We consider the three-dimensional electron gas confined by a strictly two-dimensional homogeneous positive charge density at z=0. Within the Hartree-Fock approximation, we study the mode structure in the confined direction in the metallic regime. We find, that for rs<1.3 (rs<2.5) the unpolarized (polarized) electron gas starts to populate also the first excited state in the z-direction.
@article{arxiv.1009.3789,
title = {Ground state of a quasi-two-dimensional electron gas},
author = {B. Bernu and F. Delyon and M. Holzmann},
journal= {arXiv preprint arXiv:1009.3789},
year = {2011}
}