Quantum Hall effect and the topological number in graphene

Abstract

Recently unusual integer quantum Hall effect was observed in graphene in which the Hall conductivity is quantized as ${\sigma }_{xy}=(\pm 2,\pm 6,\pm 10,>...)\times \frac{e^2}{h}$, where $e$ is the electron charge and $h$ is the Planck constant. %\cite{Novoselov2005,Zheng2005}, %although it can be explained in the argument of massless Dirac fermions, To explain this we consider the energy structure as a function of magnetic field (the Hofstadter butterfly diagram) on the honeycomb lattice and the Streda formula for Hall conductivity. The quantized Hall conductivity is obtained to be odd integer, $\pm 1,\pm 3,\pm 5,...$ times two (spin degrees of freedom) when a uniform magnetic field is as high as 30T for example. When the system is anisotropic and described by the generalized honeycomb lattice, Hall conductivity can be quantized to be any integer number. We also compare the results with those for the square lattice under extremely strong magnetic field.

4 pages, 10 figures

References in corpus (7)

• Two-Dimensional Gas of Massless Dirac Fermions in Graphene
• Experimental Observation of Quantum Hall Effect and Berry's Phase in Graphene
• Electronic Properties of Disordered Two-Dimensional Carbon
• Unconventional Integer Quantum Hall effect in graphene
• Zero modes of tight binding electrons on the honeycomb lattice
• Quantized Hall conductivity of Bloch electrons: topology and the Dirac fermion
• Band Collapse and the Quantum Hall Effect in Graphene

Cited by in corpus (40)

• A new magnetic field dependence of Landau levels on a graphene like structure
• A universal Hamiltonian for the motion and the merging of Dirac cones in a two-dimensional crystal
• Zero modes and the edge states of the honeycomb lattice
• Theory of the Magnetic-Field-Induced Insulator in Neutral Graphene
• Spontaneous Symmetry Breaking and Quantum Hall Effect in Graphene
• Hofstadter butterflies of bilayer graphene
• Proposal of a Cold-atom Realization of Quantum Maps with Hofstadter's Butterfly Spectrum
• Quantum field theory on toroidal topology: algebraic structure and applications
• Unusual field and temperature dependence of Hall effect in graphene
• Inhomogeneous magnetic catalysis on graphene's honeycomb lattice
• Periodic Landau gauge and Quantum Hall effect in twisted bilayer graphene
• Hofstadter butterflies and magnetically induced band gap quenching in graphene antidot lattices
• Self-similar occurrence of massless Dirac particles in graphene under magnetic field
• Quantum oscillations of magnetization in the tight-binding electrons on honeycomb lattice
• Quantum Hall Effect in Bilayer Graphene: Disorder Effect and Quantum Phase Transition
• Zero modes, energy gap, and edge states of anisotropic honeycomb lattice in a magnetic field
• Quantum Hall Effects with High Spin-Chern Numbers in Buckled Honeycomb Structure with Magnetic Order
• Topologically Universal Spectral Hierarchies of Quasiperiodic Systems
• The Hall conductance, topological quantum phase transition and the Diophantine equation on honeycomb lattice
• Entanglement Spectra and Entanglement Thermodynamics of Hofstadter Bilayers
• On the Role of Symmetries in the Theory of Photonic Crystals
• Magnetic vortex liquid and anomalous thermal Hall conductivity in frustrated magnets with bond-dependent interactions
• Magnonic Floquet Hofstadter Butterfly
• Three-quarter Dirac points, Landau levels and magnetization in $\alpha$-(BEDT-TTF)${}_2$I${}_3$
• Electric-field induced penetration of edge states at the interface between monolayer and bilayer graphene
• Single Dirac-Cone State and Quantum Hall Effects in Honeycomb Structure
• Driven Hofstadter Butterflies and Related Topological Invariants
• Edge states in a honeycomb lattice: effects of anisotropic hopping and mixed edges
• Hofstadter problem in higher dimensions
• De Haas-van Alphen oscillations near the Lifshitz transition from two electron pockets to one electron pocket in the two-dimensional Dirac fermion systems
• Quantum Hall conductance and de Haas van Alphen oscillation in a tight-binding model with electron and hole pockets for (TMTSF)${}_2$NO${}_3$
• Floquet engineering the Hofstadter butterfly in the square lattice and its effective Hamiltonian
• Quantized Hall conductance in graphene by nonperturbative magnetic-field-containing relativistic tight-binding approximation method
• Conventional and unconventional anomalous velocities in multiband systems
• A magnetic Hofstadter butterfly and its topologically quantized Hall conductance
• Landau levels on a surface of weak topological insulators
• Gauge fields, quantized fluxes and monopole confinement of the honeycomb lattice
• Merged Four Dirac Points at the Critical Interlayer Distance in Commensurately Twisted Bilayer Graphene: the Origin of the Zero Velocity
• Hofstadter Butterfly in Graphene
• Hofstadter Butterfly and Broken-Symmetry Quantum Hall States in α-Type Organic Dirac Fermion Systems