Free-electron approximation

In solid-state physics the opening of a gap at the zone boundary is usually studied in the free electron approximation, where the application of e.g., a ID weak periodic potential V, with period a [V x) = V x + a)], opens an energy gap at 7r/a (Madelung, 1978 Zangwill, 1988). E k) splits up at the Brillouin zone boundaries, where Bragg conditions are satished. Let us consider the Bloch function from Eq. (1.28) in ID expressed as a linear combination of plane waves ... 

Energy bands in the free-electron approximation symmorphic space groups... 

Figure 17.3. Energy bands for the simple cubic Bravais lattice in the free-electron approximation at A on rx. The symmetry of the eigenfunctions at T and at X given in the diagram satisfy compatibility requirements (Koster et al. (1963)). Degeneracies are not marked, but may be easily calculated from the dimensions of the representations.

From the form of Eq. 4.30, it is seen that 8 is a simple parabolic function of k [s(A ) = e k + G)]. The band structure describes this dependence of e k) on k, and it is an n + dimensional quantity, where n is the number dimensionality of the crystal. To visualize it, e k) is plotted along particular projectories between high-symmetry points. Hence, for a one-dimensional crystal, the band stmcture will consist of a single parabola in the free-electron approximation. The parabola shows all the degenerate (positive and negative) values for k. For three-dimensional crystals, a single paraboloid is obtained. 

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. 

Only quite small electric fields can be applied in a metal, because of the high conductivity in the time an electron moves before colliding with a defect or with the surface it can only be accelerated very slightly. Unless it happens to lie very close to a Bragg plane it will not be affected by the diffraction. Since most electrons are thus unaware of the lattice, conductivity can to a large extent be treated in the free-electron approximation, as we indicated earlier. 

These conclusions are not unexpected. Firstly, the free-electron approximation works best for the alkah metals, with significant deviations for divalent and trivalent metals. Secondly, the LCAO-MO TSH treatment requires a more complicated treatment of p and d orbitals, involving tensor... 

In the free-electron approximation, the potential energy is position-independent Ep is then an additive constant set here to zero, Ep = 0 (Figure 3) the total energy E x, y,z) is purely kinetic. A solution of Eq. 13 is... 

See, e.g., T.C. Harman and J.M. Honig, Thermoelectric and Thermomagnetic Effects and Applications, McGraw-Hill, New York, 1967. The presentation given here is of limited (but nevertheless, didactic) utility since it applies only to a metal that is modeled in the free electron approximation or to extrinsic semiconductors. For more complicated models, particularly those involving charge transport by electrons and holes in multiband materials, the more elaborate analysis presented in advanced treatises is required. 

As the name implies, observations in the field emission microscope are based on the emission of electrons from a metal into a high electric field. The electron current i in a one dimensional system, in which the free electron approximation is used to describe the metal, is given by the... 

ELECTRONIC STRUCTURE OF METALLIC SINGLE-WALL CARBON NANOTUBES TIGHT-BINDING VERSUS FREE-ELECTRON APPROXIMATION... 

In terms of the free-electron approximation, which neglects the modification of... 

At the right are shown the energy bands of silicon (after Herman, Kortum, and Kuglin, 1966). At the far left are the four levels at X from the free-electron approximation. The left panel shows their splitting due to the pseudopotential, based on essentially the same values as deduced from Table 16-1. The dashed lines identify the resulting levels with the corresponding ones in the true band structure. 

The free-electron approximation described in Chapter 15 is so successful that it is natural to expect that any effects of the pseudopotential can be treated as small perturbations, and this turns out to be true for the simple metals. This is only possible, however, if it is the pseudopotential, not the true potential, which is treated as the perturbation. If we were to start with a free-electron gas and slowly introduce the true potential, states of negative energy would occur, becoming finally the tightly bound core states these are drastic modifications of the electron gas. If, however, we start with the valence-electron gas and introduce the pseudopotential, the core states are already there, and full, and the effects of the pseudopotential are small, as would be suggested by the small magnitude of the empty-core pseudopotential shown in Fig. 15-3. 

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