Nearly-free electron approximation

In general, Eq. (2.18) cannot be solved exactly. Let us now discuss the nearly-free electron approximation where the potential V(z) can be treated as a small perturbation to the motion of a free electron. 

Since the functions V z) and Ujfz) are periodic, they can be expanded in a Fourier series 

The functions (2.27) are the eigenfunctions of the unperturbed Hamiltonian — h /lute) d /dz ) corresponding to the eigenvalues 

The first-order correction to the energy is found from perturbation theory 

This can be shown as follows. To calculate the correction to the energies accounting for the influence of the potential V(z), we first assume that the atomic chain is extended up to — L (L S a) from the side of negative z and then tend L to infinity. Direct evaluation gives the matrix elements 

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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. 

If potential barriers between wells are weak, yi < 1, energy bands are wide and spaced dose together. This is typically for metals with weakly-bound electrons, that is for alkali metals. Here the model of nearly-free electrons (NFE) works well. The nearly-free electron approximation describes well s- and sp-valent metals. 

A consequence of the cancellation between the two terms of (6.47) is the surprisingly good description of the electronic structure of solids given by the nearly-free electron approximation. The fact that many metal and semiconductor band structures are a small distortion of the free electron gas band structure suggests that the valence electrons do indeed feel a weak potential. The Phillips and Kleinman potential explains the reason for this cancellation. 

Problem 2.2. The electronic surface states are shown in Fig. 2.26 by dashed lines. The parabolic shape of the upper surface band implies that it is well described by the nearly-free electron approximation and hence this state is classified as a Shockley state. The lower surface state has the character of a surface resonance in the region where it intersects the electronic bulk band. An electronic transition between the surface states is possible if the upper state is unoccupied, i.e., it is located above the Fermi level. The minimum energy of such a transition is about 1.8 eV. 

In practice, the true crystal potential does not satisfy the criterion for the applicability of the nearly-free-electron approximation, but there are much weaker equivalent potentials, or pseudopotentials, which do. By equivalent we mean that they produce the same band structure for the valence and conduction bands (but not necessarily the same wave functions). The difference is that the potential must of necessity be strong enough to bind states at lower energies (core states, more or less) but the pseudopotential need not. The elimination of such bound states produces a potential which is much weaker in the region close to the ion cores. This cancellation of the strong inner part of the potential can be seen from many points of view but will here be accepted as a fact of life for s-p-bonded systems. 

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