Weak-Coupling BCS Theory of Skutterudite PrOs$_4$Sb$_{12}$

A weak-coupling theory of the A- and B-phases of the recently discovered heavy-fermion superconductor PrOs$_4$Sb$_{12}$ is developed. The BCS gap equation is solved assuming a p+h-wave pairing symmetry with an A-phase nodal gap function $\Delta ({\bf k}) \sim e^{\pm i\phi_{i}} (1-\hat{k}_{x}^4 - \hat{k}_{y}^4 - \hat{k}_{z}^4)$ and with a B-phase nodal gap function $\Delta({\bf k})\sim e^{\pm i\phi}(1-\hat{k}_{y}^4)$, where $e^{\pm i\phi_{i}} \sim \hat{k}_ {z} \pm i\hat{k}_{x},\,\, \hat{k}_{x} \pm i\hat{k}_{y}$ or $\hat {k}_{y} \pm i\hat{k}_{z}$ and $e^{\pm i\phi} \sim \hat{k}_{z} \pm i\hat{k}_{x}$. The B-phase order parameter has similar thermodynamic properties as the recently proposed spin-singlet s+g-wave superconductor [Maki et al, Europhys. Lett. {\bf 64} 496 (2003)]. However, there is accumulating evidence for spin- triplet pairing for superconductivity in PrOs$_4$Sb$_{12}$. Analytic limiting case results as well as numerical solutions for the whole temperature range $T \leq T_{c}$ are presented for the order parameter, the specific heat, the thermodynamic critical field, and the superfluid density. We compare these results with data from a few recent experiments.