Mott equation

The standard Mott equation for diffusion thermopower 5a in metallic systems is a function of the first derivative of the density of states at the Fermi level, and it is expressed as... 

Figure 1.19 Evaluation of the capacitance according to the Schottky-Mott equation for grains a and b for oxide layers grown potentiodynamically at different formation potentials U (100 pm electrodes, f= 1030 Hz, dU/dt —20mVs 1)[6].

Polypyrrole is comparable to inorganic semiconductors from the point of view of the dependence of its conductivity on temperature a decrease in conductivity is observed as temperature increases in the reversible potential range [96,97]. Conductivity fits the Arrhenius equation at temperatures approximately above 250 K. At temperatures (T) between 100 and 25CFK, the electrical conductivity (polypyrrole films has been reported to follow the Mott equation, based on a model of variable range hopping in three dimensions between localized states [98,99] ... 

According to Eq. 1.4, we can see that the Seebeck coefficient is not only related to carrier concentration (ti), but also in proportion to the carrier effective mass (ot ), which indicates that a large carrier effective mass will bring larger Seebeck coefficient. However, with the increase of the carrier effective mass, it is inevitable to reduce the carrier mobility, which leads to the decrease of the electrical conductivity. The exact relationship between effective mass and the mobility is very complex, and directly depends on the electronic structure of the compound. However, an intuitive expression of relationship between Seebeck coefficient and the electronic structure can be obtained based on the Mott equation [53] ... 

Ch is within 10-20 pF cm and much larger than Cgc so that its inverse may usually be neglected as shown in Equation 1.181. This result is closely related to the location of potential changes within the semiconductor as shown in Figure 1.52. One may apply the Schottky-Mott equation (Equation 1.182) to the semiconductor electrolyte combination with the Boltzmann constant k and the charge of the electron Cq. 

Eq. (14.1) is known as the Mott-Schotlky equation. We note llial for a given n-lype semiconductor, the harrier height increases as the work function of the metal increases. It is therefore expected that high work function metals will give a rectifying junction, and low work function metals an ohmic contact (it is the reverse for a p-type semiconductor). 

In Eq. (4.5.5), describing an n-type semiconductor strongly doped with electron donors, the first and third terms in brackets can be neglected for the depletion layer (Af0 kT/e). Thus, the Mott-Schottky equation is obtained for the depletion layer,... 

Semiconductors that are used in electrochemical systems often do not meet the ideal conditions on which the Mott-Schottky equation is based. This is particularly true if the semiconductor is an oxide film formed in situ by oxidizing a metal such as Fe or Ti. Such semiconducting films are often amorphous, and contain localized states in the band gap that are spread over a whole range of energies. This may give rise... 

The interfacial capacity follows the Mott-Schottky equation (7.4) over a wide range of potentials. Figure 8.4 shows a few examples for electrodes with various amounts of doping [5]. The dielectric constant of Sn02 is e 10 so the donor concentration can be determined from the slopes of these plots. 

The modification of the three and four-particle system due to the medium can be considered in cluster-mean field approximation. Describing the medium in quasi-particle approximation, a medium-modified Faddeev equation can be derived which was already solved for the case of three-particle bound states in [9], as well as for the case of four-particle bound states in [10]. Similar to the two-particle case, due to the Pauli blocking the bound state disappears at a given temperature and total momentum at the corresponding Mott density. 

In tightly bound (Frenkel) excitons, the observed peaks do not respond to the hy-drogenic equation (4.39), because the excitation is localized in the close proximity of a single atom. Thus, the exciton radius is comparable to the interatomic spacing and, consequently, we cannot consider a continuous medium with a relative dielectric constant as we did in the case of Mott-Wannier excitons. 

This equation can be transformed by Fourier conversion and Mott formulate the relation determining an electron structure amplitude ... 

Equation (3.4.28) is commonly known as Mott-Schottky equation. 

The flat band potentials of a semiconductor can be determined from the photocurrent-potential relationship for small band bending [equation (4.2.1)], or derived from the intercept of Mott-Schottky plot [equation (4.2.2)] using following equations... 

The first part of the chapter is devoted to an analysis of these correlations, as well as to the presentation of the most important experimental results. In a second part the following stage of development is reviewed, i.e. the introduction of more quantitative theories mostly based on bond structure calculations. These theories are given a thermodynamic form (equation of states at zero temperature), and explain the typical behaviour of such ground state properties as cohesive energies, atomic volumes, and bulk moduli across the series. They employ in their simplest form the Friedel model extended from the d- to the 5f-itinerant state. The Mott transition (between plutonium and americium metals) finds a good justification within this frame. 

The arguments for Edwards9 cancellation theorem are rather subtle, and we do not think that it is universally true (Mott 1989). For weak scattering, the relaxation time t must be proportional tog-1, by Fermi s golden rule. The mean free path l is given by the equation... 

The effect leading to a term in Lb described in Section 8, can be expressed by an equation first given by Kawabata (1981) and Kaveh and Mott (1982) ... 

In this section we give a proof of the Kawabata formula (52), following a method due to Kaveh (1984) and Mott and Kaveh (1985a, b). We assume that an electron undergoes a random walk, which determines an elastic mean free path l and diffusion coefficient D. If an electron starts at time t=0 at the point r0 then the probability per unit volume of finding it at a distance r, at time U denoted by n(r, t) obeys a diffusion equation... 

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