Thermal conductivity in dirty transition-metal superconductors near the upper critical field

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 $H_{c2}\left(T\right)$. It is shown that because of the existence of interband impurity scattering, characterized by the interband-impurity-scattering relaxation time ${\tau }_{ds}$, and with ${\tau }_{ds}^{-1}$ acting as a pair-breaking parameter, the d-band thermal conductivity immediately below the upper critical field $K_d^{\left(s\right)}$ is given by the following relation: $K_d^{\left(s\right)}=K_d^{\left(n\right)}-\left(\frac{1}{2e}\right)\frac{(H_{c2}-H)}{\left[2{\kappa }_2^2\right(t)-1]{\beta }_A}\rho ℒ\left(t\right)$, where $K_d^{\left(n\right)}$ is the thermal conductivity of the d band in the normal state and $\rho =\frac{{ϵ}_{d0}}{4\pi T}=\frac{2e{D}_d{H}_{c2}}{4\pi T}$. It is noted that 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.