We note that Nmod(4) ∈ 1,2,3 systems exhibit new position types, requiring further modelling. Although such investigation would greatly inform the ongoing discussion of disorder in δ-doped systems, due to computational resource constraints, they are not considered here. Models were replicated as A N , B N , C N , and undoped (for bulk properties comparison without band-folding complication) structures. Electronic relaxation was undertaken, with opposite donor spins initialised for each layer and various properties calculated. The general method of [16] using SIESTA [28], and energy convergence of 10-6 eV, was used with two exceptions: an optimised

double- ζ with polarisation (DZP) basis [19] (rather than the default) was employed for all calculations, and the C 80 model was only converged to 2 × 10-4 in density (and 10-6 eV in energy) due to intractability. Band structures had at least RAD001 research buy 25 points between high-symmetry locations. The choice of a DZP basis over a single- ζ with polarisation (SZP) basis was discussed in [16], where it was found for single δ layers to give valley splittings in far better agreement with those calculated via plane-wave

methods. In the recent study by Carter et al. [23], less resource-intensive methods were employed to approximate the disordered-bilayer Carfilzomib cell line system, however, here we employ the DZP basis to model the completely ordered system. Results and discussion Benchmarking of N = 80 model Although we used the general method of [16], as we used the optimised basis of [19], we benchmark our A 80 model with their 80 ML single- δ-layer (δ 1) calculation rather than those of [16]. (Lee et al. [18] also used the same general method.) Our supercell being precisely twice theirs, apart from having spin freedom between layers, results should be near identical. Figure 2 is the A 80 band structure. Agreement is very good; band shapes are similar, and the structure is nearly identical. A closer look reveals that A 80 has two bands to the δ 1’s one, as we should expect – A 80 has

two dopant layers to Protein tyrosine phosphatase δ 1’s one. Due to 80 ML of Si insulation, the layers behave independently, resulting in degenerate eigenspectra. Comparison of band minima shows quantitative agreement within 20 meV; the discrepancy is likely a combination of numerical differences in the calculations (generally accurate to approximately 5 meV), the additional spin degree of freedom (which may allow less repulsion between the layers), and band folding from the extension of the bilayer supercell in z. Figure 2 A 80 band structure and the δ 1 band structure of [12]. The partially occupied bilayer bands are doubly degenerate, and the valence band maximum has been set to zero energy. Band structures and splittings Band structures for other models were calculated in the same fashion. Comparisons of band minima are shown in Table 1. Within types, the band minima change drastically as N shrinks and the δ sheets come closer together.