Minibands in twisted bilayer graphene probed by magnetic focusing

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.

Probing the Anisotropic Fermi Surface in Tetralayer Graphene via Transverse Magnetic Focusing

Bernal-stacked tetralayer graphene (4LG) exhibits intriguing low-energy properties, featuring two massive sub-bands and showcasing diverse features of topologically distinct, anisotropic Fermi surfaces, including Lifshitz transitions and trigonal warping. Here, we study the influence of the band structure on electron dynamics within 4LG using transverse magnetic focusing. Our analysis reveals two distinct focusing peaks corresponding to the two sub-bands. Furthermore, we uncover a pronounced dependence of the focusing spectra on crystal orientations, indicative of an anisotropic Fermi surface. Utilizing the semiclassical model, we attribute this orientation-dependent behavior to the trigonal warping of the band structure. This phenomenon leads to variations in electron trajectories based on crystal orientation. Our findings not only enhance our understanding of the dynamics of electrons in 4LG but also offer a promising method for probing anisotropic Fermi surfaces in other materials.

Ballistic transport spectroscopy of spin-orbit-coupled bands in monolayer graphene on WSe2

Van der Waals interactions with transition metal dichalcogenides were shown to induce strong spin-orbit coupling (SOC) in graphene, offering great promises to combine large experimental flexibility of graphene with unique tuning capabilities of the SOC. Here, we probe SOC-driven band splitting and electron dynamics in graphene on WSe2 by measuring ballistic transverse magnetic focusing. We found a clear splitting in the first focusing peak whose evolution in charge density and magnetic field is well reproduced by calculations using the SOC strength of ~ 13 meV, and no splitting in the second peak that indicates stronger Rashba SOC. Possible suppression of electron-electron scatterings was found in temperature dependence measurement. Further, we found that Shubnikov-de Haas oscillations exhibit a weaker band splitting, suggesting that it probes different electron dynamics, calling for a new theory. Our study demonstrates an interesting possibility to exploit ballistic electron motion pronounced in graphene for emerging spin-orbitronics.

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.

Interlayer Electron-Hole Friction in Tunable Twisted Bilayer Graphene Semimetal

Charge-neutral conducting systems represent a class of materials with unusual properties governed by electron-hole (e-h) interactions. Depending on the quasiparticle statistics, band structure, and device geometry these semimetallic phases of matter can feature unconventional responses to external fields that often defy simple interpretations in terms of single-particle physics. Here we show that small-angle twisted bilayer graphene (SA TBG) offers a highly tunable system in which to explore interactions-limited electron conduction. By employing a dual-gated device architecture we tune our devices from a nondegenerate charge-neutral Dirac fluid to a compensated two-component e-h Fermi liquid where spatially separated electrons and holes experience strong mutual friction. This friction is revealed through the T^{2} resistivity that accurately follows the e-h drag theory we develop. Our results provide a textbook illustration of a smooth transition between different interaction-limited transport regimes and clarify the conduction mechanisms in charge-neutral SA TBG.

Moiré-Induced Transport in CVD-Based Small-Angle Twisted Bilayer Graphene

To realize the applicative potential of 2D twistronic devices, scalable synthesis and assembly techniques need to meet stringent requirements in terms of interface cleanness and twist-angle homogeneity. Here, we show that small-angle twisted bilayer graphene assembled from separated CVD-grown graphene single-crystals can ensure high-quality transport properties, determined by a device-scale-uniform moiré potential. Via low-temperature dual-gated magnetotransport, we demonstrate the hallmarks of a 2.4°-twisted superlattice, including tunable regimes of interlayer coupling, reduced Fermi velocity, large interlayer capacitance, and density-independent Brown-Zak oscillations. The observation of these moiré-induced electrical transport features establishes CVD-based twisted bilayer graphene as an alternative to "tear-and-stack" exfoliated flakes for fundamental studies, while serving as a proof-of-concept for future large-scale assembly.

Correlated States in Strained Twisted Bilayer Graphenes Away from the Magic Angle

Graphene moiré superlattice formed by rotating two graphene sheets can host strongly correlated and topological states when flat bands form at so-called magic angles. Here, we report that, for a twisting angle far away from the magic angle, the heterostrain induced during stacking heterostructures can also create flat bands. Combining a direct visualization of strain effect in twisted bilayer graphene moiré superlattices and transport measurements, features of correlated states appear at "non-magic" angles in twisted bilayer graphene under the heterostrain. Observing correlated states in these "non-standard" conditions can enrich the understanding of the possible origins of the correlated states and widen the freedom in tuning the moiré heterostructures and the scope of exploring the correlated physics in moiré superlattices.

One-dimensional quantum channel in bent honeycomb nanoribbons

The directionality of steering charge carriers is of great importance for the application of two-dimensional (2D) materials. Using the generalized Bloch theorem coupled with the self-consistent charge density-functional tight-binding method, we theoretically propose an approach to construct a one-dimensional (1D) quantum channel in honeycomb nanoribbons (NR) via in-plane bending deformation. Bending-induced pseudo-magnetic fields lead to Landau quantization and localize the electronic states along both edges of bent NR. These localized states form robust 1D quantum channels, whose energies can be linearly modulated through the bending angle. Our findings give new inspiration for the realization of transverse magnetic focusing (TMF) under zero magnetic fields and pave the way for the design of 2D material-based nano-devices via strain-engineering.

Moiré Superlattice Effects and Band Structure Evolution in Near-30-Degree Twisted Bilayer Graphene

In stacks of two-dimensional crystals, mismatch of their lattice constants and misalignment of crystallographic axes lead to formation of moiré patterns. We show that moiré superlattice effects persist in twisted bilayer graphene (tBLG) with large twists and short moiré periods. Using angle-resolved photoemission, we observe dramatic changes in valence band topology across large regions of the Brillouin zone, including the vicinity of the saddle point at M and across 3 eV from the Dirac points. In this energy range, we resolve several moiré minibands and detect signatures of secondary Dirac points in the reconstructed dispersions. For twists θ > 21.8°, the low-energy minigaps are not due to cone anticrossing as is the case at smaller twist angles but rather due to moiré scattering of electrons in one graphene layer on the potential of the other which generates intervalley coupling. Our work demonstrates the robustness of the mechanisms which enable engineering of electronic dispersions of stacks of two-dimensional crystals by tuning the interface twist angles. It also shows that large-angle tBLG hosts electronic minigaps and van Hove singularities of different origin which, given recent progress in extreme doping of graphene, could be explored experimentally.

Precision measurement of electron-electron scattering in GaAs/AlGaAs using transverse magnetic focusing

Electron-electron (e-e) interactions assume a cardinal role in solid-state physics. Quantifying the e-e scattering length is hence critical. In this paper we show that the mesoscopic phenomenon of transverse magnetic focusing (TMF) in two-dimensional electron systems forms a precise and sensitive technique to measure this length scale. Conversely we quantitatively demonstrate that e-e scattering is the predominant effect limiting TMF amplitudes in high-mobility materials. Using high-resolution kinetic simulations, we show that the TMF amplitude at a maximum decays exponentially as a function of the e-e scattering length, which leads to a ready approach to extract this length from the measured TMF amplitudes. The approach is applied to measure the temperature-dependent e-e scattering length in high-mobility GaAs/AlGaAs heterostructures. The simulations further reveal current vortices that accompany the cyclotron orbits - a collective phenomenon counterintuitive to the ballistic transport underlying a TMF setting.

Electron-electron scattering plays a crucial role in many solid state phenomena; however, the direct measurement of electron-electron scattering length is challenging. Here, the authors use transverse magnetic focusing to measure this quantity in high-mobility GaAs/AlGaAs heterostructures.

Out-of-Plane Dielectric Susceptibility of Graphene in Twistronic and Bernal Bilayers

We describe how the out-of-plane dielectric polarizability of monolayer graphene influences the electrostatics of bilayer graphene-both Bernal (BLG) and twisted (tBLG). We compare the polarizability value computed using density functional theory with the output from previously published experimental data on the electrostatically controlled interlayer asymmetry potential in BLG and data on the on-layer density distribution in tBLG. We show that monolayers in tBLG are described well by polarizability α_exp = 10.8 ų and effective out-of-plane dielectric susceptibility ϵ_z = 2.5, including their on-layer electron density distribution at zero magnetic field and the interlayer Landau level pinning at quantizing magnetic fields.

Strong Interminivalley Scattering in Twisted Bilayer Graphene Revealed by High-Temperature Magneto-Oscillations

Twisted bilayer graphene (TBG) provides an example of a system in which the interplay of interlayer interactions and superlattice structure impacts electron transport in a variety of nontrivial ways and gives rise to a plethora of interesting effects. Understanding the mechanisms of electron scattering in TBG has, however, proven challenging, raising many questions about the origins of resistivity in this system. Here we show that TBG exhibits high-temperature magneto-oscillations originating from the scattering of charge carriers between TBG minivalleys. The amplitude of these oscillations reveals that interminivalley scattering is strong, and its characteristic timescale is comparable to that of its intraminivalley counterpart. Furthermore, by exploring the temperature dependence of these oscillations, we estimate the electron-electron collision rate in TBG and find that it exceeds that of monolayer graphene. Our study demonstrates the consequences of the relatively small size of the superlattice Brillouin zone and Fermi velocity reduction on lateral transport in TBG.

Synchronizing Bloch-Oscillating Free Carriers in Moiré Flat Bands

Achieving Bloch oscillations of free carriers under a direct current, a long-sought-after collective many-body behavior, has been challenging due to stringent constraints on the band properties. We argue that the flat bands in moiré graphene fulfill the basic requirements for observing Bloch oscillations, offering an appealing alternative to the stacked quantum wells used in previous work aiming to access this regime. Bloch-oscillating moiré superlattices emit a comblike spectrum of incommensurate frequencies, a property of interest for converting direct currents into high-frequency currents and developing broadband amplifiers in terahertz domain. The oscillations can be synchronized through coupling to an oscillator mode in a photonic or plasmonic resonator. Phase-coherent collective oscillations in the resonant regime provide a realization of current-pumped terahertz lasing.

Combined Minivalley and Layer Control in Twisted Double Bilayer Graphene

Control over minivalley polarization and interlayer coupling is demonstrated in double bilayer graphene twisted with an angle of 2.37°. This intermediate angle is small enough for the minibands to form and large enough such that the charge carrier gases in the layers can be tuned independently. Using a dual-gated geometry we identify and control all possible combinations of minivalley polarization via the population of the two bilayers. An applied displacement field opens a band gap in either of the two bilayers, allowing us to even obtain full minivalley polarization. In addition, the carriers, formerly separated by their minivalley character, are mixed by tuning through a Lifshitz transition, where the Fermi surface topology changes. The high degree of control over the minivalley character of the bulk charge transport in twisted double bilayer graphene offers new opportunities for realizing valleytronics devices such as valley valves, filters, and logic gates.