of 300 Ry. In all cases, the generalized gradient approximation (GGA) PBE [@Perdew96] exchange-correlation functional was used.

The lattice parameter for bulk Si was calculated using an 8-atom cell, and found to be converged for all methods with a $12 \times 12 \times 12$ Monkhorst–Pack (MP) *k*-point mesh [@Monkhorst76]. The resulting values are presented in Table \[tab:lattice\], and were used in all subsequent calculations.

                             Method                             $a_{0}$ ()
  ------------------------------------------------------------ ------------
    PW (<span style="font-variant:small-caps;">vasp</span>)       5.469
   DZP (<span style="font-variant:small-caps;">siesta</span>)     5.495
   SZP (<span style="font-variant:small-caps;">siesta</span>)     5.580

  : Equilibrium lattice parameters for an 8-atom cubic unit cell for the different methods used in this work.[]{data-label="tab:lattice"}

In modeling $\delta$-doped Si:P, as used in Ref. , we adopted a tetragonal supercell description of the system, akin to that of Refs. & . In accordance with experiment, we inserted the P layer in a monatomic $\left(001\right)$ plane as one atom in four to achieve 25% doping. This will henceforth be referred to as 1/4 monolayer (ML) doping. In this case, the smallest repeating in-plane unit had four atoms per monolayer (to achieve 1 in 4 doping), and was a square with sides parallel to the $\left[110\right]$ and $\left[\bar{1}10\right]$ directions. The square had side length $a\sqrt{2}$ (see Fig. \[fig:c22schematic\]), where 