Accurate Band Structures

Energy bands can be calculated from first principles, without any experimental input. The main approximation required is the one-electron approximation (see Appendix A), which we use throughout this text. Then the two remaining questions are what does one use for the potential and what representation does one use to describe the wave function At present the same essential view of the potential is taken by almost all workers, based upon free-electron exchange and little, if any, modification for correlation. (This is discussed in Appendixes A and 

) The principal differences in different calculations are in the accuracy with which the potentials are determined self-consislently with the charge densities from the states that are being calculated. It may well be that the principal remaining inaccuracies are in use of the one-electron approximation itself, and that little is to be gained from further improvements within that context. 

Perhaps the most successful representation of the wave functions for band calculations for semiconductors has been the OPW method (orthogonalized plane-wave method), developed by Herring (1940). The success of the method has been due to the ease of obtaining and using realistic potentials in the calculation, in contrast to methods that utilize the muffin-tin approximation to the potential (discussed in Chapter 20). Only recently have difficulties with the application of muffin-tin potentials to semiconductors been overcome. (P or discussion and references see Johnson, Norman, and Connolly, 1973.) For any given potential, any of the accurate methods should give the same bands if the necessary effort is applied. 

To see trends with nietallicity and with polarity, notice the levels r 25 and r,s in the silicon bands. These levels in the other systems have been split by spin-orbit coupling, which will be discussed later, but it is not difficult to locate the corresponding levels and sec approximately where r 25 and F, 5 would have occurred in the absence of spin-orbit coupling. The separation between these two sets of levels was associated with the optical peak at , 2 in Section 4-C and was written as 2(1 2 -h = 21 2(1 — p) F or the homopolar materials, with K3 = 0, this 

Another important trend is the drop seen in the nondegenerate level at F in the 

---

We may substitute values for InAs from the Solid State Table to obtain 5.8 eV directly, in rough agreement with the observed peak position of 4.5 eV. We shall see soon that the ratio of estimated to observed peak energy is very much the same in tetrahedral solids other than InAs. The discrepancy comes principally from the error in our estimates of the energies of conduction bands. It does not come principally from differences in the splitting at (from accurate band structures) and the observed peak energy. The error in scale will be absorbed in the parameters that will be introduced in the discussion of in Section 4-D. 

The first accurate band structure calculations with inclusion of relativistic effects were published in the mid-sixties. Loucks published [64-67] his relativistic generalization of Slaters Augmented Plane Wave (APW) method. [68] Neither the first APW, nor its relativistic version (RAPW), were linearized, and calculations used ad hoc potentials based on Slaters s Xa scheme, [69] and were thus not strictly consistent with the density-functional theory. Nevertheless (or, maybe therefore ) good descriptions of the bands, Fermi surfaces etc. of heavy-element solids like W and Au were obtained.[3,65,70,71] With this background it was a rather simple matter to include [4,31,32,72] relativistic effects in the linear methods [30] when they (LMTO, LAPW) appeared in 1975. 

The primary reason for interest in extended Huckel today is because the method is general enough to use for all the elements in the periodic table. This is not an extremely accurate or sophisticated method however, it is still used for inorganic modeling due to the scarcity of full periodic table methods with reasonable CPU time requirements. Another current use is for computing band structures, which are extremely computation-intensive calculations. Because of this, extended Huckel is often the method of choice for band structure calculations. It is also a very convenient way to view orbital symmetry. It is known to be fairly poor at predicting molecular geometries. 

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. 

Recent calculations of hyperfine parameters using pseudopotential-density-functional theory, when combined with the ability to generate accurate total-energy surfaces, establish this technique as a powerful tool for the study of defects in semiconductors. One area in which theory is not yet able to make accurate predictions is for positions of defect levels in the band structure. Methods that go beyond the one-particle description are available but presently too computationally demanding. Increasing computer power and/or the development of simplified schemes will hopefully... 

The empirical approach [7] was by far the most fruitful first attempt. The idea was to fit a few Fourier coefficients or form factors of the potential. This approach assumed that the pseudopotential could be represented accurately with around three Fourier form factors for each element and that the potential contained both the electron-core and electron-electron interactions. The form factors were generally fit to optical properties. This approach, called the Empirical Pseudopotential Method (EPM), gave [7] extremely accurate energy band structures and wave functions, and applications were made to a large number of solids, especially semiconductors. [8] In fact, it is probably fair to say that the electronic band structure problem and optical properties in the visible and UV for the standard semiconductors was solved in the 1960s and 1970s by the EPM. Before the EPM, even the electronic structure of Si, which was and is the prototype semiconductor, was only partially known. 

Figures 4.35, 4.36, and 4.37 show the absorption spectra of the free radicals CIO, BrO, and IO, respectively (Wahner et al., 1988 DeMore et al., 1997 Laszlo et al., 1995). All have beautifully banded structures at longer wavelengths and large absorption cross sections, which allows one to measure these species in laboratory and atmospheric systems using differential optical absorption spectrometery (DOAS) (see Chapter 11.A.Id). However, as in the case of HCHO, adequate resolution is an important factor in obtaining accurate cross sections.

For example, from eqn (2.55), the average radial distance of the hydrogenic 3d and 4s wave functions are in the ratio 0.44 1. Thus we expect the band structure of transition metals to be represented accurately by a hybrid NFE-TB secular equation of the form... 

The density functional theory (DFT) [7,8] is now widely used in studying both infinite bulk crystalline materials and finite atoms, molecules, and clusters. In principle, the ground-state total energy as well as the electron density itself in interacting many-electron systems is accurately described in DFT. Therefore, the geometry optimization by minimizing the total energy should also be accurate in DFT as well. The electronic band structure is, on the other hand, a very useful but approximate physical concept based on the quasiparticle theory for inter-... 

As for the fundamental gap of semiconductors, the LDA is known to underestimate its width, typically from half to two thirds of experimental values, while the Hartree-Fock gap is again much worse, for example more than five times larger than the experimental value in the case of silicon. In the case of covalent-bond materials including carbon, the LDA band structure is in general expected to be accurate, while the fundamental gap value should be considered to be larger in semiconductors than the LDA value. 

The smallest adsorbates are atoms. For hydrogen, many quantum chemical techniques, both cluster- and band-structure types, produce the values of QH with the accuracy of a few kilocalories/mole (141). For other atoms the results may not be that accurate. For example, in the recent ab initio calculations of C/Ni(100) by Chiarello et al. (142), the calculated value of Qc = 293 kcal/mol exceeds the experimental value of 171 kcal/mol (43) by more than 120 kcal/mol. This is especially frustrating because accurate theoretical calculations of Qc for various metal surfaces appear to be the only alternative to alleviate the lack of experimental data on Qc. Anyway, in the BOC-MP approach, the values of QA are simply taken from experiment, so that there is no point for comparison. 

---