Landau-Lifshitz theory of the magnon-drag thermopower
LLandau-Lifshitz theory of the magnon-drag thermopower
Benedetta Flebus,
1, 2
Rembert A. Duine,
1, 3 and Yaroslav Tserkovnyak Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA Department of Applied Physics, Eindhoven University of Technology,PO Box 513, 5600 MB, Eindhoven, The Netherlands
Metallic ferromagnets subjected to a temperature gradient exhibit a magnonic drag of the electriccurrent. We address this problem by solving a stochastic Landau-Lifshitz equation to calculate themagnon-drag thermopower. The long-wavelength magnetic dynamics result in two contributions tothe electromotive force acting on electrons: (1) An adiabatic Berry-phase force related to the solidangle subtended by the magnetic precession and (2) a dissipative correction thereof, which is rootedmicroscopically in the spin-dephasing scattering. The first contribution results in a net force pushingthe electrons towards the hot side, while the second contribution drags electrons towards the coldside, i.e., in the direction of the magnonic drift. The ratio between the two forces is proportionalto the ratio between the Gilbert damping coefficient α and the coefficient β parametrizing thedissipative contribution to the electromotive force. The interest in thermoelectric phenomena in ferromag-netic heterostructures has been recently revived by thediscovery of the spin Seebeck effect [1]. This effect is nowunderstood to stem from the interplay of the thermally-driven magnonic spin current in the ferromagnet and the(inverse) spin Hall voltage generation in an adjacent nor-mal metal [2]. Lucassen et al. [3] subsequently proposedthat the thermally-induced magnon flow in a metallicferromagnet can also produce a detectable (longitudinal)voltage in the bulk itself, due to the spin-transfer mecha-nism of magnon drag. Specifically, smooth magnetizationtexture dynamics induce an electromotive force [4], whosenet average over thermal fluctuations is proportional tothe temperature gradient. In this Letter, we developa Landau-Lifshitz theory for this magnon drag, whichgeneralizes Ref. [3] to include a heretofore disregardedBerry-phase contribution. This additional magnon dragcan reverse the sign of the thermopower, which can havepotential utility for designing scalable thermopiles basedon metallic ferromagnets.Electrons propagating through a smooth dynamic tex-ture of the directional order parameter n ( r , t ) [such that | n ( r , t ) | ≡
1, with the self-consistent spin density givenby s = s n ] experience the geometric electromotive forceof [4] F i = (cid:126) n · ∂ t n × ∂ i n − β∂ t n · ∂ i n ) (1)for spins up along n and − F i for spins down. The resul-tant electric current density is given by j i = σ ↑ − σ ↓ e (cid:104) F i (cid:105) = (cid:126) P σ e (cid:104) n · ∂ t n × ∂ i n − β∂ t n · ∂ i n (cid:105) , (2)where σ = σ ↑ + σ ↓ is the total electrical conductivity, P =( σ ↑ − σ ↓ ) /σ is the conducting spin polarization, and e isthe carrier charge (negative for electrons). The averaging (cid:104) . . . (cid:105) in Eq. (2) is understood to be taken over the steady-state stochastic fluctuations of the magnetic orientation. The latter obeys the stochastic Landau-Lifshitz-Gilbertequation [5] s (1 + α n × ) ∂ t n + n × ( H z + h ) + (cid:88) i ∂ i j i = 0 , (3)where α is the dimensionless Gilbert parameter [6], H parametrizes a magnetic field (and/or axial anisotropy)along the z axis, and j i = − A n × ∂ i n is the magnetic spin-current density, which is proportional to the exchangestiffness A . For H >
0, the equilibrium orientation is n → − z , which we will suppose in the following. TheLangevin field stemming from the (local) Gilbert damp-ing is described by the correlator [7] (cid:104) h i ( r , ω ) h ∗ j ( r (cid:48) , ω (cid:48) ) (cid:105) = 2 παs (cid:126) ωδ ij δ ( r − r (cid:48) ) δ ( ω − ω (cid:48) )tanh (cid:126) ω k B T ( r ) , (4)upon Fourier transforming in time: h ( ω ) = (cid:82) dte iωt h ( t ).At temperatures much less than the Curie tempera-ture, T c , it suffices to linearize the magnetic dynamicswith respect to small-angle fluctuations. To that end, weswitch to the complex variable, n ≡ n x − in y , parametriz-ing the transverse spin dynamics. Orienting a uniformthermal gradient along the x axis, T ( x ) = T + x∂ x T ,we Fourier transform the Langevin field (4) also in realspace, with respect to the y and z axes. LinearizingEq. (3) for small-angle dynamics results in the Helmholtzequation: A ( ∂ x − κ ) n ( x, q , ω ) = h ( x, q , ω ) , (5)where κ ≡ q +[ H − (1+ iα ) sω ] /A , h ≡ h x − ih y , and q isthe two-dimensional wave vector in the yz plane. SolvingEq. (5) using the Green’s function method, we substitutethe resultant n into the expression for the charge currentdensity (2), which can be appropriately rewritten in the a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y following form (for the nonzero x component): j x = (cid:126) P σ e (cid:90) d q dω (2 π ) ω × Re (1 + iβ ) (cid:104) n ( x, q , ω ) ∂ x n ∗ ( x, q (cid:48) , ω (cid:48) ) (cid:105) (2 π ) δ ( q − q (cid:48) ) δ ( ω − ω (cid:48) ) . (6)Tedious but straightforward manipulations, using thecorrelator (4), finally give the following thermoelectriccurrent density: j x = αsP σ∂ x T eA k B T (cid:90) d q dω (2 π ) ( (cid:126) ω ) sinh (cid:126) ω k B T Re [(1 + iβ ) I ] , (7)where I ( κ ) ≡ κ/ | κ | (Re κ ) , having made the conventionthat Re κ > q x by noticing that, inthe limit of low damping, α → I = 2 π (cid:90) dq x iq x /α ˜ ω (˜ ω − q x − q − ξ − ) + ( α ˜ ω ) . (8)Here, we have introduced the magnetic exchange length ξ ≡ (cid:112) A/H and defined ˜ ω ≡ sω/A . After approximatingthe Lorentzian in Eq. (8) with the delta function when α (cid:28)
1, Eq. (7) can finally be expressed in terms of adimensionless integral J ( a ) ≡ (cid:90) ∞ a/ √ dx x √ x − a sinh x , (9)as j = (cid:18) − β α (cid:19) J (cid:18) λξ (cid:19) k B P σπ e (cid:18) TT c (cid:19) / ∇ T . (10)Here, T is the ambient temperature, k B T c ≡ A ( (cid:126) /s ) / estimates the Curie temperature, and λ ≡ (cid:112) (cid:126) A/sk B T is the thermal de Broglie wavelength in the absence ofan applied field. We note that α, β (cid:28) α ∼ β , intypical transition-metal ferromagnets [8].For temperatures much larger than the magnon gap(typically of the order of 1 K in metallic ferromagnets), λ (cid:28) ξ and we can approximate J ( λ/ξ ) ≈ J (0) ∼
1. This limit effectively corresponds to the gaplessmagnon dispersion of (cid:15) q ≡ (cid:126) ω q ≈ (cid:126) Aq /s . Withinthe Boltzmann phenomenology, the magnonic heat cur-rent induced by a uniform thermal gradient is given by j Q = − ∇ T (cid:82) [ d q / (2 π ) ]( ∂ q x ω q ) τ ( ω q ) (cid:15) q ∂ T n BE , where τ − ( ω q ) = 2 αω q is the Gilbert-damping decay rate ofmagnons (to remain within the consistent LLG phe-nomenology) and n BE = [exp( (cid:15) q /k B T ) − − is the Bose-Einstein distribution function. By noticing that (cid:15) q ∂ T n BE = k B (cid:20) (cid:126) ω q / k B T sinh( (cid:126) ω q / k B T ) (cid:21) , (11) r T hydrodynamic r ⌦ < geometric ˆ x ˆ z ˆ y ⌦ e e e e e e e e FIG. 1. Schematics for the two contributions to the electron-magnon drag. In the absence of decay (i.e., α → ∝ β . The (geometric) Berry-phase drag gov-erned by the magnon decay is proportional to α and acts inthe opposite direction. It is illustrated for a spin wave that isthermally emitted from the left. As the spin wave propagatesto the right, the solid angle Ω subtended by the spin preces-sion shrinks, inducing a force oriented to the left for spinsparallel to n . it is easy to recast the second, ∝ β contribution toEq. (10) in the form j ( β ) = β (cid:126) P σ eA j Q , (12)which reproduces the main result of Ref. [3].The magnon-drag thermopower (Seebeck coefficient), S = − ∂ x V∂ x T (cid:12)(cid:12)(cid:12)(cid:12) j x =0 , (13)corresponds to the voltage gradient ∂ x V induced underthe open-circuit condition. We thus get from Eq. (10): S = (cid:18) β α − (cid:19) J k B Pπ e (cid:18) TT c (cid:19) / = ( β − α ) (cid:126) P κ m eA , (14)where κ m = (2 / π ) Jk B A ( T /T c ) / /α (cid:126) is the magnoniccontribution to the heat conductivity. Such magnon-dragthermopower has recently been observed in Fe and Co[9], with scaling ∝ T / over a broad temperature rangeand opposite sign in the two metals. Note that the signdepends on β/α and the effective carrier charge e .Equations (10) and (14) constitute the main results ofthis paper. In the absence of Gilbert damping, α → S is proportional to theheat conductivity. This contribution was studied inRef. [3] and is understood as a viscous hydrodynamicdrag. In simple model calculations [8], βP > e . When P >
0, so that the ma-jority band is polarized along the spin order parameter n , the ∝ α contribution to the thermopower is oppositeto the ∝ β contribution. (Note that α is always > ∂ x Ω <
0. The first termin Eq. (1), which is rooted in the geometric Berry con-nection [10], is proportional to the gradient of this solidangle times the precession frequency, ∝ ω∂ i Ω, resultingin a net force towards the hot side acting on the spinscollinear with n .Note that we have neglected the Onsager-reciprocalbackaction of the spin-polarized electron drift on themagnetic dynamics. This is justified as including thecorresponding spin-transfer torque in the LLG equationwould yield higher-order effects that are beyond ourtreatment [11]. The diffusive contribution to the See-beck effect, ∝ T /E F , where E F is a characteristic Fermienergy, which has been omitted from our analysis, is ex-pected to dominate only at very low temperatures [9].The conventional phonon-drag effects have likewise beendisregarded. A systematic study of the relative impor-tance of the magnon and phonon drags is called upon inmagnetic metals and semiconductors.This work is supported by the ARO under ContractNo. 911NF-14-1-0016, FAME (an SRC STARnet centersponsored by MARCO and DARPA), the Stichting voorFundamenteel Onderzoek der Materie (FOM), and theD-ITP consortium, a program of the Netherlands Orga-nization for Scientific Research (NWO) that is funded bythe Dutch Ministry of Education, Culture, and Science(OCW). [1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,K. Ando, S. Maekawa, and E. Saitoh, Nature ,778 (2008); K. Uchida, J. Xiao, H. Adachi, J. Ohe,S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa,H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh,Nat. Mater. , 894 (2010).[2] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, andS. Maekawa, Phys. Rev. B , 214418 (2010).[3] M. E. Lucassen, C. H. Wong, R. A. Duine, andY. Tserkovnyak, Appl. Phys. Lett. , 262506 (2011).[4] R. A. Duine, Phys. Rev. B , 014409 (2008);Y. Tserkovnyak and M. Mecklenburg, ibid . , 134407(2008).[5] S. Hoffman, K. Sato, and Y. Tserkovnyak, Phys. Rev. B , 064408 (2013).[6] T. L. Gilbert, IEEE Trans. Magn. , 3443 (2004).[7] W. F. Brown, Phys. Rev. , 1677 (1963).[8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J.Magn. Magn. Mater. , 1282 (2008).[9] S. J. Watzman, R. A. Duine, Y. Tserkovnyak, H. Jin,A. Prakash, Y. Zheng, and J. P. Heremans, “Magnon-drag thermopower and Nernst coefficient in Fe and Co,”arXiv:1603.03736.[10] M. V. Berry, Proc. R. Soc. London A , 45 (1984);G. E. Volovik, J. Phys. C: Sol. State Phys. , L83(1987); S. E. Barnes and S. Maekawa, Phys. Rev. Lett. , 246601 (2007); Y. Tserkovnyak and C. H. Wong,Phys. Rev. B , 014402 (2009).[11] The backaction by the spin-transfer torque would be ab-sent when the longitudinal spin current, j i = σ ( P E i + F i /e ) n , vanishes, where E i is the electric field and F i is the spin-motive force (1). Understanding Eq. (10) aspertaining to the limit of the vanishing spin current j i rather than electric current j i = σ ( E i + P F i /e ) n would,however, result in higher-order (in T /T cc