Kinetics of oxide film growth on metal crystals - II. Homogenous field approximations

Abstract

Abstract Homogeneous field approximations yield simple equations relating film thickness L(t) to time t of oxidation from coupled steady-state diffusion equations of I assuming equal ionic and electronic currents. High temperature parabolic equations are obtained with rate determined by transport of the low mobility species in the presence of the electric field created by the high mobility species; these parabolic equations have a modified rate constant k′ due to a constant electrical potential difference φ across the film. Both k′ and φ increase with (concentration) × (mobility) and ratio of boundary concentrations for the electronic species, and decrease for corresponding changes for the ionic species. For tunnelling or field emission of one species with rate determined by diffusion of the second species, the expression obtained is[1 +βtt = exp(αL(t))−αL(t)], with α and β determined from emission and diffusion parameters. This equation is based on the intermediate temperature approximation that the surface charge is essentially independent of L(t) for tunnelling from the metal crystal through the film, to traps, and between traps in the oxide for film thicknesses less than some critical thickness Lcrit: For limiting cases this equation reduces to Nth root [L(t) ∝ t 1 N ] and logarithmic [L(t) ∝ log (1 + t t 0 ] laws. The final stage of growth occurs for L(t) >Lcrit and usually predominates at low temperatures; rate is controlled by tunnelling or field emission in a thickness dependent field (E ∝ 1 L(t) ) so that a limiting thickness L∞ usually results, with L∞−Lcrit equal to several monolayers only. For this region of growth, a logarithmic law and an exponential law [L(t>) ∝ 1−exp(−vt)] result for Fowler-Nordheim behavior of the current with field, a modified exponential law [L(t) ∝ 1−exp {− ( |L(t) L ∞ ) − |t τ} ] results for a linear dependence of tunnel current through the film on field, and a cubic law [L(t) ∝t 1 3 ] results for a linear dependence of tunnel current between traps in the oxide on field. Application of this work is made to published experimental data for copper monocrystals at 1173°K and between 451 and 78°K.