Empirical pseudopotential

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

Figure Al.3.16. Reflectivity of silicon. The theoretical curve is from an empirical pseudopotential method calculation [25], The experimental curve is from [31],...

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... 

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. 

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. 

Barthelat,J.C., Durand,Ph. and Serafini,A. (1977), Non-empirical pseudopotentials for molecular calculations I. The PSIBMOL algorithm and test applications , Mol.Phys. 33, 159... 

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]. 

Accurate energy bands obtained from first principles by computer calculation are available for most covalent solids. A display of the bands obtained by the Empirical Pseudopotential Method for Si, Ge, and Sn and for the compounds of groups 3-5 and 2-6 that are isoclec-tronic with Ge and Sn shows the principal trends with mctallicity and polarity. The interpretation of trends is refined and extended on the basis of the LCAO fitting of the bands, which provides bands of almost equal accuracy in the form of analytic formulae. This fitting is the basis of the parameters of the Solid State Table, and a plot of the values provides the test of the d dependence of interatomic matrix elements. 

The energy bands of Si, Ge, and Sn and the energy bands of the polar semiconductors isoclectronic with Ge and Sn, as calculated by Chclikowsky and Cohen (1976b) by means of the Empirical Pseudopotential Method. Mctallicity increases, material by material, downward polarity increases from left to right. 

The trends are the same, but the magnitudes differ by a factor of about two. This discrepancy arises from the inaccuracy of the empty-core model in fitting the pseudopotential in this range, as can be seen by considering the third column in Table 18-1, where Empirical Pseudopotential Method matrix elements are listed directly. These are in fact taken from the same calculated pseudopotential which was fitted to the empty-core model to obtain the values in the second column. A look at Fig. 16-1, where an accurate pseudopotential is plotted along with the empty-core fit indicates that the region near qfk = 1.108 (corresponding to the... 

Let us now turn to a few other informative applications of the pseudopotential model to properties (discussed extensively in Harrison, 1976a). For that purpose it will be most convenient to take the pseudopotential parameters from the LCAO values for V2 and Fj, using Eq. (18-5). We see from Table 18-1 that for V2 this is roughly equivalent to using Empirical Pseudopotential Model parameters. Let us look first at the dielectric susceptibility, which was so important in the development of the LCAO theory. For this, the most convenient form for the susceptibility is Eq. (4-5), which we rewrite in simple form,... 

Note that, in the above discussion, we have neglected methods that generate band structures from empirical data. Most band calculations before the 1970s were of this type. The considerable contributions to knowledge made through use of the empirical pseudopotential approach, for example, have been discussed by Cohen (1979). Such approaches have... 

In fact, because of its importance in solid-state science, a large variety of band-structure approaches have been used to calculate the electronic structure of sphalerite. These have included self-eonsistent and semiem-pirical orthogonalized-plane-wave (OPW) (Stukel et al., 1969), empirical-pseudopotential (Cohen and Bergstresser, 1966), tight-binding (Pantelides and Harrison, 1975), APW (Rossler and Lietz, 1966), and modified OPW (Farberovich et al., 1980), as well as KKR (Eckelt, 1967) methods. In a recent and extremely detailed study using a density-functional approach (specifically a method termed the self-consistent potential variation... 

Another method that has been used to describe the electronic structure of bulk semiconductors is the empirical pseudopotential method [55, 56]. The potential... 

Recently, Zunger and coworkers [18, 58-63] employed the semi-empirical pseudopotential method to calculate the electronic structure of Si, CdSe [60] and InP [59] quantum dots. Unlike EMA approaches, this method, based on screened pseudopotentials, allows the treatment of the atomistic character of the nanostructure as well as the surface effects, while permitting multiband and intervalley coupling. The atomic pseudopotentials are extracted from first principles LDA calculations on bulk solids. The single particle LDA equation,... 

Fig. 3.4. Band structure of germanium calculated from empirical pseudopotentials including s-o coupling as a function of the electron wave vector along selected directions of the reciprocal lattice. The s-o splitting Aso is the energy difference between Ts+ and IV. The light- and heavy-hole VBs are the IV1" to Le and IV1" to L4 +L5 bands, respectively. The energy reference is the VB maximum at the T point after [21]...

NBE NBE NEPM NVRAM near band edge nitrogen bound exciton non-local empirical pseudopotential method non-volatile random access memory... 

The empirical pseudopotential method can be illustrated by considering a specific semiconductor such as silicon. The crystal structure of Si is diamond. The structure is shown in figure A 1.3.4. The lattice vectors and basis for a primitive cell have been defined in the section on crystal structures (Al.3.4.1). In Cartesian coordinates, one can write G for the diamond structure as... 

Figure Al.3.15. Density of states for silicon (bottom panel) as calculated from empirical pseudopotential...

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