Pseudopotential Theory of Covalent Bonding

The pseudopotential formulation also provides a simple way to estimate the relative band positions at interfaces between two semiconductors. The estimate is in reasonable accord with the results of the LCAO calculations for heterojunctions moreover, it provides an approach for analysis of junctions between metals and semiconductors. 

In Chapter 2, it was mentioned that there is a strong resemblance between bands obtained from nearest-neighbor interactions in the LCAO approximation and the bands obtained from nearly-free-electron theory. In fact, formulae for the interatomic matrix elements based upon that similarity were used to estimate properties of covalent and ionic solids in the chapters that followed, Now that a 

The resemblance between LCAO bands and nearly-frec-electron bands is an assertion that at the observed interatomic distances in solids both the metallic and the atomic descriptions are tenable. This would certainly not be true at very large spacing, where the electronic states are truly atomic and the frce-electron description has no relevance. It is also not true at sufficiently high density, where the LCAO description has no relevance. The assertion applies only near the observed spacing. 

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. Pseudopotenlial perturbation theory is an expansion in which the ratio W/Ey of the pseudopotential to the kinetic energy is treated as small, whereas for covalent solids just the reverse quantity, Ey/W, should be treated as small. The distinction becomes /wimportant 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 KIcinman (1959) nor in the more recent application of the Empirical Pseudopotenlial Method u.scd 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. 

Let us now complete the derivation of formulae for the interatomic matrix elemenfis, which was described in Section 2-D, by equating band energies obtained from LCAO theory and those obtained from nearly-free-electron bands. This analysis follows a study by Froyen and Harrison (1979). The band energies obtained from nearest-neighbor LCAO theory at symmetry points were given in 

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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,... 

It is possible in principle to calculate all of these modes from the theory of the electronic structure, which is equivalent to the calculation of all the force constants. Indeed we will see that this is possible in practice for the simple metals by using pseudopotential theory. In covalent solids, even within the Bond Orbital Approximation, this proves extremely difficult because of the need to rotate and to optimize the hybrids, and it has not been attempted. The other alternative is to make a model of the interactions, which reduces the number of parameters. The most direct approach of this kind is to reduce the force constants to as few as possible by symmetry, and then to include only interactions with as many sets of neighbors as one has data to fit- for example, interactions with nearest and next-nearest neighbors. This is the Born-von Karman expansion, and it has somewhat surprisingly proved to be very poorly convergent. This simply means that in all systems there arc rather long-ranged forces. 

Fourier component of the pseudopotential, which is the largest matrix element, dominates all others leads to a semiquantitative theory of the bonding properties of the covalent systems. This theory allows us to identify the even and odd parts of that pseudopotential with the covalent and polar energies of the LCAO theory, and provides another rationalization for the d" -dependence of the covalent energy. As we deform the covalent structure into an ionic structure, we see that the covalent energy ceases to contribute to the bonding properties in this approximation. 

Pseudopolentials. See also Pseudopotential form factor Pseudopolential theory application to covalent bonding. 407-429 core radius. See Core radius in covalent solids, 41,407-429 defined,352 energy dependence, 545 formulation for metals, 360ff, 543-545 history of, 343... 

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