(a) PW (vasp), (b) DZP (siesta) and (c) SZP basis sets were used. Fermi level is shown by a solid horizontal red line. The difference between the energies of the first

two band minima (Γ1−Γ2, illustrated in Figure 5), or the valley splitting, from the PW and DZP calculations, agrees with each other to within ∼6 meV. Significantly, the value obtained using our SZP basis set differs by 52 meV, some 55% larger than the value obtained using the PW basis set. The importance of this discrepancy cannot be overstated; valley splitting is directly relatable to experimentally observable resonances in transport spectroscopy of devices made with this δ-doping technology find more (see [26]). Figure 5 Minimum band energies for tetragonal systems with 1/4 ML doping. (a) PW (vasp), (b) DZP (siesta) and (c) SZP (siesta) basis sets were used. Fermi level also shown where appropriate. Bold numbers indicate energy differences between band minima. In the smallest cells (<16 layers), less than three bands are observed. This is likely due to the lack of cladding in the z direction, leading to a significant interaction between the dopant layers, raising the energy of each band. Whilst the absolute energy of each level still varies somewhat, even with over 100 layers incorporated, we find that the Γ1–Γ2 values

are well converged with 80 layers of cladding for all methods (see Figure 5). Indeed, learn more Resminostat they may be considered reasonably 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 (Γ2–δ splittings) are also shown in Figure 5 and show good convergence (within 1 meV) for cells of 80 layers or larger. The Fermi level follows a similar pattern to the Γ- and ∆-levels.

In particular, the gap between the Fermi level and Γ1 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 Table 2 summarises 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 δ-layer. As the results from the work of Carter et al. [32] show, the use of implicit doping causes the valley splitting value to be much smaller than in an explicit case (∼7 meV vs. 120 meV).