Interfacial space charge and capacitance in ionic crystals: Intrinsic conductors

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Abstract
The conventional theory of surface charge and distributed space charge in single crystals exhibiting Frenkel or Schottky disorder is generalized in several ways. First we generalize the usual continuum theory of the diffuse double layer in the crystal by adopting a lattice gas model which restricts the mobile charges to a fixed number of lattice sites. The lattice gas activities can be used in both zero and nonzero current situations and are shown to be consistent with a three‐dimensional generalization of ordinary Langmuir adsorption. The substantial effects of the resulting limit on concentrations in accumulation regions are examined and shown to be particularly important in high bias situations and in materials in which the bulk equilibrium charge densities are very high. Second, we take explicitly into account the physical separation between the plane of the surface charges, which balance the bulk space charge, and the first normal lattice plane of the crystal and show that under most conditions of interest this separation modifies the earlier surface potential results of Poeppel and Blakely substantially. We also show that the thermodynamically generated equilibrium relation between surface charge density and surface potential is itself just a form of the Langmuir adsorption isotherm applied to occupancy of the kink sites. Finally, we investigate the response of the system when a completely blocking electrode is attached, with the physical separation between the equipotential plane of the electrode and the plane of the electrical centers of the ’’adsorbed’’ surface charges explicitly introduced. Free energy minimization in this situation leads to Langmuir adsorption at the surface. A new equivalent circuit representing the total differential capacitance of the system is derived, and competing effects of adsorption capacitance, diffuse double layer capacitance, and separation capacitances are investigated. An important result is that we find the surface adsorption capacitance to be essentially in parallel rather than in series with the diffuse double layer capacitance. Numerous analytic and numerical results of capacitance versus temperature and applied bias are presented for different limiting surface site concentrations and it is found that the surface potential plays somewhat the role of a diffusion potential, causing the minimum of the diffuse layer capacitance (occurring at the ’’flat‐band potential’’) and the maximum of the adsorption capacitance to be displaced from one another in potential. Some of the results of the present work also may be relevant to the theory of adsorption capacitance in unsupported aqueous electrolytes.