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

Real-time probing of electrons can uncover intricate relaxation mechanisms and many-body interactions in strongly correlated materials. Here, we introduce time, momentum, and energy resolved pump-probe tunneling spectroscopy (Tr-MERTS). The method allows the injection of electrons at a particular energy and observation of their subsequent decay in energy-momentum space. Using Tr-MERTS, we visualize electronic decay processes, with lifetimes from tens of nanoseconds to tens of microseconds, in Landau levels formed in a GaAs quantum well. Although most observed features agree with simple energy-relaxation, we discovered a splitting in the nonequilibrium energy spectrum in the vicinity of a ferromagnetic state. An exact diagonalization study suggests that the splitting arises from a maximally spin-polarized state with higher energy than a conventional equilibrium skyrmion. Furthermore, we observe time-dependent relaxation of the splitting, which we attribute to single-flipped spins forming skyrmions. These results establish Tr-MERTS as a powerful tool for studying the properties of a 2DES beyond equilibrium., (© 2023. The Author(s).)

Electronic states in quasicrystals generally preclude a Bloch description 1 , 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 quasicrystals 2,3 , 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 transition 4,5 , 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 crystals 6-12 . 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., (© 2023. The Author(s), under exclusive licence to Springer Nature Limited.)

We report the first electronic compressibility measurements of magic-angle twisted bilayer graphene. The evolution of the compressibility with carrier density offers insights into the interaction-driven ground state that have not been accessible in prior transport and tunneling studies. From capacitance measurements, we determine the chemical potential as a function of carrier density and find the widths of the energy gaps at fractional filling of the moiré lattice. In the electron-doped regime, we observe unexpectedly large gaps at quarter- and half-filling and strong electron-hole asymmetry. Moreover, we measure a ∼35  meV minibandwidth that is much wider than most theoretical estimates. Finally, we explore the field dependence up to the quantum Hall regime and observe significant differences from transport measurements.

A van der Waals heterostructure is a type of metamaterial that consists of vertically stacked two-dimensional building blocks held together by the van der Waals forces between the layers. This design means that the properties of van der Waals heterostructures can be engineered precisely, even more so than those of two-dimensional materials. One such property is the 'twist' angle between different layers in the heterostructure. This angle has a crucial role in the electronic properties of van der Waals heterostructures, but does not have a direct analogue in other types of heterostructure, such as semiconductors grown using molecular beam epitaxy. For small twist angles, the moiré pattern that is produced by the lattice misorientation between the two-dimensional layers creates long-range modulation of the stacking order. So far, studies of the effects of the twist angle in van der Waals heterostructures have concentrated mostly on heterostructures consisting of monolayer graphene on top of hexagonal boron nitride, which exhibit relatively weak interlayer interaction owing to the large bandgap in hexagonal boron nitride. Here we study a heterostructure consisting of bilayer graphene, in which the two graphene layers are twisted relative to each other by a certain angle. We show experimentally that, as predicted theoretically, when this angle is close to the 'magic' angle the electronic band structure near zero Fermi energy becomes flat, owing to strong interlayer coupling. These flat bands exhibit insulating states at half-filling, which are not expected in the absence of correlations between electrons. We show that these correlated states at half-filling are consistent with Mott-like insulator states, which can arise from electrons being localized in the superlattice that is induced by the moiré pattern. These properties of magic-angle-twisted bilayer graphene heterostructures suggest that these materials could be used to study other exotic many-body quantum phases in two dimensions in the absence of a magnetic field. The accessibility of the flat bands through electrical tunability and the bandwidth tunability through the twist angle could pave the way towards more exotic correlated systems, such as unconventional superconductors and quantum spin liquids.

The single-particle spectral function measures the density of electronic states in a material as a function of both momentum and energy, providing central insights into strongly correlated electron phenomena. Here we demonstrate a high-resolution method for measuring the full momentum- and energy-resolved electronic spectral function of a two-dimensional (2D) electronic system embedded in a semiconductor. The technique remains operational in the presence of large externally applied magnetic fields and functions even for electronic systems with zero electrical conductivity or with zero electron density. Using the technique on a prototypical 2D system, a GaAs quantum well, we uncover signatures of many-body effects involving electron-phonon interactions, plasmons, polarons, and a phonon analog of the vacuum Rabi splitting in atomic systems., (Copyright © 2017, American Association for the Advancement of Science.)

Helical 1D electronic systems are a promising route towards realizing circuits of topological quantum states that exhibit non-Abelian statistics. Here, we demonstrate a versatile platform to realize 1D systems made by combining quantum Hall (QH) edge states of opposite chiralities in a graphene electron-hole bilayer at moderate magnetic fields. Using this approach, we engineer helical 1D edge conductors where the counterpropagating modes are localized in separate electron and hole layers by a tunable electric field. These helical conductors exhibit strong non-local transport signals and suppressed backscattering due to the opposite spin polarizations of the counterpropagating modes. Unlike other approaches used for realizing helical states, the graphene electron-hole bilayer can be used to build new 1D systems incorporating fractional edge states. Indeed, we are able to tune the bilayer devices into a regime hosting fractional and integer edge states of opposite chiralities, paving the way towards 1D helical conductors with fractional quantum statistics.

We report on electronic transport measurements of dual-gated nanodevices of the low-carrier density topological insulator (TI) Bi_{1.5}Sb_{0.5}Te_{1.7}Se_{1.3}. In all devices, the upper and lower surface states are independently tunable to the Dirac point by the top and bottom gate electrodes. In thin devices, electric fields are found to penetrate through the bulk, indicating finite capacitive coupling between the surface states. A charging model allows us to use the penetrating electric field as a measurement of the intersurface capacitance C_{TI} and the surface state energy-density relationship μ(n), which is found to be consistent with independent angle-resolved photoemission spectroscopy measurements. At high magnetic fields, increased field penetration through the surface states is observed, strongly suggestive of the opening of a surface state band gap due to broken time-reversal symmetry.

Low-dimensional electronic systems have traditionally been obtained by electrostatically confining electrons, either in heterostructures or in intrinsically nanoscale materials such as single molecules, nanowires and graphene. Recently, a new method has emerged with the recognition that symmetry-protected topological (SPT) phases, which occur in systems with an energy gap to quasiparticle excitations (such as insulators or superconductors), can host robust surface states that remain gapless as long as the relevant global symmetry remains unbroken. The nature of the charge carriers in SPT surface states is intimately tied to the symmetry of the bulk, resulting in one- and two-dimensional electronic systems with novel properties. For example, time reversal symmetry endows the massless charge carriers on the surface of a three-dimensional topological insulator with helicity, fixing the orientation of their spin relative to their momentum. Weakly breaking this symmetry generates a gap on the surface, resulting in charge carriers with finite effective mass and exotic spin textures. Analogous manipulations have yet to be demonstrated in two-dimensional topological insulators, where the primary example of a SPT phase is the quantum spin Hall state. Here we demonstrate experimentally that charge-neutral monolayer graphene has a quantum spin Hall state when it is subjected to a very large magnetic field angled with respect to the graphene plane. In contrast to time-reversal-symmetric systems, this state is protected by a symmetry of planar spin rotations that emerges as electron spins in a half-filled Landau level are polarized by the large magnetic field. The properties of the resulting helical edge states can be modulated by balancing the applied field against an intrinsic antiferromagnetic instability, which tends to spontaneously break the spin-rotation symmetry. In the resulting canted antiferromagnetic state, we observe transport signatures of gapped edge states, which constitute a new kind of one-dimensional electronic system with a tunable bandgap and an associated spin texture.

van der Waals heterostructures constitute a new class of artificial materials formed by stacking atomically thin planar crystals. We demonstrated band structure engineering in a van der Waals heterostructure composed of a monolayer graphene flake coupled to a rotationally aligned hexagonal boron nitride substrate. The spatially varying interlayer atomic registry results in both a local breaking of the carbon sublattice symmetry and a long-range moiré superlattice potential in the graphene. In our samples, this interplay between short- and long-wavelength effects resulted in a band structure described by isolated superlattice minibands and an unexpectedly large band gap at charge neutrality. This picture is confirmed by our observation of fractional quantum Hall states at ± 5/3 filling and features associated with the Hofstadter butterfly at ultrahigh magnetic fields.

Increases in the gate capacitance of field-effect transistor structures allow the production of lower-power devices that are compatible with higher clock rates, driving the race for developing high-κ dielectrics. However, many-body effects in an electronic system can also enhance capacitance. Onto the electron system that forms at the LaAlO(3)/SrTiO(3) interface, we fabricated top-gate electrodes that can fully deplete the interface of all mobile electrons. Near depletion, we found a greater than 40% enhancement of the gate capacitance. Using an electric-field penetration measurement method, we show that this capacitance originates from a negative compressibility of the interface electron system. Capacitance enhancement exists at room temperature and arises at low electron densities, in which disorder is strong and the in-plane conductance is much smaller than the quantum conductance.

The two-dimensional electron system is a powerful laboratory for investigating the physics of interacting particles. Application of a large magnetic field produces massively degenerate quantum levels known as Landau levels; within a Landau level the kinetic energy of the electrons is suppressed, and electron-electron interactions set the only energy scale. Coulomb interactions break the degeneracy of the Landau levels and can cause the electrons to order into complex ground states. Here we observe, in the high energy single particle spectrum of this system, salient and unexpected structure that extends across a wide range of Landau level filling fractions. The structure appears only when the two-dimensional electron system is cooled to very low temperatures, indicating that it arises from delicate ground state correlations. We characterize this structure by its evolution with changing electron density and applied magnetic field, and present two possible models for understanding these observations. Some of the energies of the features agree qualitatively with what might be expected for composite fermions, which have proven effective for interpreting other experiments in this regime. At the same time, a simple model with electrons localized on ordered lattice sites also generates structure similar to that observed in the experiment. Neither of these models alone is sufficient to explain the observations across the entire range of densities measured. The discovery of this unexpected prominent structure in the single particle spectrum of an otherwise thoroughly studied system suggests that there exist core features of the two-dimensional electron system that have yet to be understood.

Spectroscopic methods involving the sudden injection or ejection of electrons in materials are a powerful probe of electronic structure and interactions. These techniques, such as photoemission and tunnelling, yield measurements of the 'single-particle' density of states spectrum of a system. This density of states is proportional to the probability of successfully injecting or ejecting an electron in these experiments. It is equal to the number of electronic states in the system able to accept an injected electron as a function of its energy, and is among the most fundamental and directly calculable quantities in theories of highly interacting systems. However, the two-dimensional electron system (2DES), host to remarkable correlated electron states such as the fractional quantum Hall effect, has proved difficult to probe spectroscopically. Here we present an improved version of time-domain capacitance spectroscopy that allows us to measure the single-particle density of states of a 2DES with unprecedented fidelity and resolution. Using the method, we perform measurements of a cold 2DES, providing direct measurements of interesting correlated electronic effects at energies that are difficult to reach with other techniques; these effects include the single-particle exchange-enhanced spin gap, single-particle lifetimes in the quantum Hall system, and exchange splitting of Landau levels not at the Fermi surface.

We use a scanning capacitance probe to image transport in the quantum Hall system. Applying a dc bias voltage to the tip induces a ring-shaped incompressible strip (IS) in the 2D electron system (2DES) that moves with the tip. At certain tip positions, short-range disorder in the 2DES creates a quantum dot island in the IS. These islands enable resonant tunneling across the IS, enhancing its conductance by more than 4 orders of magnitude. The images provide a quantitative measure of disorder and suggest resonant tunneling as the primary mechanism for transport across ISs.

Using single electron capacitance spectroscopy, we study electron additions in quantum dots containing two potential minima separated by a shallow barrier. Analysis of the addition spectra in the magnetic field allows us to distinguish between electrons delocalized over the entire dot and those localized in either of the potential minima. We demonstrate that a high magnetic field abruptly splits up a low-density droplet into two smaller fragments, each residing in a potential minimum. An unexplained cancellation of electron repulsion between electrons in these fragments gives rise to paired electron additions.

A scanning probe technique was used to obtain a high-resolution map of the random electrostatic potential inside the quantum Hall liquid. A sharp metal tip, scanned above a semiconductor surface, sensed charges in an embedded two-dimensional (2D) electron gas. Under quantum Hall effect conditions, applying a positive voltage to the tip locally enhanced the 2D electron density and created a "bubble" of electrons in an otherwise unoccupied Landau level. As the tip scanned along the sample surface, the bubble followed underneath. The tip sensed the motions of single electrons entering or leaving the bubble in response to changes in the local 2D electrostatic potential.

Single-electron capacitance spectroscopy precisely measures the energies required to add individual electrons to a quantum dot. The spatial extent of electronic wave functions is probed by investigating the dependence of these energies on changes in the dot confining potential. For low electron densities, electrons occupy distinct spatial sites localized within the dot. At higher densities, the electrons become delocalized, and all wave functions are spread over the full dot area. Near the delocalization transition, the last remaining localized states exist at the perimeter of the dot. Unexpectedly, these electrons appear to bind with electrons in the dot center.