  Monolayer \delta-doped structures in silicon have attracted renewed interest
with their recent incorporation into atomic-scale device fabrication strategies
as source and drain electrodes and in-plane gates. Modeling the physics of
\delta-doping at this scale proves challenging, however, due to the large
computational overhead associated with ab initio and atomistic methods. Here,
we develop an analytical theory based on an effective mass approximation. We
specifically consider the Si:P materials system, and the limit of high donor
density, which has been the subject of recent experiments. In this case,
metallic behavior including screening tends to smooth out the local disorder
potential associated with random dopant placement. While smooth potentials may
be difficult to incorporate into microscopic, single-electron analyses, the
problem is easily treated in the effective mass theory by means of a jellium
approximation for the ionic charge. We then go beyond the analytic model,
incorporating exchange and correlation effects within a simple numerical model.
We argue that such an approach is appropriate for describing realistic,
high-density, highly disordered devices, providing results comparable to
density functional theory, but with greater intuitive appeal, and lower
computational effort. We investigate valley coupling in these structures,
finding that valley splitting in the low-lying \Gamma band grows much more
quickly than the \Gamma-\Delta band splitting at high densities. We also find
that many-body exchange and correlation corrections affect the valley splitting
more strongly than they affect the band splitting.
