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.

Spontaneous spin-valley polarization in NbSe2 at a van der Waals interface.

A proximity effect at a van der Waals (vdW) interface enables creation of an emergent quantum electronic ground state. Here we demonstrate that an originally superconducting two-dimensional (2D) NbSe2 forms a ferromagnetic ground state with spontaneous spin polarization at a vdW interface with a 2D ferromagnet V5Se8. We investigated the anomalous Hall effect (AHE) of the NbSe2/V5Se8 magnetic vdW heterostructures, and found that the sign of the AHE was reversed as the number of the V5Se8 layer was thinned down to the monolayer limit. Interestingly, the AHE signal of those samples was enhanced with the in-plane magnetic fields, suggesting an additional contribution to the AHE signal other than magnetization. This unusual behavior is well reproduced by band structure calculations, where the emergence of the Berry curvature along the spin-degenerate nodal lines in 2D NbSe2 by the in-plane magnetization plays a key role, unveiling a unique interplay between magnetism and Zeeman-type spin-orbit interaction in a non-centrosymmetric 2D quantum material.

Science of 2.5 dimensional materials: paradigm shift of materials science toward future social innovation.

The past decades of materials science discoveries are the basis of our present society - from the foundation of semiconductor devices to the recent development of internet of things (IoT) technologies. These materials science developments have depended mainly on control of rigid chemical bonds, such as covalent and ionic bonds, in organic molecules and polymers, inorganic crystals and thin films. The recent discovery of graphene and other two-dimensional (2D) materials offers a novel approach to synthesizing materials by controlling their weak out-of-plane van der Waals (vdW) interactions. Artificial stacks of different types of 2D materials are a novel concept in materials synthesis, with the stacks not limited by rigid chemical bonds nor by lattice constants. This offers plenty of opportunities to explore new physics, chemistry, and engineering. An often-overlooked characteristic of vdW stacks is the well-defined 2D nanospace between the layers, which provides unique physical phenomena and a rich field for synthesis of novel materials. Applying the science of intercalation compounds to 2D materials provides new insights and expectations about the use of the vdW nanospace. We call this nascent field of science '2.5 dimensional (2.5D) materials,' to acknowledge the important extra degree of freedom beyond 2D materials. 2.5D materials not only offer a new field of scientific research, but also contribute to the development of practical applications, and will lead to future social innovation. In this paper, we introduce the new scientific concept of this science of '2.5D materials' and review recent research developments based on this new scientific concept.

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.

Dirac Fermion Kinetics in 3D Curved Graphene.

3D integration of graphene has attracted attention for realizing carbon-based electronic devices. While the 3D integration can amplify various excellent properties of graphene, the influence of 3D curved surfaces on the fundamental physical properties of graphene has not been clarified. The electronic properties of 3D nanoporous graphene with a curvature radius down to 25-50 nm are systematically investigated and the ambipolar electronic states of Dirac fermions are essentially preserved in the 3D graphene nanoarchitectures, while the 3D curvature can effectively suppress the slope of the linear density of states of Dirac fermion near the Fermi level are demonstrated. Importantly, the 3D curvature can be utilized to tune the back-scattering-suppressed electrical transport of Dirac fermions and enhance both electron localization and electron-electron interaction. As a result, nanoscale curvature provides a new degree of freedom to manipulate 3D graphene electrical properties, which may pave a new way to design new 3D graphene devices with preserved 2D electronic properties and novel functionalities.

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.

Landau Quantization in Coupled Weyl Points: A Case Study of Semimetal NbP.

Weyl semimetal (WSM) is a newly discovered quantum phase of matter that exhibits topologically protected states characterized by two separated Weyl points with linear dispersion in all directions. Here, via combining theoretical analysis and magneto-infrared spectroscopy of an archetypal Weyl semimetal, niobium phosphide, we demonstrate that the coupling between Weyl points can significantly modify the electronic structure of a WSM and provide a new twist to the protected states. These findings suggest that the coupled Weyl points should be considered as the basis for analysis of realistic WSMs.

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.

Three-dimensional porous graphene networks expand graphene-based electronic device applications.

In recent years, there has been increasing demand for 3D porous graphene structures with excellent 2D characteristics and great potential. As one avenue, several approaches for fabricating 3D porous graphene network structures have been proposed to realize multi-functional graphene materials with 2D graphene structures. Herein, we overview characteristics of 3D porous graphene for applications in future electronic devices along with physical insights into "2D to 3D graphene", in which the characters of 2D graphene such as massless Dirac fermions are well preserved. The present review thus summarizes recent 3D porous graphene studies with a perspective for providing new and board applications of graphene in electronic devices.

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.

Electric Properties of Dirac Fermions Captured into 3D Nanoporous Graphene Networks.

Nanoporous graphene- based electric-double-layer transistors (EDLTs) are successfully fabricated. Transport measurements of the EDLTs demonstrate that the ambipolar electronic states of massless Dirac fermions with a high carrier mobility are well preserved in 3D nanoporous graphene along with anomalous nonlinear Hall resistance and exceptional transistor on/off ratio. This study may open a new avenue for device applications of graphene.

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.

Insulating state in tetralayers reveals an even-odd interaction effect in multilayer graphene.

Close to charge neutrality, the electronic properties of graphene and its multilayers are sensitive to electron-electron interactions. In bilayers, for instance, interactions are predicted to open a gap between valence and conduction bands, turning the system into an insulator. In mono and (Bernal-stacked) trilayers, which remain conducting at low temperature, interactions do not have equally drastic consequences. It is expected that interaction effects become weaker for thicker multilayers, whose behaviour should converge to that of graphite. Here we show that this expectation does not correspond to reality by revealing the occurrence of an insulating state close to charge neutrality in Bernal-stacked tetralayer graphene. The phenomenology-incompatible with the behaviour expected from the single-particle band structure-resembles that observed in bilayers, but the insulating state in tetralayers is visible at higher temperature. We explain our findings, and the systematic even-odd effect of interactions in Bernal-stacked layers of different thickness that emerges from experiments, in terms of a generalization of the interaction-driven, symmetry-broken states proposed for bilayers.

Evidence for a fractional fractal quantum Hall effect in graphene superlattices.

The Hofstadter energy spectrum provides a uniquely tunable system to study emergent topological order in the regime of strong interactions. Previous experiments, however, have been limited to low Bloch band fillings where only the Landau level index plays a role. We report measurements of high-mobility graphene superlattices where the complete unit cell of the Hofstadter spectrum is accessible. We observed coexistence of conventional fractional quantum Hall effect (QHE) states together with the integer QHE states associated with the fractal Hofstadter spectrum. At large magnetic field, we observed signatures of another series of states, which appeared at fractional Bloch filling index. These fractional Bloch band QHE states are not anticipated by existing theoretical pictures and point toward a distinct type of many-body state.

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.

Accessing the transport properties of graphene and its multilayers at high carrier density.

We present a comparative study of high carrier density transport in mono-, bi-, and trilayer graphene using electric double-layer transistors to continuously tune the carrier density up to values exceeding 10(14) cm(-2). Whereas in monolayer the conductivity saturates, in bi- and trilayer filling of the higher-energy bands is observed to cause a nonmonotonic behavior of the conductivity and a large increase in the quantum capacitance. These systematic trends not only show how the intrinsic high-density transport properties of graphene can be accessed by field effect, but also demonstrate the robustness of ion-gated graphene, which is crucial for possible future applications.

Magnetic field screening and mirroring in graphene.

The orbital magnetism in spatially varying magnetic fields is studied in monolayer graphene within the effective-mass approximation. We find that, unlike the conventional two-dimensional electron system, graphene with a small Fermi wave number kF works as a magnetic shield where the field produced by a magnetic object placed above graphene is always screened by a constant factor on the other side of graphene. The object is repelled by a diamagnetic force from the graphene, as if there exists its mirror image with a reduced amplitude on the other side of graphene. The magnitude of the force is much greater than that of the conventional two-dimensional system. The effect disappears with the increase of kF.

Quantum Hall effect of massless dirac fermions in a vanishing magnetic field.

The effect of strong long-range disorder on the quantization of the Hall conductivity sigma{xy} in graphene is studied numerically. It is shown that increasing Landau-level mixing progressively destroys all plateaus in sigma{xy} except the plateaus at sigma{xy}=-/+e{2}/2h (per valley and per spin). The critical state at the Dirac point is robust to strong disorder and belongs to the universality class of the conventional plateau transitions in the integer quantum Hall effect. We propose that the breaking of time-reversal symmetry by ripples in graphene can realize this quantum critical point in a vanishing magnetic field.