Creating and Steering Highly Directional Electron Beams in Graphene

Electron Collimation in Twisted Bilayer Graphene via Gate-Defined Moiré Barriers

Electron collimation via a graphene p-n junction allows electrostatic control of ballistic electron trajectories akin to that of an optical circuit. Similar manipulation of novel correlated electronic phases in twisted-bilayer graphene (tBLG) can provide additional probes to the underlying physics and device components toward advanced quantum electronics. In this work, we demonstrate collimation of the electron flow via gate-defined moiré barriers in a tBLG device, utilizing the band-insulator gap of the moiré superlattice. A single junction can be tuned to host a chosen combination of conventional pseudo barrier and moiré tunnel barriers, from which we demonstrate improved collimation efficiency. By measuring transport through two consecutive moiré collimators separated by 1 μm, we demonstrate evidence of electron collimation in tBLG in the presence of realistic twist-angle inhomogeneity.

Electron wave and quantum optics in graphene

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

Exchange Coupling Effects on the Magnetotransport Properties of Ni-Nanoparticle-Decorated Graphene

We characterize the effect of ferromagnetic nickel nanoparticles (size ∼6 nm) on the magnetotransport properties of chemical-vapor-deposited (CVD) graphene. The nanoparticles were formed by thermal annealing of a thin Ni film evaporated on top of a graphene ribbon. The magnetoresistance was measured while sweeping the magnetic field at different temperatures, and compared against measurements performed on pristine graphene. Our results show that, in the presence of Ni nanoparticles, the usually observed zero-field peak of resistivity produced by weak localization is widely suppressed (by a factor of ∼3), most likely due to the reduction of the dephasing time as a consequence of the increase in magnetic scattering. On the other hand, the high-field magnetoresistance is amplified by the contribution of a large effective interaction field. The results are discussed in terms of a local exchange coupling, J∼6 meV, between the graphene π electrons and the 3d magnetic moment of nickel. Interestingly, this magnetic coupling does not affect the intrinsic transport parameters of graphene, such as the mobility and transport scattering rate, which remain the same with and without Ni nanoparticles, indicating that the changes in the magnetotransport properties have a purely magnetic origin.

Giant orbital magnetic moments and paramagnetic shift in artificial relativistic atoms and molecules

Materials such as graphene and topological insulators host massless Dirac fermions that enable the study of relativistic quantum phenomena. Single quantum dots and coupled quantum dots formed with massless Dirac fermions can be viewed as artificial relativistic atoms and molecules, respectively. Such structures offer a unique testbed to study atomic and molecular physics in the ultrarelativistic regime (particle speed close to the speed of light). Here we use a scanning tunnelling microscope to create and probe single and coupled electrostatically defined graphene quantum dots to unravel the magnetic-field responses of artificial relativistic nanostructures. We observe a giant orbital Zeeman splitting and orbital magnetic moment up to ~70 meV T^-1 and ~600μ_B (μ_B, Bohr magneton) in single graphene quantum dots. For coupled graphene quantum dots, Aharonov-Bohm oscillations and a strong Van Vleck paramagnetic shift of ~20 meV T^-2 are observed. Our findings provide fundamental insights into relativistic quantum dot states, which can be potentially leveraged for use in quantum information science.

Angular-Dependent Klein Tunneling in Photonic Graphene

The Klein paradox consists in the perfect tunneling of relativistic particles through high potential barriers. It is responsible for the exceptional conductive properties of graphene. It was recently studied in atomic condensates and topological photonics and phononics. While in theory the perfect tunneling holds only for normal incidence, so far the angular dependence of the Klein tunneling and its strong variation with the barrier height were not measured experimentally. In this Letter, we capitalize on the versatility of atomic vapor cells with paraxial beam propagation and index patterning by electromagnetically induced transparency. We report the first experimental observation of perfect Klein transmission in a 2D photonic system (photonic graphene) at normal incidence and measure the angular dependence. Counterintuitively, but in agreement with the Dirac equation, we observe that the decay of the Klein transmission versus angle is suppressed by increasing the barrier height, a key result for the conductivity of graphene and its analogs.

Gate-tunable Veselago interference in a bipolar graphene microcavity

The relativistic charge carriers in monolayer graphene can be manipulated in manners akin to conventional optics. Klein tunneling and Veselago lensing have been previously demonstrated in ballistic graphene pn-junction devices, but collimation and focusing efficiency remains relatively low, preventing realization of advanced quantum devices and controlled quantum interference. Here, we present a graphene microcavity defined by carefully-engineered local strain and electrostatic fields. Electrons are manipulated to form an interference path inside the cavity at zero magnetic field via consecutive Veselago refractions. The observation of unique Veselago interference peaks via transport measurement and their magnetic field dependence agrees with the theoretical expectation. We further utilize Veselago interference to demonstrate localization of uncollimated electrons and thus improvement in collimation efficiency. Our work sheds new light on relativistic single-particle physics and provide a new device concept toward next-generation quantum devices based on manipulation of ballistic electron trajectory.

Graphene Whisperitronics: Transducing Whispering Gallery Modes into Electronic Transport

When confined in circular cavities, graphene relativistic charge carriers occupy whispering gallery modes (WGMs) in analogy to classical acoustic and optical fields. The rich geometrical patterns of the WGMs decorating the local density of states offer promising perspectives to devise new disruptive quantum devices. However, exploiting these highly sensitive resonances requires the transduction of the WGMs to the outside world through source and drain electrodes, a yet unreported configuration. Here, we create a circular p-n island in a graphene device using a polarized scanning gate microscope tip and probe the resulting WGM signatures in in-plane electronic transport through the p-n island. Combining tight-binding simulations and the exact solution of the Dirac equation, we assign the measured device conductance features to WGMs and demonstrate mode selectivity by displacing the p-n island with respect to a constriction. This work therefore constitutes a proof of concept for graphene whisperitronic devices.

Abrikosov vortex corrections to effective magnetic field enhancement in epitaxial graphene

Here, we report the effects of enhanced magnetic fields resulting from type-II superconducting NbTiN slabs adjacent to narrow Hall bar devices fabricated from epitaxial graphene. Observed changes in the magnetoresistances were found to have minimal contributions from device inhomogeneities, magnet hysteresis, electron density variations along the devices, and transient phenomena. We hypothesize that Abrikosov vortices, present in type-II superconductors, contribute to these observations. By determining the London penetration depth, coupled with elements of Ginzburg-Landau theory, one can approximate an upper bound on the effect that vortex densities at low fields (< 1T) have on the reported observations. These analyses offer insights into device fabrication and how to utilize the Meissner effect for any low-field and low-temperature applications using superconductors.

Optical Excitations with Electron Beams: Challenges and Opportunities

Free electron beams such as those employed in electron microscopes have evolved into powerful tools to investigate photonic nanostructures with an unrivaled combination of spatial and spectral precision through the analysis of electron energy losses and cathodoluminescence light emission. In combination with ultrafast optics, the emerging field of ultrafast electron microscopy utilizes synchronized femtosecond electron and light pulses that are aimed at the sampled structures, holding the promise to bring simultaneous sub-Å-sub-fs-sub-meV space-time-energy resolution to the study of material and optical-field dynamics. In addition, these advances enable the manipulation of the wave function of individual free electrons in unprecedented ways, opening sound prospects to probe and control quantum excitations at the nanoscale. Here, we provide an overview of photonics research based on free electrons, supplemented by original theoretical insights and discussion of several stimulating challenges and opportunities. In particular, we show that the excitation probability by a single electron is independent of its wave function, apart from a classical average over the transverse beam density profile, whereas the probability for two or more modulated electrons depends on their relative spatial arrangement, thus reflecting the quantum nature of their interactions. We derive first-principles analytical expressions that embody these results and have general validity for arbitrarily shaped electrons and any type of electron-sample interaction. We conclude with some perspectives on various exciting directions that include disruptive approaches to noninvasive spectroscopy and microscopy, the possibility of sampling the nonlinear optical response at the nanoscale, the manipulation of the density matrices associated with free electrons and optical sample modes, and appealing applications in optical modulation of electron beams, all of which could potentially revolutionize the use of free electrons in photonics.

Anomalous Cyclotron Motion in Graphene Superlattice Cavities

We consider graphene superlattice miniband fermions probed by electronic interferometry in magnetotransport experiments. By decoding the observed Fabry-Pérot interference patterns together with our corresponding quantum transport simulations, we find that the Dirac quasiparticles originating from the superlattice minibands do not undergo conventional cyclotron motion but follow more subtle trajectories. In particular, dynamics at low magnetic fields is characterized by peculiar, straight trajectory segments. Our results provide new insights into superlattice miniband fermions and open up novel possibilities to use periodic potentials in electron optics experiments.

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.

Chemical identification through two-dimensional electron energy-loss spectroscopy

We explore a disruptive approach to nanoscale sensing by performing electron energy loss spectroscopy through the use of low-energy ballistic electrons that propagate on a two-dimensional semiconductor. In analogy to free-space electron microscopy, we show that the presence of analyte molecules in the vicinity of the semiconductor produces substantial energy losses in the electrons, which can be resolved by energy-selective electron injection and detection through actively controlled potential gates. The infrared excitation spectra of the molecules are thereby gathered in this electronic device, enabling the identification of chemical species with high sensitivity. Our realistic theoretical calculations demonstrate the superiority of this technique for molecular sensing, capable of performing spectral identification at the zeptomol level within a microscopic all-electrical device.

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.

Graphene-Based THz Absorber with a Broad Band for Tuning the Absorption Rate and a Narrow Band for Tuning the Absorbing Frequency

In this paper, we propose a broadband absorption-controllable absorber based on nested nanostructure graphene and a narrowband frequency-tunable absorber utilizing gold-graphene hybrid structure in the terahertz regime. The numerical simulation results showed that the absorption of the broadband absorber can be changed from 27% to more than 90% over 0.75 to 1.7 THz by regulating the chemical potential of graphene. With the same regulation mechanism, the absorbing peak of the narrowband absorber can be moved from 2.29 to 2.48 THz continuously with absorption of 90%. Furthermore, via the cascade of the two types of absorbers, an independently tunable dual-band absorber is constituted. Its absorption spectrum is the superposition of absorption-controllable absorber and frequency-tunable absorber. The absorptivity and operating frequency of the two absorbing bands can be tuned independently without mutual inference. Moreover, it is insensitive to the polarization and it maintains high absorption over a wide range of incident angle. For the flexibility, tunability as well as the independence of polarization and angle, this design has wide prospects in various applications.

A corner reflector of graphene Dirac fermions as a phonon-scattering sensor

Dirac fermion optics exploits the refraction of chiral fermions across optics-inspired Klein-tunneling barriers defined by high-transparency p-n junctions. We consider the corner reflector (CR) geometry introduced in optics or radars. We fabricate Dirac fermion CRs using bottom-gate-defined barriers in hBN-encapsulated graphene. By suppressing transmission upon multiple internal reflections, CRs are sensitive to minute phonon scattering rates. Here we report on doping-independent CR transmission in quantitative agreement with a simple scattering model including thermal phonon scattering. As a signature of CRs, we observe Fabry-Pérot oscillations at low temperature, consistent with single-path reflections. Finally, we demonstrate high-frequency operation which promotes CRs as fast phonon detectors. Our work establishes the relevance of Dirac fermion optics in graphene and opens a route for its implementation in topological Dirac matter.

Graphene transistor based on tunable Dirac fermion optics

We present a quantum switch based on analogous Dirac fermion optics (DFO), in which the angle dependence of Klein tunneling is explicitly utilized to build tunable collimators and reflectors for the quantum wave function of Dirac fermions. We employ a dual-source design with a single flat reflector, which minimizes diffusive edge scattering and suppresses the background incoherent transmission. Our gate-tunable collimator-reflector device design enables the quantitative measurement of the net DFO contribution in the switching device operation. We obtain a full set of transmission coefficients between multiple leads of the device, separating the classical contribution from the coherent transport contribution. The DFO behavior demonstrated in this work requires no explicit energy gap. We demonstrate its robustness against thermal fluctuations up to 230 K and large bias current density up to 10² A/m, over a wide range of carrier densities. The characterizable and tunable optical components (collimator-reflector) coupled with the conjugated source electrodes developed in this work provide essential building blocks toward more advanced DFO circuits such as quantum interferometers. The capability of building optical circuit analogies at a microscopic scale with highly tunable electron wavelength paves a path toward highly integrated and electrically tunable electron-optical components and circuits.

Controlling electron flow in anisotropic Dirac materials heterojunctions: a super-diverging lens

Ballistic heterojunctions of Dirac materials offer the opportunity of exploring optics-like phenomena in electronic systems. In this paper, a new perfect lens through special positive refraction is predicted with omnidirectional Klein tunneling of massless Dirac fermions. The novel optics component called a super-diverging lens (SDL) is the counterpart of a Veselago lens (VL). The use of SDL and VL creates a device that simulates the ocular vision. This atypical refraction is due to electrons obeying different Snell's laws of pseudo-spin and group velocity in heterojunctions with elliptical Dirac cones. These findings pave the way for an electron elliptical Dirac optics and open up new possibilities for the guiding of electrons.

Large-scale tight-binding simulations of quantum transport in ballistic graphene

Graphene has proven to host outstanding mesoscopic effects involving massless Dirac quasiparticles travelling ballistically resulting in the current flow exhibiting light-like behaviour. A new branch of 2D electronics inspired by the standard principles of optics is rapidly evolving, calling for a deeper understanding of transport in large-scale devices at a quantum level. Here we perform large-scale quantum transport calculations based on a tight-binding model of graphene and the non-equilibrium Green's function method and include the effects of p-n junctions of different shape, magnetic field, and absorptive regions acting as drains for current. We stress the importance of choosing absorbing boundary conditions in the calculations to correctly capture how current flows in the limit of infinite devices. As a specific application we present a fully quantum-mechanical framework for the '2D Dirac fermion microscope' recently proposed by Bøggild et al (2017 Nat. Commun. 8 10.1038), tackling several key electron-optical effects therein predicted via semiclassical trajectory simulations, such as electron beam collimation, deflection and scattering off Veselago dots. Our results confirm that a semiclassical approach to a large extend is sufficient to capture the main transport features in the mesoscopic limit and the optical regime, but also that a richer electron-optical landscape is to be expected when coherence or other purely quantum effects are accounted for in the simulations.

Visualization and Control of Single-Electron Charging in Bilayer Graphene Quantum Dots

Graphene p-n junctions provide an ideal platform for investigating novel behavior at the boundary between electronics and optics that arise from massless Dirac Fermions, such as whispering gallery modes and Veselago lensing. Bilayer graphene also hosts Dirac Fermions, but they differ from single-layer graphene charge carriers because they are massive, can be gapped by an applied perpendicular electric field, and have very different pseudospin selection rules across a p-n junction. Novel phenomena predicted for these massive Dirac Fermions at p-n junctions include anti-Klein tunneling, oscillatory Zener tunneling, and electron cloaked states. Despite these predictions there has been little experimental focus on the microscopic spatial behavior of massive Dirac Fermions in the presence of p-n junctions. Here we report the experimental manipulation and characterization of massive Dirac Fermions within bilayer graphene quantum dots defined by circular p-n junctions through the use of scanning tunneling microscopy-based (STM) methods. Our p-n junctions are created via a flexible technique that enables realization of exposed quantum dots in bilayer graphene/hBN heterostructures. These quantum dots exhibit sharp spectroscopic resonances that disperse in energy as a function of applied gate voltage. Spatial maps of these features show prominent concentric rings with diameters that can be tuned by an electrostatic gate. This behavior is explained by single-electron charging of localized states that arise from the quantum confinement of massive Dirac Fermions within our exposed bilayer graphene quantum dots.

Commensurability Oscillations in One-Dimensional Graphene Superlattices

We report the experimental observation of commensurability oscillations (COs) in 1D graphene superlattices. The widely tunable periodic potential modulation in hBN-encapsulated graphene is generated via the interplay of nanopatterned few-layer graphene acting as a local bottom gate and a global Si back gate. The longitudinal magnetoresistance shows pronounced COs when the sample is tuned into the unipolar transport regime. We observe up to six CO minima, providing evidence for a long mean free path despite the potential modulation. Comparison to existing theories shows that small-angle scattering is dominant in hBN/graphene/hBN heterostructures. We observe robust COs persisting to temperatures exceeding T=150  K. At high temperatures, we find deviations from the predicted T dependence, which we ascribe to electron-electron scattering.

Chaos in Dirac Electron Optics: Emergence of a Relativistic Quantum Chimera

We uncover a remarkable quantum scattering phenomenon in two-dimensional Dirac material systems where the manifestations of both classically integrable and chaotic dynamics emerge simultaneously and are electrically controllable. The distinct relativistic quantum fingerprints associated with different electron spin states are due to a physical mechanism analogous to a chiroptical effect in the presence of degeneracy breaking. The phenomenon mimics a chimera state in classical complex dynamical systems but here in a relativistic quantum setting-henceforth the term "Dirac quantum chimera," associated with which are physical phenomena with potentially significant applications such as enhancement of spin polarization, unusual coexisting quasibound states for distinct spin configurations, and spin selective caustics. Experimental observations of these phenomena are possible through, e.g., optical realizations of ballistic Dirac fermion systems.

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.

Absorptive pinhole collimators for ballistic Dirac fermions in graphene

Ballistic electrons in solids can have mean free paths far larger than the smallest features patterned by lithography. This has allowed development and study of solid-state electron-optical devices such as beam splitters and quantum point contacts, which have informed our understanding of electron flow and interactions. Recently, high-mobility graphene has emerged as an ideal two-dimensional semimetal that hosts unique chiral electron-optical effects due to its honeycomb crystalline lattice. However, this chiral transport prevents the simple use of electrostatic gates to define electron-optical devices in graphene. Here we present a method of creating highly collimated electron beams in graphene based on collinear pairs of slits, with absorptive sidewalls between the slits. By this method, we achieve beams with angular width 18° or narrower, and transmission matching classical ballistic predictions.

Shaping and guiding the flow of ballistic electrons is at the core of electron optics; however in graphene this is hindered by chiral tunneling. Here, the authors experimentally demonstrate an electron collimator based on hBN-encapsulated ballistic graphene, capable of emitting narrow electron beams.