Magnetocrystalline anisotropy energy of transition-metal thin films: A nonperturbative theory

The magnetocrystalline anisotropy energy $E_{\mathrm{anis}}$ of free-standing monolayers and thin films of Fe and Ni is determined using two different semiempirical schemes. Within a tight-binding calculation for the $3d$ bands alone, we analyze in detail the relation between band structure and $E_{\mathrm{anis}},$ treating spin-orbit coupling (SOC) nonperturbatively. We find important contributions to $E_{\mathrm{anis}}$ due to the lifting of band degeneracies near the Fermi level by SOC. The important role of degeneracies is supported by the calculation of the electron temperature dependence of the magnetocrystalline anisotropy energy, which decreases with the temperature increasing on a scale of several hundred K. In general, $E_{\mathrm{anis}}$ scales with the square of the SOC constant ${\lambda }_{\mathrm{so}}.$ Including 4$s$ bands and $s$-$d$ hybridization, the combined interpolation scheme yields anisotropy energies that quantitatively agree well with experiments for Fe and Ni monolayers on Cu(001). Finally, the anisotropy energy is calculated for systems of up to 14 layers. Even after including $s$ bands and for multilayers, the importance of degeneracies persists. Considering a fixed fct-Fe structure, we find a reorientation of the magnetization from perpendicular to in-plane at about 4 layers. For Ni, we find the correct in-plane easy axis for the monolayer. However, since the anisotropy energy remains nearly constant, we do not find the experimentally observed reorientation.