Advanced Atomistic Modelling of Magnetisation Dynamics with Spin and Lattice Degrees of Freedom

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 a term that 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], Spin-Lattice dynamics (SLD). The transfer of energy and angular momentum between the lattice and the spin system is realised by the pseudo-dipolar coupling [3], which arises from the spin-orbit interaction and can be parameterised by magneto-elastic experiments. Our results show that the equilibrium magnetisation is independent of the thermostat used 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. The autocorrelation of the coupling field (which acts as an induced noise on the spin systems) shows well defined excitation peaks at nonzero frequencies corresponding to both phonon and magnon modes, proving a successful coupling between the two subsystems (Fig.1). The magnon-phonon damping is evaluated within this framework (Fig. 2) and agrees well with the damping measured in magnetic insulators where it depends more on magnon-phonon coupling than electronic effects. Our model opens the possibility to describe the distinct dynamics of spins and phonons, necessary for the understanding of ultrafast magnetisation dynamics experiments.