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

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 $⟨-\ln G⟩$ is proportional to the length of the wire $L$ for all values of $\alpha$, providing the localization length $\xi$ from $⟨-\ln G⟩=2L/\xi$. Then, we show that the probability distribution function $P(G)$ is fully determined by the exponent $\alpha$ and $⟨-\ln G⟩$. 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.

5 pages, 5 figures