An analytic solution of the "conventional" multiple-trapping problem is obtained for a small quantity of charge moving through a spatially varying, but time-independent, electric field and an arbitrary distribution of traps. The solution is shown to apply to cases of microscopic hopping as well as free-translation through extended states. The solution for a discrete set of traps, simply characterized by their mean times for capture and release (${\tau }_i \mathrm{and} {\tau }_{r,i}$, respectively) appears in the form of convolutions of modified Bessel functions of order unity, but for a uniform electric field and a continuum of traps satisfying the relation ${\tau }_i^{-1\mathrm{}}={\tau }_{r,i}^{-\alpha }$ with the ${\tau }_{r,i}$ uniformly spaced on a logarithmic scale, the solution reduces to a simple algebraic form which is identical to the one obtained by Scher and Montroll for their power-law waiting-time distribution function $\psi \mathrm{}\left(t\right)\mathrm{}\sim t^{-(1+\alpha )}$. A general equivalence between trapping and continuous-time random walk (CTRW) is further established which shows that $\psi \mathrm{}\left(t\right)\mathrm{}$ can always be constructed from capture and release kinetics, and vice versa. The new trapping solution (and its equivalence to CTRW) is illustrated by reinterpreting transit-time data on $a-{\mathrm{As}}_2{\mathrm{Se}}_3$. A trap density satisfying $N_i\propto {\nu }_i^{\alpha -1\mathrm{}}$ for ${10}^{11}<{\nu }_i<{10}^{14}$ ${\sec }^{-1\mathrm{}}$ is obtained, where ${\nu }_i$ is the coefficient of ${\tau }_{r,i}={\nu }_i^{-1\mathrm{}}\exp \left(\frac{{\epsilon }_i}{\mathrm{kT}}\right)$ and ${\epsilon }_i=0.65\mathrm{}$ eV (for all traps) is the activation energy for release. With the plausible assumption that the microscopic mobility and capture processes are similarly activated (if at all), a trap density for the half-decade interval of ${\nu }_i$ around 7 × ${10}^{11}$ ${\sec }^{-1\mathrm{}}$ is found to satisfy $\left(\frac{N_i}{N_{\nu }}\right)=4\mathrm{}\times {10}^{-6\mathrm{}}{\mu }_0$ (${\mathrm{cm}}^2$/V sec), where $N_i$ and $N_{\nu }$ are concentrations of traps and transport states, respectively, and ${\mu }_0$ is the prefactor in $\mu ={\mu }_0\exp (-\frac{{\Delta }_{\mu }}{\mathrm{kT}})$. Both hopping and extendedstate motion are compatible with these results, but the most plausible tentative view is hopping with $N_i\sim {10}^{13}$ ${\mathrm{cm}}^{-3\mathrm{}}$ and ${\mu }_0=0.2\mathrm{} \mathrm{to} 0.002$ ${\mathrm{cm}}^2$/V sec. Additional photo- and dark-conduction data could significantly reduce the range of plausible values.

• Received 13 January 1977

DOI:https://doi.org/10.1103/PhysRevB.16.2362