One-dimensional proximity superconductivity in the quantum Hall regime.

Extensive efforts have been undertaken to combine superconductivity and the quantum Hall effect so that Cooper-pair transport between superconducting electrodes in Josephson junctions is mediated by one-dimensional edge states1-6. This interest has been motivated by prospects of finding new physics, including topologically protected quasiparticles7-9, but also extends into metrology and device applications10-13. So far it has proven challenging to achieve detectable supercurrents through quantum Hall conductors2,3,6. Here we show that domain walls in minimally twisted bilayer graphene14-18 support exceptionally robust proximity superconductivity in the quantum Hall regime, allowing Josephson junctions to operate in fields close to the upper critical field of superconducting electrodes. The critical current is found to be non-oscillatory and practically unchanging over the entire range of quantizing fields, with its value being limited by the quantum conductance of ballistic, strictly one-dimensional, electronic channels residing within the domain walls. The system described is unique in its ability to support Andreev bound states at quantizing fields and offers many interesting directions for further exploration.

Mixing of moiré-surface and bulk states in graphite.

Van der Waals assembly enables the design of electronic states in two-dimensional (2D) materials, often by superimposing a long-wavelength periodic potential on a crystal lattice using moiré superlattices1-9. This twistronics approach has resulted in numerous previously undescribed physics, including strong correlations and superconductivity in twisted bilayer graphene10-12, resonant excitons, charge ordering and Wigner crystallization in transition-metal chalcogenide moiré structures13-18 and Hofstadter's butterfly spectra and Brown-Zak quantum oscillations in graphene superlattices19-22. Moreover, twistronics has been used to modify near-surface states at the interface between van der Waals crystals23,24. Here we show that electronic states in three-dimensional (3D) crystals such as graphite can be tuned by a superlattice potential occurring at the interface with another crystal-namely, crystallographically aligned hexagonal boron nitride. This alignment results in several Lifshitz transitions and Brown-Zak oscillations arising from near-surface states, whereas, in high magnetic fields, fractal states of Hofstadter's butterfly draw deep into the bulk of graphite. Our work shows a way in which 3D spectra can be controlled using the approach of 2D twistronics.

Proton transport through nanoscale corrugations in two-dimensional crystals.

Defect-free graphene is impermeable to all atoms1-5 and ions6,7 under ambient conditions. Experiments that can resolve gas flows of a few atoms per hour through micrometre-sized membranes found that monocrystalline graphene is completely impermeable to helium, the smallest atom2,5. Such membranes were also shown to be impermeable to all ions, including the smallest one, lithium6,7. By contrast, graphene was reported to be highly permeable to protons, nuclei of hydrogen atoms8,9. There is no consensus, however, either on the mechanism behind the unexpectedly high proton permeability10-14 or even on whether it requires defects in graphene's crystal lattice6,8,15-17. Here, using high-resolution scanning electrochemical cell microscopy, we show that, although proton permeation through mechanically exfoliated monolayers of graphene and hexagonal boron nitride cannot be attributed to any structural defects, nanoscale non-flatness of two-dimensional membranes greatly facilitates proton transport. The spatial distribution of proton currents visualized by scanning electrochemical cell microscopy reveals marked inhomogeneities that are strongly correlated with nanoscale wrinkles and other features where strain is accumulated. Our results highlight nanoscale morphology as an important parameter enabling proton transport through two-dimensional crystals, mostly considered and modelled as flat, and indicate that strain and curvature can be used as additional degrees of freedom to control the proton permeability of two-dimensional materials.

Giant magnetoresistance of Dirac plasma in high-mobility graphene.

The most recognizable feature of graphene's electronic spectrum is its Dirac point, around which interesting phenomena tend to cluster. At low temperatures, the intrinsic behaviour in this regime is often obscured by charge inhomogeneity1,2 but thermal excitations can overcome the disorder at elevated temperatures and create an electron-hole plasma of Dirac fermions. The Dirac plasma has been found to exhibit unusual properties, including quantum-critical scattering3-5 and hydrodynamic flow6-8. However, little is known about the plasma's behaviour in magnetic fields. Here we report magnetotransport in this quantum-critical regime. In low fields, the plasma exhibits giant parabolic magnetoresistivity reaching more than 100 per cent in a magnetic field of 0.1 tesla at room temperature. This is orders-of-magnitude higher than magnetoresistivity found in any other system at such temperatures. We show that this behaviour is unique to monolayer graphene, being underpinned by its massless spectrum and ultrahigh mobility, despite frequent (Planckian limit) scattering3-5,9-14. With the onset of Landau quantization in a magnetic field of a few tesla, where the electron-hole plasma resides entirely on the zeroth Landau level, giant linear magnetoresistivity emerges. It is nearly independent of temperature and can be suppressed by proximity screening15, indicating a many-body origin. Clear parallels with magnetotransport in strange metals12-14 and so-called quantum linear magnetoresistance predicted for Weyl metals16 offer an interesting opportunity to further explore relevant physics using this well defined quantum-critical two-dimensional system.

Imaging hydrodynamic electrons flowing without Landauer-Sharvin resistance.

Electrical resistance usually originates from lattice imperfections. However, even a perfect lattice has a fundamental resistance limit, given by the Landauer1 conductance caused by a finite number of propagating electron modes. This resistance, shown by Sharvin2 to appear at the contacts of electronic devices, sets the ultimate conduction limit of non-interacting electrons. Recent years have seen growing evidence of hydrodynamic electronic phenomena3-18, prompting recent theories19,20 to ask whether an electronic fluid can radically break the fundamental Landauer-Sharvin limit. Here, we use single-electron-transistor imaging of electronic flow in high-mobility graphene Corbino disk devices to answer this question. First, by imaging ballistic flows at liquid-helium temperatures, we observe a Landauer-Sharvin resistance that does not appear at the contacts but is instead distributed throughout the bulk. This underpins the phase-space origin of this resistance-as emerging from spatial gradients in the number of conduction modes. At elevated temperatures, by identifying and accounting for electron-phonon scattering, we show the details of the purely hydrodynamic flow. Strikingly, we find that electron hydrodynamics eliminates the bulk Landauer-Sharvin resistance. Finally, by imaging spiralling magneto-hydrodynamic Corbino flows, we show the key emergent length scale predicted by hydrodynamic theories-the Gurzhi length. These observations demonstrate that electronic fluids can dramatically transcend the fundamental limitations of ballistic electrons, with important implications for fundamental science and future technologies.

Long-range nontopological edge currents in charge-neutral graphene.

Van der Waals heterostructures display numerous unique electronic properties. Nonlocal measurements, wherein a voltage is measured at contacts placed far away from the expected classical flow of charge carriers, have been widely used in the search for novel transport mechanisms, including dissipationless spin and valley transport1-9, topological charge-neutral currents10-12, hydrodynamic flows13 and helical edge modes14-16. Monolayer1-5,10,15-19, bilayer9,11,14,20 and few-layer21 graphene, transition-metal dichalcogenides6,7 and moiré superlattices8,10,12 have been found to display pronounced nonlocal effects. However, the origin of these effects is hotly debated3,11,17,22-24. Graphene, in particular, exhibits giant nonlocality at charge neutrality1,15-19, a striking behaviour that has attracted competing explanations. Using a superconducting quantum interference device on a tip (SQUID-on-tip) for nanoscale thermal and scanning gate imaging25, here we demonstrate that the commonly occurring charge accumulation at graphene edges23,26-31 leads to giant nonlocality, producing narrow conductive channels that support long-range currents. Unexpectedly, although the edge conductance has little effect on the current flow in zero magnetic field, it leads to field-induced decoupling between edge and bulk transport at moderate fields. The resulting giant nonlocality at charge neutrality and away from it produces exotic flow patterns that are sensitive to edge disorder, in which charges can flow against the global electric field. The observed one-dimensional edge transport is generic and nontopological and is expected to support nonlocal transport in many electronic systems, offering insight into the numerous controversies and linking them to long-range guided electronic states at system edges.

Capillary condensation under atomic-scale confinement.

Capillary condensation of water is ubiquitous in nature and technology. It routinely occurs in granular and porous media, can strongly alter such properties as adhesion, lubrication, friction and corrosion, and is important in many processes used by microelectronics, pharmaceutical, food and other industries1-4. The century-old Kelvin equation5 is frequently used to describe condensation phenomena and has been shown to hold well for liquid menisci with diameters as small as several nanometres1-4,6-14. For even smaller capillaries that are involved in condensation under ambient humidity and so of particular practical interest, the Kelvin equation is expected to break down because the required confinement becomes comparable to the size of water molecules1-22. Here we use van der Waals assembly of two-dimensional crystals to create atomic-scale capillaries and study condensation within them. Our smallest capillaries are less than four ångströms in height and can accommodate just a monolayer of water. Surprisingly, even at this scale, we find that the macroscopic Kelvin equation using the characteristics of bulk water describes the condensation transition accurately in strongly hydrophilic (mica) capillaries and remains qualitatively valid for weakly hydrophilic (graphite) ones. We show that this agreement is fortuitous and can be attributed to elastic deformation of capillary walls23-25, which suppresses the giant oscillatory behaviour expected from the commensurability between the atomic-scale capillaries and water molecules20,21. Our work provides a basis for an improved understanding of capillary effects at the smallest scale possible, which is important in many realistic situations.

Limits on gas impermeability of graphene.

Despite being only one-atom thick, defect-free graphene is considered to be completely impermeable to all gases and liquids1-10. This conclusion is based on theory3-8 and supported by experiments1,9,10 that could not detect gas permeation through micrometre-size membranes within a detection limit of 105 to 106 atoms per second. Here, using small monocrystalline containers tightly sealed with graphene, we show that defect-free graphene is impermeable with an accuracy of eight to nine orders of magnitude higher than in the previous experiments. We are capable of discerning (but did not observe) permeation of just a few helium atoms per hour, and this detection limit is also valid for all other gases tested (neon, nitrogen, oxygen, argon, krypton and xenon), except for hydrogen. Hydrogen shows noticeable permeation, even though its molecule is larger than helium and should experience a higher energy barrier. This puzzling observation is attributed to a two-stage process that involves dissociation of molecular hydrogen at catalytically active graphene ripples, followed by adsorbed atoms flipping to the other side of the graphene sheet with a relatively low activation energy of about 1.0 electronvolt, a value close to that previously reported for proton transport11,12. Our work provides a key reference for the impermeability of two-dimensional materials and is important from a fundamental perspective and for their potential applications.

Electronic phase separation in multilayer rhombohedral graphite.

Of the two stable forms of graphite, hexagonal and rhombohedral, the former is more common and has been studied extensively. The latter is less stable, which has so far precluded its detailed investigation, despite many theoretical predictions about the abundance of exotic interaction-induced physics1-6. Advances in van der Waals heterostructure technology7 have now allowed us to make high-quality rhombohedral graphite films up to 50 graphene layers thick and study their transport properties. Here we show that the bulk electronic states in such rhombohedral graphite are gapped8 and, at low temperatures, electron transport is dominated by surface states. Because of their proposed topological nature, the surface states are of sufficiently high quality to observe the quantum Hall effect, whereby rhombohedral graphite exhibits phase transitions between a gapless semimetallic phase and a gapped quantum spin Hall phase with giant Berry curvature. We find that an energy gap can also be opened in the surface states by breaking their inversion symmetry by applying a perpendicular electric field. Moreover, in rhombohedral graphite thinner than four nanometres, a gap is present even without an external electric field. This spontaneous gap opening shows pronounced hysteresis and other signatures characteristic of electronic phase separation, which we attribute to emergence of strongly correlated electronic surface states.

Unexpected catalytic activity of nanorippled graphene.

Graphite is one of the most chemically inert materials. Its elementary constituent, monolayer graphene, is generally expected to inherit most of the parent material's properties including chemical inertness. Here, we show that, unlike graphite, defect-free monolayer graphene exhibits a strong activity with respect to splitting molecular hydrogen, which is comparable to that of metallic and other known catalysts for this reaction. We attribute the unexpected catalytic activity to surface corrugations (nanoscale ripples), a conclusion supported by theory. Nanoripples are likely to play a role in other chemical reactions involving graphene and, because nanorippling is inherent to atomically thin crystals, can be important for two-dimensional (2D) materials in general.

Molecular streaming and its voltage control in ångström-scale channels.

Over the past decade, the ability to reduce the dimensions of fluidic devices to the nanometre scale (by using nanotubes1-5 or nanopores6-11, for example) has led to the discovery of unexpected water- and ion-transport phenomena12-14. More recently, van der Waals assembly of two-dimensional materials15 has allowed the creation of artificial channels with ångström-scale precision16. Such channels push fluid confinement to the molecular scale, wherein the limits of continuum transport equations17 are challenged. Water films on this scale can rearrange into one or two layers with strongly suppressed dielectric permittivity18,19 or form a room-temperature ice phase20. Ionic motion in such confined channels21 is affected by direct interactions between the channel walls and the hydration shells of the ions, and water transport becomes strongly dependent on the channel wall material22. We explore how water and ionic transport are coupled in such confinement. Here we report measurements of ionic fluid transport through molecular-sized slit-like channels. The transport, driven by pressure and by an applied electric field, reveals a transistor-like electrohydrodynamic effect. An applied bias of a fraction of a volt increases the measured pressure-driven ionic transport (characterized by streaming mobilities) by up to 20 times. This gating effect is observed in both graphite and hexagonal boron nitride channels but exhibits marked material-dependent differences. We use a modified continuum framework accounting for the material-dependent frictional interaction of water molecules, ions and the confining surfaces to explain the differences observed between channels made of graphene and hexagonal boron nitride. This highly nonlinear gating of fluid transport under molecular-scale confinement may offer new routes to control molecular and ion transport, and to explore electromechanical couplings that may have a role in recently discovered mechanosensitive ionic channels23.

Imaging work and dissipation in the quantum Hall state in graphene.

Topology is a powerful recent concept asserting that quantum states could be globally protected against local perturbations1,2. Dissipationless topologically protected states are therefore of major fundamental interest as well as of practical importance in metrology and quantum information technology. Although topological protection can be robust theoretically, in realistic devices it is often susceptible to various dissipative mechanisms, which are difficult to study directly because of their microscopic origins. Here we use scanning nanothermometry3 to visualize and investigate the microscopic mechanisms that undermine dissipationless transport in the quantum Hall state in graphene. Simultaneous nanoscale thermal and scanning gate microscopy shows that the dissipation is governed by crosstalk between counterpropagating pairs of downstream and upstream channels that appear at graphene boundaries as a result of edge reconstruction. Instead of local Joule heating, however, the dissipation mechanism comprises two distinct and spatially separated processes. The work-generating process that we image directly, which involves elastic tunnelling of charge carriers between the quantum channels, determines the transport properties but does not generate local heat. By contrast, the heat and entropy generation process-which we visualize independently-occurs nonlocally upon resonant inelastic scattering from single atomic defects at graphene edges, and does not affect transport. Our findings provide an insight into the mechanisms that conceal the true topological protection, and suggest routes towards engineering more robust quantum states for device applications.

Publisher Correction: Imaging work and dissipation in the quantum Hall state in graphene.

An Amendment to this paper has been published and can be accessed via a link at the top of the paper.

Ballistic molecular transport through two-dimensional channels.

Gas permeation through nanoscale pores is ubiquitous in nature and has an important role in many technologies1,2. Because the pore size is typically smaller than the mean free path of gas molecules, the flow of the gas molecules is conventionally described by Knudsen theory, which assumes diffuse reflection (random-angle scattering) at confining walls3-7. This assumption holds surprisingly well in experiments, with only a few cases of partially specular (mirror-like) reflection known5,8-11. Here we report gas transport through ångström-scale channels with atomically flat walls12,13 and show that surface scattering can be either diffuse or specular, depending on the fine details of the atomic landscape of the surface, and that quantum effects contribute to the specularity at room temperature. The channels, made from graphene or boron nitride, allow helium gas flow that is orders of magnitude faster than expected from theory. This is explained by specular surface scattering, which leads to ballistic transport and frictionless gas flow. Similar channels, but with molybdenum disulfide walls, exhibit much slower permeation that remains well described by Knudsen diffusion. We attribute the difference to the larger atomic corrugations at molybdenum disulfide surfaces, which are similar in height to the size of the atoms being transported and their de Broglie wavelength. The importance of this matter-wave contribution is corroborated by the observation of a reversed isotope effect, whereby the mass flow of hydrogen is notably higher than that of deuterium, in contrast to the relation expected for classical flows. Our results provide insights into the atomistic details of molecular permeation, which previously could be accessed only in simulations10,14, and demonstrate the possibility of studying gas transport under controlled confinement comparable in size to the quantum-mechanical size of atoms.

Electrically controlled water permeation through graphene oxide membranes.

Controlled transport of water molecules through membranes and capillaries is important in areas as diverse as water purification and healthcare technologies1-7. Previous attempts to control water permeation through membranes (mainly polymeric ones) have concentrated on modulating the structure of the membrane and the physicochemical properties of its surface by varying the pH, temperature or ionic strength3,8. Electrical control over water transport is an attractive alternative; however, theory and simulations9-14 have often yielded conflicting results, from freezing of water molecules to melting of ice14-16 under an applied electric field. Here we report electrically controlled water permeation through micrometre-thick graphene oxide membranes17-21. Such membranes have previously been shown to exhibit ultrafast permeation of water17,22 and molecular sieving properties18,21, with the potential for industrial-scale production. To achieve electrical control over water permeation, we create conductive filaments in the graphene oxide membranes via controllable electrical breakdown. The electric field that concentrates around these current-carrying filaments ionizes water molecules inside graphene capillaries within the graphene oxide membranes, which impedes water transport. We thus demonstrate precise control of water permeation, from ultrafast permeation to complete blocking. Our work opens up an avenue for developing smart membrane technologies for artificial biological systems, tissue engineering and filtration.

Nanoscale thermal imaging of dissipation in quantum systems.

Energy dissipation is a fundamental process governing the dynamics of physical, chemical and biological systems. It is also one of the main characteristics that distinguish quantum from classical phenomena. In particular, in condensed matter physics, scattering mechanisms, loss of quantum information or breakdown of topological protection are deeply rooted in the intricate details of how and where the dissipation occurs. Yet the microscopic behaviour of a system is usually not formulated in terms of dissipation because energy dissipation is not a readily measurable quantity on the micrometre scale. Although nanoscale thermometry has gained much recent interest, existing thermal imaging methods are not sensitive enough for the study of quantum systems and are also unsuitable for the low-temperature operation that is required. Here we report a nano-thermometer based on a superconducting quantum interference device with a diameter of less than 50 nanometres that resides at the apex of a sharp pipette: it provides scanning cryogenic thermal sensing that is four orders of magnitude more sensitive than previous devices-below 1 µK Hz-1/2. This non-contact, non-invasive thermometry allows thermal imaging of very low intensity, nanoscale energy dissipation down to the fundamental Landauer limit of 40 femtowatts for continuous readout of a single qubit at one gigahertz at 4.2 kelvin. These advances enable the observation of changes in dissipation due to single-electron charging of individual quantum dots in carbon nanotubes. They also reveal a dissipation mechanism attributable to resonant localized states in graphene encapsulated within hexagonal boron nitride, opening the door to direct thermal imaging of nanoscale dissipation processes in quantum matter.

Molecular transport through capillaries made with atomic-scale precision.

Nanometre-scale pores and capillaries have long been studied because of their importance in many natural phenomena and their use in numerous applications. A more recent development is the ability to fabricate artificial capillaries with nanometre dimensions, which has enabled new research on molecular transport and led to the emergence of nanofluidics. But surface roughness in particular makes it challenging to produce capillaries with precisely controlled dimensions at this spatial scale. Here we report the fabrication of narrow and smooth capillaries through van der Waals assembly, with atomically flat sheets at the top and bottom separated by spacers made of two-dimensional crystals with a precisely controlled number of layers. We use graphene and its multilayers as archetypal two-dimensional materials to demonstrate this technology, which produces structures that can be viewed as if individual atomic planes had been removed from a bulk crystal to leave behind flat voids of a height chosen with atomic-scale precision. Water transport through the channels, ranging in height from one to several dozen atomic planes, is characterized by unexpectedly fast flow (up to 1 metre per second) that we attribute to high capillary pressures (about 1,000 bar) and large slip lengths. For channels that accommodate only a few layers of water, the flow exhibits a marked enhancement that we associate with an increased structural order in nanoconfined water. Our work opens up an avenue to making capillaries and cavities with sizes tunable to ångström precision, and with permeation properties further controlled through a wide choice of atomically flat materials available for channel walls.

Square ice in graphene nanocapillaries.

Bulk water exists in many forms, including liquid, vapour and numerous crystalline and amorphous phases of ice, with hexagonal ice being responsible for the fascinating variety of snowflakes. Much less noticeable but equally ubiquitous is water adsorbed at interfaces and confined in microscopic pores. Such low-dimensional water determines aspects of various phenomena in materials science, geology, biology, tribology and nanotechnology. Theory suggests many possible phases for adsorbed and confined water, but it has proved challenging to assess its crystal structure experimentally. Here we report high-resolution electron microscopy imaging of water locked between two graphene sheets, an archetypal example of hydrophobic confinement. The observations show that the nanoconfined water at room temperature forms 'square ice'--a phase having symmetry qualitatively different from the conventional tetrahedral geometry of hydrogen bonding between water molecules. Square ice has a high packing density with a lattice constant of 2.83 Å and can assemble in bilayer and trilayer crystallites. Molecular dynamics simulations indicate that square ice should be present inside hydrophobic nanochannels independently of their exact atomic nature.

High-order fractal states in graphene superlattices.

Graphene superlattices were shown to exhibit high-temperature quantum oscillations due to periodic emergence of delocalized Bloch states in high magnetic fields such that unit fractions of the flux quantum pierce a superlattice unit cell. Under these conditions, semiclassical electron trajectories become straight again, similar to the case of zero magnetic field. Here, we report magnetotransport measurements that reveal second-, third-, and fourth-order magnetic Bloch states at high electron densities and temperatures above 100 K. The recurrence of these states creates a fractal pattern intimately related to the origin of Hofstadter butterflies. The hierarchy of the fractal states is determined by the width of magnetic minibands, in qualitative agreement with our band-structure calculations.

Indirect Excitons and Trions in MoSe2/WSe2 van der Waals Heterostructures.

Indirect excitons (IX) in semiconductor heterostructures are bosons, which can cool below the temperature of quantum degeneracy and can be effectively controlled by voltage and light. IX quantum Bose gases and IX devices were explored in GaAs heterostructures where an IX range of existence is limited to low temperatures due to low IX binding energies. IXs in van der Waals transition-metal dichalcogenide (TMD) heterostructures are characterized by large binding energies giving the opportunity for exploring excitonic quantum gases and for creating excitonic devices at high temperatures. TMD heterostructures also offer a new platform for studying single-exciton phenomena and few-particle complexes. In this work, we present studies of IXs in MoSe2/WSe2 heterostructures and report on two IX luminescence lines whose energy splitting and temperature dependence identify them as neutral and charged IXs. The experimentally found binding energy of the indirect charged excitons, that is, indirect trions, is close to the calculated binding energy of 28 meV for negative indirect trions in TMD heterostructures [Deilmann, T.; Thygesen, K. S. Nano Lett. 2018, 18, 1460]. We also report on the realization of IXs with a luminescence line width reaching 4 meV at low temperatures. An enhancement of IX luminescence intensity and the narrow line width are observed in localized spots.