Dissecting the mechanisms of magnetocrystalline anisotropy in alloys: Electronic structure analysis tools and applications to (Fe$_{1-x}$Co$_x$)$_2$B and Li$_{3-x}$Fe$_x$N alloys

We describe a first-principles code and a set of tools providing detailed information about the mechanisms of the magnetocrystalline anisotropy (MCA) in alloys. The spin-orbit coupling (SOC) is included in the Green's function-based linear muffin-tin orbital (LMTO) method combined with the coherent potential approximation. Third-order correspondence with the LMTO Hamiltonian is formally demonstrated. The analysis tools include the identification of contributions from different spin channels, single-ion and two-ion terms and alloy components by computing the SOC energy with scaled SOC parameters, as well as a full reciprocal-space resolution of MCA in the Brillouin zone. Application of these tools is illustrated for the (Fe$_{1-x}$Co$_x$)$_2$B system, where the complicated non-monotonic concentration dependence of MCA is attributed to the combination of band filling and SOC selection rules. For Li$_{3-x}$Fe$_x$N we demonstrate the interplay between chemical disorder, orbital polarization, and correlation effects in a doubly degenerate impurity band.