Valence band theory Semiconductors

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 Schottky-Mott theory predicts a current / = (4 7t e m kB2/h3) T2 exp (—e A/kB 7) exp (e n V/kB T)— 1], where e is the electronic charge, m is the effective mass of the carrier, kB is Boltzmann s constant, T is the absolute temperature, n is a filling factor, A is the Schottky barrier height (see Fig. 1), and V is the applied voltage [31]. In Schottky-Mott theory, A should be the difference between the Fermi level of the metal and the conduction band minimum (for an n-type semiconductor-to-metal interface) or the valence band maximum (for a p-type semiconductor-metal interface) [32, 33]. Certain experimentally observed variations of A were for decades ascribed to pinning of states, but can now be attributed to local inhomogeneities of the interface, so the Schottky-Mott theory is secure. The opposite of a Schottky barrier is an ohmic contact, where there is only an added electrical resistance at the junction, typically between two metals. 

Chapter 4 discussed semiconductivity in terms of band theory. An intrinsic semiconductor has an empty conduction band lying close above the filled valence band. Electrons can be promoted into this conduction band by heating, leaving positive holes in the valence band the current is carried by both the electrons in the conduction band and by the positive holes in the valence band. Semiconductors, such as silicon, can also be doped with impurities to enhance their conductivity. For instance, if a small amount of phosphorus is incorporated into the lattice the extra electrons form impurity levels near the empty conduction band and are easily excited into it. The current is now carried by the electrons in the conduction band and the semiconductor is known as fl-type n for negative). Correspondingly, doping with Ga increases the conductivity by creating positive holes in the valence band and such semiconductors are called / -type (p for positive). 

The energy states of gaseous atoms split because of the overlap between electron clouds. Obviously, therefore, atoms must come much closer before the clouds of the core electrons begin to overlap compared with the distance at which the clouds of outer (or valence) electrons overlap (Fig. 6.119). Hence, at the equilibrium interatomic distances, the energy levels of the core electrons (in contrast to the valence electrons) do not show any band structure and therefore will be neglected in the following discussion. This simplified picture of the band theory of solids will now be used to explain the differences in conductivity of metals, semiconductors, and insulators. 

From the theorist s point of view, the work of Sommerfeld on the Electron Theory of Metals was most seminal. It was eventually reviewed on a quantum mechanical basis in a famous article in the Handbuch der Physik , Vol. XXIV/2 [A. Sommerfeld, H. Bethe (1933)]. Two years before, Heisenberg had introduced the electron hole . A. H. Wilson worked on the Lheory of semiconductors, and it was understood that at T - OK their valence band was completely filled with electrons, whereas the conduction band was empty. At T> 0 K, electrons are thermally excited from the valence band into the conduction band. 

In a metal, the Fermi level is located within the conduction band. In a semiconductor, this level usually is found in the forbidden gap between the valence band and the conductivity band by doping it can be shifted up or down relative to the band edges. The activation energy of a catalyzed reaction depends on the distance of the Fermi level from the band edges for acceptor reactions it is related to the distance from the conduction band, for donor reactions to the distance from the valence band. The exact theory will not be presented here it has been given by Hauffe (6) and by Steinbach (9). 

The electrical properties of any material are a result of the material s electronic structure. The presumption that CPs form bands through extensive molecular obital overlap leads to the assumption that their electronic properties can be explained by band theory. With such an approach, the bands and their electronic population are the chief determinants of whether or not a material is conductive. Here, materials are classified as one of three types shown in Scheme 2, being metals, semiconductors, or insulators. Metals are materials that possess partially-filled bands, and this characteristic is the key factor leading to the conductive nature of this class of materials. Semiconductors, on the other hand, have filled (valence bands) and unfilled (conduction bands) bands that are separated by a range of forbidden energies (known as the band gap ). The conduction band can be populated, at the expense of the valence band, by exciting electrons (thermally and/or photochemically) across this band gap. Insulators possess a band structure similar to semiconductors except here the band gap is much larger and inaccessible under the environmental conditions employed. 

It is the Peierl s instability that is believed to be responsible for the fact that most CPs in their neutral state are insulators or, at best, weak semiconductors. Hence, there is enough of an energy separation between the conduction and valence bands that thermal energy alone is insufficient to excite electrons across the band gap. To explain the conductive properties of these polymers, several concepts from band theory and solid state physics have been adopted. For electrical conductivity to occur, an electron must have a vacant place (a hole) to move to and occupy. When bands are completely filled or empty, conduction can not occur. Metals are highly conductive because they possess unfilled bands. Semiconductors possess an energy gap small enough that thermal excitation of electrons from the valence to the conduction bands is sufficient for conductivity however, the band gap in insulators is too large for thermal excitation of an electron accross the band gap. 

In LCAO1 theory the valence band maximum of a tetrahedrally bonded semiconductor is derived from the anion p-levels and its energy given by [89] ... 

In the analyses of conventional ZB semiconductors, we frequently assume a symmetric parabolic band for the conduction band state, and the Luttinger-Kohn Hamiltonian is used to describe the valence band states. In general, the effective Hamiltonian is derived from a k.p perturbation theory or from the theory of invariants developed by Pikus and Bir. In the latter theory, the operator form of the effective Hamiltonian can easily be constructed from symmetry consideration alone. Within this framework, the lowest two conduction bands and the upper six valence bands are described to the second order of k. The invariant forms of the Hamiltonians are written as follows [26,27] ... 

The band theory of solids provides a clear set of criteria for distinguishing between conductors (metals), insulators and semiconductors. As we have seen, a conductor must posses an upper range of allowed levels that are only partially filled with valence electrons. These levels can be within a single band, or they can be the combination of two overlapping bands. A band structure of this type is known as a conduction band. 

We have seen in the previous section that the bulk semiconductor is characterised by a filled valence band and an empty conduction band with a distribution of acceptor and donor levels. Provided that there is no nett charge on the semiconductor, the position of the Fermi level in the semiconductor is then determined solely by the semiconductor statistics as shown above. However, once immersed in an electrolyte, the interfacial potential change, and therefore the charge on the semiconductor, can be externally controlled and the distribution of energy levels becomes more complex [7-15]. The general theory will be developed below, but some important... 

Wc saw the general form of the valence bands in Fig. 6-1 for a number of semiconductors, and discussed the general features there. In Fig. 6-6 we show another version of the bands for GaAs, which will be useful for reference as we construct the conduction bands from LCAO theory. 

The spectra may also be described in the language of solid state theory. The atomic excited states are the same as the excitons that were described, for semiconductors, at the close of Chapter 6. They are electrons in the conduction band that are bound to the valence-band hole thus they form an excitation that cannot carry current. The difference between atomic excited states and excitons is merely that of different extremes the weakly bound exciton found in the semiconductor is frequently called a Mott-Wannier exciton-, the tightly bound cxciton found in the inert-gas solid is called a Frenkel exciton. The important point is that thecxcitonic absorption that is so prominent in the spectra for inert-gas solids does not produce free carriers and therefore it docs not give a measure of the band gap but of a smaller energy. Values for the exciton energy are given in Table 12-4. 

Thus, when a particle jumps, it leaves behind a hole. So then, instead of saying that a transport process occurs by particles hopping along, one could equally well say that the transport processes occur by holes moving. The concept is commonplace in semiconductor theory, where the movement of electrons in the conduction band is taken as being equivalent to a movement of so-called holes in the valence band. It has in fact already been assumed at the start of the viscosity treatment (Section 5.7.1) that the viscous flow of fused salts can be discussed in terms of the momentum transferred between liquid layers by moving holes. 

In the theory of non-equilibrium processes at solid state junction and also semiconductor-liquid interfaces, as developed in the previous section, frequently quasi-Fermi levels have been used for the description of minority carrier reactions [90, 91], A concept for a quantitative analysis for reactions at n- and p-type electrodes has been derived [92, 93], using the usual definition of a quasi-Fermi level (Eqs. (3a) and (3b)). Taking a valence band process as an example, the quasi-Fermi level concept can be illustrated as follows ... 

The electrical property of a material is determined by its electronic structure, and the relevant theory that explains the electronic structure in the solid state is band theory. This theory, however, does not fully explain conductivity in polymers. It is noteworthy that the energy spacing between the highest occupied and lowest unoccupied bands is called the bandgap the highest occupied band is called the valence band and the lowest unoccupied band is called the conduction band. The bandgaps of insulators and semiconductors are wide and narrow, respectively there are no band-gaps in metals. 

According to band theory, the electrical properties of a solid depend on how the bands are filled. There is no conduction when the bands are filled or empty. If the material has a narrow bandgap, thermal excitations of electrons from the valence band to the conduction band occur, giving rise to conductivity in classical semiconductors. 

In crystalline semiconductors, it is relatively easy to understand the formation of gaps in energy states of electrons and hence of the valence and conduction bands using band theory (see Ziman, 1972). Band structure arises as a consequence of the translational periodicity in the crystalline materials. For a typical crystalline material which is a periodic array of atoms in three dimensions, the crystal hamiltonian is represented by a periodic array of potential wells, v(r), and therefore is of the form, 7/crystai = ip l2m) + v(r), where the first term p l2rri) represents the kinetic energy. It imposes the eondition that the electron wave functions, which are solutions to the hamiltonian equation, H V i = E, Y, are of the form... 

---