Atomistic-model study of temperature-dependent domain walls in the neodymium permanent magnet ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$

We studied the properties of domain walls (DWs) of the Neodymium magnet, ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$. Applying an atomistic model, in which the magnetic moments of all atoms and exchange interactions were determined by a first-principles calculation (Korringa-Kohn-Rostoker Green's function method), we performed a Monte Carlo simulation for two types of DW, i.e., moving along the $a$ axis and along the $c$ axis, which are classified into a Bloch-type wall and a Neel-type wall, respectively. We found that the shapes of the DWs of both types are described well by those derived from the continuum model used in micromagnetics. We show that the estimated DW widths are very close to the experimentally evaluated ones. Furthermore, we discovered that the width of the latter type is smaller than that of the former type. We also investigated the temperature dependence of the DW width and found that at higher temperatures it becomes larger and the magnitude of the magnetization becomes smaller, which agrees with experimental observations.

Figure 1

Unit cell of ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ magnet, which consists of 68 atoms. Red, blue, and yellow balls denote Nd, Fe, and B atoms, respectively.

Figure 2

(a) DW moving along the $a$ axis (type I, Bloch-type wall); (b) DW moving along the $c$ axis (type II, Neel-type wall).

Figure 3

Temperature dependence of ${\tilde{M}}_z$ (closed circles) and ${\tilde{M}}_{xy}$ (open triangles) in the bulk.

Figure 4

(a) $M_z$ along the $a$ axis (type I) at 300 K. The unit of the vertical axis is ${\mu }_{\mathrm{B}}$/atom. (b) $M_{xy}$ along the $a$ axis (type I) at 300 K. (c) $M_z$ along the $c$ axis (type II) at 300 K. (d) $M_{xy}$ along the $c$ axis (type II) at 300 K. The analytical functions $m_z\left(x\right)$ and $m_y\left(x\right)$ are given by black lines in (a) and (b) [(c) and (d)], respectively. Here symbols denote $M_z\left(x\right)\left[M_{xy}\right]$ at different MCS.

Figure 5

Temperature dependence of $M_z$ along the $a$ axis (type I) (a)–(c) and along the $c$ axis (type II) (d)–(f). (a),(d) 200 K; (b),(e) 400 K; (c),(f) 600 K. The unit of the vertical axis is ${\mu }_{\mathrm{B}}$/atom.

Figure 6

Temperature dependence of $\delta W$ in type I (blue circles) and type II (green circles). The up-and-down arrow (Exp. 1) denotes the range of the experimentally estimated $\delta W$ at room temperature from the values of $K_1$ and $A$, and the down arrow (Exp. 2) denotes that obtained by electron microscopes at room temperature.

Figure 7

$M_z$ along the $a$ axis (type I) for the model with the dipole-dipole interaction by the MC method (symbols). The black solid line denotes the function (4) in the text with $\delta =2.57$ and $m\left(T\right)=1.57$.