Origin of the Magnetic Spin Hall Effect: Spin Current Vorticity in the Fermi Sea
Alexander Mook, Robin R. Neumann, Annika Johansson, Jürgen Henk, Ingrid Mertig
OOrigin of the Magnetic Spin Hall E ff ect: Spin Current Vorticity in the Fermi Sea Alexander Mook,
1, 2
Robin R. Neumann, Annika Johansson, J¨urgen Henk, and Ingrid Mertig
1, 3 Institut f¨ur Physik, Martin-Luther-Universit¨at Halle-Wittenberg, D-06099 Halle (Saale), Germany Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Max-Planck-Institut f¨ur Mikrostrukturphysik, D-06120 Halle (Saale), Germany (Dated: October 30, 2019)The interplay of spin-orbit coupling (SOC) and magnetism gives rise to a plethora of charge-to-spin conversionphenomena that harbor great potential for spintronics applications. In addition to the spin Hall e ff ect, magnetsmay exhibit a magnetic spin Hall e ff ect (MSHE), as was recently discovered [Kimata et al. , Nature , 627-630(2019)]. To date, the MSHE is still awaiting its intuitive explanation. Here we relate the MSHE to the vorticityof spin currents in the Fermi sea, which explains pictorially the origin of the MSHE. For all magnetic Lauegroups that allow for nonzero spin current vorticities the related tensor elements of the MSH conductivity aregiven. Minimal requirements for the occurrence of a MSHE are compatibility with either a magnetization or amagnetic toroidal quadrupole. This finding implies in particular that the MSHE is expected in all ferromagnetswith su ffi ciently large SOC. To substantiate our symmetry analysis, we present various models, in particular atwo-dimensional magnetized Rashba electron gas, that corroborate an interpretation by means of spin currentvortices. Considering thermally induced spin transport and the magnetic spin Nernst e ff ect in magnetic insulators,which are brought about by magnons, our findings for electron transport can be carried over to the realm ofspincaloritronics, heat-to-spin conversion, and energy harvesting. I. FROM THE CONVENTIONAL TO THE MAGNETICSPIN HALL EFFECT
The spin Hall e ff ect (SHE) [1] and its inverse are withoutdoubt important discoveries [2–6] in the field of spintronics[7, 8]. They serve not only as ‘working horses’ for generatingand detecting spin currents [9] but also as key ingredients inspin-orbit torque devices for electric magnetization switching[10–12]. Compared to spin-transfer torque devices [13–17],spin-orbit torque devices are faster, more robust, and consumeless power upon operation [18–20]; for a recent review seeRef. 21.While the anomalous Hall e ff ect (AHE) in a magnet [22]produces a transverse charge current density upon applying anelectric field E , the SHE in a nonmagnet produces a transversespin current density (cid:104) j γ (cid:105) = σ γ E ( γ = x , y , z indicates thetransported spin component). Mathematically, the SHE isquantified by the antisymmetric part of the spin conductivitytensor σ γ . For example, the σ zxy element comprises z -polarizedspin currents in x direction as a response to an electric field in y direction.In a simple picture, the intrinsic SHE [23, 24] is explained byspinning electrons that experience a spin-dependent Magnusforce caused by spin-orbit coupling (SOC). It appears as if‘built-in’ spin-dependent magnetic fields evoke spin-dependentLorentz forces that result in a transverse pure spin current. Theextrinsic SHE [25–27] is covered by Mott scattering at defects[28].Since the SHE does not rely on broken time-reversal symme-try (TRS), it is featured in nonmagnetic metals [29] or semicon-ductors [2]. Imposing few demands on a material’s properties,a SHE can be expected in any material with su ffi ciently largeSOC (or, instead of SOC, with a noncollinear magnetic texture[30]). From a mathematical perspective the existence of anSHE can be traced to the transformation behavior of the sum σ zxy − σ zyx + σ xyz − σ xzy + σ yzx − σ yxz , (1) of antisymmetric spin conductivity tensor elements tradition-ally associated with a SHE (applied field, current flow direction,and transported spin component are mutually orthogonal). Tak-ing time-reversal evenness for granted, this sum behaves likean electric monopole (space-inversion even, scalar). For thereare no crystalline symmetries (reflections, rotations, inversions)that could render such an object zero, a SHE can basically oc-cur in any material. For the rest of this Paper, we refer to thisSHE as ‘conventional SHE.’To combine the virtues of transverse spin transport with mag-netic recording, the conventional SHE was studied in magneticmaterials with broken TRS (ferromagnets [31–44] or antifer-romagnets [30, 45–48]), which revealed various phenomenaassociated with the interplay of SOC and magnetism. Forexample, ferromagnetic metals exhibit an (inverse) conven-tional SHE [31]; una ff ected by magnetization reversal [33] itis time-reversal even.Since charge currents in ferromagnets are intrinsically spin-polarized, transverse AHE currents are spin-polarized as welland are used to generate spin torques [32, 35]. This e ff ect issometimes referred to as anomalous SHE [35], but is funda-mentally a conventional SHE. Since these spin currents are tiedto the AHE charge currents, the spin accumulations broughtabout by this e ff ect can be manipulated by varying the magne-tization direction [38, 42, 43]. This finding can be understoodby considering symmetries. For a nonmagnetic cubic mate-rial only the components in Eq. (1) are allowed nonzero. Incontrast, a ferromagnetic material magnetized along a genericdirection has a lower symmetry: there are no constraints thatprohibit ‘populating’ the entire spin conductivity tensor. Uponmanipulation of the magnetization, an electric field in, say, y direction causes an arbitrary spin polarization flowing in, e. g., x direction. This o ff ers greater versatility for spin torque ap-plications than the conventional SHE in nonmagnetic cubicmaterials (nonmagnetic and noncubic materials also admit ofgreater versatility and nontraditional tensor elements [49–52],but they do not o ff er external means, such as magnetization, to a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t manipulate spin polarizations).Since TRS is intrinsically broken in magnets, one expectsthat spin accumulations brought about by transverse spin cur-rents have two components, one that does not reverse undermagnetization reversal (we will refer this e ff ect as SHE, a sub-set of which is the conventional SHE) and a second that isreversed under magnetization reversal. This opposite behaviorunder time reversal causes di ff erent restrictions imposed by themagnetic point-group symmetry [45, 50] on the two types ofspin accumulations. In particular, the latter magnetism-inducedaccumulations do not have to be parallelly polarized to the SHEspin accumulations. Such signatures were observed in Ref. 36.The disentanglement of spin current contributions odd oreven under time reversal has been elucidated in Ref. 45. Inessence, the spin conductivity tensor in Kubo transport the-ory is decomposed into a time-reversal even part and a time-reversal odd part. Upon disregarding spin-dependent scattering,skew scattering, and side jumps, the time-reversal even partis associated with ‘intrinsic’ contributions to the spin conduc-tivity, a contribution given solely in terms of band structureproperties (in the so-called clean limit) [45]. Likewise, thetime-reversal odd part is associated with ‘extrinsic’ contribu-tions that depend on relaxation times [45]. The latter givesrise to the magnetism-induced e ff ects. In systems with lowsymmetry both parts contribute to all components of σ γ and,in particular, to its antisymmetric part: the time-reversal evenpart gives rise to the SHE and the odd part to the magnetic spinHall e ff ect (MSHE) [45, 47, 48].The MSHE has recently been experimentally detected in thenoncollinear antiferromagnet Mn Sn [48], and the aforemen-tioned results of Ref. 36 on ferromagnets can also be consid-ered proof of the MSHE (in Ref. 53 referred to as ‘transverseSHE with spin rotation’). Although instances of the MSHEhave been identified, an intuitive picture that explains how andunder which circumstances this e ff ect comes about is missing. II. CHIRAL VORTICES OF SPIN CURRENTS: SUMMARYOF THIS PAPER
We o ff er a vivid microscopic picture of the MSHE by relat-ing it to the spin current vorticity (SCV) of the Fermi sea or,equivalently, to the circulation of spin currents about the Fermisurface.In a rough draft, magnetic materials feature spin currentwhirlpools (or vortices) in reciprocal space for each of thethree spin directions γ = x , y , z ; as usual for angular quantities,we denote the axis of a vortex by a vector ω γ . Similar to wa-ter whirlpools (in real space), whose handedness leads to anasymmetric deflection of plane water waves, the spin currentwhirlpools (in reciprocal space) cause an asymmetric deflec-tion of the respective spin component. Since the spin currentvortices occur in reciprocal space, they are delocalized in realspace and, hence, do not act as scattering centers (like defects)but rather like an overall vortical background. To rephrase thisstatement in mathematical terms: although spin transport istreated within the constant relaxation time approximation thatdoes not capture asymmetric scattering at defects (thereby rul- (a) M E y xz
MSHESHE (b) × M E
FIG. 1. Conventional and magnetic spin Hall e ff ect in ferromagnetswith magnetization M along (a) z direction and (b) − z direction (e. g.,MLG 4 / mm (cid:48) m (cid:48) ). Upon application of an electric field E in z direction(and a charge current in the same direction) spin accumulates at theboundaries of the sample. Those associated with the conventionalspin Hall e ff ect (SHE) are indicated by blue arrows, those with theMSHE by red arrows. Magnetization reversal acts only on the MSHEaccumulations. ing out extrinsic skew scattering and side jump contributions),the MSHE is captured, because the spin current itself—but notthe scattering—is chiral.In terms of SCVs, the time-reversal odd nature of the MSHEis easily understood as a reversal of a vortex’s handedness thatresults in opposite deflection. Then, a reversal of the magnetictexture has to reverse the spin accumulations brought about bythe MSHE spin currents as well; recall that SHE spin currentsremain una ff ected.In order to show that these spin current vortices may existwe analyze all magnetic Laue groups (MLGs) with respect totheir compatibility with a nonzero SCV, thereby identifying allpossible MSHE scenarios.One especially simple scenario is a ferromagnet with SOC:assuming a tetragonal ferromagnet with magnetization M in z direction we find the SCVs ω x (cid:20) ˆ y , ω y (cid:20) − ˆ x , and ω z = . (2)For the z spin component, the Fermi sea is loosely speaking‘calm’ and does not cause an MSHE. In contrast, the x and y spin components ‘experience a rough chiral Fermi sea’: thenonzero vorticities cause MSHEs. More precisely, the MSHEfor the x ( y ) spin component takes place in the xz ( yz ) plane.Consequently, such ferromagnets exhibit nonzero antisymmet-ric parts of σ γγ z (and σ γ z γ ) as long as γ = x , y ; the transportedspin component, the electric field, and the flow direction ofthe spin current lie within a plane that contains the magne-tization. This is why the MSHE spin currents are pure: thetransverse AHE charge currents compatible with a magnetiza-tion in z direction flow within the xy plane (i. e., normal to themagnetization).To elaborate on the di ff erence to the SHE, let us assume thatan electric field E (cid:107) M (cid:107) ˆ z is applied, as depicted in Fig. 1. Dueto the (conventional) SHE spin conductivity tensor elements( σ xyz = − σ yxz ) spin is accumulated within the surface planesof the sample (blue arrows). The polarization of these spinaccumulations is orthogonal to both E and the surface normal.Being time-reversal even, it does not flip under magnetizationreversal [compare panel (a) vs. (b)]. In contrast, the MSHE( σ xxz = σ yyz ) causes additional accumulations polarized normalto the surface planes (red arrows). Their time-reversal odd na-ture forces a flip upon magnetization reversal, as is representedby the reversed red arrows in panel (b).Thus, the MSHE allows to generate a spin accumulationorthogonal to the conventional SHE spin accumulation. Inscenarios in which the magnetization of the ferromagnet isfixed this feature may result in the decisive spin accumulationdirection necessary to perform a particular spin torque switch-ing. For example, the field-free magnetization switching ofa perpendicularly magnetized film observed in Ref. 40 maybe explained in terms of MSHE spin accumulations. We em-phasize that the existence of spin current vortices is a bulkproperty that gives rise to bulk MSHE spin currents, which,in turn, cause spin accumulations at the edges or interfaces.Additional interface e ff ects, as those accounted for in Refs. 37,40, and 54, come on top.That the SHE and MSHE cause spin accumulations pointingin orthogonal directions is a speciality of the MLGs 4 / mm (cid:48) m (cid:48) ,4 (cid:48) / mm (cid:48) m , and m (cid:48) m (cid:48) m , which allow for the ‘clearest’ disen-tanglement of MSHE and SHE. For other MLGs there is atleast one element of the spin conductivity tensor that carriessimultaneously contributions from the SHE and the MSHE,leaving the behavior under time-reversal (texture reversal) asthe only distinguishing characteristic.Apart from three-dimensional ferromagnets, two-dimensional electron gases (2DEGs) appear highly attractive.2DEGs are well known for their e ffi cient charge-to-spinconversion due to large Rashba SOC [55], magnetism incombination with superconductivity [56–61], and electricalcontrollability [62]. Recent progress in achieving roomtemperature magnetism in 2DEGs [63] suggests to investigatethe MSHE in these systems. To do so we consider a minimalRashba Hamiltonian with warping and an exchange fieldwhose direction provides a handle to switch between di ff erentMLGs. It turns out that upon in-plane rotation of the field, theMSHE of in-plane polarized spins is manipulated but also thatof out-of-plane polarized spins. Similar conclusions hold fortopological Dirac surface states, as in Sn-doped Bi Te [64](e. g., in the presence of exchange fields due to proximity to aferromagnetic normal insulator [65]), and for the noncollinearantiferromagnet Mn Sn.Instead of an electric field, a temperature gradient may be uti-lized to cause thermodynamic non-equilibrium and spin trans-port. As above for the SHE and the MSHE, time-reversal eventransverse spin transport is then referred to as spin Nernst e ff ect(SNE), and the time-reversal odd partner is termed magneticSNE (MSNE). Since the spin current vortices in reciprocalspace exist irrespective of the driving force, the existence of anMSHE immediately implies that of an MSNE. Recalling thatthe symmetry analysis is independent of the type of spin carri-ers, it applies just as well to magnetic insulators, in which spinis transported by magnons. We present a proof of principle byconsidering antiferromagnetic spin textures, as in Mn Sn, anddemonstrate that the magnetic excitations give rise to nonzero spin current vortices and, thus, to a magnonic MSNE. There-fore, the results of this Paper can be carried over to the realmof spincaloritronics in magnetic insulators, where they mayinspire studies of novel heat-to-spin conversion mechanismsand energy harvesting concepts.The remainder of the Paper is organized as follows. InSec. III the theoretical framework within which we describespin transport is introduced. We disentangle time-reversal evenfrom odd contributions in Sec. III A, isolate the MSHE andintroduce the SCV interpretation in Sec. III B. Then, we turnto the symmetry analysis of all MLGs and summarize keyfindings in Sec. III C. These are elaborated on in Sec. IV byconsidering specific toy models; the latter serve to underlinethe minimal requirements for a nonzero MSHE (Sec. IV A), tomake connection to magnetized Rashba materials (Sec. IV B,and to demonstrate the magnonic MSNE (Sec. IV C). We dis-cuss the relation of our work to literature in Sec. V and sum-marize in Sec. VI.
III. LINEAR-RESPONSE THEORY OF THE MAGNETICSPIN HALL EFFECT
The elements of the optical spin conductivity tensor read[50] σ γµν ( (cid:36) ) = V (cid:90) ∞ d t e i (cid:36) t (cid:90) β d κ (cid:68) J ν J γµ ( t + i (cid:126) κ ) (cid:69) (3)in Kubo linear-response theory [66, 67]. J ν , J γµ , (cid:36) , V , and β are the total charge and spin current operators, the frequencyof E ( (cid:36) ), the system’s volume, and the inverse temperature,respectively. The shape of σ γ was derived for all MLGs bysymmetry arguments in Ref. 50. However, such a superordinatesymmetry approach neither provides insights into the characterof the MSHE nor does it identify the MSHE contributions tothe tensor elements. The latter requires to decompose σ γ . A. Decomposition of the spin conductivity tensor
We work in the limit of non-interacting electrons describedby the Hamiltonian H = (cid:88) k Ψ † k H k Ψ k (4)in crystal momentum ( k ) representation. Ψ † k ( Ψ k ) is a vectorof electronic creation (annihilation) operators; its index runsover spin, orbitals, and basis lattice sites. The eigenenergies ε n k and corresponding eigenvectors | n (cid:105) = | u n k (cid:105) , which representthe lattice-periodic part of a Bloch wavefunction with bandindex n , are obtained from diagonalizing the Hamilton kernel H k .The dc spin conductivity then reads σ γµν ≡ Re (cid:20) lim (cid:36) → σ γµν ( (cid:36) ) (cid:21) = σ γ, odd µν + σ γ, even µν , (5)with the two contributions [45] σ γ, odd µν = (cid:126) Γ V (cid:88) n , m , k f m k − f n k ε n k − ε m k Re (cid:16) (cid:104) n | J γ k ,µ | m (cid:105)(cid:104) m | J k ,ν | n (cid:105) (cid:17) ( ε n k − ε m k ) + ( (cid:126) Γ ) , (6a) σ γ, even µν = − (cid:126) V (cid:88) n , m , k ( f m k − f n k ) Im (cid:16) (cid:104) n | J γ k ,µ | m (cid:105)(cid:104) m | J k ,ν | n (cid:105) (cid:17) ( ε n k − ε m k ) + ( (cid:126) Γ ) . (6b) f n k = (e β ( ε n k − ε F ) + − is the Fermi distribution function withFermi energy ε F . (cid:126) Γ is an artificial spectral broadening andthe total currents are decomposed into their Fourier kernels J γ k ,µ and J k ,ν = − ev k ,ν = − e (cid:126) − ∂ H k /∂ k ν (in the Ψ k basis),respectively.The superscripts of σ γ, odd µν and σ γ, even µν indicate their behaviorunder time-reversal. We recall that spin (charge) current istime-reversal even (odd) and that the TR operator comprisescomplex conjugation [45]. The behavior under time rever-sal can be addressed by a reversal of the magnetic texture (acollection { m i } of magnetic moments): σ γ, odd µν [ { m i } ] = − σ γ, odd µν [ −{ m i } ] ,σ γ, even µν [ { m i } ] = σ γ, even µν [ −{ m i } ] . (7)In an experiment these contributions can be disentangled bymeasuring the spin accumulations brought about by the spincurrents for both the original and the reversed texture.Following up on Eq. (6a), a MSHE was identified by sym-metry arguments for Mn X ( X = Sn , Ga , Ge) in Ref. 45 (seethe first entry in the right column of Tab. I of that paper andconsider σ xxy (cid:44) σ xyx , which makes the antisymmetric part of σ x nonzero). The term MSHE was coined in Refs. 47 and 48,the latter of which reported on its experimental observation inMn Sn.In what follows we concentrate on σ γ, odd µν , because σ γ, even µν isrelated to the intrinsic SHE [24] which is of minor interest inthis Paper. We decompose σ γ, odd µν into intraband contributions( n = m ) σ γ, odd µν, intra = Γ V (cid:88) n , k J γ n k ,µ J n k ,ν (cid:32) − ∂ f n k ∂ε (cid:33) (8)and interband contributions given by Eq. (6a) with the sumrestricted to n (cid:44) m . J γ n k ,µ ≡ (cid:104) n | J γ k ,µ | n (cid:105) is the spin and J n k ,ν ≡ − e (cid:104) n | v k ,ν | n (cid:105) = − ev n k ,ν the charge current expectationvalue; v n k ,ν = (cid:126) − ∂ε n k /∂ k ν is the group velocity and e > Γ → σ γ, odd µν, intra di-verges while the interband contributions converge to a constant.From here on, we thus drop the specifiers ‘odd’ and ‘intra’,work in the ‘almost clean’ limit ( (cid:126) Γ >
B. Identification of the MSHE contributions
The antisymmetric part σ γ, (a) ≡ σ γ − σ γ, T γ -spin MSHE vector as [ σ γ MSHE ] × = σ γ, (a) ,written compactly as [cf. eq. (8)] σ γ MSHE ≡ σ γ, (a) yz σ γ, (a) zx σ γ, (a) xy = Γ V (cid:88) n , k J γ n k × J n k (cid:32) − ∂ f n k ∂ε (cid:33) . (10)At zero temperature, − ∂ f n k /∂ε = δ ( ε n k − ε F ) allows to replacethe k summation by an integral over the Fermi surface, σ γ MSHE = e (cid:126) Γ (2 π ) (cid:88) n (cid:73) ε n = ε F ˆ v n k × J γ n k d S (11)(ˆ v n k = v n k / v n k is the local normal of the Fermi surface). Thisintegral measures the tangential vector flow of J γ n k on the Fermisurface. σ γ MSHE is nonzero if there is an integrated sense ofrotation of the spin current about the Fermi surface.Alternatively, we write Eq. (11) as a Fermi sea integral, σ γ MSHE = e (cid:126) Γ (2 π ) ω γ ( ε F ) , (12)over the net spin current vorticity (SCV) ω γ ( ε F ) ≡ (cid:88) n (cid:36) ε n ≤ ε F ω γ n k d k , ω γ n k ≡ ∇ k × J γ n k , (13)that is defined in analogy to the vorticity of a fluid [68]. ω γ n k describes the local rotation, shear or curvature of J γ n k . Figu-ratively speaking, the vorticity of a vector field is nonzero atthose points at which a paddle wheel would start to rotate (notethat integrals over fully occupied bands are zero, i. e., eachband has a vanishing total SCV).Equations (11) – (13) are our main findings. They show thata MSHE is a result of the spin current circulation about theFermi surface [Eq. (11)] or, put di ff erently, a result of a finiteSCV in the Fermi sea [Eq. (12)].For illustration we stretch the analogy to fluid vortices andrecall the time-reversal asymmetric propagation of an acousticwave through a fluid with a vortex [69], briefly laid out in theintroduction. The broken TRS in magnets causes SCVs ω γ for each spin component γ = x , y , z in the Fermi sea, which isexperienced by the γ spin component of an electron’s Blochwave propagating through the crystal. A consequence is aHall-like deflection within the plane normal to ω γ of that spincomponent. Time reversal is equivalent to inversion of thevortex’s circulation direction ( ω γ → − ω γ ), which signifies thetime-odd signature of the MSHE.Considering reciprocal space, a simple picture may be help-ful. In a two-dimensional crystal with a single Fermi line, the (a) 00 k x k y (b) 0 k x (c) 0 k x FIG. 2. Vorticities of spin current vector fields in reciprocal space.Red / white / blue color indicates positive / zero / blue vorticity in a regionabout the origin. The integral over the vorticity within the Fermisurface, indicated by black circles, is proportional to the magneticspin Hall conductivity. (a) The irrotational source field has zerovorticity. (b) The quadrupolar field has locally nonzero vorticity butzero integral. (c) A general vortex with vorticity of varying sign. spin current vector field J γ k may look as depicted in Fig. 2 (wesuppressed the band index). The integral of the k -dependentvorticity over the Fermi sea is proportional to the magnetic spinHall conductivity. In scenario (a), J γ k is irrotational and, thus,has zero vorticity. In (b) J γ k shows local vorticity that integratesto zero due to symmetry. And in (c), the Fermi surface cutsout a region with nonzero vorticity causing a MSHE. To checkthe behavior under time reversal recall the mapping J γ k to J γ − k ,which reverses the circulation direction.One may discuss the e ff ect in terms of a shift of the Fermisurface that is caused by the redistribution of electrons. Anelectric field E along − x direction produces a shift in positive k x direction (accounting for the negative electron charge). Forthe situation in Fig. 2(a), this displacement does not yield atransversal response since (cid:104) j γ (cid:105) (cid:107) ˆ x . However, for (b) and (c) (cid:104) j γ (cid:105) (cid:107) ˆ y and (cid:104) j γ (cid:105) (cid:107) − ˆ y , respectively. If E is along the − y direction (shift in positive k y direction), (cid:104) j γ (cid:105) (cid:107) ˆ x for both (b)and (c). Hence, only a finite SCV, as depicted in (c), fixesthe sign of (cid:104) j γ (cid:105) × E , thereby causing a nonzero antisymmetricpart of σ γ , that is a MSHE. The scenario (b) gives rise toa symmetric part of σ γ , conceivably referred to as ‘planarmagnetic spin Hall e ff ect’. C. Symmetry analysis
Although breaking of TRS is necessary for a nonzero localSCV ω γ n k , it is not su ffi cient because symmetries of the mag-netic crystal may render the SCV integral in Eq. (13) zero [cf.Fig. 2(b)]. In what follows we derive which MLGs do or donot allow for a MSHE. The restriction to MLGs—instead tomagnetic point groups—is feasible because σ γ is related tothe correlation function of a spin current and a charge current[Eq. (3)]. Both currents change sign upon inversion; thus, thepresence or absence of inversion symmetry does not imposerestrictions on the shape of the spin conductivity tensor. Onemay then augment each magnetic point group with the elementof space inversion to map it onto the set of MLGs. Recall thatthe considerably smaller number of MLGs facilitates the anal-ysis. This argumentation is in line with Refs. [50, 70–72]. We like to refer the reader to Ref. [50] for mappings of magneticpoint groups onto MLGs.We combine the three spin-dependent MSHE vectors ofEq. (12) to the MSHE tensor σ MSHE ≡ (cid:16) σ x MSHE , σ y MSHE , σ z MSHE (cid:17) ∝ ω ≡ ( ω x , ω y , ω z ) . (14)Equation (14) links the elements of σ MSHE to those of the SCVtensor ω , the latter itself constructed from the three spin currentvorticity vectors given in Eq. (13) (argument ε F suppressed). ω can be decomposed into three contributions: ω = ω I + [ Ω ] × + W , (15)that is a scalar ω = Tr( ω ) /
3, a vector [ Ω ] × = ( ω − ω T ) /
2, anda traceless symmetric tensor W = ( ω + ω T ) / − ω I , with I the3 × ω is time-reversal odd butspace-inversion even and calling to mind the transformationproperties of electromagnetic multipoles [73], one finds that ω , Ω , and W transform as a magnetic toroidal monopole (Jahnsymbol a , Ref. 74), a magnetic dipole ( eaV ), and a magnetictoroidal quadrupole ( a [ V ]), respectively. Such a multipoledecomposition is in line with Ref. 75 [cf. Eqs. (D20) and (D21)of that publication].Utilizing the M tensor application [76] of the Bilbao Crystal-lographic Server [77–79], we identified all MLGs permittingthese multipoles. By virtue of Eqs. (10) and (14) these resultsare carried forward to σ MSHE and σ γ, (a) ; a summary is given inTable I. The results of the symmetry analysis are not restrictedto the intraband approximation (i. e., the interpretation in termsof SCVs) but apply to Eq. (6a) as well [one may consider thesummand in Eq. (6a) as a generalization of the SCV to inter-band contributions]. We now list and discuss illustrative keyfindings.(i) Any MLG that contains pure time-reversal 1 (cid:48) (reversalof the magnetic texture maps the crystal onto itself modulo atranslation) is incompatible with a SCV and a MSHE because ω , Ω , and W transform as magnetic multipoles.(ii) The MLG m ¯3 m of cubic systems does not allow for amagnetization ( Ω = ) but for ω and W xx = W yy = W zz , fromwhich Eq . (15) → ω xx = ω yy = ω zz Eq . (14) → σ x MSHE , x = σ y MSHE , y = σ z MSHE , z Eq . (10) → σ x , (a) yz = σ y , (a) zx = σ z , (a) xy follows. MSHEs with mutually orthogonal spin, flow, andforce directions are expected, a situation known from the SHEin nonmagnetic cubic materials. The SHE and MSHE can bedisentangled by their opposite time-reversal signature whichcan be probed by a reversal of the magnetic texture [48].(iii) The MLG 4 / mm (cid:48) m (cid:48) admits of a magnetization, Ω = (0 , , Ω z ) T , but neither of ω nor of W . We find W yx = , Ω z (cid:44) Eq . (15) → ω yx = − ω xy Eq . (14) , (10) → σ y , (a) yz = − σ x , (a) zx . In contrast to the anomalous Hall e ff ect (AHE), a magnetization(along z ) does not cause transverse transport within a planeperpendicular to it ( xy plane), but in planes that contain itself TABLE I. Symmetry analysis of the MSHE. The columns give magnetic Laue groups (MLGs), components of the spin current vorticity (SCV) ω , and elements of spin conductivity σ γ that are compatible with an MSHE.MLG a Admitted elements of SCV tensor ω MSHE part of spin conductivity tensor σ γ ω Ω W b σ x , (a) σ y , (a) σ z , (a) ¯1 ω Ω x , Ω y , Ω z W xx , W yy , W zz , W yx , W zx , W zy σ xxy σ xxz − σ xxy σ xyz − σ xxz − σ xyz σ yxy σ yxz − σ yxy σ yyz − σ yxz − σ yyz σ zxy σ zxz − σ zxy σ zyz − σ zxz − σ zyz / m ω Ω y W xx , W yy , W zz , W zx σ xxy − σ xxy σ xyz − σ xyz σ yxz − σ yxz σ zxy − σ zxy σ zyz − σ zyz mmm ω − W xx , W yy , W zz σ xyz − σ xyz σ yxz − σ yxz σ zxy − σ zxy / m ω Ω z W xx = W yy , W zz σ xxz σ xyz − σ xxz − σ xyz − σ xyz σ xxz σ xyz − σ xxz σ zxy − σ zxy / m ¯34 / mmm ω − W xx = W yy , W zz σ xyz − σ xyz − σ xyz σ xyz σ zxy − σ zxy / mmm ¯3 mm ¯3 ω − W xx = W yy = W zz σ xyz − σ xyz − σ xyz σ xyz σ xyz − σ xyz m ¯3 m (cid:48) / m (cid:48) − Ω x , Ω z W yx , W zy σ xxz − σ xxz σ yxy − σ yxy σ yyz − σ yyz σ zxz − σ zxz m (cid:48) m (cid:48) m − Ω z W yx σ xxz − σ xxz σ yyz − σ yyz − (cid:48) / m − − W yx , W xx = − W yy σ xxz σ xyz − σ xxz − σ xyz σ xyz − σ xxz − σ xyz σ xxz − (cid:48) / mm (cid:48) m − − W yx σ xxz − σ xxz − σ xxz σ xxz − / mm (cid:48) m (cid:48) − Ω z − σ xxz − σ xxz σ xxz − σ xxz − / mm (cid:48) m (cid:48) ¯3 m (cid:48) a The MLGs ¯11 (cid:48) , 2 / m (cid:48) , mmm (cid:48) , 4 / m (cid:48) , 6 / m (cid:48) , 4 / mmm (cid:48) , 6 / mmm (cid:48) , ¯31 (cid:48) , ¯3 m (cid:48) , m ¯31 (cid:48) , and m ¯3 m (cid:48) contain pure time-reversal symmetry and are incompatiblewith a spin current vorticity and, thus, a MSHE. The MLGs 6 (cid:48) / m (cid:48) , 6 (cid:48) / m (cid:48) mm (cid:48) , and m ¯3 m (cid:48) are MSHE-incompatible as well. b Since W is symmetric, admittance of W ji implies admittance of W ij . ( xz and yz planes). Only the transported spin component hasto be normal to the magnetization ( x and y ). Moreover, it hasto lie within the plane of transport. Thus, although tetragonalferromagnets allow both for the AHE and the MSHE, thespin current attributed to the MSHE is a pure spin currentbecause the AHE-induced current flows in a di ff erent plane.This scenario was outlined in Sec. II via Eq. (2).(iv) A magnetic toroidal quadrupole W also allows for thespin transport discussed in (iii). Consider the MLG 4 (cid:48) / mm (cid:48) m that permits only a nonzero W yx , which translates to σ y , (a) yz = σ x , (a) zx . Compared to (iii), only the relation of the signs ofnonzero components has changed. For a geometry as depicted in Fig. 1 (four-fold rotational axis aligned with the z direction),this reversed sign translates into a MSHE spin accumulationwith a polarization that alternates between pointing paralleland antiparallel to the surface normal.(v) The MLG m (cid:48) m (cid:48) m combines the scenarios of (iii) and(iv): Ω z , W yx , σ y , (a) yz , and σ x , (a) zx may be nonzero but there is noadditional symmetry-imposed relation between the latter two.(vi) An MSHE has been experimentally established in thenoncollinear antiferromagnet Mn Sn [48] that—depending onthe spin orientation—belongs either to the MLG 2 (cid:48) / m (cid:48) or to2 / m [80]. These are the same MLGs we shall discuss in thecontext of a magnetized Rashba electron gas with warping (a) (c) (e)(b) (d) (f) xyz k x k y k z k x k y k z k x k y k z k x k y k z FIG. 3. MSHE in the pyrochlorelattice for (a) ferromagnetic (MLG4 / mm (cid:48) m (cid:48) ) and (b) antiferromag-netic textures (MLG 4 (cid:48) / mm (cid:48) m ).The pyrochlore lattice is projectedonto the xy plane, such that thetetrahedra appear as squares. (c)and (d) depict Fermi surfaces witharrows indicating J x k and the colorscale depicts the y component ofˆ v k × J x k (blue / white / red indicatesnegative / zero / positive values; thevalue range is symmetric aboutzero). (e) and (f) as (c) and (d)but for J y k and the x component ofˆ v k × J y k . (Sec. IV B). Please note that the MLG 2 / m allows for σ zxy (cid:44) xy plane with out-of-plane polarizedspins currents (a geometry similar to the conventional SHE),whereas 2 (cid:48) / m (cid:48) does not. Thus, upon rotation of the coplanarmagnetic texture of Mn Sn, one could switch the MLGs andthereby engineer the transport of out-of-plane polarized spins;this e ff ect awaits experimental verification (the experimentalsetup in Ref. 48 was sensitive to in-plane spin polarizations). IV. EXAMPLES
With the above results at hand, we now address selectedexamples for various MLGs. Section IV A focuses on minimalrequirements for a MSHE and illustrates its interpretation interms of spin current vorticities. In Sec. IV B we make contactwith Rashba materials whose MLGs cover Mn Sn. Finally, weconsider a magnetic spin Nernst e ff ect in insulating materials(Sec. IV C). A. Minimal requirements for a MSHE
According to the points (iii) and (iv) in Section III C, com-patibility of a MLG with either a magnetization (e. g., MLG4 / mm (cid:48) m (cid:48) ) or a magnetic toroidal quadrupole (e. g., MLG4 (cid:48) / mm (cid:48) m ) su ffi ces for a MSHE. To show explicitly the spin cur-rent vortex about the Fermi surface [in the sense of Eq. (11)],we consider the sd Hamiltonian H = (cid:88) (cid:104) i j (cid:105) c † i (cid:16) t + i α τ · ˆ d i j (cid:17) c j + J (cid:88) i c † i ( τ · ˆ m i ) c i (16) on the pyrochlore lattice [Fig. 3(a)] which consists of corner-sharing tetrahedra [81]. c † i ( c i ) creates (annihilates) an electronspinor at site i , τ T = ( τ x , τ y , τ z ) is the vector of Pauli matrices.The hopping (with amplitude t ) of electrons is accompaniedby a spin rotation due to SOC (with amplitude α ). The unitvectorsˆ d = √ − , ˆ d = √ − , ˆ d = √ − , ˆ d = √ , ˆ d = √ − − , ˆ d = √ (17)specify the directions of the e ff ective SOC. Each of the ˆ d i j isorthogonal to the i – j bond and lies within a face of a cube thatencloses a tetrahedron. ˆ d i j = − ˆ d ji implies (cid:80) j = ˆ d i j = foreach i = , . . . , J , the electron spins are connected withthe local magnetic moments ˆ m i (black arrows in Fig. 3). Theferromagnetic texture in (a), with ˆ m i = (0 , , T , belongs tothe MLG 4 / mm (cid:48) m (cid:48) , while the antiferromagnetic texture in (b)belongs to 4 (cid:48) / mm (cid:48) m (the notation of the MLGs matches thatof the respective magnetic point groups):ˆ m = √ − , ˆ m = √ − − , ˆ m = √ − , ˆ m = √ . (18)The latter MLG 4 (cid:48) / mm (cid:48) m is compatible with a magnetictoroidal quadrupole ( W yx (cid:44) , , , T ∝ (cid:88) i = r i × ˆ m i , (19)in which coordinates r i are taken with respect to the tetrahe-dron’s center of mass. We obtain T = (0 , , T z ) T with T z < T z > (cid:48) / mm (cid:48) m because the Jahn sym-bol aV of T ) but a nonzero net magnetic toroidal quadrupole[82].We have diagonalized the Hamiltonian (16) in reciprocalspace. Since the magnetic unit cell contains the same numberof sites as the structural unit cell, there are four electronic bands(not shown). Fig. 3(c)–(f) show representative iso-energy cuts(Fermi surfaces) with the arrows depicting J x k [in panels (c)and (d); band index suppressed] and J y k [in panels (e) and (f)],respectively. The color scales visualize particular integrands inEq. (11): the y component of ˆ v k × J x k in (c) and (d) and the x component of ˆ v k × J y k in (e) and (f). The spin current is definedas usual by J γ k ,µ = { s γ , v k ,µ } ( s γ γ -spin operator).The spin current circulation about the Fermi surface, that isthe spin current vortex, is clearly identified in panels (c) and(e). The respective integrals σ x , (a) zx ∝ (cid:88) n (cid:73) ε n = ε F ˆ v n k × J xn k (cid:12)(cid:12)(cid:12) y d S (20)and σ y , (a) yz ∝ (cid:88) n (cid:73) ε n = ε F ˆ v n k × J yn k (cid:12)(cid:12)(cid:12) x d S (21)are nonzero [either red (c) or blue color (e) dominates]. Due tothe symmetry of the value range, one finds − σ x , (a) zx = σ x , (a) xz = σ y , (a) yz , (22)as was confirmed numerically. These findings agree fully withpoint (iii) of Sec. III C and with the 4 / mm (cid:48) m (cid:48) row of Table I.A similar, albeit less striking observation can be made forpanels (d) and (f). Red dominates slightly over blue in bothcases, visualizing nonzero integrals for σ xzx and σ yyz . In accor-dance with point (iv) of Sec. III C and the entry for 4 (cid:48) / mm (cid:48) m in Table I, σ x , (a) zx = − σ x , (a) xz = σ y , (a) yz (23)holds.For all other components of ˆ v k × J γ k that are not shown inFig. 3, the color distribution on the Fermi surface—blue andred appear equally—indicate magnetic spin Hall conductivitiesof zero, in agreement with Table I. (c) B k ˆ y E ne r g y ε ( e V ) M o m en t u m k x ( Å − ) M o m e n t u m k y ( Å − ) ε d ε s (d) B k ˆ y E ne r g y ε ( e V ) M o m en t u m k x ( Å − ) M o m e n t u m k y ( Å − ) ε d ε s (a) B k ˆ x E ne r g y ε ( e V ) M o m en t u m k x ( Å − ) M o m e n t u m k y ( Å − ) ε d ε s (b) B k ˆ x E ne r g y ε ( e V ) M o m en t u m k x ( Å − ) M o m e n t u m k y ( Å − ) ε d ε s FIG. 4. Two-dimensional electron gas with Rashba SOC and in-planemagnetic field along x [(a) and (b)] or y [(c), (d)]. ε d and ε s are theenergy of the degeneracy point and of the saddle point. The colorscales represent spin current vorticities ω x k , z [(a), (c)] and ω y k , z [(b),(d); blue / white / red color indicates negative / zero / positive values]. Fordetails, see text. B. MSHE in Rashba materials
For the three-dimensional models addressed in the precedingsection, we concentrated on the spin current circulation aboutthe Fermi surface. Similar conclusions can be drawn fromcalculated SCVs which one could represent as ‘vortex lines’of the field J γ k . Since this makes for hardly interpretable three-dimensional pictures, we focus now on a two-dimensionalmodel for which the SCV is clearly identified.
1. Magnetized Rashba model and its band structure
Recent progress on magnetism in two-dimensional electrongases motivates to demonstrate the existence of SCVs in anin-plane magnetized Rashba model with Hamiltonian H = (cid:126) k m + α R (cid:16) k x τ y − k y τ x (cid:17)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) H R + µ B B · τ (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) H Zee , (24)( τ i Pauli matrices, m e ff ective mass, and α R Rashba parameter);hexagonal warping is accounted for later. The continuous rota-tional symmetry of H R is broken by an in-plane exchange field B = ( B x , B y , T ( µ B Bohr’s magneton). The set of parameters( m = . m e , α R = .
95 eV ˚A, with m e electron mass) cor-responds to those of the ordered ( √ × √ R ◦ Bi / Ag(111)surface alloy [83–86]; we set µ B B = . σ xx ( − / Ω ) B ∥ ˆ x , λ = eV Å B ∥ ˆ x , λ = eV Å B ∥ ˆ x , λ = eV Å B ∥ ˆ y , λ = eV Å B ∥ ˆ y , λ = eV Å B ∥ ˆ y , λ = eV Å − − − − − σ x , ( a ) x y ( − / Ω ) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 ε F (eV) σ y , ( a ) x y ( − / Ω ) − σ z , ( a ) x y ( − / Ω ) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 ε F (eV)(a) (c)(b) (d) ε d ε s ε d ε s FIG. 5. Charge conductivity [(a)]and MSHE [(b)-(d)] in a Rashbasystem with in-plane magneticfield B and hexagonal warping.Model parameters as for Fig. 4,except for the warping strength λ =
18 eV ˚A . For λ =
18 eV ˚A ,the model is applicable only for ε < . λ =
0) and with re-duced warping ( λ = ) areconsidered. Circles (squares) for B (cid:107) ˆ x ( B (cid:107) ˆ y ); σ γ, (a) multipliedby e (cid:126) to better compare with thecharge conductivity. For B in x direction, the two bands are degenerate at a pointon the k y axis [panels (a) and (b) of Fig. 4; at energy ε d ];likewise for B along y , the bands are degenerate at a point onthe k x axis [(c) and (d)]. On top of that, the lower band has asaddle point at energy ε s . Overall, the band structure merelyexhibits a k x → − k x [(a), (b)] or a k y → − k y symmetry [(c),(d)], rendering iso-energy lines anisotropic.
2. Symmetries, spin current vorticities, and spin conductivity
With B in x direction ( y direction), the model shows nonzero Ω x ( Ω y ) and, thus, nonzero σ y , (a) xy ( σ x , (a) xy ). Recall that transporttakes place in a plane containing the magnetization and that thetransported spin component is orthogonal to the magnetization.Since we focus on transport in the xy plane, we consider neither σ z , (a) xz nor σ z , (a) yz although allowed by Ω x or Ω y .The momentum-resolved SCVs, shown by color in Fig. 4,exhibit a band antisymmetry, ω x , k , z = − ω x , k , z and ω y , k , z = − ω y , k , z (1 and 2 band indices), which is a feature of a two-bandmodel. Moreover, the SCVs exhibit the following reflection(anti-)symmetries: B (cid:107) ˆ x : ω xn , k x , k y , z = − ω xn , − k x , k y , z , ω yn , k x , k y , z = ω yn , − k x , k y , z , (25a) B (cid:107) ˆ y : ω xn , k x , k y , z = ω xn , k x , − k y , z , ω yn , k x , k y , z = − ω yn , k x , − k y , z . (25b)Even without explicit calculations one verifies that any Fermisea integral over ω x k , z for B (cid:107) ˆ x [panel (a)] or ω y k , z for B (cid:107) ˆ y [panel (d)] equals zero due to these antisymmetries. Equa-tion (12) gives σ x , (a) xy = B (cid:107) ˆ x and σ y , (a) xy = B (cid:107) ˆ y .In contrast, the integrals over ω y k , z for B (cid:107) ˆ x (b) or ω x k , z for B (cid:107) ˆ x (c) are nonzero, as becomes especially plausible for lowenergies, at which only red (b) or blue (c) shows up.For a quantitative analysis, we address the energy depen-dence of the magnetic spin Hall conductivity. First, we recallthat of the charge conductivity σ xx , as shown in Fig. 5(a).For low Fermi energies ε F , the direction of the magnetic fieldstrongly a ff ects the shape of the iso-energy lines, leading to di ff erent σ xx for B (cid:107) ˆ x and B (cid:107) ˆ y , respectively [compare blueand black symbols in Fig. 5(a)]. At higher energies SOC dom-inates over exchange and, thus, the shape of the iso-energylines depends marginally on the direction of B : there is barelya di ff erence in σ xx for B (cid:107) ˆ x and B (cid:107) ˆ y .For understanding better the energy dependence of theMSHE depicted in Figs. 5(b)–(d), we inspect the SCVs ω x k , z , ω y k , z , and ω z k , z for B (cid:107) ˆ x and B (cid:107) ˆ y , respectively (Figs. 6 and7). As expected, σ x , (a) xy ( σ y , (a) xy ) vanishes for B y = B x = σ y , (a) xy ( σ x , (a) xy ) occurs due toincomplete cancellation [Figs. 6(b) and 7(a)].As sketched in Fig. 4, the bands are degenerate at ε d ≈ σ y , (a) xy ( σ x , (a) xy ) increases [cf. Figs. 5(c)and (b)] due to the growing number of states contributing tothe MSHE [cf. black, blue, and green Fermi lines in Figs. 6 or7].Around the saddle point at ε s ≈ −
83 meV the SCV changessign and the states contribute oppositely to the magnetic spinHall conductivity, leading to an extremum of σ y , (a) xy ( σ x , (a) xy ) closeto ε s .Above ε d , both bands are occupied (in Figs. 6 and 7, theinner iso-energy line corresponds to the upper band, the outerto the lower band). Only the SCV of the lower band is shown,but the upper band’s SCV di ff ers from the lower band’s onlyby sign. Thus, regions in k space in which both bands areoccupied do not contribute to σ γ, (a) xy .With increasing ε F the additional contribution of the lowerband’s states to the MSHE is compensated by the states in theupper band, thus, σ y , (a) xy and σ x , (a) xy are almost independent of ε F .The magnetic spin Hall angle, defined as α γ SH = e (cid:126) × σ γ, (a) xy / ( σ xx + σ yy ) is sizable (up to 70%) near the band edgeand decreases for larger ε F . Above ε d , it is in the order of 10%,which may be considered substantial. Its energy dependenceis dominated by the almost linear energy dependence of thecharge conductivity.0 FIG. 6. Spin current vorticity (red / blue color scale) and iso-energylines (colored lines) of the lower band of Rashba systems with andwithout hexagonal warping in the presence of an in-plane magneticfield B (cid:107) ˆ x . Model parameters as in Fig. 5.FIG. 7. As Fig. 6 but for B (cid:107) ˆ y .
3. E ff ect of hexagonal warping To come closer to realistic materials, a hexagonal warpingterm H w = i λ (cid:16) k + − k − (cid:17) τ z , k ± = k x ± i k y , (26)is added to the Hamiltonian (24). For ( √ × √ R ◦ Bi / Ag(111), the strength of the warping is λ =
18 eV ˚A [86].Similar to the exchange field, H w breaks the continuous rota-tion symmetry of H R , leaving the xz plane as a mirror plane.For B (cid:107) ˆ y this mirror plane is retained (MLG 2 / m ; Table I).Since H w introduces a spin- z component [notice τ z in Eq. (26)], we expect nonzero σ z , (a) xy [associated with W zz , as explained in(ii); cf. Fig. 5(b) and (d)]. For weak warping ( λ = ),the band structure as well as the SCV ω x k , z appear mildly af-fected by the additional SOC [Fig. 7(d)]. Thus, σ x , (a) xy is weaklyinfluenced by warping. However, a finite spin current vorticity ω z k , z gives rise to a nonzero σ z , (a) xy . The sign changes of σ z , (a) xy for ε F < ε s are due the anisotropy of the Fermi lines and thealternating sign of ω z k , z in reciprocal space [Fig. 7(f) and (i)].For stronger warping ( λ =
18 eV ˚A ), the energy dispersionand ω x k , z are remarkably modified, which leads to a slightlyenhanced MSH conductivity σ x , (a) xy at any ε F . Furthermore, theabsolute value of σ z , (a) xy is increased. B (cid:107) ˆ x makes the xz plane a time-reversal mirror plane (MLG2 (cid:48) / m (cid:48) ; Table I). Instead of σ x , (a) xy and σ z , (a) xy , only σ y , (a) xy is nownonzero [Fig. 5(b)], and its energy dependence is equivalent tothat of σ x , (a) xy for B (cid:107) ˆ y . Note in particular that Figs. 6 (f) and(i) demonstrate that the Fermi sea integral over ω z k , z vanishesfor symmetry reasons.
4. Concluding remarks and applicability to real materials
To conclude, in-plane magnetized Rashba 2DEGs exhibit aMSHE with in-plane polarized spin current ( σ x , ( a ) xy and σ y , ( a ) xy )if warping is negligibly small. By rotating the in-plane B field,the transported spin components can be manipulated. On topof that, the interplay of warping and the direction of B allowsfor nonzero σ z , ( a ) xy , thereby causing out-of-plane polarized spinaccumulations at the edges of the sample (similar to the con-ventional SHE). Upon continuous rotation of B , the magneticspin Hall conductivity σ z , ( a ) xy alternates from positive ( B (cid:107) ˆ y )via zero ( B (cid:107) ± ˆ x ) to negative ( B (cid:107) − ˆ y ) values.A similar e ff ect is expected for the warped topological Diracsurface states in Sn-doped Bi Te [64]; the exchange fieldcould be induced by proximity to a ferromagnetic insulator[65], a strategy giving rise to an AHE [87]. Another exampleis the noncollinear antiferromagnet Mn Sn with spin texturesas shown in Figs. 8(a) and (d). While the manipulation ofin-plane polarized MSHE spin currents by an in-plane field hasbeen successfully demonstrated in Ref. 48, the manipulationof the out-of-plane polarized MSHE awaits its experimentalconfirmation.The Rashba model for a 2DEG is easily extended to threedimensions, in order to cover multiferroic Rashba semiconduc-tors with bulk Rashba SOC, an example being like (GeMn)Te[88–90]. In equilibrium, the magnetization of (GeMn)Te is par-allel to the direction of the ferroelectric polarization [88]; it isconceivable that an in-plane field causes considerable in-planecanting and a MSHE.
C. Magnetic spin Nernst e ff ect The symmetry considerations of Sec. III C apply also to themagnetic spin Nernst e ff ect (MSNE) (cid:104) j γ (cid:105) = α γ ( − ∇ T ) ( ∇ T (b) (c) / m (d)1 23 (e) (f) / m (a)1 23 FIG. 8. Magnon SCVs for two antiferromag-netic textures on the kagome lattice. Top row:MLG 2 (cid:48) / m (cid:48) . (a) Colored arrows at the ver-tices and black arrows at the bond centersindicate the spin texture and the DMI vec-tors for counter-clockwise circulation, respec-tively. (b) and (c) Dispersion relation of thelowest band in the vicinity of the Brillouinzone center. Color scales indicate the value ofthe SCVs (b) ω x k , z and (c) ω y k , z (red / white / bluestands for positive / zero / negative SCV). Bot-tom row: as top row but for the MLG 2 / m . (e)and (f) depict ω x k , z and ω y k , z , respectively. temperature gradient) which is determined by the antisym-metric part of the magnetothermal conductivity α γ . Withinlinear-response theory α γµν ( (cid:36) ) = T V (cid:90) ∞ d t e i (cid:36) t (cid:90) β d κ (cid:68) Q ν J γµ ( t + i (cid:126) κ ) (cid:69) + ˜ α γµν ( (cid:36) )(27)( Q ν total heat current) [91]. ˜ α γµν ( (cid:36) ) accounts for circulatingequilibrium currents that do not contribute to transport [92].As far as the intraband contribution is concerned, we write α γ, odd µν, intra = Γ T V (cid:88) n , k J γ n k ,µ Q n k ,ν (cid:32) − ∂ f n k ∂ε (cid:33) (28)and derive the MSNE vector α γ MSNE ≡ Γ T V (cid:88) n , k J γ n k × Q n k (cid:32) − ∂ f n k ∂ε (cid:33) = − (cid:126) Γ T (2 π ) (cid:88) n (cid:90) ∞−∞ d ε ( ε − ε F ) ω γ ( ε ) (cid:32) − ∂ f n k ∂ε (cid:33) (29)using Q n k = ( ε n k − ε F ) v n k . The SCV ω γ ( ε ) at energy ε isobtained from Eq. (13) by replacing ε F by ε . Overall, α γ MSNE and σ γ MSHE obey the Mott relation α γ MSNE = − T (cid:90) ∞−∞ d ε (cid:32) − ∂ f n k ∂ε (cid:33) ε − µ e σ γ MSHE ( T = , ε ) . (30)Thus, the symmetry restrictions on σ γ MSHE also apply to α γ MSNE and, in particular, the MSNE is also related to the SCV. Con-sequently, a nonzero MSHE implies a nonzero MSNE. We now demonstrate the MSNE in magnetic insulators inwhich ∇ ν T causes magnonic spin currents. Although the Fermi-Dirac distribution f n k in Eq. (30) has to be replaced by the Bose-Einstein distribution ρ n k = (e βε n k − − and the charge currenthas to be replaced by the particle current, the connection to theSCV remains. Thus, our aim is to show explicitly the existenceof a magnonic SCV.Inspired by the antiferromagnetic magnetic texture of thekagome-lattice compound Mn Sn—for which a MSHE wasdemonstrated in Ref. 48—we consider the spin-wave excita-tions of this texture. Assuming that the kagome plane is not amirror plane of a surrounding crystal, a minimal spin Hamilto-nian reads [93] H = (cid:126) (cid:88) (cid:104) i , j (cid:105) (cid:16) J S i · S j + D (cid:107) ˆ d i j · S i × S j + D z ˆ z i j · S i × S j (cid:17) . (31) J > D (cid:107) parametrize the antiferromagnetic exchange andthe in-plane Dzyaloshinskii-Moriya interaction [DMI; the unitvectors ˆ d i j are depicted in Fig. 8(a)], respectively. D z > z i j = ± ˆ z ; the upper(lower) sign is for (anti-)cyclic indices i j .The out-of-plane DMI D z > D (cid:107) as long as | D (cid:107) / D z | is smaller thana critical value (otherwise a canted all-in–all-out texture be-comes the energetic minimum [93]). The irrelevance of D (cid:107) asfar as the classical energy is concerned imposes an accidentalrotational degeneracy: the spins can be rotated about the z axis without a classical energy penalty, in particular by π / ff erent magneticpoint groups—(a) 2 (cid:48) / m (cid:48) and (d) 2 / m —there seems to be a‘classical’ ambiguity concerning spin transport. This ambiguityis lifted upon performing linear spin-wave theory about thetwo classical magnetic ground states. Following Ref. 94, wefind that the order-by-quantum-disorder mechanism (harmoniczero-point fluctuations contribute to the ground state energy)prefers texture (a) over (d). Nonetheless, it is instructive tostudy the SCV for both textures to appreciate the e ff ect of mag-netic point group symmetries (details of the linear spin wavetheory calculation are given in Appendix A).We concentrate on low energies because these are most rele-vant when accounting for thermal occupation (Bose-Einsteindistribution). The SCVs ω x k , z and ω y k , z of the lowest magnonband in the vicinity of the Brillouin zone center are given forthe MLG 2 (cid:48) / m (cid:48) in Fig. 8(b) and (e) as well as for the MLG2 / m in (c) and (f).Inspection of panels (b) and (f) tells that for each state k thereis a state k (cid:48) with the same energy but with opposite SCV ( ε , k = ε , k (cid:48) , ω γ , k , z = − ω γ , k (cid:48) , z ), a symmetry also found in the Rashbamodel (Sec. IV B). Irrespective of the distribution function, thelocal contributions to the integrated ω γ z ( ε ) [Eq. (29)] cancelout: α x , ( a ) xy = (cid:48) / m (cid:48) texture and α y , ( a ) xy = / m texture. In contrast, the SCVs in Fig. 8(c) and (e) do notexhibit such an antisymmetry and thus have nonzero integral: α y , ( a ) xy (cid:44) (cid:48) / m (cid:48) texture and α x , ( a ) xy (cid:44) / m texture. These findings agree with the 2 × xy subtensors of σ x , ( a ) and σ y , ( a ) for both MLGs (Table I), α x , ( a )2 (cid:48) / m (cid:48) = (cid:32) (cid:33) and α y , ( a )2 (cid:48) / m (cid:48) = (cid:32) α yxy − α yxy (cid:33) , (32a) α x , ( a )2 / m = (cid:32) α xxy − α xxy (cid:33) and α y , ( a )2 / m = (cid:32) (cid:33) . (32b)One could expect a finite α zxy for 2 / m , as is the case for theelectronic Rashba model studied in Sec. IV B. For the texturehas no out-of-plane component and the magnon spin is definedwith respect to the spin directions o ff ered by the ground statetexture (Ref. 95 and Appendix A), this component cannot becaptured within the present framework. One may regard thisa shortcoming of the definition of magnon spin that has to betreated in the future. V. DISCUSSION
As known from the Barnett e ff ect [96], a rotating magneticobject is magnetized due to the coupling of angular velocityand spin. Similarly, the vorticity of a fluid couples to spin.Such e ff ects are studied, for example, in nuclear physics [97]or in the context of spin hydrodynamic generation [98–101].Concerning the latter, spin currents brought about by the vor-ticity of a confined fluid generate nonequilibrium spin voltages.These examples have in common that a vorticity of a fluid inreal space is involved. In contrast, the SCV studied here ‘lives’in momentum space. To put the SCV into a wider context,we show that the concept of vorticity is tightly connected toextrinsic contributions to Hall e ff ects. Within semiclassical Boltzmann transport theory (e. g.,Ref. 102) the extrinsic skew scattering contribution to the AHEis given by σ skewAHE = V (cid:88) n k J n k × Λ n k (cid:32) − ∂ f n k ∂ε (cid:33) , (33)for which an AHE vector is constructed similar to the MSHEvector in Eq. (10). For a small electric field and a linearizedBoltzmann equation, the vectorial mean free path Λ n k is ob-tained from [102] Λ n k = Γ n k v n k + (cid:88) n (cid:48) , k (cid:48) P n ← n (cid:48) k ← k (cid:48) Λ n (cid:48) k (cid:48) . (34) Γ n k = (cid:80) k (cid:48) P n (cid:48) ← n k (cid:48) ← k is the relaxation rate and P n ← n (cid:48) k ← k (cid:48) is the scatter-ing rate from a state ( n (cid:48) , k (cid:48) ) into a state ( n , k ). The same stepsthat lead to the SCV yield σ skewAHE = − e (cid:126) (2 π ) (cid:88) n (cid:73) ε n = ε F ˆ v n k × Λ n k d S (35a) = − e (cid:126) (2 π ) (cid:88) n (cid:36) ε n ≤ ε F ∇ k × Λ n k d k (35b)for zero temperature and J n k = − e v n k . ∇ k × Λ n k is the vorticityof the mean free path ( Λ vorticity for short). This meansthat a scattering process contributes to the AHE if it causes avorticity in the mean free path; the latter is brought about bythe scattering-in terms [sum in Eq. (34)]. To capture the skewscattering contributions to the AHE, the scattering-in termshave to be taken into account, since one finds σ skewAHE = if theseterms are neglected (e. g. in relaxation time approximation Λ n k = v n k /Γ n k ). This reasoning complies with establishedresults for the AHE [22].A skewness of the scattering is not necessary for a nonzeroSCV, for the latter may be nonzero even in case of a constantrelaxation rate Γ n k = Γ (this is the case considered so far). Ifthe relaxation rate depends on momentum one can still writethe MSH conductivity in the form of Eqs. (12) and (13) butwith a renormalized SCV ω γ n k Γ → ˜ ω γ n k = Γ n k ω γ n k − J γ n k × ∇ k Γ n k . (36)The original SCV ω γ n k can hence be considered the backboneof the MSHE, on top of which come corrections from skewscattering, side jump or a momentum-dependent relaxationtime. Future work may address the MSHE within a quantumkinetic approach, thereby taking into account the electrons’SU(2) nature and spin-dependent scattering.In order to avoid the ill definition of spin current, the authorsof Ref. 48 considered the MSHE in terms of spin-accumulationrather than of spin-current responses. Such a reasoning fits topresent experiments in which spin accumulations rather thanspin currents are measured. Nonetheless, the observed spinaccumulations may arise from two contributions: a local pro-duction (as for the Edelstein e ff ect [103, 104]) and a transportof spins from the bulk toward the edges of the sample.3Compared to the spin current operator J γ , the velocity doesnot appear in the time-reversal odd spin operator s γ . Replacing J γ by s γ in the Kubo formula implies then that the time-reversalodd and even parts change roles; consequently, spin accumula-tions brought about by the MSHE appear in the intrinsic part[48] and stay finite for Γ →
0. The latter finding is to becontrasted with the present theory which predicts a divergenceof the bulk spin current in this limit [Eq. (8)]. This variancein one and the same limit suggests that the two underlyingmechanisms are fundamentally distinct. To disentangle theirrelative contribution we propose that future experiments mayclarify the role of relaxation processes when taking the cleanlimit.
VI. CONCLUDING REMARKS
We identified spin current vortices in the Fermi sea as ori-gin of the MSHE. Spin current whirls in reciprocal spaceprovide not only a vivid interpretation of the MSHE but alsocorroborate that the MSHE has a bulk contribution. Futureinvestigations in which the importance of the bulk and theinterfacial contributions is considered could tell how to max-imize the MSHE signal. It goes without saying that, due toOnsager’s reciprocity relation [105], the SCV also covers aninverse MSHE, that is a transverse charge current caused by aspin bias.Having identified all magnetic Laue groups that allow forspin current vortices, we demonstrated that any ferromagnetpotentially features an MSHE; furthermore, antiferromagnetswhose MLG permits a magnetic toroidal quadrupole exhibitan MSHE as well. Two pyrochlore models served as examples(Sec. IV A) with magnetic textures exhibiting the multipoleassociated with the MSH conductivities. To issue a caveat,we note that compatibility with a magnetic multipole doesnot necessitate the presence of the multipole. For example, acompletely compensated antiferromagnetic texture may stillexhibit symmetries that permit a magnetization, an observa-tion that was appreciated in the context of the AHE for bothcollinear [106] as well as noncollinear antiferromagnets [107–109]. To name two examples: the kagome magnet discussed inSec. IV C admits of a magnetization without exhibiting a netmoment, and the magnetic warped Rashba model in Sec. IV Badmits of magnetic toroidal quadrupoles although the magnetictexture is collinear.Besides three-dimensional materials, in-plane magnetizedRashba 2DEGs with warping provide a playground for investi-gating a MSHE. A feature they have in common with Mn Snis the option to manipulate out-of-plane polarized spin currentsby rotation of the in-plane magnetic texture. The magnetizationprovides thus an external means to engineer spin accumula-tions.Turning to magnons and replacing the electric field by atemperature gradient, our approach supports that a MSNE isexpected but awaits experimental detection. The magnonicMSNE extends the family of magnonic pendants of electronictransport phenomena [110], its potential for energy harvestingand nonelectronic spin transport remains to be investigated. A candidate material for a proof-of-principle is the ferromag-netic pyrochlore Lu V O . It realizes the magnonic versionof the ferromagnetic pyrochlore model of Sec. IV A [111] andis known for a thermal Hall e ff ect [112, 113] and for Weylmagnons [114]. For a magnetization and temperature gradi-ent along the [001] direction, we expect magnon-mediatedaccumulations of magnetic moments, in analogy to the spinaccumulations shown in Fig. 1. ACKNOWLEDGMENTS
This work is supported by CRC / TRR 227 of DeutscheForschungsgemeinschaft (DFG).
Appendix A: Linear spin wave theory
We provide some background information on the resultspresented in Sec. IV C.The directions ˆ z i of the spins in the classical ground statedefine a local coordinate system { ˆ x i , ˆ y i , ˆ z i } . After a (truncated)Holstein-Primako ff transformation [115] S i (cid:126) ≈ (cid:114) S (cid:104)(cid:16) ψ i + ψ † i (cid:17) ˆ x i − i (cid:16) ψ i − ψ † i (cid:17) ˆ y i (cid:105) + (cid:16) S − ψ † i ψ i (cid:17) ˆ z i , (A1)from spin operators to bosonic creation and annihilation opera-tors ( ψ † i and ψ i ), the bilinear Hamiltonian reads H = (cid:88) k Ψ † k H k Ψ k (A2)after a Fourier transformation. The vector Ψ † k = (cid:16) ψ † , k , ψ † , k , ψ † , k , ψ , − k , ψ , − k , ψ , − k (cid:17) (A3)comprises the Fourier transformed bosonic operators ψ ( † ) n , k ( n = , , H k = S (cid:32) A k B k B † k A ∗ k (cid:33) (A4)is built from the submatrices A k = (cid:16) √ D z + J (cid:17) q c q c q ∗ c (cid:16) √ D z + J (cid:17) q c q ∗ c q ∗ c (cid:16) √ D z + J (cid:17) (A5)and B k = q c q c q c q c q c q c , (A6)4with the k -dependent cosines c = cos (cid:16) ak y (cid:17) , (A7a) c = cos a (cid:16) √ k x − k y (cid:17) , (A7b) c = cos a (cid:16) √ k x + k y (cid:17) . (A7c)Since the classical ground state is (accidentally) degenerate—the spins can be rigidly rotated within the xy plane withoutenergy cost—the q i ( i = , . . . ,
6) depend on the chosen groundstate.Here, we consider two textures. The first texture, shown inFig 8(a), is a representative of the MLG 2 (cid:48) / m (cid:48) ; its q i read q = − √ D z + D (cid:107) + J , (A8a) q = − √ D z + i D (cid:107) + J , (A8b) q = − √ D z − i D (cid:107) + J , (A8c) q = − (cid:16) √ D z + J (cid:17) , (A8d) q = − (cid:16) √ D z + D (cid:107) + J (cid:17) , (A8e) q = − √ D z + D (cid:107) − J . (A8f)The second, rotated texture, shown in Fig 8(d), belongs to theMLG 2 / m , q = − √ D z + J , (A9a) q = − √ D z − √ D (cid:107) + J , (A9b) q = − √ D z − √ D (cid:107) + J , (A9c) q = − √ D z + √ D (cid:107) − J , (A9d) q = − (cid:16) √ D z + √ D (cid:107) + J (cid:17) , (A9e) q = q . (A9f) Next, we diagonalize the bilinear Hamiltonian H = (cid:88) k Φ † k E k Φ k . (A10) E k = diag( ε , k , ε , k , ε , k , ε , − k , ε , − k , ε , − k ) contains the eigen-values, and Φ k = (cid:16) Φ , k , Φ , k , Φ , k , Φ † , − k , Φ † , − k , Φ † , − k (cid:17) is alinear combination of the old bosonic operators, Φ † k = Ψ † k T † k . (A11) T k diagonalizes H k and retains the bosonic commutation rules, T † k Σ T k = Σ, Σ = diag(1 , , , − , − , − . (A12)This procedure follows Ref. 116.The expectation value of the magnonic spin current of the n -th band is defined as [117] J γ n k ,µ = T † k (cid:16) v k ,µ Σ s γ + s γ Σ v k ,µ (cid:17) T k (cid:12)(cid:12)(cid:12)(cid:12) n , n , (A13)in which v k ,µ = (cid:126) − ∂ H k /∂ k µ is the velocity operator and s γ = diag( z γ , z γ , z γ , z γ , z γ , z γ ) contains the γ = x , y , z coordinate ofthe local spin directions for the ground state. The magnonicSCV ∇ k × J γ n k is computed numerically; cf. Fig. 8. [1] Jairo Sinova, Sergio O. Valenzuela, J. Wunderlich, C. H. Back,and T. Jungwirth, “Spin Hall e ff ects,” Rev. Mod. Phys. ,1213–1260 (2015).[2] Yuichiro K. Kato, Roberto C. Myers, Arthur C. Gossard, andDavid D. Awschalom, “Observation of the spin Hall e ff ect insemiconductors,” Science , 1910–1913 (2004).[3] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth,“Experimental observation of the spin-Hall e ff ect in a two-dimensional spin-orbit coupled semiconductor system,” Phys.Rev. Lett. , 047204 (2005).[4] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, “Conversionof spin current into charge current at room temperature: Inversespin-Hall e ff ect,” Applied Physics Letters , 182509 (2006).[5] S. O. Valenzuela and M. Tinkham, “Direct electronic measure-ment of the spin Hall e ff ect,” Nature , 176–179 (2006).[6] Hui Zhao, Eric J. Loren, H. M. van Driel, and Arthur L. Smirl,“Coherence control of Hall charge and spin currents,” Phys. Rev. Lett. , 246601 (2006).[7] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton,S. Von Molnar, M. L. Roukes, A. Yu Chtchelkanova, and D. M.Treger, “Spintronics: a spin-based electronics vision for thefuture,” Science , 1488–1495 (2001).[8] Igor ˇZuti´c, Jaroslav Fabian, and S. Das Sarma, “Spintronics:Fundamentals and applications,” Rev. Mod. Phys. , 323–410(2004).[9] Tomas Jungwirth, J¨org Wunderlich, and Kamil Olejn´ık, “SpinHall e ff ect devices,” Nature materials , 382 (2012).[10] Alexandr Chernyshov, Mason Overby, Xinyu Liu, Jacek K. Fur-dyna, Yuli Lyanda-Geller, and Leonid P. Rokhinson, “Evidencefor reversible control of magnetization in a ferromagnetic ma-terial by means of spin–orbit magnetic field,” Nature Physics ,656 (2009).[11] Luqiao Liu, Takahiro Moriyama, D. C. Ralph, and R. A.Buhrman, “Spin-torque ferromagnetic resonance induced by the spin Hall e ff ect,” Phys. Rev. Lett. , 036601 (2011).[12] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A.Buhrman, “Spin-torque switching with the giant spin Hall e ff ectof tantalum,” Science , 555–558 (2012).[13] J.C. Slonczewski, “Current-driven excitation of magnetic mul-tilayers,” Journal of Magnetism and Magnetic Materials ,L1–L7 (1996).[14] L. Berger, “Emission of spin waves by a magnetic multilayertraversed by a current,” Phys. Rev. B , 9353–9358 (1996).[15] D.C. Ralph and M.D. Stiles, “Spin transfer torques,” Journal ofMagnetism and Magnetic Materials , 1190–1216 (2008).[16] Arne Brataas, Andrew D. Kent, and Hideo Ohno, “Current-induced torques in magnetic materials,” Nature Materials ,372–381 (2012).[17] A. V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R. S.Beach, A. Ong, X. Tang, A. Driskill-Smith, W. H. Butler, P. B.Visscher, D. Lottis, E. Chen, V. Nikitin, and M. Krounbi,“Basic principles of STT-MRAM cell operation in memoryarrays,” Journal of Physics D: Applied Physics , 074001(2013).[18] Pietro Gambardella and Ioan Mihai Miron, “Current-inducedspin–orbit torques,” Philosophical Transactions of the RoyalSociety A: Mathematical, Physical and Engineering Sciences , 3175–3197 (2011).[19] Kotb Jabeur, Gregory Di Pendina, Fabrice Bernard-Granger,and Guillaume Prenat, “Spin orbit torque non-volatile flip-flopfor high speed and low energy applications,” IEEE ElectronDevice Letters , 408–410 (2014).[20] Guillaume Prenat, Kotb Jabeur, Pierre Vanhauwaert, Gre-gory Di Pendina, Fabian Oboril, Rajendra Bishnoi, MojtabaEbrahimi, Nathalie Lamard, Olivier Boulle, Kevin Garello,Juergen Langer, Berthold Ocker, Marie-Claire Cyrille, PietroGambardella, Mehdi Tahoori, and Gilles Gaudin, “Ultra-fastand high-reliability SOT-MRAM: From cache replacement tonormally-o ff computing,” IEEE Transactions on Multi-ScaleComputing Systems , 49–60 (2016).[21] Yucai Li, Kevin William Edmonds, Xionghua Liu, HouzhiZheng, and Kaiyou Wang, “Manipulation of magnetizationby spin-orbit torque,” Advanced Quantum Technologies ,1800052 (2018).[22] Naoto Nagaosa, Jairo Sinova, Shigeki Onoda, A. H. MacDon-ald, and N. P. Ong, “Anomalous Hall e ff ect,” Rev. Mod. Phys. , 1539–1592 (2010).[23] N. Nagaosa S. Murakami and S.-C. Zhang, “Dissipationlessquantum spin current at room temperature,” Science , 1348–1351 (2003).[24] Jairo Sinova, Dimitrie Culcer, Q. Niu, N. A. Sinitsyn, T. Jung-wirth, and A. H. MacDonald, “Universal intrinsic spin Halle ff ect,” Phys. Rev. Lett. , 126603 (2004).[25] M.I. Dyakonov and V.I. Perel, “Current-induced spin orien-tation of electrons in semiconductors,” Physics Letters A ,459–460 (1971).[26] J. E. Hirsch, “Spin Hall e ff ect,” Phys. Rev. Lett. , 1834–1837(1999).[27] Shufeng Zhang, “Spin Hall e ff ect in the presence of spin di ff u-sion,” Phys. Rev. Lett. , 393–396 (2000).[28] N. F. Mott, “The scattering of fast electrons by atomic nuclei,”Proceedings of the Royal Society A: Mathematical, Physicaland Engineering Sciences , 425–442 (1929).[29] G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa, “Intrin-sic spin Hall e ff ect in platinum: First-principles calculations,”Phys. Rev. Lett. , 096401 (2008).[30] Yang Zhang, Jakub ˇZelezn`y, Yan Sun, Jeroen van den Brink,and Binghai Yan, “Spin Hall e ff ect emerging from a non- collinear magnetic lattice without spin–orbit coupling,” NewJournal of Physics , 073028 (2018).[31] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, “Inversespin Hall e ff ect in a ferromagnetic metal,” Phys. Rev. Lett. ,066602 (2013).[32] Tomohiro Taniguchi, J. Grollier, and M. D. Stiles, “Spin-transfer torques generated by the anomalous Hall e ff ect andanisotropic magnetoresistance,” Phys. Rev. Applied , 044001(2015).[33] Dai Tian, Yufan Li, D. Qu, S. Y. Huang, Xiaofeng Jin, and C. L.Chien, “Manipulation of pure spin current in ferromagneticmetals independent of magnetization,” Phys. Rev. B , 020403(2016).[34] H. Wu, X. Wang, L. Huang, J.Y. Qin, C. Fang, X. Zhang, C.H.Wan, and X.F. Han, “Separation of inverse spin Hall e ff ect andanomalous Nernst e ff ect in ferromagnetic metals,” Journal ofMagnetism and Magnetic Materials , 149–153 (2017).[35] K. S. Das, W. Y. Schoemaker, B. J. van Wees, and I. J. Vera-Marun, “Spin injection and detection via the anomalous spinHall e ff ect of a ferromagnetic metal,” Phys. Rev. B , 220408(2017).[36] Alisha M. Humphries, Tao Wang, Eric R. J. Edwards, Shane R.Allen, Justin M. Shaw, Hans T. Nembach, John Q. Xiao, T. J.Silva, and Xin Fan, “Observation of spin-orbit e ff ects with spinrotation symmetry,” Nature Communications , 911 (2017).[37] V. P. Amin, J. Zemen, and M. D. Stiles, “Interface-generatedspin currents,” Phys. Rev. Lett. , 136805 (2018).[38] Jonathan D. Gibbons, David MacNeill, Robert A. Buhrman,and Daniel C. Ralph, “Reorientable spin direction for spincurrent produced by the anomalous Hall e ff ect,” Phys. Rev.Applied , 064033 (2018).[39] Arnab Bose, D. D. Lam, S. Bhuktare, S. Dutta, H. Singh,Y. Jibiki, M. Goto, S. Miwa, and A. A. Tulapurkar, “Obser-vation of anomalous spin torque generated by a ferromagnet,”Phys. Rev. Applied , 064026 (2018).[40] Seung-heon C. Baek, Vivek P. Amin, Young-Wan Oh, Gyung-choon Go, Seung-Jae Lee, Geun-Hee Lee, Kab-Jin Kim, M. D.Stiles, Byong-Guk Park, and Kyung-Jin Lee, “Spin currentsand spin-orbit torques in ferromagnetic trilayers,” Nature Mate-rials , 509–513 (2018).[41] Yasutomo Omori, Edurne Sagasta, Yasuhiro Niimi, MartinGradhand, Luis E. Hueso, F`elix Casanova, and YoshiChikaOtani, “Relation between spin Hall e ff ect and anomalous Halle ff ect in 3 d ferromagnetic metals,” Phys. Rev. B , 014403(2019).[42] V. P. Amin, Junwen Li, M. D. Stiles, and P. M. Haney, “Intrin-sic spin currents in ferromagnets,” (2019).[43] Guanxiong Qu, Kohji Nakamura, and Masamitsu Hayashi,“Magnetization direction dependent spin Hall e ff ect in 3d ferro-magnets,” (2019), arXiv:1901.10740.[44] Wenrui Wang, Tao Wang, Vivek P. Amin, Yang Wang, AnilRadhakrishnan, Angie Davidson, Shane R. Allen, T. J. Silva,Hendrik Ohldag, Davor Balzar, Barry L. Zink, Paul M. Haney,John Q. Xiao, David G. Cahill, Virginia O. Lorenz, and XinFan, “Anomalous spin-orbit torques in magnetic single-layerfilms,” Nature Nanotechnology , 819–824 (2019).[45] Jakub ˇZelezn´y, Yang Zhang, Claudia Felser, and Binghai Yan,“Spin-polarized current in noncollinear antiferromagnets,” Phys.Rev. Lett. , 187204 (2017).[46] J. ˇZelezn´y, P. Wadley, K. Olejn´ık, A. Ho ff mann, and H. Ohno,“Spin transport and spin torque in antiferromagnetic devices,”Nature Physics , 220–228 (2018).[47] Hua Chen, Qian Niu, and Allan H. MacDonald, “Spin Halle ff ect without spin currents in magnetic insulators,” (2018), arXiv:1803.01294.[48] Motoi Kimata, Hua Chen, Kouta Kondou, Satoshi Sugimoto,Prasanta K. Muduli, Muhammad Ikhlas, Yasutomo Omori,Takahiro Tomita, Allan H. MacDonald, Satoru Nakatsuji, andYoshichika Otani, “Magnetic and magnetic inverse spin Halle ff ects in a non-collinear antiferromagnet,” Nature (2019),10.1038 / s41586-018-0853-0.[49] S. Wimmer, M. Seemann, K. Chadova, D. K¨odderitzsch, andH. Ebert, “Spin-orbit-induced longitudinal spin-polarized cur-rents in nonmagnetic solids,” Phys. Rev. B , 041101 (2015).[50] M. Seemann, D. K¨odderitzsch, S. Wimmer, and H. Ebert,“Symmetry-imposed shape of linear response tensors,” Phys.Rev. B , 155138 (2015).[51] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A.Buhrman, J. Park, and D. C. Ralph, “Control of spin-orbittorques through crystal symmetry in wte2 / ferromagnet bilay-ers,” Nature Physics , 300 EP – (2016), article.[52] Jiaqi Zhou, Junfeng Qiao, Arnaud Bournel, and WeishengZhao, “Intrinsic spin Hall conductivity of the semimetalsMoTe and WTe ,” Phys. Rev. B , 060408 (2019).[53] Angie Davidson, Vivek P. Amin, Wafa S. Aljuaid, Paul M.Haney, and Xin Fan, “Perspectives of electrically gen-erated spin currents in ferromagnetic materials,” (2019),arXiv:1906.11772.[54] Albert H¨onemann, Christian Herschbach, Dmitry V. Fedorov,Martin Gradhand, and Ingrid Mertig, “Spin and charge currentsinduced by the spin Hall and anomalous Hall e ff ects uponcrossing ferromagnetic / nonmagnetic interfaces,” Phys. Rev. B , 024420 (2019).[55] E. Lesne, Yu Fu, S. Oyarzun, J. C. Rojas-S´anchez, D. C. Vaz,H. Naganuma, G. Sicoli, J.-P. Attan´e, M. Jamet, E. Jacquet,J.-M. George, A. Barth´el´emy, H. Ja ff r`es, A. Fert, M. Bibes, andL. Vila, “Highly e ffi cient and tunable spin-to-charge conversionthrough Rashba coupling at oxide interfaces,” Nature Materials , 1261–1266 (2016).[56] A. Brinkman, M. Huijben, M. van Zalk, J. Huijben, U. Zeitler,J. C. Maan, W. G. van der Wiel, G. Rijnders, D. H. A. Blank,and H. Hilgenkamp, “Magnetic e ff ects at the interface betweennon-magnetic oxides,” Nature Materials , 493–496 (2007).[57] N. Reyren, S. Thiel, A. D. Caviglia, L. F. Kourkoutis, G. Ham-merl, C. Richter, C. W. Schneider, T. Kopp, A.-S. Ruetschi,D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, andJ. Mannhart, “Superconducting interfaces between insulatingoxides,” Science , 1196–1199 (2007).[58] Julie A. Bert, Beena Kalisky, Christopher Bell, Minu Kim, Ya-suyuki Hikita, Harold Y. Hwang, and Kathryn A. Moler, “Di-rect imaging of the coexistence of ferromagnetism and super-conductivity at the LaAlO3 / SrTiO3 interface,” Nature Physics , 767–771 (2011).[59] D. A. Dikin, M. Mehta, C. W. Bark, C. M. Folkman, C. B.Eom, and V. Chandrasekhar, “Coexistence of superconductiv-ity and ferromagnetism in two dimensions,” Phys. Rev. Lett. , 056802 (2011).[60] Lu Li, C. Richter, J. Mannhart, and R. C. Ashoori, “Coexis-tence of magnetic order and two-dimensional superconductiv-ity at LaAlO3 / SrTiO3 interfaces,” Nature Physics , 762–766(2011).[61] Arjun Joshua, S Pecker, J Ruhman, E Altman, and S Ilani, “Auniversal critical density underlying the physics of electrons atthe LaAlO / SrTiO interface,” Nature communications , 1129(2012).[62] A. D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schnei-der, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M.Triscone, “Electric field control of the LaAlO3 / SrTiO3 inter- face ground state,” Nature , 624–627 (2008).[63] Hongrui Zhang, Xi Yan, Hui Zhang, Fei Wang, Youdi Gu,Xingkun Ning, Tahira Khan, Rui Li, Yuansha Chen, Wei Liu,Shufang Wang, Baogen Shen, and Jirong Sun, “Magnetictwo-dimensional electron gases with high Curie temperaturesat LaAlO / SrTiO :fe interfaces,” Phys. Rev. B , 155150(2018).[64] Liang Fu, “Hexagonal warping e ff ects in the surface states ofthe topological insulator Bi Te ,” Phys. Rev. Lett. , 266801(2009).[65] S. V. Eremeev, V. N. Men’shov, V. V. Tugushev, P. M.Echenique, and E. V. Chulkov, “Magnetic proximity e ff ectat the three-dimensional topological insulator / magnetic insula-tor interface,” Phys. Rev. B , 144430 (2013).[66] Ryogo Kubo, Mario Yokota, and Sadao Nakajima, “Statistical-mechanical theory of irreversible processes. II. Response tothermal disturbance,” Journal of the Physical Society of Japan , 1203–1211 (1957).[67] Gerald D. Mahan, Many-Particle Physics (Springer Science + Business Media, 2000).[68] L D Landau and E. M. Lifshitz,
Fluid Mechanics: Landau andLifshitz: Course of Theoretical Physics, Volume 6 (Pergamon,2013).[69] Ph. Roux and M. Fink, “Experimental evidence in acoustics ofthe violation of time-reversal invariance induced by vorticity,”EPL (Europhysics Letters) , 25 (1995).[70] W. H. Kleiner, “Space-time symmetry of transport coe ffi cients,”Phys. Rev. , 318–326 (1966).[71] W. H. Kleiner, “Space-time symmetry restrictions on transportcoe ffi cients. ii. Two theories compared,” Phys. Rev. , 726–727 (1967).[72] W. H. Kleiner, “Space-time symmetry restrictions on transportcoe ffi cients. iii. Thermogalvanomagnetic coe ffi cients,” Phys.Rev. , 705–709 (1969).[73] Stefan Nanz, “Why another multipole family?” in ToroidalMultipole Moments in Classical Electrodynamics (SpringerFachmedien Wiesbaden, 2016) pp. 5–11.[74] H. A. Jahn, “Note on the Bhagavantam–Suranarayana methodof enumerating the physical constants of crystals,” Acta Crys-tallographica , 30–33 (1949).[75] Satoru Hayami, Megumi Yatsushiro, Yuki Yanagi, and HiroakiKusunose, “Classification of atomic-scale multipoles undercrystallographic point groups and application to linear responsetensors,” Phys. Rev. B , 165110 (2018).[76] Samuel V. Gallego, Jesus Etxebarria, Luis Elcoro, Emre S.Tasci, and J. Manuel Perez-Mato, “Automatic calculation ofsymmetry-adapted tensors in magnetic and non-magnetic ma-terials: a new tool of the Bilbao Crystallographic Server,” ActaCrystallographica Section A , 438–447 (2019).[77] Mois Ilia Aroyo, Juan Manuel Perez-Mato, Cesar Capillas,Eli Kroumova, Svetoslav Ivantchev, Gotzon Madariaga, AsenKirov, and Hans Wondratschek, “Bilbao crystallographicserver: I. Databases and crystallographic computing programs,”Zeitschrift f¨ur Kristallographie - Crystalline Materials (2006), 10.1524 / zkri.2006.221.1.15.[78] Mois I. Aroyo, Asen Kirov, Cesar Capillas, J. M. Perez-Mato,and Hans Wondratschek, “Bilbao crystallographic server. II.Representations of crystallographic point groups and spacegroups,” Acta Crystallographica Section A Foundations ofCrystallography , 115–128 (2006).[79] Mois I. Aroyo, J.M. Perez-Mato, D. Orobengoa, E. Tasci,G. De La Flor, and A. Kirov, “Crystallography online: Bilbaocrystallographic server,” Bulgarian Chemical Communications , 183–197 (2011). [80] If the spin orientation breaks all crystal symmetries, Mn Snbelongs to the MLG ¯1, which is, however, unlikely in theabsence of a magnetic field, because of anisotropies along highsymmetry directions of the lattice.[81] Tom´aˇs Bzduˇsek, Andreas R¨uegg, and Manfred Sigrist, “Weylsemimetal from spontaneous inversion symmetry breaking inpyrochlore oxides,” Phys. Rev. B , 165105 (2015).[82] M.-T. Suzuki, T. Nomoto, R. Arita, Y. Yanagi, S. Hayami, andH. Kusunose, “Multipole expansion for magnetic structures:A generation scheme for a symmetry-adapted orthonormalbasis set in the crystallographic point group,” Phys. Rev. B ,174407 (2019).[83] Christian R. Ast, J¨urgen Henk, Arthur Ernst, Luca Moreschini,Mihaela C. Falub, Daniela Pacil´e, Patrick Bruno, Klaus Kern,and Marco Grioni, “Giant spin splitting through surface alloy-ing,” Phys. Rev. Lett. , 186807 (2007).[84] Fabian Meier, Hugo Dil, Jorge Lobo-Checa, Luc Patthey, andJ¨urg Osterwalder, “Quantitative vectorial spin analysis in angle-resolved photoemission: Bi / Ag(111) and Pb / Ag(111),” Phys.Rev. B , 165431 (2008).[85] H. Bentmann, F. Forster, G. Bihlmayer, E. V. Chulkov,L. Moreschini, M. Grioni, and F. Reinert, “Origin and ma-nipulation of the Rashba splitting in surface alloys,” EPL (Eu-rophysics Letters) , 37003 (2009).[86] Emmanouil Frantzeskakis and Marco Grioni, “Anisotropy ef-fects on Rashba and topological insulator spin-polarized sur-face states: A unified phenomenological description,” Phys.Rev. B , 155453 (2011).[87] Masataka Mogi, Taro Nakajima, Victor Ukleev, AtsushiTsukazaki, Ryutaro Yoshimi, Minoru Kawamura, Kei S. Taka-hashi, Takayasu Hanashima, Kazuhisa Kakurai, Taka-hisaArima, Masashi Kawasaki, and Yoshinori Tokura, “Largeanomalous Hall e ff ect in topological insulators with proximi-tized ferromagnetic insulators,” Phys. Rev. Lett. , 016804(2019).[88] J. Krempask´y, S. Mu ff , F. Bisti, M. Fanciulli, H. Volfov´a, A. P.Weber, N. Pilet, P. Warnicke, H. Ebert, J. Braun, F. Bertran,V. V. Volobuev, J. Min´ar, G. Springholz, J. H. Dil, and V. N.Strocov, “Entanglement and manipulation of the magnetic andspin–orbit order in multiferroic Rashba semiconductors,” Na-ture Communications (2016), 10.1038 / ncomms13071.[89] J. Krempask´y, G. Springholz, J. Min´ar, and J. H. Dil, “ α -GeTe and (GeMn)Te semiconductors: A new paradigm for spin-tronics,” AIP Conference Proceedings , 020026 (2018),https: // aip.scitation.org / doi / pdf / / , eaat9989 (2018).[91] Jung Hoon Han and Hyunyong Lee, “Spin chirality and Hall-like transport phenomena of spin excitations,” Journal of thePhysical Society of Japan , 011007 (2017).[92] Tao Qin, Qian Niu, and Junren Shi, “Energy magnetization andthe thermal Hall e ff ect,” Phys. Rev. Lett. , 236601 (2011).[93] M. Elhajal, B. Canals, and C. Lacroix, “Symmetry breakingdue to Dzyaloshinsky-Moriya interactions in the kagom´e lat-tice,” Phys. Rev. B , 014422 (2002).[94] Alexander Mook, J¨urgen Henk, and Ingrid Mertig, “ThermalHall e ff ect in noncollinear coplanar insulating antiferromag-nets,” Phys. Rev. B , 014427 (2019).[95] Nobuyuki Okuma, “Magnon spin-momentum locking: Variousspin vortices and Dirac magnons in noncollinear antiferromag-nets,” Phys. Rev. Lett. , 107205 (2017).[96] S. J. Barnett, “Magnetization by rotation,” Phys. Rev. , 239– 270 (1915).[97] D.E. Kharzeev, J. Liao, S.A. Voloshin, and G. Wang, “Chiralmagnetic and vortical e ff ects in high-energy nuclear collisionsastatus report,” Progress in Particle and Nuclear Physics , 1 –28 (2016).[98] R. Takahashi, M. Matsuo, M. Ono, K. Harii, H. Chudo,S. Okayasu, J. Ieda, S. Takahashi, S. Maekawa, and E. Saitoh,“Spin hydrodynamic generation,” Nature Physics , 52 (2016).[99] Mamoru Matsuo, Eiji Saitoh, and Sadamichi Maekawa, “Spin-Mechatronics,” Journal of the Physical Society of Japan ,011011 (2017).[100] M. Matsuo, Y. Ohnuma, and S. Maekawa, “Theory of spinhydrodynamic generation,” Phys. Rev. B , 020401 (2017).[101] Ruben J. Doornenbal, Marco Polini, and Rembert A. Duine,“Spin–vorticity coupling in viscous electron fluids,” Journal ofPhysics: Materials , 015006 (2019).[102] Voicu Popescu, Peter Kratzer, Peter Entel, Christian Heiliger,Michael Czerner, Katarina Tauber, Franziska T¨opler, Chris-tian Herschbach, Dmitry V Fedorov, Martin Gradhand, In-grid Mertig, Roman Kov´aˇcik, Phivos Mavropoulos, DanielWortmann, Stefan Bl¨ugel, Frank Freimuth, Yuriy Mokrousov,Sebastian Wimmer, Diemo K¨odderitzsch, Marten Seemann,Kristina Chadova, and Hubert Ebert, “Spin caloric transportfrom density-functional theory,” Journal of Physics D: AppliedPhysics , 073001 (2018).[103] A. G. Aronov and Yu B. Lyanda-Geller, “Nuclear electric res-onance and orientation of carrier spins by an electric field,”Soviet Journal of Experimental and Theoretical Physics Letters , 431 (1989).[104] V.M. Edelstein, “Spin polarization of conduction electronsinduced by electric current in two-dimensional asymmetricelectron systems,” Solid State Communications , 233–235(1990).[105] Lars Onsager, “Reciprocal relations in irreversible processes.I.” Phys. Rev. , 405–426 (1931).[106] Libor ˇSmejkal, Rafael Gonz´alez-Hern´andez, Tom´aˇs Jungwirth,and Jairo Sinova, “Crystal Hall e ff ect in collinear antiferromag-nets,” (2019), arXiv:1901.00445.[107] Hua Chen, Qian Niu, and A. H. MacDonald, “Anomalous Halle ff ect arising from noncollinear antiferromagnetism,” Phys.Rev. Lett. , 017205 (2014).[108] J¨urgen K¨ubler and Claudia Felser, “Non-collinear antiferro-magnets and the anomalous Hall e ff ect,” EPL (EurophysicsLetters) , 67001 (2014).[109] M.-T. Suzuki, T. Koretsune, M. Ochi, and R. Arita, “Clustermultipole theory for anomalous Hall e ff ect in antiferromagnets,”Phys. Rev. B , 094406 (2017).[110] Alexander Mook, B¨orge G¨obel, J¨urgen Henk, and Ingrid Mer-tig, “Taking an electron-magnon duality shortcut from electronto magnon transport,” Phys. Rev. B , 140401 (2018).[111] Maged Elhajal, Benjamin Canals, Raimon Sunyer, and Clau-dine Lacroix, “Ordering in the pyrochlore antiferromagnetdue to Dzyaloshinsky-Moriya interactions,” Phys. Rev. B ,094420 (2005).[112] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, andY. Tokura, “Observation of the magnon hall e ff ect,” Science , 297–299 (2010).[113] T. Ideue, Y. Onose, H. Katsura, Y. Shiomi, S. Ishiwata, N. Na-gaosa, and Y. Tokura, “E ff ect of lattice geometry on magnonHall e ff ect in ferromagnetic insulators,” Phys. Rev. B ,134411 (2012).[114] Alexander Mook, J¨urgen Henk, and Ingrid Mertig, “Tunablemagnon Weyl points in ferromagnetic pyrochlores,” Phys. Rev.Lett. , 157204 (2016). [115] T. Holstein and H. Primako ff , “Field dependence of the intrinsicdomain magnetization of a ferromagnet,” Phys. Rev. , 1098–1113 (1940).[116] J.H.P. Colpa, “Diagonalization of the quadratic boson hamil-tonian,” Physica A: Statistical Mechanics and its Applications , 327–353 (1978).[117] Alexander Mook, Robin R. Neumann, J¨urgen Henk, and IngridMertig, “Spin Seebeck and spin Nernst e ff ects of magnons innoncollinear antiferromagnetic insulators,” Phys. Rev. B100