Hall Effect in Dirty Transition-Metal Superconductors near the Upper Critical Field

Abstract

The two-band model is used to explain the Hall effect in dirty type-II transition-metal super-conductors immediately below the upper critical field ${H}_{c2}(T)$. It is shown that because of the existence of interband impurity scattering, which corresponds to an interband impurity-scattering relaxation time ${\ensuremath{\tau}}_{\mathrm{ds}}$, the $d$-band Hall angle ${\ensuremath{\alpha}}_{d}(T)$ is given by $tan{\ensuremath{\alpha}}_{d}(T)={\ensuremath{\eta}}_{d}(H)+{\ensuremath{\eta}}_{d}({H}_{c2})\frac{X(t)}{{\ensuremath{\tau}}_{\mathrm{ds}}{\ensuremath{\epsilon}}_{d0}}\frac{4{K}_{1}^{2}(0)(1\ensuremath{-}\frac{H}{{H}_{c2}})}{[2{K}_{2}^{2}(t)\ensuremath{-}1]{\ensuremath{\beta}}_{A}}$ ;where ${\ensuremath{\epsilon}}_{d0}=2{D}_{d}e{H}_{c2}$, with ${D}_{d}$ being the $d$-band diffusion coefficient; ${\ensuremath{\eta}}_{d}(H)=(\frac{e{\ensuremath{\tau}}_{t,rd}}{{m}_{d}c})H$, which is the tangent of the Hall angle for a $d$ band in the normal phase; and $X(t)$ is a quantity of the order of unity. This relation explains qualitatively the experimental data for a dirty niobium super-conductor reported by Niessen and Staas.