converged even at the 40-layer level (0.5 meV or less difference to the largest models considered). The differences between the energies of the second and third band minima ($\Gamma_{2}$–$\Delta$ splittings) are also shown in Fig. \[fig:energies\], and show good convergence (within 1 meV) for cells of 80 layers or larger.

The Fermi level follows a similar pattern to the $\Gamma$- and $\Delta$-levels. In particular, the gap between the Fermi level and $\Gamma_{1}$ level does not change by more than 1 meV from 60 to 160 layers.

Given that the properties of interest are the differences between the energy levels, rather than their absolute values (or position relative to the valence band), in the interest of computational efficiency we observe that using the DZP basis with 80 layers of cladding is sufficient to achieve consistent, converged results.

Valley splitting {#sec:valley}
----------------

Table \[tab:valley\] summarizes the valley splitting values of 1/4 ML P-doped silicon obtained using different techniques, showing a large variation in the actual values. In order to make sense of these results, it is important to note two major factors that affect valley splitting: the doping method and the arrangement of phosphorus atoms in the $\delta$-layer. As the results from 