Critical Behavior and Extended States in 2D and 3D Systems with Gas-like Disorder

TL;DR

This study investigates localization and extended states in amorphous 2D and 3D systems with gas-like disorder using finite size scaling and IPR, revealing phase diagrams with mobility edges and critical exponents.

Contribution

It introduces a combined finite size scaling approach to map phase diagrams and identify mobility edges in disordered amorphous conductors with exponential hopping.

Findings

States are localized below a critical length scale in 2D.

Extended states exist for hopping range above a threshold.

Mobility edges are characterized as critical points with calculated exponents.

Abstract

With a tight binding treatment we examine amorphous conductors with gas-like disorder, or no correlations among the site positions. We consider an exponentially decaying hopping integral with range $l$, and the Inverse Participation Ratio (IPR) is used to characterize carrier wave functions with respect to localization. With the aid of two complementary finite size scaling techniques to extrapolate to the bulk limit (both methods exploit critical behavior in different ways to find the boundary between domains of extended and localized wave functions) which nevertheless yield identical results, we obtain phase diagrams showing regions where states are extended and domains of localized states. In the 2D case, states are localized below a threshold length scale $l_{c}$ on the order of the interparticle separation $\rho^{-1/2}$ with a finite fraction of states extended for $l > l_{c}$. For $D = 3$, the extended phase is flanked by regions of localized states and bounded by two mobility edges. The swath of extended states, broad for $l \sim 1$, becomes narrower with decreasing $l$, though persisting with finite width even for $l < (1/5)\rho^{-1/3}$. Mobility edges are interpreted as lines of critical points, and we calculate the corresponding critical exponents.