supercell (8 atoms) aligned with the $\langle1 0 0\rangle$ crystallographic directions. In this case, we experience some of the common labelling; the $\Delta$ direction is defined in the same manner for both BZs, although we see band folding (in a similar manner to that discussed above) due to the size difference of the reciprocal cells (see Fig. \[fig:bandstructure\]). We also see a difference in that although the $\Sigma$ direction is consistent, the points at the BZ boundaries have different symmetries and therefore, labels ($K_{\rm{FCC}}$, $M_{\rm{SC}}$). (The $L_{\rm{FCC}}$-point and $\Lambda_{\rm{FCC}}$-direction have no equivalent for tetragonal cells, and hence we do not consider bandstructure in that direction here)

Consider now the $\delta$-doping case discussed above (see Sec. \[sec:methods\]), where we wish to align our cell with the $\left[1 1 0\right]$ and $\left[\bar{1} 1 0\right]$ directions (by rotating the cell 45$^{\circ}$ anticlockwise about $z$; this will also require a resizing of the cell in the plane to maintain periodicity – see Fig. \[fig:rotation\]), to allow us to include precisely four atoms per monolayer (as required for the minimal representation of 1/4 ML doping). We now have a situation where the $X_{\rm{TET}}$ point in the new tetragonal BZ (see Fig. \[fig:tetBZ\]) is no longer 