As mentioned in Ch. 1, simulatingab initioa sufficiently large sample of disorder realisations, for a number of concentrations and systems sizes, is simply not feasible, even with the efficient scheme implemented in ONETEP. Since they have the same structure, however, we can translate the DFT matrices into a TBM that is more efficient to build.
The idea of combining DFT and TBM is far from new, with early work including Andersen and Jepsen (1984), Sankey and Niklewski (1989) or Porezag et al. (1995). In those works the goal is to calculateab initiospecific tight-binding elements, an operation that usually requires the transformation from the standard- DFT basis of plane waves, to the localised basis set of the TBM.
In our approach, instead, we recognise the tight-binding structure of the Kohn-Sham Hamiltonian, when written in a basis of localised support functions, and we use it to build effective Hamiltonians that can reproduce the results ob- tained from the DFT. The focus is on building a DFT-like system rather than a TBM. This is achieved by simulating prototypes of pure or doped Si, as de- scribed in Ch. 2. The idea is the following: when we build the Hamiltonian² of a disordered sample, we assume that the self-energies and hopping blocks for each impurity are those calculated with ONETEP for a doped system. The final Hamiltonian, therefore, looks like the Hamiltonian for pure Si, expect for those blocks that describe the presence of a defect. The matrix blocks that we reuse from the DFT simulations are stored incatalogues, whose construction and use we now describe (see Fig. 3.2).
The assumption behind using catalogues to build effective models is that the potential around each impurity islocally the same. It is up to us to decide the extent of this ‘locality’: the higher the range of the hopping, the more accurately we capture the effect of the impurity potential. For our effective models, we keep all hopping terms up to the 10th shell of nearest neighbours,³ since this is the extent of the “near-sighted” description via the local orbitals (see Sec. 2.1.5).
This picture should be completed by treating pairs of impurity with a separate catalogue, unless we can justify that (i) two defects do not interact at any distance or (ii) the formation of pairs is discouraged by an associated energy cost, as de- scribed in Sec. 2.4. For this reason, we have computed a catalogue of pairs only
2In the rest of the section, everything that is said about the Hamiltonian also holds for the
overlap matrix.
3.2. TRANSLATION FROM DFT TO EFFECTIVE MODELS 41 a) Catalogue 0 2 4 6 8 y / a 0 2 4 6 8 0 2 4 6 8 x/a y / a
0
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b) Effective tight-binding model
Figure 3.2: Construction of the effective tight-binding model. Part (a) represents the catalogue of prototypes. For clarity we show a projection on thexy plane and distances in units ofa, the Si lattice parameter. The upper plot depicts one impurity (yellow) and the neighbouring Si atoms (green); the lower plot shows two impurities at distancea and their Si neighbours (dark green). Gray sites indicate Si atoms unaffected by the impurity potential. In (b) we show how we build an effective tight-binding model with 29 impurities. The colour code is the same as in (a) and indicates which catalogue is used. Due to the projection on the xyplane some impurities appear closer than they are. Reproduced from Carnio et al. (2017).
for sulphur, and not for phosphorus. Analogously to how we use the catalogue for isolated impurities, when a site is near a pair the Hamiltonian blocks are taken from the DFT simulation of that pair. For Si sites that are close to two impurities that are not considered a pair, we use the matrix blocks that connect it to the closest of the defects.
In this work, two defects are a pair when they are no more than a unit cell apart. This cut off can be decided by comparing the ONETEP runs of pairs at increasing distance to their effective model using the single-impurity catalogue. Because each defect induces a state in the band gap of Si, a parameter we can compare is the difference in energy between the defect states, see Fig. 3.3. When S defects are first nearest neighbours, they form bonding and anti-bonding states, where the former descend into the valence band while the latter remain in the band gap. In this case we compare the distance of the anti-bonding state to the lowest-energy conduction band state. While the 1-impurity catalogue manages to capture this feature of first nearest neighbours, the corresponding 2-impurity catalogues gives
1-impurity catalogue Extended catalogue D ef ec t st at es g ap m is es m a on 0.1 1.0 10.0
Distance between impuri es [a0]
0.4 0.6 0.8 1.0 1.2 1.4
Figure 3.3: Ratio of the energy difference between the two impurity states ap- pearing when a system of 4096 atoms is doped with a pair of defects at increasing distance. The ‘1-impurity catalogue’ is built from systems with only one impu- rity, while the ‘extended catalogue’ includes the description of pairs of defects. For situations where only one state in the gap appears, discussed in the text, the gap is taken between said state and the lowest conduction band state.
a better description by definition. Moreover, the 1-impurity catalogue predicts a similar situation when S defects are second nearest neighbours. This contradicts the results from ONETEP and is correctly rectified in the extended catalogue.
Of course, restricting the catalogues to pairs is arbitrary: like in diagrammatic theories, we could include triplet, quadruplets,et cetera. To gauge how our ap- proximation is performing, we have simulated a system with 4096 atoms and 29 S impurities, both with our effective model and ONETEP. As shown in Fig. 3.4, we obtain a spectrum of impurity states that extends roughly over the same range of energies, especially towards the valence band. The number of states in the spectrum also matches between the effective model and the ONETEP simulation, i.e. we obtain the correct number of bonding and anti-bonding states, as discussed in the previous paragraph. We note, however, that the effective model slightly underestimates the energy in the upper portion of the spectrum, resulting in a larger band gap than computed by the DFT, where the impurity and conduction band seem to have already joined.
3.3. TECHNICAL CONSTRUCTION 43