A Study of Spin–Phonon Interactions in Ferromagnetic Insulators. I. The Spin Wave Spectrum

Abstract

A field-theoretical perturbation method is used to study the effects of spin-phonon interactions in a Heisenberg ferromagnet. The calculations are carried out by employing an expansion in powers of the high-density parameter l/z, where z is the number of spins interacting with any given spin. I n lowest order ( 1 / z ) O this leads to molecular field theory, and the effects of spin fluctuations and spin-phonon interactions are then introduced in higher orders of the perturbation expansion. The renormalised energy and damping of the spin wave excitations are evaluated to second order in the spin-phonon coupling, and the dependence of these quantities on temperature, wave vector, and magnetic field is obtained.The calculations are valid over a wide range of temperature below Tc.Le couplage spins-phonons dans un ferromagnhtique de Heisenberg est ittudii! par une mkthode de perturbation de la thitorie des champs, en ntilisant un d6veloppement en puissances d'un parametre de haute densiti: l / z ( z -le nombre de spins voisins avec lesquels chaque spin interagit). Le terme en (l/z)O reproduit les resultats de la thitorie du champ mol6culaire; les effets des fluctuations des spins et du couplage spins-phonons sont introduits en considitrant les ordres supitrieurs du dkveloppement en sitrie. La renormalisation de l'knergie e t l'amortissement des ondes de spin sont i!valui!es en second ordre de perturbation en couplage spins-phonons, et la ditpendance de la temphature, du champ magnbtique et du vecteur d'onde sont examinites. On trouve que les rksultats sont valables dans un domaine large des tempitratures au-dessous de la tempitrature de Curie Tc.

Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment

“…This behavior indicates that the stronger effect of the spin-phonon coupling comes via the <S i •S j > term in Eq. (8). By contrast, the B and C coefficients for CoF 2 are approximately equal for both modes.…”

“…In fact, the four Raman-active modes leave the magnetic atoms stationary, while producing displacements in r n that can be written down following [29]. Then a Taylorseries expansion in terms of these displacements yields a leading-order spin-phonon coupling Hamiltonian that is quadratic in the spin operators and linear in the phonon operators (see, e.g., [6][7][8][9][10][11]). The effects of the weaker intra-sublattice exchange [1] on the phonon modes in the rutile antiferromagnets will be neglected here, but it is important to include the magnetic single-ion anisotropy terms in the full Hamiltonian, since these are relatively large for some of the materials studied here.…”

“…The basic mechanism assumed for the spin-phonon coupling in the rutile-structure antiferromagnets arises from a spatial modulation of the dominant exchange parameter due to the relative displacements of the atoms corresponding to the phonon modes, as first proposed by Akhiezer [6]. This effect can lead to a modification of both the phonon and magnon dispersion relations, and has been studied theoretically for ferromagnets [7][8][9] and antiferromagnets [10,11]. In this work an extended version of the approach used in our previous spin-phonon calculations [12,13], and applied primarily to FeF 2 and MnF 2 , will be followed.…”