Stochastic ferrimagnetic Landau-Lifshitz-Bloch equation for finite magnetic structures

Precise modeling of the magnetization dynamics of nanoparticles with finite-size effects at fast varying temperatures is a computationally challenging task. Based on the Landau-Lifshitz-Bloch (LLB) equation we derive a coarse-grained model for disordered ferrimagnets, which is both fast and accurate. First, we incorporate stochastic fluctuations to the existing ferrimagnetic LLB equation. Further, we derive a thermodynamic expression for the temperature-dependent susceptibilities, which is essential to model finite-size effects. Together with the zero-field equilibrium magnetization the susceptibilities are used in the stochastic ferrimagnetic LLB to simulate a ferrimagnetic Gd-30 (FeCo)(70) particle with a diameter of 5 nm and a height of 10 nm under various external applied fields and heat pulses. The obtained trajectories agree well with those of an atomistic model, which solves the stochastic Landau-Lifshitz-Gilbert equation for each atom. Finally, we apply a 50-fs heat pulse with a maximum temperature of 1193 K to a Gd-24 (FeCo)(76) particle of the same size to test the proposed model for all-optical switching (AOS). We observe switching with a ferromagneticlike state, which was identified to be the key for AOS in GdFeCo. Although the coarse-grained model in general shows remarkably good agreement with atomistic simulations, the computed switching probability is lower than expected. The magnetization trajectories seem to be too far away from equilibrium for ultrashort and very high heat pulses, which are typical for AOS. This leads to the conclusion that either the model needs higher-order corrections to accurately describe AOS, or that the two-spin model is not capable of describing deterministic AOS at all.