Further Information
The functionality of semiconductors is closely connected to their point defects, controlling the electronic and optical behavior of the material. This has motivated the evolution of defect physics, which has by now become a substantial part of both theoretical and experimental condensed matter physics. In the past decade, the emphasis has shifted from microelectronics to optoelectronics and photovoltaics, i.e., from traditional semiconductors with relatively narrow gaps, to wide band gap materials. While the standard local (LDA) and semi-local (GGA) approximations of density functional theory (DFT) have played an important role in understanding the electrical and optical properties of defects in the former case, these approximatons often fail completely in the latter [1,2]. On the one hand, the total energy, as a function of the occupation numbers, has no derivative discontinuity at integer values, which leads to serious underestimation of the band gap. Consequently it often happens that actual gap levels are hidden in the bands and are incorrectly unoccupied (donors) or occupied (acceptors). On the other hand, the convex nature of the total energy between integer occupation numbers leads to artificial delocalization of defect states. Since the localization of a defect state usually determines the position of the corresponding level in the gap, this error precludes accurate prediction of gap states. In addition, small polaron states, which often occur in wide band gap materials, are usually missed. First-principles total energy calculations beyond (semi)local DFT are still struggling with the system sizes necessary for defect calculations. Even though algorithms for RPA and GW calculations with cubic scaling have been developed [3], in lack of vertex correction, this method could not properly account for polaron localization in TiO2 [4]. In recent years hybrid functionals have emerged as a useful alternative, especially screened hybrids [5,6]. Most often the semi-empirical HSE functional of Heyd, Scuseria and Ernzerhof [7] is applied, with the standard parameters (HSE06 [8]) chosen to work well for a set of semiconductors with medium screening. HSE06 breaks down in metals [9], and is not so good either, if the screening is weak (ionic insulators) [10,11]. for the latter case, the HSE parameters are often tuned to reproduce the band gap (see, e.g., Refs.[12–15], with uncertain consequences to defect state localization. To maintain proper localization, it has been suggested, that such a tuning must make the functional compliant with the generalized Koopman’s theorem (following from the linearity of the total energy with fractional occupation numbers) [16,17]. In any event, tuning leads to material specific parameters.
A computationally less demanding approach is the use of DFT+U [18], which partially corrects for the self-interaction error in case of the strongly correlated d (an f) electrons, by employing an on-site Hubbard-like U potential, which has also the effect of shifting occupied states down and unoccupied ones up, i.e., increasing the gap in some systems. The value of U can be determined from first principles in a perfect crystalline system but that leads to material specific results, e.g., even to slightly different values in rutile and anatase [19], and is not really transferable to defects. At the ab initio value, the gap is still too small, while semi-empirical values tuned to reproduce the latter cause over-localization of defect states [20]. In addition, the application of DFT+U leads to less accurate geometries and thermodynamic data. Therefore, quite often, U corrections are also applied to s- and p-states, more for practical reasons than by physical justification, even though – with appropriate care – quite good results can be achieved [21,22]. Obviously, such parameterizations pertain only to the chosen system.
To these semi-empirical methods one might also add the use of non-local empirical potentials for improving the gap [23], and the polaron correction [24] to make a GGA calculation compliant with the generalized Koopmans’ theorem. Like with all the other methods so far, one ends up with strongly material specific, semi-empirical parameters, with limited transferability and general predictive power. One has to realize that all these methods try to do the task of firstprinciple many-body theories at a fraction of the cost. It is, therefore, not surprising that we are getting what we paid for. However, it is still a long way before ab initio many-body methods will be applicable to supercells of several hundred atoms, and there is definitely room for improving the semi-empirical approximations in order to capture more and more of the real physics. The way to do that is to test and compare the existing methods in systems with only slightly different properties, e.g., in structurally or compositionally similar semiconductors. Testing requires reliable reference data. One possibility is to consider Ga-based semiconductors. It should be noted that, e.g., HSE06, so successful in Group IV semiconductors [25] (with the calculated band gaps of 1.17, 3.21, and 5.42 eV reproducing accurately the experimental values 1.17, 3.23 and 5.48 eV in Si, 4H-SiC and diamond, respectively), underestimates the gap of all Ga-based semiconductors, resulting in 1.18 eV in GaAs [8], 2.22 eV in CuGaS2 [26], 3.24 eV in GaN [27], and 4.25 eV in ß-Ga2O3 [28], as compared to the experimental (optical) gaps of 1.42 eV, 2.44 eV, 3.51 eV, and 4.70 eV, respectively. Therefore, these materials provide the possibility to understand the systematics in the optimized parameters. At the same time, these materials are of immense practical significance in many application areas and, thereby, also interesting for experimentalists.
Among gallium based semiconductors, the blue laser diodes and LEDs have made GaN the most important material for optoelectronics, while ß-Ga2O3, with a wider band gap than GaN but readily available as good quality single crystal or thin film, has a great potential for application as n-type layer in power electronic devices [29–32]. It is also considered for deep-UV transparent electrodes [33,34] or UV-blind photodetectors [35]. The Ga-based chalcopyrites, CuGaS(Se)2 (and their indium containing analogues, the so called CIGS materials) are the most promising absorbers for low-cost high-efficiency thin film solar cells [36,37]. CuGaS2, which can be doped p-type [38], has the widest band gap among the CIGS materials, and Sn-doping allows to produce an intermediate defect band in the gap [39,40] for enhanced light utilization [41]. While there are still many open questions regarding defects even in GaN (like the assignment of the charge transition level 0.3 eV above the valence band maximum (VBM) [42] to the CN substitutional, for which theory predicts 0.9 eV [43]), defect physics of Ga2O3 has barely started. Recently, deep level spectra have become available [44,45], and new information is also available from photoluminescence experiments [46,47]. Still, theoretical work with hybrid functionals has provided conflicting results [14,16,48] and the basic intrinsic defects are not yet identified. Defects in CIGS materials are being under study for a long time now, but the problems of LDA/GGA make comparison with experiment very difficult [49,50,51]. The situation is aggravated by the fact that experimental data are mostly available on mixed CuInxGa1-x crystals [52], which makes it difficult to interpret them even on the basis of hybrid calculations [53,54,55]. Experimental defect spectroscopy is plagued by several difficulties: in capacitance based methods it is difficult to distinguish between signals caused by defects or by barriers and other contributions to the capacitance [56,57,58] whereas photoluminescence in Cu-poor material is dominated by electrostatic potential fluctuations[59]. Nevertheless, combining recent hybrid calculations with the available experimental data, a consistent model starts of emerge [60]. Most defect calculations in the field of CIGS are done on the indium compound. There is little information about the Ga compound, although the technologically relevant material is an alloy of the indium and the gallium compound.
Considering the depicted general situation of defects in gallium oxide and gallium based chalcogenides, it seems to be a good idea to bring together theorists and experimentalist working in the field.
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