Concentrations of Charge Carriers

The concentration of electric charge carriers in intrinsic semiconductors are represented by the particle density of electrons, n, and holes, p, both expressed in m They can be calculated from the energy band theory in solids, assuming that the energy difference between the Fermi level and the band edges is larger than the thermal motion kT (0.025 eV at room temperature). Therefore, the statistical distribution of charge carrier follows a classical Boltzmann distribution  

The product of the electron and hole densities does not depend on the Fermi level and can be expressed as  

In the case of intrinsic semiconductors n = p = n. thus the charge-carrier density can be 

For instance, for pure silicon the charge-carrier density is roughly 1.4 x lO cm while for germanium it is 1.8 x 10 cm  

In the case of extrinsic semiconductors for which traces of impurities are added intentionally by doping in order to modify their electrical properties, the concentration of donors (e.g., P, As, or Sb) is denoted hyN, while the concentration of acceptors (e.g., B, Al, or Ga) is denoted by Nj. To calculate the carrier concentration in this kind of semiconductor, it is necessary to use the equation of electrical-charge neutrality  

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The variations of dielectric constant and of the tangent of the dielectric-loss angle with time provide information on the mobility and concentration of charge carriers, the dissociation of defect clusters, the occurrence of phase transitions and the formation of solid solutions. Techniques and the interpretation of results for sodium azide are described by Ellis and Hall [372]. 

The basic difference between metal-electrolyte and semiconductor-electrolyte interfaces lies primarily in the fact that the concentration of charge carriers is very low in semiconductors (see Section 2.4.1). For this reason and also because the permittivity of a semiconductor is limited, the semiconductor part of the electrical double layer at the semiconductor-electrolyte interface has a marked diffuse character with Debye lengths of the order of 10 4-10 6cm. This layer is termed the space charge region in solid-state physics. 

The data in Figure 5 can now be considered in light of the conduction model developed above. As stated previously, conduction in reduced poly-I behaves like an activated process. There are two sources that potentially could be responsible for this behavior. The first is the Boltzmann type concentration dependence of the 1+ and 1- states discussed above. The number of charge carriers is expected to decrease approximately exponentially with T. The second is the activation barrier to self-exchange between 1+ and 0 sites and 0 and 1- sites. For low concentration of charge carriers both processes are expected to contribute to the measured resistance. 

A hopping semiconductor such as an oxide is often best described by Eq. (7.1) rather than by classical band theory. In these materials the conductivity increases with temperature because of the exponential term, which is due to an increase in the successful number of jumps, that is, the mobility, as the temperature rises. Moreover, the concentration of charge carriers increases as the degree of nonstoichiometry increases. 

A particular important property of silicon electrodes (semiconductors in general) is the sensitivity of the rate of electrochemical reactions to the radius of curvature of the surface. Since an electric field is present in the space charge layer near the surface of a semiconductor, the vector of the field varies with the radius of surface curvature. The surface concentration of charge carriers and the rate of carrier supply, which are determined by the field vector, are thus affected by surface curvature. The situation is different on a metal surface. There exists no such a field inside the metal near the surface and all sites on a metal surface, whether it is curved not, is identical in this aspect. 

The recombination of electrons and holes is a rather complicated process. We have to distinguish between (a) the direct recombination of electrons and holes, occurring in particular at high concentrations of charge carriers (b) recombination via defect states which depends, among other factors, on the densities and capture cross-sections of the defects (recombination centers) located in the bulk of the solid or on its surface. 

When one examines the value of n = p, it turns out that the density of charge carriers in an intrinsic semiconductor (Table 6.16) at room temperature is in the range of 10 to 10 cm, compared with about 10 cm in a metal. It is this relatively low concentration of charge carriers in intrinsic semiconductors that is responsible for the most important differences between semiconductor electrodes and metal electrodes. 

Impurity Semiconductors, n-Type andp-Type. The discussion has been restricted so far to pure intrinsic semiconductors exemplified by germanium and silicon. In these substances, there is a low concentration of charge carriers (compared with metals). Further, the hole and electron concentrations are equal, and their product is a constant given by the law of mass action... 

The approach based on the LH equation, or the related Eley-Rideal mechanism [30], is common in the literature today [5]. The equation is generally derived assuming equal steady-state concentrations of charge carriers or their concentration independent of substrate and the electron scavengers concentration. The independence of the actual concentration of active species (either h + or ecu) is also usually assumed by taking a mean time of survival of these species independent on other system concentrations. However, it is obvious from the oldest literature that the concentrations of the active species are mutually dependent. The first factor is the recombination in bulk. Bahnemann et al. derived the transient... 

A description of charge transport in molecular conductors has been adapted from the band theory of semiconductors (79MI11300). The conductivity is given by the product of the concentration of charge carriers, expressed in the format of an activation energy, and the carrier mobility which is inversely proportional to an exponent of the absolute temperature. Both expressions contain parameters specific to each sample and the general approach is of little use in the design and synthesis of new materials. 

Semiconductors like silicon or germanium are an intermediate case. Their electrons are not as tightly bound as in insulators so that at any given time a small fraction of them will be mobile. In a perfect germanium crystal, for instance at 25°C, about 3 x 1019 electrons per m3 are free. This corresponds to a concentration of 5 x 10-8 M or 50 nM. It is much lower than the concentration of charge carriers (cat- and anions) in an aqueous electrolyte solution. Despite this small concentration, the conductivities are of the same order of magnitude, because the electrons in a semiconductor are typically 108 times more mobile than ions in solution. 

The bulk conductivity a depends on the concentration of charge carriers and on their mobility, either of which can be modulated by exposure to the gas. The first prerequisite of such an interaction is the penetration of the analyte to the interior of the layer. The second is the ability of the gas to form a charge-transfer complex with the selective layer. This process then constitutes secondary doping, which affects the overall conductivity. For a mixed semiconductor, the overall conductivity is determined by the combined contribution from the holes (p) and electrons (n), as given by the general conductivity equation. 

The impedance for the study of materials and electrochemical processes is of major importance. In principle, each property or external parameter that has an influence on the electrical conductivity of an electrochemical system can be studied by measurement of the impedance. The measured data can provide information for a pure phase, such as electrical conductivity, dielectrical constant or mobility of equilibrium concentration of charge carriers. In addition, parameters related to properties of the interface of a system can be studied in this way heterogeneous electron-transfer constants between ion and electron conductors, or capacity of the electrical double layer. In particular, measurement of the impedance is useful in those systems that cannot be studied with DC methods, e.g. because of the presence of a poor conductive surface coating. 

In ceramics containing transition metal ions the possibility of hopping arises, where the electron transfer is visualized as occurring between ions of the same element in different oxidation states. The concentration of charge carriers remains fixed, determined by the doping level and the relative concentrations in the different oxidation states, and it is the temperature-activated mobility, which is very much lower than in band conduction, that determines a. 

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