Electronic conduction theory free electron

Within the theory presented here, the onset of superconductivity, Tc, is when the condition, v = co, takes place. At this point, the normal conduction through free-electrons becomes zero and therefore is frozen . As stated before, the phonon (antisymmetric normal mode) vibration is written as ... 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

Slater s Xa method is now regarded as so much history, but it gave an important stepping stone towards modem density functional theory. In Chapter 12, I discussed the free-electron model of the conduction electrons in a solid. The electrons were assumed to occupy a volume of space that we identified with the dimensions of the metal under smdy, and the electrons were taken to be non-interacting. 

We should point out that up to now we have considered only polycrystals characterized by an a priori surface area depleted in principal charge carriers. For instance, chemisorption of acceptor particles which is accompanied by transition-free electrons from conductivity band to adsorption induced SS is described in this case in terms of the theory of depleted layer [31]. This model is applicable fairly well to describe properties of zinc oxide which is oxidized in air and is characterized by the content of surface adjacent layers which is close to the stoichiometric one [30]. 

According to the electronic theory, a particle chemisorbed on the surface of a semiconductor has a definite affinity for a free electron or, depending on its nature, for a free hole in the lattice. In the first case the chemisorbed particle is presented in the energy spectrum of the lattice as an acceptor and in the second as a donor surface local level situated in the forbidden zone between the valency band and the conduction band. In the general case, one and the same particle may possess an affinity both for an electron and a hole. In this case two alternative local levels, an acceptor and a donor, will correspond to it. 

The nature of light absorption in a crystal is of no significance for theory. What is important here is that this absorption be photoelectrically active, i.e., results in a change of the concentration of free carriers in a crystal. This process may take the form either of the so-called intrinsic absorption accompanied by the transition of an electron from the valency to the conduction band, or of the so-called impurity absorption caused by an electronic transition between the energy band and the impurity local level. 

In band theory the electrons responsible for conduction are not linked to any particular atom. They can move easily throughout the crystal and are said to be free or very nearly so. The wave functions of these electrons are considered to extend throughout the whole of the crystal and are delocalized. The outer electrons in a solid, that is, the electrons that are of greatest importance from the point of view of both chemical and electronic properties, occupy bands of allowed energies. Between these bands are regions that cannot be occupied, called band gaps. 

The free electron (FEMO) theory had its origins in work on the conduction electrons of metals in the 1940s, when several workers independently recognised the close analogy between these and the delocalised Jt-electrons of polyene dyes. The method was extended to many other classes of dyes, notably by Kuhn in the 1950s, but it has not found general acceptance for spectroscopic calculations, since it lacks adaptability by simple parameter adjustment. 

This notion of occasional ion hops, apparently at random, forms the basis of random walk theory which is widely used to provide a semi-quantitative analysis or description of ionic conductivity (Goodenough, 1983 see Chapter 3 for a more detailed treatment of conduction). There is very little evidence in most solid electrolytes that the ions are instead able to move around without thermal activation in a true liquid-like motion. Nor is there much evidence of a free-ion state in which a particular ion can be activated to a state in which it is completely free to move, i.e. there appears to be no ionic equivalent of free or nearly free electron motion. 

Classical Free-Electron Theory, Classical free-electron theory assumes the valence electrons to be virtually free everywhere in the metal. The periodic lattice field of the positively charged ions is evened out into a uniform potential inside the metal. The major assumptions of this model are that (1) an electron can pass from one atom to another, and (2) in the absence of an electric field, electrons move randomly in all directions and their movements obey the laws of classical mechanics and the kinetic theory of gases. In an electric field, electrons drift toward the positive direction of the field, producing an electric current in the metal. The two main successes of classical free-electron theory are that (1) it provides an explanation of the high electronic and thermal conductivities of metals in terms of the ease with which the free electrons could move, and (2) it provides an explanation of the Wiedemann-Franz law, which states that at a given temperature T, the ratio of the electrical (cr) to the thermal (k) conductivities should be the same for all metals, in near agreement with experiment ... 

The simplest model of metals is the Sommerfeld theory of free-electron metals (Ashcroft and Mermin 1985, Chapter 2), where a metal is described by a single parameter, the conduction electron density n. A widely used measure of... 

Because n can take all integral values, this means an infinite number of energy levels exist with larger and larger gaps between each level. Most solids of course are three-dimensional (although we shall meet some later where conductivity is confined to one or two dimensions) and so we need to extend the free electron theory to three dimensions. 

We know that not all solids conduct electricity, and the simple free electron model discussed previously does not explain this. To understand semiconductors and insulators, we turn to another description of solids, molecular orbital theory. In the molecular orbital approach to bonding in solids, we regard solids as a very large collection of atoms bonded together and try to solve the Schrodinger equation for a periodically repeating system. For chemists, this has the advantage that solids are not treated as very different species from small molecules. 

The dielectric function of a metal can be decomposed into a free-electron term and an interband, or bound-electron term, as was done for silver in Fig. 9.12. This separation of terms is important in the mean free path limitation because only the free-electron term is modified. For metals such as gold and copper there is a large interband contribution near the Frohlich mode frequency, but for metals such as silver and aluminum the free-electron term dominates. A good discussion of the mean free path limitation has been given by Kreibig (1974), who applied his results to interpreting absorption by small silver particles. The basic idea is simple the damping constant in the Drude theory, which is the inverse of the collision time for conduction electrons, is increased because of additional collisions with the boundary of the particle. Under the assumption that the electrons are diffusely reflected at the boundary, y can be written... 

An expression for the electrical conductivity of a metal can be derived in terms of the free-electron theory. When an electric field E is applied, the free carriers in a solid are accelerated but the acceleration is interrupted because of scattering by lattice vibrations (phonons) and other imperfections. The net result is that the charge carriers acquire a drift velocity given by... 

Lorentz1 advanced a theory of metals that accounts in a qualitative way for some of their characteristic properties and that has been extensively developed in recent years by the application of quantum mechanics. He thought of a metal as a crystalline arrangement of hard spheres (the metal cations), with free electrons moving in the interstices.. This free-electron theory provides a simple explanation of metallic luster and other optical properties, of high thermal and electric conductivity, of high values of heat capacity and entropy, and of certain other properties. 

The three-dimensional particle in a box corresponds to the real life problem of gas molecules in a container, and is also sometimes used as a first approximation for the free conduction electrons in a metal. As we found for one dimension (Section 2.3), the allowed energy levels are extremely closely spaced in macroscopically sized boxes. For many purposes they can be regarded as a continuum, with no discernible energy gaps. Nevertheless, there are problems, for example in the theory of metals and in the calculation of thermodynamic properties of gases, where it is essential to take note of the existence of discrete quantized levels, rather than a true continuum. 

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