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.

• Received 7 November 2011

DOI:https://doi.org/10.1103/PhysRevB.85.155419