  Using a first-principles fully relativistic scheme based on ultrasoft
pseudopotentials and density functional perturbation theory, we study the
magnetocrystalline anisotropy free energy of the ferromagnetic binary compound
MnBi. We find that differences in the phonon dispersions due to the different
orientations of the magnetization (in-plane and perpendicular to the plane)
give a difference between the vibrational free energies of the high-temperature
and low-temperature phases. This vibrational contribution to the
magnetocrystalline anisotropy energy (MAE) constant, $K_u$, is non-negligible.
When the energy contribution to the MAE is calculated by the PBEsol exchange
and correlation functional, the addition of the phonon contribution allows to
get a $T = 0$ K $K_u$ and a spin-reorientation transition temperature in
reasonable agreement with experiments.
