Calculation of the magnetocrystalline anisotropy energy using a torque method

Abstract

We calculate the torque on a magnetic moment with arbitrary orientation in a solid and use it to calculate the magnetocrystalline anisotropy energy (MAE) of bulk materials by integrating the torque along an angular path connecting the easy and hard magnetization directions. We apply this approach to the calculation of the MAE of elemental ferromagnets Ni and Fe using a tight-binding model with an added spin-orbit-coupling (SOC) term λSOL⋅S. The MAE for both Fe and Ni is computed for a wide range of values of the SOC strength. Our results for the MAE calculated with the torque method agree with the MAE determined from energy differences. The convergence rate as a function of k-vector needed for integrating over the Brillouin zone for the torque method is comparable to the convergence rate of conventional energy difference schemes. We compare the calculated torque as function of angle to the lowest order term in the expansion in anisotropy constants.