Band structure zincblende

Zinc antimonide, 3 44, 53—54 Zinc atomizing process, 26 598 Zinc baths, 9 828-829, 830t Zincblende semiconductors, 22 141 band structure of, 22 142-144 transport properties of, 22 148, 149t Zinc borates, 4 282-283 Zinc brass... 

The theoretical calculations of the band structure of InN can be grouped into semi-empirical (pseudopotential [10-12] or tight binding [13,14]) ones and first principles ones [15-22], In the former, form factors or matrix elements are adjusted to reproduce the energy of some critical points of the band structure. In the work of Jenkins et al [14], the matrix elements for InN are not adjusted, but deduced from those of InP, InAs and InSb. The bandgap obtained for InN is 2.2 eV, not far from the experimentally measured value. Interestingly, these authors have calculated the band structure of zincblende InN, and have found the same bandgap value [14]. 

FIGURE 1 Electronic band structures of (a) wurtzite and (b) zincblende GaN. 

FIGURE 8 Schematic band structures around the top of the valence bands of zincblende GaN (a) excluding the spin-orbit interaction, and (b) including it. The figure is not drawn to scale. 

Pantelides, S., and W. A. Harrison (1975). Structure of the valence band of zincblende-type semiconductors. Phys. Rev. Bll, 3006-21. 

The band structure of solids has been studied theoretically by various research groups. In most cases it is rather complex as shown for Si and GaAs in Fig. 1.5. The band structure, E(kf is a function of the three-dimensional wave vector within the Brillouin zone. The latter depends on the crystal structure and corresponds to the unit cell of the reciprocal lattice. One example is the Brillouin zone of a diamond type of crystal structure (C, Si, Ge), as shown in Fig. 1.6. The diamond lattice can also be considered as two penetrating face-centered cubic (f.c.c.) lattices. In the case of silicon, all cell atoms are Si. The main crystal directions, F —> L ([111]), F X ([100]) and F K ([110]), where Tis the center, are indicated in the Brillouin zone by the dashed lines in Fig. 1.6. Crystals of zincblende structure, such as GaAs, can be described in the same way. Here one sublattice consists of Ga atoms and the other of As atoms. The band structure, E(k), is usually plotted along particular directions within the Brillouin zone, for instance from the center Falong the [Hl] and the [HX)] directions as given in Fig. 1.5. 

Generally, the lowest two conduction bands are well described within the second order of k. The energy dispersion around the conduction band minimum (CBM) which is the T point in zincblende (ZB) and wurtzite (WZ) structures is given by... 

In the analyses of conventional zincblende (ZB) semiconductors, we frequently assume a parabolic band for the conduction bands, and the 6 x 6 Luttinger-Kohn Hamiltonians are used to describe the upper valence bands [1,2], In treating the valence bands together with the conduction bands on an equal footing, as when estimating the momentum matrix elements, we often make use of the 8 x 8 Kane Hamiltonian [3], However, the form of the Hamiltonians reflects the crystal symmetry, and Kane Hamiltonians are constructed under the condition of cubic symmetry. For wurtzite (WZ) materials, therefore, we must consider hexagonal symmetry in the effective Hamiltonian. Let us consider the 8 x 8 k.p Hamiltonian for WZ structure [4,5],... 

Band energy will vary in proportion to the square of k near the conduction-band minimum and, for the zincblende structure, must by symmetry be independent of the direction of k. Thus, near the conduction-band minimum, we can write... 

Poole, Liesegang, Leckey, and Jenkin (1975) have reviewed published band calculations for the alkali halides and tabulated the corresponding parameters obtained by various methods. Pantclidcs (1975c) has used an empirical LCAO method that is similar to that described for cesium chloride in Chapter 2 (see Fig. 2-2), to obtain a universal one-parameter form for the upper valence bands in the rocksalt structure. This study did not assume only one important interatomic matrix clement, as we did in Chapter 2, but assumed that all interatomic matrix elements scale as d with universal parameters. Thus it follows that all systems would have bands of exactly the same form but of varying scale. That form is shown in Fig. 14-2. Rocksalt and zincblende have the same Brillouin Zone and symmetry lines, which were shown in Fig. 3.6. The total band width was given by... 

Pantiledes s values of r. arc given in Table 14-2. He went on to study the relation between these bands and the zincblende bands (Pantiledes, 1975c), as discussed in Chapter 6. In addition, he has carried out the corresponding analysis for the valence bands in the lluorite structure, to obtain universal bands for that system (unpublished). A more recent band calculation for fluorite itself has been made by Albert, Jouain, and Gout (1977). 

At the same time, calling this a. fit to the bands is very much understating the accomplishment. The set of four parameters in Table 2-1 and the term values in Table 2-2 (all in the Solid State Table) allow calculation of energy bands for any of the homopolar semiconductors or any of the zincblende-structure compounds, as simply for one as for the other, without computers, with consistent accuracy, and without need for a previous accurate calculation for that compound. Only in first-row compounds is there indication of significant uncertainty in the results. Furthermore, as we noted in Table 2-1, the theoretical matrix elements are very nearly equal to the ones obtained by fitting bands thus, if we had plotted bands in Fig. 3-8,a that were based upon purely theoretical parameters, the curves would have been hardly distinguishable. 

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