Flatbands and Perfect Metal in Trilayer Moiré Graphene

The influence of interlayer bias and crystal field on the electronic characteristics of twisted tri-layer graphene

Twisted van der Waals multilayers have proven to be highly effective in solid-state systems, facilitating the emergence of unique quantum behaviors. By utilizing a real-space tight-binding model, we demonstrate that in twisted trilayer graphene (t-TLG), both localized and dispersive modes can be significantly altered through adjustments in the interlayer bias and crystal field. Interestingly, the interlayer bias results in Dirac crossings above and below the charge neutrality point, alongside several anti-crossings. In contrast, the crystal field creates asymmetry between the inner and outer layers by applying differing electrostatic potentials, which in turn inherently induces an interlayer bias. Our findings indicate that an accurate prediction of the electronic characteristics of t-TLG requires accounting for the effects of both interlayer bias and crystal fields.

Striped Twisted State in the Orientational Epitaxy on Quasicrystals

The optimal "twisted" geometry of a crystalline layer on a crystal has long been known, but that on a quasicrystal is still unknown and open. We predict analytically that the layer equilibrium configuration will generally exhibit a nonzero misfit angle. The theory perfectly agrees with numerical optimization of a colloid monolayer on a quasiperiodic decagonal optical lattice. Strikingly different from crystal-on-crystal epitaxy, the structure of the novel emerging twisted state exhibits an unexpected stripe pattern. Its high anisotropy should reflect on the tribomechanical properties of this unconventional interface.

Energy spectrum theory of incommensurate systems

Because of the lack of translational symmetry, calculating the energy spectrum of an incommensurate system has always been a theoretical challenge. Here, we propose a natural approach to generalize energy band theory to incommensurate systems without reliance on the commensurate approximation, thus providing a comprehensive energy spectrum theory of incommensurate systems. Except for a truncation-dependent weighting factor, the formulae of this theory are formally almost identical to that of Bloch electrons, making it particularly suitable for complex incommensurate structures. To illustrate the application of this theory, we give three typical examples: one-dimensional bichromatic and trichromatic incommensurate potential models, as well as a moiré quasicrystal. Our theory establishes a fundamental framework for understanding incommensurate systems.

Topological minibands and interaction driven quantum anomalous Hall state in topological insulator based moiré heterostructures

The presence of topological flat minibands in moiré materials provides an opportunity to explore the interplay between topology and correlation. In this work, we study moiré minibands in topological insulator films with two hybridized surface states under a moiré superlattice potential created by two-dimensional insulating materials. We show the lowest conduction (highest valence) Kramers' pair of minibands can beZ 2 non-trivial when the minima (maxima) of moiré potential approximately form a hexagonal lattice with six-fold rotation symmetry. Coulomb interaction can drive the non-trivial Kramers' minibands into the quantum anomalous Hall state when they are half-filled, which is further stabilized by applying external gate voltages to break inversion. We propose the monolayer Sb2 on top of Sb2Te3 films as a candidate based on first principles calculations. Our work demonstrates the topological insulator based moiré heterostructure as a potential platform for studying interacting topological phases.

Local atomic stacking and symmetry in twisted graphene trilayers

Moiré superlattices formed by twisting trilayers of graphene are a useful model for studying correlated electron behaviour and offer several advantages over their formative bilayer analogues, including a more diverse collection of correlated phases and more robust superconductivity. Spontaneous structural relaxation alters the behaviour of moiré superlattices considerably and has been suggested to play an important role in the relative stability of superconductivity in trilayers. Here we use an interferometric four-dimensional scanning transmission electron microscopy approach to directly probe the local graphene layer alignment over a wide range of trilayer graphene structures. Our results inform a thorough understanding of how reconstruction modulates the local lattice symmetries crucial for establishing correlated phases in twisted graphene trilayers, evincing a relaxed structure that is markedly different from that proposed previously.

Emerging Characteristics and Properties of Moiré Materials

In recent years, scientists have conducted extensive research on Moiré materials and have discovered some compelling properties. The Moiré superlattice allows superconductivity through flat-band and strong correlation effects. The presence of flat bands causes the Moiré material to exhibit topological properties as well. Modulating electronic interactions with magnetic fields in Moiré materials enables the fractional quantum Hall effect. In addition, Moiré materials have ferromagnetic and antiferromagnetic properties. By tuning the interlayer coupling and spin interactions of the Moiré superlattice, different magnetic properties can be achieved. Finally, this review also discusses the applications of Moiré materials in the fields of photocurrent, superconductivity, and thermoelectricity. Overall, Moiré superlattices provide a new dimension in the development of two-dimensional materials.

Magic-angle helical trilayer graphene

We propose magic-angle helical trilayer graphene (HTG), a helical structure featuring identical rotation angles between three consecutive layers of graphene, as a unique and experimentally accessible platform for realizing exotic correlated topological states of matter. While nominally forming a supermoiré (or moiré-of-moiré) structure, we show that HTG locally relaxes into large regions of a periodic single-moiré structure realizing flat topological bands carrying nontrivial valley Chern number. These bands feature near-ideal quantum geometry and are isolated from remote bands by a very large energy gap, making HTG a promising platform for experimental realization of correlated topological states such as integer and fractional quantum anomalous Hall states.

Superconductivity and strong interactions in a tunable moiré quasicrystal

Electronic states in quasicrystals generally preclude a Bloch description1, rendering them fascinating and enigmatic. Owing to their complexity and scarcity, quasicrystals are underexplored relative to periodic and amorphous structures. Here we introduce a new type of highly tunable quasicrystal easily assembled from periodic components. By twisting three layers of graphene with two different twist angles, we form two mutually incommensurate moiré patterns. In contrast to many common atomic-scale quasicrystals2,3, the quasiperiodicity in our system is defined on moiré length scales of several nanometres. This 'moiré quasicrystal' allows us to tune the chemical potential and thus the electronic system between a periodic-like regime at low energies and a strongly quasiperiodic regime at higher energies, the latter hosting a large density of weakly dispersing states. Notably, in the quasiperiodic regime, we observe superconductivity near a flavour-symmetry-breaking phase transition4,5, the latter indicative of the important role that electronic interactions play in that regime. The prevalence of interacting phenomena in future systems with in situ tunability is not only useful for the study of quasiperiodic systems but may also provide insights into electronic ordering in related periodic moiré crystals6-12. We anticipate that extending this platform to engineer quasicrystals by varying the number of layers and twist angles, and by using different two-dimensional components, will lead to a new family of quantum materials to investigate the properties of strongly interacting quasicrystals.

Pair Density Waves from Local Band Geometry

A band-projection formalism is developed for calculating the superfluid weight in two-dimensional multiorbital superconductors with an orbital-dependent pairing. It is discovered that, in this case, the band geometric superfluid stiffness tensor can be locally nonpositive definite in some regions of the Brillouin zone. When these regions are large enough or include nodal singularities, the total superfluid weight becomes nonpositive definite due to pairing fluctuations, resulting in the transition of a BCS state to a pair density wave (PDW). This geometric BCS-PDW transition is studied in the context of two-orbital superconductors, and proof of the existence of a geometric BCS-PDW transition in a generic topological flat band is established.

Functional Renormalization Group Study of Superconductivity in Rhombohedral Trilayer Graphene

We employ a functional renormalization group approach to ascertain the pairing mechanism and symmetry of the superconducting phase observed in rhombohedral trilayer graphene. Superconductivity in this system occurs in a regime of carrier density and displacement field with a weakly distorted annular Fermi sea. We find that repulsive Coulomb interactions can induce electron pairing on the Fermi surface by taking advantage of momentum-space structure associated with the finite width of the Fermi sea annulus. The degeneracy between spin-singlet and spin-triplet pairing is lifted by valley-exchange interactions that strengthen under the RG flow and develop nontrivial momentum-space structure. We find that the leading pairing instability is d-wave-like and spin singlet, and that the theoretical phase diagram versus carrier density and displacement field agrees qualitatively with experiment.

A primer on twistronics: a massless Dirac fermion's journey to moiré patterns and flat bands in twisted bilayer graphene

The recent discovery of superconductivity in magic-angle twisted bilayer graphene (TBLG) has sparked a renewed interest in the strongly-correlated physics ofsp²carbons, in stark contrast to preliminary investigations which were dominated by the one-body physics of the massless Dirac fermions. We thus provide a self-contained, theoretical perspective of the journey of graphene from its single-particle physics-dominated regime to the strongly-correlated physics of the flat bands. Beginning from the origin of the Dirac points in condensed matter systems, we discuss the effect of the superlattice on the Fermi velocity and Van Hove singularities in graphene and how it leads naturally to investigations of the moiré pattern in van der Waals heterostructures exemplified by graphene-hexagonal boron-nitride and TBLG. Subsequently, we illuminate the origin of flat bands in TBLG at the magic angles by elaborating on a broad range of prominent theoretical works in a pedagogical way while linking them to available experimental support, where appropriate. We conclude by providing a list of topics in the study of the electronic properties of TBLG not covered by this review but may readily be approached with the help of this primer.

Observation of Coexisting Dirac Bands and Moiré Flat Bands in Magic-Angle Twisted Trilayer Graphene

Moiré superlattices that consist of two or more layers of 2D materials stacked together with a small twist angle have emerged as a tunable platform to realize various correlated and topological phases, such as Mott insulators, unconventional superconductivity, and quantum anomalous Hall effect. Recently, magic-angle twisted trilayer graphene (MATTG) has shown both robust superconductivity similar to magic-angle twisted bilayer graphene and other unique properties, including the Pauli-limit violating and re-entrant superconductivity. These rich properties are deeply rooted in its electronic structure under the influence of distinct moiré potential and mirror symmetry. Here, combining nanometer-scale spatially resolved angle-resolved photoemission spectroscopy and scanning tunneling microscopy/spectroscopy, the as-yet unexplored band structure of MATTG near charge neutrality is systematically measured. These measurements reveal the coexistence of the distinct dispersive Dirac band with the emergent moiré flat band, showing nice agreement with the theoretical calculations. These results serve as a stepstone for further understanding of the unconventional superconductivity in MATTG.

Promotion of superconductivity in magic-angle graphene multilayers

Graphene moiré superlattices show an abundance of correlated insulating, topological, and superconducting phases. Whereas the origins of strong correlations and nontrivial topology can be directly linked to flat bands, the nature of superconductivity remains enigmatic. We demonstrate that magic-angle devices made of twisted tri-, quadri-, and pentalayer graphene placed on monolayer tungsten diselenide exhibit flavor polarization and superconductivity. We also observe insulating states in the tril- and quadrilayer arising at finite electric displacement fields. As the number of layers increases, superconductivity emerges over an enhanced filling-factor range, and in the pentalayer it extends well beyond the filling of four electrons per moiré unit cell. Our results highlight the role of the interplay between flat and more dispersive bands in extending superconducting regions in graphene moiré superlattices.

Moiré bands in twisted trilayer black phosphorene: effects of pressure and electric field

Twist-induced moiré bands and accompanied correlated phenomena have been extensively investigated in twisted hexagonal lattices with weak interlayer coupling. However, the formation of moiré bands in strongly coupled layered materials and their controlled tuning remain largely unexplored. Here, we systematically study the moiré bands in twisted trilayer black phosphorene (TTbP) and the influences of pressure and electric field on them. Moiré states can form in various TTbPs even when the twist angle is larger than 16° similar to that of twisted bilayer bP. However, different TTbPs show different localization patterns depending on the twisting layer, leading to distinct dipolar behaviors. While these moiré states become quasi-one-dimensional (1D) as the twist angle decreases, external pressure causes the crossover of moiré states from quasi-1D to 0D with a dramatic change in localization areas and greatly reduced bandwidth. Interestingly, compared to twisted bilayer and pristine bP, TTbPs show a much larger electric-field induced Stark effect, controllable by either the twist angle or twist layer. Our work thus demonstrates TTbP as an attractive platform to explore moiré-controlled electronic and optical properties, as well as tunable optoelectronic applications.

Electronic Properties of Oxidized Graphene: Effects of Strain and an Electric Field on Flat Bands and the Energy Gap

A multiscale modeling and simulation approach, including first-principles calculations, ab initio molecular dynamics simulations, and a tight binding approach, is employed to study band flattening of the electronic band structure of oxidized monolayer graphene. The width of flat bands can be tuned by strain, the external electric field, and the density of functional groups and their distribution. A transition to a conducting state is found for monolayer graphene with impurities when it is subjected to an electric field of ∼1.0 V/Å. Several parallel impurity-induced flat bands appear in the low-energy spectrum of monolayer graphene when the number of epoxy groups is changed. The width of the flat band decreases with an increase in tensile strain but is independent of the electric field strength. Here an alternative and easy route for obtaining band flattening in thermodynamically stable functionalized monolayer graphene is introduced. Our work discloses a new avenue for research on band flattening in monolayer graphene.

Correlated Insulating States and Transport Signature of Superconductivity in Twisted Trilayer Graphene Superlattices

Layers of two-dimensional materials stacked with a small twist angle give rise to beating periodic patterns on a scale much larger than the original lattice, referred to as a "moiré superlattice." Here, we demonstrate a higher-order "moiré of moiré" superlattice in twisted trilayer graphene with two consecutive small twist angles. We report correlated insulating states near the half filling of the moiré of moiré superlattice at an extremely low carrier density (∼10^{10}  cm^{-2}), near which we also report a zero-resistance transport behavior typically expected in a 2D superconductor. The full-occupancy (ν=-4 and ν=4) states are semimetallic and gapless, distinct from the twisted bilayer systems.

In-Plane Critical Magnetic Fields in Magic-Angle Twisted Trilayer Graphene

It has recently been shown [Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Pauli-limit violation and re-entrant superconductivity in moiré graphene, Nature (London) 595, 526 (2021).NATUAS0028-0836] that superconductivity in magic-angle twisted trilayer graphene survives to in-plane magnetic fields that are well in excess of the Pauli limit, and much stronger than the in-plane critical magnetic fields of magic-angle twisted bilayer graphene. The difference is surprising because twisted bilayers and trilayers both support the magic-angle flat bands thought to be the fountainhead of twisted graphene superconductivity. We show here that the difference in critical magnetic fields can be traced to a C_{2}M_{h} symmetry in trilayers that survives in-plane magnetic fields, and also relative displacements between top and bottom layers that are not under experimental control at present. An gate electric field breaks the C_{2}M_{h} symmetry and therefore limits the in-plane critical magnetic field.

Superradiant Phase Transition in Electronic Systems and Emergent Topological Phases

We derive a general criterion for determining the onset of superradiant phase transition in electronic bands coupled to a cavity field, with possibly electron-electron interactions. For longitudinal superradiance in 2D or genuine 1D systems, we prove that it is always prevented, thereby extending existing no-go theorems. Instead, a superradiant phase transition can occur to a nonuniform transverse cavity field and we give specific examples in noninteracting models, either through Fermi surface nesting or parabolic band touching. Investigating the resulting time-reversal symmetry breaking superradiant states, we find in the former case Fermi surface lifting down to four Dirac points on a square lattice model, with topologically protected zero modes, and in the latter case topological bands with nonzero Chern number on an hexagonal lattice.

Electronic Raman Scattering in Twistronic Few-Layer Graphene

We study electronic contribution to the Raman scattering signals of two-, three- and four-layer graphene with layers at one of the interfaces twisted by a small angle with respect to each other. We find that the Raman spectra of these systems feature two peaks produced by van Hove singularities in moiré minibands of twistronic graphene, one related to direct hybridization of the Dirac states, and the other resulting from band folding caused by moiré superlattice. The positions of both peaks strongly depend on the twist angle, so that their detection can be used for noninvasive characterization of the twist, even in hBN-encapsulated structures.

Twisted Trilayer Graphene: A Precisely Tunable Platform for Correlated Electrons

We introduce twisted trilayer graphene (tTLG) with two independent twist angles as an ideal system for the precise tuning of the electronic interlayer coupling to maximize the effect of correlated behaviors. As established by experiment and theory in the related twisted bilayer graphene system, van Hove singularities (VHS) in the density of states can be used as a proxy of the tendency for correlated behaviors. To explore the evolution of VHS in the twist-angle phase space of tTLG, we present a general low-energy electronic structure model for any pair of twist angles. We show that the basis of the model has infinite dimensions even at a finite energy cutoff and that no Brillouin zone exists even in the continuum limit. Using this model, we demonstrate that the tTLG system exhibits a wide range of magic angles at which VHS merge and that the density of states has a sharp peak at the charge-neutrality point through two distinct mechanisms: the incommensurate perturbation of twisted bilayer graphene's flatbands or the equal hybridization between two bilayer moiré superlattices.

Ultraheavy and Ultrarelativistic Dirac Quasiparticles in Sandwiched Graphenes

Electrons in quantum materials exhibiting coexistence of dispersionless (flat) bands piercing dispersive (steep) bands give rise to strongly correlated phenomena and are associated with unconventional superconductivity. We show that in twisted sandwiched graphene (TSWG)-a three-layer van der Waals heterostructure with a twisted middle layer-steep Dirac cones can coexist with dramatic band flattening at the same energy scale, if twisted by 1.5°. This phenomenon is not stable in the simplified continuum models. The key result of this Letter is that the flat bands become stable only as a consequence of lattice relaxation processes included in our atomistic calculations. Further on, external fields can change the relative energy offset between the Dirac cone vertex and the flat bands and enhance band hybridization, which could permit controlling correlated phases. Our work establishes twisted sandwiched graphene as a new platform for research into strongly interacting two-dimensional quantum matter.

Moiré Flat Bands in Twisted Double Bilayer Graphene

We investigate twisted double bilayer graphene (TDBG), a four-layer system composed of two AB-stacked graphene bilayers rotated with respect to each other by a small angle. Our ab initio band structure calculations reveal a considerable energy gap at the charge-neutrality point that we assign to the intrinsic symmetric polarization (ISP). We then introduce the ISP effect into the tight-binding parametrization and perform calculations on TDBG models that include lattice relaxation effects down to very small twist angles. We identify a narrow region around the magic angle characterized by a manifold of remarkably flat bands gapped out from other states even without external electric fields. To understand the fundamental origin of the magic angle in TDBG, we construct a continuum model that points to a hidden mathematical link to the twisted bilayer graphene model, thus indicating that the band flattening is a fundamental feature of TDBG and is not a result of external fields.

Twists and the Electronic Structure of Graphitic Materials

We analyze the effect of twists on the electronic structure of configurations of infinite stacks of graphene layers. We focus on three different cases: an infinite stack where each layer is rotated with respect to the previous one by a fixed angle, two pieces of semi-infinite graphite rotated with respect to each other, and finally a single layer of graphene rotated with respect to a graphite surface. In all three cases, we find a rich structure, with sharp resonances and flat bands for small twist angles. The method used can be easily generalized to more complex arrangements and stacking sequences.