Nonlinear hopping transport in ring systems and open channels

Abstract

We study the nonlinear hopping transport in one-dimensional rings and open channels. Analytical results are derived for the stationary current response to a constant bias without assuming any specific coupling of the rates to the external fields. It is shown that anomalous large effective jump lengths, as observed in recent experiments by taking the ratio of the third-order nonlinear and the linear conductivity, can occur already in ordered systems. Rectification effects due to site energy disorder in ring systems are expected to become irrelevant for large system sizes. In open channels, in contrast, rectification effects occur already for disorder in the jump barriers and do not vanish in the thermodynamic limit. Numerical solutions for a sinusoidal bias show that the ring system provides a good description for the transport behavior in the open channel for intermediate and high frequencies. For low frequencies temporal variations in the mean particle number have to be taken into account in the open channel, which cannot be captured in the more simple ring model.

Theoretical Description of Ion Conduction in Disordered Systems: From Linear to Nonlinear Response

“…For a typical experimental value of w ≈ 0.05 the Taylor expansion thus breaks down if N significantly exceeds the very small scale of 10 2 sites. As explicitly derived in [55] (see also [52] for similar work) one obtains Interestingly, this result factorizes into a term, reflecting the homogeneous system without any disorder (sinh(w)) and one term, reflecting the disorder. Without disorder one has b 0 = b 1 = 1.…”

“…Here we restrict ourselves to the special case of a point-symmetric energy landscape, corresponding to a strictly antisymmetric dependence of the current on the electric field (see Ref. [52] for results without intrinsic pointsymmetry). One obtains for large N (in practice N > 40)…”

“…Stated differently, the pre-averaging as defined above is equivalent to the disorder average. We note in passing that this is not true for open boundary conditions as discussed in [52].…”

“…Of course, in general this does not need to be the case and has been studied in the context of rectification effects, see [52] and references therein. A very simple model to rationalize the emergence of nonlinear effects is a hopping model with identical sites.…”

“…A very simple model to rationalize the emergence of nonlinear effects is a hopping model with identical sites. In this case the current can be written as [52][53][54] …”