Metrics details Van der Waals moiré heterostructure have been found to exhibits a robust interfacial ferroelectricity down to atomic thickness and discovering and understanding the complex polarization state in moiré systems is of fundamental interest to condensed-matter research we examine the moiré ferroelectricity in twisted h-BN heterostructure by piezoresponse force microscopy and we directly observe sliding ferroelectricity in the center of triangular moiré patterns as well as robust in-plane polarization topology emerging at the boundary of adjacent triangles The edge polarization possesses non-trivial and robust vortex polarization topology Our calculations trace the origin of this phenomenon to joined piezoelectric effects with sliding ferroelectricity This work provides intuitive insights to explore the unique moiré ferroelectricity in non-polar background matrix and the inherent stability of the topological structures ensures reliable and durable performance of electronic devices The nanoscale characterization based on scanning probe microscopy (SPM) is capable of detecting the localized moiré domains when the twist angles are decreased below the magic angle (<1.05°) more complex and diverse physical behaviors typically occur identifying the continuous polar structures in the moiré system at microscopic twist angles remains a significant challenge A Top view of the relaxed atomic structure of t-BN C Top view of IP polarization with clockwise and counterclockwise like merons and anti-merons OOP polarization at the edge is displayed in the form of blue and red colors The color bar represents the direction of OOP polarization D Unusual Bloch-type domain walls with down-up-down-up polarization transitions of moiré domains at a tip-sample angle of 0° signals measured at the same position with a tip-sample angle of 60° responses obtained at a tip-sample angle of −60°.The gray tip represents the direction of the cantilever beam and the red and blue dashed arrows in the phase image indicate the direction of IP polarization images obtained at different tip-sample angles (from 0° to 360° with a step of 30°) along the red dashed line in (D) and the same position in (E) A V-Amp. and B V-Pha. images at the same region in Fig. 2 at the regions within the black dashed boxes in (A) and (B) at the edge labeled by black dash squares in (C) and (D) extracted along the red dashes highlighted in (E) and (F) indicating that such configuration is suitable for different moiré systems A IP polarization distribution in relaxed h-BN with twist angle of 0.99° The length and color intensity of the arrows represent the magnitude of polarization in relaxed h-BN with a twist angle of 0.99° C OOP polarization induced by piezoelectric effects and thus leaving no effect on the global IP polarization the tested moiré pattern twist angles are all less than 1° B Expansion (composed of BA stacking) and contraction (composed of AB stacking) of ferroelectric domains under an electric field of 0.5 V/Å while the area of AA stacking does not show significant change C In-plane atomic displacement of upper layer after structural reconstruction under the effect of electric field we propose a robust edge topology that features IP like meron-antimeron polarization The edge consists of two splitting regions with opposite OOP polarizations and a complete IP topological structure The adjacent triangular moiré patterns in the center only possess opposite OOP polarizations we identify that the combined sliding ferroelectricity and piezoelectric effects are the physical origin of the complex polarization characteristics in twisted BN The stability of vortex-antivortex structures with unique edges is significantly influenced by their topological characteristics which is essential for the development of devices that rely on these topological features ensuring remarkable reliability and durability even under varying external perturbations Our discovery provides new insights into unconventional interfacial ferroelectrics in moiré systems opening up new opportunities for fundamental research and applications in electronics and data storage Based on the method training the intrinsic BN energy model we trained an energy model under an applied electric field of 0.5 V/Å First-principles datasets of the structure and energy were obtained by running AIMD simulations in VASP with an 0.5 V/Å electric field applied along the out-of-plane axis The DP model was trained over 1,200,000 steps Same relaxation method was then used to get the moire structure under the influence of electric field thereby assigning charges of \(2{{{\rm{e}}}}^{-}\) to wannier centres where \({{\rm{e}}}\) is the unit electronic charge \({{{\rm{Z}}}}_{{{\rm{i}}}}\) are atomic numbers \({{{\bf{r}}}}_{{{\bf{i}}}}\) are the position vectors of the nuclei The Wannier centers are obtained from a unitary transformation that minimizes the spatial spread in the occupied orbital subspace We use a valence-only pseudopotential approach where the nuclear charges \({{{\rm{e}}}Z}_{i}\) represent the ions consisting of the nuclei and the frozen core electrons while the Wannier centers correspond to the valence electrons which contains boron (\({{{\bf{r}}}}_{{{{\bf{B}}}}_{{{\bf{i}}}}}\)) and nitride (\({{{\bf{r}}}}_{{{{\bf{N}}}}_{{{\bf{i}}}}}\)) ions wannier centers are only associated with N the most electronegative atoms during molecular evolution also aligns well with the results obtained using the Berry phase method it is reasonable to expect that our model will perform well on relaxed moiré systems as well The piezoelectric polarizations in the moiré superlattice can be expressed as the function of strain tensor \({{\rm{u}}}\) where \({u}_{{xx}},{u}_{{yy}},{u}_{{xy}}\) arise from IP atomic displacement under the relaxation effects the piezoelectric charge density is the divergence of the piezoelectric polarization Due to the equal magnitude and opposite sign of the charge densities at the same IP position for the two layers vertical polarization is created by the interlayer potential drop where \(d\) is the interlayer distance at the local unit cell in the moiré structure This is the polarization induced by piezoelectric effects The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information files Additional data are available from the corresponding author upon reasonable request High-density switchable skyrmion-like polar nanodomains integrated on silicon Unconventional superconductivity in magic-angle graphene superlattices Correlated insulator behaviour at half-filling in magic-angle graphene superlattices Intrinsic quantized anomalous Hall effect in a moiré heterostructure Negative capacitance in multidomain ferroelectric superlattices High-temperature topological superconductivity in twisted double-layer copper oxides All Magic Angles in Twisted Bilayer Graphene are 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electron Stacking Domains and Dislocation Networks in Marginally Twisted Bilayers of Transition Metal Dichalcogenides Multifaceted moiré superlattice physics in twisted WSe 2 bilayers DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics DeePMD-kit v2: A software package for deep potential models Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems In: Proceedings of the 32nd International Conference on Neural Information Processing Systems 4441–4451 (Curran Associates Inc. LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set From ultrasoft pseudopotentials to the projector augmented-wave method Generalized Gradient Approximation Made Simple Semiempirical GGA-type density functional constructed with a long-range dispersion correction and interatomic force constants from density-functional perturbation theory Maximally localized generalized Wannier functions for composite energy bands Maximally localized Wannier functions: Theory and applications Intrinsic Piezoelectricity in Two-Dimensional Materials A beginner’s guide to the modern theory of polarization Linear optical properties in the projector-augmented wave methodology Ab initio calculation of the macroscopic dielectric constant in silicon Download references National Key Research and Development Program of China grant 2024YFA1409700 and 2022YFA1402902; National Natural Science Foundation of China grant 12474084 12304218; Chenguang Program of Shanghai Education Development Foundation and Shanghai Municipal Education Commission; Shanghai Pujiang Program grant 23PJ1402200; ECNU (East China Normal University) Multifunctional Platform for Innovation (006); Fundamental Research Funds for the Central Universities These authors contributed equally: Wen-Cheng Fan Key Laboratory of Polar Materials and Devices Ministry of Education & Shanghai Center of Brain-inspired Intelligent Materials and Devices Key Laboratory of MEMS of Ministry of Education Collaborative Innovation Center of Extreme Optics performed the PFM experiments supervised by N.Z performed the DFT calculations supervised by W.Y.T discussed the results and contributed to the writing of the manuscript The authors declare no competing interests Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Download citation DOI: https://doi.org/10.1038/s41467-025-58877-1 Anyone you share the following link with will be able to read this content: a shareable link is not currently available for this article Sign up for the Nature Briefing newsletter — what matters in science Metrics details Two-dimensional electron systems in both magnetic fields and periodic potentials are described by the Hofstadter butterfly a fundamental problem of solid-state physics While moiré systems provide a powerful method to realize this type of spectrum previous experiments have been limited to fractional flux quanta regime due to the difficulty of building ~ 50 nm periodic modulations we demonstrate a super-moiré strategy to overcome this challenge By aligning monolayer graphene (G) with 1.0° twisted hexagonal boron nitride (t-hBN) a 63.2 nm bichromatic G/t-hBN super-moiré is constructed made possible by exploiting the electrostatic nature of t-hBN potential magnetic Bloch states at \(\phi /{\phi }_{0}=1-9\) are achieved and observed as integer Brown-Zak oscillations expanding the flux quanta from fractions to integers Theoretical analysis reproduces these experimental findings This work opens promising avenues to study unexplored Hofstadter butterfly explore emergent topological order at integer flux quanta and engineer long-wavelength periodic modulations a Illustration of the two single moirés and super-moiré The red (blue) indicates the top (bottom) layer of twisted hBN (t-hBN) and black is graphene G/hBN and G/t-hBN super-moiré are \({\lambda}_{1}\sim14.4\) nm \({\lambda}_{2}\sim13.0\) nm and \({\lambda }_{{sm}}=63.2\) nm in device D1 b Schematic illustration of the superimposition of potentials from t-hBN (\({V}_{t-{hBN}}\)) and G/hBN (\({V}_{G/{hBN}}\)) layers which generates a super-moiré potential (\({V}_{{sm}}\)) with a much larger wavelength c Schematics for the formation of super-moiré in reciprocal space \({{{{\bf{G}}}}}_{2}\) and \({{{{\bf{G}}}}}_{{{{\rm{s}}}}{{{\rm{m}}}}}\) are the reciprocal lattice vectors for t-hBN and the three hexagons are corresponding Brillouin zones d Calculated \({\lambda }_{{sm}}\) plotted as a function of \({\theta }_{1}\) and \({\theta }_{2}\) Only commensurate super-moiré configurations with wavelengths larger than 14 nm are considered and shown The super-moiré wavelength realized in this experiment is denoted by the red box \({\theta }_{2}=0.4^\circ\) and \({\lambda }_{{sm}}=64.6\) nm where monolayer graphene is aligned with t-hBN substrate f Longitudinal resistance \({R}_{{xx}}\) as a function of carrier density \({n}_{{tot}}\) at 1.5 K are related to the full fillings of the t-hBN and G/hBN moiré potentials where the magenta curve is the second derivative of \({R}_{{xx}}\) \({{{{{\rm{d}}}}}^{2}R}_{{xx}}/{{{\rm{d}}}}{n}_{{tot}}^{2}\) The total charge density \({n}_{{tot}}\) induced by electrostatic gating is calculated from \({n}_{{tot}}=1/e[{c}_{{TG}}\left({V}_{{TG}}-{V}_{{TG},0}\right)+{c}_{{BG}}\left({V}_{{BG}}-{V}_{{BG},0}\right)]\) Here \({c}_{{BG}}\) (\({c}_{{TG}}\)) is the back-gate (top-gate) capacitance and \({V}_{{BG}}\) (\({V}_{{TG}}\)) is the back-gate (top-gate) voltage \({V}_{{BG},0}\) and \({V}_{{TG},0}\) are the voltage offsets from back and top gate due to channel impurities As these parameters are consistent with our device fabrication procedure it is very clear that G/t-hBN super-moiré is constructed in our sample successfully and results in those small satellite peaks in longitudinal resistance Below we demonstrate that this super-moiré potential can induce magnetic Bloch states at integer flux quanta a \({R}_{{xx}}\) as a function of \(B\) and \({n}_{{tot}}\) at 142 K The fractal features appear at \(B=p({\phi }_{0}/S)\) with \(p=1-9\) suggesting the integer BZ oscillations periodic in \(B\) Inset is the same data but plotted in a narrower \({n}_{{tot}}\) range and brighter color filling b Second derivative \(\varDelta {R}_{{xx}}={\partial }^{2}{R}_{{xx}}/\partial {B}^{2}\) of the same data in a plotted as a function of normalized magnetic field \(\phi /{\phi }_{0}\) and \({n}_{{tot}}\) which removes the smooth background and highlights the oscillating features c \(\varDelta {R}_{{xx}}\) as function of \(\phi /{\phi }_{0}\) at fixed \({n}_{{tot}}=4.15\times {10}^{12}\) cm–2 which is a line cut from b marked by the blue arrow The integer BZ oscillations periodic in \(B\) are clearly seen and the \(B\) spacing of \(\varDelta B=1.20\) T implies a super-moiré wavelength of ~ 63.2 nm a \({R}_{{xx}}\) measured versus \({n}_{{tot}}\) under zero magnetic field at 1.5 K The satellite peaks around the main Dirac point (MDP) are related to the superlattice potential the size of t-hBN unit cell can be calculated as \(S=4/{n}_{{tot}}\) giving rise to a wavelength of ~ 62.5 nm Left inset is the moiré pattern at the t-hBN interface The red (blue) indicates the top (bottom) layer of t-hBN Right inset is the side view of the stack configuration in device D2 where graphene is misaligned with the t-hBN layers purposely b Landau fan diagram of device D2 measured at 75 K Besides the typical Shubnikov-de Haas (SdH) oscillations for monolayer graphene The STM samples were assembled with similar processes of fabricating transport devices but had different stack orders hBN and the remaining half of hBN were picked up by PC successively and then the three-layer stack was placed on the highly ordered pyrolytic graphite (HOPG) substrate while even numbers of layers do not show a measurable SHG signal (blue curve) here we use the STM to conduct the visualization The STM experiments were conducted with a Unisoku ultrahigh-vacuum STM at 4.8 K The tungsten tips were flashed by electron-beam bombardment for several minutes before use Typical imaging parameters were sample voltages between 100 mV and 500 mV and tunneling currents between 10 pA and 500 pA Transport measurements were performed in a cryogenic system which provides stable temperatures ranging from 1.4 to ∼ 300 K and fields up to 14 T AC bias voltage was applied to the source probe through Stanford Research Systems DS360 The current and voltage were measured with low-frequency lock-in technique (SR 830 with SR550 as the preamplifier) Relevant data supporting the key findings of this study are available within the article and the Supplementary Information file All raw data generated during the current study are available from the corresponding authors upon request Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields Bloch electrons in a uniform magnetic field Emergence of superlattice Dirac points in graphene on hexagonal boron nitride Superlattice-induced insulating states and valley-protected orbits in twisted bilayer graphene Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices Cloning of Dirac fermions in graphene superlattices High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices High-order fractal states in graphene superlattices High-order fractal quantum oscillations in graphene/BN superlattices in the extreme doping limit Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices Observation of fractional Chern insulators in a van der Waals heterostructure Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride Band structure engineering of 2D materials using patterned dielectric superlattices Band conductivity oscillations in a gate-tunable graphene superlattice Multiple flat bands and topological Hofstadter butterfly in twisted bilayer graphene close to the second magic angle Heteromoiré engineering on magnetic Bloch transport in twisted graphene superlattices Hofstadter topology: noncrystalline topological materials at high flux Reentrant correlated insulators in twisted bilayer graphene at 25 T (2 π flux) Observation of reentrant correlated insulators and interaction-driven Fermi-surface reconstructions at one magnetic flux quantum per moiré unit cell in magic-angle twisted bilayer graphene Universal superlattice potential for 2D materials from twisted interface inside h-BN substrate Moiré band structure engineering using a twisted boron nitride substrate Zhang, T., Xiao, C., Xue, H., Yao, W. & Ki, D.-K. Emergence of moiré superlattice potential in graphene by twisted-hBN layers. Preprint at https://arxiv.org/abs/2405.10079 (2024) Composite super-moiré lattices in double-aligned graphene heterostructures In situ manipulation of van der Waals heterostructures for twistronics New generation of moiré superlattices in doubly aligned hBN/graphene/hBN heterostructures Tunable crystal symmetry in graphene–boron nitride heterostructures with coexisting moiré superlattices Double moiré with a twist: Supermoiré in encapsulated graphene Enhanced electron-phonon coupling in doubly aligned hexagonal boron nitride bilayer graphene heterostructure Correlated states in doubly-aligned hBN/graphene/hBN heterostructures Controlled alignment of supermoiré lattice in double-aligned graphene heterostructures Higher order gaps in the renormalized band structure of doubly aligned hBN/bilayer graphene moiré superlattice Topological Flat bands in graphene super-moiré lattices Interferences of electrostatic moiré potentials and bichromatic superlattices of electrons and excitons in transition metal dichalcogenides Raman fingerprint of aligned graphene/h-BN superlattices Transport through a network of topological channels in twisted bilayer graphene Giant oscillations in a triangular network of one-dimensional states in marginally twisted graphene Engineering band structures of two-dimensional materials with remote moiré ferroelectricity Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice Landau levels in twisted bilayer graphene and semiclassical orbits Highly Tunable Moiré Superlattice Potentials in Twisted Hexagonal Boron Nitrides Mapping the twist-angle disorder and Landau levels in magic-angle graphene Commensurate and incommensurate double moiré interference in twisted trilayer graphene Modeling mechanical relaxation in incommensurate trilayer van der Waals heterostructures Correlated insulating states and transport signature of superconductivity in twisted trilayer graphene superlattices Topological bands and correlated states in helical trilayer graphene Twisted trilayer graphene: A precisely tunable platform for correlated electrons Ab initio theory of moiré superlattice bands in layered two-dimensional materials Deep quantum-dot arrays in moiré superlattices of non-van der Waals materials Moiré excitons: From programmable quantum emitter arrays to spin-orbit–coupled artificial lattices Classification of topological phase transitions and van Hove singularity steering mechanism in graphene superlattices Dirac edges of fractal magnetic minibands in graphene with hexagonal moiré superlattices One-dimensional electrical contact to a two-dimensional material Optical second harmonic generation at interfaces Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation Programming twist angle and strain profiles in 2D materials Imaging graphene moiré superlattices via scanning Kelvin probe microscopy Download references Device fabrication was conducted at the MCPF and WMINST of HKUST with great technical support from Mr These authors contributed equally: Yaqi Ma Department of Physics and Center for Quantum Materials The Hong Kong University of Science and Technology Department of Physics and HK Institute of Quantum Science & Technology ShanghaiTech Laboratory for Topological Physics fabricated the devices and performed the transport measurements under the instruction of M performed theoretical computations under the supervision of Z conducted the STM measurements under the supervision of S conducted the SHG measurements under the supervision of X analyzed the data and wrote the manuscript provided technical support in the device fabrication process finalized the manuscript with contributions from all authors Nature Communications thanks Yashika Kapoor and the other reviewers for their contribution to the peer review of this work Download citation DOI: https://doi.org/10.1038/s41467-025-57111-2 Metrics details Moiré superlattices in two-dimensional stacks have attracted worldwide interest due to their unique electronic properties A typical example is the moiré ferroelectricity where adjacent moirés exhibit opposite spontaneous polarization that can be switched through interlayer sliding in contrast to ideal regular ferroelectric moiré domains (equilateral triangles) built in most theoretical models the unavoidable irregular moiré supercells (non-equilateral triangles) induced by external strain fields during the transfer process have been given less attention Manipulation of controllable polarization evolutions is also a big challenge due to an interlinked network of polarized domains we employ a sliding-disturb measurement to examine and modulate these irregular moirés via mechanical force and three distinct types of moiré domains with different patterns are identified and modulated by external mechanical force disturbing They exhibit reduced pinning forces when the shear direction is not aligned with the strain direction The shift of the moirés is observed to be orthogonal to the shear direction This work offers an effective pathway for the controlled switch of the polarization in interfacial ferroelectricity that substantial experimental evidence remains largely desired showing that potential corrugation decreases with increasing strain when the sliding direction is misaligned with the strain direction the evolution of domain walls can be clearly visualized and demonstrated A Schematic of a t-BN sample produced through a tear-and-stack method with the aid of a curved PDMS C EFM phase image of the moiré pattern extracted from the black square in (B) D EFM image of the enlarged region zoomed from the region 4 in (C) E Line profile of the EFM image labeled by the black dashed line in (D) To discern the supercell topologies within the moiré pattern one can count the number of nodes present along the boundary surrounding a particular supercell Some representative nodes are indicated by dashed triangles and circles with the direction of force and edges marked by blue and black arrows When the disturbing mechanical force is applied those approximately equilateral triangular moirés remain largely unchanged (light gray masks) accompanied with slight migration perpendicular to the strain direction The distorted and stretched triangular moirés are partially pinned by the grains or particles in the sample They shift and merge vertically to the direction of strain The irregular long strips are formed under extreme strain and shift forward and backward accompanied with annihilation and reconstruction under the function of continuous mechanical force forming a larger hexagon and a new local pinning center Those unique moirés form due to large strain at the expense of collapsing triangular moirés Clear shifts along the direction of the long sides of these strips are observed and the distance between the adjacent parallel lines also experiences dynamic evolutions The labeled blue dashes are the boundary position of the previous process blue dashes in III is the boundary position in II labeled by black dashes and the yellow and black arrows highlight the shifts and deformations It is not difficult to find that the number of boundaries fluctuates during the sliding-disturb measurements which might be due to the competition between lattice mismatch and commensurate condition Moiré patterns can be elongated along the stretch direction while the polarization move vertical to the disturbing force direction we find an effective pathway to controllably manipulate the t-BN moiré patterns at the nanoscale three types of moiré patterns/edges with similar evolution dynamics are observed by PFM: (I) almost unchanged patterns due to large pinning force (III) collapsing moirés and formed parallel edges Theoretical calculations of the moiré pattern evolution under strain confirm the experimental results regarding the reduction of the pinning force and the shift directions of the moiré patterns shed light on the regulation of polarization switching through interlayer sliding moiré polarization could be switched by applying a disturbed mechanical force which offers a promising platform for modulating the moiré ferroelectricity of sophisticated twisted van der Waals structures The curved Polydimethylsiloxane (PDMS) is fabricated onto a glass with specified dimensions A Poly (bisphenol A carbonate) (PC) film is then affixed to the curved PDMS a 100 nm thick evaporated Au thin film is transferred onto the substrate at 90 °C to enhance the adhesion of the PC film resulting in the glass/PDMS/PC/Au substrate Graphite with more than a hundred nanometers is transferred onto the substrate as the bottom electrode followed by the sequential placement of hexagonal boron nitride (h-BN) flakes These h-BN flakes are exfoliated on SiO2 (300 nm)/Si substrates and cut using a sharp nanoscale tip with a high scanning frequency (~50 Hz) The target region measures approximately 100 µm ∗ 50 µm with a thickness of ~2 nm verified by atomic force microscopy (AFM) The substrate with the bottom electrode is used to pick up the two parts of h-BN in parallel using the tear-and-rotate technique It’s important to note that tiny rotation angles are typically unavoidable during the transfer process even with efforts to minimize them to nearly 0° Dry transfer techniques are employed throughout the fabrication process Slow transfer rates and in-situ heating (90 °C for 15 min) help mitigate the formation of bubbles and wrinkles Although some contaminants remain on the sample they serve as pinning centers during sliding-disturb measurements adding intrigue to the dynamic evolution observed SPM measurements are carried out on commercial SPM system Cypher S and Park AFM NX10 containing the functions of AFM MicroMasch) are utilized during the entire measurement The electromechanical responses in PFM were detected by driving the tip at a vertical resonance frequency of ~340 kHz and an AC voltage of 0.8 V The surface potential was detected ~20 nm far away from the sample surface by conductive probes with +3 V positive charges 2D materials and van der Waals heterostructures Shaping and structuring 2D materials via kirigami and origami Structural superlubricity and ultralow friction across the length scales Switchable moiré potentials in ferroelectric WTe2/WSe2 superlattices Observation of Van Hove singularities in twisted graphene layers Single-layer behavior and its breakdown in twisted graphene layers Direct imaging of topological edge states at a bilayer graphene domain wall Binary compound bilayer and multilayer with vertical polarizations: two-dimensional ferroelectrics Sliding ferroelectricity in 2D van der Waals materials: related physics and future opportunities Ferroelectricity in untwisted heterobilayers of transition metal dichalcogenides Spatially resolved polarization manipulation of ferroelectricity in twisted hBN Twistable electronics with dynamically rotatable heterostructures Recent progress in two‐dimensional ferroelectric materials Advance in two-dimensional twisted moiré materials: fabrication Mixed-stacking few-layer graphene as an elemental weak ferroelectric material Spontaneous electric polarization in graphene polytypes Molino, L. et al. Ferroelectric switching at symmetry‐broken interfaces by local control of dislocations networks. Adv. Mater. 2207816 https://doi.org/10.1002/adma.202207816 (2023) Symmetry breaking and anomalous conductivity in a double-moiré superlattice Global control of stacking-order phase transition by doping and electric field in few-layer graphene Nanoscale lattice dynamics in hexagonal boron nitride moiré superlattices Interfacial ferroelectricity in marginally twisted 2D semiconductors Domino-like stacking order switching in twisted monolayer–multilayer graphene Strain solitons and topological defects in bilayer graphene Tunnel junctions based on interfacial two dimensional ferroelectrics Strain-induced quasi-1D channels in twisted moiré lattices Zhong, N. et al. Edge polarization topology integrated with sliding ferroelectricity in moiré system. https://doi.org/10.21203/rs.3.rs-5377031/v1 (2024) Engineering interfacial polarization switching in van der Waals multilayers Interlayer registry dictates interfacial 2D material ferroelectricity Robust superlubricity in graphene/ h -BN heterojunctions Interlayer commensurability and superlubricity in rigid layered materials Stacking and registry effects in layered materials: the case of hexagonal boron nitride Colloquium: sliding and pinning in structurally lubric 2D material interfaces Shear strain-induced two-dimensional slip avalanches in rhombohedral MoS2 Fast parallel algorithms for short-range molecular dynamics Characterization of thermal transport in low-dimensional boron nitride nanostructures Interlayer potential for homogeneous graphene and hexagonal boron nitride systems: reparametrization for many-body dispersion effects Inter-layer potential for hexagonal boron nitride Assessment and optimization of the fast inertial relaxation engine (fire) for energy minimization in atomistic simulations and its implementation in lammps Methods of conjugate gradients for solving linear systems Download references National Key Research and Development Program of China (Grant No National Natural Science Foundation of China (Grant Nos Chenguang Program of Shanghai Education Development Foundation and Shanghai Municipal Education Commission ECNU (East China Normal University) Multifunctional Platform for Innovation (006) Fundamental Research Funds for the Central Universities thanks the support from National Natural Science Foundation of China (No Natural Science Foundation of Jiangsu Province (No and the Jiangsu Specially Appointed Professors Program We are grateful to the High Performance Computing Center of Nanjing Tech University for supporting the computational resources thanks the support by Postgraduate Research & Practice Innovation program of Jiangsu province KYCX22_0228 These authors contributed equally: Zhao Guan Key Laboratory of Polar Materials and Devices (Ministry of Education) Shanghai Center of Brain-Inspired Intelligent Materials and Devices State Key Laboratory of Materials-Oriented Chemical Engineering and carried out the SPM measurements including PFM Nature Communications thanks Jiong Zhao and the other anonymous reviewer(s) for their contribution to the peer review of this work Download citation DOI: https://doi.org/10.1038/s41467-025-56073-9 Metrics details we observed displacement field-induced quantum phase transitions from the EQAH states to the Fermi liquid FQAH liquid and the likely composite Fermi liquid Our observations established a new topological phase of electrons with quantized Hall resistance at zero magnetic field and enriched the emergent quantum phenomena in materials with topological flat bands Prices may be subject to local taxes which are calculated during checkout The data shown in Figs. 1–4 are available from https://doi.org/10.7910/DVN/YPO5VS Other data that support the findings of this study are available from the corresponding authors upon reasonable request Fractional quantum anomalous Hall effect in multilayer graphene Theory of quantum anomalous Hall phases in pentalayer rhombohedral graphene moiré structures Fractional quantum anomalous Hall effects in rhombohedral multilayer graphene in the moiréless limit and in Coulomb imprinted superlattice Anomalous Hall crystals in rhombohedral multilayer graphene I: Interaction-driven Chern bands and fractional quantum Hall states at zero magnetic field Theory of fractional Chern insulator states in pentalayer graphene moiré superlattice Kwan, Y. 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Preprint at https://arxiv.org/abs/2402.05456 (2024) Evidence for a liquid-to-crystal phase transition in a classical Observation of a magnetically induced Wigner solid Quantum liquid versus electron solid around v=1/5 Landau-level filling Observation of a reentrant insulating phase near the 1/3 fractional quantum Hall liquid in a two-dimensional hole system Direct observation of a magnetic-field-induced Wigner crystal Wigner crystallization in (DI − DCNQI)2 Ag detected by synchrotron radiation X-ray diffraction Observation of narrow-band noise accompanying the breakdown of insulating states in high Landau levels Evidence for two different solid phases of two-dimensional electrons in high magnetic fields Moving crystal phases of a quantum Wigner solid in an ultra-high-quality 2D electron system Imaging stacking order in few-layer graphene Enhanced optical conductivity induced by surface states in ABC-stacked few-layer graphene Large quantum anomalous Hall effect in spin-orbit proximitized rhombohedral graphene Orbital multiferroicity in pentalayer rhombohedral graphene Correlated insulator and Chern insulators in pentalayer rhombohedral-stacked graphene Spontaneous broken-symmetry insulator and metals in tetralayer rhombohedral graphene Xie, J. et al. 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Preprint at https://arxiv.org/abs/2405.16944 (2024) Stacking-dependent band gap and quantum transport in trilayer graphene Electronic phase separation in multilayer rhombohedral graphite Topological valley transport at bilayer graphene domain walls Observation of a Chern insulator in crystalline ABCA-tetralayer graphene with spin-orbit coupling Band structure of ABC-stacked graphene trilayers Gate-induced interlayer asymmetry in ABA-stacked trilayer graphene Transport studies of dual-gated ABC and ABA trilayer graphene: band gap opening and band structure tuning in very large perpendicular electric fields The experimental observation of quantum Hall effect of l = 3 chiral quasiparticles in trilayer graphene Competition between spontaneous symmetry breaking and single-particle gaps in trilayer graphene Large tunable intrinsic gap in rhombohedral-stacked tetralayer graphene at half filling Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice Layer-polarized ferromagnetism in rhombohedral multilayer graphene Ding, J. et al. Electrical switching of chirality in rhombohedral graphene Chern insulators. Preprint at https://arxiv.org/abs/2406.14289 (2024) Half- and quarter-metals in rhombohedral trilayer graphene Signatures of tunable superconductivity in a trilayer graphene moiré superlattice Superconductivity in rhombohedral trilayer graphene Download references We acknowledge the helpful discussions with T acknowledges support from a Sloan Fellowship acknowledges support from a Mathworks Fellowship This study was funded by the US Department of Energy Materials Sciences and Engineering Division under contract no Device fabrication was performed at the Harvard Center for Nanoscale Systems and MIT.nano acknowledge support from the JSPS KAKENHI (grant nos 21H05233 and 23H02052) and the World Premier International Research Center Initiative These authors contributed equally: Zhengguang Lu Research Center for Electronic and Optical Materials Research Center for Materials Nanoarchitectonics helped in installing and testing the dilution refrigerator All authors discussed the results and wrote the paper Optical image of an exfoliated graphene flake on SiO2/Si substrate The majority of the flake (outlined by dashed lines) is tetra-layer No contrast can be seen within the tetra-layer region Domains with different contrasts can be seen which correspond to different stacking orders The rhombohedral stacking domain (ABCA) has a contrast between the Bernal stacked domain (ABAB) and an intermediate stacked domain (ABCB) These stacking orders have been confirmed by Raman scattering measurement We cut 6 rectangles out of the ABCA domain using a laser The InGaAs camera image of 3 (out of 6 ABCA) flakes that are picked up The two flakes on the left have partially changed to other stacking orders while the right-most flake remains in ABCA stacking The InGaAs camera image of the three flakes in c after they are dropped down to hBN and bottom graphite gate The right-most flake in c remained in the ABCA stacking It has a clear contrast with the middle flake in c The ABCA flake in the dashed box was made into Device 3 InGaAs camera image and near-field infrared nanoscopy image of an exfoliated multilayer graphene flake on SiO2/Si substrate The latter reveals more domains (labeled as 1-4) with clear contrast than the former does Near-field infrared nanoscopy images of two exfoliated trilayer graphene flakes Both images show domain and domain walls that have dimensions much smaller than 1 μm which is well-below the diffraction-limit of the far-field imaging based on the InGaAs camera Temperature dependence of Rxx in a graphene superconducting state The density range of the superconducting dome remains expanding below 40 mK Temperature dependence of Rxx at ne = 1.9 × 1012 cm−2 Schematics of the gates and contacts layout in device 1 and 2 Top and bottom graphite gate shifted relative to each other creating a n-p-n junction on one side of the contacts Gates and contacts layout in device 3 with optimized geometry No p-n junction will be formed on either side of the device Schematics of the layer polarization of the low energy electrons in the conduction band Rxx versus the filling factor v and displacement field D/ε0 at the base temperature Rxx versus top and bottom gate voltages at the base temperature Rxx and Rxy versus DC bias at v = 1/2 & D/ε0 = 0.951 V/nm (a&b) and v = 3/5 & D/ε0 = 0.950 V/nm (e&f) the transition can be induced by increasing the temperature or the bias DC current Temperature evolution of the EQAH Region 1-3 Overlapping EQAH contour plots at selective temperatures to demonstrate the shrinking of the range as the temperature increases Temperature evolution of the Rxx and Rxy line cuts versus filling across v = 1 Temperature dependence of Rxx and Rxy at v = 1 and EQAH (v = 0.96) states Rxy versus displacement field at moiré filling factor 2/3 The EQAH state become FQAH as the temperature increases to about 340 mK Rxy versus displacement field at moiré filling factor 3/5 the FQAH state occupies the displacement field range from 0.920 V/nm to 0.930 V/nm and the EQAH state occupies from 0.945 V/nm to 0.960 V/nm EQAH in displacement field range 0.945 V/nm to 0.950 V/nm will lose the quantization at h/e2 and become quantized at 5 h/3e2 which indicates the recovery of FQAH at this temperature Rxy versus displacement field at moiré filling factor v = 4/7 and 5/9 FQAH states stays from the base temperature up to more than 400 mK in the displacement field range from 0.920 V/nm to 0.940 V/nm The EQAH states near 0.95 V/nm become the phase boundary between FQAH at lower D side and Fermi liquid at higher D side as the temperature increases Rxy versus displacement field at moiré filling factor v = 5/11 FQAH states persist from base temperature up to more than 400 mK in a range of displacement field The displacement field range will shrink as the phase boundary between FQAH and Fermi liquid broadened by temperature a&b. Mapping of Rxx and Rxy in a large range of v and D at a mixing chamber temperature of 10 mK. The main features are similar to that of Device 1 shown in Fig. 2 Three regions show quantized Rxy at h/e2and vanishing Rxx These three regions are almost connected into one big region with Rxy = h/e2 that swamps the FQAH states at v > 1/2 Line-cut of Rxy at D/\({{\rm{\varepsilon }}}_{0}\) = 0.96 V/nm and varied bias current featuring the phase transition from CFL to EQAH driven by displacement field Rxx and Rxy along the dashed lines in a&b showing plateaus and dips of FQAHE at fractional fillings The Rxx at fractional fillings and the neighborhood of v = ½ are several times smaller than in Device 1 Mapping of Rxx and Rxy in a large range of v and D at IDC = 0 nA and IAC = 0.1 nA which is beyond the break-down threshold current of EQAH states Landau fan corresponding to the line-cut in c where the dashed lines are derived from the Streda’s formula for states with Rxy = 5 h/3e2 The dispersions of dips in Rxx agree well with the dashed lines as expected for fractional and integer Chern insulators at v = 3/5 the EQAH region extends significantly beyond v = 1 and reaches v = 1.3 Mapping of Rxy in a large range of v and D at IDC = 0 A and IAC = 0.2 nA Five states at ‘stars’ positions show quantized Rxy at h/e2 and vanishing Rxx Rxx and Rxy as a function of IDC at the light and dark blue ‘star’ positions corresponding to v = 1 in a showing the break-down behavior of the C = 1 Chern insulator at large current (>50 nA) Rxx and Rxy as a function of IDC at the green showing the break-down behavior of the EQAH states at small current (<1 nA) in contrast to the large break-down current of the C = 1 Chern insulator in b&c Inset: zoom-in of curves at around IDC = 0 nA which reveal the threshold of EQAH break-down at ~ 1 nA Note: Spike-like rapid changes in b-f are artifacts from voltage source meter when switching the output range a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law Download citation DOI: https://doi.org/10.1038/s41586-024-08470-1 Metrics details The stacking order and twist angle provide abundant opportunities for engineering band structures of two-dimensional materials The inversion symmetry breaking in rhombohedral-stacked transitional metal dichalcogenides endows them with an interfacial ferroelectricity associated with an out-of-plane electric polarization By utilizing twist angle as a knob to construct rhombohedral-stacked transitional metal dichalcogenides antiferroelectric domain networks with alternating out-of-plane polarization can be generated we demonstrate that such spatially periodic ferroelectric polarizations in parallel-stacked twisted WSe2 can imprint their moiré potential onto a remote bilayer graphene This remote moiré potential gives rise to pronounced satellite resistance peaks besides the charge-neutrality point in graphene which are tunable by the twist angle of WSe2 Our observations of ferroelectric hysteresis at finite displacement fields suggest the moiré is delivered by a long-range electrostatic potential The constructed superlattices by moiré ferroelectricity represent a highly flexible approach as they involve the separation of the moiré construction layer from the electronic transport layer This remote moiré is identified as a weak potential and can coexist with conventional moiré Our results offer a comprehensive strategy for engineering band structures and properties of two-dimensional materials by utilizing moiré ferroelectricity current widely used technologies for constructing moiré superlattices require the constituent materials to function as both moiré construction layers and electronic transport layers thus limiting the range of applicable materials these moiré superlattices typically exhibit short-range interactions arising from interlayer hybridizations or atomic reconstructions The blue and red shadow areas mark the top and bottom bilayer WSe2 The red dashed box denotes the parallel-stacked interface b TEM images of reconstructed twisted bilayer WSe2 Three different stacking configurations are labeled Two kinds of rhombohedral-stacked (MX and XM) domains dominate in the system arising from lattice reconstructions c Schematic of the domain configurations in parallel-stacked TMDCs The vertical alignments of M and X atoms for MX and XM domains are shown The rhombohedral stackings endow MX and XM domains with out-of-plane polarizations d Vertical PFM image in a phase channel of a parallel-stacked twisted bilayer WSe2 e Illustration of the process of a ferroelectric moiré superlattice imprinting its potential onto a remote bilayer graphene This strategy separates the moiré construction layer from the electronic transport layer f Longitudinal resistance \({R}_{{xx}}\) as a function of carrier density \(n\) measured at \(T=1.6\) K for Device D1 (60°-0.61°) Two satellite resistance peaks symmetrically appear near CNP suggesting that the constructed moiré bands in bilayer graphene are tunable by the twist angle of WSe2 We observe that the moiré ferroelectric potential present in twisted WSe2 can be imprinted onto bilayer graphene manifesting as pronounced satellite resistance peaks that are tunable by the twist angle of WSe2 We chose double bilayer WSe2 as the building block due to its ability to demonstrate that remote moiré potential can be imprinted onto a target layer without direct contact a prominent moiré effect was observed even at a separation distance of 1.4 nm corresponding to the thickness of the bilayer WSe2 Two symmetric satellite resistance peaks with respect to the charge-neutrality point (CNP) can be observed in all samples The electrons in bilayer graphene experience a moiré electrostatic potential generated in the parallelly twisted double bilayer WSe2 via ferroelectric polarization resulting in the formation of a moiré miniband by zone folding The twist angle θ of WSe2 gives rise to a moiré pattern with a long wavelength λ described by λ = a/2sin(θ/2) where a is the in-plane lattice constant of WSe2 The carrier density at the full filling of a moiré band in graphene is \({n}_{{\mbox{s}}}=4/A\) where \(A=\sqrt{3}{\lambda }^{2}/2\) is the area of a moiré unit cell The pre-factor of 4 in \({n}_{{\mbox{s}}}\) arises from the four-fold degeneracy in the graphene band structure which includes two-fold spin and two-fold valley degeneracy a \({R}_{{xx}}\) as a function of \(n\) measured at \(D=0\) V nm−1 b Temperature dependent \({R}_{{xx}}\) as a function of \({V}_{{\mbox{t}}}\) at a fixed \({V}_{{\mbox{b}}}=0\) V c The color plot of the \(n-D\) map of \({R}_{{xx}}\) d The color plot of \({R}_{{xx}}\) as a function of \(n\) and \(B\) at a fixed \(D=0\) V nm−1 The black dashed lines mark the Landau levels with corresponding filling factors Figure 2c shows a color plot of the longitudinal resistance \({R}_{{xx}}\) as functions of n and D two vertical peaks appear at fixed \({n}_{{\mbox{s}}}=\pm 4.9\times {10}^{11}\) cm−2 although they gradually merged with the CNP peak at high D This is because the positions of satellite peaks are only determined by the periodicity of the moiré superlattice Although the D can alter the basis of a moiré unit cell the relative areas of the MX and XM domains the periodicity of the moiré superlattice remains unchanged from a long-range viewpoint the moiré effect in our structure exhibits weak interaction the remaining origin is the periodic electrostatic potential arising from alternating out-of-plane polarization in twisted WSe2 This can be confirmed by examining the ferroelectric hysteresis in our samples a The enlarged plot of the \(n-D\) map of \({R}_{{xx}}\) near CNP b Temperature dependent \({R}_{{xx}}\) as a function of \({V}_{{\mbox{t}}}\) at a fixed \({V}_{{\mbox{b}}}=-0.6\) V c \({R}_{{xx}}\) as a function of \({V}_{{\mbox{b}}}\) by sweeping \({V}_{{\mbox{b}}}\) forward (black solid line) and backward (red dashed line) at a fixed \({V}_{{\mbox{t}}}=2\) V d \({R}_{{xx}}\) as a function of \({V}_{{\mbox{t}}}\) by sweeping \({V}_{{\mbox{t}}}\) forward (black solid line) and backward (red dashed line) at a fixed \({V}_{{\mbox{b}}}=-1.3\) V e The color plot of the difference in \({R}_{{xx}}\) between forward and backward sweeps of \({V}_{{\mbox{b}}}\) at each fixed \({V}_{{\mbox{t}}}\) \({V}_{{\mbox{b}}}\) is the fast-scan axis and \({V}_{{\mbox{t}}}\) is the slow-scan axis \({V}_{{\mbox{t}}}\)) is converted to \({R}_{{xx}}\)(n the two domains induced equal-density carriers of opposite types resulting in no change in total n originating from the polarization of twisted WSe2 and the CNP of graphene appears as a normal shape The application of finite D opens a gap at CNP in bilayer graphene resulting in the broadening of resistance peak Besides this typical feature of bilayer graphene the peak splitting at CNP under finite D is quite unusual The key difference between our samples and previously reported ones is that the twist angle in our Device D1 is much larger This allows us to achieve a more uniform moiré structure The small lattice mismatch between graphene and h-BN produces a C-moiré superlattice on the scale of around 10 nm where graphene functions as both the moiré construction layer and the electronic transport layer a \({R}_{{xx}}\) as a function of n measured at \(D=0\) V nm−1 Two sets of satellite peaks symmetrically appear near CNP The color plot of the \(n-D\) map of \({R}_{{xx}}\) showing two groups of vertical lines besides the CNP d The color plot of \({R}_{{xx}}\) (c) and \({R}_{{xy}}\) (d) as a function of n and B at a fixed \(D=0\) V nm−1 The red dashed boxes show the fine maps of Landau fan diagrams near ferroelectric moiré induced resistance peaks The right y axis shows the normalized magnetic flux \(\phi /{\phi }_{0}\) where the commensurate states (\(1/q\)) are marked Prominent Landau fans emerge from the CNP and \({n}_{{\mbox{s}}}^{{\prime} }\) Their intersections form Hofstadter’s fractal structures and Brown-Zak oscillations From the period in 1/B of the Brown-Zak oscillation we can alternatively calculate the twist angle between graphene and h-BN according to \(\frac{\phi }{{\phi }_{0}}=1/q\) where \(\phi={BA}\) is the magnetic flux per moiré unit cell which agrees well with the one calculated from \({n}_{{\mbox{s}}}^{{\prime} }\) F-moiré exhibits a weaker magnetic response manifesting as positive magnetoresistance accompanied by faint sign reversal near \({n}_{{\mbox{s}}}\) No prominent Brown-Zak oscillation associated with F-moiré is observed in this device Our strategy offers a promising approach to study correlated topological states with controllable moiré wavelengths in such systems The period of the moiré potential can be easily tuned by adjusting the twist angle of the remote TMDCs while its strength can be controlled by selecting different thicknesses of TMDCs as the building blocks This allows the distance between the target layer and the moiré construction layer to be precisely adjusted graphene and h-BN flakes were mechanically exfoliated from bulk crystals onto SiO2 (285 nm)/Si substrates The layer numbers of graphene and WSe2 were initially identified from their optical contrasts and later confirmed by Raman spectroscopy To assemble the van der Waals heterostructures Bilayer WSe2 flakes were cut into two parts using a sharp tungsten tip The heterostructures were assembled in the sequence of top h-BN/bilayer graphene/twisted double bilayer WSe2/bottom h-BN/bottom graphite using a standard dry transfer technique at temperatures between 90 °C and 120 °C with the assistance of a poly(bisphenol A carbonate) (PC)/polydimethylsiloxane (PDMS) stamp we carefully picked up the first part of the pre-cut flake then intentionally rotated the remaining part on the stage to achieve a target twist angle of 0.5° − 2° and subsequently picked up the second part of the flake and bubble-free regions were selected as the channel area of the devices to prevent inhomogeneity and strain One-dimensional electrical contacts to the graphene were achieved by dry etching with CHF3/O2 plasma and the deposition of Cr/Au (3/60 nm) The top gate was defined by electron-beam lithography (EBL) followed by the deposition of Cr/Au (3/50 nm) The final Hall bar was shaped through additional steps of EBL and etching process The samples for PFM characterizations were prepared using a similar process to that for transport devices Few-layer graphite and twisted bilayer WSe2 flakes were subsequently picked up using a PC/PDMS stamp The PC film with the stack was exfoliated from the PDMS and flip onto fresh PDMS The PC film was then dissolved in N-Methyl-2-pyrrolidone (NMP) solution the stack was transferred onto another substrate The twisted WSe2 flakes prepared by this method were clean enough for PFM characterization The vertical PFM measurements were conducted with the Asylum Research Cypher ES AFM at room temperature The measured contact resonance frequency was around 340 kHz we used dry-transfer to assemble a h-BN/twisted bilayer WSe2/h-BN structures Two thin h-BN ( ~ 5 nm) flakes were used to protect ultrathin WSe2 flakes and reduce the strain induced during the procedure The whole stack was transferred onto a 50 nm thick SiN membrane supported TEM grid The stack was suspended on the hole of SiN membrane Spectra Ultra STEM with acceleration voltage of 200 kV were used for TEM imaging Annular dark-field detector was used to image the sample To unveil the non-centrosymmetric structure of parallelly twisted WSe2 we resorted to the second-harmonic generations We intentionally prepared a twisted bilayer WSe2 (with twist angle ~1°) attached by an intrinsic bilayer WSe2 (2H phase) which allows us to in-situ compare their SHG signals Witec Alpha 300RAS with UHTS 300 spectrometer (grating with 300 lines/mm blazed at 500 nm) was used to perform SHG The sample was excited by a 1064 nm ultrafast fiber laser (Rainbow 1064 Polarization-dependent SHG measurements were conducted by a motorized achromatic half-wave plate inserted between beam splitter and the objective lens The emitted signal from the sample was analyzed by a linear polarizer before entering the spectrometer We carried out the transport measurements in a cryostat (Oxford TeslatronPT) at a base temperature of 1.6 K Standard low-frequency lock-in techniques were used to acquire the data The temperature-dependent measurements were performed using the VTI temperature controller We identified the twist angle of WSe2 from the corresponding carrier density at which pronounced resistance peaks symmetrically appearing at both sides of graphene’s CNP We assigned them as the full filling carrier density \({n}_{{\mbox{s}}}\left(\nu=\pm 4\right)=4/A\) by considering 2-fold spin and 2-fold valley degeneracy in graphene where \(A\) is the unit cell area of moiré superlattices The moiré superlattice is created from parallelly twisting WSe2 with an angle of \(\theta\) giving \(A=\sqrt{3}{a}^{2}/8{\sin }^{2}(\theta /2)\) where \(a=0.3297\) nm is the lattice constant of WSe2 The dual-gate configuration allows us to independently tune the carrier density \(n\) and displacement field \(D\) through \(n=({C}_{{\mbox{b}}}{V}_{{\mbox{b}}}+{C}_{{\mbox{t}}}{V}_{{\mbox{t}}})/e\) and \(D=({C}_{{\mbox{b}}}{V}_{{\mbox{b}}}-{C}_{{\mbox{t}}}{V}_{{\mbox{t}}})/2{\varepsilon }_{0}\) where \({C}_{{\mbox{b}}}\) (\({C}_{{\mbox{t}}}\)) is the back (top) gate capacitance per unit area \({V}_{{\mbox{b}}}\) (\({V}_{{\mbox{t}}}\)) is the back (top) gate voltage \(e\) is the elementary charge and \({\varepsilon }_{0}\) is the vacuum permittivity \({C}_{{\mbox{b}}}\) and \({C}_{{\mbox{t}}}\) were extracted through Hall density measurement at \(\pm 1\) T with anti-symmetrized treatment to remove longitudinal components Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article Relevant data supporting the findings of this study are available within the article and the Supplementary Information file All raw data are available from the corresponding authors upon request Boron nitride substrates for high-quality graphene electronics Hofstadter's butterfly and the fractal quantum hall effect in moiré superlattices Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene Unconventional ferroelectricity in moiré heterostructures Fractional quantum anomalous hall effect in multilayer graphene Intrinsic quantized anomalous hall effect in a moiré heterostructure and layer polarization in marginally twisted MoS2 bilayers Room-temperature ferroelectricity in 1T′-ReS2 multilayers On electrically tunable stacking domains and ferroelectricity in moiré superlattices Theory of polar domains in moiré heterostructures Engineering correlated insulators in bilayer graphene with a remote Coulomb superlattice Visualizing moiré ferroelectricity via plasmons and nano-photocurrent in graphene/twisted-WSe2 structures Transport evidence of superlattice Dirac cones in graphene monolayer on twisted boron nitride substrate Imaging two-dimensional generalized Wigner crystals Band alignment in WSe2–graphene heterostructures Spin-orbit-driven band inversion in bilayer graphene by the van der Waals proximity effect Atomic reconstruction in twisted bilayers of transition metal dichalcogenides Topological and stacked flat bands in bilayer graphene with a superlattice potential Gate-tunable topological phases in superlattice modulated bilayer graphene Commensurate–incommensurate transition in graphene on hexagonal boron nitride Control of electron-electron interaction in graphene by proximity screening Zhou, B., Yang, H. & Zhang, Y.-H. Fractional quantum anomalous Hall effects in rhombohedral multilayer graphene in the moiréless limit and in Coulomb imprinted superlattice. arXiv https://doi.org/10.48550/arXiv.2311.04217 (2023) Dong, Z., Patri, A. S. & Senthil, T. Theory of fractional quantum anomalous hall phases in pentalayer rhombohedral graphene moiré structures. arXiv https://doi.org/10.48550/arXiv.2311.03445 (2023) Dong, J. et al. Anomalous hall crystals in rhombohedral multilayer graphene I: interaction-driven chern bands and fractional quantum hall states at zero magnetic field. arXiv https://doi.org/10.48550/arXiv.2311.05568 (2023) Download references This work was funded by National Natural Science Foundation of China (Grant No the Zhejiang Provincial Natural Science Foundation of China (Grant No and Westlake Education Foundation at Westlake University We thank Chao Zhang and Qike Jiang from the Instrumentation and Service Center for Physical Sciences (ISCPS) at Westlake University for technical support in data acquisition We also thank Westlake Center for Micro/Nano Fabrication and the Instrumentation and Service Centers for Molecular Science for facility support acknowledge support from the JSPS KAKENHI (Grant Numbers 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI) Key Laboratory for Quantum Materials of Zhejiang Province conceived the idea and supervised the project fabricated the devices with the assistance of H.X. performed the transport measurements with the assistance of W.Z prepared TEM sample and performed TEM characterization All the authors contributed to the discussions Nikhil Tilak and the other anonymous reviewer(s) for their contribution to the peer review of this work Download citation DOI: https://doi.org/10.1038/s41467-024-53440-w Metrics details An Author Correction to this article was published on 17 February 2025 This article has been updated A moiré potential—the superposition of two periodic potentials with different wavelengths—will either introduce a new periodicity into a system if the two potentials are commensurate or force the system to be quasiperiodic if they are not Here we demonstrate that quasiperiodicity can change the ground-state properties of one-dimensional moiré systems with respect to their periodic counterparts We show that although narrow bands play a role in enhancing interactions for both commensurate and incommensurate structures only quasiperiodicity is able to extend the ordered phase down to an infinitesimal interaction strength the state enabled by quasiperiodicity has contributions from electronic states with a very large number of wavevectors This quasi-fractal regime cannot be stabilized in the commensurate case even in the presence of a narrow band These findings suggest that quasiperiodicity may be a critical factor in stabilizing non-trivial ordered phases in interacting moiré structures and highlight that multifractal non-interacting phases might be particularly promising parent states The code used in this work is available via GitHub at https://github.com/gmiguel17/Phd_codes/tree/main/Incommensurability_enabled_quasi-fractal_order_in_1D_narrow-band_moire_systems A Correction to this paper has been published: https://doi.org/10.1038/s41567-025-02824-w Analyticity breaking and Anderson localization in incommensurate lattices Anderson localization of a non-interacting 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Preprint at https://arxiv.org/abs/2212.08024 (2022) Exact new mobility edges between critical and localized states Predicted critical state based on invariance of the Lyapunov exponent in dual spaces Many-body localization in a quasiperiodic system Many-body localization and thermalization in disordered Hubbard chains Interaction instability of localization in quasiperiodic systems Butterfly effect in interacting Aubry-Andre model: thermalization Many-body delocalization dynamics in long Aubry-André quasiperiodic chains Fermionic many-body localization for random and quasiperiodic systems in the presence of short- and long-range interactions Finite-size scaling analysis of the many-body localization transition in quasiperiodic spin chains Many-body critical phase: extended and nonthermal Enhanced compressibility due to repulsive interaction in the Harper model Detecting a many-body mobility edge with quantum quenches Critical properties of the ground-state localization-delocalization transition in the many-particle Aubry-André model Moiré versus Mott: incommensuration and interaction in one-dimensional bichromatic lattices Oliveira, R., Gonçalves, M., Ribeiro, P., Castro, E. V. & Amorim, B. Incommensurability-induced enhancement of superconductivity in one dimensional critical systems. Preprint at https://arxiv.org/abs/2303.17656 (2023) Short-range interactions are irrelevant at the quasiperiodicity-driven Luttinger liquid to Anderson glass transition Superconductivity and strong interactions in a tunable moiré quasicrystal One dimensional bosons: from condensed matter systems to ultracold gases Phase diagram of the half-filled one-dimensional t-v-\({V}^{{\prime} }\) model Fidelity approach to quantum phase transitions Fidelity at Berezinskii-Kosterlitz-Thouless quantum phase transitions Majorana fermions in superconducting 1D systems having periodic Magic-angle semimetals with chiral symmetry Fishman, M., White, S. R. & Stoudenmire, E. M. Codebase release 0.3 for ITensor. SciPost Phys. Codebases https://doi.org/10.21468/SciPostPhysCodeb.4-r0.3 (2022) Fishman, M., White, S. R. & Stoudenmire, E. M. The ITensor software library for tensor network calculations. SciPost Phys. Codebases https://doi.org/10.21468/SciPostPhysCodeb.4 (2022) Gonçalves, M. et al. Data of figures in arXiv:2305.03800. Zenodo https://doi.org/10.5281/zenodo.8082294 (2023) Download references acknowledge partial support from Fundação para a Ciência e Tecnologia (FCT-Portugal; Grant No UID/CTM/04540/2019 and to the Research Unit UID/04540: CeFEMA financed by FCT-Portugal) acknowledge partial support from FCT-Portugal (Grant No UID/04650 - Centro de Física das Universidades do Minho e do Porto) acknowledges further support from FCT-Portugal (Grant No We also acknowledge the Tianhe-2JK cluster at the Beijing Computational Science Research Center the Bob∣Macc supercomputer (Computational Project CPCA/A1/470243/2021) and the OBLIVION supercomputer (Projects HPCUE/A1/468700/2021 2022.15834.CPCA.A1 and 2022.15910.CPCA.A1) The OBLIVION supercomputer is at the High Performance Computing Center and is funded by the ENGAGE SKA Research Infrastructure (Reference POCI-01-0145-FEDER-022217 - COMPETE 2020) the BigData@UE project (Reference ALT20-03-0246-FEDER-000033 - FEDER) and the Alentejo 2020 Regional Operational Program Computer assistance was provided by the Computational Science Research Center Miguel Gonçalves, Flavio Riche & Pedro Ribeiro Centro de Física das Universidades do Minho e do Porto (CF-UM-UP) Laboratório de Física para Materiais e Tecnologias Emergentes (LaPMET) Centro de Física das Universidades do Minho e do Porto Beijing Computational Science Research Center Contains all the source data for Fig. 1 text files containing the raw data and a ‘READ_ME.md’ file containing a detailed description of all the text files Contains all the source data for Fig. 2 Contains all the source data for Fig. 3 Contains all the source data for Fig. 4 Download citation DOI: https://doi.org/10.1038/s41567-024-02662-2 Metrics details The unique physics in moiré superlattices of twisted or lattice-mismatched atomic layers holds great promise for future quantum technologies twisted configurations are thermodynamically unfavourable making accurate twist angle control during growth implausible lattice-mismatched moirés such as WSe2/WS2 can be synthesized they lack the critical moiré period tunability and their formation mechanisms are not well understood we report the thermodynamically driven van der Waals epitaxy of moirés with a tunable period from 10 to 45 nanometres using lattice mismatch engineering in two WSSe layers with adjustable chalcogen ratios where lattice-mismatch-induced stress hinders high-quality growth we reveal the key role of bulk stress in moiré formation and its unique interplay with edge stress in shaping the moiré growth modes the superlattices display tunable interlayer excitons and moiré intralayer excitons Our studies unveil the epitaxial science of moiré synthesis and lay the foundations for moiré-based technologies Source data are provided with this paper Other supporting data and very large files are available from the corresponding authors upon reasonable request The codes that support the findings of this study are available from the corresponding authors upon reasonable request Electronic effects in scanning tunneling microscopy: moiré pattern on a graphite surface Moiré bands in twisted double-layer graphene Van der Waals heterostructures with high accuracy rotational alignment Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures Observation of moiré excitons in WSe2/WS2 heterostructure superlattices Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers Evidence for moiré excitons in van der Waals heterostructures Moiré excitons: from programmable quantum emitter arrays to spin-orbit–coupled artificial lattices Moiré intralayer excitons in a MoSe2/MoS2 heterostructure Tuning superconductivity in twisted bilayer graphene Moiré heterostructures as a condensed-matter quantum simulator Intelligent infrared sensing enabled by tunable moiré quantum geometry Wafer-scale growth of single-crystal monolayer graphene on reusable hydrogen-terminated germanium Dirac electrons in a dodecagonal graphene quasicrystal Hetero-site nucleation for growing twisted bilayer graphene with a wide range of twist angles Designed growth of large bilayer graphene with arbitrary twist angles Epitaxial growth of single-domain graphene on hexagonal boron nitride Atomically thin resonant tunnel diodes built from synthetic van der Waals heterostructures Direct vapor phase growth and optoelectronic application of large band offset SnS2/MoS2 vertical bilayer heterostructures with high lattice mismatch Two-dimensional GaSe/MoSe2 misfit bilayer heterojunctions by van der Waals epitaxy Molecular beam epitaxy of highly crystalline monolayer molybdenum disulfide on hexagonal boron nitride Van der Waals epitaxial growth and optoelectronics of large-scale WSe2/SnS2 vertical bilayer p–n junctions Excitonic processes in atomically-thin MoSe2/MoS2 vertical heterostructures One-step synthesis of metal/semiconductor heterostructure NbS2/MoS2 Van der Waals epitaxial growth of atomically thin 2D metals on dangling-bond-free WSe2 and WS2 Vapor growth of WSe2/WS2 heterostructures with stacking dependent optical properties Direct vapor growth of 2D vertical heterostructures with tunable band alignments and interfacial charge transfer behaviors General synthesis of two-dimensional van der Waals heterostructure arrays Van der Waals epitaxial growth of air-stable CrSe2 nanosheets with thickness-tunable magnetic order Step-edge-guided nucleation and growth of aligned WSe2 on sapphire via a layer-over-layer growth mode Defect-controlled nucleation and orientation of WSe2 on hBN: a route to single-crystal epitaxial monolayers Wafer-scale epitaxial growth of unidirectional WS2 monolayers on sapphire Dual-coupling-guided epitaxial growth of wafer-scale single-crystal WS2 monolayer on vicinal a-plane sapphire Uniform nucleation and epitaxy of bilayer molybdenum disulfide on sapphire Edge stresses of non-stoichiometric edges in two-dimensional crystals Interpretable molecular models for molybdenum disulfide and insight into selective peptide recognition Dynamics of antimonene–graphene van der Waals growth Growth of epitaxial graphene: theory and experiment First-principles comparative study on the interlayer adhesion and shear strength of transition-metal dichalcogenides and graphene Growth-substrate induced performance degradation in chemically synthesized monolayer MoS2 field effect transistors Strain and structure heterogeneity in MoS2 atomic layers grown by chemical vapour deposition Substrate-induced strain and charge doping in CVD-grown monolayer MoS2 Approaching the intrinsic photoluminescence linewidth in transition metal dichalcogenide monolayers Ultrafast charge transfer in atomically thin MoS2/WS2 heterostructures Band offsets and heterostructures of two-dimensional semiconductors Composition-induced type I and direct bandgap transition metal dichalcogenides alloy vertical heterojunctions Synthesis of WS2xSe2–2x alloy nanosheets with composition-tunable electronic properties Electronic and magnetic characterization of epitaxial CrBr3 monolayers on a superconducting substrate Experimental realization of atomic monolayer Si9C15 Silicene: compelling experimental evidence for graphenelike two-dimensional silicon Halide-assisted atmospheric pressure growth of large WSe2 and WS2 monolayer crystals A universal etching-free transfer of MoS2 films for applications in photodetectors Prismatic 2.0 – simulation software for scanning and high resolution transmission electron microscopy (STEM and HRTEM) Nano-“squeegee” for the creation of clean 2D material interfaces Download references acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Fonds de Recherche du Québec (FRQNT) acknowledge the partial support from the Government of Israel and Yale University acknowledge support from Japan Society for the Promotion of Science KAKENHI (grant nos 19H05790 Matthieu Fortin-Deschênes & Fengnian Xia International Center for Materials Nanoarchitectonics conceived and carried out the experiments and simulations analysed the results and wrote the manuscript Plot of moiré lattice parameter versus ΔSe Download citation DOI: https://doi.org/10.1038/s41563-023-01596-z Metrics details A moiré is an interference pattern that appears when two different periodic structures are overlaid The image created is extremely sensitive to small variations in the original layers and is thus very interesting for anti-counterfeit protection We present a microfabricated 1D moiré enabling complex high-resolution patterns as a significantly improved security feature that cannot be reproduced using standard printing methods that a microscopic deviation from the original design results in a macroscopic variation in the moiré that is clearly visible to the naked eye The record resolution achieved in the elements fabricated and the increased design freedom make these high-resolution moirés excellent candidates for a variety of visually appealing security applications the size of 2D moiré design elements is severely limited since they must reside within a single period of the 2D pattern array it would be necessary to increase the pattern array period but this would drastically reduces the possibility of developing sophisticated design elements at high frequencies and a revealing layer placed over it to sample these graphical features Usually the base layer is created using printing techniques and the revealing layer is made by printing a line grating on a transparent film or by using an array of cylindrical microlenses Such printing techniques provide a minimal line width ranging between 10 and 20 µm This yields an effective offset printed resolution between 1270 and 2540 dpi this is not sufficient to prevent counterfeiting using desktop scanners and printers fabrication and characterization of visually attractive 1D moirés with resolutions up to 9754 dpi Our new mathematical model determines the geometrical transformations necessary to design esthetic 1D moirés with a circular layout The moiré elements move along radial or spiral trajectories when the base and the revealing layer are displaced relative to each other In order to achieve the high resolutions proposed using reliable both the base and the revealing layers were manufactured using scalable microfabrication techniques (a) Representation of a 1D moiré rectilinear base and revealing (lenticular array) layers The moiré image is revealed when both elements are superimposed (b) Schematic view of the moiré shapes (in cyan) generated by the superposition of a base layer with replicated base bands (‘VALID’ in green) and of transparent revealing layer lines (dashed red lines) (c) Example of a base band grating made of a replicated with the moiré revealed by a transparent line grating (revealing layer) As described in Supplementary Information 1 the height H of the corresponding moiré can be expressed as: where ty corresponds to the vertical component of the base band repetition vector (tx, ty). The moiré presented in Figure 1c has been designed with a Tr of 20 pixels and a band repetition vector of (−2 both the base band shapes and the moiré shapes have the same orientation When displacing the revealing layer upwards with respect to the base layer base band shapes and moiré shapes have inverse orientations Moving the revealing layer up has the effect of moving the moiré upwards This mathematical model is used here to design two moirés representing the letters ‘ABC’ and the numbers ‘123’ In order to observe these moirés with the naked eye their height has been set to H=5 mm the respective base layouts are reproduced at a print resolution of 9754 dpi The revealing layer period for both moirés is Tr=62.5 µm corresponding to a line grating of 160 lines cm−1 The base layer height is ty=63.3 µm for the ‘ABC’ moiré and ty=61.7 µm for the ‘123’ moiré In both cases the base layer period ty and the revealing layer period Tr are close one to another even a microscopic variation of the base or revealing layer periods results in a macroscopic modification of the height of the revealed moiré images The robustness of these high-resolution moirés against counterfeiting attempts is shown further below we first describe the new concept of curvilinear 1D moirés whose curvilinear layout follows the level lines of a desired continuous function These curvilinear moirés moving along radial or spiral trajectories offer new (a) a desired moiré in the original space with (b) its corresponding enlarged base layer section; (c) shows the same moiré in a circularly transformed space together with (d) its corresponding enlarged base layer section Moiré base layers with resolutions up to 9754 dpi (corresponding to a pixel size of 2.6 µm) were fabricated by means of an Au lift-off process14 using UV lithography This is a higher resolution than those achievable with standard printing When the channels are fully filled the SU-8 is cross-linked the poly-dimethylsiloxane stamp is removed mechanically and the cylindrical microlens array is ready to be used (d) An optical picture of a bent lenticular array fabricated over a flexible substrate and SEM images of the lenticular arrays fabricated from (e) a commercial master (f) a positive photoresist master fabricated in our laboratory and (g) a magnified image of the lenses shown in (f) Cylindrical microlens array contour shape Contour shape of a cylindrical microlens array in the orthogonal direction (perpendicular to the lenses) The inset shows the superimposed contour shape of five different lenses Optical images of the base layers: (a) rectilinear ‘ABC’ and ‘123’ moirés Below are corresponding moiré images when base and appropriate revealing layers are super-imposed Counterfeiting robustness of high-resolution moirés Top: optical images of the rectilinear 1D moiré for scaling factors ranging from 96% to 104% of the base layer period Bottom: graphical representation of both the calculated and experimental height (cm) of the ‘123’ moiré and of the inverted ‘ABC’ moiré as a function of the base layer scaling factor (%) This paper describes the first use of 1D moirés to create complex moving features with resolutions close to 10 000 dpi fabricated and characterized for use as esthetic security features New geometrical transformations were developed to create circularly laid out moiré images Microfabrication techniques were developed to fabricate the base layer in Au on glass substrates with resolutions of up to 9754 dpi The features obtained at such a resolution cannot easily be replicated with standard lithography or printing methods these base layers are robust against counterfeiting attempts relying on desktop scanners and printers were made by MIMIC using a transparent epoxy polymer over flexible substrates with frequencies up to 200 lenses cm−1 three different moirés showing different symbols and letters in one single layout were studied: (i) a rectilinear moiré with its elements distributed along parallel horizontal lines; (ii) a moiré with circularly laid out star symbols aligned along radial lines; and (iii) circularly laid out ‘EPFL’ moiré shapes that shift relative to each other according to their distance from the center point exhibiting new moiré displacement patterns along radial and spiral lines Such features are of great interest for applications requiring both visual attractiveness and counterfeit prevention elements Authentication can be performed thanks to structures easily recognizable to the naked eye the moiré images created are extremely sensitive to reproduction variations We specifically demonstrated that microscopic variations (around ±1 µm of the period) result in strong macroscopic changes in the reconstructed moiré images providing a new and easily identifiable component This ensures that replication of the 1D moiré elements is very difficult since copies must be perfect at a scale of 1∶1 down to the micrometer range such copies cannot be easily achieved since the moirés have been fabricated using microtechnologies at resolutions up to 9754 dpi These features confirm that the high-resolution moirés presented in this work are excellent candidates for providing both improved protection against counterfeits and additional visual attractiveness RDH and JB conceived the proposed high-resolution moirés based on micro-technology fabrication methods; SC and RDH designed the moirés elements and elaborated the mathematical model; VJC and KS fabricated the base layers; VJC fabricated the revealing layers and characterized the moirés elements; VJC wrote the manuscript text supported by RDH and JB All the authors reviewed and edited the manuscript Supplementary Information for this article can be found on Light: Science and Applications’ website (http://www.nature.com/lsa) Sui fenomeni che si producono colla sovrapposizione di due reticoli e sopra alcune loro applicazioni Theoretical Interpretation of Moiré Patterns Sharpening and multiplication of moiré fringes A generalized fourier-based method for the analysis of 2D moiré envelope-forms in screen superpositions Optical Security and Counterfeit Deterrence Techniques V Diffractive moiré features for optically variable devices Optical Security and Counterfeit Deterrence Techniques VI Polymer microlenses with modified micromolding in capillaries (MIMIC) technology IEEE Photon Technol Lett 2005; 17: 2628–2630 Surface profiles of reflow microlenses under the influence of surface tension and gravity negative-tone near-UV photoresist and its applications for MEMS Download references The authors are pleased to acknowledge the EPFL Center of MicroNano Technology (CMI) for their valuable discussions and help This work was partially funded by projects 200020-105119/1 and 200021_143501/1 of the Swiss National Science Foundation École Polytechnique Fédérale de Lausanne (EPFL) SI 1: Rectilinear 1D moiré mathematical model (DOC 26 kb) SI 2: Curvilinear 1D moiré mathematical model (DOC 283 kb) SI 3: Curvilinear 1D moiré moving along spiral trajectories (DOC 26 kb) This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 3.0 Unported License visit http://creativecommons.org/licenses/by-nc-sa/3.0/ Download citation periodic structures of slightly different sizes The resulting interference pattern shows clearly defined features that do not appear in either of the original structures Cadarso and co-workers at the Swiss Federal Institute of Technology in Lausanne (EPFL) have now used a variant of this approach to create micro-images at a resolution close to 10,000 dots per inch which are revealed as moiré patterns upon superposition with an array of cylindrical microlenses Such a high resolution means that the structures cannot be copied using standard desktop scanners and printers reproduction inaccuracies on the micrometre-scale change the moiré pattern in ways that are clearly visible to the naked eye make this technique attractive for producing secure anti-counterfeit features Metrics details An alloy engineering approach is developed to reliably grow atomically thin bilayers with predictable and tunable moiré patterns Fortin-Deschênes, M., Watanabe, K., Taniguchi, T. & Xia, F. Nat. Mater. https://doi.org/10.1038/s41563-023-01596-z (2023) Download references This work was supported by the US Department of Energy Materials Sciences and Engineering Division Reprints and permissions Download citation DOI: https://doi.org/10.1038/s41563-023-01618-w Metrics details Large-scale two-dimensional (2D) moiré superlattices are driving a revolution in designer quantum materials The electronic interactions in these superlattices strongly dependent on the periodicity and symmetry of the moiré pattern critically determine the emergent properties and phase diagrams the relative twist angle between two layers has been the primary tuning parameter for a given choice of constituent crystals we establish strain as a powerful mechanism to in situ modify the moiré periodicity and symmetry We develop an analytically exact mathematical description for the moiré lattice under arbitrary in-plane heterostrain acting on any bilayer structure We demonstrate the ability to fine-tune the moiré lattice near critical points such as the magic angle in bilayer graphene or fully reconfigure the moiré lattice symmetry beyond that imposed by the unstrained constituent crystals Due to this unprecedented simultaneous control over the strength of electronic interactions and lattice symmetry 2D heterostrain provides a powerful platform to engineer and probe strongly correlated moiré materials restricting the capability to fine-tune parameters near critical points and broadly explore phase diagrams This limitation constrains the usefulness of moiré materials as quantum simulators We thus show that strain control is a promising strategy to provide the critical degree of freedom required to realize reconfigurable quantum materials and achieve a fully tunable on-chip quantum simulator The most general moiré pattern is formed between two monoclinic lattices with Bravais lattice vectors ai for the lower layer and bi for the upper one, as sketched in Fig. 1a These primitive lattice vectors can be written as follows: where ai are the lattice constants of a general 2D monoclinic lattice and \({R}_{{\psi }_{{{{\rm{i}}}}}}\) is the 2D rotation matrix with an angle ψi a Sketch of the different TMD layers that compose the moiré lattice The lattice parameters of the first (second) layer are a1 (b1) and a2 (b2) the angle between those vectors is β for both layers and the relative angle between the lattices is θ b Moiré lattice parameters A1 and A2 and the internal angle α between those vectors in a bilayer formed by the hexagonal monolayer crystals both lattices are subject to a geometric deformation due to an applied strain on the bottom layer We restrict our discussion to a global strain that is uniform across the whole layer To account for any slippage between layers we introduce the strain transfer parameter μ This strain transforms the Bravais lattice vectors ai and bi of the individual layers in the form the strain on the lower layer is not transferred to the upper one the layers are more commensurate such that 0 < μ < 1 A full transfer of strain μ = 1 is referred to as homostrain The strain tensor ϵ is commonly written as a symmetric 2 × 2 matrix with three independent parameters32 \({\epsilon }_{{{{\rm{c}}}}}=\frac{{\epsilon }_{{{{\rm{xx}}}}}+{\epsilon }_{{{{\rm{yy}}}}}}{2}\) as biaxial strain \({\epsilon }_{{{{\rm{s}}}}}=\sqrt{{\left(\frac{{\epsilon }_{{{{\rm{xx}}}}}-{\epsilon }_{{{{\rm{yy}}}}}}{2}\right)}^{2}+{\epsilon }_{{{{\rm{xy}}}}}^{2}}\) as shear strain \({S}_{{\phi }_{{{{\rm{s}}}}}}=\cos ({\phi }_{{{{\rm{s}}}}}){\sigma }_{{{{\rm{x}}}}}+\sin ({\phi }_{{{{\rm{s}}}}}){\sigma }_{{{{\rm{z}}}}}\) as shear matrix \({\phi }_{{{{\rm{s}}}}}=\arccos \left(\frac{{\epsilon }_{{{{\rm{xy}}}}}}{{\epsilon }_{{{{\rm{s}}}}}}\right)\) as shear strain angle ϵyy = −νϵu) is a mixture of biaxial and shear strain with a shear angle set to ϕs = 90° the crystal is elongated along one direction the crystal deforms proportional to the Poisson ratio − ν Using the previous definitions, it is possible to calculate the real space moiré lattice under general strain, as presented in Supplementary Note 1 The new moiré lattice vectors \({{{{{\bf{A}}}}}^{{\prime} }}_{{{{\rm{i}}}}}\) take the form \({c}_{1}={(1+{\epsilon }_{{{{\rm{c}}}}})}^{2}-{\epsilon }_{{{{\rm{s}}}}}^{2}\) and \({c}_{\mu }={(1+\mu {\epsilon }_{{{{\rm{c}}}}})}^{2}-{\mu }^{2}{\epsilon }_{{{{\rm{s}}}}}^{2}\) we calculate the moiré unit cell area M as follows: The full calculation is provided in Supplementary Note 2 This effect occurs when the deformation of one layer due to shear strain aligns the lattice sites in one direction while leaving a mismatch in the perpendicular direction We emphasize that this effect is not dependent on the orientation of the lattice θ0 the shear angle ϕs or the shape of the layer lattices In addition to the size of the moiré pattern, shear strain is also able to change the moiré lattice geometry since ϵs changes the shape of the individual lattice depending on the orientation of ϕs and θ0. The main defining parameter for the geometry of the moiré pattern is the angle α between the moiré vectors (see Fig. 1b) 2D strain has the potential to tune the size and shape of the moiré pattern via the three independent strain parameters ϵc we will focus on the case of pure heterostrain (μ = 0) on homo- and heterobilayer structures and show how the different types of strain can affect the moiré lattice geometry a Sketch of the moiré lattice formation The upper layer is stacked with an angle θ respect to the lower one b Moiré lattice parameters \({A}_{1}^{{\prime} }\) and \({A}_{2}^{{\prime} }\) as function of biaxial strain and stacking angle for homo- (left) and heterobilayer (right) structures The scale bar in II has a length of 30 nm and is valid for all real space moiré patterns I–III a Scheme of the moiré lattice formation under shear heterostrain ϵs The deformation of the lower layer due to shear strain along the zigzag direction is depicted through arrows The upper layer was stacked with an angle θ respect to the lower one b Relative angle α (upper panels) and moiré lattice area M (lower panels) as a function of shear strain ϵs and stacking angle θ for homobilayer (left panels) and heterobilayer (right panels) systems Gray dotted lines denote configurations in which the moiré lattice is rectangular (α = 90°) we have presented a general geometrical description of the effect of strain in homo- and heterobilayer systems We show that heterostrain can be used to form a vast variety of moiré lattice geometries independent of the underlying lattice size and shape We demonstrate how an initial moiré lattice geometry can be tuned into a variety of particular moiré patterns a hexagonal lattice can be made rectangular or even 1D where we also display how to apply pure biaxial or shear strain as well as their possible combinations the capability to in situ tune the geometry and the interaction strength in highly correlated moiré quantum systems is expect strain tuning of moiré materials will have a major impact on the exploration of highly interacting quantum systems from fine-tuning magic-angle graphene to the realization of moiré quantum simulators for Luttinger liquids The data presented were generated from the mathematical algorithm outlined in the main text Code for geometrical illustrations of (hetero)strained moiré patterns with the calculated moiré lattice as those presented in Fig. 6 can be found in https://github.com/QuantumPhotonicsLab/Strained-Moire-Visualization Strongly correlated electrons and hybrid excitons in a moiré heterostructure Correlated electronic phases in twisted bilayer transition metal dichalcogenides One-dimensional Luttinger liquids in a two-dimensional moiré lattice Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices Mott and generalized Wigner crystal states in wse2/ws2 moiré superlattices Quantum anomalous hall effect from intertwined moiré bands Exciton-polarons in the presence of strongly correlated electronic states in a MoSe2/WSe2 moiré superlattice Signatures of moiré-trapped valley excitons in mose2/wse2 heterobilayers Highly energy-tunable quantum light from moiré-trapped excitons Direct visualization of magnetic domains and moiré magnetism in twisted 2d magnets Twist engineering of the two-dimensional magnetism in double bilayer chromium triiodide homostructures Coexisting ferromagnetic-antiferromagnetic state in twisted bilayer cri3 Tunable strain soliton networks confine electrons in van der Waals materials Topological polaritons and photonic magic angles in twisted α-moo3 bilayers Configurable phonon polaritons in twisted α-moo3 Electric field-tunable superconductivity in alternating-twist magic-angle trilayer graphene Superconductivity in the doped Hubbard model and its interplay with next-nearest hopping t Quantum simulation of the Hubbard model with ultracold fermions in optical lattices Quantum simulations with ultracold atoms in optical lattices Hubbard model physics in transition metal dichalcogenide moiré bands Quantum phase diagram of a Moiré-Hubbard model Moiré pattern as a magnifying glass for strain and dislocations in van der Waals heterostructures Electronic spectrum of twisted graphene layers under heterostrain Topological mosaics in moiré superlattices of van der Waals heterobilayers Twistronics versus straintronics in twisted bilayers of graphene and transition metal dichalcogenides Twist versus heterostrain control of optical properties of moiré exciton minibands Moiré metrology of energy landscapes in van der Waals heterostructures Maximized electron interactions at the magic angle in twisted bilayer graphene Heterostrain determines flat bands in magic-angle twisted graphene layers Heterostrain-enabled dynamically tunable moiré superlattice in twisted bilayer graphene Interchain conductivity of coupled Luttinger liquids and organic conductors Anomalous superperiodicity in scanning tunneling microscope images of graphite Commensurate-incommensurate transition in graphene on hexagonal boron nitride Large-scale mapping of moiré superlattices by hyperspectral Raman imaging Interlayer friction and superlubricity in bilayer graphene and MoS2/MoSe2 van der Waals heterostructures Robust microscale superlubricity in graphite/hexagonal boron nitride layered heterojunctions Excitons in a reconstructed moiré potential in twisted WSe2/WSe2 homobilayers Stacking domains and dislocation networks in marginally twisted bilayers of transition metal dichalcogenides Robust superlubricity by strain engineering Stretching and breaking of ultrathin MoS 2 Excitons in strain-induced one-dimensional moiré potentials at transition metal dichalcogenide heterojunctions Enhanced tunable second harmonic generation from twistable interfaces and vertical superlattices in boron nitride homostructures Tunable crystal symmetry in graphene-boron nitride heterostructures with coexisting moiré superlattices Materials data on wse2 by materials project (2020) Crystal structures of tungsten disulfide and diselenide Strain-tunable single photon sources in WSe2 monolayers Mobility enhancement in graphene by in situ reduction of random strain fluctuations Piezoelectric-based apparatus for strain tuning Correlated states in strained twisted bilayer graphenes away from the magic angle In situ strain tuning in hBN-encapsulated graphene electronic devices Approaching the Schottky-Mott limit in van der Waals metal-semiconductor junctions Correlation-induced valley splitting and orbital magnetism in a strain-induced zero-energy flatband in twisted bilayer graphene near the magic angle Download references gratefully acknowledge the German Science Foundation (DFG) for financial support via grants FI 947/8-1 as well as the clusters of excellence MCQST (EXS-2111) and e-conversion (EXS-2089) acknowledges the Royal Society for support via a University Research Fellowship acknowledges the Royal Society for a Wolfson Merit Award and the Royal Academy of Engineering for a Chair in Emerging Technology Institute of Photonics and Quantum Sciences School of Engineering and Physical Sciences Walter Schottky Institut and TUM School of Natural Sciences developed an algorithm for specific strain tunings developed the mathematical algorithm for the exact description of a universal Moire strain tuning interpreted the calculations and wrote the manuscript equally Download citation DOI: https://doi.org/10.1038/s41699-023-00382-4 Metrics details Berry curvature physics and quantum geometric effects have been instrumental in advancing topological condensed matter physics in recent decades Although Landau level-based flat bands and conventional 3D solids have been pivotal in exploring rich topological phenomena they are constrained by their limited ability to undergo dynamic tuning moiré systems have risen as a versatile platform for engineering bands and manipulating the distribution of Berry curvature in momentum space These moiré systems not only harbour tunable topological bands modifiable through a plethora of parameters but also provide unprecedented access to large length scales and low energy scales they offer unique opportunities stemming from the symmetry-breaking mechanisms and electron correlations associated with the underlying flat bands that are beyond the reach of conventional crystalline solids encompassing quantum electron transport in both linear and nonlinear response regimes and optical excitation techniques provide direct avenues for investigating Berry physics in these materials This Review navigates the evolving landscape of tunable moiré materials highlighting recent experimental breakthroughs in the field of topological physics we delineate the most pressing challenges and offer insights into promising avenues for future research This review discusses using the tunability of moiré heterostructures to experimentally simulate different fundamental many-body quantum models in condensed matter Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds This theoretical study identifies possible 2D materials This review provides an early overview of 2D materials research This review provides an overview of the emerging 2D materials research Van der Waals heterostructures and devices This paper provides the first experimental demonstration of a correlated insulator arising in a moiré superlattice This paper reports the first observation of superconductivity in flat bands of a moiré superlattice This paper reports the observation of the quantized anomalous Hall effect in a moiré superlattice This paper reports the observation of the anomalous Hall effect in a moiré superlattice This review covers the different mechanisms responsible for the anomalous Hall effect observed in non-moiré materials This review discusses the quantum Hall effect Berry phase effects on electronic properties This review provides a pedagogical introduction to Berry physics and its experimental effects Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure Quantum metric-induced nonlinear transport in a topological antiferromagnet and anomalous Hall effect in twisted multilayer graphene systems Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials This paper theoretically proposes second-harmonic Hall voltage generation owing to the Berry curvature dipole Berry curvature dipole senses topological transition in a moiré superlattice This paper reports the experimental detection of a topological transition in a moiré material Resonant second-harmonic generation as a probe of quantum geometry Riemannian geometry of resonant optical responses Anomalous Hall effect in ferromagnetic semiconductors Topological phases in two-dimensional materials: a review Berry Phases in Electronic Structure Theory: Electric Polarization Orbital Magnetization and Topological Insulators (Cambridge Univ Topological Insulators and Topological Superconductors (Princeton Univ Valley-contrasting physics in graphene: magnetic moment and topological transport This paper presents a theoretical study of the Berry phase inducing a valley-dependent Hall transport Bulk valley transport and Berry curvature spreading at the edge of flat bands Topological Bloch bands in graphene superlattices This theoretical study highlights that moiré bands can be topological Substrate-induced topological minibands in graphene Nearly flat Chern bands in moiré superlattices Gate-tunable topological flat bands in trilayer graphene boron-nitride moiré superlattices and field-tunable topological transitions in twisted multilayer graphene systems Voltage-controlled magnetic reversal in orbital Chern insulators Electrical switching of magnetic order in an orbital Chern insulator Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bands Chiral response of twisted bilayer graphene Hofstadter butterfly and the quantum Hall effect in twisted double bilayer graphene Band structure and topological properties of twisted double bilayer graphene Flat bands in twisted double bilayer graphene Higher-order band topology in twisted moiré superlattice Symmetry-broken Chern insulators in twisted double bilayer graphene This review highlights the reproducibility issues and challenges in the fabrication of moiré superlattice devices Nonlinear Hall effects in strained twisted bilayer WSe2 Giant nonlinear Hall effect in strained twisted bilayer graphene Nonlinear anomalous Hall effects probe topological phase-transitions in twisted double bilayer graphene Ferroelectric order in van der Waals layered materials Electronic flexoelectricity in low-dimensional systems Van der Waals engineering of ferromagnetic semiconductor heterostructures for spin and valleytronics Zero-field superconducting diode effect in small-twist-angle trilayer graphene Superconductivity in metallic twisted bilayer graphene stabilized by WSe2 Spin-orbit-driven ferromagnetism at half moiré filling in magic-angle twisted bilayer graphene Quantum anomalous Hall effect from inverted charge transfer gap This theoretical paper proposes TMDCs as a Hubbard model simulator Topological insulators in twisted transition metal dichalcogenide homobilayers This theoretical study highlights that TMDC moiré materials are a platform to realize correlated and topological states Anomalous Hall metal and fractional Chern insulator in twisted transition metal dichalcogenides Topological exciton bands in moiré heterojunctions Correlated insulator of excitons in WSe2/WS2 moiré superlattices orbital magnets and correlated states in magic-angle bilayer graphene Correlation-driven topological phases in magic-angle twisted bilayer graphene Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening Emergence of correlations in alternating twist quadrilayer graphene Bhowmik, S., Ghosh, A. & Chandni, U. Emergent phases in graphene flat bands. Preprint at http://arxiv.org/abs/2309.08938 (2023) Electrical control of the valley Hall effect in bilayer MoS2 transistors Generation and detection of pure valley current by electrically induced Berry curvature in bilayer graphene Gate-tunable topological valley transport in bilayer graphene Tunable and giant valley-selective Hall effect in gapped bilayer graphene The valley Hall effect in MoS2 transistors This experimental work is the first demonstration of the valley Hall effect Detecting topological currents in graphene superlattices Tunable bandwidths and gaps in twisted double bilayer graphene on the verge of correlations Bulk and edge properties of twisted double bilayer graphene Moiré band topology in twisted bilayer graphene Long-range nontopological edge currents in charge-neutral graphene The quantum anomalous Hall effect: theory and experiment Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly” This theoretical paper outlines the Haldane model that demonstrates the possibility of having a quantum Hall effect in the absence of a magnetic field This paper reports the first experimental observation of the quantum anomalous Hall effect in a topological insulator Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4 Proximity-induced ferromagnetism in graphene revealed by the anomalous Hall effect Quantum anomalous Hall effect from intertwined moiré bands Signatures of fractional quantum anomalous Hall states in twisted MoTe2 Thermodynamic evidence of fractional Chern insulator in moiré MoTe2 this paper experimentally demonstrates the fractional quantum anomalous Hall effect in moiré heterostructures Fractional Chern insulator states in twisted bilayer graphene: an analytical approach Chern bands of twisted bilayer graphene: fractional Chern insulators and spin phase transition Interplay of fractional Chern insulator and charge density wave phases in twisted bilayer graphene Fractional Chern insulator in twisted bilayer MoTe2 Fractional quantum anomalous Hall states in twisted bilayer MoTe2 and WSe2 Toward a global phase diagram of the fractional quantum anomalous Hall effect Magic in twisted transition metal dichalcogenide bilayers Spontaneous fractional Chern insulators in transition metal dichalcogenide moiré superlattices Fractional Chern insulators in magic-angle twisted bilayer graphene Dong, Z., Patri, A. S. & Senthil, T. Theory of fractional quantum anomalous Hall phases in pentalayer rhombohedral graphene moiré structures. Preprint at http://arxiv.org/abs/2311.03445 (2023) Zhou, B., Yang, H. & Zhang, Y.-H. Fractional quantum anomalous Hall effects in rhombohedral multilayer graphene in the moiréless limit and in Coulomb imprinted superlattice. Preprint at http://arxiv.org/abs/2311.04217 (2023) Dong, J. et al. Anomalous Hall crystals in rhombohedral multilayer graphene I: interaction-driven Chern bands and fractional quantum Hall states at zero magnetic field. Preprint at http://arxiv.org/abs/2311.05568 (2023) Topological charge density waves at half-integer filling of a moiré superlattice Broken-symmetry states at half-integer band fillings in twisted bilayer graphene Enhanced superconductivity in monolayer Td-MoTe2 Coupled ferroelectricity and superconductivity in bilayer Td-MoTe2 Phase-engineered low-resistance contacts for ultrathin MoS2 transistors Ultra-strong spin-orbit coupling and topological moiré engineering in twisted ZrS2 bilayers Composite fermi liquid at zero magnetic field in twisted MoTe2 Zero-field composite fermi liquid in twisted semiconductor bilayers composite fermions form and flow without a magnetic field Topology and geometry under the nonlinear electromagnetic spotlight Observation of the nonlinear Hall effect under time-reversal-symmetric conditions This paper represents the first experimental report of the nonlinear Hall effect Nonlinear anomalous Hall effect in few-layer WTe2 Strain engineering of the Berry curvature dipole and valley magnetization in monolayer MoS2 orbital magnetization and nonlinear Hall effect in WSe2 monolayer Giant nonlinear Hall effect in twisted bilayer WSe2 Intrinsic nonlinear Hall effect and gate-switchable Berry curvature sliding in twisted bilayer graphene Orderly disorder in magic-angle twisted trilayer graphene Tunable large Berry dipole in strained twisted bilayer graphene Giant nonlinear Hall effect in twisted bilayer WTe2 Strongly enhanced Berry dipole at topological phase transitions in BiTeI Graphene moiré superlattices with giant quantum nonlinearity of chiral Bloch electrons Giant second-order nonlinear Hall effect in twisted bilayer graphene Disorder-induced nonlinear Hall effect with time-reversal symmetry Nonlinear magnetoresistivity in two-dimensional systems induced by Berry curvature Berry curvature memory through electrically driven stacking transitions Ferroic Berry curvature dipole in a topological crystalline insulator at room temperature Observation of chiral and slow plasmons in twisted bilayer graphene Tunable interband and intraband plasmons in twisted double bilayer graphene Observation of interband collective excitations in twisted bilayer graphene Efficient Fizeau drag from Dirac electrons in monolayer graphene Quantum plasmonic nonreciprocity in parity-violating magnets Plasmonic nonreciprocity driven by band hybridization in moiré materials Intrinsic nonreciprocal bulk plasmons in noncentrosymmetric magnetic systems Quantum metric and correlated states in two-dimensional systems Mandal, D., Sarkar, S., Das, K. & Agarwal, A. Quantum geometry induced third order nonlinear transport responses. Preprint at https://doi.org/10.48550/arXiv.2310.19092 (2023) Intrinsic nonlinear conductivities induced by the quantum metric Komissarov, I., Holder, T. & Queiroz, R. The quantum geometric origin of capacitance in insulators. Preprint at https://doi.org/10.48550/arXiv.2306.08035 (2023) Quantum fluctuation of the quantum geometric tensor and its manifestation as intrinsic Hall signatures in time-reversal invariant systems Superfluidity in topologically nontrivial flat bands These theoretical papers connect the idea of quantum metric to that of flat-band superconductivity Geometric origin of superfluidity in the Lieb-lattice flat band this theoretical paper connects the idea of quantum metric to that of flat-band superconductivity superfluidity and quantum geometry in twisted multilayer systems Evidence for Dirac flat band superconductivity enabled by quantum geometry Bounds on the superconducting transition temperature: applications to twisted bilayer graphene and cold atoms Diamagnetic response and phase stiffness for interacting isolated narrow bands Strain-induced large injection current in twisted bilayer graphene Control of valley polarization in monolayer MoS2 by optical helicity and nonlinear optical response in moiré transition metal dichalcogenide heterobilayers Intrinsic spin Hall torque in a moiré Chern magnet Giant spin Hall effect in AB-stacked MoTe2/WSe2 bilayers Twisted photovoltaics at terahertz frequencies from momentum shift current Shift-current response as a probe of quantum geometry and electron-electron interactions in twisted bilayer graphene Absence of edge states in the valley Chern insulator in moiré graphene Fractional quantum Hall physics in topological flat bands in Encyclopedia of Condensed Matter Physics 2nd edn Higher-order topological insulator in twisted bilayer graphene Second-order and real Chern topological insulator in twisted bilayer α-graphyne Low-frequency and Moiré-Floquet engineering: a review Topological Floquet engineering of twisted bilayer graphene Floquet-engineered topological flat bands in irradiated twisted bilayer graphene Valley-selective Floquet Chern flat bands in twisted multilayer graphene Floquet engineering of topological transitions in a twisted transition metal dichalcogenide homobilayer Tunable phases of moiré excitons in van der Waals heterostructures Excitons in mesoscopically reconstructed moiré heterostructures Moiré phonons in magic-angle twisted bilayer graphene Moiré magnons in twisted bilayer magnets with collinear order Visualization of moiré magnons in monolayer ferromagnet Topological charge pumping by a sliding moiré pattern Topological charge pumping in twisted bilayer graphene Theory of tunable flux lattices in the homobilayer moiré of twisted and uniformly strained transition metal dichalcogenides Competing phases of interacting electrons on triangular lattices in moiré heterostructures Topological chiral superconductivity with spontaneous vortices and supercurrent in twisted bilayer graphene Optical control of topological memory based on orbital magnetization Onishi, Y. & Fu, L. High-efficiency energy harvesting based on nonlinear Hall rectifier. Preprint at http://arxiv.org/abs/2211.17219 (2023) Layered materials as a platform for quantum technologies Quantum frequency doubling in the topological insulator Bi2Se3 Hall effects in artificially corrugated bilayer graphene without breaking time-reversal symmetry Direct electronic measurement of the spin Hall effect Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene Anomalous Hall effect at half filling in twisted bilayer graphene Spontaneous time-reversal symmetry breaking in twisted double bilayer graphene Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects Geometry and the anomalous Hall effect in ferromagnets 8th International Symposium on Foundations of Quantum Mechanics in the Light of New Technology Berry phase modification to the energy spectrum of excitons Download references acknowledges the Department of Science and Technology (DST) of India for the J.C and DST SUPRA grant SPR/2019/001247 along with the Department of Atomic Energy of the Government of India 12-R&D-TFR-5.10-0100 for the support thanks the Department of Science and Technology of the Government of India for project number DST/NM/TUE/QM-6/2019(G)-IIT Kanpur acknowledges support by the National Science Foundation under grant number OMA-2328993 These authors contributed equally: Pratap Chandra Adak Department of Condensed Matter Physics and Materials Science ideated and led the writing of this Review All authors discussed and contributed to the writing Nature Review Materials thanks the anonymous reviewers for their contribution to the peer review of this work Download citation DOI: https://doi.org/10.1038/s41578-024-00671-4 Metrics details The study of moiré materials has so far been limited to structures comprising no more than a few van der Waals sheets because a moiré pattern localized to a single two-dimensional interface is generally assumed to be incapable of appreciably modifying the properties of a bulk three-dimensional crystal we perform transport measurements of dual-gated devices constructed by slightly rotating a monolayer graphene sheet atop a thin bulk graphite crystal We find that the moiré potential transforms the electronic properties of the entire bulk graphitic thin film transport is mediated by a combination of gate-tuneable moiré and graphite surface states as well as coexisting semimetallic bulk states that do not respond to gating the moiré potential hybridizes with the graphitic bulk states due to the unique properties of the two lowest Landau bands of graphite These Landau bands facilitate the formation of a single quasi-two-dimensional hybrid structure in which the moiré and bulk graphite states are inextricably mixed Our results establish twisted graphene–graphite as the first in a new class of mixed-dimensional moiré materials Source data are provided with this paper All other data that support the findings of this study are available from the corresponding author upon request Flat bands in slightly twisted bilayer graphene: tight-binding calculations and correlated states in magic angle bilayer graphene Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene Electric field tunable superconductivity in alternating-twist magic-angle trilayer graphene Robust superconductivity in magic-angle multilayer graphene family Promotion of superconductivity in magic-angle graphene multilayers Electrical switching of magnetic order in an orbital chern insulator Tunable van Hove singularities and correlated states in twisted monolayer-bilayer graphene Competing correlated states and abundant orbital magnetism in twisted monolayer-bilayer graphene Correlated states in twisted double bilayer graphene Tunable spin-polarized correlated states in twisted double bilayer graphene Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene Correlated insulating states in twisted double bilayer graphene Symmetry breaking in twisted double bilayer graphene Twists and the electronic structure of graphitic materials Observation of van Hove singularities in twisted graphene layers Magnetic field dependence of the hall effect and magnetoresistance in graphite single crystals quantum Hall effect and layer parity in graphite films Linear magnetoresistance in the quantum limit in graphite Hofstadter’s butterfly and the fractal quantum Hall effect in moire superlattices Multilayered atomic relaxation in van der Waals heterostructures Isospin magnetism and spin-polarized superconductivity in Bernal bilayer graphene Evidence of a gate-tunable mott insulator in a trilayer graphene moiré superlattice Independent superconductors and correlated insulators in twisted bilayer graphene Analysis of multicarrier galvanomagnetic data for graphite First-principles study of the electronic properties of graphite Effective continuum model for relaxed twisted bilayer graphene and moiré electron-phonon interaction Existence and topological stability of fermi points in multilayered graphene Interlayer screening effect in graphene multilayers with aba and abc stacking Download references This work was supported by National Science Foundation (NSF) CAREER award no acknowledges support from the State of Washington-funded Clean Energy Institute was supported by an appointment to the Intelligence Community Postdoctoral Research Fellowship Program at University of Washington administered by Oak Ridge Institute for Science and Education through an interagency agreement between the US Department of Energy and the Office of the Director of National Intelligence Electrical transport calculations were supported by the Department of Energy Pro-QM Energy Frontier Research Center (grant no acknowledge support from the Elemental Strategy Initiative conducted by the MEXT JPMXP0112101001) and JSPS KAKENHI (grant nos This research acknowledges usage of the millikelvin optoelectronic quantum material laboratory supported by the M.J These authors contributed equally: Dacen Waters Dacen Waters, Ellis Thompson, Di Xiao & Matthew Yankowitz Intelligence Community Postdoctoral Research Fellowship Program Department of Materials Science and Engineering Esmeralda Arreguin-Martinez, Manato Fujimoto, Yafei Ren, Ting Cao, Di Xiao & Matthew Yankowitz fabricated the devices and performed the measurements performed the magnetotransport calculation analysed the data and wrote the paper with input from all authors Nature thanks the anonymous reviewers for their contribution to the peer review of this work Longitudinal (top) and Hall (bottom) resistance measurements acquired in steps of B = 50 mT as indicated in the top left of each column The zig-zag transport behaviour first becomes evident at fields as low as 50 mT and becomes more obvious as the field is raised Source Data Landau fan diagram acquired by sweeping Vm at Vgr = 0 The purple (pink) dashed lines denote selected QOs that project to ν = 0 (ν = ± 4) Landau fan diagrams acquired by sweeping Vgr at various fixed values of Vm The black dashed lines denote selected QOs that project to Vgr ≈ 0 at B = 0 The blue lines denote selected QOs that project to a Vgr ≠ 0 that depends on Vm Longitudinal (left) and Hall (right) resistance maps acquired at B = 0.5 T Zero-field projections of the ν = 0 and ± 4 states from the Vm Landau fans are overlaid on the Rxx map Zero-field projections of QOs from the Vgr Landau fans are overlaid on the Rxy map The blue curve is averaged over all values of Vgr for the Landau fan in (a) acquired at Vm/dm = − 0.11 V/nm The red curve is averaged over a range of Vm values corresponding to ∣ν∣ < 4 for the Landau fan in (b) acquired at Vgr = 0 Brown-Zak oscillations can be seen upon sweeping either gate (Inset) Longitudinal resistance map acquired at B = 0 T Source Data Landau fan diagrams from a 24-layer graphite device (different from the one shown in the main text) acquired by sweeping the top gate voltage at various fixed values of the bottom gate voltage The black dashed lines denote selected QOs that project to Vt ≈ 0 at B = 0 and the pink lines denote QOs that project to a Vt ≠ 0 that depends on the value of Vb The QOs projecting to Vb ≠ 0 are denoted in blue The QOs projecting to approximately zero gate voltage in each Landau fan are overlaid on the Rxx map and form a cross The QOs projecting to non-zero gate voltages are overlaid on the Rxy map and closely track the condition of overall charge neutrality Dashed black lines denote selected QOs that depend only on a single gate which arise from localized states on either the top or bottom graphite surfaces The blue/pink dashed lines denote QOs that depend on both gates which evolve parallel to the line of overall charge neutrality Source Data Dual-gate Rxx maps (top) and corresponding numerical second derivative (bottom) acquired for the (a) t1+10 device at B = 9 T Solid black bars at the top of each map denote regions of gate voltage dominated by vertical QOs which correspond to surface-localized states contain diagonal QOs which depend on the value of both gate voltages These features correspond to extended bulk standing wave states Data in panel a was acquired in a dilution refrigerator with a nominal base temperature of T ≈ 20 mK Source Data Similar fitting using the unconstrained and constrained fitting procedures but for the sequence of QOs that project to Vgr ≠ 0 Projection points determined from the fits shown in c and d The shaded pink region contains data points and associated error bars from the unconstrained fit \({V}_{i}^{{\rm{u}}}\pm {{\rm{\sigma }}}_{i}\) corresponding to the dashed lines in c of the same color The vertical black line and surrounding shaded grey bar denote the weighted average and associated error of the unconstrained fit \(\bar{{V}_{0}^{{\rm{u}}}}\pm \bar{{{\rm{\sigma }}}^{{\rm{u}}}}\) The shaded blue region contains the result of the constrained fit from d The single black data point with error bars is the projection point determined by the constrained fit \({V}_{0}^{{\rm{c}}}\pm {{\rm{\sigma }}}^{{\rm{c}}}\) Note that \(\bar{{V}_{0}^{{\rm{u}}}}\pm \bar{{{\rm{\sigma }}}^{{\rm{u}}}}\) and \({V}_{0}^{{\rm{c}}}\pm {{\rm{\sigma }}}^{{\rm{c}}}\) are consistent with one another Note that the extracted QO projection points in g and h differ significantly from one another allowing us to unambiguously identify two distinct sets of QOs corresponding to the surface states and bulk states since the moiré is located at the center of the structure (Right) The density of states integrated over the moiré Brillouin zone The red filtered curve corresponds to the four central graphene sheets whereas the black corresponds to the total density of states Source Data The blue curve is averaged over all values of Vgr for the Landau fan acquired at Vm/dm = − 0.05 V/nm The red curve is averaged over a range of Vm values corresponding to ∣ν∣ < 4 for the Landau fan in b acquired at Vgr = 0 Brown-Zak oscillations case be seen upon sweeping either gate Source Data a, Landau fan diagram of Rxx in the t1+10 device, acquired by sweeping Vm with Vgr/dgr = 0.01 V/nm. b, Numerical second derivative of the data in a. The color scale is saturated to only show positive values, as in Extended Data Fig. 5 Results of the constrained fit overlaid on the second derivative data from b Solid segments denote the range of magnetic field over which the QOs were fit and the dashed segments denote the projection over the entire range of magnetic field Same plots as in Extended Data Fig. 8 Landau fan acquired by sweeping Vgr with Vm/dm = 0.14 V/nm Landau fan acquired by sweeping Vm with Vgr/dgr = 0 Source Data Landau fan acquired by sweeping Vgr with Vm/dm = 0.09 V/nm Landau fan acquired by sweeping Vm with Vgr/dgr = − 0.02 V/nm Source Data The left shows the Landau fan diagrams from the t1 + 10 device acquired by sweeping Vgr at the indicated values of Vm The black and blue dots correspond to the B = 0 projection points of the QO sequences the blue dots align with the value of Vgr corresponding to the highest resistance over a wide range of B in the map The right shows the Rxy map acquired at B = 0.5 T with the corresponding black and blue dots overlaid The left shows Landau fan diagrams from the t1 + 10 device acquired by sweeping Vm at the indicated values of Vgr The pink and purple dots correspond to the B = 0 projection points of the QO sequences The right shows the Rxx map acquired at B = 0.5 T with the corresponding pink (ν = 0) and purple (ν = ± 4) dots overlaid Same as Supplementary Video 1 Same as Supplementary Video 2 Same as Supplementary Video 1 This video only includes Landau fans with Vgr/dgr ≥ −0.18 V nm−1 The fans acquired for Vgr/dgr ≤ −0.18 V nm−1 were taken only up to B = 5 T due to technical constraints in those particular measurements and are not included in the video for the sake of continuity these lower-field fans still enable unambiguous QO projections Download citation DOI: https://doi.org/10.1038/s41586-023-06290-3 Metrics details Two-dimensional moiré materials are formed by overlaying two layered crystals with small differences in orientation or/and lattice constant where their direct coupling generates moiré potentials Moiré materials have emerged as a platform for the discovery of new physics and device concepts but while moiré materials are highly tunable Here we demonstrate the electrostatic imprinting of moiré lattices onto a target monolayer semiconductor The moiré potential—created by a lattice of electrons that is supported by a Mott insulator state in a remote MoSe2/WS2 moiré bilayer—imprints a moiré potential that generates flat bands and correlated insulating states in the target monolayer and can be turned on/off by gate tuning the doping density of the moiré bilayer we studied the interplay between the electrostatic and structural relaxation contributions to moiré imprinting Our results demonstrate a pathway towards gate control of moiré lattices Source data are provided with this paper Additional data that support the findings of this study are available from the corresponding authors upon reasonable request Hubbard model physics in transition metal dichalcogenide moire bands Imaging moiré flat bands in three-dimensional reconstructed WSe2/WS2 superlattices Creation of moiré bands in a monolayer semiconductor by spatially periodic dielectric screening Spin-conserving resonant tunneling in twist-controlled WSe2-hBN-WSe2 heterostructures Dielectric catastrophe at the Wigner-Mott transition in a moiré superlattice Bilayer WSe2 as a natural platform for interlayer exciton condensates in the strong coupling limit A tunable bilayer Hubbard model in twisted WSe2 Excitonic insulator in a heterojunction moiré superlattice Gate-tunable heavy fermions in a moiré Kondo lattice Dipolar excitonic insulator in a moiré lattice Correlated interlayer exciton insulator in heterostructures of monolayer WSe2 and moiré WS2/WSe2 Excitons in semiconductor moiré superlattices Emerging exciton physics in transition metal dichalcogenide heterobilayers Excitons and emergent quantum phenomena in stacked 2D semiconductors Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides Exciton density waves in Coulomb-coupled dual moiré lattices Coulomb engineering of the bandgap and excitons in two-dimensional materials Strongly correlated excitonic insulator in atomic double layers Optical signatures of periodic charge distribution in a Mott-like correlated insulator state Charge-order-enhanced capacitance in semiconductor moiré superlattices Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet Stripe phases in WSe2/WS2 moiré superlattices Moiré quantum chemistry: charge transfer in transition metal dichalcogenide superlattices Colloquium: excitons in atomically thin transition metal dichalcogenides Disorder in van der Waals heterostructures of 2D materials Download references This work was supported by the Gordon and Betty Moore Foundation grant DOI: 10.37807/GBMF11563 (experimental design and analysis) the Air Force Office of Scientific Research MURI under award number FA9550-18-1-0480 (device fabrication) and the National Science Foundation grant DMR-2114535 (optical sensing measurement) The study was performed in part at the Cornell NanoScale Facility a member of the National Nanotechnology Coordinated Infrastructure supported by National Science Foundation grant NNCI-2025233 The growth of the hBN crystals was supported by the Elemental Strategy Initiative of the Ministry of Education Japan and the Core Research for Evolutional Science and Technology programme of the Japan Science and Technology Agency (grant no acknowledges a postdoc fellowship from the Swiss National Science Foundation Laboratory of Atomic and Solid State Physics Kavli Institute at Cornell for Nanoscale Science oversaw the project and cowrote the manuscript All authors discussed the results and commented on the manuscript The target MoSe2 monolayer (white) and the moiré bilayer with monolayer MoSe2 (blue) and WS2 (red) are labeled and outlined The effective area of the devices is shaded in grey \({\nu }_{m}=0\) and \({\nu }_{t} > 0\) (c) and \({\nu }_{m}=1\) and \({\nu }_{t} > 0\) (d) Incompressible states are labeled by their filling factors The 1 s and 2 s excitons of the WSe2 sensor layer as well as the 2 s exciton of the MoSe2 target layer are labeled The robust 1 s exciton resonance of the WSe2 layer demonstrates the charge neutrality of the sensor Source data Gate voltage \(({V}_{t}+{V}_{b})\) dependent reflectance contrast spectrum at electric field 0.045 V/nm (a) and 0.12 V/nm (b) Five different exciton species can be identified: the fundamental moiré exciton MX1 the neutral (X) and charged (X-) excitons of the target layer The red dashed arrows denote the onset of electron doping in the device The moiré (target) layer is first doped at low (high) electric field Source data Electric-field (E) and gate \(({V}_{t}+{V}_{b})\) dependences of the spectral feature corresponding to the sensor 2 s exciton (a) The analysis of the reflectance contrast spectra is described in Methods The X− reflectance (b) determines the boundary between the doped (high) and neutral (low) target layer (black solid line) The MX1 reflectance (c) determines the boundary between the doped (low) and neutral (high) moiré layer (pink solid line) The MX2 reflectance (d) determines the boundaries of the \(({\nu }_{m},{\nu }_{t})=(1,{\nu }_{t})\) region with enhanced contrast (pink dashed lines) Source data a,b, Gate voltage (\({V}_{t}+{V}_{b}\)) dependent reflectance contrast spectrum at electric field 0.07 V/nm (a) and 0.16 V/nm (b). Similar to Extended Data Fig. 3 five different exciton species can be identified Source data It also determines the boundaries (black dashed lines) of the \(({\nu }_{m},{\nu }_{t})=({\nu }_{m},1)\) region Source data Calculated electrostatic potential profile in a plane that is 1 nm above a triangular electron lattice we set the period to 8 nm in the calculation and used the dielectric constant of hBN The triangular electron lattice imprints a honeycomb potential profile (black line denotes the unit cell and dots denote the sublattice sites) Electrostatic potential along the dashed line in a It shows a trapping potential depth of about 4 meV Source data Second energy derivative of the 2 s exciton reflectance contrast \((\frac{{d}^{2}R}{d{\epsilon }^{2}})\) as function of the gate voltage \(({V}_{t}+{V}_{b})\) and the electric field (E) for device 2 Extracted 2 s peak intensity as a function of the electric field along the red dashed line in a The blue and green regions correspond to \({\nu }_{m}=1\) and \({\nu }_{m}=0\) The enhanced 2 s intensity in the blue region suggests a more robust correlated insulating state at \({\nu }_{t}=1\) due to the electrostatic imprinting effect Source data First energy derivative of the sensor 2 s spectrum \((\frac{dR}{d{\epsilon }})\) as a function of the gate voltage \(({V}_{t}+{V}_{b})\) for device 3 with a 4-layer hBN spacer (a) and device 4 with a 2-layer hBN spacer (b) The gate voltage \(({V}_{t}+{V}_{b})\) controls the total filling factor under a constant electric field (controlled by \({V}_{t}-{V}_{b}\)) We label the incompressible states \(({\nu }_{m},{\nu }_{t})=(\mathrm{0,1})\) and \((\mathrm{1,1})\) on the left and right panels Source data Download citation DOI: https://doi.org/10.1038/s41563-023-01709-8 Metrics details Recent advances in surface-patterning techniques of liquid crystals have enabled the precise creation of topological defects which promise a variety of emergent applications the manipulation and application of these defects remain limited we harness the moiré effect to engineer topological defects in patterned nematic liquid crystal cells we combine simulation and experiment to examine a nematic cell confined between two substrates of periodic surface anchoring patterns; by rotating one surface against the other we observe a rich variety of highly tunable These defects are shown to guide the three-dimensional self-assembly of colloids which can conversely impact defects by preventing the self-annihilation of loop-defects through jamming we demonstrate that certain nematic moiré cells can engender arbitrary shapes represented by defect regions the proposed simple twist method enables the design and tuning of mesoscopic structures in liquid crystals facilitating applications including defect-directed self-assembly we hypothesize that the moiré effect can also serve as an alternative method to engineer the mesoscopic structure of a material we demonstrate that moiré patterns can be used to manipulate the mesoscopic director field and the emerging topological defects in a nematic cell we consider a nematic cell confined by two surfaces with an identical periodic pattern imposing orientational preference (namely anchoring) to the nematic; we combine simulation and experiment to study the moiré effect of different periodic anchoring patterns Our nematic moiré patterns can give rise to a rich variety of periodic disclination structures Their three-dimensional (3D) topological structures are modelled by continuum simulations and then further confirmed by confocal microscope experiments The geometry of these emerging periodic topological structures is sensitive to twist angles and cell gaps and reveals both low- and high-frequency modes of the geometric moiré the latter of which are often difficult to see in conventional moiré systems The cross-polarized optical patterns from nematic moirés contain grain- and ring-like features which are distinct from the optical appearance of isotropic moiré patterns periodic moiré patterns arising from geometric patterns without anisotropy The versatility of the nematic moirés is further demonstrated by the fact that a certain one-dimensional (1D) surface pattern can even form a two-dimensional (2D) lattice of disclination loops and optical patterns a 2D surface pattern formed by a lattice of ±1 defects can give rise to heterogeneous defect structures corresponding to different regimes in the geometric moiré pattern The 3D defect networks generated in thick nematic moiré cells are capable of guiding 3D self-assembly and nucleation of colloidal structures when a colloidal particle-laden defect loop in a nematic moiré cell undergoes shrinkage those particles can prevent self-annihilation through jamming we show that certain judiciously designed nematic moiré patterns can engender pixelated shapes represented by defect regions our proposed nematic moiré pattern offers a versatile platform to investigate the interplay of topology and ordering in LCs and other soft materials systems facilitates defect-based emergent applications and opens the door for inverse design of mesoscopic structures of materials helical topological structure of the C-state (colored by angle \(\beta\)) H Topological structure of the S-state and W-state (colored by angle \(\alpha\)) \approx \,{12}^{\circ}\) showing good agreement with the simulation Using confocal microscopy to scan the cell from top to bottom we observe L disclination lines (the first group) close to the top substrate and M the disclination lines (the second group) appearing near the bottom substrate O Two 3D rotation views of the sample to visualize the two groups at the same time \({z}_{c}\) is the spacing between the two groups in the scanning process \({z}_{c}=40{{{{{\rm{\mu }}}}}}{{{{{\rm{m}}}}}}\) N When the two groups of disclinations are visualized from the cell top at the same time the bright lines are the first group disclination lines Both groups of lines are shown due to the scattering effect and the second group of lines close to the bottom substrate is in gray because of lower contrast Note that as soon as the angle \(\Psi\) starts to deviate from \({0}^{\circ}\) the disclination lines in the above states come from infinity one by one with their distance decreasing with increasing \(\Psi\) A 1D cosinusoidal geometric moiré lattice showing the (±1 B Simulation and experimental results of the nematic moiré at \(\Psi={11}^{\circ}\) Disclinations in the simulation are colored by twist angle \(\beta\) \(T\) is the spacing distance for neighboring defect curves and \(\omega\) is their tilting angle with respect to the \(x\) axis The defect helical structure diameter \({A}_{{xy}}\) and pitch \({T}^{*}\) are introduced to characterize defect shapes C The periodicity and orientation of the colloidal assembly by the defects can be tuned in the nematic moiré D Moiré period \(T/L\) as a function of rotation angle \(\Psi\) E Moiré tilting angle \(\omega\) as a function of rotation angle \(\Psi\) connected by pseudolines (black dashed lines) F \({T}^{*}/{T{{\hbox{'}}}}\) and \({A}_{{xy}}/L\) as functions of \(\Psi\) while fixing \(H/L=0.43\) G \({T}^{*}/{T{{\hbox{'}}}}\) and \({A}_{{xy}}/T\) as functions of \(H/L\) at \(\Psi={12}^{\circ}\) Source data are provided as a Source Data file \(\beta\) approaches \(\pi /2\) (pure-twist); when the curve passes through the midplane of the cell with \(\beta\) approaching \(0\) and \(\pi\) alternatively (wedge-twist) the C-state defect belongs to the wedge-twist type wedge disclinations of winding number \(\pm 1/2\) can be transformed through a continuous transformation with the symmetric twist state found in between This continuous transformation of the local profiles can also be used to explain the state transitions (between the S- This is potentially helpful for reversible reprogramming of colloids as building blocks to achieve multiple functions our nematic moirés provide a simple method to tune the topology and geometry (shapes etc.) of the disclinations using the geometry of moiré patterns which can further guide the colloidal assembly A Schematic of the system with the dashed thick rods indicating the geometry of the pattern The inset is the sinusoidal pattern in a period \(L\) B Superposed surface-preferred director field of the two anchoring patterns and C its mapped geometric pattern with identical period \(L\) and rotation angle \(\Psi\) D Theoretically predicted defect prediction for \(\Psi={3}^{\circ}\) showing defect period \(T\) and tilting angle \(\omega\) E From left to right: the simulation of \(H/L=0.1\) has two groups of defect loops \(H/L=0.3\) only has surviving larger loops and \(H/L=1.0\) gives rise to web-like defects (\(\Psi={30}^{\circ}\)) and Loop-III in the simulations and the classical wedge-twist loop POM images (H) and the corresponding simulated images (I) for thin cells (left) and medium-gap cells (right) at \(\Psi={30}^{\circ}\) K Moiré tilting angle \(\omega\) in theory Scale bar: \(50{{{{{\rm{\mu }}}}}}{{{{{\rm{m}}}}}}\) Since there is no more than a \(\pi\) rotation from the top to the bottom patterned substrate These defects can emerge when neighboring defect loops coalesce into large ones A The nucleation and expansion of the disclination loops can collect nearby particles B The colloidal assembly undergoes jamming as defect loops shrink and prevent their self-annihilation. C In a similar scenario without colloids the defect loop is eventually self-annihilated Scale bar: \(50\ {{{{\mu }}}}{{{{{\rm{m}}}}}}\) A Schematic of the system with the patterned top substrate in pink and the bottom substrate in dark green The inset is a schematic of the 2D pattern B The mapping from the dot-screen pattern to the \(\pm 1\) defect lattice pattern (left) and the superposed 2D geometric square gratings formed at \(\Psi={5}^{\circ}\) C Superposed 2D geometric moiré pattern at \(\Psi\ D Simulated defect structure in the nematic moiré at \(\Psi={5}^{\circ}\) F Experimental bright field snapshots corresponding to two boxed regions in the dashed box in (B G Superposed period \(T/L\) from geometric moiré pattern theory H Superposed tilting angle \(\omega\) from geometric moiré pattern theory Yellow is for the \(\Psi \le {21}^{\circ }\) regime with \(({{{{\mathrm{1,0}}}}},-{{{{\mathrm{1,0}}}}})\) as the dominating moiré and blue is for the \(\Psi \in [{34}^{\circ },{40}^{\circ }]\) regime with dominating moiré \(\left({{{{\mathrm{1,2}}}}},-2,-1\right)\) with a singular point (infinitely large period) at \(\Psi={\tan }^{-1}(3/4).\) For the other angle ranges \(\Psi \in [{21}^{\circ },{34}^{\circ }]\) and \(\Psi\ no dominating moiré is observed (using green) I Nematic moiré in simulation for \(\Psi \ Its period and tilting angle are \(\widetilde{T}\) and \(\widetilde{\omega }.\) J Experimental coloured POM snapshots (top) and bright field image (bottom) of a periodic defect structure at \(\Psi\ \approx \,{36.8}^{\circ}\) corresponding to the boxed region in (C) K Fast rotation simulation snapshots at \(\Psi={15}^{\circ}\) (left) and \(\Psi={180}^{\circ}\) (right) with the simulation time ratio being \({\tau }_{{{{{{\rm{s}}}}}}}\)/\({\tau }_{{{{{{\rm{cell}}}}}}}=35.4\) (“Methods”) Beyond the versatility of the proposed simple twist method we proceed to demonstrate the applications of nematic moiré cells existing research efforts have been devoted to the forward design problem: which topological structures with what properties can be formed from a given geometry or a given pattern all the possible defect structures are often quite limited to a few metastable states This greatly limits the applications of nematic disclinations we propose using the moiré effect to address the inverse problem of designing arbitrary pixelated defect regions various Latin letter-shaped defects following the moiré period and tilting angle can be generated A By putting a “revealer” pattern (circular) on top of a “document” pattern (T-shape) an array of amplified T shapes can be revealed in the geometric moiré One of the generated T shapes is noted by a dashed blue frame The inset is the circular LC pattern mapped from the circular geometric pattern B Map the “document pattern” at the bottom to the LC pattern (in the shape of U C Results of three simulations for \(\Psi=4^\circ\): Overlapping the circular LC pattern on the top and the three Latin letter LC patterns on the bottom sequentially and T comprising defect loops are generated in the three simulations the nematic moiré pattern exhibits a threshold voltage approximately \(10\%\) lower than that of TNCs showing the promise of nematic moiré patterns in applications such as displays and responsive materials We first consider a one-dimensional (1D) splay-bend anchoring pattern the two surface anchoring patterns are aligned Upon twisting one surface against the other the periodicity and shape geometry of which can be explained by the corresponding geometric moiré pattern and dimensionality of these defect structures are highly tunable and sensitive to the rotation angle and the cell gap Based on the preferred orientation difference between the two surfaces a simple theory can correctly predict these disclinations in the thin-cell limit but fails for thicker cells The simulations and experiments agree very well not only on defect configurations but also on optical structures (POM images) which appear distinct from the optical images of the corresponding isotropic moiré patterns the confocal microscope experiment further confirms the simulation prediction The engendered defect networks can be used to guide the 3D self-assembly of colloidal particles these particles can also alter the defect structure by preventing self-annihilation of loop defects through jamming but the rotation operation over two overlapping periodic patterns was not investigated We also find that the resulting nematic structure is sensitive to the rotation speed: slow rotation can bring the system back to the ground state after a π-rotation whereas fast rotation can drive the system into disordered defect states which are stuck in a state comprised of surface defects our calculations show that the Frederiks transition voltage of the nematic moiré is \(\sim 10\%\) lower than that of a planar twist cell with the same twisting angle which can facilitate display-related applications This can be understood by the finer elastic distortions incurred by the anchoring pattern The proposed nematic system equipped with the moiré effect shows the following features and advantages: (1) distinct periodic topological structures with highly tunable periodicities can be systematically realized in one system by a simple twist method and their geometries can be well understood by the moiré theory-based analytic model which can enable the inverse design of these structures; (2) the unique POM images of nematic moirés could enrich their applications in microscopic imaging and strain analysis; and (3) the interplay between the engendered disclinations and colloidal particles contains rich physics and can lead to emerging mesoscopic patterns If incorporating the nematic moiré effect in the ongoing defect-based application research we can foresee much richer applications of the moiré effect in liquid crystals The LC director is aligned with the orientations of azo dye molecules the polarization pattern of light is imprinted into the photosensitive substrate that is used to align the liquid crystal Please note that this maskless patterning technique by projecting display can produce any pattern of the director field with the typical scale of spatial gradients ranging from approximately tens of micrometers to centimeters To reduce the influence of light irradiation on the patterned substrate, an additional layer of liquid crystal polymer is coated on the top of the pattern. Monomer RM257, Fig. S26 is mixed with toluene at a concentration of \(7\) wt% with photoinitiator Irgacure 651 (from Ciba Inc.) at a concentration of \(5\) wt% RM257 This solution was spin-coated onto the patterned SD1 substrates at \(3000\) rpm for \(30\) s The substrates were photopolymerized under unpolarized ultraviolet light with an intensity of \(1.4\) mW/cm2 for \(30\) min The polymer pattern replicates the pattern of SD1 alignment beneath it and 5CB is used in the rest of the experiments The colloidal dispersion in the LC is injected into the photopatterned cell at room temperature (22 °C). The colloids are manipulated by using a laser tweezers (JCOPTIX We used a 50X-1000X Advanced Upright Polarized light Microscope from Amscope with both a 10x Plan Optical microscopy images were captured by a 20 MP USB3.0 BSI C-mount Microscope Camera from Amscope The bright-field optical micrographs were taken by a 40X-1000X Upright Fluorescence Microscope with Rotating Multifilter Turret from Amscope with both 10x Plan The anisotropic fluorescent dye N,N′-bis(2,5-di-tert-butylphenyl)-3,4,9,10-perylenedicarboximide (BTBP) (from Sigma‒Aldrich) was mixed in methanol at 0.01 wt% Due to the low birefringence of LC JXLC-20000 it is used in the confocal measurement to reduce the scattering effect during the imaging process The BTBP solution is then mixed with JXLC-20000 at a weight ratio of 1:1 methanol was evaporated overnight on a hotplate at 90°C Imaging was performed with a Nikon A1 laser scanning confocal fluorescence microscope using lasers with excitation wavelength of 488 nm and emission wavelength of 530 nm We apply two substrate patterns and let the system relax to equilibrium The total free energy F consists of three bulk terms: where parameter \({A}_{0}\) is the energy density scale and parameter \(U\) controls the magnitude of \({S}_{0}\) of a homogeneous static system through The elastic energy density \({f}_{{{{{{\rm{el}}}}}}}\) in our simulations is expressed in terms of the \({{{{{\bf{Q}}}}}}\)-tensor form as For simplicity, we use a one-constant assumption without special notation (Supplementary Material 6.2.1) The electric field-induced energy density takes the following form: where \({\left[...\right]}^{{{{{{\rm{st}}}}}}}\) is a symmetric and traceless operator The evolution for bulk points is governed by our simulation can be mapped to the nematic 5CB at room temperature and atmospheric pressure by choosing \({\xi }_{N} \ \({A}_{0}=1.17\times {10}^{5}\,{{{{{\rm{J}}}}}}/{{{{{{\rm{m}}}}}}}^{3}\) This gives rise to \({\tau }_{{{{{{\rm{LC}}}}}}} \ 0.667\ {{{{\mu }}}}{{{{{\rm{s}}}}}}\) and \({\tau }_{{{{{{\rm{cell}}}}}}} \ surface pattern period \(L=30\) in simulation units which is compared to 75 μm in the experiment the simulation and experiment agree very well Write \({{{{{\bf{n}}}}}}\) in terms of \({{{{{\bf{t}}}}}}\) and \({{{{{\bf{m}}}}}}\) Vector \({{{{{{\bf{e}}}}}}}_{1}\) is on the plane that is normal to \({{{{{\bf{t}}}}}}\) pointing from the defect core center along \(\varphi=0\) and \({{{{{{\bf{e}}}}}}}_{2}\) is written as \({{{{{{\bf{e}}}}}}}_{2}{{{{{\boldsymbol{=}}}}}}{{{{{\bf{t}}}}}}{{{{{\boldsymbol{\times }}}}}}{{{{{{\bf{e}}}}}}}_{1}\) vector \({{{{{\bf{m}}}}}}\) can be expressed as and \(\alpha\) is the phase shift angle between vector m and radial line \(\varphi=0\). As \(\alpha=1/2\pi\) or \(\alpha=3/2\pi\), the local profile is of the tangential twist type, and \(\alpha=0\) or \(\pi\) corresponds to the radial twist type. 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manuscript Download citation DOI: https://doi.org/10.1038/s41467-024-45529-z Metrics details The superlattice obtained by aligning a monolayer graphene and boron nitride (BN) inherits from the hexagonal lattice a sixty degrees periodicity with the layer alignment the properties of the heterostructure must be identical for 0° and 60° of layer alignment using dynamically rotatable van der Waals heterostructures that the moiré superlattice formed in a bilayer graphene/BN has different electronic properties at 0° and 60° of alignment Although the existence of these non-identical moiré twins is explained by different relaxation of the atomic structures for each alignment the origin of the observed valley Hall effect remains to be explained A simple Berry curvature argument is not sufficient to explain the 120° periodicity of this observation Our results highlight the complexity of the interplay between mechanical and electronic properties in moiré structures and the importance of taking into account atomic structure relaxation to understand their electronic properties These corrugations have the same periodicity as the moiré pattern both the long-wavelength pattern and the commensurate state are observed every time one of the layers is rotated by sixty degrees very little is known about how the graphene/BN alignment affects systems with more than one layer Our measurements reveal distinct behaviors for 0° and 60° which we attribute to different electronic band structures generated by different atomic displacements inside the moiré superlattice only present for 0° of alignment remains to be explained given that the current theoretical model fail to explain this hundred and twenty degrees periodicity a Schematic representation (left) of a dynamically rotatable van der Waals heterostructure the boron nitride between the graphite and graphene layers has been omitted for clarity The central circular shape on the graphene represents the range of action of the graphite gate and the moiré superlattice Cross section of the same heterostructure is presented in the right pannel b Four probes resistance measurement as a function of carrier density for several angular alignments of the bilayer graphene and the BN handle from misaligned (15° - brown curve) to fully aligned (0° - light blue) at room temperature c Resistance of the charge neutrality point as a function of the angular alignment d Four probes resistance measurement as a function of carrier density for a selection of angular alignments of b at 20 K The angular alignment is calculated from the position in energy of the satellite peaks and to be consistent among all our samples we establish that the aligned position with the highest resistance at the CNP will be named 0° of alignment such as the observation of the valley Hall effect are all consistently observed in the alignment with highest resistant at room temperature g Non-local resistance as a function of the local resistivity Solid lines are linear fits to the experimental data meant to extract the power law dependence g) are measurements between T = 1.4 K and T = 110 K This suggest that the non-local signal is independent of the local one In other words that this is not a simple ohmic response when \({\sigma }_{{{{{{{{\rm{xy}}}}}}}}}^{{{{{{{{\rm{v}}}}}}}}}\ll \sigma\) \({\sigma }_{{{{{{{{\rm{xy}}}}}}}}}^{{{{{{{{\rm{v}}}}}}}}}\) is the valley Hall conductivity ρ = 1/σ is the local resistivity and W and L are the width and length of the sample where the non-local signal is independent of the local response is characteristic of a system where the valley conductivity is larger than the local conductivity which implies a fully developed valley Hall effect For T > 40 K we do not have a clear picture of why we observe a nearly quadratic behavior we attribute this to a mixture of regimes where both the valley Hall effect and ohmic response compete This regime needs more experimental and theoretical investigation Further combinations of scanning gates and electron transport will be necessary to clarify our observation a Electronic band structures for a relaxed bilayer graphene aligned with BN with 0.1 V/nm of displacement field for 0° b Local energy gaps obtained by thermal activation at different crystallographic alignments (for sample I and II) and theoretical values obtained from (a) divided by a factor of 4.5 Error bars represent the standard deviation of our measurements a moiré pattern of λ ~ 14 nm is clearly observed Note that the asymmetry on the AFM image and large value on the height of the deformation is an artifact given by the width of the AFM tip being comparable to the size of the features we want to measure (~5 nm) This confirms the existence of a commensurate state and the transmission of the corrugations to the second layer in aligned bilayer graphene/BN heterostructures the explanation of these is out of the scope of this manuscript A feature that we do not recover in our experimental measurements is the presence of an energy gap in the valence band for 0° of alignment Our hypothesis is that this energy gap is too small to be observed in our sample Even when the quality of our samples is remarkable the calculated energy gap is about four times smaller than the energy gap at the CNP which will place it out of reach in our temperature range At low temperature the values of the resistance are inverted now 0° of alignment has a resistance that is about six times lower than the 60° alignment This can be explain by the presence of the valley Hall effect which will reduce the scattering creating a much better conduction in the 0° case these measurements can only be taken as a signal of a larger strain but cannot be used to calculate the pseudo-magnetic field of the system given that they represent an average over the whole device local measurements such as scanning tunneling microscopy If this effect is at the origin of the valley Hall effect in monolayer graphene the picture becomes more complicated when dealing with bilayer graphene Following the results of our numerical simulations we can say that the spatial variations of broken sublattice symmetry will be different between the two layers and it will always exist for the first layer It is then not evident why the valley effect is observed for only one of the two layer alignments further numerical investigations would be needed to clarify the situation our experimental results show the existence of non-identical moirés in bilayer graphene aligned with BN We attribute this difference to the atomic structure relaxation of the commensurate state which modifies the band structure of bilayer graphene in different ways for 0° and 60° of alignment The observation of the valley Hall effect with a hundred and twenty degrees periodicity cannot be explained by current theoretical model We hope that our experimental results inspire further theoretical and experimental developments to address the existence of the valley Hall effect in this system Low temperature transport measurements were performed at 10 nA using lock-in amplifiers and low frequency In the case of non-local measurements a high impedance amplifier was also implemented to avoid any leaking currents we use an OPA to keep the potential of the sample constant The large tight-binding Hamiltonian matrix of moiré structures is diagonalized using conventional eigenvalue calculations of large-sparse matrices The Source Data underlying the figures of this study are available at https://doi.org/10.5281/zenodo.7982000 Codes used to support our findings are available upon request to Viet-Hung Nguyen and Jean-Christophe Charlier Massive Dirac fermions and hofstadter butterfly in a van der Waals heterostructure Evidence for a fractional fractal quantum Hall effect in graphene superlattices Origin of band gaps in graphene on hexagonal boron nitride New generation of Moiré superlattices in doubly aligned hBN/Graphene/hBN heterostructures High-quality electrostatically defined hall bars in monolayer graphene Proximity-induced ferromagnetism in graphene revealed by the anomalous hall effect Gate-tunable topological valley transport in bilayer graphene - nature physics Generation and detection of pure valley current by electrically induced Berry curvature in bilayer graphene - Nature Physics Evidence for the ballistic intrinsic spin Hall effect in HgTe nanostructures Observation of the quantum valley hall state in ballistic graphene superlattices Topological valley currents via ballistic edge modes in graphene superlattices near the primary Dirac point - communications physics Topological valley currents in bilayer graphene/hexagonal boron nitride superlattices Intrinsic valley Hall transport in atomically thin MoS2 Nonlocal charge transport mediated by spin diffusion in the spin Hall effect regime Nonlocal topological valley transport at large valley Hall angles Valley hall effect in two-dimensional hexagonal lattices Long-range nontopological edge currents in charge-neutral graphene - Nature Coherent jetting from a gate-defined channel in bilayer graphene Gap opening in twisted double bilayer graphene by crystal fields Transport spectroscopy of ultraclean tunable band gaps in bilayer graphene Shintaku, T. et al. Berry curvature induced valley Hall effect in non-encapsulated hBN/Bilayer graphene heterostructure aligned with near-zero twist angle. arXiv https://doi.org/10.48550/arXiv.2301.02358 (2023) Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering Global strain-induced scalar potential in graphene devices Effects of strain on electronic properties of graphene Valley Hall effect and nonlocal resistance in locally gapped graphene Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene Registry-dependent interlayer potential for graphitic systems Download references The authors acknowledge discussions with Ulf Gennser Xuan-Hoang TRINH for his helps in implementation of numerical codes to compute the lattice atomic structure relaxation This work was done within the C2N micro nanotechnologies platforms and partly supported by the RENATECH network the General Council of Essonne and the DIM-SIRTEQ This work was supported by: ERC starting grant N° 853282 - TWISTRONICS (R.R-P.) Fédération Wallonie-Bruxelles through the ARC Grant N° 21/26-116 (V.-H.N Flag-Era JTC projects “TATTOOS” N° R.8010.19 (V.-H.N and R.R.-P.) and “MINERVA” N° R.8006.21 (V.-H.N Pathfinder project “FLATS” N° 101099139 (V.-H.N and J.-C.C.) and EOS project “CONNECT” N° 40007563 (V.-H.N and from the Belgium F.R.S.-FNRS through the research project N° T.029.22F (V.-H.N Computational resources have been provided by the CISM supercomputing facilities of UCLouvain and the CE CI consortium funded by F.R.S.-FNRS of Belgium N° 2.5020.11 (V.-H.N These authors contributed equally: Everton Arrighi Centre de Nanosciences et de Nanotechnologies (C2N) Dominique Mailly & Rebeca Ribeiro-Palau Institute of Condensed Matter and Nanosciences Université catholique de Louvain (UCLouvain) Viet-Hung Nguyen & Jean-Christophe Charlier fabricated the devices for electron transport measurements fabricated the samples for structural characterization and performed the AFM measurements and R.R-P performed the electron transport experiments and analyzed the data grew the crystals of hexagonal boron nitride performed the numerical simulations and participated to the data analysis All authors participated to writing the paper Nature Communications thanks Shaffique Adam and the other, anonymous, reviewers for their contribution to the peer review of this work. A peer review file is available Reprints and permissions Download citation DOI: https://doi.org/10.1038/s41467-023-43965-x Metrics details Overlaying two atomic layers with a slight lattice mismatch or at a small rotation angle creates a moiré superlattice which has properties that are markedly modified from (and at times entirely absent in) the ‘parent’ materials Such moiré materials have progressed the study and engineering of strongly correlated phenomena and topological systems in reduced dimensions The fundamental understanding of the electronic phases requires a precise control of the challenging fabrication process involving the rotational alignment of two atomically thin layers with an angular precision below 0.1 degrees Here we review the essential properties of moiré materials and discuss their fabrication and physics from a reproducibility perspective References 1,2 report the observation of a correlated insulating state and superconductivity in twisted bilayer graphene Tunable van Hove singularities and correlated states in twisted monolayer–bilayer graphene This reference shows that the flat bands in moiré TMD structures can be described by generalized Hubbard models and that a number of many-body ground states are possible References 15,16 report a Mott insulator in WSe2/WS2 bilayer superlattices at half filling and generalized Wigner crystal and ferromagnetic states at fractional fillings This reference reports a gate-tunable insulator state in an r-TLG/hBN moiré superlattice at half filling Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene Topological winding number change and broken inversion symmetry in a Hofstadter’s butterfly High-precision twist-controlled bilayer and trilayer graphene The hot pick-up technique for batch assembly of van der Waals heterostructures Study of local anodic oxidation regimes in MoSe2 Zhou, H. et al. 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Preprint at https://arxiv.org/abs/2107.01796 (2021) Raman characterization of ABA- and ABC-stacked trilayer graphene Self-retracting motion of graphite microflakes Observation of microscale superlubricity in graphite Observation of high-speed microscale superlubricity in graphite Macroscopic self-reorientation of interacting two-dimensional crystals Stacking transition in bilayer graphene caused by thermally activated rotation Superlubric sliding of graphene nanoflakes on graphene This reference reports the commensurate–incommensurate transition in monolayer graphene at very small angles to the underlying hBN substrates Twinning and twisting of tri- and bilayer graphene This reference maps the strain fields and structural relaxation in t-BLG using Bragg interferometry This reference demonstrates strong three-dimensional buckling reconstruction and large in-plane strain redistribution in WSe2/WS2 moiré heterostructures This reference theoretically shows the presence of flat bands in t-BLG and predicts the magic angle of 1.05° Electrically tunable gauge fields in tiny-angle twisted bilayer graphene Strain-engineering of twist-angle in graphene/hBN superlattice devices Rotational disorder in twisted bilayer graphene and 2D electronic superlattices in MoS2/WSe2 hetero-bilayers Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene Electronic correlations in twisted bilayer graphene near the magic angle This reference maps the twist-angle disorder in t-BLG using scanning SQUID microscopy Cascade of phase transitions and Dirac revivals in magic-angle graphene Sainz-Cruz, H., Cea, T., Pantaleón, P. A. & Guinea, F. High transmission in twisted bilayer graphene with angle disorder. Preprint at https://arxiv.org/abs/2105.03383 (2021) Graphene transfer in vacuum yielding a high quality graphene Autonomous robotic searching and assembly of two-dimensional crystals to build van der Waals superlattices Vertical and in-plane heterostructures from WS2/MoS2 monolayers Orbital magnetic states in moiré graphene systems All magic angles in twisted bilayer graphene are topological Faithful tight-binding models and fragile topology of magic-angle bilayer graphene Failure of Nielsen–Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle Geometric and conventional contribution to the superfluid weight in twisted bilayer graphene Superfluid weight and Berezinskii–Kosterlitz–Thouless transition temperature of twisted bilayer graphene Topology-bounded superfluid weight in twisted bilayer graphene Tian, H. et al. Evidence for flat band Dirac superconductor originating from quantum geometry. Preprint at https://arxiv.org/abs/2112.13401 (2021) This reference reports the observation of a correlated insulating state orbital magnetism and superconductivity at every integer filling of t-BLG Stepanov, P. et al. Competing zero-field Chern insulators in superconducting twisted bilayer graphene. Preprint at https://arxiv.org/abs/2012.15126 (2020) Interaction effects and superconductivity signatures in twisted double-bilayer WSe2 Rodan-Legrain, D. et al. Highly tunable junctions and nonlocal Josephson effect in magic angle graphene tunneling devices. Preprint at https://arxiv.org/abs/2011.02500 (2020) Vries, F. K. D. et al. Gate-defined Josephson junctions in magic-angle twisted bilayer graphene. 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Preprint at https://arxiv.org/abs/2103:09850 (2021) Origin of magic angles in twisted bilayer graphene Non-Abelian gauge potentials in graphene bilayers Magic angle hierarchy in twisted graphene multilayers Γ valley transition metal dichalcogenide moiré bands Download references acknowledges support from the Department of Energy under grant number DOE DE-SC0020187 acknowledges support from the National Science Foundation under grant numbers DMR 2004801 and DMR-2105028 acknowledges support from the Army Research Office under grant number W911NF-18-1-0416 and the National Science Foundation under grant numbers DMR-1945351 acknowledges support from the Air Force Office of Scientific Research under award number FA9550-20-1-0219 and School of Applied and Engineering Physics Nature thanks Roman Gorbachev and the other Download citation DOI: https://doi.org/10.1038/s41586-021-04173-z Metrics details Moiré systems formed by 2D atomic layers have widely tunable electrical and optical properties and host exotic strongly correlated and topological phenomena correlated insulator states and orbital magnetism researchers studying different aspects of moiré materials discuss the most exciting directions in this rapidly expanding field acknowledges support from the Welch Foundation grant F-2018-20190330 the National Science Foundation grant DMR-1720595 and the Army Research Office grant W911NF-17-1-0312 acknowledges support from the US Department of Energy (DOE; DOE-FG02-99ER45742) and the Gordon and Betty Moore Foundation (GBMF9453) acknowledges support from DOE grant DE-FG02-02ER45958 acknowledges support by the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems (EPiQS) Initiative through grant GBMF9643 and the Fundación Ramón Areces acknowledges the support of the US Office of Naval Research under award N00014-20-1-2609 acknowledges funding from the Gordon and Betty Moore Foundation as part of the EPiQS initiative GBMF 9469 acknowledges the support of the Air Force Office of Scientific Research under award number FA9550-20-1-0219 was supported by NSF grant DMR-1911666 and partially through a Simons Investigator Award from the Simons Foundation acknowledges support from the Ministry of Economy and Competitiveness of Spain through the Severo Ochoa programme for centres of excellence in research and development (SE5-0522) the Generalitat de Catalunya through the CERCA program and the La Caixa Foundation The Barcelona Institute of Science and Technology Departments of Physics and Applied and Engineering Physics Department of Electrical and Computer Engineering Andrei is Board of Governors Chaired Professor in the Department of Physics and Astronomy at Rutgers University She studies the effects of moiré structures on the electronic properties of stacked 2D crystals by using transport and scanning tunnelling microscopy and spectroscopy Her group discovered that the twist angle between two superposed graphene crystals controls their electronic properties She is currently exploring new pathways for engineering flat bands in 2D materials using buckling transformations and substrate patterning Efetov joined Institut de Ciencies Fotoniques in Barcelona as a Professor and Group Leader in 2017 His group investigates moiré materials with electronic transport and optoelectronic techniques and is credited for the development of the highest-quality magic-angle twisted bilayer graphene devices demonstrated so far Pablo Jarillo-Herrero is currently Cecil and Ida Green Professor of Physics at Massachusetts Institute of Technology and is the recipient of the American Physical Society 2020 Oliver E Buckley Condensed Matter Physics Prize and the 2020 Wolf Prize in Physics for the discovery of correlations and superconductivity in magic-angle graphene His research interests lie in the area of experimental condensed matter physics quantum electronic transport and optoelectronics in novel 2D materials with special emphasis on investigating their superconducting Richardson Foundation Regents Chair Professor of Physics at The University of Texas at Austin his primary research interests centre on the influence of electron–electron interactions on the electronic properties of metals and semiconductors on the understanding of moiré superlattice systems Kin Fai Mak is an Associate Professor of Physics and of Applied and Engineering Physics at Cornell University His research group uses optical and electrical probes to explore condensed matter phenomena in atomically thin materials and their heterostructures Senthil is a Professor of Physics at Massachusetts Institute of Technology He is a theorist interested in correlated materials their novel phases and the associated phase transitions Emanuel Tutuc is a Professor of Electrical and Computer Engineering with a courtesy appointment in Physics at The University of Texas at Austin His research interests are the electronic properties of low-dimensional systems and the realization of novel devices His group introduced the experimental techniques used to realize rotationally controlled van der Waals heterostructures Ali Yazdani is the Class of 1909 Professor of Physics and the Director of the Princeton Center for Complex Materials He specializes in the development and the application of high-resolution microscopy and spectroscopy to quantum materials His group has uncovered a wide range of novel correlated and topological phenomena Young is Associate Professor at the University of California His group combines nanofabrication and electronic measurement techniques to investigate the properties of electronic states in quantum materials Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Reprints and permissions Download citation DOI: https://doi.org/10.1038/s41578-021-00284-1 Metrics details Lattice reconstruction and corresponding strain accumulation plays a key role in defining the electronic structure of two-dimensional moiré superlattices including those of transition metal dichalcogenides (TMDs) Imaging of TMD moirés has so far provided a qualitative understanding of this relaxation process in terms of interlayer stacking energy while models of the underlying deformation mechanisms have relied on simulations we use interferometric four-dimensional scanning transmission electron microscopy to quantitatively map the mechanical deformations through which reconstruction occurs in small-angle twisted bilayer MoS2 and WSe2/MoS2 heterobilayers We provide direct evidence that local rotations govern relaxation for twisted homobilayers while local dilations are prominent in heterobilayers possessing a sufficiently large lattice mismatch Encapsulation of the moiré layers in hBN further localizes and enhances these in-plane reconstruction pathways by suppressing out-of-plane corrugation We also find that extrinsic uniaxial heterostrain which introduces a lattice constant difference in twisted homobilayers leads to accumulation and redistribution of reconstruction strain demonstrating another route to modify the moiré potential to directly map the intralayer mechanical deformations driving reconstruction in TMD moiré systems We identify distinct reconstruction mechanisms for moiré homobilayers versus heterobilayers and examine their twist angle dependence distinguishing the relative importance of local lattice rotations and dilations in both systems as well as the critical role that encapsulation layers play in affecting the balance between these in-plane deformations and out-of-plane corrugations We also measure reconstruction-induced strain fields and demonstrate how application of an external mechanical force can be leveraged to manipulate strain distributions a Schematic of Bragg interferometry measurement TMD1 and TMD2 are MoS2 for homobilayers or MoS2 and WSe2 for heterobilayers Dashed box and corresponding inset illustrate formation of a moiré superlattice between TMD1 and TMD2 ky indicate real space and reciprocal space coordinate systems c Average convergent beam electron diffraction patterns for a MoS2/MoS2 moiré homobilayer and WSe2/MoS2 moiré heterobilayer with moiré twist angle θm and lattice constant percent difference δ Overlapping TMD Bragg disks are highlighted in the insets e Representative displacement maps for moiré bilayers with parallel (P) and anti-parallel (AP) orientations The moiré twist angle and heterostrain are labelled as θm and ϵ g High-symmetry stacking sequences and corresponding displacement vectors for P and AP H-phase TMD moiré bilayers h 2D displacement hexagon legend for the displacement field maps in d and e signifying the magnitudes and directions of the local displacement vectors in pixel hues and values Here ux and uy represent interlayer displacements in the x and y directions and SP3 represent the three unique saddle point stacking directions although the hBN encapsulation layers are colocalized with the TMD layers in real space the hBN Bragg disks are sufficiently offset from those of the TMD layers in reciprocal space and thus do not impede the structural analysis in Bragg interferometry This 4D-STEM approach is therefore not restricted by buried interfaces as in the case of scanning probe methods (which often require the sample surface to be exposed) and conventional real-space electron imaging methods (that can be obscured by encapsulating layers) ostensibly suggesting that reconstruction is relatively weak compared to the twisted homobilayers the strain mapping and geometric analyses that follow show that reconstruction remains strong even in these cases a–c Maps of local reconstruction rotation (θR) and (g–i) maximum shear strain (γmax) for P-stacked MoS2/MoS2 with θm = 1.23∘ and AP-stacked MoS2/MoS2 with θm = 0.77∘ and 1.69∘ Hexagonal insets show the average reconstruction rotation (〈θR〉) and shear strain (〈γmax〉) as a function of interlayer displacement (ux The overlaid dashed lines correspond to the moiré unit cell geometry d–f Average dilations (〈Dil〉) as a function of displacement j Rotation-driven reconstruction schematics for a P-stacked and two AP-stacked moiré homobilayer cases (θm < θc and θm > θc where θc is the critical twist angle separating two reconstruction regimes) Yellow indicates accumulation of shear strain (γmax > 0) and blue indicates no shear strain Arrows illustrate the measured direction of θR with counterclockwise rotation defined as θR > 0 Arrow sizes depict relative growth or shrinkage of local stacking domain n relative stacking areas as a function of twist angle (θm) for AP and P stacking orientations n) indicate the relative stacking areas in a rigid moiré based on the chosen partitioning of the displacement space horizontal error bars represent standard deviations and vertical error bars represent standard errors Dotted trend lines are polynomial fits to the experimental data fewer stacking sequences in the AP case correspond to the energy extremes While reconstruction in a P moiré bilayer is driven by a preference for both MX and XM stacking over MMXX reconstruction in AP moiré bilayers is driven predominantly by a preference for XMMX over XX stacking with relatively little preference for size of MM regions This produces a stronger driving force for reconstruction in the P moiré bilayer Relative stacking area percentages for (a) AP (θm = 0.13∘) Light blue-gray bars indicate stacking areas calculated for the rigid (not reconstructed) moiré and dark blue bars represent experimentally measured values d–f Maps of local dilations for the samples from a–c Hexagonal insets show show the average dilation (〈Dil〉) as a function of interlayer displacement (ux h–j Corresponding 2D plots of average reconstruction rotation (〈θR〉) as a function of displacement g Schematics of dilation-driven reconstruction and accumulation of volumetric strain Orange indicates a positive dilation and purple indicates a negative dilation Arrows illustrate volumetric expansion (pointing outward) or compression (pointing inward) Arrow sizes represent relative growth or shrinkage of the local stacking domain a–c Example convergent beam electron diffraction patterns for an AP WSe2/MoS2 sample with three regions: fully encapsulated with hBN (a Dashed and solid boxes highlight representative hBN and TMD diffraction disks d–f Average dilations (〈Dil〉) as a function of interlayer displacement (ux g–i Corresponding 2D plots of average reconstruction rotation (〈θR〉) as a function of displacement j Schematic (exaggerated) depicting the effects of out-of-plane corrugation on projected interlayer displacements Magnified pictures of select regions are provided in boxes 1 and 2) k Difference in dilation (ΔDil) as a function of stacking parameter for suspended versus fully encapsulated (dark blue-gray curve) and suspended versus partially encapsulated (light blue-gray curve) structures with θm ≈ 0.1–0.2∘ Dilation values for each case obtained by taking line cuts through the average dilation plots in d–f Gray dashed line represents the theoretically calculated ΔDil comparing the corrugated versus rigid heterobilayer l–n Maps of local dilations for fully capped (l The experimental ΔDil profile for the encapsulated versus suspended case displays clear similarities to that for the theoretical corrugated structure indicating that the suspended heterobilayer has undergone a combination of in-plane and out-of-plane reconstruction while the fully encapsulated structure is reconstructed almost entirely in-plane The residual differences between the experimental and theoretical curves suggest that some out-of-plane corrugation may remain in the encapsulated structure since the thin hBN is not perfectly rigid; however overall the corrugations have been largely suppressed by the presence of hBN on both sides the ΔDil profile for the partially encapsulated versus suspended case differs more substantially from that of the corrugated model due to further mixing of in-plane and out-of-plane reconstruction when only one side of the heterobilayer is encapsulated which were measured for partially encapsulated structures are likely systematically smaller than those in analogous fully encapsulated structures Average reconstruction rotation (〈θR〉) and maximum shear strain (〈γmax〉) as a function of interlayer displacement (ux uy) for P (a–c) and AP (g–i) MoS2/MoS2 moiré homobilayers with varying amounts of heterostrain Corresponding maps of maximum shear strain (γmax) shown in d–f and j–l White arrows indicate moiré unit cell extension (e Scan regions affected by sample charging during data acquisition have been removed for clarity in j–l Bragg interferometry is also distinctively compatible with both freely suspended and encapsulated moiré structures we found that hBN capping layers suppress out-of-plane relaxation modes and subsequently promote in-plane deformations implicating critical connections between sample design The versatility of this methodology could also enable extension to more complex opening avenues for direct correlative measurements between lattice reconstruction and electrically controllable emergent (opto)electronic phenomena Electron microscopy was performed at the National Center for Electron Microscopy in the Molecular Foundry at Lawrence Berkeley National Laboratory Four-dimensional STEM datasets were acquired using a Gatan K3 direct detection camera located at the end of a Gatan Continuum imaging filter on a TEAM I microscope (aberration-corrected Thermo Fisher Scientific Titan 80–300) The microscope was operated in energy-filtered STEM mode at 80 kV with a 10 eV energy filter centred around the zero-loss peak an indicated convergence angle of 1.71 mrad and a typical beam current of 45–65 pA depending on the sample These conditions yield an effective probe size of 1.25 nm (full-width at half-maximum value) Diffraction patterns were collected using a step size of either 0.5 nm or 1 nm with 50 x 50 to 300 x 300 beam positions covering an area ranging from 25 nm x 25 nm to 300 nm x 300 nm The K3 camera was used in full-frame electron counting mode with a binning of 4 × 4 pixels and an energy-filtered STEM camera length of 800 mm Each diffraction pattern had an exposure time of 13 ms which is the sum of multiple counted frames In this equation and all subsequent analysis gj are the reciprocal space vectors associated with each Bragg peak a1 and a2 are the average real space lattice vectors from the two TMD layers This procedure involves using one of several strategies to partition the displacements into zones of constant lattice vector offsets (n m) such that u + na1 + ma2 is continuously oriented followed by use of a mixed-integer program to refine zone boundaries The following analysis therefore assumes a large moiré pattern λ ≫ a0 and that the local variation of utop in reconstructed materials preserves ∣∣ ∇ utop∣∣∞ ≪ 1 we can decompose the displacement gradient ∇ utop as the sum of a symmetric infinitesimal strain tensor \(\underline{\underline{\epsilon }}\) and a skew-symmetric infinitesimal rotation matrix \(\underline{\underline{\omega }}\) where we have defined the total local rotation in the top layer ∇ × utop as \({\theta }_{T}^{top}\) and finite width of the electron beam may soften the observed strain and stacking features but do not affect the overall trends or conclusions drawn The code used for data processing and strain analysis is publicly available on GitHub via Zenodo56 Tunable angle-dependent electrochemistry at twisted bilayer graphene with moiré flat bands Intralayer charge-transfer moiré excitons in van der Waals superlattices Hyperspectral imaging of exciton confinement within a moiré unit cell with a subnanometer electron probe Signatures of moiré trions in WSe2/MoSe2 heterobilayers Electrically Tunable localized versus delocalized intralayer moiré excitons and trions in a twisted MoS2 bilayer Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice Twist angle-dependent atomic reconstruction and moiré patterns in transition metal dichalcogenide heterostructures Torsional Periodic Lattice Distortions and Diffraction of Twisted 2D Materials Deep moiré potentials in twisted transition metal dichalcogenide bilayers Origin and evolution of ultra flat bands in twisted bilayer transition metal dichalcogenides: Realization of triangular quantum dots Lattice reconstruction induced multiple ultra-flat bands in twisted bilayer WSe2 Ultraflatbands and shear solitons in moiré patterns of twisted bilayer transition metal dichalcogenides Band energy landscapes in twisted homobilayers of transition metal dichalcogenides Interferometric 4D-STEM for lattice distortion and interlayer spacing measurements of bilayer and trilayer 2D materials Four-dimensional scanning transmission electron microscopy (4D-STEM): From scanning nanodiffraction to ptychography and beyond van der Waals heterostructures with high accuracy rotational alignment Imaging the crystal orientation of 2D transition metal dichalcogenides using polarization-resolved second harmonic-generation Strain mapping of two-dimensional heterostructures with subpicometer precision Strain mapping at nanometer resolution using advanced nano-beam electron diffraction Solid Mechanics Lecture Notes (University of Auckland McGinty, B. Continuum Mechanics. https://continuummechanics.org (2012) In The Linearized Theory of Elasticity 97-155 (Brikhäuser Relaxation and domain formation in incommensurate two-dimensional heterostructures Twisted bilayer graphene: Moiré with a twist Structural and electron diffraction scaling of twisted graphene bilayers Coexisting ferromagnetic-antiferromagnetic state in twisted bilayer CrI3 py4DSTEM: A software package for four-dimensional scanning transmission electron microscopy data analysis estimation and predictive control in APMonitor SciPy 1.0: fundamental algorithms for scientific computing in Python Ab-initio simulations of materials using VASP: Density-functional thoery and beyond Van Winkle, M., Craig, I. M., Bustillo, K. C., Ciston, J. & Bediako, D. K. Rotational and dilational reconstruction in transition metal dichalcogenide moiré bilayers [Data sets]. Zenodo, https://doi.org/10.5281/zenodo.7779104 (2023) Craig, I. M. pyInterferometry. GitHub. https://doi.org/10.5281/zenodo.7863859 (2023) Download references This material is based upon work supported by the U.S National Science Foundation (NSF) under award no acknowledges support from a University of California Berkeley Berkeley Fellowship and a National Defense Science and Engineering Graduate (NDSEG) Fellowship under contract FA9550-21-F-0003 sponsored by the Air Force Research Laboratory (AFRL) the Office of Naval Research (ONR) and the Army Research Office (ARO) acknowledges support from the National Science Foundation under grant no Department of Energy (DOE) Early Career Research Award acknowledges support from the Presidential Early Career Award for Scientists and Engineers through the U.S LBNL was supported by the Office of Science Computational studies were carried out using supercomputing resources of the National Energy Research Scientific Computing Center (NERSC) and the TMF clusters managed by the High-Performance Computing Services Group at LBNL and were supported by the Laboratory Directed Research and Development Program of LBNL under the same Contract No Other instrumentation used in this work was supported by grants from the Gordon and Betty Moore Foundation EPiQS Initiative (Award no Canadian Institute for Advanced Research (CIFAR-Azrieli Global Scholar and the 3M Foundation through the 3M Non-Tenured Faculty Award (no acknowledge support from the Elemental Strategy Initiative conducted by the Ministry of Education JPMXP0112101001) and Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research (KAKENHI; grant nos These authors contributed equally: Madeline Van Winkle created the data analysis code and strain calculation framework with input from C.O carried out DFT and relaxation simulations interpreted the data and wrote the manuscript All the authors contributed to the overall scientific discussion and edited the manuscript Download citation DOI: https://doi.org/10.1038/s41467-023-38504-7 Metrics details Moiré superlattices (MSLs) are modulated structures produced from homogeneous or heterogeneous 2D layers stacked with a twist angle and/or lattice mismatch Expanding the range of available materials and realization of unique emergent properties are key challenges Here we report a facile bottom-up synthesis of homogeneous MSL based on a wide-gap 2D semiconductor using a one-pot solvothermal approach with robust reproducibility Unlike previous MSLs usually prepared by directly stacking two monolayers direct way through chemical growth of spiral-type nanosheets driven by screw-dislocations We find emergent properties including large band gap reduction (∼0.6 eV) and strongly enhanced photocatalytic activity First-principles calculations reveal that such unusual properties can be ascribed to the locally enhanced inter-layer coupling associated with the Moiré potential modulation Our results demonstrate the promise of MSL materials for chemical and physical functions The lack of diverse set of fabricated MSL materials impedes understanding of properties of these materials as well as applications It is therefore desirable to find ways of fabricating the MSLs consisting of alternative 2D material sublayers and exploring their emergent unique properties Furthermore we find a more than twofold increase in carrier lifetime which facilitates carriers separation for optoelectronics As the result the nanosheets show much enhanced photocatalytic activity by comparison with bulk BiOCl The experimental observations are supported by first-principles calculations which ascribe the underlying mechanism to the locally enhanced interlayer coupling associated with the Moiré pattern leading to a spatially modulated electronic structure feature including demonstration of enhanced chemical functionality in an MSL show the promise of exploiting the MSLs to modulate properties of 2D materials for practical electronic optoelectronic and photocatalytic applications Morphological and structural characterization of BiOCl spiral nanosheets a SEM images of BiOCl nanosheet with the spiral shape (either clockwise or anti-clockwise) b AFM height-sensor micrograph of the nanosheet (scale bar: 500 nm) One can see two sets of distributed centrosymmetric screw related fringes (marked in red and white) The cracks and irregular edges may be associated with the damage caused by the external forces during ultrasonic treatment and stirring in the process of sample preparation or the result of relaxation of residual strains possibly existing in the nanosheets Selected marginal region (in red circle) is measured by high-resolution TEM (e) (scale bar: 5 nm) and selected centered spiral region (in blue) is zoomed in to clearly show periodicity of Moiré pattern (g) (scale bar: 20 nm) h Selected area electron diffraction measurements for the regions of (e) and (g) two sets of diffraction spots and formed intersection angle of 2.0° indicates adjacent nanosheets are twisted by 2.0° i High-resolution TEM cross-sectional image of single nanosheet which indicates each sheet consists of seven BiOCl atomic monolayers (scale bar: 2 nm) j Schematic of the grown BiOCl spiral nanosheets driven by the screw-dislocation Each nanosheet consists of seven BiOCl monolayers (only O atoms are shown k 3D scheme and 2D perpendicular view of the bilayer Moiré superlattice model used in first-principles calculations only O atoms are shown and the upper and lower layers are shown in dodgerblue and purple The period of Moiré superlattice L is indicated This captures the key structural features that are responsible for the spatial modulation of physical properties of interest MSL percentage is evaluated based on the coverage of the spiral nanosheet region on the TEM images The error bar represents the variation range of the data from multiple times of measurements d Time-resolved photoluminescence spectra of BiOCl bulk (in black) BiOCl MSL nanosheets measured at the reaction time of 30 min (in red) The measurements were taken via time-correlated single-photon counting (monitored at 455 nm) with an excitation wavelength of 365 nm The average carrier lifetimes inferred from the emission decay (see main text) are indicated indicating uniaxial-oriented growth of BiOCl MSLs Some peaks missing in the XRD spectra for 40 min and above is ascribed to the broken in-plane crystal periodicity with the emergence of Moiré pattern in the incommensurately twisted stacking of the nanosheets This shows a similar band gap with our nanosheets at the initial growth stage before formation of the MSL The average carrier lifetime τavg is further evaluated by τavg = A1τ1 + A2τ2 with component proportion magnitudes A1 = B1τ1/(B1τ1 + B2τ2) and A2 = B2τ2/(B1τ1 + B2τ2) We find the high proportion of short component A1 for all the samples measured (>95%) consistent with the indirect band gap of BiOCl the photo-induced carriers decay more slowly we find that the average lifetime τavg changes from 2.86 ns in the bulk sample to 3.94 ns in the spiral nanosheet at 30 min and to 6.21 ns in the spiral nanosheet at 120 min This suggests charge separation of photogenerated carriers in the MSLs Electronic structure calculations of BiOCl MSL a Perpendicular view of the simulated BiOCl spiral bilayer structure with the twist angle of 1.7° forming a MSL with the periodicity L = 13.1 nm the upper and downer layers are shown in dodgerblue and purple b HRTEM image of an experimentally synthesized BiOCl spiral nanosheet with the twist angle of ~1.7° c The bilayer structures with different stacking patterns which dominate three distinct regions observed in (a) (indicated by red d Variation of calculated band gaps throughout the MSL shown within the yellow squared region of (a) The inset shows the dependence of band gap on the interlayer distance d for the HH-stack d0 = 0.736 nm represents the experimentally determined distance between the two adjacent nanosheets e Distinct band structures of the HH-stack and AA-stack structures The band energies are aligned with respect to the vacuum energy level The contributions from Cl-px/y and Cl-pz orbitals are depicted by the sizes of red and blue circles f Schematic of how the valence bands of HH-stack and AA-stack bilayers are formed by the states of composing monolayers g Variation of band gap (upper panel) and band-edge positions (lower panel) evolving from the HH-stack to the AA-stack regions [i.e. the path indicated by the black arrow in (d)] In the upper panel the Cl-orbitals projected density of states for each point is shown and in the lower panel the chemical bonding character [as described in (f)] of the VBM state is given d The near-field amplitude image from s-SNOM measurement showing distribution of free carriers One observes that the centered MSL region shows the lower local carrier density indicated by the weaker near-field amplitude (scale bar: 500 nm) e Photocatalytic degradation rate of Rhodamine B versus time without catalyst and with BiOCl bulk Negative time denotes dark condition before light irradiation The measurements are taken under simulated sunlight equipped with an AM 1.5 G filter (UV-Vis) and visible-light (Vis) irradiation f Transient photocurrent response versus time without bias measured in a 0.5 mol L−1 Na2SO4 aqueous solution and the Ag/AgCl reference electrode under visible-light irradiation the VBMs vary gradually from one region to the other The gradually varying energy of hole states with the existing concentration gradient among different regions would be expected to result in reasonable hole transfer out of the AA-stack region which have almost the same area of the central screw-dislocation region The results show substantially enhanced photocatalytic activity in the BiOCl MSLs (120 min) This further evidences that the enhancement of photocatalytic activity is attributed to the emergence of MSL inducing strong modulation of optoelectronic properties as discussed above the MSL clearly shows enhanced photocatalytic activity for degrading the colorless TC while only 12.1% TC is degraded by the bulk and edge sites may be the reaction sites on BiOCl MSLs during catalytic degradation though the detection of explicit reaction sites is challenging and beyond the scope of this study we have demonstrated the bottom-up synthesis of 2D Moiré superlattices (MSLs) based on a wide-gap oxychloride semiconductor BiOCl The controlled growth was conducted in facile solvothermal reaction condition producing spiral-type nanosheets under a screw-dislocation driven mechanism The homogeneous MSL based on one semiconductor is of high crystalline quality the fabricated BiOCl MSLs leads to strong modulation of optoelectronic properties including up to 0.6 eV band gap reduction and resulted strongly enhanced visible-light photocatalytic activity We exploit the idea of formation of 2D MSLs and provide a clean modification of optoelectronic properties in pristine BiOCl without introducing any external impurities/materials This is distinct from the usual strategies for modifying the optoelectronic properties of BiOCl encapsulation involving metal nanoparticles heterostructures with other semiconductors First-principles calculations reveal that it is the locally enhanced interlayer coupling upon appearance of the Moiré pattern that results in modification of electronic band structure and optoelectronic properties By achieving such highly scalable MSLs with modulated photophysical properties our work advocate the promise of 2D MSL family in optoelectronics photocatalytic and practical environmental purifications our synthesized MSLs are formed by the stacking of two spiraling nanosheets with one twist angle It is possible to synthesize multiple spiraling nanosheets with the constant or changed twist angles between two adjacent sheets The former will result in multiple MSLs stacked along the perpendicular direction with the same periodicity whereas the latter will give rise to the MSLs with the changed periodicity the optical spectral properties will be contributed by individual MSLs and there could be a gradient in properties such as bands edges carrier concentration depending on the details Powder X-ray diffraction patterns (XRD) were obtained by using a Bragg–Brentano diffractometer (D8-tools Scanning electron microscopy (SEM) images were taken by using the Ultra-High Resolution (1.0 nm) Scanning Electron Microscope SU-8010 (HITACHI Co. The powders of BiOCl spiral nanosheet dispersed in water were dropped on 300 mesh carbon-coated copper grids and dried for the characterizations of transmission electron microscopy (TEM) high-resolution transmission electron microscopy (HRTEM) and selected area electron diffraction (SAED) These were acquired by using a JEM-2100F transmission electron microscope (JEOL Co. High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images and energy-disperse X-ray (EDX) elemental mapping were measured by using a FEI Tecnai G2 F20 transmission electron microscope equipped with a high-angle annular dark-field detector (FEI Company Atomic force microscopy (AFM) measurements were performed on the XE-100 multimode AFM from Park Systems (USA) Ultraviolet–visible diffused reflectance spectra were collected by using an ultraviolet–visible spectrophotometer (Shimadzu UV-2550 with BaSO4 as a reference) and were then converted from reflection to absorbance by the Kubelka-Munk method The ultraviolet photoelectron spectra (UPS) measurements were carried out using an ESCALAB MK II instrument (Thermo Fisher Scientific The X-ray photoelectron spectroscopy (XPS) was obtained over Thermos ESCALAB 250 spectrometer Electron paramagnetic resonance (EPR) spectra were recorded at 123 K using EPR spectrometer (A300-10/12 Aberration-corrected scanning transmission electron microscopy (STEM) images of BiOCl MSL were obtained by a JEOL ARM200F (JEOL Japan) operated at 200 kV with cold field emission gun and double hexapole Cs correctors (CEOS GmbH The fluorescence decay processes were recorded with the time-correlated single-photon counting (TCSPC) technique on an Edinburgh FLS920 phosphorescence lifetime system equipped with a 450 W Xe lamp and a time-correlated single-photon counting card at room temperature Nanoscale mapping of free carriers in the spiral nanosheets were performed by scattering-type scanning near-field optical microscopy (s-SNOM) The s-SNOM system is built based on an atomic force microscope (AFM) that is illuminated by a focused infrared (IR) laser beam Free-carrier mapping is achieved by recording the amplitude and phase of light backscattered from the metalized AFM tip X-ray absorption spectra (XANES) were collected at Beijing Synchrotron Facility (BSRF) on beamline 4B7B The BSRF storage ring is operated at the electron energy of 2.2 GeV with beam current of 250 mA A Si (111) double crystal monochromatic was applied The beam size used at the sample position is about 900 × 300 μm2 All the data were collected with the transmission mode at ambient temperature Curve fitting and data analysis were performed with Artemis and IFEFFIT software Rhodamine B (RhB) for photocatalytic degradation were purchased from Sinopharm Chemical Reagent Co The photocatalytic activity of the BiOCl MSL and bulk samples has been evaluated for photodecomposition of RhB under simulated sunlight from a 300 W Xenon lamp equipped with an AM 1.5 G filter (UV-Vis) or a 400 nm cutoff filter (Vis) The reaction temperature was controlled to be 24 °C by a thermostat bath Typically 0.02 g photocatalyst was dispersed in 50 mL RhB solution (10 mg L−1) in a reaction cell with a Pyrex jacket by sonication for 5 min the suspension was stirred in the dark for 30 min for adsorption–desorption equilibrium Three milliliters of the suspensions were withdrawn and centrifuged (12,000 × g 10 min) to separate the photocatalyst for UV-Vis spectrophotometer measurements (Shimadzu UV-2550) The concentration of RhB was determined by monitoring its characteristic absorption peak at 553 nm from UV-vis absorption spectra The degradation efficiency (E) of RhB was calculated by the formula of \(E = \frac{{C_0 - C}}{{C_0}} \times {\mathrm{100\% }}\) where C0 and C are the initial concentration (10 mg L−1) and the instant degradation concentration of Rhodamine B during the degradation process The photocatalytic tetracycline (TC) solution (20 mg L−1) measurements are implemented at ambient temperature (24 °C) Fifteen milligrams of catalyst is dispersed in 50 ml TC solution by sonication the reactant mixture is continously stirred in the dark for 30 min then illuminated by visible light (λ > 400 nm) A two milliliters suspension were withdrawn and centrifuged during the reaction the concentration variation of TC was examined by a UV-Vis spectrophotometer The photocatalyst powder was dispersed in H2O to form a 10 mg mL−1 solution under sonication for 30 min was cleaned by sonication in cleanout fluid The as-prepared solution was dropped onto the pretreated ITO surface and allowed to dry under vacuum conditions for 24 h at 60 °C the uncoated part of the ITO glass was isolated with epoxy resin The photocurrent was measured by an electrochemical analyzer (CHI660D Instruments) in a conventional three-electrode electrochemical cell with the working electrode a platinum foil counter electrode and Ag/AgCl (3 M KCl) as reference electrode A 300 W Xenon lamp with 400 nm cutoff filter was utilized as a light source A 0.5 M Na2SO4 aqueous solution was used as the electrolyte The Mott–Schottky measurements were carried out using a Bio-Logic SP-150 potentiostat equipped with a VMP3B-20 20A booster Spin-orbit coupling (SOC) is included in the electronic structure calculations The other data support the findings of this study are available from the corresponding author upon request 2D materials and Van der Waals heterostructures Electron quantum metamaterials in van der Waals heterostructures Observing the quantization of zero mass carriers in graphene Unraveling the intrinsic and robust nature of van hove singularities in twisted bilayer graphene by scanning tunneling microscopy and theoretical analysis Materials science: graphene moire mystery solved Quasicrystalline 30° twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling Atomic layers of hybridized boron nitride and graphene domains Observation of an intrinsic bandgap and Landau level renormalization in graphene/boron-nitride heterostructures Gaps induced by inversion symmetry breaking and second-generation Dirac cones in graphene/hexagonal boron nitride Dynamic band-structure tuning of graphene moiré superlattices with pressure Emergence of tertiary Dirac points in graphene Moiré superlattices Hofstadter butterfly and many-body effects in epitaxial graphene superlattice Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure Gate-dependent pseudospin mixing in graphene/boron nitride moiré superlattices Ballistic miniband conduction in a graphene superlattice Evidence of local commensurate state with lattice match of graphene on hexagonal boron nitride Strong interlayer coupling in Van der Waals heterostructures built from single-layer chalcogenides Atomically thin resonant tunnel diodes built from synthetic Van der Waals heterostructures Multicomponent fractional quantum Hall effect in graphene Micrometer-scale ballistic transport in encapsulated graphene at room temperature I) nanostructures for highly efficient photocatalytic applications Vacancy associates promoting solar-driven photocatalytic activity of ultrathin bismuth oxychloride nanosheets Solar water splitting and nitrogen fixation with layered bismuth oxyhalides Bismuth oxyhalide layered materials for energy and environmental applications The crystal structure of the bismuth oxyhalides Electrically active screw dislocations in helical ZnO and Si nanowires and nanotubes Imperfect oriented attachment: dislocation generation in defect-free nanocrystals Large‐scale growth of two‐dimensional SnS2 crystals driven by screw dislocations and application to photodetectors Screw dislocation-driven growth of two-dimensional nanoplates Screw-dislocation-driven bidirectional spiral growth of Bi2Se3 nanoplates Mechanism and kinetics of spontaneous nanotube growth driven by screw dislocations Screw dislocation driven growth of nanomaterials Broken symmetry induced strong nonlinear optical effects in spiral WS2 nanosheets Edge overgrowth of spiral bimetallic hydroxides ultrathin-nanosheets for water oxidation Controlled synthesis of layered double hydroxide nanoplates driven by screw dislocations Spontaneous growth of hexagonal ZrB2 nanoplates driven by a screw dislocation mechanism Screw dislocation-driven t-Ba2V2O7 helical meso/nanosquares: microwave irradiation assisted-SDBS fabrication and their unique magnetic properties Surface reorganization leads to enhanced photocatalytic activity in defective BiOCl Synthesis and facet-dependent photoreactivity of BiOCl single-crystalline nanosheets A facet-dependent Schottky-junction electron shuttle in a BiVO4{010}–Au–Cu2O Z-scheme photocatalyst for efficient charge separation Interlayer coupling in two-dimensional semiconductor materials First-principle high-throughput calculations of carrier effective masses of two-dimensional transition metal dichalcogenides Evolution of electronic structure as a function of layer thickness in group-VIB transition metal dichalcogenides: emergence of localization prototypes Soft x-ray absorption spectra of alkali halides Nanoscale free-carrier profiling of individual semiconductor nanowires by infrared near-field nanoscopy Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set Generalized gradient approximation made simple Chemical accuracy for the van der Waals density functional Influence of the exchange screening parameter on the performance of screened hybrid functionals Facile composition-controlled preparation and photocatalytic application of BiOCl/Bi2O2CO3 nanosheets Download references This work was financially supported by the National Key R&D Program of China (Grants 2016YFA0200400) the National Natural Science Foundation of China (21275064 Jilin province science and technology development program (20190201233JC and 20190201016JC) and Program for JLU Science and Technology Innovative Research Team (JLUSTIRT Bao acknowledges the support from Australian Research Council (ARC gratefully acknowledges financial support from Singapore Ministry of Education via two Tier1 grants (RG 113/16 and RG 194/17) We thank the Beijing Synchrotron Radiation Facility (BSRF) for providing us the beam time on beamline 1W1B for the XAS measurement Calculations were performed in part at the high performance computing center of Jilin University We express our sincere thanks to Shengye Jin and Shitan Wang for their discussions and help These authors contributed equally: Lulu Liu State Key Laboratory of Automotive Simulation and Control State Key Laboratory of Superhard Materials Key Laboratory of Automobile Materials of MOE School of Materials Science and Engineering College of Electronic Science & Technology and Institute for Advanced Study Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province ARC Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET) School of Physical and Mathematical Sciences carried out the first-principles electronic structure calculations performed materials properties characterization wrote the paper with inputs from all authors Peer review information Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work Download citation DOI: https://doi.org/10.1038/s41467-019-12347-7 Metrics details The atomic structure at the interface between two-dimensional (2D) and three-dimensional (3D) materials influences properties such as contact resistance Moiré engineering is yet to be utilized for tailoring this 2D/3D interface despite its success in enabling correlated physics at 2D/2D interfaces Using epitaxially aligned MoS2/Au{111} as a model system we demonstrate the use of advanced scanning transmission electron microscopy (STEM) combined with a geometric convolution technique in imaging the crystallographic 32 Å moiré pattern at the 2D/3D interface This moiré period is often hidden in conventional electron microscopy where the Au structure is seen in projection via ab initio electronic structure calculations that charge density is modulated according to the moiré period illustrating the potential for (opto-)electronic moiré engineering at the 2D/3D interface Our work presents a general pathway to directly image periodic modulation at interfaces using this combination of emerging microscopy techniques We show that iDPC and 4D STEM are able to decouple higher order moiré periods to form real space images of the moiré pattern at the 2D/3D interface of MoS2/Au{111} revealing the crystallographic 32 Å period We explain the difference compared to conventional (S)TEM in terms of projection effects of the ABC stacking of the 3D metal We then use ab initio electronic structure calculations to corroborate that MoS2/Au{111} charge density modulation is concentrated at the interface and follows the 32 Å moiré periodicity Together these findings demonstrate the utility of direct imaging via iDPC and 4D STEM for understanding the structure and electronic properties of 2D/3D heterostructures a Schematic of epitaxially aligned Au deposited on suspended MoS2 supported on a SiNx TEM grid The SiNx membrane is shown in dark blue b Reciprocal space model and experimental selected area electron diffraction pattern of the Au {111} zone aligned on MoS2 {0001} with weak intensity 1/3{422}Au spots (see text) aligned with \(\{10\overline 10\}_{\rm{MoS}_{2}}\) and higher intensity {220}Au spots aligned with \(\{2\overline {11}0\}_{\rm{MoS}_{2}}\) Orange dots represent frequencies from Au crystal planes while purple represent frequencies from MoS2 crystal planes showing apparent 18 Å-period moiré pattern d High angle annular dark field (HAADF) STEM image Coalescence boundaries are marked by blue arrows a Atomic model [100] zone axis for the 32 Å moiré with the Au atoms indicated (orange b Plan view atomic model for the 32 Å moiré Boxed areas represent three inequivalent sites in the 32 Å moiré c Close up plan-view image of each of the sites highlighted in (b) r is the projected distance from the central aligned sites and the dotted circles show two representative r values d Corresponding RDFs of the three inequivalent sites in the 32 Å moiré e HAADF and f iDPC STEM images showing the (apparent) 18 Å and 32 Å moiré cells h Relative intensity distributions and statistical variation of the three inequivalent sites in the island immediately above in the corresponding images The equivalent disk radius for each spot was calculated and partitioned to inequivalent sites (red The histograms were smoothed using a gaussian kernel of radius 0.5 Å for visual clarity Illustrative orange dots represent frequencies from Au crystal planes d Simulated FFT for Au/MoS2 generated via the geometric convolution technique with each spot colored to show its origin (orange: Au Area of spots is proportional to absolute intensity but with inner moiré spots magnified 2x for clarity a Schematic of 4D STEM technique showing rastered beam (red) on MoS2/Au{111} with corresponding CBED pattern at each point The green and blue scattered beams are centred on the spots of the annuli shown in (c and d) b CBED pattern formed by averaging patterns collected over the entire scan area c 4D STEM annulus used to isolate 18 Å moiré periodicity (angular range 31–43 mrad green) and d 4D STEM annulus used to isolate 32 Å periodicity (angular range 11–24 mrad f Virtual ADF STEM images revealing 18 (green) and 32 Å (blue) period moirés a Charge density difference viewed down a [100] cross section of the 32 Å commensurate moiré (11Au x 10\(_{\rm{MoS}_{2}}\) superstructure) Purple denotes negative and green positive charge density isosurface contours b Calculated charge density difference at the MoS2/Au{111} interface showing electronic modulation following the 32 Å moiré periodicity Black line indicates the 32 Å crystallographic moiré unit cell c Unfolded band structure for MoS2/Au{111} system (left) and corresponding density of states (right) Color corresponds to the band’s spectral weight the combination of several different imaging techniques with electronic calculations provides a clearer picture of the moiré structure than is possible with any single measurement In HRTEM the 18 Å moiré is the strongest visible leading to the possibility of erroneously predicting that electronic properties should be modulated with this period Our calculations reveal that electronic modulation instead follows the true crystallographic 32 Å periodicity This periodicity is hidden in conventional TEM due to projection effects combined with a geometric convolution analysis of the full moiré spectrum allows a direct real-space observational link between atomic structure and moiré-induced electronic modulation at this 2D/3D interface The combination of analysis techniques also explains the discrepancy between moiré patterns observed by TEM and STM at this 2D/3D interface These results highlight electronic modulation at the 2D/3D interface and showcase the growing opportunities for advanced STEM techniques for direct imaging of moiré structures at the atomic scale suggest that such control of interfacial orientation is increasingly feasible extending opportunities of 2D/3D moiré engineering a scalpel is used to cut the CAB around the desired flake A drop of deionised water can then be intercalated between the CAB and SiO2/Si surface and the entire flake transferred to the TEM grid with the CAB polymer handle using a tweezers The transferred flakes were baked at 140 °C for 5–10 min to improve adhesion After dissolving the CAB in acetone for 15 min the flakes were dipped in isopropanol and dried using a critical point dryer MoS2/SiNx substrates were loaded into a UHV sample preparation chamber and cleaned of residual polymer by heating resistively in UHV to ~550 °C for several hours Au deposition was carried out in the same multichamber UHV system (base pressure 2 × 10−9 Torr) The deposited thickness was calibrated by measuring the evaporation rate with a quartz crystal microbalance immediately before and after deposition AFM analyses of island thickness were performed in a Veeco Metrology Nanoscope V in tapping mode There is no intentional heating during deposition but thermocouple measurements show that the sample temperature rises to 50–60 °C STEM image simulations in Supplementary Fig. 4 were performed using an orthorhombic supercell consisting of 3 Au layers on an MoS2 monolayer (7956 atoms) A repeating unit from the supercell was cropped and simulated using custom Python-based STEM image simulation software Simulation parameters similar to experiments were used with an accelerating voltage of 60 kV and collection angles of 25–153 mrad (ADF) and 6–24 (iDPC) Simulated ADF and iDPC images were convolved with a gaussian kernel having FWHM of 80 pm approximately accounting for the finite effective source size The ground-state charge density difference (Δρ) between the Au/MoS2 heterostructure \((\rho _{{\mathrm{Au}}/{\mathrm{MoS}}_2})\) and pristine Au (ρAu) and MoS2 \((\rho _{{\mathrm{MoS}}_2})\) is given by The authors declare that the main data supporting the findings of this study are available within the article and its Supplementary Information files Code available upon request from the authors Hofstadter’s butterfly and the fractal quantum hall effect in moiré superlattices Hesp, N. C. H. et al. Collective excitations in twisted bilayer graphene close to the magic angle. Preprint at https://arxiv.org/abs/1910.07893 (2019) Photonic crystals for nano-light in moiré graphene superlattices Infrared interlayer exciton emission in MoS2/WSe2 heterostructures Moiré engineering of electronic phenomena in correlated oxides Electrical contacts to two-dimensional semiconductors Integration of bulk materials with two-dimensional materials for physical coupling and applications Gate-tunable semiconductor heterojunctions from 2D/3D van der Waals interfaces Moiré minibands in graphene heterostructures with almost commensurate √3×√3 hexagonal crystals In situ nanoscale imaging of moiré superlattices in twisted van der Waals heterostructures Moire pattern in LEED obtained by van der Waals epitaxy of lattice mismatched WS2/MoTe2(0001) heterointerfaces Measuring the local twist angle and layer arrangement in van der Waals heterostructures Convergent beam electron holography for analysis of van der Waals heterostructures Structure and electronic properties of in situ synthesized single-layer MoS2 on a gold surface Graphene on Ir(111): physisorption with chemical modulation Controlling excitons in an atomically thin membrane with a mirror Suppression of intrinsic roughness in encapsulated graphene Correlating the three-dimensional atomic defects and electronic properties of two-dimensional transition metal dichalcogenides Direct observation of epitaxial alignment of Au on MoS2 at atomic resolution Monolayer and thin hBN as substrates for electron spectro-microscopy analysis of plasmonic nanoparticles Deterministic coupling of quantum emitters in 2D materials to plasmonic nanocavity arrays Phase contrast STEM for thin samples: Integrated differential phase contrast Atomic electric fields revealed by a quantum mechanical approach to electron picodiffraction Four-dimensional scanning transmission electron microscopy (4D-STEM): from scanning nanodiffraction to ptychography and beyond A high-speed area detector for novel imaging techniques in a scanning transmission electron microscope Development of diffraction imaging for orientation analysis of grains in scanning transmission electron microscopy Visualization of light elements using 4D STEM: the layered‐to‐rock salt phase transition in LiNiO2 cathode material Sub-Ångstrom electric field measurements on a universal detector in a scanning transmission electron microscope Direct electric field imaging of graphene defects Electron ptychography of 2D materials to deep sub-ångström resolution Atomic electrostatic maps of 1D channels in 2D semiconductors using 4D scanning transmission electron microscopy Simultaneous identification of low and high atomic number atoms in monolayer 2D materials using 4D scanning transmission electron microscopy Catalytically mediated epitaxy of 3D semiconductors on van der Waals substrates On the interpretation of the forbidden spots observed in the electron diffraction patterns of flat Au triangular nanoparticles Quantitative measurements of internal electric fields with differential phase contrast microscopy on InGaN/GaN quantum well structures Imaging of built-in electric field at a p-n junction by scanning transmission electron microscopy What are the possible moiré patterns of graphene on hexagonally packed surfaces Universal solution for hexagonal coincidence lattices Single-layer MoS2 on Au(111): band gap renormalization and substrate interaction A new metal transfer process for van der Waals contacts to vertical Schottky-junction transition metal dichalcogenide photovoltaics Improved contacts to MoS2 transistors by ultra high vacuum deposition Moiré structures in twisted bilayer graphene studied by transmission electron microscopy LeBeau, J. M. 4D STEM Explorer. https://doi.org/10.5281/zenodo.1325482 (2018) Van der Waals density functionals applied to solids Effects of extrinsic and intrinsic perturbations on the electronic structure of graphene: retaining an effective primitive cell band structure by band unfolding Download references This work was carried out with the use of facilities and instrumentation supported by NSF through the Massachusetts Institute of Technology Materials Research Science and Engineering Center DMR - 1419807 This work was carried out in part through the use of MIT.nano’s facilities acknowledge funding from NSF Award DMR-1905295 and from the Office of Naval Research (ONR) Grant Number N00014-18-1-2691 for the theory and computation in this work is a Moore Inventor Fellow supported through Grant GBMF8048 from the Gordon and Betty Moore Foundation acknowledges funding from a MIT MathWorks Engineering Fellowship and an OGE MIT Fellowship acknowledges support from Independent Research Fund Denmark though Grant Number 9035-00006B The authors would like to acknowledge Michael Tarkanian for help in manufacturing TEM sample holders; Professor Pierre Stadelmann Qiong Ma for helpful discussions; a manuscript reviewer for the suggestion of additional iDPC measurements; and Profs Jeehwan Kim and Silvija Gradecak for equipment access This research was primarily conducted on the traditional We acknowledge the painful history of forced removal from this territory and we respect the many diverse indigenous people connected to this land These authors contributed equally: Kate Reidy Massachusetts Institute of Technology (MIT) Paulson School of Engineering and Applied Sciences Department of Brain and Cognitive Sciences developed the epitaxial deposition and fabricated samples extended the geometric convolution model in 2D/3D systems performed geometric convolution analysis and analysed the data performed SAED and bright field TEM imaging performed 4D STEM imaging and STEM multislice simulations and multislice calculations with input from A.K performed the density functional theory calculations and made atomic models The manuscript was written with contributions from all authors Peer review information Nature Communications thanks Jordi Arbiol and the other anonymous reviewer(s) for their contribution to the peer review of this work Download citation DOI: https://doi.org/10.1038/s41467-021-21363-5 Metrics details Misalignment-induced moiré patterns and chirality in two-dimensional materials offer vast opportunities for manipulating their properties but they face challenges in synthesis and structural control Two-dimensional (2D) materials consist of a single or a few atomic layers Misalignment between the adjacent layers can modify the interlayer and intralayer interactions and hence allow the manipulation of the structure and functions of 2D materials A typical misalignment arises from the mismatch of lattice constants between the two layers exemplified by diverse 2D heterostructures investigated for electronics the burgeoning field of ‘twistronics’ has grown rapidly exploring the exotic electronic states and quantum transport phenomena in twisted 2D materials that form moiré patterns marked by the absence of mirror symmetry in atomic lattices Although studies on the properties of moiré and chiral structures are expanding synthetic methods for reliably obtaining these intricate structures with precise control present great difficulties because many of these configurations are thermodynamically unfavourable we present a Focus consisting of research articles News & Views pieces and a Perspective that discusses the growth and characterization of twisted structures moiré patterns and chirality in 2D layered materials which forms orientation-aligned moirés with thermodynamic stability offers distinct advantages for large-scale manufacturing and moiré design leveraging angle-resolved transport measurements identify spontaneous broken symmetries in mirror-symmetric twisted trilayer graphene and a series of momentum-polarized states that might explain the zero-field superconducting diode effect in twisted trilayer graphene Beyond the moiré patterns that have attracted substantial interest in 2D materials research, structural chirality emerges as another feature that can be harnessed through different strategies, such as the adsorption of chiral molecules on 2D layers, the rolling-up of 2D sheets into quasi-1D tubes, and geometrical rotation or bending of 2D lattices. In a Perspective article Hanyu Zhu and Boris Yakobson discuss the approaches for constructing nearly 2D chiral materials the emergent properties arising from structural chirality and the opportunities for manipulating chiral functionalities in 2D layered materials which would potentially lead to unconventional optoelectronic materials beyond the current landscape of 2D moiré structures these nanotubes can be grown at any pre-selected place where gold nanoparticles are pre-deposited presenting opportunities for applications such as field-emission probes electrical conduits or waveguiding polaritonic modes A fascination with 2D materials lies in the abundant parameter space of stacking and misalignment granting tremendous freedom to play with atomic structures and electronic properties But this potential must be underpinned by precise structural control and characterization of these atomic structures laying the fundamental basis for exploring their physics and applications Reprints and permissions Download citation DOI: https://doi.org/10.1038/s41563-024-01839-7 Metrics details the discovery of electron states that fractionalize in the presence of a time-reversal symmetry breaking magnetic field opened up new directions in condensed matter physics evidence has accumulated that a version of these states in which the time-reversal symmetry breaking is spontaneous appears in moiré materials Experimentalists have developed and refined techniques for flexible stacking of atomic scale 2D materials isolated from van der Waals layer compounds Theorists have improved understanding of how ideal quantum geometry can lead to electronic correlations in the Bloch bands of a two-dimensional crystal that are similar to those favoured by strong magnetic fields Twisted bilayer MoTe2 and graphene multilayer moiré materials identified theoretically as attractive targets have now been definitively established as fractional quantum anomalous Hall state hosts Model for a quantum Hall effect without Landau levels: condensed matter realization of the “parity anomaly” Band geometry of fractional topological insulators Download references Reprints and permissions Download citation DOI: https://doi.org/10.1038/s42254-024-00718-z Metrics details The ability to precisely control moiré patterns in two-dimensional materials has enabled the realization of unprecedented physical phenomena including Mott insulators the application of independent strain in each layer of stacked two-dimensional materials—termed heterostrain—has become a powerful means to manipulate the moiré potential landscapes Recent experimental studies have demonstrated the possibility of continuously tuning the twist angle and the resulting physical properties the dynamic control of heterostrain that allows the on-demand manipulation of moiré superlattices has yet to be experimentally realized by harnessing the weak interlayer van der Waals bonding in twisted bilayer graphene devices we demonstrate the realization of dynamically tunable heterostrain of up to 1.3% Polarization-resolved Raman spectroscopy confirmed the existence of substantial heterostrain by presenting triple G peaks arising from the independently strained graphene layers Theoretical calculations revealed that the distorted moiré patterns via heterostrain can significantly alter the electronic structure of twisted bilayer graphene allowing the emergence of multiple absorption peaks ranging from near-infrared to visible spectral ranges Our experimental demonstration presents a new degree of freedom towards the dynamic modulation of moiré superlattices holding the promise to unveil unprecedented physics and applications of stacked two-dimensional materials controlling heterostrain dynamically has yet to be experimentally realized limiting the feasibility of heterostrain-enabled on-demand manipulation of moiré superlattices which played a key role in achieving a large heterostrain of up to ~ 1.3% The induced heterostrain in TBG devices was dynamically tunable by continuously changing the bending radius of the PET substrate Raman spectroscopy provided unambiguous evidence for the generation of a substantial heterostrain by presenting triple G peaks that arise from both unstrained and strained graphene layers having a single degenerate mode of the G peak and two split G+ and G− peaks Strong polarization dependence of the triple G peaks further confirmed the formation of heterostrain We also verified the influence of the experimentally obtained heterostrain on the modulation of moiré superlattices and the electronic band structures of TBG through tight-binding simulations The 1.3% heterostrain was predicted to allow achieving multiple absorption peaks including one in the near-infrared spectral range that is attributed to the optical transition between the two saddle points which is forbidden in the absence of heterostrain Heterostrain-enabled dynamically tunable moiré superlattice in TBG b) Schematic illustration of dynamically tunable moiré superlattices in TBG on a flexible substrate before and after bending (c) Optical microscope image of the TBG device with a 13.2° twist angle used in our study (d) Calculated (red line) strain on PET surface and experimental (black squares) uniaxial strain in bottom graphene Inset: photo showing the flexibility of our substrate with the TBG device on top Raman spectra of graphene 2D peak give evidence of weak electronic coupling in TBG and heterostrained TBG (a) Raman spectrum of 2D peak in unstrained bottom graphene (b) Raman spectrum of 2D peak in bottom graphene under 1.3% uniaxial strain (blue dots) 2D spectrum of unstrained graphene was added for reference (c) Raman spectrum of 2D peak in unstrained TBG (d) Raman spectrum of 2D peak in TBG under 1.3% heterostrain Dots are experimental data and curves are obtained by Lorentz fitting which results in the heterostrained TBG device we conclude that a weak interlayer coupling effect still exists when 1.3% heterostrain is applied on TBG with 13.2° misorientation Tight-binding simulations of TBG with and without heterostrain b) Moiré patterns of TBG with 0% and 1.3% heterostrain d) Band structure of TBG with 0% and 1.3% heterostrain TBG (blue) and TBG under 1.3% heterostrain (red) (f) Dynamic conductivity spectra of TBG with (red) and without (blue) heterostrain the matrix element is non-zero due to symmetry breaking see the Supplementary Note 7 for more details) in 1.3% heterostrained TBG which allows an enhanced optical absorption at the technologically important near-infrared spectral range Graphene flakes were prepared by mechanical exfoliation of Kish graphite (Natural Kish graphite Graphene Supermarket) by scotch tape and deposited onto Si/SiO2 substrate (285 nm SiO2 for good optical contrast) Number of graphene layers was identified by analyzing optical contrast and Raman spectra Long graphene strips were selected as bottom layers of TBG while smaller monolayer flakes were selected as tops layers Both the selected bottom and top layers has straight edges such that crystal orientation could be identified from flake edges A 135-μm thick PET substrate was cut into 1 cm × 5 cm rectangles isopropanol alcohol (IPA) and deionized (DI) water About 400 nm polymethyl methacrylate (PMMA MICROCHEM) was spin-coated onto PET at 4500 rpm for 90 s Twisted bilayer graphene was mechanically stacked on PMMA-coated PET with a home-made transfer stage long edge of bottom graphene was aligned along horizontal direction by rotating the sample stage PMMA-coated PET thin film was attached to a PDMS stamp of ~ 4 mm × 2 mm × 0.6 mm Bending direction of PET rectangle was also aligned to horizontal direction such that uniaxial strain will be applied along long side of bottom graphene Then Glass/PDMS/PET/PMMA stack was used to pick up the bottom graphene layer bottom graphene is firmly adhered to PET substrate Next top graphene was aligned 13.2° to horizontal which was subsequently picked up by Glass/PDMS/PET/PMMA/Bottom graphene detach PET substrate from glass slide and PET/PMMA/TBG was obtained Cured PMMA can successfully transfer strain from PET to bottom layer of TBG during the bending process Graphene G peaks were measured by Raman spectroscopy (Alpha300 M+ WITec) with a 100× objective and 1800 g/mm grating laser power was adjusted to low level to avoid heating effect Lorentz fitting was applied to determine peak position and FWHM Heteromoiré Engineering on magnetic Bloch transport in twisted graphene Superlattices Tunable macroscale structural superlubricity in two-layer graphene via strain engineering Electron interactions in strain-induced zero-energy flat band in twisted bilayer graphene near the magic angle Strain-induced excitonic instability in twisted bilayer graphene Adhesion and friction in mesoscopic graphite contacts Direct determination of the crystallographic orientation of graphene edges by atomic resolution imaging Uniaxial strain in graphene by Raman spectroscopy: G peak splitting Raman spectrum of graphene and graphene layers Raman mapping investigation of graphene on transparent flexible substrate: The strain effect Optical absorption in twisted bilayer graphene Optical absorption of twisted bilayer graphene with interlayer potential asymmetry Strong mid-infrared photoresponse in small-twist-angle bilayer graphene Luo, M. et al. Triaxially strained suspended graphene for large-area pseudo-magnetic fields. Preprint at https://arxiv.org/abs/2110.04205 (2021) Pseudo-magnetic field-induced slow carrier dynamics in periodically strained graphene Download references The research of the project was in part supported by Ministry of Education under grant AcRF TIER 1 2019-T1-002-050 (RG 148/19 (S)) The research of the project was also supported by Ministry of Education under grant AcRF TIER 2 (MOE2018-T2-2-011 (S)) This work is also supported by National Research Foundation of Singapore through the Competitive Research Program (NRF-CRP19-2017-01) This work is also supported by National Research Foundation of Singapore through the NRF-ANR Joint Grant (NRF2018-NRF-ANR009 TIGER) This work is also supported by the iGrant of Singapore A*STAR AME IRG (A2083c0053).The authors would like to acknowledge and thank the Nanyang NanoFabrication Centre (N2FC) These authors contributed equally: Xuejiao Gao and Hao Sun School of Electrical and Electronic Engineering fabricated the samples under the guidance of Q.W performed the Raman measurements and analysis Download citation DOI: https://doi.org/10.1038/s41598-021-00757-x Create a reading list by clicking the Read Later icon next to the articles you wish to save Dan Page for Quanta Magazine When two very similar grids with light and dark elements overlap new sinuous patterns emerge that seem to shimmer and flow you have probably seen or at least heard of these moiré patterns The name moiré (pronounced mwa-ray) is etymologically related to the French word for mohair and entered the language centuries ago moiré patterns are seen not only in visual fields such as optics but also in seemingly far-flung areas like marine engineering and the detection of forged banknotes a sheet of carbon crystal with atoms arranged in a hexagonal one-atom-thick lattice is dropped on another one and rotated to just the correct angle of about 1.1 degrees the graphene magically acquires the ability to become superconductive when the requisite number of electrons are added That interesting things would happen at around this magic angle had been mathematically predicted by the theorists Rafi Bistritzer and Allan H Jarillo-Herrero’s team made their discovery while trying to create graphene twists that matched a bunch of magic rotation angles from this earlier prediction 5W Infographics for Quanta Magazine Why does rotated graphene exhibit this behavior (Hint: It has something to do with moiré patterns.) The figure depicts what two layers of carbon atoms might look like when arranged in a hexagonal lattice and the skewed overlap between the two creates a larger hexagonal moiré pattern (outlined in blue) This large hexagonal superlattice enables large numbers of electrons to interact especially when the layers are rotated at specific angles at which The large moiré pattern of the superlattice is both visually striking and electrically special as the emergence of superconductivity shows Superconductivity is a state that exhibits quantum behavior macroscopically allowing electric currents to circulate indefinitely without any resistance whatsoever This concordance between the visual and the electrical in graphene almost seems to be an example of life imitating art right down to the quantum level This discovery has sparked an explosion of interest in a new subfield of materials science The exact reason why specific angles produce the observed quantum effects is an active area of research in graphene twistronics. The answer depends on the details of atomic spacing and complex calculations of electron interaction and quantum tunneling; Freedman’s follow-up blog post “What’s the Magic Behind Graphene’s ‘Magic’ Angle?” offers an update on this subject we’ll focus on the geometry of how moiré visual effects arise and how factors such as the periods of and discrepancy between the two grids affect the size of the resulting moiré pattern it’s worth noting that these patterns are inherently dynamic and static pictures do not do them justice the easiest and most enjoyable way to explore moiré patterns is to sit back and watch one of the many excellent moiré videos which gives us a feel for how patterns evolve and how surprises abound when we’re dealing with moiré effects: Let’s start with the simplest example of how two similar patterns interfere with each other — the phenomenon of acoustic beats the two interfering patterns are sound waves (mathematically sine waves) of slightly different frequencies which together produce the acoustically dominant recurring beat pattern at a frequency that is the difference between the two original frequencies the waxing and waning of the beats is caused by the fact that the original sine waves are in phase in the middle of the beat Assuming that the original frequencies are centered around 500 hertz Notice that as the differences between the frequencies diminish Now let’s move from beats to line moiré patterns Here the waxing and waning of the beats is replaced by patterns of light and dark lines Notice that the light and dark areas in the moiré pattern are much larger than the original dark and light lines much like the acoustic beat durations and the large hexagons in the graphene Assume that the distance from the center of one dark line to the next in the square on the upper right is 10 units What is the distance between the centers of the dark areas in the moiré pattern if the corresponding distance between lines in the lower left square is a) 12 units or b) 11 units a smaller difference between the interfering patterns produces a larger grain in the moiré pattern When we move to moiré patterns caused by rotation the two overlapping patterns can be absolutely identical The misalignment that was caused in the previous cases by a difference in frequency or distance now comes from the fact that one of the patterns is rotated The approximate scale of the moiré pattern is inversely proportional to the angle of rotation You can see this very clearly in this video which shows the moiré patterns for overlapping graphene sheets rotated between 0 and 30 degrees: PMende Try stepping through the video manually or in slow motion to get a feel for how the pattern evolves and how the moiré hexagons shrink in size as the angle increases Or you can step backward and see how the hexagons increase as the rotation angle decreases Here are the images for rotations of 8.8 degrees 5W Infographics for Quanta Magazine at 8.8 degrees the moiré pattern that emerges has a central superlattice hexagon surrounded by six others of the same size similar to the ones outlined in blue in the first figure above Each of these moiré hexagons occupies about a quarter to a third of the entire hexagonal field in each direction the central superlattice hexagon expands to about twice its previous size covering about one-half to two-thirds of the entire field the central superlattice hexagon covers the entire original bounding hexagon When you halve the angle one more time to 1.1 degrees the angle that produces the interesting effects in graphene the superlattice seems to have disappeared completely and the pattern looks almost like a single sheet without any moiré What do you think has happened to the large moiré superlattice at this angle What happens when you halve the angle again a few more times to match some of the even smaller angle predictions When you advance the angle in the rotation video above you notice that moiré patterns change in a smooth and continuous manner The superlattice hexagons get smaller and more numerous and their size varies continuously with the angle the electron tunneling phenomenon is discrete and recurs periodically at specific angles only Can you venture an entirely geometric explanation of how the seemingly continuous rotation profile could possibly produce a discrete physical phenomenon that only recurs periodically It might help to look at the Problem 1 scenario again We’ve dissected the geometry of moiré patterns but let’s not forget how beautiful and artistic they are Now that we have explored some moiré science I encourage readers to post links to their favorite moiré patterns and briefly describe why they like them Come up with your best couplets to explain the graphene story while parodying Dean Martin’s signature song “That’s Amore,” replacing those words with “That’s a Moiré.” Here’s one to start you off Here’s wishing you a happy and creative fusion of art and science Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a Quanta Magazine T-shirt or one of the two new Quanta books, Alice and Bob Meet the Wall of Fire or The Prime Number Conspiracy (winner’s choice) And if you’d like to suggest a favorite puzzle for a future Insights column clearly marked “NEW PUZZLE SUGGESTION.” (It will not appear online so solutions to the puzzle above should be submitted separately.) Note that we may hold comments for the first day or two to allow for independent contributions by readers Update: The solution has been published here Quanta Magazine moderates comments to facilitate an informed incoherent or off-topic comments will be rejected Moderators are staffed during regular business hours (New York time) and can only accept comments written in English We’ll email you instructions to reset your password Metrics details Two-dimensional (2D) van der Waals (vdW) superstructures are formed by the precise restacking of 2D nanosheet lattices which can lead to unique physical and electronic properties that are not available in the parent nanosheets Moiré patterns formed by the crystalline mismatch between adjacent nanosheets are the most direct features for vdW superstructures under microscopic imaging transmission electron microscopy (TEM) observation of hexagonal Moiré patterns with unusually large micrometer-sized lateral areas (up to ~1 μm2) and periodicities (up to ~50 nm) from restacking of liquid exfoliated hexagonal boron nitride nanosheets (BNNSs) is reported This observation was attributed to the long range crystallinity and the contaminant-free surfaces of these chemically inert nanosheets Parallel-line-like Moiré fringes with similarly large periodicities were also observed The simulations and experiments unambiguously revealed that the hexagonal patterns and the parallel fringes originated from the same rotationally mismatched vdW stacking of BNNSs and can be inter-converted by simply tilting the TEM specimen following designated directions This finding may pave the way for further structural decoding of other 2D vdW superstructure systems with more complex Moiré images because the overlapped elemental maps are only indications of whether the different nanosheets are in the same path of the electron beam such results can only serve as indirect proof for vdW superstructure formation The Moiré patterns often have a periodicity sometimes referred to as “Moiré wavelength” or “Moiré lattice constant” measured by the center distance of the neighboring unit features Because of the ordered periodic characteristics Moiré patterns can only form when the nanosheets are stacked at the vdW distance and thus are direct proof of vdW superstructure presence the comparatively less time-consuming TEM technique is rather useful to analyze objects through thicknesses up to tens of nanometers due to electron penetration Such preparations typically involve the use of sonication which often results in the size reduction and the loss of long range crystallinity of the resultant exfoliated nanosheets Sonication can also partially decompose the solvent or other reagents used (such as surfactants) and the nanosheet surfaces are thus prone to contamination TEM observations of Moiré features with unusually large lateral dimensions and periodicities from vdW homostructures formed by the restacking of liquid exfoliated BNNSs are reported The Moiré features could be either hexagonal spots or parallel lines but were found to be inter-convertible by tilting the projections of the nanosheets The mechanistic origins and evolutions of such Moiré features were corroborated with simulations as discussed in detail in the following text During solvent evaporation of the dispersions when preparing TEM specimens Although not within the scope of the current work a general observation was that the likelihood to observe vdW superstructures from an aged dispersion was typically higher corresponding to the calculated values from periodicities measured in (a,b) in order to form such mesoscale ordered patterns both nanosheets that contributed to the pattern must be highly crystalline and the interacting surfaces must be contaminant-free the stacking of two nanosheets must be highly ordered over a long range including both uniform rotational lattice mismatch and uniform intersheet distance The rotational lattice mismatch is required to form the hexagonal Moiré patterns and the intersheet distance uniformity can be readily attributed to the vdW restacking The folding back of a nanosheet onto itself to form an ordered superstructure is mechanistically and conceptually similar to the vdW restacking of two nanosheets for which the previously described rules would also apply in order to form Moiré patterns The BNNS restacking process that led to small rotational faults was likely a result of an interplay between the thermodynamic preference of perfectly AA′ restacked nanosheets and the kinetic formation of random agglomerations in a liquid dispersion (either spontaneously or upon solvent evaporation) Although the kinetic nature determines that it is highly unlikely for two exfoliated nanosheets to restack in a perfect order the strong B-N interactions that resulted in AA′ stacking might have made the small lattice mismatch much more prone to occur than the case of restacked graphene nanosheets (much less stacking preference) for which large inter-sheet rotational angles were commonly observed Although hexagonal Moiré pattern observation is not uncommon what is really striking is that the liquid exfoliated BNNS allowed the formation of highly ordered Moiré patterns at such ultra-large lateral scale suggesting that the nanosheet surfaces must be extremely “clean” ordered mesoscale Moiré patterns were seldom observed in few-layered graphene and MoS2 nanosheets similarly prepared from liquid exfoliation experiments under similar conditions likely because the nanosheets were less crystalline and more prone to contaminations in addition to their lack of strong lip-lip inter-sheet interactions Preliminary experiments were also conducted by solvent exfoliating a mixture of h-BN and graphite only the stand-alone few-layered BNNSs (identified by HR-TEM imaging showing a single set of hexagon lattice with constant of ~0.25 nm) exhibited ordered Moiré features TEM images of parallel-line like Moiré fringes from BNNS vdW superstructures Shown in the insets are the corresponding zoom-out images at lower magnifications Evolution of mesoscale hexagonal Moiré patterns from BNNS vdW superstructures to parallel line fringes by tilting: (a) 0°; (b) + 4°; (c) −1° The black-colored arrow indicates the tilting axis that slightly deviates from [110] (orange-colored dash line) The red and blue arrows indicate tilting toward θ > 0 and θ < 0 directions The above experimental results unambiguously demonstrated that the parallel fringes and the hexagonal patterns are indeed inter-related and from the same origin: while the hexagonal patterns are from vdW superstructures with rotational mismatches that are vertical to the detector view the occurrence of parallel fringes is the optical projection of the same superstructure but with a slightly tilted 2D plane Complete evolution profile in a 2D angular coordinate system of Moiré features of BNNS vdW superstructures: (a) simulation results of various fringe lines obtained by tilting a hexagonal lattice (D0 = 5.54 nm) against the various axes shown (tilting directions follow the right-hand rule); (b) schematic illustration of fringe lines corresponding to the original hexagonal lattice where θ is the true tilt angle defined by: the two periodicity values could be simplified into: Therefore, parallel fringes from small angle tilting along [100], [110] and [010] directions share the same derived Moiré periodicity of D1, while tilting along the other three distinct directions also share the same periodicity of D2. The simplified schematics of this evolution profile is shown in Fig. 4b Tilting the hexagonal pattern along the axes between the above discussed distinct directions resulted in various pattern shapes that are intermediates of hexagons and parallel fringes from rotation along nearby distinct axes. Detailed information is given in the Supporting Information (Figure S3) Comprehensive evolution of Moiré pattern and fringes of a BNNS vdW superstructure via free tilting experiments The arrows in (d) point to the desired fringe directions. All tilting coordinates are superimposed onto the ideal schematics in (e). Images corresponding to the red-colored tilting coordinates are shown in this Figure, while those corresponding to the grey-colored ones are presented in Figure S4 in the Supporting Information the various Moiré fringes and patterns observed could all be readily assigned to the same vdW BNNS superstructure slightly tilted along arbitrary axes Similar tilting-induced fringe-pattern inter-conversions were achieved in multiple locations of the same specimen as well as other liquid exfoliated BNNS specimens These results suggested that, while the |ϕ| value for hexagonal patterns could be straightforwardly obtained via Eq. (1) those for the parallel fringes from vdW BNNS superstructures could fall in either the D1 or D2 category A straightforward tilting experiment should help accurately determine the correct category and thus the correct |ϕ| value considering that most nanosheets viewed in TEM are likely not placed perfectly horizontally but with a small tilting angle Finally, it should be noted that, the preliminary simulation data on the dependence of fringe-pattern inter-conversions on nanosheet layer numbers suggested that the parallel Moiré fringes exhibited higher “optical” contrast with thicker nanosheet layers, while the hexagonal patterns with no or little titling were less affected (Figure S5) More theoretical and experimental studies are needed to fully address this correlation Lot HZ010PA4.$06) was provided by UK Abrasives (Northbrook The majority of TEM experiments were conducted on a JEOL 2100 field emission HR-TEM system at an accelerating voltage of 200 kV. Tilting experiments were conducted using a double-tilt specimen holder (JEOL Model EM-31640). The tilting directions were calibrated using an independent sample (see Figure S2) Some low-magnification TEM images were also acquired on a Hitachi S-5200 field-emission SEM system at an accelerating voltage of 30 kV the as-obtained h-BN powder (~20 mg) was sonicated in a selected solvent (DMF or water in this work; 20 mL) for 6–24 h using a bath sonicator (Branson 2510 Ultrasonic Cleaner The reaction flask was capped with a rubber stopper to avoid loss of volatile reagents the resultant slurry was subjected to centrifugation at 3000 × g for 10 min to separate the supernatant dispersion from the residue The supernatant was collected as the exfoliated BNNS a few drops of the as-prepared dispersion was placed onto a holey carbon–coated copper grid followed by solvent evaporation under ambient conditions Material Studios (Accelrys) was used in all simulation experiments Two 10-layer (unless otherwise specified) BNNSs with inter-layer distance of 0.33 nm and AA′ stacking order The vdW distance between the two sheets was set at 0.33 nm which is also the inter-layer value for each nanosheet The atomic radii for B and N atoms were both set at 0.06 nm imitating the unresolved relative contrast of B/N atoms in TEM imaging while the top nanosheet was rotated to a specific angle ϕ in a counterclockwise fashion the top nanosheet was first rotated counterclockwise to a desired angle as above and then tilted in designated directions expressed through a 2D angular coordinate system (θx,θy) following the right-hand rule Evolution of Moiré Profiles from van der Waals Superstructures of Boron Nitride Nanosheets challenges and opportunities in two-dimensional materials beyond graphene Recent advances in two-dimensional materials beyond graphene Direct growth of graphene/hexagonal boron nitride stacked layers Artificially stacked atomic layers: toward new van der Waals solids van der Waals epitaxy of MoS2 layers using graphene as growth templates Lateral and vertical two-dimensional layered topological insulator heterostructures Local electronic properties of graphene on a BN substrate via scanning tunneling microscopy Effect of rotational stacking faults on the Raman spectra of folded graphene Probing interlayer interactions in transition metal dichalcogenide heterostructures by optical spectroscopy: MoS2/WS2 and MoSe2/WSe2 Moiré superlattice effects in graphene/boron-nitride van der Waals heterostructures Electronic effects in scanning tunneling microscopy: Moiré pattern on a graphite surface Chemical vapor deposition and characterization of aligned and incommensurate graphene/hexagonal boron nitride heterostack on Cu (111) Moiré patterns as a probe of interplanar interactions for graphene on h-BN Graphene on Rh (111): Scanning tunneling and atomic force microscopies studies Precisely aligned graphene grown on hexagonal boron nitride by catalyst free chemical vapor deposition Unraveling the intrinsic and robust nature of van Hove singularities in twisted bilayer graphene by scanning tunneling microscopy and theoretical analysis Transmission Electron Microscopy-A Textbook for Materials Science Moiré-like fringes in transmission electron microscopy images of coherently strained semiconductor islands Direct imaging of rotational stacking faults in few layer graphene Atomic-scale observation of rotational misorientation in suspended fewlayer graphene sheets simulation study of aberration-corrected highresolution transmission electron microscopy imaging of few-layer-graphene stacking Growth and properties of few-layer graphene prepared by chemical vapor deposition Atomic resolution imaging of rotated bilayer graphene sheets using a low kV aberration-corrected transmission electron microscope Superstructural defects and superlattice domains in stacked graphene Structure of chemically derived mono- and few-atomic-layer boron nitride sheets Structure and catalytic properties of hexagonal molybdenum disulfide nanoplates Chemical vapor deposition of twisted bilayer and few-layer MoSe2 over SiOx substrates van der Waals epitaxial growth of atomically thin Bi2Se3 and thicknessdependent topological phase transition The hide-and-seek of grain boundaries from Moiré pattern fringe of two-dimensional graphene Grain boundary mapping in polycrystalline graphene Two-dimensional nanosheets produced by liquid exfoliation of layered materials Large-scale fabrication of boron nitride nanosheets and their utilization in polymeric composites with improved thermal and mechanical properties “Chemical Blowing” of thin-walled bubbles: high-throughput fabrication of large-area Aqueous dispersions of few-layered and monolayered hexagonal boron nitride nanosheets from sonication-assisted hydrolysis: Critical role of water.” J Moiré pattern in scanning tunneling microscopy: Mechanism in observation of subsurface nanostructures Moiré pattern in scanning tunneling microscopy of monolayer graphite Advances in 2D boron nitride nanostructures: nanosheets Evidence from Moiré patterns of packing faults in boron nitride crystals Moiré patterns on electron micrographs and their application to the study of dislocations in metals Observation and interpretation of adjacent Moiré patterns of different shapes in bilayer graphene Structural analysis of multilayer graphene via atomic moiré interferometry Tuning structures and electronic spectra of graphene layers with tilt grain boundaries Download references Lin acknowledges the support by the National Institute of Aerospace acknowledges the support by the Department of Defense (Grant W911NF-12-1-0083) and NASA (Grant Nos NNX10AM80H and NNX13AB22A) Liao is partially supported by the fellowship awarded via the Institute for Functional Nanomaterials (IFN) at the University of Puerto Rico Lin carried out the synthesis and co-wrote the paper Lin conducted the microscopy characterizations Lin discussed the results and commented on the manuscript The authors declare no competing financial interests Download citation Metrics details The intrinsic atomic mechanisms responsible for electronic doping of epitaxial graphene Moirés on transition metal surfaces is still an open issue To better understand this process we have carried out a first-principles full characterization of the most representative Moiré superstructures observed on the Gr/Pt(111) system and confronted the results with atomically resolved scanning tunneling microscopy experiments We find that for all reported Moirés the system relaxes inducing a non-negligible atomic corrugation both at the graphene and at the outermost platinum layer a mirror “anti-Moiré” reconstruction appears at the substrate giving rise to the appearance of pinning-points We show that these points are responsible for the development of the superstructure while charge from the Pt substrate is injected into the graphene mostly localized at these specific pinning-point positions since normally a large number of rotational domains forming different Moirés coexist on the same single crystal surface and can form polycrystalline graphene domains a microscopic characterization of their different adsorption geometries is needed some of the Moiré domains analyzed in the present study the question of the dependence of charge carrier doping on the rotation angle of the Moiré is still open a characterization by means of first-principles density functional theory (DFT) of the structural and electronic properties of the different Moirés becomes necessary However this turns into a very challenging task due to large size of the unit cells involved in the present work we analyze the nature of the graphene—substrate interaction for several Moiré superstructures appearing for Gr/Pt(111) by using an adequate combination of local ultra-high vacuum (UHV) STM experiments and first-principles calculations accounting for an accurate – given the high amount of atoms involved in some of the calculations – van der Waals (vdW) interaction We show hereafter that the Pt surface atoms tend to relax out of plane approaching towards the graphene layer and originating a sort of low-dimensional draining points where charge can efficiently flow between the two materials Thermal drift was corrected using a custom program in order to reduce errors in measurements of angles and distances This program corrects the images for a given unit cell keeping the fast scan axis distances as the reference one obtaining with both vdW schemes a good comparison with those presented in this study which justifies our choice of this DFT+D vdW framework The ion–electron interaction is modeled by ultrasoft pseudopotentials38 and the one-electron wave-functions are expanded in a basis of plane-waves with energy cut-offs of 400 and 500 eV for the kinetic energy and for the electronic density The Brillouin zone (BZ) was sampled by means of [4 × 4 × 1] [2 × 2 × 1] and [1 × 1 × 1] Monkhorst-Pack grids for the Optimized ground-state geometries of the five Gr/Pt(111) Moiré superstructures considered in this study: (a) or μR0.8° C atoms of the graphene monolayer are depicted in different colors depending on their relative height (the lowest C atom is set as a baseline at 0 Å) (f) Topmost Pt layer “anti-moiré” pattern for the βR19° case is also shown (the topmost Pt atom is set as a baseline at 0 Å) Light-colored dashed lines indicate the unit cell used in the calculations Side views of the Gr/Pt(111) structures are also included showing the minimum perpendicular distance between the Pt(111) surface and graphene a partial scaling can be found between the charge transfer and the graphene Dirac-cone shift w.r.t with the exception of the domains where the charge transfer is more pronounced This effect will be analysed in detail below This extremely corrugated topmost Pt slab can be seen as a limit case of the corrugation-rotation correlation Comparison between UHV experimental (large pictures) and theoretical (top-right corner insets) constant-current STM images for the (a) μR0.8°, (b) εR8.9°, (c) βR19° and (d) ζR7.2° Gr/Pt(111) Moiré superstructures (see Fig. 1) for Vs = −0.2–0.2 V and Itunnel = 0.1 nA Experimental (solid symbols) and computed (empty symbols) apparent Moiré corrugation (in Å) as a function of the tunnelling bias voltage (in V) for the different Gr/Pt(111) superstructures shown in Fig. 1 along the same selected scanning paths Theoretical STM images were obtained at constant-current conditions with Itunnel = 0.1 nA mimicking the experimental tunnelling conditions (a) Experimental STM images corresponding to the Moiré for Vs = −0.15 and 0.15 V, with Itunnel = 0.1 nA. Theoretical constant-current STM images of Gr/Pt(111) for Vs = −0.15 and 0.15 V with Itunnel = 0.1 nA, for the Moiré superstructures shown in Fig. 1: (b) μR0.8° charge transfers and density of state (DOS) profiles projected on the different graphene sheets This electronic information allow us to analyze the character of each interfacial interaction as well as the relative shift of the Dirac-cone for the different graphene domains involved in each Moiré superstructure w.r.t non-deformed graphene at the equilibrium distance on Pt(111) in order to estimate the Dirac-cone left-shift (and the local n-doping level) by effect of the formation of the different Moirés we have designed a theoretical procedure consisting in obtaining the equilibrium geometries of flat graphene on Pt(111) (forcing the graphene layer to keep flat but with the possibility of relaxing the G-Pt perpendicular distance) for each periodicity analyzed in this study one can just integrate downwards the PDOS profile from the Fermi energy up to a point in the density of states profiles fulfilling the condition of integrating just up to the charge transfer obtained by the Bader analysis for all the cases This theoretical strategy permits a direct comparison between the projected DOS on flat graphene on Pt for every domain with the projected DOS on graphene on Pt once the Moirés have been formed this direct comparison yields the Dirac-cone shift by effect of just the formation of the different corrugated Moirés eliminating in the shift values so obtained any possible contribution coming from the tendency of Gr and Pt chemical potentials to align (a) Calculated DOS (in an energy window of [−2, 2] eV) projected on graphene (PDOS) involved in each Gr/Pt Moiré of Fig. 1 as well as for non-deformed graphene on Pt(111) all the DOS profiles have been normalized to the number of C atoms in each unit cell (b) Scheme of the Dirac-cone for pristine graphene and for graphene in each Gr/Pt representing the left-shift of the Dirac-cone to accommodate the charge transferred from the Pt substrate (c) Projected density of states (PDOS) on the s and orbitals of a Pt atom forming a pinning-point in the Gr/Pt system (red line) as compared to a surface Pt atom of the clean Pt(111) surface (black line) it is important to have in mind that our DFT-based model imposes commensurability the low mismatch between substrate and graphene makes that the atoms at the pinning-points are still the closer to each other and that they will be the point from where to inject charge until the separation between C and Pt atoms stop the coherence of the Moiré which reinforces our interpretation about the localized n-doping mechanism in Gr/Pt(111) proposed here we have shown that the structural and electronic properties of graphene superstructures on Pt(111) surfaces are dictated by the existence of pinning-points which can be understood as the best coincident atoms between substrate and overlayer for a given rotational angle At these pinning-points there is a significant atomic relaxation approaching the C and Pt atoms migrated electronic charge from the s towards the orbitals in the Pt atoms increases the orbital directionality which favours the hybridization with the emerging pz orbitals from the buckled Gr C atoms at the pinning-points every one of the Moiré domain superstructures manifests a different n-doping and the multidomain graphene/Pt(111) can be regarded as true graphene-laboratory for testing fundamental properties of doped graphene Role of the Pinning Points in epitaxial Graphene Moiré Superstructures on the Pt(111) Surface Roll-to-roll production of 30-inch graphene films for transparent electrodes The surface science of graphene: Metal interfaces Long-range magnetic order in a purely organic 2D layer adsorbed on epitaxial graphene Controlling graphene corrugation on lattice-mismatched substrates Role of dispersion forces in the structure of graphene monolayers on Ru surfaces Lattice-matched versus lattice-mismatched models to describe epitaxial monolayer graphene on Ru(0001) Graphene on Ni(111): Strong interaction and weak adsorption Defects of graphene on Ir(111): Rotational domains and ridges Quantitative atomic resolution force imaging on epitaxial graphene with reactive and nonreactive AFM probes strain and lattice parameter in graphene on Iridium Local deformations and incommensurability of high-quality epitaxial graphene on a weakly interacting transition metal Effect of preparation on the commensurabilities and thermal expansion of graphene on Ir(111) between 10 and 1300 K Homogeneous optical and electronic properties of graphene due to the suppression of multilayer patches during CVD on copper foils Orientation-dependent work function of graphene on Pd(111) STM investigation of single layer graphite structures produced on Pt(111) by hydrocarbon decomposition Epitaxial growth and structural property of graphene on Pt(111) Strain-driven Moiré superstructures of 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functional study Graphene on the Ir(111) surface: from van der Waals to strong bonding Insight into the description of van der Waals forces for benzene adsorption on transition metal (111) surfaces Effect of van der Waals Forces on the Stacking of Coronenes Encapsulated in a Single-wall Carbon Nanotube and Many-body Excitation Spectrum Self-interaction correction to density-functional approximations for many-electron systems Soft self-consistent pseudopotentials in a generalized eigenvalue formalism First-principles simulations of STM images: from tunneling to the contact regime Improvement of STM resolution with H-sensitized tips Multi-oriented moiré superstructures of graphene on Ir(111): experimental observations and theoretical models First-principles study of the graphene/Ru(0001) interface work function and reactivity of graphene on Ru(0001) from first principles Ordered vacancy network induced by the growth of epitaxial graphene on Pt(111) Graphene on Pt(111): growth and substrate interaction Understanding the origin of band gap formation in graphene on metals: graphene on Cu/Ir(111) In Atoms in molecules - A quantum theory (ed Bader Tug-of-war between corrugation and binding energy: revealing the formation of multiple moiré patterns on a strongly interacting graphene–metal system Vacancy formation on C60/Pt(111): Unraveling the complex atomistic mechanism Download references We acknowledge funding from the Spanish MINECO (Grant MAT2014-54231-C4-1-P) the EU via the ERC-Synergy Program (Grant ERC-2013-SYG-610256 Nanocosmos) and computing resources from CTI-CSIC The research leading to these results has received funding from the European Union Seventh Framework Programme under Grant agreement No acknowledges funding from both the CSIC-JaeDoc Fellowship Program (co-funded by the European Social Fund) and Nanocosmos was supported by the “Rafael Calvo Rodés” Program Institute of Material Science of Madrid (ICMM-CSIC) Center for Mechanical Technology and Automation (TEMA) conducted and designed all the theoretical protocols and calculations wrote the manuscript with inputs from all co-authors Download citation Metrics details Moiré superlattices formed by stacking two-dimensional crystals have reinvigorated the pursuit for emergent functionalities of engineered superlattices Unique optical characteristics can be realized from the interplay between the electronic excitations and the atomic rearrangements owing to their intrinsic softness Although large-scale reconstructions have been identified at small twist angles they have been treated as being rigid at large twist angles we report that moiré superlattices made from single layers of MoS2 and WSe2 exhibit a pair of torsional strains with opposite chirality irrespective of the twist angle The whirlpool-shaped periodic lattice distortions introduce fuzziness in the Raman spectra and universal redshifts to the intralayer excitons for all twist angles We show that both of these modulations become weaker as the twist angle increases but do not disappear whereas they are turned off when the constituent layers are not tightly coupled thus establishing an essential structure–property relationship for moiré superlattices The data that support the plots within this paper, and other findings of this study, are available from the corresponding authors upon reasonable request. Source data are provided with this paper Superlattice and negative differential conductivity in semiconductors Anisotropic behaviours of massless Dirac fermions in graphene under periodic potentials Momentum-space indirect interlayer excitons in transition-metal dichalcogenide van der Waals heterostructures and exciton binding in 2D semiconductor heterostructures Tailoring the electronic structure in bilayer molybdenum disulfide via interlayer twist Flat bands and mechanical deformation effects in the moiré superlattice of MoS2-WSe2 heterobilayers Moiré excitons: from programmable quantum emitter arrays to spin–orbit coupled artificial lattices Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping Group theory analysis of phonons in two-dimensional transition metal dichalcogenides Observation of interlayer phonon modes in van der Waals heterostructures Large‐scale mapping of moiré superlattices by hyperspectral Raman imaging Anomalous excitonic resonance Raman effects in few-layered MoS2 Excitonic resonance effects and Davydov splitting in circularly polarized Raman spectra of few-layer WSe2 Anomalous lattice vibrations of single- and few-layer MoS2 Evolution of high-frequency Raman modes and their doping dependence in twisted bilayer MoS2 Bandgap engineering of strained monolayer and bilayer MoS2 Strain engineering and Raman spectroscopy of monolayer transition metal dichalcogenides Atomic corrugation and electron localization due to moiré patterns in twisted bilayer graphenes Relaxation of moiré patterns for slightly misaligned identical lattices: graphene on graphite Lattice relaxation and energy band modulation in twisted bilayer graphene Flat band properties of twisted transition metal dichalcogenide homo- and heterobilayers of MoS2 Observability of charge-density waves by neutron diffraction The Bethe–Salpeter equation formalism: from physics to chemistry Rotational strain in Weyl semimetals: a continuum approach Efficient pseudopotentials for plane-wave calculations Density-functional method for very large systems with LCAO basis sets First principles phonon calculations in materials science Download references This work was supported by the National Research Foundation (NRF) grant funded by the Korean government (MSIT) (2019R1A2C3006189 SRC program: vdWMRC center and 2021R1C1C1010924) was partly supported by a KIAS individual grant (no was supported by a KIAS individual grant (no Computations were supported by the CAC of KIAS These authors contributed equally: Jungcheol Kim Department of Materials Science and Engineering and Research Institute of Advanced Materials prepared the samples and carried out optical measurements The spectroscopic data were analysed by H.C All the authors discussed the results and wrote the manuscript together Nature Materials thanks Kristiaan De Greve Compilation of Raman spectra for all samples measured Source data Twist-angle dependence of the ILB mode frequency (a) and force constant between MoS2 and WSe2 layers (b) Source data The pink circle and red circle denote S and Mo atoms The magnitude of dv (length of the red arrow) is enlarged from one hundred times to three hundred times as indicated Source data SAED pattern of a MoS2/WSe2 HS with a twist angle of 1.0° \(\bar 2110\) DF images obtained by selecting electron diffraction peaks marked with dashed squares in a c,e,g are fast-Fourier-transforms (FFT) of the DF images shown in b,d,f The peaks marked with black dashed circles in FFTs correspond to the frequency component of 0.25 nm−1 Such frequency is associated with the moiré fringe obtained by simply interfering two sets of {\(11\bar 20\)} planes of MoS2 and WSe2 twisted with 1.0° The other set of peaks marked with red dashed circles correspond to the frequency of 0.14 nm−1 that are associated with the periodic structures visualized due to the lattice modulation at the interface Spectra and spectral parameters extracted from spectra Spectral parameters extracted from spectra Download citation DOI: https://doi.org/10.1038/s41563-022-01240-2 Metrics details Grain boundaries (GBs) commonly exist in crystalline materials and affect various properties of materials The facile identification of GBs is one of the significant requirements for systematical study of polycrystalline materials including recently emerging two-dimensional materials Previous observations of GBs have been performed by various tools including high resolution transmission electron microscopy we choose graphene bilayers with a GB as a model system and investigate the effects of interlayer rotations to the identification of GBs We provide a critical condition between adjacent moiré fringe spacings which determines the possibility of GB recognition for monolayer graphene with a grain boundary we demonstrate that low-angle GBs can be distinguished easily by inducing moiré patterns deliberately with an artificial reference overlay GBs can be estimated from compiling and marking the transition peaks of DPs which is a tedious and time consuming method these methods have serious limitations in the case of low angle GBs because the adjacent peaks are shaded by the volume of peaks which prevent the following of transition points we elucidate the effects of interlayer rotation angle to the identification of GBs and provide the critical condition to determine whether GB recognition is possible or not this paper proposes a method to distinguish GB readily by inducing moiré patterns with an artificial reference overlay The reason why the moiré pattern is effective at distinguishing the GBs is also explained using a concept in visual perception (a) An atomic model of bilayer graphene Bottom layer is a single grain and top layer is composed of two tilt grains The denoted angles represent rotational angles between grains (b,c) Two atomic models with different θ1−2 value The θ1−2 of (b) and (c) are 0.5° and 19.5° The GB can be detected more readily with large θ1−2 Insets of (b) and (c) are FFTs of atomic models where d is the lattice plane spacing of graphene, which is 2.13 Å and β is the rotation angle between the upper and bottom layers in radian. θ is rotation angle in degree, which has different condition at 30°. (a–d) Bilayer graphene atomic models showing various values with identical θ1−2 When the intra-misorientation angle (θ1−2) are same the detection of GB is more difficult when the θ1−2 goes to 0 or the sum of interlayer rotation angle becomes 60° (e) Inter-rotation angle dependence of rotational moiré fringe spacings in graphene twisted bilayers (f) Moiré fringe differences as a function of intra-misorientation angle and reference rotation angle The GB distinguishability is shown in color Red color represents the easier identification of GBs The atomic models in panel (a-d) are marked The observation of GB becomes more difficult from (a) to (d) It is obvious that the determination of GB using the moiré pattern is closely related to the difference of moiré fringe spacings The subgrain boundary is a network of discrete dislocations, shown as Fig. 3(a) is rotated by angle θ around polar vector θ Moiré fringe spacing depends on θ and b(c) is the same way different moiré fringe spacings have different numbers of Burgers vectors The magnitude of the Burgers vectors is the same as the lattice parameter of graphene we can’t distinguish the GBs when the difference of moiré fringe spacings is below the graphene lattice parameter (b) Demonstration of Frank’s formula with a hexagonal lattice Difference vector can be described as the sum of discrete burgers vectors (c) Invisible GB region is shown as red color according to represented criterion (a,c) The processed TEM images of graphene with two tilt grains The images are obtained with inverse FFT of bilayer graphene (b,d) The TEM images overlaid with reference layer (a,b) have small θ1−2 and (c,d) have relative big θ1−2 Insets are color overlay images to show the grains more clearly (a) Comparison of moiré patterns between homostructure and heterotructure With a heterostructure (hexagonal boron nitride on graphene) as moiré pattern periodicity is not in line (b) The comparison of rotational moiré fringe spacings and general moiré fringe spacings according to the which are different whether lattice mismatch is 0 or 0.02 (BN/Gr bilayer) (f) The relationship of the lattice mismatch and moiré fringe difference The grain distinguishability is more proper when using zero lattice mismatch reference layer we present why GB observation is difficult in low GB in terms of the difference in moiré fringe spacings between superlattices which have different The lower the difference moiré fringe spacings exhibit the harder it is to distinguish grains due to the combined moiré pattern we can calculate the critical indistinguishable point according to Frank’s formula GBs can be shown obviously by controlling we present an image to represent GBs obviously using an artificial reference layer to maximize the difference of moiré fringe spacings This research is meaningful in that understanding and application of GB detection is established which had difficulty in determining GBs using diffraction patterns and FFT Our results will be useful for intrinsic GB studies GB diffusion and enhanced chemical reactivity at the boundary not only for single layers but also multilayers The atomic model simulation images (Fig. 1,2 and Reference layer overlaid in Fig. 4(b,d)) were produced mainly by Jmol Jmol is an open-source Java viewer for three-dimensional chemical structures Jmol can read a variety of file types and output from quantum chemistry programs and make animations of multi-frame files and computed normal modes from quantum programs The main frame of the model was made by Chemsketch which is produced by ACD/Labs for free TEM images and simulated images were overlaid using Adobe Photoshop The Hide-and-Seek of Grain Boundaries from Moiré Pattern Fringe of Two-Dimensional Graphene Control and characterization of individual grains and grain boundaries in graphene grown by chemical vapour deposition Effects of Polycrystalline Cu Substrate on Graphene Growth by Chemical Vapor Deposition Grain boundaries in graphene grown by chemical vapor deposition Anomalous Strength Characteristics of Tilt Grain Boundaries in Graphene Domain (Grain) Boundaries and Evidence of “Twinlike” Structures in Chemically Vapor Deposited Grown Graphene Grains and grain boundaries in single-layer graphene atomic patchwork quilts Intrinsic Strength and Failure Behaviors of Graphene Grain Boundaries chemical and dynamical trends in graphene grain boundaries energy and structural transformations of graphene grain boundaries from atomistic simulations Topological defects in graphene: Dislocations and grain boundaries Mechanical properties of bilayer graphene with twist and grain boundaries Automatic grain boundary detection and grain size analysis using polarization micrographs or orientation images Grain Boundary Mapping in Polycrystalline Graphene Characterization of Strain-Rate Sensitivity and Grain Boundary Structure in Nanocrystalline Gold-Copper Alloys High-resolution electron microscopy characterization of nanocrystalline grain boundaries in gold-copper alloys Lord On the Manufacture and Theory of Diffraction-gratings Unification of formulation of moiré fringe spacing in parametric equation and Fourier analysis methods Applications of the Optical Moiré Technique in Hrem Controlling the electronic structure of bilayer graphene Novel hetero-layered materials with tunable direct band gaps by sandwiching different metal disulfides and diselenides Observation of an electrically tunable band gap in trilayer graphene Twinning and Twisting of Tri- and Bilayer Graphene Graphene on boron-nitride: Moiré pattern in the van der Waals energy Atomic Resolution Imaging of Rotated Bilayer Graphene Sheets Using a Low kV Aberration-corrected Transmission Electron Microscope Atomic-scale observation of rotational misorientation in suspended few-layer graphene sheets Direct Imaging of Rotational Stacking Faults in Few Layer Graphene Electronic structure of graphene twist stacks Graphene bilayer with a twist: Electronic structure Quantum Hall Effect in Twisted Bilayer Graphene Symmetry breaking in commensurate graphene rotational stacking: Comparison of theory and experiment Dislocation Networks-Subgrain Boundaries in CRYSTAL DEFECTS AND CRYSTALLINE INTERFACES Ch Spatial vision in Basic vision: an introduction to visual perception Ch On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images Download references This work was supported by Nano Material Technology Development Program and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science Ulsan National Institute of Science and Technology (UNIST) All authors discussed the results and reviewed the manuscript Download citation Metrics details Moiré superlattices in graphene supported on various substrates have opened a new avenue to engineer graphene’s electronic properties the exact crystallographic structure on which their band structure depends remains highly debated In this scanning tunneling microscopy and density functional theory study we have analysed graphene samples grown on multilayer graphene prepared onto SiC and on the close-packed surfaces of Re and Ir with ultra-high precision We resolve small-angle twists and shears in graphene and identify large unit cells comprising more than 1,000 carbon atoms and exhibiting non-trivial nanopatterns for moiré superlattices which are commensurate to the graphene lattice a general formalism applicable to any hexagonal moiré is presented to classify all reported structures the conical electronic bands around the K and K′ points of the Brillouin zone are called Dirac cones tuning moiré superlattices is a mean to tailor graphene’s electronic properties Moiré superlattice and beatings: (a) Ball model of a chain of (small) carbon atoms in graphene on top of a chain of (large) atoms from the support both having different lattice parameters agr and as whose commensurability defines a moiré superlattice with period am For 6 graphene periods matching 5 support periods (orange) a single beating occurs within the moiré period and the fast Fourier transform (FFT) reveals a fundamental harmonic defined by k = (kgr − ks)/(6 − 5) (b) For 11 graphene periods matching 9 support periods (purple) two beatings occur within the moiré period with similar stacking configurations at the edges and at the middle of the ball model The corresponding FFT reveals a fundamental harmonic at k = (kgr − ks)/(11 − 9) (b) such as a metallic substrate whose deformations induced by graphene can be neglected is not accounted for to our best knowledge with the help of geometrical transformations we provide a fully-consistent description of the full complexity of commensurate moiré superlattices for the general case of an arbitrary strain (including shear biaxial) and of superlattices having any number of beatings This transformation is expressed within a matrix formalism and in an extension of the so-called Wood’s notation which gives the angles formed between the unit cell vectors of graphene and the moiré as well as the ratio between these vectors’ length We use the latter notation to construct maps of the possible commensurate moiré superlattices and to revisit previously published analysis of experimental observations We show that supported graphene is subjected to strain levels far below what is usually assumed We apply this description to resolve the structure of the moiré superlattices in graphene on multilayer graphene prepared on SiC and in monolayer graphene on Re(0001) and Ir(111) we resort to scanning tunneling microscopy (STM) in both direct and reciprocal (Fourier) space in the latter case achieving better than 0.1 pm precision on the lattice parameter determination owing to distortion-less imaging with atomic resolution across several 10 nm fields of view We find rotated and sheared moiré superlattices shear appears more obviously in the moiré than in the graphene as confirmed by our density functional theory (DFT) calculations Some of these moirés comprise several moiré beatings in the case of metal substrates commensurability between graphene and moiré superlattices provides a fine description of even very large moiré supercells supported graphene and its substrate do not share the same lattice parameter and/or graphene lies twisted by some angle with respect to its support Assuming commensurability between the two lattices a supercell can be defined which comprises the smallest integer numbers of unit cells of both graphene and the support This supercell defines the moiré superlattice the moiré superlattice parameter am is an integer number times graphene’s (agr) or the support’s (as) lattice parameters: am = i agr = m as The Fourier description of moirés naturally makes the distinction between both the i − m = 1 moiré containing only one beating and the i − m = 2 comprising two distinct ones The analysis of the moiré superlattices presented below will be performed by expressing the moiré superlattice unit vectors as function of those of the graphene and support unit cells The analysis will also be expressed as function of elementary geometrical deformations Structural interpretation of a moiré superlattice: (a) The transformation relating graphene lattice vectors to those of its support can be decomposed into four steps (1) Graphene vectors are isotropically rescaled with respect to those of the support (light red) (2) Graphene is rotated with respect to its support (red) in order to determine the direction in which (3) a horizontal rescaling is applied (dark red) (b) The lattice vectors of the moiré superlattice decompose into both graphene and support bases (c) Corresponding extended Wood’s notation: (p1 Rφ1 × p2 Rφ2) where p1 and p2 are scaling factors and φ1 and φ2 are rotation angles one can equivalently describe the graphene-substrate relation by explicitly writing the commensurate relation defining the moiré superlattice In order to account for its structural complexity in two dimensions which are determined through atomically-resolved microscopy is then necessary (only four are needed to describe graphene maintaining the D6h symmetry when it is only strained biaxially and rotated): This translates into reciprocal space as (the “T” superscript denotes the transpose operation): At this point we can generalize to the two-dimensional limit the concept of number of beatings N in a moiré cell. One can then define number of beatings N1 and N2 along and (see Supplementary information): The number of beatings N within a moiré cell is then simply given by the product N = N1N2 The same can be done to relate the moiré unit vectors to those of the support The geometrical formalism developed so far proves necessary to properly interpret the refined twist angles and shearings observed in atomically-resolved microscopy images The experimental uncertainty on the identification of the (i r) integers is here discussed to justify this necessity l) can be lowered by precisely determining and we measure the distance between the moiré spots and the graphene spots in the Fourier transform image (each spot corresponding to a Fourier component) The sharpness of the spots is inversely proportional to the size of the atomically resolved image and the number of spots increases with the contrast of the moiré with respect to the atomic lattice The former effect sets a precision in the determination of the spacing between two spots of 6% in the case of gr/Ir for which the image field of view is ~500 nm2 The latter effect translates into an uncertainty as low as in the case of gr/Ir (see Results below) which corresponds to ~50 Fourier component spacings along one direction The same is true around the graphene harmonics this precision translates through propagation of uncertainty into 2% atomic resolution imaging can artificially produce sheared images Such shears may result from imaging artefacts thermal drift of the piezoelectric scanners or inequivalent calibration of these scanners along the two scan directions these artefacts have no influence on the decomposition of and onto and STM analysis of multilayer graphene on C-face SiC: (a) (3.2 × 3.8 nm2) STM topograph (Itunnel = 10 nA Vbias = 100 mV) with emphasized upper graphene lattice (black) moiré superlattice cell (blue rhombus) and lattice vectors of upper graphene and moiré (red and blue respectively) (b) Corresponding FFT-image with emphasized moiré reciprocal lattice (black) and lattice vectors of moiré and upper and lower layers of graphene (blue STM analysis of gr/Re: (a) (5.6 × 5.2 nm2) STM topograph (Itunnel = 6 nA Vbias = 30 mV) with overlaid graphene lattice (black) and lattice vectors of graphene and N = 2 superlattice (red and blue arrows respectively) Moiré cell (blue full line) and its closest unsheared approximation with N = 1 beating (green dashed line) with the coordinates of its corners in the graphene basis The “odd-even” transition along lines of carbon atoms is also emphasized as well as the either 6 or 3 C atoms observed in a moiré hill or valley (b) Corresponding FFT-image with emphasized moiré reciprocal lattice (black) and lattice vectors of moiré Inset shows the harmonics surrounding the center of the FFT-image with improved contrast Its origin is well illustrated in the case of the two distinctive moiré valleys they differ only in the site of the remaining visible C atom: hcp or fcc Depending on whether the site is hcp or fcc the corresponding C atom belongs to sub-lattice A or B of graphene the C atoms that are observed switch continuously from one sub-lattice to the other this induces an apparent oscillation of the row these two effects are related to a modulation of the electronic density of states on the two sub-lattices of graphene which is correlated with the moiré periodicity which is another clue that indicates graphene structure is sheared on this STM topograph the positions of the graphene spots with respect to the moiré reciprocal network vary for the three main directions the moiré structure considered here is sheared The commensurability relation of this structure reads as: where the superstructure lattice vectors are explicitly decomposed on the graphene lattice A more physical description of such a structure is given by comparing the graphene overlayer with its HOPG counterpart and decomposing the strain in terms of a uniaxial and a biaxial contributions Using Equation (S21a,b) in the case of gr/Re graphene is biaxally compressed by εb ~ −0.14% and uniaxially compressed by εu ~ −0.84% This shows a moiré is actually related to a non-trivial distortion of the graphene lattice (b) FFT-image obtained from a 15.6 × 30 nm2 STM topograph and overlaid with the lattice paved with vectors and lattice vectors of moiré Inset shows moiré spots surrounding the center of the FFT-image with improved contrast Three support lattices have been considered so far revealing that a moiré structure can be rotated It also demonstrates that moiré superlattices comprising more than one beating are commonly encountered Three equivalent ways have been presented to describe moiré superlattices with ease: Using an extended Wood’s notation for a pictorial description using two scaling factors and two angles In more physical terms with rotation angles and uniaxial and biaxial strains With eight integers that decompose independently the two moiré lattice vectors onto those of graphene and of its support these techniques will detect the fundamental component of N > 1 beatings superlattices The geometrical analysis presented here is thus a tool towards the quantitative prediction of such effects Out of these two equivalent equations and using only the support lattice constant as, one can express the moiré lattice constant am, the biaxial strain ε (assuming the lattice parameter of HOPG as a zero-strain situation) and the twist angle φ between graphene and its support (see Supplementary information): The link with their Wood’s notation (p × p) Rφ can be established straightforwardly (see Supplementary information) as: Moiré lattice constant am versus angle φ between graphene and its support where and N the number of beatings given by Equation (3) with N1 = N2 The analysis performed here demonstrates the rich variety of moiré superlattices to be expected for graphene on a substrate well beyond the simple case of N = 1 unsheared cases Although many structures are possible from the geometrical point of view few of them have actually been reported in the literature This state of fact can be interpreted in two different ways: either differentiating some very similar structures has not been considered or is not possible due to too limited space resolution or only a few of them are stable enough to actually exist Gr/Ir and gr/Pt are typical of the first situation. Numerous moiré phases have been reported for them, as shown on Fig. 6a,b The majority of them is identified as N = 1 moiré superlattices this description appears sometimes unrealistic graphene tends to align its zigzag rows to the close-packed rows of the metal (φ ~ 0°) even in growth conditions quite far from thermodynamic equilibrium the strong bonds of covalent character between carbon and metal atoms inside the growing flake are not readily broken Such similar behaviours may lead to the conclusion that the mechanism presented here is common to every system where graphene is in strong interaction with its substrate the existence of a multitude of commensurate superlattices discretely spanning the 0 − 60° twist range could as well account for the observation due to the finite size of the diffraction spot (set by the domain size or the instrumental resolution) that they yield different supported graphene systems have been studied with STM A consistent analysis of moiré superlattices involving both direct and FFT STM images has been presented has been rationalized by calculating electronic density maps derived from DFT calculations A spatial precision of a tenth of 1 pm is achieved revealing that graphene lying on a substrate is actually twisted A geometrical model enables to classify all moiré superlattices This model gives a global picture assuming commensurability between graphene and its substrate (and consequently between graphene or the substrate and the moiré) While a very large number of structures is possible In the case of strong graphene-substrate interaction it is unlikely that all predicted superlattices are discovered since for instance phases corresponding to a substantial rotation of graphene with respect to the substrate do not tend to form For low interaction graphene-substrate systems the complexity of the moiré superlattices has been undetected or overlooked leading to possibly simplified interpretations We anticipate that moiré superlattices with N > 1 number of beatings will produce rich electronic modulations in graphene SiC surface was first cleaned in H2 and Ar atmosphere at 1,560 °C and subsequently annealed in Ar atmosphere at the same temperature by saturating the Re(0001) surface with C2H4 at room temperature (introduced with a 3 · 10−8 mbar pressure) and two subsequent cycles of flash-annealing/cooling at 750 °C with a 5 · 10−7 mbar C2H4 pressure An Ir single crystal cut in the (111) surface purchased from Surface Preparation Laboratory was cleaned in the same UHV chamber as for gr/Re by cycles of Ar+ ion bombardment at 1 keV and subsequent annealing at 1,200 °C The gr/Ir was prepared by exposing to 10−8 mbar of C2H4 at 1,000 °C for 15 minutes STM measurements were performed at 4 K in a home-made He-cooled STM using a commercial Pt/Ir tip bought from Bruker STM measurements were performed at room temperature under UHV thermal drift and miscalibrations have been corrected The Methfessel Paxton method is used to calculate the total energy with a smearing of 0.2 The supercell consists in four Re layers and one C layer with an empty space of 9 Å to avoid spurious interactions Re atoms are kept fixed in the bottom second Re layer calculations are performed using the K point only strained and sheared graphene moiré superlattices Condensed-matter simulation of a three-dimensional anomaly Two-dimensional gas of massless Dirac fermions in graphene Structural and electronic properties of epitaxial graphene on SiC(0001): a review of growth transfer doping and hydrogen intercalation Graphene growth and properties on metal substrates Chemical origin of a graphene moiré overlayer on Ru(0001) General approach to understanding the electronic structure of graphene on metals Structural properties of the multilayer graphene/4H-SiC system as determined by surface x-ray diffraction Cross-sectional imaging of individual layers and buried interfaces of graphene-based heterostructures and superlattices CRC Handbook 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corrugated epitaxial graphene grown on Re(0001) Scanning tunneling microscopy of graphene on Ru(0001) single-crystalline graphene monolayer formed on Ru(0001) Graphene on Rh(111): Scanning tunneling and atomic force microscopies studies AFM-STM studies of carbonization and graphitization of polyimide films Giant superstructures formed on graphite surface treated with NaOH solutions studied by scanning tunneling microscopy Bulk defects in graphite observed with a scanning tunneling microscope Strains induced by point defects in graphene on a metal Topography of the graphene/Ir(111) moiré studied by surface x-ray diffraction Resonance effects on the Raman spectra of graphene superlattices Strain-induced modulation of Dirac cones and van Hove singularities in a twisted graphene bilayer Scalable synthesis of graphene on single crystal Ir(111) films band gap and group velocities for graphene on close-packed metal surfaces Small-angle lattice rotations in graphene on Ru(0001) Why multilayer graphene on 4H-SiC behaves like a single sheet of graphene Rotational disorder in few-layer graphene films on 6H-SiC(000-1): A scanning tunneling microscopy study Towards wafer-size graphene layers by atmospheric pressure graphitization of silicon carbide Ab initio molecular dynamics for liquid metals General gradient approximation made simple Download references Financial support from Agence Nationale de la Recherche through contract ANR-14-OHRI-0004-01 2DTransformers is acknowledged Calculations were performed using HPC resources from GENCI-IDRIS (Grant 2015-097015) We thank Jean-Yves Veuillen and Karim Ferhat for fruitful discussions was supported by a CIBLE fellowship from Région Rhône-Alpes Download citation UTokyo FOCUS Material behaviors depend on many things including not just the composition of the material but also the arrangement of its molecular parts researchers have found a way to coax carbon nanotubes into creating moiré patterns Such structures could be useful in materials research in particular in the field of superconducting materials Professor Hiroyuki Isobe from the Department of Chemistry at the University of Tokyo and his team create nanoscopic material structures Their aim is to explore new ways to create carbon nanostructures and to find useful applications for them The most recent breakthrough from their lab is a new form of carbon nanotube with a very specific arrangement of atoms that has attracted much attention in the field of nanomaterials “We successfully created different kinds of atom-thick carbon nanotubes which self-assemble into complex structures,” said Isobe “These nanotubes are made from rolled up sheets of carbon atoms arranged hexagonally We made wide ones and narrow ones which fit inside them This means the resulting complex tube structure has a double-layered wall The hexagonal patterns of these layers are offset such that the two layers together create what is known as a moiré pattern And this is significant for materials researchers.” You may see moiré patterns in your everyday life When repeating patterns overlay one another a new resultant pattern emerges this resultant pattern will change slightly if you look at a screen door through a mesh curtain In the case of the team’s moiré patterns they are made when one hexagonal grid of carbon atoms is rotated slightly relative to another similar hexagonal grid they can imbue materials with functional properties Two areas that might especially benefit from the properties created here are synthetic chemistry as the moiré carbon bilayer tubes could be challenging yet attractive targets of molecular self-assembly which could lead to a generational leap in electrical devices which require far less power to run and would be far more capable than current devices The purple outer layer and blue inner layer each have a similar arrangement of carbon atoms but are rotated relative to each other by just a few degrees When repeating patterns overlap and are rotated they create these mesmerizing arrangements called moiré patterns Here hexagonal grids are rotated in either direction Inquiries about the content of this page: Division for Strategic Public Relations Send inquiry Kashiwa Campus Hongo Campus Komaba Campus Access and Campus Maps Back Metrics details Two-dimensional hexagonal arrays of Pt nanoparticles (1.5 nm diameter) have been obtained by deposition of preformed and size selected Pt nanoparticles on graphene This original self-organization is induced by the 2D periodic undulation (the moiré pattern) of graphene epitaxially grown on the Ir(111) surface By means of complementary techniques (scanning tunneling microscopy the Pt clusters shapes and organization are characterized and the structural evolution during annealing is investigated The soft-landed clusters remain quasi-spherical and a large proportion appears to be pinned on specific moiré sites The quantitative determination of the proportion of organized clusters reveals that the obtained hexagonal array of the almost spherical nanoparticles is stable up to 650 K which is an indication of a strong cluster-surface interaction the main limitations of platinum clusters produced by atomic deposition on g/Ir(111) regarding the catalytic applications for instance are their small height of 1 to 2 MLs and the impossibility to tune independently cluster coverage and sizes If the organization of small NPs is expected in a way similar to the organization of clusters formed by physical evaporation the organization of much larger NPs on templates such as g/Ru(0001) or g/Ir(111) is expected to be more difficult because of the much smaller ratio between interfacial NP atoms and bulk or surface ones we report on the investigation of a diluted two dimensional (2D) array of preformed and size selected Pt clusters (c.a 80 atoms and 1.5 nm mean diameter) that are soft-landed at room temperature (RT) on g/Ir(111) by means of three complementary techniques: STM grazing incidence X-ray diffraction (GIXD) and grazing incidence small-angle X-ray scattering (GISAXS) The morphological and structural evolutions of this system are also investigated during annealings above RT An efficient and original method to quantify the organization order of the cluster super-lattice at a macroscopic scale is proposed (a,b) Typical STM topographs of size-selected (1.5 nm in diameter) platinum clusters supported on g/Ir(111) The coverage of the NPs corresponds to a low (a) and medium (b) density (c) Height distribution of the Pt clusters measured by STM (d) Nearest neighbor distance (center to center) distribution of Pt clusters deposited on g/Ir(111) with a medium density The distribution is fitted by two Gaussian functions STM does not provide precise information on several characteristics of this system the epitaxial relationship of the NPs with the substrate their precise shape or the correlation length of the NPs organization GIXD and GISAXS studies were then conducted to answer these questions In-plane scans along the h direction in the neighborhood of the (200) Ir rod The temperatures are (a) RT and (b) RT to 650 K bare g/Ir(111) (black circles) and g/Ir(111) covered with a medium density of Pt clusters (red triangle) and (b) g/Ir(111) covered with a medium density of Pt clusters the inset shows a zoom of the scan in linear scale Measurable intensity from the NPs along such scans is expected only if a significant fraction of the NPs are in epitaxy with the substrate the much wider peak appearing around the (200) rod upon NP deposition shows that some NPs are in epitaxy with Ir(111) The decrease of the graphene Bragg peak (h ≈ 2.23) can be understood as an effect of the Pt deposition inducing inhomogeneous distortions of the graphene Such distortions can also induce a peak broadening but this is hardly detected because of the low signal compared to the background The last observation is the increase of the moiré peak intensity which implies that some NPs are pinned on a specific site of the moiré cell a qualitative analysis by GIXD indicates that a significant fraction of the NPs are in epitaxy with the Ir(111) substrate; and at least part of them are perfectly anchored to a specific moiré site; the rest possibly being in different places (i.e with no coherence from one unit cell to the other) We then define Θ as the proportion of clusters pinned on the moiré The indications of the organization and epitaxy of the clusters given by GIXD have been followed during an annealing of the samples. In Fig. 2b the vicinity of the (200) rod of Ir is reported for the same sample before and after annealing the width of the (200) rod of Ir is reduced to the value found for the bare g/Ir(111) The intensity of the moiré peak around h ≈ 1.9 decreases and reaches the background level at T = 650 K At this point, two preliminary conclusions can be drawn. Between RT and 460 K the clusters that were in epitaxy but are not positioned on specific sites have lost their epitaxial relationship with Ir(111). The decrease of the moiré peak with temperature above 560 K (see Fig. 2c) suggests either a loss of epitaxy or a decrease of Θ or a combination of these two features (a) Map of the reciprocal space at RT at αf = αc = 7 mrad (left) and the corresponding simulation (right) with the incident beam along the Ir[100] (left) Ir[110] (center) directions and out of azimuth (right) (c) Perpendicular (left) and parallel (right) line cuts of the 2D GISAXS pattern shown in (d) the continuous lines correspond to the best fits and the symbols (circles squares and diamonds) correspond to the experimental data points (d) 2D experimental GISAXS pattern at RT with the incident beam along the Ir[100] direction (left) and the corresponding simulation (right) (a,b) correspond to a sample of Pt clusters deposited on g/Ir(111) with a high density (in order to have a better visibility of the correlation peak in the direction Ir[110]) while (c,d) correspond to a medium cluster density I(q) is assumed to be the incoherent sum (since non-organized particles lie at random locations their phases cancel on average and the summation can be made incoherently) of the intensity scattered by NPs lying at random locations (IR) and that scattered by the NPs anchored on the graphene moiré (IL): S(q) is the total interference function which describes the statistical distribution of the Pt clusters on the surface and thus their lateral correlations (a) Fit of the RT experimental GISAXS correlation peak (black dots) with various fraction Θ of Pt clusters lying on the moiré lattice Difference of 2D GISAXS pattern line cuts (αf = αc ≈ 7 mrad) with the incident beam 10° off azimuth and in the Ir[100] direction The intermediates values of Θ are represented in dashed black lines (b) Best fits of experimental GISAXS correlation peaks at various temperatures (c) Plot of Θ versus the annealing temperature Pt NPs have been deposited on g/Ir(111) with a medium density We deduce that half of the preformed Pt clusters deposited on g/Ir(111) self-organize as an incomplete (i.e Let us emphasize that the occupation ratio of the moiré sites are 8 and 27% for the samples covered with a medium and a high densities of NPs respectively although the clusters are still anchored at specific moiré site This emphasizes the strength of the GISAXS technique which is sensitive to all NPs whether they are in epitaxy or not For temperatures higher than 650 K, the shape of the clusters is modified (see Figure S2 in supporting information) These morphological modifications are probably due to the coalescence or sintering of NPs or possibly to intercalation of Pt below the graphene Anyway the system is no more a diluted 2D array of size selected clusters and the modification is irreversible as confirmed by measurements after cooling down to RT the homogeneity observed from STM images of various areas seems to refute this hypothesis (ii) The cluster adsorption energy landscape could present one principal potential well able to attract clusters in an area corresponding to ca while there exists many smaller potentials wells distributed over the rest of the unit cell A theoretical study should help understanding the kinetics and thermodynamics of NP trapping on g/Ir(111) This could eventually give clues to enhance the ratio of organized clusters for instance by increasing the temperature during the deposition process The mass-selected low energy cluster beam deposition (MS-LECBD) technique where preformed clusters having a 3D morphology are soft-landed on a surface The clusters were then soft landed at RT on a g/Ir(111) under UHV (10−11 to 10−10 mbar) The kinetic energy of the incident clusters is around 0.4 eV/atom ensuring that there will neither be cluster fragmentation on the surface nor surface defect creation The incident flux of NPs (cations) has been determined using a Faraday cup connected to a pico-amperemeter the density and size of the clusters are set independently medium (3 × 104 NPs μm−2) and high (105 NPs μm−2) densities of incidents clusters were deposited (relative uncertainties of 8%) The deposition chamber is connected to an Omicron UHV STM allowing in situ STM observation of the samples at RT the samples were transferred back under UHV to the synchrotron facility in Grenoble (France) Moiré induced organization of size-selected Pt clusters soft landed on epitaxial graphene The Structure of Catalytically Active Gold on Titania Why gold nanoparticles are more precious than pretty gold: noble metal surface plasmon resonance and its enhancement of the radiative and nonradiative properties of nanocrystals of different shapes Experimental Observation of Superparamagnetism in Manganese Clusters 5255–5257; 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BM32. at &lt; http://www.esrf.eu/UsersAndScience/Experiments/CRG/BM32&gt;,(2014) (Date of acces 09/10/2014) PLYRA. Plateforme PLYRA. Plateforme PLYRA at&lt; http://plyra.univ-lyon1.fr/&gt;,(2014) (Date of acces 09/10/2014) Application of a static quadrupole deviator to the deposition of size-selected cluster ions from a laser vaporization source Download references Geaymond and the staff of the BM32 beamline Research supported by French ANR Contract No Present address: Laboratoire des Multimatériaux et Interfaces (LMI) involved in the clusters synthesis and deposition; T.Z. designed and build the UHV transfer device wrote the manuscript and prepared the figures Download citation