Space Charge in Growing Oxide Films. V. Perturbation Coefficients and Second-Order Treatment

The coefficients in a previously developed perturbation treatment of space-charge effects for growing oxide films are explicitly evaluated in terms of the boundary concentrations of the diffusing species. The quadratic forms for the coefficients in terms of concentrations of ions and electrons makes quantitative comparison of theory and experiment possible, and facilitates the development of analytical approximations for limiting cases. The perturbation treatment itself is extended to include higher-order terms. The resulting equations reduce to quadratures; numerical computations for a specific example are presented to compare first- and second-order results. An approximate equation L(t)=L∞′(1−exp{−(t/τ′)−[L(t)/L∞′]}) is thereby obtained relating film thickness L(t) to time t for the later stages of growth, with the parameters L′∞ and τ′ determined by the space-charge concentrations in the film. Dimensionless curves are presented which illustrate the relationship between this equation and the first-order equation L(t)2=L∞2[1−exp(−t/τ)].