Empirical Pseudopotential Method

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

The solution at these k-points gives the band structure of the nanocrystals. 

Ramakrishna and Friesner [57] calculated the band structure of CdS and GaP nanocrystals up to 30 A radius in the zinc-blende phase. The calculated variation agrees quite well with the experimental data points. The band gap variation curve in the case of GaP shows a red shifted band gap in the case of very small nanocrystals. This has not yet been experimentally verified. 

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, 

---

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... 

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. 

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... 

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... 

The remarkable conclusion of this argument is that though pseudopotentials can be used to describe semiconductors as well as metals, the pseudopotential perturbation theory which is the essence of the theory of metals is completely inappropriate in semiconductors. Pseudopotential perturbation theory is an expansion in which the ratio W/Ep of the pseudopotential to the kinetic energy is treated as small, whereas for covalent solids just the reverse quantity, EpiW, should be treated as small. The distinction becomes unimportant if we diagonalize the Hamiltonian matrix to obtain the bands since, for that, we do not need to know which terms are large. Thus the distinction was not essential to the first use of pseudopotentials in solids by Phillips and Kleinman (1959) nor in the more recent application of the Empirical Pseudopotential Method used by M. L. Cohen and co-workers. Only in approximate theories, which are the principal subject of this text, must one put terms in the proper order. 

SOURCE of values for the Empirical Pseudopotential Method (EPM) Cohen and Bergstresser (1966). 

Prior to 1975 or so, the words ab initio did not exist in the scientific vocabulary for methods describing the electronic structure of the solid state. At that time, a number of very powerful and successful methods had been developed to describe the electronic structure of solids, but these methods did not pretend to be first principles or ab initio methods. The foremost example of electronic structure methods at that time was the empirical pseudopotential method (EPM) [1]. The EPM was based on the Phillips-Kleinman cancelation theorem [2], which justified the replacement of the strong, allelectron potential with a weak pseudopotential [1]. The pseudopotential replicated only the chemically active valence electron states. Physically, the cancelation theorem is based on the orthogonality requirement of the valence states to the core states [1]. This requirement results in a repulsive part of the pseudopotential which cancels the strongly attractive part of the core potential and excludes the valence states from the core region. Because of this property, simple bases such as plane waves can be used efficiently with pseudopotentials. 

An empirical pseudopotential method (EPM) calculation (23) was done to reproduce the gap and the reflectivity spectrum adjusting the pseudopotential form factors. This study led to a minimum direct gap at L and the lowest conduction state was obtained at Ff. A band stmcture calculation using a semi-ab initio approach (10) obtained an indirect gap (Hs Ff) of 2.0 eV and a comparable direct gap (Hs Tj) of approximately 2.0 eV. The minimum band gaps of BP have been reliably estimated from the experimental optical absorption. However, the direct band gaps and other excitation energies must be estimated from structure in the optical response versus frequency. The accuracy of the resulting experimental values depends on the correct identification of features in, e.g., the reflectivity with particular transitions between band states. Then the GW results may be more reliable estimates than the experimental direct band gaps. 

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. 

---