Effect of impurities on one-dimensional hopping conductivity: Saturation of current

Hopping conductivity is considered in a one-dimensional (1D) system with a finite density of impurities which present a finite barrier to hopping. The nonlinear hopping equations, which exclude jumps to occupied sites, are solved for steady-current conditions in an applied electric field $E$. The current saturates at a value independent of $E$ and of $c$ for $c\lesssim \frac{\mathrm{eaE}}{k_{B}T}\ll 1$ where $c$ is the impurity concentration and $a$ the lattice spacing.