Nonlinear Landau Fan Diagram for Graphene Electrons Exposed to a Moiré Potential.

Due to Landau quantization, the conductance of two-dimensional electrons exposed to a perpendicular magnetic field exhibits oscillations that generate a fan of linear trajectories when plotted in the parameter space spanned by density and field. This fan looks identical, irrespective of the dispersion and field dependence of the Landau level energy. This is no surprise because the position of conductance minima depends solely on the level degeneracy that is linear in flux. The fractal energy spectrum that emerges within each Landau band when electrons are also exposed to a two-dimensional superlattice potential produces numerous additional oscillations, but they also create just linear fans for identical reasons. Here, we report conductance oscillations of graphene electrons exposed to a moiré potential that defy this general rule and form nonlinear trajectories in the density-field plane. We attribute this anomalous behavior to the simultaneous occupation of multiple minibands and magnetic breakdown-induced open orbits.

Interlayer Interactions in 1D Van der Waals Moiré Superlattices.

Studying two-dimensional (2D) van der Waals (vdW) moiré superlattices and their interlayer interactions have received surging attention after recent discoveries of many new phases of matter that are highly tunable. Different atomistic registry between layers forming the inner and outer nanotubes can also form one-dimensional (1D) vdW moiré superlattices. In this review, experimental observations and theoretical perspectives related to interlayer interactions in 1D vdW moiré superlattices are summarized. The discussion focuses on double-walled carbon nanotubes (DWNTs), a model 1D vdW moiré system, and the authors highlight the new optical features emerging from the non-trivial strong interlayer coupling effect and the unique physics in 1D DWNTs. Future directions and questions in probing the intriguing physical phenomena in 1D vdW moiré superlattices such as, correlated physics in different 1D moiré systems beyond DWNTs are proposed and discussed.

Orbital Gating Driven by Giant Stark Effect in Tunneling Phototransistors.

Conventional gating in transistors uses electric fields through external dielectrics that require complex fabrication processes. Various optoelectronic devices deploy photogating by electric fields from trapped charges in neighbor nanoparticles or dielectrics under light illumination. Orbital gating driven by giant Stark effect is demonstrated in tunneling phototransistors based on 2H-MoTe2 without using external gating bias or slow charge trapping dynamics in photogating. The original self-gating by light illumination modulates the interlayer potential gradient by switching on and off the giant Stark effect where the dz 2-orbitals of molybdenum atoms play the dominant role. The orbital gating shifts the electronic bands of the top atomic layer of the MoTe2 by up to 100 meV, which is equivalent to modulation of a carrier density of 7.3 × 1011 cm-2 by electrical gating. Suppressing conventional photoconductivity, the orbital gating in tunneling phototransistors achieves low dark current, practical photoresponsivity (3357 AW-1 ), and fast switching time (0.5 ms) simultaneously.

Robust Quantum Oscillation of Dirac Fermions in a Single-Defect Resonant Transistor.

The massless nature of Dirac Fermions produces large energy gaps between Landau levels (LLs), which is promising for topological devices. While the energy gap between the zeroth and first LLs reaches 36 meV in a magnetic field of 1 T in graphene, exploiting the quantum Hall effect at room temperature requires large magnetic fields (∼30 T) to overcome the energy level broadening induced by charge inhomogeneities in the device. Here, we report a way to use the robust quantum oscillations of Dirac Fermions in a single-defect resonant transistor, which is based on local tunneling through a thin (∼1.4 nm) hexagonal boron nitride (h-BN) between lattice-orientation-aligned graphene layers. A single point defect in the h-BN, selected by the orientation-tuned graphene layers, probes local LLs in its proximity, minimizing the energy broadening of the LLs by charge inhomogeneity at a moderate magnetic field and ambient conditions. Thus, the resonant tunneling between lattice-orientation-aligned graphene layers highlights the potential to spectroscopically locate the atomic defects in the h-BN, which contributes to the study on electrically tunable single photon source via defect states in h-BN.

Trigonal quasicrystalline states in [Formula: see text] rotated double moiré superlattices.

We study the lattice configuration and electronic structure of a double moiré superlattice, which is composed of a graphene layer encapsulated by two other layers in a way such that the two hexagonal moiré patterns are arranged in a dodecagonal quasicrystalline configuration. We show that there are between 0 and 4 such configurations depending on the lattice mismatch between graphene and the encapsulating layer. We then reveal the resonant interaction, which is distinct from the conventional 2-, 3-, 4-wave mixing of moiré superlattices, that brings together and hybridizes twelve degenerate Bloch states of monolayer graphene. These states do not fully satisfy the dodecagonal quasicrystalline rotational symmetry due to the symmetry of the wave vectors involved. Instead, their wave functions exhibit trigonal quasicrystalline order, which lacks inversion symmetry, at the energies much closer to the charge neutrality point of graphene.

Odd Integer Quantum Hall States with Interlayer Coherence in Twisted Bilayer Graphene.

We report on the quantum Hall effect in two stacked graphene layers rotated by 2°. The tunneling strength among the layers can be varied from very weak to strong via the mechanism of magnetic breakdown when tuning the density. Odd-integer quantum Hall physics is not anticipated in the regime of suppressed tunneling for balanced layer densities, yet it is observed. We interpret this as a signature of Coulomb interaction induced interlayer coherence and Bose-Einstein condensation of excitons that form at half filling of each layer. A density imbalance gives rise to reentrant behavior due to a phase transition from the interlayer coherent state to incompressible behavior caused by simultaneous condensation of both layers in different quantum Hall states. With increasing overall density, magnetic breakdown gains the upper hand. As a consequence of the enhanced interlayer tunneling, the interlayer coherent state and the phase transition vanish.

Anisotropic band flattening in graphene with one-dimensional superlattices.

Patterning graphene with a spatially periodic potential provides a powerful means to modify its electronic properties1-3. In particular, in twisted bilayers, coupling to the resulting moiré superlattice yields an isolated flat band that hosts correlated many-body phases4,5. However, both the symmetry and strength of the effective moiré potential are constrained by the constituent crystals, limiting its tunability. Here, we have exploited the technique of dielectric patterning6 to subject graphene to a one-dimensional electrostatic superlattice (SL)1. We observed the emergence of multiple Dirac cones and found evidence that with increasing SL potential the main and satellite Dirac cones are sequentially flattened in the direction parallel to the SL basis vector, behaviour resulting from the interaction between the one-dimensional SL electric potential and the massless Dirac fermions hosted by graphene. Our results demonstrate the ability to induce tunable anisotropy in high-mobility two-dimensional materials, a long-desired property for novel electronic and optical applications7,8. Moreover, these findings offer a new approach to engineering flat energy bands where electron interactions can lead to emergent properties9.

Emergence of orbital angular moment at van Hove singularity in graphene/h-BN moiré superlattice.

Bloch electrons lacking inversion symmetry exhibit orbital magnetic moments owing to the rotation around their center of mass; this moment induces a valley splitting in a magnetic field. For the graphene/h-BN moiré superlattice, inversion symmetry is broken by the h-BN. The superlattice potential generates a series of Dirac points (DPs) and van Hove singularities (vHSs) within an experimentally accessible low energy state, providing a platform to study orbital moments with respect to band structure. In this work, theoretical calculations and magnetothermoelectric measurements are combined to reveal the emergence of an orbital magnetic moment at vHSs in graphene/h-BN moiré superlattices. The thermoelectric signal for the vHS at the low energy side of the hole-side secondary DP exhibited significant magnetic field-induced valley splitting with an effective g-factor of approximately 130; splitting for other vHSs was negligible. This was attributed to the emergence of an orbital magnetic moment at the second vHS at the hole-side.

Observation of Drastic Electronic-Structure Change in a One-Dimensional Moiré Superlattice.

We report the first experimental observation of a strong-coupling effect in a one-dimensional moiré superlattice. We study one-dimensional double-wall carbon nanotubes (DWCNTs) in which van der Waals-coupled two single nanotubes form a one-dimensional moiré superlattice. We experimentally combine Rayleigh scattering spectroscopy and electron beam diffraction on the same individual DWCNTs to probe the optical transitions of the structure-identified DWCNTs in the visible spectral range. Among more than 30 structure-identified DWCNTs examined, we experimentally observed and identified a drastic change of the optical transition spectrum in a DWCNT with chirality (12,11)@(17,16). The origin of the marked change is attributed to the strong intertube coupling effect in the moiré superlattice formed by two nearly armchair nanotubes. Our numerical simulation is consistent with the experimental findings.

Resonant Tunneling Spectroscopy to Probe the Giant Stark Effect in Atomically Thin Materials.

Each atomic layer in van der Waals heterostructures possesses a distinct electronic band structure that can be manipulated for unique device operations. In the precise device architecture, the subtle but critical band splits by the giant Stark effect between atomic layers, varied by the momentum of electrons and external electric fields in device operation, has not yet been demonstrated or applied to design original devices with the full potential of atomically thin materials. Here, resonant tunneling spectroscopy based on the negligible quantum capacitance of 2D semiconductors in resonant tunneling transistors is reported. The bandgaps and sub-band structures of various channel materials could be demonstrated by the new conceptual spectroscopy at the device scale without debatable quasiparticle effects. Moreover, the band splits by the giant Stark effect in the channel materials could be probed, overcoming the limitations of conventional optical, photoemission, and tunneling spectroscopy. The resonant tunneling spectroscopy reveals essential and practical information for novel device applications.

Dirac electrons in a dodecagonal graphene quasicrystal.

Quantum states of quasiparticles in solids are dictated by symmetry. We have experimentally demonstrated quantum states of Dirac electrons in a two-dimensional quasicrystal without translational symmetry. A dodecagonal quasicrystalline order was realized by epitaxial growth of twisted bilayer graphene rotated exactly 30°. We grew the graphene quasicrystal up to a millimeter scale on a silicon carbide surface while maintaining the single rotation angle over an entire sample and successfully isolated the quasicrystal from a substrate, demonstrating its structural and chemical stability under ambient conditions. Multiple Dirac cones replicated with the 12-fold rotational symmetry were observed in angle-resolved photoemission spectra, which revealed anomalous strong interlayer coupling with quasi-periodicity. Our study provides a way to explore physical properties of relativistic fermions with controllable quasicrystalline orders.

Band structure engineering of 2D materials using patterned dielectric superlattices.

The ability to manipulate electrons in two-dimensional materials with external electric fields provides a route to synthetic band engineering. By imposing artificially designed and spatially periodic superlattice potentials, electronic properties can be further altered beyond the constraints of naturally occurring atomic crystals1-5. Here, we report a new approach to fabricate high-mobility superlattice devices by integrating surface dielectric patterning with atomically thin van der Waals materials. By separating the device assembly and superlattice fabrication processes, we address the intractable trade-off between device processing and mobility degradation that constrains superlattice engineering in conventional systems. The improved electrostatics of atomically thin materials allows smaller wavelength superlattice patterns relative to previous demonstrations. Moreover, we observe the formation of replica Dirac cones in ballistic graphene devices with sub-40 nm wavelength superlattices and report fractal Hofstadter spectra6-8 under large magnetic fields from superlattices with designed lattice symmetries that differ from that of the host crystal. Our results establish a robust and versatile technique for band structure engineering of graphene and related van der Waals materials with dynamic tunability.

Emergence of Tertiary Dirac Points in Graphene Moiré Superlattices.

The electronic structure of a crystalline solid is largely determined by its lattice structure. Recent advances in van der Waals solids, artificial crystals with controlled stacking of two-dimensional (2D) atomic films, have enabled the creation of materials with novel electronic structures. In particular, stacking graphene on hexagonal boron nitride (hBN) introduces a moiré superlattice that fundamentally modifies graphene's band structure and gives rise to secondary Dirac points (SDPs). Here we find that the formation of a moiré superlattice in graphene on hBN yields new, unexpected consequences: a set of tertiary Dirac points (TDPs) emerge, which give rise to additional sets of Landau levels when the sample is subjected to an external magnetic field. Our observations hint at the formation of a hidden Kekulé superstructure on top of the moiré superlattice under appropriate carrier doping and magnetic fields.

Charge Inversion and Topological Phase Transition at a Twist Angle Induced van Hove Singularity of Bilayer Graphene.

van Hove singularities (VHS's) in the density of states play an outstanding and diverse role for the electronic and thermodynamic properties of crystalline solids. At the critical point the Fermi surface connectivity changes, and topological properties undergo a transition. Opportunities to systematically pass a VHS at the turn of a voltage knob and study its diverse impact are however rare. With the advent of van der Waals heterostructures, control over the atomic registry of neighboring graphene layers offers an unprecedented tool to generate a low energy VHS easily accessible with conventional gating. Here we have addressed magnetotransport when the chemical potential crosses the twist angle induced VHS in twisted bilayer graphene. A topological phase transition is experimentally disclosed in the abrupt conversion of electrons to holes or vice versa, a loss of a nonzero Berry phase and distinct sequences of integer quantum Hall states above and below the singularity.