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Temperature-dependent magnetic properties of Nd$_2$Fe$_{14}$B permanentmagnets, i.e., saturation magnetization $M_\text{s}(T)$, effective magneticanisotropy constants $K_i^\text{eff}(T)$ ($i=1,2,3$), domain wall width$\delta_w(T)$, and exchange stiffness constant $A_\text{e}(T)$, are calculatedby using \textit{ab-initio} informed atomistic spin model simulations. Weconstruct the atomistic spin model Hamiltonian for Nd$_2$Fe$_{14}$B by usingthe Heisenberg exchange of Fe$-$Fe and Fe$-$Nd atomic pairs, the uniaxialsingle-ion anisotropy of Fe atoms, and the crystal-field energy of Nd ionswhich is approximately expanded into an energy formula featured by second,fourth, and sixth-order phenomenological anisotropy constants. After applying atemperature rescaling strategy, we show that the calculated Curie temperature,spin-reorientation phenomenon, $M_\text{s}(T)$, $\delta_w(T)$, and$K_i^\text{eff}(T)$ agree well with the experimental results. $A_\text{e}(T)$is estimated through a general continuum description of the domain wall profileby mapping atomistic magnetic moments to the macroscopic magnetization.$A_\text{e}$ is found to decrease more slowly than $K_1^\text{eff}$ withincreasing temperature, and approximately scale with normalized magnetizationas $A_\text{e}(T) \sim m^{1.2}$. This work provokes a scale bridge between\textit{ab-initio} calculations and temperature-dependent micromagneticsimulations of Nd-Fe-B permanent magnets.

Calculating temperature-dependent properties of Nd$_2$Fe$_{14}$B permanent magnets by atomistic spin model simulations