  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.
