Session J29: Focus Session: Carbon Nanotubes and Related Materials VII: Electronic Properties

Biased bilayer graphene: Hall effect and zero-energy Edge States

We demonstrate that the electronic gap of a graphene bilayer can be controlled externally by applying a gate bias. From the magneto-transport data (Shubnikov-de Haas measurements of the cyclotron mass), and using a tight binding model, we extract the value of the gap as a function of the electronic density. We show that the gap can be changed from zero to mid-infrared energies by using fields of $\alt 1$ V/nm, below the electric breakdown of SiO$_2$. The opening of a gap is clearly seen in the quantum Hall regime. We further report the existence of zero energy surface states localized at zigzag edges of bilayer graphene. It is shown that zero energy edge states in bilayer graphene can be divided into two families: (i) states living only on a single plane, equivalent to surface states in monolayer graphene; (ii) states with finite amplitude over the two layers, with an enhanced penetration into the bulk.   

Cyclotron resonance in bilayer graphene

The hyperbolic dispersion of bilayer graphene leads to a Landau level (LL) spectrum that is linear in the magnetic field, B, at low energies but shifts to a $\sqrt{\textrm{B}}$ dependence with increasing energy. Here we present the first infrared transmission measurements of the unique B-field dependence of LL transitions in bilayer graphene, in a gated $400 \mu$m$^2$ sample in fields up to $\textrm{B}=18$ T. Eight intraband transitions are observed among LL indices $|n| \leq 4$, including the unusual zero-energy $n = 0$ level, and are found to follow a selection rule of $\Delta n = +1$. We find the change in field dependence is plainly visible between the behavior of the transition energies for $n = -1 \to 0$ and $n = 0 \to +1$, which are close to linear in B, as compared with all other transitions which display a clear $\sqrt{\textrm{B}}$ behavior. However, the shift in field dependence occurs at energies well below where it is expected based on nearest-neighbor tight-binding calculations, and a single set of fitting parameters within this theory fails to describe our results.   

Graphene as an electronic membrane

Experiments are finally revealing intricate facts about graphene which go beyond the ideal picture of relativistic Dirac fermions in pristine two dimensional (2D) space, two years after its first isolation. While observations of rippling [1, 2, 3] added another dimension to the richness of the physics of graphene, scanning single electron transistor images displayed prevalent charge inhomogeneity [4]. The importance of understanding these non-ideal aspects cannot be overstated both from the fundamental research interest since graphene is a unique arena for their interplay, and from the device applications interest since the quality control is a key to applications. We investigate the membrane aspect of graphene and its impact on the electronic properties. We show that curvature generates spatially varying electrochemical potential. Further we show that the charge inhomogeneity in turn stabilizes ripple formation.[5]\newline [1] Meyer, J.C., et al., Nature 446, 60 (2007).\newline [2] Stolyarova E. et al., PNAS, 104, 9209 (2007).\newline [3] Ishigami, M. et al., Nano Letters 7, 1643 (2007).\newline [4] Martin, J. et al., unpublished, cond-mat/0705.2180 (2007).\newline [5] E.-A, Kim and A. Castro Neto, cond-mat/0702562   

Demonstration of one-parameter scaling at the Dirac point in graphene

We numerically calculate the conductivity $\sigma$ of an undoped graphene sheet (size $L$) in the limit of vanishingly small lattice constant. We demonstrate one-parameter scaling for random impurity scattering and determine the scaling function $\beta(\sigma)=d\ln\sigma/d\ln L$. Contrary to a recent prediction, the scaling flow has no fixed point ($\beta>0$) for conductivities up to and beyond the symplectic metal-insulator transition. Instead, the data supports an alternative scaling flow for which the conductivity at the Dirac point increases logarithmically with sample size in the absence of intervalley scattering --- without reaching a scale-invariant limit.   

Electron localization in gapped bilayer graphenes with disorder

The bilayer graphene is a zero-gap semiconductor, and it is known that the width of energy gap can be controlled by the electric field perpendicular to the layer [1,2]. Even in zero magnetic field, the electronic states carry the non-zero Hall conductivity in presence of the gap, while the Hall currents cancel out in summation over two valleys (K and K' points). Here we numerically calculate the electronic states in gapped bilayer graphenes with the smooth disorder potential, and estimate the localization length as a function of the gap width $\Delta$. We find that the conductivity at zero Fermi energy does not simply goes down as $\Delta$ increases, but has a maximum at a certain finite $\Delta$, and the localization length diverges there. We show that this can be interpreted as a ``Hall plateau transition'' in each decoupled valley, even though the total Hall conductivity remains zero. [1] E. McCann Phys. Rev. B 74, 161403(R) (2006) [2] Eduardo V. Castro, et al, Phys. Rev. Lett. 99, 216802 (2007)   

Effect of electron interactions on the infrared conductivity of a monolayer graphene

Recent experiments on the infrared spectroscopy of a monolayer graphene has revealed an unexpected non-Lorentzian form of the Drude peak and an anomalously large broadening of the interband absorption edge in this material. We present a theoretical investigation that attributes these features to Coulomb scattering between Dirac quasiparticles. This scattering shows up in the dynamical conductivity because relativistic collisions lead to the current relaxation despite the conservation of the total momentum. This is in contrast to the conventional case where electron-electron scattering along does not cause a finite resistivity.   

Excitonic Effects in the Optical Spectra of Graphene Nanoribbons

We present a first-principles calculation of the optical properties of graphene nanoribbons (GNRs) with many-electron effects included, employing the GW-BSE approach. The reduced dimensionality of GNRs gives rise to an enhanced electron-hole binding energy for both bright and dark exciton states and changes the optical spectra significantly. The characteristics of the excitons of different types of GNRs are compared and discussed. The enhanced excitonic effects found here are expected to be of importance in considering possible applications (such as optoelectronics) of graphene-based nanostructures. This work was supported by National Science Foundation Grant No. DMR07-05941, the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Computational resources have been provided by Datastar at the San Diego Supercomputer Center.   

Localized states at zigzag edges of graphene multilayers and graphite steps

Among the uncommon features of graphene monolayer we find the presence of zero energy states localized at zigzag edges, leading to the self-doping phenomenon and inducing edge magnetization. Here we report the existence of zero energy surface states localized at zigzag edges of bilayer graphene and stacks with any number of layers. Working within the tight-binding approximation we derive an analytic solution for the wavefunctions of these peculiar surface states. It is shown that zero energy edge states in bilayer graphene can be divided into two families: (\emph{i})~states living only on a single plane, equivalent to surface states in monolayer graphene; (\emph{ii})~states with finite amplitude over the two layers, with an enhanced penetration into the bulk. The effect of edge states on the electronic structure and magnetic order of bilayer graphene nanoribbons is also studied. We show that edge states measured through scanning tunneling microscopy and spectroscopy of graphite step edges belong to family~(\emph{i}) or~(\emph{ii}) mentioned above, depending on the way the top layer is cut.   

Minimal conductivity of graphene: role of the Coulomb interaction

The effect of the Coulomb interaction on the zero-temperature low-frequency conductivity in undoped graphene is studied. We will show that the Coulomb interaction introduces a universal and positive leading logarithmic correction to the gaussian value of the dc conductivity [1]. This finding suggests that the origin of the unusually large minimal conductivity observed in graphene may be intrinsic, and arises from the Coulomb correction effectively cut off by finite temeperature/disorder/size effects. A mechanism of such a cutoff based on the non-trivial interplay between the Coulomb interaction and the ripples, both unavoidably present in the graphene sheet, will be briefly discussed. References: [1] I. F. Herbut, V. Juricic, and O. Vafek, arXiv.0707.4171.   

Low density ferromagnetism in a biased bilayer graphene

We compute the phase diagram of a biased graphene bilayer. The existence of a ferromagnetic phase is discussed with respect both to carrier density and temperature. We find that the ferromagnetic transition is first order, lowering the value of $U$ relatively to the usual Stoner criterion. We show that in the ferromagnetic phase the two planes have unequal magnetization and that the electronic density is hole like in one plane and electron like in the other.   

Minimal conductivity of rippled graphene with topological disorder.

We study the transport properties of a neutral graphene sheet with curved regions induced or stabilized by topological defects. The proposed model gives rise to Dirac fermions in a random magnetic field and a random space dependent Fermi velocity induced by the curvature. This last term leads to singular long range correlated disorder with special characteristics. The Drude minimal conductivity at zero energy is found to be inversely proportional to the density of topological disorder, a signature of diffusive behavior.   

Weak Localization of Dirac Fermions in Graphene

In the presence of the charged impurities with screened Coulomb potentials, we study the weak localization (WL) effect by evaluating the quantum interference correction (QIC) to the conductivity of Dirac fermions in graphene. With the inelastic scattering rate due to electron-electron interactions obtained from our previous calculation, we investigate the dependence of QIC on the carrier concentration, the temperature and the size of the sample. It is found that WL is present in large size samples at finite carrier doping where the strength of the intervalley scatterings due to the charged impurities is not weak. In addition, we argue that the system is delocalized at very low doping. We also analyze the absence of WL in experiment. It is found that WL is quenched at low temperature for small size samples as studied in the experiments.   

Electric Transport Theory of Dirac Fermions

The self-consistent Born approximation is employed to calculate transport properties in graphene with finite-range impurity potentials. The current-current correlation function is determined by a system of four coupled integral equations, unlike the case of short-range impurity scattering, and yet the results for the latter can exactly be reproduced in our formalism. As a test, we numerically calculate the dc electric conductivity of graphene for charged impurities with screened Coulomb potential, the linear dependence of the dc conductivity on the carrier concentration and the extrapolated value for zero-doping is shown to be finite, in a qualitative agreement with the experimental observations.