region, and were then extended to their tractable limit (200 and 160 layers, respectively) to study convergence past the capability of PW.

![Ball & stick model of a $\delta$-doped Si:P layer, viewed along the $[110]$ direction; 32 layers in the $[001]$ direction are shown. Si atoms (small gray spheres), P atoms (large dark gray spheres), covalent bonds (gray sticks), repeating cell boundary (solid line).[]{data-label="fig:diagram_P_layer_pbc"}](diagram_P_layer_pbc){width="0.6\linewidth"}

For the tetragonal cells the *k*-point sampling was set as a $9\times9\times N$ $\Gamma$-centred MP mesh, as we have found that failing to include $\Gamma$ in the mesh can lead to anomalous placement of the Fermi level on bandstructure diagrams. $N$ varied from 12 to 1 as the cells became more elongated (see Table \[tab:bulkSi\] in App. \[sec:bravais\]).

Although it has been previously found that relaxing the positions of the nuclei gave negligible differences ($<$ 0.005 ) to the geometry,[@Carter09] this was for a 12-layer cell and may not have included enough space between periodic repetitions of the doping plane for the full effect to be seen. We have performed a test relaxation on a 40-layer cell using the PW basis (<span style="font-variant:small-caps;">vasp</span>). The maximum subsequent ionic displacement was 0.05 , with most being an order 