Mott-Schottky theory

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. 

Thus, we have to conclude that, without knowing the physical nature of the frequency dependence of the differential capacitance of a semiconductor electrode, the donor (or acceptor) concentration in the electrode cannot be reliably determined on the basis of the Schottky theory, irrespective of the Mott-Schottky plot presentation format. Therefore, the reported in literature acceptor concentrations in diamond, determined by the Schottky theory disregarding the frequency effect under discussion, must be taken as an approximation only. However, we believe that the o 2 vs. E plot (the more so, when the exponent a approaches 1), or the Ccaic 2 vs. E plot, are more convenient for a qualitative comparison of electrodes made of the same semiconductor material. 

At present there is no unambiguous method for determining the energetic position of the conduction band edge of nanocrystalline semiconductor films under experimental conditions relevant to dye sensitization. Clearly, the lack of a depletion layer precludes traditional capacitance measurements with Mott-Schottky analysis described in Section 1.1. The onset of photocurrents in photocurrent-voltage plots can provide a crude working estimate under some conditions, but is problematic for the reasons described previously. This is unfortunate as accurate energetics are necessary to compare experimental results with even the simplest theory [4[. 

The selection of a reference potential at the Ohmic contact is arbitrary and was chosen to emphasize the degree of band bending and straightening in the semiconductor. The development of Mott-Schottky theory in Section 12.3.2 employs a potential referenced to the Ohmic contact. A difference in sign will be seen if the potential is referenced instead to a reference electrode located in the electrolyte. The potential of the electrolyte has been fovmd to be independent of current and illumination intensity when referenced to an external quantity such as the Fermi energy of an electron in vacuum. This concept has proved useful for predicting the interaction between semiconductors and a variety of redox couples. The lUPAC standard for photoelectrochemical systems, in fact, is that the potential is referred to a reference electrode in the electrolyte. ... 

D. B. Bonham and M. E. Orazem, "A Mathematical Model for the Influence of Deep-Level Electronic States on Photoelectrochemical Impedance Spectroscopy 2. Assessment of Characterization Methods Based on Mott-Schottky Theory," Journal of The Electrochemical Society, 139 (1992) 126-131. 

The nuclei form both at the surface and in the interior of the cry tal, but the theory developed by Mott (49) assumes that the defects are interstitial Ba - ions and that the nuclei grow at the surface of the grains only. As in the formation of the latent image in sensitized Ag halide grains, the assumption of pure Frenkel defects could only lead to the formulation of surface nucleation, but recent developments have led Mitchell to consider the role of Schottky defects (F-centres), which may forn internal aggregates ultimately bi caking away from the parent lattice as minute nuclei of the new metal phase. It is possible that a similar state of affairs exists in azides (36) unfortunately, nothing is known of the nature of... 

Mott-Schottky theory can be used to determine the flat band potential (see Sect. 2.1.3.1). This then allows one to calculate the band-bending for any value of the applied potential, if the Helmholtz potential remains constant (i.e. the band edges are pinned). The band-bending determines the concentration of majority carriers at the surface (see Eqs. 25 and 26). In a p-type electrode these are holes, which are essential for the dissolution reaction. In an n-type electrode, the band-bending determines the surface electron concentration and thus, the rate of recombination... 

The X-ray excitation process frequently is analyzed in terms of an excitonic electron hole pair (e.g. Cauchois and Mott 1949). The excitonic approach to X-ray absorption spectra accounts for the fact that the excited state is a hydrogen-like bound state. The X-ray exciton is different from the well-known optical excitons. In the latter cases the ejected electron polarizes a macroscopic fraction of the crystal-fine volume because the lifetime of optical excitations is in the order of lO s. The lifetime of the excited deep core level state, however, is in the order of 10 — 10 s, much too short to p-obe more than the direct vicinity of excited atom. Following Haken and Schottky (1958) the distance r between the ejected electron and core hole of an excited atom for E = 1 turns out to be r oc [h/(2m 0))] Here m denotes the effective mass of the ejected electron, to is the phonon frequency and is the dielectric constant. A numerical estimate yields r 10 A. Thus the information obtainable in an L, spectrum of the solid is very local the measurement probes essentially the 5d state of the absorbing atom as modified from the atomic 5d states by its immediate neighbors only. It is not suited to give information about extended Bloch states. On the other hand it is well suited to extract information about local correlations within the 5d conduction electrons, whose proper treatment is at the heart of the difficulty of the theory of narrow band materials and about chemical binding effects. 

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