Image-potential band-gap narrowing at a metal/semiconductor interface

GW approximation is used to systematically revisit the image-potential band-gap narrowing at metal/semiconductor interfaces proposed by Inkson in the 1970's. Here we have questioned how the narrowing as calculated from quasi-particle energy spectra for the jellium/Si interface depends on $r_s$ of the jellium. The gap narrowing is found to only weakly depend on $r_s$ (i.e., narrowing $\simeq 0.3$ eV even for a large $r_s = 6)$. Hence we can turn to smaller polarizability in the semiconductor side as an important factor in looking for larger narrowing.