Unambiguous Simulation of Diffusive Charge Transport in Disordered Nanoribbons

TL;DR

The paper addresses the challenge of numerically observing the diffusive transport regime in disordered 2D systems, which requires fully coherent transport in large samples where the mean-free path $\ell$ and localization length $\xi$ are well separated. It introduces a linear-scaling time-resolved framework that combines bandwidth compression in finite leads, Chebyshev time evolution, and stochastic trace evaluation to compute dc transport in two-terminal disordered 2D nanoribbons, including the leads. The authors demonstrate ballistic, diffusive, and localized transport, with a clear diffusive plateau where the conductance obeys $G=\sigma S/L$ and $g=GL/S$ becomes constant, and they validate their results by comparing with Landauer and Kubo-Greenwood formalisms, showing good agreement in the diffusive regime. This approach provides a scalable, physically transparent tool to study 2D diffusive transport and connects mesoscopic and bulk transport pictures, with potential extensions to nanowires and other transport observables.

Abstract

Charge transport in disordered two-dimensional (2D) systems showcases a myriad of unique phenomenologies that highlight different aspects of the underlying quantum dynamics. Electrons in such systems undergo a crossover from ballistic propagation to Anderson localization, contingent on the system's effective coherence length. Between the extended and localized phases lies a diffusive crossover in which the charge conductivity is properly defined. The numerical observation of this regime has remained elusive because it requires fully coherent transport to be simulated in systems whose dimensions are sufficiently large to meaningfully split the mean-free path and localization length scales. To address this challenge, we employed a novel linear scaling time-resolved approach that enabled us to derive the dc-transport characteristics and observe the three expected 2D transport regimes - ballistic, diffusive, and localized.

Figures (9)

• Figure 1: a) Geometry used in our calculations: two-dimensional tight-binding lattice divided into disordered sample and left and right leads, with applied potential $V\left(x\right)$. The sample's longitudinal length is noted by L, whereas its width is described by S. Both finite sized contacts possess a lateral length of $L_{l}$ sites. b) Longitudinal cross-section of the spatial profile of the applied potential.
• Figure 2: Schematics of the operations required to compute the first term of a local current's time-derivative.
• Figure 3: In$\,\left(a\right)$ we represent the hopping term for two different 1D tight-binding chains. Whereas the blue plot corresponds to an unchanged system, on the red curve we are altering the value of the hopping term at half the sites of the chain. In$\,\left(b\right)$ we compute the density of states (DoS) for these two models. On one hand, we retain the DoS for the unchanged system (with half of the total number of states), while on the other hand we have the density of states of a system whose hopping term is $t_{asy}$. This result shows that we are capable of increasing the DoS on a region within the energy spectrum that is solely controlled by $t_{asy}$.
• Figure 4: Schematics of the bandwidth compression introduced by the hopping's modulation within a one dimensional system. The density of states increases in the neighbourhood of the Fermi energy, and the separation between the its maxima is approximately $4t_{asy}$, where $t_{asy}$ corresponds to the chosen asymptotic value of the hopping term.
• Figure 5: Representation of the current time-evolution for the 1D limiting case. The simulations were performed with $\varepsilon_{F}\,=\,-0.2w,L\,=\,128,\tau\,=\,100,\beta\,=\,1024\,w^{-1}$ and $W\,=\,0.3w$ for $\left(a\right)\,L_{l}\,=\,4096$ and $\left(b\right)\,L_{l}\,=\,512$.It is shown that a decrease in the asymptotic value of the hopping enables the extension of the current's plateau. Whenever the chosen value of $t_{asy}$ is too small, we are no longer able to ensure that all the states covering the transmission band contribute to the maintenance of the quasi-steady state. Consequently, in$\,\left(b\right)$ an earlier reflection is observed when $t_{asy}\,=\,0$. The normalization, $I_{Land}$, was computed using the RGF method.