Low Temperature Thermal Conductivity of High Purity YBa${}_2$Cu${}_3$O${}_{6.99}$ in the Vortex State

The low temperature thermal conductivity of a d-wave superconductor is considered for arbitrary strength of the impurity potential. A random distribution of vortices is included within the usual semiclassical approach. The vortex-quasiparticle scattering is accounted for phenomenologically, in addition to impurity scattering. Application of the theory to recent data on high purity YBa${}_2$Cu${}_3$O${}_{6.99}$ shows excellent agreement with experiment, both in zero field and in the presence of an applied magnetic field. We find that the strength of the impurity potential required to describe the data deviates qualitatively from either the unitary or Born limits.