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.