Ce and Dy substitutions in Nd$_{2}$Fe$_{14}$B: site-specific magnetic anisotropy from first-principles

A first-principles approach combining density functional and dynamical mean-field theories in conjunction with a quasi-atomic approximation for the strongly localized 4$f$ shell is applied to Nd$_{2}$Fe$_{14}$B-based hard magnets in order to evaluate crystal-field and exchange-field parameters at rare-earth sites and their corresponding single-ion contribution to the magnetic anisotropy. In pure Nd$_2$Fe$_{14}$B, our calculations reproduce the easy-cone to easy axis transition; theoretical magnetization curves agree quantitatively with experiment. Our study reveals that the rare-earth single-ion anisotropy in the "2-14-1" structure is strongly site-dependent, with the $g$ rare-earth site exhibiting a larger value. In particular, we predict that increased $f$ and $g$-site occupancy of $R=$ Ce and Dy, respectively, leads to an increase of the magnetic anisotropy of the corresponding (Nd,$R$)$_{2}$Fe$_{14}$B substituted compounds.