Superconductivity, magnetic order, and quadrupolar order in the filled skutterudite system Pr${}_{1-x}$Nd${}_x$Os${}_4$Sb${}_{12}$

Superconductivity, magnetic order, and the high field ordered phase have been investigated in the filled skutterudite system Pr${}_{1-x}$Nd${}_x$Os${}_4$Sb${}_{12}$ as a function of composition $x$ in magnetic fields up to 9 T and at temperatures between 50 mK and 10 K. Electrical resistivity measurements indicate that the high field ordered phase, which has been identified with antiferroquadruoplar order, persists to $x$ $~$ 0.5. The superconducting critical temperature $T_c$ of PrOs${}_4$Sb${}_{12}$ is depressed linearly with Nd concentration to $x$ $~$ 0.55, whereas the Curie temperature $T_{\mathrm{FM}}$ of NdOs${}_4$Sb${}_{12}$ is depressed linearly with Pr composition to ($1-x$) $~$ 0.45. In the superconducting region, the upper critical field $H_{c2}(x,0)$ is depressed quadratically with $x$ in the range 0 $<$ $x$ $\lesssim$ 0.3, exhibits a kink at $x$ $\approx$ 0.3, and then decreases linearly with $x$ in the range 0.3 $\lesssim$ $x$ $\lesssim$ 0.6. The behavior of $H_{c2}(x,0)$ appears to be a result of pair breaking caused by the applied magnetic field and the exchange field associated with the polarization of the Nd magnetic moments in the superconducting state. From magnetic susceptibility measurements, the correlations between the Nd moments in the superconducting state appear to change from ferromagnetic in the range 0.3 $\lesssim$ $x$ $\lesssim$ 0.6 to antiferromagnetic in the range 0 $<$ $x$ $\lesssim$ 0.3. Specific-heat measurements on a sample with $x$ $=$ 0.45 indicate that magnetic order may occur in the superconducting state, as is also inferred from the depression of $H_{c2}(x,0)$ with $x$.

Figure 1

The dc molar magnetic susceptibility ${\chi }_{\mathrm{dc}}$ as a function of temperature $T$ from 2 to 8 K for single crystals of Pr${}_{1-x}$Nd${}_x$Os${}_4$Sb${}_{12}$ at various Nd concentrations $x$. Inset: Nd concentration $x$ dependence of the lattice parameter.

Figure 2

(a) Temperature $T$ dependence of the ac magnetic susceptibility ${\chi }_{\mathrm{ac}}$ in arbitrary units for various Nd concentrations $x$. (b) SC transition temperature $T_{\mathrm{c}}$ and FM transition (Curie) temperature $T_{\mathrm{FM}}$ vs $x$ for Pr${}_{1-x}$Nd${}_x$Os${}_4$Sb${}_{12}$ determined from measurements of the ac magnetic susceptibility ${\chi }_{\mathrm{ac}}$, electrical resistivity $\rho$, and specific heat $C$. Vertical bars indicate transition widths as defined in the text. For $x$ $=$ 0.45, sample “a”, which exhibits both SC and FM, $T_c$ is defined as the onset temperature for SC.

Figure 3

Specific heat $C$ of Pr${}_{0.55}$Nd${}_{0.45}$Os${}_4$Sb${}_{12}$ (batch “a”) vs $T$ at $H=$ 0, 0.043, 0.086, and 0.429 T. As $H$ increases, the broad peak moves to higher $T$. Inset: the corresponding ${\chi }_{\mathrm{ac}}(T)$ measured on single crystals from the same batch.

Figure 4

Electrical resistivity $\rho$ vs magnetic field $H$ at various temperatures $T$. The rapid drop in $\rho (T,H)$ to zero is due to the SC transition, while the shoulders in $\rho (H)$ above 4 T are due to the HFOP (Ref. 4).

Figure 5

Temperature dependence of upper critical field $H_{c2}$ for Nd concentrations $x⩽0.5$. The horizontal and vertical bars represent the transition widths, defined from the $10\%$ and $90\%$ values of the drop in $\rho (T,\mathrm{fixed}\hspace{0.5em}H)$ at $T_c$ and $\rho (\mathrm{fixed}\hspace{0.5em}T,\mathcal{H})$ at $H_{c2}$.

Figure 6

Nd concentration $x$ dependence of the Curie-Weiss temperature (left axis) $\Theta$ and the Curie constant $C$ (right axis) of the Nd${}^{3+}$ ion in Pr${}_{1-x}$Nd${}_x$Os${}_4$Sb${}_{12}$ after the magnetic susceptibility ${\chi }_{\mathrm{dc}}(T)$ of PrOs${}_4$Sb${}_{12}$ has been subtracted. The fitting range is between 2 and 10 K. Solid and dashed lines are guides to the eye.

Figure 7

(a) Magnetic field $H-x$ phase diagram of Pr${}_{1-x}$Nd${}_x$Os${}_4$Sb${}_{12}$ at $~0$ K. (b) Zero-kelvin extrapolation of the experimentally determined upper critical field $H_{c2}(x,0)$. The solid line is a curve based on Eqs. (3). Inset: Normalized $H$${}_{c2}$($x$,0) and $x$ with respect to $H$${}_{\mathrm{c}2}$(0,0) and $x_{\mathit{cr},1}$ for Pr${}_{1-x}$Nd${}_x$Os${}_4$Sb${}_{12}$ and La${}_{3-x}$Gd${}_x$In. Note the difference between the curvature at low concentration.