Magnon-polaron transport in magnetic insulators
Benedetta Flebus, Ka Shen, Takashi Kikkawa, Ken-ichi Uchida, Zhiyong Qiu, Eiji Saitoh, Rembert A. Duine, Gerrit E. W. Bauer
MMagnon-polaron transport in magnetic insulators
Benedetta Flebus, Ka Shen, Takashi Kikkawa,
3, 4
Ken-ichi Uchida,
3, 5, 6, 7
ZhiyongQiu, Eiji Saitoh,
3, 4, 7, 8
Rembert A. Duine,
1, 9 and Gerrit E. W. Bauer
3, 4, 2, 7 Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan National Institute for Materials Science, Tsukuba 305-0047, Japan PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan Department of Applied Physics, Eindhoven University of Technology,PO Box 513, 5600 MB Eindhoven, The Netherlands (Dated: February 9, 2017)We theoretically study the effects of strong magnetoelastic coupling on the transport propertiesof magnetic insulators. We develop a Boltzmann transport theory for the mixed magnon-phononmodes (”magnon polarons”) and determine transport coefficients and spin diffusion length. Magnon-polaron formation causes anomalous features in the magnetic field and temperature dependence ofthe spin Seebeck effect when the disorder scattering in the magnetic and elastic subsystems issufficiently different. Experimental data by Kikkawa et al. [PRL , 207203 (2016)] on yttriumiron garnet films can be explained by an acoustic quality that is much better than the magneticquality of the material. We predict similar anomalous features in the spin and heat conductivityand non-local spin transport experiments.
I. INTRODUCTION
The magnetoelastic coupling (MEC) between magneticmoments and lattice vibrations in ferromagnets stemsfrom spin-orbit, dipole-dipole and exchange interactions.This coupling gives rise to magnon-polarons, i.e., hy-bridized magnon and phonon modes in proximity of theintersection of the uncoupled elastic and magnetic dis-persions [1–4]. Interest in the coupling of magnetic andelastic excitations emerged recently in the field of spincaloritronics [5], since it affects thermal and spin trans-port properties of magnetic insulators such as yttriumiron garnet (YIG) [6–11].In this work we address the spin Seebeck effect (SSE)at low temperatures – which provides an especially strik-ing evidence for magnon-polarons in the form of asym-metric spikes in the magnetic field dependence [12]. Theenhancement emerges at the magnetic fields correspond-ing to the tangential intersection of the magnonic disper-sion with the acoustic longitudinal and transverse phononbranches that we explain by phase-space arguments andan unexpected high acoustic quality of YIG.Here we present a Boltzmann transport theory for cou-pled magnon and phonon transport in bulk magnetic in-sulators and elucidate the anomalous field and tempera-ture dependencies of the SSE in terms of the compositenature of the magnon-polarons. The good agreement be-tween theory and the experiments generates confidencethat the SSE can be used as an instrument to character-ize magnons vs. phonon scattering in a given material.We derive the full Onsager matrix, including spin andheat conductivity as well as the spin diffusion length. We predict magnon-polaron signatures in all transportcoefficients that await experimental exposure.This work is organized as follows: In Sec. II we start byintroducing the standard model for spin wave and phononband dispersions of a magnetic insulator and the magne-toelastic coupling. In Sec. III, we describe the magnon-polaron modes and their field-dependent behavior in re-ciprocal space. The linearized Boltzmann equation isshown to lead to expressions for the magnon-polarontransport coefficients. In Sec. IV, we present numeri-cal results for the spin Seebeck coefficient, spin and heatconductivity, and spin diffusion length for YIG. We alsoderive approximate analytical expressions for the fieldand temperature dependence of the anomalies emergingin the transport coefficients and compare our results withthe experiments. In Sec. V we present our conclusionsand an outlook.
II. MODEL
In this section we introduce the Hamiltonian describ-ing the coupling between magnons and phonons in mag-netic insulators. The experimentally relevant geometryis schematically depicted in Fig. 1.
A. Magnetic Hamiltonian
We consider a magnetic insulator with spins S p = S ( r p ) localized on lattice sites r p . The magnetic Hamil-tonian consists of dipolar and (Heisenberg) exchange in- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b teractions between spins and of the Zeeman interactiondue to an external magnetic field B = µ H ˆz [13–15]. Itreads as H mag = µ ( gµ B ) (cid:88) p (cid:54) = q | r pq | S p · S q − r pq · S p ) ( r pq · S q ) | r pq | − J (cid:88) p (cid:54) = q S p · S q − gµ B B (cid:88) p S zp . (1)Here, g is the g-factor, µ the vacuum permeability, µ B the Bohr magneton, J the exchange interaction strength,and r pq = r p − r q . By averaging over the complex unitcell of a material such as YIG, we define a coarse-grained,classical spin S = | S p | = a M s / ( gµ B ) on a cubic latticewith unit cell lattice constant a , with M s being the zerotemperature saturation magnetization density. The crys-tal anisotropy is disregarded, while the dipolar interac-tion is evaluated for a magnetic film in the yz -plane, seeFig. 1. We employ the Holstein-Primakoff transformationand expand the spin operators as [16] S − p = √ Sa † p (cid:115) − a † p a p S ≈ √ S (cid:34) a † p − a † p a † p a p S (cid:35) ,S zp = S − a † p a p , (2)where S − p = S xp − iS yp , and a p / a † p annihilate/create amagnon at the lattice site r p and obey Boson commuta-tion rules [ a p , a † q ] = δ pq . Substituting the Fourier repre-sentation a p = 1 √ N (cid:88) k e i k · r p a k , a † p = 1 √ N (cid:88) k e − i k · r p a † k , (3)where N is the number of lattice sites, and retaining onlyquadratic terms in the bosonic operators and disregard-ing a constant, the Hamiltonian (1) becomes H mag = (cid:88) k A k a † k a k + 12 (cid:16) B k a − k a k + B ∗ k a † k a †− k (cid:17) , (4)with A k (cid:126) = D ex F k + γµ H + γµ M s sin θ k ,B k (cid:126) = γµ M s sin θ k e − iφ k . (5)Here, D ex = 2 SJa is the exchange stiffness, γ = gµ B / (cid:126) the gyromagnetic ratio, θ k = arccos ( k z /k ) the polar an-gle between wave-vector k with k = | k | and the magneticfield along ˆz and φ k the azimuthal angle of k in the xy plane. The form factor F ( k ) = 2(3 − cos k x a − cos k y a − cos k z a ) /a can be approximated as F ( k ) ≈ k in thelong-wavelength limit ( ka (cid:28) (cid:20) a k a †− k (cid:21) = (cid:20) u k − v k − v ∗ k u k (cid:21) (cid:20) α k α †− k (cid:21) , (6) YIG Pt L r T V j m ˆ x ˆ y H ˆ z FIG. 1: Pt | YIG bilayer subject to a thermal gradient ∇ T (cid:107) ˆ x and a magnetic field H (cid:107) ˆ z . The thermal bias gives rise to aflow of magnons, i.e., a magnonic spin current j m , in the YIGfilm of thickness L . In the Pt lead, the spin current is thenconverted into a measurable voltage V via the inverse SpinHall effect. with parameters u k = (cid:114) A k + (cid:126) ω k (cid:126) ω k , v k = (cid:114) A k − (cid:126) ω k (cid:126) ω k e iφ k . (7)The Hamiltonian (4) is then simplified to H mag = (cid:88) k (cid:126) ω k α † k α k , (8)where (cid:126) ω k = (cid:112) A k − | B k | is the magnon dispersion. Forbulk magnons in the long-wavelength limit [18, 19] ω k = (cid:112) D ex k + γµ H (cid:113) D ex k + γµ ( H + M s sin θ k ) . (9)We disregard Damon-Eshbach modes [20] localized at thesurface since, in the following, we focus on transport inthick films normal to the plane, i.e., in the x -directionin Fig. 1. For thick films the backward moving volumemodes are relevant only for wave numbers k very close tothe origin and are disregarded as well. Higher order termsin the magnon operators that encode magnon-magnonscattering processes have been disregarded as well inEq. (4), which is allowed for sufficiently low magnon-densities or temperatures (for YIG (cid:46)
100 K [21]). In thisregime, the main relaxation mechanism is magnon scat-tering by static disorder [6] with Hamiltonian H mag-imp = (cid:88) k , k (cid:48) v mag k , k (cid:48) α † k α k (cid:48) , (10)where v mag k , k (cid:48) is an impurity-scattering potential. In thefollowing, we employ the isotropic, short-range scatteringapproximation v mag k , k (cid:48) = v mag . B. Mechanical Hamiltonian
We focus on lattice vibrations or sound waves withwavelengths much larger than the lattice constant thatare well-described by continuum mechanics. The Hamil-tonian of an elastically isotropic solid reads [23] H el = (cid:90) d r (cid:88) i,j Π i ( r )2¯ ρ δ ij + ( c (cid:107) − c ⊥ ) ¯ ρ ∂R i ( r ) ∂x i ∂R j ( r ) ∂x j + c (cid:107) ¯ ρ ∂R i ( r ) ∂x j ∂R i ( r ) ∂x j , (11)where ¯ ρ is the average mass density, R i is the i -th compo-nent of the displacement vector R of a volume element at r with respect to its equilibrium position, Π i is the conju-gate phonon momentum and c (cid:107) and c ⊥ are the velocitiesof the longitudinal acoustic (LA) and transverse acoustic(TA) lattice waves, respectively. The Hamiltonian (11)can be quantized by the phonon creation (annihilation)operators c † λ k ( c λ k ) as R i ( r , t ) = (cid:88) k ,λ (cid:15) iλ ( k ) (cid:18) (cid:126) ρV ω λ k (cid:19) / ( c † λ k + c λ − k ) e i kr , (12)Π i ( r , t ) = i (cid:88) k ,λ (cid:15) iλ ( k ) (cid:18) ¯ ρ (cid:126) ω λ k V (cid:19) / (cid:16) c † λ k − c λ − k (cid:17) e − i kr , (13)where λ = 1 , k (TA phonons), while λ = 3 representsa pressure wave (LA phonons). Here ω λ k = c λ | k | is thephonon dispersion and (cid:15) iλ ( k ) = ˆ x i · ˆ (cid:15) ( k , λ ) are Cartesiancomponents i = x, y, z of the unit polarization vectorsˆ (cid:15) ( k ,
1) = (cos θ k cos φ k , cos θ k sin φ k , − sin θ k ) , (14a)ˆ (cid:15) ( k ,
2) = i ( − sin φ k , cos φ k , , (14b)ˆ (cid:15) ( k ,
3) = i (sin θ k cos φ k , sin θ k sin φ k , cos θ k ) , (14c)that satisfy ˆ (cid:15) ∗ ( k , λ ) = ˆ (cid:15) ( − k , λ ) [6]. In terms of the oper-ators c λ k and c † λ k , Eq. (11) becomes H el = (cid:88) k ,λ (cid:126) ω λ k (cid:16) c † λ k c λ k + (cid:17) . (15)Analogous to magnons, at low temperatures phononrelaxation is dominated by static disorder H imp = (cid:88) λ (cid:88) k , k (cid:48) v ph k , k (cid:48) c † λ k c λ k (cid:48) , (16)where v ph k , k (cid:48) is the phonon impurity-scattering potential,in the following assumed to be isotropic and short-range,i.e., v ph k , k (cid:48) = v ph . C. Magnetoelastic coupling
The magnetic excitations are coupled to the elastic dis-placement via magnetoelastic interactions. In the long-wavelength limit, to leading order in the magnetization M i = ngµ B S i ( n = 1 /a ) and displacement field R i , themagnetoelastic energy reads as [2, 17] H mec = (cid:126) nM s (cid:90) d r (cid:88) ij [ B ij M i ( r ) M j ( r )+ B (cid:48) ij ∂ M ( r ) ∂r i · ∂ M ( r ) ∂r j (cid:21) R ij ( r ) , (17)where B ij = δ ij B (cid:107) + (1 − δ ij ) B ⊥ and B (cid:48) ij = δ ij B (cid:48)(cid:107) +(1 − δ ij ) B (cid:48)⊥ are the phenomenological magnetoelastic con-stants and R ij ( r ) = 12 (cid:20) ∂R i ( r ) ∂r j + ∂R j ( r ) ∂r i (cid:21) , (18)is the displacement gradient R ij .The exchange term ∼ B (cid:48) ij in Eq. (17) contains magne-tization gradients and predominantly affects short wave-length magnons. We disregard this term since we areinterested in capturing low temperature features. Lin-earizing with respect to small nonequilibrium variables – R i , M x , M y – Eq. (17) then becomes H mec = (cid:126) nB ⊥ (cid:18) γ (cid:126) M s ¯ ρ (cid:19) / (cid:88) k ,λ kω − / k λ e − iφ a k ( c λ − k + c † λ k ) × ( − iδ λ cos 2 θ k + iδ λ cos θ k − δ λ sin 2 θ k ) + H . c . , (19)where δ λi is the Kronecker delta. III. MAGNON-POLARONS
Here we introduce magnon-polarons and formulatetheir semiclassical transport properties.
A. Magnon-polaron modes
We rewrite the Hamiltonian H = H mag + H el + H mec as H = 12 (cid:88) k (cid:2) β † k β − k (cid:3) · H k · (cid:104) β k β †− k (cid:105) T (20)where β † k ≡ (cid:16) α † k c † k c † k c † k (cid:17) and the Bogoliubov-deGennes Hamiltonian H k is an 8 × T k that diagonalizes H k as H k T k = ν T k (cid:20) E k − E − k (cid:21) , (21) ω / π ( T H z ) k (10 /m) µ H = 1T (a) magnon polaronmagnonphonon ω / π ( T H z ) k (10 /m) µ H = 2.64T (b) magnon polaronmagnonphonon
FIG. 2: Magnon, TA phonon ( λ = 1), and magnon-polaronmode dispersions for YIG (see Table I for parameters) with H (cid:107) ˆ z and k (cid:107) ˆ x ( θ = π/ φ = 0). (a) For µ H = 1 T,the magnon and transverse phonon dispersions intersect attwo crossing points k , . The mixing between magnons andphonons (see insets) is maximized at these crossings. (b) For µ H ⊥ = 2 .
64 T, the phonon dispersion becomes a tangent tothe magnon dispersion which maximizes the phase space ofmagnon-polaron formation (see inset). where [ ν ] jm = δ jm ν j with ν j = +1 for j = 1 , .., ν j = − j = 5 , ..,
8, and E k is a diagonalmatrix, whose i -th element (cid:126) Ω i k represents the disper-sion relation of the hybrid mode with creation opera-tor Γ † i k = (cid:80) j =1 [ β † k β − k ] j ( T − k ) ∗ ij that is neither a purephonon or magnon, but a magnon-polaron.Let us focus our attention to waves propagating per-pendicularly to the magnetic field, i.e., k = k ˆ x (seeFig. 1). It follows from Eq. (19) that magnon-polaronsinvolve only TA phonons. Disregarding the dipolar inter-actions, the TA phonon branch is tangent to the magnondispersion for µ H ⊥ = c ⊥ / D ex γ at k ⊥ = c ⊥ / D ex .This estimate holds for M s (cid:28) H ⊥ ; otherwise the dipolarinteraction shifts the magnon dispersion to higher values,leading to a smaller critical field H ⊥ . For H < H ⊥ , theTA phonon dispersion intersects the spin wave spectrumat two crossing points, k and k , k , = k ⊥ ∓ (cid:114) k ⊥ − γµ HD ex , (22)where the minus (plus) corresponds to the label 1 (2).In the vicinity of k , , the modes corresponding to thedispersions Ω , k are strongly coupled, as shown in theinset of Fig. 2(a). The magnetoelastic coupling changes the crossing at k , into an anti-crossing with energysplitting ∆Ω k , = Ω k , − Ω k , . For k (cid:28) k , theΓ † k (Γ † k ) mode resembles closely a pure lattice vibration(spin wave) whilst for k (cid:28) k (cid:28) k these roles are re-versed, returning to their original character for k (cid:29) k .At the critical magnetic field H ⊥ , the magnon disper-sion shifts upwards such that the TA phonon branch be-comes tangential. Figure 2(b) shows that this “touch-ing” condition generates the strongest effects of the MEC,since the magnon and phonon modes are strongly cou-pled over a relatively large volume in momentum space.At higher magnetic fields, the uncoupled magnonic andTA phononic curves no longer cross, hence the MEC doesnot play a significant role, and T k reduces to the identitymatrix.An analogous physical picture holds when consider-ing the magnon-polaron modes arising from the cou-pling between magnons and LA phonons for sin 2 θ k (cid:54) = 0,with critical field µ H (cid:107) = c (cid:107) / D ex γ and touch point k (cid:107) = c (cid:107) / D ex (for M s (cid:28) H (cid:107) ). B. Magnon-polaron transport
We proceed to assess the magnetoelastic coupling ef-fects on the transport properties of a magnetic insulatorin order to model the spin Seebeck effect and magnoninjection by heavy metal contacts.A non-equilibrium state at the interface between themagnetic insulator and the normal metal generates a spincurrent that can be detected by the inverse spin Halleffect, as shown in Fig. 1. The spin current and spin-mediated heat currents are then proportional to the in-terface spin mixing conductance that is governed by theexchange interaction between conduction electrons in themetal and the magnetic order in the ferromagnet. In thepresence of magnon-polarons, the excitations at the in-terface have mixed character. Since the spin-pumpingand spin torque processes are mediated by the exchangeinteraction, only the magnetic component of the magnon-polaron in the metal interacts with the conduction elec-trons. We focus here on the limit in which the smallerof the magnon spin diffusion length and magnetic filmthickness is sufficiently large such that the spin currentis dominated by the bulk transport and the interface pro-cesses may be disregarded. We therefore calculate in thefollowing the spin-projected angular momentum and heatcurrents in the bulk of the ferromagnet, assuming thatthe interface scattering processes and subsequent con-version into an inverse spin Hall voltage do not changethe dependence of the observed signals on magnetic field,temperature gradient, material parameters, etc.. Sincethe phonon specific heat is an order of magnitude largerthan the magnon one at low temperatures [25], we mayassume that the phonon temperature and distribution isnot significantly perturbed by the magnons. T is thephonon temperature at equilibrium and we are inter-ested in the response to a constant gradient ∇ T (cid:107) ˆx . The TABLE I: Selected YIG parameters [26–33].Symbol Value UnitMacrospin S 20 -g-factor g 2 -Lattice constant a γ π ×
28 GHz/TSaturation magnetization µ M s D ex . × − m / sLA-phonon sound velocity c (cid:107) . × m/sTA-phonon sound velocity c ⊥ . × m/sMagnetoelastic coupling B ⊥ π × ρ . × Kg / m Gilbert damping α − - spin-conserving relaxation of the magnon distribution to-wards the phonon temperature is assumed to be so effi-cient that the magnon temperature is everywhere equalto the phonon temperature. Also the magnon-polarontemperature profile is then T ( x ) = T + | ∇ T | x . Assum-ing efficient thermalization of both magnons and phononsand weak spin-non-conserving processes as motivated bythe small Gilbert damping, a non-equilibrium distribu-tion as injected by a metallic contact can be parameter-ized by a single parameter, viz. the effective magnon-polaron chemical potential µ [34]. This approximationmight break down at a very low temperatures, but todate there is no evidence for that.In equilibrium the chemical potential of magnons andphonons vanishes since their number is not conserved.The occupation of the i -th magnon-polaron in equilib-rium is therefore given by the Planck distribution func-tion f (0) i k = (cid:18) exp (cid:126) Ω i k k B T − (cid:19) − . (23)Note that here we have assumed the i -th magnon po-laron scattering rate to be sufficiently smaller than thegap between the magnon-polaron mode dispersions, i.e., τ − i k i (cid:28) ∆Ω k i for every k i , which guarantees the i -thmagnon-polaron to not dephase and hence its distri-bution function to be well-defined. We focus on filmswith thickness L (cid:29) Λ mag , Λ ph ,λ , (cid:96) m , (cid:96) ph ,λ , where Λ mag =(4 π (cid:126) D ex /k B T ) / and Λ ph ,λ = (cid:126) c λ /k B T are the thermalmagnon and phonon (de Broglie) wavelengths, respec-tively, and (cid:96) m ( (cid:96) ph ,λ ) the magnon (phonon) mean freepath. The bulk transport of magnon-polarons is thensemiclassical and can be treated by means of Boltzmanntransport theory. In the relaxation time approximationto the collision integral, the Boltzmann equation for theout-of-equilibrium distribution function f i k ( r , t ) reads ∂ t f i k + ∂ r f i k · ∂ k Ω i k = − ( f i k − f (0) i k ) /τ i k , (24)where τ i k is the relaxation time towards equilibrium. Inthe steady state, the deviation δf i k ( r ) = f i k ( r ) − f (0) i k encodes the magnonic spin, j m , and heat, j Q, m , current densities j m = (cid:90) d k (2 π ) (cid:88) i W i k ( ∂ k Ω i k ) δf i k , (25) j Q, m = (cid:90) d k (2 π ) (cid:88) i W i k ( ∂ k Ω i k )( (cid:126) Ω i k ) δf i k . (26)Here, W i k = | ( U k ) i | + | ( U k ) i | is the magnetic ampli-tude of the i -th quasi-particle branch with U k = T − k .For small temperature gradients, Eqs. (25) and (26) canbe linearized j m (cid:39) − σ · ∇ µ − ζ · ∇ T , (27) j Q, m (cid:39) − ρ (m) · ∇ µ − κ (m) · ∇ T , (28)where the tensors σ , κ (m) , ζ , and ρ (m) (= T ζ by theOnsager-Kelvin relation) are, respectively, the spin and(magnetic) heat conductivities, and the spin Seebeck andPeltier coefficients. In the absence of magnetoelastic cou-pling, Eqs. (27) and (28) reduce to the spin and heatcurrents of magnon diffusion theory [34].The total heat current j Q carried by both magnon andphonon systems does not invoke the spin projection W i k ,i.e., j Q = (cid:90) d k (2 π ) (cid:88) i ( ∂ k Ω i k )( (cid:126) Ω i k ) δf i k , (cid:39) − κ · ∇ T , (29)where κ is the total heat conductivity.In terms of the general transport coefficients L mnαγ = β (cid:90) d k (2 π ) (cid:88) i ( W i k ) m τ i k ( ∂ k α Ω i k )( ∂ k γ Ω i k ) × e β (cid:126) Ω i k ( e β (cid:126) Ω i k − ( (cid:126) Ω i k ) n , (30)(with β = 1 /k B T ), we identify σ αγ = L αγ , ζ αγ = L αγ /T , κ (m) αγ = L αγ /T and κ αγ = L αγ /T .At low temperatures, the excitations relax dominantlyby elastic magnon- and phonon-disorder scattering asmodelled here by Eqs. (10) and (16), respectively. TheFermi Golden Rule scattering rate τ − i k of the i -thmagnon-polaron reads τ − i k = 2 π (cid:126) (cid:88) l =1 (cid:88) j k (cid:48) (cid:2) ( U k (cid:48) ) ∗ jl ( U k ) il +( U k (cid:48) ) ∗ jl +4 ( U k ) il +4 (cid:3) | v l | δ ( (cid:126) Ω i k − (cid:126) Ω j k (cid:48) ) , (31)where v = v mag and v , , = v ph , while the purelymagnonic and phononic scattering rates are given by τ − k , mag = L | v mag | π (cid:126) D ex k , τ − k , ph λ = L | v ph | π (cid:126) c λ k . (32) -9 -8 -7 -6 -5 -4 -3
1 2 3 4 5 6 τ ( s ) k (10 /m) µ H = 1T (a) magnon polaron (L)magnon polaron (H)magnonTA phonon -9 -8 -7 -6 -5 -4 -3
1 2 3 4 5 6 τ ( s ) k (10 /m) µ H = 2.6T (b) magnon polaron (L)magnon polaron (H)magnonTA phonon
FIG. 3: (a) Scattering times of magnons, TA phonons ( λ = 1),and lower (L)/upper (H) branch magnon-polarons in YIG for µ H = 1 T ( H (cid:107) ˆ z ) as a function of wave vector k (cid:107) ˆ x for η = 100. (b) Same as (a) but µ H ⊥ = 2 .
64 T.
IV. RESULTS
In this section we discuss our numerical results for thetransport coefficients, in particular the emergence of fieldand temperature dependent anomalies, and we comparethe thermally induced spin current with measured spinSeebeck voltages [12].
A. Spin and heat transport
We consider a sufficiently thick ( > µ m) YIG filmsubject to a temperature gradient ∇ T (cid:107) ˆ x and mag-netic field H (cid:107) ˆ z , as illustrated in Fig. 1. The parame-ters we employ are summarized in Table I. A scatteringpotential | v mag | = 10 − s − (with v mag in units of (cid:126) )reproduces the observed low-temperature magnon meanfree path [25]. We treat the ratio between magneticand non-magnetic impurity-scattering potentials, η = | v mag /v ph | , as an adjustable parameter. With the de-ployed scattering potentials τ − k i (cid:28) ∆Ω k i for all magnon-polaron modes, ensuring the validity of our treatment.We compute the integrals appearing in Eq. (30) numeri-cally on a fine grid ( ∼ k -points) to guarantee accurateresults.Figure 3(a) shows the magnon-polaron scattering timesand how they deviate from the purely phononic andmagnonic ones close to the anticrossings. At the “touch- ζ xx [ ( m ⋅ s ⋅ K ) - ] µ H (T)1K ( × ( × ( × ( × η = 10010.01 FIG. 4: Magnetic field and temperature dependence of thespin Seebeck coefficient ζ xx for different values of the ratio η between magnon and phonon impurity-scattering potentials. ing” fields the phase space portion over which the scat-tering times are modified with respect to the uncoupledsituation is maximal (see Fig.2(b)) as are the effects onspin and heat transport properties as discussed below.In Fig. 4, we plot the (bulk) spin Seebeck coefficient ζ xx as a function of magnetic field for different valuesof η . For η = 1, ζ xx decreases monotonously with in-creasing magnetic field, while for η (cid:54) = 1 two anoma-lies are observed at µ H ⊥ ∼ .
64 T and µ H (cid:107) ∼ . η = 100(0 .
01) atthe same magnetic fields but with amplitudes that de-pend on temperature. The underlying physics can beunderstood in terms of the dispersion curves plotted inthe inset of Fig. 5(a). The first (second) anomaly oc-curs when the TA (LA) phonon branch becomes a tan-gent of the magnon dispersion, which maximizes the in-tegrated magnon-polaron coupling. The group velocityof the resulting magnon-polaron does not differ substan-tially from the purely magnonic one, but its scatteringtime can be drastically modified, depending on the ratiobetween the magnonic and phononic scattering potentials(see Fig. 3(b)). The spin currents can therefore be bothenhanced or suppressed by the MEC. When the magnon-impurity scattering potential is larger than the phonon-impurity one, the hybridization induced by the MEC low-ers the effective potential perceived by magnons, givingrise to an enhanced scattering time and hence larger cur-rents. This can be confirmed by comparing the blue solid( η = 100) and the black dash-dotted ( η = 10 ) linesin Fig. 5(a), showing that the magnitude of the peaksincreases with increasing η . When magnetic and non-magnetic scattering potentials are the same, i.e., η = 1,the anomalies vanish as illustrated by the dashed blueline in Fig. 5(a), and agrees with the results obtained inthe absence of MEC (triangles).The frequencies at which magnon and phonon disper-sions are tangential for uncoupled transverse and longitu-dinal modes are 0.16 THz ( ˆ=8 K)and 0.53 THz ( ˆ=26 K).Far below these temperatures, the magnon-polaron statesare not populated, which explains the disappearance ofthe second anomaly and the strongly reduced magnitude ζ xx [ ( m ⋅ s ⋅ K ) - ] µ H (T)T = 10 K H ⊥ H || (a) no MEC η = 10 ω / ( π ) ( T H z ) k (10 /m) TALA θ = π /2 H ⊥ H || -4-3-2-1 0 1 2 3 4 0 1 2 3 4 5 6 7 8 9 10 M a gnon - po l a r on ζ xx [ ( m ⋅ s ⋅ K ) - ] µ H (T)
Temperature increase (b)
Temperature increase η = 10010.01 M a gnon - po l a r on ζ xx [ ( m ⋅ s ⋅ K ) - ] µ H (T)T = 10 K η = 100 (c) H ⊥∇ T ( ϑ T = π /2) H " " || " " ∇ T ( ϑ T = 0) FIG. 5: (a) Spin Seebeck coefficient ζ xx of bulk YIG as a func-tion of magnetic field at T = 10 K. The black dash-dotted,blue solid, blue dashed, blue dotted lines are computed for,respectively, η = 10 , , , .
01. The triangles are obtainedfor zero MEC. The inset shows the dispersions of uncoupledtransverse (TA) and longitudinal (LA) acoustic phonons andthe magnons shifted by H (cid:107) and H ⊥ magnetic fields. (b) Themagnetic field and temperature dependence of the magnon-polaron contribution for different values of the ratio η betweenmagnon and phonon impurity-scattering potentials. (c) ζ xx as function of magnetic field for H ⊥ ∇ T (blue solid line)and H (cid:107) ∇ T (green dashed line) at T = 10 K for η = 100. of the first one at 1 K in Fig. 4. In the opposite limit, thehigher energy anomaly becomes relatively stronger [seethe solid curve at 50 K in Fig. 4]. The overall decay ofthe spin Seebeck coefficient with increasing magnetic fieldis explained by the freeze-out caused by the increasingmagnon gap opened by the magnetic field [see the insetof Fig. 5(a)]. This strong decrease has been observed insingle YIG crystals [22, 35], but it is suppressed in thin-ner samples or even enhanced at low temperatures [12].The effect is tentatively ascribed to the paramagnetic σ xx [ ( m ⋅ s ⋅ J ) - ] µ H (T)1K ( ×
50) 10K ( × (a) η = 10010.01 κ ( m ) xx ( W m - K - ) µ H (T)1K ( × )10K ( × (b) η = 10010.01 FIG. 6: (a) The magnetic field and temperature dependenceof the magnon spin conductivity σ xx for different values of theratio η between magnon and phonon impurity-scattering po-tentials. (b) The magnetic field and temperature dependenceof the magnon heat conductivity κ (m) xx for different values ofthe ratio η between magnon and phonon impurity-scatteringpotentials. GGG substrate that becomes magnetically active a lowtemperatures [12] and is beyond the scope of the presenttheory. We therefore subtract the pure magnonic back-ground (triangles in Fig. 5(a)) from the magnon-polaronspin currents, which leads to the net magnon-polaroncontribution shown in Fig. 5(b).The dipolar interaction is responsible for theanisotropy in the magnon dispersion in Eq. (9), whichis reflected in the magnetic field dependence of the heatand spin currents. In Fig. 5(c) we plot ζ xx as functionof the angle ϑ T between magnetic field and transport di-rection for η = 100 and T = 10 K. The magnon-polaroncontributions for magnetization parallel and perpendic-ular to the transport are plotted as the green dashedand blue solid curves, respectively. The anisotropy shiftsthe magnon-polaron peak positions, but does not sub-stantially modify their amplitude. On these grounds, weproceed with computing other transport coefficients forthe configuration H ⊥ ∇ T only.Figure 6(a) shows the magnon spin conductivity σ xx as function of the magnetic field and temperature fordifferent values of η . Two peaks (dips) appear at H ⊥ and H (cid:107) for η = 100 ( η = 0 .
01) at 10 K and 50 K, whilethey disappear for η = 1. At very low temperatures, T = 1 K, the anomalies are not visible anymore. Thedependence of the spin conductivity on the temperature,on the angle between the magnetic field and temperaturegradient, and on the scattering potentials ratio η is thesame as reported for the spin Seebeck coefficient ζ xx .In Fig. 6(b), we plot the dependence of the magnonheat conductivity κ (m) xx on the magnetic field and on thetemperature for different values of η . The only differ-ence with respect to the coefficient ζ xx is in the ratiobetween the amplitudes of the two anomalies at T = 10K, at which the magnon modes contributing to the low-field ( H ⊥ ) anomaly are thermally excited, in contrastto high field (cid:0) H (cid:107) (cid:1) modes. In ζ xx the anomaly at H ⊥ should therefore by better visible, as is indeed the case.The magnon heat conductivity from Eq. (30) contains anadditional factor in the integrand which is proportionalto the energy of the magnon-polaron modes. The lat-ter compensates for the lower thermal occupation, whichexplains why the anomaly at H (cid:107) is more pronounced incomparison with the spin Seebeck effect.Perhaps surprisingly, the total heat conductivity κ xx in Fig. 7(a) displays only dips for η (cid:54) = 1 at the spe-cial fields H ⊥ , (cid:107) . This can be explained as follows. For η (cid:29)
1, the phonon contribution to the heat conductiv-ity is larger than the magnon contribution. Except atthe critical fields H ⊥ , (cid:107) , the magnetic field dependenceof κ xx is therefore very weak (solid blue line). Whenphonons mix with magnons with a short scattering time,the thermal conductivity is suppressed, causing the dipsclose to H ⊥ , (cid:107) . For η (cid:28)
1, on the other hand, the magnoncontribution to heat conductivity prevails, as is seen bythe strong magnetic field dependence of κ xx (dotted blueline). Since now | v mag | < | v ph | , the heat conductivityof the resulting magnon-polaron mode is lower than thepurely magnonic one. Again dips appear close to the“touching” magnetic fields.Experimentally, the magnon heat conductivity κ (m , exp) xx at a given temperature was referred to the differ-ence between finite-field value κ xx ( H ) and κ xx ( ∞ ), i.e., κ (m , exp) xx ( H ) = κ xx ( H ) − κ xx ( ∞ ) [25]. The latter, κ xx ( ∞ ), corresponds to the saturation value of the heatconductivity at high-field limit, above which it becomesa constant function of the magnetic field, suggesting thatthe magnon contribution has been completely frozen outand only the phonon contribution remains. In general, κ (m) xx and κ (m , exp) xx differ in the presence of magnetoelastic-ity. The magnon heat conductivity κ (m , exp) xx in Fig. 7(b),evaluated by subtracting the high-field limit for T = 10K, shows dips for both η = 0 .
01 and η = 100 , in con-trast to the magnon heat conductivity κ (m) xx in Fig. 6(b)with peaks for η = 100. The disagreement stems from κ xx ( ∞ ) , which is the (pure) phonon contribution to theheat conductivity at infinite magnetic fields, but is notthe same as the phonon heat conductivity at ambientmagnetic fields when the MEC is significant. In the lat-ter case, the phonon heat conductivity itself depends onthe magnetic field and displays anomalies at H ⊥ , (cid:107) ; hence κ (m , exp) xx (cid:54) = κ (m) xx .Nonetheless κ (m , exp) xx can be useful since its fine struc- κ xx ( W m - K - ) µ H (T) η = 100 η = 1 ( × η = 0.01 ( × (a) -80-60-40-20 0 20 40 0 1 2 3 4 5 6 7 8 9 10 κ xx ( H )- κ xx ( ∞ ) ( W m - K - ) µ H (T)T = 10 K η = 100 η = 1 (0.01) (b) FIG. 7: (a) The magnetic field dependence of the heat con-ductivity κ xx at T = 10 K for different scattering parameters η . (b) Magnetic field dependence of the heat conductivitydifference κ xx ( H ) − κ xx ( ∞ ) simulating the experimental pro-cedure [25] at T = 10 K. ture contains information about the ratio between themagnon-impurity and phonon-impurity scattering poten-tials | v mag | and | v ph | . Also, κ xx ( ∞ ) for η = 100 is muchlarger than for η = 0 .
01, and its value gives additional in-formation about the relative acoustic and magnetic qual-ity of the sample. For example, the results reportedby Ref. [25] can be interpreted, within our theory, assuggesting a much higher acoustic than magnetic qual-ity of the samples, i.e., η (cid:29)
1. The authors, however,have not investigated the magnetic field dependence ofthe heat conductivity but rather the temperature depen-dence, which is beyond the scope of this work. It is worthto mention that already the work of Ref. [36] suggeststhat impurity scattering plays a key role in determiningthe magnetic field dependence of the heat conductivity.The appearance of the anomalies can be understoodanalytically with few straightforward simplifications. Letus consider a one-dimensional system along ˆ x and H =(0 , , H ) . According to Eq. (19) only the TA phononscouple to the magnons leading to the magnon-polarondispersionΩ , k = ω k + ω k ± (cid:112) ( ω k − ω k ) + ˜ ω k , (33)where ˜ ω k = ( S ⊥ k ) / and S ⊥ = ( nB ⊥ ) ( γ (cid:126) / M s ¯ ρc ⊥ ).The magnon-polaron spin amplitudes W , k are W k = ω k − ω k + (cid:112) ( ω k − ω k ) + ˜ ω k (cid:112) ( ω k − ω k ) + ˜ ω k , (34)and W k = 1 − W k . Disregarding the small dipolarinteractions ( M s (cid:28) H ⊥ ) the uncoupled dispersions touchat µ H ⊥ = c ⊥ / D ex γ . We focus on the contribution ofthe k ⊥ – mode (with k ⊥ = c ⊥ / D ex ) to the transportcoefficients (30) close to the touching field and expandin δH = H − H ⊥ . As in Fig. 2(b), for k = k ⊥ and δH (cid:28) H ⊥ , the energies and group velocities of the upperand lower magnon-polarons are approximately the same,i.e., Ω k ⊥ (cid:39) Ω k ⊥ and ∂ k Ω | k = k ⊥ (cid:39) ∂ k Ω | k = k ⊥ . Eq. (34)then reads W k ⊥ = 12 kδH (cid:113) kδH ) , (35)with ˜ k = µ γ/ (4 S ⊥ k ⊥ ) / . The scattering times (31) canbe approximated as τ , k ⊥ ∼ ∂ k Ω , k | k = k ⊥ | v ph | − W , k ⊥ ) + ηW , k ⊥ . (36)Hence L nmxx ∼ βL | v ph | ( ∂ k Ω k ) e β (cid:126) Ω k ( e β (cid:126) Ω k − ( (cid:126) Ω k ) n (cid:12)(cid:12)(cid:12)(cid:12) k = k ⊥ ,H = H ⊥ × y m ( δH ) , (37)where y ( δH ) = 4 (cid:104) kδH ) (cid:105) (1 + η )1 + η (cid:104) kδH ) + η (cid:105) , and y ( δH ) = 2 (cid:104) kδH ) + η (cid:105) η (cid:104) kδH ) + η (cid:105) . The indices n and m correspond to those in Eq. (30).Both y ( δH ) and y ( δH ) have a single extremum at H = H ⊥ , i.e., y (cid:48) ( δH ) | δH =0 = y (cid:48) ( δH ) | δH =0 = 0 , (38) y (cid:48)(cid:48) ( δH ) | δH =0 ∝ (1 − η ) , (39) y (cid:48)(cid:48) ( δH ) | δH =0 ∝ (1 − η ) . (40)Eqs. (38) and (39) prove that y has a minimum at H = H ⊥ for η (cid:54) = 1, while for η = 1 it is a constant. Thisexplains our numerical results for the heat conductivity κ xx , which is unstructured for η = 1 and always displaydips for both η < η > y is also stationary at H = H ⊥ , but it has a minimum only for η <
1, whilean inflection point for η = 1, and a maximum otherwise.The resulting dependence on η of Eq. (37) explains thespin Seebeck coefficient ζ xx , the spin conductivity σ xx and magnon heat conductivity κ (m) xx , in Figs. 4, 6(a) and6(b) respectively. As we have discussed in detail in thereporting of the numerical results, the anomalies can beunderstood physically in terms of the scattering time ofthe magnon-polaron. This scattering time is the sum ofmagnonic and phononic scattering times, so, dependingon the value of η , the spin transport is enhanced ( η > η <
1) close to the touching point.
B. Spin diffusion length
Integrating the spin-projection of Eq. (24) over mo-mentum leads to the spin conservation equation:˙ n s + ∇ · j s = − g µ µ , (41)where n s = (cid:90) d k (2 π ) (cid:88) i f i k ( r ) , (42)is the total magnon density (in units of (cid:126) ), and g µ = β (cid:90) d k (2 π ) (cid:88) i W i k τ nc i k e β (cid:126) Ω i k ( e β (cid:126) Ω i k − , (43)is the magnon relaxation rate, and we have introducedthe relaxation time τ nc i k . Elastic magnon-impurity scat-tering processes discussed in the previous sections do notcontribute to τ nc i k . However, we parameterize the spinnot-conserving processes as1 τ nc i k = 2 α Ω k i , (44)in terms of the dimensionless Gilbert damping constant α . In the non-equilibrium steady-state Eq. (41) becomes ∇ µ = 1 λ n µ , (45)in terms of the magnon diffusion length λ n ≡ (cid:112) σ xx /g µ that is plotted in Fig. 8. At 10 K and 50 K, the spindiffusion length decreases monotonously with the mag-netic field for η = 1, in agreement with observations atroom temperature [34]. For η = 100 ( η = 0 .
01) the spindiffusion length displays two peaks (dips) at the criticalfields H ⊥ and H (cid:107) , which become more pronounced whenlowering the temperature. At T = 1 K only the peak(dip) at H ⊥ is visible for η = 100 ( η = 0 . η = 1,the spin diffusion length monotonically decreases with in-creasing magnetic field. The curve for η = 0 .
01 behavessimilar except for the dip at H = H ⊥ . On the otherhand, for η = 100, the spin diffusion length behaves very0
10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 7 8 9 10 λ n ( µ m ) µ H (T)1K 10K50K η = 10010.01 FIG. 8: Magnetic field and temperature dependence of themagnon diffusion length λ n for different values of the scatter-ing parameter η . differently showing strong enhancement at both low andhigh magnetic fields. This strong increase of the diffusionlength (for constant Gilbert damping) happens when σ xx ( H , η ) σ xx ( H , η ) > g µ ( H ) g µ ( H ) , (46)where H , are two given values of the applied magneticfield, with H > H . To understand the dependence ofthe ratio σ xx ( H , η ) /σ xx ( H , η ) on η and on the tempera-ture, we recall that the main contribution to the magnonspin conductivity σ xx arises from magnon-like branches.At relatively high temperature, the magnon-like branchesare sufficiently populated to overcome the phonon contri-bution to the magnon spin conductivity at all η . Indeed,Fig. 6(a) shows that, at relatively high temperatures, theratio σ xx ( H , η ) /σ xx ( H , η ) hardly depends on η . Onthe other hand, when the temperature decreases belowthe magnon energy, the contribution of the magnon-likebranches are quickly frozen out by a magnetic field. Themagnitude of η then becomes very relevant. On the otherhand, while the right-hand side of Eq. (46) depends ontemperature, it is not affected by η . For η <
1, thephonon mobility is smaller than the magnon one andhence the phonons are short circuited by the magnons.For η >
1, the phonons prevail, leading to a higher ratio σ xx ( H , η ) /σ xx ( H , η ) because the phonon dispersion isnot affected by the magnetic field. When η (cid:29)
1, thecondition (46) is therefore satisfied. While in this regimethe spin current is very small, it is perhaps an interestinglimit for studying fluctuation and shot noise in the spincurrent [9].
C. Comparison with experiments
The spin Seebeck effect was measured in Pt | YIG | GGGstructures in the longitudinal configuration, i.e., by ap-plying a temperature difference normal to the interfaces( x -direction) and subjecting the sample to a magnetic field H (cid:107) ˆ z [12]. The thermal bias induces a spin cur-rent into the Pt layer that by the inverse Spin Hall effect(ISHE) leads to the detected transverse voltage V overthe contact, see Fig. 1. The bottom of the GGG sub-strate and the top of the Pt layer are in contact withheat reservoirs at temperature T L and T H , respectively.Disregarding phonon (Kapitza) interface resistances, thephonon temperature gradient is ∇ T = ( T H − T L ) /L , with L being the thickness of the stack, and average temper-ature T = ( T H + T L ) /
2. As discussed, we assume thatthe magnon and phonon temperatures are the same anddisregard the interface mixing conductance. The mea-sured voltage is then directly proportional to the bulkspin Seebeck coefficient.In the experimental temperature range of 3 . −
50 K thethermal magnon, Λ mag , and phonon, Λ ph ,µ , wavelengthsare of the order of 1 −
10 nm. Even if the magnon andphonon thermal mean free paths have been estimated tobe of the order of ∼ µ m at very low temperatures [25],here we assume that the transport in the YIG film ofthickness L (cid:39) µ m can be treated semiclassically. Notethat scattering at the interfaces can make the transportdiffusive even when the formal conditions for diffusivetransport are not satisfied. The bulk spin Seebeck coeffi-cient is then well-described by Eq. (30) and proportionalto the observed voltage V . These assumptions are en-couraged by the good agreement for the observed andcalculated peak structures at H ⊥ and H (cid:107) with a singlefitting parameter η = 100 [12]. We may therefore con-clude that the disorder potential scatters the magnonsmore than the phonons and is therefore likely to be mag-netic. V. CONCLUSION AND OUTLOOK
We have established a framework which captures theeffects of the magnetoelastic interaction on the transportproperties of magnetic insulators. In particular, we showthat the magnon-phonon coupling gives rise to peak-likeor dip-like structures in the field dependence of the spinand heat transport coefficients, and of the spin diffusionlength.Our numerical evaluation reproduces the peaks in theobserved low temperature longitudinal spin Seebeck volt-ages of YIG | Pt layers as a function of magnetic field.We quantitatively explain the temperature-dependentbehavior of these anomalies in terms of hybrid magnon-phonon excitations (“magnon-polarons”). The peaks oc-cur at magnetic fields and wave numbers at which thephonon dispersion curves are tangents to the magnondispersion, i.e., when magnon and phonon energies aswell as group velocities become the same. Under theseconditions the effects of the magnetoelastic interactionare maximized. The computed angle dependence showsa robustness of the anomalies with respect to rotations ofthe magnetization relative to the temperature gradient.The agreement between the theory and the experimental1results confirms that elastic magnon(phonon) impurity-scattering is the main relaxation channel that limits thelow temperature transport in YIG. Our theory containsone adjustable parameter that is fitted to the large setof experimental data, consistently finding a much betteracoustic than magnetic quality of the samples. The spinSeebeck effect is therefore a unique analytical instrumentnot only of magnetic, but also mechanical material prop-erties. The predicted effects of magnon-polaron effectson magnonic spin and heat conductivity call for furtherexperimental confirmation.We believe that the presented results open new av-enues in spin caloritronics. We focused here on the lowenergy magnon dispersion of cubic YIG, which is wellrepresented by the magnetostatic exchange waves of ahomogeneous ferromagnet [21]. However, the theoreticalframework can be easily extended to include anisotropiesas well as ferri- or antiferromagnetic order. The magne-toelastic coupling in YIG is relatively small and the con-spicuous magnon-polaron effects can be destroyed eas-ily. However, in materials with large magnon-phononcouplings these effects should survive in the presence oflarger magnetization broadening as well as higher tem- peratures.
VI. ACKNOWLEDGEMENTS
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