Calculating temperature-dependent properties of ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ 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 $M_{\text{s}}\left(T\right)$, effective magnetic anisotropy constants $K_i^{\text{eff}}\left(T\right)\left(i=1,2,3\right)$, domain-wall width ${\delta }_w\left(T\right)$, and exchange stiffness constant $A_{\text{e}}\left(T\right)$, 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 $\mathrm{Fe}-\mathrm{Fe}$ and $\mathrm{Fe}-\mathrm{Nd}$ 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, $M_{\text{s}}\left(T\right),{\delta }_w\left(T\right)$, and $K_i^{\text{eff}}\left(T\right)$, agree well with the experimental results. $A_{\text{e}}\left(T\right)$ 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 $A_{\text{e}}\left(T\right)\sim m^{1.2}$. Specifically, the possible domain-wall configurations at temperatures below the spin-reorientation temperature and the associated ${\delta }_w$ 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.

Figure 1

Exchange parameters $J_{ij}$ as a function of interatomic distance, with the nearest neighbor considered. Inset: Unit cell of ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ showing different kinds of crystallographically equivalent atoms. The results of the $2\times 1\times 1$ supercell are also presented to show the independence of $J_{ij}$ on the calculated cell size.

Figure 2

Temperature dependence of (a, b) the magnetization amplitude, (c) the magnetization components $M_z$ and $M_{xy}$, and (d) the magnetic moment per atom in Nd and Fe sublattices. The corrected curves are plotted by $\alpha =1.802$. The experimental results are taken from [43].

Figure 3

(a) Internal torque density $T_y\left(\theta \right)$ and (b) free-energy density $F\left(\theta \right)$ at different temperatures. (c) Temperature-dependent experimental and calculated effective magnetic anisotropy constants $K_i^{\text{eff}}\left(i=1,2,3\right)$. The experimental results are taken from [42].

Figure 4

(a, d, g) Three types of possible low-temperature (easy axis tilted from the $z$ axis with angle ${\theta }_0)$ domain-wall configurations displayed by the distribution of atomistic magnetic moments. The distribution of macroscopic magnetization components along the $x$ axis in the case of (b, c) domain wall ${\theta }_0\to -{\theta }_0$, (e, f) domain wall ${\theta }_0\to \pi +{\theta }_0$, and (h, i) domain wall ${\theta }_0\to \pi -{\theta }_0$.

Figure 5

(a) High-temperature (easy axis along $z$ axis) domain-wall configuration displayed by the distribution of atomistic magnetic moments. Macroscopic (b) $M_z$ and (c) $M_{xy}$ distribution along the $x$ axis at $T=300$ K. (d) Domain-wall width ${\delta }_w$ at different temperatures. Inset in (d): ${\delta }_w$ scaling with magnetization as a function of $m^{2.92}$. ${\delta }_{w0}$ is the wall width at zero temperature.

Figure 6

(a) $M_z$ distribution along the $x$ axis at different temperatures. (b) Calculated temperature-dependent exchange stiffness. (c) Scaling behavior of the exchange stiffness, with the solid lines showing the scaling law with normalized magnetization. $A_{\text{e0}}$ is the exchange stiffness at zero temperature.