Generalized form of the magnetic anisotropy field in micromagnetic and atomistic spin models

We present a general approach to the derivation of the effective anisotropy field which determines the dynamical behavior of magnetic spins according to the Landau-Lifshitz-Gilbert equation. The approach is based on the gradient in spherical polar coordinates with the final results being expressed in Cartesian coordinates as usually applied in atomistic and micromagnetic model calculations. The approach is generally valid for all orders of anisotropies including higher-order combinations of azimuthal and rotational anisotropies often found in functional magnetic materials such as permanent magnets and an emerging class of antiferromagnetic materials with applications in spintronics. Anisotropies are represented in terms of spherical harmonics which have the important property of rational temperature scaling. Effective field vectors are given for anisotropies up to sixth order, presenting a unified framework for implementing higher-order magnetic anisotropies in numerical simulations.