Magnetic moment of rare earth elements in R2Fe14B estimated with {\mu}^+SR

TL;DR

This study uses muon spin rotation ({}^+SR) measurements combined with DFT calculations to estimate the magnetic moments of rare earth elements in R2Fe14B, providing insights into their ferromagnetic properties and temperature dependence.

Contribution

It introduces a method to estimate rare earth magnetic moments in R2Fe14B using {}^+SR data and theoretical calculations, extending previous approaches.

Findings

Estimated Nd moment at 5 K is 3.31 μ_B.

M_R values in various R2Fe14B compounds agree with other methods.

Ferromagnetic order persists above 600 K in Nd2Fe14B.

Abstract

The ferromagnetic (FM) nature of Nd2Fe14B has been investigated with muon spin rotation and relaxation ({\mu}^+SR) measurements on an aligned, sintered plate-shaped sample. A clear muon spin precession frequency (f_{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 (~T_C). This indicated that the implanted muons are static in the Nd2Fe14B lattice even at temperatures above around 600 K. Using the predicted muon site and local spin densities predicted by DFT calculations, the ordered Nd moment (M_{Nd}) was estimated to be 3.31 {\mu}_B at 5 K, when both M_{Fe} and M_{Nd} are parallel to the c-axis and M_{Fe} = 2.1 {\mu}_B. Furthermore, M_R in R2Fe14B with R = Y, Ce, Pr, Sm, Gd, Tb, Dy, Ho, Er, and Tm was estimated from f_{\mu} values reported in earlier {\mu}+SR work, using the FM structure proposed by neutron scattering and the same muon site and local spin density as in Nd2Fe14B. Such estimations yielded M_R values consistent with those obtained by the other methods.

Tables

Table 1. Table 1: Crystallographic parameters of ferromagnetic Nd 2 ​ Fe 14 ​ B subscript Nd 2 subscript Fe 14 B {\rm Nd_{2}Fe_{14}B} . Space group: P ​ 4 2 / m ​ n ​ m 𝑃 subscript 4 2 𝑚 𝑛 𝑚 P4_{2}/mnm  (No. 136). Lattice constants: a = 8.797 𝑎 8.797 a=8.797  Å, c = 12.149 𝑐 12.149 c=12.149  Å (Calc.), and a = 8.795 𝑎 8.795 a=8.795  Å, c = 12.188 𝑐 12.188 c=12.188  Å (Expt.).

		Calc.				Expt.111Reference Li et al. (1996)
	site	$x$	$y$	$z$		$x$	$y$	$z$
Nd1	$4g$	0.2313	0.7687	0		0.2313	0.7687	0
Nd2	$4f$	0.3570	0.3570	0		0.3585	0.3585	0
Fe1	$16k$	0.0373	0.3599	0.3239		0.0379	0.3587	0.3237
Fe2	$16k$	0.0675	0.2754	0.1270		0.0671	0.2765	0.1269
Fe3	$8j$	0.0980	0.0980	0.2950		0.0979	0.0979	0.2951
Fe4	$8j$	0.3180	0.3180	0.2542		0.3174	0.3174	0.2535
Fe5	$4e$	0	0	0.1143		0	0	0.1144
Fe6	$4c$	0	1/2	0		0	1/2	0
B	$4f$	0.1236	0.1236	0		0.1243	0.1243	0

Table 2. Table 2: The ordered magnetic moment of each element in Nd 2 ​ Fe 14 ​ B subscript Nd 2 subscript Fe 14 B {\rm Nd_{2}Fe_{14}B} predicted by DFT calculations without and with U = 5 𝑈 5 U=5~{} eV.

		GGA		GGA+$U$
	site	$M$ (${\mu }_B$)		$M$ (${\mu }_B$)
Nd1	$4g$	2.92		2.74
Nd2	$4f$	3.01		2.72
Fe1	$16k$	2.25		2.28
Fe2	$16k$	2.20		2.22
Fe3	$8j$	2.09		2.17
Fe4	$8j$	2.68		2.68
Fe5	$4e$	2.03		2.03
Fe6	$4c$	2.32		2.36
B	$4f$	-0.25		-0.26

Table 3. Table 3: The Fe moment at each site in Nd 2 Fe 14 B and the Nd moment ( M Nd subscript 𝑀 Nd M_{\rm Nd} ) estimated from the μ + superscript 𝜇 \mu^{+} SR data.

case	$16k1$	$16k2$	$8j1$	$8j2$	$4e$	$4c$	average	$M_{\mathrm{Nd}}$
average	2.1	2.1	2.1	2.1	2.1	2.1	2.1	3.31
exp1Fruchart et al. (1987)	2.08	2.16	2.06	2.43	2.28	1.97	2.16	3.52
exp2Onodera et al. (1987)	2.24	2.30	2.21	2.55	2.00	2.17	2.28	4.27
exp3Noort et al. (1986)	2.27	2.41	2.19	2.70	2.20	2.10	2.38	4.61
calc1Jaswal (1990)	2.15	2.18	2.12	2.74	2.13	1.59	2.20	3.33
calc2Hummler and Fähnle (1996)	2.22	2.28	2.67	2.16	1.96	2.43	2.29	3.80
calc3Moriya et al. (2009)	2.28	2.37	2.32	2.74	2.19	2.46	2.38	4.27

Table 4. Table 4: The internal magnetic field detected with μ + superscript 𝜇 \mu^{+} SR Niedermayer et al. ( 1991 ) , the saturated magnetization Herbst ( 1991 ) , the magnetic moment of R 𝑅 R ( M R subscript 𝑀 𝑅 M_{R} ) estimated with μ + superscript 𝜇 \mu^{+} SR ( M R μ ​ SR superscript subscript 𝑀 𝑅 𝜇 SR M_{R}^{\mu\rm SR} ), and M R subscript 𝑀 𝑅 M_{R} proposed with magnetization measurements at 4 K ( M R Mag superscript subscript 𝑀 𝑅 Mag M_{R}^{\rm Mag} ) Herbst ( 1991 ) , and g ​ J 𝑔 𝐽 gJ , where g 𝑔 g is the Land e ´ ´ e \acute{\rm e} g 𝑔 g -factor and J 𝐽 J is the quantum number of the total angular momentum.

$R_2$Fe14B	$H_{\mu }$ (MHz)	$H_{\mu }$ (kOe)	3$H_L=4\pi M_s$ (kOe)	$M_R^{\mu \mathrm{SR}}$ (${\mu }_B$)	$M_R^{\mathrm{Mag}}$ (${\mu }_B$)	$gJ$
$R=$Y	204.5	15.07	15.9	0.11	0	—
La	—	—	14.8	—	—	0
Ce	189.6	14.0	14.7	0.66	—	2.14
Pr	162.5	11.97	18.4	2.79	3.1	3.20
Nd	152.6	11.26	18.5	3.31	3.2	3.27
Pm	—	—	—	—	—	2.40
Sm	63.0	4.65	16.7	$\sim 0$	1.0	0.72
Eu	—	—	—	—	—	0
Gd	374.0	27.60	9.2	-9.48	-6.8	7.0
Tb	405.2	29.90	6.6	-11.4	-9.1	9.0
Dy	429.0	31.65	5.7	-12.6	-10.1	10.0
Ho	388.0	28.60	5.7	-10.3	-10.1	10.0
Er	157.2	11.58	6.6	-9.94	-9.3	9.0
Tm	154.6	11.41	9.2	-9.57	-6.7	7.0
Yb	—	—	$\sim 12$	—	-4.2	4.0
Lu	—	—	14.7	—	—	0

Equations

$$A_0{P}_{\mathrm{ZF}}(t)$$

$$+$$

$$A_0{P}_{\mathrm{TF}}(t)$$

$$+$$

$${\mathbf{H}}_{\mathrm{FM}}$$

$$=$$

$$f_{\mathrm{FM}}$$

$$=$$

$${\mathbf{H}}_{\mathrm{dip}}$$

$${\mathbf{H}}_{\mathrm{L}}$$

$${\mathbf{H}}_{\mathrm{hf}}$$

Full text

††preprint: APS/123-QED

I Introduction

Among many permanent magnet materials, Nd2Fe14B Sagawa et al. (1984) and related intermetallic compounds Herbst (1991) are known to be very suitable for industrial applications, due to their high saturation magnetization ($M_{\rm s}=16~{}$kG), large energy product ($H_{\rm c}M_{\rm s}=64~{}$MGOe) and relatively low cost compared with that of Sm2Fe17Nx Coey and Sun (1990). Furthermore, although the Curie temperature ($T_{\rm C}$) is 592 K for Nd2Fe14B, the Nd2Fe14B phase does not decompose until 1428 K, resulting in flexibility of its synthesis process. Therefore, Nd2Fe14B and related compounds are widely used for high performance motors in many devices, electric vehicles and audio speakers.

In the ferromagnetic (FM) phase, past neutron scattering measurements suggested a collinear spin structure at room temperature Herbst et al. (1984), in which both Fe and Nd moments ($M_{\rm Nd}$ & $M_{\rm Fe}$) are aligned parallel along the [001] direction. The magnitude of the ordered $M_{\rm Fe}$ was almost saturated even at 300 K, i.e. $\sim 2.2~{}\mu_{\rm B}$, while $M_{\rm Nd}$ was initially thought to be below 1 $\mu_{\rm B}$ Herbst et al. (1984). The other neutron work reported that $M_{\rm Fe}\sim 2.32(3)~{}\mu_{\rm B}$ and $M_{\rm Nd}\sim 2.2~{}\mu_{\rm B}$ Givord et al. (1985), but the recent work revealed that $M_{\rm Fe}=1.9(1)~{}\mu_{\rm B}$ and $M_{\rm Nd}=1.5(1)~{}\mu_{\rm B}$ Teplykh et al. (2013). Then, more detailed magnetization measurements at 4 K on $R_{2}$Fe14B with $R={\rm La,Y,...}$ revealed that $M_{\rm Fe}=2.1~{}\mu_{\rm B}$ Herbst (1991), leading to $M_{\rm Nd}=3.2~{}\mu_{\rm B}$. In addition, Nd-NMR measurements suggested that $M_{\rm Nd}=2.7~{}\mu_{\rm B}$ at 4.2 K Potenziani II (1985). An X-ray magnetic circular dichroism (XMCD) study on $R_{2}$Fe14B Chaboy et al. (1996); Miguel-Soriano et al. (2000) implied that the ordered $M_{R}$s are very close to the values obtained from $gJ$ of $4f$ electrons, where $J$ is the quantum number of the total angular momentum and $g$ is the Land$\acute{\rm e}$ factor. This means that $M_{\rm Nd}\sim 3.3~{}\mu_{\rm B}$.

Furthermore, the FM spin structure in Nd2Fe14B was found to change at $135~{}{\rm K}(=T_{\rm SRT})$ due to a spin reorientation transition from a high-temperature phase with ${\bm{M}}\parallel[001]$ to a low-temperature phase with ${\bm{M}}$ canted along the [110] direction by magnetization measurements Givord et al. (1984); Abache and Oesterreicher (1985); Sagawa et al. (1985); Hirosawa et al. (1986). Initially, a collinear FM structure with a canting angle $\theta=30.6^{\circ}$ at 4.2 K was proposed based on magnetization measurements on a single crystal sample Tokuhara et al. (1985), where $\theta$ is the angle of ${\bm{M}}$ from the [001] direction to the [110] direction. However, both M$\ddot{\rm o}$ssbauer Onodera et al. (1987) and XMCD Chaboy et al. (1998) measurements suggested a non-collinear spin structure below $T_{\rm SRT}$. That is, $\theta_{\rm Fe}^{\rm M\ddot{\rm o}ss}=27^{\circ}$ and $\theta_{\rm Nd}^{\rm M\ddot{\rm o}ss}=58^{\circ}$ at 4.2 K, while $\theta_{\rm Fe}^{\rm XMCD}=28^{\circ}$ and $\theta_{\rm Nd}^{\rm XMCD}=40^{\circ}$ at 4.2 K. The continuation of XMCD work Bartolome et al. (2000) indicated the formation of a further noncollinear spin structure among the Nd moments at temperatures between 80 K and $T_{\rm SRT}$, at which $\theta_{\rm Nd,4f}\sim 80^{\circ}$ and $\theta_{\rm Nd,4g}\sim 25^{\circ}$.

In order to further elucidate the FM ground state of Nd2Fe14B, we need another technique sensitive to internal magnetic field(s) ($\vec{H}_{\rm int}$) in solids. Unfortunately, neutron scattering is unlikely to be useful for investigating the magnetic nature of ferromagnets, because relatively weak magnetic diffraction peaks always overlap with strong nuclear Bragg peaks. Indeed, the estimated $M_{\rm Nd}$ with neutron ranges from 1 to 2.2 $\mu_{\rm B}$ Herbst et al. (1984); Givord et al. (1985); Teplykh et al. (2013), which is rather small compared with those obtained with the other techniques. On the other hand, a positive muon spin rotation and relaxation ($\mu^{+}$SR) provides information on the local magnetic environments at the site(s) of the implanted muons, which usually locate at the interstitial site with the minimum electrostatic potential, regardless of magnetic order and/or disorder Kalvius et al. (2001); Yaouanc and de R$\acute{\rm e}$otier (2011).

In fact, immediately after the discovery of the Nd2Fe14B system, a $\mu^{+}$SR experiment was performed at the Paul Sherrer Institut Yaouanc et al. (1987); Niedermayer et al. (1991) using powder $R_{2}$Fe14B samples with $R=$ Y, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, and Tm in the temperature range between 300 and 4.2 K. The $\mu^{+}$SR spectra obtained in zero external field (ZF) exhibited a clear oscillation with one precession frequency for all the samples, indicating both the formation of static FM order and a single muon lattice site. However, since it was very difficult to determine the correct muon site(s) in the lattice, the muon site was assumed to be a tetrahedral site with two Fe and two Nd nearest neighbors, based on the M$\ddot{\rm o}$ssbauer and neutron data of hydrated $R_{2}$Fe14B Ferreira et al. (1985); Niedermayer et al. (1991). In addition, the lack of information on the local spin density at the muon site made it eventually impossible to estimate the magnitude of $M_{R}$. As a result, the past $\mu^{+}$SR result is unlikely to be recognized as a crucial work for elucidating the magnetic ground state of Nd2Fe14B and $R_{2}$Fe14B.

We have therefore attempted to measure the $\mu^{+}$SR spectra for Nd2Fe14B up to above $T_{\rm C}$ to know the variation of $H_{\rm int}$ with temperature and to predict muon site(s) in the lattice with density functional theory (DFT) calculations. Using the predicted muon site and the measured local spin density at the muon site, the magnitude of $M_{\rm Nd}$ was clearly estimated even below $T_{\rm SRT}$. Furthermore, using the past $\mu^{+}$SR data for $R_{2}$Fe14B and the predicted muon site, we have obtained a systematic change in $M_{R}$ with the number of $4f$ electrons in $R$.

II Experimental

Aligned sintered plates of Nd2Fe14B were prepared from jet-milled fine powder with the composition of 31.8Nd-0.98B-0.10Cu-0.90Co-0.15Al-0.05Ga-66.02Fe (wt%). The mean particle size of the powder was about $6~{}\mu$m. The powders were then pressed under a magnetic field of 1.8 T followed by uniaxial pressing with 15 MPa. The pressed powders were sintered at 1293-1353 K for 4 hours in vacuum $(<10^{-2}~{}$Pa). Finally, the sintered powder $8\times 8\times 8~{}{\rm mm}^{3}$ cube was sliced into 1 mm thick plates with the aligned $c$-axis perpendicular to the plane. The preparation and characterization of the sintered sample are explained in more detail elsewhere Takada et al. (2012).

The $\mu^{+}$SR time spectra were measured on the M20 surface muon beam line using the LAMPF spectrometer of the CMMS facility at TRIUMF in Canada. Four plates with $8\times 8\times 1~{}$mm3 were arranged onto a sample holder with their $\hat{\bm{c}}$ axes parallel to the beam direction (${z}$) as defined in Fig. 2. For measurements in the $T$ range between 1.8 and 300 K, the samples were attached to a low-background sample holder in a liquid-He flow-type cryostat with 0.05 mm thick Al-coated Mylar tape. For measurements in the $T$ range between 300 and 600 K, the samples were fixed onto a silver plate by a $50~{}\mu$m thick titanium foil, which is sandwiched between a second silver plate with a $16\times 16$ mm2 square aperture through which incoming muons passed. For the former setup, there is essentially no background signal, while for the latter case the $\mu^{+}$SR signal naturally includes a background signal from muons stopped in the surrounding silver plate.

The $\mu^{+}$SR spectra were obtained in either zero applied field (ZF) or transverse field (TF) with four positron detectors [backward (B), forward (F), up (U) and down (D)] arranged as shown in Fig. 2. The initial direction of the muon polarization [${\bm{S}}_{\mu}(0)$] relative to the plane of the plates was set by a Wien filter spin rotator. Here TF means the applied field is perpendicular to ${\bm{S}}_{\mu}(0)$, i.e. TF$\parallel y$ in this study. The experimental techniques are described in more detail elsewhere Kalvius et al. (2001); Yaouanc and de R$\acute{\rm e}$otier (2011). The resulting $\mu^{+}$SR data were analyzed with musrfit Suter and Wojek (2012).

The distributions of electrostatic potential and local spin density were predicted by DFT calculations with a generalized gradient approximation (GGA) plus on-site Coulomb interaction ($U$), as described in Sec. III.2.

III Results

III.1 $\mu^{+}$SR

Figure 3 shows the ZF-$\mu^{+}$SR time spectra for the Nd2Fe14B sample recorded at 300 and 2 K. The spectrum obtained with SR mode [${\bm{S}_{\mu}}\perp\hat{\bm{c}}$] exhibits a clear oscillation, while that obtained with NSR mode [${\bm{S}_{\mu}}\parallel\hat{\bm{c}}$] shows mainly a non-oscillatory relaxation together with an oscillation with a very small amplitude. Since the Fourier transform frequency spectrum of the ZF-$\mu^{+}$SR time spectrum shows the presence of only one component, the two spectra were fitted by a combination of an exponentially relaxing cosine signal and an exponentially relaxing non-oscillatory signal:

[TABLE]

Here $A_{0}$ is the initial asymmetry, $P_{\rm ZF}(t)$ is the muon spin depolarization function in ZF, $A_{\rm FM}$ and $A_{\rm tail}$ are the asymmetries associated with the two signals, $\lambda_{\rm FM}$ and $\lambda_{\rm tail}$ are their exponential relaxation rates, $f_{\rm FM}(\equiv\omega_{\rm FM}/2\pi)$ is the muon Larmor frequency corresponding to the quasi-static internal FM field, and $\phi_{\rm FM}$ is the initial phase. At each temperature, the two spectra were fitted using common $\lambda_{\rm FM}$ and $f_{\rm FM}$.

Such fits yielded $A_{\rm FM}^{S\perp c}=0.208(7)$, $A_{\rm FM}^{S\parallel c}=0.021(5)$, $\lambda_{\rm FM}=30.6(1.3)~{}\mu{\rm s}^{-1}$, $f_{\rm FM}=153.3(2)~{}$MHz, $\phi_{\rm FM}^{S\perp c}=1(2)~{}$deg, $\phi_{\rm FM}^{S\parallel c}=10(14)~{}$deg, $A_{\rm tail}^{S\perp c}=0.0159(2)$, $A_{\rm tail}^{S\parallel c}=0.2160(4)$, and $\lambda_{\rm tail}=0.0141(6)~{}\mu{\rm s}^{-1}$ at 300 K. Thus, the deviation from the $c$-axis of the magnetic field at the muon site, i.e. the magnetic anisotropy at the muon site is estimated to be $\Theta(300~{}{\rm K})=\tan^{-1}\left(A_{\rm FM}^{S\parallel c}/A_{\rm FM}^{S\perp c}\right)=7(4)~{}$deg. The same fit to the data at 2 K yielded $\Theta(2~{}{\rm K})=6(4)$ deg. This means that $\Theta$ is almost zero below 250 K within the accuracy of $\mu^{+}$SR.

Figure 4 shows the temperature dependences of $f_{\rm FM}$, $\lambda_{\rm FM}$, and $\lambda_{\rm FM}/f_{\rm FM}$ for the Nd2Fe14B sample. The $f_{\rm FM}(T)$ curve exhibits an order parameter-like temperature dependence and $f_{\rm FM}$ disappears at temperatures above around 582 K ($=T_{\rm C}^{\rm\mu SR}$), which is slightly lower than $T_{\rm C}$ in literatures, i.e. 592 K Sagawa et al. (1984); Herbst (1991). Here it should be noted that $T_{\rm C}^{\rm\mu SR}$ is estimated from the data obtained in ZF, while the other techniques require the application of a large external magnetic field, which naturally enhances FM order. The $f_{\rm FM}(T)$ curve also shows a sharp local maximum at 135 K $(=T_{\rm SRT})$, indicating a change in the local FM environment caused by a spin reorientation transition.

As temperature increases from 2 K, $\lambda_{\rm FM}$ decreases slightly up to 100 K, then suddenly increases up to 150 K, and then decreases again towards $T_{\rm C}$ with an increasing slope (d$\lambda_{\rm FM}$/d$T$). However, below the vicinity of $T_{\rm C}$, $\lambda_{\rm FM}$ rapidly increases with temperature, and then suddenly drops to zero at $T_{\rm C}$; that is, a critical behavior is observed below the vicinity of $T_{\rm C}$.

It should be noted that $\lambda_{\rm FM}/f_{\rm FM}$, which corresponds to the normalized field distribution width, is almost temperature independent at temperatures below 100 K and at temperatures between 150 and 550 K. This means that besides the temperatures around a spin reorientation transition and below the vicinity of $T_{\rm C}$, $H_{\rm int}$ in the FM phase depends only on the magnitude of the ordered moments. These results suggest that muons are stable in the Nd2Fe14B lattice until $T_{\rm C}^{\rm\mu SR}$. The present result reproduces those in past $\mu^{+}$SR work carried out below room temperature Yaouanc et al. (1987); Niedermayer et al. (1991).

In order to estimate $T_{\rm C}$ more correctly, Fig. 5 shows the temperature dependence of the weak transverse asymmetry ($A_{\rm TF}$) measured with TF=50 Oe in the vicinity of $T_{\rm C}$, together with that of $f_{\rm FM}$. Here, “weak” means that the applied TF is very small compared with $H_{\rm int}$ caused by FM order. The wTF-$\mu^{+}$SR spectrum was fitted by a combination of an exponentially relaxing cosine oscillation due to muon spin precession in TF and Eq. (2):

[TABLE]

At temperatures $T\gg T_{\rm C}$, $A_{\rm FM}=A_{\rm tail}=0$; at temperatures $T\ll T_{\rm C}$, $A_{\rm TF}=0$. From the middle point of a step-like change in the $A_{\rm TF}(T)$ curve, $T_{\rm C}$ is estimated as 581.57(14) K, because $A_{\rm TF}$ is proportional to the volume fraction of paramagnetic phases in a sample. The finite value of $A_{\rm TF}$ below $T_{\rm C}$ ($\sim 0.06$) is from muons stopped in the surrounding silver plate.

III.2 DFT calculations

First-principles calculations based on a density functional theory (DFT) Hohenberg and Kohn (1964); Kohn and Sham (1965) have been performed to determine the muon site in ${\rm Nd_{2}Fe_{14}B}$. A self consistent field (SCF) calculation is carried out using the ultrasoft pseudopotential methodVanderbilt (1990); Miwa (2011a), where the on-site Coulomb interaction for localized Nd-$4f$ electrons is taken into consideration using the ${\rm DFT}+U$ method Liechtenstein et al. (1995); Miwa (2018a). The obtained pseudo SCF charge density is transformed into an all electron form with the projector augmented wave operatorsBlöchl (1994), from which the muon occupation site is estimated by the electrostatic potential analysis.

The program used for the DFT calculations is an original code developed by one of the authors (K. M.), which has been successfully applied for various materials Miwa and Fukumoto (2002); Miwa et al. (2004); Miwa (2011b); Sugiyama et al. (2013); Miwa (2018b).

The cutoff energies of plane waves are set to be 25 and 200 hatrees for the pseudo wavefunctions and the charge density, respectively. The $4\times 4\times 4$ $k$-point mesh is adopted for the Brillouin zone integration. The generalized gradient approximation Perdew et al. (1996) is used for the exchange-correlation functional. The effective Coulomb and exchange parameters for Nd-$4f$ orbitals are assumed to be $U=5$ eVAlam et al. (2013) and $J=0.5$ eV, respectively.

Table 1 shows the result of the structural relaxation in which atomic positions as well as lattice constants are fully optimized. The calculated parameters are in good agreement with the experimental ones Li et al. (1996). Figure 6(a) depicts the electrostatic potential: The muon site is found to be $8i~{}(0.6745,0.8838,0)$ which is located near the center of a square base of a pyramid composed of Nd-3Fe-B atoms. As shown in Fig. 6(b), the spin density at the muon site is negligibly small, $\rho_{\rm spin}=-2\times 10^{-3}~{}\mu_{B}/\textrm{bohr}^{3}$, which is eventually zero.

It should be noted that the DFT calculations with $U=0$ provides very similar muon site and local spin density to those predicted with $U=5~{}$eV. This means that the two significant parameters, i.e. the muon site and $\rho_{\rm spin}$, are not sensitive to $U$ in the Nd2Fe14B lattice.

On the contrary, the ordered magnetic moment of each element varies with $U$ (Table 2). More correctly, the introduction of $U=5~{}$eV reduces $M_{\rm Nd}$ by 10%, while the change in $M_{\rm Fe}$ is about 1%. The magnitude of $M_{\rm Fe}$ at each site is comparable to the reported ones (see Table 3). This indicates the importance of the magnitude of $U$ for estimating $M_{\rm Nd}$ by DFT calculations.

IV Discussion

IV.1 Nd2Fe14B

For non-magnetized ferromagnetic materials in zero applied field, the internal magnetic field at a muon site (${\bm{H}}_{\mu}$) is represented by Barth et al. (1986); Schenck and Gygax (1995); Yaouanc and de R$\acute{\rm e}$otier (2011); Sugiyama et al. (2012);

[TABLE]

This field is connected to the muon-spin precession frequencies through the muon gyromagnetic ratio [$f=H\gamma_{\mu}/(2\pi)=0.013553~{}$(MHz/Oe)$\times H~{}$(Oe)] leading to

[TABLE]

where ${\bm{H}}_{\rm dip}$ is the dipolar field, ${\bm{H}}_{\rm L}$ is the Lorentz field, ${\bm{H}}_{\rm hf}$ is the hyperfine field, and $f_{\mu}$, $f_{\rm L}$, and $f_{\rm hf}$ are the corresponding muon spin precession frequencies. Furthermore, ${\bm{H}}_{\rm L}$ and ${\bm{H}}_{\rm hf}$ are connected to the saturated magnetization (${\bm{M}}_{\rm s}$) and the local spin density at the muon sites ($\rho_{\rm spin}$) as follows:

[TABLE]

In order to estimate ${\bm{H}}_{\rm dip}$ $(f_{\rm dip})$, we use the results of neutron diffractionHerbst et al. (1984) and M$\ddot{\rm o}$ssbauerRosenberg et al. (1985) measurements for the magnitude and direction of the Fe moments. Assuming that the magnitude of the ordered $M_{\rm Fe}$ is $2.1~{}\mu_{\rm B}$ Herbst (1991), ${\bm{H}}_{\rm dip}$ at the muon site is easily calculated as a function of the Nd moment using crystal structural data with dipelec Kojima et al. (2004).

We start by considering a collinear FM structure along the $c$-axis, that is, ${\bm{M}}_{\rm Fe}\parallel[001]$ and ${\bm{M}}_{\rm Nd}\parallel[001]$. Since $4\pi M_{\rm s}=18.5~{}$kOe at 5 K (see Table 4) Herbst (1991); Niedermayer et al. (1991), ${\bm{H}}_{\rm L}=(0,0,6.2$ kOe) from Eq. (12). Moreover, ${\bm{H}}_{\rm hf}=(0,0,0)$ because of the absence of any local spin density at the muon site. Consequently, we obtain the relationship between $|{\bf H}_{\mu}|=H_{\mu}^{\rm calc}$ and the magnitude of the Nd moment ($M_{\rm Nd}$), as seen in Fig. 7(a). Here, the measured value of $f_{\mu}$ ($f_{\mu}^{\rm exp}$) is 152.6(2) MHz at 2.2 K, which is very close to the reported value (156 MHz) at 5 K. Thus, in order to explain $H_{\mu}^{\rm exp}$, $M_{\rm Nd}$ is uniquely determined as 3.31 $\mu_{\rm B}$. This is almost equivalent to $M_{\rm Nd}$ estimated from magnetization measurements, i.e. $M_{\rm Nd}=3.2~{}\mu_{\rm B}$ Herbst (1991), confirming the reliability of the predicted muon site from DFT calculations. From the data at room temperature, i.e. $4\pi M_{\rm s}=16.0~{}$kOe at 295 K and $H_{\mu}^{\rm exp}=151(2)~{}$MHz at 300 K, we also obtain that $M_{\rm Nd}=3.01~{}\mu_{\rm B}$.

Although we assumed that $M_{\rm Fe}=2.1~{}\mu_{\rm B}$,

$M_{\rm Nd}$ estimated with the above procedure is found to increase linearly with $M_{\rm Fe}$ (see Fig. 8). On the contrary, Fig. 8 provides an acceptable range for $M_{\rm Fe}$ as $2.0\leq M_{\rm Fe}\leq 2.15~{}\mu_{\rm B}$, when $M_{\rm Nd}$ ranges between 3.0 and $3.5~{}\mu_{\rm B}$. Furthermore, we assumed that $M_{\rm Fe}$ is identical for all the Fe sites. However, experimental studies and DFT calculations reported that $M_{\rm Fe}$ at each site deviates slightly from $2.1~{}\mu_{\rm B}$. In order to know the effect of such deviations on the estimation of $M_{\rm Nd}$, the relationship between $H_{\mu}$ and $M_{\rm Nd}$ is also shown for the two cases in Fig. 7(a) and six cases in Table 3. This indicates that the four estimations for $M_{\rm Fe}$, i.e. exp2, exp3, calc2, and calc3, provide unusually large $M_{\rm Nd}$ under the collinear FM structure along the $c$-axis.

By contrast, at low temperatures the spin orientation is reported to change from the [001] to the [110] direction below $T_{\rm SRT}=135~{}$K Givord et al. (1984); Abache and Oesterreicher (1985); Sagawa et al. (1985); Hirosawa et al. (1986). The corresponding anomaly is clearly seen in the $f_{\rm FM}(T)$ and $\lambda_{\rm FM}(T)$ curves [Fig. 4]. More correctly, both Fe and Nd moments are thought to be canted towards the [110] direction from the [001] direction, based on both first principles calculations and Fe $K$-edge x-ray magnetic circular dichroism (XMCD) measurements Chaboy et al. (1998). The canting angle ($\theta$) was estimated to be 27*∘* for Fe ($\theta_{\rm Fe}=27^{\circ}$) and 58*∘* for Nd ($\theta_{\rm Nd}=58^{\circ}$) at 4.2 K. Figure 7(b) shows the relationship between $H_{\mu}$ and $M_{\rm Nd}$ for several $\theta_{\rm Nd}$ values. The $\mu^{+}$SR result clearly excludes a collinear structure, in which $\theta_{\rm Fe}=\theta_{\rm Nd}=27^{\circ}$, as an FM ground state. On the other hand, non-collinear structures provide a more plausible $M_{\rm Nd}$, particularly when $\theta_{\rm Nd}\sim 60^{\circ}$. If we assume that $M_{\rm Nd}=3.2~{}\mu_{\rm B}$, $\theta_{\rm Nd}$ should be 63*∘, which is very close to the value reported by XMCD (58∘*).

Dipole field calculations provide that the magnetic anisotropy at the muon site ($\Theta$) is 16 deg at temperatures below $T_{\rm SRT}$, while $\Theta=0~{}$deg at temperatures above $T_{\rm SRT}$. Making comparison with the experimental result [$\Theta(300~{}{\rm K})=7(4)$deg and $\Theta(2~{}{\rm K})=6(4)$deg], the experimental accuracy of $\Theta$ was likely to be overestimated. This is probably due to the fact that ${\bm{S}}_{\mu}(0)$ for NSR mode is deviated from the $z$ direction by about 10 deg to eliminate the other particles in the muon beam. Nevertheless, we should note that the above estimation for $M_{\rm Nd}$ is based only on the magnitude of $f_{\mu}$, and as a result, the estimated value is not affected by the alignment of the sample.

Another XMCD study at low temperatures Bartolome et al. (2000) proposed the possibility of a non-collinear spin arrangement among the Nd moments. That is, $\theta_{\rm Nd}\sim 25^{\circ}$ for the Nd ions at the $4g$ site, but $\theta_{\rm Nd}\sim 80^{\circ}$ for the Nd ions at the $4f$ site. Figure 7(c) shows the dependence of $H_{\mu}$ on $M_{\rm Nd}$ as $\theta_{\rm Nd(4f)}$ changes from 73 to 84*∘. The calculations also predict that $\theta_{\rm Nd}=82^{\circ}$ for $M_{\rm Nd}=3.2~{}\mu_{\rm B}$, which looks consistent with the proposed arrangement. However, we should note that there are eight crystallographically equivalent muon sites ($8i$) in the Nd2Fe14B lattice. Moreover, such a non-collinear spin arrangement among the Nd moments produces two different $H_{\mu}$s at each $8i$ site — namely, $H_{\mu}=11270~{}$Oe for four of the sites and 11655 Oe for the other four sites. Although the difference of the two $H_{\mu}$s (about 4%) is too small to observe two distinct muon precession frequencies in the ZF-$\mu^{+}$SR spectrum, such a split naturally increases the field distribution width, resulting in an increased relaxation rate $\lambda_{\rm FM}$. In reality, $\lambda_{\rm FM}$ and $\lambda_{\rm FM}/f_{\rm FM}$ at 2 K are smaller than those at room temperature [Fig. 4(b)]. This clearly excludes the model of a non-collinear spin arrangement among the Nd moments from the FM ground state for Nd2Fe14*B. Since the $\lambda_{\rm FM}(T)$ curve exhibits a broad maximum at around $T_{\rm SRT}$ [see Fig. 4(b)], such a non-collinear spin arrangement among the Nd moments could appear in a limited temperature range particularly below the vicinity of $T_{\rm SRT}$.

Even for this case, the predicted $\Theta$ is the same to that for the collinear spin arrangement among the Nd moments, i.e. 16 deg. Therefore, $\Theta$ provides no crucial information on the spin arrangement in Nd2Fe14B within the present accuracy.

IV.2 $R_{2}$Fe14B

Although we have measured $\mu^{+}$SR spectra only for Nd2Fe14B, both ${\bm{H}}_{\mu}$ and ${\bm{M}}_{\rm s}$ were reported for the other $R_{2}$Fe14B compounds with $R=$ Y, Ce, Pr, Sm, Gd, Tb, Dy, Ho, Er, and Tm (see Table 4) Yaouanc et al. (1987); Niedermayer et al. (1991). Since $4f$ electrons are well localized at the $R$ site, it is reasonable to assume the same muon site in $R_{2}$Fe14B as in Nd2Fe14B. Concerning the spin arrangement in the FM phase, the easy direction of magnetization at base temperature Herbst (1991) revealed that both ${\bm{M}}_{\rm Fe}$ and ${\bm{M}}_{R}$ are parallel to the [001] direction in $R_{2}$Fe14B with $R=$ Y, Ce, Pr, Nd, Gd, Tb, Dy, and Ho, but they are parallel to the [100] direction in $R_{2}$Fe14B with $R=$ Sm, Er, and Tm. We also assume that $M_{\rm Fe}=2.1~{}\mu_{\rm B}$ in $R_{2}$Fe14B regardless of $R$.

Using the structural data of each compound, Fig. 9 shows the relationship between $H_{\mu}$ and $M_{R}$. For Y2Fe14B, $M_{\rm Y}$ is estimated to be almost zero (0.11 $\mu_{\rm B}$), as expected for Y3+. In fact, the recent photoelectron spectroscopic analysis result on Nd2Fe14B Wang et al. (2014); Min et al. (1993) revealed that the valence state of Nd ions is very close to 3+, while there is, to our knowledge, no XPS work on Y2Fe14B. As the atomic number increases, $H_{\mu}^{\rm exp}$ decreases systematically. From the intersection between $H_{\mu}^{\rm exp}$ and $H_{\mu}^{\rm calc}$, $M_{\rm Ce}$ and $M_{\rm Pr}$ are estimated to be 0.66 and 2.79 $\mu_{\rm B}$, respectively (Table 4).

For Sm2Fe14B, Er2Fe14B, and Tm2Fe14B, since ${\bm{M}}_{\rm Fe}\parallel[100]$ and ${\bm{M}}_{R}\parallel[100]$, the $H_{\mu}^{\rm calc}(M_{R})$ curve exhibits a parabolic shape with a minimum at $M_{R}=0$ [Fig. 9(b)]. For Sm2Fe14B, $H_{\mu}^{\rm exp}<H_{\mu}^{\rm calc}$ in the whole possible range of $M_{\rm Sm}$, leading tentatively to $M_{\rm Sm}=0$. This implies that the FM spin structure is slightly different from the proposed one Hiroyoshi et al. (1985). For Er2Fe14B, and Tm2Fe14B, there are two intersections between the $H_{\mu}^{\rm exp}(M_{R})$ and $H_{\mu}^{\rm calc}(M_{R})$ curves. This means that two values are available for $M_{\rm Er}$ and $M_{\rm Tm}$. However, neutron diffraction measurements proposed that ${\bm{M}}_{R}$ is antiparallel to ${\bm{M}}_{\rm Fe}$ Yelon and Herbst (1986); Davis et al. (1985); Yamada et al. (1985). Therefore, a negative value is selected for $M_{\rm Er}$ and $M_{\rm Tm}$, that is, -9.94 and -9.54 $\mu_{\rm B}$, respectively.

For Gd2Fe14B Herbst (1991), Tb2Fe14B Herbst et al. (1993), Dy2Fe14B Herbst and Yelon (1985), and Ho2Fe14B Wolfers et al. (1990), ${\bm{M}}_{\rm Fe}\parallel[001]$, ${\bm{M}}_{R}\parallel[001]$, and ${\bm{M}}_{R}$ is antiparallel to ${\bm{M}}_{\rm Fe}$. Indeed, $H_{\mu}^{\rm exp}$ is reproduced only when $M_{R}<-9~{}\mu_{\rm B}$ [Fig. 9(c)]. As a result, we obtain that $M_{\rm Gd}=-9.48~{}\mu_{\rm B}$, $M_{\rm Tb}=-11.4~{}\mu_{\rm B}$, $M_{\rm Dy}=-12.6~{}\mu_{\rm B}$, and $M_{\rm Ho}=-10.3~{}\mu_{\rm B}$.

Finally, Fig. 10 shows the relationship between $M_{R}$ and the expected magnetic moment ($gJ$) derived from Land$\acute{\rm e}$ $g$ factor and the quantum number of the total angular momentum ($J$) for free $R^{3+}$ ions. $M_{R}$ estimated with the magnetization measurements ($M_{R}^{\rm Mag}$) is almost equivalent to $gJ$ Herbst (1991), suggesting the presence of stronger exchange field to the $4f$ moments than the crystal field Herbst (1991). On the other hand, the slope of the $M_{R}^{\mu\rm SR}(gJ)$ curve estimated with $\mu^{+}$SR is steeper than that for the $M_{R}^{\rm Mag}(gJ)$ curve, mainly because $\mid M_{R}^{\mu\rm SR}\mid>\mid M_{R}^{\rm Mag}\mid$ for the heavy rare earth elements. Although the reason for this discrepancy is not clear at present, we should note that $\mu^{+}$SR is very sensitive to local magnetic environments. Recently, not only for Nd2Fe14B but also for Ho2Fe14B, a non-collinear spin structure for the Ho moment is proposed with neutron using a single crystal sample Wolfers et al. (1990). This implies the possibility that such non-collinear structure appears in the other $R_{2}$Fe14B at low temperatures, which would affect the magnitude of $M_{R}^{\mu\rm SR}$. It would be thus an interesting subject to reconfirm the magnetic structure in $R_{2}$Fe14B at low temperatures using a high quality sample. Finally, this work clearly demonstrates the unique power of a combination of $\mu^{+}$SR and DFT calculations for determining the magnetic moments of rare earth elements through the observation of local $H_{\rm int}$.

V Summary

We have studied the internal magnetic field in a sintered Nd2Fe14B permanent magnet sample with a positive muon spin rotation and relaxation ($\mu^{+}$SR) technique, which provides microscopic magnetic information at the muon site. Combining the $\mu^{+}$SR data with the result of DFT calculations for predicting the muon site in the lattice, the magnitude of the ordered Nd moment was clearly estimated both for a collinear ferromagnetic structure at room temperature and a canted ferromagnetic structure at 2 K. Furthermore, a similar estimation for the ordered moment of the rare earth elements in $R_{2}$Fe14B provided reasonable values consistent with those reported by magnetization and Mössbauer measurements. $\mu^{+}$SR has been widely used for investigating a magnetic nature in antiferromagnetic, spin-glass, and/or paramagnetic materials, in which both the Lorentz field and hyperfine field are usually zero and, as a result, the dipole field is predominant. On the contrary, the present work demonstrates that a combination of $\mu^{+}$SR and DFT calculations further expands the research field into ferromagnetic materials.

VI acknowledgments

We thank the staff of TRIUMF (especially the CMMS) for help with the $\mu^{+}$SR experiments. This work was supported by Japan Society for the Promotion Science (JSPS) KAKENHI Grant No. JP18H01863.