Effects of the electric field shape on nano-scale oxidation

An axisymmetric continuum model for oxide growth by the scanned probe oxidation technique is presented. The model includes equations describing the electric fields, hydroxyl and hydrogen ion concentrations, and oxide free boundaries defining the system. The governing system of partial differential equations tracks ion transport in the liquid and oxide layers and includes reactions at the substrate/oxide interface. Further, space charge effects are considered near the substrate/oxide interface. Two liquid configurations, semi-infinite layer and hemispherical drop of liquid, are examined to determine the potential in the liquid region. The AFM tip is modeled as either a point or a ring source of charges. The asymptotic limit of a small aspect ratio (height to width) oxide feature is used to reduce the governing equations to a quasi-one-dimensional system to determine the ion transport. The solution of the reduced system leads to an evolution equation for the oxide thickness and radius. Numerical simulations of the evolution equation predict oxide features that qualitatively agree with experimental observations. A parametric study is conducted to determine the influence of key operating and material parameters on the thickness and radius of the oxide, and the electric and ion concentration fields in the system.