---
abstract: 'The differences in energy between electronic bands due to valley splitting are of paramount importance in interpreting transport spectroscopy experiments on state-of-the-art quantum devices defined by scanning tunneling microscope lithography. We develop a plane-wave density functional theory description of these systems which is size-limited due to computational tractability. We then develop a less resource-intensive alternative via localized basis functions, retaining the physics of the plane-wave description, and extend this model beyond the capability of plane-wave methods to determine the *ab initio* valley splitting of well-isolated $\delta$-layers. In obtaining agreement between plane-wave and delocalized methods, we show that the valley splitting has been overestimated in previous *ab initio* calculations by more than 50%.'
author:
- 'D. W. Drumm'
- 'A. Budi'
- 'M. C. Per'
- 'S. P. Russo'
- 'L. C. L. Hollenberg'
title: ' *Ab initio* calculation of valley splitting in monolayer $\delta$-doped phosphorus in silicon '
---

Introduction {#sec:intro}
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Quantum devices in silicon have been the subject of concentrated interest, both experimental and theoretical, in recent years. Efforts to make such devices have led to atomically precise fabrication methods which incorporate phosphorus atoms in a single monolayer of a silicon crystal [@Tucker98; @OBrien01; @Shen02; @Fuechsle07]. 