Calculating temperature-dependent properties of Nd2Fe14B permanent magnets by atomistic spin model simulations

Temperature-dependent magnetic properties of ${\mathrm{Nd}}_{2}{\mathrm{Fe}}_{14}\mathrm{B}$ permanent magnets, i.e., saturation magnetization ${M}_{\text{s}}(T)$, effective magnetic anisotropy constants ${K}_{i}^{\text{eff}}(T)\phantom{\rule{4pt}{0ex}}(i=1,2,3)$, domain-wall width ${\ensuremath{\delta}}_{w}(T)$, and exchange stiffness constant ${A}_{\text{e}}(T)$, are calculated by using ab initio informed atomistic spin model simulations. We construct the atomistic spin model Hamiltonian for ${\mathrm{Nd}}_{2}{\mathrm{Fe}}_{14}\mathrm{B}$ by using the Heisenberg exchange of $\mathrm{Fe}\ensuremath{-}\mathrm{Fe}$ and $\mathrm{Fe}\ensuremath{-}\mathrm{Nd}$ 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, ${M}_{\text{s}}(T),{\ensuremath{\delta}}_{w}(T)$, 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 ${A}_{\text{e}}(T)\ensuremath{\sim}{m}^{1.2}$. Specifically, the possible domain-wall configurations at temperatures below the spin-reorientation temperature and the associated ${\ensuremath{\delta}}_{w}$ and ${A}_{\text{e}}$ are identified. This work provokes a scale bridge between ab initio calculations and temperature-dependent micromagnetic simulations of Nd-Fe-B permanent magnets.