Tight-Binding Band Theory

In this paper we present preliminary results of an ab-initio study of quantum diffusion in the crystalline a-AlMnSi phase. The number of atoms in the unit cell (138) is sufficiently small to permit computation with the ab-initio Linearized Muffin Tin Orbitals (LMTO) method and provides us a good starting model. Within the Density Functional Theory (DFT) [15,16], this approach has still limitations due to the Local Density Approximation (LDA) for the exchange-correlation potential treatment of electron correlations and due to the approximation in the solution of the Schrodinger equation as explained in next section. However, we believe that this starting point is much better than simplified parametrized tight-binding like s-band models. 

Chapter 2 introduces the band theory of solids. The main approach is via the tight binding model, seen as an extension of the molecular orbital theory familiar to chemists. Physicists more often develop the band model via the free electron theory, which is included here for completeness. This chapter also discusses electronic condnctivity in solids and in particular properties and applications of semiconductors. 

In this section we show how the general form of Renner-Teller interaction matrices can be obtained at any order in the phonon variables and with electron orbital functions of different symmetry (p-like, < like, /-like, etc.). For this purpose, we use an intuitive approach [18] based on the Slater-Koster [19] technique and its generalization [20] to express crystal field or two-center integrals in terms of independent parameters in the tight-binding band theory [21] then we apply standard series developments in terms of normal coordinates. 

W. A. Harrison and J. Tersoff, Tight-binding theory of heterojunction band lineups and interface dipoles, J. Vac. Sci. and Technol. B4, 1068 (1986). 

Electronic band structures were calculated for several polymeric chains structurally analogous to polyacetylene (-CH-CH) and carbyne (-CbC). Ihe present calculations use the Extended Huckel molecular orbital theory within the tight binding approximation, and values of the calculated band gaps E and band widths BW were used to assess the potential applic ilitf of these materials as electrical semiconductors. Substitution of F or Cl atoms for H atoms in polyacetylene tended to decrease both the E and BW values (relative to that for polyacetylene). Rotation about rhe backbone bonds in the chains away from the planar conformations led to sharp increases in E and decreases in BW. Substitution of -SiH or -Si(CH,) groups for H in polyacetylene invaribly led to an increase in E and a decrease in BW, as was generally the case for insertion of Y ... 

The band structure of a three-dimensional solid, such as a semiconductor crystal, can be obtained in a similar fashion to that of a polyene. Localized molecular orbitals are constructed based on an appropriate set of valence atomic orbitals, and the effects of delocalization are then incorporated into the molecnlar orbital as the number of repeat units in the crystal lattice is increased to infinity. This process is widely known to the chemical conununity as extended Hiickel theory (see Extended Hiickel Molecular Orbital Theory). It is also called tight binding theory by physicists who apply these methods to calcnlate the band structures of semiconducting and metallic solids. 

The relative importance of the Coulomb interaction versus the band structure is a classic problem of the field. In tight-binding theory, the ir-electron band structure extends over a band width. 

Fig. 4.15. Band structure for Mo as obtained from tight-binding calculations and from density functional theory (from Mehl and Papaconstantopoulos (1996)). The notation F, N, H, etc. refers to particular i-points within the BriUourn zone.

There are many theories on the mechanism of the segregation effect that suggest either a chemical or an electronic mechanism or both types of mechanisms. However, it seems that the most reliable mechanism is electronic as proposed by Mukheijee and Moran [35]. This electronic model calculates the chemical properties of the pure constituents from their physical parameters and then estimates those of the alloys. It employs the tight-binding electronic theory, the band filling of the density of states, and the bandwidth of the pure components for the calculations. However, it seems that the 2D Monte Carlo simulations produce better results by using the embedded atom and superposition methods. The latter allows for the calculation of the compositions from the relative atom positions, and the strain and the vibrational energies has been reviewed for 25 different metal combinations in [36]. It was also possible to predict composition oscillations as a consequence of the size mismatch. 

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