Transport Properties of Anisotropic Superconductors.

In some heavy-fermion compounds (U Pt_3, and U Be_{13 } under pressure) the normal-state transport properties at low temperatures behave qualitatively as one would expect if electron-impurity interactions were the dominant source of scattering. The experimental data on ultrasonic attenuation and thermal conduction in the superconducting phase of U Pt_3 do not behave as one would expect if the electron-impurity phase shift were small, and therefore, assuming electron -impurity scattering to be the dominant process, we have calculated transport coefficients for a range of phase shifts, and for three anisotropic superconducting states, the axial and polar p-wave states, and a d-wave state consistent with cubic and hexagonal crystal symmetries. We have calculated the amplitude for an impurity to scatter a quasiparticle by taking into account repeated scattering of a quasiparticle by a single impurity, and we find that for phase shifts which are not small the superconducting correlations can give rise to a very different energy dependence of the quasiparticle mean free path in the superconducting state, as compared with its behavior in the normal state. The calculated ultrasonic attenuation and thermal conductivity agree much better with the data for U Pt _3 if the normal-state phase shift is taken to be resonant (close to pi/2), and if the superconducting state has a nodal line at the equator of the Fermi surface. For phase shifts which are neither small, nor resonant, we find first, that the mean free times for quasiparticle -like excitations differ from the quasihole-like ones, which we show leads to an enhancement of the thermoelectric effect in the superconducting state by a factor of order T _{F}/T_{c}, and second, an angular asymmetry of the scattering amplitude with respect to reflection of the final momentum in the plane determined by the incoming momentum and the symmetry axis of the gap. This latter effect can occur if the superconducting state has non-trivial phase variations over the Fermi surface. The angular asymmetry leads to new tensor components of transport coefficients which are non-zero only if the parity of the superconducting order parameter is the same as that of the current or currents entering the transport coefficient. Experimental detection of these new coefficients would therefore serve to determine the parity of the order parameter.