Single Carrier Steady‐State Theory for Formation of Anodic Films Under Conditions of High Space Charge in Very Large Electric Fields

Abstract

The kinetics of growth of anodic films are modeled in the high space charge limit; the predictions of the model are examined analytically and numerically. The results illustrate that space charge due to the mobile ionic species can lead to a marked retardation of the rate of growth as measured under constant voltage conditions. Time increases by factors as large as 3000 are found to be required for growth to a given thickness. For a given film thickness the time factors are somewhat greater at larger voltages; at a given voltage the time factors increase nearly linearly with increasing thickness. These observations lead to the following major conclusion: Space charge effects do not disappear and do not saturate with increasing electric field value; on the contrary, the space charge retardation is even more pronounced in the realm of very high electric fields, large applied voltages, and very thick anodic films. For the case of growth under constant current conditions in the presence of space charge, a given growth rate was found to require a significantly larger voltage than would be predicted in the usually considered homogeneous field limit; however, the increase in voltage is relatively less in the higher voltage regime (e.g., a factor of 2) than it is for growth at very low voltages (where factors as large as 10 to 100 are not uncommon). The voltage at constant current was found to increase quasi‐linearly with thickness in the thick film high voltage realm, whereas at low voltages the space charge produced marked curvature in the voltage‐thickness plots. The quasi‐linear variation of voltage with thickness found for the thick film high voltage realm constitutes proof sine qua non that present experimental data in the literature indicating experimental constancy of the field at constant current cannot be employed to disqualify anodic film growth models having uncompensated space charge.