Muon spin rotation and relaxation in ${\text{Pr}}_{1-x}$${\text{Nd}}_x$${\text{Os}}_4$${\text{Sb}}_{12}$: Magnetic and superconducting ground states

Muon spin rotation and relaxation ($\mu$SR) experiments have been carried out to characterize magnetic and superconducting ground states in the ${\text{Pr}}_{1-x}$${\text{Nd}}_x$${\text{Os}}_4$${\text{Sb}}_{12}$ alloy series. In the ferromagnetic end compound ${\text{NdOs}}_4$${\text{Sb}}_{12}$ the spontaneous local field at positive-muon (${\mu }^{+}$) sites below the ordering temperature $T_C$ is greater than expected from dipolar coupling to ferromagnetically aligned Nd${}^{3+}$ moments, indicating an additional indirect RKKY-like transferred hyperfine mechanism. For $0.45\le x\le 0.75$, ${\mu }^{+}$ spin relaxation rates in zero and weak longitudinal applied fields indicate that static fields at ${\mu }^{+}$ sites below $T_C$ are reduced and strongly disordered. We argue this is unlikely to be due to reduction of Nd${}^{3+}$ moments, and speculate that the Nd${}^{3+}$-${\mu }^{+}$ interaction is suppressed and disordered by Pr doping. In an $x=0.25$ sample, which is superconducting below $T_c=1.3$ K, there is no sign of “spin freezing” (static Nd${}^{3+}$ magnetism), ordered or disordered, down to 25 mK. Dynamic ${\mu }^{+}$ spin relaxation is strong, indicating significant Nd-moment fluctuations. The ${\mu }^{+}$ diamagnetic frequency shift and spin relaxation in the superconducting vortex-lattice phase decrease slowly below $T_c$, suggesting pair breaking and/or possible modification of Fermi-liquid renormalization by Nd spin fluctuations. For $0.25\le x\le 0.75$, the $\mu$SR data provide evidence against phase separation; superconductivity and Nd${}^{3+}$ magnetism coexist on the atomic scale.

Figure 1

Phase diagram of ${\text{Pr}}_{1-x}$${\text{Nd}}_x$${\text{Os}}_4$${\text{Sb}}_{12}$ from resistivity ($\rho$), ac susceptibility (${\chi }_{\text{ac}}$), and specific heat ($C$) measurements [7], LF-$\mu$SR quasistatic relaxation rates ${\Delta }_{\text{eff}}$ ($0.45\le x\le 1$), and TF-$\mu$SR quasistatic rate ${\sigma }_T$ and ZF-$\mu$SR dynamic rate ${\lambda }_L$ ($x=0.25$). See also Sec. 4.

Figure 2

Static zero-field Gaussian-broadened Gaussian and Lorentzian KT polarization functions (relaxation rates ${\Delta }_{\text{eff}}$ and $a$, respectively).

Figure 3

Early-time ${\mu }^{+}$ spin polarization in ${\text{NdOs}}_4$${\text{Sb}}_{12}$, $H_L=6.1$ Oe, and $T=25$ mK. Curve: fits of Eq. (6) to the data. Inset: Fourier transforms $g\left(\omega \right)$ of data and fit.

Figure 4

Early-time ${\mu }^{+}$ spin polarization at various temperatures $\lesssim T_C$ from weak-LF-$\mu$SR in ${\text{NdOs}}_4$${\text{Sb}}_{12}$, $H_L=6.1$ Oe. Data for $T=25$ mK from Fig. 3. Curves: fits of Eq. (6) to the data.

Figure 5

Temperature dependencies of weak-LF-$\mu$SR parameters in ${\text{NdOs}}_4$${\text{Sb}}_{12}$, $H_L=6.1$ Oe. (a) Spontaneous ${\mu }^{+}$ spin precession frequency ${\omega }_{\mu }$ and static transverse spin relaxation rate ${\lambda }_T$. Inset: fraction $f_L$ of longitudinal component. (b) Longitudinal spin relaxation rate ${\lambda }_L$. Arrow: Curie temperature $T_C$.

Figure 6

Early-time ${\mu }^{+}$ spin polarization from weak-LF-$\mu$SR in ${\text{Pr}}_{0.25}$${\text{Nd}}_{0.75}$${\text{Os}}_4$${\text{Sb}}_{12}$, $H_L=14.8$ Oe. Curve: fit of Eq. (7) to the data.

Figure 7

Temperature dependencies of parameters from fits of Eq. (7) to weak-LF-$\mu$SR data from ${\text{Pr}}_{0.25}$${\text{Nd}}_{0.75}$${\text{Os}}_4$${\text{Sb}}_{12}$, $H_L=14.8$ Oe. See text for details. Circles: effective ${\mu }^{+}$ spin relaxation rate ${\Delta }_{\text{eff}}$. Triangles: longitudinal spin relaxation rate ${\lambda }_L$. Arrow: magnetic transition temperature $T_C$ from Ref. [7]. Inset: Temperature dependence of the ratio $R$ of distribution widths.

Figure 8

(a) Early-time ${\mu }^{+}$ spin polarization from LF-$\mu$SR in ${\text{Pr}}_{1-x}$${\text{Nd}}_x$${\text{Os}}_4$${\text{Sb}}_{12}$, $x=0.45$ ($H_L=15.9$ Oe), 0.50 ($H_L=16.1$ Oe), and 0.55 ($H_L=16.3$ Oe), $T=25$ mK. Curves ($x\ne 1$): fits of Eq. (7) to the data. (b) $x=0.75$ and 1.00 (data and fits of Figs. 6 and 4, respectively) for comparison.

Figure 9

Temperature dependencies of LF-$\mu$SR parameters in ${\text{Pr}}_{1-x}$${\text{Nd}}_x$${\text{Os}}_4$${\text{Sb}}_{12}$, $x=0.45$, 0.50, and 0.55, from fits of the damped GbG KT polarization function [Eq. (7)] to the data. See text for details. (a) Effective ${\mu }^{+}$ spin relaxation rates ${\Delta }_{\text{eff}}$. Inset: ratios $R$ of distribution widths. (b) Longitudinal spin relaxation rates ${\lambda }_L$.

Figure 10

Early-time LF-$\mu$SR spin polarization functions at $T=25$ mK in ${\text{Pr}}_{0.55}$${\text{Nd}}_{0.45}$${\text{Os}}_4$${\text{Sb}}_{12}$ at various applied fields.

Figure 11

Weak-TF-$\mu$SR asymmetry data in ${\text{Pr}}_{0.75}$${\text{Nd}}_{0.25}$${\text{Os}}_4$${\text{Sb}}_{12}$, $H_T=107.5$ Oe, $T=1.604$ K and 25 mK. The signal from muons that stop in the silver cold finger (see text) has been subtracted. Curves: fits of Eq. (8) to the data.

Figure 12

Temperature dependencies of weak-TF-$\mu$SR parameters in ${\text{Pr}}_{0.75}$${\text{Nd}}_{0.25}$${\text{Os}}_4$${\text{Sb}}_{12}$, $H_T=107.5$ Oe. (a) Average ${\mu }^{+}$ fields ${\overline{B}}_{\mu }$ from ${\mu }^{+}$ precession in sample and Ag reference. (b) Static Gaussian transverse relaxation rate ${\sigma }_T$ and exponential transverse relaxation rate ${\lambda }_T$. Curve: fit of Eq. (9) to the data. Arrows: superconducting transition temperature $T_c$.

Figure 13

ZF ${\mu }^{+}$ spin polarization in ${\text{Pr}}_{0.75}$${\text{Nd}}_{0.25}$${\text{Os}}_4$${\text{Sb}}_{12}$, $25\text{mK}\le T\le 2.50$ K. Curves: fits of Eq. (11) to the data.

Figure 14

Temperature dependencies of static Gaussian KT ${\mu }^{+}$ spin relaxation rate $\Delta$ and dynamic longitudinal spin relaxation rate ${\lambda }_L$ from ZF-$\mu$SR in ${\text{Pr}}_{0.75}$${\text{Nd}}_{0.25}$${\text{Os}}_4$${\text{Sb}}_{12}$. Line: average $\overline{\Delta \left(T\right)}$. Arrow: superconducting transition temperature $T_c$.

Figure 15

Comparison of spin relaxation rates ${\lambda }_L$ ($H=0$) and ${\lambda }_T$ ($H_T=107.5$ Oe) in ${\text{Pr}}_{0.75}$${\text{Nd}}_{0.25}$${\text{Os}}_4$${\text{Sb}}_{12}$. Arrow: superconducting transition temperature $T_c$.

Figure 16

${\mu }^{+}$ spin polarization in ${\text{Pr}}_{0.75}$${\text{Nd}}_{0.25}$${\text{Os}}_4$${\text{Sb}}_{12}$, $T=25$ mK. Curves: fits of Eq. (13) ($H_L=0$) and Eq. (12) ($H_L=10$ and 200 Oe) to the data. Inset: power exponent $\beta$ vs longitudinal field.

Figure 17

ZF- and weak-LF-$\mu$SR GbG effective spin relaxation rates ${\Delta }_{\text{eff}}$ ($x<1$, squares) and experimental ${\mu }^{+}$ precession frequency ${\omega }_{\mu }^{\text{expt}}$, $x=1$ (circle), vs Nd concentration $x$ in ${\text{Pr}}_{1-x}$${\text{Nd}}_x$${\text{Os}}_4$${\text{Sb}}_{12}$, $T=25$ mK. Cross: calculated ${\mu }^{+}$ precession frequency ${\omega }_{\mu }^{\text{dip}}$ in maximum ${\text{NdOs}}_4$${\text{Sb}}_{12}$ dipolar local field, assuming saturation Nd${}^{3+}$ moment ${\mu }_{\text{sat}}=1.73{\mu }_B$ (Ref. [5]) and uniform (ferromagnetic) moment alignment. Curve: calculated ${\Delta }_{\text{eff}}^{\text{dip}}\left(x\right)$ from dipolar ${\mu }^{+}$ local fields [=Van Vleck rate ${\sigma }_{\text{VV}}^{\text{dip}}\left(x\right)$; see text], assuming the saturation Nd${}^{3+}$ moment of $1.73{\mu }_B$ and random moment orientations.