 
Summary: Statistics of random voltage fluctuations and the lowdensity residual conductivity of graphene
Victor M. Galitski, Shaffique Adam, and S. Das Sarma
Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 207424111, USA
Received 9 February 2007; revised manuscript received 12 August 2007; published 7 December 2007
We consider a graphene sheet in the vicinity of a substrate, which contains charged impurities. A general
analytic theory to describe the statistical properties of voltage fluctuations due to the longrange disorder is
developed. In particular, we derive a general expression for the probability distribution function of voltage
fluctuations, which is shown to be nonGaussian. The voltage fluctuations lead to the appearance of randomly
distributed density inhomogeneities in the graphene plane. We argue that these disorderinduced density fluc
tuations produce a finite conductivity even at a zero gate voltage in accordance with recent experimental
observations. We determine the width of the minimal conductivity plateau and the typical size of the electron
and hole puddles. We also propose a simple selfconsistent approach to estimate the residual density and the
nonuniversal minimal conductivity in the lowdensity regime. The existence of inhomogeneous random
puddles of electrons and holes should be a generic feature of all graphene layers at low gate voltages due to the
invariable presence of charged impurities in the substrate.
DOI: 10.1103/PhysRevB.76.245405 PACS number s : 73.40. c, 72.10. d, 81.05.Uw
I. INTRODUCTION
Charge inhomogeneities are known to play an important
role in understanding transport properties of semiconductors.
The randomly positioned impurity ions give rise to a random
