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Here, a deterministic algorithm is proposed, that is capable of constructing a common supercell between two similar crystalline surfaces without scanning all possible cases. Using the complex plane, the 2D lattice is defined as the 2D complex vector. Then, the relationship between two surfaces becomes the eigenvector-eigenvalue relation where an operator corresponds to a transformation matrix. It is shown that this transformation matrix can be directly determined from the lattice parameters and rotation angle of the two given crystalline surfaces with O(log Nmax) time complexity, where Nmax is the maximum index of repetition matrix elements. This process is much faster than the conventional brute force approach ( O ( N max 4 ) $O(N_{\mathrm{max}}^4)$ ). By implementing the method in Python code, experimental 2D heterostructures and their moiré patterns and additionally find new moiré patterns that have not yet been reported are successfully generated. According to the density functional theory (DFT) calculations, some of the new moiré patterns are expected to be as stable as experimentally-observed moiré patterns. Taken together, it is believed that the method can be widely applied as a useful tool for designing new heterostructures with interesting properties.

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Optical tracking of head pose via fiducial markers has been proven to enable effective correction of motion artifacts in the brain during magnetic resonance imaging but remains difficult to implement in the clinic due to lengthy calibration and set up times. Advances in deep learning for markerless head pose estimation have yet to be applied to this problem because of the sub-millimetre spatial resolution required for motion correction. In the present work, two optical tracking systems are described for the development and training of a neural network: one marker-based system (a testing platform for measuring ground truth head pose) with high tracking fidelity to act as the training labels, and one markerless deep-learning-based system using images of the markerless head as input to the network. The markerless system has the potential to overcome issues of marker occlusion, insufficient rigid attachment of the marker, lengthy calibration times, and unequal performance across degrees of freedom (DOF), all of which hamper the adoption of marker-based solutions in the clinic. Detail is provided on the development of a custom moiré-enhanced fiducial marker for use as ground truth and on the calibration procedure for both optical tracking systems. Additionally, the development of a synthetic head pose dataset is described for the proof of concept and initial pre-training of a simple convolutional neural network. Results indicate that the ground truth system has been sufficiently calibrated and can track head pose with an error of <1 mm and <1°. Tracking data of a healthy, adult participant are shown. Pre-training results show that the average root-mean-squared error across the 6 DOF is 0.13 and 0.36 (mm or degrees) on a head model included and excluded from the training dataset, respectively. Overall, this work indicates excellent feasibility of the deep-learning-based approach and will enable future work in training and testing on a real dataset in the MRI environment.

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Moiré patterns caused by aliasing between the camera's sensor and the monitor can severely degrade image quality. Image demoiréing is a multi-task image restoration method that includes texture and color restoration. This paper proposes a new multibranch wavelet-based image demoiréing network (MBWDN) for moiré pattern removal. Moiré images are separated into sub-band images using wavelet decomposition, and demoiréing can be achieved using the different learning strategies of two networks: moiré removal network (MRN) and detail-enhanced moiré removal network (DMRN). MRN removes moiré patterns from low-frequency images while preserving the structure of smooth areas. DMRN simultaneously removes high-frequency moiré patterns and enhances fine details in images. Wavelet decomposition is used to replace traditional upsampling, and max pooling effectively increases the receptive field of the network without losing the spatial information. Through decomposing the moiré image into different levels using wavelet transform, the feature learning results of each branch can be fully preserved and fed into the next branch; therefore, possible distortions in the recovered image are avoided. Thanks to the separation of high- and low-frequency images during feature training, the proposed two networks achieve impressive moiré removal effects. Based on extensive experiments conducted using public datasets, the proposed method shows good demoiréing validity both quantitatively and qualitatively when compared with the state-of-the-art approaches.

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Twisted van der Waals materials featuring Moiré patterns present new design possibilities and demonstrate unconventional behaviors in electrical, optical, spintronic, and superconducting properties. However, experimental exploration of thermal transport across Moiré patterns has not been as extensive, despite its critical role in nanoelectronics, thermal management, and energy technologies. Here, the first experimental study is conducted on thermal transport across twisted graphene, demonstrating a phonon polarizer concept from the rotational misalignment between stacked layers. The direct thermal and acoustic measurements, structural characterizations, and atomistic modeling, reveal a modulation up to 631% in thermal conductance with various Moiré angles, while maintaining a high acoustic transmission. By comparing experiments with density functional theory and molecular dynamics simulations, mode-dependent phonon transmissions are quantified based on the angle alignment of graphene band structures and attributed to the coupling among flexural phonon modes. The agreement confirms the dominant tuning mechanisms in adjusting phonon transmission from high-frequency thermal modes while having negligible effects on low-frequency acoustic modes near Brillouin zone center. This study offers crucial insights into the fundamental thermal transport in Moiré structures, opening avenues for the invention of quantum thermal devices and new design methodologies based on manipulations of vibrational band structures and phonon spectra.

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It is crucial to deeply understand how the interlayer interaction acts on controlling the structural and electronic properties of shifted patterns of bilayer phosphorene. A comprehensive first-principles study on the bilayer phosphorene through relative translation along different directions has revealed that there is a direct correlation between the potential energy surface and the interlayer equilibrium distance. The shorter the interlayer distance, the lower the potential energy surface. The shifted patterns with the most stable state, the metastable state, and the transition state (with energy barrier of ∼1.3 meV/atom) were found associated with the AB, the Aδ, and the TS stacking configurations, respectively. The high energy barriers, on the other hand, are ∼9.3 meV/atom at the AA stacking configuration along the zigzag pathway, ∼5.3 meV/atom at the AB' stacking configuration along the armchair pathway, and ∼11.2 meV/atom at the AA' stacking configuration along the diagonal pathway, respectively. The character of electronic bandgap with respect to the shifting shows an anisotropic behavior (with the value of 0.69-1.22 eV). A transition from the indirect to the direct bandgap occurs under the shifting, implying a tunable bandgap by stacking engineering. Furthermore, the orbital hybridization at the interfacial region induces a redistribution of the net charge (∼0.002-0.011e) associated with the relative shifting between layers, leading to a strong polarization with stripe-like electron depletion near the lone pairs and accumulation in the middle of the interfacial region. It is expected that such interesting findings will provide a fundamental reference to deeply understand and analyze the complex local structural and electronic properties of twisted bilayer phosphorene and will make the shifted patterns of bilayer phosphorene promising for nanoelectronics as versatile shiftronics materials.

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Twisting bilayers of transition metal dichalcogenides gives rise to a moiré potential resulting in flat bands with localized wave functions and enhanced correlation effects. In this work, scanning tunneling microscopy is used to image a WS2 bilayer twisted approximately 3° off the antiparallel alignment. Scanning tunneling spectroscopy reveals localized states in the vicinity of the valence band onset, which is observed to occur first in regions with S-on-S Bernal stacking. In contrast, density functional theory calculations on twisted bilayers that have been relaxed in vacuum predict the highest-lying flat valence band to be localized in regions of AA' stacking. However, agreement with experiment is recovered when the calculations are performed on bilayers in which the atomic displacements from the unrelaxed positions have been reduced, reflecting the influence of the substrate and finite temperature. This demonstrates the delicate interplay of atomic relaxations and the electronic structure of twisted bilayer materials.

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The physical properties of graphene nanoribbons (GNRs) are closely related to their morphology; meanwhile GNRs can easily slide on surfaces (e.g., superlubricity), which may largely affect the configuration and hence the properties. However, the morphological evolution of GNRs during sliding remain elusive. We explore the intriguing tail swing behavior of GNRs under various sliding configurations on Au substrate. Two distinct modes of tail swing emerge, characterized by regular and irregular swings, depending on the GNR width and initial position relative to the substrate. The mechanism can be explained by the moiré effect, presenting both symmetric and asymmetric patterns, resembling a mesmerizing nanomillipede. We reveal a compelling correlation between the tail swing mode and the edge wrinkle patterns of GNRs induced by the moiré effect. These findings provide fundamental understanding of how edge effects influence the tribomorphological responses of GNRs, offering valuable insights for precise manipulation and operation of GNRs.

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Transition metal dichalcogenide heterobilayers feature strong moiré potentials with multiple local minima, which can spatially trap interlayer excitons at different locations within one moiré unit cell (dubbed moiré locales). However, current studies mainly focus on moiré excitons trapped at a single moiré locale. Exploring interlayer excitons trapped at different moiré locales is highly desirable for building polarized light-emitter arrays and studying multiorbital correlated and topological physics. Here, via enhancing the interlayer coupling and engineering the heterointerface, we report the observation and modulation of high-temperature interlayer excitons trapped at separate moiré locales in WS2/WSe2 heterobilayers. These moiré-locale excitons are identified by two emission peaks with an energy separation of ∼60 meV, exhibiting opposite circular polarizations due to their distinct local stacking registries. With the increase of temperature, two momentum-indirect moiré-locale excitons are observed, which show a distinct strain dependence with the momentum-direct one. The emission of these moiré-locale excitons can be controlled via engineering the heterointerface with different phonon scattering, while their emission energy can be further modulated via strain engineering. Our reported highly tunable interlayer excitons provide important information on understanding moiré excitonic physics, with possible applications in building high-temperature excitonic devices.

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The stacking state of atomic layers critically determines the physical properties of twisted van der Waals materials. Unfortunately, precise characterization of the stacked interfaces remains a great challenge as they are buried internally. With conductive atomic force microscopy, we show that the moiré superlattice structure formed at the embedded interfaces of small-angle twisted multilayer graphene (tMLG) can noticeably regulate surface conductivity even when the twisted interfaces are 10 atomic layers beneath the surface. Assisted by molecular dynamics (MD) simulations, a theoretical model is proposed to correlate surface conductivity with the sequential stacking state of the graphene layers of tMLG. The theoretical model is then employed to extract the complex structure of a tMLG sample with crystalline defects. Probing and visualizing the internal stacking structures of twisted layered materials is essential for understanding their unique physical properties, and our work offers a powerful tool for this via simple surface conductivity mapping.

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Semiconducting ferroelectric materials with low energy polarization switching offer a platform for next-generation electronics such as ferroelectric field-effect transistors. Recently discovered interfacial ferroelectricity in bilayers of transition metal dichalcogenide films provides an opportunity to combine the potential of semiconducting ferroelectrics with the design flexibility of 2D material devices. Here, local control of ferroelectric domains in a marginally twisted WS2 bilayer is demonstrated with a scanning tunneling microscope at room temperature, and their observed reversible evolution is understood using a string-like model of the domain wall network (DWN). Two characteristic regimes of DWN evolution are identified: (i) elastic bending of partial screw dislocations separating smaller domains with twin stackings due to mutual sliding of monolayers at domain boundaries and (ii) merging of primary domain walls into perfect screw dislocations, which become the seeds for the recovery of the initial domain structure upon reversing electric field. These results open the possibility to achieve full control over atomically thin semiconducting ferroelectric domains using local electric fields, which is a critical step towards their technological use.

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Structural moiré superstructures arising from two competing lattices may lead to unexpected electronic behavior. Sb is predicted to show thickness-dependent topological properties, providing potential applications for low-energy-consuming electronic devices. Here we successfully synthesize ultrathin Sb films on semi-insulating InSb(111)A. Despite the covalent nature of the substrate, which has dangling bonds on the surface, we prove by scanning transmission electron microscopy that the first layer of Sb atoms grows in an unstrained manner. Rather than compensating for the lattice mismatch of -6.4% by structural modifications, the Sb films form a pronounced moiré pattern as we evidence by scanning tunneling microscopy. Our model calculations assign the moiré pattern to a periodic surface corrugation. In agreement with theoretical predictions, irrespective of the moiré modulation, the topological surface state known on a thick Sb film is experimentally confirmed to persist down to small film thicknesses, and the Dirac point shifts toward lower binding energies with a decrease in Sb thickness.

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The interlayer interaction in Pt-dichalcogenides strongly affects their electronic structures. The modulations of the interlayer atom-coordination in vertical heterostructures based on these materials are expected to laterally modify these interlayer interactions and thus provide an opportunity to texture the electronic structure. To determine the effects of local variation of the interlayer atom coordination on the electronic structure of PtSe2, van der Waals heterostructures of PtSe2 and PtTe2 have been synthesized by molecular beam epitaxy. The heterostructure forms a coincidence lattice with 13 unit cells of PtSe2 matching 12 unit cells of PtTe2, forming a moiré superstructure. The interaction with PtTe2 reduces the band gap of PtSe2 monolayers from 1.8 eV to 0.5 eV. While the band gap is uniform across the moiré unit cell, scanning tunneling spectroscopy and dI/dV mapping identify gap states that are localized within certain regions of the moiré unit cell. Deep states associated with chalcogen pz-orbitals at binding energies of ∼ -2 eV also exhibit lateral variation within the moiré unit cell, indicative of varying interlayer chalcogen interactions. Density functional theory calculations indicate that local variations in atom coordination in the moiré unit cell cause variations in the charge transfer from PtTe2 to PtSe2, thus affecting the value of the interface dipole. Experimentally this is confirmed by measuring the local work function by field emission resonance spectroscopy, which reveals a large work function modulation of ∼0.5 eV within the moiré structure. These results show that the local coordination variation of the chalcogen atoms in the PtSe2/PtTe2 van der Waals heterostructure induces a nanoscale electronic structure texture in PtSe2.

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

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A methodology to test the interlayer bonding strength of two-dimensional (2D) surfaces and associated one (1D)- and two (2D)- dimensional surface defects using scanning tunneling microscope tip-induced deformation, is demonstrated. Surface elastic deformation characteristics of soft 2D monatomic sheets of graphene and graphite in contrast to NbSe2indicates related association with the underlying local bonding configurations. Surface deformation of 2D graphitic moiré patterns reveal the inter-layer van der Waals strength varying across its domains. These results help in the understanding of the comparable interlayer bonding strength of 1D grain boundary as well as the grains. Anomalous phenomena related to probing 2D materials at small gap distances as a function of strain is discussed.

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Bioresponsive hydrogels are smart materials that respond to various external stimuli and exhibit great potential as biosensors owing to their capability of real-time and label-free detection. Here, we propose a sensing platform based on bioresponsive hydrogels, employing the concept of moiré patterns. Two sets of line patterns with different pitch sizes are prepared; a hydrogel grating whose pitch size changes according to external stimuli and a reference grating with constant pitch size. The volume changes of the hydrogel caused by external stimuli changes the pitch size of the hydrogel grating, and subsequently, the pitch sizes of the moiré patterns (moiré signal), whose values can be obtained in a real-time and label-free manner through customized moiré microscopy and signal processing. After confirming that the pH-induced swelling of hydrogel could be monitored using moiré patterns, we performed moiré pattern-based detection of specific proteins using protein-responsive hydrogel that underwent shrinking via interaction with target proteins. Brain-derived neurotrophic factor and platelet-derived growth factor were selected as the model proteins, and our proposed system successfully detected both proteins at nanomolar levels. In both cases, the pitch size change of hydrogel grating was monitored much more sensitively using moiré patterns than through direct measurements. The changes in the moiré signals caused by target proteins were detected in ex-vivo environments using a custom-made intraocular lens incorporating the hydrogel grating, demonstrating the capability of the proposed system to detect various markers in intraocular aqueous humor, when implanted in the eye.

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Monolayer hexagonal boron nitride (hBN) has attracted interest as an ultrathin tunnel barrier or environmental protection layer. Recently, wafer-scale hBN growth on Cu(111) was developed for semiconductor chip applications. For basic research and technology, understanding how hBN perturbs underlying electronically active layers is critical. Encouragingly, hBN/Cu(111) has been shown to preserve the Cu(111) surface state (SS), but it was unknown how tunneling into this SS through hBN varies spatially. Here, we demonstrate that the Cu(111) SS under wafer-scale hBN is homogeneous in energy and spectral weight over nanometer length scales and across atomic terraces. In contrast, a new spectral feature─not seen on bare Cu(111)─varies with atomic registry and shares the spatial periodicity of the hBN/Cu(111) moiré. This work demonstrates that, for some 2D electron systems, an hBN overlayer can act as a protective yet remarkably transparent window on fragile low-energy electronic structure below.

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We show that the anisotropic energy of a 2D antiferromagnet is greatly enhanced via stacking on a magnetic substrate layer, arising from the sublattice-dependent interlayer magnetic interaction that defines an effective anisotropic energy. Interestingly, this effective energy couples strongly with the interlayer stacking order and the magnetic order of the substrate layer, providing unique mechanical and magnetic means to control the antiferromagnetic order. These two types of control methods distinctly affect the sublattice magnetization dynamics, with a change in the ratio of sublattice precession amplitudes in the former and its chirality in the latter. In moiré superlattices formed by a relative twist or strain between the layers, the coupling with stacking order introduces a landscape of effective anisotropic energy across the moiré, which can be utilized to create nonuniform antiferromagnetic textures featuring periodically localized low-energy magnons.

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The search for artificial topological superconductivity has been limited by the stringent conditions required for its emergence. As exemplified by the recent discoveries of various correlated electronic states in twisted van der Waals materials, moiré patterns can act as a powerful knob to create artificial electronic structures. Here, we demonstrate that a moiré pattern between a van der Waals superconductor and a monolayer ferromagnet creates a periodic potential modulation that enables the realization of a topological superconducting state that would not be accessible in the absence of the moiré. The magnetic moiré pattern gives rise to Yu-Shiba-Rusinov minibands and periodic modulation of the Majorana edge modes that we detect using low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS). Moiré patterns and, more broadly, periodic potential modulations are powerful tools to overcome the conventional constraints for realizing and controlling topological superconductivity.

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Here, we introduce and explore Scintillating Starbursts, a stimulus type made up of concentric star polygons that induce illusory scintillating rays or beams. We test experimentally which factors, such as contrast and number of vertices, modulate how observers experience this stimulus class. We explain how the illusion arises from the interplay of known visual processes, specifically central versus peripheral vision, and interpret the phenomenology evoked by these patterns. We discuss how Starbursts differ from similar and related visual illusions such as illusory contours, grid illusions such as the pincushion grid illusion as well as moiré patterns.

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A real-space approach for the calculation of the moiré lattice parameters for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides is presented. Apparent moiré lattices continuously form for all rotation angles, and their lattice parameter to a good approximation follows a hyperbolical angle dependence. Moiré crystals, i.e. moiré lattices decorated with a basis, require more crucial assessment of the commensurabilities and lead to discrete solutions and a non-continuous angle dependence of the moiré-crystal lattice parameter. In particular, this lattice parameter critically depends on the rotation angle, and continuous variation of the angle can lead to apparently erratic changes of the lattice parameter. The solutions form a highly complex pattern, which reflects number-theoretical relations between formation parameters of the moiré crystal. The analysis also provides insight into the special case of a 30° rotation of the constituting lattices, for which a dodecagonal quasicrystalline structure forms.