Impurity State and Variable Range Hopping Conduction in Graphene

TL;DR

This paper investigates impurity states in graphene, revealing power-law decay of wave functions and revising variable range hopping theory to align with experimental conductivity data.

Contribution

It introduces a new understanding of impurity state localization in graphene and modifies the variable range hopping theory accordingly.

Findings

Impurity wave functions decay as a power law in graphene.

Conductivity follows a power-law temperature dependence.

Results agree with experimental observations.

Abstract

The variable range hopping theory, as formulated for exponentially localized impurity states, does not necessarily apply in the case of graphene with covalently attached impurities. We analyze the localization of impurity states in graphene using the nearest-neighbor, tight-binding model of an adatom-graphene system with Green's function perturbation methods. The amplitude of the impurity state wave function is determined to decay as a power law with exponents depending on sublattice, direction, and the impurity species. We revisit the variable range hopping theory in view of this result and find that the conductivity depends as a power law of the temperature with an exponent related to the localization of the wave function. We show that this temperature dependence is in agreement with available experimental results.