Electronic States in One, Two, and Three Dimensional Highly Amorphous Materials: A Tight Binding Treatment

TL;DR

This study investigates electronic states in highly disordered amorphous materials across one, two, and three dimensions using a tight binding model, revealing localization and extension behaviors dependent on dimensionality and decay range.

Contribution

It provides a comprehensive analysis of localization phenomena in amorphous systems with exponential decay tunneling, including phase diagrams and scaling behaviors across dimensions.

Findings

All states are localized in 1D.

In 2D, a threshold decay length determines extended states.

In 3D, mobility edges separate localized and extended states.

Abstract

In a tight binding framework, we analyze the characteristics of electronic states in strongly disordered materials (hopping sites are placed randomly with no local order) with tunneling matrix elements decaying exponentially in the atomic separation with various decay ranges l examined. We calculate the density of states (DOS) and the Inverse Participation Ratio (IPR) for amorphous atomic configurations in one, two, and three dimensions. With a finite size scaling analysis of the IPR statistical distributions, it is shown that states are either extended or localized for a particular energy, and phase portraits for wave functions are obtained showing extended and localized behavior in the thermodynamic limit. While we conclude that all states are localized in 1D, in the 2D case there is a threshold for l above which some eigenstates appear to be extended and below which wave functions are entirely localized. For 3D geometries, there are two mobility boundaries flanking an intermediate range of energies where states are extended with eigenstates localized for energies above or below this range. While a zone of extended states persists even for very short l, the width of the region tends to zero exponentially (i.e. scaling as exp{-A/l}) for very small decay length scales.