Low- T phononic thermal conductivity in superconductors with line nodes

The phonon contribution to the thermal conductivity at low temperature in superconductors with line nodes is calculated assuming that scattering by both nodal quasiparticles and the sample boundaries is significant. It is determined that, within the low-$T$ regime in which the quasiparticles are in the universal limit and the phonon attenuation is in the hydrodynamic limit, there exists a wide temperature range over which the phonon thermal conductivity varies as ${T}^{2}$. This behavior comes from the fact that transverse phonons propagating along certain high symmetry directions do not interact with nodal quasiparticles. Since this is required by the symmetry of the crystal and superconducting gap, the result is independent of the model used for the electron-phonon interaction. The ${T}^{2}$ dependence of the phonon thermal conductivity occurs over a well-defined intermediate temperature range: at higher $T$ the temperature dependence is found to be linear while at lower $T$ the usual ${T}^{3}$ (boundary-limited) behavior is recovered. Results are compared to recent measurements of the thermal conductivity of Tl2201, and are shown to be consistent with the data.