Band structure calculations pseudopotentials

Two band-structure calculations using a linear APW method7 and a pseudopotential local-orbital method8 produced similar band-structures... 

The pseudopotential concept was advanced a long ago [1] and is based on the natural energetic and spatial separation of core and valence electrons. The concept allows a significant reduction in computational efforts without missing the essential physics of phenomena provided the interaction of core and valence electrons is well described by some effective (model) Hamiltonian. Traditionally, pseudopotentials are widely used in the band structure calculations [2], because they allow convenient expansions of the wavefunctions in terms of plane waves suited to describing periodical systems. For molecular and/or nonperiodical systems, the main advantage of pseudopotentials is a... 

The methodology of band structure calculations is diverse< >. In general, however, the band structure determined in reciprocal space does not give a straightforward indication of the charge density in real space. Electronic charge densities have recently been calculated, however, for several semicon-ductors,< > with wavefunctions calculated using pseudopotential theory. 

Band structure calculations have been performed with the valence effective Hamiltonian (VEH) nonempirical pseudopotential technique. The VEH method yields one-electron energies of ab initio double-zeta quality and has been demonstrated to provide accurate estimates of essential electronic properties such as ionization potentials (IP), bandwidths (BW), bandgaps (Eg), and electron affinities (EA) in the context of conducting polymers. All the calculations have been carried out using the VEH parameters previously reported for sulfur, oxygen, and nitrogen atoms and those recently obtained for carbon and hydrogen atoms,... 

Let us suppose an infinite nondegenerate polymer chain (e.g., polythiophene) doped heavily with electron acceptors. At a high dopant content, the polymer-chain structure and electronic structure of the doped polymer are radically different from those of the intact polymer. As typical cases, we will describe two kinds of lattice structures of doped polythiophene (dopant content, 25 mole% per thiophene ring) a polaron lattice and a bipolaron lattice. They are the regular infinite arrays of polarons and bipolarons. The schematic polymer-chain structures are shown in Figure 4-16. Band-structure calculations have been performed for polaron and/or bipolaron lattices of poly(p-phenylene) [124], polypyrrole [124], polyaniline [125], polythiophene [124, 126], and poly( p-phenylenevinylene) [127], with the valence-effective Hamiltonian pseudopotential method on the basis of geometries obtained by MO methods. The schematic electronic band structures shown in Figure 4-17... 

Several factors have contributed to the present success of ab initio calculations for real materials systems. These include the development of approximations to the density functional formalism, refinements in band structure calculational techniques, invention of the ab initio pseudopotentials, and development of techniques for calculating total energies. Equally important, of course, is the availability of modern high speed computers. 

The organization of the lectures is as follows. A brief review of the theoretical techniques is given in Sec. II. This includes a discussion on the density functional formalism, generation of ab Initio pseudopotentials, and techniques for band structure calculations. The bulk systems are discussed in Sec. III. The static structural properties are presented in Sec. IIIA. These results establish the accuracy of the calculations. Examples will be given for semiconductors, insulators, and transition metals. The vibrational properties are discussed in Sec. IIIB. Phonon frequencies are calculated using the frozen phonon technique. 

Band structure calculations of fee crystalline Cgo using a pseudopotential local density formalism predict C(io to be a semiconductor with a band gap in the range 0.9-1.5 eV, and widths of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) derived bands in the range 0.4-... 

Any of the existing band structure methods can be adapted for use as a semiempirical scheme, or an interpolative scheme to facilitate the calculation of quantities which depend on interband integrals and the like. Tight binding theory, reduced to its bare essentials, with the overlap parameters used to fit experimental data or as an interpolation scheme in band structure calculation is generally referred to as Slater-Koster theory. Pseudopotential theory used in this way has been dubbed the empirical pseudopotential method (EPM) and has been the subject of a recent comprehensive review. Some comparisons of parameters t>(g), which have been fitted to experiment, with theoretical calculations have already been shown in Figure 12. 

Figure Al.3.14. Band structure for silicon as calculated from empirical pseudopotentials [25],...

It is possible to identify particular spectral features in the modulated reflectivity spectra to band structure features. For example, in a direct band gap the joint density of states must resemble that of critical point. One of the first applications of the empirical pseudopotential method was to calculate reflectivity spectra for a given energy band. Differences between the calculated and measured reflectivity spectra could be assigned to errors in the energy band... 

Figure B3.2.1. The band structure of hexagonal GaN, calculated using EHT-TB parameters detemiined by a genetic algorithm [23]. The target energies are indicated by crosses. The target band structure has been calculated with an ab initio pseudopotential method using a quasiparticle approach to include many-particle corrections [194].

For H at T in Ge, Pickett et al. (1979) carried out empirical-pseudopotential supercell calculations. Their band structures showed a H-induced deep donor state more than 6 eV below the valence-band maximum in a non-self-consistent calculation. This binding energy was substantially reduced in a self-consistent calculation. However, lack of convergence and the use of empirical pseudopotentials cast doubt on the quantitative accuracy. More recent calculations (Denteneer et al., 1989b) using ab initio norm-conserving pseudopotentials have shown that H at T in Ge induces a level just below the valence-band maximum, very similar to the situation in Si. The arguments by Pickett et al. that a spin-polarized treatment would be essential (which would introduce a shift in the defect level of up to 0.5 Ry), have already been refuted in Section II.2.d. 

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... 

In order to calculate the band structure and the density of states (DOS) of periodic unit cells of a-rhombohedral boron (Fig. la) and of boron nanotubes (Fig. 3a), we applied the VASP package [27], an ab initio density functional code, using plane-waves basis sets and ultrasoft pseudopotentials. The electron-electron interaction was treated within the local density approximation (LDA) with the Geperley-Alder exchange-correlation functional [28]. The kinetic-energy cutoff used for the plane-wave expansion of... 

FIGURE 6.8 Band structure of graphite calculated by a mixed-basis pseudopotential method. The solid and dotted lines denote the O- and re-bands, respectively. The Fermi level is indicated by EF. (From Holzwarth, N.A.W., et al., Phys. Rev. B, 26, 5382, 1982. With permission.)... 

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]. 

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. 

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