Metal and Oxygen Ion Transport during Ionic Conduction in Amorphous Anodic Oxide Films

A mathematical model was developed for ionic conduction in amorphous oxide films. The model is based on a hypothesized “defect cluster” mechanism in which both metal and oxygen ions are involved in transport. Defect clusters are created by inward displacement of oxygen ions around an oxygen vacancy‐like defect in response to the vacancy's electric field. Metal ions are assumed to migrate easily in the gap between the first and second layer of oxygen ions around the vacancy. The model includes the polarization of the conductive gap in the applied electric field, the exchange of mobile metal ions in the cluster with stationary metal ions in the surrounding oxide, and space charge generated in the film by clusters and oxide nonstoichiometry. The rate‐limiting step is the jump of the oxygen vacancy in the cluster. It was found that polarization of the cluster leads to a stoichiometric excess of metal ions in the cluster and that this excess produces a net transport of metal ions due to the motion of the cluster. The metal ion transport number was found to increase with electric field and to depend on the dielectric constant and cluster size. The field dependence follows that found experimentally. The calculated transport numbers are in quantitative agreement with experimental values for tantalum, niobium, and tungsten oxide but smaller than experimental values for aluminum oxide. The field coefficient in the high‐field‐conduction‐rate expression is also predicted and agrees with experimental values to within 10%. Disciplines Chemical Engineering Comments This article is from Journal of the Electrochemical Society 146 (1999): 3741–3749, doi:10.1149/1.1392543. Posted with permission. This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/cbe_pubs/68 Journal of The Electrochemical Society, 146 (10) 3741-3749 (1999) 3741 S0013-4651(98)09-078-8 CCC: $7.00 © The Electrochemical Society, Inc. Anodic oxidation of valve metals in solutions which do not dissolve the oxide produces compact, nonporous oxide films which find use in dielectric applications. These anodic oxides include the films on aluminum, tantalum, tungsten, niobium, zirconium, and titanium,1 the first four of which have amorphous structures.2 Transport processes in anodic films have been extensively investigated experimentally. The anodic oxides are electronic insulators; during ionic conduction, the current density is proportional to exp(BE), in which E is the electric field in the film. The field coefficient B is approximately 5 cm/MV for all these systems.1 The contributions of metal vs. oxygen ions to conduction have been studied by depositing immobile chemical species onto the films as markers.2,3 The fraction of current carried by metal ions is denoted by tM, the metal ion transport number. tM is found to vary from 0.1 to 0.5 in the various systems2 and to increase with electric field.4,5 The mechanisms of ionic mass transport have been investigated using tracers, usually O18 isotopes for studying oxygen transport6,7 and bilayer metal films to study metal ion transport.6,8,9 Oxygen transport occurs via exchange between transported oxygen ions and those constituting the film. For crystalline oxides, this exchange would result from vacancy or exchange interstitial transport, although such a distinction is less meaningful in amorphous materials. In the case of metal ion transport, the available measurements suggest that exchange between transported and constituent ions also occurs, although there may be a contribution from interstitial-like transport as well. Models for conduction in amorphous oxide films have evolved over time to include new concepts derived from experimental results. The earliest ion-conduction models for anodic films assumed that the current was carried exclusively by metal ions, via interstitial or vacancy-type defects.6,10,11 This assumption was shown to be erroneous by the later marker studies of amorphous anodic films. These models take the rate-limiting step in conduction to be a jump of an ion into an interstitial site or vacancy, either within the oxide or at the interface. They yield an exponential current-field relation in which the field coefficient is related to the jump distance. While the form of the conduction law agrees with experiment, the calculated jump distance is found to be significantly larger than the average oxygen-oxygen distance.12 More recently, detailed defect-conduction models have been developed for crystalline oxide films, for example, those on passive metals such as iron. These models include features such as interface reactions with the metal and the solution, defect equilibria in the oxide, and space-charge effects, which are also relevant for amorphous anodic films.13,14 One of them, Macdonald’s point-defect model, has also been applied to conduction in amorphous films.13 However, it does not yield a high-field conduction law, and only oxygen ions contribute to film growth. Fromhold pointed out that the comparable magnitude of metal and oxygen transport numbers is an important distinguishing feature for conduction in amorphous vs. crystalline films.15 He argued that if transport occurred through independent metal and oxygen defects, the variability of the activation energies of motion of these defects would dictate that the transport of only one type of defect should predominate at a given temperature. In this case, the measured metal transport number would be either zero or one. For this reason, the comparable metal and oxygen transport rates in amorphous oxides suggests a cooperative mechanism for ion migration, as originally proposed by Pringle.16 A number of authors have discussed transport in these films in terms of such cooperative mechanisms.2,15,17-19 Cooperative transport is thought to arise from the amorphous structure of the anodic films, since in crystalline films, metal and oxygen transport numbers are not of comparable magnitude. As pointed out by Despic and Parkhutik,20 while it seems important to incorporate the features of the amorphous oxide structure in quantitative transport models, few efforts have done so.21,22 The cooperative transport processes cited previously were described, for the most part, in qualitative terms. Pringle originally proposed that both metal and oxygen ions participate in the elementary transport event, during which the oxide structure locally loses its rigidity; details of this event were not specified, nor were transport numbers calculated.16 Fromhold suggested that the cooperative transport event involves place exchange of metal and oxygen ions,15 and calculated the transport number directly from the stoichiometry of the place exchange event, assuming that only these events contribute to charge transport. The calculated transport numbers agreed with experimental values only when several ions were involved in the place exchange. Fromhold’s calculation did not consider, for example, the effect of the transported charged defects on the local electric field in the film, and he did not offer a rate expression for charge transport. Mott proposed that local excitations may be possible in amorphous oxide films involving on the order of 10 ions, in which, for a time on the order of 1 ps, the ions vibrate with a liquid-like amplitude about their positions.18 During this brief time, liquid-type transport is possible within this “liquid-like cluster,” causing both metal or oxygen ions to migrate Metal and Oxygen Ion Transport during Ionic Conduction in Amorphous Anodic Oxide Films Mei-Hui Wanga and Kurt R. Hebert* Department of Chemical Engineering, Iowa State University, Ames, Iowa 50011, USA A mathematical model was developed for ionic conduction in amorphous oxide films. The model is based on a hypothesized “defect cluster” mechanism in which both metal and oxygen ions are involved in transport. Defect clusters are created by inward displacement of oxygen ions around an oxygen vacancy-like defect in response to the vacancy’s electric field. Metal ions are assumed to migrate easily in the gap between the first and second layer of oxygen ions around the vacancy. The model includes the polarization of the conductive gap in the applied electric field, the exchange of mobile metal ions in the cluster with stationary metal ions in the surrounding oxide, and space charge generated in the film by clusters and oxide nonstoichiometry. The rate-limiting step is the jump of the oxygen vacancy in the cluster. It was found that polarization of the cluster leads to a stoichiometric excess of metal ions in the cluster and that this excess produces a net transport of metal ions due to the motion of the cluster. The metal ion transport number was found to increase with electric field and to depend on the dielectric constant and cluster size. The field dependence follows that found experimentally. The calculated transport numbers are in quantitative agreement with experimental values for tantalum, niobium, and tungsten oxide but smaller than experimental values for aluminum oxide. The field coefficient in the high-field-conduction-rate expression is also predicted and agrees with experimental values to within 10%. © 1999 The Electrochemical Society. S0013-4651(98)09-078-8. All rights reserved. Manuscript submitted September 24, 1998; revised manuscript received May 7, 1999. * Electrochemical Society Active Member. a Present address: Industrial Technology Research Institute, Hsinchu, Taiwan. ecsdl.org/site/terms_use address. Redistribution subject to ECS license or copyright; see 129.186.176.91 Downloaded on 2014-02-10 to IP 3742 Journal of The Electrochemical Society, 146 (10) 3741-3749 (1999) S0013-4651(98)09-078-8 CCC: $7.00 © The Electrochemical Society, Inc. in the field with roughly equal probability. Other investigators have mentioned the place-exchange or liquid-like cluster mechanisms in discussing their experimental results,5,19 but these ideas have not yet been developed further as quantitative transport models. In this paper, a cooperative ion conduction mechanism for an amorphous oxide film is described and then developed as a mathematical transport model for the film. The model is used to predict mass transport of metal and oxygen ions during steady-state anodic growth of amorphous oxide films on aluminum, tantalum, niobium, and tungsten. These predictions are compared with experimental measurements, with particular regard to the transport number, its dependence on electric field, and the field coefficient B. The model views the elementary process of conduction as the jump of an oxygen ion into a vacancy-like site, as suggested by the tracer experiments. However, metal ions in the amorphous oxide near the vacancy are transported along with the vacancy. The calculations show that this hypothesis results in quantitatively reasonable predictions for conduction phenomena. The model is limited to steady-state oxide growth and does not consider transients produced by current or potential modulation.23,24 Additionally, the influence of mechanical strain energy in the film on conduction is not included in the model.25