Theory of trap-controlled transient photoconduction

Abstract

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 (${\ensuremath{\tau}}_{i} \mathrm{and} {\ensuremath{\tau}}_{r,i}$, 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 ${\ensuremath{\tau}}_{i}^{\ensuremath{-}1}={\ensuremath{\tau}}_{r,i}^{\ensuremath{-}\ensuremath{\alpha}}$ with the ${\ensuremath{\tau}}_{r,i}$ 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 $\ensuremath{\psi}(t)\ensuremath{\sim}{t}^{\ensuremath{-}(1+\ensuremath{\alpha})}$. A general equivalence between trapping and continuous-time random walk (CTRW) is further established which shows that $\ensuremath{\psi}(t)$ 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 $a\ensuremath{-}{\mathrm{As}}_{2}{\mathrm{Se}}_{3}$. A trap density satisfying ${N}_{i}\ensuremath{\propto}{\ensuremath{\nu}}_{i}^{\ensuremath{\alpha}\ensuremath{-}1}$ for ${10}^{11}<{\ensuremath{\nu}}_{i}<{10}^{14}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ is obtained, where ${\ensuremath{\nu}}_{i}$ is the coefficient of ${\ensuremath{\tau}}_{r,i}={\ensuremath{\nu}}_{i}^{\ensuremath{-}1}\mathrm{exp}(\frac{{\ensuremath{\epsilon}}_{i}}{\mathrm{kT}})$ and ${\ensuremath{\epsilon}}_{i}=0.65$ 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 ${\ensuremath{\nu}}_{i}$ around 7 \ifmmode\times\else\texttimes\fi{} ${10}^{11}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ is found to satisfy $(\frac{{N}_{i}}{{N}_{\ensuremath{\nu}}})=4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}{\ensuremath{\mu}}_{0}$ (${\mathrm{cm}}^{2}$/V sec), where ${N}_{i}$ and ${N}_{\ensuremath{\nu}}$ are concentrations of traps and transport states, respectively, and ${\ensuremath{\mu}}_{0}$ is the prefactor in $\ensuremath{\mu}={\ensuremath{\mu}}_{0}\mathrm{exp}(\ensuremath{-}\frac{{\ensuremath{\Delta}}_{\ensuremath{\mu}}}{\mathrm{kT}})$. Both hopping and extendedstate motion are compatible with these results, but the most plausible tentative view is hopping with ${N}_{i}\ensuremath{\sim}{10}^{13}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ and ${\ensuremath{\mu}}_{0}=0.2 \mathrm{to} 0.002$ ${\mathrm{cm}}^{2}$/V sec. Additional photo- and dark-conduction data could significantly reduce the range of plausible values.