Upper critical field of strongly disordered three-dimensional superconductors: Localization effects

We calculate the influence of localization on the upper critical field, $H_{c2}$(T), of strongly disordered superconductors in three dimensions. The present work expands upon our previous paper [L. Coffey, K. A. Muttalib, and K. Levin, Phys. Rev. Lett. 52, 783 (1984)] on this topic. It is our conclusion that studying these field-dependent effects may be the ‘‘cleanest’’ way to analyze the interplay of superconductivity and localization phenomena. Our approach is based on the exact impurity eigenstate method which has been used previously to examine both (zero-field) normal-state and superconducting properties of strongly disordered materials. This approach has the advantage over diagrammatic schemes of being nonperturbative in the disorder although it is fundamentally phenomenological in nature. The most striking qualitative effect of extreme disorder is an enhancement of $H_{c2}$(T) relative to the standard Werthamer, Helfand, and Hohenberg curve. This occurs at low temperatures corresponding to the high-field suppression of localization. This enhancement has been observed in recent experiments on transition-metal-based alloys. In our approach localization effects in $H_{c2}$(T) arise mainly from the field dependence of the Coulomb pseudopotential ${\mu }^{\mathrm{*}}$. The changes with H in the density of states and electron-phonon interaction are found to be relatively less important. The pseudopotential depends on the field-dependent particle-hole function $g_H$(rr’;ω) which is closely related to the density-density correlation function. This same function arises in all exact impurity eigenstate calculations for normal as well as superconducting properties. For weak disorder the Fourier transform $g_H$(q;ω)≊$D_H$ $q^2$/(${\omega }^2$+$D_H$ ${}_{}{}^2{}_{}{}^{}$ ${)}^2$ where $D_H$ is the diffusion constant at finite H.