Competition between spontaneous symmetry breaking and single-particle gaps in trilayer graphene

Fractional quantum anomalous Hall effect in multilayer graphene

The fractional quantum anomalous Hall effect (FQAHE), the analogue of the fractional quantum Hall effect1 at zero magnetic field, is predicted to exist in topological flat bands under spontaneous time-reversal-symmetry breaking2-6. The demonstration of FQAHE could lead to non-Abelian anyons that form the basis of topological quantum computation7-9. So far, FQAHE has been observed only in twisted MoTe2 at a moiré filling factor v > 1/2 (refs. 10-13). Graphene-based moiré superlattices are believed to host FQAHE with the potential advantage of superior material quality and higher electron mobility. Here we report the observation of integer and fractional QAH effects in a rhombohedral pentalayer graphene-hBN moiré superlattice. At zero magnetic field, we observed plateaus of quantized Hall resistance [Formula: see text] at v = 1, 2/3, 3/5, 4/7, 4/9, 3/7 and 2/5 of the moiré superlattice, respectively, accompanied by clear dips in the longitudinal resistance Rxx. Rxy equals [Formula: see text] at v = 1/2 and varies linearly with v, similar to the composite Fermi liquid in the half-filled lowest Landau level at high magnetic fields14-16. By tuning the gate-displacement field D and v, we observed phase transitions from composite Fermi liquid and FQAH states to other correlated electron states. Our system provides an ideal platform for exploring charge fractionalization and (non-Abelian) anyonic braiding at zero magnetic field7-9,17-19, especially considering a lateral junction between FQAHE and superconducting regions in the same device20-22.

Spontaneous broken-symmetry insulator and metals in tetralayer rhombohedral graphene

Interactions among charge carriers in graphene can lead to the spontaneous breaking of multiple degeneracies. When increasing the number of graphene layers following rhombohedral stacking, the dominant role of Coulomb interactions becomes pronounced due to the significant reduction in kinetic energy. In this study, we employ phonon-polariton-assisted near-field infrared imaging to determine the stacking orders of tetralayer graphene devices. Through quantum transport measurements, we observe a range of spontaneous broken-symmetry states and their transitions, which can be finely tuned by carrier density n and electric displacement field D. Specifically, we observe a layer-antiferromagnetic insulator at n = D = 0 with a gap of approximately 15 meV. Increasing D allows for a continuous phase transition from a layer-antiferromagnetic insulator to a layer-polarized insulator. By simultaneously tuning n and D, we observe isospin-polarized metals, including spin-valley-polarized and spin-polarized metals. These transitions are associated with changes in the Fermi surface topology and are consistent with the Stoner criteria. Our findings highlight the efficient fabrication of specially stacked multilayer graphene devices and demonstrate that crystalline multilayer graphene is an ideal platform for investigating a wide range of broken symmetries driven by Coulomb interactions.

Chemical Potential Characterization of Symmetry-Breaking Phases in a Rhombohedral Trilayer Graphene

Rhombohedral trilayer graphene has recently emerged as a natural flat-band platform for studying interaction-driven symmetry-breaking phases. The displacement field (D) can further flatten the band to enhance the density of states, thereby controlling the electronic correlation that tips the energy balance between spin and valley degrees of freedom. To characterize the energy competition, chemical potential measurement─a direct thermodynamic probe of Fermi surfaces─is highly demanding to be conducted under a constant D. In this work, we characterize D-dependent isospin flavor polarization, where electronic states with isospin degeneracies of one and two can be identified. We also developed a method to measure the chemical potential at a fixed D, allowing for the extraction of energy variation during phase transitions. Furthermore, symmetry breaking could also be invoked in Landau levels, manifesting as quantum Hall ferromagnetism. Our work opens more opportunities for the thermodynamic characterization of displacement-field tuned van der Waals heterostructures.

Observation of competing, correlated ground states in the flat band of rhombohedral graphite

In crystalline solids, the interactions of charge and spin can result in a variety of emergent quantum ground states, especially in partially filled, topological flat bands such as Landau levels or in "magic angle" graphene layers. Much less explored is rhombohedral graphite (RG), perhaps the simplest and structurally most perfect condensed matter system to host a flat band protected by symmetry. By scanning tunneling microscopy, we map the flat band charge density of 8, 10, 14, and 17 layers and identify a domain structure emerging from a competition between a sublattice antiferromagnetic insulator and a gapless correlated paramagnet. Our density matrix renormalization group calculations explain the observed features and demonstrate that the correlations are fundamentally different from graphene-based magnetism identified until now, forming the ground state of a quantum magnet. Our work establishes RG as a platform to study many-body interactions beyond the mean-field approach, where quantum fluctuations and entanglement dominate.

Gate-Tunable Magnetism and Giant Magnetoresistance in Suspended Rhombohedral-Stacked Few-Layer Graphene

Conventionally, magnetism arises from the strong exchange interaction among the magnetic moments of d- or f-shell electrons. It can also emerge in perfect lattices from nonmagnetic elements, such as that exemplified by the Stoner criterion. Here we report tunable magnetism in suspended rhombohedral-stacked few-layer graphene (r-FLG) devices with flat bands. At small doping levels (n ∼ 10¹¹ cm^-2), we observe prominent conductance hysteresis and giant magnetoconductance that exceeds 1000% as a function of magnetic fields. Both phenomena are tunable by density and temperature and disappear at n > 10¹² cm^-2 or T > 5 K. These results are confirmed by first-principles calculations, which indicate the formation of a half-metallic state in doped r-FLG, in which the magnetization is tunable by electric field. Our combined experimental and theoretical work demonstrate that magnetism and spin polarization, arising from the strong electronic interactions in flat bands, emerge in a system composed entirely of carbon atoms.

Developing Graphene-Based Moiré Heterostructures for Twistronics

Graphene-based moiré heterostructures are strongly correlated materials, and they are considered to be an effective platform to investigate the challenges of condensed matter physics. This is due to the distinct electronic properties that are unique to moiré superlattices and peculiar band structures. The increasing research on strongly correlated physics via graphene-based moiré heterostructures, especially unconventional superconductors, greatly promotes the development of condensed matter physics. Herein, the preparation methods of graphene-based moiré heterostructures on both in situ growth and assembling monolayer 2D materials are discussed. Methods to improve the quality of graphene and optimize the transfer process are presented to mitigate the limitations of low-quality graphene and damage caused by the transfer process during the fabrication of graphene-based moiré heterostructures. Then, the topological properties in various graphene-based moiré heterostructures are reviewed. Furthermore, recent advances regarding the factors that influence physical performances via a changing twist angle, the exertion of strain, and regulation of the dielectric environment are presented. Moreover, various unique physical properties in graphene-based moiré heterostructures are demonstrated. Finally, the challenges faced during the preparation and characterization of graphene-based moiré heterostructures are discussed. An outlook for the further development of moiré heterostructures is also presented.

Charge Neutral Current Generation in a Spontaneous Quantum Hall Antiferromagnet

The intrinsic Hall effect allows for the generation of a nondissipative charge neutral current, such as a pure spin current generated via the spin Hall effect. Breaking of the spatial inversion or time reversal symmetries, or the spin-orbit interaction is generally considered necessary for the generation of such a charge neutral current. Here, we challenge this general concept and present generation and detection of a charge neutral current in a centrosymmetric material with little spin-orbit interaction. We employ bilayer graphene, and find enhanced nonlocal transport in the quantum Hall antiferromagnetic state, where spontaneous symmetry breaking occurs due to the electronic correlation.

Signatures of tunable superconductivity in a trilayer graphene moiré superlattice

Understanding the mechanism of high-transition-temperature (high-T_c) superconductivity is a central problem in condensed matter physics. It is often speculated that high-T_c superconductivity arises in a doped Mott insulator¹ as described by the Hubbard model^2-4. An exact solution of the Hubbard model, however, is extremely challenging owing to the strong electron-electron correlation in Mott insulators. Therefore, it is highly desirable to study a tunable Hubbard system, in which systematic investigations of the unconventional superconductivity and its evolution with the Hubbard parameters can deepen our understanding of the Hubbard model. Here we report signatures of tunable superconductivity in an ABC-trilayer graphene (TLG) and hexagonal boron nitride (hBN) moiré superlattice. Unlike in 'magic angle' twisted bilayer graphene, theoretical calculations show that under a vertical displacement field, the ABC-TLG/hBN heterostructure features an isolated flat valence miniband associated with a Hubbard model on a triangular superlattice^5,6 where the bandwidth can be tuned continuously with the vertical displacement field. Upon applying such a displacement field we find experimentally that the ABC-TLG/hBN superlattice displays Mott insulating states below 20 kelvin at one-quarter and one-half fillings of the states, corresponding to one and two holes per unit cell, respectively. Upon further cooling, signatures of superconductivity ('domes') emerge below 1 kelvin for the electron- and hole-doped sides of the one-quarter-filling Mott state. The electronic behaviour in the ABC-TLG/hBN superlattice is expected to depend sensitively on the interplay between the electron-electron interaction and the miniband bandwidth. By varying the vertical displacement field, we demonstrate transitions from the candidate superconductor to Mott insulator and metallic phases. Our study shows that ABC-TLG/hBN heterostructures offer attractive model systems in which to explore rich correlated behaviour emerging in the tunable triangular Hubbard model.

Signatures of tunable superconductivity in a trilayer graphene moiré superlattice

Understanding the mechanism of high-transition-temperature (high-T_c) superconductivity is a central problem in condensed matter physics. It is often speculated that high-T_c superconductivity arises in a doped Mott insulator¹ as described by the Hubbard model^2-4. An exact solution of the Hubbard model, however, is extremely challenging owing to the strong electron-electron correlation in Mott insulators. Therefore, it is highly desirable to study a tunable Hubbard system, in which systematic investigations of the unconventional superconductivity and its evolution with the Hubbard parameters can deepen our understanding of the Hubbard model. Here we report signatures of tunable superconductivity in an ABC-trilayer graphene (TLG) and hexagonal boron nitride (hBN) moiré superlattice. Unlike in 'magic angle' twisted bilayer graphene, theoretical calculations show that under a vertical displacement field, the ABC-TLG/hBN heterostructure features an isolated flat valence miniband associated with a Hubbard model on a triangular superlattice^5,6 where the bandwidth can be tuned continuously with the vertical displacement field. Upon applying such a displacement field we find experimentally that the ABC-TLG/hBN superlattice displays Mott insulating states below 20 kelvin at one-quarter and one-half fillings of the states, corresponding to one and two holes per unit cell, respectively. Upon further cooling, signatures of superconductivity ('domes') emerge below 1 kelvin for the electron- and hole-doped sides of the one-quarter-filling Mott state. The electronic behaviour in the ABC-TLG/hBN superlattice is expected to depend sensitively on the interplay between the electron-electron interaction and the miniband bandwidth. By varying the vertical displacement field, we demonstrate transitions from the candidate superconductor to Mott insulator and metallic phases. Our study shows that ABC-TLG/hBN heterostructures offer attractive model systems in which to explore rich correlated behaviour emerging in the tunable triangular Hubbard model.

Quantum parity Hall effect in Bernal-stacked trilayer graphene

The quantum Hall effect has recently been generalized from transport of conserved charges to include transport of other approximately conserved-state variables, including spin and valley, via spin- or valley-polarized boundary states with different chiralities. Here, we report a class of quantum Hall effect in Bernal- or ABA-stacked trilayer graphene (TLG), the quantum parity Hall (QPH) effect, in which boundary channels are distinguished by even or odd parity under the system's mirror reflection symmetry. At the charge neutrality point, the longitudinal conductance [Formula: see text] is first quantized to [Formula: see text] at a small perpendicular magnetic field [Formula: see text], establishing the presence of four edge channels. As [Formula: see text] increases, [Formula: see text] first decreases to [Formula: see text], indicating spin-polarized counterpropagating edge states, and then, to approximately zero. These behaviors arise from level crossings between even- and odd-parity bulk Landau levels driven by exchange interactions with the underlying Fermi sea, which favor an ordinary insulator ground state in the strong [Formula: see text] limit and a spin-polarized state at intermediate fields. The transitions between spin-polarized and -unpolarized states can be tuned by varying Zeeman energy. Our findings demonstrate a topological phase that is protected by a gate-controllable symmetry and sensitive to Coulomb interactions.

High-Magnetic-Field Tunneling Spectra of ABC-Stacked Trilayer Graphene on Graphite

ABC-stacked trilayer graphene (TLG) was predicted to exhibit novel many-body phenomena due to the existence of almost dispersionless flat bands near the charge neutrality point. Here, using high-magnetic-field scanning tunneling microscopy, we present Landau Level (LL) spectroscopy measurements of high-quality ABC-stacked TLG on graphite. We observe an approximately linear magnetic-field scaling of valley splitting and spin splitting in the ABC-stacked TLG. Our experiment indicates that the spin splitting decreases dramatically with increasing the LL index. When the lowest LL is partially filled, we find an obvious enhancement of the spin splitting, attributing to strong many-body effects. Moreover, we observe linear energy scaling of the inverse lifetime of quasiparticles, providing an additional evidence for the strong electron-electron interactions in the ABC-stacked TLG. These results imply that interesting broken-symmetry states and novel electron correlated effects could emerge in the ABC-stacked TLG in the presence of high magnetic fields.

A family of finite-temperature electronic phase transitions in graphene multilayers

Suspended Bernal-stacked graphene multilayers up to an unexpectedly large thickness exhibit a broken-symmetry ground state whose origin remains to be understood. We show that a finite-temperature second-order phase transition occurs in multilayers whose critical temperature (T _c) increases from 12 kelvins (K) in bilayers to 100 K in heptalayers. A comparison of the data with a phenomenological model inspired by a mean-field approach suggests that the transition is associated with the appearance of a self-consistent valley- and spin-dependent staggered potential that changes sign from one layer to the next, appearing at T _c and increasing upon cooling. The systematic evolution with thickness of several measured quantities imposes constraints on any microscopic theory aiming to analyze the nature of electronic correlations in this system.

Landau Level Diagram and the Continuous Rotational Symmetry Breaking in Trilayer Graphene

The sequence of the zeroth Landau levels (LLs) between filling factors ν=-6 to 6 in ABA-stacked trilayer graphene (TLG) is unknown because it depends sensitively on the nonuniform charge distribution on the three layers of ABA-stacked TLG. Using the sensitivity of quantum Hall data on the electric field and magnetic field, in an ultraclean ABA-stacked TLG sample, we quantitatively estimate the nonuniformity of the electric field and determine the sequence of the zeroth LLs. We also observe anticrossings between some LLs differing by 3 in LL index, which result from the breaking of the continuous rotational to C_{3} symmetry by the trigonal warping.

Tunable Lifshitz Transitions and Multiband Transport in Tetralayer Graphene

As the Fermi level and band structure of two-dimensional materials are readily tunable, they constitute an ideal platform for exploring the Lifshitz transition, a change in the topology of a material's Fermi surface. Using tetralayer graphene that host two intersecting massive Dirac bands, we demonstrate multiple Lifshitz transitions and multiband transport, which manifest as a nonmonotonic dependence of conductivity on the charge density n and out-of-plane electric field D, anomalous quantum Hall sequences and Landau level crossings that evolve with n, D, and B.

Electrically Controllable Magnetism in Twisted Bilayer Graphene

Twisted graphene bilayers develop highly localized states around AA-stacked regions for small twist angles. We show that interaction effects may induce either an antiferromagnetic or a ferromagnetic (FM) polarization of said regions, depending on the electrical bias between layers. Remarkably, FM-polarized AA regions under bias develop spiral magnetic ordering, with a relative 120° misalignment between neighboring regions due to a frustrated antiferromagnetic exchange. This remarkable spiral magnetism emerges naturally without the need of spin-orbit coupling, and competes with the more conventional lattice-antiferromagnetic instability, which interestingly develops at smaller bias under weaker interactions than in monolayer graphene, due to Fermi velocity suppression. This rich and electrically controllable magnetism could turn twisted bilayer graphene into an ideal system to study frustrated magnetism in two dimensions.

Strong electronic interaction and multiple quantum Hall ferromagnetic phases in trilayer graphene

Quantum Hall effect provides a simple way to study the competition between single particle physics and electronic interaction. However, electronic interaction becomes important only in very clean graphene samples and so far the trilayer graphene experiments are understood within non-interacting electron picture. Here, we report evidence of strong electronic interactions and quantum Hall ferromagnetism seen in Bernal-stacked trilayer graphene. Due to high mobility ∼500,000 cm²V⁻¹s⁻¹ in our device compared to previous studies, we find all symmetry broken states and that Landau-level gaps are enhanced by interactions; an aspect explained by our self-consistent Hartree–Fock calculations. Moreover, we observe hysteresis as a function of filling factor and spikes in the longitudinal resistance which, together, signal the formation of quantum Hall ferromagnetic states at low magnetic field.

Few-layered graphene offers a powerful platform to investigate electronic interactions beyond the non-interacting electron picture approximation. Here, the authors report the signature of strong electronic interactions and quantum Hall ferromagnetism in trilayer graphene with ABA stacking.

Tunable Symmetries of Integer and Fractional Quantum Hall Phases in Heterostructures with Multiple Dirac Bands

The copresence of multiple Dirac bands in few-layer graphene leads to a rich phase diagram in the quantum Hall regime. Using transport measurements, we map the phase diagram of BN-encapsulated ABA-stacked trilayer graphene as a function charge density n, magnetic field B, and interlayer displacement field D, and observe transitions among states with different spin, valley, orbital, and parity polarizations. Such a rich pattern arises from crossings between Landau levels from different subbands, which reflect the evolving symmetries that are tunable in situ. At D=0, we observe fractional quantum Hall (FQH) states at filling factors 2/3 and -11/3. Unlike those in bilayer graphene, these FQH states are destabilized by a small interlayer potential that hybridizes the different Dirac bands.

Rhombohedral Multilayer Graphene: A Magneto-Raman Scattering Study

Graphene layers are known to stack in two stable configurations, namely, ABA or ABC stacking, with drastically distinct electronic properties. Unlike the ABA stacking, little has been done to experimentally investigate the electronic properties of ABC graphene multilayers. Here, we report on the first magneto optical study of a large ABC domain in a graphene multilayer flake, with ABC sequences exceeding 17 graphene sheets. ABC-stacked multilayers can be fingerprinted with a characteristic electronic Raman scattering response, which persists even at room temperatures. Tracing the magnetic field evolution of the inter Landau level excitations from this domain gives strong evidence for the existence of a dispersionless electronic band near the Fermi level, characteristic of such stacking. Our findings present a simple yet powerful approach to probe ABC stacking in graphene multilayer flakes, where this highly degenerated band appears as an appealing candidate to host strongly correlated states.

Stacking-Dependent Interlayer Coupling in Trilayer MoS₂ with Broken Inversion Symmetry

The stacking configuration in few-layer two-dimensional (2D) materials results in different structural symmetries and layer-to-layer interactions, and hence it provides a very useful parameter for tuning their electronic properties. For example, ABA-stacking trilayer graphene remains semimetallic similar to that of monolayer, while ABC-stacking is predicted to be a tunable band gap semiconductor under an external electric field. Such stacking dependence resulting from many-body interactions has recently been the focus of intense research activities. Here we demonstrate that few-layer MoS2 samples grown by chemical vapor deposition with different stacking configurations (AA, AB for bilayer; AAB, ABB, ABA, AAA for trilayer) exhibit distinct coupling phenomena in both photoluminescence and Raman spectra. By means of ultralow-frequency (ULF) Raman spectroscopy, we demonstrate that the evolution of interlayer interaction with various stacking configurations correlates strongly with layer-breathing mode (LBM) vibrations. Our ab initio calculations reveal that the layer-dependent properties arise from both the spin-orbit coupling (SOC) and interlayer coupling in different structural symmetries. Such detailed understanding provides useful guidance for future spintronics fabrication using various stacked few-layer MoS2 blocks.

Direct Observation of a Gate Tunable Band Gap in Electrical Transport in ABC-Trilayer Graphene

Few layer graphene systems such as Bernal stacked bilayer and rhombohedral (ABC-) stacked trilayer offer the unique possibility to open an electric field tunable energy gap. To date, this energy gap has been experimentally confirmed in optical spectroscopy. Here we report the first direct observation of the electric field tunable energy gap in electronic transport experiments on doubly gated suspended ABC-trilayer graphene. From a systematic study of the nonlinearities in current versus voltage characteristics and the temperature dependence of the conductivity, we demonstrate that thermally activated transport over the energy-gap dominates the electrical response of these transistors. The estimated values for energy gap from the temperature dependence and from the current voltage characteristics follow the theoretically expected electric field dependence with critical exponent 3/2. These experiments indicate that high quality few-layer graphene are suitable candidates for exploring novel tunable terahertz light sources and detectors.