Magnon-phonon damping calculations in the spin-lattice dynamics model

Abstract

The relaxation of the magnetisation in magnetic materials is of great importance from both an applied and fundamental point of view. Conventional modelling of magnetisation dynamics employs the Landau-Lifshitz-Gilbert (LLG) equation [1], where the damping is included phenomenologically by the Gilbert damping term. This describes the coupling of the magnetic modes (given primarily by the atomic spin) to the non-magnetic modes (lattice vibrations and electron orbits) which are assumed to be in equilibrium. Recent studies of the dynamics induced by THz laser pulses has highlighted the necessity of understanding magnetisation relaxation beyond this assumption.In reality the spin and lattice dynamics mutually influence one another, hence it is necessary to employ a unified model of molecular and spin dynamics [2,3,4], Spin-Lattice dynamics (SLD). In the present work we employ a model where the transfer of energy and angular momentum between the lattice and the spin system (Fig. 1) is realised by the pseudo-dipolar coupling [3,4], which arises from the spin-orbit interaction and can be parameterised by magneto-elastic experiments. The spin system exchange parameters are taken from ab-initio parameterisation for BCC iron [2], while for the phonon system interactions potential we compare the Harmonic and Morse potentials. Our results in Fig. 1 show that equilibration of both sub-systems can be obtained on a sub-nanosecond timescale in both the microcanonical and canonical ensembles, the relaxation time being governed by the energy initially deposited in the system. We also observe that the equilibrium magnetisation is independent of the thermostat and by coupling the spin system only to the lattice vibrations the magnetisation temperature dependence can be reproduced without the need of a phenomenological spin damping [3].Our model allows the evaluation of the effective magnon-phonon damping (Fig. 2) which agrees well with the values measured in magnetic insulators where they depend predominantly on magnon-phonon coupling rather than on electronic effects. The functional form of the damping variation is quadratic, in accordance with the form of the pseudo-dipolar coupling term. The magnon-phonon damping has been also calculated for an alternate form of the coupling, where the pseudo-dipolar coupling is replaced by an on-site form, i.e a Néel-like anisotropy term. The latter leads to much smaller damping values, as shown in Ref. [3], suggesting that the magnon-phonon damping can clearly have complex behaviour depending on the properties of the system, especially the coupling term. Based on this remark, no universal behaviour of damping as a function of temperature can be deduced for spin lattice models.The model developed in this work [3] opens the possibility to describe the distinct dynamics of spins and phonons, necessary for the understanding of ultrafast magnetisation dynamics experiments and the subsequent angular momentum transfer between the two subsystems. We have demonstrated that the model works well in the absence of a phenomenological Gilbert damping, which consists mainly of electronic contributions, hence the SLD model can be employed to study magnetic insulators, such as YIG, where the principal contribution to damping is via magnon-phonon interactions. **