The thick-film parabolic growth law

Consider the limit of thick oxides and moderate values of the electric field where both ion transport and electron transport take place in accordance with the linear diffusion equation [see eqn. (96)]. In essence, this means that we are assuming that the inequality 

The coupled-currents condition [eqn. (172)] for this case can be written as 

Substituting the ionic current (species 1) and the electronic current (species 2) given by eqn. (96) into the coupled-currents condition gives the relation 

If local space-charge neutrality can be assumed, which has been justified by extensive numerical calculations [46] for the present case, then we can write 

Equation (180) for local space-charge neutrality requires that the oppositely charged mobile defect species concentrations have a very special relationship. Solving for C2 gives 

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Therefore, we obtain from the integration of eqn. (188) the thick-film parabolic growth law... 

This completes our development of the thick-film parabolic growth law. This particular theory has been presented in some detail because it is an extremely important domain of metal oxidation. In addition, it provides an excellent example of the way the coupled-currents approach [10,11] can be used to obtain oxide growth kinetics and built-in voltages in thermal oxidation. 

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