Stable and fast semi-implicit integration of the stochastic Landau–Lifshitz equation

Abstract

S ta b le and fast sem i-im p licit in teg r a tio n o f th e sto ch a stic L andau-L ifshitz eq u a tio n J .H . M e n tin k 1, M .V . T retyak ov2, A . F a so lin o 1, M .I. K a tsn e lso n 1, T h . R a sin g 1 A b s t r a c t . We propose new sem i-im plicit num erical m ethods for th e integ ratio n of th e sto ch astic L andau-L ifshitz equation w ith b u ilt-in angular m om entum conservation. T he perform ance of th e proposed in teg rato rs is te ste d on th e 1D H eisenberg chain. For th is system , our schem es show b e tte r stab ility properties and allow us to use considerably larger tim e steps th a n sta n d a rd explicit m ethods. A t th e sam e tim e, these sem i-im plicit schem es are also of com parable accuracy to and co m putationally m uch cheaper th a n th e sta n d a rd m idpoint im plicit m ethod. T he results are of key im p o rtan ce for atom istic spin dynam ics sim ulations and th e stu d y of spin dynam ics beyond th e m acro spin approxim ation.

Complex magnetic phase diagram and skyrmion lifetime in an ultrathin film from atomistic simulations

“…The integration was performed using the so-called semi-implicit B method discussed in Ref. [65], which conserves the length of the spins.…”

Thermally induced magnetization switching in Gd/Fe multilayers

“…These results were obtained by simulating a system of 32 × 32 × 128 fcc unit cells (524,288 spins) with periodic boundary conditions. For each of the Fe and Gd spins we write a Landau-Lifshitz-Gilbert equation and solve it using the Heun numerical integration scheme [27][28][29][30] . The system is equilibrated until there is only a small change in the magnetization for each temperature point.…”

Classical spin model of the relaxation dynamics of rare-earth doped permalloy

“…28, which is a symplectic integrator that conserves the magnitude of the spin length, a requirement of the model. The basis of the scheme is the implicit midpoint scheme as given in Ref.…”