@Lee11], and effective mass theories (EMT) [@Scolfaro94; @Rodriguez06; @Drumm11]. We note that several of these papers are based upon the assumption that the effective masses of $\delta$-doped P in Si remain unchanged from bulk-doped values;[@Scolfaro94; @Rodriguez06] an assumption which has been challenged.[@Qian05; @Cartoixa05] Others assume doping over a multi-atomic plane band[@Cartoixa05; @Scolfaro94] which no longer represents the state of the art in fabrication. We also note that Ref. represents the first attempt to model these devices by considering explicitly doped $\delta$-layers with DFT, using a relatively small localized basis set with the assumption that a basis set sufficient to describe bulk silicon would also adequately describe P-doped Si. There is currently little agreement between the valley splitting values obtained using these methods, with predictions ranging between 5 to 270 meV, depending on the arrangement of dopant atoms within the $\delta$-layer. Density functional theory has been shown to be a useful tool in predicting how quantum confinement and/or doping perturbs the bulk electronic structure in silicon- and diamond-like structures,[@Delley93; @Delley95; @Ramos01; @Zhou03; @Barnard03c] and it might be expected that the removal of the basis set assumption will lead to the best estimate of the valley splitting available.

In this paper we 