Band structure of crystals

Theoretical calculations of the band structure of crystals belong to solid state physics and are not discussed here. Quantitative ab initio prediction of a band gap is a problem of great complexity. However, empirical and semi-empirical estimates of Eg, using the concepts of structural chemistry, are sufficient for most purposes of physical chemistry and materials science. Indeed, since the valence band of a compound usually involves primary orbitals of the anions (nonmetal atoms), and the conduction band involves primary orbitals of the cations (metal atoms), the energy of the transition between the two (i.e.. Eg) must be related to some atomic properties. 

The effect of strains on the band structure of crystals has been intensively investigated by Bir and Pikus, using both the theory of perturbation and a group-theoretical method, namely the theory of invariants. The approach can be applied to deduce the strain derivatives of the wavefunctions appearing in Eq. (5). Following the procedure of the authors of Ref. 8, the Kohn-Sham operator for a valence electron in a strained crystal can be written in the form... 

In practical calculations of the band structure of crystals by the Hartree-Fock method, the one-electron density matrix Prr (k) in the reciprocal space can be calculated only for a rather small finite set of special points (see Sect. 4.2). Let us consider such a set of special points kj, j = 1,2,..., Nq. Then, in the calculation of the density matrix of the infinite crystal the integration over the BriUouin-zone volume Vg is changed by the sum over the special points chosen... 

Band Structure of crystals Table 4.1. Equations that define the TBA model. 

Figure 6-4. Qualitative energy level diagram of the 1 Bu excinm band structure of T<, at A =0 derived by the Ewald dipole-dipole sums for excitation light propagating along the a crystal axis.

Fig. 8.12 Crystal structure (a), band structure of La3(B2N4) (b), and orbital interactions along [B2N4] stacks (c) (interactions with lanthanum orbitals are omitted for clarity).

The valence band structure of very small metal crystallites is expected to differ from that of an infinite crystal for a number of reasons (a) with a ratio of surface to bulk atoms approaching unity (ca. 2 nm diameter), the potential seen by the nearly free valence electrons will be very different from the periodic potential of an infinite crystal (b) surface states, if they exist, would be expected to dominate the electronic density of states (DOS) (c) the electronic DOS of very small metal crystallites on a support surface will be affected by the metal-support interactions. It is essential to determine at what crystallite size (or number of atoms per crystallite) the electronic density of sates begins to depart from that of the infinite crystal, as the material state of the catalyst particle can affect changes in the surface thermodynamics which may control the catalysis and electro-catalysis of heterogeneous reactions as well as the physical properties of the catalyst particle [26]. 

In a supercell geometry, which seems to have become the method of choice these days, the impurity is surrounded by a finite number of semiconductor atoms, and what whole structure is periodically repeated (e.g., Pickett et al., 1979 Van de Walle et al., 1989). This allows the use of various techniques that require translational periodicity of the system. Provided the impurities are sufficiently well separated, properties of a single isolated impurity can be derived. Supercells containing 16 or 32 atoms have typically been found to be sufficient for such purposes (Van de Walle et al., 1989). The band structure of the host crystal is well described. 

Another approach that provides a good desciption of the band structure of the host crystal is based on the Green s function determined for the perfect crystal. This function is then used to calculate changes induced by the presence of the defect (e.g., Rodriguez et al., 1979 Katayama-Yoshida and Shindo, 1983). The Green s function approach seems to be more cumbersome and less physically transparent that the supercell technique. 

The impurity interacts with the band structure of the host crystal, modifying it, and often introducing new levels. An analysis of the band structure provides information about the electronic states of the system. Charge densities, and spin densities in the case of spin-polarized calculations, provide additional insight into the electronic structure of the defect, bonding mechansims, the degree of localization, etc. Spin densities also provide a direct link with quantities measured in EPR or pSR, which probe the interaction between electronic wavefunctions and nuclear spins. First-principles spin-density-functional calculations have recently been shown to yield reliable values for isotropic and anisotropic hyperfine parameters for hydrogen or muonium in Si (Van de Walle, 1990) results will be discussed in Section IV.2. 

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. 

V. L. Bonch-Bruevich, Effect of Heavy Doping on the Semiconductor Band Structure Donald Long, Energy Band Structures of Mixed Crystals of III-V Compounds Laura M. Roth and Petros N. Argyres, Magnetic Quantum Effects... 

SILAR has been used for the synthesis of CdS/ZnS coatings for CdSe quantum dots. The precursor solutions were prepared by dissolving CdO, ZnO, and S in oleic acid and octadecane. The final coating consisted of three layers of CdS and three additional layers of ZnS. The photonic band structure of the photonic crystal had a modifying influence on the photoluminescence of the embedded quantum dots.90... 

Electron transitions in transition-metal ions usually involve electron movement between the d orbitals (d-d transitions) and in lanthanides between the / orbitals (/-/ transitions). The band structure of the solid plays only a small part in the energy of these transitions, and, when these atoms are introduced into crystals, they can be represented as a set of levels within the wide band gap of the oxide (Fig. 9.15). 

The simple energy-gap scheme of Figure 4.6 seems to indicate that transitions in solids should be broader than in atoms, but still centered on defined energies. However, interband transitions usually display a complicated spectral shape. This is due to the typical band structure of solids, because of the dependence of the band energy E on the wave vector k ( k =2nl a, a being an interatomic distance) of electrons in the crystal. 

It is reported that the band structure of ZnS doped with transition metal ions is remarkably different from that of pure ZnS crystal. Due to the effect of the doped ions, the quantum yield for the photoluminescence of samples can be increased. The fact is that because more and more electron-holes are excited and irradiative recombination is enhanced. Our calculation is in good correspondence with this explanation. When the ZnS (110) surface is doped with metal ions, these ions will produce surface state to occupy the valence band and the conduction band. These surface states can also accept or donate electrons from bulk ZnS. Thus, it will lead to the improvements of the photoluminescence property and surface reactivity of ZnS. 

Fig. 1. Representation of the band structure of GaAs, a prototypical direct band gap semiconductor. Electron energy, E> is usually measured in electron volts relative to the valence, v, band maximum which is used as the zero reference. Crystal momentum, k, is in the first Brillouin zone in units of 2%/a...

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