Ref.  show, the use of implicit doping causes the valley splitting value to be much smaller than in an explicit case ($\sim$7 meV vs. 120 meV). It is also shown that the use of random P coverage on the $\delta$-layer reduces the valley splitting value by only 40–50 meV compared to the fully ordered placement, leaving a large discrepancy between the valley splitting results from implicit and explicit doping. This massive decrease in valley splitting due to implicit doping can be explained by the smearing of the doping layer in the direction normal to the $\delta$-layer, thereby decreasing the quantum-confinement effect responsible for breaking the degeneracy in the system. Ref.  also shows that the arrangement of the phosphorus atoms in the $\delta$-layer strongly influences the valley splitting value. In particular, they showed that there is a difference of up to 220 meV between P doping along the $[110]$ direction and along the $[100]$ direction. It must be noted, however, that most of their patterns are not yet physically realizable due to the P incorporation mechanism currently employed.

  ------------------------------------------------------------------ ----------- -----------
                              Technique                                No. of      Valley
                                                                       layers     splitting
                                                                                    (meV)
              Planar Wannier orbital[^1]$^{,}$[@Qian05]                 1000         20
                Tight binding (4 K)[^2]$^{,}$[@Lee09]                 $\sim$150   $\sim$17
                  Tight binding 