Short Series Superconductor Density of States Integral Approximations

Abstract

Accurate and fast Fermi-Dirac integral approximations are used in semiconductor device simulators to compute carrier concentrations where Boltzmann statistics cannot be applied. A similar integral can be used to compute carrier densities in the subgap region of a superconductor. The primary difference between the two integrals is the density of states (DOS) functional used. Electrons and holes in semiconductors use the spherical band approximation yielding a DOS functional proportional to the square root of state energy ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$E$</tex-math></inline-formula> ). Superconductors in the subgap region can be modeled with a DOS of the form <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(1-\frac{E^{2}}{\Delta ^{2}})^{-0.5}$</tex-math></inline-formula> . A short series approximation using Gaussian quadrature is computed for the superconductor DOS integral. Short series approximations are also applied to the Fréchet derivatives of the integral with respect to its parameters. All short series approximations will be compared against numeric integration solutions and results in a six hundred fold reduction in integration time. A table containing the short series approximation roots and weights is given. Error plots are shown for the short series approximations at different temperatures for the niobium pair-breaking potential energy.