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 $K_{\mathrm{u}}(T)\mathrm{/}{K}_{\mathrm{u}}(0)=[M(T)\mathrm{/}M(0){]}^{\alpha }$. 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.