Parallel Micromagnetic Monte Carlo Method for Computation of Thermodynamic Equilibrium States in One and Two-Sublattice Systems

Abstract

An efficient method for computing thermodynamic equilibrium states at the micromagnetic length scale is introduced [1], based on the Markov chain Monte Carlo method. Trial moves include not only rotations of spins, but also a change in their magnetization length. The method is parameterized using the longitudinal susceptibility, reproduces the same Maxwell-Boltzmann distribution as the stochastic Landau-Lifshitz-Bloch equation, and is applicable both below and above the Curie temperature. The algorithm has been implemented in Boris [2], is fully parallel [3], can be executed on graphical processing units, and efficiently includes the long range dipolar interaction. Applications to finite-temperature hysteresis loop modelling, chiral magnetic thin films, granular magnetic media, and artificial spin ices are discussed. A typical application is exemplified in relation to Figure 1, where finite-temperature hysteresis loops in a 2 nm thin Co film with interfacial Dzyaloshinsky-Moriya interaction are computed. Simulation of such hysteresis loops in chiral films is problematic with quasi-zero temperature methods such as steepest descent energy minimizer, since the nucleation and annihilation of skyrmions and labyrinth domain structure is principally a thermally activated process.Furthermore, the method is also extended to two-sublattice systems, allowing modelling of ferrimagnetic and antiferromagnetic materials at the micromagnetic length scale. Using a multi-layered computation, which includes both ferromagnetic and antiferromagnetic layers, exchange bias may be simulated in bilayers, as well as more complex multi-layered structures including synthetic antiferromagnetic stacks. Moreover we show the method may also be applied to study of thermal and athermal training effects in exchange bias.  Figure 1 - Hysteresis loops in a 2 nm thin Co film on Pt, using a uniform 300 K temperature (thin dashed line), and a Gaussian temperature profile varying from 300 K at extremities to 400 K at the centre (thick dashed line). The insets show the perpendicular magnetization component on the increasing field sweep, with blue denoting magnetization into the plane, and red out of the plane.