In the late 1990s, Daniel Vanmaekelbergh, Laurie Peter, and others observed that electron diffusion coefficient in dye-sensitized solar cells is density dependent. It was recognized that the electron lifetime is also density dependent, i.e., it depends on the specific position of the steady state Fermi level of electrons. These variations where shown to be a consequence of the multiple trapping model, insofar as a displacement of the Fermi level changes the kinetics of trapping and detrapping. Thus, the dependence of the diffusion coefficient and lifetime does not affect the microscopic parameters of either phenomenon, namely, the jump rate, and the probability of charge transfer of an electron in a trap (J. Bisquert and V. S. Vikhrenko, J. Phys. Chem. B 108, 2313 (2004) http://dx.doi.org/10.1021/jp035395y). However, there as some mechanisms as the Langevin recombination, where the speed of an electron during diffusion does determine recombination rates, since an encounter with a recombination center makes the electron disappear from the structure. It has been for some time a puzzle, what is happening in the case of DSCs, and how both types of regimes could be generally distinguished. It is an involved problem as diffusion and charge transfer are occurring at the same time. Furthermore it is not enough to run some simulations and see the results. It is needed to make a clear cut physical criterion, since we want to have an interpretation of the macroscopic time constants as the lifetime, that are effectively measured.
A clear solution to this problem has been completed by my friends Mehdi Ansari-Rad and Juan Antonio Anta, building on a previous paper that laid general definitions of the measurable quantities from basic stochastic behaviour (M. Ansari-Rad, J. A. Anta and J. Bisquert, J. Phys. Chem. C 117, 16275 (2013). http://dx.doi.org/10.1021/jp403232b). In the new paper they show the structure of the lifetime from basic kinetic parameters, in both a reaction limited and diffusion limited regime. In the first one the lifetime is not dependent on any transport properties, and it occurs when the electrons are rapidly thermalized. This result supports the definition of the electron lifetime that has been applied for many years, as Rrec Cchem, where Rrec is recombination resistance and Cchem is the chemical capacitance.
I think this paper brings a lot of insight to the fundamental electronic processes in nanostructures, and it will be helpful for the interpretation of experiments.
Conditions for diffusion-limited and reaction-limited recombination in nanostructured solar cells
Mehdi Ansari-Rad, Juan A. Anta and Ezatollah Arzi
J. Chem. Phys. 140, 134702 (2014); http://dx.doi.org/10.1063/1.4869748
The performance of Dye-sensitized solar cells (DSC) and related devices made of nanostructured semiconductors relies on a good charge separation, which in turn is achieved by favoring charge transport against recombination. Although both processes occur at very different time scales, hence ensuring good charge separation, in certain cases the kinetics oftransport and recombination can be connected, either in a direct or an indirect way. In this work, the connection between electron transport and recombination in nanostructured solar cells is studied both theoretically and by Monte Carlo simulation. Calculations using the Multiple-Trapping model and a realistic trap distribution for nanostructured TiO2 show that for attempt-to-jump frequencies higher than 1011–1013 Hz, the system adopts a reaction limited (RL) regime, with a lifetime which is effectively independent from the speed of the electrons in thetransport level. For frequencies lower than those, and depending on the concentration of recombination centers in the material, the system enters a diffusion-limited regime (DL), where the lifetime increases if the speed of free electrons decreases. In general, the conditions for RL or DL recombination depend critically on the time scale difference between recombination kinetics and free-electron transport. Hence, if the former is too rapid with respect to the latter, the system is in the DL regime and total thermalization of carriers is not possible. In the opposite situation, a RL regime arises. Numerical data available in the literature, and the behavior of the lifetime with respect to (1) density of recombination centers and (2) probability of recombination at a given center, suggest that a typical DSC in operation stays in the RL regime with complete thermalization, although a transition to the DL regime may occur for electrolytesor hole conductors where recombination is especially rapid or where there is a larger dispersion of energies of electron acceptors.