Alloy Fermi-Surface Topology Information from Superconductivity Measurements under Pressure

A modified BCS gap equation is used to obtain an approximate solution for the pressure derivative of the superconducting transition temperature, $\frac{d{T}_c}{\mathrm{dp}}$, at values of the Fermi energy $E_F$ near critical points in the density of states. $\frac{d{T}_c}{\mathrm{dp}}$ is a functional of the energy derivative of the density of states, and therefore reflects strong structure at values of $E_F$ near the van Hove singularities in the density of states associated with a Fermi-surface topology change. The model is extended to include the case of a dilute random alloy system in which impurity scattering broadens the singularities. A Lorentzian spectral distribution is assumed, and an analytic equation is obtained for the broadened van Hove-singularity contributions to the density of states. The model is applied to the case of indium doped with cadmium, the rate of change of the Fermi energy with concentration being estimated from a free-electron formula, and the rate of change of the life-time estimated from the residual resistivity. A qualitative fit between the model calculation and the structure of available $\frac{d{T}_c}{\mathrm{dp}}$ data requires the existence of two singularities: an electron saddle point disconnecting at 0.9 at.% Cd, and an electron sphere vanishing at about 1.6 at.% Cd. A quantitative calculation is made of the variation of $\frac{d{T}_c}{\mathrm{dp}}$ with alloy concentration, assuming only the empirically determined singular points, and deriving all other quantities from available indium Fermi-surface information on the third-band ring. The model calculation gives surprising quantitative agreement with the data, suggesting that valuable information on the Fermi-surface topology of dilute alloys can be obtained from measurements of $\frac{d{T}_c}{\mathrm{dp}}$ as a function of concentration.