Kinetics of Oxide Film Growth on Metal Crystals: Electronic and Ionic Diffusion in Large Surface-Charge and Space-Charge Fields

Abstract

The coupled-currents approach is utilized to study oxidation kinetics for the case of electron and ion transport by field-modified diffusion. The expression developed for the particle current $J$ for a given species is $J=(\frac{\ensuremath{\nu}}{{S}_{N}})\mathrm{exp}(\frac{\ensuremath{-}W}{{k}_{B}T})[{n}_{0}\ensuremath{-}{n}_{N}\mathrm{exp}(\frac{\mathrm{Ze}{V}_{N}}{{k}_{B}T})]$, with ${S}_{N}=\ensuremath{\Sigma}{k=1}^{N}\mathrm{exp}[\frac{Ze({V}_{k\ensuremath{-}1}+{V}_{k})}{2{k}_{B}T}]$, where $\ensuremath{\nu}$, $W$, ${n}_{0}$, ${n}_{N}$, and $Z$ are, respectively, the attempt frequency, activation barrier, areal density at the metal-oxide interface, areal density at the oxide-oxygen interface, and charge per particle in units of $e$ for the diffusing defect species in question; ${k}_{B}$, $T$, and $e$ denote the Boltzmann constant, the temperature, and the electronic-charge magnitude. The macroscopic electrostatic potential at the position of the potential minimum following the $k\mathrm{th}$ potential maximum due to the ordinary lattice periodicity is denoted by ${V}_{k}$. In the low-space-charge high-field limit for equal magnitudes of $Z$ for the oppositely charged diffusing species, the electrostatic potential developed across the film is a constant, and the resulting kinetics have the Mott-Cabrera form. The time $t$ as a function of film thickness $L$ is given by a series of second-order exponential integrals ${E}_{2}$ with successively increasing values of the argument: $\frac{t}{\ensuremath{\tau}}=2\frac{L}{{L}_{\mathrm{crit}}}\ensuremath{\Sigma}\stackrel{\ensuremath{\infty}}{m=0} {E}_{2}\left((2m+1)\frac{{L}_{\mathrm{crit}}}{L}\right),$ where $\ensuremath{\tau}$ and ${L}_{\mathrm{crit}}$ are determined by the transport parameters for the system in question. This expression reduces to the previously derived homogeneous-field parabolic growth law ${L}^{2}=2{{L}_{\mathrm{crit}}}^{2}\frac{t}{\ensuremath{\tau}}$ whenever nonlinear effects become inappreciable. Space charge can retard, enhance, or provide no modification of the growth rate, depending on the potential developed across the film and the sign of the space charge relative to the rate-limiting species. For nonzero potentials with the sign of the space charge opposite to that of the rate-limiting species, the growth rate is found to be enhanced; for nonzero potentials with the sign of the space charge the same as that of the rate-limiting species, the growth rate is found to be retarded.