Observation of electron–hole puddles in graphene using a scanning single-electron transistor

Abstract

The electronic density of states of graphene is equivalent to that of relativistic electrons [1-3]. In the absence of disorder or external doping the Fermi energy lies at the Dirac point where the density of states vanishes. Although transport measurements at high carrier densities indicate rather high mobilities [4-6], many questions pertaining to disorder [7-14] remain unanswered. In particular, it has been argued theoretically, that when the average carrier density is zero, the inescapable presence of disorder will lead to electron and hole puddles with equal probability. In this work, we use a scanning single electron transistor to image the carrier density landscape of graphene in the vicinity of the neutrality point. Our results clearly show the electron-hole puddles expected theoretically [13]. In addition, our measurement technique enables to determine locally the density of states in graphene. In contrast to previously studied massive two dimensional electron systems, the kinetic contribution to the density of states accounts quantitatively for the measured signal. Our results suggests that exchange and correlation effects are either weak or have canceling contributions. The kinetic energy of Dirac particles in graphene increases linearly with momentum [1,15,16]. The total energy per particle however, also referred to as the chemical potential, µ , contains additional exchange and correlations contributions that arise from the Coulomb interactionHere K E , ex E and c E are the kinetic, exchange and correlation terms respectively. Therefore, the chemical potential and its derivative with respect to density, known as the inverse compressibility or density of states, provide direct insight into the properties of the Coulomb interaction in such a system. Compressibility measurements of conventional, massive, two dimensional electron systems made for example of Si or GaAs have been carried out by several groups [19,[23][24][25][26]. It has been shown that at zero magnetic field the chemical potential may be described rather accurately within the Hartree-Fock approximation. Quantitatively it has been found that Coulomb interactions add a substantial contribution to the compressibility and become dominant at low carrier densities. In a perfectly uniform and clean graphene sample, the inverse compressibility is expected to diverge at the Dirac point in view of the vanishing density of states. This divergence, however, is expected to be rounded off by disorder on length scales smaller than our tip size. Long range disorder on the other hand will cause local shifts in the Dirac point indicative of a non-zero local density. In this work we measure the spatial dependence of the local compressibility versus carrier density across the sample. Fluctuations in the Dirac point across the sample are translated into

Hybrid Surface-Phonon-Plasmon Polariton Modes in Graphene/Monolayer h-BN Heterostructures

“…1,7 The background doping observed in our samples has been shown previously to be mainly caused by the FeCl etchant that is used to remove the copper foil from the as-grown CVD graphene, 7 although atmospheric impurities, and charge traps in the substrate can also play a role. 8,9 Interband transition measurements and E F fittings performed on bare graphene areas showed a similar gate vs. carrier density dependence to the patterned graphene areas, as was also observed in previous works. 7 We note that the carrier density we extract by monitoring the interband transitions is lower that what would be obtained from a simple parallel plate capacitance calculation for our device.…”

Structure of graphene and its disorders: a review

“…The charged impurities adsorbed on graphene or located at the interface between graphene and substrate induce Coulomb scattering [259], and are responsible for the electron-hole puddles at the neutrality point [260,261]. Both the resonant scattering [262] and Coulomb scattering significantly influence the drift mobility and electron mean free path, and therefore change the electrical properties of graphene [263].…”

Approaching ballistic transport in suspended graphene

“…For example, the carrier mobility in graphene deposited on a substrate such as Si/SiO 2 deteriorates due to trapped charges in the oxide or to contaminants that get trapped at the graphene-substrate interface during fabrication. The substrate-induced charge inhomogeneity is particularly deleterious near the DP where screening is weak, 14,15 leading to reduced carrier mobility there. In addition, the atomic roughness of the substrate introduces short range scattering centers and may contribute to quench-condensation of ripples within the graphene layer 16 .…”

“…The charge inhomogeneity introduces electron-hole puddles 14 close to the DP which cause spatial fluctuations in doping levels. These fluctuations define an energy bandwidth, , for the average deviation of local DP from the Fermi energy, .…”