Session D21: Focus Session: Graphene: Correlated States

Field-Induced Kosterlitz-Thouless Transition in the N=0 Landau Level of Graphene

Graphene displays an unconventional quantization of the Hall conductivity when subjected to a magnetic field. The Hall conductivity is measured to be a half-integer in units of 4 times the conductance quantum. In the presence of a strong magnetic field, graphene's 4 fold degeneracy is lifted by the exchange Coulomb interaction. Recent experiments indicate that high quality graphene samples exhibit a very unusual high-resistance metallic state and a transition to a complete insulating phase at the charge neutral point. We propose that the current carriers in this state are charged vortices of the XY valley-pseudospin order parameter, a situation which is dual to a conventional thin superconducting film. We study energetic and the stability of this phase in the presence of disorder. A phase diagram as a function of magnetic fields and the sample mobility is determined. K. Nomura, S. Ryu, D.-H. Lee, arXiv:0906.0159 (to appear in Phys. Rev. Lett).   

Skyrmions in a graphene bilayer at filling factors $\nu=-3,-1$

In a quantizing magnetic field, the Landau level $N=0$ of a graphene bilayer in the Bernal stacking consists of an octet of degenerate states if Zeeman coupling is neglected. The quantum numbers of an electron are its spin, its layer (or valley) index and its $\it{orbital}$ index $n=0,1$. It was shown recently that, in the Hartree-Fock approximation, the ground states of the bilayer at integer filling factors $\nu \in [-3,4] $ and with an applied electrical bias can be described as different kinds of quantum Hall ferromagnets (QHF's) [1]. In this talk, we discuss the single-particle excitations of the QHF states at filling factors $\nu=-3,-1$ when the bias is such that the ground states are orbital ferromagnets. In particular, we look for the possibility of exciting quasiparticles with orbital-pseudospin textures (orbital skyrmions) at these filling factors. In these orbital skyrmions, a topological charge $Q=2$ would be associated with a $q=1$ electronic charge. \newline \newline $\lbrack 1\rbrack$ Yafis Barlas, R. C\^ot\'e, K. Nomura, and A. H. MacDonald, Phys. Rev. Lett. {\bf 101},097601 (2008).   

Observation of the fractional quantum Hall effect in graphene

Only a glimpse of correlated electron physics has been observed in graphene so far, mostly due to the strong electron scattering caused by charged impurities in the substrate. To overcome this limitation,we fabricate devices where electrically contacted and electrostatically gated graphene samples are suspended over a substrate. The measured low-temperature sample mobility is found to exceed 100,000 cm2/Vs in such devices. The very high mobility of our specimens allows us to observe previously inaccessible transport regimes in graphene. We report the observation of the fractional quantum Hall effect, supporting the existence of interaction induced correlated electron states in the presence of a magnetic field. In addition, at low carrier density graphene becomes an insulator with an energy gap tunable by magnetic field.   

Magnetically induced correlated states in suspended graphene

Quantum Hall effect (QHE) plays an important role in understanding the electronic properties of graphene. The anomalous QHE provides the first experimental evidence of the ``ultra-relativistic'' nature of graphene's charge carriers. Because of the theoretical expectation that interactions and correlations should be important in this material, there have been long-standing yet baffled efforts in observing the manifestation of the collective quantum behavior, the fractional QHE (FQHE), in graphene. We show that such difficulty arises from the obscuring effect of the disorders which are mainly associated with the substrates supporting the graphene films. Using suspended graphene devices probed by two-terminal charge transport measurements, we are able to isolate the samples from substrate-induced perturbations and to avoid effects of finite geometry. FQHE at filling factor 1/3 has been successful observed. At low carrier density, we find a field-induced transition to an insulator that competes with the FQHE, allowing its observation only in the highest quality samples.   

Fractional Quantum Hall Effect in Suspended Graphene: Transport Coefficients and Electron Interaction Strength

Fractional quantum Hall effect (FQHE), observed recently in suspended graphene (SG) [1], was found to persist up to temperatures much higher than in previously studied systems, such as GaAs. This suggests strong electron interactions in SG. Can the interaction strength be inferred from the measurements? The best results on FQHE were obtained on micron-size SG flakes, where only two-terminal measurements could be performed. This talk will address the problem of determining transport coefficients from the two-terminal conductance in the FQHE regime. A general approach, which relies on the conformal invariance of two-dimensional magnetotransport, is used to extract $\sigma_{xx}$ and $\sigma_{xy}$. From the temperature dependence of $\sigma_{xx}$ we estimate the energy gap of quasiparticle excitations in the $\nu = 1/3$ state. The gap value, which is found to be well above that measured in GaAs systems [2], is compared to theoretical predictions. Our approach provides a new tool for the studies of quantum transport in suspended graphene and other nanoscale systems. [1] X. Du et al, Nature 462, 192 (2009); K. Bolotin et al, ibid., 196 (2009). [2] G. S. Boebinger et al, Phys. Rev. Lett. 55, 1606 (1985).   

Broken-Symmetry States and Divergent Resistance in Suspended Bilayer Graphene

We report the fabrication of suspended bilayer graphene devices with very little disorder. Transport measurements at zero magnetic field indicate that charge inhomogeneity in these flakes reaches as low as 10$^{10}$ cm$^{-2}$. We observe quantum Hall states that are fully quantized at a magnetic field of 0.2 T, as well as broken-symmetry states at intermediate filling factors \textit{$\nu $} = 0, $\pm $1, $\pm $2 and $\pm $3. In the \textit{$\nu $} = 0 state, the resistance of the flakes increases exponentially with applied magnetic field and scales as magnetic field divided by temperature. This resistance is predominantly affected by the perpendicular component of the applied field and the extracted gap size is larger than expected from Zeeman splitting, indicating that the broken-symmetry states arise from many-body interactions and underscoring the importance of Coulomb interactions in bilayer graphene.   

The metal-insulator transition of the half integer quantum-Hall effect in epitaxial graphene

The observation of the half integer quantum-Hall effect (QHE) in Hall resistance along with the pronounced Shubnikov-de Haas (SdH) oscillations confirms that the electrical properties of epitaxial graphene on SiC share the same relativistic physics as those in exfoliated graphene films. The temperature-dependent half-width \textit{$\Delta $B(T) }of the SdH peaks and the maximum of the slope of the Hall resistance \textit{$\partial \rho $}$_{xy}$\textit{/$\partial $B} of gated epitaxial graphene are investigated at temperatures between 0.4 K to 300K. The preliminary data shows \textit{$\Delta $B(T)} for the first Laudau level of electrons in epitaxial graphene on SiC (0001) display a power-law behavior with a scaling exponent \textit{$\kappa $}$\approx $0.43, being consistent with the previously reported results from 2DES formed at AlGaAs/GaAs or InGaAs/InP heterojunctions [1,2] and the exfoliated graphene [3]. More detailed results on \textit{$\kappa $} for high Laudau levels and the study of size-dependence of the quantum-Hall plateau-plateau transition in epitaxial graphene will also be presented. [1] H.P. Wei, D.C. Tsui, M.A. Paalanen, and A.M.M. Pruisken, Phys. Rev. Lett. 61, 1294 (1988). [2] S. Koch, R.J. Haug, K. von Klitzing, and K. Ploog, Phys. Rev. Lett. 67, 883 (1991). [3] A.J.M. Giesbers, U. Zeitler, L.A. Ponomarenko, R. Yang, K.S. Novoselov, A.K. Geim, and J.C. Maan, arXiv:0908.0461v1.   

Insulating state of a graphene edge in the spin-polarized quantum Hall effect regime

The edge of an undoped graphene in the quantum Hall effect regime that supports spin-polarized bulk and gapless counter-propagating edge states is studied using one-dimensional bosonization technique. The attractive long-range Coulomb interactions between electron and hole modes result in a strong tendency towards excitonic pairing with the associated spin-density wave structure. In the absence of backscattering, however, the edge excitations remains gapless, no order is formed, and the edge remains metallic. Backscattering on random spin-orbital defects that involves spin flips leads to the localization of the edge states. The localization effects can be significantly enhanced by the Coulomb interactions.   

Quantum Hall ferromagnetic states of a graphene bilayer at $\nu=-1$

It was shown recently [1] that Coulomb interaction can lift the degeneracy of the octet of states in Landau level $N=0$ of a graphene bilayer by forming different kinds of quantum Hall ferromagnetic states. In this talk, we study the sequence of phase transitions induced by an external potential difference, $\Delta_{B}$ between the layers at filling factor $\nu=-1$. With $\Delta_{B}$, the system evolves from an interlayer coherent state at small $\Delta_{B}$, to a state with mixed interlayer and inter-orbital coherence at intermediate $\Delta_ {B}$, and then into a state with inter-orbital coherence only at larger $\Delta_{B}$. We discuss the nature of the ground state of these three phases and compute the dispersion of their collective excitations in the generalized random-phase approximation. For the inter-orbital coherent state, we develop an effective pseudospin model and explain that the finite wave- vector instability of the pseudospin mode at some critical bias $\Delta_{B}^{*}$. is due to the presence of a Dzyaloshinskii- Moriya term in the Hamiltonian. This term may drive the system into a spiral state for $\Delta_{B} > \Delta_{B}^{*}$. \newline \newline $\lbrack 1\rbrack$ Yafis Barlas, R. C\^ot\'e, K. Nomura, and A. H. MacDonald, Phys. Rev. Lett. {\bf 101},097601 (2008).   

Deconfined fractional electric charges in graphene at high magnetic fields

The resistance at the charge neutral (Dirac) point was shown by Checkelsky \textit{et al} in Phys.\ Rev.\ B \textbf{79}, 115434 (2009) to diverge upon the application of a strong magnetic field normal to graphene. We argue that this divergence is the signature for a Kekul\'e instability of graphene, which is induced by the magnetic field. We show that the strong magnetic field does not remove the zero modes that bind a fraction of the electron around vortices in the Kekul\'e dimerization pattern, and that quenched disorder present in the system makes it energetically possible to separate the fractional charges. These findings, altogether, indicate that graphene can sustain deconfined fractionalized electrons.   

Thermal and high current bias study of the quantum Hall effect in Graphene

What is the underlying mechanism responsible for the smooth breakdown of the Quantum Hall Effect in graphene at finite energy? For that purpose we study the longitudinal resistance $\rho _{xx}$ as a function of temperature and bias current of Hall bars on exfoliated graphene in magnetic field up to 17 Tesla. Temperature ranges from 1.4K to 300Kand current from 10nA to 100$\mu $A. Similarly to conventional 2DEGs, our measurements show Efros Shklovskii Variable Range Hopping transport followed by thermal activation. On the Hall plateaus $\rho _{xx }\sim $ 1/T*exp[(-T$_{0}$/T)$^{1/2}$], where T$_{0}$ is the characteristic VRH temperature linked to the localisation length $\xi $. A similar relation is observed as a function of the bias current. It leads to an effective temperature. Comparison of the characteristic VRH temperature and current can give information on the decay of Hall voltage over the localization length. At higher energy we observe activated behaviour of $\rho _{xx}$ from which we can obtain the Energy gaps of the quantum Hall effect, comparing these values to theoretical values gives information on the Broadening of Landau Levels by disorder.   

Half integer quantum Hall effect in high mobility single layer epitaxial graphene

We present results on the quantum Hall effect for single layer epitaxial graphene grown on the C-face of 4H silicon carbide. Hall plateaus at half integer values and vanishing resistivity are observed for high mobility samples ($\mu= 20,000 cm^{2}/V.s$ at 4 K and $15,000 cm^{2}/V.s$ at 300 K) despite contamination and substrate steps. This is comparable to the best exfoliated graphene flakes on SiO2 and an order of magnitude larger than Si-face epitaxial graphene monolayers. Splitting of the n=0 Landau level ($\nu=1$) is observed at high field. These and other properties indicate that C-face epitaxial graphene is an ideal platform for graphene-based electronics.   

Splitting of critical energies in the $n$=0 Landau level of graphene driven by a random hopping disorder

The lifting of the degeneracies of the states from the graphene $n$=0 Landau level (LL) was recently observed experimentally and is discussed in many theoretical works, with various different proposed explanations for the origins of the observed splittings. In this work, valley/sublattice symmetry breaking is investigated through a non-interacting tight-binding model with a random hopping disorder. A disorder-driven splitting of two bands and of two critical energies is observed by means of density of states and participation ratio calculations, elucidating the interplay of lattice and disorder effects on the splitting process. The analysis of the probability densities of the states within the $n$=0 LL shows that there is a region, for the states closer to $E$=0, where there is an important asymmetry in the distribution of the wave function amplitudes between the two sublattices (where there is no matching of the spatial positions of the amplitudes over both sublattices). Furthermore, it is shown that as the splitting is increased, the two split levels also get increasingly broadened, in such a way that the proportion of the overlapped states keeps approximately constant for a wide range of disorder or magnetic field variation.