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There have recently been several experiments studying inducedsuperconductivity in semiconducting two-dimensional electron gases that arestrongly coupled to thin superconducting layers, as well as probing possibletopological phases supporting Majorana bound states in such setups. We showthat a large band shift is induced in the semiconductor by the superconductorin this geometry, thus making it challenging to realize a topological phase.Additionally, we show that while increasing the thickness of thesuperconducting layer reduces the magnitude of the band shift, it also leads toa more significant renormalization of the semiconducting material parametersand does not reduce the challenge of tuning into a topological phase.

Proximity effect in a two-dimensional electron gas coupled to a thin superconducting layer