Magnetic moment of rare-earth elements in $R_2{\mathrm{Fe}}_{14}\mathrm{B}$ estimated with ${\mu }^{+}\mathrm{SR}$

The ferromagnetic (FM) nature of ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ has been investigated with muon spin rotation and relaxation (${\mu }^{+}\mathrm{SR}$) measurements on an aligned, sintered plate-shaped sample. A clear muon spin precession frequency ($f_{\mathrm{FM}}$) corresponding to the static internal FM field at the muon site showed an order parameter-like temperature dependence and disappeared above around 582 K ($\sim T_{\mathrm{C}}$). This indicated that the implanted muons are static in the ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ lattice even at temperatures above around 600 K. Using the muon site and local spin densities predicted by DFT calculations, the ordered Nd moment ($M_{\mathrm{Nd}}$) was estimated to be 3.31 ${\mu }_{\mathrm{B}}$ at 5 K, when both ${\mathit{M}}_{\mathrm{Fe}}$ and ${\mathit{M}}_{\mathrm{Nd}}$ are parallel to the $\widehat{\mathbf{c}}$ axis and $M_{\mathrm{Fe}}=2.1{\mu }_{\mathrm{B}}$. Furthermore, $M_R$ in $R_2{\mathrm{Fe}}_{14}\mathrm{B}$ with $R=\mathrm{Y}$, Ce, Pr, Sm, Gd, Tb, Dy, Ho, Er, and Tm was estimated from $f_{\mu }$ values reported in earlier ${\mu }^{+}\mathrm{SR}$ work, using the FM structure proposed by neutron scattering and the same muon site and local spin density as in ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$. Such estimations yielded $M_R$ values consistent with those obtained by the other methods.

Figure 1

The crystal structure of ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ in tetragonal symmetry with space group $P{4}_2/mnm$ drawn by VESTA [11]. Large red and yellow spheres show Nd at two different sites, medium blue and green spheres show Fe at six different sites, and small orange spheres show B. Very small pink spheres represent the muon site (0.6745,0.8838,0) predicted by first-principles calculations (see text).

Figure 2

Geometry of the ${\mu }^{+}\mathrm{SR}$ experiment in TRIUMF: Four counters [backward (B), forward (F), up (U) and down (D)] detect decay positrons emitted in the $-z$, $+z$, $+x$, and $-x$ directions, respectively. The initial muon spin direction ${\mathit{S}}_{\mu }\left(0\right)$ is in the $+x$ direction ($\parallel \widehat{\mathit{a}}$ of the plates) for spin-rotated (SR) mode (a) or in the $-z$ direction ($\parallel \widehat{\mathit{c}}$) for non-spin-rotated (NSR) mode (b). Thus if the internal magnetic field (${\mathit{H}}_{\mathrm{int}}$) is parallel to $\widehat{\mathit{c}}$, only U and D counters will detect a muon spin oscillation, and that only in SR mode; but if ${\mathit{H}}_{\mathrm{int}}\perp \widehat{\mathit{c}}$, only B and F counters in NSR mode will show an oscillatory signal. Using both configurations, one can estimate the magnetic anisotropy in the sample.

Figure 3

The ZF-${\mu }^{+}\mathrm{SR}$ spectrum for the sintered align ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ plate sample recorded at (a) 300 K and (b) 2 K in two different configurations: a non-spin-rotated (NSR) mode $[{\mathit{S}}_{\mu }\left(0\right)\parallel \widehat{\mathit{c}}]$ shown in red and a spin-rotated (SR) mode $[{\mathit{S}}_{\mu }\left(0\right)\perp \widehat{\mathit{c}}]$ shown in green. The solid lines represent the best fits using Eq. (1).

Figure 4

The temperature dependencies of (a) the muon spin precession frequency ($f_{\mathrm{FM}}$), (b) the magnification of the $f_{\mathrm{FM}}\left(T\right)$ curve to show the anomaly at around 135 K, (c) the exponential relaxation rate (${\lambda }_{\mathrm{FM}}$), and (d) the ratio between ${\lambda }_{\mathrm{FM}}$ and $f_{\mathrm{FM}}$ for the ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ sample. The data were obtained by fitting the ZF-${\mu }^{+}\mathrm{SR}$ spectrum with Eq. (1).

Figure 5

The temperature dependencies of $f_{\mathrm{FM}}$ and the weak transverse field asymmetry ($A_{\mathrm{TF}}$) with TF = 50 Oe. The $f_{\mathrm{FM}}\left(T\right)$ curve is the same as that in Fig. 4. Here $T_{\mathrm{C}}$ corresponds to the midpoint of a step-like change in the $A_{\mathrm{TF}}\left(T\right)$ curve.

Figure 6

Contour plots for ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ in the (001) plane. (a) Electrostatic potential ${\mathrm{\Phi }}_{\text{E}}$ and (b) spin density $m(={\rho }^{\uparrow }-{\rho }^{\downarrow }$). The muon site is indicated by black circles.

Figure 7

The relationship between the calculated $H_{\mu }$ and $M_{\mathrm{Nd}}$ in ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ using the model that (a) ${\mathit{M}}_{\mathrm{Fe}}\parallel \left[001\right]$ and ${\mathit{M}}_{\mathrm{Nd}}\parallel \left[001\right]$, (b) ${\mathit{M}}_{\mathrm{Fe}}$ and ${\mathit{M}}_{\mathrm{Nd}}$ are both canted from the [001] direction to the [110] direction with a canting angle ($\theta$) of ${27}^{\circ }$ for Fe and 55–${66}^{\circ }$ for Nd, and (c) $\theta ={27}^{\circ }$ for Fe and Nd at the $4g$ site, but $\theta =73$–${84}^{\circ }$ for Nd at the $4f$ site. In (b), a collinear FM spin arrangement—i.e., ${\theta }_{\mathrm{Fe}}={\theta }_{\mathrm{Nd}}={27}^{\circ }$—is also shown with a broad black line.

Figure 8

The relationship between $M_{\mathrm{Nd}}$ and $M_{\mathrm{Fe}}$ in ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ using the model that ${\mathit{M}}_{\mathrm{Fe}}\parallel \left[001\right]$ and ${\mathit{M}}_{\mathrm{Nd}}\parallel \left[001\right]$.

Figure 9

The relationship between the calculated $H_{\mu }$ and $M_{\mathrm{Nd}}$ in $R_2{\mathrm{Fe}}_{14}\mathrm{B}$ using the model that (a) ${\mathit{M}}_{\mathrm{Fe}}\parallel \left[001\right]$ and ${\mathit{M}}_R\parallel \left[001\right]$, (b) ${\mathit{M}}_{\mathrm{Fe}}\parallel \left[100\right]$ and ${\mathit{M}}_R\parallel \left[100\right]$, and (c) ${\mathit{M}}_{\mathrm{Fe}}\parallel \left[001\right]$ and ${\mathit{M}}_R\parallel \left[001\right]$. In (a)–(c), the magnitude of $M_{\mathrm{Fe}}$ is assumed to be $2.1{\mu }_{\mathrm{B}}$.

Figure 10

The relationship between the magnetic moment of the rare-earth element ($M_R$) and expected magnetic moments ($gJ$). For heavy rare-earth elements, negative value of $gJ$ is used, because ${\mathit{M}}_R$ is antiparallel to ${\mathit{M}}_{\mathrm{Fe}}$.