Topological Bloch bands in graphene superlattices

Quantum Geometric Moment Encodes Stacking Order of Moiré Matter

Exploring the topological characteristics of electronic bands is essential in condensed matter physics. Moiré materials featuring flat bands provide a versatile platform for engineering band topology and correlation effects. In moiré materials that break either time-reversal symmetry or inversion symmetry or both, electronic bands exhibit Berry curvature hotspots. Different stacking orders in these materials result in varied Berry curvature distributions within the flat bands, even when the band dispersion remains similar. However, experimental studies probing the impact of stacking order on the quantum geometric quantities are lacking. 1.4° twisted double bilayer graphene (TDBG) facilitates two distinct stacking orders (AB-AB, AB-BA) and forms an inversion broken moiré superlattice with electrically tunable flat bands. The valley Chern numbers of the flat bands depend on the stacking order, and the nonlinear Hall (NLH) effect distinguishes the differences in Berry curvature dipole (BCD), the first moment of Berry curvature. The BCD exhibits antisymmetric behavior, flipping its sign with the polarity of the perpendicular electric field in AB-AB TDBG, while it displays a symmetric behavior, maintaining the same sign regardless of the electric field's polarity in AB-BA TDBG. This approach electronically detects stacking-induced quantum geometry, while opening a pathway to quantum geometry engineering and detection.

Higher order gaps in the renormalized band structure of doubly aligned hBN/bilayer graphene moiré superlattice

This paper presents our findings on the recursive band gap engineering of chiral fermions in bilayer graphene doubly aligned with hBN. Using two interfering moiré potentials, we generate a supermoiré pattern that renormalizes the electronic bands of the pristine bilayer graphene, resulting in higher order fractal gaps even at very low energies. These Bragg gaps can be mapped using a unique linear combination of periodic areas within the system. To validate our findings, we use electronic transport measurements to identify the position of these gaps as a function of the carrier density. We establish their agreement with the predicted carrier densities and corresponding quantum numbers obtained using the continuum model. Our study provides strong evidence of the quantization of the momentum-space area of quasi-Brillouin zones in a minimally incommensurate lattice. It fills important gaps in the understanding of band structure engineering of Dirac fermions with a doubly periodic superlattice spinor potential.

Non-identical moiré twins in bilayer graphene

The superlattice obtained by aligning a monolayer graphene and boron nitride (BN) inherits from the hexagonal lattice a sixty degrees periodicity with the layer alignment. It implies that, in principle, the properties of the heterostructure must be identical for 0° and 60° of layer alignment. Here, we demonstrate, using dynamically rotatable van der Waals heterostructures, that the moiré superlattice formed in a bilayer graphene/BN has different electronic properties at 0° and 60° of alignment. Although the existence of these non-identical moiré twins is explained by different relaxation of the atomic structures for each alignment, the origin of the observed valley Hall effect remains to be explained. A simple Berry curvature argument is not sufficient to explain the 120° periodicity of this observation. Our results highlight the complexity of the interplay between mechanical and electronic properties in moiré structures and the importance of taking into account atomic structure relaxation to understand their electronic properties.

Intrinsic Nonlinear Hall Effect and Gate-Switchable Berry Curvature Sliding in Twisted Bilayer Graphene

Though the observation of the quantum anomalous Hall effect and nonlocal transport response reveals nontrivial band topology governed by the Berry curvature in twisted bilayer graphene, some recent works reported nonlinear Hall signals in graphene superlattices that are caused by the extrinsic disorder scattering rather than the intrinsic Berry curvature dipole moment. In this Letter, we report a Berry curvature dipole induced intrinsic nonlinear Hall effect in high-quality twisted bilayer graphene devices. We also find that the application of the displacement field substantially changes the direction and amplitude of the nonlinear Hall voltages, as a result of a field-induced sliding of the Berry curvature hotspots. Our Letter not only proves that the Berry curvature dipole could play a dominant role in generating the intrinsic nonlinear Hall signal in graphene superlattices with low disorder densities, but also demonstrates twisted bilayer graphene to be a sensitive and fine-tunable platform for second harmonic generation and rectification.

An image interaction approach to quantum-phase engineering of two-dimensional materials

Tuning electrical, optical, and thermal material properties is central for engineering and understanding solid-state systems. In this scenario, atomically thin materials are appealing because of their sensitivity to electric and magnetic gating, as well as to interlayer hybridization. Here, we introduce a radically different approach to material engineering relying on the image interaction experienced by electrons in a two-dimensional material when placed in proximity of an electrically neutral structure. We theoretically show that electrons in a semiconductor atomic layer acquire a quantum phase resulting from the image potential induced by the presence of a neighboring periodic array of conducting ribbons, which in turn modifies the optical, electrical, and thermal properties of the monolayer, giving rise to additional interband optical absorption, plasmon hybridization, and metal-insulator transitions. Beyond its fundamental interest, material engineering based on the image interaction represents a disruptive approach to tailor the properties of atomic layers for application in nanodevices.

Graphene moiré superlattices with giant quantum nonlinearity of chiral Bloch electrons

Graphene-based samples have shown a plethora of exotic characteristics and these properties may help the realization of a new generation of fast electronic devices. However, graphene's centrosymmetry prohibits second-order electronic transport. Here, we show giant second-order nonlinear transports in graphene moiré superlattices at zero magnetic field, both longitudinal and transverse to the applied current direction. High carrier mobility and inversion symmetry breaking by hexagonal boron nitride lead to nonlinear conductivities five orders of magnitude larger than those in WTe₂. The nonlinear conductivity strongly depends on the gate voltage as well as on the stacking configuration, with a giant enhancement originating from the moiré bands. Longitudinal nonlinear conductivity cannot originate from Berry curvature dipoles. Our theoretical modelling highlights skew scattering of chiral Bloch electrons as the physical origin. With these results, we demonstrate nonlinear charge transport due to valley-contrasting chirality, which constitutes an alternative means to induce second-order transports in van der Waals heterostructures. Our approach is promising for applications in frequency-doubling and energy harvesting via rectification.

Tunable and giant valley-selective Hall effect in gapped bilayer graphene

Berry curvature is analogous to magnetic field but in momentum space and is commonly present in materials with nontrivial quantum geometry. It endows Bloch electrons with transverse anomalous velocities to produce Hall-like currents even in the absence of a magnetic field. We report the direct observation of in situ tunable valley-selective Hall effect (VSHE), where inversion symmetry, and thus the geometric phase of electrons, is controllable by an out-of-plane electric field. We use high-quality bilayer graphene with an intrinsic and tunable bandgap, illuminated by circularly polarized midinfrared light, and confirm that the observed Hall voltage arises from an optically induced valley population. Compared with molybdenum disulfide (MoS₂), we find orders of magnitude larger VSHE, attributed to the inverse scaling of the Berry curvature with bandgap. By monitoring the valley-selective Hall conductivity, we study the Berry curvature's evolution with bandgap. This in situ manipulation of VSHE paves the way for topological and quantum geometric optoelectronic devices, such as more robust switches and detectors.

Magic in twisted transition metal dichalcogenide bilayers

The long-wavelength moiré superlattices in twisted 2D structures have emerged as a highly tunable platform for strongly correlated electron physics. We study the moiré bands in twisted transition metal dichalcogenide homobilayers, focusing on WSe2, at small twist angles using a combination of first principles density functional theory, continuum modeling, and Hartree-Fock approximation. We reveal the rich physics at small twist angles θ < 4∘, and identify a particular magic angle at which the top valence moiré band achieves almost perfect flatness. In the vicinity of this magic angle, we predict the realization of a generalized Kane-Mele model with a topological flat band, interaction-driven Haldane insulator, and Mott insulators at the filling of one hole per moiré unit cell. The combination of flat dispersion and uniformity of Berry curvature near the magic angle holds promise for realizing fractional quantum anomalous Hall effect at fractional filling. We also identify twist angles favorable for quantum spin Hall insulators and interaction-induced quantum anomalous Hall insulators at other integer fillings.

Ultrafast Interlayer Charge Separation, Enhanced Visible-Light Absorption, and Tunable Overpotential in Twisted Graphitic Carbon Nitride Bilayers for Water Splitting

Moiré pattern superlattice formed by 2D van der Waals layered structures have attracted great attention for diverse applications. In experiments, the enhancement of catalytic performance in twisted bilayer systems is reported while its mechanism remains unclear. From high-accuracy first-principles and time-dependent ab initio nonadiabatic molecular dynamics calculations, ultrafast interlayer charge transfer within 120 fs, excellent charge separation, improved visible-light absorption, and satisfactory overpotentials for the hydrogen evolution and oxygen evolution reactions in twisted graphitic carbon nitride (g-C₃ N₄ ) bilayers are found, which are beneficial to photocatalytic, photo-electrocatalytic, or electrocatalytic water splitting. This work provides insightful guidance to advanced nanocatalysis based on twisted layered materials.

Localization transitions and mobility edges in quasiperiodic ladder

We investigate localization properties of two-coupled uniform chains (ladder) with quasiperiodic modulation on interchain coupling strength. We demonstrate that this ladder is equivalent to two Aubry-André chains when two legs are symmetric. Analytical and numerical results indicate the appearance of mobility edges in asymmetric ladder systems. We propose an easy-to-engineer quasiperiodic Moiré superlattice ladder system comprising two-coupled uniform chains. An irrational lattice constant difference results in a quasiperiodic structure. Numerical simulations indicate that such a system supports the existence of mobility edges. Furthermore, we demonstrate that the mobility edges can be detected through a dynamical method, that is based on the measurement of survival probability in the presence of a single imaginary negative potential. The results provide insights into localization transitions and mobility edges in experiments.

Moiré superlattices and related moiré excitons in twisted van der Waals heterostructures

Recent advances in moiré superlattices and moiré excitons, such as quantum emission arrays, low-energy flat bands, and Mott insulators, have rapidly attracted attention in the fields of optoelectronics, materials, and energy research. The interlayer twist turns into a degree of freedom that alters the properties of the systems of materials, and the realization of moiré excitons also offers the feasibility of making artificial exciton crystals. Moreover, moiré excitons exhibit many exciting properties under the regulation of various external conditions, including spatial polarisation, alternating dipolar to alternating dipolar moments and gate-dependence to gate voltage dependence; all are pertinent to their applications in nano-photonics and quantum information. But the lag in theoretical development and the low-efficiency of processing technologies significantly limit the potential of moiré superlattice applications. In this review, we systematically summarise and discuss the recent progress in moiré superlattices and moiré excitons, and analyze the current challenges, and put forward relevant recommendations. There is no doubt that further research will lead to breakthroughs in their application and promote reforms and innovations in traditional solid-state physics and materials science.

Imaging orbital ferromagnetism in a moiré Chern insulator

Electrons in moiré flat band systems can spontaneously break time-reversal symmetry, giving rise to a quantized anomalous Hall effect. In this study, we use a superconducting quantum interference device to image stray magnetic fields in twisted bilayer graphene aligned to hexagonal boron nitride. We find a magnetization of several Bohr magnetons per charge carrier, demonstrating that the magnetism is primarily orbital in nature. Our measurements reveal a large change in the magnetization as the chemical potential is swept across the quantum anomalous Hall gap, consistent with the expected contribution of chiral edge states to the magnetization of an orbital Chern insulator. Mapping the spatial evolution of field-driven magnetic reversal, we find a series of reproducible micrometer-scale domains pinned to structural disorder.

Evidence of Orbital Ferromagnetism in Twisted Bilayer Graphene Aligned to Hexagonal Boron Nitride

We have previously reported ferromagnetism evinced by a large hysteretic anomalous Hall effect in twisted bilayer graphene (tBLG). Subsequent measurements of a quantized Hall resistance and small longitudinal resistance confirmed that this magnetic state is a Chern insulator. Here, we report that when tilting the sample in an external magnetic field, the ferromagnetism is highly anisotropic. Because spin-orbit coupling is weak in graphene, such anisotropy is unlikely to come from spin but rather favors theories in which the ferromagnetism is orbital. We know of no other case in which ferromagnetism has a purely orbital origin. For an applied in-plane field larger than 5 T, the out-of-plane magnetization is destroyed, suggesting a transition to a new phase.

Observation of Time-Reversal Invariant Helical Edge-Modes in Bilayer Graphene/WSe2 Heterostructure

Topological insulators, along with Chern insulators and quantum Hall insulator phases, are considered as paradigms for symmetry protected topological phases of matter. This article reports the experimental realization of the time-reversal invariant helical edge-modes in bilayer graphene/monolayer WSe₂-based heterostructures-a phase generally considered as a precursor to the field of generic topological insulators. Our observation of this elusive phase depended crucially on our ability to create mesoscopic devices comprising both a moiré superlattice potential and strong spin-orbit coupling; this resulted in materials whose electronic band structure could be tuned from trivial to topological by an external displacement field. We find that the topological phase is characterized by a bulk bandgap and by helical edge-modes with electrical conductance quantized exactly to 2e²/h in zero external magnetic field. We put the helical edge-modes on firm ground through supporting experiments, including the verification of predictions of the Landauer-Büttiker model for quantum transport in multiterminal mesoscopic devices. Our nonlocal transport properties measurements show that the helical edge-modes are dissipationless and equilibrate at the contact probes. We achieved the tunability of the different topological phases with electric and magnetic fields, which allowed us to achieve topological phase transitions between trivial and multiple, distinct topological phases. We also present results of a theoretical study of a realistic model which, in addition to replicating our experimental results, explains the origin of the topological insulating bulk and helical edge-modes. Our experimental and theoretical results establish a viable route to realizing the time-reversal invariant Z2 topological phase of matter.

Electrical switching of magnetic order in an orbital Chern insulator

Magnetism typically arises from the joint effect of Fermi statistics and repulsive Coulomb interactions, which favours ground states with non-zero electron spin. As a result, controlling spin magnetism with electric fields-a longstanding technological goal in spintronics and multiferroics^1,2-can be achieved only indirectly. Here we experimentally demonstrate direct electric-field control of magnetic states in an orbital Chern insulator^3-6, a magnetic system in which non-trivial band topology favours long-range order of orbital angular momentum but the spins are thought to remain disordered^7-14. We use van der Waals heterostructures consisting of a graphene monolayer rotationally faulted with respect to a Bernal-stacked bilayer to realize narrow and topologically non-trivial valley-projected moiré minibands^15-17. At fillings of one and three electrons per moiré unit cell within these bands, we observe quantized anomalous Hall effects¹⁸ with transverse resistance approximately equal to h/2e² (where h is Planck's constant and e is the charge on the electron), which is indicative of spontaneous polarization of the system into a single-valley-projected band with a Chern number equal to two. At a filling of three electrons per moiré unit cell, we find that the sign of the quantum anomalous Hall effect can be reversed via field-effect control of the chemical potential; moreover, this transition is hysteretic, which we use to demonstrate non-volatile electric-field-induced reversal of the magnetic state. A theoretical analysis¹⁹ indicates that the effect arises from the topological edge states, which drive a change in sign of the magnetization and thus a reversal in the favoured magnetic state. Voltage control of magnetic states can be used to electrically pattern non-volatile magnetic-domain structures hosting chiral edge states, with applications ranging from reconfigurable microwave circuit elements to ultralow-power magnetic memories.

Bulk valley transport and Berry curvature spreading at the edge of flat bands

2D materials based superlattices have emerged as a promising platform to modulate band structure and its symmetries. In particular, moiré periodicity in twisted graphene systems produces flat Chern bands. The recent observation of anomalous Hall effect (AHE) and orbital magnetism in twisted bilayer graphene has been associated with spontaneous symmetry breaking of such Chern bands. However, the valley Hall state as a precursor of AHE state, when time-reversal symmetry is still protected, has not been observed. Our work probes this precursor state using the valley Hall effect. We show that broken inversion symmetry in twisted double bilayer graphene (TDBG) facilitates the generation of bulk valley current by reporting experimental evidence of nonlocal transport in a nearly flat band system. Despite the spread of Berry curvature hotspots and reduced quasiparticle velocities of the carriers in these flat bands, we observe large nonlocal voltage several micrometers away from the charge current path - this persists when the Fermi energy lies inside a gap with large Berry curvature. The high sensitivity of the nonlocal voltage to gate tunable carrier density and gap modulating perpendicular electric field makes TDBG an attractive platform for valley-twistronics based on flat bands.

Shedding light on moiré excitons: A first-principles perspective

Moiré superlattices in van der Waals (vdW) heterostructures could trap long-lived interlayer excitons. These moiré excitons could form ordered quantum dot arrays, paving the way for unprecedented optoelectronic and quantum information applications. Here, we perform first-principles simulations to shed light on moiré excitons in twisted MoS₂/WS₂ heterostructures. We provide direct evidence of localized interlayer moiré excitons in vdW heterostructures. The interlayer and intralayer moiré potentials are mapped out based on spatial modulations of energy gaps. Nearly flat valence bands are observed in the heterostructures. The dependence of spatial localization and binding energy of the moiré excitons on the twist angle of the heterostructures is examined. We explore how vertical electric field can be tuned to control the position, polarity, emission energy, and hybridization strength of the moiré excitons. We predict that alternating electric fields could modulate the dipole moments of hybridized moiré excitons and suppress their diffusion in moiré lattices.

Realization of asymmetric spin splitting Dirac cones in antiferromagnetic graphene/CrAs2/graphene heterotrilayer

Nonmagnetic graphene-based van der Waals heterotrilayers exhibit peculiar electronic features such as energetically and/or spatially resolved Dirac rings/cones. Here, using first-principles calculations we study the effect of magnetic proximity effect and mirror symmetry of antiferromagnetic CrAs2monolayer sandwiched between graphene on the Dirac cones. We clearly identify the common vertical shift of the Dirac bands in the spin up channel. While in the spin down channel, we surprisingly observe the remarkable transverse splitting Dirac cones. The underling mechanism can be attributed to the static electric field caused by the charge transfer between the interlayers, and the polarized field arising from the weakly magnetized graphene. Both fields collectively give rise to an inequivalent space inversion broken between graphene and CrAs2layers. Such unique Dirac states are absent in its nonmagnetic or ferromagnetic counterpart, ferromagnetic heterotrilayer with the glide symmetry, and graphene/CrAs2heterobilayer. Our findings would provide a new insight into the correlation between Dirac cones and magnetic monolayer sandwiched between graphene.

Intrinsic quantized anomalous Hall effect in a moiré heterostructure

The quantum anomalous Hall (QAH) effect combines topology and magnetism to produce precisely quantized Hall resistance at zero magnetic field. We report the observation of a QAH effect in twisted bilayer graphene aligned to hexagonal boron nitride. The effect is driven by intrinsic strong interactions, which polarize the electrons into a single spin- and valley-resolved moiré miniband with Chern number C = 1. In contrast to magnetically doped systems, the measured transport energy gap is larger than the Curie temperature for magnetic ordering, and quantization to within 0.1% of the von Klitzing constant persists to temperatures of several kelvin at zero magnetic field. Electrical currents as small as 1 nanoampere controllably switch the magnetic order between states of opposite polarization, forming an electrically rewritable magnetic memory.

Superlattices based on van der Waals 2D materials

Two-dimensional (2D) materials exhibit a number of improved mechanical, optical, and electronic properties compared to their bulk counterparts. The absence of dangling bonds in the cleaved surfaces of these materials allows combining different 2D materials into van der Waals heterostructures to fabricate p-n junctions, photodetectors, and 2D-2D ohmic contacts that show unexpected performances. These intriguing results are regularly summarized in comprehensive reviews. A strategy to tailor their properties even further and to observe novel quantum phenomena consists in the fabrication of superlattices whose unit cell is formed either by two dissimilar 2D materials or by a 2D material subjected to a periodic perturbation, each component contributing with different characteristics. Furthermore, in a 2D material-based superlattice, the interlayer interaction between the layers mediated by van der Waals forces constitutes a key parameter to tune the global properties of the superlattice. The above-mentioned factors reflect the potential to devise countless combinations of van der Waals 2D material-based superlattices. In the present feature article, we explain in detail the state-of-the-art of 2D material-based superlattices and describe the different methods to fabricate them, classified as vertical stacking, intercalation with atoms or molecules, moiré patterning, strain engineering and lithographic design. We also aim to highlight some of the specific applications of each type of superlattices.

Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers

We show that moiré bands of twisted homobilayers can be topologically nontrivial, and illustrate the tendency by studying valence band states in ±K valleys of twisted bilayer transition metal dichalcogenides, in particular, bilayer MoTe_{2}. Because of the large spin-orbit splitting at the monolayer valence band maxima, the low energy valence states of the twisted bilayer MoTe_{2} at the +K (-K) valley can be described using a two-band model with a layer-pseudospin magnetic field Δ(r) that has the moiré period. We show that Δ(r) has a topologically nontrivial skyrmion lattice texture in real space, and that the topmost moiré valence bands provide a realization of the Kane-Mele quantum spin-Hall model, i.e., the two-dimensional time-reversal-invariant topological insulator. Because the bands narrow at small twist angles, a rich set of broken symmetry insulating states can occur at integer numbers of electrons per moiré cell.

Observation of moiré excitons in WSe2/WS2 heterostructure superlattices

Moiré superlattices enable the generation of new quantum phenomena in two-dimensional heterostructures, in which the interactions between the atomically thin layers qualitatively change the electronic band structure of the superlattice. For example, mini-Dirac points, tunable Mott insulator states and the Hofstadter butterfly pattern can emerge in different types of graphene/boron nitride moiré superlattices, whereas correlated insulating states and superconductivity have been reported in twisted bilayer graphene moiré superlattices^1-12. In addition to their pronounced effects on single-particle states, moiré superlattices have recently been predicted to host excited states such as moiré exciton bands^13-15. Here we report the observation of moiré superlattice exciton states in tungsten diselenide/tungsten disulfide (WSe₂/WS₂) heterostructures in which the layers are closely aligned. These moiré exciton states manifest as multiple emergent peaks around the original WSe₂ A exciton resonance in the absorption spectra, and they exhibit gate dependences that are distinct from that of the A exciton in WSe₂ monolayers and in WSe₂/WS₂ heterostructures with large twist angles. These phenomena can be described by a theoretical model in which the periodic moiré potential is much stronger than the exciton kinetic energy and generates multiple flat exciton minibands. The moiré exciton bands provide an attractive platform from which to explore and control excited states of matter, such as topological excitons and a correlated exciton Hubbard model, in transition-metal dichalcogenides.

Multiple parameter dynamic photoresponse microscopy for data-intensive optoelectronic measurements of van der Waals heterostructures

Quantum devices made from van der Waals (vdW) heterostructures of two dimensional (2D) materials may herald a new frontier in designer materials that exhibit novel electronic properties and unusual electronic phases. However, due to the complexity of layered atomic structures and the physics that emerges, experimental realization of devices with tailored physical properties will require comprehensive measurements across a large domain of material and device parameters. Such multi-parameter measurements require new strategies that combine data-intensive techniques-often applied in astronomy and high energy physics-with the experimental tools of solid state physics and materials science. We discuss the challenges of comprehensive experimental science and present a technique, called Multi-Parameter Dynamic Photoresponse Microscopy (MPDPM), which utilizes ultrafast lasers, diffraction limited scanning beam optics, and hardware automation to characterize the photoresponse of 2D heterostructures in a time efficient manner. Using comprehensive methods on vdW heterostructures results in large and complicated data sets; in the case of MPDPM, we measure a large set of images requiring advanced image analysis to extract the underlying physics. We discuss how to approach such data sets in general and in the specific case of a graphene-boron nitride-graphite heterostructure photocell.

Gate-Tunable Topological Flat Bands in Trilayer Graphene Boron-Nitride Moiré Superlattices

We investigate the electronic structure of the flat bands induced by moiré superlattices and electric fields in nearly aligned ABC trilayer graphene (TLG) boron-nitride (BN) interfaces where Coulomb effects can lead to correlated gapped phases. Our calculations indicate that valley-spin resolved isolated superlattice flat bands that carry a finite Chern number C=3 proportional to the layer number can appear near charge neutrality for appropriate perpendicular electric fields and twist angles. When the degeneracy of the bands is lifted by Coulomb interactions, these topological bands can lead to anomalous quantum Hall phases that embody orbital and spin magnetism. Narrow bandwidths of ∼10  meV achievable for a continuous range of twist angles θ≲0.6° with moderate interlayer potential differences of ∼50  meV make the TLG-BN systems a promising platform for the study of electric-field tunable Coulomb-interaction-driven spontaneous Hall phases.

Electron quantum metamaterials in van der Waals heterostructures

In recent decades, scientists have developed the means to engineer synthetic periodic arrays with feature sizes below the wavelength of light. When such features are appropriately structured, electromagnetic radiation can be manipulated in unusual ways, resulting in optical metamaterials whose function is directly controlled through nanoscale structure. Nature, too, has adopted such techniques-for example in the unique colouring of butterfly wings-to manipulate photons as they propagate through nanoscale periodic assemblies. In this Perspective, we highlight the intriguing potential of designer structuring of electronic matter at scales at and below the electron wavelength, which affords a new range of synthetic quantum metamaterials with unconventional responses. Driven by experimental developments in stacking atomically layered heterostructures-such as mechanical pick-up/transfer assembly-atomic-scale registrations and structures can be readily tuned over distances smaller than characteristic electronic length scales (such as the electron wavelength, screening length and electron mean free path). Yet electronic metamaterials promise far richer categories of behaviour than those found in conventional optical metamaterial technologies. This is because, unlike photons, which scarcely interact with each other, electrons in subwavelength-structured metamaterials are charged and strongly interact. As a result, an enormous variety of emergent phenomena can be expected and radically new classes of interacting quantum metamaterials designed.

A 50/50 electronic beam splitter in graphene nanoribbons as a building block for electron optics

Based on the investigation of the multi-terminal conductance of a system composed of two graphene nanoribbons, in which one is on top of the other and rotated by [Formula: see text], we propose a setup for a 50/50 electronic beam splitter that neither requires large magnetic fields nor ultra low temperatures. Our findings are based on an atomistic tight-binding description of the system and on the Green function method to compute the Landauer conductance. We demonstrate that this system acts as a perfect 50/50 electronic beam splitter, in which its operation can be switched on and off by varying the doping (Fermi energy). We show that this device is robust against thermal fluctuations and long range disorder, as zigzag valley chiral states of the nanoribbons are protected against backscattering. We suggest that the proposed device can be applied as the fundamental element of the Hong-Ou-Mandel interferometer, as well as a building block of many devices in electron optics.