Parabolic Growth Law for Coherent Oxides

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Abstract
The Wagner derivation of the parabolic growth law is shown to be inconclusive, since the rate constant can be a function of film thickness. The additional criterion to be met is that the dependence of concentrations C(x, L) on position x and film thickness L(t) be a function of x/L(t) only. This can be examined for any particular case by solving the appropriate diffusion equation and applying the proper boundary conditions. The condition that C(x, L) be a function of x/L(t) for parabolic growth is in accordance with the results of previous work showing that surface charge due to two diffusing species enhances the parabolic rate constant but space charge causes deviations from the parabolic law. The functional form of C(x, L) is examined further by solving the diffusion equation for the general cases in which mobilities μ vary arbitrarily with temperature and concentration. The parabolic law with modified rate constants is shown to be valid for these general cases, unless the values of | μC | for the two oppositely charged species do not differ by more than an order of magnitude and the mobilities depend strongly on several powers of the concentration.

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