Gate-defined confinement in bilayer graphene-hexagonal boron nitride hybrid devices

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.

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.

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.

Sublattice Dependence and Gate Tunability of Midgap and Resonant States Induced by Native Dopants in Bernal-Stacked Bilayer Graphene

The properties of semiconductors can be crucially impacted by midgap states induced by dopants, which can be native or intentionally incorporated in the crystal lattice. For Bernal-stacked bilayer graphene (BLG), which has a tunable band gap, the existence of midgap states induced by dopants or adatoms has been investigated theoretically and observed indirectly in electron transport experiments. Here, we characterize BLG midgap states in real space, with atomic-scale resolution with scanning tunneling microscopy and spectroscopy. We show that the midgap states in BLG-for which we demonstrate gate tunability-appear when the dopant is hosted on the nondimer sublattice sites. We further evidence the presence of narrow resonances at the onset of the high-energy bands (valence or conduction, depending on the dopant type) when the dopants lie on the dimer sublattice sites. Our results are supported by tight-binding calculations that agree remarkably well with the experimental findings.

Visualization and Manipulation of Bilayer Graphene Quantum Dots with Broken Rotational Symmetry and Nontrivial Topology

Electrostatically defined quantum dots (QDs) in Bernal stacked bilayer graphene (BLG) are a promising quantum information platform because of their long spin decoherence times, high sample quality, and tunability. Importantly, the shape of QD states determines the electron energy spectrum, the interactions between electrons, and the coupling of electrons to their environment, all of which are relevant for quantum information processing. Despite its importance, the shape of BLG QD states remains experimentally unexamined. Here we report direct visualization of BLG QD states by using a scanning tunneling microscope. Strikingly, we find these states exhibit a robust broken rotational symmetry. By using a numerical tight-binding model, we determine that the observed broken rotational symmetry can be attributed to low energy anisotropic bands. We then compare confined holes and electrons and demonstrate the influence of BLG's nontrivial band topology. Our study distinguishes BLG QDs from prior QD platforms with trivial band topology.

Single-Carrier Transport in Graphene/hBN Superlattices

Graphene/hexagonal boron nitride (hBN) moiré superlattices have attracted interest for use in the study of many-body effects and fractal physics in Dirac fermion systems. Many exotic transport properties have been intensively examined in such superlattices, but previous studies have not focused on single-carrier transport. The investigation of the single-carrier behavior in these superlattices would lead to an understanding of the transition of single-particle/correlated phenomena. Here, we show the single-carrier transport in a high-quality bilayer graphene/hBN superlattice-based quantum dot device. We demonstrate remarkable device controllability in the energy range near the charge neutrality point (CNP) and the hole-side satellite point. Under a perpendicular magnetic field, Coulomb oscillations disappear near the CNP, which could be a signature of the crossover between Coulomb blockade and quantum Hall regimes. Our results pave the way for exploring the relationship of single-electron transport and fractal quantum Hall effects with correlated phenomena in two-dimensional quantum materials.

Gap Opening in Twisted Double Bilayer Graphene by Crystal Fields

Crystal fields occur due to a potential difference between chemically different atomic species. In van der Waals heterostructures such fields are naturally present perpendicular to the planes. It has been realized recently that twisted graphene multilayers provide powerful playgrounds to engineer electronic properties by the number of layers, the twist angle, applied electric biases, electronic interactions, and elastic relaxations, but crystal fields have not received the attention they deserve. Here, we show that the band structure of large-angle twisted double bilayer graphene is strongly modified by crystal fields. In particular, we experimentally demonstrate that twisted double bilayer graphene, encapsulated between hBN layers, exhibits an intrinsic band gap. By the application of an external field, the gaps in the individual bilayers can be closed, allowing to determine the crystal fields. We find that crystal fields point from the outer to the inner layers with strengths in the bottom/top bilayer [Formula: see text] = 0.13 V/nm ≈ [Formula: see text] = 0.12 V/nm. We show both by means of first-principles calculations and low energy models that crystal fields open a band gap in the ground state. Our results put forward a physical scenario in which a crystal field effect in carbon substantially impacts the low energy properties of twisted double bilayer graphene, suggesting that such contributions must be taken into account in other regimes to faithfully predict the electronic properties of twisted graphene multilayers.

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.

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.

Electrically Inert h-BN/Bilayer Graphene Interface in All-Two-Dimensional Heterostructure Field Effect Transistors

Bilayer graphene field effect transistors (BLG-FETs), unlike conventional semiconductors, are greatly sensitive to potential fluctuations because of the charged impurities in high- k gate stacks because the potential difference between two layers induced by the external perpendicular electrical filed is the physical origin behind the band gap opening. The assembly of BLG with layered h-BN insulators into a van der Waals heterostructure has been widely recognized to achieve the superior electrical transport properties. However, the carrier response properties at the h-BN/BLG heterointerface, which control the device performance, have not yet been revealed because of the inevitably large parasitic capacitance. In this study, the significant reduction of potential fluctuations to ∼1 meV is achieved in an all-two-dimensional heterostructure BLG-FET on a quartz substrate, which results in the suppression of the off-current to the measurement limit at a small band gap of ∼90 meV at 20 K. By capacitance measurement, we demonstrate that the electron trap/detrap response at such heterointerface is suppressed to undetectable level in the measurement frequency range. The electrically inert van der Waals heterointerface paves the way for the realization of future BLG electronics applications.

Gate-Defined Electron-Hole Double Dots in Bilayer Graphene

We present gate-controlled single-, double-, and triple-dot operation in electrostatically gapped bilayer graphene. Thanks to the recent advancements in sample fabrication, which include the encapsulation of bilayer graphene in hexagonal boron nitride and the use of graphite gates, it has become possible to electrostatically confine carriers in bilayer graphene and to completely pinch-off current through quantum dot devices. Here, we discuss the operation and characterization of electron-hole double dots. We show a remarkable degree of control of our device, which allows the implementation of two different gate-defined electron-hole double-dot systems with very similar energy scales. In the single-dot regime, we extract excited state energies and investigate their evolution in a parallel magnetic field, which is in agreement with a Zeeman-spin-splitting expected for a g-factor of 2.

Coupled Quantum Dots in Bilayer Graphene

Electrostatic confinement of charge carriers in bilayer graphene provides a unique platform for carbon-based spin, charge, or exchange qubits. By exploiting the possibility to induce a band gap with electrostatic gating, we form a versatile and widely tunable multiquantum dot system. We demonstrate the formation of single, double and triple quantum dots that are free of any sign of disorder. In bilayer graphene, we have the possibility to form tunnel barriers using different mechanisms. We can exploit the ambipolar nature of bilayer graphene where pn-junctions form natural tunnel barriers. Alternatively, we can use gates to form tunnel barriers, where we can vary the tunnel coupling by more than 2 orders of magnitude tuning between a deeply Coulomb blockaded system and a Fabry-Pérot-like cavity. Demonstrating such tunability is an important step toward graphene-based quantum computation.

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.

Large tunable valley splitting in edge-free graphene quantum dots on boron nitride

Coherent manipulation of the binary degrees of freedom is at the heart of modern quantum technologies. Graphene offers two binary degrees: the electron spin and the valley. Efficient spin control has been demonstrated in many solid-state systems, whereas exploitation of the valley has only recently been started, albeit without control at the single-electron level. Here, we show that van der Waals stacking of graphene onto hexagonal boron nitride offers a natural platform for valley control. We use a graphene quantum dot induced by the tip of a scanning tunnelling microscope and demonstrate valley splitting that is tunable from -5 to +10 meV (including valley inversion) by sub-10-nm displacements of the quantum dot position. This boosts the range of controlled valley splitting by about one order of magnitude. The tunable inversion of spin and valley states should enable coherent superposition of these degrees of freedom as a first step towards graphene-based qubits.

Synthesis, structure and applications of graphene-based 2D heterostructures

With the profuse amount of two-dimensional (2D) materials discovered and the improvements in their synthesis and handling, the field of 2D heterostructures has gained increased interest in recent years. Such heterostructures not only overcome the inherent limitations of each of the materials, but also allow the realization of novel properties by their proper combination. The physical and mechanical properties of graphene mean it has a prominent place in the area of 2D heterostructures. In this review, we will discuss the evolution and current state in the synthesis and applications of graphene-based 2D heterostructures. In addition to stacked and in-plane heterostructures with other 2D materials and their potential applications, we will also cover heterostructures realized with lower dimensionality materials, along with intercalation in few-layer graphene as a special case of a heterostructure. Finally, graphene heterostructures produced using liquid phase exfoliation techniques and their applications to energy storage will be reviewed.

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.

Gate-Defined One-Dimensional Channel and Broken Symmetry States in MoS2 van der Waals Heterostructures

We have realized encapsulated trilayer MoS₂ devices with gated graphene contacts. In the bulk, we observe an electron mobility as high as 7000 cm²/(V s) at a density of 3 × 10¹² cm^-2 at a temperature of 1.9 K. Shubnikov-de Haas oscillations start at magnetic fields as low as 0.9 T. The observed 3-fold Landau level degeneracy can be understood based on the valley Zeeman effect. Negatively biased split gate electrodes allow us to form a channel that can be completely pinched off for sufficiently large gate voltages. The measured conductance displays plateau-like features.

Realisation of topological zero-energy mode in bilayer graphene in zero magnetic field

Bilayer graphene (BLG) gapped by a vertical electric field represents a valley-symmetry-protected topological insulating state. Emergence of a new topological zero-energy mode has been proposed in BLG at a boundary between regions of inverted band gaps induced by two oppositely polarized vertical electric fields. However, its realisation has been challenged by the enormous difficulty in arranging two pairs of accurately aligned split gates on the top and bottom surfaces of clean BLG. Here we report realisation of the topological zero-energy mode in ballistic BLG, with zero-bias differential conductance close to the ideal value of 4 e²/h (e is the electron charge and h is Planck’s constant) along a boundary channel between a pair of gate-defined inverted band gaps. This constitutes the bona fide electrical-gate-tuned generation of a valley-symmetry-protected topological boundary conducting channel in BLG in zero magnetic field, which is essential to valleytronics applications of BLG.

Split-gated point-contact for electrostatic confinement of transport in MoS2/h-BN hybrid structures

Electrostatically defined nanoscale devices on two-dimensional semiconductor heterostructures are the building blocks of various quantum electrical circuits. Owing to its atomically flat interfaces and the inherent two-dimensional nature, van der Waals heterostructures hold the advantage of large-scale uniformity, flexibility and portability over the conventional bulk semiconductor heterostructures. In this letter we show the operation of a split-gate defined point contact device on a MoS₂/h-BN heterostructure, the first step towards realizing electrostatically gated quantum circuits on van der Waals semiconductors. By controlling the voltage on the split-gate we are able to control and confine the electron flow in the device leading to the formation of the point contact. The formation of the point contact in our device is elucidated by the three characteristic regimes observed in the pinch-off curve; transport similar to the conventional FET, electrostatically confined transport and the tunneling dominated transport. We explore the role of the carrier concentration and the drain-source voltages on the pinch-off characteristics. We are able to tune the pinch-off characteristics by varying the back-gate voltage at temperatures ranging from 4 K to 300 K.

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.

Signatures of single quantum dots in graphene nanoribbons within the quantum Hall regime

We report on the observation of periodic conductance oscillations near quantum Hall plateaus in suspended graphene nanoribbons. They are attributed to single quantum dots that are formed in the narrowest part of the ribbon, in the valleys and hills of a disorder potential. In a wide flake with two gates, a double-dot system's signature has been observed. Electrostatic confinement is enabled in single-layer graphene due to the gaps that are formed between the Landau levels, suggesting a way to create gate-defined quantum dots that can be accessed with quantum Hall edge states.

Guiding of Electrons in a Few-Mode Ballistic Graphene Channel

In graphene, the extremely fast charge carriers can be controlled by electron-optical elements, such as waveguides, in which the transmissivity is tuned by the wavelength. In this work, charge carriers are guided in a suspended ballistic few-mode graphene channel, defined by electrostatic gating. By depleting the channel, a reduction of mode number and steps in the conductance are observed, until the channel is completely emptied. The measurements are supported by tight-binding transport calculations including the full electrostatics of the sample.

Controlled growth of few-layer hexagonal boron nitride on copper foils using ion beam sputtering deposition

Ion beam sputtering deposition (IBSD) is used to synthesize high quality few-layer hexagonal boron nitride (h-BN) on copper foils. Compared to the conventional chemical vapor deposition, the IBSD technique avoids the use of unconventional precursors and is much easier to control, which should be very useful for the large-scale production of h-BN in the future.

Science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems

We present the science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems, targeting an evolution in technology, that might lead to impacts and benefits reaching into most areas of society. This roadmap was developed within the framework of the European Graphene Flagship and outlines the main targets and research areas as best understood at the start of this ambitious project. We provide an overview of the key aspects of graphene and related materials (GRMs), ranging from fundamental research challenges to a variety of applications in a large number of sectors, highlighting the steps necessary to take GRMs from a state of raw potential to a point where they might revolutionize multiple industries. We also define an extensive list of acronyms in an effort to standardize the nomenclature in this emerging field.

Graphene on hexagonal boron nitride

The field of graphene research has developed rapidly since its first isolation by mechanical exfoliation in 2004. Due to the relativistic Dirac nature of its charge carriers, graphene is both a promising material for next-generation electronic devices and a convenient low-energy testbed for intrinsically high-energy physical phenomena. Both of these research branches require the facile fabrication of clean graphene devices so as not to obscure its intrinsic physical properties. Hexagonal boron nitride has emerged as a promising substrate for graphene devices as it is insulating, atomically flat and provides a clean charge environment for the graphene. Additionally, the interaction between graphene and boron nitride provides a path for the study of new physical phenomena not present in bare graphene devices. This review focuses on recent advancements in the study of graphene on hexagonal boron nitride devices from the perspective of scanning tunneling microscopy with highlights of some important results from electrical transport measurements.

The electronic properties of bilayer graphene

We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energies. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects.