Local compressibility measurements of correlated states in suspended bilayer graphene

Energy Gap of the Even-Denominator Fractional Quantum Hall State in Bilayer Graphene

Bernal bilayer graphene hosts even-denominator fractional quantum Hall states thought to be described by a Pfaffian wave function with non-Abelian quasiparticle excitations. Here, we report the quantitative determination of fractional quantum Hall energy gaps in bilayer graphene using both thermally activated transport and by direct measurement of the chemical potential. We find a transport activation gap of 5.1 K at B=12  T for a half filled N=1 Landau level, consistent with density matrix renormalization group calculations for the Pfaffian state. However, the measured thermodynamic gap of 11.6 K is smaller than theoretical expectations for the clean limit by approximately a factor of 2. We analyze the chemical potential data near fractional filling within a simplified model of a Wigner crystal of fractional quasiparticles with long-wavelength disorder, explaining this discrepancy. Our results quantitatively establish bilayer graphene as a robust platform for probing the non-Abelian anyons expected to arise as the elementary excitations of the even-denominator state.

Impact of Electric Field Disorder on Broken-Symmetry States in Ultraclean Bilayer Graphene

Bilayer graphene (BLG) has multiple internal degrees of freedom and a constant density of states down to the charge neutrality point when trigonal warping is ignored. Consequently, it is susceptible to various competing ground states. However, a coherent experimental determination of the ground state has been challenging due to the interaction-disorder interplay. Here we present an extensive transport study in a series of dually gated freestanding BLG devices and identify the layer-antiferromagnet as the ground state with a continuous strength across all devices. This strength correlates with the width of the state in the electric field. We systematically identify electric-field disorder─spatial variations in the interlayer potential difference─as the main source responsible for the observations. Our results pinpoint for the first time the importance of electric-field disorder on spontaneous symmetry breaking in BLG and solve a long-standing debate on its ground state. The electric-field disorder should be universal to all 2D materials.

Domain Formation Driven by the Entropy of Topological Edge Modes

In this Letter we study interacting systems with spontaneous discrete symmetry breaking, where the degenerate symmetry-broken states are topologically distinct gapped phases. Edge modes appear at domain walls between the two topological phases. In the presence of a weak disorder field conjugate to the order parameter, we find that the entropy of the edge modes drives a thermal transition between a gapped uniform phase and a phase with a disorder-induced domain structure. We characterize this transition using a phenomenological Landau functional, and corroborate our conclusions with a concrete microscopic model. Finally, we discuss the possibilities of experimental signatures of this phase transition, and propose graphene-based moiré heterostructures as candidate materials in which such a phase transition can be detected.

Fractional Chern insulators in magic-angle twisted bilayer graphene

Fractional Chern insulators (FCIs) are lattice analogues of fractional quantum Hall states that may provide a new avenue towards manipulating non-Abelian excitations. Early theoretical studies1-7 have predicted their existence in systems with flat Chern bands and highlighted the critical role of a particular quantum geometry. However, FCI states have been observed only in Bernal-stacked bilayer graphene (BLG) aligned with hexagonal boron nitride (hBN)8, in which a very large magnetic field is responsible for the existence of the Chern bands, precluding the realization of FCIs at zero field. By contrast, magic-angle twisted BLG9-12 supports flat Chern bands at zero magnetic field13-17, and therefore offers a promising route towards stabilizing zero-field FCIs. Here we report the observation of eight FCI states at low magnetic field in magic-angle twisted BLG enabled by high-resolution local compressibility measurements. The first of these states emerge at 5 T, and their appearance is accompanied by the simultaneous disappearance of nearby topologically trivial charge density wave states. We demonstrate that, unlike the case of the BLG/hBN platform, the principal role of the weak magnetic field is merely to redistribute the Berry curvature of the native Chern bands and thereby realize a quantum geometry favourable for the emergence of FCIs. Our findings strongly suggest that FCIs may be realized at zero magnetic field and pave the way for the exploration and manipulation of anyonic excitations in flat moiré Chern bands.

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.

Tunable Layer Circular Photogalvanic Effect in Twisted Bilayers

We develop a general theory of the layer circular photogalvanic effect (LCPGE) in quasi-two-dimensional chiral bilayers, which refers to the appearance of a polarization-dependent, out-of-plane static dipole moment induced by circularly polarized light. We elucidate the geometric origin of the LCPGE as two types of interlayer coordinate shift weighted by the quantum metric tensor and the Berry curvature, respectively. As a concrete example, we calculate the LCPGE in twisted bilayer graphene, and find that it exhibits a resonance peak whose frequency can be tuned from visible to infrared as the twisting angle varies. The LCPGE thus provides a promising route toward frequency-sensitive, circularly polarized light detection, particularly in the infrared range.

Landau Levels of Topologically-Protected Surface States Probed by Dual-Gated Quantum Capacitance

Spectroscopy of discrete Landau levels (LLs) in bulk-insulating three-dimensional topological insulators (3D TIs) in perpendicular magnetic field characterizes the Dirac nature of their surface states. Despite a number of studies demonstrating the quantum Hall effect (QHE) of topological surface states, quantitative evaluation of the LL energies, which serve as fundamental electronic quantities for study of the quantum states, is still limited. In this work, we explore the density of states of LLs by measuring quantum capacitance (C_Q) in a truly bulk insulating 3D TI via a van der Waals heterostructure configuration. By applying dual-gate voltages, we access the individual surface states' LLs and extract their chemical potentials to quantify the LL spacings of each surface. We evaluate the LLs' energies at two distinguished QH states, namely, dissipationless (ν = ±1) and dissipative (ν = 0) states in the 3D TI.

Tunable Coupling between Surface States of a Three-Dimensional Topological Insulator in the Quantum Hall Regime

The paired top and bottom Dirac surface states, each associated with a half-integer quantum Hall (QH) effect, and a resultant integer QH conductance (νe^{2}/h), are hallmarks of a three-dimensional topological insulator (TI). In a dual-gated system, chemical potentials of the paired surface states are controlled through separate gates. In this work, we establish tunable capacitive coupling between the surface states of a bulk-insulating TI BiSbTeSe_{2} and study the effect of this coupling on QH plateaus and Landau level (LL) fan diagram via dual-gate control. We observe nonlinear QH transitions at low charge density in strongly coupled surface states, which are related to the charge-density-dependent coupling strength. A splitting of the N=0 LL at the charge neutrality point for thin devices (but thicker than the 2D limit) indicates intersurface hybridization possibly beyond single-particle effects. By applying capacitor charging models to the surface states, we explore their chemical potential as a function of charge density and extract the fundamental electronic quantity of LL energy gaps from dual-gated transport measurements. These studies are essential for the realization of exotic quantum effects such as topological exciton condensation.

Tunneling Spectroscopy of Quantum Hall States in Bilayer Graphene p-n Junctions

We report tunneling transport in spatially controlled networks of quantum Hall (QH) edge states in bilayer graphene. By manipulating the separation, location, and spatial span of QH edge states via gate-defined electrostatics, we observe resonant tunneling between copropagating QH states across incompressible strips. Employing spectroscopic tunneling measurements and an analytical model, we characterize the energy gap, width, density of states, and compressibility of the QH edge states with high precision and sensitivity within the same device. The capability to engineer the QH edge network also provides an opportunity to build future quantum electronic devices with electrostatic manipulation of QH edge states, supported by rich underlying physics.

Effective Landau Level Diagram of Bilayer Graphene

The E=0 octet of bilayer graphene in the filling factor range of -4<ν<4 is a fertile playground for many-body phenomena, yet a Landau level diagram is missing due to strong interactions and competing quantum degrees of freedom. We combine measurements and modeling to construct an empirical and quantitative spectrum. The single-particlelike diagram incorporates interaction effects effectively and provides a unified framework to understand the occupation sequence, gap energies, and phase transitions observed in the octet. It serves as a new starting point for more sophisticated calculations and experiments.

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.

Interaction-Induced Dirac Fermions from Quadratic Band Touching in Bilayer Graphene

We revisit the effect of local interactions on the quadratic band touching (QBT) of the Bernal honeycomb bilayer model using renormalization group (RG) arguments and quantum Monte Carlo (QMC) simulations. We present a RG argument which predicts, contrary to previous studies, that weak interactions do not flow to strong coupling even if the free dispersion has a QBT. Instead, they generate a linear term in the dispersion, which causes the interactions to flow back to weak coupling. Consistent with this RG scenario, in unbiased QMC simulations of the Hubbard model we find compelling evidence that antiferromagnetism turns on at a finite U/t despite the U=0 hopping problem having a QBT. The onset of antiferromagnetism takes place at a continuous transition which is consistent with (2+1)D Gross-Neveu criticality. We conclude that generically in models of bilayer graphene, even if the free dispersion has a QBT, small local interactions generate a Dirac phase with no symmetry breaking and that there is a finite-coupling transition out of this phase to a symmetry-broken state.

Graphene homojunction: closed-edge bilayer graphene by pseudospin interaction

Depending on the sublattices they are propagated in, low-energy electrons or holes are labeled with pseudospin. By engineering pseudospin interactions, we propose that two critical features of a junction, i.e., band gap opening and spatial charge separation, can be realized in graphene layers with proper stacking. We also demonstrate theoretically that such a graphene diode may play a role in future pseudospin electronics such as for harvesting solar energy.

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.

Transport in disordered monolayer MoS2 nanoflakes--evidence for inhomogeneous charge transport

We study charge transport in a monolayer MoS2 nanoflake over a wide range of carrier density, temperature and electric bias. We find that the transport is best described by a percolating picture in which the disorder breaks translational invariance, breaking the system up into a series of puddles, rather than previous pictures in which the disorder is treated as homogeneous and uniform. Our work provides insight to a unified picture of charge transport in monolayer MoS2 nanoflakes and contributes to the development of next-generation MoS2-based devices.

Spontaneous layer-pseudospin domain walls in bilayer graphene

Bilayer graphene is susceptible to a family of unusual broken symmetry states with spin and valley dependent layer polarization. We report on a microscopic study of the domain walls in these systems, demonstrating that they have interesting microscopic structure related to the topological character of the ordered states. We use our results to show that the metal-insulator transition temperature in bilayer graphene is reduced from mean-field estimates by thermal excitation of domain walls.

Superconductivity on the brink of spin-charge order in a doped honeycomb bilayer

Using a controlled weak-coupling renormalization group approach, we establish the mechanism of unconventional superconductivity in the vicinity of spin or charge ordered excitonic states for the case of electrons on the Bernal stacked bilayer honeycomb lattice. With one electron per site, this system, physically realized in bilayer graphene, is unstable towards a spontaneous symmetry breaking. Repulsive interactions favor excitonic order, such as a charge nematic and/or a layer antiferromagnet. We find that upon adding charge carriers to the system, the excitonic order is suppressed, and unconventional superconductivity appears in its place, before it is replaced by a Fermi liquid. We focus on firmly establishing this phenomenon using the renormalization group formalism within an idealized model with parabolic touching of conduction and valence bands.

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.

Fast energy relaxation of hot carriers near the Dirac point of graphene

We investigate energy relaxation of hot carriers in monolayer and bilayer graphene devices, demonstrating that the relaxation rate increases significantly as the Dirac point is approached from either the conduction or valence band. This counterintuitive behavior appears consistent with ideas of charge puddling under disorder, suggesting that it becomes very difficult to excite carriers out of these localized regions. These results therefore demonstrate how the peculiar properties of graphene extend also to the behavior of its nonequilibrium carriers.

Valley Chern numbers and boundary modes in gapped bilayer graphene

Electronic states at domain walls in bilayer graphene are studied by analyzing their four- and two-band continuum models, by performing numerical calculations on the lattice, and by using quantum geometric arguments. The continuum theories explain the distinct electronic properties of boundary modes localized near domain walls formed by interlayer electric field reversal, by interlayer stacking reversal, and by simultaneous reversal of both quantities. Boundary mode properties are related to topological transitions and gap closures, which occur in the bulk Hamiltonian parameter space. The important role played by intervalley coupling effects not directly captured by the continuum model is addressed using lattice calculations for specific domain wall structures.

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.

In situ observation of electrostatic and thermal manipulation of suspended graphene membranes

Graphene is nature's thinnest elastic membrane, and its morphology has important impacts on its electrical, mechanical, and electromechanical properties. Here we report manipulation of the morphology of suspended graphene via electrostatic and thermal control. By measuring the out-of-plane deflection as a function of applied gate voltage and number of layers, we show that graphene adopts a parabolic profile at large gate voltages with inhomogeneous distribution of charge density and strain. Unclamped graphene sheets slide into the trench under tension; for doubly clamped devices, the results are well-accounted for by membrane deflection with effective Young's modulus E = 1.1 TPa. Upon cooling to 100 K, we observe buckling-induced ripples in the central portion and large upward buckling of the free edges, which arises from graphene's large negative thermal expansion coefficient.

Antiferromagnetism in the Hubbard model on the Bernal-stacked honeycomb bilayer

Using a combination of quantum Monte Carlo simulations, functional renormalization group calculations and mean-field theory, we study the Hubbard model on the Bernal-stacked honeycomb bilayer at half-filling as a model system for bilayer graphene. The free bands consisting of two Fermi points with quadratic dispersions lead to a finite density of states at the Fermi level, which triggers an antiferromagnetic instability that spontaneously breaks sublattice and spin rotational symmetry once local Coulomb repulsions are introduced. Our results reveal an inhomogeneous participation of the spin moments in the ordered ground state, with enhanced moments at the threefold coordinated sites. Furthermore, we find the antiferromagnetic ground state to be robust with respect to enhanced interlayer couplings and extended Coulomb interactions.

Evidence for a spontaneous gapped state in ultraclean bilayer graphene

At the charge neutrality point, bilayer graphene (BLG) is strongly susceptible to electronic interactions and is expected to undergo a phase transition to a state with spontaneously broken symmetries. By systematically investigating a large number of single-and double-gated BLG devices, we observe a bimodal distribution of minimum conductivities at the charge neutrality point. Although σ(min) is often approximately 2-3 e(2)/h (where e is the electron charge and h is Planck's constant), it is several orders of magnitude smaller in BLG devices that have both high mobility and low extrinsic doping. The insulating state in the latter samples appears below a transition temperature T(c) of approximately 5 K and has a T = 0 energy gap of approximately 3 meV. Transitions between these different states can be tuned by adjusting disorder or carrier density.

Distinguishing spontaneous quantum Hall states in bilayer graphene

Chirally stacked N-layer graphene with N≥2 is susceptible to a variety of distinct broken symmetry states in which each spin-valley flavor spontaneously transfers charge between layers. In mean-field theory, one of the likely candidate ground states for a neutral bilayer is the layer antiferromagnet that has opposite spin polarizations in opposite layers. In this Letter, we analyze how the layer antiferromagnet and other competing states are influenced by Zeeman fields that couple to spin and by interlayer electric fields that couple to layer pseudospin, and comment on the possibility of using Zeeman responses and edge state signatures to identify the character of the bilayer ground state experimentally.

Spontaneously gapped ground state in suspended bilayer graphene

Bilayer graphene bears an eightfold degeneracy due to spin, valley, and layer symmetry, allowing for a wealth of broken symmetry states induced by magnetic or electric fields, by strain, or even spontaneously by interaction. We study the electrical transport in clean current annealed suspended bilayer graphene. We find two kinds of devices. In bilayers of type B1 the eightfold zero-energy Landau level is partially lifted above a threshold field revealing an insulating ν=0 quantum-Hall state at the charge neutrality point. In bilayers of type B2 the Landau level lifting is full and a gap appears in the differential conductance even at zero magnetic field, suggesting an insulating spontaneously broken symmetry state. Unlike B1, the minimum conductance in B2 is not exponentially suppressed, but remains finite with a value G is < or approximately equall to e(2)/h even in a large magnetic field. We suggest that this phase of B2 is insulating in the bulk and bound by compressible edge states.

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.

Enhanced thermoelectric power in dual-gated bilayer graphene

The thermoelectric power of a material, typically governed by its band structure and carrier density, can be varied by chemical doping that is often restricted by solubility of the dopant. Materials showing large thermoelectric power are useful for many industrial applications, such as the heat-to-electricity conversion and the thermoelectric cooling device. Here we show a full electric-field tuning of thermoelectric power in a dual-gated bilayer graphene device resulting from the opening of a band gap by applying a perpendicular electric field on bilayer graphene. We uncover a large enhancement in thermoelectric power at a low temperature, which may open up a new possibility in low temperature thermoelectric application using graphene-based device.

Polar Kerr effect and time reversal symmetry breaking in bilayer graphene

The unique sensitivity of optical response to different types of symmetry breaking can be used to detect and identify spontaneously ordered many-body states in bilayer graphene. We predict a strong response at optical frequencies, sensitive to electronic phenomena at low energies, which arises because of nonzero interband matrix elements of the electric current operator. In particular, the polar Kerr rotation and reflection anisotropy provide fingerprints of the quantum anomalous Hall state and the nematic state, characterized by spontaneously broken time-reversal symmetry and lattice rotation symmetry, respectively. These optical signatures, which undergo a resonant enhancement in the near-infrared regime, lie well within reach of existing experimental techniques.

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.