Phonon-Magnon Processes with Relevance to Acoustic Spin Pumping

PHYSICAL REVIEW B 90, 224421 (2014)

Phonon-magnon resonant processes with relevance to acoustic spin pumping

P. A. Deymier,1 J. O. Vasseur,2 K. Runge,1 A. Manchon,3 and O. Bou-Matar4 1Department of Materials Science and Engineering, University of Arizona, Tucson, Arizona 85721, USA 2Institut d’Electronique, de Micro-electronique´ et de Nanotechnologie, UMR CNRS 8520, Cite´ Scientiﬁque, 59652 Villeneuve d’Ascq Cedex, France 3King Abdullah University of Science and Technology (KAUST), Physical Science and Engineering Division, Thuwal 23955-6900, Saudi Arabia 4International Associated Laboratory LEMAC/LICS: IEMN, UMR CNRS 8520, PRES Lille Nord de France, Ecole Centrale Lille, 59652 Villeneuve d’Ascq, France (Received 24 September 2014; revised manuscript received 3 December 2014; published 23 December 2014)

The recently described phenomenon of resonant acoustic spin pumping is due to resonant coupling between an incident elastic wave and spin waves in a ferromagnetic medium. A classical one-dimensional discrete model of a ferromagnet with two forms of magnetoelastic coupling is treated to shed light on the conditions for resonance between phonons and magnons. Nonlinear phonon-magnon interactions in the case of a coupling restricted to diagonal terms in the components of the spin degrees of freedom are analyzed within the framework of the multiple timescale perturbation theory. In that case, one-phonon-two-magnon resonances are the dominant mechanism for pumping. The effect of coupling on the dispersion relations depends on the square of the amplitude of the phonon and magnon excitations. A straightforward analysis of a linear phonon-magnon interaction in the case of a magnetoelastic coupling restricted to off-diagonal terms in the components of the spins shows a one-phonon to one-magnon resonance as the pumping mechanism. The resonant dispersion relations are independent of the amplitude of the waves. In both cases, when an elastic wave with a fixed frequency is used to stimulate magnons, application of an external magnetic field can be used to approach resonant conditions. Both resonance conditions exhibit the same type of dependency on the strength of an applied magnetic field.

DOI: 10.1103/PhysRevB.90.224421 PACS number(s): 43.20.+g, 63.20.kk, 75.30.Ds

I. INTRODUCTION also been used recently to coherently excite the magnetization of ferromagnets, induce ultrafast demagnetization, or reorient The production of spin polarized currents is of tremendous the magnetization direction [11,12]. importance for the transport of information in the form of In this context, the role of phonon-magnon coupling is the spin instead of the charge of electrons. The process of particularly inspiring. Indeed, it has recently been shown that generation of spin currents is an essential component in the the resonant absorption of elastic waves by a ferromagnet development of spintronics. To date, several approaches have can also drive a spin current [13]. This approach has been been investigated to generate spin currents. The first one utilizes nonlocal spin injection [1,2]: spin current is created by termed acoustic spin pumping (ASP) [14]. Acoustic spin electrons flowing in a ferromagnet and diffuses in an adjacent pumping is a resonant equivalent of the SSE. The fundamental normal metal. A second set of methods generates pure spin mechanism underlying ASP results from the coupling between currents from spin pumping. A precessing magnetization (e.g., an elastic wave (phonons) and spin waves (magnons) through driven into ferromagnetic resonance by an external magnetic the magnetoelastic effect [15]. Of other significant relevance field) injects a nonequilibrium spin current into an adjacent is the role played by phonon-magnon interactions and the normal metal via spin angular momentum transfer [3–5]. An- magnetoelastic effect in the phenomenon of phonon drag other popular method that currently fosters intense efforts is the enhancement of SSE [16] or the development of the field of spin Hall effect (SHE), where unpolarized electrons flowing spin caloritronics [17,18]. in a normal metal are asymmetrically scattered via spin-orbit Several theories of the magnetoelastic effect have relied on coupled impurities, resulting in the generation of a transverse (a) approximating the ferromagnet as a continuum and (b) ap- pure spin current [6,7]. All the above methods seek to generate plying the quasistatic approximation whereby magnons are at pure spin currents from electrical voltages and electron flows. equilibrium with respect to the slower phonons [19–21]. More In fact, recent developments in the field of spintronics have recently, methods based on molecular and spin-dynamics have demonstrated that heat (i.e., acoustic phonons) could be been employed to investigate numerically phonon-magnon used to pump spin currents from (insulating or metallic) interactions. These numerical methods simultaneously solve ferromagnets. The exploration of the spin Seebeck effect the equations of motion for atoms and spins [22–24]. Some of (SSE) [8]—i.e., the generation of spin currents using heat—has these studies have shown softening and damping of magnon revealed that chargeless spin currents can be efficiently carried modes due to lattice vibrations and the existence of coupled by magnons in magnetic insulators such as yttrium iron magnon-phonon modes with identical frequencies [22]. In garnet (YIG) [9]. Combining elastic waves with magnons in light of the importance of magnon-phonon interactions for a magnetic insulators offers opportunities for the development wide range of emerging scientific and technical fields, we have of chargeless information control since no electrical voltages developed an analytical approach to shed light on resonant or charge current flows are needed [10]. Elastic waves have magnon-phonon processes in models of ferromagnetic media.

We focus on a one-dimensional model composed of atomic made in the more complex nonlinear case. Finally, in Sec. V, sites that support lattice vibrations and spin precession. We we summarize our findings and draw conclusions concerning solve the coupled magnetoelastic equations of motion for the the observed phonon-magnon resonances in the context of spin degrees of freedom and the atomic displacement using acoustic spin pumping. analytical mathematical methods. We have considered two cases, namely a case with a nonlinear magnetoelastic coupling II. MODEL and another one with linear coupling. While analyzing the linear system is straightforward, the nonlinear system requires A. Hamiltonian the use of an approach based on the multiple time scale We consider a one-dimensional discrete model of a medium perturbation theory [25]. This approach enables us to obtain an- that can support coupled spin waves and elastic waves (see alytical solutions for the wave representation of the nonlinearly Fig. 1). This is used as a model of a ferromagnetic material interacting phonons and magnons. We derive the conditions for with magnetoelastic coupling. This model is constituted of a resonance between a phonon and a magnon with an identical chain of atoms. Each atomic site n is characterized by a spin wave number in both cases. Phonon-magnon resonance leads Sn and a displacement un. We assume that the model is limited to shifts in the dispersion bands of both excitations. We also to first nearest neighbor interactions. The Hamiltonian for this demonstrate that the condition for resonance is modified by system is taking the form the application of an external magnetic field. The strength of i,j i j H =−2 J (u + ,u ) S S the applied field can be used to tune the ferromagnetic medium n,n+1 n 1 n n n+1 = to achieve resonance. n i,j x,y,z This paper is organized as follows. In Sec. II, we introduce 1 2 + β(u + − u ) . (1) the Hamiltonian for a discrete one-dimensional atomic model 2 n 1 n with two forms of magnetoelastic coupling. One form leads n to a nonlinear coupling between phonons and magnons. The In this equation, the atoms are interacting elastically via a nonlinear phonon-magnon interactions arise from a coupling harmonic potential with a uniform, constant stiffness β.The contribution to the Hamiltonian restricted to diagonal terms remaining part of the Hamiltonian is that of a Heisenberg in the components of the spin degrees of freedom. A linear model where the nearest neighbor spin exchange coupling phonon-magnon interaction is obtained in the case of a i,j constant Jn,n+1 (un+1,un) depends on the displacement of the coupling restricted to off-diagonal terms in the components of atoms. The superscripts i and j run over all directions x, y, the spins. The coupled equations of motion for the spin degrees and z. of freedom and the lattice vibrations in both cases are derived in We consider two cases for the magnetoelastic coupling, that same section. The multiple timescale perturbation method namely case I [Eq. (2a)] and case II [Eq. (2b)] is applied to analyzing the nonlinear equations of motion in the Appendix. In the Appendix, we derive the sets of i,j = Jn,n+1 (un+1,un) Jn,n+1 (un+1,un) equations of motion to zeroth, first, and second order in the = − − phonon-magnon coupling constant. We also find analytical Jδij K (un+1 un) δij , (2a) solutions to these sets of nonlinear equations. We show i,j = − − − Jn,n+1(un+1,un) Jδij KT (un+1 un)(1 δij ). (2b) that the mechanism for energy pumping between phonons and magnons is a resonance involving one phonon and two In the first case, the coupling between the displacement and magnons. This leads to frequency shifts and modifications of the spins is limited to terms in the Hamiltonian of the form i i the dispersion relations that are proportional to the square of SnSn+1. In case I, the magnetoelastic coupling is restricted the amplitude of the elastic and magnetic waves as discussed to diagonal terms in the components of the spin degrees in Sec. III. The effect of the magnetic and elastic physical of freedom to the same coordinates. Case II corresponds to characteristics of the systems on the dispersion bands are interactions limited to cross terms with respect to components also discussed in terms of their softening and hardening in of the spin degrees of freedom. The magnetoelastic coupling Sec. III. We also consider the effect of an externally applied term in the Hamiltonian involves only terms of the form i j=i magnetic field on the phonon-magnon resonance condition. SnSn+1. Here, J , K, and KT are positive constants. The minus Even though an external field adds only a linear term in sign in Eqs. (2a) and (2b) indicates that the coupling between the equations of motion of the spin degrees of freedom, adjacent spin weakens as the separation distance increases. we show that the condition for phonon-magnon resonance is sensitive to the strength of the external field. In Sec. IV, we solve the set of equations of motions for the linearly z z z coupled phonons and magnons. We find in that case that a S single phonon interacts with a single magnon. The resonance n-1 Sn Sn+1 z that drives energy pumping between phonons and magnons y is involving one phonon and one magnon. As expected, it x occurs when both excitations have the same phase velocity. n-1 n n+1 Modifications to the dispersion relations due to the resonance are here independent of the amplitude of the waves. The effect FIG. 1. Schematic illustration of the one-dimensional discrete of an external magnetic field on the one-phonon-one-magnon atomic model supporting coupled spin waves and elastic waves. The resonance condition leads to observations similar to those x axis is defined along the chain of atoms.

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The Hamiltonian for case I takes the specific form B. Equations of motion for the spin degrees of freedom The dynamics of a spin are given by the Landau-Lifshitz 1 2 H =−2 J + (u ,u + ) S .S + + β(u + − u ) . equation n,n 1 n n 1 n n 1 2 n 1 n n n ∂Sn (3) =−γ Sn × hn, (6a) ∂t This Hamiltonian, then, simplifies to where 1 ∂H h = . n (6b) H =−2J Sn.Sn+1 + 2K (un+1 − un) Sn.Sn+1 γ ∂Sn n n is the effective magnetic field at site n. In Eq. (6), γ (>0) 1 2 and stand for the atomic gyromagnetic ratio and the reduced + β(u + − u ) . (4) 2 n 1 n Planck’s constant, respectively. For the sake of simplicity, we n neglect magnetic damping. The second term in Eq. (4) represents the magnetoelastic Using the Hamiltonian of Eq. (4) that corresponds to case interactions between spin and displacement. As discussed I, the spin equation of motion becomes previously, magnetoelastic coupling is composed only of terms ∂Sn J K diagonal in the components of the spin degrees of freedom; = 2 Sn × (Sn−1 + Sn+1) − 2 [(un+1 − un)Sn ∂t that is, these interactions involve only pairs of components: × + − × xx, yy, and zz by virtue of the dot product. Sn+1 (un un−1)Sn Sn−1]. (7) In case II considered here, the Hamiltonian is written as We seek a solution in the form ⎛ ⎞ ⎛ ⎞ =− + − 0 εn,x H 2J Sn.Sn+1 2KT (un+1 un) ⎝ ⎠ ⎝ ⎠ Sn = So +εn = 0 + εn,y with n n 0 S εn,z × x y + y x + z x + x z z Sn Sn+1 Sn Sn+1 SnSn+1 Sn Sn+1 ε ,ε ,ε S0. (8) 1 n,x n,y n,z z + z y + y z + − 2 SnSn+1 Sn Sn+1 β(un+1 un) . (5) 2 Here, So is the magnetic moment of the spin and is oriented n along the z axis in the positive direction. Also, εn,z must be a The second term in Eq. (5) involves products of different second-order term to satisfy the conservation of the norm of Sn. components of the spin degrees of freedom; that is, these We expand terms of the form Sn × Sn−1 up to second order interactions involve only pairs of mixed components of the in ε ⎛ ⎞ type: xy, yz, and xz. − 0 − Sz (εn−1,y εn,y ) To relate the Hamiltonians given by Eqs. (4) and (5)tomore ⎝ ⎠ S × S − ∼ + 0 − familiar forms, we refer to the continuum limit of the mag- n n 1 Sz (εn−1,x εn,x ) 0 netoelastic energy for ferromagnetic materials (see Eq. (8) of ⎛ ⎞ Ref. [26]). In general, this energy involves all possible products 0 of two components of the spins and all possible polarizations + ⎝ 0 ⎠ . (9) of the displacement. For a cubic material, the magnetoelastic εn,x εn−1,y − εn,y εn−1,x energy reduces to two terms (see Eq. (10) of Ref. [26]). The first term corresponds to the interaction between the diagonal With this, Eq. (6a) expressed in component form becomes components of the strain and spins. This term is equivalent ∂εn,x J 0 =−2 S (ε + − 2ε + ε − ) to the discrete form of the magnetoelastic energy of Eq. (4). ∂t z n 1,y n,y n 1,y The second term represents interactions between the shear K 0 components of the strain with the mixed components of the + S u + − u ε + − ε 2 z [( n 1 n)( n 1,y n,y ) spins. This second term is isomorphic to that of Eq. (5). For + − − − − a cubic material, the scalar quantity un in Eq. (4) refers to a (un un 1)(εn 1,y εn,y )], (10a) longitudinal displacement, while in Eq. (5), it corresponds ∂εn,y J 0 =+2 S (ε + − 2ε + ε − ) to a transverse displacement (perpendicular to the chain). ∂t z n 1,x n,x n 1,x For cubic symmetry, the Hamiltonian of Eq. (4) is therefore K 0 modeling longitudinal waves/spin waves interactions, while − S u + − u ε + − ε 2 z [( n 1 n)( n 1,x n,x ) the Hamiltonian of Eq. (5) is used to describe shear waves/spin + − − waves interactions. We have separated these two contributions (un un−1)(εn−1,x εn,x )], (10b) in two Hamiltonians for the sake of analytical simplicity and ∂εn,z J = 2 [ε (ε − + ε + ) − ε (ε − + ε + )]. to unravel distinct characteristics of interactions between spin ∂t n,x n 1,y n 1,y n,y n 1,x n 1,x waves and transverse and longitudinal phonons. We would (10c) like to note that for noncubic symmetries, the Hamiltonian of Eq. (5) may also be representative of the interactions between The first term on the right-hand side of Eqs. (10a) and (10b) longitudinal displacements and spin waves. are the usual first-order terms that appear in the equations of

224421-3 DEYMIER, VASSEUR, RUNGE, MANCHON, AND BOU-MATAR PHYSICAL REVIEW B 90, 224421 (2014) motion of spin waves. All other terms that involve the constant When K = 0, one recovers the usual equation of motion K are second-order terms resulting from the coupling between for a monoatomic one-dimensional harmonic chain. The displacement and spin. Equation (10c) is second order and second term on the right-hand side of Eq. (15) models the couples the directions x and y to direction z. magnetoelastic coupling to the second order. For case II, application of Eq. (6b) to the Hamiltonian given Equations (10a)–(10c) and (15) form the complete set of by Eq. (5) results in equations describing the coupled motion of spins and atoms in case I. In this case, the magnetoelastic coupling is nonlinear, J KT h =− S − + S + + { u + − u V + n 2 ( n 1 n 1) 2 ( n 1 n) n 1 and we will solve these equations within the context of multiple timescale perturbation theory [25]. The method and results are + (un − un−1)Vn−1}, (11) detailed in the Appendix. y + z In case II, limiting the interaction terms to the first order in Sn Sn x + z where the vector Vn is defined as (Sn Sn ). Using Eq. (8), we the spin degrees of freedom, one gets the equation of motion x + y Sn Sn for the displacement approximate to the zeroth order this vector by the constant 0 Sz ∂2u S0 n vector ( z ). This vector is independent of location along the m = β(un+1 − 2un + un−1) 0 ∂t2 chain of atoms. Using this, we obtain the equations of motion 0 + 2K S ((ε + − ε − ) for the x and y components to first order in displacement T z n 1,x n 1,x + (εn+1,y − εn−1,y )). (16) ∂εn,x J 0 =−2 S (εn+1,y − 2εn,y + εn−1,y ) ∂t z Equations (12a), (12b), and (16) are used to model the KT 0 2 dynamics of the displacement and the spin degrees of freedom + S u + − u − , 2 z ( n 1 n 1) (12a) in case II. Because these equations are linear, solutions are straightforward and will be derived in Sec. IV.

∂εn,y J 0 =+2 S (ε + − 2ε + ε − ) ∂t z n 1,x n,x n 1,x III. DISCUSSION OF RESULTS FOR CASE I KT 0 2 A. Dispersion with magnon-phonon interactions − S u + − u − , 2 z ( n 1 n 1) (12b) First of all, we summarize the findings of the Appendix. We We do not have to consider an equation of motion for the have seen that the dispersion relations of the magnons and of component εn,z since, in case II, all equations can be expressed the phonons are shifted due to their mutual interaction. Both to first order only. dispersion relations are re-expressed in the form 2 C. Equations of motion for the displacement ∗ = − 2 0 2 ω0 (k) ω0(k) K Sz α0[P (k)]pv, (17) In case I, the force acting on an atom n is obtained as ∂H 4 S0 F =− =+β(u + − u ) − β(u − u − ) ∗ 2 z 2 n n 1 n n n 1 ω (k) = ω0 (k) + K λ [P (k) Q (k)] . (18) ∂un 0 2 0 pv mωM + 2K(Sn.Sn+1 − Sn−1.Sn). (13) Here, ω and ω0 are the frequencies of the magnons and 0 ∗ ∗ Assuming that each atom has the same mass m, their motion phonons in absence of interactions. Also, ω0 and ω0 are is described by the equation the frequencies of magnons and phonons when they interact 2 nonlinearly. In Eq. (18), ω stands for β . The frequency shift ∂ un M m m = β(u + − 2u + u − ) + 2K(S .S + − S − .S ). ∂t2 n 1 n n 1 n n 1 n 1 n of the phonons depends on the square of the amplitude of the (14) magnons and vice versa. The amplitudes of the magnons and phonons are given by λ0 and α0, respectively. In Eqs. (17) and Utilizing Eq. (8), Eq. (14) takes its final form (18), the corrected dispersion relations involve two functions 2 of the wave number ∂ un 0 m = β(u + − 2u + u − ) + 2K S (ε + − ε − ) 2 n 1 n n 1 z n 1,z n 1,z sin (2ka) − 2sin (ka) ∂t P (k) = , sin2 (ka) − sin2 ka − ρ sin ka + εn,x (εn+1,x − εn−1,x ) + εn,y (εn+1,y − εn−1,y ) . 2 2 (15) and

sin (2ka) − 2sin (ka) 1 (cos (ka) − cos (2ka)) sin (ka) + ρsin ka (sin (2ka) − sin (ka)) Q (k) = × 2 2 , 2 − 2 ka − ka 2 ka sin (ka) sin 2 ρ sin 2 sin 2

√ = 2 β/m of the nonlinear coupled system are softened or hardened with the dimensionless quantity ρ deﬁned as ρ 8J 0 .The Sz compared to the dispersion of the noninteracting phonons sign of these functions determines whether the frequencies

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√ 1 = 2 β/m = 1 TD TABLE I. Values of the parameter ρ 8J 0 for some Sz 4 TC 0 representative ferromagnetic materials. -1 -2 Material TD(K) TC (K) ρ -3 -4 (a) YIG 600 550 0.2727 Ni 450 627 0.1794 -5 DENOMINATOR Fe 470 1043 0.1126 -6 ka Co 385 1388 0.0693

5 1.047). This resonance condition is illustrated in the inset of Fig. 2(a). The function P remains positive but exhibits

k 0 a large maximum near ka = 1.047. The function Q also P shows a positive maximum near the resonance condition but -5 (b) changes sign with increasing wave number. The frequencies of the interacting magnons are softened over the complete BZ -10 ka [Eq. (17)]. The frequency of the phonons increases at small wave number and decreases at larger ones in comparison to 10 the frequency of the noninteracting phonons [Eq. (18)]. As ρ decreases below the value of 1, the denominator possesses two 5 zeros, one at low ka and one at large ka. These correspond to two resonance conditions. The low ka resonance evolves k 0 ρ ka

Q towards a longer wavelength as decreases. The high

-5 (c) resonance moves towards the upper edge of the BZ. The functions P and Q change sign several times over the complete -10 BZ. These are the conditions that may be representative of most ka common ferromagnets, as will be discussed now. The parameter ρ can be related phenomenologically to FIG. 2. (Color online) (a) Denominator of functions P and Q the Curie temperature TC and the Debye temperature TD. versus reduced wave number ka for different values of the parameter Ferromagnetic order breaks down at nonzero temperature for ρ. The inset represents the magnon (dotted line) and the phonon = a one-dimensional Ising model with nearest neighbor interac- (solid line) dispersion curves at ρ 1. The condition for resonance tions. However, a one-dimensional Ising model in the presence ω (2k) = (ω (k) + ω (k)) is highlighted with vertical lines. (b) P 0 0 0 of a small external field exhibits ferromagnetism at finite and (c) Q as functions of ka and ρ. temperature. At high temperature, the magnetic susceptibility follows a Curie-Weiss form with a Curie temperature defined 2J as TC = where kB is Boltzmann’s constant [27]. In the and magnons. In Fig. 2, we plot these functions along with kB the denominator of these very same functions. We vary ka preceding expression, the factor 2 relates the number of nearest neighbors of a given spin site. This is the form of the Curie in the positive region√ of the Brillouin zone (BZ): [0,π].The = 2 β/m temperature that would be obtained erroneously when the parameter ρ 8J 0 measures the relative value of the cutoff Sz one-dimensional system is treated with a mean field approach. frequencies of the magnons and the phonons in the chain of In spite of the inability of the mean field model to capture = atoms. We investigate the cases: ρ 4,2,1,0.4,0.04. the physics of the one-dimensional system, we employ this Figure 2(a) illustrates the condition for phonon-magnon temperature as a phenomenological measure of the thermal resonance as a function of the reduced wave number ka and equivalent of the energy of our magnetic excitations. Defining ratio of cutoff frequencies ρ. When ρ>1, at a given ka,the the Debye frequency υ = 1 2 β enables us to introduce the frequency of a phonon is always larger than that of a magnon. D 2π m hνD 0 The denominator of the functions P and Q is always nonzero Debye temperature TD = . With these, and taking S = 1, kB z and negative. P is positive over the entire BZ, leading to a the parameter ρ can be expressed in terms of Curie and Debye softening of the magnon frequency [see Eq. (17)]. Also, Q temperatures as ρ = 1 TD . In Table I, we report some values 4 TC is positive for small ka and changes sign beyond a threshold of the parameter ρ for some representative ferromagnetic wave number. Phonon-magnon coupling hardens the phonons materials. at low ka and softens them beyond the threshold. At ρ = 1, We note that these values are less than 1 and fall within a transition occurs. This corresponds to phonon and magnon the lower range of the cases we discuss in Fig. 2.Again, bands with the same cutoff frequency. The denominator takes we have used a one-dimensional-system mean field formula on a value approaching 0 at a nonvanishing wave number for the Curie temperature. Should we have considered three- zJ (ka = 1.047). A near resonance condition is reached when dimensional structures, TC = where z is the number of kB a phonon with frequency ω0(ka = 1.047) and a magnon nearest neighbors. This number would range from 6 to 8 to = with frequency ω0(ka 1.047) interact to create a magnon 12 for simple cubic, body-centered-cubic, and face-centered- = = = + = of frequency ω0(ka 2.094) ω0(ka 1.047) ω0(ka cubic structures, effectively increasing the value of ρ by factors

224421-5 DEYMIER, VASSEUR, RUNGE, MANCHON, AND BOU-MATAR PHYSICAL REVIEW B 90, 224421 (2014) of 3, 4, or 6, respectively. Even with such multiplicative the SAW was constant and determined by the interdigitated factors, the parameter ρ would remain less than 1 for common transducer that emitted it. These authors observed a resonant ferromagnetic media. absorption of the SAW at a distinct value of the magnetic field strength in accordance with our one-phonon-two-magnon B. Effect of an external magnetic field resonant process prediction. In Ref. [15], a similar study was conducted for a nickel film. The strength of the magnetic We can also consider the nonlinear magnon-phonon in- field necessary to achieve resonance was shown to increase teractions in the presence of an external magnetic field b with increasing SAW frequency. This could be explained in oriented along the z direction. In this case, Eq. (6b) becomes the context of our one-phonon-two-magnon resonant process. = + 1 ∂H | | hn b . . This adds the linear terms γ bz εn,y and γ ∂Sn Indeed, in the long wavelength limit, ω0 (k) increases rapidly −γ |bz| εn,x to the right-hand sides of Eqs. (10a) and (10b), with increasing wave number owing to the quadratic form respectively. Here, |bz| is the intensity of the external field. of the magnon dispersion relation. In that same limit, the These linear terms do not affect the form of the results frequency of the phonons increases linearly with the wave concerning the frequency shifts in the dispersion relations number. One therefore needs to increase the value of γ |bz| obtained with the multiple timescale perturbation approach (i.e., the strength of the external field) as one increases the wave used previously. To account for these linear terms, one number (i.e., phonon frequency) to maintain the condition for simply needs to replace the magnon dispersion relation given resonance. = 8J 0 2 ka + | | by Eq. (A7)byω0 Sz sin 2 γ bz . The external field shifts the noninteracting magnon dispersion curve by a IV. ANALYSIS OF CASE II constant proportional to the zth component of the field. By employing this zeroth-order magnon dispersion relation, In Sec. III, we have addressed the coupling between we can rewrite the function d (k), defined by Eq. (A35), in the phonons and magnons in case I, whereby the interactions form between these excitations are nonlinear. In this section, we d (k) = ω (2k) − γ |b |−(ω (k) + ω (k)) + iψ. (19) derive the solutions of the equations of motion in Eqs. (12a), 0 z 0 0 (12b), and (16), which correspond to a linear coupling between In the absence of damping and a magnetic field, this phonons and magnons. The linear interactions arose from the expression gives a condition for the divergence (resonance) of fact that we limited the magnetoelastic coupling to cross terms the function f (k). This divergence corresponds to a process in the components of the spin in the system’s Hamiltonian. In involving a phonon with wave number k and frequency ω0 (k) this case, we assume that the spin degrees of freedom and the interacting with a magnon with the same wave number but a displacement take a plane wave form frequency ω (k) to produce a magnon with wave number 2k 0 ikna iωt = + εn,x = εo,xe e , (21a) and frequency ω0(2k) (ω0(k) ω0(k)). This corresponds to a single-phonon-two-magnon resonant process [28]. Keeping = ikna iωt this picture of phonon-magnon interactions as a basis for εn,y εo,ye e , (21b) our discussion, the application of an external magnetic field u = A eiknaeiωt. (21c) effectively modifies the phonon resonant frequency to a value n o + | | given by ω0 (k) γ bz . Therefore, using a fixed frequency Inserting these solutions into the equations of motion leads phonon to excite magnons, one can apply a variable external to a simple eigenvalue problem. The frequency of the linearly field to reach resonance. On the other hand, one can apply coupled phonon and magnon is subsequently obtained as a fixed magnetic field and change the phonon frequency to attain the resonance condition. This is illustrated by rewriting ω 2 + ω2 1 8 S0B2 Eq. (19) in the form of the following resonance condition ω2 = 0 0 ± ω2 − ω2 2 + z ω , (22) 2 2 0 0 m 0 = − = + | | ω0 ω0 (2k) ω0 (k) ω0 (k) γ bz . (20) where B = 4K S0 sin (ka) . Note that, in this relation, the left-hand term is independent T z In Eq. (22), we have used the expressions for the eigenvalues of an applied magnetic field. Consider now that the phonon of the phonons and magnons in the absence of interactions frequency ω (k) is fixed. That is, we excite the system with 0 given by Eqs. (A7) and (A10) (see Appendix). If an external a device such as an interdigitated transducer that emits elastic magnetic field is applied, the frequency ω is simply aug- waves over a narrow band (i.e., produces monochromatic 0 mented by the constant term γ |b0|. Equation (22) indicates elastic waves with a nearly fixed frequency and wave number). z that a resonance between a single magnon and a single phonon In that case, the quantity ω0 is also a constant. Let us = 0 will occur when ω0 (k) ω0 (k); that is, when the elastic assume that the system is subjected to an applied field bz such = wave and the magnetic wave have identical phase velocities. that the resonance condition is not met; that is, ω0 (k) + | 0| This resonance is more easily seen when performing a Taylor ω0 (k) γ bz . It is therefore possible to detune positively | 0|± expansion of Eq. (22) to the second order in KT or negatively the magnitude of the external field bz b to approach the condition for resonance given by Eq. (20). In 2 + 2 2 ω0 ω0 1 2 2 2 2 et al. ω = ± sgn ω − ω ω − ω Ref. [13], Weiler have studied the attenuation of a surface 2 2 0 0 0 0 acoustic wave (SAW) launched in a ferromagnetic cobalt 2 S0B2 ω film as a function of the magnitude of an external magnetic ± sgn ω2 − ω2 z 0 . (23) 0 0 2 − 2 field applied in some specific direction. The frequency of m ω0 ω0

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2 − 2 { 2 − 2 } the phenomena of resonant spin pumping with coherent elastic Depending on the sign of ω0 ω0 (i.e., sgn (ω0 ω0) ), 2 2 waves [13,15], variation of the strength of an externally applied the first two terms give ω0 and ω0. The third terms represents the effect of the resonance. The effect of magnetoelastic magnetic field at a fixed frequency of the elastic wave can coupling on the dispersion relations of the phonons and be used to tune the system toward resonance. In addition, magnons in case II is independent of the amplitude of the the strength of the external field at resonance increases with waves. increasing acoustic wave frequency. We can also consider the application of an external magnetic In Ref. [24], molecular dynamics simulations of coupled field. In that case, an external magnetic field can be applied lattice and spin dynamics in body-centered-cubic iron have to the ferromagnetic medium to tune the dispersion curve shown a softening of the magnons for all the wave vectors stud- of the magnons with the aim of achieving resonance under ied. In contrast, for iron (see Table I), we have observed soft- the condition of a stimulating acoustic wave with a fixed ening of the spin waves for long and short wavelength waves frequency. The observations concerning the dependence of but a hardening for intermediate wavelengths [Fig. 2(b)]. strength of the applied magnetic field on the frequency of The computational studies also conjectured the existence of the acoustic wave made in Sec. III for the single-phonon- a resonance between longitudinal lattice vibrations and spin two-magnon resonance also transpose to the case of a single- waves of the same frequency and wave number. We have phonon-single-magnon resonance. shown that such a resonant condition arises in the case of linear interaction between phonons and magnons (case II). However, in this case, for cubic symmetries, the phonon polarization V. CONCLUSIONS is transverse and not longitudinal. We noted in Sec. II that, We considered the effect of coupling between elastic waves for noncubic materials, such a coupling could arise with and magnetic waves in a one-dimensional discrete model longitudinal waves. Such a situation may occur in molecular system of a ferromagnetic medium. Two cases are addressed. dynamics simulations through the instantaneous breaking of The first case corresponds to a system where the magnons the symmetry of the cubic model. However, direct comparison and phonons interact nonlinearly. The nonlinear interaction with numerical simulations of ferromagnetic materials with is due to a displacement-dependent spin exchange coupling interactions between spin/lattice vibrations [22–24] is clouded constant that involves only diagonal terms in the components by the fact that computational results arise from the collective of the spin, i.e., to the interaction between longitudinal effects of multiple phonon-magnon scattering processes with components of the strain and spins in a cubic material. differing wave numbers. Our analytical solutions are focused The second case leads to linear interactions resulting from on the interactions between elastic and spin waves with specific a displacement-dependent spin exchange coupling constant wave numbers. Nevertheless, by providing analytical solutions that involves off-diagonal terms in the components of the for the spin degrees of freedom and the lattice dynamics in spin, i.e., to the interaction between shear components of model ferromagnetic atomic systems, our paper enables a the strain and spins in a cubic material. We have developed deeper insight into the complex interplay between magnons an approach based on the multiple timescale perturbation and phonons interacting via the magnetoelastic effect as a method to calculate analytically the effect of the nonlinear mechanism for spin pumping. This insight offers perspective magnetoelastic coupling on the dispersion relation of magnons in the design of ferromagnetic media that may optimize and phonons. For the sake of mathematical tractability, we the energy conversion between phonons and magnons. In have limited our paper to the interaction between magnons particular, one may be able to engineer the phonon and magnon and phonons with the same wave number. We have determined band structures of the ferromagnetic medium to control the the conditions that lead to magnon-phonon resonance. In the conditions of resonance. We have in mind composite structures absence of an external magnetic field, the resonance for the first composed of ferromagnetic and normal materials that can be case is associated with a single-phonon-two-magnon process. tailored to exhibit band gaps in their magnon band structures Such one-phonon-two-magnon processes are characteristic of [30]. Similarly, these types of composites may also act as the interaction between longitudinal phonons with magnons phononic crystals that can simultaneously exhibit band gaps in nearest neighbor Heisenberg models where the exchange in their phonon band structure [31]. Wave localization resulting coupling constant is modulated with respect to the lattice from these anomalous band structures may subsequently displacements [29]. Our two-magnon-one-phonon resonance be employed to enhance magnetoelastic interactions and leads to a frequency shift of the magnon and phonon bands that therefore the efficiency of energy conversion between phonon depends on the magnitude of the elastic and magnetic waves. and magnons in acoustic spin pumping devices. The frequency shift also depends on the relative magnitude of the cutoff frequencies of the two excitations. For the second case (i.e., linear coupling), the resonance includes only one phonon and one magnon. The effect of magnetoelastic APPENDIX coupling results in modifications to the dispersion relations that do not depend on the magnitude of the waves. We have 1. Multiple timescale perturbation method applied to case I also investigated the effect of an external field on that linear We apply the multiple timescale perturbation method [25]to resonant condition. The behavior of the linear system with the nonlinear equations of motion of case I, given by Eq. (15). respect to the strength of the magnetic field is similar to We assume that the constant K that couples the spins and that of the nonlinear system. We have demonstrated that, the displacement is small. Subsequently, we expand the spin similarly to observations reported in the literature concerning degrees of freedom and the displacement in a power series of

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K up to the second order We insert Eqs. (A2a), (A2b), and (A1a), (A1b) into Eqs. (10) and (15) and separate the terms that are independent of K = (0) + (1) + 2 (2) = εn,i εn,i Kεn,i K εn,i ,i x,y,z, (A1a) (zeroth-order terms in K), from the first-order terms in K, from the terms in K2. This operation leads to three sets of four u = u(0) + Ku(1) + K2u(2). (A1b) n n n n equations. (0) (1) (2) (0) (1) (2) The equations of motion to the zeroth order in K are The quantities εn,i ,εn,i ,εn,i ,un ,un ,un are functions of three time-related variables defined as τ = t, τ = Kt, and 0 1 ∂ε(0) J τ = K2t [25]. n,x =−2 S0 ε(0) − 2ε(0) + ε(0) , (A3a) 2 z n+1,y n,y n−1,y The first- and second-order time derivatives on the left- ∂τ0 hand side of Eqs. (10a)–(10c) and (15) are then rewritten ∂ε(0) J n,y =+2 S0 ε(0) − 2ε(0) + ε(0) , (A3b) as z n+1,x n,x n−1,x ∂τ0 2 2 (0) 2 (1) 2 (0) ∂ un ∂ un ∂ un ∂ un ∂ε(0) = + K + 2 n,z J (0) (0) (0) 2 2 2 =+2 ε ε + ε ∂t ∂τ0 ∂τ0 ∂τ0∂τ1 ∂τ n,x n+1,y n−1,y 0 2 (1) 2 (1) 2 (0) 2 (0) (0) (0) 2 ∂ un ∂ un ∂ un ∂ un − (0) + + K + 2 + 2 + , εn,y εn+1,x εn−1,x , (A3c) 2 2 ∂τ0 ∂τ0∂τ1 ∂τ0∂τ2 ∂τ1 ∂2u(0) (A2a) n = ω2 u(0) − 2u(0) + u(0) . (A3d) ∂τ 2 M n+1 n n−1 (0) (1) (0) 0 ∂ε = ∂ε + ∂ε + ∂ε K We have introduced in Eq. (A3d) the characteristic frequency ∂t ∂τ0 ∂τ0 ∂τ1 ω = β . ∂ε(2) ∂ε(1) ∂ε(0) M m + K2 + + , (A2b) To the first order in K, we obtain ∂τ0 ∂τ1 ∂τ2

∂ε(1) ∂ε(0) J 2 n,x + n,x =− 0 (1) − (1) + (1) + 0 (0) − (0) (0) − (0) − (0) − (0) (0) − (0) 2 Sz εn+1,y 2εn,y εn−1,y Sz un+1 un εn+1,y εn,y un un−1 εn,y εn−1,y , ∂τ0 ∂τ1 (A4a) ∂ε(1) ∂ε(0) J 2 n,y + n,y =+ 0 (1) − (1) + (1) − 0 (0) − (0) (0) − (0) − (0) − (0) (0) − (0) 2 Sz εn+1,x 2εn,x εn−1,x Sz un+1 un εn+1,x εn,x un un−1 εn,x εn−1,x , ∂τ0 ∂τ1 (A4b) ∂ε(1) ∂ε(0) J n,z + n,z = (1) (0) + (0) + (0) (1) + (1) − (1) (0) + (0) − (0) (1) + (1) 2 εn,x εn+1,y εn−1,y εn,x εn+1,y εn−1,y εn,y εn+1,x εn−1,x εn,y εn+1,x εn−1,x , ∂τ0 ∂τ1 (A4c) ∂2u(1) ∂2u(0) 2 n + 2 n = ω2 u(1) − 2u(1) + u(1) + S0 ε(0) − ε(0) 2 M n+1 n n−1 z n+1,z n−1,z ∂τ0 ∂τ0∂τ1 m 2 + ε(0) ε(0) − ε(0) + ε(0) ε(0) − ε(0) , (A4d) m n,x n+1,x n−1,x n,y n+1,y n−1,y

The equations to the order of K2 are ∂ε(2) ∂ε(1) ∂ε(0) J 2 n,x + n,x + n,x =−2 S0 ε(2) − 2ε(2) + ε(2) + S0 u(0) − u(0) ε(1) − ε(1) + u(1) − u(1) ε(0) − ε(0) ∂τ ∂τ ∂τ z n+1,y n,y n−1,y z n+1 n n+1,y n,y n+1 n n+1,y n,y 0 1 2 − (0) − (0) (1) − (1) − (1) − (1) (0) − (0) un un−1 εn,y εn−1,y un un−1 εn,y εn−1,y , (A5a)

∂ε(2) ∂ε(1) ∂ε(0) J 2 n,y + n,y + n,y =+2 S0 ε(2) − 2ε(2) + ε(2) − S0 u(0) − u(0) ε(1) − ε(1) ∂τ ∂τ ∂τ z n+1,x n,x n−1,x z n+1 n n+1,x n,x 0 1 2 + (1) − (1) (0) − (0) − (0) − (0) (1) − (1) − (1) − (1) (0) − (0) un+1 un εn+1,x εn,x un un−1 εn,x εn−1,x un un−1 εn,x εn−1,x , (A5b)

∂ε(2) ∂ε(1) ∂ε(0) J n,z + n,z + n,z = 2 ε(2) ε(0) + ε(0) + ε(0) ε(2) + ε(2) + ε(1) ε(1) + ε(1) − ε(2) ε(0) + ε(0) ∂τ ∂τ ∂τ n,x n+1,y n−1,y n,x n+1,y n−1,y n,x n+1,y n−1,y n,y n+1,x n−1,x 0 1 2 − (0) (2) + (2) − (1) (1) + (1) εn,y εn+1,x εn−1,x εn,y εn+1,x εn−1,x , (A5c)

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∂2u(2) ∂2u(1) ∂2u(0) ∂2u(0) 2 2 n + 2 n + 2 n + n = ω2 u(2) − 2u(2) + u(2) + S0 ε(1) − ε(1) + ε(0) ε(1) − ε(1) ∂τ 2 ∂τ ∂τ ∂τ ∂τ ∂τ 2 M n+1 n n−1 m z n+1,z n−1,z m n,x n+1,x n−1,x 0 0 1 0 2 1 + (1) (0) − (0) + (0) (1) − (1) + (1) (0) − (0) εn,x εn+1,x εn−1,x εn,y εn+1,y εn−1,y εn,y εn+1,y εn−1,y . (A5d)

2. Solutions of the equations of motion in case I doing this, we restrict our calculation to solutions whereby the a. Zeroth order in K magnons and phonons that interact with each other have the same wave number k. For instance, Eq. (A4d) can be rewritten Here, we solve the set of Eqs. (A3a)–(A3d). Equations as (A3a)–(A3c) and (A3d) are not coupled, so we can solve them ∂2u(1) separately. For Eqs. (A3a) and (A3b), we assume that the x n − u(1) − 2u(1) + u(1) 2 n+1 n n−1 and y components of the spin degree of freedom are expressed ∂τ0 as ∂A ∂A¯ 0 ikna −iω0τ0 0 −ikna iω0τ0 − − = 2iω0 e e + e e . (A11) (0) = ikna iω0τ0 + ¯ ikna iω0τ0 εn,x (τ0,τ1,τ2) X(τ1,τ2)e e X(τ1,τ2)e e , ∂τ1 ∂τ1 (A6a) The solution of the homogeneous equation (left-hand side − − of the equation equal to zero) is of the form (0) = ikna iω0τ0 + ¯ ikna iω0τ0 εn,y (τ0,τ1,τ2) Y (τ1,τ2)e e Y (τ1,τ2)e e . (1) (A6b) un,h(τ0,τ1,τ2) − − = ikna iω0τ0 + ¯ ikna iω0τ0 In Eqs. (A6a) and (A6b), a stands for the spacing between B0(τ1,τ2)e e B0(τ1,τ2)e e . (A12) ¯ ¯ two atoms in the chain. Here, X and Y are the complex The term in parenthesis on the right-hand side of Eq. (A11) conjugates of X and Y . Inserting Eqs. (A6a) and (A6b)into leads to secular terms unless one sets it to zero by imposing Eqs. (A3a) and (A3b) yields two sets of linear equations of X ∂A0 ∂A¯0 = 0 and = 0. This implies that the quantities A0 and ¯ ¯ ∂τ1 ∂τ1 and Y and X and Y . The eigenvalues associated with these sets ¯ of equations are given by A0 are functions of τ2 only. This constrains the zeroth-order solution [Eq. (A9)] to have the form J ka − − ω = 8 S0sin2 . (A7) u(0)(τ ,τ ) = A (τ )eiknae iω0τ0 + A¯ (τ )e iknaeiω0τ0 . (A13) 0 z 2 n 0 2 0 2 0 2 Similarly, the same constraint applies to the solution of the In the limit of small wave number k, one recovers the usual homogeneous equation [Eq. (A12)] quadratic dispersion relation for spin waves or magnons in the − − long wavelength. The eigenvectors are found to be Y =−iX (1) = ikna iω0τ0 + ¯ ikna iω0τ0 un,h(τ0,τ2) B0(τ2)e e B0(τ2)e e . and Y¯ = iX¯ . Then, the solutions in Eqs. (A6a) and (A6b)take the specific form (A14) (0) ikna −iω τ −ikna iω τ We now solve the first-order equations of motion for the ε = ε (τ ,τ )e e 0 0 + ε¯ (τ ,τ )e e 0 0 , (A8a) n,x o 1 2 0 1 2 spin degrees of freedom. We begin with Eq. (A4a). We rewrite − − (0) =− ikna iω0τ0 + ikna iω0τ0 it in the form εn,y iεo(τ1,τ2)e e iε¯0(τ1,τ2)e e , ∂ε(1) J (A8b) n,x + 0 (1) − (1) + (1) 2 Sz εn+1,y 2εn,y εn−1,y ∂τ0 where εo is some yet unknown function, andε ¯0 its complex ∂ε(0) 2 conjugate. =− n,x + 0 (0) − (0) (0) − (0) Sz un+1 un εn+1,y εn,y If one inserts Eqs. (A8a) and (A8b) into Eq. (A3c), the ∂τ1 right-hand side term is analytically zero, and one finds that − (0) − (0) (0) − (0) (0) un un−1 εn,y εn−1,y . (A15) εn,z is constant. It can be taken to be equal to zero. Finally, we also chose a general solution for the zeroth order The solution of the homogeneous equation is given by the displacement given by expression (0) (1) − − u (τ0,τ1,τ2) = ikna iω0τ0 + ikna iω0τ0 n εn,x,h η0(τ1,τ2)e e η¯0(τ1,τ2)e e . − − = ikna iω0τ0 + ¯ ikna iω0τ0 A0(τ1,τ2)e e A0(τ1,τ2)e e . (A9) Inserting Eq. (A8a) into the first term of the right-hand side Inserting this expression into the zeroth-order equation of of Eq. (A15) will lead to secular terms unless one imposes = = motion in Eq. (A3d) yields the well-known eigenvalues for the constraints εo(τ1,τ2) εo(τ2) andε ¯0(τ1,τ2) ε¯0(τ2). This elastic waves or phonons in a monoatomic harmonic chain results in the reformulation of the zeroth-order solution − − (0) = ikna iω0τ0 + ikna iω0τ0 ka εn,x εo(τ2)e e ε¯0(τ2)e e (A16) ω = 2ω sin . (A10) 0 M 2 as well as the reformulation of the homogeneous solution (1) − − = ikna iω0τ0 + ikna iω0τ0 b. First order in K εn,x,h η0(τ2)e e η¯0(τ2)e e . (A17) We can now insert the zeroth-order solutions obtained in the We finally rewrite Eq. (A15) by calculating the term between preceding subsection into the first-order Eqs. (A4a)–(A4d). By the square brackets. After lengthy algebraic manipulations,

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Eq. (A15) becomes Particular solutions of Eqs. (A18) and (A21) can be introduced (1) in the form ∂εn,x J 0 (1) (1) (1) + 2 S ε + − 2ε + ε − − + − + z n 1,y n,y n 1,y (1) = 2ikna i(ω0 ω0)τ0 + ¯ 2ikna i(ω0 ω0)τ0 ∂τ0 εn,x,p X (τ2)e e X (τ2)e e ,

(A22a) J 0 2ikna −i(ω +ω )τ = S ( ka − ka ) ε A e e 0 0 0 2 z 2 sin(2 ) 2sin( ) ( 0 0 (1) 2ikna −i(ω +ω )τ −2ikna i(ω +ω )τ − + = 0 0 0 + 0 0 0 2ikna i(ω0 ω )τ0 εn,y,p Y (τ2)e e Y (τ2)e e . + ε¯0A¯0e e 0 ). (A18) (A22b) We proceed in the same way for the equation of motion in Eq. (A4b). We ﬁnd that we need to reformulate the zeroth-order Inserting these particular solutions into Eqs. (A18) and solution (A21) and solving for X,Y ,X,Y yields − − =− (0) =− ikna iω0τ0 + ikna iω0τ0 X iε0A0f (k), (A23a) εn,y iεo(τ2)e e iε¯0(τ2)e e (A19) as well as the homogeneous solution Y =−iX =−ε0A0f (k), (A23b)

(1) − − =− ikna iω0τ0 + ikna iω0τ0 = ¯ εn,y,h iη0(τ2)e e iη¯0(τ2)e e . (A20) X iε¯0A0f (k), (A23c) The equation of motion in Eq. (A4b) is rewritten in the Y = iX =−ε¯ A¯ f (k), (A23d) form 0 0 4 S0(sin(2ka)−2 sin(ka)) (1) z where we define f (k) = .Wehavein- ∂εn,y J 8J S0sin2(ka)−(ω +ω )+iψ − 0 (1) − (1) + (1) z 0 0 2 Sz εn+1,x 2εn,x εn−1,x ∂τ0 troduced a damping term in the definition of f (k) since its denominator may become zero. We will take the limit ψ → 0 J 0 2ikna −i(ω +ω )τ =− S i( ka − ka ) ε A e e 0 0 0 at the end of our derivation. 2 z 2 sin(2 ) 2sin( ) ( 0 0 Adding the homogeneous and particular solutions gives the −2ikna i(ω0+ω )τ0 − ε¯0A¯0e e 0 ). (A21) first-order solutions

− − − + − + (1) = ikna iω0τ0 + ikna iω0τ0 − 2ikna i(ω0 ω0)τ0 − ¯ 2ikna i(ω0 ω0)τ0 εn,x η0(τ2)e e η¯0(τ2)e e f (k)(iε0A0e e iε¯0A0e e ), (A24a)

− − − + − + (1) =− ikna iω0τ0 + ikna iω0τ0 − 2ikna i(ω0 ω0)τ0 + ¯ 2ikna i(ω0 ω0)τ0 εn,y iη0(τ2)e e iη¯0(τ2)e e f (k)(ε0A0e e ε¯0A0e e ). (A24b)

(0) ∂εn,z (0) We can now address the solutions of the ﬁrst-order Eq. (A4c). We have to set = 0 (i.e., make ε independent of τ1)to ∂τ1 n,z eliminate secular solutions. Inserting the zeroth- and ﬁrst-order solutions into Eq. (A4) results in ∂ε(1) 8J n,z ikna −iω0τ0 −ikna iω0τ0 = f (k)ε0ε¯0(cos(ka) − 2 cos(2ka))(A0e e + A¯0e e ). (A25) ∂τ0 Solutions of Eq. (A25) are then easily obtained to within a constant of integration which may be taken equal to zero 8J 1 − − (1) = − − ikna iω0τ0 + ¯ ikna iω0τ0 εn,z f (k)ε0ε¯0(cos(ka) 2 cos(2ka)) ( A0e e A0e e ). (A26) iω0

c. Second order in K

We first deal with the Eq. (A5d) for phonons. The terms with derivatives with respect to τ1 are equal to zero. Inserting the zeroth- and first-order solutions determined previously and after extensive algebraic manipulations we obtain 2 (2) 2 (0) ∂ u ∂ u 8 − − n − ω2 u(2) − 2u(2) + u(2) =−2 n − f (k)g(k)ε ε¯ A eiknae iω0τ0 + A¯ e iknaeiω0τ0 , (A27) 2 M n+1 n n−1 0 0 0 0 ∂τ0 ∂τ0∂τ2 m where the function g(k) is given by the expression 4J 1 = 0 − + − g(k) Sz (cos(ka) cos(2ka)) sin(ka) sin(2ka) sin(ka). (A28) ω0 We subsequently define

−iϕ(τ2) A0(τ2) = α(τ2)e , and

+iϕ(τ2) A¯0(τ2) = α(τ2)e .

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With this deﬁnition, the term 2 (0) ∂ u ∂A0(τ2) − ∂A¯0(τ2) − − 2 n =−2 −iω eiknae iω0τ0 + iω e iknaeiω0τ0 ∂τ ∂τ 0 ∂τ 0 ∂τ 0 2 2 2 ∂ϕ ∂α ∂ϕ ∂α ikna −iω0τ0 −iϕ −ikna iω0τ0 iϕ = e e 2ω0e α + i + e e 2ω0e α − i . (A29) ∂τ2 ∂τ2 ∂τ2 ∂τ2 − The homogeneous equation (i.e., left-hand term of Eq. (A27) set equal to zero) has solutions of the form eiknae iω0τ0 − and e iknaeiω0τ0 ; so to avoid secular terms, the right-hand side of Eq. (A27) must be equal to zero. This gives the two relations −iϕ ∂ϕ ∂α 8 −iϕ 2ω0e α + i = f (k)g(k)ε0ε¯0αe , ∂τ2 ∂τ2 m and iϕ ∂ϕ ∂α 8 iϕ 2ω0e α − i = f (k)g(k)ε0ε¯0αe . ∂τ2 ∂τ2 m

∂α Since ε0ε¯0 is real, one must have = 0, and α = α0 is a constant. Both preceding equations give the same relation ∂τ2 ∂ϕ 4 = f (k)g(k)ε0ε¯0. ∂τ2 mω0

This equation leads to a linear dependency of ϕ on the time τ2 to within an arbitrary constant (which is taken to be zero)

4 4 2 2 ϕ = f (k)g(k)ε0ε¯0τ2 = f (k)g(k)ε0ε¯0K τ0 = K h(k)τ0. (A30) mω0 mω0 This implies that the zeroth-order solution given by Eq. (A13) for the displacement includes a frequency shift. The complete zeroth-order solution for the displacement is now

− + 2 − + 2 (0) = ikna i(ω0 K h(k))τ0 + ikna i(ω0 K h(k))τ0 un (τ0,τ2) α0e e α¯ 0e e . (A31) 2 The frequency of the phonon is shifted from ω0 to ω0 + K h(k). (2) (2) We now address the second-order equations of motion for εn,x and for εn,y ; that is, Eqs. (A5a) and (A5b). For this, we need to insert the expressions for the complete zeroth-order displacement [Eq. (A31)] and the ﬁrst-order displacement given by Eq. (A14). We also need the zeroth-order solutions for ε [Eqs. (A16) and (A19)] and their ﬁrst-order expressions given by Eqs. (A24a) and (A24b). Both equations of motion become

∂ε(2) J n,x + 0 (2) − (2) + (2) 2 Sz εn+1,y 2εn,y εn−1,y ∂τ0

(0) ∂εn,x 4 − + − + =− − 0 − + i2kna i(ω0 ω0)τ0 − ¯ + ¯ i2kna i(ω0 ω0)τ0 Sz (sin(2ka) 2sin(ka))[(η0A0 ε0B0)e e (¯η0A0 ε¯0B0)e e ] ∂τ2 4 0 2 i3kna −i(2ω +ω )τ 2 −i3kna i(2ω +ω )τ − i S f k ( ka − ka − ka ) ε A e e 0 0 0 − ε A¯ e e 0 0 0 z ( ) sin(3 ) sin(2 ) sin( ) 0 0 ¯0 0

4 0 ikna −iω τ −ikna iω τ − i S f k A A¯ ( ka − ka ) −ε e e 0 0 + ε e e 0 0 , z ( ) 0 0 sin(2 ) 2sin( ) ( o ¯0 ) (A32a)

∂ε(2) J n,y − 0 (2) − (2) + (2) 2 Sz εn+1,x 2εn,x εn−1,x ∂τ0

(0) ∂εn,y 4 − + − + =− − 0 − + i2kna i(ω0 ω0)τ0 − ¯ + ¯ i2kna i(ω0 ω0)τ0 i Sz (sin(2ka) 2sin(ka))[(η0A0 ε0B0)e e (¯η0A0 ε¯0B0)e e ] ∂τ2 4 0 2 i3kna −i(2ω +ω )τ 2 −i3kna i(2ω +ω )τ − S f k ( ka − ka − ka ) ε A e e 0 0 0 + ε A¯ e e 0 0 0 z ( ) sin(3 ) sin(2 ) sin( ) 0 0 ¯0 0

4 0 ikna −iω τ −ikna iω τ + S f k A A¯ ( ka − ka ) ε e e 0 0 + ε e e 0 0 . z ( ) 0 0 sin(2 ) 2sin( ) ( o ¯0 ) (A32b)

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ikna −iω τ −ikna iω τ Only the terms in e e 0 0 and e e 0 0 will give secular solutions. To eliminate these solutions, we set (0) ∂εn,x 4 − − =− − 0 ¯ − − ikna iω0τ0 + ikna iω0τ0 0 i Sz f (k)A0A0(sin(2ka) 2sin(ka))( εoe e ε¯0e e ), ∂τ2 (0) ∂εn,y 4 − − =− + 0 ¯ − ikna iω0τ0 + ikna iω0τ0 0 Sz f (k)A0A0(sin(2ka) 2sin(ka))(εoe e ε¯0e e ). ∂τ2 (0) (0) Using the expressions previously derived for εn,x and εn,y , these two conditions reduce to ∂ε 4 0 = 0 ¯ − i Sz f (k)A0A0(sin(2ka) 2sin(ka))ε0(τ2), ∂τ2 ∂ε¯ 4 0 =− 0 ¯ − i Sz f (k)A0A0(sin(2ka) 2sin(ka))ε¯0(τ2). ∂τ2

−iϕ(τ2) iϕ(τ2) Similarly to ﬁnding the second-order solution for the displacement, we deﬁne ε0(τ2) = λ(τ2)e andε ¯0(τ2) = λ(τ2)e . Inserting these expressions into the preceding conditions leads to taking λ(τ2) = λ0 (i.e., a constant) and 4 ϕ τ =− S0f k α2( ka − ka )τ = K2h k τ . ( 2) z ( ) 0 sin(2 ) 2sin( ) 2 ( ) 0 (A33) To arrive at that equation, we have assumed that the constant of integration is zero. (0) (0) Equation (A33) introduces a correction to the zeroth-order solutions for εn,x and εn,y which become − + 2 − + 2 (0) = ikna i(ω0 K h (k))τ0 + ¯ ikna i(ω0 K h (k))τ0 εn,x λoe e λ0e e , (A34a)

− + 2 − + 2 (0) =− ikna i(ω0 K h (k))τ0 + ¯ ikna i(ω0 K h (k))τ0 εn,y iλoe e iλ0(τ2)e e . (A34b) Similarly to the phonons, the magnons are frequency shifted. The denominator of the function f (k) was written as J d k = S0 2 ka − ω + ω + iψ ( ) 8 z sin ( ) ( 0 0) (A35) The introduction of the small damping iψ enables us to overcome the divergence at the resonance. Indeed, we can take the following limit using Sokhotski’s formula [32] 1 1 limψ→ = limψ→ 0 0 J 0 2 d(k) 8 S sin (ka) − (ω0(k) + ω (k)) + iψ z 0 1 J = − iπδ 8 S0sin2(ka) − (ω (k) + ω (k) . J 0 2 − + z 0 0 8 Sz sin (ka) (ω0(k) ω0(k)) pv

In the previous equation, the bracket []pv means Cauchy’s principal value. The shift in frequency of the magnons and phonons becomes ﬁnite with a damping at the resonance. The imaginary term with the delta function leads to a ﬁnite lifetime for the phonon and magnon excitations at the resonance.

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