DC electric field effect on the anomalous exponent of the hopping conduction in the one-dimensional disorder model

TL;DR

This paper investigates how a DC electric field influences the anomalous exponent in hopping conduction within a one-dimensional disorder model, revealing field-dependent behaviors and transitions verified by simulations.

Contribution

It provides analytical expressions for the anomalous exponent under various field and disorder conditions, extending understanding of hopping conduction in disordered systems.

Findings

Weak field: exponent matches diffusive systems and supports existing theories.

Strong field and disorder: exponent aligns with the Multiple Trapping Model.

Strong field and weak disorder: diffusion becomes normal with exponent 1.

Abstract

DC electric field effect on the anomalous exponent of the hopping conduction in the disorder model is investigated. First, we explain the model and derive an analytical expression of the effective waiting time for the general case. We show that the exponent depends on the external field. Then we focus on a one-dimensional system in order to illustrate the features of the anomalous exponent. We derive approximate expressions of the anomalous exponent of the system analytically. For the case of a weak field, the anomalous exponent is consistent with that of diffusive systems. This is consistent with the treatments of Barkai et al. [Phys. Rev. E {\bf 63}, 046118 (2001)] and our result supports their theory. On the other hand, for the case of a strong field and a strong disorder, the time evolution of the exponent clearly differs from that in the weak field. The exponent is consistent with the well-known expression of the anomalous exponent in the Multiple Trapping Model at mesoscopic time scales. In the long time limit, a transition of the anomalous exponent to the same value of the weak field occurs. For the case of a strong field and a weak disorder, the exponent is equal to 1 and thus the diffusion is normal. These findings are verified by the Monte Carlo simulation.