Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level

Aharonov-Bohm interference and statistical phase-jump evolution in fractional quantum Hall states in bilayer graphene

In the fractional quantum Hall effect, quasiparticles are collective excitations that have a fractional charge and show fractional statistics as they interchange positions. While the fractional charge affects semi-classical characteristics such as shot noise and charging energies, fractional statistics is most notable through quantum interference. Here we study fractional statistics in a bilayer graphene Fabry-Pérot interferometer. We tune the interferometer from the Coulomb-dominated regime to the Aharonov-Bohm regime, both for integer and fractional quantum Hall states. Focusing on the fractional quantum Hall state with a filling factor ν = 1/3, we follow the evolution of the Aharonov-Bohm interference of quasiparticles while varying the magnetic flux through an interference loop and the charge density within the loop independently. When their combined variation is such that the Landau filling remains 1/3, the charge density in the loop varies continuously. We then observe pristine Aharonov-Bohm oscillations with a period of three flux quanta, as expected for quasiparticles of one-third of the electron charge. Yet, when the combined variation leads to discrete events of quasiparticle addition or removal, phase jumps emerge and alter the phase evolution. Notably, across all cases with discrete and continuous charge variation, the average phase consistently increases by 2π with each addition of one electron to the loop, as expected for quasiparticles, obeying fractional statistics.

Emergent phases in graphene flat bands

Electronic correlations in two-dimensional materials play a crucial role in stabilising emergent phases of matter. The realisation of correlation-driven phenomena in graphene has remained a longstanding goal, primarily due to the absence of strong electron-electron interactions within its low-energy bands. In this context, magic-angle twisted bilayer graphene has recently emerged as a novel platform featuring correlated phases favoured by the low-energy flat bands of the underlying moiré superlattice. Notably, the observation of correlated insulators and superconductivity, and the interplay between these phases have garnered significant attention. A wealth of correlated phases with unprecedented tunability was discovered subsequently, including orbital ferromagnetism, Chern insulators, strange metallicity, density waves, and nematicity. However, a comprehensive understanding of these closely competing phases remains elusive. The ability to controllably twist and stack multiple graphene layers has enabled the creation of a whole new family of moiré superlattices with myriad properties. Here, we review the progress and development achieved so far, encompassing the rich phase diagrams offered by these graphene-based moiré systems. Additionally, we discuss multiple phases recently observed in non-moiré multilayer graphene systems. Finally, we outline future opportunities and challenges for the exploration of hidden phases in this new generation of moiré materials.

Tunable even- and odd-denominator fractional quantum Hall states in trilayer graphene

Fractional quantum Hall (FQH) states are exotic quantum many-body phases whose elementary charged excitations are anyons obeying fractional braiding statistics. While most FQH states are believed to have Abelian anyons, the Moore-Read type states with even denominators - appearing at half filling of a Landau level (LL) - are predicted to possess non-Abelian excitations with appealing potential in topological quantum computation. These states, however, depend sensitively on the orbital contents of the single-particle LL wavefunctions and the LL mixing. Here we report magnetotransport measurements on Bernal-stacked trilayer graphene, whose multiband structure facilitates interlaced LL mixing, which can be controlled by external magnetic and displacement fields. We observe robust FQH states including even-denominator ones at filling factors ν = - 9/2, - 3/2, 3/2 and 9/2. In addition, we fine-tune the LL mixing and crossings to drive quantum phase transitions of these half-filling states and neighbouring odd-denominator ones, exhibiting related emerging and waning behaviour.

Electron wave and quantum optics in graphene

In the last decade, graphene has become an exciting platform for electron optical experiments, in some aspects superior to conventional two-dimensional electron gases (2DEGs). A major advantage, besides the ultra-large mobilities, is the fine control over the electrostatics, which gives the possibility of realising gap-less and compact p-n interfaces with high precision. The latter host non-trivial states,e.g., snake states in moderate magnetic fields, and serve as building blocks of complex electron interferometers. Thanks to the Dirac spectrum and its non-trivial Berry phase, the internal (valley and sublattice) degrees of freedom, and the possibility to tailor the band structure using proximity effects, such interferometers open up a completely new playground based on novel device architectures. In this review, we introduce the theoretical background of graphene electron optics, fabrication methods used to realise electron-optical devices, and techniques for corresponding numerical simulations. Based on this, we give a comprehensive review of ballistic transport experiments and simple building blocks of electron optical devices both in single and bilayer graphene, highlighting the novel physics that is brought in compared to conventional 2DEGs. After describing the different magnetic field regimes in graphene p-n junctions and nanostructures, we conclude by discussing the state of the art in graphene-based Mach-Zender and Fabry-Perot interferometers.

Fingerprints of Anti-Pfaffian Topological Order in Quantum Point Contact Transport

Despite recent experimental developments, the topological order of the fractional quantum Hall state at filling ν=5/2 remains an outstanding question. We study conductance and shot noise in a quantum point contact device in the charge-equilibrated regime and show that, among Pfaffian, particle-hole Praffian, and anti-Pfaffian (aPf) candidate states, the hole-conjugate aPf state is unique in that it can produce a conductance plateau at G=(7/3)e^{2}/h by two fundamentally distinct mechanisms. We demonstrate that these mechanisms can be distinguished by shot noise measurements on the plateaus. We also determine distinct features of the conductance of the aPf state in the coherent regime. Our results can be used to experimentally single out the aPf order.

Strongly coupled magneto-exciton condensates in large-angle twisted double bilayer graphene

Excitons, pairs of electrons and holes, undergo a Bose-Einstein condensation at low temperatures. An important platform to study excitons is double-layer two-dimensional electron gases, with two parallel planes of electrons and holes separated by a thin insulating layer. Lowering this separation (d) strengthens the exciton binding energy, however, leads to the undesired interlayer tunneling, resulting in annihilation of excitons. Here, we report the observation of a sequences of robust exciton condensates (ECs) in double bilayer graphene twisted to ~ 10° with no insulating mid-layer. The large momentum mismatch between two graphene layers suppresses interlayer tunneling, reaching a d ~ 0.334 nm. Measuring the bulk and edge transport, we find incompressible states corresponding to ECs when both layers are in half-filled N = 0, 1 Landau levels (LLs). Theoretical calculations suggest that the low-energy charged excitation of ECs can be meron-antimeron or particle-hole pair, which relies on both LL index and carrier type. Our results establish a novel platform with extreme coupling strength for studying quantum bosonic phase.

Full Classification of Transport on an Equilibrated 5/2 Edge via Shot Noise

The nature of the bulk topological order of the 5/2 non-Abelian fractional quantum Hall state and the steady state of its edge are long-studied questions. The most promising non-Abelian model bulk states are the Pfaffian (Pf), anti-Pffafian (APf), and particle-hole symmetric Pfaffian (PHPf). Here, we propose to employ a set of dc current-current correlations (electrical shot noise) in order to distinguish among the Pf, APf, and PHPf candidate states, as well as to determine their edge thermal equilibration regimes: full vs partial. Using other tools, measurements of GaAs platforms have already indicated consistency with the PHPf state. Our protocol, realizable with available experimental tools, is based on fully electrical measurements.

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.

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.

Charge Oscillations in Bilayer Graphene Quantum Confinement Devices

Quantum confinement structures are building blocks of quantum devices in fundamental physics exploration and technological applications. In this work, we fabricate dual-gated bilayer graphene Fabry-Pérot quantum Hall interferometers employing two different gating strategies and conduct finite element simulations to understand the electrostatics of the confinement structures and to guide device design and fabrication. We observe two types of resistance oscillations arising from the charging of quantum dots formed inside the interferometers. We obtain the size, location, and charging energy of the dots by measuring the dependence of the oscillations on the magnetic field, gate voltages, and dc bias. We analyze and discuss the origin of the quantum dots and their impact on quantum Hall edge state backscattering and interference. Insights gained in these studies shed light on the construction of van der Waals quantum confinement devices.

Universal chiral Luttinger liquid behavior in a graphene fractional quantum Hall point contact

One-dimensional conductors are described by Luttinger liquid theory, which predicts a power-law suppression of the single-electron tunneling density of states at low voltages. The scaling exponent is predicted to be quantized when tunneling into a single isolated chiral edge state of the fractional quantum Hall effect. We report conductance measurements across a point contact linking integer and fractional quantum Hall edge states (at fillings 1 and [Formula: see text], respectively). At weak coupling, we observe the predicted universal quadratic scaling with temperature and voltage. At strong coupling, we demonstrate perfect Andreev reflection of fractionalized quasiparticles at the point contact. We use the strong coupling physics to realize a nearly dissipationless direct current voltage step-up transformer, whose gain arises directly from topological fractionalization of electrical charge.

Gate-Defined Josephson Weak-Links in Monolayer WTe2

Systems combining superconductors with topological insulators offer a platform for the study of Majorana bound states and a possible route to realize fault tolerant topological quantum computation. Among the systems being considered in this field, monolayers of tungsten ditelluride (WTe2 ) have a rare combination of properties. Notably, it has been demonstrated to be a quantum spin Hall insulator (QSHI) and can easily be gated into a superconducting state. Measurements on gate-defined Josephson weak-link devices fabricated using monolayer WTe2 are reported. It is found that consideration of the 2D superconducting leads are critical in the interpretation of magnetic interference in the resulting junctions. The reported fabrication procedures suggest a facile way to produce further devices from this technically challenging material and the results mark the first step toward realizing versatile all-in-one topological Josephson weak-links using monolayer WTe2 .

Highly Anisotropic Even-Denominator Fractional Quantum Hall State in an Orbitally Coupled Half-Filled Landau Level

The even-denominator fractional quantum Hall states (FQHSs) in half-filled Landau levels are generally believed to host non-Abelian quasiparticles and be of potential use in topological quantum computing. Of particular interest is the competition and interplay between the even-denominator FQHSs and other ground states, such as anisotropic phases and composite fermion Fermi seas. Here, we report the observation of an even-denominator fractional quantum Hall state with highly anisotropic in-plane transport coefficients at Landau level filling factor ν=3/2. We observe this state in an ultra-high-quality GaAs two-dimensional hole system when a large in-plane magnetic field is applied. By increasing the in-plane field, we observe a sharp transition from an isotropic composite fermion Fermi sea to an anisotropic even-denominator FQHS. Our data and calculations suggest that a unique feature of two-dimensional holes, namely the coupling between heavy-hole and light-hole states, combines different orbital components in the wave function of one Landau level, and leads to the emergence of a highly anisotropic even-denominator fractional quantum Hall state. Our results demonstrate that the GaAs two-dimensional hole system is a unique platform for the exploration of exotic, many-body ground states.

Superconductivity and strong interactions in a tunable moiré quasicrystal

Electronic states in quasicrystals generally preclude a Bloch description1, rendering them fascinating and enigmatic. Owing to their complexity and scarcity, quasicrystals are underexplored relative to periodic and amorphous structures. Here we introduce a new type of highly tunable quasicrystal easily assembled from periodic components. By twisting three layers of graphene with two different twist angles, we form two mutually incommensurate moiré patterns. In contrast to many common atomic-scale quasicrystals2,3, the quasiperiodicity in our system is defined on moiré length scales of several nanometres. This 'moiré quasicrystal' allows us to tune the chemical potential and thus the electronic system between a periodic-like regime at low energies and a strongly quasiperiodic regime at higher energies, the latter hosting a large density of weakly dispersing states. Notably, in the quasiperiodic regime, we observe superconductivity near a flavour-symmetry-breaking phase transition4,5, the latter indicative of the important role that electronic interactions play in that regime. The prevalence of interacting phenomena in future systems with in situ tunability is not only useful for the study of quasiperiodic systems but may also provide insights into electronic ordering in related periodic moiré crystals6-12. We anticipate that extending this platform to engineer quasicrystals by varying the number of layers and twist angles, and by using different two-dimensional components, will lead to a new family of quantum materials to investigate the properties of strongly interacting quasicrystals.

Orbitally Controlled Quantum Hall States in Decoupled Two-Bilayer Graphene Sheets

The authors report on integer and fractional quantum Hall states in a stack of two twisted Bernal bilayer graphene sheets. By exploiting the momentum mismatch in reciprocal space, the single-particle tunneling between both bilayers is suppressed. Since the bilayers are spatially separated by only 0.34 nm, the stack benefits from strong interlayer Coulombic interactions. These interactions can cause the formation of a Bose-Einstein condensate. Indeed, such a condensate is observed for half-filling in each bilayer sheet. However, only when the partially filled levels have orbital index 1. It is absent for partially filled levels with orbital index 0. This discrepancy is tentatively attributed to the role of skyrmion/anti-skyrmion pair excitations and the dependence of the energy of these excitations on the orbital index. The application of asymmetric top and bottom gate voltages enables to influence the orbital nature of the electronic states of the graphene bilayers at the chemical potential and to navigate in orbital mixed space. The latter hosts an even denominator fractional quantum Hall state at total filling of -3/2. These observations suggest a unique edge reconstruction involving both electrons and chiral p-wave composite fermions.

Hunting for Majoranas

Over the past decade, there have been considerable efforts to observe non-abelian quasiparticles in novel quantum materials and devices. These efforts are motivated by the goals of demonstrating quantum statistics of quasiparticles beyond those of fermions and bosons and of establishing the underlying science for the creation of topologically protected quantum bits. In this Review, we focus on efforts to create topological superconducting phases that host Majorana zero modes. We consider the lessons learned from existing experimental efforts, which are motivating both improvements to present platforms and the exploration of new approaches. Although the experimental detection of non-abelian quasiparticles remains challenging, the knowledge gained thus far and the opportunities ahead offer high potential for discovery and advances in this exciting area of quantum physics.

Dynamic Response of Wigner Crystals

The Wigner crystal, an ordered array of electrons, is one of the very first proposed many-body phases stabilized by the electron-electron interaction. We examine this quantum phase with simultaneous capacitance and conductance measurements, and observe a large capacitive response while the conductance vanishes. We study one sample with four devices whose length scale is comparable with the crystal's correlation length, and deduce the crystal's elastic modulus, permittivity, pinning strength, etc. Such a systematic quantitative investigation of all properties on a single sample has a great promise to advance the study of Wigner crystals.

Composite Fermion Pairing Induced by Landau Level Mixing

Pairing of composite fermions provides a possible mechanism for fractional quantum Hall effect at even denominator fractions and is believed to serve as a platform for realizing quasiparticles with non-Abelian braiding statistics. We present results from fixed-phase diffusion Monte Carlo calculations which predict that substantial Landau level mixing can induce a pairing of composite fermions at filling factors ν=1/2 and ν=1/4 in the l=-3 relative angular momentum channel, thereby destabilizing the composite-fermion Fermi seas to produce non-Abelian fractional quantum Hall states.

Aharonov-Bohm Oscillations in Bilayer Graphene Quantum Hall Edge State Fabry-Pérot Interferometers

Bernal-stacked bilayer graphene exhibits a wealth of interaction-driven phenomena, including robust even-denominator fractional quantum Hall states. We construct Fabry-Pérot interferometers using a split-gate design and present measurements of the Aharonov-Bohm oscillations. The edge state velocity is found to be approximately 6 × 10⁴ m/s at filling factor ν = 2 and decreases with increasing filling factor. The dc bias and temperature dependence of the interference point to electron-electron interaction induced decoherence mechanisms. These results pave the way for the quest of fractional and non-Abelian braiding statistics in this promising device platform.

Robust Interlayer-Coherent Quantum Hall States in Twisted Bilayer Graphene

We introduce a novel two-dimensional electronic system with ultrastrong interlayer interactions, namely, twisted bilayer graphene with a large twist angle, as an ideal ground for realizing interlayer-coherent excitonic condensates. In these systems, sub-nanometer atomic separation between the layers allows significant interlayer interactions, while interlayer electron tunneling is geometrically suppressed due to the large twist angle. By fully exploiting these two features we demonstrate that a sequence of odd-integer quantum Hall states with interlayer coherence appears at the second Landau level (N = 1). Notably the energy gaps for these states are of order 1 K, which is several orders of magnitude greater than those in GaAs. Furthermore, a variety of quantum Hall phase transitions are observed experimentally. All the experimental observations are largely consistent with our phenomenological model calculations. Hence, we establish that a large twist angle system is an excellent platform for high-temperature excitonic condensation.

Enhanced superconductivity in spin-orbit proximitized bilayer graphene

In the presence of a large perpendicular electric field, Bernal-stacked bilayer graphene (BLG) features several broken-symmetry metallic phases^1-3 as well as magnetic-field-induced superconductivity¹. The superconducting state is quite fragile, however, appearing only in a narrow window of density and with a maximum critical temperature T_c≈ 30 mK. Here we show that placing monolayer tungsten diselenide (WSe₂) on BLG promotes Cooper pairing to an extraordinary degree: superconductivity appears at zero magnetic field, exhibits an order of magnitude enhancement in T_c and occurs over a density range that is wider by a factor of eight. By mapping quantum oscillations in BLG-WSe₂ as a function of electric field and doping, we establish that superconductivity emerges throughout a region for which the normal state is polarized, with two out of four spin-valley flavours predominantly populated. In-plane magnetic field measurements further reveal that superconductivity in BLG-WSe₂ can exhibit striking dependence of the critical field on doping, with the Chandrasekhar-Clogston (Pauli) limit roughly obeyed on one end of the superconducting dome, yet sharply violated on the other. Moreover, the superconductivity arises only for perpendicular electric fields that push BLG hole wavefunctions towards WSe₂, indicating that proximity-induced (Ising) spin-orbit coupling plays a key role in stabilizing the pairing. Our results pave the way for engineering robust, highly tunable and ultra-clean graphene-based superconductors.

Sensing Remote Bulk Defects through Resistance Noise in a Large-Area Graphene Field-Effect Transistor

The substrate plays a crucial role in determining the transport and low-frequency noise behavior of graphene field-effect devices. Typically, a heavily doped Si/SiO₂ substrate is used to fabricate these devices for efficient gating. Trapping-detrapping processes close to the graphene/substrate interface are the dominant sources of resistance fluctuations in the graphene channel, while Coulomb fluctuations arising due to any remote charge fluctuations inside the bulk of the substrate are effectively screened by the heavily doped substrate. Here, we present the electronic transport and low-frequency noise characteristics of a large-area CVD graphene field-effect transistor (FET) prepared on a lightly doped Si/SiO₂ substrate (N_A ≈ 10¹⁵ cm^-3). Through a systematic characterization of transport, noise, and capacitance at various temperatures, we reveal that the remote Si/SiO₂ interface can affect the charge transport in graphene severely and any charge fluctuations inside the bulk of the silicon substrate can be sensed by the graphene channel. The resistance (R) vs back-gate voltage (V_bg) characteristics of the device show a hump around the depletion region formed at the SiO₂/Si interface, confirmed by the capacitance (C)-voltage (V) measurement. A low-frequency noise measurement on these fabricated devices shows a peak in the noise amplitude close to the depletion region. This indicates that due to the absence of any charge layer at the Si/SiO₂ interface, the screening ability decreases, and as a consequence, any fluctuations in the deep-level Coulomb impurities inside the silicon substrate can be observed as noise in resistance in the graphene channel via mobility fluctuations. The noise behavior on ionic liquid-gated graphene on the same substrate exhibits no such peak in noise and can be explained by the interfacial trapping-detrapping processes close to the graphene channel. Our study will definitely be useful for integrating graphene with the existing silicon technology, in particular, for high-frequency applications.

High-Temperature Quantum Hall Effect in Graphite-Gated Graphene Heterostructure Devices with High Carrier Mobility

Since the discovery of the quantum Hall effect in 1980, it has attracted intense interest in condensed matter physics and has led to a new type of metrological standard by utilizing the resistance quantum. Graphene, a true two-dimensional electron gas material, has demonstrated the half-integer quantum Hall effect and composite-fermion fractional quantum Hall effect due to its unique massless Dirac fermions and ultra-high carrier mobility. Here, we use a monolayer graphene encapsulated with hexagonal boron nitride and few-layer graphite to fabricate micrometer-scale graphene Hall devices. The application of a graphite gate electrode significantly screens the phonon scattering from a conventional SiO2/Si substrate, and thus enhances the carrier mobility of graphene. At a low temperature, the carrier mobility of graphene devices can reach 3 × 105 cm2/V·s, and at room temperature, the carrier mobility can still exceed 1 × 105 cm2/V·s, which is very helpful for the development of high-temperature quantum Hall effects under moderate magnetic fields. At a low temperature of 1.6 K, a series of half-integer quantum Hall plateaus are well-observed in graphene with a magnetic field of 1 T. More importantly, the ν = ±2 quantum Hall plateau clearly persists up to 150 K with only a few-tesla magnetic field. These findings show that graphite-gated high-mobility graphene devices hold great potential for high-sensitivity Hall sensors and resistance metrology standards for the new Système International d'unités.

Imaging the Breakdown of Ohmic Transport in Graphene

Ohm's law describes the proportionality of the current density and electric field. In solid-state conductors, Ohm's law emerges due to electron scattering processes that relax the electrical current. Here, we use nitrogen-vacancy center magnetometry to directly image the local breakdown of Ohm's law in a narrow constriction fabricated in a high mobility graphene monolayer. Ohmic flow is visible at room temperature as current concentration on the constriction edges, with flow profiles entirely determined by sample geometry. However, as the temperature is lowered below 200 K, the current concentrates near the constriction center. The change in the flow pattern is consistent with a crossover from diffusive to viscous electron transport dominated by electron-electron scattering processes that do not relax current.

Microwave Synthesized 2D Gold and Its 2D-2D Hybrids

Xenes, i.e., monoelemental two-dimensional atomic sheets, are promising for sensitive and ultrafast sensor applications owing to exceptional carrier mobility; however, most of them oxidize below 500 °C and therefore cannot be employed for high-temperature applications. 2D gold, an oxidation-resistant plasmonic Xene, is extremely promising. 2D gold was experimentally realized by both atomic layer deposition and chemical synthesis using sodium citrate. However, it is imperative to develop a new facile single-step method to synthesize 2D gold. Here, liquid-phase synthesis of 2D gold is demonstrated by microwave exposure to auric chloride dispersed in dimethylformamide. Microscopies (AFM and high-resolution TEM), spectroscopies (Raman, UV-vis, and X-ray photoelectron), and X-ray diffraction establish the formation of a hexagonal crystallographic phase for 2D gold. 2D-2D hybrids of 2D gold have also been synthesized and investigated for electronic/optoelectronic behaviors and SERS-based molecular sensing. DFT band structure calculation for 2D gold and its hybrids corroborates the experimental findings.

Bilayer WSe2 as a natural platform for interlayer exciton condensates in the strong coupling limit

Exciton condensates (ECs) are macroscopic coherent states arising from condensation of electron-hole pairs¹. Bilayer heterostructures, consisting of two-dimensional electron and hole layers separated by a tunnel barrier, provide a versatile platform to realize and study ECs^2-4. The tunnel barrier suppresses recombination, yielding long-lived excitons^5-10. However, this separation also reduces interlayer Coulomb interactions, limiting the exciton binding strength. Here, we report the observation of ECs in naturally occurring 2H-stacked bilayer WSe₂. In this system, the intrinsic spin-valley structure suppresses interlayer tunnelling even when the separation is reduced to the atomic limit, providing access to a previously unattainable regime of strong interlayer coupling. Using capacitance spectroscopy, we investigate magneto-ECs, formed when partially filled Landau levels couple between the layers. We find that the strong-coupling ECs show dramatically different behaviour compared with previous reports, including an unanticipated variation of EC robustness with the orbital number, and find evidence for a transition between two types of low-energy charged excitations. Our results provide a demonstration of tuning EC properties by varying the constituent single-particle wavefunctions.

High precision, low excitation capacitance measurement methods from 10 mK to room temperature

Capacitance measurement is a useful technique in studying quantum devices, as it directly probes the local particle charging properties, i.e., the system compressibility. Here, we report one approach that can measure capacitance from mK to room temperature with excellent accuracy. Our experiments show that such a high-precision technique is able to reveal delicate and essential properties of high-mobility two-dimensional electron systems.

Isospin magnetism and spin-polarized superconductivity in Bernal bilayer graphene

In conventional superconductors, Cooper pairing occurs between electrons of opposite spin. We observe spin-polarized superconductivity in Bernal bilayer graphene when doped to a saddle-point van Hove singularity generated by a large applied perpendicular electric field. We observe a cascade of electrostatic gate-tuned transitions between electronic phases distinguished by their polarization within the isospin space defined by the combination of the spin and momentum-space valley degrees of freedom. Although all of these phases are metallic at zero magnetic field, we observe a transition to a superconducting state at finite magnetic field B_‖ ≈ 150 milliteslas applied parallel to the two-dimensional sheet. Superconductivity occurs near a symmetry-breaking transition and exists exclusively above the B_‖ limit expected of a paramagnetic superconductor with the observed transition critical temperature T_C ≈ 30 millikelvins, consistent with a spin-triplet order parameter.

Donor-Acceptor Pair Quantum Emitters in Hexagonal Boron Nitride

Quantum emitters are needed for a myriad of applications ranging from quantum sensing to quantum computing. Hexagonal boron nitride (hBN) quantum emitters are one of the most promising solid-state platforms to date due to their high brightness and stability and the possibility of a spin-photon interface. However, the understanding of the physical origins of the single-photon emitters (SPEs) is still limited. Here we report dense SPEs in hBN across the entire visible spectrum and present evidence that most of these SPEs can be well explained by donor-acceptor pairs (DAPs). On the basis of the DAP transition generation mechanism, we calculated their wavelength fingerprint, matching well with the experimentally observed photoluminescence spectrum. Our work serves as a step forward for the physical understanding of SPEs in hBN and their applications in quantum technologies.

Quantum anomalous Hall effect from intertwined moiré bands

Electron correlation and topology are two central threads of modern condensed matter physics. Semiconductor moiré materials provide a highly tuneable platform for studies of electron correlation^1-12. Correlation-driven phenomena, including the Mott insulator^2-5, generalized Wigner crystals^2,6,9, stripe phases¹⁰ and continuous Mott transition^11,12, have been demonstrated. However, non-trivial band topology has remained unclear. Here we report the observation of a quantum anomalous Hall effect in AB-stacked MoTe₂ /WSe₂ moiré heterobilayers. Unlike in the AA-stacked heterobilayers¹¹, an out-of-plane electric field not only controls the bandwidth but also the band topology by intertwining moiré bands centred at different layers. At half band filling, corresponding to one particle per moiré unit cell, we observe quantized Hall resistance, h/e² (with h and e denoting the Planck's constant and electron charge, respectively), and vanishing longitudinal resistance at zero magnetic field. The electric-field-induced topological phase transition from a Mott insulator to a quantum anomalous Hall insulator precedes an insulator-to-metal transition. Contrary to most known topological phase transitions¹³, it is not accompanied by a bulk charge gap closure. Our study paves the way for discovery of emergent phenomena arising from the combined influence of strong correlation and topology in semiconductor moiré materials.

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.

Charge-order-enhanced capacitance in semiconductor moiré superlattices

Van der Waals moiré materials have emerged as a highly controllable platform to study electronic correlation phenomena^1-17. Robust correlated insulating states have recently been discovered at both integer and fractional filling factors of semiconductor moiré systems^10-17. In this study we explored the thermodynamic properties of these states by measuring the gate capacitance of MoSe₂/WS₂ moiré superlattices. We observed a series of incompressible states for filling factors 0-8 and anomalously large capacitance in the intervening compressible regions. The anomalously large capacitance, which was nearly 60% above the device's geometrical capacitance, was most pronounced at small filling factors, below the melting temperature of the charge-ordered states, and for small sample-gate separation. It is a manifestation of the device-geometry-dependent Coulomb interaction between electrons and phase mixing of the charge-ordered states. Based on these results, we were able to extract the thermodynamic gap of the correlated insulating states and the device's electronic entropy and specific heat capacity. Our findings establish capacitance as a powerful probe of the correlated states in semiconductor moiré systems and demonstrate control of these states via sample-gate coupling.

Correlated electron-hole state in twisted double-bilayer graphene

[Figure: see text].

Half- and quarter-metals in rhombohedral trilayer graphene

Ferromagnetism is most common in transition metal compounds where electrons occupy highly localized d orbitals. However, ferromagnetic order may also arise in low-density two-dimensional electron systems^1-5. Here we show that gate-tuned van Hove singularities in rhombohedral trilayer graphene⁶ drive spontaneous ferromagnetic polarization of the electron system into one or more spin and valley flavours. Using capacitance and transport measurements, we observe a cascade of transitions tuned to the density and electronic displacement field between phases in which quantum oscillations have fourfold, twofold or onefold degeneracy, associated with a spin- and valley-degenerate normal metal, spin-polarized 'half-metal', and spin- and valley-polarized 'quarter-metal', respectively. For electron doping, the salient features of the data are well captured by a phenomenological Stoner model⁷ that includes valley-anisotropic interactions. For hole filling, we observe a richer phase diagram featuring a delicate interplay of broken symmetries and transitions in the Fermi surface topology. Finally, we introduce a moiré superlattice using a rotationally aligned hexagonal boron nitride substrate^5,8. Remarkably, we find that the isospin order is only weakly perturbed, with the moiré potential catalysing the formation of topologically nontrivial gapped states whenever itinerant half- or quarter-metal states occur at half- or quarter-superlattice band filling. Our results show that rhombohedral graphene is an ideal platform for well-controlled tests of many-body theory, and reveal magnetism in moiré materials^4,5,9,10 to be fundamentally itinerant in nature.

Stripe phases in WSe2/WS2 moiré superlattices

Stripe phases, in which the rotational symmetry of charge density is spontaneously broken, occur in many strongly correlated systems with competing interactions^1-11. However, identifying and studying such stripe phases remains challenging. Here we uncover stripe phases in WSe₂/WS₂ moiré superlattices by combining optical anisotropy and electronic compressibility measurements. We find strong electronic anisotropy over a large doping range peaked at 1/2 filling of the moiré superlattice. The 1/2 state is incompressible and assigned to an insulating stripe crystal phase. Wide-field imaging reveals domain configurations with a preferential alignment along the high-symmetry axes of the moiré superlattice. Away from 1/2 filling, we observe additional stripe crystals at commensurate filling 1/4, 2/5 and 3/5, and compressible electronic liquid crystal states at incommensurate fillings. Our results demonstrate that two-dimensional semiconductor moiré superlattices are a highly tunable platform from which to study the stripe phases and their interplay with other symmetry breaking ground states.

Fractional charge and fractional statistics in the quantum Hall effects

Quasiparticles with fractional charge and fractional statistics are key features of the fractional quantum Hall effect. We discuss in detail the definitions of fractional charge and statistics and the ways in which these properties may be observed. In addition to theoretical foundations, we review the present status of the experiments in the area. We also discuss the notions of non-Abelian statistics and attempts to find experimental evidence for the existence of non-Abelian quasiparticles in certain quantum Hall systems.

Realization of Supersymmetry and Its Spontaneous Breaking in Quantum Hall Edges

Supersymmetry (SUSY) relating bosons and fermions plays an important role in unifying different fundamental interactions in particle physics. Since no superpartners of elementary particles have been observed, SUSY, if present, must be broken at low-energy. This makes it important to understand how SUSY is realized and broken, and study their consequences. We show that an N=(1,0) SUSY, arguably the simplest type, can be realized at the edge of the Moore-Read quantum Hall state. Depending on the absence or presence of edge reconstruction, both SUSY-preserving and SUSY broken phases can be realized in the same system, allowing for their unified description. The significance of the gapless fermionic Goldstino mode in the SUSY broken phase is discussed.

Aharonov-Bohm effect in graphene-based Fabry-Pérot quantum Hall interferometers

Interferometers probe the wave-nature and exchange statistics of indistinguishable particles-for example, electrons in the chiral one-dimensional edge channels of the quantum Hall effect (QHE). Quantum point contacts can split and recombine these channels, enabling interference of charged particles. Such quantum Hall interferometers (QHIs) can unveil the exchange statistics of anyonic quasi-particles in the fractional quantum Hall effect (FQHE). Here, we present a fabrication technique for QHIs in van der Waals (vdW) materials and realize a tunable, graphene-based Fabry-Pérot (FP) QHI. The graphite-encapsulated architecture allows observation of FQHE at a magnetic field of 3T and precise partitioning of integer and fractional edge modes. We measure pure Aharonov-Bohm interference in the integer QHE, a major technical challenge in small FP interferometers, and find that edge modes exhibit high-visibility interference due to large velocities. Our results establish vdW heterostructures as a versatile alternative to GaAs-based interferometers for future experiments targeting anyonic quasi-particles.

Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening

Controlling the strength of interactions is essential for studying quantum phenomena emerging in systems of correlated fermions. We introduce a device geometry whereby magic-angle twisted bilayer graphene is placed in close proximity to a Bernal bilayer graphene, separated by a 3-nanometer-thick barrier. By using charge screening from the Bernal bilayer, the strength of electron-electron Coulomb interaction within the twisted bilayer can be continuously tuned. Transport measurements show that tuning Coulomb screening has opposite effects on the insulating and superconducting states: As Coulomb interaction is weakened by screening, the insulating states become less robust, whereas the stability of superconductivity at the optimal doping is enhanced. The results provide important constraints on theoretical models for understanding the mechanism of superconductivity in magic-angle twisted bilayer graphene.

Scattering of Magnons at Graphene Quantum-Hall-Magnet Junctions

Motivated by recent nonlocal transport studies of quantum-Hall-magnet (QHM) states formed in monolayer graphene's N=0 Landau level, we study the scattering of QHM magnons by gate-controlled junctions between states with different integer filling factors ν. For the ν=1|-1|1 geometry we find that magnons are weakly scattered by electric potential variation in the junction region, and that the scattering is chiral when the junction lacks a mirror symmetry. For the ν=1|0|1 geometry, we find that kinematic constraints completely block magnon transmission if the incident angle exceeds a critical value. Our results explain the suppressed nonlocal-voltage signals observed in the ν=1|0|1 case. We use our theory to propose that valley waves generated at ν=-1|1 junctions and magnons can be used in combination to probe the spin or valley flavor structure of QHM states at integer and fractional filling factors.

Tunnel field-effect transistors for sensitive terahertz detection

The rectification of electromagnetic waves to direct currents is a crucial process for energy harvesting, beyond-5G wireless communications, ultra-fast science, and observational astronomy. As the radiation frequency is raised to the sub-terahertz (THz) domain, ac-to-dc conversion by conventional electronics becomes challenging and requires alternative rectification protocols. Here, we address this challenge by tunnel field-effect transistors made of bilayer graphene (BLG). Taking advantage of BLG's electrically tunable band structure, we create a lateral tunnel junction and couple it to an antenna exposed to THz radiation. The incoming radiation is then down-converted by the tunnel junction nonlinearity, resulting in high responsivity (>4 kV/W) and low-noise (0.2 pW/[Formula: see text]) detection. We demonstrate how switching from intraband Ohmic to interband tunneling regime can raise detectors' responsivity by few orders of magnitude, in agreement with the developed theory. Our work demonstrates a potential application of tunnel transistors for THz detection and reveals BLG as a promising platform therefor.

Magnetic field detection limits for ultraclean graphene Hall sensors

Solid-state magnetic field sensors are important for applications in commercial electronics and fundamental materials research. Most magnetic field sensors function in a limited range of temperature and magnetic field, but Hall sensors in principle operate over a broad range of these conditions. Here, we evaluate ultraclean graphene as a material platform for high-performance Hall sensors. We fabricate micrometer-scale devices from graphene encapsulated with hexagonal boron nitride and few-layer graphite. We optimize the magnetic field detection limit under different conditions. At 1 kHz for a 1 μm device, we estimate a detection limit of 700 nT Hz^−1/2 at room temperature, 80 nT Hz^−1/2 at 4.2 K, and 3 μT Hz^−1/2 in 3 T background field at 4.2 K. Our devices perform similarly to the best Hall sensors reported in the literature at room temperature, outperform other Hall sensors at 4.2 K, and demonstrate high performance in a few-Tesla magnetic field at which the sensors exhibit the quantum Hall effect.

The development of high-performance magnetic field sensors is important for magnetic sensing and imaging. Here, the authors fabricate Hall sensors from graphene encapsulated in hBN and few-layer graphite, demonstrating high performance over a wide range of temperature and background magnetic field.

Observation of Terahertz-Induced Magnetooscillations in Graphene

When high-frequency radiation is incident upon graphene subjected to a perpendicular magnetic field, graphene absorbs incident photons by allowing transitions between nearest Landau levels that follow strict selection rules dictated by angular momentum conservation. Here, we show a qualitative deviation from this behavior in high-quality graphene devices exposed to terahertz (THz) radiation. We demonstrate the emergence of a pronounced THz-driven photoresponse, which exhibits low-field magnetooscillations governed by the ratio of the frequency of the incoming radiation and the quasiclassical cyclotron frequency. We analyze the modifications of generated photovoltage with the radiation frequency and carrier density and demonstrate that the observed photoresponse shares a common origin with microwave-induced resistance oscillations discovered in GaAs-based heterostructures; however, in graphene it appears at much higher frequencies and persists above liquid nitrogen temperatures. Our observations expand the family of radiation-driven phenomena in graphene, paving the way for future studies of nonequilibrium electron transport.

Odd- and even-denominator fractional quantum Hall states in monolayer WSe2

Monolayer semiconducting transition-metal dichalcogenides (TMDs) represent a unique class of two-dimensional (2D) electron systems. Their atomically thin structure facilitates gate tunability just like graphene does, but unlike graphene, TMDs have the advantage of a sizable band gap and strong spin-orbit coupling. Measurements under large magnetic fields have revealed an unusual Landau level (LL) structure^1-3, distinct from other 2D electron systems. However, owing to the limited sample quality and poor electrical contact, probing the lowest LLs has been challenging, and observation of electron correlations within the fractionally filled LL regime has not been possible. Here, through bulk electronic compressibility measurements, we investigate the LL structure of monolayer WSe₂ in the extreme quantum limit, and observe fractional quantum Hall states in the lowest three LLs. The odd-denominator fractional quantum Hall sequences demonstrate a systematic evolution with the LL orbital index, consistent with generic theoretical expectations. In addition, we observe an even-denominator state in the second LL that is expected to host non-Abelian statistics. Our results suggest that the 2D semiconductors can provide an experimental platform that closely resembles idealized theoretical models in the quantum Hall regime.

Tunable Valley Splitting due to Topological Orbital Magnetic Moment in Bilayer Graphene Quantum Point Contacts

In multivalley semiconductors, the valley degree of freedom can be potentially used to store, manipulate, and read quantum information, but its control remains challenging. The valleys in bilayer graphene can be addressed by a perpendicular magnetic field which couples by the valley g factor g_{v}. However, control over g_{v} has not been demonstrated yet. We experimentally determine the energy spectrum of a quantum point contact realized by a suitable gate geometry in bilayer graphene. Using finite bias spectroscopy, we measure the energy scales arising from the lateral confinement as well as the Zeeman splitting and find a spin g factor g_{s}∼2. g_{v} can be tuned by a factor of 3 using vertical electric fields, g_{v}∼40-120. The results are quantitatively explained by a calculation considering topological magnetic moment and its dependence on confinement and the vertical displacement field.

Widely Tunable Quantum Phase Transition from Moore-Read to Composite Fermi Liquid in Bilayer Graphene

We develop a proposal to realize a widely tunable and clean quantum phase transition in bilayer graphene between two paradigmatic fractionalized phases of matter: the Moore-Read fractional quantum Hall state and the composite Fermi liquid metal. This transition can be realized at total fillings ν=±3+1/2 and the critical point can be controllably accessed by tuning either the interlayer electric bias or the perpendicular magnetic field values over a wide range of parameters. We study the transition numerically within a model that contains all leading single particle corrections to the band structure of bilayer graphene and includes the fluctuations between the n=0 and n=1 cyclotron orbitals of its zeroth Landau level to delineate the most favorable region of parameters to experimentally access this unconventional critical point. We also find evidence for a new anisotropic gapless phase stabilized near the level crossing of n=0/1 orbits.

The electronic thickness of graphene

When two dimensional crystals are atomically close, their finite thickness becomes relevant. Using transport measurements, we investigate the electrostatics of two graphene layers, twisted by θ = 22° such that the layers are decoupled by the huge momentum mismatch between the K and K' points of the two layers. We observe a splitting of the zero-density lines of the two layers with increasing interlayer energy difference. This splitting is given by the ratio of single-layer quantum capacitance over interlayer capacitance C _m and is therefore suited to extract C _m. We explain the large observed value of C _m by considering the finite dielectric thickness d _g of each graphene layer and determine d _g ≈ 2.6 Å. In a second experiment, we map out the entire density range with a Fabry-Pérot resonator. We can precisely measure the Fermi wavelength λ in each layer, showing that the layers are decoupled. Our findings are reproduced using tight-binding calculations.

Reliable Postprocessing Improvement of van der Waals Heterostructures

The successful assembly of heterostructures consisting of several layers of different 2D materials in arbitrary order by exploiting van der Waals forces has truly been a game changer in the field of low-dimensional physics. For instance, the encapsulation of graphene or MoS₂ between atomically flat hexagonal boron nitride (hBN) layers with strong affinity and graphitic gates that screen charge impurity disorder provided access to a plethora of interesting physical phenomena by drastically boosting the device quality. The encapsulation is accompanied by a self-cleansing effect at the interfaces. The otherwise predominant charged impurity disorder is minimized, and random strain fluctuations ultimately constitute the main source of residual disorder. Despite these advances, the fabricated heterostructures still vary notably in their performance. Although some achieve record mobilities, others only possess mediocre quality. Here, we report a reliable method to improve fully completed van der Waals heterostructure devices with a straightforward postprocessing surface treatment based on thermal annealing and contact mode atomic force microscopy (AFM). The impact is demonstrated by comparing magnetotransport measurements before and after the AFM treatment on one and the same device as well as on a larger set of treated and untreated devices to collect device statistics. Both the low-temperature properties and the room temperature electrical characteristics, as relevant for applications, improve on average substantially. We surmise that the main beneficial effect arises from reducing nanometer scale corrugations at the interfaces, that is, the detrimental impact of random strain fluctuations.

Intrinsic quantized anomalous Hall effect in a moiré heterostructure

The quantum anomalous Hall (QAH) effect combines topology and magnetism to produce precisely quantized Hall resistance at zero magnetic field. We report the observation of a QAH effect in twisted bilayer graphene aligned to hexagonal boron nitride. The effect is driven by intrinsic strong interactions, which polarize the electrons into a single spin- and valley-resolved moiré miniband with Chern number C = 1. In contrast to magnetically doped systems, the measured transport energy gap is larger than the Curie temperature for magnetic ordering, and quantization to within 0.1% of the von Klitzing constant persists to temperatures of several kelvin at zero magnetic field. Electrical currents as small as 1 nanoampere controllably switch the magnetic order between states of opposite polarization, forming an electrically rewritable magnetic memory.