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J.-S. Kim, J. Fleig, J. Maier:
"Space charge conduction: Simple analytical solutions for ionic and mixed conductors and application to nanocrystalline ceria";
Physical Chemistry Chemical Physics, 5 (2003), 2268 - 2273.

In this paper we present a set of simple analytical solutions for the parallel and the serial contributions of space charge zones to the overall conduction in bicrystals and (brick layered) ceramics for a variety of situations. The different situations are characterized by signs of the space charge potential, by different charge numbers of the carriers and by the presence or absence of acceptor or donor dopants. We can give solutions for arbitrary charge numbers as long as we restrict to sufficiently high space charge potentials and can presuppose equilibrium for the mobile carriers. A recent analysis of impedance measurements of nanocrystalline ceria based on a space charge model is briefly mentioned as a specific example, as it reflects the occurrence of highly n-conducting accumulation layers in addition to strongly blocking oxygen vacancy depletion layers.

http://dx.doi.org/10.1039/B300170A