A theoretical and numerical study of thermal oxidation kinetics is presented for a model based on ionic diffusion (including high-electrostatic-field nonlinear effects) and thermionic emission over metal-oxide work-function barriers (appropriately modified by the macroscopic electric field in the oxide and image effects in the parent metal and adsorbed oxygen). The kinetics are obtained by coupling the transport equations through the surface-charge field, imposing the condition of equal charge currents, and numerically integrating the system of equations. The expressions utilized for both the ionic and the electronic currents inherently contain the capability for a current equilibrium of either species. Conditions favorable for the establishment of a virtual ionic current equilibrium are low temperatures, a high ionic mobility, and a large metal-oxide work function. Rate is determined in this case by the Schottky emission of electrons from the parent metal into the conduction band of the oxide under the influence of a fixed positive ionic-diffusion potential ${V_D}^{\mathrm{}\left(i\right)\mathrm{}}$. The intermediate situation in which neither current is in a virtual equilibrium is investigated numerically; the results show that the electrostatic potential $V_K$ across the growing oxide varies markedly with film thickness and temperature. The transition toward the other extreme of a virtual electronic current equilibrium is favored by higher temperatures, a low ionic mobility, and a small metal-oxide work function. The negative electron-equilibrium potential $V_{\mathrm{Eq}}$ deviates significantly from the Mott potential $V_M$ (which represents equalization of the electronic energy levels in adsorbed oxygen with the metal Fermi level), because of a factor which depends upon film thickness and temperature. The corresponding oxidation rate is determined by the high-field transport of ions under the film-thickness-dependent surface-charge field established by the electrons. The Mott-Cabrera expression integrated in terms of the exponential integral provides an analytical expression for the kinetics for large negative potentials, provided that a temperature-dependent effective kinetic potential $〈V_K〉$ is employed instead of $V_M$. The results are utilized to correlate published data for several experimental studies of the oxidation of aluminum and tantalum.

• Received 19 July 1967

DOI:https://doi.org/10.1103/PhysRev.163.650