Session B40: QHE in Graphene

The Nature of Quantum Hall States near the Charge Neutral Dirac Point in Graphene

We investigate the quantum Hall (QH) states near the charge neutral Dirac point of a high mobility graphene sample in high magnetic fields ($B$). We find that the QH states at filling factors $\nu=\pm2$ show thermally activated behavior with an energy gap as large as $\sim890$ K at $B=45$ T. This large energy gap between the $n=0$ Landau level (LL) and the $n=1$ LL, enables us to observe a well-defined QH effect in graphene over a wide temperature range and even up to room temperature. In addition, the data reveal an activation energy gap at filling factor $\nu=1$, which is considerably larger than the previous studied spin states at $\nu=\pm4$ and shows a square root dependence on $B$, suggesting a many-body origin of this state. Such an origin is further supported by tilted field measurements, in which the $\nu=\pm1$ gaps are found to depend only on the normal component of the field with respect to the graphene plane. We therefore propose that the $\nu=\pm1$ states arise from the lifting of the sublattice degeneracy of the $n=0$ LL.   

Room Temperature Quantum Hall Effect in Graphene

The quantum Hall effect (QHE) is an example of those few quantum phenomena that occur on a truly macroscopic scale, and it has been attracting intense interest since its discovery in 1980. As many other quantum phenomena, the observation of the QHE requires temperatures $T$ typically below 4K. Efforts to extend the QHE temperature range by using semiconductors with small effective masses of charge carriers have so far failed to reach temperatures above 30K. We show that in graphene -- a single layer of carbon atoms densely packed in a honeycomb crystal lattice -- the QHE can be observed even at room temperature. This is due to the highly unusual nature of charge carriers in graphene, which behave as massless relativistic particles (Dirac fermions) and move with little scattering under ambient conditions.   

Quantum transport of massless Dirac fermions in graphene

Graphene is a two-dimensional carbon material with a honeycomb lattice and Dirac-like low-energy excitations. Motivated by recent graphene transport experiments, we have undertaken a numerical study of the conductivity of disordered two-dimensional massless Dirac fermions. It has been found that in the long-range Coulombic scattering case the conductivity depends linearly on the carrier density and that the minimum conductivity $\sigma_{\rm min}$ $\stackrel{\approx}{\_\_}$ e$^{2/h}$, consistent with experiments if inter-valley scattering is assumed to be negligible. We study the transport properties of graphene in the presence of a perpendicular magnetic field. In a strong field the conductivity reveals the massless Dirac fermion Landau level structure and quantized Hall conductivities. On the other hand, in the strongly disordered (weak field) regime in which Landau level maxing is important, the conductivity closes to e$^{2/h}$ for the long-range scattering case. K. Nomura, A. H. MacDonaold, cond-mat/0606589.   

Dirac and Normal Fermions in Graphite and Graphene: Implications to the Quantum Hall Effect

Spectral analysis of Shubnikov de Haas (SdH) oscillations of magnetoresistance and of Quantum Hall Effect (QHE) measured in quasi-2D highly oriented pyrolytic graphite (HOPG) reveals two types of carriers: normal (massive) electrons with Berry phase $0$ and Dirac-like (massless) holes with Berry phase $\pi$. We demonstrate that recently reported integer- and semi-integer QHE for bi-layer and single-layer graphenes take place simultaneously in HOPG samples.   

Delocalization of electrons in the lowest Landau level of disordered graphene

We investigate analytically and numerically the effects of disorder on the density of states and localization of relativistic two dimensional fermions in the lowest Landau level. We have used the numerical technique pioneered by Huckestein, to establish the localization-delocalization transition and calculate the localization length exponents for different types of disorder. For some combinations of disorder the localization-delocalization transition is shown to belong to a different universality class compared to the localization-delocalization transition in the lowest Landau level of nonrelativistic fermions. Our results have direct relevance to the integer quantum hall effect observed in graphene.   

Quantum Hall Stripes in Graphene

We study unmodulated stripes in graphene with a high, half integer filling factor. The ground-state energy and stripe density are calculated in the Hartree-Fock approximation to determine the effect of the differing Landau indices on the two sublattices in the single-particle wavefunctions. For appropriate filling factors, the stripes should support a spontaneous polarization of the valley degree of freedom in the limit where electron-electron interactions may be treated as SU(2) symmetric. The implications of the groundstate structure for collective modes and stability of the stripe state will be considered.   

Localization and resistance oscillations in n-layer graphene

Single and double-layer graphene have attracted much attention recently because of their unusual electronic band structures and novel physical properties. Theoretical calculations on n-layer graphene (nLG) revealed varying electronic properties, either semiconducting or semimetallic, depending on $n$ as well as the stacking pattern. We have prepared nLG devices, with $n$ ranging from 1 to 5, using a lithography-free, ``all-dry'' process, and measured the conductance of these devices as a function of the temperature, magnetic field, and the gate voltage. The conductance was found to exhibit quantum oscillations and magnetic field dependence that appear to have resulted from weak localization effects. Results of other measurements will also be presented.   

Quantum Hall ferromagnetism in monolayer and bilayer graphene

Graphene is a two-dimensional carbon material with a honeycomb lattice and Dirac-like low-energy excitations. Recent experiments and theoretical studies have clarified the unconventional quantum Hall effects that occur in both graphene systems because of their chiral band structures. In this contribution we address the influence of interaction on the quantum Hall effect in single-layer[1] and bilayer graphene, concentrating on the competition between disorder and interactions and on the variety of broken symmetry states that occur at integer filling factors. We also comment on the unusual fractional quantum Hall effect in bilayer graphene, which is strongest at filling factor $\nu$=2/5 rather than at filling factor $\nu$=1/3. \newline \newline [1] K. Nomura, A. H. MacDonald, Phys. Rev. Lett. 96, 256602 (2006).   

Magnetic field effects in bilayer graphene systems

Recently, it has been shown that bilayer graphene systems display interesting electronic properties: examples include the quantum Hall effect with additional plateaus at zero hall conductivity, and electronic gaps tunable by a bias voltage. Here, we study an experimentally relevant problem of the effect of an external applied magnetic flux density on the bulk band structure and edge dynamics of bilayer graphene. We combine exact diagonalization studies with analytical transfer matrix methods and systematically study the quantized Hall conductivity, edge state structure, and possible topological phases of the system over a range of externally applied fields.   

Biased bilayer graphene: semiconductor with a gap tunable by electric field effect

A graphene bilayer with an electrostatic potential difference between layers~--~biased bilayer~--~has been experimentally realized recently. Using a tight binding description we demonstrate that the externally applied gate bias effectively controls the electronic gap between the valence and the conduction bands of bilayer graphene. Applying the theory to the description of magneto-transport data (Shubnikov-de Haas measurements of the cyclotron mass) we extract the value of the gap as a function of the electronic density. We show that the gap can be tuned between zero and mid-infrared energies using fields still below the electric breakdown of SiO$_2$. The opening of a gap is clearly seen in the quantum Hall regime, where the zero-energy double step characteristic to the anomalous quantum Hall effect in unbiased bilayer graphene, splits into two, giving rise to an additional plateau at zero Hall conductivity, besides the standard quantum Hall sequence.   

Quantum Hall effect in carbon nanotubes and curved graphene strips

We show the development of Landau levels for thick carbon nanotubes in a transverse magnetic field, making use of a long-wavelength description in terms of Dirac fermion fields. For values of the magnetic length smaller than the nanotube radius, quantized longitudinal currents are carried by states localized at the flanks of the nanotube. We find that the Hall conductivity is given by even multiples of $2 e^2 /h$, and clarify the transition to the typical odd-integer quantization of graphene as the nanotube is unrolled to form a curved strip. We also show that the absence of significant backscattering interactions opens the possibility to observe a robust chiral liquid at the flanks of the nanotube.   

Long-range interactions and Pseudo-relativistic Phenomena in Disordered Graphene: The Zero-bias Anomaly

Two-dimensional graphene creates a window into new and unusual transport phenomena, which can be described in terms of the propagation of non-interacting Dirac quasiparticles (DQP). In such a system, Coulomb interactions also remain unscreened, and it is of interest how such long-ranged correlations might significantly affect DQP excitations. Using single-particle tunneling measurements, the DQP densities of states are computed analytically, in the presence of mild impurities and for energies ranging between the diffusive and the ballistic limits. Interesting interplay between the Coulomb interactions and the DQP is best revealed in the ballistic regime, whereas in the diffusive limit we recover what is essentially the conventional 2DEG. The evolution of the anomalous exponent characterizing the ``zero-bias'' anomaly in the ballistic regime is discussed.