($\Delta_{\rm{FCC}}$) direction, below the value at $\Gamma$, while the SZP bandstructure shows the reverse – the minimum at $\Gamma$, a similar amount below a secondary minimum in the $\Sigma_{\rm{TET}}$ direction.

The difference between the energies of the first two band minima ($\Gamma_{1}$–$\Gamma_{2}$, illustrated in Fig. \[fig:energies\]), or the valley splitting, from the PW and DZP calculations agree with each other to within $\sim$6 meV. Significantly, the value obtained using the SZP basis set differs by 52 meV, some 55% larger than the value obtained using the PW basis set. The importance of this discrepancy cannot be overstated; this valley splitting is directly relatable to experimentally observable resonances in transport spectroscopy of devices made with this $\delta$-doping technology (see Ref. ).

In the smallest cells ($<$ 16 layers), less than three bands are observed. This is likely due to the lack of cladding in the $z$-direction, leading to significant interaction between the dopant layers, raising the energy of each band. Whilst the absolute energy of each level still varies somewhat, even with over 100 layers incorporated, we find that the $\Gamma_{1}$–$\Gamma_{2}$ values are well-converged with 80 layers of cladding for all methods (see Fig. \[fig:energies\]). Indeed, they may be considered reasonably 