Friedel oscillations in the two-dimensional electron gas in the low-density regime

We calculate the screened Coulomb interaction in real space for two-dimensional electron gases in a plane and in a superlattice. We include anomalous screening due to many-body effects via the local-field correction. We find that many-body effects lead to a strong attractive intraplane potential. The strong attractive part is interpreted as a many-body effect in the low-carrier-density regime. Our interpretation for a superlattice is supported by the results for a single plane. The instability of the electron gas is of Kohn-Luttinger type. For r s > 2 we find that the attraction is much larger than the Fermi energy which should lead to a Bose-Einstein condensation.