Double superconducting transition in the filled skutterudite ${\text{PrOs}}_4{\text{Sb}}_{12}$ and sample characterizations

Through the characterization of many samples of the filled skutterudite compound ${\text{PrOs}}_4{\text{Sb}}_{12}$, we found that even though the double superconducting transition in the specific heat ($T_{c1}\sim 1.89\text{K}$ and $T_{c2}\sim 1.72\text{K}$) appears in samples of good quality, there are various pieces of evidence that cast doubt on its intrinsic nature. First, three samples exhibit a single sharp transition of $\sim 15\text{mK}$ width at $T_c\sim 1.7\text{K}$. The quality of the samples exhibiting a single transition is clearly better; the width of the transition is smallest and the normalized specific heat jump ${\phantom{(}\left(C-C_{\text{normal}}\right)/C_{\text{normal}})}_{T_c}$ is larger than the sum of the two specific heat jumps when a double transition exists. Second, the ratio of the two specific heat jumps $\frac{\Delta C\left(T_{c1}\right)}{\Delta C\left(T_{c2}\right)}$ exhibits a wide variation with samples not only among different batches but also even within a batch. Finally, this ratio was strongly reduced by polishing a sample down to $\sim 100\mu \text{m}$, which points to bulk inhomogeneities as an origin of the spurious transition. Our x-ray diffraction analysis points to Pr-vacancy problem in the samples, but it does not show that they are the origin of the double transition. We provide the superconducting phase diagram under magnetic field of a sample exhibiting a single transition and fit the curve $H_{c2}\left(T\right)$ with a two-band model taking into account the appropriate values for the gap as deduced from thermal conductivity measurements.

Figure 1

(Color online) Definition of the criteria we use in this paper depending on the kind of superconducting transition: (a) double, (b) single and sharp, or (c) single and broad; the superconducting transition $T_c$, $T_{c1}$, and $T_{c2}$ obtained on the onset, the specific heat jumps $\Delta {\left(C/T\right)}_1$ at $T_{c1}$, $\Delta {\left(C/T\right)}_2$ at $T_{c2}$, and $\Delta \left(C/T\right)$ for the whole jump, and the width of the transition $\Delta \left(T_{c1}\right)$ at $T_{c1}$ and $\Delta \left(T_{c2}\right)$ at $T_{c2}$ or $\Delta \left(T_c\right)$ when a single transition appears.

Figure 2

(Color online) Specific heat as $C/T$ versus $T$ for selected samples of ${\text{PrOs}}_4{\text{Sb}}_{12}$ from different batches. The names of the samples are labeled following the decreasing magnitude of $C/T$ at 2 K where the contribution from the Schottky anomaly predominates. The inset gives a zoom around the superconducting transition for the samples having a double superconducting transition. The absolute value of $C$ clearly depends on the sample. A double superconducting transition appears in samples S1-1, L1-1, L3-1, and L2-1, while samples C1-1 and C3 exhibit a broad single transition.

Figure 3

(Color online) Superconducting transition temperature [determined by specific heat (circle) or resistivity (cross) measurements] versus ${\text{RRR}}_{2\text{K}}$. The open red circles show the transition temperature for the samples exhibiting a single and broad superconducting transition. They exist in the low ${\text{RRR}}_{2\text{K}}$ region. The full red and blue circles are $T_{c1}$ and $T_{c2}$, respectively, of the double transition samples. The large purple open circles and crosses show the single sharp transitions of samples L1-5 and L1-1A, which are determined by the $C$ and $\rho$ measurements, respectively. Please note the definition of the $T_c\left(C\right)$ in the Fig. 1. $T_c\left(\rho \right)$ is the onset temperature. As a general trend, the double transition is realized in the sample with relatively large ${\text{RRR}}_{2\text{K}}$. NB: sample L1-6 with a single sharp transition is not included because the resistivity measurement was not performed.

Figure 4

(Color online) $C/T$ at 2 K and the total specific heat jump $\Delta \left(C/T\right)$ defined in Fig. 1 versus ${\text{RRR}}_{2\text{K}}$. There is apparently a positive correlation between the three quantities. All the samples exhibiting clear double superconducting transitions (batches S1, L1, and L3) are within the large circle. They are of high quality: they are characterized by a large ${\text{RRR}}_{2\text{K}}$, a large $C/T$ at 2 K, and a large specific heat jump at the superconducting transition.

Figure 5

(Color online) Normalized specific heat (defined in Introduction) of selected samples of ${\text{PrOs}}_4{\text{Sb}}_{12}$. Sample L1-1 exhibits a double transition. Most remarkable finding is the existence of three samples, L1-5, L1-6, and L1-1A, exhibiting only a single sharp transition, which casts serious doubt on the intrinsic origin of the double superconducting transition. The specific heat jumps for L1-5 and L1-6 measured by ac method are underestimated. The specific heat of sample L1-5 was also determined semiquantitatively by relaxation method (cf. Fig. 9). All properties are included in Table .

Figure 6

(Color online) Normalized specific heat, ac susceptibility, and resistivity versus temperature of sample L1-5. This sample has a single and sharp superconducting transition in all measurements with $T_c$ of 1.733 K. $T_c$ is consistent among all the measurements ($C$, $\rho$, and $\chi$) and is similar to $T_{c2}$ in the samples exhibiting a double transition.

Figure 7

(Color online) Comparison of ac specific heat versus temperature between S1-2 and S1-2A. The latter was obtained by polishing S1-2 down to $120\mu \text{m}$. The ratio $\Delta {\left(C/T\right)}_1/\Delta {\left(C/T\right)}_2$ changes clearly from 1.9 to $\sim 1.0$.

Figure 8

(Color online) Fit of the superconducting transitions in the specific heat by the model described in Ref. 28 (cf. text).

Figure 9

(Color online) Normalized specific heat versus temperature for three selected samples including L1-5 (measured with the PPMS) and with different ratios of the two specific heat jumps. The names of the samples are labeled following the decreasing magnitude of normalized $C$ at $\sim 1.8\text{K}$. Results of four circle x-ray diffraction on these samples are presented in Table .

Figure 10

(Color online) (a) Superconducting phase diagram $H\text{−}T$ for single transition sample L1-5, which was obtained by resistivity measurements, and for the sample with a double transition S1 (Ref. 9). The inset shows the small positive curvature at low magnetic field both for L1-5 and $T_{c2}\left(H\right)$ of the sample with a double transition. Clearly, $H_{c2}\left(T\right)$ for L1-5 follows that for the lower-temperature one of the double transition sample. The small positive curvature in low fields is more clearly resolved in L1-5. (b) Fit of the superconducting phase diagram $H\text{−}T$ of L1-5. The error bars indicate the width of the transition. A two-band model like in Ref. 9 was used. The new set of parameters is described in the text. The dashed dotted line shows the fit without the Pauli limitation (with $g=0$) (Ref. 34).