Direct and inverse problems in dispersive time-of-flight photocurrent revisited

Using the fact that the continuous time random walk (CTRW) scheme is a random process subordinated to a simple random walk under the operational time given by the number of steps taken by the walker up to a given time, we revisit the problem of strongly dispersive transport in disordered media, which first lead Scher and Montroll to introducing the power law waiting time distributions. Using a subordination approach permits to disentangle the complexity of the problem, separating the solution of the boundary value problem (which is solved on the level of normal diffusive transport) from the influence of the waiting times, which allows for the solution of the direct problem in the whole time domain (including short times, out of reach of the initial approach), and simplifying strongly the analysis of the inverse problem. This analysis shows that the current traces do not contain information sufficient for unique restoration of the waiting time probability densities, but define a single-parametric family of functions that can be restored, all leading to the same photocurrent forms. The members of the family have the power-law tails which differ only by a prefactor, but may look astonishingly different at their body. The same applies to the multiple trapping model, mathematically equivalent to a special limiting case of CTRW.