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Consistent Asymptotic Expansion of Mott's Solution for Oxide Growth

Many relatively thick metal oxide films grow according to what is called the parabolic law L = 2At + . . . . Mott explained this for monovalent carriers by assuming that monovalent ions and electrons are the bulk charge carriers, and that their number fluxes vary as t^{-1/2} at sufficiently long t. In this theory no charge is present in the bulk, and surface charges were not discussed. However, it can be analyzed in terms of a discharging capacitor, with the oxide surfaces as the plates. The theory is inconsistent because the field decreases, corresponding to discharge, but there is no net current to cause discharge. The present work, which also includes non-monovalent carriers, systematically extends the theory and obtains the discharge current. Because the Planck-Nernst equations are nonlinear (although Gauss's Law and the continuity equations are linear) this leads to a systematic order-by-order expansion in powers of t^{-1/2} for the number currents, concentrations, and electric field during oxide growth. At higher order the bulk develops a non-zero charge density, with a corresponding non-uniform net current, and there are corrections to the electric field and the ion currents. The second order correction to ion current implies a logarithmic term in the thickness of the oxide layer: L = (2At)^{1/2} + B ln t + . . . . It would be of interest to verify this result with high-precision measurements.