Thermal Conductivity in Dirty Transition-Metal Superconductors near the Upper Critical Field

Abstract

The two-band model is used to explain the thermal conductivity in dirty type-II transitionmetal superconductors, such as niobium and vanadium, immediately below the upper critical field ${H}_{c2}(T)$. It is shown that because of the existence of interband impurity scattering, characterized by the interband-impurity-scattering relaxation time ${\ensuremath{\tau}}_{\mathrm{ds}}$, and with ${\ensuremath{\tau}}_{\mathrm{ds}}^{\ensuremath{-}1}$ acting as a pair-breaking parameter, the $d$-band thermal conductivity immediately below the upper critical field ${K}_{d}^{(s)}$ is given by the following relation: ${K}_{d}^{(s)}={K}_{d}^{(n)}\ensuremath{-}(\frac{1}{2e}){\frac{({H}_{c2}\ensuremath{-}H)}{[2{\ensuremath{\kappa}}_{2}^{2}(t)\ensuremath{-}1]{\ensuremath{\beta}}_{A}}}\ensuremath{\rho}\mathcal{L}(t)$, where ${K}_{d}^{(n)}$ is the thermal conductivity of the $d$ band in the normal state and $\ensuremath{\rho}=\frac{{\ensuremath{\epsilon}}_{d0}}{4\ensuremath{\pi}T}=\frac{2e{D}_{d}{H}_{c2}}{4\ensuremath{\pi}T}$. It is noted that $\mathcal{L}(t)$ as a function of temperature $t=\frac{T}{{T}_{c}}$ is smaller when the transition-metal superconductor is dirtier. This relation is in qualitative agreement with the experimental findings of Lowell and Sousa.