The nearly free electron approximation

The bandstructure of fee aluminium is shown in Fig. 5.9 along the directions and TL respectively. It was computed by solving the Schrodinger equation selfconsistently within the local density approximation (LDA). We see that aluminium is indeed a NFE metal in that only small energy gaps have opened up at the Brillouin zone boundary. We may, therefore, look for an approximate solution to the Schrodinger equation that comprises the linear combination of only a few plane waves, the so-called NFE approximation. 

In particular, let us consider the band structure along where kr = (0,0,0) and kx = (2n/a)(l, 0,0) with a the edge length of the face-central cubic unit celL (Note that the X point for fee is In/a not nfa like for simple cubic.) In this direction the two lowest free-electron bands correspond to Ek = (H2/2m)k2 and k+I = (H2/2m)(k + g)2 respectively. The term g is the reciprocal lattice vector (2n/a)(2,0,0) that folds-back5 the free-electron states into the Brillouin zone along so that Ek and k+l 

Substituting eqn (5.35) into the Schrodinger equation, premultiplying by or 2), and integrating over the volume of the crystal yields the NFE secular equation 

The element (200) is the (2 / )(2,0,0) Fourier component of the crystalline potential normalized by the volume of the crystal, namely 

Therefore, at the zone boundary X where k2 = (k + g)2, the eigenvalues are given by 

---

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. 

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. 

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. 

In the simplest form, the Thomas-Fermi-Dirac model, the functionals are those which are valid for an electronic gas with slow spatial variations (the nearly free electron gas ). In this approximation, the kinetic energy T is given by... 

Although the nearly-free-electron construction is absolutely precise, it gives only an approximate description of the electron orbits in the real crystal. The approximation lies in the assumption that pseudopotentials are of sufficient... 

Valence energy bands for cesium chloride in (a) the simplest LCAO approximation and (b) in the nearly-free-electron limit. Parameters e, b, V2, etc., are chosen for convenience and are not realistic. 

The important physical properties of simple metals and, in particular, the alkali metals can be understood in terms of a free electron model in which the most weakly bound electrons of the constituent atoms move freely throughout the volume of the metal (231). This is analogous to the free electron model for conjugated systems (365) where the electrons are assumed to be free to move along the bonds throughout the system under a potential field which is, in a first approximation, constant (the particle-in-a-box model). The free electron approach can be improved by replaeing the constant potential with a periodic potential to represent discrete atoms in the chain (365). This corresponds to the nearly free electron model (231) for treating electrons in a metal. 

The above discussions concentrated on the N ( p) values derived from specific heat data. However, there is an alternative way of thinking about the properties. The key equation is m = 3fi y[fI/(3w Z)] /kg, where m is the efifective electron mass. Cl the volume per unit cell, and Z the number of electrons per unit cell [72]. This equation is valid in the nearly-free-electron-mass approximation, i.e., when m and N ( p) is approximately the value derived if all of the... 

The Nearly-Free Electron (NFE) Approximation ( Two-Band Model )... 

LCAO matrix. In short, even for a virtual crystal the larger the difference in atomic state energies, the larger the bovraig, as expected from the above arguments. The virtual crystal approximation also leads to parabolic decreases in the density of states around the band edges as in the nearly free electron model. 

Fig. 1.10 Plot of energy against kx in the approximation of nearly free electrons A is the...

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... 

---