Thermal Conductivity in the Two-Band Model of Superconducting Transition Metals Containing Nonmagnetic Impurities

Abstract

Using the Green's-function formulation, an expression has been derived for the thermal conductivity of transition-metal superconductors containing nonmagnetic impurities in the two-band model of Suhl, Matthias, and Walker (SMW). The calculations have been carried out neglecting the interband electron-phonon coupling and taking the interband-scattering collision time ${\ensuremath{\tau}}_{\mathrm{sd}}\ensuremath{\approx}{10}^{\ensuremath{-}12}$ sec. The thermal conductivity $K$ is found to be the sum of two terms ${K}_{1}$ and ${K}_{2}$. ${K}_{1}$ is the dominating term and depends in a complicated manner on both energy gap parameters ${\overline{\ensuremath{\Delta}}}_{s}$ and ${\overline{\ensuremath{\Delta}}}_{d}$, due to $s$ and $d$ bands, respectively. On the other hand, ${K}_{2}$ depends only on ${\overline{\ensuremath{\Delta}}}_{d}$ and is smaller by a factor of \ensuremath{\sim}${10}^{\ensuremath{-}7}$. A log-log plot of thermal conductivity vs temperature turns out to be a straight-line curve with a slight change in its slope at a temperature which depends upon the impurity concentration. Furthermore, $K$ is found to decrease with increase in impurity concentration. These qualitative features of the present study are in very good agreement with the recent experimental investigations of Anderson et al. on niobium and lend support to the validity of the SMW two-band-model theory.