Size quantization of Dirac fermions in graphene constrictions

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.

Quantum Transport in Large-Scale Patterned Nitrogen-Doped Graphene

It has recently been demonstrated how the nitrogen dopant concentration in graphene can be controlled spatially on the nano-meter scale using a molecular mask. This technique may be used to create ballistic electron optics-like structures of high/low doping regions; for example, to focus electron beams, harnessing the quantum wave nature of the electronic propagation. Here, we employ large-scale Greens function transport calculations based on a tight-binding approach. We first benchmark different tight-binding models of nitrogen in graphene with parameters based on density functional theory (DFT) and the virtual crystal approximation (VCA). Then, we study theoretically how the random distribution within the masked regions and the discreteness of the nitrogen scattering centers impact the transport behavior of sharp n-p and n-n' interfaces formed by different, realistic nitrogen concentrations. We investigate how constrictions for the current can be realized by patterned high/low doping regions with experimentally feasible nitrogen concentrations. The constrictions can guide the electronic current, while the quantized conductance is significantly washed out due to the nitrogen scattering. The implications for device design is that a p-n junction with nitrogen corrugation should still be viable for current focusing. Furthermore, a guiding channel with less nitrogen in the conducting canal preserves more features of quantized conductance and, therefore, its low-noise regime.

Gate-Defined Electron Interferometer in Bilayer Graphene

We present an electron interferometer defined purely by electrostatic gating in an encapsulated bilayer graphene. This minimizes possible sample degradation introduced by conventional etching methods when preparing quantum devices. The device quality is demonstrated by observing Aharonov-Bohm (AB) oscillations with a period of h/e, h/2e, h/3e, and h/4e, witnessing a coherence length of many microns. The AB oscillations as well as the type of carriers (electrons or holes) are seamlessly tunable with gating. The coherence length longer than the ring perimeter and semiclassical trajectory of the carrier are established from the analysis of the temperature and magnetic field dependence of the oscillations. Our gate-defined ring geometry has the potential to evolve into a platform for exploring correlated quantum states such as superconductivity in interferometers in twisted bilayer graphene.

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.

Super-Resolution Nanolithography of Two-Dimensional Materials by Anisotropic Etching

Nanostructuring allows altering of the electronic and photonic properties of two-dimensional (2D) materials. The efficiency, flexibility, and convenience of top-down lithography processes are, however, compromised by nanometer-scale edge roughness and resolution variability issues, which especially affect the performance of 2D materials. Here, we study how dry anisotropic etching of multilayer 2D materials with sulfur hexafluoride (SF₆) may overcome some of these issues, showing results for hexagonal boron nitride (hBN), tungsten disulfide (WS₂), tungsten diselenide (WSe₂), molybdenum disulfide (MoS₂), and molybdenum ditelluride (MoTe₂). Scanning electron microscopy and transmission electron microscopy reveal that etching leads to anisotropic hexagonal features in the studied transition metal dichalcogenides, with the relative degree of anisotropy ranked as: WS₂ > WSe₂ > MoTe₂ ∼ MoS₂. Etched holes are terminated by zigzag edges while etched dots (protrusions) are terminated by armchair edges. This can be explained by Wulff constructions, taking the relative stabilities of the edges and the AA' stacking order into account. Patterns in WS₂ are transferred to an underlying graphite layer, demonstrating a possible use for creating sub-10 nm features. In contrast, multilayer hBN exhibits no lateral anisotropy but shows consistent vertical etch angles, independent of crystal orientation. Using an hBN crystal as the base, ultrasharp corners can be created in lithographic patterns, which are then transferred to a graphite crystal underneath. We find that the anisotropic SF₆ reactive ion etching process makes it possible to downsize nanostructures and obtain smooth edges, sharp corners, and feature sizes significantly below the resolution limit of electron beam lithography. The nanostructured 2D materials can be used themselves or as etch masks to pattern other nanomaterials.

Upstream modes and antidots poison graphene quantum Hall effect

The quantum Hall effect is the seminal example of topological protection, as charge carriers are transmitted through one-dimensional edge channels where backscattering is prohibited. Graphene has made its marks as an exceptional platform to reveal new facets of this remarkable property. However, in conventional Hall bar geometries, topological protection of graphene edge channels is found regrettably less robust than in high mobility semi-conductors. Here, we explore graphene quantum Hall regime at the local scale, using a scanning gate microscope. We reveal the detrimental influence of antidots along the graphene edges, mediating backscattering towards upstream edge channels, hence triggering topological breakdown. Combined with simulations, our experimental results provide further insights into graphene quantum Hall channels vulnerability. In turn, this may ease future developments towards precise manipulation of topologically protected edge channels hosted in various types of two-dimensional crystals.

2D-3D integration of hexagonal boron nitride and a high-κ dielectric for ultrafast graphene-based electro-absorption modulators

Electro-absorption (EA) waveguide-coupled modulators are essential building blocks for on-chip optical communications. Compared to state-of-the-art silicon (Si) devices, graphene-based EA modulators promise smaller footprints, larger temperature stability, cost-effective integration and high speeds. However, combining high speed and large modulation efficiencies in a single graphene-based device has remained elusive so far. In this work, we overcome this fundamental trade-off by demonstrating the 2D-3D dielectric integration in a high-quality encapsulated graphene device. We integrated hafnium oxide (HfO₂) and two-dimensional hexagonal boron nitride (hBN) within the insulating section of a double-layer (DL) graphene EA modulator. This combination of materials allows for a high-quality modulator device with high performances: a ~39 GHz bandwidth (BW) with a three-fold increase in modulation efficiency compared to previously reported high-speed modulators. This 2D-3D dielectric integration paves the way to a plethora of electronic and opto-electronic devices with enhanced performance and stability, while expanding the freedom for new device designs.

Here, three-dimensional hafnium oxide and two-dimensional hexagonal boron nitride are integrated in the insulating section of double-layer graphene optical modulators, leading to a maximum bandwidth of 39 GHz and enhanced modulation efficiency.

Interfacial Charge Transfer and Gate-Induced Hysteresis in Monochalcogenide InSe/GaSe Heterostructures

Heterostructures of two-dimensional (2D) van der Waals semiconductor materials offer a diverse playground for exploring fundamental physics and potential device applications. In InSe/GaSe heterostructures formed by sequential mechanical exfoliation and stacking of 2D monochalcogenides InSe and GaSe, we observe charge transfer between InSe and GaSe because of the 2D van der Waals interface formation and a strong hysteresis effect in the electron transport through the InSe layer when a gate voltage is applied through the GaSe layer. A gate voltage-dependent conductance decay rate is also observed. We relate these observations to the gate voltage-dependent dynamical charge transfer between InSe and GaSe layers.

Pseudo chiral anomaly in zigzag graphene ribbons

As the three-dimensional analogs of graphene, Weyl semimetals display signatures of chiral anomaly which arises from charge pumping between the lowest chiral Landau levels of the Weyl nodes in the presence of parallel electric and magnetic fields. In this work, we study the pseudo chiral anomaly and its transport signatures in graphene ribbon with zigzag edges. Here 'pseudo' refers to the case where the inverse of width of zigzag graphene ribbon plays the same role as magnetic field in three-dimensional Weyl semimetals. The valley chiral bands in zigzag graphene ribbons can be introduced by edge potentials, giving rise to the nonconservation of chiral current, i.e. pseudo chiral anomaly, in the presence of a longitudinal electric field. Further numerical results reveal that pseudo magnetoconductivity of zigzag graphene ribbons is positive and has a nearly quadratic dependence on the pseudofield, which is regarded as the transport signature of pseudo chiral anomaly.

Sub-10 nm nanogap fabrication on suspended glassy carbon nanofibers

Glassy carbon nanofibers (GCNFs) are considered promising candidates for the fabrication of nanosensors for biosensing applications. Importantly, in part due to their great stability, carbon electrodes with sub-10 nm nanogaps represent an attractive platform for probing the electrical characteristics of molecules. The fabrication of sub-10 nm nanogap electrodes in these GCNFs, which is achieved by electrically stimulating the fibers until they break, was previously found to require fibers shorter than 2 µm; however, this process is generally hampered by the limitations inherent to photolithographic methods. In this work, to obtain nanogaps on the order of 10 nm without the need for sub-2 µm GCNFs, we employed a fabrication strategy in which the fibers were gradually thinned down by continuously monitoring the changes in the electrical resistance of the fiber and adjusting the applied voltage accordingly. To further reduce the nanogap size, we studied the mechanism behind the thinning and eventual breakdown of the suspended GCNFs by controlling the environmental conditions and pressure during the experiment. Following this approach, which includes performing the experiments in a high-vacuum chamber after a series of carbon dioxide (CO₂) purging cycles, nanogaps on the order of 10 nm were produced in suspended GCNFs 52 µm in length, much longer than the ~2 µm GCNFs needed to produce such small gaps without the procedure employed in this work. Furthermore, the electrodes showed no apparent change in their shape or nanogap width after being stored at room temperature for approximately 6 months.

Quantum nanoconstrictions fabricated by cryo-etching in encapsulated graphene

We report on a novel implementation of the cryo-etching method, which enabled us to fabricate low-roughness hBN-encapsulated graphene nanoconstrictions with unprecedented control of the structure edges; the typical edge roughness is on the order of a few nanometers. We characterized the system by atomic force microscopy and used the measured parameters of the edge geometry in numerical simulations of the system conductance, which agree quantitatively with our low temperature transport measurements. The quality of our devices is confirmed by the observation of well defined quantized 2e²/h conductance steps at zero magnetic field. To the best of our knowledge, such an observation reports the clearest conductance quantization in physically etched graphene nanoconstrictions. The fabrication of such high quality systems and the scalability of the cryo-etching method opens a novel promising possibility of producing more complex truly-ballistic devices based on graphene.

Recovery of the scattering symmetry looking at the conductance in asymmetric graphene nano-ribbons systems using pseudo-spin filters

In this work we study some applications for pseudo-spin filters. The filters are potential barriers with hyperboloid sub-band contributions that are locally applied over graphene nano-ribbons. These filters modulate the pseudo-spin and the quirality of the wave-function allowing the recovery of the conductance loss due to imperfections, bends, or constrictions (asymmetries) found in the system. The recovery of the conductance is fulfilled by a direct manipulation of the pseudo-spin polarization at both sides of the device by localizing the filters at the system's entrance and exit points. This procedure allows the recovery of the wave-function symmetry at these points with the consequent recovery of the conductance, even when it is zero, regardless of the different internal regions that affect the transmission, i.e. the filters are used as patches for damaged regions. Our results can be extrapolated for spatially asymmetrical potentials generated by electrical (or magnetic) impurities.

Lithographic band structure engineering of graphene

Two-dimensional materials such as graphene allow direct access to the entirety of atoms constituting the crystal. While this makes shaping by lithography particularly attractive as a tool for band structure engineering through quantum confinement effects, edge disorder and contamination have so far limited progress towards experimental realization. Here, we define a superlattice in graphene encapsulated in hexagonal boron nitride, by etching an array of holes through the heterostructure with minimum feature sizes of 12-15 nm. We observe a magnetotransport regime that is distinctly different from the characteristic Landau fan of graphene, with a sizeable bandgap that can be tuned by a magnetic field. The measurements are accurately described by transport simulations and analytical calculations. Finally, we observe strong indications that the lithographically engineered band structure at the main Dirac point is cloned to a satellite peak that appears due to moiré interactions between the graphene and the encapsulating material.

Low-Magnetic-Field Regime of a Gate-Defined Constriction in High-Mobility Graphene

We report on the evolution of the coherent electronic transport through a gate-defined constriction in a high-mobility graphene device from ballistic transport to quantum Hall regime upon increasing the magnetic field. At a low field, the conductance exhibits Fabry-Pérot resonances resulting from the npn cavities formed beneath the top-gated regions. Above a critical field B* corresponding to the cyclotron radius equal to the npn cavity length, Fabry-Pérot resonances vanish, and snake trajectories are guided through the constriction with a characteristic set of conductance oscillations. Increasing further the magnetic field allows us to probe the Landau level spectrum in the constriction and unveil distortions due to the combination of confinement and deconfinement of Landau levels in a saddle potential. These observations are confirmed by numerical calculations.

Topologically Nontrivial Valley States in Bilayer Graphene Quantum Point Contacts

We present measurements of quantized conductance in electrostatically induced quantum point contacts in bilayer graphene. The application of a perpendicular magnetic field leads to an intricate pattern of lifted and restored degeneracies with increasing field: at zero magnetic field the degeneracy of quantized one-dimensional subbands is four, because of a twofold spin and a twofold valley degeneracy. By switching on the magnetic field, the valley degeneracy is lifted. Because of the Berry curvature, states from different valleys split linearly in magnetic field. In the quantum Hall regime fourfold degenerate conductance plateaus reemerge. During the adiabatic transition to the quantum Hall regime, levels from one valley shift by two in quantum number with respect to the other valley, forming an interweaving pattern that can be reproduced by numerical calculations.

Valley Subband Splitting in Bilayer Graphene Quantum Point Contacts

We report a study of one-dimensional subband splitting in a bilayer graphene quantum point contact in which quantized conductance in steps of 4e^{2}/h is clearly defined down to the lowest subband. While our source-drain bias spectroscopy measurements reveal an unconventional confinement, we observe a full lifting of the valley degeneracy at high magnetic fields perpendicular to the bilayer graphene plane for the first two lowest subbands where confinement and Coulomb interactions are the strongest and a peculiar merging or mixing of K and K^{'} valleys from two nonadjacent subbands with indices (N,N+2), which are well described by our semiphenomenological model.

Fabrication of comb-drive actuators for straining nanostructured suspended graphene

We report on the fabrication and characterization of an optimized comb-drive actuator design for strain-dependent transport measurements on suspended graphene. We fabricate devices from highly p-doped silicon using deep reactive ion etching with a chromium mask. Crucially, we implement a gold layer to reduce the device resistance from ≈51.6 kΩ to ≈236 Ω at room temperature in order to allow for strain-dependent transport measurements. The graphene is integrated by mechanically transferring it directly onto the actuator using a polymethylmethacrylate membrane. Importantly, the integrated graphene can be nanostructured afterwards to optimize device functionality. The minimum feature size of the structured suspended graphene is 30 nm, which allows for interesting device concepts such as mechanically-tunable nanoconstrictions. Finally, we characterize the fabricated devices by measuring the Raman spectrum as well as the a mechanical resonance frequency of an integrated graphene sheet for different strain values.

Edge-Limited Valley-Preserved Transport in Quasi-1D Constriction of Bilayer Graphene

We investigated the quantization of the conductance of quasi-one-dimensional (quasi-1D) constrictions in high-mobility bilayer graphene (BLG) with different geometrical aspect ratios. Ultrashort (a few tens of nanometers long) constrictions were fabricated by applying an under-cut etching technique. Conductance was quantized in steps of ∼4 e²/ h (∼2 e²/ h) in devices with aspect ratios smaller (larger) than 1. We argue that scattering at the edges of a quasi-1D BLG constriction limits the intervalley scattering length, which causes valley-preserved (valley-broken) quantum transport in devices with aspect ratios smaller (larger) than 1. The subband energy levels, analyzed in terms of the bias-voltage and temperature dependences of the quantized conductance, indicated that they corresponded well to the effective channel width of a physically defined conducting channel with a hard-wall confining potential. Our study in ultrashort high-mobility BLG nano constrictions with physically tailored edges clearly confirms that physical edges are the major source of intervalley scattering in graphene in the ballistic limit.

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.

Gate-Defined Quantum Confinement in InSe-Based van der Waals Heterostructures

Indium selenide, a post-transition metal chalcogenide, is a novel two-dimensional (2D) semiconductor with interesting electronic properties. Its tunable band gap and high electron mobility have already attracted considerable research interest. Here we demonstrate strong quantum confinement and manipulation of single electrons in devices made from few-layer crystals of InSe using electrostatic gating. We report on gate-controlled quantum dots in the Coulomb blockade regime as well as one-dimensional quantization in point contacts, revealing multiple plateaus. The work represents an important milestone in the development of quality devices based on 2D materials and makes InSe a prime candidate for relevant electronic and optoelectronic applications.

Quantum transport through MoS2 constrictions defined by photodoping

We present a device scheme to explore mesoscopic transport through molybdenum disulfide (MoS₂) constrictions using photodoping. The devices are based on van-der-Waals heterostructures where few-layer MoS₂ flakes are partially encapsulated by hexagonal boron nitride (hBN) and covered by a few-layer graphene flake to fabricate electrical contacts. Since the as-fabricated devices are insulating at low temperatures, we use photo-induced remote doping in the hBN substrate to create free charge carriers in the MoS₂ layer. On top of the device, we place additional metal structures, which define the shape of the constriction and act as shadow masks during photodoping of the underlying MoS₂/hBN heterostructure. Low temperature two- and four-terminal transport measurements show evidence of quantum confinement effects.

Low-field magnetotransport in graphene cavity devices

Confinement and edge structures are known to play significant roles in the electronic and transport properties of two-dimensional materials. Here, we report on low-temperature magnetotransport measurements of lithographically patterned graphene cavity nanodevices. It is found that the evolution of the low-field magnetoconductance characteristics with varying carrier density exhibits different behaviors in graphene cavity and bulk graphene devices. In the graphene cavity devices, we observed that intravalley scattering becomes dominant as the Fermi level gets close to the Dirac point. We associate this enhanced intravalley scattering to the effect of charge inhomogeneities and edge disorder in the confined graphene nanostructures. We also observed that the dephasing rate of carriers in the cavity devices follows a parabolic temperature dependence, indicating that the direct Coulomb interaction scattering mechanism governs the dephasing at low temperatures. Our results demonstrate the importance of confinement in carrier transport in graphene nanostructure devices.

Conductance quantization suppression in the quantum Hall regime

Conductance quantization is the quintessential feature of electronic transport in non-interacting mesoscopic systems. This phenomenon is observed in quasi one-dimensional conductors at zero magnetic field B, and the formation of edge states at finite magnetic fields results in wider conductance plateaus within the quantum Hall regime. Electrostatic interactions can change this picture qualitatively. At finite B, screening mechanisms in narrow, gated ballistic conductors are predicted to give rise to an increase in conductance and a suppression of quantization due to the appearance of additional conduction channels. Despite being a universal effect, this regime has proven experimentally elusive because of difficulties in realizing one-dimensional systems with sufficiently hard-walled, disorder-free confinement. Here, we experimentally demonstrate the suppression of conductance quantization within the quantum Hall regime for graphene nanoconstrictions with low edge roughness. Our findings may have profound impact on fundamental studies of quantum transport in finite-size, two-dimensional crystals with low disorder.

Conductance quantization is the hallmark of non-interacting confined systems. The authors show that the quantization in graphene nanoconstrictions with low edge disorder is suppressed in the quantum Hall regime. This is explained by the addition of new conductance channels due to electrostatic screening.

Electrostatically Induced Quantum Point Contacts in Bilayer Graphene

We report the fabrication of electrostatically defined nanostructures in encapsulated bilayer graphene, with leakage resistances below depletion gates as high as R ∼ 10 GΩ. This exceeds previously reported values of R = 10-100 kΩ.1-3 We attribute this improvement to the use of a graphite back gate. We realize two split gate devices which define an electronic channel on the scale of the Fermi-wavelength. A channel gate covering the gap between the split gates varies the charge carrier density in the channel. We observe device-dependent conductance quantization of ΔG = 2e²/h and ΔG = 4e²/h. In quantizing magnetic fields normal to the sample plane, we recover the four-fold Landau level degeneracy of bilayer graphene. Unexpected mode crossings appear at the crossover between zero magnetic field and the quantum Hall regime.

Inhomogeneous screening of gate electric field by interface states in graphene FETs

The electronic states at graphene-SiO₂ interface and their inhomogeneity is investigated using the back-gate-voltage dependence of local tunnel spectra acquired with a scanning tunneling microscope. The conductance spectra show two, or occasionally three, minima that evolve along the bias-voltage axis with the back gate voltage. This evolution is modeled using tip-gating and interface states. The energy dependent interface states' density, [Formula: see text], required to model the back-gate evolution of the minima, is found to have significant inhomogeneity in its energy-width. A broad [Formula: see text] leads to an effect similar to a reduction in the Fermi velocity while the narrow [Formula: see text] leads to the pinning of the Fermi energy close to the Dirac point, as observed in some places, due to enhanced screening of the gate electric field by the narrow [Formula: see text]. Finally, this also demonstrates STM as a tool to probe the density of interface states in various 2D Dirac materials.

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.

Tunable transmission of quantum Hall edge channels with full degeneracy lifting in split-gated graphene devices

Charge carriers in the quantum Hall regime propagate via one-dimensional conducting channels that form along the edges of a two-dimensional electron gas. Controlling their transmission through a gate-tunable constriction, also called quantum point contact, is fundamental for many coherent transport experiments. However, in graphene, tailoring a constriction with electrostatic gates remains challenging due to the formation of p–n junctions below gate electrodes along which electron and hole edge channels co-propagate and mix, short circuiting the constriction. Here we show that this electron–hole mixing is drastically reduced in high-mobility graphene van der Waals heterostructures thanks to the full degeneracy lifting of the Landau levels, enabling quantum point contact operation with full channel pinch-off. We demonstrate gate-tunable selective transmission of integer and fractional quantum Hall edge channels through the quantum point contact. This gate control of edge channels opens the door to quantum Hall interferometry and electron quantum optics experiments in the integer and fractional quantum Hall regimes of graphene.

Quantum point contacts are gate-tunable constrictions allowing for control of charge carrier transmission in 2D electron gases. Here, the authors fabricate a hBN/graphene/hBN van der Waals heterojunction to enable quantum point contact devices in the integer and fractional quantum Hall regimes.

Veselago lens and Klein collimator in disordered graphene

We simulate electron transport through graphene nanoribbons of realistic size containing a p-n junction patterned by electrostatic gates. For a sharp p-n interface, Klein tunneling leads to refocusing of a divergent beam forming a Veselago lens. Wider transition regions allow only electrons with near-perpendicular incidence to pass the junction, forming a Klein collimator. Using a third nearest neighbor tight binding description we explore the influence of interface roughness and bulk disorder on guiding properties. We provide bounds on disorder amplitudes and p-n junction properties to be satisfied in order to experimentally observe the focusing effect and compare our predictions to very recent realizations.

Electrostatically Confined Monolayer Graphene Quantum Dots with Orbital and Valley Splittings

The electrostatic confinement of massless charge carriers is hampered by Klein tunneling. Circumventing this problem in graphene mainly relies on carving out nanostructures or applying electric displacement fields to open a band gap in bilayer graphene. So far, these approaches suffer from edge disorder or insufficiently controlled localization of electrons. Here we realize an alternative strategy in monolayer graphene, by combining a homogeneous magnetic field and electrostatic confinement. Using the tip of a scanning tunneling microscope, we induce a confining potential in the Landau gaps of bulk graphene without the need for physical edges. Gating the localized states toward the Fermi energy leads to regular charging sequences with more than 40 Coulomb peaks exhibiting typical addition energies of 7-20 meV. Orbital splittings of 4-10 meV and a valley splitting of about 3 meV for the first orbital state can be deduced. These experimental observations are quantitatively reproduced by tight binding calculations, which include the interactions of the graphene with the aligned hexagonal boron nitride substrate. The demonstrated confinement approach appears suitable to create quantum dots with well-defined wave function properties beyond the reach of traditional techniques.