Polarization of a Rotationally Degenerate Pair of Bands Causing the Uniaxial Magnetism of Permanent Magnets

By analogy with the selection rule governing the uniaxial magnetism of magnetic insulators, we found that the uniaxial magnetism of permanent magnets (i.e., itinerant or metallic ferromagnets) requires the presence of a strong polarization (i.e., unequal populations of the two members constituting a pair) in one of the three rotationally degenerate pairs, (xz², yz²), [xyz, (x² - y²)z], and [x(x² - 3y²), y(y² - 3x²)], of the f-bands in rare-earth (RE)-based ones, and in one of the two rotationally degenerate pairs, (xz, yz) and (xy, x² - y²), of the d-bands in RE-free ones. This requirement was confirmed by determining the magnetocrystalline anisotropy energies (MAEs) of permanent magnets SmCo₅ and MnAl as well as the partial density of states (PDOS) plots of their rotationally degenerate pairs of bands on the basis of density functional theory (DFT) calculations. As a quantitative measure for the strength of the pair-polarization in each rotationally degenerate pair of bands, we defined the pair-polarization index Δp, which is associated with the calculated PDOS plots. Finally, we presented qualitative arguments with which to explain why a strong pair-polarization occurs in a particular rotationally degenerate pair in both RE-based and RE-free permanent magnets.