The Energy-Band Problem

Schlosser, H. and Marcus, R (1963). Composite wave variational method for solution of the energy-band problem in solids, Phys. Rev. 131, 2529-2546. 

First principles MO calculations were done by the Hartree-Fodc-Slater (HFS) method using the DV-Xa method. The DV-Xa method was first developed to solve the energy-band problems by D. E. Ellis and G. S. Painter [18] and was then applied to the cluster calculation by F. W. Averill and D. E. Ellis [19]. Self-consistent calculations were developed by A. Rosen et al. 

Before presenting the linear method, let us briefly review how the energy-band problem has been tackled in the past. In this context we note that the traditional methods may be divided into those which express the wave functions as linear combinations of some fixed basis functions, say plane waves or atomic orbitals, and those like the cellular, APW, and KKR methods which employ matching of partial waves. As we shall see, both approaches have their strong and weak points. 

In Chap.2 I deal with the simplest aspects of the LMTO method based upon the KKR-ASA equations. The intention is to familiarise the reader with the concepts and language used in linear theory. This is where I introduce structure constants, potential functions, canonical bands, and potential parameters, and where it is shown that the energy-band problem may be separated into a potential-dependent part and a crystal-structure-dependent part. 

At this stage we have completed the presentation of the LMTO method as an efficient procedure for solving the energy-band problem. Inter alia we have introduced a number of concepts which allow us to interpret the complete calculations. 

The physical interpretation of the KKR-ASA equations is that the energy-band problem may be approximated by a boundary-value problem in which the surrounding lattice through the structure constants imposes a k-dependent and non-spherically symmetric boundary condition on the solutions P (E) inside the atomic Wigner-Seitz sphere. This interpretation is illustrated in Fig.2.1. 

In the following sections we shal] discuss the structure- and potential-dependent parts of the energy-band problem separately and introduce the concepts of canonical band theory. 

We have now completed the parametrisation of the non-structurally dependent part of the energy-band problem. In the process, we found that for each i quantum number only four parameters D, 9 and were needed to pro-... 

The energy band formalism is exact. The other two describe the corresponding valence and geometrical schemes (which are, as usual, more or less conditional and approximate). These schemes are, however, very useful and are conveniently employed in a number of problems. As a rule, each problem permits of a translation from one formalism to another. 

The problem of detailed modeling can be divided into three parts (1) What is the shape of the energy bands within the device, particularly close to the electrodes (2) What is the mechanism for injection of charges through any barriers which are formed at the interface (3) To what extent are the characteristics of the device determined by transport and recombination within the bulk of the polymer, rather than by injection at the interfaces ... 

Many models have been presented to explain quantitatively the dependence of exciton energy on the cluster size [7, 11, 21-30]. This problem was first treated by Efros et al. [7], who considered a simple particle in a box model. This model assumes that the energy band is parabolic in shape, equivalent to the so-called effective mass approximation. The shift in absorption threshold, AE, is dependent upon the value of the cluster radius, R, Bohr radius of the electron, ae ( = h2v.jmce2), and Bohr radius of the hole, ah (= h2e/mhe2). When (1) R ah and R ae, and (2) ah R ac,... 

The STR may be used to calculate the canonical structure constants defined by (6.7-9) or (8.23,24). In a typical application the programme is executed once for a given crystal structure. It produces and stores on disk or tape a set of structure-constant matrices distributed on a suitable grid in the irreducible wedge of the Brillouin zone. Whenever that particular crystal structure is encountered, the structure constant matrices may be retrieved and used to set up the LMTO eigenvalue problem which, in turn, leads to the energy bands of the material considered. 

The next problem is the band filling and the position of the Fermi level Ep at zero Kelvin. We have seen that the supposed uniform degree of ionicity p can be defined from the stoichiometry (cf. Table 2). It allows us to define the number of charge carriers per molecule, (Ng/N), and therefore, from the energy band dispersion E(k), the wave vector kp which is associated with the Fermi level position. It follows that every orbital can hold two electrons ... 

Different approaches to the problem of excitations in correlated electron systems are considered in [183]. They are based on a quasiparticle description when electron correlations are weak and on a Green s function or projection operator approach when they are strong. In both cases, intersite correlation contributions to the energy bands require special attention. 

One of the early triumphs of quantum mechanics in the area of solid state materials was the ability to explain why certain materials are metallic conductors while others are insulating or semiconducting. The energy band structure, which can be calculated by ab initio or semiempirical methods, provides access to these electrical properties. As will be shown below, DFT methods are well suited to treat metallic systems whereas there are problems in the accurate prediction of energy band gaps in semiconductors and insulators. Furthermore, certain transition metal oxides and solids containing rare-earth and actinide elements present serious theoretical challenges which have not been completely resolved yet,... 

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