A model system of nanoporous TiO 2 sensitized with dye molecules at the outer surface is investigated by time resolved surface photovoltage ͑SPV͒ at temperatures between −120°C and 270°C to get information about electron diffusion over more than 10 orders of magnitude in time. The SPV transients increase in time due to independent electron diffusion and reach a maximum at a certain peak time due to reaching the screening length. The increasing parts of the SPV transients are characterized by a power law while the SPV power coefficient amounts to half of the dispersion parameter of anomalous diffusion. Anomalous diffusion is observed for times down to the duration time of the laser pulse ͑150 ps͒. With increasing temperature, the SPV power coefficient increases to its saturation value of 0.5 corresponding to normal diffusion. At lower temperatures, the SPV power coefficients decrease with increasing intensity of the exciting laser pulses. The decay of the SPV transients is determined by thermally activated normal diffusion. The minimal charge transfer time of an electron back to the positively charged dye molecule amounts to 2 ps which is obtained from thermally activated logarithmic decays. DOI: 10.1103/PhysRevB.73.045407 PACS number͑s͒: 72.40.ϩw, 73.63.Bd
INTRODUCTIONSpatial charge separation as well as electron diffusion play a crucial role in many biological, chemical, and physical systems. Charge separation takes place at very different time and length scales depending on the investigated system. For example, charge separation may proceed within tens of fs during ultrafast injection from dye molecules into TiO 2 ͑Ref. 1͒ or over long times during diffusion in a porous semiconductor.2 In devices such as nanostructured solar cells, 3,4 it is important to clarify the time and length scales at which the related processes of initial charge separation, electron diffusion, and recombination take place, especially considering the strong energy disorder that affects both the diffusion and recombination of electrons 5 in these structures. It is recognized that the dynamics of electrons photoinjected from dye molecules into nanoscale semiconductor networks is composed of a hierarchy of processes spanning many orders of magnitude in time. Normal and anomalous diffusion belong to these processes.Techniques operating in the low frequency range ͑10 mHz-1000 Hz͒ such as impedance spectroscopy or itensity-modulated photocurrent spectroscopy 6 provide macroscopic parameters that are time and spatial averaged with respect to the microscopic processes. The latter are resolved separately in fast optical pump-probe experiments ͑10 ns-10 ms͒, for example, monitoring the decay of the photoinduced dye cation excited by a laser pulse.5 However, direct information about the spatial separation of carriers cannot be obtained from an optical measurement itself. In contrast, the surface photovoltage ͑SPV͒ technique 7 is a locally sensitive technique and it can be used to study the spatial charge separation induced by diffusion even over ...