Quantifying temperature-induced magnetic disorder effects on the anisotropy of MnBi

MnBi is remarkable for having a magnetocrystalline anisotropy (MCA) that increases with temperature. This unusual behavior has been attributed to the thermal expansion of the lattice and, more recently, to an anisotropic vibrational free energy. However, the effect of magnetic fluctuations on the MCA has not yet been quantified. Here, first-principles density-functional theory calculations based on the disordered local moment picture (DFT-DLM) are used to calculate the MCA of MnBi in the presence of magnetic disorder. The MCA is obtained from the magnetic torque, calculated as a function of magnetization angle and temperature T. At fixed ionic positions, the MCA decays monotonically with increasing T. The DFT-DLM torques provide access to the individual anisotropy constants kappa(l) (which parametrize the relation between magnetic energy and magnetization angle), and their dependence on order parameter m. The lowest order constant kappa(2) follows single-ion-like behavior at low T but decays as m(4) as T increases, while the higher-order constants deviate strongly from single-ion predictions. Zero-T calculations show that a spin reorientation transition can be triggered by removing 0.1 electrons, resulting in c-axis magnetization once 0.25 electrons have been removed. The zero-temperature calculations are cross-validated using an alternative implementation of DFT based on wave functions, plane waves, and pseudopotentials.