Our thinking about electron transport is usually rooted in Ohm's law, J(x) = σ(x)E(x), which is a conveniently local relation between the current density J(x) and the electric field E(x) at some point x.Underlying Ohm's law is an assumption that the rate of momentum relaxation, and therefore of energy dissipation, is determined by spatially local processes.Even though we often analogize the flow of electric current to the flow of water, such locality is not generally how actual fluids behave.Instead, fluids typically relax their momentum by viscous shearing, which arises from gradients of current density.Thus, viscosity generates a "nonlocality" in transport, where the local electric field depends on gradients of the current.Where viscous effects exist, the linearized equation of motion (for steady-state, incompressible flow) becomes

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.

Van der Waals heterostructures display a rich variety of unique electronic properties. To identify novel transport mechanisms, nonlocal measurements have been widely used, wherein a voltage is measured at contacts placed far away from the expected classical flow of charge carriers. This approach was employed in search of dissipationless spin and valley transport, topological charge-neutral currents, hydrodynamic flows and helical edge modes. Monolayer, bilayer, and few-layer graphene, transition-metal dichalcogenides, and moire superlattices were found to display pronounced nonlocal effects. However, the origin of these effects is hotly debated. Graphene, in particular, exhibits giant nonlocality at charge neutrality, a prominent behavior that attracted competing explanations. Utilizing superconducting quantum interference device on a tip (SQUID-on-tip) for nanoscale thermal and scanning gate imaging, we demonstrate that the commonly-occurring charge accumulation at graphene edges leads to giant nonlocality, producing narrow conductive channels that support long-range currents. Unexpectedly, while the edge conductance has little impact 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 both at charge neutrality and away from it produces exotic flow patterns in which charges can flow against the global electric field. We have visualized surprisingly intricate patterns of nonlocal currents, which are sensitive to edge disorder. The observed one-dimensional edge transport, being generic and nontopological, is expected to support nonlocal transport in many electronic systems, offering insight into numerous controversies in the literature and linking them to long-range guided electronic states at system edges.

In 1665 Christiaan Huygens first noticed how two pendulums, regardless of their initial state, would synchronize. It is now known that the universe is full of complex self‐organizing systems, from neural networks to correlated materials. Here, graphene flakes, nucleated over a polycrystalline graphene film, synchronize during growth so as to ultimately yield a common crystal orientation at the macroscale. Strain and diffusion gradients are argued as the probable causes for the long‐range cross‐talk between flakes and the formation of a single‐grain graphene layer. The work demonstrates that graphene synthesis can be advanced to control the nucleated crystal shape, registry, and relative alignment between graphene crystals for large area, that is, a single‐crystal bilayer, and (AB‐stacked) few‐layer graphene can been grown at the wafer scale.

Graphene flakes, nucleated over a polycrystalline graphene film, are shown by Slava V. Rotkin, Mark H. Rümmeli, and co-workers in article number 2002755 to rotate and synchronize their crystal orientation globally early on during growth. This process causes the flakes to adopt a variety of fractal like shapes. As the flakes continue to grow, they eventually merge to form large-area single-crystal graphene.

s by alphabetic order Spin-charge relaxation in a Silicon MOS transistor Agostino Aprà ∗ , Alessandro Crippa , Marco Tagliaferri , Rami Ezzouch ∗ , Simon Zihlmann ∗ , Nicolas Piot ∗ , Iulian Matei , Louis Hutin , Maude Vinet , Xavier Jehl ∗ , Romain Maurand ∗ , Silvano De Franceschi ∗

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.

A variety of physical phenomena associated with nanoscale electron transport often results in non-trivial spatial voltage and current patterns, particularly in nonlocal transport regimes. While numerous techniques have been devised to image electron flows, the need remains for a nanoscale probe capable of simultaneously imaging current and voltage distributions with high sensitivity and minimal invasiveness, in a magnetic field, across a broad range of temperatures and beneath an insulating surface. Here we present a technique for spatially mapping electron flows based on a nanotube single-electron transistor, which achieves high sensitivity for both voltage and current imaging. In a series of experiments using high-mobility graphene devices, we demonstrate the ability of our technique to visualize local aspects of intrinsically nonlocal transport, as in ballistic flows, which are not easily resolvable via existing methods. This technique should aid in understanding the physics of two-dimensional electronic devices and enable new classes of experiments that image electron flow through buried nanostructures in the quantum and interaction-dominated regimes.

Van der Waals materials and their heterostructures offer a versatile platform for studying a variety of quantum transport phenomena due to their unique crystalline properties and the exceptional ability in tuning their electronic spectrum. However, most experiments are limited to devices that have lateral dimensions of only a few micrometres. Here, we perform magnetotransport measurements on graphene/hexagonal boron-nitride Hall bars and show that wider devices reveal additional quantum effects. In devices wider than ten micrometres we observe distinct magnetoresistance oscillations that are caused by resonant scattering of Landau-quantised Dirac electrons by acoustic phonons in graphene. The study allows us to accurately determine graphene’s low energy phonon dispersion curves and shows that transverse acoustic modes cause most of phonon scattering. Our work highlights the crucial importance of device width when probing quantum effects and also demonstrates a precise, spectroscopic method for studying electron-phonon interactions in van der Waals heterostructures.

When two-dimensional (2D) atomic crystals are brought into close proximity to form a van der Waals heterostructure, neighbouring crystals may influence each other's properties. Of particular interest is when the two crystals closely match and a moiré pattern forms, resulting in modified electronic and excitonic spectra, crystal reconstruction, and more. Thus, moiré patterns are a viable tool for controlling the properties of 2D materials. However, the difference in periodicity of the two crystals limits the reconstruction and, thus, is a barrier to the low-energy regime. Here, we present a route to spectrum reconstruction at all energies. By using graphene which is aligned to two hexagonal boron nitride layers, one can make electrons scatter in the differential moiré pattern which results in spectral changes at arbitrarily low energies. Further, we demonstrate that the strength of this potential relies crucially on the atomic reconstruction of graphene within the differential moiré super cell.

Hydrodynamics, which generally describes the flow of a fluid, is expected to hold even for fundamental particles such as electrons when inter-particle interactions dominate1. Although various aspects of electron hydrodynamics have been revealed in recent experiments2–11, the fundamental spatial structure of hydrodynamic electrons—the Poiseuille flow profile—has remained elusive. Here we provide direct imaging of the Poiseuille flow of an electronic fluid, as well as a visualization of its evolution from ballistic flow. Using a scanning carbon nanotube single-electron transistor12, we image the Hall voltage of electronic flow through channels of high-mobility graphene. We find that the profile of the Hall field across the channel is a key physical quantity for distinguishing ballistic from hydrodynamic flow. We image the transition from flat, ballistic field profiles at low temperatures into parabolic field profiles at elevated temperatures, which is the hallmark of Poiseuille flow. The curvature of the imaged profiles is qualitatively reproduced by Boltzmann calculations, which allow us to create a ‘phase diagram’ that characterizes the electron flow regimes. Our results provide direct confirmation of Poiseuille flow in the solid state, and enable exploration of the rich physics of interacting electrons in real space. The emergence of a liquid-like electronic flow from ballistic flow in graphene is imaged, and an almost-ideal viscous hydrodynamic fluid of electrons exhibiting a parabolic Poiseuille flow profile is observed.

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

Aside from unusual properties of monolayer graphene, bilayer has been shown to have even more interesting physics, in particular allowing bandgap opening with dual gating for proper interlayer symmetry. Such properties, promising for device applications, ignited significant interest in understanding and controlling the growth of bilayer graphene. Here we systematically investigate a broad set of flow rates and relative gas ratio of CH4 to H2 in atmospheric pressure chemical vapor deposition of multilayered graphene. Two very different growth windows are identified. For relatively high CH4 to H2 ratios, graphene growth is relatively rapid with an initial first full layer forming in seconds upon which new graphene flakes nucleate then grow on top of the first layer. The stacking of these flakes versus the initial graphene layer is mostly turbostratic. This growth mode can be likened to Stranski-Krastanov growth. With relatively low CH4 to H2 ratios, growth rates are reduced due to a lower carbon supply rate. In addition bi-, tri-, and few-layer flakes form directly over the Cu substrate as individual islands. Etching studies show that in this growth mode subsequent layers form beneath the first layer presumably through carbon radical intercalation. This growth mode is similar to that found with Volmer-Weber growth and was shown to produce highly oriented AB-stacked materials. These systematic studies provide new insight into bilayer graphene formation and define the synthetic range where gapped bilayer graphene can be reliably produced.

While optical properties of graphene in the visible region are solely defined by the frequency-independent fine structure constant, an onset of absorption has been observed in the infrared region due to Pauli blocking of interband transitions. Here, we report a complete absorption quenching in the infrared region by coating graphene with bis(trifluoromethanesulfonyl)amine (TFSA), an optically transparent p-type chemical dopant. The Fermi level downshift due to TFSA doping results in enhanced transmission in the infrared region proportional to the doping concentration. An absorption quenching onset method, developed in our work, to extract the Fermi level shift in pristine and doped graphene agrees with values extracted from Raman G-band and 2D-band shifts, Hall measurements and the binding energy shift observed in X-ray photo-electron spectroscopy. Performing simple UV-visible transmittance spectroscopy to obtain the absorption quenching onset of graphene also allows detection of environmental and substrate effects via Fermi level shift. Our method opens up the practical implementation of this unique phenomenon of graphene in future optoelectronic devices.

Single-crystalline artificial AB-stacked bilayer graphene is formed by aligned transfer of two single-crystalline monolayers on a wafer-scale. The obtained bilayer has a well-defined interface and is electronically equivalent to exfoliated or direct-grown AB-stacked bilayers.

Atomically smooth van der Waals materials are structurally stable in a monolayer and a few layers but are susceptible to oxygen-rich environments. In particular, recently emerging materials such as black phosphorus and perovskite have revealed stronger environmental sensitivity than other two-dimensional layered materials, often obscuring the interesting intrinsic electronic and optical properties. Unleashing the true potential of these materials requires oxidation-free sample preparation that protects thin flakes from air exposure. Here, we fabricated few-layer hafnium disulfide (HfS2) field effect transistors (FETs) using an integrated vacuum cluster system and study their electronic properties and stability under ambient conditions. By performing all the device fabrication and characterization procedure under an oxygen- and moisture-free environment, we found that few-layer AA-stacking HfS2-FETs display excellent field effect responses (Ion/Ioff ≈ 10(7)) with reduced hysteresis compared to the FETs prepared under ambient conditions. Oxidation of HfS2 occurs uniformly over the entire area, increasing the film thickness by 250% at a prolonged oxidation time of >120 h, while defects on the surface are the preferential initial oxidation sites. We further demonstrated that the stability of the device in air is significantly improved by passivating FETs with BN in a vacuum cluster.

Grain boundaries in monolayer transition metal dichalcogenides have unique atomic defect structures and band dispersion relations that depend on the inter-domain misorientation angle. Here, we explore misorientation angle-dependent electrical transport at grain boundaries in monolayer MoS2 by correlating the atomic defect structures of measured devices analysed with transmission electron microscopy and first-principles calculations. Transmission electron microscopy indicates that grain boundaries are primarily composed of 5-7 dislocation cores with periodicity and additional complex defects formed at high angles, obeying the classical low-angle theory for angles <22°. The inter-domain mobility is minimized for angles <9° and increases nonlinearly by two orders of magnitude before saturating at ∼ 16 cm(2) V(-1) s(-1) around misorientation angle ≈ 20°. This trend is explained via grain-boundary electrostatic barriers estimated from density functional calculations and experimental tunnelling barrier heights, which are ≈ 0.5 eV at low angles and ≈ 0.15 eV at high angles (≥ 20°).

Nature Communications 6, Article number: 7809 (2015); Published 30 July 2015; Updated 28 January 2016 In Fig. 3 of this article, there are a number of errors in the colours used for the data points and curves. In Fig. 3b, the blue data should be green, referring to a thickness of ‘3.5 nm’, and the green data should be blue, referring to a thickness of ‘8 nm’.