Interpretation of electron diffusion coefficient in organic and inorganic semiconductors with broad distributions of states

The carrier transport properties in nanocrystalline semiconductors and organic materials play a key role for modern organic/inorganic devices such as dye-sensitized (DSC) and organic solar cells, organic and hybrid light-emitting diodes (OLEDs), organic field-effect transistors, and electrochemical sensors and displays. Carrier transport in these materials usually occurs by transitions in a broad distribution of localized states. As a result the transport is dominated by thermal activation to a band of extended states (multiple trapping), or if these do not exist, by hopping via localized states. We provide a general view of the physical interpretation of the variations of carrier transport coefficients (diffusion coefficient and mobility) with respect to the carrier concentration, or Fermi level, examining in detail models for carrier transport in nanocrystalline semiconductors and organic materials with the following distributions: single and two-level systems, exponential and Gaussian density of states. We treat both the multiple trapping models and the hopping model in the transport energy approximation. The analysis is simplified by thermodynamic properties: the chemical capacitance, C-mu, and the thermodynamic factor, chi(n), that allow us to derive many properties of the chemical diffusion coefficient, D-n, used in Fick's law. The formulation of the generalized Einstein relation for the mobility to diffusion ratio shows that the carrier mobility is proportional to the jump diffusion coefficient, D-J, that is derived from single particle random walk. Characteristic experimental data for nanocrystalline TiO2 in DSC and electrochemically doped conducting polymers are discussed in the light of these models.