Estimate of time-scale for the current relaxation of percolative Random Resistor cum Tunneling Network model

TL;DR

This paper analyzes the relaxation dynamics of the Random Resistor cum Tunneling Network model, identifying a single dominant time-scale that governs the current relaxation process towards steady-state.

Contribution

It provides an analytical understanding of the relaxation time-scales in RRTN, revealing that only one independent time-scale controls the entire relaxation process.

Findings

Identification of two phenomenological time-scales, $ au_t$ and $ au_s$.

Demonstration that only one of these time-scales is independent.

Mapping the relaxation problem to a Gauss-Seidel type convergence analysis.

Abstract

The Random Resistor cum Tunneling Network (RRTN) model was proposed from our group by considering an extra phenomenological (semi-classical) tunneling process into a classical RRN bond percolation model. We earlier reported about early-stage two inverse power-laws, followed by large time purely exponential tail in some of the RRTN macroscopic current relaxations. In this paper, we investigate on the broader perspective of current relaxation. We present here an analytical argument behind the strong convergence (irrespective of initial voltage configuration) of the bulk current towards its steady-state, mapping the problem into a special kind of Gauss-Seidel method. We find two phenomenological time-scales (referred as $\tau_t$ and $\tau_s$), those emerge from the variation of macroscopic quantities during current dynamics. We show that not both, only one of them is independent. Thus there exists a {\it single} scale in time which controls the entire dynamics.