Kinetics of oxide film growth on metal crystals—I.: Formulation and numerical solutions

Abstract

A system of coupled diffusion equations can describe oxide film growth. Ideal treatment of surface processes leads to linear-type growth for surface controlled rate, and provides boundary values for the coupled equations for diffusion controlled rate. A comparison of the space charge relaxation time with the monolayer formation time, as well as direct numerical computation, justifies steady-state equations for 100 Å films with defect concentrations less than 1 per cent of the lattice sites. Fourier analysis of the fixed boundary problem provides an iterative solution for Stefan's problem. Numerical solutions for time-dependent interfacial charge densities show a parabolic-type growth for charged particles with an enhanced rate constant ξ k, where k is the parabolic rate constant for the diffusion of uncharged particles. The enhancement factor ξ is determined by an electrical potential difference Đindependent of L( t) developed across the film. The parameters ξ and Đ depend on relative values of the products of concentration C and mobility μ, increasing with increasing μ C for the electronic species and decreasing with increasing μ C for the ionic species. Typical values of Đ and ξ are 0·3 V and 3. For equal values of μ C, Đ is nearly zero and ξ is approximately one in the small space chargelimit. Non-negligible space charge relative to interfacial charge density decreases ξ. In certain cases, Đ and ξ depend on L( t), so departures from parabolic growth occur, especially in the transient stages of growth. Numerical solutions for constant external or surface charge fields yield Nth root and logarithmic growth laws. Space charge can aid or hinder growth ; this is determined by the contribution of the space charge field to transport and the change in ion distribution for this field. Experimental data for growth of Cu 2O on Cu crystals at 50 and 130°C were fit assuming small space charge.