Magnetoresistance measurements of graphene at the charge neutrality point

We report on transport measurements of the insulating state that forms at the charge neutrality point of graphene in a magnetic field. Using both conventional two-terminal measurements, sensitive to bulk and edge conductance, and Corbino measurements, sensitive only to the bulk conductance, we observed a vanishing conductance with increasing magnetic fields. By examining the resistance changes of this insulating state with varying perpendicular and in-plane fields, we probe the spin-active components of the excitations in total fields of up to 45?T. Our results indicate that the ν=0 quantum Hall state in single layer graphene is not spin-polarized.     

Landau level spectrum for bilayer graphene in a tilted magnetic field

We consider a graphene bilayer in a constant magnetic field of arbitrary orientation, i.e., tilted with respect to the graphene plane. In the low energy approximation to the tight-binding model with Peierls substitution, we find the Landau level spectrum analytically in terms of spheroidal functions and the respective eigenvalues. We compare our result to the perpendicular and purely in-plane field cases. In the limit of perpendicular field we reproduce the known equidistant spectrum for Landau levels. In the opposite limit of large in-plane field this spectrum becomes two-fold degenerate, which is a consequence of Dirac point splitting induced by the in-plane field.     

Quantum Hall activation gaps in bilayer graphene

We have measured the quantum Hall activation gaps in bilayer graphene at filling factors $\nu =\pm 4$ and $\nu =\pm 8$ in high magnetic fields up to 30 T. We find that energy levels can be described by a 4-band relativistic hyperbolic dispersion. The Landau level width is found to contain a field independent background due to an intrinsic level broadening and a component which increases linearly with magnetic field.     

Charge density excitations in bilayer graphene in high magnetic field

The charge-density excitations in bilayer graphene at the filling-factor ν 1 at small momenta are considered in the frame of the Hartree-Fock approximation. The presence of small asymmetry of graphene layers is included. The dependence of the magnetoplasmon energy on the bilayer ground state is shown. The energy splitting proportional to $ \sqrt H $ for the symmetric case with half-filled zero-energy levels is found both for bilayer and monolayer graphene.     

Energy gaps in graphene nanoribbons

Based on a first-principles approach, we present scaling rules for the band gaps of graphene nanoribbons (GNRs) as a function of their widths. The GNRs considered have either armchair or zigzag shaped edges on both sides with hydrogen passivation. Both varieties of ribbons are shown to have band gaps. This differs from the results of simple tight-binding calculations or solutions of the Dirac's equation based on them. Our ab initio calculations show that the origin of energy gaps for GNRs with armchair shaped edges arises from both quantum confinement and the crucial effect of the edges. For GNRs with zigzag shaped edges, gaps appear because of a staggered sublattice potential on the hexagonal lattice due to edge magnetization. The rich gap structure for ribbons with armchair shaped edges is further obtained analytically including edge effects. These results reproduce our ab initio calculation results very well.     

QHE and far infra-red properties of bilayer graphene in a strong magnetic field

We describe the quantum Hall effect (QHE) and far infra-red (FIR) absorption properties of bilayer graphene in a strong magnetic field and contrast them with the weak field regime. This includes a derivation of the effective low energy Hamiltonian for this system and the consequences of this Hamiltonian for the sequencing of the Landau levels in the material: The form of this effective Hamiltonian gives rise to the presence of a level with doubled degeneracy at zero energy. The effect of a potential difference between the layer of a bilayer is also investigated. It is found that there is a density-dependent gap near the K points in the band structure. The consequences of this gap on the QHE are then described. Also, the magneto-absorption spectrum is investigated and an experiment proposed to distinguish between model ground states of the bilayer quantum Hall effect system based on the different absorption characteristics of right-handed and left-handed polarisation of FIR light. Finally, the effects of trigonal warping are taken into account in the absorption picture.     

Landau levels in deformed bilayer graphene at low magnetic fields

We review the effect of uniaxial strain on the low-energy electronic dispersion and Landau level structure of bilayer graphene. Based on the tight-binding approach, we derive a strain-induced term in the low-energy Hamiltonian and show how strain affects the low-energy electronic band structure. Depending on the magnitude and direction of applied strain, we identify three regimes of qualitatively different electronic dispersions. We also show that in a weak magnetic field, sufficient strain results in the filling factor ν=±4 being the most stable in the quantum Hall effect measurement, instead of ν=±8 in unperturbed bilayer at a weak magnetic field. To mention, in one of the strain regimes, the activation gap at ν=±4 is, down to very low fields, weakly dependent on the strength of the magnetic field.     

Effect of a nonuniform magnetic field on the Landau states in a biased AA-stacked graphene bilayer

We study the Landau states in the biased AA-stacked graphene bilayer under an exponentially decaying magnetic field along one spatial dimension. The results show that the energy eigenvalues of the system are strongly dependent on the inhomogeneity of the magnetic field and the bias voltage between the graphene layers, and in particular the reordering and mixing of finite Landau states could occur. Moreover, we also demonstrate that the current carrying states induced by the decaying magnetic field propagate vertically to the magnetic-field gradient within the graphene sample and can be further modulated by the bias voltage between the layers.     

Luttinger liquid at the edge of undoped graphene in a strong magnetic field

We demonstrate that an undoped two-dimensional carbon plane (graphene) whose bulk is in the integer quantum Hall regime supports a nonchiral Luttinger liquid at an armchair edge. This behavior arises due to the unusual dispersion of the noninteracting edge states, causing a crossing of bands with different valley and spin indices at the edge. We demonstrate that this stabilizes a domain wall structure with a spontaneously ordered phase degree of freedom. This coherent domain wall supports gapless charged excitations, and has a power law tunneling I-V with a nonintegral exponent. In proximity to a bulk lead, the edge may undergo a quantum phase transition between the Luttinger liquid phase and a metallic state.     

Zero-energy state in graphene in a high magnetic field

The fate of the charge-neutral Dirac point in graphene in a high magnetic field H has been investigated at low temperatures (T approximately 0.3 K). In samples with small gate-voltage offset V0, the resistance R0 at the Dirac point diverges steeply with H, signaling a crossover to a state with a very large R0. The approach to this state is highly unusual. Despite the steep divergence in R0, the profile of R0 vs T in fixed H saturates to a T-independent value below 2 K, consistent with gapless charge-carrying excitations.     

Electrical properties of bilayer heterojunctions in a strong magnetic field

The electrical properties of bilayer heterojunctions in a strong magnetic field at low temperatures have been considered. It has been shown that both the ohmic and Hall conductivities decrease exponentially due to the formation of neutral pairs if the electric fields in the two layers are parallel. In the antiparallel fields, the Hall conductivity is still determined by the activation energy of the excited electrons and decreases exponentially, but the ohmic conductivity decreases much slower, proportional to the temperature square.     

Valley filtering due to orbital magnetic moment in bilayer graphene

We investigate the effect of valley-dependent orbital magnetic moment on the transmission of quasiparticles through biased bilayer graphene npn and pnp junctions in the presence of out-of-plane magnetic field. It is shown that the valley-polarized Zeeman-like energy splitting, due to the interaction of orbital magnetic moment with magnetic field, can suppress the transmission of quasiparticles of one valley while transmitting those of the other valley. This valley-selective transmission property can be exploited for valley filtering. We demonstrate that the npn and pnp junction, respectively, filters off the $K^{\prime }$-valley and K-valley particles, with nearly perfect degree of filtration.     

Tunable quantum capacitance and magnetic oscillation in bilayer graphene device

We address the quantum capacitance of a bilayer graphene device in the presence of Rashba spin–orbit interaction (SOI) by applying external magnetic fields and interlayer biases. Quantum capacitance reflects the mixing of the spin-up and spin-down states of Landau levels and can be effectively modulated by the interlayer bias. The interplay between interlayer bias and Rashba SOI strongly affects magnetic oscillations. The typical beating pattern changes tuned by Rashba SOI strength, interlayer bias energy, and temperature are examined as well.     

Reversal of Klein reflection by magnetic barriers in bilayer graphene

Massless Dirac fermions in monolayer graphene exhibit total transmission when normally incident on a scalar potential barrier, a consequence of the Klein paradox originally predicted by O Klein for relativistic electrons obeying the 3 + 1 dimensional Dirac equation. For bilayer graphene, charge carriers are massive Dirac fermions and, due to different chiralities, electron and hole states are not coupled to each other. Therefore, the wavefunction of an incident particle decays inside a barrier as for the non-relativistic Schr?dinger equation. This leads to exponentially small transmission upon normal incidence. We show that, in the presence of magnetic barriers, such massive Dirac fermions can have transmission even at normal incidence. The general consequences of this behavior for multilayer graphene consisting of massless and massive modes are mentioned. We also briefly discuss the effect of a bias voltage on such magnetotransport.     

Impurities in a biased graphene bilayer

We study the problem of impurities and midgap states in a biased graphene bilayer. We show that the properties of the bound states, such as localization lengths and binding energies, can be controlled externally by an electric field effect. Moreover, the band gap is renormalized and impurity bands are created at finite impurity concentrations. Using the coherent potential approximation, we calculate the electronic density of states and its dependence on the applied bias voltage.     

Magnetoplasmons in gapped graphene in a periodically modulated magnetic field

Motivated by recent experiments on long‐lived magnetoplasmons in the presence of a perpendicular magnetic field, we investigate the dynamical dielectric response function of graphene in contact with a substrate using the random phase approximation. We add a periodically modulated magnetic field within the graphene plane and address both the inter and intra Landau band magnetoplasmons. Verification of the predicted magnetic modulation effects is possible by experiments analogous to those for the zero gap limit.     

Effect of electric field and magnetic field on spin transport in bilayer graphene armchair nanoribbons: A Monte Carlo simulation study

In this article we study the effect of external magnetic field and electric field on spin transport in bilayer armchair graphene nanoribbons (GNR) by employing semiclassical Monte Carlo approach. We include D'yakonov-Perel' (DP) relaxation due to structural inversion asymmetry (Rashba spin-orbit coupling) and Elliott-Yafet (EY) relaxation to model spin dephasing. In the model we neglect the effect of local magnetic moments due to adatoms and vacancies. We have considered injection polarization along z-direction perpendicular to the plane of graphene and the magnitude of ensemble averaged spin variation is studied along the x-direction which is the transport direction. To the best of our knowledge there has been no theoretical investigation of the effects of external magnetic field on spin transport in graphene nanoribbons. This theoretical investigation is important in order to identify the factors responsible for experimentally observed spin relaxation length in graphene GNRs.