determine a consistent value of the valley splitting in explicitly $\delta$-doped structures by obtaining convergence between distinct DFT approaches in terms of basis set and system sizes. We perform a comparison of DFT techniques, involving localized numerical atomic orbitals and delocalized plane-wave (PW) basis sets. Convergence of results with regard to the amount of Si “cladding” about the $\delta$-doped plane is studied. This corresponds to the normal criterion of supercell size, where periodic boundary conditions may introduce artificial interactions between replicated dopants in neighboring cells. A benchmark is set via the delocalized basis for DFT models of $\delta$-doped Si:P against which the localized basis techniques are assessed. Implications for the type of modeling being undertaken are discussed, and the models extended beyond those tractable with plane-wave techniques. Using these calculations, we obtain converged values for properties such as bandstructures, energy levels, valley splitting, electronic densities of state and charge densities near the $\delta$-doped layer.

The paper is organized as follows: Sec. \[sec:methods\] outlines the parameters used in our particular calculations; we present the results of our calculations in Sec. \[sec:blah\]; and conclusions are drawn in Sec. \[sec:conclusion\]. An elucidation of effects modifying the bulk bandstructure follows in App. \[sec:bravais\] & 