Cabrera-Mott field

Growth of surface oxide films takes place only if cations, anions, and electrons can diffuse through the oxide layer. The growth kinetics of very thin films ( 10-50 A) often follow the Mott or Cabrera-Mott mechanisms in which electrons tunnel through the film and associate with oxygen atoms to produce oxide ions at the surface. A large local electric field (10 -10 V/cm) results at the surface which facilitates cation diffusion from the metal-oxide interface to an interstitial site of the oxide. The film thickness Z at time t is given by... 

The generally accepted model for passive film growth, illustrated in Fig. 3-14, is of field-assisted film formation, which is essentially a modified Cabrera-Mott model originally established for gaseous oxidation and the formation of thin oxide films in a gas at low temperature (Cabrera and Mott, 1948-1949 Fehlner and Mott, 1970). This classical theory describes the growth, in the direction perpendicular to the surface, of an oxide layer completely covering the substrate surface, by a hopping mechanism. The... 

A strong electric field is formed in very thin films (with a thickness of about 10 5 cm) during current flow. If the average electrochemical potential difference between two neighbouring ions in the lattice is comparable with their energy of thermal motion, kTy then Ohm s law is no longer valid for charge transport in the film. Verwey, Cabrera, and Mott developed a theory of ion transport for this case. 

Numerous other models have been proposed to explain the deviation of dry oxidation from linear-parabolic kinetics. For example, field-assisted oxidant diffiision during the oxidation of metals was proposed by Cabrera and Mott (75) and used by Deal and Grove (69) to explain the results for thin oxides. Ghez and van der Meulen (76) proposed the dissociation of molecular oxygen into atomic oxygen at the Si-Si02 interface and the re-... 

This constancy of the field implies that either electroneutrality prevails in the bulk of the film or the concentration of ions is not controlled by electroneutrality, but that the films are so thin, and the density of carriers so low, that the space charge due to the carriers causes negligible change in field across the film. The first case was that considered implicitly by Verwey, who was the first to apply Frenkel s theory to these systems. The second was proposed by Mott, and was discussed by Mott and Gurney, and by Cabrera and Mott. ... 

The model of Cabrera and Mott is discussed in die book on low-temperature oxidation by Fehiner [5b]. The basis of the model is die quantum mechanical concept of electron tunneling. An electron can penetrate an energy barrier without die requirement for thermal activation. As soon as a three-dimensional oxide forms on a metal, electrons tunneling through the oxide are captured by adsorbed oxygen on die oxide surface. The charge separation thus established between die oxide surfece and the metal sets up an electric field across the oxide. The proposed mechanism is illustrated in Figure 1. 

The second assumption is due to Grimley and Trappnell [GRI 56], which considers that the layer has a constant charge, that is, an electric field that does not depend on the thickness of the layer. Now, we will use the Mott and Cabrera approximation. 

We now study layer thickness mnch lower than Xq (defined by [15.5]). The electric field is thus prevalent and the concentration gradient has a negligible effect. The diffusion flux is thus given by eqnation [15.6], keeping the Mott and Cabrera assumption. Consider then ... 

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