Session X36: Graphene: Quantum Hall Effect

Graphene in a periodically alternating magnetic field: an unusual quantization of the anomalous Hall effect

We study the energy spectrum and electronic properties of graphene in a periodic magnetic field of zero average with a symmetry of triangular lattice. The periodic field leads to formation of a set of minibands separated by the gaps, which can be manipulated by external field. The Berry phase, related to the motion of electrons in $k$ space, and the corresponding Chern numbers characterizing topology of the energy bands are calculated analytically and numerically. In this connection, we discuss the anomalous Hall effect in the insulator state, when the Fermi level is located in the minigap. The results of calculations show that in the model of gapless Dirac spectrum of graphene the anomalous Hall effect can be treated as a sum of fractional quantum numbers, related to the nonequivalent Dirac points.   

Quantum Hall Edge States in Bilayer Graphene Ribbons

We study the low energy edge states of bilayer graphene ribbons subject to a strong perpendicular magnetic field $B$, and show that they can be described within a continuum model (the Dirac equation). We are mainly interested in investigating the energy- band structure of ribbons with a zigzag termination. At the zero Landau Level there are eight degenerate bands, whose degeneracy can be broken and controlled by an external inter- layer voltage bias $V$. This leads to the opening of a gap in the bulk. On the edges, due to a mixture of hole- and particle- like bands (from the same valley), an avoided crossing occurs which can be understood within a perturbative expansion in the inter-layer hopping. On the other hand, edge states from different valleys are protected from mixing by a long-range disorder potential. Hence, hole- and particle-like states can cross without mixing, and the system has properties of a topological insulator. In the presence of interactions, the rich behavior of crossing single-electron edge states may lead to a variety of collective edge-modes, whose properties dominate the transport behavior of this system.   

Hofstadter's Fractal Energy Spectrum in Twisted Bilayer Graphene

Hofstadter butterfly, the fractal spectrum of 2D lattice electrons in a magnetic field, has been studied theoretically for a few prototypical systems. However, due to the small unit cell in traditional materials, it is difficult to directly observe such a structure in the experiment. In this work we demonstrate that the Hofstadter butterfly structure can be detected in twisted bilayer graphene with a reasonable strength of the magnetic field. Based on the recursive tight-binding method, we have systematically studied the landau level dependence on the magnetic field as a function of the twist angle, with the underlying electronic structure ranging from the parabolic dispersion of Bernal bilayer graphene to the linear dispersion of decoupled graphene layers. The signature of transition is characterized by some low-lying landau levels in slightly twisted bilayer graphene, which are related to the flat bands induced in the layer decoupling process.   

Multi-Domain Model of Bilayer Graphene with Broken Time Reversal Symmetry

Recent experiments suggest strange new states for bilayer graphene at zero electric and magnetic fields~[1]. We consider recent models for the ground state of bilayer graphene with interactions~[2,3]. These models predict domains with non-zero local Hall conductance but an expected overall average Hall conductance of $\sigma_{xy}=0$, and should exhibit broken time reversal symmetry. We examine theoretical models for random four-probe measurements for various domain geometries in bilayer graphene at zero electric and magnetic fields. We find non-zero Hall conductance of magnitude dependent on domain geometry and network structure. \\[4pt] [1] Weitz, R.T., Allen, M.T., Feldman, B.E., Martin, J. Yacoby, A. Science. {\bf330}, 6005 (2010). \\[0pt] [2] Nandkishore, R., Levitov, LPhys. Rev. Lett. {\bf104}, 156803 (2010) \\[0pt] [3] Zhang F., Jung J., Fiete, G.A., Niu, Q., MacDonald, A.H. arXiv:1010.4003v1 (2010)   

Fractional quantum Hall effect in graphene: multicomponent states and tunable interactions

We study the fractional quantum Hall (FQH) states in graphene using exact diagonalization and taking into account the multicomponent degrees of freedom and the possibility of tuning the interaction potential. The recently observed graphene FQH state at a filling factor $\nu_G=1/3$ is found to be adiabatically connected to the 1/3 Laughlin state in the upper spin branch, with SU(2) valley-isospin ferromagnetic ordering and a completely filled lower spin branch. At the experimentally relevant values of the Zeeman field, however, the state possesses characteristic low-energy spin-flip excitations (different from the magneto-roton expected at large Zeeman fields) that may be unveiled in inelastic light-scattering experiments. We also discuss the possibility of realizing other Abelian and non-Abelian FQH states in graphene by modifying the effective interaction potential using a combination of insulating substrates.   

Decomposition into half-integer quantum Hall numbers from Dirac cones in a graphene-related lattice model

In field theory it is well-known that the Hall conductivity of a massive Dirac particle is equal to $1/2\,\,{\rm sgn}\, (m) $ (in units of $-e^2/h$) when the Fermi energy lies in the mass gap. By contrast, any lattice model must have an integer quantum Hall number when we consider noninteracting electrons, as dictated by the TKNN formula arising from the periodicity in the Brillouin zone. Usually, this is understood to be consistent with the fact that in lattice models such as graphene honeycomb lattice Dirac dispersions tend to appear in pairs (as dictated by the Nielsen-Ninomiya theorem for chiral cases), but this does obscure a half-integer contribution from each Dirac cone. To resolve this, here we show that it is possible to identify half-integer contributions by constructing a lattice model with systematically shifted Dirac points [1]. Edge-state spectrum also confirms this. \\[4pt] [1] H. Watanabe, Y. Hatsugai and H. Aoki, arXiv:1008.0130, to appear in Phys. Rev. B (R).   

Lattice Theory of Pseudospin Ferromagnetism in Bilayer Graphene: Competing Orders and Interaction Induced Quantum Hall States

In mean-field-theory bilayer graphene's massive Dirac fermion model has a family of broken inversion symmetry ground states with charge gaps and spin-valley flavor dependent spontaneous charge transfers between layers. We use a lattice Hartree-Fock model to explore some of the physics which controls whether or not this type of broken symmetry state, which can be viewed as a pseudospin ferromagnet, occurs in nature. We find that inversion symmetry is still broken in the lattice model and estimate that transferred areal densities are $\sim 10^{-5}$ per carbon atom, that the associated energy gaps are $\sim 10^{-2} {\rm eV}$, the ordering condensation energies are $\sim 10^{-7} eV$ per carbon atom and the energy differences between competing orders at the neutrality point to be of the order of $\sim 10^{-9} eV$ per carbon atom. We explore the quantum phase transitions between different states induced by external magnetic fields and by externally controlled electric potential differences between the layers. We find, in particular, that in an external magnetic field coupling to spontaneous orbital moments favors broken time-reversal-symmetry states that have spontaneous quantized anomalous Hall effects. Our theory predicts a non monotonic behavior of the band gap as a function of electric field in qualitative agreement with recent experiments.   

$\nu=0$ quantum Hall ferromagnet in a monolayer graphene: bulk ground states and charged edge excitations

The $\nu=0$ quantum Hall state in a defect-free graphene sample is studied within the framework of the quantum Hall ferromagnetism. Starting from the low-energy electron Hamiltonian, in which all allowed by symmetry sublattice- and valley-anisotropic terms due to the Coulomb and leading electron-phonon interactions are taken into account, the energy functional for the quantum Hall ferromagnet is derived. Paying special attention to the signs of anisotropies, we find that the anisotropy due to the repulsive Coulomb interactions always favors the spin-polarized pseudospin-singlet state. On the other hand, the anisotropy due to the phonon-mediated attractive interactions favors the $XY$ pseudospin-polarized spin-singlet state. It is then demonstrated that, in the case of the $XY$ pseudospin bulk order and armchair boundary, the Skyrmion-type charged excitations are gapped at the edge, which makes the whole sample insulating. These findings suggest that the experimentally observed insulating $\nu=0$ state is an $XY$ pseudospin ferromagnet favored by electron-phonon interactions.   

Disorder-induced magnetooscillations in bilayer graphene at high bias

Energy spectrum of biased bilayer graphene near the bottom has a ``Mexican-hat''-like shape. For the Fermi level within the Mexican hat we demonstrate that, apart from conventional magnetooscillations which vanish with temperature, there are additional magnetooscillations of capacitance and conductance which are weakly sensitive to temperature. These oscillations are also insensitive to a long-range disorder. Their period in magnetic field scales with bias, $V$, as $V^2$. The origin of these oscillations is the disorder-induced scattering between electron-like and hole-like Fermi-surfaces, specific for Mexican hat. At low temperatures, oscillations transform into quantum Hall plateaus in $\sigma_{xy}$. We predict that evolution of $\sigma_{xy}$ with magnetic field is highly non-trivial. This is because the contributions to $\sigma_{xy}$ from electron-like and hole-like Landau levels have opposite signs.   

Dynamical scaling analysis of the optical Hall conductivity in the quantum Hall regime

We study the optical Hall conductivity $\sigma_{xy}(\varepsilon_F, \omega)$ in two-dimensional electron gas (2DEG) and in graphene in the quantum Hall regime, which is measurable by the Faraday rotation. It was previously demonstrated that both conductivities retain their plateau structure at finite frequency, up to the optical frequency regime. Physically, the robustness of the plateau structure in the ac optical regime can be attributed to the localization of electrons in the QHE. To quantify this picture, a dynamical scaling analysis of $\sigma_{xy}(\varepsilon_F, \omega)$ is performed for the $n=0$ Landau level in graphene as well as for the conventional quantum Hall system. This analysis examines whether the system size dependence of $\sigma_{xy}(\varepsilon_F, \omega)$ can be captured with a universal scaling function that involves the localization exponent $\nu$ and the dynamic critical exponents $z$. Based on exact diagonalization of these systems with potential disorder, employing the Kubo formula, it is shown that $\sigma_{xy}(\varepsilon_F,\omega)$ obeys a well-defined dynamical scaling behavior. For both systems, the static exponents $\nu$ are similar and the dynamical exponents $z$ are found to be $\approx 2$. Our quantitative analysis indicates that the plateau structure in the ac Hall conductivity should be robust and experimentally testable in the THz regime.   

Disorder Effect of Quantum Anomalous Hall effect in Graphene

We investigate the possibility of realizing quantum anomalous Hall effect in graphene. We show that a bulk energy gap can be opened in the presence of both Rashba spin-orbit coupling and an exchange field. We calculate the Berry curvature distribution and find a nonzero Chern number for the valence bands and demonstrate the existence of gapless edge states. Inspired by this finding, we also study, by first-principles method, a concrete example of graphene with Fe atoms adsorbed on top, obtaining the same result. We further study the disorder effect of this quantum anomalous Hall effect and show how this state is localized in the presence of strong disorders.   

Multiferro\"ic-like behavior in the quantum Hall ferromagnetic states of a graphene bilayer

In a quantizing magnetic field, a graphene bilayer has an octet of degenerate states in the Landau level $N=0$. An electron in this level must be described by three quantum numbers: its spin, its valley index $K$ or $K'$ and an orbital quantum number $n=0,1$. In the Hartree-Fock approximation, the ground states of the graphene bilayer at integer filling factors $\nu \in [-3,4]$ can be described as different kinds of quantum Hall ferromagnets (QHF's) with finite interlayer, inter- orbital, or inter-spin coherence. In this talk, we present the phase diagram of the two-dimensional electron gas (2DEG) in $N=0 $ when the filling factor or a finite interlayer voltage, $\Delta_{B}$ is varied. A finite density of $\it electric$ dipoles is either spontaneously present in the QHF phases with inter-orbital coherence or can be generated by applying an external electric field in the plane of the layers. We show that by changing the strength of this electric field, and so the coupling with the electric dipoles, it is possible to control the degree of $\it magnetic$ polarisation of the 2DEG.   

Longitudinal Conductivity of Bilayer Graphene in the Integer Quantum Hall Regime

We investigate the frequency dependent conductivity of disordered bilayer graphene near neutral filling, in a strong magnetic field. Absent Zeeman coupling, and with two independent valleys, a graphene bilayer's lowest Landau level is eightfold-degenerate, comprising two Landau orbitals of equal energy. Its spectral properties are altered by an inter-layer bias potential, which can open a gap between the constituent orbitals. We establish the dependence of the one and two-particle disorder-averaged Greens' functions on inter-layer bias, and show that the longitudinal conductivity exhibits the signature of disorder-induced Landau orbital mixing.   

Transverse Thermoelectric Conductivity of Bi-layer graphene in quantum Hall Regime

We performed electric and thermoelectric transport measurements of bilayer graphene in a magnetic field up to 15 Tesla. The transverse thermoelectric conductivity $\rm\alpha_{xy}$, determined from four transport coefficients, attains a peak value of $\rm\alpha_{xy, peak}$ whenever chemical potential lies in the center of a Landau level. The temperature dependence of $\rm\alpha_{xy, peak}$ is dictated by the disorder width $\rm W_L$. For $\rm k_BT/W_L\leq$0.2, $\rm\alpha_ {xy, peak}$ is nominally linear in temperature, which gives $\rm\alpha_{xy,peak}/T=0.19 \pm 0.03 n A/K^2$ independent of the magnetic field, temperature and Landau Level index. At $\rm k_BT/W_L\geq$0.5, $\rm\alpha_{xy, peak}$ saturates to a value close to the predicted universal value of $\rm 4\times(ln2) k_Be/h$ according to the theory of Girvin and Jonson. We remark that an anomaly is found in $\rm\alpha_{xy}$ near the charge neutral point, similar to that in single-layer graphene.   

Interface Landau levels in graphene monolayer-bilayer junction

Electronic structure of graphene monolayer-bilayer junction in a magnetic field is studied within an effective-mass approximation. The energy spectrum is characterized by interface Landau levels, i.e., the locally flat bands appearing near the boundary region, resulting in a series of characteristic peaks in the local density of states. Their energies are independent of boundary types such as zigzag or armchair. In the atomic scale, the local density of states shows a Kekul\'{e} pattern due to the valley mixing in the armchair boundary, while does not in the zigzag boundary.