn this Appendix, we present a detailed derivation of the spin excitation Hamiltonian HSE≡ HSE\nkαn,k′βn′(Q) in\nEq. (6). HSEis a Hermitian matrix, which is an effective Hamiltonian in the sense that it gives the spin excitation\nspectrum upon diagonalization. Let us define the fundamental variables under consideration. The momentum of spin\nexcitation is denoted as Q, its energy as E(Q), and the gapless magnon energy as Em(Q). Following Eq. (S3), we\nbegin by defining the occupied band electron creation operator as\nϕ†\nnk=NorbX\nαc†\nkα↑|un(k)⟩α, (S6)\nwhere n∈ {1, . . . , N occ}. The ground state is given by\n|Ω⟩=NoccY\nnNcY\nkϕ†\nnk|0⟩, (S7)\nwhere |0⟩indicates the vacuum state. Consequently, spin excitations |Q⟩with momentum Qare created by acting\non|Ω⟩\nΨ†\nQ=NcX\nkNorbX\nαNoccX\nnzkαn(Q)c†\nk+Qα↓ϕnk. (S8)\nThis describes a transition from occupied bands to spin ↓orbitals, where zkαn(Q) are arbitrary complex numbers to\nbe determined.\nThe spin excitations are obtained by forming a proper choice of zkαn(Q). Although it seems natural to solve for\nH|Q⟩= (E(Q) +EGS)|Q⟩, (S9)\nan exact solution to it does not exist [41]. To see this, note that the left hand side contains 2- and 3-body terms,\nwhile the right hand side contains a 1-body term. To satisfy the equality, one has to reduce the many body terms to\na 1-body term using anticommutation relations and the ground state properties. However, direct calculation shows\nthat Hint|Q⟩cannot be simplified into a 1-body operator unless the system is half-filled ; that is, the spin excitation\nspectrum is exactly solvable only at half filling. As a resolution, we define the spin excitations through a two-step\napproach: performing a mean-field decoupling of\nE(Q) =⟨Q|H|Q⟩\n⟨Q|Q⟩− ⟨Ω|H|Ω⟩, (S10)10\nand employing a variational method with respect to z∗\nkαn(Q). We demonstrate shortly after that this yields\nHSE\nkαn,k′βn′(Q).\nWe emphasize that the utilization of mean-field theory is not a mere approximation, but rather mathematically\nrigorous, as elaborated in the main text. Furthermore, it is worth mentioning that Eq. (S10) must be calculated using\nHinstead of Hmf. This is because a Goldstone mode appears when a continuous symmetry of the effective action,\nin our case the Hamiltonian, does not exist in the ground state. However, since Hmfis obtained after spontaneous\nsymmetry breaking, the SU(2) symmetry is already broken in the Hamiltonian level. As a result, the spin excitation\nspectrum obtained from Hmfis gapped.\nLet us present the mean-field decoupling of Eq. (S10). To do so, we make use of the following identities,\n⟨ϕ†\nn1k1ϕn2k2⟩=δn1n2δk1k2,(ni∈[1, Nocc])\n⟨ck1α1↓c†\nk2α2↓⟩=δα1α2δk1k2\n⟨c†\nkα↑ck′β↑⟩=δkk′NoccX\nnα⟨un(k)|un(k)⟩β. (S11)\nThe third identity is obtained from\nNorbX\nnβ⟨un(k)|ϕ†\nnk=NorbX\nnαβ⟨un(k)|un(k)⟩αc†\nkα↑=NorbX\nαδαβc†\nkα↑=c†\nkβ↑. (S12)\nIt is worth noting that this is different from δkk′δαβ, since the sum runs over (1 , . . . , N occ). From now on, we assumeP\nα=PNorb\nαandP\nn=PNocc\nnfor any orbital and band indices, unless specified otherwise.\nLet⟨Q|H|Q⟩=I1+I2+I3, where I1,2=⟨Q|P\nkαβh(k)αβc†\nkα↑,↓ckα↑,↓|Q⟩andI3=⟨Q|Hint|Q⟩. Denoting\n⟨. . .⟩=⟨Ω|. . .|Ω⟩, we obtain\nI1=X\nkαβ⟨Q|h(k)αβc†\nkα↑ckβ↑|Q⟩\n=X\nkαβX\nn1α1k1X\nn2α2k2z∗\nn1α1k1(Q)zn2α2k2(Q)h(k)αβ⟨ϕ†\nn1k1ck1+Qα1↓c†\nkα↑ckβ↑c†\nk2+Qα2↓ϕn2k2⟩. (S13)\nWe apply mean-field decoupling to ⟨. . .⟩and decompose it into a product of 1-body correlations. Upon doing so, two\nnonzero contractions survive, which we denote as ⟨. . .⟩=I′\n1+I′′\n1. The first term\nI′\n1=−⟨ϕ†\nn1k1ckβ↑⟩⟨ck1+Qα1↓c†\nk2+Qα2↓⟩⟨c†\nkα↑ϕn2k2⟩\n=−δk1kδk1k2X\nα3α4α4⟨un2(k2)|un1(k1)⟩α3⟨c†\nk1α3↑ckβ↑⟩⟨c†\nkα↑ckα4↑⟩\n=−δk1kδk1k2X\nα3α4X\nn3n4α4⟨un2(k)|un1(k)⟩α3α3⟨un3(k)|un3(k)⟩βα⟨un4(k)|un4(k)⟩α4\n=−δk1kδk1k2X\nα3α4X\nn3n4α4⟨un2(k)|un4(k)⟩α4α3⟨un3(k)|un1(k)⟩α3α⟨un4(k)|un3(k)⟩β\n=−δk1kδk1k2α⟨un2(k)|un1(k)⟩β (S14)\nupon summation yields\n−X\nn1n2α1kαβz∗\nn1α1k(Q)zn2α1k(Q)h(k)αβα⟨un2(k)|un1(k)⟩β=−X\nnαk|znαk(Q)|2En(k). (S15)\nThe second term is given by\nI′′\n1=⟨ϕ†\nn1k1ϕn2k2⟩⟨ck1+Qα1↓c†\nk2+Qα2↓⟩⟨c†\nkα↑ckβ↑⟩\n=NoccX\nn3α⟨un3(k)|un3(k)⟩βδn1n2δα1α2δk1k2, (S16)11\nwhich evaluates to\nX\nn1α1k1kαβ|zn1α1k1(Q)|2X\nn3α⟨un3(k)|un3(k)⟩βh(k)αβ=X\nnαk|znαk(Q)|2X\nn′k′En′(k′). (S17)\nThus, I1=P\nnαk|znαk(Q)|2P\nn′k′En′(k′)−P\nnαk|znαk(Q)|2En(k).\nSimilarly, the second term results in\nI2=X\nkαβh(k)αβ⟨Q|c†\nkα↓ckβ↓|Q⟩\n=X\nkαβX\nn1α1k1X\nn2α2k2z∗\nn1α1k1(Q)zn2α2k2(Q)h(k)αβ⟨ϕ†\nn1k1ck1+Qα1↓c†\nkα↓ckβ↓c†\nk2+Qα2↓ϕn2k2⟩\n=X\nn1k1αβz∗\nn1αk1(Q)zn1βk1(Q)h(k1+Q)αβ, (S18)\nThe third term is expressed as\nI3=U\nNcX\nkk′qα⟨Q|c†\nk+qα↑c†\nk′−qα↓ck′α↓ckα↑|Q⟩\n=U\nNcX\nn1α1k1X\nn2α2k2X\nkk′qαz∗\nn1α1k1(Q)zn2α2k2(Q)⟨ϕ†\nn1k1ck1+Qα1↓c†\nk+qα↑c†\nk′−qα↓ck′α↓ckα↑c†\nk2+Qα2↓ϕn2k2⟩.(S19)\nThis contains two nonzero contractions. Again, we denote ⟨. . .⟩=I′\n3+I′′\n3, and calculate these terms. Upon doing so,\nthe first term evaluates to\nI′\n3=−⟨ϕ†\nn1k1ckα↑⟩⟨c†\nk+qα↑ϕn2k2⟩⟨ck1+Qα1↓c†\nk′−qα↓⟩⟨ck′α↓c†\nk2+Qα2↓⟩\n=−X\nα3α4α4⟨un2(k2)|un1(k1)⟩α3⟨c†\nk1α3↑ckα↑⟩⟨c†\nk+qα↑ck2α4↑⟩⟨ck1+Qα1↓c†\nk′−qα↓⟩⟨ck′α↓c†\nk2+Qα2↓⟩\n=−X\nα3α4X\nn3n4α4⟨un2(k2)|un1(k1)⟩α3α3⟨un3(k)|un3(k)⟩αα⟨un4(k+q)|un4(k+q)⟩α4\n×δkk1δk2,k+qδk1+Q,k′−qδk′,k2+Qδαα1δαα2\n=−X\nα3α4X\nn3n4α3⟨un3(k)|un1(k)⟩α3α4⟨un2(k+q)|un4(k+q)⟩α4α⟨un4(k+q)|un3(k)⟩α\n×δkk1δk2,k+qδk′,k+q+Qδαα1δαα2\n=−X\nn3n4α⟨un4(k+q)|un3(k)⟩αδn1n3δn2n4δkk1δk2,k+qδk′,k+q+Qδαα1δαα2\n=−α⟨un2(k+q)|un1(k)⟩δkk1δk2,k+qδk′,k+q+Qδαα1δαα2, (S20)\nwhere n3, n4are summed over 1 , . . . , N occ. As usual, the orbital indices are summed over 1 , . . . , N orb. The second\ncontraction is given by\nI′′\n3=⟨ϕ†\nn1k1ϕn2k2⟩⟨c†\nk+qα↑ckα↑⟩⟨ck1+Qα1↓c†\nk′−qα↓⟩⟨ck′α↓c†\nk2+Qα2↓⟩\n=NoccX\nn′α⟨un′(k+q)|un′(k)⟩αδαα1δαα2δn1n2δk1k2δk+q,kδk1+Q,k′−qδk′,k2+Q\n=NoccX\nn′α⟨un′(k)|un′(k)⟩αδαα1δαα2δn1n2δk1k2δq,0δk1+Q,k′. (S21)\nSumming over, one obtains\nI3=U\nNcX\nαX\nnk|znαk(Q)|2X\nn′k′α⟨un′(k′)|un′(k′)⟩α−U\nNcX\nnn′αkk′z∗\nnαk(Q)zn′αk′(Q)α⟨un′(k′)|un(k)⟩α. (S22)12\nThe remaining terms in Eq. (S10) are given by\n⟨Q|Q⟩=X\nn1α1k1X\nn2α2k2z∗\nn1α1k1(Q)zn2α2k2(Q)⟨ϕ†\nn1k1ck1+Qα1↓c†\nk2+Qα2↓ϕn2k2⟩\n=X\nn1α1k1X\nn2α2k2z∗\nn1α1k1(Q)zn2α2k2(Q)⟨ϕ†\nn1k1ϕn2k2⟩⟨ck1+Qα1↓c†\nk2+Qα2↓⟩\n=X\nn1α1k1X\nn2α2k2z∗\nn1α1k1(Q)zn2α2k2(Q)δn1n2δα1α2δk1k2δk1+Q,k2+Q\n=X\nnαk|znαk(Q)|2, (S23)\nand\nEGS=⟨Ω|X\nkαβX\nσh(k)αβc†\nkασckβσ+U\nNcX\nkk′qαc†\nk+qα↑c†\nk′−qα↓ck′α↓ckα↑|Ω⟩=X\nnkEn(k). (S24)\nUsing these results, Eq. (S10) becomes\nE(Q)X\nnαk|znαk(Q)|2=X\nnkαβz∗\nnαk(Q)h(k+Q)αβznβk(Q)−X\nnαk|znαk(Q)|2En(k)\n+U\nNcX\nαX\nnk|znαk(Q)|2X\nlqα⟨ul(q)|ul(q)⟩α−U\nNcX\nnn′αkk′z∗\nnαk(Q)zn′αk′(Q)α⟨un′(k′)|un(k)⟩α.\n(S25)\nNext, we take a partial derivative with respect to z∗\nnαk(Q) and obtain\nE(Q)znαk(Q) =X\nβh(k+Q)αβznβk(Q)−znαk(Q)En(k) +U\nNcznαk(Q)X\nlqα⟨ul(q)|ul(q)⟩α\n−U\nNcX\nn′k′zn′αk′(Q)α⟨un′(k′)|un(k)⟩α. (S26)\nThis equation can be solved by converting it to an eigenvalue problem,\nE(Q)zkαn(Q) =X\nk′βn′HSE\nkαn,k′βn′(Q)zk′βn′(Q), (S27)\nwhere\nHSE\nkαn,k′βn′(Q) =h(k+Q)αβδkk′δnn′−En(k)δkk′δαβδnn′+U\nNcX\nqlα⟨ul(q)|ul(q)⟩αδkk′δαβδnn′\n−U\nNcα⟨un′(k′)|un(k)⟩αδαβ (S28)\nis the spin excitation Hamiltonian. In practice, the index-6 quantity HSE\nkαn,k′βn′(Q) has to be restructured into an\nindex-2 matrix with dimension NcNorbNocc×NcNorbNocc. As illustrated in Fig. 1(c), the eigenvalues E(Q) show the\nStoner continuum and the spin waves.\nS4. SPIN EXCITATIONS AND THE GAPLESS MAGNON MODE\nIn this Appendix, we introduce noteworthy aspects of the spin excitation spectrum. As illustrated in Fig. 1(c) of\nthe main text, spin excitations consist of two parts. The first part is the quasiparticle excitations which form the\nStoner continuum, whose energy scales linearly with U. The second part encompasses the Norbcollective excitations,\nalso known as spin waves , which exist below the Stoner continuum. Contrary to the quasiparticle excitations, the spin\nwave energy reaches an upper bound determined by h(k) asUincreases. The spin waves further consist of Norb−113\ngapped modes and 1 gapless mode, where the latter corresponds to a Goldstone boson generated by spontaneous\nsymmetry breaking of the SU(2) group.\nLet us solve for the gapless mode at Q=0. First, note that Em(0) = 0 indicates that the gapless mode at Q=0\ncorresponds to another ground state of Hwith Sz\ntot=Smax−1. Since the ground states are given by S−\ntot|GS⟩, we can\nintuitively predict that |Q⟩will be given by S−\ntot|Ω⟩. Let ψ†\nnkσ=PNorb\nα|un(k)⟩αc†\nkασdenote the creation operator of\nthen-th band of the kinetic Hamiltonian. In terms of this operator, we obtain\nS−\ntot=X\niαc†\niα↓ciα↑=X\nkαc†\nkα↓ckα↑=X\nkαmnψ†\nmk↓ψnk↑·α⟨um(k)|un(k)⟩α=X\nknψ†\nnk↓ψnk↑, (S29)\nwhere the sum over α, m, n runs through 1 , . . . , N orb. Combined with Eq. (S7) and Eq. (S8), we obtain\nzkαn(0) =|un(k)⟩α (S30)\nfor the gapless mode. We verify that this does indeed correspond to the gapless mode by substituting into Eq. (S27)\nas follows.\nX\nn′k′βHSE\nkαn,k′βn′(0)|un′(k′)⟩β=X\nn′k′βh\b\nh(k)αβ−En(k)δαβ\t\nδkk′δnn′+U\nNc\bX\nqlα⟨ul(q)|ul(q)⟩αδkk′δnn′\n−α⟨un′(k′)|un(k)⟩α\t\nδαβi\n|un′(k′)⟩β\n=X\nβh(k)αβ|un(k)⟩β−En(k)|un(k)⟩α+U\nNcX\nqlα⟨ul(q)|ul(q)⟩α|un(k)⟩α\n−U\nNcX\nn′k′α⟨un′(k′)|un(k)⟩α|un′(k′)⟩α\n=En(k)|un(k)⟩α−En(k)|un(k)⟩α\n+U\nNc\u0010X\nqlα⟨ul(q)|ul(q)⟩α−X\nn′k′α⟨un′(k′)|un′(k′)⟩α\u0011\n|un(k)⟩α\n= 0, (S31)\nwhere n′, lare summed over 1 , . . . , N occ. Thus, Eq. (S30) solves the gapless magnon mode at Q=0.\nNote that spin excitations with negative energy imply that we have assumed the wrong ground state when calculating\nHSE. Conversely, for a saturated ferromagnet, the gapless magnon mode is the nondegenerate ground state of HSE(0),\nsince the other spin excitations are gapped. Furthermore, Em(Q̸=0)>0 is required, since any ground state of H\nmust have maximal total spin, which is impossible for an arbitrary Q. This implies that Q=0corresponds to the\nglobal minimum of Em(Q). Expanding around Γ, we obtain\nEm(Q) =E(0) +Qµ∂µEm(Q)|Q=0+QµQν\n2∂µ∂νEm(Q)|Q=0+O(Q3), (S32)\nwhere the repeated indices are summed over. Thus, the linear term must vanish, and the spin stiffness Dµν≡\n∂µ∂νEm(Q)|Q=0must be a positive definite tensor. Using the Feynman-Hellmann theorem, we show that the first\ncondition is always satisfied:\n∂µEm(Q)|Q=0=P\nkαnP\nk′βn′∂µHSE(Q)|Q=0z∗\nnαk(0)zn′βk′(0)P\nnαk|znαk(0)|2\n=1\nNtotX\nnk⟨un(k)|∂h(k+Q)\n∂Qµ\f\f\f\f\nQ=0|un(k)⟩\n=1\nNtotX\nnk⟨un(k)|∂µh(k)|un(k)⟩\n=1\nNtotX\nnk∂µEn(k)\n= 0, (S33)\nwhere the first and fourth equality comes from the Feynman-Hellman theorem, and the last equality comes from the\nfact that En(k) is periodic throughout the 1st Brillouin zone (BZ). Therefore, a necessary but not sufficient condition\nfor saturated ferromagnetism is that Dµνis a positive definite tensor.14\nS5. UPPER BOUND OF THE GAPLESS MAGNON MODE\nIn this Appendix, we calculate the upper bound of the gapless magnon mode assuming a saturated ferromagnetic\nground state.\nA. Upper bound from first order perturbation\nWe begin by proving that nondegenerate first order perturbation theory overestimates the true ground state energy.\nWe prove this statement in two different ways as follows.\n1. Intuitive argument from perturbation theory\nLet us briefly review some results from nondegenerate perturbation theory. Let\nH(λ) =H0+λH′, (S34)\nwhere H0andH(λ) are the unperturbed and perturbed Hamiltonians, respectively. The eigenstates and eigenvalues\nare given by\nH(λ)|n(λ)⟩=εn(λ)|n(λ)⟩, H 0|n⟩=εn|n⟩, (S35)\nwith n= 0,1, . . .andεn≤εn+1. For a nondegenerate energy level, one readily obtains\nεn(λ) =εn+λ∆(1)\nn+λ2∆(2)\nn+. . .\n=εn+λ⟨n|H′|n⟩+λ2X\nm̸=n|⟨n|H′|m⟩|2\nεn−εm+O(λ3), (S36)\nwhere ε(i)\nn=εn+λ∆(1)\nn+···+λi∆(i)\nnis the energy obtained up to i-th order correction. Applying Eq. (S36) to the\nground state, we see that ∆(2)\n0≤0. Assuming that the O(λ3) contribution is smaller than λ2∆(2)\n0, one can intuitively\nunderstand that first order perturbation provides an upper bound of the ground state energy.\n2. Rigorous proof of the statement\nIn this Appendix, we rigorously prove that applying nondegenerate perturbation to the ground state always gives an\nupper bound, even if (i) the ground state is degenerate, or the (ii) perturbation is much larger than the unperturbed\nHamiltonian, i.e., even in cases where the O(λ3) term cannot be ignored. This is done by noticing that for any\nnormalized |ψ⟩,\nε0(λ) =⟨0(λ)|H(λ)|0(λ)⟩ ≤ ⟨ψ|H(λ)|ψ⟩, (S37)\nwhich is just the definition of a ground state. Setting |ψ⟩=|0⟩, Eq. (S37) becomes\nε0(λ)≤ ⟨0|H0+λH′|0⟩=ε0+λ∆(1)\n0=ε(1)\n0, (S38)\nwhich proves the statement.\nB. Upper bound of the gapless mode\nWe apply first order perturbation to the gapless mode to obtain an upper bound of the energy. Starting from\nEq. (9) and Eq. (S30), we obtain\nE(1)\nm(Q) =1\nNtotNcX\nkNoccX\nn⟨un(k)|h(k+Q)−h(k)|un(k)⟩. (S39)15\nUsing the results of Sec. S4, the leading order expansion in Qµreads\nE(1)\nm(Q) =1\n2QµQν∂µ∂νEm(Q)|Q=0\n=QµQν\n2NtotNcX\nkNoccX\nn⟨un(k)|∂2h(k+Q)\n∂Qµ∂Qν\f\f\f\f\nQ=0|un(k)⟩\n=QµQν\n2NtotNcX\nkNoccX\nn⟨un(k)|∂µ∂νh(k)|un(k)⟩. (S40)\nInserting a resolution of identity, we obtain\nE(1)\nm(Q) =QµQν\n2NtotNcX\nkNoccX\nnNorbX\nm⟨un(k)|h\n∂µ∂ν|um(k)⟩⟨um(k)|Em(k)i\n|un(k)⟩\n=QµQν\n2NtotNcX\nkNoccX\nnNorbX\nm⟨un(k)|\nh\n|∂µ∂νum(k)⟩⟨um(k)|Em(k) +|∂µum(k)⟩⟨∂νum(k)|Em(k) +|∂µum(k)⟩⟨um(k)|∂νEm(k)\n+|∂νum(k)⟩⟨∂µum(k)|Em(k) +|um(k)⟩⟨∂µ∂νum(k)|Em(k) +|um(k)⟩⟨∂µum(k)|∂νEm(k)\n+|∂νum(k)⟩⟨um(k)|∂µEm(k) +|um(k)⟩⟨∂νum(k)|∂µEm(k) +|um(k)⟩⟨um(k)|∂µ∂νEm(k)i\n|un(k)⟩\n=QµQν\n2NtotNcX\nkNoccX\nnn\n⟨un(k)|∂µ∂νun(k)⟩En(k) +⟨un(k)|∂µun(k)⟩∂νEn(k) +⟨∂µ∂νun(k)|un(k)⟩En(k)\n+⟨∂µun(k)|un(k)⟩∂νEn(k) +⟨un(k)|∂νun(k)⟩∂µEn(k) +⟨∂νun(k)|un(k)⟩∂µEn(k) +∂µ∂νEn(k)\n+NorbX\nm\u0000\n⟨un(k)|∂µum(k)⟩⟨∂νum(k)|un(k)⟩+⟨un(k)|∂νum(k)⟩⟨∂µum(k)|un(k)⟩\u0001\nEm(k)o\n. (S41)\nTo simplify Eq. (S41), we use the following identities:\n⟨um(k)|un(k)⟩=δµν,⟨∂µum(k)|un(k)⟩+⟨um(k)|∂µun(k)⟩= 0,and\n⟨∂µ∂νum(k)|un(k)⟩+⟨um(k)|∂µ∂νun(k)⟩+⟨∂µum(k)|∂νun(k)⟩+⟨∂νum(k)|∂µun(k)⟩= 0. (S42)\nThen, we obtain\nE(1)\nm(Q) =QµQν\n2NtotNcX\nkNoccX\nnh\n∂µ∂νEn(k)−\u0000\n⟨∂µun(k)|∂νun(k)⟩+⟨∂νun(k)|∂µun(k)⟩\u0001\nEn(k)\n+NorbX\nm\u0000\n⟨∂µum(k)|un(k)⟩⟨un(k)|∂νum(k)⟩+⟨∂νum(k)|un(k)⟩⟨un(k)|∂µum(k)⟩)Em(k)i\n=QµQν\n2NtotNcX\nkNoccX\nnh\n∂µ∂νEn(k) +NorbX\nm\u0000\n⟨∂µum(k)|un(k)⟩⟨un(k)|∂νum(k)⟩+µ↔ν\u0001\nEm(k)\n−NorbX\nm\u0000\n⟨∂µun(k)|um(k)⟩⟨um(k)|∂νun(k)⟩+⟨∂νun(k)|um(k)⟩⟨um(k)|∂µun(k)⟩\u0001\nEn(k)i\n=QµQν\n2NtotNcX\nkNoccX\nnh\n∂µ∂νEn(k) +NorbX\nm\u0000\n⟨∂µum(k)|un(k)⟩⟨un(k)|∂νum(k)⟩+µ↔ν\u0001\nEm(k)\n−NorbX\nm\u0000\n⟨un(k)|∂µum(k)⟩⟨∂νum(k)|un(k)⟩+⟨un(k)|∂νum(k)⟩⟨∂µum(k)|un(k)⟩\u0001\nEn(k)i\n=QµQν\n2NtotNcX\nkhNoccX\nn∂µ∂νEn(k) +NorbX\nmNoccX\nn(Em(k)−En(k))\u0000\n⟨∂µum(k)|un(k)⟩⟨un(k)|∂νum(k)⟩+µ↔ν\u0001i\n,\n(S43)16\nwhere we have inserted a resolution of identity to obtain the second equality.\nWe further simplify Eq. (S43) by introducing the fidelity tensor χmn\nµν(k)≡ ⟨∂µum(k)|un(k)⟩⟨un(k)|∂νum(k)⟩, which\nis a gauge-invariant quantity that characterizes the interband transition probability [10, 30] as\nPk→k+δk\nm→n =|⟨un(k+δk)|um(k)⟩|2\n=⟨um(k)|un(k+δk)⟩⟨un(k+δk)|um(k)⟩\n=\u0000\n⟨um(k)|∂µun(k)⟩⟨∂νun(k)|um(k)⟩+µ↔ν\u0001\nδkµδkν+O(δk3\nµ)\n=\u0000\n⟨∂µum(k)|un(k)⟩⟨un(k)|∂νum(k)⟩+µ↔ν\u0001\nδkµδkν+O(δk3\nµ)\n= (χmn\nµν(k) +χmn\nνµ(k))δkµδkν+O(δk3\nµ). (S44)\nThen, we obtain\nE(1)\nm(Q) =QµQν\n2NtotNcX\nkhNoccX\nn∂µ∂νEn(k) +NorbX\nmNoccX\nn(Em(k)−En(k))(χmn\nµν(k) +χmn\nνµ(k))i\n=QµQν\n2NtotNcX\nkhNoccX\nn∂µ∂νEn(k) + 2NorbX\nmNoccX\nn(Em(k)−En(k))χmn\nµν(k)i\n, (S45)\nwhere we have interchanged the summation indices µandνof the last term to obtain the second equality. We further\nnote that χmn\nµν(k) =χnm\nνµ(k). Then, the terms with m∈ {1, . . . , N occ}in the second summation become 0 from the\nfollowing relation:\ndX\nµνNoccX\nmNoccX\nn(Em(k)−En(k))χmn\nµν(k)QµQν\n=dX\nµνNoccX\nmNoccX\nn(En(k)−Em(k))χnm\nνµ(k)QµQν\n=−dX\nµνNoccX\nmNoccX\nn(Em(k)−En(k))χmn\nµν(k)QµQν. (S46)\nThus, we finally arrive at\nE(1)\nm(Q) =QµQν\n2NtotNcX\nkhNoccX\nn∂µ∂νEn(k) + 2NorbX\nm>N occNoccX\nn(Em(k)−En(k))χmn\nµν(k)i\n. (S47)\nC. Upper bound of the spin stiffness\nIn this Appendix, we prove that\nEm(Q)|U<∞≤ Em(Q)|U→∞≤ E(1)\nm(Q), (S48)\nimplies\nDµµ(U <∞)≤Dµµ(U→ ∞ )≤D(1)\nµµ. (S49)\nThe proof follows directly in the following way. Consider two smooth analytic functions fandg, where f(0) =g(0) = 0\nandf(Q)≤g(Q). Let F(Q) =g(Q)−f(Q)≥0, with F(0) = 0. This implies that Hessian of F(Q) defined by\nDµν≡∂2F\n∂Qµ∂Qνis positive semidefinite at Q=0. This implies that the diagonal entries are greater than or equal to\n0. Applying this to Eq. (S48) finishes the proof.\nS6. NUMERICAL CALCULATIONS ON THE NO-GO THEOREM\nIn this Appendix, we demonstrate the validity of the no-go theorem by calculating the magnon dispersion of\nhalf-filled models. Fig. S1(a) describes the magnon spectrum of the NN kagome lattice at half filling. The magnon17\nbands with negative energy indicate the instability of saturated ferromagnetism. Fig. S1(b) illustrates a representative\nexample of a half-filled system with a single orbital, which is obtained from a 2D square lattice. To show the generality\nof the no-go theorem, we introduce arbitrary complex hopping amplitudes up to next-nearest-neighbor (NNN), and\nbreak every symmetry other than SU(2) symmetry. The noninteracting Hamiltonian is given by\nh(k) =t1eikx+t2eiky+v1ei(kx−ky)+v2ei(kx+ky)+c.c., (S50)\nwhere tiandvidenote the NN and NNN hopping amplitudes, respectively. The magnon spectrum with negative\nenergy shown in Fig. S1(b) clearly indicates that saturated ferromagnetism is prohibited.\na\nℰ(𝑸)\nQxQy\n0\n-0.05\n-0.1\n-4\n0\n4-202U=100\n0\n-0.5\n-1\n-π\n0\nπ-π0π\nQxQyℰ(𝑸) U=100b\nFIG. S1. Negative spin-wave energy of half-filled Hubbard models. (a) The kagome lattice with t= 1 and U= 100.\n(b) The square lattice model introduced in Eq. (S50) with t1=−1.2−0.7i, t2=−0.5 + 2.6i, v1=−1.7−0.4i, v2=−1.6 + 0.9i,\nandU= 100.\nS7. LOWER BOUNDS OF THE QUANTUM METRIC\nIn this Appendix, we elaborate on the lower bounds of the quantum metric briefly mentioned in the main text. In\n2D systems with Cnzsymmetry ( n= 3,4,6), the magnon energy must inherit such symmetry. Consequently, the spin\nstiffness has the form D(1)\nµν=δµν∆\nNcNoccPNc\nkTr[g(k)], where the trace is performed on the spatial indices. In this case, a\nlower bound of D(1)\nµνimposed by the Chern number of occupied bands, indicating that the band topology may stabilize\nsaturated ferromagnetism [13, 17]. Similarly, fragile topology or the existence of obstructed atomic insulators (OAIs)\ncan also give another lower bound that can be obtained from real space invariants developed recently [31, 32]. In fact,\nthe nontrivial metric model is an OAI, whose Wannier center is at1\n3(a1+a2). This shows that fragile topology or\nOAIs may also have an influence on saturated ferromagnetism.\nS8. VALIDITY OF THE RESULTS\nLet us comment on the relation between this work and previously studied theorems of saturated ferromagnetism.\nOur results do not contradict Stoner ferromagnetism in electron gases, since the electron dispersion is not periodic in\nsuch case. The same holds for Nagaoka’s ferromagnetism, which deals with the case where the number of electrons\nper unit cell is not an integer.\nNext, we comment on the assumptions used in this study. The first assumption we used in this was that the\nmagnon dispersion is correctly described by the variational principle. As elaborated in SM Sec. S3 and [41], this is\nnecessary due to the fact that general spin excitations cannot be exact eigenstates of the Hamiltonian. However,\nspin excitations can be exact eigenstates at half filling, and the spin excitation spectrum obtained from variational\nprinciple is exact in this case. The second assumption used is the validity of the mean-field decoupling of Eq. (4). As\nmentioned in SM Sec. S3 and [3], this is mathematically rigorous when the kinetic Hamiltonian is nondegenerate in\nthe ground state manifold, in which case the ground state can always be chosen as a single Slater determinant. This\nis the case for half-filled models or insulators, where the occupied states of Hkinare energetically separated from the\nunoccupied states. Thus, the no-go theorem of half-filled Hubbard models is mathematically rigorous to any level of\napproximation. Also, the results obtained for ferromagnetic insulators are justified in that the variational principle\nmust be used to define spin excitations."    },    {        "title": "1108.4807v1.Ferromagnetism_and_Superconductivity_in_Uranium_Compounds.pdf",        "content": "arXiv:1108.4807v1  [cond-mat.str-el]  24 Aug 2011Typeset with jpsj3.cls <ver.1.1> Full Paper\nFerromagnetism and Superconductivity in Uranium Compound s\nDaiAoki1∗and Jacques Flouquet1†\n1INAC/SPSMS, CEA-Grenoble, 17 rue des Martyrs, 38054 Grenob le, France\nRecentadvancesonferromagnetic superconductors,UGe 2,URhGeandUCoGearepresented.\nThe superconductivity (SC) peacefully coexists with the fe rromagnetism (FM), forming the\nspin-triplet state of Cooper pairs. The striking new phenom ena, such as SC reinforced by the\nmagnetic ﬁeld, are associated with Ising-type ferromagnet ic ﬂuctuations. A variety of ferromag-\nnetic ordered moments between UGe 2, URhGe and UCoGe aﬀords to understand the relation\nbetween FM, tricriticality and SC.\nKEYWORDS: ferromagnetism, superconductivity, spin-trip let state, magnetic ﬂuctuation, UGe 2, URhGe,\nUCoGe\n1. Introduction\nTheferromagnetismhadbeenthoughttobeantagonis-\ntic to superconductivity (SC) in the framework of singlet\ns-wave pairing up to the discovery of new class of ma-\nterials like the Chevrel phase compounds (REMo 6Se8,\nRE: rare earth).1)Antiferromagnetism (AF) obviously\ncoexists peacefully with SC in these RE intermetallic\ncompounds where the 4 felectrons are localized on the\nRE site and the magnetic interaction between RE ions\nare mediated indirectly by the conduction electrons via\nso-called RKKY (Ruderman, Kittel, Kasuya, Yosida) in-\nteraction. The simple image is that under the large su-\nperconducting coherence length ξ0, the average value of\nthe magnetization is zero as the AF periodicity is gener-\nally smaller than ξ0.2)In contrast, no microscopic coex-\nistence of FM and SC has been observed. In HoMo 6Se83)\nor ErRh 4B4,4)SC is observed in the intermediate tem-\nperature range approximately from 2 to 10K, but when\nit is cooled further, FM destroys SC at low tempera-\ntures. For example in ErRh 4B4, SC appears below 8 .7K.\nBecause of the internal conﬂicts between SC and FM,\nan intermediate phase exists in the narrow temperature\nrange between 0 .8K and 1K with the establishment of a\nmodulated magnetic structure which can be regarded as\na domain-like arrangement with the periodicity d≪ξ0.\nHowever,further cooling,FM isoverwhelmedand s-wave\nSC is destroyed as basically the energy gained by the RE\natoms due to the magnetic transition ( kBTCurie) is far\nhigher than the energy gained by the electrons as they\nform Cooper pairs at the SC transition (( kBTsc)2ρ(εF)),\nwhereρ(εF) is the density of states of the conduction\nelectrons at the Fermi level. In the case of AF–SC sys-\ntems, the unusual situation can happen under magnetic\nﬁeld, when the applied magnetic ﬁeld may cancel the\ntotal internal ﬁeld and thus the ﬁeld re-entrant SC is ob-\nserved as predicted by the so-called Jaccario-Peter com-\npensation eﬀect.5)This was ﬁrst observed in the Chevrel\nphase compound6)and recently in the organic systems\nλ(BETS) 2FeCl4.7,8)\n∗E-mail address: dai.aoki@cea.fr\n†J.F. is also “directeur de recherche ´ em´ erite” in CNRS1.1 Unconventional superconductivity\nIn the previous case, the electrons which are responsi-\nble for the magnetism are basically localized. In strongly\ncorrelated electron systems such as heavy fermion com-\npounds, the high Tccuprates and the new Fe-pnictide\nfamilies, at ﬁrst approximation, the electrons must be\nregarded as itinerant and thus the interplay between\nmagnetism and SC will be strong.9)Furthermore, the\nimportance of the Coulomb repulsion between the elec-\ntrons wipes out the possibility of s-wavepairingbased on\nthe BCS theory because the attractive force mediated by\nphonon cannot overcome the strong Coulomb repulsive\nforce. This situation opens the opportunity to search for\nunconventional superconductivity with d-wave or p-wave\npairings possessing ﬁnite angular momentum leading to\nnovelSC propertiessuchas anisotropicSC gapand order\nparameters with speciﬁc temperature and magnetic ﬁeld\nresponse. Here, an important reference is the triplet p-\nwave superﬂuidity of the Fermi liquid3He10)with exotic\nAphase and Bphase; the former one characterized by\nthe so-called equal spin pairing with separation between\n↑↑and↓↓spin carriers. Magnetic ﬁeld can even give rise\nto two successive superﬂuid transition. In contrast to the\ncase of the uranium ferromagnetic superconductors, the\nliquid phase of3He never reach a ferromagnetic insta-\nbility as shown in the weak pressure dependence of i) its\nLandau parameter Fa\n0describingthe enhancement of the\nsusceptibility by comparison to the speciﬁc heat and ii)\nits weak Gr¨ uneisen parameter11)\nThe great advantage of the heavy fermion compounds\nbased on the presence of 4 f(cerium, ytterbium) or 5 f\n(uranium, neptunium, plutonium) electrons, which are\nquite ready to become magnetic, is that moderate pres-\nsure and even magnetic ﬁeld can drive them to mag-\nnetic instability i.e. from long range magnetic order (AF\nor FM) to paramagnetic (PM) ground state at a criti-\ncal pressure Pc. As now the Cooper pair mechanism is\nlinked to the electron correlation itself, SC will be often\nobserved in a dome shape centered around Pc, where the\nmagnetic ﬂuctuation is enhanced.\nStarting with the discovery of CeCu 2Si2superconduc-\ntivity in 197912)there are now many macroscopic as well\nas microscopic evidences of d-wavespin singlet supercon-\n12 J. Phys. Soc. Jpn. Full Paper D. Aoki and J. Flouquet\nductivity in the heavy fermion compounds interplaying\nbetween SC and AF.\nThe coexistence of FM and SC was ﬁrst discovered in\nUGe213)under pressure in 2000, almost two decades af-\nter the discovery of SC in CeCu 2Si2. Soon afterward, the\nSC was found in the weak ferromagnet URhGe for the\nﬁrst time at ambient pressure.14)Recently UCoGe with\nidentical crystal structure of URhGe was found to be a\nferromagnetic superconductor, as well.15)In all of these\ncompounds, Tscis lower than TCurie, indicating that SC\nphase exists in the FM phase, which is contrary to the\nprevious case such as ErRh 4B4or Chevrel phase com-\npounds where Tscis higher than TCurie. Furthermore, the\nordered moments of uranium ferromagnetic supercon-\nductorsaremuchlowerthanthoseexpected fromthefree\nuranium ion. Therefore 5 felectrons are naively believed\nto be itinerant. To date, all the ferromagnetic supercon-\nductors are uranium compounds. The well-known weak\nferromagnet ZrZn 2was ﬁrst reported to reveal SC,16)\nhowever after careful sample preparations and charac-\nterizations, SC was found to be extrinsic most likely\ndue to the Zr alloys on the surface.17)SC is observed in\nthe ferromagnet UIr with non-inversion symmetry of the\ncrystal structure in FM3 phase in narrow pressure range\n(2.6/lessorsimilarP/lessorsimilar2.8GPa) with the maximum Tsc∼0.15K.\nThe bulk SC has not been established yet. The upper\ncritical ﬁeld of SC is quite small ( ∼0.026T) compared\nto those for above mentioned three ferromagnetic super-\nconductors.\nIn this paper, ﬁrst we review experimental results of\nthree ferromagnetic superconductors UGe 2, URhGe and\nUCoGe. Next we describe some theoretical views for FM\nand SC. Finally the conclusion and remarks are given. A\nvery recent complementary our review of ferromagnetic\nsuperconductors can be found in Ref. 18\n2. UGe 2–the ﬁrst ferromagnetic superconduc-\ntor: superconductivity and phase diagram\nUGe2crystallizes in the orthorhombic structure, as\nshown in Fig. 1.19)The U zig-zagchain with the distance\nof the next nearest neighbor dU-U= 3.85˚A is formed\nalonga-axis,which is similarto α-U with CDW. The FM\ntransition had been observed at TCurie= 52K20)and the\nordered moment is relatively large, M0∼1.5µB. The\nproperties of UGe 2, together with URhGe and UCoGe\nare summarized in Table 2. The magnetic moment is di-\nrected along a-axis. With increasing pressure TCuriecol-\nlapses and ﬁnally PM ground state is realized above the\ncritical pressure Pc∼1.5GPa.21)Surprisingly, SC ap-\npears around 1 .2GPa with Tsc∼0.7K as a maximum.\nAsshownin Fig.2,atthispressure TCurie∼35Kismuch\nhigher than TscandM0is also large ( ∼1µB).\nBothTCurieandM0collapse at Pc. Complementary\nmeasurements23,24)show later that the maxima of Tsc\ncorresponds to the pressure just at Px∼1.2GPa where\nthe system switches from largemoment ( M0∼1.5µB) at\nlow pressure phase (FM2) to small moment ( M0∼1µB)\nat high pressure phase (FM1), through a ﬁrst order tran-\nsition. The transition from FM1 to PM is also associ-\nated with the ﬁrst order transition at Pcwith an abruptb\nUGe\nUGe2 URhGe, UCoGecac\nU\nRh or CoGe\nba\nFig. 1. Crystal structure of UGe 2, URhGe and UCoGe. The ar-\nrows on the U site denote the direction of the moment.\n60\n50\n40\n30\n20\n10\n0T (K)\n2.0 1.5 1.0 0.5 0\nP (GPa)5 x TSCTCurie\nTCP\nCEP\nPcFM2FM1\nPxPM\nSC\nFig. 2. (Color online) Temperature–pressure phase diagram of\nUGe2. FM1, FM2 and PM represent the ferromagnetic state\nwith large moment ( ∼1.5muB), the ferromagnetic state with\nsmall moment ( ∼1µB) paramagnetic state, respectively. Above\nthe pressure of the tricritical point (TCP), the ﬁrst order f er-\nromagnetic transition is observed. Below the pressure of cr it-\nical end point for Tx, the crossover occurs between FM1 and\nFM2.13,22,23)\ndrop ofsublattice magnetization (∆ M0∼0.8µB).25)Ev-\nidences for homogeneous coexistence of FM and SC at\nP∼Pxwere given by the persistence of FM in the\nSC phase observedin neutron diﬀraction experiments,26)\nthe temperature dependence of nuclear spin-lattice re-\nlaxation rate in NQR measurements (see Fig. 3)27)and\nthe speciﬁc heat jump at Tsc.28)Open question is how\nthe homogeneous coexistence of FM and SC is realized\nbelow and above Px. Figure 4 shows the pressure varia-\ntion ofTscobtained by the resistivity measurements and\nthe ∆C/γTscby the speciﬁc heat measurements.28,29)\nTscshows the maximum at Px. The speciﬁc heat jump is\nmuch smaller than the value expected for weak coupling\nBCS scheme. Furthermore the residual Sommerfeld co-\neﬃcient ( γ-value) is quite large, 70% of γ-value in theJ. Phys. Soc. Jpn. Full Paper D. Aoki and J. Flouquet 3\nTable I. Characteristic properties of UGe 2, URhGe and UCoGe.\n(dU–U: distance of the ﬁrst nearest neighbor of U atom, M0:\nordered moment, Hint: internal ﬁeld associated with M0,Pc:\ncritical pressure between FM and PM, Ha,b,c\nc2: upper critical ﬁeld\nforH/bardbla,b,c -axis)†the values of Hc2in UGe 2are at∼1.2 GPa.\nUGe2 URhGe UCoGe\nStructure Ortho. Ortho. Ortho.\nSpace group Cmmm Pnma Pnma\ndU–U(˚A) 3.85 3.50 3.48\nTCurie (K) 52 9.5 ∼3\nM0(µB) 1.48 0.4 ∼0.05\nMag. easy-axis a c c\nHint(T) 0.28 0.08 0.01\nγ(mJ/K2mol) 34 160 55\nPc(GPa) 1.5 <0∼1.2\nTsc(K) 0.8 0.26 0.7\n∆C/γTsc ∼0.3 0.6 0.7\nHa\nc2(T) 1.4†2.5 >30\nHb\nc2(T) 2.4†2 ∼18\nHc\nc2(T) 4.8†0.7 0.6\n0.010.111010010-310-210-1100101102103\nTemperature (K)1 / T1 (sec-1)\nTsc = 0.7 KTcurie = 31 K\n             polar\n2Δ/kBT = 3.6\nNres/N0=0.37SC1.2GPa\nSC1 / T T (sec   K  )1-1 -1 \nT\n1 0.1\nT (K)0.010.17.75 MHz\n8.50 MHz\n9.12 MHz1\nFig. 3. Temperature dependence of the spin-lattice relaxat ion\nrate 1/T1by NQR experiments at 1 .2 GPa in UGe 2, as an evi-\ndence of microscopic coexistence of FM and SC.27)\nnormal state, in spite of very high quality single crystal.\nThis might be related with the large sublattice moment\nand the self-induced vortex state. Another possible rea-\nson is the ﬁrst order transition between FM1 and FM2,\nwhich can induce the phase separation of FM1 and FM2\ndue to the the small pressuregradientin the pressurecell\nas well as in the the sample.\nAnother tool to modify the PxandPcboundary is to\napply the magnetic ﬁeld along the magnetization easy-\naxis (a-axis). At the pressure of FM1 phase at zero ﬁeld\n(Px< P < P c), the FM2 phase is recovered through the\nmetamagnetic transition at H=Hx. In the PM phase,\njustabove Pc,acascadeofﬁrstordermetamagnetictran-\nsition occurs at Hc(from PM to FM1) and Hx(from\nFig. 4. Pressure variation of Tscby resistivity measurements and\nthe ∆C/γTscby speciﬁc heat measurements in UGe 2.29)\nFM1 to FM2). Careful studies for H∝bardblM0(a-axis) was\nrecently realized in order to clarify the FM-PM border of\nUGe2asitisanexcellentexamplefortricriticalityinitin-\nerant ferromagnet.30,31)Under pressure, at H= 0, the\nphase transition changes from second order to ﬁrst order\nat a tricritical point TTCP= 24K, PTCP∼1.42GPa,\nwhich is very close to Pc= 1.49GPa.22,31)Under mag-\nnetic ﬁelds above Pc, the ﬁrst order metamagnetic tran-\nsition will terminate at a critical end point ( PQCEP∼\n3.5GPa,HQCEP∼16T, see Fig. 5)30,32)For the tran-\nsition between FM1 and FM2, the critical end point at\nH= 0 is located at Tx\nCEP∼7K and Px\nCEP∼1.16GPa,\nwhich is very near the pressure where SC dome is sup-\npressed. Below Px\nCEP,Txis a crossoverbetween FM1 and\nFM2.\nFig. 5. (Color online) Temperature-pressure-ﬁeld phase di agram\nof UGe 2forH/bardblM0(a-axis).30,32)\nA striking point on the SC phase is that the temper-\nature dependence of the superconducting upper critical4 J. Phys. Soc. Jpn. Full Paper D. Aoki and J. Flouquet\nﬁeldHc2forH∝bardbla-axis (easy magnetization axis) indi-\ncates the ﬁeld-enhanced SC phase when the pressure is\ntuned just above Px, as shown in Fig. 6.33)This pecu-\nliar shape of Hc2curve is associated with the crossing of\nthe metamagnetic transition at Hx. If there is no doubt\non the strong “S”-shaped curvature of Hc2(T), the open\nquestion will be whether this phenomena is characteris-\ntic of FM2 phase or whether it is a combined eﬀect of the\nvolume expansion in the FM1 phase through the positive\nmetamagnetic feedback, leading to an increase of Tsc(P)\njust right at the maximum at Px.\n5\n4\n3\n2\n1\n0Hc2 (T)\n0.50.40.30.20.10\nT (K)UGe2 \nH // a-axis \n1.35 GPa\nFM1 SCFM2\nFig. 6. (Color online) Temperature dependence of Hc2forH/bardbla-\naxis in UGe 2at 1.35 GPa, which is just above Px. The metam-\nagnetic transition is detected at Hxbetween FM1 and FM2.33)\nThe de Haas-van Alphen (dHvA) experiments under\npressure reveal that Fermi surfaces between FM2, FM1\nand PM are quite diﬀerent each other. For H∝bardblb-axis,\nwhere FM2, FM1 and PM phases are not aﬀected by the\nmagneticﬁelds,dHvAbranchesofFM2phasedisappears\nin FM1 and new branches exhibit in FM2.34,35)In PM\nphase, completely new branches are observed again. The\ncyclotron eﬀective mass graduallyincreases with increas-\ning pressure up to Pxand in PM phase quite large eﬀec-\ntive masses ranging from 20 to 60 are detected, in agree-\nment with the pressure dependence of the γ-value.28)For\nH∝bardbla-axis, the ﬁeld re-entrant FM1 and FM2 phases oc-\ncurs, as shown in Fig. 5, thus the results are more com-\nplicated.36,37)Nevertheless, the drastic change of Fermi\nsurfacesisdetectedbycrossingFM1,FM2andPMphase\nboundary. The cyclotron eﬀective mass increases, ap-\nproaching to Hx. The change of Fermi surface is also\nfound for H∝bardblc-axis, as well.38)\nAn interesting theoretical scenario proposed for the\nweak itinerant ferromagnet ZrZn 2is the quantum meta-\nmagnetic transition associated with the topological\nchange of Fermi surfaces, as proposed for a Lifshitz tran-\nsition.39)The Fermi surface study under pressure can\nbe found in Ref. 40. In UGe 2, there is an evidence by\nthe combined resistivity and Hall eﬀect measurements32)\nthat the topological change of Fermi surface from PM toFM1canbetuned bythepressureandﬁeld,followingthe\nwing-shaped ( T,P,H) phase diagram predicted in “con-\nventional” spin ﬂuctuation approaches of FM-QCEP.\n3. URhGe: a ferromagnetic superconductor at\nambient pressure and ﬁeld-reentrant SC\nAlthough the discovery of pressure induced SC in\nUGe2can be a major breakthrough, the ambient pres-\nsure case provides much variety of experimental meth-\nods which goes deep inside the understanding of un-\nconventional SC. The discovery of SC at ambient pres-\nsure in the weak ferromagnet URhGe with Tsc= 0.26K,\nTCurie= 9.5K and M0= 0.4µBopened the new oppor-\ntunities.14)\nThe properties of URhGe is summarized in Table 2.\nThe crystal structure is orthorhombic TiNiSi-type, as\nshown in Fig. 1. The U atom forms the zig-zag chain\nalonga-axis with the distance of dU-U= 3.50˚A, which\nis close to the so-called Hill limit associated with the\ndirect overlap of 5 f-wave function.41)Figure 7 shows\ntheγ-value and the magnetic ordered temperature as\na function of the distance of the next nearest neighbor\non U atom dU-Uin UTGe (T: transition element) fam-\nily.42)The systematic variation can be seen. The PM\nground state is realized for the small dU-U, while the\nlargedU-Uinduces the magnetic order with the large\nmoment. URhGe as well as UCoGe are located on the\nboundary between PM and AF with large moments ac-\ncompanied with long range magnetic ordering. The max-\nimum and moderately enhanced γ-values are observed in\nURhGe and UCoGe. The similar trend is also known in\nUX3and NpX 3(X: group 13 and 14 elements).43,44)\nFM AF PM250\n200\n150\n100\n50\n0γ (mJ/K2mol)\n3.6 3.5 3.4\ndU-U ( Å )70\n60\n50\n40\n30\n20\n10\n0\nTN / Curie (K)\nFig. 7. (Color online) γ-value of speciﬁc heat and the magnetic\nordered temperature as a function of the distance of the next\nnearest neighbor on U atom for UTGe (T: transition element)\nfamily. URuGe is paramagnet, UCoGe and URhGe are ferromag-\nnets, the other UTGe are antiferromagnets.\nIn URhGe, TCurieis much lower than than the band\nwidth,TCurie≪W. In the speciﬁc heat measurements,\nit is estimated that the contribution to the γ-value from\nthe ﬁt above TCurieamounts to γB∼110mJ/K2mol,J. Phys. Soc. Jpn. Full Paper D. Aoki and J. Flouquet 5\nwhile the FM ﬂuctuation contributes ∼50mJ/K2mol to\nthe total γ-value (160mJ /K2mol).45)Finally at low tem-\nperatures the relatively enhanced γ-value 160mJ /K2mol\nis achieved. Contrary to UGe 2, with increasing pressure,\nTCurieincreases monotonously at least up to 12GPa, as\nshown in Fig. 8, indicating the system is pushed far from\nthe FM instability.46,47)Correspondingly the positive\n∂TCurie/∂Pis obtained from the Ehrenfest relation.48)\nThe decrease of Tscwith pressure is associated with the\ndecrease of m∗∗, which also implies that the system is\ngetting away from the FM instability.\n1.0\n0.5\n0Tsc (K)\n543210\nP (GPa)15\n10\n5\n0\nTCurie (K)\nURhGeTscTCurie\nFig. 8. (Color online) Pressure dependence of TCurie andTscin\nURhGe.46,47)\nThe new feature is that the slope of magnetization\ncurve∂M/∂H for the ﬁeld along the hard magnetization\naxis (b-axis) is larger than that along the easy axis ( c-\naxis), leading to the spin reorientation at HR∼12T for\nH∝bardblb-axis, as shown in Fig. 9. The magnetization curve\nforH∝bardblb-axis displays that the extrapolation of M(H)\nfromH > H RtoH= 0 gives a ﬁnite value, ( ∼0.15µB).\nThis conﬁrms that the system remains in FM side with\nthe decrease of M0from 0.5µBto 0.15µB, as if the ﬁeld\nsweep along b-axis leads to approach the FM instability.\nThis spin reorientation gives rise to the ﬁeld reentrant\nSC (RSC) around HRat low temperatures.49)Figure 10\nshowsthetemperature-ﬁeldphasediagramfor H∝bardblb-axis\nin URhGe.50)Applying ﬁeld, SC is suppressed around\n2T, further increasing ﬁeld, RSC appears approximately\nbetween 11T and 14T. Interestingly the maximum of\nTscfor RSC phase ( ≈0.42K) is higher than Tscfor low\nﬁeld SC phase ( ≈0.26K). At high temperatures, TCurie\ndecreases with increasing ﬁelds as it is phenomenologi-\ncally described by means of Landau free energy.51)The\nreducedTCurieis connected to the spin reorientationﬁeld\nHRat low temperatures. The RSC as well as low ﬁeld\nSC are very sensitive to the sample quality, indicating\nthat both SCs are unconventional.50)When the ﬁeld is\nslightlytilted to the magnetizationeasy-axis( c-axis),the\nRSC phase immediately shifts to higher ﬁelds and col-\nlapses.52,53)This is attributed to the rapid suppression\nof longitudinal magnetic ﬂuctuation by tilting ﬁeld. On0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0M (µB/U)H // c-axis\nb-axis\na-axisURhGe\nT → 0 K\n0.20\n0.15\n0.10\n0.05\n0dM/dH (µB//T)\n15 10 5 0\nH (T)H // b-axisc-axis\na-axis\nFig. 9. (Color online) Magnetization curves and ﬁeld deriva tive\nof magnetization in URhGe.45)\ntheotherhand,RSCisveryrobustwhentheﬁeldistilted\nfrombtoa-axis, i.e. maintaining the hard-magnetization\naxis.HRincreases as a function of 1 /cosθ, whereθis\nthe ﬁeld angle from btoa-axis. Accompanying with the\nincrease of HR, RSC is sustained even above 28T.52)\n0.1110T (K)\n15 10 5 0\nH (T)URhGe\nH // b-axis\nRSC SCHR\nFMTCurie\n10\n5\n0ρ (µΩ⋅cm)\n151050\nH (T)\nFig. 10. (Color online) Temperature-ﬁeld phase diagram for H/bardbl\nb-axis in URhGe. SC, RSC and FM denote superconductivity,\nreentrant superconductivity and ferromagnetism, respect ively.\nThe inset shows the ﬁeld dependence of resistivity at low tem -\nperatures ( ≈80 mK).50)It is noted that the ﬁeld range of RSC\nis very sensitive against the small mis-orientation to c-axis.6 J. Phys. Soc. Jpn. Full Paper D. Aoki and J. Flouquet\nThe origin of RSC is the enhancement of eﬀective\nmassm∗aroundHRwhich is ascribed by approach-\ning FM instability when a transverse ﬁeld is applied\nin this Ising ferromagnets.45,50,54)Figure 11 shows the\nﬁeld dependence of γ-value obtained by the thermody-\nnamic Maxwell relation via temperature dependence of\nthe magnetization, i.e. ∂γ/∂H=∂2M/∂T2. The validity\nof this analysis via Maxwell relation was already demon-\nstrated for CeRu 2Si255,56)and CeCoIn 5.57)The inset\nshows the results of direct speciﬁc heat measurements\nforH∝bardblbandc-axis at 0 .4K. The similar results were\nalso obtained by the ﬁeld dependence of Acoeﬃcient of\nT2-term of resistivity based on the so-called Kadowaki-\nWoods relation.50,53)Theγ-value is enhanced around\nHRforH∝bardblb-axis, while it is suppressed for H∝bardblc-axis,\nindicatingthattheIsing-typeFMﬂuctuationisenhanced\naroundHRforH∝bardblb-axis.\n250\n200\n150\n100γ (mJ/K2mol)\n15 10 5 0\nH (T)URhGeH // b-axis\na-axis\nc-axisHR 200\n150\n100C/T (mJ/K2mol)\n1.00.50\nH/HR0.4 Kb-axis\nc-axis\nFig. 11. (Color online) Field dependence of γ-values obtained by\nmagnetization measurements via Maxwell relation. The init ial\nγ-value at 0 T is taken as 160 mJ /K2mol.45)The inset shows\nthe results of direct speciﬁc heat measurements at 0 .4 K.53)It\nis noted that the spin-reorientation ﬁeld HRon speciﬁc heat\nmeasurements slightly shifts to 15 .2 T, because of the small mis-\norientation to c-axis within two degrees.\nThe simple picture is that SC is related to the eﬀec-\ntive massofconductionelectron m∗which hastwomajor\ncontributions: the renormalized band mass mBand the\neﬀective mass gained through ferromagnetic or antifer-\nromagnetic correlations m∗∗, namely\nm∗=mB+m∗∗. (1)\nUsing McMillan-like formula, the superconducting tran-\nsition temperature Tscis described by\nTsc=T0exp(−m∗/m∗∗), (2)\nwhereT0is the same as characteristic cut oﬀ energy.\nSincem∗∗is strongly enhanced around HR,Tscunder\nﬁelds is enhanced as well. In ferromagnetic superconduc-\ntors, the formation of spin-triplet state with equal-spin\npairing is realized. Hc2is free from the Pauli limit basedon the spin-singlet state, instead Hc2is governed by the\norbital limit Horb. Since the superconducting coherence\nlengthξis described by ξ≈/planckover2pi1vF/kBTsc, we obtain a sim-\nple relation as Horb∼(m∗Tsc)2, wherevFis Fermi veloc-\nity. Ifm∗is enhanced, both TscandHorbincrease, and\nconsequently RSC is observed at high ﬁelds. It should\nbe noted that enhancement of Tscis aﬀected by mBand\nthe Fermi surface. To date, no drastic change of Fermi\nsurface is inferred at HRby thermopower measurements\nwhich will be published elsewhere.58)\nApplying pressure, RSC phase shifts to higher ﬁelds\nassociated with the increase of HR, and eventually dis-\nappears above ∼1.5GPa, as shown in Fig. 12.47)On the\nother hand, low-ﬁeld SC will survive above 3GPa. The\nsuppressionofboth RSC andlow-ﬁeldSC isexplained by\nthe decreaseofmassenhancement,Interestingly,the sim-\nilar behavior is observed at ambient pressure when the\nﬁeld is tilted to c-axis,53)where the longitudinal mag-\nnetic ﬂuctuation is suppressed due to the Ising-type fer-\nromagnetism.\n20\n15\n10\n5\n0H (T)\n3.0 2.0 1.0 0\nP (GPa)SCRSCURhGe\nH // b-axis\nT → 0 KHR\nFig. 12. (Color online) Pressure dependence of Hc2for low ﬁeld\nSC and the critical ﬁelds for RSC for H/bardblb-axis in URhGe. HR\ndenotes the spin reorientation ﬁeld.47)\nAt ambient pressure, Hc2of low ﬁeld SC exceeds the\nPauli limit for all three direction. From the anisotropy\nofHc2, the line node gap in the bc-plane is inferred,\nassuming the equal-spin pairing.,59)where the attrac-\ntive interaction between ↑↑electrons are described by\nVσσ′(k,k′)∝δ↑σδ↑σ′kak′\na, corresponding to an order pa-\nrameterka| ↑↑∝angbracketright.59,60)Itwasdemonstratedthatthisorder\nparameter will remain when the FM moment rotates in\nthebcplane.59,61)\n4. UCoGe: strong interplay between FM and SC\nA new breakthrough was given by the discovery that\nthe weak itinerant ferromagnet UCoGe ( TCurie∼3K,\nM0∼0.05µB) becomes SC at Tsc∼0.7K.15)Theγ-\nvalue is moderately enhanced as 55mJ /K2mol The crys-\ntal structure is identical to that of URhGe with or-\nthorhombic TiNiSi-type, as shown in Fig. 1 The distance\nof next nearest neighbor of U atom is slightly smallerJ. Phys. Soc. Jpn. Full Paper D. Aoki and J. Flouquet 7\nthan that of URhGe. The magnetic moment is directed\nalongc-axis, which is also identical to URhGe, however\nthe ordered moment ( ∼0.05µB) is much smaller that\nof URhGe. At high ﬁelds for H∝bardblc-axis, the magnetic\nmoment is induced on the Co site with antiparallel di-\nrection.62)The characteristic properties are summarized\nin Table 2. According to the band calculations based\non the 5 f-itinerant model with/without spin polariza-\ntion, the carrier number in PM state is small with semi-\nmetallic type Fermi surface, while the carrier number in-\ncreases in FM state, but is still small.63)The Shubnikov-\nde Haas experiments were carriedout and a small pocket\nFermi surface ( F∼1kT) was detected with large cy-\nclotron mass (25 m0),64)implying that UCoGe is a low\ncarrier system with heavy quasi-particles. This is also\nsupported by the large Seebeck coeﬃcient.58)These sit-\nuationsareresembletothoseofwell-knownsemi-metallic\nheavy fermion superconductor URu 2Si2.65–69)\nIn UCoGe the interplay between FM and SC is strong\nsinceTCurieis already close to Tsc. Figure 13 represents\nthe results ofresistivityand speciﬁc heat. Twoanomalies\nare clearly observed at TCurieandTscboth in resistivity\nand in speciﬁc heat. Contrary to URhGe, TCurieis sen-\nsitive to the sample quality, while Tsccan detected for\nthe poor quality sample with the residual resistivity ra-\ntio RRR = 3 at least by resistivity measurements. This\nmight be related with the fact that TCurieis of ﬁrst order\nand SC survives even above Pcin PM state, as described\nbelow.\n30\n25\n20\n15\n10\n5\n0ρ (µΩ⋅cm)\nUCoGe\nJ // c-axis\nRRR = 54TCurie\nTsc\n90\n80\n70\n60\n50\n40C/T (mJ/K2mol)\n543210\nT (K)(a)\n(b)\nFig. 13. (Color online) Temperature dependence of resistiv ity\nand speciﬁc heat in UCoGe.\nApplying the pressure, TCurieandTscare merged\naround the critical pressure Pc∼1GPa.70–72)AC sus-\nceptibility and AC calorimetry measurements clearly es-\ntablished that SC is very robust through Pc. The simpleidea is that when TCurieis close to Tsc, FM collapses via\nthe ﬁrst order transition and the associated volume dis-\ncontinuity gives rise to the system with comparable FM\nﬂuctuations and thus to comparable Tscon both sides\nofPc. The phase diagram shown in Fig. 14(a) is contra-\ndictory to the theoretical prediction near FM ﬂuctuation\nby Fay and Appel,73)assuming a second order quantum\ncritical point.73)A new theory through symmetry ap-\nproach has been proposed for the ( T,P) phase diagram\nof UCoGe.74)Experimentally the domain of coexistence\nbetween SC and FM with Tsc< TCuriehas not been de-\ntected.\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0T (K)UCoGe\nPM FM\nFM+SC SCTCurie\nTscPc\n10\n5\n0Hc2 (T)\n2.52.01.51.00.50\nPressure (GPa)H // a-axis\nb-axis\nc-axisT → 0 K(a)\n(b)\nFig. 14. (Color online) (a)Temperature–pressure phase dia -\ngram70–72)and (b)pressure dependence of Hc2extrapolated to\n0 K obtained by resistivity measurements down to 90 mK in\nUCoGe. It is noted that the values of Hc2foraandb-axis are\nreduced by comparison to those of perfectly aligned ﬁeld dir ec-\ntion. The mis-orientation to c-axis is estimated to be within two\ndegrees.\nPressure dependence of Hc2extrapolated to 0K is\nshown in Fig. 14(b). It should be noted that the values\nofHc2are reduced, compared to those for the perfectly\nﬁeld-aligned case, since the values of Hc2foraandb-axis\nare very sensitive to the ﬁeld mis-orientation to c-axis,\nas mentioned later. Nevertheless, measured Hc2curves\nfor all ﬁeld direction reveal almost linear increase with\nslight upward curvature with decreasing temperature for\nallthepressurerange(notshown),thuswecandetermine\nthe pressure dependence of Hc2at 0K. For aandb-axis,\nbroad maxima of Hc2are observed around Pc, associated8 J. Phys. Soc. Jpn. Full Paper D. Aoki and J. Flouquet\nwith the broad maximum of Tsc. On the other hand, Hc2\nforH∝bardblc-axis increases monotonously with pressure, in-\ndicatingthattheanisotropiesbetween aandcorbetween\nbandc, i.e.Ha\nc2/Hc\nc2andHb\nc2/Hc\nc2are reduced above Pc.\nThis may be due to the fact that longitudinal magnetic\nﬂuctuation is suppressed above Pc. The ﬁrst order na-\nture ofTCurieand the acute Hc2enhancement for H∝bardbla\nandb-axis impede the precise determination of Hc2as\na function of pressure at the moment. Another supple-\nment event might be the change of Fermi surface at the\nFM–PM transition.\nThe new striking point is that Hc2at ambient pres-\nsure is strongly enhanced when the ﬁeld is applied along\nthe hard magnetization axis ( aandb-axis), as shown in\nFig. 15(a).53,75)At 0,K,Hc2forH∝bardblbanda-axisreaches\nHb\nc2∼18T and Ha\nc2>30T, which considerably exceed\nthe Pauli limit estimated from Tsc∼0.6K. On the other\nhand,Hc2forH∝bardblc-axis is 0 .6T, which is comparable or\neven less than the Pauli limit. The acute enhanced Hc2\ncan be seen for H∝bardbla-axis, as shown in Fig. 15(b). The\nfact that Hc2isstronglydamped bytilting the ﬁeld angle\nslightlyto c-axiscannotbe explainedbythe conventional\neﬀective mass model associated with the Fermi surface\ntopology, but should be ascribed by the anisotropic mag-\nnetic ﬂuctuation. The huge anisotropy of Hc2including\nan “S”-shaped curve for b-axis is qualitatively explained\nby the anisotropic ﬁeld response of eﬀective mass. The\nﬁeld dependence of Acoeﬃcient of T2term of resistiv-\nity, which is linked to the γ-value and eﬀective mass by\nKadowaki-Woodsrelationassumingthe stronglocalﬂuc-\ntuation ( A∝γ2∝m∗2), shows that AforH∝bardblc-axis is\nsuppressed with ﬁeld as usual weak itinerant ferromag-\nnets, while AforH∝bardblbanda-axis remains at high value,\nin addition, AforH∝bardblb-axis reveals the maximum at\nﬁeld where the “S”-shaped Hc2is observed. The results\nare similar to those obtained in URhGe, as shown in\nFig. 11.\n25\n20\n15\n10\n5\n0Hc2 (T)\n1.0 0.5 0\nT / TscUCoGe\nH // b-axis\na-axis\nc-axis\n9060300\nField Angle (deg)a-axis c-axis0.09 K(a) (b)\nFig. 15. (Color online) (a)Temperature dependence of Hc2and\n(b)angular dependence of Hc2fromatoc-axis at 0 .09 K in\nUCoGe. Tscis∼0.65 K at 0 T .75)\nNMR and NQR experiments are very good probes tostudy magnetic ﬂuctuations associated with dynamic as\nwellasstaticsusceptibilities,because59Coisanexcellent\nnuclear in UCoGe. An evidence for the ﬁrst order tran-\nsition at TCurieat ambient pressure was given from the\nabrupt change of the resonant frequency below TCurie.76)\nThis implies that UCoGe is already above the tricritical\npoint. The nuclear spin-lattice relaxation rate 1 /T1mea-\nsurements microscopicallyconﬁrmthe coexistence ofFM\nand SC, and the strong Ising character of the ferromag-\nnetic ﬂuctuation.76)The dynamic susceptibility shows\nthe remarkable anisotropy, implying that the longitudi-\nnal FM magnetic ﬂuctuation is dominated.77)\nThe results of experiments on three ferromagnetic su-\nperconductors, UGe 2, URhGe and UCoGe conﬁrm that\nIsing-type FM with longitudinal ﬂuctuation mode is fa-\nvorable for SC, Up to now, there are no other cases of\ncoexistence of FM and SC despite the attempt to ﬁnd Ce\nbased heavy fermion compounds. Let us remark that SC\nnearAF criticalityseemstobe favoredbytransversespin\nﬂuctuations as it occurs in CeCu 2Si2, CePd 2Si2, Ce and\nPu-115systems and NpPd 5Al2,78–82)while for Ising-type\nAF system, such as CeRu 2Si2, no SC has been observed.\nIn contrast to URhGe, no spin reorientation is ex-\npected in UCoGe for H∝bardblb-axis as the slope of magneti-\nzationχbis smaller than χc. The key criteria will be the\nsizeoftheﬁeld induced magnetization Mbbycomparison\nto the ordered moment M0at zero ﬁeld. When the mag-\nnetic ﬁeld reaches HbwhereMbis comparable to M0,\nnamelyM0≈Mb=χbHb, the drastic change of eﬀective\nmass is expected. Table II summarizes the parameters of\nthree uranium ferromagnetic superconductors.\nTable II. Susceptibilities and characteristic ﬁelds of UGe 2,\nURhGe and UCoGe.\nχa χbχcHaHbHc\n(µB/T) (T)\nUGe2 0.006 0.0055 0.011 230 250 122\nURhGe 0.006 0.03 0.01 66 13 40\nUCoGe 0.0024 0.006 0.029 29 12 2.5\nAn interesting unique point in ferromagnetic super-\nconductors is that the internal ﬁeld associated with the\nordered moment M0is large compared to the expected\nvalue of superconducting lower critical ﬁeld Hc1. Thus\nspontaneous vortex state must be realized at zero ﬁeld.\nSo far there is no direct observation of the correspond-\ning vortex lattice, but clear marks are obtained by NQR\nmeasurements in UCoGe76)and the unusual initial slope\nofHc2in URhGe.59)\nAn interesting macroscopic observation is the modiﬁ-\ncation of the hysteresis loop of magnetization through\nthe SC transition, as shown in Fig. 16.57)Careful anal-\nysis of the DC magnetization shows that no Hc1exists\nat least for H∝bardblc-axis. The similar experiments can be\nfound in Ref. 83.\n5. Theoretical view\nThe ﬁrst prediction of triplet SC in metallic system\nnear FM instability was reported by Fay and Appel.73)J. Phys. Soc. Jpn. Full Paper D. Aoki and J. Flouquet 9\n-0.8-0.6-0.4-0.200.20.40.60.8M (emu/g)\n-150 0 150\nH (Oe)UCoGe\nH // c-axis\n0.50 K\n-150 0 150\nH (Oe)0.075 K\nFig. 16. (Color online) Hysteresis loops of magnetizations under\nvarious ﬁeld-sweep windows above and below TscforH/bardblc-axis\nin UCoGe.57)\nTscreveals two maxima both in PM and in FM phase.\nThe very slow energy ﬂuctuation gives rise to the pair-\nbreaking in the vicinity of Pc, whilep-wave SC in nearly\nFM itinerant system was already calculated by Layzer\nand Fay in 1971.84)Recent discussion for nearly ferro-\nmagnetic system can be found in Ref. 85–87. A more\ncomplicated treatment wipes out the collapse of Tscat\nPc, instead only a minimum of Tscwill occur. However,\nthe experimental observationforUCoGe is that Tschasa\nbroad maximum at Pcaccompanied with the ﬁrst order\ntransition of TCurie, instead of second order (see Fig. 14).\nDiscussiononthecoexistenceofSCandFMcanbefound\nin Ref. 88 An interesting point is also that in the FM\nstateTscfor the majority spin ( ↑) diﬀers from that for\nthe minority spin ( ↓):T↓↓> T↑↑. Thus SC in the FM\nregion is a two-band superconductor with the possibility\nthat only one type of band is gapped.\nThe possible order parameters in FM phase have been\nclassiﬁed on general symmetry arguments.89–91)On the\nbasis of the report of SC in the cubic ZrZn 2,16)which\nwas found to be extrinsic afterward,17)it was predicted\nthat the gap nodes will change when the magnetization\nis rotated by magnetic ﬁeld.92)It was also proposed that\nin the weak Heisenberg ferromagnet Tscwill be enhanced\non the FM side due to the development of transverse\nmagnetic ﬂuctuation. To date, the evidences of SC in\nFM materials is limited to Ising-type FM.\nFor UGe 2, the striking point is that Tschas maximum\natPx. It was proposed that the CDW/SDW ﬂuctuation\nmay occur,93)but up to now no extra superstructure was\nfound experimentally. Phenomenological model was de-\nveloped assuminga twin-peak in the electronic density of\nstates.94)It was even proposed that for UGe 2close toPx\nasM0isstillhighandthusthemagnetismisbasedonthe\nstrongly localized case that the coupling of two electrons\nvia localized spin can be attractive,95)and demonstrated\nthat this s-waveattractionholdsforthe wholeFermi sur-\nface.96)However, this hypothesis is questionable because\nthe Fermi surfaces calculations clearly show that 5 felec-\ntron must be considered as itinerant and this character\nis strongly reinforced in FM1 phase.Interesting new features are predicted as the SC or-\nder parameter is linked to M0with the possibility that\ndomain walls play a role of weak links.90,97)It was also\nstressed that SC may appear locally at a domain wall\nnot inside a magnetic domain.98)\nRecently it was stressed in good agreement with\nthe study made on URhGe that when the ﬁeld is ap-\nplied along b-axis (⊥M0),TCurie(H) will decrease in a\nquadratic dependence (∆ TCurie∝ −H2).51)Analysis of\nthe unusual temperature dependence and the anisotropy\nofHc2(T) was made close to Pc. Due to the long range\nnatureoftheFMinteraction,non-analyticcorrectioncan\nenhance the SC transition.99)The position of nodes with\nrespecttothemagneticﬁelddirectioncanexplaintheun-\nusual angular dependence of Hc2. For UCoGe it was pro-\nposed that the large anisotropy between Hc2forH∝bardbla-\naxis (Ha\nc2) andHc\nc2, namely Ha\nc2≫Hc\nc2implies a point\nnode gap, not an horizontal line node gap, with respect\nto the vertical M0direction.\n6. Conclusion\nDiscovery of three uranium ferromagnetic supercon-\nductors has open the interesting frontiers for the inter-\nplay between two major ground states of condensed mat-\nter, FM and SC. Experimentally, a key challenge is to\ndiscover an ideal simple case, such as Ce-115 systems for\nthe interplayofAF and SC, where the high quality single\ncrystals with large size can be obtained.\nUGe2is unfortunately not an ideal example for SC\ndespitetheavailabilityofhighqualitysinglecrystal,since\nthe external pressure is required and there is not the\nsole transition from FM to PM, but also the switch from\nFM1 to FM2. Furthermore the changes of ground state\noccurs at marked ﬁrst order transitions. It is an excellent\nexample to study the tricriticality and the properties at\nQCEP.\nURhGe and UCoGe suﬀer from the unavailability of\nlarge and high quality single crystals. Thus, careful tests\nare required to be sure that the measurements character-\nize the bulks homogeneous SC. At least clear new phe-\nnomena have emerged such as the link for ﬁeld-reentrant\nSC in transverse ﬁeld response with respect to the easy\nmagnetization axis in URhGe and UCoGe.\nAcknowledgments\nWe thank J. P. Brison, A. Buzdin, S. Fujimoto, Y.\nHaga,F. Hardy,H. Harima,K.Hasselbach,E. Hassinger,\nL. Howald, K. Ishida, S. Kambe, W. Knafo, G. Knebel,\nH. Kotegawa, L. Malone, T. D. Matsuda, C. Meingast,\nV. Michal,V.Mineev, A.Miyake,K.Miyake,C.Paulsen,\nS. Raymond, R. Settai, I. Sheikin, Y. Tada, V. Taufour,\nfor fruitful discussion. This work was supported by ERC\nstartinggrant(NewHeavyFermion), FrenchANR project\n(CORMAT, SINUS, DELICE) and REIMEI program in\nJAEA.\n1) Ø. Fischer: Magnetic Superconductors in Ferromagnetic Ma-\nterials (Science Publishers BV, Amsterdam, 1990).\n2) J. Flouquet and A. Buzdin: Physics World (2002) 41.\n3) M. Ishikawa and O. Fischer: Solid State Commun. 23(1977)10 J. Phys. Soc. Jpn. Full Paper D. Aoki and J. Flouquet\n37.\n4) W. A. Fertig, D. C. 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Conduit1,2,∗\n1Department of Condensed Matter Physics, Weizmann Institut e of Science, Rehovot 76100, Israel\n2Physics Department, Ben Gurion University, Beer Sheva 8410 5, Israel\n(Dated: June 20, 2018)\nMotivated by the ﬁrst experimental evidence of ferromagnet ic behavior in a three-dimensional\nultracold atomic gas, we explore the possibility of itinera nt ferromagnetism in a trapped two-\ndimensional atomic gas. Firstly, we develop a formalism tha t demonstrates how quantum ﬂuc-\ntuations drive the ferromagnetic reconstruction ﬁrst orde r, and consider the consequences of an\nimposed population imbalance. Secondly, we adapt this form alism to elucidate the key experimen-\ntal signatures of ferromagnetism in a realistic trapped geo metry.\nPACS numbers: 03.75.Ss, 75.20.En, 64.60.Kw, 75.45.+j\nI. INTRODUCTION\nItinerant ferromagnetism is a ubiquitous strongly cor-\nrelated phase of matter in the solid state. The theo-\nretical study of itinerant ferromagnetism dates back to\nthe pioneering work of Stoner [1] and Wohlfarth, which\nshowed that ferromagnetism emerges as repulsive pair-\nwise interactions between electrons overcome the kinetic\nenergy penalty of polarization. Subsequent theoretical\nwork has determined that soft transverse magnetic ﬂuc-\ntuations have the potential to drive the ferromagnetic\ntransition ﬁrst order before the quantum critical point\nis reached [3–8]. Phenomena consistent with a ﬁrst or-\nder transition have been observed in the solid state;\nthough it is diﬃcult to determine whether they are due\nto soft magnetic ﬂuctuations or the coupling of the mag-\nnetic moment to phonon degrees of freedom. However,\nJoet al. [9] have recently presented the ﬁrst tentative\nevidence [10, 11] of itinerant ferromagnetism in an ultra-\ncold atomic gas. The cold atom gas is a clean system\nin which to study ferromagnetism, completely devoid of\nthe interfering phonon degrees of freedom encountered in\nthe solid state, so gifts investigators with a valuable tool\nwith which to answerlong-standingquestions about solid\nstate ferromagnets. Furthermore, ultracold atoms exper-\niments also present a unique opportunity to explore fun-\ndamentally new physics associated with ferromagnetism\nincluding the consequences of population imbalance [7],\na conserved net magnetization [12], the damping of ﬂuc-\ntuations by three-body loss [13], spin drag [14], and mass\nimbalance. Here we aim to take advantage of the high\nlevels of control investigators can exercise over the exter-\nnal potential trapping the gas and turn to study ferro-\nmagnetism in a two-dimensional thin ﬁlm.\nItinerant ferromagnetism is diﬃcult to observe in two\ndimensions in the solid state [15, 16]. However, it could\nbe realized in an ultracold atom gas by using counter-\npropagating lasers to create one-dimensional potential\n∗Electronic address: gjc29@cam.ac.ukwhich will lead to a stacked two-dimensional gas. The\nsystem also oﬀers investigators the opportunity to study\nthepossibilityforasuperconductinginstabilitytoemerge\nnear to the ferromagnetic phase transition [17]. The two-\ndimensionalsystem is ofparticularinterest in this caseas\nitcouldshedlightonhightemperaturesuperconductivity\nwhere antiferromagnetism competes with d-wave super-\nconductivitytoformthegroundstate. Hereweadaptthe\nformalismintroducedforthethree-dimensionalcase[7]to\nexpose the contrasting behavior of the two-dimensional\nferromagnet. We develop a formalism that captures the\neﬀects of transverse quantum ﬂuctuations and explore\nhow they renormalize the eﬀective interaction strength.\nWe then address how population imbalance modiﬁes the\nbehavior of the atomic gas before studying ferromagnetic\nordering in a trapped geometry.\nII. FIELD INTEGRAL FORMALISM\nIt has been long established that quantum ﬂuctuations\nin a three-dimensional fermionic gas with repulsive in-\nteractions have the potential to drive the ferromagnetic\ntransition ﬁrst order [4–8]. To investigate the impact\nof quantum ﬂuctuations in a two-dimensional fermionic\ngas we explore ferromagnetic reconstruction within the\nsetting of an atomic gas, adapting the phenomenology\ndeveloped for the three-dimensional case in Ref. [7]. We\nadopt this formalism because unlike the Eliashberg the-\nory [3] it provides an exact expression for the free energy\nwhich then allows us to make a prediction of the criti-\ncal interaction strength for the onset of ferromagnetism\nand study the atomic gas within a harmonic well. More-\nover,ab initio Quantum Monte Carlo calculations [8, 18]\nhave recently been used to verify the three-dimensional\nformalism, which should therefore provide a solid foun-\ndation from which to study the two-dimensional case.\nAlthough the atoms do not carry spin, we discriminate\nbetween the two fermionic species with a pseudospin\nσ∈ {↑,↓}. The species cannot interconvert so separate\nchemical potentials µσtune the population imbalance,\nwhich in turn pins the net polarization along the pseu-\ndospin direction. However, when the spontaneous mag-2\nnetization formed exceeds the population imbalance, a\nnonzero in-plane magnetization emerges. To study the\npotential for ferromagnetic ordering we express the par-\ntition function as a fermionic coherent state path integral\nZ= Tre−β(ˆH−µˆN)=/integraltext\nDψe−Swith the action\nS=/integraldisplay/summationdisplay\nσ={↑,↓}¯ψσ/parenleftbig\n∂τ+ǫˆk−µσ/parenrightbig\nψσ+/integraldisplay\ng¯ψ↑¯ψ↓ψ↓ψ↑.\n(1)\nHere/integraltext\n≡/integraltextβ\n0dτ/integraltext\nd2rwith reduced temperature β=\n1/kBT, andǫˆkdenotes the dispersion. As we wish\nto investigate two-dimensional ferromagnetism we have\nconstrained the spatial integral to a plane. A two-\ndimensional atomic gas could be realized experimentally\nusing counter-propagating laser beams whose antinodes\nat half-wavelength spacing bwill deﬁne stacked quasi-\ntwo-dimensionallayers. Thoughatﬁnitetemperaturethe\nferromagnetic ordering is only marginally stable, long-\nrange order should be stabilized by the weak inter-plane\ncoupling [12]. The repulsive contact interaction param-\neterg=gδ3(r) that can be tuned with a Feshbach res-\nonance [2] is linked to the s-wave scattering length ain\nthree dimensions through g=/radicalbig\n2/πa/b[23]. Unique to\ntwo dimensions, the interaction strength is independent\nof density. This means that within a trapped geometry\nthe entire atomic gas will experience the same eﬀective\ninteraction strength and therefore adopt the same polar-\nization.\nTo develop an eﬀective Landau theory of the mag-\nnetic transition, Hertz introduced a scalar Hubbard-\nStratonovichdecouplingofthetwo-bodyinteractionterm\nin the spin channel [2]. However, this form of de-\ncoupling neglects the potential impact of soft trans-\nverse ﬁeld ﬂuctuations, which in three dimensions are\nresponsible for driving the second order transition ﬁrst\norder [7, 8]. Therefore, we will introduce a general\nHubbard-Stratonovich decoupling that incorporates ﬂuc-\ntuations in all of the spin φand charge ρsectors. In-\ntegrating over the fermion degrees of freedom yields\nZ=/integraltext\ne−SDφDρwith the action\nS=/integraldisplay\ng(φ2−ρ2)−Trln[∂τ+ǫˆk−µσz+gρ−gσ·φ].(2)\nAt this stage a saddle point analysis would determine the\nmean-ﬁeldvaluesof ρandφ. However,quadraticﬂuctua-\ntionsintheseauxiliaryﬁeldsrenormalizetheseequations.\nTherefore we introduce the putative saddle point values\nρ0for density and mfor magnetization, integrate out\nﬂuctuations in the auxiliary ﬁelds, and ﬁnally minimize\nthe energy to determine ρ0andm. It is also convenient\ntorotatethe z-axisfromthequantizationaxistolie along\nthe direction of the saddle point magnetization m, with\ncomponents labeled by s∈ {+,−}. After integrating\nover ﬂuctuations in both the density ρand magnetiza-\ntion channels φto Gaussian order, an expansion of theaction to second order in gleads to\nZ= exp/bracketleftBigg\n−/integraldisplay\ng(m2−ρ2)+Trln ˆG−1\n−g2\n2Tr(ˆΠ+−ˆΠ−+−ˆΠ++ˆΠ−−)/bracketrightBigg\n,(3)\nwhere we have deﬁned the spin-dependent polarization\noperator ˆΠss′=ˆGsˆGs′, andˆG−1\n±=∂τ+ǫˆk−µ±+gρ0∓\ng|m|. The contact interaction means that an unphysical\nultraviolet divergence arises from the term in the action\nthat is second order in g. To remove it we must aﬀect\nthe standard regularizationofthe linear term g(m2−ρ2),\nsettingg∝ma√sto→/radicalbig\n2/πa/b−2(/radicalbig\n2/πa/b)2A−1/summationtext′\nk3,4(ǫk1+\nǫk2−ǫk3−ǫk4)−1[19], where the prime indicates that\nthe summation is subject to the momentum conservation\ncondition k1+k2=k3+k4, andAdenotes the total area\nof one stacked layer.\nFinally, after carrying out the remaining Matsubara\nsummations, one obtains the following expression for the\nfree energy:\nF=/summationdisplay\nk,s=±ǫs\nkns(ǫk)+/radicalbigg\n2\nπa\nbAN+N−\n−2/parenleftBigg/radicalbigg\n2\nπa\nbA/parenrightBigg2/summationdisplay\nk1,2,3,4′n+(ǫk1)n−(ǫk2)[n+(ǫk3)+n−(ǫk4)]\nǫk1+ǫk2−ǫk3−ǫk4,(4)\nwherens(ǫ) = 1/[1+eβ(ǫ−µs−s|m|√\n2/πa/b)] is the Fermi\ndistribution, and Ns=/summationtext\nkns(ǫk). To evaluate the ﬁnal\nnine-dimensional integral in Eq. (4) numerically we em-\nploythere-parameterizationoutlinedinApp.Atoreduce\nit to a four-dimensional integral. Moreover, as we are in-\nterested in searching for extrema in the free energy with\nchanging polarization we can diﬀerentiate our expression\nwithrespecttomagnetization,whichatzerotemperature\nfurther reduces the integral to just three dimensions.\nTo highlight the potential importance of ﬂuctuation\ncorrections we brieﬂy study the contribution to the en-\nergy from particle-hole excitations around momentum\n2kF. At zero temperature a non-analytic contribution to\nthe free energyof the form |m|3lnm2emerges. The same\nnon-analyticity was found diagrammatically in Refs. [3].\nThe formation of a ﬁnite magnetization increases the\nphase-space available for the formation of virtual inter-\nmediate pairs of particle-hole pairs, and this phase space\nenhancement donates a non-analytic term to the free en-\nergy giving the transition the potential for ﬁrst order\ncharacter. In the next section we study the eﬀect that\nthis non-analyticity has on the phase diagram.\nIII. PHASE BEHAVIOR\nWith the formal development of the theory complete\nwe will now apply the formalism to explore the implica-\ntions of ferromagnetism in the two-dimensional atomic3/BC\n/BC/BA/BE/BC/BA/BG/BF/BA/BL /BF/BA/BL/BH /BG /BG/BA/BC/BH/CC /BP/CC\n/BY/CP/BP/CQ\n/B4/CQ/B5/CC/BV/C8/BY/CX/D6/D7/D8\n/D3/D6/CS/CT/D6\n/CB/CT\r/D3/D2/CS\n/D3/D6/CS/CT/D6/C8 /CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /C9/D9/CP/D7/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r\n/BC\n/BC/BA/BH\n/BD/D1\n/B4/CP/B5\n/CC /BP /BC\n/CC /BP /BC /BM /BD /CC\n/BY/CC\n/BP\n/BC\n/BM\n/BE\n/CC\n/BY/CC\n/BP\n/BC\n/BM\n/BF\n/CC\n/BY/BX/D1\n/B4/CX/B5\n/CC /BP /BC/CC /BP /BC /BM /BD /CC\n/BY/CC\n/BP\n/BC\n/BM\n/BE\n/CC\n/BY/CC\n/BP\n/BC\n/BM\n/BF\n/CC\n/BY/BX/D1\n/B4/CX/CX/B5\n/CC /BP /BC/CC /BP /BC /BM /BD /CC\n/BY/CC\n/BP\n/BC\n/BM\n/BE\n/CC\n/BY/CC\n/BP\n/BC\n/BM\n/BF\n/CC\n/BY\nFIG. 1: (Color online) (a) The growth of magnetization m\nwith scattering length for diﬀerent temperatures. The in-\nset ﬁgures show the energy landscape with magnetization for\nT= 0 either side of the ﬁrst order transition. (b) the phase\ndiagram of temperature with scattering length shows the ﬁrs t\norder (dashed red line) and second order (solid red line)\n(quasi)ferromagnetic ordering from the paramagnetic phas e.\nFermi gas, and critically compare the results with the\nthree-dimensional case [7]. Before we study the phase\ndiagram of the ﬂuctuation corrected free energy, to make\ncontact with the conventional Stoner theory we ﬁrst con-\nsider the result of a direct saddle point approximation\nscheme in which the second order term in the free en-\nergy is neglected. In this approximation at zero temper-\nature the free energy is F= (1 +a/b√\n2π3/2)µ2/2π+\n(1−a/b√\n2π3/2)m2/2πµ2. This expression is exact, and\nwith magnetization featuring only as the lowest available\nterm in a Landau expansion its analysis is straightfor-\nward. For a <√\n2π3/2b≈7.874bthis model predicts\nthat the gas is paramagnetic, whereas for a >√\n2π3/2b\nthe system is fully polarized, a scenario that remains un-\naltered with the introduction of population imbalance.\nAn immediate corollary is that the spontaneous magne-\ntization formed is independent of the local density, which\nalso holds true when ﬂuctuation corrections are taken\ninto account. Therefore, within a trap, the entire atomic\ngas adopts the same polarization.\nHaving studied the mean-ﬁeld limit, we now consider\nthe repercussions of ﬂuctuation corrections on the be-\nhavior of the magnetization. To orient our discussion,\nwe ﬁrst consider a gas with equal populations of up and\ndown-spin atoms. As shown in Fig. 1(ai) at scatter-\ning lengths below a≈3.945bthe energy proﬁle pos-\nsesses a single minimum at zero magnetization. With\nrising interaction strength a second minimum in the en-\nergy landscape develops at m≈0.6, which, with ris-\ning scattering length, deepens in Fig. 1(aii) to become\nthe global minimum at a≈3.953bandm≈0.8. At01\n3.9 3.95 4 4.05p\na/bUnM, m⊥= 0\nPM,m⊥ /negationslash= 0\nFM,m⊥ /negationslash= 0(b)\n01\n3.9 3.95 4 4.05p\na/bUnM, m⊥= 0\nPM,m⊥ /negationslash= 0\nFM,m⊥ /negationslash= 0(b)01m(a)\nFIG. 2: (Color online) (a) The growth of magnetization m\nwith scattering length at T= 0. (b) the T= 0 phase diagram\nfor imposed population imbalance pwith scattering length\na/bshows the ﬁrst order (dashed red line) and second order\n(solid red line) ferromagnetic ordering from the unmagneti zed\n(UnM)tothepartiallymagnetized(PM)andfullymagnetized\n(FM) regions, here unmagnetized refers to having no in-plan e\nmagnetization.\nthis scattering length the system undergoes a ﬁrst order\ntransition from m= 0 into the polarized regime with\nm≈0.8. As shown in Fig. 1(a) with a further increase\nin the interaction strength the magnetization saturates\nat a scattering length a≈4.048b. Fluctuation correc-\ntions have had signiﬁcant impact: they have driven the\nferromagnetic transition to a signiﬁcantly weaker inter-\naction strength ( a≈3.953b) compared to the mean-ﬁeld\ncase (a≈7.874b). At this weaker interaction strength\nthem2term in the free energy has a positive coeﬃcient,\nand the ordering is driven by the non-analytic |m|3lnm2\nterm. The abetment of the transition by ﬂuctuation cor-\nrections and reduction in interaction strength at which\nferromagnetism is seen is common to both the two and\nthree-dimensional cases, though in two dimensions the\ntransition is immediately to full polarization at mean-\nﬁeld level and ﬂuctuation corrections drive a ﬁrst order\ntransition at a weaker interaction strength.\nWe nowturnto addressthe behaviorofthe phasetran-\nsition at ﬁnite temperature in Fig. 1(a). Increasing tem-\nperaturedulls theﬂuctuationcorrectionsandthescatter-\ning length of the ﬁrst order transition rises and the mag-\nnetization following the transition is reduced. Fig. 1(b)\nshows that at T≈0.28TFa tricritical point emerges\nand the system reverts to second order behavior. The\nMermin-Wagner-Hohenberg theorem [20] states that al-\nthough an ordered phase can exist in two dimensions at\nzero temperature, at any ﬁnite temperature ﬂuctuations\nwill destroy long range correlations in the system, and\nthe state will be characterized by exponentially decaying4\ncorrelation functions. Therefore we denote the ferromag-\nnetic state as a “quasiferromagnet” (ferromagnet with\nﬂuctuating polarization direction). However, since the\ntwo-dimensional gas is experimentally realized in a se-\nries of disks, each one can couple to its neighbors and\ntunneling should stabilize the phase [12, 24]. Further-\nmore, the Mermin-Wagner-Hohenberg theorem is valid\nonly in the thermodynamic limit and does not apply to\nﬁnite-sizedsystems. Foratwo-dimensionalBosegaswith\nattractive interactions it has been shown that a poten-\ntial trap restricts the system and stabilizes a quasi-Bose\nEinstein condensate [21]. In a similar way the harmonic\ntrap should stabilize a ferromagnetic phase. So far we\nhave focused on how equilibrium properties can stabilize\nthe ferromagnetic phase, however, there are also non-\nequilibrium aspects to consider. Within the current ex-\nperimental realization of cold atom gas ferromagnetism\nthree-body losses necessitate that the experiment be per-\nformed out of equilibrium. Following a quench small\nferromagnetic domains are formed [22] which then grow\nsteadily [10]. The ﬁnal size of these ferromagnetic do-\nmains∼6/kF[10] atT= 0.1TFandkFa= 2 is small\ncompared to the length-scale of the thermal ﬂuctuations\ngiven byaexp[2π(2kFa/π−1)TF/T]≈107/kF[25]. This\nmeans that at suﬃciently low temperature ﬂuctuations\nwill not disrupt the ferromagnetic state and so in exper-\niments a true ferromagnetic phase should be observed as\nshown in Fig. 1.\nHavingaddressedthe situation without populationim-\nbalance, we now consider how a ﬁxed spin population\nimbalance inﬂuences the phase diagram. The two con-\nstituent species cannot interconvert so an initial popu-\nlation imbalance is maintained by the diﬀerence in their\nchemical potentials. However, if energetically favorable,\nthe gas can become more polarized either by phase sep-\naration or the development of an in-plane magnetic mo-\nment. As shown in Fig. 1(ai and aii), at weak inter-\nactions such that a/lessorapproxeql3.945b, the energy monotonically\nincreases with magnetization, but the magnetization re-\nmains pinned to the minimum value deﬁned by the pop-\nulation imbalance p. However, with rising interaction\nstrength a second minimum develops in the free energy\nlandscape from a≈3.945bandm≈0.6. If that magneti-\nzation exceeds the population imbalance, then as shown\nin Fig. 2(b) the system will phase separate between this\nminimum and that at zero magnetization, with relative\nfractions governed by the Maxwell construction. When\nthe emerging minimum becomes the global minimum at\nm≈0.8, then gases with a lower population imbalance\nenter this global minimum with an appropriate in-plane\nmagnetic moment. As the magnetization of the mini-\nmum rises it envelops systems with higher population\nimbalance, and tracks the magnetization curve shown in\nFig. 2(a) until it reaches full polarization at a/b≈4.048.\nLike the three-dimensional case, the population imbal-\nance renders the characteristic interaction strength of\nthe transition to be almost constant up to an imposed\npopulation imbalance of p≈0.8, which could be a keyexperimental signature of ﬁrst order behavior.\nIV. TRAPPED GEOMETRY\nHaving addressed the phase behavior of a uniform sys-\ntem, to make contact with the experiment we now turn\nto address the atomic gas trapped within the spherical\npotentialV(r) =ωr2/2. Following the program devel-\noped in Refs. [10–12] we aim to minimize the free en-\nergy within the local density approximation using the\nkernelf(r) =F(r) +V(r)[n+(r) +n−(r)]−γ+n+(r) +\nγ−n−(r), hereF(r) denotes the energy kernel Eq. (4)\nevaluated with the local chemical potential at r. The\nLagrange multipliers γ±enforce the constraints of con-\nstant number of atoms imposed by the trap geometry\nNtot=/integraltext\n[n+(r) +n−(r)]d2rand population imbalance\np≤/integraltext\n[n+(r)−n−(r)]d2r/Ntot; without loss of generality\nwe assume that p≥0 and therefore γ+≥γ−. To study\nthe eﬀects of spatial density variations we invoke a local\ndensity approximation that enables the variational min-\nimizationδf/δn s(r) and yields the simultaneous equa-\ntions for the eﬀective local chemical potentials µ±(r) for\nthe species in the rotated spin basis\nµ±(r)=γ±−V(r)−/radicalbigg\n2\nπa\nbAn∓(r)+2/bracketleftBigg/radicalbigg\n2\nπa\nbA/bracketrightBigg2/summationdisplay\nk1,2,3,4′\nn∓(ǫk2)\n×n±(ǫk1)δ(ǫk3−µ±)+[n±(ǫk3)+n∓(ǫk4)]δ(ǫk1−µ±)\nǫk1+ǫk2−ǫk3−ǫk4.(5)\nThese equations can be understood as having been con-\nstructed out of three orders of perturbation theory. The\nlowest, independent of the scattering length a, corre-\nsponds to the Thomas-Fermi approximation within the\nconﬁning potential, the term ﬁrst order in aintroduces\nthe mean-ﬁeld energy penalty of the interaction whereas\nthe second order term introduces the energy associated\nwith magnetic quantum ﬂuctuations. The detailed study\nof the uniform system revealed that the polarization de-\npends only on the interaction strength and not spatial\ndensity variations, meaning that the ratio n+(r)/n−(r)\nand therefore µ+(r)/µ−(r) is constant across the trap.\nTherefore, the two equations reduce to just one that is\nsolved by iteration.\nA. Heuristic observations\nWe ﬁrst address what can be determined about the\nbehavior of the atomic gas heuristically before present-\ning the results of the full solution of Eq. (5). To de-\nvelopourintuition we focuson perhapsthe most physical\nquantity that can be measured by experiment, namely\nthe cloud size. To start the analysis we consider the\nnon-interacting limit a= 0 where the system is un-\npolarized and the eﬀective chemical potentials given by\nEq. (5) follow the familiar Thomas-Fermi form. The root5\nmean square (RMS) radius would increase with popula-\ntionimbalanceas[1+(1−p\n1+p)3/2]1/2/√\n2due tothe increas-\ning Fermi degeneracy pressure. With weak interactions\na≪√\n2π3/2bwe need consider Eq. (5) only to ﬁrst or-\nder inawhich yields µ±(r) =γ±−V(r)−amax[γ∓−\nV(r),0]/21/2π3/2b. The ﬁrst order term reduces the ef-\nfectivechemicalpotentialsotoconservethe totalnumber\nof trapped atoms we renormalize the Lagrange multipli-\ners upwards from the Thomas Fermi value by a factor of\n1+a(1−p)/(2π)3/2b(1+p) for the majority spin species\nand 1 +a[2−(1−p\n1+p)1/2]/(2π)3/2bfor the minority spin\nspecies. This reduction in the eﬀective chemical poten-\ntial and corresponding fall in local density can be under-\nstood in terms of an increase in the local pressure within\nthe cloud due to the repulsive interactions between the\natoms. This pressure inﬂates the cloud causing the RMS\nradius to rise through a factor of 1+ a(1−p)(3√1+p−√1−p)/25/2π3/2b[(1+p)3/2+(1−p)3/2]. Having ana-\nlyzed the weakly interacting regime it is natural to also\nexamine the strongly interacting limit. Here the atomic\ngas is fully polarized so µ−= 0 andµ+(r) =γ+−V(r),\nmeaning that the system is ﬁrmly in the Thomas Fermi\nregime. We again require that the number of particles is\nconserved which sets the majority spin Lagrange multi-\nplier torescalebyafactorof21/2fromits originalvalue if\nthere wereno population imbalance. Consequentiallythe\nenhanced Fermi degeneracy pressure dilates the RMS ra-\ndiusofacloudwithzeropopulationimbalancebyafactor\nof 21/4. The key limits of weak and strong interactions\nhold true whatever the true theory of ferromagnetism so\nprovide two valuable handles for potential experiments.\nB. Exact analysis of trapped behavior\nHaving completed the overview of the trapped behav-\nior we now turn to consider the ramiﬁcations of ﬂuctu-\nation corrections and self-consistently solve Eq. (5) for\nthe chemical potentials µ±. We then integrate over the\ntrap to extract the full behavior of the experimental ob-\nservables, namely cloud size, kinetic energy, and three-\nbody loss rate, which for the mean-ﬁeld limit are shown\nin Fig. 3. The same calculation repeated for ﬂuctuation\ncorrectionsisshowninFig.4. Ausefulreferencethrough-\nout will be the complementary analysis in three dimen-\nsions [10, 11]. The orthodox Stoner mean-ﬁeld theory\npredicts that at the onset of ferromagnetic ordering the\nsystem immediately fully polarizes across the entire trap\nata/b=√\n2π3/2≈7.874,whereasﬂuctuationcorrections\nallowthecloudtoadoptpartialpolarizationoverthewin-\ndow of scattering lengths 3 .953/lessorapproxeqla/b/lessorapproxeql4.048. In two-\ndimensionsastheentiregaspolarizesatthesameinterac-\ntion strength striking features emerge at these respective\ninteraction strengths. Current experiments [9] can probe\nscattering lengths to ∼10% accuracy, therefore in cur-\nrent experiments the ﬂuctuation corrected transition will\nalso appear to immediately give complete polarization.\nTo developour intuition we ﬁrst examinethe projected11.051.11.151.2R/R 0(a) Cloud size\np= 0p= 0.5Small a/bMean-ﬁeld\n0.20.3EK/E0\nF(b) Kinetic energy\n10010−310−310−3\n0 2 4 6 8Γ/Γ0\na/b(c) Loss ratep= 0\np= 0.5\nSmall a/b\nFIG. 3: (Color online) The variation of (a) cloud size, (b) ki -\nnetic energy, and (c) atom loss rate on ferromagnetic orderi ng\nwith increasing scattering length a/bfor the orthodox Stoner\nmean-ﬁeld theory case. The thin blue dashed line highlights\nthe small a/bbehavior. The solid lines are at zero popula-\ntion imbalance whereas the dotted line is with an imposed\npopulation imbalance of 0 .5.\ncloud size . In the mean-ﬁeld limit (Fig. 3(a)) with weak\ninteractions the RMS radius grows linearly with scat-\ntering length as the atoms repel each other within the\ntrap. Theradiusgrowsfollowingthe universalscalingde-\nscribed above. Population imbalance causes the cloud to\nhave an initially larger radius due to the increased Fermi\ndegeneracy pressure. The entire cloud becomes fully po-\nlarized at the same scattering length, a/b=√\n2π3/2, and\nat this point the cloud size immediately adopts its ﬁ-\nnal inﬂated radius R/R0= 21/4, maintained by Fermi\ndegeneracy pressure. This is in contrast to the three-\ndimensional case [10, 11] in which the transition takes\nplace over a range of interaction strengths, thus making\nthe transition less distinct. Fig. 4(a) shows that ﬂuctua-\ntion corrections drive the cloud expansion faster, causing\nit to dilate rapidly. In contrast to the three-dimensional\ncase [10, 11], this pressure cannot drive the cloud to\ngrow larger than the fully polarized size 21/4RRMS\n0. As\nthe interaction strength is unaﬀected by the density of\natoms,thetransitionoccursatthesamescatteringlength6\n11.051.11.151.2R/R 0(a) Cloud size\np= 0p= 0.5Small a/bFluctuation corrected\n0.20.3EK/E0\nF(b) Kinetic energy\n10010−510−4\n0 2 4 6 8Γ/Γ0\na/b(c) Loss rate\np= 0\np= 0.5\nSmall a/b\nFIG. 4: (Color online) The variation of (a) cloud size, (b)\nkinetic energy, and (c) atom loss rate on ferromagnetic or-\ndering with increasing scattering length a/bwhen ﬂuctuation\ncorrections are taken into account. The thin blue dashed lin e\nhighlights the small a/bbehavior. The solid lines are at zero\npopulation imbalance whereas the dotted line is with an im-\nposed population imbalance of 0 .5.\na≈3.954bseen in the uniform case.\nThetotal kinetic energy is probed experimentally by\nreleasing the atoms from the trap and imaging them fol-\nlowing a ballistic expansion. Starting from the mean-\nﬁeld analysis in Fig. 3(b), at zero interactions an initial\npopulation imbalance increases the kinetic energy due\nto the enlarged majority spin Fermi surface by a fac-\ntor of [1 + (1−p\n1+p)3/2]/2. The weak interactions dilate\nthe cloud, causing local density and kinetic energy to\nfall with the universal scaling 1 −a(1−p)(3√1+p+√1−p)/25/2π3/2b[(1 +p)3/2+ (1−p)3/2]. When the\nscattering length is increased beyond a/b=√\n2π3/2the\nentire gas becomes ferromagnetic and the atoms all enter\nthe same Fermi surface. This Fermi surface is inﬂated\nand the kinetic energy plateaus at the ﬁnal value that is\n21/2times that for the non-interacting gas. When ﬂuctu-\nation corrections are taken into account one recovers the\nvariation of kinetic energy shown in Fig. 4(b). The ﬂuc-\ntuations drive the transition to take place at a reducedinteraction strength a≈3.954bseen in the uniform case.\nTheatom loss rate due to three-body recombination\nis Γ = Γ 0(a/b)6/integraltext\nn+(r)n−(r)[n+(r)+n−(r)]d2r[26]. In\nthe recent experiment [9] the three-body loss was signiﬁ-\ncant and forced the experiment to be performed rapidly\nand out of equilibrium, and here we study the situation\nin two-dimensions. We start by examining the mean-\nﬁeld limit in Fig. 3(c), which shows the three-body loss\nintegrated over the entire trap. At weak interaction\nstrengthsthelossraterisesrapidlyasΓ = Γ 0(a/b)6µ4(1−\np2)/8π2ω. At a scattering length a/b=√\n2π3/2the gas\nacross the entire trap becomes fully polarized so n−= 0\nand therefore the three-body loss is completely cut oﬀ.\nThis immediate elimination of loss contrasts the three-\ndimensional case where loss remains until high interac-\ntionstrengths, whereit forcestheexperiment outofequi-\nlibrium [10], and also renormalizes the eﬀective interac-\ntion strength [13]. Fig. 3 highlights how these eﬀects are\nreducedin the two-dimensionalcasewhich couldaid with\nthe positive identiﬁcation of the ferromagnetic phase. It\ncan also be seen that population imbalance reduces atom\nloss primarily through reduction of the n+(r)n−(r) term.\nHaving studied the mean-ﬁeld limit we now look at the\nimpact of ﬂuctuation corrections on three-body loss in\nFig. 4(c). The ﬂuctuation corrections drive the ferro-\nmagnetic transition to take place at a reduced scattering\nlength ofa/b= 3.954. This in turn means that the peak\nthree-bodyloss( ∝a6\ncrit)issigniﬁcantlyreduced. Thisfall\nin loss rate will mean that an experiment searching for\nsignatures of ferromagnetism can be performed nearer to\nthe equilibrium regime which should yield clearer results.\nV. DISCUSSION\nIn conclusion, on the repulsive side of the Feshbach\nresonance coupling of transverse magnetic ﬂuctuations\ndrives ferromagnetic ordering ﬁrst order. We studied the\nspeciﬁc variation of three experimental signatures of fer-\nromagnetism: cloud size, release energy, and atom loss\nrate. The formalism highlighted the beneﬁts of studying\nferromagnetism in two rather than three dimensions. In\ntwo-dimensions the eﬀective interaction strength is inde-\npendent of density and therefore radius in the harmonic\nwell. As the interaction strength is ramped upwards the\nentire gas will enter into the ferromagnetic phase at the\nsame Feshbach ﬁeld, whereas in three-dimensions the gas\nﬁrst enters the ferromagnetic state at the center. There-\nfore the signatures of the ferromagnetic phase are en-\nhanced in two-dimensions, which should aid the exact\ncharacterization of the state. At weak interactions these\nobservables displayed universal scaling, and the variation\nwith an imposed population imbalance was also consid-\nered.\nOne intriguing possibility opened up by the new for-\nmalism developed to study ﬂuctuation corrections is fer-\nromagneticreconstruction into a spin textured state, in a\nmatter analogous to the FFLO state in superconductors.7\nThis has already been shown to be possible in three di-\nmensions [8] and, with enhanced Fermi surface nesting in\ntwo dimensions, poses an interesting direction for future\nresearch.\nTheauthorthanksEhudAltman, AndrewGreen, Gyu-Boong Jo, Wolfgang Ketterle, Ben Simons, and Joseph\nThywissen for useful discussions. The author acknowl-\nedges the ﬁnancial support of the Royal Commission for\nthe Exhibition of 1851 and the Kreitman Foundation.\n[1] E.C. Stoner, Proc. R. Soc. London 165, 372 (1957).\n[2] J.A. Hertz, Phys. Rev. B 14, 1165 (1976); A.J. Millis,\nPhys. Rev. B 48, 7183 (1993); T. Moriya, Solid State\nScience56(Springer, Berlin, Heidelberg, 1985).\n[3] D.L. Maslov, A.V. Chubukov and R. Saha, Phys. Rev. B\n74, 220402(R) (2006). D.V. Efremov, J.J. Betouras and\nA. Chubukov, Phys. Rev. B 77, 220401(R) (2008).\n[4] R.A. Duine and A.H. MacDonald. Phys. Rev. Lett. 95,\n230403 (2005).\n[5] For a review, see D. Belitz, T. Kirkpatrick, T. Vojta,\nRev. Mod. Phys. 77, 579 (2005).\n[6] A.A. Abrikosov and I.M. Khalatnikov, Soviet Phys.\nJETP6, 888 (1958).\n[7] G.J. Conduit and B.D. Simons, Phys. Rev. A 79, 053606\n(2009)\n[8] G.J. Conduit, A.G. Green and B.D. Simons. Phys. Rev.\nLett.103, 207201 (2009).\n[9] G.-B. Jo, Y.-R. Lee, J.-H. Choi, C.A. Christensen, T.H.\nKim, J.H. Thywissen, D.E. Pritchard and W. Ketterle,\nScience325, 1521 (2009).\n[10] G.J. Conduit and B.D. Simons. Phys. Rev. Lett. 103,\n200403 (2009).\n[11] L.J. LeBlanc, J.H. Thywissen, A.A. Burkov and A.\nParamekanti, Phys. Rev. A 80, 013607 (2009).\n[12] I. Berdnikov, P. Coleman and S.H. Simon, Phys. Rev. B\n79224403 (2009).\n[13] G.J. Conduit and E. Altman. arXiv:0911.2839.\n[14] R.A. Duine, M. Polini, H.T.C. Stoof and G. Vignale,\nPhys. Rev. Lett. 104, 220403 (2010).\n[15] T. Chatterji, Journal of Alloys and Compounds 326, 15\n(2001).\n[16] Y.J. Chang, C.H. Kim, S.H. Phark, Y.S. Kim, J. Yu and\nT.W. Noh, Phys. Rev. Lett. 103, 057201 (2009).\n[17] H. v. L¨ ohneysen et al., Rev. Mod. Phys. 79, 1015 (2007).\n[18] S. Pilati et al., arXiv:1004.1169v1; S.-Y. Chang, M. Ran-\nderia, and N. Trivedi, arXiv:1004.2680v1.\n[19] R.K. Pathria, Statistical Mechanics, Pergamon Press\n(1996).\n[20] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 22, 1133\n(1966); P.C. Hohenberg, Phys. Rev. 158, 383 (1967).\n[21] J.O. Andersen, U. Al Khawaja and H.T.C. Stoof, Phys.\nRev. Lett. 88, 070407 (2002); U. Al Khawaja, J.O. An-\ndersen, N.P. Proukakis and H.T.C. Stoof, Phys. Rev. A\n66, 013615 (2002); C. Gies, B.P. van Zyl, S.A. Mor-\ngan and D.A.W. Hutchinson, Phys. Rev. A 69, 023616\n(2004).\n[22] M. Babadi, D. Pekker, R. Sensarma, A. Georges and\nE. Demler, arXiv:0908.3483.\n[23] R.K. Bhaduri, S.M. Reimann, S. Viefers, A. Ghose\nChoudhury and M.K. Srivastava, J. Phys. B 33, 3895\n(2000).\n[24] D.S. Petrov, M. Holzmann and G.V. Shlyapnikov, Phys.\nRev. Lett. 84, 2551 (2000).\n[25] A. Auerbach, Interacting Electrons and Quantum Mag-netism, Springer (1994).\n[26] D.S. Petrov, Phys. Rev. A 67, 010703(R) (2003).\n[27] W. Zwerger, Science 325, 1507 (2009).8\nk3p34k4θ12k2\nk1p12\nFIG. 5: The re-parameterization of the momenta k1,2,3,4.θ12\nrepresents the angle between k1andk2. The two momenta\np12=k1+k2andp34=k3+k4are constrained to be equal,\np12=p34, by the Dirac delta function in Eq. (A1).\nAppendix A: Computing the momentum space\nintegral\nAn important integral Eq. (4) encountered in this pa-\nper has the form\n/integraldisplay/integraldisplay/integraldisplay/integraldisplay\ndk1dk2dk3dk4F(k1,k2,k3,k4)δ(k1+k2−k3−k4).\n(A1)\nTo evaluate this integral one could substitute k4=\nk1+k2−k3, andthen integrateoverthe threeparameters\nrepresenting the lengths of vectors k1,k2, andk3, and\na minimum of three relative angles between these vec-\ntors, giving a total of six integration parameters. How-\never, since numerical integration generally becomes pro-\nhibitive with increasing number of dimensions we outline\na scheme that takes advantage of the fact that the func-\ntionFdepends only on the magnitude ofthe momenta to\nperform the angular integrals and leave a numerical inte-\ngral over just the four vector lengths. A similar scheme\nhas been developed in the three-dimensional case [7].\nThe integral is re-parameterized according to Fig. 5.\nThe angular integral associated with vectors k1andk2\nis/integraltext2π\n02πk1k2dθ12, whereθ12is the angle between k1andk2. We now change the variable of the angular integral\noverθ12to the vector p12=k1+k2through the relation-\nship cosθ12= (k2\n1+k2\n2−p2\n12)/2k1k2and so/integraltext2π\n0dθ12=/integraltextk1+k2\n|k1−k2|dp128πk1k2p12[4k2\n1k2\n2−(k2\n1+k2\n2−p2\n12)2]−1/2. This\nexpression, andan analogousonein p34=k3+k4, allows\nus to rewrite the original integral Eq. (A1) in terms of\nthe parameters p12andp34. The momentum conserva-\ntion requirement is imposed by δ(k1+k2−k3−k4) which\nnowintroducesanewconservationlaw δ(p12−p34). This\nsets the two integration parameters equal, p12=p34=p,\nso there is just one integral over parameter premaining,\nand since the delta function constrainsthe angle between\np12andp34we must also divide by the phase space as-\nsociated with the angular integration of 2 πp. We then\nobtain\n32π/integraldisplay/integraldisplay/integraldisplay/integraldisplay\ndk1dk2dk3dk4/integraldisplaymin(k1+k2,k3+k4)\nmax(|k1−k2|,|k3−k4|)dp×\nF(k1,k2,k3,k4)k1k2k3k4p/radicalbig\n4k2\n1k2\n2−(k2\n1+k2\n2−p2)2/radicalbig\n4k2\n3k2\n4−(k2\n3+k2\n4−p2)2.\nFinally, we note that the integral over variable pis Carl-\nson’s standard elliptic integral of the ﬁrst kind, which we\ndenote byRF. This yields the ﬁnal result\n32π/integraldisplay/integraldisplay/integraldisplay/integraldisplay\ndk1dk2dk3dk4F(k1,k2,k3,k4)×\nΘ(k1+k2−|k3−k4|)Θ(k3+k4−|k1−k2|)×\nRF/parenleftBig\n0,1+/vextendsingle/vextendsingle/vextendsingle[(k1+k2)2−(k3−k4)2][(k1−k2)2−(k3+k4)2]\n[(k1+k2)2−(k3+k4)2][(k1−k2)2−(k3−k4)2]/vextendsingle/vextendsingle/vextendsingle,1/parenrightBig\n/radicalbig\n|[(k1−k2)2−(k3−k4)2][(k1+k2)2−(k3+k4)2]|.\nThe term introduced to compensate for the angular in-\ntegrals can be eﬃciently computed by a suitable nu-\nmerical library. This four-dimensional integral is now\nbetter suited to computational evaluation than the six-\ndimensional form of the original expression Eq. (A1)."    },    {        "title": "2104.07937v1.Ferromagnetism_in_2D_Vanadium_Diselenide.pdf",        "content": "Ferromagnetism in 2D Vanadium D iselenide  \n \nXiong Wang 1, Dian Li 1, Zejun L i 2, Changzheng Wu 2, Gang Chen 1, Xiaodong Cui 1* \n \n1. Physics Department, University of Hong Kong, Hong Kong, China  \n2. Hefei National Laboratory for Physical Sciences at the Microscale, CAS Center for \nExcellence in Nanoscience, and CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei, China \n*e-mail: xdcui@hku.hk  \n \nTwo -dimensional  (2D) Van der W aals ferromagnets carry the promise of ultimately \nminiature  spintronics and information storage devices.\n1,2 Among t he newly \ndiscovered 2D  ferromagnets all inherit the magnetic ordering from their bulk \nancestors.3-5 Here we report a new 2D ferromagnetic semiconductor at room \ntemperature, 2H  phase v anad ium diselenide (VSe 2) which show ferromagnetic at 2D \nform  only. This unique 2D ferromagnetic semiconductor manifests an enhanced \nmagnetic ordering owing to structural anisot ropy at 2D limit.  \nMagnetism has been a long -lasting fascinating topic in condensed matter physics and \nbeen carrying a potential for next generation informatics and electronics1,2. Although the \nMermin -Wagner theorem indicates the absence of the long -range magnetic order for the \ntwo-dimensional (2D) isotropic Heisenberg model at the finite temperatures6, magne tic \norder still could survive in the anisotropic 2D systems where symmetry breaking is \nmaterialized either by the finite system size, defect effects (breaking the lattice translation symmetry) or the anisotropic spin exchange interactions (breaking the spi n rotational \nsymmetry). With the growing enthusiasm in the emerging V an der Waals crystals, 2D \nferromagnetic materials including Cr\n2Ge2Te6, CrI 3 and Fe 3GeTe 2 have been discovered \nand construct a new family  of 2D Van der Waals ferromagnets3-5. As yet these newly \ndiscovered 2D magnets inherit the magnetic properties of their bulk crystals as the \nmagnetic order sustains from multiple layers down to monolayers. Al though monol ayer \n1T-VSe 2 with ferromagnetism w as theoretically predicted and experimentally reported  \nrecently7-11, the controversial  results with both theoretical and experimental evidences \nraised ambiguity12-16. Here  we study the magnetic properties of the 2D VSe 2 single \ncrystal samples with the polar magnetic circular dichroism (MCD) microscopy and the \nsecond harmonic generation (SHG) technique . A strong ferromagnetism is observed in \nthe 2H  phase of VSe 2 with a Curie -Weiss temperature up to  425 K, and this \nferromagnetism softens with the increased sample thickness. We attribute the change of \nthe magnetic properties to the changing of the magnetic anisotropy that fundamentally \narises from the spin -orbit coupling. Our f inding provides a new ferromagnetic semiconductor with a potential of building blocks for spintronics and valleytronics and a \nmicroscopic understanding of the internal interaction in 2D ferromagnet s.  \nVanadium diselenide  is among the family of transition metal dichalcogenides (TMD Cs) \nwhich are believed to be paramagnetic and feature strong light -matter interactions, spin \nand valley degrees of freedom and giant spin- valley coupling at the 2D limit. Most \nTMD C materials show  the polymorphism and could exist in two different structural \nphases  as illustrated in Fig 1a , the 2H  phase with a trigonal prismatic cell and a point \ngroup symmetry 𝐷𝐷6ℎ4 and the 1T  phase with an octahedral cell and a point group \nsymmetry 𝐷𝐷3𝑑𝑑3, respecti vely 17. In contrast to its celebrated sister compounds MoS 2 and \nWSe 2, the VSe 2 crystals usually exist in the 1T  form  instead of 2H  18. While the \ndimensionality decreases, the 2D VSe 2 would thermodynamically favor s the 2H  phase \nand the structural phase transition from 1T  to 2H  phase could be irreversibly materialized \nby annealing19. Vanadium is among the group V  element with electron configuration of \n3d34s2 and is well -known for the strong electron correlation owing to its 3d- shell orbits.  \nIn the unit cell of monolayer H  phase VSe 2, the Se -V-Se bonds yield one d- shell electron \nper cell for the V4+ ion, making VSe 2 a promising candidate of  the Mott insulator with \nferromagnetism. The spin- orbit coupling is usually active for the d1 electron \nconfiguration of the V4+ ions , and our exper imental study here does indicate the \nimportance of the spin- orbit coupling in the understanding of the magnetic properties.  \nThe phase transition occurs in VSe 2 from 1T  to 2H  by annealing the multi -layer flake, \naccompanying with the metal -insulator  transition as shown in Fig. 1b &  1c.19 Fig. 1d \nsummarizes the MCD data  of the 1T  phase and 2H  phase 2D V Se2 at the room \ntemperature. The 1T  phase VSe 2 both at bulk form and  2D cases  (down to 20 -layer) \nshows a paramagnetic response  which  is consistent with the recent report12 where no \nferromagnetism was observed at 1T  VSe 2 monolayer down to 10 K. In contrast, robust \nferromagn etism displays in the 2H  phase VSe 2 multilayers , which is consistent with the \nprevious calculations20-22. These contrasting magnetic behaviors fit the intuition of solid -\nstate physics. Although the V4+ ion carries single electron , for such a VSe 2 compound, \nferromagnetism mo st likely is realized via super -exchange interaction and magnetic \nanisotropy. At each unit layer, the inversion symmetry is explicitly broken in 2H  phase. \nThis inversion symmetry breaking induces a spin- orbit coupling as well as a charge band \ngap. The mirr or symmetry with respect to the vanadium atom plane and the three -fold of \nrotation symmetry secure the spin orientation along the out -of-plane direction. In \naddition, the partially filled d- orbital  electron shell allows the atomic spin -orbit coupling \nto be  active. These factors increase the magnetic anisotropy of the 2H  phase against the \n1T phase and explain the magnetic difference between 2H  and 1T  phases.  \nFerromagnetism in the 2H -VSe 2 clearly shows a thickness dependence. It monotonically \nsoftens with the increased sample thickness within the sample range where the reproducible 2H  phase samples span from 15 nm (about 20 layer s) to 47 nm  in our \nexperiments, until the ferromagnetism disappears at the sample thicker than 50 nm . The thickness dependent coercive force and the saturated  MCD signal are summarized in F ig. \n2b &  2c.  \nThe thickness dependence could also be attributed to the magnetic anisotropy. There \ncould be two kinds of anisotropy playing a role in the ferromagnetism here. The first lies \nin the layer -layer stacking order. The 2H  phase crystals follow a Bernal packing order \nand the A -B-A stacking restore s the spatial inversion symmetry as a whole in the Van der \nWaals crystal s. As the thickness thin s into 2D, the inversion s ymmetry is gradually \nbroken at multilayer level. This is demonstrated by the characterization of second harmonic generation (SHG) which is a second order nonlinear optical effect displayed in \nsystems without structural inversion symmetry. Fig. 3b presents a SHG mapping on 2H \nVSe\n2 nano- flakes . The SHG intensity monotonically increases with the reduced \nthickness, implying the increased spatial asymmetry at thinner crystals. The other origin of the anisotropy is the interface effect. The interface breaks the i nversion symmetry that \nagain induces magnetic anisotropy and renders the out -of-plane easy axis for \nferromagnetic thin films. For the 2D VSe\n2, the anisotropy owing to interface effect \ndramatically increase as the sample shrinks  to atomic layers . \nThe saturated  MCD signal s well follow the power -law form of 𝛼𝛼(1−𝑇𝑇/𝑇𝑇𝐶𝐶)𝛽𝛽where 𝛼𝛼, 𝛽𝛽 \nand 𝑇𝑇𝐶𝐶 as simultaneous fitting parameters. For the representative multilayer sample of a \n16.9 nm thick 2H -VSe 2 flakes, 𝛽𝛽 is given at 0.332 ± 0.045, consistent with 𝛽𝛽=0.326 for \nthe 3D Ising model (Adj. R -Square is 0.995). The deviation from the Heisenberg \nuniversality class indicates the presence of the magnetic anisotropy that lowers the spin symmetry from SO(3)  down to Z2 . The Ising universality class is consistent  with the \nexpectation for the spin- orbit coupling that breaks the spin rotational symmetry and \ngenerates the magnetic anisotropy. \nIn summary, we demonstrated the 2H  phase V Se2 multilayers as a high temperature  2D \nferromagnetic semiconductor . The ferromagnetic ordering exists only at 2D form owing \nto the enhanced structural anisotropy.  \nMethod s \nSample preparation.  The chemical vapor transport method was used to grow VSe 2 \nsingle crystals with iodine as the transport agent. The typical method is to put a mixture \nof vanadium, selenium powder and iodine powder with a stoichiometric ratio of 1:2 into a vacuum quartz tube. Then, the tube was put into a muffle furnace. Warm up to 850 °C, store for 2 days, slowly cool to 500 °C within 2 days, and finally cool to room \ntemperature. A few millimeters of VSe\n2 crystals can be obtained.  \nThe VSe 2 single crystals are grown using chemical vapor transport method and the \nnanoflakes with the thickness ranging from about  10 to 50 nm are mechanically \nexfoliated onto silicon substrates capped with a 300- nm oxide layer in a glove box (H 2O \nand O 2 < 0.1 ppm). We cannot obtain samples thinner than 8 nm in our mechanical \nexfoliation. The thickness of flakes was visually pre -screened under optical microscope \nand precisely measured with atomic force microscope (AFM). The thermal annealing was carried out in a tube furnace in argon atmosphere. After annealing at 650 K, the samples \nwere cooled down to ambient temperature in the same argon environment. The sample was placed in a high vacuum chamber (about 10\n-6 mbar) to prevent the influence of air \nand moisture. \nMCD microscopy.  Materials with non -zero magnetic moments can exhibit magnetic \ncircular dichroism  (MCD). MCD arises from the differential absorption of left -circularly \npolarized (LCP) light and right -circularly polarized (RCP) light. The MCD measurements \nwere performed at  a home -made heating stage in a vacuum with a temperature range \nfrom 300 K to 425 K unde r an out -of-plane magnetic field. An incident beam from the \npower -stabilized 633 nm laser diode of about 80 µW was parallel to the magnetic vector \nand normal to the reflecting surface. The beam  was focused through an objective lens \nonto the VSe 2 flakes wit h a spot size of about 1 µm and the reflected beam was collected \nby the same objective lens . The incident beam was modulated by a photoelastic \nmodulator (PEM). Then the MCD signal carried by the reflected beam was detected by  \nbalanced photodiode .  \nSHG meas urement.  The second harmonic generation (SHG) measurement was carried \nout by an excitation pulse from a Ti: sapphire oscillator (120fs, 80MHz).  \nAcknowledgement:  \nThe work was supported by Croucher foundation, GRF  #17304518, CRF  C7036- 17W  of \nthe Research Grant Council of Hong Kong and MOST 2020YFA0309603.   \nAuthor Contribution  \nXC conceived and supervised the project. XW and DL conducted the experiments and \nanalyzed the data. ZL and CW fabricated the VSe 2 single crystals. GC provided the \ntheory support. XW, GC and XC wrote the paper.  \nAdditional information  \nThe authors declare no competing financial interests. Correspondence  and requests for \nmaterials should be addressed to X.C. \n \nReference  \n1 Soumyanarayanan, A., Reyren, N., Fert, A. & Panagopoulos, C. Emergent phenomena \ninduced by spin –orbit coupling at surfaces and interfaces. Nature  539, 509 -517, \ndoi:10.1038/nature19820 (2016).  \n2 Žutić, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamental s and applications. Reviews \nof Modern Physics  76, 323 -410, doi:10.1103/revmodphys.76.323 (2004).  \n3 Fei, Z.  et al.  Two -dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2. Nat \nMater  17, 778 -782, doi:10.1038/s41563 -018-0149- 7 (2018).  \n4 Gong, C.  et al.  Discovery of intrinsic ferromagnetism in two -dimensional van der Waals \ncrystals. Nature  546, 265 -269, doi:10.1038/nature22060 (2017).  5 Huang, B.  et al.  Layer -dependent ferromagnetism in a van der Waals crystal down to the \nmonolayer limit. Nature  546, 270 -273, doi:10.1038/nature22391 (2017).  \n6 Mermin, N. D. & Wagner, H. Absence of Ferromagnetism or Antiferromagnetism in One-  \nor Two -Dimensional Isotropic Heisenberg Models. Physical Review Letters  17, 1133 -\n1136, doi:10.1103/physrevlett.17.1133 (1966).  \n7 Bonilla, M.  et al.  Strong room -temperature ferromagnetism in VSe2 monolayers on van \nder Waals substrates. Nat Nanotechnol  13, 289 -293, doi:10.1038/s41565 -018-0063 -9 \n(2018).  \n8 Yu, W.  et al.  Chemically Exfoliated VSe2 Monolayers with Room -Temperature \nFerromagnetism. Adv Mater , e1903779, doi:10.1002/adma.201903779 (2019).  \n9 Ma, Y.  et al.  Evidence of the Existence of Magnetism in Pristine VX2 Monolayers (X = S, \nSe) and Their Strain -Induced Tuna ble Magnetic Properties. ACS Nano  6, 1695 -1701, \ndoi:10.1021/nn204667z (2012).  \n10 Pan, H. Electronic and Magnetic Properties of Vanadium Dichalcogenides Monolayers \nTuned by Hydrogenation. The Journal of Physical Chemistry C  118, 13248 -13253, \ndoi:10.1021/jp503030b (2014).  \n11 Pushkarev, G. V., Mazurenko, V. G., Mazurenko, V. V. & Boukhvalov, D. W. Structural \nphase transitions in VSe2: energetics, electronic structure and magnetism. Physical \nChemistry Chemical Physics  21, 22647 -22653, doi:10.1039/c9cp03726h (20 19). \n12 Feng, J.  et al.  Electronic Structure and Enhanced Charge -Density Wave Order of \nMonolayer VSe2. Nano Lett  18, 4493 -4499, doi:10.1021/acs.nanolett.8b01649 (2018).  \n13 Duvjir, G.  et al.  Emergence of a Metal– Insulator Transition and High -Temperature \nCharge-Density Waves in VSe2 at the Monolayer Limit. Nano Letters  18, 5432- 5438, \ndoi:10.1021/acs.nanolett.8b01764 (2018).  \n14 Fumega, A. O.  et al.  Absence of Ferromagnetism in VSe2 Caused by Its Charge Density \nWave Phase. The Journal of Physical Chemistry C  123, 27802 -27810, \ndoi:10.1021/acs.jpcc.9b08868 (2019).  \n15 Coelho, P. M.  et al.  Charge Density Wave State Suppresses Ferromagnetic Ordering in \nVSe2 Monolayers. The Journal of Physical Chemistry C  123, 14089 -14096, \ndoi:10.1021/acs.jpcc.9b04281 (2019).  \n16 Wong,  P. K. J.  et al.  Evidence of Spin Frustration in a Vanadium Diselenide Monolayer \nMagnet. Advanced Materials  31, 1901185, doi:10.1002/adma.201901185 (2019).  \n17 Chhowalla, M.  et al.  The chemistry of two -dimensional layered transition metal \ndichalcogenide nan osheets. Nature Chemistry  5, 263 -275, doi:10.1038/nchem.1589 \n(2013).  \n18 Xu, K.  et al.  Ultrathin nanosheets of vanadium diselenide: a metallic two -dimensional \nmaterial with ferromagnetic charge -density -wave behavior. Angew Chem Int Ed Engl  52, \n10477 -10481, doi:10.1002/anie.201304337 (2013).  \n19 Li, D.  et al.  Structural Phase Transition of Multilayer VSe2. ACS Applied Materials & \nInterfaces  12, 25143 -25149, doi:10.1021/acsami.0c04449 (2020).  \n20 Tong, W. -Y., Gong, S. -J., Wan, X. & Duan, C. -G. Concepts of ferrov alley material and \nanomalous valley Hall effect. Nature Communications  7, 13612, \ndoi:10.1038/ncomms13612 (2016).  \n21 Fuh, H. R.  et al.  Newtype single -layer magnetic semiconductor in transition -metal \ndichalcogenides VX2 (X = S, Se and Te). Sci Rep  6, 32625, doi:10.1038/srep32625 (2016).  \n22 Liu, J.  et al.  Intrinsic valley polarization of magnetic VSe2 monolayers. J Phys Condens \nMatter  29, 255501, doi:10.1088/1361 -648X/aa6e6e (2017).   \nFigures and figure  captions  \n \n  \nFigure 1 . Comparisons of structural  symmetry , electric al and magnetic  properti es \nbetween two phases  of 2D VSe 2. a, The crystal structures of the 1T - and 2H -VSe 2. b, c, \nThe crystal unit cell s are represented by  the red  dashed rectangular boxes. The \nrepresentative temperature dependent electric resistivity of 1T -VSe 2 (b) and 2H -VSe 2 (c). \nThe t emperature dependence  show s the contrasting behaviors: metallic 1T  phase vs. \nsemiconducting 2H  phase  correspondingly. d , Representative MCD signals for the \ncorresponding multilayer VSe 2 at 300 K.  Error bars indicate the standard deviation (SD) \nof sample signals.  \n \n   \n \n \n \nFigure 2. Magnetic properties of m ultilayer 2H -VSe 2 on Si/SiO 2 substrate s at \ndifferent thicknesses . a, MCD hysteresis loops of 2H -VSe 2 of 15.3 and 29.7 nm , \nrespectively . b, c,  The t hickness dependence of coercive field Hc (b) and saturated  MCD \nsignal  (c) for 2H-VSe 2. The  Hc, MCD signal  and thickness error bars  indicate  \nuncertaintie s in calibration of measurement s. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3. The change of i nversion symmetry  with various thickness  of 2H -VSe 2. a, b, \nThe AFM topography ( a) and the  SHG mapping ( b) of the 2H -VSe 2 nano- flakes . c, The \nsample t hickness dependence of SHG intensity for multilayer 2H -VSe 2. The  error bars \ncorrespond to the standard deviations of SHG signals from four regions  of various \nthickness . d, The SHG intensity as a function of the excitation intensity .  \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4.  The representative temperature dependent MCD of a 16.9 nm -thick 2H -\nVSe 2. a, The hysteresis loops of the 2H -VSe 2 at 300 and 400 K. b , c, The coercive field \nHc (b) and the s aturated  MCD signal ( c) as a function of temperature . The fitting curve \nfollows  the form  of 𝛼𝛼(1−𝑇𝑇/𝑇𝑇𝐶𝐶)𝛽𝛽 where  𝑇𝑇𝐶𝐶 = 418.5 ± 7.8 K is extracted.  \n \n  \n"    },    {        "title": "1402.1148v1.Re_entrant_superspin_glass_phase_in_La___0_82__Ca___0_18__MnO__3__ferromagnetic_insulator.pdf",        "content": "arXiv:1402.1148v1  [cond-mat.str-el]  5 Feb 2014Re-entrant superspin glass phase in La 0.82Ca0.18MnO 3ferromagnetic insulator\nP. Anil Kumar,1,2R. Mathieu,2P. Nordblad,2Sugata Ray,1,3\nOlof Karis,4Gabriella Andersson,4and D. D. Sarma1,4,5,6, ∗\n1Centre for Advanced Materials, Indian Association for the C ultivation of Science, Kolkata 700032, India\n2Department of Engineering Sciences, Uppsala University, P .O. Box 534, SE-751 21 Uppsala, Sweden\n3Department of Materials Science,Indian Association for th e Cultivation of Science, Kolkata 700032, India\n4Department of Physics and Astronomy, Uppsala University, B ox - 516, 75120 Uppsala, Sweden\n5Solid State and Structural Chemistry Unit, Indian Institut e of Science, Bangalore 560012, India\n6Also at Jawaharlal Nehru Centre for Advanced Scientiﬁc Rese arch,\nBangalore and Council of Scientiﬁc and Industrial Research -Network of Institutes for Solar Energy (CSIR-NISE).\n(Dated: November 23, 2021)\nWe report results of magnetization and ac susceptibility me asurements down to very low ﬁelds\non a single crystal of the perovskite manganite, La 0.82Ca0.18MnO3. This composition falls in the\nintriguing ferromagnetic insulator region of the manganit e phase diagram. In contrast to earlier\nbeliefs, our investigations reveal that the system is magne tically (and in every other sense) single-\nphase with a ferromagnetic ordering temperature of ∼170 K. However, this ferromagnetic state is\nmagnetically frustrated, and the system exhibits pronounc ed glassy dynamics below 90 K. Based on\nmeasured dynamical properties, we propose that this quasi- long-ranged ferromagnetic phase, and\nassociated superspin glass behavior, is the true magnetic s tate of the system, rather than being a\nmacroscopic mixture of ferromagnetic and antiferromagnet ic phases as often suggested. Our results\nprovide an understanding of the quantum phase transition fr om an antiferromagnetic insulator to\na ferromagnetic metal via this ferromagnetic insulating st ate as a function of xin La1−xCaxMnO3,\nin terms of the possible formation of magnetic polarons.\nI. INTRODUCTION\nHole-doped perovskite manganites of general formula\nA1−xBxMnO3(A= trivalent Lanthanide, B= divalent\nalkalimetal)withalowdoping, generally0.05 < x <0.22\nforA= La and B= Ca, are of fundamental interest be-\ncause they constitute the few examples of ferromagnetic\ninsulators, unlike the ones with higher xor larger band-\nwidthwhichareferromagneticbutalsometallic,orwitha\nlowerxwhichareinsulating, but alsoantiferromagnetic.1\nWhile the origin of ferromagnetic, insulating state in\nsome undoped compounds, such as La 2NiMnO 6,2,3has\nbeen explained in the past, chargedoped systems present\nadditional diﬃculties in comprehending a ferromagnetic\ninsulating ground state. Speciﬁcally, the origin of the\ncoexistence of ferromagnetic and insulating properties\nin La1−xCaxMnO3(LCMO), has not been established.\nThe ferromagnetic metal (FMM) phase of the perovskite\nmanganites with larger doping has been successfully ex-\nplained by considering the double exchange mechanism\nproposed by Zener.4,5This mechanism, while explaining\nthe ferromagnetism, invariably requires the system to be\nmetallic due to hopping of electrons between Mn3+and\nMn4+and hence failing to explain the ferromagnetic in-\nsulating state. Further it is to be noted that electronic\nphase separation of varying length scales has been re-\nported at higher Ca doping levels.6\nA major obstacle in establishing the true ground state\nin case of manganites has been the diﬃculty in preparing\nsinglephasesamples, whichhasleadto the speculation7,8\nthat the ferromagnetic insulating (FMI) state is the re-\nsultofspatiallydistinctcoexistenceofseparateferromag-\nnetic metallic and antiferromagnetic insulating phasesin a single sample. However, investigations performed\nusing other single crystal samples have suggested mi-\ncroscopically homogeneous electronic properties of the\nsamples.9–11Interestingly, even the reports on these ho-\nmogeneous samples are interpreted to promote contrast-\ning pictures for the magnetic ground state. For exam-\nple, the authors of Refs.[ 9,10] observed a non-diverging\nmagnetic correlationlength and signaturesof short range\nmagnetic polarons using neutron scattering experiments,\nwhile a much more recent work11suggested an ideal\n3-dimensional Heisenberg ferromagnetic ground state,\nbased on the values of critical exponents that they ob-\ntain for the FMI composition of LCMO. Nevertheless,\nboth theory and experiments suggest formation of local\nlattice distortions or lattice polarons in the ordered state\nleadingtoananoscaleinhomogeneitywhichisstarklydif-\nferent from the chemical or macroscopic electronic phase\nseparation.10,12–16\nAC susceptibility technique can be useful in deter-\nmining the true magnetic state of a material. An en-\nlightening report on the magnetic properties of a sin-\ngle crystal of La 0.8Ca0.2MnO3, which is closer to the\nFMM composition, using ac-techniques is found in Ref. [\n17] however without considering non-equilibrium (aging)\nand non-linear ﬁeld eﬀects. We here investigate in detail\nthe static and dynamical magnetic properties of a single-\ncrystal of La 0.82Ca0.18MnO3(LCMO18) whose composi-\ntion lies within the FMI regime (0.05 < x <0.22). It\nis found that the material orders ferromagnetically, al-\nbeit short ranged and displaying glassy dynamics. The\nobserved glassy dynamics suggest that frustration eﬀects\ngovern the ferromagnetic conﬁguration, as in re-entrant\nspin glass systems (re-entrant ferromagnet in the present2\ncase). However, with the signiﬁcant diﬀerence that the\nmagnetic entities in the present case are groups of spins\n(magnetic polarons), instead of the single spins of con-\nventional re-entrant magnets or spin glasses; i.e. a re-\nentrant superspin glass state. The results indicate that\nthe re-entrant ferromagnetic insulating state is the in-\ntrinsic magnetic state of this composition, rather than a\n(macroscopically) phase separated one. We discuss the\nmicroscopic nature and origin of the ferromagnetic insu-\nlating phase, as well as the transition to ferromagnetic\nmetal for larger hole doping.\nII. EXPERIMENTAL DETAILS\nLa0.82Ca0.18MnO3single crystal pieces were grown by\nﬂoating zone technique (Crystal Systems Corporation,\nJapan) starting from a single phase polycrystalline sam-\nple of the same composition. To compensate for the\nMn evaporation during crystal growth 1% MnO is ad-\nditionally added. The phase purity of the ﬁnal grown\ncrystal is veriﬁed by powder x-ray diﬀraction (XRD) us-\ning a Bruker D8 advance diﬀractometer. Spot analysis\nof energy dispersive x-ray spectroscopy on a Jeol FE-\nSEM reveals that the cation distribution in the plane\nperpendicular to the growth direction is uniform, and\nthat the cation concentrations are also close to the nom-\ninal composition. The correct stoichiometry of the single\ncrystals was additionally veriﬁed by Inductively Coupled\nPlasma-opticalemissionspectroscopy(ICP-OES)usinga\nPerkin Elmer instrument. The electronic transport mea-\nsurements are carried out using a laboratory setup and\nPPMS from Quantum Design Inc. PPMS is also used for\nheat capacity measurements. Magnetization Mand ac\nsusceptibility (in-phase component χ′and out-of-phase\ncomponent χ′′) data are collected, on a roughly rectan-\ngular piece of the crystal with 5 mm ×1.2 mm×1.5 mm\nsize, using Quantum Design MPMS XL SQUID mag-\nnetometer. Magneto-optic Kerr eﬀect (MOKE) images\nwere recorded at temperatures between 80 K and 273\nK using a polarizing microscope with an external ﬁeld\napplied in the plane of the sample, and parallel to the\nscattering plane of the light (longitudinal MOKE conﬁg-\nuration).\nIII. RESULTS AND DISCUSSION\nIn Figure1(a), wepresentthe temperaturedependence\noftheelectricalresistance, R, oftheLCMO18singlecrys-\ntal. A weak signatureof insulatorto metal transition can\nbe observed in the zero magnetic ﬁeld data at about 170\nK before re-entering an insulating state on further low-\nering the temperature below ∼140 K. It is in agreement\nwith the earlier literature data18for this composition,\nthis fact also ensures the quality of the sample, since the\nelectronic transport behavior is the most sensitive prop-\nerty to the oxygen non-stoichiometry and compositional\nFIG. 1. (Color) (a) Temperature dependence of the elec-\ntrical resistance Rof LCMO18 recorded under zero and 40\nkOe ﬁelds. The third curve (right axis) shows the associ-\nated magnetoresistance MR% = 100 ×(R(H= 0)-R(H=\n40 kOe))/ R(H= 0) (b) The behavior of Rwith tempera-\nture,Tfor diﬀerent excitation currents is shown to exemplify\nthe electroresistance behavior. The inset in panel (b) show s a\ntypicalI-Vcurve recorded at T= 50 K. The ﬂat region in the\ncurve is due to the current limit, 10 mA, of the electrometer.\nvariation. The associated magnetoresistance ( MR) ob-\nserved in 40 kOe applied ﬁeld is also indicated in the ﬁg-\nure. Interestingly, this composition also exhibits colossal\nelectroresistance (CER)19,20as demonstrated in Figure\n1(b) and the inset. The resistance of the crystal is in-\ndeed highly sensitive to the magnitude of current that\npassesthroughthesample. Theinsetshowsthereversible\nswitching of sample resistance with applied voltage. Cor-\nresponding results on MR and CER of LCMO have been\nreportedin the literature, howeverthe detailsofthe large\nelectroresistance are diﬀerent.19,20\nIn Figure2(a) we showthe zero-ﬁeld-cooled(ZFC) and\nﬁeld-cooled (FC) magnetization as a function of temper-\nature. In the ZFC curve, a sharp rise in Mresembling\na magnetic transition is observed near 170 K, coincid-\ning with the insulator-metal transition temperature (c.f.\nﬁg. 1(a)). In addition, a drop in ZFC magnetization is\nobserved at ∼70 K on lowering the temperature. This\nsecond anomaly is not reﬂected in the heat capacity ( C)\ncurve depicted in the inset to Figure 2(a). However a\npeak is observed in the C(T) curve in the vicinity of the\nﬁrst magnetic transition, conﬁrming the enhanced mag-\nnetic correlations in the system. The peak in heat ca-\npacity is relatively broad, suggesting that the long-range\nordering of the ferromagnetic state established at 170 K\nishindered.21Interestinglythemagnetization-ﬁeld(hys-\nteresis) loops measured at diﬀerent temperatures across3\nFIG. 2. (Color) (a) Temperature dependence of the zero-ﬁeld -\ncooled (ZFC) and ﬁeld-cooled (FC) magnetization in 30 Oe\nﬁeld. Inset shows the temperature dependence of the heat ca-\npacityCmeasured on heating. (b) Magnetic ﬁeld dependence\nof the magnetization (hysteresis loops) measured at diﬀere nt\ntemperatures (10, 50, 100, 150, 200 and 250 K).\nthe two anomalies in M(T) curve, presented in Figure\n2(b), show ferromagnetic behavior at and below 150 K\nin line with the magnetic ordering at 170 K. The satu-\nration moment per Mn ( ∼3.78µB) at 10 K is very close\nto the value (3.82 µB) expected for a full ferromagnetic\narrangement of Mn ions in the system. The tempera-\nture and magnetic ﬁeld dependence of the magnetization\nhence suggests that the system is ferromagnetically or-\ndered below 170 K. However, with hindered critical di-\nvergenceat TC. Yet, the high ﬁeld magnetic moment (for\nﬁelds largerthan 10 kOe) correspondsto full polarization\nof that ferromagneticstate, suggesting the lack of macro-\nscopic phase separation into ferromagnetic and antiferro-\nmagnetic regions. Further, Kerr microscopy images col-\nlected on our crystal at 100 K did not show any domain\nstructure within the instrumental resolution limit of 0.5\nµm. However, the magneto-optic-Kerr-eﬀect (MOKE)\nintensity follows the bulk magnetization value as shown\nin Figure 3. The MOKE data corresponds to a 425 ×\n325µm2area of the crystal with the ﬁeld applied in the\nsame plane as for the bulk magnetization measurement.\nThe result of temperature dependent ac-susceptibility\nmeasurements is presented in Figure 4. Data collected\nusing three diﬀerent ac-ﬁeld amplitudes, h, are shown.\nAs seen in the ﬁgure, the temperature dependence of the\nin- and out-of-phase components of the susceptibility is\nqualitatively similar for all amplitudes of the ac prob-\nFIG. 3. Comparison of MOKE intensity to the bulk magne-\ntization curve measured at 100 K.\nFIG. 4. (Color) Temperature dependence of the (a) in-phase\nχ′and(b)out-of-phase χ′′componentsoftheac-susceptibility\nrecorded using diﬀerent amplitudes of the probing ac ﬁeld ( h\n= 0.4, 1.25 and 4 Oe). The frequency fis 170 Hz. χ′′(T)\ndata obtained for 1.7 and 17 Hz are also plotted for h= 4 Oe\nas indication of dynamical behavior.\ning ﬁeld. For example the χ′(T) curves include a sharp\npeak near 170 K, and a drop in the susceptibility below\n90 K, akin to the features observed in the ZFC magne-\ntization presented in Figure 2(a). Sharp peaks near 170\nK and drops below 90 K are also observed for all am-\nplitudes in χ′′(T) curves. The low ﬁeld susceptibility is\nstrongly non-linear and enhanced by the amplitude of\nthe ac ﬁeld, h(see e.g. the 100 - 150 K temperature\ninterval for diﬀerent values of h). Additional measure-\nments as a function of amplitude hshow that the re-\nsponse of the system is essentially linear at least up to\n0.5 Oe at temperatures below 140 K. We hence choose\nan amplitude of 0.4 Oe for all subsequent ac magnetic\nmeasurements in order to study the intrinsic magnetism\nof the material. For this amplitude, the ac-susceptibility\ncurves (see left panels of Fig. 5) are reminiscent to those\nof re-entrant ferromagnets, which exhibit two peaks in\ntheχ′′(T) curves reﬂecting ferromagnetic and spin-glass4\nphase transitions respectively,3,22,23the higher tempera-\nturetransitionwithnofrequencydispersionwhilethelow\ntemperature transition showing clear frequency disper-\nsion. Spin glass states result from frustration eﬀects due\nto the presence of competing magnetic interactions (the\nfrustration may also be geometric). Further, particulates\nof ferromagnetic domains may also exhibit glassy mag-\nnetic features when the particulate size is low enough.24\nIt is also important to note that the spin correlation\nlength does not diverge at the ferromagnetic transition\ntemperature in the FMI composition of LCMO, suggest-\ningformationofnanoscopicferromagneticdomainsatthe\ntransition temperature.9\nIn the susceptibility curves presented in the left pan-\nels [(a) and (b)] of Figure 5, the onset of the high tem-\nperature peak is frequency independent, as expected for\na ferromagnetic transition, while the lower temperature\npeak is quite frequency dependent, suggesting a low-\ntemperature glassy behavior. It is remarkable that, as\nobserved in systems with blocked magnetic clusters, the\nmagnitude of χ′′(T) drops very rapidly below the low\ntemperature peak and remains at and near zero, with\nnegligible frequency dependence, on further lowering of\nthe temperature. This behavior suggests that the fun-\ndamental entities of this glassy transition are groups of\ncoherentspins(superspins)ratherthanindividualatomic\nspins.22,25The low temperature ac-susceptibility of spin\nglasses is weakly aﬀected by low superimposed dc mag-\nnetic ﬁelds. This is also the case in LCMO18, as seen in\ntherightpanels[(c)and(d)]ofFigure5. Amagneticﬁeld\nof 50 Oe minorly aﬀects the susceptibility curve in the\nwhole measured range of temperatures. However, a ﬁeld\nof500Oesigniﬁcantlyreducestheac-susceptibilityabove\n80 K. The ac-susceptibility of the low-temperature phase\nis essentially not aﬀected by the superimposed magnetic\nﬁeld, akin to re-entrant ferromagnets.22\nAlthough disordered, spin glass materials undergo a\nmagnetic phase transition at a temperature Tgwith well-\nestablished critical exponents. If a spin glass is cooled\ndown below Tg, it will always be out of equilibrium,\nand its spin conﬁguration will rearrangeitself toward the\nequilibrium conﬁguration for that temperature.26,27This\nequilibration is referred to as aging.27–29Aging can be\nobserved in relaxation experiments, which can employ dc\nor ac excitation to record the isothermal magnetization\nor susceptibility as a function of time.27,30If the temper-\nature is changed, the system will again rearrange itself\ntowardtheequilibriumconﬁgurationofthenewtempera-\nture; the system will be reinitialized or rejuvenated. Yet,\nitcanbeshownthatthespinconﬁgurationresultingfrom\nthe ﬁrst equilibration is kept in memory while the second\nproceeds.27,31If the new temperature is lower than the\ninitial one, the systemequilibratesto conﬁgurationcorre-\nsponding to new temperature and keeps the equilibrated\nconﬁguration of the higher temperature in memory. On\nthe other hand, if the new temperature is higher than\nthe initial one, the system will be reinitialized, even for\nshort durations at the new temperature.Some systems will ﬁrst order magnetically as long-\nrangedferro-orantiferromagnets,tobecome(orre-enter)\na disordered phase at low temperatures. This is the case\nof re-entrant spin glasses, or in the case of ferromag-\nnets, re-entrant ferromagnets.23The disordered phase is\na consequence of the magnetic frustration in the ordered\nphase, hindering the perfect ordering. It has been shown\nthat the low-temperature spin glass phase of such mate-\nrials has similar dynamical behavior as those of ordinary\nspin glasses, and also that the ferromagnetic phase ex-\nhibits glassyfeaturesand aging,althoughthe spin conﬁg-\nuration of this phase reinitializes upon both positive and\nnegative temperature cycling, unlike in spin glasses.32\nWhile the critical slowing down at the spin glass phase\ntransition can be investigated by e.g. scaling analyses of\nthe onset of χ′′in ac-susceptibility measurements,26such\nanalyses are diﬃcult in re-entrant ferromagnets due to\nthe “parasitic” ferromagnetic phase contributing to the\nsusceptibility.22,23Nevertheless, the onset of low temper-\nature glass phase is indicated by the frequency depen-\ndence of the low temperature peak of the χ-Tcurves.\nAging phenomena are observed in LCMO18 at tem-\nperatures below 170 K. As illustrated in Figure 6(a), χ′′\nrelaxes downwards at constant temperature after being\ncooled from a reference temperature in the paramagnetic\nregion (Tref= 220 K). Yet, the shape and magnitude\nof the relaxation of the χ′′(t) curves are quite diﬀerent\nat temperatures above and below the low temperature\npeak. The inset in Figure 6(a) shows the relaxation at\n60 K in an expanded scale to clearly illustrate the dif-\nference in shape of the relaxation in the ferro and glass\nphases of the sample. The time-dependent susceptibility\ndatarecordedatvarioustemperaturesfrom50to180Kis\nplotted (vertical lines) in Figure 6(b) and (c) as function\nof temperature. For comparison, a conventional temper-\nature dependent ac-susceptibility measurement recorded\nusing the same frequency and excitation ( f= 1.7 Hz, h\n= 0.4 Oe) is also plotted. We can see in those panels\nthat the weak downward relaxation observed at 50, 60\nand 70 K start from about the same susceptibility values\nasthose obtainedin the ordinarytemperature-dependent\nmeasurement. This natureoftheagingrelatedrelaxation\nis similar to what is expected and observed to occur in\nspin glass states.27,30For the temperatures above 70K,\nthe behavior is quite diﬀerent, and one can see (more\nclearlyin ﬁg. 6 (c)) that the relaxationstartsfromhigher\nvalues of the ac-susceptibility, and ﬁnishes below the T-\ndependent values on the longest time observed in our ex-\nperiments (3600 s). This behavior is reminiscent of that\nof the ferromagnetic phase of a re-entrant ferromagnet.32\nThe diﬀerence in relaxation behavior between the two\ntemperature regions (above and below 80 K) is again an\nevidence of the phase change from a frustrated ferromag-\nnetically dominated response to a glassy one.\nComing back to the origin of the magnetic be-\nhavior in these compounds, there are some plausible\nexplanations16,33,34such as the two electron ﬂuid lb\nmodel; onetypeofelectronsareessentiallylocalizedcom-5\nFIG. 5. (Color) Temperature dependence of the (upper panels ) in-phase χ′and (lower panels) out-of-phase χ′′components\nof the ac-susceptibility recorded for (a,b) diﬀerent frequ enciesf= 1.7, 17, and 170 Hz; h= 0.4 Oe, and (c,d) under diﬀerent\nsuperimposed dc magnetic ﬁelds H= 0, 10, 50 and 500 Oe; f= 170 Hz; h= 0.4 Oe.\nbined with a distortion of MnO 6octahedra (polaronic)\nwhile the other type are characterised by ﬁnite hopping\nandnon-distortedlattice. Oneoftheexperimentalproofs\nforsuchascenariocomesfromtheextendedx-rayabsorp-\ntion ﬁne structure (EXAFS) analysis.35–37The micro-\nscopiccrystalstructureisprobedusingEXAFS inseveral\ndoped manganite samples/crystals which focuses on the\nfractionofJahn-Teller(JT) distortedMnO 6octahedrain\nthe lattice; Mn3+is JT active while the Mn4+is JT inac-\ntive and in doped manganites there is a mixture of both.\nIt is concluded that in the FMM samples the JT distor-\ntion is completely removed in the fully magnetized state\nof the sample which is achieved either by lowering the\ntemperature or by application of external magnetic ﬁeld.\nHowever, in the case of FMI samples there exist JT dis-\ntorted MnO 6octahedra even in the ferromagnetic state,\nnevertheless the application of relatively large magnetic\nﬁeld leads to reduction in the fraction of JT distorted\nMnO6octahedra.36These observations are linked to the\nfact thatin FMM samplesthe egelectronsarecompletely\nitinerant and hence no Mn ions are JT active whereas\nin the FMI samples the egelectrons are localized be-\ntween neighbouring Mn ions and hence some Mn ions are\nJT active. Due to such a combination of polaronic and\nnon-polaronic MnO 6octahedra in the FMI compositions\nthe magnetic properties are also expected to be diﬀerent\nfrom the ferromagnetic ground states that are observed\nin metallic compositions. The solid state nuclear mag-\nnetic resonance (NMR) experiments on LCMO samples\nhave also pointed to the presence of diﬀerent fractions\nof FMI and FMM phases depending on the composition\nand temperature.38Our results suggest that the mag-\nnetic and electrical properties with hole doping in low\nbandwidth manganites evolve from an antiferromagnetic\ninsulator (AFI) at lowest doping levels via a frustratedferromagnetic insulator (FFMI) for intermediate doping\nto the ferromagnetic metal (FMM) at higher doping.\nIV. CONCLUSIONS\nWe have reported the results of dc and ac magnetic\ncharacterization of a single crystal of La 0.82Ca0.18MnO3,\nwhich falls in the fundamentally interesting composition\nrange of a ferromagnetic insulating phase in the family\nof manganites. The dc and ac magnetic measurements\ncombinedly suggest that the low temperature magnetic\nstate of this composition is a re-entrant ferromagnet.\nI.e. LCMO18 enters a frustrated but all ferromagnetic\nstate at 170 K followed by a re-entrant superspin glass\nstate at about 80 K that is transformed to a fully mag-\nnetized ferromagnetic state by a large enough magnetic\nﬁeld. This intrinsic glassy magnetic state is in accor-\ndance with the results of theoretical models and EXAFS\nexperiments on the low doped manganites, which predict\nmagnetic polaron formation. A correlated group of such\npolarons which make up the superspins can be termed as\na nanoscopic phase separation.\nACKNOWLEDGMENTS\nAuthors thank the Department of Science and Tech-\nnology, Government of India and Swedish Foundation for\nInternational Cooperation in Research and Higher Edu-\ncation (STINT) for supporting this research. PAK, RM\nand PN thank the Swedish Research Council (VR) and\nthe G¨ oran Gustafsson Foundation, Sweden for funding.6\nFIG. 6. (Color) (a): time-dependence of the normalized out-\nof-phase susceptibility for diﬀerent temperatures record ed, af-\nter aquenchfrom 220K, at T=50, 60, 70, 80, and90K. Inset\nshows a similar curve measured at 60 K but using a higher\nﬁeld amplitude, h= 4 Oe. (b) and (c): the time-dependent\ndata presented in the upper panel is plotted as function of\ntemperature, for the in-phase and out-of phase components\nof the susceptibility, without normalization. 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