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Angular momentum conservation and phonon spin in magnetic insulators

Rückriegel, Andreas; Streib, Simon; Bauer, Gerrit E.W.; Duine, Rembert A. DOI 10.1103/PhysRevB.101.104402 Publication date 2020 Document Version Final published version Published in Physical Review B

Citation (APA) Rückriegel, A., Streib, S., Bauer, G. E. W., & Duine, R. A. (2020). Angular momentum conservation and phonon spin in magnetic insulators. Physical Review B, 101(10), [104402]. https://doi.org/10.1103/PhysRevB.101.104402 Important note To cite this publication, please use the final published version (if applicable). Please check the document version above.

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Angular momentum conservation and phonon spin in magnetic insulators

Andreas Rückriegel,1 Simon Streib ,2 Gerrit E. W. Bauer ,2,3 and Rembert A. Duine1,4,5 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 & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan 4Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 5Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

(Received 23 December 2019; accepted 17 February 2020; published 2 March 2020)

We develop a microscopic theory of spin-lattice interactions in magnetic insulators, separating rigid-body rotations and the internal angular momentum, or spin, of the phonons, while conserving the total angular momentum. In the low-energy limit, the microscopic couplings are mapped onto experimentally accessible magnetoelastic constants. We show that the transient phonon spin contribution of the excited system can dominate over the magnon spin, leading to nontrivial Einstein-de Haas physics.

DOI: 10.1103/PhysRevB.101.104402

I. INTRODUCTION existence of a sink for angular momentum. The magnetocrys- talline anisotropy breaks the spin rotational invariance by The discovery of the spin Seebeck effect led to renewed in- imposing a preferred magnetization direction relative to the terest in spin-lattice interactions in magnetic insulators [1,2], crystal lattice, while the lattice dynamics itself is described i.e., the spin current generation by a temperature gradient, in terms of spinless phonons. The resulting loss of angular which is strongly affected by lattice vibrations [3–7]. The momentum conservation is justified when the spin-phonon spin-lattice interaction is also responsible for the dynamics of Hamiltonian does not possess rotational invariance in the the angular momentum transfer between the magnetic order first place [11], e.g., by excluding rigid-body dynamics of and the underlying crystal lattice that supports both rigid-body the lattice and/or by boundary conditions that break rota- dynamics and lattice vibrations, i.e., phonons. In the Einstein- tional invariance. In the absence of such boundary condi- de Haas [8] and Barnett effects [9], a change of magnetization tions, the angular momentum must be conserved in all spin- induces a global rotation and vice versa. While both effects lattice interactions. Phenomenological theories that address have been discovered more than a century ago, their dynamics this issue [10,11,18,29–33] incorporate infinitesimal lattice and the underlying microscopic mechanisms are still under rotations due to phonons but do not allow for global rigid- debate [10–13]. body dynamics and therefore cannot describe the physics of In 1962 Vonsovskii and Svirskii [14,15] suggested that the Einstein-de Haas and Barnett effects. Conversely, theories circularly polarized transverse phonons can carry angular mo- addressing specifically Einstein-de Haas and Barnett effects mentum, analogous to the spin of circularly polarized photons. usually disregard effects of phonons beyond magnetization This prediction was confirmed recently by Holanda et al. [16] damping [34–37]. by Brillouin light scattering on magnetic films in magnetic Here we develop a theory of the coupled spin-lattice dy- field gradients that exposed spin-coherent magnon-phonon namics for sufficiently large but finite particles of a magnetic conversion. Ultrafast demagnetization experiments can be insulator allowing for global rotations. We proceed from explained only by the transfer of spin from the magnetic a microscopic Hamiltonian that conserves the total angular system to the lattice on subpicosecond time scales in the momentum. We carefully separate rigid-body dynamics and form of transverse phonons [12]. Theories address the phonon phonons, which allows us to define a phonon spin and to spin induced by Raman spin-phonon interactions [17], by obtain the mechanical torques exerted by the magnetic order the relaxation of magnetic impurities [11,18], by temperature on the rigid-body and the phonon degrees of freedom (and gradients in magnets with broken inversion symmetry [19], vice versa). The theory of magnetoelasticity is recovered as and phonon spin pumping into nonmagnetic contacts by fer- the low-energy limit of our microscopic model in the body- romagnetic resonance dynamics [20]. The phonon Zeeman fixed frame and thereby reconciled with angular momentum effect has also been considered [21]. The quantum dynamics conservation. We compute the nonequilibrium spin dynamics of magnetic rigid rotors has been investigated recently in of a bulk ferromagnet subject to heating and spin pumping the context of levitated nanoparticles [22–25]. Very recently in linear response and find that in nonequilibrium the phonons ferromagnetic resonance experiments have shown coherent carry finite spin, viz. a momentum imbalance between the two magnon-phonon coupling over millimeter distances [26]. circularly polarized transverse phonon modes. We also show Most theories of spin-lattice interactions do not con- that the phonon spin can have nontrivial effects on the rigid- serve angular momentum [11,27,28], thereby assuming the body rotation; in particular, it can lead to an experimentally

2469-9950/2020/101(10)/104402(14) 104402-1 ©2020 American Physical Society RÜCKRIEGEL, STREIB, BAUER, AND DUINE PHYSICAL REVIEW B 101, 104402 (2020) observable, transient change in the sense of rotation during (see Sec. II B). The interactions J(rij) and K(rij) in principle equilibration. include dipolar interactions. A Hamiltonian of the form of The rest of the paper is organized as follows: The spin- Eq. (2) has been used recently to compute the relaxation of lattice Hamiltonian and the decoupling of rigid-body dynam- a classical spin system [38]. ics and phonons is presented in Sec. II. The spin transfer in a Ultimately, the origin of the Hamiltonian (2) lies in the bulk ferromagnet is studied in Sec. III within linear response spin-orbit coupling of the electrons: The anisotropic contri- theory. Section IV contains a discussion of our results and bution (2b) arises from the dynamical crystal field that affects concluding remarks. Expressions for the total angular momen- the electronic orbitals and thereby the spin states, whereas the tum operator in terms of the Euler angles of the rigid-body position dependence of the exchange contribution (2a) is due rotation are presented in Appendix A, while Appendix B to the dependence of the electronic hopping integrals on the details the phonon commutation relations in finite systems. interatomic distances. For ultrafast processes that occur on Finally, Appendix C addresses the relaxation rates computed the timescales of the orbital motion, a description of these by linear response. intermediate, electronic stages of the spin-lattice coupling might be necessary; however, this is beyond the scope of this II. MICROSCOPIC SPIN-LATTICE HAMILTONIAN work. We address here a finite magnetic insulator of N atoms A. Rigid-body rotations and phonon spin (or clusters of atoms) with masses mi and spin operators Si governed by the Hamiltonian The Hamiltonian (1) commutes with and thereby conserves the total angular momentum, i.e., the sum of intrinsic elec- N 2 pi tron spin and mechanical angular momentum. In a solid, the H = + V ({ri}) + Vext ({ri}) + HS, (1) 2mi mechanical angular momentum arises from the rotation of i=1 the rigid lattice and the internal phonon dynamics. We may − where ri and pi are the canonical position and momentum decouple the 6 rigid-body and the 3N 6 phonon degrees of operators of the ith atom, and the potential V ({ri})isassumed freedom by the following transformation: to be (Euclidean) invariant to translations and rotations of the 3N−6 whole body. V ({r }) accounts for external mechanical forces f (Ri ) ext i r = R + R(φ,θ,χ) R + √n q , (3) acting on the body. Because of translational and rotational i CM i m n n=1 i invariance, the spin Hamiltonian HS depends only on rij = R φ,θ,χ = ri − r j and must be a scalar under simultaneous rotations of where RCM is the center-of-mass position, ( ) R φ R θ R χ lattice and spin degrees of freedom. Since Si are pseudovec- z( ) y( ) x ( ) is a three-dimensional rotation φ θ χ R α tors and rij are vectors, the spin-lattice interaction depends parametrized by the Euler angles , , and ( μ( ) denoting a rotation by an angle α around an axis eˆμ), R is only on r = r2 , S · S and even powers of S · r .1 Con- i ij ij i j i ij the body-fixed equilibrium position of the ith particle, and the sidering only pair interactions between two spins, we arrive qn are the normal coordinates of the lattice, i.e., the phonons, at HS = HEx + HA + HZ, with exchange (Ex), anisotropy (A), with eigenfunctions f n(Ri ) that diagonalize the energy to and Zeeman (Z) contributions second order in qn: N − 1 1 3N 6 HEx =− J(rij)Si · S j, (2a) { } = { } + ω2 2 + O 3 . 2 V ( ri ) V ( Ri ) nqn (q ) (4) , = 2 i j 1 n=1 i = j In molecular physics this decoupling of rotations and vibra- N 1 tions is referred to as Eckart convention [39–41]. Neglecting H =− K(r )(S · r )(S · r ), (2b) A 2 ij i ij j ij surface effects of the external forces on the phonons, we also i, j=1 have Vext ({ri}) ≈ Vext (RCM,φ,θ,φ). N Since we describe the phonons within a rotating reference HZ =−γ B · Si. (2c) frame, it is advantageous to also rotate the spin operators i=1 globally by the unitary transformation: Here J(rij) is an isotropic and K(rij) an anisotropic ex- − i φ N Sz − i θ N Sy − i χ N Sx U (φ,θ,χ) = e h¯ j=1 j e h¯ j=1 j e h¯ j=1 j , (5) change interaction, B is the external magnetic field, and γ = gμ /h † φ,θ,χ φ,θ,χ = R φ,θ,χ B ¯ is the (modulus of the) gyromagnetic ratio, defined in so that U ( )SiU ( ) ( )Si. As a result, − f R terms of the g factor and Bohr magneton μB, and Planck’s con- + 3N 6 √n ( i ) (3) and (5) transform ri into Ri n=1 qn and B into stanth ¯. H encodes the interaction of the spins with the crystal mi A RT (φ,θ,χ)B in the spin Hamiltonian (2) and change the lattice or crystalline anisotropy, which in the long-wavelength lattice kinetic energy to [40–42] limit reduces to the conventional crystal field Hamiltonian − in terms of anisotropy and magnetoelastic constants [27,28] N p2 P2 1 1 3N 6 i → CM + · I · + p2 + O(I−2 ). (6) 2m 2M 2 2 n i=1 i n=1

1 Broken inversion symmetry would allow Dzyaloshinskii-Moriya Here, PCM =−ih¯∂/∂RCM and pn =−ih¯∂/∂qn are the interactions of the form (ril × r jl ) · (Si × S j ). momentum operators of center-of-mass translation and

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(a) (b) Care has to be exercised when interpreting the phonon operators (9) and (8) in a finite system. The exclusion of the 6 degrees of freedom of the rigid-body dynamics breaks the canonical commutation relations of the phonon position, momentum, and angular momentum operators. Corrections of O(I−1)[42] are important for nanoscale systems. For details of the derivation of the kinetic energy (6) and the finite size corrections, we refer to Refs. [40–42]. In the following, we focus on systems large enough, i.e., N  1asshownin Appendix B, to disregard finite size corrections to L’s thermal FIG. 1. Illustration of the different kinds of angular momenta that or quantum fluctuations and treat the phonon operators (9) and are relevant for the angular momentum balance (7) of a magnetic (8) canonically. insulator: (a) Rigid rotation with angular velocity of a cylinder The equations of motion of the relevant angular momentum around its axis with moment of inertia I and total spin S, (b) sketch operators are now of a phonon mode with angular momentum L, showing the motion of four different volume elements without a global rotation as in (a). 1 T ∂t S + ( × S − S × ) = S × R (φ,θ,χ)γ B The total angular momentum is J = I · + L + S. 2 N ∂H + (R + u ) × S , (10a) = N i i ∂ phonons, respectively, M i=1 mi is the total mass, = ui αβ = i 1 and I is the equilibrium moment of inertia tensor I N αβ 2 α β N m (δ R − R R ). The latter is defined in the frame at- 1 ∂V ∂HS i=1 i i i i ∂ L + ( × L − L × ) =− u × + , tached to and rotating with the body, referred to as “rotating,” t i ∂ ∂ 2 = ui ui “molecular,” or “body-fixed” frame. O(I−2 ) denotes correc- i 1 tion terms originating from instantaneous phonon corrections (10b) to the moment of inertia and quantum-mechanical commuta- 1 ∂ J + ( × J − J × ) = S × RT (φ,θ,χ)γ B + T , tors generated by the nonlinear coordinate transformation (3) t 2 ext [40–42]. Finally, (10c) = −1 · − − I (J L S) (7) where T ext =−i[J,Vext]/h¯ is the external mechanical torque that acts on the magnet in the body-fixed frame. Thus, the is the vector of the angular velocity of the rigid rotation in the angular momentum I · of the rigid rotation satisfies body-fixed reference frame. Here the total and phonon angular momentum operators J and L, and the total spin operator ∂ · + 1 × · − · × N t (I ) [ (I ) (I ) ] S = = Si are also in the body-fixed coordinate system. The 2 i 1 three angular momenta in a magnetic insulator are sketched in N ∂HS ∂V Fig. 1. The total angular momentum in the laboratory frame = T ext − Ri × − ui × . (11) ∂ui ∂ui Jlab = R(φ,θ,χ)J obeys the standard angular momentum i=1 algebra [Jx , Jy ] = ihJ¯ z and is conserved in the absence of lab lab lab Equations (10) and (11) constitute the microscopic equations external torques, i.e., for B = 0 and [J,V ] = 0. The total ext for the Einstein-de Haas [8] and Barnett [9] effects in mag- angular momentum in the body-fixed frame depends only netic insulators. The left-hand sides are covariant derivatives on the Euler angles, i.e., it is carried solely by the rigid- that account for the change in angular momentum in the body- body rotation [40] and the angular momentum commutation fixed frame [41], whereas the right-hand sides are the internal relations are anomalous (with negative sign) [Jx, Jy] =−ihJ¯ z mechanical (V ), spin-lattice (H ), and external magnetic (B) [40–43]. Explicit expressions for the total angular momentum S and mechanical (T ) torques. The spins exert a torque on operators in the body-fixed and laboratory frame are relegated ext the lattice by driving the rigid-body rotation and exciting to Appendix A. The “phonon spin” is the phonon angular phonons. The torques on the right-hand sides depend on the momentum in the body-fixed frame: microscopic phonon and spin degrees of freedom that act as N N thermal baths and thereby break time-reversal symmetry. We L = l i = ui × πi, (8) disregard radiative damping, so energy is conserved and en- i=1 i=1 tropy cannot decrease, in contrast to conventional approaches to the Einstein-de Haas effect that demand angular momentum π where ui and i are, respectively, the displacement and linear conservation only and do not include thermal baths. Hence, momentum operators: energy is not conserved in these approaches and entropy can − decrease. 1 3N 6 u = √ f (R )q , (9a) i m n i n i n=1 B. Derivation of the phenomenological theory √ 3N−6 of magnetoelasticity π = . i mi f n(Ri )pn (9b) Our general model for spin-lattice interactions can n=1 be parametrized by a small number of magnetic and

104402-3 RÜCKRIEGEL, STREIB, BAUER, AND DUINE PHYSICAL REVIEW B 101, 104402 (2020) magnetoelastic constants at low energies. In the long wave- length continuum limit Si → S(r)/n and ui → u(r), where n is the number density of magnetic moments. To lowest order in the gradients of spin and phonon operators ⎡ ⎤ n 3 ⎣Jμν αβ ⎦ H ≈ d r + Aμναβ (r) + ... Ex 2 2 h¯ s μν 2 αβ ∂ ∂ × S(r) · S(r), ∂ μ ∂ ν (12a) r r ⎡ ⎤ n 3 ⎣ Kμν αβ ⎦ H ≈ d r − + Bμναβ (r) + ... A 2 2 h¯ s μν 2 αβ

× S˜μ(r)S˜ν (r), (12b) where s = Sn is the saturation spin density in units ofh ¯,   1 ∂uβ (r) ∂uα (r) ∂u(r) ∂u(r) αβ (r) = + + · (13) 2 ∂rα ∂rβ ∂rα ∂rβ is the elastic strain tensor, FIG. 2. The system under consideration in Sec. III: A macro-   scopic ferromagnet with moment of inertia I and volume V at μ μ ∂u(r) temperature T . The total spin S of the ferromagnet is aligned parallel S˜ (r) = S (r) + S(r) · (14) ∂rμ to an external magnetic field B. In addition, the ferromagnet may rotate with angular velocity and supports a phonon spin L.The are the projections of the spin density on the elastically de- system is assumed to be at rest with = 0 initially. Finite and formed anisotropy axes, and the ellipses stand for higher pow- L are induced by exciting the system into a nonequilibrium state at ers of the strain tensor. Exchange, anisotropy, and magnetoe- time t = 0, e.g., by heating the phonons or by pumping the magnons lastic constants can be expressed as moments of the isotropic with an rf field. In this case both and L are parallel to the total spin (J) and anisotropic (K) exchange interactions and their spatial S because of the conservation of total angular momentum. derivatives J(R) = ∂J(R)/∂R and K(R) = ∂K(R)/∂R: 2 2 h¯ s μ ν B and B⊥ are known for many magnets [27]. The Jμν = J(R )R R , (15a) 2n2 i i i exchange-induced magnetoelastic constant can be estimated i 3  ≈ = ∂ /∂ as A 2 mSJs [7], where m ln TC ln V is the 2 2 = h¯ s μ ν , magnetic Grüneisen parameter that quantifies the change of Kμν K(Ri )Ri Ri (15b) n2 Curie temperature TC with the volume V . i 2 2  h¯ s J (Ri ) μ ν α β III. THERMAL SPIN TRANSFER Aμναβ = R R R R (15c) 4n2 R i i i i i i In the remainder of this paper, we focus on a particular 2 2  application of the general theory, viz. the angular momentum =−h¯ s K (Ri ) μ ν α . Bμναβ R Ri Ri Ri (15d) transfer by thermal fluctuations in the bulk of a macroscopic, 2n2 R i i i externally excited, levitated ferromagnetic particle that does The continuum limit (12) agrees with the standard, phe- not rotate ( =0) initially. We assume a simple cubic lattice nomenological theory of magnetoelasticity [27]. Equation at low temperatures. The average magnetic order parameter, (12b) includes the spin-lattice coupling by rotational strains i.e., the total spin S, is aligned to an external magnetic field μ [11] via the spin density projections S˜ (r). B = Beˆz. For convenience we chose an axially symmetric The exchange, anisotropy, and magnetoelastic constants setup as sketched in Fig. 2. Local spin fluctuations are de- (15) reflect the microscopic crystal symmetries. For a scribed via the leading order Holstein-Primakoff transforma- simple cubic lattice with lattice constant a = n−1/3, tion [44] √ and nearest-neighbor isotropic as well as next-nearest- + = − † = + O / , Si (Si ) h¯ 2S[bi (1 S)] (16a) neighbor anisotropic exchange we find Jμν = SJsδμν = δ = 2 2 z = − † , and Kμν K μν , with spin stiffness Js h¯√SJ(a)a Si h¯(S bi bi ) (16b) = 2 2 + 2 and anisotropy constant K 2¯h S [K(a) 4K( 2a)]a . † where bi (bi ) is the magnon creation (annihilation) operator The latter may be disregarded because it only adds a , constant to the Hamiltonian. The magnetoelastic coupling on site i which satisfies the Boson commutation relations [b , b†] = δ . constants become Aμναβ = Aδμν δναδαβ, and Bμναβ = i j ij 3 1 In a macroscopic magnet the time scales between the (B − B⊥)δμν δναδαβ + B⊥(δμνδαβ + δμαδνβ + δμβ δνα), 2 2 √ √ rigid-body rotation and the internal magnon and phonon h¯2S2  2 2   3  =  =− + with A 2√J (a), B √ h¯ S [K (a) 2K ( 2a)]a , dynamics are decoupled: For a system with volume V ,the 2 2  3 5/3 and B⊥ =−2 2¯h S K ( 2a)a . The anisotropy parameters moment of inertia I ∼ V , whereas the angular momentum

104402-4 ANGULAR MOMENTUM CONSERVATION AND PHONON SPIN … PHYSICAL REVIEW B 101, 104402 (2020) operators J, L, and S are extensive quantities, proportional a circular basis λ =±for the transverse phonons [11,15,45] to V . According to Eq. (7) the angular velocity scales and express the momentum k in spherical coordinates, so that −2/3 as V . On the other hand, the lowest phonon frequency 1 −1/3 = √ θ φ ∓ φ ωmin ∼ V , while the magnon gap is controlled by external eˆk± [eˆx (cos k cos k i sin k ) magnetic and internal anisotropy fields and is typically of 2 the order of 10 GHz independent of V . For sufficiently large + eˆy(cos θk sin φk ± i cos φk ) − eˆz sin θk], (19a) systems and weak driving, inertial forces of the rigid-body eˆk = i[eˆx sin θk cos φk + eˆy sin θk sin φk + eˆz cos θk] rotation therefore affect the dynamics of both magnons and k phonons only negligibly and can be disregarded. By the same = i . (19b) 1 · · ∼ 1/3 k argument, the energy 2 I V of the rigid-body rotation is small compared to the total magnon and phonon In this basis the phonon spin (8) is diagonal [11,15,45]: energies ∼V . Energy is then (almost) exclusively equilibrated k † † by spin-phonon interactions, under the constraint of angular L =−h¯ (a a + − a a −), (20) k k+ k k− k momentum conservation that includes the rigid-body rotation. k For example, consider the change in energy of the magnet where we dropped terms that have vanishing expectation at rest when a single magnon with frequency is removed values for noninteracting phonons. Analogous to photons, cir- from the system, which increases the spin by S = h¯. If this cularly polarized phonons with λ =±carry one spin quantum angular momentum is fully transferred to the rigid rotation of ∓h¯ parallel to their wave vector that is carried exclusively a sphere with scalar moment of inertia I, L =−h¯ = Iz. R by transverse phonons. Mentink et al. [13] report that only For a macroscopic magnet the change of rotational energy longitudinal phonons contribute to the electron-phonon spin E = h¯2/2I is negligible compared to the magnetic energy R transfer. This is not a contradiction, because they define change Em =−h¯ , since the typical magnon frequencies are − phonon angular momentum different from Eq. (8) as adhered in the GHz-THz range, whereas both z and I 1 are small by to in most papers [11,15,45]. On the other hand, that definition some power of the inverse volume. Consequently, the energy appears similar to the field (or pseudo) angular momentum of the magnon cannot be transferred completely to the rigid introduced independently by Nakane and Kohno [18]. rotation, since both energy and angular momentum cannot be The leading one-phonon/one- and two-magnon contribu- conserved simultaneously. The Einstein-de Haas effect can tions to the magnetoelastic Hamiltonian (12) read in momen- therefore not exist without an intermediate bath, which in tum space magnetic insulators can only be the lattice vibrations. At temperatures sufficiently below the Curie and Debye =  + ∗ † · Hmp ( kbk −kb−k ) u−k temperatures and weak external excitation, only the long- k wavelength modes are occupied and Eq. (12) is appropriate. 1 † At not too low temperatures we may also disregard magne- + √  ·  ˆ  U k,k uk−k bkbk N  todipolar interactions [7]. We assume again that the magnet kk is sufficiently large that surface effects are small and the 1 1 ∗ † † eigenmodes of the system may be approximated by plane + V ,  · u− −  b b  + V  · u +  b b  , (21) − / − · k k k k k k k,k k k k k = 1 2 ik Ri 2 2 waves. Then the Fourier transform bi N k e bk leads to the magnetic Hamiltonian: with interaction vertices † iB⊥ z x y Hm = h¯ kb bk, (17) k =−√ [(eˆx − ieˆy )k + eˆz(k − ik )], (22a) k S k 2 iB x y z 2 U k+q,k = (eˆxq + eˆyq − 2eˆzq ) where k = γ B + Jsk /h¯ is the magnon frequency dispersion S b† b relation and k( k ) are creation (annihilation) operators of a 2iA α α α α magnon with wave vector k. + eˆα (k + q )k q , (22b) S Analogously, the finite size of a sufficiently large sys- α 1/3 tem only affects phonons with wavelengths O(V ) and iB B⊥ =− x − y − y + x . = V k+q,−k (eˆxq eˆyq ) (eˆxq eˆyq ) a small density of states. We may then expand ui S S −1/2 −ik·Ri N k e uk, with (22c)    The first line of the magnetoelastic Hamiltonian (21) h¯ † u = eˆ λ a λ + a . (18) describes the hybridization of magnons and phonons or k ω k k −kλ λ 2m kλ magnon polaron [5], while the second and third line are, respectively, magnon-number conserving Cherenkov scatter- † Here, akλ(akλ) creates (annihilates) a phonon with momentum ing and magnon-number nonconserving confluence processes k, polarization vector eˆkλ, and frequency ωkλ, and Bose com- [28] as illustrated by Fig. 3. We disregard the weak two- † λ,  = δ  δλλ mutation relations [ak ak λ ] kk . An isotropic elastic phonon one-magnon scattering processes [7]. Angular mo- solid supports three acoustic phonon branches: two degenerate mentum is transferred between the magnetic order and the transverse (λ =±) and one longitudinal (λ =) one, with lattice by the magnon-number nonconserving hybridization ωkλ = cλk, where the cλ are the sound velocities. We choose and confluence processes, while magnon-number conserving

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× +  + − +  (a) [(1 nk )nk (1 N−qλ) nk(1 nk )N−qλ] ∗ π| ·  |2 1 eˆqλ V k,k + δ  δ +  − ω k+k ,q ω ( k k qλ) N  mh¯ qλ k qλ (b) × + +  −  + , [(1 nk )(1 nk )Nqλ nknk (1 Nqλ)] (24) and π| ∗ ·  |2 eˆqλ q N˙qλ = δ( q − ωqλ)(nq − Nqλ) mh¯ωqλ

π| ·  |2 1 eˆqλ U k,k + δ  δ −  − ω k−k ,q ω ( k k qλ) N  mh¯ qλ (c) kk × +  + − +  [nk(1 nk )(1 Nqλ) (1 nk )nk Nqλ] ∗ π| ·  |2 1 eˆqλ V k,k + δ  δ +  − ω k+k ,q ω ( k k qλ) N  2mh¯ qλ kk

×  + − + +  . [nknk (1 Nqλ) (1 nk )(1 nk )Nqλ] (25) The first term on the right-hand side of both Eqs. (24) FIG. 3. Diagrams of the leading magnon-phonon scattering pro- and (25) is caused by the direct magnon-phonon conversion cesses in Eq. (21), with corresponding change of electron spin S. process in Fig. 3(a) that gives rise to magnetoelastic waves Solid, directed lines denote magnons and dashed lines phonons. (magnon polarons) [5,46]. It diverges because perturbation (a) Magnon-phonon hybridization, (b) magnon-number noncon- theory breaks down at the crossing of magnon and phonon serving confluence processes, and (c) magnon-number conserving modes. The singularity can be removed by choosing a basis Cherenkov scattering. that diagonalizes the Hamiltonian [47]. Here we regularize it with a finite broadening [6] that is larger than the magnon- polaron gap, which leads to well-behaved integrated quantities scatterings transfer energy only. Energy conservation requires such as energy, momentum, and spin densities. that phonons have frequencies larger than γ B (2γ B), i.e., in the GHz range, in order to participate resonantly (by conflu- ence) to the spin transfer. The applied magnetic field is an B. Linear response important control parameter since above the critical value We capture the dynamics of energy and spin relaxation in

2 linear response to weak perturbations, assuming that magnons hc¯ λ and phonons stay close to a common thermal equilibrium Bc,λ = (23) 4γ Js at temperature T . The spin-lattice interaction transfers both hybridization and confluence processes are forbidden for energy and angular momentum which changes magnon and λ > , phonon energy phonons with polarization .ForBext Bc,⊥ Bc, other   spin transfer mechanisms must be invoked, such as δ = 1 − h¯ k , interface/surface [20] or higher order magnon-phonon scat- Em(t ) h¯ k nk(t ) fB (26a) V kBT tering [7]. k   1 h¯ωkλ δEλ(t ) = h¯ωkλ Nkλ(t ) − fB , (26b) V kBT A. Kinetic equations k Treating the magnon-phonon interaction Hamiltonian (21) as well as spin densities by Fermi’s golden rule leads to rate equations for the bulk   † h¯ h¯ magnon and phonon distribution functions nk = b bk and k k δs(t ) = nk(t ) − fB , (27a) = † V kBT Nkλ akλakλ [28]: k z z π| ∗ ·  |2 L (t ) h¯ k eˆkλ k δ = =− + − − , n = δ( − ω λ) N λ − n l(t ) [Nk (t ) Nk (t )] (27b) ˙k ω k k ( k k ) V V k λ mh¯ kλ k = / x − π| ·  |2 where fB(x) 1 (e 1) is the Bose distribution function. 1 eˆqλ U k,k + δ  δ −  − ω k−k ,q ω ( k k qλ) Since a precessing magnetic moment can inject a transverse, N  mh¯ qλ k qλ circularly polarized momentum current into an adjacent non- magnetic insulator [20] we also consider transverse phonon × [(1 + n )n  N λ − n (1 + n  )(1 + N λ)] k k q k k q momentum densities π| ·  |2 1 eˆqλ U k,k 1 + δ − , δ( −  + ω λ) ρ = z , k k q ω k k q ±(t ) hk¯ Nk±(t ) (28) N  mh¯ qλ V k qλ k

104402-6 ANGULAR MOMENTUM CONSERVATION AND PHONON SPIN … PHYSICAL REVIEW B 101, 104402 (2020) with ρ+ =−ρ−at equilibrium. In a driven system δρ = ρ+ − allow for different temperatures of longitudinal and transverse ρ− can be finite as can be seen from the interaction vertex phonons, δT(t ) and δT⊥(t ), and associated energy densi- between magnons and transverse phonons in equations (24) ties. Global linear momentum conservation requires v+(t ) = and (25): −v−(t ) ≡ v(t ) and v(t ) = 0. Just as for the magnon distri- bution function (30), the parametrization (31) of the phonon B2 k2 | ∗ · |2 = ⊥ 2 θ − 2 θ ∓ θ 2. eˆk± (cos k sin k cos k ) (29) distribution function contains some tacit assumptions about 4S the relative importance of different scattering mechanisms: For phonons propagating along ±eˆz, i.e., with cos θk =±1, In particular, it should be applicable when polarization- and this expression is only finite for polarization direction λ =∓. momentum-conserving phonon-phonon scattering dominates Hence, magnons couple to λ =+(λ =−) phonons traveling over the nonconserving scattering mechanisms. In YIG, the in the −eˆz (+eˆz) direction. An imbalance in the magnon acoustic quality is much better than the magnetic one [5], distribution thus creates transverse phonon polarizations δρ which supports our shifted-distribution ansatz (31). Also, a and a finite phonon spin polarization (27b). finite drift velocity v implies existence of a phonon current The coupled kinetic equations (24) and (25) can be simpli- on relatively large time scales, which requires a system size fied by assuming that magnon-magnon and phonon-phonon ∼V 1/3 larger than the phonon relaxation length. interactions thermalize the distributions to a quasiequilibrium The response to leading order in the nonequilibrium pa- form that can be parametrized by slowly varying variables rameters reads conjugate to the macroscopic observables of interest [6,48], δ μ i.e., the energy and spin densities given in Eqs. (26a) and δ =−1  h¯ k h¯ k Tm(t ) + (t ) , Em(t ) fB h¯ k (27a), respectively. These conjugate variables are a temper- V kBT kBT T kBT k ature deviation δT (t ) and a magnon chemical potential μ(t ) m (32a) [48], such that 1 h¯ω λ h¯ω λ δTλ(t ) h¯ − μ(t ) δ =−  k ω k , = k . Eλ(t ) fB h¯ kλ (32b) nk(t ) fB (30) V kBT kBT T k (T + δT (t )) k B m δ μ This parametrization of the magnon distribution is accurate δ =−h¯  h¯ k h¯ k Tm(t ) + (t ) , s(t ) fB (32c) for thermal magnons when the number-conserving exchange V kBT kBT T kBT k interaction is the dominant scattering mechanism, which is 2¯h h¯ω ⊥ h¯v(t ) usually the case in magnetic insulators [28,48]. δρ =−  k z 2 , (t ) fB (k ) (32d) A parametrization such as Eq. (30) of the phonon distribu- V kBT kBT k tion fails because a phonon chemical potential does not lead δρ  = ∂ /∂ to a finite phonon spin polarization because the angular where fB(x) fB(x) x, and the nonequilibrium phonon dependence, kz ∝ cos θk in Eq. (28), averages to zero when spin density is the distribution Nk± is isotropic in momentum space. We ω z 2 therefore focus on the leading anisotropic term, which is a δ = 2¯h  h¯ k⊥ (k ) h¯v(t ). l(t ) fB (33) Bose distribution rigidly shifted by a polarization-dependent V kBT k kBT k phonon drift velocity vλ: z According to Eq. (7), the angular momentum of the rigid h¯ωkλ − h¯vλ(t )k Nkλ(t ) = fB . (31) body rotation around a principal axis of the tensor of inertia kB(T + δTλ(t )) z is I (t ) = V [δ j0 + δs(t ) − δl(t )], where δ j0 is an angular Because the transverse phonon modes are degenerate, we set momentum density injected by external torques. δT+(t ) = δT−(t ) ≡ δT⊥(t ) without loss of generality, but we The linear response can be summarized by

⎛ ⎞ ⎛ ⎞ + − − − − − ⎛ ⎞ δEm(t )/kBT ⊥  ⊥  ⊥μ μ ⊥v δTm(t )/T ⎜ ⎟ ⎜ ⎟⎜ ⎟ δ / ⎜ − ⊥ ⊥ ⊥μ ⊥ ⎟ δ / ⎜ E⊥(t ) kBT ⎟ ⎜ 0 v ⎟⎜ T⊥(t ) T ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ∂ ⎜ δ / ⎟ =−⎜ −  0  μ 0 ⎟⎜ δ / ⎟, t ⎜ E(t ) kBT ⎟ ⎜ ⎟⎜ T(t ) T ⎟ (34) ⎜ ⎟ ⎜− − ⎟⎜ ⎟ ⎝ δs(t )/h¯ ⎠ ⎝ ⊥μ μ ⊥μ μ μ vμ ⎠⎝μ(t )/kBT ⎠ − c⊥δρ(t )/kBT ⊥v ⊥v 0 vμ v v(t )/c⊥

where δE⊥ = δE+ + δE−is the change in the transverse in which energy injected into the lattice, e.g., by a femtosec- phonon energy density. The Onsager-reciprocal relaxation ond laser pulse at an optical phonon resonance, relaxes very rates αβ from Eqs. (24) and (25) are listed in Appendix C. quickly to a distribution of the form (31), which subsequently In the following, we discuss the solutions for the material releases energy to the magnetic system. In this case there parameters of yttrium-iron garnet in Table I. We discuss three is no angular momentum transfer from the environment and scenarios: (i) heating, (ii) parametric pumping by microwaves, δ j0 = 0. Figure 4 shows the calculated dynamics when the and (iii) optical spin injection. First, we consider the scenario magnetic order is perturbed by a sudden increase of the

104402-7 RÜCKRIEGEL, STREIB, BAUER, AND DUINE PHYSICAL REVIEW B 101, 104402 (2020)

TABLE I. Magnetic and elastic material parameters of yttrium- the lattice. We assume that the pumped magnons thermalize iron garnet, adopted from Refs. [7,28,48–50]. If not indicated other- quickly to a distribution with increased temperature and finite wise, the parameters are measured at room temperature. magnon chemical potential, while the lattice is initially in equilibrium and plot the results in Fig. 5. The third scenario Symbol Value Unit addresses the direct injection of angular momentum into the lattice constant a 12.376 Å phonons. This can be achieved by exposing the magnet to effective spin per unit cell S 20 circularly polarized light that couples only to phonons with for T  50 K a certain spin polarization, or by phonon spin pumping from a −40 2 exchange stiffness constant Js 8.458 × 10 Jm thin film of another magnet attached to the system [20]. Since g factor g 2 the phonon spin is supplied by the external environment in this −24 mass per unit cell m 9.800 × 10 kg case, we have δ j0 = δl(t = 0), while the magnons are initially −1 longitudinal sound velocity c 7209 m s in equilibrium. The response to such an external torque is −1 transverse sound velocity c⊥ 3843 m s plotted in Fig. 6. longitudinal critical field Bc, 9.21 T The cases (i) and (ii) share many features. Figures 4(a) transverse critical field Bc,⊥ 2.62 T and 5(a) show that the energy relaxes in two stages: First, − . magnetic Grüneisen m 3 2 the longitudinal-phonon and magnon temperatures converge, parameter after which they both equilibrate with the transverse-phonon −22 diagonal magnetoelastic B 6.597 × 10 J temperature. The longitudinal phonons and the magnons equi- constant librate faster than the transverse phonons, because the specific . × −21 off-diagonal magnetoelastic B⊥ 1 319 10 J heat of the former is an order of magnitude smaller than that constant of the latter. In parallel, the magnetoelastic coupling builds − . × −38 2 exchange magnetoelastic A 8 120 10 Jm up transient, counterpropagating currents of the two circular constant phonon modes, i.e., a phononic spin, on a time scale similar to the magnon chemical potential or spin accumulation, see Figs. 4(b) and 5(b). The phonon spin density generated by phonon temperature δT(0) = δT⊥(0). Parallel microwave phonon heating is typically an order of magnitude smaller pumping is the nonlinear process in which a microwave mag- than the magnon spin. The induced rigid rotation in Fig. 4(c) netic field parallel to the magnetization parametrically excites is therefore mainly a magnonic effect. However, when the the Kittel mode above a certain threshold intensity. In contrast system is excited by pumping the magnons, the phonon spin to the (linear) ferromagnetic resonance, the linearly polarized transiently dominates the magnon contribution, see Fig. 5(c). radiation does not inject angular momentum into the magnet, The angular velocity z temporarily changes sign, i.e., the so also in this case δ j0 = 0. The angular momentum needed body rotates in the opposite direction, seemingly breaking to excite the magnetization is therefore provided only by the angular conservation law. After the magnon-dominated

FIG. 4. Nonequilibrium dynamics of (a) magnon and phonon temperatures, (b) magnon chemical potential and phonon drift velocity, (c) spin densities, and (d) torque densities of magnons, phonons, and rigid rotation, for heating initial conditions δT⊥(t = 0) = δT(t = 0) = 1K,andδTm (t = 0) = μ(t = 0) = v(t = 0) = 0. Temperature and external magnetic field are T = 10 K and B = 1T.

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FIG. 5. Nonequilibrium dynamics of (a) magnon and phonon temperatures, (b) magnon chemical potential and phonon drift velocity, (c) spin densities, and (d) torque densities of magnons, phonons, and rigid rotation, with initial conditions δTm (t = 0) = 1K, μ(t = 0) = 0.1 × h¯γ B,andδT⊥(t = 0) = δT(t = 0) = v(t = 0) = 0, corresponding to magnon pumping, e.g., by applying a parallel parametric pumping field. Temperature and external magnetic field are T = 10 K and B = 1T.

first microsecond, the torques exerted by both phonons and which means that the system at large times must relax to magnons in Figs. 4(d) and 5(d) are very similar. the initial temperature T. However, spin may be transferred Figure 6 sketches the even more dramatic effect when the from the phonons to the magnons, which heats the magnons injected phonons initially carry a spin without excess energy, and endows them with a finite chemical potential, see

FIG. 6. Nonequilibrium dynamics of (a) magnon and phonon temperatures, (b) magnon chemical potential and phonon drift velocity, (c) spin densities, and (d) torque densities of magnons, phonons, and rigid rotation, with initial conditions v(t = 0) =−0.1 × c⊥,andδT⊥(t = 0) = δT(t = 0) = δTm (t = 0) = μ(t = 0) = 0. corresponding to a finite phonon spin. Temperature and external magnetic field are T = 10 K and B = 1T.

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Figs. 6(a)–6(c), which is only possible by transient cooling torques and total volume is predicted to be constant as long as of the transverse phonons. Actually only a small fraction the excitation is more or less homogeneous. of the spin is transferred from the phonons to the magnons Our treatment of angular momentum transfer in spin-lattice in Fig. 6(c): The loss of phonon spin is accommodated by interactions should be useful in the study of a variety of the rigid rotation of the entire magnet. The overall torque in problems. Of particular interest would be the application Fig. 6(d) is dominated by the phonons at almost all times. to the magnetic nanosystems like cantilevers [51–53] and nanoparticles in polymer cavities [54] or levitated in traps [22–25]. It could also be extended to study the role of the IV. DISCUSSION AND CONCLUSIONS phonon spin in transport phenomena like the spin Seebeck We present a microscopic theory of spin-lattice interactions effect [1]. Moreover, the microscopic spin-lattice Hamiltonian and angular momentum conservation in magnetic insulators. that we proposed could be used to extend computations of After separating the mechanical degrees of freedom into magnon-phonon interactions [7] into the high-temperature rigid-body and internal vibrations, we find that also phonons regime where magnetoelastic theory is no longer valid and carry internal angular momentum. We derive equations of to determine material constants from ab initio computations. motion for the spin, rigid-body, and phonon spin operators Another extension of the formalism would allow addressing that govern the Einstein-de Haas and Barnett effects, and finite-size corrections, quantum effects, and time scales at show that the torque generated by spin-lattice interactions which rigid-body and internal phonon dynamics cannot be drives both the rigid-body rotation and the phonon spin. In separated. the long-wavelength limit, we recover the phenomenological theory of magnetoelasticity. ACKNOWLEDGMENTS We apply the formalism to a linear response analysis of a levitated magnet that is large enough that surface effects can This work is supported by the European Research Council be disregarded but small enough that rotations are observable. via Consolidator Grant No. 725509 SPINBEYOND and JSPS In contrast to the magnon chemical potential or accumulation, KAKENHI Grant No. 19H006450. R.A.D. is a member of the the phonon chemical potential does not couple to the total D-ITP consortium, a program of the Netherlands Organisation rotation. It is rather an internal phonon current that governs the for Scientific Research (NWO) that is funded by the Dutch phonon contribution to the Einstein-de Haas effect. Depend- Ministry of Education, Culture and Science (OCW). This ing on the driving protocol, the transient Einstein-de Haas work is also part of the normal research program of NWO. dynamics can involve a change in the sense of rotation. When This research was supported in part by the National Science the system is not levitated but fixed, e.g., on a substrate, Foundation under Grant No. NSF PHY-1748958. the torques exerted by the magnon and phonon spins on the sample are in principle measurable [51–53]. Brillouin light APPENDIX A: TOTAL ANGULAR MOMENTUM scattering [16] can resolve the phonon spin; our prediction CONSERVATION of a momentum imbalance between the two circularly polar- φ θ χ ized phonon modes should therefore also be experimentally In terms of the three Euler angles , , and as defined in accessible. Eq. (3) the body-fixed total angular momentum operator reads Several assumptions and approximations imply that the (see, e.g., Refs. [40,41] for details) ⎛ ⎞ present results are valid for a limited temperature and size of ∂χ the system. The adoption of the magnetoelastic limit implies h¯ ⎜ sin χ ⎟ J = ⎝ ∂φ + cos χ∂θ + tan θ sin χ∂χ ⎠. (A1) that temperatures should not exceed the frequencies for which i cos θ cos χ a continuum mechanics and magnetism holds, roughly T < ∂φ − χ∂θ + θ χ∂χ cos θ sin tan cos 100 K. The decoupling of internal phonon modes from the total rotation introduces errors that we estimate to vanish In the laboratory frame when the number of spins is much larger than unity which Jlab = R(φ,θ,χ)J (A2) is not very restrictive. More drastic is the assumption that the ⎛ ⎞ φ θ∂ − φ∂ + cos φ ∂ phonon relaxation length should be much smaller than the sys- cos tan φ sin θ θ χ ⎜ cos ⎟ tems size, in order to allow the flow of transient phonon spin h¯ sin φ = ⎝sin φ tan θ∂φ + cos φ∂θ + ∂χ ⎠. (A3) currents. This is a material specific and temperature dependent i cos θ ∂ parameter that is not well known. When the phonon relaxation φ length is much larger than the particle size, a phonon spin does These operators obey the commutation relations not build up, strongly suppressing the phonon contribution to the Einstein-de Haas effect. For materials with extremely [Jx, Jy] =−ihJ¯ z, (A4) low acoustic attenuation such as YIG, the phonon propaga- tion length at GHz frequencies can be centimeters. Thermal x , y = z , Jlab Jlab ihJ¯ lab (A5) phonons at not too low temperatures are more strongly scat- 2 = 2 tered, which leads us to believe that YIG spheres that can be including their cyclic permutations. Also, J Jlab, and fabricated for diameters 0.5 mm are suitable model systems 2 , = , to test our predictions. The size estimates for other materials Jlab J 0 (A6) can be substantially smaller, however. For particles larger z , = , than the phonon relaxation length, the ratio between predicted Jlab J 0 (A7)

104402-10 ANGULAR MOMENTUM CONSERVATION AND PHONON SPIN … PHYSICAL REVIEW B 101, 104402 (2020) whereas Ri → Ri + φ × Ri, with |φ|1, yielding N = mi × . , † = R φ,θ,χ × γ f R,μ(Ri ) nˆ μ Ri (B4) [Jlab U HSU] ih¯ [ ( )Si] B Iμ i=1 Here the nˆ μ with μ = 1, 2, 3 are the principal axes of the + ih¯R(φ,θ,χ)T . (A8) ext system in the body-fixed frame, and Iμ the corresponding principal moments of inertia. The commutator (B2) becomes Hence, in the absence of torques by an external magnetic  2 field B or external mechanical forces, the absolute value Jlab α β αβ m j αβ z ,π = δ δ − δ z J ui j ih¯ ij and the component lab of the total angular momentum are M conserved.  3 α β m (nˆ μ × R ) (nˆ μ × R ) − j i j . (B5) APPENDIX B: PHONON COMMUTATION RELATIONS Iμ μ=1 The commutator of the phonon displacement and position We can estimate the order of magnitude of the corrections operators introduced in Eq. (9)isgivenby to the commutation relations by introducing a unit cell around  = / ≈ ρ 3N−6 each particle with volume V V N and mass mi V , α,πβ = m j α β where ρ = M/V is the mass density. The maximum distance ui ih¯ fn (Ri ) fm (R j )(B1) j m 1/3 i n=1 Ri of a particle from the origin is of the order V ,sothe 5/3 moment of inertia Iμ ∼ ρV . Hence m = ih¯ δ δαβ − j f α (R ) f β (R ) , m ρV 1 ij m n i m j j ∼ = , (B6a) i n∈zero modes M ρV N (B2) α β 2/3 m (nˆ μ × R ) (nˆ μ × R ) ρVV 1 j i j ∼ = . (B6b) ρ 5/3 where we used the completeness relation Iμ V N 3N α β = δ δαβ n=1 fn (Ri ) fm (R j ) ij for the combined phonon Therefore, the noncanonical corrections to the commutator and rigid-body zero modes. (B2) scale with the inverse of the number of particles N in the Explicit expressions for the 6 zero mode eigenfunctions system; hence the phonon operators (9) and (8) in the body- of center-of-mass translation and rigid rotation must obey fixed frame can be treated as canonical whenever N  1, so the translational and rotational invariance. The transformation one has to worry about corrections only for small molecules. Ri → Ri + a, where a is a constant vector, leaves the poten- tial V ({r }) invariant. Comparing to the definition (3)ofthe i APPENDIX C: LINEAR RESPONSE RELAXATION RATES phonon eigenmode expansion in the body-fixed frame, we find the 3 (normalized) zero mode eigenfunctions of center- The linear response relaxation rates of-mass translation = (1) + (2), αβ αβ αβ (C1) m = i , (1) (2) f CM,μ(Ri ) eˆμ (B3) where and are due to one-magnon one-phonon and M αβ αβ two-magnon one-phonon processes, respectively, follow from where the eˆμ with μ = 1, 2, 3 are an arbitrary set of or- inserting the ansätze (30) and (31) for the magnon and phonon thonormal basis vectors. Similarly, we obtain the 3 zero distribution functions into the kinetic equations (24) and (25). modes of rigid rotation by considering infinitesimal rotations Explicitly,   π (1) h¯ ∗ 2 h¯ k h¯ k = ω λδλ,⊥|eˆ ·  | δ( − ω λ) 1 + f f , (C2a) ⊥ 2 2 k kλ k k k B B mkBT V kBT kBT k   π (1) h¯ ∗ 2 h¯ k h¯ k = ω λδλ,|eˆ ·  | δ( − ω λ) 1 + f f , (C2b)  2 2 k kλ k k k B B mk T V kBT kBT B k   πhc¯ 2   |eˆ∗ ·  |2 h¯ h¯ (1) = ⊥ δ z 2 kλ k δ − ω + k k , v λ,⊥ k ( k kλ) 1 fB fB (C2c) 2 2 ω λ mkBT V λ k kBT kBT k   π (1) hc¯ ⊥ z ∗ 2 h¯ k h¯ k = δλ,⊥λk |eˆ ·  | δ( − ω λ) 1 + f f , (C2d) ⊥v 2 2 kλ k k k B B mkBT V λ kBT kBT k   π h¯ h¯ (1) =− δ | ∗ ·  |2δ − ω + k k , ⊥μ λ,⊥ eˆkλ k ( k kλ) 1 fB fB (C2e) mkBTV λ kBT kBT k   π h¯ h¯ (1) =− δ | ∗ ·  |2δ − ω + k k , μ λ, eˆkλ k ( k kλ) 1 fB fB (C2f) mkBTV kBT kBT kλ

104402-11 RÜCKRIEGEL, STREIB, BAUER, AND DUINE PHYSICAL REVIEW B 101, 104402 (2020)   ∗ 2 πc⊥ |eˆ ·  | h¯ h¯ (1) =− δ λ z kλ k δ − ω + k k , vμ λ,⊥ k ( k kλ) 1 fB fB (C2g) mkBTV ωkλ kBT kBT kλ   π | ∗ ·  |2 (1) eˆkλ k h¯ k h¯ k μ = δ( k − ωkλ) 1 + fB fB , (C2h) hmV¯ ωkλ kBT kBT kλ and   π 3  ω (2) ha¯ 2 h¯ k h¯ k h¯ qλ = δλ,⊥ω λ δ − , |eˆ λ · U ,  | δ( −  − ω λ) 1 + f f f ⊥ 2 2 2 q k k q q k k k k q B B B mk T V  kBT kBT kBT B kk qλ    ∗ 2 | ·  |  ω eˆqλ V k,k h¯ k h¯ k h¯ qλ + δ  δ +  − ω + + , k+k ,q ( k k qλ) 1 fB 1 fB fB (C3a) 2 kBT kBT kBT   π 3  ω (2) ha¯ 2 h¯ k h¯ k h¯ qλ = δλ,ω λ δ − , |eˆ λ · U ,  | δ( −  − ω λ) 1 + f f f  2 2 2 q k k q q k k k k q B B B mk T V  kBT kBT kBT B kk qλ    ∗ 2 | ·  |  ω eˆqλ V k,k h¯ k h¯ k h¯ qλ + δ  δ +  − ω + + , k+k ,q ( k k qλ) 1 fB 1 fB fB (C3b) 2 kBT kBT kBT   π 3 2 | ·  |2  ω (2) ha¯ c⊥ z 2 eˆqλ U k,k h¯ k h¯ k h¯ qλ = δλ,⊥(q ) δ − , δ( −  − ω λ) 1 + f f f v 2 2 2 k k q ω k k q B B B mk T V  qλ kBT kBT kBT B kk qλ    ∗ 2 | ·  |  ω eˆqλ V k,k h¯ k h¯ k h¯ qλ + δ  δ +  − ω + + , k+k ,q ( k k qλ) 1 fB 1 fB fB (C3c) 2ωqλ kBT kBT kBT   π 3  ω (2) ha¯ c⊥ z 2 h¯ k h¯ k h¯ qλ = δλ,⊥λq δ − , |eˆ λ · U ,  | δ( −  − ω λ) 1 + f f f ⊥v 2 2 2 k k q q k k k k q B B B mk T V  kBT kBT kBT B kk qλ    ∗ 2 | ·  |  ω eˆqλ V k,k h¯ k h¯ k h¯ qλ + δ  δ +  − ω + + , k+k ,q ( k k qλ) 1 fB 1 fB fB (C3d) 2 kBT kBT kBT    π 3  ω (2) a ∗ 2 h¯ k h¯ k h¯ qλ =− δλ,⊥δ + , |eˆ · V ,  | δ( +  − ω λ) 1 + f 1 + f f , (C3e) ⊥μ 2 k k q qλ k k k k q B B B mkBTV  kBT kBT kBT kk qλ    π 3  ω (2) a ∗ 2 h¯ k h¯ k h¯ qλ =− δλ,δ + , |eˆ · V ,  | δ( +  − ω λ) 1 + f 1 + f f , (C3f) μ 2 k k q qλ k k k k q B B B mkBTV  kBT kBT kBT kk qλ    ∗ 2 π 3 |eˆ · V  |  ω (2) a c⊥ z qλ k,k h¯ k h¯ k h¯ qλ =− δλ,⊥δ + , λq δ( +  − ω λ) 1 + f 1 + f f , (C3g) vμ 2 k k q ω k k q B B B mkBTV  qλ kBT kBT kBT kk qλ    ∗ 2 π 3 |eˆ · V  |  ω (2) 2 a qλ k,k h¯ k h¯ k h¯ qλ = δ + , δ( +  − ω λ) 1 + f 1 + f f , (C3h) μ 2 k k q ω k k q B B B hmV¯  qλ kBT kBT kBT kk qλ where δλ,⊥ = δλ,+ + δλ,−.

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