A Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems

TL;DR

This study numerically investigates conductance fluctuations in one-dimensional disordered systems with long-tailed on-site energy distributions, extending Lloyd's model and revealing unique bimodal conductance distributions and their dependence on disorder parameters.

Contribution

It generalizes Lloyd's model by analyzing conductance in 1D disordered wires with heavy-tailed distributions, revealing new distribution behaviors and their relation to disorder characteristics.

Findings

Ensemble average of -ln G is proportional to wire length for all alpha.

Conductance distribution P(G) is determined by alpha and average -ln G.

Distribution P(ln G) follows a G^beta law with beta ≤ alpha/2.

Abstract

We perform a detailed numerical study of the conductance $G$ through one-dimensional (1D) tight-binding wires with on-site disorder. The random configurations of the on-site energies $\epsilon$ of the tight-binding Hamiltonian are characterized by long-tailed distributions: For large $\epsilon$, $P(\epsilon)\sim 1/\epsilon^{1+\alpha}$ with $\alpha\in(0,2)$. Our model serves as a generalization of 1D Lloyd's model, which corresponds to $\alpha=1$. First, we verify that the ensemble average $\left\langle -\ln G\right\rangle$ is proportional to the length of the wire $L$ for all values of $\alpha$, providing the localization length $\xi$ from $\left\langle-\ln G\right\rangle=2L/\xi$. Then, we show that the probability distribution function $P(G)$ is fully determined by the exponent $\alpha$ and $\left\langle-\ln G\right\rangle$. In contrast to 1D wires with standard white-noise disorder, our wire model exhibits bimodal distributions of the conductance with peaks at $G=0$ and $1$. In addition, we show that $P(\ln G)$ is proportional to $G^\beta$, for $G\to 0$, with $\beta\le\alpha/2$, in agreement to previous studies.