Hopping solid state properties

Solid mixed ionic-electronic conductors (MIECs) exhibit both ionic and electronic (electron-hole) conductivity. Naturally, in any material there are in principle nonzero electronic and ionic conductivities (a i, a,). It is customary to limit the use of the term MIEC to those materials in which a, and 0, 1 do not differ by more than two orders of magnitude. It is also customary to use the term MIEC if a, and Ogi are not too low (o, a i 10 S/cm). Obviously, there are no strict rules. There are processes where the minority carriers play an important role despite the fact that 0,70 1 exceeds those limits and a, aj,i< 10 S/cm. In MIECs, ion transport normally occurs via interstitial sites or by hopping into a vacant site or a more complex combination based on interstitial and vacant sites, and electronic (electron/hole) conductivity occurs via delocalized states in the conduction/valence band or via localized states by a thermally assisted hopping mechanism. With respect to their properties, MIECs have found wide applications in solid oxide fuel cells, batteries, smart windows, selective membranes, sensors, catalysis, and so on. 

The Monte-Carlo simulation of these models reproduces well the charge carrier transport properties in the mesophases, i.e., the mobility independence of both temperature and electric field in the temperature above room temperature, if a small sigma (40 60 meV) is taken for the Gaussian width of the distribution of localized states in Eq. (2.2). In this equation p, is the mobility, a is the Gaussian width of the distributed energy states for hopping sites, E is an index of the positional disorder, k is the Boltzman constant, T is the temperature, E is the electric field and C is a constant. The constants a and n depend on the type of mesophase, e.g., 0.8 and 2 for the SmB phase and 0.78 and 1.5 for the SmE phase, respectively. This value of cr (40-60 meV) is half that for typical amorphous solids [61]. 

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