Mott electronic charge

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. 

Here q is the electronic charge, e is the dielectric constant of semiconductor, and eo is the permittivity of free space. The carrier density of holes1 is denoted by p and the electric field by F. Mott showed that the diffusion component of the current is negligible and... 

Equation (13) represents an ordinary differential equation requiring two boundary conditions and a reference point for the potential, commonly set to zero in the bulk. If ionic defects are frozen in place and unable to move, the charge density in Poisson s equation is determined only by redistribution of electronic charge carriers that are ionized from the dopants. This is known as the Mott-Schottky approximation [18], which results in a simplification to Poisson s equation by setting Cj(x) = Cj ... 

The famous Mott-Schottky relationship [25,26] in Eq. 5-21 represents a different potential-dependent surface capacitive case. This relationship was derived to express the electronic properties of passive capacitive films of constant thickness formed on metals. The methods based on the Mott-Schottky equation have been widely used as a valid tool to determine semiconductive character and dopant density of the surface films in the semiconductor industry and in corrosion studies. The change of the space-charge layer capacitance of the passive film (or space charge distribution) depends on the difference between the applied DC potential V and flat band potential V g characteristic of the surface film, where Np = concentration of donors (or acceptors) or "doping density" ( 10 - lO cm" ), and Cg = 1.6 KT C electron charge ... 

The incorporation of phosphorus yields fourfold-coordinated P atoms, which are positively charged, as phosphorus normally is threefold coordinated. This substitutional doping mechanism was described by Street [52], thereby resolving the apparent discrepancy with the so-called S N rule, with N the number of valence electrons, as originally proposed by Mott [53]. In addition, the incorporation mechanism, because charge neutrality must be preserved, leads to the formation of deep defects (dangling bonds). This increase in defect density as a result of doping explains the fact that a-Si H photovoltaic devices are not simple p-n diodes (as with crystalline materials) an intrinsic layer, with low defect density, must be introduced between the p- and n-doped layers. 

Since the discovery of the first organic conductors based on TTF, [TTF]C1 in 1972 [38] and TTF - TCNQ in 1973 [39], TTF has been the elementary building block of hundreds of conducting salts [40] (1) charge-transfer salts if an electron acceptor such as TCNQ is used, and (2) cation radical salts when an innocent anion is introduced by electrocrystallization [41]. In both cases, a mixed-valence state of the TTF is required to allow for a metallic conductivity (Scheme 5), as the fully oxidized salts of TTF+ cation radicals most often either behave as Mott insulators (weakly interacting spins) or associate into... 

We consider, now, an electron-depleted space charge layer that is gradually polarized in the anodic direction. As long as the Fermi level is located away from the surface state, the interfacial capacity is determined by the capacity of the depletion layer that obeys a Mott-Schottlsy relation as shown in Fig. 5-61. 

It was noted earlier that the charge density of a narrow resonance band lies within the atoms rather than in the interstitial regions of the crystal in contrast to the main conduction electron density. In this sense it is sometimes said to be localized. However, the charge density from each state in the band is divided among many atoms and it is only when all states up to the Fermi level have contributed that the correct average number of electrons per atom is produced. In a rare earth such as terbium the 8 4f electrons are essentially in atomic 4f states and the number of 4f electrons per atom is fixed without reference to the Fermi level. In this case the f-states are also said to be locaUzed but in a very different sense. Unfortunately the two senses are often confused in literature on the actinides and, in order not to do so here, we shall refer to resonant states and Mott-localized states specifically. 

Figure 6.52 Schematic electron addition and removal spectra representing the electronic structure of transition-metal compounds for different regimes of the parameter values (a) charge-transfer insulator with U > A (b) Mott-Hubbard insulator A> U (From Rao et al, 1992).

If a positive charge ze is immersed in a degenerate electron gas, the Coulomb field is screened by the electrons. The screening was first estimated by the present author using the Thomas-Fermi approximation (Mott 1936, Mott and Jones 1936, p. 86), and by this method one finds for the potential energy of an electron... 

More recent work has shown, however, that an exponential decay of the screening potential as in (21) is not correct, and that round any scattering centre the charge density falls off as r 3 cos 2kFr. This we shall now show, by introducing the phase shifts t/, defined as follows (cf. (13)). Consider the wave functions Fx of a free electron in the field of an impurity. These behave at large distances from the impurity as (Mott and Massey 1965)... 

Perhaps the earliest system to be described as an Anderson transition was cerium sulphide (Cutler and Mott (1969), on the basis of observations by Cutler and Leavy (1964)). The material in question can be written Ce3 xvxS4, where v is a cerium vacancy, the vacancies being distributed at random. The field near a cerium vacancy repels electrons, because they are negatively charged. Variation of x, then, changes the number of electrons and the number of scatterers. Figure 1.25 shows some results on the conductivity. At that time the present author believed... 

The electron transport mechanism in mesoporous Ti02 film is modeled mainly by using diffusion theory, except in the report by Augustinski et al.,45) who proposed the explanation that the initial film charging by dye-sensitization, in terms of the self-doping, causes an insulator-metal (Mott) transition in a donor band of Ti02, accompanied by a sharp rise in conductivity of the nanoparticles. 

In the Gurney-Mott mechanism, the trapped electron exerts a coulombic attraction for the interstitial silver ion. This attraction would be limited to a short distance by the high dielectric constant of the silver bromide. Slifkin (1) estimated that the electrostatic potential of a unit point charge in silver bromide falls to within the thermal noise level at a distance of "some 15 interatomic spacings." The maximum charge on the sulfide nucleus would be 1 e. The charge on a positive kink or jog site after capture of an electron would not exceed e/2. An AgJ would have to diffuse to within the attraction range before coulombic forces could become a factor. 

---