Intra-Landau-level cyclotron resonance in bilayer graphene

Intra-Zero-Energy Landau Level Crossings in Bilayer Graphene at High Electric Fields

The highly tunable band structure of the zero-energy Landau level (zLL) of bilayer graphene makes it an ideal platform for engineering novel quantum states. However, the zero-energy Landau level at high electric fields has remained largely unexplored. Here we present magnetotransport measurements of bilayer graphene in high transverse electric fields. We observe previously undetected Landau level crossings at filling factors ν = -2, 1, and 3 at high electric fields. These crossings provide constraints for theoretical models of the zero-energy Landau level and show that the orbital, valley, and spin character of the quantum Hall states at high electric fields is very different from low electric fields. At high E, new transitions between states at ν = -2 with different orbital and spin polarization can be controlled by the gate bias, while the transitions between ν = 0 → 1 and ν = 2 → 3 show anomalous behavior.

Weak-Field Hall Resistivity and Spin-Valley Flavor Symmetry Breaking in Magic-Angle Twisted Bilayer Graphene

Near a magic twist angle, the lowest energy conduction and valence bands of bilayer graphene moiré superlattices become extremely narrow. The band dispersion that remains is sensitive to the moiré's strain pattern, nonlocal tunneling between layers, and filling-factor-dependent Hartree and exchange band renormalizations. In this Letter, we analyze the influence of these band-structure details on the pattern of flavor symmetry breaking observed in this narrow band system and on the associated pattern of Fermi surface reconstructions revealed by weak-field Hall and Shubnikov-de Haas magnetotransport measurements.

Quantum anomalous Hall octet driven by orbital magnetism in bilayer graphene

The quantum anomalous Hall (QAH) effect-a macroscopic manifestation of chiral band topology at zero magnetic field-has been experimentally realized only by the magnetic doping of topological insulators^1-3 and the delicate design of moiré heterostructures^4-8. However, the seemingly simple bilayer graphene without magnetic doping or moiré engineering has long been predicted to host competing ordered states with QAH effects^9-11. Here we explore states in bilayer graphene with a conductance of 2 e² h^-1 (where e is the electronic charge and h is Planck's constant) that not only survive down to anomalously small magnetic fields and up to temperatures of five kelvin but also exhibit magnetic hysteresis. Together, the experimental signatures provide compelling evidence for orbital-magnetism-driven QAH behaviour that is tunable via electric and magnetic fields as well as carrier sign. The observed octet of QAH phases is distinct from previous observations owing to its peculiar ferrimagnetic and ferrielectric order that is characterized by quantized anomalous charge, spin, valley and spin-valley Hall behaviour⁹.

Exact Diagonalization for Magic-Angle Twisted Bilayer Graphene

We report on finite-size exact-diagonalization calculations in a Hilbert space defined by the continuum-model flat moiré bands of magic angle twisted bilayer graphene. For moiré band filling 3>|ν|>2, where superconductivity is strongest, we obtain evidence that the ground state is a spin ferromagnet. Near |ν|=3, we find Chern insulator ground states that have spontaneous spin, valley, and sublattice polarization, and demonstrate that the anisotropy energy in this order-parameter space is strongly band-filling-factor dependent. We emphasize that inclusion of the remote band self-energy is necessary for a reliable description of magic angle twisted bilayer graphene flat band correlations.

Quantum Hall valley nematics

Two-dimensional electron gases in strong magnetic fields provide a canonical platform for realizing a variety of electronic ordering phenomena. Here we review the physics of one intriguing class of interaction-driven quantum Hall states: quantum Hall valley nematics. These phases of matter emerge when the formation of a topologically insulating quantum Hall state is accompanied by the spontaneous breaking of a point-group symmetry that combines a spatial rotation with a permutation of valley indices. The resulting orientational order is particularly sensitive to quenched disorder, while quantum Hall physics links charge conduction to topological defects. We discuss how these combine to yield a rich phase structure, and their implications for transport and spectroscopy measurements. In parallel, we discuss relevant experimental systems. We close with an outlook on future directions.

Fractional and Symmetry-Broken Chern Insulators in Tunable Moiré Superlattices

We study dual-gated graphene bilayer/hBN moiré superlattices. Under zero magnetic field, we observe additional resistance peaks as the charge density varies. The peaks' resistivities vary approximately quadratically with an applied perpendicular displacement field D. Data fit to a continuum model yield a bilayer/hBN interaction energy scale ∼30 ± 10 meV. Under a perpendicular magnetic field, we observe Hofstadter butterfly spectra as well as symmetry-broken and fractional Chern insulator states. Their topology and lattice symmetry breaking is D-tunable, enabling the realization of new topological states in this system.

Counterpropagating Fractional Hall States in Mirror-Symmetric Dirac Semimetals

The Landau bands of mirror symmetric 2D Dirac semimetals (e.g., odd layers of ABA graphene) can be identified by their parity with respect to mirror symmetry. This symmetry facilitates a new class of counterpropagating Hall states. We predict the presence of a Laughlin-like correlated liquid state, at the charge neutrality point, with opposite but equal electron and hole filling factors |ν_{±}|=1/m (m odd). This state exhibits fractionally charged quasiparticle-hole pair excitations and counterpropagating edge states with opposite parity. Using a bosonized one-dimensional edge state theory, we show that the fractionally quantized two-terminal longitudinal conductance, σ_{xx}=2e^{2}/(mh), is robust to short-ranged intermode interactions.

Observation of Anomalous Resistance Behavior in Bilayer Graphene

Our measurement results have shown that bilayer graphene exhibits an unexpected sharp transition of the resistance value in the temperature region 200~250 K. We argue that this behavior originates from the interlayer ripple scattering effect between the top and bottom ripple graphene layer. The inter-scattering can mimic the Coulomb scattering but is strongly dependent on temperature. The observed behavior is consistent with the theoretical prediction that charged impurities are the dominant scatters in bilayer graphene. The resistance increase with increasing perpendicular magnetic field strongly supports the postulate that magnetic field induces an excitonic gap in bilayer graphene. Our results reveal that the relative change of resistance induced by magnetic field in the bilayer graphene shows an anomalous thermally activated property.

Direct measurement of discrete valley and orbital quantum numbers in bilayer graphene

The high magnetic field electronic structure of bilayer graphene is enhanced by the spin, valley isospin, and an accidental orbital degeneracy, leading to a complex phase diagram of broken symmetry states. Here, we present a technique for measuring the layer-resolved charge density, from which we directly determine the valley and orbital polarization within the zero energy Landau level. Layer polarization evolves in discrete steps across 32 electric field-tuned phase transitions between states of different valley, spin, and orbital order, including previously unobserved orbitally polarized states stabilized by skew interlayer hopping. We fit our data to a model that captures both single-particle and interaction-induced anisotropies, providing a complete picture of this correlated electron system. The resulting roadmap to symmetry breaking paves the way for deterministic engineering of fractional quantum Hall states, while our layer-resolved technique is readily extendable to other two-dimensional materials where layer polarization maps to the valley or spin quantum numbers.

The phase diagram of bilayer graphene at high magnetic fields has been an outstanding question, with orders possibly between multiple internal quantum degrees of freedom. Here, Hunt et al. report the measurement of the valley and orbital order, allowing them to directly reconstruct the phase diagram.

Thermal Transport Signatures of Broken-Symmetry Phases in Graphene

In the half filled zero-energy Landau level of bilayer graphene, competing phases with spontaneously broken symmetries and an intriguing quantum critical behavior have been predicted. Here we investigate signatures of these broken-symmetry phases in thermal transport measurements. To this end, we calculate the spectrum of spin and valley waves in the ν=0 quantum Hall state of bilayer graphene. The presence of Goldstone modes enables heat transport even at low temperatures, which can serve as compelling evidence for spontaneous symmetry breaking. By varying external electric and magnetic fields, it is possible to determine the nature of the symmetry breaking. Temperature-dependent measurements may yield additional information about gapped modes.

Layer Polarizability and Easy-Axis Quantum Hall Ferromagnetism in Bilayer Graphene

We report magnetotransport measurements of graphene bilayers at large perpendicular electric displacement fields, up to ∼1.5 V/nm, where we observe crossings between Landau levels with different orbital quantum numbers. The displacement fields at the studied crossings are primarily determined by energy shifts originating from the Landau level layer polarizability or polarization. Despite decreasing Landau level spacing with energy, successive crossings occur at larger displacement fields, resulting from decreasing polarizability with orbital quantum number. For particular crossings we observe resistivity hysteresis in displacement field, indicating the presence of a first-order transition between states exhibiting easy-axis quantum Hall ferromagnetism.

Energy Gaps and Layer Polarization of Integer and Fractional Quantum Hall States in Bilayer Graphene

Owing to the spin, valley, and orbital symmetries, the lowest Landau level in bilayer graphene exhibits multicomponent quantum Hall ferromagnetism. Using transport spectroscopy, we investigate the energy gaps of integer and fractional quantum Hall (QH) states in bilayer graphene with controlled layer polarization. The state at filling factor ν=1 has two distinct phases: a layer polarized state that has a larger energy gap and is stabilized by high electric field, and a hitherto unobserved interlayer coherent state with a smaller gap that is stabilized by large magnetic field. In contrast, the ν=2/3 quantum Hall state and a feature at ν=1/2 are only resolved at finite electric field and large magnetic field. These results underscore the importance of controlling layer polarization in understanding the competing symmetries in the unusual QH system of BLG.

Multicomponent Quantum Hall Ferromagnetism and Landau Level Crossing in Rhombohedral Trilayer Graphene

Using transport measurements, we investigate multicomponent quantum Hall (QH) ferromagnetism in dual-gated rhombohedral trilayer graphene (r-TLG) in which the real spin, orbital pseudospin, and layer pseudospins of the lowest Landau level form spontaneous ordering. We observe intermediate QH plateaus, indicating a complete lifting of the degeneracy of the zeroth Landau level (LL) in the hole-doped regime. In charge neutral r-TLG, the orbital degeneracy is broken first, and the layer degeneracy is broken last and only in the presence of an interlayer potential U⊥. In the phase space of U⊥ and filling factor ν, we observe an intriguing "hexagon" pattern, which is accounted for by a model based on crossings between symmetry-broken LLs.

Fractional Quantum Hall States in Bilayer Graphene Probed by Transconductance Fluctuations

We have investigated fractional quantum Hall (QH) states in Bernal-stacked bilayer graphene using transconductance fluctuation measurements. A variety of odd-denominator fractional QH states with νQH → νQH + 2 symmetry, as previously reported, are observed. However, surprising is that also particle-hole symmetric states are clearly resolved in the same measurement set. We attribute their emergence to the reversal of orbital states in the octet level scheme induced by a strong local charge imbalance, which can be captured by the transconductance fluctuations. Also the even-denominator fractional QH state at filling -1/2 is observed. However, contrary to a previous study on a suspended graphene layer [ Ki et al. Nano Lett. 2014, 14 , 2135 ], the particle-hole symmetric state at filling 1/2 is detected as well. These observations suggest that the stability of both odd and even denominator fractional QH states is very sensitive to local transverse electric fields in bilayer graphene.

Tunable fractional quantum Hall phases in bilayer graphene

Symmetry-breaking in a quantum system often leads to complex emergent behavior. In bilayer graphene (BLG), an electric field applied perpendicular to the basal plane breaks the inversion symmetry of the lattice, opening a band gap at the charge neutrality point. In a quantizing magnetic field, electron interactions can cause spontaneous symmetry-breaking within the spin and valley degrees of freedom, resulting in quantum Hall effect (QHE) states with complex order. Here, we report fractional QHE states in BLG that show phase transitions that can be tuned by a transverse electric field. This result provides a model platform with which to study the role of symmetry-breaking in emergent states with topological order.

Transport measurement of Landau level gaps in bilayer graphene with layer polarization control

Landau level (LL) gaps are important parameters for understanding electronic interactions and symmetry-broken processes in bilayer graphene (BLG). Here we present transport spectroscopy measurements of LL gaps in double-gated suspended BLG with high mobilities in the quantum Hall regime. By using bias as a spectroscopic tool, we measure the gap Δ for the quantum Hall (QH) state at filling factors ν = ±4 and -2. The single-particle Δ(ν=4) scales linearly with magnetic field B and is independent of the out-of-plane electric field E⊥. For the symmetry-broken ν = -2 state, the measured values of Δ(ν=-2) are ∼1.1 meV/T and 0.17 meV/T for singly gated geometry and dual-gated geometry at E⊥ = 0, respectively. The difference between the two values arises from the E⊥. dependence of Δ(ν=-2), suggesting that the ν = -2 state is layer polarized. Our studies provide the first measurements of the gaps of the broken symmetry QH states in BLG with well-controlled E⊥ and establish a robust method that can be implemented for studying similar states in other layered materials.

Stacking boundaries and transport in bilayer graphene

Pristine bilayer graphene behaves in some instances as an insulator with a transport gap of a few millielectronvolts. This behavior has been interpreted as the result of an intrinsic electronic instability induced by many-body correlations. Intriguingly, however, some samples of similar mobility exhibit good metallic properties with a minimal conductivity of the order of 2e(2)/h. Here, we propose an explanation for this dichotomy, which is unrelated to electron interactions and based instead on the reversible formation of boundaries between stacking domains ("solitons"). We argue, using a numerical analysis, that the hallmark features of the previously inferred many-body insulating state can be explained by scattering on boundaries between domains with different stacking order (AB and BA). We furthermore present experimental evidence, reinforcing our interpretation, of reversible switching between a metallic and an insulating regime in suspended bilayers when subjected to thermal cycling or high current annealing.

Observation of even denominator fractional quantum Hall effect in suspended bilayer graphene

We investigate low-temperature magneto-transport in recently developed, high-quality multiterminal suspended bilayer graphene devices, enabling the independent measurement of the longitudinal and transverse resistance. We observe clear signatures of the fractional quantum Hall effect with different states that are either fully developed, and exhibit a clear plateau in the transverse resistance with a concomitant dip in longitudinal resistance or incipient, and exhibit only a longitudinal resistance minimum. All observed states scale as a function of filling factor ν, as expected. An unprecedented even-denominator fractional state is observed at ν = -1/2 on the hole side, exhibiting a clear plateau in Rxy quantized at the expected value of 2h/e(2) with a precision of ∼0.5%. Many of our observations, together with a recent electronic compressibility measurement performed in graphene bilayers on hexagonal boron-nitride (hBN) substrates, are consistent with a recent theory that accounts for the effect of the degeneracy between the N = 0 and N = 1 Landau levels in the fractional quantum Hall effect and predicts the occurrence of a Moore-Read type ν = -1/2 state. Owing to the experimental flexibility of bilayer graphene, which has a gate-dependent band structure, can be easily accessed by scanning probes, and can be contacted with materials such as superconductors, our findings offer new possibilities to explore the microscopic nature of even-denominator fractional quantum Hall effect.

Topological phases in the zeroth Landau level of bilayer graphene

We analyze the phase diagram of the zeroth Landau level of bilayer graphene, taking into account the realistic effects of screening of the Coulomb interaction and strong mixing between two degenerate sublevels. We identify robust quantum Hall states at filling factors ν=-1, -4/3, -5/3, -8/5, -1/2 and discuss the nature of their ground states, collective excitations, and relation to the more familiar states in GaAs using a tractable model. In particular, we present evidence that the ν=-1/2 state is non-Abelian and described by either the Moore-Read wave function or its particle-hole conjugate, while ruling out other candidates such as the 331 state.

The electronic properties of bilayer graphene

We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energies. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects.

Quantum Hall to charge-density-wave phase transitions in ABC-trilayer graphene

ABC-stacked trilayer graphene's chiral band structure results in three (n=0, 1, 2) Landau level orbitals with zero kinetic energy. This unique feature has important consequences on the interaction-driven states of the 12-fold degenerate (including spin and valley) N=0 Landau level. In particular, at many filling factors ν(T) = ±5, ±4, ±2, ±1 a quantum phase transition from a quantum Hall liquid state to a triangular charge-density wave occurs as a function of the single particle-induced Landau level orbital splitting Δ(LL). Experimental signatures of this phase transition are also discussed.

Canted antiferromagnetic phase of the ν=0 quantum Hall state in bilayer graphene

Motivated to understand the nature of the strongly insulating ν=0 quantum Hall state in bilayer graphene, we develop the theory of the state in the framework of quantum Hall ferromagnetism. The generic phase diagram, obtained in the presence of the isospin anisotropy, perpendicular electric field, and Zeeman effect, consists of the spin-polarized ferromagnetic (F), canted antiferromagnetic (CAF), and partially (PLP) and fully (FLP) layer-polarized phases. We address the edge transport properties of the phases. Comparing our findings with the recent data on suspended dual-gated devices, we conclude that the insulating ν=0 state realized in bilayer graphene at lower electric field is the CAF phase. We also predict a continuous and a sharp insulator-metal phase transition upon tilting the magnetic field from the insulating CAF and FLP phases, respectively, to the F phase with metallic edge conductance 2e(2)/h, which could be within the reach of available fields and could allow one to identify and distinguish the phases experimentally.

Quantum Hall effects in graphene-based two-dimensional electron systems

In this article we review the quantum Hall physics of graphene-based two-dimensional electron systems, with a special focus on recent experimental and theoretical developments. We explain why graphene and bilayer graphene can be viewed respectively as J D 1 and 2 chiral two-dimensional electron gases (C2DEGs), and why this property frames their quantum Hall physics. The current status of experimental and theoretical work on the role of electron-electron interactions is reviewed at length with an emphasis on unresolved issues in the field, including the role of disorder in current experiments. Special attention is given to the interesting low magnetic field limit, and to the relationship between quantum Hall effects and the spontaneous anomalous Hall effects that might occur in bilayer graphene systems in the absence of a magnetic field.

Transport spectroscopy of symmetry-broken insulating states in bilayer graphene

Bilayer graphene is an attractive platform for studying new two-dimensional electron physics, because its flat energy bands are sensitive to out-of-plane electric fields and these bands magnify electron-electron interaction effects. Theory predicts a variety of interesting broken symmetry states when the electron density is at the carrier neutrality point, and some of these states are characterized by spontaneous mass gaps, which lead to insulating behaviour. These proposed gaps are analogous to the masses generated by broken symmetries in particle physics, and they give rise to large Berry phase effects accompanied by spontaneous quantum Hall effects. Although recent experiments have provided evidence for strong electronic correlations near the charge neutrality point, the presence of gaps remains controversial. Here, we report transport measurements in ultraclean double-gated bilayer graphene and use source-drain bias as a spectroscopic tool to resolve a gap of ∼2 meV at the charge neutrality point. The gap can be closed by a perpendicular electric field of strength ∼15 mV nm(-1), but it increases monotonically with magnetic field, with an apparent particle-hole asymmetry above the gap. These data represent the first spectroscopic mapping of the ground states in bilayer graphene in the presence of both electric and magnetic fields.

Tunable electron interactions and fractional quantum Hall States in graphene

The recent discovery of fractional quantum Hall (FQH) states in graphene raises the question of whether the physics of graphene offers any advantages over GaAs-based materials in exploring strongly correlated states of two-dimensional electrons. Here we propose a method to continuously tune the effective electron interactions in graphene and its bilayer by the dielectric environment of the sample. Using this method, the charge gaps of prominent FQH states, including ν=1/3 or ν=5/2 states, can be increased several times, or reduced to zero. The tunability of the interactions can be used to realize and stabilize various strongly correlated phases and explore the transitions between them.

Spin-polarized to valley-polarized transition in graphene bilayers at ν=0 in high magnetic fields

We investigate the transverse electric field (E) dependence of the ν=0 quantum Hall state (QHS) in dual-gated graphene bilayers in high magnetic fields. The longitudinal resistivity ρ(xx) measured at ν=0 shows an insulating behavior which is strongest in the vicinity of E=0, as well as at large E fields. At a fixed perpendicular magnetic field (B), the ν=0 QHS undergoes a transition as a function of the applied E, marked by a minimum, temperature-independent ρ(xx). This observation is explained by a transition from a spin-polarized ν=0 QHS at small E fields to a valley- (layer-)polarized ν=0 QHS at large E fields. The E field value at which the transition occurs follows a linear dependence on B.

Spontaneous quantum Hall states in chirally stacked few-layer graphene systems

Chirally stacked N-layer graphene systems with N≥2 exhibit a variety of distinct broken symmetry states in which charge density contributions from different spins and valleys are spontaneously transferred between layers. We explain how these states are distinguished by their charge, spin, and valley Hall conductivities, by their orbital magnetizations, and by their edge state properties. We argue that valley Hall states have [N/2] edge channels per spin valley.

Effect of magnetic field on the electronic transport in trilayer graphene

The perpendicular magnetic field dependence of the longitudinal resistance in trilayer graphene at various temperatures has been systematically studied. For a fixed magnetic field, the trilayer graphene displays an intrinsic semiconductor behavior over the temperature range of 5-340 K. This is attributed to the parabolic band structure of trilayer graphene, where the Coulomb scattering is a strong function of temperature. The dependence of resistance on the magnetic field can be explained by the splitting of Landau levels (LLs). Our results reveal that the energy gap in the trilayer graphene is thermally activated and increases with √B.

Local compressibility measurements of correlated states in suspended bilayer graphene

Bilayer graphene has attracted considerable interest due to the important role played by many-body effects, particularly at low energies. Here we report local compressibility measurements of a suspended graphene bilayer. We find that the energy gaps at filling factors ν= ± 4 do not vanish at low fields, but instead merge into an incompressible region near the charge neutrality point at zero electric and magnetic field. These results indicate the existence of a zero-field ordered state and are consistent with the formation of either an anomalous quantum Hall state or a nematic phase with broken rotational symmetry. At higher fields, we measure the intrinsic energy gaps of broken-symmetry states at ν=0, ± 1, and ± 2, and find that they scale linearly with magnetic field, yet another manifestation of the strong Coulomb interactions in bilayer graphene.

Broken-symmetry states in doubly gated suspended bilayer graphene

The single-particle energy spectra of graphene and its bilayer counterpart exhibit multiple degeneracies that arise through inherent symmetries. Interactions among charge carriers should spontaneously break these symmetries and lead to ordered states that exhibit energy gaps. In the quantum Hall regime, these states are predicted to be ferromagnetic in nature, whereby the system becomes spin polarized, layer polarized, or both. The parabolic dispersion of bilayer graphene makes it susceptible to interaction-induced symmetry breaking even at zero magnetic field. We investigated the underlying order of the various broken-symmetry states in bilayer graphene suspended between top and bottom gate electrodes. We deduced the order parameter of the various quantum Hall ferromagnetic states by controllably breaking the spin and sublattice symmetries. At small carrier density, we identified three distinct broken-symmetry states, one of which is consistent with either spontaneously broken time-reversal symmetry or spontaneously broken rotational symmetry.

Anomalous exciton condensation in graphene bilayers

In ordinary semiconductor bilayers, exciton condensates appear at total Landau-level filling factor nu{T}=1. We predict that similar states will occur in Bernal stacked graphene bilayers at many nonzero integer filling factors. For nu{T}=-3, 1 we find that the superfluid density of the exciton condensate vanishes and that a finite-temperature fluctuation-induced first order isotropic-smectic phase transition occurs when the layer densities are not balanced. These anomalous properties of bilayer graphene exciton condensates are due to the degeneracy of Landau levels with n=0 and n=1 orbital character.

Symmetry breaking in the zero-energy Landau level in bilayer graphene

The quantum Hall effect near the charge neutrality point in bilayer graphene is investigated in high magnetic fields of up to 35 T using electronic transport measurements. In the high-field regime, the eightfold degeneracy in the zero-energy Landau level is completely lifted, exhibiting new quantum Hall states corresponding to filling factors nu=0, 1, 2, and 3. Measurements of the activation energy gaps for the nu=2 and 3 filling factors in tilted magnetic fields exhibit no appreciable dependence on the in-plane magnetic field, suggesting that these Landau level splittings are independent of spin. In addition, measurements taken at the nu=0 charge neutral point show that, similar to single layer graphene, the bilayer becomes insulating at high fields.

Charge 2e skyrmions in bilayer graphene

Quantum Hall states that result from interaction induced lifting of the eightfold degeneracy of the zeroth Landau level in bilayer graphene are considered. We show that at even filling factors electric charge is injected into the system in the form of charge 2e Skyrmions. This is a rare example of binding of charges in a system with purely repulsive interactions. We calculate the Skyrmion energy and size as a function of the effective Zeeman interaction and discuss the signatures of the charge 2e Skyrmions in the scanning probe experiments.

Long-range Coulomb interaction in bilayer graphene

We report on our studies of interacting electrons in bilayer graphene in a magnetic field. We demonstrate that the long-range Coulomb interactions between electrons in this material are highly important and account for the band asymmetry in recent optical magneto-absorption experiments [E. A. Henriksen, Phys. Rev. Lett. 100, 087403 (2008)10.1103/PhysRevLett.100.087403]. We show that in the unbiased bilayer (where both layers are at the same electrostatic potential), the interactions can cause mixing of Landau levels in moderate magnetic fields. For the biased bilayer (when the two layers are at different potentials), we demonstrate that the interactions are responsible for a change in the total spin of the ground state for half-filled Landau levels in the valence band.

Open and closed edges of graphene layers

Edge structures of thermally treated graphite have been studied by means of atomically resolved high-resolution TEM. The method for the determination of a monolayer or more than one layer graphene sheets is established. A series of tilting experiments proves that the zigzag and armchair edges are mostly closed between adjacent graphene layers, and the number of dangling bonds is therefore minimized. Surprisingly bilayer graphene often exhibits AA stacking and is very hard to distinguish from a single graphene layer. Open edge structures with carbon dangling bonds can be found only in a local area where the closed (folding) edge is partially broken.