We propose Landau levels as a probe for the topological character of electronic bands in two-dimensional moiré superlattices. We consider two configurations of twisted double bilayer graphene (TDBG) that have very similar band structures, but show different valley Chern numbers of the flat bands. These differences between the AB-AB and AB-BA configurations of TDBG clearly manifest as different Landau level sequences in the Hofstadter butterfly spectra calculated using the tight-binding model. The Landau level sequences are explained from the point of view of the distribution of orbital magnetization in momentum space that is governed by the rotational $C_2$ and time-reversal $\mathcal{T}$ symmetries. Our results can be readily extended to other twisted graphene multilayers and $h$-BN/graphene heterostructures thus establishing the Hofstadter butterfly spectra as a powerful tool for detecting the non-trivial valley band topology.
Different atomistic registry between the layers forming the inner and outer nanotubes can form one-dimensional (1D) van der Waals (vdW) moiré superlattices. Unlike the two-dimensional (2D) vdW moiré superlattices, effects of 1D vdW moiré superlattices on electronic and optical properties in 1D moiré superlattices are not well understood, and they are often neglected. In this Perspective, we summarize new experimental observations and theoretical perspectives related to interlayer interactions in double-walled carbon nanotubes (DWNTs), a representative 1D vdW moiré system. Our discussion will f…
Recently, moiré superlattices have attracted considerable attentions because they are found to exhibit intriguing electronic phenomena of tunable Mott insulators and unconventional superconductivity. These phenomena are highly related to the physical mechanism of the interlayer coupling. However, up to now, there has not existed any theory that can completely interpret the experimental results of the interlayer conductance of moiré superlattice. In order to solve this problem, the superposition of periods and the corresponding coherence, which are the essential characteristics of moiré superla…
Moiré materials, with superlattice periodicity many times the atomic length scale, have enabled the studies of strong electronic correlations and band topology with unprecedented tunability. However, nonvolatile control of the moiré potentials, which could allow on-demand switching of the superlattice effects, has not been achieved to date. Here we demonstrate the switching of the correlated and moiré band insulating states and the associated nonlinear anomalous Hall effect by the ferroelectric effect. This is achieved in a ferroelectric WTe2 bilayer of the Td structure with a centered-rectang…
Van der Waals heterostructures of graphene and hexagonal boron nitride feature a moiré superlattice for graphene's Dirac electrons. Here, we review the effects generated by this superlattice, including a specific miniband structure featuring gaps and secondary Dirac points, and a fractal spectrum of magnetic minibands known as Hofstadter's butterfly.
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. When the stacking involves more than two layers with independent twist angles between adjacent layers, it generates moiré of moiré superlattices, with multiple length scales that control the system's behavior. Here we demonstrate these effects of a high-order moiré superlattice in twisted trilayer graphene with two consecutive small twist angles. We report correlated insulating states near the half fil…
Moiré superlattices provide a powerful tool to engineer novel quantum phenomena in two-dimensional (2D) heterostructures, where 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 can emerge in different types of graphene/boron nitride moiré superlattices, while correlated insulating states and superconductivity have been reported in twisted bilayer graphene moiré superlattices. In addition to their dramatic effects on the single pa…
Moiré superlattices in van der Waals (vdW) heterostructures have given rise to a number of emergent electronic phenomena due to the interplay between atomic structure and electron correlations. A lack of a simple way to characterize moiré superlattices has impeded progress in the field. In this work we outline a simple, room-temperature, ambient method to visualize real-space moiré superlattices with sub-5 nm spatial resolution in a variety of twisted vdW heterostructures including but not limited to conducting graphene, insulating boron nitride and semiconducting transition metal dichalcogeni…
In long-wavelength moiré superlattices of stacked transition metal dichalcogenides (TMDs), structural reconstruction ubiquitously occurs, which has reported to impact significantly their electronic properties. However, complete microscopic understandings of the interplay between the lattice reconstruction and alteration of electronic properties, and their further response to external perturbations in the reconstructed TMDs moiré superlattice are still lacking. Here, using scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) combined with first-principles calculation, w…
The specific rotational alignment of two-dimensional lattices results in a moiré superlattice with a larger period than the original lattices and allows one to engineer the electronic band structure of such materials. So far, transport signatures of such superlattices have been reported for graphene/hBN and graphene/graphene systems. Here we report moiré superlattices in fully hBN encapsulated graphene with both the top and the bottom hBN aligned to the graphene. In the graphene, two different moiré superlattices form with the top and the bottom hBN, respectively. The overlay of the two superl…
While Fabry-Pérot resonances and Moiré superlattices are intensively studied in graphene on hexagonal boron nitride (hBN), the two effects have not been discussed in their coexistence. Here we investigate the FP oscillations in a ballistic pnp-junctions in the presence and absence of a Moiré superlattice. First, we address the effect of the smoothness of the confining potential on the visibility of the FP resonances and carefully map the evolution of the FP cavity size as a function of densities in and outside the cavity in the absence of a superlattice, when the cavity is bound by regular pn-…
Moiré superlattices are excellent platforms to realize strongly correlated quantum phenomena, such as Mott insulation and superconductivity. In particular, recent research has revealed stripe phases and generalized Wigner crystals at fractional fillings of moiré superlattices. But these experiments have not focused on the influence of electronic crystallization on the excitonic and valleytronic properties of the superlattices. Here we report excitonic and valleytronic signatures of correlated states at fractional fillings in a WSe$_2$/WS$_2$ moiré superlattice. We observe reflection spectral m…
Moiré superlattices can induce correlated-electronic phases in twisted van-der-Waals materials. Strongly correlated quantum phenomena emerge, such as superconductivity and the Mott-insulating state. However, moiré superlattices produced through artificial stacking can be quite inhomogeneous, which hampers the development of a clear correlation between the moiré period and the emerging electrical and optical properties. Here we demonstrate in twisted-bilayer transition-metal dichalcogenides that low-frequency Raman scattering can be utilized not only to detect atomic reconstruction, but also to…
Moiré superlattice created by twist stacking has multiple physical properties. These physical properties depend on the twist angle, hence investigation of the twist angle dependency is important for the deep understanding of physical phenomena in moiré superlattice. In this work, negative magnetoresistance owing to weak localization (WL) was investigated in twisted double bilayer graphene (TDBG) as a function of the twist angle. The ratio of the intervalley scattering time to the intravalley scattering time, estimated using the WL formula for bilayer graphene, tended to decrease as the twist a…
We theoretically study in-plane acoustic phonons of graphene/hexagonal boron nitride moiré superlattice by using a continuum model. We demonstrate that the original phonon bands of individual layers are strongly hybridized and reconstructed into moiré phonon bands consisting of dispersive bands and flat bands. The phonon band structure can be effectively described by a spring-mass network model to simulate the motion of moiré domain walls, where the flat-band modes are interpreted as vibrations of independent, decoupled strings. We also show that the moiré phonon has angular momentum due to th…
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 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 consequenc…
In twisted homobilayer transition metal dichalcogenides, intra- and inter-layer valley excitons hybridize with the layer configurations spatially varying in the moiré. The ground state valley excitons are trapped at two high-symmetry points with opposite electric dipoles in a moiré supercell, forming a honeycomb superlattice of nearest-neighbor dipolar attraction. We find that the spatial texture of layer configuration results in a luminescence anomaly of the moiré trapped excitons, where a tiny displacement by interactions dramatically increases the brightness and changes polarization from ci…
It is well known that stacking domains form in moiré superlattices due to the competition between the interlayer van der Waals forces and intralayer elastic forces, which can be recognized as polar domains due to the local spontaneous polarization in bilayers without centrosymmetry. We propose a theoretical model which captures the effect of an applied electric field on the domain structure. The coupling between the spontaneous polarization and field leads to uneven relaxation of the domains, and a net polarization in the superlattice at nonzero fields, which is sensitive to the moiré period. …
Moiré superlattices are emerging as a new route for engineering strongly correlated electronic states in two-dimensional van der Waals heterostructures, as recently demonstrated in the correlated insulating and superconducting states in magic-angle twisted bilayer graphene and ABC trilayer graphene/boron nitride moiré superlattices. Transition metal dichalcogenide (TMDC) moiré heterostructures provide another exciting model system to explore correlated quantum phenomena, with the addition of strong light-matter interactions and large spin-orbital coupling. Here we report the optical detection …
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 Blo…
Moiré patterns with a superlattice potential can be formed by vertically stacking two layered materials with a relative twist or lattice constant mismatch. The moiré superlattice can generate flat bands that result in new correlated insulating, superconducting, and topological states. Strong electron correlations, tunable by the fractional filling, have been observed in both graphene and transition metal dichalcogenide (TMD) based systems. In addition, in TMD based systems, the moiré potential landscape can trap interlayer excitons (IX) at specific atomic registries. Here we report that spatia…
Graphene-metal nanoparticle hybrid materials potentially display not only the unique properties of metal nanoparticles and those of graphene, but also additional novel properties due to the interaction between graphene and nanoparticles. This study shows that gold nanoislands can be used to tailor the local electronic properties of graphene. Graphene on crystalline gold nanoislands exhibits moiré superlattices, which generate secondary Dirac points in the local density of states. Conversely, the graphene covered gold regions undergo a polycrystalline to Au(111) phase transition upon annealing.…
Moiré superlattices provide a powerful way to engineer properties of electrons and excitons in two-dimensional van der Waals heterostructures. The moiré effect can be especially strong for interlayer excitons, where electrons and holes reside in different layers and can be addressed separately. In particular, it was recently proposed that the moiré superlattice potential not only localizes interlayer exciton states at different superlattice positions, but also hosts an emerging moiré quasi-angular momentum (QAM) that periodically switches the optical selection rules for interlayer excitons at …
Many-body interactions between carriers lie at the heart of correlated physics. The ability to tune such interactions would open the possibility to access and control complex electronic phase diagrams on demand. Recently, moiré superlattices formed by two-dimensional materials have emerged as a promising platform for quantum engineering such phenomena. The power of the moiré system lies in the high tunability of its physical parameters by tweaking layer twist angle, electrical field, moiré carrier filling, and interlayer coupling. Here, we report that optical excitation can drastically tune th…
When atomically thin two-dimensional (2D) materials are layered they often form incommensurate non-crystalline structures that exhibit long-period moir{\' e} patterns when examined by scanning probes. In this paper we present an approach which uses information obtained from {\it ab initio} calculations performed on short-period crystalline structures to derive effective Hamiltonians that are able to efficiently describe the influence of the moir{\' e} pattern superlattices on electronic properties. We apply our approach to the cases of graphene on graphene (G/G) and graphene on hexagonal boron…
In graphene moiré superlattices, electronic interactions between layers are mostly hidden as band structures get crowded because of folding, making their interpretation cumbersome. Here, the evolution of the electronic band structure as a function of the interlayer rotation angle is studied using Density Functional Theory followed by unfolding bands and then comparing them to their corresponding individual components. We observe interactions at regions not theoretically elucidated so far, where for small interlayer angles, gaps turn into discrete-like states that are evenly spaced in energy. W…
We have studied the electronic excitation spectrum in periodically rippled graphene on Ru(0001) and flat, commensurate graphene on Ni(111) by means of high-resolution electron energy loss spectroscopy and a combination of density functional theory and tight-binding approaches. We show that the periodic moiré superlattice originated by the lattice mismatch in graphene/Ru(0001) induces the emergence of an extra mode, which is not present in graphene/Ni(111). Contrary to the ordinary intra-band plasmon of doped graphene, the extra mode is robust in charge-neutral graphene/metal contacts, having i…
In moiré crystals formed by stacking van der Waals (vdW) materials, surprisingly diverse correlated electronic phases and optical properties can be realized by a subtle change in the twist angle. Here, we discover that phonon spectra are also renormalized in MoS$_2$ twisted bilayers, adding a new perspective to moiré physics. Over a range of small twist angles, the phonon spectra evolve rapidly due to ultra-strong coupling between different phonon modes and atomic reconstructions of the moiré pattern. We develop a new low-energy continuum model for phonons that overcomes the outstanding challe…
Moiré superlattices in twisted van der Waals materials constitute a promising platform for engineering electronic and optical properties. However, a major obstacle to fully understanding these systems and harnessing their potential is the limited ability to correlate the local moiré structure with optical properties. By using a recently developed scanning electron microscopy technique to image twisted WSe$_2$/WSe$_2$ bilayers, we directly correlate increasing moiré periodicity with the emergence of two distinct exciton species. These can be tuned individually through electrostatic gating, and …
We investigate the optical absorption spectrum and the selection rule for the Hofstadter butterfly in twisted bilayer graphene under magnetic fields. We demonstrate that the absorption spectrum exhibits a self-similar recursive pattern reflecting the fractal nature of the energy spectrum. We find that the optical selection rule has a nested self-similar structure as well, and it is governed by the conservation of the total angular momentum summed over different hierarchies.
Twist-Tuned Magnonic Nanocavity Mode in a Trilayer Moiré Superlattice
Fert Beijing Institute, MIIT Key Laboratory of Spintronics, School of Integrated Circuit Science and Engineering, Beihang University（北京航空航天大学集成电路科学与工程学院）；CNR-Istituto Officina dei Materiali (IOM)（意大利国家研究委员会材料研究所）；Dipartimento di Fisica e Geologia, Università di Perugia（佩鲁贾大学物理与地质学系）；International Quantum Academy, Shenzhen（深圳量子创新研究院）
We find a systematic reappearance of massive Dirac features at the edges of consecutive minibands formed at magnetic fields B_{p/q}= pϕ_0/(qS) providing rational magnetic flux through a unit cell of the moire superlattice created by a hexagonal substrate for electrons in graphene. The Dirac-type features in the minibands at B=B_{p/q} determine a hierarchy of gaps in the surrounding fractal spectrum, and show that these minibands have topological insulator properties. Using the additional $q$-fold degeneracy of magnetic minibands at B_{p/q}, we trace the hierarchy of the gaps to their manifesta…
In heterostructures consisting of atomically thin crystals layered on top of one another, lattice mismatch or rotation between the layers results in long-wavelength moiré superlattices. These moiré patterns can drive significant band structure reconstruction of the composite material, leading to a wide range of emergent phenomena including superconductivity, magnetism, fractional Chern insulating states, and moiré excitons. Here, we investigate monolayer graphene encapsulated between two crystals of boron nitride (BN), where the rotational alignment between all three components can be varied. …
We investigate the electronic structure of the flat bands induced by moiré superlattices and electric fields in nearly aligned ABC trilayer graphene-boron nitride 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 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 …
We theoretically investigate the electronic structures of moiré superlattices arising in monolayer / bilayer graphene stacked on hexagonal boron nitride (hBN) in presence and absence of magnetic field. We develop an effective continuum model from a microscopic tight-binding lattice Hamiltonian, and calculate the electronic structures of graphene-hBN systems with different rotation angles. Using the effective model, we explain the characteristic band properties such as the gap opening at the corners of the superlattice Brillouin zone (mini-Dirac point). We also investigate the energy spectrum a…
Van der Waals structures formed by aligning monolayer graphene with insulating layers of hexagonal boron nitride exhibit a moiré superlattice that is expected to break sublattice symmetry. Despite an energy gap of several tens of millielectron volts opening in the Dirac spectrum, electrical resistivity remains lower than expected at low temperature and varies between devices. While subgap states are likely to play a role in this behavior, their precise nature is unclear. We present a scanning gate microscopy study of moiré superlattice devices with comparable activation energy but with differe…
Heterostructures of atomically-thin materials have attracted significant interest owing to their ability to host novel electronic properties fundamentally distinct from their constituent layers. In the case of graphene on boron nitride, the closely-matched lattices yield a moiré superlattice that modifies the graphene electron dispersion and opens gaps both at the primary Dirac point (DP) and the moiré-induced secondary Dirac point (SDP) in the valence band. While significant effort has focused on controlling the superlattice period via the rotational stacking order, the role played by the mag…
Moiré structures in van der Waals heterostructures lead to emergent phenomena including superconductivity in twisted bilayer graphene and optically accessible strongly-correlated electron states in transition metal dichalcogenide heterobilayers. Dual periodicity moiré structures (DPMS) formed in layered structures with more than two layers have been shown to lead to ferromagnetism and multiple secondary Dirac points in TBG. Whilst in principle it is possible to obtain DPMS in bilayers there has not been clear experimental evidence of this yet. In this paper we present signatures of DPMS in a t…
The superlattice of alternating graphene/h-BN few-layered heterostructures is found to exhibit strong dependence on the parity of the number of layers within the stack. Odd-parity systems show a unique flamingo-like pattern, whereas their even-parity counterparts exhibit regular hexagonal or rectangular superlattices. When the alternating stack consists of seven layers or more, the flamingo pattern becomes favorable, regardless of parity. Notably, the out-of-plane corrugation of the system strongly depends on the shape of the superstructure resulting in significant parity dependence of its mec…
Monolayer graphene aligned with hexagonal boron nitride (hBN) develops a gap at the charge neutrality point (CNP). This gap has previously been extensively studied by electrical transport through thermal activation measurements. Here, we report the determination of the gap size at the CNP of graphene/hBN superlattice through photocurrent spectroscopy study. We demonstrate two distinct measurement approaches to extract the gap size. A maximum of ~ 14 meV gap is observed for devices with a twist angle of less than 1 degree. This value is significantly smaller than that obtained from thermal acti…
Recent experimental progresses have demonstrated the great potential of electronic and excitonic moiré superlattices in transition metal dichalcogenides (TMDs) for quantum many-body simulations and quantum optics applications. Here we reveal that the moiré potential landscapes in the TMDs heterostructures have an electrostatic origin from the spontaneous charge transfer across the heterointerfaces dependent on the atomic registry. This allows engineering tunable multi-chromatic superlattices through the interference of moiré potentials from independently configurable heterointerfaces in multil…
Non-trivial Berry phase of graphene leads to unusual quantum correction to the conductivity. Berry phase of pi in single layer graphene (SLG) and 2pi in bi-layer graphene (BLG) is expected to reveal weak anti-localization (WAL) and weak localization (WL), respectively. However, experimentally both WAL and WL have been observed in graphene devices depending on the strength of different scattering mechanisms. Graphene superlattice having multiple Dirac cones is expected to exhibit pi to 2pi Berry phase transition from primary Dirac cone (PDC) to cloned Dirac cone (CDC). However, its effect on lo…
Emergent electronic insulating states in a one-dimensional moiré superlattice
Zhejiang University（浙江大学）；The University of Osaka（大阪大学）；Dalian University（大连大学）；Inha University（仁荷大学）；National Institute for Materials Science (NIMS)（国立材料研究所）；Stony Brook University（石溪大学）
