Semiconductors pseudopotential theory

We might also expect the matrix elements of djdx to scale inversely with d among the homopolar semiconductors (and correspondingly, for the matrix elements of X to scale with d) and this is in fact predicted by the pseudopotential theory of Chapter 18. However, that does not describe the trends in Xi(0) well, and in Section 4-C, we shall allow the proportionality constant to vary from row to row in the Periodic Table. 

This completes the specification of the pseudopotential as a perturbation in a perfect crystal. We have obtained all of the matrix elements between the plane-wave states, which arc the electronic states of zero order in the pseudopotcntial. We have found that they vanish unless the difference in wave number between the two coupled states is a lattice wave number, and in that case they are given by the pseudopotential form factor for that wave number difference by Eq. (16-7), assuming that there is only one ion per primitive cell, as in the face-centered and body-centered cubic structures. We discuss only cases with more than one ion per primitive cell when we apply pseudopotential theory to semiconductors in Chapter 18. Tlicn the matrix element will be given by a structure factor, Eq. (16-17),... 

The remarkable conclusion of this argument is that though pseudopotentials can be used to describe semiconductors as well as metals, the pseudopotential perturbation theory which is the essence of the theory of metals is completely inappropriate in semiconductors. Pseudopotential perturbation theory is an expansion in which the ratio W/Ep of the pseudopotential to the kinetic energy is treated as small, whereas for covalent solids just the reverse quantity, EpiW, should be treated as small. The distinction becomes unimportant if we diagonalize the Hamiltonian matrix to obtain the bands since, for that, we do not need to know which terms are large. Thus the distinction was not essential to the first use of pseudopotentials in solids by Phillips and Kleinman (1959) nor in the more recent application of the Empirical Pseudopotential Method used by M. L. Cohen and co-workers. Only in approximate theories, which are the principal subject of this text, must one put terms in the proper order. 

In reality, typical solids of interest occupy a spectrum between these two extremes, from the alkali halides, which are well described by TB theories, to Al, which is well described by NFE theories. (It is not, in fact, straightforward NFE theory, but rather pseudopotential theory, which is successful for Al, but for the moment this distinction will be ignored.) Intermediate between the two cases are, for instance, group IV semiconductors, in which both points of view have proved useful, and transition metals, in which different parts of the band structure conform to NFE and TB descriptions. 

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... 

What has been accomplished is a very simple relation between the pseudopotential and the important gap in the band structure. What is more, we have provided such a simple representation of the band structure that we may use it to calculate other properties of the semiconductor, just as we did with the LCAO theory once we had made the Bond Orbital Approximation. 

In order to treat polar semiconductors, we must make some assumption as to how the odd part of the pseudopotential, K, varies with distortion. The assumption that it is independent of shear, which was used in the LCAO theory, gives a polar value different from the homopolar value by a factor 2/( 2 + V Y = ae, or values for the isoelectronic series of Ge, GaAs, and ZnSe of 0.87,0.81, and 0.74, respectively, compared to the experimental values of 0.80, 0.65, and 0.32. The trend is right, though it is not quantitatively very accurate. To estimate a, we used the empty-core polarities from Table 18-2. The agreement is better if LCAO values are used but not significantly so. 

In Section 10-F, the description of the interface between two semiconductors came quite naturally in terms of LCAO theory. That is probably the best way to treat interfaces, but the pseiidopotenlial method also provides a correct way to proceed, and it is of some interest to make the comparison, particularly since that is presumably the way that semiconductor-metal interfaces should be formulated, Pseudopotential treatments of. semiconductor interfaces have in fact been made by Chelikowsky and Cohen (1976c), Frensicy and Kroenier (1976), and Baraff ct al. 

In order to treat polar semiconductors, we must make some assumption as to how the odd part of the pseudopotential, K3, varies with distortion. The assumption that it is independent of shear, which was used in the LCAO theory, gives a polar value different from the homopolar value by a factor V2j V + — a, ... 

If the reader has actually made it up to this point, he or she will have the impression that the whole universe of solid-state materials, i.e., insulators, semiconductors, metals, and intermetallic compounds can nowadays be studied by electronic-structure theory, and predictive conclusions are really in our own hands. Indeed, the numerical limitations of most classical approaches - in particular, the ionic model of everything - have been overcome. While the computational methods of today include very different quantum-chemical methods, their varying levels of accuracy and speed are due to differences in their atomic potentials and the choice of the basis sets that are involved. The latter may either be totally delocalized (plane waves) or localized (atomic-like), adapted to the valence electrons only (pseudopotentials) or to all the electrons. In order to understand structures and compositions of solid-state materials, the results of electronic-structure theory are typically investigated in terms of some quantum-chemical analysis. 

The concept of quasi-free conduction electrons implies that their scattering by the ion core potential in the solid is rather weak. From here the modem theory of the pseudopotential has been developed. This theory shows that it is possible to reproduce the scattering of electron waves by replacing the deep potential at each site of the ionic core by a very much weaker effective potential, the pseudopotential. Thus the total pseudopotential in the metal or the semiconductor, which the conduction electrons feel, is fairly uniform, and the replacement of the real potential by the pseudopotential is a perfectly rigorous procedure. Furthermore, the fact that the total pseudopotential is fairly flat means that one can apply perturbation theory in order to calculate electron energies, cohesion, optical properties, etc. [20]. 

Furthermore, for treating semiconductors (and also the stucture of simple metals), one has to leave the second order perturbation theory. The self-consistent pseudopotentials which are then needed, are not discussed in these lecture notes at all. They wOl be discussed by several lecturers at this ASI, in connection with several other topics, not even mentioned in these notes. 

This set of lectures reviews some of the many recent theoretical accomplishments in this area. Experimental results are discussed only in so far as they bear on a theoretical result. In addition, in order to limit the scope of this review, emphasis will be placed on a few theoretical techniques—primarily the pseudopotential approach. Specific prototype systems are considered to illustrate the accomplishments of the theory for semiconductors, insulators, and transition metals. Some details of the calculations and results will be given, but the reader should go to the original papers for more specifics. 

As in any semiconductors, point defects affect the electrical and optical properties of ZnO as well. Point defects include native defects (vacancies, interstitials, and antisites), impurities, and defect complexes. The concentration of point defects depends on their formation energies. Van de WaHe et al. [86,87] calculated formation energies and electronic structure of native point defects and hydrogen in ZnO by using the first-principles, plane-wave pseudopotential technique together with the supercell approach. In this theory, the concentration of a defect in a crystal under thermodynamic equilibrium depends upon its formation energy if in the following form ... 

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