Defects affect physical properties of solids in a dramatic way. This is particularly apparent in the case of semiconductors, such as Si or GaAs, in which small concentrations of impurities - donors or acceptors - cause huge changes in conductance. Almost all electronic and optoelectronic devices contain layers of either n-doped or p-doped semiconductors, and both conductivity types are often required in the same device, e.g. in light-emitting diodes, semiconductor lasers, bipolar transistors, optical detectors, solar cells, etc . While technology advances, new materials are being used and new applications are being implemented. One class of materials which experience a rapid increase of interest are wide-gap semiconductors and oxides [2-4]: ZnO [5-14], In2O3 , SnO2 , NiO , ZnSe , GaN [18,19], etc. For instance, a large band gap is needed for optoelectronic devices working in the green, blue, and UV range. Contrary to conventional semiconductors, those materials often exhibit a pronounced doping asymmetry: ZnO or SnO2 can be readily n-doped, but p-doping has proved extremely challenging; the opposite is true for NiO. Electronic structure calculations have been invaluable for understanding the role of defects . One can calculate formation energy of defects in different charge states, which gives information about their abundance, charge activity (i.e. acceptor or donor character), and possible compensation mechanisms which counteract doping. Such calculations provide deep insight into the nature of unintentional doping, and, moreover, can guide experimental efforts in the attempt to overcome the "doping bottleneck" in these materials . Problems similar to those in wide-gap oxides arise in connection with new photovoltaic materials such as chalcopyrite semiconductors [21,22], as well as nanocrystalline semiconductors [23,24].
Theoretical modelling is also crucial in the field of microelectronics [25-31]. The aggressive scaling of complementary metal-oxide-semiconductor field-effect transistors based on silicon will not last much longer: alternative materials have to replace silicon and its (nitro-)oxide. On the oxide side [32-37], high-k dielectrics, such as HfO2, Al2O3 and ZrO2 are being investigated, with hafnium-based devices already on the market. On the semiconductor side [31,33,38-40], high mobility Ge and GaAs are currently studied, while SiC is finding its place in power applications. Alternative semiconductors also allow a larger freedom in the choice of the dielectric. Integrating these new materials has been very challenging with yet many problems to solve. Flat-band and threshold voltage shifts, carrier mobility degradation, charge trapping, gate dielectric wear-out and breakdown, as well as temperature instabilities are thought to mainly originate from various defects forming at (or close to) the semiconductor-oxide interface. Since interfaces are notoriously difficult to characterize experimentally, theoretical simulation can be extremely helpful in identifying relevant defects, the position of their charge states, and explaining (or predicting) the pertinence of certain defects to particular experimental observations. Similar conclusions hold in all fields where interfaces are important, such as in photovoltaic devices, semiconductor lasers, etc.
The theoretical investigation of defects, i.e. the study of formation energies, charge-state transition levels, equilibrium concentrations, resulting charge carrier densities, as well as the alignment of defect levels at interfaces, mainly resorts to density functional theory (DFT) to describe the electronic structure. However, standard semi-local approximations to DFT, i.e. the local density approximation (LDA) and the generalized gradient approximation (GGA), severely underestimate the band gaps of bulk materials . For example, LDA band gaps of Si and ZnO are ~0.6 and ~0.7 eV, to be compared with experimental values of 1.1 and 3.4 eV, respectively. In the most extreme cases, certain semiconductors, such as Ge or CuInSe2, are predicted to have vanishing or negative band gaps in semi-local approximations. This "band-gap problem" of LDA and GGA seriously affects the predictive power of DFT calculations [42,43]. Since the fundamental gap is the relevant energy scale for electron chemical potentials and charge transition levels, a proper description of defects requires that the band gap is described accurately.
Recently, significant progress has been achieved in developing advanced electronic structure methods which go beyond standard DFT. Many-body perturbation theory in the GW approximation (and related methods) [44-48] is in principle the most accurate scheme to calculate bulk band gaps of semiconductors and insulators. One can combine quasi-particle calculations in GW with total-energy calculations in DFT to obtain very accurate defect levels and formation energies [43,49,50]. This line of research is theoretically appealing but requires further development. GW calculations are very demanding and are not always practical. On a more pragmatic level, one resorts to simpler, sometimes more intuitive, schemes. DFT+U is one of such schemes, in which it is assumed that the self-interaction of metal d states is at the core of the band-gap problem, as in ZnO, SnO2 or In2O3 [3,10,13,15,16,21,22]. DFT+U schemes do not fully correct the band gap, and, when applied to defects, further extrapolation is needed. The latter schemes, however, differ in the way they are performed [3,10], and the optimal method is yet to be found. Another approach to improve the band gap consists in using hybrid density functionals, in which a fraction of Hartree-Fock exchange is admixed to GGA exchange [51-54]. These functionals are increasingly being used for defect calculations [32-34,40,42,57]. It has been observed that in order to bring calculated band gaps in agreement with experiment different fractions of Hartree-Fock exchange have to be used for different materials [14,33,37,40,57,58]. In addition to varying this fraction one can vary the screening length of Hartree-Fock exchange, achieving a similar effect . The theoretical justification of this band-gap tuning has so far not much been discussed and requires further research. A method which is related to hybrid functionals is the optimized effective potential (OEP), in which the exact exchange potential has a local form .
Some methods to relieve the "band-gap problem" are more specific, and are usually only applied to defect calculations. One is the use of modified pseudopotentials [60-63]. Here, one replaces an ab initio-derived ionic pseudopotential with a modified one, ensuring that the relevant structural properties and energy differences remain unaffected, but bringing the calculated band gap of the material in agreement with experiment. In the past, the scissor-operator scheme has been popular to correct for the band-gap error in semiconductors . Applied to defects this scheme consists of shifting defects levels in accordance to their valence band or conduction band character. As recently suggested , the "band-gap problem" affects indirectly even formation energies of neutral defects. For these systems accurate quantum Monte Carlo studies provide reliable benchmark results [56,65].
In addition to electrical properties mentioned above, optical properties of defects are also very important. Not only do these determine the coloration of natural crystals, but they are also crucial for a wide range of applications, e.g. in solid-state laser technologies. Many defects in wide-gap materials are experimentally identified only via their typical optical signatures. Thus, studies of excited states of defects are very important. The description of optical properties goes beyond the single-particle picture of DFT and GW. There exist three major ways to study excited states of defects: (i) quantum chemical electron correlation methods if the defect can be described as a small cluster [72,73]; (ii) time-dependent DFT which relies on approximations to the time-dependent exchange-correlation potential ; and (iii) calculation of GW quasi-particle energies followed by the solution of the Bethe-Salpeter equation (BSE), which accounts for electron-hole interactions . All these approximations provide complementary insight into the nature of the physical problem and are presently the subject of intensive research.
DFT calculations of defects are usually performed employing the supercell formalism, in which finite-size effects are very important [19,21,66-71]. Different proposals for corrections may lead to different quantitative and even qualitative results. The optimal correction scheme is an issue which will also be discussed, since it critically affects the comparison with experiment.