Thermal conductivity of a two-gap superconductor MgB$_{2 }$in High Magnetic Field and Low Temperatures

We calculate the thermal conductivity for a single MgB$_{2}$ crystal in a mixed state, placed in high magnetic field at zero temperature. We plot the thermal conductivity in the superconducting state $\kappa $(H) rescaled by the normal state value $\kappa _{N}$ as a function of magnetic field for a realistic $i.e.$ disordered MgB$_{2}$ superconductor. Our theoretical curve exhibits good qualitative agreement with the experimental data. We report on the self-consistent calculation of the influence disorder has on the thermal conductivity.