Pseudopotential band

As described briefly in Chapter 3, a promising new method of electronic structure calculation utilizing combined molecular-dynamics and density-functional theory has recently been developed by Car and Pari-nello (1985). This approach has recently been applied to cristobalite, yielding equilibrium lattice constants within 1% of experiment (Allan and Teter, 1987), as shown in Table 7.2. New oxygen nonlocal pseudopotentials were also an important part of this study. Such a method is a substantial advance upon density-functional pseudopotential band theory, since it can be efficiently applied both to amorphous systems and to systems at finite temperature. 

We have determined, both experimentally and theoretically, the one-electron density of states surrounding the Fermi level in polymeric sulfur nitride, (SN)X. The experimental measurements were performed using X-ray and ultraviolet photoemission spectroscopy (XPS and UPS), while the theoretical studies employed calculations based on OPW and pseudopotential band structures of (SN)X. 

Here C ° and E denote a zeroth-order approximation for the quasi-particle states. In our Si calculation this zeroth-order approximation was extracted from an empirically fitted pseudopotential band-structure (see ref.4 and 35). This bandstructure is fitted in terms of a fourth-nearest neighbor (in the fcc-lattice sites) overlap model of bonding and antibond ng orbitals as described n our earlier work on optical properties and impurity screening. Also the calculation of the two-particle Green s function is based on this bandstructure and follows closely the impurity studies (for details see in particular, ref.35). 

The first reliable energy band theories were based on a powerfiil approximation, call the pseudopotential approximation. Within this approximation, the all-electron potential corresponding to interaction of a valence electron with the iimer, core electrons and the nucleus is replaced by a pseudopotential. The pseudopotential reproduces only the properties of the outer electrons. There are rigorous theorems such as the Phillips-Kleinman cancellation theorem that can be used to justify the pseudopotential model [2, 3, 26]. The Phillips-Kleimnan cancellation theorem states that the orthogonality requirement of the valence states to the core states can be described by an effective repulsive... 

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 Al.3.24. Band structure of LiF from ab initio pseudopotentials [39],...

Figure Al.3.27. Energy bands of copper from ab initio pseudopotential calculations [40].

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].

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

The systems discussed in this chapter give some examples using different theoretical models for the interpretation of, primarily, UPS valence band data, both for pristine and doped systems as well as for the initial stages of interface formation between metals and conjugated systems. Among the various methods used in the examples are the following semiempirical Hartree-Fock methods such as the Modified Neglect of Diatomic Overlap (MNDO) [31, 32) and Austin Model 1 (AMI) [33] the non-empirical Valence Effective Hamiltonian (VEH) pseudopotential method [3, 34J and ab initio Hartree-Fock techniques. 

The charge density in a (110) plane for neutral H at the bond-center site in Si, as obtained from pseudopotential-density-functional calculations by Van de Walle et al. (1989), is shown in Fig. 7a. In the bond region most of the H-related charge is derived from levels buried in the valence band. It is also interesting to examine the spin density that results from a spin-polarized calculation, as described in Section II.2.d. The difference between spin-up and spin-down densities is displayed in Fig. 7b. It is clear... 

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... 

Figure 21 shows the photoelectron spectra of 34 (M = Ge) at 42 and 272 °C. The evident change in the shape of the spectrum clearly indicates the decomposition of the trimer. Band 1 was attributed to ionization of the sulphur lone pair of the monomeric species 33 (M = Ge), band 2 is related to the ji Ge=S bond, bands 4 and 6 to the a Ge—S and a Ge—C bonds respectively. Bands 3 and 5 were assigned to the ionization of a dimeric species. The assignment was supported by pseudopotential calculations. Also, the photoelectron spectrum of the dimeric species ( -Bu2GeS)2 was detected. 

The empirical approach [7] was by far the most fruitful first attempt. The idea was to fit a few Fourier coefficients or form factors of the potential. This approach assumed that the pseudopotential could be represented accurately with around three Fourier form factors for each element and that the potential contained both the electron-core and electron-electron interactions. The form factors were generally fit to optical properties. This approach, called the Empirical Pseudopotential Method (EPM), gave [7] extremely accurate energy band structures and wave functions, and applications were made to a large number of solids, especially semiconductors. [8] In fact, it is probably fair to say that the electronic band structure problem and optical properties in the visible and UV for the standard semiconductors was solved in the 1960s and 1970s by the EPM. Before the EPM, even the electronic structure of Si, which was and is the prototype semiconductor, was only partially known. 

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... 

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