Validity of perturbation theory in calculations of magnetocrystalline anisotropy in Co-based layered systems

Validity of second-order perturbation theory (PT) is examined for magnetocrystalline anisotropy (MCA) energy in Co films with enhanced spin-orbit coupling (SOC) and Co/Pt bilayers. Comparison with accurate results obtained with the force theorem (FT) reveals significant discrepancies in the dependence of the MCA energy on the Co thickness. For systems with strong SOC, the PT fails to correctly describe the oscillations of the MCA energy, largely overestimating their amplitude and even failing (for Co/Pt bilayers) to reproduce their specific periodicity. These failures specifically concern the dominating oscillations with the 2-monolayer period which arise from pairs of quantum well (QW) minority-spin d states in the Co layer, degenerate at the centre of the Brillouin zone (BZ). A simplified model of such states demonstrates that the large discrepancies between PT and FT predictions arise from the breakdown of the PT in a region around the BZ centre where the energy spacing between states within each pair is small compared to the SOC strength. The oscillation amplitude of the MCA energy calculated with the FT is limited by the finite energy spacing between consecutive QW pairs, whereas this amplitude grows quadratically with the SOC strength in the PT calculations. Furthermore, for weak and moderate SOC strengths, the accuracy of the PT diminishes with increasing the ratio of the SOC constant to temperature. This explains why the PT overestimates the amplitude of MCA energy oscillations at low temperatures, even for the Co film with a relatively weak nominal SOC. For the Co/Pt bilayer, the strong temperature dependence of the oscillation amplitude in the PT approach leads to the MCA energies markedly different at low and zero temperatures, of opposite sign and several times larger in magnitude, compared to the FT results.