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

## TL;DR

This study uses atomistic spin model simulations informed by ab-initio calculations to accurately predict temperature-dependent magnetic properties of Nd$_2$Fe$_{14}$B permanent magnets, bridging microscopic and macroscopic modeling approaches.

## Contribution

It introduces a comprehensive atomistic spin model for Nd$_2$Fe$_{14}$B that captures temperature effects and aligns well with experimental data, enabling better micromagnetic simulations.

## Key findings

- Calculated Curie temperature matches experiments
- Magnetization and anisotropy constants agree with observed data
- Exchange stiffness decreases slowly with temperature

## Abstract

Temperature-dependent magnetic properties of Nd$_2$Fe$_{14}$B permanent magnets, i.e., saturation magnetization $M_\text{s}(T)$, effective magnetic anisotropy 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 calculated by using \textit{ab-initio} informed atomistic spin model simulations. We construct the atomistic spin model Hamiltonian for Nd$_2$Fe$_{14}$B by using the Heisenberg exchange of Fe$-$Fe and Fe$-$Nd atomic pairs, the uniaxial single-ion anisotropy of Fe atoms, and the crystal-field energy of Nd ions which is approximately expanded into an energy formula featured by second, fourth, and sixth-order phenomenological anisotropy constants. After applying a temperature 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 profile by mapping atomistic magnetic moments to the macroscopic magnetization. $A_\text{e}$ is found to decrease more slowly than $K_1^\text{eff}$ with increasing temperature, and approximately scale with normalized magnetization as $A_\text{e}(T) \sim m^{1.2}$. This work provokes a scale bridge between \textit{ab-initio} calculations and temperature-dependent micromagnetic simulations of Nd-Fe-B permanent magnets.