Multiscale simulations toward calculating coercivity of Nd-Fe-B permanent magnets at high temperatures

Abstract

A computational scheme integrating the atomistic spin model (ASM) and micromagnetic simulations is proposed to predict the coercivity of Nd-Fe-B permanent magnets at high temperatures. ASM simulations are applied to calculate the temperature-dependent intrinsic parameters of ${\mathrm{Nd}}_{2}{\mathrm{Fe}}_{14}\mathrm{B}$, including the saturated magnetization, magnetocrystalline anisotropy, and exchange stiffness, which are shown to agree well with the experimental values. With the ASM results as input, finite-temperature micromagnetic simulations based on the stochastic Landau-Lifshitz-Gilbert equation are performed to calculate the magnetic reversal and coercivity (${H}_{\text{c}}$) at high temperatures. It is found that in addition to the decrease of anisotropy field with temperature, thermal fluctuations further reduce ${H}_{\text{c}}$ by 5--10% and $\ensuremath{\beta}$ (temperature coefficient of coercivity) by 0.02--0.1% ${\text{K}}^{\ensuremath{-}1}$ in the presence of a defect layer. The computed thermal-activation volume, which increases with temperature, is shown to be enhanced by several times due to the defect layer with strong magnetization (e.g., 1 T), but can be decreased by introducing a hard shell. Both ${H}_{\text{c}}$ and $\ensuremath{\beta}$ can be enhanced by adding the Dy-rich shell, but saturate at a shell thickness (${t}^{\text{sh}}$) around 6--8 nm after which further increasing ${t}^{\text{sh}}$ or adding Dy into the core is not essential.