One of the most notorious difficulties associated with ab initio defect calculations is the band gap problem of DFT. It is well known, that the electronic gaps of semiconductor and insulator materials are significantly underestimated by Kohn-Sham DFT [107]. Two sources of error are typically cited: the self-interaction present in the Kohn-Sham Hartree energy (see Section2.1) and the lack of discontinuity in the functional-derivative of the exchange-correlation energy with respect to the particle number [108]. Such derivative discontinuity associated errors have been demonstrated to persist even if the exact exchange-correlation functional, able to counter the self-interaction energy, was used [109].
Errors in defect calculations resultant from a too small band gap are two fold in nature: (i) if the true band gap and the calculated one differ by ∆EG, the energy of the
DLS induced by a defect can be altered by up to the same amount. In cases where this state is occupied by an electron, the formation energy of the defect can be significantly affected by the shift. This scenario is illustrated in Fig.3.4(a) for a charge neutral TeSn impurity in SnO2. In compounds where the band edge shifts are extreme, the DLS can even be erroneously predicted to lie resonant with the valence or conduction bands, resulting in fictitious hybridization between the bulk and defect states, and an incorrect characterization of the defect; (ii) the determination of thermodynamic transition levels (described in Section3.3.1) with respect to the host band edges is prohibited. Fig.3.4(b) illustrates how the inaccurate assessment of the conduction band minimum can lead to the prediction of shallow donor character for a deep donor defect. As a consequence, only semi-quantitative interpretation of the electric and optical activity of a defect is provided by LDA and GGA calculations [110].
The self-interaction error of DFT presents an additional challenge as it tends to result in charge delocalization [111]. This effect can lead to changes in local atomic geometries surrounding a defect and the prediction of fictitiously delocalized states.
3.4. The Band Gap Problem ε(+/0) VBM CBM (b) PBE PBE0 (a) EG EGDFT EG EG
Figure 3.4. The effect of band gap changes on defect calculations: (a) Shift of the
defect level inside the fundamental band gap, associated with the band gap opening re- sultant from PBE0 calculations, illustrated for Te-doped SnO2 and (b) misclassification of a deep donor based on inaccurate band gap estimation. The light-blue shaded area shows the calculated and the dark-blue – the true conduction band edge.
Various approaches to overcome the above shortcomings of DFT have been proposed [97,
105] and are briefly summarized below.
The simplest approach suggests recovering the correct electronic band gap through the use of scissor-operators [112]. Scissor-operator based schemes shift rigidly only the respective band edges in order to open up the band gap [113]. Within this approach the structure of the bands (their width and dispersion) is assumed to be accurately represented by the underlying DFT calculations. A more sophisticated approach at- tempts to also shift the defect states accordingly [105,114] by decomposing them based on their valence and conduction band character [115]. Such a correction, while simple and, hence, computationally favorable does nothing to account for the self-interaction error and further offers no capacity to deal with situations where fictitious hybridization between the host and the defect states occurs [116].
The self-interaction error of density functional theory affects most prominently states that are strongly localized, such as the semicore d- or f-electrons of an atom. The method known as DFT + U [117,118] attempts to reduce this interaction by introduc- ing an on-site attractive Coulombic potential, U. To obtain the parameter U a variety of first-principles [117, 119] and empirical fitting [120, 121] approaches have been pro- posed. This, however, makes the method problematic to apply in general as it is not straightforwardly transferable between systems. In particular, systems with no a priori experimental information cannot be studied if fitting approaches are to be used. Worse still, chemical reference phases cannot be treated on the same footing as the host since the parameter U should reflect the dielectric screening of the material [119] and, thus, differs between systems. Additionally, although the self-interaction error may be coun- tered, the band gap, while improved, typically remains underestimated [97, 120, 121], requiring further extrapolation schemes to be adopted.
Modified pseudopotentials, similar in spirit to the DFT + U approach, offer a further alternative to account for the self-interaction. Here, a self-interaction correction (SIC) is incorporated directly into the construction of a pseudopotential [122]. The great advantage of the SIC approach is that once fitted the calculations are as cost efficient
as standard LDA/GGA methods [75, 123]. The drawbacks are the lack of predictive power outside of the fitted system and the lack of transferability, in particular, to reference phase calculations.
Many-body methods, such as GW [86, 87], are also available and can predict accu- rately the quasiparticle energies of a system. In this way, accurate calculations of not only the band gap, but also of the defect levels within it are permitted. However, such approaches are challenging to apply to large supercells, which are frequently required to converge defect calculations, due to their hefty computational cost. Furthermore, local relaxations cannot yet be performed using GW approaches, therefore, geometry inaccuracies inherent to self-interaction of DFT would still need to be circumvented by an alternative method [124]. Nevertheless, significant progress has been made in this direction and such calculations are usually informative in the evaluation of “lower- level” calculation performance. If the high computational cost can be afforded, quantum Monte Carlo simulations [125] for point defects in solids [126] offer a final alternative.
In this thesis, we chose to use the hybrid-functional approach to overcome the band gap problem. Hybrid-functionals are discussed at length in Section 2.1.5, where we demonstrate how the admixture of the exact-exchange of Hartree-Fock theory helps to counter the self-interaction problem of DFT. Despite having no fitted parameters, functionals such as PBE0 and HSE06, seem to result in electronic band gaps that offer astonishingly accurate agreement with experimental results, as demonstrated in Chapter 4. Furthermore, hybrid functionals have been ubiquitously cited to compare well with both experimental observations and the predictions provided by higher levels of theory, such as GW or quantum Monte Carlo [97].