Noncollinearity effects on magnetocrystalline anisotropy for $R_2$Fe$_{14}$B magnets

We present a theoretical investigation of the magnetocrystalline anisotropy (MA) in $R_2$Fe$_{14}$B ($R$ is a rare-earth element) magnets in consideration of the non-collinearity effect (NCE) between the $R$ and Fe magnetization directions. In particular, the temperature dependence of the MA of Dy$_2$Fe$_{14}$B magnets is detailed in terms of the $n$th-order MA constant (MAC) $K_n(T)$ at a temperature $T$. The features of this constant are as follows: $K_1(T)$ has a broad plateau in the low-temperature range and $K_2(T)$ persistently survives in the high-temperature range. The present theory explains these features in terms of the NCE on the MA by using numerical calculations for the entire temperature range, and further, by using a high-temperature expansion. The high-temperature expansion for $K_n(T)$ is expressed in the form of $K_n(T)=\kappa_1(T)\left[1+\delta(T)\right][-\delta(T)]^{n-1}$, where $\kappa_1(T)$ is the part without the NCE and $\delta(T)$ is a correction factor for the NCE introduced in this study. We also provide a convenient expression to evaluate $K_n(T)$, which can be determined only by a second-order crystalline electric field coefficient and an effective exchange field.