Pseudopotentials matrix elements

Defining the normalized ion-core pseudopotential matrix element by... 

The band structure of nonmagnetic fee and bcc iron is shown in Fig. 7.5, being computed from the hybrid NFE-TB secular equation with resonant parameters Ed = 0.540 Ry and = 0.088 Ry. The NFE pseudopotential matrix elements were chosen by fitting the first principles band structure derived by Wood (1962) at the pure p states Nv (tiuo = 0.040 Ry), L2> ( U1 = 0.039 Ry), and X (t 200 = 0.034 Ry). Comparing the band structure of iron in the 100> and 111> directions with the canonical d bands in Fig. 

Jaros(1977) 0 ap2 Yes Matrix elements approximated via Penn model pseudopotential approach and Calculated... 

The relaxation of the structure in the KMC-DR method was done using an approach based on the density functional theory and linear combination of atomic orbitals implemented in the Siesta code [97]. The minimum basis set of localized numerical orbitals of Sankey type [98] was used for all atoms except silicon atoms near the interface, for which polarization functions were added to improve the description of the SiOx layer. The core electrons were replaced with norm-conserving Troullier-Martins pseudopotentials [99] (Zr atoms also include 4p electrons in the valence shell). Calculations were done in the local density approximation (LDA) of DFT. The grid in the real space for the calculation of matrix elements has an equivalent cutoff energy of 60 Ry. The standard diagonalization scheme with Pulay mixing was used to get a self-consistent solution. In the framework of the KMC-DR method, it is not necessary to perform an accurate optimization of the structure, since structure relaxation is performed many times. 

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]. 

The last contribution to Vsoiv. conies from a repulsive term, which we jrsu-ally call the non-electrostatic perturbation or sometimes the pseudo-potential, Vnei.. First the formulation of Vnei. is given. After that, previous applications of pseudopotentials are reviewed, which leads to the purpose and justification of this potential in QMSTAT. For the same general basis set as above, a matrix element of Vnei. equals... 

We conclude here that the matrix elements of d/dx are constant. This conclusion will follow also from the pseudopotential theory of covalent bonding in Chapter 18, and was found to Idc true of the matrix elements in the nonlocal pscudopotential calculations of Chclikowsky and Cohen (quoted by Phillips,... 

We might also expect the matrix elements of djdx to scale inversely with d among the homopolar semiconductors (and correspondingly, for the matrix elements of X to scale with d) and this is in fact predicted by the pseudopotential theory of Chapter 18. However, that does not describe the trends in Xi(0) well, and in Section 4-C, we shall allow the proportionality constant to vary from row to row in the Periodic Table. 

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. 

This matrix clement is the Fourier component of the pseudopotential with wave number equal to the difference q. In the more complete pseudopotential theory of Appendix D, the pseudopotential becomes an operator, so that IT (r) and c cannot be interchanged in the final step of Eq. (16-2), and the matrix element depends upon k also. This is not a major complication, but we shall utilize the simpler form given in the last step of Eq. (16-2), called the local approximation to the pseudopotential. 

This completes the specification of the pseudopotential as a perturbation in a perfect crystal. We have obtained all of the matrix elements between the plane-wave states, which arc the electronic states of zero order in the pseudopotcntial. We have found that they vanish unless the difference in wave number between the two coupled states is a lattice wave number, and in that case they are given by the pseudopotential form factor for that wave number difference by Eq. (16-7), assuming that there is only one ion per primitive cell, as in the face-centered and body-centered cubic structures. We discuss only cases with more than one ion per primitive cell when we apply pseudopotential theory to semiconductors in Chapter 18. Tlicn the matrix element will be given by a structure factor, Eq. (16-17),... 

How to proceed with these matrix elements will depend upon which property one wishes to estimate. Let us begin by discussing the effect of the pseudopotential as a cause of diffraction by the electrons this leads to the nearly-free-electron approximation. The relation of this description to the description of the electronic structure used for other systems will be seen. We shall then compute the screening of the pseudopotential, which is necessary to obtain correct magnitudes for the form factors, and then use quantum-mechanical perturbation theory to calculate electron scattering by defects and the changes in energy that accompany distortion of the lattice. 

This view runs into difficulties that have only recently been completely resolved. The principal one is that the pseudopotential form factor happens to be very small for this particular diffraction. In Fig. 18-4 is sketched the pseudopotenlial form factor for silicon obtained from the Solid Stale Table the form factor that gives the [220] diffraction is indicated. Because it lies so close to the crossing, it is small and the diffraction is not expected to be strong. Heine and Jones (1969) noted, however, that a second-order diffraction can take an electron across the Jones Zone this could be a virtual diffraction by a lattice wave number of [1 ll]27t/fl followed by a virtual diffraction by [I lT]27c/a. (Virtual diffraction is an expression used to describe terms in perturbation theory it can be helpful but is not essential to the analysis here.) This second-order diffraction would involve the large matrix elements associated with the [11 l]27t/a lattice wave number indicated in Fig. 18-4, and Heine and Jones correctly indicated that these are the dominant matrix elements. 

In the covalent solids, the Jones Zone gap should be identified with the principal optical absorption peak previously identified with LCAO interatomic matrix elements. Thus it allows a direct relation between the parameters associated with the LCAO and with the pseudopotential theories. It is best, however, to simplify the pseudopotential analysis still further before making that identification. 

There the relation was made in terms of the splitting at F rather than X, since the corresponding formulae are simpler.) The first comparison we make is between the LCAO values and the empty-core pseudopotential. We shall find only qualitative correspondence between the values because of errors in the empty-core model, which become large here. We shall then go on to consider other properties, using pseudopotential matrix elements obtained without resort to the empty-core model. 

Making first a comparison of the covalent energy, notice that in homopolar semiconductors, Wy, becomes simply w, The various geometrical factors in the empty-core pseudopotential may be directly evaluated. Then, the pseudopotential matrix element becomes... 

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... 

We have seen, particularly in the discussion of covalent crystals in terms of pseudopotentials, the importance of recognizing which matrix elements or effects are dominant and which should be treated as corrections afterward. Tliis is also true in transition-metal systems, and different effects arc dominant in different transition-metal systems thus the correct ordering of terms is of foremost importance. For many transition-metal systems, we find that band calculations, particularly those by L. F. Mattheiss, provide an invaluable guide to electronic structure. Mattheiss uses the Augmented Plane Wave method (APW method), which is analogous to the OPW method discussed in Appendix D. 

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