Rotational motion of magnon and thermal Hall effect
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Rotational motion of magnon and thermal Hall effect
Ryo Matsumoto and Shuichi Murakami ∗ Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan (Dated: May 22, 2018)Due to the Berry curvature in momentum space, the magnon wavepacket undergoes two types oforbital motions in analogy with the electron system: the self-rotation motion and a motion alongthe boundary of the sample (edge current). The magnon edge current causes the thermal Hall effect,and these orbital motions give corrections to the thermal transport coefficients. We also apply ourtheory to the magnetostatic spin wave in a thin-film ferromagnet, and derive expression for theBerry curvature.
PACS numbers: 85.75.-d, 66.70.-f, 75.30.-m, 75.47.-m
I. INTRODUCTION
Recently in the spintronics field, the magnon (spinwave ) transport in an insulating magnet attracts muchattention as a candidate of a carrier of the spin informa-tion with good coherence and without dissipation of theJoule heating. In particular, magnon can propagate overcentimeter distance in some magnets , e.g. yttrium-iron-garnet (YIG), and this is long enough compared with thespins in metals and doped semiconductors. The magnoncurrent can be experimentally generated by the spin Halleffect , and its motion can be observed by the time- andspace-resolved measurement methods . Besides, a pre-cise control of the spin information is necessary for theapplication in spintronics devices.The thermal Hall effect (Righi-Leduc effect) of themagnon, which is useful to control the magnon trans-port, is predicted theoretically by Katsura et al. andobserved experimentally by Onose et al. . Katsura et al. considered a ferromagnet with the Kagom´e lattice struc-ture, and calculated a thermal Hall conductivity by theKubo formula; Onose et al. measured the thermal Hallconductivity using an insulating ferromagnet Lu V O which has a pyrochlore structure with the Dzyaloshinskii-Moriya (DM) interaction. On the other hand, we foundin our recent study that there are correction terms to thethermal Hall conductivity in the linear response theory,and showed that the thermal Hall effect of the magnonarise from the edge current of the magnon in the semi-classical picture. Our theory is applicable not only to thequantum mechanical spin wave, e.g. in Lu V O , butalso to the classical magnetostatic spin wave, where thewave length is long and exchange coupling is negligible,e.g. in YIG film.In the present paper, we develop the transport theoryof magnons with detailed calculations. Some parts of thetheory has been published in Ref. 7. There are two ap-proaches; the semiclassical theory and the linear responsetheory. From the semiclassical equation of motion, themagnon edge current are described by the Berry curva-ture in momentum space and does not depend on thedetails of the system such as the shape of the boundaryof the sample. From this magnon edge current, we obtainthe magnon current and energy current density under a spatial variation of the temperature or the chemical po-tential, resulting in the thermal Hall effect of the magnon.Since our result of the thermal Hall conductivity doesnot agree with the previous works in Refs. 5,6, we refor-mulated the linear response theory in analogy with theelectron system , by noting that the temperature gradi-ent is not a dynamical force but a statistical force. Itis identified that the difference from the previous workarises from orbital motions of the magnon, and that themagnon rotates around itself besides the magnon edgecurrent.We apply our theory to Lu V O , and calculate theorbital angular momenta of the rotational motions of themagnon. For another application, the expression of theBerry curvature for the magnetostatic forward volumewave in YIG is derived. In this case the Berry curva-ture of the highest energy band enhances and that of theother bands converges to 0 at k = 0. Besides the Berrycurvature becomes larger as the magnetic field becomessmall.This paper is organized as follows. We present thesemiclassical theory for the magnon and consider thethermal Hall effect of the magnon in Section II. Thelinear response theory with a temperature gradient andthe orbital motions of the magnon are discussed in Sec-tion III. Section IV and Section V are devoted to ap-plications of our theory to Lu V O and YIG, respec-tively. We conclude with a summary in Section VI, anda brief review of the linear response theory for the elec-tron system and some useful equations are presented inAppendix A.Throughout this paper we consider two-dimensional in-sulating magnetic systems for simplicity, and assume thatmagnons do not interact with each other. Generalizationto three-dimensional magnets is straightforward. II. SEMICLASSICAL THEORY
Our approach is based on the semiclassical theory,in analogy with the electron system . We consider amagnon wavepacket which is well localized around thecenter ( r c , k c ) in the phase space: | W n i = Z d k a n ( k , t ) | φ n k i , (1)where | φ n k i is the Bloch wave function in the n th magnonband, a n ( k , t ) satisfies Z d k | a n ( k , t ) | = 1 , (2) Z d k | a n ( k , t ) | k = k c . (3)and | W n i satisfies h W n | ˆ r | W n i = r c . (4)Hereafter we omit the index c for brevity. The dynam-ics of the wavepacket is described by the semiclassicalequation of motion, which includes the topological Berryphase term: ˙ r = 1 ~ ∂ε n k ∂ k − ˙ k × Ω n ( k ) , (5) ~ ˙ k = −∇ U ( r ) . (6)Here n is the band index, ε n k is the energy of the magnonin the n th band, Ω n ( k ) is the Berry curvature in momen-tum space: Ω n ( k ) = i (cid:28) ∂u n ∂ k (cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂ k (cid:29) , (7)with | u n ( k ) i being the periodic part of Bloch waves inthe n th band defined as φ n k ( r ) = u n ( k , r ) e i k · r . U ( r ) isa confining potential which exists only near the boundaryof the sample. This potential U ( r ) forbids the magnonwavepacket going outside of the sample, and its gradientexerts a force on magnons. Such approach of the confin-ing potential is successful in describing the edge pictureof the quantum Hall effect in electron systems . Thuswe have similarly introduced the confining potential formagnons. Strictly speaking, for the validity of Eqs. (5)and (6), the spatial variation of U ( r ) should be muchslower, compared with the size of the wavepacket. Nev-ertheless, as we can see from the quantum Hall effect asan example, many of the results for the slowly varying U ( r ) are expected to carry over to the case of rapidlychanging U ( r ) as well.Near the edge of the sample, there exists an edge cur-rent of magnons due to the anomalous velocity term − ˙ k × Ω n ( k ) = ∇ U ( r ) / ~ × Ω n ( k ) in Eq. (5). For ex-ample, the magnon edge current for the edge along the y direction is expressed as I y = Z ba dx V X n, k ρ ( ε n k + U ( r )) [ ∇ U ( r ) / ~ × Ω n ( k )] y , = − ~ V X n, k Z ∞ ε n k dερ ( ε )Ω n,z ( k ) , (8) FIG. 1: Coordinate of the ferromagnet, used for the calcula-tion of the edge current. U ( r ) is a confining potential. where x = a and x = b are chosen well inside and outsideof the sample so that U ( a ) = 0 and U ( b ) = ∞ , V is thearea of the sample, ρ ( ε ) is the Bose distribution function ρ ( ε ) = ( e β ( ε − µ ) − − , β = 1 /k B T , k B is the Boltz-mann constant, µ is the chemical potential, and T is thetemperature. Henceforth the magnon current means thecurrent of the magnon number. We used in Eq. (8) thefact that Ω n ( k ) in the two-dimensional system is per-pendicular to the plane. Similarly we obtain the edgecurrent for the edge along the x direction I x , which isidentical to I y . Thus the edge current does not dependon the edge direction or the expression for the confiningpotential U ( r ). Therefore, the magnon moves even alongthe curved edge. Here we should note that in addition tothe velocity along the edge (i.e. the second term in ther.h.s. of Eq.(5)), there exists a group velocity (the firstterm in the r.h.s. of Eq.(5)). Because of this group ve-locity, a single wavepacket does not go purely along theedge. What we have shown is that there is an additionalvelocity along the edge due to Berry curvature, and thetotal magnon edge current is given by Eq. (8) when allthe magnons in thermal equilibrium are summed over.If the chemical potential µ or temperature T variesspatially, the thermal Hall effect will occur because themagnon edge current no longer cancels between one edgeand the opposite edge, and a net current will appear. Inthe following we show the details and calculate thermaltransport coefficients. We focus on the edge current inthe x direction with small temperature gradient in the y direction as an example, and set the coordinate systemshown in Fig. 1. Here w is the width of the system and a , b , b is defined as U ( a ) = 0, U ( b ) = U ( b ) = ∞ and b < − w/ < a < w/ < b . The current density isobtained by summing up the local current density j x ( y )and dividing it by the width: j x = 1 w Z b b dyj x ( y ) = 1 w Z b a dyj x ( y ) + 1 w Z ab dyj x ( y ) . (9)Here we defined j x ( y ) as the following: j x ( y ) = 1 ~ V X n, k ρ ( ε n k + U ( r ); T ( y )) ∂U ( r ) ∂y Ω n,z ( k ) . (10)This quantity is nonzero when ∂U ( r ) /∂y = 0, i.e., y ∼± w/
2. At these points, ρ ( ε n k + U ( r ); T ( y )) (11) ≈ ( ρ ( ε n k + U ( r ); T ( w )) ( y ∼ w ) ,ρ ( ε n k + U ( r ); T ( − w )) ( y ∼ − w ) . Thus Eq. (9) is written as j x = 1 w ~ V X n, k Z ∞ ε n k dε × (cid:16) ρ (cid:16) ε ; T (cid:16) w (cid:17)(cid:17) − ρ (cid:16) ε ; T (cid:16) − w (cid:17)(cid:17)(cid:17) Ω n,z ( k )= ∂∂y ~ V X n, k Z ∞ ε n k ρ ( ε ; T ( y ))Ω n,z ( k ) dε . (12)The edge current along the y direction with temperaturegradient in the x direction is similarly written as j y = − ∂∂x ~ V X n, k Z ∞ ε n k ρ ( ε ; T ( x ))Ω n,z ( k ) dε . (13)In the presence of the chemical potential gradient theedge current can be written as the same form likeEqs. (12) and (13). Therefore, if the spatial variationis well gradual, the edge current density is written as: j = ∇ × ~ V X n, k Z ∞ ε n k ρ ( ε ) Ω n ( k ) dε. (14)In the same way, the energy current from the edge currentdensity is written as j E = ∇ × ~ V X n, k Z ∞ ε n k ερ ( ε ) Ω n ( k ) dε. (15)From Eqs. (14) and (15), we can derive various transversetransport coefficients. For instance, in the presence of thetemperature gradient in the y direction again, the edgecurrent and energy current density in the x direction arewritten as( j ) ∇ Tx = T ∂ y (cid:18) T (cid:19) X n, k Z ∞ ε n k ε − µ ~ V (cid:18) dρdε (cid:19) Ω n,z ( k ) dε, (16)( j E ) ∇ Tx = T ∂ y (cid:18) T (cid:19) X n, k Z ∞ ε n k ε ( ε − µ ) ~ V (cid:18) dρdε (cid:19) Ω n,z ( k ) dε. (17)Here we note that the temperature gradient affectsthese currents through the Bose distribution function as ∂ T ρ ( ε ) = ( ε − µ ) ( ∂ρ ( ε ) /∂ε ) T ∂ T (1 /T ). Similarly weobtain these currents in the presence of the gradient of the chemical potential in the y direction:( j ) ∇ µx = − ∂ y µ ~ V X n, k Z ∞ ε n k (cid:18) dρdε (cid:19) Ω n,z ( k ) dε, (18)( j E ) ∇ µx = − ∂ y µ ~ V X n, k Z ∞ ε n k ε (cid:18) dρdε (cid:19) Ω n,z ( k ) dε. (19)Now we define a heat current as j Q ≡ j E − µ j , and writedown the linear response of the magnon current and heatcurrent as j = L [ −∇ U − ∇ µ ] + L (cid:20) T ∇ (cid:18) T (cid:19)(cid:21) , (20) j Q = L [ −∇ U − ∇ µ ] + L (cid:20) T ∇ (cid:18) T (cid:19)(cid:21) . (21)From Eqs. (16)-(19), the transverse thermal transportcoefficients L xyij can be derived as L xyij = − ~ V β q X n, k Ω n,z ( k ) c q ( ρ n ) , (22)where i, j = 1 , c q ( ρ n ) = R ∞ ε n k dε ( β ( ε − µ )) q (cid:16) − dρdε (cid:17) = R ρ n (cid:0) log tt (cid:1) q dt , q = i + j −
2, and ρ n ≡ ρ ( ε n k ). For ex-ample, c ( ρ ) = ρ , c ( ρ ) = (1 + ρ ) log (1 + ρ ) − ρ log ρ , and c ( ρ ) = (1 + ρ ) (cid:16) log ρρ (cid:17) − (log ρ ) − ( − ρ ), whereLi ( z ) is the polylogarithm function. Finally we derivethe thermal Hall conductivity in a clean limit by substi-tuting Eq. (22) to κ xy = L xy /T , κ xy = 2 k T ~ V X n, k c ( ρ n )Im (cid:28) ∂u n ∂k x (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k y (cid:29) . (23)Thus the thermal Hall conductivity is expressed as theBerry curvature in momentum space, which is sensitiveto the magnon band structure. Since the Berry curvaturepart is expressed asIm (cid:28) ∂u n ∂k x (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k y (cid:29) = X m ( = n ) Im D u n (cid:12)(cid:12)(cid:12) ∂H∂k x (cid:12)(cid:12)(cid:12) u m E D u m (cid:12)(cid:12)(cid:12) ∂H∂k y (cid:12)(cid:12)(cid:12) u n E ( ε n k − ε m k ) . (24)Hence κ xy in Eq. (23) is enhanced if there is an avoidedband crossing. III. LINEAR RESPONSE THEORY
Compared with our result in Eq. (23), the expressionfor the thermal Hall conductivity obtained in the previ-ous works ,¯ κ xy = 2 ~ V T X n, k ρ n Im (cid:28) ∂u n ∂k x (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) H + ε n k (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k y (cid:29) , (25)is different in some points. As we see later in this sec-tion, in the linear response theory, our result (Eq. (23))consists of two parts ( S ) xyij + ( M ) xyij (defined in Eqs. (32)-(34)), while the result in Refs. 5,6 (Eq. (25)) containsonly ( S ) xyij . The correction term ( M ) xyij comes from or-bital motions of magnons. In the following, we apply thelinear response theory to the magnon system in analogywith the electron system . In Appendix A, we brieflyreview the linear response theory under the temperaturegradient in the electron system, and derive some usefulexpression for the thermal transport coefficients.Here we shortly discuss the linear response theory withexternal fields . In the presence of the temperaturegradient, it is convenient to introduce a fictitious gravi-tational potential ψ ( r ), which exerts a force proportionalto the energy of the particle . This is because in orderto obtain the thermal transport coefficients by the linearresponse theory, we need to take the temperature gradi-ent into the Hamiltonian as an external field. However,it is not possible to directly incorporate the temperaturegradient into the linear response theory, since the tem-perature gradient is not a dynamical force which exertsforce to the particles, but a statistical force which affectsthe particles through the distribution function. There-fore, to avoid this difficulty, the fictitious potential ψ ,giving a dynamical force, has been introduced. As we seein Appendix A, the thermal transport coefficients fromthe temperature gradient are derived by calculating thecoefficient from the gradient of the fictitious potential ψ . This is analogous to the situation that the transportcoefficients from the gradient of the chemical potentialcan be obtained by calculating the coefficients from theelectric field.Now we consider the magnon system. Since themagnon has no charge, we cannot use the electric field E as an external field. Instead, we again use the gradi-ent of the confining potential −∇ U ( r ), which appearedin Eq. (6). The perturbation Hamiltonian is written as H ′ = P j U ( r j ) + n H, c P j r j · ∇ ψ ( r ) o , where r j isthe position of the j th magnon, H is the unperturbedHamiltonian, and { ˆ A, ˆ B } = ˆ A ˆ B + ˆ B ˆ A represents the an-ticomutator. In equilibrium, the magnon current density and energy current density are written as j (0) ( r ) = 12 X j { v j , δ ( r − r j ) } , (26) j (0) E ( r ) = 12 n H, j (0) ( r ) o , (27)where v j is the velocity operator of the j th magnon. Inthe presence of the external fields H ′ , they acquire addi-tional terms, j ( r ) = j (0) ( r ) + 12 j (0) ( r ) , c X j r j · ∇ ψ ( r ) , (28) j E ( r ) = j (0) E ( r ) + 12 X j n U ( r j ) , j (0) ( r ) o + 14 c X j (cid:16)n { H, r j · ∇ ψ ( r ) } , j (0) ( r ) o + nn j (0) ( r ) , r j · ∇ ψ ( r ) o , H o(cid:17) , (29)where c is the speed of light. Correspondingly, the ther-mal transport coefficients consist of two parts: ( L ) αβij =( S ) αβij + ( M ) αβij . Here α, β = x, y , i, j = 1 ,
2, and inthis section the thermal transport coefficients ( L ) αβij aredefined as: J = ( L ) h −∇ U − T ∇ (cid:16) µT (cid:17)i + ( L ) (cid:20) T ∇ (cid:18) T (cid:19) − ∇ ψc (cid:21) , (30) J E = ( L ) h −∇ U − T ∇ (cid:16) µT (cid:17)i + ( L ) (cid:20) T ∇ (cid:18) T (cid:19) − ∇ ψc (cid:21) . (31)A deviation of the distribution function from the equi-librium state generates ( S ) αβij , which is calculated by theKubo formula; a deviation of the current operator dueto external fields from the equilibrium state generates( M ) αβij . In the clean limit they are expressed as( S B ) αβij = i ~ V Z ρ ( η )Tr (cid:18) j αi dG + dη j βj δ ( η − H ) − j αi δ ( η − H ) j βj dG − dη (cid:19) dη, (32)( M B ) αβ = 0 , ( M B ) αβ = 12 V Z ρ ( η )Tr[ δ ( η − H )( r α v β − r β v α )] dη, (33)( M B ) αβ = 1 V Z ηρ ( η )Tr δ ( η − H )( r α v β − r β v α ) dη + i ~ V Z ρ ( η )Tr δ ( η − H )[ v α , v β ] dη. (34)Here G ± is the Green’s function G ± ( η ) = ( η − H ± iǫ ) − with ǫ being the positive infinitesimal, ρ ( η ) is the Bose distribution function ρ ( η ) = (cid:0) e β ( η − µ ) − (cid:1) − , j = v , j = ( H v + v H ), v is the velocity of magnons, and thelabel of the superscript “B” means a boson. By usingEqs. (A10)-(A13), these thermal transport coefficients forthe magnon system can be written by the wave functionof magnons:( S B ) αβij = 2 ~ V Im X n, k ρ n (cid:28) ∂u n ∂k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) H + ε n k (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k β (cid:29) , (35)( M B ) αβij = − ( S B ) αβij + 2 ( k B T ) q ~ V Im X n, k c q ( ρ n ) (cid:28) ∂u n ∂k α (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k β (cid:29) , (36)where we have taken µ = 0 since the magnon num-ber is not conserved. We note that q = i + j − M B ) αβ = 0. Thus the total thermal transport coeffi-cients are written as( L B ) αβij = ( S B ) αβij + ( M B ) αβij , = − ( k B T ) q ~ V X n, k c q ( ρ n )Ω n,z ( k ) , (37)and from this equation we again derive the same thermalHall conductivity as Eq. (23). The result in Refs. 5,6,shown in Eq. (25), contains only the contribution from( S B ) αβ . Therefore the difference between the resultsof Refs. 5,6 and ours arise from the correction terms( M B ) αβij .As we can see from Eqs. (33) and (34), these correctionterms ( M B ) αβij are related to the orbital motion of theparticle, namely, a reduced orbital angular momentum h r × v i . Equation. (36) means that ( M B ) αβij are expressedas the Berry curvature in momentum space, which is gen-erally nonzero. Hence, in this case, the magnon has fi-nite orbital angular momentum due to the Berry cur-vature. This orbital angular momentum consists of twoparts: the edge current and the self-rotation motion ofthe wavepacket. The reduced angular momentum for theedge current per unit area is derived from Eq. (8), l edge z = − ~ V X n, k Z ∞ ε n k dερ ( ε )Ω n,z ( k ) , (38)and that for the self-rotation motion is calculated in anal-ogy with the electron system as l self z = − ~ V Im X n, k ρ n (cid:28) ∂u n ∂k x (cid:12)(cid:12)(cid:12)(cid:12) ( H − ε n k ) (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k y (cid:29) . (39)It is easy to show that l edge z + l self z = 2( M B ) xy . Thisresult is expected from the Eq. (33), i.e., the correctionterms comes from the orbital angular momentum of themagnon.Therefore, due to the Berry curvature, the magnongenerally has a nonvanishing orbital angular momentumin equilibrium. This orbital motion of magnon can be regarded as a generalized cyclotron motion. However,since the magnon has no charge, it feels no Lorentz forceand cannot have a cyclotron motion in the same senseas that of electrons. In this respect, this motion purelyreflects the magnon band structure. A similar effect canbe found in various wave phenomena such as electrons ,photons , and so on.As mentioned earlier, within the semiclassical theory,the result for the edge current is derived under the as-sumption that the spatial variation of the confining po-tential is slow. Nevertheless, as we have seen in thissection, the linear response theory, which does not needassumptions for confining potential, gives the same trans-port coefficients as the semiclassical theory. This stronglysuggests that the edge-current picture carries over to theabrupt spatial variation of the confining potential. Inthe quantum Hall systems, this idea is indeed true; atthe abrupt edges of the quantum Hall system, the elec-trons undergo a skipping motion. Namely, near the edge,electrons undergo a cyclotron motion and when electronshit the edge they are bounced. As a whole the electronsgo along the edge with skipping orbitals, which are re-garded as the chiral edge current in the quantum Hallsystem. Therefore, we can similarly expect that in theferromagnets with edges, the magnon will undergo a su-perposition of a skipping motion along the edge and amotion along the group velocity of the magnon.The coherence length of the magnons is important inthe orbital motions and transport of magnons. For thevalidity of the linear response theory developed above,it is implicitly assumed that the coherence length of themagnons is sufficiently short compared with the systemsize. By this assumption, when we apply temperaturedifference between the two opposite sides of the system,we can define a local temperature, and the temperaturegradient becomes uniform. The linear response theoryis then justified. Otherwise, when the coherence lengthis as long as the system size, the magnon transport isdescribed in the similar way as the Landauer formula,and the linear response theory no longer applies.Orbital motions of electrons generate magnetic mo-ments by their charge. On the other hand, magnons donot have charge, but have magnetic moments. There-fore we can regard the rotating magnon as a circulatingspin current. As is similar to the magnetoelectric ef-fect in noncollinear spin structure , the rotation of themagnon is expected to generate an electric polarization.For this effect, the spin-orbit coupling interaction, suchas Dzyaloshinskii-Moriya (DM) interaction, is necessary. IV. EXAMPLE 1: LU V O In this section we apply our results to the ferromag-netic Mott-insulator Lu V O with pyrochlore structure,for which the thermal Hall effect has been measured andanalyzed in Ref. 6. Following Ref. 6, we briefly reviewthe magnetic properties of the material. The magne-tization comes from spin-1/2 V ions with the DMinteraction. The ground state is a collinear ferromag-net, because at the ferromagnetic ground state the to-tal DM vectors for the six bonds sharing a single site iszero . The DM interaction affects the spin-wave disper-sion, and the effective spin-wave Hamiltonian is writtenas H eff = P h i,j i − J S i · S j + D ij · ( S i × S j ) − gµ B H · P i S i ,where h i, j i denotes the nearest neighbor pairs, J is theexchange interaction, D is the DM vector, g is the g-factor, µ B is Bohr magneton, and H is the magnetic fieldin the z direction. The temperature is assumed to bemuch lower than the Curie temperature T C = 70[K], forexistence of well-defined Bloch waves of magnons. Thereare four magnon bands, and the lowest band is well sep-arated from the other higher bands, with the separationmuch larger than k B T . Actually, the differences of theenergy between the lowest band and other bands near k = 0 are written as ε − ε ≃ JS p f ( k ) ≃ JS and ε − ε = ε − ε ≃ JS + 2 JS p f ( k ) ≃ JS , where f ( k ) = cos(2 k x A ) cos(2 k y A ) + cos(2 k y A ) cos(2 k z A ) +cos(2 k z A ) cos(2 k x A ) and 8 JS ≃ . ,z ≃ − A √ DJ H z H ( k x + k y + 2 k z ) ascalculated in Ref. 6, with A being a quarter of the latticeconstant. We can estimate the orbital angular momen-tum of the magnon from both the self-rotation motion L self z and the edge current L edge z . Near k = 0, the lowest-band dispersion is quadratic and we can introduce theeffective mass of the magnon of the lowest band m ∗ , de-fined as m ∗ n ≡ ~ ( ∂ ε n k /∂k ) − . The orbital angularmomentum of the self-rotation motion is analytically cal-culated from Eq. (39): L self z ≃ m ∗ l self z = − JSm ∗ ~ V Im X k ρ ( ε k ) (cid:28) ∂u ∂k α (cid:12)(cid:12)(cid:12)(cid:12) ∂u ∂k β (cid:29) = − JSm ∗ ~ A DJ π (cid:18) k B TJS (cid:19) / Z ∞ x / e ( x + βgµ B H ) − dx = − JSm ∗ ~ A DJ π / (cid:18) k B TJS (cid:19) / Li (cid:18) e − gµ B Hk B T (cid:19) . (40)We obtain L self z ≃ − . ~ and L edge z = m ∗ l edge z ≃ +0 . ~ per unit cell. The thermal Hall conductivity κ xy is also calculated, by assuming that the contributionof the lowest band dominates. Figure 2 show the resultof the thermal Hall conductivity which is calculated from( S ) αβij + ( M ) αβij (solid curve) and ( S ) αβij (broken curve).They correspond to our results and the previous resultsin Ref. 6, respectively. Our result (solid curve in Fig. 2)roughly agrees with the experimental data in Ref. 6. V. EXAMPLE 2: MAGNETOSTATIC SPINWAVE
In the following, we apply our theory to the magneto-static spin waves in a ferromagnet. In the magnetostatic
FIG. 2: (Color online) Dependence of the thermal Hall con-ductivity on a magnetic field. The red (broken) curve denotesthe result which is calculated from only ( S ) αβij calculated inRef. 6; the green (solid) curve denotes the result which iscalculated from ( S ) αβij + ( M ) αβij .FIG. 3: Geometry of the coordinate axes. The magnetization M precesses around M . spin wave, the wavelength is sufficiently long and theexchange coupling between spins is negligible. The mag-netic anisotropy comes from the demagnetizing field de-termined by the sample shape. This magnetic anisotropydue to the demagnetizing field plays the similar role asthe spin-orbit coupling in electronic systems. It then in-duces the Berry curvature, and the Hall effect of spinwaves appears.Let us consider the yttrium-iron-garnet (YIG) filmwhich is magnetized by an external magnetic field. Weintroduce two coordinate systems xyz and ξηζ , shownin Fig. 3. The film is taken to be infinite in the η - and ζ -direction, and perpendicular to the ξ direction. ζ axisis chosen to be along the magnon wave vector k . The z direction is parallel to the saturation magnetization M and the internal static magnetic field H . We assumethat the spin wave mode has a form of the plane wave: m ( ξ, ζ, t ) = m ( ξ ) exp( i ( kζ − ωt )), where ω is a frequencyof the spin wave . The equation of motion of the magne-tization is written as the following integral equation : ω H m ( ξ ) − ω M Z L/ − L/ dξ ′ ˆ G ( ξ, ξ ′ ) m ( ξ ′ ) = ωσ y m ( ξ ) . (41)Here we use the SI units, L is the thickness of the film, σ y = (cid:0) − ii (cid:1) is the Pauli matrix, m ( ξ ) = (cid:16) m x ( ξ ) m y ( ξ ) (cid:17) isthe vector Fourier amplitude of the spin wave which isperpendicular to M , ω H = γH , ω M = γM , and γ is the gyromagnetic ratio. ˆ G ( ξ, ξ ′ ) is the 2 × G ( ξ, ξ ′ ) = (cid:18) G xx G xy G yx G yy (cid:19) , (42) G xx = ( G P − δ ( ξ − ξ ′ )) sin θ − iG Q sin 2 θ cos ϕ − G P cos θ cos ϕ, (43) G xy = G yx = − iG Q sin θ sin ϕ − G P cos θ sin ϕ cos ϕ, (44) G yy = − G P sin ϕ, (45)where G P = k − k | ξ − ξ ′ | ) , (46) G Q = G P sign( ξ − ξ ′ ) , (47)and θ , ϕ are the spherical coordinates of M in the ξηζ -space (see Fig. 3). We note that we adopt the defini-tions of θ and ϕ used in Ref. 21, and they are differ-ent from the standard definition of the spherical coordi-nates. The integral equation (41) is equivalent to the lin-earized Landau-Lifshitz equation d M /dt = − γ ( M × H ),Maxwell equation in the magnetostatic limit ∇ × H = 0, ∇ · B = 0, and the usual boundary conditions for H and B . Since the equation (41) is a generalized eigenvalue problem, we have to modify the prescription of ourtheory of the Berry curvature. Similar to the previouswork , the Berry curvature is defined asΩ n,γ ( k ) = iǫ αβγ (cid:28) ∂ m n, k ∂k α (cid:12)(cid:12)(cid:12)(cid:12) σ y (cid:12)(cid:12)(cid:12)(cid:12) ∂ m n, k ∂k β (cid:29) , (48)where ǫ αβγ is the antisymmetric tensor, n is the bandindex of the spin wave mode, and the bra-ket productmeans an usual inner product of vectors and integral over z . In some cases, Eq. (48) becomes zero because of thesymmetry of the system. This occurs when the saturationmagnetization M is in the film ( θ = π/ θ = π/
2, we can show Ω n,γ ( k ) = 0 explicitly by performinga gauge transformation m ′ ≡ U − m = (cid:18) i (cid:19) m . ThenEq. (41) becomes a generalized eigen value problem withreal coefficients: ω H m ′ ( ξ ) − ω M Z L/ − L/ dξ ′ ˆ G ′ ( ξ, ξ ′ ) m ′ ( ξ ′ ) = − ωσ x m ′ ( ξ ) , (49)where ˆ G ′ ( ξ, ξ ′ ) isˆ G ′ ( ξ, ξ ′ ) = U − ˆ G ( ξ, ξ ′ ) U = (cid:18) G ′ xx G ′ xy G ′ yx G ′ yy (cid:19) , (50) G ′ xx = G P − δ ( ξ − ξ ′ ) , G ′ xy = − G Q sin ϕ, (51) G ′ yx = G Q sin ϕ, G ′ yy = − G P sin ϕ. (52) Since all the terms in Eq. (49) are real, the eigen vector m ′ is also real. Correspondingly, the Berry curvatureEq. (48) becomesΩ n,γ ( k ) = iǫ αβγ (cid:28) ∂ m ′ n, k ∂k α (cid:12)(cid:12)(cid:12)(cid:12) U − σ y U (cid:12)(cid:12)(cid:12)(cid:12) ∂ m ′ n, k ∂k β (cid:29) , (53)= ǫ αβγ Im (cid:28) ∂ m ′ n, k ∂k α (cid:12)(cid:12)(cid:12)(cid:12) σ x (cid:12)(cid:12)(cid:12)(cid:12) ∂ m ′ n, k ∂k β (cid:29) . (54)Because m ′ is real and there is no imaginary part, thisBerry curvature vanishes. Thus when M is in the film,we cannot expect either an orbital rotational motion ofspin wave packet or the thermal Hall effect due to theBerry curvature effect. In other words, in the magneto-static backward volume wave (MSBVW) and the mag-netostatic surface wave (MSSW), the effects of the Berrycurvature do not appear.On the other hand, the Berry curvature is finite ifthe saturation magnetization is perpendicular to the film( θ = 0), i.e., in the magnetostatic forward volume wave(MSFVW). In the following, we demonstrate the calcu-lation of the Berry curvature for MSFVW. We note that ξ coincides with z direction when θ = 0. The solution ofthe integral equation (41) of the n -th band for θ = 0 iswritten as m n k ( z ) = (cid:18) m xn k ( z ) m yn k ( z ) (cid:19) = √ N (cid:18) iκ ν − ν iκ (cid:19) (cid:18) k x k y (cid:19) cos (cid:16) √ pkz + nπ (cid:17) , (55)where κ = ω M ω H / ( ω H − ω n ), ν = ω M ω n / ( ω H − ω n ), p = − − κ > ω n is the n -th band energy for n = 0 , , , . . . ,which is determined by √ p tan (cid:18) √ pkL + nπ (cid:19) = 1 , (56)and N is a normalization factor which is determined by h m n, k | σ y | m n, k i = 1 . (57)To obtain Eq. (55), we have rewritten the solution inRef. 23 in the polar coordinate into the form of the planewave. The dispersion determined by (56) is shown inFig. 4(a) for H /M = 1 .
0. We use the normalizationEq. (57), because m † n, k σ y m n, k is proportional to the en-ergy density for the magnon .Substituting this solution to Eq. (48), we can cancalculate the Berry curvature for the n -th MSFVWmode. For simplicity of the notation, we set F n ( k, z ) ≡√ N cos (cid:0) √ pkz + nπ (cid:1) . Then the Berry curvature is writ-ten from Eq. (48),Ω n,z ( k ) /
2= Re "(cid:28) ∂m xn, k ∂k x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂m yn, k ∂k y + − (cid:28) ∂m xn, k ∂k y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂m yn, k ∂k x + = ( κ + ν ) k Z L/ − L/ dz ∂F n ∂k F n + (cid:26) k (cid:18) κ ∂κ∂k + ν ∂ν∂k (cid:19) + κ + ν (cid:27) Z L/ − L/ dzF n . (58)Using the normalization condition of Eq. (57), one obtain2 κνk Z L/ − L/ dzF n = 1 . (59)Derivative of Eq. (59) in terms of k leads to Z L/ − L/ dzF n ∂F n ∂k = − (cid:18) κ ∂κ∂k + 12 ν ∂ν∂k + 1 k (cid:19) Z L/ − L/ F n dz. (60)Thus Eq. (58) is rewritten asΩ n,z ( k ) / κνk ω M ω H − ω n (cid:18) κ ∂κ∂k − ν ∂ν∂k (cid:19) . (61)Since κ and ν satisfy the following relation1 κ ∂κ∂k − ν ∂ν∂k = − ω n ∂ω n ∂k , (62)the Berry curvature is derived asΩ n,z ( k ) = 12 ω H k ∂ω n ∂k (cid:18) − ω H ω n (cid:19) . (63)Figure 4(b)-(d) shows the numerical results of Eq. (63)for various magnitude of the magnetic field. It is surpris-ing that the Berry curvature for any MSFVW mode isalways positive, because ω H < ω n and the group velocity ∂ω/∂k is positive. In the vicinity of k = 0, we can cal-culate asymptotic forms of the Berry curvature. When k ∼ ω n is close to ω H . If we set ω n = ω H + ∆ ω n , p can be written as p ≃ ω M / ω n ≫ ω n is smallnear k = 0. Using an approximation tan x ≃ x ( x ≪
1) ,we find ∆ ω n = ω M kL ( n = 0) ω M (cid:18) kLnπ (cid:19) ( n >
0) (64)Therefore, the Berry curvature near k = 0 can be ob-tained from Eq. (63) and (64):Ω n,z ( k ) /L ≃ (cid:18) M H (cid:19) ( n = 0)12 (cid:18) nπ (cid:19) (cid:18) M H (cid:19) ( kL ) ( n > FIG. 4: (Color online) (a) Dispersion relation for the MSFVWmode with n = 0 , , . . . ,
5, and the Berry curvature for theMSFVW mode for (b) H /M = 0 .
1, (c) H /M = 1 .
0, and(d) H /M = 2 . It is easy to see that Ω n,z ( k = 0) = 0 for n > n = 0 mode Ω ,z ( k ) enhances up to (cid:16) L M H (cid:17) but does not diverge at k = 0. VI. CONCLUSIONS
In summary, we found that magnon wavepacket hastwo types of orbital motions due to the Berry curvaturein momentum space: the magnon edge current and theself-rotation motion. The magnon edge current causesthe thermal Hall effect of magnon, and the self-rotationmotion of magnon without Lorentz force is expected toaccompany an electric polarization. We showed that ourtheory is applied to not only the exchange spin wave(quantum-mechanical magnon) e.g. in Lu V O , but alsothe classical magnetostatic waves e.g. in YIG. In bothcases, the Berry curvature is enhanced near the bandcrossings, where the magnon frequency in a focused bandis close to those of other bands. We expect to control theBerry curvature by designing magnonic crystals . Acknowledgments
We would like to thank B. I. Halperin, Q. Niu,T. Ono, and E. Saitoh for discussions. This work ispartly supported by Grant-in-Aids from MEXT, Japan(No. 21000004 and 22540327), and by the Global Cen-ter of Excellence Program by MEXT, Japan throughthe ”Nanoscience and Quantum Physics” Project of theTokyo Institute of Technology.
Appendix A
Here we briefly review the linear response theory forthe electron system with a temperature gradient, devel-oped in Refs. 11–15. In equilibrium, the electric currentoperator j (0) ( r ) and the energy current operator j (0) E ( r )are given by j (0) ( r ) = − e X j { v j , δ ( r − r j ) } , (A1) j (0) E ( r ) = − e n H, j (0) ( r ) o , (A2)where r j denotes the position of the j th electron, − e ( e >
0) is the electron charge, v j is the velocity operatorof the j th electron, H is the unperturbed Hamiltonianof the system, and { ˆ A, ˆ B } = ˆ A ˆ B + ˆ B ˆ A denotes the an-ticomutator. We note that r j is a quantum mechanicaloperator, while r is a c-number.Under the electric potential φ ( r ) and the fictitiousgravitational potential ψ ( r ), the Hamiltonian is writtenas H tot = H + e P j r j · ∇ φ + n H, c P j r j · ∇ ψ ( r ) o ,where H is an unperturbed Hamiltonian and c is thespeed of light. Subsequently, the electric current opera-tor j ( r ) and the energy current operator j E ( r ) deviatefrom the equilibrium state. They are written as: j ( r ) = j (0) ( r ) + j (1) ( r )= j (0) ( r ) + 12 j (0) ( r ) , c X j r j · ∇ ψ ( r ) , (A3) j E ( r ) = j (0) E ( r ) + j (1) E ( r )= j (0) E ( r ) + 12 n φ ( r j ) , j (0) ( r ) o + 14 c X j (cid:16)n { H, r j · ∇ ψ ( r ) } , j (0) ( r ) o + nn j (0) ( r ) , r j · ∇ ψ ( r ) o , H o(cid:17) . (A4) The linear response for the electric current and energycurrent is written as J = ( L F ) (cid:20) E + Te ∇ (cid:16) µT (cid:17)(cid:21) + ( L F ) (cid:20) T ∇ (cid:18) T (cid:19) − ∇ ψc (cid:21) , (A5) J E = ( L F ) (cid:20) E + Te ∇ (cid:16) µT (cid:17)(cid:21) + ( L F ) (cid:20) T ∇ (cid:18) T (cid:19) − ∇ ψc (cid:21) , (A6)where “F” is a label which means a fermion, E is anelectric field, µ is the chemical potential, and ( L F ) ij isthe transport coefficients ( i, j = 1 , J and J E are obtained by taking av-erage over the volume of the sample and the quantum-mechanical and thermodynamic averages of the currentoperators j ( r ) and j E ( r ), respectively. Due to the devi-ations j (1) ( r ) and j (1) E ( r ), the thermal transport coeffi-cients ( L F ) αβij ( α, β = x, y ) consist of two parts, ( S F ) αβij and ( M F ) αβij :( S F ) αβij = i ~ V Z f ( η )Tr (cid:18) j αi dG + dη j βj δ ( η − H ) − j αi δ ( η − H ) j βj dG − dη (cid:19) dη, (A7)( M F ) αβ = 0 , ( M F ) αβ = − e V Z f ( η )Tr[ δ ( η − H )( r α v β − r β v α )] dη, (A8)( M F ) αβ = 1 V Z ηf ( η )Tr δ ( η − H )( r α v β − r β v α ) dη + i ~ V Z f ( η )Tr δ ( η − H )[ v α , v β ] dη. (A9)Here G ± is the Green’s function G ± ( η ) = ( η − H ± iǫ ) − which is introduced in Eq. (A7) via δ ( η − H ) = − ( G + − G − ) / πi , f ( η ) is the Fermi distribution function f ( η ) = (cid:0) e β ( η − µ ) + 1 (cid:1) − , j = − e v , j = ( H v + v H ),and v is the velocity of electrons. ( S F ) αβij is calculated0from the current operators in equilibrium state, j (0) ( r )and j (0) E ( r ), with the deviation of the distribution func-tion from the equilibrium state; ( M F ) αβij is calculatedfrom the deviation of the current operators, j (1) ( r ) and j (1) E ( r ), with the equilibrium distribution function. Actu-ally, ( S F ) αβij is the Kubo formula, and ( M F ) αβij representcorrection terms. The total thermal transport coefficientsare their sums: ( L F ) αβij = ( S F ) αβij + ( M F ) αβij .From these results, we can derive some useful equationsfor later calculations. First, we can write down ( S F ) αβij interms of the Berry phase. For example, ( S F ) αβ is writtenas:( S F ) αβ = − e ~ V Im X n, k f ( ε n k ) (cid:28) ∂u n ∂k α (cid:12)(cid:12)(cid:12)(cid:12) ( H + ε n k ) (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k β (cid:29) . (A10)Second, because the Fermi distribution function f ( η ) be-comes the step function Θ( µ − η ) in the zero temperaturelimit, ( L F ) αβ and ( S F ) αβ is written as( L F ) αβ = µ − e ( L F ) αβ = − eµ ~ V Im X n, k Θ( µ − ε n k ) (cid:28) ∂u n ∂k α (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k β (cid:29) , (A11) and ( S F ) αβ = − e ~ V Im X n, k Θ( µ − ε n k ) × (cid:28) ∂u n ∂k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) H + ε n k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k β (cid:29) . (A12)Thus the relation ( L F ) αβ = ( S F ) αβ + ( M F ) αβ andEq. (A8) in the zero temperature limit lead to the fol-lowing useful formula:Tr[ δ ( µ − H )( r α v β − r β v α )]= ddµ Z µ −∞ Tr[ δ ( η − H )( r α v β − r β v α )] dη = 2 V − e ddµ ( M F ) αβ (cid:12)(cid:12)(cid:12) T → = − ~ ddµ X n, k Θ( µ − ε n k ) × Im (cid:28) ∂u n ∂k α (cid:12)(cid:12)(cid:12)(cid:12) ( H + ε n k − µ ) (cid:12)(cid:12)(cid:12)(cid:12) ∂u n ∂k β (cid:29) . (A13)We note that this equation does not depend whether theparticles are fermion or boson. Therefore we can applythis equation to the magnon system as well. ∗ Electronic address: [email protected] C. Kittel,
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