Probing the superconductivity of ${PrPt}_4{\mathrm{Ge}}_{12}$ through Ce substitution

We report measurements of electrical resistivity, magnetic susceptibility, specific heat, and thermoelectric power on the system Pr${}_{1-x}$Ce${}_x$Pt${}_4$Ge${}_{12}$. Superconductivity is suppressed with increasing Ce concentration up to $x=0.5$, above which there is no evidence for superconductivity down to 1.1 K. The Sommerfeld coefficient $\gamma$ increases with increasing $x$ from $\sim$48 mJ/mol K${}^2$ up to $\sim$120 mJ/mol K${}^2$ at $x=$ 0.5, indicating an increase in strength of electronic correlations. The temperature dependence of the specific heat at low temperatures evolves from roughly $T^3$ for $x=$ 0 to e${}^{-\Delta /T}$ behavior for $x=$ 0.05 and above, suggesting a crossover from a nodal to a nodeless superconducting energy gap or a transition from multiband to single-band superconductivity. Fermi-liquid behavior is observed throughout the series in low-temperature magnetization, specific heat, and electrical resistivity measurements.

Figure 1

Powder XRD pattern for a representative concentration of Pr${}_{1-x}$Ce${}_x$Pt${}_4$Ge${}_{12}$ ($x=$ 0.5). Tick marks below the pattern indicate the position of expected Bragg reflections for the refined filled skutterudite crystal structure. The inset shows a linear increase of lattice parameter $a$ with $x$, which is consistent with Vegard's law.

Figure 2

Electrical resistivity data for selected Pr${}_{1-x}$Ce${}_x$Pt${}_4$Ge${}_{12}$ samples with Ce concentrations $x$ indicated in figure.

Figure 3

Electrical resistivity normalized to the normal state resistivity (${\rho }_N$) just above $T_c$. $T_c$ decreases with increasing $x$, where the onset of superconductivity is observable up to $x\sim 0.5$. The transition width broadens with increasing $x$, which may be the result of additional chemical disorder.

Figure 4

(a) Selected $\rho \left(T\right)$ data for Pr${}_{1-x}$Ce${}_x$Pt${}_4$Ge${}_{12}$ plotted as a function of $T^2$, with offsets added for clarity. Power law fits of the form $\rho \left(T\right)=$ ${\rho }_0+A{T}^2$ were performed up to roughly 250 K${}^2$. The solid lines represent best fits to the data. (b) Coefficient $A$ plotted across the entire series. $A$ decreases from 5 n$\Omega$ cm/K${}^2$ at $x=$ 0, to a minimum value at $x=$ 0.5, after which it increases. (c) The residual resistivity ${\rho }_0$, extracted from the power law fits, increases with increasing $x$ across the series. ${\rho }_0$ starts increasing from $\sim$1.7 $\mu \Omega$ cm for $x=$ 0 and approaches a maximum of 44 $\mu \Omega$ cm at $x=$ 0.75, after which it drops to 13 $\mu \Omega$ cm at $x=$ 1, due to the great degree of atomic order of the CePt${}_4$Ge${}_{12}$ end-member compound.

Figure 5

$M/H$ as a function of temperature measured in an applied magnetic field of $H=$ 1 T except for the $x=$ 0 and 1 samples which were measured in a $H=$ 0.5 T applied magnetic field. The inset highlights the Meissner effect in samples with $x\le 0.2$, for which $M/H$ was measured in an applied field of $H=$ 1 mT. The superconducting volume fraction appears to be approximately 100$\%$ for $x\le 0.1$ down to 2 K; however, corrections using the demagnetization factor were not made, which may account for the volume fraction achieving values greater than 1. $T_c$ decreases with $x$ until $x\sim 0.2$, where no signatures of superconductivity are observed down to 1.8 K.

Figure 6

Effective magnetic moment (${\mu }_{\mathrm{eff}}/{\mu }_B$) and ${\Theta }_{\mathrm{CW}}$ obtained from Curie-Weiss fits to $M/H$ data. The Pr and Ce parent compounds have ${\mu }_{\mathrm{eff}}=$ 3.69${\mu }_B$ and ${\mu }_{\mathrm{eff}}=$ 2.69${\mu }_B$, respectively, which are close to the expected values for Pr${}^{3+}$ and Ce${}^{3+}$ free ions (3.58${\mu }_B$ and 2.54${\mu }_B$, respectively). The red solid line represents the ${\mu }_{\mathrm{eff}}$ extracted from C-W fits that follows closely the expected values calculated by the relation ${\mu }_{\mathrm{eff}}\left(x\right)=$ $\sqrt{{\left({\mu }_{Pr{3}^{+}}\right)}^2(1-x)+{\left({\mu }_{Ce{3}^{+}}\right)}^2\left(x\right)}$, which governs the decrease of ${\mu }_{\mathrm{eff}}$ with increasing $x$ when ${\Theta }_{\mathrm{CW}}$ is $x$ independent.

Figure 7

Specific heat data displayed as $C/T$ vs $T^2$. (a) Samples with $x\le 0.2$ exhibit a jump associated with the transition to the superconducting state. (b) Samples with $x\ge 0.25$ exhibit no evidence of superconductivity down to 2 K.

Figure 8

The Sommerfeld coefficient $\gamma$, and Debye temperature ${\Theta }_D$, obtained from linear fits of $C\left(T\right)/T=$ $\gamma +\beta T^2$ to the $C/T$ vs $T^2$ data. The electronic specific heat coefficient $\gamma$ exhibits a moderate enhancement as $x$ increases up to $x=0.5$, and saturates above this concentration. The Debye temperature ${\Theta }_D$ increases in magnitude with $x$ throughout the series.

Figure 9

(a) Electronic contribution to specific heat plotted as $log(C_e/\gamma T_c)$ vs $T_c/T$. The data were offset for visual clarity. The solid lines represent best fits to the data using 3.46 $T_c/T^{-3.04}$ for $x=$ 0 and $a{e}^{-\Delta /T}$ for the other concentrations. (b) $a$ vs $x$. (c) $\Delta /T_c$ vs $x$. The solid line represents the BCS prediction of $\Delta /T_c=$ 1.76.

Figure 10

(a) Thermoelectric power $S$ of CePt${}_4$Ge${}_{12}$ and PrPt${}_4$Ge${}_{12}$ as a function of temperature in the absence of applied magnetic field. The compound CePt${}_4$Ge${}_{12}$ exhibits a broad peak at $\sim$80 K as well as a shoulderlike feature at lower temperatures. The presence of the feature suggests that CePt${}_4$Ge${}_{12}$ is on the border of Kondo lattice and intermediate valence. For PrPt${}_4$Ge${}_{12}$, $S$ is roughly 0 $\mu$V/K below $T_c$, as expected for superconductors. (b) A 2-T magnetic field suppresses the superconducting state in PrPt${}_4$Ge${}_{12}$ and leads to a nonzero $S$. The inset displays $S/T$ for CePt${}_4$Ge${}_{12}$, which extrapolates to $\sim 0.63\mu$V/K${}^2$ in the limit $T\to 0$ K.

Figure 11

Phase diagram of $T_c$ as a function of Ce concentration $x$ obtained from magnetization, specific heat, and electrical resistivity measurements. The error bars for $\rho$ were taken as the 90$\%$ and 10$\%$ drop in resistivity. $T_c$ is suppressed by Ce substitution up to $x\sim$ 0.5 with a small positive curvature, above which, evidence for superconductivity is not observed down to $\sim$120 mK. In $x=$ 0.3, 0.4, and 0.5 only a portion of the superconducting transition was observed. The solid line is a guide to the eye.

Figure 12

Kadowaki-Woods ratio, $R_{\mathrm{KW}}=$ $A/{\gamma }^2$, and the dimensionless Sommerfeld-Wilson ratio, $R\propto {\chi }_0/\gamma$, plotted as functions of $x$. The uncertainty in $R_{\mathrm{KW}}$ and $R$ were propagated through in calculations from the errors of $A$, $\gamma$, and ${\chi }_0$ employing standard error analysis [56]. (a) $R_{\mathrm{KW}}$ is of order ${10}^{-6}$ $\mu \Omega$ cm (mol K mJ${}^{-1}$)${}^2$, which is close to expected values for heavy-fermion systems [58]. ($x=$ 0.45, 0.625, and 0.875 were omitted as explained in the text.) (b) $R$ decreases with increasing $x$, suggesting that the Ce parent behaves similarly to a free electron system, while the higher values for small $x$ may be due to magnetic exchange enhancement.