Imaging Cyclotron Orbits of Electrons in Graphene

Specular Electron Focusing between Gate-Defined Quantum Point Contacts in Bilayer Graphene

We report multiterminal measurements in a ballistic bilayer graphene (BLG) channel, where multiple spin- and valley-degenerate quantum point contacts (QPCs) are defined by electrostatic gating. By patterning QPCs of different shapes along different crystallographic directions, we study the effect of size quantization and trigonal warping on transverse electron focusing (TEF). Our TEF spectra show eight clear peaks with comparable amplitudes and weak signatures of quantum interference at the lowest temperature, indicating that reflections at the gate-defined edges are specular, and transport is phase coherent. The temperature dependence of the focusing signal shows that, despite the small gate-induced bandgaps in our sample (≲45 meV), several peaks are visible up to 100 K. The achievement of specular reflection, which is expected to preserve the pseudospin information of the electron jets, is promising for the realization of ballistic interconnects for new valleytronic devices.

Measurement of current distribution using infrared thermography

Current distribution measurement methods are widely used in medical examinations, predicting faults in semiconductor devices and assessing structural integrity. Several methods for measuring current distribution are available, such as electrode arrays, coils, and magnetic sensors. However, these measurement methods are unable to obtain images of current distribution with high spatial resolution. Therefore, there is a need to develop a non-contact method to measure current distribution that is capable of capturing images with high spatial resolution. In this study, a non-contact current distribution measurement method based on infrared thermography is proposed. The method uses thermal variations to quantify the current amplitude and reconstructs the current direction based on the passivity of the electric field. For quantification of low frequency current amplitude, the experimental results show that the method can provide accurate current measurement results, for example, at the power frequency (50 Hz), in the range of 1.05-3.45 A, its relative error can be improved to ±3.66% when the calibration fitting method is used. For the high-frequency current, an effective estimate of the current amplitude is obtained using the first-order derivative of temperature variation. When applied to the eddy current detection (256 KHz), it achieves a high-resolution image of the current distribution, and the effectiveness of the method is verified through simulation experiments. The experimental results show that the proposed method not only measures the current amplitude accurately but also improves the spatial resolution in acquiring two-dimensional current distribution images.

Magnetotransport in graphene nanoribbons sandwiched by superconductors at side edges

Magnetotransport properties of the graphene nanoribbons (GNR) that are in contact with superconductors at side edges are investigated numerically with respect to oscillations caused by the cyclotron motion. In terms of the modelling, the superconductors are incorporated as superconducting GNRs to make the Andreev reflection at the graphene-superconductor interface almost perfect. The classical commensurability oscillation appears at low magnetic fields where the cyclotron radius is larger than the width of the nanoribbons. A transition to the circumstance dominated by the quantum interference between Andreev- and normal-reflected components takes place when the Andreev reflection probability is reduced by introducing a barrier at the interface. The near perfection of the Andreev reflection enlarges the period of the oscillation associated with skipping orbits a few orders of magnitude in the quantum limit. Chaotic fluctuations emerge furthermore in the regime of Hofstadter's butterfly. The periodicity of a transmission modulation at the onset of the chaos is revealed to change continuously over eight orders of magnitude of the magnetic-field variation. The commensurability and edge-state oscillations are examined additionally for the situations with specular Andreev reflection.

Electron Optics and Valley Hall Effect of Undulated Graphene

Electron optics is the systematic use of electromagnetic (EM) fields to control electron motions. In graphene, strain induces pseudo-electromagnetic fields to guide electron motion. Here we demonstrate the use of substrate topography to impart desirable strain on graphene to induce static pseudo-EM fields. We derive the quasi-classical equation of motion for Dirac Fermions in a pseudo-EM field in graphene and establish the correspondence between the quasi-classical and quantum mechanical snake states. Based on the trajectory analysis, we design sculpted substrates to realize various "optical devices" such as a converging lens or a collimator, and further propose a setup to achieve valley Hall effect solely through substrate patterning, without any external fields, to be used in valleytronics applications. Finally, we discuss how the predicted strain/pseudo-EM field patterns can be experimentally sustained by typical substrates and generalized to other 2D materials.

Transport Simulation of Graphene Devices with a Generic Potential in the Presence of an Orthogonal Magnetic Field

The effect of an orthogonal magnetic field is introduced into a numerical simulator, based on the solution of the Dirac equation in the reciprocal space, for the study of transport in graphene devices consisting of armchair ribbons with a generic potential. Different approaches are proposed to reach this aim. Their efficiency and range of applicability are compared, with particular focus on the requirements in terms of model setup and on the possible numerical issues that may arise. Then, the extended code is successfully validated, simulating several interesting magnetic-related phenomena in graphene devices, including magnetic-field-induced energy-gap modulation, coherent electron focusing, and Aharonov–Bohm interference effects.

Coherent Jetting from a Gate-Defined Channel in Bilayer Graphene

Graphene has evolved as a platform for quantum transport that can compete with the best and cleanest semiconductor systems. Here, we report on the observation of distinct electronic jets emanating from a narrow split-gate-defined channel in bilayer graphene. We find that these jets, which are visible via their interference patterns, occur predominantly with an angle of 60° between each other. This observation is related to the trigonal warping in the band structure of bilayer graphene, which, in conjunction with electron injection through a constriction, leads to a valley-dependent selection of momenta. This experimental observation of electron jetting has consequences for carrier transport in two-dimensional materials with a trigonally warped band structure in general, as well as for devices relying on ballistic and valley-selective transport.

Quantum magnetotransport oscillations in graphene nanoribbons coupled to superconductors

Magnetotransport properties of zigzag and armchair graphene nanoribbons that are in contact with superconductors are investigated using a tight-binding model. The cyclotron orbital motion together with the quantum interference under the coexistence of Andreev and normal reflections gives rise to a number of oscillations in characteristic magnetic-field regimes when the superconducting coupling is weak. The oscillations become irregular and/or suppressed as the coupling is made strong. The period of the oscillations differs from that when a nonrelativistic two-dimensional electron gas is employed rather than the graphene sheet. The modifications of the oscillations are attributed to the phase shift associated with the reflection from the graphene-superconductor interface. The presence of a magnetic field suppresses the quantum blocking of Andreev transmission, which occurs for the edge mode of zigzag nanoribbons, in the same way regardless of it being induced by the Andreev retro- or specular reflection.

Unveiling Electron Optics in Two-Dimensional Materials by Nonlocal Resistance Mapping

We propose a technique based on nonlocal resistance measurements for mapping transport in electron optics experiments. Utilizing tight-binding transport methods, we show how to use a four-terminal measurement to isolate the ballistic transport from a single lead of interest and reconstruct its contribution to the local density of states. This enables us to propose an experimentally tractable four-terminal device with via contacts for measuring Veselago lensing in a graphene p-n junction. Furthermore, we demonstrate how to extend this method as a scanning probe technique, implementing mapping of complex electron optics experiments including angled junctions, collimation optics, and beam steering. Our results highlight the fundamental importance of electron dephasing in ballistic transport and provide guidelines for isolating electron optics signals of interest. These findings unveil a fresh approach to performing electron optics experiments, with a plethora of two-dimensional material platforms to explore.

Heat capacity in doped graphene under magnetic fields: the role of spin splitting

We study the electronic heat capacity in doped graphene under magnetic fields. The partition function is calculated considering only the thermal excitations in the last occupied energy levels. Due to the large energy separation between the Landau levels (LLs) and the Zeeman splitting, at low temperatures the heat capacity is dominated by the spin excitations in the last occupied LL. Correspondingly the heat capacity oscillates with maximum amplitude at half filling of each LL. At higher temperatures the inter-LLs excitations dominate the heat capacity, with maximum amplitude at full filling factors. The oscillation amplitudes are compared with the phonon heat capacityCp. It is shown that the spin induced heat capacity oscillations have a maximum amplitude approaching 3% ofCp, whereas for the inter-LLs excitations the maximum amplitude is only 0.1% ofCp. These amplitudes decrease in the presence of impurities, although the effect is appreciable if the LLs broadening is bigger than the excitation energies.

Imaging Andreev Reflection in Graphene

Coherent charge transport along ballistic paths can be introduced into graphene by Andreev reflection, for which an electron reflects from a superconducting contact as a hole, while a Cooper pair is transmitted. We use liquid-helium cooled scanning gate microscopy (SGM) to image Andreev reflection in graphene in the magnetic focusing regime, where carriers move along cyclotron orbits between contacts. Images of flow are obtained by deflecting carrier paths and displaying the resulting change in conductance. When electrons enter the superconductor, Andreev-reflected holes leave for the collecting contact. To test the results, we destroy Andreev reflection with a large current and by heating above the critical temperature. In both cases, the reflected carriers change from holes to electrons.

Transient lensing from a photoemitted electron gas imaged by ultrafast electron microscopy

Understanding and controlling ultrafast charge carrier dynamics is of fundamental importance in diverse fields of (quantum) science and technology. Here, we create a three-dimensional hot electron gas through two-photon photoemission from a copper surface in vacuum. We employ an ultrafast electron microscope to record movies of the subsequent electron dynamics on the picosecond-nanosecond time scale. After a prompt Coulomb explosion, the subsequent dynamics is characterized by a rapid oblate-to-prolate shape transformation of the electron gas, and periodic and long-lived electron cyclotron oscillations inside the magnetic field of the objective lens. In this regime, the collective behavior of the oscillating electrons causes a transient, mean-field lensing effect and pronounced distortions in the images. We derive an analytical expression for the time-dependent focal length of the electron-gas lens, and perform numerical electron dynamics and probe image simulations to determine the role of Coulomb self-fields and image charges. This work inspires the visualization of cyclotron dynamics inside two-dimensional electron-gas materials and enables the elucidation of electron/plasma dynamics and properties that could benefit the development of high-brightness electron and X-ray sources.

Concurrent and dual N-doping of graphene/ZnO nanocomposites for enhanced Cr(vi) photoreduction activity under visible-light irradiation

Simultaneous and dual N-doping of two components in reduced graphene oxide/ZnO nanocomposites were successfully achieved by thermally annealing a GO/Zn(HCO₃)₂ precursor in an NH₃ atmosphere. In this facile preparative procedure, NH₃ was used not only as the reagent for in situ reduction of GO but also as the source for N-doping. Detailed characterizations showed that the nitrogen element was successfully and simultaneously incorporated into the crystal lattice of ZnO and graphene phases in the composites and the formation of oxygen vacancies was also achieved. The photocatalytic tests indicated that N-doping of graphene/ZnO (NG/NZO) nanocomposites exhibited the higher Cr(vi) photoreduction activity than graphene/ZnO, virgin N-doped ZnO (NZO) and the sample prepared via simple physically mixing. The mechanistic study demonstrated that the remarkable photocatalytic activity of NG/NZO photocatalysts was due to the synergistic effect of simultaneously N-doping two phases in the composites and the existence of oxygen vacancies, and mainly included the increased electrical conductivity of N-doped graphene (NG), the expansion of visible light harvesting capability of NZO and the effective separation of electron–hole pairs of the oxygen vacancies.

Simultaneous and dual N-doping of two components in reduced graphene oxide/ZnO nanocomposites were successfully achieved by thermally annealing a GO/Zn(HCO₃)₂ precursor in an NH₃ atmosphere.

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 perturbations^1,2. 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.

Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene

When two sheets of graphene are stacked at a small twist angle, the resulting flat superlattice minibands are expected to strongly enhance electron-electron interactions. Here, we present evidence that near three-quarters (34) filling of the conduction miniband, these enhanced interactions drive the twisted bilayer graphene into a ferromagnetic state. In a narrow density range around an apparent insulating state at34, we observe emergent ferromagnetic hysteresis, with a giant anomalous Hall (AH) effect as large as 10.4 kilohms and indications of chiral edge states. Notably, the magnetization of the sample can be reversed by applying a small direct current. Although the AH resistance is not quantized, and dissipation is present, our measurements suggest that the system may be an incipient Chern insulator.

Simultaneous voltage and current density imaging of flowing electrons in two dimensions

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.

Multi-scale approach to first-principles electron transport beyond 100 nm

Multi-scale computational approaches are important for studies of novel, low-dimensional electronic devices since they are able to capture the different length-scales involved in the device operation, and at the same time describe critical parts such as surfaces, defects, interfaces, gates, and applied bias, on a atomistic, quantum-chemical level. Here we present a multi-scale method which enables calculations of electronic currents in two-dimensional devices larger than 100 nm2, where multiple perturbed regions described by density functional theory (DFT) are embedded into an extended unperturbed region described by a DFT-parametrized tight-binding model. We explain the details of the method, provide examples, and point out the main challenges regarding its practical implementation. Finally we apply it to study current propagation in pristine, defected and nanoporous graphene devices, injected by chemically accurate contacts simulating scanning tunneling microscopy probes.

Electron Transport in Nanoporous Graphene: Probing the Talbot Effect

Electrons in graphene can show diffraction and interference phenomena fully analogous to light thanks to their Dirac-like energy dispersion. However, it is not clear how this optical analogy persists in nanostructured graphene, for example, with pores. Nanoporous graphene (NPG) consisting of linked graphene nanoribbons has recently been fabricated using molecular precursors and bottom-up assembly (Moreno et al. Science 2018, 360, 199). We predict that electrons propagating in NPG exhibit the interference Talbot effect, analogous to photons in coupled waveguides. Our results are obtained by parameter-free atomistic calculations of real-sized NPG samples based on seamlessly integrated density functional theory and tight-binding regions. We link the origins of this interference phenomenon to the band structure of the NPG. Most importantly, we demonstrate how the Talbot effect may be detected experimentally using dual-probe scanning tunneling microscopy. Talbot interference of electron waves in NPG or other related materials may open up new opportunities for future quantum electronics, computing, or sensing.

Breakdown of the Law of Reflection at a Disordered Graphene Edge

The law of reflection states that smooth surfaces reflect waves specularly, thereby acting as a mirror. This law is insensitive to disorder as long as its length scale is smaller than the wavelength. Monolayer graphene exhibits a linear dispersion at low energies and consequently a diverging Fermi wavelength. We present proof that for a disordered graphene boundary, resonant scattering off disordered edge modes results in diffusive electron reflection even when the electron wavelength is much longer than the disorder correlation length. Using numerical quantum transport simulations, we demonstrate that this phenomenon can be observed as a nonlocal conductance dip in a magnetic focusing experiment.

Independent Band Modulation in 2D van der Waals Heterostructures via a Novel Device Architecture

Benefiting from the technique of vertically stacking 2D layered materials (2DLMs), an advanced novel device architecture based on a top-gated MoS2/WSe2 van der Waals (vdWs) heterostructure is designed. By adopting a self-aligned metal screening layer (Pd) to the WSe2 channel, a fixed p-doped state of the WSe2 as well as an independent doping control of the MoS2 channel can be achieved, thus guaranteeing an effective energy-band offset modulation and large through current. In such a device, under specific top-gate voltages, a sharp PN junction forms at the edge of the Pd layer and can be effectively manipulated. By varying top-gate voltages, the device can be operated under both quasi-Esaki diode and unipolar-Zener diode modes with tunable current modulations. A maximum gate-coupling efficiency as high as ≈90% and a subthreshold swing smaller than 60 mV dec-1 can be achieved under the band-to-band tunneling regime. The superiority of the proposed device architecture is also confirmed by comparison with a traditional heterostructure device. This work demonstrates the feasibility of a new device structure based on vdWs heterostructures and its potential in future low-power electronic and optoelectronic device applications.

A two-dimensional Dirac fermion microscope

The electron microscope has been a powerful, highly versatile workhorse in the fields of material and surface science, micro and nanotechnology, biology and geology, for nearly 80 years. The advent of two-dimensional materials opens new possibilities for realizing an analogy to electron microscopy in the solid state. Here we provide a perspective view on how a two-dimensional (2D) Dirac fermion-based microscope can be realistically implemented and operated, using graphene as a vacuum chamber for ballistic electrons. We use semiclassical simulations to propose concrete architectures and design rules of 2D electron guns, deflectors, tunable lenses and various detectors. The simulations show how simple objects can be imaged with well-controlled and collimated in-plane beams consisting of relativistic charge carriers. Finally, we discuss the potential of such microscopes for investigating edges, terminations and defects, as well as interfaces, including external nanoscale structures such as adsorbed molecules, nanoparticles or quantum dots.

Theoretical Prediction of a Giant Anisotropic Magnetoresistance in Carbon Nanoscrolls

Snake orbits are trajectories of charge carriers curving back and forth that form at an interface where either the magnetic field direction or the charge carrier type are inverted. In ballistic samples, their presence is manifested in the appearance of magnetoconductance oscillations at small magnetic fields. Here we show that signatures of snake orbits can also be found in the opposite diffusive transport regime. We illustrate this by studying the classical magnetotransport properties of carbon tubular structures subject to relatively weak transversal magnetic fields where snake trajectories appear in close proximity to the zero radial field projections. In carbon nanoscrolls, the formation of snake orbits leads to a strongly directional dependent positive magnetoresistance with an anisotropy up to 80%.

Scanning gate microscopy of magnetic focusing in graphene devices: quantum versus classical simulation

We compare classical versus quantum electron transport in recently investigated magnetic focusing devices (Bhandari et al 2016 Nano Lett. 16 1690) exposed to the perturbing potential of a scanning gate microscope (SGM). Using the Landauer-Büttiker formalism for a multi-terminal device, we calculate resistance maps that are obtained as the SGM tip is scanned over the sample. There are three unique regimes in which the scanning tip can operate (focusing, repelling, and mixed regime) which are investigated. Tip interacts mostly with electrons with cyclotron trajectories passing directly underneath it, leaving a trail of modified current density behind it. Other (indirect) trajectories become relevant when the tip is placed near the edges of the sample, and current is scattered between the tip and the edge. We point out that, in contrast to SGM experiments on gapped semiconductors, the STM tip can induce a pn junction in graphene, which improves contrast and resolution in SGM. We also discuss possible explanations for spatial asymmetry of experimentally measured resistance maps, and connect it with specific configurations of the measuring probes.

Quantum imaging of current flow in graphene

Since its first discovery in 2004, graphene has been found to host a plethora of unusual electronic transport phenomena, making it a fascinating system for fundamental studies in condensed matter physics as well as offering tremendous opportunities for future electronic and sensing devices. Typically, electronic transport in graphene has been investigated via resistivity measurements; however, these measurements are generally blind to spatial information critical to observing and studying landmark transport phenomena in real space and in realistic imperfect devices. We apply quantum imaging to the problem and demonstrate noninvasive, high-resolution imaging of current flow in monolayer graphene structures. Our method uses an engineered array of near-surface, atomic-sized quantum sensors in diamond to map the vector magnetic field and reconstruct the vector current density over graphene geometries of varying complexity, from monoribbons to junctions, with spatial resolution at the diffraction limit and a projected sensitivity to currents as small as 1 μA. The measured current maps reveal strong spatial variations corresponding to physical defects at the submicrometer scale. The demonstrated method opens up an important new avenue to investigate fundamental electronic and spin transport in graphene structures and devices and, more generally, in emerging two-dimensional materials and thin-film systems.

Creating and Steering Highly Directional Electron Beams in Graphene

We put forward a concept to create highly collimated, nondispersive electron beams in pseudorelativistic Dirac materials such as graphene or topological insulator surfaces. Combining negative refraction and Klein collimation at a parabolic pn junction, the proposed lens generates beams, as narrow as the focal length, that stay focused over scales of several microns and can be steered by a magnetic field without losing collimation. We demonstrate the lens capabilities by applying it to two paradigmatic settings of graphene electron optics: We propose a setup for observing high-resolution angle-dependent Klein tunneling, and, exploiting the intimate quantum-to-classical correspondence of these focused electron waves, we consider high-fidelity transverse magnetic focusing accompanied by simulations for current mapping through scanning gate microscopy. Our proposal opens up new perspectives for next-generation graphene electron optics experiments.

Unusual renormalization group (RG) flow and temperature-dependent phase transition in strongly-insulating monolayer epitaxial graphene

By changing the measurement temperature (T), one can vary the effective sample size so as to study the renormalization group (RG) (or T-driven) flow of a semiconductor, a topological insulator, or a graphene device in the complex conductivity plane.