Infrared spectroscopy of hole doped ABA-stacked trilayer graphene

Using infrared spectroscopy, we investigate bottom gated ABA-stacked trilayer graphene subject to an additional environment-induced p-type doping. We find that the Slonczewski-Weiss-McClure tight-binding model and the Kubo formula reproduce the gate voltage-modulated reflectivity spectra very accurately. This allows us to determine the charge densities and the potentials of the {\pi}-band electrons on all graphene layers separately and to extract the interlayer permittivity due to higher energy bands.Comment: 6 pages, 6 figures Corrected sign of fig 3 and visibilty of fig

Energy spectrum and Landau levels in bilayer graphene with spin-orbit interaction

We present a theoretical study of the bandstructure and Landau levels in bilayer graphene at low energies in the presence of a transverse magnetic field and Rashba spin-orbit interaction in the regime of negligible trigonal distortion. Within an effective low energy approach (L\"owdin partitioning theory) we derive an effective Hamiltonian for bilayer graphene that incorporates the influence of the Zeeman effect, the Rashba spin-orbit interaction, and inclusively, the role of the intrinsic spin-orbit interaction on the same footing. Particular attention is spent to the energy spectrum and Landau levels. Our modeling unveil the strong influence of the Rashba coupling $\lambda_R$ in the spin-splitting of the electron and hole bands. Graphene bilayers with weak Rashba spin-orbit interaction show a spin-splitting linear in momentum and proportional to $\lambda_R$, but scales inversely proportional to the interlayer hopping energy $\gamma_1$. However, at robust spin-orbit coupling $\lambda_R$ the energy spectrum shows a strong warping behavior near the Dirac points. We find the bias-induced gap in bilayer graphene to be decreasing with increasing Rashba coupling, a behavior resembling a topological insulator transition. We further predict an unexpected assymetric spin-splitting and crossings of the Landau levels due to the interplay between the Rashba interaction and the external bias voltage. Our results are of relevance for interpreting magnetotransport and infrared cyclotron resonance measurements, including also situations of comparatively weak spin-orbit coupling.Comment: 25 pages, 5 figure

The influence of interlayer asymmetry on the magnetospectroscopy of bilayer graphene.

We present a self-consistent calculation of the interlayer asymmetry in bilayer graphene caused by an applied electric field in magnetic fields. We show how this asymmetry influences the Landau level spectrum in bilayer graphene and the observable inter-Landau level transitions when they are studied as a function of high magnetic field at fixed filling factor as measured experimentally in Ref. [1]. We also analyze the magneto-optical spectra of bilayer flakes in the photon-energy range corresponding to transitions between degenerate and split bands of bilayers

The electronic properties of bilayer graphene

We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energy. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects.Comment: review, 31 pages, 15 figure

Quantum Hall effect and Landau level crossing of Dirac fermions in trilayer graphene

We investigate electronic transport in high mobility (\textgreater 100,000 cm$^2$/V$\cdot$s) trilayer graphene devices on hexagonal boron nitride, which enables the observation of Shubnikov-de Haas oscillations and an unconventional quantum Hall effect. The massless and massive characters of the TLG subbands lead to a set of Landau level crossings, whose magnetic field and filling factor coordinates enable the direct determination of the Slonczewski-Weiss-McClure (SWMcC) parameters used to describe the peculiar electronic structure of trilayer graphene. Moreover, at high magnetic fields, the degenerate crossing points split into manifolds indicating the existence of broken-symmetry quantum Hall states.Comment: Supplementary Information at http://jarilloherrero.mit.edu/wp-content/uploads/2011/04/Supplementary_Taychatanapat.pd

Stacking-Dependent Band Gap and Quantum Transport in Trilayer Graphene

In a multi-layer electronic system, stacking order provides a rarely-explored degree of freedom for tuning its electronic properties. Here we demonstrate the dramatically different transport properties in trilayer graphene (TLG) with different stacking orders. At the Dirac point, ABA-stacked TLG remains metallic while the ABC counterpart becomes insulating. The latter exhibits a gap-like dI/dV characteristics at low temperature and thermally activated conduction at higher temperatures, indicating an intrinsic gap ~6 meV. In magnetic fields, in addition to an insulating state at filling factor {\nu}=0, ABC TLG exhibits quantum Hall plateaus at {\nu}=-30, \pm 18, \pm 9, each of which splits into 3 branches at higher fields. Such splittings are signatures of the Lifshitz transition induced by trigonal warping, found only in ABC TLG, and in semi-quantitative agreement with theory. Our results underscore the rich interaction-induced phenomena in trilayer graphene with different stacking orders, and its potential towards electronic applications.Comment: minor revision; published versio

Continuous-distribution puddle model for conduction in trilayer graphene

An insulator-to-metal transition is observed in trilayer graphene based on the temperature dependence of the resistance under different applied gate voltages. At small gate voltages the resistance decreases with increasing temperature due to the increase in carrier concentration resulting from thermal excitation of electron-hole pairs. At large gate voltages excitation of electron-hole pairs is suppressed, and the resistance increases with increasing temperature because of the enhanced electron-phonon scattering. We find that the simple model with overlapping conduction and valence bands, each with quadratic dispersion relations, is unsatisfactory. Instead, we conclude that impurities in the substrate that create local puddles of higher electron or hole densities are responsible for the residual conductivity at low temperatures. The best fit is obtained using a continuous distribution of puddles. From the fit the average of the electron and hole effective masses can be determined.Comment: 18 pages, 5 figure

Strong Suppression of Electrical Noise in Bilayer Graphene Nano Devices

Low-frequency 1/f noise is ubiquitous, and dominates the signal-to-noise performance in nanodevices. Here we investigate the noise characteristics of single-layer and bilayer graphene nano-devices, and uncover an unexpected 1/f noise behavior for bilayer devices. Graphene is a single layer of graphite, where carbon atoms form a 2D honeycomb lattice. Despite the similar composition, bilayer graphene (two graphene monolayers stacked in the natural graphite order) is a distinct 2D system with a different band structure and electrical properties. In graphene monolayers, the 1/f noise is found to follow Hooge's empirical relation with a noise parameter comparable to that of bulk semiconductors. However, this 1/f noise is strongly suppressed in bilayer graphene devices, and exhibits an unusual dependence on the carrier density, different from most other materials. The unexpected noise behavior in graphene bilayers is associated with its unique band structure that varies with the charge distribution among the two layers, resulting in an effective screening of potential fluctuations due to external impurity charges. The findings here point to exciting opportunities for graphene bilayers in low-noise applications

Intrinsic Zeeman Effect in Graphene

The intrinsic Zeeman energy is precisely one half of the cyclotron energy for electrons in graphene. As a result a Landau-level mixing occurs to create the energy spectrum comprised of the $4j$-fold degenerated zero-energy level and 4-fold degenerated nonzero-energy levels in the $j$-layer graphene, where $j=1,2,3$ for monolayer, bilayer and trilayer, respectively. The degeneracy manifests itself in the quantum Hall (QH) effect. We study how the degeneracy is removed by the Coulomb interactions. With respect to the zero-energy level, an excitonic gap opens by making a BCS-type condensation of electron-hole pairs at the filling factor $\nu =0$. It gives birth to the Ising QH ferromagnet at $\nu =\pm 1$ for monolayer, $\nu =\pm 1,\pm 3$ for bilayer, and $\nu =\pm 1,\pm 3,\pm 5$ for trilayer graphene from the zero-energy degeneracy. With respect to the nonzero-energy level, a remarkable consequence is derived that the effective Coulomb potential depends on spins, since a single energy level contains up-spin and down-spin electrons belonging to different Landau levels. The spin-dependent Coulomb interaction leads to the valley polarization at $\nu =\pm 4, \pm 8, \pm 12, ...$ for monolayer, $\nu =\pm 2, \pm 6, \pm 10, >...$ for bilayer, and $\nu =\pm 2,\pm 4, \pm 8, \pm 12, ...$ for trilayer graphene.Comment: 24 pages, 9 figures, to appear in J. Phys. Soc. Jp

Electronic properties of bilayer and multilayer graphene

We study the effects of site dilution disorder on the electronic properties in graphene multilayers, in particular the bilayer and the infinite stack. The simplicity of the model allows for an easy implementation of the coherent potential approximation and some analytical results. Within the model we compute the self-energies, the density of states and the spectral functions. Moreover, we obtain the frequency and temperature dependence of the conductivity as well as the DC conductivity. The c-axis response is unconventional in the sense that impurities increase the response for low enough doping. We also study the problem of impurities in the biased graphene bilayer.Comment: 36 pages, 42 figures, references adde