Plasmon effect on the Coulomb pseudopotential $mu^*$ in the McMillan equation

We examine the Coulomb pseudopotential $mu^*$ in the McMillan equation applying to the superconductivity of heavily doped semiconductors. Systematic calculation using the first-principles calculation suggests that $mu^*$ should be considered as a variable quantity depending on carrier density $n$ in semiconductors, although it is usually considered as a constant about 0.1. To clarify $n-$dependence of $mu^*$, we solve the McMillan equation inversely for $mu^*$ by combining the result of the first-principles calculation and that of experiments. It indicates that $mu^*$ decreases with $n$ and becomes negative under $n sim 5 times 10^{-21}[{rm cm^{-3}}]$. This reduction is explained by the effect of plasmon which may play an important role in the superconductivity of low carrier systems such as heavily doped semiconductors.