Magnon-polaron transport in magnetic insulators
Citation for published version (APA): Flebus, B., Shen, K., Kikkawa, T., Uchida, K. I., Qiu, Z., Saitoh, E., Duine, R. A., & Bauer, G. E. W. (2017). Magnon-polaron transport in magnetic insulators. Physical Review B, 95(14), 1-11. [144420]. https://doi.org/10.1103/PhysRevB.95.144420
DOI: 10.1103/PhysRevB.95.144420
Document status and date: Published: 14/04/2017
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Download date: 01. Oct. 2021 PHYSICAL REVIEW B 95, 144420 (2017)
Magnon-polaron transport in magnetic insulators
Benedetta Flebus,1 Ka Shen,2 Takashi Kikkawa,3,4 Ken-ichi Uchida,3,5,6,7 Zhiyong Qiu,4 Eiji Saitoh,3,4,7,8 Rembert A. Duine,1,9 and Gerrit E. W. Bauer2,3,4,7 1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 2Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 5National Institute for Materials Science, Tsukuba 305-0047, Japan 6PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan 7Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 8Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan 9Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (Received 7 February 2017; published 14 April 2017)
We theoretically study the effects of strong magnetoelastic coupling on the transport properties of magnetic insulators. We develop a Boltzmann transport theory for the mixed magnon-phonon modes (“magnon polarons”) and determine transport coefficients and the spin diffusion length. Magnon-polaron formation causes anomalous features in the magnetic field and temperature dependence of the spin Seebeck effect when the disorder scattering in the magnetic and elastic subsystems is sufficiently different. Experimental data by Kikkawa et al. [Phys. Rev. Lett. 117, 207203 (2016)] on yttrium iron garnet films can be explained by an acoustic quality that is much better than the magnetic quality of the material. We predict similar anomalous features in the spin and heat conductivity and nonlocal spin transport experiments.
DOI: 10.1103/PhysRevB.95.144420
I. INTRODUCTION dispersions of a magnetic insulator and the magnetoelastic coupling. In Sec. III, we describe the magnon-polaron modes The magnetoelastic coupling (MEC) between magnetic and their field-dependent behavior in reciprocal space. The moments and lattice vibrations in ferromagnets stems from linearized Boltzmann equation is shown to lead to expressions spin-orbit, dipole-dipole, and exchange interactions. This cou- for the magnon-polaron transport coefficients. In Sec. IV,we pling gives rise to magnon-polarons, i.e., hybridized magnon present numerical results for the spin Seebeck coefficient, spin and phonon modes in proximity of the intersection of the and heat conductivity, and spin diffusion length for YIG. We uncoupled elastic and magnetic dispersions [1–4]. Interest also derive approximate analytical expressions for the field in the coupling of magnetic and elastic excitations emerged and temperature dependence of the anomalies emerging in the recently in the field of spin caloritronics [5], since it affects transport coefficients and compare our results with the exper- thermal and spin transport properties of magnetic insulators iments. In Sec. V, we present our conclusions and an outlook. such as yttrium iron garnet (YIG) [6–11]. In this work, we address the spin Seebeck effect (SSE) at low temperatures—which provides an especially striking II. MODEL evidence for magnon-polarons in the form of asymmetric In this section, we introduce the Hamiltonian describing spikes in the magnetic field dependence [12]. The enhance- the coupling between magnons and phonons in magnetic insu- ment emerges at the magnetic fields corresponding to the lators. The experimentally relevant geometry is schematically tangential intersection of the magnonic dispersion with the depicted in Fig. 1. acoustic longitudinal and transverse phonon branches that we explain by phase-space arguments and an unexpected high A. Magnetic Hamiltonian acoustic quality of YIG. = Here we present a Boltzmann transport theory for coupled We consider a magnetic insulator with spins Sp S(rp) magnon and phonon transport in bulk magnetic insulators localized on lattice sites rp. The magnetic Hamiltonian and elucidate the anomalous field and temperature depen- consists of dipolar and (Heisenberg) exchange interactions dencies of the SSE in terms of the composite nature of the between spins and of the Zeeman interaction due to an external = magnon-polarons. The good agreement between theory and magnetic field B μ0Hzˆ [13–15]. It reads as the experiments generates confidence that the SSE can be 2 2 μ0(gμB ) |rpq| Sp · Sq − 3(rpq · Sp)(rpq · Sq ) used as an instrument to characterize magnons versus phonon Hmag = 2 |r |5 scattering in a given material. We derive the full Onsager p =q pq matrix, including spin and heat conductivity as well as the − J S · S − gμ B Sz . (1) spin diffusion length. We predict magnon-polaron signatures p q B p = in all transport coefficients that await experimental exposure. p q p This work is organized as follows. In Sec. II,westartby Here, g is the g factor, μ0 is the vacuum permeability, μB is introducing the standard model for spin wave and phonon band the Bohr magneton, J is the exchange interaction strength, and
2469-9950/2017/95(14)/144420(11) 144420-1 ©2017 American Physical Society BENEDETTA FLEBUS et al. PHYSICAL REVIEW B 95, 144420 (2017) rpq = rp − rq . By averaging over the complex unit cell of a material such as YIG, we define a coarse-grained, classical spin =| |= 3 S Sp a0 Ms /(gμB ) on a cubic lattice with unit cell lattice constant a0, with Ms being the zero-temperature saturation magnetization density. The crystal anisotropy is disregarded, while the dipolar interaction is evaluated for a magnetic film in the yz plane, see Fig. 1. We employ the Holstein-Primakoff transformation and expand the spin operators as [16] † † † √ a a √ a a a S− = 2Sa† 1 − p p ≈ 2S a† − p p p , p p 2S p 4S (2) z = − † Sp S apap, − = x − y † where Sp Sp iSp, and ap/ap annihilate/create a magnon FIG. 1. Pt|YIG bilayer subject to a thermal gradient ∇T xˆ and a at the lattice site rp and obey Boson commutation rules † magnetic field H zˆ. The thermal bias gives rise to a flow of magnons, [ap,aq ] = δpq. Substituting the Fourier representation i.e., a magnonic spin current jm, in the YIG film of thickness L.In 1 · † 1 − · † the Pt lead, the spin current is then converted into a measurable = √ ik rp = √ ik rp ap e ak,ap e ak, (3) voltage V via the inverse spin Hall effect. N k N k where N is the number of lattice sites, and retaining only quadratic terms in the bosonic operators and disregarding a We disregard Damon-Eshbach modes [20] localized at the constant, the Hamiltonian (1) becomes surface since, in the following, we focus on transport in thick films normal to the plane, i.e., in the x direction in Fig. 1. † 1 ∗ † † For thick films, the backward moving volume modes are H = A a a + (B a− a + B a a ), (4) mag k k k 2 k k k k k −k relevant only for wave numbers k very close to the origin k and are disregarded as well. Higher-order terms in the magnon with operators that encode magnon-magnon scattering processes 2 have been disregarded as well in Eq. (4), which is allowed for Ak γμ0Ms sin θk = DexFk + γμ0H + , sufficiently low magnon-densities or temperatures (for YIG h¯ 2 100 K [21]). In this regime, the main relaxation mechanism 2 Bk γμ0Ms sin θk − is magnon scattering by static disorder [6] with Hamiltonian = e 2iφk . (5) h¯ 2 mag † H = = 2 = mag-imp vk,k αkαk , (10) Here, Dex 2SJa0 is the exchange stiffness, γ gμB /h¯ the k,k gyromagnetic ratio, θk = arccos (kz/k) the polar angle be- =| | tween wave vector k with k k and the magnetic field along mag where v is an impurity-scattering potential. In the follow- zˆ and φk the azimuthal angle of k in the xy plane. The form k,k factor F(k) = 2(3 − cos k a − cos k a − cos k a )/a2 can ing, we employ the isotropic, short-range scattering approxi- x 0 y 0 z 0 0 mag = mag be approximated as F(k) ≈ k2 in the long-wavelength limit mation vk,k v . (ka0 1). Equation (4) is diagonalized by the Bogoliubov transformation [17] B. Mechanical Hamiltonian − ak = uk vk αk We focus on lattice vibrations or sound waves with † − ∗ † , (6) a−k vk uk α−k wavelengths much larger than the lattice constant that are well-described by continuum mechanics. The Hamiltonian of with parameters an elastically isotropic solid reads [22]
Ak +hω¯ k Ak −hω¯ k = = 2iφk 2 uk ,vk e . (7) 3 i (r) 2 2 ρ¯ ∂Ri (r) ∂Rj (r) 2¯hω 2¯hω Hel = d r δij + (c − c⊥) k k 2¯ρ 2 ∂x ∂x i,j i j The Hamiltonian (4) is then simplified to ρ¯ ∂R (r) ∂R (r) + 2 i i H = † c⊥ , (11) mag hω¯ kαkαk, (8) 2 ∂xj ∂xj k √ whereρ ¯ is the average mass density, Ri is the ith component = 2 −| |2 wherehω ¯ k Ak Bk is the magnon dispersion. For bulk of the displacement vector R of a volume element at r with magnons in the long-wavelength limit [18,19], respect to its equilibrium position, is the conjugate phonon i momentum, and c and c⊥ are the velocities of the longitudinal ω = D k2 + γμ H D k2 + γμ (H + M sin2 θ ). k ex 0 ex 0 s k acoustic (LA) and transverse acoustic (TA) lattice waves, (9) respectively. The Hamiltonian (11) can be quantized by the
144420-2 MAGNON-POLARON TRANSPORT IN MAGNETIC INSULATORS PHYSICAL REVIEW B 95, 144420 (2017)
† phonon creation (annihilation) operators c (c )as becomes λk λk 2 1/2 1/2 γh¯ −1/2 −iφ † H = ⊥ − + h¯ † ikr mec hnB¯ kωkλ e ak(cλ k cλk) R r,t = k c + c − e , 4M ρ¯ i ( ) iλ( ) ( λk λ k) (12) s k,λ 2¯ρVωλk k,λ × − + − + 1/2 ( iδλ1 cos 2θk iδλ2 cos θk δλ3 sin 2θk) H.c., ρ¯hω¯ λk † −ikr (r,t) = i (k) (c − c − )e , (13) i iλ 2V λk λ k (19) k,λ where δλi is the Kronecker delta. where λ = 1,2 labels the shear waves polarized normal to the wave vector k (TA phonons), while λ = 3 represents a III. MAGNON-POLARONS pressure wave (LA phonons). Here, ωλk = cλ|k| is the phonon dispersion and iλ(k) = xˆi · ˆ(k,λ) are Cartesian components Here we introduce magnon-polarons and formulate their i = x,y,z of the unit polarization vectors semiclassical transport properties.
= − ˆ(k,1) (cos θk cos φk, cos θk sin φk, sin θk), (14a) A. Magnon-polaron modes = − ˆ(k,2) i( sin φk, cos φk,0), (14b) We rewrite the Hamiltonian H = Hmag + Hel + Hmec as = ˆ(k,3) i(sin θk cos φk, sin θk sin φk, cos θk), (14c) 1 † † T H = β β− · Hk · β β , (20) ∗ = − 2 k k k −k that satisfy ˆ (k,λ) ˆ( k,λ)[6]. In terms of the operators k † c and c ,Eq.(11) becomes λk λk β† ≡ † † † † where k (αk c1k c2k c3k) and the Bogoliubov-de Gennes × Hamiltonian Hk is an 8 8 Hermitian matrix. Following † 1 H = + Ref. [23], we introduce the para-unitary matrix T k that el hω¯ λk cλkcλk . (15) 2 diagonalizes Hk as k,λ Analogous to magnons, at low temperatures, phonon relax- T = νT Ek 0 Hk k k − , (21) ation is dominated by static disorder: 0 E−k ph † where [ν] = δ ν with ν =+1forj = 1,...,4 and ν = H = jm jm j j j imp vk,k cλkcλk , (16) −1forj = 5,...,8, and Ek is a diagonal matrix, whose ith λ k,k elementh ¯ ik represents the dispersion relation of the hybrid † 8 † −1 ∗ ph mode with creation operator = [β β− ] (T ) where vk,k is the phonon impurity-scattering potential, in ik j=1 k k j k ij the following assumed to be isotropic and short-range, i.e., that is neither a pure phonon or magnon, but a magnon-polaron. ph ph Let us focus our attention to waves propagating perpendic- v = v . k,k ularly to the magnetic field, i.e., k = kxˆ (see Fig. 1). It follows from Eq. (19) that magnon-polarons involve only TA phonons. C. Magnetoelastic coupling Disregarding the dipolar interactions, the TA phonon branch 2 The magnetic excitations are coupled to the elastic displace- is tangent to the magnon dispersion for μ0H⊥ = c⊥/4Dexγ at ment via magnetoelastic interactions. In the long-wavelength k⊥ = c⊥/2Dex. This estimate holds for Ms H⊥; otherwise limit, to leading order in the magnetization Mi = ngμB Si the dipolar interaction shifts the magnon dispersion to higher = 3 (n 1/a0 ) and displacement field Ri , the magnetoelastic values, leading to a smaller critical field H⊥.ForH
144420-3 BENEDETTA FLEBUS et al. PHYSICAL REVIEW B 95, 144420 (2017)
0.4 the excitations at the interface have mixed character. Since magnon polaron μ H = 1T the spin-pumping and spin torque processes are mediated 0 magnon by the exchange interaction, only the magnetic component of 0.3 phonon the magnon-polaron in the metal interacts with the conduction 0.0355 electrons. We focus here on the limit in which the smaller of the magnon spin diffusion length and magnetic film thickness is
(THz) 0.2
π 0.035 sufficiently large such that the spin current is dominated by the
/2 0.56 0.58 4.44 4.46 bulk transport and the interface processes may be disregarded. ω 0.1 0.276 We therefore calculate in the following the spin-projected 0.274 angular momentum and heat currents in the bulk of the ferromagnet, assuming that the interface scattering processes 0 0 1 2 3 4 5 and subsequent conversion into an inverse spin Hall voltage do (a) 8 not change the dependence of the observed signals on magnetic k (10 /m) field, temperature gradient, material parameters, etc. Since the phonon specific heat is an order of magnitude 0.4 larger than the magnon one at low temperatures [24], we μ 0H = 2.64T may assume that the phonon temperature and distribution 0.3 0.16 is not significantly perturbed by the magnons. T is the phonon temperature at equilibrium and we are interested in the response to a constant gradient ∇T xˆ.Thespin-
(THz) 0.2 0.156 conserving relaxation of the magnon distribution towards π 2.55 2.6
/2 the phonon temperature is assumed to be so efficient that ω 0.1 magnon polaron the magnon temperature is everywhere equal to the phonon magnon temperature. Also the magnon-polaron temperature profile is phonon then T (x) = T +|∇T |x. Assuming efficient thermalization 0 of both magnons and phonons and weak spin-non-conserving 1 2 3 4 5 processes as motivated by the small Gilbert damping, a non- 8 (b) k (10 /m) equilibrium distribution as injected by a metallic contact can be parameterized by a single parameter, viz. the effective magnon- = FIG. 2. Magnon, TA phonon (λ 1), and magnon-polaron mode polaron chemical potential μ [33]. This approximation might dispersions for YIG (see Table I for parameters) with H zˆ and k xˆ break down at a very low temperatures, but to date there is no = = = (θ π/2andφ 0). (a) For μ0H 1 T, the magnon and transverse evidence for that. phonon dispersions intersect at two crossing points k1,2. The mixing In equilibrium, the chemical potential of magnons and between magnons and phonons (see insets) is maximized at these phonons vanishes since their number is not conserved. The crossings. (b) For μ H⊥ = 2.64 T, the phonon dispersion becomes a 0 occupation of the ith magnon-polaron in equilibrium is tangent to the magnon dispersion which maximizes the phase space of magnon-polaron formation (see inset). therefore given by the Planck distribution function
− h¯ 1 f (0) = exp ik − 1 . (23) momentum space. At higher magnetic fields, the uncoupled ik k T magnonic and TA phononic curves no longer cross, hence B T the MEC does not play a significant role, and k reduces Note that here we have assumed the ith magnon polaron to the identity matrix. An analogous physical picture holds scattering rate to be sufficiently smaller than the gap between when considering the magnon-polaron modes arising from the −1 the magnon-polaron mode dispersions, i.e., τ k for = iki i coupling between magnons and LA phonons for sin 2θk 0, every k , which guarantees the ith magnon-polaron to not = 2 i with critical field μ0H c/4Dexγ and touch point dephase and hence its distribution function to be well-defined. k = c/2Dex (for Ms H). We focus on films with thickness L mag,ph,λ,m,ph,λ, 1/2 where mag = (4πhD¯ ex/kB T ) and ph,λ = hc¯ λ/kB T are B. Magnon-polaron transport the thermal magnon and phonon (de Broglie) wavelengths, re- spectively, and ( ) the magnon (phonon) mean free path. We proceed to assess the magnetoelastic coupling effects m ph,λ The bulk transport of magnon-polarons is then semiclassical on the transport properties of a magnetic insulator in order to and can be treated by means of Boltzmann transport theory. model the spin Seebeck effect and magnon injection by heavy In the relaxation time approximation to the collision integral, metal contacts. A nonequilibrium state at the interface between the Boltzmann equation for the out-of-equilibrium distribution the magnetic insulator and the normal metal generates a spin function f (r,t) reads current that can be detected by the inverse spin Hall effect, ik as shown in Fig. 1. The spin current and spin-mediated heat + · =− − (0) currents are then proportional to the interface spin mixing ∂t fik ∂rfik ∂kik fik fik /τik, (24) conductance that is governed by the exchange interaction between conduction electrons in the metal and the magnetic where τik is the relaxation time towards equilibrium. In the = − (0) order in the ferromagnet. In the presence of magnon-polarons, steady state, the deviation δfik(r) fik(r) fik encodes the
144420-4 MAGNON-POLARON TRANSPORT IN MAGNETIC INSULATORS PHYSICAL REVIEW B 95, 144420 (2017) magnonic spin, jm, and heat, jQ,m, current densities TABLE I. Selected YIG parameters [25–32].
d3k Symbol Value Unit j = W (∂ )δf , (25) m (2π)3 ik k ik ik i Macrospin S 20 - d3k g factor g 2 - j = W (∂ )(¯h )δf . (26) ˚ Q,m 3 ik k ik ik ik Lattice constant a0 12.376 A (2π) − i Gyromagnetic ratio γ 2π × 28 GHz T 1 =| |2 +| |2 Saturation magnetization μ0Ms 0.2439 T Here, Wik (U k)i1 (U k)i5 is the magnetic amplitude × −6 2 −1 −1 Exchange stiffness Dex 7.7 10 m s of the ith quasiparticle branch with U = T . For small 3 −1 k k LA-phonon sound velocity c 7.2 × 10 ms temperature gradients, Eqs. (25) and (26) can be linearized 3 −1 TA-phonon sound velocity c⊥ 3.9 × 10 ms Magnetoelastic coupling B⊥ 2π × 1988 GHz − jm −σ · ∇μ − ζ · ∇T, (27) Average mass densityρ ¯ 5.17 × 103 Kg m 3 Gilbert damping α 10−4 - (m) (m) jQ,m −ρ · ∇μ − κ · ∇T, (28) where the tensors σ,κ(m), ζ , and ρ(m)(= T ζ by the Onsager- Kelvin relation) are, respectively, the spin and (magnetic) heat thermally induced spin current with measured spin Seebeck conductivities, and the spin Seebeck and Peltier coefficients. voltages [12]. In the absence of magnetoelastic coupling, Eqs. (27) and (28) reduce to the spin and heat currents of magnon diffusion A. Spin and heat transport theory [33]. We consider a sufficiently thick (>1 μm) YIG film subject The total heat current jQ carried by both magnon and to a temperature gradient ∇T xˆ and magnetic field H zˆ, phonon systems does not invoke the spin projection Wik, i.e., as illustrated in Fig. 1. The parameters we employ are summa- | mag|2 = −5 −2 d3k rized in Table I. A scattering potential v 10 s (with mag jQ = (∂kik)(¯hik)δfik, v in units ofh ¯) reproduces the observed low-temperature (2π)3 i magnon mean free path [24]. We treat the ratio between = −κ · ∇T, (29) magnetic and nonmagnetic impurity-scattering potentials, η |vmag/vph|2, as an adjustable parameter. With the deployed where κ is the total heat conductivity. scattering potentials τ −1 for all magnon-polaron ki ki In terms of the general transport coefficients modes, ensuring the validity of our treatment. We compute the integrals appearing in Eq. (30) numerically on a fine grid d3k Lmn = β (W )mτ (∂ )(∂ ) (∼106k points) to guarantee accurate results. αγ (2π)3 ik ik kα ik kγ ik i Figure 3(a) shows the magnon-polaron scattering times and
βh¯ ik how they deviate from the purely phononic and magnonic ones e n × (¯hik) , (30) close to the anticrossings. At the “touching” fields, the phase βh¯ ik − 2 (e 1) space portion over which the scattering times are modified with 10 11 respect to the uncoupled situation is maximal [see Fig. 2(b)]as (with β = 1/kB T ), we identify σαγ = L ,ζαγ = L /T, αγ αγ are the effects on spin and heat transport properties as discussed κ(m) = L12 /T and κ = L02 /T. αγ αγ αγ αγ below. At low temperatures, the excitations relax dominantly by In Fig. 4, we plot the (bulk) spin Seebeck coefficient ζ elastic magnon- and phonon-disorder scattering as modelled xx as a function of magnetic field for different values of η. here by Eqs. (10) and (16), respectively. The Fermi golden rule − For η = 1,ζ decreases monotonously with increasing mag- scattering rate τ 1 of the ith magnon-polaron reads xx ik netic field, while for η = 1 two anomalies are observed at 4 μ0H⊥ ∼ 2.64 T and μ0H ∼ 9.3 T. More precisely, peaks −1 2π ∗ τ = [(U ) (U ) (dips) appear for η = 100(0.01) at the same magnetic fields ik h¯ k jl k il l=1 jk but with amplitudes that depend on temperature. ∗ The underlying physics can be understood in terms of the + 2| |2 − (U k )jl+4(U k)il+4] vl δ(¯hik h¯ jk ), (31) dispersion curves plotted in the inset of Fig. 5(a). The first (second) anomaly occurs when the TA (LA) phonon branch where v = vmag and v = vph, while the purely magnonic 1 2,3,4 becomes a tangent of the magnon dispersion, which maximizes and phononic scattering rates are given by the integrated magnon-polaron coupling. L3|vmag|2 L3|vph|2 The group velocity of the resulting magnon-polaron does τ −1 = k, τ−1 = k2. (32) k,mag 2πh¯ 2D k,phλ πh¯ 2c not differ substantially from the purely magnonic one, but ex λ its scattering time can be drastically modified, depending on the ratio between the magnonic and phononic scattering IV. RESULTS potentials [see Fig. 3(b)]. The spin currents can therefore be In this section, we discuss our numerical results for the both enhanced or suppressed by the MEC. When the magnon- transport coefficients, in particular, the emergence of field impurity scattering potential is larger than the phonon-impurity and temperature dependent anomalies, and we compare the one, the hybridization induced by the MEC lowers the
144420-5 BENEDETTA FLEBUS et al. PHYSICAL REVIEW B 95, 144420 (2017)
-3 10 5 η = 100 magnon polaron (L) ] 1 -4 -1 10 magnon polaron (H) 4 0.01 K)
magnon ⋅ 50K -5 s 10 TA phonon ⋅ 3 20K (×2.5) -6 (m (s) 10 23 10K (×5) τ 2 -7 10 [10 5K (×10)
xx 1
-8 ζ 10 μ 1K (×50) 0H = 1T 0 10-9 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 μ H (T) (a) k (108/m) 0 10-3 FIG. 4. Magnetic field and temperature dependence of the spin magnon polaron (L) Seebeck coefficient ζxx for different values of the ratio η between -4 10 magnon polaron (H) magnon and phonon impurity-scattering potentials. -5 magnon 10 TA phonon -6 beyond the scope of the present theory. We therefore subtract (s) 10 the pure magnonic background [triangles in Fig. 5(a)] from τ the magnon-polaron spin currents, which leads to the net 10-7 magnon-polaron contribution shown in Fig. 5(b). -8 The dipolar interaction is responsible for the anisotropy in 10 μ 0H = 2.6T the magnon dispersion in Eq. (9), which is reflected in the 10-9 magnetic field dependence of the heat and spin currents. In 1 2 3 4 5 6 8 Fig. 5(c),weplotζxx as function of the angle ϑT between (b) k (10 /m) magnetic field and transport direction for η = 100 and T = 10 K. The magnon-polaron contributions for magnetization FIG. 3. (a) Scattering times of magnons, TAphonons (λ = 1), and = parallel and perpendicular to the transport are plotted as lower (L)/upper (H) branch magnon-polarons in YIG for μ0H 1T the green dashed and blue solid curves, respectively. The (H zˆ) as a function of wave vector k xˆ for η = 100. (b) Same as anisotropy shifts the magnon-polaron peak positions, but does (a) but μ H⊥ = 2.64 T. 0 not substantially modify their amplitude. On these grounds, we proceed with computing other transport coefficients for the effective potential perceived by magnons, giving rise to an configuration H⊥∇T only. enhanced scattering time and hence larger currents. This can Figure 6(a) shows the magnon spin conductivity σxx as be confirmed by comparing the blue solid (η = 100) and the function of the magnetic field and temperature for different 3 black dash-dotted (η = 10 ) lines in Fig. 5(a), showing that values of η. Two peaks (dips) appear at H⊥ and H for η = 100 the magnitude of the peaks increases with increasing η. When (η = 0.01) at 10 and 50 K, while they disappear for η = 1. magnetic and nonmagnetic scattering potentials are the same, At very low temperatures, T = 1 K, the anomalies are not i.e., η = 1, the anomalies vanish as illustrated by the dashed visible anymore. The dependence of the spin conductivity on blue line in Fig. 5(a), and agrees with the results obtained in the temperature, on the angle between the magnetic field and the absence of MEC (triangles). temperature gradient, and on the scattering potentials ratio η The frequencies at which magnon and phonon dispersions is the same as reported for the spin Seebeck coefficient ζxx. are tangential for uncoupled transverse and longitudinal modes In Fig. 6(b), we plot the dependence of the magnon heat = = (m) are0.16THz(ˆ8 K)and 0.53 THz ( ˆ 26 K). Far below these conductivity κxx on the magnetic field and on the temperature temperatures, the magnon-polaron states are not populated, for different values of η. The only difference with respect to which explains the disappearance of the second anomaly and the coefficient ζxx is in the ratio between the amplitudes of the strongly reduced magnitude of the first one at 1 K in Fig. the two anomalies at T = 10 K, at which the magnon modes 4. In the opposite limit, the higher-energy anomaly becomes contributing to the low-field (H⊥) anomaly are thermally relatively stronger [see the solid curve at 50 K in Fig. 4]. The excited, in contrast to high field (H) modes. In ζxx,the overall decay of the spin Seebeck coefficient with increasing anomaly at H⊥ should therefore by better visible, as is indeed magnetic field is explained by the freeze-out caused by the the case. The magnon heat conductivity from Eq. (30) contains increasing magnon gap opened by the magnetic field [see the an additional factor in the integrand which is proportional to the inset of Fig. 5(a)]. energy of the magnon-polaron modes. The latter compensates This strong decrease has been observed in single YIG for the lower thermal occupation, which explains why the crystals [34,35], but it is suppressed in thinner samples anomaly at H is more pronounced in comparison with the or even enhanced at low temperatures [12]. The effect is spin Seebeck effect. tentatively ascribed to the paramagnetic GGG substrate that Perhaps surprisingly, the total heat conductivity κxx in becomes magnetically active a low temperatures [12] and is Fig. 7(a) displays only dips for η = 1 at the special fields
144420-6 MAGNON-POLARON TRANSPORT IN MAGNETIC INSULATORS PHYSICAL REVIEW B 95, 144420 (2017)
9 6 H⊥ ] ] η 8 0.6 H|| 5 = 100 -1 -1 1 H⊥ θ = π/2 J) ) (THz) ⋅ K)
π 0.01 ⋅ 7 s 4 ⋅ s 0.3 /(2 ⋅
ω TA 6 (m (m LA 3 46
22 0 50K 5 T = 10 K 0 2 4 6 k (108/m) 2 [10 [10 no MEC 4 H|| xx xx 1 ×
3 σ 10K ( 5)
ζ η = 10 1K (×50) 3 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 μ (a) μ (a) 0H (T) 0H (T) ]
-1 4 0.7
K) η = 100 ⋅ ) s
⋅ 3 0.6 50K
1 -1 (m 2 0.01 K
21 0.5 -1 1 Temperature increase [10 0.4
xx ×
Wm 10K ( 20) ζ 0 3 0.3 -1 Temperature increase 0.2 (10 η -2 = 100 (m) xx 0.1 × 3 1 κ 1K ( 10 ) -3 0.01 0
Magnon-polaron -4 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 μ (b) μ H (T) (b) 0H (T) 0 ]
-1 3 FIG. 6. (a) The magnetic field and temperature dependence K)
⋅ of the magnon spin conductivity σxx for different values of the s ⋅ H⊥∇ ϑ π T ( T = /2) ratio η between magnon and phonon impurity-scattering potentials. (m
21 H ∇ ϑ (b) The magnetic field and temperature dependence of the magnon 2 || T ( T = 0) (m) heat conductivity κxx for different values of the ratio η between [10
xx magnon and phonon impurity-scattering potentials.
ζ T = 10 K 1 η = 100
time, the thermal conductivity is suppressed, causing the dips 0 close to H⊥,.Forη 1, on the other hand, the magnon contribution to heat conductivity prevails, as is seen by the Magnon-polaron 0 1 2 3 4 5 6 7 8 9 10 strong magnetic field dependence of κxx (dotted blue line). μ mag ph (c) 0H (T) Since now |v | < |v |, the heat conductivity of the resulting magnon-polaron mode is lower than the purely magnonic FIG. 5. (a) Spin Seebeck coefficient ζxx of bulk YIG as a one. Again dips appear close to the “touching” magnetic function of magnetic field at T = 10 K. The black dash-dotted, blue fields. (m,exp) solid, blue dashed, blue dotted lines are computed for, respectively, Experimentally, the magnon heat conductivity κxx at 3 η = 10 , 100, 1, and 0.01. The triangles are obtained for zero MEC. a given temperature was referred to the difference between The inset shows the dispersions of uncoupled transverse (TA) and (m,exp) finite-field value κxx(H ) and κxx(∞), i.e., κxx (H ) = longitudinal (LA) acoustic phonons and the magnons shifted by H κxx(H ) − κxx(∞)[24]. The latter, κxx(∞), corresponds to and H⊥ magnetic fields. (b) The magnetic field and temperature the saturation value of the heat conductivity at high-field dependence of the magnon-polaron contribution for different values of the ratio η between magnon and phonon impurity-scattering limit, above which it becomes a constant function of the magnetic field, suggesting that the magnon contribution has potentials. (c) ζxx as function of magnetic field for H ⊥ ∇T (blue solid line) and H∇T (green dashed line) at T = 10 K for η = 100. been completely frozen out and only the phonon contribution (m) (m,exp) remains. In general, κxx and κxx differ in the presence (m,exp) of magnetoelasticity. The magnon heat conductivity κxx H⊥,. This can be explained as follows. For η 1, the phonon in Fig. 7(b), evaluated by subtracting the high-field limit for contribution to the heat conductivity is larger than the magnon T = 10 K, shows dips for both η = 0.01 and 100, in contrast (m) contribution. Except at the critical fields H⊥,, the magnetic to the magnon heat conductivity κxx in Fig. 6(b) with peaks field dependence of κxx is therefore very weak (solid blue for η = 100. The disagreement stems from κxx(∞), which line). When phonons mix with magnons with a short scattering is the (pure) phonon contribution to the heat conductivity at
144420-7 BENEDETTA FLEBUS et al. PHYSICAL REVIEW B 95, 144420 (2017) infinite magnetic fields, but is not the same as the phonon 3 heat conductivity at ambient magnetic fields when the MEC ) T = 10 K is significant. In the latter case, the phonon heat conductivity -1 itself depends on the magnetic field and displays anomalies at K η = 100 -1 (m,exp) = (m) H⊥,; hence κxx κxx . (m,exp) 2
Nonetheless, κxx can be useful since its fine structure Wm 3 η = 1 (×40) contains information about the ratio between the magnon- | mag| impurity and phonon-impurity scattering potentials v and (10 ph η = 0.01 (×40) | | ∞ = xx v .Also,κxx( )forη 100 is much larger than for 1 η = 0.01, and its value gives additional information about κ the relative acoustic and magnetic quality of the sample. For example, the results reported by Ref. [24] can be interpreted, 0 1 2 3 4 5 6 7 8 9 10 μ within our theory, as suggesting a much higher acoustic than (a) 0H (T) magnetic quality of the samples, i.e., η 1. The authors, ) 40
however, have not investigated the magnetic field dependence -1 η of the heat conductivity but rather the temperature dependence, K = 1 (0.01)
-1 20 which is beyond the scope of this work. It is worth to mention that already the work of Ref. [36] suggests that impurity 0 scattering plays a key role in determining the magnetic field η = 100 ) (Wm
dependence of the heat conductivity. ∞ -20 The appearance of the anomalies can be understood ( xx analytically with few straightforward simplifications. Let us κ -40 T = 10 K consider a one-dimensional system along xˆ and H = (0,0,H ).
(H)- -60 According to Eq. (19) only the TA phonons couple to the xx magnons leading to the magnon-polaron dispersion κ -80 0 1 2 3 4 5 6 7 8 9 10 + ± − 2 + 2 (b) μ ωk ω1k (ωk ω1k) 4˜ωk 0H (T) = , (33) 1,2k 2 FIG. 7. (a) The magnetic field dependence of the heat con- = 1/2 = 2 2 whereω ˜ k (S⊥k) and S⊥ (nB⊥) (γh¯ /4Ms ρc¯ ⊥). The ductivity κxx at T = 10 K for different scattering parameters η. magnon-polaron spin amplitudes W1,2k are (b) Magnetic field dependence of the heat conductivity difference κxx(H ) − κxx(∞) simulating the experimental procedure [24]at − + − 2 + 2 T = 10 K. ωk ω1k (ωk ω1k) 4˜ωk W1k = , (34) − 2 + 2 2 (ωk ω1k) 4˜ωk where = − + kδH˜ 2 + η and W2k 1 W1k. Disregarding the small dipolar interac- = 4[1 ( ) ](1 ) = y0(δH) tions (Ms H⊥) the uncoupled dispersions touch at μ0H⊥ 1 + η[2 + 4(kδH˜ )2 + η] 2 c /4D γ . We focus on the contribution of the k⊥ mode ⊥ ex and (with k⊥ = c⊥/2Dex) to the transport coefficients (30)close 2 to the touching field and expand in δH = H − H⊥.As 2[1 + 2(kδH˜ ) + η] y1(δH) = . in Fig. 2(b),fork = k⊥ and δH H⊥, the energies and 1 + η[2 + 4(kδH˜ )2 + η] group velocities of the upper and lower magnon-polarons are | The indices n and m correspond to those in Eq. (30). Both approximately the same, i.e., 1k⊥ 2k⊥ and ∂k1 k=k⊥ = | y0(δH) and y1(δH) have a single extremum at H H⊥, i.e., ∂k2 k=k⊥ . Equation (34) then reads | = | = ˜ y0(δH) δH=0 y1(δH) δH=0 0, (38) = 1 + kδH W1k⊥ 1 , (35) 2 1 + (kδH˜ )2 | ∝ − 2 y0(δH) δH=0 (1 η) , (39) 1/2 with k˜ = μ0γ/(4S⊥k⊥) . The scattering times (31) can be y (δH)| = ∝ (1 − η). (40) approximated as 1 δH 0 | Equations (38) and (39) prove that y0 has a minimum at ∂k1,2k k=k⊥ 1 τ ∼ . (36) H = H⊥ for η = 1, while for η = 1 it is a constant. This 1,2k⊥ | |2 − + vph (1 W1,2k⊥ ) ηW1,2k⊥ explains our numerical results for the heat conductivity κxx, Hence which is unstructured for η = 1 and always display dips for both η<1 and η>1[seeFig.7(a)]. According to Eqs. (38) βh¯ 1k β e = Lnm ∼ (∂ )3 (¯h )n and (40) the function y1 is also stationary at H H⊥,butit xx 2 2 k 1k βh¯ 2 1k = L |vph| (e 1k − 1) k k⊥, has a minimum only for η<1, while an inflection point for H = H⊥ η = 1, and a maximum otherwise. The resulting dependence × ym(δH), (37) on η of Eq. (37) explains the spin Seebeck coefficient ζxx,the
144420-8 MAGNON-POLARON TRANSPORT IN MAGNETIC INSULATORS PHYSICAL REVIEW B 95, 144420 (2017)
(m) spin conductivity σxx and magnon heat conductivity κxx ,in 50 Figs. 4, 6(a), and 6(b), respectively. As we have discussed in 45 η detail in the reporting of the numerical results, the anomalies = 100 40 1 can be understood physically in terms of the scattering time 0.01 of the magnon-polaron. This scattering time is the sum of 35 m) magnonic and phononic scattering times, so, depending on the μ 30 ( value of η, the spin transport is enhanced (η>1) or suppressed n 1K (η<1) close to the touching point. λ 25 20 10K B. Spin diffusion length 15 50K Integrating the spin-projection of Eq. (24) over momentum 10 leads to the spin conservation equation: 0 1 2 3 4 5 6 7 8 9 10 μ 0H (T) n˙s + ∇ · js =−gμμ, (41) where FIG. 8. Magnetic field and temperature dependence of the magnon diffusion length λ for different values of the scattering n d3k parameter η. n = f (r), (42) s (2π)3 ik i is the total magnon density (in units ofh ¯), and recall that the main contribution to the magnon spin con- ductivity σ arises from magnonlike branches. At relatively xx d3k 1 eβh¯ ik high temperature, the magnonlike branches are sufficiently gμ = β Wik (43) 3 nc βh¯ ik − 2 populated to overcome the phonon contribution to the magnon (2π) τik (e 1) i spin conductivity at all η. Indeed, Fig. 6(a) shows that, at is the magnon relaxation rate, and we have introduced relatively high temperatures, the ratio σxx(H1,η)/σxx(H2,η) nc the relaxation time τik . Elastic magnon-impurity scattering hardly depends on η. On the other hand, when the temperature processes discussed in the previous sections do not contribute decreases below the magnon energy, the contribution of the nc to τik . However, we parametrize the spin not-conserving magnon-like branches are quickly frozen out by a magnetic processes as field. The magnitude of η then becomes very relevant. On the other hand, while the right-hand side of Eq. (46) depends 1 = on temperature, it is not affected by η.Forη<1, the phonon nc 2αki , (44) τik mobility is smaller than the magnon one and hence the phonons in terms of the dimensionless Gilbert damping constant α.In are short circuited by the magnons. For η>1, the phonons the nonequilibrium steady-state, Eq. (41) becomes prevail, leading to a higher ratio σxx(H1,η)/σxx(H2,η) because the phonon dispersion is not affected by the magnetic field. 1 ∇2μ = μ, (45) When η 1, the condition (46) is therefore satisfied. While λn in this regime the spin current is very small, it is perhaps an interesting limit for studying fluctuation and shot noise in the ≡ in terms of the magnon diffusion length λn σxx/gμ that is spin current [9]. plotted in Fig. 8. At 10 K and 50 K, the spin diffusion length decreases monotonously with the magnetic field for η = 1, in agreement with observations at room temperature [37]. C. Comparison with experiments For η = 100 (η = 0.01) the spin diffusion length displays two The spin Seebeck effect was measured in Pt|YIG|GGG peaks (dips) at the critical fields H⊥ and H, which become structures in the longitudinal configuration, i.e., by applying a more pronounced when lowering the temperature. At T = 1K, temperature difference normal to the interfaces (x direction) only the peak (dip) at H⊥ is visible for η = 100 (η = 0.01). and subjecting the sample to a magnetic field H zˆ [12]. For η = 1, the spin diffusion length monotonically decreases The thermal bias induces a spin current into the Pt layer that with increasing magnetic field. The curve for η = 0.01 behaves by the inverse spin Hall effect (ISHE) leads to the detected similarly except for the dip at H = H⊥. On the other hand, transverse voltage V over the contact, see Fig. 1. The bottom for η = 100, the spin diffusion length behaves very differently of the GGG substrate and the top of the Pt layer are in showing strong enhancement at both low and high magnetic contact with heat reservoirs at temperature TL and TH , respec- fields. tively. Disregarding phonon (Kapitza) interface resistances, This strong increase of the diffusion length (for constant the phonon temperature gradient is ∇T = (TH − TL)/L, with Gilbert damping) happens when L being the thickness of the stack, and average temperature T = (T + T )/2. As discussed, we assume that the magnon σxx(H1,η) gμ(H1) H L > , (46) and phonon temperatures are the same and disregard the σ (H ,η) g (H ) xx 2 μ 2 interface mixing conductance. The measured voltage is then where H1,2 are two given values of the applied magnetic directly proportional to the bulk spin Seebeck coefficient. field, with H1 >H2. To understand the dependence of the In the experimental temperature range of 3.5–50 K the ratio σxx(H1,η)/σxx(H2,η)onη and on the temperature, we thermal magnon, mag, and phonon, ph,μ, wavelengths are
144420-9 BENEDETTA FLEBUS et al. PHYSICAL REVIEW B 95, 144420 (2017) of the order of 1–10 nm. Even if the magnon and phonon confirms that elastic magnon(phonon) impurity scattering is thermal mean free paths have been estimated to be of the the main relaxation channel that limits the low temperature order of ∼100 μm at very low temperatures [24], here transport in YIG. Our theory contains one adjustable parameter we assume that the transport in the YIG film of thickness that is fitted to the large set of experimental data, consistently L 4 μm can be treated semiclassically. Note that scattering finding a much better acoustic than magnetic quality of at the interfaces can make the transport diffusive even when the samples. The spin Seebeck effect is therefore a unique the formal conditions for diffusive transport are not satisfied. analytical instrument not only of magnetic, but also mechanical The bulk spin Seebeck coefficient is then well-described by material properties. The predicted effects of magnon-polaron Eq. (30) and proportional to the observed voltage V . These effects on magnonic spin and heat conductivity call for further assumptions are encouraged by the good agreement for the experimental confirmation. observed and calculated peak structures at H⊥ and H with We believe that the presented results open new avenues a single fitting parameter η = 100 [12]. We may therefore in spin caloritronics. We focused here on the low energy conclude that the magnons are more strongly scattered than magnon dispersion of cubic YIG, which is well represented the phonons. This points towards a relatively high crystalline by the magnetostatic exchange waves of a homogeneous quality of the sample and towards the presence of magnetic ferromagnet [21]. However, the theoretical framework can impurities. Experimentally [38], it was indeed found that that be easily extended to include anisotropies as well as ferri- the magnon and phonon scattering rate could be relatively or antiferromagnetic order. The magnetoelastic coupling in tuned, which provides a possible route to experimentally YIG is relatively small and the conspicuous magnon-polaron investigate our predictions. effects can be destroyed easily. However, in materials with large magnon-phonon couplings these effects should survive V. CONCLUSION AND OUTLOOK in the presence of larger magnetization broadening as well as higher temperatures. We have established a framework which captures the effects of the magnetoelastic interaction on the transport properties of ACKNOWLEDGMENTS magnetic insulators. In particular, we show that the magnon- phonon coupling gives rise to peaklike or diplike structures in This work was supported by the Stichting voor Fun- the field dependence of the spin and heat transport coefficients, damenteel Onderzoek der Materie (FOM), the European and of the spin diffusion length. Research Council (ERC), the DFG Priority Programme Our numerical evaluation reproduces the peaks in the 1538 “Spin-Caloric Transport”, Grant-in-Aid for Scientific observed low temperature longitudinal spin Seebeck voltages Research on Innovative Area “Nano Spin Conversion Science” of YIG | Pt layers as a function of magnetic field. We (Grants No. JP26103005 and No. JP26103006), Grant-in-Aid quantitatively explain the temperature-dependent behavior of for Scientific Research (A) (Grants No. JP25247056 and No. these anomalies in terms of hybrid magnon-phonon excitations JP15H02012), and (S) (Grant No. JP25220910) from JSPS (“magnon-polarons”). The peaks occur at magnetic fields KAKENHI, Japan, PRESTO “Phase Interfaces for Highly and wave numbers at which the phonon dispersion curves Efficient Energy Utilization” and ERATO “Spin Quantum are tangents to the magnon dispersion, i.e., when magnon Rectification Project” from JST, Japan, NEC Corporation, and and phonon energies as well as group velocities become the The Noguchi Institute. It is part of the D-ITP consortium, same. Under these conditions the effects of the magnetoelastic a program of the Netherlands Organization for Scientific interaction are maximized. The computed angle dependence Research (NWO) that is funded by the Dutch Ministry of shows a robustness of the anomalies with respect to rotations Education, Culture and Science (OCW). T.K. is supported by of the magnetization relative to the temperature gradient. The JSPS through a research fellowship for young scientists (Grant agreement between the theory and the experimental results No. 15J08026).
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