Temperature Dependence of Magnetocrystalline Anisotropy in Itinerant Ferromagnets

We theoretically investigated magnetocrystalline anisotropy (MA) at a finite temperature $T$ in ferromagnetic metals. Assuming a Rashba-type ferromagnet with uniaxial MA, we defined the MA constants $K_\mathrm{u}(T)$ derived from several different concepts. Our purpose was to examine the equality between them and to confirm a power law between $K_\mathrm{u}(T)$ and magnetization $M(T)$ in the form of $K_\mathrm{u}(T)/K_\mathrm{u}(0)=[M(T)/M(0)]^\alpha$. We demonstrate that $\alpha$ equals 2 in the itinerant-electron limit and increases with the localized feature of electrons passing through $\alpha=3$, predicted for the single-ion MA in spin models.