Nonreciprocal spin Seebeck effect in antiferromagnets
NNonreciprocal spin Seebeck e ff ect in antiferromagnets Rina Takashima, Yuki Shiomi,
1, 2 and Yukitoshi Motome Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan ∗ RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan (Dated: July 16, 2018)We theoretically propose a nonreciprocal spin Seebeck e ff ect, i.e., nonreciprocal spin transport generated bya temperature gradient, in antiferromagnetic insulators with broken inversion symmetry. We find that nonre-ciprocity in antiferromagnets has rich properties not expected in ferromagnets. In particular, we show that polarantiferromagnets, in which the crystal lacks the spatial inversion symmetry, exhibit perfect nonreciprocity —one-way spin current flow irrespective of the direction of the temperature gradient. We also show that nonpolarcentrosymmetric crystals can exhibit nonreciprocity when a magnetic order breaks the inversion symmetry, andin this case, the direction of the nonreciprocal flow can be controlled by reversing the magnetic domain. As theirrepresentatives, we calculate the nonreciprocal spin Seebeck voltages for the polar antiferromagnet α -Cu V O and the honeycomb antiferromagnet MnPS , while varying temperature and magnetic field. The reciprocal relation is a fundamental principle in ther-modynamics assured by the symmetry of the system. It is,however, violated when a certain symmetry is broken, e.g.,by crystal structures, electronic orderings, and external fields.Such violation of the reciprocity has attracted much interestfrom both fundamental physics and application. An archetypeis the Faraday e ff ect of light, in which the breaking of time-reversal symmetry causes a rotation of the polarization planein an opposite direction when the propagation direction oflight is switched. This nonreciprocal property has been usedfor an optical isolator and optical data storage. Another exam-ple is found in a p - n junction, which allows a one-way flow ofan electric current. A similar diode e ff ect can also occur in abulk crystal when time-reversal and spatial-inversion symme-tries are simultaneously broken [1].The nonreciprocity has also been studied for the propa-gation of spin waves in magnetic materials. The most pro-nounced example is the Damon-Eshbach mode, in which spinwaves propagate on a material surface only in one direc-tion [2]. Also in a bulk magnet, the breaking of spatial-inversion symmetry gives rise to nonreciprocal propagation ofspin waves. There, an asymmetric exchange interaction calledthe Dzyaloshinskii-Moriya (DM) interaction [3, 4] bringsabout asymmetry in the spin-wave dispersion with respect tothe propagation direction. This has been experimentally ob-served in hetero-multilayer films of ferromagnets [5], ferro-magnets having noncentrosymmetric crystal symmetries [6–8], and a polar antiferromagnet (AFM) α -Cu V O [9]. Sincea spin wave can carry a spin current, the asymmetric disper-sion may give rise to a nonreciprocal spin current. However,such an e ff ect remains elusive thus far, despite the relevanceto applications in spintronics as well as magnonic devices.In this Rapid Communication, we propose a nonrecipro-cal response of a spin current in antiferromagnetic insula-tors, which stems from the asymmetric spin-wave dispersion.Specifically, we consider the spin Seebeck e ff ect (SSE), amagnetothermal phenomenon in which a temperature gradi-ent causes a spin voltage [10–13]. We show that a nonrecip- ∗ [email protected] rocal spin current can be generated as a nonlinear responseto a temperature gradient [Fig. 1(a)]. A nonreciprocal spincurrent response to an electric field was recently discussed innoncentrosymmetric metals [14, 15], which su ff er from Jouleheating. We here discuss the nonreciprocal SSE, mainly forantiferromagnetic insulators, as they have drawn considerableinterest in recent spintronics owing to less stray field and ul-trafast spin dynamics [16, 17]. We find that the AFMs showremarkable properties in the nonreciprocal SSE, which are notexpected in ferromagnets. We demonstrate that the nonrecip-rocal SSE appears in a di ff erent manner for two di ff erent typesof AFMs: One is a polar AFM on a noncentrosymmetric lat-tice and the other is a zigzag AFM on a centrosymmetric lat-tice. The polar AFMs exhibit perfect nonreciprocity : A spincurrent flows only in one direction irrespective of the directionof the temperature gradient [Fig. 1(a)]. On the other hand, inthe zigzag AFM, the nonreciprocity can be controlled by re-versing magnetic domains. For the experimental observations,we calculate the spin Seebeck voltages for candidate materialsfor the two cases, the polar AFM α -Cu V O and the honey-comb (two-dimensional zigzag) AFM MnPS , and clarify thedependence on temperature, the magnetic field, and the direc-tion of the temperature gradient.We consider the spin current generated parallel to a temper-ature gradient up to the second order: j s z x = S xx ( ∂ x T ) + S xx ( ∂ x T ) , (1)where j s z x is the spin current that flows in the x direction car-rying the spin along the z axis, and T is the local temperatureof the sample. The first term in Eq. (1) corresponds to theconventional SSE [10–13], and the second one is the nonlin-ear term, which we will discuss in this work. When S xx isnonzero, the magnitude of j s z x changes depending on the signof ∂ x T . Thus, the nonlinear contribution in the SSE gives riseto a nonreciprocal spin current j s z x . We note that, from thesymmetry point of view, such nonreciprocity is not allowedwhen the system is symmetric under the spatial inversion I ormirror reflection with respect to the xz plane, denoted by M y .Meanwhile, the linear component S xx vanishes when M x ( yz mirror) or M y symmetry exists. a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l (b) (c) (a) A BA B ho t co l d N on r ec i p r oca l r esponse ho t ho t cold cold P e f ec t non r ec i p r oc it y hot hotcold cold s p i n c u rr en t po l a r A F M z i gzag A F M FIG. 1. (a) Schematic picture of a nonreciprocal spin current under athermal gradient in an AFM. (b), (c) Schematic pictures of (b) a polarAFM with a uniform DM interaction and (c) a zigzag AFM with astaggered DM interaction. The color gradient in (b) represents thebreaking of mirror symmetry with respect to the xz plane. I M x M y B z dep. of S xx B z dep. of S xx domainpolar AFM × (cid:88) × odd even indep.zigzag AFM × × (cid:88) odd odd dep.TABLE I. Symmetry arguments on the magnetic field dependence( B z dep.) of the SSE coe ffi cients for the one-dimensional spin mod-els for polar and zigzag AFMs [Eqs. (2)-(4); see also Figs. 1(b) and1(c)]. The domain dependence (dep.) or independence (indep.) ofthe nonreciprocal SSE on the magnetic domains are also shown. I is the inversion symmetry, and M x and M y are the mirror reflectionsymmetry with respect to the yz and zx planes, respectively. Notethat each mirror symmetry represents the mirror reflection combinedwith half translation in the x direction. In this study, we calculate the spin current in Eq. (1) forAFMs, which exhibit richer nonreciprocal properties com-pared to ferromagnets [18]. We consider two types of non-centrosymmetric AFMs. One is an AFM on a noncentrosym-metric lattice, and the other is an AFM in which the inversionsymmetry is broken by the magnetic order. As their typicalexamples, we first study one-dimensional spin models for thetwo types, which we call polar AFMs [Fig. 1(b)] and zigzagAFMs [Fig. 1(c)], respectively [19]. Their Hamiltonians aregiven by H = H + H polar / zigzagD , where H = (cid:80) r (cid:44) r (cid:48) (cid:104) J rr (cid:48) S r · S r (cid:48) + G rr (cid:48) ( S zr S zr (cid:48) − S xr S xr (cid:48) − S yr S yr (cid:48) ) (cid:105) + g s µ B (cid:126) B z (cid:80) r S zr . (2)Here S r = ( S xr , S yr , S zr ) is the spin operator at r = ( i , (cid:96) ), where i denotes the unit cell and (cid:96) denotes the sublattice. We as-sume that a (magnetic) unit cell has two sites: (cid:96) = { A , B } . J rr (cid:48) and G rr (cid:48) denote the coupling constants for the isotropicand anisotropic exchange interactions, respectively; the latteroriginates from the spin-orbit coupling. g s is the electron sping-factor (we take g s = µ B is the Bohr magneton, (cid:126) is the re-duced Planck constant, and B z is the magnetic field along the z direction. H polarD and H zigzagD represent the DM interactions inthe polar and zigzag systems, respectively: H polarD = D (cid:88) i ˆ z · (cid:0) S i , A × S i , B + S i , B × S i + , A (cid:1) , (3) H zigzagD = D (cid:88) i ˆ z · (cid:0) S i , A × S i + , A − S i , B × S i + , B (cid:1) , (4)where ˆ z is the unit vector along the z direction. Here, tak-ing the chain direction as x , we assume that the polar systemlacks M y symmetry while preserving M z symmetry ( xy mir-ror) [Fig. 1(b)]; hence, we include a uniform DM interactionfor all the nearest neighbors with the DM vector D (cid:107) ˆ z inEq. (3). On the other hand, in the zigzag system, there isno inversion symmetry at the centers of the second neighborbonds, while the system is inversion symmetric with respectto the centers of the nearest neighbor bonds. Therefore, weinclude a staggered DM interaction for the second neighborswith the DM vector D (cid:107) ˆ z in Eq. (4).Assuming a collinear antiferromagnetic ground state,namely, (cid:104) S zi , A (cid:105) = −(cid:104) S zi , B (cid:105) = S , we consider magnon excita-tions by using the Holstein-Primako ff transformation as S + i , A = (cid:126) (2 S − a † i a i ) / a i , S zi , A = (cid:126) (cid:16) S − a † i a i (cid:17) , (5) S + i , B = (cid:126) b † i (2 S − b † i b i ) / , S zi , B = (cid:126) (cid:16) b † i b i − S (cid:17) , (6)where S + i ,(cid:96) = S xi ,(cid:96) + iS yi ,(cid:96) = ( S − i ,(cid:96) ) † . By substituting Eqs. (5)and (6) into the Hamiltonian and using the linear spin waveapproximation, we obtain the magnon Hamiltonian in the bi-linear form of the operators of a i and b i . Diagonalizing theHamiltonian by the Bogoliubov transformation, we obtain H = (cid:80) σ k x ε σ k x α † σ k x α σ k x , where α σ k x ( α † σ k x ) is the annihilation(creation) operator of a magnon with the spin angular momen-tum σ = {↑ , ↓} and the momentum k x . ε σ k x ≥ k x [19]. This is the crucial feature to produce the nonrecip-rocal SSE as discussed below.In the present systems, the total spin along the z direction S z tot ≡ (cid:80) i (cid:16) S zi , A + S zi , B (cid:17) = (cid:80) k x (cid:16) − α †↓ k x α ↓ k x + α †↑ k x α ↑ k x (cid:17) is con-served, as the DM vectors point along the z direction. Sinceeach magnon excitation carries the spin angular momentum ± (cid:126) , the local spin current density is given by J s z x = (cid:126) (cid:90) dk x π (cid:16) (cid:51) x ↑ k x n ( ε ↑ k x ) − (cid:51) x ↓ k x n ( ε ↓ k x ) (cid:17) (7)where the velocity is defined by (cid:51) x σ k x = (1 / (cid:126) ) ∂ε σ k x /∂ k x , and n ( ε σ k x ) = (cid:104) α † σ k x α σ k x (cid:105) denotes the magnon distribution at a fi-nite temperature.To analyze the SSE, we use the Boltzmann transport the-ory [20]. We assume that the temperature of the system hasa linear gradient, T ( x ) = T + α x , where the coe ffi cient α issmall enough to allow us to define the equilibrium distributionof magnons by n ( ε σ k x ) = (exp( ε σ k x / T ( x )) − − . With the re-laxation time approximation, the Boltzmann theory gives (cid:51) x σ k x ∂ n ( ε σ k x ) ∂ x = − ∂ n ∂ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) col . = − n ( ε σ k x ) − n ( ε σ k x ) τ , (8)where we have neglected the energy and momentum depen-dence of the relaxation time τ . Substituting the solution ofEq. (8) into Eq. (7) and averaging over the space, we obtainthe net component of spin current in Eq. (1) with the coe ffi -cients of S xx = − (cid:126) τ (cid:82) dk x π (cid:104) ( (cid:51) x ↑ k x ) f (1) ( ε n ↑ k x ) − ( (cid:51) x ↓ k x ) f (1) ( ε ↓ k x ) (cid:105) , (9) S xx = (cid:126) τ (cid:82) dk x π (cid:104) ( (cid:51) x ↑ k x ) f (2) ( ε ↑ k x ) − ( (cid:51) x ↓ k x ) f (2) ( ε ↓ k x ) (cid:105) , (10)where f (1) ( ε ) = ∂ n /∂ T | T = T and f (2) ( ε ) = ∂ n /∂ T | T = T [18]. Eq. (10) indicates that the nonlin-ear component originates in the asymmetry in the magnondispersion, whose measure is given by the cube of the velocityaveraged over a constant energy surface as (cid:104) ( (cid:51) x σ k x ) (cid:105) ε kx = ε (cid:66) (cid:90) ε σ kx = ε dk x π ( (cid:51) x σ k x ) . (11)This value vanishes when the magnon dispersion for each spincomponent is symmetric with respect to k x .As mentioned above, the polar and zigzag AFMs have theasymmetric magnon dispersions with (cid:104) ( (cid:51) x σ k x ) (cid:105) ε kx = ε (cid:44)
0, andhence, they exhibit the nonreciprocal SSE. Due to the di ff erentsymmetry, however, the SSE appears in a di ff erent mannerbetween the two cases. As noted below Eq. (1), the linearSSE coe ffi cient S xx can be nonzero only when both M x and M y symmetries are broken, whereas the nonlinear one S xx can be nonzero when M y symmetry is broken in addition tothe inversion symmetry I . In polar AFMs, where M y ( M x ) is(un)broken, S xx vanishes but S xx may become nonzero at B z =
0. When the magnetic field B z , which breaks M x , is applied, S xx is induced as an odd function of B z , while S xx is an evenfunction of B z . On the other hand, in the zigzag AFMs, where M y is preserved, both S xx and S xx are odd functions of B z .The results are summarized in Table I.From the symmetry arguments, an interesting phenomenonis readily concluded for the polar AFMs. When S xx is nonzeroat B z = ffi cients given by Eqs. (9) and(10) for real materials. First we consider a candidate for thepolar AFMs, α -Cu V O , whose lattice structure breaks themirror symmetry with respect to the ab plane [see Fig. 2(a)].Below T N = . α -Cu V O shows an antiferromagneticorder, where Cu + spins ( S = /
2) align antiparallel along[100] [Fig. 2(a)] with a small canting along [001] [9, 21–23].The magnon bands obtained by a recent neutron scattering ex-periment indicate the presence of the strong uniform DM in-teraction [9] similar to the polar AFMs discussed above. Inthe following calculation, we use the model Hamiltonian, ob-tained via the inelastic neutron scattering experiment [9]. Ithas the isotropic exchange interactions between the first, sec-ond, and third neighbors, J = . J = .
99, and J = . G = .
282 meV, and the nearest-neighbor DMinteraction D = .
79 meV; note that while the ( x , y , z ) axes (a) (b)(c) (000) (010) momentum e n e r g y [ m e V ] (010) µ FIG. 2. (a) Schematic picture of the lattice structure and thespin configuration on the Cu + ions in α -Cu V O . The crystallo-graphic axes are also shown. (b) Magnon bands in the polar AFM α -Cu V O . The blue (red) bands carry the spin angular momentum S z tot = − k y . The parametersare given in the main text. (c) Dependence of the spin Seebeck volt-ages on the temperature and the applied field along [100]: the linearterm V SSE1 (left) and the nonlinear term V SSE2 (right). are taken along the crystallographic ( a , b , c ) axes, they corre-spond to ( z , x , y ) in the model in Eqs. (2) and (3) (the totalspin angular momentum along the x direction is conserved).Since each unit cell has 16 Cu + spins, the magnon bandshave eight branches per spin [9], as reproduced in Fig. 2(b).The magnon dispersions are asymmetric along the k y direc-tion resulting in (cid:104) ( (cid:51) yn σ k ) (cid:105) ε n σ k = ε (cid:44) n being the band in-dex. Hence, the system exhibits the SSE along the y direction, j s x y = S yy ( ∂ y T ) + S yy ( ∂ y T ) .In experiments, the spin current generated by the SSE canbe measured by the inverse spin Hall e ff ect of Pt attached tothe sample. We assume that the induced voltage in Pt is sim-ply given by the sum of linear and nonlinear components as V SSE = V SSE1 + V SSE2 , where V SSE1 = − ρ Pt θ sh 2 e (cid:126) L S yy ( ∂ y T ) , (12) V SSE2 = − ρ Pt θ sh 2 e (cid:126) L S yy ( ∂ y T ) , (13) ρ Pt is the electrical resistivity of Pt, θ sh is the spin Hall an-gle of Pt, and L is the length of the sample along the volt-age direction. Recently, V SSE1 was measured for α -Cu V O ,and τ ∝ T − fits the experimental data well [13]. In ouranalysis, assuming the power-law behavior, we estimate themagnitude of τ using the experimental data in Ref. [13]. Weuse ρ Pt = . × − Ω m and θ sh = .
021 [24], and set ∂ y T = Km − and L = × − m based on the experi-mental setup [13]. With the above assumptions, we obtain therelaxation time τ (cid:39) c / T with c = × − K sec.Using the obtained relaxation time, we calculate V SSE1 and hot hotcold cold sp i n current (a) KM Γ energy θ[°] [ a r b . u n i t ][ a r b . u n i t ] e n e r g y [ m e V ] = (c) (b) K’ [meV] K Γ K’ θ =0 FIG. 3. (a) Schematic picture of the lattice structure and the spinconfiguration on the Mn + ions in MnPS . Crystallographic axes arealso shown. (b) Energy dispersion of the magnons. The energies ofthe magnons with S z tot = ± B z is normal to thehoneycomb plane. The right panels show the directional dependenceof the linear term V SSE1 (top) and the nonlinear term V SSE2 (bottom). V SSE2 as functions of the temperature and the field along the x direction, B x . The results are shown in Fig. 2(c). We find thatthe nonlinear component V SSE2 appears as an even function of B x , whereas the linear one V SSE1 is odd. Furthermore, V SSE2 isnonzero at B x =
0, i.e., the system exhibits the perfect nonre-ciprocal spin transport. These behaviors are exactly what weexpected for the polar AFM; in the present material, instead ofthe mirror symmetry, the C rotational symmetry along [001]makes S yy zero, while the breaking of both inversion and M z symmetries results in nonzero S yy .With regard to the temperature dependence, both V SSE1 and V SSE2 exhibit peaks at finite temperatures, and decay at highertemperatures, as shown in Fig. 2(c). Note that the calculatedcurve of V SSE1 reproduces the experimental data well [13]. Thepeak structure comes from the competition between the ther-mal excitations of magnons and the scattering rate. At a verylow temperature, the SSE is enhanced by the thermal excita-tions of magnons as increasing temperature. With a furtherincrease of temperature, however, the scattering processes,characterized by τ , begin to suppress the SSE, leaving thepeak structure at an intermediate temperature. We note thatthe peak temperatures are lower and the peaks are sharper for V SSE2 compared to V SSE1 . This arises from the dependence on τ ∝ T − : V SSE1 and V SSE2 depend on τ and τ , respectively, asshown in Eqs. (9) and (10).Next, we discuss a candidate for the zigzag AFMs, the hon-eycomb AFM MnPS . Note that the two-dimensional honey-comb structure is composed of one-dimensional zigzag chainsrunning in three di ff erent directions. MnPS has a layeredhoneycomb structure with the weak interlayer van der Waals interaction, as shown in Fig. 3(a). A neutron di ff raction studyshows that Mn + spins ( S = /
2) align in a staggered way be-low T N =
78 K, whose moment directions are almost normalto the honeycomb plane [25] [Fig. 3(a)]. Hereafter, we labelthe crystallographic coordinate ( a , b , c ∗ ) by ( x , y , z ), where c ∗ is normal to the ab plane. The spin model obtained by an in-elastic neutron scattering [26] includes J = . J = . J = .
36, and G = . × − in units of meV. Since the inter-layer exchange interaction is much smaller than the intralayerexchange interactions, we calculate the SSE for a single hon-eycomb layer.From the lattice symmetry, the system has a staggeredDM interaction between the second neighbors along the threetypes of zigzag chains, as in Eq. (4) [27]. This leads to theasymmetry in the magnon bands, as shown in Fig. 3(b) [19].This staggered DM interaction is reported to show an interest-ing magnon transport, the “Nernst” e ff ect of a magnon spincurrent [28–30]. In the experiment of the magnon Nernste ff ect [30], the magnitude of D has been estimated as D ∼ . S z tot = ± (cid:104) ( (cid:51) xn σ k ) (cid:105) ε n ,σ, k = ε (cid:44)
0, e.g., along the K- Γ -K’ line. In this sit-uation, the nonreciprocal SSE appears when a magnetic fieldlifts the degeneracy of the two magnon bands ( S z tot = ± V SSE1 and V SSE2 are odd functions of the magnetic fieldnormal to the honeycomb plane B z .To experimentally detect the nonreciprocal SSE in this hon-eycomb system, we can exploit the directional dependence of V SSE1 and V SSE2 . Figure 3(b) shows that the energy dispersionalong the M- Γ -M cut is symmetric, which suggests that thenonreciprocal SSE does not occur along this direction. In-deed, we find the directional dependence with threefold ro-tational symmetry at nonzero temperature under a finite B z ,as shown in the lowerright panel of Fig. 3(c). (Note that themagnitudes depend on τ , of which we do not have a quantita-tive estimate for the present compound.) The nonlinear spinSeebeck voltage V SSE2 vanishes in the directions correspond-ing to M- Γ -M (e.g., θ = ◦ ), whereas the linear one V SSE1 (theupper panel) is always nonzero for B z (cid:44) S xx . When the inversion symmetry is brokenby a magnetic order as in the zigzag AFMs (e.g., MnPS ), S xx changes its sign between two di ff erent magnetic domainsTherefore the nonreciprocal SSE can be controlled by revers-ing magnetic domains [33]. On the other hand, when the in-version symmetry is broken by the crystal structure as in thepolar AFM (e.g., α -Cu V O ), S xx is not changed by mag-netic domain reversal. The results are shown in Table I.In summary, we have theoretically investigated the nonre-ciprocal response of a spin current in AFMs under a thermalgradient. 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Matter , 6417 (1998).[27] In a honeycomb lattice, it is allowed by the symmetry to have D (cid:80) (cid:96) = { , , } (cid:80) i z · (cid:16) S i , A × S i + a (cid:96) , A − S i , B × S i + a (cid:96) , B (cid:17) , where r i + a = r i + a (cid:16) , √ (cid:17) , r i + a = r i + a ( − , r i + a = r i + a (cid:16) , − √ (cid:17) .Here A and B are the sublattice indices and a is the length of theprimitive vector.[28] R. Cheng, S. Okamoto, and D. Xiao, Phys. Rev. Lett. ,217202 (2016).[29] V. A. Zyuzin and A. A. Kovalev, Phys. Rev. Lett. , 217203(2016).[30] Y. Shiomi, R. Takashima, and E. Saitoh, Phys. Rev. B ,134425 (2017).[31] We note that the sign of D has not been estimated.[32] In our calculation within the linear spin wave theory, V SSE1 doesnot show any angular dependence.[33] E. Ressouche, M. Loire, V. Simonet, R. Ballou, A. Stunault,and A. Wildes, Phys. Rev. B , 100408(R) (2010). Supplemental Material for“Nonreciprocal spin Seebeck e ff ect in antiferromagnets” I. NONRECIPROCAL SPIN SEEBECK EFFECT IN A POLAR FERROMAGNET
Here we discuss the spin Seebeck e ff ect in ferromagnets on polar lattices, which we call polar ferromagnets. We consider aone-dimensional model, whose Hamiltonian is given by H = H FM0 + H FMD , where H FM0 = (cid:88) i (cid:104) J S i · S i + + J S i · S i + + G ( S zi S zi + − S xi S xi + − S yi S yi + ) (cid:105) + g s µ B (cid:126) B z (cid:88) i S zi , (S1) H FMD = D (cid:88) i ˆ z · ( S i × S i + ) . (S2)Here S i = ( S xi , S yi , S zi ) is the spin operator at site i . J ( J ) is the coupling constants for the isotropic exchange interactions forthe first (second) neighbors, and G is the anisotropic one for the first neighbors. H FMD represents the uniform DM interaction forall the nearest neighbors with the DM vector D (cid:107) ˆ z as in the polar AFM in the main text.Assuming a collinear ferromagnetic ground state, namely, (cid:104) S zi (cid:105) = S , we calculate the magnon dispersion ε k x within the linearspin wave approximation by using the Holstein-Primako ff transformation as S + i = (cid:126) (2 S − a † i a i ) / a i , S zi = (cid:126) (cid:16) S − a † i a i (cid:17) , (S3)where S + i = S xi + iS yi = ( S − i ) † . Because of the DM interaction, the magnon dispersion becomes asymmetric with respect to k x ,which results in a nonzero (cid:104) ( (cid:51) xk x ) (cid:105) ε kx = ε defined in Eq. (11) in the main text. Noting that each magnon carries the spin angularmomentum − (cid:126) , the local spin current density is given by J s z x = − (cid:126) (cid:90) dk x π (cid:51) xk x n ( ε k x ) , where (cid:51) xk x = (1 / (cid:126) ) ∂ε k x /∂ k x is the velocity and n ( ε k x ) is the distribution function of the magnon. Using the relaxation timeapproximation and averaging over the space, the coe ffi cients in Eq. (1) in the main text are given by S xx = (cid:126) τ (cid:82) dk x π ( (cid:51) xk x ) f (1) ( ε k x ) , (S4) S xx = − (cid:126) τ (cid:82) dk x π ( (cid:51) xk x ) f (2) ( ε k x ) , (S5)where the functions f (1) ( ε ) and f (2) ( ε ) are defined in the main text. In polar ferromagnets, the inversion, M x , and M y symmetriesare all broken. Therefore, both the linear coe ffi cient S xx and nonlinear one S xx can be nonzero even without a magnetic field,and they are even functions of B z . II. BOLTZMANN THEORY FOR THE SPIN SEEBECK EFFECT
In this section, we show the derivation of Eqs. (9) and (10) from Eqs. (1), (7), and (8) in the main text. In the small temperaturegradient T ( x ) = T + α x , where we assume that T /α is larger than the magnon mean free path, the equilibrium distributionfunction of magnons can be expanded as n σ k ( x ) = ε σ k / T ( x )) − (cid:39) n σ k (cid:12)(cid:12)(cid:12) T = T + α x ∂ n σ k ∂ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = T + α x ∂ n σ k ∂ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = T . The distribution function n σ k ( r ) can also be expanded in the series of τ as n σ k ( x ) (cid:39) n σ k ( x ) − τ v x σ k x ∂ n σ k ( x ) ∂ x + ( τ v x σ k x ) ∂ n σ k ( x ) ∂ x . Then the spin current in Eq. (7) in the main text is given by J s z x = (cid:126) (cid:90) dk x π (cid:16) (cid:51) x ↑ k x n ( ε ↑ k x ) − (cid:51) x ↓ k x n ( ε ↓ k x ) (cid:17) = − (cid:88) σ (cid:126) τα (cid:90) dk x π s σ ( v x σ k ) ∂ n σ k ∂ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = T − (cid:88) σ (cid:126) τα x (cid:90) dk x π s σ ( v x σ k ) ∂ n σ k ∂ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = T + (cid:88) σ (cid:126) τ α (cid:90) dk x π s σ ( v x σ k x ) ∂ n σ k ∂ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = T , (S6)where s ↑ ( s ↓ ) = + − (cid:90) dk x π f ( ε σ k ) v x σ k = (cid:90) dk x π f ( ε σ k ) (cid:32) d ε σ k dk x (cid:33) = (cid:90) dk x π ddk x F ( ε σ k ) = π [ F ( ε σ k )] k x = π k x = − π = , (S7)where f ( ε σ k ) is an arbitrary function of an energy with which we can introduce F ( ε ) that satisfies dd ε F ( ε ) = f ( ε ). Averaging J s z x in Eq. (S6) over the space, we obtain Eq. (1) with the coe ffiffi