The Band Gap Problem

We have already mentioned the inaccuracy of LDA-KS when determining the band gap of semiconductors and insulators. This failure is intimately related to a pathological non-analytical behaviour of the xc energy functional, as shown by J. P. Perdew and M. Levy and by L. J. Sham and M. Schliiter [47,48], Namely, the xc potential may be increased by a finite constant of the order of 1 eV as a result of the addition of an extra electron to an extended system, that is, after an infinitesimal change of the electron density. 

As is well known [49,50], the band gap Egg,p of an N electron system is defined as the difference between the electron affinity A = 

For a non-interacting system, the gap can be readily written in terms of its orbital energies. Therefore, for the fictitious N electron KS system we have 

From (5.16) and (5.17), we immediately get that the actual and KS gaps are related through 

The existence of a discontinuity in v c is, then, more plausible than an error in the LDA band-structm e. 

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The problem of the cohesive energy, in principle, can be corrected with a better exchange-correlation functional. The band gap problem is more difficult because density functional formalism does not explicitly provide information on the excitation energies. The quasiparticle energies should, in general, be obtained from the one-particle Green s function.However, recently Sham and Schliiter have shown that there is a formal relationship between the minimum gap Eg of an insulator and the Kohn-Sham gap g obtained from a difference in eigenvalues ... 

Rex W. Godby and Pablo Garcla-GonzAlez 5.3.1 The Band Gap Problem... 

Usually, the LDA conduction bands are shifted from the correct bands by a quantity that is only weakly dependent on k. A common solution to the band-gap problem is then to rigidly shift upward the Kohn-Sham conduction bands. This is called the scissors operator . 

While the experimental investigation of surface electronic properties is still in its infancy, a good picture of surface electronic properties has emerged from ah initio DFT-LDA calculations. Characteristic of the (0001) and (0001) surfaces are N- or In(Ga)-adatom-terminated structures, and for very group-III-rich surface conditions, a metalHc Ga or In bilayer evolves (Section 13.4.2). For the Ga- or In-adatom-stabihzed GaN or InN(OOOl) surfaces, calculated band structures are shown in Figure 13.41. In this study, modified pseudopotentials have been applied to solve the band gap problem inherent to LDA and GGA [115]. Thus, the energy positions of both occupied and unoccupied electronic bands should be reliable. 

As it has been described in various other review articles before, the conversion efficiencies of photovoltaic cells depend on the band gap of the semiconductor used in these systems The maximum efficiency is expected for a bandgap around Eg = 1.3eV. Theoretically, efficiencies up to 30% seem to be possible . Experimental values of 20% as obtained with single crystal solid state devices have been reported " . Since the basic properties are identical for solid/solid junctions and for solid/liquid junctions the same conditions for high efficiencies are valid. Before discussing special problems of electrochemical solar cells the limiting factors in solid photovoltaic cells will be described first. 

Numerous papers have dealt with the problem of simultaneous production of hydrogen and oxygen on illuminated colloidal or suspended Ti02 particles. It is not intended to review this problem in a comprehensive manner, but a few important details may be mentioned. H2 and Oj were reported to be generated in the band gap illumination of platinized powdered TiOj in contact with water, although the mecha-... 

The band-gap excitation of semiconductor electrodes brings two practical problems for photoelectrochemical solar energy conversion (1) Most of the useful semiconductors have relatively wide band gaps, hence they can be excited only by ultraviolet radiation, whose proportion in the solar spectrum is rather low. (2) the photogenerated minority charge carriers in these semiconductors possess a high oxidative or reductive power to cause a rapid photocorrosion. 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

The most important feature in Fig. 8.4 is the existence of states in the Au-doped Si within the band gap of pure Si. These new states make the recombination of electron-hole pairs far more rapid in the doped material than in pure Si. That is, the existence of Au impurities in crystalline Si has serious negative effects on the properties of the material as a solar cell. This problem does not only occur with Au a large number of metal impurities are known to cause similar problems. More information about this topic is available from the sources in the Further Reading section at the end of the chapter. 

Electron correlation plays an important role in determining the electronic structures of many solids. Hubbard (1963) treated the correlation problem in terms of the parameter, U. Figure 6.2 shows how U varies with the band-width W, resulting in the overlap of the upper and lower Hubbard states (or in the disappearance of the band gap). In NiO, there is a splitting between the upper and lower Hubbard bands since IV relative values of U and W determine the electronic structure of transition-metal compounds. Unfortunately, it is difficult to obtain reliable values of U. The Hubbard model takes into account only the d orbitals of the transition metal (single band model). One has to include the mixing of the oxygen p and metal d orbitals in a more realistic treatment. It would also be necessary to take into account the presence of mixed-valence of a metal (e.g. Cu ", Cu ). 

The corrosion reactions cause another serious problem. They produce a number of surface reaction iniermediates such as dangling bonds and atomic vacancies, having electronic levels within the band gap. Namely, corrosion reactions produce a number of mid-gap surface states, which act as surface recombination centers for photogenerated carriers. The Voc is decreased largely by the production of such surface states.3,41... 

Another potential image sensor is the CCD discussed in Chapter 9 by Matsumura. The major problem in using a-Si H for CCD applications is the high density of traps or localized states in the band gap. A theoretical analysis assuming an exponential distribution of gap states permits us to predict the transfer characteristics (residual electron density as a function of time). With state-of-the-art material it should be possible to make usable image sensors. Early experimental results with novel test structures have yielded transfer inefficiencies of less than 1% at clock frequencies between 1 and 200 kHz. 

For insulating materials, such as ceramics, this normally does not pose problems, because the conduction band is empty and the band gap error is not reflected energetically. However, for M/C interfaces there may be some reason to worry, since mixing between oxide conduction bands and metal states will influence the magnitude of adhesion and charge transfer. These states may become filled, depending on the Fermi level on the metal side. Further research on this issue is warranted. 

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