Allowed energy bands

The free electron resides in a quantized energy well, defined by k (in wave-numbers). This result Ccm be derived from the Schroedinger wave-equation. However, in the presence of a periodic array of electromagnetic potentials arising from the atoms confined in a crystalline lattice, the energies of the electrons from all of the atoms are severely limited in orbit and are restricted to specific allowed energy bands. This potential originates from attraction and repulsion of the electron clouds from the periodic array of atoms in the structure. Solutions to this problem were... 

We have shown the least complicated one which turns out to be the simple cubic lattice. Such bands are called "Brilluoin" zones and, as we have said, are the allowed energy bands of electrons in any given crystalline latttice. A number of metals and simple compounds have heen studied and their Brilluoin structure determined. However, when one gives a representation of the energy bands in a solid, a "band-model is usually presented. The following diagram shows three band models ... 

With insulators or semiconductors, the valence band is completely filled and is separated from the next highest allowable energy band (conduction band) by a gap (or forbidden band). In order for valence electrons to be raised into the conduction band, they must be given a sizable amount of energy. With insulators this gap corresponds to several electron volts. For semiconductors the gap is sufficiently small so that a significant number of electrons can acquire the necessary additional energy by thermal means at room temperature. 

Fig. 1-3. Probability density of electron energy distribution, fli), state density, D(t), and occupied electron density. Die) fit), in an allowed energy band much higher than the Fermi level in solid semiconductors, where the Boltzmann function is applicable.

Figure 3.12. Schematic electron occupancy of allowed energy bands for a metal (a), an insulator (b), and a semiconductor (c). The shaded areas indicate the regions filled with electrons. The vertical extent of the boxes indicates the energy regions allowed.

Fig. 4.5. Surface states. By solving the Schrodinger equation for a cut-off Kronig-Penney potential, it is found that in the energy gaps of the corresponding Kronig-Penney solid, there are surface states that decay exponentially into the vacuum and into the solid (Tamm, 1932). The explicit wavefiinction of a Tamm state with P = 15 and a = 3Aat = 5eV below the vacuum level is shown. The shaded areas represent allowed energy bands in the bulk.

In a semiconductor, as discussed in the previous section, localisation can also occur as the width of the allowed energy band is reduced, and this was defined in terms of a limiting mobility. The Anderson model shows that disorder can lead to localisation in metals as well as semiconductors. In metals, since conduction is due only to electrons within a partially filled band, the energy in the band tail that separates localised from delocalised electron states is termed the mobility edge. The onset of localisation in a metal occurs at a minimum conductivity. This can be seen as follows. For an electron at the Fermi energy its mean free path, l, is just the scattering time, r, multiplied by the electron velocity at the Fermi energy, vF. Then, from Equations (4.1) and (4.2) it follows that ... 

The index p labels the allowed energy bands. The number of values which p can achieve is given by a sum of the considered exciton bands + two j = 1,2- two branches of the transverse photons). 

Fig. 7.28 Schematic depiction of electon occupancy of allowed energy bands for a classical metal, a semiconductor, an insulator, and a semimetal. The energy of the highest occupied level is called the Fermi energy. The unoccupied energy levels are white, the occupied levels black. [After Marks.]...

Fig. 8. Schematic illustration of the occupation of the allowed energy bands (rectangles) for a metal, semiconductor, insulator, and semimetal.

A solid behaves as an insulator if the allowed energy bands are either filled or empty, for then no electrons can move in an... 

Between two allowed energy bands is a forbidden band whose width is represented by a gap in energy. 

An electron located in the lower half of an allowed energy band has a positive effective mass. In the presence of a positive external force, it will be subjected to a positive acceleration value, which is in accordance with the findings of classical mechanics. 

Figure 23.16 Effect of doping on the occupancy of the allowed energy levels in silicon, (a) Pure silicon. The valence-shell electrons just fill the lower-eneigy allowed energy band. 

Ceramic materials are usually defined by specifying what they are not. They are called inorganic, because they do not consist of electrically neutral molecules— which are typical for organic compounds. They are also called non-metallic, because ceramics are characterised by a gap between allowed energy bands of valence electrons, a feature not existing in metallic materials. 

Let us consider the forbidden gap between the first and the second allowed energy bands in some more detail. In this case one has n = 1 and quasidegeneracy of the energies at the first Brillouin zone boundaries, k = nfa. 

Fig. 2.6 The dispersion of eiectronic energy in a 1D crystai within the first and second Brii-iouin zones. The energy intervais corresponding to the buik states (soiid iines) form the allowed energy bands. They are separated by...

Allowed energy band An energy band consisting of electronic states available for electrons in the bulk crystal. 

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